𝕿𝖊𝖈𝖍𝖗𝖎𝖌𝖍𝖙𝖘 Bulletin for Wednesday, February 09, 2022
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Generated Thu 10 Feb 02:41:12 GMT 2022
Created by Dr. Roy Schestowitz (𝚛𝚘𝚢 (at) 𝚜𝚌𝚑𝚎𝚜𝚝𝚘𝚠𝚒𝚝𝚣 (dot) 𝚌𝚘𝚖)
Full hyperlinks for navigation omitted but are fully available in the originals
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Latest in 𝒉𝒕𝒕𝒑://𝒕𝒆𝒄𝒉𝒓𝒊𝒈𝒉𝒕𝒔.𝒐𝒓𝒈/𝒕𝒙𝒕 and older bulletins can be found at 𝒉𝒕𝒕𝒑://𝒕𝒆𝒄𝒉𝒓𝒊𝒈𝒉𝒕𝒔.𝒐𝒓𝒈/𝒕𝒙𝒕-𝒂𝒓𝒄𝒉𝒊𝒗𝒆𝒔
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Gemini index for the day: gemini://gemini.techrights.org/2022/02/09/


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

⦿ [Meme] Team UPC and Team Battistelli Above the Law, Ignore the Law, Break the Law | Techrights

⦿ The EPO’s Overseer/Overseen Collusion — Part XXXXVI: An Erosion of Fundamental Rights Protection? | Techrights

⦿ Response to Gemini Misconceptions and Mythology (It’s Not Ugly or Just for Geeks) | Techrights

⦿ Despite Occasional Shutdowns, Geminispace Has a Net Gain of About 100 Working Capsules Per Month (It Has Accelerated Further in 2022) | Techrights

⦿ Staff Holding Management at the EPO Accountable | Techrights

⦿ [Meme] Drunk Driving, Like Son of António Campinos | Techrights

⦿ IRC Proceedings: Tuesday, February 08, 2022 | Techrights

 ䷼ Bulletin articles (as HTML) to comment on (requires login):
		http://techrights.org/2022/02/09/breaking-the-law-in-nl/#comments
		http://techrights.org/2022/02/09/fundamental-rights-epo/#comments
		http://techrights.org/2022/02/09/gemini-not-ugly/#comments
		http://techrights.org/2022/02/09/geminispace-1705-working-capsules/#comments
		http://techrights.org/2022/02/09/holding-management-accountable/#comments
		http://techrights.org/2022/02/09/iloat-ruling-no-4430-and-iloat-ruling-no-4482/#comments
		http://techrights.org/2022/02/09/irc-log-080222/#comments

 ䷞ Followed by Daily Links (assorted news picks curated and categorised):
		http://techrights.org/2022/02/09/epic-wants-rootkits-in-linux/#comments
		http://techrights.org/2022/02/09/gulag-stadia-failing/#comments

                    ䷩ 𝚕𝚒𝚗𝚎 66

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

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/02/09/breaking-the-law-in-nl/#comments
    Gemini version at gemini://gemini.techrights.org/2022/02/09/breaking-the-law-in-nl/

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Gemini_version_available_♊︎


✐ [Meme]_Team_UPC_and_Team_Battistelli_Above_the_Law,_Ignore_the_Law,_Break_the
Law⠀✐


Posted in Europe, Patents at 8:52 am by Dr. Roy Schestowitz


Much can be said about how Team UPC too flagrantly breaks_the_law with the
EPO's_complicity


            🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇Law? Sorry, not applicable to us.⦈


Summary: In the_Netherlands, EPO management is effectively above the law, so
Benoît_Battistelli and António_Campinos can ignore court rulings, gaslight an
International_Criminal_Court_(ICC)_judge_looking_to_rectify_things_at_the_EPO,
and bribe_Dutch_media to defame actual judges with integrity


          🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇António Campinos fronting for Team UPC⦈
    Tony fronting for ‘Team UPC’ only 3 days after he became EPO President.


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

                    ䷩ 𝚕𝚒𝚗𝚎 1269

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

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    Gemini version at gemini://gemini.techrights.org/2022/02/09/fundamental-rights-epo/

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Gemini_version_available_♊︎


✐ The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXXXVI:_An_Erosion_of
Fundamental_Rights_Protection?⠀✐


Posted in Courtroom, Europe, Law, Patents at 8:40 am by Dr. Roy Schestowitz


Series parts:


   1. The_EPO’s_Overseer/Overseen_Collusion_—_Part_I:_Let_the_Sunshine_In!
   2. The_EPO’s_Overseer/Overseen_Collusion_—_Part_II:_A_“Unanimous”
      Endorsement?
   3. The_EPO’s_Overseer/Overseen_Collusion_—_Part_III:_Three_Missing_Votes
   4. The_EPO’s_Overseer/Overseen_Collusion_—_Part_IV:_The_Founding_States
   5. The_EPO’s_Overseer/Overseen_Collusion_—_Part_V:_Germany_Says_“Ja”
   6. The_EPO’s_Overseer/Overseen_Collusion_—_Part_VI:_A_Distinct_Lack_of_Dutch
      Courage
   7. The_EPO’s_Overseer/Overseen_Collusion_—_Part_VII:_Luxembourgish_Laxity
   8. The_EPO’s_Overseer/Overseen_Collusion_—_Part_VIII:_Perfidious_Albion_and
      Pusillanimous_Hibernia
   9. The_EPO’s_Overseer/Overseen_Collusion_—_Part_IX:_More_Holes_Than_Swiss
      Cheese
  10. The_EPO’s_Overseer/Overseen_Collusion_—_Part_X:_Introducing_the
      Controversial_Christian_Bock
  11. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XI:_“General_Bock”_–
      Battistelli’s_Swiss_Apprentice?
  12. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XII:_The_French_Connection
  13. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XIII:_Battistelli’s_Iberian
      Facilitators_–_Spain
  14. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XIV:_Battistelli’s_Iberian
      Facilitators_–_Portugal
  15. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XV:_Et_Tu_Felix_Austria…
  16. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XVI:_The_Demise_of_the
      Austrian_Double-Dipper
  17. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XVII:_The_Non-Monolithic
      Nordic_Bloc
  18. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XVIII:_Helsinki’s_Accord
  19. The_EPO’s_Overseer/Overseen_Collusion_—_Part_IXX:_The_Baltic_States
  20. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XX:_The_Visegrád_Group
  21. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXI:_The_Balkan_League_–_The
      Doyen_and_His_“Protégée”
  22. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXII:_The_Balkan_League_–
      North_Macedonia_and_Albania
  23. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXIII:_The_Balkan_League_–
      Bulgaria
  24. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXIV:_The_Balkan_League_–
      Romania
  25. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXV:_The_Balkan_League_–
      Fresh_Blood_or_Same_Old,_Same_Old?
  26. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXVI:_A_Trojan_Horse_on_the
      Budget_and_Finance_Committee
  27. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXVII:_Cypriot_Complicity
  28. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXVIII:_Benoît_and_António’s
      Loyal_“Habibi”
  29. The_EPO’s_Overseer/Overseen_Collusion_—_Part_IXXX:_The_EPOnian_Micro-
      States_–_Monaco_and_Malta
  30. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXX:_San_Marino_and_the
      Perfidious_Betrayal_of_Liberty
  31. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXXI:_The_Abstentionists
  32. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXXII:_“Plucky_Little
      Belgium”?
  33. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXXIII:_Swedish_Scepticism
  34. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXXIV:_An_“Extremely
      Dubious”_Proposal
  35. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXXV:_Slovakian_Scruples
  36. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXXVI:_Serbian_Sour_Grapes
  37. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXXVII:_Stubbornly
      Independent_Slovenia
  38. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXXVIII:_Ensnared_in_the
      Tentacles_of_the_SAZAS_Octopus
  39. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXXIX:_On_the_Slippery_Slope
      to_Capture
  40. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXXX:_The_Idiosyncratic
      Italians
  41. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXXXI:_Public_Service_or
      Self-Service?
  42. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXXXII:_A_Parcel_of_Rogues?
  43. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXXXIII:_A_Legal_No-Man’s
      Land
  44. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXXXIV:_Immunity_=_Impunity?
  45. The_EPO’s_Overseer/Overseen_Collusion_—_Part_XXXXV:_In_the_Shadow_of
      “Waite_and_Kennedy”
  46. YOU ARE HERE ☞ An Erosion of Fundamental Rights Protection?
                🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇Kicking_Fundamental_Rights⦈_
 Recent judgments of the European Court of Human Rights appear to have eroded
the obligation of ECHR states to protect fundamental rights in cases where the
           alleged violations involve an international organisation.


Summary: What the European Court of Human Rights means to EPO workers in light
of more recent developments, especially 5 years ago (Supreme Court of the
Netherlands)


In the last part we saw how the European Court of Human Rights (ECtHR) has
ruled that states which are signatories of the European Convention on Human
Rights (ECHR) have a responsibility to ensure that their membership of
international organisations does not lead to a breach of their ECHR
obligations.


“…it has become evident that the overriding priority of the Court is to protect
the autonomy of international organisations at the expense of the fundamental
rights of their staff (and other individuals adversely affected by the acts of
such organisations).”In other words, ECHR signatory states must ensure that
adequate provision is made for an “equivalent protection” of fundamental rights
inside any international organisation which they join.


This has implications for the internal justice systems of international
organisations on which staff are obliged to rely for legal redress, because of
their lack of access to national courts.


According to the “Waite_and_Kennedy_v._Germany” judgment, such internal justice
systems must provide staff of an international organisation with “reasonable
alternative means to protect effectively their rights under the [European]
Convention [on Human Rights]“.


However, following the delivery of ECtHR judgments in the cases of Klausecker
v. Germany (application_no._415/07) and Perez_v._Germany [PDF] (no._15521/08)
in January 2015, it has become evident that the overriding priority of the
Court is to protect the autonomy of international organisations at the expense
of the fundamental rights of their staff (and other individuals adversely
affected by the acts of such organisations).


“There was now a distinct possibility that international organisations and
their member states could get away with applying a different and less rigorous
standard of human rights protection to the acts and omissions of an
international organisation.”The Klausecker and Perez judgments prompted the
legal scholar Anne-Marie Thévenot-Werner to express concern about an_erosion_of
the_obligation_of_states_to_protect_the_fundamental_rights_enshrined_in_the
ECHR.


In an article published in the legal journal Revue de droit allemande in 2015
Thévenot-Werner acknowledged that the ECtHR had formally reaffirmed its case
law requiring the provision of “reasonable alternative means” for the
protection of fundamental rights inside international organisations.


However, at the same time the Court signalled that it would not hold a state to
account for a breach of the ECHR involving an international organisation unless
the protection available via the organisation’s internal justice system was
shown to be “manifestly deficient”.


Thévenot-Werner warned that the approach taken by the Court in these cases
risked creating a loophole with respect to the protection of fundamental rights
enshrined in the ECHR.


“It deserves to be emphasised at this point that these are matters which are
not merely of academic and theoretical interest.”There was now a distinct
possibility that international organisations and their member states could get
away with applying a different and less rigorous standard of human rights
protection to the acts and omissions of an international organisation. In such
situations, states would only be called to account if it could be established
to the satisfaction of the Court that the level of protection available was
“manifestly deficient”.


The risk of such an erosion of legal protection exists even in the case of
organisations where all participating states are signatories to the ECHR (which
is the case for the EPO).


It deserves to be emphasised at this point that these are matters which are not
merely of academic and theoretical interest.


The saga of Benoît_Battistelli‘s “Strike Regulations” at the EPO provides a
striking illustration of their practical significance for the staff of
international organisations.


“The Gerechtshof judgment showed that at least some judges in the Netherlands
were serious about setting limits to the “immunity” enjoyed by an international
organisation in order to prevent abuses involving clear-cut violations of
fundamental rights.”As noted in an_earlier_part_of_this_series, Battistelli’s
controversial “Strike Regulations” were one of a long litany of grievances
which the EPO staff union SUEPO attempted to litigate before the national
justice system in the Netherlands.


During the litigation procedure, the Gerechtshof den Haag (Appeal Court of The
Hague) decided that breaches of fundamental rights by EPO management were so
severe that they justified lifting the organisation’s immunity from
jurisdiction.


This lead to a judgment, delivered on 17_February_2015, in which the
Gerechtshof ordered_the_EPO [PDF] to rescind several amendments to the
organisation’s staff regulations, including the undue restrictions on
industrial actions imposed by the impugned “Strike Regulations”.


In addition to this, the EPO was ordered by the court to uphold the rule-of-law
and social dialogue standards in_line_with_the_European_Convention_on_Human
Rights_and_ILO_Conventions_No._87,_on_the_right_to_organise_in_trade_unions,
and_No._98_on_the_right_to_collective_bargaining. [PDF]


The Gerechtshof judgment showed that at least some judges in the Netherlands
were serious about setting limits to the “immunity” enjoyed by an international
organisation in order to prevent abuses involving clear-cut violations of
fundamental rights.


The European Public Service Union (EPSU) reported on the judgment under the
headline “European_Patent_Office_does_not_live_in_a_Dutch_no_man’s_land”.


“The problem with the Gerechtshof judgment was that it set a worrying precedent
which had political ramifications going far beyond the EPO.”Unfortunately, both
SUEPO’s legal victory and EPSU’s optimism turned out to be short-lived.


Less than two years later, the Gerechtshof judgment was set aside, thereby
confirming what many had long suspected: the European Patent Office does indeed
live in a Dutch no man’s land!


                   🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇Gerechtshof_den_Haag⦈_
 An independent judicial review of violations of fundamental rights by the EPO
  carried out by the Gerechtshof den Haag led to a short-lived lifting of the
 organisation’s immunity in February 2015. This was overruled in January 2017
when the Supreme Court proclaimed that Dutch courts had “no jurisdiction in the
                                 EPO dispute”.


The problem with the Gerechtshof judgment was that it set a worrying precedent
which had political ramifications going far beyond the EPO.


“The Hoge Raad attempted to justify its findings on the basis that the
fundamental rights of EPO staff “were sufficiently protected by the internal
dispute settlement procedure provided for by EPOrg”.”It was a veritable thorn
in the flesh not only for Battistelli and his clique at the EPO, but also for
the Dutch government which reaps significant economic benefits from the
country’s status as a host to a large number of international organisations.


This led to the Dutch government joining forces with Team Battistelli in
support of an application for “cassation” of the judgment.


Finally, on January 2017, the Hoge Raad (Supreme Court of the Netherlands)
issued a ruling to the effect that Dutch courts had no_jurisdiction_in_the_EPO
dispute:


     “According to the European Court of Human Rights (ECtHR), granting
     jurisdictional immunity to an international organisation constitutes
     a limitation of the right of access to a court as referred to in
     article 6 of the Convention for the Protection of Human Rights and
     Fundamental Freedom Rights (ECHR). This limitation is acceptable
     provided that litigants have a reasonable alternative means of
     protecting their rights effectively.


     The Supreme Court found that such alternative means exist. The rights
     of VEOB [the Netherlands branch of SUEPO] and SUEPO are sufficiently
     protected by the internal dispute settlement procedure provided for
     by EPOrg, under which individual employees and staff representatives
     can ultimately take their complaint to the Administrative Tribunal of
     the International Labour Organization in Geneva. According to the
     Supreme Court, this means that the essence of their right of access
     to a court has not been impaired.”


The Hoge Raad attempted to justify its findings on the basis that the
fundamental rights of EPO staff “were sufficiently protected by the internal
dispute settlement procedure provided for by EPOrg”.


But is this really the case?


“In the upcoming parts, we will take a closer look at the EPO’s internal
justice system and the role which it played in “Strike Regulations” affair.”Not
everybody would agree. Indeed, many would argue that the facts tell a very
different story and that the judgment delivered by the Hoge Raad in January
2017 was driven by political expediency and a misplaced desire to tell the
Dutch government and the EPO what they wanted to hear rather than what they
needed to hear.


In the upcoming parts, we will take a closer look at the EPO’s internal justice
system and the role which it played in “Strike Regulations” affair.


In particular, we will see how a former judge of the European Court of Human
Rights played a bizarre and incongruous role in prolonging the suppression of
the fundamental right to “freedom of association” at the EPO. █


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⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉⠋⠛⠍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠛⠚⠓⠛⠛⢿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣠⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠎⠀⠀⠀⠘⠙⠿⡤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠐⠤⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣽⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⠛⠛⢹⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢤⡂⠸⠦⣠⣄⣈⣿⣿⣿⣿⣿⣿⣿⣿⣯⠆⠀⠀⠀⠀⠀⠀⣨⣧⣶⣿⣿⣿⣿⣵⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣿⣿⣿⣿⣿⣿⡛⠈⠀⠀⠀⠀⠀⠀⠀⠐⠹⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⠃⠀⢰⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢹⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⣠⣿⣿⣿⣿⣿⣿⣿⣟⠿⠃⠈⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡟⢀⡇⢸⣽⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⣿⣽⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡼⡏⠽⠟⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⡀⠸⣿⣿⣿⣿⣿⣿⣿⠁⣔⠀⣼⣿⣧⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣴⡠⠀⠀⠀⠃⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣔⣶⣟⡀⢛⣿⣿⣿⣿⣿⡅⣰⣿⠀⣿⣿⣿⣤⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢱⣿⣿⣿⡿⡿⠿⠟⠛⠁⠀⣀⣀⣤⣶⣯⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠁⠈⠉⠉⠙⠃⠈⢿⣿⣿⡟⠛⠀⠙⠉⠈⠉⠉⠉⠙⠂⠙⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⣿⣿⣯⡉⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢴⣾⠗⣀⣀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⣿⣤⡄⡀⠀⠀⠀⠀⠀⣀⢠⢼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡚⠐⠳⠧⠀⠀⠙⠁⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡣⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣀⣀⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣷⣶⣿⣿⣿⣿⣿⣿⠿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣴⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠝⠛⠛⠀⠸⡿⠛⠙⠻⣿⣿⣿⡇⠀⢄⠀⢀⠽⠋⠛⠟⡻⡻⠛⠛⢟⠀⠘⠙⢯⠟⠉⠙⢿⠏⠙⠫⢿⠛⠛⠛⠻⢯⠙⠙⠙⡟⠛⠛⠏⡽⠉⠉⠻⢏⡇⢘⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢐⡠⠀⠰⠀⠀⠁⠀⣿⣿⣿⡇⠀⡀⠀⣼⠀⠀⠀⠀⠂⠀⠤⡄⣞⠀⢠⡂⢨⣠⠀⠰⣾⡀⠈⠐⢾⠀⠠⡤⠀⢹⠀⢰⣤⠆⠈⠀⠀⡵⠀⠈⠀⢸⡇⠀⠀⢼⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣠⣼⣧⣄⣀⣴⣿⣿⣿⣇⣤⣷⣄⣸⣦⣄⣀⣤⣴⣦⣀⣠⣴⣤⣜⣇⣸⣿⣄⣀⣼⣄⣀⣤⡼⠀⢠⣀⣠⣝⣤⣜⣧⣤⣀⣀⣤⣧⣄⣀⣤⣼⣇⣸⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠻⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⡟⢛⣿⣿⢿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣟⡎⢛⣻⣿⠛⠛⠟⢿⢿⣿⣿⣿⣿⣿⣿⣟⡟⢻⣿⠻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠠⣟⠓⠺⠍⠉⠉⠍⢉⠍⠩⠉⠟⢉⣉⠁⢈⡉⣫⠀⡍⠉⠁⠇⢈⠉⡋⠉⠍⠹⠄⢨⣽⣷⠀⣿⡄⢨⠋⠍⠉⠍⢡⠙⣿⣿⡇⢘⡛⢸⡍⠩⠙⠏⠩⠉⡟⢩⡍⢹⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣄⣉⣁⣸⡂⢀⢠⣇⣨⣤⣀⣀⡃⡈⡉⣁⣸⣂⣙⣀⡡⣀⣀⣁⣨⣀⡧⣀⠁⢸⣂⣾⣿⣿⣀⣉⣠⢾⣀⣀⣴⣂⣊⣀⣿⣿⣃⣸⣟⢈⢁⣀⣀⣇⣈⣀⡆⢈⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣰⣽⣿⣿⣿⣿⣿⣿⣿⣿

                    ䷩ 𝚕𝚒𝚗𝚎 1632

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

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/02/09/gemini-not-ugly/#comments
    Gemini version at gemini://gemini.techrights.org/2022/02/09/gemini-not-ugly/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 02.09.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Response_to_Gemini_Misconceptions_and_Mythology_(It’s_Not_Ugly_or_Just_for
Geeks)⠀✐


Posted in Deception, Free/Libre_Software, FUD, Protocol, Standard at 7:54 am by
Dr. Roy Schestowitz


Video_download_link | md5sum 420621d8ea77533494fdde5e187ca23a
Gemini Misconceptions
Creative Commons Attribution-No Derivative Works 4.0


http://techrights.org/videos/gemini-myths.webm


Summary: Another gemini:// FUD response; this time we deal with this_new
article, a polite article, which nevertheless makes Gemini as a whole seem
crude and ugly; in reality, depending how it’s presented, Gemini can be
beautiful, expressive, and feature-rich (depending on the creativity of
pertinent capsules)


THE RISE of Gemini isn’t a subjective thing. It is measurable. It’s objective.


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇1700 Gemini capsules approached⦈ As shown in the image to
the right, Gemini is growing fast. On January 16th we_wrote_about_1,600_active
capsule and now we’re flirting with 1,700. Almost 100 active capsules in net
gain in 3.5 weeks is nothing to sneeze at (not the same as 100 new capsules as
it’s possible that 10 of the existing ones shut down while 110 new ones
emerged).


“Almost 100 active capsules in net gain in 3.5 weeks is nothing to sneeze at
(not the same as 100 new capsules as it’s possible that 10 of the existing ones
shut down while 110 new ones emerged).”Over the past few months I’ve seen a
number of Gemini-bashing posts that were mostly based on misunderstandings.
Sometimes there’s almost outdated information or misinformation. “I had a
discussion with someone about Gemini,” one person told me this morning, “a
couple of years ago.” That’s like the time Gemini was only months old. A lot
has changed since then. The number of capsules grew from a couple hundred to
over 2,000, with probably around a million pages in them (Lupa alone has
retrieved and analysed almost half a million pages).


That same person then asked me and another Gemini proponent: “What are the
advantages you two see in Gemini?”


“The number of capsules grew from a couple hundred to over 2,000, with probably
around a million pages in them (Lupa alone has retrieved and analysed almost
half a million pages).”“Speaking for myself,” I replied, “self-hosting is one.
The Web became far too bloated, so it’s difficult to host from one’s
residential perimeter, hence people move to centralised platforms and
datacentres.”


“But people’s connections have gotten faster, it’s just the Web that changed.
Why can’t people take back control of the Internet? In his talks about
Freedombox, Eben Moglen raised this point.”


The person then asked: “Does Gemini use HTML? Could we post the same pages HTML
pages we have now on a gemini server, or would they need to be modified and
adapted?”


Some people wrongly believe that Gemini Protocol is only for transmission or
only for ‘markup’. It’s actually both. HTML is very complicated for a lot of
people. Setting up a Web server is also hard compared to Gemini. So lesser
technical people cannot easily participate, unless they create some accounts in
Blogspot (Gulag) or Facebook…


“There are already dozens of Gemini clients (like Web browsers) and over a
dozen Gemini server implementations.”Gemini uses GemText, which is simple
enough to teach to 6 year olds. It’s also a lot simpler to program with, owing
to this simplicity.


There are many good HTML→GemText (and vice verse) converters. Anyone with basic
coding skills can write one in a few hours, as many people have. “XSLT might
also be an option for HTML→GemText,” an associate of ours notes.


There are already dozens_of_Gemini_clients_(like_Web_browsers)_and_over_a_dozen
Gemini_server_implementations.


Our hope is that better understanding of Gemini, owing to media coverage (the
mainstream media is paid to push ads and lies, so it won’t help Gemini grow),
can help without it resembling “advocacy”/”evangelism” (religious term) or
“marketing”. As it turns out, some mainstream media already_embraces_Gemini and
it has become big enough for Microsoft_to_attempt_to_hijack/undermine_it. █


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣇⢿⣿⡟⡇⡇⡷⣹⡜⡇⢿⢸⢸⡱⣼⢬⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣾⣀⣇⣾⣈⣿⣟⣼⣷⣇⣸⣿⣏⣾⣃⣿⣸⣽⣿⣯⣺⣢⣗⣺⣐⣟⣸⣻⣧⣽⣼⣟⣿⣿⣉⣇⣸⣿⣧⣻⣇⣵⣯⣾⣠⣗⣺⣐⣇⣿⣃⣇⣿⣸⣞⣿⣿⣒⣗⣺⣇⣿⣸⣬⣵⣷⣺⣡⣗⣿⣿
⣿⣿⣿⣿⣿⢿⡿⣿⡿⣿⢿⡿⡿⣿⢻⢿⣿⣿⣿⣿⢿⣿⣟⣿⢻⢿⣿⢿⣿⡿⣿⢿⣿⣿⢻⣿⠿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣾⣷⣿⣿⣯⣿⣷⣷⣿⣼⣧⣿⣿⣯⣿⣿⣿⣿⣿⣵⣿⣿⣾⣿⣿⣿⣿⣿⣿⣾⣿⣶⣽⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

                    ䷩ 𝚕𝚒𝚗𝚎 1753

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

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/02/09/geminispace-1705-working-capsules/#comments
    Gemini version at gemini://gemini.techrights.org/2022/02/09/geminispace-1705-working-capsules/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 02.09.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Despite_Occasional_Shutdowns,_Geminispace_Has_a_Net_Gain_of_About_100_Working
Capsules_Per_Month_(It_Has_Accelerated_Further_in_2022)⠀✐


Posted in Site_News at 9:37 am by Dr. Roy Schestowitz


 🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇January 2021: ~400; June 2021: 1000; December 2021: 1500;
               January 2022: 1600; Today (Feb. 9th): Over 1700⦈


                      🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇1705_capsules⦈_


Summary: With almost 2,100 known Gemini capsules, about 82% of which are online
and accessible, the_growth_continues_unabated (net gain of about 100 in just 3
weeks or roughly 5 per day, on average)


