𝕿𝖊𝖈𝖍𝖗𝖎𝖌𝖍𝖙𝖘 Bulletin for Sunday, January 23, 2022
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Gemini index for the day: gemini://gemini.techrights.org/2022/01/23/


╒═══════════════════ 𝐑𝐄𝐂𝐄𝐍𝐓 𝐁𝐔𝐋𝐋𝐄𝐓𝐈𝐍𝐒 ════════════════════════════════════╕
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╒═══════════════════ 𝐈𝐍𝐃𝐄𝐗 ═══════════════════════════════════════════════════╕

⦿ Peak Code — Part III: After Code | Techrights

⦿ The Demonisation and Stereotyping of Coders Not Working for Big Corporations (or ’The System’) | Techrights

⦿ [Meme] Team UPC Congratulating Itself | Techrights

⦿ IRC Proceedings: Saturday, January 22, 2022 | Techrights

⦿ On the ’Peak Hacker’ Series | Techrights

 ䷼ Bulletin articles (as HTML) to comment on (requires login):
		http://techrights.org/2022/01/23/after-code/#comments
		http://techrights.org/2022/01/23/criminal-stigma/#comments
		http://techrights.org/2022/01/23/illegal-upc/#comments
		http://techrights.org/2022/01/23/irc-log-220122/#comments
		http://techrights.org/2022/01/23/peak-hacker/#comments

 ䷞ Followed by Daily Links (assorted news picks curated and categorised):
		http://techrights.org/2022/01/23/pgbadger-11-7-released/#comments
		http://techrights.org/2022/01/23/sway-1-7-released/#comments

                    ䷩ 𝚕𝚒𝚗𝚎 60

╒═══════════════════ 𝐀𝐑𝐓𝐈𝐂𝐋𝐄 ═════════════════════════════════════════════════╕

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/01/23/after-code/#comments
    Gemini version at gemini://gemini.techrights.org/2022/01/23/after-code/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 01.23.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Peak_Code_—_Part_III:_After_Code⠀✐


Posted in Free/Libre_Software, GNU/Linux at 3:58 am by Guest Editorial Team


Article/series by Dr. Andy Farnell


This work is licensed under version 4.0 of the Creative Commons CC-BY-SA
license


Series parts:


   1. Peak_Code_—_Part_I:_Before_the_Wars
   2. Peak_Code_—_Part_II:_Lost_Source
   3. YOU ARE HERE ☞ After Code
                          🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇Robot⦈


Summary: “Surveillance perimeters, smart TVs (Telescreens built to Orwell’s
original blueprint) watched over our living rooms. Mandatory smart everything
kept us ‘trustless’. Safe search, safe thoughts. We withdrew. Inside, we went
quietly mad.”


The mind can rule the body. Psychogenic death happens when the software gives
up. After a cancer diagnosis or a voodoo curse some people age ten years in a
month and pass away even while quite healthy. We think empires fall from over-
reach, but that is only half a truth. History books say the edges crumbled, but
in fact Rome died first and things fell apart from the inside. The centre can
only hold strong where there is uncorrupted will. The fish rots from its head.


“Before this self-devouring monster, the imperfect technological society of the
twenty first century had built itself on the energy and love of those they
called the “hackers”, many of whom lived and worked in the twentieth
century.”People always asked me, before the collapse, “But how will we build a
Utopia”. I laughed. Real people are not driven by visions of Utopia, but by the
simple faults at the edges of life. At heart we are explorers who must find new
risks, healers who seek out our injured, teachers who seek the attention of
those hungry for knowledge, and protectors who see danger and vulnerable others
to put ourselves between. Those are our itches at which we scratch. It is in
service of each others’ suffering that we find meaning.


A Utopia? A society so perfect, so abstracted, so managed, so owned, so
tranquil, so fair, so incorruptible in its smartness, so beyond mere technology
– is a society not worth living in or for. It is a simulation and performance
of a society that offers no reason for itself but that everything be in service
of it.


Before this self-devouring monster, the imperfect technological society of the
twenty first century had built itself on the energy and love of those they
called the “hackers”, many of whom lived and worked in the twentieth century.
Simply, they were workers, though some elevated themselves to titles like
“developer”, “CTO” or “computer scientists”. We were always workers.


“Hackers were all sorts of things, but foremost they were people. Software is
people.”Some were intrinsically motivated autotelic personalities engaged in
the love of problem solving. Some were quiet moral labourers committed to the
challenge of building a “better world”. Some were social refuseniks who hid
behind avatars and anonymity. Others were dissatisfied and restive women and
men, motley crews of misfits and egotists hungry to be known and admired.
Hackers were all sorts of things, but foremost they were people. Software is
people.


People built and ran the machine. Now, there it was, at last. Tamed. A system
so slick, elegant, and otherly. It could provide everything we asked for,
except for one thing – its absence. Any whim, impulse or fantasy…
satisfied now. Delivered instantly, no pause, no shame, pay later. Anything you
like, except for one simple request, that it shut itself off and leave us
alone. Soon, all people talked about was the machine, it’s constant demands for
our happiness, acquiescence and smiling fealty.


“A dearth of new coders was eclipsed by the crisis of maintainers, growing old
and retiring.”Nobody noticed the shrinking number of human accounts at the
Ministry of Code. A dearth of new coders was eclipsed by the crisis of
maintainers, growing old and retiring. Schools had not taught programming for
decades. Universities had dumbed down computer science courses to certificate
training. Hackers were brought into the fold, domesticated and made “ethical”.


By the time the BigTeks realised that education was a real hard and
unprofitable profession they had trained a generation of mindless, and mostly
useless digital mechanics who could not think beyond their parochial,
proprietary instructions.


By the time the tech press reported just how serious the recruitment problem
was, it was 20 years too late. Some old guy named Linus, someone of importance
if I recall correctly, retired, leaving half-hearted replacements and a
bloated, fragmented project that soon faltered. Your aunt Alice fought in the
Kernel Wars you know, but we don’t talk about Alice.


“Trillionaire philanthropists opened their bank accounts, “Money no object”
they said. But it was no use. Old hackers want walks in the park with the
grandkids.”The “war” that never happened began when BigTek and the governments
needed a narrative to explain what was happening. They wanted the old hackers
back. But they were dying-off by then. Forty years too late, they offered to
pay for Free software. “Code whatever you like” they said, “We’ll pay whatever
it takes”.


Trillionaire philanthropists opened their bank accounts, “Money no object” they
said. But it was no use. Old hackers want walks in the park with the grandkids.
“Not my business any longer”, they said. We tried to train new hackers, but
those kids weren’t interested, ambivalent now towards technology that only
seemed to control their lives. It was “no fun to compute”. The excitement of
possibility, of “making a difference”, was gone.


So the powers needed an enemy, a scapegoat. Goldberg’s Neo-Luddite subversives,
saboteurs, and malcontents were now blamed for the
software crisis. Supposedly “a secret resistance” were coding new systems that
would save us. Was that a worry or a wish? But beyond the fantasies of the
press, nobody could find Eponymous. The CIA, adept at entrapment and honeypots,
created fake “militant hacker groups” to draw in and reorient high-IQ
candidates, but nobody joined.


“We tried to train new hackers, but those kids weren’t interested, ambivalent
now towards technology that only seemed to control their lives.”The sun was
setting on the age of software. We’d enjoyed all we dreamed of. Drones flew in
the skies over our houses, protecting us. Fingerprint door handles kept out the
bad men. Everything was on the Face-Chain. Surveillance perimeters, smart TVs
(Telescreens built to Orwell’s original blueprint) watched over our living
rooms. Mandatory smart everything kept us ‘trustless’. Safe search, safe
thoughts. We withdrew. Inside, we went quietly mad. We went Meta. And then Meta
metastasised, into the heart of our care, until there was care no more. Nobody
cared. Nobody coded. And the machine whimpered. █


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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣾⣿⣿⣿⣿⣿⣿⣧⣦⣐⣾⡽⠟⠋⣡⣴⣾⣿⢿⣦⠀⢀⡦⣗⠀⠀⠀⠃⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⠀⣾⣿⣿⣿⣿⣿⣿⣆⠀⠘⡚⣇⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣛⣿⣄⣀⣨⣵⣶⣠⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⠿⣿⡿⢛⣫⣿⣻⣿⣿⣿⣻⢿⡟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣶⣿⣿⣿⣿⣿⢿⣧⣴⣾⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⡿⡗⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢐⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣿⣿⣿⣿⣿⣿⣿⣷⣗⡧⡇⡅⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠈⠙⠛⠛⠟⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠈⠉⠉⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡣⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⢶⣶⣶⣶⣦⣤⣤⣄⣀⣀⣀⡀⠈⣿⣷⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣟⣛⣻⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠛⠛⠛⠛⠛⠛⠻⠻⠓⠋⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣏⣿⡄⢯⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣷⣿⣿⣾⣿⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⣿⠿⣿⣷⣶⣶⣶⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⣿⠟⠛⠛⠻⠿⣟⣛⣣⢀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⠘⠛⠿⠿⠶⠶⢶⣶⣦⣤⣤⣤⣤⣀⣀⣀⣀⡀⠋⠀⠐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣷⡼⢻⣿⣿⣿⢹⠿⠟⠛⠏⢈⠀⠀⠐⠀⣶⠞⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿
⠉⠙⠛⠛⠛⠁⠀⢸⣿⣶⣶⣮⣭⣭⣭⣽⣛⣛⣛⣛⣛⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣎⣼⣏⢹⡏⢇⠈⠦⠂⠌⠂⡃⠂⠀⠄⠀⠉⠁⢸⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣦⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠄⠀⠀⠈⠉⠋⠉⠉⠉⠉⠙⢓⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣶⣶⣶⣶⣤⣤⣤⣤⣦⣀⣀⣸⣾⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣯⣾⣽⣿⣧⢸⣯⢸⣿⣭⣟⢲⠢⣏⣩⡉⡏⠉⠙⢻⡿⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⡷⣿⣿⣿⣿⣾⣇⣸⣾⢬⣏⣹⠐⡗⢴⠠⣏⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⢂⠆⠀⢽⣧⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠾⢿⣿⣿⣿⣿⣿⣿⠭⠽⠿⣿⣿⣿⣿⣿⣛⣿⣿⣿⠃⣿⡟⣿⣿⣿⢻⠶⡇⢨⡈⡅⣸⠀⡇⠀⠀⢠⠐⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠂⠘⠋⠀⠿⠽⠿⠿⠿⢄⣤⣴⣶⣾⣿⣿⣿⡏⢉⣭⣩⣭⣿⣿⣿⣿⠁⣤⣶⣿⣿⣿⣿⣏⣿⢿⢿⣿⠐⢻⣧⡝⣿⣿⣿⣶⣷⣾⣶⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⡀⠀⡙⣀⣀⣀⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⢻⣿⣿⣾⣿⣸⣈⣻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠄⠀⠀⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⠻⠿⠿⡌⣿⠛⠿⠟⠒⠓⠛⠒⠚⠛⠛⠛⠛⡻⠛⠃⢸⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡉⢸⡿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⠀⠀⠀⣧⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠛⡜⢻⣿⣿⣿⣿⢷⠀⢘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣰⢀⣠⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠰⠏⢻⣿⣿⡟⠋⢈⠀⠀⠈⡿⢿⡛⠿⢿⠿⠿⠿⣿⣿⣿⣿⣿⠿⠟⠹⣿⣿⣧⣤⣬⣿⣿⣶⣼⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠠⠄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⢸⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠉⠉⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⣿⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡑⡄⣀⣀⣀⣸⣇⣀⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢰⡀⠀⢴⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠈⠻⣿⣄⠀⠹⣿⣆⠀⠻⣷⣄⠈⠻⣷⡄⠘⠻⠆⣀⢄⣂⠀⠠⣽⣿⣿⣿⣿⣿⣿⣿⣿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠈⠉⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢗⡾⠎⠤⢼⣿⣿⣿⣿⣿⣿⡿⠋⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠁⢳⡿⠟⢻⣿⣿⣿⣿⣿⣿⣻⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⢐⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠈⠠⢸⣿⣿⣿⣿⣿⣿⣃⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⢧⣿⢿⡇⢐⣲⣤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⣿⣿⣿⣿⣿⣿⣧⠈⠀
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⣜⣷⣤⣤⣶⣶⣶⣾⣆⠀⠀⠐⠡⠐⣲⡿⠒⠈⠾⡇⢠⣿⡯⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡨⣷⣿⢿⣽⣟⠇⠀⠀
⣿⣿⣿⣿⢿⣿⢿⠿⠿⠿⠿⠿⠟⠙⠙⠟⠛⠛⠛⠋⠉⠀⠀⠐⢀⡀⢅⣗⠀⠀⠀⠀⢸⣿⠇⣪⣿⡦⠀⡀⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠷⣿⣟⣿⣿⠛⡆⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢲⣦⣸⣷⣲⣴⣤⣴⣶⡮⠀⣸⣿⠀⣽⡿⠃⠨⠀⠀⠄⠀⠀⠀⠀⡃⡁⠄⠄⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣟⣿⣟⣿⣿⣾⠳⠍⠉⠀⠀

                    ䷩ 𝚕𝚒𝚗𝚎 266

╒═══════════════════ 𝐀𝐑𝐓𝐈𝐂𝐋𝐄 ═════════════════════════════════════════════════╕

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/01/23/criminal-stigma/#comments
    Gemini version at gemini://gemini.techrights.org/2022/01/23/criminal-stigma/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 01.23.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ The_Demonisation_and_Stereotyping_of_Coders_Not_Working_for_Big_Corporations_
(or_‘The_System’)⠀✐


Posted in Deception at 10:23 am by Dr. Roy Schestowitz


It’s not about crime prevention, it’s about control


🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇RIAA_meme:_Media_tells_you_online_anonymity_means_wearing_a
                         mask_and_committing_crimes⦈_
Does anyone seriously think cyber criminals wear a mask in front of the screen?


Summary: The war on encrypted communication (or secure communications) carries
on despite a lack of evidence that encryption stands in the way of crime
investigations (most criminals use none of it)


Encryption is evil[1-4]
It leads to upheaval


Security with limits
For all of you dimwits


Accountability for thy
But not_for_me


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇Git hacking/Man coding⦈
Coding for Meta
Collecting your data


Spying for Gulag
Apple and Dehomag


Microsoft professional
Ethics just occasional


The free hackers online
Must be put in line


Freedom is fine
Let democracy shine


Resistance is disruptive
Unlike Twitter manipulative


COVID on the loose
But go drink some booze


Lads, keep hopping
Ladies, resume shopping


Digital activism afoul
Peasants, be more banal!


Coding permitted
If to the state submitted


Leaks for transparency
But not our own, that’s heresy!


Studying paramount
But corporate training tantamount


Go get the app
Two buttons, “back” and “stop”


Applications are “old school”
For every citizen is a fool


Functionality limited
DRM decides what’s permitted


Computing for trust
In villains of antitrust


Web sites for all
Logs we recall


Accessibility first
Except JavaScript burst


Cyberspace you roam
With clones of Gulag Chrome


Telnet_a_crime
Bailed on a dime


Warning received
‘Suspicious behaviour’ perceived


Fearing unknowns
People who aren’t pawns


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇Oh, look, I don't use Apple and Microsoft so I must be a
criminal⦈ Alternatives esoteric
Use something genetic


The state shall rejoice
Amazon and the boys


Olympic app installed
When COVID tracings fold


Share on_the_Clown
Upload your frown


Big Brother awaits
You and your mates


Trendy “tech” afoot
Antagonism is moot


Coding is GitHub
Censoring Alice and Bob


Sanitised behaviour
Impeded coding endeavour


Pluton for plutocrats
Insurrectionists and Democrats


For national security strive
Back doors for the beehive


“Crypto” is currency
Planet-burning entity


Encryption is centralised
E2EE_is_for_the_'radicalised'


Plenty of choices
They capture all voices


Brands suck alike
So what’s not to like?


Disagreement is futile
Pry Minister infantile


Get on with the programme
Be a docile lamb █
________
Related/contextual items from the news:


   1. ⚓ UK_Government_to_Launch_PR_Campaign_Undermining_End-to-End_Encryption⠀⇛


   2. ⚓ Revealed:_UK_Gov’t_Plans_Publicity_Blitz_to_Undermine_Chat_Privacy_–
      Rolling_Stone⠀⇛


   3. ⚓ Privacy_is_for_paedophiles,_UK_government_seems_to_be_saying_while
      spending_£500k_demonising_online_chat_encryption⠀⇛


