𝕿𝖊𝖈𝖍𝖗𝖎𝖌𝖍𝖙𝖘 Bulletin for Saturday, January 08, 2022
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Generated Sun 9 Jan 02:40:39 GMT 2022
Created by Dr. Roy Schestowitz (𝚛𝚘𝚢 (at) 𝚜𝚌𝚑𝚎𝚜𝚝𝚘𝚠𝚒𝚝𝚣 (dot) 𝚌𝚘𝚖)
Full hyperlinks for navigation omitted but are fully available in the originals
The corresponding HTML versions are at 𝒕𝒆𝒄𝒉𝒓𝒊𝒈𝒉𝒕𝒔.𝒐𝒓𝒈
Latest in 𝒉𝒕𝒕𝒑://𝒕𝒆𝒄𝒉𝒓𝒊𝒈𝒉𝒕𝒔.𝒐𝒓𝒈/𝒕𝒙𝒕 and older bulletins can be found at 𝒉𝒕𝒕𝒑://𝒕𝒆𝒄𝒉𝒓𝒊𝒈𝒉𝒕𝒔.𝒐𝒓𝒈/𝒕𝒙𝒕-𝒂𝒓𝒄𝒉𝒊𝒗𝒆𝒔
Full IPFS index in 𝒉𝒕𝒕𝒑://𝒕𝒆𝒄𝒉𝒓𝒊𝒈𝒉𝒕𝒔.𝒐𝒓𝒈/𝒊𝒑𝒇𝒔 and as plain text in 𝒉𝒕𝒕𝒑://𝒕𝒆𝒄𝒉𝒓𝒊𝒈𝒉𝒕𝒔.𝒐𝒓𝒈/𝒊𝒑𝒇𝒔/𝒕𝒙𝒕
Gemini index for the day: gemini://gemini.techrights.org/2022/01/08/


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

⦿ Gopher and Gemini Can Co-exist, They’re Not the Same | Techrights

⦿ What Gemini Clients to Use in 2022 | Techrights

⦿ IRC Proceedings: Friday, January 07, 2022 | Techrights

⦿ Installing and Configuring/Using Kristall (Gemini Client) on GNU/Linux Distros Using AppImage | Techrights

 ䷼ Bulletin articles (as HTML) to comment on (requires login):
		http://techrights.org/2022/01/08/gemini-and-gopher-compared/#comments
		http://techrights.org/2022/01/08/gemini-clients/#comments
		http://techrights.org/2022/01/08/irc-log-070122/#comments
		http://techrights.org/2022/01/08/kristall-appimage/#comments

 ䷞ Followed by Daily Links (assorted news picks curated and categorised):
		http://techrights.org/2022/01/08/easyos-3-2/#comments
		http://techrights.org/2022/01/08/kde-frameworks-5-90-0/#comments

                    ䷩ 𝚕𝚒𝚗𝚎 57

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

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/01/08/gemini-and-gopher-compared/#comments
    Gemini version at gemini://gemini.techrights.org/2022/01/08/gemini-and-gopher-compared/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 01.08.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Gopher_and_Gemini_Can_Co-exist,_They’re_Not_the_Same⠀✐


Posted in Free/Libre_Software, Protocol, Site_News, Standard at 2:40 pm by Dr.
Roy Schestowitz


Video_download_link | md5sum 605608feb13aaba90bda86094d6fcd6c
Gemini and Gopher Compared
Creative Commons Attribution-No Derivative Works 4.0


http://techrights.org/videos/gemini-gopher-2022.webm


Summary: There are some misconceptions about what Gemini is compared to Gopher;
the video above demonstrates the difference by browsing/navigating through some
Gopher pages


Little less than a year ago people expressed scepticism and also ridiculed us
for embracing Gemini and adding presence in Geminispace. Some people who had
relied on hearsay decided to dismissively say that Gemini_Protocol is little
but a “love letter to Gopher” (as if it’s just another Gopher). Since then
adoption of the protocol — in terms of the number of capsules — has more than
trebled. Yes, in less than a year!


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇Castor with gemini⦈ Software with support for Gemini is
increasing in terms of size and in terms of number (e.g. number of Gemini
clients and scripts/converters for GemText). There are already several search
engines for Geminispace (Totally_Legit_Gemini_Search_(TLGS) continues to grow)
and Gemini’s founder came back from a long hiatus just_over_two_months_ago. In
his own words: “Despite my total lack of involvement for several months and the
lack of any progress on the spec, Geminispace *itself*, which is our real goal,
has neither stagnated nor shrunk. It has only gotten better. Awesome things
like smol.pub have turned up. All the time there are more and more people
setting up little digital homes in Geminispace, who accept and embrace Gemini
as it is right now, and many of them are very happy with the status quo. They
are writing truly wonderful content, and I have not come across a single thing
written there yet which made me think “right now this is merely good, but it
could be excellent if only Gemini supported X, Y or Z”. And all of this is
hosted on diverse servers and compatible with diverse clients, including
clients which have not been updated in months. All of this says we have gotten
the most important things right or close enough to right already, and there is
no risk of catastrophically messing anything up if we simply resolve
outstanding technical issues with the minimum possible change.”


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇I'm NOT Gemini (meme)⦈ Any further changes to the protocol
(GemText in particular) would likely entail a massive overhaul in capsules,
e.g. regenerating all pertinent pages, which is time-consuming and risky
(cannot test things exhaustively to assure sanity in our case because we have
close to 40,000 pages in Gemini).


On a separate note, it may help to think of Gemini as a modernised Gopher
rather than profoundly more advanced (read: complicated) variant of Gopher. As
Gopher is mostly an historical thing from another era we can leave it aside and
focus on Gemini instead. No need for Gopher to go away, it can co-exist with a
sort of ‘successor’, made to adopt encryption and broader character sets.


An associate reminds us that “in Gemini, TLS is obligatory. Also it has links
and some limited structural elements.”


Contrariwise, “Gopher is just a plain file + a directory structure, without
encryption. The main downside to Gemini is the bug where it does not send the
size of the file in advance.” █