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⢰⣶⡆⢰⣶⣶⡏⠉⣷⣦⢰⣶⡆⣶⣶⢱⣶⡏⢹⣶⣶⡏⢻⣶⣶⣶⣬⢱⣶⡄⣶⡾⢻⡟⣵⣶⣶⣍⢋⣵⣶⣶⣍⢫⣴⣶⣦⡙⣛⣩⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟⣵⣶⣶⠈⣽⣶⣶⣭⠛⣽⣶⣶⣌⠻⣿
⠿⡇⢸⣿⡇⢸⣿⣿⣿⠀⣿⣿⣾⣿⡇⣿⣿⢸⣿⡇⣼⣿⣿⣷⢸⣿⣿⣽⣿⠀⣿⣷⣿⢇⣿⣾⠿⢇⣿⣿⢸⣿⡇⣿⣿⠸⠿⢠⣿⡇⠛⣿⣿⢈⣿⣿⣿⢟⣛⡛⠛⡛⢱⡿⣿⣿⢸⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠉
⠀⠀⢸⣿⡇⣿⣿⣸⣿⡄⣿⣿⣿⣿⡇⣿⣿⢸⣿⡇⣿⣿⣹⣿⢸⣿⣿⢿⣷⠀⢸⣿⡟⣸⣿⣿⢃⣾⡿⠃⢸⣿⡇⣿⣿⠀⣠⣿⡟⢰⡇⣿⣿⠘⣛⣿⣷⣟⣉⡛⢛⠃⣿⣇⣿⣿⣸⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠀
⠀⢠⣼⣿⡇⣿⣿⢻⣿⡇⣿⡇⢿⣿⡇⢿⣿⣼⣿⢧⣿⡿⢻⣿⣿⣿⣿⢸⣿⠀⢸⣿⡇⠹⡿⢡⣿⣿⣥⣤⠸⣿⣧⣿⡟⢰⣿⣯⣤⡄⠁⣿⣿⢰⣶⡏⢿⣿⣿⠿⠟⠘⠛⠛⣿⣿⠋⢿⣧⣿⡿⠘⣿⣧⣿⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⣿⣆⠀⠀⠀⢀⣰⣾⣿⣿⣿⡆⠀⠀⠀⠀⢰⣟⣷⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⣠⣶⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀
⠀⠠⠠⠤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⣄⣀⣀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠠⠄⠀
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⠤⠤⠤⠶⠶⠶⠒⣲⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⠍⠀⠀⠀⠈⠉⠉⠐⠶⢦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠉⠉⠛⠙⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠿⠋⠙⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣷⣶⣶⡒⠲⠤⠤⠤⠤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣴⣿⠷⢿⣿⣿⣿⣿⣿⡿⠟⠉⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀
⠄⠠⠒⠒⠚⠛⡛⠋⢹⣟⠁⠀⣻⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⣠⣿⣿⣿⣿⣿⣴⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢀⠠⠤⣶⣾⣿⣿⣿⣿⣿⡿⠋⡌⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀
⠀⠖⠶⣶⠏⢶⠄⠱⠶⠶⣆⡹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠉⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⢰⣶⡆⢰⣶⣶⡏⠉⣷⣦⢰⣶⡆⣶⣶⢱⣶⡏⢹⣶⣶⡏⢻⣶⣶⣶⣬⢱⣶⡄⣶⡾⢻⡟⣵⣶⣶⣍⢋⣵⣶⣶⣍⢫⣴⣶⣦⡙⣛⣩⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟⣵⣶⣶⠈⣽⣶⣶⣭⠛⣽⣶⣶⣌⠻⣿
⠻⡇⢸⣿⡇⢸⣿⣿⣿⠀⣿⣿⣾⣿⡇⣿⣿⢸⣿⡇⣼⣿⣿⣷⢸⣿⣿⣹⣿⠀⣿⣷⣿⢇⣿⣾⠿⢇⣿⣿⢸⣿⡇⣿⣿⠸⠿⢠⣿⡇⠛⣿⣿⢈⣿⣿⣿⢟⣛⡛⠛⡛⢰⡿⣿⣿⢰⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠉
⠀⠀⢸⣿⡇⣿⣿⣸⣿⡄⣿⣿⣿⣿⡇⣿⣿⢸⣿⡇⣿⣿⣹⣿⢸⣿⣿⢿⣷⠀⢸⣿⡟⣸⣿⣿⢃⣾⡿⠃⢸⣿⡇⣿⣿⠀⣠⣿⡟⢰⡇⣿⣿⠘⣛⣿⣷⣟⣉⡛⠛⠃⣿⣇⣿⣿⣸⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠀
⠀⢠⣼⣿⡇⣿⣿⢻⣿⡇⣿⡇⢿⣿⡇⢿⣿⣼⣿⢧⣿⡿⢻⣿⣿⣿⣿⢸⣿⠀⢸⣿⡇⠹⡿⢡⣿⣿⣥⣤⠸⣿⣧⣿⡟⢰⣿⣯⣤⡌⠁⣿⣿⢰⣶⡏⢿⣿⣿⠿⠟⠘⠛⠛⣿⣿⠋⢿⣧⣿⡿⠘⣿⣧⣿⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⣿⣆⠀⠀⠀⢀⣰⣾⣿⣿⣿⡄⠀⠀⠀⠀⢰⣞⣷⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⣠⣶⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀
⠀⠄⠠⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⣀⣀⣀⠀⠀⠀⠠⠤⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⡤⠤⠤⠴⠶⠶⠖⣲⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠉⠀⠀⠀⠈⠉⠉⠒⠶⢦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠂⠉⠉⠉⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠿⠋⠉⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣷⣶⣶⡒⠦⠤⠤⠤⢤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣴⣿⡷⢶⣿⣿⣿⣿⣿⡿⠟⠉⠉⠛⠋⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀
⠄⠠⠒⠒⠚⠛⠛⠋⢽⡿⠁⠁⣿⣶⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⣠⣿⣿⣿⡿⣋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⢀⠠⠤⣶⣾⣿⣿⣿⣿⣿⡿⠋⡀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀
⠀⠖⠲⣶⠏⢶⠄⠱⠶⢶⣆⡹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠿⠛⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⢰⣶⡆⢰⣶⣶⡏⠉⣷⣦⢰⣶⡆⣶⣶⢱⣶⡏⢹⣶⣶⡏⢻⣶⣶⣶⣬⢱⣶⡄⣶⡾⢻⡟⣵⣶⣶⣍⢋⣵⣶⣶⣝⢫⣴⣶⣦⡙⣛⣩⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟⣵⣶⣶⠈⣽⣶⣶⣭⠛⣽⣶⣶⣌⠻⣿
⠿⡇⢸⣿⡇⢸⣿⣿⣿⠀⣿⣿⣾⣿⡇⣿⣿⢸⣿⡇⣼⣿⣿⣷⢸⣿⣿⣽⣿⠀⣿⣷⣿⢇⣿⣾⠿⢇⣿⣿⢸⣿⡇⣿⣿⠸⠿⢠⣿⡇⠛⣿⣿⢈⣿⣿⣿⢟⣛⡛⠟⡛⢱⡿⣿⣿⢸⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠁
⠀⠀⢸⣿⡇⣿⣿⣸⣿⡄⣿⣿⣿⣿⡇⣿⣿⢸⣿⡇⣿⣿⣹⣿⢸⣿⣿⢿⣷⠀⢸⣿⡟⣸⣿⣿⢃⣾⡿⠃⢸⣿⡇⣿⣿⠀⣠⣿⡟⢰⡇⣿⣿⠘⣛⣿⣷⣟⣉⡛⠛⠃⣿⣇⣿⣿⣸⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠀
⠀⢠⣼⣿⣇⣿⣿⢻⣿⡇⣿⡇⢿⣿⡇⢿⣿⣼⣿⢧⣿⡿⢻⣿⣿⣿⣿⢸⣿⠀⢸⣿⡇⠹⡿⢡⣿⣿⣥⣤⠸⣿⣧⣿⡟⢰⣿⣯⣤⡄⠁⣿⣿⢰⣶⡏⢿⣿⣿⠿⠟⠘⠛⠛⣿⣿⠋⢿⣧⣿⡿⠘⣿⣧⣿⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⠀⠀⠀⠀⠀⠠⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⢿⣿⣄⠀⠀⠀⢀⣲⣿⣿⣿⣿⡄⠀⠀⠀⠀⢰⣾⣧⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣠⣾⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣁⣀⣀⣀⠀⠀⠠⠤⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⡤⠤⠤⠴⠶⠶⠖⣲⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠉⠀⠀⠀⠈⠉⠉⠒⠶⢤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠉⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠿⠋⠉⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣷⣶⣶⡒⠦⠤⠤⠤⢤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣴⣾⡶⢾⣿⣿⣿⣿⣿⡿⠏⠉⠈⠛⠛⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀
⠤⠠⠔⠒⠚⠛⠛⠋⢹⡏⠁⠁⣲⣶⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⣿⡿⣋⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⢀⠠⠤⣶⣶⣿⣿⣿⣿⣿⡿⠋⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀
⠀⠖⠶⣶⠏⢶⠄⠱⠶⢶⣆⡹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠁⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⢰⣶⡆⢰⣶⣶⡏⠉⣷⣦⢰⣶⡆⣶⣶⢱⣶⡏⢹⣶⣶⡏⢻⣶⣶⣶⣭⢱⣶⡄⣶⡾⢻⡟⣵⣶⣶⣍⢋⣵⣶⣶⣝⢫⣴⣶⣦⡙⢛⣩⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟⣵⣶⣶⠈⣽⣶⣶⣭⠛⣽⣶⣶⣬⠻⣿
⠻⡇⢸⣿⡇⢸⣿⣿⣿⠀⣿⣿⣾⣿⡇⣿⣿⢸⣿⡇⣼⣿⣿⣷⢸⣿⣿⣽⣿⠀⣿⣷⣿⢇⣿⣾⠿⢇⣿⣿⢸⣿⡇⣿⣿⠸⠿⢠⣿⡇⠛⣿⣿⢈⣿⣿⣿⢟⣛⡛⠛⡛⢰⡿⣿⣿⢰⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠉
⠀⠀⢸⣿⡇⣿⣿⣸⣿⡄⣿⣿⣿⣿⡇⣿⣿⢸⣿⡇⣿⣿⣹⣿⢸⣿⣿⢿⣷⠀⢸⣿⡟⣸⣿⣿⢃⣾⡿⠃⢸⣿⡇⣿⣿⠀⣠⣿⡟⢰⡇⣿⣿⠘⣛⣿⣷⣟⣉⡛⠛⠃⣿⣇⣿⣿⣸⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠀
⠀⢠⣼⣿⣇⣿⣿⢻⣿⡇⣿⡇⢿⣿⡇⢿⣿⣼⣿⢧⣿⡿⢻⣿⣿⣿⣿⢸⣿⠀⢸⣿⡇⠹⡿⢡⣿⣿⣥⣤⠸⣿⣧⣿⡟⢰⣿⣯⣤⡄⠁⣿⣿⢰⣶⡏⢿⣿⡿⠿⠟⠘⠛⠛⣿⣿⠋⢿⣧⣿⡿⠘⣿⣧⣿⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣛⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡄⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣷⡄⠀⠀⠀⢐⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⣲⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⣠⣾⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀
⠀⠀⠀⢄⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⣀⣀⣀⣀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⡤⠤⠤⠤⠶⠶⠖⣲⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡝⠉⠀⠀⠀⠈⠉⠉⠒⠶⢤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠂⠘⠛⠋⠙⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠋⠙⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣷⣶⣶⡒⠦⠤⠤⢤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣦⣶⡶⢶⣿⣿⣿⣿⣿⡿⠟⠉⠙⠛⠛⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀
⠀⠠⠰⠒⠚⠛⠛⠋⠻⠏⠉⠀⣩⣶⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⣠⣿⣿⣿⣿⣏⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠤⣶⣶⣿⣿⣿⣿⣿⡿⠋⠈⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀
⠀⠖⠶⣶⠏⢶⠄⠱⠶⢶⣆⠹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠿⠉⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⢰⣶⡆⢰⣶⣶⡏⠉⣷⣦⢰⣶⡆⣶⣶⢱⣶⡏⢹⣶⣶⡏⢻⣶⣶⣶⣬⢱⣶⡄⣶⡾⢻⡟⣵⣶⣶⣍⢋⣵⣶⣶⣝⢫⣴⣶⣦⡙⣛⣩⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟⣵⣶⣶⠈⣽⣶⣶⣭⠛⣽⣶⣶⣬⠻⣿
⠿⡇⢸⣿⡇⢸⣿⣿⣿⠀⣿⣿⣾⣿⡇⣿⣿⢸⣿⡇⣼⣿⣿⣷⢸⣿⣿⣽⣿⠀⣿⣷⣿⢇⣿⣾⠿⢇⣿⣿⢸⣿⡇⣿⣿⠸⠿⢠⣿⡇⠛⣿⣿⢈⣿⣿⣿⢟⣛⡛⠛⡛⢱⡿⣿⣿⢸⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠉
⠀⠀⢸⣿⡇⣿⣿⣸⣿⡄⣿⣿⣿⣿⡇⣿⣿⢸⣿⡇⣿⣿⣹⣿⢸⣿⣿⢿⣷⠀⢸⣿⡟⣸⣿⣿⢃⣾⡿⠃⢸⣿⡇⣿⣿⠀⣠⣿⡟⢰⡇⣿⣿⠘⣛⣿⣷⣟⣉⡛⠛⢃⣿⣇⣿⣿⣸⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠀
⠀⢠⣼⣿⣇⣿⣿⢻⣿⡇⣿⡇⢿⣿⡇⢿⣿⣼⣿⢧⣿⡿⢻⣿⣿⣿⣿⢸⣿⠀⢸⣿⡇⠹⡿⢡⣿⣿⣥⣤⠸⣿⣧⣿⡟⢰⣿⣯⣤⡄⠁⣿⣿⢰⣶⡏⢿⣿⡿⠿⠟⠘⠛⠛⣿⣿⠋⢿⣧⣿⡿⠘⣿⣧⣿⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⢀⣼⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣧⡀⠀⠀⠀⢐⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣰⣲⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢻⣿⣷⣤⣴⣾⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣤⣿⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀
⠀⠀⠠⢤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣀⣀⣀⣀⠀⠄⠀⠄⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⠤⠤⠤⠶⠶⠖⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡙⠋⠀⠀⠀⠈⠉⠉⠒⠶⣤⣄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠐⠚⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠋⠙⠛⠋⠋⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣷⣶⣶⡒⠤⠤⠤⠤⣤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣴⣷⡾⢿⣿⣿⣿⣿⣿⡿⠟⠉⠛⠛⠛⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀
⠀⠤⠔⠒⠒⠛⠟⠛⢻⡛⠉⠁⣵⣶⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⣠⣿⣿⣿⣿⣏⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⠠⠤⣶⣶⣿⣿⣿⣿⣿⡿⠋⠈⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀
⠀⠖⠲⣶⠏⢶⠄⠱⠶⢶⣖⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠟⠋⠁⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⢰⣶⡆⢰⣶⣶⡏⠉⣷⣦⢰⣶⡆⣶⣶⢱⣶⡏⢹⣶⣶⡏⢻⣶⣶⣶⣬⢱⣶⡄⣶⡾⢻⡟⣵⣶⣶⣍⢋⣵⣶⣶⣝⢫⣴⣶⣦⡙⢛⣩⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟⣵⣶⣶⠈⣵⣶⣶⣭⠛⣽⣶⣶⣌⠻⣿
⠻⡇⢸⣿⡇⢸⣿⣿⣿⠀⣿⣿⣾⣿⡇⣿⣿⢸⣿⡇⣼⣿⣿⣷⢸⣿⣿⣽⣿⠀⣿⣷⣿⢇⣿⣾⠿⢇⣿⣿⢸⣿⡇⣿⣿⠸⠿⢠⣿⡇⠛⣿⣿⢈⣿⣿⣿⢟⣛⡛⠛⡛⢱⡿⣿⣿⢰⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠉
⠀⠀⢸⣿⡇⣿⣿⣸⣿⡄⣿⣿⣿⣿⡇⣿⣿⢸⣿⡇⣿⣿⣹⣿⢸⣿⣿⢿⣷⠀⢸⣿⡟⣸⣿⣿⢃⣾⡿⠃⢸⣿⡇⣿⣿⠀⣠⣿⡟⢰⡇⣿⣿⠘⣛⣿⣷⣟⣉⡛⠛⠃⣿⣇⣿⣿⣸⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠀
⠀⢠⣼⣿⣇⣿⣿⢻⣿⡇⣿⡇⢿⣿⡇⢿⣿⣼⣿⢧⣿⡿⢻⣿⣿⣿⣿⢸⣿⠀⢸⣿⡇⠹⡿⢡⣿⣿⣥⣤⠸⣿⣧⣿⡟⢰⣿⣯⣤⡄⠁⣿⣿⢰⣶⡏⢿⣿⡿⠿⠟⠘⠛⠛⣿⣿⠋⢿⣧⣿⡿⠘⣿⣧⣿⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣾⣄⠀⠀⠀⠀⢺⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣔⣲⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⣿⣦⣠⣴⣾⣿⣿⣿⣿⣿⣿⣶⣤⣀⢀⣠⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀
⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠒⠒⠂⠂⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣀⣀⣀⣀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠄⠀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⡤⠤⠤⠶⠶⠶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡙⠃⠀⠀⠀⠉⠉⠉⠒⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠒⠘⠋⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠋⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣶⣶⡤⠤⠤⠤⢤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣶⡶⢶⣾⣿⣿⣿⣿⡿⠛⠉⠉⠛⠛⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀
⠠⠤⠔⠒⠚⠛⠛⠛⢻⡋⠉⠉⣬⣶⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⣀⣠⣿⣿⣿⡿⣋⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠤⣴⣶⣿⣿⣿⣿⣿⡿⠋⠩⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀
⠀⠖⠲⣶⠏⢶⠄⠱⠶⢶⣆⡙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠿⠓⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⢱⣶⡆⢰⣶⣶⡏⠉⣷⣦⢰⣶⡆⣶⣶⢱⣶⡏⢹⣶⣶⡏⢻⣶⣶⣶⣬⢱⣶⡄⣶⡾⢻⡟⣵⣶⣶⣍⢋⣵⣶⣶⣝⢫⣴⣶⣦⡙⣛⣩⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟⣵⣶⣶⠈⣵⣶⣶⣭⠛⣽⣶⣶⣬⠻⣿
⠿⡇⢸⣿⡇⢸⣿⣿⣿⠀⣿⣿⣾⣿⡇⣿⣿⢸⣿⡇⣼⣿⣿⣷⢸⣿⣿⣹⣿⠀⣿⣷⣿⢇⣿⣾⠿⢇⣿⣿⢸⣿⡇⣿⣿⠸⠿⢠⣿⡇⠛⣿⣿⢈⣿⣿⣿⢟⣛⡛⠛⡛⢱⡿⣿⣿⢰⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠉
⠀⠀⢸⣿⡇⣿⣿⣸⣿⡄⣿⣿⣿⣿⡇⣿⣿⢸⣿⡇⣿⣿⣹⣿⢸⣿⣿⢿⣷⠀⢸⣿⡟⣸⣿⣿⢃⣾⡿⠃⢸⣿⡇⣿⣿⠀⣠⣿⡟⢰⡇⣿⣿⠘⣛⣿⣷⣟⣉⡛⠛⠃⣿⣇⣿⣿⣸⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠀
⠀⢠⣼⣿⣇⣿⣿⢻⣿⡇⣿⡇⢿⣿⡇⢿⣿⣼⣿⢧⣿⡿⢻⣿⣿⣿⣿⢸⣿⠀⢸⣿⡇⠹⡿⢡⣿⣿⣥⣤⠸⣿⣧⣿⡟⢰⣿⣯⣤⡄⠁⣿⣿⢰⣶⡎⢿⣿⡿⠿⠟⠘⠛⠛⣿⣿⠋⢿⣧⣿⡿⠘⣿⣧⣿⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣦⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣯⣿⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣔⢶⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣿⣿⣄⣀⣤⣶⣿⣿⣿⣿⣿⣿⣶⣄⡀⢀⣤⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀
⠀⠀⠀⣀⣀⣀⣠⣤⣄⣄⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣄⣀⣀⣀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⡤⠤⠤⠤⠶⠶⣲⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡙⠋⠀⠀⠀⠉⠉⠉⠒⠶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠂⠚⠛⠛⠛⠙⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠋⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣶⣶⣖⡠⠤⠤⠤⢤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣴⣶⡶⣶⣶⣿⣿⣿⣿⡿⠟⠉⠙⠛⠛⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⣶⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀
⠀⠤⠤⠖⠒⠛⠛⠛⢛⡉⠉⠠⣤⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⣰⣿⣿⣿⣿⣏⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠤⣴⣶⣾⣿⣿⣿⣿⡿⠋⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀
⠀⠖⠶⣶⠏⢶⠄⠱⠶⢶⣖⡹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠟⠁⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⢱⣶⡆⢱⣶⣶⡏⠉⣷⣦⢰⣶⡆⣶⣶⢱⣶⡏⢹⣶⣶⡏⢻⣶⣶⣶⣬⢱⣶⡄⣶⡾⢻⡟⣵⣶⣶⣍⢋⣵⣶⣶⣝⢫⣴⣶⣦⡙⣛⣩⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟⣵⣶⣶⠈⣵⣶⣶⣭⠛⣽⣶⣶⣬⠻⣿
⠿⡇⢸⣿⡇⢸⣿⣿⣿⠀⣿⣿⣾⣿⡇⣿⣿⢸⣿⡇⣼⣿⣿⣷⢸⣿⣿⣹⣿⠀⣿⣷⣿⢇⣿⣾⠿⢇⣿⣿⢸⣿⡇⣿⣿⠸⠿⢠⣿⡇⠛⣿⣿⢈⣿⣿⣿⢟⣛⡛⠛⡛⢱⡿⣿⣿⢸⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠉
⠀⠀⢸⣿⡇⣿⣿⣸⣿⡄⣿⣿⣿⣿⡇⣿⣿⢸⣿⡇⣿⣿⣹⣿⢸⣿⣿⢿⣷⠀⢸⣿⡟⣸⣿⣿⢃⣾⡿⠃⢸⣿⡇⣿⣿⠀⣠⣿⡟⢰⡇⣿⣿⠘⣛⣿⣷⣟⣉⡛⠛⠃⣿⣇⣿⣿⣸⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠀
⠀⢠⣼⣿⣇⣿⣿⢻⣿⡇⣿⡇⢿⣿⡇⢿⣿⣼⣿⢧⣿⡿⢻⣿⣿⣿⣿⢸⣿⠀⢸⣿⡇⠹⡿⢡⣿⣿⣥⣤⠸⣿⣧⣿⡟⢰⣿⣯⣤⡄⠁⣿⣿⢰⣶⡏⢿⣿⡿⠿⠟⠘⠛⠛⣿⣿⠋⢿⣧⣿⡿⠘⣿⣧⣿⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡆⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⡤⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⢻⣿⣦⣄⣀⣠⣴⣿⣿⣿⣿⣿⣿⣦⣄⠀⢀⣠⣿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀
⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠒⠒⠒⠂⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠠⠄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⡤⠤⠤⠴⠶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠛⠋⠀⠀⠀⠉⠉⠉⠒⠶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠒⠒⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠻⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⣤⡤⠤⠤⢤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣦⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⣶⡶⢶⣶⣿⣿⣿⣿⣿⠟⠉⠙⠛⠛⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⣶⣦⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀
⠀⠤⠤⠖⠲⠛⠛⠛⣿⠋⠉⠡⣦⣶⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⣰⣿⣿⣿⣿⣋⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠤⣤⣶⣾⣿⣿⣿⣿⡿⠋⡈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀
⠀⠖⠶⣶⠏⢶⠄⠱⠶⢶⣖⡙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠿⠇⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⢱⣶⡆⢰⣶⣶⡎⠉⣷⣦⢰⣶⡎⣷⣾⢱⣶⡏⢹⣶⣶⡏⢻⣶⣶⣶⣭⢱⣶⡄⣶⡾⢻⡟⣵⣶⣶⣍⢋⣵⣶⣶⣝⢫⣴⣶⣦⡙⣛⣩⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟⣵⣶⣶⠈⣵⣶⣶⣭⠛⣽⣶⣶⣌⠻⣿
⠿⡇⢸⣿⡇⢸⣿⣿⣿⠀⣿⣿⣾⣿⡇⣿⣿⢸⣿⡇⣼⣿⣿⣷⢸⣿⣿⣽⣿⠀⣿⣷⣿⢇⣿⣾⠿⢇⣿⣿⢸⣿⡇⣿⣿⠸⠿⢠⣿⡇⠛⣿⣿⢈⣿⣿⣿⢟⣛⡛⠛⡛⢱⡿⣿⣿⢸⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠉
⠀⠀⢸⣿⡇⣿⣿⣸⣿⡄⣿⣿⣿⣿⡇⣿⣿⢸⣿⡇⣿⣿⣹⣿⢸⣿⣿⢿⣷⠀⢸⣿⡟⣸⣿⣿⢃⣾⡿⠃⢸⣿⡇⣿⣿⠀⣠⣿⡟⢰⡆⣿⣿⠘⣛⣿⣷⣟⣉⡛⠛⢃⣿⣇⣿⣿⣸⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠀
⠀⢠⣼⣿⣇⣿⣿⢻⣿⡇⣿⡇⢿⣿⡇⢿⣿⣼⣿⢧⣿⡿⢻⣿⣿⣿⣿⢸⣿⠀⢸⣿⡇⠹⡟⢡⣿⣿⣥⣤⠸⣿⣧⣿⡟⢰⣿⣯⣤⡄⠀⣿⣿⢰⣶⡎⢿⣿⡿⠿⠟⠘⠛⠛⣿⣿⠋⢿⣧⣿⡿⠘⣿⣧⣿⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡆⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣲⡄⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⢿⣿⣤⡀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⣠⣿⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⡀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠐⠒⠒⠒⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠠⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⠤⠤⠤⠤⠶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡻⠏⠀⠀⠀⠉⠉⠙⠒⠶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠒⠚⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠻⠟⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣷⣤⣴⡤⠤⠤⢤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣤⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣴⣦⣶⢶⣶⣿⣿⣿⣿⡿⠟⠉⠹⠟⠛⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣶⣦⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀
⠀⠤⠤⠶⠞⠟⠛⠛⢿⠍⠉⢻⣧⣶⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣸⣿⣿⣿⡿⠋⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠤⣴⣶⣾⣿⣿⣿⣿⡿⠋⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀
⠀⠖⠶⣶⠏⢶⠄⠱⠶⢶⣖⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⢿⡅⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⢱⣶⡆⢱⣶⣶⡎⠋⣷⣦⢰⣶⡎⣷⣾⢱⣶⡏⢹⣶⣶⡏⢻⣶⣶⣶⣭⢱⣶⡄⣶⡾⢻⡟⣵⣶⣶⣍⢋⣵⣶⣶⣝⢫⣴⣶⣦⡙⣛⣩⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟⣵⣶⣶⠈⣴⣶⣶⣬⠛⣵⣶⣶⣌⠻⣿
⠿⡇⢸⣿⡇⢸⣿⣿⣿⠀⣿⣿⣾⣿⡇⣿⣿⢸⣿⡇⣼⣿⣿⣷⢸⣿⣿⣽⣿⠀⣿⣷⣿⢇⣿⣾⠿⢇⣿⣿⢸⣿⡇⣿⣿⠸⠿⢠⣿⡇⠛⣿⣿⢈⣿⣿⣿⢟⣛⡛⠛⡛⢱⡿⣿⣿⢰⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠉
⠀⠀⢸⣿⡇⣿⣿⣸⣿⡄⣿⣿⣿⣿⡇⣿⣿⢸⣿⡇⣿⣿⣹⣿⢸⣿⣿⢿⣷⠀⢸⣿⡟⣸⣿⣿⢃⣾⡿⠃⢸⣿⡇⣿⣿⠀⣠⣿⡟⢠⡆⣿⣿⠘⣛⣿⣷⣟⣉⡛⠛⢃⣿⣇⣿⣿⣸⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠀
⠀⢠⣼⣿⣇⣿⣿⢻⣿⡇⣿⡇⢿⣿⡇⢿⣿⣼⣿⢧⣿⡿⢻⣿⣿⣿⣿⢸⣿⠀⢸⣿⡇⠙⠿⢡⣿⣿⣥⣤⠸⣿⣧⣿⡟⢰⣿⣯⣤⡄⠀⣿⣿⢰⣶⡎⢿⣿⡿⠿⠛⠘⠛⠛⣿⣿⠋⢿⣧⣿⡿⠘⣿⣧⣿⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡄⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣯⢽⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⡄⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣷⣶⣷⡀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣷⣤⡀⠀⠀⣠⣿⣶⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢹⣿⣿⣦⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀
⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⠀⠀⠀⠀⠒⠒⠒⠂⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠤⠤⠤⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠠⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⡤⠤⠤⠤⠴⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠟⠃⠀⠀⠉⠉⠙⠒⠶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠒⠒⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠋⠛⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣶⣤⣤⡤⠤⠤⢤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣦⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣦⣴⣶⣶⣾⣿⣿⣿⡿⠟⠉⠙⠿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣷⣶⣦⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀
⠀⠠⠤⠴⠶⠟⠛⠛⣻⠉⠉⢉⣥⣶⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⣸⣿⣿⣿⡿⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠠⣤⣴⣶⣿⣿⣿⣿⡿⠉⠑⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀
⠀⠖⠲⣶⠏⢶⠄⠱⠶⢶⣆⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠟⠋⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⢱⣶⡎⢰⣶⣶⡎⠉⣷⣦⢰⣶⡎⣷⣾⢱⣶⡏⢹⣶⣶⡏⢻⣶⣶⣶⣭⢱⣶⡄⣶⡾⢻⡟⣵⣶⣶⣍⢋⣵⣶⣶⣝⢫⣴⣶⣦⡙⣛⣩⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟⣵⣶⣶⠈⣴⣶⣶⣭⠛⣵⣶⣶⣌⠻⣿
⠿⡇⢸⣿⡇⣸⣿⣿⣿⠀⣿⣿⣾⣿⡇⣿⣿⢸⣿⡇⣼⣿⣿⣷⢸⣿⣿⣽⣿⠀⣿⣷⣿⢇⣿⣾⠿⢇⣿⣿⢸⣿⡇⣿⣿⠸⠿⢠⣿⡇⠛⣿⣿⢈⣿⣿⣿⢟⣛⡛⠛⡛⢱⡿⣿⣿⢰⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠉
⠀⠀⢸⣿⡇⣿⣿⣸⣿⡄⣿⣿⣿⣿⡇⣿⣿⢸⣿⡇⣿⣿⣹⣿⢸⣿⣿⢿⣷⠀⢸⣿⡟⣸⣿⣿⢃⣾⡿⠃⢸⣿⡇⣿⣿⠀⣠⣿⡟⢠⡆⣿⣿⠘⢛⣿⣷⣟⣉⡛⠛⠃⣿⣇⣿⣿⣸⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠀
⠀⢠⣼⣿⣇⣿⣿⢻⣿⡇⣿⡇⢿⣿⡇⢿⣿⣼⣿⢧⣿⡿⢻⣿⣿⣿⣿⢸⣿⠀⢸⣿⡇⠙⠟⢡⣿⣿⣥⣤⠸⣿⣧⣿⡟⢰⣿⣯⣤⡄⠀⣿⣿⢰⣶⡊⢿⣿⠿⠿⠟⠘⠛⠛⣿⣿⠋⢿⣧⣿⡿⠘⣿⣧⣿⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡆⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢷⢼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠛⢿⣿⣦⡀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⣠⣿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠒⠒⠒⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⣤⠤⠤⠤⠤⠤⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠤⠤⠤⠤⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡙⠋⠁⠀⠀⠉⠉⠓⠒⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠒⠒⠒⠚⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠋⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣶⣤⣤⡤⠤⠤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣤⡀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⠟⠋⠙⠛⠻⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣶⣶⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀
⡀⠠⠤⠴⠖⠿⠛⠛⡛⠉⠉⢉⣥⣶⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠅
⠀⠀⠀⠀⠀⢠⣤⣴⣶⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀
⠀⠖⠲⣶⠏⢶⠄⠱⠶⢶⡆⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⠿⠁⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⢱⣶⡆⢱⣶⣶⡎⠋⣷⣦⢰⣶⡎⣷⣾⢱⣶⡏⢹⣶⣶⡏⢻⣶⣶⣶⣭⢱⣶⡌⣶⡾⢻⡟⣵⣶⣶⣍⢋⣵⣶⣶⣝⢫⣴⣶⣦⡙⣛⣩⣾⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟⣵⣶⣶⠈⣴⣶⣶⣬⠛⣵⣶⣶⣌⠻⣿
⠿⡇⢸⣿⡇⢸⣿⣿⣿⠀⣿⣿⣾⣿⡇⣿⣿⢸⣿⡇⣼⣿⣿⣷⢸⣿⣿⣽⣿⠀⣿⣷⣿⢇⣿⣾⠿⢇⣿⣿⢸⣿⡇⣿⣿⠸⠿⢠⣿⡇⠛⣿⣿⢈⣿⣿⣿⢟⣛⡛⠛⡛⢱⡿⣿⣿⢸⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠉
⠀⠀⢸⣿⡇⣿⣿⣸⣿⡄⣿⣿⣿⣿⡇⣿⣿⢸⣿⡇⣿⣿⣹⣿⢸⣿⣿⢿⣷⠀⢸⣿⡟⣸⣿⣿⢃⣾⡿⠃⢸⣿⡇⣿⣿⠀⣠⣿⡟⢠⡆⣿⣿⠘⢛⣿⣷⣟⣉⣛⠛⢃⣿⣇⣿⣿⣸⣿⡇⣿⣿⢸⣿⡇⣿⣿⠀⠀
⠀⢠⣼⣿⣇⣿⣿⢻⣿⡇⣿⡇⢿⣿⡇⢿⣿⣼⣿⢧⣿⡿⢻⣿⣿⣿⣿⢸⣿⠀⢸⣿⡇⠹⠟⢡⣿⣿⣥⣤⠸⣿⣧⣿⡟⢰⣿⣯⣤⡄⠀⣿⣿⢰⣶⡎⢿⣿⠿⠿⠟⠘⠛⠛⣿⣿⠋⢿⣧⣿⡿⠘⣿⣧⣿⡿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡆⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠻⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⡆⠀⠀⠀⠀⠀⢰⣶⣶⣿⣿⣇⠀⠀⠀⠀⠀⣤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢻⣿⣦⡀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⣠⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠒⠒⠒⠂⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣥⣾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⣀⣀⣀⣀⣤⣠⣄⣀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⡤⠤⠤⠤⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣯⡛⠋⠀⠀⠈⠉⠉⠓⠲⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠒⠒⠒⠚⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠟⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣶⣤⣤⡤⠤⢤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣤⡀⠀⠀⠀⠉⠉⠉⠉⠉⠑⠐⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣤⣤⣶⣶⣿⣿⣿⣿⣿⠟⠻⠿⠿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣶⣶⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀
⠀⡀⠤⠴⠶⠿⠛⢻⠛⠋⠉⢹⣥⣶⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣼⣿⣿⣿⡿⢋⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣤⣴⣶⣾⣿⣿⣿⡿⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀
⠀⠖⠲⣶⠏⢶⠄⠱⠶⢶⡆⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠟⠁⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛
⠀⠈⠁⠈⠉⠛⠛⠛⠿⠿⠟⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠈⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠋⠉⠉⠈⠉⠛⠛⠿⠿⢿⢿⣿⠿⠿⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠯⠧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢷⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⣻⡀⠀⠀⠀⠀⢀⣴⣤⣿⣿⣷⠀⠀⠀⠀⠀⢠⠴⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⣷⣤⣀⣤⣴⣾⣿⣿⣿⣿⣿⣷⣤⣀⣀⣤⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠒⠂⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⣩⣥⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡤⠤⠤⠤⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠁⠀⠀⠉⠉⠓⠒⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠒⠒⠒⠚⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠉⠉⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣶⣤⣤⡤⠤⢄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣤⡀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣶⣾⣿⣿⣿⡿⠛⠋⠛⠟⠻⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣶⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠠⠤⠶⠾⠟⠛⠛⠉⠉⠉⣁⣶⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣟⠁⠀
⠀⠖⠲⣶⠏⢶⠄⠱⠶⢶⠖⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⡿⠃⠄⠀⠀
⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠻⠟⠛⠻⠿⠿⠿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠛⡛⠻⢿⣿⠿⢿⡿⢿⣿⡿⠿⣻⠿⢿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⡿⠿⢿⡿⠿⣛⠿⢿⣿⡿⠛⡛⠛⢿⣿⠿⢿⡻⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⡈⣿⣿⢸⣿⣿⣿⣏⣿⡏⣾⣿⢻⣿⣆⢰⣿⡿⢿⣷⡌⣴⣿⠿⣿⣧⣩⣴⣾⣿⠸⣿⣿⣿⢯⣥⣾⣿⠁⣾⣿⠿⣿⣦⠉⣾⣿⢻⣿⣆⢠⣾⣿⢻⣿⡎⣿⣿⣿⣿⣿
⠿⣿⣿⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⣧⣿⣿⢸⣿⣿⣠⣿⣿⣾⣛⠛⣼⣿⡏⢸⣿⡇⢸⣿⡇⣛⡛⣰⣿⡿⣉⢹⣿⣿⢠⣬⡝⣿⣮⡉⣿⣿⠀⣿⣿⠀⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡗⠹⣿⠿⠿⠋
⠀⠈⠁⠉⠁⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡿⣿⣿⣿⢸⣿⣿⠛⣳⣿⣿⢏⣼⣿⠟⠀⢸⣿⡇⢸⣿⡇⠛⣴⣿⡿⣱⣿⢸⣿⣿⠘⠛⢣⣿⣿⡇⣿⣿⠀⣿⣿⠀⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣿⠘⣿⣿⣼⣿⡿⢸⣿⡇⢻⣿⣿⢸⣿⣿⣤⡌⠻⢁⣾⣿⣯⣤⡄⢸⣿⣇⣼⣿⠇⣼⣿⣯⣤⣭⠉⢸⣿⣿⢠⣤⡄⠙⠛⠀⣿⣿⠀⢿⣿⣤⣿⡿⠈⣿⣿⣸⣿⡿⠸⣿⣿⣸⣿⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠙⠛⠉⠀⠈⠉⠁⠈⠉⠉⠈⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠁⠀⠉⠛⠛⠁⠀⣉⣉⣉⠉⠉⠀⠈⠉⠉⠈⠉⠁⠀⠀⠀⠉⠉⠁⠈⠙⠛⠋⠁⠀⠈⠙⠛⠉⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢷⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⣻⠀⠀⠀⠀⠀⢀⣴⣤⣿⣿⣷⠀⠀⠀⠀⠀⣠⠴⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⣷⣤⣀⣤⣶⣾⣿⣿⣿⣿⣿⣶⣤⣀⣀⣤⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠒⠒⠒⠂⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠉⠁⠈⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠁⠈⠀⠀⣩⣥⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠤⠤⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠁⠀⠈⠉⠉⠓⠒⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠉⠉⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⣶⣤⣤⡤⠤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣤⡀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠐⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣤⣤⣤⣶⣾⣿⣿⣿⣿⠟⠛⠛⠿⠻⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣶⣦⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀
⡀⠠⠤⠤⠶⠿⠟⠛⠛⠛⠉⢩⣥⣶⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃
⠀⠀⠀⠀⠀⠀⣤⣤⣶⣶⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡯⡋⠀
⠀⠖⠶⣶⠏⢶⠄⠱⠶⣶⡖⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⡿⠛⠀⠀
⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠛⠟⠛⠻⠿⠿⠿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠛⡛⠻⢿⣿⠿⢿⡿⠿⣿⡿⠿⣻⠿⢿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⡿⠿⢿⡿⠿⣛⠿⢿⣿⡿⠟⡛⠛⢿⣿⠿⢿⡻⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⡈⣿⣿⢸⣿⣿⣿⣏⣿⠏⣾⣿⢻⣿⣆⢰⣿⡿⢿⣷⡌⣼⣿⠿⣿⣧⣩⣴⣿⣿⠸⣿⣿⣿⢯⣥⣾⣿⠁⣾⣿⠿⣿⣦⠉⣾⣿⢻⣿⣦⢠⣾⣿⢻⣿⡎⣿⣿⣿⣿⣿
⠿⣿⣿⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⣧⣿⣿⢸⣿⣿⣠⣿⣿⣾⣛⠛⣼⣿⡏⢸⣿⡇⢸⣿⡇⢛⠛⣰⣿⡿⣉⢹⣿⣿⢠⣬⡝⣿⣮⡉⣿⣿⠀⣿⣿⠀⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡇⠹⣿⠿⠟⠋
⠀⠈⠁⠉⠁⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡿⣿⣿⣿⢸⣿⣿⠛⣳⣿⣿⢏⣼⣿⠟⠀⢸⣿⡇⢸⣿⡇⠛⣴⣿⡿⣡⣿⢸⣿⣿⠈⠛⢣⣿⣿⡇⣿⣿⠀⣿⣿⠀⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣿⠘⣿⣿⣼⣿⡿⢸⣿⡇⢻⣿⣿⢸⣿⣿⣤⡌⠛⢁⣾⣿⣯⣤⡄⢸⣿⣇⣾⣿⠇⣼⣿⣯⣤⣭⠉⢸⣿⣿⢠⣤⡆⠙⠛⠀⣿⣿⠀⢿⣿⣤⣿⡿⠈⣿⣿⣸⣿⡿⠸⣿⣿⣸⣿⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠙⠛⠉⠀⠈⠉⠁⠈⠉⠉⠈⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠁⠀⠉⠛⠛⠉⠀⢉⣉⣉⠉⠉⠀⠈⠉⠉⠈⠉⠁⠀⠀⠀⠉⠉⠁⠈⠙⠛⠋⠁⠀⠈⠙⠛⠉⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡁⠀⠀⠀⠀⠀⠸⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣇⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⡀⠀⠀⠀⠀⢠⣚⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣤⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⣴⣾⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠀⠀⠐⠒⠒⠒⠒⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈⠈⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣩⣭⣿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⠀⢀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠤⠤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡻⠛⠁⠀⠈⠉⠙⠓⠂⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠒⠒⠒⠶⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠟⠉⠉⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣤⣤⡤⢄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣤⡀⠀⠈⠉⠉⠉⠉⠉⠛⠐⠒⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣴⣾⣿⣿⣿⡿⠟⠛⠻⠿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣶⣦⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀
⢀⣀⣤⡤⠶⠶⠾⠟⠛⠛⠋⢉⣍⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣾⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣿⣿⣿⡟⠋⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁
⠀⠖⠲⣶⠏⢶⠄⠱⠶⣶⡖⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⡿⡿⠍⠀⠀
⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠛⠟⠛⠻⠿⠿⠿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠛⡛⠻⢿⣿⠿⢿⡿⠿⣿⡿⠿⣻⠿⢿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⡿⠿⣿⡿⠟⣛⠿⢿⣿⡿⠟⡛⠛⢿⣿⠿⢿⡻⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⡈⣿⣿⢸⣿⣿⣿⣏⣿⠏⣾⣿⢻⣿⣆⢱⣿⡿⣿⣷⡌⣼⣿⠿⣿⣧⣩⣴⣿⣿⠸⣿⣿⣿⢯⣥⣾⣿⠁⣾⣿⠿⣿⣦⠉⣾⣿⢻⣿⣦⢠⣾⣿⢻⣿⡎⣿⣿⣿⣿⣿
⠿⣿⣿⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⣧⣿⣿⢸⣿⣿⣠⣿⣿⣾⣛⠛⣼⣿⡏⢸⣿⡇⢸⣿⡇⢛⡛⣰⣿⡿⣉⢹⣿⣿⢠⣤⡝⣿⣮⡉⣿⣿⠀⣿⣿⠀⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡗⠹⣿⠿⠟⠋
⠀⠈⠁⠈⠁⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡿⣿⣿⣿⢸⣿⣿⠛⣳⣿⣿⢏⣼⣿⠟⠀⢸⣿⡇⢸⣿⡇⠛⣴⣿⡿⣱⣿⢸⣿⣿⠈⠛⢣⣿⣿⠇⣿⣿⠀⣿⣿⠀⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣿⠘⣿⣿⣼⣿⡿⢸⣿⡇⢻⣿⣿⢸⣿⣿⣤⡌⠛⢁⣾⣿⣯⣤⡄⢸⣿⣇⣼⣿⠇⣼⣿⣯⣤⣭⠉⢸⣿⣿⢠⣤⡆⠙⠛⠀⣿⣿⠀⢿⣿⣤⣿⡿⠈⣿⣿⣸⣿⡿⠸⣿⣿⣸⣿⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠙⠛⠉⠀⠈⠉⠁⠈⠉⠉⠈⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠁⠀⠉⠛⠛⠁⠀⠉⣉⣉⡉⠉⠀⠈⠉⠉⠈⠉⠁⠀⠀⠀⠉⠉⠁⠈⠙⠛⠋⠁⠀⠈⠙⠛⠉⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠹⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡿⣧⠀⠀⠀⠀⠀⠀⠿⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⢿⣷⡀⠀⠀⠀⢀⣀⣀⣿⣿⣿⡀⠀⠀⠀⠀⣴⣌⣧⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣤⣴⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣄⣤⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠒⠒⠒⠒⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣁⣬⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠤⠤⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠁⠀⠈⠉⠙⠒⠢⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠲⠲⠶⠒⠒⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠁⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⠟⠿⠟⠛⠉⠉⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣶⣤⡀⠀⠈⠉⠉⠉⠉⠉⠒⠒⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣴⣿⣿⣿⣿⡿⠟⠛⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣶⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⢀⣤⡤⢶⠶⠾⠿⠛⠛⠛⢩⣭⣷⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢀⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠅
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⠟⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣯⠏⠀
⠀⠖⠶⣶⠏⢶⠄⠱⠶⣶⡖⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⡿⠿⠋⠁⠀
⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠛⠛⠛⠻⠿⠿⠿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠛⡛⠻⢿⣿⠿⢿⡿⠿⣿⡿⠿⣻⠿⢿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⡿⠿⢿⡿⠿⣛⠿⢿⣿⡿⠟⡛⠛⢿⣿⠿⢿⡻⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⡈⣿⣿⢸⣿⣿⣿⣏⣿⠏⣾⣿⢻⣿⣆⢱⣿⡿⣿⣷⡌⣼⣿⠿⣿⣧⣩⣴⣿⣿⠸⣿⣿⣿⢯⣥⣾⣿⠁⣾⣿⠿⣿⣦⠉⣾⣿⢻⣿⣦⢠⣾⣿⢻⣿⡎⣿⣿⣿⣿⣿
⠿⣿⣿⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⣧⣿⣿⢸⣿⣿⣠⣿⣿⣾⣛⠛⣼⣿⡏⢸⣿⡇⢸⣿⡇⢛⠛⣰⣿⡿⣉⢹⣿⣿⢠⣤⡝⣿⣯⡉⣿⣿⠀⣿⣿⠀⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡗⠹⣿⡿⠟⠋
⠀⠈⠁⠉⠁⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡿⣿⣿⣿⢸⣿⣿⠛⣷⣿⣿⢏⣼⣿⠟⠀⢸⣿⡇⢸⣿⡇⠛⣴⣿⡿⣡⣿⢸⣿⣿⠈⠛⢣⣿⣿⠇⣿⣿⠀⣿⣿⠀⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣿⠘⣿⣿⣼⣿⡿⢸⣿⡇⢻⣿⣿⢸⣿⣿⣤⡌⠛⢁⣾⣿⣯⣤⡄⢸⣿⣇⣼⣿⠇⣼⣿⣯⣤⣭⠉⢸⣿⣿⢠⣤⡄⠙⠛⠀⣿⣿⠀⢿⣿⣤⣿⡿⠈⣿⣿⣸⣿⡿⠸⣿⣿⣸⣿⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠙⠛⠉⠀⠈⠉⠁⠈⠉⠉⠈⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠁⠀⠉⠛⠛⠉⠀⠉⣉⣉⣉⠉⠀⠈⠉⠉⠈⠉⠁⠀⠀⠀⠉⠉⠁⠈⠙⠛⠋⠁⠀⠈⠙⠛⠉⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡻⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⢿⣿⡄⠀⠀⠀⢀⣀⡀⣼⣿⣿⡄⠀⠀⠀⠀⢴⣭⣧⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣾⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠒⠒⠒⠂⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⣩⣬⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠤⠤⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠁⠀⠈⠉⠙⠒⠢⢤⣄⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠒⠖⠖⠒⠒⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⠛⠿⠻⠟⠉⠙⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣶⣤⣤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣶⣄⡀⠀⠉⠉⠉⠉⠉⠉⠐⠒⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣠⣤⣤⣴⣾⣿⣿⣿⡿⠟⠛⠿⠿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀
⣀⣀⠠⠤⠶⠶⠾⠿⠛⠛⠛⣯⣅⣾⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⠟⠋⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀
⠀⠖⠶⣶⠏⢶⠄⠱⠶⣶⡖⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⡿⠟⠀⠀
⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠛⠛⠛⠻⠿⠿⠿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠛⡛⠻⢿⣿⠿⢿⡿⠿⣿⡿⠿⣻⠿⢿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⡿⠿⣿⡿⠿⣛⠿⢿⣿⡿⠟⡛⠛⠿⣿⠿⢿⡻⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⡈⣿⣿⢸⣿⣿⣿⣏⣿⡏⣾⣿⢻⣿⣆⢱⣿⡿⣿⣷⡌⣼⣿⠿⣿⣧⣩⣴⣿⣿⠸⣿⣿⣿⢯⣥⣾⣿⠁⣾⣿⠿⣿⣦⠉⣾⣿⢻⣿⣆⢠⣾⣿⢻⣿⡎⣿⣿⣿⣿⣿
⠿⣿⣿⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⣧⣿⣿⢸⣿⣿⣠⣿⣿⣾⣛⠛⣼⣿⡏⢸⣿⡇⢸⣿⡇⢛⠛⣰⣿⡿⣉⢹⣿⣿⢠⣬⡝⣿⣯⠉⣿⣿⠀⣿⣿⠀⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡗⠸⣿⠿⠟⠋
⠀⠈⠁⠉⠁⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡿⣿⣿⣿⢸⣿⣿⠛⣷⣿⣿⢏⣼⣿⠟⠀⢸⣿⡇⢸⣿⡇⠛⣴⣿⡿⣱⣿⢸⣿⣿⠈⠛⠃⣿⣿⠇⣿⣿⠀⣿⣿⠀⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣿⠘⣿⣿⣼⣿⡿⢸⣿⡇⢻⣿⣿⢸⣿⣿⣤⡌⠛⢁⣾⣿⣯⣤⡄⢸⣿⣇⣾⣿⠇⣼⣿⣯⣤⣬⠉⢸⣿⣿⢠⣤⡄⠉⠋⠀⣿⣿⠀⢿⣿⣤⣿⡿⠈⣿⣿⣸⣿⡿⠸⣿⣿⣸⣿⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠙⠛⠉⠀⠈⠉⠁⠈⠉⠉⠈⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠁⠀⠉⠛⠛⠁⠀⠉⢉⣉⣉⡉⠀⠈⠉⠉⠈⠉⠁⠀⠀⠀⠉⠉⠁⠈⠙⠛⠋⠁⠀⠈⠙⠛⠉⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣔⣄⠀⠀⠀⠀⠀⠀⢰⣟⡻⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⣽⡄⠀⠀⠀⠀⠀⠚⠻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠟⠛⣿⣷⡀⠀⠀⢀⡀⠀⠀⣿⣿⡇⢀⠀⠀⠀⣴⣛⣿⣴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣦⣤⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠤⠄⠀⠐⠒⠒⠒⠒⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⢀⣠⢾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡤⠤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⠁⠀⠈⠉⠙⠒⠢⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠴⠶⠶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠏⠉⠛⠛⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣦⣤⣤⢤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣶⣤⡀⠀⠈⠉⠉⠉⠙⠓⠒⠒⠒⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣾⣿⣿⣿⡿⠟⠛⠻⠟⠻⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣀⣠⣤⣴⠶⠶⠿⠟⠛⠋⢋⣉⣽⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢐⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣿⣿⣿⠟⠉⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀
⠀⠖⠶⣶⠏⢶⠄⠱⠶⣶⡖⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⠿⡉⠀⠀
⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠛⠛⠛⠻⠿⠿⠿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠛⡛⠻⢿⣿⠿⢟⡿⠿⣿⡿⠿⣛⠿⢿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⡿⠿⣿⡿⠿⣛⠿⢿⣿⡿⠟⡛⠛⠿⣿⠿⢿⡻⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⡈⣿⣿⢸⣿⣿⣿⣏⣿⠏⣾⣿⢻⣿⣆⢱⣿⡿⢿⣷⡌⣴⣿⠿⣿⣧⣩⣴⣾⣿⠸⣿⣿⣿⢯⣥⣾⣿⠃⣾⣿⠿⣿⣦⠉⣾⣿⢻⣿⣦⢠⣾⣿⢻⣿⡎⣿⣿⣿⣿⣿
⠿⣿⣿⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⣧⣿⣿⢸⣿⣿⣠⣿⣿⣾⣛⠛⣼⣿⡏⢸⣿⡇⢸⣿⡇⢛⠛⣰⣿⡿⣉⢹⣿⣿⢠⣤⡝⣿⣯⠉⣿⣿⠀⣿⣿⠀⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡗⠸⣿⠿⠟⠋
⠀⠈⠁⠈⠁⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡿⣿⣿⣿⢸⣿⣿⠛⣷⣿⣿⢏⣼⣿⠟⠀⢸⣿⡇⢸⣿⡇⠛⣴⣿⡿⣱⣿⢸⣿⣿⠘⠛⠃⣿⣿⠇⣿⣿⠀⣿⣿⠀⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣿⠘⣿⣿⣼⣿⡿⢸⣿⡇⢻⣿⣿⢸⣿⣿⣤⡌⠙⢁⣾⣿⣯⣤⡄⢸⣿⣇⣾⣿⠇⣼⣿⣯⣤⣬⠁⢸⣿⣿⢠⣤⡄⠉⠋⠀⣿⣿⠀⢿⣿⣤⣿⡿⠈⣿⣿⣸⣿⡿⠸⣿⣿⣸⣿⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠙⠛⠉⠀⠈⠉⠁⠈⠉⠉⠈⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠁⠀⠉⠛⠛⠁⠀⠉⢉⣉⣉⡉⠀⠈⠉⠉⠈⠉⠁⠀⠀⠀⠉⠉⠁⠈⠙⠛⠋⠁⠀⠈⠙⠛⠉⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⡤⠀⠀⠀⠀⠀⠀⣰⣟⠒⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠚⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠟⠻⣿⣦⡀⠀⠀⣀⣀⣂⣸⣿⣿⡇⣀⡄⠀⠀⢰⣎⣿⣶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤⠀⠀⠀⠒⠒⠒⠒⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢀⣤⣿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠈⠉⠙⠒⠢⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠴⠶⠶⠶⠶⠾⠟⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡿⠛⠻⠛⠋⠈⠙⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣦⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⡿⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣤⡀⠀⠉⠉⠉⠉⠙⠓⠒⠒⠒⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣴⣾⣿⣿⣿⡿⠟⠛⠻⠟⠻⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣀⣀⣤⣴⠶⠶⠿⠟⠛⠋⠋⢉⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣾⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⣿⣿⠟⠋⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀
⠀⠖⠶⣶⠏⢶⠄⠱⠶⢶⠆⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⡿⠇⠀⠀
⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠛⠻⠿⠿⠿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠛⡛⠻⢿⣿⠿⢟⡛⠿⣿⡿⠿⣛⠿⢿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⡿⠿⢿⡿⠟⣛⠿⢿⣿⡿⠟⡛⠛⠻⡿⠿⢿⡻⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⡈⣿⣿⢸⣿⣿⣿⣏⣿⠏⣾⣿⢻⣿⣆⢱⣿⡿⣿⣷⡌⣼⣿⠿⣿⣧⣩⣴⣾⣿⠘⣿⣿⣿⢯⣥⣾⣿⠃⣾⣿⠿⣿⣦⠉⣾⣿⢻⣿⣆⢠⣾⣿⢻⣿⡎⢿⣿⣿⣿⣿
⠻⣿⣿⣿⣷⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⣧⣿⣿⢸⣿⣿⣠⣿⣿⣾⣛⠛⣼⣿⡏⢸⣿⡇⢸⣿⡇⢛⠛⣰⣿⡿⢉⢹⣿⣿⢠⣤⡝⣿⣯⠉⣿⣿⠀⣿⣿⠀⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣷⠸⠿⠟⠛⠉
⠀⠈⠀⠀⠀⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡿⣿⣿⣿⢸⣿⣿⠛⣷⣿⣿⢇⣼⣿⠟⠀⢸⣿⡇⢸⣿⡇⠛⣴⣿⡿⣡⣿⢸⣿⣿⠈⠛⠃⣿⣿⠆⣿⣿⠀⣿⣿⠀⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡷⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣿⠘⣿⣿⣼⣿⡿⢸⣿⡇⢻⣿⣿⢸⣿⣿⣤⡌⠙⢁⣾⣿⣯⣤⡄⢸⣿⣧⣾⣿⠇⣼⣿⣿⣤⣬⠁⢸⣿⣿⢠⣤⡄⠈⠉⠀⣿⣿⠀⢿⣿⣤⣿⡿⠘⣿⣿⣸⣿⡿⠸⣿⣿⣸⣿⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠙⠛⠉⠀⠈⠉⠁⠈⠉⠉⠈⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠁⣀⠉⠛⠛⠉⠀⠉⣉⣉⣉⠉⠀⠈⠉⠉⠈⠉⠁⠀⠀⠀⠉⠉⠁⠈⠙⠛⠋⠁⠀⠈⠙⠛⠉⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⣰⣛⣳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⣿⣿⡇⠀⠀⠀⣽⣾⣿⣿⠀⢠⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⡾⢛⠀⠀⠀⣤⣾⣿⣿⣿⣷⣾⣿⣄⡀⢠⣶⣻⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠁⠘⣿⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡀⠀⣀⣀⣀⡀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠄⣀⣠⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠋⠁⠈⠉⠛⠒⠠⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⠀⠒⠒⠒⠒⠒⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⡿⠛⠛⠟⠋⠈⠙⠛⠿⠿⠿⠿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣦⣀⣀⡀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣶⣄⡀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣾⣿⣿⣿⡿⠟⠿⠿⠿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠴⠖⠒⠛⠋⠉⢉⣴⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⣀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀
⠀⠖⠶⣶⠏⢶⠄⠰⢶⣶⠖⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡿⡿⠿⠁⠀⠀⠀
⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠛⠻⠿⠿⠿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠛⡛⠻⢿⣿⠿⢟⡛⠿⣿⡿⠟⣛⠿⢿⣿⣿⠿⠟⢿⣿⣿⣿⣿⣿⡿⠿⣿⡿⠟⣛⠿⢿⣿⡿⠟⡛⠛⠻⠿⠿⢿⡛⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⡈⣿⣿⢸⣿⣿⣿⣏⣿⠏⣾⣿⢻⣿⣆⢱⣿⡿⣿⣷⡌⣴⣿⠿⣿⣧⣩⣴⣿⣿⠈⣿⣿⣿⢯⣥⣾⣿⠃⣾⣿⠿⣿⣦⠉⣾⣿⢻⣿⣦⢠⣾⣿⢻⣿⡎⢿⣿⣿⣿⣿
⠛⣿⡿⣿⡷⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⣧⣿⣿⢸⣿⣿⣠⣿⣿⣾⣛⠛⣼⣿⡏⢸⣿⡇⢸⣿⡇⢛⠛⣰⣿⡿⢉⢹⣿⣿⢠⣤⡝⣿⣮⠉⣿⣿⠀⣿⣿⠀⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣷⠘⠟⠛⠋⠁
⠀⠈⠀⠀⠀⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡿⣿⣿⣿⢸⣿⣿⠛⣷⣿⣿⢇⣼⣿⠟⠀⢸⣿⡇⢸⣿⡇⠊⣴⣿⡿⣡⣿⢸⣿⣿⠈⠛⠃⣿⣿⠆⣿⣿⠀⣿⣿⠀⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣿⠘⣿⣿⣼⣿⡿⢸⣿⡇⢻⣿⣿⢸⣿⣿⣤⡌⠙⢁⣾⣿⣯⣤⡄⢸⣿⣧⣾⣿⠇⣼⣿⣿⣤⣬⠁⢸⣿⣿⢠⣤⡄⠈⠁⠀⣿⣿⠀⣿⣿⣤⣿⡿⠈⣿⣿⣸⣿⡿⠸⣿⣿⣸⣿⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⠙⠛⠉⠀⠈⠉⢁⡈⠉⡉⠈⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠁⣀⠉⠛⠛⠉⠀⠉⣉⣉⣉⠉⠀⠈⠉⠉⠈⠉⠁⠀⠀⠀⠉⠉⠁⠈⠙⠛⠋⠁⠀⠈⠙⠛⠉⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢟⠻⣿⡟⡯⠈⠁⠀⣒⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣰⣛⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡐⠠⠀⠀⠀⢺⣿⣿⣿⣿⣿⡿⣿⣿⡧⠀⠀⠀⣽⣾⣿⣿⢠⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⣿⣿⣿⣿⣿⣷⡿⣿⢡⡀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣷⣄⢠⣶⣻⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠻⠝⠚⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠏⠁⠈⠉⠓⠒⠠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠒⠒⠒⠒⠒⠒⠚⠛⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠛⠿⠿⠋⠉⠙⠛⠿⠿⠿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⠿⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣾⣿⣿⣿⣿⠿⠻⠿⠟⠻⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠀⠒⠒⠚⠉⠉⢁⣴⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⣿⣿⠟⠋⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡛⠀⠀⠀
⠀⠖⠶⣶⠏⢶⠄⠰⢶⣶⡖⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⡿⠿⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠛⠻⠿⠿⠿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠛⡛⠻⢿⣿⠿⢛⡛⠿⣿⡿⠟⣛⠿⢿⣿⣿⠿⠛⢿⣿⣿⣿⣿⣿⠿⠿⢿⡿⠟⣛⠻⢿⣿⡿⠟⡛⠉⠀⠿⠻⢿⡛⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⡈⣿⣿⢸⣿⣿⣿⣏⣿⠏⣾⣿⢻⣿⣆⢱⣿⡿⣿⣷⡌⣼⣿⠿⣿⣧⣩⣴⣿⣿⠀⣿⣿⣿⢫⣥⣾⣿⠃⣾⣿⠿⣿⣦⠉⣾⣿⢻⣿⣦⢠⣾⣿⢻⣿⡎⢻⣿⣿⣿⣿
⠛⣿⡿⣿⡷⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⣧⣿⣿⢸⣿⣿⣠⣿⣿⣾⣛⠛⣼⣿⡏⢸⣿⡇⢸⣿⡇⢛⠛⣰⣿⡿⢉⢹⣿⣿⢠⣤⡌⣿⣮⠉⣿⣿⠀⣿⣿⠀⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣷⠘⠛⠋⠉⠀