   4. ⚓ The_UK_Has_A_Voyeuristic_New_Propaganda_Campaign_Against_Encryption⠀⇛


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢨⣾⡇⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢰⠆⠀⠀⠀⠀⣸⣿⣿⣀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢸⣿⡆⠀⠀⢰⡍⢻⣿⢛⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠻⠿⣿⣿⡇⠀⠀⢸⣿⣇⠀⠀⢸⢧⣎⣽⢸⡡⣮⠱⡆⣿⡹⣛⢹⣿⢨⢫⣝⡇⡇⡟⢿⣿⡟⣟⡻⣛⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⢸⣿⣿⠀⢠⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣾⣷⣿⣿⣷⣯⣵⣿⡿⣽⣧⣿⣾⣼⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣉⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡖⢩⡝⣭⡍⣿⣼⢺⢛⡛⢟⡟⣿⣿⡿⡿⠿⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⣟⡿⣯⣉⣉⣩⠉⣿⣿⣿⠏⣠⠞⣁⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣼⣿⣿⣿⣶⣾⣾⣷⣷⣿⣼⣼⣧⣮⣿⣿⣇⣶⣧⣿⣢⣻⣺⣸⣇⣧⢣⡇⠾⢰⠸⠆⡇⣷⠘⠑⠓⠀⣿⣿⡏⢰⠇⢸⣿⡇⣿⢸⣿⢠⣿⠈⣧⠀⠀
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣥⠛⠛⠛⠛⠟⣿⣷⣟⣀⠀⠀⠀⣿⣿⡇⢸⠀⢸⣿⣆⡏⡾⣿⣸⣿⠀⣼⠀⠀
⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⣧⣧⣓⣏⣂⣃⣻⣸⣮⣹⣿⣈⡌⣱⠸⢬⠲⢸⣯⣼⢸⢸⡍⡋⠍⣤⡂⠀⠀⡜⢛⢛⠻⣟⠳⠖⣳⣿⣿⣷⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣶⣱⢆⡽⠃⠀⠀⠣⣬⣼⣤⣤⣄⣑⣪⣷⣝⣿⠓⠈⠳⣤⣀⣀⣀⣀⣠⠶⠋⠀⠀⠀
⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣶⢸⠶⠆⣿⠫⣯⢻⡏⡏⡍⢫⢹⢛⡛⡛⣟⢸⢙⡋⣛⣙⢛⢿⣿⠿⣄⣀⠀⡠⣾⣻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡿⠋⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⡿⠿⠿⢿⠷⣶⣷⣾⣷⣷⠾⠬⠬⢥⡧⢧⣼⣼⣯⣏⣛⣻⣘⢊⡙⣿⣅⢻⠃⡿⠸⠰⠆⡆⡏⢒⣭⡚⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⠿⠛⠁⠀⠈⠘⠿⢿⣦⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠈⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡬⠀⠀⠀⡀⠀⠈⠉⠉⠉⠉⠉⣩⠀⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣯⣂⣀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠇⠀⠀⠀⠀⠀⠀⣿⣿⣦⣀⠀⠠⠄⠀⠀⠀⢸⣿⣿⣿⣿⣿⢰⡾⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣽⣶⢶⣿⣶⣶⣦⣸⣿⣿⣿⠑⢿⣿⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣴⣶⣶⣶⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢀⠀⣆⠈⠄⠀⠀⠀⠀⠀⠀⠀⣿⡽⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠴⠆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣤⡙⠛⠿⡿⠛⠛⠿⠟⢹⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡊⠈⠙⠆⠊⢰⠦⣠⡄⣀⣀⣤⣴⣾⣿⡿⠿⣆⣠⣦⣄⠀⠀⠀⢸⣥⣄⣀⠀⠈⠉⠙⠓⠒⠨⠭⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿
⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⠀⠉⠋⢯⣭⣭⣙⣛⣛⣛⣛⣛⣫⣽⣿⣿⣤⣀⣀⠀⠀⠉⠒⢸⣿⣿⣿⣿⣟⣋⣴⣾⣿⣿⣿⣿⣏⠙⠻⢿⢿⣿⣃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⡟⠛⠛⠿⠿⣿⣿⣿⣿
⠀⠆⠲⣶⠄⠿⠖⠰⠰⠶⠆⠀⠀⠀⠈⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠀⠀⠀⠀⣉⣛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⡘⠀⠀⠀⠀⣰⣿⣿⣀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣇⠀⠀⢠⡍⢻⣿⢛⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠉⠉⠉⠙⠛⠿⣿⡿⠀⠀⢸⣿⣿⠀⠀⢸⢧⣎⣽⢸⡡⣮⠱⡆⣿⡽⣛⢹⣿⢨⢫⣝⡇⡇⣟⢿⣿⡟⣿⡻⣛⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⣿⡄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣾⣷⣿⣷⣷⣯⣵⣿⡿⣽⣧⣿⣾⣼⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣉⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡗⢩⡝⣍⡍⣿⣼⢺⢛⡛⢟⡟⣿⣿⡿⡿⠿⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⣟⡿⣏⣉⣉⣉⡍⣿⣿⣿⠏⣰⠞⣁⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀
⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣼⣿⣿⣿⣶⣾⣾⣷⣷⣿⣼⣼⣧⣮⣿⣿⣇⣴⣇⣿⣢⣻⣸⣸⣇⣧⢫⡇⠾⢰⠸⠆⡇⣷⠘⠙⠓⠀⢺⣿⡏⢰⠇⢸⣿⡇⣿⢸⣿⢠⣿⠈⣧⠀⠀
⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢛⡻⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣥⡛⠛⠛⠛⠿⣿⣷⣟⣀⠀⠀⠀⣾⣿⡇⢸⠀⢸⣿⣆⡏⣾⣿⣸⣿⠀⣼⠀⠀
⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⣧⣧⣛⣏⣂⣃⣻⣸⣮⣹⣿⣈⡌⣱⠸⢬⠳⢸⣯⣼⢸⢹⡝⡋⠍⣵⡀⠀⠀⡀⢛⢛⠻⣟⠳⡖⣲⣾⣿⣷⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣶⣱⣆⣽⠇⠀⠀⠡⣬⣼⣤⣤⣄⣕⣪⣳⣝⣿⡗⠈⠳⣤⣀⣀⣀⣀⣠⠶⠋⠀⠀⠀
⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠇⣶⢸⠶⠆⣿⠫⣯⢻⡏⡏⣍⢫⢹⢛⡛⡛⣟⢸⢙⡋⣛⣙⢛⢿⣿⠿⣄⣀⠀⡠⢾⣻⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡟⠉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⠿⠿⢿⠷⣾⣷⣾⣷⣷⠾⠬⠬⢥⡧⢧⣼⣼⣯⣏⣛⣻⣘⢊⡙⣿⣇⢻⡃⡿⠸⠰⠆⡆⡏⢖⣭⡛⣿⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⠿⠋⠁⠀⠈⠸⠿⣿⣦⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢸⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⡤⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⣩⠁⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣯⡢⡀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠇⠀⠀⠀⠀⠀⠀⣿⣿⣧⣀⠀⠀⠄⠀⠀⠀⢐⣿⣿⣿⣿⣿⠰⡶⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣽⣶⠶⣿⣷⣶⣦⣼⣿⣿⣿⡑⢿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣴⣶⣶⣶⣬
⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠄⠀⠀⠀⠀⠀⠀⠀⣿⡝⣿⣿⣿⡿⣿⣿⣿⣷⣼⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⡀⠀⠀⢰⠆⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢉⣀⡘⠛⠿⡿⠛⠛⠻⠟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⠐⠋⠚⠆⠂⢠⠦⣤⣄⣀⣀⣠⣤⣶⣿⣿⢿⣆⣴⣶⣄⠀⠀⠀⢸⣭⣄⣀⠀⠈⠉⠛⠓⠒⠬⠍⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿
⠀⠀⠀⠘⢳⡀⠀⠀⠀⠀⠀⠈⠙⠹⣿⣛⣛⠛⠛⠛⠛⠻⣛⣻⣿⣿⣿⣤⣀⠀⠀⠀⠉⠟⢨⣿⣿⣿⣿⣿⣋⣶⣾⣿⣿⣿⣿⣯⠙⠻⠿⣿⣿⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣿⡟⠛⠻⠿⢿⣿⣿⣿⣿
⠀⠆⠲⣶⠈⠷⠆⠠⠰⠶⠆⠀⠀⠀⠘⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠈⠀⠀⠀⠀⢉⣛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⡀⠀⠀⠀⠀⢸⣿⣿⣀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣷⡆⠀⠀⢰⡍⢻⣿⢛⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠙⠻⢿⡇⠀⠀⢸⣿⣧⠀⠀⢸⢇⣎⣽⢸⡡⣮⠱⡆⣿⡹⣛⢹⣿⢨⢫⣝⡇⡇⡟⢿⣿⡟⣿⡻⣛⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⣿⠀⣠⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣾⣷⣿⣿⣷⣯⣵⣿⡿⣽⣧⣿⣾⣼⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣉⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢸⣿⣿⡖⢩⡝⣍⡍⣿⣼⢺⢛⡛⢟⡟⣿⣿⡿⡿⠿⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⣟⡿⣯⣉⣉⣉⠉⣿⣿⣿⠏⣠⠞⣁⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀
⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣸⣿⣿⣿⣶⣾⣾⣷⣷⣿⣼⣼⣧⣮⣿⣿⣅⣶⣇⣿⣢⣻⣸⣸⣇⣧⢫⡇⠾⢰⠸⠆⡇⣷⠘⠑⠓⠀⣿⣿⡏⢰⠇⢸⣿⡇⣿⢸⣿⢠⣿⠈⣧⠀⠀
⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣥⡛⠛⠛⠛⢿⣿⣷⣟⡀⠀⠀⠀⣿⣿⡇⢸⠀⢸⣿⣆⡏⡾⣿⣸⣿⠀⣼⠀⠀
⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⣥⣧⣧⣓⣏⣂⣃⣻⣸⣮⣹⣿⡈⡌⣰⠸⢬⠲⢸⣯⣼⢸⢸⡍⡋⠍⣴⡄⠀⠀⡐⢛⢛⠻⣟⠳⡞⣻⣿⣿⣷⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣶⣱⢆⣽⡇⠀⠀⠡⣬⣼⣤⣤⣄⣑⣨⣷⣝⣿⠗⠈⠳⣤⣀⣀⣀⣀⣠⠶⠋⠀⠀⠀
⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠄⠇⣶⢸⠶⠆⣿⠫⣯⢻⡏⡏⣍⢫⢹⢛⡛⡛⣟⢸⢙⡋⣛⣙⢛⢻⣿⠿⣄⣀⠀⡤⢾⣻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡟⠁⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⡿⠿⠿⢿⠷⣾⣷⣾⣷⣷⠾⠬⠬⣥⡧⢥⣼⣼⣯⣏⣛⣻⣘⢊⡑⣿⣃⢻⠃⡿⠸⠰⠆⡆⡏⢲⣭⡛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⠛⠁⠀⠀⠀⠙⢛⣿⣶⣤⣤⣤⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⢻⣿⣿⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡤⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⢉⣭⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⣿⣿⣯⣢⡀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⣿⣿⣆⡀⠀⠠⠄⠀⠀⠀⢸⣿⣿⣿⣿⡇⢶⠾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣽⡶⢶⣿⣾⣶⣤⣸⣿⣿⣯⠺⣿⣿⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣶⣶⣤⣶⣽
⠀⠀⠀⠀⠀⠀⠀⠀⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢠⡆⠰⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⣿⣿⣿⢿⣿⣿⣿⣦⣿⣿⣿⣿⣧⣬⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⡀⠀⠠⠾⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠛⠻⣿⡟⠛⠛⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣔⠠⠉⠛⠂⠣⡴⣦⣤⣀⣀⣀⣠⣤⣶⣿⣿⣷⣤⣶⣦⡀⠀⠀⠀⣽⣅⣀⡀⠈⠉⠙⠛⠒⠬⠭⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿
⠀⠂⠀⠘⢆⠀⠀⠀⠀⠀⠀⠈⠙⠽⣟⣛⣛⠛⠛⠛⠿⠿⣛⣻⣿⣿⣧⣄⣀⠀⠀⠀⠹⠛⣽⣿⣿⣿⣿⣟⣻⣷⣿⣿⣿⣿⣿⡏⠛⠿⣿⣿⣟⣁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⠛⠿⠿⣿⣿⣿⣿⣿⣿
⠀⠆⠲⣶⠌⠷⠄⠠⠰⠶⠖⠀⠀⠐⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠃⠀⠀⠀⠉⣙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡤⠀⠀⠀⠀⠀⣾⣿⣇⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠻⠿⣿⣿⣿⣿⠃⠀⠀⣥⣦⠀⠀⠀⣸⡏⢻⡿⢛⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⠀⠀⠠⣿⣿⡄⠀⠀⣿⣇⣎⣵⢸⡡⣮⠱⡆⣿⡽⣛⢹⣿⢨⢫⣝⡇⡇⡟⢿⣿⡟⣿⡻⣛⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀
⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢠⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣾⣷⣿⣿⣷⣯⣵⣿⡿⣼⣧⣿⣾⣼⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣉⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡀⠀⣿⣿⡇⠀⢩⡝⣭⡍⡿⣸⢺⢛⡛⢟⡟⣿⣿⡿⡿⠿⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⣟⡿⣟⣉⣉⣉⢹⣿⣿⣿⠏⣠⠞⣁⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀
⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢀⣿⣿⣧⣾⣶⣾⣾⣷⣷⣿⣼⣼⣧⣮⣿⣿⣴⣶⣇⣿⣢⣻⣺⣸⣇⣧⢣⠇⠾⢰⠸⠆⡇⣷⠸⠑⠃⢸⣿⣿⡏⢰⠇⢸⣿⡇⣿⢸⣿⢠⣿⠈⣧⠀⠀
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣥⡛⠛⠋⠛⢿⣿⣷⣟⣀⠀⠀⢸⣿⣿⡇⢸⠀⢸⣿⣆⡏⡾⣿⣸⣿⠀⣼⠀⠀
⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣬⣥⣧⣧⣚⣏⣂⣃⣻⣸⣮⣹⣿⡈⡌⣰⠡⢬⠲⢸⢨⣾⢸⢸⡍⡋⠍⣥⡀⠀⠀⡔⢛⢛⠛⣟⠳⡞⣻⣿⣿⣧⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⢿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣶⣱⣆⡽⠃⠀⠀⠀⣬⣼⣤⣤⣄⣑⣸⣷⣝⣿⠁⠈⠳⣤⣀⣀⣀⣀⣠⠶⠋⠀⠀⠀
⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠄⠇⣶⢸⠶⢾⣿⠫⣯⢻⡏⡏⡍⢫⢹⢛⡛⡛⣟⢻⢙⡋⣛⣙⢛⢻⣿⠿⢄⣀⠀⡰⢾⢻⣿⣿⣿⣿⣿⣿⣿⡯⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⣿⡿⠇⠀⠠⢴⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⠿⠿⠿⡿⣷⣾⣷⣾⣷⡷⠾⠬⢬⡥⠧⣥⣼⣼⣏⣏⣛⣻⣘⢊⡑⣿⣃⠻⠃⠿⠸⠰⠆⡆⡏⣶⣭⡊⡟⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠀⠀⠀⠀⠀⠉⢩⣿⣿⣿⣶⣶⣤⣀⠀⠀⢀⣶⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠄⠀⠀⠉⠉⠈⠉⠉⠉⠉⠉⢩⡉⠿⢿⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣵⣄⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⣿⣿⣧⣄⠀⠀⠄⠀⠀⠀⠀⣿⣿⣿⣿⣷⢰⡶⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣷⢶⣿⣷⣶⣦⣤⣿⣿⣿⡵⢿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣦⣤⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣆⠈⡆⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⡿⣿⣿⣿⣷⣼⣿⣿⣿⣧⣤⣽⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⣼⠂⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⡁⠙⠛⢿⣿⠛⠛⢿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣔⠈⠛⠆⠘⠀⣦⣤⡄⣀⣀⣠⣤⣶⣿⣿⣿⣤⣴⣶⡄⠀⠀⠀⢸⣟⣀⡀⠈⠉⠙⠛⠒⠤⠭⢙⣛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢿⣏⣉⣛⡛⠛⠛⢛⣻⣭⣿⣿⣧⣄⣀⠀⠀⠉⠿⢹⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣿⣏⠛⠿⢿⢿⣿⣋⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⡟⠿⠿⢿⣿⣿⣿⣿⣿⣿
⠀⠆⠲⣶⠷⢶⠄⠠⠰⠶⠆⠀⠀⠀⠀⠛⠻⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠀⠀⠀⠈⢙⡛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⡇⠀⠀⠀⠀⢠⣿⣿⣀⢀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠙⠛⠛⠛⠿⣿⣿⣿⣿⡿⠀⠀⣴⣿⡄⠀⠀⠀⣼⡏⢻⡿⢛⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋
⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⡇⠀⠀⢹⣿⣿⠀⠀⢀⣿⣧⣎⣵⢸⡡⣮⠱⡆⣿⡽⣛⢹⣿⢨⢫⣝⡇⡇⡟⢟⣿⡟⣿⡻⣛⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀
⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢸⣿⣿⠀⠀⢨⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣾⣷⣿⣷⣷⣯⣵⣿⡿⣼⣧⣿⣾⣼⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣉⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⢸⣿⣿⠀⠀⢹⡝⣍⡍⡿⣸⢺⢛⡻⠟⡿⣿⣿⠿⡿⠿⢿⠿⢿⠿⠿⠿⠿⢿⣿⣿⣿⣟⡿⣟⣋⣉⡉⣿⣿⣿⣿⠏⣠⠞⣁⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡧⠀⢸⣿⣿⣄⣿⣶⣾⣾⣷⣷⣿⣬⣼⣧⣮⣿⣟⣴⣶⣇⣿⣢⣻⣸⣸⣇⣧⢫⡇⠾⢰⠸⠆⡇⣷⠸⠐⠂⣿⣿⣿⡏⢰⠇⢸⣿⡇⣿⢰⣿⢠⣿⠈⣧⠀⠀
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⢰⢛⡻⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⢿⣿⣿⣷⣿⣼⣿⣿⣿⢿⣿⣿⣿⣅⠛⠛⠙⠛⠟⣿⣷⣟⡀⠀⠀⣿⣿⣿⡅⢸⠀⢸⣿⣆⡏⡾⣿⣸⣿⠀⣼⠀⠀
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣼⣥⣧⣧⣒⣏⣂⣇⣻⣸⣮⣹⣿⡈⡌⣱⠠⢬⢲⢸⢨⣼⢸⢸⡍⡋⠍⣤⡀⠀⠀⡔⢙⢛⠛⣟⠲⡟⣻⣿⠿⣧⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣶⣱⢆⡽⠃⠀⠀⠁⣬⣼⣤⣤⣄⣑⣾⣷⣝⡏⠰⠈⠳⣤⣀⣀⣀⣀⣠⠶⠋⠀⠀⠀
⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠇⣶⢘⠶⠆⣿⠫⣯⢫⡏⡏⡍⢩⢹⢛⡛⡛⣟⢺⢙⡋⣛⣙⢛⢻⣿⠿⢀⣀⠀⡠⢿⢻⣿⣿⣿⣿⣿⣿⣿⡇⡅⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⣿⠏⠀⠀⠀⠒⡶⢦⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠻⣿⡿⠿⠿⣿⢿⣷⣾⣷⣶⡷⠗⠺⠬⢬⠥⢧⣭⣽⣼⣏⣏⣋⣻⣘⢊⡙⣿⡃⠛⠃⠿⠸⠐⠆⡆⡏⣶⣭⡂⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣦⣀⠀⣾⡇⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡤⠀⠀⠀⠉⠁⠉⠉⠉⠉⠉⠉⠍⠹⢿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⠀⠀⠠⠀⠀⠀⠀⢸⣿⣿⣿⣿⠂⣶⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣽⡶⢾⣿⣶⣶⣤⣼⣿⣿⣏⠾⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣤⣤⣴⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢰⡀⠱⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⢿⣿⣿⣿⣦⣿⣿⣿⣿⣤⣬⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⢰⡇⠸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠀⠛⠻⣿⡟⠛⠻⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠉⠑⠀⠀⡰⢀⣤⢀⣀⣀⣤⣴⣾⣿⣿⣧⣶⣴⣦⠀⠀⠀⠀⣯⣃⣀⠀⠉⠉⠛⠓⠢⠬⠍⣛⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣛⣛⣛⣛⣯⣿⣿⣿⣧⣄⣀⠀⠀⠐⠈⣿⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⣷⠙⠻⠿⡿⣿⣟⣁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠆⠲⣶⠿⢷⠄⠠⠰⠖⠆⠀⠀⠀⠀⠁⠘⠻⠛⠛⠻⠿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠃⠀⠀⠀⠉⣛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛
⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⢃⡆⠀⠀⠀⠀⣿⣿⣆⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⡟⠀⠀⢰⣷⣇⠀⠀⠀⢰⣿⡏⢻⡿⢛⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉
⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⡇⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⣧⣎⣵⢸⡡⣮⡱⡆⣿⡽⣛⢹⣿⢨⢫⣝⡇⡇⡟⢿⣿⡟⣟⡻⣛⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀
⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⠀⢸⣿⣿⠀⠀⢀⣾⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣾⣷⣿⣿⣷⣯⣵⣿⡿⣼⣧⣿⣾⣼⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣉⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠁⠀⠀⢸⣿⣿⠀⠀⢸⢹⡝⣍⡍⡿⣸⢺⢻⡻⢟⡿⣿⡿⠿⡿⠿⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⣟⡿⣟⣋⡉⣽⣿⣿⣿⡟⠉⣠⠞⣁⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢦⠀⠀⢸⣿⣿⣄⣠⣾⣶⣾⣾⣷⣷⣿⣼⣼⣯⣮⣯⣿⣴⣷⣇⣯⣢⣻⣹⣸⣇⣧⢣⡇⠾⢰⠸⠆⡇⣷⠸⡀⣿⣿⣿⣿⡇⢰⠇⢸⣿⡇⣿⢸⣿⢠⣿⠈⣧⠀⠀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⢸⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⢿⣿⣿⣯⣵⣿⣿⣿⣿⢿⣿⣿⣿⣥⠛⠙⠛⠛⡻⣿⣷⣟⣀⡀⣿⣿⣿⣏⠁⢸⠀⢸⣿⣆⡏⡾⣿⣸⣿⠀⣼⠀⠀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣼⣧⣧⣧⣛⣏⣆⣇⣻⣸⣪⣸⣿⠈⡌⣱⠠⢬⢳⢸⢢⣾⢸⢨⡍⡋⠍⣥⡀⠀⠀⡐⢛⢛⠛⣟⠻⣿⣿⣯⠿⢠⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣶⣱⢆⡽⠇⠀⠀⠡⣬⣼⣤⣤⣄⣵⣾⣗⣝⢸⠁⠈⠳⣤⣀⣀⣀⣀⣠⠶⠋⠀⠀⠀
⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⡅⠇⣶⢸⠶⠦⣿⡯⣯⢫⡏⡏⡍⢩⢹⠛⡟⡙⣟⢸⢙⡋⣛⣙⢛⢻⡿⠿⢄⣀⠀⠰⢿⢻⣿⣿⣿⣿⣿⣿⣿⣸⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠟⠁⠀⠀⠀⠀⠲⠢⣤⣤⣤⣤⣄⡀⠀⠀⠀⡀⠀⠀⠀⠘⣿⡿⠿⠿⢿⢿⣿⣷⣾⣷⡶⠷⠷⠾⢬⡬⢥⣧⣯⣼⣼⣏⣏⣛⣻⣘⢊⡙⣿⠇⠻⠃⠿⠸⠰⠆⡆⡏⣶⡭⡂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣷⣄⣸⣇⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡤⠀⠀⠈⡁⠉⠉⠉⠉⠉⠉⠋⠹⠹⣿⣿⣷⠆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⢀⣿⣿⡦⡀⠀⠠⠄⠀⠀⠀⢸⣿⣿⣿⣿⡶⣶⡶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠠⠀⠀⠿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡶⢶⣿⣶⣶⣤⣸⣿⣿⣿⠰⣿⣿⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣜⣿⣿
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⢰⡄⠰⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿⣿⣿⢿⣿⣿⣿⣦⣿⣿⣿⣿⣦⣤⣽⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠰⠎⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⢈⣥⣄⡙⠛⣿⡿⠛⠛⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢓⠀⠇⢴⣄⣤⢀⣀⣠⣤⣶⣿⣿⣿⣿⣤⣄⣄⠀⠀⠀⠀⣿⣃⣀⠀⠉⠉⠛⠛⠢⠬⢝⣛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣭⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⣈⠑⠘⣿⣿⣿⣿⣿⣛⣡⣴⣿⣿⣿⣿⣿⡙⠛⠿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠆⠲⣶⠏⢶⠶⠰⠴⠶⠆⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡍⠻⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠃⠀⠀⠠⠉⠛⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿
⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣬⣨⠀⠀⠀⠀⢸⣿⣿⣀⣀⡀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⡇⠀⠀⣯⣿⡇⠀⠀⠀⣸⣿⡏⢻⡿⢛⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⣧⣎⣵⢸⡡⣮⡱⡆⣿⡽⣛⢹⣿⢨⢫⣝⡇⡇⡟⢟⣿⡟⣿⡻⣛⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀
⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⣿⣿⡇⠀⠀⢠⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣾⣷⣿⣷⣷⣯⣵⣿⡿⣼⣧⣿⣾⣼⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣉⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⣿⣿⡇⠀⠀⣿⢹⡝⣍⡍⡿⣸⢺⢻⡻⢟⡿⡿⠿⠿⠿⠿⢿⠿⢿⠿⠿⠿⡿⢿⣿⣿⣿⡟⡿⣿⣛⢻⣿⣿⣿⣿⠋⠉⣠⠞⣁⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀⠀⣿⣿⣧⣀⣠⣿⣾⣾⣾⣷⣷⣿⣼⣼⣧⣮⣯⣿⣴⣷⣇⣿⣢⣻⣹⣸⣇⣧⢫⠏⠾⢰⠸⠆⡇⣷⠸⢸⣿⣿⣿⣿⠈⢰⠇⢸⣿⡇⣿⢸⣿⢠⣿⠈⣧⠀⠀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣿⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⢿⣿⣿⣅⣼⣿⣿⣿⣿⢿⣿⣿⣿⣥⠛⠛⠛⠛⢿⣿⣷⣟⣰⢸⣿⣿⣿⡏⡀⢸⠀⢸⣿⣆⡏⡾⣿⣸⣿⠀⣼⠀⠀
⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣥⣧⣧⣛⣏⣆⣇⣻⣸⣪⣸⣿⠈⡎⣱⠠⢬⠳⢸⢢⣾⢸⢨⡍⡋⠍⣴⡄⠀⠀⡄⢛⢛⠻⣟⢻⣿⣿⡏⠗⡧⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣾⣶⣱⢆⡽⠇⠀⠘⠡⣬⣼⣤⣤⣼⣷⣾⣃⣍⡇⠀⠈⠳⣤⣀⣀⣀⣀⣠⠶⠋⠀⠀⠀
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡫⡅⠇⣶⢸⠶⢾⣿⡯⣯⢫⡏⡍⡍⢫⢹⢙⡙⡛⣟⢸⢙⡋⣛⣙⢛⢿⡿⠿⢄⡀⠀⠼⢾⢻⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠏⠀⠀⠀⠚⠖⠦⠤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠿⠿⠿⡿⣿⣿⣷⣶⣷⠶⠓⠷⠾⣬⠸⠥⣧⢯⣽⣼⣏⣏⣛⣻⣘⢊⡙⣿⠃⠛⠃⡿⠸⠰⠆⡆⡏⣶⡍⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⣽⣿⣿⣿⣿⣿⣦⣄⠀⢠⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠄⠀⠈⢈⠁⠉⠉⠉⠉⠉⠛⠙⠋⢿⣿⣿⡷⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢘⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠀⠀⠀⠀⠐⠀⣸⣿⣷⣀⡀⠀⠤⠀⠀⠀⠀⣿⣿⣿⣿⣷⢴⣶⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠈⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠶⣾⣷⣶⣦⡄⣿⣿⣿⡃⢾⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣤⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢀⠀⣦⠀⠆⠀⠀⠀⠀⠀⠀⠀⣿⡏⣿⣿⣿⡿⣿⣿⣿⣷⣼⣿⣿⣿⣧⣤⣽⣿⣿⣿⣿⣶⣶⣶⣦⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠾⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⡘⠛⢿⣿⠛⠛⢿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠛⠂⢀⢰⣆⣤⣀⣀⣀⣤⣤⣶⣿⡿⢿⣆⣠⣤⣄⠀⠀⠀⢸⣛⣀⠈⠉⠉⠛⠛⠿⠭⢝⣛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠉⠙⢸⣿⣿⣿⣿⣟⣋⣴⣾⣿⣿⣿⣿⣯⠛⠿⢿⢿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠖⠶⣶⠿⢷⠄⠠⠰⠶⠆⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠙⠀⠀⠀⠈⠙⠻⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠉⠙⠛⠻⠿⠿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⢀⡀⠀⠀⠀⠀⣿⣿⡇⢀⣀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠿⢿⣿⣿⡟⠀⠀⢸⣿⣇⠀⠀⠀⢰⣿⣿⡏⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿⡇⣎⣵⢸⡱⣮⡱⡆⣿⡽⣛⢹⣿⢨⢫⣝⡇⡇⡟⢟⣿⡟⣿⡻⣛⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀
⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣼⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣾⣷⣿⣷⣷⣯⣵⣿⡿⣼⣧⣿⣾⣼⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣉⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣸⣿⡿⠀⠀⢰⣿⢹⡝⣍⡍⡿⣸⢺⢻⡻⡟⡿⠿⡿⠿⠿⠿⢿⠿⢿⠿⠿⠿⡿⢿⣿⣿⣿⡟⡿⣿⣛⣿⣿⣿⣿⡟⠉⠀⣠⠞⣁⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀
⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣭⣭⣽⣿⣿⣿⣷⣀⢀⣸⣿⣾⣾⣾⣷⣷⣿⣼⣼⣯⣦⣿⣿⣵⣷⣧⣿⣢⣻⣹⣸⣇⣧⢫⡇⠾⢰⠸⠆⡇⣷⠐⣽⣿⣿⣿⣧⡇⢰⠇⢸⣿⡇⣿⢰⣿⢠⣿⠈⣧⠀⠀
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣯⣼⣿⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⢿⣿⡯⣤⣿⣿⣿⣿⣿⢿⣿⣿⣿⣥⠛⠛⠛⠛⢿⣿⣷⣟⣅⣿⣿⣿⣿⢹⠅⢸⠀⢸⣿⣆⡏⡾⣿⣸⣿⠀⣼⠀⠀
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⣿⣼⣥⣧⣧⣛⣏⣆⣇⣻⣸⠪⣸⣿⢘⡆⣱⠠⣬⢲⢸⢢⡾⢸⢨⡍⡋⠍⣤⡀⠀⠀⡟⢛⢛⠛⣟⣻⣿⣻⡆⢾⠠⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⣿⢿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣾⣶⣱⢆⡽⠁⠀⠀⠡⢬⣼⣤⣤⣽⣷⣾⣁⣜⠀⠀⠈⠳⣤⣀⣀⣀⣀⣠⠶⠋⠀⠀⠀
⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⢸⡫⡅⠇⣶⢸⠶⢾⣿⡯⣮⢩⡏⡏⡍⢫⢹⢛⡙⡟⣟⢸⢙⡋⣛⣙⢛⢿⡿⠷⢄⡀⠀⠼⢾⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠋⠀⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠟⠿⠿⣿⢿⣿⣿⣷⡶⡷⠒⠓⠷⢾⠌⠬⢥⡧⣯⣽⣼⣏⣏⣛⣻⣘⢊⡙⣿⠃⠛⠃⠿⠸⠰⠆⡆⡏⣶⡍⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢁⣤⣶⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢛⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡤⠀⠀⠉⡈⠁⠉⠉⠉⠉⠛⠋⠛⢹⣿⣿⣷⠇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠙⠀⠀⠀⠀⠀⠂⢸⣿⣿⣆⣀⠀⠠⠄⠀⠀⠀⢸⣿⣿⣿⣿⡇⣶⡖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⡶⢶⣿⣶⣶⣤⣸⣿⣿⡟⢴⣷⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⣤⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⡀⢰⡀⢡⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⣿⣿⣿⢿⣿⣿⣿⣤⣿⣿⣿⣿⣄⣉⣿⣿⣿⣷⣶⣶⣤⣤⣤⣴⣾⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠰⡇⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠛⠛⣿⡟⠛⠻⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠲⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠓⠛⠒⠀⠄⠶⣤⡄⢀⣀⣀⣤⣴⣶⣿⣿⣦⣠⣴⣦⡀⠀⠀⠀⣟⡋⠉⠉⠙⠛⠻⠿⠭⣙⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⣿⣿⣿⠿⢿⣿⣷⣆⡀⠀⠀⠀⠀⠹⠋⣿⣿⣿⣿⣿⣛⣹⣷⣿⣿⣿⣿⣿⡟⠻⢿⡿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠺⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠖⠲⣶⠀⠶⠄⠰⠶⠶⠶⢢⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠃⠀⠀⠌⠉⢛⠿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠉⠉⠙⠛⠻⠿⠿⣿⣿⣿
⣿⣿⣿⣿⡁⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⠀⠀⠀⠀⢸⣿⣧⠀⣀⣀⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠻⠿⠿⣿⣿⠁⠀⠀⣌⡒⠀⠀⠀⠀⢸⣿⣿⡌⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀
⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣽⣷⡇⠀⠀⠀⣿⣿⣿⠇⣎⡵⢸⡱⣮⡱⡆⣿⡽⣛⢹⣿⢨⢫⣝⡇⡇⡟⢟⣿⡟⣟⡻⣛⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀
⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢀⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣶⣷⣿⣷⣷⣯⣵⣿⡿⣼⣧⣿⣾⣼⣼⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣉⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣿⣶⣦⣤⡀⣿⣿⠇⠀⠀⢸⣿⢹⡝⢍⡍⡿⣸⢺⢻⡿⠟⡟⠿⡿⠿⠿⠿⢿⠿⢿⠿⠿⠿⠿⢿⣿⣿⣿⣟⡿⣿⣻⣿⣿⣿⣿⠋⠁⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠼⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣄⢀⢀⣼⣿⣾⣾⣾⣷⣷⣿⣼⣼⣯⣦⣿⣿⣴⣷⣇⣿⣢⣻⣸⣸⣇⣧⢣⠇⠾⢰⠸⠆⡇⣷⠸⣽⣿⣿⣿⡄⠀⢰⠇⢸⣿⡇⣿⢰⣿⢠⣷⠈⣧⠀⠀
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⡯⢿⣿⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⣿⢿⠁⣼⣿⣿⣧⣿⣿⢿⣿⣿⣝⡀⠀⠈⠉⠛⠛⢿⣷⣟⣸⣿⣿⣿⡇⡇⠀⢸⠀⢸⣿⡆⡏⡾⣿⣸⣿⠀⣼⠀⠀
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣃⣴⣯⣬⣥⣧⣧⣻⣏⣂⣃⣻⣘⠀⣸⣷⣘⡌⣰⠸⢬⢲⢸⢠⡔⢸⢨⠌⡋⠍⣤⡀⠀⠀⡀⢘⢛⠻⣟⣻⣿⣻⡄⠇⠠⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠐⠉⣿⣿⢿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣧⣮⣿⣹⣏⣿⣿⣿⣿⣶⣷⣶⣾⣶⡁⢆⡹⠀⠀⠀⠀⢬⣬⣤⣼⣽⣗⣪⣁⡍⠀⠀⠈⠳⣤⣀⣀⢀⣀⣠⠶⠋⠀⠀⠀
⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⢿⡃⠄⠇⣶⢘⠶⢾⣿⠫⣮⢩⡍⡍⡍⢫⢹⠉⡙⡛⣟⢸⢙⡋⣛⢙⢛⢻⡿⠶⠀⠀⠀⠀⢶⢛⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠏⠀⠀⠀⢸⠿⠛⠻⠿⡿⢿⣿⣿⣷⠶⠗⠒⠓⠳⢾⠸⢬⣥⢧⣯⣽⣼⣏⣏⣛⣹⣘⢊⡑⡟⠃⠛⠃⠿⠸⠀⠆⡆⡏⢶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠄⠀⠀⢉⠈⠁⠉⠉⠉⠉⠙⠙⠁⢿⣿⣿⠷⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠠⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⣿⣿⣿⣿⣿⣿⣶⣤⣠⡀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠐⠀⣸⣿⣧⡀⠀⠀⠤⠀⠀⠀⠀⣿⣿⣿⣿⡷⢲⡶⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤
⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣭⣶⠶⣾⣷⣶⣦⡄⣿⣿⣿⡡⣶⣦⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⣠⡴⠾⣿⣿⣿
⠀⠀⠀⠀⠀⠁⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣿⣿⣿⡿⣿⣿⣿⣷⣴⣿⣿⣿⣧⣈⣻⣿⣿⣶⣶⣶⣤⣤⣤⣤⣤⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠴⠂⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢉⣁⡈⠛⢻⣿⠛⠛⢻⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠙⠂⠐⠠⠂⢀⠀⣀⣀⣀⣤⣶⣿⣿⣿⣦⣤⣄⡀⠀⠀⠀⠘⠏⠉⠉⠙⠛⠿⠿⠭⣙⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠳⠷⠶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠈⢡⣿⣿⣿⣿⣟⣛⣹⣾⣿⣿⣿⣿⣯⠛⠿⣿⢿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠉⠒⠺⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠆⠲⣶⠏⢶⠦⠴⠶⢶⣦⣀⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠆⠀⠀⠈⠙⠻⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠈⠉⠉⠛⠛⠿⠿⢿⣿⣿⣿⣿
⣿⣿⣿⣿⡀⠉⠍⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⢶⣲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣻⣿⠿⠿⠿⠻⠿⠿⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡌⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀
⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠐⢱⢦⡀⠀⠀⠀⣿⣿⣿⠇⣎⡵⢸⡱⣮⡱⡆⣿⡽⣛⢹⣿⢨⢫⣝⡇⡇⡟⢟⣿⡟⣟⡻⣛⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢀⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣶⣷⣿⣷⣷⣯⣵⣿⡿⣴⣧⣿⣾⣼⣼⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⣁⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⣿⣿⡇⠀⠀⠀⠀⠀⢀⣤⣤⣤⣶⣦⣤⣀⣀⠀⠀⠀⣼⣿⠂⠀⠀⢸⣿⢹⡝⢉⡍⡿⣸⢺⢻⡻⠟⡟⢿⡿⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣟⡿⣿⣻⣿⣿⣿⣿⠉⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⣿⣿⠇⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⣿⣿⡀⠀⡀⣼⣿⣶⣾⣾⣷⣷⣯⣼⣼⣯⢠⣿⣿⣴⣷⣇⣿⣠⣻⣸⣸⣇⣧⢣⠇⠾⢰⠸⠆⡇⣷⠸⣽⣿⣿⡇⡇⠀⢰⠇⢸⣿⡇⣿⢰⣿⢀⣷⠈⣧⠀⠀
⣿⣟⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⣿⢸⠁⣼⣿⣿⣦⣿⣿⢯⣍⣟⣟⡀⠀⠈⠉⠉⠙⠻⣷⣞⣸⣿⣿⣿⡅⡇⠀⢸⠀⢸⣿⡆⡏⡾⣿⣸⣿⠀⣸⠀⠀
⣿⣿⠀⠀⠀⠀⠀⠀⢀⣤⣬⣭⣽⣿⣿⣿⣿⠋⠀⠀⣿⣯⣬⣥⣧⣧⣻⣏⣂⣃⣻⣘⠀⣸⢿⣘⠈⣰⠰⢬⢲⢸⠠⡔⢸⢨⡌⡋⠅⣤⠀⠀⠀⠀⢐⢛⠛⣟⣻⣿⣻⣦⠇⠀⠘⣯⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⢿⣛⡩⠤⠀⠀⣿⣿⢿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣠⣸⣿⢸⣇⣿⣿⣿⣶⣶⣷⣶⣾⣶⡐⢆⡑⠀⠀⠀⠀⠨⣬⣤⣼⣽⣗⣪⣑⡄⠀⠀⠈⠳⢤⣀⣀⢀⣀⣠⠖⠋⠀⠀⠀
⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠄⠇⢶⢘⠶⢾⣿⠫⣮⢩⡍⡍⡍⢫⢹⠉⡛⡛⣟⢸⢙⡋⣛⢙⢛⢻⠿⠃⢀⡀⠀⠀⢴⢚⣿⣿⣿⣿⣿⣹⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⢸⠟⠙⠛⠿⠿⢿⣿⣿⣷⠲⠖⠒⠓⠳⠾⠨⢤⣥⢧⣯⣼⣼⣏⣏⣛⣹⣘⢊⡑⡿⠃⠛⠃⠿⠸⠀⠆⡆⡏⢖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠀⣠⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠉⠈⠀⠉⠉⠉⠉⠉⠉⠁⢿⣿⣿⠗⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣤⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⣜⣿⣿⣿⣿⣶⣤⣀⠀⡀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣸⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⣸⣿⣷⠀⠀⠀⠤⠀⠀⠀⠀⣿⣿⣿⣿⡗⢰⡶⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤
⠀⠀⠀⠘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣬⣶⠶⣾⣷⣶⣦⡄⣿⣿⣿⢡⣶⣬⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⣤⡶⠾⣿⣿⣿
⠀⠀⠀⠀⠀⠠⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠄⠀⠀⠀⠀⠀⠀⠀⣿⡏⣿⣿⣿⡿⣿⣿⣿⣷⣴⣿⣿⣿⣇⣈⣻⣿⣿⣶⣶⣶⣤⣤⣤⣤⣤⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠴⠂⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⡈⠛⢻⣿⠛⠛⢻⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠐⠂⠀⠀⠀⢀⢀⣀⣀⣀⣤⣾⣿⣿⣿⣧⣤⣀⡀⠀⠀⠀⠸⠋⠉⠉⠛⠛⠿⠿⢍⣙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣨⣙⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣟⣋⣱⣴⣿⣿⣿⣿⣯⠛⠿⢿⢿⣷⡀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠺⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠆⠲⣶⠄⢶⠄⠠⠼⠿⠏⢉⣉⣉⣩⣥⣴⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠂⠀⠀⠈⠙⢳⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠈⠉⠙⠛⠛⠿⠿⢿⣿⣿⣿⣿
⣿⣿⣿⣗⠈⡄⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣾⡗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⠀⠀⢠⣱⠔⠀⠀⠀⠀⣿⣿⣿⡄⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡟⠀⠀⢸⣿⣶⠀⠀⠀⠀⣿⣿⣿⠇⣎⡵⢸⡱⣮⡹⡆⣿⡹⣛⢹⣿⢨⢫⣝⡇⡇⡟⢟⣿⡟⣟⡻⣛⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣿⣿⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣶⣷⣿⣷⣷⣯⣵⣿⡿⣼⣧⣿⣾⣼⣼⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⣁⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⢹⡝⢉⡍⡟⣸⢺⢻⡿⠟⡿⠿⡿⠿⠿⠿⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⣟⡿⣿⣿⣿⣿⣿⡟⠉⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢀⣾⣿⣶⣾⣾⣷⣷⣿⣾⣼⣧⢤⣿⣿⣴⣷⣇⣿⣠⣻⣸⣸⣇⣧⢣⠇⠾⢰⠸⠆⡇⣷⠸⣽⣿⣿⡇⡇⠀⢰⠇⢸⣿⡇⣿⢰⣿⢠⣿⠈⣧⠀⠀
⠀⠀⠀⠀⠉⠉⠿⠿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⣿⣿⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⣿⢸⠡⣿⣿⣿⣿⣿⣿⢽⣿⣿⣝⡀⠈⠉⠛⠙⠛⠻⣷⣟⣼⣿⣿⣿⣁⡇⠀⢸⠀⢸⣿⡆⡏⡾⣿⣸⣿⠀⣸⠀⠀
⠀⣀⡀⠀⠀⣀⣀⣤⣾⣿⣿⣿⣍⠀⠀⠀⠀⠀⠀⠀⣿⣯⣬⣥⣧⣧⣻⣏⣆⣃⣻⣘⠀⣘⢿⡘⠈⣰⠰⢬⢲⢸⠠⡖⢸⢨⡌⡋⠅⡄⠀⠀⠀⠀⢘⢛⠛⣟⣻⡿⣻⣼⠇⠀⠘⣯⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠷⠄⠀⣰⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣭⣑⣸⣿⣸⣿⣿⣿⣿⣾⣶⣷⣶⣾⣶⡁⢆⡙⠁⠀⠀⠀⢨⣬⣤⣼⣽⣗⣪⣑⡄⠀⠀⠈⠳⢤⣀⣀⢀⣀⣠⠖⠋⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣿⡃⠄⠇⣶⢘⠶⢾⣿⠫⣮⢩⡍⡍⡍⢫⢹⠉⡛⡋⣟⢸⢙⡋⣛⣙⢛⢻⡿⠆⢀⠀⠀⠀⢼⢛⣿⣿⣿⣿⣿⣸⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⡀⠀⠀⠀⣾⣿⠀⠀⠀⠀⢸⠟⠛⠛⢿⠿⢿⣿⣿⣷⠶⠓⠒⠓⠳⠾⠨⢬⣥⢧⣯⣼⣼⣏⣏⣋⣹⣘⢊⡑⡿⠃⠓⠃⠿⠸⠀⠆⡆⡏⢖⡈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⣠⢻⣿⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠉⠈⠀⠉⠉⠉⠉⠉⠙⠁⢿⣿⣿⠗⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣠⣤⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⣂⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠿⣿⣿⣿⣶⣬⣐⠠⡀⠀⢀⡖⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⡿⢶⣿⣿⣽⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⣿⣿⣗⢀⠀⠀⠄⠀⠀⠀⠀⣿⣿⣿⣿⠧⢰⡶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⣷⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣮⣶⠶⣾⣷⣶⣦⡄⣿⣿⣿⢰⣶⣬⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⣤⠶⡿⣿⣿⣿
⠀⠀⠀⠀⠙⠛⠛⠻⢿⣿⣿⣿⣿⣷⣬⣿⡟⣿⣧⠀⠀⠀⠀⠀⠀⠀⣆⠀⠄⠀⠀⠀⠀⠀⠀⠀⣿⡏⣿⣿⣿⡿⣿⣿⣿⣷⣼⣿⣿⣿⣇⣈⣿⣿⣿⣶⣶⣶⣤⣤⣤⣴⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⢿⣷⡀⠀⠀⠀⠲⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⡘⠛⢻⣿⠛⠛⢿⠟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡧⠀⠀⠀⠀⠀⠀⠀⠂⠀⢄⣀⣀⣀⣤⣴⣿⡿⠿⣆⣠⣤⣄⠀⠀⠀⢘⡋⠉⠉⠙⠛⠻⠿⠭⣉⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣟⣋⣰⣶⣿⣿⣿⣿⣯⠛⠿⢿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠆⠲⣶⠉⢷⠔⠺⠾⠿⠿⠛⠛⠛⢛⣋⣭⣷⣾⣿⣿⣿⣿⣽⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠙⠃⠀⠀⠈⠙⠻⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠉⠉⠙⠛⠻⠿⠿⣿⣿⣿⣿
⣾⡿⠿⠦⠀⠽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⢀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠀⠀⢀⡀⠀⠀⠀⠀⠀⣾⣿⣿⡌⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡏⠀⠀⠰⣞⣿⠀⠀⠀⠀⣿⣿⣿⠇⣎⡵⢸⡡⣮⡱⡆⡿⡽⣛⢹⣿⢨⢫⣝⡇⡇⡟⢟⣿⡟⣟⡻⣛⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀
⣴⣶⣶⣶⣿⣿⣿⣶⣦⣄⠀⠀⠀⣿⣿⣿⡇⠀⠀⠘⣿⣿⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣶⣷⣿⣷⣷⣯⣵⣿⡿⣴⣧⣿⣾⣼⣼⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⣁⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢻⣿⣿⠇⠀⠀⠀⣿⡿⠀⠀⠀⢸⣿⢹⡝⢉⡍⡟⢸⢺⢻⡻⠟⠟⠿⠿⠻⠿⠿⠿⠿⠿⠿⠿⠿⡿⣿⣿⣿⣿⣟⡿⣟⣿⣿⣿⣿⡟⠉⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠉⢙⣟⡛⠿⣿⣿⣿⣿⣿⣟⠃⠀⢸⣿⣿⠀⠀⠀⠀⣿⣧⠀⠀⢀⣾⣿⣶⣾⣾⣷⣷⣿⣼⣼⣧⢤⣿⣿⣵⣥⣇⣯⣠⣻⣸⣸⣇⣇⢣⠇⠶⠰⠸⠆⡇⣧⠸⣽⣿⣿⡇⡇⠀⢰⠇⢸⣿⡇⣿⢰⣿⢀⣿⠈⣧⠀⠀
⠀⠀⢀⣀⢰⣾⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⣿⢸⠄⣿⣿⣿⣧⣾⣽⢽⣯⣿⣝⡁⠉⠉⠉⠋⠛⠛⢷⣟⣽⣿⣿⣿⢡⡇⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣼⠀⠀
⠀⠀⠀⠉⠉⠛⠻⠛⠛⠛⠃⠀⠀⣸⣿⣿⠀⠀⠀⠀⣿⣯⣬⣥⣧⣧⣛⣏⣂⣃⣻⣘⠀⡠⢹⡘⡄⣰⠠⠬⠲⢸⠠⡖⢸⢨⠌⡋⠅⣀⠀⠀⠀⠀⢐⢛⠛⣟⣻⡿⣻⢸⠇⠀⠘⣯⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⡿⢿⣿⣿⣿⣿⡛⣿⣿⣿⣿⢝⣀⣼⣿⣼⣉⣿⣿⣿⣶⣶⣷⣶⣾⣶⡑⠆⠑⠀⠀⠀⠀⠈⢬⣤⣽⣽⣗⣪⣀⠈⠀⠀⠈⠳⢤⣀⣀⢀⣀⣠⠖⠋⠀⠀⠀
⠀⠀⠀⠀⢰⣶⣿⣿⡆⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⡃⠀⠇⢶⢘⠶⢾⣿⠫⣮⢩⡍⡍⡍⢫⢹⠉⡟⡉⣟⢸⢙⡋⣛⢙⢛⢻⡿⠆⢀⡀⠀⠀⢤⢚⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠈⠻⣿⡇⠀⠀⠀⠀⢿⠋⠙⠛⢿⠿⢿⣿⣿⡷⠒⠂⠒⠑⠳⠾⠬⠤⠥⠧⢯⣼⣬⣏⣏⣋⣹⣘⢊⡑⣿⠃⠛⠃⠿⠸⠀⠆⡆⡏⢖⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠬⠇⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠸⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡤⠀⠀⠀⠉⠈⠀⠉⠉⠉⠉⠉⠉⠀⣿⣿⣿⠇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⡤⣶⣶⣂⡠⠄⠀⠀⠀⠀⣼⡀⢺⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠘⢿⣿⣿⣿⣷⣬⣐⢴⣾⣿⣿⢸⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠤⣿⣿⣿⣿⣿⣿⣿⣛⡛⢿⣿⣧⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⣿⣿⡆⢀⠀⠀⠄⠀⠀⠀⢸⣿⣿⣿⣿⠂⢰⠶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣬⣶⢶⣿⣶⣶⣦⢸⣿⣿⣿⠰⣿⣮⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣠⣤⣞⡻⣿⣿⣿
⠀⠀⠀⠀⠉⠛⠛⣛⣿⣿⣿⣿⣿⣿⣷⣬⢧⡹⣿⣧⡀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡽⣿⣿⣿⢿⣿⣿⣿⣷⣼⣿⣿⣿⣦⣬⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⠻⣷⠀⠀⠀⠞⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠛⠿⡿⠛⠛⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣽⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⣀⣀⣠⣤⣶⣿⣿⣷⣤⣴⣶⣄⠀⠀⠀⢚⣁⡀⠉⠉⠛⠛⠿⠤⠍⣛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣧⣄⣀⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣟⣋⣷⣿⣿⣿⣿⣽⣯⠛⠿⢿⢿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠂⠲⣶⠋⢶⠔⠰⠶⠿⠟⠛⠛⠛⢛⣉⣭⣶⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠈⠀⠀⠀⠈⠙⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠙⠛⠻⠿⠿⣿⣿⣿
⠶⠾⠶⠐⠀⠱⢡⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣢⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡄⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀
⢀⣀⣀⣀⣀⣄⣀⡀⠀⠀⠀⢹⣿⣿⣿⣿⡇⠀⠀⢀⠀⣶⠀⠀⠀⢠⣿⣿⣿⠇⣎⡵⢸⡡⣮⡱⡆⡿⡽⣛⢹⣿⢨⢫⣝⡇⡇⡟⢟⣿⡟⣟⡻⣛⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⣮⣽⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣶⣷⣿⣷⣧⣯⣵⣿⡿⣴⣧⣿⣾⣼⣼⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⣁⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⠀⢹⣿⣿⣿⠀⠀⠀⠀⣽⡟⠀⠀⠀⢸⣿⢹⡝⢉⡍⡟⣸⢺⢻⡿⠟⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⣿⣿⣿⣿⣟⡿⣿⣿⣿⣿⣿⡟⠉⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠶⠶⠖⢉⣙⣿⣿⣿⣿⠿⢦⠀⠘⣿⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⣾⣿⣶⣾⣾⣷⣷⣿⣾⣼⣧⣮⣯⣿⣵⣥⣇⣯⣩⣻⣸⣸⣇⣇⢣⠇⠶⢰⠸⠆⡇⣧⠸⣽⣿⣿⣇⠀⠀⢰⠇⢸⣿⡇⣿⢰⣿⢀⣿⠈⣧⠀⠀
⠀⠀⠶⠎⢿⣿⡿⠿⠯⠥⠈⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⣿⡿⡇⠀⣿⣼⣿⣀⣾⣿⢸⣿⣶⣝⡀⠈⠉⠉⠛⠛⢛⢷⡟⣽⣿⣿⣿⢸⠀⠀⢸⠀⢸⣿⡆⡏⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⠀⣿⣯⣬⣥⣧⣧⣛⣏⣂⣃⣻⣘⠀⡨⢿⡈⡄⣰⠠⠬⠲⢸⠀⡄⢸⢨⠌⡋⠅⡀⠀⠀⠀⠀⢘⢛⠛⣟⡻⡿⣻⠈⠆⠀⠘⣯⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⡿⢿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣯⣡⣽⣇⡇⣃⣿⣿⣿⣶⣶⣷⣶⣾⣶⡀⢆⡑⠁⠀⠀⠀⠈⢬⣤⣽⣽⣗⣪⣀⠀⠀⠀⠈⠳⢤⣀⣀⢀⣀⣠⠖⠋⠀⠀⠀
⠀⠀⠀⠀⠹⢿⣿⡿⠇⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⣿⡃⠀⠇⣶⢘⠶⢾⣿⠫⣮⢩⡍⡍⡍⢫⢹⠉⡟⡉⣟⢸⢙⡋⣛⣙⢛⢻⡿⠆⢀⡀⠀⠀⢬⢸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠛⣿⡇⠀⠀⠀⠀⡿⠉⠙⠛⢿⠻⣿⣿⣿⡷⠒⠂⠀⠓⠷⠞⠬⠬⠥⠧⠯⣼⣬⣏⣇⣋⣹⣘⢊⡑⠿⠃⠛⠃⠿⠸⠰⡆⡆⡏⢖⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⡇⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⠈⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⡤⠀⠀⠀⠉⠀⠀⠉⠉⠉⠉⠉⠉⠸⣿⣿⡿⠇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⢖⣻⣿⣤⣔⡂⠄⡀⠀⢀⣶⣑⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⣝⣿⣿⣿⣿⣶⣌⡒⢿⣿⣿⡸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣭⣝⠻⣿⣷⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠠⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⣶⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶
⠀⠀⠀⠁⢿⣿⣿⣿⣿⣿⣿⣿⣷⣌⡋⢻⡯⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣼⡶⢶⣿⣶⣶⣤⢸⣿⣿⣏⠸⣿⣯⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣴⣛⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠉⢉⣽⣿⣿⣿⣿⣿⣿⣿⣦⣹⣎⠻⣷⡀⠀⠀⠀⠀⢠⡄⠐⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿⣿⣿⢿⣿⣿⣿⣦⣿⣿⣿⣿⣤⣬⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠫⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡼⣿⠀⠀⠀⠊⡸⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡙⠛⠿⠟⠋⠛⠿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠋⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣄⣤⣤⣴⣾⣿⣿⣿⣤⣤⣤⡀⠀⠀⠀⣛⣀⡀⠈⠉⠙⠛⠓⠤⠍⣙⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣧⣄⣀⡀⣀⡀⠀⠀⠀⣼⣿⣿⣿⣿⣟⣩⣷⣿⣿⣿⣿⣿⡉⠛⠿⢿⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠲⣶⠁⢶⠄⠰⠶⠿⠟⠉⠉⠉⣉⣉⣥⣶⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠋⠀⠀⠀⠈⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠉⠛⠛⠿⠿⣿⣿⣿
⣵⢟⠿⠄⠀⠄⠰⡲⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣕⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⡄⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀
⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢹⣿⣿⠁⠀⠀⠀⢀⣆⠀⠀⠀⢸⣿⣿⣿⠇⣎⡵⢸⡱⣮⡹⡆⣿⡹⣛⢹⣿⢨⢫⣝⡇⡇⡟⢟⣿⡟⣟⡻⣛⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⣿⣿⠀⠀⠀⢈⣷⡟⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣶⣷⣿⣷⣷⣯⣵⣿⡿⣴⣧⣿⣾⣼⣼⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⣁⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠾⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⣿⠀⠀⠀⠨⠞⠇⠀⠀⠀⢸⣿⢹⡍⢉⡍⡟⣸⢺⢻⡟⠟⠛⠿⠟⠻⡟⠟⢿⢿⢿⠿⢿⠿⣿⣿⣿⣿⣿⣟⡿⣯⣿⣿⣿⣿⡏⠁⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠐⡒⠒⠻⠿⣋⣻⣿⣿⣿⣿⣆⠀⢹⡏⠀⠀⠀⢐⣵⡄⠀⠀⠀⣾⣿⣶⣾⣶⣷⣷⣿⣾⣼⡧⣮⡯⣿⣵⣥⣇⣏⣢⣻⣸⣸⣇⣇⢣⠇⠶⢰⠸⠆⡇⣧⠸⣽⣿⣿⢳⠀⠀⢰⠇⢸⣿⡇⣿⢰⣿⢀⣷⠈⣧⠀⠀
⠀⠀⠀⠒⠶⠶⣿⣿⣿⣿⣿⣿⣈⠀⢸⡇⠀⠀⠀⢸⣿⣷⢚⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⡇⡇⠀⣿⣾⣿⣤⣦⣶⢼⣵⣿⣭⡀⠈⠈⠉⠉⠙⢿⢷⣟⣽⣿⣿⣏⢸⠀⠀⢸⠀⢸⣿⡆⡏⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣼⡇⠀⠀⠀⢸⣿⣯⣬⣥⣧⣧⣛⣏⣂⣃⣻⣘⠀⠈⣿⡀⡄⣰⠠⠬⠲⢸⠀⡄⢸⢨⡌⡋⠅⡀⠀⠀⠀⠀⢐⢛⠻⣟⡻⡿⣻⡈⠀⠀⠘⣯⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⣿⡿⢿⣿⣿⣿⣿⡛⣿⣿⣿⣟⣍⣀⣽⡇⡏⢸⣿⣿⣿⣶⣶⣷⣶⣾⣶⡁⠆⠑⠀⠀⠀⠀⠈⢼⣤⣽⣽⣕⣪⣁⡀⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠖⠋⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠂⠀⠀⣸⣿⡇⠀⠀⠀⢸⣿⡃⠀⠇⣶⢘⠶⢾⣿⠋⣮⢩⡍⡍⡍⢣⢹⢙⡟⡍⣟⢸⢙⡋⣛⣙⢛⢻⣿⠆⢀⣀⠀⠀⠴⢸⣿⣿⣿⣿⡏⡟⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡃⠀⠀⠀⠘⡏⠁⠙⠛⠿⠻⣿⣿⣿⠗⠒⠀⠀⠐⠷⠜⠬⠬⠥⠧⠭⠽⠬⢏⣏⣉⣸⣘⢊⡑⣿⡃⠛⠃⠿⠸⠰⡆⡆⡏⢖⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠐⠸⢿⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠃⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⠄⠀⠀⠀⠀⠀⠀⠁⠁⠈⠉⢉⡉⠸⠿⣿⠿⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠙⣽⣷⣷⣬⣐⠠⡀⠀⠀⡀⢠⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⣿⣷⣬⣑⣾⣿⡄⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣍⢻⣿⣹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⠀⠀⢐⡀⠀⠀⠀⢸⣿⣿⣿⠗⠀⠶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶
⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣷⣽⠻⣿⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣛⢿⣿⣷⣶⠦⣿⣿⣿⡏⠾⣷⣯⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣖⣛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠐⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢹⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣤⣬⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠠⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡀⠀⠀⠐⠃⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⡉⠩⠿⠋⠉⠙⠟⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⣤⣤⣶⣾⣿⠿⠛⣿⣶⣦⣤⡀⣀⣀⣀⣭⣀⡀⠀⠉⠙⠛⠓⠤⠍⣛⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠽⣿⣿⣿⣿⡿⠿⣟⣻⣿⣿⣦⣠⣀⣀⣀⣀⡀⠀⢠⣿⣿⣿⣿⣯⣥⣼⣿⣿⣿⣿⣿⣧⡉⠛⠻⢟⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠲⣶⠏⢶⠄⠰⠶⠶⠦⢀⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠃⠀⠀⠀⢈⠹⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠉⠙⠛⠻⠿⠿⣿⣿⣿
⢦⠠⠀⠀⠀⠠⢐⣠⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣯⠿⠀⠀⠀⠀⠀⠉⠘⠛⠛⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⡄⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⡄⢆⡄⠀⠀⠀⣸⣿⣿⣿⠇⣎⡵⢸⡱⣮⡱⡆⣿⡹⣛⢹⣿⢨⢫⣝⡇⡇⡟⢟⣿⡟⣟⡻⣛⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⢬⣿⡃⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣶⣷⣿⣷⣧⣯⣵⣿⡿⣴⣧⣿⣾⣼⣼⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⡼⡽⠁⠀⠀⠀⣿⣿⢹⡝⢉⡍⡿⣸⢺⢻⡛⠟⠛⠿⠟⠻⠟⠟⢿⠿⢿⠿⠿⠿⣿⣿⣿⣿⣿⣟⡿⣿⣿⣿⣿⣿⠋⠁⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠤⠶⠶⣿⣿⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣯⣤⠀⠀⠀⣠⣿⣿⣶⣾⣾⣷⣷⣿⣾⣼⣧⣮⣯⣿⣥⣥⢇⣯⣠⣻⣸⣸⣇⣇⢣⠇⠶⢰⠸⠆⡇⣶⠸⣽⣿⡿⡇⠀⠀⢰⠇⢸⣿⡇⣿⢰⣿⢀⣷⠈⣧⠀⠀
⠀⠀⠀⠠⠀⠠⢶⣤⣽⣶⣾⣿⣿⣟⠋⡀⠀⠀⠀⣿⣿⣀⢒⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⢿⠀⢸⣯⣿⣧⣦⣾⣿⢿⣿⣿⣿⡁⠚⠋⠉⠛⠛⢿⣷⣟⣽⣿⣿⡇⡇⠀⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠑⠒⠀⠀⠀⠀⢿⣟⠋⣬⣥⣧⣧⣛⣏⣂⣃⣻⣘⠀⣰⣿⢈⠀⣰⠠⠬⠲⢸⠀⡄⢸⢨⠌⡋⠍⣄⡀⠀⠀⠀⢙⢛⢻⣟⡻⡟⣻⡇⠆⠀⠘⣯⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣇⠠⢽⣿⣿⣿⣿⡛⣿⣿⣿⡟⢉⣠⣿⢸⢈⣹⣿⣽⣿⣶⣶⣷⣶⣾⣶⡀⢆⡙⠀⠀⠀⠀⢬⣬⣼⣽⣵⣕⣺⣁⠀⠀⠀⠈⠳⢤⣀⣀⢀⣀⣠⠖⠋⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢲⣶⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠇⣶⢸⠶⢾⣿⠋⣦⠩⡍⡍⡍⢩⢹⢹⡟⡍⣟⢸⢙⡋⣛⣙⢛⢻⡿⠦⢀⡀⠀⠀⢴⢻⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⣰⡇⠀⠀⠀⠀⢸⠀⠈⠉⠹⠟⢿⣿⣿⡿⠓⠂⠀⠀⠳⠷⠸⠬⠬⠥⠧⠯⠼⣬⣏⣏⣉⣹⣘⢊⡑⡿⠃⠛⠃⠿⠸⠰⡆⡆⡏⢒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡤⠄⠀⠀⠀⠈⠀⠁⠉⠉⠉⠉⠩⠁⢿⣿⣿⠟⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⢁⣿⣿⣷⣬⣐⠀⡀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⣿⣿⣿⣿⣿⣷⣮⣦⣤⠀⣿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠚⣿⣿⣿⣿⣿⣿⣿⣷⣍⢷⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠃⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⣀⠀⠀⠀⠀⣿⣿⣿⣿⠇⢰⠶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤
⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣞⠲⣿⣿⣶⣦⣼⣿⣿⣿⠱⣿⣾⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣠⣤⣶⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠁⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠄⠀⠀⠀⠀⠀⠀⠀⣿⡎⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣧⣬⣿⣿⣿⣷⣶⣶⣦⣤⣤⣶⣾⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠼⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠞⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠛⠽⠿⠛⠙⠻⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⣀⣠⣤⣴⣾⣿⠿⢿⣆⣤⣤⡀⠀⣀⡀⣘⣃⠉⠉⠉⠛⠛⠿⠭⢉⣛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠻⠿⠟⢛⣫⣭⣿⣿⣷⣄⣀⣀⣀⡀⠀⠀⠀⣼⣿⣿⣿⣿⣯⣥⣴⣿⣿⣿⣿⣿⡌⠙⠻⢿⣿⣟⡁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠂⠲⣶⠏⢶⠄⠰⠶⠶⠶⢶⠾⡾⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠀⠀⠀⠈⠙⠟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠉⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿
⣺⡐⠀⠀⠀⣀⣠⣷⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣽⡏⠀⠈⠑⠓⠋⠛⠛⠻⠿⢿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⢀⡌⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⡀⠰⠀⠀⠀⠀⣿⣿⣿⢸⠇⣎⣵⢸⡱⣮⡹⡆⣿⡹⣛⢹⣿⢨⢫⣝⡇⡇⡟⢟⣿⡟⣟⡻⣛⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣀⣴⣶⣾⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⢠⢟⡍⠀⠀⠀⢐⣿⣿⣿⣼⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣶⣷⣿⣷⣧⣯⣵⣿⡿⣴⣧⣿⣾⣼⣼⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢀⡙⡏⠀⠀⠀⢸⣿⣿⢩⡝⢉⡍⡿⣼⢺⢛⠛⠟⠛⠿⠟⠻⠟⠛⢿⢿⢿⠿⠿⠿⣿⣿⣿⣿⣿⣟⡿⣿⣿⣿⣿⡟⠉⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⣚⡿⠷⣿⣿⢿⣿⣿⣿⣿⠇⠀⠀⠀⢐⢗⡀⠀⠀⠀⣼⣿⣿⣶⣾⣾⣷⣷⣿⣾⣼⣯⣮⣯⣿⣥⣵⣇⣯⣢⣻⣸⣸⣇⣇⢣⠇⠶⢰⠸⠆⡇⣷⠸⣽⣿⢻⠀⠀⠀⢰⠇⢸⣯⡇⣿⢰⣿⢀⣿⠈⣧⠀⠀
⠀⠀⠀⠠⠀⠲⢶⣤⣽⣶⣾⣽⣿⣿⠋⠀⠀⠀⢀⣻⣇⢀⢐⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⠇⠠⣿⣿⣷⣀⢤⣶⣿⢯⣽⣾⣟⡁⠉⠈⠉⠙⠛⢿⣷⣟⣽⣿⣿⣸⠀⠀⠀⢸⠀⢸⣿⡆⡏⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠥⠊⠀⠀⠀⠀⢻⠁⠀⣬⣥⣧⣧⣛⣏⣂⣃⣻⣘⠀⣸⡇⡀⠀⣱⠰⠬⠲⢸⠀⡄⢸⢨⠌⡋⠅⡀⠀⠀⠀⠀⢘⢛⣻⣟⡻⡟⣻⡄⠀⠀⠘⣯⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⣿⣿⣿⣿⡛⣿⣿⡟⠉⢅⣸⡇⡇⣘⣿⣝⣿⣿⣶⣶⣷⣶⣾⣶⡑⢆⡑⠀⠀⠀⠀⢬⣼⣿⣽⣥⣕⣮⣀⠀⠀⠀⠈⠳⢤⣀⣀⢀⣀⣠⠖⠋⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⣿⣿⣶⣶⠀⠀⢀⡄⠀⠀⠀⠀⡿⠁⠀⠀⠇⣶⢸⠶⢆⣿⠋⣦⠡⡍⡍⡅⢫⢸⣟⡝⡝⣟⢸⢙⡋⣛⣙⢛⢻⡿⠗⢀⠀⠀⠀⢾⢺⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⢀⣾⡆⠀⠀⠀⠀⡇⠀⠀⠉⠿⠻⣿⣿⣿⠟⠃⠀⠀⠀⠳⠇⠾⠬⠬⠥⠧⠯⠼⢬⣏⣏⣉⣹⣘⢊⡑⡿⠃⠛⠃⠿⢸⠰⡆⡆⡏⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣦⠄⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉⡍⠸⢿⣿⠿⠻⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣿⣿⣿⣦⣅⠂⢀⠀⠀⠀⢀⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣹⣿⣿⣿⣿⣿⣿⣶⣷⣤⡤⢹⡦⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣦⣹⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⡀⠀⢐⡀⠀⠀⠀⣼⣿⣿⣿⡿⠀⡶⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤
⠀⠀⠀⢹⣿⣻⣿⣿⣿⣿⣿⣿⣿⣻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣛⢿⣿⣷⣶⣦⣿⣿⣿⡏⢾⣶⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⣤⣶⡾⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⡀⢼⠀⠠⠀⠀⠀⠀⠀⠀⠀⢸⣷⢿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣤⣽⣿⣿⣿⣶⣶⣦⣤⣤⣶⣶⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠠⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣘⠃⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠉⠻⠿⠛⠉⠙⠟⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣤⣠⣤⣴⣶⣿⣿⢿⣧⣴⣿⣦⡀⣀⣀⣐⣛⡉⠉⠉⠙⠛⠿⠯⢍⣛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⠿⠿⠟⢛⣻⣭⣿⣿⣤⣠⣀⣀⣀⣀⠀⠀⢰⣿⣿⣿⣿⣿⣭⣴⣿⣿⣿⣿⣿⣧⠉⠛⠿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠲⣶⠏⢶⠔⠰⠶⠶⢶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠈⠀⠀⠀⠀⠈⠹⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠉⠉⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿
⣬⠦⠀⠀⠄⡠⣤⣦⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠽⠓⠈⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⢀⡄⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⠇⣎⣵⢸⡱⣮⡱⡆⣿⡽⣛⢹⣿⢨⢫⣝⡇⡇⡟⢟⣿⡟⣟⡻⣛⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣰⣾⣿⣿⣿⣷⣶⣶⣤⣀⠀⠀⠀⠀⠀⠈⣛⡁⠀⠀⠀⢸⣿⣿⣇⣾⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣶⣷⣿⣷⣧⣯⣵⣿⡿⣼⣧⣿⣾⣼⣼⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠰⡉⠀⠀⠀⠀⢸⣿⣿⢩⡍⢉⡍⡿⣼⢺⢛⠛⠟⠛⠿⠟⠛⠟⠛⢿⢟⢿⠿⢿⠿⣿⣿⣿⣿⣿⣟⡿⣿⣿⣿⣿⠛⠉⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⣒⡚⠿⢿⣿⠿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠰⣷⠀⠀⠀⠀⣾⣿⣿⣶⣾⣾⣷⣷⣿⣾⡬⣯⠮⣯⣿⣥⣵⢇⣏⣠⣻⣸⣸⣇⣇⢣⠇⠶⢰⠸⠆⡇⣷⠸⣽⣿⢸⠀⠀⠀⢰⠇⢸⣿⡇⣿⢰⣿⢀⣷⠈⣧⠀⠀
⠀⠀⠀⠀⠚⠷⣶⣿⣿⣿⣿⣿⣟⠃⡀⠀⠀⠀⢸⣿⡀⠀⢒⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡇⠇⢠⣿⣿⣧⡤⢄⣾⣿⢹⣽⣿⣽⠀⠉⠉⠉⠙⠛⢿⣿⣟⣽⣿⡏⣾⠀⠀⠀⢸⠀⢸⣿⡆⡏⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠓⠀⠀⠀⠀⠸⣾⠀⠈⣬⣥⣧⣧⣻⣇⣂⣃⣻⣘⠀⣸⡇⡀⠀⣱⠰⠬⠲⢸⠀⡄⢸⢨⡌⡋⠅⡀⠀⠀⠀⠀⢘⢛⣻⣟⡻⡟⣻⡀⠀⠀⠘⣯⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠠⢿⣿⣿⣿⣿⡛⣿⣿⡋⠉⢉⣽⡇⡇⣸⣿⣉⣽⣿⣶⣶⣷⣶⣾⣶⡀⢆⡑⠀⠀⠀⠀⢬⢼⣿⣽⣥⣕⣪⡀⠀⠀⠀⠈⠳⢤⣀⣀⢀⣀⣠⠖⠋⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠄⠀⠀⣼⠀⠀⠀⠀⠨⡷⠀⠀⠀⠇⢶⢸⠶⢆⣿⠋⢦⢁⡍⡍⡍⢫⢸⣭⡝⡍⣟⢸⢙⡋⣛⣙⢛⢻⡿⠆⢀⠀⠀⠀⢼⢺⣿⣿⣿⣿⣿⠍⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⣼⣿⠀⠀⠀⠀⠀⠃⠀⠀⠉⡟⠻⣿⣿⣿⠟⠃⠀⠀⠐⠷⠓⠾⠼⠬⠥⠧⠭⠼⠬⣏⣇⣉⣹⣘⢊⡑⡿⠃⠛⠃⡿⢸⠰⡆⡆⠏⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣀⣄⠀⢀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠣⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⡶⠄⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⢉⠁⠿⣿⣿⠿⠹⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣒⣿⣿⣷⣮⣐⠠⡀⠀⠀⢀⣤⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢘⣿⣿⣿⣿⣿⣿⣷⣾⣶⣴⣤⢻⣗⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⡀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠐⠛⣿⣿⣿⣿⣿⣿⣿⣷⣍⢿⣾⣿⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⡀⠀⢂⡀⠀⠀⠀⣿⣿⣿⡿⠇⠰⠾⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤
⠀⠀⠈⠟⣻⣿⣿⣿⣿⣿⣿⣷⣟⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣾⣛⣿⣿⣷⣶⠄⣿⣿⣿⡥⣿⣶⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⣤⡶⢾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣴⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇⢿⣿⣿⣟⣿⣿⣿⣷⣾⣿⣿⣿⣧⣤⣽⣿⣿⣿⣶⣶⣦⣤⣴⣶⣾⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠨⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠘⠀⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠉⠹⡿⠋⠉⠻⠏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠐⠑⠀⠀⣤⣤⣤⣶⣾⣿⡿⣿⣷⣾⣿⣦⣀⣀⣀⣘⣃⡀⠉⠉⠙⠛⠿⠭⢍⣛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⠿⠿⠛⣛⣯⣿⣿⣿⣤⣤⣀⣀⣀⣀⡀⠀⣸⣿⣿⣿⣿⣯⣥⣾⣿⣿⣿⣿⣿⣧⠉⠛⠿⣻⣿⡁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠂⠲⣶⠏⢶⠔⠰⠶⠶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠐⡀⠉⠝⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⠉⠉⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠐⠀⠰⣼⡹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣽⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢠⡄⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣿⣿⠀⢹⠇⣎⣵⢸⡱⣮⡱⡆⣿⡹⣛⢹⣿⢨⢫⣝⡇⡇⡟⢟⣿⡟⣟⡻⣛⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⣿⣿⡇⠀⠀⠀⢠⢉⡄⠀⠀⠀⢼⣿⣿⣇⣾⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣶⣷⣿⣷⣧⣯⣵⣿⡿⣼⣧⣿⣾⣼⣼⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⣿⣿⣿⢩⡍⢉⡍⡿⣼⢺⢛⡛⠟⡛⣛⣛⡛⡛⡛⢿⢛⢿⠿⢿⠿⣿⣿⣿⣿⣿⡟⡿⣿⣿⣿⣿⠋⠉⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠉⠉⠉⢹⣬⣿⣿⣿⣿⡿⠇⠀⣻⣿⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀⣿⣿⣿⣶⣾⣶⣷⣷⣿⣾⣼⠧⢮⣯⣿⢕⣅⡃⣟⣠⣻⣸⣸⣇⣇⢣⠇⠶⠰⠸⠆⡇⣷⠸⣽⡿⡇⠀⠀⠀⢰⠇⢸⣯⡇⣿⢰⣿⢀⣷⠈⣧⠀⠀
⠉⠙⠛⠿⠿⣿⡿⠿⢿⣯⠄⠀⣿⣿⠀⠀⠀⠀⡀⡆⠀⠀⢐⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⠀⢸⡿⣿⣆⣠⣤⣮⣤⢄⣱⣾⢍⡀⠈⠈⠉⠙⠛⢿⣿⣟⣽⣿⡇⡇⠀⠀⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠓⡇⠀⠀⣬⣥⣧⣧⣛⣏⣂⣃⣻⣘⠀⣹⢸⠀⠈⣱⠸⠬⠲⢸⠀⡄⢸⢨⠌⡋⠅⡀⠀⠀⠀⠀⢘⢛⣻⣟⡻⡟⣳⠀⠀⠀⠘⣏⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⠀⠀⢀⣀⣠⣤⣀⡀⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⢚⠂⠀⠀⢿⣿⣿⣿⣿⡛⣿⣟⠋⠉⢉⣿⢸⠀⣸⣏⢉⣹⣾⣶⣶⣷⣶⣾⣶⡀⠂⠑⠀⠀⠀⠀⡬⣼⣿⣵⣥⣑⣨⡀⠀⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠖⠋⠀⠀⠀
⠀⠀⠀⠚⠿⣿⠿⠇⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠇⣶⢸⠶⠆⣿⠉⢦⠀⡌⡍⡌⢠⢹⣭⡝⡍⣟⢸⢙⡋⣛⣙⢛⢻⣷⠄⠀⠀⠀⠀⢧⣻⣿⣿⣿⣿⣯⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢹⠛⢿⣿⣿⡿⠛⠁⠀⠀⠐⠓⠑⠾⠮⠬⠥⠧⠭⠼⠬⢅⣇⣉⣹⣘⢊⡑⠟⠃⠛⠃⡾⢸⠰⠆⡆⡏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⠶⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⢉⠁⠿⣿⡿⠏⠏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣾⣶⣦⣅⡀⢄⠀⠀⣰⡆⠒⣂⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣽⣿⣿⣿⣿⣶⣬⣾⣿⣷⡄⣿⣼⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢻⣿⣿⣿⣿⣿⣷⣦⡝⢿⣿⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⣂⠀⠀⠀⢰⣿⣿⣿⠿⠇⠰⠞⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣮⡋⢿⡫⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣟⣻⣿⣿⣶⡶⢸⣿⣿⣯⠱⣷⣾⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⣠⣴⠶⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣷⣿⣌⠻⣷⡀⠀⠀⠀⠀⠀⢠⡇⠈⠀⠀⠀⠀⠀⠀⠀⠠⣿⠜⣿⣿⣿⣻⣿⣿⣿⣶⣾⣿⣿⣿⣦⣬⣿⣿⣿⣷⣶⣶⣤⣤⣶⣶⣾⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡸⢷⡀⠀⠀⠀⠁⠘⠏⢀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⣌⠉⢿⡟⠉⠉⠿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣶⣾⣿⡿⣋⣿⣿⣿⣿⣆⢀⣀⣀⣛⣁⠀⠈⠉⠛⠛⠯⠥⣉⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⡿⠿⢛⣿⣿⣿⣤⣀⣀⣀⣀⣀⡀⠀⠀⢾⣿⣿⣿⣿⣭⣴⣿⣿⣿⣿⣿⣿⡈⠙⠻⠟⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠺⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠖⠲⣶⠏⢷⠄⠰⠼⠿⣯⣤⣤⣶⣿⣿⣿⣿⣿⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠁⠀⠀⢀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠈⠉⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢨⣽⣿⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡇⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⠀⠀⡄⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀
⣶⣶⣶⣶⣶⣦⣄⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⠇⣎⣵⢸⡱⣮⡱⡆⣿⡹⣛⢹⣿⢨⢫⣝⡇⡇⡟⢟⣿⡟⣟⡻⣛⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣆⠀⠈⣿⣿⣿⣿⠀⠀⠀⠀⢤⡧⠀⠀⠀⠀⣿⣿⣿⢠⣼⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣶⣷⣿⣷⣧⣯⣵⣿⡿⣴⣧⣿⣾⣼⣼⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⣻⠿⠿⣿⣿⣿⣿⣿⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⣿⣿⣿⢩⡍⢉⡍⡿⣼⢺⢛⡛⠛⡛⣛⣛⡛⡛⡛⣿⢟⢻⢿⢿⠿⣿⣿⣿⣿⣿⡟⡿⣿⣿⣿⡟⠉⠀⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⣁⣰⣶⣿⣿⣿⣿⠏⠱⠀⢸⣿⣿⡟⠀⠀⠀⠀⡎⠂⠀⠀⠀⢰⣿⣿⣿⣶⣾⣾⣷⣷⣿⣾⣼⠧⣮⣯⣿⠕⢅⢃⣇⣠⣛⣰⣸⣇⣇⢣⠇⠶⠰⠸⠆⡇⣷⠸⣽⣧⠀⠀⠀⠀⢰⠃⢸⣯⡇⣿⢰⣿⢀⣷⠈⣧⠀⠀
⠙⠋⠙⠻⠛⠛⠛⠋⠁⠀⢸⣿⣿⡇⠀⠀⠀⠀⠁⠀⠀⢀⢘⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡄⠀⣿⠿⣿⣀⠀⢀⣰⣾⢤⣵⣎⣙⠀⠉⠈⠉⠉⠙⣿⣿⣟⣽⣿⢸⠀⠀⠀⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢀⠀⠀⠈⣬⣥⣧⣧⣓⣇⣂⣃⣺⣐⠨⡉⡇⠀⡎⣰⠠⠌⠲⢸⠀⡄⢸⢨⠌⡋⠅⡀⠀⠀⠀⠀⢛⢛⣻⣟⡻⡿⣲⠀⠀⠀⠘⣯⠛⠛⠛⠓⠓⠻⠿⠿⣤⠇⠀⠀
⠀⠀⢤⣶⣶⣶⡄⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠇⠀⠀⠀⢿⣿⣿⣿⣿⡛⣿⡿⠋⢉⣽⡇⡇⠀⣿⠉⢁⣽⣾⣶⣶⣷⣶⣾⣶⡀⠆⠑⠀⠀⠀⠠⢨⣼⣿⣥⣄⣕⣠⣀⡀⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠖⠋⠀⠀⠀
⠀⠀⠉⠛⠿⠛⠁⠀⠀⠰⣿⣿⣿⠁⠀⠀⠀⠀⠅⠀⠀⠀⠀⠇⣶⢸⠶⠆⣿⠁⢤⠀⡍⡇⡅⢡⢻⢭⡝⡍⣟⢸⢙⡋⣛⣙⢛⢻⣷⠆⢀⡀⠀⠰⢼⣻⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⣿⣿⣿⠟⠛⠁⠀⠀⠒⠃⠓⠚⠎⠬⠥⠧⠭⠼⠬⠥⣇⣉⣸⣘⢈⡑⣟⠃⠓⠃⢾⢸⠰⠆⡆⡏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠤⡤⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⡙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡉⠸⠿⠿⠿⠹⠙⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠽⣿⣿⣷⣦⣀⠠⡀⣀⣾⣄⠻⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣾⣿⣿⣿⣿⣷⡾⢿⣿⣿⡆⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⢿⣿⣿⣿⣿⣿⣿⣷⣜⢿⣶⣼⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⢸⣿⣿⡀⠀⠀⠐⣀⠀⠀⠀⢸⣿⣿⡿⠟⠀⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤
⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣌⣫⡫⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣾⣟⣻⣿⣿⣿⠦⣾⣿⣿⡇⢾⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣴⡺⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⣿⣿⡄⠀⠀⠀⠀⠀⢼⡀⠐⠀⠀⠀⠀⠀⠀⠀⠸⠿⠺⣿⣿⣿⣻⣿⣿⣿⣷⣿⣿⣿⣿⣤⣽⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠨⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⣿⡄⠀⠀⠈⠁⠘⠇⢀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠈⠙⣻⠋⠉⠙⠟⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⣾⣿⡿⣛⣿⣿⣿⣷⡀⣀⣀⣀⣉⣀⠀⠈⠉⠛⠛⠧⠤⣙⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⢿⣿⠿⠿⢛⣻⣽⣿⣷⣦⣤⣀⣀⣀⣀⣀⠀⠠⢿⣿⣿⣿⣿⣭⣾⣿⣿⣿⣿⣭⣿⠈⠙⠛⠛⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠺⠿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠆⠲⣶⠏⢶⠤⠰⠶⠶⣶⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠈⠀⠀⠀⠂⠀⠩⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠉⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠰⠦⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣠⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠈⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⡌⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡠⠀⠀⠀⠀⢸⣿⣿⡇⠀⢠⠇⣎⣵⢸⡱⣮⡱⡆⣿⡹⣛⢹⣿⢨⢫⣝⡇⡇⡟⢟⣿⡟⡟⡻⣛⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⡇⢀⣼⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣶⣷⣿⣷⣧⣯⣵⣿⡿⣴⣧⣿⣾⣼⣜⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⣿⣿⡗⢩⡍⢉⡍⡿⣼⢺⢛⡛⢟⡛⣛⣛⡛⡛⡛⣿⢭⢯⢭⢽⡯⣭⣯⣿⣿⣿⡟⡿⣿⣿⣿⣏⠀⠀⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠈⠙⠛⣻⣿⣿⣿⣿⣿⣿⠷⠻⡟⠀⠀⠀⠀⠊⠀⠀⠀⠀⣸⣿⣿⣿⣶⣾⣾⣷⣷⣿⡮⡼⠧⣮⢫⡿⠑⠁⠃⡋⣀⣛⣸⣸⣇⣇⢣⠇⠶⠰⠘⠆⡇⣷⠸⣽⢻⠀⠀⠀⠀⢰⠃⢸⣯⡇⣿⢰⣿⢀⣷⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⢀⢚⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⠇⡇⠀⣿⢿⣇⡀⠀⢀⣶⣾⢭⣉⣘⣍⠀⠀⠈⠉⠉⠛⣿⣿⣟⣽⣿⣸⠀⠀⠀⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠈⣬⣥⣧⣧⣓⣇⣂⣃⣺⣐⠈⡀⡇⠀⡌⣐⠡⠬⠲⢸⠀⡄⢸⢨⠌⡋⠅⡀⠀⠀⠀⠀⢛⢛⣻⣟⠻⡟⡫⠀⠀⠀⠘⣏⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⣿⡛⣿⡟⠋⠉⣿⡇⡇⢈⣿⢀⠡⣹⣾⣶⣶⣷⣶⣾⣶⡀⠆⠁⠀⠀⠀⢠⢨⣼⣿⣥⣄⣕⣠⣀⡀⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠔⠋⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠁⠀⠀⠀⠅⠇⣶⢸⠶⠆⣿⠁⢄⠀⡍⡅⡍⢩⢹⢽⡝⡝⣟⢸⢙⡋⣋⣽⢛⢿⣷⠦⣀⣀⠀⠸⢸⣻⣿⣿⣿⣿⡓⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠻⣿⣿⣿⠿⠋⠀⠀⠀⠲⠃⠓⠚⠊⠈⠁⠧⠥⠼⠬⠥⠧⣤⣸⣘⢈⡑⣏⣁⢓⠃⢾⢸⠰⠆⡆⠏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠸⠿⠿⠿⠹⠙⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢐⣿⣿⣿⣿⣿⣦⣔⣄⣄⠀⣺⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠙⣿⣿⣿⣿⣿⣿⣿⣿⡻⣷⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⡀⢀⣄⡀⠀⠀⣿⣿⣿⡿⠛⠀⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶
⠀⠀⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣧⠀⠀⠀⠀⠀⠈⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣷⣾⣝⣻⣿⣿⣿⠦⣿⣿⣿⡆⠿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣠⣴⣦⣤⣴⣶⣛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⡀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠻⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠊⠀⠘⠃⢀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡈⠙⠛⠉⠀⠙⠋⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣲⣶⣶⣿⣿⣿⣿⣿⣿⣷⣿⣶⣦⡀⣠⣤⣬⣥⡀⠀⠈⠉⠛⠛⠲⠤⢍⣛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⣛⣭⣽⣿⣿⣶⣤⣤⣀⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⠉⠛⠿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠂⠲⣶⠏⢶⠖⠰⠶⠶⠾⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠈⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠉⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⢀⡌⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣷⣦⠀⢠⡄⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⠇⣎⣵⢸⡱⣮⡱⡆⣿⡹⣛⢹⣿⢨⢫⣝⡇⡇⡟⢟⣿⡟⣟⡻⣛⢿⢿⢿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⣿⣿⣿⣿⠀⢸⠗⠀⠀⠀⠀⣿⣿⣿⠀⣀⣼⣿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣾⠾⢿⣶⣷⣿⣷⣧⣯⣵⣿⠿⣱⣧⣻⣾⣼⣼⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⣿⣿⣿⣿⠀⢸⠁⠀⠀⠀⠀⣿⣿⣿⡇⢩⡝⢉⡍⡿⣼⢺⢻⡻⠟⡟⣿⣿⡻⡿⡿⣿⢿⢿⠿⢿⠿⡿⣿⣿⣿⣿⡟⡿⣿⣿⣿⢰⠀⠀⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠄⢸⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣾⣾⣷⣷⣿⣬⡬⢤⣦⣩⡏⠀⠀⣃⣃⣠⣘⣠⣸⣇⣇⢣⠇⠴⢰⠨⠆⡇⣷⠸⣹⢸⠀⠀⠀⠀⢰⠃⢸⣯⡇⣿⢰⣿⢀⣷⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⠀⢰⢛⡻⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⠀⢸⣿⣿⣿⣶⣶⣦⣾⣿⢿⣿⣿⣿⣀⠙⠛⠛⠛⢿⢿⣿⣟⣵⡏⣿⠀⠀⠀⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠀⠀⠁⠀⠀⢨⣬⣥⣧⣧⣚⣇⣂⣃⣺⣘⠈⢸⠀⢘⡆⣰⠠⠬⠰⢸⠀⡄⢸⢨⠌⡋⠍⣀⡀⠀⠀⡘⢙⢛⣻⣟⡛⡟⣢⠀⠀⠀⠘⣏⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠸⠀⠀⠀⢸⢿⣿⣿⣿⣿⡛⣿⣿⢉⢹⣿⢸⠀⣸⣿⣯⣽⣿⣿⣶⣶⣷⣶⣾⣶⣀⢆⡹⠀⠀⠀⢠⢸⣼⣿⣥⣄⣕⣨⠁⠀⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠖⠋⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠁⠄⠇⣶⢸⠶⠆⣿⠉⢦⢩⡏⠈⣬⢩⢻⢛⡛⡛⣟⢻⢙⡋⡛⠛⢛⢿⠿⠿⠿⡿⠿⠿⢾⣻⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⢀⣲⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠛⢿⣿⣿⣿⠿⠋⠀⠀⠐⠲⠖⠷⠼⣬⣤⣥⣥⣥⣼⠈⠅⢃⠉⢸⠘⢊⠑⡿⠃⠳⠂⣾⢸⠰⠆⡆⡏⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣸⣿⣿⣿⣿⣷⣦⣠⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠋⠁⠆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⣿⠃⠈⠉⠉⠉⠛⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠐⠀⣼⣿⣷⠀⠀⠀⠄⠀⠀⠀⠀⣿⣿⣿⣿⡷⢠⡴⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣾⠶⣾⣷⣶⣦⡄⣿⣿⣿⠃⣼⣾⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣄⣴⣶⣿⣿⣿⣿⣿
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⡇⠈⡀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣇⣈⣻⣿⣿⣷⣶⣶⣦⣤⣤⣤⣴⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⡾⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⣼⣿⣿⣿⣟⠛⢿⠟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠈⠀⠀⠀⠀⠀⠂⢤⠀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⠗⠀⠀⢸⣟⠛⠙⠛⠻⠿⣿⣿⣙⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⠿⠿⠿⢿⠿⠿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⠛⠿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠈⠁⠒⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠀⠀⠀⠈⡛⡿⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠟⠉⠙⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠲⣶⠀⠶⠄⠰⠶⠶⠶⠉⠉⠉⠙⠉⢻⠟⣻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⣱⣾⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⢿
⠀⠀⠔⠀⠚⠶⠟⠿⠿⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⣿⣿⣿⠀⠀⢀⡍⢻⣿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⠇⣎⣵⢸⡱⣮⡱⡆⣿⡽⣛⢹⣿⢨⢫⣝⡇⡇⡟⢟⣿⡟⣟⡻⣛⢿⢿⢿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⠀⠀⠀⠀⠀⣿⣿⣿⠀⣠⣾⣿⣿⡿⣿⣿⣿⣯⣿⣿⣯⣼⣼⣿⣶⣷⣿⣷⣥⣭⣥⣿⡷⣴⣧⣯⣼⣼⣼⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡅⠀⠀⠀⠀⣿⣿⣿⢸⢩⡍⢉⡍⡿⣼⢺⢛⠛⠟⡛⡟⠛⠛⠛⠛⠿⠿⢿⠿⠿⠿⡿⢿⣿⣿⣿⡟⡿⣿⣿⣿⢸⠀⠀⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣰⡄⠀⠀⢀⢀⣿⣿⣿⣿⣶⣾⣶⣷⣥⣿⢬⠬⢥⣦⣭⣇⣀⣀⣃⣃⣀⣘⣀⣸⣇⣇⢣⠇⠴⢰⠨⠆⡇⣷⠸⣹⣸⠀⠀⠀⠀⢰⠇⢸⣿⡇⣿⢰⣿⢀⣷⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢸⢛⡻⡛⡿⣛⠿⡿⡿⠿⢿⠿⢸⠀⢸⣿⣿⡟⢟⣏⣥⣾⣿⢽⣿⣿⣿⡄⠉⠙⠛⠛⣿⢿⣿⣟⣵⡇⣧⠀⠀⠀⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣬⣥⣧⣧⣓⣇⣂⣃⣺⣘⠨⢸⠀⢘⡄⣰⠠⠬⠲⢸⠀⡄⢸⢨⡌⡋⠍⣤⡀⠀⠀⡘⢙⢛⡻⣟⠛⡟⣲⠄⠄⠀⠘⣯⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⢿⣿⣿⣿⣿⡛⣿⣟⡯⣽⣿⢸⣀⣼⣿⣿⣿⣿⣿⣶⣶⣷⣶⣾⣶⣰⣆⣽⣧⣀⣀⡥⣼⣼⣿⣥⣤⣕⣨⠀⠀⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠖⠋⠀⠀⠀
⠀⠀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠇⣶⢸⠶⠆⣿⠁⣦⢩⡍⡈⡍⢫⢹⠛⡛⡛⡟⢺⢙⡋⡻⠽⢟⢿⡿⠯⢯⣍⠙⡇⢿⣻⣿⣿⣿⣿⣥⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⢠⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠿⠛⠻⠿⠟⠉⠀⠀⠀⠲⠓⠷⢼⡬⣬⣥⣧⢯⣼⠬⠅⢧⠀⠸⡘⢈⡐⣯⣅⣵⠄⠿⠸⠰⠆⡆⡏⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠄⠀⠀⠀⠀⠀⠀⠉⠙⣿⠁⠀⠉⠁⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⣿⣧⣧⣄⣀⣧⣀⡀⡀⢀⣿⣿⣿⡟⢳⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿
⠀⠙⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⢁⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⢾⣿⣿⣿⡄⠘⢿⣿⣷⣦⣤⣄⣀⡀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣶⣿⣶⣶⣷⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⡄⢸⣇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠿⠚⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠈⠁⠈⠃⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣶⡉⢛⣏⢁⣀⣛⠁⢸⣿⠿⠿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠁⠀⠀⣰⣶⣿⣿⣿⣿⠟⣫⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠉⠙⠛⠛⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣋⣉⣛⣛⣿⣿⣿⣿⣿⣶⣤⣄⣤⣤⣤⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⢛⠿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣠⣏⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⠿⢿⣿⣿⣿
⠀⠀⠰⣶⠀⠰⠄⠰⠶⠶⠖⠀⠀⠀⠀⠹⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⣰⣾⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠛⠉⠉⠉⠉⠉⠙⠛⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣿⣿⡇⠀⢠⡌⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢸⠇⣎⣵⢸⡱⢮⡱⡆⣟⡹⣙⢹⣿⢨⢋⣝⡇⡇⡟⢟⣿⡟⣟⡻⣛⢿⢻⢿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⢸⣿⣿⡇⣤⣾⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⠾⠾⠿⠶⢷⢿⡷⣧⣯⡵⠿⠷⣰⣧⣻⣼⣘⣼⣿⣿⣿⢸⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢣⠀⠀⠀⠀⢸⣿⣿⡷⢩⡍⠉⡍⡟⢸⢺⢛⠙⠟⡛⣿⣀⠀⠀⠒⢶⠶⢿⠷⠶⠶⡾⢾⣿⣿⣿⡟⡿⣿⣿⣟⠋⡟⠀⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⡀⣸⣿⣿⣿⣶⣾⣶⣧⣥⣭⡬⡬⠤⣮⣩⡿⢁⠄⢃⣃⣀⣘⣀⣸⣇⣇⢡⠇⠴⢰⠈⠆⡇⣷⠸⣹⢁⡇⠀⠀⠀⢰⠇⢸⣏⡇⣿⢰⣿⢀⣷⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⠇⡇⠠⣿⠀⡀⢀⠠⣢⣾⣿⢭⣟⣝⣯⡀⠈⠈⠋⠙⠻⣿⣿⣟⣽⣿⢼⡇⠀⠀⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢨⣬⣥⣧⣧⣛⣇⣂⣃⣺⣐⠨⡀⡇⢈⣎⣴⠠⣬⢰⢸⣤⣤⢸⢨⡌⡋⢅⣤⡀⠀⠀⡀⢛⢛⣻⣟⠻⠟⣻⠀⠀⠀⠘⣏⠛⠛⠛⠓⠓⠻⠿⠿⣤⠇⠀⠀
⠀⠀⣴⣾⣿⣿⣷⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⢿⣿⣿⣿⣿⡛⣿⣿⠯⢭⣿⡁⢷⠾⠿⠿⢿⣿⣿⣿⣿⣿⣶⣾⣷⣑⣎⣿⣃⠃⠛⢣⣬⣼⣿⣥⣄⣕⣨⡀⠀⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠔⠋⠀⠀⠀
⠀⣺⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠇⠶⠘⠷⠆⣿⠁⣤⢩⡍⡍⡍⢫⢹⢛⡛⡟⣟⢺⢙⡋⡋⠝⢛⢻⠿⠏⢉⡉⠉⡿⢽⣻⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀
⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠛⠋⠀⠀⠀⠲⠓⠓⠘⠈⠨⠥⠧⠥⠼⠬⣥⣇⣀⣸⣘⢈⠐⣿⣇⢷⡇⠾⠸⠀⠆⠆⡏⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⣀⣀⣀⣀⠀⠀⢻⠇⠀⠈⠉⠁⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⡀⠀⢐⣀⠀⠀⠀⣸⣿⣿⣿⡿⢿⣷⡇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾
⠀⠸⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣟⣿⣿⣿⣷⠦⣿⣿⣿⣏⠀⠻⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿
⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⢀⠀⣿⡀⠁⠀⠀⠀⠀⠀⠀⠀⠸⠷⢻⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠙⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⣉⠀⠙⠁⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡉⠹⠿⠋⠉⠙⠟⠉⣿⣿⠿⠿⢿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠁⠄⠂⣴⣶⣶⣾⣿⣿⠟⣛⣿⣾⣿⣷⣤⣶⣶⣼⣿⣷⣦⡄⠀⠀⠉⠉⠛⠓⠒⠨⠉⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⡿⣹⣟⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⠰⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣤⠘⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠪⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀
⡨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠜⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡣⠀⠀⠀⠀⣿⣿⣿⠀⢠⡌⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡁⠀⠀⠀⠀⣿⣿⣿⠀⢻⠇⣎⣵⢸⡱⣮⡱⡆⡿⡽⣓⢹⣿⢨⢫⣍⡇⡇⡟⠟⣷⡖⡶⡲⣒⢶⢶⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡀⠀⠀⠀⠀⣿⣿⣿⣠⣿⣿⣿⡿⢿⣿⣿⡟⢿⣿⣿⡞⠚⠛⠒⠳⣻⠷⣥⣭⡥⠿⠷⣰⣧⣛⣼⣘⣘⣿⣿⣿⡟⡇⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⠇⠀⠀⠀⠀⣿⣿⣿⢹⡍⢉⡍⡟⢸⢺⢛⡙⠞⡛⣿⣇⡀⡐⡒⢶⠶⠿⠷⠶⠶⡾⢾⢿⣿⣿⡟⡿⣿⣿⣿⡏⢸⡇⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢠⣀⣿⣿⣿⣶⣾⣶⣧⣥⣬⣬⣼⡇⢬⡯⠿⠁⠀⢃⣃⣀⣘⣠⣸⣇⣇⢣⠇⠴⠰⠈⠆⡇⣷⠸⣽⡇⢸⠀⠀⠀⢰⠇⢸⣯⡇⣿⢰⣿⢀⣷⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⢚⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⠸⠀⢼⡧⣀⠀⣀⣠⣺⣿⢉⣃⣚⣉⠀⠀⠈⠉⠙⠛⢿⣿⣟⣽⣿⣇⣼⠀⠀⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⣤⣶⣦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢈⣬⣥⣧⣧⣛⣇⣂⣃⣻⣘⠀⣸⢸⠀⡌⣷⠰⣬⢲⢸⣤⣴⢸⢨⡌⡯⢅⣤⡄⠀⣠⣀⢛⢛⣻⣟⡻⡟⣻⡄⠀⠀⠘⣯⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠰⢿⣿⣿⣿⣿⡛⣿⣿⠏⢉⣼⣿⣈⣙⣛⣛⣛⣿⣿⣾⣶⣷⣶⣾⣶⡡⣆⣽⣧⣅⣙⣭⣭⣼⣿⣽⣥⣕⣪⣀⠀⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠖⠋⠀⠀⠀
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠇⠶⠘⠲⠂⣿⠉⣦⠡⡍⡍⡍⢩⢹⢛⡛⡛⣟⢺⢙⡋⡋⠹⠛⢻⠿⠄⠀⡀⠬⡼⣿⢻⣿⣿⣿⣿⣿⡷⠂⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠛⠁⠀⠀⠐⠷⣧⡘⠈⠈⠁⠃⠁⠸⠨⢇⣧⣂⣸⣘⢀⡐⡿⠄⢷⡇⠿⠸⠠⠆⡆⡏⢒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣤⣤⣤⣄⢀⣘⡟⠀⠈⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡶⣀⣀⣥⣄⡀⠀⠀⣿⣿⣿⡿⠛⣿⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿
⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣽⣿⣿⣿⡿⢲⣿⣿⣿⡄⠀⠹⢿⣿⣿⣷⣦⣤⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⢠⡆⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣖⠤⠀⠉⢀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣦⡀⣙⠋⠀⠀⠙⠁⣸⣿⠟⠛⠛⠻⠿⢿⣿⣿⣛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠈⢸⣿⣿⣿⣿⡿⠟⢛⣽⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣷⠂⠀⠀⠀⠀⠀⠈⠉⠑⠲⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⠟⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣯⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠿⢶⠄⠠⠤⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠈⠉⠛⠿⠿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡫⣿⡇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⡧⠀⠀⠀⢸⣿⣿⣇⠀⢀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣮⡇⠀⠀⠀⢸⣿⣿⣿⡍⢻⡿⢻⣿⣿⣿⡟⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢟⣽⣇⠁⠀⠀⠀⢸⣿⣿⣿⠇⣎⡵⢸⡱⣮⡱⠆⣷⡸⣓⢹⣿⢨⢫⣝⡇⡇⡟⢟⣿⡟⡟⡻⣛⢿⢻⠻⡅⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠼⢻⣟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡗⣿⣾⣾⠛⠲⣶⣷⣶⣥⣥⡥⠿⠷⢰⣧⣫⣼⣘⡀⢸⣿⣿⣿⠿⢿⡇⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢸⣿⢹⡝⢍⡍⡟⢸⢺⢛⡛⠟⡀⣿⣟⡳⡶⡾⢿⠾⢿⠷⠶⠿⡿⢶⢶⣿⣿⡟⡿⣦⣸⣿⣿⡿⠀⣸⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⣀⣤⣼⣿⣶⣾⣾⣷⣧⣭⣬⣼⣧⣦⠉⣿⡀⣀⣃⣃⣀⣘⣀⣸⣇⣇⢣⠇⠴⢰⠈⠆⡇⣦⠸⣽⣿⡇⠀⡟⠀⢰⠇⢸⣿⡇⣿⢰⣿⢠⣷⠈⣧⠀⠀
⠀⠀⠀⠀⠀⣀⣦⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢍⡿⠿⣿⠴⣿⣄⣀⣬⣿⣿⢿⣿⣿⣽⡄⠈⠈⠉⠋⠛⢿⣷⣞⣼⣿⣿⣧⣀⡇⠀⢸⠀⢸⣿⡆⡏⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⢸⡟⢩⣬⣥⣧⣧⣛⣏⣂⣃⣻⣘⠀⣠⣿⡈⡄⣰⠹⢼⠳⢸⢫⣿⢹⢹⡝⡫⠍⣵⡶⠄⣶⣖⢚⢛⡻⣟⡻⡿⣻⡄⠇⠀⠘⣯⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠈⡃⡼⢿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣝⣠⣿⣧⣼⣯⣭⣽⣿⣿⣿⣷⣶⣾⣷⡑⢆⠿⠧⠤⠼⠥⡬⣼⣤⣽⣽⣗⣪⣁⣄⠀⠀⠈⠳⢤⣀⣀⢀⣀⣠⠶⠋⠀⠀⠀
⠀⠀⠠⢮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢿⠃⠀⠇⣶⠘⠰⠀⠙⠫⣮⢩⡍⡍⡍⢫⢹⢛⡛⡛⣟⢻⢙⡋⡛⠙⢛⢻⠷⠖⠀⠀⠠⠴⢾⣻⣿⣿⣿⣿⡿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠉⠙⠉⠉⠀⠁⠀⢀⡴⠖⠀⠐⠓⠷⣾⣬⡄⠁⠡⠅⠼⣬⢅⠇⣂⣸⣘⢊⡐⡷⡄⠳⠃⡾⠸⠀⠆⡆⡏⢐⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⠀⠀⢀⠀⠀⠀⠀⢸⣿⣶⣶⣷⣶⣦⣼⡿⠏⠉⠉⠛⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⣴⣶⣤⣤⣀⢸⣿⣿⣿⠋⠹⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⣀⣰⣾
⠀⠐⣄⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⡀⣴⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣍⣿⣿⣿⣿⣤⣤⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠂⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣖⠃⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣈⠻⢿⣿⡿⠿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠣⢤⣤⣄⣀⣤⣤⣶⣿⣿⣿⣷⣄⣀⣄⠀⠀⠀⠀⣿⣿⣥⣄⣀⠈⠉⠛⠛⠛⠲⠬⢍⣛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣷⣥⣿⣿⣿⣿⣿⣟⣩⣶⣿⣿⣿⣿⣿⣾⣿⣾⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠀⠀⠀⠉⠉⠛⠛⠻⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠛⠿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠎⢶⠔⠰⡶⠶⠗⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣷⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⢬⣿⣿⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣸⣿⣿⠀⠀⠀⢨⣿⣿⣿⢄⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣵⢩⣿⡇⠀⠀⠀⢸⣿⣿⡏⢻⡿⢛⣿⣿⣿⡟⣏⢭⣭⣭⣭⣭⣭⣭⡭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡭⢹⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢸⣿⡇⠀⠀⠀⢸⣿⣿⡇⣎⣵⢸⡡⣮⡱⡆⡿⠹⢛⢹⣿⢨⢋⣝⡇⡇⡟⠟⣿⡛⡛⡛⣛⢟⢻⠛⠛⠃⢸⣿⣿⣿⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⢸⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠾⢾⣾⣿⣶⣶⣷⣷⣥⣭⣥⣿⠷⣰⣧⣫⣼⣘⡀⠀⠀⢸⣿⣿⡟⠋⡿⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢸⣿⠀⠀⠀⢰⣿⢹⡝⣍⡍⡿⢸⢺⢛⡛⠟⡛⢄⣿⡻⡟⠛⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⡟⡿⣧⡀⠀⣿⣿⣿⠁⢀⠇⣠⠞⣁⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⣀⣀⣾⣿⣾⣾⣿⣷⣷⣭⣬⣼⣧⣦⣩⠛⣤⣄⣃⣇⣠⣙⣠⣸⣇⣇⢣⠇⠶⢰⠨⠆⡇⣶⠐⠀⣿⣿⣿⠀⠀⢰⠇⢸⣿⡇⣿⢸⣿⢠⣿⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⢸⣿⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡭⢿⣿⣿⣤⣿⣿⣿⣿⣿⢿⣿⣿⣿⣥⣿⣉⣙⣛⣻⣿⣷⣟⣰⢸⣿⣿⣷⡄⠀⢸⠀⢸⣿⣆⡏⡾⣿⣸⣿⠀⣼⠀⠀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠸⣯⣬⣥⣧⣧⣓⣏⣂⣇⣻⣸⠨⣸⡷⣈⡌⣰⠰⢬⠳⢸⠩⡭⢹⢩⡍⡋⢍⣽⡟⠙⣛⡟⢛⢛⠻⣟⢻⣿⣻⡇⠗⡧⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⣿⢿⣿⣿⣿⣿⠛⡿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣷⣷⣶⣾⣶⣱⣆⡙⡁⠁⠙⢁⣬⣼⣥⣤⣼⣷⣾⣁⣌⠃⠀⠈⠳⣤⣀⣀⢀⣀⣠⠶⠋⠀⠀⠀
⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⡃⠅⠇⠶⠘⠰⠀⠛⠀⣮⢩⡏⡍⡍⢩⢹⢛⡛⡛⡟⢺⢙⡋⡿⠽⢟⢿⡿⠿⢀⡀⠀⠸⢿⢻⣿⣿⣿⣿⣿⡍⠙⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⠁⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠙⠛⠛⠉⠉⠁⢺⠠⣴⡶⢷⣶⣷⣷⣾⣼⣼⣥⣧⠥⠼⢨⣧⣇⣂⣸⣘⢎⡑⣿⣄⢓⠃⡾⠸⠰⠆⡆⡏⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡇⠀⠒⠀⠆⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢸⠀⠈⠁⠀⠀⠀⠈⠃⢸⣿⣿⣿⣦⠀⠐⣃⠍⠉⠀⢸⣾⣷⣿⣿⣿⣿⣿⣿⣏⣉⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢧⠀⠸⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠈⠁⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⣻⣿⣿⣶⣦⢸⣿⣿⣿⠀⠀⣨⣛⣿⣿⣿⣿⣦⣤⣄⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶
⠀⠀⠈⠣⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢠⡆⣿⡄⠐⠀⠀⠀⠀⠀⠀⠀⠸⠿⢻⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠂⠀⠈⠢⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⠸⠃⣤⡀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣯⠙⢿⡋⠉⠙⠟⠉⣿⣿⣿⠏⠉⠙⠛⠻⠿⣿⣿⣉⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿
⠀⠀⠀⠈⠉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣉⠹⢿⣿⣿⣶⣿⣿⣿⡿⠟⣋⣿⣶⣿⣿⣦⣀⣀⣀⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠉⠐⠲⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠋⠉⠙⠛⠻⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⢿⣿⣿⣿⣿⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠿⢷⠶⠴⠶⠿⠷⠀⠀⠀⠀⠀⠀⢸⣷⡤⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⢿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢩⣿⣿⠁⠀⠀⠀⣿⣿⣿⣄⣀⣀⣠⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣿⣿⡏⠀⠀⠀⢐⣿⣿⡏⢻⡿⢻⣿⣿⣿⡟⣧⢾⣶⣿⣿⣷⣶⣾⡿⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣖⣖⣒⠂⣿⣿⣿⣿⡟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡇⠀⠀⠀⠈⣿⣿⣇⣎⣵⢸⡡⣮⠱⡆⡇⠹⠁⢹⣿⢨⢋⣍⡇⡇⡟⠟⣿⡛⡛⡛⣛⢟⢻⠋⠁⠀⠀⣿⣿⣿⣿⢇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⢀⢠⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡿⣶⣾⣿⣶⣶⣷⣷⣥⣭⣥⣿⠷⣰⣧⣫⣼⣘⣀⠀⠀⢰⣿⣿⣿⠋⢸⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠈⣿⠀⠀⠀⢸⢽⢹⡝⣍⡍⡿⣸⢺⢛⡛⠟⡛⢧⣼⠻⡟⠟⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⡟⡿⣿⡀⠀⢸⣿⣿⡏⠀⠈⣠⠞⣁⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀
⠀⠀⠀⠒⠖⠤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⣀⣀⣼⣿⣾⣾⣿⣷⣷⣯⣬⣼⣧⣦⣩⣻⢰⣄⣃⣇⣠⣙⣰⣸⣇⣇⢣⠇⠶⢰⠸⠆⡇⣷⠐⡀⣾⣿⣿⠁⠀⢰⠇⢸⣿⡇⣿⢸⣿⢠⣿⠈⣧⠀⠀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣷⣄⠀⢀⡀⠀⠀⠀⢸⣿⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⢿⣿⢾⣴⣿⣿⣿⣿⣿⢿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣟⣴⣇⣿⣿⣿⣤⠀⢺⠀⢸⣿⣆⡏⡾⣿⣸⣿⠀⣼⠀⠀
⠀⠀⠀⠀⢨⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠇⠀⠀⠀⢸⣯⣬⣥⣧⣧⣛⣏⣆⣇⣻⣸⠪⣸⣷⣘⡌⣰⠡⢬⢳⢸⢩⣜⢸⢹⡝⡋⠍⣽⡗⠅⢿⡟⢛⢛⡻⣟⠻⣿⣿⡏⠿⡴⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠈⣿⢿⣿⣿⣿⣿⠛⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣷⣱⢆⣽⠇⠀⠸⢧⣬⣼⣥⣤⣼⣗⣾⣃⡍⡇⠑⠈⠳⣤⣀⣀⣀⣀⣠⠶⠋⠀⠀⠀
⠀⠀⠀⠀⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠰⡃⠅⠇⠶⢘⠰⠂⠿⠀⣮⢫⡏⡏⡍⢫⢻⢝⡛⡛⣟⢸⢙⡋⡛⠹⢿⢿⡿⠿⢦⡄⠀⠼⢿⢻⣿⣿⣿⣿⣿⡏⢠⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠙⠛⠉⠉⠉⠉⢹⠺⢷⣶⣷⣶⡷⢷⣾⣼⣼⣧⣧⣥⠼⢬⣧⣧⣦⣸⣸⢎⡑⣿⣇⣷⡇⣾⢸⠰⠆⡆⡏⢖⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠠⡀⠀⠸⠀⠀⠁⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⣦⣨⣿⣷⣒⠀⢸⣾⣷⣿⡿⠿⢿⡿⢿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠐⠀⠀⠀⠀⠀⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⣴⠀⠆⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⣿⣿⣿⣽⣿⣿⣿⡛⣸⣿⣿⣿⡀⢠⣼⣤⣤⣽⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠁⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⡇⢿⡆⠐⠀⠀⠀⠀⠀⠀⠀⠘⠛⠙⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠿⡺⢧⣿⣦⣤⣀⣀⣀⣀⣠⣴⣾⣷⣦⢈⣉⣁⠀⠀⠉⠀⣿⣿⣿⣯⣦⣄⠀⠀⠈⠉⠉⠛⠒⠢⠭⢝⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⢿⣿⣿⣿⡿⣛⣻⣿⣿⣿⣦⣈⣈⣟⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⣿⣿⣿⣿⣷⣶⣶⣤⣿⣿⣿⣿⣿⡅⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⡛⠻⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣷⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⡿⠛⠁⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠒⠺⣿⣭⣉⠉⠛⢶⣾⣿⣿⣿⣿⣿⣿⣿⡄⠹⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠶⣶⠿⢷⠶⣷⣶⠶⠷⠀⠀⠤⣤⣶⣶⣖⡀⠀⠀⠩⢭⣽⣷⣆⠈⢿⣿⣿⣿⣿⣿⣿⣷⠀⠈⢿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣦⣤⣀⠀⠀⠀⠀⠀⠀
⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⠂⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀
⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡟⠻⣿⠃⠀⠀⢰⣿⣿⣿⣤⣄⣀⣤⣿⣿⣿⡟⠛⣛⣛⣛⣛⡛⢛⣛⣛⣛⣛⡛⢛⣛⣛⣛⣛⣛⣛⣛⣛⡛⢻⣿⣿⣿⣿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⣿⣿⠀⠀⠀⢸⣿⣿⡏⢻⡿⢻⣿⣿⣿⡟⣧⢿⣿⣿⣿⣿⣿⣿⠿⡿⢿⡿⠿⠿⠽⠭⠿⠿⠿⠿⠭⠭⠅⣾⣿⣿⣿⣿⣼⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠀⠀⠀⠀⣿⣿⣧⣎⣵⢸⡡⢮⠹⡆⡗⡹⠀⢹⣿⢨⢋⣍⡇⡇⡟⠟⣷⡂⡚⡻⣛⢿⢲⠀⠀⠀⠀⣿⣿⣿⡿⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⢠⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⢶⣾⣿⣶⣷⣷⣷⣥⣭⣥⣿⠷⣰⣧⣻⣼⣘⣀⠀⠀⠀⣿⣿⣿⣋⢰⠇⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣿⡇⠀⠀⢸⣿⢹⡝⣭⡍⡿⣸⢺⢛⡛⠟⡛⣿⣞⠻⡟⠟⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⡟⡿⣿⡄⠀⢸⣿⣿⡿⠁⠈⣰⠞⣁⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀
⠀⠀⠙⠿⢶⣶⣴⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⣠⣼⣿⣾⣾⣿⣷⣷⣯⣬⣼⣧⣮⣩⣿⣴⣤⣃⣇⣠⣛⣰⣸⣇⣇⢣⡇⠶⢰⠸⡆⡇⣷⠘⡀⢸⣿⣿⠇⡀⢰⠇⢸⣿⡇⣿⢸⣿⢠⣿⠈⣧⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⢄⠀⠀⠀⠀⠀⠀⢠⣿⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⢿⣿⣿⣽⣾⣿⣿⣿⣿⢿⣿⣿⣿⣽⣿⡿⢿⣿⣿⣿⣿⣟⣵⡇⣿⣿⣿⣦⠄⢻⠀⢸⣿⣆⡏⣾⣿⣸⣿⠀⣼⠀⠀
⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧⣴⡇⠀⠀⠀⠈⣯⣬⣥⣧⣧⣛⣏⣆⣇⣻⣸⣪⣸⣿⣘⡌⣰⠠⢬⢲⢸⣠⣼⢹⢩⡍⡋⢍⣿⡟⠕⢻⡟⢛⢛⡻⣟⠻⣿⣿⡏⠿⣠⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⣿⢿⣿⣿⣿⣿⠛⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣱⣎⣿⠇⠀⠸⢧⣬⣼⣵⣴⣤⣕⣺⣓⠝⡏⠐⠈⠳⣤⣀⣀⣀⣀⣠⠶⠋⠀⠀⠀
⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢸⠃⠅⠇⣶⢘⠰⢠⣿⠀⣮⢫⡏⡏⡍⢫⢹⢟⡟⡛⣟⢸⢙⡋⡛⢝⢟⢿⡿⠿⢗⣢⠤⠾⢿⢻⣿⣿⣿⣿⣿⡏⠀⠃⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠙⠛⠉⠉⠙⠋⢹⡿⠳⣶⣷⣾⣷⠷⣾⣼⣾⣧⣧⣥⣼⣬⣧⣧⣯⣸⣘⢎⡳⣿⣇⣷⡇⣾⢸⢰⡆⡆⡏⢖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠂⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠐⠆⠀⠈⡀⠀⠈⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣿⣿⣿⣟⢸⣾⣷⣿⡿⠛⢻⡿⠻⠿⣿⣿⣿⣷⣅⣐⠂⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤
⡄⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⡀⣸⡄⠰⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⣿⣿⣿⣾⣿⣿⣿⣏⣹⣿⣿⣿⣇⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⡄⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⣇⡻⡇⢀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣀⡛⢿⣿⡿⠿⠿⣿⠿⢿⣿⣿⣿⣟⠛⠛⠛⠿⠿⣿⣿⣛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠢⠀⠀⠀⠀⠀⠘⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⢻⣼⣷⣶⣤⣤⣤⣤⣴⣶⣿⣿⣿⣦⣠⣀⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠔⠀⠀⠀⠀⠀⠀⠉⠉⠒⠺⠭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣶⣭⣍⣛⣽⣿⣿⣿⣿⣶⣬⣑⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣞⣿⣶⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⠏⠉⠛⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣭⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣬⣿⠛⠛⠿⢿⣛⣿⣿⣿⣿⣿⣿⣿⣯⠻⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⠿⠟⠛⠀⠀⠀⠉⠛⠛⠿⢿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣂⣉⣠⠤⠌⠛⣛⠳⠦⣤⡈⠻⣿⣿⣿⣿⣿⣿⣿⡆⠘⢿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠶⣶⠯⢶⠖⣴⠶⠶⠆⠀⠀⠈⠡⠿⢿⣿⣿⣶⣤⡀⠀⠉⢻⣟⠓⣦⢹⣿⣿⣿⣿⣿⣿⡷⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣷⣶⣤⣀⡀⠀⠀⠀
⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣏⣿⠀⠀⠀⢠⣿⣿⣿⣅⣀⣀⣠⣿⣿⣿⡏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⢹⣿⣿⣿⣿⢏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠘⣿⣿⡏⢻⡿⢻⣿⣿⣿⡟⣇⢿⡟⣿⣻⣿⣛⣛⠿⡿⢛⠛⠒⠛⠒⠒⠚⠒⠒⠒⠒⠒⠂⣼⣿⣿⣿⣿⣸⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⣿⣿⣇⣎⣵⢸⡡⢮⠱⡆⡇⠹⠀⢹⣿⢨⢋⣍⡇⡇⡟⠟⣦⡒⡒⡓⣛⢿⢲⠀⠀⠀⠀⣿⣿⣿⢟⠋⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⢰⢀⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡿⣶⣾⣻⣶⣷⣿⣷⣥⣭⣥⣿⠷⣰⣧⣻⣼⣘⣀⠀⠀⠀⣿⣿⣿⡏⢠⠇⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣿⠀⠀⠀⢸⣾⢹⡝⣭⡍⡿⣸⢺⢛⡛⠟⡛⣷⣟⡻⡟⠟⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⡟⡿⣿⡂⠀⢸⣿⣿⣟⠁⠈⣰⠞⣁⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀
⠀⠀⠈⠀⢸⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⣀⣠⣼⣿⣾⣾⣿⣷⣷⣯⣬⣼⣧⣮⣩⣿⣴⣤⣇⣧⣠⣙⣰⣸⣇⣇⣣⡇⠶⢰⠨⡆⡇⣷⠘⡀⢸⣿⣿⠇⡀⢰⠇⢸⣿⡇⣿⢸⣿⢠⣿⠈⣧⠀⠀
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢸⣿⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⢿⣿⣿⣮⢿⣿⣿⣿⣿⢿⣿⣿⣿⣽⣿⡿⠿⢿⣿⣿⣿⣟⣴⡇⣿⣿⣿⣦⠀⢺⠀⢸⣿⣆⡏⡾⣿⣸⣿⠀⣼⠀⠀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⡇⠀⠀⠀⠘⣯⣬⣥⣧⣧⣛⣏⣆⣇⣻⣸⣪⣸⣿⣘⡌⣰⠠⢬⠲⢸⣤⣬⢸⢩⡍⡋⢍⣝⡋⠉⢻⡟⢛⢛⠻⣟⠻⣿⣿⡏⠿⢰⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⣿⢿⣿⣿⣿⣿⡛⡿⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⣱⣎⣿⡃⠀⠘⢧⣬⣼⣷⣴⣤⣕⣺⣁⠉⡃⠐⠈⠳⣤⣀⣀⣀⣀⣠⠶⠋⠀⠀⠀
⠀⠀⠀⣁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠰⡃⠅⠇⠶⢘⠰⢀⣷⠠⣮⢫⡏⡏⣍⢫⢹⢿⡟⡛⣟⢸⢙⡋⡛⢽⢟⢿⡿⠿⢭⡥⠶⠾⢿⢻⣿⣿⣿⣿⣿⡇⠀⠁⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠰⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠉⠛⠉⠉⠉⠉⢸⠿⠷⣶⣷⢶⠷⢷⣾⣼⣾⣧⣧⣥⣼⢬⣧⣇⣊⣸⣸⢎⡳⣿⣇⣷⡇⣾⢸⢲⡆⡆⡏⢖⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠈⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠐⠂⢄⠈⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢶⣿⣿⣿⣯⢹⣾⣷⣿⢿⠛⢻⡟⠛⠻⢿⣿⣿⣷⣧⣈⣁⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣴⣶
⣆⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⢀⡄⣼⡀⢡⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿⣿⣿⢿⣿⣿⣿⣭⣽⣿⣿⣿⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⢧⡀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣿⣇⢻⡇⣈⠀⠀⠀⠀⠀⠀⠀⢀⣥⣤⣝⠻⣿⡟⠛⠛⣿⠟⣿⣿⣿⣿⣏⠙⠋⠙⠛⠿⠿⢿⣏⣛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿
⠀⠀⠙⠄⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠼⠇⣿⣿⣶⣶⣤⣤⣶⣾⣿⣿⡿⢿⣧⣴⣶⣄⠀⠀⣠⣿⣿⣿⣿⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠲⠭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣴⣾⣿⣿⣿⣿⣿⣿⣿⣯⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠇⠈⠉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣶⣶⣮⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢤⣶⡒⠚⠻⢯⣝⣻⣿⣿⣿⣿⣿⣿⣷⠙⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣐⣂⣡⡤⠤⠬⠙⣛⡓⠶⣤⡈⠛⣿⣿⣿⣿⣿⣿⣿⡇⠈⢿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠏⢶⡶⠿⠿⠶⠆⠀⠀⠈⠉⠿⠿⣿⣿⣶⣦⣀⠀⠉⢻⣿⠛⢦⠸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣶⣤⣄⣀⠀⠀⠀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣿⡇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⣼⣿⣿⣇⣀⣄⣀⣼⣿⣿⣿⢉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠉⣿⣿⣿⣿⡿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⣿⣿⠀⠀⠀⠀⢿⣿⣿⡏⢻⡿⢻⣿⣿⣿⡟⣸⢿⣻⣟⣿⠛⣻⣻⠿⡿⢛⡛⠛⠛⠒⠛⠓⠚⠒⠒⠒⠒⢰⣿⣿⣿⣿⣇⡇⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠀⠀⠀⠀⢸⣿⣿⡇⣎⣵⢸⡡⢮⠹⡆⣿⠸⢘⢹⣿⢨⢋⣍⡇⡇⡟⠟⣶⡒⡒⡒⣚⢿⢲⠀⠀⠀⢸⣿⣿⣿⡟⢸⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⡆⣸⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡾⣾⡞⣿⣶⣷⣿⣷⣥⣭⣥⣿⠷⣰⣧⣻⣼⣘⡀⠀⠀⣸⣿⣿⣷⠃⡾⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⢤⢤⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⡇⠀⠀⠀⣇⣿⢹⡝⣍⡍⡿⣸⢺⢛⡛⠟⡛⢵⣟⠻⡟⡛⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⡟⡿⣿⣄⠀⣿⣿⣿⡑⠁⠇⣰⠞⣁⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀
⠀⠀⠈⢠⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢀⣀⣠⣷⣿⣾⣾⣿⣷⣷⣯⣬⣼⣧⣦⣯⣻⣥⣤⣧⣧⣠⣻⣨⣸⣇⣇⣣⡇⠶⢰⠸⡆⡇⣷⠘⣀⣿⣿⡟⠀⠀⢰⠇⢸⣿⡇⣿⢸⣿⢠⣿⠈⣧⠀⠀
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⡀⠀⠀⠀⠀⣿⣿⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⢿⣿⢷⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣽⣿⣟⠿⠿⣿⣿⣿⣟⣵⢸⣿⣿⣷⡄⡀⢺⠀⢸⣿⣆⡏⣾⣿⣸⣿⠀⣼⠀⠀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢻⣯⣬⣥⣧⣧⣻⣏⣆⣇⣻⣸⠪⣸⣿⣘⡌⣰⠠⢬⢲⢸⣤⣼⢸⢨⡍⡋⢍⣽⡋⠉⢻⡟⢛⢛⠻⣟⢻⣿⣻⡏⠇⡧⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⡛⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣷⣱⣆⣽⡃⠀⠘⣧⣬⣼⣿⣦⣄⣕⣪⣁⢜⠁⠑⠈⠳⣤⣀⣀⣀⣀⣠⠶⠋⠀⠀⠀
⠀⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢸⡃⠅⠇⠶⢘⠴⠆⣿⡨⣮⢫⡏⡏⣍⢫⢹⢟⡟⡛⣟⢸⢙⡋⡛⠽⢛⢿⡿⠿⢿⡵⠾⠿⢿⣻⣿⣿⣿⣿⣿⡏⡸⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠫⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠙⠉⠉⠉⠋⢉⡿⠛⢷⡶⠷⠾⠷⣷⣾⣾⣾⣧⣧⣥⣼⢬⣧⣇⣋⣸⣸⢎⡳⣿⣇⣷⣇⣾⢸⢲⡆⡆⡏⢒⠁⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠘⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠲⠀⡀⢃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠷⣾⣿⣿⣿⣍⣿⣾⣷⣿⡟⠙⣿⠛⠛⠿⣿⣿⣿⣷⣅⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣶
⣄⠀⠀⠀⠈⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣤⢰⡇⠈⡄⠀⠀⠀⠀⠀⠀⠀⣿⡯⣿⣿⣿⡿⣿⣿⣿⣯⣥⣿⣿⣿⣿⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠙⢷⡀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣸⣟⠘⡿⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣌⡛⢿⣿⠛⠛⢻⡿⢻⣿⣿⣿⣿⡟⠛⠛⠛⠻⠿⣿⣿⣛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿
⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣟⢵⡟⣼⣿⣶⣶⣤⣤⣴⣶⣿⣿⣿⣿⣷⣤⣴⣦⡀⠀⠀⢸⣿⣿⣿⣿⣷⡦⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠲⠭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣶⣿⣿⣿⣿⣿⣿⣿⣭⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⠏⠉⠛⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣷⣶⣶⣬⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠒⠒⠿⢿⣍⣹⣿⣿⣿⣿⣿⣿⣿⡎⠻⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠁⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣤⡶⢖⠀⠀⠀⠀⠠⢐⣈⣀⠀⠈⢛⣛⠳⠦⣤⡈⠻⣿⣿⣿⣿⣿⣿⣿⣷⠀⠙⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣷⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⢿⣷⠿⠿⠶⠾⠖⠀⠐⠈⡿⣯⣿⣷⣦⣄⡀⠀⠙⢻⣟⠻⣦⠘⣿⣿⣿⣿⣿⣿⣿⠂⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢸⣿⣿⡇⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣾⣿⡿⠀⠀⠀⠀⣿⣿⣿⣄⣀⣀⣠⣿⣿⣿⣟⣛⣙⣛⣛⣛⣛⣛⣛⣛⣛⣛⣋⣉⣙⣛⣛⣛⣉⣉⣛⣛⡛⢻⣿⣿⣿⣿⠻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢹⣿⡇⠀⠀⠀⠀⣿⣿⣿⡏⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⢿⣿⣿⠿⡿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠯⠅⣼⣿⣿⣿⣿⣼⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⣿⣿⣿⢇⣎⡵⢸⡡⢯⠹⡆⡗⠸⣓⢹⣿⢨⢋⣍⡇⡇⡞⠟⣶⡂⡒⡒⣓⢖⢲⠀⠀⠀⣿⣿⣿⣿⠇⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢰⣿⠇⠀⠀⢠⠀⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡿⣾⣾⠾⣿⣶⣶⣷⣷⣥⣭⣥⣿⠷⣰⣧⣻⣼⣘⣀⠀⢀⣿⣿⣿⡟⢰⠃⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠤⠄⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⢸⠀⣿⢹⡝⣍⡍⡟⣸⢺⢛⡛⠟⡃⢼⣟⠋⡟⠛⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⡟⡿⣿⣀⢸⣿⣿⣯⠂⢸⠀⣠⠞⣁⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠆⢀⣀⣸⣾⣿⣾⣾⣾⣷⣷⣭⣬⣼⣧⣬⡉⣿⣤⣄⣃⣇⣠⣙⣠⣸⣇⣇⢣⠇⠶⢰⠸⡆⡇⣷⠐⣸⣿⣿⠃⠀⡇⢰⠇⢸⣿⡇⣿⢸⣿⢠⣿⠈⣧⠀⠀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣴⡆⠀⠀⠀⠈⣿⣷⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⢿⢿⠤⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣽⡿⠿⢿⠿⣿⣿⣷⣟⣵⣿⣿⣿⣦⢀⠃⢺⠀⢸⣿⣆⡏⡾⣿⣸⣿⠀⣼⠀⠀
⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡏⣯⣬⣥⣧⣧⣻⣏⣆⣃⣻⣸⠨⣸⣿⣘⡄⣰⠠⢬⢲⢸⣤⣬⢸⢩⡍⡋⠍⣽⡗⠁⢲⡟⢛⢛⠻⣟⣻⣿⣻⡆⠼⢠⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⣧⡿⢿⣿⣿⣿⣿⠛⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣶⣱⣆⡿⠇⠀⠸⣧⣬⣼⣦⣤⣭⣑⣪⣁⡌⠈⠐⠈⠳⣤⣀⣀⣀⣀⣠⠶⠋⠀⠀⠀
⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢻⡃⠄⠇⠶⢈⠰⠀⣷⠨⣮⢩⡏⡏⡍⢫⢻⢛⡙⡛⣟⢸⢙⡋⡛⠙⢛⢿⡿⠿⢷⡤⠴⠿⢿⣻⣿⣿⣿⣿⣿⣯⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠉⠉⠉⠉⠉⢹⠿⠻⣷⠶⠷⠶⠷⣷⢾⣾⣼⣧⣧⡥⢼⣬⣧⡇⣂⣸⣸⢎⡳⣿⣇⣷⡇⣾⢸⠠⠆⡆⡏⢒⠉⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠐⠦⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣶⣿⣿⣿⣏⢸⣿⣾⣷⡿⠛⢻⡟⠛⢿⣿⣿⣿⣿⣀⠑⠂⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⡀⠀⠀⠀⠀⠺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⣾⠀⢡⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⣿⣿⣿⣾⣿⣿⣿⣯⣹⣿⣿⣿⡆⢀⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠙⢆⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣺⠃⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠈⠻⢿⣿⡿⠟⠿⣿⠿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡟⣨⢰⣿⣶⣤⣤⣠⣤⣤⣶⣾⣿⣿⣷⣠⣤⣠⡀⠀⠀⠀⣿⣿⣿⣿⣷⣤⡤⠀⠀⠀⠉⠉⠉⠒⠒⠬⢍⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣛⣩⣿⣿⣿⣿⣿⣿⡖⠶⣶⣿⣿⣿⠿⡿⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⠻⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⠖⠀⠀⠀⠀⠀⠀⠐⠈⠙⢿⣶⣤⣀⡉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢻⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠶⣶⠿⣷⡶⠿⠿⠷⠷⠀⠠⣄⣖⣶⣦⣁⠀⠀⠐⠲⣼⣿⣷⡀⠹⣿⣿⣿⣿⣿⣿⣿⡇⠀⠹⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡁⢸⣿⣿⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⠄⢸⣿⡏⠀⠀⠀⢸⣿⣿⣿⢀⣀⣀⣠⣿⣿⣿⡟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛⣻⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢸⣿⡇⠀⠀⠀⢸⣿⣿⣿⡍⢻⡿⢻⣿⣿⣿⡇⣿⢿⣿⣿⣿⠿⣿⣿⠿⡿⢿⡿⠿⠯⠯⠭⠯⠯⠯⠭⠭⠁⣿⣿⣿⣿⡟⡇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⣿⠇⣎⣵⢸⡱⢮⡹⠆⡇⠸⣓⢹⣿⢨⢋⣍⡇⡇⡞⠟⣶⡀⡒⡒⣚⢗⢲⠀⠀⠀⣿⣿⣿⠿⢣⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢸⣿⠁⠀⠀⢀⠀⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡿⢾⣾⢾⣿⣶⣶⣷⣷⣥⣭⣥⣿⠷⣰⣧⣻⣼⣘⡀⠀⢸⣿⣿⡏⠁⢸⠁⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⢀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⠨⣿⢹⡝⣍⡍⡿⢸⢺⢛⡛⠟⡓⢼⣟⠙⡟⠛⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⡟⡿⣷⡄⢸⣿⣿⡇⠀⡞⠀⣰⠞⣁⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀
⠀⠀⠀⣥⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣀⣤⣴⣿⣾⣾⣾⣷⣷⣭⣬⣼⣧⣬⡉⣿⣄⣄⣃⣇⣠⣙⣠⣸⣇⣇⢣⠇⠶⢰⠸⠆⡇⣷⠐⣼⣿⣿⠀⠀⡇⢰⠇⢸⣿⡇⣿⢸⣿⢠⣿⠈⣧⠀⠀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⣴⣾⠀⠀⠀⠀⢸⣿⣶⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⢿⢼⣬⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣵⡿⠿⣿⣿⣿⣿⣷⣟⣵⣿⣿⣿⣤⢰⠀⢺⠀⢸⣿⣆⡏⡾⣿⣸⣿⠀⣼⠀⠀
⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠸⡟⣯⣬⣥⣧⣧⣻⣏⣆⣃⣻⣸⠠⣸⣿⣈⡌⣰⠠⢬⢲⢸⣨⣍⢹⢩⡍⡋⠍⣿⡒⠀⢶⡟⢛⢛⠻⣟⣻⣿⣻⡆⠾⢠⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⠀⠠⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⣧⡿⢿⣿⣿⣿⣿⠛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣶⣱⣆⡿⠆⠀⠸⣧⣬⣼⣧⣼⣝⣗⣺⣁⡍⠁⠀⠈⠳⣤⣀⣀⣀⣀⣠⠶⠋⠀⠀⠀
⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣿⡃⠄⠇⠶⢈⠴⠀⣷⠩⣮⢩⡏⡍⣍⢫⢻⢙⡙⡛⣟⢸⢙⡋⡛⠙⢟⢿⡿⠷⢦⡤⠴⠿⢿⣻⣿⣿⣿⣿⣿⣽⡃⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠈⠉⠉⠉⠉⠉⢹⠿⢻⡷⠶⢷⠶⢷⣷⢾⣾⣾⣧⣧⠥⢼⣬⢧⣧⣂⣸⣸⢎⡳⣿⣆⢷⡇⣾⢸⠠⠆⡆⡏⢒⠉⡂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠠⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣤⣿⣿⣩⣆⢸⣿⣾⣷⡟⠛⢻⠟⠿⣿⣿⣿⣿⣛⠀⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠩⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⡀⣴⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⣿⣿⣷⣾⣿⣿⣿⣛⣸⣿⣿⣿⡄⠀⣼⣶⣦⣬⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢀⣾⠃⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⠈⠿⢿⣿⡾⠿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠇⠘⢠⣦⣤⣄⣀⣀⣀⣤⣴⣾⣿⣿⣦⣀⣍⣁⠀⠀⠉⠀⣿⣿⣿⣿⣧⣀⠀⠀⠉⠙⠛⠛⠲⠤⢍⣙⡻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣝⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣬⣽⣿⣻⣿⣿⣿⣿⣿⣟⢋⣭⣿⣿⣿⣿⣷⡲⢶⣶⣿⣿⣿⡿⠿⢏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣾⣿⣿⣿⣿⣷⣶⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠃⠀⠀⠈⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠄⢶⣖⣈⠉⠛⠻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠛⠛⠃⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠶⣶⠿⢷⣶⡾⠿⠷⠖⠀⠀⢀⣀⣤⠄⠀⠈⠉⢉⣛⣲⣦⡄⠙⣿⣿⣿⣿⣿⣿⣿⣿⡀⠘⢿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣍⡙⠃⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⠀⣿⣿⠇⠀⠀⠀⣿⣿⣿⡇⢀⣀⣠⣼⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⢻⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣷⡍⢻⡿⢻⣿⣿⣿⡜⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⢸⣿⣿⣿⣿⢳⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⢻⣿⣿⣿⠇⣎⡵⢸⡱⣮⡱⠆⡇⠸⡃⢹⣯⢠⢋⣍⡇⡇⡞⠝⣶⡂⡒⡒⣒⢟⢲⠀⠀⢸⣿⣿⣿⡿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡃⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡿⣿⣾⣾⣿⣶⣶⣷⣷⣥⣭⣥⣿⠷⣰⣧⣫⣼⣘⣀⠀⣿⣿⣿⠇⠀⡏⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⢸⣿⢹⡝⢍⡍⡟⢸⢺⢛⡛⠟⡂⣿⡗⠛⡚⠛⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⡟⡿⣿⡆⣿⣿⣿⠀⢠⠇⠀⣠⠞⣁⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀
⠀⠐⡒⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⡅⠀⠀⣠⣦⣼⣿⣾⣾⣾⣷⣧⣭⣬⣼⡧⣦⠉⣿⡀⣄⣃⣇⣠⣘⣠⣸⣇⣇⢣⠇⠶⢰⠨⠆⡇⣶⠸⣽⣿⡏⠀⢸⠀⢰⠇⢸⣿⡇⣿⢸⣿⢠⣿⠈⣧⠀⠀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣗⢄⠀⣴⣿⡇⠀⠀⠀⠀⣿⠁⣴⢚⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⠧⣷⢴⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣷⣟⣸⣿⣿⣷⡄⡄⠀⢸⠀⢸⣿⣆⡏⡾⣿⣸⣿⠀⣼⠀⠀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡇⠀⠀⠀⠀⠟⠀⢫⣬⣥⣧⣧⣛⣏⣂⣃⣻⣘⠈⣸⣿⡈⡌⣱⠡⢬⠲⢸⢈⣜⢸⢸⡍⡫⠍⣵⡤⠄⢤⣟⢛⢛⡻⣟⣻⡿⣻⡆⠷⣦⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⠀⡼⢿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣶⣱⢆⡽⠅⠀⢼⣧⣬⣼⣤⣼⣽⣗⣪⣁⢌⠋⠁⠈⠳⣤⣀⣀⣀⣀⣠⠶⠋⠀⠀⠀
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠘⠸⠃⠄⠇⠶⠈⠰⠀⢹⠫⣮⢩⡍⡍⡍⢩⢹⢙⡙⡛⣟⢸⢙⡋⡛⠽⢟⢿⡿⠶⢄⡀⠴⠿⢿⣻⣿⣿⣿⣿⣗⣾⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠁⠉⠉⠉⡇⠾⣿⠷⠶⣷⠶⣷⡷⣾⣾⣬⣥⠧⠥⠼⣬⣧⣇⣀⣸⣸⢎⡑⣿⣄⢷⠇⣾⠸⠠⠆⡆⡏⢒⡍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠁⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⡀⠀⠀⡇⠀⠈⠀⠀⠀⠀⠈⠀⣻⣿⣿⣿⣷⣀⣩⣗⣖⣂⠀⢹⣿⣾⣷⡿⠿⣿⢿⣿⣿⣭⣻⠛⠋⠀⠧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⡄⠰⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣭⣿⣿⣿⣿⠳⣿⣿⣿⣿⠀⠀⣥⣤⣙⣿⣿⣿⣷⣶⣤⣤⣀⣀⣀⣠⣤⣤⣤⣤⣶⣶⣾⣷⣾⣿⣿⣿
⠀⠀⠀⠈⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⢠⡦⢸⣧⠀⠃⠀⠀⠀⠀⠀⠀⠀⠿⠇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠿⣷⠆⢯⢠⣄⡀⠀⠀⠀⠀⣀⣠⣾⣿⣷⣆⠘⠛⠁⠀⠘⠋⠀⣿⣿⣿⣿⠛⠛⠛⠻⠿⣿⣿⣟⡛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠻⢟⣿⣿⣿⣿⣿⢿⣛⣿⣿⣿⣿⣧⣀⠀⠀⠈⣿⣿⣿⣷⣿⣿⣿⣿⡿⠟⣻⣿⣶⣾⣿⣦⣤⣤⣼⣿⣿⣿⡿⣿⠆⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣷⣿⣇⣭⣿⣿⣿⣿⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⡟⠛⠻⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⢻⣿⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠟⠛⣻⣿⣿⣿⣿⣷⣶⣮⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⣶⣟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠙⠻⠿⠃⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠯⢶⣤⠴⠶⠶⠆⠀⠀⠀⠀⠂⠈⠿⣦⣤⣀⣀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡃⠀⢿⣿⡟⠀⠀⠀⢸⣿⣿⣿⠀⢀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡗⠀⢸⣿⠅⠀⠀⠀⢸⣿⣿⣿⣸⡍⢻⡿⢻⣿⣿⣯⡍⣭⢭⣭⣤⣤⣤⣤⣤⠤⡤⣤⣤⣤⣦⣤⣤⣴⣶⣶⣤⢀⣿⣿⣿⣿⡇⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⣎⣵⢸⡱⣯⠱⠆⡿⠹⡓⢹⣿⢨⢋⣍⡇⡇⡟⠟⣿⡛⡛⡛⣛⢟⢻⠛⢸⣿⣿⣿⣿⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣾⣮⣿⣶⣶⣷⣷⣥⣭⣥⣿⠷⣰⣧⣻⣼⣘⣀⢸⣿⣿⡏⠉⢸⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⢹⡝⢉⡍⡟⢸⢺⢛⡙⠟⡘⣿⣛⠻⡛⠛⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⡟⡿⣿⣾⣿⣿⡇⠀⡏⠀⠀⣠⠞⣁⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀
⡀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⢀⣤⣠⣿⣿⣶⣾⣶⣷⣧⣬⣬⣼⣤⢄⣩⡟⢀⣀⣃⣃⣠⣘⣠⣸⣇⣇⢣⠇⠶⢰⠈⠆⡇⣧⠸⣽⣿⠁⠀⠇⠀⢰⠇⢸⣿⡇⣿⢸⣿⢠⣿⠈⣧⠀⠀
⠀⢠⣴⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡗⠀⢰⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⢸⢀⣿⣷⣶⣶⣿⣿⣿⢿⣿⣿⣿⣥⣛⣛⣛⣻⣿⣿⣷⣟⣽⣿⣿⣦⢰⠀⠀⢸⠀⢸⣿⡆⡏⡾⣿⣸⣿⠀⣼⠀⠀
⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⣴⣿⠀⠀⠀⠀⢸⠁⠀⢨⣬⣥⣧⣧⣻⣏⣂⣃⣻⣘⠀⣸⡿⠘⠆⣰⠩⠬⠳⢹⢩⡭⢹⢩⡍⡋⢍⣽⡛⠉⢻⡟⢛⢛⢻⣟⡻⡟⣻⡌⠆⠠⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠀⠀⡼⢿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣶⣶⣷⣶⣾⣶⣁⢆⡙⠁⠁⣉⣩⣬⣼⣼⣽⣵⣕⣪⣁⣜⠃⠀⠈⠳⣤⣀⣀⢀⣀⣠⠶⠋⠀⠀⠀
⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠠⠃⠀⠇⠶⠈⠰⠀⣿⠋⣮⢩⡍⡍⡍⢩⢹⠛⡛⡛⡟⢻⢙⡋⡛⣹⢛⢿⡿⠇⠀⠀⠽⡿⣿⣻⣿⣿⣿⣿⣽⡟⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠉⠉⠁⢂⠀⣴⣿⠗⠰⠲⢶⡷⣷⣾⣬⡤⠁⠧⠅⣼⣬⣇⣇⣋⣼⣘⢊⡑⣿⣀⠳⠇⣿⠸⠠⠆⡆⠏⢐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⠀⠀⠀⠃⢸⣿⣿⣿⣆⡀⠀⣦⠍⠉⠀⠘⣿⣿⣾⣷⣿⣿⣿⣿⣏⠉⠉⠛⠋⠉⢐⠃⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠐⠀⢀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢿⣿⣿⣶⣦⣸⣿⣿⣿⠁⠀⢨⣁⡻⣿⣿⣶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤
⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⢀⡀⣼⠀⠡⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⡇⠀⣴⣾⣷⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢤⠾⣀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡙⠻⠿⠟⠋⠙⠿⠋⢻⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠙⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⠁⠀⠉⠜⣷⣶⣤⣤⣤⣤⣶⣾⣿⣿⣿⣆⣠⣴⣆⠀⠀⠀⣿⣿⣿⣿⡁⠀⠀⠈⠉⠙⠛⠿⠶⠤⣉⣙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣶⣤⣤⣤⣘⣁⣼⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⠛⠿⢿⣿⣿⣛⣂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠃⠉⠉⠛⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣶⣾⣿⡿⠿⢿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣼⣿⣿⣿⣿⣿⣿⣾⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠆⠶⠄⠠⠰⠶⠆⠀⠀⠀⠀⠀⢠⣔⠐⠘⠛⠿⣭⣤⣽⣿⣿⣿⣿⣿⣿⣿⣷⠙⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠈⠉⠀⠈⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⣿⣿⡇⠀⠀⠀⣸⣿⣿⡇⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⢿⣿⣿⡇⣴⡍⢻⡿⢻⣿⣿⣟⡛⣛⢛⣛⣛⣋⣉⣹⣻⡿⣭⣭⣉⣉⣉⣉⣉⣉⣩⣩⣉⢩⣿⣿⣿⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣰⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⣿⠇⣎⣵⢸⡱⣯⠱⠆⡿⡹⡓⢹⣿⢨⢋⣝⡇⡇⡟⠟⣿⡟⡛⡻⣛⢿⢻⠓⢸⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⠀⠀⠀⠀⢸⣿⣿⣧⣿⣿⣿⡿⢿⣿⣿⣿⡿⢿⣿⣶⠎⡿⣶⣶⣷⣷⣥⣭⣥⣿⠷⣰⣧⣫⣼⣘⣀⢸⣿⣿⡟⠋⣼⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⡀⠸⣿⣿⢩⡍⢉⡍⡟⢸⢺⢛⡙⠟⡛⡇⡿⠻⡿⠟⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⡟⡿⣷⣿⣿⣿⠃⠀⡇⠀⠀⣠⠞⣁⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⡏⠀⠀⢀⣇⣰⣿⣿⣶⣾⣶⣷⣧⣭⣬⣼⣤⣄⣭⡟⣁⣀⣃⣃⣀⣘⣠⣸⣇⣇⢣⠇⠶⢰⠨⠆⡇⣷⠸⣽⣿⠀⠀⠃⠀⢰⠇⢸⣿⡇⣿⢸⣿⢠⣿⠈⣧⠀⠀
⠀⠀⢄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⢠⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⠽⢠⣿⣥⣤⣴⣿⣿⣿⢿⣿⣿⣿⣥⣛⡛⠛⠻⢿⣿⣷⣟⣽⣿⣿⣶⢸⠀⠀⢸⠀⢸⣿⣆⡏⡾⣿⣸⣿⠀⣸⠀⠀
⠀⣐⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢨⣬⣥⣧⣧⣻⣏⣂⣃⣻⣘⠀⣸⡟⠈⡄⣽⠹⢼⢳⢹⢫⣿⢹⢩⡝⡋⠍⣽⡟⠝⠿⡟⢛⢛⢻⣟⡻⡟⣻⡈⠆⡀⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢠⣤⠀⠀⠀⠀⢸⠀⠀⡼⢿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣽⣷⣾⣶⣾⣿⣿⣿⣾⣶⣷⣶⣾⣷⡑⢆⠿⠃⠃⠙⠣⢬⣼⣼⣽⣵⣕⣪⣁⣜⠂⠀⠈⠳⣤⣀⣀⢀⣀⣠⠶⠋⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠀⢀⠃⠀⠇⠶⢈⠰⠀⣿⠋⣮⢩⡍⡏⡍⢫⢹⢛⡛⡛⣟⢻⢙⡋⡛⢙⢛⢻⠿⠛⠀⠀⠾⠿⣿⣻⣿⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⡅⠀⢀⠿⠓⠀⠀⠒⠷⣷⣾⣬⠀⠁⠡⠥⣼⣬⣇⡇⣋⣸⣘⢊⡑⡿⠀⢳⠇⣿⠸⠠⠆⡆⡏⢐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠄⢸⣿⣿⣧⠀⠀⠀⠈⠈⠀⠀⢨⣿⣿⣾⣷⣿⣿⣿⠿⠟⠉⠉⠛⠉⠉⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣶⣿⣶⣤⣄⢸⣿⣿⣿⠋⠉⠘⠁⣾⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣀⣀
⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣶⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⣿⣿⣿⣾⣿⣿⣿⣯⣹⣿⣿⣿⡆⠀⢈⣉⣋⣙⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠼⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣈⠻⢿⣿⡿⠿⠿⣿⠿⢿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡻⠦⠨⠄⣦⣀⣀⣀⠀⣀⣀⣤⣾⣿⣿⣷⡄⠈⡁⠀⠀⠀⠀⢸⣿⣿⣿⠟⠛⠛⠻⠿⢿⣿⣿⣟⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⡀⠀⠀⠙⠻⠛⣿⣿⣿⣿⣿⣿⠋⣉⣿⣿⣿⣿⣿⣶⠶⣶⣿⣿⣿⠿⠆⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒⠬⢭⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡄⣀⣽⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣫⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣭⣭⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠲⣶⠀⠴⠄⠠⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⠟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⡆⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀⢸⡇⠀⠀⠀⣿⣿⣿⡇⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣾⠃⠀⠀⠀⣿⣿⣿⡇⢰⡍⢻⡿⢻⣿⣿⢿⡟⢿⠿⠿⠿⠿⠿⠿⣿⠿⡿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⣭⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⡇⣻⠇⣎⣵⢸⡱⣯⠱⡆⡿⠹⣓⢹⣿⢨⢫⣝⡇⡇⡟⠟⣿⡟⡟⡻⣛⢿⢿⠇⣼⣿⣿⣿⣿⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠃⠀⠀⠀⠀⠀⢻⣿⣿⣧⣿⣿⣿⡿⢿⣿⣿⣿⠟⣿⣿⣶⠈⠻⠶⣶⣷⣷⣥⣭⣥⣿⠷⣰⣧⣫⣼⣘⡀⣿⣿⣿⠿⠇⡇⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠓⠂⠀⠀⠀⠀⢸⣿⣿⢹⡍⢉⡍⡟⢸⢺⢛⡛⠗⡛⣯⣷⡲⡶⠾⢿⠿⢿⠿⠿⠿⡿⢿⣿⣿⣿⡟⡿⣧⣿⣿⣿⠀⢀⠇⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣸⣿⣿⣶⣾⣶⣧⣥⣬⣬⣬⡧⢄⣯⠗⠀⣀⣃⣃⣀⣘⣠⣸⣇⣇⢣⠇⠴⢰⠈⠆⡇⣦⠸⣽⣿⠀⢸⠀⠀⢰⠇⢸⣿⡇⣿⢰⣿⢀⣿⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡗⠀⣿⡀⠀⢠⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡇⡇⢠⣿⣤⣤⣤⣷⣾⣿⢿⣿⣿⣿⡅⠛⠉⠙⠛⢿⣿⣷⣟⣽⣿⣿⣄⣸⠀⠀⢸⠀⢸⣿⡆⡏⡾⣿⣸⣿⠀⣸⠀⠀
⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣿⠀⠀⢨⣬⣥⣧⣧⣛⣏⣂⣃⣻⣘⠀⣸⡇⣀⡄⣿⠹⢼⠳⢸⢮⣾⢸⢸⠝⡯⠅⣴⡶⠄⣦⣔⢛⢛⢻⣟⡻⡟⣻⡆⠆⠀⠘⣯⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣴⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠨⢿⣿⣿⣿⣿⡛⣿⣿⣯⣿⣿⣿⣇⣽⣭⣉⣍⣙⣿⣿⣿⣷⣶⣾⣶⡰⢆⠿⠧⠤⢤⣥⣬⣼⣼⣽⣵⣕⣢⣁⣌⡀⠀⠈⠳⢤⣀⣀⢀⣀⣠⠖⠋⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠇⠶⠈⠱⠂⣿⠋⣦⢩⡍⡍⡍⢫⢹⢛⡛⡛⣟⢻⢙⡋⡛⠙⢛⢻⠷⠦⠀⡠⠴⠶⣾⣛⣿⣿⣿⣿⣿⣯⡯⠅⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠟⠃⠀⠀⠒⠷⢷⣜⠈⠈⠁⠃⠁⠘⠈⠇⣇⣇⣸⣘⢊⡐⡾⠄⠷⠇⣿⠸⠀⠆⡆⠏⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠇⠀⠀⠀⠀⢀⠀⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣶⣶⣄⣀⣻⡏⠈⠉⠛⠉⠉⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣗⣤⣀⣨⣥⣀⣀⠀⢸⣿⣿⣿⠛⠛⠛⠋⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠰⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⣿⣿⣯⣽⣿⣿⣿⠛⣸⣿⣿⣿⡀⠀⠈⠬⠻⣿⣿⣷⣦⣄⣀⡀⠀⡀⠀⠀⠀⠀⠀⣄⣀⣠⣴⣶⣦⣶⣶⣶⣶
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⣤⢸⣧⠀⠃⠀⠀⠀⠀⠀⠀⠀⠻⠓⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠶⠠⠠⡁⠀⠉⢀⡀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣄⠘⠛⠁⠀⠈⠋⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠈⠜⠿⢓⣶⣶⣾⣿⣿⣿⠟⠉⣙⣦⣴⣶⣦⣤⣤⣤⣿⣿⣿⡏⠁⠀⠀⠀⠉⠙⠛⠿⠦⠤⣉⣛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣿⣿⣯⣭⣭⣭⣶⣾⣿⣿⣿⣷⣤⣤⣤⣤⣤⣀⣸⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣇⠉⠛⢻⣿⣯⣴⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠿⠿⢿⡿⡿⠿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣉⣽⣴⣶⣿⣿⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠏⠉⠛⠛⠻⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⣴⣦⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⠲⠄⠠⠠⠖⠆⠀⠀⠀⠈⠛⠛⠛⠛⠉⠉⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⡄⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢐⠀⠀⠀⠀⠀⣿⣿⣿⠀⢠⡌⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⡿⠿⢼⡀⠀⠀⠀⠀⣿⣿⣿⠀⢸⠇⣎⣵⢸⡱⡦⠱⡆⡷⡸⣒⢰⣶⢠⢊⣝⡇⡇⡖⠖⣶⡖⡒⡒⣒⢶⢲⠂⣿⣿⣿⣿⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣐⣿⣿⣿⣿⣿⣿⣯⡭⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣠⣾⣿⣿⡿⢿⣿⣿⡟⠛⣿⣾⡮⠈⠙⠂⠒⢳⣷⣥⣭⡥⠋⠗⢰⣧⣋⣼⣘⢀⣿⣿⣿⣿⢇⠃⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⠿⢧⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⢩⡍⠉⡍⡟⢸⢺⢋⡓⠞⡛⢻⣅⠀⡀⡒⠤⠬⢜⠥⠦⠴⠤⢶⢶⣿⣿⡟⡿⣾⣿⣿⡇⠀⢸⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⣿⣿⣿⣶⣾⣶⣧⣥⣬⣬⣬⡥⢠⡭⠩⠁⠀⢃⠃⣀⣘⣀⣸⣇⡆⢡⠇⠰⠰⠈⠆⡇⣶⠸⣽⡇⠀⠘⠀⠀⢰⠃⢰⣯⡇⣿⢰⣿⢀⣶⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⣦⠄⠀⠀⠀⠀⠀⠀⢀⢚⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⠸⠀⢸⠆⠀⠀⠠⢠⣴⣶⢀⣡⣀⣈⠀⠀⠀⠀⠁⠛⠛⣷⣞⣽⣿⣷⡄⡇⠀⠀⢸⠀⢸⣿⡄⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⣬⣥⣧⣧⣛⣇⣂⣃⣺⣐⠀⣸⢸⠀⠈⣷⠰⣤⢲⢸⣤⣴⢸⢠⡌⡫⢄⣤⡄⡄⣀⣀⢐⢛⣻⣟⡻⡟⣳⡅⠀⠀⠘⣏⠙⠛⠛⠓⠓⠛⠿⠾⢤⠇⠀⠀
⢶⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡛⣿⡟⠁⢉⢸⣿⣈⣉⡛⣛⠛⢛⠛⠶⣶⣷⣶⣶⣶⡠⢆⣭⣅⣁⣉⣁⣨⡬⣿⣽⣥⣕⣨⣀⡄⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠔⠋⠀⠀⠀
⣿⣿⣿⣿⣿⣦⡐⢄⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠇⠶⠘⠲⠆⣿⠁⢤⠠⡍⡍⡍⢩⢹⠛⡛⡛⣟⢺⢙⡋⡋⠙⠛⢻⠧⠀⠀⡀⠀⡤⣼⢈⣿⣿⣿⣿⣿⣸⣂⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⣷⣄⠀⠀⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠋⠀⠀⠀⠲⠷⣁⠈⠈⠈⠁⠃⠁⠘⠌⠇⣥⣀⠸⣐⢀⡀⠴⠀⠷⠇⡿⠸⠀⠆⡆⠈⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡓⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣍⣀⣀⣀⡀⠀⠀⣿⠇⠈⠀⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣗⠀⠀⠀⢂⡀⠀⠀⠀⣾⣿⣿⡿⠟⠉⡿⠋⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣶⣿⣛⣿⣿⣿⡷⠆⣿⣿⣿⡆⠀⠀⠦⣐⢿⣿⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣤⣤⣤⣤
⠀⠐⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⢀⡀⢸⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠿⠗⢿⣿⣿⣟⣿⣿⣿⣿⣶⣿⣿⣿⣿⠀⢀⣤⣤⣤⣽⣿⣿⣿⣿⣷⣶⣾⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠉⠻⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠠⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⠉⠻⠟⠉⠉⠹⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠁⠐⢠⣤⣤⣴⣶⣿⣿⣿⢿⣧⣀⣤⣶⣀⣀⣀⣸⣿⣿⣿⠛⠉⠉⠉⠛⠛⠿⢿⣿⣋⡙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠿⠛⢛⣋⣭⣽⣿⣿⣦⣤⣀⣀⣀⣀⣀⠀⠸⣿⣿⣿⣿⣿⣭⣤⣾⣿⣿⣿⣿⣿⡏⠛⠿⣿⣿⣛⣅⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠂⠬⣭⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠦⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠓⣀⣤⣾⣿⣿⣿⣿⣿⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⡟⠻⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣩⣤⣼⡿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣋⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉
⠀⠀⠰⣶⠀⠰⠄⠰⠰⠶⠖⠈⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⣀⣴⣿⣿⣿⣿⣷⣶⣯⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣶⣿⣿⣷⣶⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀
⠒⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡇⠀⠀⡄⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠉⠙⡻⡿⢿⣿⣿⣟⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠘⠇⡎⡵⢸⡡⠮⠡⠆⡯⠩⠁⢩⣭⢨⢩⣍⡇⡇⡌⠍⣤⡄⡄⡠⣀⢤⢤⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠚⠶⣾⣿⣿⣿⣷⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣀⣠⣿⣿⡿⢿⣿⣿⠟⢻⣿⠛⠚⠚⠛⠒⠓⠛⠷⠥⠍⠁⠛⠓⠰⠧⢛⣼⣘⢐⣿⣿⣿⣿⢹⠀⠀⠀⠀⠀⣀⡤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠷⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⢩⡌⠉⡍⡟⢸⢺⢙⠀⠘⡃⢻⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢴⣶⡞⡿⢼⣿⣿⡏⠉⢸⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣿⣿⣿⣶⣾⣖⣧⡥⣬⣬⢸⠀⢨⡍⣛⠀⠀⠃⠁⠀⠈⠀⠘⠇⡆⠁⠇⠰⠰⠈⠆⡇⣶⠸⣽⠇⠀⡇⠀⠀⢰⠃⢰⣏⡇⣿⢰⣷⢀⣶⠈⣧⠀⠀
⠀⠀⠀⣶⣶⣶⣦⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡛⡛⡿⣛⠿⡿⡷⠿⢤⠼⢻⠸⠀⢸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡷⣟⣽⣿⣦⠀⡇⠀⠀⢸⠀⢸⢿⡄⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠉⠛⠛⠛⠁⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣥⣧⣧⣓⣇⣂⣃⣺⠀⠀⡀⢸⠀⠈⣰⠠⣤⢰⢠⢠⣤⢠⢠⡄⡂⢄⣀⡀⠀⠀⠀⢀⢓⡻⣟⡻⡟⣺⠀⠀⠀⠘⣏⠙⠛⠛⠓⠓⠛⠿⠾⢤⠇⠀⠀
⣷⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠛⣿⠅⠀⠀⢨⣁⠘⠒⠻⠛⠿⠿⠿⢿⣿⣿⣶⣼⣷⣁⣈⡙⠃⠁⠙⠃⢬⢬⣿⣵⣅⣑⣨⣀⠀⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠔⠃⠀⠀⠀
⣿⣿⣿⣷⣦⣀⢄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠶⠘⠶⠀⣿⠀⢠⠀⡌⡍⡍⢩⢹⠐⡒⡖⣖⢲⢐⡉⡋⠘⠛⢉⠉⠉⠉⠉⠉⠉⠏⢸⣿⣿⣿⣿⡟⡹⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⠐⠗⠓⠘⠈⠀⠁⠃⠥⠼⠬⢄⣁⣀⣸⣀⠀⠀⠀⠀⠳⠇⡿⠸⠀⠆⠆⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢻⣿⣿⣿⣿⣿⣿⣷⣮⡙⢦⣸⣷⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠺⠛⢿⣿⣿⣿⣿⣿⣿⣷⣮⡿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠄⠀⠀⠀⠀⣿⣿⣿⣿⠇⢠⡴⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⣲⢶⣿⣶⣶⣦⡄⣿⣿⣿⡁⠀⠀⣀⠠⣾⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⠀⣀
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣄⠀⠀⠀⠀⠀⠀⣤⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿⣿⣿⢿⣿⣿⣿⣷⣼⣿⣿⣿⡇⠀⠀⣀⣁⣈⣻⣿⣿⣷⣶⣤⣤⣄⣀⣀⣀⣤⣤⣬⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠉⠝⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⠛⢿⡿⠛⠛⢿⠟⢸⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠁⠹⠿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣾⣿⣿⣷⡄⢀⣀⣄⠀⠀⠀⢸⣿⣿⣿⠟⠛⠛⠻⠿⢿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠚⠛⠻⠟⠛⢛⣩⣭⣷⣿⣷⣤⣀⣀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣋⣉⣬⣿⣿⣿⣿⣿⡻⠿⣿⣿⡿⠟⡀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠢⠤⢉⣛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣇⣠⣶⣾⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣩⣭⠷⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠀⠀⠉⠉⠛⠛
⠀⠀⠰⢦⠀⠐⠄⠰⠲⠶⠖⠀⠀⠀⠛⠛⠛⠛⠋⠉⠀⠀⠀⢈⣿⣿⣿⣯⣭⣭⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀
⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⡌⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣽⣿⣛⠛⠻⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠘⠇⣎⣵⢸⡡⢬⠡⡆⡯⠩⣉⢩⣭⢨⢉⣍⡇⡇⡝⢍⣭⡍⡍⡩⣉⢭⢭⠹⣿⣿⣿⣿⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠻⠀⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⡀⣸⣿⣿⡿⢿⣿⢸⠟⢻⡟⢟⠚⠚⠛⠒⠓⠳⠷⠇⠍⠁⠛⠓⠐⢧⣛⣼⣈⢘⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⣀⡤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠉⠉⠉⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢡⡍⠉⡍⡟⢸⢺⢘⠀⠘⡃⣗⣀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⢄⠤⣼⣾⡞⡯⣾⣿⣿⠋⠁⡆⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣶⣾⣖⣣⡥⢬⣬⢬⠀⢨⡅⣛⠁⠀⠃⠁⠀⠘⡀⠘⠇⡆⢀⠇⠰⠰⠈⠆⡇⣷⠸⣹⠀⠀⡇⠀⠀⢰⠃⢰⣏⡇⣿⢰⣷⢀⣶⠈⣧⠀⠀
⠀⠠⣽⣿⣿⣶⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⢓⡛⡛⡿⣛⠿⡻⡿⠷⢥⠸⠃⠈⠀⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡗⣟⣽⣿⣦⢰⠀⠀⠀⢸⠀⢸⢿⡄⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠨⣬⣥⣧⣧⣒⣆⣂⣃⣺⡐⠀⡀⡄⠀⡆⣠⠠⣤⢠⢰⣀⣄⢨⢠⡈⡃⢀⣀⡀⠀⠀⠀⢐⢓⣻⣟⡻⡟⣸⠀⠀⠀⠘⣏⠙⠛⠛⠓⠓⠛⠿⠾⢤⠇⠀⠀
⣿⣷⣖⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⠛⣿⠁⠀⠀⢸⡁⠳⠤⠯⠿⠿⠿⢿⣿⣿⣿⣖⣺⣷⡑⠊⠛⠓⠃⠛⠣⢬⢸⣿⣵⣄⣑⣨⣀⠄⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠔⠃⠀⠀⠀
⣿⣿⣿⣿⣶⣔⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠶⠸⠶⠀⣿⠀⠀⠀⡍⡅⡄⢠⢰⠀⡐⡖⣖⢰⢀⡈⡋⠘⠋⠛⠉⠉⠉⠉⠓⠐⠚⢸⣿⣿⣿⡇⣇⣟⡃⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠋⠁⠀⠀⠀⠀⠒⠓⠘⠈⠀⠁⠧⠥⢼⠬⢀⣁⣀⣘⡈⠀⠀⠀⠀⠳⠇⡿⠸⠀⠆⠆⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣻⣿⣿⣿⣿⣿⣿⣿⣦⡙⢷⣼⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠘⣿⠁⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⡿⠿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠠⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⢠⡤⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠨⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⡶⢶⣿⣶⣤⣤⢰⣿⣿⣽⠃⠀⠰⠀⢠⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢀⡆⠈⠄⠀⠀⠀⠀⠀⠀⠀⢸⣯⢿⣿⣿⣿⢾⣿⣿⣿⣥⣸⣿⣿⣿⡆⠀⠀⠉⠁⠙⣿⣿⣿⣶⣤⣤⣄⣄⣀⣀⣀⣀⣀⣠⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀⠛⢛⣿⡿⠛⠻⣿⠏⢹⣿⣿⣿⣷⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣠⣤⣾⣿⣿⣷⡄⢀⡀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠛⠿⠿⠿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⢛⣩⣭⣷⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣋⢁⣬⣿⣿⣿⣿⣶⠾⢶⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠤⠍⣙⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣾⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣤⣤⡾⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠃⠀⠈⠉⠙⠛⠻
⠀⠀⠰⢢⠀⠀⠄⠠⠰⠶⠖⠀⠀⠀⠘⠛⠛⠛⠛⠉⠀⠀⠀⣈⣻⣿⣿⣭⣭⣝⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀
⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀
⠉⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣛⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⡌⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠶⣶⣶⣿⣿⣿⣿⣿⣿⡟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⠀⠀⠘⠇⣎⣵⢸⡡⢮⡱⡆⣟⡙⣉⢙⣻⢨⢋⣉⡇⡇⡟⢟⣟⡛⡛⡛⣛⢛⢋⢻⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠉⠙⠛⠛⠟⠿⠿⠥⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣸⣿⣿⡿⢿⣿⡿⠿⣿⡿⠿⠾⠾⠿⠶⠶⠷⠷⠧⢭⡵⠿⠷⢰⣧⢻⣼⡀⣘⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⣀⡤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢸⢡⡍⠉⡍⡟⢸⢺⢋⠐⠟⡑⠆⠀⠀⠀⠂⠠⠀⠀⠠⠴⠄⠠⢤⢦⣾⣿⡟⡯⣿⣿⣿⠟⢃⡇⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⢠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣼⣶⣾⣶⣧⣥⣭⡌⡌⠀⣦⠨⠿⠄⠄⢃⠃⠀⠘⡀⠘⡇⡆⢁⠇⠰⠰⠈⠆⡇⣷⠸⣱⡇⢸⠀⠀⠀⢰⠃⢰⣯⡇⣿⢰⣷⢀⣶⠈⣧⠀⠀
⠀⠀⠀⠀⠙⠿⢿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢐⡛⡛⡿⣛⠿⡻⡾⠷⢷⠿⠇⡇⠀⣏⠈⠁⠀⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⢹⢿⣟⣽⣿⡇⣸⠀⠀⠀⢸⠀⢸⢿⡄⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣬⣥⣧⣧⣒⣆⣂⣃⣺⣐⠀⠀⡇⢀⡄⣠⠠⣤⢰⢰⢀⣄⢨⢠⠈⡃⠀⡀⠀⠀⠀⠀⢘⢚⣻⣟⠻⡟⣻⠀⠀⠀⠘⣏⠙⠛⠛⠓⠓⠛⠿⠾⢤⠇⠀⠀
⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⠛⣿⠁⠁⢠⣿⠀⠤⠼⠿⢿⣿⣿⣿⣿⣟⣿⣶⣾⣷⡑⠆⠿⠷⠦⠶⠦⡌⣼⣿⣥⣄⣑⣢⣀⠀⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠔⠃⠀⠀⠀
⣿⣿⣿⣿⣿⣦⣤⣄⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢶⢘⠴⠀⣿⠀⠤⠨⡍⡄⡄⢠⢰⢀⡀⡄⣖⢰⢈⠉⡋⠉⠛⠻⠟⠓⠒⠒⠒⠲⠇⣻⣿⣿⣿⢿⣿⡷⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠙⠋⠁⠀⠀⠀⠒⠒⠓⠘⠈⠈⠥⠧⠥⣼⣈⠀⡁⣀⠘⠈⠀⠐⡷⡀⢷⡇⠖⠸⠀⠆⠆⠉⢐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡙⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⢹⡏⠀⠀⠈⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⡆⠀⣄⡄⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣤⡴⠦⣼⣷⣤⣤⡀⣼⣿⣿⡏⠁⠀⠉⠀⣴⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢰⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⣿⡝⣿⣿⣿⣷⣿⣿⣿⣯⣍⣿⣿⣿⣷⠀⠀⠉⠑⠈⢙⣿⣿⣿⣦⣤⣤⣠⣀⣀⣀⣀⣀⣠⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠂⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠻⣿⣷⠿⠟⢿⡿⠋⣿⣿⣿⣿⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⣤⣶⣿⣿⣦⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠛⠛⠛⠿⠿⣿⣿⣿⡿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠛⠛⠛⠻⠿⠿⢛⣛⣭⣿⣿⣤⣀⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠿⠛⣉⣿⣶⣿⣿⣷⡶⢶⣾⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠠⠌⣙⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣬⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣬⣴⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⣋⣩⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⠛⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠀⠀⠀⠉⠉⠛⠛
⠀⠀⠰⣶⠀⠀⠄⠠⠰⠶⠆⠀⠀⠀⠀⠺⠿⠿⠿⠟⠛⠁⠼⣙⣿⣿⣟⣛⣛⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀
⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢀⡌⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠛⠿⣿⣙⢿⣿⣏⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠘⠇⣎⣵⢸⡡⢮⠱⠆⡟⠙⠛⢙⣛⢈⢋⣙⡇⡇⡟⠟⣛⡛⡛⡛⣛⢛⢛⢿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠁⠲⢽⣿⣿⣿⣿⣿⣿⣿⡻⢠⡀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣸⣿⣿⡿⢿⣿⣿⢿⣿⢿⡿⠾⠾⠿⠶⠷⢿⠷⣥⣭⡥⠿⠷⣰⣧⣯⣼⣘⣜⣿⣿⣿⣇⠂⠀⠀⠀⠀⠀⣀⡤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠟⠛⠁⠿⠏⠀⠀⠀⠀⠀⠀⣿⣿⡇⡐⢡⡍⠉⡍⡟⣸⢺⢉⠘⠟⡐⠆⠀⠀⠀⠀⠡⠀⠂⠠⠤⠄⠠⢤⢤⣾⣿⡟⡿⣿⣿⣿⠟⢸⠀⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣿⢶⣾⣖⣧⣥⣭⢬⠌⢤⣦⠨⠿⠤⠄⠂⠃⠀⡘⡀⢘⡇⡆⢡⠇⠰⠰⠈⠆⡇⣷⠸⣱⡇⡘⠀⠀⠀⢰⠃⢰⣯⡇⣿⢰⣷⢀⣶⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢚⡛⡛⡿⡛⠿⡻⡿⠿⢿⠿⢸⠀⢸⠟⠈⠀⠀⠀⠀⠀⣀⢀⡀⡀⠀⠀⠀⠀⠀⠈⠙⡿⣿⣟⣽⣿⡇⡇⠀⠀⠀⢸⠀⢸⢿⡄⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣬⣥⣧⣧⣒⣆⣂⣃⣺⣘⠀⢸⠀⢘⡄⣠⠠⢄⠰⢰⠀⡄⢨⢠⠈⡃⠀⠀⠀⠀⠀⠀⢓⢛⣻⣟⠻⡗⣓⠀⠀⠀⠘⣏⠙⠛⠛⠓⠓⠛⠿⠾⢤⠇⠀⠀
⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢿⣿⣿⣿⣿⠛⡟⠈⠀⢸⣿⢸⣤⣼⣟⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⡱⠆⠿⠶⠦⢴⣦⠬⣼⣯⣥⣄⣑⣨⣀⠀⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠔⠃⠀⠀⠀
⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠇⣶⢘⠰⠀⡿⠀⠤⢨⡍⡀⡅⢠⢠⢀⡀⡌⣟⢘⢙⠉⡋⠙⠛⠻⠷⠖⠲⠦⠤⠴⢸⣻⣿⣿⣿⣽⣿⡷⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⣤⣤⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠉⠛⠋⠀⠀⠀⠀⠒⠒⠓⠘⠈⠨⠥⠧⢭⡼⠈⠀⠁⠀⠸⠐⠀⠐⣶⣄⢶⡆⠾⠸⠀⠆⠆⡏⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠉⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⠀⠀⡀⠀⠀⠀⠀⣿⣶⣶⣶⡆⢀⣀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣀⣤⣤⣶⣤⣄⣀⠀⣿⣿⣿⠏⠀⠈⠁⢠⣶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣠⠀⠂⠀⠀⠀⠀⠀⠀⠀⢨⣿⢻⣿⣿⣿⣾⣿⣿⣿⣏⣱⣿⣿⣿⡇⠀⠀⠒⠂⠙⢿⣿⣿⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣿⣷⣶⣿⣿⣿⣿⣿
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠰⠀⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠻⠿⣿⣾⠿⠿⣿⡿⢹⣿⣿⣿⣧⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⠻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⢀⣀⣤⣶⣶⣦⡀⠉⢁⣀⠀⠈⠀⢸⣿⣿⣿⢟⠛⠛⠻⠿⢿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠟⢛⣻⣷⣾⣿⣿⣶⣶⣶⣾⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠠⠬⢉⣛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⣋⣭⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣛⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠿⡇⣈⣩⣿⣷⣶⣶⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣋⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃⠀⠀⠉⠉⠙⠛⠻
⠀⠀⠲⣶⠀⠶⠄⠰⠶⠶⠞⠉⠉⠉⠉⠈⠉⢉⣩⣼⣿⠿⣿⣿⣿⣿⣟⣛⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀
⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⢠⡌⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⢸⠇⣎⣵⢸⡱⢮⠱⠆⡟⠙⠛⢙⣛⢈⢋⣙⡇⡇⡟⠟⣛⡛⡛⡛⣛⢛⢻⢿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠸⠿⠿⠿⣿⡿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⣸⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⠾⠾⠿⠶⢷⣿⣷⣧⣭⣥⣿⠷⣰⣧⣯⣼⣘⣜⣿⣿⣿⢰⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣼⣿⣶⣴⣷⣷⣼⣿⡟⠏⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⡆⢠⡍⠉⡍⡿⣸⢸⢙⠙⠏⡛⠀⠀⠀⠀⠚⠥⠄⠄⠶⠾⠶⠼⢶⢷⣿⣿⡟⡿⣿⣿⣿⡟⡌⠀⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⢿⣿⣿⠿⠏⣠⡾⠀⠀⠀⠀⠀⢰⣿⣿⣧⣿⣶⣾⣶⣧⣥⣭⢬⠬⢤⡆⠯⠿⠤⠄⠂⡃⡀⡘⣀⣸⣇⣇⢡⠇⠰⠰⠈⠆⡇⣷⠸⣰⠇⡇⠀⠀⠀⢰⠃⢰⣯⡇⣿⢰⣿⢀⣶⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣇⢛⡛⡛⡿⣛⠿⡻⡿⠿⢿⠿⢸⠀⢼⠃⠉⠀⠀⡀⠀⣠⣴⢀⡁⣖⢄⠀⠀⠀⠀⠉⠛⣿⣿⣟⣽⣿⢰⡃⢀⠀⠀⢸⠀⢸⣿⡄⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣬⣥⣧⣧⣒⣆⣂⣃⣺⣘⠀⡐⠀⣈⡄⣠⠀⢄⠰⢸⠀⡄⢨⢨⠌⡋⠄⡀⠀⠀⠀⡀⢙⢛⡻⣟⠻⠾⣳⠀⠀⠀⠘⣏⠙⠛⠛⠓⠓⠻⠿⠾⢤⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⢿⣿⣿⣿⣿⠛⡏⠙⠉⢹⡇⢥⣤⣟⣛⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⡱⢆⠷⠶⢤⣴⣦⢬⣼⣧⣤⣄⣕⣨⡀⠀⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠔⠋⠀⠀⠀
⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠇⣶⢘⠰⠀⣷⠀⢤⢩⡅⡌⡄⢠⢠⢈⡉⡉⣟⢘⢙⡋⡛⠙⠛⢻⠷⠖⠲⠤⠤⠼⢼⣻⣿⣿⣯⣿⣿⡦⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀
⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠛⠋⠀⠀⠀⠀⠒⠒⠓⠸⠨⠬⠥⣧⢥⣼⠈⠀⠁⠀⠸⠐⠀⠐⡶⣄⢶⡇⠾⠸⠀⠆⠆⠏⢒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣶⣷⣄⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠈⠁⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣟⠳⢤⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠄⠀⠀⢀⠀⠀⠀⠀⠀⣷⣶⣶⣶⠀⢀⣀⠤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣠⣤⣤⣶⣤⣠⣀⢠⣿⣿⣿⠋⠀⢨⠁⣴⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣍⣉⣛⣛⣻⣿⣿⣿⣿⣿⣽⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⡀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣻⣿⣿⣷⣾⣿⣿⣿⣋⣸⣿⣿⣿⡄⠀⠀⠒⠊⠙⢿⣿⣿⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⣿⣷⣾⣿⣿⣿⣿⣷
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢀⠀⠐⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠿⠿⣿⡾⠿⢿⣿⡟⣿⣿⣿⣿⣧⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⠂⠀⠀⠂⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣦⠀⢉⣁⣀⠀⠉⠀⣿⣿⣿⣿⡟⠛⠛⠿⠿⣿⣿⣿⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠉⢉⣿⣾⣿⣿⣿⣶⢶⣶⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠤⠍⣙⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣼⣿⣭⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠿⢿⣻⣿⣷⣤⣤⣤⣤⣤⣄⣀⣘⣻⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣷⣿⡄⢈⣹⣿⣷⣶⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⢿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣥⣴⣾⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠉⠉⠙⠛⠻
⠀⠆⠲⣶⠟⢷⠖⠺⠾⠿⠿⠋⠉⠉⠛⠛⠛⢋⣻⣿⡾⠻⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡇⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣀⣴⣶⣾⣿⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢠⡍⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⢸⠇⣎⣵⢸⠡⢮⡱⠆⡟⡹⡛⢙⣿⢨⢋⣝⡇⡇⡟⠟⣻⡛⡛⡛⣛⢟⢻⢿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠒⠚⠯⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣸⣿⣿⡿⣿⣿⣿⣿⡿⣿⠾⠾⠾⠿⠶⡶⢿⣷⡧⣯⠥⠿⠷⣰⣧⣿⣼⣸⣜⣿⣿⣿⠸⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣀⠉⠙⠋⠙⠻⢿⣯⣿⢛⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⡂⢠⡍⠉⡍⡿⢸⢸⢉⠙⠟⡛⠁⠀⠀⠀⠒⠦⠤⠿⠷⠦⠴⠾⢾⢿⣿⣿⡟⡿⣿⣿⣿⠏⡇⠀⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠐⠲⠬⢽⣿⣶⣿⣿⣷⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣷⣴⣾⣶⣧⣥⣭⣬⠨⣤⡆⠯⠿⠤⠄⠃⣃⢀⣘⣀⣘⣇⣆⢡⠇⠰⠰⠈⠆⡇⣷⠸⣰⠃⠇⠀⠀⠀⢰⠃⢰⣯⡇⣿⢰⣿⢀⣶⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠥⠴⠶⠁⠀⠀⠀⠀⠀⣇⢛⡛⡛⡿⡛⠿⡻⡿⠿⢿⠇⠇⠀⠿⠁⠁⠀⠀⠀⢀⣰⣮⢨⣁⣀⣉⠀⠈⠀⠀⠉⠛⢻⣿⣟⣵⣿⢸⡀⠀⠀⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣬⣥⣧⣧⣒⣆⣂⣃⣺⣘⠀⡀⠀⣈⡄⣠⠠⣄⠰⢸⢀⡄⢨⢠⠈⡋⠅⡀⠀⠀⠀⠘⢘⢛⡻⣟⠻⠞⣲⠀⠀⠀⠘⣏⠛⠛⠛⠓⠓⠛⠿⠾⢤⠇⠀⠀
⠀⠀⠀⠀⠀⢤⣴⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⠛⡿⠉⠁⣽⡇⠧⢤⣿⣿⣿⣿⣟⣿⣿⣿⣿⣶⣾⣷⡱⢆⠿⠶⠦⠴⡦⢸⣼⣧⣤⣄⣕⣨⡀⠀⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠔⠋⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠇⣶⢘⠰⠀⡷⠀⢤⢩⡅⡄⡄⢠⢰⢀⡄⡌⣟⢘⢙⠋⡛⠙⠛⢻⠿⠖⠒⠲⠰⠆⢾⣻⣿⣿⣷⣿⣿⠖⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀
⣿⣶⣄⡀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠛⠁⠀⠀⠀⠀⠒⠒⠓⠸⠨⠬⠥⢧⣥⣜⠈⡀⢃⠀⠘⠘⠀⠐⣷⣆⣶⠆⠾⠸⠀⠆⠆⠉⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣶⣷⣄⠀⠀⠀⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠉⠀⠀⠀⠀⠀⢸⡿⠀⠀⠈⠁⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣍⠛⢦⣴⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⢀⠀⠀⠀⠀⢸⣷⣶⣶⣶⠀⣀⡄⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⣬⡛⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣥⣠⢦⣴⣶⣦⣤⣀⢸⣿⣿⡟⠁⠀⠉⠀⢾⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⣬⣭⣭⣭⣿⣿⣿⣿⣿⣾⣿⣻⣿⡀⠀⠀⠀⠀⠀⠀⢀⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⣿⣿⣿⣷⣾⣿⣿⣿⣍⣾⣿⣿⣿⠀⠀⣀⣒⣀⣙⣿⣿⣿⣶⣦⣤⣤⣀⣀⣀⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠆⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⡘⠻⢿⣷⡿⠿⢿⣿⠏⣿⣿⣿⣿⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠹⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣤⣶⣿⣿⠿⣆⠀⣀⣀⡀⠀⠀⠀⣿⣿⣿⠿⠋⠛⠛⠛⠻⠿⣿⣿⡛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠫⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡟⠋⠀⣀⣼⣿⣿⣿⣿⡶⢶⣶⣿⣿⡿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠠⢭⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠐⠿⠿⠿⠿⠿⠛⣛⣭⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣀⣰⣿⣾⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⡟⠿⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣠⣠⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣋⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠙⠛
⠀⠆⠲⣶⠉⢷⠄⠸⠿⠿⠗⠀⠀⢀⣈⣀⣤⣶⡿⠛⠛⠛⣿⣿⣿⣟⣛⣛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀
⠀⢀⣤⣤⣴⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢠⡍⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⠇⣎⣵⢸⠡⣬⡱⡆⣏⡹⣉⢙⣻⢨⢋⣉⡇⡇⡟⢟⣟⡛⡛⡛⣛⠛⢻⢿⣿⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠉⠉⠀⠙⠛⠻⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⠀⠀⠀⣸⣿⣿⣿⢿⣿⣿⣿⡿⡿⠾⠾⠾⠿⠶⢶⣷⣷⣧⢯⠵⠿⠷⣰⣧⣻⣴⣸⣜⣿⣿⡟⡆⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⡀⠀⠲⣄⣠⣠⣦⣤⣽⣭⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢡⡍⠉⡍⡿⢸⢺⢙⡛⠟⠛⠀⠀⠀⠀⠀⠶⠼⢿⠷⠾⠷⠶⢾⢿⣿⣿⡟⡿⣿⣿⣿⠃⡇⠀⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣸⣆⣴⣾⣶⣧⣥⡭⡬⠬⣧⡤⠯⠿⠄⠀⢃⣃⣀⡘⣀⢸⣇⣇⢣⠇⠰⢰⠈⠆⡇⣷⠸⣰⢸⠀⠀⠀⠀⢰⠃⢸⣯⡇⣿⢰⣿⢀⣶⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⢀⢛⡛⡛⡿⣛⠿⡻⡿⠿⢿⠇⠇⠀⡿⠉⠁⠀⠀⠀⢤⣰⣦⢬⣥⣊⣍⠀⠀⠀⠁⠉⢻⢿⣿⣟⣵⣿⣸⠀⠀⠀⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣼⣥⣧⣧⣒⣆⣂⣃⣺⣘⠀⡀⠀⣈⡄⣰⠠⣤⢰⢸⢀⣄⢨⢠⡌⡋⠅⡀⠀⠀⠀⡘⢘⢛⡻⣟⠛⡟⣲⠀⠀⠀⠘⣏⠛⠛⠛⠓⠓⠻⠿⠿⢤⠏⠀⠀
⠀⠀⠀⠈⠛⠿⡿⠇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⢿⣿⣿⣿⣿⠛⡿⠉⢉⣿⠁⠧⠤⠿⠿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⡱⠆⠿⠷⠦⠶⡇⣼⣼⣧⣤⣄⣕⣨⣀⠀⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠖⠋⠀⠀⠀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠇⣶⢘⠰⠀⡿⠀⢤⢩⡅⡄⡄⢠⢰⢁⡅⡌⣟⢘⢙⡋⡋⠙⠛⢻⠿⠖⠒⠒⠒⠃⣿⣻⣿⣿⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀
⣿⣷⣦⣀⣀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠒⠓⠓⠸⠬⠬⠥⢧⣥⣜⠈⡀⣃⠀⠸⠘⠈⠐⣿⣆⣷⠂⠾⠸⠀⠆⡆⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠷⣄⡀⠀⣨⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠈⢹⡟⠀⠀⠈⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⣆⠱⣾⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⡆⠀⣀⠄⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⠿⠿⣿⣿⣿⣿⣷⣧⣹⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣤⡴⠶⣾⣷⣦⣤⡀⣸⣿⣿⡏⠁⠀⠋⠰⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⢀⣀⡀⠰
⠲⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣏⣿⣿⡆⠀⠀⠀⠀⠀⠀⢰⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣿⣿⣿⣷⣿⣿⣿⣿⣍⣿⣿⣿⣷⠀⠀⣉⣁⣀⣙⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠁⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⠻⢻⣿⠿⠛⢿⡿⠋⣿⣿⣿⣿⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⠉⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⣀⣀⣤⣶⣾⣿⣿⣆⠀⢀⠀⠀⠀⠀⢀⣿⣿⣿⠟⠉⠉⠉⠛⠻⠿⢿⣿⣋⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢉⢝⣻⣿⣿⣿⣿⣿⣿⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠛⢉⣩⣿⣾⣿⣿⣷⡲⢶⣾⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠠⠭⣝⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣉⣉⣤⣶⣿⣿⣿⣿⣶⣶⣶⣶⣦⣤⣤⣤⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⣀⣥⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⠛⠻⠿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠷⠿⠿⠟⠛⠿⠿⠿⠿⠿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣫⣵⣾⣿⣿⣿⣿⣿⣿⣿⣟⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠉⠛
⠀⠆⠲⣶⠆⢶⠄⠰⠶⠶⢶⣠⣤⣴⣤⣶⡾⠿⠋⠙⠟⢻⣿⣿⣿⣟⣛⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣀⣤⣶⣶⣶⣶⣶⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀
⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⠀⠀⠀⢠⡍⢻⣿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠉⣛⡛⠛⢻⣟⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⡿⠀⠀⠀⢸⠇⣎⣵⠸⡡⣬⡱⡆⣏⡙⣉⢙⣯⢨⢋⣉⡇⡇⡟⢝⣏⡉⣉⡛⣛⠛⢻⢿⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠺⠿⢷⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣸⣿⣿⣿⢿⣿⣿⣿⣿⡿⠾⠺⠾⠿⠶⢶⢷⣷⣧⣯⡵⠿⠷⣰⣧⣻⣤⣼⣜⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢠⠀⠠⡍⠉⡍⡿⢨⢺⢉⡛⠟⠋⠀⠀⠀⠀⠂⠤⠀⠊⠴⠦⠴⠴⢶⢿⣿⣿⡟⡿⣿⣿⣿⢡⠀⠀⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣷⣸⣀⠄⢾⣶⣧⣧⡌⡬⠨⣧⠠⠭⠿⠄⠀⠁⡃⠀⡘⣀⣘⣇⡆⢡⠇⠰⠰⠈⠆⡇⣷⠸⣰⢸⠀⠀⠀⠀⢰⠃⢰⣯⡇⣿⢰⣿⢀⣷⠈⣧⠀⠀
⠀⠀⠀⢰⣾⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⢛⡛⡛⡿⣛⠿⡻⡿⠿⢿⠇⠇⠀⠏⠈⠁⠀⠀⠀⣀⣠⣀⢀⡁⣀⡀⠀⠀⠀⠀⠉⠹⢿⣿⣟⣵⡇⡆⠀⠀⠀⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⣥⣧⣧⣒⣆⣂⣃⣺⣘⠀⠀⢸⣈⡄⣰⠠⣤⢰⢸⢀⣄⢨⢠⠈⡋⠄⡀⠀⠀⠀⡄⢘⢛⡻⣟⠓⡟⡢⠀⠀⠀⠘⣏⠙⠛⠛⠓⠓⠻⠿⠾⢤⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠛⡏⠈⢈⣿⠸⠤⠼⠿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⡱⠆⠿⠶⠦⠶⡇⣼⣼⣧⣤⣴⣑⣠⣀⠀⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠔⠋⠀⠀⠀
⣷⣦⣀⠀⠀⠀⠀⠀⠀⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣶⢘⠰⠀⠵⠀⢤⢩⡄⡄⡄⢠⢰⢀⡅⡅⣟⢘⢙⠉⡋⠙⠛⢻⠿⠒⠒⠒⠲⠃⣿⣻⣿⣿⣿⣿⡿⠅⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⣶⣦⡀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠐⠒⠓⠸⠌⠬⠥⢧⣥⣘⢈⣀⡇⠀⠸⠘⢀⠐⣿⣆⣷⠀⠾⠸⠀⠆⡆⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡛⢶⣬⣿⣿⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⠈⠀⠀⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢻⣿⣿⣿⣿⣿⣿⣦⡹⣿⣿⣷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⡆⠀⡄⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⣿⣽⣿⣿⣟⣿⣿⣿⣯⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣴⡴⠦⣴⣶⣤⣤⡀⣿⣿⣿⡋⠀⠀⡋⠰⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⢀⣀⡀⢴
⠐⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣰⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣿⡝⣿⣿⣿⣷⣿⣿⣿⣯⣁⣿⣿⣿⣧⠀⢀⣉⣂⣈⣻⣿⣿⣿⣶⣦⣤⣤⣄⣀⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠸⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠿⣻⣿⠿⠻⢿⡿⢻⣿⣿⣿⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠛⠉⠋⠙⠛⠿⠿⣿⣟⡋⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠛⢋⣹⣿⣶⣿⣿⣷⡲⢶⣾⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⠠⢭⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⣠⣴⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⠛⠻⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠛⠛⠿⠿⠿⢿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣩⣤⣾⣿⣿⣿⣿⣿⣿⣿⣟⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠙⠛
⠀⠀⠰⣶⠀⢶⠄⠰⠶⠶⠶⠀⠐⠶⣶⣶⡾⠿⠛⠉⠙⢛⣿⣿⣿⣛⣛⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣀⣴⣾⣿⣿⣿⣿⣿⣷⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀
⠘⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⠀⠀⠀⢠⡍⢻⣿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⠙⣻⣿⣿⣿⣽⣾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⢸⠇⣎⣵⢸⠡⣬⡱⡆⣟⡹⣉⢙⣿⢨⢋⣙⡇⡇⡟⠟⣿⡛⡛⡛⣛⠛⢻⢿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠛⠛⠛⠻⣿⣿⣿⠿⠃⣰⡶⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⣸⣿⣿⡿⢿⣿⣿⣿⣿⡿⠾⠾⠾⠿⠶⢶⣷⣷⡧⣯⠥⢿⠷⢰⣧⣿⣴⣼⣜⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⢻⣿⣏⢠⠀⢨⡍⠉⡍⡿⣸⢺⢉⡛⠟⠛⠀⠀⠀⠐⠒⠶⠤⠿⠿⠷⠾⠿⢶⢿⣿⣿⡟⡿⣿⣿⣿⢣⠇⠀⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣸⡀⠀⣾⣶⣧⣧⡍⡬⠬⣧⡤⠯⠿⠄⠀⢁⣃⣀⣘⣀⣸⣇⣇⢡⠇⠰⠰⠈⠆⡇⣷⠸⣰⢸⠀⠀⠀⠀⢰⠃⢸⣯⡇⣿⢰⣿⢀⣶⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⢛⡛⡛⡿⣛⠿⡻⡿⠿⢿⠇⠇⠀⠟⠈⠁⠀⠀⠀⢀⣴⣶⢀⡁⣜⣉⠀⠀⠀⠈⠉⢹⢿⣿⣟⣵⡧⣾⠀⠀⠀⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠈⣼⣥⣧⣧⣒⣆⣂⣃⣺⣘⠀⡀⢰⣈⡄⣰⠠⣤⢰⢸⢀⣄⢨⢠⠌⡋⠄⡀⠀⠀⠀⡀⢙⢛⡻⣟⠃⡟⣲⠀⠀⠀⠘⣏⠛⠛⠛⠓⠓⠻⠿⠾⢤⠏⠀⠀
⣂⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢿⣿⣿⣿⣿⠛⡟⠈⢈⣿⠀⠥⠼⡿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⡱⢆⡷⠶⠦⠴⡇⣼⣼⣧⣤⣄⣑⣠⣀⠀⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠖⠋⠀⠀⠀
⣿⣿⣶⣄⡀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠄⠇⣶⢘⠰⠀⠿⠀⢤⢩⡄⡄⡄⢠⢸⢁⡌⡄⣟⢘⢙⡋⡋⠙⠛⢻⠷⠖⠖⠖⠢⠃⣿⣻⣿⣿⣿⣿⡿⠅⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣾⣶⣄⠀⢄⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠐⠓⠳⠸⠬⠬⠥⠧⣥⣘⠈⡀⡃⠀⠘⠘⢀⠐⣿⣆⣶⠂⠿⠸⠀⠆⡆⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣝⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠁⠀⠀⠀⠀⢹⡟⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠻⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⡆⠀⣀⡄⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⡤⠶⣼⣶⣤⣤⡀⣸⣿⣿⡏⠁⠀⡉⠰⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⣀⡀⠰
⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢠⡄⠰⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣿⣿⣿⣷⣿⣿⣿⣿⣍⣿⣿⣿⣷⠀⠀⣉⣐⣈⣻⣿⣿⣿⣶⣶⣤⣤⣄⣀⣠⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠂⠸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⠘⠻⣿⣷⠿⠛⢿⡿⠃⣿⣿⣿⣿⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠊⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⣀⣀⣀⣤⣴⣿⣿⣿⣦⠀⢀⣀⡀⠀⠀⠀⣿⣿⣿⠟⠉⠙⠙⠛⠻⠿⣿⣿⣛⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠒⠙⠿⠿⢿⣿⣿⠿⠿⣛⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠛⢉⣩⣿⣿⣿⣿⢿⡶⢶⣾⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠠⢭⣝⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠀⣀⣥⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣿⣿⡟⠻⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣡⣤⣶⣿⣿⣿⣿⣿⣿⣿⣟⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠛
⠀⠀⠰⣶⠀⠲⠄⠰⠶⠶⠖⠀⠀⠀⠰⠶⠶⠿⠟⠛⠉⢛⣿⣿⣿⣟⣛⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣷⣦⣄⠀⠘⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠿⢿⣿⣿⣿⣿⣧⣸⠆⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⠾⡇⠀⠀⠀⢸⣿⣿⡇⠀⠀⢠⡍⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣅⡀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢸⠇⣎⣵⢸⡡⣬⡱⡆⣏⡹⣉⢹⣿⢨⢋⣍⡇⡇⡟⢝⣿⡋⣛⡛⣛⢛⢻⢿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠂⠀⠀⠀⠀⢸⣿⣿⡇⠀⢀⣼⣿⣿⡿⣿⣿⣿⣿⣿⣿⠿⠞⠚⠿⢶⣷⣷⣷⣧⣯⣵⡿⠷⣰⣧⣻⣼⣸⣜⣿⣿⣿⢹⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⡞⢩⡍⠉⡍⡟⣸⢸⢙⠙⠟⡛⠃⠀⠀⠒⠚⢿⠾⢿⠷⠿⠷⠿⢾⣿⣿⣿⡟⡿⣿⣿⣿⡛⣸⠀⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⡦⠀⠀⠀⠀⠀⡀⢰⣿⣿⣷⣿⣶⣾⣶⣧⣥⣭⢬⠬⢤⡆⠩⠟⠀⠄⣃⣃⣀⣘⣠⣸⣇⣇⢡⠇⠴⢰⠈⠆⡇⣷⠸⣱⡇⡇⠀⠀⠀⢰⠇⢸⣿⡇⣿⢰⣿⢀⣷⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠘⠀⠀⠀⠀⣇⢛⡛⡛⡿⣛⠿⡻⡿⠿⢿⠿⢸⠀⢼⠅⠉⢀⠀⣤⣤⣾⣿⢭⣿⣿⣿⡀⠉⠉⠉⠛⠛⣿⣿⣟⣽⣿⣅⡇⠀⠀⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⣯⣬⣥⣧⣧⣒⣇⣂⣃⣺⣘⠀⢸⠀⣘⡄⣰⠠⣬⢰⢸⣀⣄⢸⢨⡌⡋⠍⣀⠀⠀⠀⡘⢛⢛⣻⣟⠻⠿⣳⠀⠀⠀⠘⣯⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣶⣶⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⣿⣿⣿⣿⡛⣿⡛⠋⣿⣏⢳⡤⠿⠿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣷⡱⠆⡿⠷⠦⠶⢦⢬⣼⣿⣥⣄⣕⣨⣀⡀⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠖⠋⠀⠀⠀
⣿⣿⣿⣷⣌⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠇⠶⢸⠶⠀⣷⠀⢤⢩⡍⡅⡅⢢⢸⢁⡝⡍⣟⢸⢙⡋⡛⠙⠛⢻⠿⠿⠒⠒⠒⡞⢾⣻⣿⣿⣿⣿⣿⡯⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣾⣷⣄⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠛⠋⠀⠀⠀⠐⠒⠓⠓⠼⠬⠬⠥⠧⢥⣼⣈⢁⣅⣀⣸⠘⢈⡐⣿⣆⣷⡇⠾⠸⠠⠆⡆⡏⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣍⠻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠁⠘⣿⠁⠀⠈⠁⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀⠀⠀⠠⠀⠀⠀⠀⠀⣿⣶⣾⣿⠀⢠⣠⢐⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣤⡶⢶⣿⣦⣤⣄⢸⣿⣿⣿⠁⠀⠘⠀⠺⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣀⣀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣧⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⣯⢽⣿⣿⣿⢿⣿⣿⣿⣭⣸⣿⣿⣿⡆⠀⢈⣭⣤⣬⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⡀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⠛⢻⣿⡿⠛⠻⣿⠋⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠁⠁⠀⠄⢀⣀⣠⣤⣶⣿⣿⠿⢷⣀⣠⣤⡄⠀⠀⠀⣿⣿⣿⣟⠋⠉⠉⠉⠛⠛⠿⠿⣭⣉⡙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠛⡿⣿⡿⣿⣿⣿⣿⡿⢿⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣟⣋⣁⣤⣾⣿⣿⣿⡿⣾⠿⢿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⢬⣥⣴⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣄⣀⣤⣽⣿⣿⣿⣿⣶⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⠛⠛⠻⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠰⣶⠀⠰⠄⠠⠲⠶⠖⠀⠀⠀⠀⣴⣶⣾⣿⠟⢻⣿⣿⣿⣿⣿⣭⣭⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢆⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣜⣼⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡋⠀⠀⠀⠀⣿⣿⣿⠀⠀⢠⡍⢻⣿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢫⠃⠀⠀⠀⢠⣿⣿⣿⠀⠀⢸⠇⣎⣵⢸⡡⣦⡱⡆⡷⡼⣒⢸⣿⢠⢫⣝⡇⡇⡟⠟⣿⡟⡿⡻⣛⢷⢮⢿⣿⣿⣿⡏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⠇⠀⠀⠀⠘⣿⣿⣟⢀⠀⣼⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⠊⠈⠛⠶⣷⣿⣷⣥⣭⣥⣿⠷⣰⣧⣻⣼⣘⣼⣿⣿⣿⣇⠇⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣿⢩⡍⠉⡍⡟⣸⢺⢙⡙⠟⡛⣧⣄⡀⡖⠚⢿⠿⢿⠿⠿⠿⡿⢿⣿⣿⣿⡟⡿⣿⣿⣿⡍⢸⠀⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣶⣾⣶⣧⣥⣭⣬⢬⢤⡦⠩⠋⠁⣀⣃⣃⣠⣘⣠⣸⣇⣇⢣⠇⠶⢰⠨⠆⡇⣷⠸⣹⡇⡜⠀⠀⠀⢰⠇⢸⣿⡇⣿⢰⣿⢠⣿⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⣿⢚⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢸⠀⢸⠧⠀⣀⢀⣠⣤⣾⣿⢿⣽⣿⣟⡁⠉⠉⠙⠛⠿⣿⣿⣟⣽⣿⡇⣇⠀⠀⠀⢸⠀⢸⣿⡆⡏⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠀⠀⡯⣬⣥⣧⣧⣓⣇⣂⣃⣻⣘⠪⢸⠀⢘⡎⣷⠰⣬⢲⢸⣤⣴⢸⢨⡌⡫⢅⣤⡀⠀⣀⡂⢛⢛⣻⣟⡻⡟⣻⡀⠀⠀⠘⣯⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣷⣶⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⡷⢿⣿⣿⣿⣿⡛⣿⣿⢋⣽⣿⣘⣛⣛⣛⣛⢻⣿⣿⣿⣿⣷⣶⣾⣷⣱⣎⣽⣃⣁⣙⢫⣬⣼⣿⣵⣤⣑⣨⣁⡌⠀⠀⠈⠳⢤⣀⣀⢀⣀⣠⠶⠋⠀⠀⠀
⣿⣿⣿⣷⣌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠄⠇⠶⠘⠶⠂⣿⠁⢦⢩⡏⡍⡍⢩⢹⢛⡛⡟⣟⢺⢙⡋⡛⠙⢛⢻⠿⠿⢈⣉⠉⡯⢼⣻⣿⣿⣿⣿⣿⡋⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⣮⣦⡀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⠋⠉⠀⠀⠰⠦⣁⠑⠘⠈⠈⠁⠣⠥⠼⠬⣧⣇⡀⢸⣘⢈⡐⣿⣇⢿⡇⠾⠸⠠⠆⡆⡏⢒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣯⡙⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣃⣀⣈⣀⡀⠀⠉⣿⠁⠀⠈⠉⠈⠘⠻⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡦⡄⣀⣀⣤⣀⠀⠀⠀⣿⣿⣿⡿⠟⢹⠞⢨⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣻⣿⣿⣿⡷⢴⣿⣿⣿⡀⠀⠐⣀⡈⠛⢿⣿⣿⣶⣤⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣴⣦⣤⣴⣤⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣇⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠉⠉⠀⠙⡀⡤⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⣄⠙⠛⠋⠉⠉⠛⠁⣹⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠂⠈⠹⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠊⣴⣶⣾⣿⣿⣿⠿⠛⢻⣦⣤⣶⣶⣀⣀⣀⣿⣿⣿⣷⢶⡤⠀⠀⠀⠀⠉⠉⠛⠛⠒⠠⠌⢙⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠻⠛⠿⠿⠿⠿⠿⠿⠿⢟⣻⣿⣿⣷⣦⣤⣤⣤⣤⣀⣀⣀⣿⣿⣿⣿⣿⣶⣦⣶⣿⣿⣿⣿⣿⣿⡉⠛⢻⣿⣯⣭⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⣻⣷⣾⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠃⠀⠀⠉⠉⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠋⠉⠉⠉⣉⣉⣩⣿⡿⢿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠲⣶⠀⠶⠄⠰⠶⠶⠶⠔⠾⠿⠿⠿⠛⠛⢉⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⢠⣿⣿⣿⠀⠀⢠⡍⢻⣿⢻⣿⣛⣛⡛⣛⢛⣛⣛⣛⣛⣛⣛⠛⡛⣛⣛⠛⠛⠛⠛⠛⠛⠛⣻⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡣⠀⠀⠀⢸⣿⣿⣿⠀⠀⢺⠇⣎⣵⢸⡡⣮⠹⠆⡿⠹⠛⢹⣿⢨⢫⣝⡇⡇⡟⠟⣿⡟⡟⡻⣛⢿⢳⢿⣿⣿⣿⡟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡤⠀⠀⠀⠈⣿⣿⣯⢠⡀⣼⣿⣿⡿⢿⣿⣿⣿⣿⣿⡿⠢⠚⠻⢶⣶⣷⣷⣥⣭⣥⣿⠷⣰⣧⣻⣼⣘⣼⣿⣿⣿⠇⡇⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣿⣿⣿⣿⢩⡍⢉⡍⡟⢸⢺⢛⡙⠟⡛⣿⣗⠲⡞⠛⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⡟⡿⣿⣿⣿⣆⢸⠁⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣶⣾⣶⣷⣥⣭⣬⣬⢥⡦⠉⠛⠀⣀⣃⣃⣠⣘⣠⣸⣇⣇⢣⠇⠶⢰⠸⠆⡇⣷⠸⣹⡟⡸⠀⠀⠀⢰⠇⢸⣿⡇⣿⢰⣿⢠⣿⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⣷⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢸⠀⢸⣧⣤⣤⣤⣼⣾⣿⣿⢿⣿⣿⣿⣅⠛⠛⠛⠛⢿⣿⣿⣟⣽⣿⡇⡇⠀⠀⠀⢸⠀⢸⣿⣆⡏⡾⣿⣸⣿⠀⣸⠀⠀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣯⣼⣥⣧⣧⣓⣇⣂⣃⣻⣘⠨⢸⣤⣘⡎⣻⠹⢼⠳⢸⢯⡿⢹⢹⡝⡯⢍⣴⡶⠄⣤⣝⢛⢛⣻⣟⡻⡟⣻⡄⠄⠠⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣿⣮⣕⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡿⢿⣿⣿⣿⣿⡛⣿⣿⣯⣿⣿⣬⣥⣤⣬⣭⣭⣿⣿⣿⣿⣷⣶⣾⣶⡱⢆⡿⠧⠤⠬⢧⢬⣼⣿⣵⣥⣑⣨⣁⡌⠀⠀⠈⠳⣤⣀⣀⢀⣀⣠⠖⠋⠀⠀⠀
⣿⣿⣿⣷⣌⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠇⠶⠈⠲⠆⣿⠉⢦⢩⡏⡍⡍⢩⢹⢛⡛⡛⣟⢸⢙⡋⡛⠙⠛⢻⠿⠿⢴⣶⠴⡿⢿⢻⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⢦⡀⢠⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⢀⠿⠋⠉⠀⠀⠲⢲⣷⡗⠘⠈⠈⠁⠧⠥⠼⠬⢇⣇⣂⣸⣘⢊⡐⣿⣇⣿⠇⠾⠸⠠⠆⡆⡏⢒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⡹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠄⠀⣿⣿⡆⠀⠀⠀⡀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣀⣹⣿⡀⠀⠉⠉⠉⠙⠛⠷⠗⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣗⣢⣤⣤⣶⣤⣀⣀⠀⣿⣿⣿⠟⠛⠙⠋⠐⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⣿⣿⣷⣾⣿⣿⣿⣛⢺⣿⣿⣿⡄⠀⠐⣲⣦⣀⣙⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠐⠂⠀⣀⣴⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠻⠿⣿⣾⠿⢿⣿⡿⢿⣿⣿⣿⣧⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢘⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠁⠈⢂⠶⢀⣀⣀⠀⣀⣠⣤⣶⣿⣿⣧⡀⠉⢁⡀⠀⠉⠀⣸⣿⣿⣿⡛⠉⠉⠉⠉⠛⠻⠿⢿⣭⣉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿
⠀⠀⠀⠸⠿⠿⡿⢿⡿⣿⣿⣿⣿⡿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠈⠀⠁⣾⣿⣿⣿⣿⠿⠛⢉⣿⣷⣾⣿⣿⣶⢶⣶⣿⣿⣿⡿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠲⠯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣤⣄⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢈⣹⣿⣷⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⠙⠛⠻⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠉⠛⠛⠻⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⣾⣿⡿⡿⠿⠿⠟⠋⠁⠻⢻⣿⣿⣿⣿⣭⣽⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⠲⠄⠠⠰⠶⠆⠀⠀⠀⢀⣀⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⠀⠀⢰⡍⢻⣿⢻⣿⣩⣭⡍⣩⢭⣉⣉⣉⣉⣩⣩⡭⣩⣉⣉⣉⣉⣉⣉⣉⣉⣉⢹⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⠁⠀⠀⠀⠀⣿⣿⣿⠀⠀⢹⠇⣎⣵⢸⡡⢮⠱⠆⡟⠹⠓⢹⣿⢨⢋⣝⡇⡇⡟⠟⣿⡟⡟⡻⣛⢿⢿⢸⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠃⠀⠀⠀⠀⣿⣿⣿⢰⣠⣼⣿⣿⡿⢿⣿⣿⣿⣿⡿⣿⡐⠚⠿⣶⣶⣷⣷⣥⣭⣥⣿⠷⣰⣧⣫⣼⣘⣸⣿⣿⡿⠟⡞⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣾⢩⡍⢉⡍⡟⢸⢺⢛⡙⠟⡛⠿⡟⠻⡟⠛⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⡟⡿⣿⣿⣿⡇⠀⡇⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣶⣾⣶⣷⣥⣭⣬⣬⣧⣮⠉⢛⢀⣀⣃⣃⣠⣙⣠⣸⣇⣇⢣⠇⠶⢰⠸⠆⡇⣷⠸⣽⡯⢸⠁⠀⠀⢰⠇⢸⣿⡇⣿⢰⣿⢠⣿⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡧⠀⠀⠀⣸⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⠧⡇⢰⣿⣤⣤⣦⣶⣿⣿⣿⢿⣿⣿⣿⣥⠛⠛⠛⠛⣿⣿⣿⣟⣽⣿⣿⣸⠀⠀⠀⢸⠀⢸⣿⣆⡏⡾⣿⣸⣿⠀⣼⠀⠀
⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⢯⣬⣥⣧⣧⣓⣇⣂⣃⣻⣘⠪⡸⠧⠈⠌⣹⠹⢬⢳⢸⢋⣟⢹⢹⡝⡋⠍⣿⡟⠔⢶⡟⢛⢛⣻⣟⡻⡟⣻⡄⠆⡀⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣿⣿⣷⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣾⣶⣶⣷⣶⣾⣷⣑⢆⡿⠆⠢⠶⢧⣬⣼⣿⣵⣥⣕⣪⣁⠌⠀⠀⠈⠳⣤⣀⣀⢀⣀⣠⠶⠋⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠇⠶⠈⠰⠂⣿⠉⣦⢩⡏⡍⡍⢫⢹⢛⡛⡛⣟⢺⢙⡋⡛⠙⢛⢻⠿⠿⢶⣶⠶⣾⢾⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣎⢷⣦⣶⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⣀⡼⠟⠉⠀⠀⠒⠲⣷⣷⡘⠈⠈⠁⢡⣥⣼⠬⢇⣇⣢⣸⣘⢊⡐⡿⣇⢷⡇⠾⠸⠠⠆⡆⡏⢒⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂⠀⣿⣿⡿⡄⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣶⣶⣿⡬⢿⣧⡀⠉⠋⠉⠉⠛⠓⠓⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣥⡴⠶⣾⣷⣤⣤⡀⣿⣿⣿⡟⠉⠉⡋⠁⠙⢿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⡀⣸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣿⡽⣿⣿⣿⣷⣿⣿⣿⣯⣅⣿⣿⣿⣷⠀⠀⣰⣶⣤⣤⣭⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠒⠠⠄⠐⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣅⣙⠻⣿⣿⠿⠻⢿⡿⢻⣿⣿⣿⣿⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠘⠷⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠁⠀⠀⠀⠀⢞⣆⢠⢀⣀⣀⣤⣴⣾⣿⣿⣿⣷⢀⣠⣤⠀⠀⠀⢸⣿⣿⣿⣿⣁⡀⠀⠀⠈⠉⠛⠛⠷⠦⠬⣉⡛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡛⣿⣿
⠀⠀⠠⢠⣔⣜⣦⣤⣤⣤⣭⣭⣶⣾⣿⣿⣿⣷⣦⣤⣀⣀⣀⡀⠀⠁⠁⣼⣿⣿⣿⣿⣟⣋⣡⣶⣿⣿⣿⣿⣷⠲⢶⣾⣿⣿⡿⠏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠉⠛⠛⠿⠻⠟⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡧⣀⣀⣿⣿⣿⣿⣿⣶⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠟⠈⠉⠙⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣶⣾⣾⣶⣶⣶⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⢁⣩⣿⣿⣿⣿⣿⣿⣭⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⠶⠄⠠⠰⠶⠆⠀⠀⠀⢐⣲⣷⣿⣿⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣸⡃⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⠝⠛⠛⠆⠀⠀⠀⠀⣿⣿⣿⠃⠀⢠⡍⢻⣿⢻⣿⣥⣤⡌⣭⢭⣭⣭⣭⣭⣭⣭⡭⣭⣭⣭⣭⣭⣭⣭⣍⣉⣉⠉⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣶⡀⠆⠀⠀⠀⠀⣿⣿⣿⠄⠀⢹⠇⣎⣽⢸⡱⢬⠱⠆⡟⠹⠛⢹⣿⢨⠋⣝⡇⡇⡟⠟⣿⡛⡛⡻⣛⢿⢻⢸⣿⣿⣿⣿⢧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⢹⣿⣿⢤⣦⣼⣿⣿⡿⢿⣿⣿⣿⣿⡿⣿⣒⠘⠻⢶⣶⣷⣷⣥⣭⣥⣿⠷⣰⣧⣫⣼⣘⣘⣿⣿⡿⠟⢸⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠀⠀⠀⠀⠀⢸⣿⣿⣿⢩⡍⢉⡍⡟⢸⢺⢛⡙⠟⡃⣛⡛⠛⡟⠛⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⡟⡿⣿⣿⣿⡇⠀⡎⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⡿⠀⠀⠀⢰⠀⣸⣿⣿⣿⣶⣾⣶⣷⣥⣭⣬⣬⠤⣦⠉⠈⢀⣀⣃⣃⣠⣘⣠⣸⣇⣇⢣⠇⠶⢰⠨⠆⡇⣷⠸⣽⡇⠀⠇⠀⠀⢰⠇⢸⣿⡇⣿⢰⣿⢠⣿⠈⣧⠀⠀
⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⠇⡇⠀⣿⣶⣤⣤⣤⣿⣿⣿⢿⣿⣿⣿⣅⣋⠛⠛⠛⢿⣿⣷⣟⣽⣿⣿⢸⠀⠀⠀⢸⠀⢸⣿⣆⡏⡾⣿⣸⣿⠀⣼⠀⠀
⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢨⣬⣥⣧⣧⣛⣇⣂⣃⣺⣘⢨⡀⠓⠈⠌⣱⠩⢬⠳⢸⢉⣽⢹⢩⡝⡋⠍⣿⡛⠝⠿⡟⢛⢛⣻⣟⡻⡟⣺⡄⠆⡀⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣿⣿⣿⡛⣿⣿⣯⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣶⣷⣶⣾⣷⡑⢆⡽⠃⠁⠚⢧⣬⣼⣿⣵⣥⣕⣪⣁⠌⠀⠀⠈⠳⣤⣀⣀⢀⣀⣠⠶⠋⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠇⠶⠈⠰⠆⣿⠉⣦⢩⡏⡍⡍⢫⢹⢛⡛⡛⡟⢺⢙⡋⡛⢙⢛⢻⠿⠷⢶⣶⠿⣿⢿⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣠⣾⠟⠉⠀⠀⠐⠶⣷⣷⣜⠈⠈⠁⠣⣥⣼⡬⠇⢇⣠⣸⣘⢊⡐⣿⣇⢷⡇⠾⠸⠠⠆⡆⡏⢖⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠐⠀⣿⣿⣿⢦⡀⠀⠤⠁⠀⠀⠀⢸⣿⣿⣿⣾⣷⣿⡾⢿⣷⣦⣉⠛⠉⠉⠟⠐⠓⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣴⠶⣾⣷⣦⣤⣄⣼⣿⣿⡟⠉⠁⠉⠀⠀⠙⢿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤
⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⡀⣸⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⣿⡎⣿⣿⣿⡿⣿⣿⣿⣯⣌⣿⣿⣿⣷⠀⠀⢩⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠁⠀⠀⠨⠁⢘⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣌⠛⢻⣿⠛⠛⢻⡿⠛⣿⣿⣿⣿⣴⠿⠿⠿⢿⣿⣿⣿⣿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⣍⠉⡹⣉⡉⠛⠛⠻⠛⣛⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⠅⠀⢠⣤⣤⣤⣶⣾⣿⡿⠿⣷⣀⣤⣀⠀⠀⠀⢠⣿⣿⣿⣿⣤⣄⡀⠀⠀⠀⠈⠉⠙⠓⠒⠤⠭⣙⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣛⣿
⠀⠀⠀⠀⠀⠙⠺⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣄⣀⣀⣼⣿⣿⣿⣿⣟⣉⣠⣶⣾⣿⣿⣿⣿⠛⠿⣿⣿⣿⠿⠋⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡛⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣦⣀⣴⣿⣿⣿⣿⣿⣷⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠃⠀⠀⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠛⠻⠿⠿⠿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣰⣾⣿⣿⣿⣷⣶⣾⣝⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⠴⠄⠠⠰⠶⠆⠀⠀⠀⣀⣀⡾⠿⢿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⠟⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⢿⠇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⠿⢿⣿⣿⠃⠀⠀⠀⢸⣿⣿⡇⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣴⣾⡄⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⣠⡌⢻⡿⢻⣿⣧⣶⡆⣤⢤⣤⣤⣤⣤⣤⣤⡤⡤⣤⣤⣤⣤⣤⣤⣤⣭⣭⠍⣿⣿⣿⣿⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠗⠋⠀⠀⠀⠀⢸⣿⣿⡇⡀⢻⠇⣎⣽⢸⡱⢮⠱⠆⡟⠹⠋⢹⣿⢨⢋⣍⡇⡇⡟⠟⣿⡛⡛⡛⣛⢟⢻⠂⣿⣿⣿⣿⡇⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣿⣿⣇⣧⣼⣿⣿⡿⢿⣿⣿⣿⣿⡿⣿⣦⣞⣻⣶⣶⣷⣷⣥⣭⣥⣿⠷⣰⣧⣫⣼⣘⣐⣿⣿⡿⠛⢱⠃⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠃⢾⠁⠀⠀⠀⠀⢸⣿⣿⣿⢩⡍⢉⡍⡟⢸⢺⢙⡙⠟⡟⣛⡛⠚⡟⠛⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⡟⡿⣿⣿⣿⣇⠀⢸⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀
⣿⣿⣷⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⠀⣸⣿⣿⣿⣶⣾⣶⣷⣧⣬⣬⣬⣤⣮⡁⠉⣀⣀⣃⣃⣠⣙⣠⣸⣇⣇⢣⠇⠶⢰⠨⠆⡇⣷⠸⣽⡿⠀⡇⠀⠀⢰⠇⢸⣿⡇⣿⢰⣿⢠⣿⠈⣧⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢸⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⠏⡯⠀⣿⣷⣶⣶⣾⣿⣿⣿⢿⣿⣿⣿⣅⣛⡛⠛⠛⢿⣿⣷⣟⣽⣿⣿⢠⠃⠀⠀⢸⠀⢸⣿⡆⡏⡾⣿⣸⣿⠀⣸⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢠⠀⠀⠀⠀⠀⡇⠀⠀⠀⢨⣬⣥⣧⣧⣓⣇⣂⣃⣻⣘⠨⡀⠷⠘⠎⣱⠡⠬⠳⢸⢩⣭⢹⢩⡍⡋⢍⣿⡛⠙⢻⡟⢛⢛⣻⣟⡻⡟⣻⡄⠧⠤⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⡛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⣷⣶⣾⣷⡑⢏⡽⠇⠁⠙⠧⣬⣼⣿⣽⣵⣕⣪⣁⠌⠀⠀⠈⠳⣤⣀⣀⢀⣀⣠⠶⠋⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠇⠶⠈⠰⠆⣿⠉⣦⢩⡏⡍⡍⢩⢹⠛⡛⡛⡟⢺⢙⡋⡛⢙⢛⢻⠿⠿⢤⣾⠿⣿⣿⢻⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢤⣾⠿⠙⠀⠀⢰⠶⢷⣷⣼⠈⠈⠁⠧⣥⣼⡬⠇⢇⣠⣸⣘⢊⡐⣿⣇⢷⡇⡾⠸⠠⠆⡆⡏⢒⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠀⢸⣿⣿⣷⠀⠀⠠⠄⠉⠀⠀⢸⣿⣿⣿⣾⣷⣿⣷⠿⣿⣿⣯⣛⠉⠉⠹⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣖⡳⢾⣿⣷⣶⣄⢸⣿⣿⣿⠁⠀⠘⠀⠀⠈⠻⢿⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⡀⢸⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢿⣿⣿⡿⣿⣿⣿⣿⣤⣿⣿⣿⣿⠄⣀⣨⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠒⠨⠅⠘⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⠙⠻⢿⠛⠛⠻⡿⠃⣿⣿⣿⣿⣶⡟⠻⠟⠿⠿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠫⢻⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣧⡀⠀⠀⠀⠊⠂⠐⢲⣤⣤⣴⣶⣾⣿⣿⣿⣧⢀⣀⣀⡀⠀⠀⠀⣿⣿⣿⣿⣷⣤⣄⡀⠀⠀⠀⠀⠉⠉⠑⠒⠢⠬⢉⣛⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣼⣿⣿⣿⣿⣿⣍⣠⣴⣿⣿⣿⣿⣿⡙⠻⢿⡿⣿⢿⣋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠷⣀⣤⣼⣿⣿⣿⣿⣿⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢃⣴⣿⣿⣿⣿⣿⣿⣶⣮⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠻⢿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠶⢶⠄⠠⠰⠶⠆⠀⠀⢀⣀⣠⡤⠴⠾⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣯⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⡿⠟⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣷⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣀⣿⡏⠀⠀⠀⢸⣿⣿⣿⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⢸⣿⣿⡿⠀⣰⡍⢻⣿⢻⣿⣧⣶⡖⣶⢤⣤⣤⣤⣤⣤⣤⡤⡤⣤⣤⣤⡤⣤⣤⣤⣤⣤⡄⣽⣿⣿⣿⣿⢉⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢸⣿⣿⡏⢀⣿⠇⣎⣽⢸⡱⣧⠹⠆⡿⠹⡉⢹⣿⢨⢋⣍⡇⡇⡟⠟⣿⡛⡛⡛⣛⢟⢻⠃⣿⣿⣿⣿⡟⡎⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣿⣾⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣶⣜⣻⣶⣶⣷⣷⣥⣭⣥⣿⠷⣰⣧⣻⣼⣘⣀⣿⣿⣿⠻⢃⡇⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⠀⠀⠀⡄⠈⣿⣿⣿⢩⡍⢉⡍⡟⢸⢺⢹⡙⠟⡟⣽⡏⠛⡟⠛⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⡟⡿⣿⣿⣿⣿⠀⢸⠁⠀⠀⣠⠞⣁⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀
⣶⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⡇⣰⣿⣿⣿⣶⣾⣶⣷⣥⣭⣬⣼⡧⣬⣍⠋⣀⣀⣃⣇⣠⣙⣠⣸⣇⣇⢣⠇⠶⢰⠸⠆⡇⣷⠸⣽⣿⠀⡘⠀⠀⢰⠇⢸⣿⡇⣿⢰⣿⢠⣿⠈⣧⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢰⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡼⠈⣼⣷⣶⣶⣿⣿⣿⣿⢿⣿⣿⣿⣅⣛⣛⠛⠻⣿⣿⣿⣟⣽⣿⣿⡇⡇⠀⠀⢸⠀⢸⣿⣆⡏⡾⣿⣸⣿⠀⣼⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⢨⣬⣥⣧⣧⣛⣏⣂⣃⣻⣘⢨⣸⠻⠘⡎⣱⠡⠬⠳⢸⢩⣭⢹⢩⡍⡋⢍⣽⡛⠙⢛⡟⢛⢛⣻⣟⡻⡟⣻⡄⠯⡤⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣷⣶⣾⣶⣑⢎⣽⠇⠁⠙⠣⣼⣼⣿⣽⣥⣕⣨⣁⠜⠀⠀⠈⠳⣤⣀⣀⢀⣀⣠⠶⠋⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⢳⡀⠀⠀⠀⠀⠀⠀⠃⠀⠇⠶⠈⠰⠂⣿⠉⣮⢩⡏⡏⡍⢫⢹⠛⡛⡛⡟⢺⢙⡋⣛⢙⢛⢻⠿⠿⢅⣿⠿⣿⣿⢻⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣾⠿⠛⠐⠠⢶⡶⢷⣷⣾⡌⠈⠁⠧⣥⣼⣬⠇⣇⣂⣼⣘⢊⡑⣿⣃⢷⣇⠿⠸⠠⠆⡆⡏⢲⡅⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠉⠀⠀⠀⠀⠘⠁⢸⣿⣿⣷⣄⠀⠠⡌⠉⢉⠀⢠⣿⣿⣿⣾⣷⣿⣿⠿⢿⣿⣯⣟⡋⠉⠹⠢⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠁⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣛⢶⣿⣿⣶⣤⢸⣿⣿⣿⠋⠀⠸⠀⠀⠀⠙⢿⣿⣿⣷⣤⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⡀⢰⣇⠀⠄⠀⠀⠀⠀⠀⠀⠀⣼⡗⢿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⡆⢀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠤⢅⠈⠟⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠙⠻⢿⠟⠛⠛⡿⠋⣿⣿⣿⣿⡷⡞⠛⠛⠛⠿⢿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣷⡀⠀⠀⠀⠆⠸⢰⣒⣦⣤⣶⣶⣿⣿⣿⡿⣷⢀⣀⣤⣄⠀⠀⠀⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠠⠭⣙⣛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣴⣤⣤⣀⣸⣿⣿⣿⣿⣿⣭⣠⣶⣿⣿⣿⣿⣿⣏⠛⠿⡿⣿⣿⣋⡈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⡿⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⣆⣠⣤⣿⣿⣿⣿⣿⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠿⢿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠠⠶⠶⣶⢤⢶⠄⠠⠰⠶⠆⠀⠀⢀⣀⣀⡤⠤⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⠿⠟⠃⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣶⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣞⣩⡟⠀⠀⠀⢸⣿⣿⣿⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⢸⣿⣿⣿⠀⣴⡍⢻⡿⢻⣿⣧⣶⡖⣶⢦⣤⣤⣤⣤⣤⣤⠤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⢸⣿⣿⣿⣿⢛⠀⢠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⢸⣿⣿⣿⢐⣿⠇⣎⣵⢸⡱⣧⠹⠆⡿⠹⠋⢹⣯⢨⢋⣍⡇⡇⡟⠟⣿⡛⡛⡛⣛⢿⢻⠃⣾⣿⣿⣿⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣾⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣾⣶⣼⣿⣶⣶⣷⣷⣥⣭⣥⣿⠷⣰⣧⣻⣼⣘⣀⣿⣿⣿⢟⠃⡇⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⣿⣿⣿⢩⡍⢉⡍⡟⢸⢺⢛⡙⠟⡛⢸⡟⠺⡟⠛⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⡟⡿⣿⣿⣿⣿⡆⢰⠃⠀⠀⣠⠞⣁⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀
⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⣇⣠⣿⣿⣿⣶⣾⣶⣷⣧⣭⣬⣼⢧⣬⡩⠛⣀⣄⣃⣃⣠⣙⣠⣸⣇⣇⢣⠇⠶⢰⠸⠆⡇⣷⠸⣽⣿⡇⢸⠀⠀⢰⠇⢸⣿⡇⣿⢸⣿⢠⣿⠈⣧⠀⠀
⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣠⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⠽⠌⢸⣷⣶⣾⣿⣿⣿⣿⢿⣿⣿⣿⣥⣛⣛⠛⠻⣿⣿⣿⣟⣽⣿⣿⡇⡆⠀⠀⢸⠀⢸⣿⣆⡏⡾⣿⣸⣿⠀⣼⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⣵⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢩⣬⣥⣧⣧⣻⣏⣂⣃⣻⣘⠨⣸⠸⠐⡎⣱⠡⢬⠳⢸⢩⣭⢹⢩⡍⡋⢍⣽⡟⠙⢻⡟⢛⢛⣻⣟⡻⡿⣳⡇⠮⡤⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣷⣶⣾⣷⡱⢎⣽⠇⠁⠛⠧⣼⣼⣿⣽⣵⣕⣺⣁⠌⠀⠀⠈⠳⣤⣀⣀⣀⣀⣠⠶⠋⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠳⡄⠀⠀⠀⠀⠀⠀⠀⠃⠁⠇⠶⠈⠰⠂⣿⠉⣦⢩⡏⡏⡍⢫⢹⠛⡙⡛⣟⢸⢙⡋⣛⢙⢛⢻⠿⠿⢱⣿⠿⣿⣿⢻⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠠⣾⡿⠟⠓⠲⢶⡶⢷⣷⣾⡌⠨⠁⠧⣥⣼⣬⣇⣇⣂⣾⣘⢊⡑⣿⣁⢷⣇⡿⠸⠠⠆⡆⡏⢲⡅⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠁⢸⣿⣿⡷⣤⠀⠀⡅⠉⢉⠀⠀⣿⣿⣿⣾⣷⣿⣿⠿⢿⣿⣿⣿⡛⠉⠹⠖⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣛⢶⣿⣾⣶⣤⣠⣿⣿⣿⠋⠀⠸⠃⠀⠀⠙⠻⣿⣿⣷⣦⣅⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤
⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣠⢰⣇⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⣷⢹⣿⣿⣿⢿⣿⣿⣿⣧⣼⣿⣿⣿⡇⢀⣈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠤⢅⠀⡿⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣙⠻⠿⠟⠛⠛⡿⠛⣻⣿⣿⣿⡷⡞⠛⠛⠛⠻⠿⣿⣿⣿⡿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣦⠀⠀⠀⠈⠌⠼⠠⣖⣦⣤⣶⣶⣿⣿⣿⠿⣿⣆⣠⣤⣄⠀⠀⠀⣿⣿⣿⣿⣷⣶⣤⣄⠀⠀⠀⠀⠀⠉⠉⠑⠒⠢⠬⢝⣛⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣀⣸⣿⣿⣿⣿⣿⣭⣠⣶⣾⣿⣿⣿⢿⡏⠛⠿⣿⣿⡿⣟⡀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⡿⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⣆⣀⣤⣿⣿⣿⣿⣿⣶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣝⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠻⢿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⠶⠄⠠⠰⠶⠆⠀⠀⢀⣀⣠⣤⠤⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡽⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⠿⠟⠃⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⠃⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣥⡞⠀⠀⠀⣿⣿⣿⠀⣾⡍⢻⡿⢻⣿⣿⣴⡖⣶⢦⣤⣤⣤⣤⣤⣤⠤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⢩⣿⣿⣿⣿⡏⡁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⣿⣿⣿⢠⣿⠇⣎⣵⢸⡹⣯⠹⠆⡿⠹⡛⢹⣿⢨⢋⣝⡇⡇⡟⠟⣿⡛⡛⡛⣛⢟⢻⠛⢸⣿⣿⣿⣿⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⣿⣿⣼⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣾⣬⣿⣶⣶⣷⣷⣥⣭⣥⣿⠷⣰⣧⣫⣼⣘⣀⣼⣿⣿⡟⠛⣸⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣿⣿⣿⢹⡝⢉⡍⡟⢸⢺⢛⡛⠟⡟⣿⡟⠻⡟⠛⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⡟⡿⣿⣿⣿⣿⢀⠀⡇⠀⠀⣠⠞⣁⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀
⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⣰⣀⣿⣿⣿⣶⣾⣶⣷⣧⣭⣬⣬⡧⣤⣯⠟⣀⣄⣃⣇⣠⣙⣠⣸⣇⣇⢣⡇⠶⢰⠸⡆⡇⣷⠸⣽⣿⠁⢠⠃⠀⢰⠇⢸⣿⡇⣿⢸⣿⢠⣿⠈⣧⠀⠀
⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⠏⢼⣿⣶⣾⣿⣿⣿⣿⢿⣿⣿⣿⣅⣛⣛⡛⠻⣿⣿⣿⣟⣽⣿⣿⣷⣸⠀⠀⢸⠀⢸⣿⣆⡏⡾⣿⣸⣿⠀⣼⠀⠀
⣿⣿⣿⣿⣿⣿⣷⡕⢄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢩⣬⣥⣧⣧⣻⣏⣂⣃⣻⣘⠨⣸⡝⠘⠎⣰⠡⢬⠳⢸⢩⣭⢹⢩⡍⡋⢍⣽⡛⠙⢛⡟⢛⢛⣻⣟⡻⡟⣻⡆⠟⡤⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣦⣧⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡸⢿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣾⣶⣱⢎⣹⡇⠁⠙⠧⣼⣼⣽⣽⣵⣕⣪⣁⡌⠁⠀⠈⠳⣤⣀⣀⣀⣀⣠⠶⠋⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠇⠶⠈⠰⠀⣿⠋⣦⢩⡏⡏⡍⢫⢹⢛⡛⡛⣟⢺⢙⡋⣛⢙⢛⢻⠿⠿⢠⣼⠿⡿⣿⢻⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠠⣶⡿⠟⠓⠲⢶⣶⠷⣷⣾⣬⠌⠥⠣⢥⣼⣬⣇⢇⣃⣼⣘⢊⡐⣿⡇⢷⣇⡿⠸⠠⠆⡆⡏⢖⡅⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠃⠀⣿⣿⣿⣦⡀⠀⡮⠉⠉⡁⠀⣿⣿⣿⣾⣷⣿⣷⡿⠿⣿⣿⣿⣟⠉⠙⠆⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣟⠶⣿⣿⣶⣦⡄⣿⣿⣿⡟⠀⠀⠃⠀⠀⠈⠻⣿⣿⣿⣶⣄⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤
⠤⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⣀⠄⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⣿⣿⣿⢿⣿⣿⣿⣧⣼⣿⣿⣿⣷⠀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠉⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡐⠆⠠⠭⠀⢻⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣍⠛⠿⡟⠛⠛⢿⠟⢹⣿⣿⣿⣿⣾⠛⠛⠛⠻⠿⢿⣿⣿⡿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠡⢃⡖⣶⣤⣤⣶⣶⣿⣿⠿⠻⣷⣠⣀⣄⠀⠀⠀⢸⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠬⢍⣛⣻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣽
⠀⠀⠀⠀⠙⠶⣶⣶⣶⣤⣤⣤⣤⣤⣴⣶⣾⣿⣿⣿⣷⣤⣤⣤⣀⣀⣀⡀⠿⣿⣿⣿⣿⣿⣍⣴⣶⣿⣿⣿⣿⣯⠛⠿⣿⢿⣿⢟⡉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣿⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡆⣀⣠⣿⣿⣿⣿⣿⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢶⣶⣶⣶⣶⣶⣶⣶⣾⠿⠟⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⣸⣠⣵⣿⣿⣿⣿⣿⣿⣶⣯⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠛⠿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⠶⠄⠠⠰⠶⠆⠀⠀⢀⣀⣠⣤⠤⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⠿⠛⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀
⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇⠀⠀⣿⣿⣿⡇⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠟⢛⠃⠀⠀⣿⣿⣿⡇⣾⡍⢻⡿⢻⣿⣿⣴⡖⣦⢤⣤⣤⣤⣤⣤⣤⡤⡤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⠉⣿⣿⣿⣿⡿⡉⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⣦⠀⠀⠀⣿⣿⣿⢃⣿⠇⣎⣵⢸⡱⣯⠱⠆⡿⠹⡛⢹⣿⢨⢋⣝⡇⡇⡟⠟⣿⡛⡛⡛⣛⢿⢻⠛⢸⣿⣿⣿⣿⣇⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⣿⣿⣿⣾⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡿⣾⣾⣿⣶⣶⣷⣷⣥⣭⣥⣿⠷⣰⣧⣻⣼⣘⣀⢸⣿⣿⡿⠟⢹⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢻⣿⣿⢹⡝⢉⡍⡟⢸⢺⢛⡛⠟⡛⣧⣟⠻⡟⠛⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⡟⡿⣿⣾⣿⣿⡯⠀⣼⠀⠀⣠⠞⣁⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀
⣦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸⣀⣾⣿⣿⣶⣾⣶⣷⣷⣭⣬⣬⣧⣄⣯⣟⣀⣄⣃⣇⣠⣙⣠⣸⣇⣇⢣⡇⠶⢰⠸⡆⡇⣷⠸⣽⣿⠑⠀⠇⠀⢰⠇⢸⣿⡇⣿⢸⣿⢠⣿⠈⣧⠀⠀
⣿⣿⣿⣿⣿⣷⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡇⠧⢠⣿⣶⣶⣿⣭⣿⣿⢿⣿⣿⣿⣥⣻⣟⡛⠻⣿⣿⣷⣟⣽⣿⣿⣷⢸⠀⠀⢸⠀⢸⣿⣆⡏⡾⣿⣸⣿⠀⣼⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣮⣢⡀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢨⣬⣥⣧⣧⣻⣏⣂⣃⣻⣘⢨⣸⡇⠘⡎⣱⠡⢬⠳⢸⢩⣭⢹⢩⡍⡋⢍⣽⡟⠙⢻⡟⢛⢛⣻⣟⡻⡿⣻⡌⠇⣠⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣾⣶⣱⢎⡽⠇⠁⠹⠧⣬⣼⣼⣽⣵⣕⣪⣃⡌⠭⠀⠈⠳⣤⣀⣀⣀⣀⣠⠶⠋⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠐⠃⠁⠇⠶⢈⠰⠀⣿⠋⣦⢩⡏⡏⡍⢫⢹⢛⡛⡛⣟⢺⢙⡋⡛⢙⢛⢻⠿⠿⢄⣽⠿⡿⣿⢻⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠠⣴⡿⠿⠃⠲⢶⡶⠷⣷⣾⣌⠬⠡⠧⢥⣼⣬⣇⣇⣋⣸⣘⢊⡑⡿⢇⢷⠇⡿⠸⠠⠆⡆⡏⢲⡅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠁⠀⠀⠀⠀⠃⠀⣿⣿⣿⣦⡀⠀⠬⠉⠉⡀⠀⣿⣿⣿⣾⣷⣿⣿⡿⠿⣿⣿⣿⣟⡉⠉⠒⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⠀⢀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣞⠶⣿⣿⣶⣦⡄⣿⣿⣿⡟⠉⠁⡋⠀⠀⠀⠙⢿⣿⣿⣷⣤⣀⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣀⣤⣤⣤
⢼⡺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⣠⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⣿⣿⣿⢿⣿⣿⣿⣧⣤⣿⣿⣿⣷⠀⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠐⠛⠽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣔⠀⠤⢭⠀⢻⡁⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⠛⢿⡿⠛⠛⢿⠟⢻⣿⣿⣿⣿⣴⡟⠛⠛⠛⠿⢿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠁⠀⡿⣧⣭⣭⣉⣉⣙⠛⠛⣛⣛⣻⣿⣿⣶⡀⠀⠀⠀⠁⠃⠀⢰⢶⣤⣤⣶⣶⣿⣿⣿⡿⣧⣀⣀⣀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣦⣤⡀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠤⠭⣙⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣀⣀⠾⣿⣿⣿⣿⣿⣍⣤⣶⣿⣿⣿⣿⣯⠛⠿⣿⢿⣿⠿⡛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⡿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡌⠉⣛⢻⣋⣉⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡆⣀⣠⣭⣿⣿⣿⣿⣶⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠿⠿⠿⠿⠟⠻⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣼⣿⣿⣿⣿⣿⣿⣷⣮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠛⠿⢿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⠶⠄⠠⠰⠶⠆⠀⠀⢀⣀⣠⣤⡴⠾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⡿⠟⠁⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⣿⣿⣿⡇⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⡸⠀⠀⠀⣿⣿⣿⡇⣾⡍⢻⡿⢻⣿⣿⣧⡖⣶⢤⣤⣤⣤⣤⣤⣤⠤⡤⣤⣥⣤⣭⣭⣭⣭⣭⣭⣉⡉⣿⣿⣿⣿⣿⠉⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡟⠀⠀⠀⢿⣿⣿⡇⣿⠇⣎⣵⢸⡱⣯⠱⠆⡿⠹⡛⢹⣿⢨⢋⣝⡇⡇⡟⠟⣿⡛⡛⡛⣛⢿⢻⠛⠀⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣶⣶⣷⣷⣥⣭⣥⣿⠷⣰⣧⣻⣼⣘⣀⢰⣿⣿⡿⠟⢳⠇⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⢸⣿⣿⢹⡝⢉⡍⡟⢸⢺⢛⡛⠟⡛⣇⣟⠻⡟⠛⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⡟⡿⣿⣸⣿⣿⠇⠀⢸⠀⠀⣠⠞⣁⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀
⣿⢿⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢸⣀⣼⣿⣿⣶⣾⣶⣷⣧⣭⣬⣬⣧⣤⣯⢋⢀⣄⣃⣇⣠⣙⣠⣸⣇⣇⢣⠇⠶⢰⠸⡆⡇⣷⠸⣽⣿⠀⠀⡇⠀⢰⠇⢸⣿⡇⣿⢰⣿⢠⣿⠈⣧⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⢰⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡇⡇⢠⣿⣷⣶⣿⣿⣿⣿⢿⣿⣿⣿⣥⣛⡛⠛⠻⣿⣿⣷⣟⣽⣿⣿⣶⢠⠁⠀⢸⠀⢸⣿⣆⡏⡾⣿⣸⣿⠀⣼⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢨⣬⣥⣧⣧⣻⣏⣂⣃⣻⣘⠠⣸⡇⠘⠎⣳⠡⠬⠳⢸⢩⣭⢹⢩⡍⡋⢍⣽⡟⠙⢻⡟⢛⢛⣻⣟⡻⡟⣻⡈⠷⣤⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⡸⢿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣷⣶⣾⣶⣑⢎⡽⠇⠁⠙⠧⣬⣼⣼⣽⣵⣕⣪⣁⡌⠒⠀⠈⠳⣤⣀⣀⢀⣀⣠⠶⠋⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠐⠃⠀⠇⠶⠈⠰⠀⣿⠋⣦⢩⡏⡏⡍⢫⢹⠛⡛⡛⣟⢺⢙⡋⡛⢙⢛⢻⠿⠿⠀⡼⠿⡿⣿⣻⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠈⣴⡿⠟⠃⠐⢶⡶⠷⣷⣾⣌⠬⠅⠧⢥⣼⣬⣇⣇⣋⣸⣘⢊⡑⣿⠂⢷⠇⡿⠸⠠⠆⡆⠏⢲⡅⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠁⠀⠀⠀⠀⠃⠀⣿⣿⣿⣦⠀⠀⠬⠉⠉⡁⠀⣹⣿⣿⣾⣷⣿⣿⡿⠿⢿⣿⣿⣿⡋⠋⠖⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣟⠶⣾⣷⣶⣦⡄⣿⣿⣿⡟⠉⠀⡋⠀⠀⠀⠙⢿⣿⣿⣷⣦⣄⣀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣤⣠⣤⣤⣤
⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠸⣿⣿⣿⣿⣿⣿⣿⣧⣤⣿⣿⣿⣷⠀⢀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⠴⠭⠀⠹⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣉⠛⠿⡿⠛⠛⢿⠟⢻⣿⣿⣿⣿⣴⠟⠛⠛⠛⠻⠿⢿⣿⣟⡛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠊⠻⠿⣿⣿⣿⣷⣿⣭⣭⣿⣿⣿⣿⣷⡀⠀⠀⠀⠁⠈⠒⣲⣤⣤⣶⣾⣿⣿⣿⢿⣦⣀⣤⣦⡀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣦⡀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠬⢍⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⣿⣿⣿⣿⣿⣯⣥⣼⣾⣿⣿⣿⢿⣿⠙⠿⣿⢿⣿⠿⡛⠛⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⡟⠿⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠇⢀⣠⣽⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⠛⠿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠄⢶⠄⠠⠰⠶⠆⠀⠀⣀⡠⠤⣤⡒⠚⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣯⠀⠀⠀⢸⣿⣿⣿⠀⢀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡟⠀⠀⠀⣸⣿⣿⡇⣴⡍⢻⡿⢻⣿⣿⣧⡄⣤⢤⣤⣤⣤⣬⣭⣭⠭⣭⣭⣭⣭⣭⣭⣭⣭⣉⣉⣉⡉⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢻⣿⣿⡏⣿⠇⣎⣵⢸⡱⣯⠱⠆⡿⠹⡙⢹⣿⢨⢋⣝⡇⡇⡟⠟⣿⡛⡛⡛⣛⢿⢻⠿⠀⣿⣿⣿⣿⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣿⣿⣿⡿⢿⣿⣿⣿⣿⢿⣿⣶⣞⣿⣶⣶⣷⣷⣥⣭⣥⣿⠷⣰⣧⣻⣼⣘⣀⢰⣿⣿⡿⠿⢣⠇⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠂⢸⣿⣿⢹⡝⢉⡍⡟⢸⢺⢛⡙⠟⡚⡿⣛⠛⡟⠛⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⡟⡿⣿⣸⣿⣿⡇⠀⢸⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⢀⣽⣿⣿⣿⣷⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣼⣿⣿⣶⣾⣾⣷⣧⣭⣬⣬⣦⣤⣭⡟⢀⣄⣃⣇⣠⣙⣠⣸⣇⣇⢣⠇⠶⢰⠸⡆⡇⣷⠸⣽⣿⠀⠀⡇⠀⢰⠇⢸⣿⡇⣿⢸⣿⢠⣿⠈⣧⠀⠀
⣬⣿⣿⣿⣿⣿⣿⣿⣿⣗⢄⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢰⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡇⡇⢠⣿⣷⣶⣿⣿⣿⣿⢿⣿⣿⣿⣥⣛⡛⠛⠛⣿⣿⣷⣟⣽⣿⣿⣦⢠⠁⠀⢸⠀⢸⣿⣆⡏⡾⣿⣸⣿⠀⣼⠀⠀
⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⡇⠀⠀⢩⣼⣥⣧⣧⣻⣏⣂⣃⣻⣘⠨⣸⡇⠘⡂⣳⠡⢬⠳⢸⢩⣭⢹⢩⡍⡋⢍⣽⡛⠙⠿⡟⢛⢛⣻⣟⡻⡟⣻⡈⠧⣤⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠁⠀⠀⡸⢿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣷⣶⣾⣷⣱⢆⡹⠇⠁⠙⠇⣬⣼⣼⣽⣵⣕⣪⣁⡜⠓⠀⠈⠳⣤⣀⣀⣀⣀⣠⠶⠋⠀⠀⠀
⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⣆⠀⠀⠀⠀⠀⠁⠀⠇⠶⠈⠰⠀⣿⠋⣮⢩⡍⡏⡍⢫⢹⠛⡛⡛⣟⢺⢙⡋⡛⠙⢛⢻⠿⠟⢀⡸⠿⡿⣿⣻⣿⣿⣿⣿⣿⣯⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⡆⠀⠀⠀⠀⠀⠀⠉⠸⠀⢴⣿⠟⠃⠲⠶⣶⠷⣷⣾⣬⠌⠥⠧⢥⣼⣬⣇⣇⣋⣸⣘⢊⡑⡿⠄⢷⠇⡿⠸⠠⠆⡆⡏⢲⠅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⣿⣿⣿⣷⡀⠀⠬⠉⠉⡁⠀⢸⣿⣿⣾⣷⣿⣿⡽⠿⢿⣿⣿⣿⣋⠋⠖⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠚⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣗⡷⣾⣷⣶⣦⡄⣾⣿⣿⡟⠉⠉⡋⠀⠀⠀⠙⢻⣿⣿⣷⣶⣄⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠐⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⢀⡄⣾⡆⠘⠀⠀⠀⠀⠀⠀⠀⠀⣿⡮⣿⣿⣿⡿⣿⣿⣿⣯⣤⣿⣿⣿⣷⠀⢠⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠺⠀⠘⡃⣀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣌⠛⠿⡿⠛⠛⢻⠟⢻⣿⣿⣿⣿⡴⠟⠛⠛⠛⠛⠿⢿⣿⣟⡛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠁⢻⠲⢶⣶⣶⣶⣿⣿⣿⠿⠻⣇⣠⣤⣤⡀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⡄⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠨⢭⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣾⣿⣿⣿⣿⣭⣤⣼⣶⣿⣿⣿⣿⣿⠙⠻⢿⣿⣿⠿⠻⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠿⣿⣿⣿⡟⠻⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠛⠛⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢷⣀⣤⣾⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⣀⣼⣿⣿⣿⣿⣿⣿⣿⣶⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⠛⠿⢿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠐⠖⠶⣶⠄⢶⠄⠠⠰⠶⠆⠀⠀⢀⡠⠤⢤⣖⠒⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡆⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠛⠆⠀⠀⢸⣿⣿⣿⠀⢀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣾⠁⠀⠀⢸⣿⣿⣿⣼⡍⢻⡿⢻⣿⣿⣏⡍⣭⢭⣭⣭⣭⣭⣭⣭⡭⣭⣭⣭⣉⣉⣉⣉⣉⣉⣉⣉⡉⣹⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⣎⣵⢸⡱⣯⡱⠆⡟⠹⡓⢹⣿⢨⢋⣝⡇⡇⡟⠟⣿⡛⡛⡻⣛⢿⢿⠿⠆⣿⣿⣿⣿⡿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⢾⣚⡿⣶⣶⣷⣷⣥⣭⣥⣿⠷⣰⣧⣫⣼⣘⣀⠀⣿⣿⣿⡿⠇⡇⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡆⠘⣿⣿⢹⡝⢉⡍⡟⢸⢺⢛⡙⠟⡚⣿⣚⠛⡟⠛⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⡟⡿⣿⣾⣿⣿⡟⠆⢸⠀⠀⣠⠞⣁⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀
⠈⢈⣻⣿⣷⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣸⣿⣿⣶⣾⣾⣷⣧⣭⣬⣼⣥⣤⣩⣏⢀⣀⣃⣃⣠⣙⣠⣸⣇⣇⢣⠇⠶⢰⠸⡆⡇⣷⠸⣽⣿⡇⠀⡜⠀⢰⠇⢸⣿⡇⣿⢰⣿⢠⣿⠈⣧⠀⠀
⣀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⢤⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⢼⢤⣿⣷⣴⣾⣿⣿⣿⢿⣿⣿⣿⣥⡛⠛⠛⠛⣿⣿⣷⣟⣽⣿⣿⣿⡄⡇⠀⢸⠀⢸⣿⣆⡏⡾⣿⣸⣿⠀⣼⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣵⣄⠀⠀⠀⠀⠀⠟⠀⠀⢩⣼⣥⣧⣧⣻⣏⣂⣃⣻⣘⠨⣸⡿⠘⡆⣱⠩⢬⠳⢸⢩⣽⢹⢩⡝⡋⠍⣿⡟⠝⢿⡟⢛⢛⡻⣟⡻⡟⣻⡀⠷⣦⠘⣿⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠼⢿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣾⣷⣱⢆⡽⠇⠂⠐⠣⣬⣼⣼⣽⣵⣕⣪⣀⡜⠓⠀⠈⠳⣤⣀⣀⢀⣀⣠⠶⠋⠀⠀⠀
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⢣⡀⠀⠀⠀⠀⠃⠀⠇⠶⠈⠰⠀⣿⠋⣦⢩⡏⡍⡍⢫⢹⠛⡛⡛⣟⢻⢙⡋⡛⢙⢛⢻⠿⠿⠀⡠⠿⡿⣿⣻⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢧⠀⠀⠀⠀⠀⠀⠉⠘⠁⢠⣾⠿⠃⠂⠶⡶⠷⣷⣾⣬⠌⠁⠧⢥⣼⣬⣇⣇⣋⣸⣘⢊⡑⣿⠆⢷⠇⣿⠸⠠⠆⡆⡏⢒⡅⡂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠘⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠀⢸⣿⣿⣧⡀⠀⠠⠍⡉⠀⠀⢸⣿⣿⣾⣶⣿⣿⣬⠿⢿⣿⣿⣟⡋⠉⠐⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢽⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⣴⡶⣾⣷⣦⣤⣄⢸⣿⣿⣟⠉⠉⢉⠁⠀⠀⠙⢿⣿⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⢀⡀⢸⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣿⣿⣿⡿⣿⣿⣿⣿⣍⣿⣿⣿⣿⠀⢀⣀⣾⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠚⠁⠘⢇⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⠛⠻⣿⠟⠛⢻⡿⠃⣿⣿⣿⣿⡤⡞⠛⠛⠛⠿⢿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠛⢻⣿⠿⠿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠈⠈⠼⡔⢖⣦⣤⣴⣶⣾⣿⣿⣿⣧⢀⣀⣀⡀⠀⠀⢀⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠤⠍⣙⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣄⣸⣿⣿⣿⣿⣯⣉⣠⣶⣿⣿⣿⣿⣿⡟⠿⢿⣿⣿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣿⡿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢷⢀⣀⣭⣿⣿⣿⣿⣶⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣦⣴⣿⣿⠿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣀⣠⣿⣿⣿⣿⣿⣿⣷⣾⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠻⢿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠧⠶⣶⠀⢶⠄⠠⠰⠶⠆⠀⠀⠀⢀⣀⣤⡤⠴⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀
⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣛⣻⣿⡿⠀⠀⠀⣿⣿⣿⠀⢀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣁⠀⠀⢰⣿⣿⣿⣰⡍⢻⡿⢻⣿⣿⣿⡍⣭⢭⣭⣉⣉⣉⣉⣉⡙⣛⣋⣉⣉⣛⣛⣛⣛⣛⣛⣛⡛⢻⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠇⠀⠀⠘⣿⣿⣿⣻⠇⣎⣵⢸⡱⣮⡱⠆⡟⡹⠓⢹⣿⢨⢋⣝⡇⡇⡟⠟⣿⡟⡟⡻⣛⢿⢿⠿⡅⣼⣿⣿⣿⣿⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡿⠷⣿⢾⣞⣿⣶⣶⣷⣶⣥⣭⣥⣿⠷⣰⣧⣫⣼⣘⣀⠀⣿⣿⣿⡿⠇⡇⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠂⠀⠀⠀⠀⣿⣿⢹⡍⢉⡍⡟⢸⢺⢛⡛⠟⡚⣿⣘⠻⡟⠟⢿⠿⢿⠿⠿⠿⡿⢾⣿⣿⣿⡟⡿⣿⣦⣿⣿⡏⠁⢰⠃⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠻⡶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⣆⣠⣿⣿⣶⣾⣶⣷⣥⣭⣬⣼⢧⠄⢩⡏⢀⣀⣃⣃⣀⣘⣠⣸⣇⣇⢣⠇⠶⢰⠨⠆⡇⣶⠸⣽⣿⡇⠀⡸⠀⢰⠇⢸⣿⡇⣿⢰⣿⢠⣿⠈⣧⠀⠀
⠀⣤⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢛⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡿⢸⢤⢼⣧⣤⣼⣿⣿⣿⢿⣿⣿⣿⣅⠛⠛⠛⠛⢿⣿⣷⣟⣼⣿⣿⣷⡄⡇⠀⢸⠀⢸⣿⡆⡏⡾⣿⣸⣿⠀⣸⠀⠀
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠨⣬⣥⣧⣧⣻⣏⣂⣃⣻⣘⠀⣸⡿⠘⡄⣼⠩⢬⠳⢸⢉⣟⢹⢹⡝⡋⠍⣿⡟⠕⢶⡞⢛⢛⠻⣟⡻⡿⣻⣤⠗⡠⠘⣯⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣷⣶⣾⣷⡑⢆⡻⠂⠂⠰⠧⣬⣼⣤⣽⣵⣕⣪⣀⠌⠃⠀⠈⠳⣤⣀⣀⢀⣀⣠⠶⠋⠀⠀⠀
⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢷⡄⠀⠀⠀⠀⠀⠀⠃⠄⠇⠶⠈⠰⠀⣿⠋⣦⢩⡍⡍⡍⢫⢹⢛⡛⡛⣟⢻⢙⡋⡛⠙⢛⢻⠿⠖⠀⠀⠴⡶⣾⣻⣿⣿⣿⣿⣿⣯⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡸⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⡆⢀⣴⠟⠃⠀⠀⠲⠷⢷⣾⣌⠈⠁⠡⠥⣼⣬⣇⣇⣃⣸⣘⢊⡐⡾⠄⠷⠇⣿⠸⠠⠆⡆⡏⢒⠁⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠺⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠐⠀⢸⣿⣿⢧⡀⠀⠠⠀⠀⠀⠀⢸⣿⣿⣶⣶⣷⣖⣠⣿⣿⣾⣿⣛⠛⠉⠐⠁⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠚⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣮⣷⠶⢶⣷⣦⣤⣀⢸⣿⣿⣿⠍⠉⠹⠁⠀⠈⠛⢿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠘⠋⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣿⣿⣿⣿⣾⣿⣿⣿⣍⣼⣿⣿⣿⡄⢀⠐⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠛⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠐⠇⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⠛⢻⣿⠿⠛⢿⣿⠏⣿⣿⣿⣿⣤⡾⠿⠿⠿⢿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠈⠁⠂⠂⣤⣤⣤⣶⣾⣿⣿⣿⣧⠀⣀⣤⡄⠀⠀⠀⣿⣿⣿⣿⣷⣤⣤⡀⠀⠀⠀⠉⠉⠛⠓⠒⠤⠍⣙⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢷⣷⣶⣶⣦⣴⣶⣾⣿⣿⣿⣷⣦⣤⣤⣀⣀⡀⣸⣿⣿⣿⣿⣟⣉⣁⣴⣿⣿⣿⣿⢿⡗⠶⣶⣿⣿⡿⠟⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢄⣀⣵⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠃⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣷⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢀⣿⣿⣿⣿⣿⣷⣶⣶⣯⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⠴⠄⠠⠠⠶⠆⠀⠀⠀⢀⣀⣠⡤⠴⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⣛⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⠀⠀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀
⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣿⣿⣿⢴⡌⢻⡿⢻⣿⣿⣿⡉⣉⢉⣉⣉⣉⣙⣛⣛⡛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛⠛⠛⢻⣿⣿⣿⣿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⡿⢠⡆⠀⠀⣿⣿⣿⢺⠇⣎⡵⢸⡱⣮⡱⠆⡟⠹⠓⢹⣿⢨⠋⣝⡇⡇⡟⠟⣿⡟⡟⡻⣛⢿⢿⠿⡇⣼⣿⣿⣿⣿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣯⡷⠗⣿⢶⡞⢙⢶⣶⣷⣷⣥⣭⣥⡿⠷⣰⣧⣫⣼⣘⡀⠀⣿⣿⣿⣿⠏⡇⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⣿⣿⢹⡝⢉⡍⡟⢸⢺⢛⡛⠏⡀⣿⠸⠻⠿⠟⢷⠿⢿⠿⠿⠿⠿⢾⣿⣿⣿⡟⡿⣷⣄⣿⣿⡇⠀⢠⠃⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠃⠀⠀⠀⣄⣠⣿⣿⣶⣾⣶⣧⣧⣬⣬⣬⢥⠄⢩⡏⠀⢀⣃⣃⣀⣘⣠⣸⣇⣇⢣⠇⠴⢰⠈⠆⡇⣶⠸⣽⣿⠇⠀⠸⠀⢰⠇⢸⣿⡇⣿⢰⣿⢠⣿⠈⣧⠀⠀
⠀⢹⣶⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢚⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢽⡿⢸⠠⢼⣧⣤⣴⣗⣾⣿⢿⣿⣿⣽⡄⠉⠉⠉⠙⠛⣿⣷⣟⣼⣿⣿⣶⡀⡇⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠐⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⣬⣥⣧⣧⣻⣏⣂⣃⣻⣘⠀⣸⡿⠘⡄⢼⠹⢬⢳⢸⢉⣟⢹⢹⠝⡋⠍⡽⡗⠔⢶⡞⢛⢛⡛⣟⡻⡟⣻⣄⠇⡠⠘⣯⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠛⣿⣿⣿⣯⣍⣹⣧⣶⣶⣶⣿⣿⣾⣶⣶⣷⣶⣾⣷⡑⢆⡓⠃⠆⠴⠥⢬⣼⣤⣽⣵⣕⣪⣀⢈⠃⠀⠈⠳⢤⣀⡀⢀⣀⣠⠖⠋⠀⠀⠀
⠁⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠶⠈⠰⠀⣿⠋⣦⠩⡍⡍⡍⢫⢹⠛⡛⡛⣟⢻⢙⡋⡛⠙⢛⢻⠿⠒⠀⠀⠴⠶⣾⣻⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⠡⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡆⢀⣤⠛⠁⠀⠀⠒⠳⢷⣾⣌⠀⠁⠁⠡⡼⢬⠅⠇⣃⣸⣘⢊⡐⡷⠄⠳⠇⣿⠸⠠⠆⡆⠏⢐⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣖⣀⣻⣟⣶⣍⡙⠋⠉⠐⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⠶⣶⣶⣦⣤⣀⢸⣿⣿⣿⠏⠉⠹⠁⠀⠈⠻⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀
⠀⠀⠀⠡⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣿⣿⣿⣷⣾⣿⣿⣿⣉⣸⣿⣿⣿⡄⠀⠐⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢛⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠠⠅⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⣈⠛⢻⣿⠿⠟⢿⣿⠏⣿⣿⣿⣿⣆⡼⠿⠿⠿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠈⠀⠀⠀⠀⣤⣤⣤⣶⣾⣿⣿⣿⣷⠀⣀⣤⣄⠀⠀⠀⣿⣿⣿⣿⣧⣤⣄⠀⠀⠀⠀⠉⠙⠛⠒⠢⠬⢉⡛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣀⣀⣤⣬⣭⣭⣤⣶⣾⣿⣿⣦⣤⣀⣀⣀⡀⠀⢰⣿⣿⣿⣿⣟⣉⣁⣴⣿⣿⣿⣿⢿⣷⠶⣶⣿⣿⣿⠟⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠆⣀⣡⣿⣿⣿⣶⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠃⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣥⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠉⠁⢐⣿⣯⣿⣿⣿⣿⣶⣶⣭⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠻⢿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠄⠲⣶⠀⠰⠄⠠⠠⠖⠆⠀⠀⠀⠀⣀⣀⡤⠤⠶⢿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣮⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀
⣿⡇⠀⠀⠀⠀⠀⠈⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢺⣿⣿⡇⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀
⣿⡅⠀⠀⠀⠀⠀⠀⠀⠔⠛⠿⠿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣾⣿⣿⣇⡄⢻⡿⢻⣿⣿⣿⡛⣛⢛⣛⣛⣛⣛⣛⣛⠛⡛⢛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠒⢶⣶⣦⣶⣾⣿⣿⣻⡿⠏⠀⠀⠀⠀⠀⣿⣿⣿⢿⠇⡎⡵⢸⡡⣮⡱⠆⡿⠹⠛⢹⣿⢨⠋⣝⡇⡇⡟⠟⣿⡟⡟⡻⣛⢿⢻⠿⡇⣸⣿⣿⣿⣿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠿⠿⠿⠛⣁⡤⠂⠀⠀⠀⠀⢿⣿⣿⣸⣿⣿⡿⢿⣿⣿⡟⡗⠒⣿⢲⠞⠈⠀⠶⢷⡶⣥⢥⡥⠫⠷⢰⣧⣋⣴⣘⡀⠀⣿⣿⣿⣿⡏⡇⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢿⣿⢹⡍⢉⡍⡟⢸⢸⢛⡃⠇⠀⣿⠘⠻⠟⠖⢶⠼⠿⠿⠦⠴⠶⢶⢷⣿⣿⡟⡿⣦⣄⣿⣿⡏⠀⢠⠃⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣾⣿⣶⣾⣶⣧⣥⣬⣬⣬⢧⠄⢨⡟⠀⠀⢀⣃⣀⣘⣀⢸⣇⣇⢡⠇⠰⢰⠈⠆⡇⣶⠸⣽⣿⠇⠀⢸⠀⢰⠃⢸⣯⡇⣿⢰⣿⢀⣷⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⢒⡛⡛⡿⣛⠿⡿⡿⠿⢿⠕⢸⡿⢸⠀⢸⣤⣀⣀⣄⣶⣦⢈⡅⣀⣍⠀⠀⠀⠀⠉⠙⢿⣷⣞⣸⣿⣿⣄⡀⡆⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣸⠀⠀
