𝕿𝖊𝖈𝖍𝖗𝖎𝖌𝖍𝖙𝖘 Bulletin for Wednesday, October 19, 2022
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Generated Thu 20 Oct 02:45:18 BST 2022
Created by Dr. Roy Schestowitz (𝚛𝚘𝚢 (at) 𝚜𝚌𝚑𝚎𝚜𝚝𝚘𝚠𝚒𝚝𝚣 (dot) 𝚌𝚘𝚖)
Full hyperlinks for navigation omitted but are fully available in the originals
The corresponding HTML versions are at 𝒕𝒆𝒄𝒉𝒓𝒊𝒈𝒉𝒕𝒔.𝒐𝒓𝒈
Latest in 𝒉𝒕𝒕𝒑://𝒕𝒆𝒄𝒉𝒓𝒊𝒈𝒉𝒕𝒔.𝒐𝒓𝒈/𝒕𝒙𝒕 and older bulletins can be found at 𝒉𝒕𝒕𝒑://𝒕𝒆𝒄𝒉𝒓𝒊𝒈𝒉𝒕𝒔.𝒐𝒓𝒈/𝒕𝒙𝒕-𝒂𝒓𝒄𝒉𝒊𝒗𝒆𝒔
Full IPFS index in 𝒉𝒕𝒕𝒑://𝒕𝒆𝒄𝒉𝒓𝒊𝒈𝒉𝒕𝒔.𝒐𝒓𝒈/𝒊𝒑𝒇𝒔 and as plain text in 𝒉𝒕𝒕𝒑://𝒕𝒆𝒄𝒉𝒓𝒊𝒈𝒉𝒕𝒔.𝒐𝒓𝒈/𝒊𝒑𝒇𝒔/𝒕𝒙𝒕
Gemini index for the day: gemini://gemini.techrights.org/2022/10/19/


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

QmenFrYXbGBu2iTqkM1ZVF7uotUuiMKBWQgMwjKRxvNmog
QmYa74Gwhe1CXD1aRApxCZPiNvwqHtoaRNRJYUXXzTAzcC
QmR2u26QZwYsd8Gmx2PtKEkxWiRY7DBNoBnQ9eCG6wYxoB
QmVLjiRjCRwAgqnB8rLKTYXEMmL6cgQMvDGgKZemCXwyc4
QmQQSCUTAMNTrxToyg9EZTr3n4PdA5SSHFH5pZzst2ZETe
QmY3hPf5WuZiFd6LH7kZWe1SBQLGYp3DzB3DvDwHxtAxpX
Qmem9pBYyW9sJYffppf7bdghegoTHn1RzdbF1Q1wjVzuGt
QmaiE9iDuXMZhkHdQ8sJYqT3PNn8BWc5agK74EKN1FFgVy
QmQGHw3z4ajT5x6NnbyGQJY26ZEzJFXRTaYxZMQjss3typ
Qmb95NfQvZvbiG3UrtcZ61z7v8dG2fT8Xta1dM4tpptx3Q
QmUWAGmBBXwstayLUmSY9GEzyUBGwmk4zqtUacYeqFcfHh
QmYL4QpoDXAuMDYpijo6ixymXcLVY36699bQ4n638FyYYW
QmWagM6RUv8w3PGimQk2Jw6kAYSZ68kiHJh6JX3FpLMDLJ
QmeujuDVZ2c9H9rLtDft62WqQQuPXjH3uNgNd1mb6hN4Z4
QmWNRczB6EwJrhHTYT8mrC5NnQ2JiW5AnotNG26HncgbeS
QmWUqtTJ5A915BjiikNX4tcP5UqeaYwu2pTVfqCwtSdmX1
QmanWSwdWKoBnEEpdQYF8hRpxJXaJXHrA8matdZ7p2g7Ui
QmNmKK6D23Ute87ocKh4uQTutC8tKr57v4KaYevBmU77dB
QmV2MXBi3SZKoVRsy5rrvEwJJ2S9T5pKs5B6Ravu4xXApS
QmSZzkk2vvg7T5pLZayi713tGeNgvw8o1pX2K38PrthWaz
QmXiY6pc7Wic9TMGYJRv1DHHFXZi8qAmMS1GDp97S7KkZL


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

⦿ IRC Proceedings: Tuesday, October 18, 2022 | Techrights

⦿ Linux Foundation, Not ’Supply Chain’ Nonsense, is Becoming the Biggest Threat to Linux Security and Computing Security (It’s Deliberate) | Techrights

⦿ Microsoft’s Open Compute Project Foundation (OCP): Embrace, Extend, Extinguish Security, Thanks to ’Linux’ Foundation | Techrights

⦿ Microsoft: Azure is Our Future. Also Microsoft: TWO Rounds of Layoffs in Azure So Far This Year (Stock Crashes Again). | Techrights

⦿ Walmart’s Employee Benefits Web Site Requires Microsoft Internet Explorer in 2022 | Techrights

 ䷼ Bulletin articles (as HTML) to comment on (requires login):
		http://techrights.org/2022/10/19/irc-log-181022/#comments
		http://techrights.org/2022/10/19/linux-foundation-ocp/#comments
		http://techrights.org/2022/10/19/linux-foundation-vs-security/#comments
		http://techrights.org/2022/10/19/microsoft-future-is-layoffs/#comments
		http://techrights.org/2022/10/19/walmart-msie-in-2022/#comments

 ䷞ Followed by Daily Links (assorted news picks curated and categorised):
		http://techrights.org/2022/10/19/antix-22-released/#comments
		http://techrights.org/2022/10/19/easyos-4-4-2/#comments

                    ䷩ 𝚕𝚒𝚗𝚎 60

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

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/10/19/irc-log-181022/#comments
    Gemini version at gemini://gemini.techrights.org/2022/10/19/irc-log-181022/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 10.19.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ IRC_Proceedings:_Tuesday,_October_18,_2022⠀✐


Posted in IRC_Logs at 2:59 am by Needs Sunlight


Also available via the Gemini protocol at:


    * gemini://gemini.techrights.org/irc-gmi/irc-log-techrights-181022.gmi
    * gemini://gemini.techrights.org/irc-gmi/irc-log-181022.gmi
    * gemini://gemini.techrights.org/irc-gmi/irc-log-social-181022.gmi
    * gemini://gemini.techrights.org/irc-gmi/irc-log-techbytes-181022.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
 QmaMyBjwyH5icSnjE3NGSrZAkQgjfcccGazLoA4DypHiAZ #boycottnovell  🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
                                                 (full IRC log
                                                 as HTML)


                                                 IRC log for
                                                 #boycottnovell
 QmVEG5QXw9G9Hs4ZotpCXWjYEU3iC8CjC215k92zoB9MpM (full IRC log   🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
                                                 as plain/ASCII
                                                 text)


                                                 IRC log for
                                                 #boycottnovell-
 QmUhTYVV3LkBkgEEE5Kn2DKruurg7PSgJooqGBLr1JaVVu social          🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
                                                 (full IRC log
                                                 as HTML)


                                                 IRC log for
                                                 #boycottnovell-
 QmZXP7XrAQzJKa9CAjAEnpZpStCseXeE8FRZox3oKRG1ed social          🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
                                                 (full IRC log
                                                 as plain/ASCII
                                                 text)


                                                 IRC log for
 QmZy2f7PCYYb1REp2RkXEZ8dQ4TgtSmxPdeMiesTQtTR45 #techbytes      🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
                                                 (full IRC log
                                                 as HTML)


                                                 IRC log for
                                                 #techbytes
 QmYS64pY5zHx2dEXHoBdsXR8ko4V66wrDyvgsrmaGA8mcb (full IRC log   🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
                                                 as plain/ASCII
                                                 text)


                                                 IRC log for
 Qmd98TSCtZbVFttTAxUqhSh37ZCjkjqftaGnx51XRccbtK #techrights     🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
                                                 (full IRC log
                                                 as HTML)


                                                 IRC log for
                                                 #techrights
 QmZwaaoeYwfXMgDEBGZ9F3EovRwG9sDCMZWAiXjkvPwU7p (full IRC log   🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
                                                 as plain/ASCII
                                                 text)


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇IPFS logo⦈


§ Bulletin for Yesterday⠀➾


Local_copy | CID (IPFS): QmXiY6pc7Wic9TMGYJRv1DHHFXZi8qAmMS1GDp97S7KkZL



                    ䷩ 𝚕𝚒𝚗𝚎 187

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

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/10/19/linux-foundation-ocp/#comments
    Gemini version at gemini://gemini.techrights.org/2022/10/19/linux-foundation-ocp/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 10.19.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Linux_Foundation,_Not_‘Supply_Chain’_Nonsense,_is_Becoming_the_Biggest_Threat
to_Linux_Security_and_Computing_Security_(It’s_Deliberate)⠀✐


Posted in Deception, Microsoft, Security at 11:27 am by Dr. Roy Schestowitz


Phoronix just_repeated_false_claims_and_reproduced_Microsoft's_marketing
material_with_laughable_openwashing:


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


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


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


Summary: The Linux_Foundation helps undermine real security by letting truly
untrustworthy companies lock systems down (via OCP, a Microsoft front group
which pretends to be vendor-neutral), just like it did for IBM with TPM/2


                      🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇LF_'security'⦈_


Ignore the scapegoat, focus on who is a proponent for imperial_back_doors. The
Linux Foundation is fronting for them.