⠀⠈⠀⠀⠀⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡿⣿⣿⣿⢸⣿⣿⠛⣷⣿⣿⢇⣼⣿⠟⠀⢸⣿⡇⢸⣿⡇⠊⣴⣿⡿⣡⣿⢸⣿⣿⠘⠛⠃⣾⣿⠀⣿⣿⠀⣿⣿⠀⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣿⠘⣿⣿⣼⣿⡿⢸⣿⡇⢻⣿⣿⢸⣿⣿⣤⡌⠙⢁⣾⣿⣯⣤⡄⢸⣿⣧⣾⣿⠇⣼⣿⣿⣤⣤⠀⢸⣿⣿⢠⣤⡆⠀⠀⠀⣿⣿⠀⣿⣿⣤⣿⡿⠘⣿⣿⣸⣿⡿⠸⣿⣿⣸⣿⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⠙⠛⠋⣀⣈⠉⣁⣈⣉⣉⣈⣉⠉⠉⢁⣤⣈⣉⣉⣉⣉⣁⣀⡉⠛⠛⠉⢀⣩⣉⣉⣉⠉⠀⠈⠉⠉⠈⠉⠁⠀⠀⠀⠉⠉⠁⠈⠙⠛⠋⠁⠀⠈⠙⠛⠉⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠝⠊⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⣾⢻⡻⣿⣿⠄⣶⡤⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠿⠿⣟⢿⡽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⣀⣽⣿⣿⣿⣿⣾⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠙⠋⡈⠫⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣴⢖⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠉⠙⠓⠒⠠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠐⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉⠛⠿⠿⠿⠿⠿⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⢿⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⠿⠿⠿⠟⢿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠿⠋⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀
⠀⠖⠶⣶⠏⢶⠄⠰⢶⣶⡖⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡇⠁⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠛⠻⠿⠿⠿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠛⡛⠻⢿⣿⠿⢛⡛⠿⣿⡿⠟⣛⠿⢿⣿⣿⠿⠛⢿⣿⣿⣿⣿⣿⠿⠿⢿⡿⠟⣛⠻⢿⣿⡿⠟⡋⠁⠀⠘⠛⢿⡛⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⡈⣿⣿⢸⣿⣿⣿⣏⣿⠏⣾⣿⢻⣿⣆⢱⣿⡿⣿⣷⡌⣼⣿⠿⣿⣧⣩⣴⣿⣿⠀⣿⣿⣿⢫⣥⣾⣿⠃⣾⣿⠿⣿⣦⠉⣾⣿⢻⣿⣦⢠⣾⣿⢻⣿⡎⢻⣿⣿⣿⡿
⠛⣿⡿⣿⡧⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⣧⣿⣿⢸⣿⣿⣠⣿⣿⣾⡛⠛⣼⣿⡏⢸⣿⡇⣾⣿⡇⢛⠛⣰⣿⡿⢉⢹⣿⣿⢠⣤⡌⢿⣮⠉⣿⣿⠀⣿⣿⠀⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣷⠘⠛⠉⠁⠀
⠀⠈⠀⠀⠀⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡿⣿⣿⣿⢸⣿⣿⠛⣷⣿⣿⢇⣼⣿⠟⠀⢸⣿⡇⣿⣿⡇⠈⣴⣿⡿⣡⣿⢸⣿⣿⠘⠛⠃⢾⣿⠀⣿⣿⠀⣿⣿⠀⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⣼⣿⣿⠘⣿⣿⣼⣿⡿⢸⣿⡇⢻⣿⣿⢸⣿⣿⣤⡌⠹⢁⣾⣿⣯⣤⡄⢸⣿⣧⣿⣿⠇⣼⣿⣿⣤⣤⠀⢸⣿⣿⢠⣤⡆⠀⠀⠀⣿⣿⡀⣿⣿⣤⣿⡿⠘⣿⣿⣸⣿⡿⠸⣿⣿⣸⣿⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⠁⢀⠈⠙⠛⠋⣀⣈⣉⣁⣈⣉⣉⣈⣉⡉⣉⣁⣴⣈⣉⣉⣉⣉⣁⣄⣉⠛⠛⢉⣀⣍⣉⣉⢉⠉⢠⠈⠉⠉⠈⠉⠁⠀⠀⠀⠉⠉⠁⠈⠙⠛⠋⠁⠀⠈⠙⠛⠉⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠙⠻⠻⠻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠎⣿⢿⣿⣿⣿⣷⣿⣿⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⣠⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢘⢏⣟⡛⡻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡍⠈⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠻⠿⢿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠁⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠋⠀⠉⠙⠓⠒⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠋⠙⠻⠿⠿⠿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⢿⣿⣿⡿⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡿⠿⡿⠿⢿⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⠓⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⠿⢋⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠽⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠏⢶⠄⠰⢶⣶⡖⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡇⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠛⠛⠻⠿⠿⠿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠛⡛⠻⢿⣿⠟⢛⡛⠿⣿⡿⠟⣛⠻⢿⣿⡿⠿⠛⠳⣤⣼⣿⣿⣿⠿⠿⢿⡿⠟⣛⠻⢿⣿⡿⠟⡁⠀⠀⠀⠘⢛⡛⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⡈⣿⣿⢸⣿⣿⣿⣏⣿⠏⣾⣿⢻⣿⣆⢱⣿⡿⣿⣷⡌⣼⣿⠿⣿⣧⣩⣴⣿⣿⠀⢿⣿⣿⢫⣥⣾⣿⠇⣾⣿⠿⣿⣧⠉⣾⣿⢿⣿⣦⢠⣾⣿⢻⣿⡎⢻⣿⣿⣿⠿
⠙⣿⡿⣿⡧⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⣧⣿⣿⢸⣿⣿⣠⡿⣿⣾⡛⠛⣼⣿⡏⢸⣿⡇⣿⣿⡇⢛⠛⣰⣿⡿⢉⢹⣿⣿⢠⣤⡌⢿⣮⠉⣿⣿⠀⣿⣿⠀⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣷⠘⠛⠉⠀⠀
⠀⠈⠀⠀⠀⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡿⣿⣿⣿⢸⣿⣿⠛⣳⣿⣿⢇⣼⣿⠟⠀⢸⣿⡇⣿⣿⡇⠈⣴⣿⡿⢡⣾⢸⣿⣿⠘⠛⠃⢸⣿⠀⣿⣿⠀⣿⣿⠀⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣼⣿⣿⠘⣿⣿⣼⣿⡿⢸⣿⡇⢻⣿⣿⢸⣿⣿⣤⡌⠹⢁⣾⣿⣯⣤⡄⢸⣿⣧⣿⣿⠇⣼⣿⣿⣤⣤⠀⢸⣿⣿⢠⣶⡆⠀⠀⠀⣿⣿⡀⣿⣿⣤⣿⡿⠘⣿⣿⣸⣿⡿⠸⣿⣿⣸⣿⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⠁⣀⣈⠙⠛⢋⣠⣌⣉⣁⣈⣉⣉⣈⣉⣉⣉⣁⣴⣌⣉⣉⣉⣉⣁⣤⣉⠛⠛⢉⣠⣍⣉⣉⣉⠉⣠⡈⠉⠉⠈⠉⠁⠀⠀⠀⠉⠉⠁⠈⠙⠛⠋⠁⠀⠈⠙⠛⠉⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢽⣿⣿⣿⣿⣿⣿⣿⣗⠄⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢫⢛⠻⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠋⢉⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡻⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠈⠩⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⠹⠛⠿⢻⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣮⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠈⠉⠚⠒⠲⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠋⠛⠿⢿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⡿⢿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠟⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣷⣶⣶⣦⣤⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⢛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⠿⡛⠴⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣯⠁⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠏⢶⠄⠰⣶⣶⣖⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡄⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⠟⠛⠻⠿⠿⠿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠛⡛⠻⢿⣿⠟⢛⡛⠿⣿⡿⠟⣛⠻⢿⣿⡿⠟⠃⠀⢀⣠⣾⣿⣿⠿⠻⢿⡿⠟⣛⠻⢿⣿⡿⠟⡀⠀⠀⠀⠀⢛⡛⠻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⡘⣿⣿⢸⣿⣿⣿⣏⣿⠏⣾⣿⢻⣿⣆⢰⣿⡿⣿⣷⡄⣼⣿⠿⣿⣧⣩⣴⣿⣿⠀⠻⣿⣿⢫⣥⣾⣿⡇⣾⣿⢿⣿⣧⠉⣾⣿⢿⣿⣦⢠⣾⣿⢿⣿⡎⢹⣿⣿⣿⠟
⠙⣿⡿⣿⡧⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⣧⣿⣿⢸⣿⣿⣠⡿⣿⣾⡛⠛⣼⣿⡏⢸⣿⡇⣾⣿⡇⢛⠛⣰⣿⡿⠉⢹⣿⣿⢠⣤⡌⢿⣮⠉⣿⣿⠁⣿⣿⢰⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣷⠈⠋⠁⠀⠀
⠀⠈⠀⠀⠀⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡿⣿⣿⣿⢸⣿⣿⠛⣷⣿⣿⢇⣼⣿⠟⠀⢸⣿⡇⣿⣿⡇⠀⣴⣿⡿⢡⣾⢸⣿⣿⠘⠛⠃⢸⣿⠀⣿⣿⠀⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣼⣿⣿⠘⣿⣿⣼⣿⡿⢸⣿⡇⢻⣿⣿⢸⣿⣿⣤⡌⢻⢁⣾⣿⣯⣤⡄⢸⣿⣧⣿⣿⠇⣼⣿⣿⣤⣤⠀⢸⣿⣿⢠⣶⡆⠀⠀⠀⣿⣿⡆⣿⣿⣼⣿⣿⠘⣿⣿⣸⣿⡿⠸⣿⣿⣸⣿⡏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⢁⣤⣈⡙⠛⣋⣠⣌⣉⣁⣌⣉⣉⣈⣉⣉⣉⣡⣾⣌⣉⣉⣉⣉⣁⣤⣉⠛⠛⢉⣤⣍⣉⣉⣉⢉⣠⠌⠉⠉⠈⠉⠁⠀⠀⠀⠉⠉⠁⠈⠙⠛⠋⠁⠀⠈⠙⠛⠋⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠿⡛⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠘⠉⠟⢯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠣⠒⡊⡟⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠁⢁⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠈⠉⠛⠒⠶⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠋⠻⠿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠟⠋⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣟⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⡿⣿⣿⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠏⢶⠄⠰⣶⣶⡖⠹⠁⠈⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⡄⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠛⡛⠻⢿⣿⠟⢛⡛⠿⣿⡿⠟⣛⠛⢿⣿⡿⠟⠃⠀⠀⢠⣼⣿⣿⠿⠻⢿⡿⠟⣛⠻⢿⣿⡿⠛⡀⠀⠀⠀⠀⢛⡛⠻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⡸⣿⣿⢸⣿⣿⣿⣏⣿⠏⣾⣿⢻⣿⣆⢰⣿⡿⣿⣷⡄⣼⣿⠿⣿⣧⣩⣴⣿⣿⠀⠀⣿⣿⢫⣥⣾⣿⡇⣾⣿⢿⣿⣦⢉⣾⣿⢿⣿⣦⢠⣾⣿⢿⣿⡎⢹⣿⣿⣿⠟
⠉⣿⡟⣿⠧⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⣧⣿⣿⢸⣿⣿⣤⡿⣿⣾⡛⠛⣼⣿⡏⢸⣿⡇⣿⣿⡇⠛⠛⣰⣿⡿⠉⢹⣿⣿⢠⣤⡌⢿⣮⠉⣿⣿⠀⣿⣿⢰⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⠈⠋⠁⠀⠀
⠀⠈⠀⠀⠀⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡿⣿⣿⣿⢸⣿⣿⠛⣳⣿⣿⢇⣼⣿⠟⠀⢸⣿⡇⣿⣿⡇⠀⣴⣿⡿⢡⣾⢸⣿⣿⠘⠛⠃⠸⣿⠀⣿⣿⠆⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣼⣿⣿⠘⣿⣿⣼⣿⡿⢸⣿⡇⢻⣿⣿⢸⣿⣿⣤⡌⢻⢁⣾⣿⣯⣤⡄⢸⣿⣧⣿⣿⠇⣼⣿⣿⣤⣤⠀⢸⣿⣿⢠⣶⡆⠀⠀⠀⣿⣿⡆⣿⣿⣼⣿⣿⠘⣿⣿⣸⣿⡿⠸⣿⣿⣸⣿⡏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⢁⣤⣈⡙⠛⣋⣠⣌⣉⣁⣌⣉⣉⣈⣉⣉⣉⣡⣾⣌⣉⣉⣉⣉⣡⣤⣉⠛⠛⣉⣤⣍⣉⣉⣉⢉⣠⠈⠉⠉⠈⠉⠁⠀⠀⠀⠉⠉⠁⠈⠙⠛⠋⠁⠀⠈⠙⠛⠉⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠨⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠟⠻⢺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⡿⢿⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⢙⠿⠻⠿⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠙⠛⠒⠶⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠋⠻⠿⣿⣿⣿⣿⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⡿⢿⣿⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠛⠋⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⣶⣖⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣟⣩⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠏⢶⠄⠰⣶⣶⡖⠙⠋⠀⠃⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⡆⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⠿⠛⠿⠿⠿⠿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⢛⡛⠻⢿⣿⠟⢛⡛⠿⣿⡿⠛⣛⠛⢿⣿⡿⠟⠃⠀⠀⠀⣤⣿⣿⠿⠛⢿⠿⠛⣛⠻⢿⣿⡿⠛⡀⠀⠀⠀⠀⢘⡛⠛⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⡘⣿⣿⢸⣿⣿⣿⣇⣿⠏⣾⣿⢻⣿⣆⢰⣿⡿⣿⣷⡄⣼⣿⠿⣿⣧⣩⣴⣿⣿⠀⠀⠐⣿⢫⣤⣾⣿⡇⣾⣿⢿⣿⣧⢁⣾⣿⢿⣿⣦⢠⣾⣿⢿⣿⡎⢹⣿⣿⣿⠟
⠙⣿⡿⠿⡧⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⣧⣿⣿⢸⣿⣿⣠⡿⣿⣾⡛⠛⣼⣿⡏⢸⣿⡇⣿⣿⡇⠛⠛⣰⣿⡿⠉⢹⣿⣿⢠⣤⡄⢻⣎⠉⣿⣿⡇⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣷⠈⠉⠁⠀⠀
⠀⠈⠁⠀⠁⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡿⣿⣿⣿⢸⣿⣿⠛⣳⣿⣿⢃⣼⣿⠟⠀⢸⣿⡇⣿⣿⡇⠀⣴⣿⡿⢡⠶⢸⣿⣿⠘⠛⠃⠸⡿⠀⣿⣿⡁⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣼⣿⣿⠘⣿⣿⣼⣿⡿⢸⣿⡇⢻⣿⣿⢸⣿⣿⣤⡌⢿⢁⣾⣿⣯⣤⡄⢸⣿⣧⣿⣿⠇⣼⣿⣿⣤⣤⠀⢸⣿⣿⢠⣶⡆⠀⠀⠀⣿⣿⡇⣿⣿⣼⣿⣿⠘⣿⣿⣸⣿⡿⠸⣿⣿⣸⣿⡏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⢉⣁⣤⣌⣙⢛⣋⣤⣬⣉⣡⣌⣉⣉⣈⣉⣉⣉⣡⣾⣮⣉⣉⣉⣉⣡⣤⣉⡛⠛⣉⣴⣍⣉⣉⣉⢉⣤⡈⠉⠉⠈⠉⠁⠀⠀⠀⠉⠉⠁⠈⠙⠛⠋⠁⠀⠈⠙⠛⠋⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠨⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠩⠂⠓⠐⠋⠊⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠙⠛⣻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠽⢧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠐⠒⠲⠦⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠻⢿⣿⣿⣿⣿⡿⠟⠟⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠟⠛⠟⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠏⢶⠄⠰⣶⣶⣖⠙⠃⠘⠉⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡆⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⠿⠟⠿⠿⠛⠛⠿⠛⠿⠿⠿⠿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠛⡛⠻⢿⣿⠟⢛⡛⠻⣿⡿⠛⣛⠛⢿⣿⡿⠛⠁⠀⠀⠀⣠⣿⣿⠟⠛⢿⠿⠛⣛⠻⢿⣿⠟⠛⡀⠀⠀⠀⠀⢘⡛⠛⢿⣿⣿⣿⣿⣿
⣿⡿⢿⣿⣷⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⡘⣿⣿⢸⣿⣿⣿⣇⣿⠏⣾⣿⢻⣿⣆⢰⣿⡿⣿⣷⡄⣼⣿⠿⣿⣧⣩⣴⣿⣿⠀⠀⠀⡛⢫⣤⣾⣿⡇⣾⣿⢿⣿⣧⢀⣾⣿⢿⣿⣦⢠⣾⣿⢿⣿⡎⢉⣿⣿⡿⠟
⠉⣿⡿⠿⡦⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⣧⣿⣿⢸⣿⣿⣤⡿⣿⣾⡛⠛⣼⣿⡏⢸⣿⡇⣿⣿⡇⠛⠛⣰⣿⡿⠉⢹⣿⣿⢠⣤⡄⠻⣎⠉⣿⣿⡇⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⠀⠉⠀⠀⠀
⠀⠈⠁⢐⠇⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⡿⣿⣿⣿⢸⣿⣿⠛⣳⣿⣿⢃⣼⣿⠟⠀⢸⣿⡇⣿⣿⡇⠀⣴⣿⡿⠡⠶⢸⣿⣿⠘⠛⠃⠸⠿⠀⣿⣿⡁⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣼⣿⣿⠘⣿⣿⣼⣿⡿⢸⣿⡇⢻⣿⣿⢸⣿⣿⣤⡌⢻⢁⣾⣿⣯⣤⡄⢸⣿⣧⣿⣿⠇⣼⣿⣿⣤⣤⠀⢸⣿⣿⢠⣶⡆⠀⠀⠀⣿⣿⡇⣿⣿⣼⣿⣿⠘⣿⣿⣸⣿⡿⠸⣿⣿⣸⣿⡏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠉⣁⣤⣌⣙⢛⣋⣤⣬⣉⣡⣌⣉⣉⣌⣉⣉⣉⣡⣿⣮⣉⣉⣉⣉⣡⣤⣉⡛⢛⣉⣤⣍⣉⣉⣉⢉⣤⡈⠉⠉⠈⠉⠁⠀⠀⠀⠉⠉⠁⠈⠙⠛⠋⠁⠀⠈⠙⠛⠋⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣾⣿⣿⣿⣿⣷⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠍⠛⠉⠫⠿⡺⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠌⠿⢷⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣀⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠑⠐⠒⠲⠦⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠿⠿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠟⠛⠛⠋⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣯⡅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠏⢶⠄⠰⣶⣶⣖⠹⠋⠈⠛⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣆⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣦⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⣿
⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⢰⣿⠾⢿⣋⣿⡿⠿⠟
⠙⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡀⠀⠀⠀⠈⣷⡶⠿⠋⠉⠁⠀⠀
⠀⠈⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⠀⠀⠀⠰⠿⠿⠿⠿⠿⠿⠿⠿⠻⠟⠛⠟⠛⠛⠃⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡝⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢚⡿⠿⠿⡻⡟⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠠⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⣛⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⠓⠲⠶⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠿⢿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠬⠛⣛⠿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⠿⡿⠟⠛⠋⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣧⡤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠶⣶⠿⢷⠖⢰⣶⣶⣶⠹⢿⣿⡿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣆⠀⠀⠀⠀⠀⠀
⣿⣿⡿⠛⠻⠿⠿⡿⠿⠿⠿⡿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠟⠛⠻⣿⠛⠛⠁⠀⠀⠀⠀⠸⢿⡿⠛⢿⣿⣿⣿⣿⠎⠷⠆⠰⠄⠀⠀⠀⠀⠰⠿⠟⠛⠉⠻⣿⣿⣿
⣽⡿⠂⣿⣿⣿⣷⢸⣿⡿⠇⣴⣿⢻⣷⡀⣿⡿⠿⣿⣿⣷⣸⣿⣿⢸⣿⣿⣷⡄⣿⣿⠿⢸⣿⡿⣿⡎⠛⢰⣿⢿⣷⡄⣾⡟⣿⣦⢰⣿⢻⣷⣤⣴⣿⡇⠿⢿⢩⣴⣾⡇⣿⣿⠿⠿⢠⣾⡟⣿⣆⣰⣿⢻⣷⡀⠿⠟
⠙⣥⠀⣿⣿⣿⣿⢸⣿⣷⡆⣿⣿⠘⠛⠃⣿⣷⣶⣹⣿⣿⣿⣿⣿⢸⣿⣿⣿⡃⣿⣿⣶⢸⣿⣷⣿⡃⢾⡎⢉⣼⡿⢻⣿⡇⣿⣿⠈⢉⣾⡿⠀⢸⣿⣷⣶⠀⡷⠈⣿⡇⠿⠿⢻⣿⣼⣿⡇⣿⣿⣿⣿⢸⣿⡇⠀⠀
⠀⠈⠀⣿⣿⣿⣿⢸⣿⣇⣀⣿⣿⣸⣿⠇⣿⣯⣀⢻⣿⢿⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣀⢸⣿⡏⣿⡇⠘⢁⣾⣿⣁⡸⣿⡇⣿⣿⢠⣿⣟⣀⠀⢸⣿⣇⣀⠸⠟⠀⣿⡇⣿⣿⢸⣿⣿⣿⡇⣿⡿⢿⣿⢸⣿⠇⠀⠀
⠀⠀⠀⠛⠛⠛⢋⣘⣛⣛⣛⣈⣛⣛⣋⣄⣛⣛⣛⣙⣛⣘⣋⣛⣛⣘⣛⣛⣛⣁⣛⣛⣛⣘⣛⣋⡛⣃⣤⣜⡛⠛⠛⢃⠙⠛⠛⠁⠙⠛⠛⠛⠀⠘⠛⠛⠛⠀⠀⠀⠛⠃⠈⠛⠛⠋⠀⠙⠛⠛⠁⠈⠛⠛⠋⠀⠀⠀
⠀⠀⠐⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡐⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠓⢛⢟⣛⠿⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠈⠈⠈⠘⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⣋⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⠓⠲⠶⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⠿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠬⠛⠛⠶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠿⡿⡟⠋⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠶⣶⠟⢷⠖⢰⣶⣶⣶⠹⢟⣿⡿⠟⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣆⠀⠀⠀⠀⠀⠀
⣿⣿⡿⠛⠻⠿⠿⠿⠿⠿⠿⡿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠟⠛⠛⠟⠙⠛⠀⠀⠀⠀⠀⠘⢻⡭⠙⢥⣭⣵⣬⣤⠄⠰⠄⠀⠀⠀⠀⠀⠀⠀⠿⠛⠛⠁⠰⢿⠟⠛
⣽⡷⠂⣿⣿⣿⣷⢸⣿⡿⠇⣴⣿⢻⣷⡀⣿⡿⠿⢻⣿⣷⣸⣿⣿⢸⣿⣿⣷⡄⣿⣿⠿⢸⣿⡿⣿⡆⠛⢰⣿⢻⣷⡄⣾⡟⣿⣦⢰⣿⢻⣷⣤⣴⣿⡇⠻⠟⢡⣴⣾⡇⣿⣿⠿⠿⢠⣾⡟⣿⣆⣰⣿⢻⣷⡀⠸⠏
⠉⣥⠀⣿⣿⣿⣿⢸⣿⣷⡆⣿⣿⠘⠛⠃⣿⣷⣶⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⡃⣿⣿⣶⢸⣿⣷⣿⡃⢺⡌⢉⣼⡿⢳⣿⡇⣿⣿⠈⢉⣾⡿⠀⢸⣿⣷⣶⠀⡀⠈⣿⡇⠿⠿⢻⣿⣼⣿⡇⣿⣿⣿⣿⢸⣿⡇⠀⠀
⠀⠈⠀⣿⣿⣿⣿⢸⣿⣇⣀⣿⣿⣸⣿⠇⣿⣯⣀⢺⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣀⢸⣿⡇⣿⡇⠘⢁⣾⣿⣁⡘⣿⡇⣿⣿⢠⣿⣟⣀⠀⢸⣿⣇⣀⠘⠟⠀⣿⡇⣿⣿⢸⣿⣿⣿⡇⣿⡿⢿⣿⢸⣿⠇⠀⠀
⠀⠀⠀⠛⠛⠛⣋⣘⣛⣛⣛⣈⣛⣛⣋⣄⣛⣛⣛⣙⣛⣘⣋⣛⣛⣘⣛⣛⣛⣁⣛⣛⣛⣘⣛⣋⡛⢃⣤⣘⣛⠛⠛⣃⠙⠛⠛⠁⠘⠛⠛⠛⠀⠘⠛⠛⠛⠀⠀⠀⠛⠃⠈⠛⠛⠋⠀⠙⠛⠛⠁⠈⠛⠛⠋⠀⠀⠀
⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠨⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠈⠀⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠻⠛⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⠒⠲⠶⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠘⠛⣻⣷⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⡿⠟⠙⠛⠋⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣯⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠿⢶⠄⢰⣶⣶⣶⠹⢛⠛⠻⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣆⠀⠀⠀⠀⠀⠀
⣿⣿⡿⠛⠛⠿⠿⠿⠿⠿⠿⡿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠟⠛⠛⠟⠙⠛⠀⠀⠀⠀⠀⠈⠙⠭⠉⢠⣬⣤⣤⡤⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠿⠋⠉⠁⠐⠿⠂⠋
⣥⡷⠂⣿⣿⣿⣷⢸⣿⡿⠇⣴⣿⢻⣷⡀⣿⡿⠿⢹⣿⣷⣸⣿⣿⢸⣿⣿⣷⡄⣿⣿⠿⢸⣿⡿⣿⡆⠛⢰⣿⢻⣷⡄⣾⡟⣿⣦⢰⣿⢻⣷⣤⣴⣿⡇⠙⠟⢡⣴⣾⡇⣿⣿⠿⠿⢠⣾⡟⣿⣆⣰⣿⢻⣷⡀⠰⠋
⠉⣥⠀⣿⣿⣿⣿⢸⣿⣷⡆⣿⣿⠘⠛⠃⣿⣷⣶⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⡃⣿⣿⣶⢸⣿⣷⣿⡃⢸⡌⢉⣼⡿⢳⣿⡇⣿⣿⠈⢉⣾⡿⠀⢸⣿⣷⣶⠀⠀⠈⣿⡇⠿⠿⢻⣿⣼⣿⡇⣿⣿⣿⣿⢸⣿⡇⠀⠀
⠀⠈⠀⣿⣿⣿⣿⢸⣿⣇⣀⣿⣿⣸⣿⠇⣿⣧⣀⢸⣿⢿⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣀⢸⣿⡏⣿⡇⠸⢁⣾⣿⣁⡘⣿⡇⣿⣿⢠⣿⣟⣀⠀⢸⣿⣇⣀⠘⠟⠀⣿⡇⣿⣿⢸⣿⣿⣿⡇⣿⡿⢿⣿⢸⣿⠇⠀⠀
⠀⠀⠀⠛⠛⠛⢋⣘⣛⣛⣛⣈⣛⣛⣋⣄⣛⣛⣛⣘⣛⣘⣋⣛⣛⣘⣛⣛⣛⣁⣛⣛⣛⣘⡛⣃⢛⣃⣤⣘⣛⡛⠛⣃⠙⠛⠛⠁⠘⠛⠛⠛⠀⠘⠛⠛⠛⠀⠀⠀⠛⠃⠈⠛⠛⠋⠀⠙⠛⠛⠁⠈⠛⠛⠋⠀⠀⠀
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⢹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⢹⢿⠿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠀⠁⠀⠘⠙⢻⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣄⣤⣂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠟⠻⢛⣛⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠑⠒⠲⠶⠤⣄⣀⣿⣿⣿⣯⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠛⠿⢿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⡿⠯⠛⠛⠉⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡿⠕⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠟⢶⠄⢰⣶⣶⣶⠹⢋⠛⠟⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣆⠀⠀⠀⠀⠀⠀
⣿⣿⡿⠛⠛⠿⠿⠿⠿⠿⠿⡿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠟⠛⠛⠏⠈⠉⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⣀⣠⣄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠉⠀⠀⠶⠀⠈
⣴⡷⠀⣿⣿⣿⣷⢸⣿⡿⠇⣴⣿⢻⣷⡀⣿⡿⠿⢸⣿⣷⣸⣿⣿⢸⣿⣿⣷⡄⣿⣿⠿⢸⣿⡿⣿⡆⠛⢰⣿⢻⣷⡄⣾⡟⣿⣆⢰⣿⢻⣷⣤⣴⣿⡇⠈⠉⢡⣴⣾⡇⣿⣿⠿⠿⢠⣾⡟⣿⣆⣰⣿⢻⣷⡀⠀⠀
⠉⣤⠀⣿⣿⣿⣿⢸⣿⣷⡆⣿⣿⠘⠛⠃⣿⣷⣶⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⡃⣿⣿⣶⢸⣿⣷⣿⡃⢰⡌⢉⣼⡿⢣⣿⡇⣿⣿⠈⢉⣾⡿⠀⢸⣿⣷⣶⠀⠀⠈⣿⡇⠿⠿⢻⣿⣼⣿⡇⣿⣿⣿⣿⢸⣿⡇⠀⠀
⠀⠈⠀⣿⣿⣿⣿⢸⣿⣇⣀⣿⣿⣸⣿⠇⣿⣧⣀⢸⣿⢿⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣀⢸⣿⡏⣿⡇⠘⢁⣾⣿⣁⡘⣿⡇⣿⣿⢠⣿⣟⣀⠀⢸⣿⣇⣀⠐⠛⠀⣿⡇⣿⣿⢸⣿⣿⣿⡇⣿⡿⢿⣿⢸⣿⠇⠀⠀
⠀⠀⠀⠛⠛⠛⣋⣘⣛⣛⣛⣈⣛⣛⣋⣄⣛⣛⣛⣘⣛⣘⣋⣛⣛⣘⣛⣛⣛⣁⣛⣛⣛⣘⣛⣃⡛⣃⣤⣘⣛⣛⠛⣃⠙⠛⠛⠁⠘⠛⠛⠛⠀⠘⠛⠛⠛⠀⠀⠀⠛⠃⠈⠛⠛⠋⠀⠙⠛⠛⠁⠈⠛⠛⠋⠀⠀⠀
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠚⠛⠉⠩⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⢛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠑⠒⠶⠶⠤⣄⣀⣿⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠟⢶⠄⢰⣶⣶⣖⠙⠃⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣆⠀⠀⠀⠀⠀⠀
⡿⣿⡿⠛⠛⠿⠿⠿⠿⠿⠿⡿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠟⠿⠿⢿⣿⡿⠟⠋⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣤⡷⠀⣿⣿⣿⣷⢸⣿⡿⠇⣴⣿⢻⣷⡀⣿⡿⠿⢸⣿⣷⣸⣿⣿⢸⣿⣿⣷⡄⣿⣿⠿⢸⣿⡿⣿⡆⠋⢰⣿⢻⣷⡄⣾⡟⣿⣦⢰⣿⢻⣷⣤⣴⣿⡇⠀⠀⢠⣴⣾⡇⣿⣿⠿⠿⢠⣾⡟⣿⣆⣰⣿⢻⣷⡀⠀⠀
⠉⣤⠀⣿⣿⣿⣿⢸⣿⣷⡆⣿⣿⠘⠛⠃⣿⣷⣶⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⡃⣿⣿⣶⢸⣿⣷⣿⡃⢰⡌⢉⣼⡿⢳⣿⡇⣿⣿⠈⢉⣾⡿⠀⢸⣿⣷⣶⠀⠀⠈⣿⡇⠿⠿⢻⣿⣼⣿⡇⣿⣿⣿⣿⢸⣿⡇⠀⠀
⠀⠋⠀⣿⣿⣿⣿⢸⣿⣇⣀⣿⣿⣸⣿⠇⣿⣧⣀⢸⣿⢿⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣀⢸⣿⡏⣿⡇⠘⢁⣾⣿⣁⡘⣿⡇⣿⣿⢠⣿⣟⣀⠀⢸⣿⣇⣀⠀⠃⠀⣿⡇⣿⣿⢸⣿⣿⣿⡇⣿⡿⢿⣿⢸⣿⠇⠀⠀
⠀⠀⠀⠛⠛⠛⣋⣘⣛⣛⣛⣌⣛⣛⣋⣄⣛⣛⣛⣘⣛⣘⣋⣛⣛⣘⣛⣛⣛⣁⣛⣛⣛⣘⣛⣃⣛⣃⣤⣘⣛⣛⡛⣃⠙⠛⠛⠁⠘⠛⠛⠛⠀⠘⠛⠛⠛⠀⠀⠀⠛⠃⠈⠛⠛⠋⠀⠙⠛⠛⠁⠈⠛⠛⠋⠀⠀⠀
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠟⠿⠿⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢻⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠑⠶⠶⠶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⢿⣿⣿⣿⣿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠘⠟⠳⢶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⡿⠟⠛⠛⠋⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡅⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠿⢶⠄⢰⣶⣶⡶⠙⠙⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⠀⠀⠀⠀⠀⠀
⣼⣿⡿⠛⠛⠿⠿⡿⠿⠿⠿⡿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠟⠛⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⣤⡷⠂⣿⣿⣿⣷⢸⣿⡿⠇⣴⣿⢻⣷⡀⣿⡿⠿⢸⣿⣷⣸⣿⣿⢸⣿⣿⣷⡄⣿⣿⠿⢸⣿⡿⣿⡆⠋⢰⣿⢻⣷⡄⣾⡟⣿⣦⢰⣿⢻⣷⣤⣴⣿⡇⠀⠀⢠⣴⣾⡇⣿⣿⠿⠿⢠⣾⡟⣿⣆⣰⣿⢻⣷⡀⠀⠀
⠉⣤⠀⣿⣿⣿⣿⢸⣿⣷⡆⣿⣿⠘⠛⠃⣿⣷⣶⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⡃⣿⣿⣶⢸⣿⣷⣿⡃⢴⡌⢉⣼⡿⢻⣿⡇⣿⣿⠈⢉⣾⡿⠀⢸⣿⣷⣶⠀⠀⠈⣿⡇⠿⠿⢻⣿⣼⣿⡇⣿⣿⣿⣿⢸⣿⡇⠀⠀
⠀⠉⠀⣿⣿⣿⣿⢸⣿⣇⣀⣿⣿⣸⣿⠇⣿⣯⣀⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣀⢸⣿⡏⣿⡇⠸⢁⣾⣿⣁⡘⣿⡇⣿⣿⢠⣿⣟⣀⠀⢸⣿⣇⣀⠀⠀⠀⣿⡇⣿⣿⢸⣿⣿⣿⡇⣿⡿⢿⣿⢸⣿⠇⠀⠀
⠀⢠⡀⠛⠛⢛⣋⣘⣛⣛⣛⣌⣛⣛⣋⣤⣛⣛⣛⣘⣛⣘⣋⣛⣛⣘⣛⣛⣛⣁⣛⣛⣛⣘⣛⣃⣛⣃⣤⣘⣛⣛⣛⡃⠙⠛⠛⠁⠘⠛⠛⠛⠀⠘⠛⠛⠛⠀⠀⠀⠛⠃⠈⠛⠛⠋⠀⠙⠛⠛⠁⠈⠛⠛⠋⠀⠀⠀
⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠂⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠑⠶⠶⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠩⠟⠳⢶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⠟⠛⠋⠛⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠣⠗⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠶⣶⠿⢷⠔⢰⣶⣶⣶⠹⠋⠛⠟⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⠀⠀⠀⠀⠀⠀
⢽⣿⡿⠛⠛⠿⠿⡿⠿⠿⢿⡿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⡿⠀⣿⣿⣿⣷⢸⣿⡿⠇⣴⣿⢻⣷⡀⣿⡿⠿⢸⣿⣷⣸⣿⣿⢸⣿⣿⣷⡄⣿⣿⠿⢸⣿⡿⣿⡆⠛⢰⣿⢻⣷⡄⣾⡟⣿⣦⢰⣿⢻⣷⣤⣴⣿⡇⠀⠀⢠⣴⣾⡇⣿⣿⠿⠿⢠⣾⡟⣿⣆⢰⣿⢻⣷⡀⠀⠀
⠉⣤⠀⣿⣿⣿⣿⢸⣿⣷⡆⣿⣿⠘⠛⠃⣿⣷⣶⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⡃⣿⣿⣶⢸⣿⣷⣿⡃⢴⡌⢉⣼⡿⢻⣿⡇⣿⣿⠈⢉⣾⡿⠀⢸⣿⣷⣶⠀⠀⠈⣿⡇⠿⠿⢻⣿⣼⣿⡇⣿⣿⢸⣿⢸⣿⡇⠀⠀
⢈⡯⠀⣿⣿⣿⣿⢸⣿⣇⣀⣿⣿⣸⣿⠇⣿⣯⣀⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣀⢸⣿⡏⣿⡇⠸⢁⣾⣿⣁⡘⣿⡇⣿⣿⢠⣿⣟⣀⠀⢸⣿⣇⣀⠀⠀⠀⣿⡇⣿⣿⢸⣿⣿⣿⡇⣿⡿⢸⣿⢸⣿⠇⠀⠀
⠀⣿⣀⠛⠛⢛⣋⣘⣛⣛⣛⣌⣛⣛⣋⣤⣛⣛⣛⣘⣛⣘⣋⣛⣛⣘⣛⣛⣛⣁⣛⣛⣛⣘⣛⣛⣛⣃⣴⣘⣛⣛⡛⣃⠙⠛⠛⠁⠘⠛⠛⠛⠀⠘⠛⠛⠛⠀⠀⠀⠛⠃⠈⠛⠛⠋⠀⠙⠛⠛⠁⠈⠛⠛⠋⠀⠀⠀
⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠹⠉⠉⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⣀⣠⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⢿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠳⠶⠶⢦⣤⣄⣀⣿⡾⣿⣿⣽⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠉⢿⣿⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠁⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠛⠻⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡟⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⠓⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠿⢶⠆⢰⣶⣶⣶⠹⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣦⠀⠀⠀⠀⠀⠀
⣿⣿⡿⠛⠛⠿⠿⡿⠿⠿⢿⡿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣵⡷⠂⣿⣿⣿⣷⢸⣿⡿⠇⣴⣿⢻⣷⡀⣿⡿⠿⢸⣿⣷⣸⣿⣿⢸⣿⣿⣷⡄⣿⣿⠿⢸⣿⡿⣿⡆⠛⢰⣿⢻⣷⡄⣾⡟⣿⣦⢰⣿⢻⣷⣤⣴⣿⡇⠀⠀⢠⣴⣾⡇⣿⣿⠿⠿⢠⣾⡟⣿⣆⢰⣿⢻⣷⡀⠀⠀
⠉⣤⠀⣿⣿⣿⣿⢸⣿⣷⡆⣿⣿⠘⠛⠃⣿⣷⣶⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⡃⣿⣿⣶⢸⣿⣷⣿⡃⢴⡌⢉⣼⡿⢳⣿⡇⣿⣿⠈⢉⣾⡿⠀⢸⣿⣷⣶⠀⠀⠈⣿⡇⠿⠿⢻⣿⣼⣿⡇⣿⣿⣿⣿⢸⣿⡇⠀⠀
⠈⡟⠀⣿⣿⣿⣿⢸⣿⣇⣀⣿⣿⣸⣿⠇⣿⣷⣀⢸⣿⢿⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣀⢸⣿⡏⣿⡇⠸⢁⣾⣿⣁⡘⣿⡇⣿⣿⢠⣿⣟⣀⠀⢸⣿⣇⣀⠀⠀⠀⣿⡇⣿⣿⢸⣿⣿⣿⡇⣿⡿⢻⣿⢸⣿⠇⠀⠀
⠀⣳⣀⠛⠛⢛⣋⣘⣛⣛⣛⣌⣛⣛⣋⣤⣛⣛⣛⣘⣛⣘⣋⣛⣛⣘⣛⣛⣛⣁⣛⣛⣛⣘⣛⣓⣛⣃⣶⣘⣛⣛⢛⠃⠙⠛⠛⠁⠘⠛⠛⠛⠀⠘⠛⠛⠛⠀⠀⠀⠛⠃⠈⠛⠛⠋⠀⠙⠛⠛⠁⠈⠛⠛⠋⠀⠀⠀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠙⠉⠉⠋⠉⠉⠯⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠳⠶⠶⢦⣤⣄⣰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠁⠉⠁⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⠿⠛⠍⠉⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⢿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠛⣛⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠿⢶⠆⢰⣶⣶⣶⠹⠋⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⠀⠀⠀⠀⠀⠀
⣿⣿⠿⠛⠻⠿⠿⡿⠿⠿⢿⡿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣴⡷⠂⣿⣿⣿⣷⢸⣿⡿⠇⣴⣿⢻⣷⡀⣿⡿⠿⢸⣿⣷⣸⣿⣿⢸⣿⣿⣷⡄⣿⣿⠿⢸⣿⡿⣿⡆⠛⢰⣿⢻⣷⡄⣾⡟⣿⣦⢰⣿⢻⣷⣤⣴⣿⡇⠀⠀⢠⣴⣾⡇⣿⣿⠿⠿⢠⣾⡟⣿⣆⢰⣿⢻⣷⡀⠀⠀
⢙⣤⠀⣿⣿⣿⣿⢸⣿⣷⡆⣿⣿⠘⠛⠃⣿⣷⣶⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⡃⣿⣿⣶⢸⣿⣷⣿⡃⢴⡌⢉⣼⡿⢳⣿⡇⣿⣿⠈⢉⣾⡿⠀⢸⣿⣷⣶⠀⠀⠈⣿⡇⠿⠿⢻⣿⣼⣿⡇⣿⣿⣿⣿⢸⣿⡇⠀⠀
⠘⣏⠀⣿⣿⣿⣿⢸⣿⣇⣀⣿⣿⣸⣿⠇⣿⣧⣀⢸⣿⢿⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣀⢸⣿⡏⣿⡇⠸⢁⣾⣿⣁⡘⣿⡇⣿⣿⢠⣿⣟⣀⠀⢸⣿⣇⣀⠀⠀⠀⣿⡇⣿⣿⢸⣿⣿⣿⡇⣿⡿⢻⣿⢸⣿⠇⠀⠀
⠀⣷⣀⠛⠛⢛⣋⣘⣛⣛⣛⣌⣛⣛⣋⣤⣛⣛⣛⣘⣛⣘⣋⣛⣛⣘⣛⣛⣛⣁⣛⣛⣛⣘⣛⣃⣛⣃⣶⣘⣛⣛⠛⢃⠙⠛⠛⠁⠘⠛⠛⠛⠀⠘⠛⠛⠛⠀⠀⠀⠛⠃⠈⠛⠛⠋⠀⠙⠛⠛⠁⠈⠛⠛⠋⠀⠀⠀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠨⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠙⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠈⠚⠛⠛⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⠦⢤⣤⣀⣰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣬⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⢉⢿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠉⠉⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⠿⠋⠉⠛⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⣝⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠟⢶⠄⢰⣶⣶⣶⠙⠋⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣦⠀⠀⠀⠀⠀⠀
⣼⣿⡿⠛⠛⠿⠿⠿⠿⠿⢿⡿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢔⡿⠂⣿⣿⣿⣷⢸⣿⡿⠇⣴⣿⢻⣷⡀⣿⡿⠿⢹⣿⣷⣸⣿⣿⢸⣿⣿⣷⡄⣿⣿⠿⢸⣿⡿⣿⡆⠙⢰⣿⢻⣷⡄⣾⡟⣿⣦⢰⣿⢻⣷⣤⣴⣿⡇⠀⠀⢠⣴⣾⡇⣿⣿⠿⠿⢠⣾⡟⣿⣆⢰⣿⢻⣷⡀⠀⠀
⢘⣤⠀⣿⣿⣿⣿⢸⣿⣷⡆⣿⣿⠘⠛⠃⣿⣷⣶⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⡃⣿⣿⣶⢸⣿⣷⣿⡃⢴⡌⢉⣼⡿⢻⣿⡇⣿⣿⠈⢉⣾⡿⠀⢸⣿⣷⣶⠀⠀⠈⣿⡇⠿⠿⢻⣿⣼⣿⡇⣿⣿⣿⣿⢸⣿⡇⠀⠀
⠈⣟⠀⣿⣿⣿⣿⢸⣿⣇⣀⣿⣿⣸⣿⠇⣿⣿⣀⢸⣿⢿⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣀⢸⣿⡏⣿⡇⠸⢁⣾⣿⣁⡸⣿⡇⣿⣿⢠⣿⣟⣀⠀⢸⣿⣇⣀⠀⠀⠀⣿⡇⣿⣿⢸⣿⣿⣿⡇⣿⡿⢿⣿⢸⣿⠇⠀⠀
⠀⢻⡀⠛⠛⢛⣋⣘⣛⣛⣛⣈⣛⣛⣋⣄⣛⣛⣛⣘⣛⣘⣋⣛⣛⣘⣛⣛⣛⣁⣛⣛⣛⣘⣛⣋⣛⣃⣶⣘⣛⡛⠛⢃⠙⠛⠛⠁⠘⠛⠛⠛⠀⠘⠛⠛⠛⠀⠀⠀⠛⠃⠈⠛⠛⠋⠀⠙⠛⠛⠁⠈⠛⠛⠋⠀⠀⠀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠊⠙⠿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⠦⢤⣤⣀⣀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⡿⠫⠉⠙⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠛⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⢻⣿⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠟⢷⠖⣰⣶⣶⣶⠙⠋⠻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⠀⠀⠀⠀⠀⠀
⣻⣿⠿⠛⠻⠿⠿⠿⠿⠿⢿⡿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢹⡿⠆⣿⣿⣿⣷⢸⣿⡿⠇⣴⣿⢻⣷⡀⣿⣿⠿⢹⣿⣷⣼⣿⣿⢸⣿⣿⣷⡄⣿⣿⠿⢸⣿⡿⣿⡆⠙⢰⣿⢿⣷⡄⣾⡟⣿⣦⢰⣿⢻⣷⣤⣴⣿⡇⠀⠀⢠⣴⣾⡇⣿⣿⠿⠿⢠⣾⡟⣿⣆⢰⣿⢻⣷⡀⠀⠀
⢈⣷⠀⣿⣿⣿⣿⢸⣿⣷⡆⣿⣿⠘⠛⠃⣿⣿⣶⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⡃⣿⣿⣶⢸⣿⣿⣿⡃⢰⡌⢉⣼⡿⢻⣿⡇⣿⣿⠈⢉⣾⡿⠀⢸⣿⣷⣶⠀⠀⠈⣿⡇⠿⠿⢻⣿⣼⣿⡇⣿⣿⣿⣿⢸⣿⡇⠀⠀
⠀⣟⠀⣿⣿⣿⣿⢸⣿⣇⣀⣿⣿⣸⣿⠇⣿⣿⣀⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣀⢸⣿⣏⣿⡇⠸⢁⣾⣿⣁⡸⣿⡇⣿⣿⢠⣿⣟⣀⠀⢸⣿⣇⣀⠀⠀⠀⣿⡇⣿⣿⢸⣿⣿⣿⡇⣿⡿⢻⣿⢸⣿⠇⠀⠀
⠀⢻⣀⠛⠛⠛⣋⣘⣛⣛⣛⣈⣛⣛⣋⣄⣛⣛⣛⣘⣛⣘⣋⣛⣛⣘⣛⣛⣛⣁⣛⣛⣛⣘⣛⣛⣛⣃⣴⣼⣛⡛⡛⢃⠙⠛⠛⠁⠘⠛⠛⠛⠀⠘⠛⠛⠛⠀⠀⠀⠛⠃⠈⠛⠛⠋⠀⠙⠛⠛⠁⠈⠛⠛⠋⠀⠀⠀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣅⣄⣐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠻⡿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⢘⢽⡻⠑⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⢶⣾⣧⣼⣧⣢⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣽⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠹⠿⣿⠿⣿⣿⠷⢦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⠋⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⠻⠛⠿⠻⠻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡝⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠶⣶⠿⢷⠶⣰⣶⣾⣶⠹⡻⣿⢿⠿⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⡀⠀⠀⠀⠀⠀
⠸⣿⠿⠛⠛⠿⠿⠿⠿⠿⢿⡿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢰⡿⠂⣿⣿⣿⣷⢸⣿⡿⠇⣴⣿⢻⣷⡀⣿⣿⠿⢸⣿⣷⣸⣿⣿⢸⣿⣿⣷⡄⣿⣿⠿⢸⣿⣿⣿⡆⠉⢰⣿⢿⣷⡄⣾⡟⣿⣦⢰⣿⢻⣷⣤⣴⣿⡇⠀⠀⢠⣴⣾⡇⣿⣿⠿⠿⢠⣾⡟⣿⣆⢰⣿⢻⣷⡀⠀⠀
⠀⣤⠀⣿⣿⣿⣿⢸⣿⣷⡆⣿⣿⠘⠛⠃⣿⣿⣶⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⡃⣿⣿⣶⢸⣿⣿⣿⡃⢰⡌⢉⣼⣿⢻⣿⡇⣿⣿⠈⢉⣾⡿⠀⢸⣿⣷⣶⠀⠀⠈⣿⡇⠿⠿⢻⣿⣼⣿⡇⣿⣿⣿⣿⢸⣿⡇⠀⠀
⠀⠟⠀⣿⣿⣿⣿⢸⣿⣇⣀⣿⣿⣸⣿⠇⣿⣿⣀⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣀⢸⣿⣿⣿⡇⠸⢁⣾⣿⣁⡸⣿⡇⣿⣿⢠⣿⣟⣀⠀⢸⣿⣇⣀⠀⠀⠀⣿⡇⣿⣿⢸⣿⣿⣿⡇⣿⡿⢻⣿⢸⣿⠇⠀⠀
⠀⢠⣀⠛⠛⢛⣋⣘⣛⣛⣛⣈⣛⣛⣋⣄⣛⣛⣛⣘⣛⣘⣋⣛⣛⣘⣛⣛⣛⣁⣛⣛⣛⣘⣛⣛⣛⣃⣶⣜⡛⡛⠛⣃⠙⠛⠛⠁⠘⠛⠛⠛⠀⠘⠛⠛⠛⠀⠀⠀⠛⠃⠈⠛⠛⠋⠀⠙⠛⠛⠁⠈⠛⠛⠋⠀⠀⠀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣐⣤⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠐⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⠛⡛⡿⠻⠿⠿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣽⠝⣍⢆⡄⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⣶⣾⣷⣿⣿⣁⣿⠇⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠹⠿⠿⠻⡟⠿⠷⣦⣤⣤⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠉⠠⠁⠐⠉⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⡟⠿⠛⠿⠟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠶⣶⠿⢷⠶⣰⣶⣾⣶⠹⢿⣿⡿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⡀⠀⠀⠀⠀⠀
⠼⣿⡿⠛⠛⠿⠿⡿⠿⠿⢿⡿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢴⡿⠆⣿⣿⣿⣷⢸⣿⡿⠇⣴⣿⢻⣷⡈⣿⣿⠿⣿⣿⣷⣼⣿⣿⢸⣿⣿⣷⡄⣿⣿⠿⢸⣿⣿⣿⡆⠉⢰⣿⢿⣿⣄⣾⡟⣿⣦⣰⣿⢻⣷⣤⣶⣿⡇⠀⠀⢠⣴⣾⡇⣿⣿⠿⠿⢠⣾⡟⣿⣆⣰⣿⢻⣷⡀⠀⠀
⠚⣶⠀⣿⣿⣿⣿⢸⣿⣷⡆⣿⣿⠘⠛⠃⣿⣿⣶⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡃⣿⣿⣶⢸⣿⣿⣿⡃⣴⡌⢉⣼⣿⢻⣿⡇⣿⣿⠈⢉⣾⡿⠀⢸⣿⣷⣶⠀⠀⠈⣿⡇⠿⠿⢻⣿⣼⣿⡇⣿⣿⣿⣿⢸⣿⡇⠀⠀
⠘⣛⠀⣿⣿⣿⣿⢸⣿⣇⣀⣿⣿⣸⣿⠇⣿⣿⣀⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣀⢸⣿⣿⣿⡇⠸⢁⣾⣿⣁⡸⣿⣇⣿⣿⢠⣿⣟⣁⠀⢸⣿⣇⣀⠀⠀⠀⣿⡇⣿⣿⢸⣿⣿⣿⡇⣿⡿⢿⣿⢸⣿⠇⠀⠀
⠀⣶⡀⠛⠛⢛⣋⣘⣛⣛⣛⣈⣛⣛⣋⣄⣛⣛⣛⣙⣛⣘⣛⣛⣛⣘⣛⣛⣛⣁⣛⣛⣛⣘⣛⣛⣛⣃⣾⣞⣛⣛⡛⣃⡙⠛⠛⣁⠙⠛⠛⠛⠀⠘⠛⠛⠛⠀⠀⠀⠛⠃⠈⠛⠛⠋⠀⠙⠛⠛⠁⠈⠛⠛⠋⠀⠀⠀
⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠛⢛⡻⣻⡻⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⠁⠀⠀⠀⠀⣀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠄⠀⠉⠺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⡛⠭⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⣿⣿⣿⣿⣿⡿⢿⣿⠿⣿⣿⣿⡿⣿⠟⣻⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠶⣶⠿⢷⠶⠰⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢭⠛⠛⠛⠻⠿⠿⠿⠿⠿⡿⠿⠿⠿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠟⠛⠿⢿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⡷⠆⣿⣿⣿⣷⢸⣿⡿⠇⣴⣿⢿⣷⡀⣿⣿⠿⣿⣿⣷⣼⣿⣿⢸⣿⣿⣿⡄⣿⣿⠿⢸⣿⣿⣿⡆⠈⢰⣿⢿⣿⣄⣾⡟⣿⣦⣰⣿⢻⣷⣤⣶⣿⡇⠀⠀⢠⣴⣾⡇⣿⣿⠿⠿⢠⣾⡟⣿⣆⣰⣿⢻⣷⡀⠀⠀
⠀⣶⠀⣿⣿⣿⣿⢸⣿⣷⡆⣿⣿⠘⠛⠃⣿⣿⣶⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡃⣿⣿⣶⢸⣿⣿⣿⡃⢰⡌⢉⣼⣿⢻⣿⡇⣿⣿⠈⢉⣾⡿⠀⢸⣿⣷⣶⠀⠀⠈⣿⡇⠿⠿⢻⣿⣼⣿⡇⣿⣿⣿⣿⢸⣿⡇⠀⠀
⠀⠛⠀⣿⣿⣿⣿⢸⣿⣇⣀⣿⣿⣸⣿⠇⣿⣿⣀⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣀⢸⣿⣿⣿⡇⠸⢁⣾⣿⣁⡸⣿⣇⣿⣿⢠⣿⣟⣁⠀⢸⣿⣇⣀⠀⠀⠀⣿⡇⣿⣿⢸⣿⣿⣿⡇⣿⡿⢿⣿⢸⣿⡇⠀⠀
⠀⢠⣀⠛⠛⣛⣋⣘⣛⣛⣛⣈⣛⣛⣋⣄⣛⣛⣛⣛⣛⣘⣛⣛⣛⣘⣛⣛⣛⣁⣛⣛⣛⣘⣛⣛⣛⣣⣶⣜⣛⣛⢛⣃⠙⠛⠛⢁⠙⠛⠛⠛⠀⠘⠛⠛⠛⠀⠀⠀⠛⠃⠈⠛⠛⠋⠀⠙⠛⠛⠁⠈⠛⠛⠋⠀⠀⠀
⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠳⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠒⠹⠚⡫⠈⠉⠟⢹⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠭⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠆⠀⠀⣾⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠂⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠐⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⠀⠉⠀⠛⠛⠙⠛⢋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠶⣶⠿⢷⠶⠰⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⠙⠛⠛⠻⠿⠿⠿⠿⠿⡿⠿⠿⠿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠿⠛⠿⠿⠟⠛⠻⠿⣯⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡷⠆⣿⣿⣿⣷⢸⣿⡿⠇⣴⣿⢿⣷⡈⣿⣿⠿⣿⣿⣷⣼⣿⣿⢸⣿⣿⣿⡄⣿⣿⠿⢸⣿⣿⣿⡆⠁⢰⣿⢿⣿⣄⣾⡟⣿⣦⣰⣿⢻⣷⣤⣶⣿⡇⠀⠀⢠⣴⣾⡇⣿⣿⠿⠿⢠⣾⡟⣿⣆⣰⣿⢻⣷⡄⠀⠀
⠀⣤⠀⣿⣿⣿⣿⢸⣿⣷⡆⣿⣿⠘⠛⠃⣿⣿⣶⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡃⣿⣿⣶⢸⣿⣿⣿⡃⢠⡈⢉⣼⣿⢻⣿⡇⣿⣿⠉⢉⣾⡿⠀⢸⣿⣷⣶⠀⠀⠈⣿⡇⠿⠿⢻⣿⣼⣿⡇⣿⣿⣿⣿⢸⣿⡇⠀⠀
⠀⠋⠀⣿⣿⣿⣿⢸⣿⣇⣀⣿⣿⣸⣿⠇⣿⣿⣀⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣀⢸⣿⣿⣿⡇⠸⢁⣾⣿⣁⡸⣿⣇⣿⣿⢠⣿⣟⣁⠀⢸⣿⣇⣀⠀⠀⠀⣿⡇⣿⣿⣸⣿⣿⣿⡇⣿⡿⢿⣿⢸⣿⡇⠀⠀
⠀⢰⣀⠛⠛⣛⣋⣘⣛⣛⣛⣈⣛⢛⣋⣔⣛⣛⣛⣛⣛⣘⣛⣛⣛⣘⣛⣛⣛⣁⣛⣛⣛⣘⣛⣛⣛⣣⣾⣾⣛⣛⠛⣃⡙⠛⠛⠁⠙⠛⠛⠛⠀⠘⠛⠛⠛⠀⠀⠀⠛⠃⠈⠛⠛⠋⠀⠙⠛⠛⠁⠈⠛⠛⠋⠀⠀⠀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠫⠅⠐⠊⠛⠻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀⡀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡒⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠰⠒⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⠶⠺⠳⠄⠀⠈⠻⠿⣿⡿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣿⣾⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠿⠟⠛⠋⠉⠉⠉⠉⠉⠛⠻⠿⣿⠛⠉⠁⠚⠈⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠑⠚⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠶⣶⠿⢷⠶⠰⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠻⠿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⠿⠛⠻⠿⠛⠛⠛⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠷⠆⣿⣿⣿⣷⢸⣿⡿⠇⣼⣿⢿⣷⡈⣿⣿⠿⣿⣿⣷⣼⣿⣿⢸⣿⣿⣿⡄⣿⣿⠿⢸⣿⣿⣿⡆⠀⢰⣿⢿⣿⣄⣾⡿⣿⣦⣰⣿⢻⣷⣤⣶⣿⡇⠀⠀⢠⣴⣾⡇⣿⣿⠿⠿⢠⣾⡟⣿⣆⣰⣿⢻⣷⡄⠀⠀
⠀⣤⠀⣿⣿⣿⣿⢸⣿⣷⡆⣿⣿⠘⠛⠃⣿⣿⣶⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡃⣿⣿⣶⢸⣿⣿⣿⡃⢠⠈⢉⣼⣿⢻⣿⡇⣿⣿⠉⢉⣾⡿⠀⢸⣿⣷⣶⠂⠀⠈⣿⡇⠿⠿⢻⣿⣼⣿⡇⣿⣿⣿⣿⢸⣿⡇⠀⠀
⠀⠋⠀⣿⣿⣿⣿⢸⣿⣇⣀⣿⣿⣸⣿⠇⣿⣿⣀⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣀⢸⣿⣿⣿⡇⠈⢀⣾⣿⣁⡸⣿⣇⣿⣿⢠⣿⣟⣁⠀⢸⣿⣇⣀⠀⠀⠀⣿⡇⣿⣿⣸⣿⣿⣿⣇⣿⡿⢿⣿⢸⣿⡇⠀⠀
⠀⢳⣀⠛⠛⣛⣋⣜⣛⣛⣛⣈⡛⠛⣋⣀⣛⣛⣛⣛⣛⣘⣛⣛⣛⣘⣛⣛⣛⣁⣛⣛⣛⣘⣛⣛⣛⣣⣶⣾⣛⡛⠛⣃⡙⠛⠛⠁⠙⠛⠛⠛⠀⠘⠛⠛⠛⠀⠀⠀⠛⠃⠈⠛⠛⠋⠀⠙⠛⠛⠁⠈⠛⠛⠋⠀⠀⠀
⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣼⣧⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢩⣝⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡻⡿⠿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣄⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡭⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⠿⠿⠿⠿⠋⠁⠀⠀⠀⠸⢿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣍⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⢝⡿⡿⡻⠛⠀⠝⠻⠻⣿⣿⣿⠿⠿⢛⠉⠈⠀⠘⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠈⢰⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢄⠰⠙⠩⣿⣷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⢉⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠙⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠶⣶⠿⢷⠶⠴⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣼⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣦⣴⣦⢉⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡲⢾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⡂⠙⡩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⠀⠀⠄⠹⠝⠟⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⢪⣜⡯⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠍⡋⠇⠪⡥⣿⠿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⣊⡥⠫⠯⠁⣩⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⡿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⡝⢯⣿⢣⣽⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣽⣹⣿⢿⣫⣻⡉⣛⢷⣶⡿⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣻⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣟⣻⡻⠀⠼⣿⢡⡰⣩⠿⠙⢫⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠰⠿⠷⣶⠿⢿⠶⡾⠾⠿⡿⠚⠻⠿⠿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠄⠳⢃⣘⠗⠑⠭⠂⠈⠰⠝⠽⣯⣿⠿⠿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⢷⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠷⢶⢠⣤⡄⢠⣤⣤⡌⢩⣤⡍⣭⣭⢩⣥⡌⣭⣬⠉⣥⣤⣤⠉⣥⣤⣤⣭⡡⣤⡄⣡⣤⠄⠀⣠⣤⣤⣄⠀⣠⣤⣤⡀⢀⣤⣤⣤⡀⣠⣤⣤⣄⠀⠀⠀⠀⠀⣠⣤⢀⣠⣤⣤⡀⢀⣤⣤⣤⡀⢀⣤⣤⣤⡀⠀⠀
⠀⣤⡄⢸⣿⡇⣼⣿⣿⣧⢸⣿⣿⣼⣿⢸⣿⡇⣿⣿⢀⣿⣿⣿⡆⣿⣿⣿⣿⡇⢻⣿⣿⡟⠀⠀⠿⠏⣽⣿⢸⣿⡏⣿⣿⠸⠿⢹⣿⡇⠿⠏⣿⣿⢀⣀⠀⠘⠻⣿⣿⢸⣿⣏⣛⡛⢸⣿⡏⣿⣷⣾⣿⣿⣿⡇⠀⠀
⠀⠈⠁⢸⣿⡇⣿⣿⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣿⣿⢸⣿⣿⣿⡇⣿⣿⣿⣿⡅⠘⣿⣿⠁⣼⡦⢀⣼⣿⠏⢸⣿⡇⣿⣿⠀⣠⣿⡿⠁⠀⣼⣿⠏⠘⠿⠀⠀⠀⣿⣿⢸⣿⡟⣿⣿⢸⣿⡇⣿⣿⣿⣿⣿⣿⡇⠀⠀
⠀⠀⢠⣼⣿⣧⣿⣿⢻⣿⣾⣿⡟⣿⣿⠸⣿⣧⣿⣿⣸⣿⡟⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⠀⣿⠁⣾⣿⣧⣤⠸⣿⣧⣿⡿⢰⣿⣿⣤⡄⣾⣿⣧⣤⢠⣤⠀⠀⠀⣿⣿⢸⣿⣧⣿⡿⢸⣿⣧⣿⡟⢻⣿⣿⣿⡇⠀⠀
⠀⠀⠀⣉⣉⣬⣭⣭⣬⣭⣍⣉⣡⣉⣉⣤⣈⣉⣉⣁⣉⣉⣥⣭⣭⣭⣭⣭⣭⣥⣶⣭⣭⣾⣿⣮⣭⣭⣭⣽⣶⣬⣉⣉⣤⣈⢉⣉⣉⡁⠉⠉⠉⠉⠈⠉⠀⠀⠀⠉⠉⠀⠈⠉⠉⠀⠀⠉⠉⠉⠀⠀⠉⠉⠉⠀⠀⠀
⠀⠀⡱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣵⣿⣗⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠉⢯⡿⢿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠩⣿⣿⣿⣿⣿⡯⠓⠶⡳⢭⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠹⠉⡉⠛⢙⠁⠉⠀⠬⠒⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⢀⣴⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡂⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠠⠀⠀⠀⠈⠙⠿⠿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⠷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣯⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⠹⣿⡟⢻⠉⠋⠛⠛⠻⣿⠿⠿⢟⠟⠋⠉⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡮⣶⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠃⠻⢇⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠶⣶⠿⢷⠶⠴⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠽⢟⢻⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠶⢶⢠⣤⡄⢠⣤⣤⡄⢠⣤⡄⣤⣤⢠⣤⡄⣤⣤⠈⣤⣤⣤⠉⣤⣤⣤⣤⡠⣤⡄⣠⣤⠄⠀⣠⣤⣤⣄⠀⣠⣤⣤⡀⢀⣤⣤⣤⡀⣠⣤⣤⣄⠀⠀⠀⠀⠀⣠⣤⢀⣠⣤⣤⡀⢀⣤⣤⣤⡀⢀⣤⣤⣤⡀⠀⠀
⠀⣤⡄⢸⣿⡇⣼⣿⣿⣧⢸⣿⣿⣼⣿⢸⣿⡇⣿⣿⢀⣿⣿⣿⡆⣿⣿⣿⣿⡇⢻⣿⣿⡟⠀⠀⠿⠏⣽⣿⢸⣿⡏⣿⣿⠸⠿⢹⣿⡇⠿⠏⣿⣿⢀⣀⠀⠘⠻⣿⣿⢸⣿⣏⣛⡛⢸⣿⡏⣿⣷⣾⣿⣿⣿⡇⠀⠀
⠀⠈⠁⢸⣿⡇⣿⣿⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣿⣿⢸⣿⣿⣿⡇⣿⣿⣿⣿⡅⠘⣿⣿⠁⣀⠂⢀⣼⣿⠏⢸⣿⡇⣿⣿⠀⣠⣿⡿⠁⠀⣼⣿⠏⠘⠻⠀⠀⠀⣿⣿⢸⣿⡟⣿⣿⢸⣿⡇⣿⣿⣿⣿⣿⣿⡇⠀⠀
⠀⠀⢀⣼⣿⣧⣿⣿⢻⣿⣾⣿⡟⣿⣿⠸⣿⣧⣿⣿⣸⣿⡟⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⠀⣿⠀⣾⣿⣧⣤⠸⣿⣧⣿⡿⢰⣿⣿⣤⡄⣾⣿⣧⣤⢠⣤⠀⠀⠀⣿⣿⢸⣿⣧⣿⡿⢸⣿⣧⣿⡟⢻⣿⣿⣿⡇⠀⠀
⠀⠀⠈⣉⣉⣬⣭⣭⣬⣍⣉⣉⠁⠉⠉⠀⡈⠉⠉⢁⣉⣉⣁⣉⣉⣍⣭⣭⣭⣥⣦⣭⣭⣴⣿⣮⣭⣭⣭⣽⣶⣬⣉⣩⣤⣉⢉⢉⣉⡁⠉⠉⠉⠉⠈⠉⠀⠀⠀⠉⠉⠀⠈⠉⠉⠀⠀⠉⠉⠉⠀⠀⠉⠉⠉⠀⠀⠀
⠀⠀⢹⣿⢿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠐⣂⠀⣢⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣼⣿⣷⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠿⠯⠽⠻⣻⣿⣿⠩⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠂⠀⠊⠩⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠍⠀⢠⣴⣾⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠍⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡇⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⢿⠎⠀⠀⠀⠀⠈⠙⠿⠿⠿⠛⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣟⠵⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠿⠿⠿⠿⠛⠟⠻⣿⡿⠟⠻⠟⠏⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠗⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠁⢋⢳⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠘⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠂⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠶⣶⠿⢷⠶⠴⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠶⢶⢠⣤⡄⢠⣤⣤⡄⢠⣤⡄⣤⣤⢠⣤⡄⣤⣤⠀⣤⣤⣤⠀⣤⣤⣤⣤⡠⣤⡄⣠⣤⠄⠀⣠⣤⣤⣄⠀⣠⣤⣤⡀⢀⣤⣤⣤⡀⣠⣤⣤⣄⠀⠀⠀⠀⠀⣠⣤⢀⣠⣤⣤⡀⢀⣤⣤⣤⡀⢀⣤⣤⣤⡀⠀⠀
⠀⣤⡄⢸⣿⡇⣼⣿⣿⣧⢸⣿⣿⣼⣿⢸⣿⡇⣿⣿⠀⣿⣿⣿⡆⣿⣿⣿⣿⡇⢻⣿⣿⡟⠀⠀⠿⠏⣽⣿⢸⣿⡏⣿⣿⠸⠿⢹⣿⡇⠿⠏⣿⣿⢀⣀⠀⠘⠻⣿⣿⢸⣿⣏⣛⡛⢸⣿⡏⣿⣷⣾⣿⣿⣿⡇⠀⠀
⠀⠈⠁⢸⣿⡇⣿⣿⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣿⣿⢸⣿⣿⣿⡇⣿⣿⣿⣿⡅⠘⣿⣿⠁⠀⠀⢀⣼⣿⠏⢸⣿⡇⣿⣿⠀⣠⣿⡿⠁⠀⣼⣿⠏⠘⠿⠀⠀⠀⣿⣿⢸⣿⡟⣿⣿⢸⣿⡇⣿⣿⣿⣿⣿⣿⡇⠀⠀
⠀⠀⠀⣼⣿⣧⣿⣿⢻⣿⣾⣿⡟⣿⣿⠸⣿⣧⣿⣿⣸⣿⡟⣿⣷⣿⣿⣿⣿⡇⠀⣿⣿⠀⠀⠀⣾⣿⣧⣤⠸⣿⣧⣿⡿⢰⣿⣿⣤⡄⣾⣿⣧⣤⢠⣤⠀⠀⠀⣿⣿⢸⣿⣧⣿⡿⢸⣿⣧⣿⡟⢻⣿⣿⣿⡇⠀⠀
⠀⠀⠈⣉⢉⣨⣍⣩⣌⣉⡉⠉⠁⠉⠉⠀⠈⠉⠉⠁⠉⣉⣁⣉⣉⣉⣩⣭⣉⣡⣤⣍⣉⣴⣿⣮⣭⣉⣭⣽⣶⣮⣉⣩⣄⠈⡉⢉⡉⠁⠉⠉⠉⠉⠈⠉⠀⠀⠀⠉⠉⠀⠈⠉⠉⠀⠀⠉⠉⠉⠀⠀⠉⠉⠉⠀⠀⠀
⠀⠀⠀⠉⠿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠛⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢆⣤⣶⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⡺⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⢻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠙⠍⠁⠀⠈⠙⠛⠛⠛⠋⠛⠍⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠽⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠽⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⠿⠛⠛⠙⠛⠛⠟⣿⡿⡿⠿⡛⠟⠛⠉⠉⠉⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢩⣗⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⢼⢙⡷⣧⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢆⢒⠌⠉⠉⠋⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠶⣶⠿⢷⠶⠴⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠲⢦⢠⣤⡄⢠⣤⣤⡄⢠⣤⡄⣤⣤⢠⣤⡄⣤⣤⠀⣤⣤⣤⠀⣤⣤⣤⣤⡠⣤⡄⣠⣤⠄⠀⣠⣤⣤⣄⠀⣠⣤⣤⡀⢀⣤⣤⣤⡀⣠⣤⣤⣄⠀⠀⠀⠀⠀⣠⣤⢀⣠⣤⣤⡀⢀⣤⣤⣤⡀⢀⣤⣤⣤⡀⠀⠀
⠀⣤⡄⢸⣿⡇⣼⣿⣿⣧⢸⣿⣿⣼⣿⢸⣿⡇⣿⣿⢀⣿⣿⣿⡆⣿⣿⣿⣿⡇⢻⣿⣿⡟⠀⠀⠿⠏⣽⣿⢸⣿⡏⣿⣿⠸⠿⢹⣿⡇⠿⠏⣿⣿⢀⣀⠀⠘⠻⣿⣿⢸⣿⣏⣛⡛⢸⣿⡏⣿⣷⣾⣿⣿⣿⡇⠀⠀
⠀⠈⠁⢸⣿⡇⣿⣿⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣿⣿⢸⣿⣿⣿⡇⣿⣿⣿⣿⡅⠘⣿⣿⠁⠀⠀⢀⣼⣿⠏⢸⣿⡇⣿⣿⠀⣠⣿⡿⠁⠀⣼⣿⠏⠘⠿⠀⠀⠀⣿⣿⢸⣿⡟⣿⣿⢸⣿⡇⣿⣿⣿⣿⣿⣿⡇⠀⠀
⠀⠀⢀⣼⣿⣧⣿⣿⢻⣿⣾⣿⡟⣿⣿⠸⣿⣧⣿⣿⣸⣿⡟⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⠀⣄⠀⣾⣿⣧⣤⠸⣿⣧⣿⡿⢰⣿⣿⣤⡄⣾⣿⣧⣤⢠⣤⠀⠀⠀⣿⣿⢸⣿⣧⣿⡿⢸⣿⣧⣿⡟⢻⣿⣿⣿⡇⠀⠀
⠀⠀⠀⢉⣉⣨⣉⣩⣈⠉⠉⠉⠁⠉⠉⠀⠈⠉⠉⠁⠉⠉⣁⣉⣉⣉⣉⣉⣉⣁⢀⣍⣩⣼⣿⣾⣭⣉⣭⣽⣶⣮⣉⣩⣄⣈⠉⠉⠉⠁⠉⠉⠉⠉⠈⠉⠀⠀⠀⠉⠉⠀⠈⠉⠉⠀⠀⠉⠉⠉⠀⠀⠉⠉⠉⠀⠀⠀
⠀⠀⠀⠘⢟⠛⡿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀⣠⡤⡂⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⢿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠘⢉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠛⠛⠋⠉⠀⠁⠈⠙⠛⠛⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢀⢈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠿⠿⠿⠛⠛⠛⠿⢿⣿⣿⣿⠿⠿⠿⠛⠛⠋⠋⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢠⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢥⠻⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⠄⡁⠆⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠌⢉⡻⣄⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠣⠂⠀⠋⠀⠀⠀⠀⠀⠀⣰⣶⣾⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢁⠂⠈⠈⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠶⣶⠿⢷⠶⠰⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠲⢦⢠⣤⡄⢠⣤⣤⡄⢠⣤⡄⣤⣤⢠⣤⡄⣤⣤⠀⣤⣤⣤⠀⣤⣤⣤⣤⡠⣤⡄⣠⣤⠄⠀⣠⣤⣤⣄⠀⣠⣤⣤⡀⢀⣤⣤⣤⡀⣠⣤⣤⣄⠀⠀⠀⠀⠀⣠⣤⢀⣠⣤⣤⡀⢀⣤⣤⣤⡀⢀⣤⣤⣤⡀⠀⠀
⠀⣤⡄⢸⣿⡇⣼⣿⣿⣧⢸⣿⣿⣸⣿⢸⣿⡇⣿⣿⠀⣿⣿⣿⡆⣿⣿⣿⣿⡇⢻⣿⣿⡟⠀⠀⠿⠏⣽⣿⢸⣿⡏⣿⣿⠸⠿⢹⣿⡇⠿⠏⣿⣿⢀⣀⠀⠘⠻⣿⣿⢸⣿⣏⣛⡛⢸⣿⡏⣿⣷⣾⣿⣿⣿⡇⠀⠀
⠀⠈⠀⢸⣿⡇⣿⣿⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣿⣿⢸⣿⣿⣿⡇⣿⣿⣿⣿⡅⠘⣿⣿⠁⠀⠀⠀⣼⣿⠏⢸⣿⡇⣿⣿⠀⣠⣿⡿⠁⠀⣼⣿⠏⠘⠿⠀⠀⠀⣿⣿⢸⣿⡟⣿⣿⢸⣿⡇⣿⣿⣿⣿⣿⣿⡇⠀⠀
⠀⠀⢀⣼⣿⣧⣿⣿⢻⣿⣾⣿⡟⣿⣿⠸⣿⣧⣿⣿⣸⣿⡟⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⠀⣄⠀⣾⣿⣧⣤⠸⣿⣧⣿⡿⢰⣿⣿⣤⡄⣾⣿⣧⣤⢠⣤⠀⠀⠀⣿⣿⢸⣿⣧⣿⡿⢸⣿⣧⣿⡟⢻⣿⣿⣿⡇⠀⠀
⠀⠀⠈⠉⠉⣌⡉⠉⠈⠉⠉⠉⠁⠉⠉⠀⠈⠉⠉⠁⠉⠉⠁⠉⣉⣉⣉⢉⢉⢁⣤⣭⣭⣾⣿⣯⣍⣉⣭⣽⣶⣮⣍⣉⣀⡈⠉⠉⠉⠁⠉⠉⠉⠉⠈⠉⠀⠀⠀⠉⠉⠀⠈⠉⠉⠀⠀⠉⠉⠉⠀⠀⠉⠉⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣿⣷⣿⣿⣿⣬⣢⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣶⢒⣤⣄⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠈⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠛⠉⠀⠀⠀⠉⠉⠛⠻⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡛⢛⠛⠋⠻⠿⠿⢿⣿⡿⡿⠟⠿⠟⠛⠛⠙⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠛⠿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠔⡠⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⠀⠀⠙⢞⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠈⠊⠈⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠶⣶⠿⢷⠶⠰⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠲⠆⢠⣤⡄⢠⣤⣤⡄⢠⣤⡄⣤⣤⢠⣤⡄⣤⣤⠀⣤⣤⣤⠀⣤⣤⣤⣤⡠⣤⡄⣠⣤⠄⠀⣠⣤⣤⣄⠀⣠⣤⣤⡀⢀⣤⣤⣤⡀⣠⣤⣤⣄⠀⠀⠀⠀⠀⣠⣤⢀⣠⣤⣤⡀⢀⣤⣤⣤⡀⢀⣤⣤⣤⡀⠀⠀
⠀⣠⡄⢸⣿⡇⣼⣿⣿⣧⢸⣿⣿⣼⣿⢸⣿⡇⣿⣿⠀⣿⣿⣿⡆⣿⣿⣿⣿⡇⢻⣿⣿⡟⠀⠀⠿⠏⣽⣿⢸⣿⡏⣿⣿⠸⠿⢹⣿⡇⠿⠏⣿⣿⢀⣀⠀⠘⠻⣿⣿⢸⣿⣏⣛⡛⢸⣿⡏⣿⣧⣼⣿⣿⣿⡇⠀⠀
⠀⠈⠀⢸⣿⡇⣿⣿⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣿⣿⢸⣿⣿⣿⡇⣿⣿⣿⣿⡅⠘⣿⣿⠁⠀⠀⠀⣼⣿⠏⢸⣿⡇⣿⣿⠀⣠⣿⡿⠁⠀⣼⣿⠏⠘⠻⠀⠀⠀⣿⣿⢸⣿⡟⣿⣿⢸⣿⡇⣿⣿⣿⣿⣿⣿⡇⠀⠀
⠀⠀⠀⣼⣿⣇⣿⣿⢻⣿⣾⣿⡟⣿⣿⠸⣿⣧⣿⣿⣸⣿⡟⣿⣷⣿⣿⣿⣿⡇⠀⣿⣿⠀⠀⠀⣾⣿⣧⣤⠸⣿⣧⣿⡿⢰⣿⣿⣤⡄⣾⣿⣧⣤⢠⣤⠀⠀⠀⣿⣿⢸⣿⣧⣿⡿⢸⣿⣧⣿⡟⢻⣿⣿⣿⡇⠀⠀
⠀⠀⠀⠉⢉⠈⠉⠉⠈⠉⠉⠉⠁⠉⠉⠀⠈⠉⠉⠁⠉⠉⠁⠉⢉⣉⣉⣉⣉⣁⣄⠉⠉⢀⣴⣆⣉⣭⣭⣽⣶⣮⣍⣉⣀⠈⠉⠉⠉⠁⠉⠉⠉⠉⠈⠉⠀⠀⠀⠉⠉⠀⠈⠉⠉⠀⠀⠉⠉⠉⠀⠀⠉⠉⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣄⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣾⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢹⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠹⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠍⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢾⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠈⠟⠍⠙⠻⠿⠛⠛⠛⠛⠋⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠶⣶⠿⢷⠶⠰⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠆⢠⣤⡄⢠⣤⣤⡄⢠⣤⡄⢠⣤⢠⣤⡄⣤⣤⠀⣤⣤⣤⠀⣤⣤⣤⣤⡠⣤⡄⣠⣤⠄⠀⣠⣤⣤⣄⠀⣠⣤⣤⡀⢀⣤⣤⣤⡀⣠⣤⣤⣄⠀⠀⠀⠀⠀⣠⣤⢀⣠⣤⣤⡀⢀⣤⣤⣤⡀⢀⣤⣤⣤⡀⠀⠀
⠀⣤⡀⢸⣿⡇⣼⣿⣿⣧⢸⣿⣿⣸⣿⢸⣿⡇⣿⣿⠀⣿⣿⣿⡆⣿⣿⣿⣿⡇⢻⣿⣿⡟⠀⠀⠿⠏⣽⣿⢸⣿⡏⣿⣿⠸⠿⢹⣿⡇⠿⠏⣽⣿⢀⣀⠀⠘⠻⣿⣿⢸⣿⣏⣛⡛⢸⣿⡏⣿⣧⣼⣿⣿⣿⡇⠀⠀
⠀⠈⠀⢸⣿⡇⣿⣿⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣿⣿⢸⣿⣿⣿⡇⣿⣿⣿⣿⡅⠘⣿⣿⠁⠀⠀⠀⣼⣿⠋⢸⣿⡇⣿⣿⠀⣠⣿⡿⠁⠀⣼⣿⠏⠘⠻⠀⠀⠀⣿⣿⢸⣿⡟⣿⣿⢸⣿⡇⣿⣿⣿⣿⣿⣿⡇⠀⠀