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⢸⣿⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀
⢸⣿⣿⠈⠀⠀⠈⠈⠀⠁⠀⠁⠀⠀⠀⠀⠀⠁⠀⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀
⢸⣿⣿⠀⠀⠀⠐⠀⠀⠀⠂⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣀⣂⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀
⢸⣿⣿⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀
⢸⣿⣿⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀
⢸⣿⣿⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⣀⣀⠀⢀⠀⠀⠀⠀⠈⣉⠉⠉⠍⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀
⢸⣿⣿⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀
⢸⣿⣿⠉⠉⠉⠉⠉⠩⠍⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀
⠸⠿⠿⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠋⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣮⠒⣺⣟⠟⠛⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⠃⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠻⠟⠋⠳⠈⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠈⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⣿⡭⠘⠍⠑⠊⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠆⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣀⣀⣤⡶⣿
⣻⣻⠻⡿⢹⠯⢝⠟⠁⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠇⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠈⠙⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⢻⣿⠰⠘⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣀⡀⢀⡀⠀⠀⠀⠀⠀⡄⠀⣶⠀⢁⣾⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠿⠉⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⡿⠀⠀⠀⡈⢷⣦⡦⠆⠉⠁⠀⠀⠁⠘⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇⡀⠀⠀⢸⣿⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⡿⢻⣿⣿⡏⠋⠀⢿⠿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠇⠀⠀⣾⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⣿⡆⠘⠈⢹⠀⠀⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⣀⣠⣴⣶⡼⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠟⠻⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠻⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣟⠀⢀⣀⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠋⢹⣿⡌⠛⣝⣿⠀⠈⠛⠛⠻⣿⣩⣿⣿⣧⡙⠇⠀⠀⠀⠀⠀⠀⠀⢠⣤⡆⠀⠀⠀⣀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠛⠙⠱⣾⣿⣿⣾⣿⠋⠀⠀⠀⣤⣴⣿⣿⣿⣿⣿⣥⣤⣶⣶⣄⠀⠀⢀⣀⣀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⠟⠛⠻⣿⡋⠉⠁⣬⣤⣼⣿⣿⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢿⡽⡏⠡⢀⠀⠘⠂⠀⢻⣟⠻⣿⣿⣿⠙⢹⣿⣿⣿⡿⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣏⣀⣀⠀⡀⠀⠀⣀⣶⣀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡤⣴⣿⣿⣼⠿⡄⢰⠁⠀⠀⢀⣾⣿⣿⠀⠈⢿⣦⡿⣼⢷⠀⢰⣿⠟⠛⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣷⣤⣤⠤⠔⠀⠀⠀⠀⠀⠀⠸⠛⠀
⣤⡤⣴⡆⠀⡄⠴⠶⠄⠀⠀⢒⣆⣶⣿⣷⣦⡟⣿⣿⣗⣀⠀⠈⠀⠀⠀⠈⣿⣿⡿⠀⠀⠀⣿⣿⢿⣿⠁⠀⠹⣦⠄⠀⠀⣠⣶⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣏⣁⣉⣿⣿⢧⢁⣠⣶⣶⣿⣤⣤⣄⣀⠀⠀⠀
⣿⣶⣿⣿⣿⣶⣶⣶⡆⢠⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⡿⠁⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⣿⣿⣾⣧⠀⢀⣺⣿⣶⣶⣾⣿⣿⢁⣿⠁⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣷⣶
⡟⢻⣿⣿⣿⣿⣿⣿⠀⠀⠁⢙⣿⣿⣿⣿⣿⡇⠀⢿⠗⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⢸⣿⣿⣿⣄⣸⣿⣿⣿⣿⣿⣟⢋⣼⡏⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣻⣿⣿⠿⣿⣿⣿⣶⣤⣬⣉⣿⣟⡏⠯⣭
⢃⣈⣛⣛⣉⣉⣛⣛⣀⣀⣀⣀⣛⣛⣛⣛⣛⣂⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣀⣀⣘⣛⣛⣛⣛⣀⣛⣛⣛⣛⣛⣛⣛⣛⣀⣀⣐⣛⣛⡛⣿⣿⣿⣿⣿⣿⣿⠁⢸⠇⢙⣿⣿⣯⣭⣭⣿⣿⣿⣿⣧⣤⣤⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⣿⠿⢿⣿⣿⡟⠉⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣭⣝
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢾⡿⣿⢿⣿⣮⣻⣸⣿⣷⣙⣾⣽⡯⡧⣷⣙⡜⣽⡸⣧⡿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⢦⡀⠀⠀⣿⡇⣧⠀⠙⠿⠧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣛⣿⣿⣿⣿⣿⣯⣉⠉
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⣴⠖⣫⣶⣿⣿⣷⣦⣄⠀⢹⠀⠀⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢹⢹⡟⣿⣿⣿⣿⡿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⠟⣱⣿⡟⠁⠀⠙⠿⠿⠿⠷⡆⠀⠀⣿⡇⣛⣁⣈⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣋⣻⣿⡟⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣭⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣿⣿⡇⠀⢀⣾⣿⣿⣿⡀⠀⠙⠛⣻⣿⣿⣿⠃⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡿⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡇⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⡛⡟⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⢏⣀⢘⡆⠋⣿⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠃⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣿⢻⢿⠛⠉⠛⠛⠉⠁⠜⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠈⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠿⡿⣿⡿⢿⠻⠋⠉⠆⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠆⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣀⣀⣤⡶⣿
⢻⣟⢻⠟⢿⡛⠩⠒⠕⠀⠀⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠘
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠇⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠈⠙⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⢻⣿⠰⠘⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣄⣀⡀⢀⡀⠀⠀⠀⠀⠀⡄⠀⣶⠀⠁⣾⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠿⠉⠉⠀⠈⠉⠀⠀⢀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⡿⠀⠀⠀⡈⢷⣦⡶⠂⠉⠁⠀⠀⠁⠘⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇⡀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⡿⢻⣿⣿⡏⠋⠀⢿⠟⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠆⠀⠀⣾⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⡿⣿⣿⡇⠸⠉⢹⠀⠀⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⣀⣠⣴⣶⡼⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠟⠹⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠻⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣟⠀⢀⣀⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠋⢹⣿⡌⠛⢙⣿⠀⠘⠛⠛⠻⣿⣹⣿⣿⣧⡙⠇⠀⠀⠀⠀⠀⠀⠀⢠⣤⡆⠀⠀⠀⣀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠱⣾⣿⣿⣶⣿⠋⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣥⣤⣶⣶⣄⠀⠀⢀⣀⣀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡿⠟⠛⠻⣿⣋⠉⠉⣠⣤⣼⣿⣿⣾⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢿⡽⡏⠡⢀⠀⠘⠃⠀⢻⣟⠻⣿⢿⡿⠙⢻⣿⣿⣿⠿⠀⠀⠀⠘⢿⣿⣷⣿⣿⣿⣿⣏⣀⣀⠀⠀⠀⠀⢀⣦⣀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡤⣴⣿⣿⣼⠿⡄⢰⠁⠀⠀⢀⣾⣿⣿⠀⠈⢿⣦⣿⣸⣻⠀⢠⣿⠿⠛⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣤⣤⠤⠔⠀⠀⠀⠀⠀⠀⠰⠛⠀
⣤⡤⣴⡆⠀⡄⠰⠶⡆⠀⠀⢒⣆⣶⣿⣷⣦⡟⢷⣿⣗⣀⠀⠈⠀⠀⠀⠈⣿⣿⡿⠀⠀⠀⣿⣿⢻⡿⠁⠀⠻⡶⠄⠀⠀⣠⣶⢸⣿⡿⣿⣿⣿⣿⣿⣯⣿⣷⣶⣿⣏⣁⣉⣿⣿⢃⢁⣠⣶⣶⣿⣤⣤⣄⣀⠀⠀⠀
⣿⣶⣿⣿⣿⣶⣶⣶⠗⢰⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⡿⠁⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⣿⣿⣾⣧⠀⢀⣻⣿⣶⣶⣾⣿⣿⢁⣿⠁⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣷⣶
⡟⢻⣿⣿⣿⣿⣿⣿⠀⠀⠁⠙⣿⣿⣿⣿⣿⡇⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⢸⣿⣿⣿⣄⣸⣿⣿⣿⣿⣿⡟⢋⣼⡏⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣛⣿⣿⣿⣿⣿⣿⣦⣦⣬⣉⣿⣟⠯⠭⣽
⢃⣈⣛⣛⣉⣉⣛⣛⣀⣀⣀⣀⣛⣛⣛⣛⣛⣂⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣀⣀⣘⣛⣛⣛⣛⣀⣛⣛⣛⣛⣛⣛⣛⣛⣀⣀⣐⣛⣛⡛⣿⣿⣿⣿⣿⣿⣿⠁⢸⠇⢹⣿⣿⣯⣭⣭⣿⣿⣿⣿⣧⣤⣤⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⣿⠿⢿⣿⣿⡟⠉⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣭⣙
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢾⡿⣿⢿⣿⣮⣻⢸⣿⣷⣙⣾⣼⡯⣧⣷⣙⡜⣽⡸⣧⡿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⢦⡀⠀⠀⣿⡇⣧⠀⠙⠿⠗⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣛⣿⣿⣿⢿⣿⣏⣉⣉
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⣴⠖⣫⣶⣿⣿⣷⣦⣄⠀⢹⠀⠀⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢹⢹⡏⣿⣿⣿⣿⡿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⠟⣵⣿⡟⠁⠀⠙⠿⠿⠿⠷⡆⠀⠀⣿⡇⣛⣁⣈⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣋⣻⣿⡟⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠏⣭⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣿⣿⡇⠀⢀⣾⣿⣿⣿⡀⠀⠙⠛⣻⣿⣿⣿⠃⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡿⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣧⣿⣼⣽⣼⣿⡇⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡆⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡷⠔⠍⢹⢏⡊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠃⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣛⣟⣟⠻⠛⠃⠛⠉⠸⠼⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠈⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⢿⠛⠭⠁⠍⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠆⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣀⣀⣤⡾⣿
⢻⣿⠻⠿⢿⠫⠋⠁⠐⠈⠈⠓⠊⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⠁⠘
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠇⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠈⠙⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠙⠻⣿⠰⠘⠏⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣄⣀⡀⢀⡀⠀⠀⠀⠀⠀⡄⠀⣶⠀⢉⣾⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠉⠉⢀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⡿⠀⠀⠀⡈⢷⣦⡶⠂⠉⠁⠀⠀⠀⠘⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⡀⠀⠀⢸⣷⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⡿⢻⣿⣿⡏⠋⠀⢿⠟⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠆⠀⠀⣾⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⡿⣿⣿⡇⠸⠈⢹⠀⠀⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣟⣀⣠⣶⣦⡼⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠟⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣟⠀⢀⣘⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⠋⢹⣿⡌⠛⣹⣿⠀⠙⠛⠛⠿⣿⣽⣿⣿⣧⡙⠇⠀⠀⠀⠀⠀⠀⠀⢠⣤⡆⠀⠀⠀⣀⡲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠛⠑⣿⣿⣷⣾⣿⠋⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣥⣤⣶⣶⣄⠀⢀⢀⣀⣀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⢒⣿⣿⣿⣿⣿⡿⠟⠛⠛⣿⣛⠉⠀⣨⣤⣽⣿⣿⣾⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢿⡿⡏⠡⢀⡀⠈⠃⠀⢿⣟⢻⣿⣿⡿⠛⢻⣿⣿⣿⠟⠀⠀⠀⠘⣿⣿⣿⣿⣿⣯⣽⣟⣀⣀⠀⣀⡀⠀⣀⣦⣀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡤⣴⣿⣿⣼⠿⡄⢰⠁⠀⠐⢀⣾⣿⣿⠀⠈⢿⣦⣿⣜⣷⠀⢸⣿⠟⠋⠀⠀⠀⠀⢿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⠤⠔⠀⠀⠀⠀⠀⠀⠸⠛⠀
⣤⡤⣶⡦⠀⡄⠴⠶⠄⠀⠐⢒⣖⣶⣿⣷⣦⡟⢿⣿⣗⣀⠀⠈⠀⠀⠀⠈⣿⣿⡿⠀⠀⠀⣿⣿⢿⣿⠁⠀⢹⡶⠀⠀⠀⣠⣶⢸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣏⣀⣉⣽⣿⢟⣁⣠⣶⣶⣿⣤⣤⣄⣀⠀⠀⠀
⣿⣶⣿⣿⣿⣶⣶⣶⠗⢰⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⡿⠁⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⣿⣿⣾⣷⠀⢐⣻⣿⣶⣶⣶⣿⡿⢠⣿⠁⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣷⣶
⡟⢻⣿⣿⣿⣿⣿⣿⠀⠀⠁⠙⣿⣿⣿⣿⣿⡇⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⢸⣿⣿⣿⣄⣼⣿⣿⣿⣿⣿⣟⣡⣾⡏⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣧⣤⣿⣉⣽⣿⡻⠯⣽
⢃⣈⣛⣛⣉⣉⣛⣛⣀⣀⣀⣀⣛⣛⣛⣛⣛⣃⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣓⣀⣀⣀⣘⣛⣛⣛⣛⣈⣛⣛⣛⣛⣛⣛⣛⣛⣀⣀⣐⣛⣛⡛⣿⣿⣿⣿⣿⣿⣿⠁⠸⠃⢹⣿⣿⣯⣭⣭⣿⣿⣿⣿⣧⣤⣤⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⣿⠿⢿⣿⣿⡟⠉⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣭⣝
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢾⡿⣿⢿⣿⣮⣻⢸⣿⣷⣙⣾⣼⡯⣧⣷⣙⡜⣽⡸⣧⡿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⢦⡀⠀⠀⣿⡇⣷⠀⠘⠿⠗⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣛⣿⣿⣿⣿⣿⣿⣉⣉
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⣴⠖⣫⣶⣿⣿⣷⣦⣄⠀⢹⠀⠀⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢹⢹⡟⣿⣿⣿⣿⡿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣰⠟⣡⣿⡟⠁⠀⠙⠿⠿⠿⠷⡆⠀⠀⣿⡇⣟⣁⣈⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣋⣻⣿⡟⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣼⡄⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠏⣽⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣿⣿⡇⠀⢀⣾⣿⣿⣿⡀⠀⠙⠛⣻⣿⣿⣿⠃⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡿⠀⣿⡀⢿⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡇⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡆⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠛⠟⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠉⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣗⣙⡓⡳⠹⠟⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⢿⠃⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠯⠋⠉⠍⠃⠇⠁⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠈⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣻⢋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠇⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣄⣀⣤⡾⣿
⢻⡟⠻⠻⢹⠋⠋⠠⠋⠀⠈⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⠁⠘
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠇⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠈⠉⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠙⢻⣿⠒⠘⠋⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⠀⣀⡀⠀⠀⠀⠀⠀⠄⠀⣶⠀⢁⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠿⠉⠁⢀⠈⠁⠀⠀⢀⠀⠀⠀⡀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣾⡟⠀⠀⢀⡙⣷⣾⠆⠂⠈⠁⠀⠀⠀⠘⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣃⡀⠀⠀⢸⣷⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⡿⠻⣿⣿⡏⠋⠀⢿⠟⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⡿⣿⣿⡇⠘⠀⢹⠀⠀⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⡿⠛⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣏⣀⣠⣶⣦⡹⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠟⠹⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣟⠀⢀⡸⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⠋⢻⣷⡍⠛⣸⡯⠀⠙⠛⠛⢿⣟⣽⣿⣿⣇⣙⠇⠀⠀⠀⠀⠀⠀⠀⢠⣤⡆⠀⠀⠀⣠⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⠏⠷⣿⣿⣷⣾⣿⠋⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣤⣤⣶⣶⣄⣀⢀⢀⣀⣠⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⢘⣿⣿⣿⣿⣿⡿⠟⠉⠻⣿⣏⠁⠀⣠⣤⣾⣿⣿⣾⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⡼⢿⡿⡏⠡⢀⡀⠈⠃⠀⢿⡏⢻⣿⣿⡟⠙⢻⣿⣿⣿⠃⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⣀⣀⣀⣀⣀⣖⣀⠀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠦⣴⣿⣿⣾⠿⠄⢸⠁⠀⠐⢀⣾⣿⣿⠀⠈⢿⣦⣿⣝⣷⠀⢼⣿⠟⠁⠀⠀⠀⠀⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣷⣤⣤⠤⠐⠀⠀⠀⠀⠀⠀⠸⠛⠀
⣤⠤⢶⡦⢀⡄⠴⠶⠄⠀⠐⢒⣖⣶⣿⣶⣦⡿⢿⣿⣗⣀⠀⠈⠀⠀⠀⠀⣿⣿⡿⠀⠀⠈⣿⣟⢿⡿⠀⠈⢻⡖⠀⠀⢀⣴⣦⢺⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣏⣁⣩⣿⣿⣏⣁⣠⣶⣶⣿⣤⣤⣄⣀⠀⠀⠀
⣿⣶⣿⣾⣿⣷⣶⣶⠇⢰⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⡿⠁⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⣿⣿⣾⣷⠀⢰⣻⣿⣖⣶⣾⣿⡟⢠⣿⠀⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣷
⡏⢻⣿⣿⣿⣿⣿⣿⠀⠀⠁⢙⣿⣿⣿⣿⣿⡇⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⢻⣿⣿⣿⣄⣼⣿⣿⣿⣿⣿⣟⣡⣿⡏⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣦⣬⣭⣉⣽⣿⠿⠯⣿
⢃⣀⣛⣛⣉⣉⣙⣛⣀⣀⣀⣀⣛⣛⣛⣛⣛⣃⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣓⣀⣀⣀⣘⣟⣛⣛⣛⣈⣛⣛⣛⣛⣛⣛⣛⣛⣀⣀⣐⣛⣛⡛⣿⣿⣿⣿⣿⣿⣿⠁⠸⠋⢙⣿⣿⣯⣭⣽⣿⣿⣿⣿⣧⣤⣤⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⡿⠟⢿⣿⣿⡟⠉⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣭⣜
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢾⡿⣿⢿⣿⣮⣻⢸⣿⣷⣙⣾⣽⡧⣧⣷⣙⡜⣽⡸⣧⡿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⢦⡀⠀⠀⣿⡇⡷⠀⠘⠿⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣛⣿⣿⣿⠿⣾⣏⣉⣉
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⣴⠖⣫⣶⣿⣿⣷⣦⣄⠀⢹⠀⠀⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢹⢹⡏⣿⣿⣿⣿⡿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢰⠟⣱⣿⡟⠁⠀⠙⠿⠿⠿⠷⡆⠀⠀⣿⡇⣟⣁⣈⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣋⣻⣿⡟⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠏⣭⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣯⠉⣿⡍⣿⣿⡇⠀⢀⣾⣿⣿⣿⡀⠀⠙⠛⣻⣿⣿⣿⠃⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡿⠀⣿⡀⢿⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣧⣿⣼⣽⣼⣿⡗⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡯⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡎⠙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡛⢭⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠉⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠭⢁⡰⠻⠮⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⢿⠃⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⠟⠙⠙⠙⠋⠄⠀⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠈⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⢿⣿⣋⡭⡉⠍⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠃⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣄⣠⣴⡾⣿
⣿⡟⠿⠛⢿⠋⠃⠒⠅⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠇⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠈⠉⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠙⢻⡷⠒⠘⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⡀⢀⠀⠀⠀⠀⠀⠀⠄⠀⣶⠀⢁⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠟⠉⠁⢀⠈⠁⠀⠀⢀⠀⠀⠀⡀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣾⡟⠀⠀⢀⡘⣷⣮⠆⠂⠉⠁⠀⠀⠀⠘⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣃⠀⠀⠀⣿⣷⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡏⢻⣿⣿⡿⢻⣿⣿⠏⠋⠀⢿⠟⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠄⠀⠀⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⡿⣿⣿⡆⠘⠈⢹⠀⠀⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⡿⠛⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣏⣀⣤⣶⣦⣻⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠟⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣮⣏⠀⢀⡸⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠉⢻⣷⡉⢋⣹⡏⠀⠙⠛⠛⢿⣯⣿⣿⣿⣇⡙⠃⠀⠀⠀⠀⠀⠀⠀⢤⣴⠆⠀⠀⠀⣠⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠷⣿⣿⣷⣾⡿⠁⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣤⡤⣾⣶⣄⠀⢀⢀⣀⣠⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣟⢘⣿⣿⣿⣿⣿⡿⠿⠋⠙⣿⣟⠉⠀⣠⣤⣾⣿⣿⣾⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⡾⠿⡿⡏⠆⢀⠀⠘⠃⠀⣿⡏⢻⣿⣿⡟⠙⢿⣿⣿⣿⠋⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⣀⣀⣀⣀⣀⢖⣀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠦⣾⣿⣿⣾⠿⠄⢸⠁⠀⠐⢠⣾⣿⣿⠀⠘⢿⣮⣿⣙⣧⠀⣼⣿⠛⠃⠀⠀⠀⠀⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣦⣤⠤⠐⠀⠀⠀⠀⠀⠀⠘⠛⠀
⣤⠤⢶⡦⢀⡄⠴⠶⠆⠀⢐⣒⣖⣶⣿⣶⣦⡿⢿⣿⣗⣀⠀⠈⠀⠀⠀⠈⣿⣿⡿⠀⠀⠘⣿⣿⢿⡿⠀⠈⢻⡶⠀⠀⢀⣴⣦⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣏⣀⣨⣿⣟⢋⣀⣤⣶⣶⣿⣦⣴⣤⣀⡀⠀⠀
⣿⣶⣿⣾⣿⣷⣶⣾⠇⢰⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⡟⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠐⣿⣿⣾⣧⠀⣐⣻⣿⣶⣶⣿⣿⡟⣸⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣿⣿⣿⣶
⡏⢻⣿⣿⣿⣿⣿⣿⠀⠀⠁⢙⣿⣿⣿⣿⣿⡇⠀⠟⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⢿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⡏⣥⣿⠏⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣷⣮⣭⣉⣽⣿⠿⠯⣿
⢃⣈⣛⣛⣉⣉⣛⣛⣀⣀⣀⣀⣛⣛⣛⣛⣛⣂⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣃⣀⣀⣀⣘⣛⣛⣛⣓⣘⣛⣛⣛⣛⣛⣛⣛⣛⣀⣀⣐⣛⣛⡛⣿⣿⣿⣿⣿⣿⣿⠁⠸⠂⢹⣿⣿⣿⣤⣭⣿⣿⣿⣿⣧⣤⣴⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⡿⠟⢿⣿⣿⡟⠈⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣭⣜
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢾⡿⣿⢿⣿⣮⣻⢸⣿⣷⣙⣾⣽⡯⣷⣷⣙⡜⣽⡸⣧⡿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⢦⡀⠀⠀⣿⡇⡷⠀⠘⠿⠧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⣉
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⣴⠖⣫⣶⣿⣿⣷⣦⣄⠀⢹⠀⠀⣿⡇⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢹⢹⡏⣿⣿⣿⣿⡿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⠟⣡⣿⡟⠁⠀⠙⠿⠿⠿⠷⡆⠀⠀⣿⡇⣏⣁⣈⣤⣀⣀⣀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣋⣻⣿⡟⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠏⣽⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣯⠉⣿⡍⣿⣿⡇⠀⢀⣾⣿⣿⣿⡀⠀⠙⠛⣿⣿⣿⣿⠃⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡿⠀⣿⡀⢿⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⣇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣧⣿⣼⣽⣼⣿⡇⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡎⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠉⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⠯⣯⡁⡑⣹⢟⠒⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⢿⠁⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⢿⠿⠏⠅⠋⡙⠋⠤⢅⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣯⠈⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡿⣿⣏⡇⠉⠉⠁⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠆⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣄⣠⣴⡾⣿
⠻⠿⠻⠻⢟⠏⠉⠀⠍⠀⠑⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠀⠙
⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠃⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠈⠉⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⠙⢻⣷⠒⠘⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢡⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠟⠉⠁⠀⠈⠁⠀⠀⢀⠀⠀⠀⠀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣠⣾⡇⠀⠀⣀⡹⣶⣾⠆⠊⠈⠁⠀⠀⠀⠘⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣃⠀⠀⠀⣿⣷⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡏⢿⣿⣿⡿⢻⣿⣿⠏⠋⠐⢿⠟⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠀⠀⠰⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡿⣿⣿⠀⠘⠈⢹⠀⠈⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⡿⠛⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢇⣀⣴⣶⣦⢻⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠃⠏⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣬⣏⠀⢀⡸⣿⣿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠉⣻⣧⠙⢋⣹⡏⠀⠙⠛⠛⢿⣯⣽⣿⣿⣇⡙⠃⠀⠀⠀⠀⠀⠀⠀⢠⣴⠆⠀⠀⠀⣠⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠷⣿⣿⣷⣾⡟⠁⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣤⣤⣾⣶⣄⠀⢀⣀⣀⣠⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣟⢸⣿⣿⣿⣿⣿⡿⠟⠉⠹⣿⣯⡀⠀⣠⣤⣿⣿⣿⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⡶⢿⡿⡏⠆⢀⠀⠘⠃⠈⣿⡟⢿⣿⣿⡏⠋⢿⣿⣿⣿⠏⠀⠀⠀⢛⣿⣿⣿⣿⣿⣯⣿⣏⣀⣀⣀⣀⣀⣀⣀⢖⣀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠦⣶⣿⣿⣾⠿⠄⢸⠁⠀⠐⢠⣾⣿⣿⠀⠘⣷⣼⣿⣟⡇⠀⣼⣿⠛⠁⠀⠀⠀⠰⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣤⣤⠤⠐⠀⠀⠀⠀⠀⠀⠘⠛⠀
⣤⠤⣶⡦⠀⡄⠴⠶⡆⠀⠐⣒⣖⣶⣿⣶⣦⡿⢿⣿⣗⣀⠀⠈⠀⠀⠀⠈⣿⣿⡟⠀⠀⠸⣿⡟⢿⡏⠀⠈⢿⡆⠀⠀⢀⣴⣦⣿⣿⡟⣿⣿⣿⣿⣿⣷⣿⣿⣶⣿⣏⣀⣩⣿⣟⢋⣀⣤⣶⣶⣿⣦⣴⣤⣀⣀⠀⠀
⣿⣶⣷⣿⣿⣷⣶⣶⠇⢰⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⡟⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠸⣿⣿⣿⡇⠀⢐⣻⣿⣖⣶⣿⣿⡟⢸⣿⠀⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣿⣿⣿⣷
⠏⢹⣿⣿⣿⣿⣿⣿⠀⠀⠁⢙⣿⣿⣿⣿⣿⡇⠀⠟⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⣿⣿⣿⣷⣄⣿⣿⣿⣿⣿⣿⣏⣡⣿⡋⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣻⣿⣿⡿⣿⣿⣿⣶⣮⣭⣩⣽⣟⠻⠯⣽
⢃⣈⣛⣛⣉⣉⣛⣛⣀⣀⣀⣀⣛⣛⣛⣛⣛⣂⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣓⣀⣀⣀⣘⣟⣛⣛⣓⣘⣛⣛⣛⣛⣛⣛⣛⣛⣀⣀⣐⣛⣛⡛⣿⣿⣿⣿⣿⣿⣻⠁⠸⠂⢙⣿⣿⣷⣬⣭⣿⣿⣿⣿⣧⣤⣴⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⡿⠟⢿⣿⣿⡏⠈⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣯⣭⣌
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢾⡾⣿⢿⣿⣮⣻⢸⣿⣷⣙⣾⣼⡯⣧⣷⣙⡜⣽⡸⣧⡿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⢦⡀⠀⠀⣿⡇⡷⠀⠘⠿⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣉⣉
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⣴⠖⣫⣶⣿⣿⣷⣦⣄⠀⢹⠀⠀⣿⡇⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢹⢹⡟⣿⣿⣿⣿⡿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢰⠟⣱⣿⡟⠁⠀⠙⠿⠿⠿⠷⡆⠀⠀⣿⡇⣋⣁⣈⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣋⣻⣿⡟⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣼⡄⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣽⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣿⣿⡇⠀⢀⣾⣿⣿⣿⡀⠀⠙⠛⣻⣿⣿⣿⠃⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡿⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡧⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣧⣿⣼⣽⣼⣿⡇⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡎⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡭⠿⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⡿⠉⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡭⡁⡑⣸⠿⠗⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⢿⠃⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⣿⠻⣋⠊⠉⡥⠋⠅⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠈⠀⠀⠀⢀
⣿⣿⣿⣿⣿⢿⣿⢟⣻⣿⡟⢿⢝⠃⠛⠉⠑⠈⠀⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣤⣤⣴⡾⣿
⢿⡿⢿⠛⢻⠫⠘⠑⠰⠂⠒⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠃⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⠈⠉⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠙⢻⣿⠒⠘⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣀⠀⣀⠀⠀⠀⠀⠀⠀⠄⠐⡶⠀⢩⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠟⠉⠁⠀⠈⠁⠀⠀⢀⠀⠀⠀⠀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣠⣾⡇⠀⠀⢀⡹⣶⣞⠆⠊⠈⠁⠀⠀⠀⠘⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣃⠀⠀⠀⣿⣷⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡏⢿⣿⣿⡿⢻⣿⣿⠏⠋⠀⢿⠟⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠀⠀⢰⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⠀⠘⠈⢹⠀⠈⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢃⣀⣴⣶⣦⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠏⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣏⠀⢀⡸⣿⣿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠉⣻⣧⠙⢋⣹⡏⠀⠙⠛⠛⢿⣯⣿⣿⣿⣄⡙⠃⠀⠀⠀⠀⠀⠀⠀⢤⣴⠆⠀⠀⠀⣠⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠛⠷⣿⣿⣷⣾⡟⠁⠀⠀⢠⣴⣾⣿⣿⣿⣿⣧⣤⣴⣾⣶⣀⣀⣀⣀⣀⣠⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⠿⠛⠉⠹⣿⣯⠈⠀⣠⣤⣿⣿⣿⡾⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⣴⠿⡿⡏⠆⢀⠀⠈⠃⠈⣿⡟⢿⣿⣿⡏⠙⢿⣿⣿⡿⠇⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⣀⣈⣀⣀⣀⢖⣀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠦⣼⣿⣿⣾⠿⠄⢸⠁⠀⠐⢠⣾⣿⣿⠀⠘⣷⣼⣿⣟⣇⠀⣼⣿⠛⠁⠀⠀⠀⠰⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣶⣶⣦⠤⠒⠀⠀⠀⠀⠀⠀⠘⠛⠀
⣤⠤⢴⡆⢀⡄⠴⠶⡆⠀⢐⣒⣶⣶⣿⣶⣦⡿⢿⣿⣗⣀⠀⠈⠀⠀⠀⠈⣿⣿⡿⠀⠀⠸⣿⡟⢿⡏⠀⠈⢷⡆⠀⠀⢀⣴⣦⣿⣿⡟⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣇⣀⣩⣿⣟⣋⣀⣤⣶⣾⣿⣤⣤⣤⣀⡀⠀⠀
⣿⣶⣷⣿⣿⣷⣶⣶⠇⢰⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⡟⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠸⣿⣿⣿⡇⠀⣐⣻⣿⣖⣶⣾⣿⡏⢸⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣋⣿⣿⣿⣷
⡏⢻⣿⣿⣿⣿⣿⣿⠀⠀⠁⢙⣿⣿⣿⣿⣿⡇⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⣿⣿⣿⣷⣄⣿⣿⣿⣿⣿⣿⡏⣠⣿⠏⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⣿⣿⣿⣿⣿⣿⣶⣤⣭⣉⣽⣟⠯⠯⣽
⢃⣈⣛⣛⣉⣉⣛⣛⣀⣀⣀⣀⣛⣛⣛⣛⣛⣂⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣓⣀⣀⣀⣘⣟⣛⣛⣓⣘⣛⣛⣛⣛⣛⣛⣛⣋⣀⣀⣐⣛⣛⡛⣿⣿⣿⣿⣿⣿⣻⠁⠸⠂⢹⣿⣿⣷⣤⣭⣿⣿⣿⣿⣧⣤⣴⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⡿⠟⢿⣿⣿⡏⠈⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣭⣍
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢾⡾⣿⢿⣿⣮⣻⣸⣿⣷⣙⣾⣽⡯⣧⣷⣙⡜⣽⡸⣧⡿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⢦⡀⠀⠀⣿⡇⡷⠀⠈⠿⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣉⣉
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⣴⠖⣫⣶⣿⣿⣷⣦⣄⠀⢹⠀⠀⣿⡇⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢹⢹⡟⣿⣿⣿⣿⡿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⠟⣱⣿⡟⠁⠀⠙⠿⠿⠿⠷⡆⠀⠀⣿⡇⣋⣁⣈⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣋⣻⣿⡟⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣼⡄⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣽⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣿⣿⡇⠀⢀⣾⣿⣿⣿⡀⠀⠙⠛⣻⣿⣿⣿⠃⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡿⠀⣿⡀⢿⣿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣧⣿⣼⣽⣼⣿⡇⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡎⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠯⠏⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡟⠉⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⡬⡱⠯⠟⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⢿⠁⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡯⠿⢛⠉⠋⣻⠁⠛⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡻⠋⢙⡋⠁⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣤⣤⣶⡿⣿
⣿⢿⠿⠻⣿⠋⠈⠠⠔⠒⠐⠓⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠉
⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠎⠃⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⠈⠉⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⠉⠻⡷⠒⠘⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣀⠀⣀⠀⠀⠀⠀⠀⠀⠄⠐⡶⠀⢩⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠏⠉⠁⠀⠈⠁⠀⠀⢀⠀⠀⠀⡀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣠⣾⠇⠀⠀⣀⡹⣶⣾⠆⠂⠈⠁⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣃⠀⠀⠀⣿⣷⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡏⢿⣿⣿⡿⢻⣿⣿⠏⠋⠀⢿⠛⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠀⠀⠠⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⠀⠘⠈⢹⠀⠘⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢃⣀⣤⣶⣦⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠋⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣼⣏⠀⢀⡸⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⣿⠉⣻⣧⡙⢋⣹⡏⠀⠙⠛⠛⢿⣯⣿⣿⣿⡄⣙⠃⠀⠀⠀⠀⠀⠀⠀⠤⣴⠆⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠋⢷⣿⣿⣷⣿⡟⠁⠀⠀⢠⣴⣾⣿⣿⣿⣿⣷⣦⣴⣾⣷⣂⢀⣀⣀⣀⣤⢀⢀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣗⢸⣿⣿⣿⣿⣿⡿⠛⠋⢙⣿⣏⠀⠀⣤⣤⣿⣿⡿⠿⠯⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣤⡾⠿⡟⠏⠃⢀⠀⠈⡃⠈⣿⡟⢿⣿⣿⡏⠋⢿⣿⣿⣿⠇⠀⠀⠀⣛⣹⣿⣿⣿⣿⣿⣿⣏⣠⣄⣀⣀⣀⣀⣀⢲⣄⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠦⣼⣿⣿⣾⠿⠄⢰⠀⠀⠀⢠⣾⣿⣿⠀⠘⣷⣼⣿⣟⣇⠀⣼⣿⠛⠃⠀⠀⠀⠰⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣦⠴⠂⠀⠀⠀⠀⠀⠀⠘⠛⠀
⣴⠶⣶⡆⢀⡄⠶⠶⡆⠀⢐⣲⣧⣶⣿⣶⣶⡟⢿⣿⣗⡀⠀⠈⠀⠀⠀⠈⣿⣿⡿⠀⠀⠸⣿⡟⢿⡏⠀⠈⢿⡆⠀⠀⢀⣴⣆⣿⣿⡟⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣯⣀⣤⣿⣟⣓⣀⣤⣾⣿⣿⣤⣤⣤⣄⣀⠀⠀
⣿⣾⣿⣿⣿⣷⣶⣿⠇⢰⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⡟⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠸⣿⣿⣿⡇⠀⣐⣻⣿⣖⣶⣿⣿⡏⣸⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣿⣿⣿⣷
⠏⢹⣿⣿⣿⣿⣿⣿⠀⠀⠁⢙⣿⣿⣿⣿⣿⡇⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⣿⣿⣿⣷⣄⣿⣿⣿⣿⣿⣿⡏⣡⡿⡏⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⡿⣿⣿⣿⣷⣶⣭⣭⣽⡿⠋⠿⣽
⢁⣈⣛⣛⣉⣈⣙⣛⣀⣀⣀⣀⣛⣛⣛⣛⣛⣂⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣓⣀⣀⣀⣙⣟⣛⣛⣓⣘⣛⣛⣛⣛⣛⣛⣛⣛⣀⣀⣐⣛⣛⡛⣿⣿⣿⣿⣿⣿⢻⡁⠸⠂⢹⣿⣿⣷⣦⣽⣿⣿⣿⣿⣷⣦⣴⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⡿⠟⣿⣿⣿⡏⠈⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣯⣥⣌
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢾⡾⣿⢿⣿⣮⣻⢸⣿⣷⣙⣾⣼⡯⣧⣷⣙⡜⣽⡸⣧⡿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⢦⡀⠀⠀⣿⡇⡷⠀⠘⠟⠧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣟⣿⣿⣟⣻⣿⣯⣁⣀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⣴⠖⣫⣶⣿⣿⣷⣦⣄⠀⢹⠀⠀⣿⡇⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢹⢹⡟⣿⣿⣿⣿⡿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⠟⣡⣿⡟⠁⠀⠙⠿⠿⠿⠷⡆⠀⠀⣿⡇⣋⣁⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣋⣻⣿⡟⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠏⣽⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣿⣿⡇⠀⢀⣾⣿⣿⣿⡀⠀⠙⠛⣿⣿⣿⣿⠃⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡿⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣧⣿⣼⣽⣼⣿⡇⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡆⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⠛⠎⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠉⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡵⠫⠿⡎⠿⡑⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⢿⠃⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡻⣻⢿⣿⠛⠏⠍⠛⠒⠰⠂⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢻⠝⠋⠛⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠇⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣤⣤⣶⡞⣿
⡿⣟⠻⡟⢿⠙⠈⠈⢱⠀⠌⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡎⠇⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⠀⠉⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠉⠻⣿⠒⠘⠋⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⣀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⡶⠀⢈⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠟⠉⠁⢀⠈⠁⠀⠀⢀⡀⠀⠀⡀⢤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣾⡇⠀⠀⢀⡹⣷⣾⠆⠈⠈⠁⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣃⠀⠀⠀⣹⣷⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⡿⠛⣿⣿⡏⠋⠀⢿⠟⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠀⠀⠀⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠘⠀⢹⠀⠀⠏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⡿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⣀⣠⣶⣤⢻⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠏⠛⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣏⠀⢀⡸⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠉⢻⣧⡙⢛⣹⡏⠀⠙⠛⠛⢿⣿⣿⣿⣿⣇⣙⠃⠀⠀⠀⠀⠠⠀⠀⠠⣤⠆⠀⠀⠀⣠⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠵⣿⣿⣷⣾⡿⠃⠀⠀⢠⣤⣾⣿⣿⣿⣿⣷⣦⣴⣾⡷⣆⣀⣀⣀⣀⣤⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣗⢸⣿⣿⣿⣿⣿⡿⠛⠋⠛⣿⣿⡁⠀⣤⣤⣼⣿⡿⠿⠿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⡶⠿⡟⠏⠑⢀⡀⠈⠃⠀⣿⡏⢻⣿⣿⡟⠙⢿⣿⣿⣿⠏⠀⠀⠀⣛⣻⣿⣿⣿⣿⣿⣿⣏⣠⣄⣀⣀⣀⣀⣀⣶⣄⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠠⣀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠦⣼⣿⣿⣾⠿⠆⠰⠀⠀⠀⠀⣾⣿⣿⠀⠘⢿⣬⣿⣝⣇⠀⣼⣿⠛⠁⠀⠀⠀⠐⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⠴⠒⠀⠀⠀⠀⠀⠀⠘⠛⠀
⣴⠴⢶⡆⢀⡄⠲⠲⡆⠀⣐⣚⣷⣶⣿⣶⣶⡿⢿⣿⣷⡀⠀⠈⠀⠀⠀⠀⣿⣿⡿⠀⠀⠘⣿⡟⢿⡏⠀⠈⢷⡆⠀⠀⢀⣴⣦⢿⣿⡟⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣧⣀⣤⣿⣿⣃⣀⣤⣴⣿⣿⣦⣤⣤⣄⣀⠀⠀
⣿⣾⣿⣿⣿⣷⣶⣿⠗⠰⣿⣿⣿⣿⣿⣿⣿⡟⠈⣿⡿⠁⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠐⣿⣿⣾⡷⠀⣰⣻⣿⣶⣶⣿⣿⡟⢸⣿⠀⢿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣿⣿⣿⣿
⡏⢹⣿⣿⣿⣿⣿⣿⠀⠀⠁⠘⣿⣿⣿⣿⣿⡇⠀⢿⠗⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇⠀⠀⠀⣿⣿⣿⣷⣠⣿⣿⣿⣿⣿⣿⡟⣡⣿⡏⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⡿⣿⣿⣿⣷⣶⣯⣭⣽⡿⠿⠯⣿
⢁⣀⣛⣛⣉⣀⣙⣛⣀⣀⣀⣀⣛⣛⣛⣛⣛⣃⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣒⣀⣀⣀⣘⣟⣛⣛⣛⣘⣛⣛⣛⣛⣛⣛⣛⣛⣀⣀⣐⣛⣛⡛⣿⣿⣿⣿⣿⣿⣿⡁⠸⠇⢹⣿⣿⣿⣦⣽⣿⣿⣿⣿⣷⣦⣴⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⡿⠟⢻⣿⣿⡟⠈⠁⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣭⣜
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢾⡾⣿⢿⣿⣮⣻⢸⣿⣷⣙⣾⣼⡯⣷⣷⣙⡜⣽⡸⣧⡿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⢦⡀⠀⠀⣿⡇⡷⠀⠘⠿⠷⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣽⣿⣿⣟⣻⣿⣿⣁⣉
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⣴⠖⣫⣶⣿⣿⣷⣦⣄⠀⢹⠀⠀⣿⡇⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢻⢹⡟⣿⣿⣿⣿⡿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⠟⣡⣿⡟⠁⠀⠙⠿⠿⠿⠷⡆⠀⠀⣿⡇⣏⣁⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣋⣻⣿⡟⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠏⣭⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣿⣿⡇⠀⢀⣾⣿⣿⣿⡀⠀⠙⠛⣻⣿⣿⣿⠃⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡿⠀⣿⡀⢿⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⣇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣧⣿⣼⣽⣼⣿⡇⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠋⠁⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠁⢀⣯⠛⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⠏⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢻⢿⠻⠩⠓⠒⠂⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠄⠁⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣛⣿⡿⠻⠛⠫⠻⢏⡛⠛⠛⠊⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠇⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣥⣀⣤⡾⢿
⢿⣻⠿⠛⠻⠂⠁⠉⠠⠀⠐⠀⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡆⠏⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠈⠉⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⡿⡿⠉⠻⣿⠒⠘⠛⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣶⠀⠈⣾⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡟⠉⠁⠀⠈⠁⠀⠀⢀⡀⠀⠀⡀⢠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣼⡟⠀⠀⢀⡹⣶⣼⠆⠘⠀⠁⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⡀⠀⠀⢸⣷⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⡿⠟⣿⣿⡏⠙⠀⢻⡟⠏⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠄⠀⠀⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⠘⠁⢹⠇⠀⠟⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠟⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⢀⣠⣤⣦⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠻⠹⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣟⠀⠀⣸⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠛⢻⣧⡙⠛⣹⡟⠀⠙⠛⠿⢿⣿⣿⣿⣿⣇⣘⠇⠀⠀⠀⠀⠀⠀⠀⠠⣤⡦⠀⠀⠀⣀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠧⣿⣿⣧⣾⡿⠋⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣤⣤⣶⣷⣄⠀⣀⣀⣀⣠⡀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣇⢐⣿⣿⣿⣿⣿⡿⠟⠛⠙⣿⣿⡉⠉⣁⣤⣼⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣴⠿⡿⢿⠱⢀⡀⠈⠓⠈⣿⡟⢿⣿⣿⡟⠛⢿⣿⣿⣿⠏⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣏⣁⣀⣀⣀⣀⢀⣀⡔⣀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣴⣿⣿⣶⡿⠇⠠⠁⠀⠀⠀⣾⣿⣿⠀⠘⢷⣌⢿⡽⢇⠀⣸⣿⠟⠃⠀⠀⠀⠀⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣶⣤⠤⠔⠀⠀⠀⠀⠀⠀⠐⠛⠀
⣤⡤⣶⣶⠀⡄⠰⠶⠆⠀⢀⣒⣶⣶⣾⣷⣶⣿⢿⣿⣷⣀⠀⠀⠁⠀⠀⠀⢿⣿⣿⠀⠀⠈⣿⡿⢿⡿⠀⠈⢻⡦⠀⠀⠀⣴⣦⢼⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣁⣈⣽⣿⣛⣄⣠⣴⣷⣿⣦⣤⣤⣀⣀⠀⠀
⣿⣶⣯⣼⣿⣷⣶⣶⠗⠀⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⢙⣿⡇⠀⠀⠀⣿⣿⣾⣗⠀⢰⣾⣿⣶⣶⣿⣿⡟⢠⣿⠁⢸⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣻⣿⣿⣷
⡟⠹⣿⣿⣿⣿⣿⣿⠀⠀⠉⠘⣿⣿⣿⣿⣿⡇⠀⢻⠿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⢿⣿⣿⣿⣄⣼⣿⣿⣿⣿⣿⡟⣡⣿⡟⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⡿⣿⣿⣿⣷⣼⣭⣉⣽⣿⠟⠩⣭
⢃⣀⣛⣛⣉⣉⣙⣛⣀⣀⣀⣀⣛⣛⣛⣛⣛⣃⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣙⣀⣀⣀⣘⣟⣛⣛⣋⣘⣛⣛⣛⣛⣛⣛⣛⣛⣁⣀⣀⣛⣛⡛⣿⣿⣿⣿⣿⣿⣿⡇⠸⠇⠙⣿⣿⣿⣥⣭⣿⣿⣿⣿⣧⣤⣤⣾
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⣿⠿⢿⣿⣿⡿⠁⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣭⣍
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢶⡾⣿⣿⣿⣮⣻⢸⣿⣷⣙⣾⣽⡯⠧⣷⣙⡜⣽⡸⣧⡿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⢦⡀⠀⠀⣿⡇⡷⠀⠈⠿⠷⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣿⣿⣿⢿⣿⣿⣋⢉
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⣴⠖⣫⣶⣿⣿⣷⣦⣄⠀⢹⠀⠀⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢻⢹⡟⣿⣿⣿⣿⡿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⠟⣱⣿⡟⠁⠀⠙⠿⠿⠿⠧⡆⠀⠀⣿⡇⣛⣁⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣋⣻⣿⡿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣽⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣿⣿⡇⠀⢀⣾⣿⣿⣿⡀⠀⠙⠛⣻⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡿⠀⣿⡀⢿⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣧⣿⣼⣽⣼⣿⡇⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡷⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡦⠙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣏⠟⡟⠛⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠏⠁⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠕⠘⠽⡗⡛⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡟⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⠿⠿⢿⠯⡿⠛⢫⠿⠃⠀⠐⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠁⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠟⠿⠿⠟⠛⠋⠙⠊⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣧⣀⣤⣴⢿
⣿⠟⠛⠟⢻⡿⡉⠱⠝⠅⠠⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠃⠘
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠛⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡄⠉⠁⢀⣼⣿⣿⣿⣿⣿⣿⣿⡿⣿⠛⠛⣿⡗⠂⠿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠆⠈⣼⣿⣿⣿⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣮⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠋⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣼⣿⠀⠀⠀⡨⣷⣾⣆⠘⠀⠁⠀⠀⠀⠈⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⢸⣿⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⠟⣿⣿⡟⠙⠁⢺⡿⠟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡅⠀⠀⣼⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⢿⣿⡇⠀⠃⠙⡇⠀⠟⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠀⢀⣤⣤⡸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠸⠹⠇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⠁⠀⠈⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠗⢺⣿⡙⠛⢹⣿⠁⠙⠛⠿⢿⣿⣿⣿⣿⣇⢘⠟⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⢀⣰⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠿⠓⣾⣿⣷⣴⣿⠏⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣭⣤⣴⣶⢆⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣶⣶⠒⣾⣿⣿⣿⣿⣿⠿⠛⠻⣿⣿⠛⠉⢉⣩⣽⣿⣿⣷⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣤⡿⣯⢿⠡⠀⡀⠀⠻⠀⢿⡿⢿⣿⣿⡿⠳⢿⣿⣿⣿⡟⠁⠀⠀⠘⢻⣿⣷⣿⣿⣿⣽⣟⣁⣀⡀⠀⣀⠀⢀⡰⣀⡀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠦⣰⣾⣿⣷⡿⠇⠠⡁⠀⠀⠀⣼⣿⣿⡆⠘⢿⣆⣿⣾⢷⠀⢠⣿⡿⠟⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣽⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠐⠛⠃
⣤⡤⢴⣦⠄⣀⠠⠶⢦⠀⠀⢐⣶⣴⣼⣷⣶⣿⢿⣿⣿⣀⠀⠀⠁⠀⠀⠀⢻⣿⣿⠃⠀⠈⣿⣿⢿⣾⠃⠈⢻⣆⠀⠀⠀⣰⣦⢸⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣉⣉⣹⣿⡿⣀⣀⣴⣶⣾⣦⣤⣤⣀⡀⠀⠀
⣿⣷⣿⣤⣿⣿⣴⣶⣾⠀⣾⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⣿⣿⣾⣷⠀⢀⣞⣿⣶⣤⣾⣿⡿⢀⣿⡇⢸⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣶
⣿⠛⣿⣿⣿⣿⣿⣿⡇⠀⠈⠈⣿⣿⣿⣿⣿⡿⠀⢹⢿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⢸⣿⣿⣿⣄⣸⣿⣿⣿⣿⣿⡿⢃⣼⣿⠁⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⣿⣿⣿⣿⣿⣷⣤⣼⣛⣿⣿⡿⠩⣽
⢛⣀⣙⣛⣋⣉⣙⣛⣃⣀⣀⣀⣙⣛⣛⣛⣛⣃⣀⣘⣂⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣀⣀⣀⣘⣛⣛⣛⣛⣉⣛⣛⣛⣛⣛⣛⣛⣛⣃⣀⣀⣚⣛⡛⣿⣿⣿⣿⣿⣿⣟⡇⠰⠇⠙⣿⣿⣿⣭⣭⣿⣿⣿⣿⣯⣤⣤⣼
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⣿⠿⢿⣿⣿⣿⠁⠁⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯⣍
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢶⡾⣿⣿⣿⣯⣻⢸⣿⣷⣙⣾⣸⡯⡧⣷⣙⡜⣽⡸⣷⡿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⢦⡀⠀⠀⣿⡇⣿⠄⠈⠿⠿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣛⣿⣿⣿⢿⣷⣿⡏⢉
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⣴⠖⣫⣶⣿⣿⣷⣦⣄⠀⢹⠀⠀⣿⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢻⢹⡟⣿⣿⣿⣿⡿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢰⠟⣱⣾⡟⠁⠀⠙⠿⠿⠿⠧⡆⠀⠀⣿⡇⣟⣋⣀⣁⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣋⣻⣿⡿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣽⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣿⣿⡇⠀⢀⣾⣿⣿⣿⡀⠀⠙⠛⣻⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡿⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡏⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣧⣿⣼⣽⣼⣿⡧⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠷⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡟⠋⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢏⣋⡔⢒⣏⡻⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡟⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡿⠛⡏⠽⠋⣋⠙⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠿⠛⠿⠙⠿⠋⠋⠈⠈⠂⠐⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣧⣀⣠⣴⢾
⣿⣿⡿⡿⠿⠎⠩⠏⠑⠒⠢⠐⠐⠓⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠏⠀
⠂⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢸⢡⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡆⠉⠉⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠛⣿⡗⠆⠻⠃⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⡆⠀⢲⣿⣿⣿⣥⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠛⠋⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣩⣿⠀⠀⠀⠘⢷⣤⣷⠀⠂⠉⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⢰⣿⣼⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⡟⢿⣿⣿⠙⠁⠸⣿⢻⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⠀⠏⠙⡏⠀⢸⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠁⢀⣤⣤⡸⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠸⠛⠇⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⠃⠀⠈⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⣿⡗⠲⣿⡘⠛⠋⣿⠁⠙⠟⠿⢿⣿⣫⣿⣿⣧⠀⠿⠀⠀⠀⠀⠀⡀⠀⠀⣤⣤⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠛⢓⣴⣿⣿⣤⣿⠟⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣭⣥⣴⣶⣦⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣶⠀⢼⣿⣿⣿⣿⣿⠿⠟⠛⢿⡿⡋⠉⠉⣉⣭⣿⣿⣷⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣯⢿⠉⠄⡀⠀⠻⠀⢹⣿⠻⣿⣿⣿⠇⠻⣿⣿⣿⡿⠉⠀⠀⠰⢟⣿⣿⣶⣿⣿⣯⣿⣃⣀⡀⠀⠀⠀⠀⣠⣂⡀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣾⣿⣧⣾⣯⠀⠎⠀⠀⠀⣰⣿⣿⡆⠈⢻⣦⣻⣿⢿⡄⢀⣿⡿⠿⠀⠀⠀⠀⢾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣥⣤⣤⠤⠀⠀⠀⠀⠀⠀⠀⠞⠃
⣠⣤⣤⣶⠄⢀⠀⠶⢦⠀⠀⠰⣶⣶⣶⣿⣶⣼⠿⣿⣿⡃⠀⠀⠉⠀⠀⠀⢹⣿⣿⠇⠀⠀⢻⣿⢿⣷⡏⠈⠻⣧⠀⠀⠀⢠⣦⣸⣿⣿⢿⣿⣿⣿⣿⣿⣽⣿⣷⣿⣿⣋⣉⣹⣿⠟⣄⣀⣤⣶⣾⣦⣤⣤⣀⡀⠀⠀
⣿⣷⣾⣽⣷⣿⣶⣶⣾⠀⣼⣿⣿⣿⣿⣿⣿⣿⠇⢻⣿⡏⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⢸⣿⣶⣿⡄⠀⣴⣿⣶⣤⣾⣿⣿⠃⣹⡏⢸⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣷⣶
⣿⠛⣿⣿⣿⣿⣿⣿⡇⠀⠘⠈⢻⣿⣿⣿⣿⣿⠀⠸⡿⠄⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⢸⣿⣿⣿⣇⢰⣿⣿⣿⣿⣿⣿⢋⣴⣿⠃⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣻⣿⣿⣿⣿⣿⣷⣤⣭⣛⣻⣿⡿⠛⣭
⢛⣀⣙⣛⣋⣉⣙⣛⣃⣀⣀⣀⣘⣛⣛⣛⣛⣛⣀⣀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣃⣀⣀⣀⣻⣛⣛⣛⣉⣛⣛⣛⣛⣛⣛⣛⣛⣋⣀⣀⣘⣛⡛⣿⣿⣿⣿⣿⣿⣿⡟⠀⡿⠈⢻⣿⣿⣯⣭⣿⣿⣿⣿⣯⣤⣤⣼
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⣿⡿⠿⣿⣿⣿⠋⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣫⣥
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢶⡾⣿⣿⣿⣮⣻⢸⣿⣷⣙⣾⣸⡯⣷⣷⣙⡜⣽⡸⣷⡿⡿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⢦⡀⠀⠀⣿⡇⣿⡄⠀⠻⠿⠍⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣿⣿⣿⡿⣿⣿⣍⣉
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣴⠖⣫⣶⣿⣿⣿⣦⣄⠀⢹⠀⠀⣿⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢻⢻⡟⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢰⠟⣡⣾⡟⠁⠀⠙⠿⠿⠿⠧⡆⠀⠀⣿⡇⣟⣛⡀⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣋⣻⣿⡿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣽⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣿⣿⡇⠀⢀⣾⣿⣿⣿⡀⠀⠙⠛⣻⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡿⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⣇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡇⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠟⡓⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡟⠋⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⣿⡗⢒⢒⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⠁⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡛⠻⣿⣿⣿⠯⠛⠿⢋⡁⢀⠠⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠊⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡻⡻⠍⠓⠈⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠂⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣀⣠⣤⢶
⣯⢻⣿⡿⠿⠿⠝⠉⠉⠉⠐⠒⠉⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢸⢡⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡆⠉⠋⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⠛⢿⡇⠆⠻⠃⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡆⠀⢲⣿⣿⣿⣧⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⡿⠃⠈⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠛⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢩⣿⠇⠀⠀⠀⣷⣤⣶⠀⠂⠉⠀⠀⠈⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠉⠀⠀⠸⣿⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⡿⢿⣿⣿⠙⠃⠸⣿⢿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣏⠀⠀⢰⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠁⠉⣿⣿⣿⣿⣿⠀⠏⠙⡏⠀⢸⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠋⠀⣠⣤⣤⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠺⠟⠏⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣽⡏⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡷⠒⣿⣌⠻⠋⣿⡇⠘⠛⠿⢿⣿⣻⣿⣿⣿⠀⠻⠀⠀⠀⠀⠀⡀⠀⠀⣠⣤⠄⠀⠀⠀⣠⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠓⢰⣿⣿⣦⣶⡿⠃⠀⠀⢀⣤⣾⣿⣿⣿⣿⣯⣥⣤⣶⣦⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣶⡆⢰⣿⣿⣿⣿⣿⣿⠿⠻⢿⣿⣟⠋⢈⣉⣩⣿⣿⣯⣶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣯⢿⡇⠄⠀⠀⠘⠁⢹⣿⠿⣿⣿⣿⠟⠻⣿⣿⣿⣿⠟⠀⠀⠠⢿⣿⣿⣿⣿⣿⣯⣿⣋⣀⡀⠀⠀⠀⠀⢠⣄⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣶⣿⣧⣾⣯⠀⠌⠀⠀⠀⢠⣿⣿⡇⠀⢻⣦⣹⣿⣿⠄⠀⣾⡿⠿⠇⠀⠀⠀⢸⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠞⠃
⣀⣤⣤⣤⠄⢀⠠⠴⣦⠀⠀⠀⢶⣤⣴⣾⣥⣼⡿⣿⣿⡅⠀⠀⢈⠀⠀⠀⢸⣿⣿⠇⠀⠀⢹⣿⢿⣿⠃⠀⠻⣧⡄⠀⠀⢀⣴⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣋⣉⣙⣿⡿⢉⣀⣠⣶⣾⣷⣤⣄⣀⠀⠀⠀
⣿⣿⣶⣥⣷⣿⣤⣶⣶⠂⢸⣿⣿⣿⣿⣿⣿⣿⠇⢹⣿⡟⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⢸⣿⣷⣿⡄⠀⠠⣻⣷⣤⣴⣿⣿⠇⢹⣯⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣶
⣿⠛⣿⣿⣿⣿⣿⣿⡇⠀⠈⠉⢻⣿⣿⣿⣿⣿⠀⠸⡿⡆⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠈⣿⣿⣿⣇⣀⣿⣿⣿⣿⣿⣿⠟⣰⣿⠃⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡿⣿⣿⣿⣿⣿⣿⣤⣬⣟⣛⣿⣿⠿⣿
⢛⣂⣘⣛⣛⣋⣙⣛⣃⣀⣀⣀⣘⣛⣛⣛⣛⣛⣀⣀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣂⣀⣀⣀⣻⣛⣛⣛⣋⣙⣛⣛⣛⣛⣛⣛⣛⣛⣀⣀⣐⣛⡛⣿⣿⣿⣿⣿⣿⣿⣟⠀⣿⠉⢻⣿⣿⣯⣝⣿⣿⣿⣿⣿⣥⣤⣬
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⣿⣿⠿⣿⣿⣿⡏⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣟⣋
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢶⣾⣿⣿⣾⣯⣻⢸⣿⣷⣙⣾⣸⡯⣧⣷⣙⡜⣽⡸⣷⡿⡿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⢦⡀⠀⠀⣿⡇⣿⡆⠀⠿⢿⡍⠀⠀⠀⠀⠀⠀⣿⣿⣿⣛⣿⣿⣿⡿⢿⣿⣫⣉
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣴⠖⣫⣶⣿⣿⣿⣦⣄⠀⢹⠀⠀⣿⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢻⢻⡿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣰⠟⣱⣾⡟⠁⠀⠙⠿⠿⠿⠧⡆⠀⠀⣿⡇⣾⠛⠀⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣏⣿⣿⡿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣼⡄⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣤⣤⣤⣼⣿⣿⣿⣯⣿⣿⣿⣟⣏
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣽⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣿⣿⡇⠀⢀⣿⣿⣿⣿⡀⠀⠛⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡿⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡗⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡧⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣏⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣻⠓⡋⠛⠙⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡟⠋⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⣹⡛⢗⡚⠙⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠁⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⡿⢿⣿⡟⠿⠺⡛⢐⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⠿⠍⠩⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣯⣀⣠⣤⣶
⣯⢻⣿⡟⠟⠉⠉⠉⠑⠉⠘⠝⠚⠚⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢸⢡⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡆⠉⠋⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⠛⢿⡇⠆⠺⠃⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡆⠀⢲⣿⣿⣿⣧⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⡿⠃⠈⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠛⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢩⣿⡇⠀⠀⠀⢿⣤⣶⠀⠂⠈⠀⠀⠈⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠉⠀⠀⠘⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⡿⢿⣿⣿⠘⠃⠸⣿⢿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣯⠀⠀⢠⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠁⠉⣿⣿⣿⣿⣿⠀⠏⠙⡏⠀⢸⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠏⠀⣠⣤⣀⢿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠻⠟⠏⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⡏⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠒⣿⣇⠛⠋⣿⡏⠘⠻⠿⠿⣿⣟⣿⣿⣿⡀⠻⠀⠀⠀⠀⠀⡀⠀⠀⣠⣤⠄⠀⠀⠀⣀⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠓⢤⣿⣿⣦⣾⡿⠃⠀⠀⢀⣤⣼⣿⣿⣿⣿⣯⣥⣤⣶⣦⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣶⡆⢰⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⣟⠙⢉⣉⣩⣿⣿⣿⣶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣯⢽⡏⠄⠀⠀⠘⠂⠸⣿⠿⣿⣿⣿⡿⠺⣿⣿⣿⣿⠏⠀⠀⠠⠺⣿⣿⣷⣿⣿⣯⣿⢋⣀⡀⠀⠀⠀⠀⢠⢄⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣶⣿⣧⣾⣯⠀⠌⠀⠀⠀⢨⣾⣿⣷⠀⠻⣧⣸⣿⣿⠄⠀⣼⡿⠿⠃⠀⠀⠀⢰⣿⣿⣛⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠞⠃
⣀⣤⣤⣤⠀⢀⠠⠴⣦⠀⠀⠠⣲⣤⣴⣾⣥⣼⡿⣿⣿⡇⠀⠀⢈⠀⠀⠀⢸⣿⣿⡇⠀⠀⢹⣿⡿⣿⡇⠀⠹⣷⡄⠀⠀⢀⣴⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣋⢉⣙⣿⡿⣅⣀⣠⣶⣶⣿⣤⣄⣀⡀⠀⠀
⣿⣿⣶⣥⣷⣿⣤⣶⣶⠂⢸⣿⣿⣿⣿⣿⣿⣿⠇⢹⣿⡟⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⢸⣿⣷⣿⡄⠀⠠⣻⣷⣤⣴⣿⣿⡇⢸⣏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣶
⣿⠛⣿⣿⣿⣿⣿⣿⡇⠀⠈⠉⢻⣿⣿⣿⣿⣿⠀⠸⡿⠆⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⣿⣿⣿⣧⡀⣿⣿⣿⣿⣿⣿⠟⣠⣿⠃⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡻⣿⣿⣿⣿⣿⣿⣤⣭⣟⣛⣿⣿⠻⣿
⢛⣂⣘⣛⣛⣋⣙⣛⣃⣀⣀⣀⣘⣛⣛⣛⣛⣛⣀⣀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣃⣀⣀⣀⣛⣛⣛⣛⣋⣙⣛⣛⣛⣛⣛⣓⣛⣋⣀⣀⣘⣛⡛⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠉⢻⣷⣿⣯⣽⣿⣿⣿⣿⣿⣥⣤⣬
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⣿⣿⠿⣿⣿⣿⡏⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣛⣋
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢶⣾⣿⣿⣾⣯⣻⢸⣿⣷⣙⣾⣸⡯⣷⣷⣙⡜⣽⡸⣷⡿⡿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⢦⡀⠀⠀⣿⡇⣿⡆⠀⠿⢿⡍⠀⠀⠀⠀⠀⠀⣿⣿⣿⣛⣿⣿⣿⡿⢿⣿⣍⣉
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣴⠖⣫⣶⣿⣿⣿⣦⣄⠀⢹⠀⠀⣿⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢻⢻⡟⣿⣿⣿⣿⡿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣰⠟⣱⣾⡟⠁⠀⠙⠿⠿⠿⠧⡆⠀⠀⣿⡇⣾⠛⠀⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣏⣿⣿⡿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣼⡄⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣤⣤⣤⣼⣿⣿⣿⣯⣿⣿⣿⣟⣏
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠏⣽⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣿⣿⡇⠀⢀⣿⣿⣿⣿⡀⠀⠛⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡿⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡷⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡧⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠝⠙⠉⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠋⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢏⣋⠺⠯⢯⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⠃⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⡻⣻⣿⣿⡿⠿⠝⡯⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣏⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣝⡻⠿⠙⠘⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠂⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣀⣠⣤⣶
⢿⣿⣭⣿⠿⡗⠉⠛⢻⠉⠙⠓⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠀
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠘⢃⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡆⠉⠋⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⠛⢿⡿⠆⠸⠇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⣦⠀⢳⣿⣿⣿⣧⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠈⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠛⠋⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢉⣽⡇⠀⠀⠀⢿⣦⣴⡄⠂⠈⠁⠀⠈⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠏⠁⠀⠀⣿⣷⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⡿⢻⣿⣿⠛⠃⠸⣿⠿⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠁⠀⠀⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⠀⠟⠉⣿⠀⢸⠻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡏⠀⣀⣤⣀⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠏⠿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡟⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⣿⠖⢾⣇⠛⠛⢻⡟⠈⠻⠿⢿⣿⣟⣿⣿⣿⡄⠻⠂⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⠀⠀⠀⣀⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⢟⢤⣾⣿⣦⣼⣿⠇⠀⠀⠀⣠⣼⣿⣿⣿⣿⣯⣥⣤⣶⣦⠄⠀⠀⠀⣀⣀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣶⡆⠐⣾⣿⣿⣿⣿⣿⡿⠛⢿⣿⣏⠁⠉⣉⣩⣿⣿⣿⣾⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣽⡇⠄⠀⠀⠘⠆⠘⣿⡟⣿⣿⣿⡿⠚⣿⣿⣿⣿⡿⠁⠀⠀⠾⣿⣿⣶⣾⣿⣯⣿⣋⣀⡀⠀⠀⠀⠀⢀⣄⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣶⣿⣧⣾⣿⠀⠌⠁⠀⠀⢨⣾⣿⣷⠀⠹⣧⣘⣿⣿⡇⠀⣸⣿⠿⠏⠀⠀⠀⠰⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠜⠛
⣀⣤⣤⣤⡄⢀⡀⠶⢶⡀⠀⠠⢶⣦⣴⣾⣥⣼⡿⣿⣿⡇⠀⠀⠈⠀⠀⠀⠘⣿⣿⡇⠀⠀⠸⣿⡿⣿⡷⠀⠹⣿⣤⠀⠀⢀⣠⡆⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣾⣿⣿⣋⣉⣉⣿⡿⣋⣀⣠⣶⣾⣷⣠⣤⣀⡀⠀⠀
⣿⣿⣶⣁⣷⣾⣥⣶⣶⠇⢰⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⡟⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⢰⣿⣷⣾⣇⠀⠠⢟⣿⣦⣤⣿⣿⡟⢸⣯⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣶
⣿⡟⢿⣿⣿⣿⣿⣿⣧⠀⠈⠉⢻⣿⣿⣿⣿⣿⠇⠘⡿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⣿⣿⣿⣷⣀⣾⣿⣿⣿⣿⣿⡟⣱⣿⡏⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡻⣿⣿⣿⣿⣿⣿⣭⣭⣛⣛⣿⣿⠿⢿
⢛⣂⣘⣛⣛⣋⣙⣛⣃⣀⣀⣀⣘⣛⣛⣛⣛⣛⣀⣀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣃⣀⣀⣀⣛⣛⣛⣛⣋⣙⣛⣛⣛⣛⣛⣓⣛⣛⣀⣀⣐⣛⡛⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⠉⢻⣿⣿⣯⣍⣿⣿⣿⣿⣿⣥⣤⣤
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⣿⣿⠿⣿⣿⣿⡏⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣟⣋
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢶⣾⣿⣿⣾⣮⣻⢸⣿⣷⣙⣾⢸⡯⣷⣷⣙⡜⣽⡸⣷⡿⡿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⢦⡀⠀⠀⣿⡇⣿⡆⠀⠹⢿⡏⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣿⣿⣿⡿⢿⣿⣟⣉
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣴⠶⣫⣶⣿⣿⣿⣦⣄⠀⢹⠀⠀⣿⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢿⢻⡿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣰⠟⣱⣾⡟⠁⠀⠙⠿⠿⠿⠧⡆⠀⠀⣿⡇⣾⠛⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣏⣿⣿⡿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣟⣟
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣽⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣿⣿⡇⠀⢀⣿⣿⣿⣿⡀⠀⠛⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡷⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡇⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣿⣽⡷⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡔
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⠿⠇⠓⠘⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠛⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⢷⡟⠈⢉⣽⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⠃⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⢿⣿⣿⣿⣿⠿⠿⠿⡁⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣟⠘⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠿⣿⢿⠿⠋⠋⠙⠈⠀⠈⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠆⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣄⣀⣤⡶
⣿⡿⠿⠿⢟⣹⠍⢯⠛⠉⠁⠂⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠃⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠈⠋⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⠦⠸⠇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⢲⣿⣿⣿⣯⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠈⠁⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡿⠛⠋⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣽⡷⠀⠀⠀⢺⣦⣴⡄⠄⠈⠁⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠁⠀⠀⣿⣷⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⢻⣿⣿⠇⠋⠐⣿⠿⠇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡁⠀⠀⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⡀⠘⠈⢻⠀⢸⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠁⢀⣤⣄⣹⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠟⠿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣬⡟⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠖⢾⣧⠛⠟⢹⡟⠀⠻⠿⠿⢿⡟⣻⣿⣿⡆⠻⠆⠀⠀⠀⠀⡀⠀⠀⣠⣤⡄⠀⠀⠀⣀⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠛⠢⣾⣿⣧⣼⣾⠏⠀⠀⠀⣀⣴⣿⣿⣿⣿⣯⣥⣤⣶⣦⠄⠀⠀⠀⡀⣀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣶⡖⠰⣾⣿⣿⣿⣿⣿⠿⠛⢿⣿⡋⠉⠉⣉⣩⣿⣿⣿⣾⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡽⡏⠄⠀⠀⠘⠇⠈⣿⡟⢿⣿⣿⡿⠓⣿⣿⣿⣿⡟⠁⠀⠀⠺⣿⣿⣶⣾⣿⣯⣿⣋⣀⡀⠀⠀⠀⠀⢀⢆⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣶⣿⣿⣾⣿⠀⠌⠁⠀⠀⢨⣾⣷⣷⠀⠙⣷⣌⣿⣿⡇⠀⢸⣿⠿⠏⠀⠀⠀⠀⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠸⠻
⣀⣤⣤⣴⠄⢠⡀⠶⢶⡀⠀⠰⣲⣦⣴⣿⢷⣼⡿⣿⣿⡇⠀⠀⢘⠀⠀⠀⠸⣿⣿⡟⠀⠀⠘⣿⡿⣿⣷⠀⠙⣿⣤⠀⠀⠀⣠⡆⣿⣿⡿⣿⣿⣿⣿⣿⣯⣿⣾⣿⣿⣋⣉⣉⣿⣿⢃⣁⣠⣶⣾⣶⣤⣄⣀⡀⠀⠀
⣿⣿⣶⣧⣶⣾⣥⣶⣶⠇⢰⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⣿⣿⣾⣇⠀⠠⢼⣿⣦⣤⣾⣿⡇⢸⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣶
⣿⠛⢿⣿⣿⣿⣿⣿⣧⠀⠈⠁⢻⣿⣿⣿⣿⣿⠇⠘⡿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⢿⣿⣿⣿⡀⣸⣿⣿⣿⣿⣿⡿⣡⣿⡏⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣭⣤⣻⣛⣿⣿⠿⢿
⢛⣂⣘⣛⣛⣉⣙⣛⣋⣀⣀⣀⣘⣛⣛⣛⣛⣛⣀⣀⣓⣁⣀⣀⣀⣀⣀⣀⣀⣈⣉⣃⣀⣀⣀⣘⣟⣛⣛⣋⣙⣛⣛⣛⣛⣛⣓⣛⣛⣀⣀⣐⣛⡛⣿⣿⣿⣿⣿⣿⣿⣿⠀⢽⠉⢻⣿⣿⣯⣭⣿⣿⣿⣿⣿⣥⣤⣤
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⣿⣿⠿⣿⣿⣿⡏⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣟⣉
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢶⣾⣿⣿⣾⣯⣻⢸⣿⣷⣙⣾⢸⡯⣷⣷⣙⡜⣽⡸⣷⡿⡿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⢦⡀⠀⠀⣿⡇⣿⡇⠀⠹⠿⡏⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⡿⢿⣿⡿⢿⣿⣏⣉
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣤⠶⣫⣶⣿⣿⣿⣦⣄⠀⢹⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢿⢻⡿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢰⠟⣡⣾⡟⠁⠀⠙⠿⠿⠿⠧⡆⠀⠀⣿⡇⣾⠛⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣏⣿⣿⡿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣤⣤⣤⣬⣿⣿⣿⣯⣿⣿⣿⣟⣟
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣽⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣻⣿⡇⠀⢀⣿⣿⣿⣿⡀⠀⠛⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡷⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⣿⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡇⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡏⠳⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡔
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠘⠃⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠛⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣃⢙⠗⠈⢉⣏⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠃⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⠿⣿⣿⣿⡹⠯⣍⠀⠐⠙⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠘⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠿⠟⠛⠸⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣀⣀⣤⡶
⢹⣿⡿⡿⠿⠹⠝⠛⢋⡛⠫⠆⠨⠄⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠁
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠇⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣦⠈⠋⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⠦⠸⠇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⣦⠀⢳⣿⣿⣿⣯⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠈⠁⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⠟⠋⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣹⣿⠀⠀⠀⢰⣦⣴⡄⠀⠈⠃⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠉⠀⠀⢲⣷⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⢻⣿⣿⠇⠋⠐⣿⠿⠇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡁⠀⠀⣼⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⡀⠙⠈⢻⠀⢸⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠁⢀⣀⣄⢸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠟⠿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⡿⠁⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠆⢾⣧⠙⠿⢻⣿⠁⠻⠿⢿⣿⡿⣻⣿⣿⡇⠺⠆⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⢀⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠛⠦⣾⣿⣧⣴⣾⠟⠀⠀⠀⣀⣤⣿⣿⣿⣿⣯⣥⣤⣶⣦⠄⠀⠀⠀⡀⣀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣶⡦⠀⣾⣿⣿⣿⣿⣿⠿⠿⢿⣿⣏⠉⠉⣉⣩⣿⣿⣿⣾⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡽⡏⠄⠀⠀⠘⠇⠈⣿⡿⢿⣿⣿⣿⠻⣿⣿⣿⣿⡿⠁⠀⠀⡞⣿⣿⣶⣾⣿⣯⣿⣋⣀⡀⠀⠀⠀⠀⢀⢄⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣶⣿⣿⣾⣿⠀⠌⠁⠀⠀⢨⣾⣿⣷⠀⠙⣷⣌⣿⣻⡧⠀⢸⣿⠿⠋⠁⠀⠀⠀⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⡤⠀⠀⠀⠀⠀⠀⠀⠸⠻
⣀⣤⣤⣴⠄⢠⡀⠶⢶⡀⠀⠠⢲⣆⣴⣿⣷⣼⡿⣿⣿⡇⠀⠀⢘⠀⠀⠀⠘⣿⣿⡟⠀⠀⠘⣿⡿⣿⣿⠂⠘⢿⣤⡀⠀⠀⣠⡆⣿⣿⡿⣿⣿⣿⣿⣿⣯⣿⣾⣿⣿⣋⣉⣉⣿⡿⢋⢀⣠⣴⣶⣿⣠⣀⣀⠀⠀⠀
⣿⣿⣾⣤⣷⣾⣵⣶⣶⠇⢰⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⣿⣿⣼⣧⠀⠀⢾⣿⣧⣤⣾⣿⡿⢸⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣶
⣿⡟⢿⣿⣿⣿⣿⣿⣧⠀⠈⠁⢻⣿⣿⣿⣿⣿⠇⠘⡿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⢻⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⡿⣣⣿⡏⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡻⣿⣿⢿⣿⣿⣿⣭⣬⣛⣛⣿⣿⠿⢿
⢛⣂⣘⣛⣛⣉⣙⣛⣋⣀⣀⣀⣘⣛⣛⣛⣛⣛⣀⣀⣓⣁⣀⣀⣀⣀⣀⣀⣀⣈⣉⣃⣀⣀⣀⣘⣟⣛⣛⣛⣙⣛⣛⣛⣛⣛⣓⣛⣛⣀⣀⣐⣛⡛⣿⣿⣿⣿⣿⣿⣿⣿⠀⢽⠉⢻⣿⣿⣯⣭⣿⣿⣿⣿⣿⣥⣤⣤
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⣿⣿⠿⣿⣿⣿⡏⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣟⣋
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢶⣾⣿⣿⣾⣮⢻⢸⣿⣷⣙⣾⢸⡯⣷⣷⣙⡜⣽⡸⣷⡿⡿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⢦⡀⠀⠀⣿⡇⣿⡇⠀⠻⠿⡏⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣻⣿⣿⡿⢿⣿⣏⣉
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣤⠶⣫⣶⣿⣿⣿⣦⣄⠀⢹⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢻⢻⡿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⠟⣡⣾⡟⠁⠀⠙⠿⠿⠿⠧⡆⠀⠀⣿⡇⣾⠛⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣏⣿⣿⡿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣷⣶⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣌⣿⣿⣿⣿⣻⣿⣿⣟⣟
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣭⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣽⣿⡇⠀⢀⣿⣿⣿⣿⡀⠀⠛⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡿⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⣗⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣷⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡏⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢹⠕⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⠉⠛⠍⠑⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣉⣏⣃⡻⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠏⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⡺⢏⠉⡉⠇⠨⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠃⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⠟⠉⠛⠙⠁⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡆⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡀⢀⣀⣴
⣿⣿⣿⡿⣿⡿⠛⠙⠻⠛⠓⠂⠾⠄⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠻⠃
⠃⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⡏⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠘⠛⠁⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⣿⢥⠰⡶⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣴⠀⠀⣾⣿⣿⣿⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠃⠀⠈⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⠛⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⣉⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⣿⠉⠀⠀⠐⣦⣤⣦⠠⠈⠃⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠉⠀⠀⢰⣶⣼⣿⣧⡀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⢿⣿⣿⡇⠟⠀⣾⡶⡆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⡆⠀⠀⢸⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⣿⡇⢸⠉⢻⠁⠀⡿⣿⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⠿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠋⠁⣀⣠⣸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⡿⢻⠁⠀⠀⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⠏⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡶⢶⣿⡸⠿⠛⣿⠏⠻⠿⢿⣿⣿⣛⣿⣿⣷⠺⠆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠛⠢⣴⣿⣷⣠⣶⡿⠆⠀⠀⢀⣠⣽⣿⣿⣿⣿⣍⣀⣴⣤⡤⠀⠀⠀⠀⠈⠉⠁⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣶⣤⠀⢰⣿⣿⣿⣿⣿⡿⠿⢿⣿⣟⠉⠃⢉⣉⣾⣿⣿⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⡷⡖⢆⠀⠀⠐⠶⠀⢻⣿⢿⣿⣿⣿⡶⢿⣿⣿⣿⣿⠋⠀⠀⠰⢿⣿⣶⣶⣿⣟⣿⡟⠉⠉⠀⠐⠀⠀⠀⡄⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣰⣾⣿⣴⣿⡉⠀⠃⠁⠠⢈⣴⣶⣶⠂⠘⢿⣄⢿⡿⣿⠀⠈⣿⣿⡿⠃⠀⠀⠀⣾⣿⣛⣻⣿⣿⣿⣿⡿⣿⣿⣶⣿⣶⣶⣾⣿⣯⣄⣈⣀⠀⠀⠀⠀⠀⠀⠀⠠⠿
⣀⣀⣤⣤⡄⢀⡀⠤⣴⡄⠀⠠⠤⣄⣤⣶⣯⣤⡿⣿⣿⡏⠀⠀⠘⠀⠀⠀⠘⣿⣿⣿⠀⠀⠈⣿⣿⣿⣿⠖⠘⢿⣥⡀⠀⠀⢀⣄⢸⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣭⣿⡟⠋⠙⣿⣿⠏⠉⢀⣤⣴⣶⣄⣀⡀⠀⠀⠀
⣿⣿⣴⣇⣲⣾⣧⣴⣶⡇⢠⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⣾⣿⣾⣿⡀⠀⣼⣿⣷⣄⣴⣿⣿⢈⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠿⢿⣷⣦
⣿⡟⢿⣿⣿⣿⣿⣿⣿⠀⠀⠋⠻⣿⣿⣿⣿⣿⡇⠀⡿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⣿⣿⡄⢰⣿⣿⣿⣿⣿⣿⠟⣸⡟⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣍⣭⣛⠛⣻⣟⡛⣟
⢛⣃⣘⣛⣛⣛⣙⣛⣛⣀⣀⣀⣈⣛⣛⣛⣛⣛⣃⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣃⣀⣀⣀⣘⣛⣛⣛⣛⣙⣛⣛⣛⣛⣛⣛⣚⣛⣁⣀⣀⣛⡛⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⡙⠻⣶⣿⣿⣙⣻⣿⣿⣿⣿⣏⣀⣀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⣿⣿⣿⣿⣿⣿⡟⠘⠀⠈⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣟⣛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣽⣾⣯⢻⢸⣿⣷⣙⣾⢹⡯⣷⣷⣙⡜⣽⡸⣷⡿⡿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⢦⡀⠀⠀⣿⡇⣿⣧⠀⢻⣿⣏⠁⠀⠀⠀⠀⠀⣹⣿⣿⡿⠿⣿⣿⣿⣿⣿⡟⣋
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣤⠶⣫⣶⣿⣿⣿⣶⣄⠀⢹⠀⠀⣿⡇⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢿⢻⡿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⠟⣥⣾⡟⠁⠀⠙⠿⠿⠿⠧⡆⠀⠀⣿⡇⣶⠗⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣏⣿⣿⣿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣷⣶⣾⣷⣶⣤⣤⣤⣤⣤⣀⣀⣈⣿⣿⣿⣿⣻⣿⣿⣿⡿
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣽⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣽⣿⡇⠀⢀⣿⣿⣿⣿⡀⠀⠛⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡷⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣶⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⣗⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣷⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡟⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⢽⠇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠿⠷⠙⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣹⣓⠁⠁⡝⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⡟⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⣿⢿⢿⠿⠿⢭⣡⠭⠛⠫⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠁⠀⠀
⣿⣿⣿⢿⣿⣿⣿⣿⣟⣻⣿⡿⣿⣿⠟⠛⠱⠺⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⡇⠀⣀⣠
⢿⣿⣿⣿⢿⣿⡿⠛⠛⠉⠁⠹⠿⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏
⠘⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠟⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠛⠃⠁⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⢥⠐⡶⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠀⠐⣰⣿⣿⣿⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠋⠀⠀⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⡿⠛⠋⠀⠒⠀⠀⠀⠀⠀⠀⠀⢉⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢋⣿⡏⠀⠀⠀⣶⣄⣤⠠⠀⠓⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⠋⠁⠀⠀⣶⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⡿⣿⣿⡗⠾⠀⣶⡶⡦⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣖⠀⠀⢠⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⣿⡇⢸⠋⢻⡇⠀⡿⣿⠇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠏⠁⢀⣀⣈⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⢻⠇⠀⠀⠈⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣽⠏⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡧⠴⣾⡜⠿⠟⣿⡏⠘⠿⢿⣿⣿⠟⣿⣿⣷⠰⠆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟⠇⢰⣿⣿⣄⣴⣿⠇⠀⠀⢀⣀⣾⣿⣿⣿⣿⣏⣀⣤⣤⡀⠀⠈⠀⠀⠈⠉⠁⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣤⡄⢰⣾⣿⣿⣿⣿⣿⠿⢿⣿⣿⠛⠛⢛⣁⣲⣿⣿⣥⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣷⣶⢠⠀⠀⠐⠶⠀⢹⣿⡿⣿⣿⣿⡗⣾⣿⣿⣿⣿⠟⠁⠀⠠⢿⣿⣶⣶⣿⣿⣻⡟⠋⠉⠁⠄⠀⠀⠀⣠⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⣶⣿⣴⣿⡉⠀⠃⠀⠀⠈⣵⣶⣶⡆⠀⢿⣦⢻⡏⣿⡀⠀⣿⣿⣿⠉⠀⠀⠀⢰⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣶⣾⣶⣶⣾⣯⣿⣀⣈⣀⡀⠀⠀⠀⠀⠀⠀⢠⠿
⣀⣀⣠⣠⡄⠀⡀⢠⢴⡄⠀⠀⠤⣤⣤⣶⣯⣤⣿⣿⣿⡯⠀⠀⠘⠃⠀⠀⠀⣿⣿⣿⠀⠀⠀⢻⣿⣿⣷⡗⠘⢿⣧⣀⠀⠀⠀⣤⣸⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣯⣿⡟⠋⠙⣿⣿⣿⠉⢁⣠⣤⣶⣆⣀⡀⠀⠀⠀
⣿⣿⣤⣏⣲⣶⣧⣤⣶⡧⢀⣾⣿⣿⣿⣿⣿⣿⡷⠸⣿⣿⠋⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⢸⣿⣦⣿⡁⠀⣠⣿⣷⣄⣠⣿⣿⡏⣻⠃⢸⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⢿⣶⣦
⣿⡟⢻⣿⣿⣿⣿⣿⣿⠀⠀⠋⠹⣿⣿⣿⣿⣿⡇⠀⢿⢧⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠘⣿⣿⣿⣇⠀⣾⣿⣿⣿⣿⣿⠟⢰⡿⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣯⣹⣿⠟⣻⣿⣿⣿
⢛⣃⣀⣛⣛⣛⣙⣛⣛⣀⣀⣀⣀⣛⣛⣛⣛⣛⣃⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣃⣀⣀⣀⣀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣒⣛⣃⣀⣀⣛⡛⣿⣿⣿⣿⣿⣿⣿⣿⡃⢸⡏⠻⣷⣿⣿⣛⣛⣿⣿⣿⣿⣏⣀⣈
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⣿⣿⣿⣿⣿⣿⣿⠋⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣽⣾⣯⢻⢸⣿⣷⣙⣾⢸⡯⣷⣷⣙⡜⣽⡸⣷⡷⡿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⢦⡀⠀⠀⣿⡇⣿⣯⠀⠸⣿⣟⠁⠀⠀⠀⠀⠀⢹⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢀⣤⠶⣫⣶⣿⣿⣿⣶⣄⠀⢹⠀⠀⣿⡇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢻⢻⡿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣠⠟⣡⣾⡟⠁⠀⠙⠿⠿⠿⠧⠆⠀⠀⣿⡇⣶⠷⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣏⣿⣿⣿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣷⣶⣾⣷⣶⣦⣤⣤⣤⣤⣀⣀⣀⣻⣿⣿⣿⣻⣿⣿⣿⢿
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣭⠙⣯⠉⣭⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣽⣿⡇⠀⢀⣿⣿⣿⣿⡀⠀⠛⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡿⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⣇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡗⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣷⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡟⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⡻⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣻⡿⣿⣿⢿⠿⠙⠍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢼⣏⡏⠒⡽⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⡟⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣻⡿⡟⢫⢿⡘⠈⠺⠈⠓⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠁⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⡟⠛⠛⠘⠁⠉⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣆⠀⣀⣠
⣿⣿⣻⡟⣿⡿⠟⠟⠙⠉⠁⠸⠿⠍⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏
⠙⠉⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠟⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠛⠃⢁⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠥⠀⡶⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣴⠀⠀⣰⣿⣿⣿⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠋⠀⠀⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡿⠛⠋⠀⠒⠀⠀⠀⠀⠀⠀⠀⢉⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢋⣿⡏⠀⠀⠀⣶⣄⣤⠠⠀⠓⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⠉⠁⠀⠰⣶⣮⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⡿⣿⣿⡗⠾⠀⣶⡶⡦⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣞⠀⠀⢠⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⣿⡇⢸⠋⢻⡇⠀⡿⣿⠇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⠿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠏⠁⢀⣀⣈⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⢻⠇⠀⠀⠈⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣽⠏⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡧⠴⣾⡜⠿⠟⣿⡏⠘⠿⢿⣿⣿⠟⣿⣿⣷⠰⠆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠛⠃⢰⣿⣿⣄⣶⡿⠇⠀⠀⠀⣀⣼⣾⣿⣿⣿⣟⣀⣤⣤⡀⠀⠈⠀⠀⠈⠉⠁⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣤⡄⢰⣾⣿⣿⣿⣿⣿⠿⠿⣿⣿⡛⠛⢛⣁⣲⣿⣿⣥⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣷⣶⢢⠀⠀⠀⠶⠀⢸⣿⡿⣿⣿⣿⡗⢼⣿⣿⣿⣿⠏⠁⠀⠰⢿⣿⣶⣶⣿⣿⣿⡟⠉⠉⠡⠠⠀⠀⠀⣠⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢠⣶⣿⣴⣿⡍⠀⠃⠀⠀⠈⣵⣶⣶⠆⠀⢻⣦⣹⣇⣿⡀⠀⣿⣿⣿⠋⠀⠀⠀⢰⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣶⣾⣶⣶⣾⣷⣿⣄⣈⣀⡀⠀⠀⠀⠀⠀⠀⢠⠿
⣀⣀⣠⣤⡄⠀⡀⢠⣴⡄⠀⠀⠤⡤⣤⣶⣯⣤⣿⣿⣿⡯⠀⠀⠐⠂⠀⠀⠀⣿⣿⣿⠀⠀⠀⢹⣿⢿⣷⡗⠐⢿⣧⣀⠀⠀⠀⣤⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡟⠋⠙⣿⣿⠿⠉⠁⣠⣤⣶⣆⣀⡀⠀⠀⠀
⣿⣿⣤⣏⣰⣶⣧⣤⣶⡇⢀⣶⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⢸⣿⣦⣿⡁⠀⣠⣿⣷⣤⣤⣿⣿⡏⣻⠃⢹⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⢿⣷⣦
⣿⡟⢻⣿⣿⣿⣿⣿⣿⠀⠀⠋⠹⣿⣿⣿⣿⣿⡇⠀⢿⢷⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠘⣿⣿⣿⣇⠀⣿⣿⣿⣿⣿⣿⠟⣰⡿⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣯⣹⣻⠟⣻⣿⣿⣿
⢛⣃⣀⣛⣛⣛⣙⣛⣛⣀⣀⣀⣀⣛⣛⣛⣛⣛⣃⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣀⣋⣓⣀⣀⣀⣀⣻⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣒⣛⣃⣀⣀⣛⡛⣿⣿⣿⣿⣿⣿⣿⣿⡃⢸⡏⠻⣷⣿⣿⣛⣻⣿⣿⣿⣿⣟⣁⣀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⣿⣿⣿⣿⣿⣿⣿⠋⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⢛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣾⣯⣻⢸⣿⣷⣙⣾⢸⡯⣷⣷⣙⡜⣽⡸⣷⡿⡿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⢦⡀⠀⠀⣿⡇⣿⣯⠀⠸⣿⣟⠁⠀⠀⠀⠀⠀⢹⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⡛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣤⠶⣫⣶⣿⣿⣿⣶⣄⠀⢹⠀⠀⣿⡇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣾⣿⣿⣶⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢻⢻⡿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⠟⣡⣾⡟⠁⠀⠙⠿⠿⠿⠧⡆⠀⠀⣿⡇⣶⠷⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣏⣿⣿⣿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣷⣶⣾⣷⣶⣦⣤⣤⣤⣤⣤⣄⣀⣻⣿⣿⣿⣻⣿⣿⣿⢻
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣭⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣽⣿⡇⠀⢀⣿⣿⣿⣿⡀⠀⠛⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡿⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⣿⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣷⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡟⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡅⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣟⡿⡻⠿⠛⠉⣛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⣽⠆⠁⡝⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⡟⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡿⢿⢿⠻⢋⠭⢀⠛⠙⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠇⠃⠀⠀
⣿⣿⣿⡿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢻⠛⠛⠙⠃⠀⠉⠁⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣇⠀⣀⣠
⠿⣻⠿⡿⣿⡿⣿⢻⠓⠍⠑⠡⠵⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏
⠋⠉⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠿⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠛⠃⢁⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⣿⡧⠀⡶⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⡀⠀⣰⣿⣿⣿⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠉⠀⠀⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⡿⠛⠋⠀⠒⠀⠀⠀⠀⠀⠀⠀⢉⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢋⣿⡏⠀⠀⠀⢲⣤⣤⠀⠈⠃⠀⠀⠀⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⠋⠁⠀⠐⣶⣮⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⡿⣿⣿⡷⠺⠂⣶⡶⡖⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣞⠀⠀⢀⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⣿⡇⢸⠋⢻⡏⠀⡿⣿⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⡿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡏⠁⢀⣀⣈⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⢻⠇⠀⠀⠀⠁⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⠟⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⠴⣾⣏⠻⠟⢿⡟⠘⠿⠿⣿⣿⡟⣿⣿⣿⠰⡧⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠛⠃⢠⣿⣿⣦⣶⣾⠧⠀⠀⠀⣀⣼⣿⣿⣿⣿⣏⣀⣤⣤⡀⠀⠈⠀⠀⠈⠉⠁⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣤⡄⢠⣾⣿⣿⣿⣿⣿⠿⠿⣿⡟⠛⠛⢚⣁⣲⣿⣿⣥⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣷⣶⢠⠀⠀⠀⠶⠀⢸⣿⡿⣿⣿⣿⡷⢺⣿⣿⣿⣿⡟⠁⠀⠠⠿⣿⣷⣶⣿⣿⣻⡟⠋⠉⠁⠠⠀⠀⠀⣠⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⣾⣿⣴⣿⡍⠀⠃⠀⠀⠈⣵⣶⣶⡆⠀⢻⣧⣹⣟⣿⡄⠈⣽⣿⣿⡿⠀⠀⠀⢰⣿⣟⣻⣿⣿⣿⣿⣿⢿⣿⣶⣾⣶⣶⣾⣿⣿⣀⣈⣀⡀⠀⠀⠀⠀⠀⠀⢠⠿
⣀⣠⣠⣤⣤⠀⡀⢠⣴⡄⠀⠀⠤⣤⣤⣶⣯⣤⣿⣿⣿⡯⠀⠀⠘⠃⠀⠀⠀⣿⣿⣿⠀⠀⠀⢹⣿⣿⣷⡗⠂⠻⣯⣀⠀⠀⠀⣤⢸⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣯⣿⡟⠋⠙⣿⣿⠟⠉⢁⣠⣤⣶⣆⣀⡀⠀⠀⠀
⣿⣿⣥⣏⣳⣶⣧⣤⣶⡶⠀⣶⣿⣿⣿⣿⣿⣿⡷⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⢸⣿⣧⣿⡃⠀⢨⣿⣿⣤⣠⣾⣿⡏⣻⠃⢹⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⢿⣶⣦
⣿⡟⢻⣿⣿⣿⣿⣿⣿⡀⠀⠉⠹⣿⣿⣿⣿⣿⡏⠀⢿⢷⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠈⣿⣿⣿⣇⠀⣾⣿⣿⣿⣿⣿⠿⢠⡿⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣏⣹⣻⠟⣻⣿⣿⣿
⢛⣃⣀⣛⣛⣛⣙⣛⣛⣀⣀⣀⣀⣛⣛⣛⣛⣛⣃⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣀⣀⣀⣀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣒⣛⣃⣀⣀⣛⡛⣿⣿⣿⣿⣿⡿⣿⣿⡃⢸⡏⠻⣷⣿⣿⣛⣻⣿⣿⣿⣿⣏⣀⣀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⣻⣿⣿⣿⣿⣿⣿⠋⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⢛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣾⣯⢻⢸⣿⣷⣙⣾⢸⡯⣷⣷⣙⡜⣽⡸⣧⡷⡿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⢦⡀⠀⠀⣿⡇⣿⣯⠀⠘⢿⣏⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣤⠶⣫⣶⣿⣿⣿⣶⣄⠀⢱⠀⠀⣿⡇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣾⣿⣿⣶⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢻⢻⡿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⢟⣥⣾⡟⠁⠀⠙⠿⠿⠿⠧⡆⠀⠀⣿⡇⣶⠷⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣏⣿⣿⡿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣷⣶⣾⣷⣶⣦⣤⣤⣤⣤⣤⣀⣀⣻⣿⣿⣿⣿⣿⣿⣿⣻
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣭⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣽⣿⡇⠀⢀⣿⣿⣿⣿⡀⠀⠛⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡿⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⣷⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣷⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡟⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡙⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⢿⣟⡟⠿⠓⠏⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠻⣿⣁⣛⢚⡽⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⡟⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⡿⢿⢿⢿⡇⡕⠂⠭⠭⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡃⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣟⢻⠛⠋⠉⠁⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣇⠀⣀⣠
⢿⡿⣿⡟⠟⣿⡿⣛⡏⠋⠈⢷⠬⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏
⠿⠙⠙⠀⠈⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠟⢠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠈⠛⠃⠁⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⢧⠀⡶⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠀⠀⣸⣿⣿⣿⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠋⠀⠀⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡿⠛⠋⠀⠒⠀⠀⠀⠀⠀⠀⠀⢉⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢋⣿⡏⠀⠀⠀⢶⣤⣤⠠⠀⠓⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⠋⠁⠀⠀⣶⣮⣿⣿⡄⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⡇⠶⠀⣶⡶⡆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣞⠀⠀⢀⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⣿⡇⢸⠋⢻⡇⠀⡿⣿⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⠿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠏⠁⢀⣀⣈⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠻⢻⠇⠀⠀⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⠏⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡧⠰⣾⣆⠻⠟⢿⡟⠘⠿⠿⣿⣿⠟⣻⣿⣿⠰⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠛⠃⢠⣿⣿⣆⣾⣾⠧⠀⠀⠀⣀⣼⣿⣿⣿⣿⣏⣀⣤⣤⡀⠀⠈⠀⠀⠈⠉⠁⠀⠀⠠⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣦⡄⢰⣾⣿⣿⣿⣿⣿⠿⠿⣿⡟⠛⠃⢚⣁⣲⣿⣿⣥⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣷⣶⢠⠀⠀⠐⠶⠀⢸⣿⡿⣿⣿⣿⡿⠺⣿⣿⣿⣿⡟⠁⠀⠠⠿⣿⣶⣶⣿⣟⣿⡟⠉⠉⠁⠐⠀⠀⠀⣠⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣰⣾⣿⣴⣿⡍⠀⠃⠀⠀⠈⣵⣶⣶⠆⠀⢻⣦⣹⣟⢿⡄⠀⣽⣿⣿⠋⠀⠀⠀⢰⣶⡟⣻⣿⣿⣿⣿⣿⣿⣿⣶⣾⣶⣶⣾⣷⣿⣄⣈⣀⡀⠀⠀⠀⠀⠀⠀⢠⠿
⣀⣠⣤⣤⣄⠀⡀⠤⢴⡄⠀⠀⠴⡤⣤⣶⣥⣤⣿⣿⣿⡯⠀⠀⠘⠂⠀⠀⠀⣿⣿⡿⠀⠀⠀⢹⣿⢿⣷⡗⠀⢻⣧⡀⠀⠀⠀⣠⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡟⠋⠙⣿⣿⠿⠉⢁⣠⣤⣶⣌⣀⡀⠀⠀⠀
⣿⣿⣶⣏⣲⣾⣥⣴⣶⡆⢠⣿⣿⣿⣿⣿⣿⣿⡷⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⢸⣿⣦⣿⡁⠀⢠⣿⣿⣄⣠⣾⣿⡏⣻⠃⢹⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣶⣦
⣿⡿⢻⣿⣿⣿⣿⣿⣿⠀⠀⠋⠹⣿⣿⣿⣿⣿⡇⠀⢿⢷⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠘⣿⣿⣿⣇⠀⣾⣿⣿⣿⣿⣿⠿⣀⣿⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣯⣻⣛⠟⣻⣿⣿⣿
⢛⣃⣈⣛⣛⣛⣉⣛⣛⣀⣀⣀⣀⣛⣛⣛⣛⣛⣃⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣃⣀⣀⣀⣀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣒⣛⣃⣀⣀⣛⡛⣿⣿⣿⣿⣿⡿⣿⣿⡃⢸⡏⠻⣷⣿⣿⣛⣻⣿⣿⣿⣿⣟⣁⣀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⣿⣿⣿⣿⣿⣿⣿⠋⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣾⣯⣻⢸⣿⣷⣙⣾⢸⡯⣷⣷⣙⡜⣽⡸⣷⡿⡿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⢦⡀⠀⠀⣿⡇⣿⣯⠀⠸⣿⣟⠁⠀⠀⠀⠀⠀⢹⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⡛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣤⠶⣫⣶⣿⣿⣿⣶⣄⠀⢹⠀⠀⣿⡇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣾⣿⣿⣶⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢿⢻⡿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣠⠟⣡⣾⡟⠁⠀⠙⠿⠿⠿⠧⡆⠀⠀⣿⡇⣶⠷⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣏⣿⣿⣿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣷⣶⣾⣷⣶⣦⣤⣤⣤⣤⣤⣄⣈⣻⣿⣿⣿⣻⣿⣿⣿⣻
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣭⠙⣯⠉⣭⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣽⣿⡇⠀⢀⣿⣿⣿⣿⡀⠀⠛⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡷⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⣿⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣷⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡏⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠄⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢽⣙⣹⣿⠿⢿⠿⠋⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣋⣝⣡⡉⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⡟⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠗⠉⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠃⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⡿⠿⠟⢋⠫⠉⠉⠃⠀⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣇⠀⣀⣠
⣿⣿⣿⣿⣿⣿⣿⠿⣿⠛⠝⡿⡥⠹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇
⡟⠻⠿⡟⠛⠚⠙⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠟⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠈⠛⠃⢁⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⢧⠐⡶⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠀⠀⣸⣿⣿⣿⣀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠋⠀⠀⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠛⠋⠀⠒⠀⠀⠀⠀⠀⠁⠀⢉⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⣿⡇⠀⠀⠀⣶⣤⣦⠠⠈⠓⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠰⣶⣮⣿⣿⡄⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡇⠾⠀⣶⡶⡆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣎⠀⠀⢠⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⡇⢸⠋⢻⡃⠀⡿⣿⠃⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋⠀⣀⣄⣀⢿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠻⢻⠇⠀⠀⠈⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣾⣿⣿⣿⣿⣾⣻⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⠏⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡷⠲⣿⣎⠻⠛⢿⡟⠘⠿⠿⣿⣿⣟⣿⣿⣿⠰⠆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟⠣⢰⣿⣿⣦⣶⣿⠇⠀⠀⢀⣀⣼⣿⣿⣿⣿⣯⣀⣴⣦⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣶⡄⢰⣿⣿⣿⣿⣿⣿⠿⠿⣿⡟⡛⠋⣉⣀⣲⣿⣿⣵⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡷⣶⢠⠀⠀⠀⠶⠀⢸⣿⠿⣿⣿⣿⡿⢾⣿⣿⣿⣿⡏⠀⠀⠰⢿⣿⣷⣶⣿⣟⣿⡟⠋⠁⠡⠰⠀⠀⠀⣠⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣰⣾⣿⣴⣿⡍⠀⠁⠀⠀⠈⣵⣶⣶⠀⠀⢻⣦⣸⡏⣿⡀⠀⣿⣿⣿⠋⠀⠀⠀⢸⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣶⣾⣶⣶⣾⣿⣿⣄⣈⣀⡠⠀⠀⠀⠀⠀⠀⢠⠿
⣀⣤⣤⣤⡄⠀⡄⠴⢶⡄⠀⠀⠤⣤⣤⣶⣥⣤⡿⣿⣿⡏⠀⠀⠘⠁⠀⠀⠈⣿⣿⡿⠀⠀⠀⢹⣿⣿⣾⡗⠈⢿⣧⣀⠀⠀⢀⣤⢸⣿⣯⢿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡟⠛⠙⣿⣿⠟⠉⢁⣠⣤⣶⣆⣀⡀⠀⠀⠀
⣿⣿⣴⣏⣲⣾⣥⣴⣶⡆⢠⣿⣿⣿⣿⣿⣿⣿⡷⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⢸⣿⣧⣿⡅⠀⣠⣿⣿⣤⣤⣿⣿⡇⣻⠃⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⢿⣶⣦
⣿⡟⢻⣿⣿⣿⣿⣿⣿⠀⠀⠋⠹⣿⣿⣿⣿⣿⡇⠀⢿⠷⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠘⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⠟⣠⡿⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣏⣿⣿⠛⣻⣿⣿⣿
⢛⣃⣈⣛⣛⣛⣉⣛⣛⣀⣀⣀⣀⣛⣛⣛⣛⣛⣃⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣃⣀⣀⣀⣀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣐⣛⣁⣀⣀⣛⡛⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⡍⠻⣷⣿⣿⣛⣻⣿⣿⣿⣿⣏⣀⣀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⢿⣿⣿⡿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⣿⣿⣿⣿⣿⣿⡟⠛⠁⠈⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣾⣯⣻⢸⣿⣷⣙⣾⢸⡯⣷⣷⣙⡜⣽⡸⣷⡷⡿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⢦⡀⠀⠀⣿⡇⣿⣯⠀⠹⣿⡏⠁⠀⠀⠀⠀⠀⢹⣿⣿⡿⠿⣿⣿⣿⣿⣿⡟⠛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣤⠶⣫⣶⣿⣿⣿⣶⣄⠀⢹⠀⠀⣿⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢻⢻⡿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣠⠟⣡⣾⡟⠁⠀⠙⠿⠿⠿⠧⡆⠀⠀⣿⡇⣶⠷⠂⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣏⣿⣿⡿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣷⣶⣿⣷⣶⣶⣦⣤⣤⣤⣤⣄⣈⣻⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣽⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣽⣿⡇⠀⢀⣿⣿⣿⣿⡀⠀⠙⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡿⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⣷⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡟⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣻⣞⡿⠍⡛⠋⠛⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣼⣫⠫⠽⣯⡟⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⡟⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡽⡽⠿⠋⢉⠝⠃⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠛⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠟⢿⢥⣿⡛⠋⠙⠉⠉⠈⠉⠘⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡄⠀⣀⣤
⣿⣿⣿⣿⢿⠿⠟⠿⡿⠏⠡⡟⠙⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇
⢼⢛⡻⠟⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠟⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠈⠛⠃⢁⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⣿⢥⠀⡶⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠀⠀⣼⣿⣿⣟⣀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠃⠀⠀⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠛⠉⠀⠒⠀⠀⠀⠀⠀⠀⠀⢉⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢩⣿⡇⠀⠀⠀⣶⣤⣦⠰⠈⠃⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠉⠀⠀⠐⣶⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⡗⠖⠀⣶⡶⡆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡎⠀⠀⢠⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⣿⡇⠸⠉⢻⠁⠀⡿⣿⠃⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠉⠀⣀⣠⣀⢿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠿⢻⠁⠀⠀⠈⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣾⣻⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⠃⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⡶⠺⣿⣌⠻⠛⣿⡏⠘⠿⠿⣿⣿⢟⣽⣿⣿⠸⠆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠛⠀⣼⣿⣿⣤⣶⡿⠃⠀⠀⠀⣠⣼⣿⣿⣿⣿⣯⣀⣤⣤⡀⠀⠀⠀⠀⢈⠉⠁⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣶⡄⢰⣿⣿⣿⣿⣿⣿⠿⠿⣿⡟⠛⠋⢋⣁⣺⣿⣿⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⡷⡖⢄⠀⠀⠐⠖⠀⢹⣿⠻⣿⣿⣿⠗⢸⣿⣿⣿⣿⠋⠀⠀⠰⢿⣿⣶⣶⣿⣟⣿⡟⠉⠉⠀⠀⠀⠀⠀⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣴⣾⣿⣴⣿⠁⠀⠁⠀⠀⢈⣴⣶⣶⠀⠈⢿⣦⣹⣏⢿⡀⠀⣿⡿⠿⠁⠀⠀⠀⢾⣾⣟⣿⣿⣿⣿⣿⡿⣿⣷⣶⣾⣶⣶⣿⣻⣯⣀⣈⣀⠀⠀⠀⠀⠀⠀⠀⢠⠿
⣀⣤⣤⣴⡄⠀⡄⠴⢶⡄⠀⠀⢴⣦⣤⣾⣥⣴⡿⣿⣿⡏⠀⠀⠘⠀⠀⠀⠘⣿⣿⡿⠀⠀⠀⢻⣿⢿⣷⡆⠈⢿⣧⡀⠀⠀⢀⣤⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⡟⠋⠙⣿⣿⡏⠉⢀⣠⣴⣶⣀⣀⡀⠀⠀⠀
⢿⣿⣴⣇⣶⣾⣥⣴⣶⠆⢠⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⢾⣿⣦⣿⠇⠀⠬⣿⣷⣄⣤⣾⣿⠏⣿⠁⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠿⣷⣶⣤
⣿⡟⢿⣿⣿⣿⣿⣿⣷⠀⠀⠋⢻⣿⣿⣿⣿⣿⡇⠈⡿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⠏⣸⡟⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣍⣉⡻⠛⣿⣿⣿⣟
⢛⣃⣘⣛⣛⣋⣙⣛⣛⣀⣀⣀⣈⣛⣛⣛⣛⣛⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣃⣀⣀⣀⣘⣛⣛⣛⣛⣙⣛⣛⣛⣛⣛⣛⣚⣛⣁⣀⣀⣛⡛⣿⣿⣿⣿⣿⣿⣿⢿⠁⢹⠉⠛⣶⣿⣟⣛⣻⣿⣿⣿⣿⢋⣀⣀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⣿⣿⣿⣿⣿⣿⡟⠘⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣾⣯⢻⢸⣿⣷⣙⣾⢸⡯⣷⣷⣙⡜⣽⡸⣷⡿⡿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⢦⡀⠀⠀⣿⡇⣿⣧⠀⢻⣿⡏⠁⠀⠀⠀⠀⠀⣽⣿⣿⡿⠿⣿⣿⣿⣿⣿⠟⠛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣤⠶⣫⣶⣿⣿⣿⣶⣄⠀⢹⠀⠀⣿⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣾⣿⣿⣾⣿⣿⣷
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢻⢻⡿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣠⢟⣥⣾⡟⠁⠀⠙⠿⠿⠿⠧⡆⠀⠀⣿⡇⣶⠗⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣏⣿⣿⡿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣷⣶⣿⣷⣶⣶⣦⣤⣤⣤⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⠿
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣭⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣿⣿⡇⠀⢀⣿⣿⣿⣿⡀⠀⠛⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡿⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡯⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⡛⣛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⡿⡟⠍⠩⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣾⡿⢷⠨⠣⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠇⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⡿⠿⠿⠯⠆⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠉⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠻⠿⠋⠟⠙⠙⠉⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠆⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡀⠀⣀⣴
⣻⡿⢿⣿⡟⡿⣿⢟⣿⣷⡏⢋⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠃
⠟⠻⠏⠻⠑⠋⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠏⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠈⠛⠁⣁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⣿⠤⠰⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣤⠀⠂⣾⣿⣿⣯⣠⣶⣾⣿⣿⣿⣿⣿⣿⢿⣿⠏⠈⠁⠀⠈⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠛⠁⠐⠃⠀⠀⠀⠀⠀⠀⠀⣉⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣩⣿⠀⠀⠀⠰⣶⣤⣦⠄⠘⠃⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠛⠉⠀⠀⢰⣾⣼⣿⣿⡄⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⢿⣿⣿⡇⠟⠀⣶⡶⡆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡤⠀⠀⢸⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⣿⡇⠸⠉⢻⠁⠘⡿⣿⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠁⢀⣠⣤⣘⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡟⠻⠀⠀⠀⠈⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡟⠀⢀⣈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠂⢻⣿⡆⠛⠛⣿⡃⠘⠛⠿⠿⢿⣫⣿⣿⣷⠺⠆⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⠛⠛⠀⣾⣿⣿⣤⣿⠿⠃⠀⠀⢀⣤⣾⣿⣿⣿⣿⣭⣠⣴⣤⠀⠀⠀⠀⡀⣈⠉⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣶⡂⣸⣿⣿⣿⣿⣿⡿⠟⠻⣿⣟⠛⠃⣉⣉⣾⣿⣿⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣯⡽⡇⠄⠀⠀⠐⠇⠀⢹⣿⢻⣿⣿⣿⠗⢹⣿⣿⣿⣿⠁⠀⠀⠸⣿⣿⣶⣶⣿⣿⣿⠟⢉⠉⠁⠀⠀⠀⠀⡄⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣴⣿⣿⣾⣿⠁⠈⠁⠀⠀⢨⣶⣷⣷⠀⠈⢻⣦⣻⣏⢿⠀⢀⣿⡿⠟⠁⠀⠀⠀⢿⣿⣛⣿⣿⣿⣿⣿⡿⣿⣿⣶⣿⣿⣿⣿⣾⣯⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠴⠿
⣠⣤⠤⣤⡄⢠⡄⠶⢶⡄⠀⠐⢲⣦⣴⣾⣥⣼⡿⣿⣿⡏⠀⠀⠘⠀⠀⠀⠘⣿⣿⡟⠀⠀⠀⣿⣷⢿⣿⠎⠈⢻⣶⠄⠀⠀⢀⣄⢸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡏⠉⢙⣿⡿⠏⠁⣀⣤⣴⣾⣀⣀⡀⠀⠀⠀
⣿⣿⣶⣦⣷⣿⣥⣶⣶⠆⢰⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⣿⣿⣶⣿⡄⠀⢾⣿⣶⣤⣴⣾⣿⢈⣿⠁⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣷⣤
⣿⠛⢿⣿⣿⣿⣿⣿⣷⠀⠈⠃⢻⣿⣿⣿⣿⣿⠇⠘⡿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⢸⣿⣿⣿⡅⢰⣿⣿⣿⣿⣿⣿⠏⣸⡏⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣍⣉⡙⢛⣻⣿⠿⣿