⠀⡀⠀⣠⠀⢀⢀⠀⢀⡀⢀⠀⢀⡀⠀⡀⡄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
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⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣬⣷⣿⣘⣜⣿⣻⣟⣭⣻⣋⣬⣿⣿⣿⣾⣿⣯⣟⣽⣫⣿⣦⣿⣬⣽⣷⣸⣳⣸⣳⣿⣮⣿⣬⣿⣯⣾⣯⣹⣿⣭⣯⣿⣷⣷⣿⣏⣬⣿⣅⣿⣿⣿⣿⣟⣍⣻⣿⣵⣿⣚⣿⣍⣿⣯⣯⣿⣇⣽⣫⣝⣟⣿⣯
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⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⠐⠀⢄⠀⡂⠠⢠⢀⠀⢄⠀⠠⠠⢀⢄⠠⠀⡂⢄⠀⠠⠀⡠⡀⠄⠀⢀⠄⠀⠀⠠⢰⠀⠀⡠⠠⠀⢐⡀⡠⡠⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠐⠈⢚⠀⠁⢐⠈⠊⠀⠂⠀⠀⠁⠈⠒⠐⠀⠃⠂⠀⠐⠀⠃⠃⠂⠀⠈⠉⠀⠀⠂⠈⠨⠁⠒⠀⠀⠀⠀⠁⠑⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢈⠰⠱⠀⡁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠠⢂⠀⢄⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠡⠀⡆⡇⡏⠠⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢂⠀⠀⠀⠠⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡄⠠⠠⠠⠠⡄⢠⠄⠄⢠⠀⣄⠄⢠⣠⠠⢀⠀⠄⠄⣄⢠⠀⣄⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠢⠀⠀⠀⠀⠂⠄⢂⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠐⠈⠀⠀⠈⠀⠉⠑⢌⠈⠈⠁⠁⠁⠈⠈⠈⠉⠀⠉⠁⠁⠈⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡃⠀⠀⠶⠀⠈⠉⠈⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡐⠀⠀⠀⠄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠐⠄⡀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢀⠊⠀⠀⠀⠀⠀⠈⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣯⣭⣿⣿⣭⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡏⢹⣿⡏⢹⡏⢹⣭⠙⡟⣩⣶⣬⢸⡇⠀⢸⣿⣿⣇⠶⡏⢉⢯⢹⣿⡰⢞⢏⢹⡍⠹⡉⢹⠏⢽⡇⣦⢹⠏⢉⢯⡩⡹⡇⣴⡹⡿⢉⡏⠩⡿⢁⢿⠍⣹⢹⡠⢼⡹⠹⡯⠉⡏⠩⡿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡇⢨⣭⡅⢸⡇⢘⣫⣴⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⡟⣭⢟⠋⠙⠻⢙⢟⠛⡛⢛⡋⢚⣟⢻⡻⠛⠙⢙⢻⢛⠘⠟⢻⣿⡏⢛⢿⢛⢟⡏⠭⢛⢿⠛⡛⢛⡟⢛⣻⣛⣞⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣃⣘⣿⣃⣘⣃⣘⣿⣿⣧⣌⣛⣛⣽⣇⣀⣸⣿⣿⡷⣶⠷⢶⢶⠶⡶⡷⢶⡷⠾⠶⣶⣿⣾⣷⣾⣶⣾⣾⣷⣶⣷⣾⣷⣶⣾⣶⣶⣷⣷⣿⣶⣷⣶⣷⣶⣷⣾⣾⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠿⠿⣿⢿⣿⡿⠿⡿⠿⡟⠿⠿⡟⠿⠿⠿⢿⣿⣿⣷⣷⣷⣷⣷⣶⣿⣷⣾⣷⣶⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣯⣭⣭⣭⣽⣭⣯⣭⣭⣭⣯⣭⣯⣽⣭⣭⣿⣿⣿⡏⠝⠋⢿⢹⠹⡯⡩⡏⠏⣿⠀⣯⡍⠛⣟⠝⠛⢛⣻⠫⠛⠛⠛⠻⡛⢻⢉⠛⠛⠛⠛⣏⡝⠻⠛⠋⠛⡟⠛⢻⡭⠝⢽⣝⠃⠈⡛⢻⡛⠟⡛⠛⠛⣙⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⠻⠟⠻⠟⢟⢟⡟⠿⢀⢉⠉⢉⡉⡉⢉⠉⢉⠉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⠉⠈⢉⢉⠉⢁⠸⣻⠿⠿⣿⣷⣿⣿⣿⣿⣿
⣿⣿⣏⢹⣿⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠿⠾⠷⢿⠾⠷⢷⠿⠷⠾⠿⠷⠿⠷⠿⠿⣷⡷⠿⡶⠆⠤⣤⡤⠤⠤⡤⡤⢤⠤⡤⠤⠤⠤⢤⠤⡤⠀⣿⡿⠿⢷⠷⡿⠾⠾⡾⠶⠿⠿⣿⠿⠿⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⠴⣶⣷⣷⣶⣾⣶⡶⣶⣶⣶⣶⣶⢶⣶⢷⣶⣾⣷⡆⠒⠋⠁⠀⠀⠁⠁⠀⠈⠈⠀⠀⠁⠙⠒⠓⠀⣾⡶⣾⣾⣿⣦⣶⠮⣷⣥⣤⢤⣤⢤⣥⣽⣿⣿
⣿⣿⣧⣼⣿⣿⣯⣭⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣮⣤⣤⣥⣤⣼⣵⣵⣤⣤⣤⣤⣤⣧⣤⣤⣤⣮⣦⣯⣦⣦⣤⣦⣤⣴⣤⣦⣶⣤⣦⣴⣶⣤⣤⣴⣤⣤⣷⣧⣤⣼⣼⣧⣧⣤⣵⣿⣤⣼⣮⣤⣬⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡌⠶⠟⢓⣬⣿⣿⣿⣿⣿⣿⣿⣿⡟⣂⠀⠀⠀⠈⠙⠻⢷⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⢀⣆⣤⣷⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣨⠿⣋⣭⣶⣶⣷⣶⣶⣾⣭⡀⠀⠀⠀⠀⠀⠀⠀⠉⠊⠙⠓⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡄⠀⢸⣿⣿⣿⣿⣄⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⢟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⢸⣻⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣦⣼⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣾⣿⣿⠉⠗⣿⣿⣿⣿⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⢿⣿⣿⣻⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⣀⣿⣿⣿⣯⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢠⣤⣌⠍⠙⠻⠿⠟⠛⠛⠋⣵⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠘⣿⣍⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣠⣶⣿⣿⣙⣷⣀⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠂⠀⢠⣿⣿⠓⡖⠒⣶⣿⣿⣧⡀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣼⣤⡴⠖⠓⠉⠉⣉⣻⢟⡓⠒⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠘⠋⠁⠀⢿⠋⠛⠃⠈⢿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠋⠈⠒⠂⠚⢟⠛⣃⢤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣧⣂⣄⣀⣀⣁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⢈⡡⠤⠘⠉⣉⣥⣶⢄⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠘⠛⠛⢩⣿⣿⡿⢂⣿⣖⣴⣤⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠄⠀⣴⣿⣿⣿⣷⣿⣿⣿⣿⣛⣶⣿⣿⣏⡉⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⡤⡄⠀⠀⢰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠬⢷⡶⠾⠏⠛⠋⠅⣼⡟⠉⠉⢸⣿⡟⠁⠉⠙⠉⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡼⣯⣼⣿⣷⣾⣿⣿⡇⠀⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠈⠉⠁⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢿⡟⡿⠿⢿⣿⣿⣿⣿⣿⡟⠐⡄⠀⣀⣈⣙⣿⣷⣤⢀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠟⠀⠐⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡀⠀⠀⣧⠀⠀⠀⠀⠀⣠⣄⣀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣛⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠁⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⢡⠀⠀⠫⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣶⣶⣦⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣤⣤⣶⣾⡿⠿⠿⠟⠛⠛⠉⠉⠉⠀⢈⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢘⣉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⢿⣿⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡟⣿⣿⠻⣿⢿⠿⢿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠀⠒⠠⠤⠀⠀⠀⠀⠀⠀⠀⠁⢠⢱⣿⡟⣧⣿⢸⣷⣀⡇⠀⠀⣴⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠠⠤⣀⠁⠀⢀⠀⠀⠀⠒⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣖⣶⣶⣿⣷⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣼⣿⣿⣾⣇⣿⣎⠗⠀⠄⢸⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⣯⢵⣖⠓⠲⢖⠈⠀⢊⠂⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠉⠉⠙⠋⠉⠉⠉⠉⠉⢩⠏⠿⠛⡛⠛⠛⠛⠛⠛⠻⠛
⠀⠖⠶⣶⠿⣷⡶⠴⠶⠶⠶⢀⠀⡇⠿⣿⣿⡏⢰⠿⠿⠆⢠⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣸⣷⠀⠉⠁⠈⠁⠀⠐⠐⠒⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⢀⣀⠄⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡌⠶⠟⢓⣬⣿⣿⣿⣿⣿⣿⣿⣿⡟⣂⠀⠀⠀⠈⠙⠻⢷⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⢀⣆⣤⣷⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣨⠿⣋⣭⣶⣶⣷⣶⣶⣾⣭⡀⠀⠀⠀⠀⠀⠀⠀⠉⠊⠙⠓⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡄⠀⢸⣿⣿⣿⣿⣄⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⢟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⢸⣹⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣦⣼⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣾⣿⣿⠉⠗⣿⣿⣿⣿⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⣀⣿⣿⣿⣯⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢠⣤⣌⠍⠙⠻⠿⠟⠛⠛⠋⣵⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠘⣿⣍⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣠⣶⣿⣿⣙⣷⣀⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠂⠀⢠⣿⣿⠓⡖⠒⣶⣿⣿⣷⡀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣸⣤⣴⠖⠓⠉⠉⣉⣛⢟⡒⠒⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠘⠉⠁⠀⢿⠋⠛⠃⠈⢿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠈⠒⠒⠺⠟⠛⣃⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣧⣂⣄⣀⣀⣁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠈⡡⠤⠘⠉⣉⣥⣶⢄⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠘⠛⠛⣩⣿⣿⡿⢂⣿⣖⣰⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠄⠀⢴⣿⣿⣿⣷⣿⣿⣿⣿⣛⣶⣿⣿⣟⡉⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⡤⡄⠀⠀⢰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠬⢷⡲⠾⠋⠛⠫⠅⣼⡿⠉⠉⢸⣿⡟⠁⠙⠙⠉⠁⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡼⣧⣼⣿⣷⣾⣿⣿⡇⠀⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠈⠉⠁⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢿⡟⡿⠿⢿⣿⣿⣿⣿⣿⡟⠐⡄⠀⣀⣈⣙⣿⣷⣤⢀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠟⠀⠐⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡀⠀⠀⣧⠀⠀⠀⠀⠀⣠⣄⣀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣛⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠁⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⢡⠀⠀⠫⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣶⣶⣦⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣤⣤⣶⣾⡿⠿⠿⠟⠛⠛⠉⠉⠉⠀⢈⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢘⣉⡉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⢿⣿⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡟⣿⣿⢻⣿⢿⠿⢿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⢀⣤⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠀⠒⠠⠤⠀⠀⠀⠀⠀⠀⠀⠁⢠⢱⣿⡟⣧⣿⢸⣷⣀⡇⠀⠀⣴⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠠⠤⣀⠁⠀⢀⠀⠀⠀⠒⠀⠀⣸⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣆⣶⣾⣿⣷⣥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣼⣿⣿⣾⣇⣿⣏⠗⠀⠄⢸⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⣯⢵⣖⠓⠲⢖⠨⠀⣂⠂⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠉⠉⠙⠋⠉⠉⠉⠉⠉⢩⠏⠿⠙⡛⠛⠛⠛⠛⠛⠻⠛
⠀⠖⠶⣶⠿⢷⡶⠴⠶⠶⠶⢀⠀⡇⠿⣿⣿⡏⢰⠿⠿⠆⢠⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣸⣷⡀⠉⠁⠈⠁⠀⠐⠐⠒⠀⠀⠀⠀⠀⡀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⢀⡀⠀⢀⣀⠄⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡰⣨⠶⠿⢃⣴⣿⣿⣿⣿⣿⣿⣿⣿⢂⡀⠀⠀⠈⠉⠻⢷⣦⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠀⠀⠀⢠⣠⣼⣆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠐⣽⠟⣋⣭⣽⣿⣿⣶⣿⣭⣁⠀⠀⠀⠀⠀⠀⠀⠈⠊⠙⠻⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣄⠀⠀⣿⣿⣿⣿⣧⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⣟⣻⣿⣿⣿⣧⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣶⣤⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠛⠹⢶⣿⡟⣿⡇⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣇⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⣿⣿⣿⡄⠀⠀⠈⠀⠀⠀⠀⠀⢰⣿⣿⣇⣀⣸⣟⣿⣿⣃⣀⣨⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣠⣄⡈⡛⠻⠿⠿⠿⠿⠿⢋⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠉⠉⢻⣯⡉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢀⣠⣶⣿⣟⣳⣄⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⡗⢲⠒⢢⣼⣿⣷⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣊⣤⣭⠝⠋⠉⠛⠫⢿⡻⠶⠦⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠈⠉⠀⠹⡏⠉⠁⠀⠙⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣼⣿⣿⣾⣶⣾⣿⣿⣿⣿⣆⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠞⠉⠀⠄⠐⠢⠼⠛⢋⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣀⣀⠀⣀⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠊⡉⠈⠓⠐⢘⣰⡄⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠛⣹⣿⣿⠗⣲⣦⢀⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⣿⣿⣿⢾⣾⢻⢿⡏⣿⣶⣿⣿⡉⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣂⡤⢠⡞⠛⠿⢃⢠⣿⠛⠛⢡⣿⡿⠋⠈⠛⠋⠉⠹⢻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣧⣾⣿⣷⣿⣿⣿⠀⠈⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠁⠂⠋⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⢸⣿⢿⠿⠿⣿⣿⣿⣿⣿⣿⡗⢤⠈⠉⢈⠙⣻⣿⣤⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠟⠋⠀⢼⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣀⡀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠉⠉⠉⠙⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⡀⠀⠈⡅⠀⠀⠀⠀⣸⣿⣿⣷⣶⣶⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣶⣆⣠⣤⣴⣶⣿⡿⠿⠿⠟⠛⠛⠋⠉⠉⡙⣿⣶⣶⣶⣶⠒⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠉⠉⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣰⢻⣿⡿⡿⣿⣿⣿⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣿⣿⡇⢀⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⠄⣀⡀⠀⠀⠀⠀⠀⠀⠈⠀⡘⣼⣿⢷⣼⡇⣷⡄⢳⠀⠀⢀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠊⠀⠀⠀⠀⠄⠀⠀⣯⣿⣿⡇⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣤⣤⣤⣴⣠⣶⣶⣾⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⣿⣴⣿⢱⣵⠙⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⣦⣤⡀⠀⠀⢰⡲⡡⡄⠤⠤⠆⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠉⠉⠉⠉⠙⠋⠉⠛⠛⡿⠛⠟⠛⢛⠛⠛⠻⠿⠿⠿⠿
⠀⠖⠶⣶⠿⢶⠶⠰⠶⠶⠶⠀⣸⢸⣿⣿⣿⠂⢼⣿⡇⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⣾⡚⠉⠓⠀⠋⠁⠁⠀⠂⠆⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⢀⡀⠀⢰⠀
⠀⠀⠀⠀⠀⠀⠄⠀⣿⡇⡐⢦⣵⠾⣉⣴⣾⣿⣿⣿⣿⣿⣷⠯⢁⠀⠀⠈⠙⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠉⠁⠀⠀⣀⣀⣤⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢐⡿⠟⣛⣭⣭⣭⣭⣭⣽⣋⠁⠀⠀⠀⠀⠀⠀⠈⠘⠙⠳⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⢠⣿⣿⣿⣷⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⠀⢸⡿⣿⣿⣿⣷⡀⢸⣿⣿⣿⣿⣿⣿⣿⡟⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣷⣌⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⢻⠛⢧⢼⣿⢿⣿⠉⠉⢡⣾⣿⣿⣿⣿⣿⣿⣿⡁⢤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⡀⠀⠀⢸⣿⣼⡀⠀⣼⢿⣿⣿⣀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⣀⠈⠛⠻⢿⣿⡿⠿⠿⠟⣱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠉⠉⠻⣿⡛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⢀⣨⣽⣿⠶⣦⡀⠀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠄⠀⠀⣿⣿⠧⠤⠤⣤⣿⣿⣆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣋⣉⡙⠿⠛⠛⠪⢳⡿⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠂⠁⠘⢟⠉⠑⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣶⣷⣦⣿⣿⣿⣿⣧⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠟⠉⠉⠡⠄⢀⠶⠶⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠂⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠐⠂⠋⠹⠅⠐⢂⣄⠀⠀⠀⠀⠀⠀⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠾⢛⣯⣶⡾⢤⣄⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣿⣯⣯⣾⡯⢿⠻⢢⣼⣿⣿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⡠⠺⣿⡿⠀⣾⠛⠘⠋⣿⣿⠟⠉⠳⠚⠋⠺⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣐⣾⣯⣽⣿⣿⡏⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠒⠀⠐⠃⠁⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠈⠁⠀⠀⠀⠀⠓⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⣾⢻⡿⠿⠿⣿⣿⣿⣿⣿⠟⠎⣍⠉⠋⠙⠻⢿⣧⣀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⣶⣾⣿⣿⣿⣯⣽⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠋⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣾⣿⣷⣦⣤⣤⣤⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⣶⡦⠀⣀⣠⣤⣶⣾⣿⣿⠿⠿⠿⠛⠛⠋⠉⢻⣷⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠟⠛⠛⠛⠻⠿⠾⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡸⣻⣿⢿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⠉⠉⠉
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣿⣿⡜⣾⢰⡀⢳⡄⠀⠀⢀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣻⣿⣿⣄⣿⣿⣷⣶⣶⣶⣴⣤⣴⣦⣤⣀⣀⡄⢠⣤⣤⣤⣤⣀⣀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣿⣏⣷⡟⣜⡟⢦⠁⠀⠀⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⢐⡒⣭⠰⢤⣠⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⡟⢻⠿⠻⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠖⠲⣶⠟⢷⠶⠰⠶⠶⠶⠀⡆⣼⣿⣿⡿⠁⣿⣿⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢀⣮⠉⠐⠂⢁⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠃⠀
⠀⠀⠀⠀⠀⢀⠀⢸⣿⡀⠄⣲⣽⡞⣉⣴⣶⣾⣿⣿⣿⣿⣿⠍⠈⠀⠈⠙⠛⢶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠋⠉⠁⢀⡀⣀⣄⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠁⠀⢾⡿⠿⣛⣻⣿⣟⣿⣟⣓⠊⠀⠀⠀⠀⠀⠀⠈⠙⠹⠺⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⢸⣿⣿⣿⣆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢈⣤⣿⣿⣿⣿⣿⣿⣿⣿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⠀⣸⢿⣿⣿⣿⣇⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣵⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡿⡿⠻⣤⣿⣾⣿⡟⠉⠉⣠⣿⣿⣿⣿⣿⣿⣿⣟⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡇⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⢟⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣿⣟⠛⠛⠛⠻⣞⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣉⣿⡶⢤⣀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⢀⣿⣿⠄⡤⠄⣠⣿⣷⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠐⠛⢛⣻⠻⠾⠿⢓⡾⢷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠘⢟⠉⠁⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢸⣿⣿⣶⣧⣤⣾⣿⣿⣿⣄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡴⠞⠛⠉⣀⠠⣤⣄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠄⠐⠖⡂⠀⠠⣀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣱⣾⣿⣾⣿⣷⣾⡇⣸⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡞⠛⣠⣄⡂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣷⣷⡿⣿⣿⠻⣤⣿⣿⡟⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢁⠸⠇⠝⠁⣰⠟⠴⠛⣿⣿⠖⠹⠶⠟⠋⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢙⣇⣡⣿⣾⣽⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠄⠠⠰⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠍⠋⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⠶⡿⣟⠛⠛⠛⠛⠛⠛⠋⠀⠂⢉⠉⠛⠋⠛⠿⣿⣇⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣶⣶⣷⣿⣿⣯⣭⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⢀⣿⣷⣦⣬⣥⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡶⢦⣤⠀⢀⣀⣤⣴⣶⣿⣿⣿⠿⠿⠟⠛⠛⠛⣿⣿⣶⣤⣤⡦⢤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣉⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠿⠿⠟⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠛⠛⠿⠿⢷⡾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⣶⣦⠀⠀⡀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢣⣿⡿⣆⣿⠸⣭⠉⡄⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⢰⣿⣿⣶⣦⣤⣤⣤⣤⣠⣄⣀⣀⣀⠀⣠⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸⣿⣧⣽⡏⣄⡈⠂⠁⠀⢀⣿⣿⣿⣿⣷⣶⣤⣀⡀⠀⠀⠀⡄⠀⣀⢀⣀⣩⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⣿⣿⠘⠛⠛⠛⠛⠛⠛⠛⠛⢻⡿⢿⠿⠿⠿⠿⠿⠿⠿⢿⠿⠿
⠀⠖⠲⣶⠏⢷⠖⠰⠶⠶⠦⠀⡆⣿⣿⣿⡏⢐⣿⣷⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⢠⡯⠈⠒⠄⠀⠐⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠁⠀⠀⠀⠀⠀⡆⠀
⠀⠀⠀⠀⠀⢀⠀⢸⣿⠀⢤⣰⣬⠞⢋⣥⣶⣶⣿⣿⣿⣿⡏⠉⠁⠀⠈⠙⠻⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠋⠉⠁⢀⠀⣠⣄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡎⠀⠉⠀⢾⡿⠿⣛⣻⣛⣛⣛⣛⡒⠈⠀⠀⠀⠀⠀⠀⠈⠙⠝⠲⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⢸⣿⣿⣿⣆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢫⣴⣿⣿⣿⣿⣿⣿⣿⡿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄⠀⣼⢿⣿⣿⣿⣆⠀⣼⣿⣿⣿⣿⣿⣿⣿⠇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣯⣖⣰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡿⡿⠿⣤⣿⣾⣿⡟⠙⠉⣠⣿⣿⣿⣿⣿⣿⣿⣏⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡇⠀⠀⡿⣿⣿⡇⠀⠀⣿⣿⣿⣟⣿⣿⣿⣿⣿⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠙⠻⠿⣾⣿⣿⣿⣿⡿⢯⣝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣿⣟⠛⠛⠛⠻⣞⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⡙⣷⣦⣄⡀⠈⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣿⣿⠄⡤⠄⣰⣿⣷⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠐⠛⠛⢟⠶⠾⠽⢲⠼⣆⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⢿⠛⠃⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢸⣿⣿⣴⣧⣤⣾⣿⣿⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⠖⠛⠃⡀⠀⣤⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠈⠀⠀⠈⠀⠀⠲⠒⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣾⣿⣾⣶⣷⣶⡇⢸⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠾⠁⡄⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⢇⣧⡿⢿⣿⠓⣤⣿⣿⡟⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣄⠸⠋⠰⠏⢠⠟⣿⣿⠋⠈⠻⠋⠀⠈⠉⣿⣿⣿⣿⣿⠿⠿⠿⠿⢠⡀⠙⣇⣡⣿⣾⣽⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠤⠠⠰⠍⠂⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠁⠐⠂⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠛⢙⡻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⡉⠛⠋⠛⠿⣿⣇⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠀⠀⠀⠀⠀⠀⠀⡀⠍⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣶⣶⣷⣿⣿⣯⣭⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣶⣮⣉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠋⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⣾⣶⣦⣌⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⡶⢤⣤⠀⢀⣀⣤⣴⣶⣾⣿⣿⠿⠿⠿⠛⠛⠛⢿⣿⣧⣤⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣈⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠿⠿⠟⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡰⠞⠛⠛⠿⠿⠷⡾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣿⣿⡄⣿⢩⡉⠹⡁⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⣿⣿⠀⣿⣷⣶⣦⣤⣤⣤⣤⣠⣆⣀⣀⣀⠀⢀⣠⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣿⡏⣷⡟⡘⠿⢄⠁⠀⠀⣾⣿⣿⣿⣷⣶⣤⣀⡀⠀⠀⠀⡠⠀⣐⢀⣀⣉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⣿⣿⠘⠛⠛⠛⠛⠛⠛⠛⠛⢻⡿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠆⠲⣶⠏⢷⠖⠰⠶⠶⠖⠀⡄⣼⣿⣿⡯⠂⣿⣷⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⢠⡯⠝⠲⠄⠀⠐⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠆⠀
⠀⠀⠀⠀⠀⢀⠀⢸⣿⠀⢤⣰⣬⡞⣋⣥⣶⣶⣿⣿⣿⣿⡏⠉⠁⠀⠈⠙⠻⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠋⠉⠁⢀⠀⣠⣄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡎⠀⠉⠀⢾⡿⠿⣛⣻⣛⣛⣛⣛⡒⠈⠀⠀⠀⠀⠀⠀⠈⠙⠝⠲⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⢸⣿⣿⣿⣆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢫⣴⣿⣿⣿⣿⣿⣿⣿⡿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄⠀⣼⢿⣿⣿⣿⣆⠀⣼⣿⣿⣿⣿⣿⣿⣿⠇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣯⣖⣰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡿⡿⠿⣤⣿⣾⣿⡟⠙⠉⣠⣿⣿⣿⣿⣿⣿⣿⣏⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡇⠀⠀⡿⣿⣿⡇⠀⠀⣿⣿⣿⣟⣿⣿⣿⣿⣿⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠉⠻⠿⣾⣿⣿⣿⣿⡿⢯⣝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣿⣟⠛⠛⠛⠻⣞⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣙⣷⣦⣄⡀⠈⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣿⣿⠄⡤⠄⣰⣿⣷⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠐⠛⠛⢟⠶⠾⠝⢲⠼⣆⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⢿⠛⠁⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢸⣿⣿⣴⣧⣤⣾⣿⣿⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡴⠖⠛⠃⠀⠀⣤⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠈⠀⠀⠈⠀⠀⠲⠒⢂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣾⣿⣾⣷⣷⣶⡇⢸⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠾⠃⡀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⢇⣿⡿⢿⣿⠓⣤⣿⣿⡟⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠉⠁⡄⠘⠋⠰⠯⢠⠟⣿⣿⠏⠀⠛⠋⠁⠈⢉⣿⣿⣿⣿⣿⠿⠿⠿⠿⢠⡀⠹⣇⣡⣿⣾⣽⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠤⠠⠰⠍⠂⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠨⠁⠐⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠛⢙⡻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⡉⠛⠋⠛⠿⣿⣇⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠀⠀⠀⠀⠀⠀⠀⡀⠍⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣶⣶⣷⣿⣿⣯⣭⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣶⣮⣉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠋⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⢀⣾⣶⣦⣌⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⡶⢤⣤⠀⢀⣀⣤⣴⣶⣾⣿⣿⠿⠿⠿⠛⠛⠛⢿⣿⣧⣤⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣈⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠿⠿⠟⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡰⠞⠛⠛⠿⠿⠷⡾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⠀⠀⡀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣿⣿⡄⣿⢩⡉⢹⡁⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⣿⣿⠀⣿⣿⣶⣦⣤⣤⣤⣤⣠⣆⣀⣀⣀⡀⢀⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣿⡏⣷⡟⡘⠿⢄⠁⠀⠀⣾⣿⣿⣿⣷⣶⣤⣀⡀⠀⠀⠀⡠⠀⣐⢀⣀⣉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿⣿⠘⠛⠛⠛⠛⠛⠛⠛⠛⢻⡿⢿⠿⠿⠿⠿⠿⠿⢿⠿⠿⠿
⠀⠖⠲⣶⠏⢷⠖⠰⠶⠶⠖⠀⡄⣼⣿⣿⡯⠂⣿⣷⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⢠⡯⠕⠲⠀⠀⠐⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠆⠀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠠⣐⣮⡗⢋⣥⣶⣶⣿⣿⣿⣿⡭⠁⠁⠀⠈⠙⠻⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡚⠛⠉⠁⠀⡀⡀⣄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠉⠁⢰⡿⠿⣛⣛⣛⣛⣛⣓⠒⠁⠀⠀⠀⠀⠀⠀⠈⠙⠹⠲⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⢸⣿⣿⣿⣦⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⠟⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⠀⢸⠿⢿⣿⣿⣧⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣧⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣽⣃⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠿⣤⣼⣿⢿⣿⠉⠉⢠⣾⣿⣿⣿⣿⣿⣿⣿⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣾⣿⣧⠀⠀⣿⢿⣾⡟⠀⠀⣾⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣭⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠛⠛⠛⠻⣿⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣻⣷⣦⣄⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⠤⡤⠤⣀⣿⣷⣄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⡰⠻⠿⠿⠶⠿⠝⠲⠼⣄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⡟⠻⠋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣿⣿⣦⣧⣤⣾⣿⣿⣿⣆⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠖⠚⠁⠀⣀⣤⠤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠁⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠈⠡⠰⠶⠒⣀⣀⠄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣾⡞⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣤⡶⠟⠁⠀⠀⢂⠀⠀⡀⠀⢀⣀⣠⠤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢝⣛⣾⣾⣿⣷⢧⣼⣿⡟⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⢑⠀⠛⢀⠦⢠⠚⣿⣿⠏⠀⠛⠉⠀⠀⣤⣿⡿⠛⠛⠉⠉⠛⠛⠁⠀⢀⡀⢉⣶⣡⣾⣞⣽⣿⣿⡏⠐⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠤⠀⠀⠎⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡔⠠⠀⠀⠆⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⡰⢄⠂⠀⠀⠀⠀⠁⠀⢀⠂⠭⡉⠛⠋⠙⠿⢿⣇⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⠀⠀⠀⠀⠀⣀⠀⠠⡑⠌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⣶⣷⣾⣿⣿⣭⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣩⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠋⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣶⣶⣤⣀⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⡶⢦⣤⡀⠀⣀⣠⣤⣶⣾⣿⣿⡿⠿⠿⠛⠛⠛⢿⣿⣥⣤⣤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣈⣉⣉
⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⠿⠿⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⠞⠛⠛⠿⠿⠷⡾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇⣿⣿⠺⣼⠋⠹⣍⠙⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣶⣶⣦⣤⣤⣤⣄⣦⣀⣀⣀⡀⢀⣤⣤⢄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⣿⣿⢦⣿⢸⣄⠘⠀⠀⠀⣼⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⢠⠀⢐⣠⣀⣈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠗⠛⠛⠛⠛⠛⠛⠛⠛⠻⡿⢿⠿⠿⠿⠿⠿⠿⢿⠿⡿⠿
⠀⠖⠲⣶⠯⢷⠶⠰⠶⠶⠖⠀⢸⢸⣿⣷⣿⠓⣺⣷⡄⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⢀⣾⠁⠲⠂⠀⠒⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠁⠀⠀⠀⠀⠀⢡⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠠⣄⣦⣵⢛⣭⣴⣾⣿⣿⣿⣿⡍⠈⠀⠀⠈⠙⠛⢶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⠉⠉⠀⡀⠀⣤⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠁⠉⠁⠠⠿⢟⣛⣛⣛⣛⣛⠓⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠹⠲⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⢀⣿⣿⣿⣷⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠳⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⠀⢸⠿⣿⣿⣿⣷⡀⢰⣿⣿⣿⣿⣿⣿⣿⡿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣟⣴⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⠿⣧⣸⣷⣿⣿⠛⠋⢁⣼⣿⣿⣿⣿⣿⣿⣿⡁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢸⣿⣸⠀⠀⢸⢿⣿⣿⠀⠀⢸⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⣿⣯⣽⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⠛⠛⠛⠛⢿⡻⠻⠿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢐⣻⣷⣤⣀⠀⠀⠈⠈⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⣿⡧⣠⠄⣀⣿⣿⣆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠴⠿⠿⠯⠶⠿⠻⠦⢿⣄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣶⣤⣾⣿⣿⣿⣧⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠶⠚⠁⢀⣠⡤⠬⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠁⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠈⠀⠈⢉⢤⣶⣶⢂⡀⠠⠄⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣶⣿⢰⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⢉⣡⣾⡟⠉⠀⡀⠀⢂⣀⣀⣀⢤⣤⡶⠚⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢽⣹⣿⢿⣿⣿⢧⣼⣿⣿⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠙⠛⢫⠀⠛⢀⠉⡰⢻⣿⣿⠁⠘⠉⠀⠀⠀⠿⠿⠋⠀⠙⣿⣿⣿⣿⠟⠉⠀⢀⣀⠈⣿⠩⣾⣗⣮⣿⣿⣿⠐⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠤⠄⠀⠮⠓⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⣕⠂⠀⠀⠄⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢌⡙⠁⠀⠀⠀⠀⠀⢀⠐⠩⡉⠛⠛⠙⠿⢿⣯⣀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢻⠀⠀⠀⠀⠀⠀⢀⠀⠰⣕⠮⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣶⣶⣶⣶⣿⣿⣯⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣩⡁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⠀⠀⠸⠀⠀⠀⠀⠀⣶⣶⣤⣀⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡶⢦⣤⡄⠀⣀⣠⣤⣶⣶⣿⣿⡿⠿⠿⠟⠛⠛⢻⣭⣭⣤⣤⣤⠤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠿⠿⠟⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠶⠛⠿⠿⠿⢿⣶⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⣶⣸⠏⢹⡍⠁⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⣟⣿⣿⡇⣾⣿⣶⣶⣤⣤⣤⣤⣄⣤⣀⣀⣀⡀⢀⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⣿⣿⢌⣿⢠⡀⠑⠈⠀⠀⣰⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠠⠠⢀⣂⣀⣉⠕⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⡇⠛⠛⠟⠛⠿⠟⠛⠿⠿⣿⠿⡿⠿⠿⠿⠿⠿⣿⡿⣿⣿
⠀⠖⠲⣶⠯⢷⠶⠰⠶⠶⠖⠀⢰⢸⣿⣿⣿⠛⢸⣿⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣾⠁⠓⠦⠀⠅⠂⠂⠀⠀⡄⠀⠀⠀⠀⠀⠀⢈⣿⣿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠂⠈⠈⠀⠀⠀⢀⠀⢰⠀
⠀⠀⠀⠀⠀⠀⡀⠀⢿⣿⠠⢔⣦⣵⣛⣭⣾⣿⣿⣿⣍⡉⠁⠀⠀⠀⠀⠉⠛⠷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠛⠛⠋⠉⢀⢀⣰⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠓⠉⠉⠀⢾⣛⣛⣛⣿⡷⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠫⢗⠷⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⣿⣿⣿⣿⣆⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢷⣶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⡿⣿⣿⣿⣿⣆⠀⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⣿⣿⣿⣿⣿⣛⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⢬⣿⣿⡿⢯⣤⣿⡿⣿⡟⠛⠉⣴⣿⣿⣿⣿⣿⣿⣿⡏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣟⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⣿⣿⡇⠀⢸⡿⣿⣿⠇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢰⡿⣿⣿⣿⣿⣿⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡛⠛⠛⠛⢿⣿⠿⠿⠿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣼⣿⣯⣆⠀⠉⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⢤⠤⢀⣸⣿⣷⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠿⣿⠿⠷⠿⢿⣶⢶⣤⣀⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣶⣾⣿⣿⣿⣟⣁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣼⣿⣧⣼⣤⣼⣿⣿⣿⣿⣆⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠛⢀⣀⣴⣶⠬⠙⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⢹⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠉⠉⢉⣭⣿⣷⣶⣯⣀⣐⣦⡆⠀⠀⠀⠀⠀⠀⠀⣸⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡆⣾⣿⣷⣿⣿⣶⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠛⣫⣯⣿⡿⠋⢁⣭⣶⣦⣤⡶⣶⣾⣿⡿⢿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠣⢹⣻⣿⣿⣿⣿⣿⣤⣿⣿⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠠⢤⣙⠋⠛⣿⠁⢉⣹⣶⣿⣿⣿⣏⠹⠋⠀⠰⠾⠿⠛⠁⠀⠈⢿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠈⠁⣄⠨⣿⣏⣾⣷⣫⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠄⠂⠤⠾⠛⠀⠀⠀⡀⠀⠀⠀⠀⠑⠿⣟⠾⠻⣶⠄⡀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢝⠻⡙⠅⠀⠀⠀⠉⠀⠀⡐⢨⢍⠙⠛⠉⠻⠿⣿⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠐⢌⠢⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⣾⣶⣿⣿⣯⣭⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣍⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠐⡀⠀⠈⠀⠀⠀⠀⢠⣶⣦⣀⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⣶⣦⡄⠀⣀⣠⣤⣶⣶⣿⣿⡿⠿⠿⠟⠛⠛⠻⣿⣿⣦⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣈⣉⣉⣉⣉
⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⠿⠿⠿⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⡷⠛⠻⠿⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣆⢀⣀⡀⣀⣀⣀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⢀⡀⠀⠀⠀⠀⠈⠀⠰⠾⠿⠃⡟⣿⣿⣾⣽⠋⠹⣏⠰⠀⠀⢀⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠴⠀⠀⠀⠀⡀⠀⠀⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣶⣶⣴⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣦
⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇⣿⣿⢮⣿⢸⣀⠹⡈⠀⠀⣸⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⢠⣆⢃⡮⣶⣌⡭⡃⠀⢂⠇⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠛⠛⠿⠛⠿⠿⠛⠻⠿⣿⠿⡿⡿⢿⠿⠿⢿⣿⣿⣿⣿
⠀⠶⠶⣶⠿⢷⠶⠴⠶⠶⠖⠀⢰⢸⣿⣿⣿⢟⢸⣿⣧⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⡽⠻⠷⠈⡴⠂⠒⠘⣄⡄⠀⠀⠀⠀⠀⠠⠂⣿⣿⣇⠀⢀⠀⡄⠀⠀⠀⠀⠀⠛⠂⠈⠠⠉⠀⠀⠀⢀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⡀⠀⢿⣿⠠⢔⣦⣵⣛⣭⣾⣿⣿⣿⣍⡉⠁⠀⠀⠀⠀⠉⠛⠷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠛⠛⠋⠉⢀⢀⣰⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠓⠉⠉⠀⢾⣛⣛⣛⣿⡷⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠫⢗⡷⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⣿⣿⣿⣿⣆⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢷⣶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⡿⣿⣿⣿⣿⣆⠀⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⣿⣿⣿⣿⣿⣛⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⢬⣿⣿⡿⢯⣤⣿⡿⣿⡟⠛⠉⣴⣿⣿⣿⣿⣿⣿⣿⡏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣟⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⣿⣿⡇⠀⢸⡿⣿⣿⠇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠐⡿⣿⣿⣿⣿⣿⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡛⠛⠛⠛⢿⣿⠿⠿⠿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣼⣿⣯⣆⠀⠉⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⢤⠤⢀⣸⣿⣷⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠿⣿⠿⠷⠿⢻⣶⢶⣤⣀⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣶⣾⣿⣿⣿⣟⣁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣼⣿⣧⣼⣤⣼⣿⣿⣿⣿⣆⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠛⢀⣀⣴⣶⠬⠙⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⢹⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠉⠉⢉⣭⣿⣷⣶⣯⣀⣐⣦⡆⠀⠀⠀⠀⠀⠀⢀⣸⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡆⣾⣿⣷⣿⣿⣶⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠬⠛⣫⣽⣿⡿⠋⢁⡭⣾⣦⣤⡶⣶⣾⣿⠿⢿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠣⢹⣻⣿⣿⣿⣿⣿⣤⣿⣿⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠢⢤⣏⠉⠛⣿⠁⢉⣹⣶⣿⣿⣿⣏⠙⠋⠀⠰⠾⠿⠛⠁⠀⠈⢿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠉⠁⣄⠨⣿⣏⣾⣷⣫⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠄⠂⠤⠾⠛⠀⠀⠀⡀⠀⠀⠀⠀⠑⠿⣟⠶⠻⣶⠆⡀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢝⠻⡉⠇⠀⠀⠀⠉⠀⠀⡐⢨⢍⠙⠛⠉⠻⠿⣿⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠐⢌⠢⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⣾⣶⣿⣿⣯⣭⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣍⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠐⡀⠀⠈⠀⠀⠀⠀⢠⣶⣦⣀⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⣶⣦⡄⠀⣀⣠⣤⣶⣶⣿⣿⡿⠿⠿⠟⠛⠛⠻⣿⣿⣦⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣈⣉⣉⣉⣉
⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⠿⠿⠿⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⠷⠛⠻⠿⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣆⠀⣀⡀⣀⡀⣀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⢀⡀⠀⠀⠀⠀⠈⠀⠐⠺⠿⠋⡟⣿⣿⢾⣽⠋⠹⣇⠰⠀⠀⢀⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⡀⠀⠀⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣾⣵⣶⣶⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣦
⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇⣿⣿⢮⣿⢸⣀⠹⠈⠀⠀⣼⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⢠⣦⢃⡪⣶⣌⡭⡃⢀⢂⠅⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⠛⠛⠿⠛⠿⠿⠛⠻⠿⣿⠿⡿⡿⢿⠿⠿⢿⣿⣿⣿⣿
⠀⠶⠶⣶⠿⢷⠶⠴⠶⠶⠖⠁⢰⢸⣿⣿⣿⢟⢸⣿⣧⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⡽⠻⠷⠈⡴⠂⠒⠘⣄⡄⠀⠀⠀⠀⠀⠠⠊⣿⣿⣇⠀⢀⠀⡀⠀⠀⠀⠀⠀⠛⠂⠈⠠⠈⠀⠀⠀⢀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠠⠀⢸⣿⣄⢲⣼⣶⣿⣿⣿⣯⣁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⠛⠛⠋⢡⣀⣦⣦⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠉⠁⣘⣛⣿⣿⠟⠛⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢟⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠀⢸⣿⣿⣿⣷⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⣿⡿⣷⡶⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⡀⣾⢿⣿⣿⣿⣷⣄⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣧⣭⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⠻⣤⣿⣿⣿⡟⠉⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⢿⣿⣿⣿⣿⣟⣛⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⡇⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⣀⣿⢿⣿⣿⣿⣿⣯⣭⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⠛⠛⠛⠻⣿⣟⠻⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⢷⣄⠈⠙⠛⠛⠛⠉⠚⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⢸⣿⣿⠤⡤⠄⣼⣿⣿⣦⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠛⢛⣛⢿⣶⠶⠄⠀⠄⠈⠐⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣶⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⣿⣾⣷⣴⣿⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣚⣂⣤⣴⣶⣢⡙⠻⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣾⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⢀⡠⢀⢀⠀⢐⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣉⣁⣼⣿⣿⣿⣯⣌⣈⠀⣻⢶⣂⣀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⡁⣀⣀⣠⣶⠇⠀⣾⣿⡧⡷⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠿⣿⣿⣿⣿⣋⣠⣤⣶⣾⣿⣿⣿⡿⠿⣿⣿⣿⣿⡟⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⡟⠀⡀⠉⣹⣇⡏⠬⣿⣿⣿⣿⣿⠿⣤⣿⣿⡟⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⠀⠀⠈⠉⠲⣿⣿⢻⣿⣬⣽⣿⣿⣿⣿⣿⡽⠋⠰⢿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠟⠋⡀⡄⢉⠀⢀⡀⠉⣠⠨⢹⣏⣻⣿⣯⣽⣿⣿⡇⠃⠀⠤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠒⠂⠔⠚⠋⠃⠀⠀⠀⠀⠻⣿⠿⢿⣿⣧⡈⠁⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡻⢔⡀⠈⡁⠀⠉⠐⠀⠄⠒⢩⡉⠉⠉⠛⠿⣿⣇⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠙⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠢⡑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⣶⣿⣿⣿⣿⣭⣽⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠒⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢻⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⡀⠀⢸⢂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠋⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⠀⢧⠀⠀⠀⠀⠀⠀⢰⣦⣄⣀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣶⡶⢀⣀⣤⣴⣶⣿⣿⡿⠿⠿⠟⠛⠛⠋⠙⣿⣷⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣀⣀⣈⣉
⠀⠀⠀⠀⢠⢄⠘⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⠿⠿⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣺⠿⠿⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢹⢅⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⢀⣠⣀⣀⣀⣄⣀⣀⣀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉
⠀⢀⣤⣀⡂⠀⠐⠀⠀⠀⠀⠀⢰⢷⣿⣿⡏⣿⢹⣏⠐⡆⠀⠀⣰⣶⣶⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠠⢄⠀⣀⢀⣉⠁⠂⠚⠥⠀⠀⠀⢤⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣴⣶⣶⣶⣶⣶⣿⣿⣷
⠁⠀⠀⠀⠂⠂⡀⠀⠀⠀⠀⠀⢸⢸⣿⡏⢷⡿⡐⣿⣄⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⢀⣖⣫⡭⢧⣽⣿⣿⠀⠬⠍⠀⠀⠀⠀⠀⠐⢽⣿⣿⣿⠛⠛⠻⠿⠻⠟⠛⠛⠛⣿⡿⣿⡿⢿⠿⠿⠿⢿⡿⢿⠿
⠀⠖⠶⣶⠿⢷⠶⠶⠶⠶⠦⠀⢸⢸⣿⣿⣿⠓⢺⣿⡇⠄⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢼⣿⢙⠛⠚⠘⠋⢑⡸⢯⡤⠀⠀⠀⠀⠀⠉⠂⣿⣿⣿⠀⠀⡄⠤⠀⠀⠀⠀⠀⡙⠓⠘⠀⢋⠁⣁⡈⡰⡀⠈⡆
⠀⠀⠀⠀⠀⠀⠠⠀⢈⣿⣇⣒⣽⣾⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠛⠛⠛⠉⣄⣤⣷⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠤⣿⡏⠉⢁⣛⣿⣿⠟⠃⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⡻⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⢸⣿⣿⣿⣿⣆⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠸⣿⣿⣿⣿⣷⡿⠓⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿⣆⢸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢰⣿⣿⣿⣿⣿⣷⣽⣦⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠠⣽⣿⣿⠛⣧⣾⣿⣿⣿⠉⠉⢩⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⣿⣿⣿⣿⣟⣛⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠈⠀⠀⣿⣿⣿⡀⠀⣿⣿⣿⣿⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣰⠿⣿⣿⣿⣿⣯⣭⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⣿⣿⡋⠉⠛⠻⣿⣿⣛⠟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣦⡀⠉⠛⠟⠛⠋⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⣠⠄⠀⢀⣿⣿⠧⡤⠤⣤⣿⣿⣷⣀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠛⠛⣋⣻⣷⡶⠤⠀⠠⠀⠈⠑⠂⠀⠀⠀⠀⠀⢀⣀⣠⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⢰⣿⣿⣶⣷⣦⣿⣿⣿⣿⣿⡆⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣈⣡⣤⣾⣷⣆⣙⠛⡷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢀⢀⣀⢄⡀⡀⢐⣂⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣬⣭⣿⣿⣿⣿⣥⣄⣉⠈⠿⣷⣿⣄⣀⣨⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣟⣯⣥⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠋⠉⠉⢿⣿⣧⣀⣀⣀⣀⣤⣴⠀⠀⡀⠈⣹⣿⡟⠾⢯⣿⣿⣿⣿⠿⣷⣿⣿⡿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⡀⠈⠛⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠯⣴⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠇⡀⡄⢈⠀⢀⡄⠈⡠⠄⣹⣟⣹⣿⣿⣿⣿⣿⣯⠀⠂⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠒⠉⠐⠺⠋⠁⠀⠀⠹⠿⠿⢿⣿⣯⡈⠀⣿⣿⣿⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢀⡠⣝⢧⡓⠈⠁⠀⠉⠀⠄⠀⠃⠏⣍⠉⠉⡙⠻⣿⣿⣀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣀⣀⣀⣤⠀⠑⢍⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣯⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⠀⠀⢠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠠⠹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⡄⡀⠘⣷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠋⠉⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠢⠀⢻⡀⠀⠀⠀⠀⠀⣤⣄⣀⡀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣶⣶⣀⣀⣤⣴⣶⣿⣿⡿⠿⠿⠟⠛⠛⠋⠉⢿⣷⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣄⣀⣈⣉
⠀⠀⠀⠀⢰⣦⡈⠅⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠿⠿⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠿⠿⠿⠿⠿⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠙⢷⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣄⣠⣄⣀⣤⣤⣀⣀⣀⣀⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉
⠀⢠⣴⣀⣤⠀⠘⠁⠀⠀⠀⠀⢸⢸⣿⡏⢿⡟⡍⢹⣇⠂⠀⠀⣼⣿⣶⣶⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢴⡐⣀⢀⣐⡁⠂⠒⢃⠀⠀⠀⠤⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣴⣶⣶⣶⣶⣾⣿⣿⣿
⠉⠀⠀⢀⡀⠃⠀⠀⠀⠀⠀⠀⠘⢸⣿⡇⣸⡇⡟⡈⢯⠁⡀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⣼⣯⣭⢷⣿⣿⢿⢀⡜⣽⠀⠀⠀⠀⠀⠂⠸⣿⣿⣿⡟⠻⠻⠿⠻⣿⠛⠛⠛⢻⡿⣿⡿⢿⡿⠿⠿⣿⣿⠿⡿
⠀⠖⠶⣶⡿⢷⡶⡶⠶⠶⠦⠀⡀⢸⣿⣿⣟⠃⣿⣿⡶⠀⡅⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢰⣿⣋⢻⢚⠻⠛⢘⡲⢿⣿⠀⠀⠀⠀⠀⢍⣒⣿⣿⣿⠃⠐⡄⠤⡤⠆⠀⠀⠀⠘⠛⢸⢅⡍⡅⢉⡈⣸⡀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣒⣼⣾⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠋⣤⣸⣾⣄⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⠉⣙⣻⣿⠟⠃⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣻⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⠀⣿⣿⣿⣿⣦⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢿⣿⣿⡿⣷⡿⠓⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣆⢰⡿⣿⣿⣿⣿⣷⣀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣾⣿⣿⣿⣿⣷⣾⣧⣾⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡟⢷⢼⣿⣿⣿⠉⠉⢩⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢸⣿⣿⣿⣟⣛⣿⣿⣯⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⢸⣿⣿⡇⠀⣸⣿⣿⣿⣁⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣼⢿⣿⣿⣿⣿⣭⡿⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠀⠀⢅⠑⠀⠠⠀⠀⠀⠀⣾⣿⡟⠛⠛⠛⢿⣿⡻⡿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢷⣄⠈⠛⠛⠛⠋⠁⠈⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⢀⡠⡄⠀⢀⣿⣿⡧⣤⠤⣤⣿⣿⣿⣆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠛⢛⣛⣹⣿⡶⠤⠄⠀⠄⠀⠉⠑⠀⠀⠀⠀⢀⢀⠀⠀⣰⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣿⣿⣷⣿⣦⣾⣿⣿⣿⣿⣧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣈⣡⣤⣿⣷⣢⣝⣻⠿⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢀⠂⣀⣀⢄⡀⡀⠀⣂⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣬⣴⣿⣿⣿⣿⣥⣉⡉⠈⢿⣷⣾⣧⣤⣤⣽⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠰⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣯⣥⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣨⣿⣹⠱⢽⣿⣿⣿⣿⣿⢧⣾⣿⣿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠈⠛⢦⣄⣀⣀⣀⡀⠀⠀⡀⢰⠀⠁⢀⣀⠉⢠⠤⣭⣿⣙⣿⣿⣽⣿⣿⣿⠘⠀⠠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠂⠉⠕⠚⠋⠙⠈⠿⢿⣿⣿⣿⠏⣸⣿⣿⣧⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⡧⢜⠧⡇⠀⢉⠀⠉⠁⠆⠠⠊⠎⣍⠉⠋⡙⠻⣿⣿⣀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠹⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⣀⣀⣤⡀⠙⢌⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⣿⣿⣯⣽⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⠣⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠠⠼⣿⣿⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⣀⣀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢆⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠋⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠂⠀⠀⠀⢲⠄⢸⡅⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣶⣶⣏⣀⣤⣴⣶⣾⣿⡿⠿⠿⠿⠛⠛⡋⠉⢿⣿⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣄⣀⣀⣉⣉⣉
⠀⠀⠀⠀⢠⢐⡄⠃⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢑⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⣠⣄⣀⣠⣄⣀⣀⣀⣀⡈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠂⢰⣴⣀⣬⡀⠀⠁⠀⠀⠀⠀⠀⢸⣿⠹⣽⡏⢩⣿⢱⠀⠀⢠⣾⣿⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⢴⡠⣀⢀⣈⡁⠓⠐⢋⠄⠀⠀⠨⠀⠂⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣶⣶⣶⣶⣾⣿⣿⣿
⠈⠀⠀⠀⡀⠑⠀⠀⠀⠀⠀⠀⠀⢸⣿⠠⡿⢁⠊⢻⡋⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⢰⣛⣭⠷⣼⣷⢿⢅⡼⣭⠀⠀⠀⠀⠀⠒⠪⣿⣿⣿⡟⠛⠻⠿⡛⣿⠛⠛⠛⢻⡿⣿⣿⢿⡿⠿⣿⣿⣿⣿⣿
⠀⠖⠶⣶⡿⢷⡶⠶⠶⠶⠆⠀⠀⢰⣿⣶⡦⠼⣶⣶⢥⢤⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢀⣿⣋⢻⠲⢻⠛⢘⡢⢿⣿⠀⠀⠀⢀⠀⠍⢒⣽⣿⣿⡇⢀⣄⢰⡦⡂⠀⠀⠀⢘⠟⢰⢇⠝⡉⡉⡈⠋⡆⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠄⢀⣸⣿⣐⣸⣿⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡛⠛⠛⠋⢡⣠⣽⣦⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⢉⣻⣿⡏⠛⠀⢐⡠⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢛⠷⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⠀⠀⣿⣿⣿⣿⣷⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡜⣿⣿⡿⣿⣿⠟⠲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⣿⣿⣿⣿⣿⣷⡀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡗⣿⣯⣿⣿⣿⣶⣷⣶⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⢬⣿⣿⡟⢻⣴⣿⡿⣿⡏⠙⠉⣽⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣛⣻⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⡀⠀⠀⢸⣿⣿⡇⠀⢸⡿⣿⣿⠇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⡿⣿⣿⣿⣿⣭⣽⠛⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⢀⠰⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠛⠛⢿⣿⡿⠿⠿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⣧⣀⠙⠛⠛⠛⠉⠀⢨⣉⠍⡉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠀⡤⠀⠀⣸⣿⡧⢤⠤⢤⣼⣿⣿⣦⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠚⠛⢛⣛⣉⣿⣗⠶⠤⠀⠤⠀⠀⠉⠙⠢⠄⠈⠐⣄⡀⡀⠀⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⣿⣿⣷⣾⣤⣼⣿⣿⣿⣿⣧⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣡⣤⣶⣶⣦⣝⠿⠿⣧⡀⠀⠀⠀⠀⢀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⡀⣀⠤⣀⡀⠀⡂⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣭⣿⣿⣿⣯⣍⡉⠉⠀⢻⣿⣶⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢿⡇⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣛⣭⣭⣭⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢉⣿⣸⠳⢿⣿⣿⣿⣿⡿⢧⣼⣿⣿⠟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣋⣽⣿⣿⣿⠿⠿⠋⠀⠀⠀⠀⠀⠙⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⡁⢀⣠⠀⢁⡄⢍⣿⣉⣿⣿⣭⣿⣿⣿⠁⠐⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠉⠑⠛⠋⢻⣿⣿⣿⡏⢁⣿⣿⣿⡈⠙⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣤⣀⠀⡀⢜⡻⡜⠂⠉⠀⠈⠁⠠⠠⠐⠸⢍⠉⠙⠉⠻⢿⣿⣄⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢡⠀⠀⠀⠀⠈⠙⠛⠋⠙⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢈⣙⣛⣫⡁⠙⢮⡓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣶⣿⣿⣿⣯⣭⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠜⣿⣿⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠎⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠂⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠌⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣶⣎⣀⣤⣤⣶⣾⣿⣿⠿⠿⠿⠛⠛⠛⠉⢻⣿⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⣀⣀⣉⣉⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⣶⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠹⠿⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⣀⣄⣀⣀⣀⣀⣀⣈⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙
⢀⠀⠴⢦⣬⡁⠀⠀⠀⠀⠀⠀⠈⢹⡟⢻⡟⡛⢫⣍⠀⠀⠀⣼⣿⣿⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢰⡀⢄⣀⣈⡁⠓⡐⠋⡄⠀⠀⠠⠄⠂⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣴⣶⣶⣶⣾⣷⣶⣶
⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠘⡇⠸⠃⠑⠸⣟⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⢀⣞⣭⠵⣬⣷⢯⡇⡴⢭⠁⠀⠀⠀⠀⠐⠢⣿⣿⣿⡟⠛⠻⠿⡛⣿⡟⠛⠛⠿⣿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠖⠶⣶⠿⢿⢶⠶⠶⠶⠆⠀⠀⢀⣄⡶⠄⢷⣶⣾⣶⡇⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣼⣋⢛⡓⡻⠛⠈⣃⠿⣵⠀⠀⠀⠀⠀⠉⣗⢸⣿⣿⡇⠀⣄⢠⠠⠂⠀⠀⠀⢸⠻⢊⡏⡜⡉⡉⠈⠂⣆⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⡤⠀⠀⠀⠈⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣦⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠛⠛⠛⠋⣄⣀⣶⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣟⠉⠀⠰⠖⠤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠽⡻⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠀⢸⣿⣿⣿⣷⣄⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⡟⠛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀⢸⢿⣿⣿⣿⣿⣆⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣽⣿⣿⣷⣾⣷⣾⣯⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⣽⣿⣿⠛⣧⣼⣿⢿⣿⠛⠋⢩⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣛⣿⣿⣿⡏⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⣿⣿⣿⠀⠀⣿⢿⣿⡿⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⠛⠁⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⡈⠆⠀⠀⠀⠀⠀⠀⣿⣿⠛⠛⠛⠻⣿⣿⣿⠿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣷⣄⠙⠛⠛⠉⠉⠀⢀⣯⡁⠀⢄⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠄⠀⢠⣿⣿⠤⡤⠤⣠⣿⣿⣧⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡐⠒⢛⣛⣛⣙⠯⠙⠲⠤⠀⠤⠀⠀⠈⠉⠙⠓⠦⠀⠁⠢⣤⣀⣀⣴⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⣿⣿⣶⣧⣤⣿⣿⣿⣿⣿⡄⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣘⣢⣶⣶⣬⡙⠿⢷⣿⣆⠀⠀⠀⠀⠀⢤⣀⢀⢀⣀⣀⣠⣤⣿⣭⣿⣿⣿⣿⣷⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡐⢀⣀⡠⢀⢀⠀⢐⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣨⣿⣿⣿⣛⡉⠉⠙⠀⠙⡿⠿⢶⣿⣦⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠰⢛⠿⠟⣛⣛⣛⣓⢠⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⣽⣏⡇⠾⣿⣿⣿⣿⣿⠿⣤⣿⣿⡟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢰⣾⠟⣿⣿⣿⣿⡿⣷⣿⡿⢿⣿⣿⣿⣿⣿⣿⠛⠉⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢈⠀⣀⡀⠁⣠⠨⣹⣏⣻⣿⣯⣽⣿⣿⡇⠂⠀⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⢹⣿⣾⡿⢹⣿⣿⡟⠿⢷⢾⣿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢀⠤⡻⢼⠀⠈⡁⠀⠉⠐⠀⠔⠲⢩⡉⠙⠉⠛⠿⣿⣇⡀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⠛⠿⠀⠜⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⢿⣿⣿⣿⣿⣷⣄⣀⣠⣄⠈⠳⣕⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣾⣿⣿⣿⣭⣭⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠙⢿⣿⣿⣿⣿⣿⣧⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢻⣿⠟⠁⠀⠀⠀⡀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠒⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣶⣮⣁⣠⣤⣶⣶⣿⣿⡿⠿⠿⠟⠛⠛⠋⠹⣿⣿⣶⣶⣶⡶⣶⣶⣴⣶⣤⣤⣤⣄⣀⣀⣈⣉⣉⣉⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠿⠿⠟⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⠻⠿⠿⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣴⢾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⢀⣀⣀⣀⣀⣀⣀⢈⢉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛
⠄⡀⠰⠦⠤⠇⠀⠀⠀⠀⠀⠉⠀⠘⠿⠜⠛⠛⢏⠁⠀⠀⣠⣿⣿⣿⣶⣶⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡠⢄⡀⣀⣈⠐⠀⠚⡤⠀⠀⠀⡄⠐⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣤⣤⣤⣴⣶⣶⣶⣶
⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠛⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⢠⣩⡽⢦⣽⡾⡇⣠⢫⠁⠀⠀⠀⠀⠐⠀⢿⣿⣿⣿⠛⡛⠿⡿⣿⡿⠛⠿⠿⣿⠿⣿⡿⣿⣿⣿⣿⣿⣿⣿
⠀⠖⠶⣶⠿⢿⢶⣴⠶⠶⠆⠀⠀⠀⠀⡀⠀⣀⣼⣿⡇⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣞⡙⣗⡚⠛⠃⣓⢾⣽⡇⠀⠀⠀⠀⠨⠑⣪⣿⣿⣿⠀⣠⠠⣄⢐⠀⠀⠀⠀⡻⠂⠻⠠⢛⢁⠈⠐⢙⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠤⠀⠀⠀⠀⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠛⠛⠛⠁⣄⣰⣼⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⠉⠀⠩⠶⠦⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠽⣛⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⣿⣿⣿⣿⣆⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠻⣿⡟⢛⠿⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀⢸⡿⣿⣿⣿⣿⣦⢀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣿⣷⣾⣷⣿⣿⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⣿⣿⠟⣧⣼⣿⣿⣿⠋⠋⢉⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣛⣿⣿⣿⡏⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⣽⣿⣿⠀⠀⣼⢿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣿⣿⣿⣿⠟⠁⠀⡀⣠⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠛⠛⠻⣿⣿⠿⠿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣧⡉⠛⠛⠉⠉⠀⠀⢠⣤⣀⠀⢤⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠄⠀⠀⣿⣿⠧⡤⠤⣀⣿⣿⣦⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⠒⠻⣛⡛⣋⡱⣌⠓⠂⠀⠀⠂⠀⠀⠈⠉⠙⠛⠧⠀⠁⠱⣤⣤⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⢰⣿⣿⣦⣧⣤⣾⣿⣿⣿⣷⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣢⣴⣶⣤⡙⠿⠿⣿⣦⡀⠀⠀⠀⠈⠡⠤⡀⢀⣀⣀⣐⣠⣤⣬⣿⣿⣿⣿⣿⡿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢀⢀⡠⢄⡀⠀⠐⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣽⣿⣿⣛⠛⠛⠁⠀⠙⢿⠟⠷⣿⣷⣤⣄⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡗⡿⢸⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠠⢚⡿⠟⠛⠛⠛⡒⣀⣀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⣹⣇⡇⠾⣯⣿⣿⣿⣿⠿⣤⣿⣿⡿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠠⣿⢿⣿⣽⣿⣿⣿⣽⣿⣿⢿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⢀⠀⡀⡀⠈⡠⠀⣹⣏⣹⣿⣯⣽⣿⣿⡇⠀⠂⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⣽⣧⣿⠟⣿⣿⡿⣾⣟⣤⣶⣿⡿⠿⠛⠛⠋⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢀⡠⣙⢮⠐⠈⠁⠀⠉⠀⠀⠄⠂⠩⡉⠉⠋⠛⠿⣿⣇⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠻⢿⠁⠀⢿⣿⣷⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣤⣄⠀⠳⣝⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣶⣾⣿⣿⣿⣭⣭⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠟⠛⠋⠉⠉⢹⣿⡟⠋⠀⠀⠀⡀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣶⣀⣠⣤⣶⣶⣿⣿⠿⠿⠿⠟⠛⠛⠋⠙⣿⣿⣶⣶⣶⡶⣶⣶⣦⣤⣤⣤⣤⣄⣀⣀⣈⣉⣉⣉⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠿⠟⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠿⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣴⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⢀⣀⣀⣀⣀⣀⡀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛
⠀⠀⠰⠦⠤⢆⠀⠀⠀⠀⠀⠀⠀⠸⠆⠻⠛⠛⠁⣀⠀⠀⣼⣿⣿⣿⣷⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⣀⣀⠐⠂⠚⡡⠀⠀⠀⠄⠀⠀⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣤⣤⣤⣤⣴⣶⣶⣶
⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠚⠋⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⣨⡽⢾⡿⠾⠇⣠⠫⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠛⠛⠿⠟⢿⠿⠛⠻⠿⣿⠿⣿⡿⣿⣿⣿⣿⣿⣿⣿
⡀⠖⠶⣶⠿⢷⢶⣴⠶⠶⠶⠀⠀⠀⠀⠀⠀⢀⣿⣾⠀⠈⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣘⠙⠟⠘⠟⠃⣑⠼⢿⡆⠀⠀⠀⠀⠠⠑⡢⣿⣿⣿⠀⣠⠀⣄⢐⠀⠀⠀⠀⠛⠂⠙⠠⠛⠁⠀⠐⢰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠄⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠛⠛⠋⠉⣤⣠⣴⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡉⠀⠩⠽⠶⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠩⢛⠷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⣿⣿⣿⣿⣆⠀⠀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⣿⠟⢛⡿⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⡀⢰⡿⣿⣿⣿⣿⣆⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣾⣷⣿⣿⣧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⢸⣿⣿⠛⢧⣼⣿⣿⣿⠋⠉⢉⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣩⣿⣿⣿⣿⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣸⣿⣿⠀⠀⣼⢿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣽⣿⣿⣿⡟⠁⠀⢀⣠⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⣿⣿⠛⠛⠛⠻⣿⡟⠛⠿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⡿⣌⠙⠋⠉⠉⠀⠀⠀⣶⣄⣄⣐⣿⣷⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠄⠀⠀⣿⣿⠧⡤⠤⣀⣿⣿⣧⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⠒⠺⣟⣛⢛⣁⢀⠙⠒⠀⠀⠂⠀⠀⠀⠀⠈⠉⠛⠁⠀⠀⠑⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⣿⣿⣶⣧⣤⣶⣿⣿⣿⣷⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣒⣴⣶⣤⡉⠻⠿⣶⡲⡀⠀⠀⢀⠀⠀⠠⡤⣀⣀⣀⣀⣢⣤⣴⣶⣿⣿⣿⣿⣿⢿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⣀⡀⢄⡀⠀⠐⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣸⣿⣿⡛⠛⠛⠁⠀⠈⢻⠿⡿⣿⣯⣤⢄⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⠏⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⡿⢸⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢘⡿⠿⠟⠛⠓⠢⠀⠀⠀⣀⣰⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣹⣇⡇⠾⢯⣷⣿⣿⣿⠿⣤⣿⣿⡿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⡶⣶⣒⣿⣿⣷⠞⣭⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆⢀⠀⠀⣀⠀⡠⠄⣹⣏⣹⣿⣯⣽⣿⣿⡇⠀⠂⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡟⢰⡿⠋⣿⣿⣫⣾⢋⣩⣥⣾⣿⣿⡏⠻⠟⠁⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢀⡠⣝⠮⠔⠈⠀⠀⠉⠀⠀⠄⠃⠫⡉⠉⠉⠙⠿⣿⣇⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠃⠀⢸⣿⣿⡀⠉⠀⠐⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣠⣤⡄⠀⠑⣝⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣶⣿⣿⣿⣿⣭⣭⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠹⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⢹⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠘⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⣀⣤⣤⣶⣾⣿⣿⠿⠿⠿⠛⠛⠛⠉⠙⣿⣷⣶⣶⣶⡶⢶⣶⣶⣦⣤⣤⣤⣄⣀⣀⣈⣉⣉⣉⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠿⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠟⠛⠻⠿⠿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⢀⣀⣀⣀⣀⡀⡀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛
⠀⠀⠐⠃⠔⠆⠀⠀⠀⠀⠀⠀⠀⠦⠀⠿⠛⠋⢀⠀⠀⢠⣼⣿⣿⣿⣷⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⣀⣀⠐⠀⠚⡠⠀⠀⠀⠄⠀⠀⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣤⣤⣤⣤⣤⣶⣶⣤
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⣰⡵⢯⡽⠞⠅⡠⠣⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠛⠛⠿⠛⢻⠿⠛⠻⠿⣿⠿⣿⡿⣿⣿⣿⣿⣿⣿⣿
⡀⠖⠶⣶⠿⢶⢶⣴⠶⠶⠖⠀⠀⠀⠀⠀⠀⣸⣷⡇⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⣐⠙⠓⠐⠟⠃⣃⠼⠹⠆⠀⠀⠀⠀⠠⠑⡢⣿⣿⡧⠀⢠⠀⣄⠐⠀⠀⠀⠀⠛⠂⠙⠀⠛⠁⠀⠐⢐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣷⣦⣄⢀⣄⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⠉⠉⠀⡀⠀⣤⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡋⠁⠈⠩⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⡟⢹⡟⣿⢤⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⠟⣶⢀⣶⢲⡆⢰⡿⣦⡀⠀⢠⣿⣿⣿⣧⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⡛⢿⣿⠛⠻⠿⢶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣼⡷⣿⠸⣿⢠⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⡇⢸⣷⣟⢇⠀⢸⠿⢿⣿⣿⣷⠀⢀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣿⣷⣾⣷⣿⣿⣿⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠳⠟⠁⠻⠼⠟⠘⢿⡼⠇⣿⡿⣿⠻⣤⣼⣿⢿⣿⠛⠛⢁⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣛⣻⣿⣿⣿⣿⠏⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⠀⣀⣀⠀⢀⣀⡀⠀⠀⠀⠀⣿⣿⣦⠀⠀⣿⢿⣾⡿⠀⠀⢸⣿⣿⣿⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣫⣭⣭⣿⡿⠋⠀⠀⠠⢀⣠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠈⠿⣦⡀⠀⢸⣧⣿⠃⣿⣇⢀⣿⣿⡄⣿⡏⣿⠀⣿⣸⠇⣿⣟⠛⠛⠛⠻⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡠⢦⠸⡛⠛⠉⠉⠀⠀⠀⢀⣷⣤⣤⣤⣶⣿⠶⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣃⣿⠀⢹⡏⢠⣿⣿⠄⡄⠀⣀⣿⣷⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠠⠤⠶⠟⠛⠛⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⢀⣠⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠉⠁⠛⠛⠋⠀⠚⠃⢸⣿⣿⣴⣦⣤⣿⣿⣿⣿⡆⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣤⣌⡉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠶⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣿⣿⠷⠆⠙⠀⠀⠀⠀⠈⠓⢶⣞⣃⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡷⠷⠆⠸⣿⡿⣿⣿⣶⣿⠃⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠺⣿⣷⠶⠦⠀⠀⠀⠀⠀⠀⠀⠠⠀⣢⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣩⣁⠂⠠⣏⣳⣷⣿⣿⡾⣤⣿⣿⡿⠿⢿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠁⢠⣤⣤⡄⠀⣤⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠁⣀⠀⢙⣎⣶⣿⣫⣽⣿⣿⠃⠂⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⠋⣠⠞⠛⠉⢛⣻⣿⠿⣿⣿⠿⠟⠻⠿⠋⠉⠉⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⢤⡳⣜⠀⠈⠀⠀⠉⠀⠀⡐⠨⢉⠉⠙⠉⠻⠿⣿⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠚⠛⠻⠟⠀⠛⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⣀⣠⡄⠈⠢⡑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣾⣶⣿⣿⣿⣭⣴⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⢿⣿⣆⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⢿⣿⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣳⣶⠆⣀⣠⣤⣴⣶⣿⣿⡿⠿⠿⠿⠛⠛⠛⠻⣭⣭⣥⣤⣤⠤⣤⣤⣤⣤⣤⣤⣀⣉⣉⣉⣉⠉⠉⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠷⠛⠻⠿⠿⠿⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣷⣶⡆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣷⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣿⣿⡇⣶⣿⣶⣶⣤⣤⣤⣤⣀⣠⡀⠀⠀⠀⠀⣀⣀⣀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠀⠀⠈⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢠⠴⠤⡤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣿⣿⡇⠛⠻⠻⠛⠻⠿⠿⠿⠿⢿⢿⣿⣿⣷⣶⣿⣿⣿⣿⣿
⠀⠆⠲⣶⠯⢶⢆⠰⠶⠶⠆⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⠛⠛⠋⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⠂⠀⠂⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⠈⠆⠐⠀⠀⠀⠀⠈⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣷⣤⡄⠀⣄⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡚⠛⠉⠁⠀⡀⠀⣄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡋⠁⠀⠨⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⡏⢹⡟⣿⣄⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⠟⣶⢀⣶⢲⡆⢸⡷⣦⡀⠀⢸⣷⣿⣿⣦⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡿⢿⣿⠛⠻⠿⣶⣶⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣼⡷⣿⢨⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⡇⣸⡷⣟⠃⠀⢸⢿⢿⣿⣿⣧⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣷⣿⣶⣾⣷⣿⣿⣿⣵⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠷⠟⠁⠻⠼⠟⠘⢿⣼⠇⣿⡿⣿⠹⣄⣿⣾⣿⡟⠛⠛⢀⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣛⣛⣿⣿⣿⣿⡟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⠀⣀⣀⠀⢀⣀⡀⠀⠀⠀⠀⣿⣇⡆⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣫⣭⣭⣿⡿⠛⠀⠀⠀⠄⢀⡤⠤⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⠿⣦⡀⠀⢸⣧⣿⠁⣿⣇⢀⣿⣿⡆⣿⡏⣿⠀⣿⣸⢧⣿⡟⠛⠛⠛⠻⣾⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⢴⡄⠟⠛⠛⠋⠁⠀⠀⠀⣼⣧⣤⣤⣜⣶⣟⠶⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣃⣿⠀⢹⡏⢸⣿⡿⢤⡄⢀⣰⣿⣧⡀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠤⢴⠾⠟⠛⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣀⣠⣴⣦⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠉⠁⠛⠛⠋⠀⠚⠃⢸⣿⣯⣼⣤⣴⣿⣿⣿⣷⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣀⡁⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣻⣷⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣾⣿⣷⠯⠙⠀⠀⠀⠀⠈⠓⢶⣖⢫⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡵⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡶⢰⠆⣆⣿⣷⣿⣿⣶⣾⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠰⣿⣿⣶⢦⡀⠀⠀⠀⠀⠀⠀⠒⡐⢲⣠⣴⣿⣿⣿⣿⣟⡿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢁⣝⣨⠀⠥⣽⣾⢯⣿⡿⢧⣤⣿⣿⠿⠿⢿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣀⣀⣀⡀⠀⣀⣀⣀⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⡿⢯⡹⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⡁⠀⠠⠀⢁⡄⠈⢯⣊⣾⣷⣯⣿⣿⣿⠂⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣾⣿⠟⢁⡾⠟⠛⠿⢻⣿⠿⣿⣿⠿⠿⠿⠿⠋⠉⢉⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⡀⢌⠢⣄⠀⠁⠀⠀⠁⠀⠀⠐⠈⢍⠙⠛⠙⠻⢿⣿⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⡏⠀⠀⠀⠀⠀⢚⡿⠻⠀⠝⠋⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢀⣀⣀⣠⡀⠘⢮⡓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣶⣶⣶⣿⣿⣯⣭⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠀⠈⢿⣿⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠈⣿⡿⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⣲⣶⢄⣀⣠⣤⣶⣶⣿⣿⡿⠿⠿⠟⠛⠛⠛⢻⣭⣭⣤⣤⣤⠤⣤⣤⣤⣤⣤⣄⣀⣉⣉⣉⡉⠉⠉⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠛⠻⠿⠿⠿⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠐⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣷⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⣾⣿⣶⣦⣤⣤⣤⣤⣀⣤⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢉⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠠⠰⢤⠠⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⠓⠛⠻⠟⠻⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿
⠀⠖⠲⣶⠯⢷⡆⠰⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠿⠛⢫⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠙⠂⠐⠂⠀⠀⠠⢠⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⡀⢀⠀⠀⠀⠀⠀⠘⠁⠈⠂⠀⠂⠀⠀⠀⠈⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢍⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣤⡄⡀⣄⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠋⠉⠁⢀⠀⠀⣄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠨⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⡏⢹⡟⣿⣄⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⠟⣶⢀⣶⢲⡆⢸⡷⣦⡀⠀⢸⣿⣿⣿⣆⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠻⠿⣾⠟⠛⣿⣷⣶⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣼⡷⣿⢨⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⡇⣼⡷⣟⠃⠀⡼⠿⣿⣿⣿⣆⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣶⣾⣷⣾⣷⣾⣿⣿⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠳⠟⠁⠻⠼⠟⠘⢿⣼⠇⣿⡿⣿⢱⣄⣿⣾⣿⡟⠛⠛⣀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡟⣛⣿⣿⣿⣿⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⠀⣀⣀⠀⢀⣀⡀⠀⠀⠀⠀⣿⣧⡆⠀⢠⡿⣿⣿⡇⠀⠀⣿⣿⣿⣟⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣛⣯⣭⣽⣿⠟⠃⠀⠀⡀⠤⠤⠴⠶⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠈⠿⣦⡀⠀⢸⣧⣿⠁⣿⣇⢀⣿⣿⡆⣿⡏⣿⠀⣿⣸⢿⣿⡛⠛⠛⠛⠻⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡠⣤⠸⠛⠛⠛⠁⠀⠀⠀⢠⣿⣦⣤⣿⣿⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣃⣿⠀⢹⡟⢸⣿⡧⢤⠄⢀⣸⣿⣦⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠠⣤⡶⠾⠬⠋⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠈⠈⢁⣠⣤⣴⣾⣿⣆⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠉⠁⠛⠛⠋⠀⠚⠃⣼⣿⣧⣼⣤⣼⣿⣿⣿⣷⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣀⠀⠒⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣶⡾⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠐⣿⠿⢭⠙⠀⠀⠀⠀⠀⠓⠲⣶⡚⡁⠀⢀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡶⢰⠆⣿⣿⣷⣿⣿⣿⣾⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣥⣦⡀⠀⠀⠀⠀⠀⠀⠐⠢⠀⣦⣴⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢈⣝⣠⠀⢽⣽⣾⣯⣿⡟⢣⣼⣿⣿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⡉⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⡳⢦⣽⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡁⠀⠀⠀⢁⡄⠈⢫⣉⣾⣗⣬⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣸⣿⡿⠁⣰⠿⠟⠿⠿⣿⡿⣿⣿⣿⠿⠿⠿⠋⠉⢈⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⢜⠢⡀⠀⠁⠀⠈⠁⠀⠠⠐⠈⡍⠉⠛⠙⠻⢿⣯⣀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡟⠀⠈⠀⠀⠀⢴⠾⢻⠀⠹⠟⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢀⣀⣀⣤⡀⠙⢮⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣶⣶⣶⣾⣿⣿⣭⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠀⠀⠘⣿⣿⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠙⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣟⣲⣶⢄⣀⣤⣤⣶⣾⣿⣿⠿⠿⠿⠟⠛⠛⠋⢿⣿⣯⣤⣤⣤⢤⣤⣤⣤⣤⣤⣄⣀⣈⣉⣉⣉⢉⠉⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠟⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠛⠻⠿⠿⠿⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣷⣶⠀⠀⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⢿⣿⣿⠆⣾⣿⣶⣦⣤⣤⣤⣄⣀⣄⠀⡀⠀⠀⠀⣀⣀⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠒⢋⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠰⢤⠤⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⠒⠛⠛⠟⠛⠿⠟⠿⠿⠿⡿⢿⢿⣿⣷⣾⣿⣿⣾⣿⣿
⠀⠖⠲⣶⠿⢷⡆⠰⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠛⢋⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠘⠂⠐⠂⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠀⠀⠂⠀⠀⠀⠈⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠝⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣤⡄⡀⣄⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠋⠉⠁⢀⠀⢀⣄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠀⠂⠖⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⡏⢹⡟⣿⣄⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⠟⣶⢀⣶⢲⡆⢸⡷⣦⡀⠀⢸⣿⣿⣿⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⠿⢷⡿⠛⣿⣷⣶⣦⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣼⡷⣿⢰⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⡇⣾⡷⣟⠃⠀⡿⠿⣿⣿⣿⣆⠀⣸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢰⣾⣿⣶⣿⣶⣿⣿⣿⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠳⠟⠁⠻⠾⠟⠘⢿⡼⠇⣿⡿⣿⢱⣄⣿⡾⣿⡟⠛⠋⣠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢛⣻⣿⣿⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⠀⣀⣀⠀⢀⣀⡀⡀⠀⠀⢠⣿⡷⡄⠀⢠⡿⣷⣿⠇⠀⠀⣿⣿⣿⣟⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢟⣻⣭⣭⣿⡿⠛⠀⠀⠀⠀⠤⠤⠴⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠈⠿⣦⡀⠀⢸⣧⣿⠁⣿⣇⢀⣿⣿⡆⣿⡏⣿⠀⣿⣸⢿⣿⡛⠛⠛⠛⢿⡟⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⡤⡄⠛⠛⠛⠋⠀⠀⠀⠀⣸⣿⣴⣶⣿⣿⢷⡽⢆⣤⣀⡀⠀⠀⠀⠀⠀⠀⠸⠿⠸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣃⣿⠀⢹⡟⢸⣿⡧⢤⠤⢀⣸⣿⣦⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣠⣤⡶⠿⠔⠃⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⣀⣭⣶⣿⣿⣿⣿⣷⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠉⠁⠛⠛⠋⠀⠚⠃⣿⣿⣷⣼⣤⣼⣿⣿⣿⣷⡀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠄⢀⡀⠐⢖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣴⣶⡾⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢴⣿⣯⡍⠑⠀⠀⠀⠀⠐⠒⢶⣖⢋⡀⢀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠾⢰⠀⢿⢿⣷⣿⣷⣿⡿⢠⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣯⣤⡀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢰⣴⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣩⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢈⣝⣰⠀⠽⣝⣾⣯⣿⡟⢣⣼⣿⣿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⠛⣷⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⢀⡀⠈⣫⣁⣾⣟⣬⣿⣿⡿⠐⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣽⣿⠟⢀⣴⡿⠿⠿⢿⣿⢿⣿⣿⣿⠿⠿⠋⠉⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⢜⠣⡁⠀⠁⠀⠈⠁⠀⠠⠐⠈⣍⠉⠋⠙⠻⢿⣧⣀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠃⠀⠉⠀⠀⠠⠴⡞⠇⠘⠿⠃⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠀⠀⢀⣀⣠⣤⠀⠙⢎⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣶⣶⣿⣿⣿⣿⣭⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠘⣿⣿⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣟⣲⣶⣀⣀⣤⣤⣶⣾⣿⣿⠿⠿⠿⠛⠛⠛⠉⢿⣿⣷⣦⣤⣤⢤⣤⣤⣤⣤⣤⣤⣀⣀⣈⣉⣉⣉⣉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠟⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠛⠛⠿⠿⠷⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣷⣶⠀⠀⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛
⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣷⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⢽⣿⣿⠄⣾⣿⣶⣦⣤⣤⣤⣤⣠⣄⠀⡀⠀⠀⠀⣀⣀⢀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠠⢤⠤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⠐⠛⠛⠛⠛⠿⠛⠻⠿⢿⡿⢿⠿⢿⣷⣾⣿⣿⣿⣿⣿
⠀⠖⠲⣶⠿⢷⡦⠰⠴⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠛⣋⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠘⠂⠐⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠭⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣷⣦⡄⠀⣄⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠉⠉⠀⢀⡀⣀⣆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠀⠒⠴⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⡏⢹⡟⣿⢤⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⠟⣶⢀⣶⢲⡆⢸⡷⣦⡀⠀⢸⣿⣿⣿⣆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠻⢷⣾⠟⣻⣿⣶⣶⣶⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣼⡷⣿⠰⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⡇⣾⡷⣟⠃⠀⣿⢿⣿⣿⣿⣆⠀⣸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣶⣿⣶⣾⣶⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠳⠟⠁⠻⠼⠟⠘⢿⡼⠇⣿⡟⣿⠱⣄⣿⡿⣿⡏⠉⠉⣠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⡛⣛⣿⣿⣿⣿⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⠀⣀⣀⠀⢀⣀⡀⡀⠀⠀⢠⣿⣷⡄⠀⢠⡿⣷⣿⠇⠀⠀⣿⣿⣿⣟⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣛⣯⣭⣿⣿⠟⠁⠀⠀⢀⠐⠠⠴⣖⠚⢒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠈⠿⣦⡀⠀⢸⣧⣿⠁⣿⣇⢀⣿⣿⡆⣿⡏⣿⠀⣿⣸⢿⣿⡏⠉⠙⠛⢿⣟⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⢤⠘⠛⠛⠛⠁⠀⠀⠀⢤⣾⣷⣶⣾⣿⣿⠷⠿⣃⣢⣄⠀⠀⠀⠀⠀⠀⠀⠸⠿⠸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣇⣿⠀⢹⡏⢸⣿⡧⢠⠤⢤⣼⣿⣷⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣠⣴⠾⠧⠞⠀⠀⠀⠠⠀⠄⠂⠀⠀⠀⠀⠀⣀⣤⣼⣶⣿⣿⣿⣿⡀⠂⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠉⠁⠛⠛⠋⠀⠚⠃⣾⣿⣷⣼⣤⣴⣿⣿⣿⣷⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡀⠐⠲⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣤⣄⣤⣶⣿⡂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⢾⣍⡳⡦⠀⠀⠀⠀⠐⠲⣶⡚⣃⢀⣤⣼⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⡎⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢸⠀⢿⢿⠿⡿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣆⣄⠀⠀⠀⠀⠀⠀⠀⠰⡆⢒⣶⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⡿⢁⠽⣫⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣼⢨⠀⠹⣽⣾⣿⣿⡏⢧⣼⣿⣿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠉⠀⠀⠀⠀⢀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⡿⠧⣩⣿⣿⣿⣿⣿⣿⣦⡖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⠀⠀⠀⠀⠄⠀⣷⣀⣿⣯⣭⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣴⣶⠂⢀⣴⡶⠾⠿⣿⣿⣿⣿⣿⠿⠿⠛⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢌⡳⠅⠀⠁⠀⠀⠀⠀⠠⠀⠈⢍⠉⠉⠈⠛⢿⣧⣀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠁⠀⠉⠀⠀⣀⠔⢻⠀⢻⠟⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⢀⣀⣤⣤⠀⠈⢪⡓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷⣾⣿⣿⣿⣯⣭⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠘⢿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⣀⣠⣤⣴⣶⣿⣿⠿⠿⠿⠛⠛⠛⠋⠉⢻⣷⣶⣶⣶⣶⢶⣶⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣉⣉⣉⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠟⠛⠛⠛⠻⠿⢾⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣶⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠂⣿⣿⣶⣶⣤⣤⣤⣤⣠⣤⢀⣀⣀⠀⢀⣀⣀⣀⣀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠘⠛⠛⠛⠛⠛⠛⠛⠛⢻⡿⢿⠿⠿⠿⠿⠿⣿⣿⣿⣿
⠀⠖⠲⣶⠿⢶⡆⠠⠴⠶⠆⠀⠀⠀⠀⠀⠀⠀⣀⣾⠿⠿⠟⣛⣻⣽⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠁⠀⠀⠀⡀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠭⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣷⣦⡄⡀⣄⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠉⠉⠀⢀⠀⣀⣆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠀⠒⠔⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⡏⢹⡟⣿⢄⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⠟⣶⢀⣶⢲⡆⢸⡷⣦⡀⠀⢸⣿⣿⣿⣆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠻⢷⣾⠟⣿⣿⣶⣶⣶⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣼⡷⣿⠰⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⡇⣾⡷⣟⠃⠀⣿⢿⣿⣿⣿⣆⠀⣸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣶⣿⣶⣾⣶⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠳⠟⠁⠻⠼⠟⠘⢿⡼⠇⣿⡟⣿⠱⣄⣿⡿⣿⡏⠉⠉⣠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⡛⣛⣿⣿⣿⣿⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⠀⣀⣀⠀⢀⣀⡀⡀⠀⠀⢠⣿⣷⡆⠀⢠⡿⣳⣿⠇⠀⠀⣿⣿⣿⣟⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣛⣯⣭⣿⣿⠟⠃⠀⠀⢀⠐⠠⠴⣖⠚⢒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠈⠿⣦⡀⠀⢸⣧⣿⠁⣿⣇⢀⣿⣿⡆⣿⡏⣿⠀⣿⣸⢿⣿⡏⠉⠙⠛⢿⣟⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⢤⡘⠛⠛⠛⠁⠀⠀⠀⢤⣾⣷⣶⣾⣿⣿⠷⠯⣃⣢⣄⠀⠀⠀⠀⠀⠀⠀⠸⠿⠸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣇⣿⠀⢹⡏⢸⣿⡧⢠⠤⢤⣼⣿⣷⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣠⣴⠾⠧⠞⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⣀⣤⣼⣶⣿⣿⣿⣿⡀⠂⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠉⠁⠛⠛⠋⠀⠚⠃⣾⣿⣷⣼⣤⣴⣿⣿⣿⣷⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡀⠐⠲⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣤⣄⣤⣶⡿⡂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⢾⣍⡣⡦⠀⠀⠀⠀⠐⠲⣶⡚⡃⢀⣤⣼⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⡎⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡿⢸⠀⢿⢿⠿⡿⣿⣿⡿⢀⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣆⣄⠀⠀⠀⠀⠀⠀⠀⠰⡆⢐⣶⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⡿⢀⠽⣫⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣾⢠⠀⠹⣝⣾⣿⣿⡯⢧⣼⣿⣿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠉⠀⠀⠀⠀⢀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⡿⠧⣩⣿⣿⣿⣿⣿⣿⣦⡖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⠀⠀⠀⠀⠄⠀⣷⣀⣿⣯⣭⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣴⣶⠂⢀⣴⡶⠾⢿⣿⣿⣿⣿⣿⠿⠿⠛⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢌⡳⠅⠀⠁⠀⠀⠀⠀⠠⠀⠈⢍⠉⠉⠈⠛⢿⣧⣀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠃⠐⠈⠀⠀⣀⡔⢻⠀⢻⠟⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⢀⣀⣤⣤⠀⠈⢪⡓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷⣾⣿⣿⣿⣯⣭⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠘⢿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⣀⣠⣤⣴⣶⣿⣿⠿⠿⠿⠛⠛⠛⠋⠉⢻⣷⣶⣶⣶⣶⢶⣴⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣉⣉⣉⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠟⠛⠛⠛⠿⠿⢾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⣶⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠂⣿⣿⣾⣶⣤⣤⣤⣤⣤⣤⣀⣀⣀⠀⢀⣀⣠⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠘⠛⠛⠛⠛⠛⠛⠛⠛⢻⡿⢿⠿⠿⠿⠷⠿⣿⣿⣿⣿
⠀⠖⠲⣶⠿⢶⡆⠠⠴⠶⠆⠀⠀⠀⠀⠀⠀⠀⣀⣶⠿⠿⠟⣛⣻⣽⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⡀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡨⠙⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣤⡄⡀⣄⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⠉⠉⠀⢀⢀⣠⡄⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠐⢒⡲⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⡏⢹⡟⣿⣄⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⠟⣶⢀⣶⢲⣆⢸⡷⣦⡀⠀⣸⣿⣿⣿⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠐⠷⣶⡶⢾⣶⣶⣶⣦⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣼⡷⣿⠰⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⡇⣾⡷⣟⠃⠀⡿⠿⣿⣿⣿⣆⠀⣸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠰⣾⣷⣶⣶⣾⣿⣿⣿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠳⠟⠁⠻⠾⠟⠘⢿⡼⠇⣿⡟⣿⢱⣄⣿⡿⣿⡏⠉⠉⣠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⡿⠛⣿⣿⣿⣿⣿⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⠀⣀⣀⠀⢀⣀⡀⡀⠀⠀⢸⣿⣷⡄⠀⢰⡿⣳⣿⠇⠀⠀⣿⣿⣿⣟⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠛⣛⣿⣽⣿⣿⠟⠁⠀⠀⢀⠐⠢⠤⣖⡓⢓⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⠿⣦⡀⠀⢸⣧⣿⠃⣿⣇⢀⣿⣿⡆⣿⡏⣿⠀⣿⣸⢿⣿⡛⠛⠛⠛⢿⡟⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⡀⡀⣀⣤⠹⠟⠛⠛⠁⠀⠀⠀⣀⣿⣷⣶⣾⣿⣿⡷⡏⠋⣂⣄⡀⠀⠀⠀⠀⠀⠀⠸⠿⠸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣇⣿⠀⢹⡏⢸⣿⡧⢠⠤⢤⣸⣿⣦⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣹⣿⡿⠵⠜⠀⠀⠀⠀⠀⠄⠀⡠⠀⠀⣀⣀⣤⣤⣿⣶⣿⣿⣿⣿⣿⠐⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠉⠁⠛⠛⠋⠀⠚⠃⣿⣿⣷⣼⣤⣴⣿⣿⣿⣷⡀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠠⠀⠈⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣤⣄⣠⣴⣶⡓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠂⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣶⡛⢦⡄⠁⠀⠀⠐⢲⠤⣴⢖⣁⣀⣤⣼⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡍⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡿⢸⠀⢿⢿⢿⢿⣿⣿⡿⢠⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⣄⡉⠀⠀⠀⠀⠀⠀⢠⠤⡉⠤⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢟⡿⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣨⣿⣠⠀⠸⣝⣾⣿⣿⡿⢧⣼⣿⣿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠿⠗⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣿⣿⣿⡿⠧⣴⣿⣿⣿⣿⣿⣿⣥⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⠀⠀⠀⠀⠄⠀⣯⣀⣾⣏⣭⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⣀⣀⠀⢠⣤⣴⣶⣿⣿⣿⣿⣿⣿⠿⠏⠉⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢌⠣⠅⠀⠁⠀⠈⠁⠀⠠⠀⠈⢍⠉⠉⠙⠻⢿⣧⣀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡟⠁⡐⠋⠁⠀⠉⠹⠇⢻⣿⠋⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⢀⣀⣠⣤⠀⠙⢮⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣶⣿⣿⣿⣯⣭⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠘⣿⣿⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿
⡾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣶⣀⣠⣤⣴⣶⣿⣿⠿⠿⠿⠟⠛⠛⠋⠉⢻⣿⣶⣶⣶⣶⢶⣦⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣉⣉⣉⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠹⠟⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠟⠛⠛⠛⠿⠿⣾⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⢿⣿⣿⠂⣿⣿⣶⣶⣤⣤⣤⣤⣠⣤⡀⣀⣀⠀⢀⣀⣤⣄⣀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠈⠀⠄⠄⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠐⠛⠛⠛⠛⠛⠛⠛⠛⢿⡿⢿⠿⠿⠿⠾⠿⣿⣿⣿⣿
⠀⠖⠲⣶⠿⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠁⠀⠀⠀⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡨⠙⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⡄⡀⣄⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠁⠀⢠⢀⣠⣄⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠂⠐⠒⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⡏⢹⡟⣿⢄⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⠟⣶⢀⣶⢲⡆⢸⡷⣦⡀⠀⣸⣿⣿⣿⣆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠐⠷⣶⠿⢛⣿⣶⣶⣦⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣼⡷⣿⠰⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⡇⣿⡷⣟⠃⠀⣿⢟⣿⣿⣿⡆⠀⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠰⣾⣷⣶⣷⣾⣿⣿⣿⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠳⠟⠁⠻⠾⠟⠘⢿⡼⠇⣿⡟⣿⠱⢤⣿⡿⣿⡏⠉⠉⣤⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠘⡿⠛⣿⣿⣿⣿⣿⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⡀⣀⣀⠀⢁⣀⡀⠀⠀⠀⢸⣿⣷⣆⠀⢸⡿⣿⣿⠃⠀⢀⣿⣿⣿⣏⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠛⣛⣿⣿⣿⣿⠟⠁⠀⠀⢀⠀⠂⠦⣖⣓⠓⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⠿⣦⡀⠀⢸⣧⣿⠃⣿⣇⢀⣿⣿⡆⣿⡏⣿⠀⣿⣸⢿⣿⡉⠉⠉⠙⢷⣟⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣀⢤⣄⣂⣤⠹⠿⠿⠛⠁⠀⠀⠀⣀⣿⣷⣶⣾⣿⣿⡿⢿⠉⢐⣤⣀⠀⠀⠀⠀⠀⠀⠸⠿⠸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣇⣿⠀⢹⡏⣸⣿⡇⠠⠠⢤⣼⣿⣧⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣩⣽⡿⠵⠜⠀⠀⠀⠀⠀⠄⠀⡠⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⠓⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠉⠁⠛⠛⠋⠀⠚⠃⣿⣿⣷⣾⣦⣼⣿⣿⣿⣷⡀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠁⠤⡂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣤⣤⣠⣶⣿⡚⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣴⣆⣛⢶⡄⠁⠀⠀⠰⠲⢤⣴⢖⣁⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⡎⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣿⢺⠀⢿⢿⢿⢿⣿⣿⡿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⣄⠉⠀⠁⠀⠀⠀⠀⡠⠤⡉⠤⢁⣴⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡿⢟⠵⡫⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⢨⠀⠸⣝⣾⣿⣿⡿⢧⣾⣿⣿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⢀⣡⣴⣿⣿⣿⣿⣿⣿⣟⣙⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡀⠠⠤⠀⣿⣐⣿⣯⣽⣿⣿⡿⠈⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣀⠀⢀⣠⣤⣴⡶⠿⣿⣿⣿⣿⣿⠿⠛⠋⠉⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢌⡳⠅⠀⠀⠀⠀⠀⠀⠠⠈⠂⣌⠉⠉⡈⠛⢿⣧⣀⡀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⠋⠀⠞⠊⠈⠉⠁⡄⢻⣿⠏⠉⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⢀⣠⣤⣤⠀⠘⢮⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣯⣽⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠘⣿⣿⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿
⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠛⠋⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣀⣠⣤⣶⣶⣿⡿⠿⠿⠿⠛⠛⠛⠉⠉⢻⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣦⣤⣤⣤⣀⣀⣀⣀⣈⣉⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠟⠛⠛⠛⠛⠿⠾⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢰⢻⣿⣿⠀⣿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣀⣀⣀⡀⢀⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠈⠛⠛⠛⠛⠛⠛⠛⠛⢻⡿⢿⠿⠿⠿⠿⠿⢿⣿⢿⣿
⠀⠖⠲⣶⠿⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠁⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⡭⠛⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⡄⡀⣄⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡉⠉⠀⠀⢠⢀⣰⡄⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠁⠐⠒⠲⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⡏⢹⡟⣿⢤⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⠟⣶⢀⣶⢲⡆⢹⡷⣦⡀⠀⣾⣿⣿⣿⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠻⣶⡞⠛⣿⣶⣶⣶⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣼⡷⣿⠰⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⡇⣿⡷⣟⠃⠀⡟⠛⣿⣿⣿⡆⠀⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠠⣾⣷⣶⣷⣶⣿⣿⣿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠳⠟⠁⠻⠾⠟⠘⢿⡼⢧⣿⡟⣿⠱⢴⣿⡟⣿⡏⠉⠉⣴⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⣿⠛⣿⣿⣿⣿⣿⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⠀⣀⣀⠀⢁⣀⡀⠀⠀⠀⢸⣿⣶⣄⣀⢸⡟⣿⣿⣃⡀⢠⣿⣿⣿⣏⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⣛⣻⣿⣿⣿⠟⠁⠀⠀⢀⠀⠒⠲⣖⣛⡓⣒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⠿⣦⡀⠀⢸⣧⣿⠃⣿⣇⢀⣿⣿⡆⣿⡏⣿⠀⣿⣸⢿⣿⡉⠉⠉⠉⢳⣟⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣀⢤⣄⣂⢤⡹⠿⠿⠟⠁⠀⠀⠀⢀⣾⣷⣴⣶⣿⣿⣿⣶⣽⣶⣶⣶⠀⠀⠀⠀⠀⠀⠸⠿⠸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⢇⣿⠀⢹⡏⣾⣿⡗⠠⠀⢠⣼⣿⣦⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣠⣤⡿⠷⠜⠁⠀⠀⠀⠀⠤⠂⡀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠉⠁⠛⠛⠋⠀⠛⠃⣿⣿⣷⣿⣶⣾⣿⣿⣿⣷⡀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣶⣤⣶⣿⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⡀⠀⠠⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣴⣦⠙⢲⡤⠁⠀⠀⠐⠲⢤⣴⢖⡁⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢰⢾⡿⢺⠀⢿⢿⢿⢿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣧⣄⠉⠁⠁⠀⠀⠀⠀⡠⢤⡉⠥⠃⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠫⠚⠩⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣯⢌⠀⠸⣼⣿⡿⣿⡿⢣⣾⣿⣿⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣉⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠠⠤⠠⣷⣰⣿⣯⣿⣿⣿⡟⠈⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠀⠀⠀⠀⣀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⡿⠿⠿⠿⠷⣲⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢌⡳⠅⠀⠀⠀⠀⠀⠀⠠⠀⠂⣌⠉⠉⡈⠛⣻⣧⣄⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣳⣿⡿⠀⠠⠛⠉⠁⠁⠀⢻⣿⡿⠛⠉⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⢀⣠⣤⣤⠀⠘⢮⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣯⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠋⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⠜⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣀⣤⣤⣶⣾⣿⡿⠿⠿⠟⠛⠛⠋⠉⠉⢻⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣀⣀⣀⣀⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠋⠙⠛⠛⠛⠻⠾⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣬⣭⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠁⠀⠀⠀⠠⠀⠀⢸⣿⣿⣿⢐⣿⣿⣶⣶⣶⣦⣤⣤⣤⣦⣀⣀⣀⡀⢀⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠈⠛⠛⠛⠛⠛⠛⠛⠛⢛⡿⣿⠿⠿⠿⠿⠿⠿⢿⠿⡿
⠀⠖⠲⣶⠟⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠁⠁⠀⠀⠀⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡮⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⣄⣀⣄⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡉⠁⠀⠀⢠⢠⣰⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠀⠈⠤⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⡏⢹⡟⣿⢤⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⠟⣶⢀⣶⢲⣆⣹⡷⣦⡀⠀⣿⣿⣿⣿⣆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠛⢿⣾⠛⢻⣿⣷⣶⣶⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣼⡷⣿⠰⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⡏⣿⡷⣟⠃⠀⡟⣛⣿⣿⣿⣆⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⣿⣶⣾⣶⣾⣿⣿⣿⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠷⠟⠁⠻⠾⠟⠘⢿⡼⢧⣿⡟⣿⠱⢴⣿⡟⣿⡏⠉⠁⣴⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⢻⡛⢛⣿⣿⣿⣿⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⠀⣀⣀⠀⢁⣀⡀⠀⠀⠀⢸⣿⣶⣀⣀⣸⡟⣿⣿⣁⣀⣠⣿⣿⣿⣯⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⢛⣛⣿⣿⣿⡿⠋⠀⠀⢀⡀⠒⠲⢔⣛⡛⣒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⠿⣦⡀⠀⢸⣧⣿⠁⣿⣇⢀⣿⣿⡆⣿⡏⣿⠀⣿⣼⢿⣿⡉⠉⠉⠉⢳⣟⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣠⢤⡌⠾⠿⠿⠋⠀⠀⠀⢀⣼⣿⣶⣶⣿⣿⣿⣶⣾⣶⣶⣾⡇⠀⠀⠀⠀⠀⠸⠿⠸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⢇⣿⠀⢹⡏⣾⣿⡗⠰⠀⢠⣼⣿⣧⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣀⣤⣿⠷⠼⠁⠀⠀⠀⠀⠠⠀⠀⠄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠂⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠉⠁⠙⠛⠋⠀⠙⠃⣿⣿⣷⣿⣶⣾⣿⣿⣿⣷⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠑⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣶⣤⣶⣿⣓⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⢀⡀⡀⢀⣠⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣴⣖⡈⢧⡤⠀⠀⠀⠀⠲⠤⣴⣶⣃⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⣿⣿⡏⠣⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⢺⠀⢿⢿⢿⢿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣯⣌⠉⠀⠀⠀⠀⠀⠀⡀⠤⡍⠩⠈⠀⠀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠊⠹⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠯⢌⠀⠸⣽⣿⡿⢿⣿⢳⣾⣿⣿⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠯⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⡀⠠⠄⠠⣷⣴⣿⣯⣷⣿⣿⡏⠈⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠂⠉⠁⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⣻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢬⡳⠅⠀⠀⠀⠀⠀⠁⠠⠀⠂⣄⠉⠁⡈⠛⣻⣧⣄⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣴⣶⠆⠀⡔⠟⠛⠛⠙⠻⣿⣿⡿⠿⠟⠛⠉⠉⢩⣭⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⢀⣤⣤⡤⠀⠘⢪⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣷⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀⠈⠿⠋⠉⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠛⠉⠉⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⠏⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣀⣤⣤⣶⣾⣿⠿⠿⠿⠛⠛⠛⠉⠉⠉⢳⣶⣶⣶⣶⣶⢲⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣄⣀⣀⣀⣀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠋⠙⠛⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣬⣭⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣻⣿⣿⢀⣿⣿⣷⣶⣶⣶⣶⣶⣴⣦⣀⣀⣀⡀⠀⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠈⠉⠛⠛⠋⠛⠛⠛⠛⢛⡟⠿⠻⠿⠿⠿⠿⠿⠿⠿⡿
⠀⠖⠲⣶⠿⢶⠄⠰⠶⠶⠆⠀⠀⠀⠀⠀⠀⠈⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠐⠲⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⣨⢀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣦⣄⣀⣤⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡉⠀⠀⠀⢠⣠⣰⣄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣬⣅⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⡟⢹⡟⣿⢤⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⠟⣶⢀⣶⢲⣆⣿⡷⣦⡀⠀⣿⣿⣿⣿⣆⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣯⡛⢿⣿⢉⣝⠛⣷⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣼⡷⣿⠰⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⣟⣿⡷⣟⠃⠀⡟⣛⣿⣿⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣯⣽⣿⣿⣿⣶⣿⣿⢷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⡀⠛⠷⠟⠁⠻⠾⠟⠘⢿⡼⢿⣿⡏⣿⠱⢾⣿⣟⣿⡏⠉⠀⣴⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣍⣽⣿⣿⡿⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⡀⣀⣀⠀⢀⣀⡀⠀⠀⠀⢸⣿⣾⣀⣀⣸⡟⣿⣿⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⡭⢭⣿⣟⠁⠀⠀⠐⠤⠴⢚⠚⣒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⠿⣦⡀⠀⢸⣧⣿⠃⣿⣇⢀⣿⣿⡆⣿⡏⣿⠀⣿⣼⣿⣿⡍⠉⠉⠉⢻⣍⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⡠⡄⠰⣄⠘⠛⠛⠛⠁⠀⠀⣠⣧⣰⣶⣦⣽⣾⣿⣧⣤⣤⣶⠀⠀⠀⠀⠀⠸⠿⠸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣇⣿⠀⢹⡏⣿⣿⡗⠰⠒⠠⣾⣿⣧⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣠⣴⡯⠶⠃⠀⠈⠓⠦⠄⠀⠀⠀⠀⢀⣭⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⠂⠂⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠉⠁⠛⠛⠋⠀⠙⠃⣿⣿⣷⣿⣶⣿⣿⣿⣿⣿⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠉⠉⠥⣖⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣶⣦⣶⣿⣟⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠠⠀⣀⣀⣀⣀⡀⣀⣠⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣤⡄⠠⣀⠈⠀⠀⠀⢰⠤⢤⣴⢶⢀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⢿⣿⣷⠑⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⢿⣻⢿⠀⢻⣿⢿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⢉⡙⠂⠈⠀⠀⠀⠀⠁⣀⣈⠉⣨⠊⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠫⠞⠉⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠧⢌⠀⠺⣼⣿⡿⣿⣿⢳⣾⣿⣿⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⣙⣛⡂⠀⠀⠀⠀⠀⠀⠃⠘⠁⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⢄⡀⠠⠄⠰⣷⣼⣿⣯⣶⣿⣿⡏⠈⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⠨⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⠿⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣔⢬⡳⠅⠀⠀⠀⠀⠀⠁⠠⠁⠂⣄⠀⠁⡈⢛⣻⣷⣄⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣀⣀⠀⠀⣀⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠯⠍⠽⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠤⠀⠈⠪⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣾⣿⠏⠀⠰⠘⠉⠁⠀⠀⠹⣿⡿⠛⠋⠉⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠋⠉⠉⠹⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣠⣤⣴⣶⣿⡿⠿⠿⠟⠛⠛⠋⠉⠉⠉⢻⣿⣶⣶⣶⣶⢲⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠙⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠋⠉⠙⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣬⣭⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⢐⠀⠀⠀⠠⠀⠀⢸⢻⣿⣿⢸⣿⣿⣿⣷⣷⣶⣶⣶⣴⣦⣤⣄⣀⡄⢠⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠈⠉⠙⠋⠋⠛⠛⠛⠛⢻⡟⠻⡛⠿⡿⠿⠿⠿⠿⠿⠿
⠀⠖⠲⣶⠿⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠈⢡⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢀⠐⠰⠶⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠄⠆⠀⠀⠀⠀⠀⠈⠀⠘⠀⠁⠀⠀⠀⡀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣷⣦⣄⣀⣤⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡉⠀⠀⠀⣠⢠⣰⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣯⣥⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⡟⢹⡟⣿⢤⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⡟⣶⢀⣶⢲⣆⣿⡷⣦⡀⠀⣿⣿⣿⣿⣆⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣏⣙⣿⣟⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣼⡷⣿⠰⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⣟⣿⡷⣟⠃⠀⡟⣛⣿⣿⣿⣆⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣯⣿⣿⣿⣷⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠷⠟⠁⠻⠾⠟⠘⢿⡼⢿⣿⡟⣿⠱⢾⣿⣿⣿⡍⠉⠀⣴⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣯⣽⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⠀⣀⣀⠀⢀⣀⡀⠀⠀⠀⢸⣿⣾⣀⣀⣸⡟⣿⣿⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣓⣀⠀⠀⠤⠴⠴⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠘⠿⣦⡀⠀⢸⣧⣿⠃⣿⣇⢀⣿⣿⡆⣿⡏⣿⠀⣿⣼⣿⣿⡉⠉⠉⠉⢳⣟⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣤⠑⣧⡀⠀⡈⠛⠛⠋⠉⠀⣰⡀⠀⣀⡀⣤⣘⣴⣤⣴⣶⠀⠀⠀⠀⠀⠸⠿⠸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣇⣿⠀⢹⡏⣿⣿⡗⠴⠂⢠⣾⣿⣧⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣀⣤⣾⣳⠟⠀⠀⠙⠳⠷⠦⠤⠄⠀⠀⣈⣭⣴⣾⣿⣿⣿⣿⣿⣿⣿⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠉⠁⠛⠛⠋⠀⠙⠃⣿⣿⣷⣿⣶⣿⣿⣿⣿⣷⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⠉⠙⠭⣤⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣦⣴⣿⡟⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⢀⣀⣀⣀⡀⢀⣠⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣤⣄⠀⡀⢀⠁⠀⠀⢀⠤⣤⣤⡶⡖⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⠑⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⢿⣻⣿⠀⢻⣿⢿⢿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⣀⠉⢉⡼⠈⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠫⠞⠩⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠧⢌⠀⠺⣼⣿⡿⢿⢿⢳⣾⣿⣿⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⠂⠚⠃⠀⠀⠀⠀⠀⠀⠈⠌⠙⠂⠁⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⢄⡀⠠⠄⠰⣷⣼⣿⣯⣶⣿⣿⡏⠈⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣘⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣯⣽⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣐⢌⡳⠅⠀⠀⠀⠀⠀⠁⠠⠁⠂⡄⠉⠁⡈⢛⣻⣧⣄⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣟⡓⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡤⠀⠘⠮⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣴⣿⡿⠀⢀⠽⠛⠋⠉⠉⢻⣿⣿⠿⠟⠛⠉⠉⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠹⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣀⣤⣴⣶⣾⡿⠿⠿⠟⠛⠛⠛⠉⠉⠉⢳⣶⣶⣶⣶⡶⢲⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠛⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠋⠉⠛⠻⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣬⣭⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⠒⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⢸⣿⣿⣿⣷⣶⣶⣶⣶⣴⣦⣀⣀⣀⡀⢠⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠈⠉⠙⠋⠋⠛⠛⠛⠛⢻⠟⡻⠛⠿⡿⠿⠿⠿⠿⠿⠿
⠀⠖⠲⣶⠿⢶⠆⠰⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⢉⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢈⠐⠰⠶⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠄⠆⠀⠀⠀⠀⠀⠈⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣷⣦⣄⣀⣄⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡉⠀⠀⠀⣠⢠⣰⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣴⣶⣾⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⡟⢹⡟⣿⢤⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⡟⣶⢀⣶⢲⣆⣿⡷⣦⡀⠀⣿⣿⣿⣿⣆⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣉⢙⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣼⡷⣿⠰⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⣟⣿⡷⣟⠃⠀⡟⣛⣿⣿⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣽⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠷⠟⠁⠻⠾⠟⠘⢿⡼⢿⣿⡟⣿⠱⢾⣿⣿⣿⡍⠉⠀⣴⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣯⣭⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⠀⣀⣀⠀⢀⣀⡀⠀⠀⠀⢸⣿⣾⣀⣀⣸⡟⣿⣿⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠻⠿⣿⣿⣿⣗⣂⡠⠰⠶⠖⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⠿⣦⡀⠀⢸⣧⣿⠃⣿⣇⢀⣿⣿⡆⣿⡏⣿⠀⣿⣼⣿⣿⡉⠉⠉⠉⢳⣝⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣸⢻⣆⡀⢀⣀⠀⠈⠙⠛⠉⡄⠀⠀⠀⠀⣀⣠⣤⣶⣆⠀⠀⠀⠀⠀⠸⠿⠸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣇⣿⠀⢹⡏⣿⣿⡗⠰⠂⢠⣾⣿⣷⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣴⣟⡾⠋⠀⠈⠙⠺⠿⠿⠶⠆⠀⠀⢉⣀⣤⣶⣿⣿⣿⣿⣿⣿⡄⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠉⠁⠛⠛⠋⠀⠙⠃⣿⣿⣷⣿⣶⣿⣿⣿⣿⣷⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠛⠛⢭⡤⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣷⣴⣾⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢀⣀⣀⣀⡀⢀⣠⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣁⠀⠠⠀⠈⠂⠀⠀⢀⢤⣤⣤⡴⠆⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⢾⣻⠿⠀⢻⣿⢿⣿⢿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⡐⠻⠂⠄⠈⠊⠀⠀⠀⠀⠀⠀⠛⣋⠐⠉⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢟⠵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠧⢌⠀⠺⣼⣿⡿⢿⢿⢳⣾⣿⣿⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠣⠠⠓⠂⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠁⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⢄⡀⠠⠄⠰⣷⣼⣿⣯⣶⣿⣿⡏⠈⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠀⠐⠐⠦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣭⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣐⢌⡳⠅⠀⠀⠀⠀⠀⠁⠠⠁⠂⣄⠉⠁⡈⢛⣻⣧⣄⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⡿⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⠀⠘⠮⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣶⣶⠂⠀⠐⠿⠟⠛⠋⢻⣿⣿⣿⠿⠛⠛⠛⠉⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠹⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣀⣤⣴⣶⣾⡿⠿⠿⠟⠛⠛⠛⠉⠉⠉⢳⣶⣶⣶⣶⣶⢲⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠻⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠋⠉⠛⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣬⣭⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⠐⠀⠀⠀⠠⠀⠀⢸⢹⣿⣿⢸⣿⣿⣿⣷⣶⣶⣶⣶⣴⣦⣄⣀⣀⡀⢠⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠈⠉⠙⠛⠋⠛⠛⠛⠛⢻⠟⡻⠛⠿⡿⠿⠿⠿⠿⠿⠿
⠀⠖⠲⣶⠿⢶⡆⠰⠴⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠌⠈⠈⠐⠰⠶⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠄⠆⠀⠀⠀⠀⠀⠈⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣷⣤⣄⣀⣄⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡉⠀⠀⠀⣠⣠⣰⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⡟⢹⡟⣿⢤⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⠟⣶⢀⣶⢲⣆⣿⡷⣦⡀⠀⣿⣿⣿⣿⣆⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣟⡀⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣼⡷⣿⠰⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⡟⣿⡷⣟⠃⠀⡟⣛⣿⣿⣿⣆⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠷⠟⠁⠻⠾⠟⠘⢿⡼⢷⣿⡟⣿⠱⢶⣿⣟⣿⡏⠉⠀⣴⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⡀⣀⣀⠀⢀⣀⡀⠀⠀⠀⢸⣿⣾⣀⣀⣸⡟⣿⣿⣀⣀⣸⣿⣿⣿⣯⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠻⢿⣿⠿⠿⠿⠿⣿⣷⣦⠶⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⠿⣦⡀⠀⢸⣧⣿⠃⣿⣇⢀⣿⣿⡆⣿⡏⣿⠀⣿⣼⢿⣿⡉⠉⠉⠉⢳⣝⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣐⣿⣷⣄⡀⢀⣀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣀⣤⣴⣾⡆⠀⠀⠀⠀⠀⠸⠿⠸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⢇⣿⠀⢹⡏⣿⣿⡗⠰⠂⢠⣼⣿⣧⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣠⣼⣕⡿⠙⡈⠉⠛⠻⠿⠿⠶⠆⠀⠀⢁⣠⣴⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠉⠁⠛⠛⠋⠀⠙⠃⣿⣿⣷⣿⣶⣾⣿⣿⣿⣷⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠐⠛⠛⣩⡤⢤⡀⠈⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣴⣶⣶⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⣀⣀⣀⣀⡀⢀⣠⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⣤⣤⣤⡴⠆⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣻⢻⠀⢻⢿⢿⢿⣿⣿⠏⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⠌⠿⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠛⢃⡘⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠟⢟⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠮⢌⠀⠸⣼⣿⡿⢿⣿⢳⣾⣿⣿⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⡨⠳⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠣⠈⠁⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⢄⡀⠠⠄⠀⣵⣴⣿⣯⣶⣿⣿⡏⠈⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠐⠃⠀⠡⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣭⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣔⢬⡳⠅⠀⠀⠀⠀⠀⠁⠰⠁⠂⣄⠀⠁⡈⢛⣻⣧⣄⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⡴⠒⠊⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣿⣿⣿⣿⣿⣿⠯⠀⠀⠀⠀⠄⠀⠀⠀⢠⣤⣤⡤⠀⠈⠪⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠰⣶⡾⠛⠛⢹⣿⣿⣿⡿⠿⠛⠛⠋⠀⢀⡀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠁⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠋⠉⠉⠹⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣿⣷⣀⣤⣴⣶⣾⡿⠿⠿⠟⠛⠛⠋⠉⠉⠉⢻⣶⣶⣶⣶⣶⢲⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠸⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠋⠉⠙⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣬⣭⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣹⣿⣿⢁⣿⣿⣿⣶⣶⣶⣶⣶⣴⣦⣤⣄⣀⡀⢀⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠈⠉⠉⠉⠉⠛⠛⠛⠛⢻⡟⠻⠻⠻⡿⠿⠿⠿⠿⠿⡿
⠀⠖⠲⣶⠿⢶⣆⠰⠴⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠬⠈⠀⠀⠰⠶⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⠿⢿⣿⣿⣿⣿⣿⡿⠿⠿⠏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣶⣄⣀⣄⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡉⠀⠀⠀⢠⣠⣰⣦⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣠⣶⣶⣶⣶⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⡟⢹⡟⣿⢤⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⡟⣶⢀⣶⢲⣆⣹⡿⣦⡀⠀⣾⣿⣿⣿⣧⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣼⡷⣿⠰⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⡏⣿⡷⣟⠇⠀⣟⣛⣿⣿⣿⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠷⠟⠁⠻⠾⠟⠘⢿⡼⢧⣿⡏⣿⠹⢶⣿⡟⣿⡇⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⢠⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⡀⣀⣀⠀⢀⣀⡀⠄⠀⠀⢸⣿⣿⣆⣀⣸⣟⣿⣿⣃⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠈⠻⢿⣿⣿⣿⣛⠿⠟⠋⢀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⣿⠘⠿⣦⡀⠀⢸⣧⣿⠃⣿⣇⢀⣿⣿⡆⣿⡏⣿⠀⣿⣼⢿⣿⡍⠉⠉⠉⢻⣏⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⣙⣷⣄⡀⢈⣍⠙⠛⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⣆⠀⠀⠀⠀⠸⠿⠸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣇⣿⠀⢹⡏⣼⣿⡗⢰⠒⢠⣼⣿⣷⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣠⣾⣧⡿⠙⡀⠈⠙⠛⡿⠿⠷⠇⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠉⠁⠙⠛⠋⠀⠙⠃⣿⣿⣿⣾⣶⣾⣿⣿⣿⣿⡄⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠐⠛⠿⣛⣭⣴⣄⡨⠩⠣⠛⠁⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⣀⣠⣀⣄⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠀⠀⠄⠀⠙⠀⠀⠀⠀⢠⣤⣤⠶⠂⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⠸⠀⣿⣿⢿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣠⠴⠟⣁⣀⠑⠀⠂⠀⠀⠀⠀⠀⠘⢃⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠋⠀⠰⠎⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠮⠼⠐⠲⢮⣿⣿⢿⣏⢳⣾⣿⣿⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣀⡀⠈⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠌⠃⠀⠀⠀⠀⠙⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠀⣀⠀⢀⡀⠠⠀⠀⢵⣤⣿⣧⣶⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠺⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢔⠪⡓⠅⠀⠀⠀⠀⠀⠁⠐⠀⠀⣦⠀⠁⣈⣙⣻⣷⣤⡀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣀⡤⠌⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣛⠋⠈⠆⠀⠀⠀⢠⣤⣤⠴⠄⠈⠢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠛⠀⠀⠀⠀⣀⣀⠀⠀⣤⣤⡶⠶⠟⣿⣿⣿⡿⠿⠿⠿⠿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⣴⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠈⠀⠘⠉⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⡃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠉⠉⠉⠙⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⣿⣿⣢⣤⣴⣶⣿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠀⣹⣿⣿⣿⣷⣿⢲⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡋⠉⠉⠙⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣟⣿⣿⡀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣤⣤⣄⣀⣀⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣾⣿⣿⣷⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠒⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠈⠉⠉⠉⠉⠉⠉⠉⠛⢻⠟⠻⠻⠛⡟⠛⠿⠿⠿⠿⡿
⠀⠖⠲⣶⠿⢶⣦⡰⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠨⠉⠀⠀⠑⠶⠀⠀⠀⠀⠀⠂⠀⢸⣿⣿⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠄⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢿⣿⣿⣿⣿⣿⡿⠿⠿⠏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣶⣄⣀⣄⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡉⠀⠀⠀⢠⣠⣰⣦⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣠⣶⣶⣶⣶⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⡟⢹⡟⣿⢤⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⡟⣶⢀⣶⢲⣆⣹⡿⣦⡀⠀⣾⣿⣿⣿⣧⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣼⡷⣿⠰⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⡏⣿⡷⣟⠇⠀⣟⣛⣿⣿⣿⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠷⠟⠁⠻⠾⠟⠘⢿⡼⢧⣿⡏⣿⠹⢶⣿⡟⣿⡇⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⢠⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⡀⣀⣀⠀⢀⣀⡀⠄⠀⠀⢸⣿⣿⣆⣀⣸⣟⣿⣿⣃⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠈⠻⢿⣿⣿⣿⣛⠿⠟⠋⢀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⣿⠘⠿⣦⡀⠀⢸⣧⣿⠃⣿⣇⢀⣿⣿⡆⣿⡏⣿⠀⣿⣼⢿⣿⡍⠉⠉⠉⢻⣏⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⣙⣷⣄⡀⢈⣍⠙⠛⠃⠀⠀⠀⠀⠀⢀⣴⣿⣿⣆⠀⠀⠀⠀⠸⠿⠸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣇⣿⠀⢹⡏⣼⣿⡗⢰⠒⢠⣼⣿⣷⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣠⣾⣧⡿⠙⡀⠈⠙⢛⡿⠿⠷⠇⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠉⠁⠛⠛⠋⠀⠘⠃⣿⣿⣿⣾⣶⣾⣿⣿⣿⣿⡀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠐⠛⠿⣛⣭⣤⣄⡈⠋⠣⠙⠁⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⣀⣠⣀⣄⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢠⣤⣤⠶⠂⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⠸⠀⣿⣿⢿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣠⠤⠿⣁⣀⠑⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠋⠀⠸⠎⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠮⠼⠐⠸⢬⣿⣷⢿⣏⢳⣾⣿⣿⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⣀⡀⠈⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⠃⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠀⣀⠀⢀⡀⠠⠀⠀⢵⣤⣿⣧⣶⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠺⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢔⠪⡓⠅⠀⠀⠀⠀⠀⠁⠐⠀⠀⣦⠀⠁⣈⣙⣻⣷⣤⡀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣀⡤⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣛⠋⠈⠆⠀⠀⠀⢠⣤⣤⠴⠄⠈⠢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠛⠀⠀⠀⠀⣀⣀⠀⠀⣤⣤⡶⠶⠟⣿⣿⣿⡿⠿⠿⠿⠿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⣴⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠈⠀⠈⠉⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⡃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠉⠉⠉⠙⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⣿⣿⣢⣤⣴⣶⣿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠀⣹⣿⣿⣿⣷⣿⢲⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡋⠉⠉⠙⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⡀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣾⣿⣿⣷⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠒⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠈⠉⠉⠉⠉⠉⠉⠉⠛⢻⠟⠻⠻⠛⡟⠛⠟⠿⠿⠿⡿
⠀⠖⠲⣶⠿⢶⣦⡰⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠨⠉⠀⠀⠑⠶⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠠⡀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⢀⠄⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣵⣾⣿⣿⣿⣿⣿⣿⠿⠿⠿⠍⡀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⣄⣀⣤⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡉⠀⠀⠀⣤⣠⣸⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡃⠀⠄⠄⠈⡉⣉⣤⣶⣶⣶⣴⣶⣆⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⡟⢹⡟⣿⢤⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⡟⣶⢀⣶⢲⣆⣿⡷⣦⡀⠀⣿⣿⣿⣿⣄⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣼⡷⣿⢸⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⣿⣿⡷⣟⠃⢠⡟⣟⣿⣿⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣸⣿⣿⣿⢿⣿⣿⣿⣦⠦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠷⠟⠁⠻⠾⠟⠘⢿⡼⢿⣿⡏⣿⠱⢾⣿⣿⣿⡁⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣤⣿⣿⣿⣿⡿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⡀⣀⣀⠀⣀⣀⡀⠀⠀⠀⢸⣿⣾⣀⣀⣸⡻⣿⣿⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣄⡀⠙⠿⣿⣿⣿⣿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⠿⣦⡀⠀⢸⣧⣿⠃⣿⣇⢀⣿⣿⡆⣿⡏⣿⠀⣿⣼⣿⣿⡉⠉⠉⠉⣻⣍⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣵⣌⣙⠶⣄⡀⢉⣙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡄⠸⠿⠸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣇⣿⠀⢹⡏⣿⣿⡗⡶⠒⠤⣿⣿⣧⡀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡯⣿⠏⠈⠐⠉⠛⠻⠿⠿⠄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠉⠁⠛⠛⠋⠀⠙⠃⣿⣿⣷⣿⣶⣿⣿⣿⣿⣷⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⠛⠯⠿⣋⣥⣤⠿⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠠⠀⣀⣀⣀⣀⡀⣀⣠⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣠⣤⠶⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣛⢿⠘⢻⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⠤⠄⠶⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⣀⠀⠀⠀⠡⢸⣿⣿⣿⣿⣿⡿⠁⡿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠯⢎⠀⠺⣽⡿⡿⢿⣿⣳⣾⣿⣟⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣀⠀⠠⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⠨⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡄⢠⡀⠀⢆⡀⠰⠀⠰⣿⣼⣿⣯⣾⣿⣿⡇⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⡀⣐⢬⡳⠅⠀⠀⠀⠀⠀⠀⠠⠁⠂⡄⠀⠁⡈⣛⣻⣧⣄⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⡷⠆⠀⠀⠀⣠⣤⣤⡤⠀⠘⠮⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢐⠆⠋⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⢾⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠘⣿⣷⡴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣾⣿⡇⠂⠈⠛⠋⠀⠀⠈⠻⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠹⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣠⣤⣴⣶⣿⡿⠿⠿⠟⠛⠛⠋⠉⠉⢀⢻⣿⣿⣶⣶⡖⢲⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡋⠉⠙⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⣿⣿⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⠀⠀⡐⠀⠀⠀⠠⠀⠀⢸⢿⣿⣿⢸⣿⣿⣿⣿⣷⣶⣶⣶⣴⣦⣤⣤⣤⣤⣠⣤⣄⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠒⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠈⠉⠉⠉⠉⠛⠉⠛⠛⠻⡟⠿⠛⡻⡿⠿⠿⠿⠿⠿⠿
⠀⠖⠶⣶⠿⢶⣦⠰⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠉⠄⠌⠀⠈⠐⠰⠶⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠄⠆⡀⠀⠀⠀⠀⠈⠀⠈⠀⢀⠀⠀⠀⣀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣥⣾⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⣄⣀⣤⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠁⠀⠀⠀⣄⣠⣶⡄⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠐⠋⡋⣉⣥⣶⣶⣦⣴⣦⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⡟⢹⡟⣿⢤⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⡟⣶⢀⣶⢲⣆⣿⡷⣦⡀⢀⣿⣿⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣼⡷⣿⢸⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⣿⣿⡷⣟⠃⢸⣛⣿⣿⣿⣿⡄⢠⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⣿⣧⠦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠷⠟⠁⠻⠾⠟⠘⢿⡼⢿⣿⡏⣿⠱⢼⣿⢻⣿⠉⠁⢠⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⠀⣀⣀⠀⢀⣀⡀⠀⠀⠀⢼⣿⣾⣀⣀⣾⣻⣿⣟⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣴⣀⠈⠛⢿⣿⣿⣿⡟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⠿⣦⡀⠀⢸⣧⣿⠃⣿⣇⢀⣿⣿⡆⣿⡏⣿⠀⣿⣼⣿⣿⠉⠉⠉⠉⣿⡍⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣐⣶⣭⣻⢦⣄⠀⢉⡙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡿⠸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣇⣿⠀⢹⡟⣿⣿⠒⡖⠒⢤⣿⣿⣧⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡯⣿⠇⠉⠊⠙⠛⠿⠷⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠉⠁⠛⠛⠋⠀⠘⢣⣿⣿⣷⣷⣶⣿⣿⣿⣿⣧⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠫⠿⠟⣩⡴⠶⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠨⢀⢀⣀⢀⣀⣀⣀⡠⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠀⠀⠀⣠⣤⠶⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢿⡓⡟⢸⣿⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⠤⠤⠴⣂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⢂⢠⠀⠀⠈⠄⠀⣿⣿⣿⣿⣿⢏⡿⢟⠆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠧⢇⠒⠷⣿⣟⡿⣿⠙⣶⣿⣿⡏⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⠀⠠⠠⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠙⠫⠃⠀⠀⠀⠀⢰⣿⣿⣿⣿⠏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠆⢠⡀⠀⣂⠀⠴⠀⠸⣦⣼⣿⣿⣾⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠀⠈⠀⠀⠠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠠⡐⢝⠢⠅⠀⠀⠀⠀⠀⠀⠠⠁⠐⡄⠈⢁⡉⣛⣿⣧⣄⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡷⠖⠀⠀⣠⣤⣤⠤⠀⠑⠌⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡦⠚⠉⠁⠀⠀⠀⠀⢀⣀⣀⣤⡴⢶⣿⣿⣿⣿⣿⣿⢿⡟⠀⠀⠀⠹⣿⣷⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇⠒⠈⠛⠋⠁⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣗⣠⣤⣶⣶⣿⡿⠿⠿⠟⠛⠛⠋⠉⠉⢈⣿⣿⣷⣶⣶⡖⣲⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢘⠋⠉⠙⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⣿⣷⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⢄⡀⠀⠀⡐⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⣿⣿⣶⣶⣶⣦⣶⣦⣤⣤⣠⣄⣤⣴⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣾⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⢀⠂⢓⠦⠂⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠉⠉⠉⠉⠙⠛⠉⠛⠛⣿⠛⠟⠟⠻⠻⠻⠿⠿⠿⢿⠿
⠀⠖⠶⣶⠿⢷⡆⠰⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠉⠐⠌⠁⡈⠘⠶⠆⠀⠀⠀⠀⠀⠀⠁⣿⣿⣿⠀⠀⠀⢄⠀⠀⠀⠀⠀⠁⠁⠈⠀⢀⠀⠀⠀⡀⢀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣎⣱⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢁⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⡄⡀⣄⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣍⠁⠀⠀⢀⣄⣄⣷⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡗⠃⠀⠠⠴⠿⠿⣋⣉⣤⣶⣶⣤⣶⡀⠀⠀⠀⠀⠀⠀⢠⣿⣼⡏⢹⡟⣿⢄⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⡟⣶⢀⣶⢲⣾⣿⡶⣦⡀⢸⣿⣿⣿⣷⡀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠶⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣾⡷⣿⢸⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⣿⣿⡷⣟⠃⢸⣛⣻⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣹⣿⣿⡧⣿⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠷⠟⠁⠻⠾⠟⠘⢿⡼⣿⣿⡏⣿⠰⣿⣿⣻⣿⠀⠁⢠⣾⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⢤⣤⣾⣿⣿⣿⣿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⡀⣀⣐⠀⢀⣀⡀⠀⠀⠀⣿⣿⣖⣀⣀⣿⢿⣿⣇⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡴⣄⡀⠙⠻⢿⣿⣿⣿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⠿⣦⡀⠀⢸⣧⣿⠃⣿⣇⢀⣿⣿⡆⣿⡏⣿⠀⣿⣼⣿⣯⠉⠉⠉⠙⣯⡉⠉⠉⠋⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣩⣔⣝⡳⢦⣀⠈⣍⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣇⣿⠀⢹⣿⣿⣿⠂⡖⠂⢴⣿⣿⣆⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡭⣿⠃⠁⠈⠙⠛⠷⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠉⠁⠛⠛⠋⠀⠘⢻⣿⣿⣿⣷⣾⣿⣿⣿⣿⣧⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⡟⠻⠯⠞⣡⣶⠶⠟⠋⠉⠁⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⢀⣀⣀⣠⣀⣀⣀⡄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠉⠀⠀⠀⠈⣠⡤⠆⠀⠀⠀⠀⠉⢁⣀⣶⣶⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡇⠇⠸⣿⡿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⠤⠴⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⣀⡀⠀⠈⠄⠀⣿⣿⣿⣿⣿⣿⡟⢣⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠴⠥⡁⠐⢧⣽⣿⠿⣿⡟⣶⣿⣿⡏⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⡈⠐⠀⠀⠀⠀⠀⠀⠙⠻⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢀⡀⠠⣀⠀⠤⠀⢬⣦⣾⣿⣵⣾⣿⣿⠁⠁⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠀⠉⠁⠀⠀⠠⢦⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡷⠆⠀⠀⠀⠀⢀⠠⡲⢝⠮⠀⠀⠀⠀⠀⠈⠀⠎⠐⢠⡀⠈⢀⠙⣛⣿⣦⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⡿⠖⠀⣤⣤⣤⠄⠀⠑⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠟⠋⠁⠀⠀⠀⠀⠀⣀⣤⣤⡴⠾⣿⣿⣿⠿⢿⠿⠿⠟⠀⠀⢻⣿⣷⡧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⢰⣿⣿⠐⠀⠙⠛⠁⠀⠀⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡓⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠋⠉⠉⠉⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣆⣠⣤⣶⣾⣿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠘⣿⣿⣿⣶⣶⡖⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢘⠉⠉⠙⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⢄⡀⠀⡀⡂⠀⠀⠀⠄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⣤⣤⣴⣂⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⣴⡒⢓⠖⠂⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠩⣿⣿⡇⠉⠉⠉⠉⠙⠋⠉⠛⠛⣿⠛⠟⢛⢻⠛⠛⠿⠿⠿⢿⠿
⠀⠖⠶⣶⠟⢶⡶⠰⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢀⣀⠀⠍⠀⠍⠁⠈⠈⠲⠆⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠠⠀⢀⠀⠀⠀⠀⠀⠁⠀⠃⠀⡀⠀⠀⢀⡀⠀⠸⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢅⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣄⡀⣤⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⠁⠀⠀⢠⣄⣤⣧⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠚⠈⠀⠤⠶⠿⢟⣉⣁⣴⣶⣦⣶⡀⠀⠀⠀⠀⠀⠀⢠⣿⣼⡏⢹⡟⣿⣀⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⡟⣶⢀⣶⢲⣾⣿⡶⣦⡀⢸⣿⣿⣿⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠠⢾⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣾⡷⣿⠸⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⣿⣿⡷⣟⠃⣾⢻⣻⣿⣿⣷⠀⣸⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⢽⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠷⠟⠁⠻⠾⠟⠘⢿⣼⣿⣿⡏⣿⠰⣿⣿⣿⡏⠈⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠤⣤⣤⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⡀⣀⣀⠀⢀⣀⡀⠀⠀⠀⣿⣷⣆⣀⣀⣟⢷⣿⣇⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣴⢦⣄⠈⠙⠻⢿⣿⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⠿⣦⡀⠀⢸⣧⣿⠃⣿⣇⢀⣿⣿⡆⣿⡏⣿⠀⣿⣼⣿⣏⠉⠉⠉⢙⣯⡉⠉⠉⠙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣀⣀⣷⣨⣝⡶⣄⡀⢩⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠼⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣇⣿⠀⢹⣿⣿⣿⢲⠖⠢⢼⣿⣿⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⣿⡍⣿⠀⠀⠉⠛⣛⠺⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠉⠁⠛⠛⠋⠀⠘⢻⣿⣿⣾⣷⣾⣿⣿⣿⣿⣆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣬⠽⠛⠫⠞⣵⡶⠿⠛⠉⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠄⢀⣀⣀⣠⣀⣀⣀⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⣤⡶⠀⠀⠀⡈⠁⢀⣀⣤⣴⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⡇⠘⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡠⠦⠶⠿⠿⢉⠀⠈⠠⠂⠀⠀⠀⠀⢃⣀⠀⠀⠡⠀⢸⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠼⠵⡁⠐⢯⣿⣿⢿⡿⣿⣶⣿⣿⡉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⡀⠀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⣄⠀⠰⣀⠀⠤⠀⢾⣧⣿⣿⣵⣿⣿⣿⠀⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠀⠁⠀⠀⠀⠀⠠⠯⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣟⠀⠀⠀⠀⢀⢠⡢⣙⠮⠀⠀⠀⠀⠀⠀⠀⠌⠐⢠⡀⠈⢀⢙⣛⣿⣤⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣋⣤⣤⣤⠄⠀⠑⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠖⠋⠁⠀⠀⢀⡀⠀⣤⣤⣴⠶⠛⢿⣿⡿⠛⠛⠛⠛⠛⠀⢻⣿⣷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⣿⣿⡇⠁⠈⠋⠁⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⢻⣟⡃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠋⠉⠉⠙⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⣿⣂⣤⣤⣶⣾⡿⠿⠿⠟⠛⠛⠋⠉⠉⠁⡘⣿⣿⣿⣷⣶⠒⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣛⠉⠉⠙⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⡇⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠤⢄⡈⠀⡀⠂⠀⠀⠀⠄⠀⠀⡏⣿⣿⡇⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣤⣤⣤⣴⣒⣴⣦⣴⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣶⣤⡀⠀⢀⡭⣴⡚⠒⠖⠂⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠉⠉⠋⠉⠙⠋⠉⠛⠛⡿⠻⠿⢛⢿⠛⢛⠿⠿⠿⠿⠿
⠀⠖⠶⣶⠟⢶⠶⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢈⣂⠌⠩⠢⠁⠀⠄⠂⠶⠆⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠠⠐⢀⠀⠀⠀⠀⠀⠁⠀⠃⠀⡀⠀⠀⢀⡀⠀⢰⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠌⠱⢿⢟⣷⣿⣿⣿⣿⣿⣿⣿⣿⢇⠀⠀⠀⠀⠀⠉⠛⠿⣤⣤⡄⠀⣄⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠀⠀⠀⢠⣀⣴⣆⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠇⠈⢉⡠⠴⠿⠛⠋⠉⣄⣰⡤⡄⠀⠀⠀⠀⠀⠀⢠⣿⣼⡟⢻⡟⣿⣀⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⡟⣶⢀⣶⢲⣾⣿⡶⣦⡀⣸⣿⣿⣿⣧⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠶⣿⣿⣿⣿⣿⣷⠁⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣾⡷⣿⢸⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⣿⣿⡷⣟⠃⣟⢛⣿⣿⣿⣇⠀⣾⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠷⠟⠁⠻⠾⠟⠘⢿⣼⣿⣿⡏⣿⠠⣿⡟⣿⡏⠉⠁⣴⣿⣿⣿⣿⣿⣿⣿⣏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠠⢤⣬⣥⣿⣿⣿⣾⡿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⠀⣀⣀⠀⢀⣀⡀⠀⠀⠠⣿⣷⣆⣀⣰⣟⣿⣿⣃⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⡲⣤⣀⠉⠛⠻⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⠿⣦⡀⠀⢸⣧⣿⠃⣿⣅⢀⣿⣿⡆⣿⡏⣿⠀⣿⣼⣿⡍⠉⠉⠉⢹⣯⡙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣀⣀⣬⣶⣌⣛⠶⣄⡀⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣇⣿⠀⢹⣿⣿⡗⢲⠒⠠⣼⣿⣷⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣟⣿⣿⣿⣥⣹⡆⠀⠉⠛⡒⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠉⠁⠛⠛⠋⠀⠘⣻⣿⣷⣾⣶⣾⣿⣿⣿⣿⣄⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠥⢾⣭⠝⠛⠻⠟⣴⠿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⣀⡀⣀⣀⢀⣀⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢻⡇⢿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠒⠺⠿⠉⠀⠀⠀⠀⠀⠀⠀⠁⣠⠄⠈⠀⢸⠿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠼⠴⡑⠺⢽⣿⢻⢿⡟⣳⣶⣿⣿⠉⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠠⠠⠀⡀⠀⠐⡀⠠⠆⠀⢷⣤⣿⣿⣷⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠀⠁⠀⠀⠀⠀⠀⠀⠰⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠷⠀⠀⠀⢀⠪⡑⠬⠀⠀⠀⠀⠀⠀⠀⠈⠐⢠⠈⠈⢈⠙⣻⣿⣤⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠔⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣦⣤⠄⠈⠢⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠍⠉⠀⠀⣀⣠⣀⡐⢶⣾⠋⠉⠈⢿⡿⠁⠀⠀⠀⢾⣿⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⣿⣿⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠋⠉⠉⠙⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣂⣠⣤⣶⣾⣿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠹⣿⣶⣶⣶⣶⠒⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠉⠉⠛⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⡆⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣤⣤⣤⣴⣲⣶⣦⣴⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣶⣄⠲⠅⠲⠚⡐⠴⠂⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠉⠉⠋⠉⠛⠋⠙⠛⠛⠟⢻⠿⠛⢟⠟⠟⠿⠿⠿⡿⠿
⠀⠖⠶⣶⠏⢶⠖⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠁⠢⠁⠀⠀⠃⠴⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠠⠰⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⡀⠀⠀⢀⡀⠀⢰⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠌⠰⠟⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠉⠻⠿⣤⣤⡄⠀⣄⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠀⠀⠀⢠⣀⣰⣆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠸⠈⠈⣁⠤⠾⠟⠛⣉⢀⣠⡄⡄⠀⠀⠀⠀⠀⠀⢰⣿⣼⡟⢻⡟⣿⢀⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⠟⣶⢀⣶⢲⣿⣷⡶⣦⡀⣼⣿⣿⣿⣦⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠴⢾⣿⣿⣿⣿⣧⠁⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣾⡷⣿⢸⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⣿⣿⡷⣟⠃⣟⣛⣿⣿⣿⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢯⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠷⠟⠁⠻⠾⠟⠘⢿⣼⣿⣿⡏⣿⠰⣿⡟⣿⡏⠉⠁⣴⣿⣿⣿⣿⣿⣿⣿⣇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢤⣤⣥⣬⣿⣿⣧⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⠀⣀⣀⠀⢀⣀⡀⠀⠀⢰⣿⣷⣆⣀⣸⣟⣿⣿⣃⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠠⠲⢦⣄⡈⠙⠻⢿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⠿⣦⡀⠀⢸⣧⣿⠃⣿⣅⢀⣿⣿⡆⣿⡏⣿⠀⣿⣼⣿⡍⠉⠉⠉⢻⣏⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⢀⣠⣥⣮⣹⡳⢦⣀⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣇⣿⠀⢹⣿⣿⡗⢲⠐⠠⣼⣿⣷⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢟⣿⣿⣿⣏⡎⣿⠀⠈⠙⣓⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠈⠁⠛⠛⠋⠀⠘⣿⣿⣷⣾⣶⣾⣿⣿⣿⣿⡄⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢡⠾⢩⡽⠟⠛⠛⠏⡾⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⣀⡀⣀⣀⢀⣀⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣿⡏⠀⠀⠀⠀⠀⠀⠀⠠⣿⡟⢺⡇⣿⣿⢿⣿⣿⣿⡿⢀⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠶⠖⠚⠛⠏⠀⠀⠀⠀⠀⠀⠀⠈⣀⡆⠈⠄⡾⣿⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠼⠼⡑⠺⢽⣿⣳⢿⡿⢳⣾⣿⣿⠉⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠠⠰⠀⡀⠀⠐⡀⠠⠆⠀⢷⣤⣿⣿⣷⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠀⠁⠁⠀⠀⠀⠀⠀⠀⠴⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡿⠃⠀⠀⢀⠪⡓⠬⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠉⢈⠛⣻⣿⣤⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠅⠂⠀⠀⠀⠀⢀⣀⣤⢾⣿⣿⠿⠿⠿⠛⣫⣤⠄⠈⠢⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠟⠉⠀⠀⣠⣤⡤⡔⠻⡿⠋⠀⠀⠻⠟⠀⠀⠀⢿⣿⣾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠋⠉⠉⠙⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⣿⣿⣷⣂⣠⣤⣶⣾⣿⠿⠿⠿⠛⠛⠛⠉⠉⠉⢹⣾⣷⣶⣶⣶⠒⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠉⠉⠛⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⡆⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣤⣤⣤⣴⣢⣶⣦⣴⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣿⣿⣷⣦⠢⠍⠶⠚⡐⠴⠂⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠉⠉⠉⠉⠛⠋⠙⠛⠛⠟⠻⠿⠛⠟⠛⠟⠿⠿⠿⡿⠿
