𝕿𝖊𝖈𝖍𝖗𝖎𝖌𝖍𝖙𝖘 Bulletin for Thursday, December 14, 2023
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Generated Fri 15 Dec 02:49:29 GMT 2023
Created by Dr. Roy Schestowitz (𝚛𝚘𝚢 (at) 𝚜𝚌𝚑𝚎𝚜𝚝𝚘𝚠𝚒𝚝𝚣 (dot) 𝚌𝚘𝚖)
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Gemini index for the day: gemini://gemini.techrights.org/2023/12/14/


╒═══════════════════ 𝐑𝐄𝐂𝐄𝐍𝐓 𝐁𝐔𝐋𝐋𝐄𝐓𝐈𝐍𝐒 ════════════════════════════════════╕
  Previous bulletins in IPFS (past 21 days, in chronological order):

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╒═══════════════════ 𝐈𝐍𝐃𝐄𝐗 ═══════════════════════════════════════════════════╕

⦿ Techrights  -  Eben Moglen in 2012: “Facebook Will Not Have Success for 10 More Years.”

⦿ Techrights  -  IRC Proceedings: Wednesday, December 13, 2023

⦿ Techrights  -  Microsoft Windows: Security Patches for a Fee. Flee to GNU/Linux (Many Already Do).

⦿ Techrights  -  Over at Tux Machines...

⦿ Techrights  -  [Video] Lessons to be Learned From Gandi's Death (Total Webhosting Solutions B.V. Shaking Down Customers)

⦿ Techrights  -  [Video] 'Modern' Computing Excessively Bloated, Wasteful, and User-Restricting

⦿ Techrights  -  [Video] The Fake Economy and the Pump-and-Dump Scheme With Useless Gadgets and False Promises/Prophecies From Companies With Fake Valuations

 ䷼ Bulletin articles (as HTML) to comment on (requires login):
https://news.techrights.org/n/2023/12/14/Eben_Moglen_in_2012_Facebook_will_Not_Have_Success_for_10_More_.shtml
https://news.techrights.org/n/2023/12/14/irc-log-131223.shtml
https://news.techrights.org/n/2023/12/14/Microsoft_Windows_Security_Patches_for_a_Fee_Flee_to_GNU_Linux_.shtml
https://news.techrights.org/n/2023/12/14/Over_at_Tux_Machines.shtml
https://news.techrights.org/n/2023/12/14/_Video_Lessons_to_be_Learned_From_Gandi_s_Death_Total_Webhostin.shtml
https://news.techrights.org/n/2023/12/14/_Video_Modern_Computing_Excessively_Bloated_Wasteful_and_User_R.shtml
https://news.techrights.org/n/2023/12/14/_Video_The_Fake_Economy_and_the_Pump_and_Dump_Scheme_With_Usele.shtml

 ䷞ Followed by Daily Links (assorted news picks curated and categorised):
https://news.techrights.org/n/2023/12/14/Links_14_12_2023_COVID_19_Rebounding_Northern_Ireland_Police_Br.shtml
https://news.techrights.org/n/2023/12/14/Links_14_12_2023_Food_Insecurity_and_Escalations_in_the_Red_Sea.shtml
https://news.techrights.org/n/2023/12/14/Links_14_12_2023_Growing_Tensions_in_the_Far_East_and_More_Crac.shtml

                    ䷩ 𝚕𝚒𝚗𝚎 67

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

(ℹ) Images, hyperlinks and comments at https://news.techrights.org/n/2023/12/14/Eben_Moglen_in_2012_Facebook_will_Not_Have_Success_for_10_More_.shtml
    Gemini version at gemini://gemini.techrights.org/n/2023/12/14/Eben_Moglen_in_2012_Facebook_will_Not_Have_Success_for_10_More_.gmi

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ Eben Moglen in 2012: “Facebook Will Not
Have Success for 10 More Years.”⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


posted by Roy Schestowitz on Dec 14, 2023,
updated Dec 14, 2023


The other day: Eben_Moglen_Correctly_Predicted_the_Demise_of_Social_Control
Media_and_How_It_Would_Happen


Video: Eben_Moglen_on_Facebook,_Google_and_Government_Surveillance


🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇Eben_Moglen_on_Facebook,_Google_and_Government_Surveillance⦈


THE FOLLOWING is a transcript of the above video from 2012, which we shared 1.5
days ago:


     [00:00]


     Title: Are you saying that Facebook is the new KGB?


     Eben Moglen: Yes. That's correct. I gave a talk to college students
     about Facebook in America. I called it from Lubyanka to Harvard Yard.
     Why do you build a prison and torture people? Why do you have
     informants and build a social networking system badly designed and
     people bring you everything? They don't think they're bringing it to
     you. They think they're bringing it to their friends but they're
     actually bringing it to Lubyanka and Lubyanka is giving it to their
     friends. This is not a good way to work.


     [00:30]


     EM: It's not bad to have social networking. Social networking is
     wonderful and important. It changes human life. It's bad to have it
     in a badly designed way.


     Title: Will Facebook still be around in 10 years from now?


     Eben Moglen: Facebook will not have success for 10 more years.
     Facebook will have success until its membership starts dropping. Once
     its membership starts dropping, it will be in deep trouble. The
     theory of Facebook is nobody can leave


     [01:00]


     EM: because they have to leave their friends behind. Once we create
     federated social networking with soft migration, Facebook will not be
     able to say that any more, and its death cycle will begin. This was
     the importance of the software called Diaspora. Diaspora is correctly
     engineered with respect to the one essential question, "How do I
     leave Facebook, Flickr, Twitter, without leaving


     [01:30]


     EM: my friends and followers behind?" This is the thing that as an
     engineering matter we need to understand. As people use more and more
     social networking, they don't just use one service. They don't just
     use Facebook, or Google Plus. They use that and Twitter or Identica
     or this or that. So that pretty soon start using feed aggregators
     because they need to aggregate the various social network feeds they
     get. The


     [02:00]


     EM: aggregator then is the crucial middleware, in the same way the
     browser is the crucial middleware in the Web. So you build an
     aggregator, which is basically what Diaspora's front side is. And you
     say, ok, you have all these feeds from your friends. If your friends
     are also using Diaspora, then you're gonna share in an encrypted way,
     privately. And for your friends who aren't using it yet, you'll
     bounce off of the public services, Facebook, Flickr, Twitter,
     whatever. Over


     [02:30]


     EM: time as more and more of your friends switch to Diaspora, you
     disappear from the public services slowly and gracefully. So you
     never lose your friends, but you shift away. Why do they start using
     Diaspora? It's not even because they are worried about Facebook
     privacy. It's just a good aggregator, and they need one because they
     do more social networking they need better aggregation. So in the
     same way that Firefox frees people from a net created by Microsoft
     the


     [03:00]


     EM: web we own. FrontPage does this, therefore the browser is broken
     if it doesn't do the same thing and various software has moved away.
     Because Firefox meant that if you got tired of IE's security or your
     IT guys got tired of IT security problems created by IE then you
     could move. The same thing will happen with social networking.
     Whether we succeed in freedom [...], whether Diaspora, whether GNU
     Social, whether many other possible


     [03:30]


     EM: varieties because we have many talented teams working on better
     social networking around the planet. Many, many. One of them will
     begin to be an aggregation system other people like to use. Maybe
     tied to a browser. Maybe not tied to a browser. And as people begin
     to adopt that aggregation system, they will softly move to secure
     sharing. At some point there will come a day when Facebook has a
     maximum number of users. A billion, a billion five, a billion seven?
     I don't know. But once it starts to fall, it will come apart quickly.
     Because then people will realize you don't have to leave your
     friends. Why do we stay with this Facebook? What is it that the
     service gives us that's worth the spying? And the numbers are not
     particularly impressive. We know now from the IPO documents for sure,
     Facebook takes in $4.40 per year per user.


     [04:00]


     EM: Huh? $4.40? 3.25€, right? That's not very much money. And they're
     not providing anything that's worth all that much, once we have a
     good substitute and people don't lose touch with their friends by
     shifting, they'll naturally shift. Ten years is a good outside bet
     about the maximum length of Facebook's life.


His predictions turned out to be accurate around 2019. █




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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

                    ䷩ 𝚕𝚒𝚗𝚎 246

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

(ℹ) Images, hyperlinks and comments at https://news.techrights.org/n/2023/12/14/irc-log-131223.shtml
    Gemini version at gemini://gemini.techrights.org/n/2023/12/14/irc-log-131223.gmi

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ IRC Proceedings: Wednesday, December 13,
2023⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


posted by Roy Schestowitz on Dec 14, 2023


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


                                              #techrights_log                                                                                           #boycottnovell_log


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


                                         #boycottnovell-social_log                                                                                        #techbytes_log


                          Enter_the_IRC_channels_now





                    ䷩ 𝚕𝚒𝚗𝚎 278

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

(ℹ) Images, hyperlinks and comments at https://news.techrights.org/n/2023/12/14/Microsoft_Windows_Security_Patches_for_a_Fee_Flee_to_GNU_Linux_.shtml
    Gemini version at gemini://gemini.techrights.org/n/2023/12/14/Microsoft_Windows_Security_Patches_for_a_Fee_Flee_to_GNU_Linux_.gmi

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ Microsoft Windows: Security Patches for a
Fee. Flee to GNU/Linux (Many Already Do).⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


posted by Roy Schestowitz on Dec 14, 2023,
updated Dec 14, 2023


  🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇Microsoft plans to charge for Windows 10 updates in the
                                   future⦈


   🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇Microsoft to offer consumers paid Windows 10 security
                         updates for the first time⦈


THE "market share" of Windows is diminishing in many countries as Android
becomes_increasingly_dominant because more people in Asia and Africa get
connected, typically not with a laptop or a desktop.


Those who do go online with a laptop or a desktop are still using Windows most
of the time (the OEMs don't give any other choice), but more people then
convert those laptops or desktops to GNU/Linux, based on this_month's_numbers,
which we've_sorted_in_ascending_order:


             🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇GNU/Linux on laptops or desktops⦈


The numbers on the left are Windows share and on the right it is GNU/Linux, not
including ChromeOS.


This represents a significant shift from Microsoft on over 90% of computers to
a lot less and it destroys Windows' business prospects - to the point where
Microsoft wants to start charging for Windows security patches. Windows_revenue
has_collapsed. █




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

                    ䷩ 𝚕𝚒𝚗𝚎 388

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

(ℹ) Images, hyperlinks and comments at https://news.techrights.org/n/2023/12/14/Over_at_Tux_Machines.shtml
    Gemini version at gemini://gemini.techrights.org/n/2023/12/14/Over_at_Tux_Machines.gmi

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ Over at Tux
Machines...⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


posted by Roy Schestowitz on Dec 14, 2023


⚓  Updated This Past Day⠀⇛


   1. ⚓ Some_New_Free_Software_Project_Mini-Overviews_at_Medevel⠀⇛


           some of the more relevant among them


   2. ⚓ Microsoft_People_Control_People_From_Within_('Secure'_Boot_and_BSOD)⠀⇛


           the same old tactic, taking over the rival


   3. ⚓ Debian_12.4_“Bookworm”_Released_with_94_Bug_Fixes_and_65_Security
      Updates⠀⇛


           The Debian Project announced today the release and general
           availability of Debian 12.4 as the third ISO update to the
           latest Debian GNU/Linux 12 “Bookworm” operating system series.


   4. ⚓ Linux_Mint_21.3_“Virginia”_–_BETA_Release⠀⇛


           This is the BETA release for Linux Mint 21.3 “Virginia”


   5. ⚓ LXD_now_re-licensed_and_under_a_CLA⠀⇛


           Per the commit message performing the re-licensing, all further
           contributions will be under the AGPLv3 license and all
           contributions from Canonical employees have been re-licensed to
           AGPLv3


   6. ⚓ Linux_Blamed_for_Everything_(Even_Unpatched_Systems,_Years-Old_CVEs)⠀⇛


           FUD tactics?


      ⚓  New⠀⇛


   7. ⚓ Increase_Reader_Engagement_with_ActivityPub_Plugin⠀⇛


           I’ve had a blog for almost two decades and in the last ten
           years I’ve noticed that the comment activity on my blog has
           decreased significantly


   8. ⚓ Programming_Leftovers⠀⇛


           R and more


   9. ⚓ today's_leftovers⠀⇛


           hardware focus


  10. ⚓ Security:_Outsourced_Computing,_OpenSSF,_Istio,_and_More⠀⇛


           Some security-related news


  11. ⚓ F39_Elections_Voting_is_now_Open!⠀⇛


           Links to candidate interviews


  12. ⚓ Carbonyl:_A_Chormium-Based_Browser_Made_for_Your_Terminal⠀⇛


           can also run through SSH


  13. ⚓ PostgreSQL_JDBC_42.7.1_Released⠀⇛


           The PostgreSQL pgJDBC project has a new release. This release
           deals with a few regressions


  14. ⚓ Debian:_Rescatux_Introduction_and_Freexian_Report⠀⇛


           a pair of Debian-related articles


  15. ⚓ GNOME:_Outreachy_Summer_of_Labour_(GSoC)⠀⇛


           GNOME and programs


  16. ⚓ Ardour_8.2_Open-Source_DAW_Adds_Note_Tupling,_Support_for_Solid_State
      Logic_UF8⠀⇛


           Ardour 8.2 open-source, free, and cross-platform DAW (Digital
           Audio Workstation) software is now available for download as a
           feature and bugfix release.


  17. ⚓ Audiocasts/Shows:_Linux_Matters_and_mintCast⠀⇛


           2 new episodes


  18. ⚓ today's_howtos⠀⇛


           many howtos for this evening


  19. ⚓ Windows_TCO_Leftovers⠀⇛


           Some Microsoft-induced issues


  20. ⚓ Android_Leftovers⠀⇛


           Desktop Chrome browsing will soon be the default on premium
           Android tablets


  21. ⚓ Maintenance_release:_Godot_4.2.1⠀⇛


           This first maintenance release fixes a number of platform
           compatibility issues introduced in Godot 4.2, which should make
           it much easier to upgrade from 4.1 or start new projects on all
           platforms


  22. ⚓ List_of_Free_Software_Licensed_Fonts_Alternatives_to_Proprietary_Ones⠀⇛


           This table is written as a part of our continuous efforts in
           helping people switching to Free Software


  23. ⚓ LibreOffice_24.2_Beta1_is_available_for_testingLibreOffice_24.2_Beta1
      is_available_for_testing⠀⇛


           LibreOffice 24.2 will be released as final at the beginning of
           February, 2024


  24. ⚓ Free,_Libre,_and_Open_Source_Software_Leftovers⠀⇛


           FOSS links and some BSD


  25. ⚓ Programming_Leftovers⠀⇛


           also some on education and licences


  26. ⚓ Linux_Devices_and_Open_Hardware:_Arduino,_Raspberry_Pi,_and_More⠀⇛


           half a dozen new stories


  27. ⚓ Web_Browsers/Web_Issues_and_Mozilla⠀⇛


           3 stories of relevance


  28. ⚓ Today_in_Techrights⠀⇛


           Some of the latest articles


  29. ⚓ Cost_of_Microsoft_Systems_(Windows_TCO)⠀⇛


           cautionary tales


  30. ⚓ Games:_DOOM,_Bundlefestive,_Steam,_and_More⠀⇛


           10 stories, mainly but not only from Liam Dawe


  31. ⚓ today's_howtos_and_technical_posts⠀⇛


           over a dozen for now


  32. ⚓ Android_Leftovers⠀⇛


           10 Emergency Alerts on Android devices within 50 minutes


  33. ⚓ Inkycal_v3_is_a_Raspberry_Pi-Powered_ePaper_Dashboard_for_Your_Desk⠀⇛


           Aceinnolab has made the software open source under the
           reciprocal GNU General Public License 3


  34. ⚓ LibreOffice_Alert:_Update_Now_to_Fix_Two_Critical_Vulnerabilities⠀⇛


           Update to the latest version of LibreOffice now (7.6.4 or
           7.5.9) to fix two critical vulnerabilities.


  35. ⚓ 25_Best_Text_Editors_for_Ubuntu_and_Other_Linux⠀⇛


           We review the best of Linux text editors available today with
           their features and installation guides.


  36. ⚓ Angie_1.4.0_Web_Server_Aligns_with_Nginx_1.25.3_Functionalities⠀⇛


           Angie 1.4.0 debuts with enhanced HTTP/3 support, MQTT handling,
           and refined server recovery, plus all Nginx 1.25.3 features


  37. ⚓ 14_Best_Free_and_Open_Source_Linux_System_Profilers⠀⇛


           For Linux, there are a number of good utilities that offer the
           same type of information


  38. ⚓ Kate_KF6_Status⠀⇛


           Thanks to the help of our contributors the current state of
           Kate for the upcoming first Qt & KF 6 release looks very
           promising


  39. ⚓ Security_Leftovers⠀⇛


           GNU/Linux patches and some new incidents


  40. ⚓ Linux_vger_mailing_list_migration_to_subspace.kernel.org⠀⇛


           mailing list moving


  41. ⚓ Geekbench_ML_Ported_to_GNU/Linux⠀⇛


           it now supports Linux


  42. ⚓ Linux_Mint_21.3_and_Zorin_17_are_beta_buddies⠀⇛


           Irish software projects gear up for latest iterations


  43. ⚓ 2024_and_the_Linux_Desktop:_Cosmic,_KDE_Plasma_6,_and_Linux_Mint_News⠀⇛


           Linux has been increasing its share of the desktop market, but
           2024 promises a number of developments that could propel it to
           its best year yet....


  44. ⚓ Microsoft:_Down_From_100%_to_7%_in_Just_Over_a_Decade_(Vietnam)⠀⇛


           the media barely ever mentions this


  45. ⚓ Lubuntu_24.04_LTS_Plans_for_Snap-Free_Minimal_Option⠀⇛


           Learn about the new minimal snap-free option in Lubuntu 24.04
           Noble Numbat with additional new features and updates.


  46. ⚓ Today_in_Techrights⠀⇛


           Some of the latest articles





                    ䷩ 𝚕𝚒𝚗𝚎 716

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

(ℹ) Images, hyperlinks and comments at https://news.techrights.org/n/2023/12/14/_Video_Lessons_to_be_Learned_From_Gandi_s_Death_Total_Webhostin.shtml
    Gemini version at gemini://gemini.techrights.org/n/2023/12/14/_Video_Lessons_to_be_Learned_From_Gandi_s_Death_Total_Webhostin.gmi

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ [Video] Lessons to be Learned From Gandi's
Death (Total Webhosting Solutions B.V. Shaking Down
Customers)⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


posted by Roy Schestowitz on Dec 14, 2023


Video_download_link | md5sum 9e444ed66e96e08087da1873c93f84b1
Lessons_From_the_End_of_Gandi
Creative Commons Attribution-No Derivative Works 4.0


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


THE previous video (and corresponding/accompanying article) mentioned companies
that_are_unable_to_make_money_but_still_exist.


3 days ago I saw this_article_about_Gandi, with the key point highlighted
below.


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇Before TWS bought Gandi, I paid about 11 EUR per year, for a
   .SE-domain. All domains also came with 3 free e-mail accounts, with 3 GB
storage for each account. That all changed when TWS took over. My expenses for
 my domain and my e-mail, went from 11 EUR per year, to about 75 EUR per year.
                      That’s a 681% (!) price increase.⦈


I didn't even know it had been sold. Maybe they were relatively quiet about
this, pretending it was business as usual. It clearly is not. As the above
person put it "a 681% (!) price increase" isn't normal. No matter which
"product" or "service"...


So we've decided to look into it and we've found this_self-congratulating_fluff
claiming that "Gandi, the leading French domain name registrar, web and email
hosting service, is joining forces with pan-European champion Total Webhosting
Solutions" ("joining forces" is misleading!)


As noted in the video, I'm quite a Web person, having dealt for over 25 years
with webhosts of various kinds (over 20 years with cPanel or VPS; I favour the
latter these days, and never clown computing, which tends to be a proprietary
trap vastly worse than cPanel). Gandi isn't entirely foreign to me, even if I
was never a client of Gandi (I did check their prices and considered them
several times since a decade ago).


People are gradually starting to realise the true cost of hosting on the Web.
There's no free hosting and no cheap (not for long anyway) hosting. The other
day this_high-profile_blogger_said_that_he_finally_realised_Microsoft_and
Google_do_not_offer_free_hosting (it took years for the coin to drop), so he's
migrating everything away from them. He said "the immediate motivation for the
move is that Google has stopped allowing updates to "classic" sites.google.com,
which I was using as my CDN for images. I figured that since I had to change my
setup anyway, I may as well do a complete migration."


Microsoft (GitHub) is mentioned there too. This year there were several waves
of GitHub layoffs, with entire teams culled and even a whole office shut down.
Microsoft_is_cost-cutting, so code and Web page hosting aren't free or won't be
free for much longer. Disregard the notion of "free hosting".


Going back to Gandi, what is the fate of existing (former) clients? Not good.
Prognosis negative.


How much will it cost to move? That too should be taken into account when
choosing a host. Don't fall into traps of "managed services", such as
WordPress.com. Automattic, the parent company or steward of WordPress(.org), is
trying to upload everything to "the clown" because it is modern [sic] and it is
billing people for all sorts of things while putting ads in their pages, even
against their will (unless/until they pay).


Not nice.


Remember: Nothing is free (gratis), especially when there are staff, hardware,
electricity, and network bills. Someone needs to pay those bills. It's either
temporarily "free" or you are the product.


Clown computing is the worst type of trap because after many companies, people
and governments outsourced to it we see the prices going up sharply... all the
time (never a price reduction, even when costs of hardware/storage/bandwidth go
down). The bills for power sure go up, but not as fast as the invoices for "the
clown" (this site's hosting will get 25% more expensive next year).


Going back to Gandi (again), in IRC we've learned that they deplatform easily
(even denying access to one's own data, after shutdown without prior warning).
Now they also rip off longtime clients, with nearly 700% price hikes.


So what will be left of Gandi's reputation? Why did Gandi sell? One person said
to us: "It was one of the good registrars going back to near the beginning. Now
it is vile."


The way we see it, the company operated at loss, then sold people to another
company (customers sacrificed), just like GitHub did (it sold developers and
projects to a very hostile Microsoft after running for about a decade without
ever making a profit, only humongous losses).


Companies like Spotify are in a similar position, as price hikes may mean
losing existing customers and thus not making up for the growing expenses. This
means layoffs and other cuts become imperative. Netflix, like ClownFlare, is
another example of this. ClownFlare is overall not profitable and one day
ClownFlare will be put under pressure (from investors) to pressure all users,
turning them into paying customers. How big would the backlash (exodus) be if
that happened?


Assuming that (like ClownFlare) Gandi operated at a loss, how big a loss? Well,
it's not easy to find this information for private companies (hard to check
financial history, except in some countries where suitable registries exist).


Remember that if it is too cheap to believe (staff gets salaries, there are
energy bills, legal bills and so on), then maybe the long-term plan is rather
hostile. Or maybe it's a long-term bear trap, waiting to shut very tightly on
your foot. Sometimes super-cheap is super-risky. Unless these firms live on
borrowed time or some bank loan (or money from shareholders/pension schemes),
they are "unofficially insolvent", not in perpetuity. I should know, as I
worked for such a company... always losing, almost every year for the past
decade or so. To the point of stealing money from the staff (pension fraud).


And Gandi?


Well, they suffered this_deathblow_in_2019. In their own words: "...we were
reinvesting everything that we earned, that we didn't have debts or take out
loans, etc. [...] and we have found a new investor in Montefiore Investment"
(opportunist, liquidator).


And with webhosts, what typically happens when there's a shutdown is, they pass
the servers to another owner, another brand. The original webhosts quietly
vanish while the vultures see how much they can "monetise" what's left of the
servers' tenants. They don't care about the customers; they have a plan to just
"monetise" or shake the tree for windfall, attempting to squeeze out some more
golden eggs from the goose.


Gandi won't be remembered kindly. People will remember what Montefiore
Investment did to it (and to Gandi's customers). █