⢰⣾⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣥⣧⣧⣛⣏⣂⣃⣺⣐⠀⢸⣿⠘⠄⣸⠹⢬⠳⢸⢋⡿⢹⢹⠝⡫⠍⡴⠶⠄⣦⣆⢙⢛⡛⣟⡻⡟⣻⠀⠗⡄⠘⣏⠛⠛⠛⠓⠓⠻⠿⠾⣤⠏⠀⠀
⢻⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⡛⣿⣿⡏⠈⣀⣸⣿⣶⣶⣶⣤⣬⣬⣶⣶⣷⣶⣾⣶⡐⠂⠲⠆⠤⠤⠤⣬⣼⣤⣽⣵⣕⣪⣀⢈⠋⠁⠈⠳⢤⣀⡀⢀⣀⣠⠔⠋⠀⠀⠀
⢨⣿⣿⣿⣿⣿⣿⣿⣷⣦⣦⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠖⠈⠠⠀⣿⠋⢦⠈⡌⡍⡍⢩⢹⠙⡙⡛⡟⢻⢙⡋⡋⠘⠛⢻⠶⠆⠀⠀⠠⠴⢶⣣⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀
⠀⢰⣽⣿⣿⣿⣿⣿⣿⣿⣯⡻⢦⣴⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣠⠚⠃⠀⠀⠀⠲⢷⣾⡌⠀⠀⠁⠁⠜⠈⠅⠃⣂⣸⣘⢂⠐⠤⠀⠳⠃⣾⠸⠀⠆⡆⠉⢀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣷⣌⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⡂⠀⠀⢀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⡄⢀⣹⣿⣤⣄⠙⠉⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⣠⢤⣤⣶⣤⣀⡀⢰⣿⣿⣿⠋⠁⠘⠉⠀⠘⠻⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⣿⣿⣷⣾⣿⣿⣿⣙⣸⣿⣿⣿⡄⠀⠘⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⢻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢿⣀⣀⡀⠀⠀⠀⠀⠀⠀⢈⣁⣌⠻⢿⣷⡾⠿⢿⣿⠏⣿⣿⣿⣿⣇⣼⢿⡿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣄⣠⣤⣴⣶⣿⣿⠿⢷⡀⣈⣀⡀⠀⠁⠀⣿⣿⣿⣿⣧⣄⣀⠀⠀⠀⠈⠉⠛⠛⠲⠤⠌⣙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠿⠿⠿⠛⣛⣻⣿⣿⣦⣀⣀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣟⣋⢀⣰⣶⣿⣿⣿⣿⡶⢶⣶⣿⣿⣿⠿⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢈⣁⣿⣶⣶⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠁⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣙⣿⡵⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣽⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠿⠿⠿⠛⠋⠁⠀⠀⣠⣿⣿⣿⣷⣶⣾⣭⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⣶⠀⠰⠄⠠⠠⠖⠆⠀⠀⠀⠀⢀⣀⡠⠠⠴⣾⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⡿⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀
⣿⡃⠀⠀⠀⠀⠠⠭⣿⣻⢿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀
⣿⡆⠀⠀⠀⠀⠀⠀⠀⠐⠊⠿⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣇⡌⢻⡿⢻⣿⣿⣿⡛⣛⢛⣛⣛⣛⢛⣛⣛⠛⡛⢛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⢻⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣟⠀⠀⠀⠀⠀⠀⠀⠓⢶⣤⣤⣤⣿⣛⡛⠿⠿⠟⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⠇⡆⡵⢸⡡⣮⡱⠆⡿⠹⠛⢹⣿⢨⠫⣝⡇⡇⡟⠟⣿⡟⡟⡻⣛⢿⢻⠿⡇⢸⣿⣿⣿⣿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣸⣿⣿⡿⢿⣿⣿⡟⡗⠒⣿⢰⠞⠙⠒⠲⢷⣶⣥⣥⡥⠿⠷⣰⣧⣫⣴⣘⡀⠀⣿⣿⣿⣿⡟⡎⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠐⠀⠀⠀⠀⠀⣿⣿⢹⡝⢉⡍⡟⢸⢸⢛⡃⠇⠀⣿⠘⠻⠟⠖⠶⠾⢿⠿⠶⠶⠶⢶⢿⣿⣿⡟⡿⣧⣀⣿⣿⡟⠀⢀⡇⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣾⣿⣶⣾⣶⣧⣥⣬⣬⣬⣧⠄⢨⡟⠀⠀⢃⣃⣀⣘⣠⣸⣇⣇⢣⠇⠰⢰⠈⠆⡇⣶⠸⣽⣿⡇⠀⢸⠀⢰⠃⢸⣯⡇⣿⢰⣿⢀⣷⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡛⡛⡿⣛⠿⡿⡿⠿⢿⠯⢰⡿⢸⠀⢸⣧⣄⣀⣄⣴⣶⢁⣭⣤⡍⠀⠈⠈⠁⠉⠛⢿⣷⣞⣸⣿⣿⣦⡀⡄⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣸⠀⠀
⣿⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣥⣧⣧⣛⣏⣂⣃⣺⣐⠀⢸⣿⠘⠄⣸⠹⢬⠳⢸⢋⡿⢹⢹⠝⡫⠍⡴⡶⠄⣤⣖⢙⢛⡛⣟⡻⡟⣻⡄⠇⠠⠘⣏⠛⠛⠛⠓⠓⠻⠿⠿⣤⠇⠀⠀
⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡛⣿⣿⡋⠉⢀⣨⣧⣴⣶⣶⣤⣼⣬⣶⣶⣷⣶⣾⣶⡐⠆⠳⠦⠤⠤⢤⣬⣬⣥⣽⣵⣕⣪⣁⠈⠃⠁⠈⠳⢤⣀⡀⢀⣀⣠⠖⠋⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⣤⣢⡀⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠇⠖⠈⠰⠀⢿⠋⣤⠀⡌⡍⡍⢩⢹⠙⡙⡛⣟⢻⢙⡋⡛⠘⠛⠻⠷⠄⠀⠀⠠⠤⢶⣲⣿⣿⣿⣿⣿⣞⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀
⠉⢿⣿⣿⣿⣿⣿⣿⣿⣟⡻⢶⣶⣄⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡠⠚⠁⠀⠀⠀⠳⢷⣾⡈⠀⠀⠁⠁⠘⠈⠁⠇⣂⣸⣘⢂⠐⡤⠀⠓⠂⣾⠸⠀⠆⡆⠏⢀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣄⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⠂⠀⠀⢀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣦⡄⢀⢸⣿⣤⣍⠙⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠤⣼⣭⣭⣿⣿⣿⣿⣿⣷⣏⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣅⣠⣤⣤⣶⣄⣀⡀⢠⣿⣿⣿⠛⠁⠘⠋⠀⠘⠻⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠸⣆⠀⠀⠀⠀⠀⢀⡇⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⣿⣿⣷⣾⣿⣿⣿⣛⢸⣿⣿⣿⡀⠀⠈⣤⣤⣤⣤⣼⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡙⢆⠀⠀⠀⠀⠀⢿⣀⣤⣤⡄⠀⠀⠀⠀⠀⠈⠁⠈⠿⢿⣿⡾⠿⢿⣿⠟⣿⣿⣿⣿⣇⣼⣿⡿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⠓⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠀⠀⠀⠀⠀⠈⠈⠉⠙⠁⣀⣀⣠⣤⣴⣾⣿⡿⣦⡀⢉⠀⠀⠀⠉⠀⣿⣿⣿⣿⣧⣄⣀⠀⠀⠀⠉⠉⠛⠓⠢⠤⢈⣙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻
⠀⠀⠀⠀⠀⠙⠛⠿⠾⠿⠿⠿⣿⡿⠿⠿⢟⣛⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡛⠁⣠⣼⣿⣿⣿⣿⡶⢶⣦⣿⣿⣿⡿⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠤⠀⢤⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⡀⠈⣉⣻⣷⣶⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠀⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠋⢛⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⠿⠿⠟⠛⠉⠀⠈⣩⣿⣿⣿⣿⣶⣭⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⣶⠀⠰⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⣀⣀⠠⣴⣶⣿⣟⡛⣿⣿⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⠃⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣈⠙⠛⠛⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣾⣿⣿⣇⡌⢻⡿⢻⣿⣿⣿⡛⣛⢛⠛⠛⠛⠛⠛⠿⠿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⠿⢿⣿⣿⣿⣭⣭⠙⠋⠀⠀⠀⠀⠀⣿⣿⣿⢻⠇⡎⡱⢸⡱⣮⡱⠆⡿⠹⠛⢹⣿⢨⢫⣍⡇⡇⡟⠟⣿⡟⡟⡻⣛⢿⢻⠓⡆⢸⣿⣿⣿⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢋⣡⡴⠃⠀⠀⠀⠀⣿⣿⣿⣼⣿⣿⡿⢿⣿⣿⡟⡇⠒⣿⢲⡾⠙⠒⠶⣷⣶⣥⣥⠥⠿⠷⣰⣧⣫⣬⣘⡀⠀⣿⣿⣿⣿⡟⡆⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢹⡝⢉⡍⡟⢸⢸⢛⡓⠇⠀⣿⢘⡲⡖⠞⠷⠾⢿⠷⠶⠶⠾⢶⢿⣿⣿⡟⡿⣦⡀⣿⣿⡟⠀⢀⠇⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⣾⣿⣶⣾⣶⣧⣥⣬⣬⣬⢧⠄⢨⣟⠀⠀⢃⣃⣀⣘⣀⡸⣇⣇⢡⠇⠰⠰⠈⠆⡇⣶⠸⣽⣿⡇⠀⢸⠀⢰⠃⢰⣯⡇⣿⢰⣿⢀⣷⠈⣧⠀⠀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢚⡛⡛⡿⣛⠿⡿⡿⠿⢿⠗⢰⡿⢸⠀⢸⣇⣀⣀⣀⣶⣴⢩⣽⣐⣉⡀⠈⠀⠁⠉⠙⢿⣷⣞⣸⣿⣿⣄⡀⡌⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣸⠀⠀
⣿⣷⣶⣀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣬⣥⣧⣧⣛⣏⣂⣃⣻⣐⠀⢸⣿⢘⡄⣸⠹⢬⠳⢸⠪⡾⢸⢸⠜⡯⠅⣴⡶⠄⣤⣄⢘⢛⡛⣟⡻⡟⣻⡄⠇⠠⠘⣏⠛⠛⠛⠓⠓⠻⠿⠿⢤⠇⠀⠀
⣿⣿⣿⣿⣷⣦⡐⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⡛⣿⣿⡏⠁⢀⣨⣧⣤⣥⣤⣥⣽⣯⣿⣛⣷⣶⣾⣶⡰⠆⠵⠤⠤⢤⣥⣬⣼⣥⣽⣵⣕⣪⣁⠈⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠔⠋⠀⠀⠀
⣿⣿⣶⣾⣿⣿⣿⣷⣌⣦⡀⠀⡠⠘⣿⣤⡀⠀⠀⠀⠀⠀⠀⠇⠶⠈⠠⠀⢿⠉⣦⠁⡌⡍⡍⢩⢹⢙⡛⡛⣟⢻⢙⡋⡛⠙⠛⢻⠧⠤⠀⠀⠠⠤⢴⣺⣿⣿⣿⣿⣿⡜⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀
⣉⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⢾⣿⣗⣿⣿⣦⠀⠀⠀⠀⠀⠁⠈⠈⠄⠀⢀⠜⠃⠀⠀⠀⠳⢷⣾⡈⠀⠀⠁⠁⠘⠈⠅⠇⢂⣸⣘⢂⡀⡀⠀⠓⠂⣾⠸⠀⠆⡆⠍⢀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣥⣤⣤⣤⠄⠀⢸⣿⣥⡀⠉⠉⠁⠐⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⡦⣬⣭⣽⣿⣟⣻⣿⣿⣷⣌⣧⠋⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣂⣀⣤⣤⣤⣀⣀⠀⢠⣿⣿⣿⠟⠉⢸⠋⠀⠛⢿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠙⣿⣧⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⣿⣶⣼⣿⣿⣿⡛⢸⣿⣿⣿⡀⠀⠨⣤⣤⣤⣤⣽⣿⣿⣿⣿⣶⣶⣤⣴⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠹⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⢻⡇⠀⠀⠀⠀⠷⠀⠀⣀⣀⣤⣄⡀⠀⠀⠘⠁⠘⠿⢿⣿⣿⠿⣿⣿⡟⣿⣿⣿⣿⣧⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠠⠑⠃⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡃⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣛⣁⣠⣤⣶⣿⣿⣆⠈⠉⢁⠀⠀⠉⠀⣿⣿⣿⣿⣍⣁⡀⠀⠀⠈⠉⠛⠛⠷⠤⠌⣉⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠻
⠀⠀⠀⠀⠀⠒⠶⠶⠶⢿⣿⣿⣿⣿⠿⠿⢟⣻⣿⣿⣄⣀⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠟⠛⣩⣿⣶⣿⣿⣷⡤⣤⣤⣿⣿⣿⡿⠛⠛⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣮⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡄⠉⢉⣿⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠏⠈⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠛⠛⠋⢛⣻⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣁⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠾⠾⠿⠟⠋⠉⠀⠈⢉⣿⣿⣿⣯⣭⣭⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⣶⠀⠰⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⠿⣿⣿⣿⣿⣿⣿⣿⣷⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢉⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⠿⠿⣿⣿⡿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⣿⣶⣤⣿⣷⣬⣿⡻⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⢠⡌⢻⡿⢻⣿⣿⣿⡟⠻⠿⠿⠿⠿⠿⠿⢿⠿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠻⠿⣿⣿⡿⢂⣴⡆⠀⠀⠀⠀⣿⣿⣿⢸⠇⡎⡵⢸⡡⣮⡱⠆⡿⠹⢓⢹⣿⢨⢫⣝⡇⡇⡟⠟⣿⡟⡛⡻⣛⢿⢲⠒⡂⣾⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣿⣿⣿⣸⣿⣿⡿⢿⣿⣿⡇⡗⠓⣿⢲⡾⠙⠒⢶⣷⣶⣥⢭⡥⠿⠷⣰⣧⣫⣼⣘⡀⠁⣿⣿⣿⣿⡇⡇⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢩⡍⢉⡍⡟⢸⢸⢛⡛⠇⠀⣿⢘⡲⡖⡾⠦⠼⠶⠾⠿⠿⠿⢾⣶⣿⣿⡟⡿⣴⣠⣿⣿⡏⠀⢠⠃⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡟⠀⠀⠀⠀⠁⠀⠀⠀⣠⠀⣿⣿⣶⣾⣶⣧⣥⣬⢬⣬⢥⠄⢨⡟⠀⠀⣁⣃⣀⣘⣀⣸⣇⣆⢡⠇⠴⠰⠈⠆⡇⣶⠸⣽⣿⠇⠀⠸⠀⢰⠇⢰⣯⡇⣿⢰⣿⢀⣶⠈⣧⠀⠀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡛⡛⡿⣛⠿⡿⡿⠿⢿⠆⢨⡿⢸⠀⢸⣅⣀⣀⣄⣬⣿⢩⣅⣑⣁⠀⠀⠈⠉⠉⠙⢿⣷⣞⣸⣿⣿⣤⡄⡇⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣸⠀⠀
⣿⣿⣶⣆⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣥⣧⣧⣛⣏⣂⣃⣻⣐⠀⢸⣿⢘⡀⣸⠹⢬⠳⢸⠮⡾⢸⢸⠌⡯⢅⣴⡶⠄⣤⣄⢚⢛⡛⣟⡻⡟⣻⡄⠇⠠⠘⣏⠛⠛⠛⠓⠓⠻⠿⠾⢤⠇⠀⠀
⢿⠿⢿⣿⣿⣶⣌⠂⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⡛⣿⣿⡏⠉⢀⣨⣥⣤⣤⣭⣭⣍⣫⣻⣿⣷⣶⣶⣶⡰⠆⠽⠤⡤⢬⣁⣨⣬⣥⣽⣵⣕⣪⣀⠈⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠔⠋⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣦⣵⣄⢀⣠⡀⣿⣷⡄⠀⠀⠀⠀⠀⠀⠇⠶⠈⠰⠀⣿⠉⢤⠀⡌⡍⡍⢩⢹⢙⡛⡛⣟⢻⢙⡋⡛⠙⠛⢻⠧⠄⠀⠀⠠⠤⢼⣻⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡛⠻⣿⣇⣻⣿⣷⠀⠀⠀⠀⠀⠁⠉⠀⡁⠀⠀⠘⠁⠀⠀⠀⠳⢷⣼⠈⠀⠁⠁⠁⠘⠈⠁⠇⣆⣸⣘⢂⠀⡀⠀⠳⠂⣾⠸⠀⠆⡆⠏⢀⠁⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣦⡹⡿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣥⣤⣤⣤⠀⠀⢘⣿⣁⠀⠉⠉⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣦⣦⣬⣭⣽⣿⣻⣿⣿⣿⣧⣽⡇⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⣀⣀⣈⣤⣀⠀⠀⢠⣿⣿⣿⠟⠁⢸⠛⠈⠻⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣧⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢿⣿⣿⣮⣽⣿⣿⣿⠛⢸⣿⣿⣿⡀⠀⠠⣤⣤⣤⣬⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⡇⠀⠀⠀⠈⢷⣀⣀⣀⣀⣤⣤⣤⣤⡄⢘⣛⡙⢿⣿⣯⣽⣿⣿⣿⡿⣿⣿⣿⣿⣇⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠚⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡇⠀⠀⠀⠀⠀⠁⠉⠛⠿⢿⣛⣛⣭⣶⣿⣿⣿⣦⠈⠉⣁⡀⠀⠋⠀⣿⣿⣿⣿⣉⡀⠀⠀⠀⠉⠙⠛⠿⠯⠤⣈⡛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡻⠿
⠀⠀⠀⠀⠀⠐⠻⠷⠾⠿⣿⣿⣿⣿⣿⠿⠿⢛⣻⣿⣧⣀⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠛⢁⣰⣼⣷⣿⣿⣷⡤⣤⣤⣿⣿⣿⡿⠿⠿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣴⣶⣿⣻⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⡄⠉⠋⣿⣭⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠇⠉⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⢿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣁⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣥⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣶⣾⠿⠟⠋⠁⠀⢛⣻⣿⣿⣿⣯⣛⡻⢿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⣶⠀⠰⠄⠠⠠⠖⠆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠒⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⣻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠸⢿⣶⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢁⡄⢻⡿⢻⣿⣿⣿⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⣿⣿⣯⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢘⠇⡆⡵⢸⡡⣮⡱⠆⡷⠸⣓⢹⣿⢠⢋⣝⡇⡇⡞⠗⣶⡖⡒⡒⣒⢶⢲⠒⡂⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣸⣿⣿⡿⢿⣿⣿⡇⡗⠀⣿⢰⡞⠁⠒⢲⢳⠶⠥⢭⠥⠛⠓⢰⣧⣋⣼⣘⠀⠀⣿⣿⣿⣿⡏⡇⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⢩⡍⢉⡍⡟⢸⢸⢛⡓⠇⠀⣿⢘⡲⡆⡄⠦⠦⠞⠤⠤⠦⠴⢶⢶⣾⣿⡟⡿⢢⢀⣿⣿⡏⠁⢠⠃⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣽⣿⣶⣾⣶⣧⣥⠬⢬⣬⢥⠀⢨⠟⠀⠀⠀⢃⢀⣘⣀⡘⣇⡆⢡⠇⠰⠰⠈⠆⡇⣦⠸⣽⣿⠇⠀⢸⠀⢰⠃⢰⣯⡇⣿⢰⣿⢀⣶⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡛⡛⡿⣛⠿⡿⡿⠿⢧⠄⠈⡿⢸⠀⢸⣄⠀⠀⠀⣀⣀⢀⡁⣀⠀⠀⠀⠀⠀⠉⠉⢛⠷⡞⣸⣿⣿⣄⡀⡆⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣸⠀⠀
⣤⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⣨⣥⣧⣧⣚⣏⣂⣃⣺⡐⠀⠰⡗⢘⡀⣸⠹⢬⠲⢸⠦⣶⢸⢸⡌⡯⢄⣤⡄⠄⣀⣀⢐⢛⡛⣟⡻⡟⣻⡄⠇⣠⠘⣏⠛⠛⠛⠓⠓⠻⠿⠾⢤⠇⠀⠀
⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡛⣿⣿⠋⠀⠀⣠⣧⣠⣨⣉⣉⣉⣙⣛⣒⡓⣲⣶⣶⡠⠆⠴⡤⣀⣉⣉⣬⣼⣥⣽⣵⣕⣪⢀⠈⠓⠀⠈⠳⢤⣀⡀⢀⣀⣠⠔⠋⠀⠀⠀
⣶⣦⣭⣭⣽⣿⣿⣦⣀⣦⡀⢀⣴⣆⢸⣿⣄⠀⠀⠀⠀⠀⠀⠇⠶⠈⠀⠀⣿⠉⢤⠀⡌⡍⡍⢩⢹⠙⡛⡛⣟⢻⢙⡋⡛⠙⠛⢫⠤⠀⠀⠀⠀⠀⢥⣛⣽⣿⣿⣿⣿⡞⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀
⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⣿⣿⣿⢸⣿⣿⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠲⢷⣬⠈⠀⠁⠁⠁⠘⠈⠁⠇⣄⣸⡐⠀⠀⠀⠀⠲⠀⣾⠸⠀⠆⡆⠉⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠻⣿⣧⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣥⣤⣤⡀⠀⠀⠘⣿⣁⠀⠈⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⢀⣠⣉⣉⣛⣻⡿⢿⣿⣿⣶⣏⠻⡿⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⢀⣀⣈⣤⣀⠀⠀⢠⣿⣿⣿⠟⠁⠰⠔⠈⠻⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⠻⣿⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡟⣿⣿⣿⣮⣽⣿⣿⣿⠳⢸⣿⣿⣿⠀⠀⠰⣤⣀⣀⣈⣻⣿⣿⣿⣷⣦⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣿
⠀⠀⠀⠀⢎⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠸⣷⠀⠀⠀⢴⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠙⢿⣿⣯⣽⣿⣿⣿⡿⣿⣿⣿⣿⣇⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠘⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣸⠀⠀⠀⠀⠀⠁⠉⠻⠏⠉⠉⣉⣠⣤⣶⣾⠷⣦⡈⣛⠁⠀⠈⠋⠀⣿⣿⣿⣿⣉⠀⠀⠀⠈⠉⠛⠛⠿⠯⢄⣉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿
⠀⠀⠀⠀⠀⠀⢤⣌⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠛⣡⣷⣴⣿⣿⣷⣦⣤⣤⣄⣿⣿⣿⡿⠿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠲⠮⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡉⣉⣉⣉⣥⣶⢾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣇⠈⠙⠋⣿⣭⣥⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⠏⠉⠛⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⠛⠻⠷⠿⠿⠿⢿⡿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⣛⣣⣶⣿⣿⣿⣿⣿⣿⣿⣥⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣶⡶⠛⠋⠛⣿⣿⣿⣟⣛⣛⡻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢶⠀⠠⠄⠠⠠⠖⠆⠀⠀⠀⠀⠈⠉⠉⢉⣠⣤⣴⡶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁
⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⡿⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⡄⢻⡿⢻⣿⣿⣿⠟⠿⠿⠿⠿⠿⠿⠿⢿⠿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣾⣿⣧⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠇⡆⡱⢸⡡⣮⠱⠆⡷⡸⣒⢱⣶⢠⢊⣜⡇⡇⡖⠖⣶⡒⡒⡒⣒⢶⢲⠶⡆⣸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣸⣿⣿⡿⢿⣿⣿⡇⡇⠀⣿⢢⡌⠁⠀⠒⠳⠶⢥⠭⠅⠋⠗⠰⢧⢉⣬⡈⠀⠁⣿⣿⣿⣿⡏⡆⠀⠀⠀⣀⡤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⢩⡍⠉⡍⡏⢘⢸⢛⡃⠇⠀⣿⢘⡢⡆⠄⠤⠀⠁⠰⠤⠄⠠⢠⢤⣶⣿⡟⡮⢠⢀⣿⣿⡏⠉⢁⠇⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣾⣿⣶⣾⣶⣧⣥⠬⠬⣬⢤⠀⢨⠟⠀⠀⠀⠁⠀⠘⠀⠘⡇⡆⢡⠇⠰⠰⠈⠆⡇⢦⠸⣽⣿⠇⠀⢸⠀⢰⠃⢰⣯⡇⣿⢰⣿⢀⣶⠈⣧⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡛⡛⡿⣛⠿⡿⡿⠿⢤⠀⠀⡬⢸⠀⢸⡄⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠳⡊⣸⣿⣿⣄⡀⡄⠀⢸⠀⢸⣿⡄⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢨⣥⣧⣧⣚⣏⣂⣃⣺⡐⠀⢰⠔⢈⡀⣸⠸⢬⢲⢸⢦⣶⢰⢰⡌⡣⢄⣤⡄⠄⣀⡀⢐⢛⠛⣟⡻⡟⣻⡄⠇⠤⠘⣏⠙⠛⠛⠓⠓⠛⠿⠾⢤⠇⠀⠀
⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡛⣿⣿⠃⠀⠀⢀⣄⣀⣉⣉⣉⣉⣉⣓⠒⠒⣒⣲⢶⡠⠄⣭⣅⣁⣉⣉⡈⣼⣥⣽⣵⣕⣨⢀⠈⠂⠀⠈⠳⢤⣀⡀⢀⣀⣠⠔⠋⠀⠀⠀
⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⣸⠀⢰⣴⠀⠀⠀⠀⠀⠀⠀⠇⠶⠈⠀⠀⣿⠉⢤⠀⡄⡉⡍⢩⢹⠙⠙⡛⡟⢺⢙⡋⡋⠐⠓⢣⠄⠀⠀⠀⠀⠈⢉⣉⣻⣿⣿⣿⣿⡜⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀
⣤⣴⣮⣿⣿⣿⣷⣤⣰⣄⢀⣼⣿⡆⢸⣿⣧⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⢧⣌⠈⠀⠁⠁⠁⠘⠈⠃⠃⣄⡸⡀⠀⠀⠀⠀⠒⠀⣾⠸⠀⠆⡆⠉⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠸⢿⣿⣿⣿⣿⣿⣿⣭⡙⠻⣿⣿⠻⠸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣁⣀⣀⡀⠀⠀⠈⣿⣁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠹⣿⣿⣿⣿⣿⣿⣿⣿⣦⡘⠛⢟⠳⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⣀⣈⣄⡀⠀⠀⢀⣿⣿⣿⠿⠁⠰⠔⠈⠻⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⣉⣙⣛⣛⣛⣛⣻⣿⣧⣌⢆⠈⠙⣿⣧⠀⠀⠀⠀⠀⣄⣀⣀⣤⣤⣀⣀⣀⣀⣠⣼⣿⣿⣿⣿⣯⣹⣿⣿⣿⠷⢸⣿⣿⣷⠀⠀⠐⣀⣀⣀⣈⣻⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣣⡀⠉⢿⣧⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠻⣿⣿⣿⣽⣿⣿⣿⡷⣾⣿⣿⣿⣇⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠐⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠹⠀⠀⠀⠀⠀⠉⠉⠛⠋⠀⠀⢀⣀⣠⣤⣶⣶⡄⠈⠛⠁⠀⠈⠛⠀⣿⣿⣿⣿⡋⠉⠉⠉⠉⠙⠛⠿⠿⣭⣉⡉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠿⠛⣿⣿⣶⣶⣶⣦⣠⣤⣀⣿⣿⣿⣿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠬⢭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣥⣤⣾⠿⠿⢛⣋⣭⣿⣿⣶⣤⣤⣤⣤⣤⣄⣀⣈⣛⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣛⣇⠈⠙⠛⢿⣯⣥⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⡟⠙⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⠟⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢛⣃⣤⣴⣶⣿⣿⣿⣿⣿⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢈⣭⠞⠻⠿⣿⣿⣿⣟⣛⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢶⠀⠰⠄⠰⠲⠶⠖⠀⠀⠀⠀⠀⠴⠞⠛⣁⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠰⣾⣿⣿⣿⣷⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀
⠀⣙⣛⡻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⡄⢺⡿⢻⣿⣿⣿⠟⠿⠿⠿⠿⠿⠿⣿⣿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠇⡆⡱⢸⡡⣮⠱⠆⡗⠸⣒⢰⣶⢠⢂⣝⡇⡇⡖⠖⣶⡒⡒⡲⣒⢶⢶⠶⠆⢸⣿⣿⣿⣿⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠀⠀⠀⠀⣠⣀⣴⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣸⣿⣿⡿⢿⣿⣿⡇⡇⠀⣿⢠⡈⠉⠀⠒⠳⠲⠥⠉⠁⠉⠁⠰⠧⣉⣬⡈⠀⠁⣿⣿⣿⣿⡟⡄⠀⠀⠀⣀⡤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠛⠻⠟⠿⠿⠶⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢩⡍⠉⡍⡇⢸⢸⢛⡃⠇⠀⣿⢈⠠⠆⠀⠄⠀⠀⠠⠤⠀⠠⢄⢠⣦⣾⡗⡧⢄⠀⣿⣿⡟⠉⠁⠇⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣾⣿⣶⣾⣶⣧⣥⠬⠬⣬⢤⠀⢨⠟⠀⠀⠀⠁⠀⠘⠀⠘⡇⡆⠁⠇⠰⠰⠈⠆⡇⢦⠰⣽⣿⡇⠀⢸⠀⢰⠃⢰⣯⡇⣿⢰⣷⢀⣶⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡛⡛⡿⣛⠿⡿⡿⠿⢦⠀⠀⠌⢸⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⡚⢸⣿⣿⣀⡀⡌⠀⢸⠀⢸⢿⡄⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣥⣧⣧⣚⣇⣂⣃⣺⡐⠀⠀⠄⠈⠀⢰⠸⢬⢲⢸⢢⣴⢰⢠⡌⡢⢄⣤⡄⠀⣀⡀⢀⢐⠛⣟⡻⡟⣻⡄⠇⠠⠘⣏⠙⠛⠛⠓⠓⠛⠿⠾⢤⠇⠀⠀
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡛⣿⡿⠁⠀⠀⢀⣀⣈⣉⣉⣉⣉⡛⠒⠒⠓⠲⡶⠶⡠⠄⣍⣁⣁⣉⠉⠈⢜⣥⣽⣵⣕⣨⢁⠀⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠔⠋⠀⠀⠀
⣶⣶⣤⣀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠇⠶⠈⠀⠀⣿⠁⠄⠀⡄⠉⡍⢩⢹⠉⡙⡛⡟⢺⢙⡋⡋⠐⠒⢡⠀⠀⠀⠀⠀⠉⢉⢉⣸⣿⣿⣿⣿⡜⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀
⣉⣙⣿⣿⣿⣦⣅⡠⡀⠀⢀⣴⡇⠀⣿⣷⡆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠒⢧⣌⠈⠀⠁⠁⠁⠘⠈⠃⠃⣄⡸⠀⠀⠀⠀⠀⠒⠀⡾⠸⠀⠆⡆⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢿⣿⣿⣿⣿⣿⣯⣙⠛⢶⡿⡿⡿⡄⢻⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣁⣀⣀⡀⠀⠀⠈⣿⡁⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢝⣿⣿⣿⣿⣿⣿⣷⣄⡱⠙⢟⠳⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⣀⣀⣀⠀⠀⠀⠀⣿⣿⣿⠟⠁⠰⠐⠀⠿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠙⠛⠛⠻⠿⠿⣿⣿⣿⣧⡌⢆⠁⠌⣿⣿⡆⠀⠀⠀⠀⠀⡄⣀⣀⣀⣤⣀⡀⠀⠀⠀⢸⡿⣿⣿⣿⣭⣹⣿⣿⣿⠶⢸⣿⣿⣿⠀⠀⢐⣀⣀⣀⣈⣻⣿⣿⣿⣶⣤⣄⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶
⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡣⡀⠛⣿⣿⡄⠀⠀⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡷⣾⣿⣿⣿⡇⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠢⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠛⢿⣧⠀⠀⠀⠀⠀⠉⠑⠈⠛⠋⠛⠛⣛⣿⣿⣿⣿⣿⠓⠛⠉⠀⠉⠛⠁⣿⣿⣿⣿⠛⠉⠉⠉⠉⠙⠛⠿⢿⣯⣉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠿⠟⣿⣿⣤⣶⣿⣧⣀⣀⣀⣿⣿⣿⣿⠿⠿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⠬⢭⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣬⣿⠿⠿⢛⣋⣽⣿⣷⣤⢄⣀⣀⣀⣀⣀⣀⣐⣻⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣽⣧⡉⠛⠟⣿⣫⣥⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⠛⠛⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⡾⠟⣹⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⢀⣁⣤⣶⣾⣿⣿⣿⣿⣿⣗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠛⠉⠉⠁⣫⡥⠛⠻⢿⣿⣿⣿⣟⣛⡻⠿⣿⣿⣿⣿⣿⣿⣿⣭⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢶⠀⠶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠤⠶⠞⠛⢁⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣷⣮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⢿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉
⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀
⠀⠈⠙⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⡄⢺⡿⢻⣿⣿⣿⠟⠿⠿⠿⠿⠿⠿⣿⣿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠉⠙⢛⡻⣞⣛⣛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠇⡆⡱⢸⡡⣮⠱⠆⡷⡰⣒⢰⣶⢠⢂⣜⡇⡇⡖⠖⣶⡖⡒⡲⣒⢶⢶⠤⠄⢸⣿⣿⣿⣿⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠠⠀⠘⠛⢷⣶⣾⣿⣿⣿⣿⠿⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣸⣿⣿⡿⢿⣿⣿⡇⡇⠀⣿⢰⡎⠉⠀⠒⠓⠳⠥⠍⠅⠉⠁⠰⢧⢉⣬⡈⠀⠁⣾⣿⣿⣿⡟⡄⠀⠀⠀⣀⡤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢩⡍⠉⡍⡏⢸⢸⢛⡃⠇⠀⣿⢈⠠⠄⠀⢄⠀⠀⠀⠤⠄⠠⠦⢤⣬⣶⡔⡧⢀⠀⣿⣿⡟⠉⠁⡇⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣾⣿⣶⣾⣶⣧⢥⠬⠬⣬⢥⠀⢠⠿⠀⠀⠁⠁⠀⠘⠀⠐⠇⡆⢁⠇⠰⠰⠈⠆⡇⢦⠐⣽⣿⠇⠀⢸⠀⢰⠃⢰⣏⡇⣿⢰⣷⢀⣶⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⡛⡿⣛⠿⡿⡿⠟⠦⠀⠀⡤⢸⠀⢸⡄⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠉⠋⠣⡚⢸⣿⣿⣆⡀⡈⠀⢸⠀⢸⢿⡄⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣥⣧⣧⣚⣇⣂⣃⣺⡐⠀⠀⠄⠈⠀⢰⠸⢬⢲⢸⢠⣴⢸⢠⡌⡢⢄⣤⡄⠀⣀⡀⢀⢐⠓⣟⡻⡟⣻⡄⠇⠀⠘⣏⠙⠛⠛⠓⠓⠛⠿⠾⢤⠇⠀⠀
⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡛⣿⣿⠂⠀⠀⢀⣀⣈⣉⣉⣉⣉⠓⠒⠒⠓⠶⠶⢶⡠⢄⣉⣁⣁⡉⠉⠈⢘⡥⣼⣵⣕⣨⢁⠀⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠔⠋⠀⠀⠀
⣾⣷⣦⣅⡂⢀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠇⠶⠈⠀⠀⣿⠁⠀⠀⡄⠈⡍⢩⢹⠉⡙⡙⣟⢺⢙⡃⡋⠐⠒⢠⠀⠀⠀⠀⠀⠉⢉⢉⣸⣿⣿⣿⣿⡜⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀
⣬⣽⣿⣿⣿⣦⣄⣢⣀⠀⢀⣴⡆⢀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠧⣌⠈⠀⠁⠁⠁⠘⠈⠃⠃⣄⣸⠀⠀⠀⠀⠀⠐⠀⣾⠸⠀⠆⡆⠉⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢿⣿⣿⣿⣿⣿⣯⣍⠛⠷⡿⣿⡟⡈⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣃⣀⣀⡀⠀⠀⠈⣿⡁⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⣻⣿⣿⣿⣿⣿⣿⣷⣦⡹⠙⢏⠳⣼⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⠀⠀⣀⠀⣀⠀⠀⠀⢀⣿⣿⣿⠿⠀⢰⠐⠀⠿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠉⢉⣛⣛⣛⣛⣛⣛⣿⣧⣌⢆⠑⢘⣿⣿⡄⠀⠀⠀⠀⠀⡄⢀⣀⣀⣀⣀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣭⣹⣿⣿⣿⠶⢸⣿⣿⣿⠀⠀⢘⣀⣀⣀⣈⣻⣿⣿⣿⣶⣤⣄⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣴⣤⣴⣶⣶
⠀⠀⠐⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣳⡈⢻⣿⣷⡀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣼⣃⠻⣿⣿⣿⣿⣿⣿⣿⡶⣾⣿⣿⣿⡇⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣧⠀⠀⠀⠀⠈⠉⠙⠎⠻⠟⠛⢿⣿⣿⣿⣿⣿⣿⡗⠛⠉⠀⠉⠛⠁⣿⣿⣿⣿⠛⠉⠉⠉⠉⠛⠻⠿⢿⣿⣉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⠿⠛⣯⣻⣥⣶⣿⣦⣀⣀⣀⣿⣿⣿⣿⠿⢿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⠬⣭⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢬⣤⣬⡿⠿⠟⣛⣩⣽⣿⣷⣤⢤⣀⣀⣀⣀⣀⣀⣘⣻⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣽⣧⡉⠛⠟⣿⣋⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⠛⠛⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣴⡽⢛⣹⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠈⣁⣤⣾⣿⣿⣿⣿⣿⣿⣗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠉⢈⣋⡥⠋⢻⢿⣿⣿⣿⣟⣛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣭⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⢶⠀⠶⠄⠰⠶⠶⠖⠀⠀⠀⠠⠴⠶⠟⠛⢁⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣷⣮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠉⠛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠉⠙⠛⣉⠛⢿⣿⠿⣿⡟⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⡄⢺⡿⢻⣿⣿⣿⠟⢿⢿⠿⡿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠙⠻⢶⣾⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠇⡆⡱⢸⡡⣮⠱⠆⡶⡰⣒⢰⣶⢠⢊⣜⡇⡇⡖⠖⣶⡖⡖⡲⣒⢦⢤⠤⠄⢸⣿⣿⣿⣿⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠻⠷⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣸⣿⣿⡿⢿⣿⣿⡇⡏⠁⣿⢰⡈⠉⠀⠀⠳⠳⠥⠩⠡⠉⠕⢰⣧⣉⣼⣘⠀⠁⣾⣿⣿⣿⡟⡄⠀⠀⠀⣀⡤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢩⡍⠉⡍⡏⢸⢸⢛⡃⠇⠀⣿⢀⠠⠄⠀⢀⠀⠀⠠⠤⠀⠀⢄⢠⣦⣿⡖⡷⢀⠀⣿⣿⡟⠉⠁⡇⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣾⣶⣧⣥⠬⢬⣬⢥⠀⢠⠿⠀⠀⠁⠃⠀⠘⠀⠘⠇⢆⢁⠃⠰⠰⠈⠆⡇⢤⠰⣽⣿⡇⠀⢸⠀⢰⠃⢰⣏⡇⣿⢰⣷⢀⣶⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⡛⡿⣛⠿⡿⡿⠿⢤⠀⠀⡭⢸⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⡚⢸⣿⣿⣅⠀⡈⠀⢸⠀⢸⢿⡄⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣥⣧⣧⣚⣇⣂⣃⣺⡐⠀⠀⠒⠈⠀⢰⠰⢬⢲⢰⢤⣴⢰⢠⡄⡢⢄⣤⡀⠀⣀⡀⢀⢀⠛⣟⡻⡟⣻⡄⠇⠀⠘⣏⠙⠛⠛⠓⠓⠛⠿⠾⢤⠇⠀⠀
⢒⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡛⣿⣟⠀⠀⠀⣀⣄⣈⣉⣉⣉⡙⠒⠒⢒⠷⠶⠶⢶⡀⢄⣉⣁⣁⠙⠋⠈⢸⣥⣼⣵⣕⣨⢀⠀⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠔⠃⠀⠀⠀
⢿⣿⣷⣦⣄⡢⢀⠀⠀⠀⠀⣄⠀⢹⣧⣄⠀⠀⠀⠀⠀⠀⠀⠇⠶⠈⠐⠀⣿⠉⠄⠀⡄⠉⡍⢩⢹⠉⡙⡛⣟⢺⢙⡃⡊⠐⢓⢱⠀⠀⠀⠀⠈⠉⢉⠉⣸⣿⣿⣿⣿⡜⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀
⣶⣶⣿⣿⣿⣿⣶⣵⣦⣠⣾⣿⣅⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⢇⡈⠈⠀⠀⠁⠁⠘⠈⠃⢇⣀⣸⡀⠀⡀⠀⠀⠐⠀⣾⠸⠀⠆⡆⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⣿⣿⣿⣿⣿⣷⣦⣍⠙⢿⣯⣭⣌⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣂⣀⣀⡀⠀⠀⠈⣿⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡃⠹⡽⡝⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠐⣀⠀⠀⠀⢀⣿⣿⣿⡿⠀⢰⠐⠐⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢩⣍⣩⣙⣛⣛⣻⣿⣿⣦⡜⣄⣌⣿⣷⡀⠀⠀⠀⠀⠀⡄⣠⣤⣤⣤⣤⣀⡀⠀⠀⢸⡿⣿⣿⣾⣍⣻⣿⣿⣶⠶⢸⣿⣿⣿⠀⠀⢈⣀⡀⠀⠈⢻⣿⣿⣿⣶⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⢘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⡻⣿⣷⡄⠀⠀⠀⣤⣿⣿⣿⡿⣿⣿⣿⣿⣶⣶⣾⣧⣻⣿⣿⣟⣿⣿⣿⣿⡶⣾⣿⣿⣿⡇⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠐⠚⠺⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⠟⡇⠀⠀⠀⠀⠈⠉⠛⠟⠈⠋⠙⠛⣻⣿⣿⣿⣿⣿⡿⠛⠋⠉⠙⠟⠁⣿⣿⣿⣿⠟⠋⠉⠉⠙⠛⠿⠿⣿⣟⡋⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣷⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠿⠋⣉⣳⣤⣶⣤⣄⣀⣀⣀⣿⣿⣿⣿⢿⣶⡦⠄⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⠠⢭⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢶⣶⣾⠿⠿⢛⣫⣽⣾⣿⣶⣴⣤⣤⣤⣤⣄⣀⣘⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣇⠙⠻⢟⣿⣛⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⠛⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⡟⣫⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⣀⣴⣾⣿⣿⣿⣿⣿⣷⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢁⣨⣽⠜⠛⢿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⣶⠀⠰⠄⠰⠰⠶⠖⠀⠀⠀⠀⠰⠶⠟⠛⢋⣀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣷⣍⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠙⠻⠿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋
⠀⠉⠛⠋⠙⠻⣿⣿⣿⣿⣿⠟⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀
⠀⠠⣤⣄⣶⣾⣿⣿⣶⣤⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⡄⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠛⠿⠿⠿⢿⣿⣭⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠇⡆⡱⢸⡡⣮⠱⠆⡖⡰⣒⢰⣶⢠⢊⣍⡇⡇⡞⠔⣦⡄⡄⡠⣀⢤⢤⠤⠄⢸⣿⣿⣿⣿⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣸⣿⣿⡿⢿⣿⣿⡇⡏⠁⣿⢰⡈⠙⠀⠒⢳⠷⠥⠭⠥⠻⠗⢰⣧⣋⣼⡘⡀⠃⣾⣿⣿⣿⡟⡌⠀⠀⠀⣀⡤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢩⡍⠉⡍⡗⢸⢸⢛⡃⠇⠀⣿⢐⡀⠄⡀⢀⠀⠀⠠⠤⠄⠤⢤⢴⣶⣿⡟⡯⣄⠀⣿⣿⡟⠛⠁⡇⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠾⣿⣿⣿⡇⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣾⣶⣧⣥⠬⢬⣬⢤⠀⢨⠿⠠⠀⠁⠃⠀⠘⡀⢘⢇⡆⢁⠇⠴⠰⠈⠆⡇⢦⠰⣽⣿⡇⠀⢸⠀⢰⠃⢰⣯⡇⣿⢰⣿⢀⣶⠈⣧⠀⠀
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⢤⡎⠛⠀⠀⠀⠀⠀⠀⠀⠀⢐⡛⡛⡿⣛⠿⡿⡿⠿⢥⠄⠀⡭⢸⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠛⠷⡞⢸⣿⣿⣇⠀⡌⠀⢸⠀⢸⢿⡄⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣥⣧⣧⣛⣇⣂⣃⣺⡐⠀⠠⠑⠈⠀⣰⠰⣤⢲⢰⢤⣤⢰⢠⡌⡂⢄⣄⡀⠀⢀⡀⢀⢚⠛⣟⡻⡟⣻⡄⠇⡠⠘⣏⠙⠛⠛⠓⠓⠛⠿⠾⢤⠇⠀⠀
⣀⣂⠠⢀⠀⠀⠀⠀⠀⠀⣿⣷⠀⢾⣆⠀⠀⠀⠀⠀⠀⠀⠼⣿⣿⣿⣿⡛⣿⣿⠁⠀⠀⢠⣄⣈⣉⣉⡙⠛⠚⠒⢒⠷⢶⡶⣶⣁⣈⣉⣁⠁⠙⠃⠈⢸⣥⣽⣵⣕⣪⢁⠀⠈⠀⠈⠳⢤⣀⡀⢀⣀⣠⠔⠋⠀⠀⠀
⠿⣿⣿⣦⣌⡂⢄⡀⢀⣼⣿⣿⡆⢸⣿⣷⠄⠀⠀⠀⠀⠀⠀⠇⠶⠈⠐⠀⣿⠁⠀⠀⡄⠉⡍⢩⢹⢉⡙⡛⣟⢺⢙⡃⡋⠘⢚⢡⠀⠀⠀⠁⠈⠉⢙⠛⢽⣿⣿⣿⣿⡼⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀
⣶⣶⣾⣿⣿⣿⣶⣬⣿⣿⣿⡿⡧⠘⣿⣿⣷⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠐⢧⡈⠈⠀⠁⠁⠁⠘⠈⠇⢇⣀⡸⡀⢀⡀⠀⠀⠐⠀⣾⠸⠀⠆⡆⠉⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣶⣄⡉⢻⣿⡟⢷⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⠀⠀⠀⠈⣿⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠹⢿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣡⠙⢦⠡⠙⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠐⣀⠀⠀⠀⢠⣿⣿⣿⡟⠀⢠⠐⠐⢿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣬⣭⣭⣭⣭⣭⣽⣿⣿⣷⣌⢳⡄⠛⣿⣷⡀⠀⠀⠀⠀⡄⣠⣴⣶⣶⣶⣦⣤⣀⣀⣸⡿⣿⣿⣾⣝⣻⣿⣿⣶⠦⢸⣿⣿⣯⠀⠀⢀⡀⠀⠀⠙⢻⣿⣿⣷⣦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤
⠀⠀⠠⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⣦⠈⢻⡧⠀⠀⠀⠼⢿⣿⣿⣟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⡶⣾⣿⣿⣿⡆⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠇⠀⠀⠀⠀⠀⠁⠉⠛⠁⠉⠀⠀⢀⣀⣬⣧⣌⠉⠙⠻⠋⠉⠙⠟⠀⣿⣿⣿⣿⠞⠋⠙⠋⠛⠛⠿⢿⣿⣟⡛⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢟⢿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⠿⢻⣇⣤⣤⣴⣄⣀⣀⣀⣿⣿⣿⣿⣿⣶⡦⠄⠀⠀⠀⠀⠀⠀⠉⠉⠒⠂⠬⢍⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬
⠀⠀⠀⠀⠀⠀⠠⠤⠶⠷⠶⠶⠿⠿⢛⣩⣭⣶⣿⣿⣿⣶⣶⣶⣶⣤⣤⣤⣘⣿⣿⣿⣿⣶⣶⣴⣿⣿⣿⣿⣿⣻⣏⠙⠻⠟⣿⣛⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⠻⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣏⣵⣿⣿⣿⣿⢏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠙⢀⣤⣶⣾⣿⣿⣿⣿⣷⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣀⣠⣤⡾⠚⠻⢿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⣦⠀⠐⠄⠰⠰⠶⠖⠀⠀⠀⠀⠊⠛⠛⠋⢁⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⠻⠿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀
⠀⠀⢀⣿⣯⣭⣉⣁⠀⠀⠀⠀⠀⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⡄⢺⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠇⡆⡱⢸⡡⣮⠱⠆⡗⠰⡒⢰⣶⢠⢪⣍⡇⡇⡖⠄⣤⡄⡄⡠⣀⢤⢤⠤⡄⢸⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠉⠛⠉⠉⠛⠋⠉⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣸⣿⣿⡿⢿⣿⣿⡇⡏⠉⣿⢸⡊⠉⠀⠒⠻⠳⠥⠭⠥⠛⠗⠰⣧⣛⣼⡘⠀⠃⣿⣿⣿⣿⡟⡄⠀⠀⠀⣀⡤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢩⡍⠉⡍⡗⢸⢸⢛⡃⠇⠀⣿⢘⡡⡄⡀⢄⠀⠀⠠⠠⠀⠠⢤⢤⣬⣷⡟⡶⣄⠀⣿⣿⡟⠋⠁⠇⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⣤⣴⣦⡄⠀⠀⠀⠀⠀⡈⠿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣾⣶⣧⣥⠬⢬⣬⢠⠀⢨⡿⠠⠀⠁⠁⠀⠘⠀⠘⡇⡆⢁⠇⠰⠰⠈⠆⡇⢤⠰⣽⣿⠇⠀⢸⠀⢰⠃⢰⣯⡇⣿⢰⣿⢀⣶⠈⣧⠀⠀
⠀⠀⠛⠿⠿⠃⠀⠀⠀⠀⠀⠗⣦⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⡛⡿⣛⠿⡿⡿⠿⢤⠀⢀⡯⢸⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠷⡚⢸⣿⣿⣄⠀⡌⠀⢸⠀⢸⢿⡄⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠙⠡⠀⠀⠀⠀⠀⠀⠀⠀⢨⣥⣧⣧⣚⣇⣂⣃⣺⡐⠀⢈⠗⠈⠀⣰⠰⣤⢲⢰⢤⣤⢠⢠⡄⡂⢄⣀⡀⠀⠀⠀⢀⢒⠛⣟⡻⡟⣻⡄⠇⠀⠘⣏⠙⠛⠛⠓⠓⠛⠿⠾⢤⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⢲⣿⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡛⣿⣟⠁⠀⠀⢨⣄⣈⣉⡉⡙⠛⠚⠒⠶⠷⠶⣾⣶⣁⣈⣉⡁⠁⠙⠃⠈⢼⣤⣽⣵⣕⣨⢀⠀⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠔⠋⠀⠀⠀
⣶⣶⣤⣀⠂⠀⠀⠀⢀⣼⣿⣿⡄⢰⣿⣿⡆⠀⠀⠀⠀⠀⠀⠇⠶⠈⠐⠀⣿⠉⠄⠀⡌⡉⡍⢩⢹⢉⡙⡛⣟⢺⢑⡂⡋⠈⢒⢂⠀⠈⠀⠁⠉⠉⠉⠛⢸⣿⣿⣿⣿⡜⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀
⣉⣙⣿⣿⣷⣤⣌⡲⣾⣿⡿⠿⢷⠘⣿⣿⣿⡆⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠒⢇⡈⠈⠀⠁⠁⠁⠘⠈⠇⢅⣀⣸⣀⢀⡀⠀⠀⠒⠀⣾⠸⠀⠆⡆⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⣭⣉⠙⢿⣰⡀⠐⢶⣻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⠀⠀⠀⠀⠙⣿⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢈⣿⣿⣿⣿⣿⣿⣿⣷⣄⡁⠘⢦⡀⠉⠝⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠐⡀⠀⠀⠀⢰⣿⣿⣿⡿⠀⢠⠀⠐⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠉⠛⠛⠛⠟⠛⣛⣻⣿⣿⣧⣄⠱⣄⠀⠉⢿⣿⡄⠀⠀⠀⠀⡄⠀⣀⣤⣤⣄⣀⠀⠀⠀⢸⡿⣿⣶⣮⣝⣻⣿⣷⣶⠦⢸⣿⣿⣯⠀⠀⢀⠀⠀⠀⠙⢿⣿⣿⣷⣦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤
⠀⠀⠨⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⢦⡀⠀⠈⣿⠀⠀⠀⠠⠷⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣇⣿⣿⣿⣟⣿⣿⣿⣿⡦⣾⣿⣿⣿⠂⣄⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠘⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣆⠀⠉⠀⠀⠀⠀⠀⠀⠉⠻⠇⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠙⠟⠀⣿⣿⣿⣿⠞⠛⠛⠉⠛⠻⠿⣿⣿⣟⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠿⣿⣿⣿⣵⣾⣦⣀⣀⣀⣿⣿⣿⣿⣷⣶⡦⠀⠀⠀⠀⠀⠀⠉⠉⠁⠒⠀⠬⣉⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣿⠿⠛⣛⣭⣿⣿⣿⣦⣦⣤⣤⣤⣤⣀⣀⣘⣿⣿⣿⣿⣷⣦⣾⣿⣿⣿⣿⣿⣿⣏⠙⠻⢿⣿⣛⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⠿⣋⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢀⣴⣿⣿⣿⣿⣿⣿⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⢉⣈⣤⠶⠛⠿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣡⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠰⣶⠀⠲⠄⠰⠶⠶⠖⠀⠀⠀⠀⠒⠛⠛⠉⢀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣶⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠙⠻⢿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀
⠠⠤⠾⠿⠿⣿⣿⣿⣿⣿⡯⠍⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⡄⢺⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣀⣀⣀⣀⣴⣶⣿⣿⣶⣦⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠇⡆⡱⢸⡡⣮⠱⠆⡗⠰⡒⢰⣶⢠⢢⣍⡇⡇⡔⠄⣤⡄⡄⡠⣀⢤⢬⠥⡄⣽⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⡿⠦⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣰⣿⣿⡿⢿⣿⣿⡇⡏⠉⣿⢘⠈⠉⠀⠒⠓⠷⠥⠭⠅⠛⠓⠐⢧⢛⣼⡘⠀⠀⣿⣿⣿⣿⡏⡆⠀⠀⠀⣀⡤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢩⡍⠉⡍⡇⢸⢸⢛⠓⠇⠀⣿⢐⡠⡀⡀⠀⠀⠈⠠⠤⠀⠠⢤⢤⣴⣾⡟⡯⢀⠀⣿⣿⡟⠛⢁⠃⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣿⣿⣶⣾⣶⣧⡡⠬⠬⣬⢠⠀⢨⡟⠠⠀⠁⠁⠀⠘⠀⠘⠇⡆⢁⠇⠰⠰⠈⠆⡇⢦⠸⣽⣿⠃⠀⢸⠀⢰⠃⢰⣏⡇⣿⢰⣷⢀⣶⠈⣧⠀⠀
⠀⠀⠀⠀⠬⣭⣿⣿⣿⣷⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⡚⡿⣛⠿⡿⡿⠯⠤⠀⠀⡭⢸⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠇⡊⣸⣿⣿⣄⠀⡆⠀⢸⠀⢸⢿⡄⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣥⣧⣧⣛⣇⣂⣃⣺⡀⠀⠀⠄⠈⠀⣰⠰⣤⢲⢰⢤⣤⢠⢠⡄⡂⢄⣀⡀⠀⠀⠀⢀⢒⠓⣟⡻⡟⣻⠀⠇⠀⠘⣏⠙⠛⠛⠓⠓⠛⠿⠾⢤⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡛⣿⡏⠀⠀⠀⢀⡀⣈⣉⠛⠑⠛⠚⠒⠶⠧⠶⣾⣶⣁⣈⣉⡁⠁⠙⠃⠈⢼⢤⣽⣵⣕⣨⢀⠀⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠔⠃⠀⠀⠀
⣀⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⣟⣂⠀⠀⠀⠀⠀⠀⠀⠀⠇⠖⠈⠐⠂⣿⠁⠀⠀⡄⡉⡍⢩⢹⠉⡙⡛⣟⢺⢐⡂⡊⠈⢓⢁⠀⠈⠁⠉⠉⠉⠉⠚⢸⣿⣿⣿⡿⡎⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀
⠿⣿⣷⣆⡀⠠⣀⠀⣴⣿⣿⣥⠀⣽⣷⣃⡀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠐⢃⡈⠈⠀⠁⠁⠁⠘⠨⠇⢅⣀⣸⣀⢀⠀⠀⠀⠒⠀⣾⠘⠀⠆⡆⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣾⣿⣿⣛⡛⠪⢿⣿⢋⠉⣙⣀⢹⣿⣟⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠘⡿⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⣭⣯⣽⣿⣿⣷⣄⡈⠛⠢⡀⠘⠾⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠐⡀⠀⠀⠀⢰⣿⣿⣿⡟⠀⢠⠀⠐⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠴⠷⠿⣿⣿⣿⣿⣿⣿⣶⣀⠐⡄⠀⠀⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣀⠀⠀⢸⡿⣷⣶⣮⣟⣻⣿⣷⣶⢦⢸⣿⣿⣯⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣶⣤⣀⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤
⠀⠀⠠⠖⣦⣶⣶⣿⣿⣿⣿⣷⣌⠢⠀⠀⢹⣿⣇⠀⠀⠀⠀⠰⠿⣿⣿⣿⡿⣿⣿⣿⣿⣶⣾⣧⣿⣿⣿⣟⣿⣿⣿⣿⡦⣿⣿⣿⣿⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠡⢿⣿⣿⣿⣿⣿⣿⣿⣷⣷⡄⠀⠈⠩⠀⠀⠀⠀⠀⠀⠈⠈⠻⠃⠙⠉⠙⠛⣿⣿⣿⣿⣿⣿⠿⠉⠉⠙⠟⠀⣿⣿⣿⣿⠾⠛⠛⠛⠛⠿⢿⣿⣿⡿⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠫⣛⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⣿⢿⣯⣤⣤⡀⣀⣀⠀⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠈⠉⠁⠒⠂⠤⢉⣛⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣉⣙⣻⣿⠿⢟⣻⣿⣷⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⢻⣿⣿⣿⣿⣭⣴⣿⣿⣿⣿⣿⣿⡏⠛⠿⡿⣿⣛⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠁⡴⠾⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⢀⣤⣶⣿⣿⣿⣿⣿⣶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠶⠟⠛⠋⢙⣉⠼⢿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠂⠲⣶⠀⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠤⠶⠞⠋⠁⣀⣀⣤⣿⣿⣿⣿⣿⣿⣷⣮⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠻⢿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢀⡄⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣛⣟⣿⣿⣿⣃⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠘⠇⡆⡱⢸⡡⣮⡱⠆⡷⡰⡒⢰⣶⢠⢪⣍⡇⡇⡞⠄⣦⡄⡄⡠⣚⢭⢭⠭⡌⣽⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣰⣿⣿⡿⢿⣿⣿⣇⡟⠛⣿⢸⡊⠙⠒⡳⣿⣷⡥⣭⠥⠿⠷⢰⣧⣛⣼⣘⡀⠀⣿⣿⣿⣿⡏⡆⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢩⡍⠉⡍⡟⢸⢸⢛⡛⠅⠘⣿⢚⡡⡄⡐⢶⠶⠿⠷⠶⠤⠴⢴⢾⣿⣿⡟⡿⣤⢀⣿⣿⡟⠛⢁⠃⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣿⣿⣶⣾⣶⣧⣥⢬⣬⣬⢠⠀⢨⡟⠤⠀⠁⠁⠀⣘⣀⢘⣇⣆⢡⠇⠰⠰⠈⠆⡇⣶⠸⣽⣿⠃⠀⢸⠀⢰⠃⢰⣯⡇⣿⢰⣿⢀⣷⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠁⠀⠀⠀⠀⠀⢐⡛⡛⡿⣛⠿⡿⡿⠿⢶⠄⢨⡯⢸⠀⢸⡄⠀⢀⠀⣀⠀⢀⣁⢀⢁⠀⠀⠀⠁⠙⠛⠻⠷⣞⣸⣿⣿⣥⡀⡆⠀⢸⠀⢸⣿⡄⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣥⣧⣧⣛⣇⣂⣃⣺⣐⠀⢸⡇⠈⠀⣰⠰⣤⢲⢸⢤⣤⢨⢠⡌⡋⢅⣀⡀⠀⠀⡀⢐⢛⠛⣟⡻⡟⣻⡄⠇⠠⠘⣏⠛⠛⠛⠓⠓⠛⠿⠿⢤⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡛⣿⣿⠃⠀⠀⣸⣇⢉⡙⠛⠛⠛⠻⠶⣶⠷⣶⣾⣷⣁⣈⣹⠃⠁⠙⠃⠨⣼⣤⣽⣵⣕⣪⣀⠈⠀⠀⠈⠳⢤⣀⡀⢀⣀⣠⠔⠋⠀⠀⠀
⣄⡂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠶⠈⠰⠂⣿⠁⢤⠀⡌⡍⡍⢩⢹⢉⡛⡛⣟⢺⢘⡋⡋⠙⢛⢻⠇⠉⠁⠉⠉⠙⠟⠚⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀
⣿⣿⣷⣮⡐⢄⡀⠀⢀⡀⠀⢠⣤⣆⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠛⠀⠀⠀⠀⠳⢇⡘⠈⠀⠁⠃⠁⠸⠬⠇⢅⣀⣸⣈⢈⡀⠀⠀⠲⠁⣾⠸⠀⠆⡆⠋⢂⠁⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡛⠿⢷⣾⣿⢆⢸⣿⣻⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠘⡿⠁⠀⠉⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣷⣄⠙⡞⢶⣾⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⠀⠀⠀⢐⡀⠀⠀⠀⢸⣿⣿⣿⡿⠀⢠⠄⠘⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠻⠿⢿⣿⣿⣿⣿⣿⣿⣷⣈⢯⠋⠻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣶⣶⣤⣄⡀⠀⢼⢿⣷⣶⣾⣛⣻⣿⣷⣶⣤⢸⣿⣿⣯⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣀⣠⣤⣤
⢀⢲⣾⣶⣶⣶⣾⣿⣿⣿⣿⣿⣦⣀⣻⣿⣷⡀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⡦⣿⣿⣿⣿⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢧⠀⠀⠀⠀⠀⠀⠀⠈⠉⠟⣇⠙⠉⠉⠛⢛⣿⣿⣿⣿⣿⠿⠋⠉⠹⠟⠀⣿⣿⣿⣿⠾⠛⠛⠛⠻⠿⣿⣿⣿⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⢻⣷⣶⣤⡀⣀⡀⢠⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠉⠉⠑⠒⠤⠌⣙⣛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣙
⠀⠀⠀⠀⠀⠀⠀⠙⠙⣟⣛⣿⣿⣿⠿⢟⣻⣿⣧⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠸⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡍⠛⠿⡿⣿⣛⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⣿⣿⣿⣿⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠙⠋⠉⠙⠉⠉⣡⣴⡾⢻⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠷⣠⣴⣾⣿⣿⣿⣿⣷⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠃⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠁⠴⠛⠛⠛⢛⣛⡹⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠎⢶⠄⠰⠶⠶⠶⠀⠀⠀⠠⢦⡶⠿⠛⠉⣀⣈⣼⣿⣿⣿⣿⣿⣿⣶⣬⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⠿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠠⠔⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠐⠅⠀⠀⠀⢀⣿⣿⣿⢠⡅⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠅⠀⠀⠀⠀⢨⣿⣿⣿⢸⠇⣎⡵⢸⡱⣮⡱⡆⡷⡸⣒⢰⣶⢠⢫⣍⡇⡇⡞⠝⣦⡆⡤⡻⣛⢿⢭⠭⡍⣽⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⡀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⡿⢿⣿⣿⣧⡿⢛⣿⣾⡎⠹⢶⣷⣿⣷⣥⣭⣥⣿⠷⣰⣧⣻⣼⣘⣀⠇⣿⣿⣿⣿⡏⡇⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠈⣿⣿⢩⡝⢉⡍⡟⢸⢺⢛⡛⠗⠘⣿⣚⡡⡖⡚⢿⠾⢿⠿⠿⠿⡿⢿⣿⣿⣿⡟⡿⣷⣠⣿⣿⡟⠛⢡⠇⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣠⣿⣿⣶⣾⣶⣧⣥⣬⣬⣼⢠⠄⢩⡟⠤⣁⣃⣃⣠⣘⣠⣸⣇⣇⢣⠇⠴⢰⠈⠆⡇⣶⠸⣽⣿⡇⠀⢸⠀⢰⠇⢸⣿⡇⣿⢰⣿⢀⣿⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢚⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢹⡯⠸⠀⢼⣇⢀⣠⣤⣾⣿⢽⣿⣟⣟⡀⠉⠉⠙⠛⠛⢿⣷⣟⣼⣿⣿⣷⡀⡆⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠈⣬⣥⣧⣧⣻⣏⣂⣃⣻⣘⠀⣸⡇⡈⡄⣰⠰⣬⢲⢸⣤⣤⢸⢨⡌⡋⢍⣤⡀⠀⢀⡀⢛⢛⠛⣟⡻⡟⣻⣆⠗⡦⠘⣏⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡛⣿⣿⣿⣙⣩⣿⣇⣉⣙⣛⡻⣻⣿⣿⣾⣷⣶⣾⣷⣑⣎⣽⡃⠁⠛⠃⢼⣼⣤⣽⣵⣕⣪⣀⠌⠉⠁⠈⠳⢤⣀⡀⢀⣀⣠⠖⠋⠀⠀⠀
⣿⣶⣤⡀⢄⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠶⠘⠰⠂⣿⠋⣦⢩⡍⡍⡍⢩⢹⢙⡛⡛⣟⢺⢙⡋⡛⠙⢛⢻⠿⠋⠉⠉⠉⠻⢿⢛⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣷⣶⣦⡀⠀⠀⣀⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠛⠁⠀⠀⠒⠷⢗⡘⠈⠈⠁⠣⠥⠼⠼⠧⣇⣀⣸⣘⢈⡑⠃⠀⠳⠇⣿⠸⠠⠆⡆⠏⢒⡁⡂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣯⣛⢶⣾⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢉⣑⠀⠀⠀⠀⠀⠀⢹⡿⠉⠉⠙⠋⠉⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣦⣽⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠄⠀⠀⢐⡀⠀⠀⠀⢸⣿⣿⣿⣿⠋⣤⠄⢚⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⣤⣤⣀⡀⠀⠀⠀⣿⢿⣿⣶⣾⣛⣿⣿⣷⣶⣤⣸⣿⣿⡏⠂⠀⠀⠀⠀⠈⠻⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣄⣀⣀⣤⣤
⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠠⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣿⡇⢿⣿⣿⣟⣿⣿⣿⣿⡦⣿⣿⣿⣿⠀⣀⣸⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢍⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠁⠉⠉⠻⠻⣿⡈⠿⠛⠿⢿⣿⣿⣿⣿⣿⣽⠿⠋⠉⠻⠟⠀⣿⣿⣿⣿⠾⠛⠛⠛⠿⠿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠩⠻⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣶⣿⣿⣿⣿⣿⡿⢿⣿⣤⣤⡀⢀⠀⢰⣿⣿⣿⣷⣶⣤⣄⠀⠀⠀⠀⠈⠉⠉⠒⠂⠬⠍⣙⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣙
⠀⠀⠀⠀⢀⣩⣭⣭⣿⣿⣿⣿⣿⡿⠿⣛⣿⣿⣿⣶⣤⣤⣤⣤⣄⣀⣀⣀⣀⢼⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⡉⠛⠿⣿⣿⣟⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣀⣡⣤⣶⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠃⣠⣾⣿⣿⣿⣿⣿⣷⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠃⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⠋⠉⠉⠉⠛⠋⢉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠋⢶⠔⠰⠶⠶⠶⠀⠀⠐⠴⠿⠿⠿⠟⠋⣱⣿⣿⣿⣿⣿⣿⣷⣶⣾⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠛⢿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⣿⣿⣿⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠐⠀⠀⠀⠀⠐⣿⣿⣿⢰⡍⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⢰⣿⣿⡇⢸⠇⣎⣵⢸⡡⣮⡱⡆⡷⡸⣒⢰⣶⢠⢫⣍⡇⡇⡞⠅⣦⡆⡆⡲⣛⢿⢭⠭⡍⣿⣿⣿⣿⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⣿⣿⣯⣾⣿⣿⡿⢿⣿⣿⣿⡿⢿⣿⣾⡞⠻⣶⣷⣿⣷⣥⣭⣥⣿⠷⣰⣧⣻⣼⣘⣀⠃⣿⣿⣿⣿⡇⡇⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⠀⠀⠀⠸⣿⣿⢩⡍⢉⡍⡟⢸⢺⢛⠙⠟⡘⣿⣟⡡⡞⡛⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⡟⡿⣿⣰⣿⣿⡟⠛⢱⠁⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣆⣸⣿⣿⣶⣾⣾⣧⣥⣬⣬⣼⢤⠄⣩⡟⢥⣁⣃⣃⣀⣘⣠⣸⣇⣇⢣⠇⠴⢰⠨⠆⡇⣷⠸⣽⣿⡇⠀⠸⠀⢰⠇⢸⣿⡇⣿⢰⣿⢠⣿⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠀⠀⠀⢚⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡯⠸⠠⣾⣇⢠⣤⣮⣿⣿⢿⣿⣿⣿⡀⠉⠉⠛⠛⢿⣿⣷⣟⣼⣿⣿⣶⡄⡇⠀⢸⠀⢸⣿⡆⡏⡾⣿⣸⣿⠀⣸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠨⣬⣥⣧⣧⣛⣏⣂⣃⣻⣘⠀⣸⡇⡀⠀⣴⠰⣬⢲⢸⣤⣤⢸⢨⡌⡋⢍⣄⡀⠀⢀⡔⢛⢛⠛⣟⡻⡟⣻⣤⠗⠦⠘⣏⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣦⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠅⠀⠀⠠⢿⣿⣿⣿⣿⡛⣿⣿⣿⣛⣭⣿⣇⣉⣛⣛⠻⣿⣿⣿⣶⣷⣶⣾⣷⣡⣏⣹⡃⠁⠙⠣⣼⣼⣤⣽⣵⣕⣪⣁⢌⠋⠁⠈⠳⢤⣀⡀⢀⣀⣠⠶⠋⠀⠀⠀
⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠶⠘⠲⠂⣿⠋⣦⢩⡍⡍⡍⢩⢹⢙⡛⡛⣟⢺⢙⡋⡛⠙⢟⢻⠿⠭⠉⠉⠉⠙⢿⢛⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣦⣀⣴⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠛⠁⠀⠀⠐⠷⣗⡘⠈⠈⠁⠧⠥⠼⠼⠧⣇⣀⣸⣘⢈⡐⠫⠀⠷⠇⣿⠸⠠⠆⡆⠏⢒⡁⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣝⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢉⣑⠀⠀⠀⠀⠀⠀⢹⡿⠉⠉⠉⠋⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠂⠀⠀⢐⡀⠀⠀⠀⢸⣿⣿⣿⣿⠋⣤⠄⢺⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢂⣠⣤⣤⣄⡀⠀⠀⠀⣿⢿⣿⣶⣾⣛⣿⣿⣷⣶⣤⣼⣿⣿⡏⠂⠀⠀⠀⠀⠈⠻⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣠⣤⣤⣤
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣄⠀⠀⠀⣼⣶⣿⣿⣿⣿⣿⣿⣿⣦⣤⣿⣇⣿⣿⣿⣟⣿⣿⣿⣿⣦⣿⣿⣿⣿⢀⣀⣼⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠂⠐⠛⠛⡛⢻⣿⠈⠿⠛⠿⢿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠻⠟⠀⣿⣿⣿⣿⠾⠛⠛⠛⠿⢿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠘⠚⢻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠁⠚⢴⣶⣿⣿⣿⣿⣿⠿⢿⣁⣤⣦⡀⢀⡀⢸⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠈⠉⠙⠒⠂⠬⠍⣛⡻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⣙
⠀⠀⠀⠀⠀⠀⠮⢿⣿⣿⣿⣿⣿⠿⠿⢛⣯⣿⣿⣷⣦⣤⣤⣤⣄⣀⣀⣀⠀⢾⣿⣿⣿⣿⣭⣼⣿⣿⣿⣿⣿⣿⡍⠛⠿⣿⣿⣛⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠛⣀⣤⣿⣿⣿⣿⣿⣷⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠃⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠉⠋⠙⠛⣋⣹⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⠶⠄⠰⠶⠶⠶⠀⠀⠀⠀⠾⣿⠿⠿⠛⣉⣿⣿⣿⣿⣿⣿⣷⣶⣾⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠻⢿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡄⠀⠀⠀⠀⣿⣿⣿⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠠⣿⣿⣿⢠⡄⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⢨⣿⣿⡇⢸⠇⣎⣵⢸⡡⣮⠱⡆⡷⡸⣒⢰⣶⢠⢫⣍⡇⡇⡞⠕⣮⡄⡾⡻⣛⢿⢭⠤⡄⣿⣿⣿⣿⡿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡈⠀⠀⠀⠀⢸⣿⣿⣧⣾⣿⣿⡿⢿⣿⣿⣿⠿⢿⣿⣾⡎⢹⣶⣷⣿⣷⣥⣭⣥⣿⠷⣰⣧⣻⣼⣘⣀⢀⣿⣿⣿⣿⡇⡇⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⡃⠀⠀⠀⠀⠈⣿⣿⢩⡍⢉⡍⡟⢸⢺⢛⠙⠏⡘⣷⣟⡡⡞⡛⢿⠿⢿⠿⠿⠿⡿⢿⣿⣿⣿⡟⡿⣿⣸⣿⣿⡟⠛⢹⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣆⣰⣿⣿⣶⣾⣶⣷⣥⣬⣬⣼⣤⠀⣨⡏⠥⣁⣃⣃⣀⣘⣠⣸⣇⣇⢣⠇⠶⢰⠈⠆⡇⣶⠸⣽⣿⠃⠀⡜⠀⢰⠇⢸⣿⡇⣿⢰⣿⢠⣷⠈⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢀⢚⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢾⡯⠼⠠⣿⣄⢀⣤⣦⣾⣿⢿⣿⣿⣿⡀⠉⠉⠛⠛⣿⣿⣷⣟⣽⣿⣿⣾⡄⡇⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣸⠀⠀
⣿⣷⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠌⠀⠀⠨⣬⣥⣧⣧⣻⣏⣂⣃⣻⣘⠀⣸⡇⡈⡀⣵⠰⣬⢲⢸⣤⣤⢸⢨⡌⡋⢍⣤⡀⠀⢀⡌⢛⢛⠛⣟⡻⡟⣻⡄⠷⡦⠘⣯⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⣴⣶⣿⡇⠀⠀⠀⠀⠁⠀⠀⠀⢿⣿⣿⣿⣿⡛⣿⣿⣿⣍⢩⣿⣇⣩⣛⣻⣻⣿⣿⣶⣶⣷⣶⣾⣷⣑⣎⣽⣁⠁⠹⠫⣼⣼⣴⣽⣵⣕⣪⣀⡜⠉⠁⠈⠳⢤⣀⡀⢀⣀⣠⠖⠋⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠶⠘⠲⠂⣿⠋⣦⢩⡍⡍⡍⢩⢹⢙⡛⡛⣿⢺⢙⡋⡛⠙⢟⢻⠿⠋⠉⠉⠉⠝⢿⢛⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠛⠁⠀⠀⠐⠷⢗⠘⠈⠈⠁⠧⠥⠼⠼⠧⣇⣀⣸⣘⢈⡐⡉⠀⠷⠇⣿⠸⠀⠆⡆⠏⢒⡁⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢉⣑⠀⠀⠀⠀⠀⠀⢹⡟⠈⠉⠙⠋⠁⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⢐⡀⠀⠀⠀⢸⣿⣿⣿⣟⠏⣤⠄⢺⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⣄⣀⣀⡀⠀⠀⠀⣿⢿⣿⣶⣾⣛⣿⣿⣷⣶⡄⣾⣿⣿⡏⠀⠀⠄⠀⠀⠉⠻⣿⣿⣷⣦⣄⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤
⣈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⠀⠠⠀⠀⠀⠀⣼⣶⣿⣿⣿⣿⣿⣿⣶⣄⣀⡿⠇⢿⣿⣿⣟⣿⣿⣿⣷⣦⣿⣿⣿⣿⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠉⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠋⠻⠛⢿⡿⠘⠿⠿⢿⣿⣿⣿⣷⣾⣭⣿⠿⠉⠉⠻⠟⢸⣿⣿⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⣿⠟⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠈⠂⣶⣶⣿⣿⣿⣿⠿⣿⣿⣿⣤⣴⣄⣀⣀⢸⣿⣿⣿⣷⣶⣦⣄⠀⠀⠀⠀⠀⠉⠉⠑⠒⠢⠬⣙⣛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⣹
⠀⠀⠀⠀⠀⠀⠉⠙⢋⠛⠛⠛⠛⠛⠛⣛⣩⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣀⣠⣿⣿⣿⣿⣯⣥⣾⣿⣿⣿⣿⣿⣿⡏⠛⠿⣿⣿⣛⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⣿⣿⣿⣿⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⢀⣤⣤⣿⣿⣿⣿⣿⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠁⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡟⣛⣿⣿⣷⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣋⣰⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠂⠲⣶⠀⠰⠄⠠⠰⠶⠆⠀⠀⠀⠀⠀⠀⠸⠿⢟⣋⣴⣿⣿⣿⣿⣿⣿⣿⣶⣯⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢘⢹⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣨⡈⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠁⠀⠀⠀⠀⣿⣿⣿⢠⡌⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠸⢧⠀⠀⠀⠀⢰⣿⣿⡏⢸⠇⣎⡵⢸⡡⣮⡱⡆⡷⡸⣒⢰⣦⢠⢫⣍⡇⡇⡞⠝⣮⡍⡬⡻⣛⢯⢭⠭⠍⣿⣿⣿⣿⡟⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢠⠈⠀⠀⠀⠀⢸⣿⣿⣯⣾⣿⣿⡿⢿⣿⣿⣿⠟⢻⣿⣾⡞⠛⠶⣷⣿⣷⣥⣭⣥⣿⠷⣰⣧⣻⣼⣘⣀⢠⣿⣿⣿⣿⡇⠇⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿⣿⢩⡍⢉⡍⡟⢸⢺⢛⠙⠏⠘⣷⣟⡡⡞⡛⢿⠿⢿⠿⠿⠿⡿⢿⣿⣿⣿⡟⡿⣿⢸⣿⣿⡟⠛⢸⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⣆⣰⣿⣿⣶⣾⣶⣧⣥⣬⣬⣬⢤⠀⢨⡏⠅⣀⣃⣃⣀⣘⣠⣸⣇⣇⢣⠇⠴⢰⠈⠆⡇⣷⠸⣽⣿⠃⠀⡜⠀⢰⠇⢸⣿⡇⣿⢰⣿⢀⣷⠈⣧⠀⠀
⣴⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢐⠀⠀⢀⢚⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡏⠬⠠⣿⡄⢠⣠⣤⣾⣯⢽⣽⣟⣭⡀⠉⠙⠙⠙⢿⣿⣷⣟⣽⣿⣿⣴⣄⡇⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣸⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠈⣬⣥⣧⣧⣛⣏⣂⣃⣻⣘⠀⣸⡇⡀⠀⣴⠰⣤⢲⢸⣤⣤⢸⢨⡌⡋⢅⣀⡀⠀⢀⡄⢛⢛⠻⣟⡻⡟⣻⠈⠷⠦⠘⣯⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⣿⡛⣿⣿⣿⣏⢩⣻⣇⣩⣛⡛⠛⣿⣿⣾⣶⣷⣶⣾⣷⣑⣎⣙⡃⠁⠙⠫⣼⣼⣼⣽⣵⣕⣪⣀⡌⠉⠁⠈⠳⢤⣀⣀⢀⣀⣠⠖⠋⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠶⠈⠲⠂⣿⠋⣦⢠⡍⡍⡍⢩⢹⢛⡛⡛⣿⢺⢙⡋⡋⠙⢟⢻⠿⠉⠉⠉⠉⠛⢿⢛⣿⣿⣿⣿⣯⡟⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠛⠁⠀⠀⠐⠷⢓⠘⠈⠈⠁⠣⠅⠼⠼⠧⣇⣀⣸⣘⢈⡐⠃⠀⠷⠇⣿⠸⠀⠆⡆⠏⢲⡁⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢉⣑⠀⠀⠀⠀⠀⠀⢹⡟⠀⠉⠉⠉⠁⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠄⠀⠀⢐⡀⠀⠀⠀⢸⣿⣿⣿⣿⠋⣤⠆⢺⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢪⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⣄⣀⣀⡀⠀⠀⠀⣿⢿⣷⣶⣾⣛⣿⣿⣷⣶⡄⣼⣿⣿⡏⠀⠀⠄⠀⠀⠉⠻⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤
⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⣼⣷⣿⣿⣿⣿⣿⣿⣶⣤⣀⣿⠗⢿⣿⣿⣟⣿⣿⣿⣷⣦⣿⣿⣿⣿⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣈⠀⠐⠋⠻⠛⣿⡿⠙⠿⠿⢿⣿⣿⣿⣿⣾⣿⣿⠿⠉⠉⠻⠟⢸⣿⣿⣿⣿⠞⠛⠛⠛⠿⢿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠿⢿⡿⠿⣻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠚⠁⣶⣶⣶⣿⣿⣿⣿⡿⢿⣿⣤⣤⣀⣀⣀⢸⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠤⠍⣛⣻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣿⣿⣿⣿⣿⣭⣼⣿⣿⣿⣿⣿⣿⡍⠛⠿⣿⣿⣛⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⣠⣶⣿⣿⣿⣿⣿⣿⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠃⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣾⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣉⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⠰⠄⠠⠰⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣐⣿⣿⣿⣿⣿⣿⣿⣶⣝⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠛⠿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠄⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠂⠀⠀⠀⠠⣿⣿⣿⢠⡌⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢞⠀⠀⠀⠀⢸⣿⣿⡏⢸⠇⣎⣵⢸⡡⣮⠱⡆⡷⡸⣒⢰⣦⢠⢫⣍⡇⡇⡟⠝⣿⡏⡭⡻⣛⢯⢽⠭⠉⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡰⡈⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⡿⢿⣿⣿⣿⡟⢻⡿⣾⡚⠹⣶⣷⣿⣷⣥⣭⣥⣿⠷⣰⣧⣻⣼⣘⣐⢸⣿⣿⣿⣿⢧⠁⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡁⠀⠀⠀⠀⠈⣿⣿⢩⡍⢉⡍⡟⢸⢺⢛⠙⠇⡘⣷⣟⡡⡞⡛⢿⠿⢿⠿⠿⠿⡿⣿⣿⣿⣿⡟⡿⣿⣸⣿⣿⡟⠛⢸⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣰⣿⣿⣶⣾⣶⣧⣥⣬⣬⣬⣤⠀⣩⡇⡅⣀⣃⣃⣀⣘⣠⣸⣇⣇⢣⠇⠶⢰⠈⠆⡇⣦⠸⣽⣿⠁⠀⡏⠀⢰⠇⢸⣿⡇⣿⢰⣿⢠⣿⠈⣧⠀⠀
⢀⣾⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⢚⡛⡛⡿⣛⠿⡿⡿⠿⢿⠿⢿⡏⠍⠀⣿⣄⣀⣠⣦⣾⣿⢿⣿⣟⣝⡁⠉⠉⠛⠛⢛⣿⣷⣟⣽⣿⣿⣤⢠⡃⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣸⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠈⣬⣥⣧⣧⣻⣏⣂⣃⣻⣘⠀⣸⡇⡀⠀⣵⠰⣬⢲⢸⣤⣤⢸⢨⡌⡋⢅⣀⡀⠀⠀⡘⢛⢛⢻⣟⡻⡟⣻⠈⠷⠦⠘⣯⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡛⣿⣿⣿⠉⣉⣿⣇⣩⣛⡛⢻⣿⣿⣿⣾⣷⣶⣾⣷⣁⣎⣝⡃⠃⠛⠣⣼⣼⣼⣽⣵⣕⣪⣀⡌⠉⠁⠈⠳⢤⣀⣀⢀⣀⣠⠖⠋⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠶⠘⠲⠂⣿⠋⣦⠩⡍⡍⡍⢫⢹⢙⡛⡛⣟⢺⢙⡋⡛⠙⢟⢻⠿⠋⠉⠉⠙⠛⠿⢻⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠛⠁⠀⠀⠐⢷⣓⠘⠈⠈⠁⠧⠥⠼⠼⠧⣇⣀⣸⣘⢈⡁⠟⠀⢷⠇⣿⠸⠀⠆⡆⠏⢒⡅⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢉⣐⠀⠀⠀⠀⠀⠀⢻⡟⠈⠉⠙⠉⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣯⠀⠀⠀⢒⡀⠀⠀⠀⢸⣿⣿⣿⡟⠉⣤⠄⢺⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⢀⠀⢀⣠⣤⣄⣀⠀⠀⠀⠀⣿⢿⣿⣶⣾⣛⣿⣿⣶⣶⡄⣾⣿⣿⡏⠀⠀⠄⠀⠀⠙⢿⣿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣄⣠⣤⣤⣤
⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀⡿⠗⢿⣿⣿⣟⣿⣿⣿⣷⣦⣿⣿⣿⣿⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠐⠛⢿⠻⣿⡷⠹⠟⠿⢿⣿⣿⣿⣷⣮⣭⡻⠟⠉⠉⠻⠟⢸⣿⣿⣿⣿⠾⠻⠿⠿⠿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠙⠙⠍⢶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣤⣤⣄⣀⣀⢸⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠉⠉⠓⠒⠢⠬⢉⣛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠙⠿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣿⣿⣿⣿⣿⣯⣵⣿⣿⣿⣿⣿⣿⣏⠛⠿⣿⣿⣛⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠙⠀⣀⣬⣽⣿⣿⣿⣷⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠃⠀⠈⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣻⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠀⢰⠄⠠⠠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣶⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⢿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢠⣿⣿⡿⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠁⠀⠀⠀⢸⣿⣿⣿⢠⡌⢻⡿⢻⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⢸⣿⣿⡇⢸⠇⣎⡵⢸⡡⣮⠱⡆⡷⡸⣒⢰⣦⢠⢫⣍⡇⡇⡞⠝⣯⡏⡽⡻⣛⢿⢯⠭⢉⣿⣿⣿⣿⡟⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢸⣿⣿⣧⣾⣿⣿⡿⢿⣿⣿⣿⡟⢻⡿⣾⡞⠻⣲⣷⣿⣷⣥⣭⣥⣿⠷⣰⣧⣻⣼⣘⣀⢸⣿⣿⣿⣿⢧⠁⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢸⠁⠀⠀⠀⠀⢸⣿⣿⢩⡍⢉⡍⡟⢸⢺⢛⠙⠗⠘⡇⣟⡡⡞⡛⢿⠿⢿⠿⠿⠿⡿⢿⣿⣿⣿⡟⡿⣿⣸⣿⣿⡟⠛⢸⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡆⣸⣿⣿⣶⣾⣶⣧⣥⣬⣬⣬⣤⠀⣩⡇⠅⢀⣃⣃⣀⣘⣠⣸⣇⣇⢣⠇⠶⢰⠈⠆⡇⣶⠸⣽⣿⠁⠀⡇⠀⢰⠇⢸⣿⡇⣿⢰⣿⢀⣷⠈⣧⠀⠀
⠀⣀⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢀⢚⡛⡛⡿⣛⠿⡿⡿⠿⢿⠟⢿⡇⠇⠀⣿⣄⣠⣠⣴⣺⣿⢝⣷⣶⣝⠀⠉⠉⠉⠛⢛⣿⣷⣟⣽⣿⣿⣤⢠⠇⠀⢸⠀⢸⣿⡆⡇⡾⣿⣸⣿⠀⣸⠀⠀
⣤⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣬⣥⣧⣧⣛⣏⣂⣃⣺⣘⠀⣸⡇⡀⠀⣵⠰⣬⢲⢸⣤⣤⢸⢨⡌⡋⢅⣄⡀⠀⢀⡀⢙⢛⢻⣟⡻⡟⣻⠈⠗⠤⠘⣯⠛⠛⠛⠓⠓⠻⠿⠿⣤⠏⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡛⣿⣿⣋⣍⣗⣿⣇⣝⣛⠛⡻⡿⣿⣶⣶⣷⣶⣾⣷⣑⣎⣙⠃⠃⠛⠣⣼⣼⣼⣽⣵⣕⣪⣀⡌⠉⠁⠈⠳⢤⣀⡀⢀⣀⣠⠖⠋⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠶⠘⠲⠂⣿⠉⣤⠨⡍⡍⡍⢫⢹⢙⡛⡛⣟⢺⢙⡋⡛⠙⢟⢻⠿⠉⠉⠉⠙⠹⢿⢻⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠛⠁⠀⠀⠐⠷⣓⠘⠈⠈⠡⠧⠥⠼⠬⠧⣇⣀⣸⣘⢈⠐⠓⠀⠷⠃⣿⠸⠀⠆⡆⠏⢰⡁⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡐⠀⠀⠀⠀⠀⠀⢻⡟⠈⠉⠙⠉⠁⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠽⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⡀⠀⠀⢂⡀⠀⠀⠀⣸⣿⣿⣿⡏⠉⣄⠄⢺⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠚⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢠⠀⢀⣠⣄⣀⣀⠀⠀⠀⠀⣿⣿⣿⣶⣿⣛⣿⣿⣶⣶⡄⣿⣿⣿⡏⠀⠀⠄⠀⠀⠙⢿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣄⣀⣠⣤⣤
⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⢠⣾⣾⣿⣿⣿⣿⣿⣷⣦⣄⣠⡿⠜⣿⣿⣿⣟⣿⣿⣿⣷⣦⣿⣿⣿⣷⢀⣀⣼⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠙⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⡸⢿⡿⣿⢿⠇⠻⠿⠿⣿⣿⣿⣿⣿⣾⣯⡿⠟⠉⠉⠻⠟⢸⣿⣿⣿⣿⡾⠿⠿⠿⠿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠉⠢⠐⠲⣤⣴⣶⣿⣿⣿⣿⣿⣿⣧⣶⣷⣄⣀⡀⢸⣿⣿⣿⣷⣤⣤⡀⠀⠀⠀⠈⠉⠙⠓⠒⠤⠌⣙⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣝⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⡿⣶⣶⣦⣤⣤⣠⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣟⠻⢿⢿⣿⣛⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⣙⣤⣤⣾⣿⣿⣿⣿⣷⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠃⠀⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠲⣶⠀⠰⠄⠠⠠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣨⣿⣿⣿⣿⣿⣿⣶⣮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠉⠉⠉⠹⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠘⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⠈⢹⣿⣿⣿⣿⠿⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣀⠀⢠⣼⣿⡿⠟⠀⠀⠀⠀⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣶⣾⣿⡟⠃⠀⠀⠀⠀⠀⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣆⣀⠀⠀⠀⠈⣿⣿⣿⣿⣷⡀⠀⠀⢀⣰⣿⣿⠏⠁⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣶⠒⢲⣶⣿⣿⣿⣿⣿⣷⣦⣤⠘⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠉⠉⠹⠿⠏⠁⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢃⣀⣿⣧⣼⣟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⢀⣾⣿⣿⣿⣿⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣾⣿⣷⣆⣀⣀⣀⣀⣀⣀⣀⣰⣀⣀⣀⣀⣀⣀⣶⣀⣀⣀⣀⣀⣀⣀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡟⢃⣤⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠟⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠛⠻⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢻⣿⡟⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⢡⣼⣿⠉⠀⠀⠀⡄⠀⠀⠉⢻⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣷⣤⣤⣤⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣛⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠃⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠁⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⠉⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣶⣶⣶⣶⣿⣿⣿⡟⠛⠛⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⠉⠁⠀⠀⠀⠀⠉⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠛⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣤⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣧⣤⣤⣠⣤⣤⣿⣿⣿⣿⣤⣤⣄⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣄⣤⣼⣧⣤⣤⣿⣿⣿⣤⣤⣤⣤⣤⣤⣀⣀⣀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⡿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠘⠛⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣤⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⠈⢹⣿⣿⣿⣿⠿⠀⠀⠀⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣀⠀⢠⣼⣿⡿⠟⠀⠀⠀⠀⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣶⣾⣿⡟⠃⠀⠀⠀⠀⣤⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣀⠀⢀⣈⣿⣿⣿⣿⣿⣇⡀⠀⢰⣾⣿⠿⠇⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡟⠛⠛⠀⣾⣿⣿⣿⣿⣿⣿⡟⢳⣶⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠿⠿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠉⠀⠀⠈⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣃⣀⣤⣤⣤⣤⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣹⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣿⣷⣶⣀⣀⣀⣀⣀⣰⣶⣶⣶⣶⣆⣶⣶⣶⣶⣀⣀⣀⣀⣀⣀⣰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡟⢃⣤⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠻⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡟⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⢡⣼⣿⠋⠀⠀⠀⡄⠀⠀⠉⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⣶⣤⣤⣭⠙⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢀⡀⠀⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠁⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⢠⣿⠀⠀⠀⣭⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣶⣿⣿⣿⡏⠙⠉⠀⢸⠓⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⠿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿
⣿⣿⣿⠛⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠰⣿⣿⣿⡀⠀⣀⣀⣀⣀⠀⡀⢀⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣀⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⢻⣿⣿⣿⣿⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⠟⠛⠃⠀⠀⠀⠀⠀⠸⠿⠀⠀⠘⠛⠛⠻⢿⡿⠀⠀⠘⠛⠛⠃⠘⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡍⠁⠀⠀⠀⠀⠀⠀⢠⡄⠀⠒⢢⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠹⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⠈⢹⣿⣿⣿⣿⠿⠀⠀⠀⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣀⢀⣠⣼⣿⡿⠟⠀⠀⠀⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⡄⠀⠀⠉⢻⣿⣿⣾⣿⣿⡟⠃⠀⠀⠀⢠⣴⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣶⠶⣶⣾⣿⣿⣿⣿⣿⣷⣆⣀⠸⠿⠿⠿⠁⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠋⠉⢠⣿⣿⠛⠛⢻⣿⡟⠋⠙⣿⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠏⠿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢃⣀⣤⣤⣤⣤⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⣰⣾⣿⣿⣿⣿⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣰⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣆⣶⣶⣶⣶⣀⣀⣀⣀⣀⣀⣰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡟⢃⣤⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠻⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡟⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⢡⣼⣿⠉⠀⠀⠀⡄⠀⠀⠉⢻⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣷⣶⣤⣤⠉⠉⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠃⠀⠀⢸⣿⠀⠀⠀⣿⡀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⢠⡀⠀⢻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⠛⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣶⣿⣿⣿⡏⠉⠉⠀⠘⠃⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣤⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠈⣿⣿⣿⡀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣀⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣄⣤⣼⣧⣤⣤⣿⣿⣿⣿⣿⣤⣤⣤⣤⣛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠛⠃⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠃⠀⣸⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠋⠁⠀⢠⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠉⠀⠀⠀⣾⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⡀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⡟⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠏⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣤⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠘⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⠈⢹⣿⣿⣿⣿⠿⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠿⣿⣿⣿⣀⢀⣸⣿⣿⡿⠟⠀⠀⠀⢀⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣾⣶⣦⣤⠀⠉⢻⣿⣿⣾⣿⣿⣿⡇⠀⠀⢠⣤⣾⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠈⠹⠉⠏⠁⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢰⣿⠛⠉⠙⠛⠛⠋⠉⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠏⠿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣀⣠⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣰⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣀⣀⣀⣀⣀⣀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣤⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠻⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡟⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⢡⣼⣿⠋⠀⠀⠀⡄⠀⠀⠉⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣷⣤⣤⣤⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠃⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⢀⠀⠀⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣶⣶⣶⣶⣿⣿⣿⡏⠙⠛⠀⢸⠂⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⣶⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣧⣤⣤⣠⣤⣤⣿⣿⣿⣿⣤⣤⣀⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⣿⣿⣧⣤⣀⣀⡀⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⣿⣿⣿⡟⠛⠛⠛⣿⣿⣿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣭⡝⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠛⠀⠀⠛⠻⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⠛⠛⠛⠛⠻⠿⠿⣿⣿⣿⡿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡍⠉⠀⠀⠀⠀⠀⠀⠉⠛⣿⠛⢻⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣾⣿⣿⣿⣿⡏⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⣀⣶⡆⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⠈⢹⣿⣿⣿⣿⠿⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠘⠿⣿⣿⣿⣀⣠⣼⣿⣿⡿⠟⠀⠀⠀⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣦⣤⣭⢻⣿⣿⣿⣿⣿⣿⣧⡄⠀⢠⣴⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠿⠿⠿⢉⣿⣿⠿⣿⣿⣿⣿⠿⢷⣿⡈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠉⠉⠉⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠏⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣃⣀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⠀⣀⠀⣀⣀⡀⠀⠀⢀⣀⡀⣀⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣿⣿⣶⣿⣶⣿⣿⣷⣶⣶⣾⣿⣷⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⢃⣼⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠻⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⢰⣾⣿⠋⠀⠀⠀⡄⠀⠀⠉⢻⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⣭⡅⠀⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠃⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡄⢀⠀⠛⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⠛⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣶⣶⣶⣶⣿⣿⣿⡟⠛⣿⠃⢸⣶⠀⠈⢹⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⡿⠿⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⣿⣆⡀⠀⠀⠀⠰⣶⣿⣿⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣤⣤⣠⣤⣤⣿⣿⣿⣿⣤⣤⣄⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⣿⣿⣿⣿⣤⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⡟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠙⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣇⠀⠀⠻⠿⠿⠟⠃⠀⠿⣿⣿⣿⡟⠀⠀⠛⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠉⠉⠉⠉⠉⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⣰⣶⡆⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⠈⢹⣿⣿⣿⣿⠿⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠘⠿⣿⣿⣿⣀⣤⣼⣿⣿⡿⠟⠀⠀⢀⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣶⣽⢻⣿⣿⣿⣿⣿⣿⣷⡆⠀⢰⣾⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠉⢹⣿⣿⠿⣿⣿⣿⠿⠏⢹⣿⡆⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠙⠋⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢛⣃⣀⣀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣶⣾⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣀⣀⣀⣀⣀⣀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣼⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠿⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⢠⣼⣿⠋⠀⠀⠀⡄⠀⠀⠉⢻⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⠉⠁⠈⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠛⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣄⠀⠘⠛⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣶⣿⣿⣿⣿⡟⣿⠋⠐⣦⠀⠘⢻⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⣿⠏⠁⠀⠀⠀⠈⠀⢀⣸⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⣶⡀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⣤⣤⣠⣤⣤⣿⣿⣿⣿⣤⣤⣀⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⣿⣿⣿⣿⣤⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠉⠉⠉⠛⠛⢻⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⡿⠛⠀⠀⠻⣿⣿⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠘⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡟⠛⠛⠀⠀⠀⠀⠀⠘⠛⣿⣿⡇⠀⣾⣿⣿⣿⡟⠃⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠘⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⢶⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⠈⢹⣿⣿⣿⣿⠿⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣼⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⡟⠀⢀⣠⣼⣿⡿⠇⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠸⢿⠿⠿⣿⣿⠿⠉⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢛⣃⣀⣀⣀⣠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⣤⣀⣀⣤⣄⣀⣀⣠⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣶⣾⣿⣿⣿⣿⣿⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣼⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠿⢿⣿⣿⠟⠛⠛⠻⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡟⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⢠⣼⣿⠋⠀⠀⠀⡄⠀⠀⠉⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠃⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠃⠀⠀⢸⣿⠀⠀⠀⣿⡄⠀⠀⠠⣿⣿⣿⣤⣤⣤⣀⠛⠻⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣶⣿⣿⣭⣿⣿⣿⣿⠀⢠⠀⠙⢻⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⣶⣿⡇⠀⠁⠀⠈⠉⠀⠸⣿⣿⣿⣿
⣿⣿⣿⠛⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣀⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠙⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣧⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠛⠻⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠿⠃⢸⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣀⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠙⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠀⠀⢸⣿⣿⣿⣿⠛⠀⠈⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠏⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠏⠁⠀⢰⣾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣉⣉⣹⣿⣿⣿⣿⣿⣿⣆⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣤⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠘⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⢀⢸⣏⡁⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⢸⣿⣿⣿⣿⣿⠿⠀⠀⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣼⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⡟⣀⣠⣤⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠸⠿⠿⠿⣿⣿⠿⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⢃⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣀⣀⣤⣀⣀⣀⣠⣄⣀⣤⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣶⣶⣾⣿⣿⣿⣿⣏⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⣀⣀⣀⣀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣼⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠛⠻⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢻⣿⡟⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⢠⣼⣿⠉⠀⠀⠀⡄⠀⠀⠉⢻⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠃⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠛⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠃⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣤⣤⣤⣀⠛⠻⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣶⣶⣶⣶⣿⣿⣭⣿⣿⣿⣿⡄⢠⠀⢹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠉⠀⠈⠿⠀⠸⣿⣿⣿⣿
⣿⣿⣿⠛⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣤⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⣾⣿⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣧⣤⣤⣠⣤⣤⣿⣿⣿⣿⣤⣄⣀⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣄⣤⣼⣧⣤⣤⣿⣿⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⠿⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠒⠀⠈⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⢀⣿⣿⣿⣿⡿⠁⠉⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣉⣉⣹⣿⣿⣿⣿⣿⣿⣆⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠘⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⢀⢸⡿⠇⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⢸⣿⣿⣿⣿⣿⠿⠀⠀⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣤⣼⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⡟⣀⣠⣤⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣭⣿⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠸⠿⠿⠿⣿⣿⠿⠁⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⢃⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣄⣀⣀⣤⣀⣀⣀⣀⣄⣀⣤⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣶⣶⣾⣿⣿⣿⣿⣏⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⣀⣀⣀⣀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣼⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠟⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠛⠻⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢻⣿⡟⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⢠⣼⣿⠉⠀⠀⠀⡄⠀⠀⠉⢻⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠃⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠠⣿⣿⣿⣤⣄⣀⡀⠛⠻⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⠛⠀⠀⠀⣿⣶⣶⣶⣶⣿⣿⣭⣿⡟⣿⣷⠀⢠⠀⢹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⠿⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠁⠀⠸⠉⠀⠸⣿⣿⣿⣿
⣿⣿⣿⠛⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣧⣤⣤⣠⣤⣤⣿⣿⣿⣿⣤⣄⣀⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣄⣤⣼⣧⣤⣤⣿⣿⣤⡀⠀⠀⠀⠀⠀⠛⠀⠸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⡀⠀⠀⠀⣀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⡟⠛⠛⠛⠋⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠉⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⡏⠀⠉⣿⣿⣿⣿⣿⠏⠿⠿⠿⠿⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣉⣉⣉⣉⣿⣿⣿⣿⣿⣁⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⡞⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠿⠿⠏⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⠰⣿⡇⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⢠⣟⡃⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⠈⢹⣿⣿⣿⣿⠿⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣄⣘⣿⣿⣿⣿⣀⣼⣿⣿⣿⣿⡟⠀⢀⣀⣤⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠉⠉⢸⣿⡿⠿⣿⣿⠿⠉⠁⠈⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡄⠀⠀⠈⠉⠁⠀⠉⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⢛⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣶⣀⣀⣀⣀⣀⣀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣼⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠛⠻⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡟⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠉⢠⣼⣿⠉⠀⠀⠀⡄⠀⠀⠉⢻⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⣿⠙⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠃⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠠⣿⣿⣿⣧⣤⣤⠀⠛⠛⠿⠿⢿⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠃⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣭⠀⠀⠀⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⠂⠀⣤⠀⠀⠈⣿⣿⣿⣿
⣿⣿⣿⣿⠇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⠉⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⠿⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⣿⡇⠀⠉⠀⠀⠈⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⠛⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⣸⣿⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣧⣤⣤⣠⣤⣤⣿⣿⣿⣿⣤⣤⣀⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠇⠛⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣷⡞⣿⣿⣿⣿⣿⡅⠀⠉⣿⣿⣿⣿⡟⣿⣿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⣿⣿⣿⣿⣿⠃⠀⠉⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠹⢿⣿⣿⣿⠇⠹⣿⣿⣿⣿⡇⠀⠀⠈⠹⠿⠉⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣶⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣉⣉⣉⣉⣉⣀⣀⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠛⠛⠛⠛⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠿⠿⠏⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣤⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠘⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⠈⢹⣿⣿⣿⣿⠿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⣘⣿⣿⣿⣿⣀⣼⣿⣿⣿⣿⡟⠀⢀⣀⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣿⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠏⠉⠉⢸⣿⡿⠿⣿⣿⠿⠏⠁⠀⣿⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠉⠁⠀⠉⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢛⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⡠⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣰⣶⣾⣿⣿⣿⣿⣏⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣀⣀⣀⣀⣀⣀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣼⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠟⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠛⠻⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡟⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⢠⣼⣿⠋⠀⠀⠀⡄⠀⠀⠉⢻⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠃⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠃⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣤⣤⣤⡀⠛⠻⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⠉⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣶⣶⣶⢲⣿⣿⣿⣿⣿⣿⡇⢠⡄⠀⠙⢻⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠈⠉⠀⠀⠈⠿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣤⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣄⣀⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠉⠉⠉⠉⠙⢻⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣶⣤⣦⣤⣤⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⡀⠀⠀⠀⢠⣼⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⣿⡿⠛⠛⠿⢿⣿⣿⣿⣿⣿⠿⠿⡿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⡟⠛⠋⠉⠁⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⢸⣿⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⠀⠘⢻⣿⣿⣿⣯⠀⠀⠁⠀⠉⠉⠉⠉⠉⢩⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⠁⠀⠀⠀⠈⠹⠿⠿⠏⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣉⡉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠒⠛⠛⠛⠛⠛⠛⠋⠛⠛⠓⠒⢺⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣤⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠘⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⠈⢹⣿⣿⣿⣿⠿⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⣘⠿⣿⣿⣿⣀⣼⣿⣿⣿⣿⡟⠀⢀⣀⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠏⠉⠉⣿⣿⠿⠿⣿⣿⠿⠉⠉⠹⠿⠆⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠀⠉⠉⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢛⣻⣀⣀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣰⣾⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣀⡀⢀⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣼⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠟⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠛⠻⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⢠⣼⣿⠉⠀⠀⠀⡄⠀⠀⠉⢻⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠛⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠃⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣤⠀⠘⠛⠿⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣶⣶⣶⣶⣿⣿⣿⡟⢻⠛⠀⢠⡄⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣤⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠸⢿⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣧⣤⣤⣠⣤⣤⣿⣿⣿⣿⣤⣤⣀⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣄⣤⣼⣧⣤⣤⣿⣿⣿⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠛⣿⣿⣿⣿⡟⠛⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡆⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠛⠿⠿⢿⣿⡿⢿⣿⣿⣿⣿⠿⠃⢀⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠛⠛⠛⠛⠿⠟⠃⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⡟⠉⠙⠛⠛⠉⠉⠙⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠈⠉⠁⠘⣿⣿⣿⣿⠀⢠⣼⣿⣿⣿⣿⠉⠀⢠⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠏⠁⠀⢸⣿⣿⣿⡿⠇⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣁⡀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠘⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⢰⣾⡇⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⢈⣹⣿⣿⣿⣿⠿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣘⠿⣿⣿⣿⣤⣿⣿⣿⣿⡿⠟⠀⠀⢀⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⣶⣾⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠏⠉⣉⣿⡿⠏⠿⣿⣿⠿⠉⠉⠱⣶⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠙⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠿⡿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢃⣀⣀⣀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣶⣾⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣀⣀⣀⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣼⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠛⠻⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡟⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⢠⣼⣿⠉⠀⠀⠀⡄⠀⠀⠉⢻⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⣭⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠃⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣤⠀⠘⠛⠿⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⠉⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣶⣶⣶⣶⣿⣿⣿⡟⢻⠛⠀⢰⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠁⠀⠀⣾⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣤⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠸⢿⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣀⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣄⣤⣼⣧⣤⣤⣿⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠋⠙⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡄⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠿⠿⠛⠛⠀⠀⠘⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣤⣤⡜⣿⣿⣿⣿⠛⢀⣸⣿⣿⣿⣿⠿⠀⠀⣿⣿⣿⣿⣿⠃⠀⠀⠀⠘⠛⠿⠿⣿⣿⣤⠀⠀⠀⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠘⢻⣿⣿⣿⣿⠛⠋⠁⠉⣿⣿⡟⠀⣼⣿⣿⣿⣿⡏⠀⠀⢠⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡁⠀⠀⠀⠸⣿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢀⣸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣶⣦⣤⡌⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠉⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣤⠀⠀⢠⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⣀⣆⡀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⢈⣹⣿⣿⣿⣿⠿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠘⠿⣿⣿⣿⣤⣼⣇⣿⣿⡿⠟⠀⠀⢀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣶⣦⣤⣭⣿⣿⣿⣿⣿⣿⣿⡇⠀⣤⣴⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⠿⠿⣿⣿⣿⠿⠿⣿⣿⡿⠿⢿⣷⣿⠿⢿⣿⠉⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣭⠀⠀⠛⠛⠋⠀⠀⠉⠉⠁⠀⠘⠛⠉⠀⠈⢹⡖⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢃⣀⣀⣀⣀⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⢀⣀⠀⢀⡀⠀⠠⠤⢄⣀⠀⠀⣀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣶⣾⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣿⣷⣾⣿⣶⣾⣷⣶⣶⣶⣾⣿⣶⣶⣿⣶⣾⣷⣀⣀⣀⣀⣀⣀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣼⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠻⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡟⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⢠⣼⣿⠋⠀⠀⠀⡄⠀⠀⠉⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⣿⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠃⠀⠀⢸⣿⠀⠀⠀⣿⡀⠀⠀⠘⣿⣿⣿⣿⣿⣀⠀⡀⠛⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⠛⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣶⣿⣿⣿⣿⡟⠛⠐⣷⠀⠀⢹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠁⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⣤⠀⢸⣿⣿⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⣿⠏⠉⠉⠉⠉⠉⠀⠀⠸⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⣤⣤⣠⣤⣤⣿⣿⣿⣿⣤⣤⣄⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⣿⣿⣿⣿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⡟⠛⠛⠻⢿⣿⠿⠇⠀⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠋⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣧⠀⠀⠀⠈⠉⠀⠐⠂⠉⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣶⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣰⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠒⠚⠛⠛⠛⣶⣶⣤⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⢠⣤⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⣈⣹⣿⣿⣿⣿⠿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣀⣀⠀⠘⠿⣿⣿⣿⣤⣿⣧⣼⣿⡿⠟⠀⠀⢀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣦⣤⣭⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣶⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⢿⣿⣿⡿⠉⠿⣿⣿⠿⠉⢿⣿⠿⠹⢿⡏⡉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣤⠀⠀⠘⠛⠋⠁⠀⠀⠉⠉⠀⠀⠘⠋⠀⠀⠈⢱⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢃⣀⣀⣀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⣀⣀⣀⣀⠀⠠⢤⣀⡀⠀⢀⣀⢀⣠⠘⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣶⣾⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣰⣀⣀⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣷⣶⣾⣿⣾⣿⣆⣀⣀⣀⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣼⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠻⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡟⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⢰⣾⣿⠋⠀⠀⠀⡄⠀⠀⠉⢻⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⣭⣿⡟⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠀⣼⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡀⠀⠀⠘⣿⣿⣿⣿⣿⠀⢀⡀⠘⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⣿⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣶⣿⣿⣿⣿⡟⠀⠘⡇⠀⠀⢻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⣉⡀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣤⣤⠀⠀⠘⣿⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠰⣿⣿⣿⠿⠿⠿⠿⠿⠿⠀⠀⠀⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣄⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⣿⣿⣿⣿⠛⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⠛⠃⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡇⠛⣿⣿⣿⣿⠇⠘⠀⠀⠠⢿⣿⡿⠟⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⣿⣿⣿⠋⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣶⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠒⠚⠛⠛⠛⣶⣶⣤⡄⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⢠⣤⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⣸⣿⣿⣿⣿⣿⠿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣀⣀⠀⠘⠿⣿⣿⣿⣤⣿⣧⣼⣿⡿⠟⠀⠀⢀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣦⣤⣭⣿⣿⣿⣿⣿⣿⣿⡇⢠⣤⣶⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠿⠿⢿⣿⣿⡿⠉⠿⣿⣿⠿⠉⣿⣿⠿⠿⢿⠿⡉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣤⠀⠀⠘⠛⠛⠁⠀⠀⠉⠉⠀⠀⠙⠛⠀⠀⠈⢰⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢃⣀⣀⣀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣀⣀⣀⣀⣀⠀⢼⡇⣀⡀⠀⢀⣀⣀⣠⠘⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣶⣾⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⣿⣷⣶⣾⣿⣿⣿⣆⣀⣀⣀⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⢃⣿⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠻⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⢰⣾⣿⠋⠀⠀⠀⡄⠀⠀⠉⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⣭⣿⡏⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠀⣼⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡀⠀⠀⠘⣿⣿⣿⣿⣿⠀⣄⡀⠛⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⣿⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣶⣿⣿⣿⣿⡟⠀⠛⠓⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⣿⣆⣀⣀⣁⣀⣀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠈⣿⣿⣿⠉⠉⠀⠀⠁⠀⠀⠈⠹⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⠛⠛⢻⣿⡟⠛⠛⠛⠛⠛⣿⣿⣿⣿⠛⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠃⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣇⠛⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡄⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠘⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⡇⠀⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡁⠈⠉⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠓⠛⠛⠛⠛⠛⠛⢳⣶⣶⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣤⠀⣼⣧⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⣎⣹⣿⣿⣿⣿⠿⠀⠀⠰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⠀⠘⠿⣿⣿⣿⣿⣿⣧⣼⣿⡿⠟⠀⢀⣠⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣶⣤⣭⣿⣿⣿⣿⣿⣿⣿⣧⣤⣶⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⢏⣿⣿⡏⠉⠿⣿⣿⠿⠉⣿⡷⠉⠹⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⠀⠘⠛⠋⠁⠀⠀⠉⠉⠀⠀⠋⠁⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢃⣀⣀⣀⣀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠘⢛⣀⣀⠀⢀⣀⣀⣠⠘⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣶⣾⣿⣿⣿⣿⣿⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣶⣾⣿⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣼⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠻⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡟⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⢠⣼⣿⠋⠀⠀⠀⡄⠀⠀⠉⢻⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⣭⣭⡍⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠀⣼⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⣄⡀⠀⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⣿⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⣭⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣶⣶⣶⣶⣿⣿⣿⣿⡏⠀⠙⠃⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠇⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⣉⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⣀⡈⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠛⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⡄⠀⠀⠘⢻⣿⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣄⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⣿⣿⣿⣿⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⠛⢻⣿⡏⠉⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠛⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⠀⠀⠀⠘⠛⠛⠛⠛⠛⠀⠀⠛⠛⢿⣿⣿⣿⡿⠛⠀⣸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠃⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠈⠿⢿⣿⠇⠀⠀⠀⠀⢀⠀⠀⣰⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠏⠹⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠓⠒⠒⠒⢲⣶⣤⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠹⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢠⣼⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⣿⡇⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⢀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⡎⢹⣿⣿⣿⣿⠿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣤⣤⣀⠀⠘⠿⣿⣿⣿⣿⣿⣧⣼⣿⡿⠟⠀⣀⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⢿⣿⣿⣿⡏⠉⠿⣿⣿⠿⣹⣿⡿⠿⠿⠿⣏⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⣿⠛⠈⠉⠉⠉⠁⠀⠀⠉⠉⠀⠉⠋⠁⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢃⣀⣀⣀⣀⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣤⣀⣀⣀⣀⣀⣀⣀⣛⢛⣃⣀⡀⢀⣀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣶⣾⣿⣿⣿⣿⣿⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣾⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣼⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠿⢿⣿⣿⠟⠻⠿⠿⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠿⠛⠛⠛⠻⠿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⢰⣾⣿⠋⠀⠀⠀⡄⠀⠀⠙⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⣭⣥⡌⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠀⢻⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡄⠀⠀⠸⣿⣿⣿⣿⣿⠀⣀⡀⠛⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠘⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣶⣿⣿⣿⣿⡟⠀⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⢈⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⢸⣿⣿⣿⠉⠁⠀⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⣿⡿⠿⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠈⣿⣿⣿⠇⠀⠀⠀⠁⠀⠀⠀⠈⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⣿⣿⣿⣿⣿⡟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⠃⠀⠻⠿⠿⠃⠀⠛⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⣿⠛⠛⠛⠛⠚⠛⠛⠓⣶⣶⣤⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠹⢿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢠⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠛⠃⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⣶⣷⡆⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠛⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⡈⢹⣿⣿⣿⣿⠿⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣀⡀⠘⠿⣿⣿⣿⣿⣧⣤⣼⣿⡿⢟⣀⣤⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠉⠀⠉⠿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢃⣀⣀⣀⣀⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣀⣀⣀⣀⣀⣀⣛⣛⣃⣀⣀⢀⣀⣀⣀⣤⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣶⣾⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣀⣀⣀⣀⣀⣀⣀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣼⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠿⢿⣿⣿⠟⠛⠿⠿⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠿⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⣴⣾⣿⠋⠀⠀⠀⡄⠀⠀⠉⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢻⣿⣿⣭⡍⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠀⣿⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠸⣿⣿⣿⣿⣧⠀⣀⡘⠛⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⢻⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⢸⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣶⣿⣿⣿⣿⡟⠀⢻⡇⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⢰⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⢸⣿⣿⠿⡿⠇⠀⠏⠁⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⣿⡿⠿⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⣿⣿⣿⣿⣿⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⠛⢻⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠛⠀⠀⠀⠀⠀⠘⠛⠛⠻⢿⣿⣿⡇⠀⠀⠛⠛⠀⠀⠀⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣼⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⣿⡿⢿⡿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠉⠁⠈⠁⠉⠓⠒⠉⡝⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣾⣿⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⠉⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠛⠋⠁⠀⠉⢻⣿⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⠿⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⣤⣴⣦⣤⠀⠉⠉⠀⠀⠀⠘⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠉⣷⣀⣰⣶⢀⣀⣀⣀⣀⣀⣀⠀⠀⢀⡀⠀⠀⠉⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣛⣃⣀⣘⣣⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣉⣀⣀⣀⣀⣰⣆⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠻⠿⣿⣿⣿⠿⠿⠛⠛⠛⠿⢿⣿⣿⠟⠻⠿⠿⠿⠿⠿⢿⣿⠿⠛⠻⠿⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠿⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⡄⠀⠀⠙⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢻⣿⣿⣭⡍⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠛⣿⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠸⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⣧⠀⣀⡛⠛⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⠀⠀⣿⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⢠⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣾⣿⣿⣿⣿⡟⠀⢻⡇⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⢰⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⢸⣿⣿⠿⠹⠇⠀⠉⠁⠀⠀⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠛⡇⠀⠀⠀⢸⡇⠀⠀⢠⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⡿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⣿⣿⣿⣿⡟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠛⠀⠀⠀⠀⠀⠀⠘⠛⠻⢿⣿⣿⠿⠛⠃⠀⠀⠀⠀⣿⣿⣿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⢘⣻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣽⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠙⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡌⠉⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣼⣿⣿⣤⣼⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⡿⠿⠛⠛⠛⠿⣿⣿⣿⣤⠿⢿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁⠀⠀⠀⠀⠀⠉⢻⣿⣿⣶⣾⣿⡟⠛⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠾⠷⠿⠿⡶⠾⠿⣿⣿⠿⠀⠈⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⣤⠀⠀⢠⠀⠀⠁⠀⠀⠉⠉⠀⠀⠀⢸⣿⣯⡅⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣀⣀⣀⣃⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣉⣀⣀⣀⣀⣰⣆⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠻⠿⣿⣿⣿⠿⠿⠛⠛⠛⠿⢿⣿⣿⠟⠻⠿⠿⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠿⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⡄⠀⠀⠙⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢻⣿⣿⣭⡟⠋⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢠⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⠀⣀⡘⠛⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠉⠛⣿⣿⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠘⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣾⣿⣿⣿⣿⡟⠀⢻⡇⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⢈⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⡀⠀⠀⢸⣿⣿⣿⡉⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿
⣿⣿⣿⠛⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠸⣿⣿⠿⠁⠀⠀⠀⠁⠀⠀⠀⠸⣿⣿⣿⣿
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣿⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣿⣿⣿⣿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⡟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠃⠀⠀⠀⠘⠛⠿⠿⠿⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⢿⣿⣿⣿⡟⠀⠘⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠏⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡿⠿⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢨⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣶⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⠆⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⣰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⠟⠛⠀⠘⠛⠿⣿⣿⣿⣛⠛⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⡍⢻⣿⣿⣶⣾⣿⡟⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠉⠉⠿⠿⠏⠉⠿⣿⣿⠿⠀⠈⢹⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣤⣴⣤⢠⣴⣤⣤⣤⣤⣤⠉⠉⢠⡄⢠⣼⣿⣯⡅⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡸⢿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣛⣀⣀⣛⣠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣷⣿⣿⣿⣿⣿⣿⣿⣁⣀⣀⣀⣀⣰⣆⣀⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠿⢿⣿⣿⠟⠿⠿⠿⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠿⠛⠛⠛⠻⠿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⠋⠀⠀⠀⡄⠀⠀⠙⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢻⣿⣷⣿⣭⡍⠉⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠀⠀⣿⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⠀⣄⠀⠀⠛⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠁⠀⠀⢻⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠘⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⠀⠛⠂⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠈⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⢸⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠘⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⣿⡿⠇⠀⠀⠀⠀⠀⠸⢿⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠸⣿⣿⠿⠀⠀⠀⠀⠁⠀⠀⠀⠸⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠏⠉⠉⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠹⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡿⠿⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢨⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣶⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⠆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⠟⠛⠀⠘⠛⠿⣿⣿⣿⣛⠛⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⡍⢻⣿⣿⣶⣾⣿⡟⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠉⠉⠿⠿⠏⠉⠿⣿⣿⠿⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣤⣴⣤⢠⣴⣤⣤⣤⣤⣤⠉⠉⢠⠀⢠⣼⣿⣯⡅⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣛⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣷⣾⣿⣿⣿⣿⣿⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⠿⠟⠛⠛⠻⠿⣿⣿⣿⠿⠿⠛⠛⠛⠿⢿⣿⣿⠟⠿⠿⠿⠿⠿⠿⢿⣿⠿⠛⠛⠿⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠿⠛⠛⠛⠻⠿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⠋⠀⠀⠀⡄⠀⠀⠙⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢻⣿⣷⣿⣭⡍⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠃⠀⣿⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢠⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⠀⣄⠀⠀⠛⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠁⠀⠀⢻⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⢸⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣾⣿⣿⣿⣿⡟⠀⠛⠂⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⢈⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⢸⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⣿⡿⠇⠀⠀⠀⠀⠀⠸⢿⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠸⣿⣿⠿⠀⠀⠀⠀⠁⠀⠀⠀⠸⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣿⣿⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠏⠉⠹⠿⠿⠿⠿⠿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠙⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠿⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣼⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣶⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣧⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠶⠆⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⣰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣤⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠃⠀⠘⠛⠿⣿⣿⣿⣛⠛⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠘⢻⣿⠁⠀⠀⠀⠀⠀⣤⠉⢻⣿⣿⣶⣾⣿⡟⢻⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠉⠉⠿⠿⠉⠉⠿⣿⣿⠿⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⣤⣴⣤⠀⣴⣦⣤⣤⣴⣤⣤⠉⢩⣤⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⢿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣃⣀⣠⣤⣠⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣶⣶⣀⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⣀⣀⣀⣀⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣣⣿⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠿⢿⣿⣿⠟⠿⠿⠿⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠿⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⠋⠀⠀⠀⡄⠀⠀⠙⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢻⣿⣿⣿⡟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⠀⣄⡀⠀⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠁⠀⠀⢻⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⢸⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣾⣿⣿⣿⡟⠛⠀⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⢸⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿
⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠘⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⡿⠇⠀⠀⠀⠀⠀⠸⢿⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠸⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠃⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⡿⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣾⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠏⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠿⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣶⡍⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣼⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠱⠆⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣼⣿⣿⣤⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠛⠃⠀⠘⠛⠿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠉⠿⠿⠿⠿⠇⠈⠿⣿⣿⠿⠀⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⢹⣤⣤⣦⠀⣤⣦⣤⣤⣤⣤⣤⠀⠉⠉⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣠⣤⣤⣤⣤⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣛⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣰⣶⣆⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⣀⣀⣀⣀⣀⣀⣀⣰⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣛⢿⡿⢿⣤⣀⣸⣿⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠿⢿⣿⣿⠟⠿⠿⠿⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠿⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠈⠁⠈⣿⣿⣿⣿⠋⠀⠀⠀⡄⠀⠀⠙⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⣭⣭⡝⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡄⠘⣿⣿⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⠀⣄⡀⠘⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⠀⠉⣿⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⢨⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣾⣿⣿⣿⣿⡟⠀⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠁⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠈⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⢸⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⣿⡿⠇⠀⠀⠀⠀⠀⠸⢿⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠆⠀⠀⠀⠈⣿⣿⣿⣿
⠿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⠀⠀⠛⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣤⣴⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⢿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡿⠿⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣶⡍⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠱⡆⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣤⣸⣿⣿⣤⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠛⠛⠃⠀⠘⠛⠿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠿⠿⠿⠿⠶⠶⠈⠿⣿⣿⠿⠉⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⢻⣤⣤⣤⠀⣤⣤⣤⣤⣤⣤⠀⠀⠉⠉⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⣿⢿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣀⣠⣤⣤⣤⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣼⣿⣛⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣾⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠿⠿⠃⢠⣿⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠿⢿⣿⣿⠟⠻⠿⠿⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠿⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠛⠛⠓⣶⣤⣤⣼⣿⠋⠀⠀⠀⡄⠀⠀⠙⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢻⣿⣿⣿⠋⠙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡀⠀⣸⣿⣿⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠠⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠘⠛⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⠀⠙⣿⣿⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⢠⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣾⣿⣿⣿⣿⡟⠒⢰⣶⠀⠀⢹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠁⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⢰⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠛⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠸⢿⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠈⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿
⠿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⢻⣿⣿⢻⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠛⠀⠀⠀⠀⠀⠀⠛⠿⣿⠿⡃⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⠉⠙⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡍⠉⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠱⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣤⣸⣿⣿⣤⣼⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠁⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠀⢶⣶⣶⣶⣆⣈⣿⣿⣿⠿⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠈⢹⣤⠀⢠⠀⠀⢨⣭⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣛⣃⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣁⣀⣀⣀⣀⣰⣶⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀⣀⣀⣀⣀⣀⣀⣀⣰⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠿⢿⣿⣿⠟⠿⠿⠿⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠿⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣭⠛⠋⠛⣿⣶⣾⣿⠋⠀⠀⠀⡄⠀⠀⠙⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢻⣿⣿⣿⣭⡍⠛⠙⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣤⡀⠠⣿⣿⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢠⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡄⠀⠀⢸⣿⣿⣿⣿⣿⠀⣤⡄⠀⠛⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⠀⠉⣿⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⢨⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣾⣿⣿⣿⣿⡏⠀⠛⠃⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠁⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⢈⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠛⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠸⢿⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠸⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿
⠿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⠛⠛⣿⣿⣿⡟⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⠀⠀⠀⠿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠁⠀⠉⠙⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⠿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣧⣼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣀⢸⣷⠀⠀⠈⢹⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣤⡇⠀⠀⠀⢠⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣤⣼⣿⣿⣟⣻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠹⠇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠛⠉⠉⠉⠉⠉⢻⣿⣿⣶⣾⣿⡟⠛⢻⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⣀⠀⢀⠈⠿⣿⣿⠿⠀⠀⠀⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡅⠀⠀⠀⠀⠛⠒⠚⠒⠒⠛⠋⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣷⣶⣶⡆⠰⣶⣶⣶⣶⣶⣶⣆⡀⢀⣀⣀⣶⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢛⣛⣃⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣰⣾⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣰⣶⣀⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣰⣶⣆⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣇⣸⣿⣿⠿⠟⠛⠛⠻⠿⣿⣿⣿⠿⠿⠛⠛⠛⠿⢿⣿⣿⠟⠻⠿⠿⠿⠿⠿⢿⣿⠿⠛⠛⠻⣿⡿⠟⠛⢿⣿⡿⠛⠛⢻⣿⡿⠿⠛⠛⠛⠻⠿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⠋⠀⠀⠀⡄⠀⠀⠙⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⣿⣶⡎⠉⠉⠙⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⠀⠛⣿⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡄⠀⠀⢸⣿⣿⣿⣿⡇⣿⣧⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠘⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣾⣿⣿⣿⣯⡅⠉⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⢸⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⢸⣿⣿⣿⣿⣷⣀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢠⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠘⢻⣿⣿⣿⣿
⣿⣿⡏⠀⠀⠀⠀⠀⠸⢿⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⣿⣿⣿⣿⡟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⢸⣿⣿⡿⠟⢿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠛⢻⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠃⠀⠈⠛⠛⠛⠉⠀⠀⠀⠀⢰⣶⣶⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠿⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⡶⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠀⣸⣷⠀⠀⠈⢹⣿⣿⡿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⣀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠤⢄⠀⠀⠀⠀⣀⣀⣀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣼⣿⣿⠟⢃⣿⡿⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠸⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⠛⠛⠛⠛⢻⣿⣿⣷⣾⣿⡟⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⠿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⠁⠀⠀⠀⠀⢠⣤⣤⣤⣯⣭⡄⠀⠀⠀⠀⠈⢹⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣀⣀⣶⣆⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠰⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣛⣛⣃⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣼⣧⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⢇⣰⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣰⣶⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠿⢿⣿⣿⠟⠿⠿⠿⠿⠿⠿⢿⣿⠿⠛⠛⠿⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠿⠛⠛⠛⠻⠿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠁⠀⢻⣿⣿⠋⠀⠀⠀⡄⠀⠀⠙⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⣿⣷⡆⠉⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠇⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⡿⠇⠿⠇⠀⠀⠛⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠋⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠐⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣾⣿⣿⣿⣷⣶⣶⣤⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠉⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠈⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿
⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠘⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⠛⠃⠛⠛⠃⠀⠀⢸⣿⣿⣿⣿⣿
⣿⡿⠇⠀⠀⠀⠀⠀⠸⢿⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⢸⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⣿⣿⣿⣿⡟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠛⢿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠸⠿⠿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠋⠉⠉⠉⠙⠉⠉⠉⠙⠉⠙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠿⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣬⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆⠀⠀⠀⠱⢶⠀⠀⠀⠀⢀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠿⠀⠀⠀⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣼⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠙⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠿⠉⠉⠉⠿⣿⣿⠿⠀⠀⠀⠉⠏⠉⣹⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢛⣻⣿⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣧⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣤⣤⣤⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣶⣾⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣰⣶⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⢃⣸⣿⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠿⢿⣿⣿⠟⠿⠿⠿⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠿⠛⠛⠛⠻⠿⣿⣿⣛⣛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠋⠀⢸⣿⣿⠋⠀⠀⠀⡄⠀⠀⠙⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢻⣿⣿⣿⣦⡌⠉⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡄⠀⠀⢸⣿⣿⣿⣟⣃⣿⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠋⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠐⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠉⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⢀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⢸⣿⣿⣿⣿⣿⠉⠉⠁⠀⠀⣾⣿⣿⣿⣿⣿
⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠘⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⡿⠇⠀⠀⠀⠀⠀⠸⢿⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢻⣿⣿⣿⣿⡟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⡟⠛⠛⠛⣿⣿⣿⣿⣿⣿⠛⠋⠉⠉⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠃⠀⠀⠀⢠⣤⣀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠉⠉⠉⠉⠉⠙⠉⠉⠋⠑⠀⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡆⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⢀⣈⣿⣿⣿⣿⣿⣿⣿⠿⢸⣿⣿⣿⣿⣿⠿⠀⠀⠀⠰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣟⣤⣀⣀⣤⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠿⠿⠏⠉⠿⣿⣿⠿⠀⠀⠀⠀⠿⠇⠉⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠃⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠶⢶⠶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢛⣿⣿⣿⣘⣃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣀⣤⣤⡄⣀⣄⣀⣠⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣶⣾⣿⣿⣿⣿⣿⣿⣷⣀⣀⣀⣀⣶⣶⣀⣀⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠛⢃⣸⣿⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠿⢿⣿⣿⠟⠻⠿⠿⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠿⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⠋⠀⠀⠀⡄⠀⠀⠙⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢻⣿⣿⣿⣯⡍⠉⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠇⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡄⠀⠀⢸⣿⣿⣿⣿⡇⣤⣀⠀⠀⠀⠛⠛⠛⠛⠛⠛
⣿⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⢸⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣾⣿⣿⣿⣿⣷⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⠉⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⢈⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶
⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠘⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⡟⠃⠛⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛
⣿⡿⠇⠀⠀⠀⠀⠀⠸⢿⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠈⠹⠿⣿⣿⣿⣿⣇⣀
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⣿
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠋⠉⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠃⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⠿⣿⣿⣿⠿⠿⠃⠛⠛⠛⠻⠟⠛⠛⠛⠛⠀⠀⠀⠿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣴⣶⣿⣧⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⠀⢀⣸⠶⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⡆⠀⠀⠰⠀⡆⠀⠀⠀⠀⢀⣈⣿⣿⣿⣿⣿⣿⣿⠿⠈⢹⣿⣿⣿⣿⠿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣀⣼⣿⣿⣿⣿⣟⣀⣀⣀⣀⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣏⠿⠿⠿⠿⠿⣿⣿⠿⠀⠀⠀⠀⠸⠏⠉⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⣻⣟⣛⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⣤⣤⣤⣄⣤⣤⣄⣀⣀⣀⣀⡀⠀⣀⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⠀⢸⣿⣿⠿⠟⠛⠛⠻⠿⣿⣿⣿⠿⠿⠛⠛⠛⠿⢿⣿⣿⠟⠛⠿⠿⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠿⠛⠛⠛⠻⠿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠋⠀⢰⣾⣿⠋⠀⠀⠀⡄⠀⠀⠙⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢻⣿⣿⣿⣧⡌⠉⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠇⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⣇⣀⣄⡀⠀⠀⠛⠛⠛⠛⠛⠛
⣿⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠘⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⠀⠀⠀⠀⠀⠀
⣿⣿⣿⠿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⢸⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⡀⠀⠀⢸⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠘⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⠛⠛⠛
⣿⡿⠇⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠸⢿⣿⣿⣿⠿⠀⠀⠀
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⡟⠛⠛⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠘⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠘⠛⠛⠃⠀⠀⠀⠀⠀⢸⣿⣿⡿⠇⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⡟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢠⣤⣿⡆⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠉⠉⠉⠙⠛⠛⠋⠉⣭⡍⢩⡍⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⢸⣿⣿⡿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⣤⣼⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡄⠀⠀⢸⣶⣦⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣏⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣧⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⠀⠀⢸⠛⠀⠀⠀⠀⠀⠀⠀⠿⣿⠟⠀⠀⠀⣀⣛⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⠈⢹⣿⣿⣿⣿⠿⠀⠀⠀⠀⢰⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⣼⣿⣿⣿⣟⠀⠀⠀⠀⣀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣴⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⠿⠏⠉⠉⠀⠹⠿⠿⠿⠹⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡗⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢻⣿⣿⣇⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣧⣄⣀⣀⣀⣀⣀⣘⡛⠛⠛⣃⡀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣷⣶⣶⣶⣿⣿⣀⣀⣀⣀⣀⣀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠿⠃⢠⣿⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠿⢿⣿⣿⠟⠻⠿⠿⠿⠿⠿⢿⣿⠿⠛⠛⠻⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠿⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⢰⣾⣿⠋⠀⠀⠀⡄⠀⠀⠉⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠃⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢠⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⡟⠛⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣿⣿⣿⡟⠃⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⢨⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣾⣿⣿⣿⡟⠃⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⠿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⢈⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⡀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣤⡇⠀⠀⠀⢸⡇⠀⠀⠘⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⡿⠇⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⣀⠀⠀⠀⠀
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣼⣿⣿⣿⡟⠛⠛⠛⠻⠿⣿⣿⣿⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⢻⣿⣿⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠀⠀⠘⠛⠛⠛⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⠃⠘⠀⠀⠀⠀⣼⣿⠃⠀⠀⢸⣿⣿⣿⣿⠟⠃⠀⠛⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⡟⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠒⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢀⣸⣿⣿⣿⠁⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⣿⡟⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⣤⣼⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡄⠀⠀⢸⣶⡆⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⠀⢀⣸⠛⠀⠀⠀⠀⠀⠀⠀⠿⣿⠛⠀⠀⠀⣀⣛⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⠈⢹⣿⣿⣿⣿⠿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣜⣛⠿⣿⣿⣿⣀⢀⣠⣼⣿⡿⠟⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣿⣿⣿⣿⣾⣿⣿⣿⡇⢠⠀⠀⢠⣤⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⣿⣹⡿⠿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⡏⠉⠉⠛⠛⠛⠉⠉⠉⠉⠉⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢛⣻⣿⣟⣃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣇⡀⣀⣀⡀⢀⠀⠠⠤⢤⡄⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣷⣿⣿⣷⣾⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣾⣇⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠿⢿⣿⣿⠟⠻⠿⠿⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠿⠛⠛⠛⠻⠿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⠋⠀⠀⠀⡄⠀⠀⠙⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢻⣿⣷⣶⣤⡌⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠛⠀⢻⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡄⠀⠀⢸⣿⣿⣿⣿⣿⠀⣤⡄⠀⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⢸⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣾⣿⣿⣿⠋⠉⠀⠙⠃⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⢰⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⢈⣿⣿⣉⣰⣶⣶⣶⣆⣀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢠⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⢻⣿⣿⣿⣿
⣿⡿⠇⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⣿⣿⣿⣿⣿⡟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠘⠛⠻⣿⣿⣿⣿⠿⠻⠿⠿⠟⠃⠀⠛⠛⠛⠛⠛⠃⢀⠛⣿⣿⣿⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣶⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠹⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠹⠎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠆⠀⠈⢹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⠈⢹⣿⣿⣿⣿⠿⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣀⠀⢠⣼⣿⡿⠟⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣤⡄⠀⠀⠉⢻⣿⣿⣶⣾⣿⡟⠃⠀⠀⠀⠀⠀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣆⣰⣾⣿⣿⣿⣿⣇⠀⠀⠀⢰⣾⣿⠿⠏⠁⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⣿⣿⣿⡟⠛⢻⣿⣿⠛⠀⠀⠀⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣟⣃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣶⣶⣶⣶⣰⣶⣶⣶⣶⣶⣶⣀⣀⣀⣀⣰⣾⣇⣀⣀⣀⣀⣀⣀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠻⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⣟⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⡄⠀⠀⠉⢻⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⣶⣶⣦⠀⠙⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡀⠀⠀⠀⣿⣿⣿⣿⣿⠀⢧⡄⠀⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠁⠀⢻⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⠛⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣶⣿⣿⣿⠉⠉⠀⠈⠁⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿
⣿⣿⣿⠛⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣏⡹⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣤⣤⣠⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⣿⣿⣿⣿⣿⡟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠘⠃⠘⠛⠻⣿⣿⣿⣿⠀⠘⠃⠛⠀⠛⠛⠛⠛⠃⠀⠀⣠⣼⣿⡟⢣⣿⣿⣿⣿⣿⠃⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠃⢸⣿⣿⣿⣿⠉⠀⠀⠀⠀⢻⣿⣿⣿⡟⠛⠀⢲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠉⠉⠉⣷⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠏⠉⠿⠹⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠉⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⡇⠀⠀⢸⣿⣿⡿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣷⣤⣤⣼⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⢸⣶⣦⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣏⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣤⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠟⠀⠀⠀⣀⣛⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⠈⢹⣿⣿⣿⣿⠿⠀⠀⠀⠀⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣀⠀⢀⣸⣿⡿⠟⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣶⣾⣿⡟⠃⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣆⣀⠀⠀⠀⠈⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣶⣤⣴⣶⣿⣿⣿⣿⣦⡄⠀⠀⢰⣾⡟⡟⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡿⠏⠉⠹⢿⠿⠉⠁⢀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣟⣃⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣶⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣷⣀⣀⣀⣀⣀⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠿⢿⣿⣿⠟⠛⠿⠿⠿⠿⠿⢿⣿⠿⠛⠛⠻⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠿⠛⠛⠛⠻⠿⣿⣿⣛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⣴⣾⣿⠋⠀⠀⠀⡄⠀⠀⠉⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⣶⣶⣦⠉⠉⠉⠛⠛⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠀⣿⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡄⠀⠀⠸⣿⣿⣿⣿⡿⠀⢿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⢹⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠈⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣶⣿⣿⣿⠉⠁⠀⠈⠁⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⢸⣿⣿⣿⣆⣀⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿
⣿⣿⣿⠛⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⣿⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢻⣿⣿⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣿⣿⣿⣿⠛⠛⢻⣿⣿⣿⠛⣿⣿⡟⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡟⠃⠛⠛⠿⠿⠿⠿⠿⠿⠃⠀⠀⣀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣀⣀⡀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠏⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠹⠿⠿⢿⣉⠹⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠟⠃⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠻⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⡟⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣷⣤⣤⣼⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡄⠀⠀⢸⣶⡆⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠱⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠛⠀⠀⠀⣀⣛⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⠿⠈⢹⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣀⠀⢀⣼⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣶⣾⣿⡟⠃⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⡉⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢩⣭⠀⠀⠀⠀⠀⠀⠀⠀⣤⡇⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣆⡀⠀⠀⠀⠀⣀⣾⣿⣷⠀⠀⠀⠀⠀⢀⣶⣿⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣛⣿⣃⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣿⣿⣿⣟⣀⣤⣤⣿⣿⣿⣿⣿⣧⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣆⣀⣀⣀⣀⣀⣀⣀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣆⣀⣀⣀⣀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⢃⣿⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠿⢿⣿⣿⠟⠻⠛⠿⠿⠿⠿⢿⣿⠿⠛⠛⠻⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠿⠛⠛⠛⠻⠿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⢰⣾⣿⠋⠀⠀⠀⡄⠀⠀⠙⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⣶⣶⣮⠉⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡃⠀⠀⠸⣿⣿⣿⣿⡿⠀⣧⡀⠀⠀⠻⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⢰⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣶⣿⣿⣿⠋⠁⠀⠋⠁⠀⠀⠀⠈⠛⣿⣿⣿
⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠰⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⢸⣿⣿⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⠛⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣼⣿⣿⠿⠻⠿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣤⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠟⢻⣿⣿⣿⣿⡟⠃⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⣼⣿⣿⣿⡟⠃⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⠛⣿⠛⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢹⣿⣿⣿⣿⡿⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣁⡀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠿⠿⠿⣿⣿⣏⣉⠉⠹⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⡛⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠠⠀⠿⢿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣤⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣤⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠹⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⠿⠈⢹⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣀⠀⢀⣸⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣶⣾⣿⡟⠃⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⡀⠀⠀⠀⠀⠀⠀⣰⣾⣷⠀⠀⠀⠀⠀⠀⢀⣶⣾⡇⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢟⣻⣿⣟⣃⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣤⣀⣀⣠⣿⣿⣿⣿⣧⣄⣀⣀⣀⣀⠻⠃⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣶⣾⣿⣿⣿⣿⣿⣿⣏⣁⣶⣀⣀⣰⣀⣀⣀⣀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣷⣆⣀⣀⣀⣀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⢇⣼⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠻⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⣛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⢡⣼⣿⠋⠀⠀⠀⡄⠀⠀⠉⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⣶⣦⡄⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠃⠀⠀⢸⣿⠀⠀⠀⣿⡀⠀⠀⠘⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠉⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠁⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⣤⣤⣠⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⡟⠛⠛⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠟⠛⠀⠀⠀⠀⠀⠛⠻⢿⠿⢻⣿⣿⣿⣿⣿⠛⠿⠿⠛⠛⠃⠀⣿⣿⣿⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠶⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣁⡀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣿⣿⣿⣿⡿⠿⠿⠿⠉⠉⠉⠉⠉⠉⠉⠉⠿⠿⠿⠿⣿⣿⣿⣉⠹⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡟⠤⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠿⢿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠹⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⣿⣿⠿⠈⢹⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣀⠀⢀⣼⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣶⣾⣿⡟⠃⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠃⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣶⡀⠀⠀⠀⠀⠀⠀⣰⣾⣷⠀⠀⠀⠀⠀⠀⢀⣶⣿⡇⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢟⣻⣿⣟⣃⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣤⣀⣠⣤⣿⣿⣿⣿⣧⣄⣀⣀⣠⣼⡿⢟⡃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣶⣾⣿⣿⣿⣿⣿⣿⣏⣁⣀⣀⣰⣶⣀⣀⣀⣀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣆⣀⣀⣀⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⢇⣿⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠿⢿⣿⣿⠟⠻⠿⠿⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠿⠛⠛⠛⠻⠿⣿⣿⣛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⢡⣼⣿⠋⠀⠀⠀⡄⠀⠀⠉⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢻⣿⣿⣶⣦⡌⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠠⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠉⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠐⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠁⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⢈⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⢈⣿⣿⣿⣿⣿⠿⠿⠉⠉⠀⠀⠈⠉⠉⣹⣿
⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢠⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⡟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛⠟⠛⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠀⠛⠛⠛⠛⠿⠿⠿⠛⠃⠀⠛⠿⠿⠿⠃⠀⠀⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣤⡞⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⣀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⠿⠿⢿⣿⣉⠉⠹⢿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠃⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠛⢿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣧⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⠈⢹⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣀⠀⢀⣼⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣶⣾⣿⡟⠃⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⠀⠀⠀⠀⠀⠀⠀⠀⣤⡇⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣀⡀⠀⠀⠀⠀⣀⣾⣿⣷⠀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣇⣀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⣿⣀⣤⣤⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⡛⠀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣉⣁⣀⣀⣰⣆⣀⣀⣀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣇⣀⣀⣀⣀⣀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡟⢣⣼⣿⠿⠟⠛⠛⠻⠿⣿⣿⣿⠿⠿⠛⠛⠛⠿⢿⣿⣿⠟⠻⠿⠿⠿⠿⠿⢿⣿⠿⠛⠛⠻⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠿⠛⠛⠛⠻⠿⣿⣿⣟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⢱⣾⣿⠋⠀⠀⠀⡄⠀⠀⠙⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢻⣿⣿⣶⣦⡌⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢠⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣀⣼⣿⣿⠿⠛⢻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠉⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⢸⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⢰⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⢸⣿⣿⣿⠏⠉⠉⠉⠉⠉⠉⠉⠁⠉⠹⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢠⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⡿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⡟⠛⢡⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⣶⣶⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣀⣆⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣀⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠹⠿⠀⠈⠹⢿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠛⢿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣧⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⣿⣿⠿⠈⢹⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣀⠀⢀⣼⣿⡿⠟⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣶⣾⣿⡟⠃⠀⠀⠀⠀⠀⢸⡗⠂⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⣈⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣽⣯⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣆⡀⠀⠀⢀⣰⣶⣿⣿⣆⠀⠀⠀⠀⣰⣾⡿⠿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣀⣀⣿⣄⣸⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠛⠛⢛⣃⣤⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣏⣁⣀⣀⣀⣰⣆⣀⣀⣀⣀⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣆⣀⣀⣀⣰⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⢣⣼⣿⠿⠟⠛⠛⠻⠿⣿⣿⣿⠿⠿⠛⠛⠛⠿⢿⣿⣿⠟⠿⠿⠿⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠿⠛⠛⠛⠻⠿⣿⣿⣟⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⢰⣾⣿⠋⠀⠀⠀⡄⠀⠀⠙⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢻⣿⣿⣶⣶⡎⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠠⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡄⠀⠀⢸⣿⣿⣿⣿⣿⣤⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⢰⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⠛⠁⠀⠉⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⢀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢠⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠸⢿⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⢸⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⡟⠛⠛⠃⠀⠀⠀⠀⠀⠿⠿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣼⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣀⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⠿⠿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠿⠿⠀⠈⠹⢿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠛⢿⣿⣿⣿⡇⠀⠀⢸⣿⣿⡿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣷⣤⣤⣼⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣷⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⠟⠀⠀⠀⣀⣛⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⠈⢹⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣀⠀⢀⣸⣿⡿⠟⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣶⣾⣿⡟⠃⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⣈⡁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣽⣯⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣆⡀⠀⠀⢀⣰⣿⣿⣿⣆⠀⠀⠀⠀⢰⣾⠿⠏⠁⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣠⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⠛⢛⣃⣤⣼⣿⣿⣿⣿⣿⣿⣧⣄⣠⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣏⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣷⣆⣀⣀⣀⣰⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠇⢠⣿⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠿⢿⣿⣿⠟⠛⠛⠻⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠿⠛⠛⠛⠻⠿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⢰⣾⣿⠋⠀⠀⠀⡄⠀⠀⠙⢻⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⣶⣯⡍⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⡀⠀⠀⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠋⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷⣶⣶⣾⣿⣿⣿⠋⠉⠀⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⢈⣿⣿⣀⡀⠀⠀⠀⠀⠉⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠸⢿⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠁⠀⠀⠀⠈⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣄⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⢻⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⡿⠀⠛⠛⠛⠛⠛⠛⠀⠀⠠⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⣿⣿⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠘⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⠈⢹⣿⣿⣿⣿⠿⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⣿⣿⣀⠀⢀⣸⣿⡿⠟⠀⠀⠀⠀⢠⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣶⣾⣿⡟⠃⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣦⣤⠀⠀⢠⣴⣶⣿⣿⡆⠀⠀⠀⠀⢠⣴⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⣀⣾⣿⠿⠿⣿⣿⡿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢛⣻⣛⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣶⣿⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣶⣷⣆⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⢛⣿⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠿⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠉⢠⣼⣿⠋⠀⠀⠀⡄⠀⠀⠉⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣷⣭⣭⡍⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠇⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠘⣿⣿⣿⣿⣧⠀⣄⡻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠋⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣶⣶⣶⣶⣿⣿⣿⠛⠋⠀⠛⢳⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⣉⠀⠀⠀⠀⠀⠉⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⣿⣇⣀⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⣤⣤⣠⣤⣤⣿⣿⣿⣿⣤⣤⣀⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄⠘⠛⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠛⠛⠛⠻⠿⣿⣿⣿⣿⠇⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⢻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠉⠛⠛⠛⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⡇⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⢻⣿⡟⠉⠉⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠱⠶⠀⠈⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⢹⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⢠⣼⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠹⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⠃⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⢈⣹⣿⣿⣿⣿⣿⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⡄⠀⠀⠀⠛⠀⠀⠀⠀⣀⣤⣤⣀⡀⠘⠿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣻⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣶⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠏⠿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣭⠙⠛⠛⠛⠋⠀⠀⠉⠉⠁⠀⠈⠉⠉⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⠀⣀⡀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⣻⣿⣃⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣀⣀⣀⠀⢀⣀⣤⣤⣤⣄⠀⠀⠀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣤⣼⣿⠿⠛⠛⠛⠛⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣶⣾⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⢀⣶⡿⠏⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡟⢃⣼⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠻⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⣿⣿⠇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⢡⣼⣿⠋⠀⠀⠀⡄⠀⠀⠉⢻⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⣿⡄⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠃⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⡀⠀⠘⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠋⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣶⣶⣶⣶⣿⣿⣿⣧⡄⠀⠃⠀⠀⠉⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣄⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣄⣤⣼⣧⣤⣤⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠋⠉⢹⣿⣿⣿⣿⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠛⠃⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠤⠀⠘⠛⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⡍⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣶⠀⠐⢲⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠘⠛⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠘⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⣿⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⣿⣿⣿⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠠⣤⣼⣤⣄⣘⠿⣿⣿⣿⣤⣿⣿⣿⣿⡿⢿⣿⣿⣿⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⢻⡟⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠉⠉⢉⣿⣿⡇⠈⠿⣿⣿⠿⠀⠁⠈⠉⢾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⣶⠀⠈⠉⠉⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠘⢻⣿⣿⣦⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣎⣉⣿⣿⣏⣉⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢛⣛⣃⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣀⣀⣀⣀⣀⣛⣿⣟⣃⣀⠀⠀⣀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣶⣶⣾⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣼⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠛⠻⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢻⣿⡟⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠛⢡⣼⣿⠉⠀⠀⠀⡄⠀⠀⠉⢻⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⣤⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠛⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠃⠀⠀⢸⠿⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⣀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣶⠀⠀⠀⣿⣶⣶⣶⣶⣿⣿⣿⡇⠀⠀⠋⠀⠀⠀⢹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⠿⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⣿⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⣤⣤⣠⣤⣤⣿⣿⣿⣿⣤⣄⣀⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣄⣤⣼⣧⣤⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠉⢹⣿⣿⣿⣿⣿⣿⡟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⠉⠙⠋⠛⠛⠉⠉⠛⠙⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠟⠛⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⠿⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠈⢹⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠘⠛⣿⣿⣿⣿⣧⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠘⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⣿⠿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⠟⢃⣸⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⣸⣿⣿⣿⣿⣿⠿⠀⢸⣿⣹⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠻⣿⣿⣤⣄⣘⣿⣿⣿⣿⣤⣿⣿⣿⣿⡿⢟⣤⣼⣿⣿⣿⣿⣿⣿⠀⠸⢿⣿⣿⡿⠀⠀⢸⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠓⢺⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠉⠉⠁⠀⠀⠘⣿⣿⡏⠁⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠶⣿⠇⠈⠿⣿⣿⠿⠀⠀⠈⠉⠈⢹⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⠏⠀⠀⠀⠉⠉⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠉⠀⠀⠀⠉⠉⠀⠀⠀⢠⣤⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢸⣿⣿⣿⡏⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣀⣀⣀⠀⢀⣀⠀⠛⠃⠀⠀⠀⠀⠀⣄⣀⣠⣃⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣷⣶⣷⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣿⣿⣿⣿⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠟⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠛⠻⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢻⣿⡟⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⣿⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⠉⠀⠀⠀⡄⠀⠀⠉⢻⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⣿⡆⠈⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠿⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠛⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠃⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠋⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣭⠀⠀⠀⣿⣶⣶⣶⣶⣿⣿⣿⣿⣷⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠁⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣶⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⡿⠿⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠈⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⣤⣤⣠⣤⣤⣿⣿⣿⣿⣤⣄⣀⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣄⣠⣼⣧⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⡟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⢩⡍⠉⠉⠉⠉⠉⠛⠛⠛⠛⠋⠉⠉⠙⠛⠛⠛⢻⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠟⠛⠀⠀⠀⠀⠀⠻⠿⠿⠿⢿⣿⣿⣿⣇⠀⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⢸⣿
⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⠘⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠘⣿
⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣷⣶⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠉⠉⠉⠉⠉⠉⠉⠛⠛⢫⣭⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣧⠀⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⢸⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠸⠀⠀⢰⠀⠷⡆⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠘⠀⠀⠃⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠹⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⣿⣿⣿⣿⣿⡿⠀⢰⣿⣷⠀⠀⠀⠀⠀⢀⣀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣧⣼⣿⣿⣟⡃⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⠛⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠿⠹⢷⣶⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠻⣿⣿⣿⠀⠀⠠⢼⣤⡤⠜⠀⠀⠀⣤⣿⣿⣿⠿⢿⡿⠟⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠉⠛⠛⠒⠀⠀⠈⠉⠁⠀⠀⣤⣤⣿⣿⡟⢫⣶⡞⠁⠀⠀⣽⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠿⠿⠁⠈⠿⣿⣿⠿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢰⣾⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠸⢿⣿⢿⡿⠉⠀⣀⣀⣀⣀⣀⠀⠉⠏⠹⠿⠿⣿⣿⣿⡇⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣤⣤⣄⣀⣀⣿⣿⣿⡃⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣉⣶⣶⣶⣶⣶⣶⣶⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠻⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⣿⣛⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⢡⣼⣿⠋⠀⠀⠀⡄⠀⠀⠉⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⣿⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠃⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⠛⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣶⣶⣶⣶⣿⣿⣿⣧⡄⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⠿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿
⣿⣿⣿⠛⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⣤⣤⣠⣤⣤⣿⣿⣿⣿⣤⣤⣀⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⢻⣿⣿⣿⣿⡟⠛⠉⠉⠉⠉⠉⠉⠉⠙⠉⠛⠛⠛⠛⠋⠉⠉⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⠟⠛⠛⠀⠀⠀⠀⠀⠀⠀⠛⡟⠃⠀⠀⢠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⢻⣿
⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠁⠀⠀⠐⢺⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⢸⣿
⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣶⣶⣶⣆⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠈⢹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⣿⡍⠙⠛⠛⢫⣭⠉⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⠘⠃⠛⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠘⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠶⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⢸⣿⣿⣿⡿⠀⣰⣶⡆⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⣟⡃⠀⢿⡟⠃⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⣾⣿⣿⣿⣿⣿⠿⢿⣷⣆⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣄⡘⠿⣿⣿⣿⣿⣿⣧⣼⣿⡿⠟⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣽⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠈⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠁⣿⡿⠇⠈⠿⣿⣿⠿⠀⠀⠀⠀⠀⠈⣿⣿⣇⣀⣀⣀⣰⣆⡀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣉⣹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠉⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣷⡎⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⣀⣀⠀⠃⠀⠀⠀⢀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣉⣰⣶⣶⣶⣶⣶⣶⣆⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣸⣿⣿⣿⣿⣿⣶⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠁⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠛⠻⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⣄⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⢡⣼⣿⠉⠀⠀⠀⡄⠀⠀⠉⢻⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⠙⣿⣿⣶⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠃⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⣤⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣶⣶⣶⣶⣿⣿⣿⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⠶⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠛⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣿⣧⡄⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣄⣀⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣄⣤⣼⣧⣤⣤⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⠛⠋⠉⠉⠉⠉⠉⠙⠛⠛⠉⠋⠙⠋⠛⠉⠉⠉⠙⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠿⠿⠛⠛⠀⠀⠀⠀⣀⣤⣼⣿⣿⡟⢛⠀⢻⣧⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⠈⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣼
⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣯⣭⠉⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣤⣤⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⡿⠿⠛⠛⢻⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠈⠙⠛⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⢸⣶⣶⠀⠀⣶⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠀⠀⠸⢿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠀⢸⡇⠘⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠘⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⠀⠀⠈⢹⣿⣿⡿⠀⣀⣶⡆⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠹⠏⠉⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⢸⣿⣿⣤⣼⣿⣿⣿⣧⠄⣿⡟⠃⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⡟⠀⠀⠀⣀⣻⣿⣤⣀⣀⣠⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⡸⢿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠘⠃⠀⠀⠀⠀⣤⣤⣤⣀⡀⠘⠿⣿⣿⣿⣿⣿⣧⣼⣿⡿⠿⠿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣶⣽⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠿⢿⣿⣿⣿⡇⠉⠿⣿⣿⠿⠀⠀⠀⠀⠰⢾⣿⣏⡀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠘⠋⠛⠉⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣶⣿⣿⣿⣆⠈⠹⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣀⠀⢀⠀⠀⠀⠘⠀⠧⠄⠀⠀⠀⠀⣀⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣶⣶⣶⣶⣶⣶⣷⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡎⠹⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⠿⠿⠛⠛⠛⠻⢿⣿⣿⠟⠛⠛⠻⠿⠿⠿⢿⣿⠿⠛⠛⠛⣿⡟⠛⠛⢿⣿⡿⠛⠛⢻⣿⡿⠟⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⢡⣼⣿⠋⠀⠀⠀⡄⠀⠀⠉⣿⡟⠀⠀⠀⢠⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢻⡇⠀⠀⢸⣿⠁⠀⠀⡄⠀⠀⠀⢹⣿⣿⣿⠛⠋⠉⠀⠐⢻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⠃⠀⠀⢸⣿⠀⠀⠀⣿⠄⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⢸⣿⣿⠀⠀⠀⠀⣧⣤⣤⣤⣼⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣶⣶⣶⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡿⠿⠿⠿⢿⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿
⣿⣿⣿⠛⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢠⡄⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⡿⠏⠀⠀⠀⠀⠀⠈⢹⣿⡀⠀⠀⠀⠁⠀⠀⢀⣾⣇⠀⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣤⣤⣠⣤⣤⣿⣿⣿⣿⣤⣤⣄⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣼⣿⣧⣤⣤⣤⣼⣧⣤⣤⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠿⠟⠛⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻
⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⣤⣦⡄⠈⣿⣿⣿⣿⣿⣿⡟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢀⣸
⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⣀⡀⢀⣀⣀⣀⠀⠀⢀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣤⣤⣤⡀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⡀⣀⡀⠀⡀⠀⠀⢀⡀⠀
⠀⠀⢸⣿⣿⣿⣿⣇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠘⠿⠿⠿⠿⣿⠾⢿⡿⣿⠿⢿⣿⠿⣿⣿⠾⠿⠿⠿⣿⢿⣿⡿⢿⠿⠟⠿⠿⠻⢿⣿⣿⣿⣿⠿⠟⠿⠿⠿⠟⠻⠟⠻⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠟⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠙⠿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⢀⣤⣤⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣽⣛⡿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣷⣿⣿⣿⣷⣴⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣦⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀
⠀⠖⠶⣿⠿⢿⠿⠿⠿⠿⠿⠙⠿⠿⠛⠻⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠀⠀⠀⠀
⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⢀⡀⢀⣀⣀⣀⠀⠀⢀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣠⣤⣤⣤⡀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⡀⣀⡀⢀⡀⠀⢀⣀⡀⠀
⠀⠀⢸⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠘⠿⠿⠿⠿⣿⠾⠿⢿⣿⠿⠿⣿⠿⣿⣿⠾⠿⠿⠿⣿⣿⣿⣿⣿⡿⠟⠻⠿⠻⢿⣿⣿⣿⣿⠿⠟⠿⠿⠿⠟⠻⠟⠻⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠟⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠙⠿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⢀⣤⣤⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣽⣛⡿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⣾⣿⣿⣷⣴⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣦⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀
⠀⠖⠶⣿⠿⢿⠿⠿⠿⠿⠿⠙⠿⠿⠛⠻⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠀⠀⠀⠀
⠀⠀⠀⢀⡀⢀⣀⠀⠀⠀⠀⢀⡀⢀⣀⣀⣀⠀⠀⢀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣤⣤⣀⡀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⡀⢀⡀⠀⡀⠀⠀⢀⠀⠀
⠀⠀⢰⣿⣿⣿⣿⣇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠘⠿⠿⠿⠿⣿⠾⠿⠿⣿⠿⠿⡿⠿⢿⣿⡞⠿⠿⠿⣿⡿⣿⡿⢿⠿⠟⠿⠿⠛⢿⣿⣿⣿⡿⠿⠟⠿⠿⠿⠟⠻⠟⠻⠿⠿⠿⠺⠿⠿⠿⠿⠿⠿⠟⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠙⠿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣦⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠿⠻⢿⣿⣿⣿⣿⣿⣿⣻⠿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣀⣘⣿⣿⣿⣿⡻⠉⠁⠀⠉⠛⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠙⣿⣿⠿⡿⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⣯⢭⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⣾⣿⣿⣷⣴⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣦⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀
⠀⠖⠶⣿⠿⢿⠿⠿⠿⠿⠿⠙⠿⠿⠛⠻⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀
⠀⠀⠀⢀⡀⢀⣀⠀⠀⠀⠀⢀⡀⢀⣀⣀⣀⡀⠀⢀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣤⣤⣀⡄⠀⢀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⡀⢀⡀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣿⣿⣿⣇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠘⠿⠿⠿⠿⢿⠾⠿⠿⢿⠿⠿⡿⠿⠿⣿⠶⠿⠟⠿⣿⠿⢿⡿⠿⠿⠟⠻⠿⠛⢿⣿⣿⣿⠿⠿⠟⠿⠿⠿⠟⠻⠟⠻⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠏⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠙⠿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣦⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣗⣻⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⡿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣷⣿⣿⣻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢴⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠟⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⣾⣿⣿⣷⣴⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀
⠀⠖⠶⣿⠿⢿⠿⠿⠿⠿⠿⠙⠿⠿⠋⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠻⠿⠿⠿⠿⠿⠿⠿⠻⠿⠿⠿⠿⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠀⠀⠀⠀
⠀⠀⠀⢀⡀⣀⣀⠀⠀⠀⠀⣀⡀⣀⣀⣀⣀⡀⠀⢀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⡄⠀⣀⣀⣀⣀⢀⡀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⡀⠀⡀⠀⡀⠀⡀⠀
⠀⠀⢸⣿⣿⣿⣿⣇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠘⠿⠿⠿⠿⣿⠾⢿⣿⣿⡿⢿⣿⢿⣿⣿⡾⠿⠿⢿⣿⣿⣿⣿⣿⡿⠟⠿⠿⠻⣿⣿⣿⡿⠿⠿⠟⠿⠿⠿⠟⠻⠟⠻⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠟⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠙⠿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣦⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀
⠀⠖⠶⣿⠿⢿⠿⠿⠿⠿⠿⠙⠿⠿⠛⠻⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠻⠿⠿⠿⠿⠿⠿⠿⠻⠿⠿⠿⠿⠛⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠀⠀⠀⠀
⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⢀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣄⡄⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⡀⠀⢀⡀⠀⠀⢀⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⠀⣀⢀⡀⠀
⠀⠀⢸⣿⣿⣿⣿⣧⢰⣿⣿⣿⣿⣿⣿⣿⣿⣏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠘⠿⠿⠿⢿⣿⠞⠿⣿⣿⡿⢿⣿⢿⣿⣿⡾⠿⠿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠻⣿⣿⢿⠿⠿⠿⠟⠿⠿⠿⠟⠻⠟⠻⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠟⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠙⠿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣦⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀
⠀⠖⠶⣿⠿⢿⠿⠿⠿⠿⠿⠙⠿⠿⠛⠻⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠀⠀⠀⠀
⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⢀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡄⠀⢀⣀⣀⡀⢀⣀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⠀⣀⣀⡀⠀
⠀⠀⢸⣿⣿⣿⣿⣇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠘⠿⠿⠿⣿⣿⠾⢿⣿⣿⠿⢿⣿⣿⣿⣿⡾⠿⠿⠿⣿⣿⣿⣿⣿⣿⠟⠿⠿⠻⣿⣿⠿⠿⢿⠿⠟⠿⠿⠿⠟⠻⠟⠻⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠟⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠙⠿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣦⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀
⠀⠖⠶⣿⠿⢿⠿⠿⠿⠿⠿⠙⠿⠿⠛⠻⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⡿⠿⠿⠿⠿⠃⠻⠿⠿⠿⠿⠿⠿⠿⠻⠿⠿⠿⠿⠛⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠀⠀⠀⠀
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⢀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣤⣤⣤⣄⠀⢀⣀⣀⣀⢀⣀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⡀⢀⣀⠀⣀⣀⡀⠀
⠀⠀⢸⣿⣿⣿⣿⣇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠘⠿⠿⠿⢿⣿⠾⢿⡿⣿⡿⢿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⡿⠟⠿⠿⠻⣿⣿⠿⠿⠿⠿⠟⠿⠿⠿⠟⠻⠟⠻⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠟⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠙⠿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣶⣆⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⣾⣿⣿⣷⣴⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣦⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀
⠀⠖⠶⣿⠿⢿⠿⠿⠿⠿⠿⠙⠿⠿⠛⠻⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⡿⠿⠿⠿⠿⠋⠻⠿⠿⠿⠿⠿⠿⠿⠻⠿⠿⠿⠿⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠀⠀⠀⠀
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⢀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣠⣄⠀⢀⣀⣀⣀⢀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⠀⣀⣀⡀⠀
⠀⠀⢸⣿⣿⣿⣿⣧⢰⣿⣿⣿⣿⣿⣿⣿⣿⣇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠸⠿⠿⠿⡿⣿⠾⠿⠿⣿⡿⢿⣿⠿⣿⣿⠾⠿⠿⠿⢿⣿⣿⣿⣿⣿⠟⠿⠿⠻⣿⣿⠿⠿⠿⠿⠟⠿⠿⠿⠟⠻⠟⠻⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠟⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠙⠿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣷⣿⣿⣿⣷⣴⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣦⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀
⠀⠖⠶⣿⠿⢿⠿⠿⠿⠿⠿⠙⠿⠿⠛⠻⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⡿⠿⠿⠿⠿⠋⠻⠿⠿⠿⠿⠿⠿⠿⠻⠿⠿⠿⠿⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠀⠀⠀⠀
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⢀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣤⣤⣄⣄⠀⢀⣀⣀⣀⢀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⠀⣀⣀⡀⠀
⠀⠀⢸⣿⣿⣿⣿⣇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠘⠿⠿⠿⠿⣿⠾⠿⡿⣿⠿⢿⣿⠿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⡿⠟⠿⠿⠻⣿⣿⠿⠿⢿⠿⠟⠿⠿⠿⠟⠻⠟⠻⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠟⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠙⠿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣶⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣦⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀
⠀⠖⠶⣿⠿⢿⠿⠿⠿⠿⠿⠙⠿⠿⠛⠻⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⡿⠿⠿⠿⠿⠋⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠀⠀⠀⠀
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣠⣤⣠⣄⠀⠀⣀⣀⣀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⢀⣀⠀⣀⣀⡀⠀
⠀⠀⢸⣿⣿⣿⣿⣇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠘⠿⠿⠿⠿⣿⠾⢿⣿⣿⡿⣿⣿⣿⣿⣿⡾⠿⠟⠿⠿⠿⣿⣿⣿⡿⠟⠿⡿⠻⣿⣿⢿⠿⠿⠿⠟⠿⠿⠿⠟⠻⠟⠻⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠟⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠙⠿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣷⣿⣿⣿⣷⣴⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣦⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀
⠀⠖⠶⣿⠿⢿⠿⠿⠿⠿⠿⠙⠿⠿⠛⠻⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⢀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣤⣤⣤⡀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⠀⣀⣀⡀⠀
⠀⠀⢸⣿⣿⣿⣿⣇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠘⠿⠿⠿⠿⣿⠾⢿⣿⣿⡿⢿⣿⣿⣿⣿⡿⠿⠟⠿⠿⠿⢿⣿⣿⣿⠟⠿⡿⠻⣿⣿⢿⡿⠿⠿⠟⠿⠿⠿⠟⠻⠟⠻⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠙⠿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⣴⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣦⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀
⠀⠖⠶⣿⠿⢿⠿⠿⠿⠿⠿⠙⠿⠿⠛⠻⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⡿⠿⠿⠿⠿⠃⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠀⠀⠀⠀
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⢀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣠⣤⣄⣀⡀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⣀⢀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⢀⣀⠀⣀⣀⡀⠀
⠀⠀⢸⣿⣿⣿⣿⣇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠘⠿⠿⠿⣿⣿⠾⢿⣿⣿⡿⢿⣿⣿⣿⣿⡿⠿⠟⠿⠿⠿⠿⢿⣿⣿⠟⠿⠿⠻⢿⣿⢿⡿⠿⠿⠿⠿⠿⠿⠟⠻⠟⠻⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠟⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠙⠿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣦⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀
⠀⠖⠶⣿⠿⢿⠿⠿⠿⠿⠿⠙⠿⠿⠛⠻⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⡿⠿⠿⠿⠿⠋⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠀⠀⠀⠀
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⢀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣠⣄⡀⣀⡀⣀⣀⣀⡀⡀⠀⠀⠀⠀⢀⡀⠀⢀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀
⠀⠀⢸⣿⣿⣿⣿⣇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠸⠿⠿⠿⡿⣿⠿⢿⣿⣿⡿⢿⣿⣿⣿⣿⡿⠿⠟⠿⠿⠿⠿⠿⣿⣿⠟⢿⡿⠻⣿⣿⢿⣿⠿⠿⠿⠿⠿⠿⠟⠻⠟⠻⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠙⠿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢿⣿⣿⣿⣷⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣦⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀
⠀⠖⠶⣿⠿⢿⠿⠿⠿⠿⠿⠙⠿⠿⠛⠻⠿⠿⠿⠿⠿⠿⣿⠿⠿⢿⡿⠿⠻⠿⠿⠃⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠀⠀⠀⠀
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣀⡀⣀⣀⣀⣀⡀⠀⢀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣄⣀⣀⡀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⣀⠀⠀⣀⡀⣀⡀⣀⣀⣀⣀⡀⣀⣀⣀⣀⢀⣀⠀⣀⣀⡀⠀
⠀⠀⢸⣿⣿⣿⣿⣇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠘⠿⠿⠿⠿⣿⠿⢿⣿⣿⡿⢿⣿⣿⣿⣿⡾⠿⠟⠿⠿⠿⠿⠿⢿⣿⠿⠿⡿⠿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠟⠻⠟⠻⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠙⠿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣷⣿⣿⣿⣷⣴⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣦⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀
⠀⠖⠶⣿⠿⢿⠿⠿⠿⠿⠿⠙⠿⠿⠛⠻⠿⠿⠿⠿⠿⠿⣿⠿⠿⢿⡿⠿⠻⠿⠿⠋⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠀⠀⠀⠀
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⢀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣄⣀⣀⡀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⡀⢀⣀⣀⣀⠀⠀⢀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⠀⣀⣀⡀⠀
⠀⠀⢸⣿⣿⣿⣿⣧⢰⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠘⠿⠿⠿⠿⣿⠾⢿⣿⣿⡿⢿⣿⣿⣿⣿⡾⠿⠟⠿⠿⠿⠿⠿⢿⣿⠟⠿⠿⠻⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠟⠻⠟⠻⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠙⠿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣷⣿⣿⣿⣷⣴⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣦⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀
⠀⠖⠶⣿⠿⢿⠿⠿⠿⠿⠿⠙⠿⠿⠛⠻⠿⠿⠿⠿⠿⠿⣿⠿⠿⢿⠿⠿⠻⠿⠿⠃⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⢀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣄⣀⣀⡀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⣀⠀⠀⢀⢀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⡀⠀
⠀⠀⢸⣿⣿⣿⣿⣇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣧⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠘⠿⠿⠿⠿⣿⠾⢿⢿⣿⡿⢿⣿⣿⣿⣿⠟⠻⠏⠿⠿⠿⠿⠿⠿⢿⠿⠿⡿⠻⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠟⠻⠟⠻⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠙⠿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣦⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀
⠀⠖⠶⣿⠿⢿⠿⠿⠿⠿⠿⠙⠿⠿⠛⠻⠿⠿⠿⠿⠿⠿⣿⠿⠿⢿⡿⠿⠻⠿⠿⠋⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠀⠀⠀⠀