⠀⠀⠀⣼⣿⣇⣿⣿⢻⣿⣼⣿⡟⣿⣿⠸⣿⣧⣿⡿⣸⣿⡟⣿⣷⣿⣿⣿⣿⡇⠀⣿⣿⠀⠀⠀⣾⣿⣧⣤⠸⣿⣧⣿⡿⢰⣿⣯⣤⡄⣾⣿⣧⣤⢠⣤⠀⠀⠀⣿⣿⠸⣿⣧⣿⡿⢸⣿⣧⣿⡟⢹⣿⣿⣿⡇⠀⠀
⠀⠀⠀⠉⡉⠈⠉⠉⠈⠉⠉⠉⠁⠉⠉⠀⠈⠉⠉⠁⠉⠉⠁⠉⢉⣉⣉⣉⣉⣁⡀⠉⠉⢀⣼⣄⣉⣩⣭⣽⣶⣮⣉⣉⣀⠉⠉⠉⠉⢁⠉⠉⠉⠉⠈⠉⠀⠀⠀⠉⠉⠀⠈⠉⠉⠀⠀⠉⠉⠉⠀⠀⠉⠉⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⣿⣿⡿⠃⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣆⣀⣿⣧⣶⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠠⡨⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⢸⡚⢀⡯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⡀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⡾⢹⡑⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠻⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡧⠀⠀⠀⠀⠈⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠋⠛⠛⠛⠛⠛⠋⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠶⣶⠿⢷⠶⠰⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠆⢠⣤⡄⢠⣤⣤⡄⢠⣤⡄⢠⣤⢠⣤⡄⣤⣤⠀⣤⣤⣤⠀⣤⣤⣤⣄⡠⣤⡄⣠⣤⠀⠀⣠⣤⣤⣄⠀⣠⣤⣤⡀⢀⣤⣤⣤⡀⣠⣤⣤⣄⠀⠀⠀⠀⠀⣠⣤⠀⣠⣤⣤⡀⢀⣤⣤⣤⡀⢀⣤⣤⣤⡀⠀⠀
⠀⣠⡀⢸⣿⡇⣸⣿⣿⣧⢸⣿⣿⣸⣿⢸⣿⡇⣿⣿⠀⣿⣿⣿⡆⣿⣿⣿⣿⡇⢻⣷⣿⡟⠀⠀⠿⠏⣽⣿⢸⣿⡏⣿⣿⠸⠿⢹⣿⡇⠿⠏⣽⣿⢀⣀⠀⠐⠻⣿⣿⢸⣿⣏⣛⡛⢸⣿⡏⣿⣧⢸⣿⢹⣿⡇⠀⠀
⠀⠈⠀⢸⣿⡇⣿⣿⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣿⣿⢸⣿⣟⣿⡇⣿⣿⣿⣿⡅⠘⣿⣿⠁⠀⠀⠀⣼⣿⠋⢸⣿⡇⣿⣿⠀⣠⣿⡟⠁⠀⣼⣿⠏⠘⠛⠀⠀⠀⣿⣿⢸⣿⡏⣿⣿⢸⣿⡇⣿⣿⢽⣿⢸⣿⡇⠀⠀
⠀⠀⠀⣼⣿⣇⣿⣿⢻⣿⣼⣿⡏⣿⣿⠸⣿⣧⣿⡿⢸⣿⡟⣿⣷⣿⣿⣿⣿⡇⠀⣿⣿⠀⠀⠀⣾⣿⣧⣤⠸⣿⣧⣿⡿⢰⣿⣯⣤⡄⣾⣿⣧⣤⢠⣤⠀⠀⠀⣿⣿⠸⣿⣧⣿⡿⢸⣿⣧⣿⡏⢸⣿⣼⣿⡇⠀⠀
⠀⠀⠀⠉⠉⠈⠉⠉⠈⠉⠉⠉⠁⠉⠉⠀⠈⠉⠉⠀⠉⠉⠁⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⢀⣄⢀⣉⣉⣩⣽⣶⣮⣉⣉⣠⣌⣉⣉⣉⣡⣍⣉⠉⠉⠈⠉⠀⠀⠀⠉⠉⠀⠈⠉⠉⠀⠀⠉⠉⠉⠀⠀⠉⠉⠉⠠⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠐⣠⡄⠀⡀⠀⢠⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡆⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇⠀⠁⠠⠅⠘⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⠀⠀⠀⠀⠈⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⣤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠈⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠯⠀⠀⠨⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣧⡦⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠫⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠉⠿⣿⣿⣿⣿⣿⣿⣔⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠈⠉⠉⠉⠁⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠈⠀⠀⠀⠀⠀⠀⠋⠂⠉⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣦⠀⠀⠀⠀⠀⡀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠿⢶⠆⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠄⢠⣤⡄⢠⣤⣤⡄⢠⣤⡄⢠⣤⢠⣤⡄⣤⣤⠀⣤⣤⣤⠀⣤⣤⣤⣄⡠⣤⡄⣠⣤⠀⠀⣠⣤⣤⣀⠀⣠⣤⣤⡀⢀⣤⣤⣄⡀⣠⣤⣤⣄⠀⠀⠀⠀⠀⣠⣤⠀⣠⣤⣤⡀⢀⣤⣤⣤⡀⢀⣤⣤⣄⡀⠀⠀
⠀⠠⠀⢸⣿⡇⣸⣿⣿⣧⢸⣿⣿⣸⣿⢸⣿⡇⣿⣿⠀⣿⣿⣿⡆⣿⣿⣹⣿⡇⢻⣷⣿⡟⠀⠀⠿⠏⣽⣿⢸⣿⡇⣿⣿⠸⠿⢹⣿⡇⠿⠏⣽⣿⢀⣀⠀⠐⠻⣿⣿⢸⣿⣇⣛⡛⢸⣿⡏⣿⣧⢸⣿⢹⣿⡇⠀⠀
⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣿⣿⢰⣿⣟⣿⡇⣿⣿⣿⣿⡅⠘⣿⣿⠁⠀⠀⠀⣼⣿⠋⢸⣿⡇⣿⣿⠀⢠⣿⡟⠁⠀⣼⣿⠏⠘⠛⠀⠀⠀⣿⣿⢸⣿⡏⣿⣷⢸⣿⡇⣿⣿⢸⣿⢸⣿⡇⠀⠀
⠀⠀⠀⣼⣿⣇⣿⣿⢻⣿⣼⣿⡏⣿⣿⠸⣿⣧⣿⡿⢸⣿⡟⣿⣷⣿⣿⣿⣿⡇⠀⣿⣿⠀⠀⠀⣾⣿⣧⣤⠸⣿⣇⣿⡿⢰⣿⣯⣤⡄⣾⣿⣧⣤⢠⣤⠀⠀⠀⣿⣿⠸⣿⣧⣿⡿⢸⣿⣧⣿⡏⢸⣿⣼⣿⡇⠀⠀
⠀⠀⠀⠉⠉⠈⠉⠉⠈⠉⠉⠉⠁⠉⠉⠀⠈⠉⠉⠀⠉⠉⠁⠉⠉⠉⠉⠉⢉⡁⡀⠉⣉⣠⣦⠀⣉⣉⣭⣭⣶⣌⣉⣉⣠⣌⣉⣉⣉⣥⣍⣉⣉⡉⠈⠉⠐⠂⠀⠉⠉⠀⠈⠉⠉⠀⠀⠉⠉⠉⠀⠀⠉⠉⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢵⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣸⣹⣄⢀⠀⢄⡖⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣼⣿⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⡿⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣤⡔⠚⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢉⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠩⠨⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣦⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠅⢹⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠏⠉⠀⠀⠀⠀⠀⠈⠫⠙⡿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⠿⣿⣿⣿⣿⣿⡿⣿⢿⠿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⣄⠀⠀⠀⠀⣄⡀⢴⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠏⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠐⠁⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠄⢠⣤⡄⢠⣤⣤⡄⢠⣤⡄⢠⣤⢠⣤⡄⣤⣤⠀⣤⣤⣤⠀⣤⣤⣤⣄⡠⣤⡄⣠⣤⠀⠀⣠⣤⣤⣀⠀⣠⣤⣤⡀⢀⣤⣤⣄⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⣠⣤⠀⣠⣤⣤⡀⢀⣤⣤⣤⡀⢀⣤⣤⣄⡀⠀⠀
⠀⢠⠀⢸⣿⡇⣸⣿⣿⣧⢸⣿⣿⣸⣿⢸⣿⡇⣿⣿⠀⣿⣿⣿⡆⣿⣿⣹⣿⡇⢻⣷⣿⡟⠀⠀⠿⠏⣽⣿⢸⣿⡇⣿⣿⠸⠿⢹⣿⡇⠿⠏⣽⣿⢀⣀⠀⠐⠻⣿⣿⢸⣿⣇⣛⡛⢸⣿⡏⣿⣧⢸⣿⢹⣿⡇⠀⠀
⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣿⣿⢰⣿⣟⣿⡇⣿⣿⣿⣿⡅⠘⣿⣿⠁⠀⠀⠀⣼⣿⠋⢸⣿⡇⣿⣿⠀⢠⣿⡟⠁⠀⣼⣿⠏⠘⠛⠀⠀⠀⣿⣿⢸⣿⡏⣿⣷⢸⣿⡇⣿⣿⢸⣿⢸⣿⡇⠀⠀
⠀⠀⠀⣼⣿⣇⣿⣿⢻⣿⣼⣿⡏⣿⣿⠸⣿⣧⣿⡿⢸⣿⡟⣿⣷⣿⣿⣿⣿⡇⠀⣿⣿⠀⠀⠀⣾⣿⣧⣤⠸⣿⣇⣿⡿⢰⣿⣯⣤⡄⣾⣿⣧⣤⢠⣤⠀⠀⠀⣿⣿⠸⣿⣇⣿⡿⢸⣿⣧⣿⡏⢸⣿⣼⣿⡇⠀⠀
⠀⠀⠀⠉⠉⠈⠉⠁⠈⠉⠉⠉⠁⠉⠉⠀⠈⠉⠉⠀⠉⠉⠁⠉⠉⠉⠉⠉⠉⢁⢀⣉⣉⣤⡄⢀⣉⣭⣭⣭⣦⣌⣉⣉⣠⣌⣉⣉⣉⣥⣍⣉⣉⠉⠈⢉⠐⠂⠀⠉⠉⠀⠈⠉⠉⠀⠀⠉⠉⠉⠀⠀⠉⠉⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣸⣻⣆⠀⠀⣠⡆⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣽⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣦⣽⣿⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⡿⠿⢿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣷⣦⣄⠚⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⢈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⢉⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠅⢹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠍⠉⠀⠀⠀⠀⠀⠉⠉⠉⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⢿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣦⡀⠀⠀⠀⠀⡀⠀⣄⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠏⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠊⠁⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠄⢠⣤⡄⢠⣤⣤⡄⢠⣤⡄⢠⣤⢠⣤⡄⣤⣤⠀⣤⣤⣤⠀⣤⣤⣤⣄⡠⣤⡄⣠⣤⠀⠀⣠⣤⣤⣀⠀⣠⣤⣤⡀⢀⣤⣤⣄⡀⣠⣤⣤⣄⠀⠀⠀⠀⠀⣠⣤⠀⣠⣤⣤⡀⢀⣤⣤⣤⡀⢀⣤⣤⣄⡀⠀⠀
⠀⠠⠀⢸⣿⡇⣸⣿⣿⣧⢸⣿⣿⣸⣿⢸⣿⡇⣿⣿⠀⣿⣿⣿⡆⣿⣿⣹⣿⡇⢻⣷⣿⡟⠀⠀⠿⠏⣽⣿⢸⣿⡇⣿⣿⠸⠿⢹⣿⡇⠿⠏⣽⣿⢀⣀⠀⠐⠻⣿⣿⢸⣿⣇⣛⡛⢸⣿⡏⣿⣧⢸⣿⢹⣿⡇⠀⠀
⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣿⣿⢰⣿⣟⣿⡇⣿⣿⢿⣿⡅⠘⣿⣿⠁⠀⠀⠀⣼⣿⠋⢸⣿⡇⣿⣿⠀⢠⣿⡟⠁⠀⣼⣿⠏⠘⠛⠀⠀⠀⣿⣿⢸⣿⡏⣿⣷⢸⣿⡇⣿⣿⢸⣿⢸⣿⡇⠀⠀
⠀⠀⠀⣼⣿⣇⣿⣿⢻⣿⣼⣿⡏⣿⣿⠸⣿⣧⣿⡿⢸⣿⡟⣿⣷⣿⣿⣿⣿⡇⠀⣿⣿⠀⠀⠀⣾⣿⣧⣤⠸⣿⣇⣿⡿⢰⣿⣯⣤⡄⣾⣿⣧⣤⢠⣤⠀⠀⠀⣿⣿⠸⣿⣇⣿⡿⢸⣿⣧⣿⡏⢸⣿⣼⣿⡇⠀⠀
⠀⠀⠀⠉⠉⠈⠉⠉⠈⠉⠉⠉⠁⠉⠉⠀⠈⠉⠉⠀⠉⠉⠁⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⢀⣠⣀⣉⣭⣭⣭⣦⣌⣉⣉⣠⣌⣉⣉⣉⣥⣍⣉⣉⠉⠈⢉⠐⠀⠀⠉⠉⠀⠈⠉⠉⠀⠀⠉⠉⠉⠀⠀⠉⠉⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣜⣹⣆⠀⠀⣤⡆⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠴⣲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣦⣾⣿⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⡿⠟⠛⢿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣷⣤⣄⠘⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢉⠀⠄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⣿⣿⣿⣿⣯⢿⣿⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠍⠀⠀⠀⠈⠉⠉⠉⠉⠉⠽⠟⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⢝⡿⢿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⣠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠏⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠋⠉⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠂⠀⢠⣤⡄⢠⣤⣤⡄⢠⣤⡄⢠⣤⢠⣤⡄⣤⣤⠀⣤⣤⣤⠀⣤⣤⣤⣄⡠⣤⡄⣠⣤⠀⠀⣠⣤⣤⣀⠀⣠⣤⣤⡀⢀⣤⣤⣄⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⣠⣤⠀⣠⣤⣤⡀⢀⣤⣤⣤⡀⢀⣤⣤⣄⡀⠀⠀
⠀⠀⠀⢸⣿⡇⣸⣿⣿⣧⢸⣿⣿⣸⣿⢸⣿⡇⣿⣿⠀⣿⣿⣿⡆⣿⣿⣹⣿⡇⢻⣷⣿⡟⠀⠀⠿⠏⣽⣿⢸⣿⡇⣿⣿⠸⠿⢹⣿⡇⠿⠏⣽⣿⢀⣀⠀⠐⠻⣿⣿⢸⣿⣇⣛⡛⢸⣿⡏⣿⣧⢸⣿⢹⣿⡇⠀⠀
⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣿⣿⢰⣿⣟⣿⡇⣿⣿⢿⣿⡅⠘⣿⣿⠁⠀⠀⠀⣼⣿⠋⢸⣿⡇⣿⣿⠀⢠⣿⡟⠁⠀⣼⣿⠏⠘⠛⠀⠀⠀⣿⣿⢸⣿⡏⣿⣷⢸⣿⡇⣿⣿⢸⣿⢸⣿⡇⠀⠀
⠀⠀⠀⣼⣿⣇⣿⣿⢻⣿⣼⣿⡏⣿⣿⠸⣿⣧⣿⡿⢸⣿⡟⣿⣷⣿⣿⣾⣿⡇⠀⣿⣿⠀⠀⠀⣾⣿⣧⣤⠸⣿⣇⣿⡿⢰⣿⣯⣤⡄⣾⣿⣧⣤⢠⣤⠀⠀⠀⣿⣿⠸⣿⣇⣿⡿⢸⣿⣧⣿⡏⢸⣿⣼⣿⡇⠀⠀
⠀⠀⠀⠉⠉⠈⠉⠁⠈⠉⠉⠉⠁⠉⠉⠀⠈⠉⠉⠀⠉⠉⠁⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠀⣸⣤⣩⣭⣭⣭⣦⣌⣉⣉⣠⣌⣉⣉⣉⣥⣍⣉⣉⠉⠈⢉⠀⠀⠀⠉⠉⠀⠈⠉⠉⠀⠀⠉⠉⠉⠀⠀⠉⠉⠉⢀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢀⣰⣀⠀⠀⢠⡄⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣧⡤⣠⢤⣿⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⢷⡞⠯⠘⠿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣶⣤⡄⠘⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠆⢈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠕⢁⠄⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣓⣾⣿⣿⣦⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠿⠟⠩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠟⠻⢿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠯⠄⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⢿⠿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠏⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠍⠃⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣤⡄⢠⣤⣤⡄⢠⣤⡄⢠⣤⢠⣤⡄⣤⣤⠀⣤⣤⣤⠀⣤⣤⣤⣄⡠⣤⡄⣠⣤⠀⠀⣠⣤⣤⣀⠀⣠⣤⣤⡀⢀⣤⣤⣄⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⣠⣤⠀⣠⣤⣤⡀⢀⣤⣤⣤⡀⢀⣤⣤⣄⡀⠀⠀
⠀⠀⠀⢸⣿⡇⣸⣿⣿⣧⢸⣿⣿⣸⣿⢸⣿⡇⣿⣿⠀⣿⣿⣿⡆⣿⣿⣹⣿⡇⢻⣷⣿⡟⠀⠀⠿⠏⣽⣿⢸⣿⡇⣿⣿⠸⠿⢹⣿⡇⠿⠏⣽⣿⢀⣀⠀⠐⠻⣿⣿⢸⣿⣇⣛⡛⢸⣿⡏⣿⣧⢸⣿⢹⣿⡇⠀⠀
⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣿⣿⢰⣿⣟⣿⡇⣿⣿⢿⣿⡅⠘⣿⣿⠁⠀⠀⠀⣼⣿⠋⢸⣿⡇⣿⣿⠀⢠⣿⡟⠁⠀⣼⣿⠏⠘⠛⠀⠀⠀⣿⣿⢸⣿⡏⣿⣷⢸⣿⡇⣿⣿⢸⣿⢸⣿⡇⠀⠀
⠀⠀⠀⣼⣿⣇⣿⣿⢻⣿⣼⣿⡏⣿⣿⠸⣿⣧⣿⡿⢸⣿⡟⣿⣷⣿⣿⣿⣿⡇⠀⣿⣿⠀⠀⠀⣾⣿⣧⣤⠸⣿⣇⣿⡿⢰⣿⣯⣤⡄⣾⣿⣧⣤⢠⣤⠀⠀⠀⣿⣿⠸⣿⣇⣿⡿⢸⣿⣧⣿⡏⢸⣿⣼⣿⡇⠀⠀
⠀⠀⠀⠉⠉⠈⠉⠁⠈⠉⠉⠉⠁⠉⠉⠀⠈⠉⠉⠀⠉⠉⠁⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠀⣾⣦⣩⣭⣭⣭⣦⣌⣉⣉⣠⣬⣉⣉⣉⣥⣍⣉⣉⠉⠈⢉⠀⠀⠀⠉⠉⠀⠈⠉⠉⠀⠀⠉⠉⠉⠀⠀⢉⠉⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⢀⣱⣀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣙⣿⣿⠤⣠⢀⣽⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⢷⡿⠟⠛⠟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣦⣤⣄⠈⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆⢌⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠕⢀⠄⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣾⣿⣿⣦⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡾⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠟⠻⢿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠄⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠲⣶⠏⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠒⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠄⢠⣤⡄⢠⣤⣤⡄⢠⣤⡄⢠⣤⢠⣤⡄⣤⣤⠀⣤⣤⣤⠀⣤⣤⣤⣄⡠⣤⡄⣠⣤⠀⠀⣠⣤⣤⣀⠀⣠⣤⣤⡀⢀⣤⣤⣄⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⣠⣤⠀⣠⣤⣤⡀⢀⣤⣤⣤⡀⢀⣤⣤⣄⡀⠀⠀
⠀⠀⠀⢸⣿⡇⣸⣿⣿⣧⢸⣿⣿⣸⣿⢸⣿⡇⣿⣿⠀⣿⣿⣿⡆⣿⣿⣹⣿⡇⢻⣷⣿⡟⠀⠀⠿⠏⣽⣿⢸⣿⡇⣿⣿⠸⠿⢹⣿⡇⠿⠏⣽⣿⢀⣀⠀⠐⠻⣿⣿⢸⣿⣇⣛⡛⢸⣿⡏⣿⣧⢸⣿⢹⣿⡇⠀⠀
⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣿⣿⢰⣿⣟⣿⡇⣿⣿⢿⣿⡅⠘⣿⣿⠁⠀⠀⠀⣼⣿⠋⢸⣿⡇⣿⣿⠀⢠⣿⡟⠁⠀⣼⣿⠏⠘⠛⠀⠀⠀⣿⣿⢸⣿⡏⣿⣷⢸⣿⡇⣿⣿⢸⣿⢸⣿⡇⠀⠀
⠀⠀⠀⣼⣿⣇⣿⣿⢻⣿⣼⣿⡏⣿⣿⠸⣿⣧⣿⡿⢸⣿⡟⣿⣷⣿⣿⣾⣿⡇⠀⣿⣿⠀⠀⠀⣾⣿⣧⣤⠸⣿⣇⣿⡿⢰⣿⣯⣤⡄⣾⣿⣧⣤⢠⣤⠀⠀⠀⣿⣿⠸⣿⣇⣿⡿⢸⣿⣧⣿⡏⢸⣿⣼⣿⡇⠀⠀
⠀⠀⠀⠉⠉⠈⠉⠁⠈⠉⠉⠉⠁⠉⠉⠀⠈⠉⠉⠀⠉⠉⠁⠉⠉⠉⠉⠉⢉⡁⣀⣉⣉⣤⢰⣄⣭⣭⣭⣭⣦⣌⣉⣉⣠⣌⣉⣉⣉⣥⣍⣉⣉⠉⠈⢉⠀⠀⠀⠉⠉⠀⠈⠉⠉⠀⠀⠉⠉⠉⠀⠀⢉⠉⠉⠀⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⣀⣰⣀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣽⣿⡤⣠⣄⣷⣶⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣔⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠓⡟⠿⠙⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣶⣤⡤⠘⠁⢠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠇⠄⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠕⢀⠀⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠩⢻⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡆⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠲⣶⠏⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⠆⠖⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠂⠄⢠⣤⡄⢠⣤⣤⡄⢠⣤⡄⢠⣤⢠⣤⡄⣤⣤⠀⣤⣤⣤⠀⣤⣤⣤⣄⡠⣤⡄⣠⣤⠀⠀⣠⣤⣤⣀⠀⣠⣤⣤⡀⢀⣤⣤⣄⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⣠⣤⠀⣠⣤⣤⡀⢀⣤⣤⣤⡀⢀⣤⣤⣄⡀⠀⠀
⠀⠀⠀⢸⣿⡇⣸⣿⣿⣧⢸⣿⣿⣸⣿⢸⣿⡇⣿⣿⠀⣿⣿⣿⡆⣿⣿⣹⣿⡇⢻⣷⣿⡟⠀⠀⠿⠏⣽⣿⢸⣿⡇⣿⣿⠸⠿⢹⣿⡇⠿⠏⣽⣿⢀⣀⠀⠐⠻⣿⣿⢸⣿⣇⣛⡛⢸⣿⡏⣿⣧⢸⣿⢹⣿⡇⠀⠀
⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⢸⣿⣿⣿⣿⢸⣿⡇⣿⣿⢰⣿⣟⣿⡇⣿⣿⣿⣿⡅⠘⣿⣿⠁⠀⠀⠀⣼⣿⠋⢸⣿⡇⣿⣿⠀⢠⣿⡟⠁⠀⣼⣿⠏⠘⠛⠀⠀⠀⣿⣿⢸⣿⡏⣿⣷⢸⣿⡇⣿⣿⢸⣿⢸⣿⡇⠀⠀
⠀⠀⠀⣼⣿⣇⣿⣿⢻⣿⣼⣿⡏⣿⣿⠸⣿⣧⣿⡿⢸⣿⡟⣿⣷⣿⣿⣿⣿⡇⠀⣿⣿⠀⠀⠀⣾⣿⣧⣤⠸⣿⣇⣿⡿⢰⣿⣯⣤⡄⣾⣿⣧⣤⢠⣤⠀⠀⠀⣿⣿⠸⣿⣇⣿⡿⢸⣿⣧⣿⡏⢸⣿⣼⣿⡇⠀⠀
⠀⠀⠀⠉⠉⠈⠉⠁⠈⠉⠉⠉⠁⠉⠉⠀⠈⠉⠉⠀⠉⠉⠁⠉⠉⠉⠉⢉⣉⣁⣄⣉⣉⣤⢶⣄⣩⣭⣭⣭⣦⣌⣉⣉⣤⣌⣉⣉⣉⣥⣉⣉⣉⠉⠈⢉⡀⠀⠀⠉⠉⠀⠈⠉⠉⠀⠀⠉⠉⠉⠀⠀⠉⠉⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⣺⣂⠀⠀⢰⡏⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢙⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⠴⣦⣤⣿⣷⣲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣷⡾⠿⠛⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣷⣦⡦⠘⠁⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⠄⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⡐⠥⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠩⢹⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠲⣶⠏⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⡒⠤⠻⡿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⣹⣧⣌⣵⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠲⣾⣷⣶⠶⢄⣰⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣀⣀⣠⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣤⡄⠠⠀⠀⢐⠻⠟⠿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣧⣀⡀⠀⠀⠀⡄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⡿⠋⠋⠁⢀⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⣐⣹⣿⣿⣷⣶⣤⣤⣤⣦⣦⣀⠀⡀⣬⣢⣿⢰⠇⠋⡝⠈⠀⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣬⡿⠃⠄⠀⠀⠀⣸⡿⠋⠀⠀⣴⣮⠇⠃⢐⣦⣤⣴⣶⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡺⣿⣽⣿⣗⡋⠴⢀⠀⡀⠀⣀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⢽⡿⡋⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⣿⠌⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣜⣿⣯⢝⣿⣾⡄⠁⠀⠅⠀
⠀⠀⠀⠀⠀⠀⠠⡈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⢿⠋⡀⠀⠓⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢯⢧⠀⣠⠀⢠⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣯⠙⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⡗⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠏⠉⠉⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⢃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡆⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣄⣦⣥⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣵⡿⣷⣘⣾⠆⣀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣏⣿⣿⣿⣿⣿⡿⠃⠀⡁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣻⠿⣟⣟⣿⣟⠛⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠙⠫⣟⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠪⢙⠃⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣲⣶⣖⣆⣴⣶⣂⣾⣞⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠶⣶⠿⢷⠶⠴⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠾⢭⢶⣿⣿⣿⣿⣿⣿⣿⡿⢙⠂⡙⠛⢫⣿⣿⣿⣟⡟⠝⣙⣛⢿⢿⣿⣿⣿⣿⠿⠿⠿⡿⣿⣿⣶⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠲⢾⣶⣶⠾⠄⠠⣦⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠷⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣤⡄⠠⠀⠄⠀⠻⠻⢿⣿⢿⣿⣿⣿⣿⡻⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⡄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⣿⠿⠟⠋⠀⠀⣠⡤⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⣹⢿⣿⣷⣤⣤⣦⣠⣤⣠⣤⡀⣐⣨⣕⣿⠰⠗⢊⡍⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⡷⠃⠔⠂⠀⠀⢾⠟⠁⠀⢀⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⣰⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡺⣿⣺⣾⣗⣹⠐⠂⠀⠀⠀⣀⠀
⠀⠀⠀⠀⠀⠀⠀⣰⡿⡛⠋⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠙⣺⣷⣻⢺⢶⡄⡀⠀⠄⠀
⠀⠀⠀⠀⠀⠀⠀⡾⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠟⠋⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠻⣏⠉⡄⠀⠛⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⠦⠀⣀⠀⠠⠄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⠙⠁⠀⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠯⠩⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢚⣍⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡮⢥⢀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣿⣿⣿⣲⣿⢔⡀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣿⣿⣿⣿⡿⠎⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣷⣤⣫⡻⠙⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠈⠸⠻⠟⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡊⣙⠇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣶⣼⣛⣦⣄⣬⣹⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⡔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⡟⢹⣿⣯⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠶⣶⠿⢷⠶⠴⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣟⠏⠍⠁⠈⠘⠭⡇⠋⣛⣿⣿⣿⣿⣿⣿⣿⡿⠫⠃⠑⣋⣻⣿⣿⣿⣿⣿⢿⠹⠛⢻⢿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⡀⠀⠀⠀⣐⣿⣅⣩⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠲⢾⣶⣶⡾⢄⣨⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣤⡄⠠⠀⠄⠔⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣦⣅⠀⠀⠀⢠⣄⡤⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⢠⡀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⣿⠏⠁⠀⢀⣦⣴⠀⠀⠀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⠈⣀⣿⣿⣿⣷⣶⣴⣦⣴⣦⣤⢤⣀⣐⣬⣓⣿⢰⡇⣃⡌⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣎⣿⡟⠂⠄⠀⠀⢸⣿⠟⠀⠀⢠⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⢿⣼⡿⣓⣻⡘⢂⠀⡀⠀⣀⠀
⠀⠀⠀⠀⠀⠀⠀⣔⣿⠟⠁⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡲⣿⠞⡿⢷⡄⠀⠀⠆⠀
⠀⠀⠀⠀⠀⠀⠀⣥⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⡂⠀⠛⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡅⠀⣠⠀⠠⠄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⢝⠙⠡⠀⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⢷⡄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⠛⠔⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣩⠦⠈⠊⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣾⣷⣽⣷⣶⡁⠀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣿⣿⣿⣿⣿⣿⠟⠌⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣝⣻⣿⠟⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠛⢿⠯⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠗⢛⠏⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣷⣿⣤⣡⣲⣗⣧⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠄⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠶⣶⠿⢷⠶⠴⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠯⡿⢿⣿⡿⠏⡻⡿⣟⣿⣿⣿⣿⣿⣿⣿⡯⠓⠑⠐⠚⡿⣿⣿⣿⣿⣿⡿⠷⠿⠿⢿⣿⢿⣿⣿⣿⣿⣿⡿⣿⣿⣗⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠆⠀⠠⠲⠄⠀⠀⠀⠀⠘⠛⠛⠛⠛⠓⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠿⣿⡿⢷⣿⢻⣦⣿⡟⣿⡆⣿⣿⡟⣿⣼⡟⠀⣼⡟⢸⣿⠿⣿⡿⠇⣿⣿⣷⠀⠀⠀⣾⣿⣷⡼⣿⡿⢿⣿⢸⣿⠻⣿⠀⠀⠀⣾⡟⣿⣼⣿⢸⣿⢸⣿⠗⣿⡿⣷⡀⢠⣶⣿⠛⢻⣿⣾⡟⣷⣰⣿⢻⣧⠀⠀
⠀⠀⠀⣿⡇⢸⣿⢸⣿⣿⡇⣿⡇⣿⣿⣷⢹⣿⠃⠀⣿⡇⢸⣿⠗⣿⡿⠇⣿⣿⣿⠀⠀⠀⢿⣿⣿⡇⣿⡇⢸⣿⢿⣿⠀⣿⠘⠃⠀⣿⡇⣿⡏⣿⣿⡿⢸⣿⠇⣿⡿⣿⡀⠀⢸⣿⠀⣿⡇⣿⡇⣿⣿⣿⢸⣿⠀⠀
⠀⠀⠀⠿⠇⠘⠿⠾⠏⠿⠷⠿⠻⠿⠹⠿⠸⠿⠀⠀⠻⠷⠸⠿⠀⠿⠷⠆⠿⠿⠿⠰⠆⠀⠻⠿⠿⠁⠿⢇⡸⠿⠼⠿⠴⠿⠰⠆⣀⠻⢷⠿⠃⠿⠿⠇⠸⠿⠶⠿⠇⠿⠇⡀⠸⠿⠸⠿⠁⠻⠷⠿⠙⠿⠾⠟⠠⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣾⣶⣶⣶⣶⡖⠀⠀⠀⣄⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡶⢿⣿⣉⣖⢘⣀⡀⠰⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣻⣿⠟⡏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠊⠇⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠉⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣒⠛⠀⠸⣿⣿⣿⣿⣿⡟⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠍⠀⠀⠀⠀⠉⠩⠋⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⢿⢿⣿⣿⣿⣿⣿⡿⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠿⢷⠖⠰⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣡⣠⣶⣿⣾⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠆⠀⠠⠲⠄⠀⠀⠀⠀⠛⠛⠛⠛⠛⠊⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡀⠿⣿⡿⢷⣿⢻⣦⣿⡟⣿⡆⣿⣿⡏⣿⣼⡟⠀⣼⡟⢸⣿⠟⣿⡿⠇⣿⣿⣷⠀⠀⠀⣾⣿⣷⡼⣿⡿⢿⣿⢸⣿⠻⣿⠀⠀⠀⣾⡟⣿⣼⣿⢸⣿⢸⣿⠗⣿⡿⣷⡀⢠⣶⣿⠛⢻⣿⣾⡟⣷⣠⣿⢻⣧⠀⠀
⠀⠀⠀⣿⡇⢸⣿⢸⣿⣿⡇⣿⡇⣿⣿⣷⢹⣿⠃⠀⣿⡇⢸⣿⠗⣿⡿⠇⣿⣿⣿⠀⠀⠀⢿⣿⣿⡇⣿⡇⢸⣿⢿⣿⠀⣿⠘⠃⠀⣿⡇⣿⡏⣿⣿⡿⢸⣿⠇⣿⡿⣿⡀⠀⢸⣿⠀⣿⡇⣿⡇⣿⣿⣿⢸⣿⠀⠀
⠀⠀⠀⠿⠇⠘⠿⠾⠏⠿⠷⠿⠻⠿⠹⠿⠸⠿⠀⠀⠻⠷⠸⠿⠀⠿⠷⠆⠿⠿⠿⠰⠆⠀⠿⠿⠿⠁⠿⢇⡸⠿⠼⠿⠴⠿⠰⠆⣀⠻⢷⠿⠃⠿⠿⠇⠸⠿⠶⠿⠇⠿⠇⡀⠸⠿⠸⠿⠁⠻⠷⠿⠙⠿⠾⠟⠠⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣾⣶⣿⣿⣷⣶⣶⣾⣶⣴⣦⣴⣖⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣖⢿⣿⣌⣒⣘⣀⠂⠰⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡁⠒⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣻⣿⢿⠧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣏⠂⠇⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⠼⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣕⠻⠴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠉⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⢛⣿⣿⣿⣿⣿⣿⣿⣋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢄⠑⠂⠀⢨⣿⣿⣿⣿⣿⣿⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠋⠉⠝⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠡⠙⠛⣿⣟⠟⠿⡿⢿⣿⣿⣿⣿⣿⡿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠿⢶⠆⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣁⣠⣶⣿⣾⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡿⢿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠆⠀⠠⠲⠄⠀⠀⠀⠀⠉⠛⠛⠛⠃⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡀⠿⣿⡿⢷⣿⢻⣦⣿⡟⣿⡆⣿⣿⡏⣿⣼⡟⠀⣼⡟⢸⣿⠟⣿⡿⠇⣿⣿⣷⠀⠀⠀⣾⣿⣷⡼⣿⡿⢿⣿⢸⣿⠛⣿⠀⠀⠀⣾⡟⣿⣼⣿⢸⣿⢸⣿⠇⣿⡿⣷⠀⢠⣶⣿⠛⢻⡿⣾⡟⣷⣠⣿⢻⣧⠀⠀
⠀⠀⠀⣿⡇⢸⣿⢸⣿⣿⡇⣿⡇⣿⣿⣷⢹⣿⠃⠀⣿⡇⢸⣿⠗⣿⡿⠇⣿⣿⣿⠀⠀⠀⢿⣿⣿⡇⣿⡇⢸⣿⢿⣿⠀⣿⠘⠃⠀⣿⡇⣿⡏⣿⣿⡟⢸⣿⠇⣿⡿⣿⠀⠀⢸⣿⠀⣿⡇⣿⡇⣿⣿⣿⢸⣿⠀⠀
⠀⠀⠀⠿⠇⠘⠿⠾⠏⠿⠷⠿⠻⠿⠹⠿⠸⠿⠀⠀⠻⠷⠸⠿⠀⠿⠷⠆⠿⠿⠿⠰⠆⠀⠻⠿⠿⠁⠿⢇⡸⠿⠸⠿⠴⠿⠰⠆⣀⠻⠷⠿⠃⠿⠿⠇⠸⠿⠶⠿⠇⠿⠀⡀⠸⠿⠸⠿⠁⠻⠷⠿⠙⠿⠾⠟⠠⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠄⠀⢤⣶⣶⣶⣶⣶⣶⣿⣾⣿⣿⣿⣶⣶⣶⣶⣷⣿⣿⣿⣶⣶⣶⣶⡐⣦⣦⣖⠀⠀⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣤⠆⣺⣿⣌⣒⡈⡀⠀⠰⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠒⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢻⣿⠋⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣉⠀⠆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⠴⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣽⠺⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠉⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠉⠁⠈⣹⣿⣻⣋⣛⣾⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠟⠂⠀⠀⠀⠀⠀⠉⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠈⠛⠿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⡿⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠏⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢂⠀⣤⣶⡔⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠆⠀⠀⠲⠀⠀⠀⠀⠀⠚⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡀⠿⣿⡿⢳⣿⢻⣦⣿⡟⣿⡆⣿⣿⡏⣿⣼⡟⠀⣼⡟⢸⣿⠟⣿⡿⠇⣿⣿⣷⠀⠀⠀⣾⣿⣷⡼⣿⡿⢿⣿⢸⣿⠛⣿⠀⠀⠀⣾⡟⣿⣼⣿⢸⣿⢸⣿⠇⣿⡟⣷⠀⠠⣶⣿⠛⢻⡿⣾⡟⣷⣠⣿⢻⣧⠀⠀
⠀⠀⠀⣿⡇⢸⣿⢸⣿⣿⡇⣿⡇⣿⣿⣷⢹⣿⠁⠀⣿⡇⢸⣿⠇⣿⡿⠇⣿⣿⣿⠀⠀⠀⢿⣿⣿⡇⣿⡇⢸⣿⢿⣿⠀⣿⠘⠃⠀⣿⡇⣿⡏⣿⣿⡟⢸⣿⠇⣿⡿⣿⠀⠀⢸⣿⠀⣿⡇⣿⡇⣿⣿⣿⢸⣿⠀⠀
⠀⠀⠀⠿⠇⠘⠿⠾⠏⠿⠷⠿⠻⠿⠹⠿⠸⠿⠀⠀⠹⠷⠸⠿⠀⠿⠷⠆⠿⠿⠿⠰⠆⠀⠻⠿⠿⠁⠿⠇⡸⠿⠸⠿⠴⠿⠰⠆⠀⠻⠷⠿⠃⠿⠿⠇⠸⠿⠶⠿⠇⠿⠀⡀⠸⠿⠸⠿⠁⠻⠷⠿⠙⠿⠾⠟⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠢⠰⠔⠀⠀⠀⠀⠀⢀⣼⣷⣶⣾⣿⣷⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⡶⠀⢴⣤⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣥⣅⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⢹⣿⣈⡐⣈⣀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠙⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡷⣍⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⠤⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣙⠪⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠉⠓⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡍⠀⠀⠀⠀⠀⠙⠟⢛⠁⣀⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠋⠃⠀⠀⠀⠀⠀⠀⠈⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⡷⠿⠿⠿⠿⠿⠿⠿⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠏⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣯⠻⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠂⠀⠠⠲⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡀⠿⣿⡿⢳⣿⢻⣦⣿⡟⣿⡆⣿⣿⡏⣿⣼⡟⠀⣼⡟⢸⣿⠟⣿⡿⠇⣿⣿⣷⠀⠀⠀⣾⣿⣷⡼⣿⡿⢿⣿⢸⣿⠛⣿⠀⠀⠀⣾⡟⣿⣼⣿⢸⣿⢸⣿⠇⣿⡟⣷⠀⢠⣶⣿⠛⢻⡿⣾⡟⣷⣠⣿⢻⣧⠀⠀
⠀⠀⠀⣿⡇⢸⣿⢸⣿⣿⡇⣿⡇⣿⣿⣷⢹⣿⠃⠀⣿⡇⢸⣿⠇⣿⡿⠇⣿⣿⣿⠀⠀⠀⢿⣿⣿⡇⣿⡇⢸⣿⢿⣿⠀⣿⠘⠃⠀⣿⡇⣿⡏⣿⣿⡟⢸⣿⠇⣿⡿⣿⠀⠀⢸⣿⠀⣿⡇⣿⡇⣿⣿⣿⢸⣿⠀⠀
⠀⠀⠀⠿⠇⠘⠿⠾⠏⠿⠷⠿⠻⠿⠹⠿⠸⠿⠀⠀⠹⠷⠸⠿⠀⠿⠷⠆⠿⠿⠿⠰⠆⠀⠻⠿⠿⠁⠿⠇⡸⠿⠸⠿⠴⠿⠰⠆⠀⠻⠷⠿⠃⠿⠿⠇⠸⠿⠶⠿⠇⠿⠀⠀⠸⠿⠸⠿⠁⠻⠷⠿⠙⠿⠾⠟⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣶⣾⣿⣷⣿⣿⣿⣿⣶⣶⣶⣾⣾⣿⣷⣶⣶⣶⣦⠀⢶⣦⡀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣥⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣷⣤⣂⢸⣿⣄⣀⣈⣀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢙⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣯⣅⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⠠⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣙⠪⠆⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⠋⠛⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠀⠀⠀⠀⠀⠹⠛⠛⠁⣀⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠝⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠐⠟⡿⡿⠿⠿⠿⠿⢿⠿⡳⢟⠯⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢦⡶⣦⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠏⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠹⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠆⠀⠠⠲⠀⠀⠀⠀⠚⠛⠛⠛⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡀⠿⣿⡿⢷⣿⢻⣦⣿⡟⣿⡆⣿⣿⡏⣿⣼⡟⠀⣼⡟⢸⣿⠿⣿⡿⠇⣿⣿⣷⠀⠀⠀⣾⣿⣷⡼⣿⡿⢿⣿⢸⣿⠛⣿⠀⠀⠀⣾⡟⣿⣼⣿⢸⣿⢸⣿⠗⣿⡟⣷⠀⢠⣶⣿⠛⢻⡿⣾⡟⣷⣠⣿⢻⣧⠀⠀
⠀⠀⠀⣿⡇⢸⣿⢸⣿⣿⡇⣿⡇⣿⣿⣷⢹⣿⠃⠀⣿⡇⢸⣿⠗⣿⡿⠇⣿⣿⣿⠀⠀⠀⢿⣿⣿⡇⣿⡇⢸⣿⢿⣿⠀⣿⠘⠃⠀⣿⡇⣿⡏⣿⣿⡿⢸⣿⠇⣿⡿⣿⠀⠀⢸⣿⠀⣿⡇⣿⡇⣿⣿⣿⢸⣿⠀⠀
⠀⠀⠀⠿⠇⠘⠿⠾⠏⠿⠷⠿⠻⠿⠹⠿⠸⠿⠀⠀⠻⠷⠸⠿⠀⠿⠷⠆⠿⠿⠿⠰⠆⠀⠿⠿⠿⠁⠿⢇⡸⠿⠼⠿⠴⠿⠰⠆⡀⠻⠷⠿⠃⠿⠿⠇⠸⠿⠶⠿⠇⠿⠄⠀⠸⠿⠸⠿⠁⠻⠷⠿⠙⠿⠾⠟⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣾⣶⣶⣶⣿⣶⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⠴⣦⣦⡄⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⠑⢂⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣶⣦⣺⣿⣤⣜⣈⣀⣀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣹⣿⠟⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠲⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠢⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣙⡒⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠉⠛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡟⢛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠁⠸⢿⠿⡓⣊⣱⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃⠀⠀⠀⠀⠀⠈⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠹⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠹⢿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠛⣻⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠟⢷⠔⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠆⠀⠠⠲⠄⠀⠀⠀⠈⠛⠛⠛⠃⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡀⠿⣿⡿⢷⣿⢻⣦⣿⡟⣿⡆⣿⣿⡏⣿⣼⡟⠀⣼⡟⢸⣿⠟⣿⡿⠇⣿⣿⣷⠀⠀⠀⣾⣿⣷⡼⣿⡿⢿⣿⢸⣿⠛⣿⠀⠀⠀⣾⡟⣿⣼⣿⢸⣿⢸⣿⠗⣿⡟⣷⠀⢠⣶⣿⠛⢻⡿⣾⡟⣷⣠⣿⢻⣧⠀⠀
⠀⠀⠀⣿⡇⢸⣿⢸⣿⣿⡇⣿⡇⣿⣿⣷⢹⣿⠃⠀⣿⡇⢸⣿⠗⣿⡿⠇⣿⣿⣿⠀⠀⠀⢿⣿⣿⡇⣿⡇⢸⣿⢿⣿⠀⣿⠘⠃⠀⣿⡇⣿⡏⣿⣿⡟⢸⣿⠇⣿⡿⣿⠀⠀⢸⣿⠀⣿⡇⣿⡇⣿⣿⣿⢸⣿⠀⠀
⠀⠀⠀⠿⠇⠘⠿⠾⠏⠿⠷⠿⠻⠿⠹⠿⠸⠿⠀⠀⠻⠷⠸⠿⠀⠿⠷⠆⠿⠿⠿⠰⠆⠀⠻⠿⠿⠁⠿⢇⡸⠿⠸⠿⠴⠿⠰⠆⡀⠻⠷⠿⠃⠿⠿⠇⠸⠿⠶⠿⠇⠿⠄⠀⠸⠿⠸⠿⠁⠻⠷⠿⠙⠿⠾⠟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣠⣶⣶⣶⣶⣿⣶⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣦⠤⣶⣦⡄⠀⠀⢀⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠙⢪⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⣽⣿⣤⣌⣌⡄⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡇⠉⡀⠀⠀⠀⠀⠂⠀⠈⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡋⣹⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⠀⠒⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡲⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣙⠓⠷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⡿⣿⣿⣿⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⠁⠀⠀⠀⠁⢸⢿⣿⠿⠊⣰⠢⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠈⠀⠉⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⢀⣰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠟⢷⠄⠰⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⢿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⠝⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠂⠀⠀⠲⠀⠀⠀⠀⠀⠛⠛⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡀⠿⣿⡿⢳⣿⢻⣦⣿⡟⣿⡆⣿⣿⡏⣿⣼⡟⠀⣼⡟⢸⣿⠟⣿⡿⠇⣿⣿⣷⠀⠀⠀⣾⣿⣷⡼⣿⡿⢿⣿⢸⣿⠛⣿⠀⠀⠀⣾⡟⣿⣼⣿⢸⣿⢸⣿⠇⣿⡟⣷⠀⢠⣶⣿⠛⢻⡿⣾⡟⣷⣠⣿⢻⣧⠀⠀
⠀⠀⠀⣿⡇⢸⣿⢸⣿⣿⡇⣿⡇⣿⣿⣷⢹⣿⠁⠀⣿⡇⢸⣿⠇⣿⡿⠇⣿⣿⣿⠀⠀⠀⢿⣿⣿⡇⣿⡇⢸⣿⢿⣿⠀⣿⠘⠃⠀⣿⡇⣿⡏⣿⣿⡟⢸⣿⠇⣿⡿⣿⠀⠀⢸⣿⠀⣿⡇⣿⡇⣿⣿⣿⢸⣿⠀⠀
⠀⠀⠀⠿⠇⠘⠿⠾⠏⠿⠷⠿⠻⠿⠹⠿⠸⠿⠀⠀⠹⠷⠸⠿⠀⠿⠷⠆⠿⠿⠿⠰⠆⠀⠻⠿⠿⠁⠿⢇⡸⠿⠸⠿⠴⠿⠰⠆⠀⠻⠷⠿⠃⠿⠿⠇⠸⠿⠶⠿⠇⠿⠀⠀⠸⠿⠸⠿⠁⠻⠷⠿⠙⠿⠾⠟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣲⣆⣰⣾⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣦⠀⣶⣦⠀⠀⠀⢀⣢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠛⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡔⣂⣸⣿⣦⣌⣄⣤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣋⢉⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠡⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡢⠾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⡒⠷⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⡛⢿⣿⣿⠿⠿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠀⠀⠀⠀⠁⠘⠿⠷⠂⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠁⠊⠉⠉⠀⠀⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣰⣾⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⡗⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠏⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠙⠻⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⢷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠆⠀⠠⠲⠀⠀⠀⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡀⠿⣿⡿⢳⣿⢻⣦⣿⡟⣿⡆⣿⣿⡏⣿⣸⡟⠀⣼⡟⢸⣿⠟⣿⡿⠇⣿⣿⣷⠀⠀⠀⣾⣿⣷⡼⣿⡿⢿⣿⢸⣿⠛⣿⠀⠀⠀⣾⡟⣿⣼⣿⢸⣿⢸⣿⠇⣿⡟⣷⠀⠠⣶⣿⠛⢻⡿⣾⡟⣷⣠⣿⢻⣧⠀⠀
⠀⠀⠀⣿⡇⢸⣿⢸⣿⣿⡇⣿⡇⣿⣻⣷⢹⣿⠁⠀⣿⡇⢸⣿⠇⣿⡿⠇⣿⣿⣿⠀⠀⠀⢿⣿⣿⡇⣿⡇⢸⣿⢿⣿⠀⣿⠘⠃⠀⣿⡇⣿⡏⣿⣿⡟⢸⣿⠇⣿⡟⣿⠀⠀⢸⣿⠀⣿⡇⣿⡇⣿⣿⣿⢸⣿⠀⠀
⠀⠀⠀⠿⠇⠘⠿⠾⠏⠿⠷⠿⠻⠿⠹⠿⠸⠿⠀⠀⠹⠷⠸⠿⠀⠿⠷⠆⠿⠿⠿⠰⠆⠀⠻⠿⠿⠁⠿⠇⡸⠿⠸⠿⠴⠿⠰⠆⠀⠻⠷⠿⠁⠿⠿⠇⠸⠿⠶⠿⠇⠿⠀⠀⠸⠿⠸⠿⠁⠻⠷⠿⠙⠿⠾⠟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣶⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣆⠀⣴⣦⡀⠀⠀⠀⣠⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⡄⡀⣸⣿⣦⣬⣄⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣟⡋⡈⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠩⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⡆⠾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣉⡒⠖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⢿⣿⣿⠟⠛⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠙⠁⠀⠀⠀⠁⠘⠿⠗⠂⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠏⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣴⣟⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠂⠀⠠⠲⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡀⠿⣿⡿⢳⣿⢻⣦⣿⡟⣿⡆⣿⣿⡏⣿⣸⡟⠀⣼⡟⢸⣿⠟⣿⡿⠇⣿⣿⣷⠀⠀⠀⣾⣿⣷⡼⣿⡿⢿⣿⢸⣿⠛⣿⠀⠀⠀⣾⡟⣷⣼⣿⢸⣿⢸⣿⠇⣿⡟⣷⠀⠠⣶⣿⠛⢻⡿⣾⡟⣷⣠⣿⢻⣧⠀⠀
⠀⠀⠀⣿⡇⢸⣿⢸⣿⣿⡇⣿⡇⣿⣻⣷⢹⣿⠁⠀⣿⡇⢸⣿⠇⣿⡿⠇⣿⣿⣿⠀⠀⠀⢿⣿⣿⡇⣿⡇⢸⣿⢿⣿⠀⣿⠘⠃⠀⣿⡇⣿⡏⣿⣿⡟⢸⣿⠇⣿⡟⣿⠀⠀⢸⣿⠀⣿⡇⣿⡇⣿⣿⣿⢸⣿⠀⠀
⠀⠀⠀⠿⠇⠘⠿⠾⠏⠿⠷⠿⠻⠿⠹⠿⠸⠿⠀⠀⠹⠷⠸⠿⠀⠿⠷⠆⠿⠿⠿⠰⠆⠀⠻⠿⠿⢁⠿⠇⡸⠿⠸⠿⠴⠿⠰⠆⠀⠻⠷⠿⠁⠿⠿⠇⠸⠿⠶⠿⠇⠿⠀⠀⠸⠿⠰⠿⠁⠻⠷⠿⠙⠿⠾⠟⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣶⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⢂⠀⣶⣤⡀⠀⠀⢀⣤⢀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣷⡆⣀⣸⣿⣤⣬⣄⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠿⢿⣟⠃⠈⢽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡷⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⡂⠾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣀⡚⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣗⣿⣿⣿⣿⠄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣿⣿⠿⣏⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⢻⡿⣿⠍⠁⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠀⠀⠀⠀⠁⠘⠻⠒⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⠙⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠏⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣳⡟⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠂⠀⠀⠲⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡀⠿⣿⡿⢳⣿⢻⣦⣿⡟⣿⡆⣿⣿⡏⣿⣸⡟⠀⣼⡟⢸⣿⠟⣿⡿⠇⣿⣿⣷⠀⠀⠀⣾⣿⣷⡼⣿⡿⢿⣿⢸⣿⠛⣿⠀⠀⠀⣾⡟⣷⣼⣿⢸⣿⢸⣿⠇⣿⡟⣷⠀⠠⣶⣿⠛⢻⡿⣾⡟⣷⣠⣿⢻⣧⠀⠀
⠀⠀⠀⣿⡇⢸⣿⢸⣿⣿⡇⣿⡇⣿⣻⣷⢹⣿⠁⠀⣿⡇⢸⣿⠇⣿⡿⠇⣿⣿⣿⠀⠀⠀⢿⣿⣿⡇⣿⡇⢸⣿⢿⣿⠀⣿⠘⠃⠀⣿⡇⣿⡏⣿⣿⡟⢸⣿⠇⣿⡟⣿⠀⠀⢸⣿⠀⣿⡇⣿⡇⣿⣿⣿⢸⣿⠀⠀
⠀⠀⠀⠿⠇⠘⠿⠾⠏⠿⠷⠿⠻⠿⠹⠿⠸⠿⠀⠀⠹⠷⠸⠿⠀⠿⠷⠆⠿⠿⠿⠰⠆⠀⠻⠿⠿⣁⠿⠇⠸⠿⠸⠿⠴⠿⠰⠆⠀⠻⠷⠿⠁⠿⠿⠇⠸⠿⠶⠿⠇⠿⠀⠀⠸⠿⠰⠿⠁⠻⠷⠿⠙⠿⠾⠟⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⣶⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⡀⠀⣴⣤⡀⠀⠀⢀⣤⠄⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⡆⣀⡸⣿⣦⣬⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠿⢿⣟⠋⠀⢽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡂⠾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⣃⡚⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠄⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⢻⣿⣿⠟⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠙⠉⠀⠀⠀⠁⠈⠻⠲⠒⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠉⠙⠛⠋⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠏⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣾⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠆⠀⠀⠲⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠿⣿⡿⢳⣿⢻⣦⣿⡟⣿⡆⣿⣿⡏⣿⣸⡟⠀⣼⡟⢸⣿⠟⣿⡿⠇⣿⣿⣷⠀⠀⠀⣾⣿⣷⡼⣿⡿⢿⣿⢸⣿⠛⣿⠀⠀⠀⣾⡟⣷⣼⣿⢸⣿⢸⣿⠇⣿⡟⣷⠀⠠⣶⣿⠛⢻⡿⣾⡟⣷⣠⣿⢻⣧⠀⠀
⠀⠀⠀⣿⡇⢸⣿⢸⣿⣿⡇⣿⡇⣿⣻⣷⢹⣿⠁⠀⣿⡇⢸⣿⠇⣿⡿⠇⣿⣿⣿⠀⠀⠀⢿⣿⣿⡇⣿⡇⢸⣿⢿⣿⠀⣿⠘⠃⠀⣿⡇⣿⡏⣿⣾⡟⢸⣿⠇⣿⡟⣿⠀⠀⢸⣿⠀⣿⡇⣿⡇⣿⣿⣿⢸⣿⠀⢀
⠀⠀⠀⠿⠇⠘⠿⠾⠏⠿⠷⠿⠻⠿⠹⠿⠸⠿⠀⠀⠹⠷⠸⠿⠀⠿⠷⠆⠿⠿⠿⠰⠆⠀⠻⠿⠿⣁⠿⠇⠸⠿⠸⠿⠴⠿⠰⠆⠀⠻⠷⠿⠁⠿⠿⠇⠸⠿⠶⠿⠇⠿⠀⠀⠸⠿⠰⠿⠁⠻⠷⠿⠙⠿⠾⠟⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⣶⣾⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶⣂⠀⠴⣦⡀⠀⠀⢀⣤⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⡆⣠⣘⣿⣦⣤⣀⡀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⡀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡞⢿⣟⣋⡀⢽⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠡⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⡆⠞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠞⣃⡚⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⢿⣿⣿⡟⠉⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠋⠀⠀⠀⠁⠈⠻⠗⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄⠀⠉⠙⠛⠻⢿⠿⠟⠻⠿⠟⠟⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠏⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠆⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠿⣿⡿⢳⣿⢻⣦⣿⡟⣿⡆⣿⣿⡏⣿⣼⡟⠀⣼⡟⢸⣿⠟⣿⡿⠇⣿⣿⣷⠀⠀⠀⣾⣿⣷⡼⣿⡿⢿⣿⢸⣿⠻⣿⠀⠀⠀⣾⡟⣿⣼⣿⢸⣿⢸⣿⠇⣿⡟⣷⠀⢠⣶⣿⠛⢻⡿⣾⡟⣷⣠⣿⢻⣧⠀⠀
⠀⠀⠀⣿⡇⢸⣿⢸⣿⣿⡇⣿⡇⣿⣿⣷⢹⣿⠁⠀⣿⡇⢸⣿⠇⣿⡿⠇⣿⣿⣿⠀⠀⠀⢿⣿⣿⡇⣿⡇⢸⣿⢿⣿⠀⣿⠘⠃⠀⣿⡇⣿⡏⣿⣿⡟⢸⣿⠇⣿⡿⣿⠀⠀⢸⣿⠀⣿⡇⣿⡇⣿⣿⣿⢸⣿⠀⢠
⠀⠀⠀⠿⠇⠘⠿⠾⠏⠿⠷⠿⠻⠿⠹⠿⠸⠿⠀⠀⠻⠷⠸⠿⠀⠿⠷⠆⠿⠿⠿⠰⠆⠀⠻⠿⠿⣁⡿⢇⠸⠿⠸⠿⠴⠿⠰⠆⠀⠻⠷⠿⠁⠿⠿⠇⠸⠿⠶⠿⠇⠿⠀⠀⠸⠿⠸⠿⠁⠻⠷⠿⠙⠿⠾⠟⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣶⣾⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⡂⠀⢶⣤⡀⠀⠀⢀⣠⠄⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⡄⣀⠐⣿⣦⣤⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⢿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⢿⣛⣇⡀⢹⠗⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣒⠚⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣂⡘⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⡿⠋⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠈⠿⠷⠖⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡁⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⠁⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀⠙⠟⠿⢿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠻⠿⠅⠀⠀⠀⠀⠀⠀⠸⠽⢟⢻⣿⣴⣦⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡀⠀⠀⠁⠩⠋⠉⠀⠈⠉⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠈⠠⢷⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣶⣄⠀⠀⠀⠀⠀⠀⠠⠉⠘⢻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠏⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠆⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠿⣿⡿⢳⣿⢻⣦⣿⡟⣿⡆⣿⣿⡏⣿⣼⡟⠀⣼⡟⢸⣿⠟⣿⡿⠇⣿⣿⣷⠀⠀⠀⣾⣿⣷⡼⣿⡿⢿⣿⢸⣿⠻⣿⠀⠀⠀⣾⡟⣿⣼⣿⢸⣿⢸⣿⠇⣿⡟⣷⠀⢠⣶⣿⠛⢻⡿⣾⡟⣷⣠⣿⢻⣧⠀⠀
⠀⠀⠀⣿⡇⢸⣿⢸⣿⣿⡇⣿⡇⣿⣿⣷⢹⣿⠁⠀⣿⡇⢸⣿⠇⣿⡿⠇⣿⣿⣿⠀⠀⠀⢿⣿⣿⡇⣿⡇⢸⣿⢿⣿⠀⣿⠘⠃⠀⣿⡇⣿⡏⣿⣿⡟⢸⣿⠇⣿⡿⣿⠀⠀⢸⣿⠀⣿⡇⣿⡇⣿⣿⣿⢸⣿⠀⠠
⠀⠀⠀⠿⠇⠘⠿⠾⠏⠿⠷⠿⠻⠿⠹⠿⠸⠿⠀⠀⠻⠷⠸⠿⠀⠿⠷⠆⠿⠿⠿⠰⠆⠀⠻⠿⠿⣁⡿⢇⠸⠿⠼⠿⠴⠿⠰⠆⠀⠻⠷⠿⠁⠿⠿⠇⠸⠿⠶⠿⠇⠿⠀⠀⠸⠿⠸⠿⠁⠻⠷⠿⠙⠿⠾⠟⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣶⣾⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣾⣿⣷⣶⣶⣶⣀⠀⢶⣦⡀⠀⠀⠀⣴⠂⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⡆⣀⣐⣿⣶⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⢿⣿⡷⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣏⣅⡀⢹⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢐⠚⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣂⡘⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⠟⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠈⠻⠷⠆⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃⠸⠀⢀⠁⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⠿⠿⠿⠁⠀⠀⠀⠀⠀⠈⢼⠺⢛⠿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⡄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠶⠙⠏⠻⡅⠉⠁⠊⠠⢠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠝⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠛⠙⠓⠟⠚⠔⡺⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠠⠀⠀⠀⠀⠀⠀⠀⠐⠂⠊⣻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠿⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠛⠻⠿⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠆⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠿⣿⡿⢷⣿⢻⣦⣿⡟⣿⡆⣿⣿⡏⣿⣼⡟⠀⣼⡟⢸⣿⠟⣿⡿⠇⣿⣿⣷⠀⠀⠀⣾⣿⣷⡼⣿⡿⢿⣿⢸⣿⠻⣿⠀⠀⠀⣾⡟⣿⣼⣿⢸⣿⢸⣿⠇⣿⡟⣷⠀⢠⣶⣿⠛⢻⡿⣾⡟⣷⣠⣿⢻⣧⠀⠀
⠀⠀⠀⣿⡇⢸⣿⢸⣿⣿⡇⣿⡇⣿⣿⣷⢹⣿⠁⠀⣿⡇⢸⣿⠇⣿⡿⠇⣿⣿⣿⠀⠀⠀⢿⣿⣿⡇⣿⡇⢸⣿⢿⣿⠀⣿⠘⠃⠀⣿⡇⣿⡏⣿⣿⡟⢸⣿⠇⣿⡿⣿⠀⠀⢸⣿⠀⣿⡇⣿⡇⣿⣿⣿⢸⣿⠀⠠
⠀⠀⠀⠿⠇⠘⠿⠾⠏⠿⠷⠿⠻⠿⠹⠿⠸⠿⠀⠀⠻⠷⠸⠿⠀⠿⠷⠆⠿⠿⠿⠰⠆⠀⠿⠿⠿⢁⡿⢇⡸⠿⠼⠿⠴⠿⠰⠆⡀⠻⠷⠿⠁⠿⠿⠇⠸⠿⠶⠿⠇⠿⠀⠀⠸⠿⠸⠿⠁⠻⠷⠿⠙⠿⠾⠟⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣶⣾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡀⠀⣶⣦⠀⠀⠀⠀⣰⠢⢀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠈⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⣿⣷⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⡯⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣏⣍⣀⢸⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠆⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⠚⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢂⣐⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⡟⠋⠛⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠈⠻⡧⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠫⢀⠀⠁⠒⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣶⣔⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠁⠀⠀⠀⠀⠀⠀⢼⢽⣿⣷⣾⣶⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⣾⣿⣿⣷⣦⣾⣿⣿⣿⣿⣿⣿⡄⠈⠐⠺⢿⢛⣿⢟⣟⠙⠯⠯⢴⣆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠸⠯⠛⠛⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠈⠍⢻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠟⢶⠆⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠙⠛⠋⠛⠟⠉⠙⠛⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠡⠊⠉⠉⠛⠛⠿⠿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠆⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠿⣿⡿⢷⣿⢻⣦⣿⡟⣿⡆⣿⣿⡏⣿⣼⡟⠀⣼⡟⢸⣿⠟⣿⡿⠇⣿⣿⣷⠀⠀⠀⣾⣿⣷⡼⣿⡿⢿⣿⢸⣿⠛⣿⠀⠀⠀⣾⡟⣿⣼⣿⢸⣿⢸⣿⠇⣿⡟⣷⠀⠠⣶⣿⠛⢻⡿⣾⡟⣷⣠⣿⢻⣧⠀⠀
⠀⠀⠀⣿⡇⢸⣿⢸⣿⣿⡇⣿⡇⣿⣿⣷⢹⣿⠁⠀⣿⡇⢸⣿⠇⣿⡿⠇⣿⣿⣿⠀⠀⠀⢿⣿⣿⡇⣿⡇⢸⣿⢿⣿⠀⣿⠘⠃⠀⣿⡇⣿⡏⣿⣿⡟⢸⣿⠇⣿⡿⣿⠀⠀⢸⣿⠀⣿⡇⣿⡇⣿⣿⣿⢸⣿⠀⠠
⠀⠀⠀⠿⠇⠘⠿⠾⠏⠿⠷⠿⠻⠿⠹⠿⠸⠿⠀⠀⠹⠷⠸⠿⠀⠿⠷⠆⠿⠿⠿⠰⠆⠀⠻⠿⠿⢁⡿⢇⡸⠿⠼⠿⠴⠿⠰⠆⠀⠻⠷⠿⠁⠿⠿⠇⠸⠿⠶⠿⠇⠿⠀⠀⠸⠿⠸⠿⠁⠻⠷⠿⠙⠿⠾⠟⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡀⠀⢶⣦⠀⠀⠀⠀⣰⠂⢄⠀⠄⠀⠀⡤⠀⠀⠀⠀⠀⠀⠈⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡄⣀⣀⣿⣷⣴⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⡯⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢻⣏⡅⡀⢸⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠆⡀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⠒⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⣊⣐⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡽⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⡿⠋⠃⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠈⠻⡧⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⠣⣀⠀⠈⠢⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣶⣼⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⢿⠿⠁⠀⠀⠀⠀⠀⠀⠯⢻⣿⣷⣶⣶⣤⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣶⣤⣾⣿⣿⣿⣿⣿⣿⠀⠈⠐⠿⠿⡿⡟⠻⡟⢻⢛⡜⣛⡥⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠙⠽⢟⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠹⠞⠛⠙⠉⠀⠀⠀⠀⠀⢡⣶⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠠⠐⠛⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠏⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⠑⠛⢿⠿⣽⠟⠃⠉⠛⠻⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠔⠀⠀⠉⠛⠛⠿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⡖⠀⠂⠀⢀⢀⡀⣀⢀⢀⢀⠀⡀⣿⢀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⠧⠀⠄⠶⠸⢸⡇⠟⠈⠳⠸⠠⠇⠿⠸⠌⠨⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢹⢰⢠⠀⠄⡄⢠⠠⠀⠀⠀⢠⠀⡄⠄⠀⢈⠅⡮⣆⢕⠨⠅⠀⢠⠠⠀⡄⠀⠠⠀⡄⡄⠁⢠⠀⠄⡀⠀⠀⠀⡤⢣⠠⠀⠀⠄⢠⢠⠀⠄⠀⢠⠀⠄⡀⢀⠀⠁⡄⡄⡇⠈⠀⢄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁⠀⠁⠀⠀⠁⠀⠀⠈⠀⠀⠈⠁⠈⠀⠁⠈⠁⠀⠀⠁⠈⠀⠈⠀⠁⠈⠁⠉⠀⠁⠈⠀⠁⠀⠀⠀⠀⠁⠀⠀⠉⠀⠁⠈⠀⠉⠀⠁⠈⠀⠉⠀⠀⠈⠀⠉⠈⠁⠈⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡔⢠⢢⠐⠂⠐⢠⠢⠔⡀⡗⢰⠂⡔⡇⠀⢰⠂⡔⢤⢂⠐⠂⡖⠐⣂⠈⠀⢤⠀⠀⢺⠀⠒⠀⠀⢸⠀⡙⠨⠁⡏⠀⠀⠐⠂⠚⠀⠀⢸⠂⠒⢠⠢⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⢃⠠⡇⡄⠄⣄⢠⠀⢀⠄⢠⠠⢠⢠⢠⢠⢠⠠⢀⠠⣠⠠⠀⢀⠀⠠⡀⠄⢄⣄⢠⠀⠀⢀⣠⢠⠀⠀⡇⠄⠄⡀⠀⡄⡄⡄⡄⡄⡄⠄⡇⠄⣄⡄⡤⠀⡀⢄⠨⠁⢠⢠⠠⠠⢠⠠⠌⡠⠀⠀⠠⠀⠠⠀⠀
⠀⠀⠐⠊⠁⠃⠑⠂⠒⠉⠂⠈⠒⠈⠒⠘⠘⠘⠘⠘⠐⠊⠒⠘⠐⠀⠈⠒⠐⠐⠐⠊⠒⠉⠂⠐⠊⠒⠐⠂⠀⠑⠂⠸⠀⠀⠃⠃⠑⠃⠃⠃⠂⠓⠂⠑⠂⠃⠀⠑⠂⠐⠀⠀⠃⠃⠐⠊⠐⠐⠘⠒⠒⠂⠢⠒⠆⠀
⠀⠀⠀⠀⢀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣀⡸⢸⠒⡁⢀⢀⢘⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡴⢡⢦⠀⠀⠒⢐⢦⢂⠆⠶⠀⠀⡰⡄⠀⠀⠀⢺⢰⣶⠐⠀⠒⡀⠀⠐⡂⠒⠀⠀⠈⠀⢝⢨⠀⡇⡇⢼⠀⠀⡇⢸⢜⠀⠀⠖⠀⠀⠐⡆⠶⢰⠂⠀⠀⢒⢐⠆⡖⡄⢖⠀⡀⢸⢠⢦⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⢀⡀⢀⠀⡀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢎⠲⠿⠈⠀⠎⠀⡇⠀⠀⢉⠰⡑⢎⠄⠀⠰⠀⢸⠀⣱⠀⠁⠉⠄⡡⠀⠀⠈⠀⠀⢈⡆⠀⠀⠀⠀⠀⠂⠐⠀⢗⠟⡺⠀⠀⢇⠸⠊⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