⣿⣿⣇⣇⣤⣲⣶⣒⢖⡂⡖⡄⡖⡤⣒⠀⢸⠭⢰⢢⢲⣰⣲⢶⡤⢸⣾⣗⣤⣖⡄⢸⠭⢵⢲⢠⢺⡇⢠⢲⣼⠃⢰⠫⣥⣴⣦⡖⣆⡟⡇⡇⣿⣿⠿⣛⣛⣿⣿⣿⡏⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⢟⡇⠩⠭⠉⠉⡉⠉⠉⠉⣡⡅⣩⣩⣤⣬⣄⣬⣬⣭⣭⣭⣭⣴⣬⣭⣥⣭⠉⠁⠈⠉⠉⠈⠈⠉⠁⠈⠉⠈⡀⣤⡭⠉⢉⣉⢥⣥⣭⣥⡁⣩⣥⣬⢹⣿⣿⣿⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡏⢀⡀⠂⠀⠡⡀⢀⢢⠀⠀⠄⠀⠀⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣶⣶⢿⣿⠀⡆⢰⡂⠒⠂⢒⠶⡆⢶⣶⣾⣿⣿⣿⡿⠋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠂⠀⠀⡀⢂⠀⠰⠒⠀⠈⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⢈⠀⣠⣸⣃⣬⠃⠬⠗⠀⠀⠘⠂⢰⡜⠫⣰⣸⣼⣿⡨⠟⠟⠃⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠥⠀⡅⢤⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢂⢂⣶⣟⡄⡏⠘⣷⠼⡇⢀⠄⠝⠀⠙⠁⡛⢾⡟⠛⣛⣽⣯⢙⡃⠀⠀⡓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠉⡰⣟⠙⡫⢠⣁⣬⡉⣀⠀⠈⣀⠁⠀⢀⠻⠛⠺⠁⠈⣁⢈⣑⢊⠁⠀⠀⢉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⣾⣷⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢩⠤⢊⠃⠉⠀⣈⠈⡁⣈⣽⢛⡅⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⢈⣁⣉⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠠⠄⠸⢧⡎⠠⢯⠀⠾⢠⣝⣫⡉⢩⡅⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⢨⡁⢨⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⢻⠗⠿⠋⠀⠀⠀⠀⠀⠀⠶⠀⣀⠳⠶⢰⡄⣬⡀⠂⠀⡎⢩⡄⠠⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠭⠠⠍⠑⣀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠂⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⢤⡄⠀⢤⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣠⣄⣠⣀⣤⣄⣤⣀⣤⣀⣴⣠⣄⣀⣀⣀⣀⣈⣁⣚⣀⣀⣠⣢⣀⣀⣀⣀⣀⢀⡀⣀⢀⢀⡀⡀⢀⡀⡀⠀⢀⣀⠀⣀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢽⠫⡟⡯⢻⠓⡟⣽⢻⠚⡓⢘⠂⠛⠃⠚⠂⠃⠀⠐⠀⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡾⢳⠛⠓⣻⣛⠟⡳⢻⠞⠻⢚⠛⠚⠃⠘⠃⠛⠋⠓⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡯⣹⣻⣟⣯⣻⠙⡟⣻⣛⠈⡏⢘⠋⢘⡃⢛⠀⡛⠁⠋⠈⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢎⡗⣼⡻⣏⡿⣹⣘⣏⢢⣹⢈⣃⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣏⣯⣙⣉⣝⣋⣽⣻⣟⣝⠉⣉⢉⣉⢈⡁⢈⡡⣈⠀⢁⠀⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣯⢼⣽⢮⡇⣿⣭⣯⡷⣹⠠⣍⠀⣭⢀⡅⢨⡁⢨⡄⢈⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣺⣼⣣⣗⣼⣬⣧⣍⣬⣀⣧⣀⣔⣠⣄⢠⣄⢠⠀⢤⠀⢀⡀⡀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠯⣼⣼⣮⣫⣮⣵⣼⣧⣷⢲⢲⡗⣶⣶⣶⣶⣶⣶⣦⠄⢠⡀⠠⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣛⢾⠺⡷⣾⢾⣷⡦⢹⢟⡿⣞⠷⠟⠿⢛⠿⠛⠟⠳⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣧⣏⣦⣤⣤⣠⢦⣄⡤⡄⡤⡤⣤⠀⢸⠭⢠⢠⢤⣠⣦⢤⡀⢰⢺⡧⣤⣤⡄⢸⠭⢥⢤⢠⢼⡆⢠⢤⣰⠁⢰⠪⣇⣤⣦⡄⣄⡟⡇⡅⣿⣿⠿⣛⣛⣿⣿⣿⡏⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⢟⡅⠩⠭⠉⠉⡉⠉⠁⠉⣥⡅⣩⣩⣤⣬⣄⣬⣬⣽⣬⣭⣭⣼⣬⣭⣥⣭⠉⠁⠈⠉⠉⠈⠈⠉⠁⠈⠉⠈⡀⣤⡭⠉⢉⣉⣡⣥⣭⣥⡁⣭⣥⣬⢹⣿⣿⣿⣆⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡏⢀⡀⠂⠀⠡⡀⢀⢢⠀⠀⠄⠀⠀⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⣶⣴⢿⣿⠀⡂⢰⡂⠒⠂⢒⠶⡆⢶⣶⣾⣿⣿⣿⡿⠃⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠂⠀⠀⡀⢂⠀⠰⠒⠀⠈⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⢈⠀⣠⣸⡳⣬⢃⠬⠗⠀⠀⠘⠂⢰⡜⠯⣸⣸⣼⣿⣨⠟⠟⠃⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠥⠀⡅⢤⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢂⢂⣶⣿⡤⡟⢈⣷⠼⡇⢀⠄⠝⠀⠙⠁⡛⢾⡟⠛⣛⣽⣯⢙⡃⠀⠀⡓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⡰⣿⠙⡩⢠⣅⣬⣉⣀⠀⠈⣠⠁⠀⢀⠻⠟⠺⠀⠈⣉⢈⣑⢊⠁⠀⠀⢉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⣾⣷⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢨⠤⢊⡣⠅⠀⢘⠈⡁⣈⣿⢛⡅⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⢈⣉⣉⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠠⠄⠸⢣⣮⠠⢯⠀⠿⢠⣝⣫⢉⠩⡅⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⢨⡁⢨⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⢻⠗⠿⠋⠀⠀⠀⠀⠀⠀⠶⠀⣄⠳⠶⢰⡄⣬⡀⠂⠠⡌⢩⡄⠠⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠭⠠⠭⠑⣀⠀⡀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡀⠂⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⢤⡄⠀⢤⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣠⣄⣠⣄⣤⣄⣤⣀⣤⣀⣴⣠⣄⣀⣀⣀⣀⣈⣁⣚⣀⣀⣠⣢⣀⣀⣀⣀⣀⢀⡀⣀⢀⢀⡀⣀⢀⡀⡀⠀⢀⣀⠀⣀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢽⠫⡟⡯⢻⢓⡟⣽⢻⠚⡓⢘⠂⠛⠃⠚⠂⠃⠀⠐⠀⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡾⢳⠛⠓⣻⣛⠿⡳⢻⠞⠻⢚⠛⠚⠃⠘⠃⠛⠋⠓⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡯⣹⣻⣟⣯⣻⠙⡟⣻⣛⠈⡏⢘⠋⢘⡃⢛⠀⡛⠁⠋⠈⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢎⡗⣼⡻⣏⡿⣹⣘⣏⢪⣹⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣟⣯⣙⣉⣝⣋⣽⣻⣟⣝⠉⣉⢉⣉⢈⡁⢈⡡⣈⠀⢁⠀⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⢼⣽⢮⡇⣿⣭⣯⡷⣹⠠⣍⢀⣭⢀⡅⢨⡁⢨⡄⢈⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣺⣼⣣⣗⣼⣬⣧⣍⣬⣀⣧⣀⣔⣠⣄⣠⣄⢠⠀⢤⠀⢀⡀⡀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠯⣼⣼⣮⣫⣮⣵⣼⣧⣷⢲⢲⡗⣶⣶⣶⣶⣶⣶⣦⠄⢠⡀⠠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣛⢾⠺⡷⣾⢾⣷⡦⢹⢟⡿⣞⠷⠟⠿⢻⠿⠛⠟⠳⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡧⣶⣤⣤⣦⣴⣦⣄⡤⡔⡤⡤⣤⠀⢰⠶⢠⢤⢤⢠⢤⢤⣔⣰⣾⡦⣦⣤⡄⢰⠶⢦⢤⢠⢴⡆⢠⢤⣰⠅⢠⠒⣂⣤⣦⡤⡄⡿⡇⣇⣿⣿⠿⠛⠛⠿⣿⢿⡏⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⢛⠡⡓⢎⠛⡉⠫⠑⠁⠙⢡⣃⢯⣾⣤⣼⣤⣼⣬⣿⣼⣾⣽⣤⣽⣾⣧⣯⠓⠁⠘⠒⠚⠈⠈⠊⠁⠈⠛⡘⡀⣬⡵⠋⢙⣏⣣⣧⣵⡧⡃⢉⣡⣤⢹⣿⣿⣿⣷⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡄⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣶⣴⢿⡿⠁⡆⢰⠂⠐⠒⢒⠖⡆⢰⣶⣾⣿⣿⣿⡿⡯⠑⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠈⠀⢄⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⣠⡾⣏⣼⠁⠼⠗⠀⠀⠘⠂⢰⡜⠃⣴⢺⣦⣿⣿⠅⠘⠃⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⢂⢂⣖⣿⡔⡏⢘⣿⠼⡇⢀⠄⠜⠀⠙⡁⠛⢚⣿⠟⠋⢼⣯⢙⡇⠀⠀⡓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠷⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠉⡰⣿⠐⡫⢀⣁⣬⡉⣀⠁⠈⣠⠁⠀⢀⠻⠟⠰⠁⠊⠁⢘⣁⢊⡇⠀⠀⢉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⢩⠄⢊⠃⡍⠀⢘⠈⡁⣨⣽⢟⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⣉⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣾⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠠⡀⠘⢣⣦⠠⢮⠀⡾⢠⣝⣋⠉⢩⡅⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡁⢈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⢻⠷⠻⠋⠀⠀⠀⠀⠀⠀⠶⠀⣀⠲⠶⢴⡆⣼⡀⠂⠀⡌⠩⡄⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠩⠠⠉⠐⣀⠀⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡀⠂⢀⠀⠀⠀⠀⠀⠠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⢤⡄⠀⢤⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣠⣄⣠⣀⣤⣄⣤⣀⣤⣀⣤⣠⣄⣀⣀⣀⣀⣈⣁⣒⣀⣀⣠⣢⣀⣀⣀⣀⣀⢀⡀⣀⢀⢀⡀⡀⢀⡀⡀⠀⢀⡀⠀⣀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢽⠫⡟⡯⢻⠓⡟⣽⢻⠚⡓⢘⠂⠛⠃⠚⠂⠃⠀⠐⠀⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡾⢳⠚⠓⣻⣛⠟⡳⢻⠞⠻⢚⠛⠚⠃⠘⠃⠛⠊⠓⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡯⣹⣻⣟⣯⣻⠙⡟⣻⣛⠈⡏⢘⠋⠘⡃⢛⠀⡛⠁⠋⠈⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢎⡗⣼⡻⣏⡿⣹⣘⣏⢢⣹⢈⣃⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣏⣯⣙⣉⣝⣋⣽⣻⣟⣝⠉⣉⢉⣉⢈⡁⢈⡡⣈⠀⢁⠀⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣯⢼⣽⢮⡇⣿⣭⣯⡷⣹⠠⣍⠀⣭⢀⡅⢨⡁⢈⡄⢈⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣺⣼⣣⣗⣼⣬⣧⣍⣬⣀⣧⣀⣔⣠⣄⢠⣄⢠⠀⢤⠀⢀⡀⡀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠮⣼⣬⣮⣫⣮⣵⣼⣧⣷⢲⢶⡗⣶⣶⣶⣶⣶⣶⣦⠄⠠⡀⠠⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣛⢾⠺⡷⣾⢾⣷⡦⢹⠟⡿⡞⠷⠟⠿⢛⠿⠛⠟⠳⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣗⣆⣤⣤⣢⡤⣦⣄⡤⡔⡤⣤⣤⠀⢰⣒⢠⢤⢤⣠⢤⣤⣠⢴⢴⡦⣆⣤⡄⢰⣒⣢⢤⢠⢴⡆⢠⢤⣰⠆⢠⢒⣂⣤⣦⡤⡄⡯⡆⡆⣿⣿⠿⠛⠛⠿⣿⣿⡏⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⢛⡃⠓⠋⠒⠙⢚⠓⠋⠓⢁⣃⢧⣚⢀⣸⣤⣼⣬⣾⣼⣽⣼⣦⣼⣾⣧⣏⠓⠁⠘⠒⠚⠘⠈⠚⠁⠈⠚⡙⡀⢌⡓⠋⢛⢋⣧⣧⣵⣧⡃⢉⣥⣤⢹⣿⣿⣿⣧⣀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡃⠀⠄⠈⠀⠈⡀⠐⠀⠀⠀⠀⠀⠈⠷⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣶⣤⣿⣿⠀⡆⢰⡂⠐⠒⢒⠖⡀⢴⣶⣾⣿⣿⣿⡿⡯⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢁⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⣠⣴⣧⣼⠃⠼⠗⠀⠀⠘⠂⢰⡜⠃⣶⠺⣬⣿⣿⠕⠞⠃⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢁⠀⠀⢌⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢂⣂⣶⣿⠐⡏⠘⣿⠼⠇⢀⠄⠜⠀⠙⠁⠛⢺⣟⡟⠋⣼⣯⢙⡇⠀⠀⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⢉⡰⣛⠐⡫⢀⣅⣬⡉⣀⠀⠈⣠⠁⠀⢀⠻⠛⠲⠁⠊⢀⢘⣁⢊⠁⠀⠀⢉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠱⣾⣿⣶⡷⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⢩⠤⢊⠃⡍⠀⢘⠈⡁⣈⣽⢛⡅⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⣉⡀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠠⡀⠸⢣⣎⠠⢎⠀⢿⢠⣜⣋⢉⢩⡅⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡁⢈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣟⢻⠗⠻⠋⠀⠀⠀⠀⠀⠀⠶⠀⣀⠣⠶⢴⡆⣬⡀⠂⠠⡌⢩⡄⠀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠭⠠⠩⠐⣠⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡀⠂⢀⠀⠀⠀⠀⢀⠠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⢀⡀⢤⡄⠀⠤⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣠⣄⣠⣄⣤⣄⣤⣀⣤⣀⣤⣠⣄⣀⣀⣀⣀⣈⣉⣺⣀⣀⣠⣢⣀⣀⣀⣀⣀⢀⡀⣀⢀⢀⡀⡀⢀⡀⡀⠀⢀⣀⠀⣀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢽⠫⡟⡯⢻⠓⡟⣽⢻⠚⡓⢘⠂⠛⠃⠚⠂⠃⠀⠐⠀⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡾⢳⠚⠓⣻⣛⠟⡳⢻⠞⠻⢚⠛⠚⠃⠘⠃⠛⠊⠓⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡯⣹⣻⣟⣯⣻⠙⡟⣻⣛⠈⡏⢘⠋⠘⡃⢛⠀⡛⠁⠋⠈⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢎⡗⣼⡻⣏⡿⣹⣘⣏⢢⣹⢈⣃⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣏⣯⣙⣉⣝⣋⣽⣻⣟⣝⠉⣉⢉⣉⢈⡁⢈⡡⣈⠀⢁⠀⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣯⢼⣽⢮⡇⣿⣭⣯⡷⣹⠠⣍⠀⣭⢀⡅⢨⡁⢈⡄⢈⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣺⣼⣣⣗⣼⣬⣧⣍⣬⣀⣧⣀⣔⣠⣄⢠⣄⢠⠀⢤⠀⢀⡀⡀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠮⣼⣬⣮⣫⣮⣵⣼⣧⣷⢲⢶⡗⣶⣶⣶⣶⣶⣶⣦⠄⢠⡀⠠⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣛⢾⠺⡷⣾⢾⣷⡦⢹⠟⡿⡞⠷⠟⠿⢛⠿⠛⠟⠳⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡗⡆⣆⠤⣡⡴⡤⠄⡤⡔⡤⣄⡤⠀⢰⣒⢠⢄⢤⢠⢤⣤⡀⢰⢶⡦⣄⠤⡀⢰⣒⣢⢤⢀⢴⡆⢀⢤⣰⠆⢠⢖⣂⣤⣦⡤⡄⡿⡆⣆⢿⣿⠿⠛⠛⠿⣿⣿⡏⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⢟⡋⠓⠚⠒⠓⠛⠛⠋⠓⣃⡃⢓⣚⣀⣘⣀⣘⣜⣚⣛⣘⣙⣖⣘⣚⣁⣏⠒⠂⠘⠒⠚⠘⠘⠚⠃⠘⠚⠙⡀⣌⡒⠋⢛⣋⢣⣣⣷⣣⡃⡩⣥⣠⢹⣿⣿⣿⣧⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡁⠀⡂⠀⠀⢀⠄⢐⢰⠀⠀⠀⠀⠈⠳⢾⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣶⣶⢿⣿⠀⡂⢰⡆⠐⠒⢒⠶⡆⢴⣶⣾⣿⣿⣿⡿⡋⠕⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠂⠐⠀⠄⠁⡀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⣠⡴⣗⣼⠃⠬⠗⠀⠀⠘⠂⢰⡜⠃⣰⠹⠬⣿⣿⠅⠞⠃⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠤⠄⠄⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢆⣂⣶⣿⡔⡏⠘⣿⠼⠇⢀⠄⠝⠀⠙⠁⠛⢺⣟⡟⢉⣼⣯⢙⡇⠀⠀⡓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡴⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠉⠠⣛⠐⡋⢀⣉⣬⡉⣀⠀⠈⣠⠀⠀⢀⠿⠛⠲⠁⠊⠀⠘⣑⢊⠅⠀⠀⢉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⢩⠠⢈⡃⠉⠀⢈⠈⡁⣈⣽⢛⡅⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡁⢈⡁⣉⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠠⡄⠘⢡⣎⠠⢯⠀⣿⢠⣝⣉⡉⢩⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡁⢨⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢿⡗⠻⠋⠀⠀⠀⠀⠀⠀⠶⠀⣀⠣⠶⢶⡆⣸⡀⠂⠀⡎⢫⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠭⠠⠩⠑⣠⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠂⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⢤⡄⠀⠤⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣄⣠⣄⣠⣄⣤⣄⣤⣀⣤⣀⣴⣠⣄⣀⣀⣀⣀⣈⣋⣲⣀⣀⣠⣢⣀⣀⣀⣀⣀⢀⡀⣀⢀⢀⡀⡀⢀⡀⡀⠀⢀⣀⠀⣀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢽⠫⡟⡯⢻⠓⡟⣽⢻⠚⡓⢘⠂⠛⠃⠚⠂⠃⠀⠐⠀⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡾⢳⠛⠓⣻⣛⠟⡳⢻⠞⠻⢚⠛⠚⠃⠘⠃⠛⠋⠓⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡯⣹⣻⣟⣯⣻⠙⡟⣻⣛⠈⡏⢘⠋⢘⡃⢛⠀⡛⠁⠋⠈⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢎⡗⣼⡻⣏⡿⣹⣘⣏⢢⣹⢈⣃⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣏⣯⣙⣉⣝⣋⣽⣻⣟⣝⠉⣉⢉⣉⢈⡁⢈⡡⣈⠀⢁⠀⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣯⢼⣽⢮⡇⣿⣭⣯⡷⣹⠠⣍⠀⣭⢀⡅⢨⡁⢨⡄⢈⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣺⣼⣣⣗⣼⣬⣧⣍⣬⣀⣧⣀⣔⣠⣄⢠⣄⢠⠀⢤⠀⢀⡀⡀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠯⣼⣼⣮⣫⣮⣵⣼⣧⣷⢲⢶⡗⣶⣶⣶⣶⣶⣶⣦⠄⢠⡀⠠⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣛⢾⠺⡷⣾⢾⣷⡦⢹⢟⡿⣞⠷⠟⠿⢛⠿⠛⠟⠳⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣟⡏⣍⣭⣉⣭⣫⡍⡭⡙⣭⣉⣤⠀⢰⣒⢠⢄⣬⢬⣭⣭⡉⢩⢿⣯⣍⣤⡀⢰⣒⣢⣤⢀⢴⡆⢀⣤⢰⠆⢠⢔⣂⣠⣮⡤⡈⡻⡆⣆⣿⣿⡿⠟⢛⣻⣿⣿⡏⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⢿⣃⠓⠚⠛⠛⢛⢛⠋⠓⣃⡃⢓⣛⣀⣘⣀⣘⣘⣚⣙⣘⣙⣎⣘⣛⣁⣛⡛⠃⠘⠒⠚⠘⠘⠚⠃⠘⠚⠙⡀⣌⡒⠋⢛⣋⢣⣣⣷⣣⡃⣩⢉⣤⢹⣿⣿⣿⣷⣄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡈⠀⡐⣂⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠠⣶⣶⣿⡿⠀⡂⢰⠂⡒⡂⢒⠶⡆⢰⣶⣾⣿⣿⣿⡿⡯⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠂⠀⠈⠀⠄⠀⠐⠖⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣯⣿⣿⣿⣷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⣠⡰⣷⣼⠃⠬⠗⠀⠀⠘⠂⣰⡜⡏⣴⣫⣬⣿⣿⠕⠘⠃⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠧⠀⡁⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢂⢂⣶⣿⠐⡏⠘⣿⠼⡇⢀⠄⠽⠀⠙⠁⠛⢺⣿⠛⢛⣿⣯⢙⡃⠈⠀⡓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⡰⣟⠘⡋⢀⣁⣬⠉⣀⠁⠈⣠⠁⠀⢀⠻⠟⠺⠁⠈⢉⢘⡡⢊⡅⠀⠀⢉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢩⠤⢪⠃⡁⠀⢈⠌⡁⣈⣽⢟⡅⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡁⢨⢁⣉⡀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠠⡄⠸⢧⣦⠠⢦⠀⡿⢠⣝⣫⢉⢩⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⡁⢨⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣟⢻⠟⡿⠋⠁⠀⠀⠀⠀⠀⠶⢀⣀⠓⠖⢶⡆⣼⡀⠂⠀⡎⠫⡄⠠⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠠⠀⠭⠠⡉⠐⣀⠀⡀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⢠⡄⠀⠠⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣄⣤⣄⣤⣄⣤⣀⣤⣀⣴⣠⣄⣀⣀⣀⣀⣈⣁⣚⣀⣀⣠⣢⣀⣀⣀⣀⣀⢀⡀⣀⢀⢀⡀⣀⢀⡀⡀⠀⢀⣀⠀⣀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢽⠫⡟⡯⢻⢓⡟⣽⢻⠚⡓⢘⠂⠛⠃⠚⠂⠃⠀⠐⠀⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡾⢳⠛⠓⣻⣛⠿⡳⢻⠞⠻⢚⠛⠚⠃⠘⠃⠛⠋⠓⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡯⣹⣻⣟⣯⣻⠙⡟⣻⣛⠈⡏⢘⠋⢘⡃⢛⠀⡛⠁⠋⠈⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢎⡗⣼⡻⣏⡿⣹⣘⣏⢪⣹⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣟⣯⣙⣉⣝⣋⣽⣻⣟⣝⠉⣉⢉⣉⢈⡁⢈⡡⣈⠀⢁⠀⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⢼⣽⢮⡇⣿⣭⣯⡷⣹⠠⣍⢀⣭⢀⡅⢨⡁⢨⡄⢈⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣺⣼⣣⣗⣼⣬⣧⣍⣬⣀⣧⣀⣔⣠⣄⣠⣄⢠⠀⢤⠀⢀⡀⡀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠯⣼⣼⣮⣫⣮⣵⣼⣧⣷⢲⢶⡗⣶⣶⣶⣶⣶⣶⣦⠄⢠⡀⠠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣛⢾⠺⡷⣾⢾⣷⡦⢹⢟⡿⣞⠷⠟⠿⢻⠿⠛⠟⠳⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣝⣋⣉⣉⣉⣉⡩⣍⡭⣉⣉⣁⠈⢩⣉⢉⣉⣉⢉⣙⣉⠩⢩⣹⣫⣉⣀⡀⢠⣀⣀⣀⢀⣰⡆⢀⣀⢠⡂⢠⠌⠄⣠⣮⣀⣈⣋⡎⡂⣿⣿⠿⣛⣛⣿⣿⣿⡏⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⢟⢃⠣⠝⠛⠛⠙⠋⠃⠓⣃⡃⢓⣛⣀⣘⣀⣘⣘⣞⣹⣸⣙⣋⣘⣝⣃⣟⡋⠃⠘⠀⠘⠘⠘⠚⠃⠘⠚⠙⡀⣘⡂⠋⢋⢋⣧⣣⣧⣇⡃⣩⣥⣬⢹⣿⣿⣿⣇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡈⠀⠀⢀⠀⠈⠁⠀⠲⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⣶⣴⢿⣿⠀⡂⢰⡂⠐⡒⢒⠶⡖⢶⣶⣾⡿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠀⠄⠆⡀⠸⠒⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⢈⠀⣠⡴⡃⣼⠃⠨⡟⠀⠀⠘⠂⢠⡜⠻⣸⢨⣾⣿⣸⠗⠞⠁⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠦⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⠀⠄⠀⠀⠀⠀⠀⢂⣂⣶⣟⡐⡏⠘⣿⠾⡇⢀⠄⠜⠀⠹⠁⠛⢿⡛⠛⣋⣿⣯⢙⡃⠀⠀⡓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⢈⡴⢛⠔⡏⢄⣁⣬⡉⣈⠀⠈⣠⠀⠀⢀⠻⠏⠾⠁⠈⣉⢘⣡⢊⠁⠀⠀⢉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⢨⠠⢪⠃⠉⠀⠈⠌⡁⣈⣽⢛⡅⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡁⢈⣁⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠠⡄⠸⢡⣎⠠⢮⠀⠿⢠⣜⣋⠉⢉⡅⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢨⡁⢨⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣟⢻⡿⠿⠋⠀⠀⠀⠀⠀⠀⠶⠀⣀⠁⠖⢴⡆⣸⡀⠂⠠⡌⢩⡄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⠀⠭⠠⠉⠐⣠⠀⡄⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠂⢀⠀⠀⠀⠀⢀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⢠⡄⠀⠤⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣤⣄⣠⣀⣤⣄⣤⣀⣤⣀⣤⣠⣄⣀⣀⣀⣀⣈⣉⣒⣀⣀⣠⣢⣀⣀⣀⣀⣀⢀⡀⣀⢀⢀⡀⡀⢀⡀⡀⠀⢀⣀⠀⣀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢽⠫⡟⡯⢻⠓⡟⣽⢻⠚⡓⢘⠂⠛⠃⠚⠂⠃⠀⠐⠀⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡾⢳⠛⠓⣻⣛⠟⡳⢻⠞⠻⢚⠛⠚⠃⠘⠃⠛⠋⠓⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡯⣹⣻⣟⣯⣻⠙⡟⣻⣛⠈⡏⢘⠋⢘⡃⢛⠀⡛⠁⠋⠈⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢎⡗⣼⡻⣏⡿⣹⣘⣏⢢⣹⢈⣃⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣏⣯⣙⣉⣝⣋⣽⣻⣟⣝⠉⣉⢉⣉⢈⡁⢈⡡⣈⠀⢁⠀⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣯⢼⣽⢮⡇⣿⣭⣯⡷⣹⠠⣍⠀⣭⢀⡅⢨⡁⢨⡄⢈⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣺⣼⣣⣗⣼⣬⣧⣍⣬⣀⣧⣀⣔⣠⣄⢠⣄⢠⠀⢤⠀⢀⡀⡀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠯⣼⣼⣮⣫⣮⣵⣼⣧⣷⢲⢶⡗⣶⣶⣶⣶⣶⣶⣦⠄⢠⡀⠠⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣛⢾⠺⡷⣾⢾⣷⡦⢹⢟⡿⣞⠷⠟⠿⢛⠿⠛⠟⠳⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣟⣍⣋⣉⢍⣉⣉⡩⣩⡭⣉⣉⣈⠉⢩⢭⢩⣍⣹⣉⣹⣉⡿⢩⣯⣍⡉⣀⡀⢠⠤⢄⣀⢀⣠⡄⢀⣀⢠⡀⢀⠬⠅⣠⣮⣁⣉⣛⡍⡍⣿⡿⠿⣛⣛⣿⣿⣿⡏⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⢟⠋⠧⠝⠭⠹⠝⠫⠇⠧⢃⡇⢟⢽⣀⣸⣀⣸⣘⣼⣹⣺⣸⣗⣸⣿⣆⡟⠯⠃⠸⠤⠼⠸⠘⠼⠃⠘⠼⠻⡀⢜⡮⠏⢫⣏⣧⣧⣧⣧⡇⣩⣥⣭⢹⣿⣿⣷⡆⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡁⠀⠀⠀⠀⠁⠄⠀⢀⠀⠀⠀⠀⠈⣿⣿⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⣶⣶⢿⣿⠁⡆⢰⡂⠒⡒⢒⠶⠰⢶⣶⣾⡿⣿⣿⡿⠁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢈⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠀⣠⣼⡿⣼⢣⡼⡗⠘⠀⠘⠂⣰⣜⢻⣾⢼⣿⣯⣹⠷⠟⠃⠂⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠂⢂⣾⣟⡴⡟⢘⣿⠼⡇⢀⠄⠝⠀⢙⠁⠛⢿⡛⠿⣋⣿⣿⢛⠇⠁⠀⣙⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡀⠀⠠⠉⡠⣿⠔⡫⢀⣉⣬⣉⣈⠁⠈⣠⠁⠀⣀⠻⠛⠾⠀⢈⣉⢘⣱⢎⠀⠀⠀⢉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣮⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢩⠤⢚⠣⡉⢀⣈⠌⣁⣈⣿⢛⡅⢀⠀⡀⠀⡀⠀⠀⠀⠀⠀⡉⢈⣁⣉⡀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠠⡄⠸⣣⣮⠠⣧⠀⡷⢠⣝⣫⣍⢩⡅⢡⠀⠈⠀⠀⠀⠀⠀⠀⢀⡁⢨⡅⢨⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣟⣻⠟⠻⠋⠀⠀⠀⠀⠀⠀⠶⠀⣔⠁⠶⢴⡆⣼⡀⠂⠠⡌⢫⡅⢠⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡰⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠠⠀⠭⠠⡉⠐⣄⠀⡄⠠⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠂⢀⠀⠀⠀⠀⢁⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⢀⡀⢠⡄⠀⠠⠀⠀⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣤⣄⣤⣄⣤⣤⣤⣀⣤⣀⣤⣠⣄⣀⣀⣀⣀⣈⣉⣒⣀⣀⣠⣢⣀⣀⡀⣀⢀⢀⡀⣀⣀⠀⡀⣀⢀⡀⣀⣀⢀⡀⠀⣀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣫⡟⣯⢿⢻⡿⣽⢿⠞⡗⢸⠃⠛⠃⠺⠂⠓⠀⠐⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⢻⡓⣿⣟⢿⡳⣿⢞⠻⢞⠟⠚⠃⠘⠣⠛⠋⠛⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣻⣻⣟⣯⣻⠛⣟⣻⣻⠙⡏⢸⡋⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢞⡗⣽⡻⣏⡿⣻⣙⣟⢺⣹⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣻⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⢼⣽⢮⡧⣿⣿⣯⣷⣻⠠⣯⢂⣽⢉⡅⢨⡅⢨⡅⢨⠀⠌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣻⣼⣫⣟⣾⣿⣧⣏⣬⣄⣯⣀⣴⣠⣄⣨⣄⢠⡀⢤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣼⣼⣯⣿⣾⣵⣿⣿⣿⢳⢶⡗⣶⣶⣶⣶⣶⣶⣦⡄⠠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣟⢾⣺⣷⣾⣾⣷⡷⢹⢟⡿⣞⢷⠟⢿⢻⠿⠛⠟⠳⠀⠠⠄⠐⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡯⣝⡍⣩⢏⣉⣉⡉⣍⡍⣉⡉⣈⠉⢩⢭⢉⡉⣉⣍⣹⣉⢙⣩⣭⣍⠁⣀⠈⢠⠤⢄⣀⠀⣠⡄⠀⣀⢠⡄⢀⡬⠅⣠⣮⣁⡉⣛⡍⣍⣿⡿⢿⣛⣛⣿⣿⣿⡏⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡻⢛⠃⠧⠝⠭⠭⠷⠭⠆⠧⠇⠇⠯⠵⠀⠸⠍⠸⢸⢽⡸⠺⣹⠯⠹⠼⠇⠟⠭⠅⠸⠭⠽⠸⠸⠼⠇⠸⠼⠻⠀⠜⠬⠏⢯⢏⢧⣧⣧⣇⡇⣯⣥⣭⢹⣿⣿⣷⡆⠠⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡡⠀⠀⠀⠀⠀⡀⠈⠠⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⣶⣴⢿⣿⠀⡆⢲⡂⠐⡐⢒⠶⠰⢶⣶⣾⣿⣿⣿⡿⠁⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠀⠀⠄⠀⠒⠒⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠁⠀⠀⠀⢠⢈⠀⣠⣴⣿⣼⠣⠨⠗⠘⠀⠘⠂⣰⣜⠛⣶⢿⣿⣯⢙⠷⠟⠃⠂⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢂⣂⣾⣿⡔⡟⢘⣿⠼⡇⢀⠄⠝⠀⢙⠁⠛⢿⡛⠿⣫⣿⣿⢛⠇⠁⠀⡓⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡀⠀⠠⠈⡰⣿⠐⡯⢀⣉⣬⡉⣈⠁⠈⣠⠁⠀⢀⠻⠟⠾⠁⢚⣋⢘⣡⢎⠀⠀⠀⢉⡀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⢩⠠⢺⠃⣍⢀⣘⠈⡁⣈⣿⢛⡅⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⡉⢈⣉⣉⡀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠙⣿⣿⣿⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠠⡄⠸⣳⣮⠠⣮⠀⣿⢠⣝⣯⣍⢩⡅⢡⠀⠈⠀⠀⠀⠀⠀⠀⢀⡀⢨⡅⢬⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣟⣻⡗⠻⠋⠀⠀⠀⠀⠀⠀⠶⠀⣄⠠⡖⢾⡆⣼⡀⠂⠀⡌⢩⡄⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣥⠸⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠭⠠⣍⠑⣤⠀⡀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡀⠂⢈⠀⠀⠀⠀⢁⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡄⠀⠤⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣄⣤⣄⣤⣄⣄⣀⣄⣀⣤⣠⣄⣀⣀⣀⣀⣈⣉⣚⣀⣤⣠⣢⣀⣀⣀⣀⣀⢀⡀⣀⣀⠀⡀⣀⢀⡀⣀⣀⢀⡀⠀⣀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣫⡟⣯⢿⢻⡿⣽⢿⠞⡗⢸⠃⠛⠃⠺⠂⠓⠀⠐⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⢻⡓⣿⣟⢿⡳⣿⢞⠻⢞⠟⠚⠃⠘⠣⠛⠋⠛⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣻⣻⣟⣯⣻⠛⣟⣻⣻⠙⡏⢸⡋⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢞⡗⣽⡻⣏⡿⣻⣙⣟⢺⣹⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣻⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⢼⣽⢮⡧⣿⣿⣯⣷⣻⠠⣯⢂⣽⢉⡅⢨⡅⢨⡅⢨⠀⠌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣻⣼⣫⣟⣾⣿⣧⣏⣬⣄⣯⣀⣴⣠⣄⣨⣄⢠⡀⢤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣼⣼⣯⣿⣾⣵⣿⣿⣿⢳⢶⡗⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣟⢾⣺⣷⣾⣾⣷⡷⢹⢟⡿⣞⢷⠟⢿⢻⠿⠛⠟⠳⠄⠠⠄⠐⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣟⣟⡉⣉⣉⣉⢋⡋⣉⠋⣉⠙⣁⠉⢩⢯⢉⡛⣹⢟⣹⣉⡹⢩⣿⣍⠉⣁⠀⢠⠤⢄⣀⠀⣠⡄⠀⣀⢠⡄⢀⡬⠅⢀⣮⣁⡈⢛⡍⡅⣿⡿⢟⣋⣏⣽⣿⠿⡏⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⢟⠉⠧⠽⠶⠯⠶⠵⠆⠧⠇⠇⠯⠶⠀⠸⠉⠸⠸⠼⠻⠸⠹⠖⠸⠽⠇⠿⠷⠇⠸⠭⠽⠸⠸⠼⠇⠸⠼⠻⠀⠜⠬⠏⠿⢏⢧⣧⣧⡧⠇⣭⣥⣭⢹⣿⣿⣿⣇⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⢀⠀⠠⡀⠠⠀⠀⢶⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣶⣴⣿⡿⢀⡢⢰⡂⠐⡀⢒⠶⡆⢲⣶⣾⣿⣿⣿⣿⡁⠕⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⣠⣶⢟⣬⠃⠸⠗⠀⠀⠘⠂⢰⡜⣫⣴⠭⣿⣿⣨⠕⠟⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠃⠀⠀⢌⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⢂⣂⣖⣟⠔⡏⠘⣷⠼⡇⢀⠄⠜⠀⠙⠁⠛⢿⡟⠛⣯⣽⣯⢙⡇⠀⠀⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠈⡰⣿⠐⡩⢄⣁⣌⡉⣈⠁⠈⣠⠁⠀⠀⠛⠋⠶⠀⠀⣉⢈⣁⢊⠁⠀⠀⠁⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⡿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⢨⠤⢪⠃⡉⠀⢙⠌⡁⣈⣝⢛⡅⢀⡀⡀⠀⡀⠀⠀⠀⠀⠀⠁⢈⣁⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⣿⡿⠃⠀⠀⠀⠀⠀⠠⡀⠘⣡⣆⠀⢎⠀⡷⢠⣜⣉⠉⢩⠅⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢨⡁⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⢿⡗⠿⠋⠀⠀⠀⠀⠀⠀⠶⠀⣀⠰⠖⢴⡄⣼⡀⠂⠀⡌⢩⡀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⠀⠩⠠⠉⠐⣀⠀⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠂⢀⠀⠀⠀⠀⢀⠠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⢠⡄⠀⠤⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣤⣄⣠⣀⣤⣄⣤⣀⣤⣀⣤⣠⣄⣀⣀⣀⣀⣈⣁⣚⣀⣀⣠⣢⣀⣀⣀⣀⡀⢀⡀⣀⢀⠀⡀⡀⢀⡀⡀⠀⢀⡀⠀⣀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢽⠣⡟⡧⢻⠒⡟⣽⢻⠚⡓⠘⠂⠛⠃⠚⠂⠃⠀⠐⠀⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⡾⢳⠚⠓⣻⣛⠟⡳⢻⠜⠻⢚⠛⠚⠃⠘⠃⠛⠊⠓⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡯⣹⣻⣟⣯⣻⠙⡟⣺⣛⠈⡏⢘⠋⠘⡃⢛⠀⡛⠁⠋⠈⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢎⡗⣼⡻⣏⡿⣹⣘⣏⢢⣹⢈⣃⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣿⣏⣯⣙⣉⣝⣋⣽⣻⣟⣝⠉⣉⢈⣉⢈⡁⢈⡡⣈⠀⢁⠀⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣯⢼⣽⢬⡇⣻⣭⣯⡷⣹⠠⣍⠀⣭⢀⡅⢨⡁⢈⡄⠈⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣺⣼⣣⣗⣼⣬⣧⣍⣬⣀⣧⢀⣔⣠⣄⢠⣄⢠⠀⢤⠀⢀⡀⡀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠬⣼⣬⣬⣣⣮⣵⣼⣧⣷⢲⢶⡖⣶⣶⣶⣶⣶⣶⣦⠄⢀⡀⠠⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣛⢾⠺⡷⣾⢾⣶⡦⢹⠟⡿⡞⠷⠟⠿⢛⠿⠛⠟⠳⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⣿⡟⢛⢛⠛⠛⠛⡻⠻⠛⠋⠉⠋⢹⢿⠛⣛⣛⢛⣻⡛⣿⣟⢿⡽⠋⠀⠀⢠⠤⠄⠀⠀⢠⡄⠀⠀⢀⠄⢀⡨⡕⢠⣮⡁⠈⠛⡍⠍⣿⡿⢟⣛⣟⣿⣿⣿⡏⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⠿⠃⠧⠼⠿⠿⠾⠽⠇⠯⠇⠏⠯⠿⠀⠸⠉⠸⠸⠽⠺⠻⠹⠆⠸⠽⠍⠿⠿⠇⠸⠭⠽⠹⠸⠽⠇⠸⠽⠺⠀⠜⠤⠇⠿⢏⢯⣧⣧⡇⠇⣭⣠⣭⢹⣿⣿⣷⡄⣠⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠈⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠠⠠⣶⣴⣿⣿⢁⡆⢰⡆⠒⡒⢒⠶⠐⢶⣾⣾⣟⣿⣿⡿⠨⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠂⠀⠀⠀⠄⠀⠓⠂⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠈⠁⠀⠀⠀⢠⢈⠀⣠⡰⣿⣼⠣⠨⠗⠈⠀⠙⠂⣰⡜⢫⣴⢻⣿⣷⣽⢟⠇⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢦⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠄⠀⠀⠀⢂⢂⣾⣿⠰⣟⢘⣿⠴⡧⢀⠄⠝⠐⢙⡁⠛⢿⣛⢻⣯⣿⣿⢛⠃⠀⠀⣓⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠁⡰⣿⠙⡫⢄⣉⣬⡉⣀⠁⠈⣀⠁⠀⢀⠻⠋⠶⠀⢂⣍⢘⣁⠋⠀⠀⠀⢉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⢩⠠⢚⠃⣉⠀⣉⠈⣁⣨⣽⢛⡅⢀⠀⡀⠀⠀⠀⠀⠀⠁⠈⡉⢈⣁⣁⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠠⡄⠸⢧⣮⠠⣮⠀⣷⢠⣝⣏⢅⢩⡅⠡⠀⠉⠀⠀⠀⠀⠀⠀⢀⡅⢨⡅⢨⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠶⠀⣌⠁⠶⢶⡆⣼⡀⠂⠀⡍⢫⡄⠀⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠸⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠭⠠⠩⠑⣄⠀⡄⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠂⢀⠀⠀⠀⠀⢀⠀⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⢠⡄⠀⠤⠀⠀⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣄⣠⣄⣤⣄⣄⣀⣤⣀⣤⣠⣄⣀⣀⣀⣀⣈⣉⣺⣀⣀⣠⣢⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⡀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣫⡟⣯⢿⢻⡿⣽⢿⠞⡗⢸⠃⠛⠃⠺⠂⠓⠀⠐⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⢻⡗⣿⣟⢿⡳⣿⢾⠻⢞⠟⠚⠃⠘⠣⠛⠋⠛⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣻⣟⣯⣻⠛⣿⣻⣻⠙⡏⢸⡏⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢞⡗⣽⡻⣏⡿⣻⣙⣟⢺⣽⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣻⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⢼⣽⢮⡧⣿⣿⣯⣷⣻⠠⣯⢂⣿⢉⡅⢨⡅⢨⡅⢨⠀⠌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣻⣾⣫⣟⣾⣿⣷⣏⣬⣄⣯⣀⣴⣠⣄⣨⣄⢠⡀⢤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣼⣼⣯⣿⣾⣵⣿⣿⣿⢳⢶⡗⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣟⢾⣺⣷⣾⣿⣿⡷⢹⢟⡿⣞⢷⠟⢿⢻⠿⠛⠟⠳⠄⠠⠄⠐⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣟⡛⣛⠛⢛⡛⠛⠛⠻⠛⠛⠛⠉⠛⢛⣛⠛⠛⣻⠛⢛⡛⡛⢛⢿⡛⠛⠃⠀⢀⣀⡀⠀⠀⢀⡀⠀⠀⢀⠄⠀⣈⡑⢲⣾⡃⠙⠛⡟⡟⣿⣿⠿⣛⣟⣿⣿⣿⡏⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⣟⠋⣇⡸⣟⠿⡻⢟⡏⢏⠇⡏⡯⣻⠀⢸⠉⢸⠹⣹⢺⢹⢹⠤⠸⣹⠏⡿⣟⠇⢸⣉⣸⠹⠸⡹⠇⠸⣽⢺⠁⠼⢔⡇⢟⣏⣯⣧⣯⡧⡇⣭⣭⣬⢹⣿⣿⣿⣇⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠠⠈⠀⡀⠄⣨⠀⠈⢠⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣶⣶⢿⣿⢀⡦⢰⠂⠒⠂⢒⠶⡆⢰⣶⣾⣿⣻⣿⡿⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠂⠀⠈⠀⠄⠀⠰⠚⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⣠⡰⣫⣼⠃⠬⡗⠀⠀⠙⠀⢰⡜⠯⣴⡮⣾⣿⣼⠟⠟⠁⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠠⠥⠀⠔⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⢂⢂⣶⣿⠐⡏⢘⣷⠼⡇⢀⠄⠜⠀⠙⠁⠛⢿⠋⠛⣏⢼⣯⢙⡃⠀⠀⡓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⡰⣻⠀⡩⢀⣁⣄⡉⣈⠁⠈⣠⠁⠀⠀⠻⠛⠲⠀⢀⣑⢈⣁⢊⠁⠀⠀⢈⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⢩⠤⢋⠃⡉⠀⠈⠈⡁⣨⣽⢻⡅⢀⠀⡀⠀⠀⠀⠀⠀⠀⠈⡁⢈⣁⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⢿⡿⠿⠀⠀⠀⠀⠀⠀⠠⡄⠸⣣⣬⠀⢎⠀⡿⢠⣝⣩⠁⠩⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢨⡁⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠙⡿⢿⠛⠛⠈⠀⠀⠀⠀⠀⠀⠖⠀⣀⠃⠖⢴⡆⣸⡀⠂⠀⡍⢫⡄⠠⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠀⠀⠭⠠⠭⠐⢠⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡀⠂⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⢤⡄⠀⠤⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣠⣀⣤⣄⣄⣀⣄⣀⣤⣠⣄⣀⣀⣀⣀⣈⣉⣚⣀⣀⣠⣢⣀⣀⣀⣀⡀⢀⡀⣀⢀⣀⡀⡀⢀⡀⡀⠀⢀⡀⠀⣀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢽⠫⡟⡯⢻⠓⡟⣽⢻⠚⡓⢘⠂⠛⠃⠚⠂⠃⠀⠐⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡾⢳⠛⠓⣻⣛⠟⡳⢻⠞⠻⢞⠛⠚⠃⠘⠃⠛⠋⠓⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡯⣹⣻⣟⣯⣻⠙⡟⣻⣛⠈⡏⢘⠋⢘⡃⢛⠀⡛⠁⠋⠈⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢎⡗⣼⡻⣏⡿⣹⣘⣏⢢⣹⢈⣃⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣏⣯⣙⣉⣝⣋⣽⣻⣟⣝⠉⣉⢉⣉⢈⡁⢈⡡⣈⠀⢁⠀⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣯⢼⣽⢮⡇⣿⣭⣯⡷⣹⠠⣍⠀⣭⢀⡅⢨⡁⢨⡄⢈⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣺⣼⣣⣗⣼⣬⣧⣍⣬⣀⣧⣀⣔⣠⣄⢠⣄⢠⠀⢤⠀⢀⡀⡀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠯⣼⣼⣮⣫⣮⣵⣼⣧⣷⢲⢶⡗⣶⣶⣶⣶⣶⣶⣦⠄⢠⡀⠠⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣛⢾⠺⡷⣾⢾⣷⡦⢹⢟⡿⣞⠷⠟⠿⢛⠿⠛⠟⠳⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⣛⣟⠛⡛⢛⠛⠛⠛⠟⠛⠛⠙⠛⢛⣛⢻⡛⢛⠛⣻⡛⡻⢛⢛⡛⠛⠀⠀⢐⣂⡀⠀⠀⢀⡀⠀⠀⢀⡀⠀⣈⡑⢢⣾⡃⢛⠛⡛⣃⣿⠿⢛⣛⣛⡿⣿⣿⡏⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⠛⠉⣇⡹⣛⢚⣫⣛⡅⣏⡇⡏⣗⣫⠀⢸⠚⢹⢹⣹⢽⢹⢹⡆⢸⣻⡏⡶⣛⡂⢸⣒⣺⢹⢸⣹⡇⢰⣹⢾⠁⠼⣔⡆⣝⣏⣯⣧⣯⡇⡇⠩⣬⣭⣻⣿⣿⣧⡈⣢⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡂⠀⠀⠀⠀⣀⡄⢀⠠⠀⠀⠀⠀⠐⣶⣶⣶⣶⣶⣷⣾⣶⣶⣶⣶⣶⣶⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⣶⣶⢾⣷⢀⡦⢰⡂⠒⡒⢒⠆⠶⢶⣶⣾⣟⣿⣿⡟⠨⠙⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠅⠀⠰⠚⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠘⠁⠀⠀⠀⢠⢈⠀⡠⡰⣷⢼⠣⠸⠗⠈⠀⠙⠂⣰⡜⢫⡴⣿⣿⣯⣽⢿⠞⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢤⠀⠅⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠁⠀⠀⠀⠀⠀⢂⢂⣾⣿⡐⡟⢈⣿⢴⠇⠀⠄⢝⠀⠙⡁⣻⢟⣻⢻⣿⣾⣿⢛⠃⠀⠀⡓⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⠀⠀⠀⠁⡰⣿⠘⡩⢄⣉⣤⡉⣀⠁⠈⣀⠁⠀⠀⠻⠻⠦⡁⢈⣉⢘⣁⠏⠀⠀⠀⢉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢩⠠⢾⠣⡍⢀⣈⠈⡁⣈⣿⢛⡅⢀⡀⡀⠀⠈⠀⠀⠀⠑⠀⠋⢈⣍⣁⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⣿⣿⣿⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠠⡀⠸⢧⣬⠠⣎⠀⣷⢠⣝⣫⢅⢩⡅⢡⠀⠈⠀⠀⠀⠀⠀⠀⢀⡅⢩⡅⢨⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠻⣟⢻⠟⠻⠉⠀⠀⠀⠀⠀⠀⠶⠀⣄⠡⡶⢶⡆⣼⡀⠂⠀⡍⢩⡄⢀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠸⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠭⠠⠩⠐⢠⠀⡀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡀⠂⢈⠀⠀⠀⠀⢁⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⢤⡄⠀⠠⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣠⣄⣤⣄⣄⣀⣤⣀⣤⣠⣄⣀⣀⣀⣀⣈⣉⣺⣀⣀⣠⣢⣀⣀⣀⣀⣀⢀⡀⣀⣀⡀⡀⣀⢀⡀⡀⢀⢀⡀⠀⣀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣫⡟⣯⢿⢻⡿⣽⢿⠞⡗⢸⠃⠛⠃⠺⠂⠓⠀⠐⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⢻⡓⣿⣟⢿⡳⣿⢞⠻⢞⠟⠚⠃⠘⠣⠛⠋⠛⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣻⣻⣟⣯⣻⠛⣟⣻⣻⠙⡏⢸⡋⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢞⡗⣽⡻⣏⡿⣻⣙⣟⢺⣹⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣻⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⢼⣽⢮⡧⣿⣿⣯⣷⣻⠠⣯⢂⣽⢉⡅⢨⡅⢨⡅⢨⠀⠌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣻⣼⣫⣟⣾⣿⣧⣏⣬⣄⣯⣀⣴⣠⣄⣨⣄⢠⡀⢤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣼⣼⣯⣿⣾⣵⣿⣿⣿⢳⢶⡗⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣟⢾⣺⣷⣾⣾⣷⡷⢹⢟⡿⣞⢷⠟⢿⢻⠿⠛⠟⠳⠄⠠⠄⠐⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⣛⡛⠛⢛⡛⠛⠛⠛⠛⠛⠛⠉⠛⢛⣛⠛⠛⢛⠛⡛⡛⠛⢛⢛⡛⠛⠋⠁⢀⣀⡀⠀⠀⢀⡀⠀⠀⢀⡀⠀⣈⡑⢂⣾⡃⠛⠛⡛⡃⣿⡿⢛⣋⣯⣽⣿⢿⡏⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⢛⠋⣇⣰⣭⣛⣭⣫⡍⣏⡆⡏⣖⣫⠀⢸⠒⢸⢱⣹⢽⣹⢹⠾⢸⣹⣋⡶⣩⡂⢸⣒⣺⢹⢰⣹⡇⢰⣹⢾⠁⠸⣑⡆⣍⣏⣯⣧⣯⡇⡇⣭⢠⣭⢹⣿⣿⣿⣆⡀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⢰⣶⣶⣶⣴⣶⣶⣶⣷⣶⣶⣶⣶⣾⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣶⣶⢶⣷⠀⠦⢰⡂⠒⡐⢒⠶⠐⣲⣶⣾⣿⣿⣿⡿⠁⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡄⠀⠀⠀⠄⠀⠒⠆⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⠀⠀⢠⢈⠀⣠⡴⣿⣼⠣⠸⠗⠈⠀⠘⠂⢰⣜⠻⣴⣿⣾⣷⣾⢟⠞⠁⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠄⠀⠀⠀⠀⠀⢂⢂⢾⣟⠔⡗⢸⣿⠴⡇⢀⠄⠝⠀⠙⠁⠛⢿⡛⡻⣯⣼⣿⢙⡃⠁⠀⣓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠀⠀⠠⠈⡰⣿⠘⣩⢄⣁⣬⡉⣄⠁⠈⣠⠁⠀⣀⠿⠋⠶⠀⠀⠉⢈⣁⢏⠀⠀⠀⢉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⢩⠠⢛⠃⡉⢀⣈⠈⡉⣈⣿⢻⡅⢀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⢈⣁⣉⡀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⡿⠂⠀⠀⠀⠀⠀⠠⡄⠘⠧⣮⠠⣆⠀⡶⢠⣝⣫⠍⢩⡅⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⢨⡅⢨⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣟⢻⡿⠻⠛⠀⠀⠀⠀⠀⠀⠶⠀⣀⠃⡖⢲⡆⣼⡀⠂⠀⡭⢩⡄⠠⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⢸⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠭⠠⠏⠑⣀⠀⡀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡀⠂⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⢀⡀⢤⡄⠀⠤⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣄⣠⣄⣤⣄⣄⣀⣤⣀⣤⣠⣄⣀⣀⣀⣀⣈⣉⣒⣀⣄⣠⣢⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⡀⣀⢀⡀⣀⣀⢀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣫⡟⣯⢿⢻⡿⣽⢿⠟⡗⢘⠃⠛⠃⠺⠂⠓⠀⠐⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣷⢻⠓⣿⣟⢿⡳⣿⢞⠻⢞⠟⠚⠃⠘⠣⠛⠋⠛⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣯⣻⣻⣟⣯⣻⠛⣟⣻⣻⠙⡏⢸⠋⢘⡃⢛⠀⡛⠁⢋⠈⠃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢞⡗⣽⡻⣏⡿⣻⣙⣟⢺⣹⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣻⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⢼⣽⢮⡧⣿⣽⣯⣷⣹⠠⣯⢂⣽⢉⡅⢨⡁⢨⡅⢨⠀⠌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣻⣼⣫⣟⣾⣿⣧⣏⣬⣄⣯⣀⣴⣠⣄⣠⣄⢠⡀⢤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣼⣼⣯⣿⣾⣵⣿⣿⣿⢳⢶⡗⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣟⢾⣺⣷⣾⣾⣷⡷⢹⢟⡿⣞⢷⠟⢿⢻⠿⠛⠟⠳⠄⠠⠄⠐⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⣿⣿⠿⢿⡻⠿⠿⠿⠿⠿⠟⠉⠛⢛⣿⠿⠿⣿⠿⣿⡿⠿⠿⢿⡟⠟⠀⠀⢐⣀⡀⠀⠀⢀⡐⠀⠀⢀⡀⠀⢈⡑⢲⣾⡆⠚⠛⡛⡗⣿⣿⠿⣛⣟⣿⣿⣿⡏⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢿⠛⠃⣏⣹⣿⣗⣽⣫⡇⣏⡆⡏⣖⣿⠀⢸⠒⢰⢱⣻⢽⢿⢻⡴⢼⣹⡟⡶⣿⡆⢸⣒⣺⢻⢰⣻⡇⢰⣻⢼⠁⠼⣑⡇⣯⣟⣯⣦⣮⡧⡇⣭⣥⣬⢹⣿⣿⣿⡆⡀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠄⠠⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠢⣶⡴⢶⣷⠀⠆⢰⡂⠒⡐⢒⠶⠄⢲⣶⣾⣿⣿⣿⡿⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⠀⠠⣄⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠈⠁⠀⠀⠀⢠⢈⠀⣠⡺⣿⣼⠣⠸⠗⠈⠁⠙⠂⣰⣜⠻⣴⣿⣼⣿⣾⢟⠟⠋⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢂⣂⣶⣟⡐⡟⢘⣿⢴⡇⢀⠄⠝⠐⠹⠁⠛⢻⡟⠛⣫⣽⣿⢛⡇⠁⠀⣓⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠠⠀⡴⣿⠑⡫⢄⣉⣬⡉⣄⠁⠈⣠⠁⠀⡀⠻⠛⠺⠁⠀⢀⢜⣁⢏⠁⠀⠀⢉⡀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⢩⠠⢺⡃⢉⠀⣈⠈⡁⣈⣿⢛⡅⢀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠉⢈⣅⣉⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠠⡄⠘⢧⣮⠠⣯⠀⠿⢠⣝⣫⡍⢩⡅⢩⡀⠈⠀⠀⠀⠀⠀⠀⠀⡄⢨⡅⢨⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢻⠟⠋⠁⠀⠀⠀⠀⠀⠀⠶⠀⣄⠃⡖⢾⡆⣼⡀⠂⠠⡌⠩⡄⢀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠨⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠩⠠⠉⠑⣄⠀⡀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡀⠂⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠄⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⢤⡄⠀⠤⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣄⣤⣤⣄⣀⣤⣀⣤⣠⣄⣀⣀⣀⣀⣈⣉⣚⣀⣤⣠⣢⣀⣀⣀⣀⢀⢀⣀⣀⣀⢀⡀⡀⢀⡀⡀⢀⣀⡀⠀⣀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⢿⣫⡟⣯⢿⢻⡿⣽⢿⠟⡗⢸⠃⠛⠃⠺⠂⠓⠀⠐⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⢻⡗⣿⣟⢿⡳⣿⢾⠻⢞⠟⠚⠃⠘⠣⠛⠋⠛⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣻⣟⣯⣻⠛⣿⣻⣻⠙⡏⢸⡏⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢞⡗⣽⡻⣏⡿⣻⣙⣟⢺⣽⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣻⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⢼⣽⢮⡧⣿⣿⣯⣷⣻⠠⣯⢂⣿⢉⡅⢨⡅⢨⡅⢨⠀⠌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣻⣾⣫⣟⣾⣿⣷⣏⣬⣄⣯⣀⣴⣠⣄⣨⣄⢠⡀⢤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣼⣼⣯⣿⣾⣵⣿⣿⣿⢳⢶⡗⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣟⢾⣺⣷⣾⣿⣿⡷⢹⢟⡿⣞⢷⠟⢿⢻⠿⠛⠟⠳⠄⠠⠄⠐⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠿⣿⠿⣿⠿⠿⠿⠿⡿⠿⠿⠉⠿⠿⠿⠿⡿⠿⠿⣿⠿⠿⠿⢿⡿⠟⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠨⠑⢲⣾⡇⠘⠻⡿⡮⣿⡿⢿⣛⣛⣿⣿⣿⡏⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⠛⠃⡏⢱⣶⣶⢶⡳⡆⣖⡆⡖⣖⣶⠀⢸⠖⢰⢲⣺⢼⢻⢳⡄⢸⢺⡗⣶⣶⡆⢸⠒⢲⢲⢰⣺⡇⢰⣲⢼⠂⢼⡁⡅⣿⣏⣷⣦⣶⡇⡆⢭⣠⣭⢹⣿⣿⣷⡍⢠⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡜⠠⣀⣋⡁⢨⠀⠁⠊⠀⠈⠀⠀⠁⣑⣶⣾⣶⣾⣶⣷⣾⣾⣶⣶⣶⣷⡶⠀⠈⠀⠀⠉⠈⠀⠀⠁⠀⠠⠡⣾⡴⢾⡿⠀⠦⢰⡂⠒⠀⢒⠖⠐⣲⣶⣾⣿⣿⣿⡿⠈⠑⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢂⢀⠈⠀⠆⡀⡐⠒⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⢈⠀⣠⡼⣇⣬⠃⠨⠗⠀⠀⠘⠂⢠⡜⠻⣼⣭⣾⣿⣼⠟⠞⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠆⠀⠄⠌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢂⢂⣴⣟⡀⡇⠘⢿⠴⡗⠀⠄⠜⠀⠙⡁⠛⢿⡟⠛⣫⢼⣯⢙⡃⠀⠀⡚⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢿⣿⣿⣿⢿⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠁⠰⢛⠐⡫⢀⣁⣬⡉⣀⠁⠈⣀⠁⠀⠀⠻⠋⠾⡀⠀⠁⢘⣁⢊⠀⠀⠀⢉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢩⠠⢪⠃⡁⠀⢈⠈⡁⣈⣟⢛⡅⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡁⢨⣉⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣟⠿⠂⠀⠀⠀⠀⠀⠠⡄⠘⢣⣬⠠⢯⠀⡾⢠⣝⣋⠁⠩⡄⢨⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢨⡁⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢿⠗⠻⠋⠀⠀⠀⠀⠀⠀⠶⠀⣀⠃⠖⢰⡆⣸⠀⠂⠀⡌⠩⡄⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠭⠠⠉⠑⣀⠀⡀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠂⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⢤⡄⠀⠤⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣄⣠⣀⣤⣄⣄⣀⣤⣀⣤⣠⣄⣀⣀⣀⣀⣈⣉⣸⣀⣄⣠⣢⣀⣀⣀⣀⣀⢀⡀⣀⢀⣀⡀⡀⢀⡀⡀⠀⢀⣀⠀⣀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢽⠫⡟⡯⢻⠓⡟⣽⢻⠚⡓⢘⠂⠛⠃⠚⠂⠃⠀⠐⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⢳⠛⠓⣻⣛⠟⡳⢻⠞⠻⢞⠛⠚⠃⠘⠃⠛⠋⠓⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡯⣹⣻⣟⣯⣻⠙⡟⣻⣛⠈⡏⢘⠋⢘⡃⢛⠀⡛⠁⠋⠈⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢎⡗⣼⡻⣏⡿⣹⣘⣏⢢⣹⢈⣃⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣏⣯⣙⣉⣝⣋⣽⣻⣟⣝⠉⣉⢉⣉⢈⡁⢈⡡⣈⠀⢁⠀⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣯⢼⣽⢮⡇⣿⣭⣯⡷⣹⠠⣍⠀⣭⢀⡅⢨⡁⢨⡄⢈⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣺⣼⣣⣗⣼⣬⣧⣍⣬⣀⣧⣀⣔⣠⣄⢠⣄⢠⠀⢤⠀⢀⡀⡀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠯⣼⣼⣮⣫⣮⣵⣼⣧⣷⢲⢶⡗⣶⣶⣶⣶⣶⣶⣦⠄⢠⡀⠠⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣛⢾⠺⡷⣾⢾⣷⡦⢹⢟⡿⣞⠷⠟⠿⢛⠿⠛⠟⠳⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠿⢿⠿⠿⣿⠿⣿⡿⠿⢿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠱⢤⣾⡆⠼⠟⠿⠬⣿⡿⢟⣛⣛⡿⣿⣿⡏⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⢛⠁⡇⢡⠶⠴⢖⠷⡂⡖⡆⡖⡦⢶⠀⢸⠭⢰⣶⢺⣸⢺⢳⠀⢸⢺⠗⣦⠶⠄⢸⠭⢵⢲⢰⢺⡇⢰⢲⣼⠃⢰⡩⡅⡻⣏⣷⣦⣶⡇⡇⣍⣤⣭⢹⣿⣿⣷⡄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡁⠈⡀⠉⠁⢩⠈⢩⠉⠁⠉⠀⠁⠁⣭⣤⣬⣤⣬⣾⣿⣿⣮⣼⣿⣾⣿⡷⠁⠉⠁⠈⠉⠉⠈⠀⠉⠁⠠⠩⣬⣤⢴⣿⠁⡍⢡⠁⠁⡉⢁⠅⢐⢶⣶⣾⣟⣿⣿⡿⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠀⠀⠀⢄⠀⠐⠚⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢈⠀⣠⡴⣃⣼⠃⠨⠗⠀⠀⠘⠀⢰⡜⢋⣴⢹⣿⣿⣽⠟⠆⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠄⢘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢂⢂⣶⣟⠔⡇⢘⣿⠴⡇⠀⠄⠜⠀⠹⠁⠛⢿⢛⠟⣏⢿⣯⢙⠃⠀⠀⡓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⡰⣛⠑⡩⢀⣁⣬⡉⣈⠀⠀⣠⠁⠀⠀⠻⠟⠶⠉⢈⣉⢘⣁⢊⠀⠀⠀⢉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⢩⠠⢺⠃⠉⠀⠈⠈⡁⣨⣽⢛⡅⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⢈⣁⣉⡀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠠⡀⠘⢣⣄⠀⢮⠀⠷⢠⢝⣋⠁⠩⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢨⡁⢈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣟⢻⠟⠿⠋⠀⠀⠀⠀⠀⠀⠶⠀⣄⠁⠖⢴⡆⣭⡀⠂⠠⡍⠩⡄⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢰⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠭⠠⠉⠐⣀⠀⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡀⠂⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⢠⡄⠀⠤⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣥⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣄⣠⣀⣤⣄⣤⣀⣄⣀⣤⣠⣄⣀⣀⣀⣀⣈⣉⣒⣀⣤⣠⣢⣀⣀⣀⣀⣀⢀⡀⣀⢀⣀⡀⡀⢀⡀⡀⠀⢀⣀⠀⣀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢽⠫⡟⡯⢻⠓⡟⣽⢻⠚⡓⢘⠂⠛⠃⠚⠂⠃⠀⠐⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⢳⠚⠓⣻⣛⠟⡳⢻⠞⠻⢞⠛⠚⠃⠘⠃⠛⠊⠓⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡯⣹⣻⣟⣯⣻⠙⡟⣻⣛⠈⡏⢘⠋⠘⡃⢛⠀⡛⠁⠋⠈⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢎⡗⣼⡻⣏⡿⣹⣘⣏⢢⣹⢈⣃⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣏⣯⣙⣉⣝⣋⣽⣻⣟⣝⠉⣉⢉⣉⢈⡁⢈⡡⣈⠀⢁⠀⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣯⢼⣽⢮⡇⣿⣭⣯⡷⣹⠠⣍⠀⣭⢀⡅⢨⡁⢈⡄⢈⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣺⣼⣣⣗⣼⣬⣧⣍⣬⣀⣧⣀⣔⣠⣄⢠⣄⢠⠀⢤⠀⢀⡀⡀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠮⣼⣬⣮⣫⣮⣵⣼⣧⣷⢲⢲⡗⣶⣶⣶⣶⣶⣶⣦⠄⢠⡀⠠⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣛⢾⠺⡷⣾⢾⣷⡦⢹⠟⡿⡞⠷⠟⠿⢛⠿⠛⠟⠳⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠨⠿⠿⠿⠿⠿⢿⠿⢿⡿⠿⠿⠿⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠱⠤⣾⡆⠼⠿⠿⠬⣿⡿⢟⣛⣛⣿⣿⣿⡏⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⠟⠁⡇⢩⠶⠴⣒⠵⡂⡖⡆⡖⡦⢖⠀⢸⠭⢰⢣⢺⣸⢺⢲⡁⢸⢺⡗⣤⠶⠄⢸⠭⢵⢲⢠⢺⡇⢠⢲⣼⠃⢴⠫⡅⣛⣏⣷⣦⣶⡇⡇⣭⣤⣭⢹⣿⣿⣷⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠈⠀⠉⠅⠉⠍⠩⠩⠁⠉⠁⠁⠉⣩⣤⣬⣤⣬⣬⣭⣭⣬⣬⣥⣬⣯⡥⠁⠉⠁⠈⠉⠉⠈⠈⠉⠁⠨⠩⣬⣤⢴⡯⠁⡉⢡⠁⠁⡉⢁⠥⠀⢶⣶⣾⣟⣻⣿⡿⠁⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡃⠀⠈⠀⠤⠀⠐⠂⠀⠈⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⣠⡘⡃⣼⠃⠨⠗⠀⠀⠘⠂⢰⡜⠋⣴⢻