⢛⣂⣘⣛⣛⣉⣙⣛⣋⣀⣀⣀⣘⣛⣛⣛⣛⣛⣀⣀⣓⣁⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣀⣀⣘⣟⣛⣛⣛⣙⣛⣛⣛⣛⣛⣛⣚⣛⣀⣀⣀⣛⡛⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⠉⢻⣶⣿⣟⣛⣻⣿⣿⣿⣿⣁⣀⣀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⢿⣿⣿⡿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⣿⣿⣿⣿⣿⣿⡟⠘⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⢶⣾⣿⣿⣾⣮⣻⢸⣿⣷⣙⣾⢸⡯⣷⣷⣙⡜⣽⡸⣧⡿⡿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⢦⡀⠀⠀⣿⡇⣿⣇⠀⢻⣿⡏⠁⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⠿⠟
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣤⠶⣫⣶⣿⣿⣿⣶⣄⠀⢹⠀⠀⣿⡇⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢻⢹⡿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣠⠟⣥⣾⡟⠁⠀⠙⠿⠿⠿⠧⡆⠀⠀⣿⡇⡾⠗⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣏⣿⣿⡿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣷⣾⣿⣶⣶⣶⣦⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣟⡟
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣽⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣽⣿⡇⠀⢀⣿⣿⣿⣿⡀⠀⠛⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡷⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⣷⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡷⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⡿⣿⡍⠀⠀⠁⠁⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⢛⠁⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣿⣿⠇⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢟⠟⣻⡭⠛⠙⠳⠉⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣇⡉⠉⠉⠉⠩⠈⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠆⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⡀⠀⣀⣴
⣻⢛⠉⠁⠚⠛⠿⢻⠓⠚⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠃
⠋⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠇⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠈⠛⠁⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⣿⠤⠰⠇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣤⠀⠂⣾⣿⣿⣿⣠⣶⣾⣿⣿⣿⣿⣿⣿⢿⣿⠟⠈⠁⠀⠀⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠿⠋⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⠽⣿⠀⠀⠀⢰⣦⣶⣦⠀⠈⠃⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⣀⣀⡀⠀⠨⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⢿⣿⣿⡇⠗⠀⣶⡶⠆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟⠁⠀⠀⢿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⣿⡀⠸⠉⢻⠀⠈⠿⣿⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⡿⠻⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠛⠉⢀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠟⠻⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠋⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣏⠀⣀⣤⢿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠉⢹⣿⡌⠛⣉⣿⠁⠈⠋⠛⠛⢿⣩⣿⣿⣷⠻⠆⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⢰⣶⠈⣿⣦⣄⢠⣤⣤⡄⠀⣠⣤⣾⣿⣿⣿⣿⣥⣠⣴⡤⠀⠀⠀⠀⠀⣈⠉⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠸⠿⢠⣿⣿⣿⢸⣿⣿⡇⢸⣿⣿⣿⣿⣿⠟⠛⠛⢿⡉⠀⠀⣉⣉⣾⣿⣿⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⢯⡽⡇⢸⣿⡏⠐⠂⢸⣿⡟⣿⣿⢿⣿⡇⠸⣿⣿⣿⠟⠀⠀⠀⠘⢻⣷⣶⣶⣿⣿⣿⠟⢈⠀⠀⠀⠀⠀⠀⡄⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣖⣿⣧⣾⣯⠀⠀⠁⠀⢸⣿⡇⢸⡇⢸⣿⡇⣿⣿⢸⣿⡇⢀⣿⠿⠃⠀⠀⠀⠀⢿⣾⣛⣿⣿⣿⣿⣿⠿⣿⣷⣶⣷⣿⣾⣿⣬⣯⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠴⠿
⣠⣤⠤⣤⠀⢀⡀⠶⢶⠀⠀⠀⢲⣆⣴⣾⢥⣼⡿⣿⣿⡍⠀⠀⠈⠀⠀⠘⠛⣃⣾⡇⠘⠻⠇⠿⠏⠸⠿⠇⠀⠹⣶⠀⠀⠀⢀⣄⢸⣿⡟⣿⣿⣿⣿⣿⣯⣿⣾⣽⣿⡏⠉⢙⣿⣿⠋⠁⣀⣤⣴⣤⣀⣀⡀⠀⠀⠀
⠿⣿⣶⣤⣶⣮⣤⣶⠆⣤⣤⠉⣥⣬⠹⢋⣭⣭⣅⡘⢋⣉⣀⣀⣀⡀⠀⠀⠀⣛⣛⠃⠀⠀⠀⠿⠿⠶⠶⠀⠀⢰⣷⡷⠤⣴⣿⣿⠈⣿⠀⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡿⠿⣿⣶⣤
⣿⠛⢿⣿⣿⣿⣿⣿⡄⣿⣿⡇⣿⣿⠀⣿⣿⢹⣿⣷⠘⢻⣿⣿⠻⠇⠀⢀⣾⣿⢻⣿⡆⢸⣿⣿⣿⡇⣿⣶⣶⡀⣶⣶⣶⠀⣶⣶⠀⣤⣤⠉⣥⣬⠉⣭⣭⠙⣿⣿⣿⣿⠟⣿⣟⣿⣿⣿⣿⣿⣭⣉⡘⢛⣻⣿⠛⢿
⢛⣂⣘⣛⣛⣉⣙⣛⡁⣿⣿⣿⣿⣿⢰⣿⣿⢸⣿⣿⢀⢸⣿⣿⢀⣀⣀⢸⣿⡿⠘⠛⠃⢸⣿⣿⣐⠂⣿⣿⣿⣧⣿⣿⣿⠀⣿⣿⠀⣿⣿⡇⣿⣿⠀⣿⣿⠀⣿⣿⣿⢿⠀⣿⠉⠻⣶⣿⣏⣛⣻⣿⣿⣿⣿⢁⣀⣀
⢸⣿⣿⣿⣿⣿⣿⣿⣇⣿⡇⢿⣿⡿⢸⣿⣿⢸⣿⡏⢿⢸⣿⣿⠸⣿⣿⢸⣿⡇⢻⣿⡇⢸⣿⡟⠛⠀⣿⣿⢻⣿⡏⣿⣿⠀⣿⣿⠀⣿⣿⣿⣿⣿⠀⣿⣿⠀⣿⣿⡟⠘⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⣸⠿⢇⣌⠿⣿⣾⡿⢃⣾⢸⣿⡇⣷⣙⡜⡸⣿⣧⣼⣿⡇⢸⣿⣧⣤⠀⣿⣿⢸⣿⠃⣿⣿⢸⣿⣿⠀⣿⡇⢿⣿⡿⢸⣿⣿⠘⠿⡍⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⠛⠛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣮⣭⣵⣭⣵⣬⣍⠉⠉⠀⢛⣻⠜⣛⣐⣛⣃⣜⣛⣛⠀⠿⠇⠸⠿⠇⠸⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣶
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢻⢹⡟⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣠⠟⣡⣾⡟⠁⠀⠙⠿⠿⠿⠧⠆⠀⠀⣿⡇⡶⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣏⣿⣿⡿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣷⣶⣿⣶⣶⣶⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣟⡟
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣭⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣽⣿⡇⠀⢀⣿⣿⣿⣿⡀⠀⠛⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡷⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⢿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⣇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡏⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢃⣿⠛⡋⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⠿⠿⠿⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣾⢾⡛⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠇⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⠿⠿⠟⠛⠿⠙⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠁⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣟⠛⠻⠛⠻⠛⠋⠃⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠆⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⡀⠀⣀⣤
⣿⣿⡫⣽⣿⢿⣿⣿⣿⡟⠿⠾⠞⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠃
⠿⠿⠏⠻⠊⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠇⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠘⠛⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⣟⠥⢰⡆⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣤⠀⠀⣾⣿⣿⣏⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠈⠃⠀⠈⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣦⡄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⣻⣿⡍⠀⠀⢠⣤⣴⣦⠀⠂⠈⠃⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠀⠉⠁⠀⠀⣿⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⢿⣿⣿⡇⠖⠀⣶⣶⠆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦⣿⣿⠗⠀⠀⢸⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⠁⠸⠙⢿⠁⠘⡿⣿⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⡿⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠃⠀⠀⠸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠟⢿⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣤⣴⣶⣶⣼⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⡁⠀⡀⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⡿⠋⠉⣷⡘⠋⢹⣿⠃⠘⠛⠻⠿⢿⣩⣾⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⢰⣶⠈⣿⣦⣄⢠⣥⣤⡄⠀⢀⣤⣾⣿⣿⣿⣿⣅⣠⣤⣄⠀⠀⠀⠀⠀⡈⠉⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠸⠿⢠⣿⣿⣿⢸⣿⣿⡇⠀⣿⣿⣿⣿⡿⠟⠛⠻⣿⠋⠁⠀⣈⣀⣾⣿⣿⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡴⡇⢸⣿⡏⠐⠆⢸⣿⡟⣿⣿⢿⣿⡇⠈⢿⣿⣿⠟⠀⠀⠀⠘⣿⣷⣶⣾⣿⣿⣿⠛⠉⠁⠀⠀⠀⠀⠀⡄⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣶⣿⣧⣾⣯⠁⠀⠁⠁⢸⣿⡇⢰⡇⢸⣿⡇⣿⣿⢸⣿⡇⠀⣾⡿⠃⠀⠀⠀⠀⢾⣿⣛⣿⣿⣿⣿⣿⠿⣿⣷⣶⣶⣿⣾⣿⣼⣯⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡴⠷
⣠⣤⠤⣴⠄⢀⡀⠶⢶⡀⠀⠀⢲⣆⣤⣾⢥⣼⡿⣿⣿⡇⠀⠀⠘⠀⠀⠘⠛⣃⣾⡇⠘⠻⠇⠿⠏⠸⠿⠇⠀⠸⢷⡆⠀⠀⢀⣤⢸⣿⡟⣿⣿⣿⣿⣿⣿⣿⣾⣽⣿⡏⠉⢛⣿⣿⢃⠁⣀⣤⣴⣤⣀⡀⠀⠀⠀⠀
⢿⣿⣶⣤⣶⣯⣤⣶⠆⣤⣤⠉⣥⣭⠙⢋⣭⣬⣄⡘⢋⣉⣀⣀⣀⡀⠀⠀⠀⣛⣛⠃⠀⠀⠀⠾⠿⠶⠶⠀⠀⢲⣿⡷⠦⣶⣿⣿⠈⣿⠀⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⡿⠿⣷⣶⣄
⣿⠛⢿⣿⣿⣿⣿⣿⡄⣿⣿⡇⣿⣿⠀⣿⣿⢹⣿⣷⠘⢻⣿⣿⠻⠇⠀⢀⣾⣿⢻⣿⡆⢸⣿⣿⣿⡇⣾⣶⣶⡀⣶⣶⣶⠀⣶⣶⠀⣤⣤⠉⣥⣤⠉⣭⣭⠙⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣍⣋⡛⠛⣿⡿⠛⣛
⢛⣂⣘⣛⣋⣉⣙⣛⡀⣿⣿⣿⣿⣿⢰⣿⣿⢸⣿⣿⢀⢸⣿⣿⢀⣀⣀⢸⣿⡿⠘⠛⠃⢸⣿⣿⣐⠂⣿⣿⣿⣧⣿⣿⣿⠀⣿⣿⠀⣿⣿⡇⣿⣿⠀⣿⣿⠀⣿⣿⣿⣿⠁⣿⠉⠻⣾⣿⣏⣛⣛⣿⣿⣿⣿⢁⣀⣀
⢸⣿⣿⣿⣿⣿⣿⣿⣇⣿⡇⢿⣿⡿⢸⣿⣿⢸⣿⡏⢿⢸⣿⣿⠸⣿⣿⢸⣿⡇⢻⣿⡇⢸⣿⡟⠛⠀⣿⣿⢻⣿⡏⣿⣿⠀⣿⣿⠀⣿⣿⣿⣿⣿⢀⣿⣿⢀⣿⣿⡟⠘⠀⠀⠀⢸⣿⣿⣿⣿⣯⣿⣿⣿⣟⣛⣛⣛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⣸⠿⢇⣌⠿⣿⣾⡿⢃⣾⢸⣿⡇⣷⣙⡜⡸⣿⣧⣼⣿⡇⢸⣿⣧⣤⠀⣿⣿⢸⣿⠃⣿⣿⢸⣿⣿⠀⣿⡇⢿⣿⡿⢸⣿⣿⠘⢿⡉⠀⠀⠀⠀⠀⠈⣿⣿⣿⠿⠿⣿⣿⣿⣿⡿⠛⠙
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣮⣭⣵⣭⣵⣬⣍⠉⠉⠀⢛⣻⠜⣛⣔⣛⣃⣜⣛⣛⠀⠿⠇⠸⠿⠇⠸⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣶
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢻⢹⡟⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⠟⣡⣾⡟⠁⠀⠙⠿⠿⠿⠧⠆⠀⠀⣿⡇⡶⠖⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣏⣿⣿⡿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣶⣶⣿⣶⣶⣶⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⡿⡟
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣭⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣿⣿⡇⠀⢀⣿⣿⣿⣿⡀⠀⠛⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡿⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡏⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣭⣽⡽⠭⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⡷⡿⢛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠏⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⢿⢟⠻⠿⠟⠑⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠁⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⠛⠛⠛⠉⠉⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⠀⠀⣀⣤
⣿⣿⣿⣛⡻⣿⣿⣿⣿⡿⠏⡛⠎⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠿⠿⠟⠋⠓⠁⠀⠁⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠇⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠘⠛⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⣯⠤⢰⡆⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣤⠀⠂⣾⣿⣿⣟⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠈⠃⠀⠈⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⢿⣿⠟⠉⠁⣀⠉⣤⡀⠀⠄⠘⠃⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣔⠚⠿⠀⠀⢠⣝⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⢿⣿⣿⡇⠖⠀⣶⣶⠆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⣤⣤⡄⠀⠈⣿⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⠁⠸⠙⢻⠁⠸⠿⣿⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⡿⠻⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠄⠀⠀⢹⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⡟⢿⠀⠀⠀⠈⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡋⢀⢀⣀⣀⣸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣜⠟⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⣿⠧⠒⢶⠹⠿⢿⣿⡟⠸⣿⣿⣿⣿⡟⣻⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⢰⣶⠀⣶⣦⣄⢠⣭⣤⡄⠀⠀⠈⣱⣿⣿⣿⣿⣁⣠⣤⣤⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠸⠿⢠⣿⣿⣿⢸⣿⣿⡇⠀⣴⣿⣿⣿⣿⡟⠛⠻⣿⠏⠑⠀⢀⣀⣶⣿⣿⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣴⡇⢸⣿⡏⠐⠆⢸⣿⡟⣿⣿⢿⣿⡇⠀⣿⣿⣿⡿⠋⠀⠀⠘⢿⣷⣶⣶⣿⣟⣿⠟⠉⠉⠀⠀⠀⠀⠀⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣶⣿⣷⣾⣯⠁⡈⠁⠀⢸⣿⡇⢰⡆⢸⣿⡇⣿⣿⢸⣿⡇⠀⣸⡿⠏⠀⠀⠀⠀⢸⣾⣛⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣼⣇⣀⣉⡀⠀⠀⠀⠀⠀⠀⠀⡴⠷
⣠⣤⠤⣴⠄⢀⡀⠴⢶⡀⠀⠀⣲⣆⣴⣾⢥⣼⡿⣿⣿⡅⠀⠀⠘⠀⠀⠘⠛⣃⣿⡇⠘⠻⠇⠿⠏⠸⠿⠇⠀⠹⡇⠀⠀⠀⠀⢀⢸⣿⡿⣿⣿⣿⣿⣿⣟⣿⣿⣭⣿⠟⠙⢻⣿⡿⠋⠉⢀⣤⣤⣤⡀⡀⠀⠀⠀⠀
⣿⣿⣶⣤⣶⣯⣤⣶⠆⣤⣤⠈⣤⣭⠙⢋⣭⣬⣅⡘⢋⣉⣀⣀⣀⡀⠀⠀⠀⣛⣛⠃⠀⠀⠀⠾⠿⠤⠶⠀⠀⢤⡷⠤⠄⢠⢴⣿⢈⣿⠀⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⠿⣷⣶⣄
⣿⠛⢿⣿⣿⣿⣿⣿⡄⣿⣿⡇⣿⣿⠀⣿⣿⢹⣿⣷⠘⢻⣿⣿⠻⠇⠀⢀⣾⣿⢻⣿⡆⢸⣿⣿⣿⡇⣾⣶⣶⡈⣶⣶⣶⠀⣶⣶⠀⣤⣤⠉⣥⣬⠉⣭⣭⠙⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣍⣋⡛⠛⣿⡿⠛⢛
⢛⣂⣘⣛⣛⣉⣙⣛⡀⣿⣿⣿⣿⣿⢰⣿⣿⢸⣿⣿⢀⢸⣿⣿⢀⣀⣀⢸⣿⡿⠘⠛⠃⢸⣿⣿⣐⠂⣿⣿⣿⣧⣿⣿⣿⠀⣿⣿⠀⣿⣿⡇⣿⣿⠀⣿⣿⠀⣿⣿⣿⢿⠀⣽⠉⠻⣾⣿⣟⣛⣻⣿⣿⣾⣿⢃⣀⣀
⢸⣿⣿⣿⣿⣿⣿⣿⣇⣿⡇⢿⣿⡿⢸⣿⣿⢸⣿⡏⢿⢸⣿⣿⠸⣿⣿⢸⣿⡇⢻⣿⡇⢸⣿⡟⠛⠀⣿⣿⢻⣿⡏⣿⣿⠀⣿⣿⠀⣿⣿⣿⣿⣿⠀⣿⣿⢀⣿⣿⡟⠘⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⠿⠛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⣸⠿⢇⣌⠿⣿⣾⡿⢃⣾⢸⣿⡇⣷⣙⡜⡸⣿⣧⣼⣿⡇⢸⣿⣧⣤⠀⣿⣿⢸⣿⠃⣿⡿⢸⣿⣿⠀⣿⡇⢿⣿⡿⢸⣿⣿⠘⠿⡍⠀⠀⠀⠀⠀⠈⣿⣿⣿⠿⠿⣿⣿⣿⣯⡿⠻⠛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣮⣭⣵⣭⣵⣬⣍⠉⠉⠀⢛⣻⠜⣛⣔⣛⣃⣜⣛⣛⠀⠿⠇⠸⠿⠇⠸⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣶
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢻⢹⡟⣿⣿⣿⣿⡿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⠟⣡⣾⡟⠁⠀⠙⠿⠿⠿⠧⠆⠀⠀⣿⡇⡶⠖⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣏⣿⣿⡿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣶⣶⣿⣶⣶⣶⣤⣤⣤⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⡿⡟
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣭⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣽⣿⡇⠀⢀⣿⣿⣿⣿⡀⠀⠛⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡿⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡟⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⢟⢿⡃⠄⠀⠒⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡩⢭⠁⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⡟⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢯⣿⠿⣛⡛⢛⡀⠢⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠁⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⢿⣻⡿⢟⠋⠙⠋⠃⠀⠈⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡀⠀⣀⣤
⣿⣿⣟⡟⠙⠛⠻⠟⠋⠢⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠻⠿⠗⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠟⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠘⠛⠁⢁⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⣿⢡⠰⡆⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣤⠀⠀⣼⣿⣿⣿⣀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠋⠀⠀⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⡿⠛⠂⠀⠃⣀⡀⠀⠀⠈⠃⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡡⠼⣿⠀⠀⢀⣿⣷⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⡇⠖⠀⣶⡶⡆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⣀⣀⡀⠀⠈⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⣿⡇⢸⠛⢻⠉⠀⡿⣿⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⡃⠀⠀⢿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⠿⢿⠁⠀⠀⠈⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠁⠁⠀⡀⢸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣗⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣿⡏⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠻⣿⣿⣤⠀⠘⣿⣿⣿⡿⢻⣿⣿⣿⣿⡿⣻⣿⣿⡶⠆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⢰⣶⠀⣶⣦⣄⢠⣭⣤⡄⠉⠉⠙⢻⣷⣿⣿⣿⣇⣀⣤⣤⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠸⠿⢠⣿⣿⣿⢸⣿⣿⡇⠀⣰⣾⣿⣿⣿⣿⠿⠿⢿⡋⠁⠀⢂⣀⣾⣿⣿⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⡵⡇⢸⣿⡏⠐⠆⢸⣿⡟⣿⣿⢿⣿⡇⠀⣿⣿⣿⣿⠟⠁⠀⠐⢿⣷⣦⣶⣿⣟⣿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣴⣾⣿⣴⣿⠁⠈⠁⠀⢸⣿⡇⢰⡆⢸⣿⡇⣿⣿⢸⣿⡇⠀⢹⣿⡟⠋⠀⠀⠀⢰⣾⣛⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣞⣻⣯⣀⣈⣀⠀⠀⠀⠀⠀⠀⠀⢰⠷
⣠⣤⠤⣴⡄⢀⡀⠴⢶⡄⠀⠠⠲⣦⣤⣾⣥⣴⡿⣿⣿⡍⠀⠀⠘⠀⠀⠘⠛⣃⣾⡇⠘⠻⠇⠿⠏⠸⠿⠇⠀⢺⡏⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⣟⣿⣿⣭⣿⡟⠉⠛⣿⣿⡏⠉⢀⣤⣤⣤⡀⣀⡀⠀⠀⠀
⢿⣿⣶⣤⣶⣮⣤⣴⠆⣤⣤⠀⣥⣭⠙⢋⣥⣬⣅⡘⢋⣉⣀⣀⣀⡀⠀⠀⠀⣛⣛⠃⠀⠀⠀⠼⠿⠎⠷⠀⠀⠠⠧⠀⠀⢠⢼⣿⠀⣿⠁⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡿⠿⢷⣶⣄
⣿⡛⢿⣿⣿⣿⣿⣿⡄⣿⣿⡇⣿⣿⠀⣿⣿⢹⣿⣷⠘⢻⣿⣿⠻⠇⠀⢀⣾⣿⢻⣿⡆⢸⣿⣿⣿⡇⣾⣶⣶⡀⣶⣶⣶⠀⣶⣶⠀⣤⣤⠉⣥⣭⠉⣭⣭⠙⣿⣿⣿⣿⡿⠿⣿⣿⣿⣽⣿⣿⣍⣋⠙⠛⣿⣿⠛⢛
⢛⣃⣘⣛⣛⣋⣙⣛⡃⣿⣿⣿⣿⣿⢰⣿⣿⢸⣿⣿⢀⢸⣿⣿⢀⣀⣀⢸⣿⡿⠘⠛⠃⢸⣿⣿⣐⠂⣿⣿⣿⣧⣿⣿⣿⠀⣿⣿⠀⣿⣿⡇⣿⣿⠀⣿⣿⠀⣿⣿⣿⢿⠃⢹⠉⠻⣾⣿⣟⣛⣻⣿⣿⣿⣿⢋⣀⣀
⢸⣿⣿⣿⣿⣿⣿⣿⣇⣿⡇⢿⣿⡿⢸⣿⣿⢸⣿⡏⢿⢸⣿⣿⠼⣿⣿⢸⣿⡇⢻⣿⡇⢸⣿⡟⠛⠀⣿⣿⢻⣿⡏⣿⣿⠀⣿⣿⠀⣿⣿⣿⣿⣿⢀⣿⣿⠀⣿⣿⡟⠘⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⡿⠛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⣸⠿⢇⣌⠿⣿⣾⡿⢃⣾⢸⣿⡇⣷⣙⡜⡸⣿⣧⣼⣿⡇⢸⣿⣧⣤⠀⣿⣿⢸⣿⠃⣿⣿⢸⣿⣿⠀⣿⡇⢿⣿⡿⢸⣿⣿⠘⢟⡏⠁⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⣿⣿⣿⣿⡿⠟⠛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣮⣭⣵⣭⣵⣬⣍⠉⠉⠀⢛⣻⠜⣛⣔⣛⣃⣘⣛⣛⠀⠿⠇⠸⠿⠇⠸⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣾⣿⣷⣾⣿⣶⣶
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢻⢻⡟⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣠⠟⣡⣾⡟⠁⠀⠙⠿⠿⠿⠧⡆⠀⠀⣿⡇⣶⠖⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣏⣿⣿⡿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣷⣶⣿⣶⣶⣶⣤⣤⣤⣤⣤⣄⣘⣿⣿⣿⣿⣿⣿⣿⣿⡟
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣭⠙⣯⠉⣭⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣽⣿⡇⠀⢀⣿⣿⣿⣿⡀⠀⠛⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡷⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡟⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢦⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⡟⠐⠑⠁⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠟⢉⠙⠈⠈⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡟⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣟⠈⠙⠊⡥⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠁⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⠟⡟⢉⠋⠙⠙⠛⠁⠩⠙⠃⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡀⠀⣀⣤
⣻⣛⡛⢍⠈⠁⠋⠙⠉⠈⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇
⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠟⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠘⠛⠁⢁⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⣿⢡⠀⡶⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠀⠀⣸⣿⣿⣿⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⢿⠿⠁⠋⠀⠀⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠿⠛⠋⠀⠉⢀⡀⠀⠀⠀⠃⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠚⠿⠇⠀⠀⣿⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡇⠶⠀⣶⡶⡆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣍⣠⣤⣄⠀⠀⢿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⣿⡇⢸⠋⢻⡋⠀⡿⣿⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⡁⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠻⢻⠁⠀⠀⠈⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠁⢀⣀⣀⢸⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣿⠏⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⢿⡶⠄⠘⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣯⣿⣿⡾⠆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⢰⣶⠀⣶⣦⣄⢠⣭⣤⡄⠀⠈⢉⣹⣿⣿⣿⣿⣇⣀⣤⣤⡀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠠⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠸⠿⢠⣿⣿⣿⢸⣿⣿⡇⢀⣼⣿⣿⣿⣿⣿⠿⠿⢿⡟⠁⠀⠀⣀⣰⣿⣿⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡵⡇⢸⣿⡏⠀⠆⢸⣿⡟⣿⣿⢿⣿⡇⢸⣿⣿⣿⣿⡟⠃⠀⠀⢻⣷⣶⣶⣿⣟⣿⡟⠉⠉⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣰⣾⣿⣴⣿⡁⠀⠁⠀⢸⣿⡇⢰⡇⢸⣿⡇⣿⣿⢸⣿⡇⠀⣹⣿⠟⠋⠀⠀⠀⠰⣶⣟⣿⣿⣿⣿⣿⡿⣿⣷⣶⣶⣶⣶⣾⣟⣟⣀⣈⣀⠀⠀⠀⠀⠀⠀⠀⢠⠷
⣀⣤⡤⣤⡄⠀⡀⠴⢶⡄⠀⠀⢲⣦⣤⣾⣧⣤⣿⣿⣿⡍⠀⠀⠘⠀⠀⠘⠛⢃⣿⡇⠘⠻⠇⠿⠏⠸⠿⠇⠀⢻⡁⠀⠀⠀⠀⡀⠀⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣭⣿⡟⠉⠛⣿⣿⡟⠉⠀⣠⣤⣤⡀⣀⠀⠀⠀⠀
⡿⣿⣴⣦⣴⣾⣥⣴⠆⣤⣤⠈⣤⣬⠙⢋⣥⣬⣅⡘⢋⣉⣁⣀⣀⡀⠀⠀⠀⣛⣛⠃⠀⠀⠀⠸⠿⠦⠷⠀⠀⢨⠧⠄⠀⢠⢿⣿⠀⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⠿⢷⣶⣤
⣿⡛⢻⣿⣿⣿⣿⣿⡄⣿⣿⡇⣿⣿⠀⣿⣿⢹⣿⣷⠘⢻⣿⣿⠻⠇⠀⢀⣾⣿⢻⣿⡆⢸⣿⣿⣿⡇⣾⣶⣶⡀⣶⣶⣶⠀⣶⣶⠀⣤⣤⠈⣥⣭⠉⣭⣭⠙⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣍⣋⡛⠛⣿⣿⠿⢿
⢛⣃⣘⣛⣛⣉⣉⣛⡃⣿⣿⣿⣿⣿⢰⣿⣿⢸⣿⣿⢐⢸⣿⣿⢀⣀⣀⢸⣿⡿⠘⠛⠃⢸⣿⣿⣐⠂⣿⣿⣿⣧⣿⣿⣿⠀⣿⣿⠀⣿⣿⡇⣿⣿⠀⣿⣿⠀⣿⣿⣿⣿⠃⢹⡉⠻⣶⣿⣿⣙⣛⣿⣿⣿⣿⣋⡀⣀
⢸⣿⣿⣿⣿⣿⣿⣿⣇⣿⡇⢿⣿⡿⢸⣿⣿⢸⣿⡏⢿⢸⣿⣿⠸⣿⣿⢸⣿⡇⢻⣿⡇⢸⣿⡟⠛⠀⣿⣿⢻⣿⡏⣿⣿⠀⣿⣿⠀⣿⣿⣿⣿⣿⢀⣿⣿⠠⣿⣿⡿⠘⠀⠈⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡿⠛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⣸⠿⢇⣌⠿⣿⣾⡿⢃⣾⢸⣿⡇⣷⣙⡜⡸⣿⣧⣼⣿⡇⢸⣿⣧⣤⠀⣿⣿⢸⣿⠃⣿⡿⢸⣿⣿⠀⣿⡇⢿⣿⡿⢸⣿⣿⠐⢿⡏⠁⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⣿⣿⣿⣿⣫⠛⠙
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣮⣭⣵⣭⣵⣬⣍⠉⠉⠀⢛⣻⠜⣛⣔⣛⣃⣘⣛⣛⠀⠿⠇⠸⠿⠇⠸⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣾⣿⣿⣾⣿⣶⣶
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢻⢹⡟⣿⣿⣿⣿⡿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣠⠟⣡⣾⡟⠁⠀⠙⠿⠿⠿⠧⡆⠀⠀⣿⡇⣶⠖⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣏⣿⣿⡿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣷⣶⣿⣷⣶⣶⣦⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣭⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣿⣿⡇⠀⢀⣿⣿⣿⣿⡀⠀⠙⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡷⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⣧⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣱⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠿⠏⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡟⢋⠩⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡟⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⠋⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡃⠀⠀⠀
⣿⣿⣿⡿⣿⣿⣿⣿⡿⠛⠋⡛⠚⠉⠛⠛⠋⠋⠀⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣀⠀⢀⣤
⣻⣟⠋⠓⡀⠘⠘⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏
⠁⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠟⢠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠈⠛⠃⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⢥⠀⡶⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠀⠀⣸⣿⣿⣿⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⢿⠿⠁⠋⠀⠀⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⣿⡿⠉⠉⢀⣈⣠⣄⠀⠀⠀⠃⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡘⠻⠇⠀⠠⣽⣻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⡿⣿⣿⡗⠒⠀⣶⡶⡦⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣤⣤⣤⠀⠀⢺⣿⣿⣿⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⡇⠸⠉⢻⠁⠀⡿⣿⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠟⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⡁⠀⠀⠘⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠻⢻⠁⠀⠀⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⣀⣀⣤⣨⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡽⠏⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠻⡿⢿⡖⠂⠘⠿⢿⣿⠋⠻⠿⣿⣿⣿⣛⣿⣿⣿⠲⠆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⢰⣶⠀⣶⣦⣄⢨⣥⣤⡄⠀⢀⣠⣿⣿⣿⣿⣿⣇⣀⣤⣤⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠠⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠸⠿⢠⣿⣿⣿⢸⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⠿⠛⢿⡋⠁⠀⠀⣀⣰⣿⣿⣥⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡵⡇⢸⣿⡏⠀⠆⢸⣿⡿⣿⣿⢿⣿⡇⢸⣿⣿⣿⣿⠟⠃⠀⠀⢻⣷⣶⣶⣿⣟⣿⡟⠉⠉⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣰⣾⣿⣴⣿⡀⠀⠁⠀⢸⣿⡇⢰⡆⢸⣿⡇⣿⣿⢸⣿⡇⠀⣿⡿⠛⠁⠀⠀⠀⠰⣶⣟⣻⣿⣿⣿⣿⡿⢿⣷⣶⣶⣶⣶⣾⣃⣻⣀⣈⣀⠀⠀⠀⠀⠀⠀⠀⢠⠿
⣀⣤⡤⣤⡄⠀⡀⠤⠶⡄⠀⠀⠰⣦⣤⣾⣥⣤⡿⣷⣿⡍⠀⠀⠐⠁⠀⠘⠛⢃⣾⡧⠘⠻⠇⠿⠏⠸⠿⠇⠀⢛⠀⠀⠀⠀⢀⡀⠀⣿⡯⢻⣿⣿⣿⣿⣏⣿⣿⣯⣿⡟⠉⠙⣻⣿⢟⠉⠀⣠⣤⣤⣀⣀⡀⠀⠀⠀
⡿⣿⣶⣦⣴⣾⣥⣴⠆⣤⣤⠀⣤⣬⠙⢋⣥⣬⣅⡘⢋⣉⣁⣀⣀⡀⠀⠀⠀⣛⣛⠃⠀⠀⠀⠸⠿⠦⠖⠀⠀⠘⠧⠀⠀⠠⠾⣿⠀⣹⠁⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⡿⠿⢿⣶⣦
⣿⡛⢻⣿⣿⣿⣿⣿⡄⣿⣿⡇⣿⣿⠀⣿⣿⢹⣿⣷⠘⢻⣿⣿⠻⠃⠀⢀⣾⣿⢻⣿⡆⢸⣿⣿⣿⡇⣾⣶⣶⠀⣶⣶⣶⠀⣶⣶⠀⣤⣤⠈⣥⣬⠉⣭⣭⠙⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣍⣋⡙⠛⣿⣿⠻⢿
⢛⣃⣘⣛⣛⣉⣉⣛⡃⣿⣿⣿⣿⣿⢰⣿⣿⢸⣿⣿⢐⢸⣿⣿⢀⣀⣀⢸⣿⡿⠘⠛⠃⢸⣿⣿⣐⠂⣿⣿⣿⣧⣿⣿⣿⠀⣿⣿⠀⣿⣿⡇⣿⣿⠀⣿⣿⠀⣿⣿⣿⡿⠃⢹⠉⠻⣾⣿⣟⣙⣛⣿⣿⣿⣿⣏⣀⣀
⢸⣿⣿⣿⣿⣿⣿⣿⣇⣿⡇⢿⣿⡿⢸⣿⣿⢸⣿⡏⢿⢸⣿⣿⠸⣿⣿⢸⣿⡇⢻⣿⡇⢸⣿⡟⠛⠀⣿⣿⢻⣿⡏⣿⣿⠀⣿⣿⠀⣿⣿⣿⣿⣿⠀⣿⣿⠀⣿⣿⡟⠘⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡿⠟
⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⣸⠿⢇⣌⠿⣿⣾⡿⢃⣾⢸⣿⡇⣷⣙⡜⡸⣿⣧⣼⣿⡇⢸⣿⣧⣤⠀⣿⣿⢸⣿⠃⣿⣿⢸⣿⣿⠀⣿⡇⢿⣿⡿⢸⣿⣿⠈⠟⡋⠁⠀⠀⠀⠀⠀⣻⣿⣿⠿⠿⣿⣿⣿⣿⢙⠛⠛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣮⣭⣵⣭⣵⣬⣍⠉⠉⠀⢛⣻⠜⣛⣔⣛⣃⣘⣛⣛⠀⠿⠇⠸⠿⠇⠸⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣾⣿⣿⣾⣿⣷⣶
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢻⢹⡟⣿⣿⣿⣿⡿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⠟⣡⣾⡟⠁⠀⠙⠿⠿⠿⠧⠆⠀⠀⣿⡇⣶⠖⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣋⣻⣿⡿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣷⣷⣿⣷⣶⣶⣦⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⡿
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣭⠙⣯⠉⣭⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣽⣿⡇⠀⢀⣿⣿⣿⣿⡀⠀⠙⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡷⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡧⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⠏⠁⠉⠿⡯⠭⠯⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⡻⢯⡿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡟⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⢿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠁⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣾⠿⠋⠋⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡀⠀⣀⣠
⣿⣿⡏⣿⢻⠿⡿⠿⠟⠯⠿⠛⠻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏
⠛⠻⠛⠙⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠟⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠛⠃⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⣿⢥⠀⡶⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠀⠐⣰⣿⣿⣿⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠋⠀⠀⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⣿⡿⠉⠉⢀⣈⣠⣄⠀⠰⠀⠓⠀⠀⠀⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡐⠻⠇⠀⠠⣽⣻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⡿⣿⣿⡗⠒⠀⢶⡶⡦⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣥⣤⣤⠀⠀⢸⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣿⣿⣿⣿⡇⠸⠋⢻⡇⠀⡿⣿⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⠟⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⡁⠀⠀⠘⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠹⠻⠇⠀⠀⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⣀⣀⣤⣸⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣽⠏⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠻⣿⢿⡖⠂⠘⠿⢿⡿⠉⠿⢿⣿⣿⣿⣛⣽⣿⣿⠲⠆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⢰⣶⠀⣶⣦⣄⢨⣥⣤⡄⠀⢀⣠⣿⣿⣿⣿⣿⣏⣀⣤⣤⡀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠸⠿⢠⣿⣿⣿⢸⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⠿⠿⢿⡟⠁⠁⢈⣉⣰⣿⣿⣷⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡥⡇⢸⣿⡏⠀⠆⢸⣿⡿⣿⣿⢿⣿⡇⢸⣿⣿⣿⣿⠟⠁⠀⠀⢿⣷⣶⣶⣿⣿⣿⡟⠉⠉⠁⠀⠀⠀⠀⡠⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣰⣾⣿⣴⣿⡅⠀⠁⠀⢸⣿⡇⢰⡦⢸⣿⡇⣿⣿⢸⣿⡇⠀⣿⡿⠛⠁⠀⠀⠀⠰⣾⣟⣿⣿⣿⣿⣿⡿⢿⣿⣶⣶⣶⣶⣾⣣⣿⣀⣈⣀⠀⠀⠀⠀⠀⠀⠀⢠⠿
⣀⣤⣤⣤⡄⠀⡀⠴⢶⡄⠀⠀⠲⣦⣤⣶⡯⣤⣿⢿⣿⡯⠀⠀⠐⠀⠀⠘⠛⢃⣾⣧⠘⠻⠇⠿⠏⠸⠿⠇⠀⢛⠀⠀⠀⠀⢀⣀⠀⣿⡿⣿⣿⣿⣿⣿⣯⣿⣿⣯⣿⡟⠉⠉⣻⣿⢟⠉⠀⣠⣴⣦⣀⣀⡀⠀⠀⠀
⣿⣿⣶⣦⣴⣾⣧⣤⠆⣤⣤⠈⣤⣬⠙⢋⣥⣬⣍⡀⢋⣉⣁⣀⣀⡀⠀⠀⠀⣛⣛⠃⠀⠀⠀⠸⠿⠦⠶⠀⠀⠸⠦⠄⠀⢠⢿⣿⠃⣹⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⡿⠿⢿⣶⣦
⣿⡟⢻⣿⣿⣿⣿⣿⡄⣿⣿⡇⣿⣿⠀⣿⣿⢹⣿⣷⠘⢻⣿⣿⠻⠇⠀⢀⣾⣿⢻⣿⡆⢸⣿⣿⣿⡇⣾⣶⣶⠀⣶⣶⣶⠀⣶⣶⠀⣤⣤⠈⣥⣭⠉⣭⣭⠙⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣍⣉⡙⠛⣻⣿⠛⢛
⢛⣃⣀⣛⣛⣛⣉⣛⡃⣿⣿⣿⣿⣿⢰⣿⣿⢸⣿⣿⢘⢸⣿⣿⢀⣀⣀⢸⣿⡿⠘⠛⠃⢸⣿⣿⣐⠂⣿⣿⣿⣧⣿⣿⣿⠀⣿⣿⠀⣿⣿⡇⣿⣿⠀⣿⣿⠐⣿⣿⣿⡿⡇⢸⡍⠛⣷⣿⣿⣙⣛⣿⣿⣿⣿⣏⣀⣀
⢸⣿⣿⣿⣿⣿⣿⣿⣇⣿⡇⢿⣿⡿⢸⣿⣿⢸⣿⡏⢿⢸⣿⣿⠸⣿⣿⢸⣿⡇⢻⣿⡇⢸⣿⡟⠛⠀⣿⣿⢻⣿⡏⣿⣿⠀⣿⣿⠀⣿⣿⣿⣿⣿⢀⣿⣿⠠⣿⣿⡿⠉⠃⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡿⠟
⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⣸⠿⢇⣌⠿⣿⣾⡿⢃⣾⢸⣿⡇⣷⣙⡜⡸⣿⣧⣼⣿⡇⢸⣿⣧⣤⠀⣿⣿⢸⣿⠃⣿⣿⢸⣿⣿⠀⣿⡇⢿⣿⡿⢸⣿⣿⠀⡿⣏⠁⠀⠀⠀⠀⠀⢹⣿⣿⡿⠿⣿⣿⣿⣿⣿⡟⠛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣮⣭⣵⣭⣵⣬⣍⠉⠉⠀⢛⣻⠜⣛⣔⣛⣃⣘⣛⣛⠀⠿⠇⠸⠿⠇⠸⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣾⣿⣷⣶
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢻⢻⡟⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⠟⣡⣾⡟⠁⠀⠙⠿⠿⠿⠧⡆⠀⠀⣿⡇⣶⠖⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣏⣿⣿⡿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣷⣷⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣭⠙⣯⠉⣯⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣽⣿⡇⠀⢀⣿⣿⣿⣿⡀⠀⠛⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡷⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣾⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⠈⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢻⢋⡉⠈⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡛⠟⠻⠒⡁⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡏⠀⠀⠀
⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢻⢿⣟⠟⠿⠛⠿⠓⠘⠘⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⡇⠀⢀⣠
⣫⠿⡏⡿⠋⠈⠻⢻⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏
⠛⠫⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⢠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠛⠃⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⡃⠀⣶⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⢰⣿⣿⣿⣁⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠋⠀⠀⠙⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠿⠉⠉⢀⣈⣠⣄⠀⠰⠀⠒⠀⠀⠀⠈⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡘⠻⠇⠀⢠⣿⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⡿⣿⣿⡷⠺⠂⢴⣶⡦⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣥⣤⣤⠀⠀⢿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⡇⠈⠋⢻⡏⠀⡿⣿⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⡁⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢸⠻⡇⠀⠀⠀⠁⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⢀⣀⣤⣠⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣻⠏⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⣿⢿⡶⠄⠸⢿⣿⣿⠋⢿⣿⣿⣿⣿⢟⣯⣿⣿⠰⡦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⢠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⢰⣶⠀⣶⣦⣄⢨⣥⣤⡄⠀⠈⣠⣿⣿⣿⣿⣿⣇⣀⣤⣤⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠸⠿⢠⣿⣿⣿⢸⣿⣿⡇⢠⣿⣿⣿⣿⣿⣿⡿⠟⢿⡟⠉⠀⠀⣀⣰⣾⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣷⣶⣶⢸⣿⡏⠀⠆⢸⣿⡿⣿⣿⢿⣿⡇⢸⣿⣿⣿⣿⠟⠃⠀⠀⢿⣿⣶⣶⣾⣿⣻⡿⠋⠉⠁⠀⠀⠀⠀⢠⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢠⣶⣿⣦⣾⡍⠀⠃⠀⢸⣿⡇⢰⡆⢸⣿⡇⣿⣿⢸⣿⡇⠀⣿⣿⠛⠉⠀⠀⠀⢰⣶⣟⣻⣿⣿⣿⣿⣿⢿⣿⣶⣶⣶⣶⣾⣓⣿⣀⣈⣁⡀⠀⠀⠀⠀⠀⠀⢠⠿
⣀⣀⣤⣤⣤⠀⡀⢠⠦⡄⠀⠀⠴⢦⣤⣴⣯⣤⣿⣿⣿⣯⠁⠀⠀⠂⠀⠘⠛⢃⣾⣧⠘⠻⠇⠿⠏⠸⠿⠇⠐⢿⠀⠀⠀⠀⢀⡀⠀⣿⡿⣿⣿⣿⣿⣿⣯⣿⣿⣭⣽⡟⠋⠙⣻⣿⣟⡉⠁⣠⣤⣤⣀⣀⡀⠀⠀⠀
⣿⣿⣧⣮⣴⣶⣧⣤⡄⣤⣤⠀⣤⣬⠙⢋⣥⣤⣍⡈⢋⣉⣁⣀⣀⡀⠀⠀⠀⣘⣛⠋⠀⠀⠀⠸⠿⠮⠶⠀⠀⠸⠧⠀⠀⢠⢿⣿⠆⣹⠃⢸⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠿⢿⣶⣦
⣿⣟⢻⣿⣿⣿⣿⣿⡄⣿⣿⡇⣿⣿⠀⣿⣿⢹⣿⣷⠘⢻⣿⣿⠻⠇⠀⢀⣾⣿⢻⣿⡆⢸⣿⣿⣿⡇⣾⣶⣶⡀⣶⣶⣶⠀⣶⣶⠀⣤⣤⠈⣭⣬⠉⣭⣭⠙⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣍⣛⡘⠛⣿⣿⠟⠟
⢛⣓⣀⣛⣛⣛⣋⣛⡃⣿⣿⣿⣿⣿⢰⣿⣿⢸⣿⣿⢐⢸⣿⣿⢀⣀⣀⢸⣿⡿⠘⠛⠃⢸⣿⣿⣐⠂⣿⣿⣿⣧⣿⣿⣿⠀⣿⣿⠀⣿⣿⡇⣿⣿⠀⣿⣿⠐⣿⣿⣿⣿⡇⢸⡏⠛⣷⣿⣿⣛⣛⣿⣿⣿⣿⣏⣀⣀
⢸⣿⣿⣿⣿⣿⣿⣿⣇⣿⡇⢿⣿⡿⢸⣿⣿⢸⣿⡏⢿⢸⣿⣿⠸⣿⣿⢸⣿⡇⢻⣿⡇⢸⣿⡟⠛⠀⣿⣿⢻⣿⡏⣿⣿⠀⣿⣿⠀⣿⣿⣿⣿⣿⢀⣿⣿⠠⣿⣿⣿⠃⠃⠀⠀⠀⣿⣿⣿⣿⣿⣽⣿⣿⣿⣛⣿⠟
⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⣸⠿⢇⣌⠿⣿⣾⡿⢃⣾⢸⣿⡇⣷⣙⡜⡸⣿⣧⣼⣿⡇⢸⣿⣧⣤⠀⣿⣿⢸⣿⠃⣿⣿⢸⣿⣿⠀⣿⡇⢿⣿⡿⢸⣿⣿⠀⡿⣋⠁⠀⠀⠀⠀⠀⢹⣿⣿⡿⠿⣿⣿⣿⣿⣟⠟⠛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣮⣭⣵⣭⣵⣬⣍⠉⠉⠀⢛⣻⠜⣛⣔⣛⣃⣘⣛⣛⠀⠿⠇⠸⠿⠇⠸⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣾⣿⣿⣾⣿⣷⣶
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢻⢹⡿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⠟⣡⣾⡟⠁⠀⠉⠿⠿⠿⠧⠆⠀⠀⣿⡇⣶⠶⠆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣏⣿⣿⣿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣶⣶⣾⣷⣦⣤⣤⣤⣤⣄⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⢿
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣭⠙⣯⠉⣭⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣹⣿⡇⠀⢀⣿⣿⣿⣿⡀⠀⠛⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡷⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣿⣦⣤⣶⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠹⡂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠿⠉⠅⠁⠁⠉⠉⠃⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟
⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠋⠩⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠁⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⡟⠚⠻⠯⠥⠀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡏⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡻⣛⣿⠝⠿⠛⠛⠁⠂⠃⠉⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⡇⠀⠀⣀
⡿⢿⣿⡟⡫⠘⠛⠿⡛⠑⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣷⡿⡟
⠉⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢼⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠻⠏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡋⡀⣴⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢠⣾⣿⣿⡃⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋⠙⠁⠀⠙⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠿⠛⠋⠈⠉⢠⡀⠀⠠⠀⠒⠀⠀⠀⠘⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠘⢿⠇⠀⢀⣿⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣶⢰⠂⢰⣶⣤⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⣠⣀⣀⠀⠈⢿⣿⣿⣿⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⡟⠈⠏⠻⡏⠁⢹⢿⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⡿⣿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢹⠿⡇⠀⠀⠀⠃⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠉⠁⣀⣀⣨⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⡏⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⣿⢿⡶⠄⠸⣿⣿⣿⠛⢿⣿⣿⣿⣿⣟⣽⣿⣿⠰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⢰⣶⠀⣶⣦⣄⢨⣥⣤⡄⠀⠉⣩⣿⣿⣿⣿⣿⣗⠀⣠⣤⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⡶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠸⠿⢠⣿⣿⣿⢸⣿⣿⡇⢠⣾⣿⣿⣿⣿⣿⣿⠿⢟⡛⠋⠀⠀⠀⢠⣴⣾⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣒⣤⢸⣿⡏⠀⠄⢸⣿⡿⣿⣿⢿⣿⡇⢸⣿⣿⣿⣿⠟⠁⠀⠀⢿⣷⣤⣤⣴⣿⣿⡿⠛⠋⠉⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⢴⣶⣦⣼⡏⠉⠛⠀⢸⣿⡇⢠⡆⢸⣿⡇⣿⣿⢸⣿⡇⠈⣿⣿⠟⠃⠀⠀⠀⢠⣦⡿⢿⣿⣿⣿⣿⣿⣶⣿⣶⣶⣶⣶⣶⠯⣒⣀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡴
⣀⣀⣀⣀⣄⠀⠀⢀⣤⣄⠀⠀⠠⠤⣠⣴⣏⣉⣿⣿⣿⣟⠉⠁⠀⠂⠀⠘⠛⢃⣾⣧⠘⠻⠇⠿⠏⠸⠿⠇⠐⣿⠁⠀⠀⠀⠀⣀⠀⣿⣿⢿⣿⣿⣿⣿⣿⣺⣯⣿⣻⣿⠛⠛⢿⣿⣟⠉⠉⢀⣠⣤⡀⠀⠀⠀⠀⠀
⣿⣿⣧⣌⣑⣤⣓⣠⡄⣤⣤⠀⣤⣤⠙⢋⣥⣤⣍⡈⢋⣉⣀⣁⣀⡀⠀⠀⠀⣘⣛⠛⠀⠀⠀⠸⠿⠮⠶⠀⠀⠘⠧⠀⠀⠀⢿⣿⠆⢹⠏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣾⣶⣾⣿⣿⣿⣿⣿⠿⢷⣶⣤
⣷⣾⠿⣿⣿⣿⣿⣿⡀⣿⣿⡇⣿⣿⠀⣿⣿⢹⣿⣷⠘⢻⣿⣿⠻⠃⠀⢀⣾⣿⢻⣿⡆⢸⣿⣿⣿⡇⣾⣶⣶⠀⣶⣶⣶⠀⣶⣶⠀⣤⣤⠈⣥⣬⠉⣤⣬⠙⣿⣿⣿⣿⣿⠿⢿⣿⣿⣟⣿⣿⣟⡛⠋⠿⢷⣾⠿⢿
⢛⣋⣀⣛⣛⣛⣛⣛⡃⣿⣿⣿⣿⣿⢰⣿⣿⢸⣿⣿⢀⢸⣿⣿⢀⣀⣀⢸⣿⡿⠘⠛⠃⢸⣿⣿⣐⠂⣿⣿⣿⣧⣿⣿⣿⠀⣿⣿⠀⣿⣿⡇⣿⣿⠀⣿⣿⠀⣿⣿⣿⣿⡇⢀⡏⠛⢯⣽⣿⡛⠻⣿⣿⣷⣿⡟⠀⠀
⢸⣿⣿⣿⣿⣿⣿⣿⣇⣿⡇⢿⣿⡿⢸⣿⣿⢸⣿⡏⢿⢸⣿⣿⠸⣿⣿⢸⣿⡇⢻⣿⡇⢸⣿⡟⠛⠀⣿⣿⢻⣿⡏⣿⣿⠀⣿⣿⠀⣿⣿⣿⣿⣿⠀⣿⣿⢠⣿⣿⣿⠇⠃⠀⠁⠀⣾⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⠿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⣸⠿⢇⣌⠿⣿⣾⡿⢃⣾⢸⣿⡇⣷⣙⡜⡸⣿⣧⣼⣿⡇⢸⣿⣧⣤⠀⣿⣿⢸⣿⠃⣿⡿⢸⣿⣿⠀⣿⡇⢿⣿⡿⢸⣿⣿⠀⡿⡟⠋⠀⠀⠀⠀⠀⢹⣿⣿⣿⢿⣿⣿⣿⣿⣛⡟⠛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣮⣭⣵⣭⣵⣬⣍⠉⠉⠀⢛⣻⠜⣛⣔⣛⣃⣜⣛⣛⠀⠿⠇⠸⠿⠇⠸⠿⠿⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣾⣿⣿⣷⣿⣷⣴
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢿⢺⡷⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⠟⣡⣾⡟⠁⠀⠉⠿⠿⠿⠧⠆⠀⠀⣿⡇⣤⡶⠆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣏⣿⣿⣿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣶⣶⣶⣶⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣭⠙⣯⠉⣭⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣹⣿⡇⠀⢀⣿⣿⣿⣿⡀⠀⠛⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡷⠀⣿⡀⢾⡾⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⢿⣿⡇⠀⣼⡏⠻⣿⣿⣷⣦⣤⣶⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣷⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡇⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⡟⠉⠉⠉⠌⠉⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠝⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⠃⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠹⠅⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡏⠁⠀⠀
⣿⣿⣿⣿⡟⣿⣿⣿⡿⣿⠿⠿⠯⠓⠾⠛⢋⠙⠘⠀⠈⠛⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡇⠀⠀⢀
⠟⡿⣿⡋⣠⠉⠾⠛⠙⠃⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣶⡾⡿
⠛⠹⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⠇⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠾⠟⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡟⡀⢤⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣶⣿⣿⡇⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠏⠙⠁⠀⠙⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠁⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠛⠛⠉⠙⣀⡀⠀⠀⠄⠐⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣍⠸⢿⡆⠀⢀⣿⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣶⠰⠆⢠⣤⣤⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣁⣀⣀⠀⠈⢿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣿⣿⠀⡟⠻⡟⠁⢹⢿⡏⠀⠀⠈⠋⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢋⠀⠀⠸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢹⡿⡟⠀⠀⠀⠃⠀⠈⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠉⠁⠀⣀⣀⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢹⡟⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⡦⠄⠸⣿⣿⣿⠛⣿⣿⣿⣿⣿⡟⣻⣿⣿⠠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⢰⣶⠀⣶⣦⣄⢨⣥⣤⡄⠀⠉⣉⣿⣿⣿⣿⣿⣗⠀⣀⣤⡀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠸⠿⢠⣿⣿⣿⢸⣿⣿⡇⢀⣾⣿⣿⣿⣿⣿⣿⠿⢿⡿⠛⠉⠐⠐⢀⣴⣾⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣖⣤⢸⣿⡏⠀⠄⢸⣿⡟⣿⣿⢿⣿⡇⢸⣿⣿⣿⣿⠟⠃⠀⠀⢴⣿⣤⣤⣴⣿⢿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣴⣶⣆⣼⡏⠉⠛⠈⢸⣿⡇⢠⡄⢸⣿⡇⣿⣿⢸⣿⡇⠈⣿⣿⠟⠋⠀⠀⠀⠀⣦⡿⢿⣿⣿⣿⣿⣿⣶⣶⣤⣴⣶⣶⣶⠶⣒⡂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴
⣀⢀⣀⣀⣀⠀⠀⢀⣤⣄⠀⠀⠀⠤⣀⣠⣞⣉⣽⣿⣿⣿⠉⠁⠀⠆⠀⠘⠛⢃⣾⣧⠘⠻⠇⠿⠏⠸⠿⠇⠀⢿⠁⠀⠀⠀⠀⢀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣺⣯⣟⣻⣿⠛⠛⢿⣿⣿⠛⠋⢁⣀⣀⡀⠀⠀⠀⠀⠀
⣿⣿⣯⣌⣉⣤⣛⣠⡄⣤⣤⠀⣤⣤⠙⢋⣥⣤⣍⡀⢉⣉⣂⣀⣀⡀⠀⠀⠀⣘⣛⠻⠀⠀⠀⠸⠿⠟⠶⠀⠀⠈⠧⠀⠀⠀⠸⣿⡇⢸⠟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⢶⣶⣤
⣾⡿⠿⣿⣿⣿⣿⣿⡀⣿⣿⡇⣿⣿⠀⣿⣿⢙⣿⣷⠘⢻⣿⣿⠻⠃⠀⢀⣾⣿⢻⣿⡆⢸⣿⣿⣿⡇⣾⣶⣶⠀⣶⣶⣶⠀⣶⣶⠀⣤⣤⠈⣥⣬⠉⣤⣬⠙⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣟⠛⠋⠿⠶⣶⡿⠿
⢛⣛⣀⣛⣛⣛⣛⣛⡃⣿⣿⣿⣿⣿⢰⣿⣿⢸⣿⣿⢀⢸⣿⣿⢀⣀⣀⢸⣿⡟⠘⠛⠃⢸⣿⣿⣐⠂⣿⣿⣿⣧⣿⣿⣿⠀⣿⣿⠀⣿⣿⡇⣿⣿⠀⣿⣿⢰⣿⣿⣿⣿⡟⠀⡟⠻⢯⣽⣿⡟⠿⢿⣿⣷⣾⣿⠀⠀
⢸⣿⣿⣿⣿⣿⣿⣿⣇⣿⡇⢿⣿⡿⢸⣿⣿⢸⣿⡏⢿⢸⣿⣿⠸⣿⣿⢸⣿⡇⢻⣿⡇⢸⣿⡟⠛⠀⣿⣿⢻⣿⡏⣿⣿⠀⣿⣿⠀⣿⣿⣿⣿⣿⠀⣿⣿⢠⣿⣿⣿⠇⠃⠀⠁⠀⢴⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⡿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⣸⠿⢇⣌⠿⣿⣾⡿⢃⣾⢸⣿⡇⣷⣙⡜⡸⣿⣧⣼⣿⡇⢸⣿⣧⣤⠀⣿⣿⢸⣿⠃⣿⣿⢸⣿⣿⠀⣿⡇⢿⣿⡿⢸⣿⣿⠀⡿⢟⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⠿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣮⣭⣵⣭⣵⣬⣍⠉⠉⠀⢛⣻⠜⣛⣐⣛⣃⣘⣛⣛⠀⠿⠇⠸⠿⠇⠸⠿⠿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣶⣿⣿⣿⣿⣷⣤
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢿⢺⣷⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⠟⣡⣾⡟⠁⠀⠉⠿⠿⠿⠧⠆⠀⠀⣿⡇⣤⡴⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣏⣿⣿⣿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣷⣶⣤⣴⣦⣤⣤⣄⣀⣀⣀⣀⡀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣭⠙⣯⠉⣭⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⢹⣿⡇⠀⢀⣿⣿⣿⣿⡀⠀⠛⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡷⠀⣿⡀⢶⡾⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣷⣦⣤⣶⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣷⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡧⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣳⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠉⠁⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠉⠉⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠏⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡙⠛⢙⣎⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠈⠀⠀
⣿⣿⣿⣿⣿⢫⢻⣿⢿⣿⣿⡿⣛⠺⠟⠿⠟⠟⠋⠛⠈⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀
⢟⢻⣯⠯⠍⠘⠘⠗⡻⠙⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣷⣶⣶⣿
⠙⠍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡇⢀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⠿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡟⠁⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣴⣾⣿⡗⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡟⠹⠋⠀⠘⠇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠟⠒⠀⠛⢀⡀⠀⠀⡄⠠⠀⠀⠀⠀⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⢿⡇⠀⠀⣸⣷⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠤⡤⢀⣤⣤⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢃⠀⠀⠀⠈⢻⣿⣿⣿⣿⣂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣿⣿⠁⡻⠻⣿⠋⢸⣿⣿⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⡿⣿⠀⠀⠀⠙⠀⠈⠈⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠓⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⢿⣿⣧⣄⠀⣿⣿⣿⡟⣿⣿⣿⣿⣿⡿⣿⣿⣿⡄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⢰⣶⠀⣶⣦⣄⢠⣭⣤⡄⠉⠉⠉⣿⣿⣿⣿⣿⣷⠈⢀⣀⡀⠀⠀⠀⠀⠀⠚⠛⠁⠀⠀⢠⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠸⠿⢠⣿⣿⣿⢸⣿⣿⡇⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⠋⠀⠀⠀⣤⣶⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⢸⣿⡏⠀⠀⢸⣿⡿⣿⣿⢿⣿⡇⢸⣿⣿⣿⣿⡿⠟⠁⠀⢰⣿⣦⣀⣤⣶⡿⣿⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⣶⣆⣤⣟⠋⠹⠉⢸⣿⡇⢠⡄⢸⣿⡇⣿⣿⢸⣿⡇⠈⢿⣿⣿⠟⠁⠀⠀⠀⣤⣹⢿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣤⡶⠰⠆⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⢠
⡀⠀⠀⠀⣀⠀⠀⠀⣀⣀⠀⠀⠀⣀⣀⣀⣼⣋⣱⣿⣿⣿⡋⠉⠀⠤⠀⠘⠛⢃⣾⣧⡘⠻⠇⠿⠏⠸⠿⠇⠀⢾⠏⠁⠀⠀⠀⠀⠀⢹⣿⣾⣿⣿⣿⣿⣿⢷⣿⣛⣻⣿⠿⠻⠿⣿⣿⠛⠋⠁⣀⣀⡀⠀⠀⠀⠀⠁
⣿⣿⣟⣩⠋⢤⣴⣀⠁⣤⣤⠀⣤⣤⠘⢋⣭⣤⣍⡁⢉⣉⣁⣀⣀⡀⠀⠀⠀⣀⣛⠻⠁⠀⠀⠀⠿⠿⠳⠆⠀⠀⠧⠀⠀⠀⠰⣾⡦⠘⡿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣯⣶⣶⣾⣿⣿⣿⣶⣶⣶⣦⣤
⣴⣾⠿⣿⣿⣿⣿⣿⠀⣿⣿⡇⣿⣿⠀⣿⣿⢹⣿⣷⠘⢻⣿⣿⠻⠃⠀⢀⣾⣿⢻⣿⡆⢸⣿⣿⣿⡇⣾⣶⣶⠀⣶⣶⣶⠀⣶⣶⠀⣤⣤⠀⣥⣬⠉⣤⣬⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠛⢿⣶⣦⣿⡿
⢛⣛⣀⣙⣛⣛⣛⣛⡃⣿⣿⣿⣿⣿⢰⣿⣿⢸⣿⣿⢀⢸⣿⣿⢀⣀⣀⢸⣿⡿⠘⠛⠃⢸⣿⣿⣐⠃⣿⣿⣿⣧⣿⣿⣿⠀⣿⣿⠀⣿⣿⡇⣿⣿⠀⣿⣿⢰⣿⣿⣿⣿⣿⠀⣾⠻⣿⣙⣿⡿⠿⢿⣿⣶⣶⣿⠇⠀
⢸⣿⣿⣿⣿⣿⣿⣿⣇⣿⡇⢿⣿⡿⢸⣿⣿⢸⣿⡏⢿⢸⣿⣿⠸⣿⣿⢸⣿⡇⢻⣿⡇⢸⣿⡟⠛⠀⣿⣿⢻⣿⡏⣿⣿⠀⣿⣿⠀⣿⣿⣿⣿⣿⠀⣿⣿⢨⣿⣿⣿⡟⠰⠀⠉⠀⢸⣿⣿⣿⣷⣿⣿⣿⣿⡿⢿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⣸⠿⢇⣌⠿⣿⣾⡿⢃⣾⢸⣿⡇⣷⣙⡜⡸⣿⣧⣼⣿⡇⢸⣿⣧⣤⠀⣿⣿⢸⣿⠃⣿⣿⢸⣿⣿⠀⣿⡇⢿⣿⡿⢸⣿⣿⠈⢿⠿⠟⠁⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡶⣿⠶
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣮⣭⣵⣭⣵⣬⡍⠉⠉⠀⢛⣻⠜⣛⣔⣛⣃⣜⣛⣛⠀⠿⠇⠸⠿⠇⠸⠿⠿⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠻⣿⣿⣶⣿⣽⣿⣿⣽⣷⣦
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢿⢺⣷⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⠟⣡⣾⡟⠁⠀⠉⠿⠿⠿⠧⠆⠀⠀⣿⡇⣤⣴⣤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣏⣿⣿⣿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣷⣦⣤⣤⣶⣄⣀⣀⣀⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣭⠙⣯⠉⣭⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⢹⣿⡏⠀⢀⣿⣿⣿⣿⡀⠀⠛⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡧⠀⣿⡀⢶⡾⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣷⣦⣤⣶⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣷⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡷⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡡⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠟⠻⠛⠙⠋⠛⠁⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⡛⠋⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡟⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⣿⡛⠿⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠛⠁⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⡿⢛⠛⠛⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠀⠀⠀
⣿⣿⠿⣟⡭⣭⡽⠹⢛⡚⡧⠃⠉⢉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⣴⣾
⠛⠉⠙⠋⠀⠈⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠃⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣶⣾⣿⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⠛⠟⠀⠙⠿⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠛⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⡶⠶⠂⠒⠂⡀⠀⠀⠀⠠⠄⠀⠀⠀⢺⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠋⣿⣯⠁⠀⢠⣶⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⣤⠀⣄⣀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠗⠀⠉⠀⠈⢹⣷⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⡇⢹⠿⣿⠛⠉⣿⣿⠃⠁⠀⠻⠋⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⡿⠃⠀⠀⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣿⣿⠂⠈⠀⠸⠀⠀⠃⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠒⠁⠀⠀⢹⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣿⣯⢀⠀⢻⣿⣿⡿⢿⣿⣿⣿⣿⣿⢟⣿⣿⡄⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⢰⣶⠀⣶⣦⣄⢠⣬⣤⡄⠉⠙⠛⣻⣿⣿⣿⣿⣷⠌⠁⢀⡀⠀⠀⠀⠀⠀⠒⠚⠃⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠸⠿⢠⣿⣿⣿⢸⣿⣿⡇⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣾⡿⠛⠀⠀⠀⣠⣴⣶⠀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⢸⣿⡏⠀⠀⢸⣿⡟⣿⣿⢿⣿⡇⢰⣿⣿⣿⣿⣿⠟⠋⠀⢨⣿⣇⣀⣠⣴⣶⣿⡿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⢀⣾⠛⠻⠇⢸⣿⡇⢠⡄⢸⣿⡇⣿⣿⢸⣿⡇⠈⢻⣿⣿⠟⠁⠀⠀⠀⣠⣹⣿⣿⣿⣿⣿⣿⣦⣴⣦⣤⣤⣤⣤⣴⡧⠤⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⠀⠀⠀⢀⡀⢀⣠⠟⢓⣾⣿⣿⠟⠛⠀⢠⠀⠘⠛⠃⣾⣧⡘⠻⠇⠿⢏⠸⠿⠇⡀⢸⡟⠁⠀⠀⠀⠀⠀⢹⣿⣶⣿⣿⣿⣿⣿⡯⣿⣿⣿⣿⡿⠿⠿⣿⣿⠖⠒⠂⠀⢀⣀⠀⠀⠀⠀⠈
⣿⣷⣿⣋⡟⣁⡰⠀⠁⣤⣤⠀⣤⣤⠘⢋⣭⣥⣍⡉⢉⣉⣁⣀⣀⡀⠀⠀⠀⣀⣛⠻⠏⠀⠀⠀⠿⢿⠿⠾⠀⠀⠿⠀⠀⠀⠀⣶⣷⠘⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣶⣿⣯⣡⣤⣶⣿⣿⣿⣶⣶⣦⣤⣤
⣬⣼⣿⣿⣿⣿⣿⣿⠁⣿⣿⡇⣿⣿⠀⣿⣿⢹⣿⣷⠘⢻⣿⣿⠻⠇⠀⢀⣾⣿⢻⣿⡆⢸⣿⣿⣿⡇⣾⣶⣶⠀⣶⣶⣶⠀⣶⣶⠀⣤⣤⠀⣬⣬⠉⣭⣬⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠿⠻⢻⣿⣥⣿⢿
⢘⣛⣃⣙⣛⣛⣛⣛⡃⣿⣿⣿⣿⣿⢰⣿⣿⢸⣿⣿⢀⢸⣿⣿⢀⣀⣀⢸⣿⡿⠘⠛⠃⢸⣿⣿⣐⠃⣿⣿⣿⣧⣿⣿⣿⠀⣿⣿⠀⣿⣿⡇⣿⣿⠀⣿⣿⢠⣿⣿⣿⣿⣿⠃⢾⠿⢿⣛⣿⡿⢿⣿⣿⣶⣴⣾⡏⠀
⢸⣿⣿⣿⣿⣿⣿⣿⣇⣿⡇⢿⣿⡿⢸⣿⣿⢸⣿⡏⢿⢸⣿⣿⠸⣿⣿⢸⣿⡇⢻⣿⡇⢸⣿⡟⠛⠀⣿⣿⢻⣿⡏⣿⣿⠀⣿⣿⠀⣿⣿⣿⣿⣿⢀⣿⣿⢸⣿⣿⣿⣿⠹⠀⠘⠀⢠⣿⣿⣿⣷⣾⣿⣿⣿⣿⣷⣶
⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⣸⠿⢇⢌⠿⣿⣾⡿⢃⣾⢸⣿⡇⣷⣙⡜⡸⣿⣧⣼⣿⡇⢸⣿⣧⣤⠀⣿⣿⢸⣿⠃⣿⡿⢸⣿⣿⠀⣿⡇⢿⣿⡿⢸⣿⣿⠈⢿⣿⠿⠃⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣶⢶⣷
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣴⣾⣿⣿⣿⣿⣮⣭⣵⣭⣵⣬⡍⠉⠉⠀⢛⣻⠜⣛⣐⣛⣃⣜⣛⣛⠀⠿⠇⠸⠿⠇⠸⠿⠿⠀⠈⠁⠃⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣭⣿⣿⣿⣿⣾⣦
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢾⢼⣯⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⠟⣡⣿⠟⠁⠀⠉⠿⠿⠿⠧⠄⠀⠀⣿⡇⣀⣠⣤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢻⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣻⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣏⣿⣿⣿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣭⠙⣯⠉⣭⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡉⢹⣿⡏⠀⢀⣿⣿⣿⣿⡄⠀⠛⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡧⠀⣿⡀⢶⡶⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣷⣦⣤⣴⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣷⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠿⣻⣻⠻⠿⡿⠿⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠭⠙⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣿⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⢿⣗⣝⣯⡯⠉⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡟⠃⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⢿⠿⠛⠻⠛⠚⠓⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⣛⣻⠽⢏⠥⠀⠁⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⣤⣶
⡟⠿⠿⣋⣉⢿⠯⠍⢙⠋⠡⠛⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡇⠉⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣶⣿⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⠟⠿⠀⠉⠿⠈⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠻⠀⠀⣾⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⠶⠖⠒⠂⠀⠀⠀⢀⠀⣄⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⣻⣿⠁⠀⢀⣶⣶⣿⣧⠀⠈⠀⠀⠀⠀⠀⠀⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⣀⠀⢀⣀⣀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠷⠀⠉⠀⠀⠹⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⢿⣿⣿⣿⣿⡏⢹⡿⣿⡟⠋⣿⣿⡇⠉⠀⠹⠏⠋⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⣾⠇⠀⠀⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢿⣿⡇⠈⠀⠸⠃⠀⠋⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠃⠂⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠘⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡄⢸⣿⡏⠀⠀⢸⣿⡿⣿⣿⢿⣿⡇⠀⣿⣿⣿⣿⣿⡿⠛⠉⢉⣿⣯⣀⢀⣠⣴⣾⣿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢀⣴⠞⠛⡇⢸⣿⡇⢀⡀⢸⣿⡇⣿⣿⢸⣿⡇⠀⢻⣿⣿⡿⠋⠀⠀⠀⢨⣙⣿⣿⣿⣿⣿⣿⣧⣤⣤⣀⣀⣀⣀⣠⣆⣤⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⡟⠓⣾⣿⣿⡿⠛⠂⢠⠀⠘⠛⠃⣾⣧⡘⠻⠇⠿⢏⠸⠿⠇⡀⢰⡿⠋⠀⠀⠀⠀⠀⠸⣿⣷⣿⣿⣿⣿⣿⣯⣿⣿⡿⣿⣿⣿⣿⣿⣿⣶⠶⠖⠀⠀⢀⠀⠀⠀⠀⠘
⣿⣶⣶⢒⡛⣇⣠⡆⠈⣤⣤⠀⣤⣤⠰⢋⣭⣭⣍⡉⢉⣉⣁⣀⣀⡀⠀⠀⠀⣀⣛⠻⠟⠀⠀⠀⠹⢿⠿⠿⠀⠀⠰⠄⠀⠀⠀⣴⣦⠀⣿⠏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣾⣟⣋⣤⣴⣾⣿⣿⣶⣶⣤⣄⣀
⣨⣭⣿⣿⣷⣿⣿⣿⠀⣿⣿⡇⣿⣿⠀⣿⣿⢹⣿⣷⠘⢻⣿⣿⠻⠇⠀⢀⣾⣿⢻⣿⡆⢸⣿⣿⣿⡇⣾⣶⣶⡀⣶⣶⣶⠀⣶⣶⠀⣤⣤⠀⣬⣭⠉⣭⣭⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣯⣭⣽⣿
⢙⣛⣃⣘⣛⣛⣛⣛⡃⣿⣿⣿⣿⣿⢰⣿⣿⢸⣿⣿⢀⢸⣿⣿⢀⣀⣀⢸⣿⡿⠘⠛⠃⢸⣿⣿⣐⠃⣿⣿⣿⣧⣿⣿⣿⠀⣿⣿⠀⣿⣿⡇⣿⣿⠀⣿⣿⢠⣿⣿⣿⣿⣿⡏⢹⠿⣿⣛⣿⣿⢿⣷⣶⣮⣤⣿⡿⠀
⢸⣿⣿⣿⣿⣿⣿⣿⣇⣿⡇⢿⣿⡿⢸⣿⣿⢸⣿⡏⢿⢸⣿⣿⠸⣿⣿⢸⣿⡇⢻⣿⡇⢸⣿⡟⠛⠀⣿⣿⢻⣿⡏⣿⣿⠀⣿⣿⠀⣿⣿⣿⣿⣿⠀⣿⣿⢸⣿⣿⣿⣿⠋⡇⠘⠃⠀⣿⣿⣿⣶⣶⣿⣿⣿⣿⣷⣶
⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⣸⠿⠇⢌⠿⣿⣾⡿⢃⣾⢸⣿⡇⣷⣙⡜⡸⣿⣧⣼⣿⡇⢸⣿⣧⣤⠀⣿⣿⢸⣿⠃⣿⣿⢸⣿⣿⠀⣿⡇⢿⣿⡿⢸⣿⣿⠈⢻⣿⣿⠏⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣴⣾⣿⣿⣿⣿⣮⣭⣵⣭⣵⣮⡍⠉⠉⠀⢛⣻⠜⣛⣐⣛⣃⣜⣛⣛⠀⠿⠇⠸⠿⠇⠸⠿⠿⠀⠈⠃⠳⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣟⣿⣆
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⢾⢼⣯⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣠⠟⣡⣾⠟⠁⠀⠙⠿⠿⠿⠧⠄⠀⠀⣿⡇⣀⣀⣄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⢛⣻⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣟⣿⣿⣿⠀⠀⢡⣾⣿⣿⠀⠀⣠⣤⣤⣤⣤⣾⡄⠀⠀⣿⡇⣿⣿⣭⣀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠋⣭⠙⣯⠉⣭⠉⣯⡍⢻⡏⢹⣯⠉⣭⠉⣿⡍⣹⣿⡏⠀⢀⣿⣿⣿⣿⡄⠀⠛⠛⣿⣿⣿⣿⠇⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣾⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡧⠀⣿⡀⢾⡿⣿⠀⣿⠀⣿⡇⢸⡇⢸⣿⠀⣿⠀⣿⡇⣿⣿⡇⠀⣼⡏⠻⣿⣿⣷⣤⣤⣼⣿⣿⡿⠋⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⢿⢿⡿⣿⡿⣿⢿⢿⢹⣿⡇⠀⠹⣿⣦⣬⣽⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣷⣧⣿⣼⣼⣼⣧⣧⣯⣿⣼⣽⣼⣿⣇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