                    ䷩ 𝚕𝚒𝚗𝚎 3574

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

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/02/09/holding-management-accountable/#comments
    Gemini version at gemini://gemini.techrights.org/2022/02/09/holding-management-accountable/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 02.09.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Staff_Holding_Management_at_the_EPO_Accountable⠀✐


Posted in Europe, Patents at 2:37 pm by Dr. Roy Schestowitz


Video_download_link | md5sum 3e57c063cb4318309a3e810ef7b45ce7
Consequences for Rogues at the EPO
Creative Commons Attribution-No Derivative Works 4.0


http://techrights.org/videos/epo-action-and-impunity.webm


Summary: It seems possible that in months to come the administrative body of
the EPO will finally acknowledge that “breaking point” has been reached and
it’s time to reboot the Office


THE immunity and impunity of EPO management has been a theme of recent coverage
here. We last wrote about it earlier_today, focusing on what Benoît_Battistelli
had done in the Netherlands. After 3 years at the Office António_Campinos is
starting to see it all crumbling or falling down, starting with last summer’s
ILOAT ruling no. 4430 and then ILOAT ruling no. 4482, which is still very
fresh. But why did it take so long? As noted in the video above, many of the
people wronged are already dead, unable to claim_or_collect_compensation_from
the_Office.


The EPO got infiltrated by white-collar criminals, who in effect fronted for
international (non-European) or multinational corporations with their patent
litigation staff (in-house) and/or outside law firms. The solution to such a
dire crisis must come from the outside, but the_EU_too_is_infiltrated. The EPO
conflict has exposed a rotting unregulated system, which extends_to_EUIPO and
even some courts. It won’t be pretty, but “ordinary” EPO staff (such as patent
examiners) are on the right side of history. █



                    ䷩ 𝚕𝚒𝚗𝚎 3626

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

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/02/09/iloat-ruling-no-4430-and-iloat-ruling-no-4482/#comments
    Gemini version at gemini://gemini.techrights.org/2022/02/09/iloat-ruling-no-4430-and-iloat-ruling-no-4482/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 02.09.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ [Meme]_Drunk_Driving,_Like_Son_of_António_Campinos⠀✐


Posted in Europe, Patents at 2:47 pm by Dr. Roy Schestowitz


Benoît_Battistelli is notorious for misusing EPO funds to build himself a_pub;
António_Campinos received a lot of negative press after his son had_gotten
drunk,_crashed_the_car,_and_wanted_to_borrow_dad's_diplomatic_immunity (his
grandpa Jorge did not live long enough to see such a disgrace; he died_in_a_car
accident_in_1993)


     🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇ILOAT ruling no. 4430 and ILOAT ruling no. 4482⦈


Summary: In the Netherlands, the drunk-on-power EPO may have gotten its way (an
above-the-law status), but ILOAT keeps delivering much-belated rulings (4430,
4482, maybe more to come) which curtail the illegal actions of the EPO albeit
without any consequences for the perpetrators


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

                    ䷩ 𝚕𝚒𝚗𝚎 4662

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

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/02/09/irc-log-080222/#comments
    Gemini version at gemini://gemini.techrights.org/2022/02/09/irc-log-080222/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 02.09.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ IRC_Proceedings:_Tuesday,_February_08,_2022⠀✐


Posted in IRC_Logs at 2:05 am by Needs Sunlight


Also available via the Gemini protocol at:


    * gemini://gemini.techrights.org/irc-gmi/irc-log-techrights-080222.gmi
    * gemini://gemini.techrights.org/irc-gmi/irc-log-080222.gmi
    * gemini://gemini.techrights.org/irc-gmi/irc-log-social-080222.gmi
    * gemini://gemini.techrights.org/irc-gmi/irc-log-techbytes-080222.gmi
Over HTTP:


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


                                         #techrights_log_as_HTML5                                                                                  #boycottnovell_log_as_HTML5


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


                                    #boycottnovell-social_log_as_HTML5                                                                               #techbytes_log_as_HTML5


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


                                          #techrights_log_as_text                                                                                   #boycottnovell_log_as_text


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


                                     #boycottnovell-social_log_as_text                                                                                #techbytes_log_as_text


                          Enter_the_IRC_channels_now


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


CID                                              Description     Object type


                                                 IRC log for
 QmSPUPU5ZqHxTQsd3apxwT4G1nMn7JEgsmxe71kfJUA86Q #boycottnovell  🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
                                                 (full IRC log
                                                 as HTML)


                                                 IRC log for
                                                 #boycottnovell
 QmZNvwnCvTaTdc3QWDJPKffdcGDFfuVYe7PBXqoLQS1iHK (full IRC log   🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
                                                 as plain/ASCII
                                                 text)


                                                 IRC log for
                                                 #boycottnovell-
 QmaY3dKt7BTd8tptvq7N6n68PgSFSJhpiWLig31U8MWYcJ social          🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
                                                 (full IRC log
                                                 as HTML)


                                                 IRC log for
                                                 #boycottnovell-
 Qmeo8oAfrK6cb5eJuYgsWEdsPeQR72AUZNht7GsS4Zrb7u social          🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
                                                 (full IRC log
                                                 as plain/ASCII
                                                 text)


                                                 IRC log for
 QmZ1bdTSeqAGWrkhHaAgv4xRVVfsmxscv5gLnSfRfN71gs #techbytes      🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
                                                 (full IRC log
                                                 as HTML)


                                                 IRC log for
                                                 #techbytes
 QmfUrmap41BA4T33JqUekFtEoo9ELRmrFCQrfgsgZ5WmQ9 (full IRC log   🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
                                                 as plain/ASCII
                                                 text)


                                                 IRC log for
 QmSpmWXjBZoyq7JvUVZzbbF5nUig5NwMPdN7FXiswiNmxQ #techrights     🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
                                                 (full IRC log
                                                 as HTML)


                                                 IRC log for
                                                 #techrights
 QmSBquW9Y8DFmjXvikR4GwDGMzHgCwVJ7hbZ9K7nRC6N2b (full IRC log   🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
                                                 as plain/ASCII
                                                 text)


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇IPFS logo⦈


§ Bulletin for Yesterday⠀➾


Local_copy | CID (IPFS): QmYBkNNgDtvjiZGy3f7RwY487FEsQEUQkXJN3yYkKnMGUa



                    ䷩ 𝚕𝚒𝚗𝚎 4789

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

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 02.09.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Links_10/2/2022:_Epic_Wants_Rootkits_in_Linux_and_Serial_GPL_Violator_VMware
Launches_Vicious_FUD_Attack_on_Linux⠀✐


Posted in News_Roundup at 8:27 pm by Dr. Roy Schestowitz


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇GNOME bluefish⦈


§ Contents⠀➾


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


          o ⚓ Chrome_OS_98_Stable_Channel_arrives:_What_you_need_to_know⠀⇛


                 According to the official Chrome Release blog, the Chrome
                 OS 98 Stable Channel was released on Monday night. All
                 supported Chromebooks, Chromeboxes and Chrome OS tablets
                 should see the update now, or in the coming days.


                 As of Tuesday morning, the main Google site did not have
                 any information as to what’s in this update. So here’s a
                 rundown on the new features and functions found so far in
                 the Chrome OS 98 Stable Channel. Many are experimental
                 features.


          o § Desktop/Laptop⠀➾


                # ⚓ Compulab’s_Fitlet3_mini-PC_offers_Elkhart_Lake_and_triple
                  M.2⠀⇛


                       Compulab has launched a highly customizable
                       “Fitlet3” industrial mini-PC that starts at about
                       $260 and runs Linux or Windows on Elkhart Lake.
                       Standard features include 2x GbE, 6x USB, HDMI,
                       mini-DP, FACET expansion, and M.2 M-, B- and E-key
                       slots.


                       Compulab has updated its five-year old, Apollo Lake
                       based Fitlet2 mini-PC. The Fitlet3 advances to
                       Intel’s 10nm Elkhart Lake. Aside from the much
                       faster Atom x6000 processor, the Fitlet3 ships with
                       up to 32GB, 3200MT/s DDR4. That is is twice the RAM
                       of the Fitlet2, which is also available as the
                       Linux Mint optimized MintBox Mini 2.


                # ⚓ Fitlet3_is_a_compact,_fanless_Elkhart_Lake_mini_PC_for_IoT
                  and_industrial_applications⠀⇛


                       Fitlet3 fanless mini PC based on a choice of three
                       Atom or Celeron Elkhart Lake processors is an
                       upgrade to the Compulab Fitlet2 mini PC introduced
                       in 2017 with Apollo Lake processors.


                       Designed to be used as a fanless industrial mini PC
                       or an IoT gateway, Fitlet3 comes with up to 32GB
                       RAM, supports SATA and NVMe storage, offers two 4K
                       video outputs, up to four Gigabit Ethernet ports,
                       optional WiFi 6E, Bluetooth 5.2, and/or 4G/5G
                       cellular connectivity, as well as a wide range 7V
                       to 42V power input.


                # ⚓ Windows,_macOS,_Chrome_OS,_or_Linux:_Which_Operating_System
                  Is_Best?⠀⇛


                       You don’t have a great many choices when it comes
                       to your computer’s operating system, but the choice
                       that you make can have wide-ranging consequences.
                       The five OSes included here are your most viable
                       options. That’s not a huge number, but these
                       operating systems differ distinctly in strengths
                       and weaknesses. Four come from gargantuan
                       commercial tech giants, while one, the Linux-based
                       Ubuntu, is a free, open-source option. Windows and
                       macOS are generally the most powerful in terms of
                       hardware and software selection as well as
                       interface conveniences and utilities, while Chrome
                       OS is more lightweight and runs on inexpensive
                       hardware.


                # ⚓ Best_desktop_Linux_for_pros_2022:_Our_top_5_choices_|
                  ZDNet⠀⇛


                       You knew Fedora, Red Hat’s community Linux
                       distribution, would be first on my list. It’s the
                       mainstream distro that pushes Linux’s limits. It
                       comes powered by the newest Linux kernel and with
                       the latest open-source software.


                       In particular, Fedora is the Linux of choice for
                       programmers. No less a figure than Linus Torvalds
                       uses Fedora for his development work. Need I say
                       more?


                       Sometimes, however, when you’re running a leading-
                       edge distro, you can cut yourself. There’s a reason
                       why Fedora’s also known as a bleeding-edge Linux.


                       On the other hand, Fedora is easy to install and
                       set up. You don’t need to be a Linux expert to get
                       up and running with it.


                       For programmers, Fedora also boasts an excellent
                       Developer Portal. It features dedicated guides on
                       developing command line, desktop, mobile, and web
                       apps.


                       The Fedora Developer Portal also comes with an
                       excellent guide for developing hardware devices
                       such as Arduino and Raspberry Pi. Last but not
                       least, it comes with such development tools as the
                       top-notch integrated development environment (IDE)
                       Eclipse for Java, C/C++, and PHP and Vagrant, a
                       tool for creating reproducible, portable container
                       or virtual machine (VM)-based development
                       environments.


                       Unless you’re working on the Debian/Ubuntu family
                       programs, Fedora should be your first choice for a
                       development operating system. For developers in
                       that group, I recommend the newest version of
                       Ubuntu.


                # ⚓ Best_Linux_distros_for_beginners_in_2022:_You_can_do_this!
                  |_ZDNet⠀⇛


                       Some people still insist that using Linux is hard.
                       Sure, it was difficult — when I started with the
                       Linux desktop back in the 1990s. But that was a
                       long time ago. Today, the easiest desktop of all,
                       Chrome OS, is simply Linux with the Chrome web
                       browser on top of it. The more full-featured Linux
                       desktop distributions are as easy to use in 2021 as
                       Windows or macOS.


                       Yes, you can get a lot more from Linux if you know
                       how to do shell programming and the like. But
                       that’s also true of Windows and PowerShell. With
                       both operating systems, you don’t need to know the
                       deep ins and outs of either one to get your work
                       done.


          o § Kernel Space⠀➾


                # ⚓ From_Prototype_to_Post_Deployment:_Linux_Decision_Points⠀⇛


                       There were four options given:


                       1) Choose roll-your-own based on Yocto Project
                       Linux, Debian, or something similar: 27 %


                       2) Choose Linux provided through a semiconductor or
                       board vendor: 17 %


                       3) Choose Linux from a commercial open source
                       provider: 10 %


                       4) Continue evaluating whether you want to use
                       Linux: 45 %


                       It is interesting that the number of respondents
                       choosing Linux from a commercial OS provider is not
                       higher than 10%. Additionally, given recent trends
                       around the migration to Linux for embedded
                       development, it is surprising that 45% of the
                       audience was still evaluating whether to use Linux
                       at all.


                # § Graphics Stack⠀➾


                      # ⚓ AMD_looking_for_more_Linux_help_to_reach_client-
                        stages_on_the_open-source_platform⠀⇛


                      # ⚓ Intel’s_Open-Source_Linux_Driver_now_offers_Vulkan
                        “ANV”_ray-tracing_support⠀⇛


                             With the impending launch of Intel Arc GPUs
                             with new ray-tracing technology approaching
                             soon, the company has introduced further
                             open-source “ANV” Vulkan support for the
                             Linux operating system. In the latest driver
                             update, this new support is filed under
                             “VK_KHR_ray_query.”


          o § Applications⠀➾


                # ⚓ DeskCut_Helps_You_to_Create_Linux_Desktop_Shortcuts_with
                  Ease⠀⇛


                       This article explains briefly how to create useful
                       and standard-compliant desktop entries on Linux by
                       using the DeskCut app.


                       Have you ever installed a program in Linux from an
                       outside source and wished that it had a convenient
                       entry in your desktop environment’s menu? While
                       most GUI programs automatically create their
                       desktop shortcut during installation, some
                       applications may require you to set up associated
                       shortcuts manually.


                       And this is where DeskCut comes on the scene. But
                       before that, let’s explain what Linux shortcuts
                       actually are.


          o § Instructionals/Technical⠀➾


                # ⚓ How_To_Install_Plex_Media_Server_on_Fedora_35_–_idroot⠀⇛


                       In this tutorial, we will show you how to install
                       Plex Media Server on Fedora 35. For those of you
                       who didn’t know, Plex Media Server is a self-hosted
                       media player system for storing your movies, shows,
                       music, and photos. Over time Plex Media Server has
                       grown much and now supports many platforms.


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


                # ⚓ How_to_Install_Nginx_on_WHM/cPanel_With_Engintron_–
                  RoseHosting⠀⇛


                       In this tutorial, we are going to explain how to
                       Install Nginx with Engintron on a server with WHM/
                       cPanel control panel installed.


                       Engintron is a free WHM/cPanel plugin that can be
                       used for installing the Nginx web server as a
                       reverse proxy of the default Apache webserver. The
                       reason for using the Nginx as a reverse proxy of
                       Apache on our WHM/cPanel server is that the Apache
                       will server the dynamic content such as PHP, Perl
                       and etc, and the Nginx will be responsible for
                       serving the static content like CSS, Javascript,
                       and images.


                       Installing Nginx with Engintron to your cPanel
                       server can take a few minutes and is a very easy
                       process. Let’s get to work!


                # ⚓ How_to_install_InfluxDB_2_on_Debian_11_–_NextGenTips⠀⇛


                       InfluxDB is an open-source time-series database. It
                       is used for storage and retrieval of time series
                       data in fields such as operation monitoring,
                       operations metrics, internet of things sensor data,
                       and real-time analytics. In this tutorial, we are
                       going to learn how to install InfluxDB on Debian
                       11.


                       By default, InfluxDB uses the following network
                       ports for communication.


                # ⚓ How_to_install_Google_PageSpeed_to_improve_NGINX
                  performance_–_News_Update⠀⇛


                       Google PageSpeed is an open-source module, created
                       by Google, to help make web pages load faster. This
                       added speed is achieved by rewriting them for
                       better latency and decreasing bandwidth. Although
                       PageSpeed has been around for some time, it’s not
                       often used. However, by adding it to your NGINX or
                       Apache websites, you’ll find they perform
                       considerably better.


                       I’m going to show you how to install both NGINX and
                       PageSpeed, so your users can enjoy a much-improved
                       experience with your web pages. We are going to
                       install everything from scratch, so you’ll want to
                       do this on a non-production machine first. Once
                       you’ve mastered this, move it to production and
                       then migrate your sites to the new web server.


                # ⚓ How_to_easily_install_minix_3.3.0⠀⇛


                       This is an screen by screen guide on how to install
                       minix 3.3.0.


                # ⚓ Install_and_use_WPscan_(WordPress_security_scanner)_in
                  Linux⠀⇛


                       WPScan is a free, for non-commercial use, black box
                       WordPress security scanner written for security
                       professionals and blog maintainers to test the
                       security of their sites.


                # ⚓ Install_Metasploit_Framework_on_Ubuntu_22.04/Ubuntu_20.04_–
                  kifarunix.com⠀⇛


                       In this tutorial, you will learn how to install
                       Metasploit Framework on Ubuntu 22.04/Ubuntu 20.04
                       LTS. Metasploit Framework provides a platform and
                       tools for performing deep system security auditing
                       as well as penetration testing to unearth, exploit
                       and validate every other would-be vulnerability.


                # ⚓ How_to_install_Peppermint_OS_2022-02-02_–_Invidious⠀⇛


                # ⚓ How_to_install_OpenVPN_Client_on_Debian_11_Bullseye_–_Linux
                  Shout⠀⇛


                       Learn the steps to install OpenVPN client on Debian
                       11 Bullseye Linux to connect VPN server for
                       establishing a secure connection. It was programmed
                       by James Yonan and was first released in 2001 under
                       the open-source license.


                       OpenVPN is a popular open-source software freely
                       available. It is available in a client-server
                       architecture and can be installed on computers,
                       servers, or network devices such as routers and
                       supports operating systems such as Windows, Linux,
                       macOS, Solaris, OpenBSD, Android, and many more.


                       It can be used to implement encrypted private
                       networks between individual clients or entire
                       networks. For the connection to be established, the
                       OpenVPN software must be available and suitably
                       configured on both sides. The software transmits
                       the user data via TCP or UDP packets. The VPN
                       software can also be used across NAT borders (NAT –
                       Network Address Translation). The data exchanged in
                       the VPN is via SSL/ TLSencrypted. The OpenSSL
                       library is used for encryption.


                # ⚓ Ubuntu_Linux_PC_Won’t_Boot?_5_Common_Issues_and_Fixes⠀⇛


                       You’ve switched on your computer, preparing to do
                       some work, edit a document, mix a composition, or
                       just play a game… but something goes wrong.


                       Ubuntu won’t boot.


                       Sadly, as reliable as Linux is in general, and as
                       popular as Ubuntu is, sometimes it runs into
                       problems, just like Windows 10 or macOS. In most
                       cases, you’ll be able to work around this.


                       Whether you’re using Ubuntu desktop or Ubuntu
                       Server, here’s what to do if Ubuntu doesn’t boot.


                # ⚓ What_Is_Log4Shell_and_How_to_Protect_Your_Linux_System
                  Against_It⠀⇛


                       The Log4Shell (CVE-2021-44228) vulnerability is
                       described by many cybersecurity researchers and
                       experts to be the most critical zero-day
                       vulnerability of all time. It affects a widely-used
                       Java utility by the name of Log4j, which has been
                       around since 2001 for logging and communicating
                       events, such as routine system operations and
                       software errors. Here we show you how to protect
                       your Linux system against it.


                # ⚓ 100%_easy_Open-source_VPN_and_Proxy_with_podman⠀⇛


                       OpenConnect VPN is not officially supported by or
                       associated in any way with Cisco Systems, Juniper
                       Networks, Pulse Secure, Palo Alto Networks, F5,
                       Fortinet, or any of the companies whose protocols
                       we may support in the future. It just happens to
                       interoperate with their equipment. Trademarks
                       belong to their owners in a somewhat tautological
                       and obvious fashion.


                       An openconnect VPN server (ocserv), which
                       implements an improved version of the Cisco
                       AnyConnect protocol, was also written.


                       OpenConnect VPN server is an SSL VPN server that is
                       secure, small, fast, and configurable. It
                       implements the OpenConnect SSL VPN protocol and has
                       also (currently experimental) compatibility with
                       clients using the AnyConnect SSL VPN protocol. The
                       OpenConnect protocol provides a dual TCP/UDP VPN
                       channel and uses the standard IETF security
                       protocols to secure it. The OpenConnect client is
                       multi-platform and available here. Alternatively,
                       you can try connecting using the official Cisco
                       AnyConnect client (Confirmed working on Android).
                       For the HAproxy tutorial, you can try the link.