⣿⣿⣭⠟⠞⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠸⠀⠁⢜⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢂⣶⣟⠐⡇⠘⣿⠴⡇⠀⠄⠝⠀⠙⠁⠛⢿⠛⠛⣏⢿⣯⢙⠃⠀⠀⡑⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠠⠀⠐⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠈⡠⣿⠑⡩⢀⣁⣬⡉⣀⠁⠀⣀⠁⠀⠀⠻⠛⠶⠀⢈⣉⢘⣁⢊⠀⠀⠀⢉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢨⠀⢮⠃⠉⠀⢈⠈⡁⣈⣽⢟⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⢈⣁⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠠⠀⠸⣥⣆⠠⢦⠀⡶⢠⢝⣫⡁⠩⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⢨⡁⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢻⠗⠿⠋⠀⠀⠀⠀⠀⠀⠶⠀⣀⠁⠖⢰⡆⣼⡐⠂⠠⠄⠩⡄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠭⠠⠋⠐⣠⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⢠⡄⠀⠤⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣄⣠⣀⣤⣄⣤⣀⣄⣀⣤⣠⣄⣀⣀⣀⣀⣈⣉⣚⣀⣄⣠⣢⣀⣀⣀⣀⢀⢀⡀⣀⢀⣀⡀⡀⢀⡀⡀⠀⢀⡀⠀⣀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢽⠣⡟⡧⢻⠒⡟⣽⢻⠚⡓⠘⠂⠛⠃⠚⠂⠃⠀⠐⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡾⢳⠘⠓⣻⣛⠟⡳⢻⠜⠻⢞⠛⠚⠃⠘⠃⠛⠊⠓⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡯⣹⣻⣟⣯⣻⠙⡟⣺⣛⠈⡏⢘⠋⠘⡃⢛⠀⡛⠁⠋⠈⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢎⡗⣼⡻⣏⡿⣹⣘⣏⢢⣹⢈⣃⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣷⣏⣯⣙⣉⣝⣋⣽⣻⣟⣝⠉⣉⢈⣉⢈⡁⢈⡡⣈⠀⢁⠀⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣯⢼⣽⢬⡇⣻⣭⣯⡷⣹⠠⣍⠀⣭⢀⡅⢨⡁⢈⡄⠈⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣺⣼⣣⣗⣼⣬⣧⣍⣬⣀⣧⢀⣔⣠⣄⢠⣄⢠⠀⢤⠀⢀⡀⡀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠬⣼⣬⣬⣣⣮⣵⣼⣧⣷⢲⢶⡖⣶⣶⣶⣶⣶⣶⣦⠄⢀⡀⠠⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣛⢾⠺⡷⣾⢾⣶⡦⢹⠟⡿⡞⠷⠟⠿⢛⠿⠛⠟⠳⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⢿⣿⢿⠿⡿⡿⡿⠿⠉⠿⠿⢿⠿⡿⡿⢿⣿⣿⠿⢿⡿⠿⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠑⢶⣾⡇⠾⠿⠿⠤⣿⡿⢟⣛⣛⣿⣿⣿⡏⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⠻⠃⡇⢠⣚⡥⣒⢕⡂⡖⡄⡖⡤⣒⠀⢸⠭⢰⢢⢺⣸⢺⢶⡀⢸⢾⡗⣤⣖⡄⢸⠭⢵⢲⢠⢺⡇⢠⢲⣸⠃⢴⠫⡇⣝⣍⣷⣤⣴⡇⡇⢩⣡⣭⢹⣿⣿⣿⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡃⠀⠀⠉⠉⢩⠉⠉⠩⠁⠉⠡⠁⠉⢩⣤⣬⣤⣬⣬⣭⣬⣬⣬⣥⣬⣭⡥⠁⠉⠁⠈⠉⠉⠈⠈⠉⠁⠨⠩⣬⣤⢼⡯⠁⡍⢡⠁⠁⠉⢁⠥⠀⢲⣶⣾⣿⣻⣿⡿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠃⢌⠀⠀⠂⠀⠐⠒⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⡠⡰⡫⣬⠃⠸⠗⠀⠀⠸⠀⢰⡜⠋⣴⠪⣿⣿⣬⠟⠞⠁⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢂⣢⣶⣟⠐⡃⠐⣷⠼⡇⢀⠄⠜⠀⠙⠁⠛⢿⡛⠻⣫⢽⣯⢙⠃⠀⠀⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⣟⠑⡫⢀⣁⣤⡉⣀⠀⠈⣠⠁⠀⠀⠻⠛⠶⠀⠈⠉⢈⣁⢊⠀⠀⠀⢉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢨⠠⢪⠃⡍⠀⠈⠌⡁⣈⣽⢛⠅⢀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⢈⣁⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠠⡄⠸⣡⣎⠠⢎⠀⡷⢠⣝⣋⠉⢩⡄⢈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⢨⡁⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⢻⠷⠿⠋⠀⠀⠀⠀⠀⠀⠶⠀⣀⠃⠖⢰⡆⣬⡀⠂⠀⡎⢩⡄⠠⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠩⠠⠩⠐⣀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡀⠂⢀⠀⠀⠀⠀⢀⠠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⢠⡄⠀⠤⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣄⣠⣀⣤⣄⣤⣀⣄⣀⣤⣠⣄⣀⣀⣀⣀⣈⣁⣒⣀⣀⣠⣢⣀⣀⣀⣀⣀⢀⡀⣀⢀⢀⡀⡀⢀⠀⡀⠀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢽⠣⡟⡧⢻⠒⡟⢽⢻⠚⡓⠘⠂⠛⠃⠚⠂⠃⠀⠐⠀⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡾⢳⠚⠓⣻⣋⠟⡳⢻⠜⠻⢞⠛⠚⠃⠘⠃⠛⠊⠓⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡯⣹⣻⣟⣧⣻⠙⡟⣺⣛⠈⡃⢘⠋⠘⡃⢛⠀⡛⠁⠋⠈⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢎⡗⣼⡻⣏⡟⣹⣘⣏⢢⣹⢈⣃⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣿⣏⣯⣙⣉⣝⣋⡵⣻⡗⣝⠉⣉⢈⣉⢈⡁⢈⡡⣈⠀⢁⠀⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣯⢼⣽⢬⡇⣹⣭⣯⡷⣹⠠⣍⠀⣭⢀⡅⢨⡁⢈⡄⠈⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣺⣼⣣⣗⣼⡬⣧⡍⣬⣀⣧⢀⣔⣠⣄⢠⣄⢠⠀⢤⠀⢀⡀⡀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠬⣼⣬⣬⣣⣮⣵⣼⣧⣷⢲⢶⡖⣶⣶⣶⣶⣶⣶⣦⠄⠀⡀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣛⢾⠺⡷⣾⢾⣶⡦⢹⠟⡿⡞⠷⠞⠿⢛⠿⠛⠟⠳⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠈⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠑⣦⣾⣧⢿⠿⢿⢯⣿⡿⢟⣛⣟⣿⣿⣿⡏⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⠛⠋⡇⢠⣤⣤⣠⢥⡄⡤⡄⡤⡤⣤⠀⢸⠭⢰⢬⣽⣨⢶⢥⡐⢱⢾⡧⣤⣤⡄⢸⠭⢥⢤⢠⢼⡇⢠⢤⣰⠁⢴⠊⡇⣍⣍⣥⣤⣤⡇⡅⢩⣤⣽⢹⣿⣿⣷⡆⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡁⠀⠀⠉⠉⠩⡍⢉⠩⠁⠉⠁⠁⠉⢩⣤⣬⣤⣬⣬⣭⣭⣬⣬⣥⣬⣭⡥⠉⠉⢁⠈⠉⠉⠈⠈⠉⠁⠨⠩⣬⡤⣼⣯⠁⡥⢡⡁⠁⡁⢁⠥⠰⢶⣶⣾⣿⣿⣿⣿⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡂⠀⠁⠠⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠁⠀⠀⠀⢠⢈⠀⣠⡾⡻⣼⡣⠨⠗⠈⠈⠘⠂⣰⣜⠻⣶⢿⣿⣯⣽⢟⠞⠓⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡠⠀⡀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⢂⢂⣶⣿⡴⡗⢘⣷⠼⡇⢀⠄⠜⠐⢹⡁⡛⢿⡛⢻⣫⣿⣿⢛⡃⠁⠀⣓⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠉⡠⣻⠐⡫⢀⣉⣬⡉⣈⠁⠈⣠⠁⠀⡀⠻⠛⠾⠁⠈⣉⢘⣁⢏⠁⠀⠀⢉⡀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⢩⡠⢾⠃⡉⢀⢉⠈⡁⣨⣿⢛⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⢈⣉⣉⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠠⡄⠸⣧⣶⠠⣇⠀⠿⢠⣝⣫⢍⢉⡅⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡀⢨⡅⢨⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⢻⠿⠻⠋⠀⠀⠀⠀⠀⠀⠶⠀⣄⠁⠖⢴⡆⣼⡐⠂⠀⡭⢫⡄⠠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠰⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⢩⠠⠉⠐⣀⠀⡀⠠⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡀⠂⢈⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⢀⡀⢤⡄⠀⠤⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣄⣠⣄⣤⣄⣤⣀⣤⣀⣤⣠⣄⣀⣀⣀⣀⣈⣉⣲⣀⣀⣠⣢⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⡀⣀⢀⡀⣀⣀⢀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣫⡟⣯⢿⢻⡿⣽⢿⠞⡗⢸⠃⠛⠃⠺⠂⠓⠀⠐⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⢻⡓⣿⣟⢿⡳⣿⢞⠻⢞⠟⠚⠃⠘⠣⠛⠋⠛⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣻⣻⣟⣯⣻⠛⣟⣻⣻⠙⡏⢸⡋⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢞⡗⣽⡻⣏⡿⣻⣙⣟⢺⣹⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣻⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⢼⣽⢮⡧⣿⣿⣯⣷⣻⠠⣯⢂⣽⢉⡅⢨⡅⢨⡅⢨⠀⠌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣻⣼⣫⣟⣾⣿⣧⣏⣬⣄⣯⣀⣴⣠⣄⣨⣄⢠⡀⢤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣼⣼⣯⣿⣾⣵⣿⣿⣿⢳⢶⡗⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣟⢾⣺⣷⣾⣾⣷⡷⢹⢟⡿⣞⢷⠟⢿⢻⠿⠛⠟⠳⠄⠠⠄⠐⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡑⣤⣾⣇⣾⢿⢿⢭⣿⡿⢟⣛⣛⣿⣿⣿⡏⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡻⠻⠃⡆⢉⣬⣡⡤⣥⡄⡤⡄⡤⣤⣤⠀⢰⣚⢠⢬⢼⣠⢬⢥⡀⢰⣴⡦⣄⣤⡄⢰⣒⣢⢤⢠⢴⡆⢠⢤⣰⠆⢤⢒⡃⣭⣍⣥⣤⣤⡆⡆⡩⣤⣭⢹⣿⣿⣷⡄⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠀⠓⠊⠒⠐⢚⠒⠃⠓⠁⠃⠃⣚⣄⣸⣄⣸⣉⣾⣼⣼⣘⣧⣼⣾⡃⠋⠒⠂⠘⠒⠚⠘⠈⠚⠃⠈⠺⣙⣀⣸⡷⠃⡓⢣⠃⠃⠓⢃⠃⠐⢶⣶⣿⣟⣻⣿⡿⠈⠑⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠂⢀⠀⠀⢅⠀⠐⠒⠁⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⣠⡸⢧⣼⠁⠨⠗⠀⠀⠘⠂⢰⡜⢋⣶⢹⣿⣷⣼⠟⠞⠃⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡁⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢂⢂⣶⣿⠔⡇⠘⢷⠴⡇⢀⠄⠝⠀⠙⡁⠛⢿⠛⠛⣋⢿⣯⢙⠃⠀⠀⡃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⡠⢿⠐⡩⢀⣁⣌⡉⣀⠀⠈⣠⠁⠀⢀⠻⠛⠶⠀⠈⣉⢈⣃⢊⠀⠀⠀⢁⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢼⣿⢿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⢨⠠⢊⠃⠉⠀⢉⠈⡁⢈⣽⢛⠅⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⢈⣁⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠠⠀⠸⠣⡆⠠⢦⠀⠿⢠⢜⣋⠉⢉⡄⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⢨⡁⢈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⢿⡗⠿⠋⠀⠀⠀⠀⠀⠀⠶⠀⣀⠳⠖⢰⡆⣸⡀⠂⠀⡌⢩⡄⠠⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠭⠠⠍⠐⣀⠀⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡀⠂⢀⠀⠀⠀⠀⢀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⢤⡄⠀⠤⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣄⣠⣀⣤⣄⣤⣀⣄⣀⣤⣠⣀⣀⣀⣀⣀⣈⣉⣚⣀⣀⣠⣢⣀⣀⣀⣀⣀⢀⡀⣀⢀⢀⠀⡀⢀⠀⡀⠀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢽⠣⡟⡧⢻⠒⡟⢽⢻⠚⡓⠘⠂⠛⠃⠚⠂⠃⠀⠐⠀⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡾⢳⠚⠓⣻⣋⠟⡳⢻⠜⠻⢞⠛⠚⠃⠘⠃⠛⠊⠓⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡯⣹⣻⣟⣧⣻⠙⡟⣺⣛⠈⡃⢘⠋⠘⡃⢛⠀⡛⠁⠋⠈⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢎⡗⣼⡻⣏⡟⣹⣘⣏⢢⣹⢈⣃⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣿⣏⣯⣙⣉⣝⣋⡵⣻⡗⣝⠉⣉⢈⣉⢈⡁⢈⡡⣈⠀⢁⠀⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣯⢼⣽⢬⡇⣹⣭⣯⡷⣹⠠⣍⠀⣭⢀⡅⢨⡁⢈⡄⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣺⣼⣣⣗⣼⡬⣧⡍⣬⣀⣧⢀⣔⣠⣄⢠⣄⢠⠀⢤⠀⢀⡀⡀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠬⣼⣬⣬⣣⣮⣵⣼⣧⣷⢲⢶⡖⣶⣶⣶⣶⣶⣶⣦⠄⠀⡀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣛⢾⠺⡷⣾⢾⣶⡦⢹⠟⡿⡞⠷⠞⠿⢛⠿⠛⠟⠳⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡱⣮⣾⣇⣿⣿⣿⣷⣿⡿⢿⣛⣟⣿⣿⣿⡍⢴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⢟⠉⡎⢉⣤⣠⡤⣥⠄⡤⡄⡤⣄⡤⠀⢰⣒⢠⢤⢽⢨⣤⢧⡐⢰⢴⡦⣄⣤⡀⢰⣒⣢⢤⢠⢴⡆⢀⢤⣰⠆⢤⢒⡂⢭⣍⣥⣤⣤⡆⡆⣩⢡⣭⢹⣿⣿⣿⣆⡀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡁⠀⠀⠓⠊⠒⠒⢚⠒⠃⠓⠃⠃⠓⠚⣀⣘⣀⣘⣘⣚⣘⣼⣘⣇⣸⣿⣇⠋⠒⠂⠘⠒⠚⠘⠘⠚⠃⠘⠚⣙⣀⣸⣗⠃⡓⢃⡃⠃⡓⢃⠃⡖⢲⣶⣾⣿⣿⣿⡿⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⠀⠑⠒⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠠⠀⠀⠁⠀⠀⠀⢠⠈⠀⣠⡸⡿⣼⠡⡨⠗⠈⠀⠘⠀⣰⡜⡻⣴⣿⣾⣿⣴⢟⠟⠑⠇⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⢂⢂⣾⣟⠔⡟⢘⣷⠼⠇⢀⠄⠝⠐⠙⠁⡛⢿⡟⠛⣏⣾⣿⢛⡃⠁⠀⣓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⠀⠀⠠⠉⡰⣿⠐⡫⢀⣉⣬⡩⣈⠀⠈⣠⠁⠀⡀⠻⠟⠲⠁⠀⣃⢘⣁⢏⠁⠀⠀⢉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢩⡠⢞⡃⣍⠀⣈⠈⡁⣨⣿⢟⡅⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡉⢈⣁⣉⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠠⡄⠘⢧⣤⠠⢦⠀⢷⢠⣝⣋⢅⢩⡅⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⢨⡅⢨⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢻⠓⠿⠋⠀⠀⠀⠀⠀⠀⠖⠀⣄⠡⠖⢴⡆⣼⡀⠂⠀⡍⢩⡄⠠⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⢠⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠭⠠⠉⠐⣀⠀⡄⠀⠄⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠂⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⢀⡀⢤⡄⠀⠤⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣄⣠⣄⣤⣄⣤⣀⣤⣀⣤⣠⣄⣀⣀⣀⣀⣈⣉⣺⣀⣄⣠⣢⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⡀⣀⢀⡀⣀⣀⢀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣫⡟⣯⢿⢻⡿⣽⢿⠞⡗⢸⠃⠛⠃⠺⠂⠓⠀⠐⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⢻⡓⣿⣟⢿⡳⣿⢞⠻⢞⠟⠚⠃⠘⠣⠛⠋⠛⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣻⣻⣟⣯⣻⠛⣟⣻⣻⠙⡏⢸⡋⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢞⡗⣽⡻⣏⡿⣻⣙⣟⢺⣹⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣻⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⢼⣽⢮⡧⣿⣿⣯⣷⣻⠠⣯⢂⣽⢉⡅⢨⡅⢨⡅⢨⠀⠌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣻⣼⣫⣟⣾⣿⣧⣏⣬⣄⣯⣀⣴⣠⣄⣨⣄⢠⡀⢤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣼⣼⣯⣿⣾⣵⣿⣿⣿⢳⢶⡗⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣟⢾⣺⣷⣾⣾⣷⡷⢹⢟⡿⣞⢷⠟⢿⢻⠿⠛⠟⠳⠄⠠⠄⠐⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡱⣴⣾⣇⢿⣿⢿⣧⣿⣿⠿⣛⣛⣻⣿⣿⡏⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠻⠟⠁⡆⢀⠤⣠⠤⡠⠄⡤⡀⡤⣀⡤⠀⢰⣒⢠⢄⢴⢠⢤⢤⢀⢰⢴⡦⣄⠤⡀⢰⣒⣢⢤⢀⢴⡆⢀⢤⣰⠆⢤⢖⡂⢨⣍⣥⣤⣤⡆⡆⣩⣥⣬⢹⣿⣿⣿⣇⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡁⠀⡀⠓⠚⠛⠒⠛⡚⠃⠓⠃⠃⠓⠛⣀⣘⣀⣘⣘⣚⣘⣝⣘⣀⣘⣾⣃⠋⠛⢃⠘⠒⠚⠘⠘⠚⠃⠘⠚⣙⣀⣸⣗⠓⡓⢃⡃⠃⡓⢃⠃⡆⢲⣶⣾⣿⣿⣿⡿⠁⠑⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡁⢀⠁⠀⠄⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⣠⡸⣿⣼⠡⠨⠗⠈⠀⠘⠂⢰⣜⡻⣴⣽⣼⣿⣼⢟⠟⠁⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢂⢃⡾⣿⠐⡇⠘⣷⠴⡇⠀⠄⠝⠀⠙⡁⡛⢿⡟⠛⣯⣿⣿⢙⡇⠁⠀⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⠀⠀⠀⠁⡠⣻⠐⡫⢀⣉⣤⡩⣀⠁⠈⣠⠁⠀⢀⠻⠛⠲⠀⢀⣃⢈⣁⢏⠀⠀⠀⢉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢨⡠⢞⠃⡉⢀⣈⠈⣁⣈⣿⢛⡅⢀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⢈⣉⣉⡀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠠⡄⠘⢡⣤⠠⣮⠀⣷⢠⣝⣩⢍⢩⡅⠠⠀⠉⠀⠀⠀⠀⠀⠀⢀⡁⢩⡁⢨⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠶⠀⣀⠀⠶⢶⡆⣼⡀⠂⠀⡭⢩⡄⢠⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠨⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠭⠠⠉⠐⣀⠀⡀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡀⠂⢀⠀⠀⠀⠀⢀⠀⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⢀⡀⢠⡄⠀⠠⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣄⣠⣄⣤⣄⣤⣀⣤⣀⣤⣠⣄⣀⣀⣀⣀⣈⣉⣺⣀⣀⣠⣢⣀⣀⣀⣀⣀⢀⡀⣀⣀⢀⡀⣀⢀⡀⣀⣀⢀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣫⡟⣯⢿⢻⡿⣽⢿⠞⡗⢘⠃⠛⠃⠺⠂⠓⠀⠐⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣷⢻⠓⣿⣟⢿⡳⣿⢞⠻⢞⠟⠚⠃⠘⠣⠛⠋⠛⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣯⣻⣻⣟⣯⣻⠛⣟⣻⣻⠙⡏⢸⠋⢘⡃⢛⠀⡛⠁⢋⠈⠃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢞⡗⣽⡻⣏⡿⣻⣙⣟⢺⣹⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣻⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⢼⣽⢮⡧⣿⣽⣯⣷⣹⠠⣯⢂⣽⢉⡅⢨⡁⢨⡅⢨⠀⠌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣻⣼⣫⣟⣾⣿⣧⣏⣬⣄⣯⣀⣴⣠⣄⣠⣄⢠⡀⢤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣼⣼⣯⣿⣾⣵⣿⣿⣿⢳⢶⡗⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣟⢾⣺⣷⣾⣾⣷⡷⢹⢟⡿⣞⢷⠟⢿⢻⠿⠛⠟⠳⠄⠠⠄⠐⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠑⣦⣾⣇⣼⢿⢿⣭⣿⣿⡿⣛⣛⣽⣿⢿⡏⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⠟⠋⡏⢉⣭⣉⢍⣩⡍⡡⡉⡩⣀⣤⠀⢰⣒⢉⢍⣭⣩⣭⣭⠛⢩⢽⣯⣁⣤⡀⢰⣒⣢⣠⢀⣴⡆⢀⣄⢰⠆⢤⢔⡇⡉⣉⣥⣤⣤⡆⡆⣩⣩⣤⢹⣿⣿⣿⣷⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡂⠀⠀⠓⠚⠛⡛⢛⠚⠃⠓⠃⠃⠓⠛⢀⣘⣀⣘⣘⣚⣙⣜⣘⣃⣘⣚⣡⠛⠛⠃⠘⠒⠚⠘⠚⠚⠃⠘⠚⣙⣀⢸⡗⠓⡛⢃⠃⠃⠓⢃⠃⡆⢴⡶⣾⣿⣿⣿⡿⡏⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠂⠀⠈⠀⠅⡀⠐⠖⠀⠀⠀⠀⠀⠀⠀⢑⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⣠⡐⠃⣬⠃⠨⠗⠀⠀⠙⠃⢰⡜⠋⣴⢽⣼⣿⣽⠉⠟⠃⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠄⠠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⢂⢂⣶⣿⠔⡇⢐⣿⠼⡇⠀⠄⠝⠀⠙⠁⠛⢺⡟⠛⣛⣼⣯⢙⡃⠈⠀⡃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣛⠁⡩⢀⣀⣌⡉⣀⠁⠈⣀⠁⠀⠀⠻⠟⠲⠀⠈⣉⢈⣑⢊⠃⠀⠀⢉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⢨⠠⢊⠃⡁⠀⢈⠈⡁⣀⣽⢛⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⢈⣁⣈⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠠⠀⠸⠡⡆⠠⢯⠀⠾⢠⢝⣫⡉⠉⠄⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⡁⢈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢿⠗⡿⠋⠀⠀⠀⠀⠀⠀⠶⠀⣄⠠⠖⢴⡄⣬⡀⠂⠠⡌⠩⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⡅⢠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠭⠠⠍⠐⣀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⢤⡄⠀⢠⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣴⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣄⣠⣄⣤⣄⣤⣀⣤⣀⣤⣠⣄⣀⣀⣀⣀⣈⣉⣒⣀⣀⣠⣢⣀⣀⣀⣀⣀⢀⡀⣀⢀⢀⡀⡀⢀⠀⡀⠀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢽⠣⡟⡧⢻⠓⡟⢽⢻⠚⡓⠘⠂⠛⠃⠚⠂⠃⠀⠐⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡾⢳⠚⠓⣻⣋⠟⡳⢻⠜⠻⢚⠛⠚⠃⠘⠃⠛⠊⠓⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡯⣹⣻⣟⣧⣻⠙⡟⣺⣛⠈⡃⢘⠋⠘⡃⢛⠀⡛⠁⠋⠈⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢎⡗⣼⡻⣏⡟⣹⣘⣏⢢⣹⢈⣃⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣿⣏⣯⣙⣉⣝⣋⡵⣻⡗⣝⠉⣉⢈⣉⢈⡁⢈⡡⣈⠀⢁⠀⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣯⢼⣽⢬⡇⣹⣭⣯⡷⣹⠠⣍⠀⣭⢀⡅⢨⡁⢈⡄⠈⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣺⣼⣣⣗⣼⡬⣧⡍⣬⣀⣧⢀⣔⣠⣄⢠⣄⢠⠀⢤⠀⢀⡀⡀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠬⣼⣬⣬⣣⣮⣵⣼⣧⣷⢲⢶⡖⣶⣶⣶⣶⣶⣶⣦⠄⠀⡀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣛⢾⠺⡷⣾⢾⣶⡦⢹⠟⡿⡞⠷⠞⠿⢛⠿⠛⠟⠳⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡱⣦⣾⣧⣿⢿⢿⣯⣿⣿⡿⢛⣛⣿⣿⣿⡏⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⢛⠋⡍⢉⣉⢉⣉⣉⡉⣉⡉⣉⣉⣉⠉⢩⠭⢉⣉⣉⣉⣩⣉⠩⢩⣩⣉⣉⣁⡀⢠⠤⢄⣀⢀⣠⡄⢀⣀⢠⡄⢄⠬⠍⣉⣉⣩⣬⣬⡭⡍⣭⣭⣬⢹⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠄⢀⠀⠧⠜⠭⠻⠕⠫⠇⠧⠃⠇⠇⠽⠀⣸⣉⣸⣘⣼⣹⣼⣸⣃⣸⣿⣇⠟⠯⠃⠸⠭⠼⠸⠘⠼⠃⠘⠼⢻⡀⠸⠯⠗⠯⢇⠇⠇⠧⢇⠇⡆⢰⣶⣾⣿⣿⣿⡿⡏⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⢈⠀⠀⠇⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⢈⠀⣠⡒⢿⣼⠣⠸⠗⠈⠀⠘⠀⢰⣜⠏⣴⣭⣼⣿⣿⢅⠞⠃⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠈⢂⢂⣾⣿⡐⡟⢘⣿⠼⡇⢀⠄⠜⠐⢙⠁⢛⢺⡟⠛⣻⣾⣿⢙⡇⠈⠀⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⠀⠀⠠⠈⡰⣟⠙⡫⢀⣉⣬⡉⣈⠁⠈⣀⠁⠀⠀⠻⠛⠲⠁⠀⢉⢜⣑⢎⠃⠀⠀⢉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢩⡤⢾⠃⡉⢀⣈⠈⡁⣈⣿⢛⡅⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⢈⣉⣉⡀⠀⠀⠀⠀⢀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠠⠄⠘⢣⡮⠠⣇⠀⡿⢠⢝⣫⣍⢩⡅⠡⠀⠉⠀⠀⠀⠀⠀⠀⠀⡄⢨⡁⢨⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⢻⠟⠛⠉⠀⠀⠀⠀⠀⠀⠶⠈⣀⠀⠶⢶⡆⣼⡐⠂⠀⡭⢫⡄⠠⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣅⠰⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠭⠠⠍⠐⣄⠀⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠂⢀⠀⠀⠀⠀⢁⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⢤⡄⠀⠠⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣠⣄⣤⣤⣤⣀⣤⣀⣤⣠⣄⣀⣀⣀⣀⣈⣉⣘⣀⣄⣠⣢⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⡀⣀⢀⡀⣀⣀⢀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣫⡟⣯⢿⢻⡿⣽⢿⠞⡗⢘⠃⠛⠃⠺⠂⠓⠀⠐⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣷⢻⠓⣿⣟⢿⡳⣿⢞⠻⢞⠟⠚⠃⠘⠣⠛⠋⠛⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣯⣻⣻⣟⣯⣻⠛⣟⣻⣻⠙⡏⢸⠋⢘⡃⢛⠀⡛⠁⢋⠈⠃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢞⡗⣽⡻⣏⡿⣻⣙⣟⢺⣹⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣻⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⢼⣽⢮⡧⣿⣽⣯⣷⣹⠠⣯⢂⣽⢉⡅⢨⡁⢨⡅⢨⠀⠌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣻⣼⣫⣟⣾⣿⣧⣏⣬⣄⣯⣀⣴⣠⣄⣠⣄⢠⡀⢤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣼⣼⣯⣿⣾⣵⣿⣿⣿⢳⢶⡗⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣟⢾⣺⣷⣾⣾⣷⡷⢹⢟⡿⣞⢷⠟⢿⢻⠿⠛⠟⠳⠄⠠⠄⠐⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡱⣦⣾⣅⢿⢿⢿⣮⣿⣿⡿⢛⣛⣿⣿⢿⡏⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⢟⠉⡍⠉⣉⢉⣉⣉⡉⣉⡉⣉⡉⣉⠉⢩⠭⢉⣉⣉⢍⣩⣉⠝⢩⣩⣍⡉⣁⠀⢠⠤⢄⣀⢀⣠⡄⢀⣀⢠⡄⢄⠬⠍⣉⣉⣩⣨⣬⡍⡍⣭⣭⣬⢹⣿⣿⣿⣷⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠀⠀⠧⠜⠭⠩⠵⠭⠆⠧⠃⠇⠏⠵⠀⠸⡉⣸⣸⠼⠸⣸⣸⢇⠸⣿⣧⡟⠭⠅⠸⠭⠽⠸⠸⠼⠇⠘⠼⠻⠀⠸⠯⠗⠮⢇⠇⠇⠧⠇⠇⡆⢰⡾⣾⣿⣿⣿⡿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠀⠀⠐⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠁⠀⠀⠀⢠⢈⠀⣠⡰⡻⣼⠣⠬⠗⠀⠁⠘⠂⢰⣜⢯⣴⣽⣼⣿⣽⢅⠞⠃⠚⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠤⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⢂⢂⣾⣟⡴⡟⢘⣿⢼⡇⠀⠄⠜⠐⠙⠁⢛⢺⡟⠛⣛⣿⣿⢛⡇⠙⠀⡑⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⢁⡰⣿⠁⡫⢀⣉⣌⡉⣈⠁⠈⣀⠁⠀⠀⠻⠟⠲⠁⠈⣉⢈⣑⢎⡇⠀⠀⢉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⢩⠠⢾⠃⡉⢀⣈⠈⡁⣀⣿⢟⡅⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⢈⣉⣍⡁⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠠⡀⠸⠣⣮⠠⣎⠀⡷⢠⣝⣋⣍⢩⡅⢁⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⢨⡅⢨⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣟⢿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠖⠀⣄⠒⠖⢶⡆⣼⡀⠂⠠⡭⠩⡄⠠⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⢠⣥⠠⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠭⠠⠯⠑⣤⠀⡄⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠂⢈⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⢤⡄⠀⠤⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣄⣠⣄⣤⣤⣤⣀⣤⣀⣤⣠⣄⣀⣀⣀⣀⣈⣉⣒⣀⣀⣠⣢⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⡀⣀⢀⡀⣀⣀⢀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣫⡟⣯⢿⢻⡿⣽⢿⠞⡗⢸⠃⠛⠃⠺⠂⠓⠀⠐⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⢻⡓⣿⣟⢿⡳⣿⢞⠻⢞⠟⠚⠃⠘⠣⠛⠋⠛⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣻⣻⣟⣯⣻⠛⣟⣻⣻⠙⡏⢸⡋⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢞⡗⣽⡻⣏⡿⣻⣙⣟⢺⣹⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣻⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⢼⣽⢮⡧⣿⣿⣯⣷⣻⠠⣯⢂⣽⢉⡅⢨⡅⢨⡅⢨⠀⠌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣻⣼⣫⣟⣾⣿⣧⣏⣬⣄⣯⣀⣴⣠⣄⣨⣄⢠⡀⢤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣼⣼⣯⣿⣾⣵⣿⣿⣿⢳⢶⡗⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣟⢾⣺⣷⣾⣾⣷⡷⢹⢟⡿⣞⢷⠟⢿⢻⠿⠛⠟⠳⠄⠠⠄⠐⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡱⣦⣾⣇⢼⢻⢿⢵⢿⣿⠿⣛⣛⣿⣿⢿⡏⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⠛⠉⡍⠉⣉⢉⣉⢉⡉⣉⠉⣉⠉⣉⠉⢩⠭⢉⡉⣉⢉⣉⣉⠍⢩⣩⣉⢉⣁⠀⢠⠤⢄⣀⠀⣠⡄⠀⣀⢠⡄⢀⡬⠍⣉⣉⣩⣠⣬⡍⡌⣭⣡⣬⢹⣿⣿⣿⣧⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠀⠧⠜⠭⠭⠶⠥⠆⠧⠇⠇⠯⠵⠀⠸⠉⠸⠸⠼⠸⢼⠸⠆⠸⠾⠇⠟⠭⠅⠸⠭⠽⠸⠸⠼⠇⠸⠼⠻⠀⠸⠯⠗⠦⠇⠇⠇⠧⠇⠇⡆⢴⣶⣾⣿⣿⣿⡿⡋⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠀⠀⠄⠀⠰⠒⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⣠⡴⢣⣼⠃⠸⠗⠀⠀⠘⠀⢰⡜⢋⣰⢽⣼⣿⣭⠍⠞⠃⠓⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠡⠀⠄⢠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠄⠀⠀⠀⠀⠀⢂⢂⣶⣟⡄⡇⢘⢷⠼⡇⠀⠄⠔⠀⠙⡁⠛⢻⡟⠛⣛⣾⣯⢙⡃⠀⠀⡃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⡰⢛⠐⡩⢀⣀⣌⡉⣀⠁⠈⣀⠁⠀⠀⠛⠛⠲⠁⠀⡁⢈⣑⢊⠃⠀⠀⢉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⢨⠠⢋⠃⡀⠀⢈⠈⠁⣈⣽⢛⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⢈⣉⣈⡀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠠⡀⠸⢡⣤⠠⢦⠀⠿⢠⢝⣋⡉⠉⡄⢀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡀⢨⡁⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡿⢿⠗⠻⠋⠀⠀⠀⠀⠀⠀⠖⠀⣀⠑⠖⢶⡆⣼⡀⠂⠠⡍⢩⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠩⠠⠩⠐⢀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡀⠂⢀⠀⠀⠀⠀⢀⠠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⢤⡄⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣄⣠⣄⣤⣀⣤⣀⣤⣀⣤⣠⣄⣀⣀⣀⣀⣈⣉⣚⣀⣀⣠⣢⣀⣀⣀⣀⣀⢀⡀⣀⢀⢀⡀⡀⢀⠀⡀⠀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢽⠣⡟⡧⢻⠓⡟⢽⢻⠚⡓⠘⠂⠛⠃⠚⠂⠃⠀⠐⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡾⢳⠚⠓⣻⣋⠟⡳⢻⠜⠻⢚⠛⠚⠃⠘⠃⠛⠊⠓⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡯⣹⣻⣟⣧⣻⠙⡟⣺⣛⠈⡃⢘⠋⠘⡃⢛⠀⡛⠁⠋⠈⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢎⡗⣼⡻⣏⡟⣹⣘⣏⢢⣹⢈⣃⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣿⣏⣯⣙⣉⣝⣋⡵⣻⡗⣝⠉⣉⢈⣉⢈⡁⢈⡡⣈⠀⢁⠀⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣯⢼⣽⢬⡇⣹⣭⣯⡷⣹⠠⣍⠀⣭⢀⡅⢨⡁⢈⡄⠈⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣺⣼⣣⣗⣼⡬⣧⡍⣬⣀⣧⢀⣔⣠⣄⢠⣄⢠⠀⢤⠀⢀⡀⡀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠬⣼⣬⣬⣣⣮⣵⣼⣧⣷⢲⢶⡖⣶⣶⣶⣶⣶⣶⣦⠄⠀⡀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣛⢾⠺⡷⣾⢾⣶⡦⢹⠟⡿⡞⠷⠞⠿⢛⠿⠛⠟⠳⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠱⣤⣾⣆⢼⢿⢿⣤⢿⣿⠿⣛⣛⣻⣿⣿⡋⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⡛⠋⡍⠉⣉⠉⣉⢉⡉⢉⠉⣉⠉⣉⠉⢩⠭⢉⡛⣛⣛⣛⣋⠛⢫⣫⣍⢙⣁⠀⢠⠤⢄⣀⠀⣠⡄⠀⣀⢠⡄⢄⡬⠅⢉⢙⣭⣠⣬⡍⡌⣭⣥⣬⢹⣿⣿⣿⣧⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡄⠀⠀⠧⠼⠾⠯⠶⠵⠆⠧⠇⠇⠯⠷⠀⠸⠉⠸⠸⠼⠺⠼⠹⠆⠸⠼⠏⠟⠷⠇⠸⠭⠽⠸⠸⠼⠇⠸⠼⠺⠀⠸⠮⠗⠷⢇⠇⠇⠧⠇⠇⡆⢶⣶⣾⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠈⠀⠀⠄⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⣠⡰⠣⣬⠃⠨⠗⠀⠀⠘⠂⢰⡜⢫⣰⢻⣼⣿⣹⠙⠟⠃⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠡⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⢂⢂⣶⣟⡐⡃⢀⡷⠼⠇⢀⠄⠜⠀⠙⠁⠛⢿⡟⠛⣟⣼⣯⢙⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⡠⣛⠁⡩⢀⣉⣌⡉⣀⠀⠈⣀⠁⠀⠀⠻⠛⠺⠁⠈⡁⢈⣑⢊⠃⠀⠀⢁⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⢩⠠⢺⠃⡍⠀⢈⠈⠁⣈⣝⢛⡄⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⢈⣁⣉⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠠⡄⠸⢣⡄⠀⢎⠀⡿⢠⣜⣋⠁⠉⠄⢁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢨⡁⢈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⢻⠗⠻⠋⠀⠀⠀⠀⠀⠀⠶⠀⣀⠀⠖⢴⡆⣼⡀⠂⠀⡌⢩⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠭⠠⠍⠐⣀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠂⢀⠀⠀⠀⠀⢀⠠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⢀⡀⢤⡄⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣾⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣠⣄⣤⣀⣤⣀⣤⣀⣴⣠⣄⣀⣀⣀⣀⣈⣉⣒⣀⣀⣠⣢⣀⣀⣀⣀⣀⢀⡀⣀⢀⢀⡀⡀⢀⠀⡀⠀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢽⠣⡟⡧⢻⠒⡟⢽⢻⠒⡓⠘⠂⠓⠃⠘⠂⠃⠀⠐⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠾⢳⠚⠓⣻⣋⠟⡳⢻⠜⠻⢚⠛⠚⠃⠘⠃⠛⠊⠒⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡯⣹⣻⣟⣧⣻⠙⡟⣺⣛⠈⡃⢘⠋⠘⡃⢛⠀⡛⠁⠋⠈⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢎⡗⣼⡻⣏⡟⣹⣘⣏⢢⣹⢈⣃⢚⡁⠈⡇⢈⠐⢉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣿⣏⣭⣉⣉⣝⣋⡵⣻⡗⣝⠉⣉⢈⣉⢈⡁⢈⡡⣈⠀⢁⠀⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣯⢼⣽⢬⡇⣹⣭⣯⡷⣹⠠⣍⠀⣭⢀⡅⢨⡁⢈⡄⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣺⣼⣣⣗⣼⡬⣧⡍⣬⣀⣧⢀⣔⣠⣄⢠⣄⢠⠀⢤⠀⢀⡀⡀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠬⣼⣬⣬⣣⣮⣵⣼⣧⣷⢲⢶⡖⣶⣶⣶⣶⣶⣶⣦⠄⠀⡀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣷⣛⢾⠪⡷⣾⢾⣶⡦⢹⠟⡿⡞⠷⠞⠿⢛⠿⠛⠟⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠑⣤⣾⣇⢼⢿⢿⢬⢿⣿⠿⣛⣛⣿⣿⣿⡏⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⢛⠛⡉⠛⠛⠋⠉⠋⠙⠋⠙⠉⠙⠁⠛⢙⣛⠛⠛⢛⠛⡛⠛⠻⢛⢻⡛⠛⠁⠀⢀⣀⡀⠀⠀⢀⡀⠀⠀⢀⠄⠄⣚⡃⠉⢛⣩⣤⣬⡍⡌⠩⣥⣬⢽⣿⣿⣿⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡁⠠⠀⣇⡸⣟⠾⡻⢟⡇⢏⠇⡏⡯⣻⠀⢸⠉⢸⠹⣹⢸⣽⢹⠆⠸⣻⠍⡿⣟⠇⢸⣉⣸⢹⠸⣹⠇⠸⣹⢺⠁⠸⢖⡶⣟⡇⡏⡇⢏⡇⡇⠀⢲⣶⣾⣿⣿⣿⡿⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡂⠈⠀⠀⡀⡀⠐⠂⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⣠⡸⠓⡴⠀⠨⠖⠀⠀⠘⠂⢰⡜⢋⣰⣩⣼⣿⣹⠝⠟⠃⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢥⠀⡁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⢂⢂⣶⣿⠐⡇⢀⢷⠼⡇⢀⠄⠜⠀⠹⠁⠛⢿⡟⠛⣛⣽⣯⢙⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢾⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⣛⠀⡫⢀⣁⣬⡉⣀⠀⠈⣀⠁⠀⠀⠛⠛⠾⠀⠈⢁⢘⢑⢊⠁⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢾⣿⡾⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⢨⠠⢚⠃⠉⠀⢈⠈⠁⣀⣽⢛⡄⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⢈⣁⣉⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⢿⣿⠿⠂⠀⠀⠀⠀⠀⠠⠀⠸⣠⡆⠠⢦⠀⣷⢠⣜⣩⠁⠉⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢨⡁⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢿⠗⠻⠋⠀⠀⠀⠀⠀⠀⠖⠀⣀⠁⠖⢴⡆⣸⡀⠂⠀⡌⢩⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⢠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠭⠠⠍⠐⣀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⠀⠂⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⢀⡀⢤⡀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣥⣤⣤⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣠⣄⣠⣄⣤⣀⣤⣀⣴⣠⣄⣀⣀⣀⣀⣈⣉⣒⣀⣀⣠⣢⣀⣀⣀⣀⣀⢀⡀⣀⢀⢀⡀⡀⢀⠀⡀⠀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢽⠣⡟⡧⢻⠒⡟⢽⢻⠒⡓⠘⠂⠓⠃⠘⠂⠃⠀⠐⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠾⢳⠚⠓⣻⣋⠟⡳⢻⠜⠻⢚⠛⠚⠃⠘⠃⠛⠊⠒⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡯⣹⣻⣟⣧⣻⠙⡟⣺⣛⠈⡃⢘⠋⠘⡃⢛⠀⡛⠁⠋⠈⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢎⡗⣼⡻⣏⡟⣹⣘⣏⢢⣹⢈⣃⢚⡁⠈⡇⢈⠐⢉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣿⣏⣭⣉⣉⣝⣋⡵⣻⡗⣝⠉⣉⢈⣉⢈⡁⢈⡡⣈⠀⢁⠀⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣯⢼⣽⢬⡇⣹⣭⣯⡷⣹⠠⣍⠀⣭⢀⡅⢨⡁⢈⡄⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣺⣼⣣⣗⣼⡬⣧⡍⣬⣀⣧⢀⣔⣠⣄⢠⣄⢠⠀⢤⠀⢀⡀⡀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠬⣼⣬⣬⣣⣮⣵⣼⣧⣷⢲⢶⡖⣶⣶⣶⣶⣶⣶⣦⠄⠀⡀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣷⣛⢾⠪⡷⣾⢾⣶⡦⢹⠟⡿⡞⠷⠞⠿⢛⠿⠛⠟⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡑⣧⣾⣇⢿⢿⢿⣧⣿⣿⠿⣛⣛⣿⣿⣿⡏⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⠛⠛⡋⠋⠉⠉⠙⠋⠙⠉⠙⠉⠙⠁⠛⢙⣋⠛⠛⠛⡛⠛⠛⠛⢛⢛⡻⠛⠁⠀⢀⣀⡀⠀⠀⢀⡀⠀⠀⢀⡀⠄⣚⡋⢉⢛⢭⣬⣭⡩⡅⣭⢡⣬⢹⣿⣿⣿⣧⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠨⠈⠀⣇⡰⣛⢚⣫⣛⡅⣏⠇⡏⡗⣻⠀⢸⠒⢸⠹⣹⢺⢽⢹⣆⢸⣹⡟⡷⣛⡂⢸⣒⣺⢹⠸⣹⡇⠸⣹⢾⠁⠸⣗⡶⣟⡇⡏⡇⣏⡇⡇⡆⢲⣶⣾⣿⣿⣿⡿⠃⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢈⠀⠂⠆⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⢈⠀⡠⡲⡷⣴⠢⠸⠖⠀⠀⠘⠂⢰⣔⢿⣰⣿⣼⣿⣼⢯⠟⠃⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠄⠀⠔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⢂⢂⣾⣟⠔⡟⢘⣷⠴⠇⢀⠄⠜⠐⠙⠁⡛⢿⡟⠛⣯⣼⣿⢙⡇⠁⠀⡑⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⡰⣻⠉⡯⢄⣁⣬⡉⣈⠁⠈⣀⠁⠀⡀⠻⠛⠾⠀⠀⣉⢈⣁⢎⠁⠀⠀⢉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢩⠠⢚⠃⠉⢀⣘⠌⣁⣈⣿⢛⡅⢀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠉⢈⣉⣉⡀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠿⠂⠀⠀⠀⠀⠀⠠⡄⠸⠣⣄⠠⣧⠀⣷⢠⣝⣫⢍⢩⡅⢠⠀⠉⠀⠀⠀⠀⠀⠀⠀⡀⢨⡁⢨⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢿⠟⠻⠋⠀⠀⠀⠀⠀⠀⠶⠀⣀⠁⠶⢶⡆⣼⡀⠂⠀⡬⢫⡄⢀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⣅⢠⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠭⠠⠉⠑⣠⠀⡄⠠⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠂⢀⠀⠀⠀⠀⢁⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⢀⡀⢤⡄⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣽⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣠⣠⣠⣄⣠⣤⣤⣀⣤⣀⣤⣠⣄⣀⣀⣀⣀⣈⣉⣒⣀⣀⣠⣢⣀⣀⣀⣀⣀⢀⡀⣀⣀⢀⡀⣀⢀⡀⣀⣀⢀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣫⡟⣯⢿⢻⡿⣽⢿⠞⡗⢘⠃⠛⠃⠺⠂⠓⠀⠐⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣷⢻⠓⣿⣟⢿⡳⣿⢞⠻⢞⠟⠚⠃⠘⠣⠛⠋⠛⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣯⣻⣻⣟⣯⣻⠛⣟⣻⣻⠙⡏⢸⠋⢘⡃⢛⠀⡛⠁⢋⠈⠃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢞⡗⣽⡻⣏⡿⣻⣙⣟⢺⣹⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣻⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⢼⣽⢮⡧⣿⣽⣯⣷⣹⠠⣯⢂⣽⢉⡅⢨⡁⢨⡅⢨⠀⠌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣻⣼⣫⣟⣾⣿⣧⣏⣬⣄⣯⣀⣴⣠⣄⣠⣄⢠⡀⢤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣼⣼⣯⣿⣾⣵⣿⣿⣿⢳⢶⡗⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣟⢾⣺⣷⣾⣾⣷⡷⢹⢟⡿⣞⢷⠟⢿⢻⠿⠛⠟⠳⠄⠠⠄⠐⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡱⣦⣾⣇⣿⢿⢿⣯⣿⣿⠿⣛⣛⣿⣿⣿⡏⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⠟⠛⡛⠛⠋⠉⠉⠛⠙⠉⠙⠉⠋⠙⠛⢛⣋⠛⠛⢛⠟⠛⠛⠛⢛⢛⡛⠛⠁⠀⢀⣀⡀⠀⠀⢀⡀⠀⠀⢀⡀⠀⣚⡃⢉⢛⢩⣠⣬⡉⡍⣭⣥⣬⢹⣿⣿⣿⣇⡀⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠀⣇⡰⣛⣚⣯⣚⡅⣏⡆⡏⣗⣫⠀⢸⠒⢸⢹⣹⣼⣽⢹⠄⢸⣽⡏⡶⣛⡂⢸⣒⣺⢹⢰⣹⡇⢰⣹⢾⠁⠸⣗⡶⣝⣇⡏⡇⣏⡇⡇⡆⢰⣾⣾⣿⣻⣿⡿⠁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢲⣶⣶⣾⣶⣾⣶⣾⣶⣷⣶⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⣠⡰⢦⣴⠠⠴⠖⠀⠀⠐⠀⢰⡔⡫⣸⣯⣼⣿⣸⢿⠟⠓⠓⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢂⢂⣾⣿⡴⡟⠘⣿⠼⡇⢀⠄⠜⠐⠙⠁⠛⢻⡟⠛⣟⣼⣿⢙⡇⠁⠀⡑⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠈⡰⣻⠀⡫⢄⣁⣬⣉⣀⠁⠈⣀⠁⠀⠀⠻⠛⠺⠁⠀⢉⢘⣁⢎⡁⠀⠀⢉⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⢩⠠⢚⡃⠉⠀⢉⠈⣁⣈⣿⢻⡅⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⢈⣉⣍⡃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠠⡄⠘⣣⣄⠠⣎⠐⡷⢠⢝⣣⢍⠩⡅⢠⡀⠉⠀⠀⠀⠀⠀⠀⠀⡄⢨⡁⢨⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢻⠟⠛⠋⠀⠀⠀⠀⠀⠀⠶⠠⣀⠂⠶⢶⡆⣼⡐⠂⠀⡭⠩⡄⠀⡄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⢠⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠭⠠⠉⠐⣀⠀⡄⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠄⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⢤⡄⠀⠠⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣬⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣠⣄⣠⣤⣤⣀⣤⣀⣴⣠⣄⣀⣀⣀⣀⣈⣉⣚⣀⣄⣠⣢⣀⣀⣀⣀⡀⢀⡀⣀⣀⣀⡀⡀⢀⡀⡀⢀⢀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣫⡟⣯⢿⢛⡿⣽⢿⠞⡗⢘⠃⠛⠃⠺⠂⠓⠀⠐⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣷⢻⠓⣿⣟⢿⡳⣿⢞⠻⢞⠟⠚⠃⠘⠣⠛⠋⠛⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣯⣹⣻⣟⣯⣻⠛⣟⣻⣻⠙⡏⢸⠋⢘⡃⢛⠀⡛⠁⢋⠈⠃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢞⡗⣽⡻⣏⡿⣻⣙⣟⢺⣹⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣻⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⢼⣽⢮⡧⣿⣽⣯⣷⣹⠠⣯⢂⣽⢉⡅⢨⡁⢨⡅⢨⠀⠌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣹⣯⣻⣼⣫⣟⣾⣿⣧⣏⣬⣄⣯⣀⣴⣠⣄⣠⣄⢠⡀⢤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣼⣼⣯⣿⣾⣵⣿⣿⣿⢳⢶⡗⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣟⢾⣺⣷⣾⣾⣷⡷⢹⢟⡿⣞⢷⠟⢿⢻⠿⠛⠟⠳⠄⠠⠄⠐⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡑⣤⣾⣇⢼⢿⢿⣬⣿⣿⡿⢛⣛⣹⣿⢿⡏⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⠛⠛⡋⠋⠛⠋⠙⠛⠛⠉⠛⠋⠙⠛⠛⢛⣛⠋⠛⠛⠛⠛⠛⠛⢛⢛⡛⠛⠁⠀⢀⣀⡀⠀⠀⢀⡀⠀⠀⢀⡀⠄⣚⡃⢉⢛⣩⣤⣭⡉⡌⢭⣩⣤⢹⣿⣿⣿⣷⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡣⠀⠀⣇⣰⣭⣞⣭⣪⡅⣏⡆⡏⣖⣭⠀⢸⠒⢸⢱⣹⢸⢹⣹⡄⢸⣹⣏⡶⣭⡆⢸⣒⣺⢹⢰⣹⡇⢰⣹⢾⠁⠸⣗⡶⣮⡆⡏⡆⣎⡇⡇⡆⢰⣶⠾⣿⣿⣿⡿⡯⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣷⡶⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣠⡴⠲⣴⠂⠤⠖⠀⠀⠐⠂⢰⡔⡋⣴⣭⣤⣿⣿⠉⠟⠃⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠥⠄⠁⠌⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢂⢂⣖⣟⠔⡃⠘⠻⠴⡇⢀⠄⠝⠀⠙⠁⠛⢺⡟⠛⣛⣿⣯⢙⡇⠀⠀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⣻⠐⡫⢀⣁⣬⡉⣀⠀⠈⣀⠁⠀⠀⠻⠛⠰⠁⠀⡁⢈⣁⢊⠁⠀⠀⢁⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢨⠀⢮⠃⡁⠀⢈⠈⡁⣈⣽⢻⡄⢀⠀⡀⠀⡀⠀⠀⠀⠀⠀⡀⠈⣁⣉⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⡿⠂⠀⠀⠀⠀⠀⠠⡄⠘⠣⡄⠠⢦⠀⡶⢠⣝⣫⠉⠩⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⡁⢈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⢿⠗⠻⠉⠀⠀⠀⠀⠀⠀⠶⠀⣀⠃⠖⢴⡆⣼⡀⠂⠀⡌⠩⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠭⠠⠩⠐⢀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⢤⡄⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣾⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣄⣠⣄⣠⣄⣤⣀⣤⣀⣤⣠⣄⣀⣀⣀⣀⣈⣁⣒⣀⣀⣠⣢⣀⣀⣀⣀⡀⢀⡀⣀⢀⢀⡀⡀⢀⠀⡀⠀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢽⠣⡟⠧⢻⠓⡟⢽⢻⠚⡓⠘⠂⠛⠃⠚⠂⠃⠀⠐⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡾⢳⠚⠓⣻⣋⠟⡳⢻⠜⠻⢚⠛⠚⠃⠘⠃⠛⠊⠓⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡏⣹⣻⣟⣧⣻⠙⡟⣺⣛⠈⡃⢘⠋⠘⡃⢛⠀⡛⠁⠋⠈⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢎⡗⣼⡻⣏⡟⣹⣘⣏⢢⣹⢈⣃⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣿⣏⣯⣙⣉⣝⣋⡵⣻⡗⣝⠉⣉⢈⣉⢈⡁⢈⡡⣈⠀⢁⠀⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢼⣯⢼⣽⢬⡇⣹⣭⣯⡷⣹⠠⣍⠀⣭⢀⡅⢨⡁⢈⡄⠈⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣺⣼⣣⣗⣼⡬⣧⡍⣬⣀⣧⢀⣔⣠⣄⢠⣄⢠⠀⢤⠀⢀⡀⡀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠬⣼⣬⣬⣣⣮⣵⣼⣧⣷⢲⢶⡖⣶⣶⣶⣶⣶⣶⣦⠄⠀⡀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣛⢾⠺⡷⣾⢾⣶⡦⢹⠟⡿⡞⠷⠞⠿⢛⠿⠛⠟⠳⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡑⣤⣾⣇⢼⢛⢿⣬⣿⣿⡿⢛⣋⣭⣿⣿⡏⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⢛⠛⠏⠋⠟⠋⠋⠉⠛⠉⠻⠛⠉⠹⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⠿⠿⠇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠄⠚⠋⢉⢛⢭⣤⣬⡭⡉⣭⣍⣥⢹⣿⣿⣿⣿⣆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠀⠀⡇⢰⣶⣶⢶⡶⡆⡖⡆⡖⣖⢶⠀⢸⠛⢰⢲⣲⢰⢲⢶⡄⢸⢺⡗⣶⡶⡆⢸⠛⢲⢲⢰⢺⡇⢰⣲⢼⠂⢸⡏⣥⡶⣆⡖⡆⡖⡇⡆⡶⢰⣖⠺⣿⣿⣿⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⢈⠀⠉⠡⠉⠐⠓⠈⠀⠈⠀⠀⠁⠉⠀⣼⣶⣾⣶⣿⣾⣾⣾⣶⣶⣿⣶⠁⠈⠀⠈⠉⠉⠀⠀⢡⠀⠀⣡⡼⣦⣴⠊⠤⠞⠁⠁⠑⠊⢱⡕⢿⢰⣴⣶⣿⣿⠅⠜⠃⠒⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠁⠁⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠂⢂⢶⣟⡔⡏⢘⠻⠴⡇⠀⠄⠕⠀⠙⠁⡛⢘⡛⠛⣋⢽⣯⢙⡇⠘⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣧⣿⣿⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⣟⠐⡫⢀⣁⣬⡉⣀⠀⠈⣀⠀⠀⠀⠻⠋⠀⠁⠀⠀⢈⣁⢊⠃⠀⠀⢁⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢿⣿⣷⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⢩⠠⢪⠃⠁⠀⠈⠈⡁⣀⣽⢛⡄⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⢈⣁⣈⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠠⡄⠘⣡⣄⠠⢮⠀⠶⢠⢝⣃⠉⠩⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢨⡁⢈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⢻⠟⠛⠋⠀⠀⠀⠀⠀⠀⠶⠀⣀⠁⠖⢰⡆⣬⠀⠂⠀⠄⢩⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠭⠠⠩⠐⣀⠀⡄⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡀⠂⢀⠀⠀⠀⠀⢀⠠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⢤⡄⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣾⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣠⣄⣤⣄⣤⣀⣤⣀⣤⣠⣄⣀⣀⣀⣀⣈⣉⣒⣀⣀⣠⣢⣀⣀⣀⣀⡀⢀⡀⣀⢀⣀⡀⡀⢀⡀⡀⠀⢀⡀⠀⣀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢽⠣⡟⡧⢻⠓⡟⣭⢻⠚⡓⠘⠂⠛⠃⠚⠂⠃⠀⠐⠀⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡾⢳⠚⠓⣻⣛⠟⡳⢻⠜⠻⢚⠛⠚⠃⠘⠃⠛⠊⠓⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡯⣹⣻⡟⣯⣺⠙⡟⣺⣛⠈⡏⢘⠋⠘⡃⢛⠀⡛⠁⠋⠈⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢎⡗⣼⡻⣎⡿⣹⣘⣏⢢⣹⢈⣃⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣿⣏⣯⣙⣉⣍⣋⣽⣻⣟⣝⠉⣉⢈⣉⢈⡁⢈⡡⣈⠀⢁⠀⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣯⠼⣽⢬⡇⣻⣭⣯⡷⣹⠠⣍⠀⣭⢀⡅⢨⡁⢈⡄⠈⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣺⣼⣣⣗⣼⣬⣧⣍⣬⣀⣧⢀⣔⣠⣄⢠⣄⢠⠀⢄⠀⢀⡀⡀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠬⣬⣬⣬⣣⣮⣵⣼⣧⣷⢲⢶⡖⣶⣶⣶⣶⣶⣶⣦⠄⢀⡀⠠⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣛⢾⠺⡷⣾⢾⣶⡦⢹⠟⡿⡞⠷⠟⠿⢛⠿⠛⠟⠳⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡑⣤⣾⣇⢼⢿⢿⣭⢿⣿⡿⢟⠛⣿⣿⣿⡏⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⠟⠏⠋⠛⠟⠋⠛⠉⠛⠉⠹⠋⠹⠯⠿⠿⠯⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠋⢉⢛⢩⣤⣬⣭⡉⣭⣭⣤⢹⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠀⠀⡇⢰⠶⠴⢖⡶⡆⡖⡆⡖⡖⢶⠀⢸⠭⢰⢲⢲⢰⢶⢲⣄⢸⢺⡗⣦⡶⡆⢸⠭⢱⢲⢰⢺⡇⢰⢲⣼⠃⢸⡯⣥⡶⡆⡖⡆⡖⡇⡇⡆⢰⣶⣾⣿⣿⣿⡿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡈⠈⠀⠉⠁⠉⠈⠁⠈⠁⠉⠀⠁⠁⠉⣠⣼⣶⣾⣶⣽⣾⣾⣾⣶⣼⣿⡇⠁⠉⠀⠈⠉⠉⠈⠀⢩⠀⠀⡩⡸⡆⣤⠉⠥⡟⠁⠁⠁⠉⢡⡕⡛⣰⢻⣼⣿⣹⠓⠞⠃⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠡⠁⠁⠄⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠂⢂⣶⣟⡄⡏⠘⣿⠴⠇⢀⠄⠕⠀⠹⠁⠛⢿⡟⠛⣋⢽⣯⢙⡃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⣻⠐⡫⢀⣀⣌⠉⣀⠀⠈⣀⠁⠀⠀⠻⠟⠾⠀⠈⣉⢘⣡⢊⠁⠀⠀⢁⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢨⠀⢊⠃⡉⠀⣈⠈⡁⣈⣽⢛⡅⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⢈⣁⣈⡀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⣿⡿⠿⠂⠀⠀⠀⠀⠀⠠⡄⠘⣢⡦⠠⢦⠀⡶⢠⢝⣋⠉⢉⠄⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⡀⢨⡁⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠻⠞⠛⠉⠀⠀⠀⠀⠀⠀⠶⠀⣀⠠⠖⢴⡆⣨⡀⠂⠀⡌⢩⠄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠭⠠⠍⠐⣀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⡄⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣥⣤⣤⣽⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣄⣠⣄⣤⣄⣤⣀⣤⣀⣤⣠⣄⣀⣀⣀⣀⣈⣁⣒⣀⣀⣀⣢⣀⣀⣀⣀⣀⢀⡀⣀⢀⢀⡀⡀⢀⠀⡀⠀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢽⠣⡟⡧⢻⠒⡟⢽⢻⠒⡓⠘⠂⠓⠃⠘⠂⠃⠀⠐⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠾⢳⠚⠓⣻⣋⠟⡳⢻⠜⠻⢚⠛⠚⠃⠘⠃⠛⠊⠒⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡯⣹⣻⣟⣧⣻⠙⡟⣺⣛⠈⡃⢘⠋⠘⡃⢛⠀⡛⠁⠋⠈⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢎⡗⣼⡻⣏⡟⣹⣘⣏⢢⣹⢈⣃⢚⡁⠈⡇⢈⠐⢉⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣿⣏⣭⣉⣉⣝⣋⡵⣻⡗⣝⠉⣉⢈⣉⢈⡁⢈⡡⣈⠀⢁⠀⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣯⢼⣽⢬⡇⣹⣭⣯⡷⣹⠠⣍⠀⣭⢀⡅⢨⡁⢈⡄⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣺⣼⣣⣗⣼⡬⣧⡍⣬⣀⣧⢀⣔⣠⣄⢠⣄⢠⠀⢤⠀⢀⡀⡀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠬⣼⣬⣬⣣⣮⣵⣼⣧⣷⢲⢶⡖⣶⣶⣶⣶⣶⣶⣦⠄⠀⡀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣷⣛⢾⠪⡷⣾⢾⣶⡦⢹⠟⡿⡞⠷⠞⠿⢛⠿⠛⠟⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡁⠄⡋⠍⠩⢙⠩⠀⠍⣋⢫⠉⢹⣏⠠⢘⠉⡏⡍⡏⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠙⡿⣛⠛⣻⡿⡛⢻⠽⠽⢿⢹⢭⢹⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⢿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⡿⣿⡿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣬⣧⣿⣤⣦⣸⣧⣼⣶⣮⣯⣤⣤⣼⣬⣤⣤⣶⣦⣤⣧⣬⣤⣤⣴⣦⣴⣾⣤⣼⣤⣥⣤⣤⣧⣴⣧⣧⣥⣴⣤⣤⣤⣬⣿⣷⣾⣤⣤⣼⣤⣼⣤⣤⣴⣼⣤⣶⣴⣤⣤⣤⣤⣿⣴⣧⣴⣾⣤⣤⣷⣇⣿⣿
⣿⣿⣿⣤⣦⣤⣧⣤⣧⣽⣢⣦⣥⣤⣰⣷⣤⣤⣤⣥⣷⣢⣿⣤⣤⣟⣀⣀⣙⣋⣁⣀⣐⣑⣃⣀⣈⣚⣛⣑⣘⣐⣐⣑⣃⣓⣙⣑⣷⣞⣧⣤⣵⣬⣤⣷⣶⣤⣿⣥⣤⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⢿⠿⠿⠿⡿⡿⠿⠿⢿⢿⢿⠿⡿⡿⢿⠿⠿⠿⣿⠿⡿⡿⠿⢿⢿⠿⢿⠿⡿⠿⠿⡿⡿⠿⢿⢿⢿⢿⠿⣿⢿⠿⡿⣿⢿⠿⢿⢿⢿⠿⠿⡿⡿⡿⢿⣿⣿⡿⣿⠿⠿⣿⠿⡿⡿⠿⠿⡿⢿⠿⠿⣿⣿⣿⣿
⣿⣿⣿⢼⣤⣧⣮⢿⣦⢤⢶⠼⢤⣵⣵⣤⣷⣴⣤⣧⣤⣿⣽⡧⣧⣾⣴⢷⣥⣼⣭⣮⣤⢥⣧⢤⣴⣷⡵⣼⣵⣯⡦⣷⣤⣴⣽⣼⣯⣤⠿⢵⣴⣭⢿⣦⣧⣤⣼⣯⣯⣮⠤⣤⣼⣴⢧⠿⣤⣤⣼⣴⣧⣤⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣼⣴⣿⣼⣵⣤⣬⣷⣼⣤⣷⣦⣤⣼⣤⣤⣼⣤⣤⣦⣧⣤⣼⣯⣧⣬⣤⣤⣽⣴⣧⣷⣤⣤⣴⣬⣦⣠⣧⣤⣴⣤⣼⣤⣤⣤⣧⣿⣦⣦⣾⣤⣴⣦⣤⣿⣥⣦⣼⣤⣼⣤⣤⣼⣤⣼⣦⣤⣿⣿⣿⣿
⣿⣿⣿⣤⣬⣬⣼⣾⣤⣼⣤⣼⣬⣼⣤⣇⣤⣷⣧⣴⣤⣤⣿⣤⣬⣦⣤⣴⣴⣷⣤⣤⣿⣴⣿⣯⣼⣮⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⢿⢿⢿⠿⠿⠿⢿⡿⣿⢿⡿⢿⠿⠿⢿⢿⠿⢿⠿⢿⢿⢿⣿⠿⢿⠿⢿⢿⢿⠿⠿⣿⠿⢿⡿⢿⠿⢿⢿⠿⠿⠿⢿⣿⠿⣿⠿⡿⢿⢿⡿⢿⡿⡿⡿⢿⣿⠿⣿⣿⠿⠿⡿⡿⠿⠿⢿⠿⠿⢿⣿⣿⣿⣿⣿
⣿⣿⣿⢼⣴⡴⣥⢤⣤⣤⣧⡷⣭⣤⣴⣥⣤⣤⣿⣴⡽⣥⡼⣷⣽⣽⣤⣤⣯⣤⣿⣵⣤⣥⣮⠤⣴⣯⣼⣬⣷⢼⣥⠤⣬⣵⣽⣤⣽⣤⣥⣽⣼⣤⣴⣯⣦⣮⣼⣽⣤⣼⢿⣤⢥⣮⡷⢽⣤⡼⣤⣤⡤⣿⣿⣿⣿⣿
⣿⣿⣿⡤⣤⣤⣼⣮⣷⣧⣤⡤⣧⣴⣦⣧⣭⡤⢦⣤⣤⣤⣤⣧⣦⠴⣤⣤⣼⣷⣤⠤⣆⣴⣤⣷⣽⢤⣆⣥⡤⣤⣥⢤⣤⣦⣼⣤⣦⡼⢦⣠⣤⡦⣧⢤⣥⣦⣤⣇⣤⣤⢤⢤⣤⢬⣧⣤⠴⣯⣵⢤⢤⣴⣦⣼⣿⣿
⣿⣿⣿⣬⣤⣤⣴⣦⣬⣦⣼⣯⣥⣥⣴⣦⣌⣧⣼⣤⣤⣤⣤⣼⣼⣴⣦⣵⣦⣤⣼⣴⣬⣽⣦⣤⣤⣤⣷⣿⣤⣤⣥⣤⣴⣦⣼⣤⣤⣧⣶⣾⣿⣯⣤⣤⣤⣦⣽⣧⣤⣤⣤⣼⣴⣮⣤⣼⣴⣽⣦⣼⣤⣴⣿⣿⣿⣿
⣿⣿⣿⣨⣇⣠⣁⣄⣻⣰⣔⣨⣶⣅⣐⣅⣌⣸⣿⣄⣕⣅⣷⣀⣪⣠⣹⣖⣵⣯⣘⣷⣮⣀⣾⣀⣐⣷⣔⣘⣧⣀⣀⣰⣖⣃⣇⣴⣿⣟⣡⣀⣭⣸⣾⣷⣦⣁⣍⣇⣀⣨⣸⣀⣨⣀⣸⣺⣪⣀⣾⣨⣺⣃⣔⣈⣿⣿
⣿⣿⣿⣉⣉⣜⣉⣉⣍⣹⣫⣉⣯⣍⣉⣩⣍⣉⣻⣉⣹⣈⣉⣉⣹⣌⣩⣏⣋⣉⣉⣹⣉⣮⣩⣉⣈⣏⣉⣽⣩⣉⣉⣹⣷⣟⣋⣧⣫⣉⣈⣏⣉⣿⣩⣉⣉⣹⣉⣉⣉⣉⣉⣹⣍⣹⣈⣉⡏⣏⣉⣁⣏⣯⣉⣿⣿⣿
⣿⣿⣿⣉⣉⣉⣋⣩⣉⣉⣉⡏⣉⣈⣉⣋⣏⣉⣯⣩⣉⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡛⣛⠿⠿⢿⡿⠿⠿⠿⠿⢿⡿⢿⠿⠻⠿⢻⠿⠿⢿⡿⠿⠻⣿⠛⠻⠿⠿⠿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠧⠿⠧⠦⠼⠧⠧⠤⠤⠤⠿⠇⠼⠼⠴⠭⠼⠧⠆⠼⠧⠤⠬⠿⠬⢤⣤⣾⠿⠯⠴⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
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                    ䷩ 𝚕𝚒𝚗𝚎 1657