                    ䷩ 𝚕𝚒𝚗𝚎 1655

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

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/01/08/gemini-clients/#comments
    Gemini version at gemini://gemini.techrights.org/2022/01/08/gemini-clients/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 01.08.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ What_Gemini_Clients_to_Use_in_2022⠀✐


Posted in Free/Libre_Software, Review, Site_News at 10:42 am by Dr. Roy
Schestowitz


Video_download_link | md5sum c0cb7c0e43bf111fb7e014c9c69804f2
A Gemini Clients Survey
Creative Commons Attribution-No Derivative Works 4.0


http://techrights.org/videos/gemini-clients-2022.webm


Summary: As another year starts it seems like a good time to revisit the
options one has in Geminispace, comparing a bunch of decent Gemini clients
(like Web browsers but for gemini://)


“My son and I have been experimenting with #gopher and #gemini,” said this
person the other day. “As a browser, we are most often using #lagrange,” he
added.


There seems to be a lot of consolidation among users around Amfora and Lagrange
[1, 2], so few have paid attention to Telescope [1, 2] in the command line (a
much lighter alternative to Amfora, and one that does not use Rust) and
Moonlander for a GUI. Both projects aren’t being developed anymore, maybe due
to a lack of broad userbase, but there_are_signs_of_revival_in_Telescope.
Kristall and Bollux are demonstrated in this_video_from_last_year, but the
above video of ours does not cover them because I never used them. All in all I
tried more than half a dozen Gemini clients and it’s hard to recommend just one
because it depends on the user and the sorts of needs one has (e.g. RAM
available, access to images, tabbing etc.) and the platform one uses. For me,
personally, Lagrange is something I can settle and stay with. It seems to have
the most features and its developer ‘gets’ software freedom. █



                    ䷩ 𝚕𝚒𝚗𝚎 1708

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

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/01/08/irc-log-070122/#comments
    Gemini version at gemini://gemini.techrights.org/2022/01/08/irc-log-070122/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 01.08.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ IRC_Proceedings:_Friday,_January_07,_2022⠀✐


Posted in IRC_Logs at 3:31 am by Needs Sunlight


Also available via the Gemini protocol at:


    * gemini://gemini.techrights.org/irc-gmi/irc-log-techrights-070122.gmi
    * gemini://gemini.techrights.org/irc-gmi/irc-log-070122.gmi
    * gemini://gemini.techrights.org/irc-gmi/irc-log-social-070122.gmi
    * gemini://gemini.techrights.org/irc-gmi/irc-log-techbytes-070122.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
 QmVXcX6fxPUuCjVXAGgiuchwQv98JcgZ5jhgGkPvKfwfDU #boycottnovell  🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
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                                                 as HTML)


                                                 IRC log for
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 QmZ7Lxa71jRQ8aFqDhNbkKZFWVgxVoLDZcXAGQKm9ojBAZ (full IRC log   🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
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                                                 IRC log for
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 QmSxPDHWqHFFjc2xXtBXXf83q1Y1HyKk7Cz3vxsCrudn4n social          🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
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                                                 IRC log for
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 QmSGesdN2gRSQffDaszYkFBi63w14x652eXy2wnVMY91rP social          🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
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                                                 IRC log for
 QmcWvuEJnTUZBEsyZ5bnkMbEqe7DroAWxuJaCmncfdmFfu #techbytes      🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
                                                 (full IRC log
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                                                 IRC log for
                                                 #techbytes
 QmRcHKkBhCbhiR4GGQDmtKSYqJdHXrMoAutWqsRWWmEE4s (full IRC log   🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
                                                 as plain/ASCII
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                                                 IRC log for
 QmQk3QFhr6ckaX9gtrnYqLEd6smbHcsRji1FFyG9CHQRh1 #techrights     🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
                                                 (full IRC log
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                                                 IRC log for
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 QmYpUcFGuxtrRWFU65S4mCL29UED9QDFXoZ65myPj22G9d (full IRC log   🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
                                                 as plain/ASCII
                                                 text)


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇IPFS logo⦈


§ Bulletin for Yesterday⠀➾


Local_copy | CID (IPFS): QmP9Us6BzAW1DtJJVXeJt8AyhFyfNcNAfWtLTvSMzp6Ltp



                    ䷩ 𝚕𝚒𝚗𝚎 1835

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

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/01/08/kristall-appimage/#comments
    Gemini version at gemini://gemini.techrights.org/2022/01/08/kristall-appimage/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 01.08.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Installing_and_Configuring/Using_Kristall_(Gemini_Client)_on_GNU/Linux
Distros_Using_AppImage⠀✐


Posted in Free/Libre_Software, GNU/Linux at 6:13 pm by Dr. Roy Schestowitz


Video_download_link | md5sum d4e05ba0bbb95b71786564dc18e02ee3
Installing and Using Kristall
Creative Commons Attribution-No Derivative Works 4.0


http://techrights.org/videos/installing-kristall.webm


Summary: The Qt-based (i.e. decent KDE integration) Kristall is now available
for download as an AppImage file; it’s rich in terms of features and it
supports gemini, http, https, gopher, finger etc.


EARLIER today I discovered that Kristall is now available as AppImage, just
like Moonlander (GTK) and Lagrange (SDL2). They’re other GUI-based browsers
(other Gemini clients), but this one is Qt-based. So I turned on video and
documented my experience installing it, then toying around with it. Earlier
today in IRC we had a discussion about the need to convince GNU/Linux
maintainers/packagers/developers to adopt these bits of software and make them
easier to install. For a lot of people a Flatpak or AppImage file would likely
suffice, but in the long run there are also downsides. Anyway, Felix “xq”
Queißner’s Web page/site is here. Nightly AppImage (x86_64) of Kristall is
here. Maybe we’ll demonstrate Kristall’s usage more and more in the coming
months. █



                    ䷩ 𝚕𝚒𝚗𝚎 1882

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

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 01.08.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Links_9/1/2022:_EasyOS_3.2_and_qBittorrent_4.4.0⠀✐


Posted in News_Roundup at 8:57 pm by Dr. Roy Schestowitz


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇GNOME bluefish⦈


§ Contents⠀➾


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


          o § Audiocasts/Shows⠀➾


                # ⚓ GNU_World_Order_442⠀⇛


                       The **f** and **k** software series of Slackware:
                       Linux FAQ and the Linux kernel.


                # ⚓ Not_All_Linux_Distro_Forks_Are_Made_The_Same_–_Invidious⠀⇛


                       When we think of a Linux distro fork it’s easy to
                       group them all together but not all Linux distro
                       forks are made the same and there’s a very
                       important distinction that commonly gets overlooked
                       and that’s the repo structure.


                # ⚓ The_BEST_LINUX_APPS_for_TAKING_NOTES_–_Invidious [Ed: Is
                  this a recommendation or paid-for ad (for proprietary
                  software like OnlyOffice)?]⠀⇛


          o § Kernel Space⠀➾


                # ⚓ Linux_5.17_to_release_improvements_&_optimizations_to
                  Intel,_AMD,_&_even_Apple_M1,_offers_AMD_P-State_capability⠀⇛


                       Tomorrow should see the release of Linux 5.16, the
                       newest and most stable kernel, delivering massive
                       improvements to start off 2022 on a strong
                       foothold. Linux users and enthusiasts are showing a
                       lot of excitement for this new update, and are even
                       more excited to see 5.17, the predecessor to
                       tomorrow’s kernel, which is to show some exciting
                       enhancements.


                # ⚓ Linux_5.17_To_Bring_AMD_P-State,_Many_AMD_&_Intel
                  Improvements,_New_Optimizations_–_Phoronix⠀⇛


                       The Linux 5.16 stable kernel is slated for release
                       tomorrow and it delivers on some grand improvements
                       to kick off 2022. But as for great as the Linux
                       5.16 features are, we are already looking forward
                       to the enhancements on deck with Linux 5.17.


                       After the Linux 5.16 kernel debuts, the Linux 5.17
                       merge window opens like clockwork. With my constant
                       monitoring of Linux mailing lists and Git
                       repositories, here is a look at some of the
                       features on trajectory for landing over the next
                       two weeks for Linux 5.17. The Linux 5.17 kernel in
                       turn will debut as stable around the end of March.
                       Linux 5.17 has a lot of work as usual on new AMD
                       and Intel hardware support, new Arm improvements
                       including the ongoing Apple M1 bring-up, new I/
                       O and network optimizations in particular are
                       exciting on the performance front, and a ton of
                       other exciting hardware driver fun.


                # ⚓ Fast_Kernel_Headers_v2_Posted_–_Speeds_Up_Clang-Built_Linux
                  Kernel_Build_By_~88%_–_Phoronix⠀⇛


                       What may end up being one of the greatest Linux
                       kernel features of 2022 is the recently published
                       “Fast Kernel Headers” effort for cleaning up the
                       kernel headers and dramatically speeding up Linux
                       kernel builds both for absolute/clean and
                       incremental builds. Fast Kernel Headers can cut the
                       Linux kernel build time in half or greater and out
                       this weekend are the v2 patches.


                       Last week Ingo Molnar sent out the initial Fast
                       Kernel Headers work to cut the Linux kernel build
                       time by 50~80%. The roughly 2,300 patches clean up
                       the kernel’s “dependency hell” and completely
                       rework the header file hierarchy. Ingo was working
                       on this patch series for more than one year and
                       likely the single ever biggest “feature” to the
                       Linux kernel.


          o § Applications⠀➾


                # ⚓ qBittorrent_4.4.0⠀⇛


                       The qBittorrent project aims to provide a Free
                       Software alternative to µtorrent. qBittorrent is an
                       advanced and multi-platform BitTorrent client with
                       a nice user interface as well as a Web UI for
                       remote control and an integrated search engine.
                       qBittorrent aims to meet the needs of most users
                       while using as little CPU and memory as possible.
                       qBittorrent is a truly Open Source project, and as
                       such, anyone can and should contribute to it.


          o § Instructionals/Technical⠀➾


                # ⚓ How_to_install_MariaDB_on_NetBSD?_|_LibreByte⠀⇛


                       NetBSD is a UNIX-like operating system with a focus
                       on security, simplicity, elegance and clean source
                       code, it is highly portable and robust.


                       MariaDB is a RDBMS created from MySQL 5.1 source
                       code by the original MySQL developers and designed
                       as a direct and improved MySQL replacement. MariaDB
                       is fast, scalable, and robust, with a rich
                       ecosystem of storage engines, plugins, and other
                       tools that make it versatile and flexible in
                       different scenarios.


                       MariaDB is available on the official NetBSD
                       repositories then we can install it using the pkgin
                       package manager.


                # ⚓ How_to_install_Java_on_Linux_Mint_|_FOSS_Linux⠀⇛


                       Whether it’s the versatile development potential or
                       its multifaceted portability, Java is one of the
                       most popular programming languages in the world. It
                       has many development-friendly features that make it
                       stand out from its competition. For starters, the
                       ability to run compiled Java code on any supported
                       platform without having to recompile it is one of
                       the defining functions that Java boasts.


                       In this article, we will learn how to install Java
                       (OpenJDK) on Linux Mint version 20. OpenJDK is a
                       free and open-source distribution of Java. There is
                       also another Java distribution called Oracle JDK,
                       but that comes commercially packaged and is not
                       required unless you have specific requirements.


                       We will be using the Cinnamon edition of Linux Mint
                       20.2, which is most widely used among the operating
                       system’s three different versions. However, the
                       installation method used here should work on the
                       other two, MATE and Xfce. Let’s get right into the
                       installation now.


                # ⚓ How_to_Install_SQLite_3_on_Ubuntu_22.04_LTS_–
                  LinuxCapable⠀⇛


                       SQLite is a free, lightweight relational database
                       management system (RDBMS) in a C library. SQLite is
                       not a client-server database engine. Instead, it is
                       embedded into the end program. Primarily all
                       programming languages support SQLite, which how
                       languages embed the program is with a file with
                       .sqlite3/.sqlite/.DB extension. The software is a
                       popular choice for local/client storage such as web
                       browsers, Android devices, and much more. The list
                       is quite extensive.


                       In the following tutorial, you will learn how to
                       install SQLite 3 with Ubuntu 22.04 LTS Jammy
                       Jellyfish.


                # ⚓ How_to_list_all_the_loaded_extensions_by_PHP_–_Linux
                  Shout⠀⇛


                       In this tutorial, we will see how to install and
                       check the PHP extensions loaded on Linux using a
                       command terminal or GUI web interface.


                       PHP is a popular computer language used by
                       thousands of web servers to run various web
                       applications. It is open source distributed under
                       the PHP license. The abbreviation PHP originally
                       stands for Personal Home Page Tools also popularly
                       known as Hypertext Preprocessor. The PHP
                       infrastructure is installed on an estimated 82% of
                       all web servers on the Internet. More than 200
                       million apps and websites developed with PHP are
                       online. Over 5 million software developers use the
                       programming language.


                # ⚓ How_To_Install_Rust_on_Fedora_35_–_idroot⠀⇛


                       In this tutorial, we will show you how to install
                       Rust on Fedora 35. For those of you who didn’t
                       know, Rust is an open-source programming language
                       that focuses on speed, memory safety, and
                       parallelism. Developers use Rust to create a wide
                       range of new software applications, such as game
                       engines, operating systems, file systems, and
                       simulation engines for virtual reality. Although it
                       is sponsored by Mozilla and Samsung, it is a
                       community project. Its focus is primarily on large
                       programs that run on the client and server-side.


                       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 Rust Programming Language on a
                       Fedora 35.


                # ⚓ Useful_Wget_Command_Examples_in_Linux_System⠀⇛


                       Wget command is one of the most used and handy
                       tools for downloading files, packages, and
                       directories from the web server in Linux and Unix-
                       like operating systems. Usually, you can download
                       any big or small-sized files through the wget tool;
                       the wget does not limit the file size. Originally
                       the Wget command was abbreviated to the combination
                       of the terms World Wide Web and Get. This handy
                       tool was built under the GNU project.


                       It can access both FTP, HTTP, HTTPS, and other
                       local servers for downloading files on Linux. Even
                       with proper configuration, the wget command can
                       access firewall-protected servers. As we will be
                       talking about the wget, so for making diversity we
                       will often use the term ‘World Wide Web and Get’
                       instead of the wget.


                # ⚓ How_to_Fix_Sudo_Command_Not_Found_in_Debian_VPS.⠀⇛


                       In this article, we will show you how to fix sudo
                       command not found in Debian 10 VPS.


                       Sudo is a program for Unix-like computer operating
                       systems that enables users to run programs with the
                       security privileges of another user, by default the
                       superuser.It can stand for “superuser do”, as
                       originally that is all it did however, now it might
                       stand for “substitute user, do”, because sudo can
                       run a command as other users as well.


                       After a fresh Debian 10 installation, you could not
                       execute the privileges tasks by running the sudo
                       command. You will get the error ‘sudo command not
                       found in Debian 10′.


                # ⚓ Stop_using_Virtualbox,_Here’s_how_to_use_QEMU_instead_–
                  Invidious⠀⇛


                       In the first 60 seconds of this video I benchmark
                       Virtualbox vs QEMU.


                # ⚓ How_to_install_Composer_on_Ubuntu_22.04_|_20.04_LTS_–_Linux
                  Shout⠀⇛


                       Learn the commands to install and use PHP package
                       manager Composer on Ubuntu 22.04 Jammy or Ubuntu
                       20.04 Focal fossa. Composer can be used to manage
                       your packages, download new packages, and update
                       existing ones.


                # ⚓ How_To_Install_VLC_Media_Player_on_Debian_11_–_idroot⠀⇛


                       In this tutorial, we will show you how to install
                       VLC Media Player on Debian 11. For those of you who
                       didn’t know, VLC is a free and open-source cross-
                       platform multimedia player and framework that plays
                       most multimedia files. VLC can play almost any
                       multimedia file, as well as DVDs, Audio CDs, VCDs,
                       and various streaming protocols.


                       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 through the step-by-
                       step installation of the VLC Media Player on a
                       Debian 11 (Bullseye).


                # ⚓ How_to_Install_Nextcloud_on_Ubuntu:_A_Step-by-Step_Setup
                  Guide⠀⇛


                       If you’re looking for a true self-hosted file share
                       and sync platform, then Nextcloud is a good place
                       to start. Here I will show you how to install
                       Nextcloud on Ubuntu in a few easy-to-follow steps.


                       Nextcloud is a self-hosted file sharing application
                       server that allows you to store your files,
                       documents, and contacts from a centralized
                       location. It is a true open source platform similar
                       to Dropbox, Google Drive, OneDrive and other
                       proprietary online storage services.


                       The problem with the big players is that you don’t
                       know where your data is exactly and whether it is
                       really safe from access by others. When it comes to
                       some classified data that you don’t want to store
                       on some third-party servers then it is good to go
                       for something that you can control completely.


                       With Nextcloud you can synchronize everything
                       between your devices and share files with others as
                       well. Furthermore, you can create multiple accounts
                       for friends/family. They will then be able to log
                       into the server and store data, very similar to
                       Dropbox, etc.


                       The server-side program of Nextcloud is meant to
                       work on Linux operating systems, therefore any
                       Linux user even the beginner one can easily install
                       it. So without further ado, let’s get down to
                       installation.


                # ⚓ How_to_Install_Zoom_Client_on_Linux_Mint_20_–
                  LinuxCapable⠀⇛


                       Zoom is a communications technology platform that
                       provides videotelephony and real-time online chat
                       services through a cloud-based peer-to-peer
                       software platform and is used for teleconferencing,
                       telecommuting, distance education, and much more.


                       In the following tutorial, you will learn how to
                       install the Zoom Cloud Meetings client on Linux
                       Mint 20 using three various methods.


                # ⚓ Interrupt_or_Suspend_a_Command_Execution_in_Linux_–_TREND
                  OCEANS⠀⇛


                       An accidentally executed sequence of command and
                       system files is in danger now. Any time they will
                       wipe out, What should I do now?


                       The situation is familiar; most Linux beginners
                       face this situation and do something
                       unintentionally which risks their files.


                # ⚓ How_to_install_TermPair_to_share_and_control_terminals_in
                  real_time_from_Web_Browser⠀⇛


                       TermPair is a web service that allows anyone to
                       view and control their Terminal sessions in real
                       time from a web browser. Simply put, it lets people
                       collaborate, view, share, all in real time.


                       You can quickly and securely share your Terminal to
                       the Web and access as well as control it.


                       TermPair is good for those who wants to collaborate
                       in real-time. It also has some security loopholes.
                       If you’re not careful, it could be a catastrophic
                       mistake.


                # ⚓ How_to_Check_Linux_Memory_Usage_–_buildVirtual⠀⇛


                       How to check memory usage on linux using commands
                       such as top and free and how to query /proc/meminfo
                       to get detailed memory usage stats


          o § Games⠀➾


                # ⚓ The_Legend_of_Tianding:_Review_on_Linux_–_Boiling_Steam⠀⇛


                       The Legend of Tianding is a Taiwanese game. Asia is
                       comprised of many small countries but very few are
                       actually powerhouses when it comes to video game
                       development. Japan more or less created the video
                       games industry in the first place (Nintendo created
                       the worldwide mass market with the NES and
                       everything derived from there), then Korea created
                       the online PC gaming market before anyone else,
                       and… that’s about it. China has its owned closed
                       market that nobody knows (or cares) about, so it’s
                       kind of irrelevant. Singapore, Malaysia, Taiwan
                       have a few devs here and there but by far and large
                       nothing major. So I had very low expectations to
                       begin with. Well, virtually anyone with half a
                       brain can make a 2D platformer, but doing it well
                       requires talent and experience. In that context,
                       The Legend of Tianding is an excellent surprise.


          o § Desktop Environments/WMs⠀➾


                # ⚓ Set_fire_to_your_applications_with_Burn_My_Windows_7_–
                  Neowin⠀⇛


                       Those effects were amazing, and not for their time.
                       It was able to minimize your windows using a Mac OS
                       X-like genie effect, dissolve them, rotate your
                       desktop on a cube, and even burn your windows
                       alive! It even inspired StarDock’s WindowFX. Many
                       of the more practical effects like genie
                       minimization, have been available in Kwin (the
                       venerable KDE’s window manager) all along, but
                       they’ve completely disappeared on the contemporary,
                       GTK powered side of the Linux desktop.


                       No longer. Open-source developer Simon Scheegans is
                       working on a project called Burn My Windows that
                       restores classic desktop effects like burning
                       windows to Gnome 3x and Gnome 40x, respectively.
                       The project debuted only 3 weeks ago and is
                       hilariously already on version 7. Version 5, which
                       introduced the compelling if not somewhat
                       terrifying T-Rex-Attack effect, was released only
                       two days ago. At this rate, it may be at version
                       2005 sometime next year.


                # § GNOME Desktop/GTK⠀➾


                      # ⚓ Libadwaita_1.0_released⠀⇛


                             Libadwaita is quite controversial, as aside
                             from dark mode and a (promised) colour API,
                             applications that use Libadwaita cannot be
                             themed. It’s all the result of developers
                             being unhappy us pesky users get to decide
                             what our computers look like, so they decided
                             to prevent users from theming their systems
                             at all. GNOME’s own applications will surely
                             transition to it, and it remains to be seen
                             if the wider Gtk developer community will opt
                             for it as well.


          o § Distributions⠀➾


                # ⚓ The_Top_8_Linux_Distros_That_Have_Adopted_Flatpak⠀⇛


                       In a market dominated by premium-grade OSes with
                       dedicated COTS (commercial off-the-shelf) software,
                       Linux users must make do with open-source variants
                       of such premium applications.


                       And a distro-agnostic way to distribute such
                       software on Linux is Flatpak. This is why flatpak
                       packages are rapidly becoming the buzz-phrase
                       amongst users with a Linux-based stack.


                       But what is Flatpak and which Linux distros have
                       transitioned to Flatpak? Let’s find out.


                # ⚓ The_8_Smallest_Linux_Distros_That_Are_Minimal_and
                  Lightweight⠀⇛


                       Strapped for hard disk space? Install one of these
                       small and lightweight Linux distros to make your PC
                       usable again.


                       Do you have an old PC lying around gathering dust?
                       Would you like to make use of the old small-
                       capacity USB flash drive sitting in your draw? You
                       can reuse your old computer and a USB flash drive
                       by installing a super small Linux distribution on
                       them.


                       Here are eight of the smallest Linux distros that
                       need almost no space!


                # § New Releases⠀➾


                      # ⚓ Clonezilla_Live_Disk_Cloning_and_Partitioning_Tool_Is
                        Now_Powered_by_Linux_5.15_LTS⠀⇛


                             Clonezilla Live 2.8.1 is here one and a half
                             months after Clonezilla Live 2.8 and it’s the
                             first release of the live Linux system to be
                             powered by the latest and greatest Linux 5.15
                             LTS kernel series. The previous release was
                             powered by Linux kernel 5.14, which reached
                             end of life in November 2021.


                             Linux kernel 5.15.5 LTS is included by
                             default in the Clonezilla Live 2.8.1 release,
                             which also improves support for detecting
                             hd1, hd2, etc. disks, adds a new
                             functionality to no longer split the image
                             file of a partition when saving an image by
                             the ocs-sr script.


                      # ⚓ EasyOS_version_3.2_released⠀⇛


                             EasyOS 3.1 was released on October 25, 2021,
                             see announcement at Distrowatch. Since then,
                             we have been steadily working toward 3.2
                             …and, oh man, so many changes, where to
                             start… Alright, an announcement blurb,
                             doesn’t cover everything, just some
                             highlights:


                             Since version 3.1, EasyOS has undergone major
                             structural changes and many new applications
                             added. Some of the structural changes include
                             a move from ALSA-only to Pulseaudio,
                             applications running as their own user,
                             improved hardware-profiling for audio, fixes
                             for samba, audio and video, more video
                             drivers, new /files top-level folder.
                             Software changes include a recompile of all
                             packages in OpenEmbedded (OE) and the
                             addition of major multimedia applications
                             such as LiVES video editor, VLC video player,
                             OBS Studio video recorder/streamer and
                             Scribus desktop publisher — all cross-
                             compiled in OE. Qt5 packages are now compiled
                             in OE. More development packages in the
                             ‘devx’ SFS, including Mercurial source-
                             control and Nemiver debugger. Numerous bug-
                             fixes and improvements.