                # ⚓ Failed_to_download_metadata_for_repo⠀⇛


                # ⚓ Bash_Scripting:_Check_if_file_exists⠀⇛


                # ⚓ Bash_Scripting:_Check_if_directory_exists⠀⇛


                       When writing a Bash script, it is common that
                       you’ll run into the need to check for the existence
                       of a directory. Based on the result, your Bash
                       script can proceed with the appropriate action.


                       This functionality can be written into a Bash
                       script or used directly from the command line,
                       without writing a script for it. In this tutorial,
                       you will see how to check if a directory exists in
                       Bash on Linux systems.


                # ⚓ MySQL:_Allow_user_access_to_database⠀⇛


                       After installing MySQL on your Linux system and
                       creating a new database, you will need to setup a
                       new user to access that database, granting it
                       permissions to read and/or write data to it.


                       It is not recommended to use the root account, but
                       rather create a new account and grant privileges as
                       needed. In this tutorial, you will see how to allow
                       a user to access a MySQL database on Linux.


                # ⚓ MySQL:_Allow_access_from_specific_IP_address⠀⇛


                       If you need to allow remote access to your MySQL
                       server, a good security practice is to only allow
                       access from one or more specific IP addresses. This
                       way, you are not needlessly exposing an attack
                       vector to the entire internet.


                       In this tutorial, we will take you through the step
                       by step instructions to allow remote connections to
                       a MySQL server from a specific IP address on a
                       Linux system. These instructions should work
                       independently of whichever Linux distro you are
                       using.


                # ⚓ MySQL:_Allow_remote_connections⠀⇛


                       After installing a MySQL server on a Linux system,
                       by default it will only accept incoming connections
                       from itself (i.e. the loopback address 127.0.0.1).


                       This default configuration works perfectly fine if
                       you are only trying to read or write information
                       from the database on the same server. So users that
                       host their website and MySQL server on the same box
                       won’t need to do any extra configuration to allow
                       remote connections.


                       If you want to allow remote connections to your
                       MySQL server, because you have other computers and/
                       or users that need to access that data, you will
                       need to bind the MySQL service to a public IP
                       address on your server, and perhaps allow incoming
                       MySQL connections through your system firewall.


                       In this tutorial, we will take you through the step
                       by step instructions to allow remote connections to
                       a MySQL server on a Linux system. These
                       instructions should work independently of whichever
                       Linux distro you are using.


                # ⚓ Building_a_Docker_image_in_Jenkinsfile_and_publishing_to
                  ECR_–_Octopus_Deploy⠀⇛


                       In this post, you learn how to build and push the
                       Octopus Deploy underwater app to Amazon Elastic
                       Container Registry (ECR) using Jenkins.


          o § Games⠀➾


                # ⚓ Lilbits:_Fortnite_for_Linux_isn’t_happening,_Vivo_NEX_5
                  specs_leaked,_MAME_turns_25_–_Liliputing⠀⇛


                       While Valve’s game client has been able to run on
                       Linux for years, as of last month just over 1% of
                       Steam users were running Linux (and fewer than 3%
                       were using macOS, with Windows holding a 96%
                       share). It’ll be interesting to see if that starts
                       to change once the Steam Deck hits the streets. And
                       if it does, maybe we’ll see more game makers add
                       support for Linux… but one of the most popular
                       games around isn’t going to add Linux support
                       anytime soon: Epic CEO Tim Sweeney says the company
                       has no plans to port Fortnite to Linux.


                       He says it’s because Epic doesn’t “have confidence
                       that we’d be able to combat cheating at scale under
                       a wide array of kernel configurations including
                       custom ones,” but it’s an interesting take since
                       Epic has already ported its anti-cheat software to
                       support Mac and Linux devices including the Steam
                       Deck.


                       That means third-party games that use Epic’s Easy
                       Anti-Cheat software should be compatible with the
                       Steam Deck. But Epic’s own Fortnite won’t be one of
                       those games.


                       Then again, Fortnite also isn’t available in the
                       Steam Store, so it’s not like it’s a huge surprise
                       that the company doesn’t want to make it too easy
                       for you to play its massively popular game on a
                       device that may lock you into a rival’s ecosystem.


                       But the Steam Deck is basically a PC, and while it
                       will ship with Steam OS, it’s also capable of
                       running Windows. And that could provide folks
                       willing to install an alternate operating system a
                       way to play Fortnite or any other games that may
                       not be available or play well under Linux.


                # ⚓ Fortnite_Is_Never_Coming_To_The_Steam_Deck_–_Invidious⠀⇛


                       With EAC and Battleye now being supported on Linux
                       and by extension the Steam Deck someone asked Tim
                       Sweeney if Fortnite is coming to the Steam Deck and
                       he had a simple answer no, but his reasoning is
                       interesting


                # ⚓ Epic_Is_Worried_About_Fortnite_Cheaters_on_Linux [Ed:
                  Worried about the freedom of the platform, Epic wants to put
                  rootkits in Linux]⠀⇛


                       Epic Games makes Fortnite, which is one of the
                       biggest video game phenomenons in recent history.
                       Because the game is so popular, fans wondered
                       whether it would make its way to Valve’s Steam
                       Deck, but it appears Epic is too worried about
                       cheaters on Linux to make it happen.


                # ⚓ Tim_Sweeney_has_a_point_about_Fortnite_EAC_support_|
                  GamingOnLinux⠀⇛


                       One of the big topics of discourse in the Linux
                       gaming sphere recently has been Tim Sweeney’s
                       statement on porting Fortnite to the Steam Deck,
                       where Sweeney argues that Linux would be too
                       difficult of a target and the market not big enough
                       to warrant the amount of resources it would take to
                       bring all of Fortnite on the platform.


                       The central crux of the issue, from Sweeney’s point
                       of view, is that making Easy Anti-Cheat, with all
                       of its capabilities, run on Steam Deck (and thus on
                       Linux) would be extremely difficult. He argues,
                       that for a game of Fortnite’s size this would open
                       the flood-gates to significant influx of cheaters.


                       There have been some responses to this from the
                       Linux side, with some accusing Sweeney of
                       exaggerating the difficulty of such a port or that
                       his statements are conflicting, because he
                       simultaneously believes the Linux market is too
                       small to be worthwhile but also would provide a way
                       for too many cheaters. I will address some of these
                       aspects a bit later, but for now let’s focus on the
                       main technical blocker, which is Easy Anti-Cheat.


                # ⚓ Proton_vs_Native:_Is_There_Really_A_Difference?_–_Boiling
                  Steam⠀⇛


                       The debate over emulation or translation and native
                       games is at least as old as projects like Wine for
                       Linux, and more generally as old as gaming
                       platforms. More recently Proton and DXVK are the
                       stars, but there have been other tools in the past,
                       like VOGL (to translate DirectX 9 to OpenGL, used
                       in CS:Go, for example). Rather than rehash the same
                       debate here, what I would like to do is explore and
                       pose some questions. Not questions like which is
                       “better” (for whatever your definition of
                       “better”), but to think a bit more deeply about
                       another aspect of this debate. To me, much of the
                       endless back and forth comes down to this: is there
                       really a difference between native and tools like
                       Proton?


                       Hold your flames and passioned arguments please!
                       I’m not here to make the case for one side or the
                       other or throw my hands up to say “it doesn’t
                       matter,” but to attempt to go beyond our surface
                       arguments and see where that may lead us.


                # ⚓ Valheim_developers_have_another_fireside_chat_filled_with
                  teasers_|_GamingOnLinux⠀⇛


                       Ready for even more content to come to Valheim?
                       Well, there’s still a bit of a wait and the
                       developers teased biome updates previously but
                       there’s more to see – and hear.


                       In a fresh fireside chat, with a much upgraded
                       seating area that’s no longer just a fire in a
                       forest, three members of Iron Gate had a talk about
                       the game now it’s one year on from the Early Access
                       release. The fun thing is that this is clearly an
                       unreleased build, as there’s multiple sneaky new
                       things surrounding them. Their official Twitter
                       account even teased “How many sneak peeks can you
                       spot?”.


                # ⚓ Godot_Engine_–_Dev_snapshot:_Godot_4.0_alpha_2⠀⇛


                       Two weeks ago we finally released Godot 4.0 alpha 1
                       ― the first official alpha build of our upcoming
                       major milestone, enabling all interested users to
                       try it out and report bugs, as well as provide
                       feedback on the new features.


                       We got lots of bug reports and many have been fixed
                       already in these two weeks! To keep iterating on
                       stabilizing the 4.0 branch, we’re going to release
                       new alpha builds every other week, so that testers
                       can always have a recent version to test the latest
                       changes.


                       So here’s Godot 4.0 alpha 2 with a great deal of
                       fixes and new features!


                       Be aware that during the alpha stage the engine is
                       still not feature-complete or stable. There will
                       likely be breaking changes between this release and
                       the first beta release. Only the beta will mark the
                       so-called “feature freeze”.


                       As such, we do not recommend porting existing
                       projects to this and other upcoming alpha releases
                       unless you are prepared to do it again to fix
                       future incompatibilities. However, if you can port
                       some existing projects and demos to the new
                       version, that may provide a lot of useful
                       information about critical issues still left to
                       fix.


                       Most importantly: Make backups before opening any
                       existing project in Godot 4.0 alpha builds. There
                       is no easy way back once a project has been
                       (partially) converted.


                # ⚓ Wordle_Reverse-Engineering_And_Automated_Solving_|
                  Hackaday⠀⇛


                       We don’t know about you, but we have mixed feelings
                       about online puzzle fads. On one hand, they are
                       great tool to help keep one sharp, but they’re just
                       everywhere. The latest social-media driven fad,
                       Wordle, may be a little bit too prevalent for our
                       liking, with social media timelines stuffed with
                       updates about the thing. [Ed Locard] was getting a
                       bit miffed with friends’ constant posts about
                       ‘Today’s Wordle’, and was hoping they’d get back to
                       posting pictures of their dogs instead, so did what
                       any self-respecting hacker would do, and wrote a
                       python script to automate solving Wordle puzzles,
                       in a likely futile attempt to get them to stop
                       posting.


          o § Desktop Environments/WMs⠀➾


                # ⚓ Give_Your_Linux_Mint_and_Xubuntu_a_Visual_Uplift_Using
                  Twister_UI⠀⇛


                       Twister UI is the easiest way to give your Linxu
                       Mint and XUbuntu a visual uplift using pre-loaded
                       themes. Here’s how.


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


                      # ⚓ KDE_Community_releases_Plasma_5.24:_It’s_eccentric,
                        just_like_many_old-timers⠀⇛


                             The new release of KDE’s Plasma desktop
                             brings a GNOME Shell-like instant overview
                             and other improvements.


                             KDE is fairly mature now. The FOSS community
                             was founded in 1996, allowing it to call last
                             year’s Plasma release, 5.23, the 25th
                             Anniversary Edition. KDE 1.0 came out in
                             1998, making it the first FOSS desktop for
                             Linux. It does also support FreeBSD, although
                             not always the latest version.


                      # ⚓ Plasma_Mobile_Gear_22.02_is_Out⠀⇛


                             The Plasma Mobile team is happy to announce
                             the Plasma Mobile updates for January to
                             February 2022.


                             Plasma 5.24 was released on February 8th,
                             with many changes to the Plasma Mobile shell.


                             The primary shell repository (and
                             corresponding package name for distributions)
                             has been renamed from plasma-phone-components
                             to plasma-mobile. Devin Lin did a lot of work
                             on the shell for this release.


                      # ⚓ KDE_Neon_vs_Kubuntu_vs_KDE_Plasma⠀⇛


                             One of the best aspects of using a Linux
                             system is that there are many popular desktop
                             environments to choose from. If you do not
                             like the style or behavior of yours, it is
                             simple enough to install a different one.


                             When it comes to the KDE Plasma desktop
                             environment, there are a few different ways
                             to experience it. KDE Plasma comes not only
                             as a graphical user interface, but also
                             includes a slew of applications (over 200;
                             some installed by default and some not) that
                             integrate into the desktop environment and
                             give the user a more unified experience that
                             flows together.


                             If you are ready to try out KDE Plasma, you
                             have a few choices. First of all, you can
                             install it the same way you would any other
                             desktop environment – right from your
                             system’s package manager. But you can also
                             choose from the KDE Neon and Kubuntu Linux
                             distributions.


                             So, which one should you choose? We will go
                             over all the details in this article and help
                             you make an informed decision.


                      # ⚓ More_wlroots_with_KWinFT_5.24_–_subdiff.org⠀⇛


                             In the last year our development focus for
                             KWinFT was on the integration with the
                             wlroots library. In a single sentence our
                             goal with this is to share underlying
                             architecture, patterns and ideas across the
                             Wayland ecosystem, instead of reimplementing
                             things again and again for each and every big
                             project while leaving smaller ones behind.


                             Only a few months ago in October a first
                             iteration of this integration was released
                             with KWinFT 5.23. I talked about it here on
                             this blog and at X.Org Developer Conference
                             (XDC) 2021 in a talk including a primer on
                             wlroots itself.


                             If you’re interested in that first part of
                             integration specifically or a wlroots primer
                             in general check out my talk on YouTube.


                # § GNOME Desktop/GTK⠀➾


                      # ⚓ GNOME_42_Adds_’Dark_Wallpaper’_Support⠀⇛


                             Simply put: when the new standardised dark
                             mode setting is enabled in GNOME 42 a ‘dark’
                             version of the default wallpaper is applied.
                             When light mode is enabled, a ‘light’ version
                             of the wallpaper is applied. If this all
                             sounds a bit familiar it’s probably because I
                             mentioned this was on the way last month when
                             I wrote about the streamlining of wallpaper
                             options in GNOME 42.


                             Naturally, you are free to override this
                             behaviour and set any image you like as the
                             desktop background, regardless of whatever
                             dark mode preference is active.


                      # ⚓ Turbocharging_GNOME_Nightly_and_GNOME_OS
                        infrastructure⠀⇛


                             Mistakes were made¹, and after recent
                             maintenance, Jordan asked Infrastructure Team
                             if there is a chance GNOME Nightly (and its
                             younger sibling GNOME OS) could be moved
                             somewhere with faster network and more
                             storage. After short discussion, we decided
                             to move it to AWS (thanks to promotional
                             credits for open source projects). The new
                             server has three times more space, uses SSD
                             instead of HDD, and should offer up to 10
                             Gbps of network bandwidth.


                             It quickly became apparent we underestimated
                             GNOME Nightly popularity by a large margin.
                             On the second day, the new server transmitted
                             almost 500 GB of data. As we wanted to limit
                             potential egress costs, we reached out to
                             CDN77, and they quickly agreed to help us.
                             CDN77 team aren’t stranger to the open source
                             world, helping other projects like KDE,
                             Gentoo and Manjaro.


                      # ⚓ 7_New_Features_That_Make_GNOME_42_an_Awesome_Release
                        –_It’s_FOSS_News⠀⇛


                             GNOME 42 will be an interesting release.


                             It includes noticeable visual changes and
                             improvements to the desktop experience. Of
                             course, the changes in GNOME 41 compliments
                             the new release as well.


                             GNOME 42 is due on March 23, 2022, but it has
                             almost reached beta (scheduled for February
                             12, 2022).


          o § Distributions⠀➾


                # ⚓ Theme_in_EasyOS_3.3⠀⇛


                       The snapshot was taken on a 1024×768 monitor, which
                       is the smallest resolution that is practical to use
                       with EasyOS. You can see the icons along the top,
                       anything less than 1024 pixels, say 800 pixels, is
                       going to be a problem.


                       In a running Easy, you will see there is a file /
                       root/.packages/default-theme. This is used in woofQ
                       to set the theme, and is included in the release
                       build so that users can see the choices made. Here
                       is the file for Easy 3.3…


                # ⚓ Icon-free_desktop_peach_theme⠀⇛


                       By using the ‘xdotool’ utility, the “Drives” menu
                       will now automatically popup whenever a drive is
                       plugged in.


                       On a screen that is smaller than 1920 pixels wide,
                       the app labels automatically do not display. Here
                       is a snapshot on a 1024×768 screen…


                # ⚓ Experimental_icon-free_desktop⠀⇛


                       I have wanted to try this for a long time, a
                       desktop completely free of icons. You could then
                       have, say, a beautiful nature scene filling almost
                       the entire screen, unmarred by icons. Also, you
                       will no longer have to shove windows aside to get
                       at underlying icons.
                       ROX-Filer is not just a file manager, it also
                       manages the desktop icons and wallpaper. However,
                       it is possible for JWM window manager to display
                       all of the icons, in one or more trays, and also be
                       responsible for the wallpaper.


                       One big problem is the drive partition icons, that
                       will dynamically change as partitions are mounted
                       and unmounted, and drives plugged in and removed.
                       Implementing dynamic drive icons in a JWM tray is
                       not feasible; however, it can be done as a menu.


                # § BSD⠀➾


                      # ⚓ Why_you_should_read_Section_8_of_the_Unix_User’s
                        Manual_•_The_Register⠀⇛


                             If, like me, you were a computer-science
                             graduate student who cut your teeth on
                             Berkeley Unix – complete with the first open-
                             source implementation of TCP/IP – you know
                             Section 8 as the cryptic System Maintenance
                             Commands section of the Unix User’s Manual.


                             It was obvious, to me, that this concluding
                             section warranted a closer look because the
                             introduction warned: “Information in this
                             section is not of great interest to most
                             users.” Judging by my taste in research
                             problems over the years, reading Section 8
                             turned out to be a pretty good investment.


                             But before getting to Section 8, you first
                             learned about the rest of Unix, where you
                             discovered how empowering it is to be able to
                             build new internet applications. Anyone
                             interested in how targeted investments in
                             open-source software, coupled with affordable
                             hardware, can spur innovation should study
                             the role of the Berkeley Software
                             Distribution (BSD) in the success of the
                             internet.


                # § PCLinuxOS/Mageia/Mandriva/OpenMandriva Family⠀➾


                      # ⚓ Mozilla_Thunderbird_update_»_PCLinuxOS⠀⇛


                             Thunderbird, the email client from Mozilla
                             has been updated to 91.6.0 and shipped to the
                             PCLinuxOS Software Repository.


                      # ⚓ Firefox_browser_update_»_PCLinuxOS⠀⇛


                             Firefox, the web browser from Mozilla has
                             been updated to 97.0 and sent to the software
                             repository.


                      # ⚓ Kernel_updates_available_»_PCLinuxOS⠀⇛


                             Current kernels:
                             kernel-5.4.178-pclos1-1-1pclos2022.x86_64.rpm
                             kernel-5.10.99-pclos1-1-1pclos2022.x86_64.rpm
                             kernel-5.15.22-pclos1-1-1pclos2022.x86_64.rpm
                             kernel-5.16.8-pclos1-1-1pclos2022.x86_64.rpm


                # § IBM/Red Hat/Fedora⠀➾


                      # ⚓ 3_ideas_for_bringing_innersource_to_your_company_|
                        Opensource.com⠀⇛


                             As workforces become more distributed and new
                             tools and processes enable more effective
                             asynchronous communication, innersource is
                             very likely to keep changing the work culture
                             as we know it.


                      # ⚓ Comparison_of_Fedora_Flatpaks_and_Flathub_remotes_–
                        Fedora_Magazine⠀⇛


                             In the previous article in this series, we
                             looked at how to get started with Fedora
                             Flatpaks and how to use it. This article
                             compares and contrasts between the Fedora
                             Flatpaks remote and the Flathub remote.
                             Flathub is the de-facto standard Flatpak
                             remote, whereas Fedora Flatpaks is the Fedora
                             Project’s Flatpak remote. The things that
                             differ between the remotes include but are
                             not limited to their policies, their ways of
                             distribution, and their implementation.


                      # ⚓ GitOps_using_Red_Hat_OpenShift_console_4.9_and_4.10_|
                        Red_Hat_Developer⠀⇛


                             This article introduces new Red Hat OpenShift
                             developer console features supporting GitOps
                             processes for applications running in the
                             cloud. Learn about improvements and updates
                             to the OpenShift GitOps Operator, then watch
                             a video showing how to manage cluster
                             configurations and deliver cloud-native
                             applications using Red Hat OpenShift GitOps
                             in the developer console.


                             [...]


                             OpenShift GitOps supports GitOps processes
                             for cloud-native applications. It is built on
                             top of Argo CD, a declarative GitOps
                             continuous delivery tool for Kubernetes, and
                             includes OpenShift’s GitOps Operator and
                             Pipelines Operator, both available from the
                             OpenShift OperatorHub. Figure 1 shows the
                             development flow and tools OpenShift GitOps
                             provides.


                      # ⚓ The_State_of_Customer_Experience_at_Red_Hat:_Where
                        we’ve_been_and_where_we_are_going⠀⇛


                             In the final part of our three-part blog
                             series on the state of customer experience at
                             Red Hat, we will cover enhancements that our
                             marketing teams made in 2021 based on
                             customer feedback. In addition, we will recap
                             where we have been and will look ahead at the
                             plans we have as an organization in 2022 to
                             continue to advance and improve the customer
                             experience.


                      # ⚓ Automating_Content_Management_in_Satellite_6.9_with
                        Ansible_Automation_Platform:_Part_2⠀⇛


                             If you recall from part one of the Automating
                             Content Management series, we are working to
                             minimize the management effort of maintaining
                             the Red Hat Satellite managed content for a
                             large fleet of servers. We developed a simple
                             plan that aligns with the guide to Building a
                             Red Hat Enterprise Linux Standard Operating
                             Environment (SOE) and allows us to automate
                             our process.


                             The simple model is: update the content,
                             publish the content, validate the content,
                             promote the content to the next life cycle
                             environment, rinse and repeat.


                      # ⚓ Agile_adoption:_6_strategic_steps_for_IT_leaders_|
                        The_Enterprisers_Project⠀⇛


                             At Thales Group, our IT organization has been
                             working on a transformation known as NextGen
                             IT for a couple years. One of the key
                             elements is to change from a waterfall
                             delivery model to an agile one.


                             As part of this enterprise initiative, my
                             North America leadership team and I took on
                             the challenge of reworking our entire
                             operation to adopt agile principles for both
                             product and service delivery. We knew what we
                             wanted to achieve: squads, operating using
                             agile methods, driven by user stories, and
                             ultimately delivering incremental value.


                             As we embarked on this journey, something
                             dawned on us early on: The only way we could
                             truly be effective in leading the
                             organization through an agile transformation
                             is if we transformed our leadership team
                             first.


                             We did this by adopting agile principles and
                             adapting the agile process to our leadership
                             team. Here’s what we learned.


                # § Debian Family⠀➾


                      # ⚓ Writing_Debian_New_Contributor_Guide_–_ひとりしずかに。⠀⇛


                             Recently, I’ve wrote Debian New Cotributor
                             Guide in Japanese.


                             Here is the photo of on-demand printed
                             version.


                             This book (JIS-B5 128p) was published for
                             online event – Techbookfest 12th. It covers
                             how to package a new software in Debian.


                # § Canonical/Ubuntu Family⠀➾


                      # ⚓ Ubuntu_Blog:_Canonical_and_CoreSpace_Announce
                        Partnership_To_Offer_Organizations_‘One-Stop_Shopping’
                        for_Private_Clouds⠀⇛


                             Canonical, the publisher of Ubuntu, and
                             CoreSpace, a leading Infrastructure-as-a-
                             Service provider, announced a partnership
                             today that makes it easier and more
                             economical for organizations to set up,
                             customize, and manage private clouds.


                             Hybrid or multi-cloud environments that allow
                             organizations to run workloads where it makes
                             the most sense have become common in the
                             modern enterprise. Because private workloads
                             remain essential for most companies, high-
                             performing private clouds are playing a
                             critical role in their containerisation,
                             cloud, and digital transformation strategies.


                             The Canonical-CoreSpace alliance enables
                             organizations to adopt and get the most out
                             of private clouds. It does this through an
                             end-to-end solution that combines Canonical’s
                             leadership in designing, building, operating,
                             and supporting private clouds with
                             CoreSpace’s infrastructure technology and
                             expertise. With the announcement, CoreSpace
                             becomes Canonical’s first cloud delivery
                             partner in the United States.


          o § Devices/Embedded⠀➾


                # ⚓ Vestel_puts_TikTok_on_Linux-based_TVs_|_Light_Reading⠀⇛


                       Vestel, the largest TV manufacturer in Europe, is
                       today announcing the arrival of TikTok on its vast
                       range of Smart TVs. The app collaboration will see
                       Vestel now bring the video sharing phenomenon to
                       more than twenty high-profile and premium TV
                       brands, including its own portfolio of brand
                       licences.


                       Especially designed for TV, the TikTok app is
                       already available on all Vestel-designed Android
                       Smart TV ranges for brands in the UK, Germany and
                       France. The roll out of TikTok for Vestel Linux-
                       based Smart TVs started on the 20th, January, with
                       the app set to launch on Vestel 2021 UHD TV ranges
                       in early February, and on Vestel 2020 UHD TVs by
                       the end of March.


                       Supporting the display of high definition TikTok
                       content, the Vestel-manufactured Smart TVs bring
                       more detail, vibrancy, and colour to the screen for
                       a larger-than-life home viewing experience.


                # § Open Hardware/Modding⠀➾


                      # ⚓ Intel_Makes_Massive_Investment_in_RISC-V⠀⇛


                             Intel has joined RISC-V International, the
                             global open hardware standards organization,
                             and has announced creation of a billion-
                             dollar fund that aims to speed innovation and
                             development of RISC-V.


                             Intel’s Nios® V processors are already based
                             on RISC-V, but according to the announcement,
                             the company’s investment “will strengthen the
                             overall ecosystem and drive commercialization
                             of RISC-V.”


                # § Mobile Systems/Mobile Applications⠀➾


                      # ⚓ 6_secret_settings_for_better_browsing_in_Chrome_on
                        Android_|_Computerworld⠀⇛


                      # ⚓ OnePlus_&_Oppo_tipped_to_merge_designs_in_Android_13
                        –_9to5Google⠀⇛


                      # ⚓ Google_patched_a_critical_Android_12_security_flaw
                        today_–_PhoneArena⠀⇛


                      # ⚓ Android_12L_is_coming_to_Samsung_tablets_–
                        9to5Google⠀⇛


                      # ⚓ OnePlus’s_first_tablet_may_come_with_Android_12L_OS_–
                        PhoneArena⠀⇛


                      # ⚓ Android_12L_Beta_3_rolling_out,_including_for_Pixel_6
                        and_6_Pro_–_9to5Google⠀⇛


                      # ⚓ Android_12L_Beta_3_now_available,_adds_support_for
                        Pixel_6_series⠀⇛


                      # ⚓ Google_releases_Android_12L_Beta_3_with_support_for
                        the_Pixel_6_series_|_Android_Central⠀⇛


                      # ⚓ Galaxy_S22_gets_Android_12′s_Material_You_design,_so
                        you_can_change_all_the_colors_–_CNET⠀⇛


                      # ⚓ How_to_install_Android_12L_Beta_on_Google_Pixel_–
                        9to5Google⠀⇛


                      # ⚓ Android_12L_beta_3_now_available_on_Pixel_phones_—
                        here’s_how_to_get_it⠀⇛


                      # ⚓ Here’s_everything_new_in_Android_12L_Beta_3_[Gallery]
                        –_9to5Google⠀⇛


                      # ⚓ Wallpaper_Wednesday:_Android_wallpapers_2022-02-09_–
                        Android_Authority⠀⇛


                      # ⚓ Google_just_added_the_Pixel_6_and_Pixel_6_Pro_to_its
                        Android_beta_program⠀⇛


                      # ⚓ Samsung_phones_will_get_an_extra_year_of_Android
                        updates_versus_Google’s_Pixel_–_The_Verge⠀⇛


                      # ⚓ Samsung_Galaxy_S22_Ultra:_stylus-equipped_Android
                        superphone_launched_|_Samsung_|_The_Guardian⠀⇛


                      # ⚓ Samsung_promises_“up_to”_four_major_Android_updates
                        for_some_devices_(and_five_years_of_security_updates)_–
                        Liliputing⠀⇛


                      # ⚓ The_best_mobile_video_editing_app_is_coming_to
                        Android,_with_a_little_help_from_Samsung⠀⇛


                      # ⚓ Samsung_leapfrogs_Google_by_announcing_4-year_Android
                        OS_upgrade_commitment⠀⇛


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


                # ⚓ syslog-ng-future.blog?_Is_this_a_fork_or_what?_–_syslog-ng-
                  future.blog⠀⇛


                       I mentioned in the previous post that I would like
                       to focus on syslog-ng and put it more into the
                       spotlight. I also mentioned that Balabit, the
                       company I was a founder of and the commercial
                       sponsor behind syslog-ng, was acquired by One
                       Identity ~4 years ago. How does this add up? Who
                       owns the Intellectual Property (IP) for syslog-ng?
                       Who am I or this blog affiliated with?


                       I felt this post was important to set things
                       straight and make it easier to understand my
                       motivation. If you are not much into Free Software
                       and Open Source licenses or not interested too much
                       in administrative nuisances of FLOSS projects, feel
                       free to skip this post.


                # § Web Browsers⠀➾


                      # § Mozilla⠀➾


                            # ⚓ Firefox_97.0_Released!_Support_New_Overlay
                              Scrollbars_in_Windows_11⠀⇛


                                   Mozilla Firefox web browser 97.0 was
                                   released a day ago with few new
                                   features and various bug-fixes.


                                   The release does not include any
                                   exciting new features for Linux, but
                                   only adds new minimal overlay scrollbar
                                   support for Windows 11. As well, system
                                   font loading on macOS has been improved
                                   to make opening and switching to new
                                   tabs faster in certain situations.


                            # ⚓ This_Week_in_Glean:_Migrating_Legacy_Telemetry
                              Collections_to_Glean⠀⇛


                                   One of the things I’ve spent a fair
                                   amount of time helping with on the
                                   Glean Team are the migrations from
                                   legacy telemetry to Glean being
                                   performed by different Mozilla
                                   products. This includes everything from
                                   internal tools and libraries to the
                                   mobile browsers and even desktop
                                   Firefox. As it turns out, there were
                                   quite a few projects and products that
                                   needed to be migrated. While we have
                                   started migrating all of our active
                                   products, each of them are at different
                                   stages and have different timelines for
                                   completion. I thought it might be
                                   helpful to take a little narrative look
                                   through what the migration process
                                   looks like, and so here we go!


                # § FSF⠀➾


                      # § GNU Projects⠀➾


                            # ⚓ GNU_Binutils_2.38_released⠀⇛


                                   Version 2.38 of the GNU Binutils tool
                                   set has been released. Changes include
                                   new hardware support (including for the
                                   LoongArch architecture), various
                                   Unicode-handling improvements, a new
                                   –thin option to ar for the creation of
                                   thin archives, and more.


                # § Programming/Development⠀➾


                      # ⚓ QtQuick3D_Realtime_Reflections⠀⇛


                             Reflections enhance the realism of a rendered
                             3D scene and without it we can not have
                             objects like mirrors or puddles of water that
                             reflect the surrounding environment. Raster
                             rendering pipelines like QtQuick3D can
                             approximately calculate reflections using
                             different methods to get good results, as
                             opposed to Ray Tracing where reflections can
                             be calculated accurately by following the
                             light rays and checking where they bounce
                             from. In this blog post, we are going to
                             discuss two different techniques to do
                             reflections in QtQuick3D.


                      # ⚓ Qt_for_MCUs_2.0.1_Released⠀⇛


                             Qt for MCUs 2.0.1 has been released and is
                             available for download. This is the first
                             patch release made on top of the new Qt for
                             MCUs 2.0 series. As a patch release, Qt for
                             MCUs 2.0.1 provides bug fixes and other
                             improvements, and maintains source
                             compatibility with Qt for MCUs 2.0.0. It does
                             not add any new functionality.


                      # ⚓ Qt_for_Android_Automotive_6.2.3_released⠀⇛


                             We are happy to announce the release of Qt
                             for Android Automotive 6.2.3!


                             This release comes with several fixes that
                             improve the stability and developer
                             experience of QAA. Big thanks to our
                             technology preview users for reporting the
                             issues!


                      # ⚓ Raspberry_Pi:_This_new_feature_could_help_you_get
                        started_more_easily⠀⇛


                             The makers of Raspberry Pi computers have
                             released a new boot loader that lets users
                             install an operating system like the official
                             OS directly on the compute board rather than
                             via a separate computer.


                             Until now, Raspberry Pi owners needed the
                             imager on a macOS, Windows or Ubuntu desktop
                             to install the OS on a Pi device. It could
                             also be done via another Pi device by
                             connecting SD card readers on each of them.
                             The new Raspberry Pi bootloader delivers
                             network installation, cutting out the need
                             for a second computer to install the OS.


                      # ⚓ Getting_Started_with_C++_and_InfluxDB [Ed:
                        "InfluxData sponsored this post." The so-called 'media'
                        or so-called 'news' sites on the Web have become
                        nothing short of webspam]⠀⇛


                      # § Perl/Raku⠀➾


                            # ⚓ gfldex:_Fuzzy_commands⠀⇛


                                   Reading can make one learned. Today I
                                   had a peak into lizmat´s new module
                                   shorten-sub-commands and learned about
                                   Parameter.constraint_list. We need this
                                   method to introspect a routines literal
                                   parameters.


                      # § Shell/Bash/Zsh/Ksh⠀➾


                            # ⚓ how_to_use_vim’s_build_in_history_(re_run_re
                              play_commands)_by_line_number
                              ⠀⇛


                            # ⚓ Using_the_Linux_fold_command_to_make_text_more
                              readable_|_Network_World⠀⇛


                                   The Linux fold takes lines of text and
                                   breaks them into chunks based on the
                                   arguments that you provide. With no
                                   arguments, fold will break lines at 80
                                   characters.


                                   The first example below uses a single-
                                   line text file that includes
                                   indications of character positions.
                                   First, we count the number of
                                   characters and lines in the file using
                                   the wc -l and wc -l command…


                      # § Rust⠀➾


                            # ⚓ Dare_to_ask_for_more_#rust2024⠀⇛


                                   Last year, we shipped Rust 2021 and I
                                   have found the changes to be a real
                                   improvement in usability. Even though
                                   the actual changes themselves were
                                   quite modest, the combination of
                                   precise capture closure and simpler
                                   formatting strings (println!(“{x:?}”)
                                   instead of println!(“{:?}”, x)) is
                                   making a real difference in my “day to
                                   day” life.1 Just like NLL and the new
                                   module system from Rust 2018, I’ve
                                   quickly adapted to these new
                                   conventions. When I go back to older
                                   code, with its clunky borrow checker
                                   workarounds and format strings, I die a
                                   little inside.2


                                   As we enter 2022, I am finding my
                                   thoughts turning more and more to the
                                   next Rust edition. What do I want from
                                   Rust, and the Rust community, over the
                                   next few years? To me, the theme that
                                   keeps coming to mind is dare to ask for
                                   more. Rust has gotten quite a bit nicer
                                   to use over the last few years, but I
                                   am not satisfied. I believe that there
                                   is room for Rust to be 22x more
                                   productive3 and easy to use than it is
                                   today, and I think we can do it without
                                   meaningfully sacrificing reliability,
                                   performance, or versatility.


                      # § Java⠀➾


                            # ⚓ Java_break_and_continue_Statements⠀⇛


                                   The continue statement is used in the
                                   body of a while, do and for loop. The
                                   break statement is used in these loops,
                                   as well as in the switch statement.
                                   Iteration can be considered the
                                   execution of a loop’s body, over and
                                   over. The break statement stops the
                                   iteration. The continue statement skips
                                   one execution (rest of the statements
                                   below) of the body. This article
                                   explains the employment of the continue
                                   and break statements in Java. The
                                   while-loop compound statement will be
                                   used. For the break statement, the
                                   switch compound statement will also be
                                   used.


          o § Standards/Consortia⠀➾


                # ⚓ MIPI_CSI-2_v4.0_adds_features_for_always-on,_low_power
                  machine_vision_applications_–_CNX_Software⠀⇛


                       While MIPI CSI-2 standard was first introduced in
                       2005 as a high-speed protocol for the transmission
                       of still and video images from image sensors to
                       application processors, the standard has evolved
                       over the years, and the latest MIPI CSI-2 v4.0
                       introduces features to better support always-on,
                       low power machine vision applications, high-
                       resolution sensors, and high-dynamic-range
                       automotive image sensors.


                       The main changes for v4.0 include support for a
                       two-wire interface (MIPI I3C) to lower cost and
                       complexity, multi-pixel compression for the latest
                       generation of advanced image sensors, and RAW28
                       color depth for better image quality and an
                       improved signal-to-noise (SNR) ratio.


    * § Leftovers⠀➾


          o ⚓ Heart-Shaped_Pizza_History:_Less_Pizza_for_the_Same_Price⠀⇛


                 It doesn’t matter that the heart symbol doesn’t look like
                 an actual human heart, and never has. It nonetheless has
                 become our general symbol for what we think of as love.
                 The symbol is everywhere—on greeting cards, in emoji. But
                 there’s one particular heart-shaped thing that I can’t
                 stop thinking about right now: The heart-shaped pizza. It
                 seems like the most crass of romantic gestures, to put
                 tomato sauce and cheese on bread, but in a way, it’s
                 genius. And seemingly every major chain that’s not
                 Domino’s is offering a heart-shaped pizza. (Hey, if two
                 people are sharing, it makes splitting it kind of easy, I
                 guess.) I can’t stop thinking about the fact that lots of
                 people are expected to buy cheap, rubbery Papa John’s
                 pizza as a way to show off their love. (Papa John’s
                 doesn’t even use heart-shaped pepperonis!) Today’s
                 Tedium, in our first Valentine’s-themed post in our
                 seven-year history, talks about our cultural tendency to
                 make heart-shaped food. Especially pizza.


          o § Hardware⠀➾


                # ⚓ China_Loves_Battery_Swapping_EVs,_But_Will_They_Ever_Make
                  It_Here?_|_Hackaday⠀⇛


                       Electric vehicles promise efficiency gains over
                       their gas-fuelled predecessors, but the issue of
                       recharging remains a hurdle for many eager to jump
                       on board with the technology. The problem is only
                       magnified for those that regularly street park
                       their vehicles or live in apartments, without
                       provision to charge a vehicle overnight at home.


                       Battery swapping promises to solve that issue,
                       letting drivers of EVs change out their empty
                       battery for a freshly charged one in a matter of
                       minutes. The technology has been widely panned and
                       failed to gain traction in the US.


                       However, as it turns out, battery swapping for EVs
                       is actually thing in China, and it’s catching on at
                       a rapid rate.


                # ⚓ Build_Your_Own_CRT_TV_|_Hackaday⠀⇛


                       There was a time following the Second World War
                       when TV sets for the nascent broadcast medium were
                       still very expensive, but there was an ample supply
                       of war-surplus electronic parts including ex-radar
                       CRTs. Thus it wasn’t uncommon at all for
                       electronics enthusiasts of the day to build their
                       own TV set, and magazines would publish designs to
                       enable them. With a burgeoning consumer electronics
                       industry the price of a new TV quickly dropped to
                       the point of affordability so nobody would consider
                       building one themselves today. Perhaps that should
                       be amended to almost nobody, because [Retro Tech or
                       Die] has assembled a small black-and-white CRT TV
                       from a kit he found on AliExpress.


                # ⚓ That_Clock_On_The_Wall_Is_Actually_A_Network_Ping_Display_|
                  Hackaday⠀⇛


                       We’ve all been online from home a bit more than
                       usual lately, in ways that often stretch the limits
                       of what our ISP can muster. You know the signs —
                       audio that drops out, video sessions that make you
                       look like [Max Headroom], and during the off-hours,
                       getting owned in CS:GO by pretty much everyone. All
                       the bandwidth in the world won’t make up for high
                       latency, and knowing where you stand on that score
                       is the point of this ping-tracking clock.


                # ⚓ It’s_Always_Floppy_Time!_|_Hackaday⠀⇛


                       On [Jan Derogee]’s desk is something that wouldn’t
                       look out of place for many of us, a pile of
                       computer magazines with a case of 3.5″ floppy disks
                       on top of it. The causal observer would see nothing
                       more than the detritus of a retrocomputer
                       enthusiast’s existence, but stick around. In fact
                       it’s a clock, and one of the most unusual ones
                       we’ve seen in a long time.


                # ⚓ A_Guide_To_Milling_PCBs_At_Home_|_Hackaday⠀⇛


                       If you keep up with various retro vacuum tube
                       projects, you probably have run across
                       [UsagiElectric] aka [David]’s various PCBs that he
                       makes on his own Bridgeport EZ-Track 3-axis milling
                       machine — massively oversized for the job, as he
                       puts it. In a recent video, [David] walks us
                       through the steps of making a sample PCB,
                       introducing the various tools and procedures of his
                       workflow. He points out that these are the tools he
                       uses, but the overall process should be similar no
                       matter what tools you use.


                # ⚓ Mystery_Effect_Causing_LEDs_To_Glow_During_Reflow_|
                  Hackaday⠀⇛


                       Sometimes you notice something small that
                       nevertheless you can’t explain. [Greg Davill] found
                       himself in just such a situation this week when he
                       noticed some green LEDs glowing dimly when
                       reflowing some boards. Naturally, [Greg] set out to
                       investigate.


                       The green LEDs were wired up as power indicators,
                       and [Greg] suspected that the polymer caps on the
                       board might be generating a small current somehow
                       that was causing the LEDs to light up ever so
                       slightly. A simple test hooked a polymer cap
                       directly up to a multimeter. When warmed with a
                       heat gun, the meter showed a small current “in the
                       5-10 uA range.”


                       [...]


                       It’s a curious case, and has many speculating as to
                       the causative mechanism on Twitter. Explanations
                       from thermoelectric effects to chemical reactions
                       inside the capacitor. If you’ve got the inside
                       scoop on what’s going on here, don’t hesitate to
                       let us know in the comments. Meanwhile, check out
                       some of [Greg]’s best work – a glowing D20 dice
                       featuring a whopping 2400 LEDs.


                # ⚓ Working_Model_Reveals_Amazing_Engineering_Of_Webb’s_Mirror
                  Actuators_|_Hackaday⠀⇛


                       We end up covering a lot of space topics here on
                       Hackaday, not because we’re huge space nerds —
                       spoiler alert: we are — but because when you’ve got
                       an effectively unlimited budget and a remit to make
                       something that cannot fail, awe-inspiring
                       engineering is often the result. The mirror
                       actuators on the James Webb Space Telescope are a
                       perfect example of this extreme engineering, and to
                       understand how they work a little better, [Zachary
                       Tong] built a working model of these amazing
                       machines.


                       The main mirror of the JWST is made of 18 separate
                       hexagonal sections, the position of each which must
                       be finely tuned to make a perfect reflector. Each
                       mirror has seven actuators that move it through
                       seven degrees of freedom — the usual six that a
                       Stewart platform mechanism provides, plus the
                       ability to deform the mirror’s curvature slightly.
                       [Zach]’s model actuator is reverse-engineered from
                       public information (PDF) made available by the
                       mirror contractor, Ball Aerospace. While the OEM
                       part is made from the usual space-rated alloys and
                       materials, the model is 3D printed and powered by a
                       cheap stepper motor.


                # ⚓ A_Tuning_Fork_Clock,_With_Discrete_Logic_|_Hackaday⠀⇛


                       [Willem Koopman aka Secretbatcave] was looking at a
                       master clock he has in his collection which was
                       quite a noisy device, but wanted to use the
                       matching solenoid slave clock mechanism he had to
                       hand. Willem is a fan of old-school ‘sector’
                       clocks, so proceeded to build his ideal time piece
                       — Vibrmatic — exactly the way he wanted. Now, since
                       most time keeping devices utilise a crystal
                       oscillator — which is little more than a lump of
                       vibrating quartz — why not scale it up a bit and
                       use the same principle, except with a metal tuning
                       fork? (some profanity, just to warn you!)


          o § Integrity/Availability⠀➾


                # § Proprietary⠀➾


                      # ⚓ Microsoft_Teams_will_soon_have_LinkedIn_and_OneDrive
                        integration [Ed: While Microsoft keeps antitrust
                        regulators busy with “GAFA” or “Big Tech” [1, 2]]⠀⇛


                      # ⚓ Microsoft_February_2022_Patch_Tuesday:_48_bugs
                        squashed,_one_zero-day_resolved [Ed: Charlie Osborne of
                        Microsoft_ZDNet failing to mention that Microsoft
                        intentionally_leaves_holes_for_the_NSA]⠀⇛


                      # ⚓ Microsoft_365_will_tell_your_boss_almost_immediately
                        if_you_send_an_inappropriate_message_|_TechRadar [Ed:
                        TechRadar is posting promotional ‘articles’ (ads) that
                        are painting spyware as “protection”]⠀⇛


                             Microsoft will soon roll out a new system
                             that will help businesses identify
                             inappropriate messages sent by employees over
                             its productivity and collaboration software.


                      # § Security⠀➾


                            # ⚓ Linux_kernel_developer_Alan_Cox_on_Microsoft
                              security_practices._“I_pointed_out_that_they_ship
                              an_OpenSSL_from_2017_and_they_won’t_even
                              respond.”._–_BaronHK’s_Rants⠀⇛


                                   Linux kernel developer Alan Cox on
                                   Microsoft security practices. “I
                                   pointed out that they ship an OpenSSL
                                   from 2017 and they won’t even
                                   respond.”.


                                   OpenSSL is a deeply troubled project
                                   that Big Tech uses extensively. It has
                                   a long history of disastrous security
                                   flaws, thousands, some of which have
                                   THEIR OWN Wikipedia articles, which
                                   routinely compromise the security of
                                   the connection between your computer
                                   and a Web server. So it’s not like
                                   shipping a recent version is important
                                   or anything.


                                   Shipping it in the first place is
                                   stupid when it has a community fork
                                   that removes poorly written and little-
                                   used code and audits and fixes the
                                   rest. But that’s Microsoft/Big Tech/
                                   what most “Linux” distributions have
                                   become under their guiding hand.


                            # ⚓ Breaking_256-bit_Elliptic_Curve_Encryption_with
                              a_Quantum_Computer_–_Schneier_on_Security⠀⇛


                                   In other words: no time soon. Not even
                                   remotely soon. IBM’s largest ever
                                   superconducting quantum computer is 127
                                   physical qubits.


                            # ⚓ The_impact_of_hardware_specifications_on
                              reaching_quantum_advantage_in_the_fault_tolerant
                              regime⠀⇛


                            # ⚓ 2021_Trends_Show_Increased_Globalized_Threat_of
                              Ransomware [Ed: Could they they type this with a
                              straight face without_mentioning_Microsoft_or
                              Windows?]⠀⇛


                                   CISA, the Federal Bureau of
                                   Investigation (FBI), the National
                                   Security Agency (NSA), the Australian
                                   Cyber Security Centre (ACSC), and the
                                   United Kingdom’s National Cyber
                                   Security Centre (NCSC-UK) have released
                                   a joint Cybersecurity Advisory (CSA)
                                   highlighting a global increase in
                                   sophisticated, high-impact, ransomware
                                   incidents against critical
                                   infrastructure organizations in 2021.
                                   This CSA provides observed behaviors
                                   and trends as well as mitigation
                                   recommendations to help network
                                   defenders reduce their risk of
                                   compromise by ransomware.


                            # § Fear, Uncertainty, Doubt/Fear-mongering/
                              Dramatisation⠀➾


                                  # ⚓ Deterrence,_diplomacy,_in_hybrid_war._UK
                                    Foreign_Office_hacked._Threats_to_Linux-
                                    based_multi-cloud_environments._Patch
                                    Tuesday. [Ed: They're calling everything
                                    "Linux", even things that have nothing to
                                    do with Linux]⠀⇛


                                         Russia stages more general
                                         purpose forces near Ukraine
                                         (notably moving amphibious
                                         assault ships from the
                                         Mediterranean through the
                                         Dardanelles and toward Ukraine’s
                                         Black Sea coast) while diplomatic
                                         efforts to reduce tension
                                         continue. Belarus continues to
                                         emerge as an important staging
                                         point for Russian conventional
                                         forces. No fresh, large-scale
                                         cyber activity, however, is being
                                         reported. The New York Times
                                         reviews the current state of
                                         multilateral negotiations and
                                         sees, if not stalemate, at least
                                         stasis. Its analysis foresees “a
                                         drawn-out and dangerous
                                         diplomatic slog toward a
                                         difficult settlement.”


                                  # ⚓ Threat_actors_view_Linux_servers_as_their
                                    ticket_to_accessing_multi-cloud
                                    environments [Ed: Only weeks after Dirk
                                    left VMware the company has launched an
                                    anti-Linux FUD campaign in the media, maybe
                                    in order to distract from the back doors in
                                    its own products and blame Linux itself
                                    (while violating GPL, too)]⠀⇛


                                  # ⚓ How_cybercriminals_are_using_malware_to
                                    target_Linux-based_operating_systems [Ed:
                                    Ransomware is a Windows issue (more than
                                    90% of the time, but VMware now pays the
                                    media to shift attention to Linux]⠀⇛


                                         As the most common cloud
                                         operating system, Linux is a core
                                         part of digital infrastructure
                                         and is quickly becoming an
                                         attacker’s ticket into a multi-
                                         cloud environment. Current
                                         malware countermeasures are
                                         mostly focused on addressing
                                         Windows-based threats, leaving
                                         many public and private cloud
                                         deployments vulnerable to attacks
                                         that target Linux-based
                                         workloads, VMware reveals.


                                  # ⚓ Cybercriminals_Are_Using_Malware_To
                                    Target_Linux-Based_Systems [Ed: Microsoft-
                                    connected sites are at it, too]⠀⇛


                                  # ⚓ Cybercriminals_Target_Linux‑based_Systems
                                    with_Ransomware_and_Cryptojacking_Attacks_–
                                    VMware_News⠀⇛


                                  # ⚓ Cybercriminals_Target_Linux-based_Systems
                                    With_Ransomware_and_Cryptojacking_Attacks
                                    [Ed: VMware is attacking Linux again with
                                    FUD as paid press releases, adding to its
                                    GPL_violations etc.]⠀⇛


                                  # ⚓ Linux_Malware_on_the_Rise [Ed: Blindly
                                    repeating the talking points]⠀⇛


          o § Environment⠀➾


                # § Energy⠀➾


                      # ⚓ The_Environmental_Costs_of_Cryptocurrency⠀⇛


                             New Scientist defines Bitcoin as a digital
                             currency that relies on peer-to-peer software
                             and cryptography. A public ledger (or
                             blockchain) records all Bitcoin transactions,
                             and copies of those transactions are held on
                             servers (or nodes) around the world.
                             “Consensus on who owns which coins is reached
                             cryptographically across these nodes rather
                             than relying on a central source of trust
                             like a bank.”


                             To acquire Bitcoin, you can buy it, you can
                             sell something and be paid in Bitcoin, or you
                             can mine it. Each cryptocurrency transaction,
                             however, involves an immense amount of
                             processing power, with this article
                             estimating that each Bitcoin transaction
                             consumes more than $100 worth of electricity.


          o § Internet Policy/Net Neutrality⠀➾


                # ⚓ #SocialMediaComplianceWatch:_analysis_of_Social_Media
                  Compliance_Reports_of_November,_2021⠀⇛


                       Google (including YouTube), Facebook, Instagram,
                       ShareChat, Snap, Twitter and WhatsApp released
                       their reports in compliance with Rule 4(1)(d) of
                       the IT Rules 2021 for the month of November, 2021.
                       The reports were made available in January 2022 and
                       they exhibit similar shortcomings. This outlines
                       lack of effort on the part of the SSMIs and the
                       government to further transparency and
                       accountability in platform governance. The SSMIs
                       have continued to not report on government
                       requests, used misleading metrics, and not
                       disclosed how they use algorithms for proactive
                       monitoring. You can read our analysis of the
                       previous reports here.


          o § Monopolies⠀➾


                # § Copyrights⠀➾


                      # ⚓ A_Paper_Archaeology:_Piranesi’s_Ruinous_Fantasias_–
                        The_Public_Domain_Review⠀⇛


                             From the vast confines of his imaginary
                             prisons to the billowy scenes that comprise
                             his grotteschi, the early works of Giovanni
                             Battista Piranesi wed the exacting details of
                             first-hand observation with the farthest
                             reaches of artistic imagination. Susan
                             Stewart journeys through this 18th-century
                             engraver-architect’s paper worlds.