⠀⠖⠶⣶⠏⣶⠶⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⡀⠀⠈⠁⠢⠁⠀⠀⠃⠶⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠠⠰⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⡀⠀⠀⢀⡄⠀⢡⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠌⠰⠟⡻⣪⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠻⠿⣤⣤⡄⠀⣄⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡉⠀⠀⠀⢠⣠⣰⣆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠐⠁⠃⠉⠤⠴⠿⠛⠉⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣼⡟⢻⡟⣿⢀⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⠟⣶⢀⣶⢲⣿⣷⡶⣦⡀⣿⣿⣿⣿⣆⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠠⠶⣿⣿⣼⣿⣧⠁⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣾⡷⣿⢸⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⣿⣿⡷⣟⠃⣟⣛⣿⣿⣿⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⡯⡏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠷⠟⠁⠻⠾⠟⠘⢿⣼⣿⣿⡏⣿⢰⣿⡟⣿⡏⠉⠀⣴⣿⣿⣿⣿⣿⣿⣿⣇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠤⣤⣭⣤⣼⣿⣷⣹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⠀⣀⣂⠀⢀⣀⡀⠀⠀⢸⣿⣷⣆⣀⣸⣟⣿⣿⣁⣀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠠⡒⠦⣄⣀⠉⠛⠿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⢿⣦⡀⠀⢸⣧⣿⠃⣿⣅⢀⣿⣿⡆⣿⡏⣿⠀⣿⣼⣿⡍⠉⠉⠉⢻⣍⠉⠉⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡀⢀⣠⣬⣴⣉⣟⡦⣄⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣇⣿⠀⢹⣿⣿⡗⢲⠒⠠⣼⣿⣷⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⣟⣿⣿⣿⣿⣱⣹⡇⠀⠉⣓⡓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠈⠁⠛⠛⠋⠀⠘⣿⣿⣷⣾⣶⣾⣿⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠃⡴⠯⣽⠿⠏⠛⠚⠳⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⣀⣀⡀⣀⣀⢀⣀⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⠀⠀⠀⠐⢀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⣀⣤⣀⣠⣴⣾⡟⠀⠀⠀⠀⠀⠀⠀⠠⡿⣟⢸⡃⣿⣿⢿⣿⣿⣿⡿⢠⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡰⠖⠒⠚⠛⠃⠀⠀⠀⠀⠀⠀⠀⢁⣠⠈⢴⣶⣿⠿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⢀⣀⠠⠾⠼⠀⠺⣽⣿⣷⣿⡿⢳⣾⣿⣿⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣫⠀⠀⠀⠀⠠⠰⠀⡀⠀⢐⡀⠠⠆⠀⣵⣤⣿⣯⣷⣿⣿⣿⠈⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠚⠶⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⡋⠀⠀⢄⠮⡓⠤⠀⠀⠀⠀⠀⠀⠐⠈⠀⣄⠀⠉⣈⢛⣻⣿⣤⡀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠋⠀⠀⠀⠀⣀⣠⣴⠾⢿⣿⡟⠛⠛⢛⣩⣤⠄⠈⠢⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠁⠀⠀⢤⣶⡖⠌⠹⠟⠁⠀⠀⠙⠃⠀⠈⢿⣿⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠋⠉⠉⠙⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⣿⣷⣂⣤⣴⣶⣾⣿⠿⠿⠟⠛⠛⠋⠉⠉⠉⢹⣿⣷⣶⣶⣶⢲⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡋⠉⠉⠛⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⡄⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠐⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⣀⠀⠀⠠⣀⠀⠀⢀⠂⠀⠀⠀⠀⠀⢀⣿⣿⣿⣧⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣤⣤⣤⣔⣠⣶⣶⣶⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣦⠀⠁⠶⠊⡲⠔⠊⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡋⠉⠉⠉⠉⠙⠋⠙⠛⢻⠟⠻⠻⠛⡟⠛⠟⠿⠿⠿⡿⠿
⠀⠖⠶⣶⠟⣶⠶⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⣿⣿⣿⣿⣿⣿⡿⠄⠀⠉⠁⠢⠁⠀⠁⠓⠶⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡁⠀⠀⠠⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡀⠀⠀⣀⠀⠀⢡⠀
⠀⠀⠀⠀⠀⠀⠂⠀⣿⣿⠌⠰⠟⢻⢾⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠉⠻⠷⣤⣤⡄⠀⣄⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡉⠀⠀⠀⢠⣠⣰⣄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠇⠋⢉⢡⠤⠾⠝⠋⠁⠀⣄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣼⡟⢻⡟⣿⢀⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⡟⣶⢀⣶⢲⣿⣷⡶⣦⡀⣿⣿⣿⣿⣆⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠴⢾⡟⣸⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣾⡷⣿⢸⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⣿⣿⡷⣟⠃⣟⣛⣿⣿⣿⣆⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠷⠟⠁⠻⠾⠟⠘⢿⣼⣿⣿⡏⣿⢰⣿⡟⣿⡏⠉⠀⣴⣿⣿⣿⣿⣿⣿⣿⣇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠤⣤⣭⣥⣤⣿⣷⣑⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⠀⣀⣂⠀⢀⣀⡀⠀⠀⢸⣿⣿⣆⣀⣸⣟⣿⣿⣁⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠠⡒⠦⣤⣀⠉⠙⠻⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⠿⣦⡀⠀⢸⣧⣿⠃⣿⣅⢀⣿⣿⡆⣿⡏⣿⠀⣿⣼⣿⡍⠉⠉⠉⢳⣟⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣀⢀⣀⣬⣷⣎⣝⡶⣄⡀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠸⣷⣿⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣇⣿⠀⢹⣿⣿⡗⢰⠀⢠⣼⣿⣷⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠾⡿⢻⣿⣿⣿⡧⣎⡇⠀⠉⣓⡓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠈⠁⠛⠛⠋⠀⠘⣿⣿⣷⣾⣶⣾⣿⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠴⠺⢽⡿⠟⠉⠓⠛⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⣀⣀⡀⣀⣀⢀⣀⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡀⠀⠀⠀⠐⢀⠖⠀⠀⠀⠀⢀⣀⠀⠠⠀⠀⠀⠠⡆⠀⢀⠀⠀⠀⣀⣀⠀⢀⣠⣴⡿⠁⠀⠀⠀⠀⠀⠀⢠⣿⡿⢸⡃⣿⢿⢿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠁⠀⠀⠨⠀⢀⡶⠒⠒⠋⠛⠃⠀⠀⠀⠀⠀⠀⠁⣀⠈⢶⣶⣿⣿⣿⣿⣿⣿⣿⣃⠀⠀⠀⠀⠀⠀⢀⣀⠠⠾⠼⠀⠺⣽⣿⣷⣿⡿⢳⣾⣿⣿⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠀⠀⢸⣿⣿⣿⣿⣯⠄⠀⠀⠀⠀⠰⠀⡀⠀⠐⡀⠠⠆⠀⣷⣤⣿⣯⣷⣿⣿⣿⠈⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⠤⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⡇⠀⠀⢀⠮⡳⠥⠀⠀⠀⠀⠀⠀⠰⠈⠀⣄⠉⠉⣈⠛⣻⣷⣄⡀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⣤⠴⠊⠀⠀⠀⣀⣠⣤⡶⠛⢿⣿⠟⠙⣉⣩⣥⠄⠈⠢⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠀⠀⠀⢴⣶⡖⠎⠉⠋⠀⠀⠀⠉⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠋⠉⠉⠹⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⣿⣷⣂⣠⣤⣶⣾⣿⠿⠿⠿⠛⠛⠛⠉⠉⠉⢹⣶⣶⣶⣶⣶⢲⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡋⠉⠙⠛⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⣿⡄⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⣀⠀⠀⠀⣄⠀⠀⠠⣐⠀⠀⢒⠀⠀⠀⠠⠀⠀⠀⣿⣿⣿⣇⣿⣿⣿⣿⣷⣶⣶⣷⣶⣶⣤⣤⣤⣴⣢⣶⣦⣶⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣧⠀⠉⠶⠊⡰⠔⠈⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠉⠉⠉⠉⠛⠋⠛⠛⢻⡟⢻⠻⠛⡟⠟⠿⠿⠿⠿⡿⠿
⠀⠖⠶⣶⠟⢷⠶⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠿⣿⣿⣿⣿⣿⡿⠂⠀⠈⠁⠢⠁⠀⠁⠓⠴⠀⠀⠀⠀⠀⠀⠂⢸⣿⣿⠁⠀⠠⠠⡀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⣀⡄⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠂⠀⣿⣧⡌⠶⠟⣻⢪⣾⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⠙⠻⠷⣤⣤⡄⠀⣤⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡉⠀⠀⠀⣤⣠⣼⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠐⠇⠋⠉⠠⠴⠯⠑⡉⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣼⡟⢻⡟⣿⢀⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⡟⣶⢀⣶⢲⣿⣷⡶⣦⡀⣿⣿⣿⣿⣆⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠖⢟⡟⢼⣿⡅⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣾⡷⣿⢸⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⣿⣿⡷⣟⢃⡟⣟⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⣿⣿⡿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠷⠟⠁⠻⠾⠟⠘⢿⣾⣿⣿⡏⣿⢸⣿⣿⣿⡉⠉⢠⣾⣿⣿⣿⣿⣿⣿⣿⣇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠐⠦⣤⣬⣥⣴⣿⣧⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⡀⣀⣂⠀⢀⣀⡀⠀⠀⢸⣿⣾⣀⣀⣸⡟⣿⣿⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠴⠓⠶⢤⣄⡈⠙⠻⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⠿⣦⡀⠀⢸⣧⣿⠣⣿⣅⢀⣿⣿⡆⣿⡏⣿⠀⣿⣼⣿⡍⠉⠉⠉⢻⣍⠉⠛⠛⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⣠⣀⣠⣤⣾⣦⣭⡳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣇⣿⠀⢹⣿⣿⡗⢰⠒⠠⣿⣿⣷⡀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠿⡟⢻⣿⣿⣿⣿⣶⣻⡀⣈⣙⡓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠈⠁⠛⠛⠋⠀⠘⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣰⠞⠋⠾⠟⠋⠉⠉⠙⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⣀⣀⣀⣀⣀⣀⣀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡸⠃⠀⠀⠀⣀⣀⡤⠀⠂⠀⠀⠀⠂⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢸⡀⣿⢿⢿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠁⠁⠀⠀⠀⠈⠀⠰⠛⠛⠉⢉⠉⠁⠀⠀⠀⠀⠀⠀⢴⡐⢤⣶⣿⣿⣿⣧⣠⣾⣿⣤⠀⠀⠀⠀⠀⠀⠀⡀⠠⠾⢬⠀⠺⣽⣿⡷⣿⣿⢳⣾⣿⣿⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠐⢽⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠐⠀⣀⠀⢀⡀⠠⠄⠠⣵⣴⣿⣯⣶⣿⣿⡟⠈⠀⠐⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠓⠒⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡿⠇⠀⠀⢔⠮⡳⠥⠀⠀⠀⠀⠀⠁⠰⠈⠀⣄⠀⠁⡈⢛⣻⣷⣤⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⡶⠟⠁⠀⠀⠀⢤⣶⣾⠛⠉⢻⡏⢀⣤⣤⡤⠄⠈⠢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠘⣿⣯⠃⠀⠁⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠙⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⣿⣷⣀⣤⣴⣶⣾⡿⠿⠿⠟⠛⠛⠋⠉⠉⠁⢻⣿⣿⣿⣷⣾⢲⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠙⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣋⠉⠉⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⣄⠀⠀⠀⣄⠀⠀⠤⣔⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣷⣤⣤⣤⣆⣶⣶⣶⣦⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⣐⣿⣿⣿⣿⣿⣿⣇⠀⠁⢶⠘⡲⠴⠈⠄⡀⠀⠀⠀⠀⠀⠀⠀⠁⢹⣿⣿⠉⠉⠙⠉⠉⠋⠉⠉⠛⢻⡟⠻⡻⠛⡟⠛⠛⠿⠿⠿⡿⠿
⠀⠖⠶⣶⠟⣷⠶⠰⠶⠶⠖⠀⠀⠀⠀⣘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢿⣿⠿⠿⡯⠂⠀⠈⠁⠢⠉⠀⠁⠑⠶⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠠⡀⠀⠀⠀⠀⠈⠉⠀⠀⢂⡀⠀⠀⣀⠄⠀⢃⡀
⠀⠀⠀⠀⠀⠀⠂⠀⣿⣧⡌⠶⠟⣻⢪⣾⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⠙⠻⠷⣤⣤⡄⠀⣤⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡉⠀⠀⠀⣤⣠⣼⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠐⠇⠋⠉⠠⠴⠯⠝⠉⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣼⡟⢻⡟⣿⢀⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⡟⣶⢀⣶⢲⣿⣷⡶⣦⡀⣿⣿⣿⣿⣆⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠖⢟⠟⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣾⡷⣿⢸⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⣿⣿⡷⣟⢃⡟⣟⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⣿⣿⡿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠷⠟⠁⠻⠾⠟⠘⢿⣾⣿⣿⡏⣿⢸⣿⣿⣿⡉⠉⢠⣾⣿⣿⣿⣿⣿⣿⣿⣇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠐⠦⣤⣬⣥⣴⣿⣧⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⡀⣀⣂⠀⢀⣀⡀⠀⠀⢸⣿⣾⣀⣀⣸⡟⣿⣿⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠴⠓⠶⢤⣄⡈⠙⠻⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⠿⣦⡀⠀⢸⣧⣿⠣⣿⣅⢀⣿⣿⡆⣿⡏⣿⠀⣿⣼⣿⡍⠉⠉⠉⢻⣍⠉⠛⠛⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⣠⣀⣠⣬⣾⣦⣭⡳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣇⣿⠀⢹⣿⣿⡗⢰⠒⠠⣿⣿⣷⡀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⠿⣻⢻⣿⣿⣿⣿⣶⣻⡀⣈⣉⡓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠈⠁⠛⠛⠋⠀⠘⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣠⠞⠋⠾⠟⠃⠉⠉⠹⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⣀⣀⣀⣀⣀⣀⣀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠸⠃⠀⠀⠀⣀⣀⡤⠀⠀⠀⠀⠀⠆⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⡀⣿⢿⢿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠁⠁⠀⠀⠀⠈⠀⠰⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢴⡐⢤⣶⣿⣿⣿⣧⣠⣾⣿⣤⠀⠀⠀⠀⠀⠀⠀⡀⠠⠾⢬⠀⠺⣼⣿⡷⣿⣿⢳⣾⣿⣿⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠐⢽⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠐⠀⣀⠀⢀⡀⠠⠄⠠⣷⣴⣿⣯⣶⣿⣿⡏⠈⠀⠐⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠒⠂⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡿⠇⠀⠀⢔⠮⡳⠥⠀⠀⠀⠀⠀⠁⠰⠈⠂⣄⠀⠁⡈⢛⣻⣷⣤⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⡶⠟⠉⠀⠀⠀⢤⣶⣾⠋⠉⢻⡏⢀⣤⣤⡤⠄⠈⠢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⣿⣯⠃⠀⠁⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠙⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⣿⣷⣀⣤⣴⣶⣾⡿⠿⠿⠟⠛⠛⠋⠉⠉⠁⢻⣿⣿⣿⣷⣾⢲⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠙⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣋⠉⠉⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⣄⠀⠀⠀⣄⠀⠀⠤⣔⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣷⣤⣤⣤⣆⣶⣶⣶⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣇⠀⠁⢶⠚⡲⠴⠈⠄⡀⠀⠀⠀⠀⠀⠀⠀⠁⢹⣿⣿⠉⠉⠙⠉⠉⠉⠉⠉⠛⢻⡟⠻⡻⠛⡟⠛⠛⠿⠿⠿⡿⠿
⠀⠖⠶⣶⠟⣷⠶⠰⠶⠶⠖⠀⠀⠀⠀⣘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢿⠿⠿⠿⡿⠀⠀⠈⠁⠢⠉⠀⠁⠑⠶⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⠀⠀⠀⠠⡀⠀⠀⠀⠀⠈⠉⠀⠀⢂⡀⠀⠀⣀⠄⠀⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⡍⠾⢟⢟⣽⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠉⠻⠷⣤⣤⡄⠀⣤⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡉⠀⠀⠀⣤⣠⣼⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠘⠘⠉⠀⠤⠖⠃⣁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣼⡟⢻⡟⣿⢀⣿⠹⣿⢸⣿⣾⠇⠀⢸⡟⣿⡆⣴⡟⣶⢀⣶⢲⣿⣷⡶⣦⡀⣿⣿⣿⣿⣆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠃⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣾⡷⣿⢸⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⣿⣿⡷⣟⢃⣟⣟⣿⣿⣿⣄⢀⣿⣿⣿⣿⣿⣿⣿⡿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣰⢼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠷⠟⠁⠻⠾⠟⠘⢿⣾⣿⣿⡏⣿⢸⣿⣿⣿⡁⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠒⠶⢶⣶⣶⣾⣯⣮⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⡀⣀⣂⠀⢀⣀⡄⠀⠀⢸⣿⣿⣂⣀⣸⡛⣾⣿⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠐⠞⠓⠶⠦⣄⣀⠉⠛⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⠿⣦⡀⠀⢸⣧⣿⠣⣿⣅⢀⣿⣿⡆⣿⡏⣿⠀⣿⣼⣿⡍⠉⠉⠉⢻⣍⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣁⣤⣠⣤⣶⣾⣷⠾⣗⠦⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠸⣧⣿⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣇⣿⠀⢹⣿⣿⡗⢰⠒⠠⣿⣿⣷⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⢛⠟⣻⣿⡿⢿⣿⣿⣟⣄⣠⣭⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠈⠁⠛⠛⠋⠀⠘⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠁⡴⠟⠃⠚⠛⠉⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⢀⢀⠠⠁⣄⣀⣀⣄⣀⣀⣀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⡴⠊⢀⢀⣀⣠⣤⡤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⡀⣿⢿⢿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠋⠉⠁⢀⡀⠈⠀⠀⠀⠀⠀⠀⠾⠂⣴⣶⠿⢿⣻⣷⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⡀⠠⠾⢼⠀⠺⣬⣿⡷⣿⣿⣳⣾⣿⣿⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣒⠋⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⢾⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠐⠀⣀⠀⢄⡀⠠⠄⠠⣷⣴⣿⣯⣶⣿⣿⡏⠈⠀⠐⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠀⠀⠀⠀⣂⡀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⠿⠁⠀⡀⢔⠮⡳⠥⠀⠀⠀⠀⠀⠁⠰⠁⠀⣆⠀⠁⡈⣛⣻⣷⣤⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢿⣿⠟⠁⠀⣀⣀⡐⠲⣿⠋⠀⠹⠇⢠⣤⣤⡤⠄⠈⠢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠃⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡟⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⡂⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠋⠉⠉⠉⠘⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣀⣤⣴⣶⣿⡿⠿⠿⠛⠛⠛⠉⠉⠉⠀⢹⣿⣿⣿⣿⣿⢳⣾⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠙⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣋⠉⠉⠙⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣶⣶⣦⠀⠀⠀⣄⡀⠀⠤⣄⠀⠀⢀⠂⠀⠀⠠⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⣷⣤⣤⣤⣶⣶⣶⣶⣦⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⡀⠇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠒⠚⡲⠴⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠁⢹⣿⣿⠉⠉⠙⠉⠉⠋⠉⠉⠉⢹⠟⠻⡻⠛⡟⠛⠛⠿⠿⠛⡿⠿
⠀⠖⠶⣶⠟⣷⠶⠰⠶⠶⠖⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⠿⠿⡧⠀⠀⠀⠁⠠⠁⠀⠁⠑⠶⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⠀⠀⠂⠰⠀⠀⠀⠀⠀⠀⠈⠀⠀⢂⠀⠀⠀⣀⠄⠀⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡉⠾⣿⢯⣾⣿⡿⠿⣿⣿⡿⠄⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣤⣤⡄⠀⣤⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡉⠀⠀⠀⣠⣠⣼⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠓⠚⠁⠠⠴⠚⠒⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣼⡟⢻⡟⣿⢀⣿⠹⣿⢸⣿⣾⠇⠀⢸⡟⣿⡆⣴⡟⣶⢀⣶⢲⣿⣷⡶⣦⡀⣿⣿⣿⣿⣦⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠐⠛⠿⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣾⡷⣿⢸⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⣿⣿⡷⣟⠃⣟⣟⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⡿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠷⠟⠁⠻⠾⠟⠘⢿⣾⣿⣿⡏⣿⢸⣿⣿⣿⡇⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠈⠒⠶⢶⣶⣶⣾⣿⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⡀⣀⣂⠀⢀⣀⡄⠀⠀⢸⣿⣿⣂⣀⣸⡟⣿⣿⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠐⠟⠓⡶⠦⣤⣀⠉⠛⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⠿⣦⡀⠀⢸⣧⣿⠣⣿⣅⢀⣿⣿⡆⣿⡏⣿⠀⣿⣼⣷⡍⠉⠉⠉⢻⣯⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣤⣄⣠⣴⣶⣿⣷⠾⣗⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠸⣧⣿⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣇⣿⠀⢹⣿⣿⡗⢰⠒⢠⣾⣿⣷⡄⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⢿⠟⣿⣿⣿⣿⣝⠿⢟⣤⣤⣭⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠈⠁⠛⠛⠋⠀⠘⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁⣶⡛⠋⠛⠛⠉⠀⠀⠀⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠁⣄⣠⣀⣄⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠼⠊⢀⣀⣀⣤⣴⣶⠂⠀⠐⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⢸⢿⣟⢸⡂⣿⣿⢿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠋⠉⠀⢀⡀⠀⠁⠀⠀⠀⠀⠀⠺⠁⣴⣾⣿⣿⣻⣷⣶⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⡀⠠⠾⠼⠀⠺⣾⣿⣿⣿⣟⣳⣾⣿⣿⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠐⠀⣀⠀⢀⡀⠠⠆⠠⢷⣤⣿⣯⣶⣿⣿⡟⠈⠀⠐⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠀⠀⠀⠀⢀⣠⡾⣿⡿⢿⡿⠿⠀⠀⡀⢔⠪⡑⠤⠀⠀⠀⠀⠀⠁⠐⠈⠀⣦⠀⠁⣈⣙⣻⣷⣤⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⠟⠋⠀⣀⣤⣐⠺⠿⠋⠀⠙⠇⢠⣤⣤⡤⠄⠈⠢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠎⠀⠀⠀⠀⢻⣷⠁⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⡂⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠋⠉⠉⠉⠘⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⣿⣿⣂⣤⣴⣶⣿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠀⣹⣿⣿⣿⣿⣿⢻⣿⣷⣿⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠙⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣋⠉⠉⠙⠛⠻⠟⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣷⣶⡀⠀⠀⣄⠀⠄⠤⣄⡀⠀⢀⠂⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣷⣤⣤⣤⣶⣶⣶⣾⣷⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠠⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠐⠘⡲⠶⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠋⠉⠉⠋⠉⠉⠉⠉⠉⢹⡟⠻⡻⢛⡟⠛⠛⠿⠿⠛⡟⠿
⠀⠖⠶⣶⠟⣷⠶⠰⠶⠶⠖⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠙⠛⠿⠟⠛⡡⠀⠀⠀⠁⠠⠉⠀⠁⠐⠶⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠠⠐⠁⠀⠀⠀⠀⠉⠀⠀⢀⡀⠀⠀⣀⠄⠀⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡍⠾⡿⣱⣿⣿⡿⢿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣤⣤⡄⠀⣤⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡉⠀⠀⠀⣠⣠⣼⣆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠘⠓⠋⠁⠤⠶⠛⢊⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣼⡟⢻⡟⣿⢀⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⡟⣶⢀⣶⢲⣿⣷⡶⣦⡀⣿⣿⣿⣿⣧⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠚⠋⠏⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣾⡷⣿⢸⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⣿⣿⡷⣟⠃⣟⣟⣿⣿⣿⣇⠀⣿⣿⣿⣿⣿⣿⣿⡿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡀⠀⢀⣀⣀⠀⢀⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠷⠟⠁⠻⠾⠟⠘⢿⣾⣿⣿⡏⣿⢸⣿⣟⣿⡇⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠑⠲⠶⣶⣶⣶⣿⣷⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⡀⣀⣂⠀⢀⣀⡄⠀⠀⢸⣿⣿⣂⣀⣸⡟⣿⣿⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠲⠛⢒⡶⠤⣄⡈⠙⠻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⠿⣦⡀⠀⢸⣧⣿⠣⣿⣅⢀⣿⣿⡆⣿⡏⣿⠀⣿⣼⣷⡍⠉⠉⠉⢻⣭⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⣠⣄⣤⣶⣾⣿⡶⠿⡗⠦⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣇⣿⠀⢹⣿⣿⡗⢰⠒⢠⣾⣿⣷⡄⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⡻⠟⣿⣿⢿⣿⣝⠿⢗⣤⣤⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠈⠁⠛⠛⠋⠀⠘⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠠⠈⠁⡶⡋⠃⠛⠋⠉⠀⠀⠀⠙⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠅⣄⣠⣀⣀⣀⣀⣀⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⠺⠊⣀⣀⣠⣤⣶⣖⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⢸⡂⣿⣿⢿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠋⠉⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠺⠠⣴⣶⣿⣿⣛⣶⣶⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠾⠼⠐⠺⢼⣿⣷⣿⣟⣳⣾⣿⣿⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠰⠀⣀⠀⢀⡀⠠⠆⠀⢷⣤⣿⣯⣶⣿⣿⣿⠈⠀⠐⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠀⠀⠀⠀⣀⣤⡾⣿⡿⢿⠿⠋⠀⠀⠀⢔⠪⡓⠤⠀⠀⠀⠀⠀⠀⠐⠈⠀⣦⠀⠁⣈⣙⣻⣿⣤⡀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⡿⠋⠀⢀⣤⣄⠻⠿⠃⠀⠙⠇⢀⣤⣤⡤⠄⠈⠢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠜⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠈⢿⣿⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣟⡃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠋⠉⠉⠉⠘⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣂⣤⣴⣶⣿⡿⠿⠿⠛⠛⠛⠉⠉⠉⠀⣹⣿⣿⣿⣿⣿⢻⣿⣷⣿⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠙⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣋⠉⠉⠉⠛⠻⠟⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⡆⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢄⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣷⣶⡀⠀⠀⣄⡀⠄⡤⣄⡀⠀⢀⠂⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣤⣴⣦⣶⣶⣶⣾⣷⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠡⠘⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠐⠚⡲⠶⠀⠄⡀⡐⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⠟⠻⡻⢛⡟⠛⠛⠿⠿⠛⡟⠿
⠀⠖⠶⣶⠟⣷⠶⠰⠶⠶⠖⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠝⠉⠛⠻⠛⠋⡡⠂⠀⠀⠀⠢⠁⠀⠁⠃⠶⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠰⢀⠀⠀⠀⠀⠀⠉⠀⠀⢀⡀⠀⠀⣀⡄⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡍⣾⣟⣾⣿⣿⠿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣤⣤⡄⠀⣤⣀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡉⠀⠀⠀⣠⣠⣼⣆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠘⠘⠋⠠⠴⠞⠓⣉⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣼⡟⢻⡟⣿⢀⣿⠹⣿⢸⣿⣾⠃⠀⢸⡟⣿⡆⣴⡟⣶⢀⣶⢲⣿⣷⡶⣦⡀⣿⣿⣿⣿⣆⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠛⠽⠅⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣾⡷⣿⢸⣿⢰⣦⢸⡟⣿⡀⠀⣼⡇⣿⡇⣿⢠⣿⢸⣿⢸⣿⣿⡷⣟⠃⣟⣟⣿⣿⣿⣧⠀⣿⣿⣿⣿⣿⣿⣿⡿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⠀⠀⣀⣀⡀⠀⣰⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠉⠀⠙⠛⠁⠘⠃⠛⠃⠀⠛⠷⠟⠁⠻⠾⠟⠘⢿⣾⣿⣿⡏⣿⢸⣿⣟⣿⡇⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠈⠒⠶⢶⣶⣶⣾⣟⣦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⢶⡄⠀⢠⣤⣤⠀⣤⣤⡀⣀⣂⠀⢀⣀⡄⠀⠀⢸⣿⣿⣆⣀⣸⡟⣿⣿⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠞⠓⠲⠶⢤⣄⡈⠙⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⠿⣦⡀⠀⢸⣧⣿⠣⣿⣅⢀⣿⣿⡆⣿⡏⣿⠈⣿⣼⣿⡍⠉⠉⠉⢻⣯⡉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣤⣀⣴⣶⣿⣷⡶⢿⡲⢴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢸⣧⣽⠇⠀⣿⡇⣿⠀⣿⣀⢸⡿⣿⡇⣿⣇⣿⠀⢹⣿⣿⡗⢰⠒⢠⣾⣿⣷⡄⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⡿⢟⣿⣿⣿⣿⡝⠷⢷⣤⣤⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠈⠁⠛⠛⠋⠀⠘⣿⣿⣿⣾⣶⣾⣿⣿⣿⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠊⢀⡾⠛⠋⠛⠋⠁⠀⠀⠈⠉⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠅⣄⣠⣀⣀⣀⣀⣀⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⡀⠀⠀⠀⣰⠟⠋⣀⣀⣠⣤⣶⣖⡒⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⢸⡃⣿⣿⢿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠋⠉⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠛⢠⣴⣶⣿⣿⣷⣶⣾⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠾⠼⠀⠺⣿⣿⣿⣿⣟⣳⣾⣿⣿⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣴⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠰⠀⣀⠀⢀⡀⠠⠆⠀⢷⣤⣿⣯⣶⣿⣿⣿⠈⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⡤⠂⠀⠀⠀⣀⣴⡿⣿⡿⠿⠟⠃⠀⠀⠀⢄⠪⡑⠤⠀⠀⠀⠀⠀⠀⠐⠈⠀⣦⠀⠁⣈⣙⣻⣿⣤⡀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⡿⠋⠀⠠⣤⣬⠙⠻⠃⠀⠙⠃⢀⣤⣤⡤⠄⠈⠢⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠾⠁⠀⠀⠀⠈⣿⡎⠀⠀⠀⠀⠀⠈⢿⣿⣄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠇⠀⠀⠀⠀⠀⠀⠈⢿⣟⡃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠋⠉⠉⠉⠘⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣽⣿⣿⣂⣤⣴⣶⣿⡿⠿⠿⠛⠛⠛⠉⠉⠉⠀⣹⣿⣿⣿⣿⣿⢻⣿⣿⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠙⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣋⠉⠉⠙⠛⠻⠟⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣧⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠂⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣯⣿⣿⡆⣠⣤⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢄⡀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣶⡄⠀⠀⣄⠀⠀⡤⣄⡀⠀⢀⠂⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣦⣶⣦⣶⣶⣶⣾⣯⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠎⠁⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠐⠘⡲⠶⠀⠄⡀⡐⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⡟⠻⡻⢛⡟⠛⠛⠿⠿⠛⡟⠿
⠀⠖⠶⣶⠟⣷⠶⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠂⠀⠀⠀⠀⠈⠙⠟⠛⠙⠛⠛⢉⡡⠀⠀⠀⠀⠠⠁⠂⠁⠂⠶⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠰⢀⠀⠀⠀⠀⠀⠉⠀⠁⢂⣀⠀⠀⣀⡄⠀⢱⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠉⠉⠉⠛⢛⠋⠉⠙⠛⠛⠉⠉⠉⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠤⢔⣒⡒⠤⣭⣟⠁⠴⢂⢀⠤⠄⠙⣿⡤⠤⣒⣂⡠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⠉⠶⣚⡭⣷⣿⡷⣞⣷⣧⢿⣷⣿⡯⣷⢯⡗⠂⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿
⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠈⠘⠫⠟⠛⠈⣹⣿⡎⠀⠉⠛⠉⠁⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⠀⠀⠙⠿⣿⡿⠟⠿⠛⠛⠉⠛⠛⠿⠿⠿⠿⠿⠿⣿⠿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠉⠗⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⠈⠀⠀⠀⠈⠛⠟⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⣀⣀⡀⠀⢀⣀⣀⣠⣄⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣤⣤⠀⠀⠛⠿⠿⠿⠿⠿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⠿⡫⠀⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠀⠀⠀⢀⢈⠉⠹⠁⠉⠉⠉⣉⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠑⢒⣂⡭⢭⣙⣿⣏⡀⢚⡤⡠⣒⠀⣸⣿⢟⣣⠭⣄⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠛⢭⢖⣽⣿⡿⣿⣿⣧⢯⠿⣿⣾⣝⡾⠍⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⢉⡛⠛⠛⠛⣿⣿⣿⣿⣿
⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠉⠚⠋⠁⢠⣻⣿⢂⠀⠈⠉⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠈⠁⠀⠀⠸⣿⣿⣿⣿
⠀⠀⠙⠻⠿⠟⠛⠛⠉⠉⠉⠙⠛⠛⠛⠛⠛⠻⢿⡿⠟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠚⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣠⣤⣤⣄⣀⣠⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠶⣶⣶⣦⠀⠀⠙⠛⠛⠛⠻⠿⠟⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠷⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢿⣿⣿⣿⣿⣿⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⣿⣿⣿⣿⣿⣿⢿⡟⢹⡇⢸⠁⣿⠥⠄⢻⠀⣥⠈⣧⠄⣠⣿⣿⡇⣿⠙⡏⣿⣿⡏⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡀⠸⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣄⣇⣸⣇⣸⣀⣧⣘⣅⣸⣀⣿⣀⣏⣀⣛⣿⣿⣧⡻⣤⣧⣿⣿⣷⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⡿⠇⠀⠹⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣧⠈⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠙⢿⣿⠋⠀⠘⢿⡿⠟⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣄⠀⠀⠀⠂⠀⠀⠠⠀⢀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⡆⡎⡦⠀⠀⠐⢰⠀⣸⡇⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⡇⠂⣀⠀⢸⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣤⡇⡈⢉⡄⢸⣦⡿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣾⠿⣧⡀⠛⢀⡀⠻⣿⣧⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢁⣤⠈⢻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣛⣛⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠇⡄⢹⡇⢸⡇⣿⣿⠛⡛⢻⣟⡛⢻⣟⡛⢻⡗⢺⣿⢧⡟⢳⠀⡟⣷⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡟⢠⣥⠈⡇⢸⡇⢻⣿⠘⡏⢸⠉⠖⠸⡟⠠⠿⡇⢸⣿⡼⡇⢸⠀⡇⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣫⣟⣯⣵⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠀⢸⣿⣿⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⢿⡿⢻⡇⢸⠉⣿⠥⠄⢻⠉⣥⠈⣧⠄⣠⣿⣿⡇⣿⠀⡇⡿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⡆⠘⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣏⣿⣿⣄⣏⣸⣇⣸⣀⣧⣘⣅⣸⣁⣿⣀⣏⣀⣛⣿⣿⣧⡻⣤⣧⣧⡽⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠷⠈⠘⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣀⡀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⣿⣿⣿⣷⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠙⢿⣿⠋⠀⠘⢿⡿⠟⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣄⠀⠀⠀⠂⠀⠀⠠⠀⢀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⡀⣎⣦⢀⠀⠀⢰⠀⣸⡇⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⡇⠂⣀⠈⢺⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⠀⣤⡇⡈⢉⡄⢸⣦⡿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣾⠿⣧⡀⠛⢀⡀⠻⣿⣧⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢁⣤⠈⢻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣛⣻⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠇⡄⢻⡇⢸⡇⣿⣿⠛⡛⢻⣟⡛⢻⣟⡛⢻⡗⢺⣿⢣⡟⢻⠘⡟⣷⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡟⢠⣥⠈⡇⢸⡇⢹⣿⠈⡏⢸⠉⠖⠸⡟⠠⠿⡇⢸⣿⡼⡇⢸⠀⡇⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣫⣟⣯⣵⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⡀⠀⢘⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣽⡿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⡇⠀⠈⣿⣿⣿⣿⣿⡿⠟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⡿⢿⡏⢹⠁⣿⠥⠄⢻⠀⣭⠈⣧⠄⣠⣿⣿⡇⣿⠀⡇⡏⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⠀⢻⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣇⣼⣿⣄⣏⣸⣇⣸⣀⣧⣘⣅⣸⣀⣿⣀⣏⣐⣛⣿⣿⣧⡻⣴⣧⣧⡼⣻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠄⠉⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⣄⡀⠸⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠙⢿⣿⠋⠀⠘⢿⡿⠟⠠⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣄⠀⠀⠀⠂⠀⠀⠠⠀⢀⡀⢠⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⠀⡌⣦⢀⠀⠀⢰⠀⣸⡇⢸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⡇⠂⣀⠈⣺⠇⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣤⡇⡈⢉⡄⢸⣦⡿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣾⠿⣧⡀⠛⢀⡀⠻⣿⣧⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢁⣤⠈⢻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣛⣻⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠇⡄⢻⡇⢸⡇⣿⣿⠛⡛⢻⣟⡛⢻⣟⡛⢻⡗⢺⣿⢣⡟⢳⠘⡟⣷⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡟⢠⣥⠈⡇⢸⡇⢽⣿⠈⡏⢸⠉⠖⠸⡟⠠⠿⡇⢸⣿⡼⡇⢸⠀⡇⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣫⣟⣯⣵⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣽⡾⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⣿⣿⣿⣿⣿⡿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⢿⣿⣿⡿⢿⣿⣿⠉⣿⠭⠌⢻⠀⣥⠈⣧⠆⣠⣿⣿⡇⣿⠀⡇⡇⢸⡿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⠀⢹⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣇⣴⣿⣋⣷⣸⣇⣽⣀⣧⣘⣅⣸⣀⣿⣀⣏⣀⣛⣿⣿⣧⡻⣤⣧⣧⡼⣳⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠇⠀⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣾⣿⣿⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⡀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢄⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠙⢿⣿⠋⠀⠈⢿⡿⠟⠠⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣄⠀⠀⠀⠂⠀⠀⠠⠀⢀⡀⢠⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⠀⠀⣤⢀⠀⠐⢰⠀⣸⡇⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠥⠿⡇⠂⣀⠈⣺⡇⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣤⡇⡈⢉⡄⢸⣦⡿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣾⠿⣧⡀⠛⢀⡀⠻⣿⣧⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢁⣤⠈⢻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣛⣛⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠇⡄⢹⡇⢸⡇⣿⣿⠛⡛⢻⣟⡛⢻⣟⡛⢻⡗⢺⣿⢣⡟⢳⠨⡟⣷⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡟⢠⣥⠈⡇⢸⡇⣹⣿⠈⡏⢸⠉⠖⠸⡟⠠⠿⡇⢸⣿⡼⡇⢸⠀⡇⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣫⣟⣯⣵⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣽⡶⣿⣝⢿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⢸⣿⣿⣿⣿⡿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠻⣿⣿⣿⢿⡿⣿⡿⣿⠥⠌⢻⠁⣥⠈⣯⠇⣠⣿⣿⡇⣿⠀⡇⡇⢸⡏⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣇⣴⣶⣉⣯⣼⣇⣿⣇⣧⣘⣅⣸⣀⣿⣀⣏⣐⣛⣿⣿⣧⡻⣴⣧⣧⡼⣳⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠷⠀⠻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣾⣿⣿⣿⣿⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠙⢿⣿⠋⠀⠈⢿⡿⠟⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣄⡀⠀⠀⠂⠀⠀⠠⠀⢀⡀⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣼⣿⠃⡈⡀⢀⠀⠐⢰⠀⣸⡇⢸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⡇⠂⣀⠈⣺⡇⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣤⡇⡈⢉⡄⢸⣦⡿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣾⠿⣧⡀⠛⢀⡀⠻⣿⣧⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢁⣤⠈⢻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣛⣛⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠇⡄⢹⡇⢸⡇⣿⣿⠛⡛⢻⣟⡛⢻⣟⡛⢻⡗⢺⣿⢣⡟⢳⠈⡟⣷⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡟⢠⣥⠈⡇⢸⡇⢸⣿⠈⡏⢸⠉⠆⠸⡟⠠⠿⡇⢸⣿⡼⡇⢸⠀⡇⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣫⣟⣯⣵⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣽⡶⣾⣝⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢀⣿⣿⣿⣿⡿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠁⠟⢿⣿⣿⡿⢿⣿⣿⠯⠍⢻⠉⣭⠈⣧⠆⣡⣿⣿⡇⣿⠀⡇⡇⢸⡏⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠈⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣇⣴⣶⣀⣟⣹⣧⣿⣏⣿⣘⣅⣸⣀⣿⣀⣏⣐⣛⣿⣿⣧⡻⣠⣧⣧⡼⣳⣿⣿⣿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠀⠙⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣾⣿⣿⣿⣿⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠻⣄⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠙⢿⣿⠋⠀⠈⢻⡿⠟⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣄⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣼⣿⠀⠈⡆⠀⠀⠀⢰⠀⣸⡇⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⠀⠇⡇⠀⣀⠈⣺⡇⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣤⡇⡈⢉⡄⢸⣦⡿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣾⠿⣧⡀⠛⢁⡀⠻⣿⣧⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢁⣤⠈⢻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣛⣛⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠇⡄⢹⡇⢸⡇⣿⣿⠛⡛⢻⣟⡛⢻⣟⡛⢻⡗⢺⣿⢣⡟⢳⢈⡟⣷⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡏⢠⣥⠈⡇⢸⡇⣾⣿⠈⡏⢸⠉⠖⠸⡟⠠⠿⡇⢸⣿⡼⡇⢸⠀⡇⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣫⣟⣯⣵⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣽⡶⣮⣝⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⣸⣿⣿⣿⡿⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠁⠇⢹⣿⣿⡿⢿⢿⣿⣿⠏⢻⠁⣭⠈⣧⠆⣠⣿⣿⡇⣿⠀⡇⡇⢸⡇⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⠀⢹⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣇⣰⣶⣀⣇⣹⣇⣾⣇⣿⣙⣅⣸⣀⣿⣀⣏⣀⣙⣿⣿⣧⡻⣤⣇⣧⡼⣳⣿⣿⣿⠘⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠇⠈⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣾⣿⣿⣿⣿⠀⠀⠹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣦⡀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⢿⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠙⢿⣿⠃⠀⠈⢻⡿⠟⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠻⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣄⠀⠀⠀⠂⠀⠀⠠⠀⢀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣼⣿⠀⠈⡤⠀⠀⠀⢰⠀⣸⡇⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠀⠀⡇⠀⣀⠀⣺⡇⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣤⡇⡈⢉⡄⢸⣦⡿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣾⠿⣧⡀⠛⢀⡀⠻⣿⣧⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢁⣤⠈⢻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣛⣛⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠇⡄⢹⡇⢸⡇⣿⣿⠛⡛⢻⣟⡛⢻⣟⡛⢻⡗⢺⣿⢣⡟⢳⠈⡟⣷⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡟⢠⣥⠈⡇⢸⡇⣿⣿⠘⡟⢸⠉⠖⠸⡟⠠⠿⡇⢸⣿⡼⡇⢸⠀⡇⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣻⣟⣯⣵⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣽⡶⣾⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⢀⣿⣿⣿⠿⠟⠛⠋⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠁⠇⢸⡟⢿⣿⣿⠿⣿⣿⣿⢻⠉⣭⠈⣧⠄⣠⣿⣿⡇⣿⠀⡇⡇⢸⡇⣿⣿⣿⠙⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠈⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣇⣰⣶⣀⣇⣸⣏⣻⣇⣯⣘⣇⣸⣄⣿⣀⣏⣀⣙⣿⣿⣧⡻⣤⣧⣧⡸⣳⣿⣿⣿⠀⠈⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠇⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣾⣿⣿⣿⣿⠀⠀⠀⠙⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢶⣶⡀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠻⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠙⢿⣿⠋⠀⠈⢻⡿⠟⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣄⠀⠀⠀⠂⠀⠀⠠⠀⢀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠑⢀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣼⣿⠀⠈⡠⠀⠀⠀⢰⠀⣸⡇⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⣀⠀⢸⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⠀⣤⡇⡈⢉⡄⢸⣦⡿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣾⠿⣧⡀⠛⢀⡀⠻⣿⣧⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢁⣤⠈⢻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣛⣛⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠇⡄⢻⡇⢸⡇⣾⣿⠛⡛⢻⣟⡛⢻⣟⡛⢻⡗⢺⣿⢣⡟⢳⢈⡟⣷⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡟⢠⣥⠈⡇⢸⡇⣿⣿⠈⡏⢸⠉⠖⠸⡟⠠⠿⡇⢸⣿⡼⡇⢸⠀⡇⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣻⣟⣯⣵⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡛⠛⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣽⡶⣮⣝⢿⣿⣿⣿⠹⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣸⣿⣿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠁⠇⠸⡇⢸⡿⣿⡟⣿⢿⣿⣿⡏⣭⠈⣯⠇⣠⣿⣿⡇⣿⠀⡇⡇⢸⡇⣿⣿⣿⠀⠘⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣇⣰⣶⣀⣇⣸⣇⣸⣅⣿⣘⣍⣻⣄⣿⣀⣏⣀⣛⣿⣿⣧⡻⣤⣧⣧⡼⣳⣿⣿⣿⡀⠀⠀⠹⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣾⣿⣿⣿⣿⠀⠀⠀⠀⠙⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸⣶⣦⡀⠈⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡀⠈⢿⣿⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠙⢿⣿⠋⠀⠈⢿⡿⠟⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠉⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣄⠀⠀⠀⠂⠀⠀⠠⠀⢀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠂⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⠀⠈⡠⠀⠀⠀⢰⠀⣸⡇⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⣀⠀⢸⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣤⡇⡈⢉⡆⢸⣦⡿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣾⠿⣧⡀⠛⢀⡀⠻⣿⣧⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢁⣤⠈⢻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣛⣛⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠇⡄⢹⡇⢸⡇⢿⣿⠛⡛⢻⣟⣛⢻⣟⡛⢻⡗⢺⣿⢣⡟⢳⢈⡟⣷⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡟⢠⣥⠈⡇⢸⡇⣿⣿⠀⡇⢸⠉⠖⠸⡟⠠⠿⡇⢸⣿⡼⡇⢸⠀⡇⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣻⣟⣿⣵⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡛⠛⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣽⡶⣾⣝⢿⣿⣿⣿⠀⠙⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⣿⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠁⠇⠸⡇⢸⡏⢻⡿⣿⠿⣿⣿⡏⣭⠈⣯⠏⣩⣿⣿⡇⣿⠀⡇⡇⢸⡇⣿⣿⣿⠀⡀⠈⠻⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣇⣰⣶⣀⣇⣸⣇⣸⣀⣯⣘⣅⣻⣇⣿⣀⣯⣀⣙⣿⣿⣧⡻⣤⣧⣧⡼⣳⣿⣿⣿⣠⡀⠀⠀⠹⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⠻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠻⣷⣶⡄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠙⢿⣿⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠙⢿⣿⠋⠀⠈⢻⡿⠟⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣄⠀⠀⠀⠂⠀⠀⠠⠀⢀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣼⣿⠀⠈⡀⠀⠀⠀⢰⠀⣸⡇⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢹⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⣀⠀⢸⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣤⡇⡈⢉⡆⢸⣦⡿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣾⠿⣧⡀⠛⢀⡀⠻⣿⣧⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢁⣤⠈⢻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣛⣛⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠇⡄⢹⡇⢸⡇⢸⣿⠛⡛⢻⣛⡛⢻⣟⡛⢻⡗⢺⣿⢣⡟⢳⠈⡟⣷⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡟⢠⣥⠈⡇⢸⡇⣹⣿⠀⡇⢸⠉⠖⠸⡟⠠⠿⡇⢸⣿⡼⡇⢸⠀⡇⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣻⣟⣿⣵⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡀⠀⢹⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡛⠛⣿⡟⢻⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣽⡶⣾⣝⢿⣿⣿⣿⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡾⠛⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠁⠇⠸⡇⢸⡇⢸⠉⣿⠿⡍⣿⡿⣯⠉⣯⠏⣩⣿⣿⡇⣿⠈⡇⡇⢸⡏⣿⣿⣿⠀⠀⢠⡄⠀⠙⣿⣿⣿⡿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣇⣴⣶⣀⣇⣸⣇⣼⣄⣧⣘⣅⣹⣇⣿⣀⣏⣰⣿⣿⣿⣧⡻⣠⣧⣧⡼⣳⣿⣿⣿⠀⠀⠶⠧⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣾⣿⣿⣿⣿⠀⠀⠀⣤⣀⣀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠻⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠠⠤⠀⠘⣿⣷⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠙⢿⣿⠋⠀⠈⢿⡿⠟⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⡈⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣄⠀⠀⠀⠂⠀⠀⠠⠀⢀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⠀⠈⡀⠀⠀⠀⢰⠀⣸⡇⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⣀⠀⢺⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣤⡇⡈⢉⡆⢸⣦⡿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣾⠿⣧⡀⠛⢀⡀⠻⣿⣧⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢁⣤⠈⢻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣛⣛⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠇⡄⢻⡇⢸⡇⣿⣿⠛⡛⢻⣟⡛⢻⣟⡛⢻⡗⢺⣿⢣⡟⢳⢸⡟⣷⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡟⢠⣥⠈⡇⢸⡇⣿⣿⠀⡇⢸⠉⠖⠸⡟⠠⠿⡇⢸⣿⡼⡇⢸⠸⡇⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣻⣟⣿⣵⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠁⡄⠀⠈⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠿⠿⠿⣄⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡛⠛⣿⡟⢻⡛⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣽⡶⣾⣝⢿⣿⣿⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠚⠋⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠁⠇⠸⡇⢸⡇⢸⠉⣿⠭⠍⣿⡝⣿⠏⣯⠟⣡⣿⣿⡇⣿⠀⡇⡇⢸⡏⣿⣿⣿⠀⠠⠈⢴⡆⠀⠈⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀
⣿⣿⣿⣇⣰⣶⣀⣇⣸⣇⣸⣀⣧⣘⣅⣸⣅⣿⣀⣏⣀⣿⣿⣿⣧⡻⣠⣧⣧⡼⣳⣿⣿⣿⠀⠀⠀⠒⠷⠀⠀⠈⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣾⣿⣿⣿⣿⠀⠀⠀⠀⣤⣄⠀⣀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⢻⣿⡿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠲⠆⠀⠹⣷⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠙⢿⣿⠋⠀⠘⢿⡿⠟⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠠⣄⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣄⠀⠀⠀⠂⠀⠀⠠⠀⢀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⠀⠈⡀⠀⠀⠀⢰⠀⣸⡇⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⣄⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⣀⠀⢸⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣤⡇⡈⢉⡆⢸⣦⡿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣾⠿⣧⡀⠛⢀⡀⠻⣿⣧⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢁⣤⠈⢻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣛⣛⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠇⡄⢻⡇⢸⡇⣿⣿⠛⡛⢻⣟⡛⢻⣟⡛⢻⡗⢺⣿⢣⡟⢳⢸⡟⣷⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡟⢠⣥⠈⡇⢸⡇⣿⣿⠀⡇⢸⠉⠖⠸⡟⠠⠿⡇⢸⣿⡼⡇⢸⠸⡇⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣻⣟⣿⣵⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠦⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⠿⠿⠿⢿⣆⡀⠀⠀⠀⠀
⣿⣿⣿⣿⡛⠛⣿⡟⢻⡛⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣽⡶⣾⣝⢿⣿⣿⣿⠀⠀⠀⢀⠀⠀⠈⢿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠘⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠁⠇⠸⡇⢸⡇⢸⠀⣿⠭⠌⢻⡍⣿⢿⣯⠟⣩⣿⣿⡇⣿⠀⡇⡇⢸⡏⣿⣿⣿⠀⠀⠐⠈⣶⡆⠀⠘⢿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣇⣰⣶⣀⣇⣸⣇⣸⣀⣧⣘⣅⣸⣄⣿⣀⣏⣐⣻⣿⣿⣷⡿⣤⣇⣧⡼⣳⣿⣿⣿⠀⠀⠀⠀⠘⠓⠀⠀⠘⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣾⣿⣿⣿⣿⠀⠀⠀⠀⢠⣦⠀⣤⣀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠙⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠐⠶⠀⠈⠛⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠙⢿⣿⠋⠀⠘⢿⡿⠟⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢤⣄⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣄⠀⠀⠀⠂⠀⠀⠠⠀⢀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⠀⠀⡀⠀⠀⠀⢰⠀⣸⡇⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢷⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⣀⠀⢺⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠀⣤⡇⡈⢉⡆⢸⣦⡿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣾⠿⣧⡀⠛⢀⡀⠻⣿⣧⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢁⣤⠈⢻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣛⣛⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠇⡄⢻⡇⢸⡇⣿⣿⠛⡛⢻⣟⡛⢻⣟⡛⢻⡗⢺⣿⢣⡟⢳⢸⡟⣷⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡟⢠⣥⠈⡇⢸⡇⣿⣿⠀⡇⢸⠉⠖⠸⡟⠠⠿⡇⢸⣿⡼⡇⢸⠸⡇⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣻⣟⣿⣵⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠁⢢⠀⠀⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠿⠿⠿⢿⣆⣀⠀⠀⠀
⣿⣿⣿⣿⡛⠛⣿⡟⢻⡛⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣽⡶⣮⣝⢿⣿⣿⣿⠀⠀⠀⠀⢘⠀⠀⠘⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀
⣿⣿⣿⣿⠁⠇⢸⡇⢸⡇⢸⠀⣿⠥⠌⢻⠉⣭⠻⣿⠟⣩⣿⣿⡇⣿⠀⡇⡇⢸⡇⣿⣿⣿⠀⠀⠀⠠⣼⣷⡆⠀⠘⢿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣇⣰⣶⣀⣇⣸⣇⣸⣀⣧⣘⣅⣸⣄⣿⣀⣏⣐⣛⣿⣿⣿⣿⣤⣇⣧⡸⣳⣿⣿⣿⠀⠀⠀⠀⠀⠛⠓⠀⠀⠘⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⠀⣤⣄⣀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠐⠂⠀⠀⢿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠙⢿⣿⠋⠀⠈⢿⡿⠟⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠰⣤⡈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣄⠀⠀⠀⠂⠀⠀⠠⠀⢀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⠁⠈⡀⠀⠀⠀⢰⠀⣸⡇⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⡇⠀⣀⠀⢺⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣤⡇⡈⢉⡆⢸⣦⡿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣾⠿⣧⡀⠛⢀⡀⠻⣿⣧⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢁⣤⠈⢻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣛⣛⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠇⡄⢻⡇⢸⡇⣿⣿⠛⡛⢻⣟⡛⢻⣟⡛⢻⡗⢺⣿⢣⡟⢳⢨⡟⣷⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡟⢠⣥⠈⡇⢸⡇⣾⣿⠀⡇⢸⠉⠖⠸⡟⠠⠿⡇⢸⣿⡼⡇⢸⠸⡇⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣫⣟⣿⣵⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠉⢰⠀⠀⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠿⠿⣿⣆⡀⠀⠀
⣿⣿⣿⣿⡛⠛⣿⡟⢻⡛⢻⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣽⡶⣾⣝⢿⣿⣿⣿⠀⠀⠀⠀⠀⢸⡀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀
⣿⣿⣿⣿⠁⠇⠸⡇⢸⡇⢸⠀⣿⠥⠄⢻⠉⣭⠹⣿⡟⣩⣿⣿⣿⣿⠀⡇⡇⢸⡏⣿⣿⣿⠀⠀⠀⠀⠀⣘⣷⣦⠀⠘⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣇⣰⣶⣀⣇⣸⣇⣸⣀⣧⣘⣅⣸⣄⣿⣀⣏⣐⣛⣿⣿⣿⣿⣷⣇⣧⡸⣳⣿⣿⣿⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⣤⣤⣄⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠆⠀⠹⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠙⢿⣿⠃⠀⠈⢿⡿⠟⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢦⣄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣄⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⡆⠈⡀⠀⠀⠀⢰⠀⣸⡇⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡇⠀⣀⠀⢸⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣤⡇⡈⢉⡆⢸⣦⡿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣾⠿⣧⡀⠛⢀⡀⠻⣿⣧⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢁⣤⠈⢻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣛⣛⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠇⡄⢹⡇⢸⡇⣿⣿⠛⡛⢻⣟⡛⢻⣟⡛⢻⡗⢺⣿⢣⡟⢳⢨⡟⣷⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡟⢠⣥⠈⡇⢸⡇⣸⣿⠀⡇⢸⠉⠖⠸⡟⠠⠿⡇⢸⣿⡼⡇⢸⠘⡇⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣻⣟⣯⣵⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⢹⡆⠀⢸⣿⣿⣿⣿⣿⣿⠃⢀⣴⣿⣒⠤⡈⠛⠁⢠⣴⣆⠠⣶⡀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣄
⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣳⣬⡉⠀⢠⡆⣶⣆⡀⣀⠀⠀⣼⣷⠀⠸⣿⣿⣿⣿⣿⡟⠀⣼⠉⠉⠛⠛⠋⠀⠀⠸⠟⠛⠀⠉⠛⠓⠃⠀⣼⣿⣿⣿⡿⢿⣿⣿⣿
⠀⣿⣼⣿⠀⠠⣄⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⣿⡿⢿⣿⣿⣿⣯⡉⠉⠉⠁⠀⠚⠁⠉⠉⠀⠘⠆⠀⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⡾⠁⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⣶⣶⣦⣆⠠⠀⢈⣉⣀⠀⠀⠛⠛⠛⠀
⠀⠀⢀⣀⣀⣤⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡾⢪⣟⣟⡃⣽⣿⣿⣿⣿⣆⠀⠀⠀⠀⣶⣾⣿⣿⣶⠀⠀⠛⠿⠿⠇⠙⠻⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠘⠟⠉⠀⠀⠀⠀⢀⡀
⠀⢠⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠉⠻⠛⢿⣿⣿⣿⣿⣿⣆⠀⠀⣾⣿⣿⣿⣿⡟⠀⠀⢠⣴⣶⣤⣤⡄⠹⠴⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡟⠀⠀⠠⢶⣤⣤⠀⠀⠀⠈⠁
⢠⣾⣿⣿⣿⠿⠋⣤⡆⠀⠀⠀⠀⠀⠈⢻⣶⡆⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣶⣽⣿⣿⣿⣿⣧⡀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠰⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⠁⠀⠶⠃⠀⠀⠀⠀⠀⠀⣀⡀
⣿⣿⣿⣿⣧⣶⣿⣿⣧⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠺⡿⣿⣏⣸⣿⣿⣿⣿⣿⣿⡿⠀⠰⠨⠿⣿⣿⡿⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡿⣿⣿⣿⣿⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁
⣿⣿⢿⣿⣿⣿⣛⠋⠁⠀⠀⠀⣠⣄⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡧⠠⣤⠀⢶⣶⣌⠙⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀
⣿⣧⣤⣴⡽⠛⠉⠁⠀⠀⠀⠀⣿⣿⣿⡛⠛⠛⠛⠛⠛⠃⢀⠐⠚⠛⠛⠛⠛⠛⠛⢛⣿⣿⣿⣿⣿⠟⠛⠛⢿⣿⢛⠿⣿⠀⠘⠛⠛⠛⠛⠛⠃⠀⠀⠀⠘⠛⠛⠛⠛⠟⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⠛⣛⠛⠛⠃⢛⣛⣛⡛⠛⠛⠛⠛⠛⠛⢻⣿⣿⠟⢛⣛⣛⠛⠻⣿⣿⣿⣛⠃⠘⠛⠛⠛⠛⠛⠛⠀⠀⠀⠛⠛⠛⢛⣻⡿⢿⣿⣿⣾⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠀⢸⣿⣿⡟⣛⣛⣛⡃⠀⠀⢘⣛⣛⡃⠀⠀⠀⠀⠀⠀⡸⠟⠁⠀⢘⣛⣛⣃⣀⣿⣿⣟⣛⠋⠀⠀⢘⣛⣛⣛⣛⣛⣀⣚⣛⣛⣛⣿⣿⣷⡾⠋⠉⠉⠉⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⢸⣿⣿⣁⣛⣛⣛⡃⠀⠀⢘⣛⣛⡃⠀⠀⠀⠀⠀⣠⣥⡔⠀⠀⢘⣛⣛⣛⣛⣻⣿⣿⡋⠀⠀⠀⢘⣛⣛⡋⣛⣛⣛⣛⣛⢻⣿⣿⣿⡭⠀⠀⠀⠀⠰⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣟⣛⣛⡃⠀⠀⢙⣛⣻⡧⠀⠀⠀⢠⣴⣿⣏⠀⠀⠀⢘⣛⣛⡛⠛⠛⢿⣿⣟⡀⠀⠀⢘⣛⣛⡃⢙⣛⣛⣛⣻⣾⣿⣿⣟⠁⠀⠀⠀⠀⠀⠈⠹⣿⣆⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢸⣿⣿⣿⣏⣉⣉⣁⣀⡀⢈⣉⣹⣷⣀⣀⣀⣼⣿⣿⣿⡃⢀⣀⣈⣉⣉⣁⣀⣀⣘⣿⣿⣇⢀⣀⣈⣉⣉⡇⠀⢉⣉⣹⣿⣿⣿⣿⣟⣀⡀⠀⠀⢠⡀⠀⠀⢻⣿⣧⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⣿⣿⣯⣉⣉⣍⣉⡁⠈⢩⣭⣿⣭⣭⣭⣿⣿⣿⣯⡅⢈⣉⣉⣉⣉⣉⣉⣩⣭⣿⣿⣷⣨⣉⣉⣉⣩⡅⠀⠈⣉⢹⣿⣭⣿⣯⣭⣭⡅⠀⢀⡘⠃⠀⠀⠐⠻⣿⣷⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⣿⣿⣿⣿⣯⣭⣿⣿⠿⠀⠀⠈⢉⣿⠛⢻⣿⣯⡍⢉⡁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⢈⣿⣿⣿⠋⠉⠉⠉⠉⢘⠛⠃⠀⠀⠀⠀⠀⠻⣿⣿⣶⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠘⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠛⢉⣁⢴⣆⣹⣿⣿⣿⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿
⣿⣦⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠛⠁⣀⣤⣄⣀⠀⠀⣀⣤⣄⡀⢠⣿⣿⣿⣿⣷⣶⣦⣄⠀⢀⣀⡀⠀⣀⣀⣀⣠⣼⢿⣛⡁⠀⣸⣿⣀⣀⣻⣿⣿⣿⣷⣤⣄⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⣿⡿⠃
⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣀⡈⠛⠀⣼⣏⣀⣈⣿⡌⢻⡿⠛⠿⠿⠿⠿⣿⠀⠀⢸⣧⠀⠀⣿⡏⠀⠙⠀⢸⡇⠀⢸⣿⠀⠈⠻⣿⡿⠿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠄⠀⠀⡀⠀⠀⢠⡉⠙⠻⣷⠀⢿⡍⠉⠉⣩⡥⢸⣧⠀⠀⢠⡄⠀⣿⡀⠀⢸⣿⠀⠀⣿⡇⠀⢀⠀⢸⡇⠀⢸⣿⠀⣀⠀⠸⣷⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⢤⠐⠚⠃⠀⠀⠘⠳⠶⠞⠋⠀⠈⠛⢶⠞⠋⠀⡠⠙⢳⠶⡛⠁⠀⠙⠿⠖⠛⠛⠒⠐⠛⠛⢂⠀⠐⠚⠓⠂⠈⠻⠶⠛⣶⡦⣹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠿⣶⠿⢶⠷⠾⠿⠿⡷⠀⠉⢀⠀⠀⠀⠈⠁⠀⠀⣀⠀⠀⡀⢀⠀⡀⠀⠀⠉⢐⠀⠀⠀⠀⠆⠰⣠⣄⠀⠁⠀⠘⠲⠷⠀⠀⠀⠀⠠⠀⠀⠀⠿⠶⠟⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡇⣿⣿⣿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡇⣿⡇⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣿⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠿⣿⡟⢻⢻⠻⡇⡇⣹⣷⢛⡳⣿⣿⣿⣿⠵⣿⣿⣿⣿⣛⢿⣿⣟⣷⣾⣿⣿⣉⣿⣴⣿⣿⣿⢏⣇⣷⣇⡇⣿⣿⢿⣿⣟⣿⢿⣿⣿⣿⠿⡯⠻⣿⣿⣿⣿⣟⣟⣩⣿⣟⡻⣱⣿⠇⠃⣚⣻⣿⣛⣋⡟⣿⣯⣿⣿
⠶⣢⣿⠿⣌⣿⡯⣿⡅⡘⣛⣲⣻⣾⢿⡇⢛⡐⢶⣟⣹⣿⣿⢿⢢⣭⣛⢻⣷⣞⣹⣼⣗⡛⠋⠯⠝⣿⠊⡟⣒⣻⡟⠖⢹⣿⣿⣉⠘⡽⣿⣿⣯⡅⠙⣻⣾⣿⣿⢻⢋⣈⣿⣿⣿⣿⠆⠈⠛⠖⣭⣿⣶⣶⡆⢘⣿⡟
⢨⣏⣿⠇⠛⠾⠁⢠⣭⣱⡎⣛⠏⣿⣿⠿⢟⡙⢩⣷⣮⣀⣹⣿⣷⣞⣯⡅⢙⠺⠛⣯⣹⣿⣿⣷⣦⡄⠴⣿⣽⣿⣟⠛⠋⠹⡿⢪⣉⠁⢙⣯⣭⣔⠺⣿⡿⠿⢁⣭⣿⣿⣿⣿⣿⡋⣃⢒⣚⣛⠻⢟⣿⣿⣣⣶⣶⣶
⣿⣿⠙⣀⣀⠶⣧⡀⠈⠙⣿⣿⣿⢫⣭⠑⠾⣿⡀⠉⢀⣹⣥⡭⣿⣿⣿⡗⠾⠛⣶⣿⣿⣿⠇⢹⣭⣥⣵⡞⢿⣿⡧⣿⣶⣿⣶⣿⣯⣥⠚⢽⡿⠏⠀⣙⣃⣀⣌⠛⣗⣿⣿⣿⠷⠛⠮⡉⠩⠋⠁⢀⣻⠟⠿⢫⣿⣿
⠻⠟⠀⠈⣙⠡⠈⠁⠶⣾⣽⣯⣽⣷⣿⡶⢴⣿⣄⠐⢛⣧⡕⠿⠟⢹⣭⣭⣷⣴⠹⣿⠿⠷⠀⠀⡦⣐⣁⠀⣨⣿⣧⡝⠳⣿⠿⠥⣈⣁⠀⠘⠻⢈⣉⠙⢙⠟⣡⣤⣸⢿⣷⠀⠀⠷⣆⣶⣏⡉⠛⠻⠛⠁⠀⠰⣿⡿
⠧⠀⠀⠀⠛⠻⢷⣤⣿⣿⣿⣿⡿⠿⡏⢠⣆⡀⠙⠁⠛⢋⣯⣶⡆⠀⢸⡌⠝⠉⠀⢻⣿⡟⠈⢲⣿⡿⠛⠢⡌⠩⠃⠀⠀⢿⣿⠆⠿⠛⠀⠀⠀⠀⢎⣠⣴⣿⠿⣿⣿⣿⣿⣶⣤⡀⠈⠀⠠⠫⠶⣶⣶⣶⣾⠿⡇⠀
⢡⣁⠀⠀⠀⠠⣿⣿⢼⣿⣿⡯⠅⠀⢨⣬⣿⠟⡦⠀⠀⠛⠟⠉⠀⢀⣸⣴⣶⣆⠀⠀⠈⠁⠀⣶⣦⣄⠀⣸⣤⣶⣿⡀⠀⠀⠈⠈⣶⣦⣠⣤⣤⣤⣿⣿⠉⠏⣂⣿⣿⣿⣟⡈⠉⠻⠀⠀⢀⠥⣀⠀⠀⢀⠎⠀⠀⠀
⣤⣼⡷⠀⠀⠸⠿⢿⠛⣿⣛⣛⣿⣍⠛⠿⣿⠿⢟⡛⠛⠁⠀⠐⠚⣿⣿⣿⠟⡛⠛⠛⠛⠃⣼⣿⣿⣿⣿⣿⠿⠟⠛⠓⠀⠀⠘⠛⠛⡛⠛⠛⠛⣻⠯⠅⠘⠛⢻⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⣰⠋⠀⣀⣠⣤⣶⡄⠀
⠿⣿⣿⡆⠀⢴⠖⠉⠀⢻⣿⣿⣿⠃⠘⠛⠛⠛⣿⣿⣿⠃⢚⣛⣛⡛⠛⠛⠛⣛⣿⣿⣿⣧⠛⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠃⣸⣿⣿⣿⡟⠛⠛⠁⣀⠀⠛⠛⠛⠻⠿⢿⣇⣀⠀⠀⢀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡷⠆
⠀⠈⠁⠀⣀⣤⣤⣴⠀⠀⠹⠿⠁⠀⠀⠀⣺⣿⣿⣏⠉⠀⢘⣛⣛⡃⠀⡀⢀⠙⠻⢿⡟⠃⠐⣳⣿⣿⣿⣟⣱⣀⣘⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⡃⠉⢻⣿⣿⢦⡈⠀⠂⣾⡿⠟⠻⠿⠿⠉⠀⠀
⠀⠀⣠⣾⣿⣿⣿⣿⡀⠀⠀⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⣘⣛⣛⡃⠳⢶⢘⠇⠀⠋⠀⠀⠀⠘⣿⣿⣿⣟⣛⣛⣛⣻⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⢻⣛⣛⡃⠀⠀⠈⠉⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣿⣿⣿⠟⠋⠉⠀⠀⠂⠀⠀⠀⠂⠀⣙⣛⣛⡛⢿⣿⣟⣙⣛⡃⠀⠀⣠⣤⣶⣾⣿⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣛⣻⣯⡘⣽⣿⣿⣿⣿⣿⣿⣛⣛⡋⢘⣛⣛⡃⠀⠀⠀⠀⠀⠠⠀⢀⣶⣶⣶⣶⣿⣿⣿⣿⣿
⡀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣉⣛⣁⣀⡡⣉⣉⣉⣁⣠⣾⣿⣿⣿⣿⣿⡇⢈⣿⣿⣿⣿⣿⣿⣿⣿⣙⣻⣛⢇⣿⣿⣿⣿⣿⣿⣟⣉⡉⠀⢚⣉⣉⣁⣀⡀⠀⠀⠀⠀⠘⠀⠉⠙⠻⢿⣿⣿⠟⠛⠛
⣿⣿⣦⣄⡀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢉⣉⣉⣉⣉⣉⣉⡁⠈⢉⣉⣩⣭⣿⣿⣿⣟⣉⣭⣁⣬⣉⣉⣙⣉⣙⣭⣭⣉⣉⡉⠱⢾⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢈⣉⣉⣉⣉⡁⠀⠀⠀⠠⠖⢦⣀⠀⠀⠀⠀⠀⢠⡒⠀
⣿⣿⣿⣿⣿⣦⣀⣀⣀⣀⣀⣀⣀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠉⠁⠻⡯⠉⠉⣩⠏⠉⠉⠉⠁⠀⠀⢸⣿⣿⣿⣿⣷⣦⡀⠀⣀⣀⠈⢉⡉⠉⠉⠁⠀⠀⠀⠀⠀⢠⡺⠋⠀⠀⠀⠀⠚⠀⠀
⠿⠟⠛⢛⠿⢿⠌⣿⣿⣿⣿⣿⢃⣄⣤⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣧⣄⣈⠉⠳⠈⠛⠂⣤⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⣀⣀⣀⣤⣤
⣿⣿⠟⠁⠀⠀⠀⠸⢿⣿⣿⠇⠾⣿⣿⠀⠀⣀⣤⣄⣀⣠⠇⣀⣤⣄⡀⢠⠀⢀⣠⣤⣀⡀⣁⣀⣠⣤⣶⣶⣿⣷⣀⣀⣠⣤⢀⣈⡙⢿⣿⣿⣿⡃⣿⣿⣿⣶⣄⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿
⠿⠃⠀⠀⠀⠀⠃⠀⠀⣀⣈⡀⠀⠀⠀⣤⠸⣧⣀⡀⠛⡇⣼⣏⣀⣈⣿⡄⢰⡿⠁⠶⢼⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠘⠀⢸⡇⠀⢸⣿⠷⠀⣾⣿⡿⣇⣼⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠉⢀⡈⠙⠛⣷⠀⢿⡍⠉⠉⢉⠁⠸⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⡇⠀⢸⣿⠀⣀⠈⠻⣿⣹⣏⠋⣧⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⢿⣿⣿⣿⣿⡟⠛
⣶⣾⣿⠟⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠞⠋⠨⠋⠛⠶⠖⠋⠀⠀⠙⠲⢶⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠓⠂⠰⠔⠚⠓⠂⠀⠻⠶⠋⠀⠀⢹⠇⠀⠁⢭⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠫⡙⠛⠃⠀
⠉⠟⠶⣶⠟⠶⢤⠰⠶⠶⠆⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡿⠿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠤⠟⠀⠀⠀⢱⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡇⣿⣿⣿⡿⡏⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠻⣿⡟⢻⢿⢻⡇⡇⣹⣷⢛⡷⣿⣿⣿⣿⠵⣿⣿⣿⣿⡻⢿⣯⡟⣷⣾⣿⣿⣌⣿⣸⣿⣿⣿⠿⣿⣿⣿⠟⣿⣿⣿⣿⣟⣿⢿⣿⣿⣿⡝⣿⠿⣿⣿⣿⣿⣟⣟⣫⣿⣿⣛⢇⣿⠹⠀⢞⣹⣿⣟⣋⡻⢿⣿⣝⣿
⠷⢤⣿⠿⢈⣾⡓⣷⡅⡘⣛⣲⣲⣿⣿⡏⢛⡐⢲⣟⣻⣿⣿⢿⢂⣭⠛⠻⣷⣞⣹⣽⣗⡛⠟⠿⠽⢿⡋⠛⢖⣹⡯⠷⠛⣿⣿⢏⠘⠣⣿⣷⣾⣋⠘⠻⣥⣿⣿⠿⡿⡉⣻⣿⣿⣿⡇⠀⠙⠵⢨⣿⣷⣶⣶⣀⣛⣿
⢈⡏⣯⠇⠟⠿⠁⠈⣩⣳⣬⠹⠏⣿⣿⠷⢟⠙⢩⣷⣮⡅⠙⣾⣷⣞⣫⡍⠙⠼⠟⣿⡻⣿⣷⣷⣦⣄⠠⣿⣿⣿⣗⡻⠒⠙⣿⢩⢉⡁⠘⢃⣯⣅⡲⢟⣿⡻⠏⢈⣙⣿⣿⣿⣿⣟⢓⠔⠲⠛⠛⠾⢟⣿⣿⢐⣴⣶
⣿⣟⡙⣀⣀⢶⣧⡀⠉⠙⣷⣿⣷⢿⣷⠖⢾⣿⡀⠉⠁⣍⣫⡭⣿⣿⣿⡷⠶⠟⢢⣿⣿⣿⡿⠹⣿⣷⣧⣶⢻⣿⠿⢴⣦⣴⣶⣼⣿⣉⡐⢋⣿⣿⡁⠘⠻⣀⣀⡹⠿⣽⣿⣿⣿⡞⢢⢉⠙⠻⠉⠀⠈⠿⠾⠿⠛⣿
⢳⡞⠀⠈⠙⢁⠈⠁⠤⣶⣻⣟⣻⣧⣿⣦⣠⣽⣅⠐⠟⣷⡨⢶⡿⢻⣿⣿⣴⣤⡟⣿⣿⣿⡤⠀⢡⠈⢋⠁⠈⣿⣟⣈⠛⣻⣿⣿⡉⢋⠁⠀⠻⡇⣀⠛⠛⢻⠟⢃⣀⢹⣿⣿⠀⠰⢦⢠⣴⠞⠛⠛⢻⠛⠀⠀⢹⣿
⠤⠀⠀⠀⠛⠿⣶⣤⣯⣾⣿⣿⣿⣻⡟⠊⣂⡈⠙⠂⠛⠟⣻⣤⣄⠀⠉⡏⠛⠟⠁⠸⣿⣷⠆⠣⣾⣿⠿⠇⣈⠉⠻⠋⠀⠸⣷⣶⢸⡿⠒⠀⠀⠀⢨⠋⣀⣤⣴⣿⣿⣿⣾⣿⣤⡀⠀⠁⠈⠠⠃⣤⣥⣤⣤⣴⣶⣾
⢃⡂⠀⠀⠀⠠⣾⣿⢹⣿⣿⡿⠋⠀⢙⣀⣿⡿⢦⠀⠀⠿⠿⠋⠉⠀⢀⣯⣤⣶⠀⠀⠉⠋⠀⢠⣯⣁⠀⠀⣎⣤⣴⣆⠀⠀⠀⠓⢰⣤⣀⢀⣀⣀⣸⣿⣿⠛⡟⢅⣽⣿⣿⣿⠙⠛⢧⠀⠀⢀⣂⠀⠉⠉⠉⣩⠉⠉
⣈⣹⣦⠀⠀⢠⣾⣿⠾⣽⠿⠷⢶⡖⠛⢿⣿⣿⣛⠛⠛⠁⠀⠐⠚⢻⣿⣿⣿⠟⠛⢛⠛⠃⠘⣻⣿⣿⣿⣿⣿⣿⠟⠛⠀⠀⠘⠛⠛⠛⠛⠻⠿⠿⠿⣟⠛⠛⠛⠛⣿⣿⣛⡛⠃⠀⠀⠀⠀⠈⢀⠜⠀⠀⠀⣁⣀⣀
⣿⣿⣷⡄⠀⣠⡩⠜⠀⢿⣿⣷⣿⡟⠘⡛⠛⠻⣻⣿⣿⠃⢺⣿⣟⡛⠛⠛⠛⡛⢛⣛⣛⣃⡸⢟⣿⣿⣿⣿⠟⠛⠛⠛⠛⡗⠘⠛⠛⠛⢻⡟⠛⠠⠋⠀⠛⠛⠛⠛⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⢀⣴⣾⣿⣿⣿⣿
⠀⠙⠋⠁⠀⠘⣀⣀⠀⠈⠿⣿⠟⠀⠀⠀⢘⣛⣛⡋⠛⠁⢘⣛⣛⡃⠀⠀⠀⠰⢿⣿⣿⡿⠋⢸⣿⣿⣿⣿⣴⣄⣘⣛⣛⠃⠀⣠⣾⣿⣿⣿⣿⣷⣶⣾⣟⣛⣛⣛⡃⠀⠙⣺⣷⣦⢀⠐⠀⠠⢠⣾⣿⣿⣿⣿⠟⠋
⠀⠀⢀⣤⣶⣿⣿⣯⠀⠀⠈⠉⠀⠀⠀⠀⣹⣿⣿⣿⣶⣄⢘⣛⣛⡃⠢⣤⡀⢦⠀⠠⠋⠉⠀⠀⢿⣿⣿⣿⣟⣛⣛⣛⣯⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣛⣛⡃⠀⠀⠀⠛⠻⢞⣧⠀⠀⠚⠉⠀⠀⠁⠉⠀⠀
⠀⢰⣿⣿⣿⣿⠿⠿⠃⠀⡀⠀⠀⠀⠀⠀⣙⣛⣻⣿⣿⣿⣿⣛⣋⡁⠀⠀⠁⠁⣀⣠⣤⡄⠀⠀⢘⣻⣿⣟⣛⣻⣿⣯⣝⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⡿⢘⣛⣛⡃⠀⠀⠀⠀⠀⠀⢘⠀⠀⢀⢀⣀⣀⣀⣀⣀⣤
⠀⠘⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣙⣋⣁⣈⡓⢉⣉⣉⣁⣀⣠⣾⣿⣿⣿⣿⣷⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣏⣉⣋⣹⣸⣿⣿⡿⣿⣿⣿⡿⠁⢈⣉⣉⣁⣀⡀⠀⠀⠀⠀⠀⠀⡠⠴⠿⢿⣿⣿⡻⣿⣿
⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣌⣉⣉⣉⣉⣉⣉⡁⠉⢉⣉⣉⣩⣿⣿⣿⣿⣿⣋⡉⢈⣩⣹⣿⣿⣿⣿⣿⣿⣛⠉⠉⢋⣩⣽⣿⣿⣿⢿⣿⣿⡇⣀⢈⣉⣉⣉⣉⡁⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠈⠙⠿⠺⠋
⣿⣿⣶⣤⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠋⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⡉⠁⠸⣿⡍⠉⠉⠉⢉⡽⠛⠁⠀⠀⠈⠛⠻⣿⣿⣿⣿⣿⡿⠿⠟⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠃⣶⡀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣦⢠⣤⣤⣄⣠⣤⣤⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡗⠀⠀⠠⠎⠁⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⣶⣦⡄⠐⠶⣄⠠⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⠁⠀⠀⠀⠀⠊
⣿⠿⠟⠛⣿⣿⠇⢻⣿⣿⣿⣿⢃⣤⣿⠀⠀⣀⣤⣄⡀⠀⠀⣀⣤⣄⡀⠀⠀⢈⣠⣤⣀⡀⣀⣏⠀⢀⣀⡀⠀⣀⣀⣀⣠⣤⢀⣈⡈⢻⣿⣿⣿⣿⣿⣣⣤⣀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣠⣤⣤
⣶⣶⡾⠊⠀⠀⠀⠘⢿⣿⢿⠏⠘⡿⠃⠀⠸⣧⣀⡀⠛⢴⣿⣏⣀⣈⣿⡄⢰⡿⠁⢀⣸⡇⠀⣿⣀⣤⣼⣷⣿⣿⣿⡏⠀⠙⠀⢸⡇⠀⢹⣿⣿⡿⠿⣿⣿⣿⣿⡷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿
⡿⠋⠀⠀⠀⠀⠆⠀⠀⣀⣀⡀⠀⠁⠀⣦⢀⡈⠙⠻⣷⠂⢿⡍⠉⠉⢉⠁⠸⣇⠀⠈⢉⣝⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⡇⠀⢸⣿⠀⣾⣧⠸⣿⣿⣿⣇⢸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢤⠀⠉⠘⠳⠶⠖⠋⠀⣈⠛⠶⠖⠋⠀⠀⠙⠳⠶⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠐⠚⠓⠂⠀⠻⠶⠋⠂⠀⢹⠟⠟⠿⣌⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠻⣿⣿⣿⣿
⣤⣶⣶⣾⠏⠷⠄⠰⠶⠖⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠛⠋⠀⠀⢤⠀⠀⠀⠀⠀⠐⠾⠴⠏⠀⠀⠀⠀⠑⢭⠀⠀⠀⣀⡀⠀⠀⠡⠀⠀⠀⠈⠐⠅⠉
⣿⣿⣿⡇⣿⣿⣿⡿⡏⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢹⢹⣿⢸⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢹⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⢿⣿⡟⢻⢿⢻⠇⡇⣹⣷⢓⡧⣽⣿⣿⣿⢳⣿⣿⣿⣿⣻⢿⣯⣿⣿⣿⣿⣿⣟⣿⡻⣿⣿⣿⡿⣹⣸⣿⢸⢸⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠝⣿⣿⣿⣟⣿⣿⣋⣿⣾⣟⠸⣼⡯⠃⢥⣛⣿⣿⣯⣹⠿⣿⣝⣻
⠷⢤⣿⢿⣈⣾⣓⣷⡅⡘⣓⣲⣲⣿⣿⡏⢛⡐⢢⣟⣋⣿⣿⣿⢇⣭⡽⠻⣿⣞⣻⣿⣗⠿⠿⢻⡿⢻⡗⢻⠧⣙⣿⠥⠞⣿⣷⣿⠉⠛⣿⣷⣾⣛⠈⠻⢧⣿⣽⡟⣿⠇⢻⣿⣿⣿⣷⠀⠭⠭⠬⣹⣿⣶⣶⣏⣙⣿
⢈⡟⣭⡅⠿⣿⠃⠉⣉⣳⣬⠻⡟⣿⣷⡷⣞⠙⢈⣷⣯⡅⠛⣿⣿⣜⣫⣉⠙⠼⠟⢿⡟⢿⣿⣷⣦⣄⠁⣾⣟⣿⣿⡶⠶⠊⣿⡍⡝⢛⠈⠿⣙⣉⡐⢾⣿⡿⢿⠁⣚⢚⣿⣾⣿⣿⠿⡤⠲⠖⠺⢷⡿⢿⣿⡗⠠⣬
⣿⣟⠻⠀⠀⣦⣤⡀⠉⠻⣷⣿⣶⢿⣛⡂⢾⣿⡀⠋⠉⣝⣛⣙⣿⣿⣿⣷⡴⠷⠃⣬⣿⣿⣿⠟⣿⣛⣲⣤⣾⣿⣿⡧⣶⣀⣴⣥⣿⣏⣀⠾⣹⣿⡏⠀⠻⡆⠀⡙⢻⣿⣽⣿⣿⣿⠱⡝⠚⠛⠟⠃⠀⠹⣿⡷⣿⠿
⢷⡾⠀⠙⠛⢂⠙⠁⣠⣴⢿⣿⣿⣆⣿⣦⣠⣼⣅⠀⠾⣿⣈⢵⣾⠛⣟⣛⣒⣄⣞⢿⣿⣿⣏⠀⠈⡍⠙⠋⠀⢻⣿⣇⡰⠿⡽⣿⣟⠉⠋⠀⠘⢷⠆⠐⠺⠿⢿⠿⠂⠈⢽⣯⣍⠉⢢⡄⣡⡴⠒⠲⠶⢶⠶⠀⠀⢲
⣄⠀⠀⠀⠿⢶⣦⣀⣖⣩⣾⣿⣿⢿⡿⠋⡠⠉⠛⠃⠸⠿⢻⣠⣤⠀⠙⢿⠛⡻⠏⠀⣿⣶⡤⠰⣼⣾⣿⠿⢄⠛⠛⡿⠃⠈⢿⣿⡍⣿⠿⠀⠀⠀⠈⣩⠂⢀⣁⣤⣶⣿⣷⣿⣏⣀⠀⠈⠁⠁⢡⠂⢀⣀⣀⣀⣀⣬
⠂⠀⠀⠀⠀⠀⣼⣿⢿⣿⣿⣿⡿⠋⠓⠂⢿⣿⣦⠀⠀⠼⣿⠟⠛⠀⠀⣽⣀⣠⡄⠀⠘⠻⠃⠀⣼⡋⠁⠀⢨⢃⣠⣤⠀⠀⠈⠹⠃⣤⣀⠀⠀⠀⠀⣰⣿⣿⠿⣿⡛⣻⣿⣿⣿⡟⢷⡄⠀⠀⠀⠨⠉⠙⠛⠿⠿⡛
⠘⢳⣤⠀⠀⢀⣬⣽⣘⣿⣿⣟⣐⣀⠟⢻⣿⣛⠻⠛⠛⠃⠀⠐⠛⢛⣿⣿⣿⣿⠿⠛⡛⠃⠘⠛⢿⣿⣿⣟⣿⣿⣿⠿⠓⠀⠘⠛⠛⠛⠻⠿⢷⣿⣿⣿⣿⡿⠛⠛⠿⣿⣿⣿⠿⠥⠀⠀⠀⠀⠘⠉⣰⠆⠀⠀⠘⠁
⣿⣿⣭⡀⠀⢈⠛⣻⠉⣿⣯⣭⣿⣷⠚⠛⠻⢿⢛⣿⣿⡂⣸⣛⣛⡛⠛⠛⠛⠛⠛⠛⠛⠃⠘⣻⢿⣿⣿⣿⣿⡟⠛⠛⠛⢓⠘⠛⠛⠛⠻⣿⣛⠈⠉⡕⠛⠛⢛⡛⠛⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠐⠃⣀⣤⣶⣶⣿
⠉⠻⠿⠃⠀⠸⠊⠀⠀⠸⣿⣿⣿⠃⠀⠁⢘⣛⣻⡛⠿⠇⢘⣛⣛⡃⠀⠀⠀⠀⣴⣾⣿⣶⣾⠉⣸⣿⣿⣿⣯⣀⣘⣛⣛⠋⠀⠀⢘⣛⣛⣟⣛⣛⣀⣸⣿⣛⣛⣛⡃⠈⠙⢻⣿⣤⣄⡀⠢⠀⢀⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣠⣶⣶⣶⠂⠀⠈⠿⠁⠀⠀⠀⣛⣿⣿⣷⣄⡀⢘⣛⣛⡃⢠⣐⡀⢠⡀⠙⢻⠯⠋⠀⢹⣿⣿⣻⣛⣛⣻⣻⣯⣀⣀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⡃⠀⠀⠀⠈⢿⣿⣿⣦⠀⠀⠸⠟⠋⠉⠙⠉⠋
⠀⢀⣴⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⣿⣿⣿⣉⣉⡁⠀⠉⠛⠠⠃⠀⠁⣀⠀⠀⢘⣻⣿⣿⡿⠋⢋⣉⣙⡈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡃⠀⠀⠀⠀⠀⠀⠈⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠿⣿⣿⣿⠟⠉⠉⠀⠀⠒⠀⠀⠀⢀⣀⣉⣙⣉⣉⣙⣿⣿⣏⣉⣁⣀⣀⣀⣤⣶⣿⣿⣿⣇⣀⣈⣉⣩⣿⣧⣤⣼⣿⣿⣷⢎⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣁⣀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣠⣴⣦⣶⣶
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣉⣩⣉⣉⡁⠋⢩⣉⣉⣉⣹⣿⣿⣿⣿⣿⡿⢿⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⢈⣽⣿⣽⣯⣿⣿⡿⣿⣿⣿⣿⣋⣉⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⡆⠉⠛⠻⢿⣿⣯
⡀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠆⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠛⠛⠿⠛⠉⠁⠈⠉⡙⠉⠙⢿⣿⢿⣟⠛⠉⠀⠈⠉⠉⠉⠹⡿⣿⣿⢿⣿⣿⡏⢉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⣤⣄⠀⠀⠀⠀⠙
⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡄⠀⠀⠀⠀⡴⠊⠉⠁⠀⣤⡄⠛⢿⣷⣶⣾⣿⣿⣿⣿⠿⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡦⠀⠀⠀
⣿⣿⣿⣿⣿⣆⢤⣤⣤⣤⣤⣤⣤⠄⣿⠀⠀⣀⣤⣄⡀⠀⠀⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⣀⣀⣘⢟⣁⡀⠀⢞⣁⣀⣠⣤⢀⣈⡀⠀⣻⣿⣿⣿⣷⣶⣦⣄⣀⣑⠀⣿⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⢾⠋⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⠆⢻⣿⣿⣿⣿⡏⣴⣄⠀⢸⣧⣀⠀⠛⠀⣼⣏⣀⣈⣿⡄⢰⡟⠁⠀⠘⠇⠀⣿⠇⠀⢸⡇⠀⠀⣸⣏⡀⠘⠀⢸⡇⠀⢸⣿⣿⣿⣿⣿⣿⣧⣼⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⣉⣡⣤⠔⠉⠉⠀⠈⢿⣿⣿⡿⠐⢿⡟⠀⢀⡈⠙⠻⣷⠘⣿⡍⠉⠉⢉⠁⠸⣧⠰⠤⢠⣤⣄⣿⡀⣠⣼⣷⣶⣿⣿⣿⣧⠀⠀⢸⡇⠀⢸⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⡷⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿
⣿⡟⠁⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⢰⡜⠓⠶⠞⠋⠉⠈⠛⠶⠖⠋⠀⠀⠙⠳⠶⠚⠁⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠐⠚⠓⠂⠀⠻⠶⠋⠉⢱⣿⡏⣿⣿⣿⣇⣸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿
⠉⠖⠶⣶⠏⢶⠄⠰⠶⠞⠏⠙⠲⢤⡀⠈⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠺⠾⢿⠁⠉⠻⠯⠿⣽⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿
⣿⣿⣿⡇⣿⣿⣿⡟⡏⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⢻⣿⡿⢻⢻⢻⠇⣧⣸⣵⢓⡧⣼⣿⣿⣽⠳⣿⣿⣿⣿⣻⠿⣯⣿⣿⣽⣿⣿⣟⣿⡟⣿⣿⣿⡿⢿⣹⣿⣿⢻⣿⡿⣿⣿⣻⡿⣿⣿⣿⣯⢿⡟⣿⣿⣿⣟⣿⣻⠛⢿⣺⡿⡻⣩⣿⠏⠤⠟⣿⣿⡿⢛⡿⣿⣿⣻
⡶⣤⣿⣽⣈⣾⣗⢿⡅⡘⣓⣲⣲⣿⣿⡏⢛⡄⢲⣿⣋⣿⣿⡿⢇⣭⡽⠻⣿⣔⡻⣿⣷⢻⠿⢻⣗⢻⣿⢻⠧⢜⣿⠭⠼⢻⣿⣶⡋⠻⢹⣷⣶⣗⡃⠻⠧⣽⣿⣿⢻⠏⠙⣹⣷⣿⣿⡆⠨⠾⠭⢉⣿⣧⣤⣮⣉⣿
⠈⡟⣷⡆⠽⣽⠃⠈⣛⣛⣭⠻⡿⣿⣷⣵⣾⠙⢊⣿⣍⡉⠛⣷⣿⣔⢛⣉⠙⠪⠿⢻⡛⣿⣿⣶⣤⣄⡁⣾⣟⢿⣿⡶⢶⠎⢻⣿⢛⠛⠃⠻⡟⣋⡁⢶⣿⣿⢿⡏⢒⠚⢺⣯⣾⣿⣿⠦⠠⠴⠶⠿⣽⣿⣿⣿⡆⠭
⣾⡿⠻⠀⠀⣤⣆⡀⠉⠻⣷⣾⣶⣾⣿⡇⣴⣿⡄⠋⠋⡝⣛⣙⣻⣿⣿⣷⣤⡷⠖⣬⣿⣷⣿⡟⢿⣿⣵⣤⡴⣿⣿⣟⣰⡄⣈⣭⣺⣟⠂⠠⡞⢿⣿⠃⠙⢿⠀⠐⠚⣷⣿⣿⣿⣿⣯⢍⠒⠒⠷⠖⠂⠀⢿⣯⣵⣾
⢿⡿⠁⠙⠻⠂⠙⠃⢀⣤⢿⡿⠿⣦⢹⣥⡄⣹⡇⠀⠶⢿⣉⣩⣼⡷⣿⣿⡆⣄⠸⢿⣿⣿⣿⠁⠈⣉⠛⠟⠃⠘⣷⡶⠈⢼⢟⣿⣿⠟⠟⠁⠈⢿⡿⠀⠠⠶⣷⣾⣿⠀⠘⣿⣟⡛⠃⣌⣈⣡⡤⠶⠶⣶⣶⡾⠀⠀
⡈⠁⠀⠀⠶⣶⣤⣀⣴⢊⣴⣿⣿⣿⣿⠯⢩⠉⠻⠆⠰⠿⢿⣂⣉⡀⠙⠻⡻⢿⡿⠃⣹⣯⣭⡍⣤⣷⣶⣾⡄⠐⠿⢿⠿⠃⠲⣿⣿⡓⣾⠧⠀⠀⠈⠷⣲⠄⠀⢁⡀⣠⣴⣯⣟⣗⠀⠈⠓⠉⠓⠲⠀⠀⢊⡀⠀⠀
⠍⠄⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⡿⢳⠄⢿⣷⣤⠀⠀⢰⣿⡿⠟⠂⠀⢸⢇⣈⣀⠀⠈⠻⡟⠀⣈⡛⠋⠀⠀⡝⢀⣀⣀⠀⠀⠙⠻⢀⣁⠀⠀⠀⠀⠀⣵⣾⣿⣿⣿⡿⢿⣿⣿⣿⣿⢦⡀⠀⠀⢁⠖⠛⠛⠿⢿⣿
⠼⢶⣄⠀⠀⠀⢉⣩⣸⣿⣿⣿⠛⠀⠛⢻⣿⡟⢻⠛⠛⠃⠀⠐⠚⠛⣿⣿⣿⣿⣿⡟⠛⠃⠘⠛⢻⣿⣿⣟⠻⣿⣿⣿⡿⠆⠘⠛⠛⠛⠿⢿⣇⣤⣶⣾⣿⣿⣿⠛⠻⢻⣿⣿⣿⣿⣇⡈⠃⠀⠀⠘⠒⠢⣤⠀⠀⠀
⣶⣮⣋⠀⠀⠐⠿⢿⡖⣾⡿⣟⢛⣿⡛⠛⢿⣿⣿⣿⣛⠁⣘⣛⣛⠛⠛⠿⠟⠛⠛⠛⠛⠃⠘⠛⣻⣿⣿⣿⣿⣿⡟⠛⡛⠃⠘⠛⠛⠛⠛⢛⣛⡈⠙⠛⣻⠟⠛⢛⠛⠛⣿⣿⣯⣭⡉⠀⠀⠀⠀⠀⠀⠞⠁⢀⣠⣤
⠻⣿⡿⠆⠀⢰⡴⠊⠁⢹⣿⣿⣿⣿⠁⠠⢘⣛⣻⡿⢿⣧⢘⣛⣛⡃⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣿⣻⣿⣿⣿⣿⣙⣛⣛⠋⠁⠀⢘⣛⣛⣛⣿⣟⠀⣚⣛⣛⣛⣛⡃⠀⠉⠻⣿⠿⣇⠀⠀⡀⠀⠀⠀⠀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⣈⣤⣤⡄⠀⠻⣻⡟⠁⠀⠀⣛⣿⣿⣃⡀⠀⢘⣛⣛⡃⢀⠀⠆⠀⡈⠛⠿⢿⣟⡋⢸⣿⣿⣿⣿⣿⣟⣛⣋⡀⠀⠀⢘⣛⣛⣫⣿⣿⣿⣿⣿⣿⣿⣛⡃⠀⠀⠀⠀⠋⢿⣿⣦⣄⠀⠀⠐⡾⡿⠟⠻⠿
⠀⠀⣠⣶⣿⣿⣿⣿⡄⠀⠀⠹⠁⠀⠀⠂⣙⣿⣿⣿⣿⣷⣜⣉⣉⡁⠀⠙⠿⠆⣹⠂⠠⠋⠀⠀⢘⣻⣿⣿⣿⣿⢛⣛⣛⡉⠚⠓⢘⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡃⡀⠀⠀⠀⠀⠈⠙⠛⢿⡄⠀⠈⠀⠀⠀⠀⠀
⠀⣾⣿⣿⣿⡿⠿⠟⠋⠀⢀⠀⠀⠀⢀⣀⣉⣉⣛⣛⣿⣿⣿⣯⣉⣁⣀⣀⣀⣀⣁⣤⣤⣶⣶⣀⣈⣉⣉⣻⣿⣇⣈⣻⣿⣧⣄⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣵⣂⡀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣉⣉⣩⣭⣉⡉⠛⢩⣉⣉⣉⣉⣽⣿⣿⣿⣿⣿⣿⣯⣉⣉⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣉⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣏⡁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢨⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠻⣿⣿⣿⠿⠋⠉⠉⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⢹⠛⣷⠀⣽⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠄⠀⠀⣤⡀⠈⠘⠿⠟⠿⠅⠉⣀⣤⡄⠘⢂⣙⢐⣿⣿⣿⣿⣿⣿⣿⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠶⢄⡀⠀
⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣤⣄⡀⠀⠀⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⢀⣀⠀⢻⣟⡀⠀⣀⣀⣰⣫⣥⢀⣈⡀⠀⣸⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠟⠀
⣿⣿⣿⣿⣷⣤⢤⣤⣤⣤⣤⣤⣴⡄⠀⠀⠸⣧⣀⠀⠛⠀⣼⣏⣀⣈⣿⡄⢠⡟⠁⠀⠘⠇⠀⣿⠀⣰⢽⡇⠀⠀⣿⡏⠀⠘⠀⢸⡇⠀⢸⣿⢶⣎⡙⣿⣩⣭⣿⣁⠀⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀
⣿⣿⣿⣿⣿⣿⠉⢿⣿⣿⣿⣿⡿⢠⣄⠀⢀⡈⠉⠛⣷⠀⣿⡍⠉⠉⢉⠁⠸⣧⠀⠀⢀⡀⠀⣿⡀⠟⢸⡇⠀⠀⢹⡇⣀⣀⠀⢸⡇⠀⢸⣿⠘⣿⣿⣿⣿⣿⣿⣿⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⠟⠛⠻⠿⠏⠀⠈⢿⣿⣿⣿⠃⠻⣿⠃⠘⠳⠶⠖⠋⠀⡯⠛⠶⠖⠋⠀⠀⠙⠳⠶⠛⠙⠓⠚⠳⠖⣛⣷⣶⣶⣿⣿⣿⣿⡗⠚⠓⠂⠀⠻⠶⠋⠻⢿⣿⣿⣿⣿⣿⣿⣿⡷⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾
⣶⣶⠿⣷⠏⢷⠄⠰⠾⠿⠞⠉⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠿⠶⠟⢟⣭⡍⣿⣿⣿⣷⡘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣿
⣿⣿⣿⣧⣿⣿⣿⣇⣿⢿⡧⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⡟⡟⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠿⢿⡟⢸⢹⡯⣇⢻⢨⣵⢃⢏⣾⣿⣿⣿⡯⢿⣿⣿⣿⣿⠿⣿⣿⣿⣶⣿⣿⣷⣻⣟⣿⣿⣿⣿⠟⡏⣿⡏⡿⢿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣻⣿⡿⣛⣯⣿⠿⢹⣼⡏⡇⠼⠻⣿⣿⠟⢿⣿⣿⣻
⠶⢢⣿⢿⢌⣾⣗⢿⡌⣈⣓⣳⣿⣾⣯⣿⠘⣇⢲⣾⣛⣿⣿⣿⠿⣨⣭⠟⢿⣧⣛⣯⣻⡼⠽⠟⣿⡚⣿⡟⡿⠤⢻⡿⢬⠟⣿⣷⣾⠙⠏⣿⣦⣿⣖⠘⢿⣙⣷⣻⣿⣿⠋⢯⣽⣶⣿⣷⠄⠛⣛⣍⠛⣿⣭⣤⣍⡹
⠈⡟⣿⡆⠽⣽⠋⠈⣙⣓⣬⠹⣽⢻⣿⣧⣾⡙⠊⣽⣬⣉⠘⢻⣿⣶⠛⣝⡉⠋⢿⠿⣟⢻⣿⣦⣦⣤⡈⢠⣿⡻⣿⣿⠶⣤⠜⣿⣟⡒⠚⠋⢿⠹⡉⠠⣴⣶⣿⣿⡃⠬⠶⠾⠯⣽⣿⣿⣄⠨⢤⠶⣿⣽⣿⣿⣿⣯
⣾⡿⠻⠀⠀⣤⣆⠀⠉⠻⣿⣼⣶⣼⣿⡧⢤⣿⡆⠙⠋⢩⠺⣋⢻⣿⣿⣷⣤⣾⠶⢠⡟⣻⣿⣿⡻⣿⡿⢴⣬⡴⣿⣿⣿⣤⡀⠈⣓⣾⡟⠂⠀⡶⣿⣿⠆⠹⣯⠀⠰⠼⢷⣶⣾⣿⣿⣿⠓⠀⠴⢴⣦⡅⠉⢻⣿⡓
⣿⡿⠁⠙⠻⠆⠙⠃⢀⣠⣾⡿⢿⡧⢻⣧⡀⣉⣟⠀⠴⠿⣏⣛⣥⣿⢿⡿⠿⢤⠠⠿⢿⣿⣿⣿⠃⠙⣛⢛⠟⠃⠹⣿⡿⠉⢠⡾⣿⣿⡿⡿⠃⠈⢿⣭⠁⠀⣨⣶⣤⣶⡆⠙⠿⣿⡿⠿⣈⠠⢂⣀⣥⣤⣄⣠⣭⣗
⡈⠀⠀⠀⣶⣾⣤⡀⣬⠛⢁⣿⣿⣿⣿⣯⢋⡙⠻⠷⠠⢾⠿⡷⢘⡉⠈⢿⡿⢾⣿⣶⠘⣿⣟⣛⠂⣠⣯⣶⣶⡀⠠⢿⣷⡶⠎⠹⣿⣿⠷⣶⣄⠀⠀⠙⢲⠬⡄⠀⠑⠁⠀⠀⢘⣿⣿⡇⠉⠳⠙⠻⠠⢄⠈⠉⠏⠁
⢉⠀⠀⠀⠀⠈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣀⢰⣶⣄⠀⠀⠀⣶⣿⡿⠷⠀⠀⢰⠄⠊⠁⠀⠹⣿⡿⠁⠙⠟⠋⠁⠀⣲⠀⠘⠁⠀⠀⠻⢿⡇⠉⠀⠀⠀⠀⠀⢚⣤⣶⣶⣿⣦⣾⣿⣿⣿⣷⣦⣄⠀⠀⠀⢁⣤⡶⣾⣶
⡰⣤⡀⠀⠀⠀⠙⠛⢹⣿⣿⣿⡿⠏⠛⠛⢛⠛⢟⠟⠛⠃⠈⠑⠚⠛⢛⣻⣿⣿⣿⣟⡛⠃⠘⠛⠛⣻⣿⣿⡛⢻⣿⣿⣿⣿⡜⠛⠛⠛⠛⣿⣧⡄⠀⢀⣰⣿⣿⣿⣿⡛⢻⣿⣍⣿⣿⣿⣿⡟⠄⠀⠀⠞⠋⣀⣀⠀
⣤⣜⠋⠀⠀⠀⣶⣿⣽⣿⣿⣿⣥⣤⡿⠛⣿⣿⣿⣟⡛⠃⣘⣛⣛⡛⢻⣿⣿⡿⠛⠛⠛⠃⠘⠛⠛⠛⣛⣿⣿⣿⣟⡿⠛⠋⠘⠛⠛⠛⠛⠛⠛⠻⣜⠛⠿⣿⣿⢿⠿⠛⠃⠠⠽⣭⣽⠿⠿⠷⠀⠀⠀⠀⠀⢀⡼⠃
⣿⣿⣷⡄⠀⢠⢉⡽⠇⢿⣿⣷⣴⣾⡇⠀⣛⣛⣟⣻⣿⡦⣘⣛⣛⡃⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⢘⣻⣿⣿⣿⣿⣿⣿⣟⠋⠀⠄⢘⣛⣛⣛⣛⣛⢷⣞⣛⣛⣿⣛⡃⠀⠀⠀⠸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠈⠁⣠