                      # ⚓ What’s_Next_for_Shift_?⠀⇛


                             What’s next for Makulu Shift ? Well, They say
                             a Picture Speaks a Thousand Words, So …


                # § BSD⠀➾


                      # ⚓ Pfsense_Box_mini_PC_with_6_Gigabit_Ethernet_from
                        $417⠀⇛


                             A new mini PC equipped with 6 Gigabit
                             Ethernet ports has been recently launched
                             powered by an Intel Core i5-1135G7 processor
                             and offering support for open source router,
                             VPN, and firewall software. Enabling you to
                             add a firewall or upgrade your existing
                             network with a number of extra options
                             depending on your preference.


                             Available in a variety of different
                             configurations the barebones model is priced
                             at $417 and is now available to purchase from
                             online retailers such as Amazon or
                             AliExpress. Powered by an Intel Celeron 6305
                             processor or Core i5-1135G7 the mini PC is
                             available with also available with 32GB of
                             RAM and 512GB of solid state storage if you
                             would prefer a system that works straight
                             out-of-the-box. The mini PC router also
                             features a configuration with an optional 4G
                             LTE and SIM card slot. The Fanless Tech
                             website explains a little more :


                # § IBM/Red Hat/Fedora⠀➾


                      # ⚓ GCC_12_+_Glibc_2.35_Planned_For_Fedora_36_–
                        Phoronix⠀⇛


                             It should hardly come as a surprise given
                             Fedora’s history of always shipping with the
                             very latest GNU Compiler Collection (GCC),
                             but with this spring’s Fedora 36 the plan is
                             to ship with the yet-to-be-released GCC 12
                             and other very latest open-source compiler
                             toolchain components.


                             Fedora 36 continues its feature development
                             for this next Fedora Linux release that
                             should be out by the end of April. One of the
                             latest change proposals is for shipping
                             Fedora 36 with GCC 12, which itself will be
                             released in March or April as usual. This
                             isn’t surprising with Fedora always shipping
                             the bleeding-edge compiler even if it means
                             initially shipping with a near-final pre-
                             release package.


                # § Canonical/Ubuntu Family⠀➾


                      # ⚓ Linux_Mint_20.3_is_Out!_Full_Dark_Mode,_Theme_&_XApps
                        Updates_|_UbuntuHandbook⠀⇛


                             The third point release of Linux Mint 20 is
                             out! Unlike Ubuntu, it has different code
                             names for each point releases. And, Linux
                             Mint 20.3 codenamed ‘Una’.


                             The release still has Kernel 5.4 though user
                             may install updated Ubuntu patched Kernels
                             using ‘Update Manager’. And, it features
                             Cinnamon 5.2, MATE 1.26, and XFCE 4.16 for
                             each desktop editions.


          o § Devices/Embedded⠀➾


                # ⚓ Raspberry_PI_Touchscreen:_reviewing_the_Uperfect_RPI_All-
                  In-One_–_peppe8o⠀⇛


                       In the last few days, I get hands on an interesting
                       Raspberry PI touchscreen display. I was wishing to
                       use it to test some projects like a retro gaming
                       console and/or a home control display, but using it
                       I thought that it is enough interesting for being
                       worth a dedicated post.


                       The display is the 10.1 inches version of RPI All-
                       In-One product, from Uperfect. Please note that the
                       Raspberry PI board is not included in the standard
                       package and you have to buy your one apart from it:


                # ⚓ Vivaldi_hits_the_Astro_Slide_5G_–_NotebookCheck.net_News⠀⇛


                       The best part about the Vivaldi-Astro Slide 5G
                       combo is that those who love the hardware/software
                       combination but are not Google fans (at all) can
                       run an alternative operating system. Astro Slide
                       accepts Linux in a multi-boot setup with Android
                       and Planet Computers, its creators, decided to use
                       a Debian-based OS with the KDE desktop on top.
                       Obviously, Vivaldi is no stranger to the Linux
                       world, since it has been available for this
                       operating system from the beginning and was also
                       chosen as the default browser on the Manjaro
                       Cinnamon edition distro back in early September
                       2021.


                # ⚓ Planet_Astro_Slide_5G_brings_slide-out_physical_keyboard
                  and_Dimensity_800⠀⇛


                       If you’ve got a nostalgic spot for smartphones with
                       slide-out keyboards then Planet Computers might
                       catch your interest with its Astro Slide 5G. The
                       device was announced at CES 2022 and brings a
                       backlit keyboard that swivels from underneath the
                       phone’s display thanks to its neat hinge mechanism.
                       The phone boots Android 11 but it can also be
                       booted with Linux distributions and Planet
                       Computers is also working on bringing support for
                       Sailfish OS.


                       Astro Slide 5G brings a 6.39″ AMOLED panel with
                       FHD+ resolution and Gorilla Glass 3 protection with
                       uniform bezels and a 13MP front-facing cam. The
                       phone is powered by the Dimensity 800 5G chipset
                       paired with 8GB RAM and 128GB storage which is
                       expandable via microSD. Apart from having a QWERTY
                       keyboard you also get programmable keyboard keys
                       that let you open an app or toggle between system
                       functions.


                # ⚓ The_Vivaldi_browser_will_ship_on_the_Astro_Slide_5G_pre-
                  installed⠀⇛


                       Vivaldi Technologies just announced that they have
                       teamed up with Planet Computers. Planet Computers,
                       the maker of the Astro Slide 5G, will ship their
                       device with the Vivaldi browser pre-installed.


                       [...]


                       Both Vivaldi and Planet Computers show their love
                       for Linux with pride. You can multi-boot Linux as
                       well as Android on the Astro Slide. The Vivaldi
                       browser just became the default browser on Arch-
                       based Manjaro’s Cinnamon edition.


                # § Open Hardware/Modding⠀➾


                      # ⚓ Modernizing_The_Game_Boy_Advance_|_Hackaday⠀⇛


                             [Zekfoo] decided to honor the Game Boy
                             Advance’s 20th birthday by redesigning it at
                             the circuit level to give it a more modern
                             twist.


                      # ⚓ Advanced_PCB_Graphics_With_KiCAD_6_And_Inkscape_|
                        Hackaday⠀⇛


                             The first step is to get your bitmap, whether
                             you create it yourself, or download it, and
                             trace it into a set of vectors using the
                             Inkscape ‘trace bitmap’ tool. If you started
                             with an SVG or similar vector shape, then you
                             can skip that stage.


                             Next simply create a PCB outline shape by
                             deleting all the details that aren’t part of
                             the outline. A little scaling here and there
                             to get the dimensions correct and you’re done
                             with the first part. [Stuart] has an earlier
                             video showing that process.


                             The usability improvements in KiCAD 6.0 are
                             many, but one greatly demanded feature is the
                             ability to group objects, just like you do in
                             Inkscape and any other vector graphics tool
                             for that matter. That means you can simply
                             import that SVG outline into the Edge.Cuts
                             PCB layer and all the curves will be nicely
                             tied together. Next you select the details
                             you want for the silkscreen layer, solder
                             mask removal layers and any non-circuit
                             copper. In Inkscape it would be wise to use
                             the layers feature to assign the different
                             material types to a uniquely named layer, so
                             they can be hidden for exporting. This allows
                             you to handle silk, mask and copper PNG
                             exports from a single master file, in
                             addition to any vector details for outline,
                             slots and holes.


                      # ⚓ DE10-Lite-Ful_FPGA_Dev_Board_Hack_Plays_The_1981
                        Classic_Defender_|_Hackaday⠀⇛


                             We’re not sure what the assignment was, but
                             the results of [Garret Carter]’s homework for
                             his Digital System Design class at Tennessee
                             Tech couldn’t help but capture our attention.
                             Below the break you can see what [Garrett]
                             describes as a “simplified stylized version”
                             of the 1981 arcade hit “Defender”.


                             With the goal of keeping the price low but
                             keeping performance as high as possible,
                             [Garrett] set forth to program the DE10-Lite
                             FPGA development board in VHDL. The results
                             are convincing, and while not perfect, came
                             in under budget.


                # § Mobile Systems/Mobile Applications⠀➾


                      # ⚓ Quectel_Launches_New_Generation_of_Flagship_Android
                        Smart_Module⠀⇛


                      # ⚓ Hisense_55U6G_55-inch_QLED_Ultra-HD_Smart_Android_TV
                        Review:_Mid-Range_Winner?_|_NDTV_Gadgets_360⠀⇛


                      # ⚓ TCL’s_newest_Android_phones_will_arrive_at_T-Mobile
                        and_Verizon_this_month_and_quickly_be_forgotten⠀⇛


                      # ⚓ Is_it_safe_to_use_an_old,_outdated_or_preowned
                        Android_phone?_–_CNET⠀⇛


                      # ⚓ Android_Basics:_15_shortcuts_you_need_to_use_on_your
                        phone_–_9to5Google⠀⇛


                      # ⚓ 5_best_auction_apps_for_Android_–_Android_Authority⠀⇛


                      # ⚓ 8_Amazing_Apps_to_Customize_Android_Like_a_Pro⠀⇛


                      # ⚓ 5_Android_apps_you_shouldn’t_miss_this_week_–_Android
                        Apps_Weekly⠀⇛


                      # ⚓ The_Most_Important_Android_Phone_to_Date_Is…⠀⇛


                      # ⚓ Lenovo_Legion_Y700_is_an_upcoming_8.8-inch_120Hz
                        gaming_Android_tablet⠀⇛


                      # ⚓ Google_starts_showing_doorbell_camera_previews_in_At
                        a_Glance_for_Pixels_on_Android_12⠀⇛


                      # ⚓ Samsung_W22_5G_and_Galaxy_S10_Lite_get_Android_12-
                        based_One_UI_4_stable_updates_–_GSMArena.com_news⠀⇛


                      # ⚓ 8_Android_12_hidden_features_you_should_know_about_|
                        Laptop_Mag⠀⇛


                      # ⚓ Latest_Moto_G_Stylus_2022_report_suggests_Motorola
                        has_not_learned_its_lesson_about_Android_updates⠀⇛


                      # ⚓ How_much_space_does_GTA_Liberty_City_Stories_take_on
                        Android_in_2022?⠀⇛


                      # ⚓ Why_GTA_5_APK_download_links_on_internet_for_Android
                        are_fake⠀⇛


                      # ⚓ Locked_your_Android_smartphone?_Can’t_open_it?_Unlock
                        in_these_easy_steps⠀⇛


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


                # ⚓ Open_source_contributors_have_edge_in_jobs [Ed: This
                  article is fake news. It says: “Black Duck is an open source
                  software company.” It’s not. It’s a_Microsoft_proxy_and
                  proprietary_software_thug,_FUD_source_etc.]⠀⇛


                       CarbonData stores and archives all sorts of complex
                       data and enables these to be accessed quickly. It
                       has features like multiple indexes to quickly
                       access the data, intelligent scanning, and most
                       importantly, it enables easy scaling. “The biggest
                       problem with most data warehouses is that the
                       storage and compute functions are clustered
                       together. We separated these two to work and scale
                       independently of each other. In case of a system
                       failure, there should be no loss of data.”
                       It took seven months to build the project before it
                       began the incubation phase at Apache where it was
                       assigned a mentor. It took another year to mature
                       the project. It was finally declared as one of the
                       top-level projects, and became a mainstream Apache
                       project.
                       “When I was a student, we didn’t have access to
                       software technologies because companies kept them
                       private. But it’s very different now, with open
                       source code available for free,” Raghunandan says.
                       He urges students and enthusiasts to contribute to
                       open source projects. A good way to begin, he says,
                       is by adding documentation to existing projects.
                       Enthusiasts can find something they are interested
                       in and improve it, build it, or maintain it.
                       Beginning with something you’re familiar with helps
                       you get started with the process.


                # § WWW/Content Management Systems (CMS)⠀➾


                      # ⚓ 4_Best_Free_and_Open_Source_Clojure_Static_Site
                        Generators⠀⇛


                             LinuxLinks, like most modern websites, is
                             dynamic in that content is stored in a
                             database and converted into presentation-
                             ready HTML when readers access the site.


                             While we employ built-in server caching which
                             creates static versions of the site, we don’t
                             generate a full, static HTML website based on
                             raw data and a set of templates. However,
                             sometimes a full, static HTML website is
                             desirable. Because HTML pages are all
                             prebuilt, they load extremely quickly in web
                             browsers.


                             There are lots of other advantages of running
                             a full, static HTML website.


                      # ⚓ Best_Free_and_Open_Source_Alternatives_to_Oracle_Web
                        Cache⠀⇛


                             Oracle is a computer technology corporation
                             best known for its software products and
                             services like Java.


                             In 2020, Oracle was the second-largest
                             software company in the world by revenue and
                             market capitalization. They employ over
                             130,000 people, and sell cloud-engineering
                             services and systems and database management
                             systems.


                             Oracle has a fairly prominent position with
                             open source. They are a supporting member of
                             the Linux Foundation, Cloud Native Computing
                             Foundation, Eclipse Foundation, and the Java
                             Community Process.


                # § Programming/Development⠀➾


                      # ⚓ Ruby_3.1_lands_with_new_debugger_in_tow⠀⇛


                             As has become the custom, the team behind
                             Ruby has used the end-of-year holiday break
                             to push out a feature update for the
                             programming language. Version 3.1 is now
                             available and mostly bestows performance and
                             debugging improvements upon developers.


                             Amongst the highlights of version 3.1 is a
                             new debugger that replaces lib/debug.rb.
                             According to Ruby committer Yui Naruse, lib/
                             debug.rb wasn’t well maintained and showed
                             some performance and feature issues.


                      # ⚓ A_simple_automated_build_pipeline_for_Node.js_|
                        InfoWorld⠀⇛


                             Build processes can be quite sophisticated
                             for enterprise applications, but even simple
                             and early-stage projects can benefit from
                             automated build pipelines. This article
                             describes a quick-to-deploy system for
                             running an automated build, test, and deploy
                             pipeline with Node.js, Jenkins, and Git.


                             You’ll need Git and Node/NPM installed on
                             your system to follow along. You’ll also need
                             a Google Cloud Platform (GCP) account.
                             (Google offers a generous free trial
                             account.)


                      # ⚓ Nibble_Stew:_Portability_is_not_sufficient_for
                        portability⠀⇛


                             Before looking into portable software, let’s
                             first examine portability from a hardware
                             perspective. When you ask most people what
                             they consider a “portable computer”, they’ll
                             probably think of laptops or possibly even a
                             modern smartphone.


                             [...]


                             Some years ago I ported a sizable fraction of
                             LibreOffice to build with Meson. It worked
                             only on Linux as it used system dependencies.
                             I rebased it to current trunk and tried to
                             see if it could be built using nothing but
                             Visual Studio by getting dependencies via the
                             WrapDB. This repo contains the code, which
                             now actually does build some code including
                             dependencies like libxml, zlib and icu.


                             The code that is there is portable in the
                             laptop sense. You only need to do a git
                             checkout and start the build in a VS x64 dev
                             tools prompt. It does cheat in some points,
                             such as using pregenerated flex + bison
                             sources, but it’s not meant to be production
                             quality, just an experiment.


                      # § Perl/Raku⠀➾


                            # ⚓ To_whom_this_MySQL_UTF-8_news_may_concern_|
                              Aristotle_[blogs.perl.org]⠀⇛


                                   This is not exactly news, given that it
                                   dates from early 2018, but I hadn’t
                                   heard of this before, so I still find
                                   it worth disseminating.


                      # § Shell/Bash/Zsh/Ksh⠀➾


                            # ⚓ Extract_content_between_the_first_\”_and_the
                              last_\”_double_quotes⠀⇛


                      # § Rust⠀➾


                            # ⚓ Dyn_async_traits,_part_7:_a_design_emerges?⠀⇛


                                   Hi all! Welcome to 2022! Towards the
                                   end of last year, Tyler Mandry and I
                                   were doing a lot of iteration around
                                   supporting “dyn async trait” – i.e.,
                                   making traits that use async fn dyn
                                   safe – and we’re starting to feel
                                   pretty good about our design. This is
                                   the start of several blog posts talking
                                   about where we’re at. In this first
                                   post, I’m going to reiterate our goals
                                   and give a high-level outline of the
                                   design. The next few posts will dive
                                   more into the details and the next
                                   steps.


                      # § Java⠀➾


                            # ⚓ Java_&_JVM_Panel⠀⇛


                                   Simone Bordet, Cay Horstmann discuss
                                   Java’s new release cadence which brings
                                   exciting new features at a more
                                   consistent pace, what have been the
                                   strongest points of Java, what are we
                                   missing?


    * § Leftovers⠀➾


          o ⚓ How_to_type_foreign_languages_without_looking_stuff_up_•_The
            Register⠀⇛


                 Smug Linux types have it built in, on pretty much every
                 Linux desktop. All you have to do is enable it, for
                 instance with GNOME Tweaks, KDE System Settings or Xfce’s
                 Settings editor.


          o § Science⠀➾


                # ⚓ Can_a_goldfish_drive_a_car_on_land?⠀⇛


                       Are animals’ innate navigational abilities
                       universal or are they restricted to their home
                       environments? Researchers designed a set of wheels
                       under a goldfish tank with a camera system to
                       record and translate the fish’s movements into
                       forward and back and side to side directions to the
                       wheels. By doing so, they discovered that a
                       goldfish’s navigational ability supersedes its
                       watery environs.


                # ⚓ Mechanism_that_helps_immune_cells_to_invade_tissues⠀⇛


                       To fight infections and heal injuries, immune cells
                       need to enter tissue. They also need to invade
                       tumors to fight them from within. Scientists have
                       now discovered how immune cells protect their
                       sensitive insides as they squeeze between tissue
                       cells. The team lays the foundation for identifying
                       new targets in cancer treatment.


          o § Health/Nutrition/Agriculture⠀➾


                # ⚓ Officials_react_to_Omicron’s_spread⠀⇛


                       Nationally, hospital admissions due to COVID-19 are
                       rising for children, the CDC reported.


                # ⚓ UT_is_Working_to_End_the_Stigma_Surrounding_Mental_Health
                  and_Offer_Support_|_The_Alcalde⠀⇛


                       When the last slice of pizza had disappeared, the
                       30 students at the NAMI on Campus meeting quieted
                       and turned their attention to their vice president,
                       psychology junior Alexis McDonald. The agenda for
                       this meeting in October 2018 included a talk about
                       depressive disorders, followed by a National
                       Alliance on Mental Illness tradition: McDonald
                       would share her own story.


                       “In high school, I experienced a lot of depression
                       and anxiety, but I had never seen a therapist or
                       had a diagnosis,” she began. “I thought things
                       would improve in college, but my freshman year was
                       the most lonely, isolating experience of my life. I
                       was so anxious I couldn’t go into social spaces and
                       introduce myself. When I walked around, I felt no
                       connection to anyone.”


          o § Integrity/Availability⠀➾


                # § Proprietary⠀➾


                      # § Security⠀➾


                            # ⚓ An_Inside_Look_at_a_K-12_Ransomware_Incident_
                              (Part_2) [Ed: Responsible teachers and pupils
                              don’t_use_Microsoft_Windows_in_schools]⠀⇛


                                   In 2020, there were 408 publicly-
                                   disclosed cyber incidents impacting K-
                                   12 school districts. Of those 408
                                   incidents, roughly 50 consisted of
                                   ransomware. These incidents often
                                   resulted in school closures and
                                   prevented districts from accessing
                                   sensitive data and critical systems
                                   because they were encrypted by
                                   cybercriminals.


                                   During an attack, school district IT
                                   teams scramble to find all the
                                   ransomware symptoms to see which
                                   systems have been impacted and assess
                                   the severity. Another threat emerging
                                   is the exfiltration of data by
                                   attackers to try and force school
                                   districts to pay the ransom. This makes
                                   data loss prevention for districts more
                                   critical to have in place as part of
                                   their cloud application security
                                   checklist.


                            # ⚓ WebSpec,_a_formal_framework_for_browser
                              security_analysis,_reveals_new_cookie_attack⠀⇛


                                   Folks at Technische Universität Wien in
                                   Austria have devised a formal security
                                   framework called WebSpec to analyze
                                   browser security.


                                   And they’ve used it to identify
                                   multiple logical flaws affecting web
                                   browsers, revealing a new cookie-based
                                   attack and an unresolved Content
                                   Security Policy contradiction.


                                   These logical flaws are not necessarily
                                   security vulnerabilities, but they can
                                   be. They’re inconsistencies between Web
                                   platform specifications and the way
                                   these specs actually get implemented
                                   within web browsers.


                                   WebSpec was developed by Lorenzo
                                   Veronese, Benjamin Farinier, Mauro
                                   Tempesta, Marco Squarcina, Matteo
                                   Maffei in an effort to bring rigor to
                                   web security through automated,
                                   verifiable rule checking rather than
                                   manual evaluation.


                            # ⚓ Wireshark_3.6.1⠀⇛


                                   Wireshark is a network packet analyzer.
                                   A network packet analyzer will try to
                                   capture network packets and tries to
                                   display that packet data as detailed as
                                   possible. You could think of a network
                                   packet analyzer as a measuring device
                                   used to examine what’s going on inside
                                   a network cable, just like a voltmeter
                                   is used by an electrician to examine
                                   what’s going on inside an electric
                                   cable (but at a higher level, of
                                   course). In the past, such tools were
                                   either very expensive, proprietary, or
                                   both. However, with the advent of
                                   Wireshark, all that has changed.
                                   Wireshark is perhaps one of the best
                                   open source packet analyzers available
                                   today.


                            # ⚓ Broward_Breach_Highlights_Healthcare_Supply-
                              Chain_Problems⠀⇛


                                   The attackers breached the Broward
                                   Health network by compromising a third-
                                   party provider on Oct. 15, according to
                                   the organization’s disclosure,
                                   accessing: patient names; dates of
                                   birth; addresses; phone numbers;
                                   financial or bank information; Social-
                                   Security numbers; insurance information
                                   and account numbers; medical
                                   information including history,
                                   treatment and diagnosis; driver’s
                                   license numbers; and email addresses.


                            # ⚓ Latest_web_hacking_tools_–_Q1_2022_|_The_Daily
                              Swig⠀⇛


                                   After our recent end-of-year
                                   retrospectives, it’s time to look back
                                   again – this time at some of the most
                                   compelling open source hacking tools
                                   released during the final quarter of
                                   2021.


                                   The arsenals of pen testers,
                                   researchers, and bug hunters have been
                                   bolstered for 2022 by new tools for
                                   detecting dependency confusion attacks,
                                   finding novel HTTP request smuggling
                                   techniques, and uncovering leaked,
                                   paired private and public keys that are
                                   potentially dangerous.


                            # ⚓ Key_Considerations_for_Canada’s_Forthcoming
                              National_Cyber_Security_Strategy⠀⇛


                                   On December 16, Prime Minister Justin
                                   Trudeau released mandate letters
                                   tasking his ministers of national
                                   defense, foreign affairs, public
                                   safety, and industry to develop a new
                                   “National Cyber Security Strategy.” He
                                   specifically highlighted the need for
                                   the strategy to “articulate Canada’s
                                   long-term strategy to protect our
                                   national security and economy, deter
                                   cyber threat actors, and promote norms-
                                   based international behavior in
                                   cyberspace,” as quoted by Global News.


          o § Defence/Aggression⠀➾


                # ⚓ Book_Review_|_Raising_a_toast_to_15_British_men_who_parted
                  veil_of_India’s_history⠀⇛


                       The first British who came to rule India were men
                       of the Enlightenment with an intense curiosity


                       It has become fashionable in India to condemn
                       British who came to India during the Raj as nothing
                       but freebooters, racists and british
                       administrators. The portrayal is entirely negative:
                       British rule is generally considered to be an
                       unmitigated disaster that led to impoverishment,
                       subjugation and dishonour. This, however, is not
                       the complete picture.


                       Authors and amateur historians, Rupa and Gautam
                       Gupta, have unveiled an entirely different aspect
                       of British rule in India, one that is far from
                       negative. Their focus is on British men long dead
                       who helped revive the forgotten glory of the Indian
                       civilisation.


                # ⚓ Gold_Mine_Collapse_Leaves_38_People_Dead_in_Sudan⠀⇛


                # ⚓ Why_Libyans_want_the_UK_ambassador_expelled⠀⇛


                       On 24 December, the United Kingdom embassy in
                       Tripoli, Libya, issued a statement on its Twitter
                       and Facebook accounts that, at first, looked like a
                       routine statement on developments in the
                       country—something major countries’ embassies,
                       including the United States, used to do. Not this
                       time. A few moments later, Libyans in their
                       thousands, were flocking to social media platforms
                       to call for the ambassador to be expelled.


                       The statement reiterated the UK’s support for
                       elections but what enraged people is a sentence
                       that says that the UK will continue to recognise
                       the current Government of National Unity (GNU) as
                       “the authority tasked with leading Libya to
                       elections and does not endorse the establishment of
                       parallel” authority. The current caretaker GNU
                       government and its Prime Minister, Abdul Hamid
                       Dbeibah, have been accused of corruption, waste of
                       resources and, above all, Mr. Dbeibah is accused of
                       using public finances for his own presidential bid.
                       There has been a debate among politicians if GNU
                       should stay or not.


                # ⚓ As_China_Fishes_in_Lankan_Waters,_India_Must_Assess_Cost
                  and_Benefit_of_Bailing_out_Neighbour⠀⇛


                       Much has been written about the economic crisis
                       staring at Sri Lanka and most of it ends up blaming
                       the incumbent government of President Gotabaya
                       Rajapaksa and excludes the role of the previous
                       regimes since the country’s Independence in 1948.
                       Question is should India bail out the strategically
                       located southern neighbour, why and how.


                       Sri Lanka’s economy was designed to implode from
                       time to time, and that has happened without fail at
                       every turn. Unlike on previous occasions, this time
                       round, the global COVID-19 pandemic hit the forex-
                       centric economy so hard that it will continue to
                       reel under its effect for years and decades to
                       come, even if it overcomes the current crisis.


          o § Environment⠀➾


                # ⚓ What_Don’t_Look_Up’s_Action_Campaign_Gets_So_Wrong⠀⇛


                # ⚓ Picture_perfect!_Pembrokeshire_youngster’s_success_at_the
                  RSPCA_Young_Photographer_Awards⠀⇛


                       Images of a gosling in the Falklands and some
                       Gentoo penguins taken by a youngster from
                       Pembrokeshire have all won top prizes in the
                       prestigious RSPCA Young Photographer Awards 2021.


                # ⚓ Royal_Navy_Makes_Rare_Visit_to_Remote_South_Atlantic_Island
                  Chain_to_Study_Penguins [Ed: The British Empire's copaganda
                  takes greenwashing to a whole new level]⠀⇛


                # ⚓ World’s_strongest_ocean_current_is_speeding_up_—_and_humans
                  are_to_blame [Ed: Which humans? The ones who fly a rocket to
                  space for a little 'space walk' (vanity), as the owner of
                  this site/paper (Bezos) recently did?]⠀⇛


          o § Finance⠀➾


                # ⚓ Northern_Ireland_aims_to_break_free_from_BT’s_27-year_reign
                  with_£125m_procurement_of_land_registry_systems⠀⇛


                       Northern Ireland’s Land and Property Services, part
                       of the Department of Finance, is planning an IT
                       procurement worth up to £125m to replace an ageing
                       BT system running since 1999.


                       Costs on the BT contract to build and run LandWeb,
                       a land registry system, more than doubled since it
                       was first signed more than two decades ago leading
                       to investigations by government auditors.


                       The Land and Property Services is now looking for a
                       “Land Registration Delivery Partner” to help build
                       a “modern digitally-enabled ICT solution that will
                       support the transformation of Land Registration
                       Services,” according to a tender document published
                       before Christmas.


          o § Internet Policy/Net Neutrality⠀➾


                # ⚓ John_Goerzen:_Make_the_Internet_Yours_Again_With_an_Instant
                  Mesh_Network⠀⇛


                       Every device on the Internet, at one time, had its
                       own globally-unique IP address. This number was its
                       identifier to the world; with an IP address, you
                       can connect to any machine anywhere. Even now, when
                       you connect to a computer to download a webpage or
                       send a message, under the hood, your computer is
                       talking to the other one by IP address.


                       Only, now it’s hard to get one. The Internet
                       protocol we all grew up with, version 4 (IPv4),
                       didn’t have enough addresses for the explosive
                       growth we’ve seen. Internet providers and IT
                       departments had to use a trick called NAT (Network
                       Address Translation) to give you a sort of fake IP
                       address, so they could put hundreds or thousands of
                       devices behind a single public one. That, plus the
                       mobility of devices — changing IPs whenever they
                       change locations — has meant that a fundamental
                       rule of the old Internet is now broken:


                       Every participant is an equal peer. (Well, not any
                       more.)


                       Nowadays, you can’t you host your own website from
                       your phone. Or share files from your house.
                       (Without, that is, the use of some third-party
                       service that locks you down and acts as an
                       intermediary.)


                       [...]


                       Or, I can join the global Yggdrasil network. Each
                       device, in addition to accepting peers it finds on
                       the LAN, can also be configured to establish
                       outbound peering connections or accept inbound ones
                       over the Internet. Put a public peer or two in your
                       configuration and you’ve joined the global network.
                       Most people will probably want to do that on every
                       device (because why not?), but you could also do
                       that from just one device on your LAN. Again,
                       there’s no need to explicitly build routes via it;
                       your other machines on the LAN will discover the
                       route’s existence and use it.


                       This is one of many projects that are working to
                       democratize and decentralize the Internet. So far,
                       it has been quite successful, growing to over 2000
                       nodes. It is the direct successor to the earlier
                       cjdns/Hyperboria and BATMAN networks, and aims to
                       be a proof of concept and a viable tool for global
                       expansion.


                       Finally, think about how much easier development is
                       when you don’t have to necessarily worry about TLS
                       complexity in every single application. When you
                       don’t have to worry about port forwarding and
                       firewall penetration. It’s what the Internet should
                       be.


          o § Digital Restrictions (DRM)⠀➾


                # ⚓ The_Year_Of_Owning_It_|_Hackaday⠀⇛


                       Talking over the year in review on the Podcast, Tom
                       Nardi and I were brainstorming what we thought was
                       the single overarching trend in 2021, and we came
                       up with many different topics: victories in the
                       right to repair, increasingly dystopian service
                       contracts, a flourishing of cyberdecks, and even
                       greater prevalence of reverse engineering style
                       hacks. And then we realized: they are all different
                       faces of the same beast — people just want to own
                       the devices that they own.


                       Like Dr. Jekyll and Mr. Hyde, our modern Internet-
                       connected-everythings have two sides. On one side,
                       we get so much additional functionality from having
                       everything on the net. But on the other, if your
                       car is always connected, it gives Toyota a means to
                       make you pay a monthly fee to use a car fob, and if
                       you have to use Cricut’s free online service to
                       upload designs to the cutter, they can suddenly
                       decide to start charging you. It allows Samsung to
                       not only spy on whatever you’re currently watching
                       on your smart TV, but to also brick it if they want
                       to. More and more, we don’t actually own (in the
                       sense of control) the devices that we own (in the
                       sense of having purchased).


                # ⚓ Keurig_‘Recyclable’_K-Cups_Not_Quite_That_Recyclable_After
                  All⠀⇛


          o § Monopolies⠀➾


                # § Copyrights⠀➾


                      # ⚓ Ethical_aspects_relating_to_cyberspace:_Copyright_and
                        privacy⠀⇛


                             In recent years, there has been a trend in
                             cyberspace ethics towards the emergence of
                             intra-net mechanisms and self-regulatory
                             systems. In particular, in many European
                             countries, information service providers have
                             started to introduce voluntary self-
                             limitation. For instance, in the UK, there is
                             an independent Electronic Frontier Foundation
                             (www.eff.org), whose representatives develop
                             rating systems for Internet resources, by
                             maintaining constant monitoring to collect
                             information that infringes moral and legal
                             standards on websites, and – where necessary
                             – block access to them.


                             A solution to the problem of the quality of
                             information provided on the Internet can
                             probably come from traditional media, which
                             in recent years have been increasingly
                             committed to acquiring an electronic version
                             of their print or radio and television
                             editions. Moreover, exclusively online
                             newspapers and magazines have already emerged
                             which, thanks to their serious and cautious
                             approach, have won the online public’s trust.
                             These publications can play an extremely
                             important role through widely applied survey
                             protocols; evaluation of electronic
                             publications; maintenance of the virtual
                             media’s reputation; and supervision of the
                             implementation of the basic rules and
                             principles of professional journalistic
                             ethics on the Internet.



                    ䷩ 𝚕𝚒𝚗𝚎 3570

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

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 01.08.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Links_8/1/2022:_KDE_Frameworks_5.90.0,_Kazakhstan’s_Tough_Times⠀✐


Posted in News_Roundup at 10:21 am by Dr. Roy Schestowitz


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇GNOME bluefish⦈


§ Contents⠀➾


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


          o § Kernel Space⠀➾


                # ⚓ Why_I_(still)_use_ext4_for_my_Linux_root_filesystems⠀⇛


                       The practical answer is that I know much more about
                       managing and troubleshooting ext4 (and software
                       RAID mirrors) than I do about either XFS or
                       especially btrfs. It’s easier for me to create and
                       operate ext4 filesystems on top of software RAID
                       mirrors, and I have a high confidence that I know
                       how to recover from problems in one way or another.
                       My most recent XFS experience is on actual SGI
                       hardware and I have no real btrfs experience.


          o § Instructionals/Technical⠀➾


                # ⚓ 7_Ways_to_Kill_Unresponsive_Programs_in_Linux⠀⇛


                       Linux software is robust enough to work without
                       causing problems, but sometimes even the best apps
                       might hang. Rather than wait for them to crash, you
                       can kill these unresponsive programs. In fact,
                       there are so many ways to kill Linux programs that
                       you might find you’re spoiled for choice!


                # ⚓ How_to_Install_Google_Chrome_on_Ubuntu_22.04_LTS_–
                  LinuxCapable⠀⇛


                       Google Chrome is the most used Internet Explorer
                       software on the earth, with a recent update in 2021
                       that Chrome is currently the primary browser of
                       more than 2.65 billion internet users. However, as
                       you would know, after installing CentOS 9 Stream,
                       only Mozilla Firefox is packaged with the
                       distribution but luckily, installing Google Chrome
                       is a straightforward task.


                       In the following tutorial, you will learn how to
                       install Google Chrome in three alternative ways,
                       with the stable, beta, or unstable versions on
                       Ubuntu 22.04 LTS Jammy Jellyfish.


                # ⚓ How_to_Install_KDE_Plasma_on_Linux_Mint_20_–_LinuxCapable⠀⇛


                       The name KDE comes from “K Desktop Environment.” It
                       is a free, open-source desktop environment for
                       those not familiar with KDE Desktop. It provides
                       Linux users with an alternative graphical interface
                       to customize their desktop environment and
                       applications for everyday use enhancement.


                       In Linux Mint’s case, this is GNOME. Besides the
                       graphical enhancements and changes, it is also a
                       lightweight, fast, smooth environment with superior
                       performance compared to native shipped desktops
                       with some Linux Distributions.


                       In the following tutorial, you will have learned
                       how to install KDE Desktop Environment on Linux
                       Mint 20.


                # ⚓ How_to_list_all_the_loaded_extensions_by_PHP_–_Linux
                  Shout⠀⇛


                       In this tutorial, we will see how to install and
                       check the PHP extensions loaded on Linux using a
                       command terminal or GUI web interface.


                       PHP is a popular computer language used by
                       thousands of web servers to run various web
                       applications. It is open source distributed under
                       the PHP license. The abbreviation PHP originally
                       stands for Personal Home Page Tools also popularly
                       known as Hypertext Preprocessor. The PHP
                       infrastructure is installed on an estimated 82% of
                       all web servers on the Internet. More than 200
                       million apps and websites developed with PHP are
                       online. Over 5 million software developers use the
                       programming language.


                       It is a scripting language runs on server side to
                       used to convert PHP coded text files into machine
                       code by the web server when they are called up. It
                       is platfrom independent and can be used on any
                       hardware.


                # ⚓ How_to_Install_OpenJDK_17_on_Linux_Mint_20_–_LinuxCapable⠀⇛


                       Java is a general-purpose, class-based, object-
                       oriented multipurpose programming language that is
                       popular due to the design of having lesser
                       implementation dependencies, meaning that the
                       compiled Java code can be run on all platforms that
                       support Java without the need for recompilation.
                       Java is also fast, secure, and reliable, therefore.
                       It is widely used for developing Java applications
                       in laptops, data centers, game consoles, scientific
                       supercomputers, cell phones, etc.


                       The tutorial will install the OpenJDK version
                       instead of the default Oracle JDK. The difference
                       between these two is licensing. OpenJDK is an
                       entirely free open-source Java with a GNU General
                       Public License, and Oracle JDK requires a
                       commercial license under the Oracle Binary Code
                       License Agreement. Other differences are release
                       schedules and other factors that come into play;
                       however, performance is pretty much the same.


                # ⚓ How_to_upgrade_to_Linux_Mint_20.3⠀⇛


                       It is now possible to upgrade Linux Mint 20, 20.1
                       and 20.2 to version 20.3.


                       If you’ve been waiting for this we’d like to thank
                       you for your patience.


                # ⚓ File_Transfer_with_SSH,_Tee,_and_Base64⠀⇛


                       Even if SCP, SFTP, port forwarding, and remote
                       command execution without a login shell are
                       forbidden, as long as we get a login shell on our
                       terminal and we can print data on the terminal, we
                       are already able to transfer data from the remote
                       system to our local system. The data is in the
                       terminal. It is now only a matter of figuring out
                       how to copy that data to a file.


                # ⚓ How_to_install_APF_on_Debian_11⠀⇛


                       Hello, friends. In this post, you will learn how to
                       install APF on Debian 11. But first, let’s talk a
                       brief about it.


                # ⚓ Install_Arduino_Libraries:_methods_to_add_libraries_with
                  Arduino_IDE_–_peppe8o⠀⇛


                       Installing libraries with Arduino IDE is important
                       to enhance your code to support external devices
                       with the help of pre-built software defining
                       communication protocols


                # ⚓ [Fixed]_username_is_not_in_sudoers_file⠀⇛


                       In this article, we will fix a common error that
                       new Linux users encounter username is not in
                       sudoers file. The problem is related to user
                       permissions and can be simply resolved with a
                       single command.


                       If you have recently established a new user on your
                       Linux distribution, you may see the error “username
                       is not in soders file” when using sudo. The error
                       occurs because the logged-in user lacks the ability
                       to execute commands as sudo.


                       If you’re in a rush and need to fix the issue
                       without knowing the context of the error, here’s
                       how. Please use any of the following methods to
                       grant user sudo access, depending on the Linux
                       distribution you’re using.


                # ⚓ [Old] How_to_set_up_SDL2_on_Linux⠀⇛


                       We need to link in the SDL2 libraries, which is why
                       we add the -lSDL2 -lSDL2main. Be aware that those
                       start with a lowercase L, not a 1. The program
                       should compile. It won’t show you anything if you
                       run it, but now you know that you’re all set up to
                       write SDL2 programs on Linux.


                # ⚓ [Old] Beginning_Game_Programming_v2.0⠀⇛


                       In this tutorial we will be setting up the SDL
                       library and creating our first window.


                # ⚓ [Old] Ryan_Gordon_gets_an_Epic_MegaGrant_to_further_improve
                  SDL,_helping_with_next-gen_APIs⠀⇛


                       What is SDL? Simple DirectMedia Layer (SDL) is a
                       cross-platform development library designed to
                       provide low level access to audio, keyboard, mouse,
                       joystick, and graphics hardware. It is used by
                       video playback software, emulators, popular games
                       and some game engines.


                       Ryan Gordon is one of the people responsible for
                       its development, and Gordon has also ported plenty
                       of games to Linux, macOS and other platforms over
                       many years. In a new post on Patreon, a fun
                       announcement was detailed about an approved Epic
                       MegaGrant and how it’s going to be used to improve
                       SDL.


                # ⚓ [Old] Intro_to_Software_Rendering_with_SDL2⠀⇛


                       Lets display some pixel data on the screen with
                       SDL2 in C. There are a few ways to do this, each
                       with their own trade-offs. I am going to assume
                       that you already have SDL2 set up. Here is the
                       basic framework that our program will use, with the
                       relevant pieces slotted in where the comments are.


          o § Games⠀➾


                # ⚓ Wordle:_What’s_the_Best_Starting_Word?⠀⇛


                       Figuring out the best first word is simply running
                       the algorithm over each (guess, solution) pair and
                       averaging the filtered words by guess. Here’s a
                       chart of the results. (It took about 20 minutes to
                       run my messy, unoptimized code on my laptop).


                # ⚓ Little_useless-useful_R_functions_–_Mastermind_board_game
                  for_R⠀⇛


                       The gameplay is simple and so are the rules. The
                       board contains 10 rows (or more) with possibilities
                       of four colours and code pegs (white or black). R
                       engine stores a secret colour combination and user
                       selects a random combination.


                       Based on selection, the R engine returns the black
                       or white pegs. Black peg represents that one colour
                       is at the right place, white that the colour
                       matches, but not the position. No pegs would mean
                       that none of selected colours matches the secret
                       colour combination.


                # ⚓ Playing_Super_Hang-On_With_Hacked_Controller_Gives_Reason
                  For_Paws_|_Hackaday⠀⇛


                       There’s a thing that happens when you’re shopping
                       at a second hand store. You know how it goes: You
                       see an item that strikes your fancy, your mind
                       immediately locks in, and the item just has to be
                       yours. [Tom Tilley] experienced this when he saw a
                       Paw Patrol kids toy at a local thrift store, and
                       you can see the results of his holiday hacking
                       sessions in the video below the break.


                       How did [Tom] put the Paw Patrol game to use?
                       Looking like a motorcycle cockpit left him with few
                       choices. Before long he’d flipped the game over
                       over, pulled the innards, and hacked together what
                       just might be the most perfect toy based interface
                       we’ve seen lately.


          o § Desktop Environments/WMs⠀➾


                # § K Desktop Environment/KDE SC/Qt⠀➾


                      # ⚓ What_is_KDE_Connect?_How_Do_You_Use_It?_[Beginner's
                        Guide]⠀⇛


                             In this article, we explain what is KDE
                             Connect, its main features, basic usage guide
                             and installation steps.


                      # ⚓ This_week_in_KDE:_better_MTP_support_–_Adventures_in
                        Linux_and_KDE⠀⇛


                             Many of us are still getting over our new
                             years’ food comas, but we managed to get some
                             cool things done anyway!


                      # ⚓ KDE_Kicks_Off_2022_With_New_Feature_Work⠀⇛


                             KDE developers have kicked off 2022 into
                             full-swing with new features and other
                             improvements now on their way to the next
                             round of KDE software releases.


                             Nate Graham is out with his first KDE
                             development summary of 2022 work. Among the
                             changes that KDE developers addressed in this
                             first week of the year include:


                             - A volume slider has been added to Plasma’s
                             Task Manager tooltips for windows playing
                             audio.