                    ䷩ 𝚕𝚒𝚗𝚎 6972

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Gemini_version_available_♊︎


✐ Links_9/2/2022:_GNU_Binutils_2.38_and_Google_Stadia_Failing⠀✐


Posted in News_Roundup at 2:01 pm by Dr. Roy Schestowitz


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇GNOME bluefish⦈


§ Contents⠀➾


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


          o § Audiocasts/Shows⠀➾


                # ⚓ Shockingly_Pragmatic_|_Coder_Radio_452_|_Jupiter
                  Broadcasting⠀⇛


                       Mike makes a shocking admission, and Chris wishes
                       he had a time machine.


          o § Applications⠀➾


                # ⚓ Best_Free_and_Open_Source_Alternatives_to_Atlassian
                  Trello⠀⇛


                       Atlassian Corporation Plc is a software company
                       founded in 2002 that develops products for software
                       developers, project managers and other software
                       development teams. It employs over 7,000 people and
                       is headquartered in Sydney, Australia.


                       Atlassian’s range of proprietary software includes
                       software for collaboration, development, and issue
                       tracking software for teams. Atlassian dominates
                       several markets where it still has intense
                       competition.


          o § Instructionals/Technical⠀➾


                # ⚓ Checking_out_a_Git_branch_further_back_than_the_head⠀⇛


                       It’s possible to move a branch (including ‘main’)
                       back to an older commit while staying on the
                       branch. This avoids Git complaints about being in a
                       detached HEAD state and makes ‘git status’ do
                       useful things like report how many commits you are
                       behind the upstream tip. As far as I know so far,
                       the way you do this is: [...]


                # ⚓ Ignore_A_Package_From_Being_Upgraded_In_Arch_Linux_–
                  OSTechNix [Ed: Updated after 5 years]⠀⇛


                       Sometimes, you may want to ignore a package from
                       being upgraded in your Arch Linux system. It is
                       true that the outdated packages might be vulnerable
                       and not safe to our system. Yes! We need to update
                       the Arch Linux system regularly to get latest
                       packages and make our system more secure. However,
                       we need to hold a package or group of packages from
                       being upgraded for certain reasons.


                # ⚓ How_to_install_WhatsApp_on_Linux⠀⇛


                       WhatsApp is a telecommunications application to
                       provide video, chat and voice communication between
                       computers, tablets, and mobile devices over the
                       Internet connection. All you need is a phone number
                       to sign up, and then you can send messages from
                       your Linux system.


                       There is no official Linux client for WhatsApp,
                       unfortunately. The good news is that it is still
                       possible to download WhatsApp clients made by
                       users, which will use WhatApp’s web interface to
                       create a seamless experience. Using this method, we
                       are able to have WhatsApp on Linux and have it
                       resemble the same client you would get on a Windows
                       PC.


                # ⚓ How_to_install_PHP_on_Ubuntu_Linux⠀⇛


                       PHP is one of the most used languages when it comes
                       to programming dynamic web sites. If you are a web
                       developer, or just hosting a website that requires
                       PHP, you will need to install the software on your
                       server in order for your website to make use of its
                       PHP code.


                       PHP is also required by various content management
                       systems, including the most popular one in the
                       world, WordPress. In this tutorial, we will take
                       you through the step by step instructions to
                       install PHP on Ubuntu Linux.


                # ⚓ How_to_install_Grub_Customizer_on_Linux_(all_major
                  distros)⠀⇛


                       Grub Customizer is a software package that does
                       exactly as the name would imply. It allows the user
                       to customize different aspects of the grub boot
                       menu – such as the order that entries appear in the
                       list, how long grub waits before selecting a
                       default system to boot to, etc.


                       The good news is that you can use this application
                       on all major Linux distros. In this tutorial, we
                       will show you how to install the Grub Customizer
                       package on all Linux systems.


                # ⚓ How_to_fix_Kubernetes_namespaces_stuck_in_the_terminating
                  state_|_Enable_Sysadmin⠀⇛


                       A Kubernetes namespace isolates specific system
                       resources usually visible to all processes. Each
                       namespace has its own services, pods, and
                       deployments in the cluster. A namespace in
                       Kubernetes is essentially the same as a project in
                       OpenShift.


                       These days, most products are containerized and can
                       easily be deployed on Kubernetes or OpenShift. This
                       requires continuous application deployment and
                       testing. I recently worked on test automation that
                       involves the creation of namespaces, resources, and
                       data in the namespace, followed by running the test
                       suite, and then removing the data and namespace
                       once the tests are complete.


                       While frequently creating and deleting namespaces
                       in Kubernetes, I stumbled upon an issue where a
                       namespace got stuck in the Terminating state and
                       just refused to delete.


                       I use minikube to run Kubernetes locally, but you
                       can use the following steps and commands in any
                       Kubernetes or OpenShift environment.


                # ⚓ How_to_Fix_“Unacceptable_TLS_certificate”_Error_in_Linux?⠀⇛


                       When it comes to SSL/TLS certificates, you may come
                       across a variety of issues, some related to the
                       browser or a problem in a website’s back-end.


                       One such error is “Unacceptable TLS certificate” in
                       Linux.


                       Unfortunately, there’s no “one-solves-it-all’
                       answer to this. However, there are some potential
                       solutions that you can try, and here, I plan to
                       highlight those for you.


                # ⚓ How_to_Add_Linux_Host_to_Nagios_Ubuntu_Server_–_Part_2⠀⇛


                       This article assumes that you already have Nagios
                       installed on your Ubuntu system. If not, follow the
                       Part 1 tutorial guide article which is succeeded by
                       this one.


                # ⚓ How_To_Install_Podman_on_Linux_Mint_20_–_idroot⠀⇛


                       In this tutorial, we will show you how to install
                       Podman on Linux Mint 20. For those of you who
                       didn’t know, Podman (POD Manager) is a tool for
                       managing OCI containers and pods. It’s an open-
                       source project that can be used in most of the
                       Linux distributions that use Demon-less container
                       Engines. It exposes the same command-line interface
                       as Docker but runs containers unprivileged by
                       default.


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


                # ⚓ Create_a_data_stream_with_Amazon_Kinesis_|_Red_Hat
                  Developer⠀⇛


                       Streaming data is key to many modern applications.
                       This tutorial walks you through setting up a data
                       stream using Amazon Kinesis and Node.js.


                # ⚓ Getting_started_with_Arduino⠀⇛


                       The Arduino is an advanced form of a
                       microcontroller which is comparatively easier to
                       work on. Similarly, it is a kind of plug and play
                       device and is much easier for the beginners to
                       learn how to make different projects related to
                       embedded systems. Arduino has different types of
                       microcontroller boards having different
                       specifications and the most common Arduino board is
                       the Arduino Uno board. All the Arduino boards can
                       be configured using the Arduino IDE software. This
                       discourse explains how to use the Arduino Uno board


                # ⚓ seife’s_assorted_rants:_dracut:_fix_hibernate_after_fresh
                  install⠀⇛


                       Just for fun, I finally installed a fresh
                       Tumbleweed on one of my machines to actually find
                       out if I’m missing out on new features that are
                       masked by just always updating the old
                       installation.


                       One feature that I had missed was, that hibernation
                       was no longer working. Or, to be more exact,
                       hibernation was working, but resume was not.
                       Investigating the issue, I found that dracut’s
                       “resume” module was not included in the initramfs,
                       which in turn lead to initramfs not even trying to
                       resume.


                       The dracut mechanism has some logic to actually
                       find out if suspend and resume is configured, and
                       when it decides it is not, it will just skip adding
                       the “useless” resume module.


                       Unfortunately, the check is faulty IMHO. It checks,
                       if the resume device is configured in the kernel,
                       and if it is, it adds the module to dracut. The
                       problem is, that this kernel config is written by
                       the resume code in the initrd… so during
                       installation this is not the case, and as a result
                       the module is not added to initrd, which leads to
                       the config not being set on next reboot…


                # ⚓ Install/Upgrade_XanMod_Kernel_LTS_on_Rocky_Linux_8_–
                  LinuxCapable⠀⇛


                       XanMod is a free, open-source general-purpose Linux
                       Kernel alternative to the stock kernel that is on
                       Rocky Linux. It features custom settings and new
                       features and is built to provide a responsive and
                       smooth desktop experience, especially for new
                       hardware.


                       Installing a third-party kernel may be for you for
                       users seeking to have their system kernel up to
                       date and not wanting to install kernels or use the
                       testing/unstable repositories manually.


                       Currently, the XanMod LTS kernel is supported and
                       not the latest bleeding-edge using the rmnscnce/
                       kernel-xanmod.


                # ⚓ Install/Upgrade_MakeMKV_on_Debian_11_Bullseye_–
                  LinuxCapable⠀⇛


                       MakeMKV is a free, open-source tool that can
                       convert video clips from DVDs and Blu-rays, which
                       are usually encrypted. The output will have most
                       information preserved but not changed in any way;
                       it’s perfect for people who want their media
                       without hassle or headache caused by software
                       limitations like those found with some other
                       transcoder apps.


                       In the following tutorial, you will learn how to
                       install or upgrade to the latest version of MakeMKV
                       on Debian 11 Bullseye using a recommended
                       repository by the MakeMKV team to provide the most
                       up-to-date version.


                # ⚓ Install_Podman_on_Debian_11_–_Unix_/_Linux_the_admins
                  Tutorials⠀⇛


                       Podman is an alternative to Docker that is backed
                       by Red Hat and IBM. The open-source project has a
                       great ease of use, which is a big draw among
                       developers.


                       Well, one of the differences is that it is not
                       based on daemons (services in the *nix world). One
                       of the advantages is that Docker has a single
                       daemon that when it handles many containers grows
                       and grows, becoming much heavier.


                       This last point is where Podman wants to be a solid
                       alternative to Docker and is where Podman bases its
                       reason.


                       As you would expect, Podman has great support for
                       all the Linux distributions out there. This is also
                       the case for Debian 11 where the installation is
                       effortless.


                # ⚓ Install_Htop_on_Debian_11_Bullseye_–_LinuxCapable⠀⇛


                       Htop is a free, open-source, cross-platform
                       interactive process viewer. It is a text-mode
                       application (for console or X terminals) and
                       requires ncurses. The terminal UI is a great way to
                       see what your system looks like inside, both in
                       terms of processes and other info. It’s also
                       completely customizable, so you can change colors
                       or add different widgets for more visual
                       representation!


                       In the following tutorial, you will learn how to
                       install Htop Interactive Process Viewer on Debian
                       11 Bullseye desktop or server.


                # ⚓ 6_Linux_metacharacters_I_love_to_use_on_the_command_line_|
                  Opensource.com⠀⇛


                       Early in my Linux journey, I learned how to use the
                       command line. It’s what sets Linux apart. I could
                       lose the graphical user interface (GUI), but it was
                       unnecessary to rebuild the machine completely. Many
                       Linux computers run headless, and you can
                       accomplish all the administrative tasks on the
                       command line. It uses many basic commands that all
                       are familiar with—like ls, ls-l, ls-l, cd, pwd,
                       top, and many more.


          o § Games⠀➾


                # ⚓ Google_Stadia’s_Failure_Is_Almost_Complete⠀⇛


                       While Google’s Stadia game streaming service
                       arrived with a lot of promise, it generally landed
                       with a disappointing thud. A limited catalog,
                       deployment issues, and a quality that couldn’t
                       match current gen game consoles meant the service
                       just never saw the kind of traction Google (or a
                       lot of other people) originally envisioned. In the
                       years since, developers have been consistently
                       abandoning the platform, and Google has
                       consistently sidelined the service, even shutting
                       down its own development efforts as a parade of
                       executives headed for the exists.


                # ⚓ How_to_Connect_Xbox_Controller_with_Raspberry_Pi⠀⇛


                       For those who want to enjoy a better gaming
                       experience, they certainly need an Xbox controller
                       in order to play games with great control. Getting
                       a wireless Xbox controller is a bit expensive so
                       most people stick with buying an affordable wired
                       Xbox controller. So, if you are one of those people
                       who have a wired Xbox USB controller and you want
                       to play games with it on your Raspberry Pi OS then
                       you certainly need help in configuring it on the
                       device.


                # ⚓ Wadjet_Eye_Games_brings_Shardlight_over_to_Linux_|
                  GamingOnLinux⠀⇛


                       Is there no stopping Wadjet Eye Games? Shardlight,
                       another quality point and click adventure from 2016
                       has now been ported over to Linux. This adds to the
                       list of titles that Wadjet Eye Games has recently
                       upgraded and ported (or re-ported) to Linux
                       including Resonance, Technobabylon, Unavowed,
                       Gemini Rue, The Blackwell Bundle.


                # ⚓ ChimeraOS_30_is_out_with_big_OneXPlayer_improvements⠀⇛


                       ChimeraOS continues to provide a full-screen
                       console-like Linux experience for those that want
                       it, and ChimeraOS 30 is now live with some great
                       upgrades.


                       First, the usual upgrades to the major parts like
                       Linux Kernel 5.16.5, Mesa drivers 21.3.5, NVIDIA
                       drivers 510.47.03 and upgrades to their own special
                       compositor and Chimera software. Plus, a bump for
                       RetroArch to version 1.10.0. The bigger changes
                       come for those who have a OneXPlayer handheld
                       device as it should work vastly better with
                       ChimeraOS now thanks to controller detection fixes,
                       screen orientation fixes, a fix for custom
                       resolution support and a default resolution set
                       properly to 1280×800. So out of the box, OneXPlayer
                       devices should work nicely.


                # ⚓ SimAirport_getting_some_great_upgrades_ready_for_the_Steam
                  Deck_|_GamingOnLinux⠀⇛


                       A little building and management on the go?
                       SimAirport from LVGameDev LLC sounds like it might
                       be a good choice, with the developer highlighting
                       upcoming changes and improvements for the Steam
                       Deck and it will help all platforms.


                       You probably don’t need a big explanation to begin
                       with on this one, the name SimAirport speaks for
                       itself really. You construct and manage an Airport
                       from a single plane to a massive international
                       terminal filled to the brim with passengers, staff
                       and it’s your job to keep everything flying
                       smoothly.


                # ⚓ WRAEK_think_they_can_change_PC_gaming_with_the_Tactonic_Pro
                  |_GamingOnLinux [Ed: Looks to me like paid-for spam; is Liam
                  Dawe selling itself out like Phoronix?]⠀⇛


                       What’s really great to see though, is how Linux has
                       clearly been making more waves – enough for this
                       hardware vendor to target it too. They said their
                       software is fully compatible with Linux!


                # ⚓ With_You_looks_like_a_sweet_co-op_experience_and_it’s_free
                  |_GamingOnLinux⠀⇛


                       Need a new casual co-op game to play with a
                       partner? With You recently released on Steam and
                       itch.io and it looks adorable. It’s free to, so
                       you’ve got nothing to lose but a few moments of
                       your time, and perhaps a fun memory to gain from
                       it.


          o § Desktop Environments/WMs⠀➾


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


                      # ⚓ KDE_Plasma_5.24_LTS_Releases_with_Updated_Breeze
                        Theme_and_New_Overview_Effect⠀⇛


                             We have been keeping an eye on KDE Plasma
                             5.24 for a while.


                             From spotting the GNOME-Style overview effect
                             to the addition of fingerprint support. If
                             you have been following our coverages, you
                             already know about the changes introduced
                             with KDE Plasma 5.24.


                             Now that KDE Plasma 5.24 stable release is
                             finally here, let me highlight the key
                             additions and improvements below.


                             [...]


                             KDE Plasma 5.24 is a long-term support
                             release that will receive updates until the
                             final Plasma 5 release (and the transition to
                             Plasma 6).


                             With this release, you do not get to see
                             massive visual changes, but you can find
                             various functional improvements and subtle
                             visual refinements.


                      # ⚓ KDE/Plasma_5.24_for_Debian⠀⇛


                             Yesterday, KDE released version 5.24 of the
                             Plasma desktop with the usual long list of
                             updates and improvements. This release will
                             be considered a LTS release. And Plasma 5.24
                             is now available for all Debian releases.
                             (And don’t forget KDE Gears/Apps 21.12!)


                      # ⚓ KDE_Plasma_5.24_LTS_desktop_environment_brings
                        fingerprint_support,_performance_and_user_interface
                        improvements_and_more⠀⇛


                             The developers of the open source KDE Plasma
                             desktop environment for Linux distributions
                             have released Plasma 5.24, which brings a
                             number of performance improvements, visual
                             tweaks, and new features.


                             It’s also a long term support (LTS) release,
                             which means it will continue to receive bug
                             fixes and other updates until the next major
                             update, which will be Plasma 6.


                      # ⚓ Plasma_5.24_available_on_Kubuntu_21.10⠀⇛


                             We are pleased to announce that Plasma 5.24
                             is now available in our backports PPA for
                             Kubuntu 21.10 (Impish Indri).


                             The release announcement detailing the new
                             features and improvements in Plasma 5.24 can
                             be found here.


                      # ⚓ Plasma_Mobile_Gear_22.02_Is_Out_with_Many_Changes_to
                        the_Plasma_Mobile_Shell_and_Apps⠀⇛


                             
                             Coming hot on the heels of the KDE Plasma
                             5.24 desktop environment, which is also
                             available for mobile devices as Plasma
                             Mobile, the Plasma Mobile Gear 22.02 software
                             suite is here to update some of your favorite
                             mobile apps, as well as the Plasma Mobile
                             shell.


                             In fact, Plasma Mobile Gear 22.02 is packed
                             with great improvements for the Plasma Mobile
                             shell, including a revamped Quick Settings
                             panel with support for media and
                             notifications widgets, as well as a new
                             landscape format for tablets, a smoother
                             gesture experience, a revamped task switcher
                             to use a single row of thumbnails with
                             gesture support, and a revamped app drawer
                             open/close gesture.


                # § GNOME Desktop/GTK⠀➾


                      # ⚓ Best_Audio_and_Video_Players_for_Gnome_Desktop⠀⇛


                             To take a break from our everyday routines,
                             most unwind by watching movies, TV shows,
                             listening to music, and indulging in other
                             forms of entertainment. Aside from that,
                             videos can be utilized for business
                             information sharing, product advertisements,
                             and a variety of other tasks in which digital
                             media is at the center of business marketing.


                             There are quite a number of video and audio
                             players. They provide features like subtitle
                             synchronization, support for a variety of
                             video formats, and the ability to play
                             YouTube videos directly without
                             advertisements.


          o § Distributions⠀➾


                # ⚓ VA_Linux⠀⇛


                       In 1999, one of the hottest IPOs was an open source
                       company named VA Linux ($LNUX). The stock opened
                       10x in its first day of trading. The stock was
                       offered at $30, opened at $300, and was $8.50 a
                       year later. The company later launched SourceForge.
                       In 2015, it become a subsidiary of Gamestop.


                       VA Linux was the largest vendor of pre-installed
                       Linux computers. Red Hat had just gone public
                       earlier that year. The company had Intel and
                       Sequoia as investors in its seed round, and the
                       founder was classmates at Stanford with Yahoo
                       founders Jerry Yang and David Filo.


                # § IBM/Red Hat/Fedora⠀➾


                      # ⚓ Comparison_of_Fedora_Flatpaks_and_Flathub_remotes⠀⇛


                             
                             In the previous article in this series, we
                             looked at how to get started with Fedora
                             Flatpaks and how to use it. This article
                             compares and contrasts between the Fedora
                             Flatpaks remote and the Flathub remote.
                             Flathub is the de-facto standard Flatpak
                             remote, whereas Fedora Flatpaks is the Fedora
                             Project’s Flatpak remote. The things that
                             differ between the remotes include but are
                             not limited to their policies, their ways of
                             distribution, and their implementatio


                # § Debian Family⠀➾


                      # ⚓ Testing_Slax_11.2_based_on_Debian_Bullseye_–_Slax
                        Linux⠀⇛


                             I made my first attempt to build Slax based
                             on Debian Bullseye (version 11.2). Here is a
                             work in progress (well, it is mostly finished
                             I guess so lets call it RC1). Some of the
                             applications are no longer available, such as
                             leafpad, wicd, and I replaced pcmanfm with
                             tuxCommander.


                             Please feel free to suggest a small simple
                             GUI app (GTK-based) for network
                             configuration.


                             This version does not include chromium
                             browser due to its size, but if you click the
                             icon, it will install automatically before
                             first use.


                             Debian also no longer supports aufs, so it
                             has been replaced by overlayfs in Slax. This
                             is internal change which affects the use of
                             ‘slax activate’ command (it will no longer
                             work). I am investigating possibilities to
                             make it work again but as far as I can see,
                             there is no way to modify the existing
                             overlay filesystem on the fly as it was
                             possible with AUFS, this means it is no
                             longer possible to add new modules on the fly
                             while running Slax. You can of course still
                             add modules to your ISO (or USB directory
                             tree). Suggestions welcome.


          o § Devices/Embedded⠀➾


                # ⚓ Raspberry_Pi_4_GPIO_Pinout⠀⇛


                       Raspberry Pi 4 has come up with more advanced
                       features as compared to the previous model of the
                       Raspberry Pi. It was launched in June 2019 and has
                       come up with a much-improved processing speed of
                       about 90% as compared to previous version due to
                       the inclusion of 4GB and 8GB RAM memory. Its
                       General-Purpose Input Output (GPIO) pins have also
                       maintained the previous standard set by the
                       Raspberry Pi models, and are now more functional
                       and performing flawlessly.


                       The Raspberry Pi 4 has 40 GPIO pins that can be
                       easily configured to read inputs or write outputs.
                       If you are unfamiliar with the operation of these
                       GPIO pins, this article will assist you in
                       understanding the operation of each pin.


                # ⚓ Raspberry_Pi_4_to_support_Network_install_to_a_blank
                  MicroSD_card⠀⇛


                       The Raspberry Pi 4 will soon be able to install
                       Raspberry Pi OS without the need for external
                       hardware to flash the image. Raspberry Pi Network
                       install will allow users to install a blank MicroSD
                       card into their board, which will trigger the
                       launch of Raspberry Pi Imager on the Raspberry Pi,
                       and flash the image downloaded from the Raspberry
                       Pi website.


                       So far in order to install Raspberry Pi OS, you had
                       to download Raspberry Pi Imager on a computer or a
                       working Raspberry Pi, insert the MicroSD card into
                       the host, and flash the image, before moving the
                       MicroSD card into the Raspberry Pi board. This
                       works just fine for most people, but if the
                       Raspberry Pi is your only computing device, as may
                       be the case in developing countries, that does not
                       work. Network Installs may also speed up the
                       installation process in the classroom where
                       multiple boards may need to be set up.


                # ⚓ How_to_List_Installed_Packages_in_Raspberry_Pi_OS⠀⇛


                       You already know that your Raspberry Pi OS has so
                       many different packages installed in it. However,
                       you don’t know what packages are installed on the
                       Raspberry Pi OS. Finding those packages used to be
                       a difficult task as it requires your time in
                       searching for those packages. As it becomes
                       necessary for you to look for those packages, there
                       should be a solution that can help you in finding
                       those packages on your Raspberry Pi OS. If you are
                       still searching for the right solution then this
                       article will provide you with the solution so that
                       you can easily list the pre-installed packages in
                       Raspberry Pi OS.


                # ⚓ How_to_Mine_Crypto_on_Raspberry_Pi⠀⇛


                       Nowadays, cryptocurrency is becoming much popular
                       among the young generation and the reason is it
                       provides them the opportunity to earn a lot of
                       money. Cryptocurrency can be obtained by two
                       methods which are purchasing and mining. The
                       purchasing method of Cryptocurrency is hard for
                       most of the people because they don’t have enough
                       money in their pockets. The second method is a
                       better one to start with but it requires some
                       effort and hard work.


                       Cryptocurrency mining has long affected the passion
                       of technology addicts, pushing them to design new
                       algorithms to defend networks, mining pools, and
                       innovative ways to build mining systems. If you are
                       among the Raspberry Pi users and you want to start
                       your Cryptocurrency mining journey, you can do it
                       on your Raspberry Pi operating system. This
                       requires your efforts and some time so that you can
                       begin mining Cryptocurrency on Raspberry Pi.


                # § Mobile Systems/Mobile Applications⠀➾


                      # ⚓ Android_vs._iOS:_Is_iPhone_Better_Than_Android_or
                        Vice_Versa?⠀⇛


                      # ⚓ Exclusive:_Material_You-style_theming_will_soon_be
                        required_for_phones_getting_Android_12⠀⇛


                      # ⚓ OnePlus_Pad_tablet_tipped_to_ship_with_Android_12L⠀⇛


                      # ⚓ Google_fixes_“critical”_Android_12_security_flaw_|
                        TechRadar⠀⇛


                      # ⚓ Which_phones_and_cars_can_use_Android_digital_car
                        key?_|_Android_Central⠀⇛


                      # ⚓ How_to_forward_calls_on_Android_–_Phandroid⠀⇛


                      # ⚓ Vivaldi_5.1_arrives_for_desktop_and_Android_–_gHacks
                        Tech_News⠀⇛


                      # ⚓ Top_10_Android_MMORPGs_of_2022:_RuneScape,_Albion,
                        Adventure_Quest_3D,_and_More_|_Tech_Times⠀⇛


                      # ⚓ Top_10_Best_Third-Party_Android_App_Stores_–_2022⠀⇛


                      # ⚓ OnePlus_Pad_rumored_to_get_Android_12L._Would_you_buy
                        it?⠀⇛


                      # ⚓ Motorola’s_first_smartphone_receiving_Android_12
                        stable_update_was_launched_in_2020_–_GSMArena.com
                        news⠀⇛


                      # ⚓ Critical_Android_12_bug_fixed_in_February_security
                        patches_•_The_Register⠀⇛


                      # ⚓ OnePlus_Unified_OS_to_Debut_in_Second_Half_of_2022,
                        OnePlus_Pad_to_Run_Android_12L:_Reports_|_Technology
                        News⠀⇛


                      # ⚓ Smartphone_Lifespan:_How_Long_Android_Phones_From
                        Samsung,_Xiaomi,_Realme_And_Others_Will_Last
                        ‘Officially’⠀⇛


                      # ⚓ YouTube_Music_For_Android_Gets_A_New_Playback
                        Shortcut⠀⇛


                      # ⚓ 6_secret_settings_for_better_browsing_in_Chrome_on
                        Android_|_Computerworld⠀⇛


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


                # § FSF⠀➾


                      # § GNU Projects⠀➾


                            # ⚓ GNU_Binutils_2.38_has_been_released⠀⇛


                                   Hi Everyone,

                                   We are pleased to announce that version
                                   2.38 of the GNU Binutils project
                                   sources have been released and are now
                                   available for download at:

                                   https://ftp.gnu.org/gnu/binutils


                                   https://sourceware.org/pub/binutils/
                                   releases/

                                   The SHA256 checksums are as follows:


                                   070ec71cf077a6a58e0b959f05a09a35015378c2d8a51e90f3aeabfe30590ef8
                                   binutils-2.38.tar.bz2

                                   6e54170356709d401f1cb781f86bb59656b99b398412047f5c44f0fbc633fa4d
                                   binutils-2.38.tar.bz2.sig

                                   b3f1dc5b17e75328f19bd88250bee2ef9f91fc8cbb7bd48bdb31390338636052
                                   binutils-2.38.tar.gz

                                   864d330b71f2b40120d96e68ebed43a07c5286d9c28f3a3893ae4ae7ec25ede9
                                   binutils-2.38.tar.gz.sig

                                   807ccfa77ccfc3e09b1f760cd73e13839180c6950a37843df1c1e7a58c777dc9
                                   binutils-2.38.tar.lz

                                   49ee3c3b5803dea2acb15a3fa4c8ad0daebf142ba284a8b3d3524a52823a0ceb
                                   binutils-2.38.tar.lz.sig

                                   e316477a914f567eccc34d5d29785b8b0f5a10208d36bbacedcc39048ecfe024
                                   binutils-2.38.tar.xz

                                   41301d67da78df1ad6df04aefe9e7bea8235484b0323cee52caa8f7435385014
                                   binutils-2.38.tar.xz.sig

                                   This release contains numerous bug
                                   fixes and improvements, along with
                                   the following new features:

                                     Assembler:
                                       General:
                                         * Add support for the LoongArch
                                   architecture.

                                         * Add an option to control how
                                   multibyte characters are handled in
                                           the assembler.  Using the
                                   option warnings can be generated when
                                           such characters are encountered
                                   in symbol names, or anywhere in
                                           the input source file(s).

                                       AArch64 and ARM:
                                         * Add support for more system
                                   registers.
                                         * Add support for Scalable Matrix
                                   Extension.
                                         * Add support for Cortex-R52+,
                                   Cortex-A510, Cortex-A710,
                                           Cortex-X2, Cortex-A710 cores.
                                         * Add support for 'v8.7-a',
                                   'v8.8-a', 'v9-a', 'v9.1-a',
                                           'armv9.2-a' and 'armv9.3-a'
                                   architecture extensions.

                                       X86:
                                         * Add a command-line option to
                                   encode aligned vector move as
                                           unaligned vector move.
                                         * Add support for Intel
                                   AVX512_FP16 instructions.
                                         * The outputs of .ds.x directive
                                   and .tfloat directive with hex
                                           input have been reduced from 12
                                   bytes to 10 bytes to match the
                                           output of .tfloat directive.

                                     Linker:
                                       * Add support for the LoongArch
                                   architecture.

                                       * Add -z pack-relative-relocs/-z no
                                   pack-relative-relocs to x86 ELF
                                        linker to pack relative
                                   relocations in the DT_RELR section.

                                       * Add -z indirect-extern-access/-
                                   z noindirect-extern-access to x86
                                         ELF linker to control canonical
                                   function pointers and copy
                                         relocation.

                                     Other Binary Tools:

                                       * elfedit: Add --output-abiversion
                                   option to update ABIVERSION.

                                       * Tools which display symbols or
                                   strings (readelf, strings, nm,
                                         objdump) have a new command line
                                   option which controls how unicode
                                         characters are handled.  By
                                   default they are treated as normal for
                                         the tool.  Using --unicode=locale
                                   will display them according to
                                         the current locale.  Using --
                                   unicode=hex will display them as hex
                                         byte values, whilst --
                                   unicode=escape will display them as
                                   escape
                                         sequences.  In addition using --
                                   unicode=highlight will display
                                         them as unicode escape sequences
                                   highlighted in red (if supported
                                         by the output device).

                                       * readelf -r dumps RELR relative
                                   relocations now.

                                       * Support for efi-app-aarch64, efi-
                                   rtdrv-aarch64 and
                                         efi-bsdrv-aarch64 has been added
                                   to objcopy in order to enable
                                         UEFI development using binutils.

                                       * ar: Add --thin for creating thin
                                   archives. -T is a deprecated
                                         alias without diagnostics. In
                                   many ar implementations -T has a
                                         different meaning, as specified
                                   by X/Open System Interface.

                                   Our thanks go out to all of the
                                   binutils contributors, past and
                                   present, for helping to make this
                                   release possible.

                                   Cheers
                                     Nick Clifton
                                     GNU Binutils Chief Maintainer
                # § Programming/Development⠀➾


                      # ⚓ Sunsetting_gzip_substitutes_availability⠀⇛


                             Starting next month (2022/03/01), the build
                             farm known as ci.guix.gnu.org will no longer
                             offer gzip-compressed binary substitutes. The
                             Guix daemon has known to use lzip for
                             substitutes since 2019; unless you are
                             running a very outdated daemon, you have no
                             need to worry about this change.


                             This idea was first discussed about a year
                             ago, when it was found that gzip-compressed
                             substitutes accounted for about only 1% of
                             the downloaded substitutes. Since then, the
                             daemon has gained support for zstd on top of
                             gzip and lzip, and the build farm has happily
                             generated compressed substitutes for all of
                             these compression schemes.


                      # ⚓ I_would_like_a_job_writing_Haskell⠀⇛


                             Perhaps someone out there wants to take a
                             chance on a senior programmer with thirty
                             years of experience who wants to make a move
                             into Haskell.


                      # § Java⠀➾


                            # ⚓ Hashset_vs_Hashmap_in_Java⠀⇛


                                   The Java programming language has two
                                   distinct yet similar types of data
                                   containers, HashMap and HashSet . Both
                                   use a hash table to store data. A table
                                   is a table of values ​​that uses a hash
                                   function to determine where to look and
                                   store data. This allowsquick access to
                                   data because a value does not have to
                                   be searched. Instead, the hash function
                                   can provide the exact location of the
                                   value. Despite so much use of hash
                                   tables, HashMap and HashSet are quite
                                   different from each other.


                            # ⚓ Java_assert⠀⇛


                                   Assertion in Java is a statement based
                                   on the reserved word, assert, having a
                                   boolean expression and ending with a
                                   semicolon. It happens that certain
                                   conditions in a program may prevent it
                                   from working properly. The assert
                                   statement checks if such conditions
                                   exist in a program. Java assert is
                                   explained in this article.


                            # ⚓ How_do_You_write_Comparable_in_Java?⠀⇛


                                   Examples of Java lists are; Vector,
                                   Stack, AttributeList, ArrayList, and
                                   LinkedList. The Java Collections class
                                   has overloaded methods to sort a list
                                   of predefined objects. The Java Arrays
                                   class has overloaded methods to sort an
                                   array of predefined objects.


                                   The predefined sort() methods are to
                                   sort predefined objects. What about
                                   user-defined objects? Java has an
                                   interface called Comparable. An
                                   interface is a kind of class whose
                                   methods are declarations without
                                   definitions (bodies). So, a class must
                                   be implemented from an interface for
                                   the methods to be defined (given
                                   bodies). From the implemented class,
                                   objects can be instantiated from a name
                                   given by the programmer.


                                   Java has a predefined Comparable class.
                                   The sort() methods in Collections and
                                   Arrays use the predefined comparable
                                   class to sort predefined objects.
                                   However, in order to sort a list or
                                   array of user-defined objects, the
                                   programmer has to implement (define) a
                                   user Comparable class from the
                                   Comparable interface. This user
                                   implemented (defined) class enables the
                                   sorting of user-defined objects in a
                                   list or array. So, a predefined
                                   Comparable class enables sorting of
                                   predefined objects, while a user-
                                   defined Comparable class enables
                                   sorting of user-defined objects. User-
                                   defined objects are not literals, so a
                                   class of user-defined objects needs its
                                   own user-defined Comparable
                                   implementation.


                                   This article illustrates how to write
                                   Comparable in Java.


                            # ⚓ Hashtable_in_Java⠀⇛


                                   A hash table is for key/value pairs.
                                   Internally, the values are stored in an
                                   array. Each key is converted into an
                                   index of the array. This conversion is
                                   known as hashing. It is done by a hash
                                   function internally. In other words,
                                   the keys do not have to be stored. All
                                   that is the hash-table, in java and in
                                   other computer languages. This article
                                   explains two constructors of the Java
                                   hastable and its commonly used methods.


                            # ⚓ How_AJAX_works⠀⇛


                                   AJAX comprises a set of useful web
                                   development techniques utilized to
                                   develop dynamic and speedy web pages.
                                   Behind the scenes, it shares small
                                   chunks of the data, permitting the web
                                   pages to be updated asynchronously.
                                   This states that by using AJAX, HTML
                                   page elements will be updated without
                                   reloading.


                                   This write-up will discuss the
                                   components of AJAX and how AJAX works.
                                   We will talk about the working of AJAX
                                   in some top web-based applications.
                                   Moreover, a comparison between the
                                   conventional and AJAX model will be
                                   provided. So, let’s start!


          o § Standards/Consortia⠀➾


                # ⚓ Gemini_Isn’t_The_Solution_To_The_Broken_Web⠀⇛


                       I’m not sure if you heard, but The Web Is F*cked
                       and techies everywhere are touting the Gemini
                       protocol as its saviour. I disagree. A lot.


                       When I wrote The Web Is F*cked, I had a number of
                       people reach out asking why I didn’t mention
                       Gemini, as it’s the saviour of the web…apparently.


                       Well, I didn’t mention Gemini because I think it
                       actually does more harm than good when it comes to
                       the web. Let me explain…


    * § Leftovers⠀➾


          o ⚓ Todd_Gitlin_Told_His_Truth_Everywhere_He_Went⠀⇛


                 I met Todd Gitlin on October 23, 1987. It was the
                 beginning of what would become a decades-long “beautiful
                 friendship.” (We shared a favorite movie in Casablanca.)
                 I can pin down the date because our meeting took place at
                 the ridiculous Second Thoughts conference in which
                 apostate liberals and New Leftists were gathered to
                 denounce their former selves and embrace the new right.
                 It took place at the Grand Hyatt hotel in Washington,
                 D.C., and was organized by left-wing Stalinists turned
                 right-wing Stalinists David Horowitz and Peter Collier,
                 with funding from a Christian conservative foundation led
                 by the son of the far-right Alabama Senator Jeremiah
                 Denton, together with the Bradley and Olin foundations.
                 Funnily enough, it took place just days after Ronald
                 Reagan’s Iran/Contra plot was revealed and the largest
                 single drop in the stock market since 1929. These week’s
                 events signaled the long-awaited end to the Reaganite
                 hegemony these folks had gathered to celebrate.


          o ⚓ Everyday_Specters⠀⇛


                 Last fall, I was in the English countryside on a research
                 trip and decided to visit Blo’ Norton Hall, the moated,
                 16th-century manor house in Norfolk where a prince I was
                 writing about had once lived. It was almost dusk when I
                 arrived, but there were several people my age walking the
                 grounds: guests, I assumed, of the bed-and-breakfast to
                 which the house had been converted after the prince’s
                 death. I tried to maintain a friendly distance.


                 As I walked around the house’s rear, the four young men,
                 holding glasses of beer, strolled directly past me, and I
                 smiled at them—but none appeared to see me. Moments
                 later, I realized that this was not, in fact, a bed-and-
                 breakfast through which I was welcome to perambulate but
                 a single, large holiday home, and I was trespassing on
                 some friends’ private vacation. Mortified, I ran back to
                 my car, reversed for several minutes down the long gravel
                 driveway, and hightailed it home. It felt like a close
                 call. The occupants could have interrogated me or even
                 called the police. But one more chilling possibility also
                 struck me: Had those men seen me at all? Or had I been
                 somehow invisible to them—perhaps as an interloper from
                 another realm, or perceived as the type of grounds staff
                 some are trained not to see? Was I, in other words, a
                 sort of ghost passing through their domain?


          o ⚓ Anatomy_of_a_Cancellation⠀⇛


                 Again with no ill will towards anyone in the DSA, I’m
                 hoping this might be a teachable moment.  In order for it
                 to be one, we need to do some unpacking.  There is a
                 diverse cast of characters involved.


                 I’ll just assume everyone knows who the Democratic
                 Socialists of America are.  If not, you can look them up
                 easily enough, but they probably represent the biggest
                 organized political group to the left of the Democratic
                 Party today (or on the left of the Democratic Party,
                 depending).  As any sensible democratic socialist
                 organization would do, they are actively supporting the
                 Portland city workers in their contract negotiations with
                 the city, and their plan for an imminent strike.


          o ⚓ CNN_Could_Face_a_Reset_Once_Under_Discovery_Control⠀⇛


                 One prominent Discovery shareholder, John Malone, has
                 already been vocal. “I would like to see CNN evolve back
                 to the kind of journalism that it started with, and
                 actually have journalists, which would be unique and
                 refreshing,” the entrepreneurial media investor told CNBC
                 in November. Malone will serve on the Discovery board,
                 but no longer has a commanding block of super-voting
                 shares as he once did, leaving Zaslav at the helm.


          o ⚓ Joe_Rogan_Calls_Backlash_Over_His_Use_of_N-Word_a_‘Political_Hit
            Job’⠀⇛


                 “That video had always been out there. It’s like, this is
                 a political hit job,” Rogan said. He continued, “And so
                 they’re taking all this stuff I’ve ever said that’s wrong
                 and smushing it all together.”


          o ⚓ ViacomCBS_Elevates_Alex_Berkett_to_Chief_Corporate_Development
            and_Strategy_Officer_Role⠀⇛


                 ViacomCBS has promoted veteran dealmaker Alex Berkett to
                 the role of chief corporate development and strategy
                 officer.


                 His previous title was executive vp, corporate
                 development & strategy. Berkett will continue to report
                 to ViacomCBS CFO Naveen Chopra and to serve on the
                 company’s senior leadership team and global inclusion
                 advisory committee.


          o § Education⠀➾


                # ⚓ Hunger-Striking_Teachers_Say_School_Closures_Will_Hurt
                  Black_and_Brown_Kids⠀⇛


          o § Hardware⠀➾


                # ⚓ CR+LF_has_a_long_history…⠀⇛


                       There is one exception: carriage return [CR]. The
                       carriage is released within the 100ms time, but the
                       carriage is on a spring, and does not get back to
                       the left within 100ms if it is too far over to the
                       right. The usual fix is to send another non
                       printing character, such as a line feed [LF], as
                       the next character, where CR and LF are used for
                       each new line. The paper advances whilst the
                       carriage is returning so giving the carriage a
                       whole 200ms to complete the return. This allows
                       enough time to get to the left, but can leave the
                       carriage still bouncing and mean the first
                       printable character is not well aligned. The fact
                       that 200ms is enough is usually fine, unless you
                       are particularly fussy. The fix to this is to send
                       another non printing character, such as a NULL,
                       another CR, or even a rub out [RO]. The standard
                       HEREIS drum coding even specifies CR, LF and RO at
                       the start.


                # ⚓ Nvidia_Deal_to_Buy_Arm_From_SoftBank_Is_Off_After
                  Setbacks⠀⇛


                       Nvidia, a fast-growing company whose chips are best
                       known for rendering images in video games, in
                       September 2020 offered cash and stock then valued
                       at $40 billion for Arm, making it the most
                       expensive deal ever among chip companies. Nvidia
                       made the offer to buy Arm from SoftBank, the
                       Japanese conglomerate that has owned the British
                       company since 2016. Nvidia’s rising stock price
                       later sent the transaction’s value much higher,
                       settling at about $60 billion on Monday.


                       But the blockbuster deal encountered setbacks that
                       included a Federal Trade Commission lawsuit in
                       December to block the acquisition, as well
                       opposition from regulators in Britain.


                       Nvidia and SoftBank said early Tuesday they agreed
                       to terminate the planned deal because of
                       “significant regulatory challenges.”


                # ⚓ SoftBank_dumps_sale_of_Arm_over_regulatory_hurdles,_to_IPO
                  instead⠀⇛


                       On a company earnings call on Tuesday, SoftBank CEO
                       Son, who had said the company initially considered
                       listing Arm but opted to sell it instead due to the
                       pandemic, sought to put a positive spin on the
                       scrapped sale.


          o § Health/Nutrition/Agriculture⠀➾


                # ⚓ Worse_than_ineffective:_Two_independent_studies_have
                  confirmed_that_EpiVacCorona_is_a_worthless_vaccine_—_and_it
                  should_be_taken_off_the_market⠀⇛


                       Despite a recent report that Russia’s Health
                       Ministry has stopped placing orders of the
                       EpiVacCorona vaccine, its manufacturer is
                       optimistic and claims to expect the next production
                       order to come in February. Meanwhile, scientists,
                       as well as journalists at Meduza, have obtained
                       convincing evidence that EpiVacCorona is
                       ineffective and gives recipients a completely false
                       sense of security (unlike Sputnik V, which has been
                       proven to protect recipients from COVID-19’s most
                       severe effects). Meduza explains the two studies
                       that led to that conclusion.


                # ⚓ Opinion_|_On_What_Just_Happened_to_Medicare_for_All_in
                  California—and_the_Organizing_We_Need_to_Win⠀⇛


                       Why is a healthcare system—Medicare for All—that
                       costs less described as “too expensive?”


                # ⚓ Covid_Survivors_at_Higher_Long-Term_Risk_of_Cardiovascular
                  Damage:_Study⠀⇛


                       Covid-19 survivors are at an elevated long-term
                       risk of developing heart disease, strokes, blood
                       clots, and other cardiovascular disorders,
                       according to new peer-reviewed research out Monday.


                       Published in Nature Medicine, the study finds that
                       individuals who survive Covid-19 are more likely to
                       suffer from cardiovascular disease one year later.
                       That’s true even if they “were not hospitalized
                       during the acute phase of the infection,” although
                       disease risks and burdens are higher for patients
                       who were hospitalized and especially for those who
                       were admitted to intensive care units (ICUs).


          o § Integrity/Availability⠀➾


                # § Proprietary⠀➾


                      # ⚓ Windows_11_–_six_months_later,_still_totally_meh⠀⇛


                             Early in July 2021, I reviewed the Dev Build
                             of Windows 11. I was underwhelmed on so many
                             levels. The new operating system, if it can
                             be called that, was raw, unfinished, and came
                             with a slew of bugs and ergonomic annoyances.
                             But that was then. Since, this thing has been
                             officially released, and it even received a
                             handful of big, critical patches, designed to
                             help resolve some of the early problems.


                      # ⚓ Vivaldi_5.1_Introduces_Horizontal_Scrollable_Tabs_and
                        a_New_Reading_List
                        ⠀⇛


                             Vivaldi is a pretty good option for Linux
                             users. They focus on Linux as one of the
                             first-party platforms, which is impressive.


                             With Vivaldi 5.0 release, it proved to be a
                             versatile Chromium-based option for many
                             Linux users. Now, Vivaldi 5.1 is finally
                             here!


                      # ⚓ Microsoft_Patch_Tuesday,_February_2022_Edition⠀⇛


                      # ⚓ How_a_Texas_[crack]_changed_the_ransomware_business
                        forever [iophk: Windows TCO]⠀⇛


                             Last year, a security analyst named Dmitry
                             Smilyanets had a long online chat with
                             someone who claimed to be a member of REvil’s
                             management team. He went by the online handle
                             ‘Unknown.’


                             “Unknown was not a hacker. He was the
                             operator. He was the manager,” Smilyanets
                             said. “His job was to control the
                             infrastructure, make sure it all works. Make
                             sure that communication lines with victims
                             were up and that payments go through.”


                      # ⚓ Protecting_your_business_in_the_age_of_ransomware
                        [iophk: Windows TCO]⠀⇛


                             The GDPI survey uncovered that 64% of leaders
                             are concerned they’ll experience a disruptive
                             event, such as data loss or downtime, in the
                             next year. With the frequency of ransomware
                             attacks on the rise, all businesses should
                             expect an attack. Whether or not you should
                             be fearful depends on how prepared you are.


                      # § Security⠀➾


                            # ⚓ Security_updates_for_Wednesday_[LWN.net]⠀⇛


                                   Security updates have been issued by
                                   CentOS (aide), Debian (connman), Fedora
                                   (perl-App-cpanminus and rust-
                                   afterburn), Mageia (glibc), Red Hat
                                   (.NET 5.0, .NET 6.0, aide, log4j,
                                   ovirt-engine, and samba), SUSE
                                   (elasticsearch, elasticsearch-kit,
                                   kafka, kafka-kit, logstash, openstack-
                                   monasca-agent, openstack-monasca-log-
                                   metrics, openstack-monasca-log-
                                   persister, openstack-monasca-log-
                                   transformer, openstack-monasca-
                                   persister-java, openstack-monasca-
                                   persister-java-kit, openstack-monasca-
                                   thresh, openstack-monasca-thresh-kit,
                                   spark, spark-kit, venv-openstack-
                                   monasca, zookeeper, zookeeper-kit and
                                   elasticsearch, elasticsearch-kit,
                                   kafka, kafka-kit, logstash, openstack-
                                   monasca-agent, openstack-monasca-
                                   persister-java, openstack-monasca-
                                   persister-java-kit, openstack-monasca-
                                   thresh, openstack-monasca-thresh-kit,
                                   spark, spark-kit, storm, storm-kit,
                                   venv-openstack-monasca, zookeeper,
                                   zookeeper-kit), and Ubuntu (bluez,
                                   linux, linux-aws, linux-aws-5.4, linux-
                                   gcp, linux-gcp-5.4, linux-hwe-5.4,
                                   linux-ibm, linux-kvm, linux-oracle,
                                   linux-oracle-5.4, nvidia-graphics-
                                   drivers-450-server, nvidia-graphics-
                                   drivers-470, nvidia-graphics-drivers-
                                   470-server, nvidia-graphics-drivers-
                                   510, python2.7, and util-linux).


                            # § Privacy/Surveillance⠀➾


                                  # ⚓ ‘Party_of_Union_Busters’:_McCarthy
                                    Opposes_Congressional_Staff_Union⠀⇛


                                         The sincerity of the GOP’s
                                         attempt to rebrand as a party of
                                         workers was called into further
                                         question Monday when Rep. Kevin
                                         McCarthy—the Republican leader in
                                         the U.S. House—spoke out against
                                         congressional staffers’ nascent
                                         unionization push, which has won
                                         the enthusiastic support of
                                         progressive lawmakers and
                                         Democratic leaders.


                                         “I don’t think it would be
                                         productive for the government,”
                                         McCarthy (R-Calif.), a top ally
                                         of former President Donald Trump,
                                         told Punchbowl News of the
                                         unionization effort by Capitol
                                         Hill aides, many of whom are paid
                                         annual salaries in the low
                                         $20,000 range and struggle to
                                         afford basic necessities in one
                                         of the nation’s most expensive
                                         cities.


                                  # ⚓ Interview_With_Eugene_Shablygin_–
                                    WWPass⠀⇛


                                         Eugene Shablygin: My background
                                         is in fundamental physics, but at
                                         some point, I realized I wasn’t
                                         really a good physicist. In my
                                         opinion, I was not on par with
                                         good physicists like Einstein,
                                         and I didn’t feel the essence of
                                         physics. However, I felt that I
                                         understood how data flows through
                                         different computers.


                                  # ⚓ Faith_Leaders_Denounce_Zuckerberg’s
                                    Instagram-for-Children_Scheme⠀⇛


                                         Saying that “children’s wellbeing
                                         must come first” above profit,
                                         dozens of faith leaders on
                                         Tuesday told Facebook chief Mark
                                         Zuckeberg to permanently end any
                                         plans for a version of the photo-
                                         and video-sharing app Instagram
                                         for kids under 13.


                                         “Childhood should be all
                                         movement, play, messiness, and
                                         wriggle—life-affirming against
                                         the sleek, flat, rapacious world
                                         of screens.”


                                  # ⚓ Private_Internet_Access_Releases_Updated
                                    Transparency_Report_in_Q1_2022⠀⇛


                                         The requests were as follows:


                                  # ⚓ UK_Government_Refreshes_Its_Terrible
                                    ‘Online_Safety_Bill,’_Adds_Even_More
                                    Content_For_Platforms_To_Police⠀⇛


                                         The UK’s internet censorship bill
                                         rebranded from “Online Harms” to
                                         “Online Safety” last spring. The
                                         name change did nothing to limit
                                         the breadth of the bill, despite
                                         supposedly shifting the focus
                                         from “harm” to “safety.” Whatever
                                         the name, it’s still being touted
                                         by supporters as a fix for
                                         anything anyone doesn’t like
                                         about the internet.


                                  # ⚓ If_EARN_IT_Passes,_What_Happens_On_Your
                                    iPhone_Won’t_Stay_On_Your_iPhone⠀⇛


                                         Now, Congress wants to force
                                         Apple’s hand—along with
                                         essentially every company that
                                         allows users to store or share
                                         messages or content—and
                                         essentially mandate such
                                         scanning. 


                                         While Apple’s plan would have put
                                         the privacy and security of its
                                         users at risk, the EARN IT Act
                                         compromises security and free
                                         speech for everyone. The bill
                                         would create serious legal risk
                                         for business that hosts
                                         content—messages, photos stored
                                         in the cloud, online backups—and,
                                         potentially, even cloud-hosting
                                         sites like those using Amazon Web
                                         Services, unless they use
                                         government-approved scanning
                                         tools. 