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

(ℹ) Images, hyperlinks and comments at https://news.techrights.org/n/2023/12/14/_Video_Modern_Computing_Excessively_Bloated_Wasteful_and_User_R.shtml
    Gemini version at gemini://gemini.techrights.org/n/2023/12/14/_Video_Modern_Computing_Excessively_Bloated_Wasteful_and_User_R.gmi

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ [Video] 'Modern' Computing Excessively
Bloated, Wasteful, and User-Restricting⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


posted by Roy Schestowitz on Dec 14, 2023


Video_download_link | md5sum 3b41aa60122290620177d6dc82e0fb40
Dude,_Where_Is_My_RAM?
Creative Commons Attribution-No Derivative Works 4.0


🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇Preview_for_Dude,_Where_Is_My_RAM?⦈_


TIME is passing, but things do not necessarily improve for computer users. I've
just had to reboot my main PC for the mere mistake of opening this_article_in_a
Web_browser (the video explains my ordeal in finer details). The Web is getting
so much worse, not just in the sense that stories/journalism are less accurate
but also in the sense that they take more resources and nag the user.
Ironically, the article entitled "YouTube uses lower quality options on
browsers running on Arm-based systems — misreporting as an x86 CPU appears to
be a widespread browser fix" caused me to reboot for the first time in 300+
days; it seems that the Web is nowadays so malicious that it spies on what
hardware you use (might_be_a_bridge_leading_to_WEI_one_day), set aside all the
other forms of malice. For now they "just" read user agent strings, in the
future they might resort to DRM with binary proprietary blobs and root (or
Admin level) access.


Things were not this bad when I was a kid in the 90s. We could play fun
computer games with 100+ less in terms of computer resources and in fact DOOM
has just turned 30 (it ran fine in the 486 era). Where are the John Carmacks of
today? Working for Mark Zuckerberg on bloatware? Not anymore?


Zuckerberg also hired the person behind Firefox, in effect using him to make
the Web's worst (and very bloated) spying machine. I used to run Firefox on a
laptops with just 32 megabytes of RAM (for the entire system, not just
Firefox). Good luck trying this today on Zuckerberg's Facebook, which quickly
takes up more than 1000 megabytes of RAM for a single tab with "infinite
scrolling" (for addiction, basically wasting your time).


In terms of news on the Web, why did I need to reboot an OK machine (bought in
2022) for merely opening an article several paragraphs in length?


It's not like there are many active news sites anymore (there used to be loads
more). My RSS feeds list is mainly blogs, as far as the interesting stuff goes,
as an associate put it. It's getting really hard to find unique and important
news online and blogs are insanely time intensive to process, this associate
said. "Each has a different non-pyramid structure."


As more sites go offline (or become inactive) we're left in the precarious
position of having to check some rather odious sites that take bribes to
produce spam, puff pieces, and disinformation. If our blacklists grows too
long, we'll have nowhere left to check for news.


Dilemma...


The issue noted above shows that machines without Windows are being
discriminated against, even at the Web browser and Web site level. Yesterday I
watched a video on a quad-core ARM processor and it turns out that YouTube has
begun throttling systems for reporting ARM CPUs in the user agent string.
"YouTube changes the quality and resolution options for 'aarch64' Arm-based
systems under Linux," the above 'webapp' says (the site became mostly a
spamfarm with malicious components and the habitual real article). It says that
"changing the user agent string to indicate an underlying [defective chip
maker] Intel CPU delivered better video quality defaults and unlocked 4K
playback on [Fashion Company] Apple systems with Arm-based architectures."


Maybe Google thinks this is helping users with slow processors, but in practice
this defies the concept of 'Web neutrality', wherein you assume nothing and do
not discriminate against the user. Those are issues we did not have more than
20 years ago. These issues have begun to emerge in the past decade or so and
their rationale is financial - as we shall cover in the next video/article.
It's all about money and deterioration is always at the users' expense. █