                      # ⚓ KDE_Frameworks_5.90_Arrives_with_More_Improvements
                        for_Your_Favorite_KDE_Apps⠀⇛


                             KDE Frameworks 5.90 is here to improve the
                             scroll (mouse and touchpad) behavior in
                             QtQuick apps, especially in the Plasma
                             Wayland session when using fractional
                             scaling, add a more modern style without
                             frames to System Settings pages that display
                             a single big grid or list, as well as to
                             improve file listing speed in directories
                             that contain lots of files and folders.


                             The scrollable controls in the Plasma desktop
                             and various QtQuick-based apps have been
                             improved as well to change their contents
                             only when the user scrolls on them when the
                             cursor is over them, not when the it passes
                             over them.


                      # ⚓ KDE_Ships_Frameworks_5.90.0⠀⇛


                             KDE today announces the release of KDE
                             Frameworks 5.90.0.


                             KDE Frameworks are 83 addon libraries to Qt
                             which provide a wide variety of commonly
                             needed functionality in mature, peer reviewed
                             and well tested libraries with friendly
                             licensing terms. For an introduction see the
                             KDE Frameworks release announcement.


                             This release is part of a series of planned
                             monthly releases making improvements
                             available to developers in a quick and
                             predictable manner.


                      # ⚓ Christmas_2021_…_–_bembel.net⠀⇛


                             … is already a few days ago and I finally
                             come around to share one of my gifts with the
                             KDE community.


                             After 7 years I replaced (actually had to
                             replace) my old mobile with a new one. The
                             new one however has a camera that sticks out
                             of the back of the mobile for a few
                             millimeters which I don’t like. My daughter
                             designed a custom cover for my wife’s phone a
                             while back, and I asked her, if she can
                             design something for me


                      # ⚓ November/December_in_KDE_PIM⠀⇛


                             Since the last summary two month ago we have
                             seen the 21.12 feature releases of Kontact,
                             and more than 1800 changes by 35 contributors
                             have been integrated. While a large focus
                             remains on preparing for a smooth transition
                             to Qt 6 and KDE Frameworks 6, there have been
                             many other additions and improvements to the
                             PIM applications as well.


                      # ⚓ KDE_Developer_Contributes_To_GNOME!?_–_Kockatoo
                        Tube⠀⇛


                # § GNOME Desktop/GTK⠀➾


                      # ⚓ 5_Tiny_Yet_Useful_Features_I_Would_Like_to_See_in
                        GNOME_in_2022_–_It’s_FOSS_News⠀⇛


                             GNOME is the default choice of desktop
                             environment on many Linux distributions. It’s
                             also my favorite one as it gives a modern
                             desktop experience.


                             But that doesn’t mean GNOME is perfect and
                             doesn’t need improvements. In fact, here are
                             a few suggestions to improve the overall user
                             experience.


                      # ⚓ GNOME_On_Wayland_Lands_Improved_Handling_For_Direct
                        Scanout_Support⠀⇛


                             Adding to the changes for GNOME 42 this
                             spring is the Mutter Wayland compositor now
                             taking into account sub-surfaces when
                             determining direct scanout capabilities.


                             GNOME’s Mutter already supports direct
                             scanout for full-screen clients to reduce
                             latency and resource use for games and other
                             full-screen software by avoiding any extra
                             screen copy of the screen contents and
                             instead sending the application/game’s
                             contents directly to the output. Just earlier
                             this week Mutter landed DMA-BUF feedback
                             support for improving its direct scanout
                             capabilities particularly for multi-GPU/
                             hybrid setups while on Friday another
                             optimization was merged.


          o § Distributions⠀➾


                # § Gentoo Family⠀➾


                      # ⚓ Gentoo_Linux_Packages_Up_AMD_ROCm,_Makes_Progress_On
                        RISC-V,_LTO+PGO_Python⠀⇛


                             Gentoo Linux developers were very busy over
                             the course of 2021 for this popular rolling-
                             release operating system choice.


                             The Gentoo project has published their 2021
                             recap this week that outlines all of the
                             achievements made over the past calendar
                             year. Some of the highlights for Gentoo in
                             2021 included:


                             - AMD’s ROCm compute driver stack is now
                             fully packaged for Gentoo. While NVIDIA CUDA
                             support is relatively easy to deploy across
                             distributions. having Gentoo Linux package
                             Radeon Open eCosystem is actually an
                             achievement… AMD with ROCm binaries only
                             officially supports enterprise Linux
                             distributions while those distributions
                             independently working to package the ROCm
                             sources have seen mixed success. Gentoo
                             packaging -all- ROCm packages is the first
                             I’ve heard any major Linux distribution
                             achieving that independent milestone.


                # § Debian Family⠀➾


                      # ⚓ An_annoyance_with_Debian_postinstall_scripts_during
                        package_upgrades⠀⇛


                             Both RPMs and Debian packages have broadly
                             the same high level features, which isn’t
                             surprising because they’re facing (and
                             solving) the same problems. For example, both
                             have postinstall scripts. However, the
                             details and the social customs that result
                             are different between the two, and so every
                             so often I find something that irritates me
                             about one or the other (usually Debian).
                             Today’s irritation is in postinstall scripts.


                # § Canonical/Ubuntu Family⠀➾


                      # ⚓ Install_Linux_Mint_20.3_‘Una’_this_weekend_if_you
                        have_absolutely_nothing_better_to_do⠀⇛


                             Well, folks, we finally made it; the weekend
                             is officially here! Hopefully you have some
                             exciting activities planned. Maybe you are
                             going to a party or taking someone out for a
                             romantic dinner date. Or maybe, just maybe,
                             you have absolutely nothing planned. You know
                             what? That’s OK. A lot of people are lonely
                             and/or have no prospects. And for them, Linux
                             exists.


                             Thankfully, Linux Mint 20.3 (code-named
                             “Una”) has finally exited beta, giving
                             countless computer nerds around the world
                             something to do this weekend. And yes, this
                             includes me — I had nothing planned other
                             than a trip to Costco on Saturday and
                             watching my New York Jets lose on Sunday. But
                             now I will be installing the stable version
                             of Linux Mint 20.3 “Una” as well. Huzzah!


          o § Devices/Embedded⠀➾


                # § Open Hardware/Modding⠀➾


                      # ⚓ 11_open_source_ideas_for_being_more_eco-friendly_in
                        2022_|_Opensource.com⠀⇛


                             For governments and large organizations, open
                             source continued to be a critical component
                             for policy decisions and sustainability goals
                             in 2021. The United Nations is one such
                             organization that is relying on open source
                             to reach its goals across a wide spectrum of
                             issues including climate change. While it is
                             crucial for world leaders to make big
                             decisions to save the planet, many average
                             citizens are also eager to contribute to the
                             cause. Last year on Opensource.com, several
                             authors shared their projects and ideas for
                             making an impact on climate change. If you
                             have a goal to make more eco-friendly choices
                             in 2022, give these articles a read.


                      # ⚓ Raspberry_PI_headless_Transmission_torrent_client
                        with_web_GUI_–_peppe8o⠀⇛


                             Even if the streaming services have changed a
                             bit the roles, internet downloads have
                             evolved during the years to optimize the
                             traffic load and keep simpler sharing larger
                             files. The BitTorrent distribution has played
                             (and still plays) a great role on file
                             sharing. Raspberry PI also can use
                             Trasmission client to join the torrent
                             advantages.


                      # ⚓ Stepper_Motor_with_Raspberry_PI_Pico:_28BYJ-48_and
                        ULN2003_wiring_and_MicroPython_code_–_peppe8o⠀⇛


                             Stepper motors can trasform your code logic
                             into move in robotic projects. The 28BYJ-48
                             stepper motor (with ULN2003 motor driver) is
                             the cheapest model and works with a simple
                             code on Rapsberry PI Pico


                             In this tutorial, I’m going to show you how
                             to wire and use the 28BYJ-48 and ULN2003 with
                             Raspberry PI Pico, with MicroPython code.


                             Please note that, if you need the code for
                             Raspberry PI computer boards, you can refer
                             to my Controlling a stepper motor with
                             Raspberry Pi Zero W tutorial.


                      # ⚓ 555_Timer_On_Its_Own_In_Electronic_Dice_|_Hackaday⠀⇛


                             One of the most common clichés around here is
                             that a piece of equipment chosen for a
                             project is always too advanced. If a
                             Raspberry Pi was used, someone will say they
                             should have used an Arduino. If they use an
                             Arduino, it should have been an ATtiny.


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


                # § Web Browsers⠀➾


                      # ⚓ Government_issues_high_severity_warning_for_Google
                        Chrome_users⠀⇛


                             The Indian Computer Emergency Response Team
                             (CERT-In) under the IT ministry has issued a
                             high severity warning for Google Chrome
                             browser users. The warning is for the users
                             who are using browser’s version prior to
                             97.0.4692.71. As per the warning, multiple
                             vulnerabilities have been reported in Google
                             Chrome which can be exploited by someone to
                             execute arbitrary code on the targeted
                             system.


                # § Education⠀➾


                      # ⚓ Top_7_Best_R_Shiny_Books_and_Courses_That_Are
                        Completely_Free⠀⇛


                             So, you want to become an R Shiny Developer?
                             2022 is the year to do it. Learning a new
                             language, library, or framework can be
                             stressful – even expensive at times! That’s
                             why we decided to share the 7 best R Shiny
                             books and courses you can follow from the
                             comfort of your home completely free of
                             charge.


                             You should have at least the basic R
                             programming skills under your toolbelt if you
                             want to become proficient in R Shiny. Some of
                             the top 7 R Shiny books and courses you’ll
                             see below provide a brief refresher in R, but
                             it shouldn’t be your first exposure to the
                             language.


                # § Funding⠀➾


                      # ⚓ Software_Freedom_Conservancy_Reaches_Milestone_in
                        Fundraiser_Ending_Jan_15_–_FOSS_Force⠀⇛


                             Good news on the fundraising front from the
                             folks at Software Freedom Conservancy.


                             When the organization began its annual
                             fundraising drive back in November it
                             announced that all donations up to $159,191
                             would be matched by “a few very generous
                             anonymous donors” — its largest match offer
                             ever. On Thursday, SFC announced that the
                             match amount had been reached, meaning that
                             the fundraiser has raised at least $318,382
                             so far.


                # § FSF⠀➾


                      # § Licensing/Legal⠀➾


                            # ⚓ Libreboot_–_New_Hampshire_(USA)_may_soon
                              enshrine_Software_Freedom_into_law._YOUR_HELP_IS
                              NEEDED!⠀⇛


                                   This event of such global importance to
                                   Free Software projects, and the
                                   movement as a whole, has made me decide
                                   to write this article. The events in
                                   question, covered by this article, will
                                   occur on 11 January 2022. This is just
                                   three days away, so if you make a
                                   decision, you should make it now,
                                   today, and prepare. Please continue
                                   reading.


                                   If you live in New Hampshire or in one
                                   of the neighbouring states, especially
                                   Massachusetts, please listen up! If you
                                   are further away and unable to reach
                                   New Hampshire all that easily, please
                                   spread the following news anyway. It’s
                                   important. As alien as it may seem to
                                   many of my readers, I’m actually
                                   writing parts of this article as though
                                   someone who has never heard of Free
                                   Software is reading it, because I
                                   expect precisely that such people will
                                   read this particular article.


                                   You will see the term Free Software
                                   used in this article, but some people
                                   call it Open Source Software. However,
                                   you should call it Free Software. The
                                   word “free” refers to freedom, not
                                   price, though the software is usually
                                   also free as in gratis / zero price.


                                   The opposite of Free Software is called
                                   proprietary software, or non-free
                                   software. Proponents of Open Source
                                   sometimes call non-free software Closed
                                   Source, but you should call it non-free
                                   or proprietary, to highlight the fact
                                   that it isn’t free.


                # § Programming/Development⠀➾


                      # ⚓ Exciting_recent_developments_for_Fortran_coders⠀⇛


                             Expert Network member Sam Harrison describes
                             some exciting recent developments for Fortran
                             coders


                             At the end of last year, a new Twitter
                             channel @FortranTip was launched, with the
                             goal of sharing bite-sized tips on Fortran
                             coding best practices, similar to the
                             @SciPyTip channel does for scientific Python
                             tips.


                      # § Shell/Bash/Zsh/Ksh⠀➾


                            # ⚓ Why_“process_substitution”_is_a_late_feature_in
                              Unix_shells⠀⇛


                                   Process substitution is a great little
                                   feature and it feels very Unixy, but it
                                   took a surprisingly long time to appear
                                   in Unix and in shells. This is because
                                   it needed a crucial innovation, namely
                                   names in the filesystem for file
                                   descriptors, names that you can open()
                                   to be connected to the file descriptor.


                            # ⚓ Shell_Eval⠀⇛


                                   In this post, we will perform a few
                                   experiments to see the usefulness of
                                   the eval command for a particular
                                   scenario in a POSIX-compliant shell. At
                                   first, we prepare a test file that
                                   contains a space in its name and define
                                   a variable as follows:


                                   $ echo lorem ipsum > "foo bar"
                                   $ cmd='cat "foo bar"'
                                   We will use this file and the variable
                                   in the experiments below. All output
                                   examples below are obtained using Dash
                                   0.5.11 on a Debian GNU/Linux 11.2
                                   (bullseye) system. Dash stands for
                                   Debian Almquist Shell which is a POSIX-
                                   compliant shell available in Debian.
                                   Any POSIX conforming shell should
                                   produce similar output. On Zsh, use the
                                   command emulate sh before running these
                                   examples to get similar output.


                      # § Java⠀➾


                            # ⚓ FTC_to_Go_After_Companies_that_Ignore_Log4j⠀⇛


                                   The Federal Trade Commission (FTC) will
                                   muster its legal muscle to pursue
                                   companies and vendors that fail to
                                   protect consumer data from the risks of
                                   the Log4j vulnerabilities, it warned on
                                   Tuesday.


                                   “The FTC intends to use its full legal
                                   authority to pursue companies that fail
                                   to take reasonable steps to protect
                                   consumer data from exposure as a result
                                   of Log4j, or similar known
                                   vulnerabilities in the future,”
                                   according to the warning.


    * § Leftovers⠀➾


          o ⚓ Some_Hazy_Cosmic_Jive⠀⇛


                 Among those authors were Roger Zelazny, Samuel R. Delany,
                 Poul Anderson, Harlan Ellison, John Brunner, Ursula K.
                 Leguin and Robert Heinlein—all of whom had sold hundreds
                 of thousands of their works in those decades. By the end
                 of the 1980s, it seemed the genre was going the way of
                 the guitar in rock music. Just like the synth was
                 replacing electric guitar in popular music, fantasy
                 fiction was replacing science fiction in popular fiction.
                 I suppose part of this transition could be attributed to
                 the overall growth in book sales and the advertising
                 business’s new trend towards what they called niche
                 marketing. As almost any cognizant person who lived in
                 the US at the time might recall, the ability to focus
                 capitalism’s consumer goods at particular audiences was
                 rapidly taking over the marketplace by 1990. This would
                 become the case even more so when the world wide web
                 advanced technologically to the point where advertisers
                 could literally send an ad to a very select group of
                 people based on their use of the internet. What this
                 often meant was that products could be sold to those most
                 likely to buy them. That was the theory, at least.
                 Ultimately, this type of marketing means that what people
                 with more specific tastes are exposed to is ever more
                 limited to those tastes. If you never read sci-fi, the
                 internet is unlikely to try and sell you scifi.


                 As an occasional reader of science fiction, I am happy to
                 say that the genre seems to be experiencing n uptick in
                 popularity. New authors like N.K. Jemison and Liu Cixin
                 are quite popular, but so are many of those who were
                 popular a few decades ago. Those who think about these
                 things speculate that some of this popularity is due to a
                 common human desire to escape. This is certainly true.
                 However, I think another reason is a desire to try and
                 understand the future we are living in. Given that so
                 much of the genre’s literature is dystopian—and our
                 present has been described as such—this makes sense.
                 However, there is also the possibility that the futures
                 presented in science fiction provide the reader with some
                 potential hope.


          o ⚓ There_Will_Be_Many_Acts_of_Kindness_on_the_Way_Down⠀⇛


                 It’s now Thursday, and trees and branches have been
                 coming down all over the place. We shook branches
                 repeatedly as the snow was first arriving, to get some of
                 it off. We still had a dogwood tree come down in the back
                 yard, and some parts of crepe myrtles on the driveway,
                 and other limbs and branches all around. We shoveled the
                 snow off the house roof and the awnings over the doors as
                 well as we could.


                 Many houses and businesses around here still have no
                 electricity. Grocery stores have empty shelves. People
                 sat in cars on Interstate-95 for over 24 hours. People
                 are renting hotel rooms, but the hotel staff can’t all
                 get there due to the road conditions. More snow is
                 predicted for tonight.


          o ⚓ The_Wave_of_Crazy⠀⇛


                 It reminds me of the Indian Ocean tsunami of 2004 that
                 killed over 200-thousand people.


                 I remember news footage of that immense wall of black
                 water hitting Japan, exploding its way into fields and
                 city, ports and roads, homes and rivers — a liquid wood
                 chipper of broken debris pulverizing people and places
                 along its dreadful way, invading every nook and cranny of
                 human construction. It was an apocalyptic horror.


          o ⚓ Spider-Man:_Doxing,_Security_Culture_and_Web-Slinging⠀⇛


                 The first two decades of the comic book superhero film
                 renaissance were definitely a mixed bag. Kicked off in
                 the summer of 2000 with Bryan Singer’s X-Men and rapidly
                 followed by Sam Raimi’s Spider-Man (2002) and Ang Lee’s
                 Hulk (2003), the genre was reborn from the ashes of the
                 Batman and Superman franchises, which had ingloriously
                 crashed and burned by the mid-Nineties after noble
                 beginnings. True, there were antecedents, such as Wesley
                 Snipes’ Blade trilogy and M. Night Shyamalan’s
                 Unbreakable, but both films were marketed as horror/
                 thrillers rather than catalysts to a new family-inclusive
                 franchise.


                 The years between 2000-2008 will be remembered as a
                 period when popular comic book titles were adapted in a
                 fashion that was, like the early days of cinema, open to
                 a type of experimentation and innovation that rapidly
                 dissipated by the end of the decade. Sin City, Road to
                 Perdition, V for Vendetta, and American Splendor remain
                 outliers in terms of style, themes, and screenwriting, By
                 the time Marvel had devised the notion of Iron Man (2008)
                 being the start of a multi-picture single-narrative
                 franchise, the Marvel Cinematic Universe, things were
                 becoming sclerotic and conventional. The summer of 2012
                 release of the MCU tentpole The Avengers proved the films
                 were becoming the cinematic equivalent of Lisa Frank’s
                 color-by-numbers paintings, with Joss Whedon’s lugubrious
                 script and direction crystalizing everything wrong with
                 the MCU.


          o ⚓ Opinion_|_Spreading_Light_and_Joy_in_2022_as_We_Fight_for_a
            Better_World⠀⇛


                 Above all is the sun—the ultimate source of all our
                 energy. But we rely on plants, algae and some bacteria to
                 obtain this energy through photosynthesis. According to a
                 Lumen Learning article, “It is the only biological
                 process that can capture energy that originates in outer
                 space (sunlight) and convert it into chemical compounds
                 (carbohydrates) that every organism uses to power its
                 metabolism.”


          o ⚓ Macedonian_Ramble:_Gallipoli_as_a_Colonial_Subdivision⠀⇛


                 From the British sector, my guide Bulant and I drove
                 toward what was called S Beach, where the French landed
                 after their earlier feint at Kim Kale across the strait.
                 Behind S Beach there was more of an empty plain than at
                 Cape Helles, although the French had no more luck than
                 did the British in reaching their inland objective of
                 Achi Baba, and soon they too were entombed in their own
                 trench lines.


                 Bulant took me to the French cemetery and explained that
                 it had, of late, generated a ministerial hissy fit, when
                 someone in the government of Turkish President Recep
                 Tayyip Erdoğan realized that under many French markers
                 (metal crosses) were muslims legionaries from places such
                 as Tunisia and Algeria.


          o ⚓ Lawsuit_says_Meta_shares_blame_in_the_killing_of_a_federal
            guard.⠀⇛


                 Facebook’s parent company, Meta Platforms, has been sued
                 over the 2020 killing of a federal security guard, a move
                 that aims to challenge a federal statute that shields
                 websites and social media platforms from liability for
                 what users post.


                 The lawsuit, filed on Wednesday by Angela Underwood
                 Jacobs, the guard’s sister, argued that Facebook was
                 responsible for connecting individuals who sought to harm
                 law enforcement officers and sow civil discord. Ms.
                 Jacobs’s brother, Dave Patrick Underwood, who served at a
                 federal building and courthouse in Oakland, Calif., was
                 shot and killed in May 2020 by an Air Force sergeant with
                 antigovernment ties, according to the F.B.I.


          o § Science⠀➾


                # ⚓ How_To_Solve_A_Rubik’s_Cube_For_Beginners⠀⇛


                       This is a simple beginner’s guide on how to solve a
                       3×3 Rubik’s Cube. I’ve split the process up into
                       steps that hopefully make it easier to learn.


                       It won’t teach you to be a speed cuber, but it will
                       teach you how to solve the Rubik’s Cube…hopefully.


          o § Education⠀➾


                # ⚓ Extremist_raids_leave_500,000_Burkina_children_without
                  schools⠀⇛


                       Armed extremists have forced the closure of 3,280
                       schools – nearly one in seven nationwide –
                       affecting 511,221 students and 14,901 teachers,
                       government spokesman Alkassoum Maiga said after a
                       cabinet meeting.


          o § Hardware⠀➾


                # ⚓ We_just_have_to_talk_about_the_BMW_that_changes_colour_at
                  will_via_an_app⠀⇛


                       The concept car is called the BMW iX Flow and its
                       exterior is wrapped in a material that allows for
                       the chameleon-like superpower. BMW says the
                       material can also be used in the interior, though
                       not showcased on the iX Flow. Allowing for on-the-
                       fly customization that even James Bond could only
                       dream about in 2021.


                # ⚓ Production_PCB_And_Pogo_Pins_Produce_A_Clever_Test_Jig_|
                  Hackaday⠀⇛


                       [Hans Summers] runs a site qrp-labs.com, selling
                       self-assembly kits mostly for radio gear and GPS
                       applications, and had some production problems with
                       his QCX+ 5W QRP transceiver kit. They were using an
                       assembly house that had some problems with a sub-
                       contractor going under during the pandemic, and the
                       replacement service was somewhat below the expected
                       level of quality, resulting in a significant number
                       of SMT populated boards coming out non-functional.
                       Obviously, not wanting to pass these on to
                       customers as a debug problem, they set to work on
                       an in-house QA test jig, to give them the
                       confidence to ship kits again. The resulting
                       functional test jig, (video, embedded below) takes
                       a fairly interesting approach. Skip the video to 9:
                       00 for the description of the test jig and detailed
                       test descriptions.


                       By taking an existing known-good PCB, stripping off
                       all the SMT parts, and moving the through hole
                       components to the rear PCB side, pogo pins could be
                       soldered to strategic locations. Building the
                       assembly into a rudimentary enclosure made from
                       sawn-up raw copper clad board, with the pogos
                       facing upwards, and a simple clamp on top, allowed
                       the PCB-under-test (let’s call it the UUT from
                       hereon) to be located and clamped in place. This
                       compressed the pogos in order to make a firm
                       electrical contact. A piece of MDF that had been
                       attacked with a dremel did duty as a pressure
                       plate, with cutouts around the SMT component areas
                       to achieve the required uniform board pressure and
                       keeping the force away from the delicate soldered
                       parts. All this means that with an UUT connected
                       via pogo pins to a through-hole only test PCB, the
                       full circuit would be completed, if and only if the
                       UUT was completely functional, and that means
                       defect-free soldering and defect-free components.


                # ⚓ Baby_Steps_Toward_DIY_Autonomous_Driving:_VW_Golf_Edition⠀⇛


                       [Willem Melching] owns a 2010 Volkswagen Golf – a
                       very common vehicle in Europe – and noticed that
                       whilst the electronic steering rack supports the
                       usual Lane Keep Assist (LKAS) system, and would be
                       theoretically capable of operating in a far more
                       advanced configuration using openpilot, there were
                       some shortcomings in VW’s implementation which
                       means that it would not function for long enough to
                       make it viable. Being very interested in and
                       clearly extremely capable at reverse engineering
                       car ECUs and hacking them into submission, [Willem]
                       set about documenting his journey to unlocking
                       openpilot support for his own vehicle.


                # ⚓ China_demos_space_station’s_robotic_arm_•_The_Register⠀⇛


                       The China Manned Space Engineering Office (CMSEO)
                       says it has completed load-bearing tests on its
                       space station’s 10m robotic arm.


                       The test involved lifting and moving the Tianzhou-
                       2 cargo ship in a 47-minute operation that assessed
                       the arm’s ability to assemble sections of the
                       station while in orbit, which is exactly what space
                       boffins want to do during upcoming construction
                       tasks on the unfinished outpost.


          o § Health/Nutrition/Agriculture⠀➾


                # ⚓ Living_in_Epoch-Defining_Times:_Food,_Agriculture_and_the
                  New_World_Order⠀⇛


                       This is the future that big agritech and
                       agribusiness envisage: a future of ‘data-driven’
                       and ‘climate-friendly’ agriculture that they say is
                       essential if we are to feed a growing global
                       population.


                       The transformative vision outlined above which is
                       being promoted by the likes of the Bill and Melinda
                       Gates Foundation amounts to a power grab. Whether
                       through all aspects of data control (soil quality,
                       consumer preferences, weather, etc), e-commerce
                       monopolies, corporate land ownership, seed
                       biopiracy and patenting, synthetic lab-made food or
                       the eradication of the public sector’s role in
                       ensuring food security and national food
                       sovereignty, the aim is for a relative handful of
                       corporations to gain full control of the entire
                       global food system.


                # ⚓ Covid_Colonizes_the_Future⠀⇛


                       It didn’t have to be this way. As countries like
                       China and New Zealand demonstrated, covid can be
                       controlled. But for this you need a strong central
                       government that cares about public health. Even
                       better would have been a U.N. on steroids, a
                       strong, central WORLD government, which, other than
                       world capitalist rule, global political and
                       military bigwigs are in no way ready for. And in
                       any event, the window for humanity to contain this
                       disease closed over a year ago. The death cult that
                       is capitalism refused financial sacrifice for human
                       well-being. Thereafter, we all live with the
                       consequences.


                       How well we manage this now endemic virus lies
                       mostly in the hands of individual national rulers.
                       And by that metric, the U.S. does quite poorly.
                       Trump pretended covid didn’t exist (killing
                       hundreds of thousands of people). Biden rode the
                       fact that it did into the white house, promising to
                       tame the virus with “science.” Then he put all his
                       eggs in one basket – vaccination. This strategy was
                       a bust, because many Americans bizarrely confuse
                       simple, sane, public health measures with insulting
                       infringements on freedom.


                # ⚓ WHO_Says_Omicron_Variant_Is_Not_“Mild”_as_ER_Doctor
                  Describes_New_COVID_Wave_Overwhelming_Hospitals⠀⇛


                       We look at the skyrocketing number of COVID
                       infections. Coronavirus cases hit record highs this
                       week, with global cases climbing 70% from last week
                       to 9.5 million and the U.S. reporting a single-day
                       record of 1 million new cases on Monday. In the
                       U.S., the extraordinary volume of cases is filling
                       up emergency rooms nationwide and exhausting
                       healthcare workers, says emergency room physician
                       Dr. Craig Spencer, who has been treating
                       coronavirus patients since the pandemic began.
                       “We’re much better at treating this disease now,”
                       says Spencer, “but the problem is that the amount
                       of volume that we’re seeing threatens to really
                       wash away any added benefit from either a milder
                       variant or even all that experience that we’ve
                       learned and those tools that we’ve built up over
                       the past few years.” Spencer also critiques the
                       U.S. government’s role in prolonging the pandemic,
                       saying, “Global vaccine inequity has been one of
                       the most profound and disappointing aspects of this
                       pandemic over the past year.”


                # ⚓ A_High-Risk_Medical_Device_Didn’t_Meet_Federal_Standards.
                  The_Government_Paid_Millions_for_More.⠀⇛


                       In 2014, when the Food and Drug Administration
                       found serious problems with a life-sustaining heart
                       pump, its warning letter to the manufacturer
                       threatened to notify other federal health agencies
                       about the inspection’s findings.


                       But for years, no such alert ever went out.
                       Instead, the agency added the warning letter to an
                       online database alongside thousands of others,
                       following its typical procedures, an FDA
                       spokesperson said.


                # ⚓ EPA’s_First_New_Air_Pollutant_Addition_in_30_Years_Reveals
                  Agency_Failings:_Critics⠀⇛


                       The Environmental Protection Agency announced this
                       week the addition—for the first time in over three
                       decades—of a new substance to its list of hazardous
                       air pollutants.


                       The chemical compound called 1-Bromopropane or 1-
                       BP—used in dry cleaning and automobile care
                       products and linked to cancer and other adverse
                       human health impacts—is now among 188 pollutants
                       including benzene, formaldehyde, and methanol on
                       the list established in 1990 under the Clean Air
                       Act.


                # ⚓ Caravans_Across_California_Set_to_Hit_the_Road_as_State’s
                  Single-Payer_Bill_Advances⠀⇛


                       As a bill to deliver single-payer healthcare to
                       Californians is set to advance to a state
                       legislative health committee next week, more than a
                       dozen automobile caravans will take to the streets
                       of cities and towns across the Golden State on
                       Saturday to promote and show support for what could
                       be a first-in-the-nation universal care program.


                       “Now is the time to realize healthcare is a human
                       right—and California will lead the way with
                       CalCare.”


                # ⚓ If_Biden_Doesn’t_Act_Quickly,_Millions_Could_Lose_Medicaid
                  as_Omicron_Surges⠀⇛


                       Unless the Biden administration extends a public
                       health emergency declaration that’s set to expire
                       in just nine days, millions of vulnerable people
                       across the U.S.—including many children—could soon
                       be booted off Medicaid amid a record surge in
                       Covid-19 cases.


                       The Families First Coronavirus Response Act, a
                       relief package that Congress approved in March
                       2020, requires states to provide “continuous
                       coverage” to Medicaid enrollees for the duration of
                       the federally declared public health emergency
                       (PHE), which has been renewed several times since
                       the start of the pandemic.


                # ⚓ Right-Wing_Justices_Appear_Poised_to_Kill_Biden_Vaccine
                  Rules_Despite_Raging_Omicron⠀⇛


                       Amid expert warnings about the dire implications
                       for public health and democracy, right-wing
                       justices on the U.S. Supreme Court on Friday
                       appeared poised to strike down the Biden
                       administration’s contested federal vaccination
                       requirements even as the coronavirus pandemic rages
                       across the country.


                       “Why shouldn’t the federal government… have a
                       national rule that will protect workers?”


                # ⚓ Not_One—But_Two—Lawyers_Who_Argued_Against_Covid_Safety
                  Rules_Before_Supreme_Court…_Had_Covid⠀⇛


                       As the right-wing majority of the U.S. Supreme
                       Court appeared ready Friday to strike down the
                       Biden administration’s Covid-19 safety guidelines
                       for workplaces, the nine justices were protected in
                       their own workplace from exposure to the disease.


                       According to Reuters, two of the attorneys arguing
                       against the vaccination, testing, and mask-wearing
                       rules spoke to the court remotely because they had
                       tested positive for Covid-19 prior to the
                       proceedings.


                # ⚓ WHO_Says_Omicron_Variant_Is_Not_“Mild”_as_New_COVID_Wave
                  Overwhelms_Hospitals⠀⇛


                # ⚓ Tech_is_finally_killing_long_lines⠀⇛


                       What’s happening: Physical lines are disappearing
                       at theme parks, doctor’s offices, clothing stores
                       and elsewhere, replaced by systems that let you
                       book a slot online and then wait to be notified
                       that it’s your turn.


                # ⚓ R.I.P.,_Gayle_DeLong⠀⇛


                       My last post of 2021 discussed how that had been
                       the year that the old antivaccine tactic of
                       “dumpster diving” in the Vaccine Adverse Events
                       Reporting System (VAERS) database had been
                       mainstreamed by antivaxxers as part of their
                       ongoing efforts to portray COVID-19 vaccines as
                       deadly. I was reminded of this earlier this week,
                       when I learned of the death of Gayle DeLong, PhD.
                       One reason is that Dr. DeLong’s passing reminded me
                       of a phenomenon that has come to dominate bad
                       science about COVID-19, specifically an expert in
                       one discipline unrelated to vaccines and infectious
                       disease thinking herself an “expert” in COVID-19
                       vaccines.


          o § Integrity/Availability⠀➾


                # ⚓ GnuPG_PIN_cache,_Smartcards,_YubiKeys_and_notifications⠀⇛


                       I am still obsessed with the OpenPGP smartcard. I
                       know, it is definitely far inferior to YubiKey. It
                       has far, far less features and it’s GnuPG
                       implementation is even riddled with serious bugs
                       that can take days to work around. It definitely
                       has it’s peak years behind. But no matter how bad
                       it is, I simply like it’s form-factor.


                       I cannot state this enough. I like how it fits to
                       my wallet, along other items with similar taxonomy,
                       like credit cards or an electronic ID card. It also
                       sticks much less intrusively out of the laptop,
                       neatly and quite subtly. It is not occupying any
                       USB ports, which is what I hate the most about
                       YubiKeys. There are million form-factors of
                       YubiKeys and all have to go to some USB port. When
                       my laptop is in the dock, I have to reach out to
                       touch it (this is important, we get to this in a
                       moment). When not docked, it is easy to touch, but
                       sticks out awkwardly and it can result it
                       accidents.


                # § Proprietary⠀➾


                      # ⚓ How_to_Set_Up_Two-Factor_Authentication_for_SSH⠀⇛


                             One way to enhance SSH login security is by
                             using two-factor authentication (2FA). This
                             approach forces an administrator to self-
                             identify with an additional security
                             verification in addition to the local admin
                             credentials.


                             This tutorial guides you through setting up
                             Google Authenticator PAM to enable 2FA for
                             users connecting to SSH on a Linux server.
                             We’ll use nano as our editor in examples.


                      # § Security⠀➾


                            # § Privacy/Surveillance⠀➾


                                  # ⚓ Stop_data_pollution_from_turning_your
                                    company’s_data_lake_into_a_swamp⠀⇛


                                         More data isn’t always better.
                                         Companies should be cautious
                                         about collecting and storing data
                                         for which they have limited
                                         tangible use. Not only does this
                                         present security, privacy, and
                                         compliance risks, storing and
                                         managing such data also
                                         represents an unnecessary
                                         expense. Instead, focus on data
                                         that has value and utility – you
                                         probably have more than enough of
                                         it already!


                                  # ⚓ How_Signal_is_playing_with_fire⠀⇛


                                         Last year, current and former
                                         Signal employees told me they
                                         were worried about what that
                                         combination would bring to the
                                         app. Anonymous transactions would
                                         likely attract criminals, they
                                         told me, and that in turn would
                                         attract regulatory scrutiny.
                                         Given that end-to-end encryption
                                         already faces legal challenges
                                         around the globe, they said,
                                         Signal’s addition of anonymous
                                         payments was a needless
                                         provocation. And it could give
                                         more ammunition to lawmakers who
                                         want to end encryption as we know
                                         it.


                                  # ⚓ Worst_of_CES_Awards:_The_least_private,
                                    least_secure,_least_repairable,_and_least
                                    sustainable [Ed: I wish Doctorow knew that
                                    EFF_had_begun_promoting_fake_privacy
                                    connected_to_Microsoft]⠀⇛


                                         Six right-to-repair advocates
                                         assembled on Friday morning to
                                         present Repair.org’s second
                                         annual Worst in Show Awards, a
                                         selection of the “the least
                                         private, least secure, least
                                         repairable, and least sustainable
                                         gadgets at CES.”


                                         In a presentation streamed on
                                         YouTube, author and activist Cory
                                         Doctorow presided over the
                                         condemnation session. He said
                                         that he has been attending the
                                         Consumer Electronics Show for
                                         decades and vendors will gladly
                                         enumerate the supposed benefits
                                         of their products.


                                         “But what none of those people
                                         will ever do is tell you how it
                                         will fail,” said Doctorow. “And
                                         that’s kind of our job here
                                         today, to talk about the hidden
                                         or maybe not so hidden and
                                         completely foreseeable failure
                                         modes of these gadgets.”


          o § Defence/Aggression⠀➾


                # ⚓ The_Myth_of_the_Good_War⠀⇛


                       I have an advantage here. When I get a communique
                       like this, I know the writer read my column in a
                       regular newspaper, not a progressive site on the
                       Internet — and that’s a good thing for multiple
                       reasons.


                       One: The mainstream media is often fearful of a
                       viewpoint like mine, which is critical of war and
                       nukes and nationalism and border cages and such, so
                       I always feel delight on learning I’ve made it into
                       mainstream print.


                # ⚓ Getting_Away_With_Murder_Wasn’t_Enough_for_Tony_Blair⠀⇛


                       My first response to finding out that Tony Blair
                       had been inducted into the Most Noble Order of the
                       Garter was to wonder out loud what on earth that
                       even was. A quick Google search followed and I was
                       navigated to the official website of the British
                       Royal Family where I learnt to my amusement that
                       “King Edward III was so inspired by tales of King
                       Arthur…that he set up his own group of honourable
                       knights.”


                # ⚓ Sudan_Protests_Demand_End_to_Military_Rule:_“No
                  Negotiation,_No_Partnership,_No_Legitimacy”⠀⇛


                       We get an update from Sudan, where at least three
                       pro-democracy protesters were killed by security
                       forces on Thursday, bringing the death toll to at
                       least 60 since the military coup on October 25.
                       Thursday’s protest came four days following Abdalla
                       Hamdok’s resignation as Sudan’s prime minister,
                       after he was deposed in the October coup and then
                       shortly restored to power by the military in
                       November. However, protesters on the ground find
                       Hamdok’s resignation insignificant and consider him
                       irrelevant to the fight for full democratic control
                       over the government, says Sudanese activist Marine
                       Alneel, who joins us from Khartoum. The civilian
                       slogan is now “no negotiation, no partnership, no
                       legitimacy,” she explains, saying protesters are no
                       longer interested in preserving the joint military-
                       civilian governing deal signed after mass protests
                       in 2019 that toppled longtime leader Omar al-
                       Bashir. “In 2019, many people were displeased with
                       the partnership, and now mostly people are outright
                       rejecting any form of partnership with the
                       military,” she says.


                # ⚓ 1/6,_9/11_and_Glenn_Greenwald⠀⇛


                       Before I go any further let me offer an
                       embarrassing confession. It wasn’t long ago that I
                       was defending Greenwald to anyone I could. What is
                       Greenwald’s appeal? Now that I’m revolted by him as
                       most healthy people are, I have to say that I must
                       have forgotten even what I liked about him in the
                       first place, and his allure remains a mystery.


                       One thing I will never forget is 1/6. Liberal
                       corporate media is telling me to never forget 1/6.
                       Conservative corporate media like Greenwald tell me
                       to forget quickly. The coverage is strikingly
                       similar to 9/11. Fox News correspondent Glenn
                       Greenwald is having a field day with that. But I’m
                       skeptical. Is the new war on terror really on the
                       American right, as Greenwald claims?


                # ⚓ January_6,_a_Year_Later⠀⇛


                       This stability was unquestionably a great
                       achievement for an ethnically and religiously
                       diverse country with a large population, but this
                       American record should be celebrated cautiously,
                       with humility, and massive qualifications that must
                       never be ignored. This U.S. rise to great power
                       status rested on genocidally driven ethnic
                       cleansing of Native Americans combined with
                       economic prosperity for a land-based settler
                       colonial white elite that owed its high standard of
                       living to the racist and exploitative benefits of
                       slavery. Even after the American Civil War and the
                       end of slavery, racism remained, was cruel in its
                       dehumanizing effects on perpetrators as well as
                       victims, and extended to the entire country. That
                       the United States could constantly invoke its own
                       exceptionalism and convince most of the world that
                       it was ‘the city on the hill,’ ‘the new Jerusalem,’
                       and ‘a light unto the nations’ remains without
                       doubt a masterful triumph of public relations and
                       state propaganda, a precursor of the capitalist
                       empires built by Madison Avenue advertising
                       ingenuity. But truth it is not, and never was!


                       What was true, which was a truthful exception to
                       the big early lies, was the widespread adherence to
                       the electoral process by which political leadership
                       was determined, and legitimized. Procedural
                       democracy at its core remains about the sanctity of
                       elections as credible expressions of citizen
                       consent. Even though there is no text it was this
                       core provision of the social contract that was
                       dangerously weakened by the January 6th assault on
                       the Capitol, and even more than the assault itself,
                       by the instigating and cheerleading role played by
                       Trump and his immediate entourage. Even more
                       telling is the commitment a year later by one of
                       the two major political parties to a manifest
                       falsehood of the greatest political consequence.
                       The Republican Party overwhelmingly supports the
                       central lie that the 2020 election was stolen, and
                       this Trump deserves to be president. We can safely
                       assume that most of the Republican leadership knows
                       that it is endorsing a falsehood, but does so
                       nevertheless for cynical reasons associated with
                       calculations about their own political futures.


                # ⚓ Six_Things_the_Media_Won’t_Tell_You_About_Ukraine⠀⇛


                       Here are six crucial pieces of background that the
                       western media will not tell you.


                       The NATO Promise


                # ⚓ The_End_of_US?⠀⇛


                       This critical moment takes place before a civil war
                       breaks out or an official ceremony of dissolution
                       is held. At some point, the citizens of the country
                       stop thinking of themselves as members of a common
                       association. At some point, the mystic chords of
                       memory transmogrify into mutual disgust and
                       incomprehension.


                       At that moment, the us is over.


                # ⚓ Opinion_|_In_Praise_of_Joe_Biden’s_Fighting_Words⠀⇛


                       The January 6 speeches by both Biden and Harris
                       were terrific. On January 5,  I published a piece
                       at The Bulwark arguing that January 6 was a day not
                       to preach about healing but to renew the fight to
                       defend democracy. Yesterday morning’s speeches,
                       especially the one by Biden, exceeded my wildest
                       expectations.


                # ⚓ ‘Intentional,_Criminal_Act’:_Fire_That_Destroyed_Tennessee
                  Planned_Parenthood_Clinic_Was_Arson⠀⇛


                       After authorities in Knoxville announced Thursday
                       that a New Year’s Eve fire at the only Planned
                       Parenthood clinic in East Tennessee was arson,
                       local and national leaders at the healthcare group
                       condemned the act and vowed to resume services as
                       soon as possible.


                       “All too often, bad actors emboldened by the
                       rhetoric of anti-abortion politicians and
                       antagonists intentionally work to interrupt
                       patients’ access to healthcare and providers’
                       ability to give it.”


                # ⚓ Opinion_|_Attacks_on_Emma_Watson_Make_Clear_the
                  Demonization_of_Palestine_Solidarity⠀⇛


                       Whatever one’s reservations about our celebrity-
                       orientated culture, there’s a lot to be said for
                       big names coming out in support of important
                       causes. In what some would say was a key moment for
                       the mainstreaming of the Palestinian liberation
                       movement, Harry Potter star Emma Watson yesterday
                       shared an image on her Instagram account—with
                       sixty-four million followers—featuring the text
                       ‘Solidarity is a verb’ against a backdrop of
                       Palestinian flags.