                                  # ⚓ Ohio:_Don’t_Give_Big_Tech_a_Pass_On
                                    Privacy⠀⇛


                                         The OPPA would enshrine privacy
                                         violating practices from Big Tech
                                         and other companies, and place
                                         the responsibility for managing
                                         privacy entirely on
                                         individuals—without actually
                                         improving protections for the
                                         people of Ohio or offering them a
                                         way to stand up for their own
                                         privacy. If it is not
                                         substantially improved before it
                                         is enacted, it risks locking in
                                         industry-friendly provisions that
                                         would directly benefit tech
                                         giants such as Facebook, Google,
                                         and Amazon. In many ways, passing
                                         this bill would be worse for the
                                         everyday consumer than passing
                                         nothing at all.


                                  # ⚓ Send_Surveillance_Robot_Dogs_to_the
                                    Pound,_Not_the_Border⠀⇛


                                         The dogs are not yet in the
                                         field, but having been pushed
                                         through early stages of testing,
                                         they are undergoing evaluation in
                                         El Paso, Texas. The robot dogs
                                         would eventually be expected to
                                         walk miles through the remote
                                         terrain at the border, all the
                                         while filming and scanning.


                                         People who live along the border
                                         are some of the most heavily
                                         surveilled people in the United
                                         States. A massive amalgamation of
                                         federal, state, and local law
                                         enforcement and national security
                                         agencies are flying drones,
                                         putting up cameras, and just
                                         generally attempting to negate
                                         civil liberties—capturing the
                                         general goings-on of people who
                                         live and work in proximity to the
                                         border. But the civil liberties
                                         of these communities are of no
                                         concern to a government with
                                         nearly limitless resources to
                                         throw at putting entire regions
                                         of the United States under
                                         surveillance.


                                         Even if you live hundreds of
                                         miles from the border, don’t
                                         think these “innovations” can’t
                                         or won’t affect you. As we saw
                                         when Customs and Border
                                         Protection drones flew over a
                                         protest in Minneapolis, the $68
                                         billion border surveillance
                                         industry won’t stop expanding if
                                         there’s money to be made further
                                         inland.


                                  # ⚓ More_Fallout_For_NSO_And_Israel:_Gov’t
                                    Police_Illegally_Deployed_Malware_Against
                                    Person_Involved_In_Netanyahu_Bribery
                                    Trial⠀⇛


                                         More troubling developments for
                                         both NSO Group and the country it
                                         calls home.


                                  # ⚓ Apple_says_a_‘small_portion’_of_iPhones
                                    recorded_interactions_with_Siri_even_if_you
                                    opted_out⠀⇛


                                         Apple’s release of iOS 15.4 beta
                                         2 fixes a bug that may have
                                         recorded interactions with Siri
                                         on some devices, regardless of
                                         whether you opted out, according
                                         to a report from ZDNet. The bug,
                                         which was first introduced in iOS
                                         15, automatically enabled the
                                         Improve Siri & Dictation setting
                                         that gives Apple permission to
                                         record, store, and review your
                                         conversations with Siri.


                                  # ⚓ How_to_delete_your_Instagram_account⠀⇛


                                         If you’ve made the decision to
                                         delete Instagram, whether because
                                         you’ve outgrown the need for a
                                         certain finsta or because its
                                         parent company Meta is courting
                                         controversy again, doing so isn’t
                                         as quick or easy as it should be.
                                         Up until recently, it couldn’t
                                         even be done from within the
                                         Instagram app.


                                         Go ahead and take a moment to
                                         make an obligatory “I’m deleting
                                         Instagram” post if you’d like.
                                         After that, you have two ways you
                                         can go about it.


                                  # ⚓ Companies_Should_Mandate_Two-Factor
                                    Authentication,_Says_Head_of_National
                                    Cybersecurity_Alliance⠀⇛


                                         The interim executive director of
                                         a non-profit that has on its
                                         board members from Lenovo,
                                         Facebook, Microsoft and a number
                                         of other prominent tech firms
                                         said that companies should
                                         mandate two-factor
                                         authentication.


                                         Lisa Plaggemier of the National
                                         Cybersecurity Alliance, which
                                         advocates for cybersecurity
                                         across the country, made the
                                         comment at an event hosted by
                                         Axios Media on Tuesday.


                                  # ⚓ A_safer_Internet_starts_with_more
                                    encryption⠀⇛


                                         The Internet has been vital to
                                         help parents balance work and
                                         parenting through a lingering
                                         pandemic, yet it’s hard to ignore
                                         the societal issues increasingly
                                         amplified online. And with
                                         children increasingly learning
                                         and socializing online, the
                                         reality is we can’t always be
                                         there to hold their hand.


                                         What’s even more frightening is
                                         when government proposals
                                         undermine people’s best efforts
                                         to keep kids safe online by
                                         ignoring a simple truth: security
                                         is crucial to a safer Internet.


                                         It might seem like an obvious
                                         statement, but it’s not.


          o § Defence/Aggression⠀➾


                # ⚓ Internal_Investigation_Confirms_Border_Patrol_Failures
                  Leading_Up_to_a_16-Year-Old’s_Death_on_the_Floor_of_His
                  Cell⠀⇛


                       A government investigation into the 2019 death of a
                       Guatemalan teenager in Border Patrol custody has
                       found serious problems with the agency’s handling
                       of sick detainees.


                       The report, obtained by ProPublica through a public
                       records request, concludes that Border Patrol
                       agents did not check on 16-year-old Carlos
                       Hernandez Vasquez, who died of the flu after
                       writhing on the floor of his cell in Weslaco,
                       Texas. The report also found that the case
                       reflected broader problems with care in a detainee
                       system that at the time was overwhelmed with
                       migrants, many of whom were ill.


                # ⚓ Wave_of_Coups_Disrupts_Africa_as_U.S.-Trained_Soldiers_Play
                  Key_Role_in_Overthrowing_Governments⠀⇛


                       The African Union is condemning a wave of coups in
                       Africa, where military forces have seized power
                       over the past 18 months in Mali, Chad, Guinea,
                       Sudan and, most recently, in January, Burkina Faso.
                       Several were led by U.S.-trained officers as part
                       of a growing U.S. military presence in the region
                       under the guise of counterterrorism, which is a new
                       imperial influence that supplements the history of
                       French colonialism, says Brittany Meché, assistant
                       professor at Williams College. Some coups have been
                       met with celebration in the streets, signaling
                       armed revolt has become the last resort for people
                       dissatisfied with unresponsive governments.
                       “Between the U.S.-led war on terror and the wider
                       international community’s fixation on ‘security,’
                       this is a context that centers, if not privileges,
                       military solutions to political problems,” adds
                       Samar Al-Bulushi, contributing editor for Africa Is
                       a Country.


                # ⚓ The_Nightmare_Continues⠀⇛


                       We Own the World


                       The United States Department State Department
                       continued in its QAnon-like claim that Russia is
                       planning to fabricate a mass casualty event as a
                       pretext to invade Ukraine – an allegation for which
                       “State” provided no, um, evidence. It was vaguely
                       reminiscent of the George W. Bush administration’s
                       unsubstantiated Weapons of Mass Destruction (WMD)
                       and al-Qaeda alliance claims against Iraq before
                       the criminal, mass-murderous US petro-imperialist
                       invasion of that country.


                # ⚓ US_Sanctions_Caused_Mass_Civilian_Deaths_in_Iraq._Afghan
                  Civilians_Are_Up_Next.⠀⇛


                # ⚓ Israeli_Policies_Satisfy_the_Definition_of_Apartheid_Under
                  International_Law⠀⇛


                # ⚓ Policing_on_Trial:_Attorney_Ben_Crump_on_Fed_Case_Against
                  Three_Cops_Involved_in_George_Floyd_Murder⠀⇛


                       The Minneapolis judge who signed the no-knock
                       warrant that led to the fatal police shooting of
                       22-year-old Black man Amir Locke also presided over
                       the trial of Derek Chauvin, the ex-police officer
                       convicted for the murder of George Floyd. The trial
                       of three officers facing lesser charges — Tou Thao,
                       Thomas Lane and J. Alexander Kueng — is currently
                       underway after being delayed when one of the
                       defendants tested positive for COVID. The trial
                       will show the importance of accountability even
                       from police who are bystanders to murder, says
                       Benjamin Crump, part of the legal team for George
                       Floyd’s family.


                # ⚓ Cover-Up_in_Minneapolis?_Police_“Executed”_Amir_Locke_in
                  “No-Knock”_Raid,_Say_His_Parents,_Activists⠀⇛


                       Protests are continuing in Minneapolis after police
                       fatally shot 22-year-old Amir Locke during an
                       early-morning “no-knock” raid on February 2.
                       Bodycam video shows that Locke appeared to be
                       asleep on the couch and wrapped in a blanket when a
                       SWAT team entered the apartment. Locke held a gun
                       he was legally licensed to carry, and was not named
                       in the warrant. Minneapolis interim city Police
                       Chief Amelia Huffman claimed Locke pointed his
                       weapon in the direction of the officers, and
                       suggested he could have been connected to crime,
                       despite not being a suspect in their investigation.
                       “It was very jarring for many people in our
                       community to see Amir painted almost like a
                       criminal,” says attorney and police accountability
                       activist Nekima Levy Armstrong. No-knock warrants,
                       which Mayor Jacob Frey promised to eliminate but
                       never did, “have deadly consequences for innocent
                       Black people like Amir Locke and Breonna Taylor and
                       so many others,” says civil rights attorney
                       Benjamin Crump, part of the legal team for Amir
                       Locke’s family. This week the Biden administration
                       responded to the raid saying it may consider a
                       federal policy that limits the use of no-knock
                       warrants.


                # ⚓ Mass_Student_Walkouts_Over_Police_Killing_of_Amir_Locke⠀⇛


                       Demanding accountability from local leaders,
                       hundreds of high school students in Minneapolis and
                       St. Paul walked out of their classes on Tuesday at
                       noon in protest of the fatal police shooting of
                       Amir Locke during a no-knock raid.


                       The youth-led group MN Teen Activists organized the
                       walkout, which included students at St. Paul
                       Central High School and Capitol Hill Magnet School
                       in St. Paul and Southwest, Roosevelt, and Washburn
                       high schools in Minneapolis, as well as other
                       schools in the surrounding suburbs.


                # ⚓ Opinion_|_America’s_Strategic_Blunders_of_the_Past_That
                  Created_This_Crisis_Over_Ukraine⠀⇛


                       Understandably enough, commentaries on the crisis
                       between Russia and the West tend to dwell on
                       Ukraine. After all, more than 100,000 Russian
                       soldiers and a fearsome array of weaponry have now
                       been emplaced around the Ukrainian border. Still,
                       such a narrow perspective deflects attention from
                       an American strategic blunder that dates to the
                       1990s and is still reverberating.


                # ⚓ Opinion_|_Memo_to_Congress:_Diplomacy_for_Ukraine_Is
                  Spelled_M-I-N-S-K⠀⇛


                       While the Biden administration is sending more
                       troops and weapons to inflame the Ukraine conflict
                       and Congress is pouring more fuel on the fire, the
                       American people are on a totally different track. 


                # ⚓ ‘Beyond_the_War_Paradigm’:_Report_Offers_Alternative_to_20
                  Years_of_Post-9/11_Failures⠀⇛


                       After more than two decades of the widely condemned
                       “global war on terror” launched by the United
                       States government in the wake of the 9/11 attacks,
                       a report released Tuesday offers 10 alternatives to
                       the nation’s militarized approach to national
                       security.


                       “A reevaluation of the U.S. approach to
                       counterterrorism is a moral and strategic
                       imperative.”


                # ⚓ ‘My_task_is_to_kill_you’_Domestic_violence_victim_who_fled
                  Dagestan_describes_family_members’_thwarted_attempt_to_kidnap
                  her_in_Moscow⠀⇛


                       Aishat Khiramagomedova fled her family home in
                       Dagestan in October 2021, after reaching out to
                       Krepost (Fortress), a charity organization that
                       assists victims of domestic abuse. She made it to
                       Moscow, where Krepost provided her with shelter and
                       access to a psychologist. Within a few months she
                       was starting to find her feet — she had a job and
                       was looking for an apartment with her boyfriend.
                       But on February 4, Aishat’s family tried to kidnap
                       her and bring her back to Dagestan. 


                # ⚓ Ukraine_and_the_Threat_of_Nuclear_War⠀⇛


                       As the crisis in Ukraine deepens, it is appropriate
                       to consider what the actual consequences of war
                       there might be. An armed conventional conflict in
                       Ukraine would be a terrible humanitarian disaster.


                # ⚓ Behind_the_Olympic_Curtain⠀⇛


                # ⚓ China’s_Olympic_Battle_for_Legitimacy:_the_Prehistory_of
                  the_2022_Beijing_Games⠀⇛


                       In 1949, the Communist Party of China decisively
                       prevailed over Chiang Kai-shek’s Kuomintang (KMT)
                       after 22 years of civil war, forcing the latter to
                       flee to Taiwan. The founding of the People’s
                       Republic of China (PRC) brought a definitive end to
                       a “century of humiliation” inaugurated by the First
                       Opium War, which had seen colonial powers reduce
                       China to the “sick man of Asia.” This sickness had
                       been a byword for the weakness, internal rupture,
                       and forced narcotic dependency of the Chinese body
                       politic—transposed inevitably onto the racialized
                       Chinese body.


                       Overcoming these scars, in all their physical and
                       psychological manifestations, was the guiding
                       principle for sports policy in the PRC. Only
                       through this lens can we understand why it fought
                       in such an obstinate, pugnacious, and unabashedly
                       political way for a place in the Olympic movement
                       on its own sovereign terms. China turned the
                       Olympics into a battleground in its contest for
                       legitimacy with the KMT regime on Taiwan and its
                       imperialist backers, elevating the dispute to “the
                       main burden of Olympism,” in the words of Otto
                       Mayer, chancellor of the International Olympic
                       Committee (IOC) from 1946 to 1964. And as with the
                       parallel struggle for recognition by the United
                       Nations, this one ended after three eventful
                       decades in unqualified triumph. University of Hong
                       Kong historian Xu Guoqi relates this fascinating
                       saga in his 2008 book Olympic Dreams: China and
                       Sports, 1895-2008.


                # ⚓ Helsinki_2.0⠀⇛


                       You might think that’s an overstatement. NATO is
                       alive and well. The Organization for Security and
                       Cooperation (OSCE) in Europe is still functioning
                       at a high level.


                       Of course, there’s the possibility of a major war
                       breaking out between Russia and Ukraine. But would
                       Russian President Vladimir Putin really take such
                       an enormous risk? Moreover, periodic conflicts in
                       that part of the world—in Ukraine since 2014, in
                       Georgia in 2008, in Transnistria between 1990 and
                       1992—have not escalated into Europe-wide wars. Even
                       the horrific bloodletting of Yugoslavia in the
                       1990s was largely contained within the borders of
                       that benighted former country, and many of the
                       Yugoslav successor states have joined both the
                       European Union and NATO.


                # ⚓ Bernie_Sanders_Says_US_War_With_Russia_Over_Ukraine_Must_Be
                  Avoided_at_All_Costs⠀⇛


                       Warning of the potentially catastrophic
                       consequences of what could be the deadliest
                       European conflict since World War II, U.S. Sen.
                       Bernie Sanders on Tuesday stressed the imperative
                       for a diplomatic solution to the Russia-Ukraine
                       crisis that’s brought the world’s two nuclear
                       superpowers perilously close to war. 


                       “Wars rarely turn out the way the experts tell us
                       they will. Just ask the officials who provided rosy
                       scenarios for the wars in Vietnam, Afghanistan, and
                       Iraq.”


                # ⚓ Aid_Groups_Warn_Biden_Not_to_Revive_Trump’s_Terror
                  Designation_for_Houthis_in_Yemen⠀⇛


                       Twenty international aid groups on Tuesday urged
                       U.S. President Joe Biden to refrain from restoring
                       the Trump administration’s terrorist designation
                       for the Houthis—warning that reinstating the label
                       would have devastating impacts on civilians already
                       struggling to survive amid a worsening humanitarian
                       catastrophe in Yemen.


                       Biden was praised last year by peace and anti-
                       hunger advocates for rejecting former Secretary of
                       State Mike Pompeo’s eleventh-hour designation of
                       Ansar Allah—as the Houthi movement is officially
                       called—as a “Foreign Terrorist Organization” (FTO)
                       and a “Specially Designated Global Terrorist”
                       (SDGT).


                # ⚓ Opinion_|_North_Carolina’s_New_Education_Bill_Promotes
                  Historical_Erasure_and_White_Supremacy⠀⇛


                       In the state of North Carolina, the legislature saw
                       fit to pass House Bill 324 in response to the
                       Critical Race Theory madness set off by a FOX News
                       interview. The bill outlaws teaching that includes
                       the following:


                # ⚓ Up_to_90_more_new_tanks_for_police_in_Germany⠀⇛


                       The Federal Police and the riot police of the
                       federal states are paying € 81 million for armoured
                       vehicles from Rheinmetall. Some procured them
                       already for their special forces


                # ⚓ Not_another_no-fly_list⠀⇛


                       In a letter first reported by Reuters and first
                       published in full by The Points Guy, CEO Edward
                       Bastian of Delta Air Lines has called on Attorney
                       General Merrick Garland to “support our efforts
                       with respect to… putting any person convicted of an
                       on-board disruption on a national, comprehensive,…
                       ‘no-fly’ list that  would bar that person from
                       traveling on any commercial air carrier.”


                       The latest letter from Delta steps up a lobbying
                       campaign the airline began last fall, and which
                       remains as misguided as ever. The letter highlights
                       the urgent need for Congress to enact the Freedom
                       to Travel Act (H.R. 6030) to make clear the rights
                       of travelers, the duties of airlines and other
                       common carriers, and the limitations on when, by
                       what authority, on what grounds, and according to
                       what procedures the right to travel can be
                       restricted.


                # ⚓ Baloch_insurgency_and_continued_tale_of_apathy_and
                  violence⠀⇛


                       The province of Balochistan is once again in the
                       vortex of turmoil and violence. On February 5,
                       2022, Pakistani military said it has killed at
                       least 20 rebels in Balochistan after two army posts
                       were attacked within hours of each other on
                       February 2.[1] The attacks targeted paramilitary
                       post in the town of Panjgur, about 450km (280
                       miles) south of the provincial capital, Quetta, and
                       another one in Noshki, about 330km (205 miles)
                       north of the first attack spot. Following the twin
                       attacks, in a statement emailed to the media, the
                       ethnic Baloch separatist group Baloch Liberation
                       Army (BLA) claimed responsibility for the attacks.
                       Jeayand Baloch, the BLA’s spokesman, said the raids
                       were carried out by attackers who were prepared to
                       “self-sacrifice”.[2]


          o § Environment⠀➾


                # ⚓ Montana_Plaintiffs_Announce_First_Children’s_Climate_Trial
                  in_History_of_US⠀⇛


                # ⚓ Climate_Expert_Debunks_Big_Oil’s_Lies_About_Carbon_Capture,
                  Nature-Based_Solutions⠀⇛


                       A world-renowned climatologist made clear to
                       Congress on Tuesday that some of fossil fuel
                       companies’ key proposals to reduce planet-heating
                       emissions are talking points rather than
                       “meaningful” solutions to the climate emergency the
                       industry created.


                       “The climate crisis… is real and it is here. In
                       order to confront it, we need real solutions that
                       are proven to work and to keep our planet safe.”


                # ⚓ Climate_Coalition_Calls_on_Biden_to_Cease_All_Gulf_of
                  Mexico_Drilling_Operations⠀⇛


                       A new legal petition filed Tuesday calls on the
                       Biden administration to stop all fossil fuel
                       exploration and extraction plans throughout the
                       Gulf of Mexico, saying its ongoing approval of such
                       activities violate federal requirements to properly
                       assess climate impacts.


                       The filing with the Interior Department, signed by
                       a diverse collection of 310 groups, begins with a
                       blunt assertion that “fossil fuels are killing us
                       and killing our planet” and points to a federal
                       judge’s ruling last month blocking the
                       administration’s planned oil and gas lease sale of
                       over 80 million acres in the Gulf of Mexico.


                # § Energy⠀➾


                      # ⚓ Footage_of_Offshore_Oil_Accident_Highlights_Risks_of
                        Deepwater_Drilling⠀⇛


                             Video footage obtained by DeSmog shows an
                             offshore oil drilling incident in which heavy
                             equipment fell from a drillship onto an oil
                             and gas wellhead deep below the ocean’s
                             surface. 


                             The three short clips offer those on land an
                             unusual window into the world of offshore
                             drilling accidents. Two experts who reviewed
                             the footage for DeSmog said that the videos
                             appear to show a blowout preventer (BOP) —  a
                             kind of heavy oilfield equipment that is used
                             to control or seal the flow of oil and gas
                             from wells — falling suddenly during a
                             relatively routine procedure. An additional
                             video appears to show the resulting
                             underwater damage filmed from a remotely
                             operated vehicle. Text displayed in one clip
                             reads, in part, “Monitoring BOP Unlatch” and
                             “Pontus,” and a second video of the incident
                             bears the date November 20, 2021.


                      # ⚓ ‘Obscene’:_BP_Profits_Hit_8-Year_High_Amid_Climate
                        Emergency⠀⇛


                             Fueled by rising oil and gas prices that have
                             left millions struggling to afford energy
                             bills, British fossil fuel giant BP reported
                             its highest yearly profits in nearly a decade
                             on Tuesday while rejecting calls for a tax on
                             its financial windfall.


                             The company raked in $12.8 billion in profits
                             in 2021—more than its annual income for the
                             past eight years. The announcement comes a
                             week after BP’s rival Shell reported $19.3
                             billion in profits last year.


                      # ⚓ PG&E_is_on_the_Loose_Again⠀⇛


                             What were the terms of probation he had been
                             asked to oversee? In Judge Alsup’s statement,
                             “Rehabilitation of a criminal offender
                             remains the paramount goal of probation.
                             During these five years of criminal
                             probation, we have tried hard to rehabilitate
                             PG&E. As the supervising district judge,
                             however, I must acknowledge failure.”


                             An ordinary citizen on probation is
                             admonished not to break any laws, deal in
                             guns, drugs, and so on. Probation officers
                             supervise the probationers with weekly
                             visits. At the discovery of a single
                             violation, the probationer is remanded to
                             prison, with an additional burden of time on
                             their sentence.


                      # ⚓ Nornickel’s_Potanin_sees_bright_future_for_NFTs,_not
                        cryptocurrencies⠀⇛


                             Risky cryptocurrencies, such as much-hyped
                             Bitcoin, will give place to asset-backed
                             digital financial assets like non-fungible
                             tokens (NFTs) in the long-term due to the
                             latter’s reliability, Vladimir Potanin, one
                             of Russia’s richest men with assets in
                             different industries and countries, including
                             Finland, told Bloomberg in an interview.


                             Last week, Atomyze Russia, a fintech startup
                             backed by Potanin’s conglomerate Interros,
                             became the first in Russia to obtain the
                             central bank’s license to issue digital
                             financial assets.


                      # ⚓ How_Much_Electricity_Does_Bitcoin_Mining_Use?⠀⇛


                             Researchers from Cambridge University
                             estimates that the mining process devours 143
                             terawatt-hours (TWh) of electricity annually,
                             which is more than the yearly electricity
                             consumption of some countries. For instance,
                             Norway consumes only 124 TWh, and Bangladesh
                             consumes 71. Also, Bitcoin mining requires
                             more energy than the largest technology
                             companies like Google and Facebook.


                      # ⚓ ‘The_Entire_Cryptocurrency_Market_Is_Basically_a
                        Ponzi_Scheme’⠀⇛


                             Janine Jackson interviewed Sohale Mortazavi
                             about cryptocurrency for the February 4,
                             2022, episode of CounterSpin. This is a
                             lightly edited transcript.


                      # ⚓ Crossing_the_Wires_of_Energy_and_Cryptocurrency
                        Policy:_U.S._Congress_Investigates_the_Environmental
                        Impact_of_Crypto_Mining⠀⇛


                             There are currently two primary ways that
                             network participants lend their processing
                             power, which are framing part of the modern
                             energy policy debates around cryptocurrency.
                             The first form is “proof of work,” which is
                             the original method that Bitcoin and Ethereum
                             1.0 employ. When a group of transactions (a
                             block) needs to be verified, all of the
                             “mining” computers race to solve a complex
                             math puzzle, and whoever wins gets to add the
                             block to the chain and is rewarded in coins.
                             The competitive nature of proof of work
                             consensus systems has led to substantial
                             increases in computing power provided by
                             institutional cryptocurrency mining
                             operations and, with that, higher energy
                             demands.


                      # ⚓ 7_countries_where_cryptocurrencies_are_banned⠀⇛


                             Besides, the report found that the number of
                             countries subjecting crypto to anti-money
                             laundering and tax laws has surged 3 times
                             since 2018. For example, all the members of
                             the European Union, except Bulgaria, have
                             currently put these regulations in place.


                             Here are some countries that decided to say
                             no to crypto and why.


                      # ⚓ A_cringe_rapper_slash_Forbes_contributor_allegedly
                        found_with_billions_in_stolen_Bitcoin⠀⇛


                             The Department of Justice has finally found
                             almost all of the billions worth of Bitcoin
                             stolen during the 2016 hack of cryptocurrency
                             exchange Bitfinex — in the hands of a
                             published Forbes and Inc. writer and rapper,
                             who calls herself the “Crocodile of Wall
                             Street,” and her startup founder husband.


                             The DOJ says it seized about $3.6 billion
                             worth of the cryptocurrency, allegedly held
                             by Heather Morgan and her husband, Ilya
                             Lichtenstein. The couple is accused of trying
                             to cover their tracks by laundering the
                             stolen funds through “a labyrinth of
                             cryptocurrency transactions.” It’s the
                             department’s “largest financial seizure
                             ever,” according to a Tuesday press release.


          o § Finance⠀➾


                # ⚓ Amazon_Dodged_$5.2_Billion_in_Taxes_Last_Year,_Report
                  Finds⠀⇛


                # ⚓ ‘Class_Warfare’_in_Action:_Amazon_Dodged_$5.2_Billion_in
                  Taxes_in_2021⠀⇛


                       An analysis released Monday shows that Amazon
                       utilized several perfectly legal mechanisms to
                       avoid paying $5.2 billion in federal corporate
                       income taxes in 2021, a year in which the online
                       retail behemoth saw its profits and sales
                       skyrocket.


                       Matthew Gardner, a senior fellow at the Institute
                       on Taxation and Economic Policy (ITEP), estimated
                       that given Amazon’s record-breaking $35 billion in
                       U.S. pretax income for fiscal year 2021, the
                       Seattle-based corporate giant paid an “effective
                       federal income tax rate of 6%”—far lower than the
                       statutory corporate tax rate of 21%.


                # ⚓ Opinion_|_Paying_Rate_of_Just_6%,_Amazon_Avoided_More_Than
                  $5_Billion_in_Taxes_Last_Year⠀⇛


                       Amazon’s latest annual financial report released
                       last Friday paints a vivid picture of a company
                       that is edging toward monopoly status—and doing so
                       at the expense of U.S. taxpayers.


          o § AstroTurf/Lobbying/Politics⠀➾


                # ⚓ Opinion_|_Will_US_Turn_Away_From_Fascism_and_Abandon_This
                  GOP_Death_Cult_Before_It’s_Too_Late?⠀⇛


                       While Republican successes in blocking legislation,
                       judges and even presidential nominations may seem
                       like the Party is on a roll, the reality is that
                       the GOP is in the midst of an existential crisis as
                       severe as any party has seen since the Whigs died
                       out in the early 19th century.


                # ⚓ Manchin_Says_He’s_Skeptical_of_Congressional_Workers’_Union
                  Drive⠀⇛


                # ⚓ Palin_Redux⠀⇛


                       Refusing the vaccine means freedom, she says, And
                       no one can ever negate hers. The shots would be
                       o’er her dead body, she says. But what if the
                       body’s the waiter’s?


                # ⚓ ALEC_Is_Driving_Laws_to_Blacklist_Companies_That_Boycott
                  the_Oil_Industry⠀⇛


                # ⚓ Is_Korea_Heading_Toward_a_Political_Crossroads?⠀⇛


                       The two candidates, who are currently running neck-
                       and-neck in opinion polls, present a stark
                       contrast. Lee Jae-myung of the ruling Democratic
                       Party advocates South Korea taking the lead on
                       inter-Korean relations, in contrast to President
                       Moon Jae-in’s unwillingness to adopt any measure
                       that would elicit Washington’s disapproval. “In
                       succeeding the Moon Jae-in administration, the Lee
                       Jae-myung government should act as a more
                       independent and active mediator and problem
                       solver,” Lee announced late last year. [1] That
                       will come as a welcome change in direction if it
                       comes to fruition.


                       Lee is also disinclined to accede to U.S. demands
                       to join the anti-China campaign, questioning why
                       South Korea should be forced to choose between
                       China, its leading trading partner, and the U.S.,
                       with whom it has a military alliance. “I think the
                       situation is coming where we can make decisions
                       independently, putting our national interests
                       first. Any thinking that we have to choose between
                       the two is a very disgraceful approach,” Lee
                       argues. [2]


                # ⚓ Biden_Claims_Credit_for_ISIS_Leader’s_Death,_But_It_Will_Do
                  Little_Damage_to_Terror_Group⠀⇛


                       Islamic State (IS) is much reduced from the all-
                       conquering force it was for several years after
                       2014 when it had surprised the world by capturing
                       the city of Mosul in northern Iraq. It went on to
                       seize an area the size of Great Britain in western
                       Iraq and eastern Syria, but lost all this territory
                       between 2016 and 2019. The fact that al-Qurayshi
                       was killed so far from previous IS strongholds may
                       indicate that the movement no longer has any bases
                       that it considers secure.


                       But as a guerrilla group that no longer tries to
                       hold territory, it has recently shown renewed
                       strength with an attack on a prison in Hasakah, a
                       Kurdish controlled city in north east Syria. IS
                       leaders who had escaped the prison were at first
                       believed by locals in Idlib to be the target of the
                       US raid.


                # ⚓ No_Attack_on_Voting_Rights_Is_Too_Racist_for_This_Supreme
                  Court⠀⇛


                       I honestly do not know what kind of attack a state
                       would have to launch against the voting rights of
                       its Black citizens to make the current Supreme
                       Court step in and stop it. I do not know what form
                       of voter-suppression law, intimidation tactic, or
                       redistricting map would be so racist that this
                       court would strike it down. We are back to the Jim
                       Crow era, not just in terms of the laws that states
                       are passing against the franchise but also in how
                       the Supreme Court refuses to enforce the
                       constitutional amendments prohibiting apartheid. We
                       have literally been here before, when the Supreme
                       Court remained inert as the 14th and 15th
                       Amendments were violated with impunity by any state
                       with enough aggrieved whites to do so. All that’s
                       missing is the violent enforcement of these racist
                       voting rules… but we’ll see what happens when Black
                       people still try to vote this fall.


                # ⚓ What_America’s_Voting_Rights_Activists_Can_Learn_From
                  Previous_Civil Rights_Movements⠀⇛


                       Senate Democrats Joe Manchin of West Virginia and
                       Kyrsten Sinema of Arizona joined Senate Republicans
                       in blocking both the Freedom to Vote Act and the
                       John Lewis Voting Rights Advancement Act. These
                       bills would have combated voter suppression by
                       creating a national automatic voter registration
                       system, and they also would have banned partisan
                       gerrymandering.


                       In the wake of the vote, President Joe Biden said
                       he was “profoundly disappointed that the United
                       States Senate has failed to stand up for
                       democracy.”


                # ⚓ The_Five_(Rosa)_Luxembergs⠀⇛


                       Today, many know Rosa Luxemburg as Rosa Luxemburg
                       but when she was born – 5th March 1871 – her real
                       name was actually Róża Luksemburg. Rosa was also
                       known as Rozalia Luksenburg. On 13th June, 1919, an
                       unimaginably large funeral procession walked
                       through Germany’s capital, Berlin almost five
                       months after Rosa Luxemburg was murdered. Germany’s
                       digital library shows thousands and thousands of
                       workers have lined Berlin’s streets to pay tribute
                       to Rosa Luxemburg.


                       Yet, among one of the largest number of people ever
                       seen at a Berlin rally were also four people with
                       whom Rosa Luxemburg knew extremely well during the
                       forty-eight years before her murder by right-wing
                       death squads known as free corps. During the 1918/
                       19 revolution, Germany’s majority social-democratic
                       party, the SPD, had, what Rosa Luxemburg would call
                       a choice between Socialism or Barbarity. It used
                       Gustav Noske – known as the bloodhound – and the
                       free corps to shoot a massive amount of workers
                       and, thereby ending the revolution.


                # ⚓ Peter_Thiel_to_Exit_Meta’s_Board_to_Support_Trump-Aligned
                  Candidates⠀⇛


                       Mr. Thiel has been on Meta’s board since 2005, when
                       Facebook was a tiny start-up and he was one of its
                       first institutional investors. But scrutiny of Mr.
                       Thiel’s position on the board has steadily
                       increased as the company was embroiled in political
                       controversies, including barring Mr. Trump from the
                       platform, and as the venture capitalist has become
                       more politically active.


                # ⚓ White_House_science_adviser_Lander_quits_over_bullying
                  claims⠀⇛


                       Ms Wallace told Politico that Professor Lander
                       “retaliated against staff for speaking out and
                       asking questions by calling them names, disparaging
                       them, embarrassing them in front of their peers,
                       laughing at them, shunning them, taking away their
                       duties, and replacing them or driving them out of
                       the agency. Numerous women have been left in tears,
                       traumatised, and feeling vulnerable and isolated.”


          o § Misinformation/Disinformation⠀➾


                # ⚓ Homeland_Security’s_Attempts_to_“Combat_Misinformation”_Are
                  Suspect⠀⇛


                # ⚓ Neil_Young_urges_Spotify_employees_to_quit⠀⇛


                       Spotify has also said it will add advisories to any
                       podcast episodes that discuss Covid-19.


                # ⚓ Joe_Rogan_Offered_$100M_to_Quit_Spotify_for_Right-Wing
                  Platform⠀⇛


                       The controversial podcaster was offered $100
                       million on Monday to quit Spotify and join a right-
                       wing platform.


                       Chris Pavlovski, the CEO of the Canadian video-
                       sharing platform Rumble, tried to entice Rogan to
                       jump ship in a letter posted to the company’s
                       Twitter account Monday.


          o § Censorship/Free Speech⠀➾


                # ⚓ Techdirt_Podcast_Episode_310:_A_Global_History_Of_Free
                  Speech⠀⇛


                       We talk a lot about free speech in different
                       countries, and about the history of free speech in
                       the US — but what about the global history of this
                       fundamental concept? A new book released today,
                       Free Speech: A History from Socrates to Social
                       Media by Jacob Mchangama, tackles exactly this
                       subject in great and insightful detail. This week,
                       Jacob joins us on the podcast to discuss the
                       sweeping story of free speech throughout the ages
                       and around the world.


                # ⚓ Book_Bans_Are_on_the_Rise._But_Librarians_and_Authors_Are
                  Fighting_Back.⠀⇛


                       The idea of banning books conjures images of piles
                       of hardcovers in the street going up in flames. But
                       over the past few decades in the United States,
                       book banning has taken on a decidedly more genteel
                       character. It has taken place in deliberative
                       school board meetings and in quick after-school
                       chats between librarians and concerned parents.


                # ⚓ Israel_Gets_Georgia_to_Strip_Free_Speech_Rights_(Again)⠀⇛


                # ⚓ Opinion_|_Divestment_Is_Wisdom,_Not_Censorship⠀⇛


                       We live in a time when bribery, in the form of
                       campaign donations, is construed by our nation’s
                       Supreme Court as free speech, and divestment of the
                       kind that helped pressure South Africa to dismantle
                       its apartheid system is equated in the corporate
                       media as censorship. These are signs that the
                       culture in our democracy is unhealthy, diseased. 


                # ⚓ Platform_vs._Publishers⠀⇛


                       How do you distinguish between a platform and a
                       publisher? The debate has been reignited with the
                       backlash against Joe Rogan’s Spotify podcast. Some
                       have accused Rogan of spreading misinformation
                       about COVID on Spotify. Spotify isn’t taking Rogan
                       down and its important to note that Spotify has a
                       $100mm+ exclusive deal with Rogan for his content.
                       Is Spotify a neutral audio platform? Or is Spotify
                       a publisher?


                # ⚓ TikTok_bans_misgendering,_deadnaming_trans_people⠀⇛


                       In updated community guidelines, the company noted
                       that it is “adding clarity on the types of hateful
                       ideologies prohibited on our platform,” stating
                       that it will ban deadnaming, or using a transgender
                       person’s pre-transition name, and misgendering,
                       using incorrect pronouns.


                # ⚓ TikTok_is_banning_misgendering,_deadnaming,_and_content
                  promoting_disordered_eating⠀⇛


                       The company announced today it would begin removing
                       videos that promote disordered eating symptoms like
                       short-term fasting and overexercising. Content
                       promoting eating disorders is already banned on the
                       platform, but news reports have repeatedly shown
                       users are exposed to videos featuring unhealthy
                       eating habits. During an October congressional
                       hearing, senators pressed TikTok representatives on
                       further protecting young users.


                # ⚓ [Updated:]_EARN_IT_Act_—_an_attack_on_free_expression_and
                  privacy_—_is_back⠀⇛


                       A civil society coalition letter voicing opposition
                       to the EARN It Act was delivered to the U.S. Senate
                       Judiciary Committee leadership in advance of the
                       Committee markup. Access Now is a signatory to the
                       letter, signed by over 60 organizations and led by
                       the Center for Democracy and Technology.


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


                # ⚓ BBC_News_Russian:_Nearly_half_of_people_labeled_‘media
                  foreign_agents’_have_left_or_are_preparing_to_leave_Russia⠀⇛


                       Russia has designated 75 individuals as “media
                       foreign agents.” And according to BBC News Russian,
                       at least 32 of these people have left or are
                       preparing to leave the country. Another 14
                       individuals had already left Russia due to pressure
                       from the authorities before they were added to the
                       “foreign agent” blacklist.


                # ⚓ Journalist_Mary_Heaton_Vorse_Spent_Decades_on_the_Front
                  Lines_of_Labor⠀⇛


                       Coverage of labor, unions, and workers’ rights
                       issues has exploded across a variety of media,
                       particularly as the Covid-19 pandemic exposes the
                       plight of vulnerable workers and changes the nature
                       of work itself. Mainstream publishing behemoths are
                       now trying to catch up to the left-leaning
                       publications that have always cared about such
                       stories. The organizing streaks of media unions
                       like the Writers Guild of America, East (for which
                       I’m a council member), and the NewsGuild have
                       resulted in a generation of labor-savvy media
                       workers with a fierce commitment to improving their
                       workplaces. (This is how I unexpectedly morphed
                       from a heavy metal journalist into a labor
                       reporter!)


                # ⚓ Update_on_Julian_Assange_Case,_Future_of_Gulf_of_Mexico
                  Drilling,_and_is_Graphika_Really_a_Fake_News_Watchdog?_–_The
                  Project_Censored_Show⠀⇛


                       Notes: Kevin Gosztola is the managing editor of
                       ShadowProof as well as other press-freedom and
                       whistleblower cases. Jayeesha Dutta is co-founder
                       of AnotherGulf.com; she also works with the Climate
                       Justice Alliance. Kevin Gosztola is the managing
                       editor of ShadowProof. He has covered the Julian
                       Assange legal proceedings in the UK from the
                       beginning. Alan MacLeod is a media scholar and a
                       senior staff writer at Mint Press News.


                # ⚓ Didn’t_Those_Now_Enraged_at_Boris_Johnson’s_‘Smears’_of
                  Keir_Starmer_Defame_Corbyn_at_Every_Turn?⠀⇛


                       The reason? Last week the British prime minister,
                       Boris Johnson, blamed Starmer, now the Labour party
                       leader, for failing to prosecute Jimmy Savile, a TV
                       presenter and serial child abuser, when his case
                       came under police review in 2009. Between 2008 and
                       2013, Starmer was head of the Crown Prosecution
                       Service (CPS). Savile died in 2011 before he could
                       face justice.


                       Johnson accused Starmer, who at the time was
                       Director of Public Prosecutions, of wasting “his
                       time prosecuting journalists and failing to
                       prosecute Jimmy Savile”.


                # ⚓ ‘They_were_mostly_shouting_about_Assange!’_Tory_MP_hits
                  back_at_Boris-Starmer_row⠀⇛


                       “I’ve also watched the clip in full, it’s about
                       three minutes long and these people were mostly
                       talking about Julian Assange for some reason that I
                       don’t understand.


                # ⚓ Keir_Starmer:_No_10_has_no_intention_of_apologising_after
                  protesters_surround_Labour_leader⠀⇛


                       “Why did you go after Julian Assange, why did you
                       go after journalists?” one man shouted.


                # ⚓ Police_rescue_Sir_Keir_Starmer_from_mob_who_surround_him
                  and_accuse_him_of_‘protecting_paedophiles’⠀⇛


          o § Civil Rights/Policing⠀➾


                # ⚓ Russia’s_Justice_Ministry_seeks_dissolution_of_charitable
                  foundation_behind_Russian_LGBT_Network⠀⇛


                       The St. Petersburg branch of the Russian Justice
                       Ministry has asked the courts to liquidate the
                       charitable foundation Sfera, the operator of the
                       Russian LGBT Network.


                # ⚓ Sanders_Unveils_Bill_to_Boost_Understaffed_US_Fire
                  Departments⠀⇛


                       Citing the “unprecedented challenges” in
                       “recruiting and retaining” career and volunteer
                       firefighters across the United States, Sen. Bernie
                       Sanders on Monday introduced legislation aimed at
                       tackling the staffing crisis in the essential
                       profession. 


                       “The difficulty in recruiting and retaining
                       personnel is an absolute crisis.”


                # ⚓ Religious_Support_for_Democratic_World_Federation⠀⇛


                       But some religious groups and leaders have rejected
                       the idea that warfare is a necessary part of human
                       nature and human history.


                       In order to eliminate the war system and solve
                       humanity’s many global problems–such as genocide,
                       terrorism, violations of human rights, and
                       environmental degradation due to global
                       warming–they have advocated for the creation of a
                       democratic world federal government that would
                       create just world laws and prosecute individuals
                       who violate them. Violent conflicts would be
                       settled and global problems would be solved through
                       a nonviolent democratic process.


                # ⚓ The_Black_Migrant_Trail_of_Tragedies⠀⇛


                       Julliana Essengue arrived in Tapachula, Mexico,
                       from São Paulo, Brazil, in March 2020. She was
                       broke but determined to reach the United States.
                       After nearly two months traversing rain forests,
                       borders, and rivers by bus, car, boat, and foot,
                       she needed money. At first, Essengue and her travel
                       companions squatted. “We slept on the floor for two
                       weeks in a hallway,” she told me. “There were
                       Africans, Haitians—everybody was sleeping on the
                       floor.”1This story was published with the support
                       of a fellowship from Columbia University’s Ira A.
                       Lipman Center for Journalism and Civil and Human
                       Rights.


                # ⚓ Ohio_Supreme_Court_Rejects_Republican-Drawn_Congressional
                  Maps_–_Again⠀⇛


                # ⚓ Affirmative_Action_on_the_Block⠀⇛


                # ⚓ Connecting_the_Dots_and_Rejecting_the_“Rule_of_Law”
                  Hypocrisy⠀⇛


                       Lindiwe Sisulu’s parents, Walter and Albertina
                       Sisulu, were leading figures with Nelson Mandela in
                       the African National Congress (ANC) movement that
                       resisted the racist apartheid regime of South
                       Africa. Their daughter’s opinion column shows that
                       she inherited her parents’ zeal for justice.


                       “Whose law is it anyway” bristles with truth about
                       economic reparation, a subject that the
                       beneficiaries of colonization always try to avoid.
                       Consider the following excerpt:


                # ⚓ ‘They_Are_Firing_the_Entire_Committee!’_Memphis_Workers_Say
                  Starbucks_Targeting_Union_Organizers⠀⇛


                       Workers at a Memphis Starbucks who were fired
                       Tuesday after launching a unionization effort vowed
                       to carry on their fight, with one employee invoking
                       the legacy of the Rev. Martin Luther King Jr.—who
                       was assassinated in the Tennessee city while
                       campaigning for workers’ rights. 


                       “Starbucks has been fighting desperately to silence
                       us because we did not back down or let them shake
                       us.”


                # ⚓ Hunger-Striking_Teachers_Say_Oakland_Plan_to_Close_Schools
                  Will_Hurt_Black_&_Brown_Communities⠀⇛


                       We go to Oakland, where a group of teachers are on
                       a hunger strike to protest a plan to close and
                       merge over a dozen schools due to under-enrollment.
                       This comes ahead of a critical school board vote
                       Tuesday that will decide whether to proceed with
                       the plan. Activists argue the move threatens to
                       divert resources to charter schools and displace
                       hundreds of Black and Brown children from their
                       neighborhood schools. The hunger strike across
                       multiple different schools has empowered many to
                       speak up against longtime systemic racism, says
                       Moses Omolade, one of the striking workers and a
                       community schools manager at Westlake Middle
                       School. “The school board is attempting to close
                       predominantly Black and Brown schools without
                       engaging with us at all.”


                # ⚓ How_a_Cooperative_Run_by_the_Formerly_Incarcerated_is
                  Reshaping_Chicago’s_Food_Industry⠀⇛


                       These school meals were supplied by
                       megacorporations like PepsiCo Inc., Tyson Foods
                       Inc., Pilgrim’s Pride Corporation, Cherry Meat
                       Packers Inc., Central Valley Meat Co. Inc.,
                       American Beef Packers Inc. and Jennie-O Turkey
                       Store LLC. As detailed in a 2020 article by
                       Jennifer E. Gaddis in the professional journal for
                       educators Phi Delta Kappan, 95 percent of U.S.
                       public schools participate in the government-
                       subsidized National School Lunch Program, and this
                       program is made up almost entirely of contracts by
                       giant corporate food brands. Gaddis writes:


                       It happened again in 2020 when the Trump
                       administration proposed making the rules more
                       flexible, Gaddis adds.


                # ⚓ The_student_forced_out_after_standing_up_for_women⠀⇛


                       Yet from the minute it was announced, it became a
                       target for militant trans activists among the
                       university students – who objected to Woman’s Place
                       UK’s stance on gender recognition issues, and
                       demanded the event be cancelled.


                       Why? “The issue they had was with the meeting
                       taking place at all,” Rosario Sanchez replies.
                       “They didn’t want women to have these
                       conversations.”


                # ⚓ Why_This_Feminist_Is_Taking_the_University_of_Bristol_to
                  Court⠀⇛


                       Next week, I am taking my university to court. To
                       my knowledge, it is the first time an academic
                       institution has been forced, at trial, to justify
                       why it prioritises trans rights over women’s
                       rights. The other party in the case is the
                       University of Bristol, which one might suppose to
                       be an unlikely defendant given its distinction as
                       the first higher-education establishment in England
                       to have admitted women on an equal basis to men.
                       Unfortunately, the university has more recently
                       become known as a hotbed for anti-feminist
                       militancy.


                # ⚓ LABOR_After_Amazon_Tragedy,_Workers_Come_Together_to_Demand
                  Safe_Working_Conditions⠀⇛


          o § Digital Restrictions (DRM)⠀➾


                # ⚓ Automakers_Can’t_Give_Up_The_Idea_Of_Turning_Everyday
                  Features_Into_Subscription_Services_With_Fees⠀⇛


                       At the same time car companies are fighting the
                       right to repair movement (and the state and federal
                       legislation popping up everywhere), they’re
                       continuing the quest to turn everyday features —
                       like heated seats — into something users have to
                       pay a recurring fee for.


                # ⚓ What_on_Earth_is_going_on_with_Peloton?⠀⇛


                       In 2020, Peloton’s supply chain was struggling to
                       keep up with the unexpected surge in demand created
                       by people suddenly eager to work out at home.
                       Facing months-long fulfillment delays, Peloton
                       decided to heavily invest in building out its
                       manufacturing capabilities. It dropped millions to
                       expedite shipping and another $420 million buying
                       Precor, one of the world’s largest commercial
                       fitness equipment makers. It spent another $400
                       million on a factory in Ohio. The company was
                       positioning itself to rapidly build bikes and
                       treadmills for a market that couldn’t get enough.


          o § Monopolies⠀➾


                # § Patents⠀➾


                      # ⚓ Pfizer_Forecasts_Record-Breaking_2022_Profits_—_as
                        Billions_Remain_Unprotected⠀⇛


                      # ⚓ Pfizer_Forecasts_Record_Vaccine_Revenue_for_2022_as
                        Billions_Remain_Unprotected⠀⇛


                             Pfizer projected Tuesday that it will
                             generate a record-shattering $32 billion in
                             revenue from its publicly funded coronavirus
                             vaccine in 2022 as the U.S. pharmaceutical
                             giant refuses to share its technology with
                             other nations, leaving billions of people
                             around the world without access to lifesaving
                             shots as the pandemic continues to rage.


                             “We’ve let Pfizer withhold this essential
                             medical innovation from much of the world.”


                # § Trademarks⠀➾


                      # ⚓ Apple_Opposes_Trademark_For_Indie_Film_‘Apple-Man’
                        Claiming_Potential_Confusion⠀⇛


                             When it comes to silly trademark disputes,
                             Apple has come up for discussion many, many
                             times. The mega-corporation is a jealous
                             defender of all of its IP, but most of our
                             stories have focused on its disputes with
                             companies that created logos that involve any
                             sort of apple or other fruit. Sometimes it’s
                             not even companies that Apple is fighting
                             with, but entire foreign political parties.
                             The idea here is that when it comes to logos
                             or trade dress, Apple appears to think that
                             it owns all the apples.


                # § Copyrights⠀➾


                      # ⚓ Appeals_Court_Can_Rule_That_DMCA’s_Anti-Circumvention
                        Rules_Are_Unconstitutional⠀⇛


                             As you hopefully know, there are two main
                             parts to the DMCA law that was passed in
                             1998. There’s DMCA 512, which is what you
                             hear about most of the time. That’s the part
                             that includes the rules for notice and
                             takedown regimes for user uploaded content
                             (among other things). It’s got problems, but
                             in its current form has also enabled many
                             important services to exist. The other part,
                             which is much more problematic, is DMCA 1201,
                             which is the anti-circumvention rules — or
                             you could call it the “DRM” part of the law.
                             This has no redeeming value whatsoever. Under
                             1201 basically any attempt to circumvent a
                             “technological” protection measure, can be
                             deemed infringing even if the underlying
                             content is never infringed upon. This part of
                             the law is not only not necessary, but it’s
                             drafted in a manner that has been regularly
                             abused — enabling everyone from printer
                             manufacturers to garage door opener companies
                             to argue that simple reverse engineering to
                             create competition is “infringement.”


                      # ⚓ Do_not_feed_the_trolls⠀⇛


                             Recently, there has been an increase in
                             threatened and actual lawsuits involving CC
                             licensed works, and in some cases, license
                             enforcement has even become a business model.
                             We have now learned that even long-time
                             friend and contributor to Creative Commons,
                             Cory Doctorow, has been targeted.


                             Put simply, “license-enforcement-as-business
                             model” is a perversion of the founding ideals
                             of Creative Commons. We condemn this
                             behavior. These aggressive enforcement
                             actions lessen trust in open licensing, and
                             they erode the good faith ecosystem that is
                             the basis of the commons.


                      # ⚓ MPA_&_ACE_Embed_Staff_at_US_Govt._IPR_Center_To_Fight
                        Movie_&_TV_Show_Piracy⠀⇛


                             The Motion Picture Association and anti-
                             piracy coalition Alliance for Creativity and
                             Entertainment will embed their own personnel
                             at the US Government’s National Intellectual
                             Property Rights Coordination Center. Under
                             the umbrella of the Department of Homeland
                             Security and Immigration and Customs
                             Enforcement, they hope to tackle movie and TV
                             show piracy more effectively.


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