⠀⠙⠉⠀⠀⠸⠃⠀⠀⠘⣿⣿⣿⡿⠁⠈⣛⣛⣛⡛⠉⠛⢙⣛⣛⡃⠀⠀⢀⠀⠀⠀⢾⣿⣿⣿⣿⣻⣻⣿⣿⣿⣿⣟⡋⠀⠀⠀⢘⣛⣛⡋⣛⣻⣟⣛⣛⣻⣿⣿⣧⠀⠀⠀⠀⠈⠙⢛⣿⣤⣐⠄⠀⠀⡀⢀⣼⣿
⠀⠀⠀⣠⣴⣾⣿⣿⠃⠀⠈⠽⠏⠀⠀⠀⣛⣻⣿⣷⣦⣄⣘⣉⣉⡁⠐⠐⣤⣤⡀⢶⡀⠉⣹⠗⢚⣛⣛⣿⣿⣛⣛⣛⣻⡔⠢⣤⣾⣛⣛⡃⠙⣻⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠈⠀⣿⣿⣱⣄⠀⠀⠜⠛⠋
⠀⣴⣿⣿⣿⣿⣿⣿⡆⠀⠀⠈⠀⠀⢀⣀⣉⣻⣿⣿⣿⣿⣿⣍⣉⣁⣀⣀⣀⣈⣁⣞⣁⡀⢁⣀⣈⣉⣉⣻⣿⣿⣿⣏⣉⣁⢀⣀⣈⣉⣉⡁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣅⡀⠄⠀⠀⠀⠀⠀⠀⠉⠙⢺⠀⠀⠀⠀⠀
⠸⣿⣿⣿⡿⠛⠋⠉⠁⠀⠠⠀⠀⠀⢈⣍⣩⣉⣍⣉⣙⡿⣿⣿⣏⣉⣉⣉⣉⣉⣩⣽⣿⣿⣿⣿⣭⣉⣉⣉⣉⣻⣿⣉⠉⣿⣿⣿⣯⣉⣉⡁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀
⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠉⡁⠈⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⠿⠉⠉⠉⠉⣩⣿⣷⣶⣿⣿⣿⣿⣿⣿⡏⠹⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⣿⠟⠁⠀⠀⠀⠈⠀⡐⢿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣁⠀⠻⢿⣏⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣤⣄⡀⠀⠀⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⣀⣀⠀⢀⣀⣀⡀⣀⣛⣻⣿⣿⣏⣉⡉⠉⣸⣿⣀⣸⣃⣹⡃⣸⣿⣿⡿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⣧⣀⡀⠛⠀⣼⣏⣀⣈⣿⡄⢠⡟⠀⠠⠘⠇⠀⣿⠀⠀⢹⣿⣷⡄⣿⡏⠀⠙⢀⣸⡧⠐⢸⣿⠀⠈⠈⣿⡀⠻⣿⣿⣿⣿⣿⡿⢿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣷⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡈⠙⠛⣷⠀⢿⡍⠉⠉⢉⠁⢸⣧⠀⠀⢀⡀⠀⣿⡀⠀⢸⣇⣹⠗⣿⡇⠀⢀⡼⢸⡇⠀⢸⣿⠀⣀⠀⠛⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣯⠻⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠘⠓⠶⠖⠋⠀⠈⣛⠶⠖⠋⠀⠀⠙⠳⠶⠛⠀⠀⠘⠳⠖⠛⣿⠋⠐⠛⠓⠂⠈⠐⠚⠓⠂⠀⠻⠶⠋⠀⠀⢿⣏⣈⠉⢉⣭⣭⣭⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⠾⣿⣿⣿⣿⣿⣿⠃⢶⣦⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⠓⠂⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⣠⣤⠀⠀⠀⠀⠀⠿⠴⠟⢻⣿⣿⣿⣿⣿⣿⣿⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣟⣿⣿⣷⢿⢿⣿⣇⣿⢽⡿⡿⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⢻⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿
⡿⠿⢿⣟⢸⢸⠻⢇⢹⢨⣽⠃⠏⣾⣿⣿⣿⡮⢯⢿⣿⣿⣟⡛⣿⣿⣿⣷⣿⣿⣿⣫⣟⢻⣿⣿⣿⠿⢹⢽⣿⢹⢽⣿⣿⣿⣟⣿⢿⣿⣾⣿⡷⣿⡻⣿⣿⢿⣿⡿⡿⢟⣯⣿⠿⢻⢻⣿⡟⢩⠟⣿⣿⡿⢻⣿⣿⣿
⠖⢲⣿⢯⢃⣿⣿⢻⡌⢀⣓⣳⣿⣾⣯⣿⠘⣇⠣⣾⣛⣹⣿⣿⠗⣸⢭⠟⢿⣷⣚⡿⣻⡞⡿⠟⣿⣛⢿⡏⠻⠤⢹⣿⢥⠜⢿⣷⣶⡋⠿⣻⣦⣼⣖⠂⠻⣏⣋⣚⣿⢿⡏⠻⣿⣧⣿⣿⡦⢚⣛⣋⡛⢿⣯⢥⣽⡊
⢈⡟⣭⠅⠿⢾⠋⠈⣉⣑⣦⠹⠿⢻⣿⣽⢾⡋⠋⣽⣾⣍⠘⠻⣾⣦⡛⣍⡉⠛⢿⠟⣟⠻⣿⣷⢧⣤⣈⠁⣾⣛⢿⣿⡶⢤⠬⢻⣟⢓⠲⠋⢻⡝⠋⠀⢤⣴⣾⣿⡟⢨⠴⠶⠾⣩⣿⣿⣧⡄⢡⣴⣴⣯⣿⣿⣿⣿
⣿⠟⠳⠀⠀⣦⣦⡀⠉⠹⣿⣾⣶⣾⣿⣗⢠⣿⣇⠈⠉⢡⠚⣋⣛⣿⣿⣿⣦⡼⠶⠂⡝⣛⣼⣿⡟⣿⡿⢵⣬⣄⣾⣿⣿⢧⡀⠈⢙⣲⡿⠂⠀⣰⢾⣿⡷⠐⣿⠅⠑⠦⠷⣶⣶⣿⣿⣿⣿⠂⠤⢤⣦⣬⠍⠉⣼⣿
⣿⡟⠀⠙⠻⠂⠙⠃⢀⣠⣾⡿⠯⡥⢻⣧⡀⣈⣟⠀⠰⠿⣯⣙⣡⣿⠿⡿⠿⡄⡀⠾⢿⣿⣿⣿⠟⠘⢛⠛⡻⠛⠹⣿⣯⠉⠀⣴⢿⣿⣿⣿⠟⠀⢸⣿⡍⠀⢀⣴⣤⣴⣦⡜⠛⢿⣿⠿⢧⡁⡐⢀⣀⣄⣀⣀⣈⣻
⡈⠀⠀⠁⣶⣶⣤⡀⣴⠋⢀⣿⣿⣿⣿⡯⠋⡉⠛⠷⠀⠾⠿⠻⢘⡉⠀⠻⡿⠷⣿⣶⠆⢻⣟⣛⠒⣀⣮⣴⣶⡄⠀⢼⣿⣶⡶⠉⢭⣿⡿⢶⣦⠀⠀⠉⢡⡤⢥⡀⠉⠝⠉⠀⠀⠹⣿⣿⠀⠛⠌⠻⠏⠄⣀⠉⠉⠋
⠡⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⣿⣯⣀⢰⣶⣤⡀⠀⠀⣴⣿⡿⠷⠀⠀⢰⠇⠘⠁⠀⠘⣿⣿⠇⠘⠿⠛⠉⠁⢐⠆⠀⠋⠀⠀⠘⢿⣿⠈⠁⠀⠀⠀⠀⠀⣋⣤⣶⣶⣶⣠⣴⣿⣿⣿⣶⣄⣀⠀⠀⠐⢁⣤⡶⣶
⠶⣤⡀⠀⠀⠀⠛⠋⢻⣿⣿⣿⣿⠋⠛⠛⠛⠛⢟⠛⠛⠃⠈⠑⠚⠛⠛⣛⣿⣿⣿⣿⡛⠃⠘⠛⠛⢻⣿⣿⡛⠛⣿⣷⣾⣿⣟⠛⠛⠛⠛⢻⣷⣄⠀⠀⢘⣻⣿⣿⣿⣟⡛⢿⣿⡏⣿⣿⣿⣿⡿⠁⠀⠠⠟⠋⣐⠀
⣤⡙⠃⠀⠀⢠⣶⣾⣸⣿⣿⣿⣖⣀⡿⠛⣿⣿⣿⣟⡛⠃⢘⣛⣛⡛⢻⣿⣿⡿⠛⠛⠛⠃⠘⠛⠛⠛⠛⣻⣿⣿⣟⢿⠟⠛⠙⠛⠛⠛⠛⠛⠛⠛⣧⡘⠿⣿⣿⣿⣿⠿⠇⠈⠡⢼⣽⣿⣿⣿⣷⠄⠀⠀⠀⠐⠉⣹
⣿⣿⣷⡀⠀⢠⣉⡽⠉⣿⣯⣭⣹⣿⡆⠀⣛⣻⣟⣋⣿⣦⣘⣛⣛⡃⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢘⣛⣿⣿⣿⣿⣿⣿⣿⠛⠀⠠⢘⣛⣛⣛⣛⣓⡰⣾⣛⣛⣛⣿⡋⠀⠀⠀⠀⠨⣿⣿⣷⣦⠀⠐⠀⠀⠀⠀⠘⠃
⠈⠋⠁⠀⠀⡿⠁⠀⠀⠸⣿⣿⣿⡿⠁⠀⣛⣛⣛⡛⠋⠛⢙⣛⣙⡃⠀⠀⢀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡋⠀⠀⠀⢘⣛⣛⡋⣛⣛⣿⣟⣛⢛⣛⣛⣧⡀⠀⠀⠀⠀⠈⠛⠿⣿⣀⣤⠀⠀⠀⡀⠀⣰
⠀⠀⠀⣀⣤⣶⣶⡶⠀⠀⠙⣽⡏⠀⠀⠀⣛⣻⣿⣷⣤⣀⢈⣉⣉⡁⠐⠐⣤⣀⣀⢠⡀⠈⠛⡻⢽⣛⣛⣻⣿⡿⢿⣿⣟⣠⠶⣀⣼⣛⣛⡃⠘⣛⣻⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡦⣄⠀⠀⡸⠛
⢀⣴⣾⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢀⣀⣉⣽⣿⣿⣿⣿⣿⣍⣩⣁⣀⣀⣀⣁⣉⣠⣋⡀⢊⣀⣈⣉⣉⣙⣿⣿⣿⣿⣏⡁⢀⣀⣉⣉⣉⡁⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠄⠀⠀⠀⠀⠀⠈⠉⠛⢽⠄⠀⠀⠀
⣾⣿⣿⣿⡿⠟⠛⠋⠁⠀⡄⠀⠀⠀⢈⣍⣉⣉⣉⣙⣻⣿⣿⣿⣏⣉⣉⣉⣉⣉⣉⣭⣭⣷⣾⣿⣯⣉⣉⣉⣉⣿⣿⣏⠉⢹⣿⣿⣭⣉⣉⡁⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀
⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠙⡋⠈⠉⠉⠉⠉⣹⣿⣿⣿⣿⣿⣿⣿⡍⠉⠉⠉⠉⠉⣥⣤⣼⣿⣿⣿⣿⣿⣷⣶⠖⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠟⠉⠁⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠸⣿⣿⡿⠿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⠀⠀⣀⣤⣄⡀⠀⠀⣀⣤⣄⡀⠀⠀⢀⣠⣬⣉⡀⣀⣀⠀⠀⣀⡀⠀⣄⣙⣻⣿⣿⣟⣻⡿⣿⣻⣿⣁⣀⣀⣈⡉⢄⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⠀⠛⠀⣼⣏⣀⣈⣿⡄⢠⡟⠁⠀⠘⠇⠀⣿⠀⠀⢸⡧⣶⡄⣿⡏⠈⠛⠃⢸⡇⠀⢸⣿⠀⠀⠈⣿⡀⠐⣼⢡⣿⣿⣿⣫⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡈⠉⠛⣷⠀⢿⡍⠉⠉⢉⠁⠸⣇⠀⠀⢀⡀⠀⣿⡀⠀⢸⡇⠈⣿⣿⡇⠀⠀⠀⢸⡟⠉⢹⣿⠀⣀⠀⠘⣷⣸⣃⣸⠟⠻⠿⡿⣽⡿⢋⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣦⣄⢠⣦⣤⣤⣤⣤⣤⣤⣤⣤⣦⠀⠀⠘⠓⠶⠖⠋⠀⠈⢛⠶⠖⠋⠀⠀⠙⠷⠶⠚⠀⠀⠘⠳⠖⠛⠓⣲⡝⠛⠓⠂⠀⢰⠞⠓⠂⠀⠻⠶⠋⠀⠀⢹⠏⣙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⣯⢿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠴⣤⣀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠏⠀⠿⣄⡀⠀⢈⣩⣵⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⢻⢿⣿⣇⣿⢹⡿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣹⣿⢻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿
⠿⠹⣿⣏⢸⢸⡛⣏⢹⢨⡾⠃⠏⣺⣿⣿⢿⡭⢯⢿⣿⣿⣟⡛⣿⣿⣿⣷⣾⣯⣿⣃⣟⣼⣿⣿⣿⡿⢹⢸⣿⢹⢸⣿⡷⣿⣟⣿⣿⣿⣿⣿⣿⢿⡻⣿⣿⣿⣟⡿⣿⠟⣿⣿⡿⠋⣇⣿⢼⠠⠿⢿⣿⣿⠻⣿⣿⣟
⢛⣶⡿⢺⣣⣿⡯⢿⣄⢀⣙⣻⣝⣿⣿⣿⠈⣃⠫⣾⣟⣹⣿⣿⢗⣨⢭⠟⢻⣷⣞⡽⣼⣗⡿⠿⢻⣏⢿⡿⢹⠧⠜⣿⠥⠼⢻⣿⣶⡏⠻⢹⣷⣼⣶⠊⠻⣏⣻⣜⣿⣿⡟⠹⣹⣷⣼⣿⣦⢘⣛⣋⡙⢻⣿⢥⣬⣟
⣈⡛⣿⠅⠛⠾⠁⠀⣉⣅⣶⠙⠿⢻⣿⣼⢾⡋⠋⣽⣷⣭⠈⠻⣾⣷⡛⣫⡉⠛⠽⠟⣿⠻⣿⣷⢶⣦⣌⡁⣾⣟⢿⣿⣶⠦⡤⠛⣿⡓⠒⠋⠹⡯⢏⠁⠤⣴⣾⣯⣟⠡⠵⠶⡿⢏⣽⣿⣿⣄⢡⣤⣶⣿⣿⣿⣿⣿
⣿⣟⠣⠀⠠⣶⣤⡀⠈⠙⣿⣾⣶⣾⣟⣗⢰⣿⣇⠈⠉⢠⣙⣋⣙⣿⣿⣿⣦⡴⠷⠂⣜⣛⣾⣿⣟⢿⣿⠧⣤⣄⢼⣿⣿⡗⣄⠀⠉⣣⣿⠓⠀⢠⡾⣿⣿⠆⢾⡏⠈⠖⠿⢶⣶⣾⢿⣿⣿⣷⠠⠤⢴⣤⠥⠉⠡⣿
⣿⠇⠀⠙⠻⢂⠘⠁⢀⣤⢿⡿⠿⡤⢹⣥⡄⣀⣏⠀⠰⠿⣯⣙⣥⣽⡿⡿⠿⠥⣄⠾⠿⣿⣿⣿⡿⠈⢛⠛⡻⠛⠈⢿⣿⡍⠁⢰⡾⣿⣿⣿⠿⠃⠘⣿⣯⠀⠀⣡⣦⣤⣶⣦⠉⠹⢿⡿⠿⢏⠀⠂⣀⣠⣄⣀⣀⣉
⡀⠀⠀⠠⠷⣶⣤⡀⣴⠊⣠⣿⣿⣿⣿⡯⠋⡍⠙⠷⠀⠾⠿⢻⣐⣉⠀⠻⣿⠷⢿⣿⠇⢚⣿⣛⣳⢂⣼⣵⣶⣆⠀⠰⢾⣷⡶⠎⠩⣽⡿⠷⣶⡀⠀⠈⠃⣤⠤⣄⠀⠩⠋⠀⠀⠀⠻⣿⣿⠘⠣⠙⠿⠯⢄⡀⠉⠹
⠠⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣟⣯⣀⢸⣷⣦⡀⠀⠀⣴⣿⡿⠿⠀⠀⢐⡦⠐⠁⠀⠈⢿⣿⡟⠈⠙⠛⠉⠁⠐⡦⠀⠊⠁⠀⠀⠻⣿⡇⠉⠀⠀⠀⠀⠀⢘⣡⣴⣶⣶⣆⣠⣴⣿⣿⣿⣷⣄⣀⡀⠀⠊⣠⣤⡶
⠶⣤⠀⠀⠀⠈⠉⠋⢿⣿⣿⣿⣿⠋⠛⠛⣛⣛⠟⡛⠛⠃⠈⠑⠚⠛⢛⣛⣿⣿⣿⣿⡛⠃⠘⠛⠛⠛⣿⣿⣟⠛⣻⣷⣾⣿⣿⡛⠛⠛⠛⠛⣷⣦⡅⠀⠘⢛⣿⣿⣿⣿⣛⡙⢿⣿⠏⣙⣿⣿⣿⣿⠇⠀⠴⠛⠛⢐
⣤⡙⠁⠀⠀⢠⣶⣇⣛⣿⣿⣿⣀⣀⡟⠛⣿⣿⣿⣟⡛⠃⢘⣛⣛⡛⠛⣿⣿⣿⠟⠛⠛⠣⠘⠛⠛⠛⠛⠛⣿⣿⣿⣻⠿⠛⠙⠛⠛⠛⠛⠛⠛⠉⠻⣄⠛⠿⣿⣿⣿⠿⠿⠇⠈⠉⣤⣽⣻⣿⣿⣿⡄⠀⠀⠀⠀⠉
⣿⣿⣦⠀⠀⡈⣉⠽⠩⣿⣭⣍⣹⡿⡆⠀⣛⣛⣟⢋⣿⣷⣘⣛⣛⡃⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢘⣛⣻⣿⣿⣿⣿⣿⣿⣟⠀⠀⢜⣛⣛⣛⣛⣛⢀⣾⣿⣛⣛⣛⡷⠀⠀⠀⠀⠀⠠⣿⣿⣷⣶⡀⠀⠀⠀⠀⠀⠀
⠉⠋⠁⠀⠀⡟⠁⠀⠀⢿⣿⣿⣿⡿⠀⠘⣛⣛⣛⡓⠋⠛⢛⣛⣛⡁⠀⠀⠀⡀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⠃⠀⠀⢘⣛⣛⡋⣛⣛⣛⣿⣛⢛⣛⣛⣃⣀⡀⠀⠀⠀⠀⠈⠻⢿⠿⣇⡄⠀⠀⠀⠀⠀
⠀⠀⠀⣠⣴⣶⣶⡶⠀⠈⢛⣿⠛⠀⠀⠀⣛⣻⣿⣷⣤⣀⢘⣉⣉⡁⠀⠐⣤⣁⠀⣀⣄⠈⠛⢻⢿⣿⣛⡻⣿⣿⣿⣿⣿⡖⡠⢄⢘⣛⣛⡃⠙⣛⣛⣻⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣀⣄⠀⠀⠰
⢀⣴⣿⣿⣿⣿⣿⣇⠀⠀⠈⠊⠀⠀⢀⣂⣉⣽⣿⣿⣿⣿⣿⣍⣉⣁⣀⣀⣈⣉⣉⣁⣟⡀⢐⣁⣈⣉⣉⣁⣻⣿⣿⣿⣿⡁⢀⣈⣉⣉⣉⡁⠀⢉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣅⣀⡀⠀⠀⠀⠀⠀⠈⠛⠻⢝⡆⠀⠀
⣿⣿⣿⣿⡿⠿⠛⠋⠀⠀⡄⠀⠀⠀⢈⣉⣉⣉⣙⣛⣿⣿⣻⣿⣭⣉⣉⣉⣉⣉⣉⣉⣭⣥⣾⣿⣿⣍⣉⣉⣉⣹⣿⣿⠏⠁⣾⣿⣯⣍⣉⡁⠀⠈⣻⠋⣩⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀
⠙⠛⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⡉⠁⠙⡛⠉⠉⠉⠉⠉⢩⣿⣿⣿⣿⣿⣿⣿⣍⠉⠉⠉⠉⠉⠩⣀⣀⣾⣿⣿⣿⣿⣥⣤⣤⣤⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠁⠀⠀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢻⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣷⣶⡞⡁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⣿⠀⠀⣀⣤⣄⡀⠀⠀⣀⣤⣄⡀⠀⠀⢀⣠⣬⣉⡉⢀⣀⠀⠀⣀⡀⠀⣀⣈⣻⣿⣿⣿⣏⣻⣿⣿⣿⣻⣿⣀⣀⡈⢻⣿⣋⠉⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⠀⠛⠀⣼⣏⣀⣈⣿⡄⢠⡟⠁⠀⠘⠇⠀⣿⠀⠀⢸⡧⢀⣤⣿⡏⠁⠹⢿⢿⡷⡅⢸⣿⠀⠀⠈⣿⡀⢀⣼⢁⠀⠀⢈⣿⣿⣿⣿⢿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⠛⣷⠀⢿⡍⠉⠉⢉⡁⠸⣧⠀⠀⢀⡀⠀⣿⡀⠀⢸⡇⠀⠻⣿⡇⠀⠀⠀⢸⡇⣠⢼⣿⠒⣈⠀⠘⣷⣸⠃⠀⠁⠀⣼⣿⡿⡀⡶⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⣀⠀⠀⣀⣀⠀⠀⠀⢀⡀⠀⠀⠘⠓⠶⠖⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠳⠶⠚⠀⠀⠘⠳⠖⠛⠓⠂⢐⣻⠗⠂⠀⠐⠚⣳⠇⠀⠻⠶⠋⠀⠀⢹⠏⠀⢶⣶⣤⣅⣈⣙⣀⣀⣬⣙⣣⣤⣴⠀⠀⠀⠀⠀⠀
⣿⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠿⠶⠟⠀⠀⠀⠮⠙⠻⠿⢿⣿⣿⣿⣿⣿⢿⠟⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⢿⢿⣿⣧⣿⢹⡿⡿⣿⠯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣏⡟⢹⣿⣿⣿⣿⣿⣿⣿
⠿⠹⣿⣏⢸⢸⡑⣇⢸⢨⡾⠋⠏⣺⣿⣿⢿⡭⢯⢿⣿⣿⣟⡛⣿⣿⣿⣷⣾⣯⣿⣁⣟⣸⣿⣿⣿⡿⢿⣰⣾⠽⢿⣿⡷⣿⣿⣿⡿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣟⡿⢿⠟⣻⣽⡿⡏⡟⣿⣿⠣⠟⢿⣿⣿⠞⣿⢿⣟
⢛⣶⡿⢺⣢⣿⡯⢻⣄⢠⣹⣽⣽⣿⣿⣿⠈⣇⠫⣾⣭⣹⣿⣿⢗⣴⢭⠟⢻⣷⣞⡹⣼⣗⡛⠟⢻⣯⢻⣿⢹⠣⠊⣿⠥⠼⠹⣿⣶⣯⠛⢻⣷⣶⣷⡒⠘⠯⢽⣴⣿⡿⣿⠙⢹⣽⣼⣿⣷⡜⠚⢟⡍⠛⣿⡭⣬⣽
⣰⡹⡿⠀⠛⠿⠁⠀⣭⣍⡦⢙⠟⢫⣿⣴⢾⡎⠋⣽⣷⣬⠈⠛⣾⣷⣋⣫⡅⠓⠽⠟⢿⡛⢿⣿⣶⣦⣤⡁⣸⣟⠻⣿⣷⡶⡶⠝⣿⡗⡒⠛⠘⢿⠓⡁⠠⢬⢶⣿⣿⡃⠕⠒⠾⠏⣡⣾⣿⣧⡬⢤⡶⢾⣽⣟⣽⣿
⣿⣋⠃⣀⡰⢶⣦⡀⠈⠙⣷⣾⣷⡾⣛⡗⠰⣿⣇⠈⠉⢠⣍⣋⡹⣿⣿⣷⣶⣴⠿⠂⣬⣛⣵⣿⣯⢽⣿⡧⣦⣥⠼⣿⣿⡿⢀⠀⠈⣑⣿⣓⠀⠀⡿⢿⣿⡧⠸⣯⠀⠰⠰⠶⢶⣶⡿⢿⣿⣿⡤⠤⠶⢶⡶⠌⠉⢲
⡶⠃⠀⠉⠛⢃⠈⠁⢠⣴⢿⠿⢿⣤⢹⣧⣄⣤⣏⠀⠰⠿⣯⣙⣥⣿⠿⣿⠿⠧⢤⠺⠛⣿⣿⣿⣿⠋⢙⡙⠻⠛⠁⠹⣿⡿⠀⢀⣾⢿⣿⣿⡿⠛⠀⢿⣭⡅⠀⢀⢶⣦⣷⣶⡎⠉⠹⣿⠿⠿⡄⠑⢀⣀⣤⣄⣤⣤
⠀⠀⠀⠰⠷⣶⣦⡀⡖⢈⣬⣿⣿⣿⣿⡮⠋⠍⠙⠷⠀⠾⠿⢻⣀⣉⠀⠻⣿⠳⠿⢿⡿⠘⣿⣛⡓⠂⣰⣵⣶⣶⠀⠠⠾⢷⡶⠖⠉⢼⣿⡿⢶⣆⠀⠀⠉⢰⡤⠤⡄⠀⠑⠁⠀⠀⠀⢻⣿⣇⠉⠟⠛⠛⠡⢠⡀⠉
⠄⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣟⣯⣀⢹⣷⣦⡀⠀⠀⣴⣿⡿⠿⠀⠀⠀⡾⠐⠁⠀⠀⠻⣿⡿⠀⠙⠛⠋⠀⠀⢲⠄⠐⠁⠀⠀⠘⢿⣿⠀⠀⠀⠀⠀⠀⠀⣫⣤⣶⣾⣷⡀⣠⣾⣿⣿⣿⣷⣦⣀⡀⠀⢃⣤⣴
⢶⣄⠀⠀⠀⠈⠙⠙⣿⣿⣿⣿⡿⠉⠛⢛⣛⣛⡛⠛⠛⠃⠀⠐⠚⠛⠛⣛⣿⣿⣿⣿⣟⠃⠘⠛⠛⠛⣻⣿⣿⡛⢛⣷⣶⣿⣿⣟⠛⠛⠛⠛⢻⣷⣦⡀⠘⠛⣻⣿⣿⣿⣟⣋⡙⠻⣿⠏⢙⣿⣿⣿⣿⡟⠀⡜⠛⠛
⣄⡛⠀⠀⠀⣴⣾⣝⣿⣿⣿⣟⣀⣀⡟⠛⣿⣿⣿⣿⡛⠃⢘⣛⣛⡛⠛⣿⣿⣿⠟⠛⠛⠃⠘⠛⠛⠛⠛⠛⢻⣿⣿⣟⣿⠟⠙⠛⠛⠛⠛⠛⠛⠉⠙⢧⣛⠛⠿⣿⣿⣿⠿⠿⠃⠀⠁⣠⣜⣛⣿⣿⣿⣤⠀⠀⠀⠀
⣿⣿⡦⠀⠀⣌⣡⠏⢻⣿⣩⣙⣿⣿⡄⠀⣛⣛⣟⡋⣿⣷⣘⣛⣛⡃⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢘⣛⣛⣿⣿⣿⣿⣿⣿⣧⡂⠀⢸⣛⣛⣛⣛⣛⠀⣘⣿⣟⣛⣛⣃⠗⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀
⠙⠉⠀⠀⢀⠋⠀⠀⠈⣿⣿⣿⣿⡟⠀⠐⣛⣛⣛⡃⠋⠛⢛⣛⣛⡁⠀⠀⠀⡀⠀⠀⠀⣴⣿⣿⣿⣿⣟⣛⣛⣿⣿⣿⣿⡟⠁⠀⢘⣛⣛⡋⣛⣛⣛⣛⣿⢛⣛⣛⣃⣀⣀⡀⠀⠀⠀⠀⠈⠻⢿⣿⣇⡀⠀⠀⠀⢀
⠀⠀⢀⣠⣴⣶⣶⠆⠀⠘⢟⣿⠏⠀⠀⠀⣛⣻⣿⣧⣤⣀⢘⣉⣉⡁⠀⠀⢆⣀⠀⠀⣠⠈⠙⠻⣟⣛⣛⡻⢿⣿⣿⣿⣿⣤⢦⣤⢘⣛⣛⡃⠙⣛⣉⣛⣫⣼⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠞⢻⣿⣆⢠⠀⠀
⣀⣴⣿⣿⣿⣿⣿⡄⠀⠀⠈⡍⠀⠀⢀⣂⣉⣽⣿⣿⣿⣿⣿⣭⣉⣁⣀⣀⣀⣉⣉⣋⣸⡃⢀⣊⣉⣉⣉⣁⣸⣿⣿⣿⣯⡍⢀⣀⣉⣉⣉⡁⠀⢉⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠈⠛⠛⢮⣧⠀
⣿⣿⣿⣿⡿⠿⠟⠋⠀⢀⠀⠀⠀⠀⢈⣉⣉⣉⣛⣻⣿⣿⢻⣿⣯⣉⣉⣉⣉⣉⣉⣩⣉⣡⣬⣽⣿⣯⣉⣉⣉⣹⣿⣿⣿⠁⢘⣿⣯⣭⣉⡁⠀⠈⣩⠛⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠝⠀
⠛⠟⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⡉⠉⠙⡗⠋⠉⠉⠉⠉⠉⣽⣿⣿⣿⣿⣿⣿⣯⠉⠉⠉⠉⠉⠙⡓⠀⠈⣿⣿⣿⣻⣷⣄⣀⣠⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠿⠛⠋⠀⠀⠄⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⣠⡄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣿⠀⠀⣀⣤⣄⡀⠀⠀⣀⣤⣄⡀⠀⠀⢀⣠⣭⣉⡙⢁⣀⠀⠀⣀⡀⠀⢀⣀⣈⣻⣿⣿⣿⣯⢿⣿⣿⣿⣿⣿⣟⡀⠀⣉⣋⢏⣿⠿⢯⣙⣛⣛⣻⣯⣿⣿⣿⣿⣿⣥⡄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⠀⠋⠀⣼⣏⣀⣈⣿⡄⢠⡟⠁⠀⠘⠇⠀⣿⠀⠀⢸⣇⡀⠀⣹⡏⠀⠙⠛⣾⡏⣶⣸⣿⠘⠉⠈⣿⡀⠀⡼⠁⠛⠁⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠙⠛⣷⠀⢿⡍⠉⠉⢉⠁⠸⣇⠀⠀⢐⡀⠀⣿⡀⠀⢸⡇⠀⠈⣿⣇⠀⠀⠀⢸⡇⠀⢹⣿⣀⣀⠤⠞⣷⣰⠃⢲⠀⠀⠱⠀⠀⢨⡫⡿⠿⠿⣻⣿⣟⠛⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠶⠖⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠳⠶⠚⠀⠀⠘⠳⠖⠛⠓⠂⠐⠛⣿⣧⠀⠐⠚⠓⠂⣸⡻⠶⠋⠀⠀⢹⠏⠀⠀⠀⠀⠀⠀⠀⠿⠋⢁⠘⠞⠿⠛⠛⠀⠀⠀⠀⠀
⣤⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣤⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠞⠁⠀⠀⠀⠀⡴⠋⠀⠀⠀⠺⠶⠏⠀⠀⠀⠀⠚⠿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⠀⠀⠀
⢟⣿⣿⣧⣿⣿⣿⣇⣿⢿⡿⡟⣾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⢹⣿⡿⢹⣿⣿⣿⣿⣿⣿⣿
⠿⠹⣿⣝⢸⣸⡹⣃⢹⢨⡾⠩⠇⣺⣿⣿⣿⡭⢯⢿⣿⣿⣟⡛⣿⣿⣿⣷⣾⣯⣿⣁⣟⣸⣿⣿⣿⡟⣿⣷⣿⡏⣿⣿⡷⣿⣿⣿⡿⣿⣿⣿⣯⢿⡟⢿⣽⣿⣿⣿⢿⡟⣻⣼⡿⣏⠼⣾⡗⠇⠽⢻⣿⣿⡞⢿⢿⣿
⣫⣾⠟⢞⡬⣿⡟⣿⣄⢌⣩⣿⣿⣿⡿⡿⢈⣇⠫⣶⣭⣹⣿⣿⢗⣰⢭⠟⢻⣷⣞⡹⣼⣗⣛⠿⠻⣯⠹⣿⠻⠣⠊⣿⠥⠾⠹⣿⣾⣿⠙⢏⣷⣶⣿⣛⠘⠿⠼⣤⣿⣿⣿⠙⢛⣉⣾⣿⣿⡦⠔⠿⡤⠛⣻⡯⣠⣭
⣘⢹⡿⠐⠓⠿⠁⢀⣭⣡⡦⢑⠟⣯⣿⣴⢿⡎⠋⣽⣷⣬⠈⠛⣾⣷⣋⣩⡅⠛⡸⠟⢿⡋⢿⣿⣾⣦⣤⣁⢸⣿⠻⣿⣷⡶⡶⠎⢻⣿⠚⠛⠈⠿⠓⣁⡠⠾⡶⣿⣿⡟⠶⠓⠺⠟⢠⣾⣿⣷⣮⠵⠶⠾⢯⣿⣹⣿
⣟⡛⢂⣀⡰⢶⣤⠀⠈⠹⣷⣾⣷⣿⣛⡖⠲⣿⣇⠈⠉⠀⣍⣋⡹⣿⣿⣷⣶⣴⠿⠂⣬⣙⣵⣿⣯⣹⣿⡯⣦⣤⡰⣿⣿⣿⢣⡀⠈⢉⣾⣗⠀⠀⢸⠻⣿⡿⠈⢿⠁⠑⠖⠓⠶⣶⣾⠿⣿⣿⣷⠴⠲⠶⢶⠶⠈⠐
⡾⠂⠈⠙⠛⢂⠉⠁⢠⣤⢿⠿⢭⡄⢹⣥⣄⣤⣏⠀⠰⠿⢿⣙⣡⣽⠯⣿⠿⠧⢤⡸⠛⢿⣿⣿⣿⠏⠙⣏⢛⠟⠃⠹⣿⣿⠡⠀⣴⡟⣿⣿⣿⠟⠁⠹⣯⡍⠀⠀⠱⣶⣾⣷⣶⠈⠉⢽⣿⠿⠭⡈⢃⣠⣤⣤⣤⣤
⠁⠀⠀⠸⢷⣶⣤⡀⡖⣁⣼⣿⣿⣿⣿⣮⠋⠍⠙⠷⠀⠾⠿⢻⣆⣉⠀⠹⣿⠓⠾⢿⡿⠐⣿⣟⡛⠒⣤⣷⣶⣶⡀⠀⠴⢿⣶⠿⠈⢨⣿⡿⠷⣆⠀⠀⠈⠃⣶⠴⢤⠀⠀⠊⠀⠀⠀⠘⢿⣿⣇⠉⠝⠛⠋⠤⣄⠀
⠄⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⢹⣷⣦⡀⠀⠀⣴⣿⡿⠿⠀⡀⠀⣺⠀⠋⠀⠀⠹⣿⣿⠃⠈⠛⠋⠁⠀⠐⡦⠀⠋⠀⠀⠀⠻⣿⠇⠀⠀⠀⠀⠀⠀⢐⣥⣴⣶⣿⣇⡀⣠⣾⣿⣿⣿⣿⣦⣀⣀⠐⢡⣴
⢦⡀⠀⠀⠀⠈⠋⢻⣿⣿⣿⣿⡟⠉⠛⢛⣛⣛⣟⢛⠛⠃⠈⠙⠚⠛⢛⣛⣻⣿⣿⣿⣟⠃⠘⠛⠛⠛⢻⣿⣿⣟⠛⣿⣶⣿⣿⣿⠛⠛⠛⠛⠃⣿⣶⣄⠘⠛⢛⣿⣛⣿⣿⣟⣁⡐⠻⣿⡟⠙⣿⣿⣿⣿⡿⠠⠞⠛
⣈⠚⠀⠀⠀⣴⣶⡸⠿⣿⣿⢋⢁⣀⡛⠛⣿⣿⣿⣟⡛⠃⢘⣛⣛⡛⠛⣿⣿⣿⡿⠛⠛⠃⠘⠛⠛⢛⠛⠛⠻⣿⣿⣿⣻⠿⠛⠛⠛⠛⠛⠛⠛⠈⠉⠻⡟⡛⠛⢿⣿⣿⣿⠿⠿⠃⠀⠉⣀⣐⣛⣿⣿⣿⣣⠀⠀⠀
⣿⣷⡄⠀⠀⡉⣉⠙⠿⡟⣛⠛⣿⣯⡄⠀⢛⣻⣟⠋⣿⣶⣘⣛⣛⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣛⣳⣿⣿⣿⣿⣿⣧⡤⠀⢘⣟⣛⣛⣛⣓⠀⣘⣻⣿⣟⣛⡃⢀⠗⠀⠀⠀⠀⠀⠀⣾⣿⣿⣭⣍⠉⠀⠀⠀
⠛⠉⠀⠀⢸⠋⠀⠀⢱⣿⣿⣿⣿⡟⠀⠠⢛⣛⣛⡃⠟⠛⢛⣋⣛⡁⠀⠀⠀⢀⠀⠀⠀⢠⣷⣿⣾⣿⣿⣛⣛⣻⣿⣿⣿⣿⠟⠀⢘⣛⣛⡋⣛⣛⣛⣛⣛⣟⣛⣛⡃⠁⣀⣀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡀⠀⠀⠀
⠀⠀⠀⣀⣤⣤⡤⠄⠀⠻⠿⡿⠏⠀⠀⠀⣛⣻⣿⣥⣄⡀⢘⣉⣉⡁⠀⠀⣄⡈⠀⠀⢀⡈⠙⠿⢿⣛⣛⡛⣻⣛⣿⣿⣿⣟⣤⣤⣘⣛⣛⡃⠙⣛⣋⣙⡛⣸⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠰⣛⣿⣷⢠⡀
⣠⣴⣿⣿⣿⣿⣷⠀⠀⠀⠚⠃⠀⠀⢀⣀⣉⣽⣿⣿⣿⣿⣿⣍⣉⣁⣀⣀⣈⣛⣙⣛⣋⡗⢀⣰⣋⣉⣉⣁⣀⣻⣿⣯⣭⣍⣁⣈⣉⣛⣛⡁⠀⢉⣉⣉⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢯⣳
⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⢈⣉⣉⣙⣿⣿⣿⣿⣿⣿⣯⣍⣉⣉⣉⣉⣉⣉⣝⣁⣈⣭⣭⣭⣩⣉⣉⣉⣿⣿⣿⣿⢈⣻⣭⣭⣉⡁⠀⠈⢉⠛⠋⢫⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⢈
⠻⠟⠟⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠺⣯⠟⠉⠉⠉⠉⠉⢩⣽⣿⣿⣿⣿⣿⣿⡉⠉⠉⠉⠉⠉⠽⠅⠈⢹⣿⣿⣿⣷⣦⠀⠀⠀⢀⣈⣉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠐⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⠃⠀⠀⠀⠀⠀⠈⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⣿⠀⠀⣀⣤⣄⡀⠀⠀⣀⣤⣄⡀⠀⠀⢀⣨⣭⣛⡛⣛⣁⠀⢀⣀⡀⠀⢀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣀⣀⠁⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⠀⠋⠀⣼⣏⣀⣈⣿⡄⢠⡟⠁⠀⠘⠇⠀⣿⡀⠀⢸⡇⠀⠀⢻⡏⠀⠙⢓⢿⣿⣿⣿⣿⠿⣿⡿⣿⡿⠃⡼⠁⠀⠈⣙⣿⠟⠙⠦⣯⣭⣭⣭⣽⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⠛⣷⠀⢿⡍⠉⠉⢩⠁⠸⣇⠀⠀⢀⡄⠀⣿⡀⠀⢸⡇⠁⠀⢸⣷⡆⠀⠀⢸⡟⠿⢺⣿⠀⣀⠀⠘⣷⣰⠃⠀⠀⠀⠈⠀⢀⡀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠶⠖⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠳⠶⠚⠀⠀⠘⠳⠖⠛⠓⠂⠐⠚⠛⢿⡆⠐⠚⠓⠂⠈⠻⣶⠟⠒⠀⢹⠏⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⢘⠝⠿⠻⠟⣹⣿⣿⠉⠉
⠀⠶⠶⣶⡧⠶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠟⠁⠀⠀⠀⠀⣰⠧⠀⠺⠤⠏⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠿⠙⠙⠀⠀
⣿⣿⣿⡏⣷⣿⣿⡏⣇⢿⡇⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡏⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣿⣿⢸⣿⣿⣿⣿⣿⣿⣯
⠿⠿⣻⣁⢻⣻⠻⣁⢿⢨⡿⢋⠏⣾⣿⡿⣿⠭⢯⣿⣿⣿⣟⡻⣿⣿⣿⣷⣿⣯⣿⣁⣞⣼⣿⣿⣿⡿⣇⣗⣷⡇⡇⣻⣷⣿⣟⣿⣿⣿⣷⣿⣯⡻⡿⢻⣿⣿⣿⡿⢿⣟⣹⣾⣿⣫⠿⢹⣷⠏⠼⢻⣿⣿⣟⣹⡿⢿
⢶⣾⠿⡿⣼⣟⠙⣯⡄⣈⣛⣳⣿⣿⣿⡟⢙⣇⠣⣾⣛⣹⣿⣿⢗⣨⢍⠟⢻⣷⣞⡹⣼⣷⣛⠿⠻⢯⠝⣿⠛⡷⠎⣻⡯⠶⠟⢿⣷⣿⡎⠋⣿⣶⣾⣟⠀⠻⠭⠥⣽⣿⣻⠏⠋⣛⣷⣽⢿⣷⡆⠸⠥⠬⢉⣿⣖⣺
⡟⣯⡇⠘⠳⠟⠀⢉⣩⣱⣆⠩⠟⣿⣷⣵⢿⡋⠋⣽⣷⣭⠘⠛⣾⣷⣚⣫⡉⠛⠸⠿⢿⡋⡿⣿⡾⣶⣤⣍⢰⣿⡻⣿⣿⣶⢶⠶⠹⣿⡟⠛⠃⠹⠏⢊⣀⡐⢷⢾⣷⣿⠳⢞⠒⠻⠛⢠⣾⣿⣿⡦⠶⠶⠿⢿⠿⣩
⠿⠻⠀⠀⣴⣤⣀⠈⠉⢻⣷⣶⣦⣿⣿⡅⢶⣿⡆⠈⠉⠁⡜⣋⣹⣿⣿⣿⣶⣴⠷⠆⢨⠛⣃⣿⣯⡭⣿⡿⢵⣬⡀⢾⣿⣿⡟⣄⠀⠉⢑⣿⡂⠀⠀⡿⢻⣿⡷⠸⣯⠁⠰⠐⠒⠶⣶⣾⣿⣿⣿⣿⡶⠚⠶⢶⠖⠀
⡿⠀⠙⠻⠗⠈⠛⠀⢀⣸⣿⣿⣏⡈⢻⣥⡄⣈⣟⠀⠰⠾⢿⠙⣁⣼⡾⣿⣿⣯⢠⡴⠿⢻⣿⣿⣿⡿⠛⢟⠛⡻⠛⠈⢿⣿⡍⠀⠀⣴⡿⣿⣿⣿⠟⠀⢹⣯⡍⠀⠈⠳⣶⣿⣷⣶⡆⠉⠩⣿⡿⠽⠥⢃⣠⣤⡄⢠
⠀⠀⠀⣶⣦⣥⡀⢀⡍⢃⣹⣶⣾⣷⡮⡙⠋⡙⠛⣷⠀⢶⡿⠿⡇⠋⠁⠹⢿⠞⢼⣿⡶⡆⢿⣿⠿⠷⢆⣸⣥⣶⣆⠀⠀⠿⣷⡶⠶⠉⠭⣿⠿⠿⣆⠀⠀⠉⠓⣦⠤⢥⠀⠀⠟⠁⠀⠀⠐⠻⣿⣷⠘⠟⠛⠛⠠⢤
⡀⠀⠀⠀⠈⢙⣿⣾⣷⣿⣿⣿⣿⣿⡏⠉⢶⣶⣄⠀⠀⠀⣠⣾⣿⡿⠄⠀⠈⢠⡆⠐⠉⠀⠘⣿⣿⡿⠀⠙⠿⠋⠉⠀⢲⡄⠀⠋⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⠀⠈⢠⣃⣤⣶⣶⣶⡀⠀⢀⣤⣾⣿⣿⣿⣦⣄⠀⠃
⡀⠀⠀⠀⠀⠛⢯⣿⣿⣿⣿⣿⣷⠞⠿⠛⠻⠿⡿⠛⠛⠃⠉⠛⠛⡛⠛⠛⢛⣻⣿⣿⣿⠃⠘⠛⠛⠛⠟⣿⣿⣟⠛⠓⣪⣴⣾⣿⡟⠛⠛⠛⠛⠃⣶⣶⣜⠛⠛⠛⣿⡿⢿⣿⠟⠋⠁⠀⠾⣿⣿⠛⢽⣿⣿⣿⣿⣷
⠹⠅⠀⠀⠀⣠⢘⣿⣿⣿⣿⠟⠁⠀⠛⢻⣿⣿⣟⡛⠛⠃⢘⣛⣛⡛⠛⢻⣿⣿⣿⡿⠿⠃⢘⠛⠛⠛⠛⠛⠻⢿⣿⣟⣻⣿⡿⠛⠛⠛⠟⠛⠛⠀⠉⠙⢻⡟⠛⠻⢿⣿⣿⣿⣿⣿⣿⡇⠀⠘⠋⠀⠨⢿⢿⣿⢟⠁
⣦⡀⠀⠀⠸⠿⣮⣿⣿⣿⣥⣬⣤⡴⠀⠈⣛⣿⣿⡿⢷⣤⣘⣛⣻⡇⠀⠈⠉⠙⠋⠀⠀⠀⠀⠀⢘⣛⣛⣃⣀⣤⣿⣿⣿⣿⣤⡀⢘⣛⣛⣛⣛⣛⠀⣘⣛⣛⣻⣟⡃⠈⠉⠙⣫⠅⠀⠀⠀⠀⠘⢹⣿⣿⣿⣛⠛⠛
⡿⠟⠀⠀⣰⡤⠃⠉⢉⣩⣭⣹⣿⣷⠀⠀⣘⣛⣛⡇⣺⣿⣿⣛⣛⡁⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⡋⣛⣛⣛⣛⣛⢛⣛⣿⡗⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡇⠀
⠀⠀⠀⠀⠏⠀⠀⠀⢿⣿⣿⣿⡿⠃⠀⠁⣉⣉⣉⡁⠀⠀⢘⣋⣉⡁⠀⠀⡀⠀⠂⠀⠀⠀⠘⢿⣿⣿⣿⣿⠻⠛⣿⣿⣿⣿⣿⡟⢘⣛⣛⡃⠙⣛⣛⣉⡉⢈⣛⣛⣁⣤⣤⣶⣶⣿⡆⠀⠀⠀⠀⠀⠀⠈⠛⣻⣷⣦
⠀⢀⣤⣶⣶⣾⠇⠀⠈⠻⣛⠛⠀⠀⢀⣀⣙⣹⣿⣿⣶⣦⣌⣉⣉⣁⣀⣀⣘⣦⣤⣤⣤⣰⣆⣀⣉⣻⣛⣋⣀⣐⣿⣭⣿⣿⣿⣷⣞⣿⣿⡧⠀⢉⣉⡉⠀⢈⣽⣿⣿⣿⣿⣿⣿⣿⣷⡤⠀⠀⠀⠀⠀⠀⠀⠁⠺⣿
⣾⣿⣿⣿⣿⣿⣇⠀⠀⠘⠍⠀⠀⠀⢈⣉⣉⣿⣿⣿⣿⣿⣿⣿⣭⣉⣉⣉⣉⣉⣉⣉⣉⡵⢋⣉⣉⣉⣉⣉⣉⣉⣽⣿⣿⣿⣿⣭⣉⣉⣉⡁⠀⠈⢉⠁⠀⣼⣿⣿⣿⣿⣿⣿⠟⠛⠉⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡿⠿⠿⠟⠀⠀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠻⢿⣟⣿⣿⣭⠉⠉⠉⠉⠉⠉⢉⣡⣬⣿⣿⣿⣿⠉⠉⠉⠉⠉⠻⣿⡿⠋⠉⣿⣿⣷⣦⣀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⣠⣶⣾⣿⣿⣿⣶⣦⣄⣀⠀⠀
⠾⠭⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠉⢻⠞⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠸⠛⣠⡄⢿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣤⣀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣤⣔⣀⠀⠀⣀⣤⣄⡀⠀⠀⢀⣸⣿⣿⣿⣿⣿⡿⠟⣛⡋⠀⢀⣀⣀⣠⣤⢀⣈⡀⣒⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠁⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠂⠀⠸⣧⣀⡀⠛⠀⣼⣏⣀⣈⣿⡄⢠⡿⠁⠀⠹⠇⠀⣿⠀⠀⢸⡇⠀⠀⣽⡏⠀⠘⠀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠹⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⠛⣷⠀⢿⡍⠉⠉⣩⠁⢸⣇⠀⠀⢀⡀⠀⣿⡀⠄⢸⡇⠀⠀⣿⡇⠀⠀⠀⢸⡟⠹⢿⣿⣿⣎⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠬⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠶⠖⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠲⠶⠚⠀⠀⠘⠳⠖⠛⠓⠂⠘⠛⠓⠂⢀⣴⠚⠓⠂⠊⢻⣿⡟⢰⠉⢹⠟⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⢈⣹⡿⠛⠯⣬⣍⣙⣉⣛
⠀⠆⠲⣶⠂⢶⠄⠰⠤⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⢂⡐⠀⠀⠀⠙⠃⠿⠷⠟⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠛⢿⣿⣿
⣿⡿⣿⡜⣧⣿⣿⡽⡇⣿⡏⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⢿⣿⣿⣿⣿⣿⡟⣿⡏⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡏⣿⢿⢹⣿⣿⣿⣿⣿⣿⣿
⡟⠿⣿⡃⢿⡿⢹⡁⣧⣨⣿⢛⠯⣿⣿⣿⣿⠭⢿⣿⣿⣿⣟⡻⣿⣿⣿⣷⣿⣯⣿⣁⣞⣼⣿⣿⣿⡿⡇⣇⣗⡝⡇⣻⣿⣿⣟⣿⣿⣿⣷⣿⣯⡻⡿⢻⣿⣿⣽⡿⠿⣟⣹⢾⣿⣫⡿⢻⣿⠸⠸⣛⣿⣿⣿⣹⡿⢿
⣲⣾⢯⠯⣸⣟⡺⣯⡀⡘⣛⣲⣲⣿⣿⡟⢛⣇⢣⣼⣛⣹⣿⣿⢷⣨⢍⠿⢻⣷⣚⡻⣽⣷⢿⠿⢻⣯⠜⣿⠝⡟⠄⢻⡯⠶⠟⢿⣷⣿⣯⠛⣿⣶⣾⣟⡁⠻⠭⠧⣼⣿⣛⡏⠛⣛⣳⣼⡿⣿⡦⠈⠯⠬⠛⣿⣗⢚
⠿⣷⡆⠹⢿⡟⠈⢉⣫⡲⣌⠫⡟⣻⣷⣱⢾⡙⠊⣽⣾⣉⠘⠻⣼⣧⡞⣫⡋⠻⠸⠿⢿⡏⣿⣿⣶⣦⣤⣍⢰⣿⡛⣿⣿⣶⠶⡶⠙⣿⡗⡛⠋⠘⠿⢋⣀⡰⢷⣶⣿⣿⡗⢾⠓⠛⠟⠀⣼⣿⣿⣷⡶⠶⠾⠽⡿⢏
⡗⠏⠀⠀⣤⣴⡀⠈⠙⢿⣧⣾⣤⣿⠿⡍⢤⣾⡆⠘⠛⠉⡒⢂⡹⢿⣿⣿⣦⣤⡷⠦⢨⠛⠃⣹⣯⡽⣿⡿⢵⣬⡀⢸⣿⣿⣿⢆⠀⠉⠓⣿⣓⠀⠀⢻⡿⣿⣷⠜⢯⠅⠱⢖⠕⠶⢶⣶⣾⡿⣿⣿⣶⡖⠳⠶⡶⠆
⡶⠀⠻⠾⠧⠙⠟⠀⢀⣸⣿⣿⣋⡈⣻⣁⡀⣉⡟⠀⢠⡶⠿⠉⢋⣹⣿⣿⣿⣟⣠⡰⠿⠟⣿⣿⣿⣿⡟⢿⡛⢻⠟⠃⢹⣯⣿⠡⠀⢠⣾⣿⣿⣿⡿⠃⠰⣿⣟⠀⠀⠱⡦⣼⣵⣦⣦⠈⠉⢼⣿⣿⣿⡜⣀⣨⣤⡀
⠀⠀⠀⣤⣤⣅⡀⢀⡽⠋⢿⣿⣿⣶⣌⠛⠟⡛⠿⣦⠀⣴⣾⠿⡏⠙⠃⠸⣶⡶⢨⣿⣶⡾⢻⣿⣿⣿⡤⣸⣧⣤⡤⠀⠀⢹⣿⣶⣶⡎⢙⣺⣿⣿⣧⠀⠀⠉⠋⣤⣤⠬⣄⠈⠩⠋⠉⠀⠀⠘⢿⣿⡆⠹⠛⠛⠋⠨
⠀⠀⠀⠉⠙⢻⣿⣾⣶⣶⣿⣿⣿⣿⠗⠓⣦⣥⡀⠀⠀⠀⢠⣴⣷⣿⡄⠀⠈⠀⣄⠩⠛⠁⠀⢿⣿⣶⠆⠛⠿⠟⠉⠀⠠⣄⠀⠙⠀⠀⠀⢻⣷⣶⠀⠀⠀⠀⠀⠈⠁⡸⣁⣤⣴⣶⡆⠀⠀⠀⢀⣼⣿⣷⣶⣤⣀⠀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣛⣛⣻⣿⠛⢻⣿⡿⠿⠛⠂⠘⠛⠛⠛⠛⠛⠛⢻⣻⣿⣿⡃⠘⠛⠻⠟⠛⢻⣿⣛⠛⠓⢐⣡⣼⣿⡟⠛⠛⠛⠋⠟⢰⣦⣜⠛⠛⠛⢛⣿⣿⣿⣿⠟⠛⠁⠀⣴⣿⣿⡿⢿⣿⣿⣿⣷
⢗⡄⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠋⠀⠛⢻⣿⣿⣿⠻⠛⠁⢘⣛⣛⡛⠛⠛⣿⣿⣿⣿⣿⠇⠘⠛⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⡿⠟⠛⢛⠛⠛⠀⠈⠛⠿⣻⡛⠛⣻⣿⣿⣿⣷⣶⣶⣿⡇⠀⠙⡿⠁⠀⣿⣿⣿⣿
⣄⠀⠀⠀⢴⣿⠸⣿⣿⣿⠳⠀⠀⣀⠀⠸⣿⣿⣿⣷⣦⣀⢘⣛⣛⣃⠀⠘⠙⠻⠟⠁⢀⠀⠈⠀⢘⣛⣛⣃⣀⣀⣙⣿⣿⣿⣍⡀⢘⣛⣛⣛⣛⣛⠀⣘⣛⣛⣛⣛⡃⠈⠙⠛⠟⢟⡉⠉⠁⠀⠀⠀⠶⣶⣿⣿⡿⠿
⣿⡷⠀⠀⡈⢉⡛⠿⠿⠿⠿⣿⣿⣏⠀⠀⢛⣛⣿⡋⢸⣿⣻⣛⣛⡃⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⢘⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣾⣟⣛⡋⣛⣛⣛⣛⣛⢛⣛⣛⣻⡦⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣷
⠁⠀⠀⠀⠟⠁⠀⢰⣴⣿⣿⣾⣿⡏⠀⠄⣉⣉⣙⡑⠚⠉⢙⣛⣉⡁⠀⠀⠀⠀⡆⠀⠀⠀⠠⣾⣿⣿⣿⣿⡟⡛⢛⣻⣿⣿⣿⣿⣿⣛⣛⡃⠙⣛⣋⣉⡁⢘⣛⣻⡃⠀⣀⣀⣠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠙⠿⡿⠿
⠀⠀⠀⣠⣀⣠⡄⠀⢻⣿⣿⣿⠋⠀⢀⣀⣋⣩⣭⣥⣄⣀⢈⣉⣉⣁⣀⣀⣸⣆⣀⣀⣀⣀⣄⣈⣛⣻⣟⣛⣁⣁⣘⣿⣿⣿⣿⣿⣘⣭⣩⣧⡀⢉⣉⡉⠀⢈⣉⣩⣷⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠸⠻
⣤⣾⣿⣿⣿⣯⠀⠀⠀⢱⣾⠁⠀⠀⢨⣉⣉⣹⣿⣿⣿⣿⣿⣯⣍⣉⣭⣍⣉⣉⣉⣉⣉⡁⣻⣋⣉⣝⣉⣉⣉⣉⣩⣿⣿⣿⣿⣏⣩⣉⣉⡁⠀⠈⢉⠁⠀⢈⣽⣿⣿⣿⣿⣿⣿⣿⡿⠿⠏⠀⢀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠃⠀⠀⠀⠈⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣯⠉⠉⠉⠉⠉⠉⠉⠙⠈⣉⣉⣉⣭⡍⠉⠉⠉⠉⢿⣿⣿⣿⠏⠹⣯⣥⣄⡀⠀⠀⠀⠨⠛⠙⠉⠙⠿⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⠟⠛⠋⠉⠁⠀⠀⠄⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠉⠙⠫⣿⣫⠟⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⢻⣯⡄⠀⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣤⣄⡠⠀⠀⣈⣥⣄⡀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠆⢀⣀⣀⣠⣤⢁⣊⡀⠀⣹⣿⣻⣿⣛⣿⣀⢀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⡈⠛⠀⣼⣏⣀⣈⣿⡄⢠⡿⠁⠘⠹⠿⠟⣿⠟⠉⢸⡇⠀⠀⢿⡏⠀⠘⠀⢸⡇⠀⣸⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⡈⠙⠛⣷⠀⢿⡍⠉⠉⢉⡁⠸⣧⠀⠀⢀⡀⠀⣿⡀⠀⢸⡇⠀⠀⢿⡇⠀⠀⠀⢸⡇⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠈⢛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠶⠖⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠳⠶⠚⠀⠀⠘⠳⠖⠛⠓⠂⠐⠟⠓⠂⠀⠐⠚⠓⠊⠋⢻⣿⣿⣿⡜⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠁⠙⢻⣿⣿⣿⣿⣿⣿⣿
⠀⠆⠲⣶⠇⢶⠄⠠⠠⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠈⠛⠻⣿⠷⠟⠛⠉⠙⠋⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠙⣛⡿⣿⣿⠿⠿
⣿⣿⣿⣼⣿⣿⣿⣸⡏⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⡏⣿⣏⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡟⣿⣿⠇⡇⡿⢽⠃⡇⣫⣷⢛⡿⣿⣿⣿⣿⢹⣟⣿⣿⣿⣿⠿⣿⣿⣿⣷⣿⣯⣿⣃⣟⣼⣿⣿⣿⡿⡇⣇⣿⣿⡇⣻⣿⢿⣿⣿⣽⣿⣿⣿⣧⡻⣿⢻⣿⣿⣿⣿⠟⣟⡹⣾⣻⣫⡧⢇⣿⡺⠸⢟⣿⣿⣿⣙⣿⢿
⣢⣾⣟⠧⣾⣷⣾⣇⡁⡙⣞⣒⣶⣿⣿⡟⠛⡅⢳⣽⣛⢻⣿⣿⡿⣨⣍⠿⢿⣷⣚⠻⢽⣧⠯⠿⢻⣿⡹⣿⠟⡿⠤⢻⡟⠴⠟⢿⣷⣾⣯⠿⢻⣶⣾⣿⡃⠹⠭⢧⣼⣿⣿⡟⠙⣛⣻⣪⣿⣿⣿⠌⠿⠬⠟⢹⣷⡐
⢿⣓⡊⡼⣼⡟⠘⠙⢓⢻⡍⢻⡿⣿⣿⣧⣾⠝⠎⣻⣯⣙⠘⠾⣼⣧⠞⢛⡋⠻⠹⡿⢿⡯⣿⣷⣶⣤⣀⣙⠁⣿⣓⣿⣿⣲⣦⣤⡝⣿⣛⠒⠟⠈⢿⠉⡁⠠⠾⣶⣿⣿⡟⠷⡕⠺⠿⠃⢁⣿⣿⣿⣴⠶⠤⠯⢽⠿
⡾⡏⠀⠀⣤⣀⠀⠘⠻⢿⣦⣮⣄⣿⣿⡁⣤⣷⡄⠙⠛⠉⡒⠖⢻⣿⣿⣿⣥⣤⣷⠦⢈⠻⠃⢹⣭⡹⣿⣿⣧⣍⡉⢸⣿⣿⣿⣧⠀⠈⠋⢾⣗⠀⠀⢹⣿⣿⣷⡦⢿⡏⠁⠆⡮⠶⢶⣶⣮⣿⣿⣿⣿⣶⡶⠶⢶⠶
⡆⠐⠶⣶⡍⠻⠷⠀⠀⢸⣿⣿⠟⠹⣿⣁⠈⠋⡷⠀⢠⣶⣷⠻⠏⣹⣽⣿⣿⠿⢁⣰⡾⠿⣷⣿⣿⣿⣿⢿⠳⠿⡿⠆⢻⣿⣟⡞⠀⠀⣠⣾⣿⣷⣶⡇⠑⣾⣟⡓⠀⠱⢦⣤⣵⣦⣤⡤⠛⢻⣾⣿⣿⣿⠐⣀⣨⡄
⠁⠀⢀⣄⣀⡂⠀⢀⣻⠛⣿⣭⣿⣅⡛⠿⡷⠿⢿⣦⠀⣤⣾⣶⡏⠻⠋⢹⣿⣽⢈⣿⣤⣴⢻⣿⣿⣿⣇⣀⣇⣨⣤⠀⠀⢙⣿⣤⣤⣤⠛⢯⢿⣿⣟⡀⠀⠈⠻⢡⣄⣀⣂⠈⠙⡻⠋⠁⠀⠀⢼⣿⣷⡄⡟⠻⠿⠋
⠀⠀⠈⠉⠛⢿⣷⣼⣤⣤⣽⣿⣿⣿⡿⠦⣤⣁⠀⠉⠀⠈⢉⣤⣷⣶⡀⠀⠙⠃⣈⠛⡿⠏⠀⣿⣿⣯⡥⠸⣿⡿⠛⠂⠀⣀⠉⢙⠏⠁⠀⠸⣿⣿⡍⠁⠀⠀⠀⠈⠙⠀⡞⢁⣀⣤⣤⠀⠀⠀⠀⠈⢹⣧⣤⣤⣀⠀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣛⡛⢻⣿⣿⣿⠛⠀⠚⠿⠟⢛⠛⠛⠛⢛⣟⣛⣛⡃⠘⠛⠿⣿⠛⠛⣛⡛⠛⠓⡀⠞⣙⣻⣿⠛⠛⠛⠛⠻⠇⣤⣜⠛⠛⠛⠛⢻⣿⣿⣿⣿⠿⠖⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿
⣦⡀⠀⠀⠀⢩⣿⣿⣿⣿⣿⣿⡶⠺⠟⠛⠛⠛⠻⢻⠛⠃⢘⣛⣛⡛⠛⠛⣻⣿⣿⣿⣿⣧⠘⠛⠛⠛⠛⠛⣿⣿⣿⣟⣿⣿⣿⣿⣿⡿⠛⠛⠛⠀⠀⠻⢿⣿⡟⠛⠛⢛⣿⣭⣯⣥⣤⣤⣤⡤⠛⢻⣿⡿⠉⢻⣿⣿
⠉⠀⠀⠀⣠⣄⣿⣿⣿⣿⡟⣋⠀⠀⠰⢻⣿⣿⣿⣧⣄⠀⢸⣛⣛⡃⠀⢠⢿⣿⣿⠛⠉⠁⠀⠂⢘⣛⣛⣃⣀⣉⣙⣿⣿⣟⡛⠋⢘⣛⣛⣛⣛⣛⠀⣚⣛⣛⣛⣛⡃⡘⠛⢿⣿⣿⡿⠟⠛⠃⠀⠀⠁⣀⣀⣘⣛⣚
⡷⡄⠀⠈⠿⢧⣿⣻⣿⣧⣥⣬⣤⡤⠀⠘⣛⣛⣿⡟⢻⣿⣞⣛⣛⡋⠒⠁⠀⠀⠀⠀⠄⠀⠀⠀⢘⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣷⣜⣛⣛⡋⣛⣛⣛⣛⣛⢛⣛⣛⡷⣦⡀⠀⠀⢀⡴⠂⠀⠁⠀⠀⠀⠀⣼⣿⣿⣯
⠋⠀⠀⢰⡢⠖⠙⠛⢉⣩⣉⢿⣿⡤⠀⠀⣈⣉⣛⣇⣺⠿⣛⣛⣉⡁⠀⠀⠀⠀⡀⠀⠀⠀⠈⣁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡃⠙⣛⣛⣋⡉⢈⣛⣛⣳⠞⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿
⠀⠀⠀⠘⠀⠀⠀⢻⣿⣿⣿⣿⡟⠁⢀⣒⣉⣉⣉⣁⣀⡀⢈⣉⣉⣁⣀⣠⣠⣀⣁⣀⣀⡀⢀⣙⣿⣿⣿⣿⣿⣨⣝⣻⣿⣿⣿⣿⣿⣟⣉⣡⡀⢉⣉⡉⠀⢈⣉⣉⣋⣀⣤⣶⣶⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⣀⣤⣷⣶⡶⠀⠈⢿⠿⡿⠟⠁⠀⢈⣍⣉⣹⣿⣿⣿⣧⣌⣩⣉⣭⣉⣉⣉⣙⣿⣿⣿⡇⢻⣍⣉⣉⣭⣛⣋⣉⣉⣿⣭⣽⣿⣿⣿⣿⣿⡏⠉⠈⢉⠁⠀⢈⣉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣧⠀⠀⠀⠾⠿⠀⠀⠀⠸⠉⠉⣽⣿⣿⣿⣿⣿⣿⣦⡉⠉⠉⠉⠉⠉⠉⠉⠁⡞⠉⠉⠋⠉⠉⠉⠉⠉⠙⣿⣿⣿⣿⣯⡍⠉⠁⠀⠀⠀⠀⠀⠈⠉⣿⣿⠿⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠂⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⠀⠀⠀⠀⠈⢻⣿⣿⣿⠀⢸⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⣤⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢿⠟⠋⠉⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿⠀⠀⣀⣤⣄⣀⠈⠙⣫⣵⣏⡀⠀⠀⢀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⡀⣀⣀⣀⣠⣤⢀⣾⡇⠀⣸⣿⣿⣿⣿⣿⣦⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⣦⣤
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣈⠟⠀⣼⣏⣀⣈⣿⡄⢠⡟⠀⢼⣿⣿⣿⣿⣿⣿⢿⡿⠟⠀⣾⡏⠀⠘⠀⢸⡇⠀⢸⣿⠀⠉⠙⢿⡿⠆⡼⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡈⠙⠛⣷⠀⢿⡍⠉⠉⢉⠁⠸⣇⠀⠈⢻⡟⠛⣿⡏⠁⢸⡇⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⢸⣿⠲⣶⣶⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠘⠓⠶⠖⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠳⠶⠚⠀⠀⠘⠳⠖⠛⠓⠂⠐⠚⠓⠂⠀⠐⠚⠓⠂⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢽⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠖⠲⣶⠀⠠⠄⠠⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠈⢿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠙⠙⠻⣿⣿⣿⣿⣿
⣿⣿⣿⢸⣿⣿⣿⣸⡏⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣟⣿⢿⣿⣿⣿⣿⣇⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣹⣿⣏⣿⣿⣿⣿⣿⣿⣿
⡿⣿⣟⠏⡏⡿⣽⠋⡇⣫⣷⢻⡣⣿⣿⣿⣽⢻⢟⣿⣿⣿⣿⠿⣿⣿⣿⣧⣿⣯⣿⣁⣟⣸⣿⣿⣿⡿⡏⡇⣿⣿⡏⢿⣿⣿⣿⣿⣽⢿⣿⣿⣿⡿⣻⠻⣿⣿⣿⣿⡿⣿⣙⣿⣻⣯⡿⠏⣷⡾⠱⠟⣿⣿⣿⣍⣿⠿
⣠⣾⡗⢧⣷⣷⢾⣇⠁⠛⠖⣞⣶⣿⣿⡏⠻⡄⢂⣽⣛⣻⣿⣿⡯⣨⣏⠿⢿⣯⣘⠻⢿⣧⠯⠿⢿⣟⡻⣿⡯⡿⠤⢻⡟⢤⠿⢽⣟⣾⣯⡿⢹⣦⣾⣶⡚⠹⠏⠿⣬⣻⣿⡿⠙⢙⣹⣎⣿⣿⣾⡆⠻⢧⠟⢫⣿⡀
⢿⣭⠈⣼⣽⠟⠈⠛⢒⢯⡝⢻⡿⣿⣿⣣⣾⠛⠜⣻⣯⣙⠹⠿⣼⣯⡾⢛⡚⠻⢙⣾⢿⡿⣿⣶⣦⣦⣄⡙⠁⣾⣒⣾⣿⣶⣦⣤⡽⢿⣿⠕⠞⠉⢿⡟⡁⠀⠬⣴⣾⣯⣿⠷⡵⠴⢾⠿⠉⣺⣿⣿⣧⣶⡦⠾⣽⣿
⣼⡋⠀⠀⣉⡄⠀⠚⠻⣿⣇⣮⣸⣿⣿⡁⣉⣷⡄⠹⠛⠛⡴⠖⢻⣿⣷⣿⣁⣀⣦⡤⢈⠳⠖⢻⣿⡚⣿⣿⣗⣍⡉⠈⣷⣿⣷⣦⠀⠘⠻⢰⣶⠀⠀⠐⣿⣿⣷⣮⢻⣟⠋⠦⢘⡠⣤⣤⣤⣹⣿⣿⣿⣦⣤⢦⣤⣥
⡌⠡⣤⣼⡙⠿⠆⠀⠀⣾⣿⣿⠟⠷⣾⠂⠉⠛⡷⠀⢀⣴⣷⠚⠗⢺⣿⣿⣿⣿⠇⣠⣾⡶⢦⣾⣿⣿⣾⣶⠦⣿⣿⡶⠹⣿⣿⡷⡁⠀⠀⣨⣿⣷⣶⡶⠉⢭⣿⡿⠀⠙⢦⣀⣚⣦⣀⣈⡘⠻⠟⣽⣿⣿⡟⠀⢀⣁
⠀⠀⢀⣉⡠⠀⠀⠠⢶⠻⣟⣛⣶⡀⠿⢿⠦⠿⣷⣄⠈⣠⣼⣶⡞⢿⠏⢻⣿⣿⠉⢷⣀⣠⢟⣾⣿⣿⡟⠀⡶⣀⣁⡀⠀⠙⢿⣃⣁⣀⡸⢿⣽⣿⣿⡏⠀⠈⠛⠇⣀⡀⠠⠉⠛⠛⡿⠛⠁⠀⠈⣽⣿⣭⡹⠿⠿⠿
⠀⠀⠘⠛⠿⣿⣆⣘⣁⠀⣿⣿⣿⣿⣷⣄⣁⡂⠀⠙⠀⠙⠋⣠⣧⣤⡄⠀⠹⠏⠘⠻⢿⡟⠐⢻⣿⣛⣛⠹⣿⣿⠿⠆⠀⢀⠙⠛⢿⠛⠁⠀⣿⣿⣟⠛⠀⠀⠀⠀⠛⠻⢩⠗⠀⢀⣁⣀⠀⠀⠀⠈⠛⢻⣁⣀⡀⠀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⣿⣿⣿⣟⠛⠀⢼⣿⡿⢛⠛⠛⠛⠛⡿⡛⠛⠃⠘⠛⢿⣿⡟⠛⠛⠛⠛⠓⠀⢨⠛⠛⣛⡛⠛⠛⠋⠿⣿⢀⣘⠛⠛⠛⠛⠛⢫⣶⣿⣿⣿⣿⡄⠀⠀⠀⣠⣴⣿⣿⣿⣿
⡄⠀⠀⠀⠨⡽⣿⣿⣿⣿⣿⣟⣛⣿⡿⠛⠛⠻⢟⢛⠛⠃⢘⣛⣛⡛⠛⠛⢛⣻⣿⣿⣿⣇⠘⠛⠛⠛⠛⠛⣿⣿⣿⣟⢛⣷⣾⣿⣿⣿⡛⠛⠛⠀⠉⠸⣿⣿⣟⠛⠛⠻⠿⠛⠿⠿⠋⣀⡀⣀⣀⠸⣿⣿⣿⠟⠯⣿
⠋⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠁⠈⢁⣤⣿⣿⣟⡋⠀⠀⢘⣛⣛⡃⠀⠀⣾⣿⣿⡿⠟⠛⠀⡄⢘⣛⣛⣃⣈⣛⣻⣿⣿⠛⠻⠿⢛⣛⣛⣛⣛⣓⠀⣚⣛⣛⣻⣟⡃⠀⠴⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠹⠉⠉⢐⣿
⡀⠀⠀⢴⣿⢼⣿⣿⣿⣿⠉⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣦⣜⣛⣛⡓⠤⠎⠈⠉⠋⠀⠀⠀⠀⠀⢘⣛⣛⣛⣛⣛⣛⣛⣿⣾⣷⣦⣘⣛⣛⡋⣛⣛⣛⣛⣛⢛⣛⣛⣃⣈⡀⠀⠙⠛⠛⢋⡀⠠⠄⠀⠀⠀⠐⠒⣶⣶
⠞⠀⠀⡘⠛⠶⣿⣾⣷⣶⣶⣶⡿⠃⠀⠀⢛⣻⣟⡋⢘⢿⣿⣋⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢘⣛⣛⣛⣛⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⡃⠙⣛⣛⣛⡋⢈⣙⣛⡓⢉⣻⠆⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿
⠀⠀⠀⡿⠊⠁⢀⣀⣴⣶⣼⣿⣿⠇⢠⣀⣈⣉⣉⣃⣞⡀⢈⣉⣉⣁⣀⣀⣀⣀⣘⣀⣀⡀⢀⣰⣿⣿⣿⣿⣿⣟⣹⣛⣻⣿⣿⣿⣿⣿⣿⡁⠀⢉⣉⡉⠀⢈⣉⣉⣁⣞⡁⠀⣀⣀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⢀⠀⢀⡀⠈⢿⣿⣿⣿⡿⠃⠀⢈⣭⣍⣩⣍⣉⣉⡁⠈⢉⣉⣉⣉⣉⣉⣿⣭⣭⣭⡅⢨⣉⣙⣿⣿⣟⣻⣉⣙⣩⡽⣿⣿⣿⣿⣿⣭⡅⠛⠊⣉⠁⠀⢈⣉⣉⣉⣩⣵⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀
⣠⣶⣿⣿⣿⠅⠀⠈⠿⣻⡛⠃⠀⠀⢈⠉⠉⣽⣿⣿⣿⣿⣶⣄⠀⠉⠉⠉⠉⠉⠉⠉⠉⠑⠘⡯⠉⢉⠝⠉⠉⠉⠉⠉⢻⣿⢿⠿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠈⠉⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃⠀⡀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⡇⠀⠀⠘⠛⠃⠀⠀⠀⣤⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⣀⡀⠀⠀⠀⠘⢷⣿⣿⣿⣷⢀⣤⣤⣀⡀⠀⠀⠀⠀⢀⠛⣿⠙⠻⠿⠿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣈⣭⣟⡛⢿⣿⣿⣿⣿⡆⠀⠀⢀⣠⣤⣀⣀⣤⣶⣿⣿⣿⣿⡇⣀⣀⣀⣠⣬⢛⣿⡿⠏⣸⣿⣿⣿⣿⣳⣦⡀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⡈⠛⠀⣼⣟⣖⣉⣿⡄⢠⡟⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣧⡀⢸⡏⠀⠘⠀⢸⣟⠂⢸⣿⠿⠿⣿⣿⡿⣷⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠙⠻⣷⠀⢿⡍⠉⠉⢉⠁⠸⣇⠀⠸⣿⣿⣿⣿⡿⢿⣿⡟⠛⠁⣼⡧⠀⠀⠀⢸⡇⠀⢸⣿⠀⣀⠀⠘⣷⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾
⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⠳⠶⠖⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠳⠶⠛⠁⠉⠙⠻⠖⠛⠓⠂⠐⠚⠓⠂⠀⠐⠚⠓⠂⠈⠻⠶⣋⣀⣀⣹⣏⣠⣤⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⢻⣶⣿⣿⣿
⠀⠄⠲⢶⠀⠰⠄⠸⠷⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣿⣿⣿⣿⣿
⣿⣿⣿⢻⣼⣿⣿⢹⡮⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⡟⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣟⠿⡟⡿⡿⠛⡇⣫⡷⢻⡧⣿⣿⣿⣿⢻⣟⣿⣿⣿⣿⠿⣿⣽⣿⣯⣿⣿⣿⣁⣟⣿⣿⣿⣿⣿⢿⡇⣿⣿⣿⢿⣿⣿⣿⢻⣽⢿⣿⣷⣿⡿⣿⡿⣿⣿⣿⣿⡿⢿⣛⣥⣿⣿⣹⠽⣾⡖⠇⠿⢻⣿⣿⣟⣿⡿
⣤⣾⡳⢇⣷⡷⣾⣇⠃⠛⠶⢾⣾⣿⣻⡇⠻⡄⢃⣽⠻⣿⣿⣿⠯⣨⣋⠿⢻⣯⡸⢻⡽⡧⠭⢿⠿⣿⣛⢿⡿⡿⠤⠻⣟⢠⡿⢻⣿⣺⣯⣻⢽⣧⣬⣶⡖⠙⠟⣹⣈⣺⣿⣿⠛⢻⣽⣦⣻⣿⣶⣷⠛⣫⡽⠛⣿⡇
⡿⢯⠑⣾⣿⠟⠘⠻⠺⣭⡙⣛⣿⣿⡟⣳⣾⠻⠜⣻⣭⠛⠹⢿⣸⣯⡾⢓⠒⠻⢙⣶⢺⣿⣿⣶⣦⣦⡅⡘⠃⣷⡶⣶⣿⣦⣤⣀⣙⢿⣿⠧⠾⠏⢻⡟⠉⠀⢨⣤⣴⣿⣿⠯⣕⠤⢶⡿⠟⠛⣿⣿⣧⣤⣦⣐⣿⣿
⣹⠋⠀⢀⣉⠄⠀⠚⠿⣿⣁⣎⢸⣿⡿⠋⣉⣷⡄⠹⠿⠛⠤⡤⢿⣿⣶⡿⣁⢀⣤⣥⠈⠿⠾⢻⣿⠺⢿⣿⣷⡉⡛⠈⣿⣾⣷⣮⡁⠈⠛⢧⣶⠄⠀⠑⢾⣷⣶⣮⡙⣿⡛⢡⠬⣂⣤⣤⣤⡌⣻⣿⣿⣿⣤⣢⣄⣌
⠈⢠⣤⣼⡛⢿⡆⠀⠀⣾⣿⣿⡿⢿⣿⠁⠙⠟⡧⠀⢀⣤⣷⢶⡞⢻⣿⣿⣿⣿⠇⢀⣼⣶⢦⣸⣿⣿⣽⣿⣯⣿⣿⣤⠩⢿⡿⢿⠖⠀⠀⢀⣻⣷⣤⣤⡜⠻⣿⠿⠇⠈⢣⣄⠰⠧⡀⠈⠋⠹⣿⢏⣻⣿⣿⡟⠀⠙
⠀⢀⠈⠉⠠⠀⠀⢠⣴⢿⡿⢿⣦⠀⣷⣶⣤⣶⣦⣄⠈⢉⣽⣤⡴⢾⣟⢻⣟⣛⠒⢿⠁⠀⣼⣿⣿⣿⣿⠏⢱⠀⠊⠁⠀⠙⠿⡿⠘⠀⠀⣿⣿⣿⣿⣿⠃⠀⠻⢿⣀⠀⠀⠐⠻⠿⠿⣿⠿⠀⠀⠐⣿⣿⣛⢲⣾⣿
⠀⠀⠺⠿⣿⣶⡄⡖⠈⠈⣿⣿⣿⣷⣤⡈⠢⠄⠈⠛⠀⠛⠛⢉⣇⣁⡀⠈⠻⡟⠀⢾⣿⣿⠏⢿⣿⠿⠿⠖⣷⣿⣿⡧⠀⠀⠰⠿⢿⣿⠟⠀⢹⣿⣿⠿⠆⠀⠀⠀⠘⠿⠶⣳⠆⠀⠘⠁⠀⠀⠀⠀⠙⠿⣿⠈⠀⠀
⠀⠀⠀⠀⣠⣽⣿⣿⣯⣤⣿⣿⣿⡿⠛⠛⣿⣿⢿⡛⠛⠀⣰⣿⣿⠿⠟⢛⠛⠛⣿⡛⠟⠋⠘⠛⣿⣿⣿⠛⠛⠛⠛⠓⠀⠐⡿⠛⠛⠛⠛⠛⠃⠻⣿⡿⠘⠛⠛⠛⠛⠛⠋⣣⣤⣶⣿⣿⣧⡀⠀⠀⠀⠀⢉⣽⣶⣶
⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣶⡟⠛⠻⢿⡿⠟⠛⠃⢙⣛⣛⡛⠛⢛⠛⢛⣿⣿⣿⣷⠘⠛⠛⠻⡟⢛⢻⣿⣿⣛⠛⢛⣽⣿⣿⣿⡛⠛⠛⠀⠘⠃⣿⣿⣿⡛⠛⠛⢳⡿⠿⣿⡿⠛⠉⠁⠀⠀⢀⣴⣿⣿⣿⣿
⠆⠀⠀⠀⣸⣿⣿⣿⣿⣿⣭⣷⠼⢿⠇⠀⢛⣛⣛⡛⠀⠀⢘⣛⣛⡃⠀⠀⣰⣿⣿⣿⣿⡿⠆⠀⢘⣛⣛⣃⣘⣿⣿⣿⣿⠟⢿⣿⣿⣛⣛⣛⣛⣓⠀⣘⣛⣛⣻⣿⡃⠀⠀⣀⣤⣦⣤⣴⣶⣶⣶⣶⠊⠙⠻⡿⠏⠈
⠀⠀⠀⣠⢈⣿⣿⣿⣿⡿⠿⠁⠀⠀⡜⣿⣿⣿⣻⣷⣶⣄⢘⣛⣛⡧⣀⡄⠛⠻⠿⠋⠀⡀⠀⠈⢘⣛⣛⣛⣛⣛⣛⣛⣛⣁⣄⣈⢘⣛⣛⡋⣛⣛⣛⣛⣛⢛⣛⣛⡃⠀⠠⠉⠛⠿⣿⣿⡿⠛⠛⠛⠀⠀⠀⠁⣀⣀
⡆⠀⠘⣿⣙⢿⣿⣿⠿⠋⢀⣀⣀⡀⠀⢻⣻⣻⣿⡿⠛⣿⣿⣉⣙⡁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢘⣛⣛⡛⠛⠋⢛⣻⣿⣿⣿⣿⣿⣿⣛⡋⠙⣛⣛⣛⡋⢈⣉⣛⡁⠐⢷⣤⠀⠀⠀⠀⣰⠂⠀⠁⠀⠀⠀⠀⠀⣼
⠀⠀⢠⣌⡜⠿⠿⠿⠿⠿⣿⣿⣏⡀⣀⣀⣉⣉⣫⣁⣶⡈⢉⣉⣉⣁⣀⣀⣀⣀⣰⣀⣀⡀⢀⣀⣈⣽⣿⣥⣶⣶⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢉⣉⡉⠀⢈⣉⣉⣁⣀⣰⠟⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⡘⠁⠀⠀⣤⣴⣿⣿⣿⣿⠟⠀⢨⣍⣉⣉⣉⣉⣉⡁⠈⢉⣉⣉⣉⣉⣭⣯⣍⣉⣩⡁⢈⣙⣿⣿⣿⣿⣿⣿⣩⣯⡍⣹⣿⣿⣿⣿⣿⡏⢰⣌⣉⠁⠀⢈⣉⣉⣉⣩⡉⢀⣠⣤⣶⣶⣾⣿⣿⡀⠀⠀⠀⠀⠀⠀
⠀⢀⣀⣀⣤⠀⠘⣿⣿⣿⣿⠟⠀⠀⠈⠙⠏⣽⣿⣭⣭⣅⡀⠀⠀⠉⠙⠉⠉⠉⠉⠉⠙⠃⠸⣯⠉⠉⠉⡫⠭⠉⠉⠁⣴⣛⣛⣿⣿⣿⣏⣥⡄⠀⠀⠀⠀⠈⠉⠉⠉⢉⣵⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀
⣾⣿⣿⣿⡇⠀⠀⠘⣋⣿⠋⠀⠀⠀⣤⠀⠀⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠜⠁⠀⠀⠀⠀⠀⢽⢟⣿⣿⣟⣉⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠁⠀⠠⠀⠀⠀
⣿⣿⣿⣿⣿⡀⠀⠀⠉⡍⠀⠀⠀⠀⣿⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣗⡀⠀⠀⢀⣠⣤⣀⡀⣈⣀⣀⣀⣀⣠⣤⣀⣀⣀⣠⣼⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⡀⠀⣀⣀⠀⣿⠈⠁⠙⠋⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⠿⠿⠟⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⡉⠛⠻⣿⣿⣟⣽⣿⡄⢠⡟⠀⠀⢘⣧⣶⣿⣿⣿⣿⣿⣿⠀⢹⡏⠀⠘⠀⢸⡟⣛⢻⣿⣸⣿⣿⣿⣽⣷⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀
⠁⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠙⠻⣷⠀⢿⡍⢋⠉⣩⠁⠸⣇⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡦⣿⡇⠀⠀⠀⢸⡗⠫⢸⣿⠈⣛⠛⠿⣷⣹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠶⠞⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠲⠶⠿⣿⣿⣿⣿⠿⡟⠛⠋⠑⠛⠓⠒⠀⠐⠚⠓⠂⠀⠻⠶⠋⠀⠀⢹⠏⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠶⠲⣶⠀⢶⠄⣰⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠶⠏⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣴⣶⣦⣤
⣿⣿⣧⢿⢹⣿⣯⢻⠫⣿⢸⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣸⢹⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⢿⣿⣿⣿⣿⣿⣿
⣿⡿⣿⢸⡾⡿⣿⢸⡇⣣⡧⢻⡣⣿⣿⣿⣿⢻⣟⣿⣿⣿⣿⠿⣿⣽⣿⣥⣿⣿⣿⣈⡽⣿⡿⣿⣿⣿⢿⡿⢽⣩⢾⠸⣿⣿⣿⠿⣿⣿⣿⣱⣿⡝⣿⣿⣿⢿⣿⣿⣿⢿⢟⢯⡿⣿⣿⣭⢫⣾⠸⠸⠟⣿⣿⣿⡻⣿
⣤⣾⠲⣦⣷⠶⣶⡂⠁⠻⠷⣶⢞⣿⣻⡏⠿⡄⢃⣭⠿⣿⣾⣿⡯⢉⣛⢿⢿⣯⡨⢻⡿⡷⢽⣽⡿⣿⡟⣾⣿⣿⣌⡸⣿⢧⣽⡿⣿⠷⣿⣹⡟⣿⣀⣵⣾⠊⢿⢛⣇⡲⣿⡿⡿⠟⠻⢧⠜⢽⣷⣮⣝⢻⣉⡿⠟⣿
⣿⡍⢂⣎⣿⠏⠛⠷⠾⣍⠙⢛⣻⣿⡙⣇⣼⠻⠔⣻⣙⠺⠹⡿⣹⣏⡼⢖⠖⠻⠚⣾⢾⣿⣳⣷⣦⣍⡋⠘⠛⢸⡶⢤⣾⣦⣤⣀⣙⡺⣿⣿⣧⡿⠛⣟⡋⠂⢀⣓⣬⣼⣿⣿⣷⡂⣨⣽⣿⠛⢫⣿⣯⣆⣬⣄⠰⣛
⠟⠃⠈⠙⠁⠀⠐⠷⢾⣿⢡⣙⢸⣿⡷⢏⡋⢶⡄⢻⡿⠛⣠⡤⢿⣿⣶⣾⠃⠀⣰⣭⠈⠿⣬⠹⣿⡏⣽⣿⣶⠞⠛⠃⢻⣯⣽⣯⣋⠀⠸⢿⣤⣤⠀⠉⢻⣷⣴⣬⣝⢻⡿⠿⣄⣙⠄⣀⣁⣀⡘⠿⢏⣿⣿⣋⢮⠉
⢃⣀⣀⡛⠿⣷⡄⠀⢠⣿⣿⣿⣿⣿⣿⠉⠻⡿⣷⠀⠀⣠⣬⣤⡬⢿⣿⣷⣿⣿⡏⠁⣠⣧⣄⡐⣻⣿⣌⣿⣿⢹⣿⣂⡸⢯⣿⣿⣏⠀⠀⠀⠸⢷⣆⣀⣀⡺⢋⣿⣿⡋⠛⣄⠀⢰⣋⠙⠛⠏⠃⣴⣿⡿⣿⣿⣿⡿
⠀⠉⠛⠉⡀⠀⠀⣀⣜⣻⣿⣝⡀⢹⣧⣄⣠⣤⣥⡀⠘⠛⣙⣈⣤⣼⣿⢿⡿⠿⡤⣶⡉⠉⣀⡾⣿⣿⣿⣿⢇⡉⠻⠟⠁⠀⣿⣶⡆⠢⠀⠀⣴⣿⣿⣿⣿⡦⠀⢿⣶⠀⠀⠈⠙⣤⣾⣥⣴⣶⡆⠁⠑⣩⣾⣿⣿⣧
⠀⠠⢿⣶⣶⣄⢠⠍⠃⢿⣶⣿⣧⣀⠉⠫⢉⠉⠛⠿⠀⠿⠿⠛⡇⠚⠁⠘⢿⣿⠃⣈⣧⣤⣭⢿⣿⣿⣿⣏⣰⣥⣶⣶⡀⠀⠀⣹⣴⣴⣶⣦⠙⢃⣾⣿⣿⣃⠀⠀⠈⢹⣶⣄⣠⡀⠉⠉⡹⠂⠀⠀⠀⠸⣿⣿⣷⠘
⠀⠀⠐⠀⢉⣿⣿⣶⡆⢸⣿⣿⣿⣿⠿⣿⣿⣿⣛⠛⠛⠁⢠⣼⣿⣿⣿⠛⠛⠛⢛⣛⢛⡿⠟⠛⣿⣿⣿⣿⠛⢿⠟⠛⠁⠀⢸⡛⠛⢻⠛⠛⠃⠠⣿⣿⣿⠛⠛⠛⠛⠛⠃⠀⠴⠋⣀⣠⣤⣤⡀⠀⠀⠀⠈⠉⢿⣤
⠀⠀⠀⠰⣿⡿⢟⣿⣷⣬⣿⣿⣯⣃⣛⠻⣿⣿⣿⡟⠛⠃⢛⣛⣛⡛⠛⠛⠛⠛⣻⠛⠛⣃⠘⠛⠛⣿⣿⡟⡛⣛⣛⠛⠛⠓⢸⠛⣛⣛⣿⡛⠛⠀⠘⠛⣿⢿⣟⡛⠛⠛⠃⠀⣰⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣭⣿⡇⠀⢙⣛⣟⣇⠀⠀⢘⣛⣛⡃⠀⠐⠀⣠⣷⣾⣿⣿⡄⠀⢘⣛⣛⣃⣃⣽⣿⣿⣿⡃⣴⣿⣿⣿⣿⣿⣛⣓⠀⣘⣛⣻⣿⣿⣧⠀⠀⠘⠛⠋⠛⠛⠋⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⢰⣲⣿⣿⣿⣿⣿⣭⣭⡜⢿⣟⣀⣀⣝⣿⣛⣃⡀⠀⢘⣛⣛⣃⠀⠀⣸⣿⣿⣿⡿⠛⠋⠀⢜⣛⣛⣛⣛⣛⣛⣛⣿⣿⠉⠛⢿⣟⣛⡋⣛⣛⣛⣛⣛⢛⣛⣛⣛⠃⠀⠀⠀⣠⣶⣶⣶⣶⣷⣾⣿⣿⡿⠈⠉⠻
⠀⢀⡈⢿⣿⣿⣿⣿⣿⡿⠃⠀⠀⡀⣿⣿⣿⣟⣿⣿⣿⣶⣌⣛⣛⡋⠓⠋⠀⠙⠛⠋⠀⢀⠀⠀⢘⣛⣛⡛⠛⠛⢛⣛⣛⣋⢀⣀⣈⣙⣛⡃⠙⣟⣛⣛⡋⢘⣋⣛⡁⠀⠀⠀⠀⠉⠙⠻⢿⡿⠿⠟⠛⠛⠃⠀⠀⠀
⠀⢿⡇⠿⣫⣾⣿⣿⡖⠀⠀⠀⠀⠁⣘⣿⣻⣻⣿⣟⣻⣿⣿⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣊⣉⣉⣁⣀⣀⣈⣉⣉⣵⣿⣿⣿⣿⣿⣅⠀⢉⣉⡉⠀⢈⣉⣉⣁⣀⡀⠐⠷⣤⡀⠀⠀⠀⢀⡖⠉⠀⠁⠀⠀⠀
⠀⣈⢁⣶⣶⣾⣷⣶⣶⣶⣿⡟⠉⠀⢈⣉⣉⣻⣍⣉⣯⡁⠈⢉⣉⣉⣉⣍⣉⣉⣩⣏⣉⡁⢈⣉⣉⣩⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣞⣉⠁⠀⢈⣉⣉⣉⣩⡅⠀⣠⠟⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀
⠰⡷⠉⠀⢈⠉⣠⣷⣬⣛⣟⣿⠂⢀⠈⠉⠉⠉⠉⠋⠉⠁⠀⠀⠀⠉⠉⠉⠉⠹⠍⠉⣉⣁⠈⠉⠻⣿⣿⣿⣿⣿⣿⠃⠹⢋⠙⣿⣿⣿⣿⣿⣿⡟⡷⠄⠀⠈⠉⠉⠉⠉⠁⠠⠉⠀⢀⣠⣤⣴⣶⣶⣿⣦⠀⠀⠀⠀
⢈⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠃⠀⠈⣬⠀⢀⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠣⣴⡀⠈⠛⠛⢟⣁⠁⠀⠃⠐⢐⣿⣿⣽⣿⣷⠈⢀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣃⡀⠀⠀
⣶⣿⣿⠇⠀⠘⣿⣿⣿⣿⠁⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⣿⣷⣄⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⢀⣨⠗⢀⣀⡰⠃⢀⣀⣀⣠⣽⢻⣾⣶⣷⣾⣿⣿⡟⣁⣀⡀⠀⣀⣀⠀⣿⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⠇⠀⢀
⣿⣿⣿⡄⠀⠀⠀⣶⣖⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣎⣿⡄⢠⡟⠀⠀⠘⠇⠀⣿⠀⠀⢸⡇⠀⡀⢸⡏⠀⠘⠑⢼⣧⣶⣿⣿⣴⠠⣬⣷⣄⡀⡼⠁⠀⠀⠀⠉⠟⠙⠻⠛⠿⠻⢟⠍⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣧⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢘⡛⠻⠿⣿⣿⣿⣽⣯⢩⣏⠁⠸⣇⠀⠀⢀⡀⠀⣿⣤⣶⣾⣿⣿⣿⣿⡇⠀⠀⠀⢸⡿⣿⣿⣿⡏⣀⢿⣿⣿⣿⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀
⠋⠙⠋⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⠒⠀⠀⠘⠳⠶⠖⠋⠈⠙⠿⣿⠞⠋⠀⠀⠙⠲⠶⢚⣴⣿⣿⣿⣿⣿⣿⣿⣿⡛⠓⠂⠀⠐⠚⠓⠊⢩⣿⠶⠋⢻⣿⣿⣯⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠰
⠀⠖⠶⣶⠀⢶⠄⠠⠰⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠾⠴⠏⠉⠑⠃⣻⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⡿⣇⣿⢽⣿⣇⢿⠻⣿⣸⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⣿⣿⣻⢹⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿
⢻⣿⣟⢹⢸⡿⡿⢸⣀⣫⡧⢻⡣⣿⣿⣿⣽⢳⢟⣿⣿⣿⣿⠿⣯⣽⣿⣥⣿⣿⣿⣘⡽⣿⡽⣿⣿⣿⢿⢹⢸⣭⣿⠸⣿⣿⣿⡿⣿⣿⣿⣹⣿⡟⢿⣿⢿⢿⣿⣿⣿⢿⡟⣫⣿⣿⡟⢭⡇⣶⣿⠹⠟⢿⢿⣿⡛⣿
⣤⣿⢺⣬⣾⢶⣾⡂⠉⠻⠶⣶⢾⣿⣻⡏⠿⡤⢃⣭⠟⣿⣾⣿⠯⢉⣛⣿⢿⣯⡨⢹⡷⡷⠽⣽⡿⣿⡏⢾⣿⣿⣏⡘⣿⢇⣼⡿⣿⡧⣿⣕⣿⢻⢅⣵⣴⠊⠺⠏⣇⣐⢿⣿⣿⠻⠛⢳⠔⠫⣷⣾⣍⡻⣏⣹⠿⣻
⣯⠙⣐⣺⡿⠃⠛⠷⠮⣍⠙⢛⣻⣟⠙⣇⡼⠻⠔⣻⣙⠺⠹⡿⣱⣏⡼⢖⠖⠻⠟⣾⣲⣿⣳⣷⣦⣨⣉⠈⠛⢸⡶⢤⢾⣶⣬⣀⢉⡱⢿⣿⣧⣿⠏⣿⡛⠃⠀⣰⣈⣵⣿⣿⣷⣆⢈⣭⣿⡟⠛⣿⣿⣷⣀⣥⡠⢞
⠟⠁⠊⠙⠁⠀⠐⠷⢾⣾⢡⣙⣾⣿⡷⢍⡫⢶⡄⢻⡿⠛⣠⡤⣿⣿⣶⣾⠛⠀⣰⣭⠈⢻⣬⠭⣿⡇⣽⣾⣶⠞⠻⠃⠹⣯⣽⣯⣟⠁⠘⢿⡆⣤⡀⠈⠡⣷⣦⣬⣝⡫⣿⠿⢇⡈⡠⢈⣊⡀⠀⠹⠻⢟⣿⣟⡁⡉
⢀⣀⣀⠛⠿⣦⠀⠀⣠⣾⣿⣿⣿⣿⣿⠘⠿⡿⡷⠀⠀⣠⣬⣤⡬⢿⣿⣵⣾⣿⡟⠁⣠⣧⣤⠐⢿⣷⣜⣿⣿⠗⣿⣃⣀⢿⣿⣿⣿⡁⠀⠀⠀⠿⣦⣀⣀⡐⠟⢹⢿⣟⠙⢧⡀⠁⣾⠉⠛⠻⠛⢀⣴⣿⠿⣿⣿⣿
⠈⠉⠻⠉⡀⠁⢀⣀⣿⣿⣟⣁⡈⢻⣥⣔⢠⣬⣅⡀⠘⠛⣛⣈⣁⣼⣿⢿⣿⣿⡧⣶⡉⠋⢀⣾⣿⣿⣿⣿⡏⡈⢛⠿⠋⠀⢿⣿⡯⠠⠀⠀⢠⣾⣿⣿⣿⣷⡄⢹⣿⡏⠀⠈⠙⢂⣤⣯⣤⣤⣶⡌⠉⠋⣁⣛⣟⣿
⠀⢴⣶⣶⣦⣀⣈⠿⠘⣷⣶⣿⣆⠈⠙⠫⢉⠉⠛⠷⠀⢾⡿⠛⡇⠝⠁⠘⢿⣿⠇⢈⣧⣤⣿⠽⡾⣿⣿⣯⣄⣫⣤⣶⡄⠀⠀⢉⣧⣴⣶⣤⡌⠟⠨⢿⣿⣿⡀⠀⠈⠁⣶⣦⡀⣂⠈⠉⠛⡭⠉⠁⠀⠀⠰⣿⣷⣦
⠀⠀⠀⠈⢙⣿⣷⣶⠀⣼⣿⣿⣿⣿⢾⣿⣿⣟⣛⠛⠛⠃⢀⣼⣿⣿⣟⠛⠛⠛⠛⡛⠟⣿⠻⠛⢻⣿⣿⣟⡛⢿⡿⠛⠋⠀⠘⡛⠛⠛⡻⠟⠃⠈⢹⣿⣽⡟⠛⠛⠛⠛⠃⠈⠁⡼⠃⠀⣈⣠⣄⠀⠀⠀⠀⠈⠛⠿
⠀⠀⠀⣴⣿⡿⣟⣻⣧⣹⣿⣿⠟⠁⠛⢻⣿⣿⣿⡟⠛⠂⢛⣿⣟⡛⠛⠛⠛⠛⢛⠟⠛⠃⠘⠛⠛⣿⣿⣿⠛⠛⠛⠛⠛⠳⠘⡟⠛⠛⣛⡛⠛⠀⠈⠛⢿⡿⣛⣛⠛⠛⠃⠀⠀⣤⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣩⣾⣿⣿⣿⣿⣿⣛⣻⡿⠁⠀⢛⣿⣛⡃⠀⠀⢘⣛⣛⡃⠀⠐⠀⢀⣸⣴⣶⣿⡇⠀⢘⣛⣛⣇⣀⣸⣿⣿⣿⡋⢠⣴⣾⣿⣿⣿⣟⣛⠀⣘⣛⣛⣿⣿⣷⡄⠀⠀⠸⠟⠻⠻⠿⠟⠁⠀⠀⠁⠀⠀⠀⣠
⠀⣠⣤⣾⣿⣿⣿⡿⠿⣛⣛⣿⣿⠏⠀⠀⣘⣛⣛⡃⠀⠀⢸⣛⣛⡃⠀⠀⢰⣿⣿⣿⣿⡿⠿⠀⢘⣛⣛⣛⣛⣛⣛⣻⣿⣿⡞⠻⢿⣿⣟⡋⣛⣛⣛⣛⣛⢛⣛⣛⡛⠻⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣴⣶⣶⡆⠛
⠀⢺⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠛⢲⣶⣶⣿⣟⣻⣿⣷⣤⣈⣛⣛⡋⠢⠖⠈⠻⠿⠟⠁⠀⠀⠀⢘⣛⣛⡛⠛⠛⢛⣛⣛⡙⠁⠀⢈⣋⣙⡃⠙⣛⣛⣛⡋⢘⣛⣙⡃⠀⠀⠀⠀⠀⠚⠿⣿⣿⣿⢿⣿⡿⠿⢿⠁⠀
⣴⡇⣿⣟⣻⣿⣯⡛⠋⠀⠀⠀⠐⠈⣿⣿⣿⣿⣿⣿⣿⡿⣻⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⡂⢀⣀⣌⣉⣉⣁⣀⣀⣈⣉⣉⡁⣠⣶⣿⣿⣭⡁⠀⢉⣉⡉⠀⢈⣉⣉⣁⣀⡀⠀⢀⣤⡀⠀⠀⠈⠉⠙⢀⠤⠄⠐⠀⠀
⠹⢇⡛⠞⢛⣟⣈⣀⣀⣠⣤⡤⠀⠀⢉⣭⣭⣍⣉⣩⣉⡉⠈⢉⣉⣉⣉⣍⣉⣉⣉⣍⣍⡁⢈⣉⣉⣉⣉⣉⣉⣉⣉⣩⣭⣿⣿⣿⣿⣿⣿⣿⣷⡌⣉⠁⠀⢈⣉⣉⣉⣉⣁⠀⠀⠈⣹⠖⠀⠀⠀⡠⠞⠀⠀⠀⠀⠀
⣖⡤⠻⠿⠿⠿⠿⠻⣿⣿⣏⡄⠀⠀⠈⠉⠉⠉⠻⠉⠋⠁⠀⠀⠀⠉⠉⠉⠉⢹⡉⠉⠉⠁⠈⠉⢩⣿⣿⣿⣿⣿⣿⣿⠠⣿⠿⣿⣿⣿⣿⣿⣿⣯⣵⢄⠀⠈⠉⠉⠉⠉⠉⠄⠀⠺⠁⠀⠀⠀⠀⣀⣀⣠⣄⠀⠀⠀
⠋⠀⠀⢠⣤⣾⣿⣷⣮⣽⡿⠁⠀⠄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⠶⢆⢀⠀⠙⢿⣿⣾⠿⠟⠋⠀⡁⠂⣬⣿⣿⢿⣿⣟⠿⠃⠂⠁⠀⠀⠀⠀⣤⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣧⠀⠀
⣀⣠⣄⠀⢿⣿⣿⣿⣿⡟⠁⠀⠀⠀⣿⠀⢰⣶⣶⣶⣦⣤⣀⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⢀⣹⡦⢀⣀⡀⢨⣖⣒⣀⣠⣤⣀⣘⡻⢿⣹⣿⣉⣀⣤⣄⡀⠀⣀⣀⠀⣿⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠄
⣿⣿⠃⠀⠀⠿⠟⡿⠟⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣄⣈⣿⡄⢠⡟⠀⠀⠘⠇⠀⣿⠍⠀⢸⡗⠉⠀⢸⡏⠀⠙⠀⢹⡟⠛⢿⣿⣛⠛⠉⣷⡀⠀⡸⠁⠀⠀⠀⠀⢰⢿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠂⠀
⣿⣿⡇⠀⠀⠀⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣽⣿⡻⣉⠁⠸⣇⠀⠀⢀⡀⠀⣿⡀⠀⢸⣇⣀⣤⣼⡇⠀⠀⠀⢸⣷⣶⣿⣿⣿⣿⠘⣿⣷⣶⣧⣄⠀⠀⠀⠀⠀⠀⠀⠈⠂⠚⠛⠋⠧⠀⠀⠀⠀⠀
⡿⣿⡯⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠟⠻⢿⡿⣿⡿⣶⠏⠀⠀⠙⠲⠶⠚⠀⢀⣼⣷⣿⣿⣿⣿⣿⡟⠓⠂⠀⠐⠚⠓⠻⣿⣿⢿⡏⠀⣹⣿⣿⣿⣨⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀
⠈⠏⠲⣶⠀⢶⠆⠠⠴⠶⠆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠃⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠀⠻⠿⠿⢿⢿⣿⠟⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⡏⣷⣿⣿⣇⣿⠿⣿⢸⢻⠏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⢿⣿⣿⣿⣿⢸⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⡿⣿⣿⣿⣿⣿⣿
⢻⣿⡗⢻⢹⠻⡟⢸⢀⣿⡏⣸⡣⣿⣿⣿⣽⢳⢟⣿⣿⣿⣿⠿⣯⣽⣿⣥⣿⣿⣿⣘⡽⣿⣽⣿⣿⣿⢿⢸⢸⣯⣿⠙⣿⣿⣿⡿⣿⡿⣿⣻⣿⡿⠿⣿⢿⢿⣿⣿⣿⡿⡿⣫⣽⣿⡟⣭⡞⣵⡏⠧⠟⢿⣿⣿⣟⢿
⣴⡯⠺⣼⡾⢶⣷⢈⠈⠻⠲⣶⢾⣿⣻⠏⠻⡠⢃⣭⠟⣿⣾⣾⠯⢉⣛⠻⢻⣯⡨⠹⡿⣷⠽⢽⡿⣿⣟⢾⣿⣿⣧⡘⣿⡇⣼⡿⢿⡯⢾⣏⣿⢻⡇⣹⣴⡗⠹⡏⢋⣐⠾⣿⣷⠻⠛⠛⡤⠋⣿⣶⣯⣹⢟⣙⡿⢻
⣭⠑⣐⣹⡿⠀⠛⠷⠮⣍⠛⣛⣿⣿⠙⣢⡾⠻⠜⣻⣙⠺⠹⡿⣹⣏⡽⢖⠖⠻⠞⣾⣲⣿⣳⣷⣦⣨⣉⠁⠛⢘⣶⠆⣼⣷⣤⣄⡈⣃⢿⣿⣿⣼⠟⢹⣟⠒⠀⢘⣈⣭⣾⣿⣟⣖⢈⣭⣽⣿⠛⢻⣿⣿⣆⣨⣀⠼
⠛⠀⠈⠉⠁⠀⠘⠷⣾⣮⢩⣃⣿⣿⠗⢉⠫⣶⡀⢻⡿⠟⣠⡤⣿⣿⣶⣾⠛⠀⣰⣭⠀⢻⣬⠭⣿⡧⢭⣿⣷⡞⠛⠛⠹⣿⣭⣿⣝⡋⠀⠿⣧⣤⡄⠈⠉⣾⣢⣤⣭⡛⢽⡿⠇⣈⢅⠄⢙⡁⠀⠈⠟⠟⢻⣿⣏⠈
⣀⣀⡐⠻⢿⣦⠀⢀⣠⣾⣿⣷⣿⣿⡯⠘⠿⣿⡇⠀⠀⣠⣮⣤⡬⢿⣿⣵⣾⣿⣟⠁⢀⣧⣥⠀⠿⣷⣆⣿⣿⡏⣾⣇⣀⠺⠹⣿⣿⣃⠀⠀⠀⠻⢶⣄⣀⣀⠾⠋⡿⣿⡛⠻⣀⠁⣰⡟⠙⠛⡿⠋⢀⣼⣿⡿⣿⣿
⠈⠙⠛⢁⠀⠁⢀⣀⣻⣿⣟⣁⡈⣿⣥⣄⣠⣬⣄⠈⠘⠛⣓⣈⣁⣼⣿⣚⣿⣯⡧⢶⡙⠋⢀⣸⣿⣿⣿⣿⣿⣈⠛⡻⠟⠀⠹⣿⣿⠁⠄⠀⠀⣠⣾⣿⣿⣿⣤⠘⣿⣿⠁⠀⠉⠳⣀⣼⣥⣤⣤⣤⡌⠉⠋⣀⣬⣯
⠀⣶⣶⣶⣄⡀⣙⠏⢹⣷⣾⣿⡀⠈⠙⠟⣉⠉⠻⠷⠀⢾⡿⠛⡏⠝⠃⠘⢿⣿⠇⠈⣣⣄⣿⡿⢲⣿⣿⣿⣅⣛⣤⣤⡄⠀⠀⠉⣻⣤⣶⣤⣤⠙⠣⠹⣿⣿⣏⠀⠀⠙⢰⣦⣄⢀⡉⠉⠙⠛⡯⠉⠁⠀⠀⢰⣿⣿
⠀⠀⠀⠉⢛⣿⣥⡤⢠⣽⣿⣿⣿⡳⣾⣿⣿⣟⠛⠛⠛⠃⢀⣴⣿⣿⣟⠛⠛⠛⠛⠻⠿⣿⣿⠛⢻⣿⣟⣛⡛⢻⣿⠟⠛⠀⠘⢛⠛⠛⢻⡿⠏⠀⠈⣿⣿⣿⡛⠛⠛⠛⠃⠈⠙⢉⡽⠂⠀⢈⣀⣀⠀⠀⠀⠀⠉⠛
⠀⠀⠀⣾⣿⡿⢿⣛⡘⣿⣿⣟⠋⠀⠛⢿⣿⣿⣿⡟⠛⠂⢻⣿⣟⡛⠛⠛⠛⠛⢛⡿⠛⠋⠘⠛⠛⣿⣿⣿⡟⠛⠛⠛⠛⠓⠘⢻⠟⠛⠛⢛⠛⠀⠀⠙⠿⣿⣟⣛⠛⠛⠃⠀⠀⠈⣡⣶⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀
⠀⠀⠀⢉⣵⣾⣿⣿⣿⣿⡿⡿⣿⣏⠁⠘⢻⣟⣛⡃⠀⠀⢘⣛⣛⡃⠀⠠⠀⠀⢸⣤⣶⣶⣆⠀⢘⣛⣛⣟⣀⣀⣿⣿⣿⣇⠀⣨⣼⣿⣿⣿⣟⣓⠀⣘⣛⣛⣛⣿⣿⣦⡀⠀⠀⡶⠿⠿⢿⢿⠟⠉⠁⠀⠂⠀⠀⠀
⠀⠀⣴⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⠎⠀⠂⣘⣛⣛⡃⠀⠀⢘⣛⣛⡃⠀⠀⢀⣾⣿⣿⣿⣿⣿⠆⢘⣛⣛⣛⣛⣛⣻⣿⣿⣿⣦⠛⢿⣿⣿⡟⣛⣛⣛⣛⣛⢛⣛⣛⡋⠛⣧⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤
⣯⣭⣿⣿⣿⣿⣷⣶⣶⠖⠙⠻⠣⣤⣤⣴⣿⣿⣿⣷⣦⣄⢘⣛⣛⡓⠦⠴⠘⠿⣿⡿⠟⠁⠀⠀⢘⣛⣛⡛⠛⠛⢛⣛⣛⡈⠟⠀⢘⣛⣛⡃⠙⣛⣛⣛⡋⢘⣛⣋⡃⠀⠀⠀⠀⠀⢀⠴⢿⣿⣿⣟⣿⣿⣿⣿⣿⣿
⢘⣿⣿⣿⣿⡿⠿⠟⠁⠀⠀⠀⠄⢿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⡤⢀⣀⣈⣉⣉⣁⣀⣀⣈⣉⣉⡁⢀⣀⣬⣭⣍⡁⠀⢉⣉⡉⠀⢈⣉⣉⣁⣀⡀⠀⠀⠀⣀⠀⠀⠈⠋⠿⠛⠏⠉⣀⡀⠀
⠉⢯⣓⣿⣿⠛⠀⠀⠀⠀⣀⠀⠀⠀⣉⣹⣯⣿⣿⣉⣉⡑⠉⢉⣉⣉⣉⣩⣉⣉⣉⣩⣩⡁⢈⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⢁⣼⣿⣿⣿⣿⣿⣶⣌⣉⠁⠀⢈⣉⣉⣉⣉⡁⠀⠀⠈⠙⢷⣆⡀⠀⠀⠀⢠⠏⠀⠀⠀
⢱⣦⣤⣶⣶⣶⣶⣶⣾⣿⠋⠁⠀⠀⠈⠉⠉⠻⡍⢻⡏⠁⠀⠀⠀⠉⠉⠉⠉⢉⠉⠉⠉⠁⠈⠉⠉⣩⣽⣿⣿⣿⣿⣿⡿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣔⡀⠈⠉⠉⠉⠉⠁⠑⡀⠀⢠⡖⠋⠀⠀⠀⠐⠁⠀⠀⠀⠀
⠉⠈⠉⠉⣩⣭⣍⡻⠿⢷⡶⠀⠀⠀⣤⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢠⣤⣄⠀⠀⢻⣿⣿⣿⣿⣿⣿⠇⠈⠛⣭⣿⣿⣾⣿⣿⣿⣿⡿⠑⠂⠀⠀⠀⣤⠀⠀⠀⠀⠏⠁⠀⢀⣠⣤⣶⣶⣶⣿⣷⡄
⠀⠀⠸⣾⣿⣿⣿⣿⣿⡿⠀⠀⠘⠈⣿⠀⣀⣀⣤⣄⣀⠀⠀⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⢁⣢⣆⢀⣘⡛⠹⣏⣀⣁⣠⣴⢀⣈⡰⣿⣻⣿⣿⣷⣉⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇
⣶⡆⠀⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣾⣏⣀⣈⣿⡄⢠⡟⠀⠀⠘⠇⠀⣿⢹⠄⢸⡇⣠⠎⢹⡏⠀⠙⠀⢾⣧⣭⣽⣿⣤⣶⣾⣿⡇⠀⡸⠁⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡀⠀⠀⠛⣫⣽⠋⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣉⠁⠸⣇⠀⠀⢀⡀⠀⣿⡃⠀⢸⡏⠁⠀⢸⡇⠀⠀⠀⢸⡇⣉⣹⣿⣭⣯⣅⠸⣷⣰⣃⠀⠀⠀⠀⠀⠀⠐⠋⠛⠻⢿⠿⡿⢿⣿⠟⠁⠈⠀
⣿⡇⠀⠀⠀⢛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣋⠀⠀⠙⠲⠶⠚⠀⠀⠘⠳⣖⣿⣷⣶⣶⣿⠓⠂⠀⠐⠚⠓⠻⣿⣿⣿⣿⣿⠀⢹⣿⣿⣿⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⠿⠲⣶⠀⢰⠄⠠⠴⠶⠆⠀⣀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢟⡻⠟⡾⠋⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⠿⠿⢿⠿⠶⢿⣿⣿⣿⣶⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣿⣿⡜⡇⣿⣿⡇⣷⡿⣿⢸⢻⠏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⢿⣿⣿⣿⣿⢹⢻⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣹⣿⣿⣿⣿⣿
⠻⣿⡇⣿⢻⠻⣇⢻⢀⣽⡏⣸⡣⣿⣿⣿⣿⠱⢟⣿⣿⣿⣻⠿⣯⣽⣿⣥⣽⣿⣿⣞⡽⣿⣼⣿⣿⣿⢿⢸⢸⣏⣿⠗⣿⣾⣿⡟⣿⡿⣿⣿⣿⣯⠻⣿⣿⡷⣿⣿⣻⡿⡿⡻⣽⣿⣟⢭⡸⢰⣯⡇⠿⠻⣿⣾⣟⢻
⣼⡏⠻⣼⡶⢶⣗⢈⠘⠻⠳⣖⣾⣿⣿⠏⠻⡀⢃⣭⠟⣿⣿⣾⠯⢉⣛⠻⢹⣯⡌⠹⡟⣷⠽⢽⣿⢿⣇⢸⣿⣿⣧⡘⢽⡏⣬⣻⢻⡿⢾⣯⣾⢿⡇⣨⣶⡗⠙⠿⢻⣐⡺⢿⣿⡟⠟⠛⠤⠜⠽⣷⣮⣝⡿⣉⡿⠛
⡅⢃⣰⣹⠟⠈⠛⠳⠮⣍⠛⣻⣿⣯⢙⣡⡿⠻⠘⣻⣙⠚⠹⡿⣸⣏⡱⢖⠒⠸⠗⣾⣒⣿⣷⣶⣦⣨⣉⠁⠻⠀⣷⠦⣴⣷⣦⣄⡈⣉⠾⣽⣿⣽⡷⠓⣟⠛⠀⢀⣂⣩⣼⣿⣿⣳⡀⣩⣽⣿⡞⠛⢻⣿⣧⣤⣅⡠
⠋⠀⢈⠋⠀⠀⠘⠷⣿⡮⣈⡃⣿⡿⠗⢉⠩⣶⡀⢻⡿⠟⣠⡤⣿⣿⣶⣾⠛⠀⣠⣩⠀⠹⣬⡍⣿⣯⢩⣿⣷⡦⠛⠟⠘⢿⣭⣽⣭⡋⠀⠹⢿⣤⣄⠀⠉⢸⣷⣤⣭⣟⢩⣿⠿⣀⡁⡤⠈⣋⠀⠀⠸⠿⠋⢻⣿⣇
⣀⣀⠘⠻⢷⡆⠀⢀⣴⣿⣿⣶⣿⣿⡏⠘⠿⠿⡆⠀⠀⣰⣬⣤⡬⢿⣿⣷⣾⣷⣟⠁⢀⣧⣥⡀⠿⣿⣦⣻⣿⡗⢾⣷⣀⡸⠫⢿⣿⣏⠀⠀⠀⠸⢷⣄⣀⣀⡲⠟⢹⣿⡿⠛⢇⡈⢀⣶⠛⠛⢛⠟⠃⢀⣼⣿⡿⣿
⠉⠛⠛⣁⠀⠀⢀⣘⣿⣿⣋⣀⠉⣿⣤⡄⣠⣬⣄⠀⠘⠛⣓⣈⣁⣬⣿⣚⣿⣿⡥⢰⡉⠋⠁⣸⣾⣿⣿⣿⣿⣅⠛⢻⠟⢁⠘⣿⣿⡍⠠⠀⠀⢀⣴⣿⣿⣿⣧⣄⠹⣿⡍⠀⠈⠙⠆⢀⣿⣤⣤⣤⣤⡌⠉⠉⢀⣨
⢠⣶⣶⣦⣄⢀⡛⠏⢿⣿⣿⣷⠀⠉⠛⠟⡉⠉⠻⠷⠀⣾⡿⠿⡏⠝⠃⠈⢿⣷⡇⠈⣹⣀⣘⣿⢃⣿⣿⣿⢏⢈⣡⣤⣤⠀⠀⠈⠙⣣⣰⣤⣤⡌⠛⠁⣽⣿⣿⠃⠀⠉⠃⣶⣤⡀⠠⠉⠉⠛⠛⡯⠍⠁⠀⠀⠩⣿
⠀⠀⠈⠙⣛⣿⣤⡄⣨⣿⣿⣿⣿⣦⣾⣿⣿⣛⠛⠛⠛⠃⢀⣰⣿⣿⣟⠛⠛⠛⠛⠻⡿⢿⣿⠟⠛⣿⣿⣛⡛⠛⣿⡿⠛⠃⠸⠛⠛⠛⠟⢛⡛⠃⠀⢘⣿⣟⣟⠛⠛⠛⠃⠈⠉⠛⢉⡗⠂⠀⠈⣀⡀⠀⠀⠀⠀⠙
⠀⠀⢠⣾⣿⣿⠿⣇⢻⣿⣿⡿⠋⠀⠻⣿⣿⣿⣿⡟⠛⠂⢺⣿⣿⡛⠛⠛⠛⠛⠛⡿⠛⠋⠙⠛⠛⢻⣿⣿⣿⠛⠛⠛⠛⠃⢘⠛⡻⠛⠛⠛⠛⠀⠀⠀⠻⣿⣿⣛⠛⠛⠃⠀⠀⠀⠈⣡⣴⣾⣿⣿⣿⣄⠀⠀⠀⠀
⠀⠀⠀⣩⣷⣾⣿⣿⣾⠿⠿⢶⣶⡖⠀⠩⢿⣛⣛⡃⠀⠀⢘⣛⣛⡃⠀⠠⠀⠀⢸⣁⣤⣴⣶⠀⢘⣛⣛⣻⣇⣀⣸⣿⣿⣇⠀⠐⣹⣿⣿⣿⣿⣛⠀⣚⣛⣛⣛⣻⣿⣶⣄⠀⠀⠠⣶⡿⢿⣿⣿⡿⠋⠉⠀⠂⠀⠀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢘⣛⣛⡃⠀⠀⢘⣛⣛⡃⠀⠀⠀⣴⣿⣿⣿⣿⣿⡆⢘⣛⣛⣛⣛⣛⣛⣿⣿⣿⣷⠼⢿⣿⣿⡿⣟⣛⣛⣛⣛⢛⣛⣛⡋⠙⠿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⢶⣿⣿⣿⣿⣟⣩⣭⣤⠽⢿⡟⣁⣀⣀⣠⣽⣿⣿⣷⣄⡀⢘⣛⣛⡓⢂⢠⠐⢿⣿⣿⡿⠋⠀⠀⢘⣛⣛⡛⠛⠛⢛⣛⣛⡉⢻⠇⢘⣛⣛⡃⠙⣛⣛⣛⡋⢘⣛⣛⡃⠀⠀⠁⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠁⢹⣿⣿⣿⣟⣽⣿⣿⣿⣿⣪⣉⣉⣁⣀⣀⣀⣀⣀⣉⣀⣀⢀⣀⣈⣉⣉⣁⣀⣀⣈⣉⣉⡁⢀⣀⣈⣉⣉⡁⠀⢉⣙⡋⠀⢈⣉⣉⣁⣀⡀⠀⠀⠀⠀⠐⠀⠁⠙⠻⢿⣿⢽⠟⠋⠉
⡟⠛⣿⣿⣯⠉⠁⠀⠀⠀⠀⠈⠈⠻⣿⣿⣿⣿⣿⣋⣙⡯⠋⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡅⢈⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠁⢈⣩⣽⣿⣿⣿⣶⣌⣉⠁⠀⢈⣍⣉⣉⣉⡁⠀⠀⠀⠀⠺⠂⣤⠀⠀⠀⠀⠀⢀⡖⠀
⠙⠚⠛⠙⢀⣀⣀⣠⣤⣶⠠⠀⠀⠀⠙⠛⢯⣉⠉⢩⡍⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠉⠁⠈⠉⠉⠉⢉⣩⣭⣭⣭⣤⣶⣶⢺⣿⣿⣿⣿⣿⣿⣿⣿⣦⢀⡈⠉⠉⠉⠉⠁⠀⠄⠀⠀⠀⣠⡿⠃⠀⠀⠀⠠⠊⠀⠀
⠿⣿⡿⠿⠿⠿⣿⣿⣿⣁⠀⠀⠀⠀⣤⠀⠀⠈⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣀⡀⠀⠀⢾⣿⣿⣿⢿⣿⣿⣿⡇⠸⣿⣿⣻⣛⣛⣿⣿⣿⣿⣭⡦⡀⠀⠀⣤⠀⠀⠈⠂⠀⠐⠟⠀⠀⠀⠀⣀⣀⣀⣤⣤
⠀⢄⣀⣴⣿⣷⣮⣽⣥⠏⠀⠀⣀⠄⣿⠀⠀⣀⣤⣄⣀⠀⠀⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡉⣛⣣⣀⢀⣛⣿⡿⣿⣛⣙⣫⣥⢀⣈⡀⢉⣻⣿⣿⣿⣿⣿⡟⠁⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿
⡀⠘⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢰⣾⣷⣶⣶⣿⣄⣼⣏⣀⣈⣿⡄⢠⡟⠀⠀⠘⠇⠀⣿⠻⡆⢸⡏⠀⢀⣿⡟⠀⠙⠀⢸⡇⠠⢽⣿⢐⠻⠟⣿⣀⣄⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠇⠀⠙⣿⣿⣿⣿⠯⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⣍⠉⢉⠁⠸⣇⠀⠀⢀⡀⠀⣿⡨⠋⢸⡇⠠⠋⢻⡇⠀⠀⠀⢸⡟⠻⢿⣿⣾⣿⣿⠿⣿⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣿⡿⣿⣿⣿⣿⣿⡿⠛
⡆⠀⠀⠈⣴⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣻⣿⢷⠋⠀⠀⠙⠲⠶⠚⠀⠀⠘⠳⠖⠛⠓⠂⠐⢚⣓⠂⠀⠐⠚⠓⠪⣬⣿⣶⣿⣿⣶⣿⢿⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠫⠉⠉⠉⠁⠀⠀
⡕⠆⠰⣶⢫⠿⠆⠠⠠⠔⠆⠀⠀⠀⠀⠘⠿⠿⠿⣿⣿⣿⣿⣿⣿⢲⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡏⠸⣿⣿⣿⣏⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣸⡏⣿⣿⡟⣧⣿⣿⢸⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⢿⣿⣿⣿⣿⣹⣿⣿⡇⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡏⣹⣿⢻⣿⣿⣿⣿⣿
⠿⡿⡇⡇⢿⠹⡇⢿⢃⣽⡟⣸⣳⣿⣿⣿⣿⠹⢟⣿⣿⣿⣻⠿⣿⣽⣿⣥⣽⣿⣿⣞⡽⣫⣼⣿⣿⣿⣿⣻⢸⣏⡇⡇⢿⣿⣿⡟⢿⡿⣿⣏⣾⣯⠻⢿⡿⡿⣿⣿⣿⡿⣿⢛⣽⣿⣿⢯⣽⢡⣾⠙⠼⠻⣿⣿⣿⣻
⣼⡏⠷⣾⡷⣾⣗⣈⠘⠛⢳⣖⣶⣿⣿⠏⠻⡐⢓⣭⠟⣿⣿⣾⠯⢍⣟⠹⢻⣯⡌⠛⣟⢷⠽⢽⣿⢿⣏⣱⣿⣿⣧⡬⢻⡿⣠⣻⠻⣿⠽⣯⣾⡟⣯⢈⣧⣶⠊⢿⠹⣁⣪⢿⣿⣿⠟⠛⢣⠔⠫⣷⣮⣭⣿⣋⣻⠟
⡌⢃⣮⣿⠏⠈⠻⠲⢮⣍⢛⣻⣿⣯⢛⣡⡿⠻⠜⣻⣙⠚⠹⡿⣸⣏⡱⢗⠒⠸⠇⣷⣒⣿⣗⣶⣦⣨⣙⠁⠛⠃⣷⠦⣰⣷⣦⣥⣀⣙⡸⣿⣿⣿⡷⠙⣽⡛⠂⠀⣃⣩⣥⣿⣿⣷⡆⢈⣭⣿⣷⠛⠫⣿⣿⣤⣥⣀
⠃⠀⢉⠋⠄⠀⠚⠿⣿⡏⣌⣱⣿⣿⠿⣉⡩⣶⡈⢻⡿⠛⣠⡤⣿⣿⣷⣾⠛⠀⣰⣩⡁⠹⣥⡍⣿⣿⢩⣷⣷⣶⠛⠟⠈⢿⣏⣽⣯⣟⠁⠘⢿⣇⣤⡀⠈⠡⣷⣄⣬⣽⡛⢽⡿⢇⡉⢠⠀⢙⡁⠀⠀⠿⠿⠟⢻⣿
⣀⣀⠚⠻⣷⡄⠀⣀⣴⣿⣿⣗⣿⣿⡏⠸⢿⢿⡆⠀⠀⣰⣬⣤⡬⢿⡿⣽⣶⣷⣿⠉⢀⣦⣤⡀⠾⢿⣶⣻⣿⣿⢰⣿⣀⣀⠯⠿⣿⣿⡃⠀⠀⠈⠿⣦⡀⣀⣀⠾⠟⣿⣿⠟⠻⡈⠁⣰⡞⠛⠛⢻⠛⠀⢀⣽⣿⣿
⠙⠫⠛⡀⠈⠀⣀⣽⣿⣿⣏⣀⢫⣯⣤⡀⣤⣬⣀⠈⠘⠛⣃⣈⣁⣼⣿⣚⣿⣿⣥⢠⡉⠛⠉⣸⣵⣿⣿⣿⣿⣧⠛⢛⠿⠋⠈⢿⣯⣭⠡⠀⠀⠀⢀⣾⣿⣿⣿⣤⡌⣿⣭⡍⠀⠙⠲⠀⣩⣃⣀⣤⣤⣤⡜⠙⠿⠣
⢰⣶⣶⣦⡀⢘⡻⠉⣿⣿⣿⡧⢀⠛⠻⠛⡉⠙⠻⡷⠀⣾⡿⠿⡏⠽⠋⠀⢿⣶⡞⠈⢛⣀⣘⣻⡟⣽⣿⣿⣿⣃⢁⣨⣤⡀⠀⠈⠙⢏⣀⣤⣤⣠⠘⡛⢍⣿⣿⣿⠃⠀⠙⢳⣦⣀⠀⠀⠉⠛⠛⠛⡛⠙⠁⠀⠀⠉
⠀⠀⠉⠻⣟⣯⣤⢄⣸⣿⣿⣿⡶⣤⣼⣿⣿⡛⠛⠛⠛⠃⠀⣰⣿⣿⣟⠛⠛⠛⠛⠻⣿⣿⡿⠿⠛⢻⣿⣿⡻⠟⢻⣿⠿⠛⠘⠛⠛⠛⠿⠿⣟⡛⠀⠀⢺⣿⣟⡛⠛⠛⠃⠀⠉⠙⠂⢉⡷⠀⠀⠈⠀⡀⠀⠀⠀⠀
⠀⠀⣴⣾⣿⣿⠿⡞⢿⣿⣿⡿⠋⠈⢻⣿⣿⣿⣿⡟⠛⠀⢾⣿⣿⡛⠛⠛⠛⠛⠛⣿⠛⠗⠙⠛⠛⢻⣿⣿⣿⡟⠛⠛⠛⠃⢘⡛⢛⡿⠛⠛⠛⠀⠀⠀⠛⢿⣿⣿⠛⠛⠃⠀⠀⠀⠀⠈⣀⣴⣶⣾⣿⣿⣀⠀⠀⠀
⠀⠀⢙⣻⣽⣴⣿⣧⣽⣿⣯⣤⣤⠆⠀⠿⣟⣿⣟⡃⠀⠀⢘⣛⣛⡃⠀⠀⠀⠀⢠⠃⣀⣤⣤⡄⢘⣛⣛⣛⣿⣀⣘⣿⣿⣇⠀⠀⢚⣻⣿⣿⣿⣿⠀⣘⣛⣛⣛⣛⣻⣷⣦⡀⠀⠀⢠⣤⣿⣿⣿⣿⡿⠟⠋⠀⠄⠀
⠀⣠⣿⣿⣿⣿⣿⣿⣶⣴⣶⣾⣯⠀⠀⠀⢘⣛⣻⡃⠀⠀⢘⣛⣛⡃⠀⠀⠀⣠⣶⣿⣿⣿⣿⣧⢘⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⣿⣛⣛⣛⢛⣛⣛⡋⠛⠻⢿⡄⠀⠀⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⡿⠿⠛⠻⣿⣿⡿⠁⠀⠀⣀⣸⣿⣿⣥⣀⠀⢘⣛⣛⡧⢄⢀⡄⢿⣿⣿⣿⠟⠋⠉⢘⣛⣛⡛⠛⠛⢛⣛⣛⡉⠻⣿⢚⣛⣛⡛⠙⣛⣛⣛⡋⢘⣛⣛⡃⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠟⠁⠈⠛⠘⣿⣿⣿⣿⣟⣽⣿⣿⣿⣷⣌⣉⣉⣁⣈⣈⣀⣀⣉⣙⣁⡀⢀⣀⣈⣉⣉⣁⣀⣀⣈⣉⣙⡁⢀⣀⣈⣉⣉⡁⠀⢋⣉⡩⠀⢈⣉⣉⣁⣀⡀⠀⠀⠀⠀⠀⠀⠆⠛⠛⢿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠿⠿⠋⠀⠀⠀⠀⠄⢿⣿⣿⣿⣿⣿⣿⣙⣻⣟⠟⢉⣉⣉⣉⣩⣍⣉⣉⣉⣉⡁⢉⣉⣉⣋⣉⣉⣉⣉⣉⣉⠉⠁⢈⣉⣉⣉⣽⣷⣦⣌⣉⠁⠀⢈⣉⣉⣉⣉⡁⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠙⠉⠉⢁
⣶⣾⣿⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⠹⠙⠉⣉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⢹⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⢀⣀⣀⣈⣩⣿⣿⣿⣿⣿⣿⣿⣦⣄⠈⠉⠉⠉⠉⠁⠀⠀⡀⠀⠀⠀⠈⠻⡧⠄⠀⠀⠀⢀⡼
⣩⣭⣥⣤⣦⣶⣶⣾⡿⠋⠀⠀⠀⠀⣤⠀⠙⢦⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡴⠂⠀⣤⠀⠀⠀⠡⠄⠀⠀⢼⠏⠁⠀⠀⠀⠀⠁⠀
⠻⠛⢛⣛⡛⢿⣿⣿⣧⡇⠀⠀⠀⠀⣿⠀⠀⣀⣥⣄⣀⠀⠀⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡻⣷⣦⠀⠘⣿⣿⣿⣿⣻⣿⣿⣿⢃⣈⡹⠛⣺⣿⣷⣿⣿⣿⣿⣿⣿⣃⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣿
⢢⣤⣿⣿⣿⣷⣾⣷⡿⠀⠀⠰⠔⠀⠀⢀⣸⣧⣀⡈⠛⠀⣼⣏⣀⣈⣿⡄⢠⡟⠀⠀⠘⠇⠀⣿⢷⡄⢸⡿⠛⠃⣿⡏⣀⣙⠀⢸⡇⠀⢸⣿⣿⠿⢟⣿⣏⠛⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿
⠈⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣿⡍⠉⠉⢉⠁⠸⣇⠀⠀⢀⡀⠀⣿⡈⣿⣸⡇⠀⠀⣿⡏⠀⠀⠀⢸⡇⣠⣸⣿⠻⣻⣟⣻⣿⣸⣧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠘⣿⡿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣖⠋⠀⠀⠙⠲⠶⠚⠀⠀⠘⢻⠖⠛⠓⠂⠚⠛⠓⠂⠀⠐⠚⠓⠂⢉⠻⠿⠿⠿⢿⣿⠟⢋⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠻⢿⣿⡿⣿⣿⣿
⠀⠀⠲⣶⣿⢷⠄⠠⠠⠔⠆⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠿⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣤⣤⠀⠀⠀⠀⠀⠐⢶⣶⣶⣿⣿⣿⣶⠘⣿⣿⣿⣶⣤⡀⠀⣀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀
⣿⣿⢹⡾⣿⣿⣸⡏⣿⣿⢹⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣻⡟⢿⣿⣿⣿⣿⣿⡯⣿⣿⡿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⢹⣿⡟⣿⣿⣿⣿⣿
⢿⡻⠘⡇⡿⠹⠇⡇⣡⣽⠏⡸⣱⣿⣿⣿⣿⠹⢿⣿⣿⣿⣻⡿⢿⣽⣿⣷⣾⣽⣿⣏⣹⣋⣾⣿⣿⢿⡿⣿⢻⣏⡿⡇⢻⣿⣿⣿⢿⣿⡿⣯⣼⣯⡜⢿⣿⢿⢿⣿⣿⣿⢿⠟⢯⣿⣿⡯⣝⡇⣶⣭⠳⠟⢿⣷⣿⣟
⣝⠏⢷⣿⡗⣾⢇⢀⡘⣓⣛⣞⣷⣟⣿⠃⠻⡐⠣⣿⠛⣿⣿⣿⠿⢄⣏⠹⠿⣿⣄⡛⣿⢾⠮⠭⣿⠿⣟⣠⣾⣿⣿⢤⠟⣿⢡⣽⡟⢿⡿⠛⣵⣿⣻⠀⣱⣶⡞⠱⡏⢏⣑⡽⢿⣿⡿⠛⠓⠤⠞⠽⣷⣯⣽⣿⣉⣻
⠘⣤⣯⡿⠃⠈⠛⢒⢢⣍⣛⣹⢿⣯⢳⣴⡯⠻⠘⣻⣭⠛⠙⡿⣨⣯⡱⠗⠒⠚⠏⣳⡖⣿⣟⣾⣶⣤⣍⡉⠘⠋⢹⡶⢄⣾⣶⣬⣄⣈⣃⡛⣿⣿⣿⡶⠑⣿⡓⠀⢘⣂⣭⣴⣿⣿⣷⠀⢉⣭⣿⣶⠛⠛⣿⣿⣅⣥
⠀⠀⣉⠡⠀⠀⠛⢿⣿⢎⣍⣰⣿⡿⠻⣉⡩⣶⠈⠻⡿⠛⣤⡤⢿⣿⣷⣾⡛⠀⢀⣭⡅⠙⢧⣭⢽⣿⡍⣽⣾⣷⡞⠻⠃⠹⣯⣉⣽⣯⣋⠀⠸⢿⣤⣄⠀⠉⢻⣦⣠⣩⣯⣛⢹⣿⣁⡁⢠⠈⠙⡁⠀⠀⠹⠿⠿⠛
⣀⡀⠛⠿⣷⡄⢀⣠⣾⣿⣿⣟⣿⣿⡇⠹⢿⢿⡆⠀⠀⣰⣬⣤⣬⢿⡿⣯⣶⣷⣿⡋⠀⣸⣬⡄⠸⢿⣷⣼⣻⣿⡗⣾⣇⣀⡘⡏⣿⣿⣿⠃⠀⠀⠸⢷⣄⠀⠀⣠⠼⠿⢻⣿⡿⠻⡈⠁⣱⡞⠛⠛⠻⡟⠃⠀⠀⣽
⠛⠝⢃⠀⠀⢀⡨⠿⣿⣟⣁⡈⢩⣯⣤⠀⣤⣬⣀⠀⠘⠟⢣⣈⣁⣨⣿⢧⣿⣿⣿⡠⣌⠛⠉⢀⣣⣴⣿⣿⣿⣿⣞⠛⢻⠿⠃⠈⣿⣯⣭⠁⠀⠀⠀⠀⣻⣿⣿⣿⣧⣄⣿⣯⡍⠀⠛⠢⠁⢉⣟⣀⣈⣤⣤⣤⡄⠉
⣶⣶⣦⣄⠀⣛⠏⢹⣯⣽⣿⠄⠘⠛⠿⡓⠉⠛⠿⡧⠠⣶⣿⠿⡏⠩⠛⠀⢻⣷⣾⠂⠙⣅⣠⣿⣿⣯⣽⣿⣿⣿⠟⢀⣡⣤⠀⠀⠈⠛⢟⣠⣤⣄⣀⡈⠻⢟⢙⣿⣿⣿⠏⠙⠧⣤⡀⠀⠀⠌⠙⠛⠿⠿⣛⠛⠃⠀
⠀⠈⠙⢿⣻⣡⡤⠀⣿⣿⣿⣷⣦⣀⣼⣿⣛⠛⠛⠛⠛⠋⠁⣰⣿⣿⣿⠛⠛⠛⠛⠛⣿⣿⣿⣿⡟⠻⢿⣿⣿⢿⡿⢻⣿⠿⠞⢛⠛⢛⠻⠿⠿⡿⠛⠃⠘⠻⢿⣿⡿⠿⠇⠀⠀⠉⠛⠆⠘⢑⡦⠀⠀⠐⠁⠀⠀⠀
⠀⢀⣴⣿⣿⣿⠿⢿⣿⣿⣿⣿⠋⠙⣿⣿⣿⣿⣿⡛⠛⠀⣾⣿⣿⡟⠛⠛⠛⠛⠛⢿⡛⢃⠜⠛⠛⠛⢿⣿⣿⣿⡟⠛⠛⡓⠘⠛⠛⠻⡟⠛⠻⠊⠀⠀⠛⠛⢻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠛⢀⣠⣴⣶⣾⣷⡀⠀
⠀⠘⠛⡟⠁⣾⡟⣰⣿⣿⣗⣀⣀⠀⠸⡿⣿⣿⣟⡃⠀⠀⢘⣛⣛⡃⠀⠀⡀⠀⠀⡞⠁⢀⣀⣀⢘⣛⣛⣻⣿⣿⣛⣻⣿⣃⡀⠈⢘⣻⣛⣛⣿⣷⣶⣚⣛⣛⣛⣛⡛⠿⣶⣤⡀⠀⠀⠀⣀⣰⣿⣿⣿⣿⣿⡿⠟⠀
⣠⣾⣿⣿⣿⣷⣽⡉⢉⣩⣭⣿⡷⠀⠀⠀⢙⣻⣻⡃⠀⠀⢘⣛⣛⡃⠀⠀⠀⠀⣠⣷⣿⣿⣿⣿⣘⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣶⣘⣿⣿⣿⣿⣿⣿⣿⣛⢛⣛⣛⡃⠀⠻⠿⣿⣧⠀⠀⠠⠋⠉⠉⠙⠛⠋⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠄⣈⣉⣛⡁⠀⠀⢘⣛⣛⣇⡀⠀⡀⣸⣿⣿⣿⣿⡿⠟⢛⣛⣋⡙⠛⠛⢉⣛⣛⣙⠛⢿⣿⣟⣛⡿⣿⣟⣛⣛⡋⢙⣛⣛⡃⠀⠀⠀⠈⠸⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⣿⣿⣿⡿⠟⠛⠛⣛⣿⣿⢿⣤⣤⣤⣤⣴⣿⣿⣿⣿⣷⣤⣈⣉⣉⣁⣈⣉⣁⣈⣻⣿⣿⡋⢀⣀⣈⣉⣉⣁⣀⣀⣈⣉⣉⣉⢀⣀⣉⣉⣉⡁⠀⢉⣙⡉⠀⣈⣉⣉⣁⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠴⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⠟⠀⠈⠛⠀⣿⣿⣿⣿⣿⣹⣿⣟⣿⣿⣿⣿⠉⣉⣉⣉⣉⣭⣉⣉⣉⣉⡁⢸⣉⣉⣩⣍⣉⣉⣉⣉⣉⠉⠁⢈⣉⣉⣉⣉⡁⠀⠈⣉⠁⠀⢈⣉⣉⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠉⠻⠿
⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠆⠙⠿⠿⢿⡏⣿⣿⠏⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⡉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⢁⣴⣿⣷⣶⣤⡈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠚⠷⣤⡄⠀⠀⠀
⣿⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣤⣀⠋⠈⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣠⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⣤⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⣈⡻⠂⠀⠀
⠃⢀⣀⣀⣀⣤⣴⣶⠒⠂⠀⠀⠀⠀⣿⠈⠙⣦⣤⣟⡀⠀⠀⣀⣤⣄⡐⠀⠀⢀⣸⣦⣀⣀⣀⣀⠀⠀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣠⠄⠀⠀⠀⠈⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀
⣿⣿⣿⣿⡿⠿⢿⣇⡆⠀⠀⠀⠀⠀⠀⠀⠸⣯⣀⡈⠛⠀⣼⣏⣀⣈⣿⡄⢠⡟⠁⠀⠘⠇⠉⣿⠣⡀⢸⡿⣿⣿⣿⣏⣿⡿⢿⢻⡇⠀⢸⣿⠿⣻⣛⣿⣛⣿⣿⣿⣿⣯⣴⠆⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⣀⣴
⣠⣶⣶⣬⣀⣴⣰⡝⠁⠀⢠⢠⠂⠀⠀⠀⢀⡈⠙⠻⣷⠀⢿⡍⠉⠉⢉⠁⠸⣇⠀⠀⢀⡀⠀⣿⡀⢸⣼⡇⠀⠛⣿⡏⠗⠀⢀⢸⡇⠀⢸⣿⠀⣈⢻⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣈⠛⠶⠖⠋⠀⠀⠙⠲⠶⠚⠀⠀⠘⠳⠖⣻⡓⠂⠐⠛⠓⣖⠋⠑⠚⠓⠂⠈⠻⠶⠋⢸⣿⣻⡏⣾⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠗⠰⠶⠖⠶⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠎⠀⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠐⠲⢶⣿⣾⣿⣥⣬⣤⣶⣶⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⠿⣿⣿
⣻⣷⢻⢸⣿⣿⢸⡏⣿⣿⢹⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⢻⡏⢿⣿⣿⣿⣿⡇⡇⣿⣿⣿⢻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣿⣿⣹⣿⣿⣿⣿
⢿⡿⠸⡆⡿⠻⠎⡇⣡⣽⠏⡸⢓⣿⣿⢿⣿⠹⢯⣿⣿⣿⣟⡛⢿⣼⡿⣷⣾⣽⣿⣏⣸⣫⣼⣿⣿⣿⣿⡧⢷⣿⣽⡇⢻⣿⡿⣿⠻⣿⣿⣿⣹⣿⣜⡿⣿⢿⢿⣿⣿⣻⡿⡟⢫⣿⣿⡯⣭⠞⣼⣎⠇⠿⠻⣿⣿⣟
⣿⠏⢷⣿⡗⣿⣇⠀⡈⣛⣛⣞⣿⣿⣿⠋⠛⡐⠣⣾⣛⣻⣿⣿⠷⢄⣭⡸⠟⣿⣦⡋⠿⢻⣮⠭⢿⠟⣿⣀⢾⣿⢿⢤⠍⢿⣃⣽⣿⢿⣿⠛⣪⣿⣹⡆⣉⣶⣶⠉⢿⢻⣐⣵⢿⣿⡿⠗⠿⢤⠜⠫⢿⣾⣿⣿⣏⣹
⠈⢬⣭⡿⠃⠊⠛⠓⢢⣍⣹⡽⣿⣥⢹⣴⡯⠛⠘⣻⣭⠛⠙⠿⣨⣿⡰⠟⡒⠊⠏⣳⣗⣿⣿⣿⣶⣤⣎⣁⠈⠛⢸⣶⠄⡴⣷⣦⣄⣀⣉⡞⢻⣿⣿⡷⠉⢺⡛⠀⢀⣃⣬⣤⣿⣿⣿⡀⠈⣭⣿⣷⡾⠛⢻⣿⣿⣬
⠀⠀⣈⡡⠀⠀⠛⢿⣿⢦⣊⣸⣿⡿⠓⣉⣩⣶⠈⠹⠿⠛⣤⣤⢽⣿⣷⣾⡏⠀⢀⣦⡅⠈⠿⣬⢩⣿⡏⣩⣾⣿⣾⠛⠛⠈⢿⣏⣼⣿⣯⡀⠈⠿⣧⣤⡀⠈⠹⣷⣴⣤⣭⣽⣋⣿⣿⣀⠐⡤⠀⣈⡀⠀⠈⠹⠟⠟
⣤⡜⠛⢿⣷⠀⢠⣴⣿⣿⣿⣏⣯⣽⡇⠙⢿⢿⡆⠀⠀⣰⣤⣤⣮⢿⡿⣯⣶⣿⣿⡏⠀⣸⣤⣄⠐⠿⠿⣧⣿⣿⡿⢺⣗⣀⣀⠻⠩⢿⣿⡟⠀⠀⠀⠿⣦⣀⡀⣐⣤⠿⠟⣿⣿⡟⢇⠉⢀⣶⡋⠙⠛⢻⠛⠁⠀⠀
⠙⠍⣁⠀⠀⣀⣘⢿⣿⣟⣁⡈⢩⣯⣤⠀⣤⣬⣀⠀⠘⠛⣃⣈⣁⣤⣿⣇⣿⣿⣟⣅⣾⠙⠋⢀⣯⣤⣾⣿⣿⣿⣷⣟⠛⡿⠛⡀⢹⣿⣭⡍⠀⠀⠀⠀⣚⣻⣿⣿⣿⣶⣬⣿⣯⡉⠘⠓⠄⠀⢨⣃⣤⣤⣤⣤⣤⡤
⣶⣶⣦⣄⢈⡛⠏⢻⣯⣽⣿⠄⡘⠛⠻⡓⡉⠙⠿⡧⠠⣾⣿⠿⡏⠩⠛⠀⢸⣷⣶⠇⠉⣣⣠⣛⣻⣻⣭⣿⣿⣿⣿⠃⣈⣤⡄⠀⠀⠙⠻⣇⣤⣄⣀⣀⡜⠻⠿⢙⣿⣿⣿⠏⠹⢥⣄⠀⠀⠀⠈⠉⠛⠻⠻⡋⠙⠇
⠀⠉⠛⣿⣿⣤⡄⢈⣿⣿⣿⣷⣦⣠⣼⣿⣛⠛⠛⠛⠛⠃⠀⣰⣿⣿⣿⡛⠛⠛⠛⠛⢿⣿⣿⡿⢿⠛⠿⣿⣿⣿⣿⠷⣿⡿⠿⠛⡛⠛⠛⠻⠿⢿⠿⠛⠛⠛⠻⢿⣿⡿⠿⠆⠀⠈⠙⠻⠄⠈⢑⡆⠀⠀⠐⠁⠀⠀
⠀⣠⣾⣿⣿⡿⠿⣿⣿⣿⣿⡿⠋⠻⣿⣿⣿⣿⣿⡛⠛⠀⣾⣿⣿⡟⠛⠛⠛⠛⠛⢻⣟⠃⡼⠛⠛⠛⠻⣿⣿⣿⣿⡟⠛⢓⠘⠛⠛⠛⢻⡟⠛⠰⠋⠀⠛⠛⠛⢻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠃⢀⣠⣴⣶⣾⣿⡂
⠀⠙⠛⠋⢱⣿⡇⠘⠽⣿⣇⣀⣀⡀⠸⠿⣿⣿⣟⡃⠀⠀⢘⣛⣛⡃⠀⠀⡀⠀⠀⢰⠃⠀⣀⣀⣘⣛⣛⣛⣻⣿⣿⣛⣿⣇⣀⠀⢘⣛⣛⣛⣛⣷⣴⣞⣛⣛⣛⣛⡋⠙⣟⣦⣄⠀⠀⠀⢀⡀⣰⣿⣿⣿⣿⣿⣿⠿
⣴⣾⣿⣿⣷⣤⡟⠙⠛⣻⣏⣿⣷⠀⠀⠈⢙⣻⣻⡃⠀⠀⢘⣛⣛⡃⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣟⣛⣛⣛⣛⣛⣟⣛⣻⣿⣿⣷⣜⣻⣿⣿⣿⣿⣿⣿⣟⢛⣛⣛⡃⠀⠀⠿⢿⡿⡆⠀⠀⠠⠋⠋⠉⠙⠛⠋⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠃⠀⠀⡀⣈⣉⣛⡁⠀⠀⢘⣛⣛⣃⡀⠀⡀⢰⣿⣿⣿⣿⣿⠿⢟⣛⣋⡉⠙⠙⢋⣙⣛⣛⠛⠿⣿⣿⣛⡻⢿⣿⣟⣛⡋⢘⣛⣛⡃⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⠿⠿⠟⢿⣿⡿⣯⣥⣤⣤⣤⣤⣿⣿⣿⣿⣷⣄⣈⣉⣉⣁⣈⣉⣃⣀⣻⣿⣿⡟⢁⣀⣈⣉⣉⣁⣀⣀⣈⣉⣉⡉⢀⣀⣘⣋⣉⡁⠀⢉⣉⡉⠀⢈⣉⣉⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣾⣿
⣿⣿⣥⣤⣶⣶⠖⠉⠻⠇⢸⣿⣿⣿⣿⣿⣩⣿⣿⣿⣿⣿⣹⠉⣉⣉⣉⣉⣩⣉⣉⣉⣉⡁⢈⣍⣉⣉⣉⣍⣉⣉⣉⣉⠉⠁⢈⣉⣉⣉⣉⡁⠀⠈⣉⠁⠀⢈⣉⣉⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⠙
⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⠀⠻⣿⣿⣿⠏⣻⣿⡟⠉⠉⠓⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⢀⣤⣶⣤⣌⡉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣶⣄⡀⠀
⣯⡙⠛⠉⠀⠀⠀⠀⠀⠈⠀⠀⣀⡀⣽⠡⠻⠛⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣶⣄⣤⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠀⠀⠩⢷⠆
⠙⠃⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⣿⠑⢦⣀⣤⣿⡀⠀⠀⣀⣤⣄⡀⠄⠀⢀⣸⣤⣀⡀⣀⣀⠀⠀⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⢀⢸⠏⠁⠀
⣤⣶⣶⣶⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⡈⠛⠀⣼⣏⣀⣈⣿⡄⢠⡟⠁⠀⠘⠇⠉⣿⠷⡄⢸⡿⢿⣿⣿⣿⣽⣿⣿⣿⣿⠿⢸⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠖⡋⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀
⠿⠛⠛⠏⠉⠁⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠙⠻⣷⠀⢿⡍⠉⠉⢉⠁⠸⣇⠀⠀⢀⡀⠀⣿⡀⠉⣾⡇⠈⠛⣿⡿⢿⡿⠋⢸⡇⠀⢸⣿⠠⣟⠑⠺⣷⣾⣷⣾⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⣿⣿⣷⣦⣶⣶⡿⠃⠀⠀⠃⠊⠀⠀⢠⣤⣼⣷⣶⣾⣋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠲⠶⠚⠀⠀⠘⠳⠖⠛⡗⠂⠐⠛⠓⠂⢀⠴⠚⠓⠂⠈⠻⠶⠋⠀⠀⢹⢟⡿⡿⢿⣿⣟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿
⣿⣿⣿⣿⣿⣿⠗⠰⠶⠖⠖⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠁⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⢀⣿⠶⠟⠀⢀⠀⠻⠛⠛⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿
⣿⣇⢿⢸⣿⣿⢸⡟⣽⣿⢹⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⢻⡧⣿⣿⣿⣿⣿⡇⡏⣿⣿⣽⢸⢻⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣇⣿⣿⢻⣿⣿⣿⣿
⢿⡿⢸⣠⡿⠻⠜⡇⢥⣿⠎⠹⢓⣿⣿⢿⣿⠽⣯⣿⣿⣿⣟⡛⢿⣼⡿⣷⣾⣽⣿⣏⣸⣫⣼⣿⣿⣿⣿⡇⡇⣿⣽⡏⢺⣿⣿⣿⠿⣿⣿⣿⣽⣿⣽⡿⣿⣿⡿⣿⣿⣿⡿⡿⢫⣿⣿⡟⢭⡺⢰⣦⡍⠿⠻⣿⢾⣿
⣿⠎⣿⣿⡛⣿⣇⠤⡈⣛⣛⣟⣿⣿⣽⠋⣛⡐⠣⣾⣛⢿⣿⣿⡷⢆⣭⡸⠟⣿⣦⡛⢿⢿⣯⠭⠿⠟⣿⣃⣼⣿⢿⠤⠜⢻⡟⣼⣿⠻⣿⠟⢒⣽⣟⡆⢈⣶⣶⡇⠻⠻⣑⣼⠾⣿⣿⠟⠽⠦⡤⠋⠿⣷⣿⣿⣿⣉
⠁⢼⣽⡟⠃⠊⠛⡛⢢⣌⣩⠽⣿⣥⢹⣴⡯⠛⠘⣻⣭⣙⠙⠿⣩⣿⡴⠟⣒⠊⠿⣱⣗⢻⣿⢿⣷⣤⣆⣥⠀⠛⢨⣶⠆⠰⣿⣦⣤⣀⣉⣸⠛⣿⣿⣷⠎⢹⣿⠃⠀⣱⣠⣥⣺⣿⣿⣇⠀⢨⣽⣿⣶⡟⠛⢿⣿⣧
⠈⢀⣀⡠⠀⠀⠛⢿⣿⣶⣈⣹⣿⡿⠓⣈⣩⣲⠈⠹⠿⠛⠤⣦⢹⣿⣷⣾⡏⠁⢀⣦⣅⠈⠿⣾⠑⣿⣿⢯⢾⣿⣶⠟⠟⠁⠻⡏⣠⣿⣿⣅⡀⠻⢿⣤⡄⠀⠙⣷⣤⣦⣬⣭⣟⣹⣿⣋⠀⠣⠄⢐⣁⠀⠀⢈⠿⠿
⣤⡘⠛⢿⣷⠀⢠⣴⣿⣿⣿⣍⣯⣽⡇⡙⢿⢿⡆⠀⠀⣸⡤⣴⣶⠸⣿⣯⣷⣿⣿⣏⠀⣸⣤⣤⣐⠻⡿⢗⣼⣿⣿⢻⣿⣀⣀⡸⠟⠽⣿⣿⡋⠀⠀⠸⢷⣄⣀⣀⣂⡸⠿⢟⣿⣿⠿⠈⠁⣰⣾⠉⠛⠛⠻⠛⠀⠀
⠚⠉⡁⠀⢀⣀⡘⠿⣿⣟⣁⡈⢭⣯⣤⠀⣤⣬⣀⠀⠙⠛⣓⣈⣁⣤⣿⡿⣺⣿⣿⣇⢼⡉⠋⠀⣮⣤⣴⣿⣿⣿⣿⣿⡟⣻⠟⢃⠈⣿⣯⣭⠉⠀⠀⠀⢀⡙⣻⣿⣿⣿⣶⣽⣿⣯⠁⠛⠦⠀⠀⢜⣀⣤⣤⣤⣤⣤
⣶⣶⣦⣀⣘⡻⠍⢿⣽⣽⣿⠄⡈⠛⠻⣓⠉⠛⠿⡧⠠⣾⣿⠿⣏⠩⠛⠁⠸⣷⣶⡗⠈⢣⣠⣟⣻⡿⣿⣭⣿⣿⣿⣿⢃⣤⣤⠀⠀⠈⠛⢿⣠⣤⣀⣀⣠⠛⠿⢟⢙⣿⣿⣿⠟⠿⣤⡀⠀⠀⠠⠉⠉⠛⠻⠻⢛⠉
⠀⠉⠛⣟⣧⣤⡄⢈⣿⣿⣿⣷⣦⣠⣼⣟⣛⠛⠛⠛⠛⠃⠀⣰⣿⣿⣿⡛⢛⠛⠛⠛⠻⣿⣿⣿⢿⠟⠻⢿⣿⣿⣿⣿⡶⣿⠿⠟⢛⠛⠛⠛⠿⠿⢿⠿⠛⠛⠛⠻⢿⣿⣿⣿⡥⠀⠙⠛⢧⠀⠈⢓⡦⠀⠀⠐⠁⠀
⢀⣴⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⠞⠻⣿⣿⣿⣿⣿⡛⠛⠀⣾⣿⣿⡿⠛⠛⠛⠛⠛⠻⣿⠃⣸⠟⠛⠛⡛⢻⣿⣿⣿⣿⠛⠃⠘⠛⠛⠛⠛⣿⠛⠀⠔⠉⠛⠛⠛⠛⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠐⠃⢀⣠⣤⣶⣶⣿
⠈⠛⠻⠋⣴⣿⠇⠻⣿⣿⡏⠁⣀⠀⢹⣿⣿⣿⣟⡃⠀⠀⢙⣛⣛⡃⠀⠀⡀⠀⠀⢰⠋⠀⢀⣀⣘⣛⣛⣋⣻⣿⣿⣟⣛⣃⣀⠀⢙⣛⣻⣛⣛⣛⣤⣾⣟⣛⣛⣛⡃⠉⠛⣟⣤⣀⠀⠀⠀⢀⡀⢰⣿⣿⣿⣿⣿⣿
⣴⣾⣷⣶⣌⡠⠞⠛⠛⣛⣻⣿⣯⠀⠈⠉⢙⣟⣻⡁⠀⠀⢘⣛⣛⡃⠀⠀⠀⠀⢀⣨⣶⣿⣿⣿⣿⣛⣛⣛⣛⣛⣻⣛⡋⣿⣿⣿⣾⣛⣻⣿⣿⣿⣿⣿⣿⣟⣛⣛⡃⠀⠀⠀⢿⣿⣷⡄⠀⠀⠸⠟⠋⠙⠛⠛⠛⠁
⣿⣿⣿⣿⣿⣿⣶⣄⣾⣿⣿⣿⠃⠀⠁⡀⣈⣉⣛⡁⠀⠀⢘⣛⣛⣃⡀⠀⠀⢀⣿⣿⣿⣿⣿⣿⢿⣟⣛⡋⠉⠉⢙⣋⣛⡙⠻⠿⣿⣿⣛⡛⢿⣿⣿⣟⡋⢘⣛⣛⡃⠀⠀⠀⠀⠈⠙⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣝⣁⣀⣀⣀⣀⣿⣿⣿⣷⣦⣀⢈⣉⣉⣁⣘⣚⣏⣹⣻⣿⣿⡿⢋⣀⣈⣉⣉⣁⣀⣀⣈⣉⣉⡉⢀⣀⣈⣻⣋⡁⠀⢉⣉⣉⠀⢈⣉⣉⣁⣀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼
⣿⠿⠛⢋⣉⣉⡽⢿⣿⠉⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⢯⠉⣉⣉⣉⣉⣩⣍⣉⣉⣉⡁⢈⣝⣉⣉⣉⣉⣉⣉⣉⣉⠉⠁⢈⣩⣉⣉⣉⡁⠀⠀⢉⠁⠀⢈⣙⣉⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈
⣿⣿⣿⣿⡿⠋⠀⠀⠉⠀⢸⣿⣿⣿⣿⠏⣿⣿⣯⠉⠛⠯⠋⠀⠀⠉⠉⠉⠉⠉⠉⠉⢉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⣀⣈⡉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀
⣿⣿⣿⠟⠀⠀⠀⠀⠀⠄⠀⠈⠛⠛⣿⠀⢿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⡄⠀⢀⣀⣀⣀⣠⣾⣿⣿⣿⣷⣦⣤⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠉⢓
⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⣿⣌⡀⣀⣴⣷⡀⠀⠀⣀⣤⣄⡀⢀⠀⢀⣠⣦⣀⡀⣀⣀⠀⠀⣀⡀⠀⣀⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣰⡖
⠁⠀⠀⢀⣀⣤⣴⡶⠄⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⡋⠛⠀⣼⣏⣀⣈⣿⡄⢠⡟⠁⠀⠸⠇⠛⣿⢶⡄⢸⡇⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⣸⠉⠀
⣾⣿⣿⣿⣿⠟⠋⡀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡈⠙⠻⣷⠀⢿⡍⠉⠉⢉⠁⠸⣇⠀⠀⢀⡀⠀⣿⡀⠙⢸⡇⠈⠙⣿⣿⣿⣷⣿⢻⡿⠿⢻⣿⠀⣀⠻⣿⣿⣻⢿⠿⠿⣿⣿⣿⣿⣿⣟⣋⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣛⡙⠉⠁⠀⢀⣶⠃⠀⢀⠀⡠⠀⠀⠀⠀⠘⠓⠶⠞⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠲⠶⠚⠀⠀⠘⠳⠖⠛⢳⡂⠐⠛⠛⠛⠙⠗⠚⢓⣂⡈⠻⠶⠛⠒⠁⢹⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣶⣶⣿⢿⠗⠰⠴⠔⠞⠁⠀⠀⢸⣿⣿⣿⣿⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠈⠿⠶⠟⠀⠨⠿⢟⠿⢿⣿⣿⡏⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⡇⣿⢸⣿⣿⢸⣽⢻⣿⢹⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⣿⣿⢻⡷⣿⣿⣿⣿⣿⣏⡟⣿⣿⢿⢸⣻⣿⣿⣿⣿⣿⣿⡿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⢯⣿⣿⣿⣿
⣿⡿⢸⣨⡿⢻⠸⡇⢤⣿⠎⠹⢓⣿⣿⣿⣿⠽⢯⣿⣿⣿⣟⡛⢿⣾⣿⣷⣾⣽⣿⣯⣸⣫⢼⣿⣿⣿⣿⡇⡇⣿⣽⠸⢘⣿⣿⣿⡿⣿⣿⣿⣵⣿⣽⣻⣿⡿⡿⣿⣿⣟⣿⣿⠫⣿⣿⣿⢭⡶⢱⣶⡙⠼⠟⣿⣾⣿
⡿⠎⣾⣟⡛⣿⣇⠤⡈⣙⣛⣿⣿⣿⣽⠃⣛⡐⠣⣾⣛⢹⣿⣿⡿⢆⣭⡩⠟⣿⣧⡛⠻⠿⣯⠽⠿⠟⢿⣃⡸⣿⢿⠧⠬⠻⣟⢬⣿⠿⣿⡟⠓⣽⣯⣷⠈⣳⣶⡎⠹⡟⣏⣪⡾⢻⣿⡿⠚⠧⡦⠚⠽⢷⣯⣿⣿⣏
⠡⣾⣼⡟⠁⠉⠛⡛⢲⣌⣹⠽⣿⣥⢹⣴⡯⠛⠈⣻⣭⣙⠉⠿⣣⣿⣔⠟⡓⠊⠿⢑⣗⢺⣿⢿⣷⣦⣴⣨⠀⠙⠃⣷⡦⢠⣾⣷⣤⣀⣉⣱⡟⣻⣿⣿⠶⠙⣿⡃⠀⢘⣂⣥⣔⣿⣿⣗⠀⢈⣽⣿⣷⣾⠛⢻⣿⡿
⠈⢀⣁⡀⠀⠐⠛⢿⣿⣶⣈⣸⣿⡿⠓⣈⣩⣲⠈⠹⠿⠛⠤⣤⢸⣿⣷⣾⡟⠁⢀⣦⣥⠈⠿⣶⠎⣿⣿⢊⣵⣿⣷⡟⠟⠃⠹⣯⢡⣾⣿⣯⡀⠘⠿⡇⣆⠀⠉⢻⣆⣶⣤⣭⣽⣏⣿⣟⡁⠘⡤⠀⢊⡀⠀⠀⠛⠿
⣤⡘⠛⢿⣦⠀⣠⣴⣿⣿⣿⣝⣯⣽⡇⡙⢿⢿⡇⠀⠀⣠⣤⣤⣶⠹⣿⢿⣷⣾⣿⣯⠀⣨⣥⣤⣀⠺⠿⠟⣙⣿⣿⡗⣿⣆⣀⣀⠿⡩⣿⣿⡟⠁⠀⠈⢷⣦⢀⡀⣐⣄⠽⠿⢻⣿⣿⠏⠉⢀⣶⡏⠙⠛⠛⡿⠛⠀
⠫⠉⡀⠀⢀⣀⠸⠿⣿⣟⣁⡘⣭⣯⣤⠀⣤⣬⣀⠀⠙⠛⢻⣀⣁⣠⣿⡿⣺⣿⣿⣟⣰⡟⠋⠁⣸⣥⣤⣾⣿⣿⣿⣿⣿⣛⡿⠟⡀⢹⣿⣯⡝⠁⠀⠀⠀⡚⢛⣿⣿⣿⣿⣦⣿⣿⡟⠘⠳⡀⠀⠠⢓⣀⣈⣤⣤⣤
⣶⣶⣤⡀⣙⡻⠍⣿⣭⣽⣿⠄⡘⠛⢿⢓⠉⡛⢿⡧⠀⣶⣿⠿⣏⠉⠝⠃⠘⣿⣿⣿⠈⠻⣀⣜⣿⣿⣿⣯⣿⣿⣿⣿⡟⢁⣤⠀⠀⠀⠙⠻⢇⣤⣀⣀⣀⣜⠺⠿⢿⢻⣿⣿⣿⡟⢧⣄⠀⠀⠀⠠⠉⠙⠛⠿⠿⠟
⠈⠙⠻⣿⣇⣤⡄⢸⣿⣿⣿⣦⣄⢀⣼⣛⡛⠛⠛⠛⠛⠋⠁⢐⣿⣿⣿⡛⠛⠛⢿⠛⢻⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣯⣼⡿⠿⠛⠛⠛⠛⠿⢿⣿⡿⠿⠿⠛⠛⠻⣿⣿⣿⣿⡥⠈⠛⠿⡄⠀⠚⢲⡦⠀⠀⠠⠋
⢀⣴⣾⣿⣿⣿⢶⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣟⠛⠛⠀⣸⣿⣿⡿⠟⠛⠛⠛⠛⠛⣿⠣⠙⡻⠛⠛⢛⠛⣿⣿⣿⣿⡛⠻⠸⠛⠛⢛⢛⢿⣟⠀⠀⠝⠛⠛⠛⠛⠛⢿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠐⠋⠀⣀⣤⣤⣶
⠘⠻⠿⠋⣵⣿⠈⢿⣿⣿⡟⡁⠀⠀⢹⣿⣿⣿⣟⡃⠀⠀⢙⣛⣛⡃⠀⠀⠀⠀⠀⠀⡿⠁⠈⠀⣘⣛⣛⣃⣘⣿⣿⣿⣛⢃⡀⠀⢘⣛⣛⣟⣛⣛⣀⣘⣿⣛⣛⣛⡃⠀⠙⠿⡿⣄⡀⠀⠀⠀⢀⠀⢠⣾⣿⣿⣿⣿
⣠⣤⣄⣀⠉⣩⠶⠶⠾⠷⢿⣿⣏⠀⠀⠉⢻⣟⣻⡃⠀⠀⢘⣛⣛⡃⠀⠀⠈⠀⠀⢀⣥⣶⣾⣿⣿⣟⣛⣛⣛⣛⣻⣿⡋⢸⣿⣿⣾⣟⣛⣫⣿⣿⣿⣿⣿⣿⣛⣛⡃⠀⠀⠈⠙⣿⣿⣶⡀⠀⠈⢡⠿⠛⠛⠿⠟⠿
⣿⣿⣿⣿⣿⣤⡀⠰⣶⣿⣿⣾⠃⠀⠂⠀⢈⣋⣛⡁⠀⠀⢘⣛⣛⣃⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣛⡛⠉⠉⢉⣛⣉⡘⠿⢿⣿⣿⣿⡛⢿⣿⣿⣿⡟⢛⣛⣻⡃⠀⠀⠀⠀⠈⠉⠳⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣦⡹⢛⡿⠃⠀⠀⢀⣀⣿⣿⣿⣷⣄⡀⢈⣉⣉⣉⣙⣢⣾⣴⣿⣿⣿⣿⣿⣋⣉⣉⣉⣙⣀⣀⣈⣉⣉⡁⢀⣀⣈⣛⣿⡃⠀⢉⣙⡉⠀⢈⣉⣉⣁⣀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡿⠿⠿⣿⣿⣷⢻⣿⣷⣶⣶⣿⣿⣟⣿⣿⣿⣿⣿⢮⠉⣉⣉⣉⣉⣉⣭⣩⣉⣉⡉⢈⣉⣍⣉⣉⣉⣉⣉⣉⣉⠉⠁⢈⣹⣉⣉⣩⡁⠀⠈⢉⠁⠈⢈⣉⣟⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⣏⣉⣤⣤⣤⡖⠋⠻⢟⠀⣿⣿⣿⣿⣿⡟⣹⣯⣍⠙⠻⣛⡿⠃⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠛⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁
⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠁⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⣀⠀⠀⠀⠀⢀⣤⣶⣶⣤⣤⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⠈⠛
⢿⠿⠛⠀⠀⠀⠀⠀⠀⠃⠀⠀⣀⠀⣿⠈⠛⣋⣤⣤⡀⠀⠀⣀⣤⣄⡀⠀⠀⢀⣠⣦⣀⡀⣀⣀⠀⢀⣀⡀⠀⣀⣀⣈⣠⣤⣀⣨⣤⣤⣼⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢼⣧⣀⡿⠛⠀⣼⣏⣀⣈⣿⡌⢠⡟⠁⠀⠹⠧⠐⣿⣤⣄⢸⣇⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⢀⣀⣤⣴⡶⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⡝⠻⣷⠀⢿⡍⠉⠉⢉⠁⠸⣇⠀⠀⢀⡀⠀⣿⡈⠙⢾⡟⠉⠿⣿⣿⣿⣿⣝⢿⣿⣿⣿⣿⠟⣀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣀⡤⠀⠀⠀⠀⠀⠀⠃
⣿⣿⣿⣿⠿⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠶⠞⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠲⠶⠚⠀⠀⠘⠳⠖⠛⣳⡂⠐⠛⠛⣿⣿⣿⣞⠓⠛⠋⠻⠶⠋⠀⠀⢹⣿⠿⣛⣿⢛⣟⣛⣿⣿⣿⣿⣿⣭⣴⠀⠀⠀⠀⠀⠀⠀
⡉⠟⠷⣶⠏⢶⡾⠰⠤⢶⠖⡄⠀⠀⠀⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠄⠀⠀⠉⠁⠀⠉⣀⡠⠤⠄⠀⠀⠿⠶⠟⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀
⣻⡆⣿⢹⣿⣿⢹⣸⢹⢿⢹⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣟⣿⣿⢹⡿⣿⣿⣿⣿⣿⣟⡿⣿⣿⣿⢻⣻⣿⣿⣿⣿⣿⣿⡽⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡸⣿⡟⣿⣿⣿⣿
⣻⡧⢻⡸⡿⢻⢸⠁⢤⡿⠎⠹⢓⣿⣿⢿⣿⠭⢯⣿⣿⣷⣟⡛⢿⣾⣿⣷⣾⣽⣿⣯⣸⣫⢼⣿⣿⣿⣿⡗⡇⣿⣽⠸⢘⣿⣿⣾⡿⣿⣿⣿⣷⣿⣥⣻⢿⡿⡿⣿⣿⣟⣿⢿⠋⣽⡿⣿⣭⡵⢣⣶⣸⠸⠟⢿⡿⣿
⡿⠈⣾⣟⡻⣿⡇⠤⡈⣛⣛⣟⣿⣿⣽⠃⠛⡐⠣⣾⣛⢹⣿⣿⡿⢆⣭⡩⠟⢿⣧⣓⠻⠿⣯⠭⠽⠿⢻⣃⡐⣿⡿⡧⡬⠙⣿⢨⣿⡟⢿⡿⠓⣸⣯⣷⡀⢡⣶⣾⠐⠟⢇⣢⣾⠻⣿⡿⠋⠣⢴⠜⠫⠿⣾⣿⣾⣯
⠡⣿⣼⡟⠀⠉⠛⡛⢲⣌⣹⠽⣿⣥⢹⣴⡯⠻⠈⣻⣭⢝⠉⠿⢇⣽⣴⠟⡓⠊⠿⢑⣖⢺⣿⣻⣷⣦⣴⣈⡀⠙⠃⣵⡦⠠⣾⣷⣤⣀⣈⣁⡿⢛⣿⣿⡷⠋⢺⡛⠀⠈⣂⣬⣄⣺⣿⣿⡃⠀⢭⣿⣷⣶⡝⠛⢿⣿
⠀⢀⣁⠄⠀⠘⠛⢿⡿⣶⣈⢸⣿⡿⠳⣈⣩⣶⠈⠹⠿⠛⠤⣤⢸⣿⣿⣶⣟⠁⢀⣢⣭⠀⠽⢦⡍⢼⣿⣍⣵⣾⣷⡿⠻⠋⠙⣯⠁⣮⣽⣿⣄⡀⠻⣿⣠⡀⠈⠹⣦⣐⣤⣥⣭⣟⣹⣿⣃⠀⢣⠀⠘⣁⠀⠀⠘⠻
⣤⠘⠻⢿⣦⠀⣀⣴⣿⣿⣿⣝⣿⣿⡇⠻⢿⢿⡆⠀⠀⣠⣥⣤⣤⠵⣿⠿⣥⣶⣿⣿⡁⢀⣧⣥⡀⠺⢿⠷⠚⣻⣟⡛⢾⣇⣀⣀⡸⠏⣿⣿⣿⠏⠀⠀⠿⣦⡀⠀⠈⣄⠈⠿⠟⣿⣿⡿⠉⠁⢠⣾⠛⠛⠻⢿⠿⠇
⠛⢋⡀⠀⢀⣀⠹⢿⣿⡛⠅⠙⣭⣯⣤⠀⣤⣼⣀⠀⠘⠻⠳⢀⣉⣠⣽⣿⢹⣿⣿⣿⣁⡟⠛⠉⢸⣡⣤⣤⣾⣿⣿⣿⣿⡿⢿⠿⠃⠈⣿⣿⣿⠋⠀⠀⠀⠀⠛⠿⣿⣿⣿⣷⣶⣿⣟⡛⠲⢆⠁⠀⡩⠂⣀⣈⣀⣀
⣶⣤⣄⡀⡛⡻⠙⣿⣿⣿⣿⠀⡘⠻⢿⢓⢛⠛⢿⣦⠀⣴⣿⡿⣏⢙⠿⠋⠈⢿⣿⣽⠅⠻⢄⣰⣿⣿⡛⢿⡙⣿⣿⣿⣿⡟⣁⣀⠀⠀⠈⠻⣻⢀⣀⡀⠀⣨⡃⠾⣿⢟⠻⣿⣿⣿⡿⣇⡀⠀⠀⠀⢠⠚⠛⠻⠿⠿
⠉⠛⠿⢿⣋⣀⡀⢹⣿⣿⣷⣤⣁⢀⣘⣛⠛⠛⠛⠛⠛⠋⠁⠐⣟⣿⣿⡟⠛⠛⠻⠟⢻⣷⣿⣿⣿⣿⡟⠛⣻⣿⣿⣿⣿⣦⣿⡿⠟⠛⠛⠛⠺⢿⣿⣿⣿⠿⠿⠛⠛⢛⣛⣛⣿⣿⣇⠘⠻⢧⠀⠀⠒⢢⣄⠀⠀⠀
⢀⣤⣾⣿⣿⣿⣴⣻⣿⣿⣿⣿⡿⣾⣿⣿⣿⣿⣟⠛⠛⠀⢸⣿⣿⣿⠿⠟⠛⠛⠛⠛⣻⡋⡙⢛⡿⠛⠛⡛⢻⣿⣿⣿⣟⡋⢝⠛⠛⠛⠛⡻⣿⠀⠈⢈⡟⠛⠛⠛⠛⠛⣿⣿⣯⣯⠁⠀⠀⠀⠀⠀⠠⠎⠁⠀⣀⣠
⠿⢿⡿⢛⣽⣶⠙⢿⣿⣿⡿⠋⠀⠀⢻⣿⣿⣿⣿⡃⠀⠀⢙⣛⣛⡃⠀⠀⠀⠀⠀⠀⢸⠗⠀⠋⢘⣛⣛⣃⣈⣻⣿⣿⣿⠃⠀⠀⢘⣛⣛⣛⣟⣛⠀⣚⣛⣛⣛⣛⡃⠀⠈⠻⠿⣿⡀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿
⠀⠀⠀⠈⠛⢃⢤⣬⣽⣽⣷⣶⡖⠀⠈⠉⢿⣛⣛⡃⠀⠀⢘⣛⣛⡃⠀⠀⠈⠀⠀⠀⢇⣠⣤⣶⣾⣟⣛⣛⣛⣛⣛⣻⡏⠀⢹⣷⣾⣟⣛⡛⣻⣿⣿⣿⣿⣿⣟⣛⡃⠀⠀⠀⠀⠻⣿⣷⣤⠀⠀⠈⢠⠾⠿⠿⢿⡿
⣿⣷⣶⣯⡜⠁⠀⢠⣴⣶⣾⣽⠍⠀⠄⠀⢈⣙⣛⡃⠀⠀⢸⣛⣛⡃⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣟⡛⠛⠋⢉⣙⣛⡀⠺⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⢿⣛⣛⣃⠀⠀⠀⠀⠀⠉⠛⢎⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⣄⠈⢻⠿⠿⠏⠀⠀⢀⣀⣾⣿⣭⣥⣀⡀⢈⣉⣉⣉⣙⣦⣶⣤⣼⣿⣿⣿⣿⣿⣛⣉⣉⣁⣆⣀⣈⣉⣉⡁⢀⣀⣈⣙⣿⣿⠀⢉⣛⡛⠋⢈⣉⣉⣁⣀⡀⠀⠀⠀⠀⠀⠀⡅⠄⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⣄⣾⣧⣄⣀⣀⣈⣭⣿⣿⣿⣿⣿⣷⣌⠉⣉⣉⣉⣉⣉⣩⣭⣭⣿⡟⢋⣉⣉⣍⣉⣉⣉⣉⣉⣉⠉⠁⢈⣉⣭⣉⣉⡅⠀⠈⢉⠁⠀⢘⣉⣉⣭⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⠿⠛⠛⠛⣿⣿⡇⢹⣿⣿⣿⣿⣿⣿⢋⡉⠛⠻⢟⣿⢿⠇⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠋⠉⠉⠩⠍⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⣶⣶⡶⠋⠈⠉⠁⠀⢿⣿⣿⣿⣿⠏⣾⣿⣧⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠈⠹⠟⠛⣿⠀⠛⣿⣥⣄⡀⠀⠀⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⣀⣀⠀⢀⣀⡀⠀⣀⣀⣀⣠⣤⢀⣈⡀⠀⣸⣿⣀⣀⣀⣀⣀⣀⣶⣾⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀
⠿⠛⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⣀⣉⡀⢸⣯⣀⣺⠟⠀⣼⣏⣀⣈⣿⡄⢠⡟⠁⠈⠹⠇⡀⣿⣀⡀⢸⡇⠀⠀⣿⣯⣶⣿⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢲⣌⠙⠿⣷⠀⢿⡍⠉⠉⢉⠁⠸⣇⠀⠀⢀⡈⠁⣿⡙⠛⢻⡇⢴⠢⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⡟⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣀⣤⣶⡆⠓⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠶⠞⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠲⠶⠚⠀⠀⠘⠳⠖⠛⠓⠚⠐⠛⢻⣿⣿⣿⣞⣻⣿⣿⣿⠿⠛⠃⠀⢹⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣠⡆⠀⠀⠀⠀
⣿⣿⣿⣿⠿⢿⠶⠠⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠉⢿⡿⠿⢯⠉⠀⠁⠀⠀⠿⠶⠟⠀⢈⣻⠟⢟⣛⡛⡛⠛⢛⣿⣿⣿⣿⣿⣥⡆⠀⠀⠀⠀
⣿⡜⡇⣿⣿⣧⢿⢸⢹⢿⠿⡏⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣟⣿⣿⢻⣿⡿⣿⣿⣿⣿⣿⢹⢸⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢇⣿⣿⢻⣿⣿⣿
⢿⠇⣷⢰⡧⠻⢸⠨⣨⣽⡆⡻⣳⣿⣿⣿⣿⡭⢯⣿⣿⣿⣟⡛⢿⣽⣿⣷⣾⣿⣿⣯⣨⣏⣹⣿⣿⣿⣿⣿⠜⢼⣯⣿⠱⣿⣿⣾⡿⢿⣿⣿⣧⢿⣧⣿⣿⣿⢿⢿⣿⣿⣿⢿⡟⢽⣿⢿⡯⣕⡞⣴⣇⡇⠾⠻⣷⢿
⡻⠱⣾⣿⢲⣿⡄⣀⡘⢓⣳⣞⣷⣿⣿⠏⠻⡀⢲⣼⢛⢹⣿⣿⡿⠅⣹⡩⠿⢿⣧⡝⠻⡗⣿⡭⠽⡿⢻⣿⣠⢾⣿⡷⢬⠝⣿⣧⣷⣷⠻⣿⠿⠊⣻⣿⣆⠙⣼⣶⡏⠺⠿⣐⣑⡵⢿⣿⡟⠙⠰⡨⠛⡽⠷⣯⣵⣿
⢢⣾⡹⡿⠀⠛⠗⠲⢵⡉⣛⣹⣿⣯⢻⣡⡾⠻⠜⣻⣙⠛⠋⢿⡏⣻⣍⠾⠒⠖⢻⠞⣶⡒⣿⣗⣷⣦⣤⣍⡉⠈⠻⢸⡶⠄⢨⣷⣦⣄⠀⠛⣼⡺⢿⣿⣿⡶⠉⣽⡯⠀⢐⣉⣉⡁⢾⣿⣿⡀⠤⢩⣿⣷⣮⡙⠛⢻
⠀⠉⠉⡀⠀⠸⠷⣾⣶⣭⠙⣲⣿⣿⠮⠙⢁⢦⡀⢻⣿⠿⢇⣤⡼⣿⣿⣴⣿⠃⠀⣘⣉⠁⢙⣧⣥⠫⣿⡯⣉⣤⣶⣾⡿⠶⠀⢿⣿⢡⣉⣻⣯⡁⠀⢿⣶⣄⠀⠙⢷⡄⣬⣉⣙⣑⣟⢹⣿⡇⠠⢤⠉⠉⠋⠁⠀⠘
⠀⠾⢿⣶⣄⠀⠀⣨⣷⣿⣿⠾⣟⣳⡖⢸⣿⣿⣦⠀⠀⠀⣆⣉⣄⡛⣻⣿⣿⣠⣴⣿⣋⠁⣒⣈⡅⠠⢿⣷⡎⠩⣿⣿⡷⡶⠀⠀⠉⢲⠟⢿⣿⣿⡿⠀⠉⣿⣄⠀⠈⠩⡉⠉⣿⡿⢿⣿⣷⡞⠂⢀⣴⡾⠿⠷⡶⣶
⡳⠆⠀⠉⠁⠀⣿⣿⣿⣿⠟⠳⠻⡿⠁⠀⢈⣸⠁⠀⠐⠿⢿⠉⠛⠃⢸⣷⣾⣿⣿⣿⡿⢸⠿⠛⠉⡖⣀⣁⡀⣀⣿⣿⣷⣷⣿⣷⣿⠇⠐⣿⣿⣿⡗⠀⠀⠀⠉⠷⢷⣶⣷⣶⣤⣤⣾⣿⡿⠦⡌⠉⠁⢒⠦⠀⠀⢃
⣀⣈⠀⠠⣶⣾⠬⣭⠿⢷⣶⠂⠸⣿⣿⣤⠻⠿⣷⣦⠀⣠⣴⣶⣿⠻⠿⣿⠓⢲⣿⣛⣿⠻⣆⠀⣴⣿⣿⠿⣿⡿⣿⣿⣿⣿⣿⡍⠀⠀⠀⠙⢿⣿⠇⠀⠀⠀⢐⠦⠀⣴⣿⣿⣿⣿⣿⣿⣿⠀⠈⠀⠀⠀⠠⣶⢿⣿
⠻⢿⣿⣶⠖⠈⠀⢿⣿⣿⣟⠁⠀⠀⠙⠛⠟⠛⠛⠛⠛⠛⠛⠛⢻⠛⠛⣛⠛⠛⢿⣿⠟⠃⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⡛⠛⠛⠃⠉⣨⣿⣾⣿⣿⣿⡟⠛⠋⠳⣤⣿⣿⣿⡏⢷⣦⡀⠀⠀⣨⡀⡀⠈
⠀⢀⣨⣿⣶⣶⠀⣠⣿⣿⣿⣷⣶⣀⣿⣿⣿⣟⣻⠛⠛⠀⢘⣻⣿⣿⣿⣿⡟⠛⠛⠛⠛⠓⠻⣿⣿⣿⡿⡟⠛⠿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠈⠉⠛⠿⢻⣛⡛⠛⠛⠃⠐⣲⣾⣿⡿⠟⠀⠈⠃⠀⠈⠀⡨⠟⠀
⣴⣿⣿⣿⠿⠟⣿⣿⣿⣿⣿⣿⠖⠛⢻⣿⣿⣿⣿⣇⠀⠀⢘⣻⣿⡟⠋⠁⠀⠀⠀⠀⠀⢶⣄⠀⣸⣛⣛⣋⣀⣀⣿⣿⣿⣿⣦⠀⢞⣛⣛⣛⣛⣛⡧⣚⣛⣟⣛⣛⡃⠀⠀⠀⠻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠁⢀⣤
⠉⠋⠋⢰⣾⡿⠘⢿⣿⣿⡏⠁⠀⠀⠸⣿⣿⣿⣿⡃⠀⠀⢘⣛⣛⡁⠀⠀⠀⠄⠀⠀⠀⡼⠁⠀⢘⣛⣛⣛⣛⣛⣛⣻⣿⣿⡏⢀⣘⣛⣛⡋⣛⣟⣛⣛⣛⣿⣿⣿⡃⠀⠀⠀⠀⠉⠛⣻⣦⣄⠀⠀⠀⠠⡀⢀⣿⣿
⠀⠀⠀⡈⠉⡱⠲⠶⠷⠾⠿⣿⣏⠁⠀⠁⢛⣿⣛⡅⠀⠀⢘⣉⣉⡃⠀⠀⠀⠀⠀⠀⢀⣡⣴⣾⣿⣿⣿⡛⠛⠛⢙⣛⣛⣟⠂⠀⣿⣿⣿⣧⡙⣹⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢿⣷⣵⡀⠀⠀⠀⠟⠛⠋
⡀⠀⠀⠗⠊⠀⠀⢰⣶⣿⣿⣾⡗⠀⢀⣀⣈⣉⣉⣁⣀⡀⢈⣙⣭⣥⣤⣄⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣈⣉⣉⡉⢀⣀⣻⣿⣿⣿⣿⣛⣛⣿⣿⣿⣿⣛⣉⣀⡂⠀⠀⠀⠀⠀⠀⠀⠑⢗⠀⠀⠀⠀⠀⠀
⣿⣿⣷⣤⣤⣴⠀⠈⢻⠿⡿⠋⠀⠀⣈⣉⣹⣿⣿⣿⣭⡁⠈⠉⣉⣉⣉⣉⣉⣋⣩⣽⣿⣿⣿⣿⣿⣋⣉⣉⣉⣙⣉⣉⠉⠁⢨⣉⣉⣉⣉⡙⢿⡏⣉⠁⠈⢉⣉⣩⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⣄⠀⠀⠚⠃⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣷⡄⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠉⠉⠉⡉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠁⠘⠀⠀⠀⠀⠈⠉⠉⠉⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣷⢰⣶⣶⣶⣶⣤⣴⣶⣾⡍⠉⠛⠿⡹⠿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⠀⠀⠀⠐⠀⣠⡄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠿⣿⣿⣿⡇⠻⣿⣿⣿⣿⣿⣿⡟⣠⣤⣄⡀⠈⠉⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⣀⣀⠀⢀⣀⡀⠀⢀⣀⣀⣠⣤⢀⣈⡀⠀⣸⣿⣀⣀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀
⠋⠉⢀⣀⣠⠴⠛⠈⠁⠀⢿⣿⣿⣿⣿⡿⢸⣿⣿⡆⠛⠀⣼⣏⣀⣈⣿⡄⢠⡿⠁⠀⢸⠇⠀⣿⠀⠀⢸⡇⠀⠀⣾⡏⠀⠸⠀⢸⡇⠀⢸⣿⠀⠀⠈⢷⡀⠀⡼⠁⠒⠶⣄⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀
⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠈⠙⠟⠛⢿⠇⢠⡉⡟⠻⣷⠀⢿⡍⠉⠉⢉⠁⠸⣇⠀⠀⢈⡇⠀⣿⣀⡀⣸⡇⠀⠀⢻⡇⠀⠀⠀⢸⡇⡀⢸⣿⣠⣤⣤⣼⣷⣴⣧⣴⣤⣤⣼⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀
⣿⡿⠋⠀⠀⠀⠀⠀⠠⠀⠀⢀⣀⡀⠈⠀⠘⠳⠶⠾⣿⠀⠈⠛⠶⠖⠋⠀⠀⠙⠲⠶⠚⠀⠀⠙⠻⠟⠛⠻⡂⢐⣿⣗⠂⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀
⠛⠗⠶⣶⠏⢶⠄⠰⠶⠶⠖⠀⠀⠈⠉⠛⢦⣄⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⢻⣿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡏⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀
⣿⡸⡇⣿⣿⣇⣿⢸⢿⣷⡯⡏⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣯⢿⣿⡏⣿⣿⣿⣿⣿⢹⢻⣿⣿⡟⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣏⣿⣿⣿
⢿⠇⡇⣑⡏⠻⢸⠨⣨⣽⡆⡻⣳⣿⣿⣿⣿⡩⢯⣿⣿⣿⣟⡟⢿⣿⣿⣧⣾⣿⣿⣯⣨⣟⣹⣿⣿⣿⣿⣿⠸⢸⣏⠧⠃⢿⣿⣿⣿⢿⣿⣿⣿⣿⣯⣿⣿⣿⢿⣿⣿⣿⣿⡿⡟⢽⣿⢿⡯⣝⣳⢱⣶⡙⠼⠟⣷⣿
⣿⠑⣿⡿⢲⣿⡌⣀⠘⠳⢳⣶⣶⣿⣿⠇⠻⡅⢒⣼⠻⢹⣿⣿⡿⡭⣙⡋⡿⢿⣯⡽⢻⡷⣷⠭⢽⣿⠿⣿⢀⣼⣿⣿⣬⡝⢿⡯⣹⣿⡿⣿⡿⠎⢻⣿⣧⠘⢤⣷⣯⠹⡟⣑⣑⣡⢾⣿⡟⢘⣀⣎⠙⢻⠷⢾⣵⣿
⢀⣒⣹⡿⠁⠛⠷⠶⢥⡉⢻⣻⣿⣏⢻⣣⢽⠻⠔⢻⣙⡓⠏⢿⡿⢺⣍⡾⠶⠶⢻⡟⢴⠦⣾⣷⣾⣤⣀⣍⣋⠀⠻⢘⣤⠄⢈⣵⣮⣥⠀⠙⢳⣗⠾⣿⣿⣷⠟⢿⣿⠀⠐⢍⣉⣉⠰⢿⣿⡇⠀⠘⢿⣯⣤⣉⠻⠿
⠊⠙⠓⡀⠀⠰⢶⣷⣾⣭⠛⢱⣿⣿⣾⠛⢓⣤⡀⢻⣿⡷⢆⣀⣁⣿⣿⣥⣾⠓⠀⢠⣙⡁⠘⣿⣉⡙⣿⣿⠝⢀⣴⣦⣿⠶⠎⢺⣿⠋⡐⠛⣻⡛⠀⢹⣿⣇⠀⠈⠻⣦⠠⣉⠙⠋⠰⠿⣻⣿⠀⡔⣉⠉⠙⠙⠀⠀
⠀⢾⣶⣦⡀⠀⠀⢈⣽⣾⣿⡯⣿⢧⡤⢹⣷⣿⣧⠀⠀⠀⡌⣙⣁⣛⠹⣿⣿⣇⣠⣿⡟⠁⢰⣈⣉⠀⣽⣿⣬⠙⢿⣿⢿⢴⡄⠀⠉⠠⣼⡿⣿⣿⣷⡆⠈⢺⣥⡀⠀⠉⢉⠙⢻⣿⣿⣿⣿⣿⠶⠀⢡⣴⠶⢷⣶⣶
⣧⡄⠀⠙⠃⠀⢶⣿⣿⣿⡿⠳⠿⣿⠃⠀⠉⠫⠋⠀⠠⣾⣿⡙⠻⠋⠘⣿⣿⣿⣿⣿⣿⠗⠿⠿⠛⢲⠄⠈⡁⠀⢘⣿⣯⣬⣶⣶⣶⣶⠈⠥⣿⡿⠿⠤⠀⠀⠉⠳⣴⣶⣶⣶⣎⣉⣠⣾⡿⠯⢄⠒⠙⠁⢢⡄⠀⠀
⡁⠐⠀⠀⣤⣶⣜⣿⣿⣧⣤⠄⠱⣶⣶⣀⢼⣿⣶⣄⠀⢀⣠⣴⣶⡶⣾⣿⡿⠭⣽⣿⣿⡖⣧⠀⢠⣾⣿⡿⠷⣴⣿⣿⣿⣿⣿⣿⡋⠁⠀⠈⢿⣿⣿⠀⠀⠀⠀⢤⡄⠀⢽⣾⣿⣿⣿⣿⣿⣧⡄⠁⠀⠀⠈⢠⣴⣶
⢿⣿⣷⣤⡭⠝⠀⣿⣾⣾⡿⠃⠠⠉⠛⠟⡟⠛⠛⠛⠛⠛⠿⠟⢛⡟⠛⠛⠛⠛⣻⣿⣿⠃⠘⠛⣛⣛⠛⢛⣛⣿⣿⣿⣿⣿⣿⣿⢿⡟⠛⠛⠃⠈⠓⢰⣾⣿⣛⣛⣿⡿⠛⠻⢿⡉⣻⣿⣿⣿⢣⣄⠀⠀⢠⠰⠉⠉
⠀⠈⢙⣿⣤⣤⠀⠈⣻⣿⣷⣦⣄⠀⣾⣿⣿⡛⢛⠛⠛⠁⢘⣛⣻⣿⣿⣿⡟⠛⠛⠛⠛⠃⢸⣿⣿⣿⣿⣿⡟⠛⣛⣟⣿⣿⣿⣿⣿⠛⠛⠛⠛⠀⡐⠻⢿⣿⣿⣿⠿⠟⠃⠈⠠⢬⣿⣿⣿⣿⠌⠙⠂⠀⠀⠈⠙⡶
⣠⣾⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⢿⠿⢿⣿⣿⣿⣇⠀⠀⣘⣿⣿⣿⠟⠛⠉⠀⠀⠀⠀⢀⣄⠉⢙⣛⣿⣛⣀⣀⣸⣿⣿⣯⣭⣉⢙⣛⣛⣛⣛⣻⢦⣜⣛⣛⣻⣟⡃⠁⠀⠀⠀⢾⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠚⠀
⠛⠿⠟⢋⣤⣴⠹⣿⣿⣿⣿⠟⠁⠀⠸⣿⣿⣿⣿⡋⠀⠀⢘⣛⣛⡃⠀⠀⠀⢀⠀⠀⠀⠈⣿⠂⢘⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⡿⢘⣛⣛⡋⣛⣛⣿⣛⣛⢛⣛⣛⣃⠀⠀⠀⠀⠀⠙⠻⢿⣇⡀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠹⠿⣏⣄⣙⣻⣿⣧⣤⣤⠄⠀⠿⣛⣿⣟⡃⠀⠀⢘⣋⣉⡁⠀⠀⠀⠀⠀⠀⠀⢸⠁⣀⣼⣿⣿⣟⠛⠛⢛⣛⣛⡛⣿⡃⢘⣿⣿⣇⡙⣛⣛⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠘⢽⣿⣠⡀⠀⠈⠁⢰⠿
⠀⠀⠀⣇⡴⠊⠈⢉⣉⣭⣭⣻⣿⠂⢀⣀⣈⣉⣋⣟⣀⡀⢈⣹⣭⣁⣀⣀⣀⣀⣀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣀⣀⣈⣉⣉⡁⢑⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⢀⠀⠀⠀⠀⠀⠈⠛⠲⣷⠀⠀⠀⠀⠀
⠀⠀⢀⠁⠀⢀⠀⠘⣿⣿⣿⣿⠏⠀⢨⣭⣩⣍⣉⣉⣉⡁⠈⠉⣉⣉⣉⣉⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣋⣉⣭⣉⠉⠁⢈⣉⣉⣉⣉⡛⠿⣿⣯⠁⠙⢻⣿⣋⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠤⠀⠀⠀⠀
⣿⣶⣶⣿⣿⣟⠁⠀⠈⢩⣭⠁⠀⠀⠈⠉⢹⣿⣿⣿⣿⣷⣤⡀⠀⠉⠉⠉⠉⠉⠉⠉⠛⠿⢿⣿⡿⠋⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠁⠀⠸⠁⠀⠀⠈⠉⠉⠉⠉⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠂⠀⠀⠀⣤⠀⠘⠿⢿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⢀⣁⠀⠀⠁⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣦⣰⣤⣤⣤⣤⣤⣿⣤⣶⣤⣤⣄⡉⠓⠿⣝⣥⣄⡀⠀⠀⢀⣠⣤⣀⡀⢀⣀⠀⢀⣀⡀⠀⢀⣃⣀⣠⣤⢀⣈⡀⠀⣸⣿⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⠷⢻⣿⣿⣿⣿⣿⣿⣿⣧⣀⡈⠛⠀⣼⣏⣀⣈⣿⡄⢠⡿⠁⠀⠸⠇⠀⣿⠀⠀⢸⡇⠀⠀⣸⡏⠀⠘⠀⢸⡇⠀⢸⣿⠀⠀⠈⣷⡀⠀⣼⠁⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀
⣿⣿⠿⠟⠛⠛⢿⠛⠂⠀⢻⣿⣿⣿⣿⣿⣇⣼⣿⡿⣷⠀⢿⡍⠉⠉⢉⠁⠸⣇⠀⠀⢀⣦⠀⣿⡀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠈⢸⡇⠀⢸⣿⠀⣀⠀⠘⣷⣰⠃⠀⠈⠀⠉⠛⠆⠰⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣁⣀⣠⣤⡤⠊⠀⠀⠀⠀⠈⠻⣿⡿⢿⣿⠘⠻⣿⠟⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠲⠶⠛⠉⠁⢘⡿⢶⣿⣓⣂⠐⠚⡓⠂⠀⠐⠚⠓⠂⠈⠻⠶⣋⣀⣤⣽⣇⣤⣤⣤⣤⣤⣤⢤⣾⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠏⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠁⠀⠀⠘⠁⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣆⠐⣿⣷⡀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀
⣿⢸⡇⣿⣿⣇⣿⢸⢿⣷⡯⡏⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡏⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣌⣿⣿⣿⣿⣿⣿⣿⢿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢺⢹⣿⡿⣽⣿⣿
⢿⠇⡇⣙⡏⠿⢻⢘⣈⣽⣆⣻⡷⣿⣿⣿⣿⡩⢟⣿⣿⣯⣟⡿⢿⣭⣿⣷⣾⣯⣿⣧⣸⣿⣡⣿⣿⣿⣿⣿⣻⢸⣏⠽⡇⢿⣿⣿⣿⢿⣿⣿⣿⡾⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⢹⣿⡿⣯⢭⣋⢧⣾⣾⢳⡿⣿⣿
⣿⠃⣷⡷⢷⣾⡈⢐⠙⠷⢶⡲⣾⣿⣻⡏⠿⡅⢒⣬⠿⢻⣿⣿⣿⡭⢹⡋⣿⢿⣯⡼⢹⣿⢷⢻⣽⣿⡟⣿⠇⣸⣿⣿⣆⣭⢻⣿⣙⣶⡿⢿⣿⠧⠹⣿⣯⠈⢃⣥⣽⡙⣿⠗⢘⣭⡴⣿⣿⠫⣂⣐⡁⠻⣷⠶⣯⣾
⠀⡶⢻⣽⠁⠿⡦⢭⢡⡙⠻⢻⣿⣟⡻⠥⣽⠿⡔⢿⣟⡷⠟⢻⣿⢰⣏⣽⠶⠤⢹⡟⢴⠦⢾⣷⣶⣤⣄⣭⠛⠃⠻⠆⣦⣄⠀⣩⣮⣍⡁⠘⠛⣖⢰⣿⣿⣿⡞⠻⣿⠇⠀⠮⢉⡉⠁⡾⣿⣶⠑⠔⠚⢿⣦⣌⠙⠿
⠌⠻⠲⠁⠀⢰⣶⣵⣮⣛⠻⢯⣿⣿⣮⠛⠷⣤⡀⠻⣿⣷⠆⢀⣁⣿⣿⣇⣼⠖⠀⢀⠝⠋⠘⣿⣘⡙⣻⣿⡟⠂⣠⣤⣿⣶⡶⠙⣿⡛⢁⠒⠶⡿⠃⠈⣿⣭⠁⠀⠛⢧⡀⢈⠛⠟⠋⣾⠟⣿⡿⠠⣊⠙⠛⠟⠃⠀
⠀⣶⣦⣅⠀⠀⠀⠉⣫⣿⣶⣾⣿⣿⣥⢱⣾⣶⣇⠀⠉⠀⠨⠙⠃⠸⢾⣿⣿⣿⠀⣾⡟⠓⢠⠍⣛⠁⢸⣿⣆⡞⢛⣿⣿⣧⣄⠀⠈⠃⣬⣷⣾⣿⣷⣦⡎⠛⣯⡀⠀⠈⢓⠛⠛⣿⣷⣾⣿⣿⣭⡆⠐⣡⣴⣶⣦⣤
⣢⡀⠉⠻⠏⠀⣤⣶⣿⣿⣿⢧⣿⣿⡟⠀⠙⢏⡟⠀⠠⣴⣿⡟⢿⠿⠉⣿⣭⣿⣿⣿⣿⣶⣾⣿⠟⢂⡌⠩⠋⠁⠈⢿⣿⣃⣣⣤⣤⣤⣬⠛⣙⣻⣿⣿⡄⠀⠉⠻⣆⣤⣤⣤⣤⡜⠋⢱⣿⣿⣿⡀⠐⠛⠁⠄⡀⠉
⠃⠠⠀⠀⣀⣠⡸⡿⣿⣟⣕⡀⢣⣭⣤⠈⢰⣶⣦⣄⠀⠀⢁⣥⣤⣤⣤⣾⣿⢟⣻⣿⣿⣯⢄⠀⠀⣼⣶⣿⣿⠄⢀⣴⣿⣿⣿⣿⣿⡟⠃⠄⠸⣿⣷⣶⠀⠀⠀⠠⢄⠉⠉⠝⣩⣾⣿⣿⣿⣷⣆⢉⠀⠀⠀⠚⢁⣤
⣿⣶⣦⣈⢛⡻⠍⣿⣭⣽⣿⠇⢀⠛⠿⡛⠛⠛⠛⠿⠟⠺⢿⡿⠛⣿⢛⠟⠛⠛⢻⣿⣿⡿⠘⠻⠛⠛⠛⠛⠛⠛⣿⡟⣿⣿⣿⣿⣿⡿⡟⠛⠂⠈⠩⠋⢰⣿⣟⠛⢛⣻⣧⢴⣿⣿⣟⢿⣿⣿⣿⣷⡀⠀⠀⢠⠛⡕
⠈⠙⠻⢿⣁⣀⠀⠘⢿⣿⣿⡀⠀⠀⣘⣛⣛⠛⠛⠛⠛⠁⢘⣛⣛⣿⣿⣿⡟⠛⠛⠛⠿⠃⠘⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⡟⠿⠛⡛⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠈⠋⢋⡾⣿⣿⣿⣇⠳⠀⠀⠀⠠⠲
⢀⣴⣾⣿⣿⣿⡀⢠⣾⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⡃⠀⠀⣘⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣁⣘⣛⣿⣿⡿⠿⢿⣟⣛⣛⣛⣛⢀⣟⣛⣛⣛⣛⡫⠀⠀⠀⠀⠀⢩⣿⣿⣿⣍⠉⠀⠀⠀⠀⠀⣠
⢿⣿⡿⠟⠋⣉⢿⣿⣿⣿⣿⣿⡾⠛⠣⣬⣽⣿⣿⡿⠃⠀⢘⣛⣛⡛⠁⠀⠀⠀⠀⠀⠀⠀⠺⣯⢘⣛⣟⣛⣛⣛⣛⣛⣿⣿⣿⣿⣾⣛⣛⡛⣛⣛⣛⣛⣟⢛⣛⣛⡃⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣰⣿⡿⢘⢻⣿⣿⣿⣫⠀⠀⠈⢻⣿⣿⣿⡇⠀⠀⢘⣋⣉⣁⠀⠀⠀⠀⠃⠀⠀⠀⢸⠃⢘⣛⣋⣉⡉⠙⢛⣛⣛⣻⣿⣿⣿⣻⣛⣃⠙⣛⣛⣛⣉⣸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠉⡿⣷⡄⠀⠀⠀⠂⠀
⠀⠀⠀⡌⠋⣳⠲⢶⣿⡿⠿⣿⣿⠋⢀⣈⣙⣻⣍⣡⣀⡀⢈⣩⣍⣁⣀⣀⣀⣀⣀⣀⣀⡀⣣⣴⣾⣿⣿⣿⣧⣀⣈⣉⣉⡉⢀⣜⣋⣉⣿⣿⣿⣯⣉⣹⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠈⠻⣧⣷⠀⠀⠀⠀
⡄⠀⠀⠓⠊⠀⠀⠰⣴⣾⣿⣿⣿⠋⢈⣍⣉⣉⣉⣉⣉⡁⠈⠉⣩⣭⣽⣿⣯⣭⣿⣭⣩⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⣉⠉⠁⢈⣉⣉⣉⣻⡿⣿⣿⣿⡏⠛⢿⣿⣿⣿⣟⡉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠙⠇⡀⠀⠀
⠀⠈⣐⣠⣤⣤⡄⠀⢻⣿⣿⡿⠃⠀⠈⠉⠉⣿⣿⣿⣯⣅⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⡿⠋⠉⠉⠉⠓⠀⠀⠈⠉⠉⠉⠉⠁⠀⠉⢻⡟⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀
⣴⣾⣿⣿⣿⣿⠀⠀⠀⠰⡿⠀⠀⠀⣤⠀⢠⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⣀⠀⠀⠀⠆⠀⠀⠀⠀⠀⣤⠁⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣉⣿⣟⣿⣟⣻⣿⣿⣄⡀⠀⠀⢀⣠⣤⣀⡀⢀⣀⠀⠀⣀⡂⠀⢀⣀⣄⣠⣤⢀⣈⡀⠀⣸⣿⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣷⡀⣈⣀⣀⣀⣀⣀⣀⣀⣸⣯⣀⡈⠛⠀⣽⣏⣀⣈⣿⡄⢠⡟⠁⠀⠘⠇⠀⣿⠀⠀⢸⡇⠀⠀⢸⡏⠀⠘⠀⢸⡇⠀⢸⣿⠀⠀⠈⢿⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣎⢻⣿⣿⣿⣿⣿⣿⣿⡿⠙⠻⣷⠀⢿⡍⠉⠉⢉⠁⠸⣇⠀⠀⢀⡀⠀⣿⡀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⢸⣿⠀⣀⠀⠘⣷⣰⠃⠀⠀⠀⠶⠴⣤⡄⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠑⠀
⣿⣿⣿⣿⣿⣿⣿⣿⠋⡁⢿⣿⣿⣿⣿⣿⣿⠳⣶⣾⡋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠳⠶⠛⠁⢷⡘⡷⠖⠛⠓⠂⠐⠛⠓⠂⠀⠐⠚⠓⠂⠀⠻⠶⠋⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠋⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠿⢿⠿⠿⠷⠶⠞⢿⣿⣿⣿⣿⡏⠀⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢬⣧⠻⣶⣶⣄⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠶⢟⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣶⣦⣤⣄⣀⠀⠀⠀
⣿⢸⡏⣿⣿⡇⣿⢸⢿⣧⡇⡏⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⡏⡇⣿⣿⣿⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⢿⣿⢻⣿⣿
⣿⠘⡇⣹⡏⢷⢿⢘⣈⣽⡇⣿⠯⣿⣿⣿⣿⣻⣟⣿⣿⣯⣿⠿⢿⣯⣿⣧⣾⣯⣿⣧⣹⡟⣣⡿⣿⣿⣽⣿⣿⢣⣿⡽⡟⢺⣿⣿⣿⣿⣿⣿⣿⡼⣿⣼⡿⣿⣿⣿⣿⣿⣿⣿⣿⠫⣿⡿⣿⢭⣛⡜⣼⣎⠇⡿⢻⣾
⣿⠂⣷⣿⢥⣶⡌⠸⠹⠯⠶⡷⣾⡿⣻⡗⠿⡅⢞⣼⠿⠿⣿⣿⣾⣭⢙⡛⣻⢿⣿⣸⠽⣟⢷⣓⣻⣿⣟⣿⠇⣠⣿⣿⣇⣉⠿⣿⢟⣷⣮⢿⣿⡧⠍⣿⣿⡌⢣⣩⣭⡋⢿⠟⠊⣋⡴⢾⣿⡏⣓⣰⣉⠟⣿⡶⢮⣶
⠉⠤⢿⣶⠎⠿⣧⣭⣣⠚⠿⢻⣿⡿⠻⡅⣰⢷⡤⠿⡟⠯⡿⢻⣷⠰⡟⣽⡶⢤⠝⣟⢴⠤⠸⣾⣶⣮⡄⢸⠛⠛⠹⠿⢢⣥⠀⢙⣬⣍⡉⠀⠻⣴⠀⣴⣿⣿⣷⠛⣿⣿⠀⠈⠉⢙⠁⢰⣿⢷⡶⠂⡒⣲⣦⣤⡛⠻
⠙⠿⠮⠁⠀⢠⣤⣦⣬⣓⠶⢏⣭⣿⣅⠲⡮⣆⡀⠾⣿⣷⣦⠀⠉⢾⣿⡷⣼⠶⠀⠀⡞⠟⠘⢿⡟⠉⢞⣿⣿⠓⢀⣠⣽⣷⣤⡜⢿⣿⠏⠰⠶⣾⠿⠀⢺⣿⡟⠀⠘⢻⡀⢀⡛⠛⠛⢢⣾⢿⣿⡆⡖⡒⠛⠛⠟⠃
⠀⣤⣤⣂⠀⠀⠈⠙⣻⣽⣶⣞⣿⣿⣏⣡⣬⣷⣇⠀⠉⠁⢘⠻⠛⠙⣿⣿⣿⣷⠆⢴⡿⠛⠀⡌⠙⠋⠘⣿⣆⣀⢻⣿⣿⣿⣥⠀⠈⠛⢠⡳⣠⣾⣿⣦⣤⡉⢷⣂⠀⠀⠛⠖⠓⠶⡿⣣⣶⣾⣿⣣⠀⠌⣡⣤⣶⣄
⢄⠉⠙⢿⠇⠀⣀⣠⣿⣿⣿⢇⢻⣿⡿⠀⠛⢟⠿⠀⢀⣤⣾⡶⠶⣶⠖⢻⣟⣻⣷⣾⣿⣦⣴⣿⡿⠷⢬⠙⠟⠋⠀⢺⣷⡖⠈⣁⣀⣀⣀⡿⠿⠺⢿⣿⣿⡁⠈⠛⢧⣀⣀⣈⣀⣤⠜⠛⢫⣿⣿⣿⠠⠐⠛⠃⣀⡈
⠍⢁⠀⠀⠀⠀⢹⣿⣿⣿⢟⠉⠏⣙⣁⠀⢠⣤⣥⣀⠀⠉⠉⣙⣈⣤⣀⣬⣿⣿⠿⢿⣿⣿⣇⠀⠀⢀⣯⣶⣶⣦⠀⠀⢈⣵⣾⣿⣿⣿⠿⠃⡀⢹⣿⣯⣽⠁⠀⠀⢀⡉⠛⠛⡿⠙⣠⣴⣿⣿⣿⡍⠑⠀⠀⠀⠔⠁
⣶⣤⣄⠐⠾⢿⡒⣿⣛⣛⣿⣏⠀⠿⣿⡿⠟⠛⠛⢿⡿⢠⣾⣿⡿⢻⣛⢛⠿⠛⠻⣿⣿⣿⡟⠻⡿⠿⠻⠛⠛⠛⠛⣷⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠙⢻⡟⣟⣛⠛⠻⠛⢃⣀⣤⣴⣿⣿⣿⣿⣿⣷⣶⣦⠀⠀⢀⡾
⠙⠻⢿⣿⠖⠉⠀⠹⣿⣿⣿⠁⠀⠀⠘⠛⠻⠛⠛⠛⠛⠃⢙⣛⣛⡻⣛⣛⣛⠛⠛⠛⢻⡟⠘⢻⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣟⣛⠛⠀⠀⠀⠛⣻⣿⣿⣿⣿⣿⣿⡏⠉⢿⢻⢊⣿⣿⣿⣿⢃⠀⠀⠈⠁
⠀⣀⣤⣶⣶⣶⠆⠀⢐⣾⣿⣶⣤⡀⣼⣷⣿⣿⣟⡃⠀⠀⣘⣻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠈⠘⣿⣿⣿⣿⣿⣿⣟⣛⣻⣿⣿⣿⣿⣿⣿⣟⣛⣛⠀⣚⣟⣛⣛⣿⡿⠟⠛⠛⠁⠀⠀⠐⢺⣿⣿⠿⠿⠶⠀⠀⠀⠀
⣾⣿⣿⡿⠿⠿⢧⣴⣿⣿⣿⣿⣿⡻⣟⠛⣻⣿⣿⣿⡆⠀⢙⣛⣻⡿⠟⠉⠀⠀⠈⠀⠀⠀⠀⣬⣘⣛⣛⣻⣟⣛⣛⣛⡋⣿⣿⣿⣽⣟⣛⡋⣻⣛⣛⣛⣛⣿⣛⣛⣃⠜⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⠀⠀⠀⠀⠀
⠙⠛⠋⢀⣠⣴⠙⢿⣿⣿⣿⣿⠟⠁⠀⢹⣿⣿⣛⡋⠀⠀⢘⣛⣛⡁⠀⠀⠀⠀⠠⠀⠀⠀⠀⢹⣟⣛⣛⡋⠉⠉⢛⣿⣛⣙⣿⣿⣿⣿⣟⡃⠙⣛⣛⣛⡿⢙⣛⣛⣃⣀⣤⡄⠀⠀⠀⠀⠀⠈⠛⠿⣷⡀⠀⠀⠀⣠
⠀⠀⠀⠺⣿⣟⢐⡞⢿⣿⣿⣼⣀⣠⣀⣈⣛⣟⣿⣃⣀⡀⢈⣉⣉⣁⣂⣀⣀⣀⣀⣀⣀⡀⢀⣏⣈⣉⣭⣥⣶⣆⣈⣉⣉⡉⢈⣛⣻⣟⣉⣷⣶⣭⣉⡉⠀⣼⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠘⠩⣿⢴⡀⠀⠀
⠀⠀⠀⡆⣠⠜⠋⠛⠛⢛⣛⡻⣿⣇⣈⣍⣉⣉⣙⣽⣏⡁⠈⠉⣽⣭⣭⣭⣭⣉⣉⣉⣉⡁⣬⣿⣿⣿⣿⣿⣿⣿⣍⣉⠉⠁⢈⣉⣉⣍⣉⣿⣿⣿⣿⣷⣾⢿⣿⣿⣿⣿⣿⠿⠛⠀⢀⠀⠀⠀⠀⠀⠀⠉⠳⣵⠀⠀