                    ䷩ 𝚕𝚒𝚗𝚎 4527

╒═══════════════════ 𝐀𝐑𝐓𝐈𝐂𝐋𝐄 ═════════════════════════════════════════════════╕

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/01/23/illegal-upc/#comments
    Gemini version at gemini://gemini.techrights.org/2022/01/23/illegal-upc/

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Gemini_version_available_♊︎


✐ [Meme]_Team_UPC_Congratulating_Itself⠀✐


Posted in Deception, Europe, Patents at 9:44 pm by Dr. Roy Schestowitz


Recent: Unified_Patent_Court_is_a_Fake_Common_Court_and_Violates_CJEU’s
Jurisprudence_of_the_Last_10_Years,_Will_Explode_at_launch,_Says_Professor
Jaeger


   🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇We're starting a system; Without even checking if it's
                                    legal⦈


Summary: The barrage_of_fake_news_and_misinformation_about_the_UPC deliberately
leaves out all the obvious and very important facts; even the EPO‘s António
Campinos and Breton (Benoît_Battistelli‘s buddy) participated in the lying


As usual, all the signal is in comments [1, 2, 3]; so-called ‘articles’ are
just fake news composed by people with a conflict of interest and EPO “media
partners”.


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣷⣾⣿⣶⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⡀⣀⡀⢀⣀⣀⣀⡀⣀⢀⣀⣀⠀⣀⣀⡀⠀⠀⣀⣀⠀⣀⣀⣀⢀⣀⣚⢛⣋⣙⡛⢛⣛⣛⡛⣛⡛⣛⡛⣂⡀⢀⣀⡀⠀⠀⢀⣀⡀⠀⠀⢀⣀⡀⢀⣀⢀⣀⢀⣀⡀⢀⣀⣀⣀⣀⣀⡀⣀⣀⢀⣀⡀⠀⠀
⠀⠀⣿⡇⣿⣧⣿⡏⣿⡟⠃⠿⢸⣿⣻⡇⣿⣿⠃⠀⢸⣯⡿⠇⢻⣿⡟⢸⡿⣿⢸⣿⣿⣿⠘⣿⣿⠃⣿⡇⣿⣧⣿⣇⣿⡿⠿⠄⠀⣾⣿⣿⠀⠀⣿⣿⠿⠘⣿⣾⠇⣿⣟⠿⠘⢻⣿⠋⣿⡟⠃⣿⣿⣸⣿⡇⠀⠀
⠀⠀⢹⣿⡿⣿⣿⡇⣿⡟⠃⠀⢸⣿⢻⡆⣿⣿⠃⠀⢨⣝⢿⡆⢸⣿⡇⣿⣧⣿⣼⣿⣿⣷⠀⣿⣿⠀⣿⡇⣿⢿⣿⡿⣿⣿⣿⡆⠀⣿⣿⣿⠀⠀⣭⡻⣷⠀⢻⣿⠀⣬⡻⣷⡄⢸⣿⠀⣿⡟⠃⣿⣿⣿⣿⡇⠀⠀
⠀⠀⠸⠿⠇⠻⠿⠁⠿⠿⠇⠀⠸⠿⠸⠇⠻⠿⠿⠀⠘⠿⠿⠃⠘⠿⠃⠿⠏⠿⠟⠿⠛⠿⠀⠻⠟⠀⠿⠇⠿⠊⠿⠃⠻⠿⠻⢃⠐⠿⠋⠿⠇⠀⠻⠿⠟⠀⠸⠟⠀⠻⠷⠟⠁⠸⠿⠀⠿⠿⠇⠿⠛⠟⠻⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⢀⡀⠀⠀⢀⣠⣾⣿⣷⣄⡀⠀⠀⠀⠰⣦⣤⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠿⠻⠋⢀⣶⡀⠀⠀⠀⠀⣠⣄⠉⠛⠻⣿⣿⣿⣿⣿⣿⣷⣿⣿⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠘⠛⠥⠀⠀⠀⠀⢿⡿⠧⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠡⢆⠈⢤⢄⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢄⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣰⣶⡄⣀⣀⣀⣀⡀⠀⠒⢀⣀⡀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣟⣧⣬⣾⠄⠶⠻⠊⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⡈⣿⣿⣿⣿⣿⣿⣏⠁⠀⠀⠙⠻⠿⠟⠊⠙⠛⠛⠛⠟⠛⣋⣉⣁⡀⢠⣤⢠⣤⣤⢠⣦⠰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀
⣃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣈⣀⡀⣀⡉⣤⡌⢀⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠘⣿⣧⣿⠀⢐⣶⡄⠀⣄⣄⣵⣧⡤⣢⣦⣴⡂⠀⠀⠀⠈⠀
⣟⠅⠀⠀⠒⢰⡶⢁⣄⡀⣀⣐⠢⡀⠀⠀⣶⡆⣶⣷⢸⣿⢸⣿⠘⢻⣿⠁⢿⣇⣿⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⢰⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣶⣶
⣷⣴⣅⠠⠑⡨⢂⡆⣰⣿⣶⢫⡢⡄⠁⠀⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣌⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠸⠿⠛⠛⠛⠛⠉⠉⠉⢉⣈⠉⢿⣿⣿⣿⣿⣿
⣿⣿⣞⡷⣴⣴⣶⣮⣿⣯⡧⢛⠆⠷⠄⠀⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⣀⣙⣛⢉⡀⠈⠉⠁⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣄⠘⡗⢠⣤⠀⢠⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣻⣿⣿⣿⣿⣿⣉⣵⡯⠂⠷⠿⠿⣦⠍⠩⢤⡤⠤⠶⠶⠞⠛⠒⣛⡛⣉⡙⢉⣥⣭⡙⠏⢠⣶⠀⣶⡶⠆⠁⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⡄⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠘⢿⣿⣿⣿⠉
⣿⣿⣿⣿⠟⠉⣀⣀⠀⢀⣀⢠⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣄⣿⠁⣿⡇⣿⣧⣹⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠀⢹⣿⠀⢿⣧⢰⡦⢈⣙⢻⣧⠀⠘⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⢸⣿⣿⣿⣤
⣿⣿⣿⣿⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⢠⡆⢸⣿⠀⢸⣿⢈⣷⡘⠿⠾⠛⣠⣄⠛⢛⣁⣋⣉⣡⣬⣭⣀⣀⣀⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿
⣿⣿⣿⣿⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⠛⢀⣄⣛⣁⣈⣉⣼⣿⣮⣤⣴⣶⣶⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿
⣿⣿⣿⣿⣇⠘⠿⠾⠟⠀⠿⠀⠛⠂⠛⠛⠃⢈⣉⣉⣤⣬⣤⣤⣤⣴⡤⣰⣶⣶⣶⣾⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣤⣤⣶⣶⣶⠶⣶⣝⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⡿⣿⢇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠙⠻⢿⣽⣿⣿⣿⣿⣿⣿⣿⢱⡟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠃⣠⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠏⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⡀⢸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣤⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀
⣿⣿⡿⠿⠿⢿⣿⣿⣿⡉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢨⡨⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣾⣿⣿⠂⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣟⢸⣿⣿⣿⡟⠀⠀
⣿⣿⡇⠀⠀⢸⣿⣿⣿⣱⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢘⣿⢿⣿⣿⣿⣿⣿⣧⣤⠀⠹⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠉⠙⠛⠿⠿⠀⠀⠀
⣿⣿⡇⠀⠀⠈⠛⠛⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⠀⣳⣾⣿⣿⣏⣿⣿⣯⣬⡆⠀⢹⣿⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣷⣠⠀⠀⠀⠠⠶⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠛⣇⣄⣹⢠⠙⣿⣿⣿⣿⣧⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣷⣾⣿⣶⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⡀⣀⡀⢀⣀⣀⣀⡀⣀⢀⣀⣀⠀⣀⣀⡀⠀⠀⣀⣀⠀⣀⣀⣀⢀⣀⣚⢛⣋⣙⡛⢛⣛⣛⡛⣛⡛⣛⡛⣂⡀⢀⣀⡀⠀⠀⢀⣀⡀⠀⠀⢀⣀⡀⢀⣀⢀⣀⢀⣀⡀⢀⣀⣀⣀⣀⣀⡀⣀⣀⢀⣀⡀⠀⠀
⠀⠀⣿⡇⣿⣧⣿⡏⣿⡟⠃⠿⢸⣿⣻⡇⣿⣿⠃⠀⢸⣯⡿⠇⢻⣿⡟⢸⡿⣿⢸⣿⣿⣿⠘⣿⣿⠃⣿⡇⣿⣧⣿⡇⣿⡿⠿⠄⠀⣾⣿⣿⠀⠀⣿⣿⠿⠘⣿⣾⠇⣿⣟⠿⠘⢻⣿⠋⣿⡟⠃⣿⣿⣸⣿⡇⠀⠀
⠀⠀⢹⣿⡿⣿⣿⡇⣿⡟⠃⠀⢸⣿⢻⡆⣿⣿⠃⠀⢨⡝⢿⡆⢸⣿⡇⣿⣧⣿⣼⣿⣿⣷⠀⣿⣿⠀⣿⡇⣿⢿⣿⡇⣿⣿⣿⡆⠀⣿⣿⣿⠀⠀⣭⡻⣷⠀⢻⣿⠀⣬⡻⣷⡄⢸⣿⠀⣿⡟⠃⣿⣿⣿⣿⡇⠀⠀
⠀⠀⠸⠿⠇⠻⠿⠁⠿⠿⠇⠀⠸⠿⠸⠇⠻⠿⠿⠀⠘⠿⠿⠃⠘⠿⠃⠿⠏⠿⠟⠿⠛⠿⠀⠻⠟⠀⠿⠇⠿⠊⠿⠇⠻⠿⠻⢃⠐⠿⠋⠿⠇⠀⠻⠿⠟⠀⠸⠟⠀⠻⠷⠟⠁⠸⠿⠀⠿⠿⠇⠿⠛⠟⠻⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣿⣿⠿⠋⠁⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⣀⣿⣿⡀⢀⡄⠀⠀⢀⣀⣾⣿⣷⣄⡀⠀⠀⠀⠰⣤⣤⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠄⡠⠀⢀⠀⣸⠗⣿⣿⣿⣿⣿⣿⣿⡿⢿⢿⢿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠒⢛⢧⢄⢐⣾⣿⣿⣿⠿⠛⠋⢀⣶⡀⠀⠀⠀⠀⣠⣄⠈⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣶⣽⣿⣿⡇⠀⠀⠀⠘⠛⠧⠀⠀⠀⠠⣿⡿⠇⠀⠀⣸⣿⣿⣿⣿⣿⣿⡟⠁⠀⡒⠀⠀⠈⠀⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣻⣿⣿⣿⢺⢿⣷⣤⣶⡄⣀⣀⣀⣀⡀⠀⠒⣀⣀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⡗⠪⠃⠀⠠⠙⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀
⣀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢻⣿⣿⣿⡌⣿⣿⣿⣿⣿⣿⣋⠀⠀⠀⠈⠛⠿⠟⠚⠛⠛⠛⠛⠟⠛⣉⣉⣀⡀⢠⣤⢠⣤⣤⢠⣦⠐⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣽⡼⡥⡀⠠⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠙⠛⠋⢁⣈⣁⡀⣀⡀⣤⡌⢀⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠈⣿⣧⣿⠀⢠⣶⡄⠀⣔⣤⣤⣦⡠⣤⣤⣵⣷⠀⠀⠀⡀⠄
⣿⣧⡂⠀⠀⢀⠃⡈⠀⠀⡄⠀⠀⠀⠀⠀⣶⡆⣶⣷⢸⣿⢸⣿⠘⢻⣿⠁⢿⣇⣿⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⢰⣿⣷⣶⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣧⣶⣿⣷
⣿⣿⣤⡀⠀⠀⠀⠠⠺⣾⡇⠴⡂⠀⠀⠀⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣈⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠸⠟⠛⠛⠛⠋⠉⠉⠉⢉⣀⠉⠿⣿⣿⣿⣿⣿
⣿⣿⣿⣗⣶⣦⣦⣷⣾⡋⡻⠊⠒⠚⢡⠀⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⢀⣙⡛⠉⠀⠈⠉⠁⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣄⠘⡗⢠⣤⠀⢠⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⢸⣿⣷⣿⣿⣿
⣿⣿⣿⣟⣿⣿⣿⣿⡇⠀⣹⡇⠢⠼⠾⠿⠦⠉⠩⠤⠀⠀⠀⠀⠀⠂⠐⣊⡛⣉⡙⢉⣤⣬⡙⠀⢠⣶⠀⣶⡶⠆⠁⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⡄⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠘⢿⣿⣿⡯⠁
⣿⣿⣿⣿⠟⠉⣀⣀⠁⢀⣀⢠⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣄⣿⠁⣿⡇⣿⣧⣸⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠀⢹⣿⠀⢻⣧⢸⡦⢈⣙⢻⣧⠀⠈⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⢸⣿⣿⣏⣁
⣿⣿⣿⣿⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⢠⡆⢸⣿⠀⢸⣿⢨⣷⡘⠿⠾⠛⣠⣄⠛⢛⣁⣋⣉⣡⣬⣍⣀⣀⣀⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿
⣿⣿⣿⣿⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⠛⠀⢄⣛⣁⣈⣉⣸⣿⣬⣤⣶⣶⣶⣾⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⠘⠿⠾⠟⠀⠿⠀⠛⠂⠛⠛⢃⣈⠉⢉⣤⡌⠀⢀⣤⣤⣶⣶⣶⣿⣿⣿⣷⣶⣶⣀⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣻⣿⣿
⣿⣿⣿⣿⣿⣶⣤⣤⣴⣶⣶⣶⠶⣦⣄⣙⣿⣿⣿⣾⣿⠃⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠈⠉⣻⣿⣿⡏⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣹⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⢐⣿⣿⣿⣿⣇⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣷⣿⡇⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀
⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⠉⠉⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡄⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣟⢸⣿⣿⣿⡏⠀⠀
⣿⣿⣇⠀⠀⣸⣿⣿⣟⢟⠋⡀⠀⣀⣼⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⡍⣟⣹⣿⣿⢿⣿⡷⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠉⠙⠛⠿⠿⠀⠀⠀
⣿⣿⣿⠀⠀⠉⠙⠛⠊⠀⠀⠁⢈⡟⢉⣸⡿⠐⠈⡻⣿⣿⣿⣿⣿⣿⣿⣿⣦⡉⠻⣶⣶⣿⣿⣿⣿⠁⣾⡟⠛⣿⢿⣿⣷⣶⣄⠈⠻⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠈⣷⣦⣷⡤⢹⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⠀⡸⠆⠀⣤⣶⡀⠀⠁⠀⢨⣾⣧⣶⣷⣿⣿⣶⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⡀⣀⡀⢀⣀⣀⣀⡀⣀⢀⣀⣀⠀⣀⣀⡋⠀⠀⣀⣀⠀⣉⣀⣀⢀⣀⣚⢛⣋⣙⡛⢛⣛⣙⡛⣛⡛⣛⡛⣂⡀⢀⣀⡀⠀⠀⢀⣀⡀⠀⠀⢀⣀⡀⢀⡀⢀⣀⢀⣀⡀⢀⣀⣀⣀⣀⣀⡀⣀⣀⢀⣀⡀⠀⠀
⠀⠀⣿⡇⣿⣧⣿⡇⣿⡟⠃⠿⢸⣿⣻⡇⣿⣟⠃⠀⢸⣯⡿⠇⢻⣿⡟⢸⡿⣿⢸⣿⣿⣿⠘⣿⣿⠃⣿⡇⣿⣧⣿⡇⣿⡿⠿⠄⠀⣾⣿⣿⠀⠀⣿⣿⠿⠘⣿⣾⠇⣿⣟⠿⠘⢻⣿⠋⣿⡏⠃⣿⣿⣸⣿⡇⠀⠀
⠀⠀⢹⣿⡿⣿⣿⡇⣿⡟⠃⠀⢸⣿⢻⡆⣿⡟⠃⠀⢨⡝⢿⡆⢸⣿⡇⣿⣧⣿⣼⣿⣿⣷⠀⣿⣿⠀⣿⡇⣿⢿⣿⡇⣿⣿⣿⡆⠀⣿⣿⣿⠀⠀⣭⡻⣷⠀⢻⣿⠀⣬⡻⣷⡄⢸⣿⠀⣿⡟⠃⣿⣿⣿⣿⡇⠀⠀
⠀⠀⠸⠿⠇⠻⠿⠁⠿⠿⠇⠀⠸⠿⠸⠇⠻⠿⠿⠀⠘⠿⠿⠃⠘⠿⠃⠿⠏⠿⠟⠿⠛⠿⠀⠻⠟⠀⠿⠇⠿⠊⠿⠃⠻⠿⠻⢃⠐⠿⠋⠿⠇⡀⠻⠿⠟⠀⠸⠟⠀⠻⠷⠟⠁⠸⠿⠀⠿⠿⠇⠿⠛⠟⠻⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⡖⠠⣴⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⠀⠘⠀⠀⠀⣀⠀⠀⢠⠉⢾⣿⣿⣿⠿⠋⠁⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⡀⠄⠄⢠⠮⣴⣦⠨⣆⡘⣿⣿⠃⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠉⠀⠐⣉⢃⠄⢩⣷⣮⡅⣐⠛⣻⣿⣿⡀⢀⡀⠀⠀⢀⣀⣾⣿⣷⣄⡀⠀⠀⠀⠰⣤⣤⣿⣿⣿⣿⣿⣿⣄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⢰⣌⣀⠉⠈⢶⣧⢉⠉⢘⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠶⠟⠀⠶⠐⠟⠋⡚⢿⠂⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⢻⣿⣿⣿⠟⠛⠋⢀⣶⡀⠀⠀⠀⠀⢠⣄⠈⠛⠛⣿⣿⣿⣿⣿⣿⣷⣿⣿⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠁⠀⠄⠤⠤⠀⢩⣴⡎⠉⠀⢙⣿⣿⡇⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⢿⡛⠆⠀⠀⢸⣿⣿⣿⣿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠘⢿⣷⡅⣈⠀⠀⢺⣧⠀⣦⡄⢀⣀⣀⣀⡀⠀⠒⢀⣀⠀⠀⣴⣿⣿⣿⣿⣿⢀⣄⣘⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⠒⠂⠀⠈⠁⠘⠛⠒⠂⠀⠀⠰⠇⠀⠀⠉⣠⣤⡌⣿⣿⣿⣿⣿⣿⣋⠀⠀⠀⠈⠛⠿⠓⠐⠙⠛⠛⠛⠟⠛⣈⣁⣀⡀⢠⣤⢠⣤⣤⢠⣦⠐⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣀⡀⣀⡀⣤⡌⢀⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠈⣿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⣶⣷⢸⣿⢸⣿⠘⢻⣿⠁⢿⣇⣿⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⢰⣷⡠⡶⡗⠢⣪⡤⡗⠀⠀⠜⡐⠆⢄⢀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣈⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠸⠿⠛⠙⠊⠁⠉⠀⠁⢀⣀⠀⢀⢆⢅⠭⠀⠓
⡁⠠⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⠀⠙⠛⢉⣀⠈⠉⠁⠀⣀⣀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣀⠘⠇⢠⣤⠀⢠⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⠀⡡⢀⡘⣅⣐
⣽⠟⠍⢁⢄⠀⠀⢠⢀⠀⠀⠢⠀⠰⠛⠻⠄⠉⠉⢤⡤⠤⠶⠖⠀⠀⠀⣀⡀⣀⡀⢀⣤⣤⡉⠛⢠⣶⠀⣶⡶⠆⠁⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⡄⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠀⠸⠀⠀⠀⠀
⣿⣿⣥⣤⠕⠀⣀⣀⠀⢀⣀⢠⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣄⣿⠁⣿⡇⣿⣧⣸⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠀⢹⣿⠀⢻⣧⢰⡷⢈⣙⢻⣧⠀⠈⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⠀⠀⢄⢀⣀
⣿⣿⣿⣿⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⠠⡇⢸⣿⠀⢸⣿⢈⣷⡘⠿⠾⠛⣠⣄⠛⢛⣁⣋⣉⣡⣬⣭⣀⣀⣀⣠⣤⣤⣤⣤⣤⡄⠀⠀⡤⠱
⣿⣿⣿⣿⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⠛⣀⣤⣛⣁⣈⣉⣸⣷⣮⣤⣴⣶⣶⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠉⠥⠓
⣿⣿⣿⣿⣆⠘⠿⠾⠟⠀⠿⠀⠛⠂⠛⠛⢃⣈⣉⢉⣤⣬⣤⣤⡄⠀⢠⣤⣶⣶⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀⢐⠀⠀
⣿⣿⣿⣿⣿⣶⣤⣤⣴⣶⣶⡆⠈⠛⠓⠶⢭⣿⣿⣿⣿⣿⣿⣿⠃⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⢿⣿⣿⣿⢻⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⢕⢕⡀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⠃⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣾⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣟⡗
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣯⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢩⣼⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠃⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⢻⣽⣿⣿⣿⣿⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀
⣿⣿⡿⠿⠿⢿⣿⣿⣿⡿⠇⣀⣤⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⡿⢹⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣟⢸⣿⣿⣿⡇⠀⠀
⣿⣿⡇⠀⠀⣸⣿⣿⠋⠀⠀⣾⣿⣿⣿⡟⢻⣿⠇⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡀⢈⣿⣿⣿⣿⣷⠀⢸⡟⠛⣿⣼⣷⣄⡹⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠉⠘⠛⠿⠷⠀⠀⠀
⣿⣿⢿⠀⠀⠉⠉⠓⠀⠀⠀⠋⢙⠋⠑⠃⠈⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣽⣿⣿⣿⣿⣿⣷⠄⠈⢱⣾⣿⠿⢿⣿⣷⣦⣍⣙⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⢾⣧⠀⠀⠀⠀⠀⠀⠂⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣸⡿⣿⣂⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⡀⠀⠁⠀⣄⠀⠀⠐⠶⠄⠀⠈⢀⣠⣤⣶⣷⣿⣿⣶⣿⣶⣦⣠⠀⣀⡆⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⡀⣀⡀⢀⣀⣀⣀⡀⣀⢀⣀⣀⠐⣀⣀⡀⠀⠀⣉⣀⠀⣀⣀⣀⢀⣀⣚⢛⣋⣙⡛⢛⣛⣛⡛⣛⡛⣛⡛⣋⡁⢀⣀⡀⠀⠀⢀⣀⡀⠀⠀⢀⣀⡀⢀⣀⢀⣀⢀⣀⡀⢀⣀⣀⣀⣀⣀⡀⣀⣀⢀⣀⡀⠀⠀
⠀⠀⣿⡇⣿⣧⣿⡏⣿⡟⠃⠿⢸⣿⣻⡇⣿⣟⠃⠀⢸⣯⡿⠇⢻⣿⡟⢸⡿⣿⢸⣿⣿⣿⠘⣿⣿⠃⣿⡇⣿⣧⣿⡇⣿⡿⠿⠄⠀⣾⣿⣿⠀⠀⣿⣿⠿⠘⣿⣾⠇⣿⣟⠿⠘⢻⣿⠋⣿⡏⠃⣿⣿⣸⣿⡇⠀⠀
⠀⠀⢹⣿⡿⣿⣿⡇⣿⡟⠃⠀⢸⣿⢻⡆⣿⡿⠃⠀⢨⡝⢿⡆⢸⣿⡇⣿⣧⣿⣼⣿⣿⣷⠀⣿⣿⠀⣿⡇⣿⢿⣿⡷⣿⣿⣿⡆⠀⣿⣿⣿⠀⠀⣭⡻⣷⠀⢻⣿⠀⣬⡻⣷⡄⢸⣿⠀⣿⡟⠃⣿⣿⣿⣿⡇⠀⠀
⠀⠀⠸⠿⠇⠻⠿⠁⠿⠿⠇⠀⠸⠿⠸⠇⠻⠿⠿⠀⠘⠿⠿⠃⠘⠿⠃⠿⠏⠿⠟⠿⠛⠿⠀⠻⠟⠀⠿⠇⠿⠊⠿⠃⠻⠿⠻⢃⠐⠿⠋⠿⠇⠀⠻⠿⠟⠀⠸⠟⠀⠻⠷⠟⠁⠸⠿⠀⠿⠿⠇⠿⠛⠟⠻⠇⠀⠀
⠀⠀⠀⠦⠀⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠆⢀⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⡇⠀⠠⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠁⠈⠀⠀⡀⠀⠐⠋⠀⠁⠀⢀⣠⠄⠀⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣇⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠀⠀⠀⠉⠀⣀⠰⢶⠆⠀⠀⠀⢀⡄⠀⠀⠀⠸⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠁⠘⠈⢿⣔⡷⠀⠈⢉⠀⠀⠠⣴⣿⣿⡀⢀⡀⠀⠀⠀⣀⣾⣿⣷⣄⡀⠀⠀⠀⠰⣤⣄⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠿⠇⠙⠀⠀⠀⠀⢀⣼⠆⠀⠄⠀⠋⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠨⡅⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⣤⠀⢼⠁⠀⠀⠀⠁⠀⠀⠀⡀⣿⣿⣿⣿⣿⣿⣿⡿⢿⢿⢿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠄⠀⠙⠂⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠿⠛⠋⢀⣶⡀⠀⠀⠀⠀⢠⣄⠈⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢛⣥⣤⡄⢀⡀⠀⠈⢠⡤⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢿⡇⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⢿⡛⠆⠀⠀⣸⣿⣿⣿⣿⡿⠿⠟⠁⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠋⠈⠀⠀⠘⠛⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠘⠿⠠⠀⢸⣧⣀⣦⡄⢀⣀⣀⣀⡀⠀⠒⢀⣀⠀⠀⣶⣿⣿⣿⣿⣿⣶⣶⣝⣠⡄⠀⠀⠀⠀⠠⠶⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣋⠀⠀⠀⠉⠻⠿⠓⠀⠙⠛⠛⠛⠿⠛⣁⣀⣀⡁⢠⣤⢠⣤⣤⢠⣦⠐⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣀⡀⣀⡀⣤⡄⢀⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠈⣿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⣶⣷⢸⣿⢸⣿⠘⢻⣿⠁⢿⣇⣿⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⢰⣦⡰⢶⣶⠲⢯⣾⢿⠇⠈⠐⢐⠠⡀⣀⠀⠠
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣈⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠸⠿⠛⠈⠁⠂⠉⠁⠀⢀⣀⠈⠤⡢⣔⠈⠒⠀
⢔⠈⢥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⡀⠙⠛⠉⠀⠈⠉⠁⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣄⠘⡗⢠⣤⠀⢠⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⠨⡒⣐⠚⠀⠄
⣯⡅⠉⠀⣀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠟⠹⠀⠉⠉⠤⡤⠤⠶⠖⠒⠒⠀⣀⡀⣀⡈⢁⣥⣤⡀⠉⢠⣶⠀⣶⡶⠆⠀⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⣆⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠀⢨⡄⠈⠁⠀
⣿⣿⣶⣦⠖⠀⣀⣀⠀⢀⣀⢠⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣄⣿⠁⣿⡇⣿⣧⣸⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠀⢸⣿⠀⢻⣧⢰⣶⢈⣙⢻⣧⠀⠈⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⠀⠘⠁⠀⠀
⣿⣿⣿⣿⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⠀⡄⢸⣿⠀⢸⣿⠈⣷⡘⠿⠾⠛⣠⣀⠛⢛⣁⣋⣉⣡⣬⣭⣤⣀⣀⣠⣤⣤⣤⣤⣴⣀⡀⠀⠀⠄
⣿⣿⣿⣿⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⠛⢀⣄⣛⣁⣈⣉⣼⣿⣌⣤⣤⣤⣴⣾⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠴⠟⠿
⣿⣿⣿⣿⣆⠘⠿⠾⠟⠀⠿⠀⠛⠂⠛⠛⢃⣈⣉⣉⣤⣬⣤⣤⣤⣤⣶⣤⣶⡆⠀⠀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣶⣤⣤⣴⣶⣶⣶⠀⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢩⣿⠇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢁⣀⣖⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠰⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⠏⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⣦⣿⣿⣿⣿⣿⣿⣿⡿⣿⠛⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⣶⠟⢿⣿⣿⣧⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀
⣿⣿⡿⠿⠿⢿⣿⣿⣿⠟⢃⢤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠙⣤⡿⣿⢸⢿⣃⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣟⢸⣿⣿⣿⡇⠀⠀
⣿⣿⣇⠀⠀⢸⣿⣿⡋⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠿⢧⣟⣿⣾⣿⣷⣦⣙⡛⠿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠛⠘⠻⠿⠿⠀⠀⠀
⣿⣿⣿⠀⠀⠉⠋⠛⠀⠀⠁⠛⣻⣿⣿⣿⡿⠛⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⡿⣿⠿⢿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⠀⠀⠀⠀⠀⠐⠖⠀⡿⢿⠋⠙⠉⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣾⡄⣿⣁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣄⡀⠀⠀⢀⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⣤⣶⣷⣾⣿⣶⣿⣶⣦⣀⠀⠀⠀⠀⠿⢋⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⡉⣀⡀⢀⣀⣀⣀⡀⣀⢀⣀⣀⠀⣀⣀⡀⠀⠀⣀⣀⠀⣀⣀⣀⢀⣀⣚⢛⣋⣙⡛⢛⣛⣛⡛⣛⡛⣛⡛⣂⡀⢀⣀⡈⠀⠀⢀⣀⡀⠀⠀⢀⣀⡀⢀⣀⢀⣀⢀⣀⡀⢀⣀⣀⣀⣀⣀⡀⣀⣀⢀⣀⡀⠀⠀
⠀⠀⣿⡇⣿⣧⣿⡏⣿⡟⠃⠿⢸⣿⣻⡇⣿⣿⠃⠀⢸⣯⡿⠇⢻⣿⡟⢸⡿⣿⢸⣿⣿⣿⠘⣿⣿⠃⣿⡇⣿⣧⣿⣇⣿⡿⠿⠄⠀⣾⣿⣿⠀⠀⣿⣿⠿⠘⣿⣾⠇⣿⣟⠿⠘⢻⣿⠋⣿⡟⠃⣿⣿⣸⣿⡇⠀⠀
⠀⠀⢹⣿⡿⣿⣿⡇⣿⡟⠃⠀⢸⣿⢻⡆⣿⣿⠃⠀⢨⡝⢿⡆⢸⣿⡇⣿⣧⣿⣼⣿⣿⣷⠀⣿⣿⠀⣿⡇⣿⢿⣿⡿⣿⣿⣿⡆⠀⣿⣿⣿⠀⠀⣭⡻⣷⠀⢻⣿⠀⣬⡻⣷⡄⢸⣿⠀⣿⡟⠃⣿⣿⣿⣿⡇⠀⠀
⠀⠀⠸⠿⠇⠻⠿⠁⠿⠿⠇⠀⠸⠿⠸⠇⠻⠿⠿⠀⠘⠿⠿⠃⠘⠿⠃⠿⠏⠿⠟⠿⠛⠿⠀⠻⠟⠀⠿⠇⠿⠊⠿⠃⠻⠿⠻⢃⠐⠿⠋⠿⠇⠀⠻⠿⠟⠀⠸⠟⠀⠻⠷⠟⠁⠸⠿⠀⠿⠿⠇⠿⠛⠟⠻⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⣀⠚⠀⠀⠀⠀⠐⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠨⠏⠁⠐⠢⠀⢀⡀⠀⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⡄⠀⠈⠭⠤⢀⡄⠀⠒⣶⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⢀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠅⠀⢀⡀⠀⢀⡀⢠⣤⢤⡤⣤⣤⣤⢰⠀⠐⠃⠀⠀⠀⠀⢀⣿⣿⡀⢀⡀⠀⠀⢀⣠⣾⣿⣷⣄⡀⠀⠀⠀⠰⣦⣤⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢴⣮⠁⠀⠄⠈⠁⠻⠻⠋⠀⠀⠀⡀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠁⠀⠀⠐⠀⠀⠀⠀⠀⡀⢀⣴⢿⠀⠀⠁⠰⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⠀⠀⠃⡀⠀⠀⢀⠀⢶⠬⠁⠀⠁⡙⠗⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⠟⠋⢀⣶⡀⠀⠀⠀⠀⣠⣄⠈⠛⠛⣿⣿⣿⣿⣿⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢾⣿⡇⠀⢀⠒⠀⢠⣤⠀⣄⡀⠘⠙⠀⠀⠁⠀⠀⠀⠀⠀⠘⢿⣿⡇⠀⠀⠀⠘⠛⠧⠀⠀⠀⠀⣻⡿⠆⠀⠀⣸⣿⣿⣿⣿⡿⢿⡿⣃⠀⠀⠀⠀⢠⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣴⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠⠀⠀⢸⣷⣰⣶⡄⣀⣀⣀⣀⣀⠀⠒⢀⣀⠀⠀⣶⣿⣿⣿⣿⣿⣾⣿⣿⣶⣿⠶⠘⠀⠨⠆⠒⠖⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠐⠀⠈⣿⣿⣿⣿⣿⣿⣏⠁⠀⠀⠉⠛⠿⠗⠂⠛⠛⠛⠛⠟⠛⣉⣀⣀⡀⢠⣤⢠⣤⣤⢠⣦⠰⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣀⡀⣀⡀⣤⡌⢀⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠈⣿⣧⣿⠀⠰⣂⠀⠀⣠⣠⣤⣶⣾⣢⣤⣤⣄⠀⠀⠀⠀⠀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣶⡆⣶⣷⢸⣿⢸⣿⠘⢻⣿⠁⢿⣇⣿⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⢰⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣽⣦⣶
⣄⠠⢬⡀⠀⠀⡀⢀⢀⣜⡆⢁⣄⡐⡀⠀⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣈⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠸⠟⠛⠛⠛⠋⠉⠉⠉⢉⣀⠉⢿⣿⣿⣿⣿⣿
⣧⢣⣾⣶⣇⣿⣴⡒⡶⢞⣶⢆⡔⣸⢠⡄⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⢀⡙⠛⢉⣀⣈⠉⠁⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣄⡘⡗⢠⣤⠀⢠⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⢸⣟⣿⣿⣿⣿
⣿⣿⢽⣯⣽⣿⢟⣻⣇⢈⣶⠄⠉⠈⠹⠷⠄⠉⠉⠤⡤⠤⠶⠖⠒⠓⠂⣈⡒⣀⡀⢀⣤⣤⡀⠛⢠⣶⠀⣶⡶⠆⠀⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⣆⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠘⢿⣿⣿⡝⠉
⣿⣿⣿⣿⠟⠉⣀⣀⠀⢀⣀⢠⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣄⣿⠁⣿⡇⣿⣧⣸⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠀⢹⣿⠀⢿⣧⢰⣶⢈⣙⢻⣧⠈⠈⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⢈⣿⣟⡲⣄
⣿⣿⣿⣿⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⠠⡆⢸⣿⠀⢸⣿⢸⣷⡘⠿⠾⠛⣠⣀⡛⢛⣁⣋⣉⣡⣬⣭⣤⣀⣀⣤⣤⣤⣤⣤⣴⣾⣿⣋⢭⣿
⣿⣿⣿⣿⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⠛⠀⠀⠛⠁⠈⠉⠀⠁⣈⣠⣶⣦⣶⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣆⠘⠿⠾⠟⠀⠿⠀⠛⠂⠛⠛⣃⣈⣉⢉⣤⣬⣤⣤⣤⣶⣴⣶⢰⣿⡆⠀⠀⢀⣤⣼⣷⣾⣿⣦⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣶
⣿⣿⣿⣿⣿⣶⣤⣤⣴⣶⣶⣶⡶⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣾⣿⡇⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⡀⠈⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣐⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⢹⣿⠿⣿⣦⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀
⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⠈⢱⡞⣿⢀⡿⣯⣇⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣟⢸⣿⣿⣿⡏⠀⠀
⣿⣿⣇⠀⠀⣸⣿⣿⣻⠏⠈⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣛⠀⠘⠓⣿⢸⣿⣾⣿⣿⣷⣤⡉⢻⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠛⠘⠻⠿⢿⠀⠀⠀
⣿⣿⣿⠀⠀⠉⠙⠛⠃⠀⠤⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡿⢿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠛⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣇⢸⣇⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣶⣿⣶⣦⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠀⣀⡉⣒⡀⢀⣀⣀⣀⡀⣘⢛⣀⣀⠀⣀⣀⡀⠀⠀⣀⣀⠀⣀⣀⣀⢀⣀⣚⢛⣋⣙⡛⢛⣛⣛⡛⣛⡛⣛⡛⣂⡀⢀⣉⡈⠀⠀⢀⣀⡀⠀⠀⢀⣀⡀⢀⣀⢀⣀⢀⣀⡀⢀⣀⣀⣀⣀⣀⡀⣀⣀⢀⣀⡀⠀⠀
⠀⠀⣿⡇⣿⣷⣿⡏⣿⡟⠃⠿⢸⣿⣻⡇⣿⣟⠃⠀⢸⣯⡿⠇⢻⣿⡟⢸⡿⣿⢸⣿⣿⣿⠘⣿⣿⠃⣿⡇⣿⣧⣿⣧⣿⡿⠿⠄⠀⣾⣿⣿⠀⠀⣿⣿⠿⠘⣿⣾⠇⣿⣟⠿⠘⢻⣿⠋⣿⡏⠃⣿⣿⣸⣿⡇⠀⠀
⠀⠀⢹⣿⡿⣿⣿⡇⣿⡟⠃⠀⢸⣿⢻⡆⣿⣿⠃⠀⢨⣝⣿⡆⢸⣿⡇⣿⣧⣿⣼⣿⣿⣷⠀⣿⣿⠀⣿⡇⣿⢿⣿⡿⣿⣿⣿⡆⠀⣿⣿⣿⠀⠀⣭⡻⣷⠀⢻⣿⠀⣬⡻⣷⡄⢸⣿⠀⣿⡟⠃⣿⣿⣿⣿⡇⠀⠀
⠀⠀⡸⠿⠇⠻⠿⠁⠿⠿⠇⠀⠸⠿⠸⠇⠻⠿⠿⠀⠘⠿⠿⠃⠘⠿⠃⠿⠏⠿⠟⠿⠛⠿⠀⠿⠟⠀⠿⠇⠿⠋⠿⠃⠻⠿⠻⢃⠐⠿⠋⠿⠇⠀⠻⠿⠟⠀⠸⠟⠀⠻⠷⠟⠁⠸⠿⠀⠿⠿⠇⠿⠛⠟⠻⠇⠀⠀
⠀⠀⠑⠶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢀⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⡇⠀⠀⠐⠀⠀⠀⠶⠀⢰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣧⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⡀⢀⡀⠀⠀⢀⣠⣾⣿⣷⣄⡀⠀⠀⠀⠰⣦⣤⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⠄⡄⡀⠀⠀⠈⠁⠀⠈⠈⠀⢁⣀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⡀⠀⠄⠀⠈⠁⠁⠈⠀⠈⠚⠃⠀⠀⠆⢀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠂⠀⠀⠀⠁⠀⠈⠃⠀⠀⠀⠰⠆⠀⠀⠀⢀⡤⠀⠞⠄⠀⢸⣿⣿⣿⡿⠟⠋⢀⣶⡀⠀⠀⠀⠀⣰⣄⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠹⢆⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠦⣬⣀⠜⢿⢿⡇⠀⠀⠀⠘⠛⠧⠀⠀⠀⠨⣿⡿⠆⠀⠀⣼⣿⣿⣿⣿⡿⠿⢟⣿⠉⣤⣀⣠⡻⠢⣤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⢴⠚⠀⠀⠀⠀⠀⣤⡐⡤⠋⠬⠦⢼⣷⣦⣶⡄⢀⣀⣀⣀⡀⠀⠒⣀⣀⡀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠿⠹⠲⠠⠟⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢟⡀⣄⡀⠄⡀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠴⠆⠀⡀⣁⣤⡄⣈⢈⣿⣿⣿⣿⣿⣿⣏⡁⠀⠀⠙⠛⠿⠟⠚⠛⠛⠛⠛⠟⠛⣉⣁⣀⡀⢠⣤⢩⣤⣤⢠⣦⠰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠄⠩⣴⠠⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠀⠀⠁⠈⠉⠁⠈⢉⣈⣀⡀⣀⡉⣤⡌⢁⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠘⣿⣧⣿⠀⠀⡆⡀⠀⣶⡧⣴⣷⣿⣶⣦⣽⣆⠤⠠⢄⡀⢀
⠀⠀⠉⠒⠀⠀⠀⠀⢀⢀⡀⣐⡉⣙⡂⠀⣶⡆⣾⣷⢸⣿⢸⣿⠘⢻⣿⠁⢿⣇⣿⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⢰⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣯
⠀⠪⡁⢠⠀⠀⠠⢂⡾⣿⣆⢒⠗⠈⣷⠀⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣌⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠸⠿⠛⠛⠛⠛⠉⠉⠉⢉⣀⠉⢿⣿⣿⣿⣿⣿
⣗⣻⣻⣶⠄⠀⠈⠮⣂⣸⣷⣶⢵⣫⣿⡄⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⣀⡙⠛⣉⠀⢈⡉⠁⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣄⠘⡗⢠⣤⠀⢰⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⢸⣿⣿⣿⣿⣿
⣿⣿⢿⡷⣿⣿⣭⡽⢼⣿⣿⠎⠈⠙⠿⠇⠄⠉⠉⢤⡤⠤⠶⠖⠚⠛⠀⣙⡉⣉⡉⢉⣤⣥⡀⠉⢠⣶⠀⣶⡶⠆⠀⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⣆⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠘⢿⣿⣿⡿⠉
⣿⣿⣿⣾⠟⠉⣀⣀⠀⢀⣀⢠⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣄⣿⠁⣿⡇⣿⣧⣹⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠀⢹⣿⠀⢿⣧⢰⣶⢈⣙⢻⣧⠈⠘⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⢸⣯⣿⣇⣦
⣿⣿⣿⣿⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⢀⡆⢸⣿⠀⢸⣿⢸⣿⡘⠿⠾⠛⣠⣄⡛⢛⣁⣋⣉⣡⣬⣭⣤⣀⣀⣤⣤⣤⣤⣤⣶⣾⣿⣿⣾⣿
⣿⣿⣿⣿⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⠛⠀⠀⠛⠁⠈⣉⣬⠿⣮⣀⣀⣤⣴⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣆⠘⠿⡾⠟⠀⠿⠀⠛⠂⠛⠛⢃⣈⠉⣉⣤⣬⣤⣤⣤⣶⣶⣶⡤⣾⣷⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠛⠀⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻
⣿⣿⣿⣿⣿⣶⣤⣤⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⣿⣿⡄⠀⠀⢠⣤⣿⣦⣿⣿⣧⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⡇⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠄⢤⣀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣧⣆⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀
⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠷⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⡽⣿⣿⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣟⢹⣿⣿⣿⡟⠀⠀
⣿⣿⣇⠀⠀⢸⣿⣿⣟⠛⠉⠁⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠇⠈⣇⢸⢠⣷⣼⣿⣿⣶⣤⡌⠻⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠛⠚⠻⠿⢿⠀⠀⠀
⣿⣿⣿⠀⠀⠉⠛⠛⠀⠄⠥⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢹⣼⡿⠿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⠀⠀⠀⠠⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⡏⢸⣟⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣠⣤⣶⣷⣾⣿⣶⣿⣶⣦⣀⠀⠀⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⡀⣀⡀⢀⣀⣀⣀⡀⣐⢀⣀⣀⠁⣀⣀⡀⠀⠀⣀⣀⠀⣀⣀⣀⢀⣈⣛⢛⣋⣙⡛⢛⣛⣛⡛⣛⡛⣛⡛⣂⡀⢀⣀⡀⠀⠀⢀⣀⡀⠀⠀⢀⣀⡀⢀⣀⢀⣀⢀⣀⡀⢀⣀⣀⣀⣀⣀⡀⣀⣀⢀⣀⡀⠀⠀
⠊⠀⣿⡇⣿⣷⣿⡏⣿⡟⠃⠿⢸⣿⣻⡇⣿⣿⠃⠀⢸⣯⡿⠇⢻⣿⡟⢸⡿⣿⢸⣿⣿⣿⠘⣿⣿⠃⣿⡇⣿⣧⣿⣧⣿⡿⠿⠄⠀⣾⣿⣿⠀⠀⣿⣿⠿⠘⣿⣾⠇⣿⣟⠿⠘⢻⣿⠋⣿⡏⠃⣿⣿⣸⣿⡇⠀⠀
⠀⠀⢹⣿⡿⣿⣿⡇⣿⡟⠃⠀⢸⣿⢻⡆⣿⣿⠃⠀⢨⡝⢿⡆⢸⣿⡇⣿⣧⣿⣼⣿⣿⣷⠀⣿⣿⠀⣿⡇⣿⢿⣿⡿⣿⣿⣿⡆⠀⣿⣿⣿⠀⠀⣭⡻⣷⠀⢻⣿⠀⣭⡻⣷⡄⢸⣿⠀⣿⡟⠃⣿⣿⣿⣿⡇⠀⠀
⠀⠀⠸⠿⠇⠻⠿⠁⠿⠿⠇⠀⠸⠿⠸⠇⠻⠿⠿⠀⠘⠿⠿⠃⠘⠿⠃⠿⠏⠿⠟⠿⠛⠿⠀⠿⠟⠀⠿⠇⠿⠊⠿⠇⠻⠿⠻⢃⠐⠿⠋⠿⠇⠀⠻⠿⠟⠀⠸⠟⠀⠻⠷⠟⠁⠸⠿⠀⠿⠿⠇⠿⠛⠟⠻⠇⠀⠀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠖⠁⠀⠀⠀⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⡇⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠐⠚⠂⠀⠈⠁⠀⠀⠁⠀⠀⠀⣁⠀⠀⠀⡀⠘⠆⣸⣿⣿⣿⠿⠋⠁⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣇⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠂⠀⠀⠙⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠀⠀⠀⠀⠀⢀⣿⣿⡀⢀⡀⠀⠀⢀⣀⣾⣿⣷⣄⡀⠀⠀⠀⠰⣤⣤⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⠀⠀⠀⠈⡶⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠆⣄⢤⠀⠀⠀⠀⠀⠀⣶⣯⡀⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣀⢤⠈⠀⠀⠀⣴⠆⣀⡀⠀⢉⡀⠀⠀⠀⠀⠀⠀⠛⠙⢸⣿⣿⣿⠿⠛⠋⢀⣶⡀⠀⠈⠀⠀⣠⣄⠉⠛⠻⢿⣿⣿⣿⣿⣿⣷⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠋⡶⠶⠁⠀⠀⠀⠀⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⢿⢿⡇⠀⠀⠀⠘⠛⠥⠀⠀⠀⠀⣿⡿⠃⠀⠀⣸⣿⣿⣿⣿⡿⢿⣟⡏⡩⠤⠀⠀⢨⠣⠀⢠⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⡄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⢒⢈⣤⣸⣷⣰⣶⡄⣀⣀⣀⣀⡀⠀⠒⣀⣀⠀⠀⣶⣿⣿⣿⣿⣿⣤⣴⣿⣿⣿⡴⠴⠆⠎⠿⠾⠟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠩⡆⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣠⣸⣯⡙⠛⢙⣿⣿⣿⣿⣿⣿⣋⡁⠀⠀⠉⠻⠿⠟⠚⠙⠛⠛⠛⠟⠛⣉⣀⣈⡉⢠⣤⢠⣤⣤⢠⣦⠰⣶⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣂⠂⠰⢠⣾⠈⠛⠀⣄⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢁⣈⣀⡀⣀⡁⣤⡌⢀⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠈⣿⣧⣿⠀⠘⡄⡀⠀⢤⣬⣲⣦⣴⣦⣜⣴⣮⣴⡆⣀⢀⠀
⠀⠂⣀⣀⡋⠂⠀⠙⢣⣔⣤⡄⡄⠀⡄⠀⣶⡆⣶⣷⢸⣿⢸⣿⠘⢻⣿⠁⢿⣇⣿⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢐⠛⢔⠀⡄⠀⠄⠀⢼⣦⡙⢈⣤⠐⠀⠀⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣌⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠸⠟⠛⠛⠛⠛⠉⠉⠉⢉⣉⠉⢻⣿⣿⣿⣿⣿
⠯⢑⣦⡀⠀⠾⠘⡃⠰⡋⣻⠶⡟⣙⣿⡄⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⣀⣙⠛⠉⠀⠈⠉⠁⠀⢀⣀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣄⡘⡗⢠⣤⠀⢠⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⢨⣿⣿⣿⣿⣿
⣿⣿⣻⣵⣆⣭⣗⣶⣦⡋⣻⠯⠁⠉⠍⠧⠀⠉⠉⠤⡤⠤⠶⠶⠞⠛⠒⣛⡛⣋⡛⢋⣭⣭⡉⠉⢠⣶⠀⣶⡶⠆⠀⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⣆⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠘⢿⣿⣿⣯⡏
⣿⣿⣿⣿⠟⠉⣀⣀⠀⢀⣀⢠⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣆⣿⠁⣿⡇⣿⣧⣹⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠀⢹⣿⠀⢿⣧⢰⣶⢈⣙⢻⣧⠈⠘⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⢸⣿⣿⣿⡧
⣿⣿⣿⣿⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⠀⠁⢸⣿⠀⢸⣿⢸⣿⡘⠿⠾⠛⣠⣀⡛⢛⣁⣋⣉⣡⣬⣭⣬⣀⣀⣤⣤⣤⣤⣤⣴⣾⣿⣿⣦⣤
⣿⣿⣿⣿⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⠛⠀⠀⠛⢁⣈⡉⠀⣤⣬⣤⣴⣦⣶⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠿
⣿⣿⣿⣿⣆⠘⠿⠾⠟⠀⠿⠀⠛⠂⠛⠛⢃⡈⢉⣉⣠⣬⣤⣤⣤⣶⣶⣶⣦⣾⣿⣷⠀⠾⣧⠷⠀⠀⠈⠻⠁⣼⣿⠻⣿⣿⡟⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⠙
⣿⣿⣿⣿⣿⣶⣤⣤⣴⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⠆⠋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢇⣿⣿⣿⡇⠀⠀⢀⣀⣶⣄⣶⣶⡏⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣾⣿⣿⣿⡇⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⡜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡁⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀
⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⡿⣿⣧⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠏⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣹⣿⣷⡆⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⢺⣿⣿⣿⡟⠀⠀
⣿⣿⣇⠀⠁⢸⣿⣿⣟⡛⠿⠃⠀⠈⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⠿⠃⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⢁⣧⣹⣿⣧⣤⣀⠈⠿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠘⠛⠘⠿⠿⣿⠁⠀⠀
⣿⣿⣿⠀⠁⠈⠛⠛⠫⠄⠈⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⡉⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⣾⢸⣿⢿⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⠀⠀⠀⠠⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡜⠙⣷⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣷⣾⣿⣶⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⡀⣀⡀⢐⣀⣀⣀⡀⣀⢀⣀⣀⠀⣀⣀⡀⠀⠀⣈⣃⠀⣀⣀⣀⢀⣀⣚⢛⣋⣙⡛⢛⣛⣙⡛⣛⡛⣛⡛⣂⡀⢀⣀⡀⠀⠀⢀⣀⡀⠀⠀⢀⣀⡀⢀⣀⢀⣀⢀⣀⡀⢀⣀⣀⣀⣀⣀⡀⣀⣀⢀⣀⡀⠀⠀
⠀⠀⣿⡇⣿⣧⣿⡏⣿⡟⠃⠿⢸⣿⣻⡇⣿⣟⠃⠀⢸⣯⡿⠇⢻⣿⡟⢸⡿⣿⢸⣿⣿⣿⠘⣿⣿⠃⣿⡇⣿⣧⣿⣇⣿⡿⠿⠄⠀⣾⣿⣿⠀⠀⣿⣿⠿⠘⣿⣾⠇⣿⣟⠿⠘⢻⣿⠋⣿⡏⠃⣿⣿⣸⣿⡇⠀⠀
⠀⠀⢹⣿⡿⣿⣿⡇⣿⡟⠃⠀⢸⣿⢻⡆⣿⣿⠃⠀⢨⡝⢿⡆⢸⣿⡇⣿⣧⣿⣼⣿⣿⣷⠀⣿⣿⠀⣿⡇⣿⢿⣿⡿⣿⣿⣿⡆⠀⣿⣿⣿⠀⠀⣭⡻⣷⠀⢻⣿⠀⣬⡻⣷⡄⢸⣿⠀⣿⡟⠃⣿⣿⣿⣿⡇⠀⠀
⡀⠀⠸⠿⠇⠻⠿⠁⠿⠿⠇⠀⠸⠿⠸⠇⠻⠿⠿⠀⠘⠿⠿⠃⠘⠿⠃⠿⠏⠿⠟⠿⠛⠿⠀⠻⠟⠀⠿⠇⠿⠊⠿⠇⠻⠿⠻⢃⠐⠿⠋⠿⠇⠀⠻⠿⠟⠀⠸⠟⠀⠻⠷⠟⠁⠸⠿⠀⠿⠿⠇⠿⠛⠟⠻⠇⠀⠀
⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠂⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠘⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠘⠀⠀⠛⠀⠀⠀⠀⠉⠀⢠⠀⠀⠉⠄⠀⠀⠀⠀⠀⢀⣿⣿⡀⢀⡀⠀⠀⢀⣠⣾⣿⣷⣄⡀⠀⠀⠀⠰⣤⣄⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠃⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠆⠐⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣶⣶⣦⣿⣿⣿⣿⠿⠟⠋⢀⣶⡀⠀⠀⠀⠀⢰⣄⠈⠛⠻⣿⣿⣿⣿⣿⣿⣷⣿⣿⠐⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠒⠀⠀⠀⠀⠀⠁⠀⠀⡘⠂⠀⣤⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢿⢿⡇⠀⠀⠀⠘⠛⠧⠀⠀⠀⠈⣿⡿⠃⠀⠀⣼⣿⣿⣿⣿⡟⡿⢿⣿⣶⢖⠁⠀⢀⠠⡠⠀⢂⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢴⡤⠄⠀⠴⠀⠀⠀⠀⠷⠄⠀⠁⠻⠶⠆⠀⠀⢐⣠⡀⣀⠀⡄⢸⣷⣢⣶⡄⣀⣀⣀⣀⣀⠀⠒⣀⣀⡀⢀⣶⣿⣿⣿⣿⣿⣀⣾⣿⣿⣿⠿⠴⠄⠅⠺⠶⠟⠊⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠑⠗⠂⠀⠀⠀⠐⠀⠀⠐⣶⠀⠀⠇⠀⠀⠡⣢⠏⣡⡉⣷⣖⡈⣿⣿⣿⣿⣿⣿⣋⡁⠀⠀⠉⠻⠿⠟⠚⠙⠛⠛⠛⠟⠛⣈⣀⣀⡀⢠⣤⢠⣤⣤⢠⣦⠐⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠴⠦⠘⡝⠿⠖⠀⠀⠀⠀⢶⡏⠀⠠⠊⠁⠀⠘⠙⠉⠈⠈⠉⠁⢀⣈⣀⡀⣀⡉⣤⡌⢀⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠈⣿⣧⣿⠀⠀⠀⠀⢐⣴⣤⣤⣼⣯⣶⣶⣦⣄⣂⣀⣀⡀⠀
⣄⠐⣊⡧⠀⠀⠀⡤⣤⣤⣈⣣⠙⠳⠆⠀⣶⡆⣶⣷⢸⣿⢸⣿⠘⢻⣿⠁⢿⣇⣿⣇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⠀⣼⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⡏⡀⠇⠀⠄⠄⣢⢵⣾⡧⠉⡘⡇⣶⣷⠀⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣈⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠼⠟⠛⠛⠙⠛⠉⠉⠉⢁⣀⠉⢿⣿⣿⣿⣿⣿
⣯⣷⣷⢂⢖⣟⣿⡬⣿⢽⠋⠋⣾⣿⣿⡄⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⣀⣙⡛⠉⠀⠈⠉⠁⠀⣀⣀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣄⡘⡗⢠⣤⠀⢰⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⢸⣿⣿⣿⣿⣿
⣿⣟⣽⣿⣽⣿⣿⣿⠟⠒⡽⠀⠉⠻⠉⠧⠀⠉⠉⠠⡤⠤⠶⠶⠞⠛⠒⣛⡛⣋⡛⢋⣭⣭⡉⠁⢠⣶⠀⣶⡶⠆⠁⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⣆⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠘⢿⣿⣿⠉⠋
⣿⣿⣷⣿⠟⠉⣀⣀⠀⢀⣀⢠⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣄⣿⠁⣿⡇⣿⣧⣹⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠀⢹⣿⠀⢻⣧⢰⣶⢈⣙⢻⣧⠈⠈⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⢸⣿⣿⣷⣴
⣿⣿⣿⣿⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⠀⡇⢸⣿⠀⢸⣿⢸⣿⡘⠿⠾⠛⣠⣀⡛⢛⣁⣋⣉⣡⣬⣭⣬⣀⣀⣠⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿
⣿⣿⣿⣿⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⠛⠀⢀⡛⠁⣈⣩⣤⣷⣮⣤⣶⣶⣴⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⣿⣿⣿⣿⣆⠘⠿⠾⠟⠀⠿⠀⠛⠂⠛⠛⢃⡈⢉⣉⣤⣬⣤⣤⣤⣶⣶⣶⣦⣶⣿⣿⣦⢺⣿⡀⠚⠁⢸⣿⣿⣿⡇⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦
⣿⣿⣿⣿⣿⣶⣤⣤⣴⣶⣶⣶⣶⣶⣾⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⡌⢿⣿⡄⠀⠀⠘⣿⠟⢀⣼⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣇⣿⣿⣿⣿⡁⠀⠈⠀⠀⠀⠀⠀⠀⣼⣿⡄⣿⢿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⡀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣼⣿⣿⣿⣿⠁⠀⠀⢀⢀⣤⠀⣤⣴⡟⠉⠁⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀
⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⢻⣿⣿⣿⡟⠀⠀
⣿⣿⣇⠀⡁⢸⣿⣿⣟⠛⠛⠁⠈⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣇⣀⠈⠻⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠘⠛⠘⠿⢿⣿⠁⠀⠀
⣿⣿⣿⠀⠁⠈⠙⠛⠃⠄⠀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠙⠛⠻⠋⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣷⣦⡈⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠋⠉⢿⢸⡧⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣷⣿⣿⣶⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⡀⣀⡀⢀⣀⣀⣀⡁⣀⢀⣀⣀⠀⣀⣀⡀⠀⠀⣀⣀⠀⣀⣀⣀⢀⣀⣚⢛⣋⣙⡛⢛⣛⣙⡛⣛⡛⣛⡛⣂⡀⢀⣀⡀⠀⠀⢀⣀⡀⠀⠀⢀⣀⡀⢀⡀⢀⣀⢀⣀⡀⢀⣀⣀⣀⣀⣀⡀⣀⣀⢀⣀⡀⠀⠀
⠀⠀⣿⡇⣿⣷⣿⡏⣿⡟⠃⠿⢸⣿⣻⡇⣿⣟⠃⠀⢸⣯⡿⠇⢻⣿⡟⢸⡿⣿⢸⣿⣿⣿⠘⣿⣿⠃⣿⡇⣿⣧⣿⣇⣿⡿⠿⠄⠀⣾⣿⣿⠀⠀⣿⣿⠿⠘⣿⣾⠇⣿⣟⠿⠘⢻⣿⠋⣿⡟⠃⣿⣿⣸⣿⡇⠀⠀
⠀⠀⢹⣿⡿⣿⣿⡇⣿⡟⠃⠀⢸⣿⢻⡆⣿⡿⠃⠀⢨⣝⢿⡆⢸⣿⡇⣿⣧⣿⣼⣿⣿⣷⠀⣿⣿⠀⣿⡇⣿⢿⣿⡿⣿⣿⣿⡆⠀⣿⣿⣿⠀⠀⣭⡻⣷⠀⢻⣿⠀⣬⡻⣷⡄⢸⣿⠀⣿⡟⠃⣿⣿⣿⣿⡇⠀⠀
⠀⠀⠸⠿⠇⠻⠿⠁⠿⠿⠇⢀⠸⠿⠸⠇⠻⠿⠿⠀⠘⠿⠿⠃⠘⠿⠃⠿⠏⠿⠟⠿⠛⠿⠀⠻⠟⠀⠿⠇⠿⠊⠿⠃⠻⠿⠻⢃⠐⠿⠋⠿⠇⠀⠻⠿⠟⠀⠸⠟⠀⠻⠷⠟⠁⠸⠿⠀⠿⠿⠇⠿⠛⠟⠻⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⢠⡄⠀⠀⢤⠀⡀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣇⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠰⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣤⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⢀⣿⣿⡀⢀⡀⠀⠀⢀⣀⣾⣿⣷⣄⡀⠀⠀⠀⠰⣤⣤⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠁⠀⠂⠈⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠰⡂⠀⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠻⡷⠀⠀⠀⣤⠀⠀⠀⠀⠀⠐⠛⠶⠀⠐⠖⠛⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⢲⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠾⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⡿⠟⠋⢀⣶⡀⠀⠀⠀⠀⣠⣄⠉⠛⠛⣿⣿⣿⣿⣿⣿⣷⣿⣿⣉⠀⠀⠀⠁⠀⠀⠀⠃⠀⢀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣧⡀⠀⠀⡀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣁⠟⠘⣿⣿⡇⠀⠀⠀⠘⠛⠥⠀⠀⠀⠀⣿⡿⠢⠀⠀⣸⣿⣿⣿⣿⡟⠿⣟⡛⠓⢢⡢⠀⠠⢀⠀⠀⡀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⠉⡛⠀⠀⡯⠀⠀⠈⠁⠀⠒⣰⠶⠂⠀⢠⠀⠀⣼⣷⣰⣶⡄⣀⣀⣀⣀⣀⠀⠒⣀⣀⠀⠀⣾⣿⣿⣿⣿⣿⣀⣼⣿⣿⣦⠍⠁⠇⠻⠏⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠤⢘⣀⠀⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡠⠉⠧⠇⠀⡾⢿⣿⣿⣿⣿⣿⣿⣋⠁⠀⠀⠈⠛⠿⠟⠚⠙⠛⠛⠛⠿⠛⣉⣁⣀⡀⢠⣤⢠⣤⣤⢠⣦⠐⣶⠀⢀⠀⠀⠐⠀⠀⡦⢆⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀
⢘⣄⠘⠁⡈⡛⣃⢀⠀⠓⠁⠃⠀⠀⠀⠀⠉⠃⠀⠈⠁⠁⠀⠀⢁⣈⣀⡀⣀⡁⣤⡌⢀⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠈⣿⣧⣿⠀⣸⡌⢀⡀⣶⡶⣧⣿⣿⣿⣾⣶⡶⢄⢂⣄⠀⠀
⣨⣕⠻⡎⠥⠀⠑⠍⢈⢹⡆⢀⠂⣊⣡⠀⣶⡆⣶⣷⢸⣿⢸⣿⠘⢻⣿⠁⢿⣇⣿⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⢰⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷
⠑⡒⣃⠁⠃⠀⠀⠀⢐⣺⡇⢨⣬⣬⣥⠀⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣈⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠸⠟⠛⠛⠛⠛⠉⠉⠉⢉⣀⠉⢿⣿⣿⣿⣿⣿
⣶⣷⣏⠠⣀⢀⢀⡄⠄⢵⣟⠛⢭⣽⣿⡆⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⣀⣙⣛⡉⠀⠈⡉⠁⠀⢀⣀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣄⠘⡗⢠⣤⠀⢠⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⢸⣿⣿⣿⣿⣿
⢹⣿⣷⣾⣿⣿⣷⣝⠘⠳⣁⠎⠿⠈⠟⠶⠀⠉⠉⠤⡤⠤⠖⠶⠞⠛⠒⣛⡛⣛⡛⢋⣥⣭⡉⠁⢠⣶⠀⣶⡶⠆⠁⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⣆⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠘⢿⣿⣿⡟⠋
⣿⣿⣷⣿⠟⠉⣀⣀⠀⢀⣀⢠⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣄⣿⠁⣿⡇⣿⣧⣹⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠀⢹⣿⠀⢻⣧⢰⣶⢈⣙⢻⣧⠀⠈⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⢸⣿⣿⣾⣶
⣿⣿⣿⣿⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⠀⡇⢸⣿⠀⢸⣿⢸⣿⡘⠿⠾⠛⣠⣀⡛⣛⣁⣋⣉⣡⣬⣭⣬⣀⣀⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿
⣿⣿⣿⣷⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⠛⣀⢀⣛⡁⢈⣩⣤⣿⣮⣤⣶⣶⣶⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿
⣿⣿⣿⣿⡆⠘⠿⠾⠟⠀⠿⠀⠛⠂⠛⠛⢃⠈⣉⣉⣤⣬⣤⣤⣤⣶⣶⣶⣦⣾⣿⣿⣦⢻⣿⡏⢸⢿⠃⣸⣿⣿⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡑
⣿⣿⣿⣿⣿⣶⣤⣤⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡇⣿⣧⡘⠂⠠⣿⣿⣿⡿⠂⢈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣿⣿⣿⡇⠹⣿⡦⠀⠀⠘⣿⠟⠀⣴⣞⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣄⣿⣿⣁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⠇⠀⠀⣀⢠⡤⢀⣤⣼⣿⠟⠉⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀
⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⡿⠀⠀⢰⣿⣿⣿⣿⣿⣿⣶⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⡿⢻⣿⣿⣿⡟⠀⠀
⣿⣿⣧⠀⠃⢸⣿⣿⡿⡿⢛⠀⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⠏⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠻⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠘⠛⠚⠿⢿⣿⠁⠀⠀
⣿⣿⣿⠀⠁⠘⠛⠛⠢⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠙⠛⠛⠛⠿⠋⠀⠠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡉⣻⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⠀⠀⠀⠠⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⡀⠐⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⡧⡈⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣷⣾⣿⣶⣿⣶⣦⣰⣄⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⡀⣀⡀⢀⣀⣀⣀⡀⣀⢀⣀⣀⠀⣀⣀⡀⠀⠀⣀⣀⠀⣀⣀⣀⢀⣀⣚⢛⣋⣙⡛⢛⣛⣛⡛⣛⡛⣛⡛⣂⡀⢀⣀⡀⠀⠀⢀⣀⡀⠀⠀⢀⣀⡀⢀⣀⢀⣀⢀⣀⡀⢀⣀⣀⣀⣀⣀⡀⣀⣀⢀⣀⡀⠀⠀
⠀⠀⣿⡇⣿⣧⣿⡏⣿⡟⠃⠿⢸⣿⣻⡇⣿⣿⠃⠀⢸⣯⡿⠇⢻⣿⡟⢸⡿⣿⢸⣿⣿⣿⠘⣿⣿⠃⣿⡇⣿⣧⣿⣇⣿⡿⠿⠄⠀⣾⣿⣿⠀⠀⣿⣿⠿⠘⣿⣾⠇⣿⣟⠿⠘⢻⣿⠋⣿⡏⠃⣿⣿⣸⣿⡇⠀⠀
⡀⠀⢹⣿⡿⣿⣿⡇⣿⡟⠃⠀⢸⣿⢻⡆⣿⣿⠃⠀⢨⣝⢿⡆⢸⣿⡇⣿⣧⣿⣼⣿⣿⣷⠀⣿⣿⠀⣿⡇⣿⢿⣿⡿⣿⣿⣿⡆⠀⣿⣿⣿⠀⠀⣭⡻⣷⠀⢻⣿⠀⣬⡻⣷⡄⢸⣿⠀⣿⡟⠃⣿⣿⣿⣿⡇⠀⠀
⠁⠀⠸⠿⠇⠻⠿⠁⠿⠿⠇⠀⠸⠿⠸⠇⠻⠿⠿⠀⠘⠿⠿⠃⠘⠿⠃⠿⠏⠿⠟⠿⠛⠿⠀⠻⠟⠀⠿⠇⠿⠊⠿⠇⠻⠿⠻⢃⡐⠿⠋⠿⠇⠀⠻⠿⠟⠀⠸⠟⠀⠻⠷⠟⠁⠸⠿⠀⠿⠿⠇⠿⠛⠟⠻⠇⠀⠀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⡀⠀⠀⠀⠀⠂⠀⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⢀⡄⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠘⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⣿⡇⠀⢐⣿⣿⡀⢀⡀⠀⠀⢀⣀⣾⣿⣷⣄⡀⠀⠀⠀⠰⣤⣤⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠂⠐⠂⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⢿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠟⠋⢀⣶⡀⠀⠀⠀⠀⣠⣄⠈⠛⠛⣿⣿⣿⣿⣿⣿⣷⣿⣿⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⢚⢿⣿⡇⠀⠀⠀⠘⠛⠡⠀⠀⠀⠀⣻⡿⠆⠀⠀⢸⣿⣿⣿⣿⡟⠿⢟⣃⡀⠠⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠛⣿⣿⣧⣠⣦⡄⢀⣀⣀⣀⣀⠀⠒⣀⣀⠀⠀⣴⣿⣿⣿⣿⣿⣀⣼⣿⣷⡅⠃⠀⠈⠀⠄⠀⠅⠀⠀⠀⢰⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⠀⠀⠘⠂⢀⠀⣠⢰⡄⠀⠀⡀⠀⠂⡀⠤⠀⣿⣳⢐⡋⠈⣿⣿⣿⣿⣿⣿⣋⠁⠀⠀⠉⠛⠿⠓⠚⠙⠛⠛⠛⠿⠛⣉⣁⣀⡀⢠⣤⢠⣤⣤⢠⣦⠐⣶⠀⠀⠀⠿⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠈⠀⠲⠀⣀⠘⠆⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠉⠉⠉⠈⢁⣈⣁⡀⣀⡁⣤⡌⢀⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠈⣿⣧⣿⠀⢠⣆⡀⢀⣠⡤⣹⣴⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀
⠀⠀⠀⢀⡄⠀⠉⠀⠀⠀⠀⣄⠀⢸⡇⠀⣶⡆⣶⣷⢸⣿⢸⣿⠘⢻⣿⠁⢿⣇⣿⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⢰⣿⣷⣾⣿⣿⣿⣷⣶⣶⣾⣶⣿⣿⣿⣾⣾⣯
⠀⣠⡀⠈⣅⡀⢄⣠⡤⣄⡀⠀⠀⢸⣶⠀⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣈⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠼⠟⠙⠛⠛⠛⠉⠉⠉⢁⣀⠈⢿⣿⣿⣿⣿⣿
⣰⡖⠀⠨⠉⢐⣛⣟⣿⡪⠺⠘⠫⠌⣿⡆⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⣀⣙⣛⡉⠀⠈⣉⠁⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣄⠘⡟⢠⣤⠀⢠⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⢸⣯⣮⣾⣿⣿
⣍⣃⣦⣾⡙⣿⣿⣭⢢⣋⡒⠖⠦⠥⠷⠷⠀⠉⠉⠠⠤⠤⠶⠴⠞⠓⠒⣛⡛⣋⡛⢋⣥⣭⡙⠁⢠⣶⠀⣶⡶⠆⠁⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⣆⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠐⢟⣿⣿⡙⠉
⣿⣿⣿⣿⠟⠉⣀⣀⠀⢀⣀⢠⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣄⣿⠁⣿⡇⣿⣧⣹⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠀⢹⣿⠀⢻⣧⢰⣶⢈⣙⢻⣧⠈⠈⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⢸⣿⣿⡠⣶
⣿⣿⣿⣿⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⠀⡇⢸⣿⠀⢸⣿⢸⣿⡘⠿⠾⠛⣠⣀⠛⢛⣁⣋⣉⣡⣬⣭⣬⣀⣀⣠⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿
⣿⣿⣿⣿⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⠛⠀⠀⣛⣁⣈⣩⣄⣿⣬⣤⣶⣶⣴⣾⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠿
⣿⣿⣿⣿⣆⠘⠿⠾⠟⠀⠿⠀⠛⠂⠛⠛⠃⢈⣉⣉⣤⣬⣤⣤⣤⣶⣶⣶⣦⣾⣿⣿⣻⣶⣦⣧⠺⠿⠛⣿⣿⣿⣏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣮
⣿⣿⣿⣿⣿⣶⣤⣤⣴⣶⣶⣴⣶⣶⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠘⣿⡟⣧⠀⣠⣿⣿⡿⣿⡮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⣿⣿⣿⣿⡇⣿⡷⠎⠀⢻⣿⣿⡿⠃⢀⣨⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⡀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣿⣿⣿⣿⣿⠁⠙⠿⠃⠀⠀⠻⠟⠀⢠⣿⣿⠀⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡇⢸⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⣿⠿⢿⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀
⣿⣿⣿⠿⡿⢿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣇⠀⠀⣠⢀⡤⣀⣴⣾⡿⠋⠁⢀⣰⣾⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡿⢹⣿⣿⣿⡟⠀⠀
⣿⣿⣏⠀⡃⢸⣿⣿⣟⠛⠁⠈⠉⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⠉⠁⣠⣿⢏⣵⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠘⠛⠚⠿⢿⣿⠁⠀⠀
⣿⣿⣿⠀⠁⠘⠛⠛⠀⠀⠀⢠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⠛⠛⠻⠿⣿⠃⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⠙⣿⣿⣿⣿⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠈⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡤⠉⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠖⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⣤⣶⣷⣾⣿⣶⣿⣶⣦⣀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⡀⣀⡀⢀⣀⣀⣀⡀⣀⢀⣀⣀⠀⣀⣀⡀⠀⠀⣀⣀⠀⣀⣀⣀⢁⣀⣚⢛⣋⣙⡛⢛⣛⣛⡛⣛⡛⣛⡛⣂⡀⢀⣀⡀⠀⠀⢀⣀⡀⠀⠀⢀⣀⡀⢀⣀⢀⣀⢀⣀⡀⢀⣀⣀⣀⣀⣀⡀⣀⣀⢀⣀⡀⠀⠀
⠀⠀⣿⡇⣿⣧⣿⡇⣿⡟⠃⠿⢸⣿⣻⡇⣿⣟⠃⠀⢸⣯⡿⠇⢻⣿⡟⢸⡿⣿⢸⣿⣿⣿⠘⣿⣿⠃⣿⡇⣿⣧⣿⣇⣿⡿⠿⠄⠀⣾⣿⣿⠀⠀⣿⣿⠿⠘⣿⣾⠇⣿⣟⠿⠘⢻⣿⠋⣿⡟⠃⣿⣿⣸⣿⡇⠀⠀
⠀⠀⢹⣿⡿⣿⣿⡇⣿⡟⠃⠀⢸⣿⢻⡆⣿⡿⠃⠀⢨⡝⢿⡆⢸⣿⡇⣿⣧⣿⣼⣿⣿⣷⠀⣿⣿⠀⣿⡇⣿⢿⣿⡟⣿⣿⣿⡆⠀⣿⣿⣿⠀⠀⣭⡻⣷⠀⢻⣿⠀⣬⡻⣷⡄⢸⣿⠀⣿⡟⠃⣿⣿⣿⣿⡇⠀⠀
⠀⠀⠸⠿⠇⠻⠿⠁⠿⠿⠇⣀⠸⠿⠸⠇⠻⠿⠟⠀⠘⠿⠿⠃⠘⠿⠃⠿⠏⠿⠟⠿⠛⠿⠀⠻⠟⠀⠿⠇⠿⠊⠿⠃⠻⠿⠻⢃⠐⠿⠋⠿⠇⠀⠻⠿⠟⠀⠸⠟⠀⠻⠷⠟⠁⠸⠿⠀⠿⠿⠇⠿⠛⠟⠻⠇⠀⠀
⠀⠀⠐⠚⠂⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⣸⣿⣿⣿⠿⠋⠁⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⠃⠀⠀⠀⠀⠀⡠⠇⠀⠀⠀⠀⠀⠀⠈⠃⠀⢀⣶⠀⠀⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⠇⠀⠀⠀⠀⠀⠇⠀⠀⠠⠄⠘⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣿⣿⡀⢀⡀⠀⠀⢀⣀⣾⣿⣷⣄⡀⠀⠀⠀⠰⣤⣤⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⢀⣤⡶⠀⠁⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠰⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⠀⠀⠀⠂⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⣿⣿⠿⠛⠋⢀⣶⡀⠀⠀⠀⠀⢠⣄⠈⠛⠻⣿⣿⣿⣿⣿⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⢀⡀⠀⠀⠀⠀⠘⢿⣿⡇⠀⠀⠀⠘⠛⠧⠀⠀⠀⠀⢻⡿⠂⠀⠀⣸⣿⣿⣿⣿⡟⠿⠟⠃⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢐⡿⠀⠀⠀⠀⠀⠀⣤⣀⠘⠃⢀⠀⠀⠀⡀⡀⢸⣧⣀⣶⡄⣀⣀⣀⣀⡀⠀⠒⣀⣀⠀⠀⣶⣿⣿⣿⣿⣿⢀⣤⣤⣰⡀⠀⠊⠐⠠⠁⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⠉⠁⠀⠀⠦⠀⠀⠿⠀⠀⠂⣼⠓⣠⡀⢄⠀⠀⠀⠘⠑⠋⣿⣿⣿⣿⣿⣿⣏⠁⠀⠀⠈⠻⠿⠟⠂⠙⠛⠛⠛⠿⠛⣀⣀⣀⡀⢠⣤⢠⣤⣤⢠⣦⠐⣶⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⠤⠙⠓⠋⠈⠙⠁⠉⠀⢀⣈⣀⡀⣀⡁⣤⡌⢀⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠈⣿⣧⣿⠀⢠⠤⠀⢠⣼⣶⣾⣅⣂⣀⣀⣢⣤⣀⣀⣀⢀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠚⡃⠀⣶⡆⣶⣷⢸⣿⢸⣿⠘⢻⣿⠁⢿⣇⣿⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⢀⣯⣄⢠⣹⡿⣿⣿⣿⣷⢿⡿⠚⣫⡽⣿⣿⣷
⣶⡆⠀⠀⠰⠇⣀⠀⠀⠀⠠⠄⡀⠀⢿⠀⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣈⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠸⠟⠓⠛⠛⠋⠉⠉⠉⢁⣀⠀⠀⠩⣞⣟⣿⣿
⠇⢆⠀⠀⠂⠀⠈⠀⠀⠀⠀⠈⠀⠠⣾⡇⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⣀⣙⣛⡉⠀⠈⣉⠁⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣀⡘⡗⢠⣤⠀⢠⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⢠⢤⣯⣿⣿⣿
⣶⡔⢠⢅⠀⣶⣶⣶⣶⡀⠂⠀⠰⣦⠾⠧⠀⠉⠉⠤⡤⠤⠶⠶⠚⠓⠒⣛⡛⣋⡛⢋⣥⣭⡙⠃⢠⣶⠀⣶⡶⠆⠁⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⣄⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠘⢛⣯⣿⡯⠉
⣿⣿⣿⣟⠟⠉⣀⣈⠉⢀⣀⢠⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣄⣿⠁⣿⡇⣿⣧⣸⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠀⢹⣿⠀⢻⣧⢰⣶⢈⣙⢻⣧⠈⠈⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⢈⣿⣻⣧⣄
⣿⣿⣿⣿⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⠠⡇⢸⣿⠀⢸⣿⢸⣿⡘⠿⠾⠛⣠⣀⡛⢛⣁⣋⣉⣡⣬⣭⣤⣀⣀⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿
⣿⣿⣿⣿⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⠛⠀⠀⣛⣁⣈⣩⣄⣿⣮⣤⣶⣶⣶⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿
⣿⣿⣿⣿⡆⠘⠿⠾⠟⠀⠿⠀⠛⠂⠛⠛⠃⣈⣉⣉⣤⣌⣠⣤⣤⣶⣶⣶⣴⣾⣿⣿⣷⣆⢐⣧⣾⣿⣿⣿⣿⣿⡏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⡆⠀
⣿⣿⣿⣿⣿⣶⣤⣤⣴⣶⣶⣦⣿⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡽⣿⣿⣿⡆⠈⢿⣿⣿⣿⢷⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣽⣿⣿⣿⣧⢸⡿⢻⠇⢠⣾⣿⣿⣿⠎⢻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⢣⣿⣿⣿⣿⣿⢸⣿⡇⠀⠸⣿⣿⡿⠃⢀⣠⣟⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣵⣶⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣿⣿⣿⣿⣿⡇⠘⠛⠁⠀⠀⠛⠋⠀⣰⣿⣿⣿⣶⡜⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀
⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠿⠿⢿⣦⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡿⢸⣿⣿⣿⡿⠀⠀
⣿⣿⣇⠀⠂⢸⣿⣿⣟⠋⠈⠉⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢿⣿⣿⣿⣿⣿⠏⠀⠀⣶⣶⠆⣠⣾⣿⡟⠃⠀⣀⣼⣾⣿⣀⠈⣻⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠘⠛⠘⠿⢿⣿⠁⠀⠀
⣿⣿⣿⠀⠁⠘⠛⠻⠅⠀⢀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠛⠛⠿⠿⢿⡟⠀⠀⣀⣼⢿⣾⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡟⠛⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⠀⠀⠀⠀⠀⠴⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠒⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡍⢙⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣠⣤⣶⣷⣾⣿⣶⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣤⡀⠀⠀⠀⠉⠀⠀⠤⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠉⠀⠛⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢰⣿⣿⡀⣀⡀⠀⠀⢀⣠⣾⣿⣷⣄⡀⠀⠀⠀⢰⣦⣤⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣤⡈⠃⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠀⠀⢀⣀⠀⠀⣄⠀⠀⠀⢀⣴⠂⠀⠀⠈⠉⠀⠀⣠⡒⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⢿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⡹⠿⠀⠀⠃⠃⠀⠀⢀⠀⠀⠀⠀⠀⠛⠂⠀⠙⠃⢸⣿⣿⣿⠿⠟⠋⢀⣶⡀⠀⠈⠀⠀⣰⣄⠈⠛⠻⣿⣿⣿⣿⣿⣿⣷⣿⣿⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⣤⠘⠀⡀⠀⠀⣶⠄⠀⠠⠀⠀⠘⢿⣿⡇⠀⠀⠀⠘⠛⠧⠀⠀⠀⠘⣿⡿⠆⠀⠀⣸⣿⣿⣿⣿⡟⠿⣿⣻⣵⢶⣾⢴⣂⡒⠄⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡁⡳⣠⣸⣷⣰⣶⡄⣀⣀⣀⣀⣀⠀⠒⣀⣀⠀⢀⣶⣿⣿⣿⣿⣿⢀⣾⣿⣿⣿⡿⠯⠽⠟⠷⠚⠃⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠁⠀⠂⠀⠲⠀⠈⠀⠀⠀⠀⠀⠀⢀⢀⠎⣈⣿⣿⣿⣾⢿⡿⣿⣿⣿⣿⣿⣿⣋⡁⠀⠀⠙⠛⠿⠟⠚⠛⠛⠛⠛⠿⠛⣈⣁⣀⡀⢠⣤⢠⣤⣤⢠⣦⠰⣶⠀⠀⠀⢀⡀⣀⢆⣾⢄⠤⢠⡄⣀⠀⠀⠀⠀⠀⠀
⠀⠀⣤⢤⣦⠀⠀⡀⠀⠀⠀⠀⠀⠀⡁⠜⠓⠐⠿⠿⠉⠉⠁⠈⢀⣈⣀⡀⣀⡉⣤⡌⢀⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠘⣿⣧⣿⠀⢨⡦⢨⣬⣿⣿⣿⣽⣿⢾⣭⣿⣾⣖⣴⣄⣠⣗
⣿⣷⡲⣟⡍⡀⠀⡔⢦⠱⠩⣖⣔⢨⠇⠀⣶⡆⣾⣷⢸⣿⢸⣿⠘⢻⣿⠁⣿⣇⣿⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⢀⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⣿⢿⠵⡄⣶⣥⢃⠇⣾⣷⢿⣿⣎⣌⣷⠀⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣌⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠸⠟⠛⠛⠛⠛⠉⠉⠉⢉⣉⠉⣿⢿⣿⣿⣿⣿
⣿⣿⣈⣭⣾⣷⢐⡔⣙⣻⣿⣿⣿⣿⣿⡆⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⢀⣙⣛⡉⠀⠈⣉⠁⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣄⡘⡗⢠⣤⠀⢰⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⢻⣿⣿⣿⣿⣿
⣻⣻⣽⣯⣿⣿⣿⣿⣿⣾⣿⣫⡿⠿⠻⣷⠄⠉⠉⠤⡤⠤⠶⠶⠞⠛⠒⣛⡛⣛⡛⢋⣭⣭⡙⠃⢠⣶⠀⣶⡶⠆⠁⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⣄⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠘⣵⣿⣿⣿⣿
⣷⣾⣿⣿⡿⠉⣀⣀⠁⢀⣀⢡⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣆⣿⠁⣿⡇⣿⣧⣹⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠀⢹⣿⠀⢿⣧⢰⣶⢈⣙⢻⣧⠈⠘⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⢸⣿⣿⣿⣿
⣿⣿⣿⣿⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⠠⡇⢸⣿⠀⢸⣿⢸⣿⡘⠿⠾⠛⣠⣄⡛⢛⣁⣋⣉⣡⣬⣭⣤⣄⣀⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿
⣿⣿⣿⣿⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⠛⠀⠀⣛⣁⣈⣩⣄⣿⣮⣤⣶⣶⣶⣾⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿
⣿⣿⣿⣿⣆⠘⠿⡾⠟⠀⠿⠀⠛⠂⠛⠛⢃⣈⣉⣉⣤⣬⣤⣤⣤⣶⣶⣶⣴⣿⣿⣿⣶⣶⣽⣮⣼⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣟
⣿⣿⣿⣿⣿⣶⣦⣶⣶⣶⣶⣶⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣾⣿⣿⣿⣿⣷⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢹⣿⣿⣿⠇⣿⣿⣿⣿⣿⡾⣧⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣹⣿⣿⣿⣿⡇⣿⣋⠏⢠⣿⡿⣿⣿⡿⠗⣿⡟⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣶⣦⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣵⣿⣿⣿⣿⣿⢳⣿⡧⠀⠸⢿⣿⠿⠋⢀⣠⣿⣿⡆⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀
⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣿⣿⣿⣿⣿⣿⡏⠈⠉⠁⠀⠀⠘⠋⠀⣴⣿⣿⣿⣿⣽⠄⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⢻⣿⣿⣿⡿⠀⠀
⣿⣿⣇⠸⡿⢹⣿⣿⣿⠏⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠘⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠿⢿⣿⣷⡄⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠚⠛⠚⠿⢿⣿⠁⠀⠀
⣿⣿⣿⡀⠁⠘⠛⠿⠟⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠐⠿⠿⠿⢿⣿⠃⠀⠀⠀⠀⠀⣴⣿⣿⣿⠃⠀⣠⣴⣿⣿⣿⣶⣦⣶⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⠀⠀⠀⠀⠠⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⣤⢀⣤⣾⣿⣿⣿⣿⣷⣾⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣷⣾⣿⣶⣿⣶⣦⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠰⠠⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢾⡷⠂⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠻⠆⠀⠀⢰⡖⢀⣿⣿⡂⢀⡀⠀⠀⢀⣠⣾⣿⣷⣄⡀⠀⠀⠀⢰⣦⣤⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠛⠷⡀⠀⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⢸⣿⣿⣿⠿⠟⠋⢀⣶⡀⠀⠀⠀⠀⣠⣄⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢿⡇⠀⠀⠀⠘⠛⠧⠀⠀⠀⠀⣿⡿⠆⠀⠀⣸⣿⣿⣿⣿⡿⠿⠛⡃⢈⣉⡦⢠⣠⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣄⠀⠀⡄⢒⣄⠀⠀⠄⣸⣷⣶⣶⡄⣀⣀⣀⣀⣀⠀⠒⣀⣀⡀⢀⣾⣿⣿⣿⣿⣿⠀⣰⣽⣿⣿⡯⠶⠃⠀⠾⠏⠆⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠠⣶⠀⠀⠀⠀⠀⠀⠸⣠⡀⠀⠈⠃⢈⡀⣠⣾⣰⣻⢾⡯⣽⣿⣿⣿⣿⣿⣿⣏⡁⠀⠀⠉⠛⠿⠟⠚⠛⠛⠛⠛⠟⠛⣈⣀⣀⡀⢠⣤⢠⣤⣤⢠⣦⠰⣶⠀⠀⠐⡀⠀⣷⡄⢀⠀⠠⠆⠀⡀⠄⠀⠀⠀⠀⠀
⣌⠋⡿⢙⣀⠀⠀⠀⣀⡀⠰⠯⠃⠘⢏⠝⠛⠃⠉⠈⠉⠉⠁⠉⢁⣉⣀⡀⣀⡉⣤⡌⢁⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠘⣿⣧⣿⠀⢰⡚⠒⣘⣿⣿⣿⣿⣦⣧⣄⣒⠦⣀⣈⠘⣀⣀
⠿⡀⡕⠞⠉⡁⡄⠄⢨⡟⠨⣗⣒⠶⡃⠀⣶⡆⣶⣷⢸⣿⢸⣿⠘⢻⣿⠁⢿⣇⣿⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣾⣿
⠇⠿⣿⣆⡀⡀⣅⣄⣘⣿⣮⣽⣿⣿⣶⠀⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣌⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠼⠿⠛⠛⠛⠛⠉⠉⠉⢉⣉⠙⢿⣿⣿⣿⣿⣿
⣯⣏⣠⣭⡆⣲⢶⣾⣿⣾⣿⣿⣿⣿⡿⠄⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⢀⣙⣛⠉⠀⠈⣉⠁⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣄⠘⡗⢠⣤⠀⢰⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⡟⣮⣿⣿⣿⣿⣿⣿⠿⣿⡿⠿⠧⠀⠉⠉⠠⡤⠤⠶⠶⠞⠛⠒⣛⡛⣛⡛⢋⣭⣭⡙⠃⢠⣶⠀⣶⡶⠆⠁⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⣄⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠘⢿⣿⣿⣿⡉
⣿⣿⣿⣿⠞⠉⣁⣀⠁⢀⣀⢠⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣄⣿⠁⣿⡇⣿⣧⣹⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠀⢹⣿⠀⢻⣧⢰⣶⢈⣙⢻⣧⠈⠘⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⢸⣿⣿⣿⣿
⣿⣿⣿⣿⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⢠⡇⢸⣿⠀⢸⣿⢸⣿⡘⠿⠾⠛⣠⣀⡛⠛⣁⣋⣉⣡⣬⣭⣬⣀⣀⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿
⣿⣿⣿⣿⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⠛⠀⢀⣛⣁⣈⣩⣸⣿⣮⣤⣶⣦⣴⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿
⣿⣿⣿⣿⡆⠘⠿⠾⠟⠀⠿⠀⠛⠂⠛⠛⣃⣈⣉⣉⣀⣬⣤⣤⣤⣴⣶⣦⣴⣿⣿⣿⣶⣖⣶⣧⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣶⣤⣤⣴⣷⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⣧⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣇⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣾⣿⣿⣿⣯⢻⣿⣿⡿⠙⣿⣿⣿⣿⣿⡎⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⣿⣿⣿⣿⣿⣼⣿⠟⠁⣰⡿⣿⣿⣿⣿⣿⣿⣷⡌⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀
⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⣿⡇⠹⠆⠀⢀⣿⣿⣏⣧⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⢻⣿⣿⣿⡟⠀⠀
⣿⣿⣷⠀⠂⢸⣿⣿⣿⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢹⣿⣿⣿⣿⣿⡟⠈⠋⠁⠀⠀⠻⠏⠀⣀⣼⣿⣿⣿⣿⡿⢷⣶⠹⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠚⠛⠚⠿⢿⣿⠃⠀⠀
⣿⣿⣿⡀⠀⠘⠛⠿⠿⢀⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠺⠿⢿⣿⣿⣿⠟⠀⠀⠀⠀⠠⠂⠀⣠⣿⣿⣿⣿⠿⣏⣁⣼⣿⣤⡈⠻⣿⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⠀⠀⠀⠠⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢀⣈⠁⠀⠀⠀⠀⠀⢠⣾⣿⣿⠏⠉⢀⣠⣬⣙⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣷⣾⣿⣶⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠿⠋⠁⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢶⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⢀⡀⠀⠀⢀⣀⣾⣿⣷⣄⡀⠀⠀⠀⢠⣤⣤⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡆⠀⠃⠀⠀⠀⠀⠃⠀⠙⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣄⠀⠘⠀⠀⢀⡄⡀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⢿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠻⠋⠀⠀⠀⠀⠀⠀⠱⠆⠈⠀⠀⠀⠠⡄⠀⠀⠀⠀⢸⣿⣿⣿⡿⠛⠋⢀⣶⡀⠀⠈⠀⠀⣠⣄⠉⠛⠛⣿⣿⣿⣿⣿⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠂⠀⢀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠘⢿⣿⡇⠀⠀⠀⠘⠛⠧⠀⠀⠀⠀⣿⡿⠆⠀⠀⣸⣿⣿⣿⣿⡿⠿⠛⣳⣰⡄⣄⣠⣦⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠘⠟⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣀⡫⠷⢠⢴⣷⡀⠖⠖⣸⣷⣰⣶⡄⣀⣀⣀⣀⡀⠀⠒⣀⣀⠀⠀⣶⣿⣿⣿⣿⣿⠀⢰⣽⣿⡿⣿⠿⠵⠹⠲⠧⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣠⣁⡀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⡀⠈⠿⣋⣿⡟⣮⡥⡾⣿⣿⣿⣿⣿⣿⣋⡁⠀⠀⠙⠛⠿⠗⠚⠛⠛⠛⠛⠿⠛⣈⣁⣀⡀⢠⣤⢠⣤⣤⢠⣦⠰⣶⠀⠀⠀⠀⠀⢎⠄⡢⡀⢑⠀⠀⣀⡀⠀⠀⠀⠀⠀
⣿⡺⢛⣵⠀⠀⠀⠨⠄⠀⡀⠀⠀⢀⠉⠚⠐⠊⠛⠉⠛⠋⠉⠈⢈⣈⣀⡀⣀⡈⣤⡌⢀⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠈⣿⣧⣿⠀⠠⡄⣆⣶⣿⣷⣾⣿⣿⣤⣶⣺⣶⣡⣄⢤⡀⣀
⣶⡶⢧⢫⡀⠉⠘⠀⠀⠀⣠⠤⣷⣶⡀⠀⣶⡆⣶⣷⢸⣿⢸⣿⠘⢻⣿⠁⢿⣇⣿⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣯⡷⣆⠆⠇⡀⢠⡤⡄⢤⣰⣤⣯⣿⣴⠀⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣌⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠸⠟⠛⠛⠻⠛⠉⠉⠉⢉⣈⠉⢿⣿⣿⣿⣿⣿
⡷⠀⢢⢁⣋⣯⣽⢿⣏⡶⣟⣿⣹⣿⣿⠀⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⢀⣙⡛⠉⣀⣈⣉⠁⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣄⡘⠗⢠⣤⠀⢠⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⢸⣿⣿⣿⣿⣿
⣿⣾⣿⣏⣿⢷⢿⣵⣿⣷⣙⠿⠿⠷⠿⠶⠀⠉⠉⠤⡤⠤⠶⠶⠞⠛⠒⣛⡛⣛⡛⢋⣭⣭⡙⠀⢠⣶⠀⣶⡶⠆⠀⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⣄⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠘⢿⣿⣿⣿⡍
⣿⣿⣿⣯⠞⠉⣉⣀⠁⢀⣀⢠⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣄⣿⠁⣿⡇⣿⣧⣹⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠀⢹⣿⠀⢿⣧⢰⣶⢈⣙⢻⣧⠈⠈⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⢸⣿⣿⣧⣧
⣿⣿⣿⣿⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⢠⡇⢸⣿⠀⢸⣿⢸⣿⡘⠿⠾⠛⣠⣀⡛⢛⣁⣋⣉⣡⣬⣭⣤⣀⣀⣤⣤⣤⣤⣤⣴⣾⣿⣻⣿⣿
⢿⣿⣿⣿⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⠛⠀⣤⣛⣁⣈⣉⣸⣿⣮⣤⣶⣶⣶⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡿
⣿⣿⣿⣿⣆⠘⠿⠾⠟⠀⠿⠀⠛⠂⠛⠛⣃⣈⣉⣉⣠⣬⣤⣤⣤⣶⣶⣤⣶⣿⣿⣿⣷⣴⣴⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣤
⣿⣿⣿⣿⣿⣶⣤⣤⣶⣶⣶⣶⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣿⣿⣿⣿⣿⣧⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⢻⣻⣿⣿⣿⣿⢿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⢸⣿⠆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣣⣿⣿⣿⣿⣿⡟⣿⣿⣿⡿⠸⣿⣿⣿⣿⣿⣷⣿⣦⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀
⣿⣿⣿⡿⡿⣧⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⡟⢀⣴⠛⣷⣼⣿⣿⣿⣿⣏⣧⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⢻⣿⣿⣿⡿⠀⠀
⣿⣿⣿⠆⠂⢸⣿⣿⡟⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠐⣿⣿⣿⣿⣿⣿⣿⢳⡿⠋⠁⣸⣿⡄⠸⠛⠉⢹⣿⣿⣿⣿⡦⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠚⠛⠚⠿⢿⣿⠃⠀⠀
⣿⣿⣿⡀⠀⠘⠛⠿⠇⣠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠿⣿⣿⣿⣿⣿⠇⠉⠀⠀⠀⢸⡿⠃⠀⢀⡔⠙⢿⣮⣿⣿⣧⣤⡀⠙⢿⣿⣿⣿⣿⣿⣇⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⠀⠀⠀⠠⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣤⡍⣻⣦⣼⣿⠋⣠⣿⣿⣿⣷⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣶⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠰⠄⠀⠀⠀⠈⠀⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡀⠈⠀⠘⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠍⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⢀⡀⠀⠀⢀⣀⣾⣿⣷⣄⡀⠀⠀⠀⠰⣤⣤⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣦⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⠿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣀⣀⢀⠇⠀⠀⠀⣠⡤⠐⠈⠃⠀⠀⠲⠀⠀⠀⢸⣿⣿⣿⠿⠟⠋⢀⣶⡀⠀⠀⠀⠀⢠⣄⠈⠛⠛⢿⣿⣿⣿⣿⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⠀⠐⠙⠛⠟⠀⢀⠀⠀⠉⠀⠀⢀⣄⠀⠀⠀⠀⠠⠀⡘⢿⣿⡇⠀⠀⠀⠘⠛⠥⠀⠀⠀⠀⣿⡿⠆⠀⠀⣸⣿⣿⣿⣿⣿⠿⠛⢁⡀⡰⣆⡠⠴⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠀⢀⡄⠀⠀⠈⠉⠀⠀⠀⢀⡒⠀⠁⠀⠀⠀⠀⠀⢻⢎⣄⢸⣷⣰⣦⡄⣀⣀⣀⣀⡀⠀⠒⣀⣀⠀⠀⣶⣿⣿⣿⣿⣿⢁⣰⣶⣾⣽⣷⠗⠝⠿⠻⠃⠔⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢬⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⢘⣇⡎⡨⣠⡐⣲⣆⣎⣿⣿⣿⣿⣿⣿⣋⡁⠀⠀⠉⠛⠿⠗⠒⠛⠛⠛⠛⠿⠛⣈⣀⣀⡀⢠⣤⢠⣤⣤⢠⣦⠐⣶⠀⠀⠀⠀⠀⠂⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣼⣥⠿⣯⡢⠀⢀⡀⡀⠀⠀⠀⠄⡀⡇⠋⠘⠛⠿⠛⠉⠉⠁⠉⢉⣉⣀⡀⣀⡉⣤⡌⢀⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠈⣿⣧⣿⠀⢠⣂⠂⢄⣷⣾⣿⣴⣤⣢⣨⣚⣇⣀⣀⣀⣀⡀
⠨⣯⠺⠛⠑⠁⠔⠀⢀⡀⠀⠀⣀⢨⠄⠀⣶⡆⣶⣷⢸⣿⢸⣿⠘⢻⣿⠁⢿⣇⣿⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⢸⣿⣽⢦⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣾⣿
⣾⣿⣍⢅⣀⡀⠀⣐⠼⠓⢢⠠⡇⣭⣤⠀⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣈⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠸⠟⠛⠛⠛⠛⠉⠉⠉⢉⣈⠉⢿⣿⣿⣿⣿⣿
⣿⣿⢟⠧⡻⣖⣤⠈⢁⠶⣰⣼⣿⣾⣿⡆⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⣀⣙⡛⠉⠀⠈⠉⠁⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣄⠘⠞⢠⣤⠀⢠⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⢸⣿⣿⣿⣿⣿
⣿⢯⠹⣟⣾⣿⣿⣿⣽⣷⡙⠼⠯⠿⠣⠧⠀⠉⠉⢤⡤⠤⠶⠶⠞⠛⠒⣛⡛⣛⡛⢋⣭⣭⡙⠀⢠⣶⠀⣶⡶⠆⠀⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⡄⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠘⢿⣿⣿⡿⠟
⣿⣿⣿⣿⠏⠉⣀⣀⠁⢀⣀⢠⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣄⣿⠁⣿⡇⣿⣧⣹⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠀⢹⣿⠀⢻⣧⢠⣶⢈⣙⢻⣧⠈⠈⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⢸⣿⣿⣧⣥
⣿⣿⣿⣿⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⢰⣆⢸⣿⠀⢸⣿⢸⣿⡘⠿⠾⠛⣠⣄⡛⠛⣁⣋⣉⣡⣬⣭⣄⣀⣀⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿
⣿⣿⣿⣿⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⠛⠀⣤⣛⣁⣈⣉⢸⣿⣮⣤⣾⣶⣶⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿
⣾⣿⣿⣿⣆⠘⠿⠾⠟⠀⠿⠀⠛⠂⠛⢛⣃⣈⣉⣉⣤⣬⣤⣤⣴⣶⣶⣴⣶⣾⣿⣷⡗⡐⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄
⣿⣿⣿⣿⣿⣷⣤⣤⣴⣶⣶⣶⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣟⣸⣿⣿⣿⣿⣿⣆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⢹⣷⠌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣶⣿⣿⣿⣿⣼⣿⣤⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀
⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⣿⣇⣿⣿⣿⣿⣿⣻⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⡿⢻⣿⣿⣿⡿⠀⠀
⣿⣿⣿⠀⠂⢸⣿⣿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⢸⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⠟⠀⡸⣿⡏⣿⣿⣿⣿⣿⣷⡄⠘⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠚⠛⠚⠿⢿⣿⠃⠀⠀
⣿⣿⣿⡀⠀⠚⠛⠿⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢼⣿⣿⣿⣿⣿⣿⠿⣻⣿⠿⠏⢠⣾⡇⢀⡿⠟⢿⣿⣟⣿⣿⣤⣄⠀⠙⣿⣿⣿⣿⣿⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⡀⠀⠀⠀⠉⠙⠻⠘⠟⠁⠀⠀⣿⣿⠇⠈⠀⢠⡄⣿⣉⢻⣿⣿⣿⣿⣦⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣷⣾⣿⣶⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠃⠀⠀⠀⠺⠇⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠊⢠⡀⠈⠁⠐⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠙⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠀⠀⠂⠙⡓⠈⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢀⡸⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⢀⡀⠘⠀⣛⣏⣿⣿⡀⢀⡀⠀⠀⢀⣠⣾⣿⣷⣄⡀⠀⠀⠀⠰⣦⣤⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⡿⠆⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠛⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠉⠀⠀⢴⡄⠀⠀⠀⣄⠀⠀⠀⠀⢸⣿⣄⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⢿⡿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠚⣿⠆⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⠟⠀⠀⢹⣿⣿⣿⠿⠟⠋⢀⣶⡀⠀⠀⠀⠀⣠⣄⠈⠛⠛⢿⣿⣿⣿⣿⣿⣷⣿⣿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠘⢿⢿⡇⠀⠀⠀⠘⠛⠥⠀⠀⠀⠀⣿⡿⠆⠀⠀⣸⣿⣿⣿⣿⣿⡿⠛⢁⠛⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⢺⣷⣲⣶⡄⣀⣀⣀⣀⣀⠀⠒⠀⣀⠀⠀⣶⣿⣿⣿⣿⣿⣋⣠⣴⣾⠪⡄⠀⠀⠁⠈⠆⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⡀⠀⠈⠀⠀⠸⠧⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⣆⣂⣀⡸⣿⣿⣿⣿⣿⣿⣋⠁⠀⠀⠉⠻⠻⠓⠚⠙⠛⠛⠛⠿⠛⣉⣁⣀⡀⢠⣤⢠⣤⣤⢠⣦⠐⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣼⣷⡀⠀⠀⠠⠀⠀⠀⠀⠀⠈⡒⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢁⣉⣀⡀⣀⡁⣤⡌⢀⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠈⣿⣧⣿⠀⠀⠄⠀⠀⢢⣴⣮⡆⠄⣤⣤⡄⡀⠀⠀⠀⠀⠀
⣾⡿⠿⣡⠀⠀⠀⠀⠀⠀⠀⢸⠿⢆⡄⠀⣶⡆⣶⣷⢸⣿⢸⣿⠘⢻⣿⠁⢿⣇⣿⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⠐⣿⣭⣦⣿⡿⣿⣷⢿⣿⣾⣿⣷⣶⣷⣶⣲⣷
⠿⢿⡂⠚⠂⠀⠀⠀⠀⠤⠀⢼⣤⣆⣣⠀⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣈⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠸⠿⠛⠛⠛⠛⠁⠉⠉⢁⣈⠉⢿⣿⣿⣿⢿⢳
⡤⡌⣿⠀⡀⣠⠀⠀⠐⠳⠋⣨⣭⣿⣿⡆⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⣀⣙⠛⠉⡀⠈⠉⠁⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣄⠘⠞⢠⣤⠀⢠⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⢸⢿⣿⣟⣿⣿
⣷⡅⠈⣘⢻⢊⣴⠀⣠⠁⠙⠙⠋⠯⠽⠥⠀⠉⠉⢤⡤⠤⠶⠶⠚⠓⠒⣛⡛⣛⡛⢈⣭⣭⡁⠀⢠⣶⠀⣶⡶⠆⠀⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⡄⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠘⢿⣿⣿⡟⠋
⣿⣾⣿⣼⡏⠀⣀⣀⠀⢀⣀⢠⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣄⣿⠁⣿⡇⣿⣧⣸⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠀⢹⣿⠀⢻⣧⢰⡷⢈⣙⢻⣧⠈⠈⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⠘⣷⢮⢧⢄
⣿⣿⣿⣿⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⢰⣆⢸⣿⠀⢸⣿⢸⣷⡘⠿⠾⠛⣠⣄⡛⢛⣁⣋⣉⣡⣬⣩⣄⣀⣀⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⡿
⣿⣿⣿⣿⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⠛⢀⣦⣛⣁⣈⣉⢸⣿⣮⣤⣶⣶⣶⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻
⣿⣿⣿⣿⣇⠘⠿⠾⠟⠀⠿⠀⠛⠂⠛⡛⢃⣈⣉⣉⣤⣬⣤⣤⣤⣶⣤⣶⣶⣶⣶⣶⡦⢠⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀
⣿⣿⣿⣿⣿⣶⣤⣤⣴⣶⣶⣶⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣃⣽⣿⣿⣿⣿⣿⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣅
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡛⣿⣿⣿⣿⡟⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⣿⡇⣿⣷⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣱⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣾⣿⣿⣿⣷⣼⣧⣄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀
⣿⣿⣿⠿⣿⣿⣿⣿⣿⣗⣼⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⢿⣸⣿⣿⣿⣿⣿⢻⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠿⢻⣿⣿⣿⡿⠀⠀
⣿⣿⣿⠀⠀⢸⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢸⣿⢹⣿⣿⣿⣿⣿⣷⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠚⠛⠚⠿⢿⣿⠃⠀⠀
⣿⣿⣿⡄⠀⠘⠛⠃⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠠⣿⣿⣿⣿⣿⣿⡿⠛⣿⠏⣿⣿⣿⡇⠀⢹⣿⠻⢻⣷⣹⣿⣷⣤⡄⠈⢻⣿⣿⣿⣿⣿⣿⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⠀⠀⠀⢼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣂⠀⠀⠀⠉⠙⠻⠷⣶⣾⣶⡟⢷⠟⢁⡄⠀⢹⣴⠘⣏⡘⢻⣿⣿⣿⣶⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣷⣾⣿⣶⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⣼⣿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⢀⡀⠀⠀⢀⣀⣾⣿⣷⣄⡀⠀⠀⠀⠰⣤⣤⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠴⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠘⠋⠈⠀⠀⠀⢠⡀⠤⣶⡆⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣽⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠀⠀⠀⠀⢀⣀⠀⢀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠟⠋⢀⣶⡀⠀⠈⠀⠀⣠⣄⠉⠛⠻⣿⣿⣿⣿⣿⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡸⡷⠄⠀⠀⠘⠋⠀⠀⠉⢉⣥⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠘⢿⢿⡇⠀⠀⠀⠘⠛⠧⠀⠀⠀⠀⢿⡟⠆⠀⠀⣸⣿⣿⣿⣿⣿⡿⠛⢙⣣⣀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠃⠫⡁⠀⠀⠀⠒⢀⡀⠀⠿⣿⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠈⣸⣷⣦⣶⡄⣀⣀⣀⣀⡀⠀⠒⢀⣀⠀⢀⣶⣿⣿⣿⣿⣿⠛⢡⣤⣾⣿⣯⠜⠳⠀⠪⠻⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣄⡐⢆⠀⠠⠏⠈⠁⠀⠀⠀⠀⠀⠀⠐⣰⢄⢀⣤⣶⣄⣊⠐⡜⣿⣿⣿⣿⣿⣿⣏⠁⠀⠀⠉⠻⠿⠟⠚⠙⠛⠛⠛⠿⠛⣀⣀⣀⡀⢠⣤⢠⣤⣤⢠⣦⠰⣶⠀⠀⠀⠀⠀⣀⢐⠀⠐⢘⠄⠀⢀⡀⠀⠀⠀⠀⠀
⣝⡿⢩⠐⢂⠨⠶⢀⠁⠀⠀⣄⠀⡐⠀⠐⠉⠙⠒⠛⠈⠉⠉⠀⢉⣉⣀⡀⣀⡁⣤⡌⢁⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠈⣿⣧⣿⠀⠠⡤⢀⡠⣶⣯⣦⡨⣬⣳⣵⣬⣥⢐⠦⡠⢀⠀
⡄⣪⣘⡆⠂⠀⠀⠀⠀⠀⠐⣾⠢⠳⠇⠀⣶⡆⣶⣷⢸⣿⢸⣿⠘⢻⣿⠁⢿⣇⣿⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⠸⣿⣷⣾⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣯⣿⣿
⢀⣃⠿⣧⠅⡃⠀⠀⣤⣰⡦⢮⣾⣛⣶⡄⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣈⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠸⠟⠛⠛⠛⠛⠉⠉⠉⢉⣉⠉⢿⣿⣿⣿⣿⣿
⢿⣟⣧⣼⠷⠣⡈⠡⠰⢿⣻⡿⣿⡿⣿⡆⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⠀⣙⠛⠉⡀⠈⠉⠁⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣄⡐⡆⢠⣤⠀⢰⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⢸⣿⣿⣿⣿⣿
⣯⣿⡅⣻⡄⣤⠁⢒⢺⣿⣻⠯⠝⠟⠿⢷⣄⠉⠉⢤⡤⠠⠶⠶⠚⠛⠒⣛⡛⣛⡛⢀⣤⣭⡀⠀⢠⣶⠀⣶⡶⠆⠀⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⡄⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠘⢿⣿⣿⣿⡟
⣿⣿⣿⣾⠝⠉⣀⣀⠀⢀⣀⢠⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣄⣿⠁⣿⡇⣿⣧⣹⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠁⢹⣿⠀⢿⣧⢰⡷⢈⣙⢻⣧⠈⠘⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⢘⣿⣿⣿⣗
⣿⣿⣿⣿⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⢠⣆⢸⣿⠀⢸⣿⢸⣷⡘⠿⠾⠛⣠⣄⡛⢛⣁⣋⣉⣡⣬⣩⣌⣀⣀⣤⣦⣤⣤⣤⣴⣾⣿⣿⣿⣿
⣿⣿⣿⣿⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⠛⢠⣦⣛⣁⣈⡉⣸⣿⣮⣤⣾⣶⣶⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿
⣿⣧⣿⣿⣦⠘⠿⠾⠟⠀⠿⠀⠛⠂⠛⠛⢃⣈⣉⣉⣤⣬⣤⣤⣤⣶⣤⣶⣶⣷⣶⡶⣷⣴⣦⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢄
⣿⣿⣿⣿⣿⣶⣤⣤⣴⣶⣶⣾⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣟⣡⣿⣿⣿⣿⣿⣿⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⢻⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⠋⣿⣦⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⡘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣷⣿⣿⣿⣿⣆⣿⣿⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀
⣿⣿⣿⠿⢿⢿⣿⣿⣿⢁⣿⣿⣶⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠸⣷⣿⣿⣿⣿⣿⡟⣇⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡿⢻⣿⣿⣿⡟⠀⠀
⣿⣿⣯⠀⠀⢈⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣿⡏⣻⣿⣿⣿⣿⣿⣦⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠚⠛⠚⠿⢿⣿⠃⠀⠀
⣿⣿⣿⡄⠀⠘⠛⠀⠀⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢼⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⠀⢉⣿⡟⠟⢹⣮⣿⣿⣇⣤⡟⠈⢿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⠀⠀⠤⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠉⠉⠛⠿⣤⣴⣿⣿⣿⣿⣿⡆⠈⢹⣷⣦⢸⣅⠙⣿⣿⣿⣿⣶⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣷⣾⣿⣶⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡀⠀⠀⠀⢀⡀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠚⠓⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠄⢀⣿⣿⡀⢀⡀⠀⠀⢀⣠⣾⣿⣷⣄⡀⠀⠀⠀⠰⣤⣄⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠷⠀⠀⠀⠀⣀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⢿⠿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠶⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠟⠋⢀⣶⡀⠀⠈⠀⠀⢠⣄⠈⠛⠛⢿⣿⣿⣿⣿⣿⣷⣿⣿⠀⠀⠀⠈⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠐⠀⠒⠂⣸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⡇⠀⠀⠀⠘⠛⠥⠀⠀⠀⠀⣻⡿⠆⠀⠀⣸⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⡀⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠄⡀⠂⠛⠛⠃⠀⣀⡠⠄⢀⣠⣶⠀⠀⠀⠀⠀⠀⣠⡰⠟⠋⢸⣷⣠⣶⡄⣀⣀⣀⣀⡀⠀⠒⣀⣀⡀⠀⣶⣿⣿⣿⣿⣿⠻⢃⠒⡜⠬⠔⠤⠁⠡⠆⠀⠀⠀⠀⣴⣄⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⡖⢤⣀⠀⠀⡿⠷⠙⠰⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⣖⠃⠀⠘⣿⣿⣿⣿⣿⣿⣋⡁⠀⠀⠉⠛⠿⠗⠒⠙⠛⠛⠛⠿⠛⣀⣀⣀⡀⢠⣤⢠⣤⣤⢠⣦⠐⣶⠀⣿⠗⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢨⣜⣰⡢⠄⢀⡦⢚⣦⠀⠀⠀⠀⠀⠷⠄⠂⠒⠁⠀⠉⠉⠙⠊⢁⣈⣀⡀⣀⡁⣤⡌⢀⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠈⣿⣧⣿⠀⠙⢀⠀⡀⣦⢾⠆⣀⢈⢤⣤⣀⢀⠀⢀⠀⠀⠀
⣸⡏⡉⠀⠀⠀⠀⠙⢵⣶⢆⡂⠀⠀⠀⠀⣶⡆⣶⣷⢸⣿⢸⣿⠘⢻⣿⠁⢿⣇⣿⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⠀⣿⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷
⠪⠚⡊⠀⠀⠀⠀⠀⣤⡵⢘⣿⡀⢀⣤⠀⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣈⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠰⠟⠛⠛⠛⠛⠉⠉⠉⢁⣀⠉⢿⣟⣿⣿⣿⣿
⣷⣖⡤⢤⡰⣄⡢⠀⠸⠑⢉⣢⠠⣚⣿⡀⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⢀⡙⠛⠉⠀⠈⠉⠁⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣀⠐⡇⢠⣤⠀⢠⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⢸⣿⣿⣿⡟⢛
⣻⣌⣕⣵⣧⣻⣿⣶⣸⣲⣽⠶⠸⠿⠿⢥⣄⠉⠉⢤⡤⠤⠶⠖⠞⠛⠒⣛⡛⣛⡋⢀⣤⣭⡀⠀⢠⣶⠀⣶⡶⠆⠀⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⡄⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠘⢾⣿⣿⡿⠆
⣿⣿⣿⣿⠋⠈⣀⣀⠉⢀⣀⢠⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣄⣿⠁⣿⡇⣿⣧⣸⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠁⢹⣿⠀⢻⣧⢰⡧⢈⣙⢻⣧⠀⠈⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⢘⣿⣿⣗⣰
⣿⣿⣿⣿⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⢠⣆⢸⣿⠀⢸⣿⢨⣷⡘⠿⠾⠛⣠⣄⠛⢛⣁⣋⣉⣡⣬⣩⣄⣀⣀⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿
⣿⣿⣿⣿⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⠛⢠⣆⣛⣁⣈⡉⣼⣿⣬⣤⣶⣶⣶⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡿⠿
⣿⣿⣿⣿⣆⠘⠿⠾⠟⠀⠿⠀⠛⠂⡛⠛⣃⣈⢉⣉⣤⣬⣤⣤⣴⣤⣴⣶⣶⣶⣦⡦⡀⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀
⣿⣿⣿⣿⣿⣶⣤⣤⣤⣶⣶⣶⣶⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣟⣸⣿⣿⣿⣿⣿⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⢻⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⢸⣿⡞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⢟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣶⣿⣿⣿⣿⣸⣿⣤⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀
⣿⣿⣿⠿⠿⢿⣿⣿⣿⢠⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⢻⣼⣿⣿⣿⣿⣿⢻⡄⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡿⢻⣿⣿⣿⡿⠀⠀
⣿⣿⣿⠀⠀⢀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⢙⣿⣿⣿⣿⣿⣷⣄⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠚⠛⠚⠿⢿⣿⠃⠀⠀
⣿⣿⣿⡄⠀⠘⠛⣀⡈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⡟⠛⣿⣿⣿⣿⣿⡇⠰⣿⡿⢻⠏⣿⣝⣿⣿⣄⣤⠀⠙⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⠀⠀⠴⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠈⠉⠙⠛⢧⣤⣾⣿⣿⣿⣿⣷⣄⠉⣿⣾⡄⣿⢈⢹⣿⣿⣿⣷⣆⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣷⣾⣿⣶⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣤⡀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣿⣿⡂⢀⡀⠀⠀⢀⣠⣾⣿⣷⣄⡀⠀⠀⠀⠰⣦⣄⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢰⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠸⠗⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠤⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠟⠋⢀⣶⡀⠀⠀⠀⠀⣠⣄⠉⠛⠻⣿⣿⣿⣿⣿⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠄⠄⠴⡾⠋⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⡀⠘⣿⢿⡇⠀⠀⠀⠘⠛⠥⠀⠀⠀⠈⣻⡟⠦⠀⠀⣼⣿⣿⣿⣿⣿⡿⠛⠃⢄⣀⠀⠀⢀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⡄⣀⠠⡀⠀⠀⠀⠀⠥⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠉⠉⠰⣂⢸⣷⣲⣶⡄⣀⣀⣀⣀⡀⠀⠒⣀⣀⠀⢀⣶⣿⣿⣿⣿⣿⠿⢣⠒⣶⠷⠾⠰⠁⠭⠹⠿⠞⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣐⣁⣯⣀⡀⢡⡀⠐⠃⠀⠀⠀⠀⠀⠀⣠⣀⠀⣀⢀⠀⣠⢰⣯⣾⣿⣿⣿⣿⣿⣿⣋⠁⠀⠀⠉⠻⠿⠟⠚⠙⠛⠛⠛⠟⠛⣀⣀⣀⡀⢠⣤⢠⣤⣤⢠⣦⠐⣶⠀⠐⠀⠀⠀⡂⣠⠃⠠⠔⡀⠀⢀⣀⠀⠀⠀⠀⠀
⣿⣿⣽⡎⠚⠎⡚⠐⠀⠀⠀⠀⠀⠀⢀⡈⠙⠓⠃⠀⠋⠉⠘⠃⢈⣈⣁⡀⣀⡁⣤⡄⢁⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠈⣿⣧⣿⠀⠠⠦⢆⣸⣿⣷⣧⣿⣧⣰⣾⣆⣠⢦⢄⢠⠀⠀
⣿⡿⡋⠁⠀⠉⠙⠓⡄⣄⣶⢒⡀⢱⡄⠁⣶⡆⣶⣷⢸⣿⢸⣿⠘⢻⣿⠁⢿⣇⣿⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⢐⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿
⣿⣷⣥⡤⠀⡤⢤⣴⣤⣿⡻⣴⣮⢨⣵⠀⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣈⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠸⠟⠛⠛⠛⠛⠉⠉⠉⢁⣈⠉⢿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣴⣶⢿⣿⣿⣿⣯⣾⣾⣿⡄⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⣀⠙⠛⠉⠀⠈⠉⠁⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣀⠐⡇⢠⣤⠀⢰⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣯⡇⠘⡯⠿⠩⠿⠿⠧⢀⠉⠉⢤⡤⠠⠶⠶⠞⠛⠒⣛⡛⣛⡉⢉⣭⣥⡀⠈⢠⣶⠀⣶⡶⠆⠀⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⡄⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠘⢿⣿⣿⣿⣿
⣿⣿⣿⣿⠟⠉⣀⣉⠀⢀⣀⢠⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣄⣿⠁⣿⡇⣿⣧⣸⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠁⢹⣿⠀⢻⣧⢰⡦⢈⣙⢻⣧⠀⠘⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⢸⣿⣿⣷⣿
⣿⣿⣿⣿⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⢠⣇⢸⣿⠀⢸⣿⢨⣷⡘⠿⠾⠛⣠⣄⠛⢛⣁⣋⣉⣡⣬⣭⣀⣀⣀⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿
⣿⣿⣿⣿⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⠛⣠⣆⣛⣁⣈⠉⣼⣿⣬⣄⣴⣶⣶⣾⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⠘⠿⠾⠟⠀⠿⠀⠛⢂⡛⠛⣃⡈⣉⣩⣤⣬⣤⣤⣴⣤⣴⣶⣶⣶⡦⠀⣠⣤⣴⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣷⣤⣤⣴⣶⣶⣶⣦⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣃⣽⣿⣿⣿⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⢹⣿⣿⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⡟⢻⣷⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢉⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢳⣾⣿⣿⣿⣧⢸⣿⣀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀
⣿⣿⣿⡿⠿⢿⣿⣿⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣧⣽⣿⣿⣿⣿⣿⢻⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡿⢻⣿⣿⣿⡿⠀⠀
⣿⣿⣿⠀⠀⢀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢿⡏⣻⣿⣿⣿⣿⣿⣧⡀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠚⠛⠚⠿⢿⣿⠃⠀⠀
⣿⣿⣿⡇⠀⠘⠃⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⠟⢛⣿⣿⣿⣿⣿⠀⢠⣿⠛⡿⢡⣧⣿⣿⣿⣀⡄⠈⠻⣿⣿⣿⣿⣿⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⠀⠀⠴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠉⠉⠛⠿⢤⣤⣿⣿⣿⣿⣿⡆⠈⢹⣶⣷⡌⣯⠸⣿⣿⣿⣿⣶⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣷⣾⣿⣶⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠉⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡂⢀⡀⠀⠀⢀⣀⣾⣿⣷⣄⡀⠀⠀⠀⢰⣦⣤⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⣿⣿⣿⠿⠟⠋⢀⣶⡀⠀⠈⠀⠀⣠⣄⠉⠛⠻⣿⣿⣿⣿⣿⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⡆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠘⢿⢿⡇⠀⠀⠀⠘⠛⠥⠀⠀⠀⠀⣿⡿⠆⠀⠀⣸⣿⣿⣿⣿⣿⡿⠛⠃⡄⣄⢀⠀⣀⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠄⢤⡤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣷⣤⣶⡄⣀⣀⣀⣀⣀⠀⠒⣀⣀⡀⠀⣶⣿⣿⣿⣿⣿⢿⣇⣴⣾⣿⠿⠿⠶⠴⠸⠺⠒⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣴⣯⣤⣇⣀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⢀⣂⣹⣱⣣⠘⣿⣿⣿⣿⣿⣿⣏⠁⠀⠀⠉⠛⠿⠟⠚⠙⠛⠛⠛⠿⠛⣀⣁⣀⡀⢠⣤⢠⣤⣤⢠⣦⠐⣶⠀⠀⠀⠀⠀⠂⡠⠛⢵⣠⣔⣀⣀⠀⡀⠀⠀⠀⠀
⣿⣿⣿⣩⠻⢣⣅⠐⠀⣠⡦⣤⣦⡄⣀⠂⠀⠒⠋⠈⠉⠉⠁⠁⢁⣈⣁⡀⣀⡉⣤⡌⢀⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠈⣿⣧⣿⠀⠀⣇⣢⣦⣿⣿⣿⣿⣾⣿⣿⣯⣿⣔⡢⣀⡀⣀
⣿⣿⣟⠀⠀⠘⠻⠁⠘⢓⠂⠈⣻⣷⣥⠀⣶⡆⣶⣷⢸⣿⢸⣿⠘⢻⣿⠁⢿⣇⣿⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣳⣀⢠⣤⣄⣤⣬⣏⢰⣮⣭⣿⠀⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣈⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠸⠿⠛⠛⠛⠛⠉⠉⠉⢉⣉⠉⢿⣿⣿⣿⣟⣻
⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡆⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⣀⠙⠛⠉⠀⠈⠉⠁⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣄⠐⡇⢠⣤⠀⢰⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠘⢿⠿⠻⠣⢄⠉⠉⢤⠤⠤⠶⠶⠞⠛⠒⣛⡛⣛⡙⢋⣭⣤⡀⠉⢠⣶⠀⣶⡶⠆⠀⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⡄⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠘⢿⣿⣿⣿⣿
⣿⣿⣿⣿⠟⠉⣁⣀⠀⢀⣀⢠⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣄⣿⠁⣿⡇⣿⣧⣸⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠀⢹⣿⠀⢿⣧⢰⡦⢈⣙⢻⣧⠀⠈⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⢈⣿⣿⣿⣿
⣿⣿⣿⣿⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⢠⣇⢸⣿⠀⢸⣿⢨⣿⡘⠿⠾⠛⣠⣄⠛⢛⣁⣋⣉⣡⣬⣭⣈⣀⣀⣠⣤⣤⣤⣤⣴⣾⣿⣿⢿⣿
⣿⣿⣿⣿⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⠛⣠⣆⣛⣁⣈⢉⣼⣿⣬⣤⣴⣶⣶⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡻
⣿⣿⣿⣿⣆⠘⠿⠾⠟⠀⠿⠀⠛⠂⠛⠛⠃⣈⣉⣉⣤⣬⣤⣤⣤⣤⣶⣶⣶⣶⡦⠸⣷⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⣿⣿⣿⣿⣿⣶⣤⣤⣶⣶⣶⣶⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣡⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⢻⣿⣿⣿⣿⡟⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⠇⣿⣷⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⡗⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡞⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣷⣿⣿⣿⣿⡄⣿⣧⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀
⣿⣿⣿⡿⠿⢿⣿⣿⣣⣼⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢹⣬⣿⣿⣿⣿⣿⡟⣷⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡿⢻⣿⣿⣿⡿⠀⠀
⣿⣿⣿⠀⠀⢀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠸⣿⢋⣿⣿⣿⣿⣿⣿⣦⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠚⠛⠚⠿⢿⣿⠁⠀⠀
⣿⣿⣿⡄⠀⠘⠃⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣇⠀⣸⣿⢻⡟⢸⣮⣻⣿⣿⣀⡀⠙⢿⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⠀⢰⢶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠉⠹⠛⠷⣤⣼⣿⣿⣿⣿⣿⠀⠉⣿⣾⣷⢸⡯⠈⣿⣿⣿⣿⣶⡈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣷⣾⣿⣶⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⢀⡀⠀⠀⢀⣠⣾⣿⣷⣄⡀⠀⠀⠀⠠⣤⣤⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⢿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠟⠋⢀⣶⡀⠀⠈⠀⠀⣰⣄⠈⠛⠛⢿⣿⣿⣿⣿⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡇⠀⠀⠀⠘⠛⠧⠀⠀⠀⠀⣿⡿⠆⠀⠀⣸⣿⣿⣿⣿⣿⡿⠛⠃⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠠⡤⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣠⣀⣀⠀⠀⠀⠀⢸⣿⣲⣦⡄⣀⣀⣀⣀⣀⠀⠒⣀⣀⠀⠀⣶⣿⣿⣿⣿⣿⣿⢧⢤⣮⢟⡬⠀⠀⠿⢾⠿⠕⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⢀⢀⢀⠀⠂⡄⠀⠀⠀⢀⡈⠛⠀⠀⠀⠀⢺⣿⠟⠑⢀⢰⢼⣿⣿⣿⣿⣿⣿⣿⣋⡁⠀⠀⠉⠛⠿⠟⠒⠙⠛⠛⠛⠿⠛⣀⣁⣀⡀⢠⣤⢠⣤⣤⢠⣦⠰⣶⠀⠀⠀⠀⠀⠀⣂⠀⠠⣰⠄⢀⡀⠀⠀⠀⠀⠀⠀
⣧⣾⣯⠘⠺⠬⠀⢄⠀⠀⠈⠀⠀⠀⠀⠀⠒⠚⠈⠂⠀⠁⠁⠁⢈⣉⣀⡀⣀⡉⣤⡌⢀⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠈⣿⣧⣿⠀⢠⡕⡀⢠⣲⡦⣽⣷⣈⢟⣶⣦⢜⣤⢀⡀⠀⠀
⡿⢘⠄⠀⠉⠀⠂⠁⠀⠀⠀⠀⠠⠀⠀⠀⣶⡆⣶⣷⢸⣿⢸⣿⠘⢻⣿⠁⢿⣇⣿⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⢸⣿⣿⣧⣷⣿⣿⢿⣷⣿⣿⢿⢷⣿⣿⣷⣴⣷
⣛⣽⣞⠀⠤⢀⣀⠠⠀⢧⣬⣤⡄⣶⣄⠀⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣈⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠸⠿⠛⠛⠛⠋⠉⠉⠉⢈⣁⠉⢾⣿⢿⡻⣟⣟
⣻⣟⣫⡄⣱⣾⣿⢳⢁⠔⢅⡸⣯⠿⢽⡄⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⡀⠙⠛⠉⠀⠈⠉⠁⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣄⠐⡆⢠⣤⠀⢠⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⢸⣿⣽⣿⣿⣟
⣻⣿⣿⣿⣿⣿⢯⣆⣴⣲⣾⣿⣿⡿⠿⠧⢄⠉⠉⠤⠤⠤⠶⠶⠞⠛⠒⣛⡛⣋⡙⢋⣭⣥⡀⠉⢠⣶⠀⣶⡶⠆⠀⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⡄⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠘⢿⣻⣝⡟⡂
⣿⣿⣿⣿⠏⠙⣁⣀⠀⢀⣀⢠⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣄⣿⠁⣿⡇⣿⣧⣸⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠀⢹⣿⠀⢻⣧⢸⡦⢈⣙⢻⣧⠀⠘⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⢸⣿⣿⡯⡻
⣿⣿⣿⣿⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⢠⡇⢸⣿⠀⢸⣿⢨⣷⡘⠿⠾⠛⣠⣄⠛⣛⣁⣋⣉⣡⣬⣭⣈⣀⣀⣤⣤⣤⣤⣤⣴⣮⣿⡿⣾⣿
⣿⣿⣿⣿⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⠛⣠⣄⣛⣁⡈⢉⣼⣿⣬⣠⣶⣶⣶⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⣿⣿⣿⣿⣆⠘⠿⠾⠟⠀⠿⠀⠛⠂⠛⠛⠃⣈⣉⣉⣤⣬⣤⣤⣤⣤⣶⣶⣶⣦⣀⠠⣶⣤⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣇
⣿⣿⣿⣿⣿⣶⣤⣤⣴⣦⣴⣶⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣾⣿⣏⣠⣿⣿⣿⣾⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡟⣿⣿⣿⣿⣿⡿⣷⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⢿⣿⣿⣿⣿⣿⠃⣿⡆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣶⣿⣿⣿⣿⡀⣿⣇⡈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀
⣿⣿⣿⠿⠿⢿⣿⡿⢠⣦⣈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡂⣿⣽⣿⣿⣿⣧⣿⡟⣇⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡿⢺⣿⣿⣿⡿⠀⠀
⣿⣿⣿⠘⠂⠀⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡟⢹⣿⣿⢿⣿⣿⣿⣆⠀⠻⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠚⠛⠚⠿⢿⣿⠃⠀⠀
⣿⣿⣿⡄⠀⠘⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⠿⠻⣿⣿⣿⣿⣿⡇⢀⣿⡟⣻⡇⢸⣎⣿⣿⣇⣠⡀⠙⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⣠⢴⠶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⡀⠉⠉⢉⠛⠻⢦⣤⣿⣿⣿⣿⣿⡇⠈⠉⣷⣿⣦⢸⡥⠉⣿⣿⣿⣷⣦⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣠⣤⣶⣷⣾⣿⣶⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⣾⠇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⢀⡀⠀⠀⢀⣠⣾⣿⣷⣄⡀⠀⠀⠀⢰⣤⣤⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⢿⢿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠘⣹⣀⢽⣿⣿⣿⡿⠟⠋⢀⣶⡀⠀⠈⠀⠀⣠⣄⠈⠛⠻⣿⣿⣿⣿⣿⣿⣷⣿⣿⠀⠀⠀⠠⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⠑⡀⠀⠀⠀⢰⡈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⢿⣿⡇⠀⠀⠀⠘⠛⠧⠀⠀⠀⠨⣿⡿⠦⠀⠀⣸⣿⣿⣿⣿⣿⡿⠛⣣⣄⡤⣬⠛⢃⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠎⡧⡄⢄⠀⠀⠀⡁⠍⠈⠀⠀⠀⠀⠀⠀⢄⠀⠀⣄⣀⣶⢲⡀⣺⣷⣰⣶⡄⣀⣀⣀⣀⣀⠀⠒⢀⣀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣭⠾⠺⠻⠿⠿⠒⠚⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣶⣿⡟⡁⠫⢰⠧⠄⣨⡄⠀⠀⠀⠀⠀⢈⣱⣼⡵⡿⣛⠐⢻⣿⣿⣿⣿⣿⣿⣋⡁⠀⠀⠙⠛⠿⠟⠚⠛⠛⠛⠛⠿⠛⣀⣁⣀⡀⢠⣤⢠⣤⣤⢠⣦⠐⣶⠀⢀⠀⠀⠀⢨⠀⣴⢀⣦⣤⣄⡀⡀⠀⠀⠀⠀⠀
⣷⣿⣿⣿⣿⠿⢽⠂⠀⠀⠉⠑⢼⣶⣉⠀⠒⠘⠛⠓⠃⠈⠈⠉⢁⣈⣁⡀⣀⡁⣤⡌⢀⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠈⣿⣧⣿⠀⢠⡾⣇⣤⣽⣷⣿⣿⣿⣯⣿⣯⣼⣄⣠⣄⣀⣀
⣿⣿⣿⣇⡲⠶⠑⠀⠐⠀⢤⢬⣿⣿⡟⠀⣶⡆⣶⣷⢸⣿⢸⣿⠘⢻⣿⠁⢿⣇⣿⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣯⡿⣳⣤⣶⣦⡌⣠⣽⣿⣿⣿⠀⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣈⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠸⠟⠛⠙⠛⠛⠉⠉⠉⢁⣈⠉⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣾⣿⣿⣿⣟⣿⣷⣿⣿⣿⣿⡄⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⡀⠙⠛⠉⠀⠈⠉⠁⠀⣀⡀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣄⠐⡆⢠⣤⠀⢠⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣾⣷⣯⣏⡻⡿⠿⠿⠿⠷⠀⠉⠉⠠⠤⠤⠶⠶⠞⠛⠒⣛⡛⣃⡛⢋⣥⣤⡀⠉⢠⣶⠀⣶⡶⠆⠀⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⡄⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠘⢿⣿⣿⣿⣿
⣿⣿⣿⣿⠟⠉⣀⣀⠁⢀⣀⢠⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣄⣿⠁⣿⡇⣿⣧⣸⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠀⢹⣿⠀⢿⣧⢰⡦⢈⣙⢻⣧⠀⠈⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⢘⣿⣿⣿⣿
⣿⣿⣿⣿⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⢠⡇⢸⣿⠀⢸⣿⢨⣷⡘⠿⠾⠛⣠⣄⠛⣛⣁⣋⣉⣡⣬⣭⣈⣀⣀⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿
⣿⣿⣿⣿⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⠛⣠⣄⣛⣁⡈⢉⣼⣿⣬⣠⣶⣶⣶⣾⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣆⠘⠿⠾⠟⠀⠿⠀⠛⠂⠛⠛⠃⣈⣉⣉⣤⣬⣤⣤⣤⣴⣶⣶⣶⣤⡀⢰⣶⣤⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢥
⣿⣿⣿⣿⣿⣶⣤⣤⣴⣶⣶⣆⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣏⣼⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢛⣿⣿⣿⣿⣿⢻⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⢿⣿⣿⣿⣿⠺⣿⡎⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⡋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢳⣾⣿⣿⣿⣧⠀⣿⣄⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀
⣿⣿⣿⣿⠿⢿⣿⡟⣶⣄⠙⢿⣿⣿⣽⣿⣿⣿⣿⣿⢿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠘⣯⣽⣿⣿⣿⣦⣿⣿⡆⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⢻⣿⣿⣿⡿⠀⠀
⣿⣿⣿⠀⠀⢸⣿⣧⣿⣿⣿⣬⣿⣿⣿⣿⣿⣿⣿⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⡟⣿⣿⣿⣿⣿⣿⣿⣄⠈⢿⣿⣿⡿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠚⠛⠚⠿⢿⣿⠃⠀⠀
⣿⣿⣿⡄⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢸⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⠀⣺⣿⠛⣿⠇⣿⣝⣿⣿⣅⣀⠀⠙⣿⣿⣿⣿⣿⣿⣷⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⠀⢰⠄⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠈⠉⠉⠉⠙⠛⠿⣄⣤⣿⣿⣿⣿⣿⠀⠉⢻⣼⣿⣤⣿⡤⢹⣿⣿⣿⣷⣆⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣷⣾⣿⣶⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠺⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡂⢀⡀⠀⠀⢀⣠⣾⣿⣷⣄⡀⠀⠀⠀⠰⣦⣤⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⢿⣿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠟⠋⢀⣶⡀⠀⠈⠀⠀⣠⣄⠉⠛⠻⣿⣿⣿⣿⣿⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠠⠡⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠘⠛⠥⠀⠀⠀⠀⣿⡿⠆⠀⠀⣸⣿⣿⣿⣿⣿⡿⢟⣣⣀⣠⡖⡢⠀⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡷⢎⢄⠄⠀⢀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠄⠀⠀⠠⢀⠀⠼⡣⢼⣷⣶⣶⡄⣀⣀⣀⣀⣀⠀⠒⣀⣀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣃⢼⣷⣿⡿⠷⠲⠶⠺⠟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡟⡽⢆⡫⢻⢃⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣀⢀⠽⡀⣀⠄⢸⡬⣹⣿⣿⣿⣿⣿⣿⣋⡁⠀⠀⠉⠻⠿⠗⠚⠙⠛⠛⠛⠿⠛⣀⣁⣀⡀⢠⣤⢠⣤⣤⢠⣦⠐⣶⠀⠀⠀⠀⠀⠤⣄⡄⠀⣠⣴⣆⡀⠀⢀⠀⠀⠀⠠
⣿⣿⣿⣿⡹⢬⣹⠑⠀⠔⠀⠀⠀⠀⠀⠘⠒⠃⠈⠉⠈⠈⠁⠈⢈⣈⣁⡀⣀⡁⣤⡌⢀⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠈⣿⣧⣿⠀⠈⣦⡠⣴⣿⣿⣷⣿⣿⣿⣿⣮⠚⣇⢄⠢⢀⡠
⣿⣿⣿⣿⣿⣇⠀⠈⠘⠟⠢⣀⣀⣈⡀⠀⣶⡆⣶⣷⢸⣿⢸⣿⠘⢻⣿⠁⢿⣇⣿⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣤⣴⣶⣻⣼⣿⣿⣷⣦⠀⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣈⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠸⠿⠛⠛⠛⠛⠉⠉⠉⢉⣈⠉⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⠀⠙⠛⠉⠀⠈⠉⠁⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣄⠐⡆⢠⣤⠀⢠⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠿⠿⠗⠀⠉⠉⠤⠤⠤⠶⠶⠾⠛⠒⣛⡛⣃⡛⢋⣥⣤⡀⠉⢠⣶⠀⣶⡶⠆⠀⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⡄⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠘⢿⣿⣿⣿⣿
⣿⣿⣿⣿⠟⠉⣀⣀⠁⢀⣀⢠⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣄⣿⠁⣿⡇⣿⣧⣹⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠀⢹⣿⠀⢿⣧⢰⡦⢈⣙⢻⣧⠀⠈⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⢸⣿⡿⣿⣿
⣿⣿⣿⣿⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⢠⡇⢸⣿⠀⢸⣿⢨⣷⡘⠿⠾⠛⣠⣄⠛⣛⣁⣋⣉⣡⣬⣭⣀⣀⣀⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿
⣿⣿⣿⣿⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⢛⣀⣄⣛⣁⡈⣉⣼⣿⣮⣤⣶⣶⣶⣾⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⣿⣿⣿⣿⣆⠘⠿⠾⠟⠀⠿⠀⠛⠂⠛⠛⢃⣈⣉⣉⣤⣬⣤⣤⣤⣴⣶⣶⣶⡄⠀⣰⣦⣴⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡡
⣿⣿⣿⣿⣿⣶⣤⣤⣶⣶⣶⡼⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣴⣿⣿⣏⣽⣿⣿⣿⣿⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⢿⣻⣿⣿⣿⣿⣿⢻⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⡿⢸⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⣿⣿⣿⣿⣿⣿⣿⡿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣷⣾⣿⣿⣿⣇⢸⣿⣀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀
⣿⣿⣿⡿⠿⢿⣿⠁⢠⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⣭⣿⣿⣿⣿⣸⣿⢻⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⢺⣿⣿⣿⡿⠀⠀
⣿⣿⣿⠀⠀⢸⣿⡤⣾⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢰⣿⠛⣿⣿⡿⣿⣿⣿⣷⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠚⠛⠚⠿⢿⣿⠃⠀⠀
⣿⣿⣿⡄⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⡏⠋⣹⡿⢻⣿⠀⣷⣹⣿⣿⣁⣀⠈⢻⣿⣿⣿⣿⣿⣿⣧⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⠀⣴⠍⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⠀⠉⠉⠉⠙⠛⠻⢧⣀⣼⣿⣿⣿⣿⡇⠀⠉⣷⣾⣷⡄⣯⠌⢹⣿⣿⣿⣦⣄⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣷⣿⣿⣶⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣀⠀⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠃⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⠰⠇⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⣠⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⢀⡀⠀⠀⢀⣀⣾⣿⣷⣄⡀⠀⠀⠀⠠⣤⣤⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⢿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠛⠋⢀⣶⡀⠀⠈⠀⠀⣠⣄⠈⠛⠛⣿⣿⣿⣿⣿⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⢿⣿⡇⠀⠀⠀⠘⠛⠧⠀⠀⠀⠀⣿⡿⠆⠀⠀⣸⣿⣿⣿⣿⣿⡿⠟⠃⠀⠀⢄⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠴⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⣷⣰⣶⡄⣀⣀⣀⣀⡀⠀⠒⢀⣀⡀⠀⣶⣿⣿⣿⣿⣿⣿⣠⠴⣷⡟⠢⠥⠠⠩⠠⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⠑⠘⠁⠀⢐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠀⡂⠠⢈⣿⣿⣿⣿⣿⣿⣏⠁⠀⠀⠙⠻⠿⠛⠚⠙⠛⠛⠛⠿⠛⣀⣁⣀⡀⢠⣤⢠⣤⣤⢠⣦⠐⣶⠀⠀⠀⠀⠀⢠⠀⠂⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀
⣿⣿⠾⠀⠀⠀⠀⡀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢈⣉⣀⡀⣀⡁⣤⡌⢀⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠈⣿⣧⣿⠀⠀⣠⢀⣄⣿⣾⣧⣤⣿⣷⣯⣦⣀⢠⣄⣀⠀⠀
⣿⡟⢠⡤⡅⡈⣉⡘⢐⠃⠚⢿⡆⠀⠀⠀⣶⡆⣶⣷⢸⣿⢸⣿⠘⢻⣿⠁⢿⣇⣿⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⢰⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣷⣿⣾⣷
⣯⣌⣰⢃⠀⣬⠄⢟⢽⣿⣶⣺⣝⣥⣄⠀⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣈⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠸⠟⠛⠛⠛⠛⠉⠉⠉⢁⣈⠙⢿⣿⣿⣿⣿⣿
⣿⣾⣿⡷⢿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣾⡄⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⠀⠙⠛⠉⠀⠈⠉⠁⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣄⠐⡆⢠⣤⠀⢠⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣿⣿⢿⡿⣿⠟⠉⠿⠍⠩⠿⠷⠀⠉⠉⠤⠤⠤⠶⠶⠾⠛⠒⣛⡛⣃⡛⢋⣥⣤⡈⠉⢠⣶⠀⣶⡶⠆⠀⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⡄⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠘⢿⣿⣿⡿⠠
⣿⣿⣿⣿⠟⠉⣀⣀⠀⢀⣀⢠⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣄⣿⠁⣿⡇⣿⣧⣹⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠀⢹⣿⠀⢿⣧⢰⡦⢈⣙⢻⣧⠀⠈⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⢸⣿⣿⡿⣳
⣿⣿⣿⣿⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⢠⡇⢸⣿⠀⢸⣿⢨⣷⡘⠿⠾⠛⣠⣄⠛⣛⣁⣋⣉⣡⣬⣭⣀⣀⣀⣤⣤⣤⣤⣤⣴⣾⣿⣿⣽⣿
⣿⣿⣿⣿⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⢛⣀⣤⣛⣁⡈⣩⣼⣿⣮⣤⣶⣶⣶⣾⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⠘⠿⠾⠟⠀⠿⠀⠛⠂⠛⠛⢃⣈⣉⣉⣤⣬⣤⣤⣤⣴⣶⣶⣦⢄⠀⣴⣦⣴⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⣿⣿⣿⣿⣿⣶⣤⣴⣶⣶⣶⡼⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣁⣿⣿⣿⣿⣿⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣟⣻⣿⣿⣿⣿⣿⢿⡎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡿⣿⣿⣿⣿⡿⢸⣷⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⣿⣿⣿⣿⣿⣿⣿⡿⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣶⣿⣿⣿⣿⡇⢸⣿⣀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀
⣿⣿⣿⠿⠿⢿⣿⢀⣦⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣹⣿⣿⣿⣧⢸⣿⢻⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⢻⣿⣿⣿⣿⠀⠀
⣿⣿⣿⠀⠀⢼⣿⣿⣿⣿⣦⡐⣿⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣼⡟⢻⣿⣿⡿⣾⣿⣿⣧⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠚⠛⠚⠿⢿⣿⠃⠀⠀
⣿⣿⣿⡄⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⠀⢀⣷⠟⣿⣯⢰⣧⡻⣿⣿⣀⡀⠈⢿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⠀⢶⠄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⢙⠉⠉⠉⠛⠻⢧⣤⣾⣿⣿⣿⣿⡄⠈⢹⣶⣿⣷⣼⣧⠌⣻⣿⣿⣷⣦⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣷⣾⣿⣶⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⢀⡀⠀⠀⢀⣠⣾⣿⣷⣄⡀⠀⠀⠀⢰⣦⣤⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⠠⠥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⠀⠀⠈⠁⠀⠀⣀⠀⠀⢀⠀⣇⡀⠀⠀⠀⠀⠀⠃⠀⠀⢸⣿⣿⣿⡿⠿⠋⢀⣶⡀⠀⠈⠀⠀⣰⣄⠈⠛⠻⣿⣿⣿⣿⣿⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠘⠛⠧⠀⠀⠀⠘⣿⡿⠆⠀⠀⣸⣿⣿⣿⣿⣿⠿⢟⠃⠀⠀⣢⣶⡐⠢⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣞⡰⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢸⣷⣶⣶⡄⣀⣀⣀⣂⣀⠀⠒⣀⣀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣤⣼⣷⣧⣷⡿⠿⠶⠿⠷⠛⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢚⣀⣿⣽⣃⡥⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⡒⠄⢦⣤⢬⣿⠞⣿⣿⣿⣿⣿⣿⣏⡁⠀⠀⠙⠻⠿⠗⠚⠙⠛⠛⠛⠿⠛⣁⣁⣀⡀⢠⣤⢠⣤⣤⢠⣦⠐⣶⠀⢀⡀⠀⠀⡀⠀⣂⡀⢠⣀⣀⣀⡀⠀⡀⠀⠀⠀
⣿⣿⣿⣿⠺⣿⣀⣀⠀⠀⠀⠀⠀⠀⠔⠋⠛⠃⠉⠙⠉⠈⠁⠉⢉⣉⣉⡁⣀⡉⣤⡌⢀⣴⣦⡄⢰⣶⢰⣶⠸⢿⣿⠃⠀⢸⣿⠛⢸⣿⢸⣿⢸⣿⣀⠈⣿⣧⣿⠀⢸⣝⣵⣵⣷⣾⣶⣧⣷⣾⣶⣿⣿⣷⣦⢄⠠⣀
⣿⣿⣿⡊⡌⢍⠉⣥⣴⣿⣆⣠⡆⠾⡁⠀⣶⡆⣶⣷⢸⣿⢸⣿⠘⢻⣿⠁⢿⣇⣿⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⢸⣿⠀⠀⢸⣿⠛⠀⣿⣾⣿⠘⣿⡍⡀⣿⠹⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣯⣽⣿⣾⣶⣤⣏⣿⣿⣿⣿⣷⣼⣧⠀⢸⣧⣿⢿⣼⣿⠸⣿⠀⢸⣿⠀⢸⣿⢹⣿⢸⣿⢸⣿⠘⣿⣈⣿⠀⢸⣿⠀⠀⠈⡿⠶⠆⠸⠿⠟⠀⠛⠛⠃⠋⠁⠉⠁⠸⠿⠛⠛⠛⠛⠉⠉⠉⢉⣉⠙⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠈⣿⣿⠘⣿⣿⠀⣿⡆⠸⠿⠀⠸⠿⠸⠛⠀⠙⠛⠉⠀⠈⠉⠁⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣄⠐⡆⢠⣤⠀⢠⣶⡶⢠⣶⢳⣦⠀⣿⣿⡆⢸⣿⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠧⣀⡉⠉⢤⠤⠤⠶⠶⠾⠛⠒⣛⡛⣃⡛⢋⣥⣤⡈⠉⢠⣶⠀⣶⡶⠆⠀⣿⡇⠛⣿⡟⠛⠇⢿⣏⡛⠃⡄⢸⣿⠀⢸⣿⣤⢸⣿⢨⣭⠀⣿⢸⣷⠘⣿⠀⠘⢿⣿⣿⣿⣿
⣿⣿⣿⣿⠟⠉⣁⣀⠈⢀⣀⢠⣤⢠⣤⣤⢀⣴⠶⣦⠀⣿⣄⣿⠁⣿⡇⣿⣧⣹⡇⢸⡇⠹⠓⠀⠘⣿⡄⣿⣧⡆⠀⢹⣿⠀⢻⣧⢰⡦⢈⣙⢻⣧⠀⠈⣿⣀⢸⣿⣀⡈⣿⣄⣿⡆⣿⡟⣿⡆⠿⠷⠆⢸⣿⣿⣿⣷
⣿⣿⣿⣀⠀⢼⡏⢹⣧⢸⣿⣸⣿⢸⣿⣤⠈⣿⡄⠛⠃⣿⣿⣿⠀⣿⡇⢹⡿⣿⡇⢸⣿⢹⣿⠀⠀⣿⡇⢿⡇⢠⡇⢸⣿⠀⢸⣿⢨⣷⡘⠿⠾⠛⣠⣄⠛⣛⣁⣋⣉⣡⣬⣭⣀⣀⣀⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿
⣿⣿⣿⣿⠀⢸⣿⢠⣤⢸⣿⠹⣿⠀⣿⣏⡀⣿⣇⣿⡇⢻⡏⢿⣇⢹⣷⠸⠇⠹⠿⠈⠻⠟⢛⣀⣤⣛⣁⡈⣉⣼⣿⣮⣤⣶⣶⣶⣾⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⠘⠿⠾⠟⠀⠿⠀⠛⠂⠛⠛⢃⣈⣉⣉⣤⣬⣤⣤⣤⣶⣶⣶⣦⣄⠀⣴⣦⣴⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣦⣤⣴⣶⣶⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣿⣿⣿⣿⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣿⣿⣿⣿⣿⢿⡎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡿⢸⣷⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⠌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣶⣿⣿⣿⣿⡇⢸⣿⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀
⣿⣿⣿⠿⠿⢿⣿⣓⣠⣀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⣿⣽⣿⣿⣿⣇⢸⣿⢻⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⢻⣿⣿⣿⡿⠁⠀
⣿⣿⣿⠀⠀⢼⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⡟⣻⣿⣿⠟⣾⣿⣿⣧⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠚⠛⠚⠻⢿⣿⠇⠀⠀
⣿⣿⣿⡆⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣿⣿⣿⣿⣿⣿⡟⠛⣿⣿⣿⣿⣿⠀⢠⣿⠟⣿⣧⢠⣧⢻⣿⣿⣁⡀⠈⢿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⠀⢶⢤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠉⣉⡙⠛⠧⣤⣾⣿⣿⣿⣿⠄⠈⢹⣼⣿⣷⣼⣧⠌⣻⣿⣿⣿⣶⣤⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀

                    ䷩ 𝚕𝚒𝚗𝚎 5366

╒═══════════════════ 𝐀𝐑𝐓𝐈𝐂𝐋𝐄 ═════════════════════════════════════════════════╕

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/01/23/irc-log-220122/#comments
    Gemini version at gemini://gemini.techrights.org/2022/01/23/irc-log-220122/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 01.23.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ IRC_Proceedings:_Saturday,_January_22,_2022⠀✐


Posted in IRC_Logs at 3:16 am by Needs Sunlight


Also available via the Gemini protocol at:


    * gemini://gemini.techrights.org/irc-gmi/irc-log-techrights-220122.gmi
    * gemini://gemini.techrights.org/irc-gmi/irc-log-220122.gmi
    * gemini://gemini.techrights.org/irc-gmi/irc-log-social-220122.gmi
    * gemini://gemini.techrights.org/irc-gmi/irc-log-techbytes-220122.gmi
Over HTTP:


                                      🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇H                                     🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇HTML5_logs⦈_


                                         #techrights_log_as_HTML5                                                                                  #boycottnovell_log_as_HTML5


                                      🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇H                                     🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇HTML5_logs⦈_


                                    #boycottnovell-social_log_as_HTML5                                                                               #techbytes_log_as_HTML5


                                      🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇t                                      🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇text_logs⦈_


                                          #techrights_log_as_text                                                                                   #boycottnovell_log_as_text


                                      🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇t                                      🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇text_logs⦈_


                                     #boycottnovell-social_log_as_text                                                                                #techbytes_log_as_text


                          Enter_the_IRC_channels_now


===============================================================================
§ IPFS Mirrors⠀➾


CID                                              Description     Object type


                                                 IRC log for
 QmXMvjAqg5QJ6DBybB6eLqm2ugh4AzhphLfFiMPzgjNmab #boycottnovell  🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
                                                 (full IRC log
                                                 as HTML)


                                                 IRC log for
                                                 #boycottnovell
 QmWFkfTK59oCSX33FhU8AkFma34mA6o2qyBp2mqPkqfKDF (full IRC log   🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
                                                 as plain/ASCII
                                                 text)


                                                 IRC log for
                                                 #boycottnovell-
 QmV3E51DVTeeXSTvtKUacq5YLXMcbkgnZiZGFgzptxzPHM social          🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
                                                 (full IRC log
                                                 as HTML)


                                                 IRC log for
                                                 #boycottnovell-
 QmafqvXcAzNzw7S28aFHdkATNBkKq1v7JDZDgdaaaxGs2q social          🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
                                                 (full IRC log
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                                                 IRC log for
 QmeaEkDo2uwTySBVZzz2x1etBWjDQrtruW9wmtNELceTSt #techbytes      🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
                                                 (full IRC log
                                                 as HTML)


                                                 IRC log for
                                                 #techbytes
 QmcGpAuFq4ZFSRMnibQvsDgqgPCrRGXstBKg31NGLKjm4M (full IRC log   🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
                                                 as plain/ASCII
                                                 text)


                                                 IRC log for
 QmeKjfoYY6FFXXSW88az8ft7eFfaKU556B7DmLHS9Rtx8C #techrights     🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
                                                 (full IRC log
                                                 as HTML)


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(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/01/23/peak-hacker/#comments
    Gemini version at gemini://gemini.techrights.org/2022/01/23/peak-hacker/

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Gemini_version_available_♊︎


✐ On_the_‘Peak_Hacker’_Series⠀✐


Posted in Free/Libre_Software, GNU/Linux at 9:01 am by Dr. Roy Schestowitz


Video_download_link | md5sum 5e72b9f7376540003a836dc02d90b570
Peak Hacker and Shunning Fake Progress
Creative Commons Attribution-No Derivative Works 4.0


http://techrights.org/videos/peak-hacker.webm


Summary: Hacker_culture, unlike Ludditism, is ultimately a movement for
justice, for equality, and for human rights through personal and collective
emancipation; Dr. Farnell has done a good job explaining where we stand and his
splendid series has come_to_a_close


TODAY we finished publishing the series from 'Digital_Vegan' Dr. Farnell, who
expects to have “his comprehensive [next book] Ethics for Hackers (scheduled
early 2022).”


I’ve had the pleasure of exchanging ideas and thoughts with him, “a British
computer scientist, writer and international visiting professor specialising in
signals and systems,” according to his bio. He’s also a proponent of BSD and
GNU/Linux, but there’s lots of us who do that. GNU/Linux is nowadays everywhere
and “Wine is a better Windows than Vista,” as David Gerard put it over a decade
ago. Even the gaming industry is moving towards GNU/Linux (like Steam Deck with
Arch Linux and KDE), so we need to look ahead and promote more than just the
brand “Linux”. Steam is DRM, so not exactly where we want to go (even though
removing_Windows_is_a_positive_step). The goal isn’t having better Windows than
Windows®.


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇Andy Farnell⦈ The video above sums up some personal thoughts
and offbeat comments. I use only Gemini in this video and I hope this will
become the standard, seeing what the Web rapidly devolves into. As of this
moment, Geminispace_approaches_2,000_known_capsules (including inactive ones)
and there seems to be growing resistance to the technological status quo. That
should not be conflated with Ludditism because this isn’t about economics
(passing workers’ jobs to machines) but about human rights or basic dignity
such as personal privacy. █


                      🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇digital-vegan⦈_


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⠀⠀⠸⠉⣮⠀⣿⣿⠂⠉⠀⠀⠀⣷⠀⠠⡄⠀⠀⢠⠀⠀⠀⠀⣶⠀⠀⣸⡀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠓⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄
⠀⠀⠀⢀⣿⠀⠀⣙⢦⡀⠀⣤⡀⢿⡇⠀⢉⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⣤⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢃⠀⢠⠄⠀⠈⠥⠂⠀⣼⠀⠀⠀⠀⠃
⠀⠀⠀⠸⣿⣇⠀⠿⠂⢙⡆⠉⣡⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢹⠀⠀⠀⠀⠀⡌⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡒⠀⠈⠀⠀⠀⠂⠀⣠⠁⠀⠀⠀⠀⠠
⠀⠀⣄⠀⢻⣿⡄⠀⢠⠘⣷⣄⠙⣇⠈⠇⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣛⡞⠀⠀⠀⠀⠀⠀⢠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⣶⠚⠀
⠀⠀⢹⡆⠀⠉⢷⠀⠈⠳⣌⠿⠀⠈⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⠁⠀⠀⠀⠀⠷⠀⠀⠀⠀⠈⠀⠀⠰⠦⠀⠀⠔⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⠀⠀
⠀⢀⣼⣷⡄⠀⢰⠄⠀⠀⠘⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠠⠀⡄⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⢤⣴⠀⠀⣀⣦⡤⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠁⠀⠀
⠀⣾⣿⣿⣷⣄⠀⠀⠱⣶⣀⠀⠁⣀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠿⠂⠀⠀⠀⠀⠈⠁⠆⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠀⠀⡂⠙⠉⠀⠀⠀⠀⠁⠀⢀⣴⠀⠀⠀⣤⠀⠐
⢠⣿⡟⠛⣻⣿⡅⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⠀⠀⠀⠀⠃⠀⠆⠄⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⠆⠀⠀⠀⠀⠀⠀⢡⢼⣿⠀⠀⠀⠉⠀⠀
⣼⣿⣷⡄⢿⣿⣧⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠤⠀⠀⠀⢻⣷⠀⠀⠀⡄⠀⠀⠂⡄⠖⠋⠈⠁⠀⠀⠀⠀⣀⡀
⣿⣿⣿⡿⠿⣿⣿⢿⠛⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠘⠀⠂⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠟⠃
⣿⢭⣿⡶⢲⣿⣿⣿⣏⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣶⠷⠹⣟⡀⠀⠀⠀⠀
⣿⣼⣿⠁⢸⣿⣿⣿⣿⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠛⠛⠒⠄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠿⠛⢛⣉⣤⠐⡆⢹⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣄⠀⢀⣴⣿⣿⣿⣷⣦⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠙⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣄⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡆⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⢰⣶⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡂⢭⣿⡿⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡧⢼⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡛⢦⣾⣿⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣙⢠⣸⣿⣇⣸⣿⣿⣿⣷⣵⠊⠿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣏⡂⣿⣿⣿⡄⠀⠀⠀
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡂⢭⣿⣿⠇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⠧⣂⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣾⡅⢍⢿⣿⣿⣿⣈⠻⣿⣿⣿⣿⣿⣿⡿⢕⣪⣿⣿⣿⣿⡀⠀⠀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⢃⢪⣿⣿⡟⢰⣿⣿⣿⣿⣿⣿⣿⠟⢋⠙⣿⣿⣿⠟⡏⠩⣿⡆⢻⣿⣿⣿⣿⣿⢃⢍⣺⣿⣿⣿⡇⢀⣉⣉⣏⡙⠻⣿⣿⣿⣷⡕⢍⣿⣿⣿⣧⠘⣿⣿⣿⣿⣿⣏⣒⣾⣿⣿⣿⣿⣧⠀⠀
⢿⣿⣿⣿⣿⣿⣿⣿⡙⢲⣿⣿⣿⡃⣼⣿⣿⣿⡟⠫⡱⣀⣭⢶⢻⢋⢣⣢⣦⣾⣿⣿⣇⢸⣿⣿⣿⣿⢋⣢⣾⣿⣿⣿⣿⡧⢼⣿⣿⣿⣿⣷⣅⠙⢿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⡿⢅⣲⣿⣿⣿⣿⡿⠸⡄⠀
⠀⠉⠛⠛⠛⠛⠛⠛⠻⢿⣿⣿⡿⢈⣿⣿⣿⣿⣧⣷⢿⠋⢣⣣⣧⣶⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⠫⢑⣾⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣶⣉⠛⢿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡯⠄⣾⣿⣿⣿⣿⠇⠀⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡇⣸⣿⡿⠟⠏⠍⣶⣤⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⢃⢨⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⡀⠌⡛⡻⢿⣿⣿⣿⣷⣾⣌⡙⢻⠿⠇⢸⣿⣿⣿⠩⢰⣿⣿⣿⣿⡿⠀⠀⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⣨⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣶⣧⣧⡞⢙⢛⡟⡿⣿⣿⣿⡄⢲⣿⣿⣿⡿⠭⣴⣿⣿⣿⣿⠇⠀⠀⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣤⣱⠩⡛⡻⡗⢸⣿⣿⣿⠇⢭⣿⣿⣿⣿⡿⠀⠀⠀⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠿⠿⢿⠛⠛⠛⡛⢉⣉⣏⡉⢀⣩⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣉⣏⣉⣉⣙⠛⠛⠻⠿⢿⣾⣿⠈⣿⣿⡟⣂⣾⣿⣿⣿⣿⠇⠀⠀⠀⣇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣥⣤⡤⠶⠶⢶⢾⢿⠿⣿⣿⣿⣿⠟⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣍⠉⣿⣿⣇⣲⣿⣿⣿⣿⣿⠀⠀⠀⠀⣸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⣾⣜⣼⣤⣸⣜⣼⣿⠿⢃⣴⣿⣿⣿⣿⣿⢿⣿⢿⢻⢻⡿⣿⢿⣿⢻⣿⣿⡟⡟⣿⣿⢻⡿⣿⣿⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⡟⣁⣴⣿⣿⣿⣿⣿⣿⣧⣂⣇⣆⣸⣸⣠⣠⣁⣿⣐⣸⣈⣀⣇⡇⣸⣸⣀⡇⢃⣿⣿⣿⣿⣿⠀⣿⠿⠟⠛⠛⠛⠛⠛⢿⠇⠀⠀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⢋⣤⣾⣿⣿⣿⣿⣿⡏⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠀⠀⣴⠷⣿⣿⣿⣿⣿⣶⣦⡀⠀⠘⠏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢡⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⠿⠿⠿⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣜⠿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠘⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣔⡨⠛⣿⣿⣿⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣔⠅⡝⢻⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠈⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠏⠀⢹⣼⠁⣹⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠙⠋⠀⠀⠀⠀⠈⠿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢻⠋⠀⠀⠈⣉⣉⡅⠀⠀⠀⠀
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                    ䷩ 𝚕𝚒𝚗𝚎 5624

╒═══════════════════ 𝐃𝐀𝐈𝐋𝐘 𝐋𝐈𝐍𝐊𝐒 ═════════════════════════════════════════════╕

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 01.23.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Links_24/1/2022:_pgBadger_11.7_Released,_Catch-up_With_Patents⠀✐


Posted in News_Roundup at 9:13 pm by Dr. Roy Schestowitz


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇GNOME bluefish⦈


§ Contents⠀➾


    * GNU/Linux
          o Distributions
          o Devices/Embedded
    * Free_Software/Open_Source
    * Leftovers
    * § GNU/Linux⠀➾


          o ⚓ Linux_Weekly_Roundup_#166⠀⇛


                 Welcome to this week’s Linux Weekly Roundup in the world
                 of Linux releases. We had another great week with plenty
                 of new releases and updates, like Deepin 20.4 (also they
                 have a nightly build ISO of Deepin V23), Qubes OS 4.1.0
                 RC4, and Linuxfx 11.1.1103.


          o § Audiocasts/Shows⠀➾


                # ⚓ ArchDI:_Arch_Linux_Desktop_Installer…_I_Guess_–_Invidious⠀⇛


                       ArchFi the Arch Linux fast installer only considers
                       getting Arch Linux installed, ArchDI is the
                       follower up desktop installation script that I
                       don’t see the point of.


          o § Kernel Space⠀➾


                # ⚓ Linux_5.17_Features_From_New_AMD_P-State_To_Xilinx_Drivers,
                  Lots_Of_New_Hardware⠀⇛


                       
                       This morning marked the release of Linux 5.17-rc1
                       that officially ends the merge window for this next
                       stable kernel series. Linux 5.17 won’t see its
                       stable debut until around the end of March but
                       there is a lot to get excited about for this open-
                       source kernel in 2022.


                       Linux 5.17 is exciting for mainlining the AMD P-
                       State driver that has been under review/testing for
                       the past several months in cooperation with Valve
                       for the Steam Deck, initial Intel Raptor Lake
                       bring-up bits, Intel Alder Lake P graphics being
                       promoted to stable, lots of preparations for future
                       AMD processors, initial support for the recently
                       launched Qualcomm Snapdragon 8 Gen 1, many tablet /
                       laptop support improvements, x86 straight line
                       speculation mitigation support, support for a low-
                       cost RISC-V platform, and a whole lot more.


                # ⚓ Linus_Torvalds_Announces_First_Linux_5.17_Kernel_Release
                  Candidate⠀⇛


                       Linus Torvalds just announced today the first
                       Release Candidate of Linux kernel 5.17, which looks
                       to be a normal release with a normal amount of
                       changes and new features. Nothing fancy, and
                       nothing that stands out. There’s a little bit of
                       everything for everyone.


                       According to Linus Torvalds, the bulk of it is
                       various driver updates, and there’s also
                       architecture, documentation, and tooling updates.
                       However, the random number generator work and the
                       rewrite of the fscache persistent local cache tool
                       stand out in the upcoming Linux 5.17 kernel series.


                # ⚓ Kernel_prepatch_5.17-rc1⠀⇛


                       The first 5.17 kernel prepatch is out for testing,
                       and the merge window is closed for this release.


          o § Applications⠀➾


                # ⚓ 5_Best_Free_and_Open_Source_Stock_Tickers⠀⇛


                       
                       A stock ticker is a report of the price of specific
                       securities, updated continuously throughout the
                       trading session by the various stock market
                       exchanges.


                       The term “tick” refers to a change in a security’s
                       price from one trade to the next. A stock ticker
                       displays these ticks, along with other relevant
                       information, like trading volume, that investors
                       and traders use to stay informed about current
                       market conditions and the interest in that
                       particular security.


          o § Instructionals/Technical⠀➾


                # ⚓ How_to_Enable_MySQL/MariaDB_Query_Logging⠀⇛


                       In this short and easy to follow guide, we cover
                       how to enable MySQL/MariaDB query logging and save
                       queries to a file.


                       When using a database, it often happens that
                       developers want to know what’s going on behind the
                       scene. Whether it is for troubleshooting,
                       performance tuning, or out of sheer curiosity.


                       General query logs remain one of the most popular
                       sources of auditing and diagnostic information in
                       MySQL/MariaDB databases. After logging is enabled,
                       the database server will write information to the
                       log file when clients connect or disconnect, and it
                       will log each SQL statement.


                # ⚓ How_to_install_Zotero_on_Ubuntu_22.04_or_20.04_LTS_–_Linux
                  Shout⠀⇛


                       Zotero is a reference management system and
                       academic network, here we learn the steps to
                       install Zotero in Linux such as Ubuntu 22.04 Jammy
                       JellyFish or Ubuntu 20.04 Focal Fossa.


                       Zotero is an open-source and free tool that helps
                       in collecting, organizing, citing, and sharing
                       research. It consists of two components: Zotero
                       standalone (Mac, Windows, Linux) with a plugin for
                       Firefox, Google Chrome, and Safari; Zotero web
                       application: computer-independent access to your
                       library, finding and setting up groups on specific
                       topics (see collaboration), making contacts with
                       others. The two components of this tool can be
                       synchronized with each other.


                       If you are often involved in extensive projects at
                       work or have to write a large-scale work during
                       your studies, you quickly lose track of the sources
                       used. With Zotero, you keep all information
                       together and create a clear library for your
                       project without much effort. It is particularly
                       suitable for literature research on the Internet
                       and the direct transfer of the researched data and
                       works in groups that want to manage literature
                       together.


                # ⚓ How_to_Copy_Directory_in_Linux:_(Beginner_to_Advance)⠀⇛


                       While working in several operating systems, we use
                       to add files and folders, cut, copy, edit and
                       delete them as per our choice. Just like all these,
                       the Linux system provides us the opportunity to
                       copy a single folder to another folder even though
                       the folder is empty or not. Therefore, we will
                       discuss some easy methods to copy one directory to
                       another in Ubuntu 20.04 system. Let’s have a fresh
                       start by using the different ways to copy a
                       directory from one place to another in Linux.


                       For this particular purpose, we must open up the
                       terminal shell as we are going to do all things in
                       it. There are two ways to open it. One is from the
                       activity search area and the other is the shortcut
                       key “Ctrl+Alt+T”. Practice one of the mentioned
                       methods to open it and start implementing the
                       commands in it one by one.


                # ⚓ How_to_Change_Default_Torrent_Client_for_Magnet_Links_in
                  Ubuntu_20.04+_|_UbuntuHandbook⠀⇛


                       When clicking a magnet link in Chrome browser, it
                       pops-up with “Open xdg-open” option allows to
                       launch the default app to start the downloading.


                       Unlike Firefox, Google Chrome does not offer an
                       option to choose which app to handle the link.
                       Instead, it launches the default app directly. For
                       those want to change this default app to handle
                       magnet links, here’s how to do the trick in Ubuntu.