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⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣻⣯⣻⣾⣫⣿⣿⣿⣷⣏⣬⣄⣯⣀⣴⣠⣤⣨⣄⢠⡀⢤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣾⣼⣯⣿⣾⣷⣿⣿⣿⢳⢶⡷⣶⣶⣶⣶⣶⣶⣦⡄⢠⡀⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡷⣟⢾⣺⣷⢿⣿⣿⡷⢹⢟⡿⠞⢷⠟⢿⢺⠟⠛⠟⠳⠄⠠⠄⠐⠀⠢⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⡻⠑⠀⡏⢳⢠⢠⡠⢼⣠⣤⠀⠀⢣⡾⣸⢸⢠⢠⣠⢠⢠⣤⡄⢺⢠⣶⠄⡿⡰⡧⣀⠄⢸⢽⠂⣞⡄⣷⣼⠐⡵⠀⠨⢑⣤⣾⣄⢸⡛⢾⢤⣿⣿⠟⣣⣶⣶⣾⣿⣆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠉⠀⠉⠉⠈⠉⠈⠉⠈⠉⠘⠀⠈⠁⠁⠈⡈⣈⢉⢈⣈⣉⣁⣨⣈⣉⣀⣅⡀⠡⠏⠀⠈⠀⠉⠀⠁⠁⠈⠀⠁⢀⡼⠃⢉⢛⢭⣤⣤⣥⡀⣭⣭⣤⢹⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣿⣧⣺⣿⣽⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣦⡄⢿⡷⠀⠂⢰⠂⠐⠀⢒⠂⠖⢢⡖⣘⡻⣿⣿⡿⠫⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⡇⠀⠐⠀⠀⠀⠀⢠⠈⠀⠠⠔⠃⢴⠁⠘⡖⠀⠀⠘⠀⢠⡈⢃⠰⢻⣶⣿⠼⡿⠞⠃⠻⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠲⣏⠀⡇⠀⠗⠸⠇⠀⠄⠄⠀⢉⠀⠛⢒⠘⠛⢋⣼⣿⣛⡇⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢿⣿⣿⣷⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⡛⠀⡡⢀⢀⡄⠉⣀⠀⠀⣀⠀⠀⠀⠛⠋⠶⠀⠈⠀⠘⣁⢈⡁⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⢀⡀⠈⠃⠀⠀⠈⠈⠁⢀⣜⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⢉⣀⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣾⣿⣿⣿⣿⠛⠁⠀⠀⠠⠀⠘⠡⡄⠀⠆⠀⠦⢠⠜⠁⠁⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢨⡅⢨⡅⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⣿⠟⢛⠓⠏⠀⠀⠀⠖⠀⡄⠀⠚⢰⡄⣨⠀⠂⠀⠄⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢨⡍⠪⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠩⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡄⠀⠠⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣤⣄⣠⣤⣤⣀⣤⣀⣤⣀⣀⣀⣀⣀⣀⣈⣉⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢿⡿⣽⢿⠿⡷⢾⠇⠻⠃⠺⢂⠛⠀⠚⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢿⡻⣿⢾⠻⢟⡟⢺⡃⠘⠣⠛⠋⠛⠀⠛⠀⠁⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣟⣯⣻⢛⣿⣿⣻⠙⡏⢹⡏⢙⡃⢛⠀⡛⠁⣋⠈⡃⠈⠁⠀⠀⠈⠁⠈⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣻⣟⣿⣻⣛⣟⢺⣿⢊⣛⢚⡃⢈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣽⣿⣿⣝⣟⣫⣏⣉⢈⣏⢈⡱⣈⡁⣉⠀⣉⠀⣀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣯⣿⣿⣯⣿⣿⡠⣯⢾⣿⢉⣅⢨⡅⢨⡅⢨⠀⣬⠀⣨⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⣿⣿⣿⣿⣿⣿⣿⣟⣬⣤⣯⣂⣵⣡⣤⣨⣅⢠⡀⢤⠀⣀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣾⣿⣿⣿⣿⢳⢶⡿⣶⣶⣶⣶⣶⣶⣦⡄⢠⡀⢠⠀⢤⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣷⢽⣟⣿⠞⢷⠟⣿⢻⠟⠛⠟⠳⠆⠰⠄⠐⠀⠦⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣧⡽⠀⡖⢲⢠⢠⡠⢼⣠⣤⠀⠀⢢⣴⣰⢸⢤⢠⣤⢠⢤⣴⡀⢰⣠⡤⠀⡶⡰⡦⣀⠄⢰⣲⠄⡶⡄⣦⣴⠐⡶⠀⠠⠑⣠⣾⣄⢘⠛⢼⢦⢾⣼⢏⣶⣷⣿⣿⣷⡍⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠉⠀⠓⠉⠘⠛⠙⠛⠈⠊⠘⠀⠘⠁⠃⠘⠘⠈⠊⠘⡈⣛⣁⣜⣘⣋⣈⠇⠁⠣⠟⠀⠘⠀⠛⠁⠑⠃⠙⠀⠃⠀⠘⠃⢉⢙⣭⣠⣤⣥⡀⢩⣡⣬⢹⣿⣿⣿⣿⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣧⣿⣿⣿⣿⣿⣿⣿⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣦⠠⢺⡷⠀⠀⢰⠂⠐⠀⠒⠀⠖⢰⣶⠜⡿⣿⣿⡿⣫⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⠈⠀⠠⠐⠂⢔⠁⠈⠖⠀⠀⠐⠀⢀⡀⠃⠰⢿⣾⣷⣿⠷⠟⠃⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠒⣏⠀⠅⠀⠒⠀⠃⠀⠄⠀⠀⠈⠀⠛⢐⢛⠛⠃⢼⡿⢚⡇⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⢿⣿⣿⣷⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠋⠀⡩⢀⢀⣄⠉⠀⠀⠀⣀⠀⠀⠀⠛⠋⠖⠀⠀⠀⢘⣁⢉⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣾⢷⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⢀⠀⠀⠂⠀⠀⠈⠈⠁⢀⣍⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡁⢀⡀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣻⣿⠛⠁⠀⠀⠀⠀⠘⠠⡄⠀⠆⠀⠦⠠⠝⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡁⢨⡅⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠟⢛⠓⠟⠀⠀⠀⠖⠀⡀⠀⠒⢰⠄⣨⠀⠂⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡌⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠩⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣄⣠⣤⣤⣀⣤⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣫⡟⣯⢿⢻⡿⣽⢿⠞⡗⢸⠃⠛⠃⠺⠂⠓⠀⠐⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⢻⡗⣿⣟⢿⡳⣿⢾⠻⢞⠟⠚⠃⠘⠣⠛⠋⠛⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣻⣟⣯⣻⠛⣿⣻⣻⠙⡏⢸⡏⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢞⡗⣽⡻⣏⡿⣻⣙⣟⢺⣽⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣻⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⢼⣽⢮⡧⣿⣿⣯⣷⣻⠠⣯⢂⣿⢉⡅⢨⡅⢨⡅⢨⠀⠌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣻⣾⣫⣟⣾⣿⣷⣏⣬⣄⣯⣀⣴⣠⣄⣨⣄⢠⡀⢤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣼⣼⣯⣿⣾⣵⣿⣿⣿⢳⢲⡷⣶⣶⣶⣶⣶⣶⣦⡄⢠⡀⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡷⣟⢾⣺⣷⢾⣿⣿⡷⢹⢟⡿⠞⢷⠟⢿⢺⠟⠛⠟⠳⠄⠠⠄⠐⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣟⣼⠄⡖⢲⢠⢠⣠⢴⣠⣤⠀⠀⢢⣴⢰⢰⢤⢠⣤⢠⢤⣴⡀⢴⢠⡦⠀⣶⣰⡦⣀⠄⢰⣲⠄⡴⡄⣦⣴⠐⡲⠀⠀⠑⣤⣾⣄⢘⠛⢷⢦⢾⡼⠟⣣⣶⣾⣿⣷⣍⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠙⠉⠀⠓⠋⠘⠚⠙⠚⠙⠚⠐⠀⠘⠃⠋⠘⠘⠘⠚⠘⠈⠒⠀⠘⠐⠞⠀⠃⠁⠣⠟⠀⠘⠀⠛⠉⠑⠋⠙⠀⠃⢄⠘⠃⠉⢙⢩⣠⣤⣤⡀⢩⣅⣤⢹⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⡤⠾⡷⠀⠂⢰⡂⠐⠀⢒⠀⠖⢠⣖⠸⡿⣿⣿⡿⡻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡇⠀⠀⠁⠀⠀⠀⢠⠈⠀⠀⠐⠂⢤⠁⠈⠖⠀⠀⠐⠀⢠⡀⠃⠰⢻⣾⣿⠿⢷⠟⠋⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠒⣏⠀⡇⠀⢓⠰⠃⠀⠄⠀⠀⠈⠀⠛⢒⠘⠛⡋⢴⡮⢜⡃⢙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⢻⣿⣿⣿⣿⣷⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⡡⢀⢀⡄⠁⠀⠀⠀⣀⠀⠀⠀⠛⠋⠖⠀⠀⠃⠀⠀⢀⡑⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈⠈⠁⢀⣌⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠁⢉⣁⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣟⣿⢿⣿⠛⠃⠀⠀⠀⠀⠐⠠⡄⠀⠆⠀⠦⠠⠝⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢨⡅⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠿⢛⠓⠏⠀⠀⠀⠶⠀⡀⠀⠒⢰⠄⣨⠀⠂⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠠⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠚⠓⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠩⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣄⣄⣠⣄⣠⣤⣄⣀⣤⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣐⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣫⡟⣯⣿⢻⡿⣽⢿⠞⡷⢸⠃⠛⠃⠺⠂⠓⠀⠐⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⢻⡗⣿⣟⢿⡳⣿⢾⠻⢞⠟⠺⠃⠘⠣⠛⠋⠛⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣻⣟⣯⣻⠛⣿⣻⣻⠙⡏⢸⡏⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢞⡗⣽⡻⣏⡿⣻⣙⣟⢺⣽⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⢾⣽⢮⡧⣿⣿⣯⣷⣻⠠⣯⢖⣿⢉⡅⢨⡅⢨⡅⢨⠀⠌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣻⣾⣫⣟⣾⣿⣷⣏⣬⣄⣯⣀⣴⣠⣄⣨⣄⢠⡀⢤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣼⣼⣯⣿⣾⣷⣿⣿⣿⢳⢶⡷⣶⣶⣶⣶⣶⣶⣦⡄⢠⡀⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡷⣟⢾⣺⣷⢿⣿⣿⡷⢹⢟⡿⠞⢷⠟⢿⢺⠟⠛⠟⠳⠄⠠⠄⠐⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⢻⢙⠀⡖⢢⢠⢠⣠⢴⣠⢄⠀⠀⢦⣴⢰⢰⢤⢠⢤⢠⢄⢤⡀⢰⢠⡴⠀⣶⢰⡦⣀⠄⢰⣲⠄⡴⡀⣦⣰⠐⡢⠀⠨⠐⣤⣾⣆⢘⠛⢾⢤⣿⣿⠟⣋⣶⣶⣿⣾⣍⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠩⠑⠀⠓⠊⠘⠚⠙⠚⠙⠒⠔⠀⠘⠃⠋⠘⠘⠘⠒⠘⠘⠓⠂⠘⠐⠻⠂⠓⠁⠃⠟⠀⠘⠈⠓⠉⠓⠋⠙⠀⠃⢀⠜⠃⢉⢛⣥⣠⣤⣤⡀⡩⣥⣠⣹⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣶⡀⢾⡷⠀⠀⢰⠀⠐⠀⠒⠂⠖⠠⣶⠘⡯⣹⣿⡿⡻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⠈⠀⠠⡔⠃⢴⠁⠘⠗⠈⠀⠘⠀⢠⡘⢋⠴⢸⣷⣿⣿⠇⣝⠗⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠬⢿⣿⣿⣿⣿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠂⢀⠂⣛⠀⡃⠀⠒⠸⠇⠀⠄⠄⠀⢈⠀⠙⢓⠊⠛⣫⣿⡯⢜⡇⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⡷⢼⠿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠋⠀⡩⢀⢀⣄⠉⣀⠀⠀⣀⠀⠀⠀⠛⠋⠖⠀⠀⠁⢐⠀⢀⡐⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣿⣷⣼⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢈⠀⠈⠂⠀⠀⠈⠈⠁⢀⣝⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⢈⣁⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡻⣿⠟⠃⠀⠀⠠⠀⠘⠡⡄⠀⠆⠀⠦⠠⠙⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⢠⡄⢨⡅⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡿⠿⢟⠛⠯⠀⠀⠀⠶⠀⡄⠠⠒⢰⡄⣨⠀⠂⠀⠄⠨⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢹⡍⠨⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠋⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠩⠠⠍⠐⢀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣄⣀⣤⣀⣤⣀⣀⣀⣀⣀⣀⣈⣉⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢿⡿⣽⢿⠿⡷⢾⠇⠻⠃⠺⢂⠛⠀⠚⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢿⣻⣿⢾⠻⢟⡟⢺⡃⠘⠣⢛⠋⠛⠀⠛⠀⠁⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣟⣯⣻⢛⣿⣿⣻⡙⡏⢹⡏⣙⡃⢛⠀⡛⠁⣛⠈⡃⠈⠁⠀⠀⠈⠁⠈⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣿⣻⣛⣟⣻⣿⢊⣛⢚⡃⢘⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣽⣟⣯⣏⣉⢈⣏⢈⡱⣈⡁⣉⠀⣉⠀⣀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣯⣿⣿⣯⣿⣿⡠⣯⢾⣿⢉⣅⢨⡅⢨⡅⢨⠀⣬⠀⣨⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣟⣬⣤⣯⣂⣵⣡⣤⣨⣅⢠⡀⢬⠀⣀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⢳⢶⡿⣶⣶⣶⣶⣶⣶⣦⡄⢠⡀⢠⠀⢤⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⢽⣟⣿⠞⢷⡟⣿⢻⠟⠛⡟⠳⠆⠰⠄⠐⠀⠦⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣟⣽⠀⡖⢦⣠⢠⣠⣴⢀⢄⠀⠀⢆⣴⢰⢰⢄⢀⢄⢠⢄⣤⡀⢸⢡⡝⠩⣿⢰⣦⣀⡄⢰⣲⠄⡴⡀⣦⣰⠠⡢⠀⠀⠐⣤⣾⣄⢸⠛⢶⢤⢾⣽⠟⢋⣦⣶⣶⣶⣍⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡻⠋⠀⠓⠚⠘⠚⠑⠚⠙⠛⠐⠀⠘⠃⠋⠘⠘⠘⠛⠘⠘⠛⠃⢘⠘⢛⠈⠇⠁⢃⡟⠀⠘⠈⠒⠉⠃⠋⠙⠀⠃⢀⡜⠃⢉⢛⣭⣤⣤⣥⡀⣩⢁⣤⢹⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣶⡀⢾⡷⠀⠂⢰⠂⠐⠀⢒⠂⠖⠢⢶⠘⡿⣿⣿⡿⡩⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⡐⠣⢤⠁⠈⠖⠀⠀⠘⠀⢠⡈⠋⡴⢸⣶⣿⣾⠇⠝⠓⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠧⢽⣿⣿⣿⣿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠂⢂⠒⣏⠀⠃⠀⠓⠠⠃⠀⠄⠄⠀⠉⠀⢛⢘⠈⠛⣻⢿⣯⢜⡃⢙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡷⢼⡿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⡩⢀⢀⡄⠁⠀⠀⠀⣀⠀⠀⠀⠛⠋⠶⠀⠀⠁⢀⠀⢀⡐⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⣾⣷⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⢀⠀⠊⠃⠈⠀⠈⠈⠁⢀⣕⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⢈⣉⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣿⣿⣿⠛⠃⠀⠀⠀⠀⠘⠠⡄⠀⠆⠀⠦⠠⠙⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⡅⢨⡅⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡿⣿⠟⢟⠛⡿⠀⠀⠀⠖⠀⡀⠀⠒⢰⠄⣨⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢨⡌⠠⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠃⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠡⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣄⣤⣄⣠⣄⣠⣤⣄⣀⣤⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣫⡟⣯⣿⢿⡿⣽⢿⠞⡷⢸⠃⠛⠃⠺⠂⠛⠀⠒⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⢻⡗⣿⣟⢿⡻⣿⢾⠻⢞⠟⠺⠃⠘⠣⠛⠋⠛⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣻⣟⣯⣻⢛⣿⣻⣻⠙⡏⢹⡏⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣻⢞⣗⣽⡻⣏⡿⣻⣙⣟⢺⣽⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣟⣻⣽⣿⣟⣝⡋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⣀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢾⣽⢮⡧⣿⣿⣯⣿⣿⠠⣯⢖⣿⢉⣅⢨⡅⢨⡅⢨⠀⢌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣻⣯⣻⣾⣫⣿⣿⣿⣷⣏⣬⣄⣯⣀⣴⣠⣤⣨⣄⢠⡀⢤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⣿⣾⣼⣯⣿⣾⣷⣿⣿⣿⢳⢶⡷⣶⣶⣶⣶⣶⣶⣦⡄⢠⡀⢠⠀⢤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡷⣟⢾⣺⣷⢿⣿⣿⡷⢹⢟⡿⠞⢷⠟⣿⢻⠟⠛⠟⠳⠄⠰⠄⠐⠀⠢⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣛⣽⠀⡅⢭⢉⢁⢈⣸⢈⣉⠁⠀⢍⣤⢩⢱⣀⢈⣉⢉⣉⣉⠉⢹⣋⣻⠙⣭⢡⣄⠀⡀⢠⢠⠄⡤⡀⣄⣠⠠⣤⠀⠈⠑⣦⣾⣄⢸⠛⢾⢦⣿⣾⢟⣣⣶⣶⣶⣶⡅⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠈⠀⠃⠚⠘⠚⠑⠚⠙⠙⠰⠀⠘⠋⠏⠘⠸⠘⠝⠘⠘⠟⠃⠸⠘⠛⢀⠇⠃⢃⡟⠀⠘⠈⠒⠉⠃⠋⠛⠀⠃⢀⠜⠃⢉⢛⣭⣤⣬⣥⠀⣩⣥⣤⢹⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣾⣦⣲⣶⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣦⡀⢾⡷⠀⠀⢰⠂⠐⠀⢒⠂⠖⢲⣾⡚⣿⣿⣿⡿⣯⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠐⠁⠀⠀⠀⢠⠈⠀⠠⡔⠃⢬⠁⠈⡗⠀⠀⠘⠀⢠⡈⠋⠲⢿⣶⣧⢿⡷⣜⠃⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠂⢀⢔⣛⠀⠗⠀⠓⠸⠃⠀⠄⠐⠀⢈⠀⠛⢐⠘⠛⡏⣼⣯⣟⡇⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡾⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⡋⠀⡁⢀⢀⡄⠉⠀⠀⠀⣀⠀⠀⠀⠛⠋⠶⠀⠀⠀⢈⣁⢜⡑⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢲⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⢀⠀⠈⠃⠁⠀⠈⠈⠁⢀⣕⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⢩⣀⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠠⠀⠘⠡⡄⠀⠆⠀⠦⠠⠙⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⢠⡄⢨⡅⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠟⢟⠛⠯⠀⠀⠀⠶⠀⡄⠠⠒⢸⠄⣨⠀⠂⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡍⠩⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠃⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠡⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣤⣄⣠⣤⣤⣀⣤⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣒⣀⣀⣀⣂⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢿⡿⣽⢿⠿⡷⢾⠇⠻⠃⠺⢂⠛⠀⠚⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢿⣻⣿⢾⠻⢟⡟⢺⡃⠘⠣⢛⠋⠛⠀⠛⠀⠁⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣟⣯⣻⢛⣿⣿⣻⡙⡏⢹⡏⣙⡃⢛⠀⡛⠁⣛⠈⡃⠈⠁⠀⠀⠈⠁⠈⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣻⣟⣿⣻⣛⣟⢺⣿⢊⣛⢚⡃⢘⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣝⣟⣫⣏⣉⢈⣏⢈⡱⣈⡁⣉⠀⣉⠀⣀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣯⣿⣿⣯⣿⣿⡠⣯⢾⣿⢉⣅⢨⡅⢨⡅⢨⠀⣬⠀⣨⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⣿⣿⣿⣿⣿⣿⣿⣟⣬⣤⣯⣂⣵⣡⣤⣨⣅⢠⡀⢬⠀⣀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣾⣿⣿⣿⣿⢳⢶⡿⣶⣶⣶⣶⣶⣶⣦⡄⢠⡀⢠⠀⢤⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⢽⣟⣿⠞⢷⠟⣿⢻⠟⠛⠟⠳⠆⠰⠄⠐⠀⠦⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣻⣵⠉⡥⢭⢉⢉⢉⣩⢉⣉⠉⠉⢍⣩⢩⢩⣀⢉⣉⢉⣉⣉⠉⣫⢉⡻⠉⣿⢡⣄⠀⡀⢠⢤⡀⣠⡀⣄⣠⠠⣤⠀⠨⠐⣤⣾⣄⢸⡛⢶⢦⣿⡾⢟⣣⣷⣶⣶⣿⡌⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠉⠀⠧⠞⠸⠼⠳⠼⠹⠝⠠⠀⠸⠏⠟⠸⠸⠘⠽⠸⠘⠭⠃⡸⠨⠕⠐⠏⠋⢇⡿⠀⠸⠉⠶⠓⠣⠏⠻⠀⠇⢀⡞⠃⢉⢛⣥⣤⣤⣥⡀⣩⣡⣬⢹⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⣦⣶⣶⣾⣿⣿⣶⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⢾⡷⠀⠀⢰⠂⠐⠀⠒⠢⠖⠲⣖⣚⡿⣿⣿⡿⣯⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⠈⠀⠠⡴⠃⣤⠡⠈⡗⠀⠀⠘⠀⢠⡘⠋⠰⢿⣶⣯⢻⡗⠔⠃⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠂⢀⠂⣋⠀⡃⠀⠓⠨⠃⠀⠄⠀⠀⠈⠀⠛⢑⠘⠛⣏⣼⣯⣟⡇⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡿⢿⣿⣿⣷⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠋⠀⡉⢀⢀⡄⠉⠀⠀⠀⣀⠀⠀⠀⠛⠋⠖⠀⠈⠁⢘⠁⢘⡁⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣿⡷⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢈⠀⢊⠃⠁⠀⠈⠈⠁⢀⣜⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⡉⣉⣈⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣾⣿⣿⣿⣿⠛⠀⠀⠀⠠⠀⢘⠡⠄⠀⠆⠀⠮⢠⢙⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⢠⡅⢨⡅⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⠛⠹⠛⠏⠀⠀⠀⠖⠀⣀⠀⠒⢰⠄⣨⠀⠂⠀⠄⠨⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢸⡍⠫⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠡⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣤⣄⣠⣤⣤⣀⣤⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢿⡿⣽⢿⠿⡷⢾⠇⠻⠃⠺⢂⠛⠀⠚⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢿⡻⣿⢾⠻⢟⡟⢺⡃⠘⠣⠛⠋⠛⠀⠛⠀⠁⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣟⣯⣻⢛⣿⣿⣻⠙⡏⢹⡏⢙⡃⢛⠀⡛⠁⣋⠈⡃⠈⠁⠀⠀⠈⠁⠈⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣻⣟⣿⣻⣛⣟⢺⣿⢊⣛⢚⡃⢈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣽⣿⣿⣝⣟⣫⣏⣉⢈⣏⢈⡱⣈⡁⣉⠀⣉⠀⣀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣯⣿⣿⣯⣿⣿⡠⣯⢾⣿⢉⣅⢨⡅⢨⡅⢨⠀⣬⠀⣨⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⣿⣿⣿⣿⣿⣿⣿⣟⣬⣤⣯⣂⣵⣡⣤⣨⣅⢠⡀⢤⠀⣀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣾⣿⣿⣿⣿⢳⢲⡿⣶⣶⣶⣶⣶⣶⣦⡄⢠⡀⢠⠀⢤⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣷⢽⣟⣿⠞⢷⠟⣿⢻⠟⠛⠟⠳⠆⠰⠄⠐⠀⠦⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣏⣴⠉⡭⢍⢉⢁⢉⣩⢉⡉⠉⠉⣍⣩⢩⢩⣀⢉⣉⢉⡉⣉⠉⢭⢉⡩⠉⣭⢡⣄⠀⡀⢠⢤⡀⣠⠀⣄⢠⠠⢤⠀⠨⠐⣤⣾⣄⢸⡛⢾⢦⢿⣽⠟⣣⣶⣿⣿⣿⣌⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠁⠀⠧⠜⠸⠼⠻⠼⠻⠭⡠⠀⠸⠏⠞⠸⠸⠸⠭⠸⠸⠿⠍⠸⠨⠾⠘⠏⠋⢇⡿⠀⠸⠙⠦⠓⠧⠟⠻⠀⠇⢀⠜⠃⢉⢛⣡⣠⣤⣠⡀⣭⣅⣤⢹⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣶⣦⣴⣦⣶⣶⣾⣶⣾⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⢦⡐⢾⡿⠀⠀⢰⠂⠐⠀⢒⠠⢖⢠⡦⠬⡿⣿⣿⡿⡫⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⠈⠀⡠⡴⠣⢤⠁⠈⡗⠀⠀⠘⠀⢰⡘⢋⠰⢻⣾⣧⣹⣟⠞⠋⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠂⢀⠐⣋⠀⡃⠀⢓⠨⠇⠀⠄⠄⠀⠈⠀⢛⢑⠘⠛⡃⢼⡿⢟⡇⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢿⣿⣿⣿⣾⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢋⠀⡉⢀⢀⡄⠉⠀⠀⠀⣀⠀⠀⠀⠛⠋⠖⠀⠈⠀⢈⣉⢛⠁⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣷⣿⡷⣾⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠈⠁⠁⠀⠈⠈⠁⢀⣝⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⡁⣈⣈⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣾⣿⣿⣿⣿⠟⠃⠀⠀⠠⠀⠘⠡⡄⠀⠆⠀⠎⠠⠜⠁⠁⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⢠⡄⢨⡅⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⠿⢟⠛⡯⠀⠀⠀⠶⢀⡄⠀⠒⢰⠄⣨⠀⠂⠀⠄⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢨⡍⠫⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠡⠠⠉⠐⢀⠀⠀⠀⠀⠀⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣄⣠⣤⣄⣀⣤⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⢿⡿⣽⢿⠿⡷⢾⠇⠻⠃⠺⢂⠛⠀⠚⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢿⡻⣿⢾⠻⢟⡟⢺⡃⠘⠣⠛⠋⠛⠀⠙⠀⠁⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣟⣯⣻⢛⣿⣻⣻⠙⡏⢹⡏⢙⡃⢛⠀⡛⠁⣋⠈⡃⠈⠁⠀⠀⠈⠁⠈⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣻⣟⣿⣻⣛⣟⢺⣿⢊⣋⢚⡃⢈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣽⣿⣿⣝⣏⣫⣏⣉⢈⣏⢈⡱⣈⡁⣉⠀⣉⠀⣀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢾⣿⣿⣯⣿⣿⣯⣿⣿⡠⣯⢾⣿⢉⣅⢨⡅⢨⡅⢨⠀⣬⠀⣨⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⣿⣿⣿⣿⣿⣿⣿⣟⣬⣄⣯⣂⣵⣡⣤⣨⣅⢠⡀⢤⠀⣀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣾⣿⣿⣿⣿⢳⢲⡿⣶⣶⣶⣶⣶⣶⣦⡄⢠⡀⢠⠀⢤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣺⣷⣿⣿⣿⣷⢽⣟⣿⠞⢷⠟⣿⢻⠟⠛⠟⠳⠆⠰⠄⠐⠀⠦⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⡿⣹⠉⡭⣍⢉⢁⢉⣩⢉⡉⠉⠉⡍⣩⢩⢩⡀⠉⡉⢉⡉⣛⠉⢻⢉⡽⠉⣯⢡⣄⠀⡀⢠⢤⡀⣠⠀⣄⢠⠠⢄⠀⠨⠑⣤⣾⣆⢘⡻⢿⢦⢾⡼⢏⣶⣷⣾⣿⣷⡄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠉⠀⠧⠼⠸⠼⠫⠼⠺⠾⡠⠀⠸⠏⠞⠸⠸⠸⠾⠸⠸⠾⠇⡺⠸⠶⡀⠏⠊⠏⡿⠁⠸⠙⠤⠓⠇⠟⠻⠀⠇⢀⡜⠃⢉⢛⣥⣤⣤⣥⡀⣭⣥⣬⢹⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣶⣶⣖⣶⣶⣶⣷⣶⣾⣴⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣦⡴⢾⡷⠀⠀⢰⠂⠐⠀⠒⠀⠖⢰⣶⡘⣿⣿⡿⡿⡻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠐⠀⠀⠀⠀⢠⠈⠀⡠⠐⠃⢤⠁⠈⠗⠀⠀⠘⠀⢰⡈⢋⠰⣶⣷⣿⣿⣷⠟⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠂⢀⠰⣏⠀⠇⠀⠓⠨⠇⠀⠄⠄⠀⠈⠀⠛⢒⢛⠛⣩⣽⣿⢟⡇⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⣿⡿⣿⣾⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢋⠀⡁⢀⢀⡄⠁⡀⠀⠈⣀⠀⠀⠀⠛⠋⠖⠀⠂⠀⢐⣀⢚⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡷⣾⣷⣾⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⢨⠀⠈⠀⠁⠀⠈⠈⠁⢀⣝⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠈⠁⣈⣨⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠠⠀⠘⠡⡄⠀⠆⠀⠎⠠⠙⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢠⡄⢨⡅⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⠿⢟⠛⠿⠀⠀⠀⠖⠀⡀⠀⠒⢰⠄⣨⠀⠂⠀⠄⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢨⡍⢬⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠚⠫⠂⠀⠀⠀⠀⠀⠀⠉⠀⠀⠡⠠⠈⠐⢀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⠁⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣤⣄⣠⣤⣄⣀⣤⣀⣴⣀⣀⣀⣀⣀⣀⣈⣁⣒⣀⣀⣀⣂⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⢿⡿⣽⢿⠿⡷⢾⠇⠻⠃⠺⢂⠛⠀⠚⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢿⡻⣿⢾⠻⢟⡟⢺⡃⠘⠣⠛⠋⠛⠀⠙⠀⠁⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣟⣯⣻⢛⣿⣻⣻⠙⡏⢹⡏⢙⡃⢛⠀⡛⠁⣋⠈⡃⠈⠁⠀⠀⠈⠁⠈⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣻⣟⣿⣻⣛⣟⢺⣿⢊⣋⢚⡃⢈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣽⣿⣿⣝⣏⣫⣏⣉⢈⣏⢈⡱⣈⡁⣉⠀⣉⠀⣀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢾⣿⣿⣯⣿⣿⣯⣿⣿⡠⣯⢾⣿⢉⣅⢨⡅⢨⡅⢨⠀⣬⠀⣨⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⣿⣿⣿⣿⣿⣿⣿⣟⣬⣄⣯⣂⣵⣡⣤⣨⣅⢠⡀⢤⠀⣀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣾⣿⣿⣿⣿⢳⢲⡿⣶⣶⣶⣶⣶⣶⣦⡄⢠⡀⢠⠀⢤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣺⣷⣿⣿⣿⣷⢽⣟⣿⠞⢷⠟⣿⢻⠟⠛⠟⠳⠆⠰⠄⠐⠀⠦⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠷⠉⡭⣍⠉⠉⠉⢩⠉⠉⠉⠈⡍⣡⢩⢩⠈⠉⠙⠙⠛⠛⠛⢻⢛⠟⠛⣿⢃⡄⠀⠀⢠⢤⡀⣠⠀⣄⢠⢀⣄⠀⠠⠑⣤⣾⣄⢘⠛⢷⢤⢾⣽⢟⣳⣷⣶⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠋⠉⠁⠧⠼⠸⠼⠪⠽⠺⠾⢠⠀⠸⠏⠞⠸⠸⠸⠾⠸⠸⠿⠇⠹⠸⠷⠀⠏⠎⠏⡶⠁⠸⠙⠤⠓⠇⠟⠻⠀⠏⢀⠈⠃⠉⢛⣡⣠⣤⣡⡀⣭⣠⣬⢹⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⣴⣦⣶⣶⣶⣶⣶⣰⣶⡆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⢾⣷⠀⠀⢰⠂⠐⠀⠒⠀⠖⢰⣖⡘⣿⣿⣯⡿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⡐⠃⢤⠡⠈⠔⠀⠀⠐⠀⢀⡈⢋⠰⢺⣾⣿⣿⢷⠜⠓⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠲⣟⠀⠄⠀⠒⠠⠃⠀⠄⠀⠀⠈⠀⠛⢒⠹⠛⣉⣽⣿⢚⡇⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡿⢿⣿⣿⣾⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠃⠀⡩⢀⢀⡄⠁⠀⠀⠀⣀⠀⠀⠀⠛⠋⠖⠀⠂⠀⢈⣡⢏⡁⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣾⢷⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈⠈⠁⢀⣍⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡁⣁⣈⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣿⣿⣿⣿⣿⠟⠃⠀⠀⠠⠀⠘⠠⡄⠀⠆⠀⠖⠠⠘⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡅⢨⡅⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠿⢛⠋⠿⠀⠀⠀⠖⠀⣀⠀⠒⢰⠆⣈⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡌⠠⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣄⣠⣄⣤⣤⣄⣀⣄⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣂⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⡀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⢿⣫⡟⣯⣿⢻⡿⣽⢿⠞⡷⢸⠃⠛⠃⠺⠂⠓⠀⠘⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⢻⡗⣿⣟⢿⡳⣿⢾⠻⢞⠟⠺⠃⠘⠣⠛⠋⠛⠀⠋⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣻⣟⣯⣻⠛⣿⣻⣻⠙⡏⢸⡏⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢞⡗⣽⡻⣏⡿⣻⣙⣟⢺⣽⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⢾⣽⢮⡧⣿⣿⣯⣷⣻⠠⣯⢖⣿⢉⡅⢨⡅⢨⡅⢨⠀⠌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣻⣾⣫⣟⣾⣿⣷⣏⣬⣄⣯⣀⣴⣠⣄⣨⣄⠠⡀⠤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣼⣼⣯⣿⣾⣷⣿⣿⣿⢳⢲⡷⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡷⣟⢾⣺⣷⢿⣿⣿⡷⢹⢟⡿⠞⢷⠟⢿⢺⠟⠛⠟⠳⠄⠠⠄⠐⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⢻⣑⠛⣙⣋⠙⠓⠙⢫⠙⠙⠛⠈⡛⢉⢛⢻⠂⠛⠛⠙⠛⠛⠛⢛⡛⠛⢛⣻⢃⡀⠀⠀⢀⣀⠀⢀⠀⡀⢀⢀⣀⠀⠨⠑⣤⣾⣄⢘⡛⢿⢤⣿⣿⠟⣣⣶⣶⣾⣿⣆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠉⠀⣇⡼⢸⣼⢮⣽⠺⡻⢠⠀⢸⡟⡞⢸⢹⠸⡻⢸⠹⣿⠆⢸⢹⡯⠀⡟⠎⡏⡶⠁⢸⠚⣥⠯⢇⡟⢿⠀⡏⢀⡜⠃⢉⢛⣩⣠⣤⣭⡀⡭⣤⣬⢹⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣤⣤⣶⣶⣾⣶⣷⣾⣦⡄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⣤⡠⢼⡷⠀⡀⢰⠀⠐⠀⢒⠀⠖⢢⡖⣘⡿⣿⣿⡿⡯⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢠⠈⠀⠠⡔⠓⣤⠁⠈⡖⠀⠀⠘⠀⢠⡈⠋⠰⢿⣶⣷⢸⡷⠖⠓⢻⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠰⣟⠀⠇⠀⠗⠘⠃⠀⠄⠄⠀⠙⠀⢛⠐⠘⠛⠏⣼⣯⢝⡇⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣷⣼⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠋⠀⡡⢀⢀⡄⠉⡀⠀⠀⣀⠀⠀⠀⠛⠋⠖⠀⠀⠁⢘⣁⢔⡁⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣷⣾⡷⣾⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⢀⠀⠈⠃⠀⠀⠈⠈⠁⢀⣔⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⢉⣀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⣿⣿⣻⣿⠛⠃⠀⠀⠀⠀⠘⠡⡄⠀⠆⠀⠶⠠⠜⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⢠⡅⢨⡅⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠿⢿⠛⡏⠀⠀⠀⠶⠀⣄⠀⠖⢰⠄⣨⠀⠂⠀⠄⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢨⡍⠨⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠡⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣄⣤⣤⣄⣀⣄⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⡀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⢿⡿⣽⢿⠿⡷⢾⠇⠻⠃⠺⢂⠛⠀⠛⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢿⡻⣿⢾⠻⢟⡟⢺⡃⠘⠣⠛⠋⠛⠀⠛⠀⠁⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣟⣯⣻⢛⣿⣻⣻⠙⡏⢹⡏⢙⡃⢛⠀⡛⠁⣋⠈⡃⠈⠁⠀⠀⠈⠁⠈⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣻⣟⣿⣻⣛⣟⢺⣿⢊⣋⢚⡃⢈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣽⣿⣿⣝⣏⣫⣏⣉⢈⣏⢈⡱⣈⡁⣉⠀⣉⠀⣀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢾⣿⣿⣯⣿⣿⣯⣿⣿⡠⣯⢾⣿⢉⣅⢨⡅⢨⡅⢨⠀⣬⠀⣨⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⣿⣿⣿⣿⣿⣿⣿⣟⣬⣄⣯⣂⣵⣡⣤⣨⣅⢠⡀⠤⠀⣀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣾⣿⣿⣿⣿⢳⢲⡿⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣺⣷⣿⣿⣿⣷⢽⣟⣿⠞⢷⠟⣿⢻⠟⠛⠟⠳⠆⠰⠄⠐⠀⠦⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⣙⡋⠙⠛⠙⢋⠙⠛⠛⠉⡛⢉⢛⢛⠂⠑⠛⠙⠛⠛⠛⢻⠛⠛⠛⣛⢃⡀⠀⠀⢀⣀⠀⢀⠀⡀⢀⢀⣀⠀⠈⠐⣤⣾⣄⢚⠛⢿⢦⣿⣼⢟⣣⣶⣿⣿⣿⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠉⠀⣇⡸⢸⣸⢾⣹⢾⣛⢀⠀⢹⡟⡼⢸⢹⢸⣛⢸⢹⣛⠂⢸⢘⡯⠑⡟⠞⡏⣶⠁⢸⠺⣥⠯⣆⡿⢾⠀⡎⢄⠜⠃⢉⢛⣥⣠⣤⣨⠀⠩⣌⣠⣹⣿⣿⣿⣿⣷⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣤⣦⣤⡤⣤⣤⣤⣤⣯⣤⣤⡄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⣤⡠⢼⡷⠀⡀⢰⡂⠐⠀⢒⠒⠖⢢⣶⠸⣯⣻⣿⡿⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⡇⠀⠐⠀⠀⠀⠀⢠⠈⠀⡠⡜⠃⢨⠡⠈⡖⠀⠀⠘⠂⢠⡈⢋⠰⢻⣶⣿⠿⣿⢟⠓⢻⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠺⢿⣿⣿⣿⣿⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠲⣟⠀⡇⠀⠓⠸⠃⠀⠄⠄⠀⠉⠀⠛⠐⠛⠛⡏⣴⡯⢜⡇⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣷⢾⣿⣿⣿⣿⣷⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⡋⠀⡡⢀⢀⣄⠉⡀⠀⠀⣀⠀⠀⠀⠛⠋⠖⠀⠀⠁⢐⣀⢘⡁⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣷⣿⣿⢿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢠⡀⠈⠃⠀⠀⠈⠈⠁⢀⣜⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⣉⣀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠠⠀⠘⠡⡄⠀⠆⠀⠦⠠⠜⠁⠁⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⢠⡅⢨⡅⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⠿⢛⠛⡏⠀⠀⠀⠖⠀⡄⠠⠒⢴⠄⣨⠀⠂⠀⠄⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡜⠤⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠚⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠡⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡄⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣄⣀⣤⣀⣀⣀⣀⣀⣀⣈⣉⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⡀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢿⡿⣽⢿⠿⡷⢾⠇⠻⠃⠺⢂⠛⠀⠛⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢿⣻⣿⢾⠻⢟⡟⢺⡃⠘⠣⢛⠋⠛⠀⠛⠀⠁⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣟⣯⣻⢛⣿⣿⣻⡙⡏⢹⡏⣙⡃⢛⠀⡛⠁⣛⠈⡃⠈⠁⠀⠀⠈⠁⠈⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣻⣟⣿⣻⣛⣟⢺⣿⢊⣛⢚⡃⢘⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣝⣟⣫⣏⣉⢈⣏⢈⡱⣈⡁⣉⠀⣉⠀⣀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣯⣿⣿⣯⣿⣿⡠⣯⢾⣿⢉⣅⢨⡅⢨⡅⢨⠀⣬⠀⣨⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⣿⣿⣿⣿⣿⣿⣿⣟⣬⣤⣯⣂⣵⣡⣤⣨⣅⢠⡀⠬⠀⣀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣾⣿⣿⣿⣿⢳⢲⡿⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⢽⣟⣿⠞⢷⠟⣿⢻⠟⠛⠟⠳⠆⠰⠄⠐⠀⠦⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣾⠛⣋⡛⠙⠛⠛⢛⠙⠙⠛⠛⡛⢉⢛⢛⠃⠀⠙⠙⠛⠛⠛⢛⠛⠛⠛⣛⢃⡀⠀⠀⢀⣀⠀⢀⠀⡀⢀⢀⡀⠀⠠⠑⣤⣾⣄⢘⠛⢷⢤⢾⣼⠿⣋⣶⣾⣿⣿⣧⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠙⢉⠀⣇⡸⢸⣸⢾⣹⢼⣑⢀⠀⢱⡟⡼⢸⢹⢸⣵⢸⢱⣽⠂⢸⢘⣭⢀⡟⠜⡏⣶⠁⢸⠺⣅⠯⣆⡿⢾⠀⡎⢀⠈⠃⠉⢙⣥⣠⣤⣬⡀⡭⢨⣤⢹⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⢄⣤⣴⣤⣤⣤⣤⣤⣥⣤⣤⡄⠀⠁⠀⠀⠀⠀⠀⠀⠠⠠⣤⠀⠸⡿⠀⠀⢰⠀⠐⠀⠒⠂⠶⢲⢴⡨⡧⣽⣿⡿⡫⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⡇⠀⠀⠁⠀⠀⠀⢠⠈⠀⠠⠰⠃⢤⠁⠈⠗⠀⠀⠘⠀⢀⡈⠋⠰⠸⣾⣿⠿⠇⠟⠓⠻⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡥⢼⣿⣿⣿⣿⢯⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠐⣟⠀⡇⠀⢖⠠⡃⠀⠄⠀⠀⠉⠀⠛⠐⠋⠛⡋⢼⡾⠘⠇⢙⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡷⢾⠿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢑⠀⡡⢀⢀⡄⡁⡀⠀⠀⣀⠀⠀⠀⠛⠋⠖⠀⠀⠁⢈⣁⣉⡁⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣿⣷⣿⣿⣿⣿⣿⣿⡿⠆⠀⠀⠀⠀⠀⢈⠂⠀⠀⠈⠀⠁⢀⣍⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠁⢈⣀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣿⣿⣿⠛⠃⠀⠀⠠⠀⠘⠠⡄⠀⠆⠀⠆⢠⠘⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⡅⢨⡅⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⢟⠛⠿⠀⠀⠀⠶⠀⡀⠀⠖⢰⠄⣈⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢨⡌⠠⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠚⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠡⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣀⣄⣀⣤⣀⣀⣀⣀⣀⣀⣈⣉⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⡀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⢿⣫⡟⣯⣿⢿⡿⣽⢿⠞⡷⢸⠃⠛⠃⠺⠂⠛⠀⠐⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⢻⡗⣿⣟⢿⡻⣿⢾⠻⢞⠟⠺⠃⠘⠣⠛⠋⠛⠀⠋⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣻⣟⣯⣻⢛⣿⣻⣻⠙⡏⢹⡏⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣻⢞⣗⣽⡻⣏⡿⣻⣙⣟⢺⣽⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣟⣻⣽⣿⣟⣝⡋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢾⣽⢮⡧⣿⣿⣯⣿⣻⠠⣯⢖⣿⢉⣅⢨⡅⢨⡅⢨⠀⢌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣻⣯⣻⣾⣫⣿⣿⣿⣷⣏⣬⣄⣯⣀⣴⣠⣤⣨⣄⢠⡀⠤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⣿⣾⣼⣯⣿⣾⣷⣿⣿⣿⢳⢲⡷⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡷⣟⢾⣺⣷⢿⣿⣿⡷⢹⢟⡿⠞⢷⠟⣿⢻⠟⠛⠟⠳⠄⠠⠄⠐⠀⠢⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⢿⡿⠷⣽⡋⠹⠛⠙⢋⠙⠛⠗⠘⡛⢉⢛⢛⠋⠀⠚⠙⠟⠻⠻⢿⠿⠿⠿⣿⢇⡀⠀⠀⢀⣀⠀⢀⠀⡀⢀⠀⡀⠀⠨⠑⣤⣾⣄⢘⠛⢷⣦⢾⣼⡿⢋⣶⣾⣿⣿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡉⠉⠀⣇⣹⢰⣸⢜⣻⢼⣵⢀⠀⢱⡟⡼⢸⢱⢰⣵⢸⢱⣿⡆⢸⢸⣧⠀⡟⠜⡟⣶⠃⢸⠺⣁⠯⡆⡿⢾⠈⡞⢀⠈⠁⢉⢛⣡⣠⣤⣤⡀⣭⣬⣥⢹⣿⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣠⣦⣤⣤⡀⠐⠃⠀⠀⠀⠀⠀⠀⠠⠠⣤⠀⢼⡯⠀⠂⢰⠀⠐⠀⠒⠀⠖⢲⣔⠐⣣⣿⣿⡿⡫⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵⣿⣿⣿⣿⣿⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⠈⠀⠠⠐⠃⢄⠡⠈⠔⠈⠀⠐⠀⢠⡘⢃⠰⠸⣶⣿⠙⢿⠟⠓⠻⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢹⣿⣿⣿⣿⠭⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠂⡋⠀⠅⠀⠒⠀⡇⠀⠄⠄⠀⠉⠀⠛⢐⠀⠛⡋⢴⡮⠚⡇⢙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢼⢿⣿⣿⣿⣷⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⡩⢀⢀⡄⡁⡀⠀⠀⣀⠀⠀⠀⠛⠋⠶⠀⠀⠁⢈⣁⣈⡅⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣷⣿⣿⣿⣿⣿⣿⡿⠆⠀⠀⠀⢀⠀⠈⠃⠀⠀⠈⠀⠁⢀⡌⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠁⢈⣀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠘⠡⡄⠀⠆⠀⠖⠠⠝⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡅⢨⡅⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⣿⢟⠛⡿⠈⠀⠀⠶⠀⡀⠀⠖⢰⠄⣨⠀⠂⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡌⠡⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠒⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠡⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣤⣤⣤⣀⣤⣀⣴⣀⣀⣀⣀⣀⣀⣈⣉⣂⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⢿⣫⡟⣯⣿⢿⡿⣽⢿⠞⡷⢸⠃⠛⠃⠺⠂⠛⠀⠐⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⢻⡗⣿⣟⢿⡻⣿⢾⠻⢞⠟⠺⠃⠘⠣⠛⠋⠛⠀⠋⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣻⣟⣯⣻⠛⣿⣻⣻⠙⡏⢹⡏⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣻⢞⣗⣽⡻⣏⡿⣻⣙⣟⢺⣽⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢾⣽⢮⡧⣿⣿⣯⣷⣻⠠⣯⢖⣿⢉⡅⢨⡅⢨⡅⢨⠀⢌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣻⣯⣻⣾⣫⣿⣿⣿⣷⣏⣬⣄⣯⣀⣴⣠⣤⣨⣄⢠⡀⠤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣾⣼⣯⣿⣾⣷⣿⣿⣿⢳⢲⡷⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡷⣟⢾⣺⣷⢿⣿⣿⡷⢹⢟⡿⠞⢷⠟⢿⢺⠟⠛⠟⠳⠄⠠⠄⠐⠀⠢⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣯⣻⠷⠿⠟⠻⠿⠟⢏⠙⠛⠿⠍⠛⠉⠟⢹⠇⠘⠟⠽⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠑⣤⣾⣄⢘⠛⢼⢤⢾⣽⡿⣣⣶⣿⣿⣿⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠑⠉⠀⡇⣹⢰⣰⢔⣺⢴⢶⢀⠀⢱⡟⡼⢸⢲⢰⢶⢰⢲⣶⠄⢸⣸⣦⠀⡟⡜⡗⣤⠂⢸⠸⣁⠯⡆⡷⣾⠈⡟⢀⠈⠃⠉⢛⣥⣠⣤⣠⡀⣭⣤⣤⢹⣿⣿⣿⣿⣷⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠉⠀⠈⠈⠀⠈⠀⠁⠈⠀⠀⠀⠁⠀⠈⣀⣥⣠⣄⣭⣄⣨⣬⣥⣤⣥⡄⠑⠃⠀⠈⠀⠉⠀⠀⠁⠀⣄⠀⢸⡷⠀⠀⢰⠀⠐⠀⠒⠀⠖⢰⡆⢀⡟⣿⣿⡟⡫⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠁⠀⠀⠀⢠⠈⠀⠠⠐⠋⣤⠀⠈⠕⠀⠀⠐⠀⢠⡈⠋⠰⢻⣾⣿⢸⠿⠟⠓⠻⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠂⢀⠲⣟⠀⠅⠀⠖⠰⠃⠀⠀⠀⠀⠉⠀⠛⢒⠘⠛⠃⢼⡿⢚⡇⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⢻⡿⣿⣷⣾⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⡁⢀⢀⡄⠁⡀⠀⠀⣀⠀⠀⠀⠛⠋⠖⠀⠀⠀⢐⣁⢊⠅⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⠴⣾⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⢀⠀⠈⠂⠀⠀⠈⠀⠁⠀⣝⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠁⢈⣀⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣾⣿⣿⣿⣿⠛⠁⠀⠀⠠⠀⠘⠠⡄⠀⠆⠀⠆⠠⠜⠁⠁⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡄⢨⡅⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⠻⢛⠛⠟⠀⠀⠀⠖⠀⡄⠀⠒⢰⠄⣨⠀⠂⠀⠄⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡌⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠩⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣄⣤⣤⣤⣀⣄⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣒⣀⣀⣀⣀⣀⣀⡀⣀⣀⢀⣀⣀⣀⣀⡀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⢿⣫⡟⣯⣿⢻⡿⣽⢿⠖⡷⢸⠃⠛⠃⠺⠂⠓⠀⠐⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⢻⡗⣿⣟⢿⡳⣿⢾⠻⢞⠟⠺⠃⠘⠣⠛⠋⠛⠀⠋⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣻⣟⣯⣻⠛⣿⣻⣻⠙⡏⢸⡏⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢞⡗⣽⡻⣏⡿⣻⣙⣟⢺⣽⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⢾⣽⢮⡧⣿⣿⣯⣷⣻⠠⣯⢖⣿⢉⡅⢨⡅⢨⡅⢨⠀⠌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣻⣾⣫⣟⣾⣿⣷⣏⣬⣄⣯⣀⣴⣠⣄⣨⣄⢠⡀⠤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣼⣼⣯⣿⣾⣷⣿⣿⣿⢳⢲⡷⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡷⣟⢾⣺⣷⢿⣿⣿⡷⢹⢟⡿⠞⢷⠟⢿⢺⠟⠛⠟⠳⠄⠠⠄⠐⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣧⡟⠿⠭⠧⠹⠛⠹⠫⠹⠹⠿⠟⠿⠉⠿⠹⠿⠏⠈⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣠⣾⣄⢺⠛⢿⢤⣾⣾⣿⢛⣧⣾⣿⣿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠡⠊⠀⡏⢱⢰⢰⣴⢺⣴⠶⠀⠀⢱⡾⣸⢸⢲⢰⠶⢰⢲⠶⠄⢸⠰⢾⠀⡿⡼⡗⣤⠂⢸⢽⣂⣞⡆⣷⣼⠈⡝⢀⠘⠁⢉⢛⢡⣠⣤⡡⡄⢍⣭⣥⢹⣿⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠉⠁⠈⠉⠈⠉⠈⠉⠊⠀⠈⠁⠁⠈⠈⠈⣉⣈⣀⣉⣀⣈⣈⣁⣀⣁⡀⠑⠋⠀⠈⠀⠉⠀⠉⠁⠈⢀⠁⢸⣷⠀⡂⢰⠀⠐⠀⠒⠀⠖⢲⡆⡐⣳⣽⣿⡿⣫⠽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⣤⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⠐⠋⣤⠡⠈⠔⠀⠀⠘⠀⢀⡘⠃⠰⠻⣾⣿⠟⠿⠟⠓⠻⠃⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⣿⣿⣿⣿⢤⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠲⣏⠀⠅⠈⠗⠰⠃⠀⠀⠄⠀⠈⠀⠛⠐⠈⠛⡋⢼⡿⢚⡇⢙⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢾⡿⢿⢿⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠋⠐⡉⢀⢀⡄⠁⠀⠀⠀⣀⠀⠀⠀⠛⠋⠶⠀⠀⠁⢈⣁⢌⡁⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣾⣷⣾⠿⣿⣿⣿⣿⣿⠂⠀⠀⠀⢀⠀⠈⠀⠀⠀⠈⠈⠁⠀⣍⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠁⢈⣀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣿⣿⣿⠟⠋⠀⠀⠠⠀⠘⠠⡄⠀⢆⠀⠆⠠⠝⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢨⡅⢨⡅⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⣿⡿⢟⠓⠿⠈⠀⠀⠖⠀⡀⠀⠒⢰⠄⣨⠀⠂⠀⠄⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢨⡌⠡⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠩⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣄⣀⣄⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⢿⣫⡟⣯⣿⢿⡿⣽⢿⠞⡷⢸⠃⠛⠃⠺⠂⠛⠀⠐⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⢹⡗⣿⣟⢿⡻⣿⢾⠻⢞⠟⠺⠃⠘⠣⠛⠋⠛⠀⠋⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣻⣟⣯⣻⢛⣿⣻⣻⠙⡏⢹⡏⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣻⢞⣗⣽⡻⣏⡿⣻⣙⣟⢺⣽⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣹⣿⣟⣻⣽⣿⣟⣝⡋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢾⣽⢮⡧⣿⣿⣯⣿⣻⠠⣯⢖⣿⢉⣅⢨⡅⢨⡅⢨⠀⢌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣻⣯⣻⣾⣫⣿⣿⣿⣷⣏⣬⣄⣯⣀⣴⣠⣤⣨⣄⢠⡀⠤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⣿⣾⣼⣯⣿⣾⣷⣿⣿⣿⢳⢲⡷⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡷⣟⢾⣺⣷⢿⣿⣿⡷⢹⢟⡿⠞⢷⠟⣿⢻⠟⠛⠟⠳⠄⠠⠄⠐⠀⠢⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⢿⠏⠹⠛⠏⠻⠛⠙⠋⠙⠿⠿⠍⠟⠉⠟⠿⠇⠀⠉⠹⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠐⣤⣾⣄⢺⠛⢴⢤⢾⣿⠿⣃⣶⣾⣿⣿⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠙⠈⠀⡏⢱⢰⢰⣴⢺⣴⠦⠀⠀⢳⡾⣸⢸⢲⢰⠶⢰⢢⠶⠄⢸⠰⣲⠀⡿⡼⡗⣤⠂⢸⢽⡂⣞⡄⣷⣼⠈⡝⢀⠈⠃⢉⢛⣥⣠⣤⣤⡄⣉⣬⣤⢹⣿⣿⣿⣿⣷⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠉⠁⠈⠉⠈⠉⠈⠉⠊⠀⠈⠁⠁⠈⠈⠈⣉⣈⣈⣉⣀⣈⣈⣁⣈⣁⡀⠑⠋⠀⠈⠀⠉⠀⠉⠁⠈⣀⡁⢸⡷⠀⡀⢰⠀⠀⠀⠒⠂⠖⢲⢄⡈⡧⣽⣿⣿⡫⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣿⣿⣿⣿⣿⣧⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡰⠃⢄⠡⠈⠖⠀⠀⠐⠀⢀⡐⢃⠰⠸⣾⣿⡿⠇⠝⠓⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡡⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠒⣛⠀⡇⠈⢗⠰⠃⠀⠄⠄⠀⠈⠀⠛⢐⠛⠛⡫⢾⡾⠜⠃⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⡷⢾⠿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠠⠀⠀⠙⠀⠡⢀⢀⡄⠁⠀⠀⠀⣀⠀⠀⠀⠛⠋⠖⠀⠀⠉⢀⣀⢀⡁⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡶⣾⣷⣾⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⢀⠀⠈⠂⠀⠀⠈⠈⠁⢀⣝⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⢉⣁⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣿⡿⣿⠛⠁⠀⠀⠀⠀⠘⠠⠄⠀⠆⠀⠦⠠⠜⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⢨⡅⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡿⣿⠿⢛⠛⠏⠀⠀⠀⠖⠀⡀⠀⠒⢰⠄⣈⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⡜⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢖⠒⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠩⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣤⣤⣄⣀⣄⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⡀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⢿⣫⡟⣯⣿⢿⡿⣽⢿⠞⡷⢸⠃⠛⠃⠺⠂⠛⠀⠐⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⢻⡗⣿⣟⢿⡻⣿⢾⠻⢞⠟⠺⠃⠘⠣⠛⠋⠛⠀⠋⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣻⣟⣯⣻⠛⣿⣻⣻⠙⡏⢹⡏⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣻⢞⣗⣽⡻⣏⡿⣻⣙⣟⢺⣽⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢾⣽⢮⡧⣿⣿⣯⣷⣻⠠⣯⢖⣿⢉⡅⢨⡅⢨⡅⢨⠀⢌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣻⣯⣻⣾⣫⣿⣿⣿⣷⣏⣬⣄⣯⣀⣴⣠⣤⣨⣄⢠⡀⠤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣾⣼⣯⣿⣾⣷⣿⣿⣿⢳⢲⡷⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡷⣟⢾⣺⣷⢿⣿⣿⡷⢹⢟⡿⠞⢷⠟⢿⢺⠟⠛⠟⠳⠄⠠⠄⠐⠀⠢⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡫⠿⠿⠭⠸⠯⠿⠻⠹⠽⠯⠏⠿⠉⠿⠻⠟⠄⠈⠹⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠑⣤⣾⣄⢘⠛⢺⢦⢾⣾⠿⢃⣦⣶⣿⣿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠉⠀⡏⢳⢰⢰⡴⢺⣰⣢⠀⠀⢣⡾⣸⢸⢢⢠⣢⢰⢢⣲⡄⢸⢰⣂⠠⡿⡸⡗⣤⠆⢸⢽⠂⣞⡄⣷⣼⠈⡕⢀⡈⠃⢉⢛⣥⣤⣤⣠⡄⢩⣨⣤⢹⣿⣿⣿⣿⣷⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠉⠉⠈⠉⠈⠉⠈⠉⠊⠀⠈⠁⠁⠈⠈⠈⣉⣈⣈⣉⣀⣈⣈⣉⣀⣃⡀⠁⠏⠀⠈⠀⠉⠀⠁⠁⠈⣀⠁⢸⣷⠀⠀⢰⠀⠐⠀⠒⠐⠖⢲⠔⠘⡯⣽⣿⣿⠫⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣵⣿⣿⣿⣿⣿⣷⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⠈⠀⠠⡐⠃⢤⠡⠈⠖⠀⠀⠘⠀⢀⡐⢋⠰⠸⣶⣷⣿⠇⠟⠃⠻⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢸⣿⣿⣿⣿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢂⠒⣏⠀⠅⠈⢒⠰⠃⠀⠄⠄⠀⠈⠀⠛⢒⠉⠛⣫⢿⣏⣙⡇⢙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡷⢼⠿⣿⣿⢿⣶⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠛⠀⠩⢀⢀⡄⠉⠀⠀⠀⣀⠀⠀⠀⠛⠋⠖⠀⠀⠁⠀⠀⢀⡁⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣾⣶⣿⣿⣾⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈⠈⠁⠀⣍⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⢈⣁⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⢿⣿⠛⠁⠀⠀⠠⠀⠘⠡⡄⠀⠆⠀⠶⠠⠜⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡁⢨⡅⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠿⢟⠛⠿⠀⠀⠀⠶⠀⡀⠀⠒⢰⠄⣈⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡜⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢖⠒⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠡⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣠⣄⣤⣤⣄⣀⣄⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⡀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⢿⣫⡟⣯⣿⢿⡿⣽⢿⠞⡷⢸⠃⠛⠃⠺⠂⠛⠀⠐⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⢻⡗⣿⣟⢿⡻⣿⢾⠻⢞⠟⠺⠃⠘⠣⠛⠋⠛⠀⠋⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣻⣟⣯⣻⠛⣿⣻⣻⠙⡏⢹⡏⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣻⢞⣗⣽⡻⣏⡿⣻⣙⣟⢺⣽⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢾⣽⢮⡧⣿⣿⣯⣷⣻⠠⣯⢖⣿⢉⡅⢨⡅⢨⡅⢨⠀⢌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣻⣯⣻⣾⣫⣿⣿⣿⣷⣏⣬⣄⣯⣀⣴⣠⣤⣨⣄⢠⡀⠤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣾⣼⣯⣿⣾⣷⣿⣿⣿⢳⢲⡷⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡷⣟⢾⣺⣷⢿⣿⣿⡷⢹⢟⡿⠞⢷⠟⢿⢺⠟⠛⠟⠳⠄⠠⠄⠐⠀⠢⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⡿⠿⡯⠿⠿⠝⠛⠫⠹⠿⠏⠿⠈⠿⠻⠇⠀⠋⠹⠿⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠑⣤⣾⣄⢘⠛⢾⢤⣿⣿⠿⢃⣶⣾⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠉⠀⡏⢳⢠⢠⡠⢼⣠⣤⠀⠀⢣⡾⣸⢸⢤⢠⣤⢠⢤⣤⡀⢸⣠⣦⢈⡿⡰⡧⣀⠄⢸⢽⠂⣞⡄⣷⣼⠐⡷⢄⠜⠃⢉⢛⣩⣤⣤⣤⣄⢩⣬⣤⢹⣿⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠉⠉⠈⠉⠈⠉⠈⠉⠘⠀⠈⠁⠁⠈⠈⢈⡉⣈⠈⢉⣁⣈⠈⣍⣀⣁⡁⠡⠏⠀⠈⠀⠉⠀⠁⠁⠈⢀⠁⢸⣷⠀⠀⢰⠀⠐⠀⠒⠂⠖⢲⠔⠘⡯⣽⣿⡿⡫⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣽⣿⣿⣿⣿⣿⣶⣿⣿⣿⡇⠀⠐⠁⠀⠀⠀⢠⠈⠀⡠⡘⠃⢬⠁⠈⠖⠀⠀⠘⠂⢠⡈⢋⠰⠾⣷⣿⣿⠇⢜⠃⠻⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢼⣿⣿⣿⣿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠂⢀⢲⣟⠀⠇⠈⢗⠨⠃⠀⠄⠐⠀⢘⠀⢛⢀⠉⠛⡏⣽⣮⣙⡇⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡷⢼⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡀⠋⠀⡩⢀⢀⣄⠉⠀⠀⠀⣀⠀⠀⠀⠛⠋⠖⠀⠀⠁⢐⣀⢐⡄⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣶⣾⣷⣿⢿⣿⣿⣿⣿⡿⠂⠀⠀⠀⢠⠀⠊⠂⠀⠀⠈⠈⠁⢀⣝⠛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⢈⣁⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⢿⣿⠛⠁⠀⠀⠀⠀⠘⠠⡄⠀⠆⠀⠶⠠⠙⠁⠁⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡅⣨⡅⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠿⢿⠛⠛⠀⠀⠀⠖⠀⡄⠠⠒⢰⡄⣬⠀⠂⠀⠄⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⡍⢬⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠡⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣄⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⡀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢿⡿⣽⢿⠿⡷⢾⠇⠻⠃⠺⢂⠛⠀⠛⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢿⣻⣿⢾⠻⢟⡟⢺⡃⠘⠣⢛⠋⠛⠀⠛⠀⠁⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣟⣯⣻⢛⣿⣿⣻⡙⡏⢹⡏⣙⡃⢛⠀⡛⠁⣛⠈⡃⠈⠁⠀⠀⠈⠁⠈⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣻⣟⣿⣻⣛⣟⢺⣿⢊⣛⢚⡃⢘⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣝⣟⣫⣏⣉⢈⣏⢈⡱⣈⡁⣉⠀⣉⠀⣀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣯⣿⣿⣯⣿⣿⡠⣯⢾⣿⢉⣅⢨⡅⢨⡅⢨⠀⣬⠀⣨⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⣿⣿⣿⣿⣿⣿⣿⣟⣬⣤⣯⣂⣵⣡⣤⣨⣅⢠⡀⠬⠀⣀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣾⣿⣿⣿⣿⢳⢲⡿⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⢽⣟⣿⠞⢷⠟⣿⢻⠟⠛⠟⠳⠆⠰⠄⠐⠀⠦⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣝⣿⣽⡷⣺⢛⠟⢏⣙⣯⣿⣭⣟⡉⣟⣻⣟⡀⣋⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠐⣤⣾⣄⢺⠛⢾⢦⢾⣿⠿⢃⣶⣾⣿⣿⣧⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠙⠋⠀⡖⢲⢠⢠⡠⢴⣠⣤⠀⠀⢢⣴⢰⢰⢤⢠⣤⢠⢤⣤⡀⢰⢠⡦⠂⣶⣰⡦⣀⠄⢰⣲⠄⡴⡄⣦⣴⠐⡲⢀⡊⠃⢉⢙⣭⣤⣤⣡⡄⡩⣬⡠⢹⣿⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠓⠊⠘⠚⠙⠚⠙⠒⠐⠀⠘⠃⠋⠘⠘⠘⠚⠘⠈⠒⠀⠘⠐⠊⠀⠃⠁⠣⠟⠀⠘⠀⠛⠉⠑⠋⠙⠀⠃⠸⡷⠀⠂⢰⠀⠀⠀⠒⠀⠒⢲⢄⠨⠯⣹⣿⡿⠩⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢠⠈⠀⠠⠐⠃⢤⠠⠈⠕⠀⠀⠘⠀⢀⡈⢃⠰⠸⣶⣿⡻⢿⠞⠃⠻⠃⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⢿⣿⣿⣿⣿⢯⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠔⣟⠀⡅⠈⢓⠰⠃⠀⠀⠀⠀⠈⠀⠛⢐⠀⠛⡋⢼⣮⢘⡇⢙⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠸⣿⣿⢿⣿⣷⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢋⠀⠉⢀⢀⡄⠉⡀⠀⠀⣀⠀⠀⠀⠛⢋⠶⠀⠀⠁⠀⡀⢀⡅⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⣾⡿⢿⣿⣿⣟⣿⣿⣿⠆⠀⠀⠀⠀⠀⠈⠃⠁⠀⠈⠀⠁⢀⣍⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⢉⣁⣉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣾⣿⣿⣿⣿⠿⡟⠀⠀⠠⠀⠘⠠⡄⠀⠆⠀⠦⢠⠙⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡄⢨⡅⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⠛⢿⠙⠁⠀⠶⠀⡀⠀⠖⢰⠄⣨⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡌⠡⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠖⠚⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠩⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣤⣤⣠⣄⣠⣤⣄⣀⣄⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⡀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⢿⣫⡟⣯⣿⢿⡿⣽⢿⠞⡷⢸⠃⠛⠃⠺⠂⠛⠀⠐⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⢻⡗⣿⣟⢿⡻⣿⢾⠻⢞⠟⠺⠃⠘⠣⠛⠋⠛⠀⠋⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣻⣟⣯⣻⠛⣿⣻⣻⠙⡏⢹⡏⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣻⢞⣗⣽⡻⣏⡿⣻⣙⣟⢺⣽⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢾⣽⢮⡧⣿⣿⣯⣷⣻⠠⣯⢖⣿⢉⡅⢨⡅⢨⡅⢨⠀⢌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣻⣯⣻⣾⣫⣿⣿⣿⣷⣏⣬⣄⣯⣀⣴⣠⣤⣨⣄⢠⡀⠤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣾⣼⣯⣿⣾⣷⣿⣿⣿⢳⢲⡷⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡷⣟⢾⣺⣷⢿⣿⣿⡷⢹⢟⡿⠞⢷⠟⢿⢺⠟⠛⠟⠳⠄⠠⠄⠐⠀⠢⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠃⣟⣵⣿⣻⠃⢹⡋⢹⣿⢿⣜⣿⡉⣟⣻⣇⠀⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠑⣤⣾⣄⢸⠛⢶⢦⢿⣿⠿⣋⣶⣾⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⠁⠀⡖⢢⢠⢠⣠⢴⣠⣤⠀⠀⢦⣴⢰⢰⢤⢠⣤⢠⢄⣤⡀⢰⢠⠴⠀⣶⣰⡦⣀⠄⢰⣲⠄⡴⡀⣦⣰⠐⡲⢀⠜⠃⢉⢙⢡⣠⣤⣥⡄⣩⠩⣤⢹⣿⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠓⠊⠘⠚⠙⠚⠙⠒⠔⠀⠘⠃⠋⠘⠘⠘⠒⠘⠘⠒⠀⠘⠐⠛⠀⠃⠁⠣⠟⠀⠘⠈⠛⠉⠓⠋⠙⠀⠃⢸⡷⠀⡂⢰⠂⠐⠀⠒⠂⠖⢲⢶⠨⠯⣿⣿⡿⠩⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⣿⣿⣿⣿⣿⣴⣿⣿⣿⡇⠀⠐⠀⠀⠀⠀⢠⠈⠀⡠⡜⠃⣬⠁⠈⡖⠀⠀⠘⠂⢠⡈⢋⠰⢸⣶⣿⠿⣿⢟⠇⠻⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢾⣿⣿⣿⣿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠂⢀⠒⣋⠀⠇⠈⢒⠘⡇⠀⠄⠄⠀⠘⠀⠛⢑⠈⠛⡋⢴⣯⢞⡧⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡷⢺⣿⣿⣿⣿⣷⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠋⠀⡉⢀⢀⣄⠉⡀⠀⠀⣀⠀⠀⠀⠛⠋⠖⠀⠀⠁⢐⣂⢚⡅⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠈⠀⠊⠃⠁⠀⠈⠈⠁⢀⣝⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⢉⣁⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⢿⣿⠛⠃⠀⠀⠠⠀⠘⠡⡄⠀⠎⠀⠮⢠⠙⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⢨⡅⢨⡅⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠿⢿⠛⡿⠀⠀⠀⠖⢀⡀⠀⠖⢴⠄⣬⠀⠂⠀⠄⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢨⡍⠨⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠓⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠡⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡄⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣄⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣒⣀⣀⣀⣂⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⡀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢿⡿⣽⢿⠿⡷⢾⠇⠻⠃⠺⢂⠛⠀⠛⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢿⣻⣿⢾⠻⢟⡟⢺⡃⠘⠣⢛⠋⠛⠀⠛⠀⠁⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣟⣯⣻⢛⣿⣿⣻⡙⡏⢹⡏⣙⡃⢛⠀⡛⠁⣛⠈⡃⠈⠁⠀⠀⠈⠁⠈⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣿⣻⣛⣟⣻⣿⢊⣛⢚⡃⢘⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣽⣟⣯⣏⣉⢈⣏⢈⡱⣈⡁⣉⠀⣉⠀⣀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣯⣿⣿⣯⣿⣿⡠⣯⢾⣿⢉⣅⢨⡅⢨⡅⢨⠀⣬⠀⣨⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣟⣬⣤⣯⣂⣵⣡⣤⣨⣅⢠⡀⠬⠀⣀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⢳⢲⡿⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⢽⣟⣿⠞⢷⡟⣿⢻⠟⠛⡟⠳⠆⠰⠄⠐⠀⠦⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡟⣟⣩⣯⢹⡿⢿⡏⣹⣿⣿⣍⣿⣉⢟⢹⣇⢈⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⣤⣾⣄⢘⠛⢶⢤⢾⣼⡿⣋⣷⣶⣿⣿⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠉⠀⡖⢢⢠⢠⣠⢴⢠⢄⠀⠀⢆⣴⢰⢰⢄⢠⢤⢠⢄⢤⡀⢰⢠⠄⠀⣶⢰⡦⣀⠄⢰⣲⠄⡴⡀⣦⣰⠐⡢⢀⠘⠃⠉⢙⣩⣠⣤⣥⡄⣩⣥⣤⢹⣿⣿⣿⣿⣷⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠓⠊⠘⠚⠑⠚⠙⠛⠔⠀⠘⠃⠋⠘⠘⠘⠛⠘⠘⠛⠀⠘⠘⠛⠀⠃⠁⠃⠟⠀⠘⠈⠓⠉⠃⠋⠙⠀⠃⠸⡿⠀⠀⢰⠀⠐⠀⠒⠂⠖⢲⣖⠨⡟⣿⣿⡿⠭⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣼⣽⣿⣿⣿⡆⠀⠀⠁⠀⠀⠀⢠⠈⠀⠠⠐⠃⢴⠀⠈⠔⠀⠀⠘⠀⢀⡀⢋⠰⢺⣾⣿⠹⢿⠞⠃⠻⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢂⠒⣋⠀⠇⠈⠗⠰⠃⠀⠀⠀⠀⠉⠀⠛⢀⠘⠛⡋⢼⣯⣘⡇⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡷⢻⣿⣿⣷⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠃⠀⡡⢀⢀⡄⠉⡀⠀⠀⣀⠀⠀⠀⠛⠋⠖⠀⠈⠀⠀⠀⢀⡁⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣾⡷⣿⣿⣿⣿⣿⣿⡿⠆⠀⠀⠀⢠⠀⠈⠃⠁⠀⠈⠀⠁⢀⣍⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⢉⣀⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⣿⣿⣿⣿⠟⠃⠀⠀⠠⠀⠐⠠⠄⠀⠆⠀⠦⠠⠘⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢨⡅⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⢟⠋⠿⠀⠀⠀⠖⠀⡀⠀⠖⢰⠄⣈⠀⠂⠀⠄⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠡⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠩⠀⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣄⣀⣄⣠⣤⣄⣀⣤⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⡀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⢿⣫⡟⣯⣿⢻⡿⣽⢿⠞⡷⢸⠃⠛⠃⠺⠂⠓⠀⠐⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⢻⡗⣿⣟⢿⡳⣿⢾⠻⢞⠟⠺⠃⠘⠣⠛⠋⠛⠀⠋⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣻⣟⣯⣻⠛⣿⣻⣻⠙⡏⢸⡏⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢞⡗⣽⡻⣏⡿⣻⣙⣟⢺⣽⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⢾⣽⢮⡧⣿⣿⣯⣷⣻⠠⣯⢖⣿⢉⡅⢨⡅⢨⡅⢨⠀⠌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣻⣾⣫⣟⣾⣿⣷⣏⣬⣄⣯⣀⣴⣠⣄⣨⣄⢠⡀⠤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣼⣼⣯⣿⣾⣷⣿⣿⣿⢳⢲⡷⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡷⣟⢾⣺⣷⢿⣿⣿⡷⢹⢟⡿⠞⢷⠟⢿⢺⠟⠛⠟⠳⠄⠠⠄⠐⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡕⡿⡏⡏⣹⡫⣿⡫⠿⠿⣿⣯⡟⣉⢟⣻⣇⠸⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠐⣤⣾⣄⢸⠛⢶⣤⣿⣾⡿⢋⣶⣶⣿⣿⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠁⠀⡖⢦⢀⢠⢠⣴⢀⢄⠀⠀⢆⣴⢰⢰⢅⢀⢄⢨⢌⣭⡁⢻⢩⡏⢩⣿⢠⣦⣀⡄⢰⣲⠄⡴⡀⣦⣰⠠⣢⢀⠜⠁⢉⢙⣥⣤⣤⣬⡄⣩⣭⣤⢹⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠓⠚⠘⠚⠑⠚⠙⠛⠐⠀⠘⠃⠋⠘⠘⠘⠛⠘⠘⠛⠃⠘⠘⠛⠀⠃⠁⢃⡟⠀⠘⠈⠒⠉⠃⠋⠙⠀⠃⠸⡷⠀⡀⢰⡂⠐⠀⠒⠒⠖⢲⡖⢸⡟⣿⣿⡿⠯⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣼⣿⣿⣿⣷⡆⠀⠐⠀⠀⠀⠀⢠⠈⠀⠠⡔⠃⢤⠁⠈⡗⠀⠀⠘⠀⢠⡈⢋⠰⢻⣼⣿⠸⡟⢞⠃⠻⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⢖⣏⠀⡇⠀⠒⠨⠃⠀⠄⠄⠀⠈⠀⠙⢑⠘⠛⡏⣼⣯⣝⡇⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢿⡿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣛⠀⠉⢀⢀⣄⠉⣀⠀⠀⣀⠀⠀⠀⠛⠋⠶⠀⠈⠀⢐⣁⢔⡁⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣾⠿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⢀⠀⠈⠃⠁⠀⠈⠈⠁⢀⣝⠛⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢉⣁⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣾⣿⣿⣿⣿⠿⡛⠀⠀⠀⠀⠘⠠⡄⠀⠆⠀⠦⠠⠝⢁⠁⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⢠⡄⢨⡅⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⢿⠛⢿⠙⠀⠀⠖⢀⡀⠀⠖⢰⡄⣨⠀⠂⠀⠄⠨⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢨⡍⢨⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠒⠊⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠡⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⢠⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣤⣠⣤⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢿⡿⣽⢿⠾⡷⢾⠇⠻⠃⠺⢂⠛⠀⠛⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢿⣻⣿⢾⠻⢟⡟⢺⡃⠘⠣⢛⠋⠛⠀⠛⠀⠁⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣟⣯⣻⢛⣿⣿⣻⡙⡏⢹⡏⣙⡃⢛⠀⡛⠁⣛⠈⡃⠈⠁⠀⠀⠈⠁⠈⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣻⣟⣿⣻⣛⣟⢺⣿⢊⣛⢚⡃⢘⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣝⣟⣫⣏⣉⢈⣏⢈⡱⣈⡁⣉⠀⣉⠀⣀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣯⣿⣿⣯⣿⣿⡠⣯⢾⣿⢉⣅⢨⡅⢨⡅⢨⠀⣬⠀⣨⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⣿⣿⣿⣿⣿⣿⣿⣟⣬⣤⣯⣂⣵⣡⣤⣨⣅⢠⡀⢬⠀⣀⡀⣀⢀⡀⠀⡀⠀⠀⣀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣾⣿⣿⣿⣿⢳⢶⡿⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⢽⣟⣿⠞⢷⠟⣿⢻⠟⠛⠟⠳⠆⠰⠄⠐⠀⠦⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣹⣧⣽⡿⣹⣧⠹⡯⣻⣷⣿⣭⣿⡉⣿⣿⡄⢘⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠐⣤⣾⣄⢺⠛⢾⢤⣿⣾⠟⣣⣶⣶⣾⣿⣆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠙⠉⠉⡭⢥⢁⢉⢁⣠⢈⣈⠈⠈⢌⣠⢩⢩⣁⢉⣉⢉⣉⣉⠉⢽⢉⣹⠉⣯⢡⣄⠀⡀⢠⢤⡀⣠⡀⣄⣠⠠⣤⢀⠜⠃⢉⢙⣩⣠⣤⣨⡄⣭⣅⣤⢹⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠧⠞⠸⠼⠳⠼⠹⠽⠠⠀⠸⠏⠎⠸⠸⠸⠽⠸⠘⠭⠁⠸⠨⡕⠄⠏⠋⢇⡿⠀⠸⠉⠶⠓⠣⠟⠻⠀⠇⢸⡷⠀⠂⢰⠂⠐⠀⢒⠂⠖⢢⣖⡸⡟⣿⣿⡿⡯⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⡆⠀⠐⠀⠀⠀⠀⢠⠀⠀⠠⡐⠂⠤⠁⠈⡗⠀⠀⠘⠀⢠⡈⢋⠰⢿⣶⣿⠺⡟⠞⠃⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢂⠖⣛⠀⡇⠀⢒⠨⡇⠀⠄⠄⠀⠘⠀⠙⠑⠘⠛⣏⣼⣯⣝⡇⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡽⢿⣿⣿⣷⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡋⠀⡡⢀⢀⣄⠁⣀⠀⠀⣀⠀⠀⠀⠛⠋⠆⠀⠈⠀⠘⣁⢚⡁⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣷⣞⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢀⠀⠚⠃⠁⠀⠈⠈⠁⢀⣍⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡁⢉⣀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠠⡀⠘⠡⡄⠀⠆⠀⠆⢠⠝⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢠⡅⢨⡅⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⡿⢟⠛⠋⠀⠀⠀⠖⢀⡄⠀⠖⢰⠄⣨⠀⠂⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣸⡍⢨⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢿⠒⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠡⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⢠⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣠⣄⣠⣤⣄⣀⣤⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢿⡿⣽⢿⠾⡷⢾⠇⠻⠃⠺⢂⠛⠀⠛⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢿⡻⣿⢾⠻⢟⡟⢺⡃⠘⠣⠛⠋⠛⠀⠛⠀⠁⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣟⣯⣻⢛⣿⣿⣻⠙⡏⢹⡏⢙⡃⢛⠀⡛⠁⣋⠈⡃⠈⠁⠀⠀⠈⠁⠈⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣻⣟⣿⣻⣛⣟⢺⣿⢊⣛⢚⡃⢈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣽⣿⣿⣝⣟⣫⣏⣉⢈⣏⢈⡱⣈⡁⣉⠀⣉⠀⣀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣯⣿⣿⣯⣿⣿⡠⣯⢾⣿⢉⣅⢨⡅⢨⡅⢨⠀⣬⠀⣨⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⣿⣿⣿⣿⣿⣿⣿⣟⣬⣤⣯⣂⣵⣡⣤⣨⣅⢠⡀⢤⠀⣀⡀⣀⢀⡀⠀⡀⠀⠀⣀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣾⣿⣿⣿⣿⢳⢲⡿⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣷⢽⣟⣿⠞⢷⠟⣿⢻⠟⠛⠟⠳⠆⠰⠄⠐⠀⠦⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣟⢍⣾⣩⡭⡹⡯⣝⠏⠻⠹⣿⣏⣛⡉⣿⣿⣇⠈⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⣤⣾⣄⢘⠛⢼⢤⢾⣾⠿⣃⣶⣶⣾⣿⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠌⠉⠈⡤⢅⢀⢉⢀⣠⢉⣀⠈⠁⢄⣠⢠⢩⣉⢈⣉⢉⣉⣉⠉⢩⢉⡩⠉⣭⢡⣄⠀⡀⢠⢤⡀⣠⡀⣄⢠⠠⢤⢀⠊⠃⢉⢝⣥⣤⣤⡡⡄⣭⣥⣤⢹⣿⣿⣿⣿⣷⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠀⠀⠧⠜⠸⠼⠻⠼⠻⠭⠠⠀⠸⠏⠞⠸⠸⠸⠭⠸⠘⠭⠁⠸⠨⠗⠀⠏⠋⢇⡿⠀⠸⠙⠶⠓⠣⠟⠻⠀⠇⠸⡷⠀⠂⢰⠂⠐⠀⠒⠂⠖⢢⡖⠘⡻⣿⣿⡟⡫⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣦⣶⣶⣴⣶⣶⣾⣧⡄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⡐⠂⣄⠠⠈⠗⠀⠀⠘⠀⢠⡀⠋⠰⢿⣴⣿⠸⠗⠆⠓⠺⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⢤⣏⠀⡀⠈⠒⠠⠇⠀⠀⠀⠀⠈⠀⠛⢐⠘⠛⣋⣼⣿⢜⡃⢙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⢿⣿⣿⣿⣿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠋⠀⠡⢀⢀⡄⠉⡀⠀⠀⣀⠀⠀⠀⠛⠋⠖⠀⠀⠀⢈⡁⢉⡁⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣾⠷⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢠⠀⠈⠃⠁⠀⠈⠀⠁⢀⣝⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢈⣀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣷⣿⣿⣿⣿⣿⠿⡏⠀⠀⠀⠀⠘⠠⡄⠀⠆⠀⠦⠠⠘⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢨⡅⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⠛⢿⠀⠀⠀⠶⠀⡀⠀⠖⢰⠄⣨⠀⠂⠀⠄⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡌⠠⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢫⠁⠚⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠡⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣀⣄⣠⣤⣀⣀⣤⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣫⡟⣯⣿⢻⡿⣽⢿⠞⡷⢸⠃⠛⠃⠺⠂⠓⠀⠐⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⢻⡗⣿⣟⢿⡳⣿⢾⠻⢞⠟⠺⠃⠘⠣⠛⠋⠛⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣻⣟⣯⣻⠛⣿⣻⣻⠙⡏⢸⡏⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣻⢞⡗⣽⡻⣏⡿⣻⣙⣟⢺⣽⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⢾⣽⢮⡧⣿⣿⣯⣷⣻⠠⣯⢖⣿⢉⡅⢨⡅⢨⡅⢨⠀⠌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣻⣯⣻⣾⣫⣟⣾⣿⣷⣏⣬⣄⣯⣀⣴⣠⣄⣨⣄⢠⡀⢤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣼⣼⣯⣿⣾⣷⣿⣿⣿⢳⢶⡷⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡷⣟⢾⣺⣷⢿⣿⣿⡷⢹⢟⡿⠞⢷⠟⢿⢻⠟⠛⠟⠳⠄⠠⠄⠐⠀⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⢛⡁⡚⢱⠏⣸⠧⢻⡛⡹⠭⡿⣌⣿⠉⡟⣿⣆⢘⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠑⣤⣾⣄⢚⠛⢾⣤⣿⣼⠟⣣⣶⣾⣾⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠉⠀⡤⢅⢈⢉⢀⣠⢀⡀⠀⠀⣅⣠⢠⢩⣁⢈⣉⢉⡉⣉⠉⣩⢉⣍⠉⣭⢡⣄⠀⡀⢠⢤⡀⣠⠀⣄⢠⠠⢤⢀⠘⠃⢉⢛⣩⣤⣤⡨⡄⣭⣡⣬⢹⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠧⠜⠸⠼⠻⠼⠻⠭⡠⠀⠸⠏⠞⠸⠸⠸⠭⠸⠸⠭⠁⠸⠨⠷⣌⠏⠋⠇⡿⠀⠸⠙⠦⠓⠧⠟⠻⠀⠇⢸⡿⠀⠀⢰⠂⠐⠀⢒⠂⠖⢠⡖⣚⡻⣿⣿⡿⣫⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣴⣤⣶⣤⣾⣶⣶⣶⣿⡆⠀⠐⠀⠀⠀⠀⢠⠀⠀⠠⡴⠢⣤⠁⠈⠖⠀⠀⠘⠀⢠⡈⢋⠰⢿⣴⣷⠸⡟⠔⠓⠻⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠒⣚⠀⡁⠀⠒⠰⡃⠀⠄⠀⠀⠈⠀⠙⢐⠘⠛⡋⣼⣟⢟⡇⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡽⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠋⠀⡡⢀⢀⡄⠉⣀⠀⠀⣀⠀⠀⠀⠛⢋⠖⠀⠀⠀⢈⡁⢉⡁⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡶⢜⢿⢿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢠⠀⠘⠃⠁⠀⠈⠈⠁⠀⣝⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢈⣀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣾⣺⣿⣿⣿⣿⠿⠟⠀⠀⠠⠀⠘⠡⡄⠀⠆⠀⠆⠠⠜⠁⠁⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⢨⡅⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⢿⠛⢿⠈⠀⠀⠖⠀⡄⠀⠒⢰⡄⣨⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢨⡍⠨⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢽⠁⠚⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠡⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣄⣠⣤⣄⣀⣤⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢿⡿⣽⢿⠾⡷⢾⠇⠻⠃⠺⢂⠛⠀⠛⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢿⡻⣿⢾⠻⢟⡟⢺⡃⠘⠣⠛⠋⠛⠀⠛⠀⠁⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣟⣯⣻⢛⣿⣿⣻⠙⡏⢹⡏⢙⡃⢛⠀⡛⠁⣋⠈⡃⠈⠁⠀⠀⠈⠁⠈⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣻⣟⣿⣻⣛⣟⢺⣿⢊⣛⢚⡃⢈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣽⣿⣿⣝⣟⣫⣏⣉⢈⣏⢈⡱⣈⡁⣉⠀⣉⠀⣀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣯⣿⣿⣯⣿⣿⡠⣯⢾⣿⢉⣅⢨⡅⢨⡅⢨⠀⣬⠀⣨⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⣿⣿⣿⣿⣿⣿⣿⣟⣬⣤⣯⣂⣵⣡⣤⣨⣅⢠⡀⢤⠀⣀⡀⣀⢀⡀⠀⡀⠀⠀⣀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣾⣿⣿⣿⣿⢳⢶⡿⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣷⢽⣟⣿⠞⢷⠟⣿⢻⠟⠛⠟⠳⠆⠰⠄⠐⠀⠦⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣯⣿⣻⡿⣽⣏⣹⣿⢿⣝⡟⡉⣟⢻⣇⠀⣛⣹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⣤⣾⣄⢚⠛⢾⢤⢾⣿⠿⣋⣦⣶⣿⣿⣧⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠋⠙⠉⡥⣅⢀⢁⢁⣡⠀⡁⠉⠉⡍⣨⢡⢩⡉⠈⡉⢉⡉⣉⠉⢹⢉⡟⠙⣿⢡⣄⠀⡀⢠⢤⡀⣠⠀⣄⢠⢠⢄⢀⠌⠃⢉⢛⣡⣠⣤⣤⡄⣭⣭⣤⢹⣿⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡈⠀⠀⠧⠼⠸⠼⠫⠼⠺⠾⡠⠀⠸⠏⠞⠸⠸⠸⠾⠸⠸⠾⠃⠸⠸⠷⠠⠏⠊⠏⡿⠁⠸⠙⠤⠓⠇⠟⠻⠀⠇⢸⡷⠀⠀⢰⠂⠐⠀⢒⠀⠖⢰⣔⠸⡟⣻⣿⡿⠫⠝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣶⣶⣶⣶⣶⣶⣴⣾⣶⣶⡄⠀⠁⠀⠀⠀⠀⢠⠀⠀⠠⡐⠂⢌⠡⠈⠔⠀⠀⠐⠀⢠⡀⠋⠰⢺⣶⣿⠸⠟⠆⠓⠺⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠽⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠔⣛⠀⡆⠈⠓⠠⠃⠀⠄⠄⠀⠈⠀⠛⠀⠘⠛⡋⣼⣿⢘⡃⢙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡴⢿⣿⣿⣷⣾⢷⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠋⠀⡉⢀⢀⡄⠁⡀⠀⠀⣀⠀⠀⠀⠛⠋⠖⠀⠀⠀⠈⠁⢈⡁⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢴⣶⣾⠿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈⠀⠁⢀⣝⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⢉⣁⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠸⣿⣿⣷⣾⣿⣿⣿⣿⠿⡋⠀⠀⠠⠀⠘⠠⡄⠀⠆⠀⠦⢠⠜⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⢨⡅⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⠿⠛⢏⠈⠁⠀⠖⠀⡄⠀⠖⢰⠄⣨⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⡌⠠⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢩⠁⠚⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠡⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣄⣠⣄⣠⣤⣀⣀⣤⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣒⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣫⡟⣯⣿⢿⡿⣽⢿⠞⡷⢸⠃⠛⠃⠺⠂⠛⠀⠐⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⢻⡗⣿⣟⢿⡻⣿⢾⠻⢞⠟⠺⠃⠘⠣⠛⠋⠛⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣻⣟⣯⣻⠛⣿⣻⣻⠙⡏⢹⡏⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣻⢞⣗⣽⡻⣏⡿⣻⣙⣟⢺⣽⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢾⣽⢮⡧⣿⣿⣯⣷⣻⠠⣯⢖⣿⢉⡅⢨⡅⢨⡅⢨⠀⢌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣻⣯⣻⣾⣫⣿⣿⣿⣷⣏⣬⣄⣯⣀⣴⣠⣤⣨⣄⢠⡀⢤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣾⣼⣯⣿⣾⣷⣿⣿⣿⢳⢶⡗⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡷⣟⢾⣺⣷⢿⣿⣿⡷⢹⢟⡿⠞⢷⠟⢿⢻⠟⠛⠟⠳⠄⠠⠄⠐⠀⠢⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣝⣮⣯⣏⣹⡿⠽⣋⣻⡟⣿⣍⣿⣈⣟⣿⣇⠀⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠑⣤⣾⣆⢚⡛⢾⢤⣿⣾⡿⢋⣦⣶⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠸⠉⠉⣁⣁⠀⠀⠈⢡⠉⠀⠉⠈⡈⢉⢉⢹⠋⠀⠈⠙⠛⠛⠛⢛⠛⠛⡛⣛⢃⡀⠀⠀⢀⣀⠀⣀⠀⡀⢀⢀⣀⢠⠜⠃⢉⢛⣭⣤⣤⣤⡄⠩⣭⣤⢹⣿⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠀⠀⣇⡼⢸⣸⢮⣹⠺⡻⢀⠀⢸⡟⡞⢸⢹⠸⡻⢸⠹⣛⠂⢸⢸⡿⢠⡿⠎⡏⡶⠁⢸⠚⣥⠯⢇⡟⢿⠀⡏⢸⡷⠀⠂⢰⠂⠐⠀⠒⠂⠶⢰⠶⠘⡟⣻⣿⡿⡫⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⣴⣤⣤⣤⣤⣤⣷⣶⣤⡄⠈⠁⠀⠀⠀⠀⢠⠀⠀⡠⡀⠀⣤⠡⠨⠗⠀⠀⠘⠀⢰⡈⢋⠰⢸⣶⣷⡾⠟⢟⠃⢺⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠂⢂⠴⣛⠀⠇⠀⠓⠨⡃⠀⠄⠔⠀⠘⠀⠛⢀⠛⠛⡏⣼⣟⣙⡇⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠷⢾⣿⣿⣿⣿⡷⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡛⠀⡩⢄⢀⡄⠉⠀⠀⠀⣀⠀⠀⠀⠛⠋⠖⠀⠈⠃⠘⠁⢍⡁⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡶⢾⠿⢿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠠⠀⠈⠂⠀⠀⠈⠈⠁⢀⣝⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡁⢉⣁⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⡿⣿⣿⣿⠿⠃⠀⠀⠠⠀⠘⠡⡄⠀⠆⠀⠦⢠⠝⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢠⡅⢨⡅⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⠿⠛⠿⠀⠀⠀⠖⠀⣄⠠⠒⢰⠄⣨⠀⠂⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡍⠨⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢩⠁⠛⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠡⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣠⣄⣠⣤⣄⣠⣄⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢿⡿⣽⢿⠾⡷⢾⠇⠻⠃⠺⢂⠛⠀⠛⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢿⣻⣿⢾⠻⢟⡟⢺⡃⠘⠣⢛⠋⠛⠀⠛⠀⠁⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣟⣯⣻⢛⣿⣿⣻⡙⡏⢹⡏⣙⡃⢛⠀⡛⠁⣛⠈⡃⠈⠁⠀⠀⠈⠁⠈⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣻⣟⣿⣻⣛⣟⢺⣿⢊⣛⢚⡃⢘⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣝⣟⣫⣏⣉⢈⣏⢈⡱⣈⡁⣉⠀⣉⠀⣀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣯⣿⣿⣯⣿⣿⡠⣯⢾⣿⢉⣅⢨⡅⢨⡅⢨⠀⢬⠀⣨⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⣿⣿⣿⣿⣿⣿⣿⣟⣬⣤⣯⣂⣵⣡⣤⣨⣅⢠⡀⢬⠀⣀⡀⣀⢀⡀⠀⡀⠀⠀⣀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣾⣿⣿⣿⣿⢳⢶⡿⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⢽⣟⣿⠞⢷⠟⣿⢻⠟⠛⠟⠳⠆⠰⠄⠐⠀⠦⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣷⣽⣾⣿⡿⡹⣿⠿⡿⣹⣽⢿⣝⣿⣉⣟⢹⣧⡀⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠑⣤⣾⣄⢘⠛⢾⢦⢾⣿⡿⢋⣦⣶⣾⣿⣦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠙⠋⠉⣀⣉⠁⠉⠁⢀⠁⠀⠈⠉⡉⢀⢁⢘⠛⠀⠉⠙⠛⠛⠛⢛⠛⠛⠛⣛⢃⡀⠀⠀⢀⣀⠀⢀⠀⡀⢀⢀⣀⢀⠘⠃⢉⢛⣥⣠⣤⣤⡄⣭⢩⣥⢹⣿⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣇⡸⢸⣸⢮⣹⢾⣛⢀⠀⢸⡟⡼⢸⢹⢸⣛⢸⠹⣛⠂⢸⢘⡻⠉⡟⠞⡏⣶⠁⢸⠚⣥⠯⢇⡿⢿⠀⡎⢸⡇⠀⠀⢰⡂⠐⠀⠒⠀⡶⢰⠖⠘⡟⣻⣿⡿⠫⠝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠈⠁⠀⠀⠀⠀⢠⠀⠀⠠⠀⠀⢤⠀⠈⠕⠀⠀⠘⠀⢠⡀⠋⠰⢸⣶⣿⠘⣿⡟⠓⠺⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⣿⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⢀⣏⠀⠆⠀⢓⠠⠇⠀⠀⠄⠀⠈⠀⠛⢐⠉⠛⡋⣼⣯⣘⡇⢙⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡶⢾⣿⣿⣿⣾⡦⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⢀⢀⡄⠉⡀⠀⠀⣀⠀⠀⠀⠛⠋⠖⠀⠀⠀⠈⠁⢈⡁⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⡽⣿⢿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢨⠀⠈⠂⠁⠀⠈⠀⠁⢀⣄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⢉⣁⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣾⣿⣿⣿⣿⠛⠃⠀⠀⠀⠀⠘⠠⡄⠀⠆⠀⠦⢠⠘⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⢨⡅⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⢿⠛⠿⠀⠀⠀⠶⠀⡀⠀⠖⢰⠆⣈⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣨⡌⠠⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠁⠊⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠩⠠⠉⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣠⣄⣠⣤⣀⣀⣄⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣫⡟⣯⣿⢿⡿⣽⢿⠞⡷⢸⠃⠛⠃⠺⠂⠛⠀⠐⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⢻⡗⣿⣟⢿⡻⣿⢾⠻⢞⠟⠺⠃⠘⠣⠛⠋⠛⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣻⣟⣯⣻⠛⣿⣻⣻⠙⡏⢹⡏⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣻⢞⣗⣽⡻⣏⡿⣻⣙⣟⢺⣽⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢾⣽⢮⡧⣿⣿⣯⣷⣻⠠⣯⢖⣿⢉⡅⢨⡅⢨⡅⢨⠀⢌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣻⣯⣻⣾⣫⣿⣿⣿⣷⣏⣬⣄⣯⣀⣴⣠⣤⣨⣄⢠⡀⢤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣾⣼⣯⣿⣾⣷⣿⣿⣿⢳⢶⡷⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡷⣟⢾⣺⣷⢿⣿⣿⡷⢹⢟⡿⠞⢷⠟⢿⢻⠟⠛⠟⠳⠄⠠⠄⠐⠀⠢⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡟⣿⣭⣿⣹⣿⣽⡊⣹⢽⣽⣬⢟⣉⣟⣸⣇⡀⣈⣹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠐⣤⣾⣄⢸⠛⢼⢦⢾⣾⡿⢋⣧⣶⣿⣿⣧⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠫⠉⠉⣁⡉⠀⠈⠀⢁⠁⠁⠉⠉⡁⢈⢉⢘⠛⠁⠈⠙⠛⠛⠛⢛⠛⠛⢛⣛⢃⡀⠀⠀⢀⣀⠀⢀⠀⡀⢀⢀⣀⢀⡈⠃⠉⢛⣩⣤⣤⣡⡄⣭⣭⣥⢹⣿⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣇⡸⢸⣸⢾⣹⢾⣓⢀⠀⢹⡟⡼⢸⢹⢸⣛⢸⢱⣛⠂⢸⢘⡿⠐⡟⠞⡏⣶⠁⢸⠺⣅⠯⣆⡿⢾⠀⡎⢸⡷⠀⠀⢰⠀⠐⠀⢒⠀⠖⢲⠄⠨⡯⣹⣿⡿⡫⠽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⡄⠈⠁⠀⠀⠀⠀⢠⠀⠀⠠⡀⠀⡤⠀⠨⠗⠀⠀⠘⠀⢠⡐⠋⠰⠸⣶⣿⡻⠗⠟⠓⠻⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢽⣿⣿⣿⣿⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢂⢒⣎⠀⡅⠀⠖⠠⡇⠀⠄⠄⠀⠈⠀⠛⢀⠊⠛⣫⣾⣮⢝⡇⢙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⢸⣿⣿⣿⣿⡷⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢛⠀⡉⢀⢀⡄⠉⡀⠁⠀⣀⠀⠀⠀⠛⠋⠶⠀⠀⠁⢀⡁⢈⡁⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡦⣶⡱⢿⡿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢠⠀⠈⠃⠁⠀⠈⠀⠁⠀⣝⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⢉⣁⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣿⣾⡻⢿⣿⣿⠿⠋⠀⠀⠠⠀⠘⠠⡄⠀⠆⠀⠆⢠⠘⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⢨⡅⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⢿⠿⠛⢿⠘⠀⠀⠖⠀⡀⠀⠒⢰⠄⣨⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡌⠠⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢩⠁⠚⠑⠀⠀⠀⠀⠀⠀⠉⠀⠀⠡⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣠⣤⣀⣀⣤⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣫⡟⣯⣿⢿⡿⣽⢿⠞⡷⢸⠃⠛⠃⠺⠂⠛⠀⠐⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⢻⡗⣿⣟⢿⡻⣿⢾⠻⢞⠟⠺⠃⠘⠣⠛⠋⠛⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣻⣟⣯⣻⠛⣿⣻⣻⠙⡏⢹⡏⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣻⢞⣗⣽⡻⣏⡿⣻⣙⣟⢺⣽⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢾⣽⢮⡧⣿⣿⣯⣷⣻⠠⣯⢖⣿⢉⡅⢨⡅⢨⡅⢨⠀⢌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣻⣯⣻⣾⣫⣿⣿⣿⣷⣏⣬⣄⣯⣀⣴⣠⣤⣨⣄⢠⡀⢤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣾⣼⣯⣿⣾⣷⣿⣿⣿⢳⢶⡷⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡷⣟⢾⣺⣷⢿⣿⣿⡷⢹⢟⡿⠞⢷⠟⢿⢻⠟⠛⠟⠳⠄⠠⠄⠐⠀⠢⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣽⣿⢿⡯⣻⡗⡝⡫⠻⠋⣿⣈⡟⣉⢟⣼⣆⡀⢈⣹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠐⣤⣾⣄⢘⠛⢾⢦⣿⣼⡿⢋⣦⣶⣿⣿⣧⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠉⠁⣀⡁⠀⠀⠀⢀⠀⠈⠈⠉⡉⢀⢀⢙⠛⠁⠀⠙⠛⠛⠛⢛⠛⠛⠛⣛⢃⡀⠀⠀⢀⣀⠀⢀⠀⡀⢀⢀⡀⢀⠼⠃⢉⢙⣩⣤⣤⣠⡄⢭⣭⣤⢹⣿⣿⣿⣿⣷⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣇⣸⢸⣸⢞⣹⢼⣵⢀⠀⢱⡟⡼⢸⢱⢸⣵⢸⢱⣽⡂⢸⢸⣥⠈⡟⠜⡟⣶⠁⢸⠺⣅⠯⡆⡿⢾⠀⡎⢸⡷⠀⠀⢰⠂⠐⠀⢒⠒⠖⢰⠄⠘⡿⡽⣿⡿⡯⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣤⣤⣤⣬⣵⡄⠀⠁⠀⠀⠀⠀⢠⠀⠀⠠⡠⠀⣤⡡⠘⠖⠀⠀⠘⠀⢠⡘⠋⠰⠻⣾⣿⣿⠇⢘⡓⠻⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠂⢂⢔⣛⠀⡇⠀⠖⠸⡃⠀⠄⠄⠀⢈⠀⠛⢘⠛⠛⠋⣽⣏⣘⡏⢛⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡷⢼⠿⣿⣿⣿⣿⣿⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠋⠀⡉⢀⢁⣄⠉⡀⠁⠀⣀⠀⠀⠀⠛⢋⠖⠀⠀⠈⢘⡀⢀⡀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡶⢶⡖⢺⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠠⠀⠈⠃⠀⠀⠈⠈⠁⢀⣜⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠈⣁⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣾⣖⣿⢾⣿⡿⠟⠀⠀⠠⠀⠘⠡⡄⠀⠆⠀⠦⢠⠝⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢠⡄⢨⡅⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⠛⢿⠘⠀⠀⠖⠀⡀⠠⠒⢰⠄⣬⠀⠂⠀⠄⠨⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡉⢈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢹⠁⠚⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠩⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡄⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣄⣤⣄⣠⣀⣤⣄⣤⣀⣀⣀⣀⣀⣀⣈⣁⣒⣀⣀⣀⣂⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢿⡿⣽⢿⠾⡷⢾⠇⠻⠃⠺⢂⠛⠀⠛⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢿⣻⣿⢾⠻⢟⡟⢺⡃⠘⠣⢛⠋⠛⠀⠛⠀⠁⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣟⣯⣻⢛⣿⣿⣻⡙⡏⢹⡏⣙⡃⢛⠀⡛⠁⣛⠈⡃⠈⠁⠀⠀⠈⠁⠈⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣻⣟⣿⣻⣛⣟⢺⣿⢊⣛⢚⡃⢘⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣝⣟⣫⣏⣉⢈⣏⢈⡱⣈⡁⣉⠀⣉⠀⣀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣯⣿⣿⣯⣿⣿⡠⣯⢾⣿⢉⣅⢨⡅⢨⡅⢨⠀⣬⠀⣨⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⣿⣿⣿⣿⣿⣿⣿⣟⣬⣤⣯⣂⣵⣡⣤⣨⣅⢠⡀⢬⠀⣀⡀⣀⢀⡀⠀⡀⠀⠀⣀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣾⣿⣿⣿⣿⢳⢶⡿⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⢽⣟⣿⠞⢷⠟⣿⢻⠟⠛⠟⠳⠆⠰⠄⠐⠀⠦⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⢿⣻⣿⣽⣿⣹⣯⢿⠯⢙⢭⣿⣞⣻⣉⣿⣻⣯⠀⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⣾⣄⢚⠛⢶⢤⢾⣼⡿⢋⣦⣶⣾⣿⣦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠋⠉⠈⠁⠁⠉⠁⢁⠈⠈⠉⠉⠉⠀⠉⢘⠟⠠⠍⠹⠿⠿⠿⠿⠿⠿⠿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠚⠁⢉⢛⣥⣤⣤⣩⡀⣭⣍⣥⢹⣿⣿⣿⣿⣷⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⡏⣹⢰⣰⢔⣺⢴⢶⢀⠀⢱⡞⡼⢸⢲⢰⢶⢰⢲⣶⠄⢸⢰⣖⠀⡟⡜⡗⣤⠂⢸⠽⣂⡟⡆⡷⣾⠈⡟⢸⡷⠀⠂⢰⠀⠐⠀⠒⠂⠖⢰⠖⠸⡟⣻⣿⡿⡫⠝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠈⠀⠉⠀⠈⠉⠈⠉⠀⠁⠈⠀⠈⠀⠁⠈⠈⢀⣥⣬⣤⣭⣄⣬⣨⣥⣤⣥⡄⠑⠃⠀⠈⠀⠉⢠⠀⠁⠈⠀⠁⣠⠁⠈⠖⠀⠀⠐⠀⢀⡈⠋⠰⠸⣶⣿⠙⢿⠖⠃⠺⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⣿⣿⣿⣿⢯⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢂⠐⣛⠀⠅⠈⠒⠈⠇⠀⠄⠄⠀⠈⠀⠛⢀⠈⠛⡋⢼⣯⢜⡃⢙⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠴⢺⣿⣿⣿⣾⢶⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠋⠀⡁⢀⢁⡄⠉⡀⠁⠀⣀⠀⠀⠀⠛⠋⠶⠀⠀⠀⢀⣀⢈⡁⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⣾⠿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠠⠀⠊⠂⠀⠀⠈⠈⠁⢀⣍⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢈⣁⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣾⣿⣿⣿⣿⠿⠋⠀⠀⠠⠀⠘⠡⡄⠀⠆⠀⠦⠠⠘⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡄⢨⡅⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⢻⣿⣿⣿⢿⠛⢟⠈⠀⠀⠖⠀⡀⠀⠒⢰⠄⣈⠀⠂⠀⠄⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢨⡌⠡⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢭⠑⠚⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠡⠠⠉⠐⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣠⣤⣀⣀⣤⣀⣤⣀⣀⣀⣀⣀⣀⣈⣁⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣫⡟⣯⣿⢿⡿⣽⢿⠞⡷⢸⠃⠛⠃⠺⠂⠛⠀⠐⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⢻⡗⣿⣟⢿⡻⣿⢾⠻⢞⠟⠺⠃⠘⠣⠛⠋⠛⠀⠉⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣻⣟⣯⣻⠛⣿⣻⣻⠙⡏⢹⡏⢘⡃⢛⠀⡛⠁⢋⠈⡃⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣻⢞⣗⣽⡻⣏⡿⣻⣙⣟⢺⣽⢈⣋⢚⡁⠈⡇⢈⠐⣉⠀⡉⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣟⣻⣽⣿⣟⣝⠋⣋⣏⣉⢈⡉⢈⡱⣈⡀⣉⠀⣉⠀⡀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢾⣽⢮⡧⣿⣿⣯⣷⣻⠠⣯⢖⣿⢉⡅⢨⡅⢨⡅⢨⠀⢌⠀⡀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣻⣯⣻⣾⣫⣿⣿⣿⣷⣏⣬⣄⣯⣀⣴⣠⣤⣨⣄⢠⡀⢤⠀⢀⡀⣀⢀⡀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣾⣼⣯⣿⣾⣷⣿⣿⣿⢳⢶⡷⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
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⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡿⣼⣼⣯⣿⣾⣷⣿⣿⣿⢳⢶⡷⣶⣶⣶⣶⣶⣶⣦⡄⢠⡄⢠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
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                    ䷩ 𝚕𝚒𝚗𝚎 2499