⠁⠠⠀⠁⠁⠀⠀⠈⣾⣿⣿⣿⣿⠃⠈⠉⠉⢉⣉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠙⠙⠉⠉⣹⣿⣿⣿⣿⣿⣿⣿⠿⠟⠀⢀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠿⣽⣧⠈⠉⠛⠛⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠖⠀
⠀⣀⣥⣶⣶⣾⠇⠀⠘⠿⢿⠿⠁⠀⣤⠀⠀⢸⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠈⠀⠀⣠⡄⠀⢀⣀⠀⠀⠀⠈⡞⠀⠀⠀⠀⣤⠀⠠⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡄⠀⠀⠘⠛⠂⠀⠀⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⢀⣠⣤⣀⡀⣀⣀⠉⠉⣀⡀⠠⣀⣀⣀⣠⣤⢀⣈⡀⠀⣸⣿⣀⣀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠿⠿⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡈⠛⠛⣼⣿⣳⣋⣿⡄⢠⡟⠀⠀⠘⠇⠀⣿⠀⠀⢸⡇⠑⠀⢹⡏⠦⠙⠀⢸⡇⠁⢸⣿⠀⠀⠈⢷⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠙⠻⣷⠀⢿⡍⠉⠉⢉⡁⠸⣇⠀⠀⢀⡀⠀⣿⡀⠀⢸⡇⠀⠀⣼⡇⠀⠀⠀⢸⡇⠀⢸⣿⠀⣀⠀⠘⣷⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣷⣤⢶⣶⣶⣿⣷⣶⣶⣶⣾⣿⡶⠞⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠳⠶⠚⠀⠀⡘⠳⠖⠛⠓⠂⠐⠛⠓⠂⠀⠐⠚⠓⠆⠀⠻⠶⠋⠀⠀⢹⠏⠀⠀⠀⠈⠀⠘⠙⠛⠶⡄⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣶⡾⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀
⣷⢹⣿⣿⣿⡇⣿⢻⣿⣧⡟⡏⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣏⡟⣿⣿⣿⢏⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡟⡟⢿⣿⢏⣿⣿
⣿⢸⡇⣻⡟⡧⣿⢘⢘⣲⡇⢿⢯⣿⣿⣿⣿⣻⣟⣿⣿⣯⣿⠿⢿⣯⣿⣯⣬⣟⣿⣧⣹⡿⣃⣿⣿⣿⣿⣿⡇⡇⣿⣽⢽⠸⣿⣿⣿⣿⣿⣿⣿⣽⣿⣼⣿⢿⣿⣿⣿⣿⣻⣿⢿⠏⣿⣿⢿⣭⣟⣵⢱⣾⡟⣸⠻⣾
⣟⢂⣧⣯⣥⣶⡘⠸⠻⠯⢧⡥⣮⣿⢻⡟⢿⡀⢾⣸⠾⠿⣿⣯⣾⣍⢙⡛⣻⣿⣿⣹⠽⣟⢵⣗⣻⣿⣿⢿⡗⢀⣿⣿⣆⣘⡹⣿⣟⣷⣮⣽⣿⣿⠜⢻⡿⡅⠙⡨⣬⣟⢻⣿⠊⣫⣥⠶⣿⣿⠕⣀⣎⡹⢻⣷⢶⣵
⠉⣬⣟⣻⠊⣯⣅⣙⣂⠲⢾⢿⣾⣷⢿⡇⣰⢷⡬⢿⡗⢶⣿⢻⣟⡣⡟⣩⣤⣤⡽⣿⢰⡬⠨⣷⣤⣮⣍⢘⡓⠞⠘⣿⢡⣭⡀⠙⣫⣭⣋⠁⠻⢧⠆⢀⣼⣿⣷⡾⢯⣿⡇⠰⡉⠙⡋⠀⣽⣿⣶⣆⢘⢒⣖⣤⣘⠛
⠲⣶⣌⠋⠀⢀⣅⣬⣜⡷⢴⡞⣿⣷⣆⢶⣭⣄⡁⠴⣿⣯⣤⠀⠙⢰⣿⣿⢠⡶⠄⠉⡐⠶⠂⢿⡍⠋⢵⣿⣷⠗⠀⢀⣸⣿⣤⡴⢻⣿⠿⠀⠤⣬⣿⠎⠙⣿⣛⠓⠀⠻⣇⠀⡖⠛⠻⠓⣴⣾⢿⣷⡤⠰⠒⠻⢿⠟
⠀⣀⣀⠀⠀⠈⠉⠻⢿⣿⣥⣼⣿⣿⡟⢢⣤⣯⣇⠀⠛⠋⢀⠒⠿⠘⣯⣽⣿⣷⡆⢠⣷⠟⠂⢐⠛⠟⠈⢿⣿⠡⠰⠿⣿⣿⣏⡀⠀⠙⢇⣐⠆⣀⣵⣶⣤⣉⠻⣒⠀⠀⠘⠷⠐⠶⢶⡶⣠⣵⣾⣟⣆⠀⢏⣠⣤⣶
⡀⠛⠻⣿⡆⠀⠀⢀⣿⣿⣿⡏⢿⣭⣿⠁⠿⡿⣿⠄⠀⣀⣼⣶⣶⣶⡮⢽⣿⣿⣇⣦⣶⣦⣠⣷⣾⠿⢄⡛⢻⠿⠃⠹⣿⣿⠍⠉⠀⠀⠀⢸⡾⣿⣽⣿⣿⣿⠃⠘⠷⡄⠀⠀⠕⣀⣠⠼⠿⠿⣿⣿⣿⠁⠸⠛⢋⡉
⡗⠂⠀⠀⠀⠀⢺⣿⣿⣿⣿⠛⡿⠻⠃⠀⠄⣈⣂⡀⠀⠙⠛⠹⢀⣉⡀⢈⣿⣿⣿⣿⣿⣿⡟⠉⠁⠀⣰⣥⣴⣶⡀⠀⠈⠛⢀⣺⣿⡿⣿⣿⠃⠐⢲⣾⣛⡻⠃⠀⠀⠀⠛⠻⠿⡟⠻⠀⢀⣙⣿⣿⡛⠃⠀⠀⠀⠰
⣄⣈⡀⠀⣤⣶⣼⣿⣿⣿⣮⡤⠈⢾⣿⣿⡛⠛⠿⣿⣿⠁⣴⣿⣿⡿⡟⠿⣿⣿⡛⣿⣟⣛⣿⠻⣟⡿⠿⠿⠟⠛⠛⠓⢀⣴⣿⣿⣿⣿⣿⣛⡃⠁⠘⠻⢿⡟⠛⠛⠛⢻⠓⠀⢀⣀⣀⣴⣿⣿⣿⣿⣷⣤⣠⠄⠀⠀
⠻⢿⣿⣶⡭⠝⠃⢻⣷⣷⣿⠟⠀⠌⠙⠻⢻⠛⠛⠛⠛⠃⢙⣛⣛⣛⠟⠛⠛⠛⠛⠛⢿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⠻⣿⣿⣿⣿⣿⡟⠀⠀⠀⠛⠛⣿⣿⣿⣛⣷⣿⣿⣿⠛⠿⢿⣿⠝⣿⣿⣿⣿⠇⠀⠐
⠀⠀⢈⣩⣤⣤⠄⠈⠛⢿⣋⣀⠀⠀⢠⣤⣽⣟⣛⡇⠀⠀⢘⣛⣻⣿⣶⣿⣿⣧⠀⠀⠀⠙⠀⠂⣿⣿⣿⣷⣶⣶⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣛⣓⠀⣘⣛⣛⣻⣿⣿⣿⣿⣿⠿⡿⠀⠀⠀⣈⣘⣿⣿⣿⣿⣄⠀⠀
⣠⣾⣿⣿⣿⣿⡆⠀⣤⣿⣿⣿⣿⣦⡾⠿⣿⣿⣿⣷⣄⠀⢙⣻⣿⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⢸⣛⣿⣿⣿⣛⣿⣛⡋⠲⢾⣿⣿⣿⣿⡿⣟⣛⣛⣛⣛⢿⣟⣛⡃⠉⢁⡖⠂⠀⠀⠀⠀⠠⣾⣿⣿⣭⡉⠁⠀⠀
⠻⠿⠿⠋⠁⢀⣶⣾⣿⣿⣿⣿⣭⡭⠻⢦⣽⣿⣟⡿⠋⠀⢘⣛⣛⡃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢿⣟⣛⣛⡿⠛⢉⣙⣛⡀⢿⣿⣿⣿⣿⣧⠙⣛⣛⣛⡋⢘⣟⣛⡃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⡇⠀⠀⠀
⠀⠀⠀⢰⣾⣿⠉⣿⣿⣿⣿⣟⠋⠀⣀⣈⣿⣿⣿⣧⣀⡀⢈⣉⣉⣁⣄⣀⣀⣀⣀⣀⣀⡀⢀⣀⣞⣉⣉⣋⣀⣀⣈⣉⣛⡁⢀⣻⣿⣿⣿⡏⣀⣉⣉⡉⠀⢈⣉⣉⣥⣶⣶⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢩⠷⡄⡀⠀
⠀⠀⠀⡘⠟⢻⣤⣙⣛⣿⣿⣿⣥⣤⣏⣉⣉⣭⣽⣉⣍⡅⠈⢩⣽⣭⣭⣭⣉⣉⣉⣉⣭⡁⢨⣩⣉⣭⣿⣿⣿⣿⣯⣉⠉⠁⢈⣉⣉⣩⣿⡅⢸⣿⣿⣷⣄⢈⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠈⢧⡽⡄
⠄⠀⠀⡇⡔⠋⠁⢉⠉⣩⣭⣝⣿⡿⠋⢉⠉⠉⠉⠙⠋⠁⠀⠀⠈⠉⠉⠉⠛⠻⢭⡿⠍⠁⢨⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠈⠉⠉⠉⠉⠁⠘⢿⣿⣿⡿⣿⠋⠛⢻⣿⣿⣿⠟⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠨⠀⠀⢀⡀⠀⢿⣿⣿⣿⣿⠃⣤⠀⠀⠀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠿⠛⠋⠀⢠⠄⣠⡄⠀⢀⣀⠀⠀⠀⠉⠻⣿⡄⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣀⣤⣶⣿⣿⣿⠃⠀⠈⠛⣛⡛⠃⠀⣿⠀⠀⣀⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⣈⣛⠿⣿⣿⡟⠃⢀⣀⣀⣠⣤⢀⣈⡀⠀⣸⣿⣀⡀⣀⣀⡀⢨⣀⣀⠀⣿⠀⠀⠀⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠩⠁⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣉⣿⡄⢠⡟⠀⠀⠸⠇⠀⣿⠀⠀⢸⡇⠀⣀⣿⡏⠀⠙⠀⢸⡇⠀⢸⣿⠀⠀⠈⢷⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⠟⠻⠛⠉⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡈⠙⠻⣷⠈⢿⡭⣋⠟⢉⠁⠸⣇⠀⠀⢀⡀⠀⣿⡀⠀⢸⡇⠀⠀⢻⡇⠀⠘⠀⢸⡇⠀⢸⣿⠀⣀⠀⠘⣷⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠞⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠲⠶⠚⠀⠀⠘⠳⠖⠛⠓⠂⠐⠛⠓⠂⠀⠐⠚⠓⠂⠀⠻⠶⠋⠀⠀⢹⠏⠀⠀⠀⠀⠀⢀⠀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⣿⢶⠄⠰⠶⣶⣆⣀⣀⣀⣀⣀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠺⠤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⠄⠺⣷⡄⠀⠀⠀⠀⠀
⡇⣿⢸⣿⣿⡛⡇⢿⣿⡇⡿⡿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⢸⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠟⣿⣿⢻⣿
⣷⢸⠘⢿⡿⡇⣿⠸⢙⣠⡇⣷⢮⣿⣿⣿⣿⣛⣟⣿⣿⣿⣿⡿⢿⣿⣿⡿⣬⣋⣿⣯⣻⣻⠂⣿⣿⣿⣿⣿⡿⡿⣼⡯⣏⠧⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣻⠿⣿⣿⣿⣿⣟⣿⣿⡟⢻⣿⡿⣯⢝⣢⡞⣶⣯⠇⡛⢿
⡷⠘⣈⣟⣃⣥⠋⢨⢻⣯⣭⢭⣬⣿⢿⣿⣿⡃⠬⣙⣻⡿⣿⣽⣶⣟⠙⠿⢻⣿⣿⠋⡿⣿⢭⣿⣛⡿⣿⣿⣿⡁⢻⣷⣾⣐⣛⢿⣿⣻⢾⣭⣿⣿⣦⠻⣿⣫⠙⢁⢭⣽⡟⢿⡗⠙⣍⡴⢶⣿⡯⡒⣠⣉⡿⢟⣶⣬
⠃⣸⡿⢯⠽⣟⡃⠒⠄⠶⣶⢷⣿⣷⣿⡧⢠⣷⡈⣿⡷⢦⣻⡿⣿⡯⡞⢻⣯⣅⣙⣿⢆⣭⠁⣣⣮⣭⣉⠁⢛⠶⠏⢻⡯⣉⣉⠈⠛⣉⣛⠛⠈⠿⡴⠄⠈⣹⣾⣶⣿⢿⣿⡇⠨⡈⠛⠛⠀⣻⣿⣶⣆⢒⠒⣖⣁⣐
⣤⣧⡹⠛⠀⢉⡋⣙⡹⠦⣠⡬⣿⣿⡞⣥⣝⡀⠁⣠⣷⣶⣬⠀⠛⠃⣼⣾⣅⣦⡄⠈⠁⠷⠶⢺⣿⠛⠋⣵⣾⣶⠆⠀⠁⢿⣥⣀⡰⣿⣿⡗⠀⣠⣼⣶⣌⠻⢿⣿⠷⠀⠿⡆⢀⠖⠻⢷⠶⢉⣴⣿⣧⣴⠐⡐⠺⠿
⠈⠙⢁⠀⠀⠘⠛⢿⣿⢳⣆⣁⣿⣿⣿⢎⣉⣝⣃⠀⠙⠛⠉⠥⢶⡖⢿⣟⣷⣷⣤⠀⣼⡶⠦⠀⡑⢶⠶⠸⣿⣟⠀⢠⣾⣿⣿⣏⠁⠘⠿⢂⠠⡦⠀⠋⣠⣄⡘⠿⡇⠀⠀⠸⢦⠠⠴⣶⣶⡆⠉⣩⣿⣷⠀⠩⢂⣀
⠀⠿⣷⣦⡀⠀⠀⠀⣾⣿⣿⣿⢿⢿⣷⡆⢿⣿⣿⡆⢀⠀⣀⣧⣤⣤⣈⣛⣿⣿⣻⣀⣈⣁⡀⣠⣷⣶⣆⠀⠾⠿⣶⠆⢺⣿⣟⡛⠃⠀⠀⠘⢻⣶⣿⣿⣿⣿⣿⣿⠿⣷⡄⠀⠀⠰⠂⣀⣁⣰⣿⣿⣿⣿⣾⡦⠿⠿
⣆⡀⠀⠙⠃⠀⢀⣤⣿⣿⣿⣿⣜⢿⡿⠁⡀⠙⠭⠁⠀⠙⠿⠟⡭⠩⠛⠀⠻⣷⣾⣷⣿⣿⣿⣿⠟⠁⠀⢸⣂⣨⣤⠀⠀⠙⠿⠁⣤⣴⣶⣶⣾⣿⠙⢣⣿⣿⣿⣿⠄⠀⠉⡠⠶⢶⣿⡿⣿⠀⠁⠤⣬⡿⡿⠆⠀⠀
⠂⠤⠀⠀⠀⣀⣸⣿⣿⣿⣿⠁⠸⢈⣙⡛⠛⣿⣿⣿⣟⠃⢀⣴⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣿⣟⡛⣻⣿⣿⡿⠟⠓⠀⠀⠘⠛⢿⣿⣿⣿⡋⠁⠀⠀⢿⣿⣿⡟⠛⠛⠃⢒⠦⠀⠀⠃⠀⠀⠀⣠⣽⣿⣿⣤⡀⠀
⣷⣶⣦⣀⠻⢿⡗⣾⣟⣛⣿⣿⡁⠸⠿⣿⠿⠛⡟⠛⠿⠃⢛⣻⣟⡛⣻⣟⢛⠟⠛⠛⢻⣷⣾⣿⠛⠻⠛⠛⠛⠛⠛⠛⡟⠃⠘⣿⣿⣿⣿⣿⣿⣦⣤⣄⠛⠛⠻⠟⣿⣿⡃⢊⣤⣶⣶⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣶
⠉⠛⠿⣿⠗⠉⠀⠸⣿⣿⣿⠏⠀⠀⠀⠀⢙⣟⣛⡃⠀⠀⢘⣛⣛⡃⢸⣡⣤⣤⡄⠀⠀⠛⢿⠃⢘⣻⣿⣧⣀⣀⣘⣛⣛⣃⢸⣿⣟⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⣛⣿⣿⣾⣿⣿⣿⣿⣷⠈⠉⠟⠻⣷⢹⣿⣿⣿
⠀⣀⣤⣶⣶⣿⡆⠀⠈⠛⠉⣀⠀⠀⢀⣿⣿⣿⣿⣁⠀⠀⢘⣛⣻⣿⣿⣿⣿⣿⡷⠀⠀⠀⠈⠀⢘⣻⣿⣿⣿⣿⣿⣿⣿⣿⠱⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⢛⣛⣛⡏⠉⠙⠻⠿⢿⠟⠛⠟⠀⠀⠈⠀⣤⣯⣿⣿⣿
⣾⣿⣿⣿⡿⠿⠇⠀⢀⣴⣿⣿⣿⣶⣌⠛⣻⣿⣿⣿⣷⡀⢈⣛⣛⡛⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⢘⣛⣛⡛⠛⠿⢿⣟⣛⣉⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⡋⢘⣛⣛⡳⣦⡀⠀⠀⢀⠎⠁⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶
⠙⠛⠋⠀⠀⠀⢤⢴⣿⣿⣿⣿⣻⣛⣻⣧⣭⣭⣭⣙⣋⡀⢈⣉⣉⣁⣀⣀⣀⣀⣀⣠⣄⡀⢀⣀⣈⣙⣿⣧⣀⣀⣼⣋⣉⡁⢀⣀⣘⣿⣿⣿⣷⣿⣭⡉⠀⢈⣉⣉⣵⣋⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠘⠿⣿⣿
⠀⠀⠀⢠⣾⣿⠛⣿⣿⣿⣿⣿⣿⠋⣉⣉⣽⣿⣿⣽⣍⡁⠈⢉⣩⣍⣉⣏⣉⣩⣉⣉⣉⡁⢈⣉⣉⣩⣟⣉⣉⣉⣉⣉⠉⠁⢈⣍⣉⣹⣿⣿⣿⣿⣿⣁⣠⣈⣉⣉⣉⣉⣥⣶⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⢱
⠀⠀⠀⠈⠿⢿⠀⢻⣯⣿⣿⣟⠴⠀⣉⡉⠉⠯⢯⡟⠋⡁⠀⠀⠈⠙⠉⢉⣍⣉⡉⠉⠉⠁⠈⠉⠉⠙⣉⣭⣿⣿⣿⣿⣿⡄⠈⠉⠉⠉⠉⠙⠛⣿⡋⢻⣿⣿⣯⡋⢹⣿⣿⣿⣿⣿⣿⣿⣿⢶⠄⠀⠀⠀⠀⠀⠀⠀
⡀⢀⠀⡇⢠⠔⠚⠷⠾⠿⠿⠿⣿⣿⣯⠀⠀⠀⠀⠈⠢⠇⠀⠀⠀⠀⠀⠀⠈⠉⠳⠰⠟⠦⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣇⠀⣠⡄⠀⢀⣀⠀⠋⡁⢸⣿⣿⣿⣿⣯⣤⠛⢿⣿⣿⣿⠛⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀
⠀⠀⢑⠀⠁⠀⠀⠀⣤⣴⣿⣿⣾⣷⣿⠀⠀⣀⣤⣄⣀⠀⠀⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢂⣈⣀⠀⣸⣿⣀⣀⣀⣀⡈⠉⣛⣭⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣀⣠⣴⣶⣶⠆⠀⠘⣿⣿⣿⣿⠃⠀⠀⠸⣯⣸⣿⣿⣿⣿⣯⣄⣈⣿⡄⢠⡟⠀⠀⠘⠇⠀⣿⠟⠻⣿⣿⣿⣿⣿⡏⠀⠙⠀⢸⡇⠀⢸⣿⠀⠀⠈⣷⡀⠀⡼⠁⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⡆⠀⠀⠈⣩⣯⠁⠀⠀⠀⢀⡈⢹⣿⣿⣿⣿⣽⣿⣿⣩⠁⠸⣇⠀⠀⢀⡀⠀⣿⡀⠀⢸⡟⠀⠀⢸⡇⡀⠀⠀⢸⡇⠀⢸⣿⠀⣀⠀⠘⣷⣰⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⠧⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⠳⠾⠟⠛⠻⠿⣟⠿⣿⢿⠆⠀⠙⠲⠶⠚⠀⠀⠘⠳⠖⠛⠓⠂⠐⠚⠓⠂⠀⠐⠚⡗⠂⠀⠻⠶⠋⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⠿⠿⣿⠛⢿⠄⠠⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⣿⢸⣿⣿⠘⡇⢿⣿⡇⡿⡟⣿⣙⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡟⣹⣿⢏⣿
⡷⢹⠘⢻⡿⡇⣗⠸⢙⣠⡇⣷⢮⣿⣿⣿⣿⣛⣛⣿⣿⣿⣿⠿⢿⣏⣿⡿⣬⣋⣿⣿⣧⣹⠚⣿⢿⣿⢿⣿⣿⢹⢸⣿⠿⡇⣿⣿⣿⣿⣿⣿⣿⣥⣿⣧⣻⡟⣿⣿⣿⣿⣟⣿⣿⣿⠹⣿⡿⣿⢕⣫⣶⢱⣿⡙⠸⠻
⡧⠸⣈⣟⣃⣥⠃⠩⢻⣏⣭⢭⣬⣿⢿⡯⣿⡃⠬⣙⣚⡿⣿⣽⣶⣟⡛⠿⢻⡏⣿⣋⡏⣿⢭⣿⣛⡿⣿⣿⣿⡎⢘⣿⣶⣅⣚⢾⣿⡟⢴⣭⣿⣿⣧⡝⢽⡯⡌⢃⡡⣭⡞⢹⡟⠍⣍⣤⠶⣿⣿⠔⢀⣌⡹⢻⣳⣦
⠃⣸⡿⢯⢹⣟⡃⠒⠄⠤⣶⢷⣽⣷⣾⡧⢠⣷⡀⣿⡷⢦⣻⡿⣿⡯⡺⢛⣯⣅⣙⢿⡗⣮⠅⢡⣦⣭⣉⡁⢛⠶⠟⢸⣯⢉⣉⠀⠛⢈⣛⠛⠁⠻⡧⠄⠀⢙⣶⣶⣾⡿⣿⣿⠠⢄⠛⠛⠂⢘⣿⣶⣶⡄⡒⣲⣃⣀
⣬⣃⠻⠛⠀⠉⠋⣙⡹⠦⣈⡬⢿⢿⡌⢡⣽⡂⠁⣠⣾⣶⣬⠁⠛⠃⣬⣾⣍⣦⣅⠈⠃⠵⡶⠸⣿⠚⠋⢡⣾⣶⡦⠀⠈⠸⣯⣀⣀⢿⣿⣿⠂⢀⣬⣷⣤⡘⠻⣿⠿⠇⠸⢷⠀⡐⠛⠿⠶⠎⣡⣾⣿⣦⡆⠂⠔⠿
⠈⠉⢁⠀⠀⠘⠻⢿⣿⢶⣖⡁⣾⣿⣿⢎⠉⣽⡳⠀⠘⠟⠉⠡⢤⡤⢿⣿⣷⣧⣤⡀⣴⣶⠦⠀⠘⢶⡶⠐⣿⣿⠂⠀⣴⣾⣿⣿⠉⠀⠻⡇⠀⢤⠄⠙⠉⣀⣀⠻⢿⠀⠀⠀⠧⡀⠥⢤⣶⣦⠈⠉⣬⣿⠇⠈⡑⢂
⠠⢿⣶⣦⡀⠀⠀⠀⣾⣿⣿⣿⢿⠿⢧⡄⢿⣿⣿⡆⠀⠀⢀⣧⣬⣄⣈⡛⣿⣿⣟⣇⡉⢁⡀⢀⣮⣴⣆⠀⠸⠿⣶⡶⠘⣿⣿⡿⠒⠀⠀⠀⠛⣧⣶⣿⡿⣿⣿⣷⣾⢷⣦⡀⠀⠀⠲⠀⣀⡀⣰⣿⡿⢿⣿⣶⡾⠿
⣀⠀⠈⠛⠃⠀⣀⣠⣾⣿⣿⢧⡸⣿⡿⠃⠈⠙⢍⠁⠀⠸⠿⠟⢫⠙⠟⠃⠹⣿⣿⣧⣾⣿⣿⣿⡿⠋⠀⠀⢗⣀⣁⡀⠀⠘⠻⠟⢈⣠⣤⣦⣴⣿⣿⠙⣂⣻⣿⣿⣯⠄⠈⠃⠠⠶⣶⣿⣿⣿⠀⠁⠤⣾⡿⠿⠄⠀
⠉⠄⠀⠀⠀⠀⢹⣿⣿⣿⠿⠋⠛⢈⣙⣛⠛⣿⣿⣿⣟⠃⠀⣰⣿⣿⣿⣛⣿⣿⣿⣟⣻⣿⣿⣿⣛⠛⢻⣿⣿⣿⠿⠗⠀⠀⠘⠛⠻⣿⣛⣛⣿⠛⠂⢀⠐⢿⣿⣿⡟⠛⠃⠀⢢⡄⠀⠠⠋⠁⠀⠀⠀⣹⣿⣷⣄⡀
⣶⣦⣄⠀⢾⣿⠦⣽⡿⠿⣿⣶⠂⠘⣿⣿⣿⠛⣛⠛⠿⠇⢻⣿⣿⠛⢛⣟⠛⡻⠟⠛⢻⣿⣿⣿⡟⠻⡟⠛⠛⠛⠛⠛⢛⡓⠘⠛⢻⣿⣿⣿⣿⣷⣦⡀⣛⠛⠛⠻⡿⣿⣃⠀⢃⣠⣴⣶⣾⣆⡀⣠⣾⣿⣿⣿⣿⣿
⠙⠻⣿⣿⠶⠊⠁⢹⣿⣿⣿⡟⠁⠐⠀⠈⢛⣿⣛⠃⠀⠀⢘⣛⣛⡃⠁⢋⣀⣀⣄⠀⠀⠹⢿⡟⢘⣻⣟⣃⣀⣀⣘⣛⣛⠃⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⡃⠻⢿⣷⣾⣿⣿⣿⣿⣯⡅⠛⠛⢿⢿⠉⢻⣿
⠀⢀⣠⣮⣴⣶⠄⠀⠙⠿⠋⠀⠀⠀⠀⣶⣾⣿⣟⡉⠀⠀⢘⣛⣛⣷⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⢘⣻⣿⣿⣿⣿⣿⣿⣿⣾⣿⠮⣛⣿⣟⣿⣿⣿⣿⣿⣿⢛⣛⣛⡃⠀⠛⠛⠿⣿⣿⡿⠿⢿⠇⠀⠀⠁⠀⣂⣸⢿
⣾⣿⣿⣿⣿⣿⡆⠀⠀⢠⣴⣶⣦⣄⡘⠿⣿⣿⣿⣿⣶⡄⢈⣛⣛⡻⢿⠿⠟⠉⠁⠀⠐⠀⠀⠀⢘⣛⣛⣟⠛⢿⣿⣿⣿⣿⠛⠀⢙⣛⣟⣛⣿⣿⣿⣿⣯⣜⣛⣛⡣⢶⣤⠀⠀⠈⠀⢠⠒⠀⠀⠀⠀⠀⠈⣽⣿⣿
⠿⠿⠟⠉⠁⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣆⣉⣉⣉⣻⣝⡃⢈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣈⣉⣹⣷⣁⣄⣈⣉⣟⡋⢀⣀⣈⣉⣽⣿⣿⣿⣿⡙⠉⢉⣉⣉⣉⣠⡜⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⠀⠀⠀⢀⣠⣤⢯⣭⣿⣿⣿⣿⣷⡶⢏⣩⣹⣿⣿⣯⣙⡁⠈⢉⣉⣉⣉⣍⣉⣍⣉⣉⣉⡁⢈⣉⣉⣉⣩⣽⣋⣩⣩⣋⠉⠁⢈⣉⣉⣉⣹⣿⣿⣿⣿⣿⡇⢈⣉⣉⣉⣉⡁⢀⣠⣤⣴⣶⣾⣶⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠸⣿⣿⠈⣿⣿⣿⣿⡟⢛⠁⠉⠉⠉⠿⣿⣹⡿⠁⠀⠀⠀⠙⡍⢉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠙⠉⢉⣉⣭⣤⣴⣦⠈⠉⠉⠉⠉⠉⢻⣿⣿⣿⢿⣿⣯⣉⠉⠉⣽⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⡆⠉⢸⣤⣘⣛⣿⣿⣯⣭⣤⣶⠂⠀⠈⠑⠲⢄⢲⠀⠀⠀⠀⠀⠀⠉⠛⠳⡤⣶⣄⠀⠀⠀⢀⣵⣾⣿⣿⣿⣿⣿⣿⡇⣠⡄⠀⢀⣀⠀⠀⢰⠷⢄⣿⣿⣿⡳⣦⠙⠿⣿⣿⣿⣿⡿⠿⠛⠋⠀⡄⠀⠀⠀⠀
⠀⠀⢀⠃⠞⠁⠀⠙⠛⢛⣻⣍⡿⣿⣿⠀⠀⣀⣤⣄⡀⠀⠀⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⢀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣎⡀⠀⣸⣿⣀⡀⣀⣀⡀⠻⣿⣿⡇⣿⡆⠀⠈⠉⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣀⡀⠀⢶⣿⣿⣿⣿⡟⠃⠀⢸⣧⣀⣤⣿⣤⣼⣏⣀⣈⣿⡄⢠⡟⠀⠀⠘⠇⠀⣿⠘⢿⣿⣿⣿⣿⣿⣿⠟⠛⠉⢸⡇⠶⢸⣿⠀⠀⠈⣷⡀⠀⡼⠁⠈⠳⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣴⣾⣿⣿⡟⠃⠀⠈⢿⡿⣿⠿⠀⠀⠀⢀⡈⠙⣿⣿⣿⣿⡿⣿⣯⣉⠁⠸⣇⠀⠀⢀⡀⠀⣿⡀⠀⢹⡿⠿⠟⣿⡇⠀⠀⠀⢸⡇⠀⢸⣿⠀⣀⠀⠘⣷⣰⠃⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠴⠷⠀⠀⠀⠀⠘⠓⠾⣿⣿⣿⣿⡿⡿⣿⡿⢀⠀⠙⠲⠶⠚⠀⠀⠘⠳⠖⠛⠓⠂⠐⠚⠓⢂⠀⠐⠚⠓⠂⠈⠻⠶⠋⠀⠀⢹⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⢿⠗⠠⠰⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⢎⠻⢯⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠖⠀⠀⠀⠀⠻⠴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⣿⢹⣿⣿⠸⣇⢿⣿⡇⡟⡟⣿⣘⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣻⣿⣿⣿⣿⣿⣿⡹⣿⣿⢻⣿⣿⣿⣿⣿⣇⣇⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⢻⢹⣿⡿⣼
⡷⢻⠘⢿⡿⡇⣟⠸⢙⣠⡇⣷⢮⣿⣿⣿⣿⣛⣛⣿⣿⣿⣿⠿⢿⣏⣿⣿⢼⣛⣿⣿⣅⣹⠛⣿⣿⣿⣯⣿⣿⢻⢣⣽⢿⡙⢿⣿⣽⣿⣿⣿⣿⣯⣿⣿⣟⡟⣿⣿⣿⣿⣟⣿⣿⣿⠿⣿⣿⢿⡵⣓⣴⢇⣾⣿⢃⠿
⡧⠸⣸⣟⢃⣥⠃⠩⢻⣯⣤⢭⣬⣿⢿⣿⣽⡃⠬⣙⣚⡿⣿⣭⣶⣟⡛⡿⢻⡏⣿⡏⠏⢿⢭⣿⣚⣿⢻⡿⣿⡏⢸⣿⣾⣇⣚⠻⣿⡟⡤⣭⣻⣿⣷⡜⢻⣟⡌⠃⡉⢭⣷⠛⣿⠊⢋⣤⠶⣾⣿⡷⠂⣤⣉⠿⣿⣦
⠃⣸⡿⢯⢹⣟⡃⡒⠄⠴⣶⢷⣽⣷⣾⡧⢠⣵⡀⣿⣷⢶⣺⡿⣿⣯⢸⢛⣮⣍⣙⢿⡗⣬⠅⢉⣦⣭⣍⡁⠙⠳⠾⠙⣯⢍⣉⡁⠙⠋⣙⡛⠃⠹⢷⠤⠀⠉⣷⣶⣶⣿⢿⣿⡟⠅⡛⠛⠓⠀⣻⣷⣶⣤⠑⢒⣚⣀
⣬⣅⠻⠋⠀⠉⠋⣉⡹⠦⣀⡬⢿⡯⡌⢡⣭⡀⠁⣠⣾⣶⣤⠁⠙⠋⣨⣿⣍⣤⣥⠈⠁⠱⡶⠖⣿⡟⠋⢡⣶⣷⡶⠀⠈⠸⣯⣁⣀⡸⣿⣿⡃⠀⣠⣿⣦⣤⠻⢿⡿⠿⠈⠿⡆⠨⠚⠻⠷⠶⠉⣰⣾⣷⣴⡄⠀⠚
⠉⠋⢁⠀⠀⠘⠻⢿⣿⢦⣖⡁⣼⣿⣿⢎⠙⢽⡳⠀⠘⠟⠋⠡⢤⣤⢿⣿⣿⣮⣤⡀⣠⣷⡶⠀⢈⠶⡶⠎⢿⣿⡗⡀⢠⣾⣿⣿⡏⠀⠙⢿⠀⠠⡄⠈⠋⢁⣀⡘⠿⡇⠀⠀⠱⢄⠈⠤⣴⣦⡤⠉⠡⣿⡿⠄⢈⠖
⠸⣷⣶⣄⠀⠀⠀⠀⣾⣿⣿⣷⢿⠿⢯⡄⢿⣿⣿⡆⠀⠀⢀⣾⣬⣄⣈⣛⣿⣿⣿⣏⣉⠁⣀⢀⣼⣤⣶⡀⠨⠿⣷⣶⠎⢻⣿⣿⠟⠂⠀⠀⠘⢻⣴⣾⣿⡿⣿⣿⣷⣾⣶⣄⠀⠀⠐⠆⠀⣈⡀⣰⣿⡿⣿⣿⣶⡾
⣀⠀⠉⠛⠃⠀⢀⣠⣿⣿⣿⣧⡸⣿⡿⠃⠈⠙⢍⠁⠀⠸⠿⠿⢫⠙⠟⠋⠘⣿⣿⡷⣼⣿⣿⣿⣿⠋⠁⠀⡸⢀⣈⣀⠀⠈⠻⡿⠀⣀⣠⣤⣤⣿⣿⡏⠛⣐⣻⣿⣿⣇⠀⠙⠀⠠⢶⣾⣿⣿⣷⠈⠉⠨⣿⣿⣿⡅
⠉⡄⠀⠀⠀⠀⣿⣿⣿⣿⡿⡋⠻⢈⣙⣛⠛⣿⣿⣿⣟⠃⠀⣐⣾⣿⣿⣛⣻⣿⣿⣿⣻⣿⣿⣿⣿⡛⠛⣿⣿⣿⡿⠷⠀⠀⠘⠛⠛⣻⡛⠛⣋⡿⠋⠀⠘⠻⣿⣿⣿⡟⠃⠀⠐⢤⠀⠀⠹⠁⠀⠀⠀⠀⢻⣿⣷⣀
⣶⣤⡀⠀⣾⣿⢭⣿⣿⣿⣥⣴⠀⠘⣿⣿⣿⠛⣛⠛⢿⠇⢸⣿⣿⡟⠛⣻⠛⢻⠿⠛⠛⣿⣿⣿⣿⠛⢟⠛⠛⠛⠛⠛⢛⡓⠘⠛⠛⣿⣿⣿⣿⣿⣷⣤⡛⠛⠛⠛⣿⢻⣇⡀⠰⢁⣠⣤⣶⣶⡄⠀⠀⣰⣿⣿⣿⣿
⠛⠿⣿⣷⠦⠊⠀⢿⣿⣷⣿⡟⠀⠤⠀⠉⢻⣿⣛⡃⠀⠀⢘⣛⣛⡃⠈⡞⢁⣁⣀⠀⠀⠙⠿⣿⢟⣛⣛⣃⣀⣀⣘⣛⣛⠃⠀⠀⢸⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣛⡃⠘⢿⣿⣶⣿⣿⣿⣿⣿⣋⣁⠘⠿⢿⣿⠟⠛
⠀⠀⣀⣭⣤⣤⠀⠈⢛⣿⠋⠀⠀⠀⠀⣤⣼⣿⣛⡁⠀⠀⢘⣛⣛⣣⣶⣿⣿⣿⣿⡄⠀⠀⠀⠉⢘⣛⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⢚⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⣛⡃⠀⠀⠚⠻⢿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠁
⣶⣿⣿⣿⣿⣿⡀⠀⠀⢀⣠⣤⣄⡀⠰⣿⣿⣿⣿⣿⣦⡀⢈⣛⣛⡻⣿⣿⡿⠛⠉⠁⠀⠂⠀⠀⢘⣛⣛⡛⠛⠿⣿⣿⣿⣿⣿⠿⢘⣛⣻⣿⣿⣿⣿⣿⣿⣿⣛⣛⡃⠀⣤⣀⠀⠀⠉⠉⢁⠤⠀⠂⠀⠀⠀⠀⠚⣻
⣿⡿⠟⠋⠉⠁⠀⢀⣶⣿⣿⣿⣿⣿⣶⣄⣉⣉⣙⣻⣿⡃⢈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣈⣉⣉⣵⣄⣀⣈⣉⣙⡯⢄⣀⣈⣉⣙⣛⣿⣿⣿⡿⠿⢿⣛⣉⣁⣀⣀⡹⠒⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⣀⣠⣖⣿⣿⣿⣿⣿⣿⣯⣽⣿⣻⣿⣿⣿⣉⡁⠈⢉⣉⣉⣉⣩⣉⣉⣉⣉⣉⡁⢈⣉⣉⣉⣉⣉⣿⣋⣉⣩⠋⠁⢈⣉⣉⣍⣉⡽⣿⣿⣿⣿⣿⣮⣉⣉⣉⣉⡝⠀⠀⢀⣀⣠⣤⣤⣦⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣾⣿⡏⢻⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⣿⣿⠛⣿⠃⠀⠀⠀⠉⠍⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⡿⠉⠉⠉⠀⣀⣀⡈⠉⠉⠉⠉⠁⠘⢿⣿⣿⣿⣿⣿⣍⠉⠉⠁⣠⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⢀⠘⠛⡇⠘⡽⣿⣿⣿⡇⠂⠀⣤⡀⠀⠁⠨⢤⡉⢠⡀⠀⠀⠀⠀⠈⠐⠶⢶⡀⣠⣄⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣷⣤⡄⠀⢀⣀⠀⠀⠙⢻⣽⡉⣿⣿⣿⣦⣠⠻⣿⣿⣿⣿⣿⣿⣿⡏⠿⠀⠀⠀⠀⠀
⠀⠀⢸⢅⠴⠋⠻⣶⣶⣿⣿⠿⡿⠿⣿⠀⠀⣀⣤⣄⡈⠊⠀⣀⣤⣄⡀⠀⠀⢀⣠⣭⣉⡈⢀⣀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⡀⠀⣸⣿⣀⡀⣀⣘⡹⠁⣿⣿⣟⣿⣽⣧⠈⠙⠛⠟⠿⠆⠀⠀⠀⠀⠈⠀⠀⠀
⠀⠀⠅⠀⠀⠀⠀⠀⣠⣴⣿⣷⣤⣮⠂⠀⢸⣧⣀⣀⠛⠀⣼⣏⣀⣈⣿⡄⢠⡟⠀⠀⠘⠇⠀⣿⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡇⠀⣸⣿⠀⠀⠈⣷⡀⠀⡼⠉⠉⠛⢤⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣠⣤⣴⣶⡶⠀⠀⢻⣿⣿⣿⣿⠃⠀⠀⢀⡈⠙⣿⣿⣿⣿⣿⣯⣉⢉⠁⠸⣇⠀⠀⢀⡀⠀⣿⡀⠈⢹⣿⢻⣿⣿⡿⠋⠀⠀⢸⡇⠀⢸⣿⠀⣀⠀⠘⣷⣰⠃⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀
⣿⣿⣿⣿⣿⣆⠀⠀⠀⠛⣛⡟⠃⠀⠀⠀⠘⠳⢶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⠲⠶⠚⠀⠀⠘⠳⠖⠛⠓⠒⠑⠛⠓⠂⠀⠐⠚⠓⠂⠈⠻⠶⠋⠀⠀⢹⠏⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⡆⠠⠰⠶⢿⠓⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⣿⡟⠛⢻⣿⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠁⠀⠀⠀⡀⠀⠀⠀⠾⠶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⣿⢹⣾⣿⡘⡇⢿⣿⡇⡇⠟⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⢿⣿⣿⣿⣿⣿⣇⡿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣯⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⢟⢇⣿⣿⢻
⡷⢻⢘⢿⡟⡇⡟⡘⢙⣠⡇⢿⢮⣿⣿⣿⣿⣛⣟⣿⣿⣿⣿⡿⢿⣯⣿⣿⢼⣿⣿⣿⣅⣿⠛⣸⣿⣿⡯⣿⣿⡏⡇⣽⣽⢿⢸⣿⣽⣿⣿⣿⣿⣿⣾⣿⣜⣿⢻⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⢿⡯⣓⣵⡜⣼⣏⡏⡚
⡗⢘⣸⣏⢥⣤⠃⠩⠻⣯⣦⢥⣬⣿⢿⡿⢽⡇⠬⣹⡻⡿⣿⣯⣶⣏⡋⡿⢻⣯⣿⣏⠯⢿⠯⣾⣚⣻⢻⡿⣾⣿⢀⣽⣾⣮⣘⡡⣿⣟⠣⣮⣭⣿⣿⡮⠛⣿⢥⠙⣈⠬⣵⣟⢿⠫⢫⣡⡶⢾⣿⣷⡖⣠⣉⡺⢟⣷
⢁⡸⡿⣭⢹⣯⣃⡒⢂⠶⣶⢷⣾⣷⣿⡧⣠⣷⡄⢿⣿⠶⣾⡿⣿⣯⢸⢋⣷⣍⣘⢿⡷⣬⠅⢉⣶⣤⣏⡉⠘⡓⠶⠋⣿⡍⣉⡁⠘⠃⣉⣙⠋⠘⠿⠠⠀⠈⢲⣶⣿⣿⣿⣿⣿⠷⡈⠛⠛⠀⢹⣿⣶⣦⣄⠒⣒⣃
⣴⣄⠻⠃⠈⢈⣋⣉⡙⠦⣠⡬⣿⣿⡆⢥⣭⣁⠉⢠⣾⣿⣮⡄⠙⠋⣠⣷⣯⣼⣤⠈⠁⡸⠶⠖⣿⡗⠛⢁⣶⣷⣶⠄⠀⠀⢿⣄⣀⡰⣻⣿⡟⠀⢀⣼⣷⣤⡌⠻⣿⡿⠧⠹⣷⠀⠖⠛⠿⠶⠎⢁⣾⣿⣦⣦⠠⡐
⠈⠉⠁⠀⠀⠘⠛⢿⡿⣲⣂⡡⣿⣿⣿⠦⣉⢼⣓⠀⠘⠛⠉⠡⢴⣶⢺⣟⣿⣾⣦⣄⢠⣷⡶⠄⠈⠷⡶⠆⢺⣿⡗⠀⢀⣴⣿⣿⣿⠁⠘⠻⡆⠀⢤⠀⠘⠁⣀⣀⡻⢿⠄⠀⠈⠶⡀⠡⢤⣶⣦⡌⠉⢡⣿⣯⠀⠡
⠺⢿⣷⣄⠀⠀⠀⢠⣾⣿⣿⡿⢿⠿⢯⡄⢻⣿⣿⡆⠀⠀⢀⣼⣬⣄⣀⣙⢿⣿⣿⣿⣈⠁⣀⣀⣸⣥⣶⡀⠀⠿⠷⣶⠆⠑⣿⣿⡿⠂⠀⠀⠘⠋⣷⣶⣿⣿⣿⣿⣿⣷⣿⣦⡄⠀⠀⠲⠀⢀⣁⠀⣴⣿⡿⢿⣿⣶
⣀⠀⠈⠛⠁⠀⢀⣤⣿⣿⣿⢧⡸⣿⡿⠁⠀⠉⢫⠁⠀⠘⠿⠿⢫⠉⠿⠋⠘⣿⣿⡿⣸⣿⣿⣿⣿⡟⠁⠀⠸⢂⣈⣁⠀⠀⠙⢿⠇⢀⣀⣤⣤⣼⣿⣿⠉⢛⣈⣿⣿⣿⣇⠀⠁⠀⠴⢶⣿⣿⣿⡇⠀⠉⢠⣟⣿⣿
⠀⠄⠀⠀⠀⠀⣿⣿⣿⣿⡿⠋⠛⢈⣙⣛⠛⣿⣿⣿⣟⠃⠀⣐⣾⣿⣿⣛⣻⣿⣿⣿⢿⡿⣿⣿⣿⡛⠛⣻⣿⣿⣿⠿⠆⠀⠘⠛⠛⢛⣟⠛⠋⣹⣝⠃⠘⠛⢻⣿⣿⣿⠃⠀⠀⠰⣤⠀⠀⠕⠁⠀⠀⠀⠀⢻⣿⣶
⣶⣦⡀⠰⣿⣯⢭⣿⣿⣷⣧⡤⠀⠘⣿⣿⣿⠟⣛⠛⢿⠇⢘⣿⣿⡟⢛⣻⡛⢛⠿⠛⠛⢿⣿⣿⣿⡟⠻⡛⠛⠛⠛⠛⠛⣃⠘⠛⠛⢻⣟⣿⣿⣿⣿⣿⣿⣛⠛⠛⠛⣿⢧⣀⡀⠘⢁⣠⣤⣶⣶⡄⠀⠀⢀⣾⣿⣿
⠻⢿⣿⣷⠔⠊⠈⣿⣾⣶⣿⡟⠀⠄⠀⠉⢻⣿⣛⡃⠀⠀⢘⣛⣛⡃⠀⡸⠁⣀⣀⡀⠀⠈⠿⢿⣿⣛⣛⣃⣀⣀⣘⣛⣛⠋⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡃⠀⠸⣿⢿⣶⣿⣿⣿⣿⣿⣉⣀⡀⠻⠿⣿⣿
⠀⢀⣠⣥⣤⡤⠀⠘⢛⡛⠋⠀⠀⠀⠀⣤⣽⣿⣛⡁⠀⠀⢘⣛⣛⣣⣴⣿⣿⣿⣿⣆⠀⠀⠀⠘⢙⣛⣻⣿⣿⣿⣿⣿⣯⣴⣶⣶⣾⣟⣻⣿⣟⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠺⠿⢿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠈
⣾⣿⣿⣿⣿⣧⠀⠀⠀⢀⣠⣤⣀⠀⠰⣿⣿⣿⣿⣿⣦⡀⢘⣛⣛⡻⣿⣿⡿⠟⠉⠁⠀⠀⠀⠀⢘⣛⣛⡛⢛⠻⣿⣿⣿⣿⣿⣿⢿⣛⣻⡿⢿⣿⣿⣿⣿⣿⣟⣟⡃⠀⢀⣤⡀⠀⠀⠉⠋⠛⡁⠠⠤⠀⠀⠀⠀⠀
⣿⡿⠟⠉⠉⠁⠀⢠⣶⣿⣿⣿⣿⣷⣦⣀⣉⣉⣛⣿⣿⡗⢈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣈⣉⣉⣁⣬⣠⣈⣉⣉⣛⣠⣤⣌⣉⣉⡱⠺⣿⣿⣿⣿⣿⣿⣿⣇⣀⡀⠈⣹⠖⠀⠀⠀⠔⠃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⡿⢿⢟⣻⣿⣿⣿⣿⣿⣩⡅⠈⢉⣉⣉⣉⣩⣉⣉⣉⣉⣩⡁⢈⣉⣉⣉⣉⣉⣹⣿⣉⣉⡩⠏⢈⣉⣉⣉⣉⡁⠸⣿⣿⣿⣷⣾⣭⣍⣉⣉⡁⠰⠃⠀⠀⠀⣀⣀⣤⣤⡄⠀⠀⠀⠀⠀
⠀⠀⢀⣿⣿⠳⣶⣿⣿⣿⣿⣿⣿⠟⠉⠉⠉⣿⣿⡏⣿⡇⠀⠀⠀⠉⡍⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⢩⡏⠉⠉⠁⠀⠀⢈⡉⠉⠉⠉⠁⠀⠘⣿⣿⣿⣿⣿⣍⡉⠉⠁⠀⢀⣴⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀
⠀⠀⢀⠿⠟⠀⡾⣿⣯⣿⣿⠛⡃⠀⣤⠀⠀⠫⢭⣀⠘⠠⡀⠀⠀⠀⠀⠀⠰⠦⣤⡀⢀⡄⠀⠀⠀⠀⠀⠈⢀⣤⣶⣶⣿⣿⣿⣧⡄⠀⢀⣀⠀⠀⠈⠻⠿⢟⡟⢿⣿⣷⣶⣄⠾⢿⣿⣿⣿⣿⣿⣿⣿⢰⡆⠀⠀⠀
⠀⠀⣷⢀⡴⠲⣤⣉⣯⣿⣾⣥⣴⣶⣿⠀⠀⣀⣤⣄⡙⠆⠃⣀⣤⣄⡀⠀⠀⢀⣠⣽⣘⡉⢂⣀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣟⡀⠀⣸⣿⣀⡀⣀⣀⡀⠟⣓⣺⣿⣿⣲⢿⣧⡐⠙⠻⢿⣿⣯⠉⠉⠀⠀⠀⠂⠀
⠀⢀⠁⠉⠀⠀⠈⠉⠋⣩⣭⣉⣉⣁⠀⠀⢸⣧⣀⡀⠛⠀⣼⣏⣀⣈⣿⡄⢠⡟⠀⠀⠘⠇⠀⣿⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠄⢸⣿⠀⠀⠈⣷⡀⠀⡼⠁⠙⠛⠻⢟⡹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣀⣀⣤⡄⠀⠸⣿⣿⣿⣿⣿⠛⠀⠀⢀⡉⠙⣿⣷⣶⣿⣯⣍⡉⢉⠁⠸⣇⠀⠀⢀⡀⠀⣿⡀⠀⢹⣿⣿⡿⣿⡿⣿⠟⠁⢸⡇⠀⢸⣿⠀⣀⠀⠘⣷⣰⠃⠀⠀⠀⠀⠀⠙⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀
⣾⣿⣿⣿⣏⠁⠀⠀⢻⣿⣿⡿⠏⠀⠀⠀⠘⠳⠶⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠙⠲⠶⠚⠀⠀⠘⠳⠖⠛⠓⠒⠒⠛⠓⠃⠀⠐⠚⠓⠂⠀⠻⠶⠋⠀⠀⢹⠏⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⠄⠠⠴⢶⣾⡆⠀⠀⠀⠀⠀⠀⠘⢿⣿⢿⣿⡿⠿⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠾⠴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⣿⢹⣿⣿⡈⣧⢹⣿⣧⡇⡟⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣹⣿⣟⣿⣿⣿⣿⣿⣿⢸⢻⣿⣿⡿⣽⣿⣿⣿⣿⣿⣿⡟⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⢻⣿⣿
⡃⢿⢨⣻⡟⡇⡯⣘⣈⣴⡇⣿⠫⣟⣻⣿⡿⡪⣞⣿⣿⣿⣿⡻⠿⣯⣿⣿⣸⣷⣿⣿⣿⡼⣅⣹⣿⣿⡿⣿⣿⣿⢸⢸⣾⡏⡇⣿⣾⣿⣿⣿⡟⣯⣷⣿⣿⣻⣻⣿⣿⣿⡿⣿⣿⣿⣿⢻⣟⡿⣿⢜⣪⣴⡎⣾⣛⠇
⠓⣸⣾⣯⣶⣦⠃⠰⠻⠧⠶⡷⢾⡿⢻⡷⠿⡅⠞⣨⣯⠿⢿⣿⣧⣯⡅⣛⠋⣿⢺⣿⡭⢻⣫⣶⣛⣹⣿⣿⢿⣿⠃⣭⣷⣷⢀⣁⢿⣿⣯⢼⣭⣵⣿⣿⠝⢻⡟⠌⢃⡡⢽⣶⣙⣿⠑⢍⣴⠶⢿⢿⣷⠂⣠⣁⠿⢛
⢠⢻⣛⡆⠹⣭⣤⣉⠂⠚⠿⢿⣾⣷⢿⡗⣰⠾⣆⠾⢿⠲⢾⡿⣿⣻⢺⣏⣵⡤⣬⠻⣿⢴⠥⢈⣼⣦⣾⣭⠀⢛⠲⠚⠹⣯⢡⣅⠀⠛⢨⣭⣙⠁⠘⠷⠄⠀⠑⢦⣾⣿⣿⣿⣿⣿⠧⡉⠛⠛⠀⢹⡿⣶⣦⡔⠒⢞
⣾⣎⠋⠁⠀⣀⣀⣤⣟⠣⢤⡞⣟⣳⡆⠶⣮⣅⡀⠰⢾⣻⣧⡄⠈⠋⢠⣾⣷⣼⡶⠀⠉⢀⠲⠖⢺⣷⠘⠁⡴⣾⣿⡆⠀⠀⠸⣧⣀⣀⢚⣿⣿⡃⠀⢠⢿⣦⣤⠟⠩⣿⣿⠧⠻⡷⠨⠞⠛⠷⠶⠂⠀⣾⣿⣶⣶⡄
⣀⡀⠄⠀⠀⠉⠙⢿⢟⣳⣥⢰⣿⣿⣿⢦⣈⣸⣝⠀⠘⠛⠉⠁⠶⡶⠘⣟⣻⣶⣶⣤⣠⣾⡶⠆⠀⡑⢺⠶⠘⣿⣿⠣⠀⢠⣾⣿⣿⣯⠀⠙⢿⠀⠀⣣⠀⠐⠁⢀⣠⣝⣿⡇⠀⠈⠶⣐⠡⠴⣶⣤⡄⠉⠡⣼⣿⣅
⠻⢿⣷⣄⠀⠀⠀⠰⣿⡿⣿⡧⣿⠿⣤⠄⢿⣿⣿⣇⠀⠀⢀⣸⣤⣤⣤⣽⢏⣿⣿⣟⣉⡄⢀⣀⣠⣿⣶⣆⠀⠈⠿⢶⡶⠆⢹⣿⣿⡗⠀⠀⠀⠘⠋⣵⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⡲⠀⠀⣁⠀⠀⣴⣿⡿⣿
⡄⠀⠈⠉⠀⣀⣠⣾⣿⣿⠿⣇⣻⣿⡿⠀⠂⠈⠂⠄⠀⠈⠟⠟⠋⡌⠹⠛⠀⢿⣿⣿⢇⣿⣿⣿⣿⣿⡋⠀⠀⡽⣀⣈⡀⠀⠀⠻⢿⠃⣀⣀⣤⣶⣾⣿⣿⡏⠹⣁⣻⣿⣿⣟⡁⢹⠀⠠⠶⣾⣿⣿⣷⠆⠁⠁⣠⣼
⠰⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠙⢃⣉⣘⣛⠛⣿⣿⣿⣿⠃⠀⣐⣾⣿⣿⣟⢛⣿⣿⣿⡿⡿⣿⣿⣿⡿⢛⢛⣿⣿⣿⡿⠷⠀⡘⠛⠛⠛⠻⣟⡋⠉⢙⢍⣛⠛⠛⠻⣿⣿⣯⣍⠁⠀⠀⢤⡀⠀⠈⠇⠈⠀⠀⠀⠀⢿
⣶⣤⣀⢺⣿⣬⣿⣿⣿⣥⣤⠄⠀⠛⣿⡿⣿⠟⣛⠛⠿⠗⢙⣿⣿⡟⠛⢛⡟⠛⡻⠟⠛⢻⣿⣟⣛⣿⠛⠿⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⢻⡟⠛⣠⣿⣿⣿⣿⣿⣟⡛⠻⢿⡧⣄⣀⠀⠎⣀⣤⣤⣶⣦⠀⠀⠀⠀⠈
⠻⣿⢿⡦⠒⠀⢻⣷⣶⣿⣿⠂⠀⠄⠀⠉⢻⣻⣛⠃⠀⠀⢘⣛⣛⡃⠈⢠⠋⢈⣀⡀⠀⠀⠻⣿⣿⣟⣛⣃⣀⣀⣘⣛⣛⠋⠀⠀⢘⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣬⠄⢿⣿⣗⣾⣿⣿⣿⣿⣿⣓⣀⣀⡀⠾
⠀⢀⣨⣤⣤⠄⠈⠿⢿⡿⠁⠀⠀⠀⢠⣤⣼⣿⣛⡁⠀⠀⢘⣛⣛⣃⣤⣾⣿⣿⣿⣇⠀⠀⠀⠀⢛⣛⣛⣿⣿⣿⣿⣿⣯⣤⣤⣤⣼⣿⣿⡛⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠩⠾⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀
⣾⣿⣿⣿⣿⡀⠀⠀⠉⢀⣀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣦⡀⢘⣛⣛⡻⢿⣿⣿⠿⠛⠋⠁⠠⠀⠀⢈⣛⣛⡛⠋⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⡃⢙⣟⣻⣟⣿⣿⣿⣻⣿⣿⠟⠀⠀⢈⡀⠀⠀⠉⠛⠓⠛⢋⣉⣉⠀⠀
⣿⡿⠛⠛⠉⠁⠀⣤⣾⣿⣿⣿⣶⣤⣀⣀⣉⣙⣻⣿⣿⡟⣈⣉⣉⣁⣀⣉⣁⣀⣠⣀⣀⡀⢀⣀⣈⣉⣉⣁⣀⣀⣉⣉⣙⡛⢛⣟⣙⣉⣉⡁⠀⢉⣙⣛⣛⣿⣿⣻⣏⣛⣀⡄⠀⠉⠙⣦⠄⠀⠀⠀⡠⠃⠀⠀⠀⠀
⠁⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣭⣭⣭⣭⡁⠈⢉⣉⣉⣉⣉⣍⣉⣉⣉⣉⡓⢈⣉⣉⣉⣉⣉⣉⣙⣿⣍⠉⠁⢨⣟⣉⣉⣉⡁⠀⠈⣉⣽⣿⣿⣿⣿⣿⣟⡛⠃⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣾⣛⣻⣿⣿⣿⣿⣟⣯⣭⡥⠛⠋⢿⣿⣿⡟⣽⣥⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⣹⠋⠁⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢀⣠⣴⣶⣷⣿⣿⣿⣆⠀⠀
⠀⠀⢿⡇⠻⣿⣿⣿⣿⣿⣿⡿⠋⠀⣤⠠⠘⠛⠛⠁⠛⠋⠀⠀⠀⠀⠈⠀⠀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢀⣀⣤⣤⣴⣶⡆⢀⣀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣶⣦⣤⡀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⡀
⠀⢰⠈⠁⠠⠻⣉⣿⣿⠷⠁⠀⠀⠀⣿⠀⠀⣉⣭⣖⣤⡀⡽⣀⣤⣄⡀⠀⠀⢀⣩⣵⣐⡿⣦⣀⠀⠀⣀⡀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣸⣿⣀⡀⣀⣀⡀⠈⣙⣛⣋⣿⣻⣿⣿⡟⢻⡍⠻⣿⣿⣿⣿⡿⠿⠿⠸⠃
⠀⡘⠴⠋⢲⣤⣬⣭⣭⣴⣶⣶⣾⠿⠁⠀⠸⣧⣀⠀⠛⠁⣼⣏⣀⣈⣿⡄⢠⡟⠀⠀⠘⠇⠀⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⠀⠀⠈⢷⡀⠀⡸⠡⠛⠓⠻⣿⣿⣿⣞⣿⡄⠀⠈⠉⠙⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠉⡉⢉⣩⣭⣍⣁⣀⠘⠂⠀⢀⡈⠉⣻⣷⣀⣿⡍⠉⠉⢉⠁⠸⣇⠀⠀⢀⡀⠀⣿⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⠻⢿⣿⠀⣀⠀⠘⣷⣰⠃⢀⠀⠀⠀⠈⠉⠉⠻⠄⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣶⣶⣶⠆⠀⢹⣿⣿⣿⣿⣿⠟⠀⠀⠀⠘⠳⠶⣿⣿⣿⣿⣿⣶⣶⡋⠀⠀⠙⠲⠶⠚⠀⠀⠘⠳⠖⠛⠛⠛⡟⣛⣻⣿⡟⡕⠚⠓⠂⠀⠻⠶⠋⠀⠀⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⠀⠀⠀⠀⠀⠀⠈⠀⠀
⣿⣿⣿⣷⡏⠶⠄⢻⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⣿⢹⣿⣿⡈⣧⢹⣿⡧⡇⡏⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣯⣿⣿⣿⣿⢻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣟⡏⣿⣿
⡇⢿⢈⣻⠙⡇⡯⣘⣈⣴⡇⣿⠣⣟⣻⣿⡿⡨⣞⣿⣿⣿⣿⡻⠿⣯⣿⣿⣰⣷⣿⣯⣿⡬⢟⣻⡿⣿⡿⣿⣿⣿⢹⢩⣿⣽⡟⣿⣾⣿⣿⢿⣟⣯⣷⡿⣷⣻⣿⣿⣿⣿⣿⣿⣟⣿⣿⠻⣿⣿⢿⣝⣖⣤⣿⢸⣟⡻
⠃⣸⣾⣏⣴⣦⠃⠐⠻⠧⠶⡶⢺⡿⢻⡷⠿⡅⠜⣨⣯⠿⢿⣿⣯⣯⡅⢛⠋⣯⢸⡿⡽⠽⣫⢶⣗⣹⣿⣿⣿⣿⡊⣨⣽⣷⢇⣉⡸⣿⡟⢴⣯⣭⢿⣿⡯⢹⣟⠍⢃⡡⠭⣶⣋⣿⠋⢨⣤⡶⠾⡿⣷⡞⢀⣉⡸⢛
⢠⢟⣛⡀⢻⣭⣠⣍⡂⠚⠿⢿⣾⣷⢿⡗⣰⠾⣆⠾⢿⠲⢾⣭⢿⣻⠺⣏⣵⡤⣤⡭⣿⢤⡥⠈⣴⣦⣴⣭⡀⢛⠓⠚⠈⣿⢁⣭⠀⠙⠋⣬⣝⠁⠘⠿⠠⠀⠈⠦⣴⣾⣿⣿⣟⣿⡷⢌⠙⠋⠀⠘⣿⠷⣶⣤⠒⠲
⣶⣌⠋⠀⠀⣈⣀⣤⣟⠃⢤⠖⣛⣳⡂⠶⣶⣅⡀⠠⢾⣻⣧⡄⠈⠋⢠⣾⣷⣼⡶⠄⠉⢀⠲⠖⢺⣿⠊⠉⢠⣾⣿⡧⠀⠀⠸⢷⣄⣀⡘⣻⣿⣏⠀⠀⣼⣷⣤⡬⠉⢽⣿⠿⠜⢿⠃⠵⠛⠻⠶⠖⠀⢐⣾⣷⣶⣦
⣈⡀⠄⠀⠀⠉⠛⢿⢟⣰⣁⢰⣿⣿⡿⢆⣈⣸⣝⠀⠘⠛⠉⠀⠶⣶⠚⣿⣻⣶⣶⣤⣀⣼⡶⠆⠀⢘⠲⡶⠂⢿⣟⡒⠀⢀⡴⣿⣿⣿⠄⠈⠿⡆⠀⣸⡄⠀⠋⠀⢀⣈⣻⣿⠀⠀⠱⢄⡈⠤⣴⣦⣤⠌⠉⠠⣹⣿
⠿⢿⣷⣄⠀⠀⠀⠰⣿⠿⣿⠾⡿⠯⢤⠄⢿⣿⣿⣇⠀⠀⢀⣨⣥⣤⣠⣝⡏⢿⣿⣿⣃⣍⠀⣀⣀⣮⣶⣆⠀⠈⠿⠷⣶⠆⠠⣿⣿⠷⠂⠀⠀⠀⠛⢭⣶⣾⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠐⠠⠀⠘⡀⠀⢀⣼⣿⣿
⡀⠀⠙⠉⠀⢀⣤⣾⣿⣿⣾⣄⢿⣿⡟⠀⠂⠈⠂⠅⠀⠈⠿⠿⠛⡍⠹⠛⠈⢿⣯⣽⠇⣼⣿⣿⣿⣿⡏⠁⠀⣸⠆⣈⣁⠀⠀⠙⢿⡟⢈⡀⢀⣤⣤⣿⣿⣿⠉⠛⠁⠾⢿⣿⡟⠉⠃⠀⠠⢴⣾⣿⣿⣶⠆⠉⠉⣀
⠠⠀⠀⠀⠘⣿⣿⣿⣿⣿⠛⠙⢂⣉⣘⣛⠛⣿⣿⣿⣟⠁⠀⢐⣺⣿⣿⣟⠛⣻⣿⣿⣿⡫⣿⣿⣿⣟⠛⠛⣻⣿⣿⣿⠿⠆⠘⠛⠛⠛⠛⢿⣃⠉⠛⢻⠝⠛⠛⠛⠻⣿⣿⣭⡉⠁⠀⠠⢄⡀⠉⢉⠏⠀⠀⠀⠀⠘
⣤⣄⠀⣾⣇⣙⣿⣿⣯⣀⣤⠄⠀⢛⣿⣿⣿⡟⣛⠛⢿⠷⢘⣿⣿⡿⠛⠛⣛⠛⢛⡿⠟⠓⣾⣿⣛⣻⡟⠿⡛⠛⠛⠛⡛⠃⠘⠛⠛⠛⠛⣻⠟⠀⠀⣡⣿⣿⣿⣿⣟⠛⠿⠿⢇⣀⠀⠠⠊⢀⣀⣤⣤⣄⠀⠀⠀⠀
⢿⣿⣷⡤⠐⠈⣿⣭⣭⣿⡿⠀⢀⠀⠈⠑⢟⣻⣛⡃⠀⠀⢘⣛⣛⡃⠀⠀⡝⠀⠁⠀⠀⠀⠹⣿⣿⣿⣛⣃⣀⣀⣘⣛⣛⠋⠀⠀⢘⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠈⢉⣿⣿⢦⣾⣿⣿⣿⣿⡷⠆⠀⠀⠀
⠀⠀⣘⣀⣀⠀⠹⣿⢿⡟⠁⠀⠀⠀⢀⣄⣘⣛⣛⠁⠀⠀⢘⣛⣛⡃⣀⣤⣶⣿⣿⣿⠀⠀⠀⠈⢿⣛⣛⣻⣿⣿⣿⣿⣯⣀⣀⣀⣘⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡎⠉⢋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣿⣿⣇⠀⠀⠐⠛⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣄⡀⢘⣛⣛⡳⣿⣿⣿⣿⠿⠛⠃⠀⠀⠀⢈⣋⣙⡛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣛⣛⣻⡏⢘⣿⣿⣿⣿⣿⣿⡿⠀⠀⠃⠀⠀⠙⠻⠿⠽⠟⠛⠛⠛
⣿⡿⠿⠛⠛⠁⠀⣀⣤⣶⣶⣦⣀⠀⢈⣉⣛⣛⣿⣿⣿⡷⣈⣉⣉⣁⣀⣙⣋⣀⣀⣀⣀⡀⢀⣀⣈⣉⣉⣁⣀⣀⣈⣉⣙⣻⣿⣿⣿⣛⣋⡉⠀⢉⣛⡛⠇⣸⣿⣿⣿⣿⣻⡏⠀⠀⠚⠓⣤⡀⠀⠀⠀⠀⣰⠀⠀⠁
⠃⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣷⣮⣉⣍⣉⣉⣉⣋⡁⠈⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡱⢈⣉⣉⣉⣉⣉⣉⣉⣿⣯⠉⠁⢈⣉⣭⣛⣉⡁⠀⠈⣉⠙⠻⣾⣿⣿⣿⣿⣿⣶⣾⠂⠀⣠⠞⠁⠀⠀⠀⠈⠁⠀⠀⠀
⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⠿⠛⠿⠙⣿⣿⣿⣿⢫⣁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⣩⡟⠁⠀⠈⠟⠉⠉⠉⠁⠀⠀⠀⠀⢺⣝⣻⢿⣯⣽⣥⡀⠀⠀⠀⠉⠀⠀⢀⣠⣤⣤⣴⣶⣦⠀
⠀⢰⣿⢥⣭⣿⣿⣿⣿⣿⣶⣾⡿⠁⣤⠀⢹⣿⢿⠃⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⣠⣄⣀⣀⣀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣇⣀⡀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⡀⠿⠀⡻⠿⠿⣿⣿⡿⠟⠋⠀⠀⣿⠈⠀⣠⣤⣄⣀⠀⢰⣄⣤⣄⡀⠀⠀⢈⣻⣧⣀⡶⣄⣀⠀⠀⣀⡀⠀⣀⣀⣤⣶⣾⣿⣿⣿⣿⣽⣿⣀⡀⣀⣀⡀⠀⣈⣻⣿⣿⣿⣿⣿⣿⣷⢤⡛⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⡇⣀⢄⠁⠠⠜⡻⠏⠃⠀⢀⣀⣤⠄⠀⠸⣧⣀⠀⠛⠂⣼⣏⣀⣈⣿⡄⢠⡟⠀⠀⠘⠇⠀⣿⠀⠀⢸⡇⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⢷⡀⠀⡸⠁⠀⢱⣶⣸⣿⣿⣿⠄⢹⣦⠈⠛⠻⠻⠟⠀⠈⠁
⠀⠘⠁⠘⢿⣶⣶⣷⣾⣿⣿⠿⠛⠁⠀⠀⢀⡈⠉⠛⣷⠀⢿⡍⠉⠉⢉⠁⠸⣇⠀⠀⢀⡀⠀⣿⡀⠀⢸⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣀⠀⠘⣧⣰⠃⠀⠀⠈⠰⠀⠛⠿⠿⣭⡜⢻⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⡀⠀⢀⣀⣤⣶⣶⣤⣤⣄⠀⠀⠀⠘⠳⠶⣶⣿⣶⣮⣿⣶⠖⠋⠀⠀⠙⠲⠶⠚⠀⠀⠘⠳⠖⠛⠛⠳⣿⣟⣻⣿⣿⡟⠛⠛⠋⠈⠻⠶⠋⠰⠄⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢨⡃⠀⠀⠀⠀⠀⠀⠄
⣿⣿⣿⣷⠇⠸⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠊⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠾⠴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀
⡜⣿⢹⣿⣿⠘⡇⢻⣿⡧⡇⡏⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣯⣿⣿⣿⣿⢿⣿⣏⣿⣿⣿⣿⣿⣿⣇⡇⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⡿⡾⣼⣿
⡃⣿⢈⣻⠹⡇⡇⣘⣈⣴⠇⣿⠣⣟⣿⣿⣿⡪⣞⣿⣿⣿⣿⡻⠿⣯⣿⣿⣰⣿⣽⣯⣿⡨⢟⣫⡿⣿⡿⣽⣿⣿⣹⠇⣿⢿⢸⢸⣾⣿⣿⢿⣿⣿⣽⣿⣿⣿⣟⡿⣿⣿⣿⢿⣿⣿⣿⡻⢿⣿⢿⣕⣗⣥⣯⢇⣿⣻
⢃⣸⣾⣏⣴⡦⠃⠑⠻⠥⠶⡷⣪⡿⢻⡷⠻⡅⠜⣨⣯⠿⢿⣿⣥⣭⡅⢛⠋⣿⢸⡿⣹⠼⣫⢵⣗⣛⡿⣿⡿⣿⡏⢨⣽⣿⡆⣉⡸⣿⣟⠵⣯⣍⢾⣿⣿⠛⣿⠣⠑⣈⠭⣶⣎⣹⡫⢪⣡⡶⠾⡿⣿⡶⢂⣈⡸⠟
⡠⣿⣭⠈⢿⣭⣀⣙⡒⠚⠿⢿⣾⣷⢿⡓⣰⠾⣆⠾⢿⠒⢾⡯⢽⣯⠴⣟⣥⡤⣤⡹⣿⢦⡬⠈⣡⣦⣴⣉⡁⠙⡓⠖⠉⣿⡍⣬⡀⠙⠃⢌⣉⡋⠀⠿⠦⠀⠈⠲⣤⣴⣿⣿⣿⣿⣿⠄⠙⠛⠃⠀⢻⣿⣶⣦⡔⠒
⣶⡜⠋⠀⠁⣈⣠⣤⣛⠧⣤⠼⣿⣷⠂⠶⣮⣥⡁⢠⢾⣿⣧⡄⠈⠛⢠⣼⣶⣤⣦⠄⠉⠀⠳⠶⠺⣿⡟⠋⢡⢶⣷⣶⠀⠀⠀⠿⣄⣀⣀⢛⣿⣿⠀⠀⢸⣷⣦⣤⡞⠫⢿⡿⠷⠿⡏⠰⠞⠻⠶⠶⠊⠁⣲⣿⣶⣶
⠉⠀⠄⠀⠀⠙⠻⢿⡟⣲⣈⢠⣿⣿⣷⢆⣈⡸⣝⠀⠘⠛⠋⠉⠴⣶⠶⣿⣿⣷⣶⣤⣀⣼⣶⠦⠀⠈⠲⡶⠆⢺⣟⡓⠀⠀⣠⣾⣿⣿⡧⠀⠻⡖⠀⠀⣦⠀⠙⠀⠀⢀⣛⣿⡇⠀⠸⠦⢐⡠⣤⣤⣤⣤⠉⠉⢈⣻
⠿⣿⣦⡀⠀⠀⠀⢰⣿⢿⣿⣸⣿⣿⣥⠄⢿⣿⣿⣇⠀⠀⠀⣠⣧⣤⣀⣘⡋⢿⣿⡿⠏⣉⠁⠀⢀⣸⣥⣤⡀⠀⠹⢷⣶⡶⠈⢼⡿⠿⠥⠀⠀⠀⠻⠳⣥⣶⣾⣿⣾⣿⣿⣿⣿⣶⡀⠀⠈⢂⠄⠀⠊⠀⠀⢀⣼⣿
⠀⠈⠛⠉⠀⠀⣠⣾⣿⣶⣿⡀⢷⣶⡿⠀⠀⠉⠉⡅⠀⠈⠿⠿⠛⣋⠛⠿⠁⢹⣯⣭⡏⢸⣷⣿⣿⣿⡟⠉⠀⠐⠖⠀⣉⡀⠀⠘⠿⣿⠀⡀⠀⣀⣤⣾⣿⣿⣿⠛⠿⠃⣿⣿⣿⡟⠙⠀⠀⣠⣶⣾⣿⣷⣶⠈⠉⠉
⠤⠀⠀⠀⢸⣽⣿⣿⣿⣿⠟⠻⠌⠒⠙⠛⠛⣿⣿⣿⣟⠁⠀⠐⣚⣿⣿⣿⠛⢛⣻⣿⣟⡻⣻⣿⣿⣿⡟⠛⢻⣿⣿⣿⣿⡧⠘⠛⠛⠛⠛⠿⣟⠉⠛⠛⢛⠛⠛⠛⠛⢻⣿⣿⣛⠛⠃⠀⠠⣀⠈⠉⢉⠍⠋⠀⠀⠀
⣄⠀⠀⣴⡆⠻⣿⢿⡏⢁⣀⡀⠈⠿⣿⣿⣿⡛⡛⠻⢿⡧⢘⣿⣿⡿⠟⠛⢛⠛⠿⢿⣻⠃⢿⣿⡿⠿⣿⢻⡟⠛⠛⠛⢛⠃⠘⠛⠛⠛⠛⢛⣻⠗⠀⠐⠛⣻⣿⣿⣿⣛⠻⣿⣿⡇⠀⠀⢠⠊⠀⣀⣠⣤⡄⠀⠀⠀
⣿⣿⣣⣉⠍⢻⣿⣛⣛⣿⣿⠀⠀⠈⠙⠻⣟⣛⣛⡃⠀⠀⢘⣛⣛⡃⠀⠁⣹⠂⠒⠁⠀⠀⠘⣿⣿⣿⣟⣃⣈⣀⣘⣛⣛⠋⠀⠀⢘⣛⣛⣛⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣷⣄⠚⢸⣿⣷⢤⣶⣿⣿⣿⣿⡷⡆⠀⠀
⠈⠉⠘⠀⠀⠀⢿⣿⣿⡿⠁⠀⠈⠀⢀⣀⣙⣛⣛⠁⠄⠀⢘⣛⣛⡃⠀⣀⣧⣶⣶⣾⠀⠀⠀⠈⢻⣟⣛⣛⣿⣿⣿⣿⣋⡀⠀⠀⢘⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣷⣶⣟⡛⠻⢛⣿⣿⣿⣿⣶⣶⣶⣶
⣴⣾⣿⣿⡏⠀⠈⠩⠯⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣀⠀⢘⣛⣛⣣⣿⣿⣿⣿⣿⠿⠷⠀⠀⠀⢈⣋⣉⡛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⡿⢟⣛⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠉⠉⠛⠿⣾⣟⣿⡿⠿
⣿⣿⡿⠿⠷⠀⠀⠀⢀⣀⣀⠀⠀⠀⢙⣛⣿⣿⣿⣿⣿⣇⣈⣉⣉⣁⣈⣻⣿⣋⣀⣀⣀⡀⢁⣀⣈⣉⣉⣁⣀⣀⣈⣉⣙⣻⣿⣿⣿⣿⣿⡿⠿⢋⣉⣻⡁⣾⣛⣻⣿⣿⣻⣿⣿⡟⠉⠀⠶⣄⡀⠀⠀⠈⠁⠀⡤⠄
⠛⠁⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣦⣄⣈⣉⣩⣉⣉⣙⣟⡋⠈⢉⣉⣉⣉⣉⣉⣉⣩⣉⣉⡁⣈⣉⣉⣉⣉⣉⣉⣉⣉⣿⣉⠁⢈⣉⣉⣉⣉⡥⠄⠈⣉⠁⠀⣨⣽⣿⣿⣿⣿⡛⣙⣁⣠⡀⠀⣘⠗⠀⠀⠀⠀⠔⠃⠀
⠀⠀⡀⢀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⠆⠈⠉⣩⠟⠉⠁⠀⠀⠀⠀⠀⠈⡹⢿⣿⣿⠿⠿⠿⠿⠛⠀⠘⠃⠀⠀⠀⠀⠀⣀⣀⣠
⠀⣠⡾⠿⣿⣿⣿⣿⣿⣿⢟⣛⣋⡉⣿⠁⢿⣿⣿⡟⣸⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⣤⡅⠀⢀⣀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣾⣿⣷⡄⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿
⠀⢹⡇⢶⣶⣿⣿⣿⣿⣿⣿⣿⠏⠀⣿⢀⠘⣛⣽⣅⣈⠋⢀⣀⣤⣄⡀⠀⠀⢐⣲⣦⣀⣀⣄⣀⠀⠀⣀⡀⠀⣀⣈⣀⣠⣤⣀⣬⣤⣴⣾⣿⣀⡀⣀⣀⡀⠀⣀⣙⢿⣿⣿⣿⣿⣷⣶⣦⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⢠⠈⠃⢨⠻⠿⣛⣿⣿⡟⠋⠁⠀⠀⠀⠀⠸⣯⣘⠚⢿⡀⣼⣏⣀⣈⣿⡄⢠⡟⠁⠀⠙⠟⠁⣿⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⢷⡀⠀⡼⠁⠀⠻⠿⠿⠿⢻⣿⣿⡟⠓⣌⠉⢿⣿⣿⣿⡟⠛
⠸⣢⠖⣄⠀⠀⢘⠋⠈⣁⣀⣀⣠⣴⠂⠀⢀⡈⠉⠛⣷⠈⢿⡍⠉⠉⢉⠁⠸⣇⠀⠀⢀⡀⠀⣿⡀⠀⢸⡇⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⠀⠘⣷⣰⠃⠀⠀⠀⠀⣼⠿⢾⣿⣿⣿⣧⣿⣧⠀⠀⠀⠈⠀⠀
⠀⠁⠀⠙⠿⣿⣿⣿⣿⠿⠿⠟⠋⠀⠀⠀⠘⠓⠶⠖⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠲⠶⠚⠀⠀⠘⠳⠖⠛⠓⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠛⠀⠀⢹⠏⠀⠀⠀⠀⠀⠁⠈⠀⠉⠛⠛⠿⣅⢸⡄⠀⠀⠀⠀⠀
⣀⣤⣶⣶⠈⢶⣴⣶⣿⣷⣶⣴⠶⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠿⠿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠾⠴⠟⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀
⠸⡇⣿⣿⣿⢸⡇⣿⣿⡇⡇⡏⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣟⣿⣿⣿⣿⣿⣿⡇⡿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⡿⢻⢻⣿
⠇⣿⢙⣿⠻⡌⡇⡘⣈⣴⠇⢾⠯⣽⣿⣿⣽⡫⡞⣿⣿⣿⣿⡿⠿⣯⣿⣿⢰⣽⣽⣯⣿⡨⢽⣣⡿⣿⡿⢽⣿⣿⡃⡇⣷⢿⢹⢹⣶⣿⣿⢿⣿⢿⣽⣿⣿⣟⣟⣿⣿⣿⣿⢟⣽⣻⣿⡿⢿⣿⠿⣧⣗⣥⡿⡟⣾⣻
⢂⣻⣽⡫⣴⡆⠃⠉⠿⡧⢦⠥⣪⣿⢻⡿⠽⡅⠼⣩⡻⡿⢿⣿⣷⣯⡉⢛⠋⣿⢹⣿⣹⠼⣯⢵⣗⣛⡿⣿⡷⣿⡇⢁⣹⣾⣮⣈⡙⣿⣿⡡⣯⢍⡴⣿⣿⠟⣽⡫⠙⣈⠬⢴⣖⣙⣟⠉⣡⣤⠶⢿⢿⣶⠆⣀⣁⠿
⡸⡿⣭⠙⣿⣽⣀⡑⠐⠲⡾⣷⣾⣷⢿⡃⣠⢾⣄⠿⣿⠲⢾⣿⢿⣿⡵⣟⣱⣬⣈⡛⣿⢦⡭⠈⢡⣦⣤⣉⡁⠙⡓⠶⠋⢿⡍⣉⡁⠘⠃⢡⣉⡛⠀⠻⡇⠀⠈⠱⣤⣠⣿⣿⣿⣟⣿⠶⡈⠛⠛⠀⠹⣿⣷⣦⣤⠒
⣤⡹⠋⠀⠀⣉⠩⣉⡻⢆⣤⠼⠿⢭⠌⣥⣭⣁⠉⢠⣾⣶⣦⡅⠘⠛⢠⣼⣯⣤⣬⡄⠉⠀⠵⠶⠒⣟⡓⠋⢁⣴⣿⣶⠀⠀⠁⠿⣤⣀⣀⠺⣻⣿⡃⠀⠀⣻⣦⣤⣴⠛⢹⣿⣿⡯⣿⢁⠶⠛⠷⠶⠶⠉⢐⣾⣷⣶
⠉⠡⠀⠀⠈⠛⠿⣿⡟⢰⡊⣨⣿⣿⣷⡄⠉⠱⣻⠀⠘⠻⠋⠉⢤⣤⡤⣿⠿⣥⣵⣄⡀⣠⣷⣦⠀⠉⠶⢶⠖⢹⣿⣿⢂⠀⢀⣴⣿⣿⣯⠀⠸⣶⠀⠀⠰⡀⠉⠃⠀⠀⠘⢿⣿⠀⠈⢶⡄⣁⢠⣤⣤⣤⡌⠉⢃⣙
⣿⣶⣄⡀⠀⠀⠀⢰⣾⣿⣿⣸⣿⣿⣁⡀⣿⣿⣿⡆⠀⠀⠀⢀⣂⣁⢀⣙⣿⢻⣿⣿⠿⢉⠁⠀⠀⣸⣧⣥⠄⠀⠹⣿⣶⣶⠎⠩⣿⣿⣯⠄⠀⠀⠸⠗⣮⣴⣶⣾⣦⣶⣿⣿⣿⣶⣤⠀⠈⠁⡠⠀⠈⠃⠀⠀⢀⣽
⠀⠙⠛⠃⠀⠀⢀⣬⣷⣬⡝⠈⣿⣿⡿⠀⠈⠛⠍⡋⠀⠀⠾⣿⠟⣛⠛⡿⠇⢲⣿⣛⣿⠈⣷⣶⣿⣿⣟⠋⠁⠀⣲⠀⠘⠁⠀⠀⠿⣿⡆⠀⠀⠀⣀⣰⣿⣿⣿⡟⠻⡿⢩⣿⣿⣿⡿⠃⠀⠀⣠⣶⣾⣿⣤⣤⠘⠛
⡀⠀⠀⠀⣤⣾⣿⣿⣿⣿⡿⢶⠙⠭⠙⠛⠛⣿⣿⣿⡛⠂⠀⠐⢚⣟⣿⣿⡛⠛⢛⣻⣛⡻⢻⣿⣿⣿⣿⠟⠛⣿⣿⣿⣿⣯⠘⠛⠛⠛⠛⢻⣿⠞⠛⠻⠿⢿⡛⠛⠛⠛⢳⣿⣿⠿⠷⠂⠀⣀⡀⠈⠛⢛⠍⠉⠀⠀
⡀⠀⢀⣤⢸⣿⣿⣿⡟⠋⠀⠈⠱⣶⣾⣿⣟⠛⠻⠿⣿⣧⢘⣿⣿⣿⡿⠿⠛⠻⣿⣿⠿⠋⣼⣻⣿⣿⣿⡿⣿⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⣻⠆⠀⠛⠛⠛⠛⣛⠛⠛⢿⣿⣿⡆⠀⠀⡠⠋⠀⠀⣀⣀⡀⠀⠀
⣿⣶⡎⠛⠶⣿⡿⠿⢷⣶⣖⠀⠀⠘⠯⢟⣛⣛⣛⡃⠈⠁⢘⣛⣛⡃⠀⠀⢸⡆⠠⠊⠉⠁⠘⣿⣿⣿⣿⣃⣉⣂⣘⣛⣛⠋⠀⠂⢘⣛⣛⣛⣛⣛⣤⣾⣿⣿⣿⣿⣿⣿⣿⣦⣉⠽⢹⣶⣦⣠⣴⣾⣿⣿⣿⣧⡄⠀
⠙⠛⡟⠉⠀⠸⣿⣿⣷⣿⠋⠀⠠⠀⠀⠀⢙⣛⣛⠁⠀⠀⢘⣛⣛⡁⠀⠀⣈⣠⣤⣤⡄⠀⠀⠘⢿⣿⣟⣛⣻⣿⣿⣛⡋⠀⠀⠀⢘⣛⣛⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠛⢻⣿⣿⢙⣻⣟⣋⣡⣠⣤
⣠⣴⣶⣶⡆⠀⠙⢛⡛⠃⠀⠀⠀⠀⢰⣶⣾⣿⣿⣁⠀⠀⢘⣛⣛⡃⣴⣿⣿⣿⣿⣿⣧⠀⠀⠀⢈⣛⣙⡙⠻⣿⣿⣿⣿⣟⣦⣶⣾⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣛⣟⣫⣽⣿⣿⣿⣿⣥⣄⠐⠛⠿⣿⣿⡿⣿⣿
⣿⣿⣿⣿⣇⠀⠀⠉⠁⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣷⣆⣈⣉⣉⣉⣻⣿⣿⣿⣋⣉⣁⡀⢐⣀⣈⣉⣉⣁⣀⣀⣈⣉⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡉⠉⢿⣛⣿⣋⣹⣿⣿⣿⣿⣿⣿⡿⢡⡀⠀⠀⠈⠙⠛⠚⠉
⠿⠛⠉⠉⠁⠀⠠⢀⣠⣤⣤⣀⠀⠀⢈⣉⣉⣉⣛⣿⣿⡿⠉⠉⣉⣉⣉⣉⣉⣉⣉⣭⣩⡁⢈⣉⣉⣉⣉⣉⣉⣉⣉⣉⡙⠁⢘⣻⣿⣿⣿⡟⠋⠉⣉⠁⠀⢯⣽⣯⣭⣽⣿⡏⣽⣿⣿⠏⠀⠈⠉⣾⡄⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣦⣈⣉⣉⣉⣉⣉⣉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠘⢿⡆⠈⠉⠉⠉⢩⠕⠒⠀⠀⠀⠀⠈⠹⢿⣿⣭⣹⣛⣭⣭⣤⣤⣴⠄⡴⠊⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⢏⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠁⣤⡄⡰⢋⣁⠀⠀⠀⠀⠀⠀⠀⢠⡜⣿⣿⣿⢛⣛⡛⠛⠉⠀⠀⠀⠀⣀⣤⣤⣴⣶
⢀⣾⠿⠿⣿⣿⣿⣿⣿⣟⣛⣋⣉⠉⣿⠀⢻⣿⣿⣏⣸⣿⠇⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⣀⣀⠀⢀⣀⡀⠀⣀⣀⣰⣧⣤⢀⣈⡀⠀⣸⣿⣀⣀⣀⣀⡀⠀⣀⣈⠻⣿⣿⣷⣿⣿⣿⡆⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿
⠀⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢸⣯⣀⡈⠛⠀⣼⣏⣀⣈⣿⡄⢠⡟⠁⠈⠿⠷⠏⣿⡀⠀⢸⡇⠀⠀⣿⡏⠀⣙⣤⣼⣷⣶⣾⣿⣗⠀⠈⢿⡀⠀⡼⠁⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣼⣻⠿⣿⣿⣿⣿⣿
⡦⠈⠀⡟⠿⠿⣻⣿⣏⠛⠋⠀⠀⠀⠀⠀⢀⡈⠉⠻⣷⢄⢿⡏⠉⠉⢉⠁⠸⣇⠀⠀⢀⡀⠀⣿⡀⠀⢸⡇⠀⠀⣺⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠘⣷⣰⠃⠀⠀⠀⠀⠛⠿⠟⡻⢻⣿⣿⣇⠘⣦⡈⠛⠛⠻⠟
⢋⠔⢂⠀⠀⠀⡛⠟⠉⠀⣀⣀⣠⡴⠀⠀⠘⠓⠶⠖⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠲⠶⠚⠀⠀⠘⠳⠖⠛⠓⠂⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣫⡄⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⣽⠿⢿⣿⣿⡿⣶⠿⡧⠀⠀⠀⠀
⠀⠀⠸⢿⣿⣿⣷⣶⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠿⠿⣿⣿⡿⠟⠛⠿⠦⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠉⠙⠛⠤⠺⠀⠀⠀⠀
⢸⡇⣿⣿⣿⢸⡇⣿⣿⡇⡏⡟⣿⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣯⣿⣿⢿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⢻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⡇⣿
⠘⡇⢟⢿⢻⠸⡇⠙⢘⣤⠇⢾⠪⣿⣿⣿⣿⣛⣞⣿⣿⣿⣿⡿⠿⣯⣽⣿⢠⣽⣻⣿⣧⡈⡽⢀⣿⣿⡿⢽⣿⣿⡃⡏⡼⣿⣿⠇⣧⣿⣿⣿⣿⢿⣽⣾⣿⣞⣟⣿⢿⣿⣿⢟⣽⢛⣿⡟⢿⣻⡿⣧⣓⣢⣾⣿⢸⣟
⠄⣧⣿⠩⣤⡀⠇⡍⠿⡩⡤⢥⢬⣿⢽⡏⢽⡇⠬⣉⣛⡿⢿⣯⣔⣿⡋⠛⠟⣿⢹⣿⣹⠹⡯⢵⣗⣛⣿⣿⡷⣿⣗⠁⣻⣾⣯⢘⡁⣿⣿⢡⣯⣝⢱⠿⣿⠞⢩⡿⠘⣈⠬⠴⣶⠋⣿⠅⣡⣤⠴⣿⡿⣶⡶⢀⣈⡿
⣸⣿⡋⠛⣟⣓⣀⠲⠰⠶⡾⣧⣿⣷⣿⡃⢠⣴⡄⣽⣿⠶⣾⣟⢿⢿⠕⡟⣡⣭⣈⡛⣿⣆⣭⠀⢃⣦⣥⣍⡁⠘⠳⠿⠏⢻⣷⢉⡉⠈⠛⢀⣙⡛⠃⠹⢿⠀⠀⠁⣥⣀⣼⣾⣿⣿⣿⡶⢔⠛⠛⠂⠘⢿⣽⣶⣤⡐
⣌⠿⠃⠀⠈⠉⠉⣉⠷⣄⣀⣽⣿⣯⠈⣥⣍⡂⠉⢠⣼⢶⣖⡁⠘⠟⢁⣹⣯⣁⣦⡅⠙⠁⠥⣤⠴⣿⡭⡛⠁⣠⣿⣶⡄⠀⠁⢺⣍⠀⢀⡸⠿⢿⡗⠀⠀⢻⣦⣀⣰⠚⠋⣻⣿⣿⣽⡍⡰⠾⠿⠦⡶⠌⠉⢬⣽⣥
⠙⢀⠀⠀⠘⠻⢿⣿⡷⢨⠙⣸⣿⣿⣦⡉⠉⣹⠷⠀⠰⠿⠛⠁⣠⣤⡌⣻⣿⡯⣥⣀⠀⣠⣧⣤⠀⠉⠧⣤⡤⠹⣿⠿⠄⠀⠀⣠⣾⣿⣿⡁⠘⢿⡍⠀⢀⣍⠉⠹⠉⠀⠀⢺⣷⡆⠀⢱⣄⠁⣀⣨⣤⣀⣠⠙⠛⢉
⣶⣤⡀⠀⠀⠀⠀⣩⣶⣿⣯⢻⣿⠿⠋⠙⣿⣶⣿⡆⠀⠀⠀⠀⡆⣈⠁⠹⢿⢿⣿⣿⡿⠋⠉⠀⠀⢠⣇⣈⡀⠀⠙⣿⣤⣤⡤⠃⣿⣿⣿⡁⠀⠈⠁⠶⢢⣋⣤⣴⣆⣀⣴⣾⣿⣷⣤⡀⠀⠉⠤⡀⠉⠙⠉⠀⠀⠀
⠉⠻⠿⠂⠀⠀⠀⣈⣉⡙⡛⢹⣿⣽⡿⠀⠙⠿⡛⠛⠀⠀⣴⣾⡿⠟⠾⣿⡏⢮⣿⣿⣶⠚⣧⣠⣴⣿⡿⠛⠃⠀⢤⡌⠩⠋⠀⠀⢺⣿⣮⠀⠀⠀⠀⠀⣼⢿⣿⣟⠻⢿⡿⢛⣿⣿⣿⣿⠂⠀⠈⣤⣴⣶⣆⣀⣀⠘
⠀⠀⠀⢀⣀⣴⣾⣿⣿⡿⢦⣄⠛⡟⠛⠛⠟⣻⣛⡛⠛⠂⠀⠐⠚⡟⣛⣛⠛⠛⢻⣿⡟⠃⢾⣿⣿⣿⣿⣿⠛⢛⣿⣷⣶⣶⡘⠛⠛⠛⠛⠛⣷⣦⠶⢿⣿⣿⣿⠿⠛⠛⢣⣬⣿⣿⣿⣥⠄⠀⠈⠉⠛⠿⠛⠛⠋⠀
⠀⠀⠀⠐⢿⣿⣿⣿⣿⠟⠁⠙⢠⣤⣼⡿⠛⠿⢿⣿⣿⡇⢘⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⡗⠘⢿⣿⣿⣿⡛⢻⡿⠿⠿⠟⡋⠙⠛⠛⠛⠛⠛⠛⡳⣤⠀⠛⠿⠛⠛⠛⠛⠃⢻⣿⣷⣶⠄⠀⠈⣙⠂⠀⠀⠈⠀⠀⠀
⣦⣤⢾⣇⣙⣶⣿⣏⣀⣀⡤⠀⠈⣻⡿⡿⣿⣟⣛⡃⠙⠋⢘⣛⣛⡋⠁⠀⢠⡄⠉⡩⠏⠀⠉⣿⣿⣿⣛⣏⣘⣄⣘⣛⣛⠋⠀⠠⢘⣛⣛⣛⣛⣿⠁⣘⣛⣿⣿⣟⣃⣤⣤⣄⣙⠻⣻⢦⣤⣀⢁⣠⣴⣶⣾⣿⡇⠀
⠿⢷⡦⠔⠉⣿⣩⣭⣽⣿⡇⠀⠀⠀⠈⠙⢛⣻⣛⠁⠀⠀⢘⣛⣛⡁⠀⠀⢰⠋⢈⣀⡀⠀⠀⠙⢿⣿⣿⣛⣛⣛⣻⣛⡋⠀⠀⠀⢘⣛⣛⡋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡸⣿⣿⣽⣟⠿⣿⡟⠛⠉⠁
⠀⢘⣀⣀⠀⠙⣿⣿⣿⠏⠀⠀⠀⠀⢠⣄⣈⣛⣋⡁⠀⠀⢘⣛⣛⡃⢀⣤⣶⣿⣿⣿⣷⠀⠀⠀⢘⣛⣋⡛⠛⣿⣿⣿⣿⣶⠦⠠⢜⣛⣛⡣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣙⡝⣠⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠀⠀⠈⠾⠄⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣤⡀⢈⣉⣉⣁⣿⣿⣿⣿⣿⣿⣛⡃⢀⣀⣈⣉⣉⣁⣀⣉⣉⣛⣿⣿⣥⣴⣾⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⣭⣯⣭⣭⣽⣿⣿⣿⣶⡦⠀⠉⠙⠿⣿⡝
⣿⡿⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⣻⣿⣿⣿⣿⣷⠌⠉⣉⣉⣉⣉⣟⣉⣉⣩⣍⡁⢈⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠉⢻⣿⣿⣿⣽⣿⣿⣿⣿⡿⠀⢈⣭⣿⣭⣭⡍⢹⣿⣿⣿⣿⣿⣿⡿⠁⣀⡀⠀⠀⠀⠀
⠁⠀⠀⠀⠀⠀⠈⣠⣴⣶⣤⣀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠨⠉⠉⠉⠉⠉⠉⠉⠉⠀⢀⡌⠈⠉⠙⠿⠛⠟⠉⠉⠀⠀⠀⠈⠙⠋⠙⠍⢁⣼⣽⣟⣷⣿⣿⠇⠀⠀⣘⠗⠂⠀⠀⠀
⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣷⣄⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣆⣠⡄⠀⢀⣠⠒⠉⠉⠀⠀⠀⠀⠀⠴⣶⣭⣥⣉⣉⣬⣤⣤⣤⣴⡾⠁⠀⠀⠀⠀⠀
⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣃⣄⡀⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⣀⣀⠀⢀⣀⡀⠀⢀⣀⣀⣠⣬⢋⣈⡀⠀⣸⣿⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠙⠿⠿⠟⢟⣛⡛⠛⠋⠀⠀⢀⣠⣴⣶⣶
⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⣉⠉⠀⠀⢻⣿⣛⡏⠻⠿⣿⣏⣀⣈⣿⡄⢰⡿⠳⠶⢼⢇⣠⣿⠀⠀⢸⡇⠀⠀⣾⡏⠸⠙⠀⢸⡇⠀⢸⣿⠀⠀⠈⣿⡀⠀⡼⠁⠀⠘⢷⣶⣦⣶⣿⣿⣿⣆⠀⠀⢰⣿⣿⣿⣿⣿
⣿⢲⣦⣦⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢠⡉⠙⠻⣷⠀⢿⡏⠉⠉⢉⠁⠸⣇⠀⠀⢈⡀⠋⣿⡂⠀⢸⡇⠀⠀⢹⡇⠀⠀⣀⣼⣷⣶⣾⣿⣿⣿⠀⠘⣷⣰⠃⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣶⣮⠛⠿⢿⣿⣿
⠀⠀⠹⣿⣿⣿⣿⡿⠿⠿⠋⠀⠀⠀⠀⠀⠘⠓⠾⠟⠻⢄⠈⠛⠶⠖⠋⠀⠀⠙⠲⠶⠚⠀⠀⠘⠳⠖⠛⠓⠂⠐⠚⣓⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢹⠇⠀⠀⠀⠀⠀⠀⠈⠻⢿⠿⠿⢻⣿⣿⣇⢘⣦⠀⠉⠙
⡠⠖⠷⣷⠿⢿⠿⠿⠷⠶⠖⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⣾⣿⣿⡗⣋⠿⡇⠀⠀
⢿⣼⣿⣿⡇⢿⡄⡏⣿⡇⡿⡟⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣯⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⢸⢻⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡏⣿⢿⢫
⢸⡇⡟⡟⢿⢸⡇⠋⢙⣤⠇⣾⢪⡿⣿⡿⣿⣛⣞⣿⣾⣿⣿⢿⢿⣧⣿⣽⢤⣝⣻⣟⣯⣋⣿⠘⣿⣾⣿⣽⣿⣿⣯⢸⢨⣿⡟⡿⢨⣿⣿⣿⣻⣿⣏⣯⣿⣿⣿⡿⣿⣿⣿⣿⣿⡾⣿⣿⣻⣟⣯⢯⣟⢲⣼⣿⡿⣘
⠈⣇⣿⢋⣬⡘⠇⡽⣿⣩⡬⢬⢌⣿⣿⡯⣽⡃⠼⢙⣛⣿⣿⣿⣒⣻⡋⢻⠟⢿⢻⣿⣹⡹⡿⣭⡿⠟⣏⣿⣿⣿⣿⠘⢘⣥⣧⡃⠊⢿⣿⣗⣸⣟⡃⣽⣿⡿⠋⣿⠏⢋⠫⢣⣤⡞⢩⡿⠅⠨⡭⢾⣿⣿⣿⡯⠁⢉
⣿⣿⡛⢾⣟⡆⠀⠦⠠⣴⣼⣆⣽⣬⣾⠅⢠⣵⡀⣿⣾⣤⢼⣿⣿⣿⡣⡾⢃⣍⠁⠻⣿⣆⣝⡁⠃⣫⣉⡜⠛⠘⠿⣬⣭⢻⣿⠛⠛⠁⠻⠇⡘⠻⠗⠀⣿⡍⠀⠙⢤⣀⣈⣒⣾⣿⣾⣿⣬⠙⠷⠶⠀⠰⣿⣢⣭⣉
⢱⣶⠃⠈⠛⠟⡛⠲⢧⡀⠉⢿⣿⡋⠛⣂⣛⠋⠋⢈⣽⣤⡬⠋⠸⣶⠃⢓⣟⡉⣍⡉⠘⠂⢁⣤⡜⣿⣿⠿⠓⠀⣸⣬⡄⠀⠙⢨⣟⡀⠀⠀⣿⢿⣶⡄⠀⠱⢦⡄⠀⢍⠿⡿⣿⣿⣿⢻⡓⣤⣶⣦⣤⣤⠘⠋⢙⣿
⠦⠀⠀⠀⠴⣶⣾⡿⣆⣘⠏⢹⣭⣿⠀⠙⠿⠻⣦⠈⢩⣾⠿⠃⠀⣀⡈⢿⣿⡟⢋⠁⠀⠀⣆⣈⠀⠙⢏⣀⣀⡘⣽⣿⣏⠀⠀⠀⢉⣷⣶⠟⠀⢺⣿⠃⠀⠀⠚⠻⢿⠻⠁⠀⢸⣿⣝⠉⢣⡀⠢⠀⢈⡁⠀⠀⠸⠿
⣀⡁⠀⠀⠀⠀⠈⢋⣤⣷⡶⣾⣿⣿⠟⠦⣤⣭⣷⡀⠀⠀⠀⠀⡍⠛⠋⢻⣿⣿⣾⣿⣦⡿⠟⠉⠁⠀⡤⠉⠋⠀⠸⢷⡆⠁⢀⡰⣿⣿⣿⣿⠇⠀⠙⢠⣤⣈⠗⢀⣀⡀⠀⠀⢈⣛⣯⣀⠈⠉⠖⠋⠙⠛⢛⠛⠃⠀
⠿⣿⣆⠀⠀⠀⠀⠿⠿⣿⡧⣿⠿⠧⡤⠀⢿⣿⣿⡷⠀⠀⣠⣴⣶⣶⣤⣤⣽⣸⣿⣿⣻⣁⣄⡀⠀⣴⣿⣿⠿⠀⠀⡘⠻⢿⠟⠃⠐⣾⣟⣻⢂⠀⠀⠀⠁⣸⣾⣿⣿⠷⣦⣾⣿⣿⣿⣿⣷⣤⡀⠠⠃⣀⣠⣤⠀⠀
⠀⠈⠀⠀⠀⢀⣤⣤⣤⡔⠁⢸⣿⣿⠟⠛⠛⠛⠻⠟⠛⠀⠙⠛⠛⣿⢛⠟⠛⠛⣿⣿⣿⠃⠙⣿⣿⣿⣿⣟⠛⠛⢛⠟⣀⣁⡘⠛⠛⠛⢿⡟⢢⣄⡀⢀⣰⣿⣿⣿⣿⣿⠛⢿⡋⢽⣿⣿⣿⠟⠀⠀⠻⣿⣿⣿⣿⣶
⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣦⡀⠑⠃⠘⠛⢛⣿⣿⣿⣟⠀⢘⣛⣛⣿⣿⣿⡛⠛⠛⣻⣛⣃⢺⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠛⠻⠿⠉⠛⠻⠿⣟⡛⠛⠃⠀⠒⣶⣾⡿⠿⠶⠀⠬⣀⠀⠉⠁⡩⠉
⠀⢀⡈⣿⣿⣿⣿⡿⠟⠁⠈⠃⣶⣶⣶⣆⣙⣛⣛⡻⢿⣷⢘⣿⣿⣿⠟⠛⠃⠰⢿⣿⣿⣿⠁⣚⣿⣿⣿⣟⣠⣾⣙⣛⣛⢃⠀⠀⢘⣛⣛⣛⣛⣛⢙⣿⣛⣛⣟⣛⡃⠀⠀⠀⠈⠻⣿⣿⣿⠀⠀⢠⠚⠁⠀⣀⣀⣀
⣿⠿⢧⣙⣻⣿⣿⣤⣤⣤⠄⠀⠙⠛⣻⠿⣟⣛⣛⡁⠀⠀⢘⣛⣛⡁⠀⠂⠀⣦⠀⡠⠊⠈⠀⠸⣿⣿⣿⣟⣛⣛⣛⣛⡋⠀⠀⠀⢙⣛⣛⡋⣛⣛⣛⣛⣻⣿⣿⣿⣷⠀⠀⠀⠀⠀⠈⠛⠭⢷⣶⣦⣴⣾⣿⣿⣿⣿
⢹⡧⠐⠈⣿⣭⣭⣽⣿⣿⠀⠀⠀⠀⠀⠙⣞⣟⣋⡁⠄⠀⢘⣋⣉⡁⠀⠀⢠⠁⣀⣀⣀⠀⠀⠀⢻⣿⣿⡿⠛⠹⣿⣿⣛⣀⣀⠀⢘⣛⣛⡃⠘⣻⣿⣿⣿⣿⣿⣿⣿⡶⣠⣴⣾⣿⣷⣦⣤⡙⠿⠿⣾⣿⠛⠿⠏⠋
⣈⣀⣀⡀⠸⣿⣿⣿⡟⠁⠀⠀⠀⠀⣤⣤⣌⣉⣉⣁⣀⡀⢈⣉⣉⣁⣀⣴⣾⣿⣿⣿⣿⡄⢀⣀⣈⣉⣻⣁⣀⣠⣿⣿⣿⣿⣧⣓⣛⣛⣉⡁⠀⢿⣛⡿⠿⢿⣟⣋⣁⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠙⠀⢀⣀⣤
⣿⣿⣿⠀⠀⠈⠽⠏⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣯⡅⠈⠉⣉⣉⣿⣿⣿⣿⣿⣿⣛⡋⢈⣩⣉⣉⣉⣉⣉⣉⣉⣉⠙⠻⢿⣯⣉⣉⣉⣁⣀⣬⣭⣥⣤⣬⣭⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣂⣛⡟⠻
⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⣿⣿⣿⣿⡆⠀⠀⠉⠉⠉⠙⠋⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⢩⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⠉⠉⢻⣿⣿⡿⠿⣽⣯⣭⣩⣭⣿⣿⣿⣿⣤⡆⠀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠙⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡍⣿⣿⣿⣾⣿⣿⣿⠿⠿⠏⠀⠀⣤⠐⠋⢡⡀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀
⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⡀⠀⠀⠀⣿⠀⠀⣀⣤⣄⡀⠀⠀⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⣀⣀⠀⠀⣀⡀⠀⢀⣀⣀⣠⣤⣴⣈⡀⠀⣸⣿⣈⣛⣀⣀⡀⠀⣀⣀⠀⣿⠳⠀⠸⠀⠀⢸⣿⣿⣟⢯⣿⣷⡈⠁⠈⠀
⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣷⣤⢤⣶⣶⣾⣧⣶⣶⡟⠀⣼⣏⣀⣈⣿⡄⢠⡟⠁⠀⠘⠇⠀⣿⠀⠀⢸⡇⠀⠀⣼⡏⠀⠘⠙⢾⣇⠀⢸⣿⠀⢀⡞⢿⡁⠀⡼⠁⠀⠀⠀⠀⣀⣀⣐⠋⠛⠿⠟⠿⠟⠉⣀⣀⠀⠀
⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠻⣿⣿⣿⣿⡿⣷⣦⣿⡍⠉⠉⢉⠁⢸⣧⠰⢤⣤⡀⠀⣿⡀⠀⢸⡇⠀⠀⣿⡇⠀⠀⢠⣽⡏⠀⢸⣿⢠⣞⠁⠘⣧⣰⠃⠀⠀⠀⠀⠀⠉⠙⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣤
⡿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⢹⣿⣿⢿⡟⠛⠿⠉⠛⠶⠖⠋⠀⠀⠙⠳⠶⠛⠓⠙⠛⠳⠖⠛⠓⠂⠐⠚⠓⠂⢀⠟⠚⠓⠂⠈⠻⠶⠋⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⣨⣵⣶⣄⠀⠀⣾⣿
⣴⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣀⠀⠀⠀⠀⠉⠁⠙⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⣤⣤⣤⣤⣿⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣷⣶⣍⠛
⣿⢸⣿⣿⡇⣿⠈⡏⣿⡏⡿⠿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣯⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⢸⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡇⡏
⢹⡞⡻⡟⢿⢸⡆⠇⢙⣥⠇⣶⡮⣿⣿⠿⣿⣛⣛⣿⣾⣿⣿⢿⢿⣗⣿⣽⢤⣝⣻⣟⣯⡃⡿⠚⣿⣾⣿⢹⣿⣿⣯⢼⠧⣿⣿⢸⢨⣽⣿⣿⣻⣿⣟⣧⣿⣷⣻⢿⣿⣿⣿⣿⣟⡿⢿⣿⣻⣿⣿⠭⣽⠖⣵⣺⣽⢃
⠘⣇⡿⢃⣬⠘⠏⠸⣟⣩⡬⣬⢌⣿⣻⡟⣿⡃⡼⢙⣟⣿⣿⣗⣒⣻⡋⠿⠽⢿⢫⣿⣻⡻⡿⣭⣿⠟⣟⣾⣿⣽⣿⠝⢘⣢⣧⡘⠘⢻⣿⡷⣹⣟⡛⢹⣿⣷⠟⣟⣏⠋⠉⣃⣤⡴⠻⡿⠇⢨⣭⢼⣿⣿⣿⣯⠉⠈
⣿⠟⠻⣾⠟⡄⠀⠤⢨⣥⣼⣆⣽⣬⣾⠃⢈⣥⡀⣿⣵⣥⣼⣿⢿⣿⡃⡾⢂⣝⠃⠻⢾⡆⣝⡉⠋⣉⣉⡘⠛⠘⢿⣥⣾⠻⣿⡿⠛⠋⠻⠏⠙⠿⡿⠈⢾⣿⠀⠈⢣⣀⠉⢓⣴⣿⣶⣿⣦⡭⠯⡾⠍⠑⣾⣗⣭⣉
⢿⣿⠀⠝⠻⠻⠛⢲⣦⠈⠙⣿⣿⠏⠟⠄⠺⠍⠃⢈⣹⣤⡜⠋⠹⣿⠃⠳⣛⠋⣍⡋⠸⠇⢋⣀⣐⣿⣿⡻⠗⠀⢐⣄⡄⠀⠙⠁⣟⡂⠀⠀⣹⣶⣶⣤⠀⠙⣶⣄⠀⠉⠩⣿⡿⣿⣿⡿⣿⢯⣤⣴⣀⣀⡈⠛⢋⣻
⠌⠀⠀⠀⣶⣦⣽⣷⣆⢲⡎⣿⣭⣯⠈⠻⡿⢲⣦⠈⢫⣿⡿⠧⠀⠀⠐⣿⣿⡿⠛⠁⠀⠀⣄⣉⠁⠘⠟⣈⣀⡐⡻⣿⣿⠀⠀⠀⠉⢻⣶⣶⠋⢹⣿⡓⠀⠀⠠⠾⢶⡾⠿⠀⠐⣷⣛⡓⠻⡈⠁⠄⠉⠉⠁⠀⠀⢿
⡐⠂⠀⠀⠀⠈⠙⢓⣠⣧⢤⣾⣿⣿⡷⣄⣤⣿⣶⡀⠀⠀⠀⠀⡘⡻⠇⢿⣿⣿⣽⣷⣤⡸⠿⠛⠁⠀⢨⠉⠟⠁⠸⣿⡿⠐⠀⠀⣾⣿⣿⣿⡦⠀⠙⠇⣤⡠⣲⠄⠀⢁⠀⠀⠀⠙⢿⣇⠀⠉⠣⠖⠛⠻⠷⠾⠿⠀
⣿⣷⡄⠀⠀⠀⠰⢿⢿⣿⣛⣻⣿⣉⣀⠸⣿⣿⣿⡧⠀⠀⣀⣤⣷⣶⣀⣀⣹⠻⣿⣿⣿⠃⣄⠀⠀⣠⣷⣶⣿⠄⠀⠈⠿⢿⣿⠟⠀⣭⣽⡿⡗⠀⠀⠀⠉⠁⣵⣾⣿⣿⣧⢀⣴⣾⣿⣿⣷⣆⣀⠀⠱⠂⢀⣀⡀⠀
⠈⠉⠁⠀⠀⠀⢀⡀⢠⠍⠂⢿⣶⣾⠟⠛⠛⠛⢟⡛⠛⠁⠙⠻⠛⣛⠛⡻⠟⠛⣿⣿⣿⡏⠚⢟⣿⣿⣿⡛⠛⠛⠛⡻⠂⠊⣙⠛⠛⠛⠿⣿⠃⣀⠀⠀⠐⢻⣛⢛⣛⣛⡃⠻⣿⠟⣻⣿⣿⣿⡿⠃⠀⣾⣿⣿⣯⣦
⠀⠀⠀⢀⣤⣾⣿⣿⣷⣦⡄⠈⠻⠋⠙⠛⢛⣿⣿⣿⡛⠀⢘⣛⣛⣟⣿⣿⡟⠛⠛⢻⡟⠃⢸⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠿⢟⡗⠻⣿⣿⣿⣿⠿⠃⠀⠉⣠⣬⣿⣿⣿⣧⠄⡀⠉⠉⠛⠛⢛
⠀⠀⢭⣿⣿⣿⣿⣿⡷⠊⠻⢀⣀⣀⣠⡀⢙⣛⣻⣿⣿⣦⣘⣿⣿⣿⣿⠿⠇⢀⣴⣾⣷⣾⠖⢹⣿⣿⣿⣿⣋⣸⣛⣛⣟⠋⠀⠀⢘⣛⣛⣛⣛⣛⢠⣾⣟⣛⣛⣻⡃⠁⠀⠀⠀⠰⣿⣿⣷⣦⠀⠀⢉⠽⠂⠀⠀⠉
⣴⣶⢸⣿⣿⣿⡿⠉⠀⠀⠀⠈⣿⣿⣿⣿⣿⣛⣛⡃⠈⠁⢙⣛⣛⡋⠀⠀⠀⣄⠙⠛⡻⠍⠀⠒⣾⣿⣿⣿⣟⣛⣟⣛⡋⠀⠁⠀⢘⣛⣛⡋⣛⣛⣛⣟⣛⢛⣛⣛⣧⡄⠀⠀⠀⠀⠈⠻⠿⣿⣤⣄⡈⢀⣤⣶⣾⣿
⣎⢉⡲⠶⠶⠿⠿⣿⣿⣏⠀⠀⠀⠉⠒⢄⣙⣯⣙⡁⠀⠀⢘⣉⣉⡁⠀⠀⠀⠟⠀⠉⠀⠀⠀⠀⢻⣿⣿⣿⠟⠙⣋⣛⣙⡀⠀⠀⢘⣛⣛⡃⠙⣋⣙⣿⣿⣿⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⡿
⠟⠁⠀⠐⣶⣿⣿⣿⡿⠁⠀⠀⠀⠀⢀⣀⣉⣉⣉⣁⣀⡀⢈⣉⣉⣁⣀⣀⣨⣴⣶⣾⣿⡇⢀⣀⣈⣙⣻⣏⣀⣀⣿⣿⣿⣿⣶⣠⣼⣿⣍⡁⠀⢉⣿⣿⣿⣿⣿⣿⣟⣛⡁⢀⣴⣾⣿⣿⣿⣶⣦⣄⠈⠙⠻⠀⠀⠀
⣴⣶⠖⠀⠙⠿⣿⠟⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣯⣍⣉⡁⠈⠉⣉⣉⣽⣿⣿⣿⣿⣿⣿⣷⢈⣉⣉⣉⣉⣉⣉⣉⣙⣿⠿⢿⣿⣿⣉⣉⣉⡁⠀⠈⣉⠁⠉⢉⣉⣉⣉⣉⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⣀⣤
⣿⣿⣀⠀⠀⠈⠝⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠉⠉⠻⢿⡿⠛⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠋⠉⠉⣭⣥⣤⣤⣤⣶⣶⣾⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⠀⠀
⠛⠛⠋⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠉⠙⠻⢿⣻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⡇⣤⠀⢻⣿⣿⣿⣿⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣤⣄⡀⠀⠀⣀⣤⣄⡀⠀⠀⢠⣠⣤⣀⡀⢀⣈⠀⠀⣀⡀⠀⢀⣀⣀⣠⣤⢀⣈⡀⠈⣻⣿⣿⣿⣙⣟⡿⠿⣟⣛⠁⣿⠀⠀⠹⠟⠋⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⡀⠛⠀⣼⣏⣀⣈⣿⡄⢠⡟⠁⠀⠘⠇⠀⣿⠀⠀⢸⡇⠀⠀⢸⡏⠀⠘⢠⣼⡇⠀⢸⣿⠛⠁⠘⣿⡀⠀⡼⢁⠀⠀⠀⠀⣦⠀⠀⠁⠀⠀⣽⣿⡿⠿⢻⣿⣯⠛
⠀⠀⢀⢀⣼⣿⣿⣿⣿⣿⣶⣤⢤⣶⣶⣶⣶⣾⣿⣿⣷⠀⢿⡍⠉⠉⢉⠁⢸⣇⠀⠀⢀⡀⠀⣿⡀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⢹⣿⡀⢸⣿⠀⣀⠀⣸⣿⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠩⡄⠀⠻⠟⠛⠀
⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠹⣿⣿⣿⣿⣿⡿⢻⣦⡌⠛⠶⠖⠋⠀⠀⠙⠳⠶⠛⢂⢀⣼⣷⡖⠛⠓⠂⠐⠛⠓⠂⠀⠐⣺⡿⠋⠈⠻⠶⠋⡴⠋⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⢿⣶⣦⣤⣤⣤⣤⣤⣶⣶
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⣿⣿⣿⣿⡇⠨⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠈⠻⠤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠈⠉⠛⠛⠻⢛⣛⠛⢻
⣿⢹⣾⣿⡇⣿⠙⣿⣿⡟⣿⢿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣯⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡟⣽
⢿⢸⠻⠿⣧⢻⡀⠇⢛⣥⠏⡶⣬⣿⣿⢿⣻⣛⣟⣿⣾⣿⣻⢿⣽⣟⣿⣽⢬⣟⣻⣟⣯⡇⣿⠿⣿⣾⣿⣹⣿⣿⣯⣿⡇⣿⣿⢸⢃⣽⣿⣿⣻⣿⣟⣣⣿⣯⣻⢿⢿⣿⣿⣿⣿⣿⢿⣿⣻⣿⣿⡽⣽⠖⣤⣟⣷⡏
⢸⠒⡟⣁⣌⢸⠘⢺⣻⣫⣌⣭⣞⣿⣿⡛⣿⡂⡼⢙⣟⣿⣿⣗⣒⣻⡋⠿⠽⣿⠫⣿⢛⡻⡷⣭⣿⠟⣟⣶⣿⣹⣿⠟⠃⢫⣯⡘⠐⢻⣿⡿⣹⣟⡛⢹⣿⣷⡿⢻⣟⠹⡙⢋⣤⣤⠛⣿⢧⢈⣭⡤⣿⣿⣿⣿⡅⠁
⣿⠟⢷⣿⠯⠀⠠⡄⢬⣤⣼⣎⣻⣡⣿⡁⢉⣥⡀⣿⣽⣅⣹⡿⠿⣿⠃⡾⠆⣜⠋⠿⣿⡦⣝⡋⠋⢩⣉⣘⠛⠈⢿⣡⣚⠻⣿⡿⠺⠂⠹⠯⠁⠿⡿⠎⢺⣿⠂⠈⠣⣄⠉⠒⣠⣿⣷⣾⣧⣭⠯⢴⠯⠉⢺⣟⣂⣉
⣭⡿⠀⠛⠿⠖⠲⢲⣄⠈⠹⣿⡿⠟⠷⠐⠾⠝⠇⠉⣹⣬⡚⠟⢹⣿⠇⠳⡞⠋⡝⡋⠹⠏⢋⣁⣐⣿⣿⣻⠟⠀⢰⣄⣀⠀⠙⠇⣿⡂⠀⠀⢩⣷⣶⣦⡀⠉⢲⣄⠀⠈⠉⣿⣿⡿⣿⣿⣿⡯⣡⣴⣆⣀⣀⠙⠛⣙
⡈⠀⠀⢀⣦⣬⣵⣶⡌⠶⡞⣛⣻⡆⠘⢿⡿⢲⡄⠈⢛⣽⣷⠧⠀⠈⠐⣾⣿⡷⠛⠋⠀⠀⢬⣙⠁⠘⠷⠈⣀⠀⢿⣿⡿⠃⠀⠀⠉⠻⣶⣶⠏⠁⣿⣿⠀⠀⠀⠼⣷⣶⢾⠎⠀⢰⣿⡿⠟⢇⠉⢍⠉⠉⠉⠀⠀⠸
⠰⠀⠀⠀⠀⠉⠻⢂⣰⣁⣤⣿⣿⣷⣦⢀⣀⣲⣦⠀⠈⠉⠀⠀⡲⢿⠇⢾⣿⣿⣸⣷⣄⡰⠿⠛⠃⠀⢀⠙⠟⠋⠹⣿⣿⠑⠀⠀⣰⣿⣿⣿⣦⠄⠙⠿⣀⡀⠐⡤⠀⠈⠀⠀⠀⠘⠿⢿⠀⠈⠛⠤⠖⠻⠷⠶⠶⠖
⣷⡤⠀⠀⠀⠀⢰⣿⣿⣿⠹⣿⣿⣋⡀⢹⣿⣟⣿⡇⠀⠀⠀⣠⣥⣤⠀⣈⣻⠻⣿⣿⣿⠇⣀⠀⠀⢀⣼⣶⣶⡂⠀⠈⢷⣶⣶⡶⠀⢼⣿⠿⠯⠄⠀⠀⠈⠉⢙⣤⣶⣾⣿⡄⠀⣠⣶⣿⣿⣦⡄⠀⠀⠲⠀⠀⣁⡀
⠙⠛⠁⠀⠀⠀⠀⠈⣘⡿⠙⣯⣭⣿⠛⠛⠛⠻⣛⠛⠛⠁⠻⣿⠟⡛⠻⣿⡟⠛⣿⣛⣻⡟⠸⢿⢛⣿⣟⠛⠛⠛⠛⢳⠆⠐⠛⠛⠛⠛⢿⣿⡟⠀⠀⠀⠐⠛⡿⠛⠿⠟⠃⠀⣾⣿⡿⢿⣿⣿⣿⣿⠇⢰⣾⣿⣿⣧
⠀⠀⠀⢀⣤⣶⣶⣦⣇⣀⠀⠙⣻⠋⠙⠛⢻⣿⣟⣛⠛⠀⢘⣛⣛⡟⣛⣛⡛⠛⠛⢿⡿⠃⢘⣿⣿⣿⣿⣿⣿⡟⢛⣿⣿⣿⣿⣟⠛⠛⠛⠛⠛⠸⣿⣷⢿⣿⣿⣿⣿⣿⣿⠇⠘⠋⢀⠘⣿⣿⣿⣍⡀⠀⠙⠙⠿⠿
⠀⢠⣴⣿⣿⣿⣟⣛⣟⢿⡆⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣄⢘⣿⣿⣿⣿⣿⡇⠀⣠⣤⣦⣤⣌⢋⣿⣿⣿⣿⣿⣁⣻⣿⣿⠟⠉⠉⢘⣛⣛⣛⣛⣓⠀⣸⣟⣛⣛⣛⣫⠕⠀⠀⠀⠀⠩⣿⣿⣿⣍⡉⠁⠓⣶⠀⠀⠀
⣠⡸⣿⣿⣿⣿⠿⠋⠀⠀⠈⢻⣿⣿⣿⣧⣛⣛⣟⡉⠻⠋⢘⣛⣛⡛⠉⠀⠀⠈⠻⠿⢿⣟⡁⠬⣽⣿⣿⣿⣿⣟⣟⣛⡋⠀⠀⠀⢘⣟⣛⡋⣛⣛⣛⣛⣿⢛⣛⣛⡃⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣃⡀⠘⠀⣀⣤⣤
⠿⢤⣙⣻⣿⣯⣤⣤⣤⠄⠀⠈⠩⠭⣟⠛⢛⣋⣙⡁⠀⠀⢈⣉⣉⡁⠀⠀⠀⣿⠄⠔⠋⠀⠀⠀⢿⣿⣿⣿⡿⠋⢛⣻⣋⡀⠀⠀⢘⣛⣛⡃⠙⣛⣙⣛⣋⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠟⢺⣿⣷⣾⣿⣿⣿
⠖⠂⠉⣉⣭⣭⣽⣿⡿⠀⠀⠀⠀⠀⢀⣑⣙⣋⣉⣁⣀⡀⢈⣉⣉⣁⣀⣀⣀⣁⣀⣤⣤⡄⢀⣀⣈⣻⣿⣿⣃⣀⣸⣿⣿⣯⣄⣤⣈⣭⣍⡁⠀⢉⣉⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⢀⠀⢀⣀⣀⣀⠈⠙⠛⠻⢿⠉⠙
⣀⣀⠀⠻⣿⣿⣿⡟⠀⠀⠀⠀⠀⢠⣬⣭⣭⣭⣉⣉⣉⡁⠈⠉⣭⣭⣭⣿⣿⣿⣿⣿⣿⣧⣈⣉⣉⣉⣉⣉⣉⣉⣹⣿⣿⣿⣿⣿⣍⣉⣉⡁⠀⠈⣿⠋⠛⢿⣿⣿⣋⣉⡁⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣶⣄⠈⠀⠀
⣿⡏⠀⠀⢉⢽⡏⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣯⡁⠀⠀⠀⠉⠙⣿⣿⣿⣿⠿⠛⠋⠉⠉⠍⠉⠉⠉⠉⠉⠉⠀⠀⠉⠙⢻⠟⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⣉⣁⡀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⣿⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠻⠿⢿⣿⣿⣿⣷⠆⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⡟⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⠀⠀⣀⣬⣕⣞⠁⠀⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⣀⣠⠄⠀⣀⡀⠀⢀⣀⣀⣠⣤⢀⣈⡀⢰⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠸⢿⣿⣿⣿⣿⠻⠿⠛⠛⠟⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⠀⠛⠀⣼⣏⣀⣈⣿⡄⢠⡿⠁⠀⠸⠇⠀⣿⠀⠀⢸⡇⠀⠀⢸⡏⠀⠘⠀⢸⡇⠀⢸⣿⣿⣿⣿⣿⡝⠟⣿⠛⠛⠋⠀⠀⠀⠀⠀⠼⠟⠁⠈⠙⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠙⠛⣷⠀⢿⡍⠉⠉⢉⡅⠸⣇⠀⠀⢀⡀⠀⣿⡀⠀⢸⡇⠀⠀⣼⡇⠀⠀⠀⢸⣷⠀⢸⣿⠀⣛⠃⠘⣷⣸⠃⠀⠀⣀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⣻⣿⣿⣿⠿
⠀⢀⣀⣴⣿⣿⣿⣿⣷⣤⡤⣤⣶⣶⣶⣶⣾⣷⣶⣾⠏⠀⠈⠛⠶⠖⠋⠀⠀⠙⠳⣶⣛⠀⠀⠘⠳⠖⠛⠓⠂⠐⠛⠓⠂⠀⠐⠚⠓⣿⡊⠻⠶⠋⠀⠀⢹⡟⠊⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣾⠟⠉⠁⠀⢾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣹⣿⣿⣿⣿⣿⣿⡟⠸⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠄⠺⠟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⠀⠿⢶⡿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣶⣤⣄⣀⣈⣀⣀⣀
⣧⣿⣿⣿⡘⣿⠻⣿⣿⣿⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⢹⣿⣽
⣿⢸⢿⠿⡇⣿⠀⠏⢓⣇⠿⡞⡼⣿⣿⢿⣻⣛⡿⣿⣿⣿⡿⢿⢿⣟⣿⣽⢮⣟⣻⣟⣿⡇⣿⡿⣿⣾⣿⣹⣿⣿⣯⡇⡇⢿⣿⢿⡇⣽⣿⣿⣻⣿⣟⣫⣿⣿⣺⠿⢟⣿⣿⣿⣿⣿⢿⣿⣿⣿⣻⡬⣽⡖⢮⣿⢷⣿
⢻⢸⡟⣀⣅⢸⢘⢾⣻⣫⣘⣭⣞⣿⣿⡛⣿⣀⣽⢙⢟⣿⣿⣓⣐⣛⡟⢿⠿⣿⠫⣿⢻⡻⣷⣏⣿⠟⣟⣾⣿⣹⣿⡇⠃⢺⣿⣜⠃⢹⣿⣿⢛⣟⡚⢙⣿⣿⣷⢻⣿⠛⡙⠝⣤⣤⡟⢿⡿⠀⣭⡥⢿⣿⣿⣿⣭⠁
⣿⠟⣷⣿⢧⠀⠠⡀⣬⣬⣾⣎⣻⣥⣿⡁⢉⣧⠀⣿⣯⣅⢽⠿⢿⡿⠣⡾⠆⣚⠋⠿⢿⡦⣝⡋⠋⢩⣙⣘⠻⠊⢿⣥⣖⡛⣿⡿⠚⠓⠹⢿⠉⠿⢿⠖⠙⣿⡇⠈⠛⣄⡈⠙⡂⢾⣿⣾⣷⣬⡭⠥⡾⠍⠱⣿⣓⣉
⣭⡏⠘⠻⠷⠖⠶⣴⣄⠉⢾⣿⣿⠻⠷⠐⠮⠙⠇⠙⣳⣈⡚⠏⢹⣿⠇⠳⠞⠋⡝⠋⠹⠏⢛⣁⣀⣿⣿⣿⠟⠂⢀⣆⣉⠀⠘⠋⢸⡃⠀⠀⢨⣧⣶⣦⡄⠈⠳⣤⡀⠈⡉⢻⣷⣿⢿⣿⣿⢿⢧⣤⣴⣀⣀⣈⠟⢋
⠋⠀⠀⢠⣤⣬⡶⣦⡩⣶⢚⣛⣳⡆⠸⢿⡷⢦⡄⠘⢛⣽⣷⡆⠀⠉⠰⣾⣿⡷⠿⠛⠀⠀⠨⠙⠉⠘⠿⠂⡀⠀⢞⣿⣿⠇⠀⠀⠈⠛⢧⣤⡾⠉⢯⣿⠇⠀⠀⢠⣵⣶⣶⣶⠀⠀⣿⣿⡿⠿⡈⠩⡉⠉⠉⠁⠀⠀
⠄⠀⠀⠀⠉⠙⠿⠎⣱⣉⣨⣿⣿⣦⣄⢀⣁⣲⣥⠀⠈⠉⠁⠀⠶⢶⠗⣿⣟⣿⣺⣧⣄⡰⣿⠟⠃⠀⢀⡚⡻⠛⠈⢿⣭⠏⠄⠀⢀⣾⣿⣿⣶⡄⠈⠻⢁⣀⠀⢧⠀⠀⠃⠀⠀⠀⠙⠿⣿⠀⠛⠣⡴⠖⠿⢷⡶⠶
⣦⡀⠀⠀⠀⠀⣴⣿⣿⡏⣿⣿⣟⠃⠈⢿⣿⣿⣿⡅⠀⠀⠀⢨⣃⣤⠀⠈⠻⢿⣿⣿⣿⠟⣀⠀⠀⢀⣸⣵⣶⣆⠀⠈⣹⣶⣶⣶⠎⠙⣻⣿⣿⡥⠀⠀⠈⠛⠁⣋⣤⣶⣾⣧⡀⠀⣠⣴⣿⣷⣦⡀⠀⢒⠄⠀⠐⢀
⠛⠛⠀⠀⠀⠀⠈⠉⢻⡋⢻⣿⣿⣿⠛⠛⠻⢟⠛⠛⠛⠃⢿⣿⡟⠛⠿⣿⣟⠻⣿⣿⣿⣗⠸⣿⠟⣛⡻⠛⠛⠛⠛⠳⡄⠀⠛⠛⠛⠛⢻⣿⣷⠀⠀⠀⠘⠛⢻⡿⠿⠿⠛⠉⢠⣾⣿⡿⢿⣿⣿⣿⣶⡍⢰⣶⣿⣿
⠀⠀⠀⢀⣤⣤⣤⣀⣋⡀⠈⠿⡿⠋⠙⠻⢛⣛⣛⡛⠛⠀⢘⣛⣛⡿⠛⠛⠛⠛⠻⣿⡿⠃⠘⣻⣿⣿⣿⣿⣿⡛⠛⣻⣿⣷⣾⣿⠛⠛⠛⠛⠛⠀⣿⣶⣿⣿⣿⣿⣿⣿⣷⣾⠆⠙⠿⠁⠲⣿⣿⣿⡟⠀⠁⠛⠻⢿
⣀⣀⣶⣿⣿⡿⠿⠿⣿⣧⠀⠀⠁⠀⠀⠀⢹⣿⣿⣿⣦⡀⢘⣛⣿⣿⣿⣿⣇⠀⠀⣈⣅⣀⣄⠟⣿⣿⣿⣿⣿⣟⣸⣿⣿⣿⠟⠛⢙⣛⣛⣛⣛⣓⠀⣙⣛⣛⣛⣛⡛⢟⠭⠉⠀⠀⠀⢚⣿⣿⣟⣛⠛⠃⠒⢤⡀⠀
⠲⣿⣿⣿⣿⣿⡿⠋⠀⠙⢰⣶⣶⣶⣖⣀⣛⣻⣛⡛⠻⠟⢘⣻⣿⡿⠛⠉⠁⠐⢿⣿⣿⣿⡏⢉⣻⣿⣿⣿⣟⣿⣿⣟⡋⠀⠀⠀⢘⣛⣛⡋⣛⣛⣛⣛⣛⣿⣛⣛⡓⠉⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡇⠀⠰⠁⠀⣀
⡛⠿⣿⣿⣿⣃⣀⣀⣀⠀⠀⢻⣻⡛⡿⠿⣛⣛⣛⡁⠀⠀⢘⣛⣋⡁⠀⠀⠀⢶⡀⢀⠜⠋⠁⠀⢽⣿⣿⣿⣟⡟⢛⣛⣉⡀⠀⠀⢘⣛⣛⡃⠙⣛⣛⣛⡛⢈⣙⣿⣷⣶⣾⡆⠀⠀⠀⠀⠀⠈⠉⡫⢽⣿⣶⣴⣿⣿
⠝⠻⠿⠛⣛⡻⣿⣿⡧⠀⠀⠀⠀⠉⢑⣄⣹⣏⣉⣁⣀⡀⢈⣉⣉⣁⣀⣀⣀⣎⣀⣀⣀⡀⢀⣀⣈⣿⣿⣿⣿⣁⣈⣭⣯⣅⣠⣤⣈⣉⣉⡁⠀⢉⣉⣉⣶⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⢿⣿⡟
⠀⠀⢲⣾⣿⣿⣿⠟⠀⠀⠀⠀⠀⣀⣈⣉⣉⣩⣉⣉⣉⡁⠈⠩⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⢈⣉⣉⣉⣉⣿⣏⣹⣉⣿⣿⣿⣿⣿⣍⣉⣉⡁⠀⠈⢉⣿⠿⢿⣿⣿⣿⣿⡉⠁⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⡀⠀⠃
⡶⠂⠈⠻⢟⡿⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣭⣉⠁⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⡿⠏⠉⢉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⢿⣿⡍⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⣧⡀⠀⠀⢫⠃⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣖⡀⠀⠀⠀⠀⠙⠿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡍⠀⢀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠟⠃⠀⢀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣿⠈⠉⣙⣿⣿⣻⠟⠁⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⣀⣀⠀⢀⣀⡀⠀⢀⣀⣀⣠⣤⢀⣈⡀⠀⣸⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⡀⠛⠀⣼⣏⣀⣈⣿⡄⢠⡿⠀⠀⠘⠇⠀⣿⠁⠀⢸⡇⠀⠀⢻⡏⠀⠘⠀⢸⡇⠀⢸⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠨⡿⠿⣿⣿⣿⡿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⠛⣷⠀⢿⡍⠉⠉⢉⡁⠸⣇⠀⠀⢀⡀⠀⣿⡀⠀⢸⡇⠀⠀⣸⡇⠀⠀⠀⢸⡇⠈⢸⣿⠘⣿⣿⣿⣷⣸⠟⠿⠿⠛⠛⠋⠀⠀⠀⠀⠀⠠⠼⠟⠁⠉⠛⢿⣿⣿
⠀⠀⠀⠊⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠶⠖⠋⠀⠈⠛⠶⠖⠋⠐⠀⠙⠳⠶⠛⠀⠀⠘⠳⠖⠛⠓⠂⠐⠛⠓⠂⠀⠐⠚⢳⣶⡈⠻⠶⠋⠙⠛⢹⠟⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⣀⠀⠀⠤⠀⠀⠀⢻⠿
⢀⣀⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣦⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢽⡇⠀⠀⠿⠶⠏⠀⢠⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢀⡎⠀
⡏⣿⣿⣿⢸⣯⣿⣿⣿⣽⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⢻⣿⡇⡿⣿⣿⣿⣿⣿⣿⢿⣻⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⢻⣿⣹
⣧⣿⠿⠿⡇⣿⠋⠏⢫⣏⠿⡞⣽⢿⡿⢿⣻⣛⡟⣿⣿⣿⡿⣿⣿⣟⣿⣽⢭⣟⣻⣟⣿⡋⣿⡿⣿⣾⣿⣹⣿⣿⣯⡷⣿⢿⣿⡇⡇⣽⣿⣿⣛⣿⣟⣻⣿⣿⣾⡿⣟⣿⣿⣿⣿⣿⠿⣿⣿⡿⣻⡯⣿⣗⢮⣼⣽⣿
⢿⢸⠏⣀⣁⢻⠐⣾⣛⣧⣐⡥⣺⣟⣿⣛⣿⢈⣼⢹⢟⣿⣿⣓⣐⣻⠟⢿⠿⣿⠫⣿⣿⡻⣿⣏⣿⠟⣟⣿⣿⣹⣿⡧⠃⢪⣿⣄⠃⠹⣿⣿⢞⣿⣚⠉⣿⣿⣦⠿⣿⠟⢋⠭⣠⣴⡾⠫⣿⠍⠭⡥⢾⣿⣷⣿⣮⠌
⡿⠺⣾⡿⠅⠀⢠⡀⣭⣬⣶⣑⣋⣥⣿⠀⣙⣥⠀⣿⣯⣁⢽⠿⢿⣿⠣⡾⠆⣚⠋⠻⣿⡧⣝⡃⠛⢩⣙⣙⠻⠂⠿⣍⣖⡉⢿⣿⠟⠒⠘⠿⠉⠻⠿⠶⠁⣾⣟⠀⠙⢤⡀⠹⢂⠼⣻⣷⣿⣶⣽⠵⢾⠏⠉⢿⣟⣈
⣽⠅⠙⠿⡧⠖⠦⣤⡀⠙⣾⣿⡿⠻⠶⠰⠯⠙⠏⠉⣳⣉⡺⠏⢹⣿⠇⠳⠾⠋⡝⠛⠹⠟⢙⣁⣀⣿⣿⣟⡿⠂⠀⣆⣈⡀⠈⠻⢸⣓⠀⠀⠈⣥⣶⣶⣤⠈⠑⣦⣄⠀⢉⠹⣿⣾⣿⢿⣿⣿⣿⣤⣴⣆⣀⣀⡙⠛
⠁⠀⠀⣠⣤⣴⣶⣦⢩⡔⢛⣛⣷⠀⠾⣿⡷⣧⡄⠘⢛⣽⣶⡆⠀⠉⠠⣾⣿⡶⠿⠛⠀⠀⢨⠙⠋⠘⢷⠃⠂⠀⠾⣿⣿⡗⠀⠀⠈⠙⢷⣤⣶⠍⢹⣽⡿⠀⠀⠀⣭⣶⣶⣶⡆⠉⠸⣿⣿⡷⢇⠉⢍⠉⠉⠋⠀⠀
⠀⠀⠀⠈⠉⠻⠿⠈⣖⣁⣸⣿⣿⣦⡀⣀⣰⣢⣍⠀⠈⠉⠁⠀⠴⣶⠞⣿⣟⣷⣺⣧⣄⠠⣾⠿⠓⠀⢀⡚⠻⠟⠈⢿⣯⡍⠆⠀⠀⣴⣿⣿⣷⣤⠀⠛⠇⣄⡀⡨⣄⠀⠐⠁⠀⠀⠀⠻⠿⡇⠀⠛⠦⡶⠺⠿⢶⡶
⣄⠀⠀⠀⠀⠀⣴⣿⣿⢽⣿⣿⡟⠁⠁⣿⣿⢿⣿⡀⢀⠀⠀⢨⣃⣡⠀⠈⠻⢿⣿⣿⣿⡟⢀⠀⠀⠀⣸⣥⣴⣆⠀⠀⢹⣥⣶⣶⡶⠙⣻⣿⣿⣯⠄⠀⠀⠙⠛⠰⣡⣴⣶⣿⣆⠀⠀⣀⣤⣿⣷⣄⠀⠀⢢⠄⠀⠘
⠿⠃⠀⠀⠀⠀⠙⠙⢿⠖⣾⣛⣻⣯⠙⠛⠿⡿⠛⠛⠛⠃⣾⣿⡿⠻⢿⣿⣿⠻⣿⣿⢿⣷⠺⣿⠟⡻⡿⠟⠛⠛⠛⠳⣆⠉⠹⠛⠛⠛⠻⣿⣷⡆⠀⠀⠘⠛⠛⣿⡿⠿⠿⠋⠉⢀⣾⣿⣿⣿⣿⣿⣿⣶⣯⣴⣶⣿
⠀⠀⠀⣀⣠⣀⡀⠸⠉⠀⠹⣿⡿⠋⠘⣛⢛⣛⡛⠛⠛⠁⢙⣛⣛⡿⠛⠛⠛⠛⠻⣿⣿⡏⠘⢛⣿⣿⣿⣿⣿⣛⠛⢛⣛⣷⣼⣿⠛⠛⠛⠛⠛⠃⢸⣶⣿⣟⣻⣿⣿⣿⣷⣶⣶⠎⠙⠿⠃⢰⣿⣿⣿⣿⠋⠈⠛⢿
⠀⣠⣾⣿⣿⣿⣿⣿⣿⡁⠀⠈⠁⠀⠀⠀⣿⣿⣿⣷⣦⡀⢘⣛⣻⣷⣾⣿⣇⠀⠀⢀⣉⣀⣀⢾⣿⣿⣿⣿⣿⣷⣚⣿⣿⣿⡿⠛⢛⣛⣛⣛⣛⣓⠀⣙⣛⣛⣛⣛⡿⠿⢟⣋⠉⠀⠀⠀⠒⣶⣾⣿⠿⠿⠿⠦⢤⡀
⣿⣿⣿⣿⣿⣶⡾⠊⠻⠇⣤⣤⣤⣤⢆⠀⣘⣟⣛⡛⢿⠿⣘⣻⣿⣿⠟⠋⠉⠀⣾⣿⣿⣿⣿⠛⢽⣿⣿⣿⣿⣿⣛⣟⡋⠉⠀⠈⢘⣛⣛⡋⣛⣛⣛⣛⣻⢻⣿⣛⡃⡠⠋⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣷⠀⢠⠊⠀
⣿⣿⣿⣿⠟⠉⠀⠀⠀⠁⠻⠿⣿⢿⣿⣷⣟⣛⣛⡁⠀⠀⢘⣛⣛⡁⠀⠀⠀⣤⠈⠉⣹⠟⠓⠈⢹⣿⣿⣛⣛⢻⢛⣛⣏⡀⠀⠀⢘⣛⣛⡃⠙⣛⣛⣛⡋⢾⣛⣛⣃⣠⣤⣴⡄⠀⠀⠀⠀⠀⠈⠛⢻⣟⣶⣦⣤⣶
⣭⣽⣿⣿⣶⣶⣾⣟⠁⠀⠀⠉⠙⠓⢅⣀⣹⣉⣉⣁⣀⡀⢈⣉⣉⣁⣀⣀⣀⣼⣃⣈⣁⡀⢀⣀⣈⣿⣿⣿⣿⣟⣈⣉⣉⣁⣀⣀⣈⣉⣉⡁⠀⢉⣉⡉⢀⣼⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠈⠀⠿⣿⣿⡿
⠀⢤⣴⣶⣶⣿⣿⠟⠁⠠⠀⠀⠀⠀⢈⣉⣉⣉⣉⣉⣉⡁⠈⢩⣭⣉⣉⣉⣉⣭⣽⣿⣿⣿⢈⣉⣉⣉⣙⣻⣿⣉⣍⣹⣿⣿⣿⣿⣭⣛⣛⡓⠀⠈⢉⠁⣼⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⡄⠘⢿⣿⣿⣿⠉⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣯⣍⡉⢩⠁⠀⠀⠈⠉⢉⣿⣿⣿⣿⣿⣿⣿⣯⠉⠉⠉⠉⠉⠉⠉⠉⠈⠻⠿⠿⣿⣿⣯⠉⠁⠀⠀⠀⠀⠉⠈⠉⠋⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣶⣦
⠀⠀⠈⡵⢾⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠈⢿⣿⣿⣿⠿⠋⠉⠁⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡝⡋⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠠⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿
⡶⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠛⠿⣿⣿⣿⣿⡼⠀⣀⣤⣄⡀⠀⠀⢉⣡⣤⣀⡀⣀⣀⠀⢀⣀⡀⠀⢀⣀⣀⣠⣤⢁⣈⡀⠀⣸⣿⣀⣀⣠⣤⣤⣤⣤⣶⣶⣿⣶⣾⣿⣿⣿⡟⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠸⣧⣈⡙⠛⠀⣼⣏⣀⣈⣿⡄⢠⡟⠀⠀⠙⠇⠀⣿⠰⠀⢸⡇⠀⠀⢿⡏⠀⠘⠀⢸⡇⠀⢸⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⡈⠉⠛⣷⠀⢿⡍⠉⠉⢉⡁⠸⣧⠃⠀⢀⡄⠀⣿⡀⠀⢸⡇⠀⠀⢽⡇⠀⠀⠀⢸⡇⠀⢸⣿⠙⣿⣿⣿⣿⣼⠻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⠿
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠶⠖⠋⠀⠈⠛⠶⠖⠋⠠⠀⠙⠳⠶⠚⠁⠀⠘⠳⠖⠛⠓⠂⠐⠛⠓⠂⠀⠐⠚⠓⠂⢈⠻⠶⠛⠻⠿⣿⣿⢗⣟⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣤⠿⠋⠈⠙
⠀⠆⠲⣶⠀⠶⠄⠠⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠆⠀⠀⠿⠶⠟⠋⠀⠘⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀
⢹⣿⣿⣧⢹⡏⣽⣿⣿⣹⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣾⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢽⡟⣿⣧
⡟⣿⡿⣿⡸⡇⡏⠟⣫⣿⢸⣞⣽⣿⡿⣿⣻⣛⣟⣿⣿⣿⡿⢿⢿⣟⣿⣽⢭⣟⣻⣟⣿⡛⣿⡟⣽⣾⣿⠽⣿⣿⣯⣽⢸⢹⣟⡻⢹⢸⣿⣿⣟⣿⣿⣛⣼⣿⣟⡿⢿⣻⣿⣿⣿⣿⡿⢿⣿⣿⢟⣿⢵⣷⡲⣵⣯⣿
⡇⣿⢁⣁⡁⠿⠳⣟⣟⣥⣠⡥⣺⣿⣟⢿⣟⢈⡜⢹⣻⡿⣿⣗⣐⣻⡋⢿⠽⢿⠫⣿⣟⣓⣷⣏⣽⠝⢻⣻⣿⣯⣾⣿⠘⠀⣷⣥⣘⠈⢿⣿⣿⣻⣟⡃⢛⣿⣿⡾⢻⡿⠛⡩⢥⣴⣶⡟⢹⡿⠡⢥⡤⣿⣿⣶⣶⣤
⠗⢿⣿⡿⡄⠀⢠⢩⣭⣭⣷⣙⣯⣼⣟⠈⣙⣦⠘⣿⣯⣁⢽⡿⣿⣿⡣⡼⠆⣞⠃⠻⢿⣶⢝⡋⠻⢩⣙⢁⠻⠓⠹⣯⣥⣪⠹⣿⡿⠛⠃⠻⠏⠙⠿⢷⠆⠑⣿⡏⠈⠛⣄⡈⠙⣂⠾⢻⣷⣿⣿⣽⡶⠾⠏⠙⢿⣿
⡏⢈⠿⢶⠴⠲⢦⣦⡈⢽⣿⣿⡟⢿⠆⠲⠯⠙⠊⠙⣫⣉⡺⠏⢿⣿⠇⠳⠞⠛⡑⠛⠙⠿⠙⣁⣀⣾⣿⣿⡿⠇⠀⣰⣈⡁⠀⠛⠇⣟⡂⠀⠀⢫⣢⣶⣶⣤⠈⠱⣦⡀⠀⡉⠻⣿⣴⣿⡿⣿⣿⡿⣤⣴⣆⣀⣀⡘
⠀⠀⢀⣤⣤⣶⢾⣌⣥⡞⣟⣛⣗⠐⢿⣿⡷⣧⡄⠚⢛⣽⣶⡆⠀⠉⠀⣾⣿⣶⠿⠛⠀⠀⢨⠉⠋⠀⠿⡎⠁⠀⢰⣿⣿⣿⠂⠀⠀⠉⠻⣦⣤⣾⠉⢽⣿⡯⠀⠀⠀⣹⣶⣶⣶⣶⠈⠉⢽⣿⡿⠷⡉⢍⠉⠉⠋⠁
⠀⠀⠈⠉⠛⠿⠇⢩⠊⣁⣿⣿⣧⣈⠀⡈⢰⣯⣅⠀⠙⠛⠁⠀⠴⣶⡞⣿⣿⣿⣒⣧⣄⠀⣾⡿⠟⠀⠀⡐⠿⡿⠇⠸⣿⣿⠋⠀⠀⠀⣴⣾⣿⣷⣤⡈⠛⡇⣀⠀⡠⣄⠀⠁⠋⠀⠀⠀⠙⢿⣿⠇⠛⢦⣴⡖⠿⢷
⡀⠀⠀⠀⠀⢀⣶⣿⡟⣿⣿⣿⠟⠉⢲⣿⣿⣿⣿⡀⠀⠀⠀⢸⣐⣁⠀⠈⠻⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣿⣤⣤⡀⠀⠈⣻⣤⣶⣤⡌⠻⣺⣿⣿⣿⡀⠀⠀⠛⠛⠲⢃⣠⣴⣶⣆⡀⠀⠀⠀⣈⣷⣤⣄⡀⠐⠤⠀
⠿⠀⠀⠀⠀⠙⠛⢿⡷⢿⡿⣿⣷⡖⠛⠿⣿⠿⠛⠛⠛⠃⣶⣿⣿⢿⣿⣿⣿⢻⣿⣿⣿⣧⠼⣿⡟⠛⠿⣿⠟⠛⠛⠓⢠⡉⠙⡻⠛⠛⠛⠻⣿⣿⡯⠀⠘⠛⠛⠛⣻⣿⣿⣿⡿⠛⠁⠀⣠⣾⣿⣿⡿⣿⣿⣶⣤⣴
⠀⠀⠀⠀⠀⠀⢠⠖⠁⠘⣿⣿⣿⠃⠘⠛⠛⠛⠟⠛⠛⠃⢛⣛⣛⣿⠻⠛⠛⠛⢻⣿⣿⣿⠙⠛⠛⣻⣿⣿⣿⣟⠛⠛⠻⣓⣘⣿⣿⡛⠛⠛⠛⠻⠇⢰⣿⣿⣛⠛⣛⣿⣧⣤⣤⣤⣤⡄⠛⢿⢟⠉⣽⣿⣿⣿⣿⠟
⣀⣴⣾⣿⣷⣦⣴⣶⡶⠀⠈⠛⠁⠀⠀⠁⣿⣿⣿⣧⣄⠀⢘⣛⣛⣧⣴⣶⣆⠀⠀⠉⢛⠁⠰⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣛⣛⣛⣓⠀⣘⣛⣛⣿⣛⡻⢿⣿⣿⡿⠟⠛⠃⠀⠈⢁⣤⣽⣿⣿⣿⣿⣴
⣿⣿⣿⣟⣛⣻⠿⣿⡟⢀⣀⣀⣀⣀⠀⠀⣙⣛⣻⣿⣿⣷⢙⣿⣿⣿⣿⠿⠛⠀⣠⣴⣾⣶⣶⣿⢿⣿⣿⣿⣿⣿⣿⣛⡏⠙⠛⠁⢘⣛⣛⡋⣛⣛⣛⣛⣛⢛⣿⣿⣃⠀⠀⢀⡴⠉⠁⠀⠀⠀⠀⠰⣿⣿⣿⣶⣦⠀
⣿⣿⣿⣿⠟⠁⠀⠀⡅⢿⣿⣿⣿⣷⣦⡀⣙⣛⣛⡁⠉⠀⢘⣛⣛⡋⠁⠠⠀⠀⠙⠿⠿⣿⣛⡋⢸⣿⣿⣿⣿⣿⣿⣟⣟⡀⠀⠀⢘⣛⣛⡃⠙⣛⣛⣛⡋⢘⣛⣿⡋⠀⠀⠉⢀⣀⡀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡿⣄
⣿⣿⡟⠀⣀⣀⣀⠀⠀⠈⠙⢛⣻⡿⣛⣉⣉⣉⣉⣁⣀⡀⢈⣉⣉⣁⣀⣀⣀⣹⣆⣀⣴⡋⢀⣀⣈⣹⣿⣿⣶⣶⣼⣉⣉⣉⢀⣀⣈⣉⣉⡁⠀⢉⣉⡉⠀⢈⣉⣉⣡⣴⣶⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠸⠓⣿
⠿⠿⠟⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠙⢪⣩⣏⣉⣉⣉⣉⡁⠈⢩⣉⣉⣉⣉⣉⣏⣉⣉⣉⣡⣌⣉⣉⣉⣹⣿⣿⣿⣏⣉⠹⣷⣾⣿⣟⣿⣭⣵⡦⡈⣉⠁⠀⢈⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣴⣿⣿⣷⣿⣿⠋⠀⠐⠈⠀⠀⢀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⢿⣭⣿⣿⣿⣿⣿⣏⠉⠉⠉⠉⠉⠹⠿⠉⠁⢠⣿⣿⣿⣿⣿⣯⠁⠀⠀⠈⠀⠀⠨⢿⠿⠿⣿⢿⣿⡿⠋⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠻⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣤⣀⠚⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⡿⠿⣿⣇⠀⠀⠀⠀⠀⠀⠉⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⠶⠆⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⣀⣤⣄⡐⠙⠻⣿⣿⣿⣋⡀⣀⣀⠀⠈⣀⡀⠀⢀⣀⣀⣠⣤⢀⣈⡀⠀⣹⣿⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣠⣾
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⣿⣿⡿⣿⠟⣼⣏⣀⣈⣿⡄⢠⡟⠁⠀⠘⠇⠀⣿⠀⠀⢸⡇⠀⠀⢻⡏⠀⠙⠀⢸⡇⠀⢸⣿⠀⠀⠈⢷⡀⠀⡼⡁⠀⢀⣀⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⡾⢱⣿⣿
⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⢀⡈⠙⠻⣷⠀⢿⡍⠉⠉⢉⠁⠸⣇⠀⠀⢀⡁⠀⣿⡀⠒⢸⡇⠀⠀⣼⡇⠀⠀⠀⢸⡇⠀⢸⣿⠀⣀⠀⢘⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠶⠖⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠳⠶⠛⠀⠀⠘⠳⠖⠛⠓⠂⠐⠚⠓⠂⠀⠐⠚⠓⠂⠀⠻⠶⢋⠐⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠙⠛
⠀⡖⠲⣶⠀⠶⠄⠰⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠿⠶⠿⠿⢿⣿⣿⣷⡌⠻⣿⣿⣿⣿⣿⡿⣿⣿⡇⠀⠀⠀⠀⠀
⢹⣿⣿⣇⢻⡞⣽⣿⣿⣽⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣹⡇⣿⣿⢻⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⡿
⡞⣿⢿⣟⢸⡇⠏⠏⣫⡟⢺⡟⣽⣿⣿⣿⣿⣛⣟⣿⣿⣿⡿⢿⢿⣟⣿⣿⣯⣟⣻⣿⣻⡜⣻⡇⣿⣾⣿⡿⣿⣿⣿⣽⣼⡯⣿⣻⢻⢷⣿⣿⡟⣿⣿⣋⣿⣿⣷⣿⢿⣻⣿⣿⣿⣿⣿⢿⣿⣿⣟⣿⡭⣷⠖⣧⣯⣷
⡇⣿⣂⣄⡀⡇⠻⣟⣟⣥⢵⡥⣿⣿⣿⢯⣟⢀⡼⢹⣻⡿⣿⣗⣒⣻⡛⠿⠽⢿⡯⣿⣟⣋⣿⣯⣽⡷⢻⣻⣿⣏⣿⣿⠝⠁⢽⣧⣜⠀⢻⣿⣿⣹⣿⡛⠙⣿⣿⣶⢿⣿⠟⣙⠅⣤⣦⡾⠫⣿⠇⠭⣤⢾⣿⣷⣷⣦
⠳⣿⣿⠿⡀⠄⣀⢠⣥⣬⣷⣙⣯⣼⣏⠈⣉⣦⠘⣿⣯⣁⢽⣿⢿⣿⡧⡼⠖⣜⡋⠻⢿⣷⢼⡋⠻⠙⣝⣁⡛⠟⠘⢿⣩⣮⡝⢿⣿⠛⠃⠙⢿⠁⠿⠿⠶⠙⢿⣿⠀⠙⢠⡈⠝⢂⡰⠿⣷⣾⣾⣿⣵⠶⠿⠉⠹⣿
⡏⠸⠻⠿⠔⠲⢦⣦⡈⢿⣿⣿⡟⠷⠆⠲⠯⠙⠊⠙⣣⣉⠺⠟⢹⣿⡏⠻⠞⠛⢙⠛⠙⠿⠙⣍⣀⣸⣿⣿⣻⠗⠀⢰⣄⣉⠀⠘⠛⢸⡃⠀⠀⢘⣗⣴⣶⣤⡄⠙⣦⣄⠀⢉⠙⣿⣧⣾⣿⣿⣿⣿⣿⣤⣦⣀⣀⣈
⠀⠀⢠⣤⣴⡶⢶⣬⣵⠻⣟⣻⣗⠐⢿⣯⡷⣧⡄⠚⢛⣽⣶⣦⠀⠉⠀⣶⣿⣷⠾⠟⠀⠀⢠⠍⠋⠁⠻⡶⠐⠀⠀⣾⣿⣿⡗⠀⠀⠈⠛⣧⣤⣼⡟⠻⣿⡿⠅⠀⠀⢈⣤⣶⣶⣶⣶⠉⠩⢿⣿⠿⠷⠉⡉⠉⠉⠋
⠀⠀⠈⠉⠛⢿⠆⠱⢊⣠⣿⣿⣧⣈⡀⡈⢲⣮⣅⠀⠙⠛⠁⠀⠴⣶⡖⢿⣿⣿⡖⣷⣄⠀⣴⣿⠿⠂⠀⣐⠿⣿⠟⠐⣿⣿⡟⠆⠀⠀⢀⣶⣿⣿⣦⣄⠙⢿⣀⡀⢀⢤⡀⠈⠩⠃⠀⠀⠀⠹⢿⣿⠈⠳⣠⣴⠶⠿
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⡟⠙⢰⣿⣿⣿⣿⡀⠀⠀⠀⢰⣐⣉⡀⠈⠻⣿⣿⣿⣿⣿⡎⠀⠀⠀⠀⣝⣠⣤⡄⠀⠈⢻⣥⣴⣤⣤⠜⢿⣿⣿⣿⣏⠀⠀⠘⠻⠂⡴⢁⣠⣴⣶⣄⠀⠀⠀⠀⢘⣲⣄⣀⡀⠁⡄
⠗⠀⠀⠀⠀⠙⢻⣿⡯⣽⠿⠿⣷⡒⠛⢿⣿⡿⠛⠛⠛⠃⣶⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⡼⣿⣟⡛⢿⣿⡿⠛⠛⠓⢀⡬⠙⢛⠿⠛⠛⠛⣿⣿⣭⡍⠜⠛⠛⠛⠛⣻⣿⣿⣿⡿⠛⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⢰⠖⠁⠸⣿⣿⣿⠃⠘⠛⠛⠛⠟⠛⠛⠃⢛⣛⣛⣿⠛⠟⠛⠛⠻⣿⣿⣿⡟⠛⠛⠟⣻⣿⣿⣛⠛⠛⢻⠟⢘⣛⣿⣟⠛⠛⠛⠛⣿⠃⣿⣿⣛⠛⠛⣛⣋⣈⣈⣀⣤⣤⣤⠚⠿⣿⣟⠟⣺⣿⣿⣿
⣠⣴⣶⣶⣤⣤⣴⣶⡆⠀⠉⡿⠁⠀⠈⠁⣿⣿⣿⣧⡀⠀⢘⣛⣛⣧⣤⣤⣦⠀⠀⠉⠻⠏⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣓⠀⣘⣛⣛⣿⣿⡷⢿⣿⣿⣿⣿⡿⠿⠏⠀⠈⠃⢉⣡⣿⣿⣿⣿
⣿⣿⣿⡿⠿⢿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⣛⣻⣿⣿⣿⣦⢛⣿⣿⣿⣿⡿⠿⠀⢀⣤⣶⣶⣶⣶⣿⣟⣿⣿⣿⣿⣿⣿⣻⠛⠛⠛⢙⣛⣛⡋⣛⣛⣛⣛⣛⢛⣛⣿⣃⠀⠀⠉⠉⣡⠒⠀⠀⠀⠀⠀⠉⣽⣿⣿⣿⣭
⣿⣿⣿⣿⠟⠁⠘⠛⢹⣿⣿⣿⣿⣯⣄⡀⣙⣻⣛⡁⠙⠁⢘⣛⣛⡛⠁⢀⠂⠀⠻⣿⣿⣿⡿⠿⢙⣛⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⢘⣛⣛⡓⠙⣛⣛⣛⡋⢈⣛⣛⡿⠋⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿
⣿⣿⠏⠉⠀⠀⠀⠈⠀⠛⠿⢿⢷⡿⣿⣃⣉⣉⣉⣁⣀⡀⢈⣉⣉⣁⣀⣁⣀⣺⣆⣀⣀⡞⢉⣉⣈⣉⣿⣿⣿⣿⣝⣉⣉⡉⢀⣀⣈⣉⣉⡁⠀⢉⣉⡉⠀⢈⣉⣙⣁⣀⣠⣤⣶⣶⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⣯
⣿⣿⣶⣶⣾⣿⠃⠀⠀⠀⠈⠉⠙⠣⣩⣹⣏⣉⣉⣉⣉⡁⠈⢉⣉⣉⣉⣉⣭⣽⣋⣭⣉⡁⢈⣉⣉⣉⣹⣿⣿⣿⣿⣏⠉⢥⣬⣽⣿⣿⣭⣁⣤⡈⣉⠁⠀⢈⣉⣭⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀
⣡⣶⣦⣿⣿⣿⠃⠀⢀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⣭⣍⣩⣽⣿⣿⣿⣿⠉⠉⠉⠉⠉⠙⢫⣟⠁⠀⢸⣿⣿⣿⣿⣯⠁⠁⠀⠁⠀⠀⠈⢉⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⠂⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣤⣄⡀⠀⣴⠂⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠂⠀⠀⠘⠻⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠈⠛⠉⣭⠙⠋⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⢛⣟⠛⠁⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣀⣤⣄⡀⢿⣿⣿⣿⣿⣿⡛⣉⣉⠀⢀⣂⡀⠀⢀⣀⣀⣠⣤⢀⣈⡀⠈⣹⣿⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠩⠉⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣻⣼⣏⣀⣈⣿⡄⢸⡿⠙⠃⠘⠇⠀⣿⠀⠀⢸⡇⠀⠀⣿⡏⠀⠙⠀⢸⡇⠀⢸⣿⠀⠀⠈⢷⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢉⡉⠻⡿⣿⠁⢿⡍⠉⠉⢉⠁⠸⣇⠀⠀⢀⡤⠀⣿⡀⢠⣼⡇⠀⠀⢿⡇⠀⠀⠀⢸⡇⠀⢸⣿⠀⣀⠀⠘⣷⣰⣃⣀⣀⣀⣀⣠⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠓⠶⠞⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠲⠶⠚⠀⠀⠘⠳⠖⠛⠓⠂⠐⠚⠓⠂⠀⠐⠚⠓⠂⠀⠻⠶⠋⠀⠀⢹⣏⣽⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠆⠲⣶⠁⢶⠄⠰⠴⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠴⠟⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀
⢹⣿⣿⣇⢻⡿⣽⣿⣿⣼⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡟⣟⣿⣿⣿⣿⣿⣿⣿⡟⡟⣿⣿⡟⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣏⣿⢻⣿
⡼⣿⢻⣿⢸⡇⠏⢏⣻⡏⠻⡟⢽⣿⣿⣿⣿⣻⣛⣿⣿⣿⣿⢿⢿⣯⣿⣿⡯⣝⣛⣿⣻⣝⢹⡧⣻⣿⣿⡿⣿⣿⣿⣽⡗⡇⣿⣛⠿⡇⣿⣿⣿⣿⣿⣟⣣⣿⣿⣻⠿⣻⣿⣿⣿⣿⣿⠿⣿⣿⣽⣿⣯⢼⡖⢿⣼⣽
⡇⣿⣢⣤⡄⠇⠿⣿⣿⣥⢭⡥⣽⣿⣿⢯⣿⡠⡮⣙⣛⡿⣿⣿⣒⣿⡛⠻⠿⢻⡯⣿⣟⣋⣿⢯⣽⡿⢻⣙⣿⣟⣽⣿⠇⠃⢸⣧⣦⡃⠹⣿⣿⣟⣿⣛⠉⢻⣿⣷⡿⣿⡿⢋⠽⢠⣴⣾⠟⢽⡿⣩⣥⠼⣿⣿⣿⣾
⠳⢽⡿⠷⠄⠄⠤⠠⣵⡴⣷⣜⣯⣾⣏⠀⣩⣦⢈⣿⣯⣥⢽⣿⢿⣿⣯⠼⠖⣜⡋⠙⠷⣷⣬⣛⠙⠋⣍⣉⡘⠟⠊⢿⣭⣦⡬⢻⣿⠿⠛⠘⠿⠏⠻⠿⡶⠎⢳⣿⡇⠈⢣⣌⠩⠓⣀⠾⠻⣶⣿⣿⣯⣴⢿⠏⠉⢻
⠇⠘⠻⠷⠞⠳⢲⣧⡀⣿⣿⣿⠟⠿⠆⠒⠯⠙⠃⠉⣳⣭⡚⠟⢹⣿⡟⠛⠞⠛⢩⠛⠉⠿⠋⣋⣀⣸⣿⣿⣿⠿⠀⢀⢆⣈⡀⠈⠻⢡⣗⠀⠀⠘⣙⣤⣶⣦⣤⠉⢳⣤⡀⠉⡙⢻⣿⣤⣾⣿⣿⣿⣿⣿⣴⣄⣀⣀
⠀⠀⢠⣤⣬⡶⢶⣬⣶⢛⣛⣻⡇⠸⢿⣿⡖⣦⡄⠛⢛⣽⣶⡦⠀⠈⠀⣶⣿⣷⡾⠟⠂⠀⠀⡍⠟⠁⠹⢷⠊⠀⠀⢼⣿⣿⣿⠀⠀⠈⠙⢷⣤⣤⡿⠛⢿⡿⠿⠀⠀⠈⣣⣴⣶⣶⣶⡆⠉⠩⢿⣿⠿⠿⢉⠉⠉⠛
⠀⠀⠈⠉⠛⠿⠆⠗⣉⣨⣿⣿⣧⣈⢀⣈⣐⣮⣅⠀⠉⠛⠁⠀⠶⣶⡖⢿⣿⣿⡖⣾⣄⠀⣴⣿⠿⠂⠀⢀⠺⢿⠿⠃⢻⣿⣿⠁⠀⠀⠀⣠⣶⣿⣷⣤⡈⠻⣇⡀⠀⠀⢤⡀⠉⠸⠀⠀⠀⠀⠻⣿⣿⠘⢆⣠⣶⢶
⠀⠀⠀⠀⠀⣰⣾⣿⢻⣿⣿⡿⠏⠙⣰⣿⣿⣿⣿⡀⠀⠀⠀⠰⢐⣉⡀⠈⠛⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢘⣡⣥⡤⠀⠈⠙⣯⣴⣤⣤⡤⠛⠫⣿⣿⣿⡃⠀⠀⠻⠷⠀⠞⣁⣤⣴⣶⡀⠀⠀⠀⠀⠘⢦⣀⣀⡁⠄
⠆⠀⠀⠀⠈⠛⢿⣿⠬⣽⠿⠿⣶⠂⠛⢿⣿⡿⠛⠛⠛⠃⣶⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⢿⣿⡛⠻⣿⣿⠟⠛⠓⠀⣈⠙⠛⣻⠟⠛⠛⢫⣿⣿⣭⠛⠛⠛⠛⠛⢛⣻⣿⣿⣿⡿⠛⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡦⠊⠀⢻⣿⣿⡿⠀⠙⠛⠛⠛⠟⠛⠛⠃⢛⣛⣛⣿⠛⠟⠛⠛⠻⣿⣿⣿⡟⠛⠛⠟⢛⣿⣿⣟⡛⠛⠛⡟⠙⣛⣛⣟⡛⠛⠛⠙⠻⣟⠻⣿⣟⡛⠛⠛⢋⣉⣉⣉⣀⣀⣀⣠⠼⠿⣿⣿⠟⠻⣿⣿
⣤⣶⣶⣦⣄⣤⣴⣶⠆⠀⠹⠏⠁⠀⠈⠁⣿⣿⣿⣧⡀⠀⢘⣛⣛⣋⣤⣤⣤⠀⠀⠈⠛⠿⠁⡐⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣓⠀⣘⣛⣻⣿⣿⣷⠴⢿⣿⣿⣿⣿⣿⡿⠿⠀⠀⠐⠀⠁⠸⠿⣿
⣿⣿⣿⣿⡿⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⣛⣻⣿⣿⣿⣦⢚⣿⣿⣿⣿⣿⠿⠆⢀⣠⣤⣷⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠻⠟⢛⣛⣛⡋⣛⣛⣛⣛⣛⢛⣛⣛⣫⡀⠀⠈⠉⠉⢋⠤⠀⠐⠀⠀⠀⠀⢛⣿⣿⣿
⣿⣿⣶⣶⠟⠉⠛⠇⣿⣿⣿⣿⡿⣃⡀⠀⣙⣿⣛⡉⠛⠁⢘⣛⣛⡟⠋⠀⡀⠀⠸⣿⣿⣿⣿⢿⢟⣛⣻⣿⣿⣿⣿⣿⣟⣰⠀⠀⢘⣛⣛⡃⠛⣛⣛⣛⡋⢘⣉⣛⡋⣹⠖⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿
⣿⡿⠛⠁⠀⠀⠀⠂⠘⠿⠿⢿⢿⣿⣿⣂⣉⣉⣉⣁⣀⡀⢈⣉⣉⣁⣀⣀⣀⣰⣆⣈⣉⡩⢓⣀⣉⣙⣻⣿⣿⣿⣿⣛⣛⡉⢂⣀⣈⣉⣉⡁⠀⢉⣉⡉⠀⢈⣉⣉⣟⣃⡀⢀⣠⣤⣴⣶⣶⡀⠀⠀⠀⠀⠀⠀⠈⠙
⣿⣧⣤⣤⣴⡶⠂⠀⠀⠈⠈⠒⠪⣄⢉⣻⣍⣉⣉⣉⣉⡁⠈⢉⣉⣉⣉⣩⣩⣩⣯⣭⣝⡁⢈⣉⣩⣍⣩⣿⣿⣿⣿⣿⠏⢁⣈⣭⣭⣭⣭⡁⢀⣈⣉⠁⠀⢈⣉⣉⣉⣽⣿⣿⣿⣿⣿⣿⣿⣇⣄⠀⠀⠀⠀⠀⠀⠀
⣻⣍⣻⣿⣿⡷⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⣉⣉⠉⣉⣭⣽⣷⣾⡏⠉⠉⠉⠉⠙⠿⣿⡟⢀⠀⣿⣿⣿⣿⣯⡉⠉⠉⠁⠀⠀⠈⠉⠉⣽⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠀⠠⠀⠀⠀⠀⠀
⣿⣿⣿⣿⠟⠀⠀⠀⠉⠠⠀⢠⣤⣤⣤⣄⣀⡀⠀⠀⣴⠆⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠘⠿⣿⠙⠻⠿⠟⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⢿⠿⠏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣦⣁⠀⠀⣀⣤⣄⡀⣿⣿⣿⣿⣿⣿⡿⣟⣛⠃⢀⣠⡀⠀⢀⣀⣀⣠⣤⢀⣈⡀⠉⣹⣿⣓⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠄
⠰⠿⠄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⢦⣼⣏⣀⣈⣿⡌⢹⡿⠻⠟⠙⠇⠀⣿⠀⠀⢸⡇⠀⠀⣿⡏⠀⠙⠀⢸⡇⠀⢸⣿⠀⠀⠈⢷⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠈⠉⠛⢻⡿⣿⡿⣷⠋⢿⡍⠉⠉⢉⠁⠸⣇⠀⠀⢀⣀⡀⣿⡀⠀⢸⡇⠀⠀⢽⡇⠀⠀⠀⢸⡇⠀⢸⣿⠀⣀⠀⠘⣧⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣤⣤
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠓⠾⠟⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠲⠶⠚⠁⠀⠘⠳⠖⠛⠓⠂⠐⠚⠓⠂⠀⠐⠚⠓⠂⠀⠻⠶⠋⠀⠀⢹⠏⠀⠰⠶⢦⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠆⠲⣶⠂⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠤⠏⠀⣠⣴⣶⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢹⣿⣿⣧⢻⣿⣿⣿⣿⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣏⣿⢹⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡼⣿⢻⣻⠸⡇⡏⣏⣵⡇⢻⡿⢿⣿⣿⣿⣽⣻⣟⣿⣿⣿⣿⡻⠿⣿⣽⣿⣧⣾⣟⣿⣽⣟⢹⣿⣫⣿⣿⣿⢿⣿⣿⣽⡧⣿⣿⣟⠟⣿⢾⣿⣿⣿⣿⣟⣻⣿⣿⣾⡿⣟⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣯⢽⣶⢫⣽⣟
⠏⣿⣬⣤⠄⠇⠙⣿⣯⣶⡭⡶⣽⢿⣿⢯⣿⡠⡮⣙⣻⣿⣿⣯⣷⣿⣛⠻⡿⢻⡿⣿⣿⣋⢿⣯⣽⣿⢛⣏⣿⣿⣽⣿⡿⠋⠨⣿⣶⣇⠘⢿⣿⣿⣿⣟⠋⠻⣽⣿⣾⢿⡿⠟⡹⡢⣴⣴⡿⠻⣿⢯⣭⣠⢿⣿⣻⣷
⡹⣿⣿⡳⠄⠀⠤⠠⢶⣶⣿⣼⣯⣾⣯⠀⣬⣦⢈⣿⣧⣤⢼⣿⣿⣿⣟⣴⠗⣌⡋⠘⠷⣿⣦⣙⠙⠋⣏⣉⣸⠛⠃⠻⣯⣭⣥⠹⣿⡿⠛⠁⠻⡏⠙⠿⢷⠖⠙⣿⣿⠀⠛⣤⡉⠛⢂⡰⠿⣿⣿⣿⣿⣷⡾⡿⠏⠙
⠀⠟⠛⠗⠋⠛⣲⣧⡀⣿⣿⣿⡟⠿⠇⠚⠷⠙⠋⢉⣷⣤⡻⠛⠸⣿⡟⠙⢞⠛⢩⣙⠃⠻⠇⣃⣀⣐⣿⣿⣿⠿⠂⠀⣆⣈⣄⠀⠛⠧⣟⡂⠀⠀⢋⣣⣴⣶⣦⡌⠙⣦⣄⠈⣍⠛⣿⣧⣬⣿⣿⣿⣿⣿⣷⣴⣀⣀
⠀⠀⣴⣶⣭⡿⢷⣴⣷⢻⣿⣿⣯⢹⠿⣿⡗⣶⡄⠛⢋⣽⣾⡧⠀⠈⠀⣷⣿⣿⡾⠟⠃⠀⠀⡍⠙⠉⠘⢿⠇⢂⠀⢸⣿⣿⣿⡆⠀⠀⠙⠻⣦⣶⣾⠟⠻⢿⡿⠧⠀⠀⠙⣤⣷⣶⣶⣶⡆⠉⠻⢿⣿⠿⠿⢉⠉⠙
⠀⠀⠀⠉⠛⠿⠃⣏⣀⣴⣿⣿⡧⣤⢀⣈⣒⣦⣅⠀⠈⠉⠁⠀⠶⢿⣿⢻⣿⣿⣷⢻⣦⡀⢴⣿⡿⠓⠀⢀⡚⢻⡿⠃⢸⣿⣿⡏⠀⠀⠀⢐⣶⣾⣿⣶⣄⡹⢿⣂⠀⠀⡨⣤⡀⠉⠜⠁⠀⠄⠀⠿⣿⣿⠱⣆⣤⡶
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡏⠙⣸⣿⣿⣿⣿⡀⢀⠀⠀⢸⣃⣁⡀⠀⠛⢿⣿⣿⣿⣿⣿⠄⠀⠀⠀⢨⣣⣤⣤⠀⠀⠙⢻⣥⣦⣤⣤⡼⠿⢿⣿⣿⣿⡃⠀⠸⠿⠆⢰⢋⣡⣤⣴⣶⡀⠀⠀⠀⠈⠙⣦⣀⣈⣡
⠀⠀⠀⠀⠈⠛⢿⣿⠯⡿⠿⢿⣷⠂⠸⢿⣿⡿⠛⠁⠈⠉⣿⣿⣿⠿⣷⣾⣷⡾⣺⣿⣿⣿⣿⣭⣧⣄⠰⣿⣿⠿⠛⠁⠀⢠⡉⠛⢛⡿⠛⠁⠈⣽⣿⣿⣿⠁⠀⠀⠀⠀⣀⣿⣿⣿⣿⠿⠛⠀⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠠⠖⠉⠀⢻⣿⣿⡿⠀⠑⠀⠈⠢⠆⠀⠀⠀⠉⠉⠁⢴⠈⡽⠛⠁⠈⣿⣿⣿⣿⠃⠉⠢⠀⢀⣤⣤⣀⠀⠀⢰⠋⢀⣉⣠⣄⠀⠀⠈⠛⢿⡟⢤⣤⣀⠀⠀⠙⢛⣉⣉⣉⣀⣀⣀⣀⣀⠿⣿⣿⣿⡿⢯
⣴⣶⣶⣤⣠⣼⣶⣶⠂⠈⠹⠏⠁⠀⠈⠃⣶⣶⣶⣤⡀⠀⠀⠀⠀⣎⣤⣤⣴⡄⠀⠈⠛⣿⠋⠰⢰⣶⣶⣿⣿⣿⣿⣷⣦⣴⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠁⠸⠿⢿⣷⣄⢴⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠿⠱⢧⣐
⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣦⠰⣶⣿⣿⣿⣿⡿⠇⠀⣀⣤⣾⣶⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠿⠟⠉⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⣫⡀⠀⠉⠛⠋⠛⡩⠄⠀⠀⠀⠀⠀⠐⣚⣿
⣿⣿⣶⣶⠞⠙⠿⢷⣿⣿⣿⣿⡿⣏⡀⠀⠘⠦⠀⠈⠛⠉⠀⠙⠿⠿⠋⠁⡠⠀⠘⢿⣿⣿⣿⣿⣿⠾⠿⣿⣿⣿⣿⣿⣿⡇⡀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠷⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿
⣿⣿⡿⠁⠀⠀⠀⠂⠻⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⣦⠀⠉⢉⡿⠓⠚⠀⠒⢷⣿⣿⡿⠿⠿⠟⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠀⠀⣀⣀⣤⣤⣴⣄⠀⠀⠀⠀⠀⠀⠈
⣿⣀⣿⣬⣤⡤⠄⠀⠀⠐⠚⠻⢿⡉⠀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠃⠀⠋⠀⠀⠀⠠⠀⠻⣿⣿⣿⣿⣶⠂⠀⢠⣤⣤⣤⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀
⣛⢻⣿⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⣀⣀⠘⢀⣠⣤⣤⣶⣶⠀⠀⠀⠀⠙⢿⣿⣿⡿⠀⠸⣿⣿⣷⣦⣉⠛⠛⠓⠂⠀⠀⠀⠀⠘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⡀⠀⠀⠀
⣿⣿⣿⣿⡟⠁⠀⠐⢃⣄⠀⣀⣀⣀⣤⣀⠀⠀⠀⠀⣴⠆⠀⠀⠀⠀⠈⢉⣽⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠹⢿⠅⠀⠰⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠐⣿⠿⠿⠿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⠏⠀⠀⠀⠀⠀⠈⢀⣿⣿⣿⣿⣿⣿⣿⣦⣄⣋⠀⠀⣀⣤⣄⡀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⢀⣀⡀⠀⢀⣀⣈⣠⣤⢀⣈⡉⠙⣻⣿⣿⣧⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣽⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣄⣼⣏⣀⣈⣿⡝⢻⣿⣿⣿⡿⠏⠀⣿⠀⠀⢸⡇⠀⠀⣿⡏⠀⠙⠀⢸⡇⠀⢸⣿⠨⡁⠈⢷⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠁⠀⠀⠀
⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⢿⣿⣿⣿⡿⣿⠟⢿⡍⠉⠉⢉⠁⢸⣧⠈⠁⢀⡀⠀⣿⡀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⢸⣿⠀⣀⠀⠘⣧⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⡀⠀⠘⠛⠿⠞⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠳⠶⠚⠁⠃⠘⠳⠖⠛⠓⠂⠐⠚⠓⠂⠀⠐⠚⠓⠂⠀⠻⠶⠋⠀⠀⢹⠏⠀⠀⠀⣀⣀⣀⣄⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣶
⠀⠆⠲⣶⠋⢶⠄⠰⠶⠶⠖⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠦⠏⠀⠀⠀⠀⠈⠋⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢻⣿⣿⣧⢻⡿⣾⣿⣿⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⢹⣿⣿⢹⢻⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿
⠸⣿⢻⡿⢸⡇⡇⣋⣽⡇⢻⡟⢾⣿⣿⣿⣽⣩⣟⣿⣿⣿⣿⣻⠿⣿⣽⣿⣷⣼⣿⣿⣽⣯⢹⣿⢫⣿⣿⣿⢿⣿⣿⣹⣿⢻⢸⣏⡿⢻⢸⣿⣿⣿⣿⣿⣟⣿⣿⣾⡿⢟⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣭⣷⢿⣿⣟
⡋⣿⣴⣦⠄⠇⠙⠿⡯⣶⡼⣖⣽⣿⣿⢳⣿⡀⠾⣸⣫⣿⢿⣯⣵⣿⣟⠙⡿⠻⣿⣾⣿⣻⢿⣯⣭⣿⠛⣏⣿⣿⣽⣿⣿⠘⠀⣿⣶⣼⡈⢿⣿⣿⣿⣿⡋⠙⣯⣿⣿⡿⣿⠿⡫⡗⢤⣦⣿⠟⢿⣿⣭⣁⣽⣿⣟⣿
⠹⣿⣟⡻⠀⠂⠦⠰⢶⣶⣿⣮⣷⣾⡯⠀⣬⣷⣈⣿⣷⣦⢼⣿⣿⣿⣿⣴⠟⣈⣉⠘⠻⣾⣦⣝⡉⠛⣋⣉⣩⠛⠋⠹⢿⣭⣥⡝⣿⣿⠟⠃⠹⢿⠙⠿⠷⡶⠞⢻⣿⡇⠙⢣⡈⠝⠳⣀⢿⢿⣿⣿⣿⣿⣾⣿⠯⠙
⠀⠋⠛⠟⠋⠛⣲⣧⡀⣿⣿⣿⡟⠿⣇⣛⠗⠙⠃⢉⣽⣤⡿⠛⠘⣿⡿⠘⢻⡛⠩⣙⡋⠹⠇⢋⣀⣀⣿⣿⣿⡿⠃⠀⡶⣀⣁⠀⠘⠻⢼⡃⠀⠀⠙⣏⣤⣶⣶⣤⡉⢲⣄⠈⠉⠙⠻⣿⣤⣿⣿⣿⢿⣿⣿⣷⣄⡀
⠀⠀⣶⣶⣿⡿⢷⡾⣿⢻⣿⣿⣯⢹⠿⣿⢟⣶⡄⠛⢫⣿⣿⡷⠀⠀⠀⣸⣿⣿⣿⠟⠃⠀⠀⡬⠙⠉⠀⠿⡞⠀⠀⠠⣿⣿⣿⣷⠀⠀⠈⠛⣧⣴⣶⣿⠟⢿⣿⠿⠄⠀⠈⢣⣼⣶⣶⣶⣶⠆⠋⠻⢿⣿⣿⡿⡉⠉
⠀⠀⠀⠍⠙⠿⣀⣋⣀⣼⣿⣿⡧⣤⢀⣈⣓⣦⣅⠀⠈⠉⠁⠀⠶⢿⣿⠘⣿⣿⣿⢚⣧⣀⢰⣿⡿⠗⠀⠀⡘⠻⡿⠟⠀⣿⣿⡟⠀⠀⠀⠀⣲⣴⣿⣿⣦⣈⠿⣷⠀⠀⢀⠤⣄⠀⠉⠟⠁⠀⠄⠀⢿⣿⣿⢰⣀⣤
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡏⠙⣸⣿⣿⣿⣿⡀⢀⠀⠀⢨⣃⣉⡀⠀⠙⢿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⣏⣬⣤⡄⠀⠈⠻⣧⣤⣤⣤⣤⠻⢿⣿⣿⣿⣿⠁⠀⠿⠷⡀⡴⢉⣠⣤⣴⣶⠀⠀⠀⠀⠉⠛⣤⠁⢉
⠀⠀⠀⠀⠈⠛⢿⣿⢯⡿⠿⢿⣷⠂⠸⢿⣿⠿⠟⠁⠀⠉⣿⣿⣿⠿⣷⣾⣷⣾⢑⣿⣿⣿⣿⣧⣷⣄⠀⣿⣿⡿⠛⠋⠀⠀⣌⠙⠛⣻⠟⠋⠀⠈⣿⣿⣿⡟⠁⠀⠀⠀⢁⣠⣿⣿⣿⣿⠿⠓⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⢰⠖⠉⠈⣿⣿⣿⡿⠀⠓⠀⠈⠢⠆⠀⠀⠀⠉⠉⠁⢴⠈⡩⠛⠉⠀⢿⣿⣿⣿⡇⠉⠣⠁⠀⢀⣤⣄⡀⠠⠀⡞⠁⣀⣁⣀⡀⠀⠀⠙⠿⣿⡇⣄⣀⠀⠀⠈⠛⠛⢉⡛⠋⣀⣀⣀⣀⣀⣰⣿⣿⣿⣿
⣴⣶⣤⣄⣤⣶⣶⡶⠀⠈⠻⠏⠁⠀⠘⢐⣶⣶⣦⣄⠀⠀⠀⠀⠀⣎⣠⣤⣤⡄⠀⠀⠛⢿⡟⠀⠆⣶⣤⣶⣿⣿⣿⣿⣶⣤⣶⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⠀⠿⣿⣿⣦⣀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠹⣿⣷
⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠈⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣦⠲⣶⣾⣿⣿⣿⣿⡷⠀⢀⣠⣤⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣋⡂⠀⠉⠛⠛⠛⢟⡩⠈⠅⠀⠀⠀⠀⠴
⣿⣯⣭⣭⠟⢿⣿⢶⣶⣾⣶⣶⡶⣊⠀⠀⠰⣦⠀⠙⢿⠋⠀⠙⠿⠿⠛⠉⡠⠀⠀⢿⣿⣿⣿⣿⣿⡯⡿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠊⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⣻⡧⠀⠀⠀⡠⠏⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⡟⠁⠀⠀⢠⠈⢿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣶⡀⠉⠛⣻⠯⠙⠃⠀⠾⣽⣿⣿⣿⣿⣿⡿⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣀⣀⣠⣤⣤⠀⠀⠀⠀⠀⠀
⡟⢙⠅⣀⣀⣀⠀⠀⠀⠩⠭⢿⣛⠋⠁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣽⠓⠀⠔⠉⠀⠀⠀⠀⠰⣿⣿⣿⣽⣧⣤⠀⠀⢠⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣤⣶⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣏⡄⠀⠀⠀⠀⠀⠀⠈⠳⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠃⠀⣀⣀⣤⣤⡄⠀⠁⠀⠘⢿⣿⣿⣿⡿⠀⢲⣿⣿⣭⣉⠓⠚⠛⠂⠀⠀⠀⠀⠀⠘⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀
⣶⣭⣿⣿⠟⠀⠀⠤⢄⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣤⡆⠀⠀⠈⠁⠉⠙⣻⣶⣾⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠙⢻⣿⠁⠀⣸⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⡿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠈⠀⠀
⣿⣿⣿⠋⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⣶⣶⣤⣤⣬⣋⠀⠀⣀⣤⣄⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⡀⠀⢀⣀⣈⣢⣤⢀⣈⡛⠻⣿⣿⣻⣿⣆⣀⡀⠀⣀⣀⠀⣿⠈⠁⠀⠀⠙⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⣼⣏⣀⣈⣿⡾⢿⣿⣿⣿⣿⡿⠋⣿⠁⠀⢸⡟⠀⠀⣿⡏⠀⠙⠀⢸⡇⠀⢸⣿⠀⠻⠋⣷⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⠀⠀
⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣲⢿⡍⠉⠉⢩⡁⢸⣯⠙⠛⢋⡀⠀⣿⡀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⢸⣿⠀⣀⠀⠘⣷⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠠⠀⠈⠉⠛⠿⣿⣿⠟⠉⠈⠛⠶⠖⠋⠀⠀⠙⠳⠶⠚⠀⠤⠙⠳⠖⠛⣗⠂⠐⠛⠓⠂⠀⠐⠚⠓⠂⠈⠻⠶⠋⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠲⣶⠇⢶⠔⠰⠶⠶⠖⠀⠀⠀⡤⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠶⠏⠀⠀⠀⠀⠀⠀⢀⣤⣧⣤⣤⣄⣄⣤⣤⣴⣶⣶⣶⣶⣶
⢿⣿⣿⣧⢻⡿⣾⣿⣿⡿⣿⡻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡇⢿⣿⡏⡗⣿⣿⣿⣿⣿⣷⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿
⢸⣿⢻⠿⢸⡇⣃⣫⣽⡇⣛⣟⢾⣿⣿⣿⣿⡭⣿⣿⣿⣿⣿⣿⠿⢿⣽⣷⣿⣼⣿⣿⣯⣧⣫⣿⣝⣿⣿⣿⠿⣿⣿⣯⣿⣿⡇⣿⣹⠯⡇⣿⣿⣿⣿⣿⣟⣻⣿⣷⣿⠿⣻⣿⣷⣿⣿⣿⡿⣿⣿⣿⢿⣟⣿⣬⣝⣿
⠘⣯⣶⣶⠆⠃⠛⠿⡯⢷⡾⣖⣾⣿⣿⠷⣿⡅⠾⣨⣫⣿⢻⣿⣵⣿⣟⠙⡿⠛⣿⣻⣿⣻⡽⣿⣭⣿⡟⣯⣽⣿⣯⣽⣿⡇⠃⡼⣷⣦⣇⠹⣿⣿⣿⣿⣛⠉⠿⣹⣿⣾⣿⣿⡟⢛⢣⣥⣴⣿⠛⣻⣟⣍⣨⣿⣿⣭
⠛⣿⣟⡻⣀⡂⠦⠰⢶⣶⣿⣮⣷⣿⠿⠀⣤⣷⡄⣽⣿⣦⡜⣿⣿⣿⣿⣳⡟⣣⣉⠉⠻⢾⣷⣝⡉⠛⢹⣉⣁⡝⠟⠉⢿⣯⣯⣤⠻⣿⡿⠛⠋⠿⡏⠻⠿⢶⠶⠙⢻⣿⠋⠛⣀⠑⠛⠂⣈⣵⣿⣾⣯⣿⣧⣼⣿⠯
⠐⠉⠛⠛⡉⠛⡾⣦⡀⣿⣿⣿⡟⠿⣃⣙⠗⡉⠋⢉⣽⣤⡏⠛⠃⢿⣿⠉⠳⡛⠋⡍⡋⠘⠿⠙⣁⣀⡺⣿⣿⣿⠿⠀⢀⢆⣈⡀⠀⠛⠧⣟⠀⠀⠀⢫⣣⣤⣴⣦⣌⡙⢤⣄⠈⢙⠛⣻⣯⣤⣼⣿⣿⣿⢿⣿⣷⣶
⠀⠁⣶⣶⣾⡻⢷⠾⡿⢻⣿⣿⡯⢸⠿⣿⠗⣶⡄⠙⢛⣽⣷⣷⠄⠀⠁⢸⣾⣿⣷⡿⠛⠀⠀⢨⠍⠛⠁⠸⢷⠇⠀⠀⣼⣿⣿⣿⣶⠀⠀⠙⠻⣤⣤⣼⣿⠛⢿⣿⣿⡁⠈⠙⠀⣠⣴⣦⣤⣽⣾⠙⠛⣛⣿⣿⣿⣟
⠀⠀⠀⠉⠻⠟⢸⣁⣀⣼⣿⣿⣧⡨⢀⣈⣲⣦⣍⠀⠘⠉⠁⠀⠰⢾⣿⡏⢿⣿⢿⡖⢷⣄⡀⣾⣿⠿⠆⠀⠀⠾⢿⡿⠟⠙⣿⣟⡳⡀⠀⠀⠀⣤⣩⣿⣷⣦⣈⣿⣷⠀⠀⠀⣰⣀⠉⠙⢻⠛⠁⠀⡀⠰⣿⣷⣶⠈
⠀⠀⠀⠀⣠⣾⣿⡿⣿⣿⣿⣿⡟⠹⣸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠇⢈⣁⠀⠘⠿⣿⣷⣿⣿⣿⣿⡅⠀⠀⠀⠸⣃⣈⣤⠀⠀⠙⠻⣇⣤⣀⣀⣠⡺⣿⣿⣿⣿⣿⡟⠈⢿⣦⠀⢡⠟⠁⣀⣠⣤⡄⠀⠀⠀⠈⠛⠿⡋
⠀⠀⠀⠀⠙⢿⣿⣧⣿⣿⣿⣯⣧⠄⢹⣿⣿⣿⠟⠃⠀⠉⣿⣾⣿⣿⣿⣴⣶⣶⡾⠸⣿⣿⣿⣿⣁⣆⡀⢰⣾⣿⡿⠟⠃⠀⠀⡘⠛⠿⢿⣿⠟⠀⠘⢺⣿⣿⡿⠗⠀⠀⠈⠃⠈⣴⣿⣿⣿⣿⡷⠆⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⣤⠜⠃⠸⣷⣶⣿⡿⠀⠂⠈⠙⠂⡄⠀⠀⠀⠛⠛⠛⢠⡌⠙⡻⠟⠁⠉⣿⣿⣿⣿⡏⠛⢍⠉⠉⠉⠀⠀⠀⠀⠈⡽⠂⠀⠊⠀⠀⠀⠀⠙⢿⣿⣿⢀⠀⠀⠀⠀⢻⠿⠛⠻⠟⠋⠀⠀⠀⠀⠀⠀⢀⣶
⣀⡀⠀⠀⣀⣥⣤⡤⠀⠙⣿⡟⠁⠀⠐⠄⣤⣤⣤⣀⠀⠀⠀⠀⠀⠰⠁⢈⣀⣀⠀⠀⠹⣿⣿⡟⢠⢀⣄⣀⣀⣴⣾⣿⣶⣤⣀⣥⣶⣾⣿⣿⣇⠀⠀⠀⠀⠘⠓⢸⣿⣶⣤⡀⠀⢀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⠈⠙
⣿⣿⣷⣾⣿⣿⣿⣷⠀⠀⠐⠀⠀⠀⠀⠰⣿⣿⣿⣿⣷⣄⠠⣀⣤⣾⣿⣿⣿⣿⡀⠀⠀⠀⢫⣀⠈⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠉⠉⠻⡷⠰⡉⠛⠻⢿⣿⣿⣿⡿⠟⠻⠏⠀⠀
⣿⣿⠿⠿⣿⣿⣏⣀⣀⣐⣀⣀⣀⠀⠀⠀⠠⣈⠙⠻⢿⠿⠀⠹⣿⣿⣿⠟⠋⡉⠀⢠⣶⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠶⣦⣄⠀⠀⠀⠀⢀⡖⠂⠀⠁⠀⠀
⣶⣶⡿⠋⠉⠻⠏⣿⣿⣿⣿⣿⣣⣦⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⡈⠻⢿⣿⣿⡿⠻⠏⠛⠻⢿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣬⠟⠁⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀
⠿⠏⠀⠀⠀⠀⠇⠘⠿⡿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⣷⡀⠀⢀⡼⠋⠉⠁⠀⠘⢻⣶⣿⣿⠿⠿⠿⠟⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠃⠀⠀⣀⣤⣤⣴⣶⣶⣆⠀⠀⠀
⠀⣀⣀⣤⡤⠄⠀⠀⠈⠀⠒⠺⣄⡀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠊⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣶⠀⢀⡾⠛⠓⢦⡤⣶⣄⠀⠀⠀⠀⠀⠀⢀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀
⣿⣿⣿⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠑⣬⠁⠀⠀⠀⠀⢀⡀⠀⠀⠑⠆⠤⠶⠶⣄⠈⢀⣠⣤⣴⣶⣶⡆⠀⠀⠀⠀⠙⢿⣿⣿⣿⣯⡄⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⣤⠀⠈⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠀
⣮⣽⣿⡟⠁⠀⠠⠄⠀⠀⠀⣀⡀⠀⣿⠀⠀⣀⣤⣄⣿⠁⠀⣀⣤⣄⡀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣗⣀⡀⠀⢀⣀⣀⣩⣿⣯⣈⡀⣿⣿⣿⣿⣿⣿⣆⡀⠀⣀⣀⠀⣿⠀⢠⢻⣿⣿⠿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀
⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣧⣄⡈⠛⠀⣼⣏⣀⣈⣿⡄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢹⡏⠀⠙⠃⢸⡇⠈⢹⣿⠻⠿⣙⣿⡄⠀⡼⠁⠀⠀⠀⠈⠀⠈⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢿⡍⠉⠉⢉⡁⢻⣿⣿⣿⣿⣿⡿⣿⡁⠁⢸⡇⠈⠃⣿⡇⠀⠀⠀⢸⡇⠀⢸⣿⠀⣀⠀⠻⣷⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠎⠛⠶⠖⠋⠀⠀⠙⠻⠿⠛⠉⠀⠘⠳⠖⠛⠓⠂⠐⠛⠓⠂⠀⠐⠚⠓⠂⠀⠻⠶⠋⠀⠈⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⢶⡄⠰⠶⠶⠖⠀⠀⠈⠉⠙⠛⠿⣿⣟⡽⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢿⣿⣿⣧⢿⡿⣾⣿⣿⡿⣿⡻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢸⣿⣿⣿⢸⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿
⢸⣯⢿⠿⢸⡇⣃⣣⣽⡧⣛⣟⢾⣿⣿⣻⣿⡯⣿⣿⣿⣿⣿⣿⠿⢿⣝⣿⣿⣼⣿⣻⣯⣿⡋⣿⣟⣽⣿⣿⡿⣿⣿⣯⣿⡅⡿⣿⣽⠏⡿⢾⣿⣿⣿⣿⣟⣣⣿⣿⣿⠿⢿⣿⣷⣿⣿⣿⡿⣿⣿⣿⢿⣟⣻⣬⢝⣸
⠸⣿⣶⡶⠆⠃⠛⠿⡯⣷⡾⣖⣿⣿⣿⡿⣿⡅⠼⣭⣻⣿⢿⣿⣵⣺⣟⠙⡿⠟⢿⣻⣿⣟⡙⣿⢍⣿⡗⢫⣹⣿⣿⣽⣿⡧⠃⢸⣷⣦⣇⡘⢿⣿⣿⣿⣟⠋⠻⢙⣿⣾⣿⣿⣿⢛⡛⣭⣅⣿⡿⢻⣿⣋⣁⣿⣿⣭
⠳⣿⣿⡻⠀⠂⠦⠰⢶⣶⣿⣮⣧⣶⡿⠀⣨⣷⡄⣽⣿⣤⡤⣿⣿⣿⣿⣳⡿⢣⣉⠁⠻⢿⣷⢜⡋⠹⢙⣍⣉⡘⠛⠃⢻⣯⣷⣥⡿⢿⣿⠟⠃⠹⣿⠙⠿⠷⡶⠎⠛⣿⡟⠙⢇⠙⡒⠢⢈⣩⣿⣾⣿⣲⣿⣤⣿⣿
⠰⠙⠻⠓⠉⠛⡶⣦⡀⣿⣿⣿⡟⠿⠫⠘⠷⡉⠋⠉⣹⣤⡜⠟⠃⢻⣿⠇⠳⠛⠋⡍⠛⠉⢿⠛⢁⣀⣺⢿⣿⣿⡿⠆⠀⡶⢀⣁⠀⠘⢿⣼⠆⠀⠀⠻⣗⣠⣤⣤⣤⣉⢣⣄⠈⢙⠚⠺⣿⣄⣰⣿⣿⣿⣿⣿⣿⣧
⠀⢠⣶⣦⣽⠿⣷⠶⣷⢛⣛⣻⣟⢸⢿⣿⡗⣶⣄⠚⢛⣭⣷⣶⡆⠀⠉⠐⣶⣿⣷⣿⠟⠀⠀⠀⡍⠛⠋⠘⢿⡏⠀⠀⢠⣿⣿⣿⣷⡄⠀⠘⠻⣆⣠⣤⣿⡟⠻⣿⣿⣟⠀⠙⠇⠀⣡⣤⣤⣭⣭⣯⠛⠛⢷⣿⣿⣿
⠀⠀⠩⠙⠻⡿⢸⠊⣀⣸⣿⣿⣇⠈⠀⡈⠲⣧⣍⠀⠘⠛⠋⠀⢠⣶⣶⡞⢿⡿⠿⡧⢶⣄⠀⣴⣶⡿⠧⠀⠀⠱⠿⣷⡾⠉⢿⣿⣿⠇⠀⠀⠀⢤⠉⣩⣿⣧⣄⡺⣷⡆⠀⠀⠀⣊⠉⠙⠛⢻⠝⠃⠀⠀⢰⣿⣿⣯
⠀⠀⠀⠀⣠⣾⣿⡿⣿⣿⣿⣿⡿⠻⠠⣿⣿⣿⣿⡂⠀⠀⠀⠀⡶⠐⠉⠀⠘⢿⣿⣿⣼⣿⣿⣾⣍⠀⠀⠀⢈⠗⢀⣁⡀⠀⠘⠿⣿⣈⣀⡀⣀⣀⢾⣿⣿⣿⣿⣿⡟⢱⣦⡄⠀⣩⠟⠀⠀⣁⣀⡀⠀⠀⠀⠙⠛⣿
⠀⠀⠀⠀⠻⣿⣿⣧⣻⣿⣿⣾⣷⡄⢸⣿⣿⣿⡿⠇⠀⠑⢲⣶⣿⣿⣿⣥⣤⣤⣽⢯⣿⣿⣿⣿⡟⢠⡀⢀⣴⣿⣿⡿⠷⠀⠀⠀⠺⠿⣿⣿⡿⠏⠈⠬⢽⣿⣿⡿⠇⠀⠀⠙⠀⠃⣴⣾⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣌⡭⠋⢹⣯⣽⣿⣿⠄⢀⠉⠛⢍⣀⠀⠀⠀⠞⠿⠟⠉⡙⠛⢿⡿⠏⠐⣺⣿⣿⣿⣿⠻⢟⠈⠉⠉⠉⠀⠀⢀⡀⢚⡦⠀⠠⠋⠁⠀⠀⠀⠻⣿⣿⣷⠀⠀⠀⠀⠐⣶⣿⠿⢿⠿⠋⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣋⣀⡀⠀⠻⣿⡿⠋⠀⢠⡀⣀⣀⣁⠀⠀⠀⠀⠀⠀⢈⠗⠀⠉⠀⠀⠀⠸⣿⣿⣿⠏⡀⣀⠀⠀⠀⣀⣴⣦⣤⡀⢙⣠⣤⣶⣶⣶⡀⠀⠀⠀⠀⠹⡇⣷⣶⣤⣀⠀⠁⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⡤
⣶⣦⣤⣶⣿⣿⣿⣇⠀⠀⠩⠁⠀⠀⠀⢠⣿⣿⣿⣿⣦⡀⠠⣀⣀⣼⣶⣾⣿⣿⡅⠀⠀⠀⠙⡇⠀⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⡆⠀⠴⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢉⡙⠛⢿⣿⣿⠆⢸⣿⣿⣿⣿⠿⠛⠋⠀⢀⣠⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⣨⣅⠀⠀⠉⠛⠛⠙⢋⡠⠤⠠⠀
⣛⣻⣯⣽⢿⣿⡾⣿⣶⣶⣶⣾⡷⡀⠀⠀⠀⠻⠂⠀⠈⠁⠀⠀⠉⠛⠋⠁⠀⠊⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢙⢿⠦⠀⠀⠀⣠⠞⠀⠀⠀⠀
⣿⡿⠋⠀⠀⠉⡁⢻⣿⣿⣿⣿⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⣴⡀⠈⠙⠛⣿⠥⠬⠠⠀⣠⣿⣟⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠉⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀
⠉⠁⠀⠀⠀⠀⠁⠀⠙⢉⣛⡻⠛⠛⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠗⠀⢀⠜⠁⠀⠀⠀⠀⠀⢾⣿⣿⣿⣭⣍⣉⠁⠀⣶⢶⣤⣄⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣷⠀
⣤⣶⣶⡶⠚⠀⠀⠀⠀⠀⠀⠉⠓⢄⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⢀⣤⣄⣀⠀⠏⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠀⣤⣤⣄⣀⠉⠉⠉⠉⠂⠀⠀⣤⠀⠀⠰⠆⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣤⣄⣰⡇⠀⣀⣤⣄⡀⠀⠉⢙⣦⣤⣶⣿⣿⣿⣿⣿⣀⡀⠀⢀⣀⣀⣹⣿⣿⣿⡃⠀⣿⣿⣿⣿⣷⣦⡀⠀⣀⣀⠀⣿⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⣷⣿⣿⠃⠀⠀⠐⠋⠠⠀⢠⣤⣤⣤⣤⣄⣸⣧⣀⡈⠟⠀⣼⣏⣀⣈⣿⡄⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⡇⠀⠀⣿⡏⠀⠙⠸⣻⡇⠀⢻⣿⣿⣿⡾⣿⣿⡄⡸⠁⠀⠀⠀⠀⠀⠟⠿⡿⠋⠙⠿⡿⠿⠿⠃⠀⠀⠀
⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢿⡍⠉⠉⢉⡁⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡇⠀⢠⣼⡇⠀⠀⠀⢸⡇⠀⢸⣿⠀⣉⠙⠺⣿⣼⠃⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀
⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠛⠶⠖⠋⠀⠈⠙⢿⣿⣿⣿⡿⠛⠳⠖⠛⠓⠂⠐⠛⠓⠂⠀⠐⠚⠓⠂⠀⠻⠶⠋⠀⠀⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠂⠲⣶⠀⠲⠄⠰⠶⠾⠿⠿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣫⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠾⠤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣇⢿⡼⣾⣿⣿⡿⣿⡻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⢸⣿⣿⢻⢺⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠹⣏⣿⠿⢸⡇⡃⣃⣽⡧⢛⡟⢾⣿⣿⣿⣿⡯⣿⣿⣿⣿⣿⣿⠿⢿⣽⣿⣿⣼⣿⣻⣯⣿⡋⡿⢟⣽⣿⣿⡿⣿⣿⣯⢽⣯⣿⢸⣟⢿⣼⢸⣿⣟⣿⣿⣟⢃⣿⣿⣿⡿⢿⣿⣷⣿⣿⣿⣿⢿⣿⣿⡿⣿⣹⣥⣅⣳
⢸⣿⣶⣦⠄⠃⠙⠯⡯⣷⠬⣖⣽⣿⣿⡯⣽⡇⠼⢍⣓⣿⢿⣿⣵⣺⣟⠛⢿⠟⢿⠿⣿⣿⡛⣿⡏⣽⡗⢻⣛⣿⣿⣿⣿⣿⠛⢈⢿⣶⣼⡘⢿⣿⣿⣿⣿⠋⠻⡟⣫⣶⣿⣿⣿⣛⡛⣩⣅⣾⣿⠟⣿⣟⣁⣹⣿⣯
⠷⣿⣿⠿⠐⠢⠤⠠⢴⡶⣿⣌⣧⣶⡿⠀⢨⣷⡄⣿⣶⣤⡤⣿⣿⡿⣿⣷⠾⢃⣙⡉⠻⢿⣷⣜⡛⠙⠋⢍⣋⣘⠻⠒⢹⣯⣱⣂⡙⢿⣿⠿⠖⠹⣯⠍⡿⢧⣤⡼⠉⢿⣿⠋⠻⡈⠒⠪⠍⢉⣵⣶⣿⡖⣿⣥⣽⣿
⠨⠛⢿⠶⠛⠓⣾⣤⠈⣿⣿⣿⡟⠿⠧⠐⠾⠙⠟⠉⣹⣬⡕⠻⠏⢿⣿⡏⠳⡾⠛⡝⠛⠋⢿⠏⠏⠀⠰⢿⣿⣿⣶⠆⠀⢨⠌⢋⠁⠘⠿⣇⠦⠀⠀⠱⠲⢀⣈⣤⣤⣈⠛⣄⡀⠙⠒⠶⠾⣇⣀⣺⣿⣻⣿⣿⣿⣿
⠀⢂⣤⣤⣵⢶⣦⣼⣶⢿⣿⣿⣗⠸⣷⣿⡷⣧⡄⠘⠛⣫⣷⣶⡆⠀⠉⠀⣦⣾⣶⣾⡟⠂⠀⠀⡛⠻⠛⠉⣿⣿⠡⠀⠀⣶⣿⣿⣷⣤⠀⠈⠿⣦⣀⣀⣼⣿⠿⢿⣿⣿⡇⠙⠻⠀⢈⣤⣌⣉⣙⣹⡙⠛⠿⠷⢼⣿