                # ⚓ How_to_install_Microsoft_Fonts_on_Elementary_OS_6.0_–
                  Invidious⠀⇛


                # ⚓ How_to_install_VCV_Rack_2_on_a_Chromebook⠀⇛


                       Today we are looking at how to install VCV Rack 2
                       on a Chromebook. Please follow the video/audio
                       guide as a tutorial where we explain the process
                       step by step and use the commands below.


                # ⚓ How_to_Install_Papirus_Icon_Theme_on_Ubuntu_–_OMG!
                  Ubuntu!⠀⇛


                       Changing the icon theme on Ubuntu is an easy way to
                       give your desktop a new look and feel. In this post
                       we show you how to install Papirus icons on Ubuntu
                       from a PPA.


                       Why use the Papirus icon set? Because it’s good —
                       in fact, it’s one of the best icon themes for
                       Ubuntu not to mention other Linux distros like
                       Linux Mint, Zorin OS, and Manjaro. Thousands of
                       users use this set, and several Linux distributions
                       ship it as their default.


                # ⚓ Exploring_GIT_Commit_Hashes_&_Generating_Cryptographic
                  Zeros_–_Jon’s_FOSS_Blog⠀⇛


                       So I was trying to research what goes into
                       generating a GIT commit hash and I thought I would
                       try to personalize the cryptographic hash output a
                       little bit. My computer isn’t that powerful but it
                       may be possible to generate more zeros!


                # ⚓ Shows_a_specific_process_memory_usage⠀⇛


                # ⚓ How_to_install_Minikube_on_Ubuntu_20.04_–_NextGenTips⠀⇛


                       In his tutorial, we are going to learn how to
                       install Minikube on Ubuntu 20.04. I am going to use
                       Docker as a containerization platform. You can
                       either VirtualBox, Podman, KVM, etc.


                       Minikube is a tool that makes it easy to run
                       Kubernetes locally. Minikube runs a single-node
                       Kubernetes cluster inside a VM on your local
                       machine. Kubernetes quickly set up a local
                       Kubernetes cluster on Linux, Windows, and macOS.


                # ⚓ How_To_Install_Brave_Browser_On_Ubuntu_/_Fedora_/_Rocky
                  Linux_&_Manjaro_Linux_|_Tips_On_UNIX⠀⇛


                       Brave is a free and open-source web browser
                       designed by the Brave team and it is based on
                       chromium.


                       Brave Browser comes with default AD blocker and
                       blocks all type of ads from every website and it
                       also has multiple extensions available from chrome
                       web store and can be added to it.


                       This tutorial will be helpful for beginners to
                       install brave browser on Ubuntu 20.04 LTS , Ubuntu
                       21.10 , Rocky Linux 8 , Fedora 35 and Manjaro
                       Linux.


                # ⚓ How_To_Install_PHP_8_on_Linux_Mint_20⠀⇛


                       In this tutorial, we will show you how to install
                       PHP 8 on Linux Mint 20. For those of you who didn’t
                       know, PHP is a popular general-purpose scripting
                       language that is especially suited to web
                       development. The PHP language is released under
                       open-source standards, also object-oriented and
                       interpreted scripting language that can be executed
                       at the server-side. With this release, there are
                       many new features, improvements, and deprecations
                       from its predecessor (PHP 7).


                       This article assumes you have at least basic
                       knowledge of Linux, know how to use the shell, and
                       most importantly, you host your site on your own
                       VPS. The installation is quite simple and assumes
                       you are running in the root account, if not you may
                       need to add ‘sudo‘ to the commands to get root
                       privileges. I will show you the step-by-step
                       installation of a PHP 8.1 on a Linux Mint 20
                       (Ulyana).


                # ⚓ Chrome_Clear_DNS_Cache⠀⇛


                       Domain Name System (DNS) is the actual directory of
                       the internet in which the domain names of the sites
                       containing information that the user requires or
                       searches for are stored in the DNS servers. All the
                       web browsers use Internet protocols (IP) addresses
                       to look for resources and services available over
                       the internet. The Domain name system interprets the
                       human-friendly domain names into Internet protocol
                       (IP) addresses, usually consisting of complex
                       numbers and alphanumeric characters so that the
                       browser can locate the services and devices present
                       on that certain address. The DNS cache acts as a
                       boost to speed up the lookup process of the Domain
                       Name System. DNS cache, also known as DNS Resolver
                       cache, is the temporary storage on independent
                       devices which stores the frequently visited sites
                       on that device and assists the Domain name system
                       by suggesting the visited sites and resources.


                # ⚓ How_Do_I_Initialize_a_Drive_in_Synology_NAS?⠀⇛


                       Each of the drives of your Synology NAS will have
                       different Allocation status depending on the state
                       the drive is in.
                       The drive Allocation statuses are:


                       i) Initialized: When you install the DSM operating
                       system on your Synology NAS, it creates a system
                       partition in all the available drives installed on
                       your NAS. A drive will have the allocation status
                       Initialized if the DSM system partition is
                       installed on the drive and the drive is not used by
                       any storage pools.


                       ii) Not Initialized: If a drive does not have a DSM
                       system partition, it will have the Not Initialized
                       status. If you install a brand new drive on your
                       NAS, it will have the Not Initialized status as
                       well.


                # ⚓ How_to_Check_Folder_Permission_Linux⠀⇛


                       Linux is a multi-user operating system. It can
                       handle multiple users accessing the system at the
                       same time. This is one of the key reasons why Linux
                       is so popular in mainframes, servers,
                       supercomputers, etc. Being a multi-user system,
                       it’s paramount to have functional user permission
                       management. It’s the fundamental that defines what
                       user/group can access what part of the system.


                       For file access, this user permission is managed as
                       the file permissions. In this guide, we’ll have a
                       deep dive into the file and folder permissions in
                       Linux.


                # ⚓ How_to_Install_Gvim_on_Ubuntu_20.04⠀⇛


                       If you are a programmer, you understand the
                       importance of a good text editor application. Many
                       of us spend a lot of time sitting in front of a PC
                       and hitting the keys of the keyboard all day long.
                       A text editor with great features can significantly
                       boost your productivity. There are plenty of text
                       editors available on the web, and choosing one that
                       suits you is generally not an easy task. But if you
                       are looking for an efficient solution, we suggest
                       you try the Vim text editor.


                       Vim, which stands for “Vi Improved”, is a classical
                       text editor which is mainly used by programmers. It
                       is generally called “programmer’s editor”. Although
                       built long ago, it is still a popular editor and
                       outruns many competitors. It can be used from
                       general text editing jobs to managing configuration
                       files.


                       Vim is available for major operating systems like
                       Mac, Windows, Linux, or Unix. Vim can run in the
                       GUI and non-GUI console mode. For example, on Mac,
                       you can install MacVim for the GUI version of Vim.
                       Similarly, for Windows, you can use the “gvim”
                       executable installer. Linux also has a graphical
                       version of Vim known as gvim. Similar to other
                       clones of the “vi” editor, Gvim on Linux has many
                       powerful GUI features.


          o § Games⠀➾


                # ⚓ More_Details_About_Half_Life_2_Running_on_Pinephone_Pro_–
                  Boiling_Steam⠀⇛


                       I came across this video earlier today, showing the
                       Pinephone Pro running Half Life 2 at very decent
                       framerates. Sure, Half Life 2 is an old game
                       (2004), but it’s still a very impressive feat to
                       see it run on a phone that does not even use the
                       same architecture. The Pinephone Pro has an ARM
                       processor, while Half Life 2 is made for x86 (Intel
                       and AMD) CPUs in the first place.


          o § Desktop Environments/WMs⠀➾


                # § GNOME Desktop/GTK⠀➾


                      # ⚓ Check_Out_GNOME_Shell’s_New_Look_in_GNOME_42_–_OMG!
                        Ubuntu!⠀⇛


                             GNOME Shell looks a little different in GNOME
                             42, which is currently in active development.


                             I wasn’t able to showcase the shell theme
                             tweaks in my GNOME 42 alpha post but, over
                             the weekend, fuelled by coffee 😉, I managed
                             to get the correct branch up and running on
                             my Fedora install. I figured I’d write a
                             short post to share some screenshots of the
                             changes I’ve spotted thus far.


                             Just keep in mind GNOME 42 is under active
                             development. Everything shown here is a work
                             in progress and very much subject to change.


                             GNOME 42 embraces libadwaita and its rounder,
                             softer look. So it’s no surprise that the
                             latest GNOME Shell theme tweaks echo these
                             changes within the Shell UI. That said, while
                             the overall effect of the changes is
                             pronounced (i.e. you’ll notice it’s change)
                             we are still talking subtle refinement on
                             what went before, rather than a wholesale
                             departure.


                             For example, most Shell elements now use a


          o § Distributions⠀➾


                # § Reviews⠀➾


                      # ⚓ Review:_UBports_on_the_PinePhone⠀⇛


                             At the end of 2021 I posted a review about
                             the PinePhone and what it was like running
                             the device’s default operating system:
                             Manjaro Linux featuring the Plasma Mobile
                             interface. While I’ve had decent experiences
                             with Plasma Mobile in the past, the
                             combination of Manjaro, the PinePhone, and
                             this interface left a lot to be desired. It
                             was slow, it was unstable, and there were
                             issues with the battery charge indicator. To
                             make matters worse, the default applications
                             were not polished or well named. It was a
                             rough experience and, though I can see where
                             there is hope for Manjaro on the PinePhone, I
                             was eager to try something different.


                             In the past I have had mostly good
                             experiences with the UBports distribution.
                             UBports is a community project which has
                             continued Canonical’s Ubuntu Touch operating
                             system for smart phones and tablets. Since
                             I’ve run UBports on devices which didn’t have
                             any more memory or processing power than the
                             PinePhone, I was optimistic about trying out
                             the PinePhone with this alternative
                             distribution.


          o § Devices/Embedded⠀➾


                # ⚓ Keyboards_and_Open-Source_–_Ignorance_is_bliss…⠀⇛


                       In my Keyboard Fun post from last year I talked a
                       bit about my interest in mechanical keyboards.


                       Since then, I played around with a few more
                       keyboards/switches/keycaps/…


                       Interesting enough, beside the actual hardware,
                       naturally there is some software component to all
                       these keyboards, too.


                       Whereas most commercial keyboards still come with
                       proprietary firmware, there is the trend within the
                       keyboard enthusiast scene to go for open-source
                       firmware.


                       This allows you to properly update the firmware
                       even from your Linux machine and do proper
                       configuration of e.g. the keymap, too.


                # ⚓ Multilingual_blink_for_Raspberry_Pi_Pico_–_Raspberry_Pi⠀⇛


                       A year ago today we launched our Raspberry Pi Pico
                       board, the first product powered by the RP2040
                       microcontroller, a brand-new chip developed right
                       here at Raspberry Pi. A year later we’ve sold
                       nearly 1.5 million Picos, and thousands of you have
                       used RP2040 in your own electronic projects and
                       products.


                # ⚓ Hackaday_Links:_January_23,_2022⠀⇛


                       Sandy Macdonald from York quickly cobbled together
                       a Raspberry Pi and a pressure/humidity sensor board
                       and added a little code to create a recording
                       barometer.


                # § Perl/Raku⠀➾


                      # ⚓ Display_Your_Speech_In_Realtime_To_Help_Lipreaders_In
                        The_Mask_Era⠀⇛


                             The system consists of a small screen that
                             can be worn on the chest or other part of the
                             body, and a lapel microphone to record the
                             wearer’s speech. Using the Deepgram AI speech
                             recognition API running on a Raspberry Pi
                             Zero W, the system decodes the speech and
                             displays it on the Hyperpixel screen.


                      # § Open Hardware/Modding⠀➾


                            # ⚓ W5100S-EVB-Pico:_A_Raspberry_PI_Pico_with
                              Ethernet_port_–_peppe8o⠀⇛


                                   Hands on a W5100S-EVB-Pico board. Let’s
                                   discover this microcontroller and its
                                   main features


                                   In this tutorial, I’m going to show you
                                   what is the W5100S-EVB-Pico
                                   microcontroller board and how to use
                                   it.


                                   I’ll explain the main features of this
                                   IoT board, then show a few usage
                                   examples with the basic Ping example
                                   and a Mosquitto (MTTQ) example to
                                   interface an MTTQ broker like
                                   Thingsboard.


                            # ⚓ Arduino_data_types⠀⇛


                                   When writing an Arduino code to perform
                                   a task or working on a project the data
                                   has to be classified in different
                                   types. The microcontroller understands
                                   the information in the form of 0 and 1
                                   and it is difficult for a user to
                                   declare the number of bites for each
                                   data. To organize and categorize the
                                   data in different forms is quite
                                   helpful for programmers to understand
                                   the data given and perform the assigned
                                   task. For these different data types
                                   are used in the Arduino programing.


                                   There are two basic data types that are
                                   used in Arduino programming: one is
                                   variable data type and the other is
                                   constant data type.


                            # ⚓ Arithmetic_Operators_in_Arduino⠀⇛


                                   Arithmetic operators in Arduino
                                   programming are used to calculate the
                                   mathematical operations taking two or
                                   more numbers as operands.The use of
                                   operators in Arduino programming plays
                                   an important role as it defines the
                                   conditions on the basis of which the
                                   program works. There are different
                                   types of arithmetic operators that are
                                   used in Arduino programming.


                            # ⚓ Arduino_if-else_statement⠀⇛


                                   In Arduino programming when there is a
                                   need to determine on which conditions
                                   the code will work, we use an if-else
                                   statement. In other words, the if-else
                                   statement determines the working flow
                                   of the Arduino program based on
                                   conditions.


                            # ⚓ Explain_the_while_loop_in_Arduino_with_the_help
                              of_examples⠀⇛


                                   In Arduino programming, many times we
                                   have to write a code in which a
                                   repeated cycle of execution of the code
                                   is needed, to execute it we use
                                   conditional loops, and the “while loop”
                                   is one of the conditional loops. In
                                   this write-up, we will understand what
                                   is the while loop and how it works in
                                   Arduino programming.


                            # ⚓ Arduino_Comparison_Operators⠀⇛


                                   In Arduino programming there are
                                   different types of operators. These
                                   operators play an important role in
                                   functioning of any specific task or
                                   working of a code. Most of the
                                   decisions in the code are taken on the
                                   basis of results given by these
                                   operators, in other words they have a
                                   direct influence on the output. The
                                   comparison operators are used where the
                                   conditions of the Arduino code are made
                                   for deciding the flow of the program by
                                   comparing the values.


                            # ⚓ Arduino_Boolean_Operators⠀⇛


                                   When writing an Arduino code for any
                                   specific project different operators
                                   will be needed for performing various
                                   functions, they can either be
                                   mathematical or logical operations and
                                   for each type of functions there will
                                   be specific operators. Here in this
                                   article the Boolean operators are
                                   discussed briefly.


                            # ⚓ Arduino_Bitwise_Operators⠀⇛


                                   There are multiple types of operators
                                   used in Arduino programming for making
                                   different conditions. These conditions
                                   then regulate the flow of the code
                                   designed to perform a specific task.
                                   One of the most important types of
                                   operators used in Arduino programming
                                   are bitwise operators and, in this
                                   article, bitwise operators and their
                                   types are explained briefly.


                            # ⚓ Arduino_Compound_Operators⠀⇛


                                   To determine the flow of a program or a
                                   code in Arduino Programming the
                                   compiler should understand how to
                                   perform multiple operations which can
                                   be either mathematical or logical. For
                                   this, different types of operators are
                                   used. Here in this article the compound
                                   operators will be briefly discussed.


                            # ⚓ Getting_started_with_the_Arduino_Web_Editor⠀⇛


                                   To access your work on an Arduino board
                                   from anywhere on any system Arduino Web
                                   Editor can be used.This web editor is
                                   just like the online version of the
                                   Arduino IDE used for writing, editing
                                   and compiling codes in C and C++
                                   languages. To use the web editor,
                                   follow the given guidelines


                            # ⚓ Printed_Circuit_Bird_Family_Calls_For_Us_To
                              Consider_Analog_|_Hackaday⠀⇛


                                   Kelly’s work takes a concept that would
                                   have many of us get the digital
                                   toolkit, and makes it wonderfully life-
                                   like using a small bouquet of simple
                                   parts. It’s a challenge to our beliefs
                                   and approaches, compelling in its
                                   grace, urging us to consider and
                                   respect analog circuits more when it
                                   comes to modelling consciousness and
                                   behaviours. If it’s this simple to
                                   model sounds and behaviour of a
                                   biological organism, a task that’d have
                                   us writing DSP and math code to
                                   replicate on a microcontroller – what
                                   else are we missing from our models?


                                   Kelly has more PCBs to arrive soon in
                                   preparation for her NYC exhibit in
                                   February, and will surely be posting
                                   updates on her Twitter page! We’ve
                                   covered her work before, and if you
                                   haven’t seen it yet, her Supercon 2019
                                   talk on Electronic Naturalism would be
                                   a great place to start! Such projects
                                   tend to inspire fellow hackers to build
                                   other non-conventional projects, and
                                   this chirping pendant follows closely
                                   in Kelly’s footsteps! The direction of
                                   this venture reminds us a lot of BEAM
                                   robotics, which we’ve recently
                                   reminisced upon as something that’s
                                   impacted generations of hackers to look
                                   at electronics we create through an
                                   entirely different lens.


                            # ⚓ Top_5_Microcontrollers_You_Should_Get_to_Know
                              in_2022⠀⇛


                                   Microcontrollers are also known as
                                   mini-computers; they consist of a
                                   microchip on which a small CPU and
                                   other components are embedded.
                                   Microcontrollers have a wide range of
                                   applications, the most common ones are
                                   the automation of electronic equipment,
                                   robotics, and industries to control the
                                   temperature. The microcontrollers are
                                   of different types like 8-bits, 16-
                                   bits, and 32-bits.


                                   In the market, there are a lot of
                                   microcontrollers that are used for
                                   different purposes and have different
                                   architectures. In this article, the top
                                   microcontrollers are discussed with
                                   their architectures and applications.


                            # ⚓ What_are_the_Conditional_Operators_in_Arduino?
                              Explain_with_Examples.⠀⇛


                                   Conditional operators in Arduino are
                                   used to decide the output on the basis
                                   of conditions. For different
                                   applications, conditional operators can
                                   be utilized. For example, they can be
                                   used in making the logic of street
                                   lights, such that the lights remain off
                                   during day time and on otherwise. In
                                   this write-up, we will discuss the
                                   conditional operators in Arduino and
                                   also understand its usage with the help
                                   of an example.


                            # ⚓ What_are_the_Arrays_in_Arduino?_Explain_with
                              the_Examples.⠀⇛


                                   An array is a data structure used to
                                   store multiple values of the same data
                                   type in it; for example an array
                                   declared with the integer data type can
                                   store multiple integers in it. In
                                   Arduino, sometimes we have to use
                                   multiple values, for example, we are
                                   supposed to blink five LEDs, we will
                                   declare an integer array that will
                                   store the 5 pin numbers where we will
                                   attach the LEDs.


                                   The use of Arrays in Arduino is similar
                                   to the other programming languages so
                                   in this write-up, we will explain the
                                   arrays and the way of utilizing them in
                                   Arduino.


                            # ⚓ What_are_the_Strings_and_String_Functions_in
                              Arduino?_Give_Examples.⠀⇛


                                   Strings are the data types that are
                                   used to store the characters and a
                                   series of characters in them. The
                                   working of Strings in Arduino is
                                   similar to “char” in C programming. The
                                   values stored in the Arduino can be
                                   displayed on the LCD as well as on
                                   serial monitor output.


                                   The Strings used in Arduino are of two
                                   types which are discussed in this
                                   write-up. We will discuss the functions
                                   of Strings with the help of examples.


                            # ⚓ How_to_Generate_the_Pseudo-Random_Numbers_in
                              Arduino⠀⇛


                                   To generate the pseudo-random numbers
                                   in Arduino, we use the built-in
                                   functions random(), first, understand
                                   why we need to generate the pseudo-
                                   random numbers? In real life, we have
                                   to generate random codes for
                                   verification purposes. Similarly, in
                                   digital gaming, we have to generate
                                   random passwords. For this purpose, we
                                   have to generate the random numbers in
                                   Arduino.


                                   In this article, we will discuss the
                                   techniques by which we can generate
                                   random numbers with the help of
                                   examples.


                            # ⚓ How_to_Convert_String_into_the_Int_Data_Type_in
                              Arduino⠀⇛


                                   In many Arduino projects, we often
                                   extract data from some devices like
                                   raspberry pi, or a web database for
                                   some arithmetic commutation, but the
                                   data is in the form of strings. There
                                   is a function in Arduino through which
                                   we can convert the strings into
                                   integers so that we can perform
                                   arithmetic operations on them.


                                   In this write-up, we will conversion of
                                   the string to integers in Arduino by
                                   using a built-in function toInt().


                      # § Mobile Systems/Mobile Applications⠀➾


                            # ⚓ 5_Android_apps_you_shouldn’t_miss_this_week_–
                              Android_Apps_Weekly⠀⇛


                            # ⚓ Realme_8_Users_in_India_Can_Now_Enjoy_Android
                              12_Features_With_New_Update⠀⇛


                            # ⚓ Flashback:_Android_4.4_KitKat_optimized_the_OS
                              for_phones_with_just_512MB_of_RAM_–_GSMArena.com
                              news⠀⇛


                            # ⚓ PUBG_Mobile_Lite_Android_update:_APK_download
                              link_for_0.22.1_version_revealed⠀⇛


                            # ⚓ Xiaomi_Mi_11_review_revisited:_Is_it_still
                              worth_buying?_–_Android_Authority⠀⇛


                # § Free, Libre, and Open Source Software⠀➾


                      # ⚓ Switching_from_OpenNTPd_to_Chrony_–_anarcat⠀⇛


                             A friend recently reminded me of the
                             existence of chrony, a “versatile
                             implementation of the Network Time Protocol
                             (NTP)”.


                      # § SaaS/Back End/Databases⠀➾


                            # ⚓ PostgreSQL:_pgBadger_v11.7_released⠀⇛


                                   pgBadger is a PostgreSQL performance
                                   analyzer, built for speed with fully
                                   detailed reports based on your
                                   PostgreSQL log files.


                                   pgBadger 11.7 was released today, this
                                   release of pgBadger fixes some issues
                                   reported by users since past five
                                   months as well as some improvements.


                      # § FSF⠀➾


                            # ⚓ Conill:_the_FSF’s_relationship_with_firmware_is
                              harmful_to_free_software_users⠀⇛


                                   Ariadne Conill writes about the FSF’s
                                   policy toward proprietary firmware and,
                                   specifically, the rules for “Respects
                                   Your Freedom” certification.


                      # § Programming/Development⠀➾


                            # ⚓ C:_getsockname_Function_Usage⠀⇛


                                   Socket programming is well-known and
                                   common in C programming between
                                   developers and users. Within this kind
                                   of programming, we tend to connect two
                                   endpoints. These endpoints can be two
                                   servers, one server, one socket, etc.
                                   As the name suggests, the “getsockname”
                                   function is utilized to get the name of
                                   a socket working in the network. It may
                                   or may not be the socket name, but the
                                   address of that particular socket would
                                   be displayed using this function.
                                   Therefore, we have tried a simple
                                   method to illustrate the getsockname()
                                   function in C within this guide. Let’s
                                   take a fresh look at its example while
                                   working on the program using Ubuntu
                                   20.04 Linux system.


                            # ⚓ How_Do_I_Use_Keyboard_Shortcuts_in_Jupyter?⠀⇛


                                   Jupyter Notebook is a client-server
                                   application in which users can edit and
                                   run their notebook documents through a
                                   browser. The notebook documents that
                                   are created in Jupyter notebook consist
                                   of python or programming code and rich
                                   text elements such as paragraphs,
                                   images, links, and equations. When you
                                   work as a programmer, you like to use
                                   quick keyboard shortcuts to solve the
                                   routine work. These shortcuts help you
                                   to complete the programming code easier
                                   and quickly.


                                   In Jupyter notebook, various functions
                                   and notebook cell options are available
                                   that you can manage efficiently using
                                   the keyboard shortcuts. Most of the
                                   users do not know how they can find and
                                   use these shortcuts in the Jupyter
                                   notebook. However, today you will learn
                                   about all the keyboard shortcuts used
                                   in the Jupyter notebook.


                            # ⚓ How_to_Enable_and_Use_Autocomplete_in_Jupyter
                              Notebook⠀⇛


                                   No matter how good you are at writing
                                   computer programs, it is hard to
                                   remember all function names, syntax,
                                   and function parameters. So,
                                   programmers are required to use the
                                   auto code completion or IntelliSense
                                   menu feature in the Jupyter notebook.
                                   Usually, programmers and developers are
                                   familiar with different auto-completion
                                   tools.


                                   The code autocomplete feature by
                                   default included in most of the source
                                   code editors through which you can view
                                   IntelliSense menu or code suggestions
                                   inside the drop-down in a program file.
                                   The code autocomplete functionality
                                   saves you time for writing complex
                                   programs, receives fewer bugs, and you
                                   can quickly include methods,
                                   attributes, and classes in your code
                                   using drop-down suggestions.


                      # § Python⠀➾


                            # ⚓ How_To_Install_Django_on_Fedora_35_–_idroot⠀⇛


                                   In this tutorial, we will show you how
                                   to install Django on Fedora 35. For
                                   those of you who didn’t know, Django is
                                   a free and open-source high-level
                                   Python Web framework built by
                                   experienced developers to encourage
                                   rapid development and pragmatic design
                                   of web applications for programmers and
                                   developers. Its main goal is to ease
                                   the creation of complex applications
                                   and take care of the internal
                                   structure.


                                   This article assumes you have at least
                                   basic knowledge of Linux, know how to
                                   use the shell, and most importantly,
                                   you host your site on your own VPS. The
                                   installation is quite simple and
                                   assumes you are running in the root
                                   account, if not you may need to add
                                   ‘sudo‘ to the commands to get root
                                   privileges. I will show you the step-
                                   by-step installation of the Django
                                   Python Framework on a Fedora 35.


    * § Leftovers⠀➾


          o § Hardware⠀➾


                # ⚓ Cables_Too_Long?_Try_Cable_Management_Via_DIY_Coiling⠀⇛


                       Annoyed by excessively-long cables? Tired of the
                       dull drudgery and ugly results of bunching up the
                       slack and wrapping it with a twist-tie? Suffer no
                       longer, because the solution is to make your own
                       coiled cables!


          o § Health/Nutrition/Agriculture⠀➾


                # ⚓ Eu­ro­pean_Par­lia­ment_signs_off_on_strength­en­ing_drug
                  reg­u­la­tor’s_abil­i­ty_to_tack­le_short­ages [Ed: They
                  should tackle artificial scarcities like patents; including
                  the patent monopolies which leave millions of people
                  needlessly dead]⠀⇛


                       The European Parliament on Thursday endorsed a plan
                       to increase the powers of the European Medicines
                       Agency, which will be better equipped to monitor
                       and mitigate shortages of drugs and medical
                       devices.


          o § Integrity/Availability⠀➾


                # § Proprietary⠀➾


                      # § Security⠀➾


                            # ⚓ Is_That_The_Moon_Worming_Its_Way_Into_Your
                              BIOS?⠀⇛


                                   When facing a malware situation, the
                                   usual “guaranteed solution” is to
                                   reinstall your OS. The new developments
                                   in malware world will also require you
                                   to have a CH341 programmer handy. In an
                                   arguably inevitable development,
                                   [Kaspersky Labs] researchers have found
                                   an active piece of malware, out in the
                                   wild, that would persist itself by
                                   writing its bootstrap code into the
                                   BIOS chip. It doesn’t matter if you
                                   shred the HDD and replace it with a new
                                   one. In fact, so-called MoonBounce
                                   never really touches the disk at all,
                                   being careful to only store itself in
                                   RAM, oh, and the SPI flash that stores
                                   the BIOS code, of course.


                                   MoonBounce is Microsoft-tailored, and
                                   able to hook into a chain of components
                                   starting from the UEFI’s DXE
                                   environment, through the Windows
                                   Loader, and finishing as a part of
                                   svchost.exe, a process we all know and
                                   love.


                                   This approach doesn’t seem to be
                                   widespread – yet, but it’s not
                                   inconceivable that we’ll eventually
                                   encounter a ransomware strain using
                                   this to, ahem, earn a bit of extra cash
                                   on the side. What will happen then –
                                   BIOS reflashing service trucks by our
                                   curbsides? After all, your motherboard
                                   built-in BIOS flasher UI is built into
                                   the same BIOS image that gets
                                   compromised, and at best, could be
                                   disabled effortlessly – at worst,
                                   subverted and used for further sneaky
                                   persistence, fooling repairpeople into
                                   comfort, only to be presented with one
                                   more Monero address a week later.


                            # ⚓ Episode_307_–_Got_vulnerabilities?_Introducing
                              GSD_–_Open_Source_Security⠀⇛


                                   Josh and Kurt talk about the Global
                                   Security Database (GSD) project. This
                                   is a Cloud Security Alliance (CSA)
                                   effort to build community around
                                   vulnerability identifiers.


          o § Finance⠀➾


                # ⚓ MetroMile:_Cheap_car_insurance,_but_the_OBDII_dongle
                  doesn’t_actually_report_Check_Engine_codes._–_BaronHK’s
                  Rants⠀⇛


                       I use MetroMile for my car insurance.


                       With what my ex did to ruin my credit, they’re the
                       only car insurance company that isn’t taking it out
                       on me with really sky high insurance premiums for
                       minimum coverage.


                       But one feature they claim their OBDII dongle can
                       do is detect engine trouble codes and report on
                       them to you.


                       Earlier this month, my Check Engine Light came on
                       and it turned out it had been storing a pending
                       code for three cylinders misfiring and just not
                       displaying the light yet, and what finally made it
                       come on was the upstream O2 sensor totally failing,
                       which caused the engine to run so poorly that the
                       computer went ahead and turned on the light.


                       For two days until I could get the car towed down
                       to the shop, I kept waiting on MetroMile’s
                       insurance dashboard to give me the engine codes,
                       but it kept saying nothing to report, and then I
                       noticed in tiny letters that it may not detect
                       everything.


                       If it can’t detect three cylinders misfiring and a
                       dead oxygen sensor, what will it detect?


                       And it’s not even the first time this has happened.
                       It also didn’t detect any problems any of the times
                       my ABS warning light was on or when my EGR valve
                       failed.


                # ⚓ Bank_of_America_Merrill_Lynch_“Wealth_Manager”_charged_with
                  Felony_Hate_Crime,_trespass,_and_disorderly_conduct_after
                  racist_tirade_at_a_coffee_shop.⠀⇛


                       with Felony Hate Crime, trespass, and disorderly
                       conduct after racist tirade at a coffee shop.


                       On the most serious offense, intimidation as a Hate
                       Crime, a Class D Felony, James Iannazzo faces up to
                       five years in prison and up to a $5,000 fine, in
                       addition to two misdemeanors on the indictment.


                       Bank of America Merrill Lynch claimed that they
                       were shocked by his behavior and took immediate
                       action. However, the pattern of big banks employing
                       people like this is disturbing.


                # ⚓ James_Iannazzo_Case:_Fairfield_Man_Arrested_for_Racist
                  Smoothie_Tirade,_Cops_Say_–_NBC_New_York⠀⇛


                       A Connecticut man in a rage over the inadvertent
                       use of peanuts in a smoothie went on a threatening,
                       racist tirade against a shop’s young employees,
                       leading to his arrest, Fairfield police said.


                       James Iannazzo, 48, of Fairfield, faces charges of
                       intimidation based on bias, breach of peace and
                       criminal trespass.


          o § Civil Rights/Policing⠀➾


                # ⚓ New_Pact_on_Migration_and_Asylum:_new_mandate,_new_start
                  for_the_European_Union_Agency_for_Asylum⠀⇛


                       Today, the new European Union Agency for Asylum
                       starts work with its reinforced mandate, building
                       on the achievements of its predecessor, the
                       European Asylum Support Office. The new agency is a
                       key deliverable under the New Pact on Migration and
                       Asylum. It will help ensure that asylum decisions
                       are taken in a fast and fair manner and that
                       reception standards converge across the EU,
                       bringing more uniformity in decision making and
                       alignment between Member States’ asylum systems.


          o § Monopolies⠀➾


                # ⚓ Market_definition_is_strongest_part_of_Epic’s_opening_brief
                  on_appeal_of_Apple’s_App_Store_antitrust_acquittal⠀⇛


                       In the Epic v. Apple App Store antitrust case, Epic
                       Games filed its opening brief with the United
                       States Court of Appeals for the Ninth Circuit
                       yesterday.


                       I already shared some thoughts on the Epic Games v.
                       Apple appeal earlier this month and it’s really
                       only going to get interesting when Apple responds,
                       but I’d like to share a few observations now.


                # ⚓ Urgent_requests_for_declaration_of_non_infringement_(DNI)?
                  Frequently_DeNIed…_–_The_IPKat [Ed: They should quit using
                  misnomers like "Intellectual Property"; even the most
                  prominent scholars increasingly reject this propaganda
                  term]⠀⇛


                       Similarly, many business cats would like to know in
                       advance whether or not one of their products could
                       be considered to infringe the intellectual property
                       rights of a competitor. For domestic cats, the only
                       way to know whether a certain activity is
                       infringing is to use a try-and-see approach. On the
                       other hand, (business cats and) commercial
                       operators, at least in Italy, have the possibility
                       of bringing an action for negative infringement
                       (being an activity doesn’t infringe) before a
                       competent Court, so that a third and impartial
                       judge can possibly settle the matter.


                       In particular, when a company wants to receive a
                       rapid response it may file a precautionary action
                       for negative declaratory relief (an urgent
                       ‘declaration of non infringement’, or ‘DNI’). In
                       Italy such a request must be supported by three
                       fundamental requirements. First of all, there must
                       be a legal interest in bringing the action (the
                       negative assessment of infringement presupposes a
                       situation of objective uncertainty caused by a
                       dispute with another party, so that a conflict may
                       arise – subjective uncertainty is not sufficient),
                       then there must be a danger of delay (the so-called
                       periculum in mora or urgency requisite, i.e. an
                       urgent need for the petitioner to obtain a
                       provisional measure in order to prevent irreparable
                       harm from occurring or consolidating in the time
                       needed for an ordinary case to reach its outcome),
                       and finally, of course, there must be a substantial
                       prerequisite (i.e. the Court must consider that the
                       product is not actually infringing).


                       In the light of these prerequisites, the path for
                       the party fearing infringement to obtain an urgent
                       DNI is not a very easy one to follow.


                # ⚓ Gruyere_cheese_and_Tennessee_whiskey:_A_trade_story_|
                  TheHill⠀⇛


                       On Dec. 15, 2021, a U.S. district judge ruled that
                       American fromagers are allowed to make gruyere
                       cheese, despite the product’s Swiss pedigree. A
                       week later, Japan became the 44th country to
                       formally recognize Tennessee whiskey as a
                       “distinctive product” of the United States. These
                       aren’t just names, they’re intellectual property.
                       But just as trade deals are helping to proliferate
                       these geographical indications (GIs) around the
                       world, the judge’s ruling on gruyere suggests that,
                       for many of them, trade may also be their
                       Achilles’s heel.


                       The battle over gruyere cheese has been heating up
                       for years. In 2020, U.S. dairy claimed a victory
                       when gruyere was denied trademark protection. Swiss
                       vendors wanted American fromagers to call their
                       product “Alpine cheese,” arguing that real gruyere
                       can only be had from Switzerland. Well, not quite.
                       The production of gruyere reached across the French
                       border long ago, and since the early 2010s,
                       Brussels has recognized it as being from both
                       Switzerland and France. Yet, Austria, the
                       Netherlands and Germany, among a host of other
                       European countries, also make the stuff. So, too,
                       do Egypt and Tunisia.


                # ⚓ Congress_Moving_Forward_with_Open_App_Marketplace
                  Legislation_|_Patently-O⠀⇛


                       Senate Judiciary Committee is considering two
                       bipartisan legislative proposals focusing on the
                       operation of “app stores” and requiring that large
                       app stores allow for a competitive marketplace
                       without the tight controls that we see today…


                # § Patents⠀➾


                      # ⚓ [Older]_Post-AIA_Patents_Are_Not_Shielded_from
                        Interferences [Ed: Interferences is the wrong term
                        because revoking phony, bad patents is something that
                        ought to have been done by USPTO before issuance]⠀⇛


                             Addressing the applicability of interference
                             proceedings to patent applications filed
                             after the Leahy-Smith America Invents Act
                             (AIA) was enacted, the Patent Trial & Appeal
                             Board (Board) found it proper to declare an
                             interference between a patent application
                             with a priority date before March 16, 2013,
                             the AIA implementation date, and a patent
                             with a priority date after March 16, 2013.
                             SNIPR Technologies Limited v. The Rockefeller
                             University, Pat. Interf. No. 106,123 (DK)
                             (PTAB Nov. 19, 2021) (Katz, APJ).


                             The AIA switched the US patent system from a
                             “first to invent” to a “first inventor to
                             file” system. In line with this change, the
                             AIA eliminated the patentability requirement
                             under 35 U.S.C. § 102(g), regarding whether
                             another inventor made the invention first,
                             and the interference proceeding under 35
                             U.S.C. §135 for determining who invented the
                             claimed invention first. Section 3(n)(2) of
                             the AIA provides a timing provision relating
                             to this change. Under this section, the
                             interference proceeding “shall apply to each
                             claim of an application for patent, and any
                             patent issued thereon, for which the
                             amendments made by this section also apply,
                             if such application or patent contains or
                             contained at any time, a claim [having a
                             priority date before March 16, 2013].”


                      # ⚓ ‘Exasperation’_over_European_deadlock_on_grace_period
                        [Ed: Rory O’Neill, salaried by patent extremists, wants
                        us to think that it’s desirable to promote patent
                        maximimalism under the old_misleading_guise_of
                        “harmonisation”]⠀⇛


                             European businesses can’t agree on how to
                             move forward with talks on harmonising
                             international patent laws


                      # ⚓ Five_Patents_In_Nanotechnology [Ed: Glorified fields
                        of science and monopolies on them as framed by liars
                        who work for Keltie]⠀⇛


                             Nanotechnology: building materials and
                             devices by controlling matter at the atomic
                             and molecular level. An interdisciplinary
                             field with far-reaching applications. Monica
                             Patel explores five particularly interesting
                             patents in this area…


                      # ⚓ Finally,_the_Unified_Patent_Court_becomes_a_reality
                        [Ed: Graham Burnett-Hall, who has long spread
                        falsehoods for his convicted corrupt employer, is
                        advancing illegal_agenda using more of the fake news
                        that floods the Web]⠀⇛


                             The Unified Patent Court has today, the 19th
                             January 2022, officially come into existence
                             as an international body, following the
                             deposit by Austria of its ratification of the
                             protocol on the provisional application of
                             the UPC Agreement.


                      # ⚓ UK:_Protecting_And_Rewarding_Inventions_Devised_By
                        Artificial_Intelligence:_UK_IPO_Calls_For_Views [Ed:
                        UKIPO will only listen to patent maximalists, including
                        liars_from_Keltie (author)]⠀⇛


                             The consultation describes how AI has and can
                             play a positive role in the development of
                             innovation and new technology. It also
                             acknowledges that AI will be capable in the
                             future of inventing and creating things that
                             have no discernible input from humans. When
                             this happens, and some feel it has happened
                             already, an IP system needs to be in place
                             that strikes a balance between protecting the
                             human benefit of the invention while also
                             supporting AI-led technical innovation and
                             artistic creativity.


                             [...]


                             The UK Government previously called for views
                             on AI and IP in September 2020, releasing a
                             follow-up response in March 2021 to the 92
                             contributions that were lodged. In that
                             response, the Government acknowledged that
                             there were mixed views on whether AI could be
                             considered an inventor. There were many
                             respondents who were both for and against a
                             change in inventorship criteria. Some felt
                             that all inventions would need human input at
                             some point and couldn’t be devised solely by
                             AI while others said that ruling out the
                             recognition of AI as an inventor would
                             discourage innovation and technological
                             advancement.


                      # ⚓ Three_Type_of_U.S._Patents_and_When_to_Use_Them [Ed:
                        So-called ‘design_patents’ should not even exist at
                        all; trademarks cover that already]⠀⇛


                             Have you heard of a “design” patent? Maybe
                             someone you know mentioned this as a less
                             expensive alternative to a “regular” U.S.
                             patent. The U.S. patent laws actually define
                             three basic types of patents: utility
                             patents, design patents, and plant patents.


                      # ⚓ UK:_Insufficient_Evidence_To_Support_Change_To_UK
                        Exhaustion_Regime [Ed: Since when is UK patent policy
                        designed to actual benefit_the_general_public?]⠀⇛


                             As reported previously, the UK government has
                             been consulting on the future regime for
                             exhaustion of intellectual property rights
                             following the UK’s departure from the EU.


                             In an update posted on 18 January 2022, it
                             was indicated that the government has
                             completed an initial analysis of the
                             responses to the consultation, but found that
                             there is not enough data to understand the
                             economic impact of any of the potential
                             alternatives to the current post-Brexit
                             system (the so-called “UK+ regime”). Given
                             the potential of any change to affect a wider
                             variety of business sectors and end
                             consumers, the government has decided to
                             defer making any changes pending further
                             evidence and policy developments.


                      # ⚓ Diamyd_Medical_patent_for_intralymphatic_injection_of
                        GAD_in_autoimmune_diabetes_to_be_granted_in_China [Ed:
                        They know there's barely any quality control for
                        patents in China; it's gaming the system (WIPO) to make
                        the system obsolete]⠀⇛


                             Patents for intralymphatic administration of
                             GAD to treat and prevent autoimmune diabetes
                             including type 1 diabetes and Latent
                             Autoimmune Diabetes in Adults (LADA) have
                             already been approved in Europe, Japan,
                             Russia, Australia and Israel, with additional
                             countries pending.


                      # ⚓ Working_today_for_the_tourism_of_the_future [Ed:
                        Promoting a fake, false ‘study’ from corrupt EPO and
                        EUIPO, reversing_causes_and_effects]⠀⇛


                             A new study published by the European Patent
                             Office (EPO) and the European Union
                             Intellectual Property Office (EUIPO) shows
                             that companies that own at least one patent,
                             registered design or trademark generate, on
                             average, 20% more income per employee than
                             companies that do not have any of these
                             industrial property rights (IPR). In
                             addition, it was found that these IPR-owning
                             companies pay 19% more wages on average than
                             other companies*.


                      # ⚓ Divisional_Applications_in_India_–_A_Guide_for
                        Applicants_and_Foreign_Practitioners [Ed:
                        "Practitioners" is a euphemism for parasitic entities
                        such as patent lawyers]⠀⇛


                      # ⚓ MAB_Discovery_to_Appeal_Against_European_Patent
                        Office’s_Favorable_Decision_on_Cantargia’s_Patent [Ed:
                        EPO corruption and maladministration will cause legal
                        chaos]⠀⇛


                      # ⚓ EU_Intellectual_Property_Office_Launches_SME_Fund_For
                        2022 [Ed: EUIPO harms SMEs and breaks_the_law; this is
                        just a PR campaign, reputation laundering by which to
                        manufacture shallow, misleading articles]⠀⇛


                             The European Union Intellectual Property
                             Office (the “EUIPO”) has implemented ‘The
                             Ideas Powered for Business SME Fund’ (the
                             “Fund”), a European Commission initiative,
                             for the year 2022. The first version of the
                             Fund was launched during 2021. Unlike the
                             previous initiative which only covered the
                             re-imbursement of administrative fees
                             relating to trademark and design
                             registrations (at both national and EU
                             level), the Fund now also covers the
                             administrative fees associated with the
                             registration of patents on a national level.


                      # ⚓ Update_on_Artificial_Intelligence_as_a_Patent
                        Inventor [Ed: Can we stop this lunacy of equating
                        computer programs with human beings? This is infantile
                        and dishonest.]⠀⇛


                             Our previous blog posts, Artificial
                             Intelligence as the Inventor of Life Sciences
                             Patents? and Update on Artificial
                             Intelligence: Court Rules that AI Cannot
                             Qualify As “Inventor,” discuss recent
                             inventorship issues surrounding AI and its
                             implications for life sciences innovations.
                             Continuing our series, we now look at the
                             appeal recently filed by Stephen Thaler
                             (“Thaler”) in his quest to obtain a patent
                             for an invention created by AI in the absence
                             of a traditional human inventor.


                      # ⚓ Update_on_Artificial_Intelligence_as_a_Patent
                        Inventor [Ed: Can we just stop collectively calling all
                        computer programs or bots "Hey Hi"? That would make
                        debates far more meaningful, but the patent extremists
                        look to advance an agenda using the deception]⠀⇛


                             Our previous blog posts, Artificial
                             Intelligence as the Inventor of Life Sciences
                             Patents? and Update on Artificial
                             Intelligence: Court Rules that AI Cannot
                             Qualify As “Inventor,” discuss recent
                             inventorship issues surrounding AI and its
                             implications for life sciences innovations.
                             Continuing our series, we now look at the
                             appeal recently filed by Stephen Thaler
                             (“Thaler”) in his quest to obtain a patent
                             for an invention created by AI in the absence
                             of a traditional human inventor.


                             As we previously reported, on September 3,
                             2021, the U.S. District Court for the Eastern
                             District of Virginia ruled that an AI machine
                             cannot qualify as an “inventor” under the
                             Patent Act, in a case that Thaler filed
                             seeking, among other things, an order
                             compelling the USPTO to reinstate his patent
                             applications. Those patent applications name
                             an AI system called “Device for Autonomous
                             Bootstrapping of Unified Sentience” aka
                             “DABUS,” as the sole inventor. Thaler, who
                             developed DABUS, remains the owner of any
                             patent rights stemming from these
                             applications. The first application is
                             directed to a design of a container based on
                             fractal geometry. The second application is
                             directed to a device and method for producing
                             light that flickers rhythmically in a
                             specific pattern mimicking human neural
                             activity. According to Thaler, he could not
                             list himself as the inventor, as he did not
                             contribute to the conception of the
                             inventions; instead DABUS performed what is
                             traditionally considered the mental part of
                             the inventive act.


                      # ⚓ Eurasian_Industrial_Design_–_The_List_Of_Countries_Is
                        Expanding [Ed: Such globalisation with patents for
                        monopolies (in effect banning competition) is an
                        international problem]⠀⇛


                             Since June 1, 2021 the Eurasian Patent Office
                             has started accepting and examining
                             applications for the issuance of Eurasian
                             patents for industrial designs, which allows
                             obtaining legal protection for an industrial
                             design simultaneously in several member
                             states of the Eurasian Patent Convention.


                             On December 31, 2021, the President of
                             Belarus signed the Law No. 137-Z “On the
                             Accession of the Republic of Belarus to the
                             Protocol on the protection of industrial
                             designs to the Eurasian Patent Convention of
                             September 9, 1994″. The Protocol on the
                             protection of industrial designs will enter
                             into force for Belarus three months after the
                             deposit of the relevant Act of accession with
                             the depositary.


                      # ⚓ PTAB_Discretionary_Denials_Escape_High_Court_Review⠀⇛


                             The U.S. Supreme Court on Tuesday refused to
                             review a highly controversial U.S. Patent and
                             Trademark Office rule allowing the Patent
                             Trial and Appeal Board to deny patent
                             challenges based on the timing of related
                             litigation.


                             The justices denied both Mylan Laboratories’
                             and Apple Inc.’s petitions asking the court
                             to evaluate whether the USPTO improperly
                             implemented a rule defining factors for when
                             PTAB judges can use their discretion to deny
                             challenges. While the Supreme Court has
                             repeatedly held that PTAB institution
                             decisions are not appealable — which doomed
                             these cases at the Federal Circuit — the
                             companies say they should be allowed to
                             challenge an…


                      # ⚓ A_brief_guide_to_patenting_surgical_inventions_in
                        Europe:_part_two [Ed: This author wants us to believe
                        that the EPO honours and follows the European Patent
                        Convention; in reality it_violates_it_all_the_time
                        without_any_consequences_for_these_violations]⠀⇛


                             The European Patent Convention excludes from
                             patentability “methods for treatment of the
                             human or animal body by surgery”. The
                             intention behind this exclusion is to leave
                             medical and veterinary practitioners free to
                             act in the best interests of their human or
                             animal patients without fear of patent
                             infringement.


                      # ⚓ Sustainable_Polymers:_Finding_the_sweet_spot [Ed:
                        Convicted corrupt firm Marks & Clerk playing along with
                        the EPO’s greenwashing campaign, which helps distract
                        from EPO crimes and the harms done to actual
                        environmentalism by such patent monopolies]⠀⇛


                             A recent report from the European Patent
                             Office (EPO) – Patents for tomorrow’s
                             plastics: Global innovation trends in
                             recycling, circular design and alternative
                             sources – provides an in-depth review of
                             patenting trends across various aspects of
                             polymer-related technologies that can help
                             guide us towards a more sustainable future.


                             It is, of course, important to reduce our
                             reliance on plastics, but the report
                             recognises that plastics are an essential
                             material for many industries – not least in
                             the health sector where the recent pandemic
                             has seen an exponential increase in the use
                             of PPE, testing kits, etc. (all relying on
                             plastics). Innovation in recycling
                             technologies and alternative plastics has a
                             critical role to play as we move towards a
                             more sustainable society.


                      # ⚓ 2021_Artificial_Intelligence_and_Automated_Systems
                        Annual_Legal_Review [Ed: Meaningless_garbage_and
                        buzzwords like “Hey Hi” have opened up loophole to
                        portray what’s illegal as needing inclusion]⠀⇛


                             2021 was a busy year for policy proposals and
                             lawmaking related to artificial intelligence
                             (“AI”) and automated technologies. The OECD
                             identified 700 AI policy initiatives in 60
                             countries, and many domestic legal frameworks
                             are taking shape. With the new Artificial
                             Intelligence Act, which is expected to be
                             finalized in 2022, it is likely that high-
                             risk AI systems will be explicitly and
                             comprehensively regulated in the EU. While
                             there have been various AI legislative
                             proposals introduced in Congress, the United
                             States has not embraced a comprehensive
                             approach to AI regulation as proposed by the
                             European Commission, instead focusing on
                             defense and infrastructure investment to
                             harness the growth of AI.


                      # ⚓ Mechanical_insufficiency:_how_to_avoid_the_elephant
                        trap [Ed: Almost all patents used to be like this; now
                        they give patents on mere thoughts… and things that
                        have existed for millions of years, predating humans;
                        this cites rigged_EPO_tribunals]⠀⇛


                             When drafting patents for mechanical
                             inventions, insufficiency issues are rare.
                             The majority of insufficiency case law
                             relates to chemical inventions, however,
                             there are still some nasty traps that one can
                             fall into accidentally. In this article, we
                             look at a few examples and discuss potential
                             issues when filing patent applications
                             directed to mechanical products and
                             processes.


                      # ⚓ Arecor_Therapeutics_Provides_Business_Update [Ed:
                        Arecor seems unaware that nowadays the EPO grants_loads
                        of_fake_patents, based on EPO insiders]⠀⇛


                             Arecor is also pleased to announce that the
                             European Patent Office has granted patent
                             EP3496734B, with claims protecting novel
                             compositions of insulin glargine with
                             improved thermostability. This grant further
                             demonstrates the strength of the patent
                             portfolio protecting the Group’s proprietary
                             ArestatTM technology.


                      # ⚓ Unified_Patent_Court:_Protocol_on_Provisional
                        Application_enters_into_force [Ed: This_is_very_clearly
                        illegal, but patent litigation companies don’t care
                        what’s legal, they just smell money and cooperate with
                        criminals]⠀⇛


                             Following yesterday’s deposit by Austria of
                             its instrument of ratification of the
                             Protocol on Provisional Application of the
                             Agreement on the Unified Patent Court, the
                             General Secretariat of the Council of the
                             European Union has just declared that the
                             Protocol has now entered into force. Similar
                             statements were issued by the European Patent
                             Office and the Preparatory Committee of the
                             Unified Patent Court.


                             [...]


                             The Preparatory Committee also announced that
                             the inaugural meetings of the governing
                             bodies of the Court, namely the
                             Administrative Committee, the Advisory
                             Committee and the Budget Committee, will soon
                             take place. It is unclear whether these
                             meetings will be preceded by the signing
                             ceremony of the draft Declaration on the
                             authentic interpretation of Article 3 of the
                             Protocol, which had been anticipated by the
                             Preparatory Committee to address the
                             uncertainties following the United Kingdom’s
                             withdrawal from the Unitary Patent System
                             (for additional details, see here) and
                             support the view that Article 3 of the
                             Protocol is to be interpreted as mirroring
                             Article 89 of the UPCA.


                      # ⚓ Brexit_boom:_UKIPO_filings_soar_but_address
                        suspicions_remain [Ed: UKIPO granting monopolies is of
                        no actual use to the British population; a lot of this
                        gets assigned to foreign companies]⠀⇛


                             As attorneys report huge increases in filing
                             numbers, suspicion over the address for
                             service provision has arisen


                      # ⚓ The_European_Unified_Patent_Court:_Coming_Soon_to_a
                        European_Patent_Near_You [Ed: More lying and fake news
                        about the UPC; they’ve been saying it since_2014]⠀⇛


                             The Unified Patent Court is a new European
                             Union court with jurisdiction over
                             infringement and revocation actions for a new
                             Unitary Patent as well as…


                      # ⚓ Exporting_Patents:_Boiled,_Broiled,_Barbecued_or
                        Fried [Ed: What an utterly, totally strange kind of
                        patents]⠀⇛


                             We know that U.S. patents are territorially
                             limited. Although Deepsouth was barred from
                             using Laitram’s patented inventions
                             throughout the United States, Deepsouth began
                             selling its deveining machine to folks
                             outside of the US in a partially constructed
                             form.


                      # ⚓ EPC,_UPCA:_First_impacts_of_the_start_of_the_UPCA
                        provisional_application_period_on_EPO_proceedings [Ed:
                        She neglects to say that this is illegal; she’s
                        cheering a clear_violation_of_the_law because she’s
                        looking to financially gain from the corruption, deeply
                        connected to the EPO and EC]⠀⇛


                             In view of the forthcoming introduction of
                             the Unitary Patent, the European Patent
                             Office (EPO) has now declared that it will
                             allow requesting a delayed grant and early
                             requests for unitary effect, and has provided
                             the rules applying to such requests.


                             By publishing two Notices and one Decision of
                             the President on 19 January 2022, the EPO
                             provided an immediate reaction to the start
                             of the provisional application of the
                             Agreement on a Unified Patent Court (UPCA)
                             that was triggered by Austria´s deposit of
                             ratification on 18 January 2022.


                      # ⚓ Sec­ondary_patents_prove_to_be_key_in_biosim­i­lar
                        block­ing_strate­gies,_re­searchers_find [Ed: As if the
                        ultimate goal is to actually reduce access to medicines
                        and to falsely claim life/nature is a human
                        "invention"]⠀⇛


                             While the US biosim­i­lars in­dus­try has
                             gen­er­al­ly been a dis­ap­point­ment since
                             its in­cep­tion, with FDA ap­prov­ing 33
                             biosim­i­lars since 2015, just a frac­tion of
                             those have im­me­di­ate­ly fol­lowed their
                             ap­provals with launch­es. And more than a
                             hand­ful of biosim­i­lars for two of the
                             biggest block­busters of all time —
                             Ab­b­Vie’s Hu­mi­ra and Am­gen’s En­brel —
                             re­main ap­proved by FDA but still have not
                             launched be­cause of le­gal set­tle­ments.


                             Part of the rea­son that many of these brand-
                             name bi­o­log­ic com­pa­nies have proven
                             suc­cess­ful at block­ing biosim­i­lar
                             com­pe­ti­tion is through their use of patent
                             fortress­es, de­signed to make it
                             im­pos­si­ble, or ex­treme­ly ex­pen­sive, to
                             wage le­gal bat­tle.


                      # ⚓ 19th_Century_Patent_Law:_Initial_Disclosures_for_the
                        Defense [Ed: Patent law before it became a laughing
                        stock and rigged_game controlled by monopolists that
                        buy the politicians, the patent office officials
                        etc.]⠀⇛


                             Patent Act of 1836, Section 15. Thus, the
                             defendant is required to automatically
                             identify the names and residence of the
                             people with prior knowledge as well as the
                             location where the item was used.


                             In Wise v. Allis, 76 U.S. 737 (1869), the
                             Supreme Court was asked to opine on the level
                             of specificity required by the statute. The
                             case involved an improvement in balancing
                             millstones for grinding grain. The defendant
                             had provided notice that the invention had
                             been previously used in “Utica, Rochester,
                             Buffalo, Albany, New York City, and Brooklyn,
                             in the State of New York.” At the time, New
                             York City had about 1,000 different mills
                             where the millstone might have been used, and
                             so that part of the notice did not really
                             provide much information. The plaintiff
                             complained that the notice was thus
                             insufficient and that the defense should be
                             barred.


                             [...]


                             Wise v. Allis, 76 U.S. 737 (1869). Today,
                             Rule 26(a)(1)(A)(i) operates in parallel to
                             this old patent rule, but for all civil
                             actions. It requires automatic disclosure of
                             the name and address and phone number “of
                             each individual likely to have discoverable
                             information … that the disclosing party may
                             use to support its claims or defenses, unless
                             the use would be solely for impeachment.”


                      # ⚓ From_disruptive_tech_to_IP_access:_top_2022_trends
                        [Ed: Sukanya Sarkar now legitimising outright frauds
                        and scams in the name of “increased democratisation of
                        IP” [sic]; this is really appalling, but only to be
                        expected when the media in this domain is controlled by
                        liars/lawyers]⠀⇛


                             Counsel list the top tech trends for 2022,
                             saying the likes of NFTs and the metaverse
                             will lead to increased democratisation of IP


                      # ⚓ IP_[sic]_owners_[sic]_urge_customs_to_embrace_new
                        tech [Ed: Buzzword puzzle for pseudo-novelty by patent
                        extremists and their media operatives]⠀⇛


                             Use of blockchain and metadata analysis to
                             track e-commerce sales could help modernise
                             customs authorities, say IP owners


                      # ⚓ Are_you_getting_ready_for_the_2022_Mobile_World
                        Congress? [Ed: Alicante and other Spanish places (like
                        Barcelona) rapidly became notorious for raids and
                        thefts by authorities in such events; this isn't about
                        the law, it's about mobster mentality at the behest of
                        corporations that aren't even European]⠀⇛


                             After the disruption brought on by the
                             pandemic, it seems that this year the Mobile
                             Word Congress (“MWC”), the largest mobile
                             trade event in the world and one of the most
                             significant trade fairs held in Barcelona,
                             will take place from 28 February to 3 March
                             2022 under normal conditions. Against this
                             background, on 10 January 2022, the Boards of
                             the Commercial Court Judges of Barcelona and
                             the Judges of the EU Trademark Courts in
                             Alicante held a joint meeting where they
                             approved a Protocol setting out a number of
                             measures aimed at expediting judicial
                             proceedings related to the MWC. This is the
                             8th Protocol adopted so far (the 4th, in the
                             case of the Alicante Judges).


                      # ⚓ Breaking:_UK_sticks_with_exhaustion_status_quo_…_for
                        now⠀⇛


                             The UKIPO opts to keep the EEA exhaustion
                             regime but does not rule out future change


                      # ⚓ Energy_storage_will_play_a_crucial_role_in_the_green
                        transition [Ed: Another appalling puff piece that helps
                        a deeply corrupt institution, EPO, with greenwashing
                        PR]⠀⇛


                             According to the European Patent Office,
                             patenting in low-carbon emitting supply
                             technologies such as renewables (wind, solar,
                             geothermal or hydroelectric power) has been
                             falling since 2012 and accounted for only 17
                             percent of all clean energy technology
                             patents in 2019. Since 2017, the key drivers
                             of innovation in this sector have been
                             technologies that cut across these fields,
                             like batteries, hydrogen and smart grids, as
                             well as carbon capture, utilization and
                             storage (CCUS). These crucial enablers of the
                             energy transition all involve some form of
                             energy storage.


                      # ⚓ Brussels_Behind_the_Scenes:_Looking_after_the_little
                        guy [Ed: This title is a disgusting lie; this is the EC
                        breaking the law and attacking_SMEs]⠀⇛


                             Innovators and entrepreneurs received some
                             welcome news this week, as a plan that
                             promises to slash costs and protect IP
                             throughout most of the EU came closer to
                             fruition. It has been a long time in the
                             making.


                             Nearly fifty years ago, the idea of a single
                             patent within the then European Communities
                             was on the drawing board of officials and
                             diplomats in Brussels. The concept was
                             simple: if you were granted protection for
                             your intellectual property in one EU member,
                             you would get it everywhere else.


                             This week, the EU finally took a step towards
                             fulfilling at least the fundamental
                             principles of that project.


                      # ⚓ The_Unified_Patent_Court_comes_back_to_life [Ed: The
                        title is false because UPC_is_still_illegal, but they
                        hope that a lot of fake news will compensate for the
                        illegality; the Team_UPC_law_firms_are_drowning_the_Web
                        with_misinformation_about_the_UPC.]⠀⇛


                             The long-awaited Unified Patent Court (the
                             UPC), which will provide a new forum for
                             patent litigation in 24 EU countries, is
                             finally getting ready to open its doors. With
                             the UPC will come the new unitary patent, a
                             single validation of a granted European
                             patent application that will take effect as a
                             single patent spanning the participating
                             countries. All European patent holders will
                             need to review their patent portfolios and
                             their patent prosecution strategies during
                             the coming months, in particular to decide
                             whether to “opt out” existing European
                             patents from the jurisdiction of the new
                             court.


                             The provisional application period of the
                             Unified Patent Court Agreement has begun,
                             with effect from 19 January 2022, following
                             Austria’s deposit of its instrument of
                             ratification of a protocol to the UPC
                             Agreement. Technical and legal judges will
                             now be appointed and trained, court buildings
                             fitted out and IT systems completed. A lot of
                             preparatory work has already been completed,
                             such as the drafting of detailed rules of
                             procedure for the new court. However, it
                             cannot be said with any certainty how long
                             the provisional application period will last.
                             It appears that it is likely to be at least 6
                             months, perhaps nearer to a year. As well as
                             completion of the administrative
                             preparations, the participating countries
                             will need to agree where the new location
                             will be for the branch of the court’s Central
                             Division that was originally to be in London.
                             Various options have been suggested,
                             including locating the branch in Italy or the
                             Netherlands, or splitting the work of the
                             branch between Paris and Munich, where the
                             two other branches of the Central Division
                             are already located.


                      # ⚓ Exclusive:_UKIPO_asked_to_assess_possible_AFS
                        breaches⠀⇛


                             Prominence of UKIPO actions emanating from
                             Chinese firms prompts fears that hard-fought
                             AFS rules could be being flouted


                      # ⚓ Fintiv_challenged_in_SCOTUS_Amicus_by_Unified_—
                        Unified_Patents⠀⇛


                             On January 14, 2022, Unified Patents filed an
                             amicus brief with the Supreme Court in
                             support of the petition for certiorari in
                             Intel Corp. v. VLSI Tech. LLC, challenging
                             the USPTO’s reliance on their own NHK-Fintiv
                             ruling, as well as the reviewability of such
                             a de facto rule on appeal.


                      # ⚓ The_Service_of_Electronic_Transmission_of_Priority
                        Documents_between_EPO_and_CNIPA_Is_Migrated_from
                        Bilateral_Exchange_Platform_to_WIPO_DAS_System⠀⇛


                      # ⚓ Patent_Awarded_to_Nanoprecise_Sci_Corp_for_its
                        Automated_Predictive_Maintenance_Solution⠀⇛


                      # ⚓ A_First:_AI_System_Named_Inventor [Ed: Misleading
                        title. Only some almost-obsolete patent office in South
                        Africa fell for this scam for serial provocateur
                        Stephen Thaler misusing buzzwords]⠀⇛


                             The South African patent office made history
                             in July when it issued a patent that listed
                             an artificial intelligence system as the
                             inventor.


                             The patent is for a food container that uses
                             fractal designs to create pits and bulges in
                             its sides. Designed for the packaging
                             industry, the new configuration allows
                             containers to fit more tightly together so
                             they can be transported better. The shape
                             also makes it easier for robotic arms to pick
                             up the containers.


                             The patent’s owner, AI pioneer Stephen L.
                             Thaler, created the inventor, the AI system
                             known as Dabus (device for the autonomous
                             bootstrapping of unified sentience).


                             The patent success in South Africa was thanks
                             to Thaler’s attorney, Ryan Abbott.


                      # ⚓ Appleton_v._Bacon_(1862)_|_Patently-O⠀⇛


                             John North was an employee of American Book
                             and Paper Folding Company, hired to design
                             and improve various paper folding machines.
                             That company ceased operation in 1857 and
                             Bacon purchased all the patents, including
                             one issued to North.


                             [...]


                             On appeal, Bacon argued that he owned the
                             invention based upon the prior employment
                             agreement. However, the Supreme Court sided
                             with Appleton, concluding that the keys to
                             the invention came-about after North’s
                             contract had ended.


                      # ⚓ Provisional_application_of_the_UPCA_to_start [Ed:
                        Criminals who hijacked the EPO, Europe's second-largest
                        organisation, add another crime to a long list of their
                        crimes]⠀⇛


                             The Protocol to the Agreement on a Unified
                             Patent Court on provisional application (PPA)
                             enters into force today, following the
                             deposit of the instrument of ratification by
                             Austria yesterday. The final preparations for
                             the Unified Patent Court will now start.


                             [...]


                             The Unitary Patent and the Unified Patent
                             Court (UPC) are the building blocks of the
                             Unitary Patent package which will supplement
                             and strengthen the existing centralised
                             European patent-granting system. Unitary
                             Patents will make it possible to obtain
                             uniform patent protection in up to 25 EU
                             Member States by submitting a single request
                             to the EPO, making the protection of
                             inventions for patent holders simpler and
                             more cost effective.


                             The UPC will be an international court with
                             jurisdiction for infringement and revocation
                             actions concerning patents granted by the
                             EPO. This specialised court will make the
                             Europe-wide enforcement of patents easier,
                             offer greater legal certainty and reduce
                             litigation costs. The Court is based on an
                             international treaty, the UPCA.


                      # ⚓ FOSS_Patents:_Ericsson_has_requested_preliminary
                        injunctions_against_Apple_in_Brazil,_declared_intent_to
                        do_so_in_the_Netherlands:_5G_patent_royalty_dispute
                        picks_up_speed⠀⇛


                             I thought I had covered the Ericsson v. Apple
                             5G patent royalty dispute pretty exhaustively
                             during the course of this week, only to find
                             multiple informative filings in the Eastern
                             District of Texas, all of which were made on
                             Wednesday (January 19). So let me recap what
                             was already known before the weekend and add
                             what I’ve just found out from the Wednesday
                             documents


                      # ⚓ FOSS_Patents:_Third_Ericsson_ITC_complaint_against
                        Apple_discovered_—_a_potentially_unprecedented_number
                        of_simultaneous_filings_with_the_U.S._trade_agency⠀⇛


                             I hadn’t seen three simultaneous ITC
                             complaints (requests for U.S. import ban) by
                             one party against another in more than ten
                             years of watching smartphone patent disputes.
                             Then I looked up the electronic docket of the
                             United States International Trade Commission,
                             a trade agency with quasi-judicial powers,
                             and just found Ericsson’s third complaint
                             against Apple (this post continues below the
                             document). So there are two non-SEP
                             complaints and one SEP complaint.


                             The number of the most recently-discovered
                             complaint is 337-TA-3597 and the title is In
                             the Matter of Certain Mobile Phones and Table
                             Computers, All With Switchable Connectivity,
                             and Products Containing Same.


                             Earlier today I noted that three patents were
                             asserted in the 22-cv-61 case in the Western
                             District of Texas but not in the 337-TA-3596
                             ITC complaint. Those three non-standard-
                             essential patents, which have not previously
                             been asserted against any party, are now at
                             the heart of Ericsson’s third ITC complaint
                             against Apple.


                      # ⚓ Apple_Countersues_Ericsson_Seeking_Base_Station
                        Import_Ban_–_The_Mac_Observer⠀⇛


                             Apple countersues Ericsson in the latest move
                             in the Ericsson v Apple battle. The company
                             seeks a U.S. import ban on mobile base
                             stations from Ericsson, reports FOSS Patents.


                      # ⚓ FOSS_Patents:_Ericsson’s_two_ITC_complaints_against
                        Apple:_one_over_four_SEPs,_another_over_five_non-SEPs_
                        (two_of_them_from_same_family)⠀⇛


                             This is my third post on Ericsson v. Apple in
                             a row. Not only does it feature an additional
                             ITC complaint I had not previously seen
                             (because it surfaced on the U.S.
                             International Trade Commission’s electronic
                             docket more than an hour after the first one)
                             but it also provides some context relating to
                             the two federal lawsuits (Western District of
                             Texas) and the other USITC complaint (the one
                             over four standard-essential patents (SEPs)).
                             This is a monumental patent dispute (that has
                             all the ingredients for further escalation),
                             so litigation watchers like me have to put
                             the mosaic together step by step.


                             This is the current landscape of Ericsson-
                             Apple cases pending in the United States,
                             just so we have a forest here and not just a
                             lot of trees…


                      # ⚓ Ericsson_sues_Apple,_alleging_5G_patent
                        infringement⠀⇛


                             Ericsson files new suits against Apple in
                             patent-friendly Western District of Texas


                             Ericsson and Apple are back in court in
                             Texas, months after firing a volley of
                             lawsuits against each other in the U.S.
                             District Court for the Eastern District of
                             Texas. Ericsson picked a different venue to
                             file two new 5G patent lawsuits against
                             Apple: the Western District of Texas.


                             In new filings on Monday, Ericsson alleged
                             that Apple is in violation of essential
                             patents for 5G. Ericsson cited a total of
                             twelve patents between the two. In a previous
                             statement provided to the press, Ericsson
                             said that Apple using its technology without
                             a license, following the expiration of a 2015
                             settlement. The companies have fired lawsuits
                             back and forth since last October, after
                             talks to renew that 2015 settlement went
                             nowhere.


                      # ⚓ Ericsson_Seeks_to_Block_U.S._Imports_of_Apple_Phone,
                        Watch_(2)⠀⇛


                             Ericsson AB filed U.S. trade complaints
                             seeking to block imports of Apple Inc.
                             devices, escalating a legal battle that shows
                             licensing talks between the two over 5G
                             telecommunications technology are going
                             poorly.


                             In complaints filed with the U.S.
                             International Trade Commission in Washington,
                             Ericsson targets a wide swath of Apple
                             products — the iPhone, tablets, smartwatches,
                             smart speakers and digital media players —
                             that it says infringe some of its patents.
                             Ericsson also lodged suits Monday in a west
                             Texas district court known to be friendly to
                             patent owners, claiming Apple…


                      # ⚓ Ericsson_escalates_patent_war_against_Apple_in
                        Europe⠀⇛


                             Swedish manufacturer of network technology
                             Ericsson has filed further lawsuits for
                             patent infringement against Apple. So far,
                             the market was only aware of patent and ITC
                             lawsuits of both companies in the US.


                             However, Ericsson has now confirmed to JUVE
                             Patent that the company also filed lawsuits
                             against Apple in Düsseldorf, Mannheim and
                             Munich earlier this week. The party also
                             filed additional lawsuits in the Netherlands
                             and Brazil, with Ericsson accusing Apple of
                             infringing several patents, including 5G
                             patents.


                      # ⚓ Intel_Wins_Backing_for_High_Court_Review_of_Patent
                        Board_Rule⠀⇛


                             Carmaker and cable industry groups are
                             joining Unified Patents LLC in supporting
                             Intel Corp.‘s bid for U.S. Supreme Court
                             review of a much-debated U.S. Patent and
                             Trademark Office rule that makes it harder to
                             initiate patent challenges.


                             Intel is challenging the agency’s Fintiv
                             rule—which allows the Patent Trial and Appeal
                             Board to deny review based on the state of a
                             parallel infringement case—and the U.S. Court
                             of Appeals for the Federal Circuit’s refusal
                             to hear appeals of denied petitions.


                             Unified Patents LLC, the Alliance for
                             Automotive Innovation, and Cable Television
                             Laboratories Inc. argue in an amicus brief in
                             support …


                      # ⚓ Priority_For_Brand_Protection_–_SME_Fund_To_Offer
                        Reimbursements [Ed: Whilst viciously_attacking_European
                        SMEs the EU does a bunch or PR stunts to pretend the
                        opposite is happening]⠀⇛


                             In today’s digital age an entity’s
                             intellectual property (IP) is of significant
                             value, which ought to be adequately
                             protected. The registration of EU and
                             national trade marks, domains, patents and /
                             or designs offers safeguards against free-
                             riding and copying by third parties, which in
                             turn brings about dilution of the brand or
                             design and unfair advantage to competitors.


                             The European Commission, the European Union
                             Intellectual Property Office and the Maltese
                             Commerce Department have issued a scheme to
                             encourage the registration of trade marks,
                             designs and patents by offering partial
                             refunds on fees.


                      # ⚓ Nicox_Provides_Fourth_Quarter_2021_Business_and
                        Financial_Highlights [Ed: Masking too much of a big
                        deal after getting European Patents, despite many_of
                        them_being_bogus]⠀⇛


                      # ⚓ Calfee’s_Intellectual_Property_Practice_Adds_Four
                        Attorneys_From_Respected_IP_Law_Firm_That [Ed:
                        Companies trying to twist the hiring of a few people as
                        major news; maybe if they pay JUVE they'll have puff
                        pieces and storytelling about the hiring of just one
                        such worker]⠀⇛


                      # ⚓ Cantargia_reports_third_party_appeal_against_EPO
                        decision_in_favor_of_Cantargia_patent⠀⇛


                      # ⚓ Cantargia_reports_third_party_appeal_against_EPO
                        decision_in_favor_of_Cantargia_patent [Ed: EPO’s nee
                        approach of just allowing patents on anything thrown at
                        it causes resentment and actions; even_the_tribunals
                        are_all_twisted_now]⠀⇛


                             Cantargia AB today reported that a third
                             party has filed a Notice of Appeal after
                             their previous unsuccessful attempt to
                             challenge the validity of one of Cantargia’s
                             European patents. The appeal concerns the
                             decision by the Opposition Division of the
                             European Patent Office (EPO) to reject the
                             opposition of Cantargia’s patent, EP 3020730,
                             for treatment of solid tumors by IL1RAP-
                             targeting antibody. This patent is part of
                             Cantargia’s robust IP portfolio and as it is
                             based on a divisional application, the parent
                             patent is not subject of the appeal.


                             “As Cantargia is reporting new results
                             supporting the commercial value of IL1RAP-
                             targeted antibody treatment of cancer, we
                             have noted increased activity from potential
                             competitors. Based on previous unsuccessful
                             attempts to invalidate our patent portfolio
                             around IL1RAP, and the clear decision by the
                             European Patent Office, we anticipate the
                             appeal to be groundless.” said Göran
                             Forsberg, CEO of Cantargia.


                      # ⚓ Balmain_x_Barbie_–_Not_Just_Your_Traditional
                        Collaboration [Ed: Convicted corrupt firm Marks & Clerk
                        legitimising online scam would be consistent with their
                        promotion of illegal software patents and illegal
                        UPC]⠀⇛


                             Barbie and Balmain recently launched a new 50
                             piece ready-to-wear collection. This is not
                             the first time that Barbie has collaborated
                             with a fashion brand, with the first designer
                             to dress Barbie being Oscar de la Renta, but
                             what is interesting here is that this
                             collaboration also sees the two brands come
                             together to launch a series of NFTs.


                             Rather than these being digital versions of
                             the ready-to-wear collection, these NFTs will
                             be Balmain-clad Barbie and Ken avatars and
                             the winners of these exclusive NFTs will
                             receive a matching Barbie-sized Balmain
                             outfit.


                      # ⚓ Provisional_Application_of_UPC_Agreement_Goes_Into
                        Effect [Ed: This is illegal and this very litigation
                        giant (Finnegan, Henderson, Farabow, Garrett & Dunner,
                        LLP) knows it; but it wants to profit from this
                        illegality]⠀⇛


                      # ⚓ Provisional_Application_of_UPC_Agreement_Goes_Into
                        Effect
                         [Ed: This is illegal and any half-decent journalist
                        would point out that it’s illegal; but this is what
                        happens when patent litigation firms (conflict of
                        interest obvious) write the so-called ‘news’]⠀⇛


                      # ⚓ European_Union:_Provisional_Application_Of_UPC
                        Agreement_Goes_Into_Effect [Ed: Spreading yet more lies
                        for Team UPC; it doesn’t seem to bother these patent
                        lawyers that this is clearly_illegal, they all sing
                        those lies in perfect harmony, as it’s their lobbying
                        strategy]⠀⇛


                             On 18 January 2022, the Austrian government
                             deposited its instrument of ratification of
                             the Protocol on Provisional Application of
                             the Unified Patent Court. Austria’s formal
                             ratification of the Protocol means that
                             sufficient member states have now ratified
                             the Protocol, and thus the Protocol has now
                             come into effect.


                             Ratification of the Protocol is a key step in
                             the creation of the Unified Patent Court
                             because it allows the Preparatory Committee
                             to interview and appoint judges to the Court
                             on a preliminary basis so that the Unified
                             Patent Court itself can come into existence.
                             The exact date when the Court will open is
                             still uncertain. However, the Preparatory
                             Committee for the Unified Patent Court has
                             previously indicated that they expected the
                             interviewing, training and appointment of
                             judges to the Court would take around 8 or 9
                             months. In practice, now that provisional
                             application has been approved, the Unified
                             Patent Court is likely to come into existence
                             at the end of 2022 or the beginning of 2023.


                      # ⚓ Gibson_Dunn_Named_a_2021_Firm_of_the_Year [Ed: Law360
                        and other sites that claim to be news sites (but
                        aren't; it's not their real product) are dishing out
                        fake awards for marketing/PR purposes; their Webspam is
                        then used like this]⠀⇛


                             Law360 named Gibson Dunn a Firm of the Year
                             for 2021 in its January 17, 2022 article “The
                             Firms That Dominated in 2021,” featuring nine
                             firms that received the most Practice Group
                             of the Year awards. The publication noted
                             that “Gibson Dunn & Crutcher LLP led the
                             pack, picking up the top Firm of the Year
                             spot with eight wins.”


                      # ⚓ Unified_Patent_Court_milestone_reached_as_preparatory
                        work_begins [Ed: This intentionally omits any mention
                        of the illegality_of_the_move_and_impending_barrier,
                        legal_challenges_etc. It only destroys the credibility
                        of the whole system]⠀⇛


                             Final preparations to establish the Unified
                             Patent Court (UPC) in Europe will ramp up in
                             2022 after an important legal treaty entered
                             into force. It is now expected the new court
                             system will become operational early next
                             year.
                             The Protocol to the Agreement on a Unified
                             Patent Court on provisional application (PAP-
                             Protocol) was formally ratified by Austria,
                             which deposited its instrument of
                             ratification on Tuesday, meaning the number
                             of ratifications from participating UPC
                             system members had reached the required
                             threshold – 13 – to enable the PAP-Protocol
                             to enter into force on Wednesday.


                      # ⚓ Patenting_Alloys_At_The_EPO:_Clarity [Ed: EPO seems
                        to_be_granting_patents_even_illegally_these_days,_and
                        nobody_is_stopping_it]⠀⇛


                             The European Patent Office has developed a
                             consistent approach to the examination of
                             patent applications relating to alloys,
                             taking into consideration the unique nature
                             of these important industrial materials.


                      # ⚓ Edging_towards_a_circular_model:_recent_innovation_in
                        the_design_and_recycling_of_plastics [Ed: Despite all
                        the greenwashing, the EPO, by granting patents on these
                        things, makes the planet worse off]⠀⇛


                             But thanks to scientific advancements, there
                             may be hope still. The recent report from the
                             European Patent Office (EPO) Patents for
                             tomorrow’s plastics describes promising
                             innovation over the last decade in two key
                             areas


                      # ⚓ The_European_Unitary_Patent_And_Unified_Patent_Court
                        –_January_2022_Update [Ed: No, UPC is not just
                        “delayed”; it’s illegal, it’s still being challenged in
                        numerous fronts, but patent_litigation_firms_keep_lying
                        about_it]⠀⇛


                             After years of delay caused by legal
                             challenges in Germany to ratification of the
                             Agreement on a Unified Patent Court (also
                             known as the “UPC Agreement” or “UPCA”), it
                             now seems that the Unified Patent Court (UPC)
                             and the European Unitary Patent (UP) will
                             come into effect in 2022. Indeed, the UPC and
                             the European Patent Office are indicating
                             that the UPC is expected to commence
                             operations in mid-2022.


                      # ⚓ IP_Funding_For_SMEs_In_The_EU [Ed: There is no such
                        thing as "IP" and whatever he actually refers to is
                        designed to protect monopolies at the expense of
                        SMEs]⠀⇛


                             Protecting your intellectual property (IP) is
                             essential to ensure that your brand, ideas,
                             products and services are not used or copied
                             without permission. IP rights, including
                             patents, trade marks and designs, can form
                             valuable assets for a business, and can be an
                             important consideration for investment in, or
                             valuation of, a business.


                             The ‘Ideas Powered for Business’ SME Fund is
                             a European Commission initiative implemented
                             by the European Union Intellectual Property
                             Office (EU IPO), through which successful
                             applicants can receive grant funding towards
                             intellectual property costs.


                      # ⚓ beDRP:_admissibility_and_probative_value_of_documents
                        from_the_Wayback_Machine⠀⇛


                             Are web archives stored in Wayback machines
                             (including Archive.org) admissible in legal
                             proceedings and, if so, what would be their
                             probative value? Lawyers specializing in
                             settlement of disputes relating to domain
                             names constantly utilize this mine of
                             information, as evidenced by the following
                             examples:


                             [...]


                             Whether it is a question of demonstrating the
                             bad faith of the holder of a domain name or,
                             on the contrary, proving the legitimate
                             right/interest of the latter, the Wayback
                             machines can be of considerable help.
                             A remarkable CEPANI decision recently
                             demonstrated, once again, the efficacy of the
                             Wayback Machine (CEPANI, 444149, La SRL ASBL
                             / La SA Perceval Technologies, December 8
                             2021, <asbl.be>, denied, sole panelist
                             Emmanuel Cornu). A Belgian company called
                             ASBL planned to obtain control of the domain
                             name <asbl.be>, held by Perceval Technologies
                             since 2000. ASBL claimed that Perceval
                             Technologies had registered the domain name
                             in bad faith. According to the complainant,
                             the passive holding of the domain name for
                             more than twenty years revealed the
                             Respondent’s bad faith. However, the latter
                             established a legitimate use of the domain
                             name, at least for a time. Indeed, Perceval
                             Technologies had effectively used <asbl.be>
                             to offer Internet services (including
                             subdomains, such as <name.asbl.be>) to non-
                             profit associations (Associations Sans But
                             Lucratif, in French).


                      # ⚓ CAFC_reigns_in_obvious-try-attack_against_dosage
                        patent_(Teva_v_Corcept) [Ed: EPO_breaks_its_own
                        rules]⠀⇛


                             The reasoning of the CAFC makes a striking
                             contrast to how the doctrine of “obvious-to-
                             try” is applied by the EPO and the national
                             courts of Europe.


                             [...]


                             As directed, Corcept conducted a clinical
                             trial of mifepristone and strong CYP3A4
                             inhibitors combination therapy (Nguyen et
                             al.). The patent (US10195214) was based on
                             the results of this clinical trial, which was
                             carried out in healthy volunteers. The patent
                             particularly claimed a method of treating a
                             patient by reducing their mifepristone dose
                             from 1200 mg or 900 mg per day, to 600 mg in
                             combination with a strong CYP3A inhibitor.


                             Teva brought post-grant review proceedings
                             against the patent, arguing that the
                             invention would have been obvious-to-try in
                             view of the KORLYM label, alone or in
                             combination with the FDA’s guidance on drug-
                             drug interaction studies. From Teva’s point
                             of view, the case was “an uncommonly clear-
                             cut obviousness case”, given that the prior
                             art disclosed the problem, the solution and
                             the way to find the solution.


                             [...]


                             In Europe, the obvious-to-try hurdle for
                             dosage regime cases is thus much higher than
                             the standard set by the CAFC in Teva vs
                             Corcept. Whilst the fact of the cases differ,
                             the PTAB and CAFC reasoning in Teva vs
                             Corcept is markedly different, for example,
                             to the reasoning of the UK High Court in
                             Bayer v Teva [2021] EWHC 2690 (Pat) (IPKat)
                             and Actavis v ICOS (IPKat), in which
                             inventions arising from multi-step drug
                             development programs were found obvious-to-
                             try in view of a known treatment effect for a
                             particular drug, regardless of whether there
                             was an unpredictable technical effect. By
                             contrast, in Teva v Corcept, a direct and
                             public instruction from the FDA to carry out
                             a clinical trial at the claimed doses was not
                             considered detrimental to the inventiveness
                             of the claimed doses.


                             From a US perspective, the decision in Teva v
                             Corcept appears to incentivise later filing
                             for dose inventions. Nonetheless, for global
                             filings, the trickier challenges to
                             patentability of dose claims in Europe cannot
                             be ignored.


                      # ⚓ Provisional_Phase_Of_The_UPC_Can_Now_Begin [Ed: UPC
                        is patently illegal, so what Team UPC boasts about here
                        is a violation of the law; they’re_just_eager_to_spread
                        fake_news_to_distract_from_simple_facts]⠀⇛


                             With the deposit on 18 January 2022 by
                             Austria of its instrument of ratification for
                             the Protocol on Provisional Application
                             Period (PAP- Protocol; see here), the UPC’s
                             Provisional Phase can now begin. The
                             Provisional Phase allows for the finalisation
                             of secondary legislation and Rules of
                             Procedure, the Court filing system and IT
                             infrastructure, financial budgets, and most
                             importantly, the hiring of the UPC’s staff,
                             including the appointment of judges.


                      # ⚓ Intelligent_digital_assistant_in_a_multi-tasking
                        environment⠀⇛


                             I held a contest and Apple wins–for having
                             the patent with the largest number of cited
                             prior art references (patents issued 2005-
                             2021). Apple’s U.S. Patent No. 11,037,565 is
                             directed to a version of Siri that works
                             across devices – an “Intelligent digital
                             assistant in a multi-tasking environment.”
                             The total, 9,441 cited references — 3 of
                             which were cited by the examiner.
                             Apple.Patent.


                             [...]


                             in accordance with a determination to
                             continue providing the media output at the
                             second electronic device: cause the second
                             electronic device to continue providing the
                             media output by resuming the media output
                             based on where the media output was
                             previously stopped at the first electronic
                             device.


                      # ⚓ Upcoming_changes_to_the_European_patent_system:_The
                        Unitary_Patent_and_Unified_Patent_Court [Ed: More
                        misleading nonsense from Team UPC; spreading fake news
                        about illegal campaigns to start something in spite of
                        many legal barriers will leave these lawyers'
                        reputation bruised as before]⠀⇛


                             The European patent system is undergoing
                             major changes that will impact all patent
                             filers in Europe. After years of legal and
                             political hurdles, the long-awaited Unitary
                             Patent and Unified Patent Court should soon
                             see the light of day, and possibly by the end
                             of 2022.


                      # ⚓ The_UK_Call_for_View_on_Standard_Essential_Patents
                        and_the_Case_for_Arbitration [Ed: “UK Intellectual
                        [sic] Property [sic] Office (UKIPO) is nakedly
                        plutocratic]⠀⇛


                             Recently, the UK Intellectual Property Office
                             (UKIPO) has launched a call for views in
                             order to better understand how effective the
                             current Standard Essential Patents (SEPs)
                             framework is in encouraging creativity,
                             innovation, and promoting competition.
                             Ultimately, the call aims to determine
                             whether legislative changes are required. An
                             interesting aspect of this call is that an
                             emphasis is made on arbitration as an
                             alternative to litigation.


                      # ⚓ KOL373_|_Against_Intellectual_Property_(audiobook
                        #2)⠀⇛


                             This is an audiobook version of my Against
                             Intellectual Property.


                      # ⚓ AutoStore,_Ocado_Fight_Over_Status_Of_High_Court_IP
                        Claim [Ed: Instead of giving of food and jobs to people
                        they give money to lawyers and patent offices, then sue
                        one another; so part of the price of food will be
                        "litigation tax"; ain't that just lovely?]⠀⇛


                             Norwegian robotics company AutoStore has said
                             it will add two more patent claims to an
                             ongoing intellectual property battle with
                             Ocado, rebutting claims by the online grocery
                             chain that it is abandoning its lawsuit.


                             AutoStore said on Wednesday that it has
                             focused on three of its patent claims in its
                             High Court row with Ocado Group PLC, which
                             has contracts with British high street chains
                             including Marks & Spencer, over automated
                             warehouse technology. The company said it
                             plans to widen the lawsuit by bringing two
                             more patents into the mix.


                      # ⚓ How_‘best_method’_became_a_major_pitfall_for_foreign_
                        (and_domestic)_patent_owners_in_Australia [Ed: Patents
                        do not have owners and they're not property; but more
                        to the point though, Australia seems to have fallen for
                        British charlatans who successfully conned a court into
                        thinking "bots" are "inventors"]⠀⇛


                             Failure to disclose the best method known to
                             the patentee of performing the invention[1]
                             has proven to be a powerful ground of attack
                             in Australia – if successful, it is capable
                             of rendering all claims of the patent
                             invalid. While other key jurisdictions have
                             abandoned the best method requirement (it was
                             abolished in the US in 2011, the UK in 1977,
                             and is not a consideration under European
                             laws), it is still a requirement under
                             Australian patent law – despite calls for its
                             abolition.


                             [...]


                             In the case of Les Laboratoires Servier v
                             Apotex Pty Ltd [2016] FCAFC 27 (Servier)
                             Servier’s patent was found to be invalid on
                             the sole basis that it did not disclose the
                             best method known to it of performing the
                             invention. The primary judge found that the
                             patentee’s description of its best method was
                             “wholly inadequate’,[2] and the Full Court of
                             the Federal Court of Australia upheld the
                             primary judge’s findings. The patentee had
                             provided a high level description that the
                             invention directed to arginine salt of
                             perindopril was ‘prepared according to a
                             classical method of salification of organic
                             chemistry’ (emphasis added).[3] While it was
                             common ground between the experts that it was
                             possible to produce an embodiment of the
                             invention claimed, this was insufficient to
                             meet the requirement of best method as the
                             skilled person would have had to engage in
                             ‘extensive trial and error experimentation’.
                             [4] The Court found that even though the
                             invention was for a product and not a
                             process, the best method obligation was not
                             met by simply identifying the claimed
                             compound.


                      # ⚓ Sigma-Aldrich_Files_Substantive_Preliminary_Motion_1
                        to_Change_the_Count_in_Interference_No._106,133 [Ed:
                        Here come the patent extremists who seriously want you
                        to believe that life and nature are human inventions
                        and some robber baron should monopolise it all]⠀⇛


                             On November 19th, Senior Party Sigma-Aldrich
                             filed its Substantive Preliminary Motion No.
                             1 in CRISPR Interference No. 106,133 (where
                             the University of California, the University
                             of Vienna, and Emmanuelle Charpentier,
                             collectively “CVC” is Junior Party) asking
                             the Board to substitute the Count pursuant to
                             37 C.F.R. §§ 41.121(a)(1)(iii) and 41.208(a)
                             (1).


                             [...]


                             Citing 37 C.F.R. § 41.201, Sigma-Aldrich
                             maintains that an interference count should
                             not encompass two patentably distinct
                             inventions. This principle is supported by
                             Judge McKelvey’s and subsequent Board
                             decisions, citing Ashurst v. Brugger, Int’f
                             No. 105,482 (McK), 2008 Pat. App. LEXIS 5953,
                             *51-52 (BPAI July 16, 2008), and Edelman v.
                             Stomp, Int’f No. 105,261 (SGL), 2006 Pat.
                             App. LEXIS 17, *18-19 (PTAB Apr. 14, 2006).
                             CVC’s portion of the Count as declared
                             encompasses “cleaving or editing the target
                             DNA molecule or modulating transcription of
                             at least one gene encoded by the target DNA
                             molecule” whereas Sigma-Aldrich’s portion of
                             the Count as declared is expressly limited to
                             “modifying a chromosomal sequence in a
                             eukaryotic cell by integrating a donor
                             sequence” (emphasis in brief). (And, facing
                             inventors who were awarded the Nobel Prize
                             for their achievements it is perhaps
                             understandable that Sigma-Aldrich
                             characterizes “its” portion of the Count as
                             “this final integration step represents a
                             milestone accomplishment in CRISPR-Cas9
                             genome engineering—not simply cleaving a DNA
                             strand, but thereafter actually modifying the
                             chromosomal sequence by integrating into that
                             cleaved strand a donor polynucleotide by
                             HDR.”) And importantly for arguments set
                             forth later in the brief (and the non-
                             obviousness argument supporting patentable
                             distinctiveness), Sigma-Aldrich states that
                             at the time of its first filing date,
                             December 6, 2012, “while . . . a POSITA would
                             have hoped that this advance in genome
                             engineering could be accomplished, whether
                             such an achievement would be accomplished
                             remained speculative.”


                      # ⚓ Netlist,_Inc._(OTCMKTS:_NLST)_Major_Reversal_brewing_
                        (912)_Seminal_Patent_(JDLA_Update)_|_Micro_Cap_Daily⠀⇛


                             The move up came after the U.S. Court of
                             Appeals for the Federal Circuit affirmed the
                             U.S. Patent Trial and Appeal Board’s decision
                             upholding the validity of Netlist’s U.S.
                             7,619,912 (‘912) patent that applies to DDR
                             server memory modules. The (‘912) patent is a
                             seminal patent; an invention so impactful
                             that it creates or shifts the technology
                             space.


                      # ⚓ After_more_than_12_years,_HTC_and_Fortress’s_IPCom
                        settle_standard-essential_patent_dispute_over_former
                        Bosch_and_Hitachi_patents [Ed: IPCom should not even
                        exist; it's a parasite and a yoke on the economy,
                        curtailing not only science but also Germany and the EU
                        at large; if the patent system submits to such an
                        agenda, no sane person will respect it anymore]⠀⇛


                             What an understatement. In the standard-
                             essential patent (SEP) space, any dispute
                             that last more than two years is already
                             “long-running” by my definition–and I doubt
                             anyone watching that space would dispute that
                             a four- or five-year patent spat is “long-
                             running.” This one here lasted more than a
                             dozen years–an eternity that has finally come
                             to an end.


                             Over the years I attended several IPCom v.
                             HTC trials in Germany, mostly in Mannheim,
                             where Deutsche Telekom recently sued for a
                             refund to the tune of 270 million euros. HTC
                             and Nokia were the first two smartphone
                             makers IPCom sued after acquiring a 3G SEP
                             portfolio from Bosch, a company that used to
                             make phones but exited that business even
                             earlier than the likes of Nokia and Ericsson.
                             The fact that HTC and Nokia had to defend
                             against many of the same patents turned those
                             two competitors–otherwise rivals–into
                             brothers-in-arms. In-house and outside
                             counsel of both companies coordinated their
                             defenses like they were one company. Even
                             when Nokia sued HTC over non-SEPs in 2012
                             (and ultimately got HTC to pay some
                             additional royalties), their friendship
                             survived. They were seen drinking beer on
                             high-speed trains from Mannheim to the
                             Cologne-Dusseldorf region just hours after
                             fighting hard in court.


                             While I occasionally disagreed with him and
                             disapproved of a couple of remarks he made at
                             the Nokia-HTC settlement party (after
                             inviting me as a surprise keynote speaker), I
                             do wish to give credit to the late Manuel
                             Chakraborty, a Hogan Lovells partner and
                             HTC’s outside counsel against IPCom and Nokia
                             at the time.


                      # ⚓ IPCom_settles_15-year_SEP_dispute_with_HTC [Ed: IPCom
                        is just a massive patent troll from Germany; of course
                        JUVE won't mention this because it's in the pocket of
                        firms that profit from the trolling]⠀⇛


                             The Bavaria-based NPE has announced that HTC
                             signed a licence agreement covering all of
                             IPCom’s assets. IPCom holds over 200 patent
                             families involving mobile communications,
                             with over 1,000 patents registered in Europe,
                             the US and Asia. Most of these patents have
                             been granted.


                             IPCom and HTC end dispute


                             The companies have not released financial
                             details of the agreement. However, it ends
                             their patent litigation which began in 2007
                             when IPCom sued Nokia and HTC. Later, the NPE
                             added lawsuits against Apple, Samsung and
                             other mobile phone manufacturers in Germany.
                             IPCom tried to enforce former UMTS patents
                             from Bosch.


                             According to a report by IAM, Apple is also
                             said to have recently accepted a licence from
                             IPCom. However, the parties have not yet
                             confirmed this.


                             About two years later, IPCom also sued Nokia
                             in the UK over the former Bosch patents.
                             Later, the NPE launched lawsuits against HTC
                             for infringement of the 100a patent.


                      # ⚓ Over_$100_million_Patent_Damages_Decision [Ed: How
                        truly absurd patent law can become]⠀⇛


                             Back in October 21, 2021, the Court of
                             Federal Claims decided to award over $100
                             million dollars concerning patent
                             infringement against the United States
                             Transportation Security Adminsitration (TSA)
                             in SecurityPoint Holdings v. United States.
                             Independent claim 1 of the patent


                      # ⚓ Engle_Grange_patent_held_invalid [Ed: USPTO issues
                        fake patents and PTAB confirms that these should never
                        have been granted in the first place (but only because
                        lawsuits were being filed; millions of fake patents
                        remain unchallenged)]⠀⇛


                             On January 19, 2022, the Patent Trial and
                             Appeal Board (PTAB) issued a final written
                             decision in Unified Patents, LLC v. Engle
                             Grange, LLC holding all challenged claims of
                             U.S. Patent 8,548,645 invalid. The ‘645
                             patent is generally directed towards a two-
                             step key fob authentication system for an
                             automobile. The patent was asserted against
                             Ford in early 2020.


                             View district court litigations by Engle
                             Grange. To read the petition and view the
                             case record, see Unified’s Portal. Unified
                             was represented by in-house counsel, David
                             Seastrunk and Roshan Mansinghani, in this
                             proceeding.


                      # ⚓ Final_UPC_preparations_begin_after_Austria_decision
                        [Ed: Of course the patent maximlaists are failing to
                        note that UPC_is_illegal_and_unconstitutional; they
                        help relay fake news for litigation conglomerates that
                        sponsor the fake news.]⠀⇛


                             The Unified Patent Court entered into its
                             final preparatory phase on Tuesday, January
                             18, meaning that the court could open by the
                             end of the year.


                             Austria confirmed on Tuesday that it had
                             completed its ratification of the protocol to
                             the UPC Agreement, allowing the court’s
                             provisional application period (PAP) to
                             begin.


                             The PAP protocol lets the UPC become a legal
                             entity and allows organisers to complete
                             their remaining work. Outstanding tasks
                             include the hiring of staff and judges, as
                             well as the completion of a new IT system.


                             The UPC Preparatory Committee estimates that
                             this process will take at least eight months.


                             The UK’s withdrawal from the UPC threatened
                             the success of the project at one point, as
                             did a series of constitutional challenges in
                             Germany. But the decision from Germany’s
                             Federal Constitutional Court last year to
                             dismiss two complaints cleared the way for
                             the UPC to move forward.


                             Not all signatories to the UPC Agreement have
                             finished ratifying it, meaning countries such
                             as Ireland are now under pressure to complete
                             this process by the time the court becomes
                             operational.


                      # ⚓ US_trends_and_events_patent_counsel_should_watch_in
                        2022 [Ed: Patent maximalist (funded by litigation
                        firms) Patrick Wingrove only ask lawyers about
                        Microsoft_Vidal; any other opinions? Not welcome…]⠀⇛


                             In-house lawyers in pharma, tech, telecoms
                             and NPE businesses say they’re keeping an eye
                             on Kathi Vidal, litigation funding and
                             western Texas


                      # ⚓ CMS_rebuilds_London_patent_practice_with_Allen_&
                        Overy_hire [Ed: Amy Sandys is marketing spam disguised
                        as “journalism”; in this case, the hiring of one
                        relatively low-level person is spun as “news”; but it’s
                        PR spam and you can follow the money to realise what
                        motivates such fake news; that’s what_JUVE_became.
                        Firms hire people all the time; but for JUVE, if that’s
                        a client, then it’s time for storytelling and marketing
                        disguised as “reporting”.]⠀⇛


                             Toby Sears (36) has joined the IP team at CMS
                             Cameron McKenna Nabarro Olswang as a partner.
                             Previously, Sears was at Allen & Overy, where
                             he was a senior associate for four years.
                             Prior to this, he spent six years at Bird &
                             Bird.


                      # ⚓ FOSS_Patents:_Ericsson_seeks_U.S._import_ban_against
                        iPhone,_iPad,_Apple_Watch;_accuses_Apple_of_‘reverse
                        hold-up’_in_4G/5G_patent_licensing_negotiations⠀⇛


                             I’ve just been able to obtain the complaint
                             Ericsson filed against Apple yesterday with
                             the United States International Trade
                             Commission, seeking a U.S. import ban of 4G
                             and 5G Apple products (iPhones, iPads, and
                             Apple Watches; I’ll provide the full list
                             further below). As I speculated in my
                             previous post on Ericsson v. Apple, one of
                             the two patent infringement lawsuits Ericsson
                             filed yesterday with the United States
                             District Court for the Western District of
                             Texas is a companion complaint to an ITC
                             complaint–and as I figured, it’s the one in
                             which Ericsson is asserting four patents:


                             U.S. Patent No. 8,102,805 on “HARQ [Hybrid
                             Automatic Repeat reQuest] in spatial
                             mutiplexing MIMO [multiple-input multiple-
                             output] system”


                             U.S. Patent No. 9,532,355 on “transmission of
                             system information on a downlink shared
                             channel”


                             U.S. Patent No. 10,425,817 on a “subscription
                             concealed identifier”


                             U.S. Patent No. 11,139,872 on “codebook
                             subset restriction signaling”


                             [...]


                             In principle, Ericsson can try to add to that
                             list any Apple products released in the
                             coming months.


                      # ⚓ FOSS_Patents:_Apple_countersues_Ericsson,_seeks_U.S.
                        import_ban_against_mobile_base_stations,_and_proposes
                        that_both_parties_withdraw_all_patent_infringement
                        actions,_let_Texas_court_set_license_fee⠀⇛


                             The flurry of Ericsson-Apple filings with the
                             United States International Trade Commission
                             (USITC, or just ITC), a U.S. trade agency
                             with the quasi-judicial power to U.S. import
                             bans, continued yesterday. For information on
                             Ericsson’s three compaints against Apple, may
                             I refer you to my most recent post on the
                             dispute, which contains a hierarchical
                             overview of the seven U.S. cases I had
                             previously discovered (one FRAND action by
                             either party, two federal lawsuits by
                             Ericsson, and three ITC complaints by
                             Erisson).


                             As I wrote, it was a given that Apple would
                             countersue, though the wide discrepancy in
                             the parties’ exposure to patent assertions
                             makes it unlikely that Apple can gain
                             decisive leverage that way. And indeed, Apple
                             has now also brought a request for a U.S.
                             import ban against Ericsson’s mobile
                             infrastructure products, asserting three
                             mmWave-related patents…


                      # ⚓ FOSS_Patents:_Ericsson_sues_Apple_in_three_German
                        courts,_the_Netherlands,_and_Brazil_over_5G_and_other
                        patents_(in_addition_to_handful_of_U.S._cases)⠀⇛


                             For an overview of Ericsson’s U.S. patent
                             infringement actions against Apple (three
                             USITC complaints and two federal lawsuits in
                             the Western District of Texas), may I refer
                             you to a very recent post. Apple filed a
                             countercomplaint with the ITC over three
                             patents and all of a sudden declared itself a
                             big fan of the United States District Court
                             for the Eastern District of Texas.


                             The choice of the three German courts is
                             unsurprising, and if I interpret Juve Patent
                             correctly, Ericsson filed two cases each in
                             Dusseldorf and Mannheim, but more than two in
                             Munich (as would I in their shoes).


                             One or more filings with the Dutch court were
                             also more than predictable, given that
                             Ericsson sought an anti-antisuit injunction
                             there last year. And just like the three
                             German courts, the Dutch court is also one of
                             the venues Ericsson chose for last year’s
                             dispute with Samsung (which got settled
                             pretty quickly).


                             In Brazil, Apple is not the first company
                             against which Ericsson enforces patents: the
                             Swedish company already did so against TCL,
                             as articles from Brazil like this one
                             confirm.