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

(ℹ) Images, hyperlinks and comments at https://news.techrights.org/n/2023/12/14/_Video_The_Fake_Economy_and_the_Pump_and_Dump_Scheme_With_Usele.shtml
    Gemini version at gemini://gemini.techrights.org/n/2023/12/14/_Video_The_Fake_Economy_and_the_Pump_and_Dump_Scheme_With_Usele.gmi

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ [Video] The Fake Economy and the Pump-and-
Dump Scheme With Useless Gadgets and False Promises/Prophecies From Companies
With Fake Valuations⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


posted by Roy Schestowitz on Dec 14, 2023


Video_download_link | md5sum dc9e410fe7e88f61052c20563f04373b
Userbase,_But_No_Business_Model
Creative Commons Attribution-No Derivative Works 4.0


🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇Preview_for_Userbase,_But_No_Business_Model⦈_


THE so-called "tech" industry isn't always about tech. It's sometimes about
conning some investment firms (or people's pension schemes among other things)
into "investing" in some farcical, unviable business plan or buying "shares" of
a company, based on paid-for puff pieces ("planted" in the media as part of the
"marketing" budget). This is how they inflate their supposed "worth" (if
they're in debt they say "that's normal" and if they never actually make any
money, as in income/profit being revenue minus expenses, they say "not yet!").
All sorts of fictional futuristic-sounding terms like "metaverse" are floated
around to maintain a much-needed flow of cash (right into the money pit, never
to be seen again). Vapourware and false promises are the norm, elevating into
management position some of the best con (wo)men, charlatans, and liars.
Plagiarism engines are mischaracterised as "chatbots" or "advanced AI" or
"generative AI" while informed critics are maligned as "Luddites" who stand in
the way of scientific and artistic progress.


What I'm saying should not sound eccentric. Do some basic research; it's rather
close to what's actually happening in Wall Street, whose indexes are dominated
by the supposed "worth" of very few companies that usually receive subsidies
from taxpayers if not bailouts (or "stimulus" or whatever they call it these
days; they keep making up new euphemisms to impede understanding).


The next video will speak about Gandi, but in this one I initially focus
automatic content recognition (ACR) in "smart TVs" and so-called "voice
assistants", which turned_out_to_be_a_passing_fad, costing some companies
billions (in losses) and resulting in device shutdowns, mass layoffs, and even
worse (bankruptcies, raids based on false pretexts, deaths, and Stasi-type
regimes with a colossal repository of recordings inside people's homes).


"That makes a strong case for requiring FOSS in appliances" according to an
associate, but I'd go further and that that it's a case for appliances without
an Internet connection, computer chips etc. There are many upsides to K.I.S.S.
(keep it simple, stupid). Repair, cost etc. I don't need a toaster or a kettle
that connects to the Internet, let alone one that comes with a computer chip.


Earlier this week one person said_in_Gemini: "I do not ever turn on WiFi. I am
happy with how the device runs and am not interested in firmware updates,
phoning home, or other shenanigans."


"Appliances will soon be unavailable without an active networked, Internet
connection," the associate noted. This may as well be true, but companies will
always exist that provide exceptions to this dire trend. "Not turning on the
appliances' wi-fi will not be an option before long," the associate said. "It
will be a prerequisite for using it. First to activate it, then later on to
even just use it."


It seems likely that a toaster or a kettle without a computer will always be
available. There are many practical benefits other than cost. If the market
demands it, that may become or perpetually remain available. It's hard to
believe they will pass laws which mandate toasters with microphones in them, to
use one outlandish concept.


The video speaks a lot about companies which over time embrace
"enshittification", looking for a way to finally make some money. One must
identity and flag services/products that cannot make money, as
"enshittification" or company shutdowns are inevitable. Either way, you los.


As a side note, US_national_debt_rose_to_unbelievable_levels_this_year_
(expected_to_cross_34_trillion_dollar_shortly,_perhaps_in_January). The clock
is ticking. To put it in some perspective, it_was_around_22_trillion_before
COVID-19. So it basically surged ~50% in about 4 years. This is not normal.


                      🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇US debt clock⦈


In the latest month on record, the US was borrowing 200,000 million dollars (in
one month). The prior month about 600,000 million or 20 BILLION DOLLAR PER DAY.
This is not a solid economy, it's a bailout economy, wherein corporations and
oligarchs suck all the capital and then moan about recession being caused by
"lazy workers" (or "quiet quitters", "great resignation" among other victim-
blaming myths).


When your government "creates" 20 billion dollars per day by merely borrowing
money, maybe you're at the cusp of Ponzi scheme gone awry. Many companies will
perish because they cannot be eternally sustained by bailouts. It deflates the
currency, causes high inflation, leaves people unable to buy basics, and in
turn fosters radical politics (see Argentina; it's_the_1990s_all_over_again)█




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

                    ䷩ 𝚕𝚒𝚗𝚎 3388

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

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ Links 14/12/2023: COVID-19 Rebounding,
Northern Ireland Police Breach Under Fresh
Scrutiny⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


posted by Roy Schestowitz on Dec 14, 2023


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇GNOME bluefish⦈


§ Contents⠀➾


    * Leftovers
          o Science
          o Education
          o Hardware
          o Health/Nutrition/Agriculture
          o Proprietary/Artificial_Intelligence_(AI)
          o Security
                # Integrity/Availability/Authenticity
                # Privacy/Surveillance
          o Defence/Aggression
          o Transparency/Investigative_Reporting
          o Environment
                # Energy/Transportation
          o Finance
          o AstroTurf/Lobbying/Politics
          o Censorship/Free_Speech
          o Freedom_of_Information_/_Freedom_of_the_Press
          o Civil_Rights/Policing
          o Internet_Policy/Net_Neutrality
          o Monopolies/Monopsonies
                # Patents
                # Copyrights
                # Gemini*_and_Gopher
                      # Personal/Opinions
                      # Technology_and_Free_Software
                            # Programming
    * § Leftovers⠀➾


          o ⚓ Project Censored ☛ Teacher's_Innovative_Approach_Humanizes
            Asylum-Seeker_Students⠀⇛


                 For every student whose first language isn’t English,
                 problems learning within the school system become
                 increasingly more common. Next to the language, there are
                 differences in family traditions, and social and cultural
                 backgrounds, along with possible trauma originating from
                 their journey to the United States.


          o ⚓ Terence Eden ☛ A_list_of_incredible_BlockChain_innovations_which
            I_would_like_funding_for_please⠀⇛


                 Dear the venture capitalists. I am a very charming white
                 man and am prepared to drop out of university if you'll
                 invest in these ideas.


          o ⚓ Digital Music News ☛ Cher_Arrives_in_Her_First_Roblox_Crossover
            Event⠀⇛


                 During the holiday-themed Cher collaboration with Warner
                 Music, players will be introduced to a new quest each
                 week in Harmony Hills, while recovering presents
                 scattered across the game world by a troublemaking elf.
                 Players can interact with Cher’s Roblox model to get in-
                 game rewards that pay homage to some of her most iconic
                 looks.


          o ⚓ Press Gazette ☛ What_The_Guardian_has_learned_from_five_years_of
            daily_news_podcasting⠀⇛


                 Safi added that they “go into every episode not assuming
                 any knowledge, so we’re going to build it for you from
                 the ground up and lead you through it in a very human way
                 as if you’re just talking to a friend at the pub or
                 something.


          o § Science⠀➾


                # ⚓ The Conversation ☛ Human_intelligence:_how_cognitive
                  circuitry,_rather_than_brain_size,_drove_its_evolution⠀⇛


                       All our organs have running costs, but some are
                       cheap and others expensive. Bones, for example, are
                       relatively cheap. Although they make up around 15%
                       of your weight, they only use 5% of your
                       metabolism. Brains are at the other end of the
                       spectrum, and at about 2% of typical human body
                       weight, running them uses around 20% of our
                       metabolism. And this without doing any conscious
                       thinking – it even happens when we’re asleep.


          o § Education⠀➾


                # ⚓ The Atlantic ☛ Why_This_Math_Professor_Objects_to_Diversity
                  Statements⠀⇛


                       Before immigrating to America from Russia as a
                       young academic, Alexander Barvinok lived under a
                       repressive regime that he experienced as “systemic
                       absurdity.” He is now a tenured mathematics
                       professor at the University of Michigan. Earlier
                       this year, he resigned his three-decade membership
                       in the American Mathematical Society in a letter
                       citing the group’s failure to oppose the growing
                       number of job openings for mathematics faculty that
                       require applicants to draft and submit a statement
                       on diversity, equity, and inclusion, or DEI. He
                       regards these statements as a gravely concerning
                       trend for his discipline, and he wanted to register
                       some sort of protest against them.


                       Painful experiences long ago convinced Barvinok
                       that requirements to affirm any ideal are corrosive
                       in academia.


                # ⚓ The Kent Stater ☛ Early_separation_plans:_Why_your
                  professor_may_have_left⠀⇛


                       The university hires an external advisor firm to
                       run data analysis and modeling based on the most
                       recent census of faculty and employees, according
                       to Polatajko. This division then calculates the
                       number of separations needed for the program and
                       how much is needed in incentives to reach their
                       desired number of separations.


          o § Hardware⠀➾


                # ⚓ Tom's Hardware ☛ AMD_says_overclocking_blows_a_hidden_fuse
                  on_Ryzen_Threadripper_7000_to_show_if_you've_overclocked_the
                  chip,_but_it_doesn't_automatically_void_your_CPU's_warranty_|
                  Tom's_Hardware [Ed: So proprietary designs = features that
                  exist only to be used against the customers, including false
                  positives]⠀⇛


                       AMD explains the hidden fuses behind Ryzen
                       Threadripper 7000 processors and how it will handle
                       warranty claims.


          o § Health/Nutrition/Agriculture⠀➾


                # ⚓ Patrick Breyer ☛ European_Health_Data_Space:_My_health_is
                  my_business,_respect_that!⠀⇛


                       The majority of citizens also do not want our
                       doctors to be able to view our entire medical
                       history, from mental disorders to abortions and
                       potency problems, without being asked.
                       Interconnected electronic health records can have
                       advantages, but as a Pirate, my conviction is:
                       Nobody other than myself has the right to decide
                       what is good for me and my health.


                # ⚓ Science Alert ☛ Woman_Who_Defied_Alzheimer's_Could_Show_Us
                  How_to_Avoid_It⠀⇛


                       Hope for millions.


                # ⚓ Latvia ☛ Number_of_hospitalized_patients_with_Covid
                  continues_to_rise⠀⇛


                       The number of Čovid-19 cases continues to increase
                       in Latvia, including the number of inpatient
                       patients, reaching 401 last week (307 a week ago),
                       the Disease Prevention and Control Center (SPKC)
                       reported on December 13.


                # ⚓ Defence Web ☛ Court_denies_Beitbridge_fence_contractors’
                  profit⠀⇛


                       That then Public Works and Infrastructure Minister
                       Patricia de Lille’s well-intentioned plan to
                       provide an extra buffer against COVID-19 in the
                       vicinity of the Beitbridge port of entry failed
                       spectacularly is well-known and a court judgement
                       this week put another nail in its coffin.


                # ⚓ Press Gazette ☛ Covid-19_origins:_A_media_conspiracy_of
                  silence⠀⇛


                       Why did so many journalists dismiss as a conspiracy
                       the idea Covid-19 leaked from a lab in Wuhan?


                # ⚓ The Straits Times ☛ Medical_experts_in_Malaysia_call_for
                  stringent_measures_as_Covid-19_cases_continue_to_rise⠀⇛


                       Most cases are mild and do not require
                       hospitalisation, though the jump in numbers is of
                       concern.


                # ⚓ The Straits Times ☛ Scientists_see_risk_of_lost_opportunity
                  for_long_Covid_research_in_China⠀⇛


                       Experts said little is known about China's
                       experience with long-term Covid-19 effects.


                # ⚓ New York Times ☛ How_a_Rare_Myocarditis_Death_Caught_the
                  Attention_of_the_Anti-Vaccine_Movement [Ed: "Anti-Vaccine
                  Movement" blamed on something other than mandating on
                  everybody COVID-19 products which turned out not to be
                  effective as advertised; this seems like a distraction from
                  the elephant in the room.]⠀⇛


                       A 24-year-old’s sudden death devastated his family
                       — and caught the attention of the movement of
                       vaccine opponents.


                # ⚓ University of Michigan ☛ Genetic_depression_‘protection’_no
                  match_for_pandemic_stress⠀⇛


                       Living through a historic pandemic while handling
                       the stress of the first year of college sent one-
                       third of students in a new U-M study into clinical
                       depression.


          o § Proprietary/Artificial Intelligence (AI)⠀➾


                # ⚓ The Register UK ☛ Adobe_warns_it_may_face_massive_fines_for
                  subscription_cancellation_practices⠀⇛


                       Adobe has revealed it may have to fork out
                       "significant monetary costs or penalties" as a
                       result of a US Federal Trade Commission (FTC)
                       investigation of its subscription cancellation
                       practices.


                       The graphics giant slipped the news into a filing
                       that details its Q4 and FY 2023 results and other
                       corporate events.


                       Adobe's results were strong. The fourth quarter saw
                       $5.05 billion of revenue, 12 percent year-over-year
                       growth and the first time Adobe has reached the $5
                       billion mark in a single quarter. Quarterly net
                       income reached almost $1.5 billion. Full year
                       revenue was $19.4 billion, up ten percent year-
                       over-year, and net profit reached $5.4 billion. CFO
                       and executive veep Dan Durn hailed "world-class
                       margins." President of digital media David Wadhwani
                       celebrated record numbers of new commercial
                       subscriptions for Adobe Creative Cloud. Execs
                       forecast Q1 2024 revenue of between $5.1 and $5.15
                       billion, and annual revenue topping $21.3 billion.


                # ⚓ Ars Technica ☛ GM_justifies_decision_to_ditch_Apple_CarPlay
                  due_to_stability_issues_|_Ars_Technica⠀⇛


                       The automaker's decision to ditch phone casting
                       interfaces remains controversial.


                # ⚓ Harvard University ☛ The_Hidden_Costs_of_Layoffs⠀⇛


                       It can take years for companies to bounce back from
                       these setbacks.


                # ⚓ Tech_Job_Losses_Soar_in_2023:_An_Alarming_Surge_Before
                  Year-End⠀⇛


                       You’ve got it right — tech layoffs are
                       unfortunately becoming a familiar trend for someone
                       like me, who’s been part of the tech industry for
                       over 16 years.


                       Having weathered through multiple rounds of
                       layoffs, including being laid off twice since 2010,
                       today brings news of another significant blow.


                       Amazon’s ongoing layoffs, which began earlier this
                       year, are extending across multiple divisions. The
                       current workforce reduction encompasses “several
                       hundred” employees from the Alexa division and the
                       recently formed Artificial General Intelligence
                       team. Moreover, the gaming division is also
                       affected, with 180 workers facing job cuts.


                # ⚓ InformationWeek ☛ IT_&_Trauma_in_the_Gaza_War⠀⇛


                       In Israel, tech workers struggle to keep things
                       afloat while their senior staff are on the front
                       lines. In Gaza, they struggle with a near-complete
                       communications shutdown.


                # ⚓ The Atlantic ☛ Why_It’s_So_Hard_to_Search_Your_Email⠀⇛


                       Comparatively speaking, web search has it easy. It
                       receives billions of search queries every single
                       day, creating data that companies can use to hone
                       their results. Even if you’re searching for
                       something very weird—say, Teenage Mutant Ninja
                       Turtles pickleball paddle—“probably dozens of
                       people before you have already done that search,”
                       Thorsten Joachims, a search expert at Cornell
                       University, told me. Google or any other search
                       company can track our behavior, seeing which search
                       results users click on as they try to find the
                       answer, and what websites they eventually end up
                       on. It can track the different queries a user tries
                       as they attempt to zero in on exactly what
                       information they are seeking.


                # ⚓ Dictionary.com LLC ☛ The_Dictionary.com_Word_of_the_Year_is
                  hallucinate.⠀⇛


                       When we look back on 2023 from whatever surreal
                       future it forks into, we’ll remember it as the year
                       that at least this much became clear: AI will
                       forever change how we work, learn, create, interact
                       with (mis)information, and think about ourselves.


                # ⚓ The Verge ☛ Extremely_softcore:_the_old_Twitter_was_an
                  idealist’s_workplace_and_a_naive_business⠀⇛


                       It’s hard to imagine the platform reaching the
                       political and cultural relevance it once had, and
                       it’s even harder to see how it could be the same
                       force for progressive social movements, especially
                       as its new owner touts a free speech policy that
                       boils down to his retweeting antisemitic
                       conspiracies. Dorsey had deluded himself into
                       thinking Twitter could be the utopian ideal of a
                       tech company. Employees wanted to believe it could
                       last.


                       “When Elon Musk took over, it was like the mask
                       slipped,” said an engineer who left just before the
                       billionaire’s arrival. “We realized we worked under
                       capitalism.”


                # ⚓ Dedoimedo ☛ Lockdown_Mode_in_iOS_-_A_fantastic_Internet
                  crap_reduction_tool⠀⇛


                       I came looking for copper, and I found gold. This
                       be a review of the Lockdown Mode in iOS 16/17 on
                       iPhone, focusing on intended purpose of security
                       and additional unintended benefits as an Internet
                       sanitization tool, activation, functionality, app
                       and site exclusion, some other observations, and
                       more. I came for security, I stayed because it
                       reduces stupidity. Yup. Do read on.


                # ⚓ Techdirt ☛ Techdirt_Podcast_Episode_373:_Could_Engineer
                  Licensing_Make_AI_Safer?⠀⇛


                       There’s plenty of discussion right now about what
                       approach, regulatory or otherwise, could rein in
                       the potential harms of AI systems. This week, we’re
                       joined by professor and researcher Chinmayi Sharma
                       to discuss her brand new article that offers an
                       interesting idea: professionalizing the AI field
                       with engineer licensing and a malpractice standard.


          o § Security⠀➾


                # § Integrity/Availability/Authenticity⠀➾


                      # ⚓ APNIC ☛ Models_of_trust_for_the_RPKI⠀⇛


                             In the 1980s, during the early days of the
                             Internet, the global address plan utilized
                             the class structure. Jon Postel, as the
                             Internet Assigned Numbers Authority (IANA)
                             record keeper, managed allocations of Class A
                             address blocks while delegating
                             administration and record-keeping for
                             numerous Class B and C address blocks to the
                             Internet Network Information Centre
                             (InterNIC), operated by Stanford Research
                             International (SRI-NIC).


                # § Privacy/Surveillance⠀➾


                      # ⚓ The Register UK ☛ Northern_Ireland_cops_count_human
                        cost_of_August_data_breach⠀⇛


                             Operational impact on the PSNI is also
                             thought to be significant, with costs
                             expected to be in the region of £24-37
                             million ($30-46.5 million) – a sum the PSNI
                             could not afford due to other financial
                             constraints. The total costs encompass
                             factors such as home security, litigation,
                             and a potential regulatory penalty.


                      # ⚓ The United Kingdom ☛ Review_into_PSNI_data_breach⠀⇛


                             Today, the NIPB have published the review's
                             findings.


                      # ⚓ Northern Ireland Policing Board UK ☛ “Protecting_From
                        Within”:_A_review_commissioned_by_the_Police_Service_of
                        Northern_Ireland_(PSNI)_and_the_Northern_Ireland
                        Policing_Board_(NIPB),_into_the_PSNI_data_breach_of_8th
                        August_2023 [PDF]⠀⇛


                             1.1 The Police Service of Northern Ireland
                             (PSNI) is the third largest police service in
                             the United Kingdom (UK) in terms of officer
                             numbers, and the second largest in terms of
                             geographic area of policing responsibility.
                             Following the establishment of the Patten
                             Commission, the Royal Ulster Constabulary
                             George Cross (RUCGC) transitioned to the PSNI
                             on the 4th November 2001. All major political
                             parties now support the PSNI, and form part
                             of the Northern Ireland Policing Board (NIPB)
                             that holds the PSNI to account for the
                             delivery of an efficient, effective, and
                             representative service.


                      # ⚓ Politico ☛ Fear_stalks_Northern_Irish_police_after
                        catastrophic_leak_of_personal_data⠀⇛


                             The unprecedented breach — committed Tuesday,
                             when the police force uploaded the wrong
                             Excel spreadsheet to a Freedom of Information
                             website — revealed details many officers had
                             kept secret even from relatives and close
                             friends.


                      # ⚓ BBC ☛ PSNI_data_breach:_Two_men_released_after
                        Terrorism_Act_arrests⠀⇛


                             In total four arrests have been made in
                             relation to the data breach.


                      # ⚓ [Old] PBS ☛ William_Binney⠀⇛


                             In '97, I became the technical director of
                             the geopolitical -- military geopolitical
                             analysis and reporting shop for the world,
                             which was about 6,000 people. Those were
                             analysts and reporters, the people, not the
                             technicians or the computer people. It was
                             more analyzing the data to figure out what
                             was going on and report it. ...


                             But part of the issue was volume, velocity
                             and variety, as the agency put it. ... This
                             is too much data. It's moving too quickly,
                             you know, too many kinds of data; we can't do
                             it.


                      # ⚓ Citizen Lab ☛ Peer-reviewed_publication:_Regulating
                        Transnational_Dissident_Cyber_espionage⠀⇛


                             In this article, Siena argues that the
                             absence of an international law to prevent
                             transnational repression permits states or
                             entities connected to states to use digital
                             technologies to suppress or silence dissent
                             from activists, human rights defenders, and
                             dissidents abroad by gathering personal data
                             that is subsequently used as a weapon against
                             the target or their networks. To protect and
                             provide safe spaces to targeted communities,
                             there is a need for states to work together
                             to define dissident cyber espionage, as well
                             as to develop a set of policies that will
                             discourage it and make it easier for victims
                             to seek justice.


                      # ⚓ uni Cambridge ☛ Regulating_Transnational_Dissident
                        Cyber_Espionage⠀⇛


                             Remote-access cyber espionage operations
                             against activists, dissidents or human rights
                             defenders abroad are increasingly a feature
                             of digital transnational repression. This
                             arises when State or State-related actors use
                             digital technologies to silence or stifle
                             dissent from human rights defenders,
                             activists and dissidents abroad through the
                             collection of confidential information that
                             is then weaponized against the target or
                             their networks. Examples include the
                             targeting of Ghanem Al-Masarir (a Saudi
                             dissident living in the United Kingdom),
                             Carine Kanimba (a United States–Belgian dual
                             citizen and daughter of Rwandan activist Paul
                             Rusesabagina living in the United States) and
                             Omar Abdulaziz (another Saudi dissident
                             living in Canada) with NSO Group's mercenary
                             spyware. This practice erodes human rights,
                             democracy and the rule of law and has a
                             negative impact on targeted communities,
                             including social isolation, self-censorship,
                             the fragmentation and impairment of
                             transnational political and social advocacy
                             networks, and psychological and social harm.
                             Despite this, international law does little
                             to restrain this practice. Building on
                             momentum around the regulation of mercenary
                             spyware and transnational repression, this
                             article elaborates on how States could
                             consider regulating dissident cyber espionage
                             and streamlines a unified approach among
                             ratifying States addressing issues such as
                             State immunity, burden of proof, export
                             control and international and public–private
                             sector collaboration.


                      # ⚓ Patrick Breyer ☛ European_Health_Data_Space:_EU
                        Parliament_opposes_mandatory_electronic_patient_records
                        for_all_citizens,_but_supports_granting_access_to
                        sensitive_health_data_without_asking_patients⠀⇛


                             ”A compulsory electronic patient file with
                             Europe-wide access entails irresponsible
                             risks of theft, hacking or loss of the most
                             personal treatment data and threatens to
                             deprive patients of any control over the
                             collection of their illnesses and disorders,”
                             emphasises Breyer, co-lead negotiator for the
                             Greens/European Free Alliance group in the EU
                             Parliament’s Civil Liberties Committee. “This
                             is nothing other than the end of medical
                             confidentiality. Have we learnt nothing from
                             the international hacker attacks on hospitals
                             and other health data?”


                      # ⚓ EFF ☛ Is_This_the_End_of_Geofence_Warrants?⠀⇛


                             Geofence warrants require a provider—almost
                             always Google—to search its entire reserve of
                             user location data to identify all users or
                             devices located within a geographic area
                             during a time period specified by law
                             enforcement. These warrants violate the
                             Fourth Amendment because they are not
                             targeted to a particular individual or
                             device, like a typical warrant for digital
                             communications. The only “evidence”
                             supporting a geofence warrant is that a crime
                             occurred in a particular area, and the
                             perpetrator likely carried a cell phone that
                             shared location data with Google. For this
                             reason, they inevitably sweep up potentially
                             hundreds of people who have no connection to
                             the crime under investigation—and could turn
                             each of those people into a suspect.


                             Geofence warrants have been possible because
                             Google collects and stores specific user
                             location data (which Google calls “Location
                             History” data) altogether in a massive
                             database called “Sensorvault.” Google
                             reported several years ago that geofence
                             warrants make up 25% of all warrants it
                             receives each year.


                             Google’s announcement outlined three changes
                             to how it will treat Location History data.
                             First, going forward, this data will be
                             stored, by default, on a user’s device,
                             instead of with Google in the cloud. Second,
                             it will be set by default to delete after
                             three months; currently Google stores the
                             data for at least 18 months. Finally, if
                             users choose to back up their data to the
                             cloud, Google will “automatically encrypt
                             your backed-up data so no one can read it,
                             including Google.”


          o § Defence/Aggression⠀➾


                # ⚓ New York Times ☛ Political_Debate_Is_Rife_on_TikTok.
                  Politicians?_Not_So_Much.⠀⇛


                       Some pundits call next year the “TikTok election”
                       because of the ballooning power and influence of
                       the video app. TikTok may have been known for viral
                       dances in 2020, but it has increasingly become a
                       news [sic] source for millennials and Gen Z-ers,
                       who will be a powerful part of the electorate.


                # ⚓ CS Monitor ☛ Waves_of_arriving_Rohingya_test_limits_of
                  Indonesia’s_welcome⠀⇛


                       Indonesia once tolerated such landings while
                       Thailand and Malaysia pushed them away. But the
                       growing hostility of some Indonesians toward the
                       Rohingya, who come from refugee camps in
                       Bangladesh, has put pressure on President Joko
                       Widodo’s government to take action.


                # ⚓ Michigan News ☛ Michigan_false_electors_appear_in_court;
                  possible_witness_dismissed_from_hearing⠀⇛


                       As part of the ongoing preliminary examinations,
                       which are for a judge to determine if there is
                       sufficient enough evidence to send a case to trial,
                       state attorneys must prove whether the group knew
                       they were committing forgery when sending an
                       alternate slate of Republican electors to the
                       National Archives and Congress.


                # ⚓ Marcy Wheeler ☛ SCOTUS_Takes_Up_January_6_Obstruction
                  Challenge_—_But_with_Unknown_Scope⠀⇛


                       Remember that defendants have always likened the
                       January 6 attack with the interruption by
                       protestors of Brett Kavanaugh’s confirmation
                       hearing (there are significant differences,
                       starting with the fact that all the protestors who
                       disrupted Kavanaugh’s hearing were in the building
                       legally). So I wouldn’t even rule out some set of
                       Republicans rejecting this application on those
                       grounds.


                # ⚓ The Scotsman ☛ COP28:_What_has_been_agreed_by_delegates?
                  What_does_'transition_away'_from_fossil_fuels_mean?⠀⇛


                       Many countries, such as small island states which
                       are severely threatened by rising seas, believe
                       there are still too many loopholes in the
                       agreement, but expressed confidence that it will
                       lead to stronger action in future.


                # ⚓ Democracy Now ☛ Phase_Down,_Not_Phase_Out:_COP28_Deal_on
                  Fossil_Fuels_Disappoints_Activists_&_Vulnerable_States⠀⇛


                       As the United Nations climate summit ends with
                       nations pledging to transition away from fossil
                       fuels instead of explicitly calling for a fossil
                       fuel phaseout, we go to the COP28 site in Dubai for
                       a debrief with Asad Rehman, spokesperson for the
                       Climate Justice Coalition. He says the deal
                       overseen by COP28 president and head of the Abu
                       Dhabi National Oil Company Sultan Al Jaber is a
                       “very weak text” with “lots of loopholes” that
                       allows rich countries to avoid responsibility. “If
                       you’re an oil and gas baron and CEO, you must be
                       rubbing your hands with glee,” says Rehman. “This
                       requires everybody to take action, but developing
                       countries can only act if they’re given the
                       support.”


                # ⚓ The Strategist ☛ A_strategy_for_defence_innovation_across
                  the_supply_chain⠀⇛


                       Lockheed Martin can’t build an F-35 fighter jet
                       without engines from Pratt & Whitney. Pratt &
                       Whitney maintains similar relationships with a
                       range of tier-two suppliers, forming a complex
                       ecosystem that took 20 years to build. Many of
                       those subcomponent suppliers rely on tier threes,
                       and so on down to the nuts and bolts that hold
                       everything together.


                # ⚓ The Strategist ☛ Australia_must_act_now_to_develop_a
                  national_security_strategy⠀⇛


                       Two-thirds of the top 50 global powers by GDP
                       produce an NSS or equivalent policy document.
                       Eighty per cent of those ranked in the top 15 have
                       an NSS; Australia (which ranks 14th), India (5th)
                       and Brazil (9th) are the outliers.


                       Focusing on those major powers that either don’t
                       produce an NSS or don’t make it publicly available,
                       it becomes clear that population size and
                       geographic region have less bearing on the decision
                       than the characterisation of the government and its
                       political approach.


                # § Russia, Belarus, and War in Ukraine⠀➾


                      # ⚓ New York Times ☛ Ukraine_Faces_Critical_Tests_as_It
                        Duels_With_Russia_for_Stamina⠀⇛


                             With Western support for Kyiv softening and
                             Congress holding up urgently needed aid,
                             Vladimir Putin’s bet on outlasting Ukraine
                             and its allies is looking stronger.


                      # ⚓ Meduza ☛ Putin_says_Russia_will_not_have_another_wave
                        of_mobilization,_echoing_promises_made_before_first
                        wave_—_Meduza⠀⇛


                      # ⚓ Atlantic Council ☛ The_US_and_NATO_must_clamp_down_on
                        Russian_fossil_fuels_to_end_the_war_in_Ukraine⠀⇛


                             The US and its EU allies have made several
                             attempts to diminish Russia's fossil fuel
                             exports, with mixed results. the West must do
                             more to staunch the flow of Russian oil and
                             gas—and restore peace for Ukraine.


                      # ⚓ PHR ☛ White_House_Must_Not_Sacrifice_U.S._Asylum
                        System_in_Funding_Negotiations:_PHR⠀⇛


                             In response to reports that the White House
                             and congressional leaders are considering
                             overhauls to the U.S. asylum system as a
                             compromise to secure funding for Ukraine,
                             Israel, and Taiwan, the following quote is
                             attributable to Michele Heisler, MD, MPH,
                             medical director at PHR and professor of
                             internal medicine and public health [...]


                      # ⚓ France24 ☛ EU_faces_challenge_from_Hungary's_Orban
                        over_plans_to_increase_Ukraine_support⠀⇛


                             EU leaders met Thursday to boost their long-
                             term support for Ukraine in its fightback
                             against Russia's invasion and secure a
                             European future, but faced a defiant
                             challenge from Hungary's Prime Minister
                             Viktor Orban.


                      # ⚓ France24 ☛ Scores_wounded in Russian_strikes on_Kyiv
                        and_Odesa as_Ukraine_seeks_more_aid_from_Europe⠀⇛


                             A barrage of Russian missiles targeted Kyiv
                             on Wednesday, wounding more than 50 people,
                             and dozens of drones targeted Odesa on
                             Thursday, wounding more than 10, said
                             Ukrainian officials, as President Volodymyr
                             Zelensky sought more military support in
                             Europe after a trip to Washington secured no
                             new pledges.


                      # ⚓ France24 ☛ More_than_50_injured_in_Russian_missile
                        strikes_on_Kyiv⠀⇛


                             Ukraine’s capital came under another
                             ballistic missile attack early Wednesday,
                             resulting in more than 50 injuries and
                             several damaged buildings, officials said.


                      # ⚓ JURIST ☛ ECHR_hears_oral_arguments_from_Ukraine_in
                        case_over_Russian_occupation_of_Crimea⠀⇛


                             The European Court of Human Rights (ECHR)
                             heard oral arguments on Wednesday in a case
                             concerning potential human rights violations
                             committed by Russia via its occupation of
                             Crimea.


                      # ⚓ JURIST ☛ US_sanctions_hundreds_of_people_and_entities
                        who_‘support_Russia’s_war’_in_Ukraine⠀⇛


                             The US Department of the Treasury, in
                             coordination with the State Department,
                             announced sanctions on over 200 individuals
                             and entities on Tuesday to target Russia’s
                             war capabilities in the ongoing war in
                             Ukraine. The sanctions target Russia’s energy
                             production, metals and mining sector,
                             munitions transfers and defense industry.


                      # ⚓ LRT ☛ Groundbreaking_summit_or_historic_fiasco_–_will
                        EU_leaders_convince_Hungary_to_relent_on_Ukraine_aid
                        and_expansion?⠀⇛


                             European Union leaders meet in Brussels on
                             Thursday for a two-day meeting to finalise
                             further support for Ukraine. In the corridors
                             of Brussels, diplomats and officials feel
                             that what’s at stake here is both the vital
                             financial and moral support for Kyiv and the
                             very unity of the EU.


                      # ⚓ LRT ☛ Lithuanian_president_says_Hungary_‘harming
                        itself’_by_objecting_to_Ukraine’s_EU_accession_talks⠀⇛


                             Lithuanian President Gitanas Nausėda has
                             criticised Hungary for objecting that the EU
                             start accession talks with Ukraine.