⠀⠈⢉⠛⠿⡷⢸⠒⠉⢘⣿⣿⣇⠘⠀⠙⠡⣥⣉⠀⠐⠛⠛⠉⢀⣤⣴⣴⢻⣿⣿⣯⢤⣄⠀⣠⣷⣶⣦⠀⠀⠡⣶⣶⣶⡏⠿⣿⡿⠯⠀⠀⠀⠈⡉⠙⣛⣯⣄⡐⢿⣿⠁⠀⠀⠀⠜⠛⠻⠿⢟⠛⠃⠀⠀⢩⣿⣿
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣯⢷⢀⣿⣶⣿⣿⠆⠀⠀⠀⠀⢤⠨⠛⠉⠀⢿⣶⣿⣠⣿⣷⣾⣟⠁⠀⠀⠀⡲⠀⠘⡁⠀⠀⠻⢿⠇⡀⠀⠀⠐⣠⣾⡿⣿⣿⣿⣷⡟⣶⣄⠀⠐⢉⡧⠀⠀⠃⠀⠀⠀⠀⠀⠙⠿
⠀⠀⠀⠀⢿⣿⣿⡇⣿⣿⣿⣭⡇⠀⢲⣿⣿⣻⣿⠇⠀⠐⢒⣦⣷⣶⣾⣇⣀⣀⣈⡸⣻⣿⣿⣿⣿⠇⡀⠀⣠⣷⣿⣿⣿⠄⠀⠀⠠⢶⣿⣿⣿⣿⠇⠙⣹⣿⣿⣿⣿⡄⠈⠙⠇⠀⠏⣠⣴⣾⣿⣿⣆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⢻⡟⢻⣟⣛⣿⣿⡅⠀⠛⠿⡛⠃⠀⠀⠀⠸⠿⠿⠛⠛⠿⢿⣿⣿⠁⢬⣿⣿⣿⣿⡶⣿⠆⠙⠛⠛⠁⠀⠀⠀⠀⢶⡄⠈⡩⠛⠁⠀⠀⠰⣿⣿⣶⡆⠀⠀⠀⠀⠠⣴⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠗⠁⠀⠈⢿⣿⣿⡟⠁⢘⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠳⠀⠔⠉⠀⠀⠈⣿⣿⣿⣿⠃⠀⠁⠀⠀⠀⠀⢀⣀⠁⠀⠏⢁⣠⣤⣤⣤⠀⠀⠀⠈⠙⢻⡗⣦⣤⣀⠀⠀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀
⣀⠀⢀⣠⣶⣾⣿⡇⠀⠈⢱⠏⠀⠀⠀⠁⣿⣿⣷⣶⣄⠀⠀⡀⠀⢀⣡⣤⣴⣶⡆⠀⠀⠈⠙⠿⠁⠀⢸⣷⣶⣦⣤⣾⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠻⠿⢿⣷⡀⠀⢀⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⣿⣿⣿⡄⠘⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⢀⣈⣀⣀⣈⣻⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠇⠉⠉⠛⢿⣿⡿⣿⠛⠛⠉
⣿⣿⣿⣿⣿⣧⣀⣀⣀⣈⣀⣀⣀⡀⠀⠀⠀⢶⠄⠈⠙⠟⠁⠀⠛⠻⠿⠛⠉⠠⠀⠀⢨⣶⣿⣿⣿⣿⣿⣿⡟⡿⣏⣻⣿⣿⣿⣿⣿⣿⡿⠁⠐⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠷⣦⣀⠀⠀⠀⠀⢀⡞⠉⠀
⣭⣤⣶⠒⠙⠿⡏⣿⣿⣿⣿⣿⢟⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠛⢿⣿⣿⣿⠟⠛⠉⠛⠛⣿⣿⣿⣿⣿⣿⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡮⠋⠀⠀⠀⠀⠊⠀⠀⠀
⣿⠋⠀⠀⠀⠀⡄⠉⢿⣿⣿⣿⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⣷⡀⠀⠀⣠⠖⠉⠉⠀⠀⠚⢳⣶⣾⣿⣾⣿⡿⠿⠓⣀⣀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠋⠀⠀⠀⣀⣤⣤⣤⣴⣶⣶
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠬⢤⣉⠁⣤⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠟⠁⠀⠘⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣾⣷⣤⢆⣀⠉⠉⠉⠓⠺⠟⠢⠀⠀⣤⠀⠀⠀⠀⡄⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣾⣿⡟⣉⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⠉⠁⣀⣤⣄⣀⡀⠀⣀⣤⣅⡀⠘⠛⢛⣧⣬⣀⣀⣀⣀⣤⣴⣶⡀⠀⢀⣀⣀⣹⣿⣿⣿⣿⣿⣸⣿⣿⣶⣦⣄⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⣿⢿⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡀⠻⠇⣼⣏⣀⣈⣿⡄⢠⡟⣁⣴⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⢿⡏⠀⠙⠛⢻⣯⠃⢸⣿⣿⣿⣿⣿⣿⣆⡼⠁⠀⠀⠀⠀⠀⠀⢠⢿⣿⣿⢿⣿⣿⣿⣿⠟⣻⠋
⣿⣿⡿⠁⠀⠀⠈⠁⠀⠀⢰⣶⣶⣶⣶⣦⣤⣈⡙⠻⣷⠀⢿⡍⠉⠉⢉⠁⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⣿⡇⠀⠀⠀⢸⡯⠁⢸⣿⠻⣿⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣀⠀⠐⠙⠛⠁⠀⠈⠋⠛⠙⠂⠀⠀
⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣏⡒⠈⠛⠶⠖⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠂⠐⠻⠓⠂⠀⠐⠚⠓⠂⠈⠻⠶⠋⠀⠀⢹⠯⠍⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⠆⠲⣶⠀⢶⠆⠰⠶⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⢀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢛⠿⢟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⡇⣿⣸⣾⣿⣿⡿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⡯⣿⣿⢸⡇⣿⣿⣿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⡿
⠹⣏⢿⡿⢸⡇⡇⣃⣹⡧⢛⡟⢼⣿⣿⣿⣿⣫⣟⣿⣿⣿⣽⣿⠿⢿⣽⣿⣿⣭⣿⣻⣯⣿⡋⣿⠇⣿⣿⣿⡿⣿⣿⣿⣽⣿⢸⢿⣟⢿⢸⢷⣿⣿⣿⣿⣟⠘⣿⣿⣿⡿⢿⣿⣷⣿⣿⣿⣿⢿⣿⣿⡿⣿⣛⣥⣭⣃
⢸⣿⣬⣤⡄⠇⠙⣿⣫⣷⠭⡔⣽⡿⣿⡯⣽⡇⡼⢍⣓⣿⡿⣿⣗⣒⣟⡋⢿⠿⢻⡯⣿⣿⣏⣿⣏⣹⡷⢻⣛⣳⣿⣿⣿⣿⡜⠈⣿⣴⣼⣈⢿⣽⣿⣿⣛⠛⠛⠟⢻⣵⣿⣿⣿⡟⡛⣫⣭⣸⣿⠾⣻⣿⣋⣸⣿⣮
⢮⣿⡿⢶⠀⠠⡄⢠⣴⣶⣿⣌⣧⣵⣿⡁⢈⣷⡄⣻⣿⣬⣄⢿⣿⣿⣿⡷⣾⠖⡞⠛⠹⢿⣷⡼⣛⠙⠏⡞⡛⢙⠿⠶⠹⣿⣕⣒⣈⢿⣿⣿⠶⠏⢿⡟⢻⣷⣤⣬⡝⢳⣿⡿⠻⡊⢂⠲⠍⠉⢫⣶⣾⣷⢛⣫⣼⣿
⢈⠻⢿⢤⠓⠶⣤⣄⠀⣿⣿⣿⡿⣿⢷⠰⢮⠝⠟⠙⢛⣏⣱⢾⠿⢹⣯⡽⠱⡾⠟⢓⠲⠂⢿⡍⠏⠀⠠⢿⣿⣿⣷⡦⠀⠨⡍⠙⠋⠀⠻⣿⠰⠀⠀⠁⢖⠆⣐⣁⣠⣀⠙⢇⡀⠘⠓⠶⢶⠿⠁⠙⣿⡛⢿⣿⣿⣿
⠀⢀⣤⣀⣲⣶⣧⣩⣥⢿⠿⢿⣧⠐⣶⣶⣦⣥⣀⠰⠿⢟⣯⣤⣄⠀⠙⠁⢀⣴⣶⣶⡾⠆⠀⠁⣐⠛⠿⠋⢿⣿⠏⠀⠀⢸⣶⣿⣷⣤⡀⠀⠻⢶⡀⠀⢠⣿⣿⠿⢿⣿⣿⠘⠛⠆⠈⣀⣤⡈⠙⠛⡟⠛⠿⠿⣿⣭
⠀⠈⠛⠻⢿⣦⢨⠭⠋⠘⣿⣿⡿⠀⠉⠛⢫⣏⡙⠀⠀⠿⠿⠋⠁⣀⣤⣤⢸⣿⣿⣿⣁⡄⠀⢀⣬⣴⣶⡀⠀⠙⣠⣶⣤⣽⠛⣾⣿⣿⡅⠀⠀⠀⡛⠛⢛⢿⣃⣀⢹⣯⡍⠁⠀⠀⠀⡻⠿⢿⣿⣿⠿⠇⠀⠀⢐⣿
⠀⠀⠀⠀⠀⣡⣼⣶⣶⣾⣿⣿⣿⣦⡀⣶⣶⣿⣿⠆⠀⠀⠀⠀⢠⠉⠻⠛⠀⢿⣿⣿⡇⣻⣦⣬⣟⠋⠀⠀⠀⢰⡄⠩⠋⠁⠀⠸⣿⣿⠀⠀⠀⠀⠌⣁⣶⣿⣿⣿⣷⣿⣷⣄⡀⠀⠰⢲⡄⠀⠀⠊⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⣾⣿⣿⡏⣿⣿⣿⡿⡟⠈⢡⣽⣿⡿⣿⣷⠀⠂⠤⢤⣬⣤⣶⣆⠀⠀⠙⠋⡾⣿⣿⣿⣿⣷⠆⠀⠀⣼⣴⣶⣿⣧⠀⠀⠀⢩⣼⣷⣶⣶⣾⠏⠛⠳⣿⣿⣿⣿⡉⠙⠻⠀⠀⠋⣀⣤⣴⣶⣿⣄⠀⠀⠀⠀
⠀⠀⠀⠀⠈⢻⣿⡷⢿⡿⠿⢿⣷⡖⠀⢿⣿⣷⠎⠁⠀⠀⢠⣿⣿⡿⠛⢿⣿⣿⣿⡿⢈⣻⣿⣿⣿⣧⣴⡦⠾⠿⠿⠟⠉⠀⠀⠀⢠⣄⠉⠛⢻⠟⠉⠀⠀⢩⣿⣿⣭⡍⠁⠀⠀⠀⣀⣠⣿⣿⣿⣿⡿⠛⠃⠀⠀⠀
⠀⠀⠀⠀⠀⡦⠊⠉⠈⣿⣿⣿⣿⠃⠀⠀⠀⠑⠰⠀⠀⠀⠀⠀⠀⠀⢰⡄⢉⠽⠛⠀⠀⢻⣿⣿⣿⣿⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⣁⣀⡀⠀⠀⠀⠙⠿⣿⠧⣄⣀⠀⠀⠀⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣠⣥⣤⡤⠀⠙⢻⡟⠃⠀⠀⠒⢰⣦⣤⣤⡀⠀⠀⠀⠀⠀⠜⢁⣀⣠⣤⠀⠀⠈⠻⢿⡿⠃⡄⢤⣤⣤⣀⣀⣠⣴⣶⣤⣠⣶⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠈⠀⢿⣿⣷⣆⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤
⣤⣠⣴⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣦⡀⠑⣶⣶⣿⣿⣿⣿⣿⡆⠀⠀⠀⠈⠃⠀⠁⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠻⠂⠀⢠⠺⠿⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡛⠋⠁⠀⠐⠀⠀⠀⠀⠂⠀⠀⢄⠉⠙⠻⣿⠟⠀⠸⣿⣿⣿⠟⠋⡉⠁⠀⢂⣀⣴⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠠⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡀⠀⠀⠈⠙⠛⠙⠋⣀
⣿⣿⠿⠿⣿⣿⣤⣤⣤⣤⣤⣤⣦⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡐⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠇⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡤⠀⠀⠀⢀⡴⠃
⣶⣶⡾⠋⠈⠛⠃⢿⣿⣿⣿⣿⢏⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢨⡄⠈⠛⠛⠻⣿⣉⠉⠀⠉⢉⣝⡿⠿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠉⠀⠀⠀⠀⠀⠀⢀
⡿⠋⠀⠀⠀⠀⠆⠈⠛⠿⢿⡿⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠄⠀⢀⡾⠃⠀⠀⠀⠀⠈⢩⣿⣿⣿⣿⣿⣛⣛⠋⠛⠛⠶⣤⣰⣦⣄⠀⣤⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⣤⣶⣾⣿⣿⣿
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠐⠒⠲⢤⡀⣿⢸⡇⣀⣤⣄⣀⠀⠀⣀⣤⣅⡀⠀⣤⣤⣤⣤⣼⡁⣀⣀⠀⠀⣀⡀⠀⢀⣀⣀⣨⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠲⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿
⣶⣾⣿⡟⣁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⢸⣧⣀⡀⢛⡶⣼⣏⣀⣈⣿⡄⢠⡟⠁⠃⠘⣇⣠⣿⣶⣶⣿⣿⠀⠀⣿⡏⠀⠙⠻⢿⣿⣿⣿⣿⢸⣿⣿⣿⡶⣄⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠿⣿⣿⡿⠇⠀⢀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡉⠙⠻⣷⠁⢿⡍⠉⠉⢉⠁⠸⣇⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢹⡇⠀⠀⠀⢸⡏⣵⣿⣿⠸⣿⣿⣿⣷⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⡞⣻⣿⡿⠿⢿⣿⣿⣿⠃
⣿⣿⡿⠁⠀⠀⠈⠁⠀⠂⢀⣶⣶⣶⣶⣶⣾⣷⣶⠞⠋⢀⡈⠛⠶⠖⠋⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⠚⠓⠂⠀⠐⠚⠓⠃⠊⠻⠶⠛⠛⠛⢻⢯⣽⡿⡀⠀⠀⠀⠀⠴⠖⠀⠈⠈⠉⠀⠀⠀⠉⠁⠁⠀
⣿⠿⠶⣶⠀⢴⠴⠰⠶⠶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣈⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠏⠉⠉⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠏⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠈⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⠯⠍⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠑⠒⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠁⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⡛⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣿⣿⠯⠀⠀⠀⠀⠋⠉⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠏⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⠏⠛⠛⠛⠁⠀⠀⠀⢿⣿⣿⣿⣿⣿
⠀⢹⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠠⠀⠈⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⢀⢙⡿⠿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠂⠀⠀⠀⠀⠰⠞⠻⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠐⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣿⣿⡿
⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⡀⣀⡀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⣀⠀⢀⣀⠀⠀⣀⡀⠀⢀⣀⠀⣀⣀⢀⣀⡀⠀⣀⣇⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⡀⠀⠃⠀⣴⠃⠀⠀⢳⠀⢠⡟⠀⠀⠈⠃⠀⢸⠀⠀⠀⡇⠀⠀⢸⠏⠀⠈⠀⢸⡇⠀⠀⡇⠀⠀⠈⢷⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠛⣦⠀⢿⡉⠉⠉⢉⠀⠸⣇⠀⠀⢀⡀⠀⢸⠀⠀⢠⡇⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠈⣧⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠃⠀⠈⠓⠒⠒⠋⠀⠀⠙⠒⠐⠊⠀⠀⠈⠓⠒⠉⠓⠂⠐⠚⠒⠂⠀⠐⠚⠓⠂⠀⠓⠒⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠈⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠻⠙⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠈⠉⠹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡏⡀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠉⠉⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠈⠠⠌⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠘⠉⠻⣿⣿⣿⠁⠀⡀⠉⠀⠀⠀⠐⢻⣿⣿⣿⣿
⠈⣿⣿⠟⠋⠉⠑⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠈⠁⠀⠀⠀⠀⠀⠀⠈⠋⠻⣿⣿
⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣿⣿⣿⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⢙⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠠⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⡀⣀⡀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⢀⣀⠀⢀⣀⠀⠀⣀⡀⠀⢀⣀⠀⣀⣀⢀⣀⡀⠀⣀⣇⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⡀⠀⠃⠀⣴⠃⠀⠀⢳⠀⢠⡟⠀⠀⠈⠃⠀⢸⠀⠀⠀⡇⠀⠀⢸⠏⠀⠈⠀⢸⡇⠀⠀⡇⠀⠀⠈⢷⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠛⣦⠀⢿⡉⠉⠉⢉⠀⠸⣇⠀⠀⢀⡀⠀⢸⠀⠀⢠⡇⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠈⣧⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠃⠀⠀⠑⠒⠒⠋⠀⠀⠉⠒⠐⠊⠀⠀⠈⠓⠒⠉⠓⠂⠐⠚⠒⠂⠀⠐⠚⠓⠂⠀⠓⠒⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠁⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡏⠛⠃⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢈⠋⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠉⠁⢻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⠭⠍⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠂⠀⠀⠀⠈⠉⠘⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠁⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⠇⠉⠉⠉⠁⠀⠀⠀⠘⣿⣿⣿⣿
⠈⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠐⠙⠻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿
⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⣙⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠠⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⡀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⠀⢀⣀⠀⠀⣀⡀⠀⢀⣀⠀⣀⣀⢀⣀⡀⠀⣀⣇⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⡀⠀⠃⠀⣼⠃⠀⠀⢳⠀⢠⡟⠀⠀⠈⠃⠀⢸⠀⠀⠀⡇⠀⠀⢸⠏⠀⠈⠀⢸⡇⠀⠀⡇⠀⠀⠈⢷⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠛⣦⠀⢿⡉⠉⠉⢉⠀⠸⣇⠀⠀⢀⡀⠀⢸⠀⠀⢠⡇⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠈⣧⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠒⠃⠀⠀⠑⠒⠒⠋⠀⠀⠉⠒⠐⠊⠀⠀⠈⠓⠒⠉⠓⠂⠐⠚⠒⠂⠀⠐⠚⠓⠂⠀⠓⠒⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡏⠃⠁⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠯⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠈⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠠⠀⠀⠀⢸⠿⣿⣿⣿⣿⣿⣿⣿⡏⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠐⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠄⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⢿⣿⠄⠀⠀⠀⢻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠈⠐⠘⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠁⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠻⣿⣿⡟⠨⠁⠨⠉⠀⠀⠀⠀⢻⣿⣿⣿
⢸⣿⡟⠉⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠸⣿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿
⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢾⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⣉⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠠⠀⠀⠄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⡀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⠀⢀⣀⠀⠀⣀⡀⠀⢀⣀⠀⣀⣀⢀⣀⡀⠀⣀⣇⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣀⠀⠃⠀⣼⠃⠀⠀⢳⠀⢠⡟⠀⠀⠈⠃⠀⢸⠀⠀⠀⡇⠀⠀⢸⠏⠀⠈⠀⢸⡇⠀⠀⡇⠀⠀⠈⢷⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠛⣦⠀⢿⡉⠉⠉⢉⠀⠸⣇⠀⠀⢀⡀⠀⢸⠀⠀⢠⡇⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠈⣧⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠃⠀⠀⠑⠒⠒⠋⠀⠀⠙⠒⠒⠊⠀⠀⠈⠓⠒⠉⠓⠂⠐⠚⠒⠂⠀⠐⠚⠓⠂⠀⠓⠒⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠩⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠁⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢛⠟⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠉⠁⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢹⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢼⡿⣿⣿⣿⣿⣿⣿⣿⡇⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠿⠒⠒⠄⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⡿⣿⣿⣿⡇⠀⠀⠀⢘⣿⣿⣿
⣿⣿⣿⣿⠿⠿⠉⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠉⠁⠀⠀⠀⠀⢉⢻⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿
⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠻⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠄⡈⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⡀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⢀⣀⠀⢀⣀⠀⠀⣀⡀⠀⢀⣀⠀⣀⣀⢀⣀⡀⠀⣀⣇⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⡀⠀⠃⠀⣴⠃⠀⠀⢳⠀⢠⡟⠀⠀⠈⠃⠀⢸⠀⠀⠀⡇⠀⠀⢸⠏⠀⠈⠀⢸⡇⠀⠀⡇⠀⠀⠈⢷⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠛⣦⠀⢿⡉⠉⠉⢉⠀⠸⣇⠀⠀⢀⡀⠀⢸⠀⠀⢠⡇⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠈⣧⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠃⠀⠀⠑⠒⠒⠋⠀⠀⠉⠒⠐⠊⠀⠀⠈⠓⠒⠉⠓⠂⠐⠚⠒⠂⠀⠐⠚⠓⠂⠀⠓⠒⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠊⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣟⣛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠝⢹⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⢀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠈⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠃⠀⠀⠀⡟⠛⣿⣿⣿⣿⣿⣿⣿⠇⠄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⡿⠻⠻⠿⠿⠀⠀⠀⢀⣿⣿⣿
⣿⣿⣿⠿⠟⢛⠀⠀⠀⠀⠀⠀⢀⠚⢿⣿⣿⣿⣿⡛⠂⠀⠀⠀⠈⠛⠻⣿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠓⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿
⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠄⠤⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠄⠀⠂⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⡀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⠀⢀⣀⠀⠀⣀⡀⠀⢀⣀⠀⣀⣀⢀⣀⡀⠀⣀⣇⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣀⠀⠃⠀⣼⠃⠀⠀⢳⠀⢠⡟⠀⠀⠈⠃⠀⢸⠀⠀⠀⡇⠀⠀⢸⠋⠀⠈⠀⢸⡇⠀⠀⡇⠀⠀⠈⢷⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠛⣦⠀⢿⡉⠉⠉⢉⠀⠸⣇⠀⠀⢀⡀⠀⢸⠀⠀⢠⡇⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠈⣧⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠃⠀⠀⠑⠒⠒⠋⠀⠀⠉⠒⠐⠊⠀⠀⠈⠓⠒⠉⠓⠂⠐⠚⠒⠂⠀⠐⠚⠓⠂⠀⠓⠒⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠐⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠓⡂⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠙⠉⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢹⣿⣿
⣿⣿⣿⣿⣿⣿⠠⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠭⠍⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠸⣿⣿
⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠈⠒⢙⣿⣿⣿⣿⣿⣿⣿⠈⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠃⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⣿⣿⡿⠙⠋⠈⠉⠁⠀⠀⠀⠘⣿⣿
⣿⡿⠛⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⠿⠛⠛⠄⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⡀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⠀⢀⣀⠀⠀⣀⡀⠀⢀⣀⠀⣀⣀⢀⣀⡀⠀⣀⣇⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣀⠀⠃⠀⣴⠃⠀⠀⢳⠀⢠⡟⠀⠀⠈⠃⠀⢸⠀⠀⠀⡇⠀⠀⢸⠋⠀⠘⠀⢸⡇⠀⠀⡇⠀⠀⠈⢷⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠛⣦⠀⢿⡉⠉⠉⢉⠀⠸⣇⠀⠀⢀⡀⠀⢻⠀⠀⢠⡇⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠈⣧⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠃⠀⠀⠑⠒⠒⠋⠀⠀⠙⠒⠐⠊⠀⠀⠈⠓⠒⠉⠓⠂⠐⠚⠒⠂⠀⠐⠚⠓⠂⠀⠓⠒⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠑⠒⠂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⠿⢿⣿⣿
⣿⣿⣿⣿⣿⡿⠉⠁⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠟⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠈⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿
⣿⣿⣿⣿⣿⡇⠆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠐⠂⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠈⠉⠈⢿⣿⣿⣿⣿⣿⠧⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠉⡙⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⠈⠉⠀⠀⠀⠀⠀⠀⠈⢹⣿
⣿⠛⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢲⢿⠿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠂⠈⢛⠻⣿⣿⣿⣿⣿⡟⠉⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠉⠛⢿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⡀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⠀⢀⣀⠀⠀⣀⡀⠀⢀⣀⠀⣀⣀⢀⣀⡀⠀⣀⣇⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣀⠀⠃⠀⣴⠃⠀⠀⢳⠀⢠⡟⠀⠀⠈⠃⠀⣼⠀⠀⠀⡇⠀⠀⢸⠋⠀⠘⠀⢸⡇⠀⠀⡇⠀⠀⠈⢷⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠛⣦⠀⢿⡉⠉⠉⢉⠀⠸⣇⠀⠀⢀⡀⠀⢻⠀⠀⢠⡇⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠈⣧⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠃⠀⠀⠑⠒⠒⠋⠀⠀⠙⠒⠒⠊⠀⠀⠈⠓⠒⠉⠓⠂⠐⠚⠒⠂⠀⠐⠚⠓⠂⠀⠓⠒⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠙⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣛⡛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠻⠛⣿⣿
⣿⣿⣿⣿⣿⠇⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠐⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠏⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠁⣿⣿
⣿⣿⣿⣿⣿⠐⠀⠀⠀⠀⡿⡿⣿⣿⣿⣿⣿⣿⣿⡇⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠂⠀⠀⠀⠀⠀⠀⠠⠟⢿⣿⣿⣿⡿⢿⡿⣿⡿⠀⠀⠀⠀⣸⣿
⣿⣿⡿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠚⢿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠯⢌⠀⠀⠀⠀⠀⠀⣉⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠟⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠟⠹⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⢀⣀⠀⢀⣀⠀⠀⣀⡀⠀⢀⣀⠀⣀⣀⢀⣀⡀⠀⣀⣇⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣀⠀⠃⠀⣴⠃⠀⠀⢳⠀⢠⡟⠀⠀⠈⠃⠀⣼⠀⠀⠀⡇⠀⠀⢸⠋⠀⠘⠀⢸⡇⠀⠀⡇⠀⠀⠈⢷⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠛⣦⠀⢿⡉⠉⠉⢉⠀⠸⣇⠀⠀⢀⡀⠀⢹⠀⠀⢠⡇⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠈⣧⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠃⠀⠀⠑⠒⠒⠋⠀⠀⠙⠒⠒⠊⠀⠀⠈⠓⠒⠉⠓⠂⠐⠚⠒⠂⠀⠐⠚⠓⠂⠀⠓⠒⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠠⠡⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠿⡯⠏⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⠏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠒⠒⢻⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠁⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢁⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣟⠈⠀⠀⠀⠸⣿⡿⣿⣿⣿⣿⣿⣿⣿⡇⠄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠂⠀⠀⠀⠀⠀⠚⠛⢿⣿⣿⡿⣿⡿⠿⠿⠃⠀⠀⠀⠠⣿
⣿⡿⠿⠛⡃⠀⠀⠀⠀⠀⢀⠀⡙⢻⣿⣿⣿⣿⣟⠂⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠒⠀⠀⠀⠀⠀⠀⣬⠿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠘⠉⠑⠃⠹⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⣻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⣀⠀⢀⣀⠀⠀⣀⡀⠀⢀⣀⠀⣀⣀⢀⣀⡀⠀⣀⣇⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣀⠀⠃⠀⣴⠃⠀⠀⢳⠀⢠⡟⠀⠀⠈⠃⠀⣼⠀⠀⠀⡇⠀⠀⢸⠋⠀⠘⠀⢸⡇⠀⠀⡇⠀⠀⠈⢷⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠛⣦⠀⢿⡉⠉⠉⢉⠀⠸⣇⠀⠀⢀⡀⠀⢸⠀⠀⢠⡇⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠈⣧⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠃⠀⠀⠑⠒⠒⠋⠀⠀⠙⠒⠒⠊⠀⠀⠈⠓⠒⠉⠓⠂⠐⠚⠒⠂⠀⠐⠚⠓⠂⠀⠓⠒⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠯⠙⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠓⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠉⠉⠹⣿
⣿⣿⣿⣿⡏⡀⠉⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠈⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠈⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣛⠛⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣇⠁⠀⠀⠀⠸⠛⠻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠛⠙⠻⣿⣿⣿⠛⠻⠛⠿⠏⠀⠀⠀⠀⢾
⣿⠿⠛⠛⠂⠀⠀⠀⠀⠀⡀⠀⢙⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⢉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠓⠢⠀⠀⠀⠀⠀⠀⠤⠿⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡇⠀⠂⠈⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⠁⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠁⠃⠹⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⣀⠀⢀⣀⠀⠀⣀⡀⠀⢀⣀⠀⣀⣀⢀⣀⡀⠀⣀⣇⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣀⠀⠃⠀⣴⠃⠀⠀⢳⠀⢠⡟⠀⠀⠈⠃⠀⣼⠀⠀⠀⡇⠀⠀⢸⠋⠀⠘⠀⢸⡇⠀⠀⡇⠀⠀⠈⢷⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠛⣦⠀⢿⡉⠉⠉⢉⠀⠸⣇⠀⠀⢀⡀⠀⢹⠀⠀⢠⡇⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠈⣧⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠃⠀⠀⠑⠒⠒⠋⠀⠀⠙⠒⠒⠊⠀⠀⠈⠓⠒⠉⠓⠂⠐⠚⠒⠂⠀⠐⠚⠓⠂⠀⠓⠒⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿
⣿⣿⣿⣿⠃⠊⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢉⠙⠉⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠁⢸
⣿⣿⣿⡇⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠂⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢉⠉⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈
⣿⣿⣿⡧⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇⠃⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠇⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠙⢻⣿⣿⠿⠿⠛⠛⠛⠁⠀⠀⠀⠐
⠟⠛⠩⠀⠀⠀⠀⠀⠀⠤⠀⢌⣹⣿⣿⡿⠿⢯⠉⠀⠀⠀⠀⠀⣀⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠶⠛⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢀⠐⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠃⢻⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⣀⠀⢀⣀⠀⠀⣀⡀⠀⢀⣀⠀⣀⣀⢀⣀⡀⠀⣀⣇⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣀⠀⠃⠀⣴⠃⠀⠀⢳⠀⢠⡟⠀⠀⠈⠃⠀⣿⠀⠀⠀⡇⠀⠀⢸⠋⠀⠘⠀⢸⡇⠀⠀⡇⠀⠀⠈⢷⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠛⣦⠀⢿⡉⠉⠉⢉⠀⠸⣇⠀⠀⢀⡀⠀⢹⠀⠀⢠⡇⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠈⣧⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠃⠀⠀⠑⠒⠒⠋⠀⠀⠙⠒⠐⠊⠀⠀⠈⠓⠒⠉⠓⠂⠐⠚⠒⠂⠀⠐⠚⠓⠂⠀⠓⠒⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠑⠲⠒⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠁⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣛⣻
⣿⣿⣿⡏⠉⠉⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠯⠙⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠁⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀
⣿⣿⣿⢠⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠈⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠩⠍⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀
⣿⣿⡿⠄⠀⠀⠀⠀⠉⠩⠙⣿⣿⣿⣿⣿⣿⣿⠀⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠁⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⢿⣿⡟⠭⠙⠛⣛⠃⠀⠀⠀⠀
⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢌⣿⣿⣿⠿⠿⡃⠀⠀⠀⠀⠀⢀⡈⣙⢿⢿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠒⠛⠻⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⢻⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⣀⠀⢀⣀⠀⠀⣀⡀⠀⢀⣀⠀⣀⣀⢀⣀⡀⠀⣀⣇⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣀⠀⠃⠀⣴⠃⠀⠀⢳⠀⢠⡟⠀⠀⠈⠃⠀⣿⠀⠀⠀⡇⠀⠀⢸⠋⠀⠘⠀⢸⡇⠀⠀⡇⠀⠀⠈⢷⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠛⣦⠀⢿⡉⠉⠉⢉⠀⠸⣇⠀⠀⢀⡀⠀⢹⠀⠀⢠⡇⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠈⣧⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠃⠀⠀⠑⠒⠒⠋⠀⠀⠙⠒⠒⠊⠀⠀⠈⠓⠒⠉⠓⠂⠐⠚⠒⠂⠀⠐⠚⠓⠂⠀⠓⠒⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠛⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⠿
⣿⣿⡿⠁⠅⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠓⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠠⠁⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠂⠂
⣿⣿⡇⠄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠐⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠥⠄⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀
⣿⣿⠧⠀⠀⠀⠀⠈⠛⠋⢹⣿⣿⣿⣿⣿⣿⣟⠈⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠿⣿⡟⠉⠙⢉⢉⠉⠀⠀⠀⠀
⠉⠙⠀⠀⠀⠀⠀⠀⠐⠐⠀⣽⣿⣿⡿⠟⠛⠂⠀⠀⠀⠀⠀⣀⠬⢙⣿⢿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠁⠂⠀⠀⠀⠀⠀⠀⠓⠛⠻⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠈⠀⠀⠈⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠊⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠽⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⣀⠀⢀⣀⠀⠀⣀⡀⠀⢀⣀⠀⣀⣀⢀⣀⡀⠀⣀⣇⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣀⠀⠃⠀⣴⠃⠀⠀⢳⠀⢠⡟⠀⠀⠈⠃⠀⣿⠀⠀⠀⡇⠀⠀⢸⠋⠀⠘⠀⢸⡇⠀⠀⡇⠀⠀⠈⢷⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠛⣦⠀⢿⡉⠉⠉⢉⠀⠸⣇⠀⠀⢀⡀⠀⢻⠀⠀⢠⡇⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠈⣧⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠃⠀⠀⠑⠒⠒⠋⠀⠀⠙⠒⠒⠊⠀⠀⠈⠓⠒⠉⠓⠂⠐⠚⠒⠂⠀⠐⠚⠓⠂⠀⠓⠒⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠡⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡻⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⠍⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠛⠟
⣿⣿⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠋⠁⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠒⠀⠂⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠈
⣿⡿⠐⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡤⠄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠐⠒⠀⠀⠀⠀⠀⠀⢼⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀
⣿⣟⠂⠀⠀⠀⠀⠀⠁⠈⢿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠹⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠛⠀⠀⠀⠀⠀⠉⡙⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡟⠁⠂⠉⠩⠍⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢲⠿⣿⡿⠟⠛⠣⠀⠀⠀⠀⠀⠀⠤⠠⢹⡿⢿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠙⡿⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⣀⠀⢀⣀⠀⠀⣀⡀⠀⢀⣀⠀⣀⣀⢀⣀⡀⠀⣀⣇⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣀⠀⠃⠀⣴⠃⠀⠀⢳⠀⢠⡟⠀⠀⠈⠃⠀⣾⠀⠀⠀⡇⠀⠀⢸⠋⠀⠘⠀⢸⡇⠀⠀⡇⠀⠀⠈⢷⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠛⣦⠀⢿⡉⠉⠉⢉⠀⠸⣇⠀⠀⢀⡀⠀⢻⠀⠀⢠⡇⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠈⣧⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠃⠀⠀⠑⠒⠒⠋⠀⠀⠙⠒⠒⠊⠀⠀⠈⠓⠒⠉⠓⠂⠐⠚⠒⠂⠀⠐⠚⠓⠂⠀⠓⠒⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡭⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠐⠐⠂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⠿⠛⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠟⠃⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠂⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠉⠛
⣿⡇⣉⠉⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠈⠠⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣁⣄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠏⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀
⣿⡉⠁⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠰⠚⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠁⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠈⠉⠂⠀⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀
⠿⠍⠀⠀⠀⠀⠀⠀⣁⠚⢿⣿⣿⣿⣿⣿⠿⠄⠀⠀⠀⠀⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠭⡍⠀⠀⠀⠀⠀⠀⣌⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⡟⠈⠉⡠⠄⠒⠀⠀⠀⠀
⠁⠀⠀⠀⠀⠀⠀⠉⠀⠘⡷⢿⠿⠟⠋⠉⠃⠀⠀⠀⠀⠀⠐⠓⠲⢽⡟⣿⣿⣿⣿⣿⣿⡿⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⠟⢿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠒⠒⠺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠠⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⡀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⠀⢀⣀⠀⠀⣀⡀⠀⢀⣀⠀⣀⣀⢀⣀⡀⠀⣀⣇⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣀⠀⠃⠀⣼⠃⠀⠀⣳⠀⢠⡟⠀⠀⠘⠃⠀⣿⠀⠀⢸⡇⠀⠀⢸⠋⠀⠘⠀⢸⡇⠀⠀⡇⠀⠀⠈⢷⡀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠛⣦⠀⢿⡉⠉⠉⢉⠀⠸⣇⠀⠀⢀⡀⠀⢿⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠘⣧⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠃⠀⠈⠑⠒⠒⠋⠀⠀⠙⠒⠒⠊⠀⠀⠈⠓⠒⠉⠓⠂⠐⠚⠒⠂⠀⠐⠚⠓⠂⠀⠓⠒⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣶⣾⣶⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠟⠛⠛⠧⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢛⡛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠈⠉⠚⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢟⣋
⢸⠿⠟⠓⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣦⣤⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⡿⠿⠛⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣵⣆⠐⠂⠀⠀⠈⢪⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠈⠛
⠮⠀⠀⠀⣀⡘⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⠉⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠇⢀⠀⠀⠀⢠⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠠⡀⢉⡙⠻⣿⣿⣿⣿⡿⢛⣛⣛⠁⠀⠀
⠁⠀⠀⠀⠀⠸⠷⣶⣶⣽⣿⣿⣿⣿⣿⣷⠉⠀⠀⠀⠘⢣⣤⣽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠇⢀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠈⠀⠀⠄⠀⠀⠢⠀⠀⠁⠀⢫⢻⡟⣻⣿⣿⡿⠛⠃⠀⠀
⠀⠀⠀⡀⠀⠀⠀⠀⠙⢿⠿⠻⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⢉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠠⠀⠀⣀⠄⠀⢀⠚⠄⠐⠂⠈⢻⠻⣿⣿⠏⠀⠀⢀⣠⣄⣀⣀⡀⠀⠈⠀⠀⠀⠀⠀⠸⠁⠋⠀⠀⠀⠐⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠒⣀⠀⢠⣠⣤⡄⠤⠀⢼⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⡀⠀⠀⠀⠀⠐⠠⠠⠤⠄⢀⠀⠀⠀⣽⡟⠒⠊⠛⠉⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⡤⠀⠀⠀⠀⠀⠀⠀⣤⡄⣄⠀⠀⠀⢸⣿⣿⣷⣴⣤⡠⣟⣯⣝⣿⣿⣿⣯⢶⠖⠛⠓⠛⠻⠓⠀⠀⢀⠀⠐⠀⠀⡀⢁⠦⠉⠳⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢄⣀⣀⠀⠀⠀⠀⠀⠀⠸⡿⠟⡇⠃⠀⠀⠉⠙⢉⠛⠿⢷⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠈⠑⠠⠀⠀⢀⣀⠀⢀⠀⠤⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⡄⠀⠀⠀⠀⢠⣤⣤⡄⠀⠤⣤⣄⠤⠈⠭⠙⠒⠂⢐⡃⠀⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠴⣶⣀⣀⣤⠀⠘⣠⠃⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠊⠉⠐⠊⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣮⣴⡶⣴⠲⣦⠀⠐⠒⠛⠉⠋⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢌⠦⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠠⠆⠀⡀⠀⠠⠦⠄⠀⠀⠀⠐⠀⠠⠀⠀⠛⠿⠟⠀⠀⠀⠀⠀⠸⠃⠒⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠸⣗⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠃⠀⠉⠀⠀⠀⠀⢠⣬⢾⣙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣤⣄⡀⠀⠀⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⢀⣀⠀⢀⣀⡀⠀⣀⣀⣀⣠⣄⢀⣈⡀⠀⣸⣿⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⠀⠃⠀⣼⣏⠀⡈⣿⡄⢠⡟⠀⠀⠘⠃⠀⣿⠀⠀⢸⡇⠀⠀⢸⡏⠀⠘⠀⢸⡇⠀⢸⣿⠀⠀⠈⢷⡀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⠛⣷⠀⢿⡍⠉⠉⢉⠁⠸⣇⠀⠀⢀⡀⠀⣿⡀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠸⣿⠀⣀⠀⠘⣧⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠓⠶⠖⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠲⠖⠊⠀⠀⠘⠳⠖⠛⠓⣂⣐⡚⠓⠂⠀⠐⠚⠓⠂⠀⠛⠶⠋⠀⠀⢸⠇⠀⠀⠀⠠⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠰⢶⠀⠰⠄⠠⠠⠔⠆⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣛⡙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠻⡿⠿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣩⣍⣉⡚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣩⣭⣩⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢁⡀⠀⠀⠃⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣥
⡛⠛⠋⠁⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢶⣿⣷⠗⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠘⠛⠟⠉⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠀⠁⠀⠀⠀⠙⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠈
⠀⠀⠀⠠⠤⣉⡉⠻⢿⣿⣿⣿⣿⣿⣿⠿⡄⠁⠀⠀⣈⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠤⠀⠂⠀⠰⢉⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⡄⠀⠀⠀⠀⠀⢦⠀⠤⠙⢿⣿⣿⣿⣟⣫⣭⣭⣅⠀⠀
⠀⠀⠀⠀⠀⠛⠻⢿⣷⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠉⠷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁⡠⠀⠀⢀⣀⡈⠛⠻⢿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠙⠀⠀⢙⣦⠁⠀⠀⠈⢞⡏⣭⡿⢿⠿⠋⠉⠀⠀
⠃⠀⠠⠀⡀⠀⠁⠀⠙⣟⠋⠛⠋⠀⠄⠀⣨⡄⠀⡄⣤⡄⠤⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠐⠁⠀⠽⠓⠀⠤⡻⢇⠈⠉⠀⠹⡙⣿⣿⠃⢀⣀⣤⡴⠶⢤⣤⣄⢀⠀⠀⠀⠀⠁⠀⠚⠃⢉⠀⠀⠀⠀⠁⠀⠀
⠂⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠤⠤⢠⣽⣧⡄⠄⠀⣽⣿⣿⣿⣿⣿⣿⣯⡤⡀⣀⡄⣀⣀⡠⠀⠈⠒⠑⠴⠥⠤⢀⡀⠀⣾⡏⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡰⠀⠊⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠐⠰⠶⠲⠦⠀⠀⠀⢸⣿⣿⣿⣿⣶⣿⣿⣧⣿⣿⣿⣿⣿⣿⠏⠉⠉⠉⠛⠉⠀⠀⠀⠀⠒⠒⡀⡄⠠⠗⠀⠛⠂⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠴⠆⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠤⣤⣀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠏⠀⠀⠀⠁⠉⣈⣙⡻⠳⡎⣁⡄⠀⡀⠀⠀⠀⠀⠀⣬⠤⠉⠉⠈⠐⠀⠀⢀⣀⠀⢀⡀⡒⠠⢤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠤⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠀⠁⠀⢠⣤⣤⡄⠀⢤⣤⡄⠤⠀⠬⠉⠙⢁⣿⡇⢀⠀⠠⡀⠀⠀⠀⠀⠉⠀⠀⠐⢻⣿⡀⢤⣿⠀⠸⣼⠋⠂⠈⠒⠀⠀⠓⠆⡐⠖⠀⠀⠈⠀⠈⠉⠐⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣦⣴⡶⣴⠳⣦⠀⢘⣛⠉⠋⠉⠀⠘⠃⠀⠀⠀⠀⢀⣤⠀⢈⠃⠉⣁⣌⣧⠀⠀⠀⠀⠀⣤⣾⠀⠀⣀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠱⠆⠄⠀⠰⢾⠿⠷⠌⠀⠸⠿⠀⠸⠇⠀⠙⠻⠿⠀⠀⠀⠀⠀⠿⠃⠿⢿⠟⠁⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⡀⠀⢀⠀⢀⣐⠀⠀⠀⠀⢸⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⠓⠾⠋⡀⠀⠀⠀⢰⡶⢿⣝⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠨⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣤⣄⡀⠀⠀⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⢀⣀⠀⢀⣀⡀⠂⣀⣀⣀⣠⣤⢀⣈⡀⠀⣸⣿⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⠀⠃⠀⣼⣏⣀⡈⣿⡄⢠⡟⠀⠀⠘⠇⠀⣿⠀⠀⢸⡇⠀⠀⢸⡏⠀⠘⠀⢸⡇⠀⢸⣿⠀⠀⠈⢷⡀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⠛⣷⠀⢿⡍⠉⠉⢉⠁⠸⣇⠀⠀⢀⡀⠀⣿⡀⠀⢸⡇⠀⠀⣻⡇⠀⠀⠀⢸⡇⠀⢸⣿⠀⣀⠀⠘⣧⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠖⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠲⠶⠚⠀⠀⠘⠳⠖⠛⠓⠂⣐⡛⠓⠂⠀⠐⠚⠓⠂⠀⠛⠶⠋⠀⠀⢸⠇⠀⠀⠀⠀⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⠰⠄⠠⠠⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣥⣦⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠘⠟⢻⣏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣤⣤⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣛⣛⣛⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢴⣶⣷⣶⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⠃⢤⠀⠀⠀⢻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠰
⠉⠁⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⠟⠿⠁⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠈⠉⠋⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃⢀⠀⠀⠀⠀⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀
⠀⠀⠀⠒⢳⣤⣌⣛⠿⣿⣿⣿⣿⣿⣟⡳⠌⠀⠀⠠⣤⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠒⠀⠁⠀⠘⢤⣬⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠖⠀⣕⡴⠐⣧⡹⡆⠰⠂⠹⣿⣿⣿⣿⣥⣶⣶⣶⡄⠉
⠀⠀⢀⣀⠈⢉⠛⢿⣿⣿⣿⣿⡿⠿⠟⠃⡀⠀⠀⠀⠁⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢉⡉⠀⣔⣄⣀⢠⣤⡄⢉⣙⠻⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⡀⠈⠀⠀⠰⠿⠦⠄⣀⡀⠓⣿⢰⡞⠻⠿⠋⢀⠀⠈
⠁⠘⠻⣷⡃⠉⠂⠀⠹⡯⠈⠉⠠⠐⠀⣸⡿⠆⠙⢿⣿⡄⢖⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠉⠀⠸⠛⠋⣀⣲⣝⠿⡄⠀⠀⠘⠌⢿⣿⣋⡠⢦⠶⠾⠛⠛⠿⠷⠆⠀⠀⠀⠀⠀⠀⠈⡉⢨⡔⠀⠀⠀⠀⠀⠀
⠉⠚⠒⠤⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠟⠷⢦⣭⣦⣤⣄⣠⣭⣯⣿⣿⣿⣿⣿⣥⣤⣤⣴⣶⣤⣤⣤⡀⠈⠩⢩⠿⠷⠂⠠⢤⣄⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠳⢁⠈⠀⠀⠀⠀⠀⠀⢠
⠀⣀⡤⠀⠟⠀⠂⠠⠄⠀⠀⠁⠛⠋⠛⠒⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠉⠁⠀⠀⢈⠘⠒⠲⣄⡆⡐⠋⠀⠉⠑⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒⠲⠦⠤⠀⠀⠀⢀⡀⡀⠸⠿⠿⠇⠀⠁⠉⠉⠠⠤⣭⣟⣻⣿⣵⡖⠠⠄⠀⢀⠀⠀⠀⠮⠶⠋⠈⢈⣀⣁⠀⢀⣀⠀⢀⣀⣈⠐⣒⣲⠤⣄⣀⠀⠀⣀⡀⠀⢀⣠⣄⣀⡀⠐⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠀⠀⠈⠉⠀⢠⣤⣤⡄⠀⣤⣤⡄⣤⡀⠤⠀⢉⣿⣿⣇⣀⠀⢸⣿⠀⠀⠀⠀⠉⢼⠀⢸⣿⣿⡤⢾⣿⠂⢸⣿⠁⠁⣛⣿⠀⠈⢿⠇⡸⠟⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣦⣴⡶⣴⠺⣶⠀⢸⣿⣛⠇⠀⠀⢸⣽⠀⠀⠀⠀⣼⣿⠀⢸⣟⠙⣷⣼⣯⠀⢨⢋⠀⠀⣿⣿⣠⡀⣀⣀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⡇⠤⠀⢸⣿⡿⢿⡿⠀⠿⠿⠀⠸⠿⠀⡙⠻⠿⠀⠀⠲⠠⠸⠿⠃⢻⣿⠿⠁⠀⠽⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⡀⠀⢀⣀⣿⣿⠇⠀⠀⠠⢿⣿⣪⣀⠀⠀⠀⢠⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⢠⢠⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⠀⠀⣶⡶⣿⣽⠇⠀⠀⠀⠀⠈⠀⠀⡅⠀⠀⠀⠀⢄⠀⠀⠰⠸⠈⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠁⠀⠀⠀⢠⢁⡿⠁⠀⠀⠈⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠀⠀⠈⠀⠋⡿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡇⠀⠀⠀⠀⠀⠀⣤⡄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣤⣄⡀⠀⠀⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⢀⣀⠀⢀⣀⣈⠇⣀⣀⣀⣠⣤⢀⣈⡀⠀⣸⣿⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⠀⠛⠀⣼⣏⣀⣈⣿⡄⢠⡟⠀⠀⠘⠇⠀⣿⠀⠀⢸⡇⠀⠀⣸⡏⠀⠘⠀⢸⡇⠀⢸⣿⠀⠀⠈⢷⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⠛⣷⠀⢿⡍⠉⠉⢉⠁⠸⣇⠀⠀⢀⡀⠀⣿⡀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⢸⣿⠀⣀⠀⠘⣷⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠖⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠲⠶⠚⠀⠀⠘⠳⠖⠛⠓⠂⣐⡛⠓⠂⠀⠐⠚⠓⠂⠀⠻⠶⠋⠀⠀⢸⠇⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⠶⠄⠠⠰⠖⠆⠀⠀⠀⠀⠉⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣾⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⡛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠛⠉⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⡿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣦⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⡿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣷⡄⠓⠀⠀⠁⠱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⣈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣯⠋⠉⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⡈⠁⡇⠀⢀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⢀⣀⣠⣄⢄⡈⢉⠛⢿⣿⣿⣿⣿⡿⠟⣟⣛⠃⣀
⠀⠀⠈⠙⢷⣶⣾⣭⣻⣿⣿⣿⣿⣿⣭⠟⠐⠀⠀⠐⣦⣩⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠇⠁⠀⠀⠀⠈⠹⢷⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠉⠀⠾⠿⢻⡿⣮⠳⡈⠋⠀⢿⣿⡿⢿⣿⣿⣿⠿⠷⠀
⢲⣲⣶⣶⢰⣤⠙⢿⣿⡿⣿⡿⠛⠉⠉⣀⣄⢠⣄⣄⡈⢙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⣤⠄⣰⣾⣾⢶⡰⣶⣦⠰⡦⠙⢿⣿⣿⣿⣿⠁⠀⠀⣀⣠⣄⣀⣀⠀⠙⠛⠛⠓⡶⠤⢬⣿⣘⡟⠿⠋⠉⠠⠆⠀
⣤⣍⡻⠿⠀⠉⠀⠀⠻⠃⠀⠀⠈⠁⢰⣿⢟⣷⣝⢿⣿⠈⠿⠂⠹⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠹⠋⠡⣾⣿⣮⣽⣇⠀⠀⠈⠇⣻⡿⠷⠞⠛⠛⠋⠉⠉⠙⠛⠉⠁⠀⠠⠀⠁⠀⠀⡤⠀⡿⠉⠠⠀⠀⠀⠀
⠉⠙⠹⢶⣴⡄⠀⠀⠀⠀⢀⢄⣠⣄⣀⠈⠛⠛⠟⣷⣮⣥⣤⣤⣤⣤⣮⣿⣿⣿⣿⣿⣴⣤⡶⠿⠷⠶⢶⠶⠤⠀⠠⣸⡿⢿⡽⣓⢲⣶⣟⣳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⡄⠈⠃⠃⠆⠀⠀⠀⠀⠀⠀⠐
⠰⠶⠡⠘⠁⠉⠁⠒⢀⠀⠀⠈⠉⠈⠉⠀⠀⠂⠠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠐⢈⢉⡖⢶⢦⢏⡌⠁⠋⠙⠋⠓⡶⢉⡀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠛⠒⠂⠀⠀⠀⣴⣤⠁⠹⠿⠿⠇⠉⠉⠉⠋⠐⠓⠾⢯⢿⣿⣿⡏⠰⠂⢀⣈⠀⠀⠐⠛⠿⠁⠀⢀⣠⣠⡀⢀⣀⠀⢀⣀⢤⣨⣉⣙⠛⣂⣀⠀⠀⣀⡀⠀⠀⠴⠶⢤⡤⡸⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⣤⣤⣄⣤⡀⣤⠀⢤⣿⣿⣧⡤⠀⢸⣿⠀⠀⠀⠀⣫⣼⠀⢸⣿⣿⡖⢻⣿⠁⢸⣿⠂⠁⣿⣿⠀⠈⢿⠇⡸⡟⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣦⣼⡶⣴⠺⣯⠀⢸⣿⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⠀⢸⣿⠙⣷⣼⣿⠀⢸⣿⠀⠀⣿⣿⢠⡂⣠⣁⠔⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡗⠒⠀⠾⣿⠿⣿⠿⠀⠿⠿⠀⠸⠿⠀⡘⠿⠿⠂⠈⠿⠷⠾⠿⠃⠺⣿⠿⠁⠘⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⡀⠀⣀⣠⣿⣿⡇⠀⠀⠠⢿⡿⣡⣀⠀⠀⠀⢠⠀⠀⠀⡄⡄⠀⠀⠀⠀⠀⠀⠀⢠⢠⡦⠮⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣿⡇⠀⠀⠀⣶⡾⣿⣽⠗⠀⠠⢈⠰⠈⠀⠁⡅⠀⡈⠠⠀⢄⠀⠀⢸⠸⠈⠀⡐⠀⠀⣸⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣴⠀⠀⠀⠈⣉⡉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠀⢨⢠⣿⠉⠀⠀⠉⣷⡀⠀⠀⠁⠉⠉⠀⠀⠀⠈⠀⠀⠈⠀⠉⣿⠁⠀⠀⠈⠈⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢟⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⣦⡄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠚⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⠀⣀⣤⣄⣘⠛⠀⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⢀⣀⠀⢀⣀⣉⢭⣀⣀⣀⣠⣼⣏⣉⡀⠀⣸⣿⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣧⣀⠈⠛⠀⣼⣏⣀⣈⣿⡄⢠⡟⠀⠀⠘⠇⠀⣿⠀⠀⢸⡟⠈⠀⣾⡏⠀⠙⠀⢸⡇⠀⢸⣿⠀⠀⠈⢷⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⠛⣷⠀⢿⡍⠉⠉⢉⠁⠸⣇⠀⠀⢀⡀⠀⣿⡀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⢸⣿⠀⣀⠀⠘⣷⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠖⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠲⠶⠚⠀⠀⠘⠳⠖⠛⠓⠂⣐⠛⠓⠂⠀⠐⠚⠓⠂⠀⠻⠶⠋⠀⠀⢹⠇⠀⠀⠀⢀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠀⢶⠄⠠⠰⠶⠆⠀⠀⠀⠀⠘⠋⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⠏⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣩⣽⣮⣭⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⢀⡈⠀⢹⠱⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢹⠿⡟⠟⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠉⠁⠀⠈⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠤⣤⣋⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⠏⠀⠁⢀⡹⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⡤⠀⠃⠀⢰⣉⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⣤⠄⢰⣧⣶⣿⡰⣦⠠⣬⠛⣿⣿⣿⣿⣿⣏⣽⣭⣍⡔
⢀⠀⡀⠘⠿⢿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠈⠀⠀⠀⠹⢳⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠃⡢⣀⡈⢀⣀⡘⡛⠿⣿⣿⣿⣿⣿⣿⡍⠁⠀⠀⠀⠈⠘⠛⠙⢯⣿⣎⣃⡈⠀⠸⣿⡟⣛⣿⣿⣿⠟⢉⠁
⠻⣿⣿⣷⢳⡶⠈⢻⣿⠻⠿⠋⣀⠤⢠⣼⣦⢳⣶⣶⡆⢤⣍⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠉⠀⠚⠁⣾⣿⠿⢿⣜⠻⣷⡘⠷⠊⢻⣿⣿⣿⡿⢀⣄⣤⣴⣾⣷⣶⣶⣤⣀⠀⠉⡽⡿⠿⣶⣞⠧⣿⣽⡃⠀⠀⠉⠀
⣷⣮⡙⠛⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⢻⣯⣽⣷⣍⡻⢿⠈⠋⠀⠹⡿⢿⣿⣿⣿⣿⣿⠃⠀⢀⣤⣄⣀⣛⠁⠘⠛⠿⣿⣾⣧⣀⣀⡀⠉⣼⣿⠉⠉⠈⠉⠀⠀⠀⠀⠁⠀⠀⠀⠸⠇⢰⠀⢰⡸⢠⠛⠁⠈⠀⢖⣆⡄
⡐⣘⢘⡿⣷⣀⣀⠀⠀⠀⠾⠵⡶⡶⠤⠀⠀⣽⠛⢿⣷⣶⣤⣤⣤⣼⣯⣿⣿⣿⣿⣷⣾⣿⡟⠋⠙⠻⠿⠿⠀⠀⠰⢻⣿⢿⠿⡿⢻⡿⣿⣿⣷⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡇⠀⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀
⠛⠣⠂⠀⠀⠛⠛⠡⢤⣤⣀⡀⠀⠀⠀⠀⠘⢁⠶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠠⠀⠀⢐⢨⣼⡟⠚⠛⠡⠃⠀⠉⠋⠑⠧⠙⠠⠤⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠁⣀⠀⠀⠐⠿⠟⠀⠸⠿⠿⠇⠉⠈⠉⠁⠈⠉⠛⠛⢛⣿⣿⡇⠈⠀⣀⣀⠀⠀⠈⠙⠛⠁⠀⢀⣴⣶⠂⣀⣀⣀⣀⣀⡸⡲⣀⣀⠈⣁⣀⠀⠀⣀⡀⠀⠀⠚⠛⠒⠖⠶⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⣤⣤⣄⣤⣄⣤⠀⢶⣿⣿⡷⠆⠀⢸⣿⠀⠀⠀⠀⣯⣼⠀⢸⣿⣿⡉⢹⣿⠁⢸⣿⠁⠀⣿⣿⠀⢘⢿⠇⡸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣦⣴⡶⣴⠺⣧⠀⢸⣿⣿⡇⠀⠀⢸⣿⡄⠀⠀⠀⣿⣿⠀⢸⣿⠙⣿⣾⣿⠀⢸⣿⠀⠀⣿⣿⢠⠋⣠⣁⠄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠁⠀⠸⢿⣿⣿⠿⠀⠿⠿⠀⠸⠿⠀⠜⠿⠿⠁⠘⠿⠷⠾⠿⠃⠠⠿⠿⠁⠘⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⡀⠀⣀⣸⣿⣿⡇⠀⠀⠠⢿⠟⣡⣀⠀⠀⠀⢠⣄⡀⠀⡄⡄⠀⠀⠀⠀⠀⠀⠀⢠⢠⡤⠬⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇⠀⠀⠀⣶⠞⣿⣽⠗⢀⢰⣤⠘⡌⢀⠁⡅⠀⡈⠠⠀⢄⠀⠀⢸⠸⠈⠀⡐⠀⠀⣸⠛⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⢸⣿⡉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠀⠈⢀⡿⠁⠀⠀⠙⣿⣷⠀⠀⠁⠉⠉⠀⠀⠀⠈⠀⠀⠈⠀⠈⠙⠀⠀⠀⠈⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⣿⣿⣧⡄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣤⣼⣛⠃⠀⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⢀⣀⠀⢀⣀⣀⠠⣀⣀⣀⣠⣿⣛⣉⡀⠀⣸⣿⣀⡀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⡈⠛⠀⣼⣏⣀⣈⣿⡄⢠⡟⠁⠀⠘⠇⠀⣿⠀⠀⢸⡇⠀⠀⣸⡏⠀⠙⠀⢸⡇⠀⢸⣿⠀⠀⠈⢷⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠙⠻⣷⠀⢿⡍⠉⠉⢉⠁⠸⣇⠀⠀⢀⡀⠀⣿⡀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⢸⣿⠀⣀⠀⠘⣷⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠶⠖⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠳⠶⠚⠀⠀⠘⠳⠖⠛⠓⠂⢐⠛⠓⠂⠀⠐⠚⠓⠂⠀⠻⠶⠋⠀⠀⢹⠏⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠲⣶⠏⢶⠄⠠⠴⠶⠆⠀⠀⠀⠀⠚⢉⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠛⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠤⠏⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⣛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠘⠟⠉⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣥⣶⣬⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢺⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡴⠛⠀⠀⠃⠱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣉⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⠛⠻⠃⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⡘⠁⡇⠈⢀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣉⠁⣵⣤⣶⣆⣄⡈⣛⠻⣿⣿⣿⣿⣿⣿⠿⣿⣛⡛
⠀⠙⣿⣶⣦⣼⣟⣿⣿⣿⣿⣿⣿⣯⡟⠚⠀⠀⠐⣶⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⡋⠀⠘⠀⠈⡷⣷⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠘⠋⣾⣿⣿⣿⣿⣌⣳⡈⠷⠊⣿⣿⣿⣿⣿⣿⣿⣿⠿
⣴⣦⣤⣬⣉⡻⣿⣿⣿⣿⣿⠟⠛⠛⣁⣤⣠⣤⣀⣛⢛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⣤⠄⣴⣾⣷⣷⡱⣷⣦⠠⣬⡙⢿⣿⣿⣿⣿⡟⠀⠀⣐⣠⣶⣚⣝⣏⡙⠙⢛⣛⣿⣿⣶⣼⣿⣷⣾⣿⡟⠋⠁⡶
⡻⢿⣿⡘⠿⠉⠈⣿⡗⠋⠁⠚⠉⢠⣿⣿⣷⡽⣿⣿⡏⢿⡷⠙⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⢀⡄⣤⢴⣿⡿⣿⣿⣿⣮⣿⣇⠘⠋⠉⢿⠿⣿⠿⠶⠟⠿⠛⠋⠉⠙⠛⠛⠉⠉⠉⣿⡏⣹⢟⢿⣿⢭⠿⠻⣳⣦⣤⣤
⢿⣷⣶⣦⠀⠀⠀⢀⡀⣀⣔⣈⣁⣚⢝⣛⣻⣿⣷⣽⣷⣀⣁⡀⢙⣿⣻⣿⣿⣿⣿⣟⣤⣤⡾⠿⢷⣿⣾⡶⠥⠴⢶⣿⣿⣿⣿⣿⣶⣾⣾⣿⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣴⡏⠈⠃⠃⠞⠈⠀⠀⠈⠙⠋
⢏⠼⠿⣿⣿⣷⠖⡘⠉⠉⠉⠛⠙⠉⠉⢁⡿⢻⣿⣿⣿⣿⣿⣷⣾⣾⣷⣬⣿⣿⣯⣭⣤⣤⡄⠀⠀⠠⢤⠤⠀⠀⡾⣣⣿⠿⢿⡟⡽⠿⢿⣿⣿⣿⣿⣖⣂⡀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠁⠁⠄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠀⠉⠛⠛⠒⠀⠀⠀⠀⣴⣤⠏⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣠⠀⠀⢸⠀⠀⠰⢨⢿⠗⠒⠀⢀⣀⡀⠀⠀⠀⠈⠉⢀⡄⠉⠙⠛⠛⠂⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⣀⢰⠆⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠤⠆⠀⠀⠈⠉⠀⠀⠸⠿⠿⠇⡀⡏⠁⠰⠇⠁⠀⡤⢸⣿⣿⣇⡁⠐⢀⣈⠀⠀⠀⠀⠀⠀⠁⢀⣘⣿⣯⣤⣴⠶⢖⣠⡦⠙⣁⣀⠀⣀⣀⠀⠀⣀⡀⠀⠀⠀⠁⠀⠉⠉⢹⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠈⣤⣬⣍⣭⣰⣾⠉⢻⣿⣿⡏⠁⠀⢸⣿⠀⠀⠀⠀⣯⣼⠀⢸⣿⣿⡀⣹⣿⡀⢸⣿⠀⠀⣿⣿⠀⢸⣿⠇⡸⡟⠁⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠠⣦⣼⡿⣵⠳⣦⠀⢸⣿⣿⡧⢀⡠⣼⣿⣷⠀⠀⠀⣿⣿⣀⣸⣿⡟⣿⣿⣿⠥⢸⣿⠀⠀⣿⣿⠀⠘⣠⣁⠔⣄⠀⠈⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠹⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠸⢿⠿⣿⠿⠀⠿⠿⢎⡻⠿⠇⡚⠿⠿⠀⠘⠿⢷⡾⠿⠋⠠⠾⡿⠃⠘⠿⠦⣀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⢹⣿⣿⣧⣄⣀⣀⣀⣀⡀⠀⣦⣸⣿⣿⡇⠀⠀⠠⢿⠞⣣⣀⠄⢠⣤⣨⢿⣆⠀⡄⡄⠠⠀⠀⠠⠀⠀⠀⢠⢀⡌⠰⡂⠠⠀⢠⢸⣧⡄⢃⠀⠀⠀⠀⠀⠀⠀⠀
⣸⣷⣧⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇⠀⠀⠀⣀⡰⣿⣽⠷⢸⢸⣿⣦⢫⣦⡇⡅⠀⡨⠠⠀⠄⢀⠀⢸⠸⠈⠀⡐⠀⠀⣸⠀⣿⣿⡄⠂⠀⠀⠀⠀⠀⠀⠀
⣿⣿⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠻⠋⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠈⠉⠀⠈⠉⠀⠀⢹⣿⣿⡆⠁⠁⠉⠉⠀⠁⠀⠈⠀⠀⠈⠀⠈⠉⠀⠀⠀⠈⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⡄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢹⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣿⣿⣋⠀⠀⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⣀⣀⠀⢀⣀⡀⠀⣀⣀⣀⣨⣿⣛⣋⡀⠀⣸⣿⣀⣀⣀⣀⡀⠀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣀⡈⠛⠀⣼⣏⣀⣈⣿⡄⢠⡿⠁⠀⠘⠇⠀⣿⠀⠀⢸⡏⠀⠀⣿⡏⠀⠙⠀⢸⡇⠀⢸⣿⠀⠀⠈⣿⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠙⠻⣷⠀⢿⡍⠉⠉⢉⠁⠸⣧⠀⠀⢀⡀⠀⣿⡀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⢸⣿⠀⣀⠀⠘⣷⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠞⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠳⠶⠚⠀⠀⠘⠳⠖⠛⠓⠂⠐⠛⠓⠚⠁⠐⠚⠓⠂⠀⠻⠶⠋⠀⠀⢹⠏⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠎⢶⠄⠰⠶⠶⠆⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣩⣭⣉⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⣀⠙⠁⢻⢣⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣶⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡿⣿⢿⠇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠋⠁⠀⠘⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣀⣌⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣿⠉⠫⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡓⣌⠁⢳⠀⢠⡞⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣥⣄⣴⣷⣿⣿⣶⣤⢨⣍⠻⣿⣿⣿⣿⣿⣿⣻⣻⣟
⠀⢹⠿⣿⣶⣿⣽⣻⣿⣿⣿⣿⣿⣿⠋⠸⠀⠀⠘⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⢅⣀⣘⣀⡀⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠀⠀⣉⢸⣿⡿⣿⣿⣷⣽⣧⣹⡋⢺⣿⣿⡿⣿⣿⣿⣿⠟
⣿⣶⣦⣦⣬⡻⣿⣿⣿⣿⠿⢋⣉⢉⣴⣦⣶⣶⣤⣥⣭⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢡⡶⠖⣼⣿⣿⣿⣜⣿⣿⡔⢶⣌⡻⣿⣿⣿⣿⡿⢀⣀⣸⣴⣿⣿⣷⣿⣬⣷⣦⣭⣿⣿⣿⣿⣿⣷⣿⣿⣿⠋⠁⠻
⣿⢿⣿⠘⠏⠀⠹⡿⠟⠁⠀⣉⡁⣾⣿⣿⣷⣟⢿⣿⡏⢿⡗⠚⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⣀⣦⣶⠺⢿⣿⣿⠿⠿⣿⣿⣷⣌⣉⡀⢹⠿⣿⡿⠟⠛⠛⠋⠉⠀⠈⠉⠉⠁⠀⠀⢸⠿⣽⡟⣿⢿⢏⡟⣻⣿⣿⣿⣿
⣿⣿⣿⣧⣀⡀⠀⣥⣠⣶⣧⣿⣯⣼⣯⣭⣿⣿⣿⣿⣷⣴⣤⡄⣸⣯⡿⣿⣿⣿⣿⣯⣶⡶⠿⠛⠿⠿⠿⠿⠚⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⡿⣀⠈⠈⠈⠀⠀⠘⠩⠩⡙
⠵⠛⢿⣿⣿⣿⠩⣍⣄⣀⠀⠁⠁⠀⠀⢰⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣽⣿⣯⣤⣤⣤⡄⠀⠀⠠⢤⠤⠀⠀⡿⣿⡿⠿⠟⠟⠼⠛⠿⣿⣿⣿⣿⡟⠬⢤⣠⣄⣀⡀⠀⠀⠀⠀⠉⠁⠀⠐⠀⡄⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠈⠀⠀⢀⠉⠋⠉⢀⠀⠀⠠⣿⣿⠁⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡵⠀⠀⢸⠀⠀⠘⢩⠻⠗⠒⠀⢠⣤⣄⡀⠀⠀⠉⠟⠁⣦⡂⠀⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣤⣶⣤⣤⣵⡖⠀⠀⠀
⠀⠀⠀⠀⠀⠐⠀⡸⠉⠀⠀⠺⠂⠀⠀⠀⠀⠀⠀⠸⠿⠿⠇⠌⠿⣇⣚⣃⠉⢀⠄⣸⣿⣿⣟⡅⠐⣀⣈⠀⠀⠀⠈⠀⠉⠉⢀⣈⣻⣯⣶⣶⠾⢛⣓⠾⠔⣃⣀⠀⣀⣀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠂⠀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⣤⣤⣄⣼⣽⣿⠉⣹⣿⣿⡏⠉⠀⢹⣿⡀⠀⠀⠀⣿⣽⠀⢸⣿⣷⣄⣿⣿⡀⢸⣿⠀⠀⣿⣿⠀⢘⣿⠇⡸⡟⠁⠻⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⡀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠀⠀⠀⠀⣀⣸⣿⣿⡇⠠⣷⣼⡿⣼⠻⣧⠀⢸⣿⣿⡷⠤⠴⣾⣿⣿⠀⠀⠀⣿⣿⣦⣼⣿⡟⣿⣿⣿⠃⢸⣿⠀⠀⣿⣿⠀⢴⣯⣃⠔⣄⠀⠀⠹⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠸⠿⠿⡿⠿⠀⠿⠿⢻⣿⡿⠇⡞⠿⠿⠁⠘⠿⢷⡾⠿⠃⠠⣾⣿⠏⠘⠿⠦⢢⡄⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀
⣬⣿⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢻⣿⣿⣧⣄⣀⣀⣀⣀⡀⠀⣶⣸⣿⣿⡇⠀⠀⠠⢾⡾⣯⣀⠄⢠⣶⣤⢻⣇⠀⡅⡤⠤⠀⠀⠤⠀⠀⠀⢠⢀⡍⠡⡄⠠⠀⢠⠸⣿⣦⡘⡄⠀⠀⠀⠀⠀⠀⠀
⣿⡿⣦⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇⠀⠀⡀⣊⡰⣿⣽⠿⢸⣿⣿⣷⠹⣶⡇⡅⠀⡨⠠⠀⢄⢀⠀⢸⠸⠈⠈⡘⠀⠀⣸⠀⣿⣿⣧⠘⠀⠀⠀⠀⠀⠀⠀
⣿⡇⢹⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢻⠋⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠈⠋⠁⠈⠉⠀⠀⢹⣿⣿⡇⠙⠁⠉⠉⠀⠁⠀⠈⠀⠀⠈⠀⠈⠉⠀⠀⠀⠈⠀⣻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢠⣿⣿⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⡄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⣿⣿⣃⠀⠀⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⣀⣀⠀⢀⣀⣀⠀⣀⣀⣀⣨⣿⣛⣋⡀⠀⣸⣿⣀⣀⣀⣀⡀⠀⣀⣀⠀⣿⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⢸⣯⣀⡈⠛⠀⣼⣏⣀⣈⣿⡄⢠⡿⠁⠀⠘⠇⠀⣿⠀⠀⢹⡟⠉⠀⣿⡏⠀⠙⠀⢸⡇⠀⢸⣿⠀⠀⠈⣿⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣙⠻⣷⠀⢿⡍⠉⠉⢉⡁⠸⣧⠀⠀⢀⡀⠀⣿⡀⠀⢸⡇⠀⠀⣿⡇⣤⣄⣀⢸⡇⠀⢸⣿⠀⣀⠀⠘⣷⣰⠃⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣶⡆⠀⠀⠀⠀⠀⠀⠀
⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠳⠶⠚⠀⠀⠘⠳⠖⠛⠓⠂⠐⠛⢓⣛⣛⡛⠛⠓⠂⠈⠻⠶⠋⠀⠀⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀
⠐⠗⠶⣶⠏⢶⠄⠰⠶⠶⠆⠀⠀⠀⠀⠀⠈⠙⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡙⠉⠀⠀⠀⠀⠀⠀⠀⠾⠶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣴⣶⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣈⠁⠘⡏⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠋⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⡿⢿⠻⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠁⠀⣀⠃⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢠⣤⣜⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⠟⠀⠛⠀⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣠⠁⢸⠀⠀⣞⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣤⡆⣴⣿⣿⣿⣿⢶⡖⢤⣝⢿⣿⣿⣿⣿⣿⣻⣿⣭
⡀⠿⠻⢿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⠉⠸⠀⠀⠸⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⣡⣀⣤⣧⣀⢏⠻⠿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⡀⢨⣼⣿⢿⣿⣿⠿⣷⣿⣤⣿⣉⣿⣿⠿⢿⣿⣿⣿⡟
⣿⣿⡿⣶⣮⡻⣿⣿⣿⡿⠛⣡⡠⢠⣶⣶⣿⣷⣦⣥⣬⣉⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠡⡾⠛⣿⣿⣿⣿⣿⣿⣿⣏⢳⣶⡝⣿⣿⣿⣿⡿⢃⣤⣤⣿⣿⣿⣶⣾⣷⡿⠷⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣈
⣿⣿⣇⠙⠃⠀⠹⠿⠉⠀⣜⣥⣠⣿⣿⣿⣿⣿⣿⣿⡏⢿⡟⠺⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⣀⣶⣿⣛⣛⣿⣿⣛⣛⣻⣿⣿⣦⣭⣄⣸⣟⣿⡟⠛⠛⠉⠉⠁⠀⠀⠀⠁⠀⠀⠀⢠⣿⢿⡿⠻⡽⡿⣹⠭⣿⣿⣿⣿
⢿⣿⣿⣧⣤⣀⣰⣷⠴⠾⣷⣾⣶⠿⠷⢶⣿⣿⣿⣿⣿⣶⣶⣄⣽⣷⡿⣿⣿⣿⣿⣯⠾⠿⠛⠛⠛⠻⠟⠛⠉⠋⢹⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⢇⠄⡀⠀⠁⠀⠀⠉⠘⢫
⠊⠛⠿⢿⣿⡏⠲⢤⣤⣄⡀⠀⠀⠀⠀⣼⣫⡿⣿⢿⣽⣿⣿⣿⣿⣿⣯⣶⣿⣿⣯⣤⣤⣤⡄⠀⠀⠠⢤⠤⠀⠀⣼⣿⣿⠿⠛⠛⠼⠛⠙⠻⠿⢿⣿⡿⠓⠲⠤⢦⣤⣄⡀⠀⠀⠀⠀⠈⠄⠐⠃⠲⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠆⠀⠀⢀⢈⠁⠀⠀⠀⠀⠰⠿⡿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡵⠂⠀⢸⠀⠀⠐⢨⠛⠓⠒⠀⠀⣶⣤⡀⠀⠀⢸⡿⠁⢴⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⢦⣤⣤⣿⠃⠀⠀
⠀⠀⠀⠀⠀⠁⠠⠌⠁⠀⠐⠚⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠧⣬⠻⣧⣽⣇⡉⢀⣟⣸⣿⣿⣯⡅⠀⣁⣈⠀⠀⠀⠈⠀⠉⠉⢀⣀⣹⠯⣶⣶⠟⢛⣛⠻⠧⣋⣀⠀⣀⣀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠁⣤⣥⣤⣬⡽⣯⠠⢽⣿⣿⡏⠈⠀⢸⣿⡄⠀⠀⠀⣿⣽⠀⢸⣿⣿⣄⣿⣿⣀⢸⣿⠀⠀⣿⣿⠀⠘⣿⠇⡸⡟⠁⠘⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠠⣀⣸⣿⣿⡇⠠⣶⣴⡿⣼⠺⣧⠀⢸⣿⣿⡗⠢⠄⣾⣿⣿⠀⠀⠀⣿⣿⣦⣾⣿⡟⣿⣿⣿⠓⢺⣿⠀⠀⣿⣿⠀⢀⣭⣃⠔⣄⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠸⠿⠿⢿⠿⠀⠿⠿⠻⣿⡿⠇⡞⠿⠿⠀⠘⠿⢷⡾⠿⠃⠀⢾⣿⡟⠸⠿⠦⢸⣄⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⣀⢻⣿⣿⣯⣀⣀⣀⣀⣀⡀⠀⣵⣸⣿⣿⡇⠀⠀⡤⢶⣶⣧⣄⠄⢀⣴⣤⠹⣿⠀⡄⡄⠤⠀⠀⠤⠀⠀⠀⢠⢀⡾⠫⡄⠠⠀⢠⠀⣿⣷⣄⢣⠀⠀⠀⠀⠀⠀⠀
⣿⣿⡆⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇⠀⠀⠀⡚⠰⣿⣽⠿⢸⢿⣿⣿⡙⣿⡇⡅⠀⡨⠠⠀⢄⠀⠀⢸⠸⠨⠈⡘⠀⠀⣸⠀⢹⣿⣿⡄⠃⠀⠀⠀⠀⠀⠀
⣿⠈⣧⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⣿⠋⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠈⠛⠁⠈⠉⠀⠀⢹⣿⣿⡇⠙⠁⠉⠉⠀⠁⠀⠈⠈⠀⠈⠀⠈⠉⠀⠀⠀⠈⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢸⣿⣿⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠈⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⣿⣿⣃⠀⠀⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⣀⣀⠀⢀⣀⣀⠀⣀⣀⣀⣸⣿⣿⣟⡀⠀⣸⣿⣀⣀⣀⣀⡀⠀⣀⣀⠀⣿⣐⣢⢠⣤⠄⠀⠸⠽⠃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠦⠤⠠⢼⣯⣉⡈⠛⠀⣼⣏⣀⣈⣿⡄⢰⡿⠁⠀⠘⠇⠀⣿⠀⠀⢺⡟⠓⠒⣿⡏⠀⣙⡃⢸⡇⠀⢸⣿⠀⠀⠈⣿⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣟⡻⣷⠀⢿⡍⠉⠉⢉⡁⢸⣧⠀⠀⢀⡀⠀⣿⡀⠀⢸⡇⠀⠀⣿⡇⣼⣿⣦⣸⡇⠀⢸⣿⠀⣀⠀⠘⣷⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡿⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠳⠶⠚⠀⠀⠘⠳⠖⠛⠓⠂⠐⠛⢛⣿⣿⣿⡟⠓⠂⠈⠻⠶⠋⠀⠀⢹⠏⠀⠀⠀⠀⡀⠀⠀⠀⠀⢾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠖⠲⣶⠏⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⠀⣹⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣟⠋⠀⠀⠀⠀⠀⠀⠾⠶⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠚⣿⡶⠍⠀⠀⠀⠀⠀⠀⠀
⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣻⣛⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣿⣶⣶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢃⣤⠀⠀⢹⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢺⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⡿⢻⠛⠏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⢇⡀⣀⣈⡀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢴⣶⣍⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⡇⠀⠉⣀⡸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡰⠀⠘⡄⠀⢮⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣴⡶⣶⣿⣿⣿⣿⣟⣷⠐⣦⣝⢿⣿⣿⣿⣿⣯⣽⣷
⡘⠟⠻⠿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⠀⢁⠀⠀⢸⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⣸⣤⣤⣳⣤⣘⣙⠛⠿⣿⣿⣿⣿⣿⣿⣿⠉⠀⢠⢿⣶⡿⠿⢿⣿⡛⠿⢿⣷⣾⣯⣼⣿⣿⣛⣿⣿⣿⡿
⣿⣿⣻⣶⡮⣻⣿⣿⣿⠟⢋⣤⠄⣰⣾⣿⣻⣷⣶⣶⣦⣭⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⣴⣿⣋⣿⣿⣿⣿⣿⣿⣿⣷⠹⣷⡬⢿⣿⣿⣿⣿⣋⣤⣤⣾⡿⠿⢿⢿⡿⠿⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤
⣿⣿⣇⠙⠀⠀⢻⡛⠁⣢⣿⣦⣼⣿⠿⣿⣿⣿⣿⣿⡇⠻⠋⠛⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣴⣿⣿⣿⣭⣿⣿⣯⣍⣿⣿⣿⣷⣾⣦⣼⣿⣿⣿⠛⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣛⣿⠏⠟⢿⠇⡟⠻⣿⣿⣿
⣿⣿⣿⣶⣤⣄⢸⡷⠾⠿⢿⠿⠷⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣷⡿⢿⣿⣿⣿⠷⠿⠟⠛⠉⠉⠛⠛⠛⠉⠉⢉⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⣶⣦⡦⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢿⡿⠰⠀⠀⠀⠀⠀⠀⠀⠈
⠉⠙⠻⢻⡿⠃⠒⠶⣦⡤⠄⠀⠀⠀⠀⣿⣽⠿⠿⢯⣿⣿⣿⣿⣿⣿⣯⣴⣿⣿⣷⣤⣤⣤⡄⠀⠀⠠⢤⠤⠀⠀⣼⣿⣿⠯⠙⠙⠑⠃⠉⠛⠿⠿⣿⣿⠋⠛⠒⠶⠶⣦⡤⠀⠀⠀⠀⠀⠀⠀⠉⠰⠦⡀⡀⠀⠀⠀
⠀⠀⠀⠘⠂⠀⠀⣀⡀⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡱⠂⠀⢸⠀⠀⠐⢨⠙⠛⠒⠀⡀⣿⣶⡤⠀⠀⢀⣛⠁⠰⣮⡄⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠶⣴⣮⡟⠀⠀
⠀⠀⠀⠀⠈⡀⠠⠃⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠸⠿⡿⠧⣸⢛⣷⣿⣧⣉⣀⣟⣼⣿⣿⣯⣥⢈⣁⣈⠀⠀⠀⠈⠀⠈⠉⢀⣀⣸⠿⣶⣳⠟⢛⣉⠛⠓⣊⣀⠀⣀⣀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠀⠀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠠⣬⣤⣤⣮⡼⣿⠂⢽⣿⣿⡇⠈⠀⢸⣿⣄⠀⠂⠀⣿⣽⠀⢸⣿⣷⣄⣿⣿⣠⢸⣿⠀⠀⣿⣿⠀⠘⢿⠗⡸⡟⠁⠈⠻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀
⣄⣀⡀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠂⠀⠀⠀⢠⣀⣸⣿⣿⡇⠰⣷⣴⡶⣼⠲⣧⠀⢸⣿⣿⡗⠒⠄⣾⣿⣿⠀⠀⠀⣿⣿⣶⣾⣿⡟⣿⣿⣿⠉⢺⣿⠀⠀⣿⣿⠀⠀⣬⣃⠔⣆⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠸⠿⠿⢿⠿⠀⠿⠿⠹⣿⣿⠇⠼⠿⠿⠈⠚⠿⢷⡾⠿⠃⠀⠾⣿⡷⠾⠿⠦⠀⣧⡀⠸⣇⠀⠀⠀⠀⠀⠀⠀
⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⢿⣿⣿⣷⣀⣀⣀⣀⣀⡀⠀⣓⣸⣿⣿⡇⠀⠀⠤⢤⣴⣯⣄⠄⢠⣸⣤⠘⣿⠀⡄⡄⠤⠀⠀⠠⠀⠀⠀⢠⢀⡴⠿⣃⠤⠀⢠⠀⣻⣿⣦⡹⡄⠀⠀⠀⠀⠀⠀
⣿⣧⡀⠀⠀⢀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇⠀⠀⠁⠚⠑⣿⣽⠿⢸⢻⣿⣿⡜⣷⡇⡅⠀⡨⠠⠀⢄⠀⠀⢸⠸⠈⠀⡘⠀⠀⣸⠀⠸⣿⣿⣇⠑⠀⠀⠀⠀⠀⠀
⡿⢹⡇⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⠹⣿⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠉⠿⠁⠈⠉⠀⠀⢹⣿⣿⣷⠘⠇⠉⠉⠀⠁⠀⠈⠀⠀⠈⠀⠈⠉⠀⠀⠀⠈⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣼⣿⣿⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠁⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⣿⣿⣿⣀⠀⠀⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⣀⣀⠀⢀⣀⣀⠀⣀⣀⣀⣸⣿⣿⣟⡀⠀⣸⣿⣀⣀⣀⣀⡀⠀⣀⣀⠀⣿⠰⢶⡀⣤⣀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⣆⣀⣀⣸⣿⣿⡏⠛⠀⣼⣏⣀⣈⣿⡄⢰⡿⠁⠀⠸⠇⠀⣿⠀⠐⢺⡿⠿⡧⣿⡯⣽⣿⣿⢿⡇⠀⢸⣿⠀⠀⠈⣿⡀⠀⡼⠁⠀⠀⠈⠉⠉⠛⠋⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⡻⣷⠀⢿⡍⠉⠉⣩⡁⢸⣧⠀⠀⢀⡀⠀⣿⡀⠀⢸⡇⠀⠀⣿⣇⣿⣿⣿⣼⡇⠀⢸⣿⠀⣀⠀⠘⣷⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⡿⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠳⠶⠚⠁⠀⠘⠳⠖⠛⠓⠂⠐⠛⢛⣿⣿⣿⣿⠓⠂⠈⠻⠶⠋⠀⠀⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠖⠲⣶⠏⢶⠄⠰⠶⠶⠖⠀⠀⠀⠀⢹⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠾⠶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀
⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣭⣍⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣶⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣴⠆⠀⠀⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⢿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠿⠙⡏⠛⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⢛⣄⣤⣤⣤⣀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢶⣶⣉⣉⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣜⡇⠀⠉⣠⡼⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢽⠂⠀⢇⠀⢹⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢰⣾⢻⣿⣿⣿⣿⣿⣽⣇⠳⣦⣽⣿⣿⣿⣿⣿⣭⣾
⣘⣋⡛⠿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠏⢀⢁⣀⡀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣉⣉⣨⣶⣶⣾⣦⣜⣌⣛⠻⢿⣿⣿⣿⣿⣿⣿⡋⠀⠀⣿⣿⣿⣟⣿⣿⣿⣛⣛⣻⣿⣿⣶⣿⣿⣟⣻⣿⣿⡿
⣿⣿⢿⣷⠮⢻⣿⣿⡿⠋⣡⠶⢂⣾⣿⣿⣻⣿⣷⣶⣶⣬⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠁⣴⣿⣉⣿⣿⣿⣿⣿⣯⣿⣿⡎⢿⡷⠽⣿⣿⣿⣿⣋⣴⣶⣾⡿⠟⠛⠿⠿⠿⠟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⣿⣿⡌⠈⠀⠈⣿⠃⠠⣾⢿⡶⠾⣿⡿⢿⣿⣿⣿⣿⣇⢻⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⢗⣀⣠⣿⣿⣿⣭⣯⣿⣿⣭⣽⣿⣿⣿⣿⣷⣶⣿⣯⣿⡍⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣹⣿⠃⠻⠽⠱⠟⠻⣿⠿
⣿⣿⣿⣶⣦⡤⠽⠷⠟⠿⠿⠿⠿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⡽⠿⣿⣿⣿⠷⠿⠟⠋⠉⠉⠙⠛⠉⠁⠀⠀⣿⣿⣿⣿⣿⣿⢻⣟⣿⣿⣿⣷⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠇⢇⡀⠀⠀⠀⠀⠀⠀
⠀⠛⠻⣿⠟⠈⠛⠿⢶⠦⠀⠀⠀⠀⢀⣯⣼⠻⠻⢣⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣷⣤⣤⣤⡄⠀⠀⠠⢤⠤⠀⠀⠰⣿⣿⡯⠁⠉⠁⠋⠈⠙⠻⠿⢿⣿⠟⠈⠙⠛⠻⠶⠶⠄⠀⠀⠀⠀⠀⠀⠈⡁⢴⠦⢀⠀⠀⠀
⠀⠀⠀⠙⠁⠀⢀⣀⠄⠀⠀⡀⠀⠀⠘⠻⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡹⠂⠀⢸⠀⠀⠐⢈⠉⠛⠒⠀⡀⢹⣷⣦⠀⠀⢄⣘⡓⣀⢶⡅⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠷⢶⡶⣽⠋⠀
⠀⠀⠀⠀⢉⠀⠼⠁⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠸⠿⡿⠷⡸⢻⣷⢿⣧⣉⣠⣟⣼⣿⣿⣿⣥⣈⣁⣈⠀⠀⠀⠈⠀⠉⠉⢀⣀⡈⠿⣷⣛⠛⢛⣉⠉⠛⣂⣁⠀⣀⣀⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡌⠠⣬⣤⣬⣾⡽⣯⠒⢸⣿⣿⡇⠈⠀⢸⣿⣇⠀⠷⠀⣿⣽⠀⢸⣿⣷⡄⣿⣿⣠⣼⣿⠀⠀⣿⣿⠀⠘⢿⠿⡻⡿⠁⠀⠈⢿⣷⡆⠀⠀⠀⠀⠀⠀⠀
⣀⣀⡀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⢤⣀⣸⣿⣿⡇⠰⣷⣴⡶⣼⠲⣧⠀⢸⣿⣿⡟⠒⠒⣾⣿⣿⠂⠀⠀⣿⣿⣶⣾⣿⡿⣿⣿⣿⠉⢹⣿⡄⠀⣿⣿⠀⠀⣬⣃⠔⣄⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠸⢿⠿⠿⡿⠀⠿⠿⠘⣿⣿⡇⠘⠿⠿⠀⠚⠿⢿⡾⠿⠃⠀⠾⢿⣧⡾⠿⠦⠀⢸⣦⠀⢻⡀⠀⠀⠀⠀⠀⠀
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⣿⣿⣿⣧⣀⣀⣀⣀⣀⡀⠀⣑⣺⣿⣿⡇⠀⠀⠤⢤⣤⣿⣄⠄⢀⢸⣦⡈⣿⡅⡄⡄⠤⠀⠀⠤⠀⠀⠀⢠⠀⠄⠹⡏⢤⠀⢠⠀⠸⣿⣷⣄⢣⠀⠀⠀⠀⠀⠀
⣿⣄⡀⠀⠀⡀⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇⠀⠀⠀⠚⠑⣿⣽⠿⢸⢸⣿⣿⡜⣷⡇⡅⠀⡨⠠⠤⠄⠀⠀⢸⠸⡀⠀⡘⠀⠀⣸⠀⠀⢿⣿⣿⡄⠃⠀⠀⠀⠀⠀
⠉⣿⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢹⣿⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠈⠶⠂⠈⠉⠀⠀⢹⣿⣿⣿⠈⡇⠉⠉⠀⠁⠀⠈⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⣿⣿⣿⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⣿⣿⣿⣀⠀⠀⣀⣤⣄⡈⠀⠀⢀⣠⣤⣀⡀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⡁⠀⣸⣿⣀⣀⣀⣀⡀⠀⣀⣀⠀⣿⣼⣻⣄⣂⣀⡀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⣆⣀⢀⣿⣿⣿⡏⠛⠀⣼⣏⣀⣈⣿⡄⢰⡿⠁⠀⠸⠇⠀⣿⠀⠰⢾⣿⣾⣧⣿⣯⣸⣿⣿⣿⡇⠀⢸⣿⠀⠀⠈⣿⡀⠀⡼⠁⠀⠀⠉⠉⠛⠛⠟⠃⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠉⣡⣿⣿⣿⡻⣷⠀⢿⡍⠉⠉⣩⡁⢸⣧⠀⠀⢀⡀⠀⣿⡀⠀⢸⡇⠀⠀⣿⣇⣿⣿⣿⣿⡇⠀⢸⣿⠀⣀⠀⠘⣷⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡟⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠳⠶⠚⠁⠀⠘⠳⠖⠛⠓⠂⠐⠛⢻⣿⣿⣿⣿⠓⠂⠈⠻⠶⠋⠀⠀⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⡇⠀⠀⠀⠀⠀
⠀⠖⠶⣶⠟⢶⠄⠰⠶⠶⠖⠀⠀⠀⢠⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡛⠀⠀⠀⠀⠀⠀⠾⠶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣶⣿⣶⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⢿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡾⡏⠀⠀⠸⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡟⠻⡿⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⣈⠃⢸⠀⠁⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣩⣥⣠⣷⣶⣾⣷⡦⣈⢙⡛⢿⣿⣿⣿⣿⣿⣿
⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠎⡇⠀⠀⢶⣞⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⡏⠀⠈⣇⠀⢹⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠀⢐⣿⣯⣿⣿⣿⣿⣿⣿⣽⣆⣻⣶⣿⣿⣿⣿⣿⣿
⣥⣬⣙⡻⣿⣿⣿⣿⣿⡿⠟⠛⠛⣁⣤⣵⣤⣄⡸⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣩⣤⣴⣴⣾⣿⣿⣏⢷⣶⣤⢭⣙⠿⣿⣿⣿⣿⣿⣿⠄⠀⢀⣿⣿⣿⣿⣽⣿⣿⣯⣭⣭⣽⣿⣿⣿⣿⣿⣭⣿⣿
⣿⡟⣿⡟⠪⣿⣿⡿⢋⣴⡟⠋⣴⣿⣿⣿⣿⣿⣿⣿⢻⣶⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⣰⣿⣷⣾⣿⣿⣿⣿⣿⣷⣿⣿⣌⣻⡗⠺⣿⣿⣿⣿⣥⣾⠿⠿⠟⠛⠉⠛⠛⠛⠋⠉⠉⢹⣿⣿⣿⣿⣿⣿⢿⣿⣿
⣿⣷⡈⠀⠀⢘⣇⢀⣼⢽⣻⣛⣻⣿⣛⣿⣿⣿⣿⣿⣇⣻⣁⠙⢿⡿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣿⣿⣿⣷⣾⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣭⣿⡟⠘⠊⠟⠸⠋⠙
⣿⣿⣿⣷⣶⠘⠟⠛⠛⠛⠛⠛⠛⠛⢛⣿⣿⣿⣿⡿⣿⣿⣿⣷⣿⣿⣞⠿⣿⣿⣿⠛⠛⠋⠉⠀⠀⠀⠉⠁⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⢹⢿⣿⣿⣿⣿⣿⣿⣖⣀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⢡⣄⠀⠀⠀⠀⠀
⠉⠉⢻⢛⠁⠉⠛⠿⠶⠂⠀⠀⠀⠀⣸⣿⡿⠛⠛⢱⣯⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣦⣤⣤⣄⣀⠀⠠⢤⠤⠀⠀⠠⣾⣿⣿⠇⠀⢈⠁⠁⠀⠉⠛⠛⢿⡿⠟⠁⡉⠉⠛⠛⠻⠶⠂⠀⠀⠀⠀⠒⠀⢉⡺⠗⠂⣄⣀
⣀⠀⠈⠃⠀⠀⣄⡤⠀⠀⢀⠀⠀⠀⠙⠛⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⠃⣀⢸⠀⠀⠐⢈⠀⠒⠛⠀⠀⠈⣿⣷⠦⠀⢀⣆⣭⣑⣁⢦⡆⠀⠼⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠿⠛⠶⠶⢿⡏
⠀⠀⠀⢈⡁⠠⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⣗⡜⢛⣛⡿⣯⣌⣠⣟⣽⣿⣿⣿⣶⡄⣤⣈⠀⠀⠀⠈⠀⢉⠉⢀⣀⡀⠸⣛⣚⠛⢛⣉⠉⠉⣛⣊⠁⣀⣀⡲⣤⣀⣁⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡌⢠⣬⣤⣤⣴⡼⣯⠁⢺⣿⣿⡇⠀⠁⢸⣿⣇⡀⢿⠗⣿⣽⠀⢸⣿⣷⡄⣿⣿⣄⣼⣿⠀⠀⣿⣿⠀⠘⢿⠗⡻⡿⠁⠀⠀⠈⢿⣿⡇⠀⠀⠀⠀⠀⠀
⣠⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⡀⠀⠀⠀⢤⣄⢀⣸⣿⣿⡇⠰⣷⣴⡶⣴⠳⣦⠀⢸⣿⣿⡟⠓⠒⣾⣿⣿⡇⠀⠀⣿⣿⣷⣾⣿⡿⣿⣿⣿⠀⢹⣿⠀⠀⣿⣿⠀⠀⣬⣃⠔⣆⠀⠀⠀⠀⠈⢿⣗⠀⠀⠀⠀⠀⠀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠸⠿⠿⠿⡿⠀⠿⠿⠘⣿⣿⡇⠘⠿⠿⠀⠸⠿⢿⣾⠿⠃⠀⠾⠿⢃⣸⡿⠧⠀⠀⢰⣦⡀⢻⣆⠀⠀⠀⠀⠀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡈⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⡀⠀⣱⣺⣿⣿⡇⠀⠀⠠⠤⣤⣴⣅⠄⠀⢰⣧⡈⣿⡯⡅⡤⠤⠀⠀⠠⠀⠀⠀⢠⠀⠄⠠⡼⢯⠀⢠⠀⠀⢸⣿⣷⣆⢻⡀⠀⠀⠀⠀
⣤⠂⠀⠀⠀⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇⠀⠀⠀⠸⠟⣿⣽⠿⢸⢸⣿⣿⣎⢿⣿⡇⠀⡨⡠⠄⢄⠀⠀⠠⠸⠀⠀⣘⠀⠀⣸⠀⠀⠀⣿⣿⣿⡄⠃⠀⠀⠀⠀
⣟⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢹⣿⠏⠉⠉⠉⢉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠀⣠⠄⠈⠉⠀⠀⢹⣿⣿⣿⠈⢿⠉⠉⠀⠁⠀⠈⠀⠀⠈⠀⠈⠉⠉⠁⠀⠈⠀⠀⠀⣻⣿⣿⣇⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣾⣿⣿⠀⢻⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡄⠘⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣿⣿⣿⣄⣀⠀⠀⣀⣤⣄⡂⠀⠀⢀⣠⣤⣀⡀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿⡃⠀⣸⣿⣀⣀⣀⣀⡀⠀⣀⣀⠀⣿⣠⣴⣴⡴⣢⢀⡀⣸⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⠀⢀⣿⣿⣿⣏⡈⠛⠀⣼⣏⣀⣈⣿⡄⢰⡿⠁⠀⠸⠇⠀⣿⠀⠠⢾⣿⣿⣧⣿⣯⣸⣿⣿⣿⡇⠀⢸⣿⠀⠀⠈⣿⡀⠀⡼⠁⠀⠀⠉⠙⠛⠻⠿⠿⠇⢻⣿⣿⡇⠀⠀⠀⠀⠀
⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢛⣿⣿⣿⠙⠻⣷⠀⢿⡍⠉⠉⣩⡁⢸⣧⠀⠀⢀⡄⠀⣿⡀⠀⢸⡇⠈⠉⣿⡏⣿⣿⣿⣿⡇⠀⢸⣿⠀⣀⠀⠘⣷⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡷⠞⠋⠀⠈⠛⠶⠶⠋⠀⠀⠙⠳⠶⠚⠁⠀⠘⠳⠖⠛⠓⠂⠐⠛⢻⣿⣿⣿⣿⡗⠂⠈⠻⠶⠋⠀⠀⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⡇⠀⠀⠀⠀
⠀⠶⠶⣶⠏⢶⠔⠰⠶⠶⠖⠀⠀⢸⣿⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠿⠶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⢿⡟⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢿⠁⠀⣀⠘⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣈⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣿⠁⠙⠏⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢤⡉⠘⡇⠀⢠⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢡⣴⣄⣶⣿⣿⣿⣷⡵⣄⢍⡙⢿⣿⣿⣿⣿⣿
⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⡆⠀⠀⢺⣮⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢹⡀⣀⣹⡀⢸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠁⠀⣜⣿⣾⣿⣿⣿⣿⣿⣿⣼⣌⣿⣶⣿⣿⣿⣿⣿
⣶⣮⣍⡻⣿⣿⣿⣿⣿⠟⣉⣉⢉⣠⣦⣴⣶⣤⣘⣏⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢱⣶⠶⣴⣿⣿⣿⣿⣽⣷⣷⡲⣬⣝⢿⣿⣿⣿⣿⣿⠟⣀⣀⣸⣿⣿⣿⣭⣽⣿⣿⣾⣭⣭⣿⣿⣿⣿⣿⣿⣾⣿
⣿⠹⣿⠛⠺⣿⡿⠟⢠⣼⣋⢁⣾⣿⣿⣿⣿⣿⣿⣿⠻⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁⢰⣷⣷⣾⣿⣿⣿⡿⣿⣿⣿⣿⣷⣸⣿⠓⢿⣿⣿⡿⠷⠾⠿⠻⠛⠋⠉⠉⠛⠛⠉⠉⠁⠉⢸⣿⣿⣿⣿⣿⣿⢿⣿
⣿⣆⠀⠀⠀⣻⣅⣠⣽⣻⣿⣝⣻⣿⣛⣿⣿⣿⣿⣿⣧⣹⣁⡘⣿⡿⣿⣿⣿⣿⣿⣿⣫⣧⣴⣿⠿⣿⣿⣿⣿⡿⠿⢷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⡏⠈⠋⠏⠸⠉
⣿⣿⣿⣷⢖⠙⠛⠉⠉⠛⠛⠛⠉⠉⢹⡿⣿⣿⣿⡿⣿⣿⣿⣷⣿⣿⣎⠻⣿⣿⣿⠋⠋⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⢠⡔⠀⠀⠀⠀
⠉⠛⣿⠟⠈⠙⠛⠿⠶⠂⠀⠀⠀⠀⣼⣿⡟⠛⠛⢱⣯⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣶⣶⣦⣠⡀⠠⢤⠤⠀⠀⠠⢰⣿⣿⡟⠀⠁⠉⠈⠁⠉⠉⠛⠻⣿⢿⠟⢀⠉⠉⠛⠛⠿⠖⠀⠀⠀⠀⠀⠀⠀⣴⡻⠻⢲⢄
⡀⠀⠉⡀⠀⢠⣤⡤⠀⠀⠀⠀⠀⠀⠛⠻⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠏⣀⢸⠀⠀⠐⢨⠀⠒⠛⠀⠀⠄⢿⣿⠦⣀⢀⣰⣴⣭⣉⡣⣶⠀⠰⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠛⠛⠶⠿⢼
⠀⠀⠀⢈⠀⠜⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣗⡟⠛⠛⣿⣯⣌⣠⣏⣽⣿⣿⡿⢷⣤⣤⣎⠀⠀⢀⡈⠀⣉⠉⢀⣀⡀⠐⣛⣛⠛⢛⣉⠉⠈⣉⣑⠉⣀⣀⠀⣤⣄⣐⠐⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠠⣬⣤⣥⣴⡽⣯⠓⢺⣿⣿⡇⠀⠁⢸⣿⣷⡄⢻⠟⣿⣽⠀⢸⣿⣷⡄⣿⣿⣤⣾⣿⠀⠀⣿⣿⠀⠘⢿⠗⡻⡿⠁⠀⠀⠀⠹⣿⣿⡆⠀⠀⠀⠀⠀
⣄⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⣤⣤⡀⣼⣿⣿⡇⠸⣷⣴⡶⣼⠳⣧⠀⢸⣿⣿⡗⠒⠲⣿⣿⣿⡿⠀⠀⣿⣿⢷⣾⣿⣿⣿⣿⣿⠀⢹⣿⠂⠀⣿⣿⠀⠀⣨⣃⠔⣆⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠹⠿⠿⠿⡿⠂⠿⠿⠈⢿⣿⣏⡸⠿⠿⠀⠘⠿⢿⣿⠿⠃⠀⠾⠿⠃⢸⣿⠧⠀⠀⠀⣸⣄⠈⢷⡀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⡀⠀⣱⣾⣿⣿⡇⠀⠀⠠⠤⣄⣤⣅⠄⢀⠀⣷⣄⣿⣿⣼⡤⠤⠀⠀⠤⠀⠀⠀⢠⠀⠄⠠⡀⢿⠃⢠⠀⠀⠀⢹⣿⣷⡌⣧⠀⠀⠀⠀
⡶⠀⠀⠀⠈⠿⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇⠀⠀⠀⠺⠟⣿⣽⠿⢸⠸⣿⣿⣯⢻⣿⡇⠀⡨⡠⠤⢄⠀⠀⢸⠸⠀⠀⣘⠀⠀⣸⠀⠀⠀⢸⣿⣿⣿⠘⡇⠀⠀⠀
⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢿⣿⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠀⣠⡄⠈⠉⠀⠀⢩⣿⣿⣿⡎⢿⠏⠉⠀⠁⠀⠈⠀⠀⠈⠀⠈⠉⠀⠁⠀⠈⠀⠀⠀⠘⣿⣿⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⣿⣿⡇⠘⡿⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠘⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣿⣿⣧⣄⣡⠀⠀⣀⣤⣄⡀⠀⠀⢀⣠⣤⣀⡀⣀⣀⠀⢀⣀⣀⠀⣀⣀⣀⣠⣿⣿⣿⡇⠀⣸⣿⣀⣀⣀⣀⡀⠀⣀⣀⠀⣿⢀⣤⣶⣶⣂⡀⣀⠀⣿⣿⣿⡗⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠔⠀⢸⣿⣿⣿⣃⡈⠛⠀⣼⣏⣀⣈⣿⡄⢰⡿⠁⠀⠸⠇⠀⣿⠀⠠⢾⣿⣿⣧⣿⣯⣼⣿⣿⣿⡇⠀⢸⣿⠀⠀⠈⣿⡀⠀⡼⠁⠀⠀⠈⠉⠙⠛⠛⠿⠟⢹⣿⣿⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢸⣿⣿⡏⠙⠻⣷⠀⢿⡍⠉⠉⣩⡁⢸⣧⠀⠀⢀⡄⠀⣿⡀⠀⢸⡇⠀⠉⣿⡏⣿⣿⣿⣿⡇⠀⢸⣿⠀⣀⠀⠘⣷⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣅⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⠶⠞⠋⠀⠈⠛⠶⠖⠋⠀⠀⠙⠳⠶⠛⠁⠀⠘⠳⠖⠛⠓⠂⠐⠛⣻⣿⣿⣿⣿⠗⠂⠈⠻⠶⠋⠀⠀⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀
⠀⠶⠶⣶⠏⢶⠖⠰⠶⠶⠖⠀⠀⣾⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠿⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡏⠀⠀⠀⠀