                      # ⚓ FOSS_Patents:_Ericsson_sues_Apple_over_infringement
                        of_5G_and_other_patents_upon_expiration_of_license
                        agreement–Apple_now_facing_an_‘Epicsson’_dilemma⠀⇛


                             Ericsson’s patent cross-license agreement
                             with Apple has expired, and no renewal has
                             been agreed upon, which is why infringement
                             litigation became inevitable. IAM was first
                             to spot two filings by Ericsson against Apple
                             in the Western District of Texas over four
                             patents in one case and eight in the other.
                             Apple will predictably retaliate, but Apple’s
                             exposure to patent assertions dwarfs
                             Ericsson’s. When there was no announcement of
                             a renewed license agreement or renewed
                             litigation after the turn of the year, I
                             thought the middle of the month was going to
                             be when we would hear more.


                             [...]


                             Having seen Ericsson enforce patents against
                             Apple and other parties in the past, I assume
                             that one of the two complaints is a companion
                             complaint to a Sec. 337 complaint with the
                             United States International Trade Commission
                             (“ITC”), seeking an exclusion order (aka
                             import ban). Most likely the 22-cv-60 case
                             (over four patents) is the companion
                             complaint as the ITC urges complainants to
                             focus on a small number of patents so it can
                             keep its schedule (it is also possible to
                             start an ITC case with many patents and to
                             then narrow it by dropping most of the
                             patents-in-suit, but I doubt that that is the
                             plan here). I’m also pretty sure that
                             Ericsson has near-simultaneously filed
                             complaints with courts in other
                             jurisdictions, among them almost certainly
                             Germany, where Munich is the new number one
                             and Mannheim continues to be a key venue.


                      # ⚓ Final_hurdle_cleared_for_UPC_preparations_to_start
                        [Ed: Everything here, including the Bristows headline,
                        is a lie; many legal obstacles remain and some are
                        entirely untenable, impossible to clear; Team UPC is
                        just engaging in criminal behaviour now, boosted by its
                        own fake news]⠀⇛


                             It appears that yesterday, as recorded here,
                             Austria finally deposited with the Council of
                             the EU its instrument of ratification of the
                             Protocol on the Provisional Application (PPA)
                             of the Unified Patent Court Agreement (UPCA).
                             This means that sufficient countries have now
                             met the requirements for the PPA to commence
                             (see here). In this provisional application
                             phase, the UPC’s legal capacity and
                             organisational capability is established and
                             final preparations for the opening of the
                             Court can be made. The Preparatory Committee
                             has estimated that these preparations may
                             take about 8 months, with the work including
                             assembly of the Court’s governing bodies,
                             adoption of all the secondary legislation,
                             and finalisation of the budget, the IT
                             systems and the recruitment of the judges.
                             The EPO will also make final preparations for
                             the introduction of the unitary patent.


                      # ⚓ Off_we_go!_The_Unified_Patent_Court_is_officially
                        born. [Ed: Even the compromised IP_Kat is promoting
                        fake news at the moment]⠀⇛


                             The entry into force of the Protocol on
                             Provisional Application of the UPC Agreement
                             has been a major milestone for the actual
                             birth of the Unitary Patent Court.


                      # ⚓ UPC_–_Provisional_Application_Period_enters_into
                        force [Ed: Thorsten Bausch (Hoffmann Eitle) could do a
                        lot more to explain — as commenters do — that what’s
                        happening here is illegal; this harms the image of law
                        firms, as they’re engaging in organised_crime]⠀⇛


                             According to the Commission’s Website,
                             Austria has deposited its instrument of
                             ratification yesterday, which means that the
                             Period for Provisional Application begins
                             today.


                      # ⚓ [Market_Eye]_Korean_tech_giants_fall_prey_to_patent
                        trolling [Ed: Korea gets Karma for KIPO; all that
                        patent maximalism in the overworked Kingdom is harming
                        the 'Kapital']⠀⇛


                             Though the pandemic has driven record-
                             breaking profits for tech firms, as demand
                             for at-home leisure and communications
                             hardware rises, it has also brought a growing
                             threat from patent trolls.


                             Companies specializing in exploiting patent
                             rights have been buying up intellectual
                             property from firms struggling with the
                             pandemic and using it as ammunition against
                             Korean firms.


                             The approach is often seen as a “file-and-
                             settle” strategy, as their targets often
                             prefer to avoid fighting a time-consuming
                             court battle and the costs associated with
                             it, even if they are likely to win the case.


                      # ⚓ UPC_preparation_phase_officially_begins [Ed: JUVE‘s
                        Mathieu Klos is still a mouthpiece of Team UPC (follow
                        the money), in effect promoting a crime using a
                        “journalist” hat]⠀⇛


                             Austria was the 13th member state that still
                             needed to deposit its instrument of
                             ratification and trigger the PAP. At the
                             beginning of December 2021, the Austrian
                             parliament ratified the UPC protocol. At the
                             time, it did not deposit the instrument with
                             the European Council. However, from today,
                             the UPC can officially begin its preparatory
                             phase.


                             Protocol dictates that 13 UPC countries must
                             ratify and deposit the instrument in
                             Brussels, in order for the protocol to enter
                             into force.


                             The next interesting decision is the
                             selection of judges. Furthermore, it must be
                             decided where the UPC Central Divisions will
                             locate its pharmaceuticals and chemicals
                             parts, following the UK’s departure from the
                             project.


                      # ⚓ UPC_Preparatory_Committee_confirms_provisional
                        application_period_has_started [Ed: Boosted by parent
                        networks, Gregory Bacon from lying_firm_Bristows is now
                        a megaphone of fake news from Mr. Ramsay, pursuing an
                        illegal agenda, based entirely on lies and abuses]⠀⇛


                             Following Austria’s deposit of its instrument
                             of ratification of the Protocol on the
                             Provisional Application (PPA) of the Unified
                             Patent Court Agreement (UPCA) (reported
                             here), the UPC Preparatory Committee has
                             reported here that the PPA came into force
                             today. The Committee has outlined the final
                             work for establishing the Court that can now
                             start and states this period will last at
                             least 8 months.


                      # ⚓ Opinion:_Irish_UPC_saga_one_to_watch_in_2022 [Ed: The
                        UPC is not pressing ahead (it_is_still_illegal), but
                        this corrupt publisher, bribed by Team UPC, is paid to
                        produce loaded headline and mislead politicians et al
                        with fake news]⠀⇛


                             The UPC is pressing ahead, but Ireland has
                             plenty left to do if it wants to take part


                      # ⚓ Watch_Party_Moves_to_N.D.Cal. [Ed: Texas has received
                        nothing short of international notoriety for its
                        crooked patent courts where the goal isn't justice but
                        money]⠀⇛


                             This was an easy mandamus transfer. The
                             plaintiffs, CA Inc. and Avago Technologies
                             Int’l, are both subsidiaries of Broadcom
                             Corp. The plaintiffs are both headquartered
                             in San Jose and Broadcom is down in Southern
                             California. Netflix is also Silicon Valley.
                             Still, the plaintiffs sued Netflix for patent
                             infringement in the Eastern District of
                             Texas. Although Netflix itself does not have
                             a place of business in EDTex, the district
                             court found that its intimate relationship
                             with ISPs to provide local content delivery
                             was sufficient. The vast majority of Netflix
                             content is delivered directly from local
                             residential Internet Service Providers
                             (ISPs).


                      # ⚓ [Older]_Patent_Owner_Loses_Everything_at_PTAB [Ed:
                        Patent bubbles bursting aplenty in recent years; it's
                        mostly a cartel system for large corporations that
                        cross-license, in bulk, a whole lot of junk]⠀⇛


                             In Mati Therapeutics, Inc. v. Ocular
                             Therapeutix, Inc., the Federal Circuit issued
                             a Rule 36 opinion affirming the Patent Trial
                             and Appeal Board’s (“PTAB’s”) Final Written
                             Decision (“FWD”) holding all challenged
                             claims unpatentable. Ocular Therapeutix, Inc.
                             v. Mati Therapeutics, Inc., IPR2019-00448,
                             Paper 56 (P.T.A.B. June 18, 2020), aff’d
                             (Fed. Cir. Nov. 8, 2021).


                      # ⚓ [Older]_Thaler_v_Comptroller_General_of_Patents,
                        Trade_Marks_and_Designs:_Court_of_Appeal_Judgment_on
                        Machine_Inventors [Ed: This provocateur is being
                        spurned and humiliated by patent offices and courts for
                        trying to insinuate patents are something a computer
                        can just spew out]⠀⇛


                             The Court of Appeal (Arnold LJ, Laing LJ and
                             Birss LJ) handed down its judgment in Thaler
                             v Comptroller General of Patents Trade Marks
                             And Designs on 21 September 2021. The court
                             held, with Birss LJ dissenting, that the
                             judge of first instance (Marcus Smith J) was
                             correct in his dismissal of the appeal.


                             The appellant, Dr. Thaler, had originally
                             applied for two patents at the UKIPO, listing
                             his artificial intelligence (“AI”) machine
                             DABUS as the inventor for both. The UKIPO
                             found that DABUS is not a person as envisaged
                             by s. 7 and s. 13 of the UK Patents Act 1977
                             (the “1977 Act”) and so cannot be considered
                             an inventor, and that Dr. Thaler had not
                             provided a satisfactory derivation of his
                             right to be granted the patents. The patent
                             applications were therefore deemed withdrawn.
                             Dr. Thaler then appealed this decision to the
                             High Court, where his appeal was dismissed by
                             Marcus Smith J.


                      # ⚓ [Older]_Federal_Circuit_Affirms_Invalidity_of_Method
                        of_Treatment_Patent_for_Lack_of_Written_Description_in
                        Hatch-Waxman_Case [Ed: More fake patents, as confirmed
                        by a regional and high court alike in the US]⠀⇛


                             On November 30, 2021, the Federal Circuit
                             affirmed the Northern District of West
                             Virginia’s invalidity finding as to a method-
                             of-treatment patent, based on a lack of
                             written description.


                             Plaintiffs Biogen International GmbH and
                             Biogen MA Inc. alleged Mylan Pharmaceuticals
                             Inc. infringed several patents listed in the
                             Orange Book for Tecfidera®, a dimethyl
                             fumarate capsule indicated for the treatment
                             of certain forms of multiple sclerosis. Among
                             the asserted patents was U.S. Patent No.
                             8,399,514, which claims a method of treating
                             MS with DMF and which was the sole patent
                             addressed in the opinion.


                      # ⚓ [Older]_China’s_Supreme_People’s_Court_Rules_No
                        Accounting_for_Profit_for_Joint_Patent_Ownership [Ed:
                        Technically patents are not owned but assigned, and
                        moreover they are never property]⠀⇛


                             In decision no. （2020）最高法知民终954号 dated
                             November 25, 2021, China’s Supreme People’s
                             Court ruled that if the co-owners of a patent
                             or patent application right do not make an
                             agreement on the exercise of the right and
                             one of the co-owners independently practices
                             the patent, the other co-owner cannot claim
                             the distribution of the proceeds from the
                             separate practicing of the patent on the
                             grounds of co-ownership of the patent right.


                      # ⚓ [Older]_Apple_wins_latest_round_of_UK_patent_dispute
                        with_Optis [Ed: Optis exists for nothing but lawsuits,
                        so no matter who it is suing, it deserves no sympathy
                        or support]⠀⇛


                             US technology giant Apple has successfully
                             argued that it is not responsible for
                             infringing a patent for technology that
                             supports continued connectivity during mobile
                             phone calls.


                             Although the Court of Appeal in London
                             rejected Apple’s claims that the patent owned
                             by telecoms company Optis Cellular Technology
                             was invalid, it accepted its case that the
                             patent was not ‘essential’ to a
                             communications standard and that it had not
                             infringed the patent.


                      # ⚓ [Older]_USPTO_To_Propose_Paperless_Patents [Ed:
                        Greenwashing_of_patents_and_patent_offices whose output
                        contributes to waste and pollution for monopolies’
                        sake]⠀⇛


                             In a Director’s Forum blog post published
                             December 10, 2021, the USPTO announced plans
                             to issue proposed rule changes relating to
                             the format in which U.S. patents are issued,
                             making electronic patents the norm, with
                             something akin to today’s ribboned copies
                             available upon request (and for a nominal
                             fee). Many of you reading this may wonder
                             what took so long, but as with many
                             government agency practices, it turns out
                             there’s a statute that had to be reckoned
                             with before the USPTO could go paperless.


                      # § Software Patents⠀➾


                            # ⚓ [Older]_The_Dust_Settles_Further_in_Relation_to
                              Patents_for_Computer_Implemented_Inventions_in
                              Australia [Ed: Australian courts have already
                              struck down, rejected software patents, so some
                              greedy law firms are adopting misleading_terms
                              from_the_EPO,_e.g._CII]⠀⇛


                                   Last month we wrote about the Full
                                   Federal Court’s decision in
                                   Commissioner of Patents v Aristocrat
                                   Technologies Australia Pty Ltd [2021]
                                   FCAFC 202 (Aristocrat), which concerned
                                   the patentability of computer
                                   implemented inventions (CIIs).


                                   This month, the Full Court determined
                                   another appeal regarding CIIs: Repipe v
                                   Commissioner of Patents [2021] FCAFC
                                   223. The decision concerned two patent
                                   applications by Repipe Pty Ltd that
                                   disclosed systems and methods for
                                   providing information to field workers
                                   by way of a central computer server
                                   connected to a GPS-enabled mobile
                                   device (i.e. a smartphone). The
                                   applications were treated as the same
                                   for all relevant purposes at trial and
                                   during the appeal.


                            # ⚓ $4,000_for_prior_art_on_another_set_of_Big_Will
                              Enterprises_patents⠀⇛


                                   On January 4, 2022, Unified Patents
                                   added two separate PATROLL contests
                                   with an opportunity to collect up to
                                   $4,000 in cash for prior art on at
                                   least claim 1 of two patents – U.S.
                                   Patent 8,559,914 and U.S. Patent
                                   8,737,951. The patents are owned by Big
                                   Will Enterprises, Inc., an NPE. The
                                   patents generally relate to an
                                   interactive personal surveillance and
                                   security (IPSS) system for users
                                   carrying wireless communication
                                   devices. The patents have been asserted
                                   against Montblanc, Unaliwear, and
                                   Kronoz.


                            # ⚓ Causam_Enterprises_patent_challenged⠀⇛


                                   On January 18, 2022, Unified filed a
                                   petition for inter partes review (IPR)
                                   against U.S. Patent 10,396,592. The
                                   ‘592 patent is owned by Causam
                                   Enterprises, Inc., an NPE. The ’592
                                   Patent relates to active load
                                   management systems and has been
                                   asserted in district court and the ITC
                                   against Ecobee, Resideo, Alarm.com,
                                   Xylem, and Itron.


                            # ⚓ WSOU_’770_patent_likely_invalid⠀⇛


                                   On January 11, 2022, the Patent Trial
                                   and Appeal Board (PTAB) instituted
                                   trial on all challenged claims in an
                                   IPR filed by Unified against U.S.
                                   Patent 7,333,770, owned by WSOU
                                   Investments, LLC. The ‘770 patent is
                                   generally related to broadcasting data
                                   in a telecommunications system and is
                                   being asserted against TP-Link
                                   Technology Co.


                            # ⚓ FOSS_Patents:_‘Access_Advance’_patent_pool_can
                              rename_itself_‘Suffer_Setback’_after_Dusseldorf
                              court_ruling:_flagrant_FRAND_abuse_concerning
                              video_codec_patents [Ed: These are abstract
                              patents or fake patents leveraged in bulk by
                              aggregate or cartel, HEVC_Advance]⠀⇛


                                   The most recent IAM Sunday Digest
                                   mentioned the Access Advance pool
                                   firm’s VVC Advance announcement as well
                                   as the Dusseldorf Regional Court’s
                                   conclusion that the HEVC Advance pool’s
                                   terms are non-FRAND. I had mentioned
                                   the latter in a recent post on a
                                   wireless SEP dispute. While I haven’t
                                   been able (yet) to obtain a copy of the
                                   Dusseldorf decision, I have meanwhile
                                   seen reports by IAM (as I mentioned)
                                   and Juve Patent, and would like to add
                                   some observations.


                                   This is not going to be my last post on
                                   that set of cases and the underlying
                                   conduct, which I find egregious. A few
                                   organizations’ greed threatens to bring
                                   the very concept of patent pools into
                                   disrepute unless we all make the proper
                                   distinction between the vast majority
                                   of pools that serve their purpose–and
                                   an abusive outlier like Access Advance.
                                   It is equally important to understand
                                   that just because a company joined that
                                   pool as a licensor doesn’t necessarily
                                   mean that it has unclean hands. They
                                   aren’t all wrongdoers, but the
                                   Dusseldorf disaster should give the
                                   more reasonable ones among those Access
                                   Advance licensors pause, above all with
                                   a view to the latest and greatest in
                                   video codec standards: VVC.


                                   There’s a certain complexity and some
                                   specialized terminology, so let’s get
                                   an overview first.


                            # ⚓ How_and_why_to_protect_algorithms_and_software
                              –_ClarkeModet [Ed: This firm profits from patent
                              litigation so of course it’ll ignore rules
                              against_this_kind_of_thing]⠀⇛


                            # ⚓ The_EPO_and_UKIPO_approaches_to_AI_and
                              patentable_subject_matter [Ed: EPO propaganda arm
                              IAM keeps promoting illegal patents on
                              algorithms, nowadays disguising these with
                              buzzwords like “Hey Hi”]⠀⇛


                                   There are important ways in which the
                                   European and the UK patent offices
                                   differ when assessing the patentability
                                   of AI inventions, write Gemma Robin,
                                   Greg Ward and Ariana Sadr-Hashemi of
                                   Haseltine Lake Kempner in this co-
                                   published article


                                   Inventions relating to Artificial
                                   Intelligence (AI), and more
                                   particularly to machine learning, have
                                   been raising questions for the patent
                                   community since they first started
                                   appearing as the subject of patent
                                   applications.


                            # ⚓ Germany:_Optimized_Load_Management:_Technical
                              [Ed: Bardehle_Pagenberg continues promoting bogus
                              patents like software patents; nothing will stop
                              such greedy lawyers ]⠀⇛


                                   This decision relates to a European
                                   patent regarding a method for
                                   optimizing a chronological developing
                                   of consumption of electric energy by a
                                   group of different consumers with
                                   regard to a supply of electric power.
                                   The patent has been unsuccessfully
                                   opposed after grant and the opponent
                                   has filed an appeal. Here are the
                                   practical takeaways from the decision T
                                   0199/16 of 27.8.2021 of Technical Board
                                   of Appeal 3.5.02:


                            # ⚓ Women_in_Business_Law_Awards_Americas_2022:
                              open_for_submissions [Ed: These fake awards are
                              the real business model of this fake news
                              propaganda mill, which takes money in exchange
                              for Webspam and phony "endorsements" (false
                              marketing)]⠀⇛


                                   We are delighted to announce the launch
                                   of the Women in Business Law Awards
                                   Americas 2022 research.


                                   Please follow the link below to
                                   nominate leading women practitioners,
                                   rising stars, firm initiatives, and in-
                                   house counsels and teams.


                            # ⚓ Dallas_Invents:_71_Patents_Granted_for_Week_of
                              Dec._21 [Ed: More bogus software patents in
                              breeding ground of patent trolls, who exercise
                              "fake economy" "free market" 'parasitisation']⠀⇛


                                   Dallas Invents is a weekly look at U.S.
                                   patents granted with a connection to
                                   the Dallas-Fort Worth-Arlington metro
                                   area. Listings include patents granted
                                   to local assignees and/or those with a
                                   North Texas inventor. Patent activity
                                   can be an indicator of future economic
                                   growth, as well as the development of
                                   emerging markets and talent attraction.
                                   By tracking both inventors and
                                   assignees in the region, we aim to
                                   provide a broader view of the region’s
                                   inventive activity. Listings are
                                   organized by Cooperative Patent
                                   Classification (CPC).


                            # ⚓ The_EPO_and_UKIPO_approaches_to_AI_and
                              patentable_subject_matter [Ed: Haseltine Lake
                              Kempner LLP is pushing software patents agenda
                              using buzzwords, in a propaganda_rag_bribed_by
                              the_EPO to spread fake news, to lie to
                              politicians, and to promote patent trolls, cover
                              up EPO crimes etc.]⠀⇛


                                   There are important ways in which the
                                   European and the UK patent offices
                                   differ when assessing the patentability
                                   of AI inventions, write Gemma Robin,
                                   Greg Ward and Ariana Sadr-Hashemi of
                                   Haseltine Lake Kempner in this co-
                                   published article


                                   Inventions relating to Artificial
                                   Intelligence (AI), and more
                                   particularly to machine learning, have
                                   been raising questions for the patent
                                   community since they first started
                                   appearing as the subject of patent
                                   applications.


                # § Trademarks⠀➾


                      # ⚓ Dutch_court_rejects_claims_for_sui_generis_database
                        protection_of_national_company_register_–_The_IPKat⠀⇛


                             In the era of fake news, reliable sources of
                             information are more valuable than ever. When
                             it comes to official registers held on a
                             governmental level, administrative bodies are
                             often tasked to ensure such registers are
                             accurate and complete. Accessing those
                             registers usually comes at a cost, prompting
                             some commercial parties to offer the same
                             information at a lower price.


                             In a recent case before the Dutch (district)
                             court of Midden-Nederland, the question arose
                             whether database rights can be invoked to
                             prevent third parties reusing and offering
                             data retrieved from the national company
                             register. For those Kats accessing a company
                             register frequently, the decision is
                             certainly worth the read.


                             [...]


                             According to VVZBI, database right protection
                             is based on the ‘economical incentive
                             theory’, entailing that database protection
                             should only be granted to risk bearing
                             investments in a database. Without database
                             protection, so the theory assumes, the
                             database would not have been created in the
                             first place, as an economical incentive to do
                             so would be missing.


                             Given that the KvK is statutorily tasked with
                             creating and maintaining the company
                             register, VVZBI argued that the investments
                             could never qualify as risk-bearing
                             economical investments. Consequently, the KvK
                             had not substantially invested in the
                             database in the sense of article 7(1)
                             Database Directive, which is a prerequisite
                             to exercise any rights over it.


                      # ⚓ TTABlog_Test:_Is_SCOOPERFEST_Confusable_With_SCOOPER
                        BOWL_for_Ice_Cream_Festivals?⠀⇛


                             The Dana-Farber Cancer Institute opposed an
                             application to register the mark SCOOPERFEST
                             for “organizing community festivals featuring
                             primarily ice cream and also providing
                             entertainment,” alleging a likelihood of
                             confusion with the registered mark SCOOPER
                             BOWL for “charitable fundraising by means of
                             an ice cream festival.” Applicant Craig
                             Marquardo claimed that the SCOOPER BOWL mark
                             is conceptually weak, that the parties
                             provide their services on opposite sides of
                             the country, and therefore that confusion is
                             not likely. How do you think this came out?
                             Dana-Farber Cancer Institute, Inc. v. Craig
                             Richard Marquardo, Opposition No. 91255611
                             (January 20, 2022) [not precedential]
                             (Opinion by Judge Melanye K. Johnson).


                      # ⚓ Anne_Gilson_LaLonde_Agrees_with_Professor_McCarthy’s
                        Criticism_of_CAFC’s_Brooklyn_Brewery_Decision_and_Adds
                        Her_Own⠀⇛


                             Trademark expert Anne Gilson LaLonde, author
                             of Gilson on Trademarks, has provided her
                             comments on the CAFC’s recent Brooklyn
                             Brewery decision [here], with errata [here],
                             agreeing with Professor J. Thomas McCarthy’s
                             criticism of the court’s view on standing
                             [TTABlogged here] and adding her own
                             criticism regarding the CAFC’s
                             misinterpretation of its role in reviewing a
                             TTAB decision.


                      # ⚓ TTABlog_Test:_Is_PISSTERINE_Confusable_with_LISTERINE
                        for_Mouthwash?⠀⇛


                             J & J opposed an application to register the
                             proposed mark PISSTERINE for non-medicated
                             mouthwash, claiming likelihood of confusion
                             with, and likely dilution of, its registered
                             mark LISTERINE for, inter alia, medicated
                             mouthwash, toothpaste, and dental floss. J &
                             J established that its mark is famous for
                             Section 2(d) purposes. The goods are
                             overlapping or related, but what about the
                             marks? What about parody? Johnson & Johnson
                             v. Pissterine, LLC, Opposition No. 91254670
                             (January 18, 2022) [not precedential]
                             (Opinion by Judge Jyll Taylor).


                             [...]


                             Applicant did not submit any testimony or
                             evidence, nor did it file a brief. In its
                             answer, however, it did raise the issue of
                             parody, claiming no likelihood of confusion
                             and fair use. The Board observed that
                             “Applicant’s intention for its mark to be a
                             parody does not necessarily negate a finding
                             of likelihood of confusion.” In short,
                             “parody is not a defense if the marks would
                             otherwise be considered confusingly similar.”
                             Nike, Inc. v. Maher, 100 USPQ2d 1018, 1023
                             (TTAB 2011) .


                      # ⚓ Precedential_No._3:_TTAB_Grants_Motion_to_Supplement
                        Notice_of_Opposition_to_Add_Post-Filing_Trademark
                        Applications⠀⇛


                             Topco Holdings opposed registration of
                             READY4LIFE for sanitizers for personal use,
                             alleging a likelihood of confusion with its
                             registered marks READY FOR LIFE and SIMPLY
                             DONE READY FOR LIFE for various personal
                             goods and cleaning products. On August 3,
                             2021, after Applicant Hand 2 Hand had
                             answered, Topco filed a motion to amend and
                             supplement its notice of opposition to add to
                             its Section 2(d) claim by pleading two
                             applications filed after this opposition was
                             commenced (March 4, 2021), and also pleading
                             common law rights, in the mark READY FOR LIFE
                             for “[a]nti-bacterial hand wipes for personal
                             use” and for “disposable sanitizing personal
                             wipes,” claiming use of its mark since 2017.
                             The Board granted the motion. Topco Holdings,
                             Inc.v. Hand 2 Hand Industries, LLC, 2020
                             USPQ2d 54 (TTAB 2022) (Order by Interlocutory
                             Attorney Katie Bukrinsky).


                             [...]


                             “Affirmative Defenses”: Finally, The Board
                             sua sponte struck the applicant’s
                             “affirmative defense” of failure to state a
                             claim because is “not a true affirmative
                             defense.” See John W. Carson Found. v.
                             Toilets.com, Inc., 94 USPQ2d 1942, 1949 (TTAB
                             2010). It allowed several other purported
                             affirmative defenses to remain in the Answer,
                             although they were merely amplifications of
                             the applicant’s denials. See, e.g., ProMark
                             Brands Inc. v. GFA Brands, Inc., 114 USPQ2d
                             1232, 1236 n.11 (TTAB 2015). And the Board
                             struck as improper the applicant’s claimed
                             “reservation of right” to assert additional
                             affirmative defenses, since such addition
                             would require a motion for leave to amend.
                             See Philanthropist.com, Inc. v. The Gen.
                             Conf. Corp. of Seventh-Day Adventists, 2021
                             USPQ2d 643, at *4 n.6 (TTAB 2021).


                      # ⚓ TTABlog_Test:_Is_FLOREO_(Spanish)_Confusable_With
                        FLOURISH_HOMES_&_Design_for_Real_Estate_Services? [Ed:
                        Dictating monopolies on words from dictionaries is a
                        slippery slope for trademark law]⠀⇛


                             The USPTO refused to register the mark FLOREO
                             for “real estate management services for
                             properties owned by third parties” and for
                             “reservation of temporary accommodations for
                             vacation real estate properties that are
                             owned by third parties,” finding confusion
                             likely with the registered mark FLOURISH
                             HOMES & Design (shown below) for “Providing
                             temporary housing accommodations” [HOMES
                             disclaimed]. The Board found that the
                             services are either related or overlapping.
                             But what about the marks? Applicant Unsalted
                             argued that the marks differ in sound and
                             appearance, and that the Spanish word
                             “floreo” has many translations and meanings
                             and in any case has not been significantly
                             used in almost 200 years. How do you think
                             this came out? In re Unsalted Vacations LLC,
                             Serial No. 90312784 (January 12, 2022) [not
                             precedential] (Opinion by Judge Christoper
                             Larkin).


                             [...]


                             The Board acknowledged that “flourish” is not
                             the only translation of “floreo,” but “it is
                             the translation that Applicant alludes to in
                             its specimen.” The Board found that “the
                             existence of meanings of ‘floreo’ in addition
                             to ‘flourish’ does not preclude the
                             application of the doctrine of foreign
                             equivalents to Applicant’s mark because
                             ‘flourish’ is the literal and direct
                             translation most likely to be associated with
                             it.”


                      # ⚓ Düsseldorf_Higher_Regional_Court_asks_CJEU_regarding
                        the_role_of_alternative_designs_in_establishing
                        functionality⠀⇛


                             While this Kat was inquiring about the role
                             of alternative designs in examination of Art.
                             8(1) Regulation 6/2002, the Düsseldorf Higher
                             Regional Court (OLG Düsseldorf) sent a
                             referral to the CJEU on just this issue (C-
                             684/21). The case goes as follows.


                             The claimant in the national case,
                             Papierfabriek Doetinchem, is the owner of
                             Community Design No 001344022-0006 for a
                             “packaging device” i.e., a holder for paper
                             rolls (see the image to the right). The
                             claimant filed an infringement suit against a
                             competitor, Sprick GmbH & Ho, in the
                             Düsseldorf District court (LG Düsseldorf).
                             The competitor, filed a counterclaim, seeking
                             a declaration that the claimant’s Community
                             design was invalid, on the ground that it is
                             dictated by its technical function per Art. 8
                             (1) Regulation 6/2002.


                      # ⚓ Turkish_Patent_And_Trademark_Office_Updates_The
                        Guidelines_Relating_To_Trademark_Examination⠀⇛


                             On August 18, 2021, TPTO published and
                             updated guidelines pertaining to examination
                             of trademark applications laying down
                             principles/standards to be followed for
                             assessment of confusing similarity between
                             rival marks including evaluation of the
                             likelihood of confusion as per Article 6(1)
                             of IP code.


                      # ⚓ Around_the_IP_Blogs⠀⇛


                             Following a public consultation that took
                             place between June and August 2021, the UKIPO
                             recently announced that the government is not
                             proposing to change the current IPRs
                             exhaustion regime in the UK. MARQUES has
                             provided further insight and commentary on
                             this announcement.


                      # ⚓ UK’s_future_exhaustion_of_intellectual_property
                        rights_regime:_Summary_of_responses_to_the_consultation
                        [Ed: It's not property and these are not rights; the
                        British government lies right from the headline...]⠀⇛


                             In June 2021 the UK government launched a
                             public consultation on the UK’s future
                             exhaustion of intellectual property (IP)
                             rights regime.


                             The system of exhaustion of rights means that
                             distributors and other traders are able to
                             move goods (including component parts that
                             make up goods) around a specified territory
                             without the rights holder’s permission. This
                             supports a market of secondary sales of
                             legitimate goods, also known as parallel
                             trade.


                             Now that the UK has left the EU, the UK has
                             an opportunity to decide its future regime
                             for the exhaustion of IP rights. This
                             decision is vitally important for the UK as
                             it will govern future rules on parallel
                             imports into the UK.


                             The purpose of the consultation was to seek
                             evidence and views from respondents to
                             understand what the most appropriate
                             exhaustion of IP rights regime would be for
                             the UK. There were 4 options tested through
                             consultation, with evidence and views
                             required on what regime should be
                             implemented, and if there were to be a
                             change, how a new regime should be
                             implemented.


                      # ⚓ UK_to_retain_current_exhaustion_regime_for_now [Ed:
                        That’s not because UKIPO cares about the actual
                        interests or needs of the British population; it is a
                        force_of_class_occupation]⠀⇛


                             The UK IPO has announced that the government
                             is not proposing to change the current
                             exhaustion regime, following a consultation
                             held last year.


                      # ⚓ EU_Trademark_Body_Hands_Louis_Vuitton_a_Loss_in_Fight
                        Over_Lookalike_Mark [Ed: Bernard Arnault thinks_he
                        ‘owns’_the_whole_world_and_its_culture, too. Rich
                        people don’t keep getting richer because they’re smart
                        or “successful” but because they write/pass/buy the
                        laws. In other words, they write all the rules in their
                        own favour.]⠀⇛


                             Louis Vuitton has been handed a loss in its
                             attempt to block the registration of a
                             trademark that it claims looks a bit too much
                             like one of its most well-known indicators of
                             source. On the heels of initiating an
                             opposition proceeding with the European Union
                             Intellectual Property Office (“EUIPO”) in
                             November on the basis that a mark that
                             consists of an interlocking “N” and “L” is
                             likely to confuse consumers about the source
                             of the products upon which it appears
                             (primarily, clothing, as well as belt clasps,
                             zippers, and closures/buttons for clothing),
                             the EUIPO’s Opposition Division has sided
                             with the filing party, an unaffiliated
                             individual named Aina Yang.


                      # ⚓ What_Indian_startups_want_from_IP_counsel,_and_how_to
                        help_them [Ed: This cult of "IP" has already
                        slaughtered hundreds of thousands of Indians, who died
                        because of the patent cartels amid pandemic; and here
                        come charlatans salaried by the patent profiteers
                        (Sukanya Sarkar) cheering for those who hurt the
                        country]⠀⇛


                             Counsel from four tech-based startups in
                             India discuss how IP is perceived in-house,
                             its role in raising funding and why
                             businesses must scale up IP capabilities


                      # ⚓ Top_IP_trends_in_LATAM’s_biggest_economies_revealed:
                        counsel_|_Managing_Intellectual_Property⠀⇛


                             Brazil’s backlogs are getting better,
                             Mexico’s IP office is preparing to decide
                             damages, and Chile, Argentina and Colombia
                             are going through other shakeups


                      # ⚓ UK:_Pharmaceutical_Trade_Marks_–_Top_Tips⠀⇛


                             Pharmaceutical products can often attract
                             patent protection. This affords the owner a
                             monopoly in that product; however, the
                             monopoly will only last a fixed period, after
                             which it is open season for competitors.
                             Choosing a strong, resilient brand name will
                             help foster brand loyalty and encourage
                             customers to continue purchasing your product
                             even when other (potentially cheaper)
                             products have become available. Obtaining a
                             trade mark registration for your brand name
                             allows you to stop other businesses cashing
                             in on your brand loyalty/success. It also
                             affords protection against inferior products
                             tainting your hard-earned brand recognition.


                             Pharmaceutical trade marks require regulatory
                             approval. In the UK, the appropriate body is
                             the Medicines and Healthcare Products
                             Regulatory Agency (MHRA). The MHRA will
                             reject a proposed pharmaceutical brand name
                             if they believe that: (i) consumers will
                             confuse the chosen name with the name of an
                             existing medicine; (ii) the chosen name is
                             misleading; or (iii) the chosen name is
                             considered unsafe. Regulatory approval is
                             required before a pharmaceutical product can
                             be sold under the chosen name.


                      # ⚓ The_Danger_Of_Using_Your_Name_As_A_Trade_Mark_–
                        Intellectual_Property_–_UK⠀⇛


                             Registering your own name as a trade mark for
                             your business may seem a good way to build up
                             a reputation in your chosen industry.
                             However, business owners should also be aware
                             of the potential danger of doing so.


                             If a business owner registers their own name
                             as a trade mark and eventually wants to, or
                             needs to sell that business, it is likely
                             this will involve selling the trade mark
                             rights in their name. This is because much of
                             the value of a business is often in the name
                             itself, particularly in the creative
                             industries, such as fashion design. As a
                             result, ex-owners may be prevented from
                             conducting business under their own name in
                             the future, as this may amount to trade mark
                             infringement.


                # § Copyrights⠀➾


                      # ⚓ Digital_Services_Act:_regulating_platforms_for_a
                        safer_online_space_for_users [Ed: This has nothing at
                        all to do with safety; it's about safeguarding the
                        wealth of the super-rich, ensuring they just get richer
                        and richer over time, at everybody else's expense]⠀⇛


                             MEPs agreed a draft set of measures to tackle
                             illegal content, to ensure platforms are held
                             accountable for their algorithms, and improve
                             content moderation.
                             The text approved today by Parliament with
                             530 votes to 78, with 80 abstentions, will be
                             used as the mandate to negotiate with the
                             French presidency of the Council,
                             representing member states.


                             After the vote, Christel Schaldemose (S&D,
                             DK), who is leading the Parliament’s
                             negotiating team, said: “Today’s vote shows
                             MEPs and EU citizens want an ambitious
                             digital regulation fit for the future. Much
                             has changed in the 20 years since we adopted
                             the e-commerce directive. Online platforms
                             have become increasingly important in our
                             daily life, bringing new opportunities, but
                             also new risks. It is our duty to make sure
                             that what is illegal offline is illegal
                             online. We need to ensure that we put in
                             place digital rules to the benefit of
                             consumers and citizens. Now we can enter into
                             negotiations with the Council, and I believe
                             we will be able to deliver on these issues”


                      # ⚓ European_Copyright_Society_(ECS):_Comment_on
                        Copyright_and_the_Digital_Services_Act_Proposal⠀⇛


                             On 15 December 2020, the European Commission
                             submitted a proposal for a Regulation on a
                             Single Market For Digital Services (Digital
                             Services Act, DSA) and amending Directive
                             2000/31/EC. In November 2021, the Council of
                             the European Union reached agreement on an
                             amended version of this proposal, and on 20
                             December 2021, the European Parliament’s
                             Committee on the Internal Market and Consumer
                             Protection (IMCO) released a draft for an EP
                             legislative resolution. The legislative
                             project “seeks to ensure the best conditions
                             for the provision of innovative digital
                             services in the internal market, to
                             contribute to online safety and the
                             protection of fundamental rights, and to set
                             a robust and durable governance structure for
                             the effective supervision of providers of
                             intermediary services”. To achieve these
                             aims, the DSA sets out numerous due diligence
                             obligations of intermediaries concerning any
                             type of illegal information, including
                             copyright infringing content.


                             Copyright enforcement online is a major issue
                             in the context of the DSA, and the DSA will
                             be of utmost importance for the future of
                             online copyright in the EU. Against this
                             background, the European Copyright Society
                             (ECS) issued on 17 January 2021 a comment on
                             “Copyright and the Digital Services Act”. We
                             reproduce below the Executive Summary of that
                             comment.


                      # ⚓ YouTube’s_first_Copyright_Transparency_Report_2021_–
                        A_step_towards_“factfulness”⠀⇛


                             In 2012, US NASA’s robotic rover touched down
                             on Mars. NASA uploaded a video of this
                             historic event onto YouTube. It was taken
                             down after one hour, as it was subject to a
                             copyright notice by a news channel relying on
                             the US DMCA. This news channel had used the
                             (open source) NASA video for its own news
                             video and ended up raising unjustified claims
                             against other copies on YouTube, including
                             NASA’s original video. A commentator noted
                             that “NASA’s new rover arrives on Mars, crash
                             lands in YouTube’s DMCA hell”. Another – more
                             recent – example involves the video of a 2020
                             copyright panel at New York University where
                             music was played to explain the application
                             of the US fair use rules. The video was
                             caught by a takedown – but ended up being put
                             back online on YouTube.


                             These and other well-known examples tend to
                             create the public impression that abusive or
                             otherwise unjustified copyright takedown
                             requests are the general rule on YouTube. But
                             is this really true? To answer this question,
                             it seems important not to rely on
                             illustrative single stories. “We are
                             naturally drawn to extreme examples, and they
                             are so easy to recall”, says Hans Rosling in
                             his book Factfulness (2018, p. 42). Rather,
                             it is necessary “to learn about the world as
                             it really is”, to quote Rosling again (p.
                             255).


                      # ⚓ Not_Invincible:_A_Cautionary_Tale_for_Creators⠀⇛


                             As a new lawsuit involving the popular comic
                             book and animated series “Invincible” shows,
                             the failure to properly document the
                             copyright status of a jointly-created work at
                             the beginning can lead to messy consequences
                             later.


                             Can someone be tricked into giving up
                             copyrights he may never have owned in the
                             first place? That’s just one of the
                             interesting questions raised by a new lawsuit
                             filed by William Crabtree, colorist of the
                             first 50 issues of the comic book series
                             “Invincible,” against the series’ writer and
                             co-creator Robert Kirkman.


                      # ⚓ Whoever_controls_the_DAO,_controls_the_book:_The
                        strange_tale_of_Jodorowsky’s_Dune⠀⇛


                             Please gather around, and listen to this
                             cautionary tale of schadenfreude and greed.
                             This is not a tale for the faint of heart; it
                             involves a cabal of crypto-enthusiasts
                             involved in a decentralised autonomous
                             organisation (DAO), a failed film adaptation
                             by Alejandro Jodorowsky of the classic book
                             Dune , and industrial amounts of
                             misunderstanding about copyright.


                             Frank Herbert’s Dune is a science fiction
                             classic that speaks to audiences across
                             cultures and time, it was published in 1965
                             and became an instant hit, winning the Nebula
                             and Hugo Awards in 1966, and it’s been
                             declared the best selling sci-fi novel of all
                             time. The intricate world building, the
                             political intrigue, the memorable characters,
                             the worms, and the ecological angle have made
                             it a fan favourite. Personally, I have loved
                             the book since I was a teenager, and I have
                             used the litany against fear in some of the
                             most stressful times of my life. It’s been
                             put on screen by none other than David Lynch
                             in a flawed masterpiece in 1984, then it was
                             turned into a miniseries in 2000, and more
                             recently it became a global hit in 2021
                             directed by Denis Villeneuve. It has spawned
                             several books, Frank Herbert wrote a number
                             of sequels, and his son Brian Herbert
                             published a much derided series of prequels
                             with author Kevin J Anderson (which I have
                             tried to remove from my memory, but I
                             digress).


                      # ⚓ Book_review:_Performers’_Rights [Ed: Richard_Arnold
                        writes a book]⠀⇛


                             This is a review of Performer’s Rights by the
                             Rt Hon Sir Richard Arnold, Judge of the Court
                             of Appeal of England and Wales. The book
                             covers the rights of performers, particularly
                             the rights conferred under Part II of the
                             Copyright, Designs and Patents Act 1988 (as
                             amended). Recognising the challenges for
                             performers in the position and enforcement of
                             their rights, the book also includes
                             discussion on other forms of protections and
                             remedies for performers, such as moral
                             rights, contracts, passing off and copyright
                             infringement.



                    ䷩ 𝚕𝚒𝚗𝚎 10370

╒═══════════════════ 𝐃𝐀𝐈𝐋𝐘 𝐋𝐈𝐍𝐊𝐒 ═════════════════════════════════════════════╕

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 01.23.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Links_23/1/2022:_First_RC_of_Linux_5.17_and_Sway_1.7_Released⠀✐


Posted in News_Roundup at 8:12 am by Dr. Roy Schestowitz


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇GNOME bluefish⦈


§ Contents⠀➾


    * GNU/Linux
          o Distributions
          o Devices/Embedded
    * Free_Software/Open_Source
    * Leftovers
    * § GNU/Linux⠀➾


          o § Desktop/Laptop⠀➾


                # ⚓ Analysts_Weigh_In:_Will_We_Ever_See_the_Year_of_the_Linux
                  Desktop? [Ed: What Slashdot means by “analyst” is Microsoft
                  pundits noted in Microsoft sites; the likes of Forrester are
                  paid_by_Microsoft_to_give_fake_information]⠀⇛


                       How popular is Linux? The Windows Central site
                       admits Linux is starting to tempt them. “It made
                       such an imprint on Windows Central that not all of
                       us even bother much with Windows anymore.”


          o § Server⠀➾


                # ⚓ GNU_Linux_hardware_reviews_–_something_between_tablet_and
                  notebook:_the_Lenovo_Ideapad_MIIX_510_12ISK_–_Can_it_Debian?
                  –_yes_it_can_(Debian_11_out_of_the_box_Debian_10_rfkill
                  reports_hardblocked_wlan)⠀⇛


          o § Audiocasts/Shows⠀➾


                # ⚓ This_Week_in_Linux_182:_Exciting_FlatHub,_Wine_7.0,_SUSE
                  Liberty_Linux,_Linux_Mint,_AppIndicators_–_TuxDigital⠀⇛


                       On this episode of This Week in Linux, Exciting
                       FlatHub Updates In Store, New AppIndicator
                       Protocol?, SUSE Announces SUSE Liberty Linux,
                       Framework to Open Source Firmware, Linux Mint 20.3
                       Edge ISO, in App News: Wine 7.0, FFMPEG 5.0,
                       Qprompt 1.0, Flameshot 11.0, Czkawka 4.0, and
                       Mumble 1.4. All that and much more on Your Weekly
                       Source for Linux GNews!


          o § Kernel Space⠀➾


                # ⚓ Linux_5.17-rc1⠀⇛


                       I've tagged the rc1 release a couple of hours
                       earlier than usual, and
                       in a timezone 10 hours before the usual one, so
                       this merge window was
                       technically a bit shorter than usual. But if
                       somebody didn't get their
                       pull request in in time, they shouldn't have left
                       it so late - and
                       there's always 5.18. Never fear - we'll not run out
                       of numbers.

                       I was nervous that this merge window would be more
                       painful than usual
                       due to my family-related travels, but I have to
                       give thanks to people:
                       a lot of you sent your pull requests early in the
                       merge window, and
                       while there were a couple of hiccups I hit early
                       on, that was all
                       before switching my main workstation to a laptop.
                       Everything seems to
                       have gone fairly smoothly.

                       Knock wood.

                       5.17 doesn't seem to be slated to be a huge
                       release, and everything
                       looks fairly normal. We've got a bit more activity
                       than usual in a
                       couple of corners of the kernel (random number
                       generator and the
                       fscache rewrite stand out), but even with those
                       things, the big
                       picture view looks very much normal: the bulk is
                       various driver
                       updates, with architectures updates, documentation,
                       and tooling being
                       the bulk of the rest. Even with a total rewrite,
                       that fscache diff
                       looks more like a blip in the big picture.

                       And hey, it may not be a huge release, but the full
                       shortlog would
                       still be much too big to post - or scan through. So
                       as is traditional,
                       I'm just appending my mergelog as a highlevel view
                       of what's been
                       going on.

                       Please give it all a test,

                                         Linus
                # ⚓ Linux_5.17-rc1_Released_A_Little_Bit_Early_But_With_Shiny
                  New_Features⠀⇛


                       Linux 5.17 also bring a lot of new hardware
                       support, prompt support for Qualcomm’s latest SoC,
                       a new real-time analysis tool, x86 straight line
                       mitigation handling, noteworthy network
                       optimizations, and much more. I’ll have out my
                       usual Linux 5.17 feature overview later today.


                       I’ve already been running benchmarks of Linux 5.17
                       Git the past few days and it’s been looking good so
                       far with no scary regressions detected yet. More
                       Linux 5.17 benchmarks on the way.


                # ⚓ AMD_Preps_for_Zen_4:_Different_Types_of_Cores_Now_Supported
                  in_Linux_|_Tom’s_Hardware⠀⇛


                       Tech sleuths are following AMD as it prepares its
                       new Zen 4-based architecture. As the prepares its
                       next-gen CPU, some eagle-eyed individuals have
                       found details about the next-gen parts on Linux and
                       other platforms.


                       AMD has quietly uploaded temperature sensor driver
                       support for Zen 4 and Zen 4C cores, reports
                       Phoronix. While these two cores share the same
                       microarchitecture, they are different and will
                       power AMD’s 96-core Genoa and 128-core Bergamo
                       processors, respectively, so it is not surprising
                       to see separate drivers. The CPUs are marked as AMD
                       Family 19h Models 10h-1Fh and A0h-AFh.


                       Perhaps, a more intriguing innovation is a new
                       Scalable Machine Check Architecture (SMCA) of some
                       future AMD platforms that could use different types
                       of SMCA and therefore cores.


                # § Graphics Stack⠀➾


                      # ⚓ Sway_1.7_Released_With_VR_Headset_DRM_Leasing,
                        Renames_“–my-next-gpu-wont-be-nvidia”⠀⇛


                             After working its way through the release
                             candidate process, Sway 1.7 is out this
                             weekend as the newest feature release for
                             this i3-inspired, lightweight Wayland
                             compositor.


                             Sway 1.7 is notable in that it offers better
                             zero-copy direct scanout support thanks to
                             making use of the DMA-BUF feedback protocol.
                             Other Wayland desktop compositors have also
                             been quick in supporting the DMA-BUF feedback
                             extension as well. The enhanced zero-copy
                             direct scanout support for full-screen
                             windows is easily one of the best features of
                             Sway 1.7.


          o § Applications⠀➾


                # ⚓ Cloud_Hypervisor_21.0_Offers_More_Efficient_Local_Live
                  Migration:_3s_Down_To_50ms_–_Phoronix⠀⇛


                       Cloud-Hypervisor 21.0 was released this past week
                       as its first feature release since this open-source
                       Intel project moved to the Linux Foundation with
                       backing from Microsoft and Arm. Cloud-Hypervisor
                       21.0 brings new features and fixes to this Rust-
                       written hypervisor.


                # ⚓ How_I_use_Linux_accessibility_settings_|_Opensource.com⠀⇛


                       When I started using Linux in the 1990s, I was in
                       my mid-40s and accessibility was not something I
                       gave much thought to. Now, however, as I’m pushing
                       70, my needs have changed. A few years ago, I
                       purchased a brand new Darter Pro from System76, and
                       its default resolution is 1920×1080, and it’s high
                       DPI, too. The system came with Pop_OS!, which I
                       found that I had to modify to be able to see the
                       icons and text on the display. Thank goodness that
                       Linux on the desktop has become much more
                       accessible than in the 1990s.


                       I need assistive technology for seeing and hearing
                       in particular. There are other areas that I do not
                       use but are useful to folks who need help typing,
                       pointing, clicking, and gesturing.


                       Various systems, like Gnome, KDE, LXDE, XFCE, and
                       others, handle these assistive technologies
                       differently. These assistive tweaks are mostly
                       available through the Settings dialog box or from
                       keyboard shortcuts.


                # ⚓ Best_Free_and_Open_Source_Alternatives_to_Base_SAS⠀⇛


                       AS Institute Inc. (“SAS”) is an American
                       multinational developer of analytics software based
                       in Cary, North Carolina. The company has around
                       14,000 employees.


                       SAS started as a project at North Carolina State
                       University to create a statistical analysis system
                       used mainly by agricultural departments at
                       universities in the late 1960s.


                       SAS is the name of their software suite that can
                       mine, alter, manage and retrieve data from a
                       variety of sources and perform statistical analysis
                       on it. It has more than 200 components covering
                       areas including statistical analysis, econometrics
                       and time series analysis, an interactive matrix
                       language, data mining and much more.


          o § Instructionals/Technical⠀➾


                # ⚓ How_to_create,_show_and_drop_Collections_in_MongoDB_|_FOSS
                  Linux⠀⇛


                       MongoDB is an open-source NoSQL database which
                       means that, unlike relational databases, it does
                       not accept input values in table format. Data is
                       stored in collections and documents since MongoDB
                       is a document-oriented database. Rows in an SQL
                       table have been replaced with documents in MongoDB.


                       This article assumes that you’ve already installed
                       the MongoDB server on your computer and connected a
                       shell to the server. If you have already done so,
                       we can explore a few features of MongoDB but first,
                       a few terminologies: If not, you can check out the
                       article on how to install MongoDB on Ubuntu.


                # ⚓ How_to_add_missing_ifconfig_command_on_Debian_|_FOSS
                  Linux⠀⇛


                       In this article, we will be tackle how to add the
                       missing ifconfig command on Debian. We will run all
                       this on Debian version 11, “bullseye.” This Debian
                       version ships with the new package ipp-usb,
                       recommended by cups-daemon, and utilizes the
                       vendor-neutral IPP-over-USB protocol reinforced by
                       multiple modern printers. This, in turn, allows a
                       USB device to be treated as a network device,
                       expanding driverless printing to include USB-
                       connected printers.


                       The ifconfig (interface configuration) command is a
                       vital utility to configure the kernel-resident
                       network interfaces. It is used to initialize the
                       interfaces as required during the boot time. On the
                       flip side, it is also used when debugging or when
                       you need system tuning. Furthermore, this command
                       is used to assign the IP address and netmask to an
                       interface or enable or disable a given interface.
                       The command is available under the net-tools
                       package.


                # ⚓ What_are_Upstream_and_Downstream_in_Linux_Terminology?⠀⇛


                       The terms: upstream and downstream are rather
                       ambiguous terms and, I think, not really used by
                       the general public. If you are a Linux user and do
                       not write or maintain software, chances are pretty
                       good that these terms will mean nothing to you, but
                       they can be instructive in how communication
                       between groups within the Linux world works.


                       The terms are used in networking, programming,
                       kernel, and even in non-computer areas such as
                       supply chains. When we talk about upstream and
                       downstream then, context is important.


                       In its simplest form, upstream and downstream is
                       the direction of the flow of information.


                       Since we are all reading this article while we’re
                       connected to the Internet, let’s look at an
                       upstream/downstream example as it applies to
                       Internet Service Providers (ISP). Here, the ISP is
                       concerned with traffic. Upstream traffic is data is
                       coming in from a user from a different ISP. For
                       example, if you have a website that offers a
                       subscription to a newsletter, the information I
                       send, to subscribe, is upstream data.


                # ⚓ How_to_create_desktop_shortcut_for_an_AppImage_–_Linux
                  Shout⠀⇛


                       Have you downloaded any AppImage and now want to
                       create a Desktop entry? Then here is the command
                       line tutorial to create a Desktop shortcut and
                       Application launcher for an AppImage of some
                       software.


                       AppImage desktop integration is not going to happen
                       automatically. It is because this is a kind of
                       portable executable file meant to run on almost all
                       Linux distros. Hence, using developers don’t need
                       to write a single software, again and again, to
                       make it compatible with various Linux systems.


                # ⚓ 5_Great_Kali_Linux_Books_–_buildVirtual⠀⇛


                       If you’re new to Kali Linux, or looking to pick up
                       some more skills then there are some great Kali
                       Linux books available. The aim of this page is to
                       show you some of the best Kali Linux books
                       currently available to help you on your journey to
                       become a Kali Linux expert! If you are looking to
                       learn Kali Linux then each Kali Linux book listed
                       here should help!


                # ⚓ Running_Ubuntu_on_Raspberry_Pi_4?_Here’s_How_to_Boost
                  Performance⠀⇛


                       Raspberry Pi 4 is a tiny device packed with useful
                       features like support for dual monitors and fast
                       ethernet connection. If you are looking for a
                       simple yet reliable solution for your next DIY
                       project, a Raspberry Pi can be an excellent base
                       for that. Moreover, you can now easily run Ubuntu
                       desktop on your Raspberry Pi 4, even with 2GB of
                       RAM.


                       Read below to learn how to improve Ubuntu’s
                       performance on a Raspberry Pi 4 with only 2GB RAM.
                       You can use this method to set up a full-fledged
                       Ubuntu environment with access to a keyboard,
                       mouse, and networking.


                # ⚓ How_to_replace_libinput_with_evdev_for_sane_mouse_control_|
                  Hund⠀⇛


                       I try to be pragmatic and open to new things, but
                       some things are just hard to embrace. Especially
                       when they’re bad. One of these things are libinput.


                       I tried giving libinput a chance, but I couldn’t
                       come close to finding some sane mouse settings for
                       it. I strongly dislike silly things like mouse
                       acceleration. If I want to quickly move my mouse
                       cursors across the screen, I’ll just quickly move
                       my arm…


                       Thankfully, it’s possible to go back to evdev, and
                       it’s not that difficult either. You just have to
                       install the package for it. In Fedora the package
                       is named xorg-x11-drv-evdev.


          o § Wine or Emulation⠀➾


                # ⚓ Enabling_EAC_Support_on_Linux_Through_Proton_is_Now_Even
                  Easier_–_Boiling_Steam⠀⇛


                       Going from this explanation, this is the probable
                       reason why there’s been so few titles with EAC to
                       work through Proton so far. But yesterday, Valve
                       wrote a new post regarding EAC. The post said that
                       the company has been working with Epic to make
                       supporting EAC titles on the Steam Deck or through
                       Proton even easier. Game binaries no longer have to
                       be updated, neither does the developer need to opt-
                       in to the latest SDK or use Epic Online Services.


          o § Desktop Environments/WMs⠀➾


                # § GNOME Desktop/GTK⠀➾


                      # ⚓ 10_Great_Apps_to_Improve_Your_GNOME_Experience_[Part
                        3]⠀⇛


                             We present the next set of great GNOME Apps
                             that brings multitude of productivity boost
                             while using your favorite GNOME desktop.


                      # ⚓ Daniel_García_Moreno:_Twitch:_GNOME_live_coding
                        streaming⠀⇛


                             This year I’ve started with something that I
                             wanted to do since the first COVID lockdown,
                             but never did. So as a first year resolution
                             I decided to start streaming and I created my
                             Twitch channel Abentogil.


                             I’ve been thinking about streaming since
                             lockdown, when a group of Spanish people
                             created a nice initiative to teach kids how
                             to code and other tech stuff. I never
                             participated on that initiative, but it was
                             the seed of this.


                             This year I’ve seen other GNOME streamers
                             doing live coding, like ebassi and Georges
                             and, at the end, I pluck up the courage to
                             start.


          o § Distributions⠀➾


                # § IBM/Red Hat/Fedora⠀➾


                      # ⚓ Vanessa’s_Blog_|_Week7_Blog_Post⠀⇛


                             Week 1-2: Learn the Fedora package maintainer
                             process, and improve existing packages via
                             pull requests to practice the workflow.


                             Week 3-5: Go over the NeuroFedora packaging
                             queue and identify a list of tools for
                             packaging


                             Week 6-9: Follow the Fedora package
                             maintaining process, and submit packages for
                             review.


                             Week 10: Follow the review process and make
                             necessary improvements to submitted packages.


                             Week 12: Submit packages for QA, test them,
                             update documentation.


                # § Canonical/Ubuntu Family⠀➾


                      # ⚓ Ubuntu_is_the_most_popular_distribution_of_2021⠀⇛


                             We finished with the review of our end-of-
                             year survey with the result related to
                             distributions, the cornerstone of this world
                             in which we live, although as expected,
                             Ubuntu is crowned one more year as the
                             average user’s favorite. No surprises, right?
                             Not completely, as we will see below delving
                             a little into the data.


                             In fact, there are some other surprises in
                             the table and more specifically, in the top
                             positions; and even in the place of honor
                             occupied by Ubuntu since we did this survey,
                             there are changes that are worth commenting
                             on, because as is the case with Firefox, the
                             parish’s unshakable totem, there are trends
                             that are not declining, but quite the
                             opposite.


          o § Devices/Embedded⠀➾


                # ⚓ Experimental_PineNote_E-Reader_Now_Available_for_$399⠀⇛


                       The 10.3-inch PineNote e-reader is now available to
                       buy, but this isn’t a device ready to compete with
                       the Kindle just yet.


                       Last year, Pine64 announced it was developing an e-
                       reader that runs Linux called the PineNote. As
                       Liliputing reports, an experimental version of the
                       PineNote has now gone on sale for $399, but it’s
                       called “experimental” and the Developer Edition for
                       a good reason.


                       A note on the PineNote product page states, “The
                       PineNote is an experimental device. PineNote
                       software is still in it’s infancy and therefore it
                       is ONLY suitable for experienced developers. At
                       present time, there is no default OS for the
                       PineNote.”


                # ⚓ PineNote_E_Ink_tablet_now_on_sale_for_all_even_as
                  developers_continue_refining_the_product⠀⇛


                       The PineNote tablet is now on sale, which means
                       anyone having $399 to spare, can order the E Ink
                       tablet right away. The company known for its
                       PinePhone and PineBook devices had put on sale
                       limited quantities of the PineNote device in
                       December though that was limited to the developers
                       only. However, the e-note is now available for all
                       to buy, and barring the Chinese New Year
                       celebrations in February, there isn’t likely to be
                       any disruption in supplies in the foreseeable
                       future, the company revealed via its Jan. 2022
                       update.


                       That said, the PineNote is still very much a work-
                       in-progress thing so that the device right now is
                       shipping without any operating system installed.
                       There is going to be just the bootloader, which
                       means buyers have the liberty to install the OS of
                       their choice. Here again, there aren’t very many
                       options as well though developers have been able to
                       run the Alpine and Debian Linux to run on the
                       tablet. A NixOS port too is being readied and
                       should be available sometime soon.


                # § Open Hardware/Modding⠀➾


                      # ⚓ Web-Centric_Gabuino_Has_Compiler,_Will_Travel_|
                        Hackaday⠀⇛


                             Arguably the biggest advantage of the Arduino
                             platform is its ease of use, especially when
                             compared to what microcontroller development
                             looked like before the introduction of the
                             open source board and its associated software
                             development environment. All you need to do
                             is download the IDE for your platform, plug
                             in your Arduino, and you can have code
                             running on the hardware with just a few
                             clicks.


                             But can it get even easier? [Gabriel Valky]
                             certainly thinks so, which is why he’s
                             developed the cloud-based Gabuino platform.
                             As of right now it only supports the DS213
                             pocket oscilloscope and LA104 logic analyzer,
                             but he says the code is lightweight enough
                             that it should work with any STM32 board that
                             has the appropriate bootloader. Using Gabuino
                             requires no software to be installed on the
                             computer, just plug in the board, and you’re
                             already half way there.


                      # ⚓ Printing_In_Silicone_|_Hackaday⠀⇛


                             When you think of making something out of
                             silicone, you usually think of using a mold
                             and injecting it with the material. Can you
                             3D print it? [Kimberly Beckett] answers that
                             very question in a recent post. The short
                             answer is yes, but you need specialized
                             printing equipment.


                             Most consumer or hobby printers use either
                             filament deposition or photoresin. Neither of
                             these processes are good for printing
                             silicone. For one thing, silicone doesn’t
                             melt and reform like a thermoplastic. After
                             all, that is why we like making hotend socks
                             and oven utensils with the material. If you
                             do melt silicone, you get a gooey mess, not a
                             nice fluid you can push through an extruder
                             nozzle. As for resin printing, silicone is
                             resistant to UV so the chances of coming up
                             with UV curable silicone are pretty small.


                             So how do you print silicone? There are a few
                             methods. Aceo is a technique that is sort of
                             like an inkjet. It deposits a solution of
                             silicone and a binder that activates on
                             exposure to UV. After placing a layer, a UV
                             light activates the binder and you repeat for
                             the next layer. There is also a technique for
                             drawing a layer of silicone liquid and then
                             curing it with a halogen lamp.


                      # ⚓ Another_DPS5005_Alternative_Firmware_|_Hackaday⠀⇛


                             The firmware has some interesting features,
                             such as programmable pre-sets intended for
                             battery charging applications. In fact, there
                             is a dedicated battery charge mode screen. We
                             want to warn, however, that charging lithium
                             ion batteries with this might not be at all
                             wise, not in the least because of a lack of
                             protection hardware in place. It would be
                             very easy to destroy the unit or overheat a
                             battery this way! However, if you must do
                             this, there are a few features to help you
                             out, such as a handy ‘counters’ screen
                             showing approximate charge delivered.


                             Remote programmability is, as usual, via the
                             easily hacked in serial port, with firmware
                             support for Bluetooth serial modules if wired
                             USB serial doesn’t suit. For those who like
                             to mount things differently, the screen can
                             be rotated by holding a key on power-up, or
                             if you hook up a MPU6050 accelerometer/gyro
                             module it will even do it automatically!


                      # ⚓ Using_4G_LTE_wireless_modems_on_a_Raspberry_Pi⠀⇛


                             There doesn’t seem to be any good central
                             resource for “4G LTE and Linux” out there,
                             just a thousand posts about the ABC’s of
                             getting an Internet connection working
                             through a 4G modem—but with precious little
                             explanation about why or how it works. (Or
                             why someone should care about random terms
                             like PPP, ECM, QMI, or MBIM, or why someone
                             would choose qmi_wwan over cdc_ether, or … I
                             could go on).


                             Hopefully you can learn something from my
                             notes. Or point out places where I’m
                             glaringly wrong 🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇:)⦈


          o § Free, Libre, and Open Source Software⠀➾


                # § Education⠀➾


                      # ⚓ Moodle:_Working_together_to_strengthen_education_–
                        our_obligation_to_future_generations⠀⇛


                             At Moodle, we support open education as
                             crucial to reimagining and renewing education
                             for our collective benefit. At its core, this
                             movement recognises that an empowered teacher
                             or educator requires access to quality
                             resources, skills and tools in order to
                             facilitate the education of a group of
                             people. This includes open methods in
                             accessing resources, an equitable approach in
                             how to use resources and tools effectively
                             and an open ed tech infrastructure that is
                             reliable and accessible to all. Moodle
                             supports the open education movement in three
                             key ways.


                # § Programming/Development⠀➾


                      # ⚓ Mold_1.0.2_Released_For_This_High_Performance_Linker
                        –_Phoronix⠀⇛


                             For those interested in compilers, Mold as
                             the “Modern Linker” is one of the interesting
                             projects to watch in 2022.


                             Mold 1.0 came at the end of 2021 for this
                             project developed by Rui Ueyama who is also
                             known as the original developer of LLVM’s LLD
                             linker. Mold 1.0 is considered stable and
                             production ready while delivering very
                             promising results compared to LLVM’s LLD or
                             GNU’s Gold linkers. Mold has been generating
                             interest among developers and now Mold 1.0.2
                             is out this Sunday with fixes and minor
                             updates.


                             While Mold is working towards native link-
                             time optimization (LTO) support, for the
                             moment Mold 1.0.2 will now fall-back to using
                             ld.bfd or ld.lld if GCC/LLVM LTO support is
                             requested. This at least will not break
                             builds and so should work out fine until
                             Mold’s LTO support is ready.


                      # ⚓ Wordle_With_Grep⠀⇛


                             i Let us solve a Wordle with the Unix grep
                             command and the Unix words file. The Wordle
                             game published on 22 Jan 2022 is solved in
                             this post. The output examples shown below
                             are obtained using the words file /usr/share/
                             dict/words, GNU grep 3.6, and GNU bash 5.1.4
                             on Debian GNU/Linux 11.2 (bullseye).


                             In this post, we will solve the Wordle in a
                             quick and dirty manner. We will not try to
                             find the most optimal strategy. The focus is
                             going to be on making constant progress and
                             reaching the solution quickly with simple
                             shell commands. The steps below show how to
                             solve a Wordle in this manner.


                      # ⚓ New_Professional_Certificate_Program_Teaches
                        Essentials_of_Open_Source_Software_Development⠀⇛


    * § Leftovers⠀➾


          o ⚓ Underwater_Drone_Films,_Is_In_Film_|_Hackaday⠀⇛


                 Having a drone that can follow you running or biking with
                 a camera isn’t big news these days. But French firm
                 Notilo Plus has an underwater drone that can follow and
                 video an underwater diver. The Seasam has been around
                 since 2019, but recently made an appearance in a French
                 film, The Deep House about a couple exploring an
                 underwater haunted house, as reported by New Atlas. You
                 can see a video about the drone — and a trailer for the
                 movie — in the videos below.


                 To follow a diver, the robot uses an acoustic signal from
                 the user’s control unit to find the approximate location
                 of the user. This works even in dark conditions. Once
                 close enough, computer vision zeros in on the diver while
                 a sonar system allows safe navigation.


          o § Science⠀➾


                # ⚓ A_proposal_for_improved_language_around_divisibility⠀⇛


          o § Education⠀➾


                # ⚓ Infosec_Skill_Sets⠀⇛


                       Everybody’s career path into and within
                       “Information Security” — about as broad and ill-
                       defined a field as any — is different. We all come
                       from different backgrounds with different
                       experiences, and even if our job titles may be the
                       same, in practice we are bound to perform different
                       tasks.[1]


                       But some skill sets tend to intersect. I’ve tried
                       to compile a list of primarily technical “core
                       competencies” before, but that only covered so-
                       called “hard skills”. “Soft skills”, on the other
                       hand, are actually a lot harder, but of course
                       there’s overlap here indicating a conceptual
                       difficulty in categorizing “technical” and “non-
                       technical” skills. Overlap, you say? To the Venn
                       Diagramminator!


          o § Hardware⠀➾


                # ⚓ Arm_releases_experimental_CHERI-enabled_Morello_board_as
                  part_of_£187M_UKRI_Digital_Security_by_Design_programme⠀⇛


                       Professor Robert N. M. Watson (Cambridge),
                       Professor Simon W. Moore (Cambridge), Professor
                       Peter Sewell (Cambridge), Dr Jonathan Woodruff
                       (Cambridge), Brooks Davis (SRI), and Dr Peter G.
                       Neumann (SRI)


                       After over a decade of research creating the CHERI
                       protection model, hardware, software, and formal
                       models and proofs, developed over three DARPA
                       research programmes, we are at a truly exciting
                       moment. Today, Arm announced first availability of
                       its experimental CHERI-enabled Morello processor,
                       System-on-Chip, and development board – an
                       industrial quality and industrial scale
                       demonstrator of CHERI merged into a high-
                       performance processor design. Not only does Morello
                       fully incorporate the features described in our
                       CHERI ISAv8 specification to provide fine-grained
                       memory protection and scalable software
                       compartmentalisation, but it also implements an
                       Instruction-Set Architecture (ISA) with formally
                       verified security properties. The Arm Morello
                       Program is supported by the £187M UKRI Digital
                       Security by Design (DSbD) research programme, a UK
                       government and industry-funded effort to transition
                       CHERI towards mainstream use.


                # ⚓ ToolsGroup:_Overcoming_supply_chain_sustainability_fears⠀⇛


                       Though we’re hearing much about the circular
                       economy today, the idea has been kicking around
                       since 1976, when it was first introduced in a
                       research report to the European Commission.
                       Recently organisations, most notably, the Ellen
                       MacArthur Foundation, have reinvigorated the model
                       and (if you’ll pardon the pun) “put wind in its
                       sails”.


                       The Ellen MacArthur Foundation states that the
                       circular economy “is based on the principles of
                       designing out waste and pollution, keeping products
                       and materials in use, and regenerating natural
                       systems”.


                       Supply chains, which manage every aspect of the
                       goods production lifecycle, are arguably in the
                       best position to make meaningful, measurable
                       contributions to a sustainable circular economy.
                       According to a report Gartner published last
                       December “Supply Chain Executive Report: Close the
                       Loop to Create Future-Fit Raw Material Strategies”,
                       51% of supply chain professionals expect the focus
                       on their circular economy strategies to increase
                       over the next two years.


                # ⚓ Running_Methanol_RC_Engines_On_Gasoline_|_Hackaday⠀⇛


                       Methanol is a popular fuel for small engines used
                       in radio-controlled models, but comes at a higher
                       price than gasoline. It’s also harder to source and
                       can be a mite corrosive, too. Gasoline comes with
                       some benefits, but running it in a methanol engine
                       usually requires some mods. [David] and [Bert]
                       worked together to build a mixture controller for
                       just this purpose.


          o § Health/Nutrition/Agriculture⠀➾


                # ⚓ Poultry_Farm_Whistleblower_Speaks_Out_On_Big_Agriculture’s
                  Unfair_Practices⠀⇛


          o § Integrity/Availability⠀➾


                # § Proprietary⠀➾


                      # ⚓ Zhengbang_Pick_&_Places_Your_Confidential_Data_In_The
                        Bag,_Slowly_|_Hackaday⠀⇛


                             It didn’t end here! After installing the
                             “clean” files, they also ran a few anti-
                             malware tools, and all seemed fine. Then,
                             they plugged the flash drive into another
                             computer again… to encounter even more alerts
                             than before. The machine was equipped with
                             some sort of mechanism to infect every .exe
                             touching it with the same aforementioned
                             malware, and the “clean” files were nothing
                             more than a distraction – the infection got
                             added to all .exe files on sight, even .exe‘s
                             of the anti-malware tools they put on that
                             USB drive. By this point, ZhengBang’s
                             intentions are pretty clear – getting data
                             from as many of your devices as possible. To
                             add to that, all of these discoveries don’t
                             count as violations of Aliexpress Terms and
                             Conditions – so if you’d like to distribute a
                             bunch of IoT malware on, say, wireless
                             routers you bought in bulk, now you know of a
                             platform that will help you!


                      # ⚓ I_Broke_My_MacBook_User_Profile_By_Deleting_A_Single
                        Folder⠀⇛


                             The OS on my M1 MacBook Air was completely
                             trashed when I tried to use it this morning.
                             All because I deleted a single folder from my
                             home directory.


                      # § Pseudo-Open Source⠀➾


                            # § Openwashing⠀➾


                                  # ⚓ Harness_CD_Platform_Now_Available_via
                                    Source-Available_License⠀⇛


                      # § Security⠀➾


                            # ⚓ Cyber_conflict_and_mixed_signals._Nation-state
                              cyberespionage,_privateering,_and_direct_theft.
                              [Ed: Ongoing efforts to shift attention away from
                              Microsoft back doors to something else]⠀⇛


                                   Russo-Ukrainian tension has moved US
                                   authorities to issue an alert on the
                                   threat of Russian cyber operations.
                                   That alert came as the continuing
                                   effort to address Log4j vulnerabilities
                                   has raised concern about open-source
                                   software security.


                            # § Privacy/Surveillance⠀➾


                                  # ⚓ Tracking_corona_infections:_German_Luca
                                    app_in_a_downfall⠀⇛


                                         Five Länder no longer renew their
                                         contracts with culture4life, the
                                         company’s new business model
                                         could also run into turbulence


                                  # ⚓ How_to_Use_SSH_Agent_Safely⠀⇛


                                         The SSH agent (ssh-agent) is an
                                         SSH key manager that stores the
                                         SSH key in a process memory so
                                         that users can log into SSH
                                         servers without having to type
                                         the key’s passphrase every time
                                         they authenticate with the
                                         server. In addition to the key
                                         management feature, SSH agent
                                         supports agent forwarding, which
                                         helps to authenticate with
                                         servers that sit behind a bastion
                                         or jump server. As the agent
                                         works as a password manager for
                                         SSH keys, incorrect usage or
                                         faulty configuration can cause
                                         security risks.


                                         In this article, we’ll explore
                                         how to avoid potential SSH agent
                                         pitfalls and recommend best
                                         practices to keep your SSH agent
                                         secure.


                                  # ⚓ Farewell,_G_Suite_legacy_free_edition⠀⇛


                                         Today I saw the news that Google
                                         are pulling on the plug on the G
                                         Suite legacy free edition (a.k.a.
                                         Google Apps for custom domains).
                                         If you don’t recall the story –
                                         originally the service was free
                                         for up to 50 users (making it
                                         quite popular with small
                                         companies), then up to 10 users,
                                         then up to 5 users, and finally
                                         it was made paid for everyone.
                                         The people who started to use the
                                         service before the terms were
                                         changed were kept on what Google
                                         called “G Suite legacy free
                                         edition”. I am (was?) one of
                                         them.


                                  # ⚓ China’s_Olympics_App_Is_Horribly
                                    Insecure⠀⇛


                                         China is mandating that athletes
                                         download and use a health and
                                         travel app when they attend the
                                         Winter Olympics next month.
                                         Citizen Lab examined the app and
                                         found it riddled with security
                                         holes.


                                  # ⚓ Israeli_police_accused_of_using_Pegasus
                                    malware_to_spy_illegally_on_citizens⠀⇛


                                         Israeli police are under fire
                                         after a report accused them of
                                         using the NSO Group’s Pegasus
                                         spyware to access the devices of
                                         the public, it was reported
                                         today.


                                         The claims were made by the
                                         Israeli newspaper Calcalist,
                                         which contends that its
                                         investigation proves that the
                                         spyware was used on critics of
                                         the former Prime Minister
                                         Benjamin Netanyahu. The
                                         technology, which has been called
                                         “the most powerful piece of
                                         spyware ever developed,” is said
                                         to have been used on activists,
                                         politicians and journalists
                                         around the world.


                                  # ⚓ Government_funds_charity_campaign_to_warn
                                    big_tech_over_the_risks_of_encryption_to
                                    children [Ed: Well, the threat to children
                                    is the_perpetrators_of_these_policies]⠀⇛


                                         The government is funding a
                                         campaign that will put pressure
                                         on Facebook and other tech
                                         companies over their plans to
                                         introduce encrypted messaging
                                         services, warning that millions
                                         of cases of child abuse could go
                                         undetected.


                                         The Home Office-backed campaign,
                                         known as No Place to Hide, warns
                                         that social media sites are
                                         “willfully blindfolding”
                                         themselves to child sexual abuse
                                         by introducing end-to-end
                                         encryption on messaging services.


                                         The campaign, coordinated by M&C
                                         Saatchi, aims to bring together
                                         charities and experts to warn
                                         parents over the risks of end-to-
                                         end encryption.


          o § Defence/Aggression⠀➾


                # ⚓ Opinion_|_Biden_Is_Failing_Yemen⠀⇛


                # ⚓ US_Bombed_Syria_Dam_on_‘No-Strike_List,’_Risked_Killing
                  Thousands_of_Civilians⠀⇛


                # ⚓ Opinion_|_Biden’s_Continuation_of_Trump’s_Western_Sahara
                  Policy_Is_Deeply_Troubling⠀⇛


                       In his final weeks in office, President Donald
                       Trump stunned the international community in
                       formally recognizing Western Sahara as part of
                       Morocco. Morocco has occupied much of its southern
                       neighbor since 1975, when it invaded and annexed
                       the former Spanish colony in defiance of the United
                       Nations Security Council and a landmark ruling of
                       the International Court of Justice.


                # ⚓ ISIS_Fighters_Attack_Syria_Prison_to_Free_Fellow
                  Jihadists⠀⇛


                       The attack on a prison in northeastern Syria and a
                       separate attack in Iraq were reminders of the
                       Islamic State’s persistent threat and the
                       unfinished business of the war against it.


                # ⚓ Man_gets_jail_for_justifying_Sialkot_lynching⠀⇛


                       He stood trial in the anti-terrorism court of
                       Gujranwala. Judge Natasha Naseem announced the
                       verdict on Friday.


                # ⚓ Afghan_Taliban_say_Oslo_talks_with_West_will_‘transform
                  atmosphere_of_war’⠀⇛


                       The group stormed back to power in August as US and
                       foreign troops began their final withdrawal from
                       the country following a stalemate on the
                       battlefield.


                       No country has yet recognised the Taliban’s
                       government — notorious for human rights abuses
                       during a first stint in power between 1996 and 2001
                       when they were ousted by a US-led invasion.


                # ⚓ Trump_Considered_Using_Military_To_Seize_Voting_Machines,
                  New_Documents_Show⠀⇛


                # ⚓ Trump_draft_executive_order_called_for_military_to_seize
                  ballots_and_voting_machines_as_part_of_coup_plot⠀⇛


                       On Friday, Politico published the draft of an
                       executive order drawn up for Donald Trump, dated
                       December 16, 2020, which authorized acting
                       Secretary of Defense Christopher Miller to deploy
                       the National Guard to seize voting machines across
                       the country.


                       The order was a brief for the military to overturn
                       the election of Joe Biden and impose a military
                       dictatorship headed by the defeated incumbent,
                       Trump. It would have given the Pentagon 60 days to
                       submit its analysis of supposed election fraud to
                       the director of national intelligence and a special
                       council to draw up criminal and civil charges
                       against Trump’s political opponents, who would
                       undoubtedly have been arrested and likely killed en
                       masse.


          o § Environment⠀➾


                # ⚓ The_“Doomsday_Glacier”_Is_Irreversibly_Melting,_Researchers
                  Say⠀⇛


                       Just as the Thwaites may unleash inland ice into
                       the ocean — much like a cork pulled from a bottle —
                       others are dumping concerning amounts of fresh
                       water near penguin populations. According to a new
                       study published by researchers at Cambridge and the
                       University of Leeds, iceberg A-68a was the world’s
                       largest before it shattered into nearly a dozen
                       mini-bergs.


                # ⚓ Defend_Chernobyl_During_an_Invasion?_Why_Bother,_Some
                  Ukrainians_Ask.⠀⇛


                       The Chernobyl zone covers about 1,000 square miles
                       straddling the shortest direct route from the
                       Belarusian border to Kyiv. While it is not
                       necessarily the most likely invasion route from the
                       north, because it is swampy and densely forested,
                       Ukraine has not ruled it out.


                # § Wildlife/Nature⠀➾


                      # ⚓ A_Mass_Extinction_Has_Already_Started,_Scientists
                        Say⠀⇛


                             The team led by Robert Cowie, a research
                             professor at the University of Hawaii, argues
                             that the International Union for Conservation
                             of Nature (IUCN) Red List of Threatened
                             Species was too focused on the loss of bird
                             and mammal species — and not focused enough
                             on invertebrates, a much larger group.


                             The organization’s Red List, founded in 1964,
                             is the world’s most comprehensive inventory
                             of endangered species and their extinction
                             risk. It has been used by other researchers
                             to argue that we are not in fact facing a
                             sixth mass extinction event.


                      # ⚓ Untangling_the_roots_of_plant_genomes|_EurekAlert!⠀⇛


                             The article, “Green Plant Genomes: What We
                             Know in an Era of Rapidly Expanding
                             Opportunities” underscores the significance
                             of this massive endeavor. “Nearly half a
                             million species of plants inhabit the Earth
                             today and the secrets to understanding nearly
                             everything about them is hidden in the
                             sequences of their DNA (the plant genome),”
                             said Dr. W. John Kress, senior author of the
                             paper and Curator Emeritus at the
                             Smithsonian. “Plants are the foundation of
                             environments across the planet and
                             deciphering their genomes will be a game
                             changer for understanding nearly all aspects
                             of our own lives, from improving foods and
                             medicines to inspiring artists and enhancing
                             ecosystem stability.”


                      # ⚓ Ecological_niche_divergence_between_extant_and
                        glacial_land_snail_populations_explained_|_Scientific
                        Reports⠀⇛


                             The presence of Last Glacial Maximum (LGM)
                             biotic communities without modern
                             counterparts is well known. It is
                             particularly evident in central European
                             fossil LGM land snails whose assemblages
                             represent an odd mix of species that are
                             currently limited to either xeric or wetland
                             habitats. Here we document a genetically
                             verified discovery of the modern calcareous
                             wetland species Pupilla alpicola on Iceland,
                             where it is limited to dry grasslands. This
                             species also represents a common European LGM
                             fossil, and its new records from Iceland help
                             explain puzzling shifts of some glacial land
                             snails of xeric grassland habitats to open
                             wetlands today. Similarities between the
                             climates of modern Iceland and LGM Eurasia
                             suggest that this species did not become
                             limited to wetlands in continental Europe
                             until after the Late Pleistocene–Holocene
                             climate transition. These results are a
                             strong reminder that assumptions of
                             ecological uniformity must be questioned and
                             that the quality and robustness of
                             palaeoecological reconstructions is dependent
                             upon adequate knowledge of the full
                             autecological range of species over time.


          o § Finance⠀➾


                # ⚓ Islamism_meets_inflation_and_Turkey_teeters⠀⇛


                       While Erdoğan and his government make these
                       terrible mistakes, many in the West are making a
                       related mistake. Our experts and opinion leaders
                       are far too accustomed to looking at leaders around
                       the world as so-called rational actors. Possibly
                       evil, possibly ignorant, but still rational.
                       History shows something different. Plenty of world
                       leaders are irrational and willing to follow
                       irrational ideologies right over a cliff, taking
                       their own and possibly other countries with them.
                       Case in point: Venezuela. The more wreckage the
                       policies of the Hugo Chavez and Nicolás Maduro
                       governments created, the more those leaders kept
                       and keep pursuing the same policies. Erdoğan may be
                       such a one.


                # ⚓ Holes_in_the_Social_Safety_Net_Leave_Millions_Without
                  Access_to_Needed_Benefits⠀⇛


                       The pandemic has shone a bright light on our
                       country’s social safety net. Record numbers of
                       Americans applied for unemployment. The federal
                       government issued hundreds of billions of dollars
                       in stimulus checks. Federal and state programs like
                       the Paycheck Protection Program (PPP), mortgage
                       relief, rental assistance, even free vaccine
                       delivery—all of which had trouble gaining
                       bipartisan support in previous years—were quickly
                       implemented as the crisis unfolded and millions of
                       Americans lost jobs, fell ill or were sidelined at
                       home caring for children whose schools and day care
                       facilities were closed due to COVID.


          o § AstroTurf/Lobbying/Politics⠀➾


                # ⚓ Biden_signs_memorandum_to_secure_sensitive_national
                  security_systems⠀⇛


                       The order specifies how national security systems,
                       the most sensitive information technology systems
                       within the government, should comply with a May
                       2021 executive order designed to improve
                       cybersecurity across the federal government.


                       While that executive order laid out a broad set of
                       tasks for managers of national security systems,
                       the new memorandum establishes specific timelines
                       and guidance for implementation. This includes
                       requiring multifactor authentication and
                       encryption.


                       The memorandum also requires agencies to develop a
                       plan to implement zero trust architectures, a move
                       considered critical to protecting systems and
                       information from both inside and outside threats.


                # ⚓ The_European_Commission_Timidly_Commits_to_Opening_it’s
                  Source_Codes_and_Contributing_to_Free_Software⠀⇛


                       Following its “Open Source Software Strategy 2020 –
                       2023″, published in October 2020, the European
                       Commission formalizes its goals and practices in
                       its decision “on the open source licensing and
                       reuse of Commission software” – an important
                       document, as the Commission can be held accountable
                       for it. Like the above-mentioned strategy, this
                       decision doesn’t demonstrate much political
                       ambition, nonetheless it sets a useful baseline and
                       confirms the Commission’s will to amplify its use
                       of free software, and contribute more to it.


                # ⚓ [Old] COMMISSION_DECISION_of_8.12.2021_on_the_open_source
                  licensing_and_reuse_of_Commission_software⠀⇛


                       This Decision determines the conditions for the
                       reuse and licensing of software produced by the
                       Commission or on its behalf, and for which the
                       Commission holds the intellectual property [sic]
                       rights [sic].


                # ⚓ Howie_Hawkins_Blasts_“Bipartisan_Biden”_on_Inauguration
                  Anniversary⠀⇛


                       The 2020 Green Party Presidential Candidate Calls
                       for Independent Progressive Politics


                       Howie Hawkins, the Green Party’s 2020 candidate for
                       President, issued the following statement on the
                       first anniversary of President Joe Biden’s
                       inauguration. It is being published on the
                       CounterPunch website today as “Bipartisan Biden
                       Builds Back Bupkis: Reflections on Biden’s
                       Inauguration Anniversary.”


                       It is not surprising that President Biden is
                       failing on his Build Back Better and voting rights
                       agendas. It is not surprising because Biden
                       campaigned as the candidate who could work with the
                       ultra-conservative, Trump-infatuated Republicans to
                       build unity and “restore the soul of America.”
                       Biden continued to pursue bipartisanship even after
                       the anti-democratic Republicans tried to overthrow
                       his election.


                       Bipartisan Biden’s political strategy has been as
                       reality-challenged as Trump’s 30,000 lies in
                       office. It is not surprising that as Biden
                       approached the January 20th anniversary of his
                       inauguration a Quinnipiac Poll showed that Biden’s
                       job approval is 33% and a 58% to 37% majority of
                       Americans believe our democracy is in danger of
                       collapsing.


          o § Censorship/Free Speech⠀➾


                # ⚓ Fact-checking_won’t_solve_much_for_content_platforms⠀⇛


                # ⚓ Internet_connectivity_knocked_out_in_Yemen_after
                  airstrike⠀⇛


                       Real-time metrics show the loss of connectivity on
                       leading network YemenNet (AS30873) on early Friday
                       morning. It is not known if telecommunication
                       infrastructure was specifically targeted, although
                       visual investigations suggest this may be the case.
                       The resulting information blackout is likely to
                       limit coverage of events on the ground.


          o § Civil Rights/Policing⠀➾


                # ⚓ Black_Mississippi_State_Senators_Stage_Walkout_as_Critical
                  Race_Theory_Ban_Passed⠀⇛


                       Every Black member of the Mississippi state Senate
                       walked out of the legislative chamber in protest on
                       Friday as the Republican-led body passed a bill
                       critics say will ban the teaching of critical race
                       theory in public schools, from kindergarten through
                       college.


                       “We did not need to spend time on this. Even the
                       author of the bill said this was not occurring in
                       Mississippi.”


                # ⚓ Amid_Existential_Threat_to_Reproductive_Rights,_Congress
                  Urged_to_Act⠀⇛


                       As reproductive freedom is imperiled by the U.S.
                       Supreme Court’s right-wing supermajority and
                       Republican-led state legislatures, advocates
                       rallied in Washington, D.C. on Saturday—the 49th
                       anniversary of the landmark Roe v. Wade
                       ruling—while abortion rights supporters called on
                       lawmakers to pass legislation to codify Roe as the
                       “law of the land.”


                       “It’s difficult to think that my daughter’s
                       generation may not have the same freedoms as mine
                       or my mother’s.”


                # ⚓ Unions_Have_the_Potential_and_the_Responsibility_to_Advance
                  a_“Just_Transition”⠀⇛


                # ⚓ Opinion_|_GOP-Led_Legislatures_in_More_Than_Half_of_U.S.
                  States_Are_Trying_to_Ban_Abortion⠀⇛


                       With the 49th anniversary of the landmark Roe v.
                       Wade case coming up, several organizations across
                       the nation including Florida are planning to make
                       an example by protesting various abortions bans
                       that threaten women’s reproductive freedom.


                # ⚓ Abortion_Funds_Are_Meeting_an_Urgent_Need_as_“Roe”_Faces
                  Dire_Threat⠀⇛


                # ⚓ Opinion_|_Is_This_the_End_of_Roe_V_Wade_and_Abortion_Rights
                  in_the_US?⠀⇛


                       Saturday 22 January marks 49 years since the US
                       Supreme Court’s landmark decision in Roe v Wade,
                       the court case established a right to abortion
                       across the US. While reproductive rights
                       organisations such as Planned Parenthood are
                       celebrating the occasion, they are also preparing
                       for the likelihood that they won’t be celebrating
                       its 50th anniversary, next year. 


                # ⚓ Opinion_|_Yes,_It’s_Easier_to_Get_Birth_Control_Than_It_Was
                  In_the_1970s—But_Women_Still_Need_Abortion_Care⠀⇛


                       A historic ruling on abortion is likely to emerge
                       from the U.S. Supreme Court this year as justices
                       consider whether Mississippi can, in fact, impose a
                       ban on abortions after 15 weeks of pregnancy.


                # ⚓ Increased_Voting_Restrictions_Put_LGBTQ_People_of_Color_at
                  Risk,_Advocates_Say⠀⇛


                # ⚓ Post-“Roe,”_Birth_Control_Is_More_Accessible,_But_Abortion
                  Is_Still_Crucial⠀⇛


                # ⚓ Covid-19_pandemic_fuels_rise_in_sexual_abuse_–_UCA_News⠀⇛


                       Murders of young women by Islamic family members
                       around the world in so-called “honor killings” are
                       growing.


                       El Salvador has the highest rate of murders of
                       women and children in Latin America.


                       In the Philippines, huge numbers of children are
                       trafficked and sexually abused online. The rape of
                       children by biological fathers and live-in partners
                       is increasing.


                       One in three girls and one in six boys are victims
                       of rape before the age of 16. This is a possible
                       cause of the predatory behavior of some males. It
                       is likely that sexual violence begets more sexual
                       violence.


          o § Digital Restrictions (DRM)⠀➾


                # ⚓ The_sky_is_falling_for_Netflix⠀⇛


                       Although price increases will probably help to
                       offset its sluggish sign ups, they could also lead
                       to more stagnation for Netflix.


                       For some consumers, price increases — even small
                       ones — are a lot to ask considering that so many
                       competitors are at Netflix’s gates. Rivals like
                       Disney+, Peacock and HBO Max from CNN parent’s
                       company, WarnerMedia, are also vying for a share of
                       consumer’s streaming budget. A dollar here or there
                       matters to consumer’s wallets.


          o § Monopolies⠀➾


                # § Patents⠀➾


                      # ⚓ AstraZeneca_and_Hoyng_avert_damages_payment_to_health
                        insurer_Menzis [Ed: AstraZeneca is a bunch of monsters,
                        but mouthpieces of AstraZeneca and the other patent
                        profiteers (Amy Sandys in this case) frame this as a
                        case of innocence ]⠀⇛


                             The Court of Appeal of The Hague has found
                             AstraZeneca not guilty of enforcing an
                             invalid patent against generic drug companies
                             in the Netherlands. Overturning a first-
                             instance decision, the court ruled that it is
                             not unlawful for a patent holder to maintain
                             an invalid patent if unaware that a court
                             could invalidate the patent. As such,
                             AstraZeneca is no longer obliged to
                             compensate third-party health insurer Menzis.


                             For Menzis, the decision is a blow in its
                             quest to gain damages from pharmaceutical
                             company AstraZeneca in the Netherlands. This
                             is in contrast to a District Court of The
                             Hague decision, which it handed down in 2020.
                             It is also the first such claim made by a
                             health insurance company.


                      # ⚓ Companies_got_fewer_patents_in_2021_–_in-house
                        explain_why [Ed: Rani Mehta, de facto lobbyist for the
                        patent cartel that sponsors her, treats this like bad
                        news; as if monopolies are desirable and they're
                        moaning about "patent eligibility challenges" (read:
                        patent quality)]⠀⇛


                             Counsel at Siemens, HPE, Honeywell and IBM
                             say COVID and patent eligibility challenges
                             are to blame for the drop in issuances last
                             year and in the future


                      # ⚓ What_are_the_challenges_in_developing_information
                        around_mixing-and-matching_COVID-19_vaccines_and
                        therapies?⠀⇛


                             The FDA has now authorized three vaccines and
                             several treatments (including both monoclonal
                             antibodies and small-molecule drugs) for the
                             prevention and treatment of COVID-19. But the
                             initial evidence supporting these products’
                             introduction into the market did not include
                             information about how they might work
                             together. Nevertheless, information about
                             mixing-and-matching COVID-19 vaccines and
                             therapies would be highly valuable not only
                             to physicians and their patients, who must
                             already make decisions about what treatment
                             options to pursue under conditions of
                             uncertainty (if the treatments are
                             available), but also for policymakers, who
                             want to know what products to prioritize for
                             investment. Why is it so difficult to obtain
                             this information? How can policymakers
                             encourage its development?


                             [...]


                             The CDC currently advises that vaccines are
                             not interchangeable for the initial two doses
                             of the Pfizer-BioNTech and Moderna mRNA
                             vaccines, but that mix-and-match dosing (more
                             formally known as “heterologous” dosing) is
                             allowed for booster shots. People who
                             received non-FDA-authorized vaccines can also
                             receive a heterologous primary dose and
                             booster. The agency cites preliminary results
                             from a pre-Omicron study by the NIH-funded
                             Mix and Match Team, which found that
                             heterologous boosters resulted in similar or
                             higher antibody responses in the first month
                             after boosting. But short-term antibody
                             responses may not indicate clinical outcomes,
                             and the study authors caution that the study
                             was not designed to compare different booster
                             regimes (given the sample size and lack of
                             controls for relevant variables). Last week,
                             the Mix and Match Team posted very
                             preliminary results that most booster
                             combinations (heterologous or homologous)
                             increase antibody response to the Omicron
                             variant, but these results have similar
                             limitations.


                      # ⚓ Medlab_receives_grant_to_develop_NanoCelle_nasal
                        vaccine_for_Covid-19 [Ed: EPO thinks or wants us to
                        believe that giving monopolies or patents on COVID-19
                        would help rather than exacerbate the crisis, ensuring
                        it never ends because it's more profitable to keep the
                        virus circulating]⠀⇛


                             Furthermore, the platform provides a market-
                             ready delivery solution with patent
                             protection until 2036 in Australia, Canada,
                             New Zealand, the UK and Europe.


                      # ⚓ Greenville_entrepreneur_helps_police_bring_down_$4.5M
                        patent_scam⠀⇛


                      # ⚓ [Older]_Texas_Patent_Litigation_Monthly_Wrap-Up_–
                        October_2021 [Ed: When patent law is turned into
                        "business" by judges, who disregard the actual law to
                        attract lawsuits]⠀⇛


                             Texas courts facing pushback on multiple
                             fronts.


                      # ⚓ [Older]_China_to_Revoke_Patent_Agencies’_Licenses_for
                        Filing_Abnormal_Patent_Applications [Ed: Patent
                        applications in China are of low quality, but to stuff
                        WIPO they’ll grant anyway]⠀⇛


                             On November 10, 2021, the China National
                             Intellectual Property Administration (CNIPA)
                             released the Notice of the CNIPA on Further
                             Severely Cracking Down on Abnormal Patent
                             Application Agency Acts
                             (国家知识产权局关于进一步严厉打击非正常专利申请代理行为的通知). CNIPA will
                             halt new business intake or even revoke
                             licenses of patent agencies that file
                             relatively large amounts of abnormal (also
                             translated as irregular) patent applications.
                             Previously, Chinese patent agencies have been
                             warned and fined such as a Shenzhen firm that
                             was recently fined 50,000 RMB for filing a
                             total of 1,264 irregular applications and a
                             Hangzhou firm was recently fined 20,000 RMB
                             for filing a total of 1,192 abnormal patent
                             application.


                      # ⚓ [Older]_Federal_Circuit_Disinfects_PTAB’s_Obviousness
                        Finding_for_MRSA_Inactivation_Patent⠀⇛


                             The University of Strathclyde recently found
                             relief at the Federal Circuit when a
                             unanimous panel reversed the Patent Trial and
                             Appeal Board’s (“the Board”) final
                             determination that their method patent for
                             photoinactivating MRSA bacteria was obvious.
                             The Federal Circuit found that due to a lack
                             of substantial evidence in the prior art, the
                             patent was not shown to be obvious/
                             unpatentable.


                      # ⚓ [Older]_Chinese_Patent_Office_Announces_Record_1.04
                        Million_RMB_Fine_for_Unauthorized_Practice_of_Patent
                        Law⠀⇛


                             On November 29, 2021, the Chinese National
                             Intellectual Property Administration (CNIPA)
                             announced a record fine of 1.04 million RMB
                             (~$163 thousand USD) against a Sichuan entity
                             for engaging in the patent agency business
                             without authorization. Per CNIPA, “The amount
                             of fines and forfeiture is the most for an
                             unqualified patent agency violation case that
                             has been investigated and punished.”


                      # ⚓ Unified_Patent_Court_and_Unitary_Patent_Moving_Ahead
                        in_Europe [Ed: Totally and intentionally ignoring the
                        fact that this_is_illegal_and_many_obstacles_remain]⠀⇛


                             Final preparations by sixteen EU member
                             states for their Unified Patent Court (UPC)
                             and Unitary Patent (UP) began on 19 January
                             2022, following Austria’s deposit on the
                             previous day of its ratification of the
                             Protocol on Provisional Application of the
                             UPC Agreement (PPA). The PPA permits the
                             organization of the UPC before it opens to
                             accept cases, including establishment of its
                             Administrative, Advisory and Budget
                             Committees, hiring of judges and staff, etc.
                             The new court is expected to open in late
                             2022 or early 2023.


                             Poland and Spain have not signed the UPC
                             Agreement and currently do not plan to
                             participate in the UP and UPC. Croatia joined
                             the EU after the Agreement was signed. Seven
                             additional EU member states have signed the
                             UPC Agreement, but have not yet ratified.
                             Those states and Croatia may participate in
                             the UP and UPC later.


                      # ⚓ The_Unitary_Patent_and_Unified_Patent_Court_Move
                        Forward [Ed: No, it_is_illegal_and_it_will_be_stopped;
                        but the lobbying tactic is to endlessly lie and make it
                        “too late” to stop something illegal]⠀⇛


                             The General Secretariat of the Council of the
                             European Union yesterday declared that the
                             Provisional Application Period (PAP) of the
                             Unified Patent Court (UPC) has started,
                             following deposit by Austria of its
                             instrument of ratification. The Unified
                             Patent Court will oversee the Unitary Patent
                             (UP) which is intended to be a single patent
                             right having effect in the territories of
                             participating EU members and European patents
                             granted in participating countries. The
                             beginning of the PAP means that the UPC
                             Preparatory Committee can begin work to
                             establish the Court, which is expected to
                             take at least eight months.


                      # ⚓ After_almost_50_years,_EU_unitary_patent_edges_closer
                        to_becoming_reality [Ed: This publisher is promoting
                        falsehoods; of_course_it’s_in_the_EPO’s_pocket,_too;
                        it’s really bad when EPO puff piece factories are also
                        producing UPC propaganda for lobbying purpose, peddling
                        unlawful agenda and to make matters worse, the EPO’s
                        propaganda outlets also cherry-pick_voices_to_help_a
                        patent_cartel_that_kills_millions_of_people_for_vaccine
                        patent_profits]⠀⇛


                      # ⚓ Intellectual_property:_Statement_by_Commissioner
                        Breton [Ed: Thierry_Breton openly and flagrantly
                        promotes illegal agenda]⠀⇛


                      # ⚓ How_John_Quinn_Built_The_World’s_Largest_Litigation
                        Specialist [Ed: Why does Forbes champion thugs and
                        bullies? They contribute nothing to the planet.]⠀⇛


                # § Copyrights⠀➾


                      # ⚓ Heuking_embarks_on_mixed_approach_with_patent
                        attorney_hire [Ed: This is truly ridiculous marketing
                        spam from Christina Schulze, who used to do actual
                        journalism. JUVE wants you to think that some firm
                        hiring a relatively ‘low-level’ person is noteworthy
                        “news” (follow the PR money)]⠀⇛


                             Engineer and patent attorney Isabella Sommer-
                             Zhang (31) has joined Heuking Kühn Lüer
                             Wojtek‘s four-lawyer-strong patent litigation
                             team in Düsseldorf. She previously worked for
                             patent attorney firms Farago and Jostarndt in
                             Aachen, passing her European patent attorney
                             examination last year


                             As part of her education and training in
                             electrical engineering and information
                             technology, Sommer-Zhang worked for clients
                             such as OLEDWorks. She also gained experience
                             with AI patents through work for technology
                             company Komet, and 3D metal printing
                             manufacturer Aixway.


                      # ⚓ [Guest_post]_The_new_German_competition_tool_in
                        action_–_Ensuring_the_effectiveness_of_the_new_press
                        publisher’s_neighbouring_right_against_Google⠀⇛


                             On 12 January 2022, the German Competition
                             Authority (Bundeskartellamt) announced that
                             it would hold consultations in the press
                             publishing sector on the Google News Showcase
                             online service (see press release here).
                             Already in June 2021, the Bundekartellamt had
                             initiated proceedings against Google
                             following a complaint by Corint Media
                             concerning Google’s licensing and
                             presentation practices i.a. in the News
                             Showcase service. Corint Media is the German
                             Collective Rights Management Organisation
                             (CMO) responsible for the management of the
                             neighbouring rights of broadcasters and press
                             publishers, including the German press
                             publisher’s neighbouring right under Section
                             87g of the German Copyright Act.


                             [...]


                             Google has now declared that it will not
                             include the Showcase content in its general
                             search results and has proposed further
                             measures to respond to the competition
                             concerns, such as to modify the Showcase
                             contracts and to clearly separate them from
                             the negotiations between Google and the press
                             publishers or their CMOs with regard to their
                             neighbouring rights. Andreas Mundt, President
                             of the Bundeskartellamt stated: “Google has
                             proposed measures to respond to our
                             competition concerns relating to Google News
                             Showcase. The company no longer plans to
                             include Showcase content in the general
                             search results. The conditions for
                             participating in Google News Showcase are not
                             intended to prevent publishers from enforcing
                             their general ancillary copyright. Access to
                             Google News Showcase is based on objective
                             criteria. We rely on the assessment of the
                             market players affected to ensure that the
                             measures proposed by Google are effective. In
                             view of the wide variety of interests the
                             publishers may have we are thus conducting
                             broader consultations in the sector.”


                             Google’s proposed commitments are not
                             publicly available, and it will be
                             interesting to see the outcome of the
                             Bundeskartellamt’s consultations in the
                             sector. Certainly, as the consultation and
                             negotiations progress, we will learn more
                             details on what Google is offering exactly.


                      # ⚓ Pirate_IPTV_Reseller_Ordered_to_Pay_TV_Companies
                        $164,000_in_Damages⠀⇛


                             A 58-year-old man has been ordered to pay
                             four Swedish TV companies more than $164,000
                             in damages after being found guilty of
                             selling illegal IPTV service subscriptions.
                             He claimed that a third party was responsible
                             for the sales but a local court found him
                             guilty of copyright infringement offenses,
                             also handing down a four-month suspended
                             prison sentence.


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