                      # ⚓ RFA ☛ Russian_governor_visit_points_to_more_worker
                        exports⠀⇛


                             Facing manpower shortage, Russia needs North
                             Korean labor due to its war in Ukraine.


                      # ⚓ RFERL ☛ Court_Cancels_Fine_For_Russian_Rights
                        Activist⠀⇛


                             The Moscow City Court on December 14
                             cancelled a lower court's decision to fine
                             veteran human rights defender Oleg Orlov
                             150,000 rubles ($1,660) for the "repeated
                             discrediting of Russia's armed forces"
                             involved in the invasion of Ukraine.


                      # ⚓ RFERL ☛ Zelenskiy_Urges_EU_Leaders_Not_To_'Betray'
                        Ukrainians'_European_Aspirations⠀⇛


                             President Volodymyr Zelenskiy called on
                             leaders attending a crucial EU summit in
                             Brussels to green-light Kyiv's opening of
                             membership negotiations with the bloc and
                             told them that a negative vote would "betray"
                             Ukrainians' European dreams and embolden
                             Russia in its aggression against Ukraine.


                      # ⚓ RFERL ☛ Downed_Russian_Drone_Falls_On_Romanian
                        Territory_Amid_Attack_On_Southern_Ukraine⠀⇛


                             Romania's Defense Ministry says a drone shot
                             down during Russia's attack on Ukraine's port
                             installations on the Danube early on December
                             14 fell on Romanian territory, leaving a 1 1/
                             2-meter deep crater near the village of
                             Grindu, which is located some 8 kilometers
                             from the Ukrainian port of Reni.


                      # ⚓ RFERL ☛ Russian_Opposition_Politician_Yashin's_Prison
                        Sentence_Upheld_On_Appeal⠀⇛


                             A court of appeals in Moscow on December 13
                             rejected an appeal filed by opposition
                             politician Ilya Yashin, who is serving an 8
                             1/2-year prison term for his criticism of
                             Moscow's invasion of Ukraine.


                      # ⚓ RFERL ☛ U.S._Sanctions_Target_Russian_Firm_That
                        Received_Dual-Use_Tech_From_Kazakhstan⠀⇛


                             The United States has imposed sanctions on a
                             Russian firm that RFE/RL found had received
                             hundreds of shipments from Kazakhstan of
                             dual-use technology that Western governments
                             say Russia is using in weapons it is
                             deploying in its war against Ukraine.


                      # ⚓ YLE ☛ Finnish_PM_Orpo_criticises_Hungary's_opposition
                        to_Ukraine_aid,_EU_membership⠀⇛


                             Hungarian Prime Minister Viktor Orban has
                             demanded concessions from the EU in exchange
                             for his country supporting a €54bn aid
                             package for Ukraine.


                      # ⚓ YLE ☛ Zelensky_makes_surprise_stop_at_Nordic_leaders'
                        meeting_in_Oslo⠀⇛


                             Finnish President Sauli Niinistö and the
                             Ukrainian president are also scheduled to
                             hold bilateral talks.


                      # ⚓ CS Monitor ☛ Does_Ukraine_aid_bolster_US_national
                        security?_Biden_makes_his_case.⠀⇛


                             Many Americans worry that U.S. support for
                             Ukraine is a waste of money. But might
                             underwriting Kyiv's fight against Russia now
                             be a kind of insurance against costlier U.S.
                             military action later?


                      # ⚓ New York Times ☛ Hungary’s_Orban_May_Hold_Up_Aid_for
                        Ukraine_at_EU_Leaders’_Meeting⠀⇛


                             The Hungarian prime minister, Viktor Orban,
                             has suggested he will block both financial
                             support and membership talks for Ukraine.
                             There are workarounds on aid, but they are
                             cumbersome.


                      # ⚓ New York Times ☛ Got_Time_for_Milk?_An_Overwhelmed
                        House,_Somehow,_Does.⠀⇛


                             With aid to Ukraine and Israel in limbo, a
                             border deal elusive and a mountain of
                             spending legislation left undone, the House
                             turned its attention to whole milk in
                             schools.


                      # ⚓ New York Times ☛ Inside_the_Border_Talks_That_Could
                        Decide_the_Fate_of_Ukraine_Aid⠀⇛


                             Negotiators are racing to reach a deal before
                             Congress is expected to head home for the
                             holidays at the end of the week.


                      # ⚓ New York Times ☛ Russian_Missile_Attack_on_Kyiv
                        Injures_Dozens⠀⇛


                             Ukraine’s military said that it had shot down
                             all the ballistic missiles targeting the
                             capital, but that falling debris caused
                             significant damage.


                      # ⚓ Latvia ☛ Russian_language_requirement_might_be_banned
                        from_job_ads_in_Latvia⠀⇛


                             It is planned that Latvian employers will not
                             be able to request mandatory knowledge of the
                             Russian language, except for certain
                             professions, as provided for by amendments to
                             the Labor Law supported by the Saeima at
                             first reading on Thursday, December 14.


                      # ⚓ New York Times ☛ With_a_New_Leader,_Poland_Prepares
                        for_a_Course_Change_in_Europe⠀⇛


                             Prime Minister Donald Tusk is sworn in and
                             vows to try to invigorate flagging European
                             support for Ukraine and push for “full
                             mobilization” against Russia’s military
                             assault.


                      # ⚓ New York Times ☛ Germany_Reaches_a_Budget_With
                        Promises_on_Ukraine⠀⇛


                             The government avoided extra borrowing for
                             2024, but must make cuts in programs to
                             address climate change.


                      # ⚓ Hong Kong Free Press ☛ South_Korea_scramble_jets_as
                        Chinese_and_Russian_warplanes_enter_air_defence_zone⠀⇛


                             South Korea’s military said Thursday it
                             scrambled fighter jets as two Chinese and
                             four Russian military planes entered its air
                             defence zone, an area wider than the
                             country’s airspace.


                      # ⚓ LRT ☛ Lithuanian_cinemas_boycott_Miyazaki’s_The_Boy
                        and_the_Heron_over_distributor’s_Russia_links⠀⇛


                             Hayao Miyazaki’s new film The Boy and the
                             Heron will not be screened in Lithuania after
                             all the country’s cinemas refused to
                             cooperate with the distributor due to its
                             links with Russia.


                      # ⚓ LRT ☛ Police_investigate_Lithuanian_citizen_caught
                        praising_Russia_on_video⠀⇛


                             The Police Commissariat in Lithuania’s
                             eastern city of Šalčininkai has launched a
                             pre-trial investigation into the statements
                             of a young Lithuanian man who was filmed
                             praising Russia in a video, Ramunas Matonis,
                             the spokesman for the Police Department, told
                             BNS.


                      # ⚓ RFERL ☛ Finland_Temporarily_Reopens_Two_Border
                        Crossings_With_Russia⠀⇛


                             Finland on December 14 reopened two border
                             crossings with Russia -- Vaalima and Niirala
                             -- for one month, the Finnish newspaper Yle
                             reported.


                      # ⚓ RFERL ☛ Soviet-Era_Dissident_Maria_Rozanova_Dies_At
                        93⠀⇛


                             Prominent Soviet-era dissident journalist
                             Maria Rozanova has died in France at the age
                             of 93, Russian writer Dmitry Bykov said on
                             December 13.


                      # ⚓ RFERL ☛ Activist's_Trial_In_Russia's_Bashkortostan_To
                        Be_Held_Behind_Closed_Doors⠀⇛


                             A court in Russia's Bashkortostan region
                             decided on December 13 that the trial of Fail
                             Alsynov, the former leader of the banned
                             Bashqort group that for years promoted the
                             Bashkir language and culture, will be held
                             behind closed doors "to avoid possible
                             provocations."


                      # ⚓ RFERL ☛ Moscow_Court_Refuses_To_Jail_Activist_Who_Was
                        Forcibly_Brought_From_Kyrgyzstan⠀⇛


                             A Moscow court on December 13 refused to
                             imprison activist Lev Skoryakin, who in
                             October went missing in Kyrgyzstan after
                             applying for asylum there and was later
                             located in a Moscow detention center.


                      # ⚓ The Straits Times ☛ Russian_court_extends_pre-trial
                        detention_for_WSJ_reporter_Gershkovich_-Ifax⠀⇛


                             A Moscow сourt ruled on Thursday that Wall
                             Street Journal reporter Evan Gershkovich must
                             remain in custody until Jan. 30, 2024.


                      # ⚓ The Straits Times ☛ Seoul_scrambles_jets_as_Russia,
                        China_warplanes_near_South_Korean_peninsula⠀⇛


                             The planes flew above waters between South
                             Korea and Japan.


                      # ⚓ RFERL ☛ Putin_Tells_Russians_War_In_Ukraine_To
                        Continue,_Goals_Remain_The_Same⠀⇛


                             President Vladimir Putin showed no signs of
                             relenting in his war against Ukraine, telling
                             Russians in his annual press conference and
                             call-in event on December 14 that his goals
                             remained the same and peace will come when
                             they are achieved.


                      # ⚓ RFERL ☛ Putin_Says_'Dialogue'_With_U.S._Over_Detained
                        Americans_Continues⠀⇛


                             President Vladimir Putin says there has been
                             "dialogue" between U.S. and Russian officials
                             over the release of two Americans currently
                             being held in Russia on espionage charges
                             both vehemently deny.


                      # ⚓ The Straits Times ☛ Ukraine's_Zelenskiy_urges_EU
                        leaders_to_back_membership_talks⠀⇛


                             Ukrainian President Volodymyr Zelenskiy urged
                             a summit of European Union leaders on
                             Thursday to open membership talks with his
                             country, warning that Europe would not
                             understand if the meeting resulted in a
                             "satisfied smile" for Vladimir Putin.


                      # ⚓ teleSUR ☛ There_Will_Be_Peace_When_We_Achieve_the
                        Goals:_Putin⠀⇛


                             "Financial, economic, and technological
                             sovereignty is the future of all countries,"
                             the Russian president said.


          o § Transparency/Investigative Reporting⠀➾


                # ⚓ The Atlantic ☛ The_Most_Consequential_Act_of_Sabotage_in
                  Modern_Times⠀⇛


                       That said, there are also ample reasons why many
                       are not eager to assign blame—even if, in the end,
                       investigators will have to come to a conclusion.
                       Officially naming Russia, the U.S., or Ukraine as
                       the saboteur would have sticky political
                       consequences all around. The belief that Russia
                       might have carried out the attack has already
                       helped swell military spending in Scandinavia,
                       spending that some in the region oppose. If Russia
                       is shown to be behind the attack, that opposition
                       could lose traction. Identifying Russia as the
                       perpetrator would also put Germany on the back
                       foot: Germany had seen Russia as a partner, and
                       German companies had invested in Nord Stream.
                       Because Germany is now aligned with the United
                       States and Ukraine in resisting Putin’s invasion,
                       pinning the attack on the U.S. or Ukraine would
                       pose its own difficulties. If Ukraine is
                       responsible, it would make that country appear
                       singularly ungrateful, because European arms and
                       ammunition have kept it in the fight. Blowing up a
                       major piece of energy infrastructure in the middle
                       of the Baltic would feel like a betrayal. At the
                       same time, it would make Russia look weak and
                       ineffectual, unable to defend a marquee
                       infrastructure project on its doorstep. The Biden
                       administration, which has worked strenuously to
                       rebuild its alliance with Europe and to rally its
                       support for Ukraine, would appear coldly
                       calculating and two-faced if it was behind the
                       sabotage.


          o § Environment⠀➾


                # § Energy/Transportation⠀➾


                      # ⚓ Bridge Michigan ☛ Whitmer_plan_would_give_rebates_to
                        Michiganders_who_buy_new_EV,_other_cars⠀⇛


                             The $25 million program, which would need
                             legislative approval, is an effort to support
                             the auto industry after a nearly seven-week
                             strike by the United Auto Workers (UAW). The
                             strike forced Ford Motor Co., General Motors
                             and Stellantis to pay higher wages, offer
                             more job security and make changes to their
                             plans to transition to electric vehicles.


                      # ⚓ The Revelator ☛ How_Low-Income_Families_Can_Reap
                        Rewards_From_Solar_Energy⠀⇛


          o § Finance⠀➾


                # ⚓ Jim Nielsen ☛ The_Largest_Money-Printing_UI_Element_Ever
                  Made⠀⇛


                       One of Evan’s assertions is that the largest source
                       of money flowing into the world of programming
                       languages comes from Google paying to be the
                       default search engine (a fascinating idea, be sure
                       to watch the entire talk to understand why).


                # ⚓ Hong Kong Free Press ☛ Hong_Kong_could_see_dimmer_economic
                  outlook_in_2024,_as_over_60%_of_companies_expect_no_growth_–
                  survey⠀⇛


                       Hong Kong could see a dimmer economic outlook in
                       2024 amid its long, post-pandemic road towards full
                       recovery, a survey conducted by the the city’s
                       oldest business group has found. Over 60 per cent
                       enterprises surveyed by the Hong Kong General
                       Chamber of Commerce (HKGCC) said they expect no
                       income growth next year.


                # ⚓ Hong Kong Free Press ☛ Hong_Kong_Tourism_Board_to_give
                  residents_HK$100_dining_vouchers_in_bid_to_boost_local
                  spending⠀⇛


                       The Hong Kong Tourism Board (HKTB) will offer
                       residents HK$100 dining vouchers to encourage them
                       to eat out in the evening in a bid to boost local
                       consumption amid a weaker than expected post-
                       pandemic economy.


          o § AstroTurf/Lobbying/Politics⠀➾


                # ⚓ Democracy Now ☛ Rep._Greg_Casar:_Biden_Must_Not_Cave_on
                  GOP’s_Hard-Line_Immigration_Demands_in_Ukraine_Funding
                  Request⠀⇛


                       President Biden appears to be caving to hard-line
                       Republican demands for a new crackdown on asylum
                       seekers and immigrants nationwide in exchange for
                       more Ukraine funding. As negotiations on the
                       emergency funding request continue, we speak with
                       Democratic Congressmember Greg Casar of Texas about
                       how he and other lawmakers oppose “some of the
                       worst changes to our immigration system in
                       decades.” Casar and Democracy Now! co-host Juan
                       González also discuss how today’s “toxic brew” of
                       border politics relates to 200 years of Monroe
                       Doctrine policies punishing Latin America, forcing
                       people to flee their home countries, and then
                       blocking them from seeking asylum in the United
                       States.


                # ⚓ The Nation ☛ We_Have_Mounting_Proof_That_Cruel_Men_Control
                  Women’s_Bodies_in_Red_States⠀⇛


                       After creepy misogynist Donald Trump’s devastating
                       inauguration, The Handmaid’s Tale, understandably,
                       took off on Hulu. I’d read the book by Margaret
                       Atwood in college, but my daughter told me not to
                       watch the television series (although she and her
                       friends were loving it). She knew I’m squeamish
                       about “entertainment” featuring oppression and
                       abuse of women, and, in those months, I was
                       overwrought generally, as were so many of us. I
                       listened to her, as I often do, and I’m glad I did.


                # ⚓ New Yorker ☛ George_Santos_and_the_Art_of_the_Scam⠀⇛


                       The ex-congressman has already pivoted from
                       politics to pop culture—and become the latest
                       beneficiary of America’s enduring fascination with
                       con artists. Are we the ones being duped?


                # § Misinformation/Disinformation/Propaganda⠀➾


                      # ⚓ NPR ☛ Fake_social_media_accounts_are_targeting
                        Taiwan's_presidential_election⠀⇛


                             An influence operation spanning Facebook,
                             TikTok and YouTube has been targeting
                             Taiwan's upcoming presidential election,
                             according to a new report from research firm
                             Graphika.


                      # ⚓ Graphika ☛ Inauthentic_Accounts_Pose_as_Taiwanese
                        Users_to_Spread_Political_Memes_and_Videos_Ahead_of
                        2024_Election⠀⇛


                             Graphika has identified a sustained and
                             coordinated effort to manipulate online
                             conversations about Taiwanese politics ahead
                             of the country’s presidential election in
                             January 2024. The operation was active
                             starting as early as May 2022, employing
                             deceptive behaviors to disseminate Chinese-
                             language videos and memes about Taiwanese
                             political parties across Facebook, YouTube,
                             and TikTok. We do not currently attribute
                             this activity to a specific actor based on
                             open-source indicators.


                             The actors behind this activity leveraged a
                             range of tactics, techniques, and procedures
                             (TTPs) associated with online influence
                             operations (IO). These TTPs included the use
                             of likely inauthentic accounts that purported
                             to be Taiwanese residents and displayed
                             profile pictures of real people that were
                             edited to alter their appearance. We also
                             observed signs of coordination, such as
                             different accounts in the operation posting
                             identical content within minutes of each
                             other.


                      # ⚓ Tom's Hardware ☛ Valve_recommends_users_not_huff
                        Steam_Deck_exhaust_vents_for_Fentanylware_(TikTok)
                        views⠀⇛


                             Valve responds to support forum post asking
                             whether it's a good idea to sniff exhaust
                             fumes from the Steam Deck's vents.


          o § Censorship/Free Speech⠀➾


                # ⚓ RFERL ☛ Teacher_Imprisoned_For_Condemning_Russia's
                  Aggression_Against_Ukraine_Placed_In_Solitary_Confinement⠀⇛


                       Supporters of Russian history teacher Nikita
                       Tushkanov, who was imprisoned for criticizing
                       Moscow's war in Ukraine online, said on December 12
                       that the activist was sent to solitary confinement
                       for seven days for lying on a bed before it was
                       officially sleeping time. [...]


                # ⚓ EFF ☛ Speaking_Freely:_Dr._Carolina_Are [Ed: No, Jillian
                  York does not stand for free speech; EFF pays her lots of
                  money as a contractor to spew out lies about people and then
                  silence them!]⠀⇛


                       Dr. Carolina Are is an Innovation Fellow at
                       Northumbria University Centre for Digital Citizens.
                       Her research primarily focuses on the intersection
                       between online abuse and censorship. Her current
                       research project investigates Instagram and
                       TikTok’s approach to malicious flagging against
                       ‘grey area’ content, or content that toes the line
                       of compliance with social media’s community
                       guidelines.


                       She is also a blogger and creator herself, as well
                       as a writer, pole dance instructor and award-
                       winning activist. Dr. Are sat down for an interview
                       with EFF’s Jillian York to discuss the impact of
                       platform censorship on sex workers and activist
                       communities, the need for systemic change around
                       content moderation, and how there’s hope to be
                       found in the younger generations.


                       Jillian York: Can you introduce yourself and tell
                       us a bit about your work? Specifically, can you
                       give us an idea of how you became a free speech
                       advocate?


                # ⚓ New Yorker ☛ Masha_Gessen_on_the_Holocaust,_Israel,_and_the
                  Politics_of_Memory⠀⇛


                       The New Yorker staff writer discusses the
                       enforcement of “memory culture” in Germany, and the
                       ongoing battle over the definition of antisemitism.


                # ⚓ Meduza ☛ Navalny’s_associates_say_unable_to_locate_him_in
                  Moscow_detention_centers_after_unconfirmed_reports_he_was
                  transferred_there_—_Meduza⠀⇛


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


                # ⚓ RFERL ☛ Media_Watchdogs_Criticize_Taliban_Over_Arrests_Of
                  Afghan_Journalists⠀⇛


                       The AFJC denounced the sentence as "unfair and
                       unacceptable due to the absence of a defense lawyer
                       during the trial" and called for a "thorough review
                       of the case, immediate and unconditional release of
                       the journalist," and reopening of Nasim Radio.


                # ⚓ ANF News ☛ Kurdish_journalist_Ezîzî_sent_to_Evin_Prison⠀⇛


                       It was learned that Pexşan Ezîzî was deprived of
                       all her rights during the interrogation process and
                       was not allowed to use her right to phone calls
                       with her family.


                # ⚓ CPJ ☛ Azerbaijani_journalist_Hafiz_Babali_latest_arrest_in
                  Abzas_Media_crackdown⠀⇛


                       “By arresting widely respected investigative
                       journalist Hafiz Babali, Azerbaijani authorities
                       are only confirming that their real aim in
                       targeting Abzas Media is to silence its
                       uncompromising reporting on official corruption
                       allegations,” said Gulnoza Said, CPJ’s Europe and
                       Central Asia program coordinator, in New York.
                       “Authorities in Azerbaijan should immediately
                       release Babali along with all other unjustly jailed
                       journalists and stop the ongoing wave of reprisals
                       against the independent press.”


          o § Civil Rights/Policing⠀➾


                # ⚓ Silicon Angle ☛ EU_agrees_to_rules_that_will_give_gig
                  workers_employee_rights⠀⇛


                       After years of negotiations, the European Union has
                       provisionally agreed to new rules to improve the
                       labor conditions of people working in such jobs as
                       delivery and ride-hailing, potentially affecting
                       millions of gig workers.


                       It’s believed that right now, there are close to
                       500 digital work platforms in the EU employing
                       about 28 million people, although it’s estimated
                       that by 2025, the number could rise to 43 million –
                       a sizable chunk of the EU’s 448 million population.


                # ⚓ New York Times ☛ The_Woman_Who_Fought_the_Texas_Abortion
                  Ban⠀⇛


                       Kate Cox is at the heart of one of the most
                       contentious cases since the overturning of Roe v.
                       Wade.


                # ⚓ New York Times ☛ Migrant_Children_Working_as_Roofers,_and_a
                  Biden_Impeachment_Inquiry⠀⇛


                       Hear the news in five minutes.


          o § Internet Policy/Net Neutrality⠀➾


                # ⚓ The Verge ☛ An_X_outage_broke_all_outgoing_links,_again⠀⇛


                       If this sounds familiar, it’s because something
                       similar happened in March, taking down links and
                       images across timelines for around an hour. Twitter
                       blamed that on an “internal change that had some
                       unintended consequences” before Platformer reported
                       the bug occurred because of a mistake by the site’s
                       single remaining site reliability engineer, who was
                       operating solo after Musk instituted massive
                       layoffs.


                # ⚓ APNIC ☛ Charting_APNIC’s_course_for_the_next_four_years⠀⇛


                       The APNIC Strategic Plan (2024-2027) was created by
                       the APNIC EC and Secretariat, and is informed by
                       feedback from Members and the community. The plan
                       sets out the future that APNIC wishes to see, the
                       objectives and priorities that need to be achieved
                       to help reach that future state, and the guiding
                       principles underpinning APNIC’s efforts.


                # ⚓ Techdirt ☛ The_‘Race_to_5G’_Wound_Up_Being_More_Of_A
                  Hobbled_Waddle_To_Nowhere⠀⇛


                       We’ve noted for years how the “race to 5G” was
                       largely just hype by telecoms and hardware vendors
                       eager to sell more gear and justify high U.S.
                       mobile data prices. While 5G does provide faster,
                       more resilient, and lower latency networks, it’s
                       more of a modest evolution than a revolution.


                # ⚓ EFF ☛ Spritely_and_Veilid:_Exciting_Projects_Building_the
                  Peer-to-Peer_Web⠀⇛


                       There are many technologies used behind the scenes
                       to create decentralized tools and platforms. There
                       has been a lot of attention lately, for example,
                       around interoperable and federated social media
                       sites using ActivityPub, such as Mastodon, as well
                       as platforms like BlueSky using a similar protocol.
                       These types of services require most individuals to
                       sign up with an intermediary service host in order
                       to participate, but they are decentralized in so
                       far as any user has a choice of intermediary, and
                       can run one of those services themselves while
                       participating in the larger network.


                       Another model for decentralized communications does
                       away with the intermediary services altogether in
                       favor of a directly peer-to-peer model. This model
                       is technically much more challenging to implement,
                       particularly in cases where privacy and security
                       are crucial, but it does result in a system that
                       gives individuals even more control over their data
                       and their online experience. Fortunately, there are
                       a few projects being developed that are aiming to
                       make purely peer-to-peer applications achievable
                       and easy for developers to create. Two leading
                       projects in this effort are Spritely and Veilid.


                       Spritely is worth keeping an eye on. Being
                       developed by the Institute of the same name,
                       Spritely is a framework for building distributed
                       apps that don’t even have to know that they’re
                       distributed. The project is spearheaded by
                       Christine Lemmer-Webber, who was one of the co-
                       authors of the ActivityPub spec that drives the
                       fediverse. She is taking the lessons learned from
                       that work, combining them with security and privacy
                       minded object capabilities models, and mixing it
                       all up into a model for peer to peer computation
                       that could pave the way for a generation of new
                       decentralized tools.


      § Monopolies/Monopsonies⠀➾


          o ⚓ CoryDoctorow ☛ An_Epic_antitrust_loss_for_Google_(permalink)⠀⇛


                 The mobile app store world is a duopoly run by Google and
                 Apple. Both use a variety of tactics to prevent their
                 customers from installing third party app stores, which
                 funnels all app makers into their own app stores. Those
                 app stores cream an eye-popping 30% off every purchase
                 made in an app.


                 This is a shocking amount to charge for payment
                 processing. The payments sector is incredibly monopolized
                 and notorious for its price-gouging – and its standard
                 (wildly inflated) rate is 2-5%: [...]


          o ⚓ India Times ☛ Apple_to_be_hit_by_EU_antitrust_order_in_fight_with
            Spotify⠀⇛


                 The decision is slated for early next year and Apple
                 could face a fine of as much as 10% of its annual sales,
                 Bloomberg reported.


                 The probe was sparked by a complaint nearly four years
                 ago from Sweden's Spotify Technology, which claimed it
                 was forced to ramp up the price of its monthly
                 subscriptions to cover costs associated with Apple's App
                 Store rules.


          o ⚓ EFF ☛ Without_Interoperability,_Apple_Customers_Will_Never_Be
            Secure⠀⇛


                 Apple’s iMessage offers end-to-end encrypted messaging
                 for its customers, but only if those customers want to
                 talk to someone who also has an Apple product. When an
                 Apple customer tries to message an Android user, the data
                 is sent over SMS, a protocol that debuted while Wayne’s
                 World was still in its first theatrical run. SMS is
                 wildly insecure, but when Apple customers ask the company
                 how to protect themselves while exchanging messages with
                 Android users, Apple’s answer is “buy them iPhones.”


                 That’s an obviously false binary. Computers are all
                 roughly equivalent, so there’s no reason that an Android
                 device couldn’t run an app that could securely send and
                 receive iMessage data. If Apple won’t make that app, then
                 someone else could. 


                 That’s exactly what Apple did, back when Microsoft
                 refused to make a high-quality MacOS version of Microsoft
                 Office: Apple reverse-engineered Office and released
                 iWork, whose Pages, Numbers and Keynote could perfectly
                 read and write Microsoft’s Word, Excel and Powerpoint
                 files.


          o § Patents⠀➾


                # ⚓ MIT Technology Review ☛ Vertex_will_pay_tens_of_millions_to
                  license_a_controversial_CRISPR_patent⠀⇛


                       The patent on CRISPR has been the fulcrum off a
                       decade-long legal fight after the Broad Institute,
                       a research center in Cambridge, Massachusetts,
                       snatched rights to the most important uses of the
                       gene-editing tool in 2014.


                       Broad’s patent claims have been opposed by the
                       University of California, Berkeley, which says
                       researchers Jennifer Doudna and Emmanuelle
                       Charpentier are the tool’s true inventors. The pair
                       won a Nobel Prize in 2020 for their work on the
                       technology.


          o § Copyrights⠀➾


                # ⚓ Digital Music News ☛ Understanding_Music_Copyright:_A_Guide
                  for_Independent_Musicians⠀⇛


                       We’ve already seen so many court cases over
                       copyright infringement. Sometimes, one artist has a
                       similar idea to another artist and inadvertently
                       makes a song that sounds similar. And sometimes it
                       may be on purpose, who knows. The point is, you
                       need to protect yourself. So let’s talk music
                       copyright.


                       (Note: I am not a lawyer or copyright expert and
                       this is not legal advice. I’m a fellow indie
                       musician who did a bunch of research so you don’t
                       have to. I’ve linked to all my sources within the
                       article).


                # ⚓ The Atlantic ☛ ChatGPT_Is_Turning_the_Internet_Into
                  Plumbing⠀⇛


                       It’s arguably a strange move for the publisher,
                       which in the old days might have seen some
                       competitive advantage in maintaining a distinctive
                       voice—that is, one that isn’t easily replicated by
                       a chatbot. But Axel Springer will be paid for
                       lending its work. That’s certainly better than
                       getting ripped off for free, which is effectively
                       what generative AI is presumed to have done to
                       publishers across the industry. Julia Sommerfield,
                       a spokesperson for Axel Springer, declined to give
                       any specific details about the deal but told me
                       over email, “Our reporters at Politico and Business
                       Insider will continue to deliver high-quality
                       journalism. The partnership introduces an
                       additional channel for distribution and revenue,
                       and enriches users’ experience on ChatGPT.” A
                       spokesperson for OpenAI declined to comment on the
                       deal.


                # ⚓ Silicon Angle ☛ OpenAI_inks_content_licensing_deal_with
                  Axel_Springer⠀⇛


                       Berlin-based Axel Springer is a media company
                       majority owned by investment firm KKR. As part of
                       the licensing deal announced today, OpenAI will
                       gain access to “select news content” from several
                       of the company’s properties including Politico and
                       Insider as well as the German tabloids Bild and
                       Welt. The contract covers not only free content but
                       also certain articles that are usually only
                       available to paid subscribers.


                       The companies didn’t disclose the deal’s financial
                       terms. However, the Wall Street Journal described
                       it as a multiyear partnership that is expected to
                       generate “substantial revenue” for Axel Springer.
                       The deal is said to be nonexclusive, which means
                       the company can ink similar partnerships with other
                       AI developers.


                # ⚓ Torrent Freak ☛ Two_Arrested_in_Pirate_IPTV_Raids,_Police
                  Obtain_Details_of_UK_Subscribers⠀⇛


                       The UK's Eastern Region Special Operations Unit
                       (ERSOU) has revealed details of raids on Tuesday
                       targeting a pirate IPTV operation. The regional
                       organized crime and counterterrorism unit says that
                       warrants were executed at addresses in Essex,
                       Hertfordshire, and Lanarkshire in Scotland. Two men
                       were arrested on suspicion of fraud, money
                       laundering, and intellectual property offenses.
                       Police say they have obtained a list of customers
                       who shouldn't rule out the possibility of criminal
                       prosecution.


                # ⚓ Torrent Freak ☛ Science_Fiction_and_Fantasy_Writers_Take
                  Aim_at_AI_Freeloading_*_TorrentFrea⠀⇛


                       Members of the Science Fiction and Fantasy Writers
                       Association have no trouble envisioning an AI-
                       centered future, but developments over the past
                       year are reason for concern. The association takes
                       offense when AI models exploit the generosity of
                       science fiction writers, who share their work
                       without DRM and free of charge.


                # ⚓ Torrent Freak ☛ France:_Streaming_&_DDL_Piracy_Fall_But
                  Cloud_&_Usenet_Boosts_Wipe_Out_Gains⠀⇛


                       French telecoms regulator Arcom has published the
                       2023 edition of its online content consumption
                       report. Covering movies, TV shows, live sports and
                       similar content, the report aims to map French
                       internet users' consumption habits from both legal
                       and illegal sources. After reductions in recent
                       years, the overall piracy rate remained static in
                       2023, with some interesting changes in access
                       methods that may signal moves to avoid detection.


          o § Gemini* and Gopher⠀➾


                # § Personal/Opinions⠀➾


                      # ⚓ LiveFromFrance_S1E05⠀⇛


                             Too bad because we were beginning to speak
                             intelligently about Ukraine-Russia and
                             Palestine-Israel conflicts. But in this time
                             of the Immigration Bill in the National
                             Assembly, any attack or aggression is a
                             pretext for controversy. A teenager was
                             stabbed in a village when people from an
                             impoverished neighborhood invited themselves
                             to a party. After a week of polemics, extreme
                             right-wing movements and groups organized
                             protests and riots in the neighborhood and in
                             the big cities. Some media drew parallels
                             with immigration. At the same time, a mosque
                             was tagged with insults. And the main far-
                             right party forgot its links with this kind
                             of violent gangs of skinheads and Neo-Nazis.
                             Some of its staff had been members in their
                             youth. The interior minister (who wrote
                             articles in a far-right newspaper in his
                             youth, and who uses the word «ensauvagement»,
                             a neologism to describe the fact to become
                             savage, like the far-right) promised to
                             disband these militias. But he didn't explain
                             why these militias were in possession of
                             information that is normally only known to
                             the police. Right-wing and far-right parties
                             (the difference is becoming so thin) are
                             using the fears of others to create a climate
                             of insurrection. And they have good examples.


                             France is looking at Netherlands, Argentina
                             or Sweden where far-right took the power or
                             at least a part of the power. Nothing was
                             done against that and with left-parties as
                             stupid as the last month in France


                # § Technology and Free Software⠀➾


                      # ⚓ Squid_service_on_gopher.zcrayfish.soy_discontinued⠀⇛


                             The precompiled version of Squid which comes
                             with my distro, Alpine Linux, seems to have
                             stopped supporting gopher recently.
                             Consequently the gopherproxy.sh cgi script on
                             the web server, which some folks were using
                             to access other gopher servers, has ceased to
                             function.


                             I don't feel like going through the
                             maintenance burden of compiling my own
                             version of squid, despite the fact that it is
                             an /extremely/ easy piece of software to
                             compile...


                      # ⚓ Finally_deciding_to_buy_a_new_laptop⠀⇛


                             I tried to subsitute my personal laptop with
                             a PineTab 2. Thinking I can just SSH into my
                             servers and do everything there. I thought
                             it'll work but ended up not. Mainly 2
                             reasons: 1. People haven't sorted out the
                             WiFi driver issue yet. And the USB dongle I
                             bought absolutely sucks. Secondly, I need
                             enough CPU power to browse the internet and
                             read documents. I was thinking a tablet would
                             be a good form factor for documentation
                             reading. But the PienTab2 is too slow to run
                             web pages smoothly. As much I don't want to
                             have any e-waste. The PineTab2 is slowing me
                             down and I won't be using it much anymore.
                             I'll probably repurpose it as a light weight
                             server or something.


                      # § Programming⠀➾


                            # ⚓ The_"can't_fail"_scripting_language,_round_2⠀⇛


                                   This is a new attempt at defining a
                                   "can't fail" scripting language. The
                                   general idea is to design a language
                                   that has no error state and can
                                   successfully parse and execute any
                                   string of binary data (subject to
                                   caveats like resource limits, not
                                   guaranteeing that any particular
                                   program can finish executing, and so
                                   on).


===============================================================================
* Gemini_(Primer) links can be opened using Gemini_software. It's like the
World Wide Web but a lot lighter.





                    ䷩ 𝚕𝚒𝚗𝚎 5369

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

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ Links 14/12/2023: Food Insecurity and
Escalations in the Red Sea⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


posted by Roy Schestowitz on Dec 14, 2023


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇GNOME bluefish⦈


§ Contents⠀➾


    * Leftovers
          o Programming/Development
          o Science
          o Hardware
          o Health/Nutrition/Agriculture
          o Proprietary/Artificial_Intelligence_(AI)
          o Defence/Aggression
          o Transparency/Investigative_Reporting
          o Environment
                # Energy/Transportation
                # Wildlife/Nature
                # Overpopulation
          o Finance
          o AstroTurf/Lobbying/Politics
          o Censorship/Free_Speech
          o Internet_Policy/Net_Neutrality
          o Monopolies/Monopsonies
                # Copyrights
    * § Leftovers⠀➾


          o § Programming/Development⠀➾


                # ⚓ Daniel Lemire ☛ Measuring_the_size_of_the_cache_line
                  empirically⠀⇛


                       Our computers do not read or write memory in units
                       of bits or even bytes. Rather memory is accessed in
                       small blocks of memory called “cache lines”. For a
                       given system, the cache line size is usually fixed
                       and small (e.g.,  16 to 256 bytes).


          o § Science⠀➾


                # ⚓ Federal News Network ☛ DARPA_kicks_off_its_10-year_plan_to
                  create_a_‘thriving_lunar_economy’ [Ed: Our economy on this
                  planet is rotting and DARPA wants more of taxpayers' money to
                  make some fantasy colony where things are thriving. "Economy"
                  is a man-made concept, not a science, and does not actually
                  exist except people's heads. But it manifests itself in
                  visible forms, such as slavery and suicides.]⠀⇛


                       In today's Federal Newscast: The Small Business
                       Administration is redefining small. The
                       Cybersecurity and Infrastructure Security Agency is
                       rolling out a new cybersecurity tool for agencies.
                       And DARPA makes plans to build a booming economy on
                       the moon.


                # ⚓ Science Alert ☛ Human_Brain_Cells_on_a_Chip_Can_Recognize
                  Speech_And_Do_Simple_Math⠀⇛


          o § Hardware⠀➾


                # ⚓ Hackaday ☛ LED_Art_Project_Is_Geometrically_Beautiful⠀⇛


                       There is no shortage of companies on the Internet
                       willing to sell you expensive glowing things to
                       stick on your walls. Many hackers prefer to make
                       their own however, and [Chris] is no exception. His
                       LED wall art is neat, tidy, and stylish, all at
                       once.


                # ⚓ Tom's Hardware ☛ US_government_doles_out_paltry_$35_million
                  of_the_$52_billion_CHIPS_Act,_warns_of_possible_delays_in
                  defective_chip_maker_Intel_and_TSMC_fab_buildouts [Ed:
                  Regimes always look for ways to 'muscle' numbers to fake
                  "growth". Nowadays they even bail out companies to fake
                  "growth". While national debt GROWS. They merely borrow from
                  the future, deflating the future. The US is GROWING. From_31
                  TRILLION_in_debt_to_almost_34_TRILLION..._since_summer!
                  Bailouts are not a viable economic strategy, but they benefit
                  the very few 'stimulus' (taxpayers' money) recipients,
                  usually those that own all the major media and manufacture
                  consent for it.] ]⠀⇛


                       The U.S. government has begun doling out portions
                       of the $52 billion CHIPS Act, beginning with just
                       $35 million to BAE Systems. It also warns of
                       possible delays in defective chip maker Intel and
                       TSMC fab buildouts.


                # ⚓ The New Stack ☛ Nvidia_GPU_Dominance_at_a_Crossroads⠀⇛


                       When Abusive Monopolist Microsoft Chaffbot was
                       released a year ago, every chip maker, except for
                       Nvidia, was caught sleeping.


                # ⚓ Hackaday ☛ ThunderScan:_The_Wild_1980s_Product_That_Turned
                  A_Printer_Into_A_Scanner⠀⇛


                       Back in the 1980s, printers were expensive things.
                       Scanners were rare, particularly for the home
                       market, because home computers could barely handle
                       basic graphics anyway. Back in these halcyon days,
                       an obscure company called Thunderware built a
                       device to convert the former into the latter. It
                       was known as the Thunderscan, and was a scanning
                       head built for the Apple ImageWriter dot matrix
                       printer. Weird enough already, but this device
                       hides some weird secrets in its design.


          o § Health/Nutrition/Agriculture⠀➾


                # ⚓ ACLU ☛ Arizona’s_High_Court_Must_Protect_Abortion_Access⠀⇛


                       Today, the Arizona Supreme Court will consider
                       whether to resurrect a more than 150-year-old
                       criminal ban on virtually all abortions. The
                       court’s decision could allow that law to take
                       precedence over Arizona’s modern abortion laws,
                       including those passed just last year by the
                       people’s current elected representatives.


                       This ban was originally struck down in 1973, thanks
                       to a lawsuit brought by Planned Parenthood and
                       physicians in Arizona, and since that time has been
                       superseded by a comprehensive scheme that regulates
                       abortion as a lawful medical procedure. But an
                       anti-abortion activist and County Attorney are now
                       asking the Arizona Supreme Court to turn back the
                       clock. No one should be forced to carry a pregnancy
                       to term against their will and face the life-
                       altering consequences of being denied essential
                       health care, but reviving this antiquated law in
                       full would do just that — and, at the same time,
                       throw Arizona’s entire contemporary legal code into
                       confusion.


                       The origins of Arizona’s near-total criminal ban at
                       issue in this case go back to 1864. To put that
                       into context: This ban was enacted during the Civil
                       War era, when women were not allowed to vote, and
                       Arizona was not even a state. Yet even though
                       Arizona’s current abortion laws permit abortion
                       through 15 weeks of pregnancy, the 1864 ban permits
                       abortion only when necessary to save the patient’s
                       life. In all other situations, performing an
                       abortion would lead to felony charges and prison
                       time.


          o § Proprietary/Artificial Intelligence (AI)⠀➾


                # ⚓ Scoop News Group ☛ AI_threats_pose_great_cyber_risks_to
                  smaller_companies,_experts_tell_House_panel⠀⇛


                       Executives from SentinelOne, Protect Hey Hi (AI)
                       and I.C.B.M. Consulting provide lawmakers on the
                       cybersecurity and infrastructure protection
                       subcommittee with a laundry list of recommendations
                       to better combat Hey Hi (AI) threats.


          o § Defence/Aggression⠀➾


                # ⚓ YLE ☛ Wednesday’s_papers:_US_soldiers_in_Rovaniemi,_girls
                  on_the_move_and_would_you_wood?⠀⇛


                       Newspapers ponder what the arrival of American
                       soldiers means to Rovaniemi and the eternal
                       question, electricity or wood?


                # ⚓ Techdirt ☛ For_Some_Reason,_LA_Drug_Cops_Received
                  Intelligence,_Training_On_‘Muslim_Extremists’_From_The
                  Israeli_Military⠀⇛


                       A few years ago, hacktivist group Anonymous
                       liberated a ton of documents from law enforcement
                       agencies all over the world. These were published
                       by transparency activists DDoS (Distributed Denial
                       of Secrets). Journalists and other activists
                       continue to dive into this 269 gigabyte treasure
                       trove, teasing out additional information law
                       enforcement agencies certainly wish was still their
                       little secret.


                # ⚓ New York Times ☛ Japan_Wants_a_Stronger_Military._Can_It
                  Find_Enough_Troops?⠀⇛


                       A shrinking, aging population poses an obstacle as
                       the nation tries to counter security threats from
                       China and North Korea.


                # ⚓ New York Times ☛ At_Least_23_Soldiers_Killed_in_Attack_in
                  Pakistan⠀⇛


                       Tensions have been rising between Pakistan and
                       Afghanistan, and violence in the border area has
                       increased since Pakistan directed all undocumented
                       foreigners to leave the country.


                # ⚓ Marcy Wheeler ☛ It_Was_Donald_Trump,_in_the_Dining_Room,
                  with_the_Ex-Twitter_Account⠀⇛


                       Expert testimony that Jack Smith's team just
                       noticed will help to prove that who sent the Tweet
                       that almost got Mike Pence killed: It was Donald
                       Trump, alone in the dining room, with the attempted
                       murder weapon.


                # ⚓ Defence Web ☛ Norwegian_tanker_Strinda_struck_by_missile_in
                  Red_Sea⠀⇛


                       A Norwegian registered and operated oil and
                       chemical products tanker, Strinda (IMO 9330771),
                       was hit by a missile launched from Yemen at about
                       midnight local time on Tuesday morning (12
                       December).


          o § Transparency/Investigative Reporting⠀➾


                # ⚓ Marcy Wheeler ☛ Former_WikiLeaks_Task_Force_Member_Charles
                  McGonigal_Didn’t_Take_Credit_for_the_Josh_Schulte
                  Investigation⠀⇛


                       It makes sense that Charles McGonigal didn't brag
                       that he oversaw investigations of Oleg Deripaska
                       before he went on to work for Deripaska. But it's
                       far more curious that McGonigal didn't mention that
                       he led not one but two WikiLeaks investigations.


          o § Environment⠀➾


                # § Energy/Transportation⠀➾


                      # ⚓ Reason ☛ Is_Climate_Change_Going_Back_to_the_Supreme
                        Court?_(Minnesota_Edition)_[UPDATED]⠀⇛


                             The justices are considering whether to grant
                             certiorari in Minnesota's lawsuit against
                             energy companies.


                      # ⚓ Defence Web ☛ More_soldiers_in_place_to_tackle
                        illegal_mining⠀⇛


                             South African President Cyril Ramaphosa used
                             last week’s Free State presidential imbizo to
                             make public the deployment of a further 395
                             soldiers in the province’s goldfields, to
                             boost efforts aimed at curbing illegal
                             mining.


                      # ⚓ The Strategist ☛ Japan’s_trust_in_Australia_erodes_in
                        response_to_energy_security_fears⠀⇛


                             For many decades, Japan has looked favourably
                             on Australia as a reliable producer of gas
                             and coal.


                # § Wildlife/Nature⠀➾


                      # ⚓ ADF ☛ Botswana_Breaks_Ground_on_Forensic_Laboratory
                        to_Combat_Wildlife_Crime⠀⇛


                             A forensic laboratory in northern Botswana
                             will play a vital role in the effort to
                             disrupt the illegal trade in wildlife,
                             according to the lab’s supporters.


                # § Overpopulation⠀➾


                      # ⚓ France24 ☛ Food_insecurity,_the_forgotten_crisis_of
                        COP28⠀⇛


                             When it comes to climate change, the world’s
                             food system is a double-edged sword. Food
                             production is both one of the biggest
                             emitters of global greenhouse gases and one
                             of the sectors hardest hit by the effects of
                             climate change. To reconcile these two
                             issues, the UN’s Food and Agriculture
                             Organization on Sunday set out an
                             unprecedented roadmap for solutions. But the
                             topic rarely makes it to the negotiating
                             table.


          o § Finance⠀➾


                # ⚓ The Straits Times ☛ Australia_govt_sees_budget_almost_in
                  balance_for_2023/24⠀⇛


                       Australia's government expects a much improved
                       budget bottom line this year as revenues outpace
                       forecasts, but is resisting calls for more cost-of-
                       living handouts to avoid inflaming inflationary
                       pressures.


                # ⚓ France24 ☛ Argentina_devalues_currency_by_50%_in_'shock'
                  measure_against_hyperinflation⠀⇛


                       Argentina devalued its currency by more than 50
                       percent Tuesday in a set of "shock" measures aimed
                       at reviving a crumbling economy and tackling
                       triple-digit inflation.


          o § AstroTurf/Lobbying/Politics⠀➾


                # ⚓ Marcy Wheeler ☛ In_Motion_to_Dismiss,_Hunter_Biden_Accuses
                  House_GOP_of_Separation_of_Powers_Violation⠀⇛


                       Two days before Hunter Biden is due to appear in
                       response to a subpoena, his attorneys argued that
                       the House GOP has improperly usurped DOJ's role in
                       his prosecution.


                # ⚓ New York Times ☛ Egypt’s_Presidential_Election_Ends,_With
                  el-Sisi_Expected_to_Win⠀⇛


                       President Abdel Fattah el-Sisi is all but certain
                       to come out on top after a three-day vote, with the
                       war in Gaza turning the country’s focus from
                       economic calamity to security.


                # ⚓ Latvia ☛ WATCH:_Rīga_Security_Forum_discussion_on_'Global
                  South_and_the_Quest_for_International_Justice'⠀⇛


                       The seventh in a season of geopolitical discussions
                       under the 'Rīga Security Forum' banner was
                       published online December 12.


                # ⚓ BIA Net ☛ Kocaeli_deputy_Bitmez_who_fainted_during_his
                  speech_in_Parliament_admitted_to_intensive_care⠀⇛


                       Bitmez received prolonged cardiac massage and was
                       subsequently transported to the hospital by
                       ambulance.


                # § Misinformation/Disinformation/Propaganda⠀➾


                      # ⚓ France24 ☛ These_photos_of_Israelis_mistreating
                        Palestinian_children_aren’t_from_the_latest_conflict
                        [Ed: About half of Palestine's Gaza is children or
                        young because they reproduce like rabbits. The media
                        needs to point that out when saying that many kids die
                        in the war. Otherwise it contributes to a misconception
                        wherein kids are being "targeted".]⠀⇛


                             More than 6,000 Palestinian children are
                             thought to have been killed in the war
                             between Israel and Hamas so far, according to
                             numbers released by Hamas.  It’s in this
                             context that photos and videos have started
                             circulating online that people say show
                             Israelis mistreating Palestinian children.
                             However, while violence against children is
                             indeed occurring, three of these images are
                             not linked to the current conflict.


          o § Censorship/Free Speech⠀➾


                # ⚓ Hong Kong Free Press ☛ Detained_Hong_Kong_activist_Chow
                  Hang-tung_awarded_human_rights_prize_by_France,_Germany⠀⇛


                       Hong Kong pro-democracy activist Chow Hang-tung,
                       who has been detained for two years under the
                       national security law, has been awarded a human
                       rights prize by the foreign ministries of France
                       and Germany.


                # ⚓ France24 ☛ French_teachers_walk_out_in_protest
                  after Renaissance_painting_sparks_row_over_nudity⠀⇛


                       Teachers at a school outside Paris refused to work
                       on Monday as the establishment grappled with a
                       crisis sparked by the showing in class of a
                       painting by a Renaissance master containing several
                       nude women.


                # ⚓ Hong Kong Free Press ☛ Ex-student_leader_seeks_to_challenge
                  Hong_Kong_election_law_over_inciting_blank_votes_in_2021
                  ‘patriots-only’_legislative_race⠀⇛


                       A former student leader in Hong Kong who pleaded
                       guilty to inciting others to cast blank votes
                       during the “patriots-only” legislative election in
                       2021 has filed an appeal, reportedly to challenge
                       the constitutionality of the election law.


                # ⚓ Reason ☛ Washington_Post_Op-Ed_Argues_That_Colleges_Should
                  'Restrict'_Speech_To_Fight_Antisemitism⠀⇛


                       In her article, University of Pennsylvania
                       professor Claire O. Finkelstein absurdly argued
                       that colleges treat free speech as "near-sacred."


                # ⚓ Reason ☛ From_Dean_Erwin_Chemerinsky_(Berkeley_Law)_and
                  Chancellor_Howard_Gillman_(UC_Irvine)_on_Free_Speech_on
                  Campus⠀⇛


                       Chemerinsky and Gillman are the co-authors of Free
                       Speech on Campus, and have long defended free
                       speech and academic freedom. (For another recent
                       item from the two of them, see here ["Free speech
                       doesn't mean hecklers get to shut down campus
                       debate"].)


                # ⚓ NYPost ☛ Kamala_Harris_heckled_by_Democratic_state_lawmaker
                  demanding_Gaza_cease-fire_during_holiday_party_speech⠀⇛


                       Vice President Kamala Harris was momentarily
                       shouted down by Democratic state lawmaker as she
                       gave remarks during a holiday party at her home on
                       Monday.


          o § Internet Policy/Net Neutrality⠀➾


                # ⚓ YLE ☛ Yle_Areena's_mobile_site_now_requires_logging_in [Ed:
                  Unless the paywall (or worse) come...]⠀⇛


                       Access to the streaming platform on a browser is
                       still open to all.


                # ⚓ YLE ☛ Nokia_to_acquire_US_defence_comms_firm_Fenix_Group⠀⇛


                       The Fenix Group markets tactical communications
                       products, as well as wireless, augmented reality-
                       enabled battlefield robot technologies.


                # ⚓ Techdirt ☛ Josh_Hawley_Back_To_Try_To_Hotline_His_Awful_AI/
                  Section_230_Bill⠀⇛


                       Last week, we wrote about the potential for Senator
                       Josh Hawley to “hotline” the bill that he put
                       together with Senator Richard Blumenthal to remove
                       Section 230 from anything touching artificial
                       intelligence. As we noted at the time, even if you
                       hate both generative AI technology and Section 230,
                       the bill was so poorly drafted that it would create
                       all kinds of problems for the internet.