                # ⚓ Opinion_|_On_Hearing_From_People_Who_Hate_That_I_Hate_War⠀⇛


                       Love thy enemy? I get a chance to do so on a
                       regular basis, thanks to the email (or nasty-mail)
                       I sometimes get in response to my column, e.g.:


                # ⚓ Opinion_|_From_Honduras_to_New_Orleans,_Without_a_Safe_Path
                  Forwards⠀⇛


                       I grew up in Honduras with my grandparents, as my
                       mom came to the United States years prior with the
                       dream of setting up a better life for me and my
                       younger brothers. Five years ago, when I was 17
                       years old, I was in the middle of school classes
                       when I received a text from her asking “Are you
                       ready to come to the US next week?” She had
                       arranged for a coyote. I managed to cross the
                       border by car, inflatable raft and walking, and
                       once stateside, immediately crossed paths with
                       border patrol who took me to a detention center in
                       McAllen, Texas. There was only one bathroom for 75
                       people, the food offered was terrible. They gave me
                       an aluminum cover wrap, but it did nothing to keep
                       away the bitter cold. I was released after a day
                       and a half and sent to a shelter in Miami, and then
                       to Metairie, New Orleans where I reunited with my
                       mom. We were able to hug for the first time since I
                       was six years old.  


                # ⚓ Trump_Was_Against_Telling_Mob_to_“Stay_Peaceful”_During
                  Jan._6,_Former_Aide_Says⠀⇛


                # ⚓ Opinion_|_The_Blood_Spilled_on_Jan._6_as_Rorschach_Test⠀⇛


                       It won’t take long — maybe a generation or two —
                       but one day, Jan. 6, 2021, will be just another day
                       of infamy just like all the rest. It will contain
                       only a shadow of its former dread. It will be
                       turned into an occasion for cosplay and winter
                       barbecue by tens of thousands of Americans who will
                       remember it fondly.


                # ⚓ January_6_Showed_Why_D.C._Deserves_Statehood⠀⇛


                       The phone call from my mom was more direct: Were my
                       partner and I safe, and did we have a plan in case
                       things went south at the Capitol?


                       The insurrection instigated by Donald Trump has
                       become a kind of political north star in the year
                       that’s passed since. An entire ecosystem of post-
                       mortem analyses has grown up around the day.


                # ⚓ PD_Whose_Officers_Brutalized_A_Black_Soldier_For_Driving_To
                  A_Well-Lit_Area_Sued_By_Virginia_Attorney_General⠀⇛


                       Windsor, Virigina was the recipient of unflattering
                       nationwide news coverage due to two police officers
                       deciding a black driver — and Army medic — needed
                       to be brutalized for seeking a well-lit area to
                       pull over. The whole thing was caught on the
                       officers’ body cameras, including their threats to
                       make Lieutenant Caron Nazario “ride the lightning”
                       (a reference to the officer’s Taser) as well as the
                       officer’s affirmation that Nazario was right to be
                       scared to exit his vehicle.


                # ⚓ Maryland_Court_Says_Baltimore_Prosecutors_Can’t_Hide_Their
                  ‘Do_Not_Call’_List_Of_Bad_Cops_From_The_Public⠀⇛


                       Changes in law, court decisions, and transparency
                       efforts have resulted in the public release of
                       names of officers prosecutors consider too
                       unreliable to ask to testify in court. Officers
                       with histories of misconduct or perjury are placed
                       on “do not call” lists by prosecutors who are
                       supposed to hand this information over to criminal
                       defendants.


                # ⚓ Father_and_Son_Who_Murdered_Ahmaud_Arbery_Given_Life_in
                  Prison_Without_Parole⠀⇛


                       This is a developing news story… Check back for
                       possible updates…


                       Three white men who murdered Ahmaud Arbery last
                       year in Georgia were sentenced Friday to life in
                       prison, and the judge denied parole to the father
                       and son who armed themselves and hunted down the
                       25-year-old Black man.


                # ⚓ Remembering_1/6⠀⇛


                # ⚓ A_dream_of_power,_an_awakening_to_destruction⠀⇛


                       In a situation where he is installed as president
                       after losing an election, Mr. Trump would vainly
                       try to control what will quickly cease to be the
                       United States. His allies who wish to destroy the
                       state will be the only winners. The precise
                       scenario of the collapse of the United States is
                       impossible to predict, but some of the following is
                       likely to happen, and quickly.


                       Tens of millions of people protest. Paramilitaries
                       on both sides emerge. Violence leads to fake and
                       real stories of deaths, and to revenge. Police and
                       armed forces will know neither whom they should
                       obey nor whom they should arrest. With traditional
                       authority broken, those wearing uniforms and
                       bearing arms will become partisans, take sides, and
                       start shooting one another. Governors will look for
                       exit strategies for their states. Americans will
                       rush to parts of the disintegrating country they
                       find safer, in a process that looks increasingly
                       like ethnic cleansing. The stock market and then
                       the economy will crash. The dollar will cease to be
                       the world currency.


                # ⚓ Taliban_to_include_suicide_bombers_in_their_army⠀⇛


                       Officials of the Islamic Emirate of Afghanistan
                       said that they will create a special battalion of
                       suicide attackers to be part of their future army.


                       Deputy Minister of information and culture and
                       spokesperson of the Taliban Zabiullah Mujahid said
                       that the battalion will be part of their special
                       forces and will be active under Defense Ministry.


                # ⚓ Boko_Haram_Fighters_Invade_Borno_Community,_Burn_Houses,
                  Loot_Shops⠀⇛


                       Militants from the Islamic State-backed faction of
                       Boko Haram, the Islamic State West Africa Province
                       (ISWAP), formerly known as Jamā’at Ahl as-Sunnah
                       lid-Da’wah wa’l-Jihād have attacked a community in
                       Borno State, destroying buildings and looting
                       shops.


          o § Environment⠀➾


                # ⚓ Tired_of_inaction,_two_students_brought_recycling_to_their
                  Jalisco_community⠀⇛


                       I live in Pinar de la Venta, a rural community in
                       the municipality of Zapopan, Jalisco, that’s
                       perched on a mountaintop not far from Guadalajara,
                       Jalisco. Not long ago, in our local WhatsApp chat,
                       I began to see notices every two weeks, inviting
                       people to collect and contribute their recyclables.


                       The posts came from two young women living in my
                       neighborhood named Xela Lloyd and Xochil
                       Vandroogenbroeck, both of them students. I asked
                       them why they had decided to start this project
                       instead of spending all day with their noses in a
                       smartphone, like so many other young moderns.


                # ⚓ Coverage_of_Tory_MPs’_Energy_Bills_Letter_Ignores_Ties_to
                  Climate_Science_Denial_Group⠀⇛


                       A media blitz by backbench Tory MPs calling for
                       cuts to green taxes, more North Sea oil and gas
                       extraction, and the return of fracking, is led by
                       politicians with close links to the UK’s main
                       climate science denial group and reflects many of
                       its demands.


                       A letter by the Net Zero Scrutiny Group (NZSG) of
                       19 MPs and one peer, published in the Sunday
                       Telegraph this week and covered on the front page
                       of Monday’s Daily Mail as well as by the BBC, has
                       fed into the ongoing debate around rising energy
                       bills. 


                # ⚓ Opinion_|_The_Biggest_Threat_in_‘Don’t_Look_Up’_Are_the
                  Capitalists_on_Earth⠀⇛


                       As the dust settles on a movie that has well and
                       truly got people talking about the climate crisis
                       in a way that no other movie has, it is worth
                       talking about one of the most important messages of
                       the movie, and one that has largely been ignored. 


                # ⚓ Greenland_Ice_Sheet_Shrunk_for_25th_Year_Straight_in_2021,
                  Report_Shows⠀⇛


                       “2021 is the 25th year in a row in which
                       Greenland’s ice sheet lost more mass during the
                       course of the melting season than it gained during
                       the winter.”


                       That’s according to the latest report from Polar
                       Portal, a website featuring observations from
                       Danish research institutions that monitor the
                       Greenland Ice Sheet and the sea ice in the Arctic.


                # ⚓ The_Climate_Emergency_Is_‘Now’:_Over_400_Weather_Stations
                  Set_New_Heat_Records_in_2021⠀⇛


                       Last year saw record-breaking high temperatures
                       recorded at more than 400 weather stations around
                       the world, with meteorologists voicing alarm over
                       what climate scientists say is the shape of things
                       to come, according to a report published Friday.


                       The Guardian reports that 10 countries—Canada,
                       Dominica, Italy, Morocco, Oman, Taiwan, Tunisia,
                       Turkey, the United Arab Emirates, and the United
                       States—set or matched their national monthly high
                       temperature records last year.


                # ⚓ When_to_Build_Sea_Walls⠀⇛


                       The most recent warning on December 30th is of
                       deteriorating conditions at the Arctic and
                       Greenland. The second warning is the threatening
                       collapse in Antarctica of one of the largest
                       glaciers in the world. As these events
                       unfortunately coincide so close together, one at
                       the top of the world, the other at the bottom,
                       should coastal cities plan to build sea walls?


                       The scale of time and material and costs to build
                       seawalls is nearly overwhelming. In fact, it is
                       overwhelming. The US Army Corps of Engineers is
                       already drafting plans for a gigantic seawall to
                       protect New York-New Jersey Harbour and Tributaries
                       from surges and flooding. It’s a multi-year study
                       that should be completed this year, 2022. The
                       estimated cost is US$119 billion built-out over a
                       period of 25 years for 6 miles of seawall. Yet,
                       already there is concern that it may prove
                       inadequate, only defending against storm surges,
                       not rising sea levels. NYC Comptroller Scott M.
                       Stringer has suggested 520 miles of exposed
                       shorelines as an alternative plan. (Source: US Army
                       Weighs Up Proposal For Gigantic Sea Wall to Defend
                       NY From Future Floods, ScienceAlert, January 20,
                       2020)


                # § Energy⠀➾


                      # ⚓ Oil_Companies_Have_Known_for_Decades⠀⇛


                      # ⚓ As_Cryptocurrency_Becomes_Mainstream,_Its_Carbon
                        Footprint_Can’t_Be_Ignored⠀⇛


                      # ⚓ Why_Alabama_Coal_Miners_Are_Still_on_Strike⠀⇛


                             “Happy miners run more coal.”


                      # ⚓ Thousands_of_Eesti_Energia_clients_set_to_receive
                        €1,000_electricity_bill⠀⇛


                             Around half of Eesti Energia’s private
                             clients have already received their bills for
                             December.


                             Spokesperson Priit Luts told ERR that some
                             clients on fixed-price contracts even saw
                             cheaper bills year-on-year for December, but
                             stock exchange price contracts went up five-
                             fold in some cases.


                # § Wildlife/Nature⠀➾


                      # ⚓ The_Lord_God_Bird_and_Dozens_of_Other_Species
                        Declared_Extinct_in_2021⠀⇛


                      # ⚓ Of_Wolves,_Chronic_Wasting_Disease_and_COVID⠀⇛


                             CWD is a fatal disease that can affect all
                             members of the cervid family, including deer,
                             elk, and moose.


                             There is a similar prion disease that infects
                             humans, called Cruetzfeldt-Jakob disease. It
                             is also fatal. These prion diseases cause
                             neurological tissue and systems degeneration,
                             so the brain has holes appearing like a
                             sponge, among other fatal problems.


                      # ⚓ Disaster_Victims_Shouldn’t_Have_to_Run_a_PR_Campaign
                        to_Get_Federal_Aid⠀⇛


          o § Finance⠀➾


                # ⚓ Toni_Morrison,_Glenn_Youngkin’s_Welfare_Queen⠀⇛


                       I agree with Eric Bohler that Youngkin’s (Net worth
                       $440 million) won Virginia by making racist
                       appeals, but whites and some Blacks on cable
                       offered other reasons. Eric Bohler is one of the
                       few who gets it. He wrote


                       When asked about The Atlantic placing the blame
                       elsewhere for Glenn Youngkin’s defeat of Terry
                       McAuliffe Democratic pollster, Cornell Belcher
                       responded angrily on Ari Melber’s show. He said:


                # ⚓ The_Revolution_Against_Low-Wage_Work⠀⇛


                       Underneath all the economic reporting of 2021,
                       there was a hidden story: Aided by the American
                       Recovery Act and other stimulus measures, low-wage
                       workers launched a small-scale revolution.
                       Employees and contractors used the additional
                       resources of the past year to successfully demand
                       better pay and working conditions. This occurred
                       alongside giant job gains—over 6 million new jobs
                       this year—creating a recovery about eight times
                       faster than the one that followed the Great
                       Recession of 2008-9. The way in which this is
                       happening is one of the more important and hopeful
                       stories about the labor market today.


                # ⚓ Roaming_Charges:_Emissions_Impossible⠀⇛


                       + With new COVID cases topping 500,000 a day in the
                       US, there are renewed calls for COVID relief, which
                       were firmly swatted down by the Biden
                       administration in terms more austere and inhumane
                       than any used by Trump:


                       + There’s no expiration date on the relief given to
                       Wall Street. It doesn’t even require a pandemic.


                # ⚓ Wall_Street_Banks_Have_an_Alibi_for_their_$11.23_Trillion
                  in_Emergency_Repo_Loans_from_the_Fed…_It’s_a_Doozy⠀⇛


                # ⚓ Unemployment_Falls,_Wages_Grow_for_Most_Disadvantaged_in
                  Strong_Jobs_Report⠀⇛


                # ⚓ 21_States_Hike_Minimum_Wage_as_Pressure_Mounts_to_Raise
                  $7.25_Federal_Wage_Floor⠀⇛


                # ⚓ ‘No_Excuse’_to_Keep_Federal_Minimum_at_$7.25,_Experts_Say
                  as_21_States_Hike_Wages⠀⇛


                       More than 20 states across the U.S. increased their
                       minimum wages on the first day of 2022, raises that
                       called additional attention to the fact that the
                       federal wage floor remains at $7.25 an hour—where
                       it has been stuck for more than a decade.


                       “The average worker, when adjusting for inflation,
                       is making $44 a week LESS today than they made in
                       the 1970s.”


                # ⚓ Elephant_Beetle:_Uncovering_an_Organized_Financial-Theft
                  Operation⠀⇛


                       Elephant Beetle operates in a well-organized and
                       stealthy pattern, efficiently executing each phase
                       of its attack plan once inside a compromised
                       environment: [...]


          o § AstroTurf/Lobbying/Politics⠀➾


                # ⚓ Democracy_Is_on_the_Ballot_in_2022._Summer_Lee_Is_Ready_for
                  This_Fight.⠀⇛


                       Americans are worried about the battered condition
                       of this country’s experiment with democracy. They
                       know that, as President Biden stated in his address
                       Thursday, one year after the January 6 US Capitol
                       attack, “We are in a battle for the soul of
                       America, a battle that by the grace of God and
                       goodness of this nation, we will win.”


                # ⚓ Republicans_Are_on_the_Retreat_in_California⠀⇛


                # ⚓ In_Search_of_Shallow_Doctrines:_Joe_Biden_and_Trumpism
                  Shorn⠀⇛


                       Much of this is often simple mythmaking for the
                       imperial minder in the White House, betraying what
                       are often shallow understandings about global
                       politics and movements.  Clarity and details are
                       often found wanting.  Variety in such doctrinal
                       matters, the Soviet Union’s veteran diplomat Andrei
                       Gromyko noted in casting his eye over the US
                       approach, meant that there was no “solid, coherent
                       and consistent policy” in the field.


                       In the case of President Joe Biden, any doctrine
                       was bound to be a readjustment made in hostility to
                       the Trump administration, at least superficially. 
                       But in so many ways, Biden has simply pulled down
                       the blinds and kept the US policy train going,
                       notably in its approach to China and its unabashed
                       embrace of the Anglosphere.  There remain
                       smatterings of nativism, doses of protectionism. 
                       There is the childlike evangelism that insists on
                       enlightened democracy doing battle with vicious
                       autocracy.  This was, according to Foreign Affairs,
                       the “everything doctrine”.


                # ⚓ Election_Reflection:_What_Does_the_Honduran_Presidential
                  Election_of_November_28_Tell_Us?⠀⇛


                # ⚓ Plutocrats_are_Working_Overtime_to_Cancel_Democracy⠀⇛


                       The European think tank International IDEA
                       certainly thinks so. International IDEA recently
                       labeled the U.S. a “backsliding democracy.” This
                       designation is in line with what most Americans
                       believe—that American democracy is “in crisis and
                       at risk of failing.”


                       This crisis of American democracy is not by
                       accident or by chance. (In politics, nothing is by
                       accident or chance.) The plutocrats who hold the
                       strings of power are working overtime to cancel
                       democracy. Real democracy means sharing power with
                       the masses, and why would powerful elites want to
                       do that? Real democracy means that the wealthy
                       would pay their fair share of taxes, and an end to
                       foreign wars that weaken America while enriching
                       arms manufacturers—and they certainly don’t want
                       that to happen. It would mean strong voting rights
                       legislation, an end to the electoral college.


                # ⚓ Biden_Warns_of_“Dagger_at_the_Throat_of_America”;_Fascism
                  Expert_Says_Trump’s_Personality_Cult_Growing⠀⇛


                       President Joe Biden warned about the looming threat
                       of autocracy during his speech marking the first
                       anniversary of the January 6 Capitol attack on
                       Thursday and denounced his predecessor Donald Trump
                       for inciting the rioters. In a statement responding
                       to Biden’s speech, Trump continued to falsely claim
                       the 2020 election was rigged. To discuss further,
                       we are joined by historian Ruth Ben-Ghiat, an
                       expert on the psychology of authoritarianism, who
                       says Trump has grown his “personality cult” since
                       his election loss and converted the GOP into “a
                       far-right authoritarian party which has enshrined
                       violence as part of the practice of power.” She
                       also discusses Trump’s recent endorsement of
                       Hungarian Prime Minister Viktor Orbán, who has been
                       recognized by European Union leadership as a threat
                       to democracy, and calls Florida Governor Ron
                       DeSantis a “mini-Trump” who is planning for “an
                       authoritarian system at the state level.”


                # ⚓ GOP_“Entitlement_Reform”:_Will_Disabled_Vets_Become_the_New
                  Welfare_Queens?⠀⇛


                       Since 2015, billions of dollars have been diverted
                       from VHA care to private doctors and for-profit
                       hospitals who treat veterans in costlier and less
                       effective fashion.  This cannibalization of the VHA
                       budget began under President Obama, escalated
                       during the Trump era, and continues under Joe
                       Biden.


                       Up until now, few Republicans, or their allies like
                       the Koch Brother-funded Concerned Veterans for
                       America (CVA), dared to attack the VA-run Veterans
                       Benefits Administration (VBA), a sacred cow even
                       for conservatives. Nearly six million veterans
                       currently receive payments for service-related
                       medical conditions that left them partially or
                       totally impaired; among them are 1.3 million men
                       and women who served in Iraq and Afghanistan. Their
                       total compensation, plus pensions, costs the public
                       about $110 billion per year.


                # ⚓ Trump_is_Winning_the_New_Nightmare_Years⠀⇛


                       So what if he tried to overthrow constitutional
                       rule of law and what’s left of civil decency and
                       democracy? The white nationalist GOP is his
                       Republifascist baby. It’s still the Amerikaner
                       Party of Trump (APoT). He’s still its grand
                       poohbah, with his Big Hitlerian Lie of a stolen
                       election as his great gaslighting club, accepted by
                       three in four Republicans.


                       While the minority rule governance order inherited
                       from the nation’s slave-owning Founders is already
                       fixed well to the right of the populace, the APoT
                       is ratcheting things farther starboard the name of
                       the Big Lie. It is purging the small number of its
                       elected and election officials who dared oppose
                       Dear Leader Orange and the attempted destruction of
                       previously normative bourgeois electoral democracy.
                       It is passing state-level measures to suppress what
                       it sees as illegitimate and un-American minority
                       and Democratic votes. It is working to cancel
                       popular presidential votes and Electoral College
                       slates that don’t align with its ethno-nationalist
                       program. It is conducting state-level
                       gerrymandering on racist and partisan steroids to
                       further ensure the emergence of a US House of
                       Representatives that will end serious legislative
                       branch investigation of the January 6, 2021, coup
                       attempt.


                # ⚓ Republicans_Are_Trying_to_Pass_More_Voting_Restrictions
                  Ahead_of_2022_Midterms⠀⇛


                # ⚓ Voting_Rights_Groups_Tell_Biden:_Don’t_Come_to_Atlanta
                  Without_a_Plan⠀⇛


                # ⚓ Chile’s_“New_Left”_Brings_Hope⠀⇛


                # ⚓ Selling_Out_Democracy_for_Political_Influence⠀⇛


                       Confederate flags were waved inside the Capitol.
                       People with zip ties and weapons were ready to do
                       harm or even kill members of Congress and the vice
                       president. These were right-wing extremists who had
                       the encouragement, and even help, from Trump allies
                       in Congress.


                       When the dust settled, many corporations rightly
                       spoke out and pledged to halt contributions to
                       lawmakers whose rhetoric and actions played a part
                       in the insurrection — including those who voted to
                       throw out the 2020 presidential election results in
                       service of the Big Lie, a group now known as the
                       “Sedition Caucus.”


                # ⚓ A_Nation_Coming_Apart_at_the_Seams⠀⇛


                       It’s easy to forget just how long this world has
                       been a dangerous place for human beings. I thought
                       about this recently when I stumbled upon a little
                       memoir my Aunt Hilda scrawled, decades ago, in a
                       small notebook. In it, she commented in passing: “I
                       was graduated during that horrible flu epidemic of
                       1919 and got it.” Badly enough, it turned out, to
                       mess up her entry into high school. She says little
                       more about it.


                       Still, I was shocked. In all the years when my
                       father and his sister were alive and, from time to
                       time, talked about the past, never had they (or my
                       mother, for that matter) mentioned the disastrous
                       “Spanish Flu” pandemic of 1918-1920. I hadn’t the
                       slightest idea that anyone in my family had been
                       affected by it. In fact, until I read John Barry’s
                       2005 book, The Great Influenza, I hadn’t even known
                       that a pandemic devastated America (and the rest of
                       the world) early in the last century — in a fashion
                       remarkably similar to, but even worse than, Covid-
                       19 (at least so far) before essentially being
                       tossed out of history and the memory books of most
                       families.


                # ⚓ Iran’s_wrestling_president_says_‘Death_to_America’_before
                  US_match⠀⇛


                       Pashaei, asked: “Why should someone (Alireza Dabir)
                       who has been one of the athletes very close to the
                       Iranian government in all the past years and part
                       of the government’s propaganda be able to travel
                       freely to the United States?”


                # ⚓ Islamophobia_envoy’s_mission_should_focus_on_Muslim
                  countries⠀⇛


                       Only 9 percent of all religious hate crimes in the
                       United States and 1 percent of all hate crimes are
                       directed at Muslims. How many government positions
                       are created to address a problem that affects 1
                       percent of the population?


          o § Misinformation/Disinformation⠀➾


                # ⚓ The_Next_Big_Lies:_Jan._6_Was_No_Big_Deal,_or_a_Left-Wing
                  Plot⠀⇛


                       By Thursday’s anniversary of the violence that has
                       been connected to at least seven deaths and left
                       some 150 police officers injured, it was an article
                       of faith among vast swaths of conservative
                       Americans that the riot was just “one day in
                       January,” in the words of former Vice President
                       Mike Pence, whose life was directly threatened. For
                       the half of Republicans who now believe the rioters
                       were at the Capitol to “protect democracy,”
                       according to the latest ABC News/Ipsos poll, any
                       talk of Jan. 6 as a singularly violent episode in
                       American democracy would likely be taken as
                       liberal, mainstream-media claptrap.


          o § Censorship/Free Speech⠀➾


                # ⚓ Veteran_actor_faces_2_years_in_prison_on_charges_of
                  insulting_the_Turkish_nation⠀⇛


                       İlyas Salman, 72, is accused in the indictment of
                       insulting the Turkish nation, the Republic of
                       Turkey, the Turkish Parliament and government and
                       legal institutions of the state in a video he
                       posted on social media on Jan. 25, 2021. The
                       charges are based on Article 301 of the Turkish
                       Penal Code (TCK), which criminalizes public
                       denigration of “Turkishness, the Republic or Grand
                       National Assembly of Turkey.”


                # ⚓ What_Kazakhstan_Isn’t⠀⇛


                       Knowledge of Kazakhstan in the West is extremely
                       slim, particularly among western media, and many
                       responses to events there have been wildly off-
                       beam.


                # ⚓ Day_six_of_Kazakhstan’s_unrest_Police_are_ordered_to_shoot
                  to_kill,_the_president_doubles_down_on_‘terrorists_and
                  killers’_rhetoric,_and_rumors_spread_about_Nazarbayev⠀⇛


                       Police and soldiers have “essentially” restored
                       order across Kazakhstan, President Kassym-Jomart
                       Tokayev declared on Friday. “The local authorities
                       control the situation, but terrorists are still
                       using weapons and damaging people’s property. That
                       is why counterterrorist actions will continue until
                       the complete destruction of the militants,” Tokayev
                       said at a meeting on January 7 with the nation’s
                       law enforcement heads.  


                # ⚓ Kazakhstan_President_Condemned_for_‘Shoot_to_Kill’_Orders
                  Against_Protesters⠀⇛


                       Human rights advocates expressed alarm Friday after
                       Kazakhstan’s president ordered security forces to
                       “shoot to kill without warning” protesters engaged
                       in ongoing demonstrations over high fuel prices,
                       economic inequality, and corruption.


                       President Kassym-Jomart Tokayev announced he had
                       given the order in a speech as he also accused
                       media outlets of encouraging unrest and claimed
                       without evidence that demonstrators have taken
                       orders from a “single command post” and have “a
                       clear plan of attacks on military, administrative,
                       and social facilities in all areas.”


                # ⚓ Kazakhstan_president_gives_shoot-to-kill_order_against
                  protesters,_dismissing_calls_for_negotiations⠀⇛


                       In contrast to this portrait of the demonstrators
                       as hardened militants, several thousand
                       demonstrated peacefully in the city of Zhanaozen,
                       one of the first hotspots of the riots, on Friday.
                       They issued the most specific list of demands to
                       date, asking for a change in power, freedom for
                       civil rights activists, and a return to a 1993
                       version of the constitution, which is considered to
                       have a more democratic tone and a clearer division
                       of power than the current one.


                # ⚓ Kazakh_president_orders_security_forces_to_shoot_to_kill
                  after_days_of_violent_protests⠀⇛


                       Kazakhstan is experiencing the worst street
                       protests since the country gained independence
                       three decades ago. The demonstrations began over a
                       near-doubling of prices for a type of vehicle fuel
                       and quickly spread across the country, reflecting
                       wider discontent over the rule of the same party
                       since independence.


                       Protests have turned extremely violent, with
                       government buildings set ablaze and scores of
                       protesters and more than a dozen law enforcement
                       officers killed. Internet across the country has
                       been shut down, and two airports closed, including
                       one in Almaty, the country’s largest city.


                # ⚓ Kazakh_leader_rejects_talks,_tells_forces_to_“shoot_to
                  kill”_as_Russia_helps_quash_anti-government_unrest⠀⇛


                       Long seen as one the most stable of the ex-Soviet
                       republics of Central Asia, energy-rich Kazakhstan
                       is facing its biggest crisis in decades. Protesters
                       stormed government buildings in Almaty on Wednesday
                       and fought running battles with police and the
                       military, with officials saying 748 security
                       officers were wounded and 18 killed.


                # ⚓ Kazakhstan’s_president_gives_shoot_to_kill_orders_against
                  anti-government_protestors⠀⇛


                       In a televised address to the nation, President
                       Kassym-Jomart Tokayev used harsh rhetoric,
                       referring to those involved in the turmoil as
                       “terrorists,” “bandits” and “militants” — though it
                       is unclear how peaceful protests gathered steam and
                       then descended into violence.


                       “I have given the order to law enforcement and the
                       army to shoot to kill without warning,” Tokayev
                       said. “Those who don’t surrender will be
                       eliminated.”


                # ⚓ Kazakhstan:_President_gives_shoot-to-kill_order_against
                  protesters⠀⇛


                       Reid Standish from Radio Free Europe told DW that
                       the president’s announcement is likely to escalate
                       the situation.


                       “It’s certainly likely to enflame the situation.
                       It’s definitely going to be taken seriously. All
                       last night there was fighting taking place across
                       Almaty, gunshots could be heard across the city
                       until the morning, so that speech shows that the
                       government is digging in deep and they have no
                       qualms about using force against people who are on
                       the streets.”


          o § Freedom of Information/Freedom of the Press⠀➾


                # ⚓ Craig_Aaron_on_Local_Journalism,_Barbara_Briggs_on
                  Workplace_Disasters⠀⇛


                       This week on CounterSpin: At FAIR, we say you can
                       change the channel all you want, but you can’t turn
                       on what isn’t there. The loss of an information
                       source—a particular place for debate, for
                       conversation, on issues relevant to you—is
                       incalculable, but very real. We talked about the
                       loss of local journalism, and why we can still be
                       hopeful, with Craig Aaron of the group Free Press.


                # ⚓ To_Redact_the_Truth_or_Not_to_Redact_the_Truth:_Is_That_the
                  Question?⠀⇛


                       My article recounted at great length the war crimes
                       revealed in a major New York Times 2-part article
                       entitled, “Hidden Pentagon Records Reveal Patterns
                       of Failure in Deadly Airstrikes.” But in the
                       section entitled, “Assange and the New York Times
                       on collateral damage,” I wrote:


                       “In contrast,” I asserted, incorrectly, “Assange’s
                       revelations, received from U.S. soldier
                       whistleblower, Chelsea Manning, were not redacted.”
                       [Emphasis added].


                # ⚓ Opinion_|_Biden_Must_Halt_the_Assault_on_Free_Press_and
                  Drop_the_Prosecution_of_Julian_Assange⠀⇛


                       Gathering threats to democracy are front and center
                       as the United States marks the first anniversary of
                       the January 6th Capitol insurrection, when
                       President Donald Trump incited thousands of
                       supporters to violently storm Congress, attempting
                       to overturn the 2020 election. While the Republican
                       Party descends into the cult of Trump, progressive
                       activists across the country are fighting to expand
                       voting rights and protect free and fair elections.
                       One of democracy’s principle bulwarks is a free
                       press. Sadly, with its ongoing prosecution of
                       Wikileaks founder Julian Assange, the Biden
                       administration is leading the attack on journalism,
                       strengthening the hand of would-be autocrats
                       everywhere.


                # ⚓ Taliban_Rules_Bring_Uncertainty_for_Provincial_Media⠀⇛


                       Uncertainty over restrictions for Afghanistan’s
                       provincial media is making it impossible for
                       journalists to work under the Taliban, local
                       journalists and station managers say.


                       Media need prior approval to cover news, in some
                       provinces women are still not allowed to work,
                       several stations have dropped entertainment
                       segments, and others have stopped broadcasting
                       altogether.


                       On Monday, representatives from 85 local radio
                       stations met with members of the Talban to ask the
                       group to clarify its position on the media laws
                       under the previous government. They also asked that
                       female journalists be allowed to work.


          o § Civil Rights/Policing⠀➾


                # ⚓ A_New_Year_One_for_Gotham⠀⇛


                       Few would vouch that de Blasio’s administration had
                       lived up to Eric Alterman’s first-year hopes that
                       the post-Bloomberg mayor would “use the power of
                       the city government to make New York a fairer and
                       more equal place for all its inhabitants.”


                       Was the relentlessness of inequality since 2014,
                       even well before the unexpected effects of COVID,
                       merely due to de Blasio being the wrong choice to
                       steer “the power of the city government,” or
                       disgraced governor and sometime de Blasio foe
                       Andrew Cuomo likewise mishandling state government,
                       not the nature of government power itself?


                # ⚓ Plans_for_Mass_Shipments_of_High-Level_Radioactive_Waste
                  Quietly_Disclosed⠀⇛


                       Last month US Nuclear Regulatory Commission (NRC)
                       staff quietly reported preparing for tens of
                       thousands of cross-country shipments of high-level
                       radioactive waste from nuclear reactors to the
                       desert Southwest. The oft-disparaged US
                       infrastructure of decrepit of roads, faulty
                       bridges, rickety rails, and rusty barges may not be
                       ready for such an onrush of immensely heavy rad
                       waste casks.


                       Diane D’Arrigo, of Nuclear Information and Resource
                       Service in Maryland, and Leona Morgan, with Nuclear
                       Issues Study Group in New Mexico, report for NIRS
                       that the transports would carry “the hottest, most
                       concentrated atomic waste from the nuclear fuel
                       chain, misleadingly dubbed “spent nuclear fuel.
                       This radioactive waste can cause death in minutes
                       if unshielded, and remains radioactive for
                       literally millions of years; it is one of the most
                       deadly materials on Earth.”


                # ⚓ Free_Jarvis_Jay_Masters!⠀⇛


                       The California Committee on the Revision of the
                       Penal Code released an advisory report in November
                       2021 that called for the state to end, once and for
                       all, a capital punishment regime that hasn’t seen
                       an execution at San Quentin State Prison since
                       January 2006, when Clarence Ray Allen was executed
                       at the age of 76. He went to the gas chamber just a
                       month after the state killed the Nobel
                       Prize–nominated Crip founder Stanley “Tookie”
                       Williams. Both men had sought and been denied
                       clemency from then-Goveror Arnold Schwarzenegger.


                # ⚓ Minneapolis_Oversight_Board_Says_Police_Department_Should
                  Ditch_‘Excited_Delirium’_Training⠀⇛


                       Excited delirium isn’t a medical condition. It’s
                       just post-death rationalization that shifts the
                       culpability for deaths at the hands of law
                       enforcement to the corpses the cops created. This
                       supposed medical diagnosis didn’t reach critical
                       mass until the introduction of one of the most
                       infamous “less-lethal” weapons ever created: the
                       Taser.


                # ⚓ Widespread_Discontent_Drove_Labor’s_Advances_in_the_Private
                  Sector_in_2021⠀⇛


                # ⚓ Full_return_to_office_is_‘dead,’_experts_say_—_and_remote
                  is_only_growing⠀⇛


                       Employees hold more cards than usual. One ace they
                       have is a two-year track record of working from
                       home without a drop in productivity — and many
                       report an increase. Workers want remote options so
                       they can cut out the commute, be their best both at
                       home and at work, have more child care flexibility
                       and reduce ongoing concerns about Covid exposure.
                       It’s a reckoning for employers.


                # ⚓ Dozens_of_protesters,_12_police_dead_in_Kazakhstan
                  protests⠀⇛


                       One police officer was found beheaded in the
                       unrest, which poses a growing challenge to
                       authoritarian rule in the former Soviet republic.


                # ⚓ Man,_41,_arrested_with_fresh_human_parts_in_Lagos⠀⇛


                       He further disclosed that an Islamic cleric, whose
                       identity he did not reveal, told him to get the
                       parts to prepare a sacrifice that would make him
                       rich.


                # ⚓ Christian_killed_after_New_Year’s_prayer_meeting⠀⇛


                       “Attacks of this nature have been ongoing for the
                       past 20 years and the silence is deafening,” he
                       said.


                       “While the government claims to be doing its best
                       to curb the violence, the reality paints a
                       different picture of a Muslim-led government
                       allowing anti-Christian violence to continue
                       without consequence.”


                # ⚓ Taliban_Orders_To_Close_All_General_Baths_For_Women_In
                  Northern_Balkh_Province:_Report⠀⇛


                       According to the head of the DPVPV in Balkh
                       province, the decision was taken after meetings
                       with religious scholars, who are also known as
                       Ulama. “Men are allowed to go to general baths as
                       people do not have access to contemporary baths at
                       home, while women are supposed to go to private
                       baths while wearing hijab,” stated the DPVPV head
                       as per Khaama News. Meanwhile, boys under the age
                       of 18 are prohibited from using public baths, and
                       body massage is prohibited in the baths. Local
                       officials in western Herat province had previously
                       temporarily closed women’s general baths, reported
                       the Afghan media outlet.


          o § Internet Policy/Net Neutrality⠀➾


                # ⚓ FCC_Revisits_Transparency_‘Nutrition_Label’_For_Broadband⠀⇛


                       For years we’ve noted how broadband providers
                       impose all manner of bullshit fees on your bill to
                       drive up the cost of service post sale. They’ve
                       also historically had a hard time being transparent
                       about what kind of broadband connection you’re
                       buying. As was evident back when Comcast thought it
                       would be a good idea to throttle all upstream
                       BitTorrent traffic (without telling anybody), or
                       AT&T decided to cap the usage of its “unlimited”
                       wireless users (without telling anybody), or
                       Verizon decided to modify user packets to track its
                       customers around the internet (without telling
                       anybody).


          o § Digital Restrictions (DRM)⠀➾


                # ⚓ How_John_Deere_created_its_autonomous_tractor⠀⇛


                       Adding new brawn to the self-driving vehicle
                       industry, John Deere unveiled a 40,000-pound
                       autonomous tractor at this year’s Consumer
                       Electronics Show that it says will be commercially
                       available by the end of 2022.


                       The system uses six pairs of stereo cameras
                       combined with GPS guidance to drive a Deere 8R
                       tractor with a chisel plow and the capability to
                       tow other equipment. A farmer can put the tractor
                       to work with a swipe of a smartphone app and then
                       walk away to spend time with family or attend to
                       other business, using the app to monitor the
                       tractor’s progress plowing a field or performing
                       some other task — and to receive alerts of
                       anomalies the software doesn’t know how to handle.
                       While it’s working, the tractor can also gather
                       data about the health of crops in the field, the
                       health and moisture content of the soil and other
                       metrics.


          o § Monopolies⠀➾


                # ⚓ Sen._Warren_&_Rep._Jayapal_Want_Google_To_Stop_Asking_For
                  DOJ’s_Kanter_Recusal⠀⇛


                       Following Kanter’s confirmation in the Senate in
                       November, Google requested the DOJ review whether
                       he should be recused from cases and investigations
                       involving its business. Google cited Kanter’s prior
                       work for its rivals like Yelp in antitrust matters
                       involving its business, and pointed to previous
                       statements about Google’s alleged dominance to
                       argue he’d already made up his mind on its
                       liability.


                # ⚓ Sen._Warren_and_Rep._Jayapal_tell_Google_to_stop_trying_to
                  ‘bully’_DOJ_antitrust_chief_into_recusal⠀⇛


                       Sen. Elizabeth Warren, D-Mass., and Rep. Pramila
                       Jayapal, D-Wash., told Google CEO Sundar Pichai on
                       Wednesday to stop trying to “bully” Department of
                       Justice antitrust chief Jonathan Kanter into
                       recusal in a new letter shared exclusively with
                       CNBC.


                       “Google should focus on complying with antitrust
                       law rather than attempting to rig the system with
                       these unseemly tactics,” the lawmakers wrote.


                # § Patents⠀➾


                      # ⚓ Senator_Tillis_Holds_Secret_Meeting_With_IP
                        Maximalists_To_Discuss_A_Single_US_‘IP’_Agency⠀⇛


                             Senator Thom Tillis is chock full of bad
                             ideas about copyrights and patents — mostly
                             focused on making things worse for the public
                             by expanding the monopoly powers granted to
                             patent and copyright holders. So I guess it
                             comes as little surprise that he held a
                             secret meeting that appears to have only been
                             attended by copyright maximalists to talk
                             about trying to merge the Copyright Office
                             into the US Patent & Trademark Office.


                      # § Software Patents⠀➾


                            # ⚓ Google_Infringed_on_Sonos_Speaker_Technology,
                              Trade_Court_Rules⠀⇛


                                   The final ruling by the United States
                                   International Trade Commission, a
                                   quasi-judicial body that decides trade
                                   cases and can block the import of goods
                                   that violate patents, closes a two-year
                                   investigation into the intellectual-
                                   property [sic] dispute.


                                   Sonos had asked the trade commission to
                                   block imports of Google products that
                                   the speaker company says infringes on
                                   its patents. They include Google Home
                                   smart speakers, Pixel phones and
                                   computers, and the Chromecast streaming
                                   video device. Those items are made in
                                   China and shipped to the United States.


                            # ⚓ Google_Infringed_On_Multiple_Sonos_Patents,
                              U.S._International_Trade_Commission_Rules⠀⇛


                                   The underlying legal battle initiated
                                   in January of 2020, when Sonos formally
                                   accused Google of infringing upon five
                                   patents involving smart-speaker
                                   technology. And as part of the firmly
                                   worded complaints – Sonos filed two
                                   separate lawsuits – the 20-year-old
                                   audio company asked the ITC to bar
                                   Google from selling laptops, phones,
                                   and smart speakers alike in the United
                                   States.


                                   Google and Sonos had partnered in 2013
                                   to bring Google Play Music support to
                                   the latter entity’s products. And
                                   Google, the patent-infringement actions
                                   claimed, had used proprietary
                                   information obtained through this
                                   partnership to lift Sonos’ multi-room
                                   speaker technology, purportedly
                                   violating a grand total of some 100
                                   patents in the process.


                            # ⚓ Google:_We_disagree_with_Sonos_patent_ruling_so
                              much,_we’ve_changed_our_code_to_avoid
                              infringement⠀⇛


                                   Those patents describe methods to
                                   manage the volume of multiple audio
                                   players on a network from a controller-
                                   like hub, pairing players, and so on.


                                   Sonos’ chief legal officer Eddie
                                   Lazarus told the New York Times: “We
                                   appreciate that the ITC has
                                   definitively validated the five Sonos
                                   patents at issue in this case and ruled
                                   unequivocally that Google infringes all
                                   five. That is an across-the-board win
                                   that is surpassingly rare in patent
                                   cases.”


                                   Google builds its gear in China and
                                   ships it to the US. Although the heavy-
                                   worded ruling seemingly affects
                                   Google’s ability to import a range of
                                   network-connected music-playing
                                   devices, from its Chromecast and Google
                                   Nest to Google Home and Pixel
                                   smartphones, in reality, Google is
                                   unlikely to be impacted by the ban,
                                   which takes effect in 60 days.


                                   That’s because after a warning shot
                                   last year from the ITC that Google was
                                   potentially infringing Sonos’s patents,
                                   Google pledged to make changes to its
                                   software and firmware to avoid
                                   violating that intellectual property.


                # § Copyrights⠀➾


                      # ⚓ Anti-Piracy_Outfit_Rightscorp’s_Corporate_Status_is
                        Void_Due_to_Unpaid_Tax_Bills⠀⇛


                             Rightscorp is a key evidence provider in
                             several multi-million dollar piracy lawsuits
                             and a trusted anti-piracy partner of the
                             RIAA. The evidence provided by Rightscorp is
                             not without controversy, however. The company
                             itself has issues too, as the state of
                             Delaware has voided its corporate status
                             after it failed to pay more than $450,000 in
                             taxes.


                      # ⚓ UK_Online_Piracy_Increases_Slightly_But_Over_Five
                        Years_Remains_Stable⠀⇛


                             A new report published by the UK’s
                             Intellectual Property Office reveals that
                             online piracy was slightly up in 2021 versus
                             the previous year. Overall, however, online
                             piracy figures have remained fairly stable
                             over the past five years, with some tentative
                             signs that hardcore pirates are still on the
                             wane.


                      # ⚓ Court_Orders_Twitter_Reveal_Anonymous_Tweeter_Over
                        Sketchy_Copyright_Claim,_Because_That_Tweeter_Won’t
                        Show_Up_In_Court⠀⇛


                             Back in November we wrote about a very
                             bizarre attempt to abuse copyright law to
                             uncover who was behind a Twitter account,
                             @CallMeMoneyBags. That account tweeted out
                             various things mocking and shaming various
                             extremely wealthy people, including
                             billionaire Brian Sheth, a private equity
                             bro. Some of the tweets in the fall of 2020
                             lightly mocked Sheth, including suggesting
                             potential infidelity. The images themselves
                             appeared to be social media-type photos of
                             young women (or possibly just one young
                             woman).


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