      § Monopolies/Monopsonies⠀➾


          o ⚓ Techdirt ☛ How_the_Grinch’s_Intellectual_Property_Stole
            Christmas⠀⇛


                 The estate of Dr. Seuss is obviously no stranger to
                 playing the intellectual property maximalist, having
                 appeared on our pages many times in the past. But more
                 specifically for this post, the estate has also,
                 ironically enough, been more than happy to stomp on the
                 Christmas joy of others in favor of jealously guarding
                 its IP when it comes to The Grinch Who Stole Christmas. I
                 have to say, I have no concept of just how much cognitive
                 dissonance one would need to have attained to take a
                 story that is all about sharing and celebrating the
                 Christmas holiday with others and use control over it to
                 do the exact opposite, but it’s impressive nonetheless.


          o ⚓ Techdirt ☛ Google_Promises_Unlimited_Cloud_Storage;_Then_Cancels
            Plan;_Then_Tells_Journalist_His_Life’s_Work_Will_Be_Deleted_Without
            Enough_Time_To_Transfer_The_Data⠀⇛


                 Over a decade ago, I pointed out that as Google kept
                 trying to worm its way deeper into our lives, a key
                 Achilles’ heel was its basically non-existent customer
                 service and unwillingness to ever engage constructively
                 with users the company fucks over. At the time, I dubbed
                 it Google’s “big, faceless, white monolith” problem,
                 because that’s how it appears to many customers. Going
                 all the way back to 2009, I had suggested that the
                 company needed not just better customer support, but
                 something like a user advocate.


          o ⚓ New York Times ☛ Google’s_Antitrust_Loss_to_Epic_Could_Preview
            Its_Legal_Fate_in_2024⠀⇛


                 A trio of antitrust cases on the docket threaten to
                 reshape Google’s business and sap its profits.


          o § Copyrights⠀➾


                # ⚓ Digital Music News ☛ Michael_Jackson_Estate_Threatens_Legal
                  Action_Over_‘Long-Lost_First_Ever_Studio_Recording’_Release⠀⇛


                       The Michael Jackson Estate threatens legal action
                       over the release of the so-called “long-lost first
                       ever studio recording” of Michael with the Jackson
                       5, warning it violates the Estate’s rights and
                       could lead to a lawsuit.


===============================================================================
* Gemini_(Primer) links can be opened using Gemini_software. It's like the
World Wide Web but a lot lighter.





                    ䷩ 𝚕𝚒𝚗𝚎 5974

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

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ Links 14/12/2023: Growing Tensions in the
Far East and More Crackdowns on Dissent⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


posted by Roy Schestowitz on Dec 14, 2023


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇GNOME bluefish⦈


§ Contents⠀➾


    * Leftovers
          o Science
          o Mobile_Systems/Mobile_Applications
          o Hardware
          o Health/Nutrition/Agriculture
          o Proprietary/Artificial_Intelligence_(AI)
          o Security
                # Privacy/Surveillance
          o Defence/Aggression
          o Environment
                # Energy/Transportation
                # Wildlife/Nature
          o Finance
          o AstroTurf/Lobbying/Politics
          o Censorship/Free_Speech
          o Freedom_of_Information_/_Freedom_of_the_Press
          o Civil_Rights/Policing
          o Internet_Policy/Net_Neutrality
          o Monopolies/Monopsonies
                # Trademarks
                # Copyrights
                # Gemini*_and_Gopher
                      # Personal/Opinions
                      # Technology_and_Free_Software
                            # Internet/Gemini
    * § Leftovers⠀➾


          o ⚓ Tedium ☛ A_Incomplete_Picture⠀⇛


                 A recent scandal around a popular YouTuber’s nonprofit
                 foundation has created a lot of drama, but what it’s
                 missing are voices that understand the nonprofit sector.


          o ⚓ CS Monitor ☛ This_Chinese_folk_artist_shares_joy_on_paper,_one
            snip_at_a_time⠀⇛


                 Women are keeping the art of paper cutting alive in
                 northwestern China. To ensure the craft’s future, they
                 must pass on their passion to the next generation.


          o § Science⠀➾


                # ⚓ Hackaday ☛ Voyager_1_In_Trouble_As_Engineers_Scramble_To
                  Debug_Issue_With_Flight_Data_System⠀⇛


                       Recently the team at JPL responsible for
                       communication with the Voyager 1 spacecraft noticed
                       an issue with the data it was returning from the
                       Flight Data System (FDS). Although normally the FDS
                       is supposed to communicate with the other
                       subsystems via the telecommunications unit (TMU),
                       this process seems to have broken down, resulting
                       in no payloads from the scientific instruments or
                       engineering sensors being returned any more, just
                       repeating binary patterns. So far the cause of this
                       breakdown is unknown, and JPL engineers are working
                       through potential causes and fixes.


                # ⚓ teleSUR ☛ Kenya:_Experts_Meet_to_Promote_Research_in
                  Nuclear_Technology⠀⇛


                       "In Africa, several countries plan to implement
                       national nuclear research reactors programs so as
                       to improve their health care, industrial and
                       agricultural productivity," said deputy director
                       general and head of the nuclear energy department
                       at the International Atomic Energy Agency (IAEA).


          o § Mobile Systems/Mobile Applications⠀➾


                # ⚓ Zimbabwe ☛ Itel_A70_review._So_much_hype_around_this_$89
                  smartphone.⠀⇛


                       So I pulled out the Itel A70 and my cousin who just
                       finished grade 7 immediately identified the phone.
                       “This phone is getting so much hype at my school”
                       were the words she said and I was not even aware of
                       its existence up until the point when I got it for
                       review.


          o § Hardware⠀➾


                # ⚓ Hackaday ☛ Cheap_Computer_PSU_Puts_On_Weight_With_Box_Of
                  Iron⠀⇛


                       Humans are funny creatures. For whatever reason,
                       when handling a piece of electronics, we tend to
                       equate heft with value.  If something feels too
                       light, it gives the impression of being cheap or
                       inferior. As such, it’s not unheard of for gadgets
                       to include a little chunk of metal that serves no
                       purpose other than to add weight.


                # ⚓ Hackaday ☛ X86_ENTER:_What’s_That_Second_Parameter?⠀⇛


                       [Raymond Chen] wondered why the x86 ENTER
                       instruction had a strange second parameter that
                       seems to always be set to zero. If you’ve ever
                       wondered, [Raymond] explains what he learned in a
                       recent blog post.


                # ⚓ Hackaday ☛ NASA_Blames_Probe_Chute_Failure_On_Wire_Labels⠀⇛


                       When NASA’s OSIRIS-REx sample return capsule
                       screamed its way through the upper atmosphere, it
                       marked the first time the space agency had brought
                       material from an asteroid back to Earth. Hundreds
                       of thousands tuned into the September 24th live
                       stream so they could watch the capsule land at the
                       Utah Test and Training Range. But about ten minutes
                       before the capsule was set to touchdown, keen eyed
                       viewers may have noticed something a bit odd — when
                       ground control called out that the vehicle’s drogue
                       parachute was commanded to deploy…nothing seemed to
                       happen.


          o § Health/Nutrition/Agriculture⠀➾


                # ⚓ University of Michigan ☛ The_Daily_Weekly:_Unraveling_ADHD
                  and_the_Social_Media_Maze⠀⇛


                       In this episode of The Daily Weekly, we delve into
                       the intricate relationship between social control
                       media, attention spans, and ADHD in college
                       students.


                # ⚓ Off Guardian ☛ Won’t_Someone_Save_Us_From_The_COVID-19
                  Inquiry?⠀⇛


                       As Scotty once said “I can’t take it Captain!” If I
                       have to read, see or hear one more legacy
                       media report about the monumental farce that is the
                       UK’s COVID-19 inquiry, I fear my brain will batter
                       my soul to death.


                # ⚓ ADF ☛ Drug_Consumption_Grows_in_West_Africa_After_Years_on
                  Transit_Route⠀⇛


                       West Africa has transitioned over the past decade
                       from a transit route for illegal drugs headed to
                       Europe from South America to a booming illegal drug
                       market with a troubling rise in domestic users.


                # ⚓ The Straits Times ☛ South_Korea_may_downgrade_Covid-19
                  alert_level_in_2024⠀⇛


                       It will go from the current second-highest level of
                       “alert” to the third-highest of “caution”.


                # ⚓ Pro Publica ☛ Idaho_Keeps_Some_Psychiatric_Patients_in
                  Prison,_Ignoring_Decades_of_Warnings_About_the_Practice⠀⇛


                       One night in March 1976, a young advocate for
                       people with mental illness arrived at the Idaho
                       statehouse with a warning.


                       Marilyn Sword urged lawmakers not to ratify a
                       system that would ultimately lock away some of
                       Idaho’s most debilitated psychiatric patients in
                       the tiny, concrete cells of a maximum security
                       prison — a kind of solitary confinement with no
                       trial, no conviction and often no charges.


                # ⚓ Pro Publica ☛ Senators_Ask_GAO_to_Probe_FDA_Medical_Device
                  Oversight⠀⇛


                       Two prominent U.S. senators are calling for a
                       government investigation into the Food and Drug
                       Administration’s oversight of medical devices
                       following revelations that the agency failed to
                       protect the public from defective breathing
                       machines capable of sending particles and fumes
                       into the masks of patients.


                       Sens. Dick Durbin, D-Ill., and Richard Blumenthal,
                       D-Conn., asked the Government Accountability Office
                       — the investigative arm of Congress — to probe how
                       the FDA tracks warnings about dangerous devices,
                       oversees recalls and takes action against companies
                       that put patients at risk.


                # ⚓ Pro Publica ☛ Thousands_of_Patients_May_Be_Undergoing
                  Vascular_Procedures_Too_Soon_or_Unnecessarily⠀⇛


                       Four years ago, researchers warned the Centers for
                       Medicare and Medicaid Services that some doctors
                       were overusing — potentially even abusing —
                       invasive vascular procedures, increasing patients’
                       chances of complications, which include amputation
                       or even death.


                       They handed the federal agency a list of outlier
                       physicians they found in the government insurer’s
                       own data. But the agency has done little to stop
                       the practice and instead continues to pay doctors
                       who exhibit this behavior millions of dollars a
                       year.


                # ⚓ Pro Publica ☛ How_ProPublica_and_CareSet_Investigated_the
                  Overuse_of_Vascular_Procedures⠀⇛


                       Over the past year, ProPublica has investigated
                       these vascular treatments, and we found some
                       doctors are earning millions of dollars conducting
                       a questionable number of procedures. For treatments
                       in outpatient clinics, federal insurers, like
                       Medicare, pay generous reimbursements to doctors,
                       who can charge tens of thousands of dollars for
                       procedures done in a single office visit.


                # ⚓ Pro Publica ☛ Researchers_Warned_of_Possible_Abuse_of
                  Lucrative_Vascular_Procedures._Doctors_Pushed_Back.⠀⇛


                       Vascular surgeon Dr. Caitlin Hicks recalls the
                       first time, about a decade ago, that a patient came
                       to her with a lower leg mottled with purple and
                       starting to decay. She searched for a pulse in the
                       damaged limb, but blood had stopped flowing into
                       it.


                       The patient had previously undergone an invasive
                       vascular procedure, using devices designed to clear
                       out clogged arteries. But when Hicks questioned the
                       patient about the symptoms that had led them to
                       initially seek treatment, they seemed mild or
                       nonexistent.


                # ⚓ Pro Publica ☛ Lawmakers_Propose_Ban_on_Insurers_Charging
                  Doctors_a_Fee_for_Electronic_Payments⠀⇛


                       A bipartisan group of six U.S. representatives has
                       introduced a bill that would prohibit insurers and
                       their intermediaries from levying fees on doctors
                       for paying them electronically. The legislation
                       comes in the wake of a ProPublica investigation
                       that detailed the toll of such fees, which add up
                       to billions of dollars that could be spent on care
                       but are instead funneled to insurers and payment
                       processors.


                       The charges are akin to having an employer deduct
                       1.5% to 5% to provide a paycheck electronically if
                       an employee prefers to receive a payment directly
                       into their bank account rather than via a paper
                       check. Yet that’s the choice many insurers are
                       increasingly forcing on doctors.


                # ⚓ Science Alert ☛ One_Simple_Trick_Could_Make_More_People_Eat
                  Vegan_Food⠀⇛


                # ⚓ Science Alert ☛ Landmark_Study_Shows_Antibody_Therapy
                  Controls_92%_of_Severe_Asthma_Cases⠀⇛


                       Reducing reliance on steroids.


                # ⚓ Science Alert ☛ Unexpected_Discovery_About_Dopamine_May
                  Help_Explain_Parkinson's⠀⇛


                # ⚓ Science Alert ☛ There's_an_Indiscriminate_Killer_Stalking
                  Your_Yard,_Study_Confirms⠀⇛


                       Another reason to keep your cat inside.


                # ⚓ RFERL ☛ WHO_Says_Hundreds_Of_Afghan_Children_Dying_Because
                  Of_Respiratory_Diseases⠀⇛


                       A new report by the World Health Organization (WHO)
                       says respiratory diseases have killed more than
                       2,000 children under the age of 5 this year in
                       Afghanistan, a problem that  may grow due to the
                       country's underfunded health-care system.


                # ⚓ Latvia ☛ 'E-health'_to_include_all_lab_results_as_of_next
                  year⠀⇛


                       As of next year, the results of laboratory
                       examinations will be readily available free of
                       charge on the eHealth Portal (eveseliba.gov.lv),
                       thus making the healthcare system more efficient
                       which has been the aim of the system all along,
                       according to amendments to cabinet regulations
                       approved by the government on Tuesday, December 12.


                # ⚓ RFA ☛ Pods_of_pollution:_COP28_art_puts_Beijing,_Delhi_smog
                  on_display⠀⇛


                       Report says China and India had the highest deaths
                       related to poor air quality.


          o § Proprietary/Artificial Intelligence (AI)⠀➾


                # ⚓ BBC ☛ Elon_Musk's_Tesla_recalls_two_million_cars_in_US_over
                  Autopilot_defect_-_BBC_News⠀⇛


                       It follows investigation into crashes which
                       occurred when firm's Autopilot system was in use.


                # ⚓ Softonic ☛ 2023_may_also_be_the_worst_year_for_video
                  games⠀⇛


                       Three weeks before the end of the year, the video
                       game industry has seen an impressive figure of
                       9,000 job layoffs.


                # ⚓ India Today ☛ Ex-Spotify_employee_calls_layoffs_an
                  emotional_roller_coaster,_says_she_has_no_words_to_express
                  herself⠀⇛


                       A former Spotify employee took to LinkedIn and
                       shared that she was laid off recently and that she
                       has no words to express how it feels.


                # ⚓ Layoffs_resume,_but_businesses_battle_for_AI_talents [Ed:
                  Hype and vapouware don't create jobs, it's just a mirage to
                  keep investors throwing money into loss-raking operations]⠀⇛


                       Despite recent job cuts, notably among US firms,
                       businesses are still battling for AI talent,
                       according to industry sources.


          o § Security⠀➾


                # § Privacy/Surveillance⠀➾


                      # ⚓ Reason ☛ House_Proposal_Would_Expand_Federal
                        Warrantless_Spying_Authority⠀⇛


                             One bill set to be considered would grow the
                             scope of federal digital surveillance and
                             would authorize the federal government to use
                             those powers against more individuals.


                      # ⚓ Reason ☛ American_Courts_Enforcing_Subpoenas_to
                        Identify_Defendants_in_Foreign_Court_Cases⠀⇛


                             There's an interesting summary of the issues
                             related to this practice from Magistrate
                             Judge Peter Kang's decision last week in U.S.
                             v. Meta Platforms, Inc. (N.D. Cal.).


                      # ⚓ Privacy International ☛ Stop_GILAB⠀⇛


                      # ⚓ Vice Media Group ☛ Congress_Pulls_Bill_That_Would
                        Massively_Expand_Surveillance_After_'Dramatic
                        Showdown'⠀⇛


                             The House was set to vote on two competing
                             bills reauthorizing a powerful mass
                             surveillance tool, but faced backlash from
                             rights groups.


                      # ⚓ ACLU ☛ The_Federal_Trade_Commission_Must_Investigate
                        Meta_and_X_for_Complicity_with_Government
                        Surveillance⠀⇛


                             When you created your Facebook, Instagram, or
                             Ex-Twitter account, did you think you were
                             signing up to have the government spy on you?
                             That your personal information — who you are,
                             what you say, what you do, your friends, and
                             your political views — could be tapped by
                             surveillance companies and sold to police,
                             deportation agencies, and other local, state,
                             and federal forces?


                             In 2016, the ACLU_of_Northern_California_blew
                             the_whistle on how private surveillance
                             vendors were using Facebook (Farcebook) and
                             Ex-Twitter to help_police_target_Black_Lives
                             Matter_activists. As a result of advocacy by
                             the ACLU of Northern California,
                             MediaJustice, and Color of Change, both
                             Facebook and Twitter strengthened their anti-
                             surveillance policies and cut off access to
                             social control media surveillance companies.


          o § Defence/Aggression⠀➾


                # ⚓ Federal News Network ☛ How_to_help_troubled_service_members
                  and_veterans_get_through_the_holidays⠀⇛


                       Tis the season to be jolly. But military service
                       members and veterans often experience a spike in
                       depression or post traumatic stress disorder -
                       - even suicide -- this time of year. For some of
                       the warning signs and how you can help, we turn to
                       the Director of Admissions at Warriors Heart,
                       Michael O'Dell.


                # ⚓ France24 ☛ French_warship_intercepts_Huthis_aerial_attack
                  on_Norwegian_tanker_near_Red_Sea⠀⇛


                       Yemen's Huthi rebels claimed responsibility on
                       Tuesday for a missile strike on a Norwegian-flagged
                       tanker, an attack the Iran-backed group said was
                       part of its military campaign against Israel.


                # ⚓ ADF ☛ Sudan_Conflict_Expanding,_‘Taking_on_a_Life_of_Its
                  Own’⠀⇛


                       Community leaders in Babanousa in Sudan’s West
                       Kordofan State hoped their truce with the Sudanese
                       Armed Forces (SAF) and paramilitary Rapid Support
                       Forces (RSF) would prevent the violence that has
                       devastated the country from consuming their
                       community.


                # ⚓ New York Times ☛ Israel-Hamas_War:_Attacks_in_Red_Sea_and
                  on_Israel_Bolster_Houthis’_Regional_Standing⠀⇛


                       Monday’s strike on a Norwegian tanker was the
                       militia’s latest flex on the fringes of the Israel-
                       Hamas war. Once a scrappy tribal group, the Houthis
                       now hold what analysts say are higher aspirations.


                # ⚓ New York Times ☛ Errant_Airstrikes_by_Nigeria’s_Military
                  Have_Killed_Worshipers,_Herders_and_Refugees⠀⇛


                       The U.S.-backed Nigerian military has been killing
                       civilians in airstrikes at an alarming pace,
                       security analysts and human rights advocates say,
                       with little accountability.


                # ⚓ RFA ☛ Calls_grow_for_release_of_man_‘tortured’_for_doxxing
                  Xi’s_daughter⠀⇛


                       Family members and activists say the post wasn’t
                       published by Niu Tengyu or his meme website,
                       VulgarWiki.


                # ⚓ The Straits Times ☛ China’s_Pooh-tin_meets_Vietnam_leaders
                  on_second_day_of_state_visit_to_Hanoi⠀⇛


                       On Dec 13, Mr Pooh-tin will pay tribute to the
                       mausoleum of the founding leader of Communist
                       Vietnam, Ho Chi Minh.


                # ⚓ RFA ☛ Vietnam_joins_China-led_vision_of_foreign_policy⠀⇛


                       During Pooh-tin Jinping’s state visit to Hanoi,
                       Vietnam agreed to build a ‘Community with a Shared
                       Future’ with China.


                # ⚓ JURIST ☛ Philippines_summons_China_ambassador_amid
                  escalating_tensions_in_South_China_Sea⠀⇛


                       The Philippine Department of Foreign Affairs
                       announced Tuesday that it summoned the Chinese
                       Ambassador to the Philippines, Huang Xilian, to
                       lodge a diplomatic protest in the wake of the most
                       recent confrontation between Chinese and Filipino
                       vessels in the South China Sea (West Philippine
                       Sea) near the Second Thomas Shoal (Ayungin Shoal)
                       [...]


                # ⚓ Defence Web ☛ South_Africa_hosts_China-South_Africa_Defence
                  Committee⠀⇛


                       South Africa has hosted the ninth meeting of the
                       China-South Africa Defence Committee, which marks a
                       continuation of the close defence ties between the
                       two BRICS nations.


                # ⚓ The Straits Times ☛ Chinese_forces_approached_close_to
                  Taiwan_coast_to_‘intimidate’_voters_before_key_elections:
                  Sources⠀⇛


                       Taiwan officials have repeatedly warned that China
                       is trying to sway voters toward candidates seeking
                       closer ties with Beijing.


                # ⚓ Atlantic Council ☛ A_maritime_blockade_of_Taiwan_by_the
                  People’s_Republic_of_China:_A_strategy_to_defeat_fear_and
                  coercion⠀⇛


                       Marek Jestrab considers a naval blockade of Taiwan
                       by the People's Republic of China and advances
                       recommendations for the United States, Taiwan, and
                       likeminded nations to resist and respond to a
                       blockade.


                # ⚓ New York Times ☛ Wednesday_Briefing:_Biden_Says_Israel_Is
                  Losing_Support⠀⇛


                       Plus, Volodymyr Zelensky pleaded for more aid in
                       Washington.


                # § Russia, Belarus, and War in Ukraine⠀➾


                      # ⚓ Meduza ☛ Preschool_director_and_teacher_in_Russia
                        ‘resign’_after_students_mistakenly_given_certificates
                        with_Ukrainian_symbols_—_Meduza⠀⇛


                      # ⚓ RFERL ☛ Biden_Hails_Ukraine's_Fight_Against_Russia_As
                        'Victory,'_Urges_Congress_To_Speed_Additional_Aid⠀⇛


                             President Joe Biden told Ukrainian leader
                             Volodymyr Zelenskiy that if U.S. Congress
                             fails to approve additional military aid to
                             Kyiv, it would represent a "Christmas gift"
                             to Russian President Vladimir Putin, as
                             Zelenskiy continued his multipronged
                             diplomatic offensive during his Washington
                             visit.


                      # ⚓ Meduza ☛ Putin_spokesman_says_Navalny’s_whereabouts
                        not_Kremlin’s_concern_—_Meduza⠀⇛


                      # ⚓ Meduza ☛ Navalny_reportedly_transferred_to_Moscow_as
                        part_of_new_criminal_investigation_—_Meduza⠀⇛


                      # ⚓ New York Times ☛ Russia-Ukraine_War:_Biden_Says_Putin
                        Is_‘Banking_On’_the_U.S._Failing_to_Deliver_Aid_to
                        Ukraine⠀⇛


                             Standing beside President Volodymyr Zelensky
                             of Ukraine, President Biden said Russian
                             propagandists were celebrating Republicans’
                             decision to demand border security changes in
                             return for aiding Kyiv’s war effort.


                      # ⚓ Latvia ☛ Latvia_plans_to_reclaim_'Moscow_House'_and
                        give_it_to_Ukraine⠀⇛


                             The Saeima will advance a law on the
                             acquisition of the so-called Moscow House
                             building in Riga, Marijas Street 7, into
                             state ownership for approval in January. If
                             this happens, the building is intended to be
                             used to assist Ukraine or to be given to
                             Ukraine, Latvian Radio reported on December
                             12.


                      # ⚓ Reason ☛ The_GOP's_Bogus_Linkage_Between_Aiding
                        Ukraine_and_"Border_Security"⠀⇛


                             The analogy between Russia's invasion of
                             Ukraine and illegal migration to the US is
                             nonsensical. And many of the GOP's demands
                             are actually intended to make legal migration
                             more difficult, a policy likely to actually
                             increase the illegal kind.


                      # ⚓ Atlantic Council ☛ Zelenskyy_visits_DC_at_Ukraine’s
                        most_dangerous_moment⠀⇛


                             Our experts analyze what Zelenskyy got from
                             the visit and what’s next with US support for
                             Ukraine's war effort.


                      # ⚓ Atlantic Council ☛ Ukraine’s_Hey_Hi_(AI)_road_map
                        seeks_to_balance_innovation_and_security⠀⇛


                             As the world grapples with the implications
                             of rapidly evolving Artificial Intelligence
                             (AI) technologies, Ukraine has recently
                             presented a national road map for Hey Hi (AI)
                             regulation that seeks to balance the core
                             values of innovation and security, writes
                             Ukraine's Minister for Digital Transformation
                             Mykhailo Fedorov.


                      # ⚓ Atlantic Council ☛ New_Polish_PM_Donald_Tusk_vows
                        “full_mobilization”_of_West_to_help_Ukraine⠀⇛


                             Poland’s newly appointed Prime Minister
                             Donald Tusk has vowed to rally Western
                             support for Ukraine as it continues to defend
                             itself against Russia’s ongoing invasion,
                             writes Peter Dickinson.


                      # ⚓ Scoop News Group ☛ Ukraine’s_largest_mobile
                        communications_provider_down_after_apparent_cyber
                        attack⠀⇛


                             Millions of customers lost mobile phone and
                             home internet service as a result of what
                             Kyivstar said was a cyber attack.


                      # ⚓ France24 ☛ Scores_injured_in_Russian_missile_strikes
                        on_Kyiv⠀⇛


                             Ukraine’s capital came under another
                             ballistic missile attack early Wednesday,
                             resulting in more than 50 injuries and
                             several damaged buildings, officials said.


                      # ⚓ France24 ☛ Zelensky,_Biden_give_joint_press
                        conference_as_US_aid_for_Ukraine_hangs_in_the_balance⠀⇛


                             President Joe Biden told President Volodymyr
                             Zelensky on Tuesday that the United States
                             will not abandon Ukraine in its desperate
                             fight against Russia, even as Republicans
                             signaled opposition to extending US war
                             funding.


                      # ⚓ France24 ☛ EU_mulls_billions_in_funding_for_Hungary
                        on_eve_of_Ukraine_aid_summit⠀⇛


                             Brussels is expected to unlock 10 billion
                             euros in EU funding for Hungary on Wednesday,
                             the eve of a summit on support for Ukraine
                             that Hungarian Prime Minister Viktor Orban
                             has threatened to derail.


                      # ⚓ France24 ☛ Ukraine's_drone_war:_the_technology_at_the
                        centre_of_the_conflict⠀⇛


                             Ukraine's top general Valery Zaluzhny
                             recently identified five areas that the
                             country’s armed forces need to work on in
                             order to regain an edge over the Russians.
                             Two of them involve drones: the drones
                             themselves, and the techniques for jamming
                             them. Big and small, in the air, on land and
                             in the sea, unmanned vehicles are absolutely
                             central to this war, and the technology is
                             evolving rapidly. But while Ukraine is a
                             hotbed of innovation, quantity matters as
                             well as quality. And that's where Ukraine is
                             lagging behind Russia. Our correspondent
                             Gulliver Cragg takes a look at the
                             constellation of companies, military units
                             and government programmes trying to provide
                             Ukraine with the drone capacity it needs.


                      # ⚓ LRT ☛ Lithuania’s_Russian_speakers_hesitant_to_use
                        their_language_in_public_amid_pressure_to_prove
                        loyalty⠀⇛


                             Since Russia’s invasion of Ukraine, Russian
                             speakers in Lithuania say they are afraid to
                             speak Russian in public and only communicate
                             in their mother tongue at home. In public,
                             questions are constantly raised about their
                             loyalties and they are readily identified
                             with the Russian government.


                      # ⚓ RFERL ☛ U.S._Intelligence_Assesses_Ukraine_War_Has
                        Cost_Russia_315,000_Casualties⠀⇛


                             A declassified U.S. intelligence report
                             assessed that the Ukraine war has cost Russia
                             315,000 dead and injured troops, or nearly 80
                             percent of the personnel it had when the
                             conflict began, a source familiar with the
                             intelligence said on December 12.


                      # ⚓ RFERL ☛ EU's_Michel_Still_Working_On_'Plan_A'_For
                        Ukraine,_Moldova,_Georgia⠀⇛


                             European Council President Charles Michel
                             said he is still working on "Plan A" when it
                             comes to giving the green light to opening
                             accession talks with Ukraine at a crucial EU
                             summit in Brussels this week despite a
                             repeated threat by Hungary to block the
                             process.


                      # ⚓ RFERL ☛ Russian_Lawmakers_Approve_In_Final_Reading
                        Bill_Allowing_National_Guard_To_Hire_Volunteers⠀⇛


                             Russian parliament's lower chamber, the State
                             Duma, on December 12 approved in its final
                             reading a bill allowing the National Guard to
                             have its own volunteer groups that can take
                             part in Russia's full-scale invasion of
                             Ukraine.


                      # ⚓ RFERL ☛ EU_Must_Convince_Hungary_Over_Ukraine_Aid_At
                        Summit,_German_Official_Says⠀⇛


                             The European Union aims to convince holdout
                             Hungary to back its proposed 50 billion-euro
                             ($54 billion) aid facility for Kyiv at its
                             summit to send a clear signal to Russia, a
                             German government official said.


                      # ⚓ RFERL ☛ Ukraine's_Largest_Phone_Operator_Says_Huge
                        Cyberattack_Disrupted_Services⠀⇛


                             Ukraine's largest mobile phone operator,
                             Kyivstar, says it was targeted on December 12
                             by a massive cyberattack that brought down
                             its cellular and Internet signal.


                      # ⚓ RFERL ☛ After_Refusing_To_Extradite_Russian
                        Ultranationalist_To_Kyiv,_Finland_Extends_His
                        Detention⠀⇛


                             After refusing to extradite to Kyiv a Russian
                             ultranationalist and former commander of the
                             Rusich sabotage group that fights alongside
                             Russia's armed forces in Ukraine, a court in
                             Finland extended his detention on December
                             11.


                      # ⚓ YLE ☛ Defence_Minister:_Russian_aggression_reinforces
                        international_community⠀⇛


                             Finland must always be on its guard, Minister
                             Antti Häkkänen (NCP) says, but the Kremlin's
                             invasion of Ukraine has brought western
                             countries and Nordic nations closer together.


                      # ⚓ CS Monitor ☛ Poland’s_new_PM_Donald_Tusk_is_pro-EU,
                        committed_to_helping_Ukraine⠀⇛


                             Centrist party leader Donald Tusk was elected
                             prime minister by Poland’s parliament after
                             eight years of conservative rule under Law &
                             Justice. He has taken a pro-European Union
                             stance and vows to keep the rest of the world
                             committed to helping Ukraine.


                      # ⚓ New York Times ☛ No_Aid_for_Ukraine_Would_Be_a_Gift
                        to_Russia,_Biden_Says_as_Zelensky_Visits⠀⇛


                             President Volodymyr Zelensky of Ukraine
                             traveled to Washington to make a last-ditch
                             appeal for more help to fight Russia. But
                             Republicans said they wouldn’t act without a
                             border deal.


                      # ⚓ New York Times ☛ Republicans_Sideline_Zelensky_With
                        Border_Demands,_Imperiling_Aid_Package⠀⇛


                             The Ukrainian president delivered an urgent
                             plea for more help for his country in its
                             fight against Russia, only to be told by
                             Republicans that his challenges were not
                             their focus.


                      # ⚓ Meduza ☛ Over_50_injured_in_Russian_nighttime_strike
                        on_Kyiv_—_Meduza⠀⇛


                      # ⚓ New York Times ☛ Russia_Strikes_Kyiv,_Injuring_53
                        People⠀⇛


                             The bombardment, which the mayor said damaged
                             an apartment building and a hospital, was at
                             least the third significant strike on
                             Ukraine’s capital in a week.


                      # ⚓ Meduza ☛ U.S._intelligence_says_315K_Russian_troops
                        killed_or_injured_in_Ukraine_—_Meduza⠀⇛


                      # ⚓ The Straits Times ☛ Ukrainian_president_Zelenskiy_on
                        surprise_visit_to_Norway⠀⇛


                             December 13, 2023 3:32 PM



                             Ukrainian President Volodymyr Zelenskiy
                             arrived in Norway on Wednesday in a
                             previously unannounced visit, the Norwegian
                             government said in a statement.


                      # ⚓ Off Guardian ☛ Russia’s_friendliest_bank_will_help
                        BRICS_implement_the_“climate_agenda”⠀⇛


                             The 28th meeting of the Conference of the
                             Parties to the UN Framework Convention on
                             Climate Change, better known as COP28, is
                             heating up in Dubai. As expected, the
                             conferencegoers are in unanimous
                             agreement that renewable energy scams are
                             very good and carbon is very, very bad.


                      # ⚓ RFERL ☛ Georgian_President_Says_No_EU_Green_Light
                        Means_A_Win_For_Russia⠀⇛


                             As the European Union heads toward a crucial
                             summit in Brussels this week, Georgia
                             President Salome Zurabishvili says she's
                             under no illusion as to what it will mean if
                             her Caucasus nation fails to be given the
                             green light to become a candidate for
                             membership in the bloc.


                      # ⚓ RFERL ☛ Finland_To_Reopen_Two_Border_Crossings_With
                        Russia_But_Will_Maintain_Other_Closures⠀⇛


                             The Finnish government on December 12 said it
                             plans to reopen two border crossings with
                             Russia by January 14. Prime Minister Petteri
                             Orpo said the large Vaalimaa crossing in
                             southeast Finland and one near Niirala
                             further north will be reopened to traffic.


                      # ⚓ RFERL ☛ Request_For_Early_Release_Of_Chechen
                        Opposition_Bloggers'_Mother_Denied⠀⇛


                             A court in the Russian North Caucasus region
                             of Chechnya has rejected a request for early
                             release filed by Zarema Musayeva, the
                             imprisoned mother of three self-exiled
                             outspoken Chechen opposition activists.


                      # ⚓ RFERL ☛ Russian_Political_Analyst_Who_Was_Charged
                        With_Online_Calls_For_Terrorism_Fined⠀⇛


                             A court in Russia's Komi region on December
                             12 fined sociologist Boris Kagarlitsky
                             600,000 rubles ($6,660) on a charge of making
                             online calls to justify terrorism and ordered
                             his release from custody.


                      # ⚓ The Straits Times ☛ North_Korea_hosts_Russia
                        delegation_for_talks_on_economic_cooperation⠀⇛


                             A North Korean minister met with the governor
                             of the Russian region bordering the two
                             countries to discuss boosting regional
                             economic cooperation, state media KCNA
                             reported on Wednesday, the latest exchange
                             between the neighbouring states.


                      # ⚓ YLE ☛ Finland_to_re-open_two_eastern_border
                        checkpoints⠀⇛


                             The Finnish government closed the entire
                             eastern border nearly two weeks ago in
                             response to a suspected Russian hybrid attack
                             involving increasing numbers of undocumented
                             migrants arriving at Finnish checkpoints.


                      # ⚓ New York Times ☛ Finland_Partly_Reopens_Its_Border
                        With_Russia⠀⇛


                             The Nordic country, which recently joined
                             NATO, has accused Moscow of funneling
                             migrants to their shared land border.


                      # ⚓ Meduza ☛ Finland_to_reopen_two_border_crossings_with
                        Russia_—_Meduza⠀⇛


                      # ⚓ Latvia ☛ Trucks_wait_on_Latvia-Belarus_border_for_up
                        to_12_days⠀⇛


                             At the Latvian-Belarusian border point
                             "Pāternieki" which is currently the only
                             place to cross, 460 vehicles were queuing on
                             Tuesday, December 12. Waiting times can take
                             as much as 12 days, Latgale regional
                             television reported.


          o § Environment⠀➾


                # ⚓ Science Alert ☛ The_Arctic_Smashed_So_Many_Terrifying
                  Records_in_2023,_What_Can_We_Even_Say?⠀⇛


                       Heating 3x faster than the global average.


                # § Energy/Transportation⠀➾


                      # ⚓ Meduza ☛ Russian_aviation_authority_says_‘incidents’
                        in_Russia_have_decreased,_while_independent_and_foreign
                        media_report_the_opposite_—_Meduza⠀⇛


                # § Wildlife/Nature⠀➾


                      # ⚓ ADF ☛ Uganda’s_Pioneering_Wildlife_Court_Is_Deterring
                        Crime⠀⇛


                             Gladys Kamasanyu frequently travels about an
                             hour outside of Uganda’s capital, Kampala, to
                             the Entebbe Zoo, where she feeds, pets and
                             even talks to rhinos, elephants, pangolins
                             and other endangered animals. She also talks
                             to the staff at the zoo, known as the Uganda
                             Wildlife Conservation Education Center [...]


          o § Finance⠀➾


                # ⚓ The Straits Times ☛ IMF_board_clears_first_review_of
                  Bangladesh's_$4.7_billion_bailout⠀⇛


                       The International Monetary Fund said its executive
                       board cleared the first review of Bangladesh's $4.7
                       billion bailout on Tuesday, providing $936.6
                       million in funding to Dhaka as it battles high
                       inflation ahead of national elections in January.


                # ⚓ New York Times ☛ Lawmakers_Call_for_Raising_Tariffs_and
                  Severing_Economic_Ties_With_China⠀⇛


                       A bipartisan report recommended stripping China of
                       the low tariffs the United States granted it two
                       decades ago, among other actions.


                # ⚓ RFA ☛ China_steps_up_positive_propaganda_in_bid_to_boost
                  economic_growth⠀⇛


                       State media pump out data to back up the claim that
                       there is 'a bright future' in store for the
                       economy.


                # ⚓ The_year_of_the_layoff:_A_timeline_of_the_biggest
                  redundancies_to_hit_the_channel_in_2023⠀⇛


                       Constrained growth and supply chain hangovers in
                       2023 sparked major layoffs across tech, here's a
                       round-up the most impactful layoff announcements


                # ⚓ Softonic ☛ The_owner_of_Dungeons_and_Dragons_announces
                  thousands_of_layoffs⠀⇛


                       No one is safe from the multinational corporations’
                       scissors. Technological transitions and high costs
                       have led to the disappearance of thousands of jobs
                       at companies like Spotify, Microsoft, or Amazon.
                       Now, the iconic toy company Hasbro joins this sad
                       trend with the recent layoff of thousands of its
                       employees.


                # ⚓ WorkReduce_founder_and_CEO_arrested_for_solicitation_of
                  minor⠀⇛


                       Remote staffing company that works with major
                       holding companies is removing Brian Dolan as an
                       officer and director


          o § AstroTurf/Lobbying/Politics⠀➾


                # ⚓ Pro Publica ☛ The_Judiciary_Has_Policed_Itself_for_Decades.
                  It_Doesn’t_Work.⠀⇛


                       For decades, judges have relied on a select group
                       to make sure the judiciary adheres to the highest
                       ethical standards: themselves.


                       The Judicial Conference, a secretive, century-old
                       council of federal judges led by the chief justice
                       of the Supreme Court, oversees the ethics and
                       financial disclosures for more than 1,700 federal
                       judges, including the nine justices of the high
                       court. Those financial disclosures, submitted
                       yearly as a list of assets and gifts, are often the
                       only window into whether judges with lifetime
                       appointments have conflicts of interest as they
                       rule on the country’s most consequential legal
                       cases.


                # § Misinformation/Disinformation/Propaganda⠀➾


                      # ⚓ Techdirt ☛ Community_Notes_Is_Great_Until_It
                        Challenges_Elon,_And_Then_It’s_Being_‘Manipulated’_By
                        State_Actors⠀⇛


                             Oh Elon. As we’ve discussed, Elon is
                             infatuated with Community Notes as a sort of
                             crowdsourced alternative to actually funding
                             a trust & safety staff and tooling. And while
                             we actually like Community Notes and think
                             more social media should use similar tools,
                             it’s simply not a full trust & safety
                             replacement.


          o § Censorship/Free Speech⠀➾


                # ⚓ RFA ☛ Hong_Kong_arrests_7_pro-democracy_activists_for
                  protesting_poll⠀⇛


                       The authorities also issue warrants for 2 overseas
                       activists who called for a boycott of ‘patriotic’
                       election.


                # ⚓ RFA ☛ Pro-Beijing_‘thugs’_tormented_Pooh-tin_protesters,
                  activists_say⠀⇛


                       US lawmaker says San Francisco police ignored
                       repeated beatings of protesters during last month’s
                       APEC summit.


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


                # ⚓ Techdirt ☛ Text_Messages,_Emails_Show_KBI_Had_Full
                  Knowledge_Of_Raid_On_Kansas_Newspaper’s_Office⠀⇛


                       On August 11, the Marion County PD — with the
                       assistance of the Kansas Department of Revenue, the
                       county sheriff’s office, and (for some fucking
                       reason) the local fire marshal — raided the office
                       of the Marion County Record, along with the home of
                       its co-owner, 98-year-old Joan Meyer.


          o § Civil Rights/Policing⠀➾


                # ⚓ The Nation ☛ Indigenous_Wisdom⠀⇛


                # ⚓ New York Times ☛ Director_of_Sciences_Po_Offers_to_Step
                  Aside_After_Domestic_Violence_Dispute⠀⇛


                       Mathias Vicherat, the director of Sciences Po, has
                       exchanged accusations of domestic violence with his
                       partner. The police are investigating.


                # ⚓ Hong Kong Free Press ☛ 9_to_face_sentencing_over_unlawful
                  assembly_near_PolyU_during_Hong_Kong_protests_in_2019⠀⇛


                       Nine people will face sentencing over unlawful
                       assembly near the Hong Kong Polytechnic University
                       in 2019, where intense clashes broke out amid the
                       extradition-bill protests and unrest. Wong Chun-
                       yin, Ronald Cheng, Tang Long-yin, Lai Hang-kei, Ng
                       Chun-fung, Vicky Chiu, Yip Pui-shan, Law Tsz-kwan
                       and Catherine Wong appeared at District Court on
                       Tuesday afternoon.


          o § Internet Policy/Net Neutrality⠀➾


                # ⚓ Techdirt ☛ The_Telecom_Industry_Is_Very_Mad_Because_The_FCC
                  MIGHT_Examine_High_Broadband_Prices⠀⇛


                       We’ve long noted how the FCC (regardless of party)
                       largely ignores how muted competition and
                       monopolization drives up prices for consumers. The
                       agency often talks a good (if ambiguous) game about
                       “bridging the digital divide,” but they don’t
                       collect and share pricing data proving market
                       failure, nor are they capable of admitting
                       monopolies exist and are harmful in public-facing
                       messaging.


      § Monopolies/Monopsonies⠀➾


          o ⚓ Public Knowledge ☛ Public_Knowledge_Welcomes_Fair_Competition_Win
            in_Epic_v._Surveillance_Giant_Google_Jury_Verdict⠀⇛


            [Ed: Public Knowledge also has Microsoft inside the board, i.e. a
            conflict of interest]
                 Jury finds that Surveillance Giant Google violated
                 antitrust laws by monopolizing the market for Android app
                 distribution.


          o ⚓ Digital Music News ☛ Epic_Games_Triumphs_Over_Surveillance_Giant
            Google_in_Antitrust_Lawsuit:_‘A_Win_for_All_App_Developers_and
            Consumers_Around_the_World’⠀⇛


                 Epic Games has officially triumphed in its antitrust
                 lawsuit, centering on fees charged to Play Store
                 developers as well as adjacent agreements, against
                 Google. >


          o ⚓ JURIST ☛ US_jury_finds_Surveillance_Giant_Google_app_store
            policies_violate_antitrust_law⠀⇛


                 A US jury decided Monday that Google’s current app store
                 policies violate Sections 1 and 2 of the Sherman
                 Antitrust Act as well as the California Cartwright Act.
                 Sometime next year, there will be a separate judicial
                 process to determine the penalty that Surveillance Giant
                 Google will suffer as a result of the jury’s Monday
                 decision.


          o § Trademarks⠀➾


                # ⚓ TTAB Blog ☛ TTABlog_Test:_Are_These_Two_Bird_Design_Marks
                  Confusingly_Similar_for_Clothing?⠀⇛


                       In this battle of the birds, frequent TTAB
                       plaintiff Abercrombie & Fitch opposed an
                       application to register the mark shown below left,
                       for various clothing items, claiming a likelihood
                       of confusion with its registered design mark shown
                       below right, for overlapping clothing items. Since
                       there were no limitations in the application or
                       pleaded registrations as to channels of trade and
                       classes of consumers, the Board presumed that the
                       goods travel in the same, normal channels to the
                       same consumer classes. So it all boiled down to the
                       marks. How do you think this case out? Abercrombie
                       &_Fitch_Trading_Co._v._Srinivasa_Rao_Gubbala,
                       Opposition No. 91255288 (December 8, 2023) [not
                       precedential] (Opinion by Judge Wendy Boldt Cohen).


                # ⚓ Dennis Crouch/Patently-O ☛ Federal_Circuit_Decision_Causes
                  Confusion_on_Trademark_Cancellation⠀⇛


                       I previously wrote about the trademark case of
                       Great Concepts v. Chutter that involves two
                       restaurants both using the same phonetic name:
                       DANTANNA’S and DAN TANA’S.  In the underlying
                       trademark cancellation proceeding, the TTAB found
                       that one party had committed fraud within its
                       declaration of incontestability; and that fraud
                       justified cancellation of the registration.  On
                       appeal the Federal Circuit reversed — concluding
                       that the statute was very specific that fraud in
                       obtaining the registration can justify
                       cancellation, but says nothing about fraud in
                       establishing incontestability.  The court
                       distinguished with some squirrely precedent finding
                       that renewal counted as “obtaining” for the fraud-
                       cancellation doctrine. Torres v. Cantine Torresella
                       S.r.l., 808 F.2d 46 (Fed. Cir. 1986).


          o § Copyrights⠀➾


                # ⚓ Digital Music News ☛ Altice_USA_Faces_Yet_Another_$1
                  Billion+_Copyright_Lawsuit_—_‘An_Attractive_Tool_and_Safe
                  Haven_for_Infringement’⠀⇛


                       New York City-headquartered internet service
                       provider (ISP) Altice USA is officially facing yet
                       another $1 billion+ copyright monopoly lawsuit
                       stemming from its alleged failure to stop
                       subscribers’ repeat infringement.


          o § Gemini* and Gopher⠀➾


                # § Personal/Opinions⠀➾


                      # ⚓ Some_of_the_music_i'm_listening_to_-_10⠀⇛


                      # ⚓ The_protocol_seems_wonderful,_but_the_community
                        result_seems_dubious_-_if_not_imaginary⠀⇛


                             What I've found instead is like a lot of
                             houses, each with a clipboard attached to a
                             board pounded into the ground in the front
                             yard, and every now and then home-dwellers
                             sneak out to replace the piece of paper
                             containing text clamped to their clipboard
                             with another piece of paper containing
                             different text, then scurry back into their
                             house before anyone notices, because
                             <deity>forbid they'd have to actually talk to
                             anyone....


                      # ⚓ The_protocol_seems_wonderful,_but_the_community
                        result_seems_dubious_-_if_not_imaginary⠀⇛


                             Same with capitalism, peoples. It's not a
                             thing. It's not like the sun. It seemingly
                             exists because you keep patronizing it by
                             buying shit, especially shit you really don't
                             need. We've learned there's a fuckload of
                             negative side-effects to that dynamic at
                             society scale.


                             Stop it - i.e. the unnecessary buying of
                             unnecessary shit - and tell others how much
                             better you feel for having been that much
                             less covetous. Let that *ease* - in contrast
                             to dis-ease - be what spreads.


                      # ⚓ Composition_Considered_Harmful⠀⇛


                             Since Alan Perlis pointed it out, y’all know
                             by now how Lisp programmers know the value of
                             everything and the cost of nothing. Same goes
                             in society. When things get neatly packaged
                             up behind a comfy interface, a neat case and
                             a button to push, it’s hard to remain aware
                             of the rusty gears and Rube Goldberg yak-
                             shaving mousetraps that happen behind the
                             scenes.


                             Like, a chicken lays two eggs every three
                             days approximately, so if you eat two eggs
                             per day, that’s three entire chickens that
                             are being kept somewhere just for your sake.


                             And modern abstraction have become recursive;
                             each of the gears in the machine are in turn
                             pushing more buttons for you, which in turns
                             start their own sets of gears, and one
                             machine suddenly turns into five which turns
                             into twenty which turns into a hundred.


                # § Technology and Free Software⠀➾


                      # ⚓ jarotsim_at_hybrid_live_coding_interfaces_2023⠀⇛


                             we presented jarotsim, a playground for the
                             discovery, exploration and livecoding of
                             Turing machines, at the Hybrid Live Coding
                             Interfaces 2023: Boundless Thinking workshop,
                             on december 11th, 2023.


                      # ⚓ Infinite_Scrolling_and_hyper_personalization_are
                        evil⠀⇛


                             I miss the old BBS and forums long gone the
                             few that exist are like ghost town. Once in a
                             blue moon I go back and I remember vividly
                             certain topics or obscure knowledge bits. The
                             ability to navigate and map the information
                             with section, broken down into topics and
                             comments.


                             The main page was the same for everybody and
                             everybody knew what the other would get.


                      # ⚓ More_Service-Breaking_Chess_Bugs⠀⇛


                             While I work in IT, I am not a programmer,
                             and my programming skills are far from
                             perfect. I've learned this the hard way with
                             the many bugs I've come across in my
                             capsule's chess service. Today, I found and
                             fixed another major one.


                             The chess service works asynchronously and
                             uses match invitations created by users to
                             find challengers to play against. A SQLite
                             database stores all match and invitation
                             information in the backend, including pending
                             invitations. As users fill in information
                             such as what color they want to play as and
                             public comments, that information is stored
                             in a pending invitation row until the
                             invitation is published.


                      # ⚓ Building_a_huge_library⠀⇛


                             While it is common for a person to
                             unconsciously open instagram or tiktok when
                             bored, I open Z-library, and scout for books.
                             Productive as this may sound, I have yet to
                             read those books I've collected in my device.
                             And then comes the question of space, both
                             storage in my device and mental space, for
                             whenever I download a book, leaf through it,
                             and decide I'm going to keep it, I am not
                             merely hoarding, but there is an implicit
                             commitment to read it sooner or later. So it
                             is that I organize my books in the filesystem
                             according to topic, and those books that I
                             store "for later" go into a backup in a hard
                             drive, perhaps for a future when the free
                             access to information becomes a thing of the
                             past. But even the few subjects that I decide
                             to keep around start growing on me, and every
                             so often I need to go and prune the tree of
                             topics and subtopics that so develops.


                      # § Internet/Gemini⠀➾


                            # ⚓ Small_Roundup_on_the_Web_and_Journalism⠀⇛


                                   Speaking of such: Sports Illustrated's
                                   publisher has fired its CEO after the
                                   scandal that had the sports icon's site
                                   include AI-generated articles by AI-
                                   generated reporters. There was an
                                   outcry over the audacity, but if you
                                   look at who owns it, you shouldn't be
                                   surprised: Authentic Brands Group is
                                   the sort of industry-agnostic value
                                   vampire that would do exactly this sort
                                   of thing, and did. If CNET can be
                                   ruined by this sort of dreck, why not
                                   Sports Illustrated, they probably
                                   reckoned. Absolutely. Why not. What's
                                   the value of creation, anyway? Why love
                                   anything? Pull the content lever and
                                   fill 'er up with sludge.


===============================================================================
* Gemini_(Primer) links can be opened using Gemini_software. It's like the
World Wide Web but a lot lighter.




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