                    ䷩ 𝚕𝚒𝚗𝚎 3487

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

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/10/19/linux-foundation-vs-security/#comments
    Gemini version at gemini://gemini.techrights.org/2022/10/19/linux-foundation-vs-security/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 10.19.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Microsoft’s_Open_Compute_Project_Foundation_(OCP):_Embrace,_Extend,
Extinguish_Security,_Thanks_to_‘Linux’_Foundation⠀✐


Posted in Deception, Hardware, Microsoft, Security at 3:53 pm by Dr. Roy
Schestowitz


Video_download_link | md5sum bce5a9a3748283875b25c7460fc00203
This is Digital Imperialism, Not Security
Creative Commons Attribution-No Derivative Works 4.0


http://techrights.org/videos/fake-lf-security.webm


Summary: The so-called ‘Linux’ Foundation is a hostage of Microsoft; it not
only badmouths Linux security (while refraining from saying any negative thing
about Windows) but it also helps Microsoft actively undermine security


THE Linux_Foundation continues to deepen_its_relationship_with_Microsoft, but
not with the GNU/Linux community or with security professionals. That’s because
it’s all about money, not science, and now it means that back_doors are chosen
over honest people, who aren’t paying Jim Zemlin and Sheela_Microsoft.


This moreover means that the so-called ‘Linux’ Foundation actively spreads_FUD
about_Linux on behalf of Microsoft. The video above discusses some of the
latest developments and condemns Phoronix for having become like a Microsoft
parrot. █



                    ䷩ 𝚕𝚒𝚗𝚎 3534

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

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/10/19/microsoft-future-is-layoffs/#comments
    Gemini version at gemini://gemini.techrights.org/2022/10/19/microsoft-future-is-layoffs/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 10.19.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Microsoft:_Azure_is_Our_Future._Also_Microsoft:_TWO_Rounds_of_Layoffs_in
Azure_So_Far_This_Year_(Stock_Crashes_Again).⠀✐


Posted in Finance, Microsoft at 8:14 am by Dr. Roy Schestowitz


Remember that this isn’t_the_first_year that “Azure” has rounds of layoffs and
"Azure"_has_been_failing_for_14_years


      🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇MSFT - Microsoft Corp Stock Performance Report -
                                Barchart.com⦈
                                    Source


Summary: As noted in passing in_Daily_Links, only a couple of months have
passed and there is yet another wave of layoffs at Microsoft, this one too
impacting Azure, which Microsoft insists is the future of the company


⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣿
⣿⣇⣭⣄⣜⣸⣄⣠⣠⣊⣀⣉⣄⣉⣂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⣶⠒⠲⠒⣾⣿
⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿
⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿
⣟⣛⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠋⠉⠉⠀⠙⠛⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡃⣿⠋⠹⠍⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿
⣿⣿⡇⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠉⠀⠀⠀⢀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿
⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣓⣒⣒⣒⣒⣂⣀⣀⣀⣀⣀⡉⠉⠐⠉⠛⠛⠛⠛⢛⣛⣛⢛⣛⣛⣛⣛⡃⠿⠉⠙⠍⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣀⣀⣀⡀⠀⠈⣙⣿⣿⡇⣀⣀⣀⣀⣸⣿
⡿⠻⢟⠛⣿⣿⠻⠿⡟⢿⣿⣟⠿⢝⠛⣿⣿⠻⢯⡟⢻⣿⡟⠟⢝⠛⣿⣿⠿⠯⡛⢻⣿⡟⠿⣝⠛⣿⣿⠿⠯⡛⢻⣿⡿⠿⡟⡟⣿⣿⠻⢛⠛⣿⣿⠿⠿⣟⠛⣿⣿⠻⢿⡻⢻⣿⣟⠿⡟⡿⣿⣷⣿⣿⣿⣿⣿⣿
⣯⣥⣬⣬⣭⣭⣭⣭⣥⣭⣭⣥⣭⣬⣬⣭⣭⣤⣭⣥⣬⣭⣭⣬⣭⣤⣭⣭⣭⣭⣥⣬⣭⣭⣬⣭⣤⣭⣭⣭⣬⣥⣬⣭⣥⣭⣥⣭⣭⣭⣤⣭⣤⣭⣭⣭⣤⣭⣤⣭⣭⣥⣬⣭⣬⣭⣥⣭⣥⣭⣭⣭⣭⣭⣭⣭⣽⣿
⣯⡭⡭⣭⣭⠭⣭⡭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⡉⣉⣉⣉⣉⣉⣉⣉⣽⣿
⡿⠷⠤⠤⠼⠴⠤⠤⠤⠤⠤⠤⠦⠤⠼⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠓⠂⠀⠓⠀⠀⠀⠚⢿⣿
⣿⣿⣙⣛⣛⢹⣿⣿⣿⣿⣉⠛⡙⣋⡙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡙⣛⠛⡏⡙⢋⣿⠛⠛⠙⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿
⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
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⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣶⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⣉⣹⣁⣈⣹⣉⣙⣉⣿⣿⣿⣿⣿⣷⣾⣶⣶⣶⣶⣿⢸⣿⣶⣶⣶⣶⣿⣶⣶⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠿⠿⡿⢿⣿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⠿⣿⠿⢿⠿⠿⣷⢰⣿⡿⢿⠿⠿⡿⢿⠿⠿⢿⣿⣿⡇⢸⣿⣿⣿⣿
⣿⣿⣙⣉⣉⣉⣹⣿⣿⣿⠷⢦⠶⣴⠶⠞⣾⣷⣾⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⢤⣤⣤⣿⣿⣷⣶⣾⡶⢷⠲⠾⠶⢲⣿⢸⣿⠶⢶⠶⠖⡲⠒⠒⣷⣼⣿⣿⡇⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣭⣭⣤⣬⣭⣽⣤⣬⣤⣿⣿⣿⣷⣿⣾⣿⣾⣶⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿

                    ䷩ 𝚕𝚒𝚗𝚎 3599

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

(ℹ) Images, hyperlinks and comments at http://techrights.org/2022/10/19/walmart-msie-in-2022/#comments
    Gemini version at gemini://gemini.techrights.org/2022/10/19/walmart-msie-in-2022/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 10.19.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Walmart’s_Employee_Benefits_Web_Site_Requires_Microsoft_Internet_Explorer_in
2022⠀✐


Posted in Microsoft, Windows at 5:00 am by Guest Editorial Team


Reprinted with permission from Ryan


OO October 15th, open enrollment for next year’s benefits started for Walmart
employees.


It was all going well until we had to set up beneficiaries.


When I got to that part, the entire page looked old, gross, and vaguely
Microsoft Internet Explorer-ish.


Sure enough, when I clicked on the buttons in a modern Web browser, nothing
happened.


Soon, I was left figuring out how to load a Windows 10 Enterprise Edition 90
day evaluation copy into VirtualBox from an image supplied by Microsoft…..


Yeah, I’m serious.


I was disgusted that it took up most of an afternoon to get the appliance
working.


Fedora, for its part, was entirely unhelpful because to set up VirtualBox, you
pretty much (1) have to have “Secure” boot turned off so it won’t reject the
kernel modules (that’s fine, I already had that crap turned off anyway), (2)
make sure that dkms and kernel-devel are installed, even though VirtualBox
doesn’t depend on them.


It uses akmods to build kernel modules for VirtualBox to use (kmods), and
finally after a while of jiggery pokery, I was able to load the virtual
appliance and run Internet Explorer to get at the beneficiaries page.


It was even more horrible than I imagined. Not just because the virtual machine
ran like shit, or because Microsoft hasn’t bothered to re-base it on a current
version of Windows 10 that has had security patches, but because in the end I
had to use Internet Explorer, which I did not ever use as my default browser.


Which I took pride in removing from Windows 98, lock, stock, and barrel, with
RoM II SE when I was a teenager.


I wasn’t especially displeased with Windows in the 1990s, until they welded
this unstable bloated piece of junk into the guts of Windows 98.


They never admitted it, of course, but it wasn’t as integrated as they said it
was, and without its bloated web-ified Active Desktop shit (RoM replaced the
shell with the one from Windows 95 OSR 2.1), it was actually very stable.


One time I even hit the 37 day bug and had Windows crash, but for some reason,
someone at Microsoft noticed that and patched it. Given that I was never able
to keep Windows 98 running for more than a day or two with Internet Explorer in
it, I wonder how anyone even found that bug.


That would be an interesting question to have answered on Raymond Chen’s blog,
but due to politics, I doubt anyone from Microsoft will ever admit, they
probably removed Internet Explorer from their computers too and then hit the
bug and decided it annoyed them.


True to form, the Windows Blog has a_post raving about their own IE Mode in
Microsoft Edge. “Dual Engine Advantage”…..LOL


They have some corporate customers, apparently Walmart wasn’t the only big
company to write an incredibly big pile of shit using Internet Explorer and
then leave it there for all time, saying how much use they’re getting out of IE
Mode for Microsoft Edge.


Of course, there is a Microsoft Edge for “Linux”, but it wouldn’t do you any
good to install it trying to get at an IE Mode.


See, IE Mode isn’t some emulation of IE.


Internet Explorer 11 is actually still there, in Windows 11, baked into the
operating system. Receiving only the bare minimum amount of patching to fix
security holes and to put new TLS certificates into it so you still can point
it at a Web server.


Windows 11 has an embarrassing oversight that Bleeping Computer revealed a
while back, where you can still start Internet Explorer 11 in browser mode, and
it’s fully functional, but you have to go through the Legacy Control Panel to
get to it.


In many respects, Windows 11 is about the furthest thing from a modern computer
operating system there is.


For Star Trek: Picard fans, it’s more like the Hellish Alternate Timeline that
Q created, where humanity became fascists and kept raping the environment and
glossed over the destruction of the biosphere with a giant shield over the
planet.


“They didn’t fix anything….”, Q explains….”Here, they just tend to the corpse.”


Windows has basically ran its course. The minute Microsoft drops IE Mode and
the other “But this mess we have needs that to run.” garbage is the minute that
the last person to leave Windows turns out the lights.


Oh they say they’ll support it until 2029 as a well tended corpse, but will
they really?


Could be more, could be less. Why trust them? Why pay them forever and beg them
to keep this mess on the road?


If you’re not fixing your applications now, you’ll be paying Microsoft a lot of
money for legacy support (because that’s what Windows is now, a failing
platform with legacy support options) and fixing the application later anyway,
probably when you make the move to GNU/Linux.


When one thing fails, Microsoft doesn’t take their lumps and learn their lesson
and design something that works. But they do design something else. And along
with that something else, they leave behind a pile of skulls from their other
abortive attempts.


Consider how many terminals and shells Windows has now, including some from the
“Linux subsystem”.


But the fact that it has three Web browsers and two of them don’t even allow
you to open them anymore (but are still there) is more amusing than that.


Windows is utter crap. It’s just a complete garbage pit, literally. They just
give up on things, leave them there forever, and go make another mess.


They themselves have a nicer euphemism. “Technical debt.”


The fact that they have their own term for the mess that they caused and still
proceed to make the mess worse is entertaining.


I have no idea what I’ll do if I need to use that page again in the future.
Maybe try to find some way to load the Windows 11 evaluation edition in the VM,
but that won’t be for at least a year. Hopefully, someone at Walmart fixes it.
█



                    ䷩ 𝚕𝚒𝚗𝚎 3784

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

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 10.19.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Links_19/10/2022:_Many_New_Videos_and_antiX-22_Released⠀✐


Posted in News_Roundup at 6:49 pm by Dr. Roy Schestowitz


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇GNOME bluefish⦈


§ Contents⠀➾


    * GNU/Linux
          o Server
          o Audiocasts/Shows
          o Kernel_Space
          o Graphics_Stack
          o Applications
          o Instructionals/Technical
          o Games
    * Distributions_and_Operating_Systems
          o New_Releases
          o BSD
          o SUSE/OpenSUSE
          o Fedora_/_Red_Hat_/_IBM
          o Canonical/Ubuntu_Family
          o Devices/Embedded
          o Open_Hardware/Modding
          o Mobile_Systems/Mobile_Applications
    * Free,_Libre,_and_Open_Source_Software
          o Web_Browsers/Web_Servers
                # Mozilla
          o Content_Management_Systems_(CMS)
          o Education
          o FSF
          o Programming/Development
                # Editors
    * Leftovers
          o Hardware
          o Pseudo-Open_Source
                # Openwashing
          o Security
                # Privacy/Surveillance
          o Finance
          o Civil_Rights/Policing
          o Internet_Policy/Net_Neutrality
    * Gemini*_and_Gopher
          o Personal
          o Technical
                # Internet/Gemini
                # Programming
    * § GNU/Linux⠀➾


          o ⚓ Make Use Of ☛ Do_You_Really_Need_to_Shut_Down_Your_Linux_PC?⠀⇛


                 Linux servers run for days, months, and even years on
                 end. Does the same hold true for Linux desktops?


                 It’s a well-known fact that Windows PCs need to be shut
                 down or restarted on a semi-regular schedule to preserve
                 their stability and apply security updates from
                 Microsoft. If you fail to do this, your machine could
                 crash, and in all likelihood will shut down by itself
                 anyway. But the same isn’t always true for Linux systems.
                 Here’s why.


          o § Server⠀➾


                # ⚓ Cockpit_278⠀⇛


                       Cockpit is the modern Linux admin interface. We
                       release regularly.


                       [...]


                       Podman now displays CPU, memory usage, ports, and
                       volumes of the containers in a pod.


                       [...]


                       Podman can now create pods, including optional port
                       and volume mappings.


                # ⚓ Red Hat Official ☛ Red_Hat_OpenShift_security_portfolio
                  grows_with_new_Red_Hat_Insights_Vulnerability_service⠀⇛


                       Red Hat Insights, which is included with Red Hat
                       subscriptions, analyzes platforms and applications
                       to help enterprises manage hybrid cloud
                       environments. Insights uses predictive analytics
                       and deep domain expertise to reduce complex
                       operational tasks from hours to minutes, including
                       identifying security and performance risks,
                       tracking licenses and managing costs.


                # ⚓ Container Journal ☛ Sysdig_Adds_Kubernetes_Cost_Monitoring
                  Tool_to_Portfolio_–_Container_Journal⠀⇛


                       Sysdig today adds a Cost Advisor module for
                       Kubernetes clusters to its portfolio of tools for
                       managing and securing container environments.


                       Aaron Newcomb, director of product marketing for
                       Sysdig, says as an extension of the existing Sysdig
                       Monitor service, the Cost Advisor module eliminates
                       the need for IT teams to deploy a separate platform
                       to track Kubernetes costs.


                # ⚓ Container Journal ☛ When_Kubernetes-as-a-Service_Doesn’t
                  Cut_It_–_Container_Journal⠀⇛


                       When it comes to installing and managing your own
                       Kubernetes service, the advice you’ll usually hear
                       is, “Don’t do it!” But sometimes, that advice is
                       wrong.


                # ⚓ FOSSLife ☛ How_to_Use_Checkmk_to_Monitor_Your_Home_Router⠀⇛


                       Every organization needs monitoring to make sure
                       that servers, networks, applications, cloud assets,
                       and other elements work as they should. Monitoring
                       also provides timely alerts, and it helps IT teams
                       track down the causes of (potential) problems.
                       Having a suitable monitoring tool will help you
                       resolve problems faster, ideally before they have
                       an impact on operational systems. Checkmk is an
                       open source monitoring tool that is ideally suited
                       for modern hybrid environments, combining
                       enterprise-grade scalability and security with the
                       extendibility of open source software.


                       In most cases, Checkmk runs on a dedicated server
                       or virtual machine (VM). As I am just running a
                       small monitoring setup for my home office
                       environment, my host will be my computer with
                       Ubuntu 18.04. The computer has four CPU cores and
                       4GB of RAM, which is more than enough to get
                       started. Checkmk also runs on other Linux
                       distributions, such as Debian, Red Hat Enterprise
                       Linux, CentOS, or SUSE Linux Enterprise Server, and
                       you can also run Checkmk on a Docker container or
                       virtual appliance. If you install Checkmk on a
                       dedicated host, the only additional steps you need
                       are transferring the files to the server.


                       Checkmk can monitor anything with an IP address,
                       including servers, cloud assets, and network
                       devices, as well as systems that belong to the
                       Internet of Things (IoT). For this article, I chose
                       to monitor my TP-Link router. Checkmk also comes
                       ready-equipped with reasonable thresholds for
                       alerts. These pre-configured thresholds will come
                       in handy, because your monitoring will be up and
                       running within a few minutes without you having to
                       worry about setting up alerts.


          o § Audiocasts/Shows⠀➾


                # ⚓ Video ☛ Solus_GNOME_4.3_Quick_overview_#linux_#SolusOS_–
                  Invidious⠀⇛


                       A Quick Overview of Solus GNOME 4.3


                # ⚓ Video ☛ Redcore_Linux_Proves_Gentoo_Doesn’t_Have_To_Be
                  Difficult_–_Invidious⠀⇛


                       Redcore Linux is a distribution based on Gentoo
                       Linux using the KDE Plasma desktop environment.
                       Redcore comes with a easy-to-use graphical
                       installer, and Redcore has it’s own easy-to-use
                       package manager named Sisyphus.


                # ⚓ Tux Digital ☛ 34:_Failed_Closed_Source_–_Linux_Out_Loud_–
                  TuxDigital⠀⇛


                       This week, Linux Out Loud chats about when closed-
                       source services fail you.


                       Welcome to episode 34 of Linux Out Loud. We fired
                       up our mics, connected those headphones as we
                       searched the community for themes to expound upon.
                       We kept the banter friendly, the conversation
                       somewhat on topic, and had fun doing it.


                # ⚓ Tux Digital ☛ 35:_Linux_Shipped_–_Linux_Out_Loud_–
                  TuxDigital⠀⇛


                       This week, Linux Out Loud chats about open software
                       paired with open hardware.


                       Welcome to episode 35 of Linux Out Loud. We fired
                       up our mics, connected those headphones as we
                       searched the community for themes to expound upon.
                       We kept the banter friendly, the conversation
                       somewhat on topic, and had fun doing it.


                # ⚓ mintCast Podcast ☛ 398_–_I_Hear_A_Leak,_Must_Be_Android_–
                  mintCast⠀⇛


                       First up in the news: Debian carries buffer
                       patches, IceWM 3, Wireshark 4, free Ubuntu Pro for
                       all, KDE winds down Plasma 5, VirtualBox 7, Proton
                       drives prices, and dreams of Forky;


                       In security and privacy: New Chinese malware for
                       everything, Intel springs a leak, and Android’s
                       leak exposed;


                       Then in our Wanderings: Joe futzes with his 3D
                       printer, while Moss continues to fight with Mint
                       21;


                       In our Innards section, we begin a historical
                       journey through Linux distros;


                       And finally, the feedback and a couple of
                       suggestions


                # ⚓ Video ☛ Ubuntu_22.10_overview_|_Upgrade_your_desktop._–
                  Invidious⠀⇛


                       In this video, I am going to show an overview of
                       Ubuntu 22.10 and some of the applications pre-
                       installed.


                # ⚓ Video ☛ Your_Raspberry_Pi_as_a_Desktop_–_DeskPi_Lite_Full
                  Review_–_Invidious⠀⇛


                       The Raspberry Pi can be transformed into a full
                       desktop PC (with the right parts, at least). In
                       this video, Jay reviews the DeskPi Lite, which is a
                       case for your Raspberry Pi that transforms it into
                       a desktop!


                # ⚓ Video ☛ Possible_Future_GitHub_Copilot_Lawsuit_–
                  Invidious⠀⇛


                       Word came out about some lawyers starting an
                       investigation on GitHub Copilot to determine if a
                       lawsuit is appropriate. In this video I’ll break
                       this down for everyone and give you my thoughts
                       too.


                # ⚓ Linux in the Ham Shack ☛ LHS_Epsiode_#485:_QSLing_Deep_Dive
                  |_Linux_in_the_Ham_Shack⠀⇛


                       Hello and welcome to Episode 485 of Linux in the
                       Ham Shack. In this episode, the hosts discuss the
                       art and science of sending and receiving QSL cards,
                       either physically or digitally. There are several
                       methods and it can be something of a chore to
                       navigate. We hope you enjoy this episode and have a
                       great week.


                # ⚓ FLOSS_Weekly_703:_Hacker’s_Ethical_Dilemma_–_Marcus
                  Sailler,_Ethical_Hacking⠀⇛


                       In cybersecurity, there isn’t an ethical dilemma
                       for bad actors, but there is for good ones making
                       and improving open source code that bad actors can
                       use too. Marcus Sailler schools Doc Searls and
                       Katherine Druckman about that dilemma, and how he
                       deals with it as Head of Offensive Security in the
                       private sector after a long military career.


          o § Kernel Space⠀➾


                # ⚓ Linux Magazine ☛ First_Release_Candidate_of_Linux_6.1
                  Kernel…_–_Linux_Magazine⠀⇛


                       The development cycle for the 6.1 Linux kernel is
                       off and running with the availability of the first
                       Release Candidate.


                       The 6.1 kernel has hit the RC stage. Although this
                       won’t bring about any massive or deal-making
                       changes to Linux, there are still a few features to
                       get excited about. Most important is the first
                       inclusion of Rust, which will be greatly expanded
                       over the coming years.


                       Other notable features include support for the new
                       Intel Arc and AMD RDNA3 graphics, Multi-Gen LUR VM
                       series (which will give the kernel a significant
                       performance boost on memory-constrained systems),
                       and the new Kernel Memory Sanitizer.


                       In addition, the x86_64 version will warn over W+X
                       mappings, the AMD Platform Management Framework has
                       been merged, five vulnerabilities with Wi-Fi
                       handling – CVE-2022-41674 (kernels up to 5.19),
                       CVE-2022-42719 (5.2 to 5.19), CVE-2022-42720 and
                       CVE-2022-42721 (both 5.1 to 5.19), and CVE-2022-
                       42722 (5.8 to 5.19) – have been fixed.


                       This release also has better support for Intel’s
                       new Gaudi2 AI accelerator chip, EUFI support for
                       LoongArch CPUs, and more.


          o § Graphics Stack⠀➾


                # ⚓ Collabora ☛ A_look_at_Vulkan_extensions_in_Venus⠀⇛


                       When it comes to Vulkan applications and libraries,
                       you’ll be hard-pressed to write them without
                       leveraging the ever-expanding list of Vulkan
                       extensions made available to take advantage of the
                       quirks and features graphics hardware has to offer.
                       In brief, Vulkan extensions are addendums to the
                       Vulkan specification that drivers are not required
                       to support. They allow vendors to expand upon the
                       existing API to allow the use of features that may
                       be unique to a particular vendor or generation of
                       devices without having to retrofit the core API.


                       Even in virtualized environments, the use of
                       extended functionality for hardware-accelerated
                       applications can be paramount for performance.
                       Indeed, it is something that virtual graphics
                       drivers, like Venus, must take into account.


                       Venus is a virtual Vulkan driver based on the
                       Virtio-GPU protocol. Effectively a protocol on top
                       of another protocol, it defines the serialization
                       of Vulkan commands between guest and host. This
                       post will cover details of the Venus driver, its
                       components, and their relations in the context of
                       extensions.


          o § Applications⠀➾


                # ⚓ Linux Links ☛ Linux_Candy:_No_More_Secrets_–_data
                  decryption_effects⠀⇛


                       Linux Candy is a series of articles covering
                       interesting eye candy software. We only feature
                       open-source software in this series.


                       No More Secrets (nms) is a command line tool that
                       recreates the famous data decryption effect seen in
                       the 1992 thriller film Sneakers. It’s free and open
                       source software.


          o § Instructionals/Technical⠀➾


                # ⚓ Fedora_Downstreams_Made_Easier_–_Think._Debate._Innovate.⠀⇛


                       The first approach was disliked by all – the second
                       one won. However, implementing this was
                       challenging, as we needed to update Fedora, CentOS,
                       and RHEL at the same time to ensure spec files and
                       their backports kept working properly.


                       Thankfully, Fedora and CentOS engineers helped me
                       push through the changes there; and Stephen
                       Gallagher additionally wrote Red Hat deinitions for
                       RHEL and pushed them through there.


                       In a surprisngly short amount of time, Fedora
                       (F35+), CentOS 9 Streams, and RHEL have these
                       macros available, as does Amazon Linux 2022.
                       Package maintainers can use these macros in Fedora,
                       and expect them to work right for any downstream
                       distributions, without having to submit modified
                       PRs for those.


                # ⚓ Citizix ☛ How_to_install_and_Configure_RabbitMQ_Server_in
                  Ubuntu_22.04⠀⇛


                       In this guide we will explore how to install the
                       latest release of RabbitMQ in Ubuntu 22.04 Server
                       or Workstation RabbitMQ is an open source message
                       broker software that implements the Advanced
                       Message Queuing Protocol (AMQP).


                # ⚓ ID Root ☛ How_To_Install_PHP_7.4_on_Ubuntu_22.04_LTS_–
                  idroot⠀⇛


                       In this tutorial, we will show you how to install
                       PHP 7.4 on Ubuntu 22.04 LTS. For those of you who
                       didn’t know, PHP 7.4 is a significant update of the
                       PHP language that was “officially” released on
                       November 28, 2019. This is a standard upgrade from
                       now on from the existing PHP 7.3 release to PHP
                       7.4, which is the last version in the 7 PHP series.
                       There are multiple uses of PHP that make it a
                       strong scripting language option for web
                       development. It operates over the web server and
                       then processes all client requests into HTML files.
                       Another good use of PHP is that it is compatible
                       with different operating systems, like macOS,
                       Windows, and Linux.


                       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 PHP 7.4 on Ubuntu 22.04 (Jammy
                       Jellyfish). You can follow the same instructions
                       for Ubuntu 22.04 and any other Debian-based
                       distribution like Linux Mint, Elementary OS,
                       Pop!_OS, and more as well.


                # ⚓ Video ☛ How_to_install_Pinta_on_Linux_Mint_21_–_Invidious
                  [Ed: Caution needed; Microsoft Mono trap]⠀⇛


                       In this tutorial, we will show you how to install
                       PHP 7.4 on Ubuntu 22.04 LTS, as well as some extra
                       required packages by PHP


                # ⚓ H2S Media ☛ 4_Ways_to_Install_FreeCad_on_Ubuntu_22.04_LTS
                  Jammy_Linux⠀⇛


                       Learn the steps to install open-source FreeCad
                       software on Ubuntu 22.04 LTS Jammy JellyFish Linux
                       to start designing your ideas.


                       For engineers and draughtsmen, dealing with CAD
                       programs has become an integral part of everyday
                       work. But this is not limited to professionals
                       only, even hobby designers use such CAD software to
                       let their fantasies run wild in the 3D environment.
                       However, the software required for this is usually
                       quite expensive. In such cases, free and open-
                       source applications are the only hope.


                       FreeCAD is one of the free and open-source
                       applications for Windows, Linux, and macOS that let
                       professionals and hobby designers digitalize their
                       ideas. FreeCAD is a free alternative to
                       professional programs and is aimed at students and
                       interested parties who want to familiarize
                       themselves with the topic of 3D CAD modeling.


                # ⚓ TechRepublic ☛ How_to_enable_Ubuntu_Pro_to_gain_expanded
                  security_maintenance_and_compliance_|_TechRepublic⠀⇛


                       Canonical has announced that its security and
                       maintenance subscription service is now free for
                       personal usage for up to five machines. This
                       service offers 10 years of security for both the
                       system and over 23,000 applications.


                # ⚓ Make Use Of ☛ How_to_Use_the_tr_and_fmt_Commands_for_Text
                  Manipulation_on_Linux⠀⇛


                       The tr and fmt commands, short for translate and
                       format, are two of the most important text
                       manipulation commands on Linux.


                       Linux makes text manipulation and processing very
                       easy with tons of commands. Command-line tools like
                       tr and fmt help you change, format, and modify text
                       files right from your terminal.


                       In this article, we will discuss how to use the tr
                       and fmt commands to format text in different use
                       cases. So what are tr and fmt? And how do you use
                       them?


                # ⚓ Ubuntu Handbook ☛ How_to_Install_KDE_Plasma_5.26_in_
                  (K)Ubuntu_22.10_via_PPA_|_UbuntuHandbook⠀⇛


                       The latest KDE Plasma desktop 5.26 is finally made
                       into PPA for (K)Ubuntu 22.10 / Ubuntu Studio 22.10,
                       the next Ubuntu edition to be released soon.


                # ⚓ Make Tech Easier ☛ How_to_Fix_an_MTD_Device_Error_in_Linux
                  –_Make_Tech_Easier⠀⇛


                       When your device freezes in the middle of a boot,
                       it can be frustrating, especially when you don’t
                       know what the problem is. This guide shows you how
                       to fix your MTD device error and get your computer
                       up and running again.


                # ⚓ TechRepublic ☛ How_to_Use_the_VirtualBox_Unattended
                  Installation_Feature_|_TechRepublic⠀⇛


                       VirtualBox 7.0 is now available to install and it
                       offers a host of very important new features. One
                       of the more impressive features is the addition of
                       unattended install configuration within the GUI.
                       Now you no longer have to run those unattended
                       installations from the command line.


                # ⚓ Linux Capable ☛ How_to_Install_Stellarium_on_Manjaro_21
                  Linux⠀⇛


                       Stellarium is a free, open-source planetarium
                       software package licensed under the GNU General
                       Public License, version 2. It is available for
                       Linux, Windows, and macOS. A port of Stellarium
                       called Stellarium Mobile is available for Android,
                       iOS, and Symbian as a paid version. It was
                       initially developed by Fabien Chéreau and is now
                       maintained by the Stellarium team. The software
                       renders the night sky in 3D, allowing users to see
                       stars, constellations, planets, nebulae, and other
                       astronomical objects in their correct positions
                       relative to each other. Users can also set the time
                       and date to see how the night sky changes.


                       In addition to being a helpful tool for astronomers
                       and amateur stargazers alike, Stellarium is also
                       used in educational settings, such as planetariums
                       and science museums. With its realistic graphics
                       and accurate simulations, Stellarium provides an
                       immersive experience that can help people to
                       understand the night sky better.


                # ⚓ UNIX Cop ☛ How_to_run_scripts_without_typing_the_full
                  path?⠀⇛


                       Hello, friends. In this post, you will learn How to
                       run scripts without typing the full path? So, you
                       can use these scripts as commands from a terminal
                       or launcher.


                # ⚓ TecAdmin ☛ How_to_Install_VirtualBox_7.0_on_Ubuntu_22.04⠀⇛


                       Did you know that you can run virtual machines on
                       your Linux computer? It’s true. Thanks to
                       virtualization software like VirtualBox, you can
                       set up a separate environment for any operating
                       system — even something as unusual as Windows or OS
                       X — without risk to your primary operating system.
                       You’ll find many uses for virtual machines, from
                       testing software before you release it to other
                       users to safely trying out experimental new
                       versions of Linux.


                       In this article, we show you how to install
                       VirtualBox on Ubuntu 22.04 and other common
                       versions of Linux.


                # ⚓ Red Hat Official ☛ How_to_monitor_information_about_your
                  system_with_Osquery_|_Enable_Sysadmin⠀⇛


                       Install and use Osquery to gather performance,
                       capacity, and other essential data about your
                       system using a SQL query language.


                # ⚓ TecMint ☛ Install_VirtualBox_7.0_in_Debian,_Ubuntu,_and
                  Linux_Mint⠀⇛


                       In this article, we will explain how to install
                       VirtualBox 7.0 on Debian-based distributions such
                       as Debian, Ubuntu, and Linux Mint distributions
                       using VirtualBox’s own repository with an apt
                       package manager.


                       VirtualBox is a powerful, general-purpose, and
                       cross-platform full virtualization software,
                       targeted at server, desktop, and embedded use. It
                       can be installed on any operating system (Linux,
                       Windows, Mac, etc) and allow you to install and run
                       multiple guest operating systems on the same
                       computer.


                       Recently Oracle has released the latest stable
                       version of Virtualbox 7.0, the newest major update
                       that ships with lots of major changes and new
                       features.


                # ⚓ DebugPoint ☛ How_to_Install_Viber_in_Ubuntu_and_Other
                  Linux⠀⇛


                       Here’s a quick guide on how you can install Viber
                       in Ubuntu and other Linux systems.


                       Viber is a free, secure calling and messaging
                       program for all popular mobile platforms and
                       operating systems.


                       It has a rich set of features such as voice/video
                       calls, text messages with GIFs, stickers, photos,
                       and videos. In addition, Viber features group
                       chats, group calls and disappearing messages.


                       Viber is a closed-source program, but available as
                       free for Linux distributions with native executable
                       clients.


                # ⚓ ID Root ☛ How_To_Install_Android_Studio_on_Rocky_Linux_9_–
                  idroot⠀⇛


                       In this tutorial, we will show you how to install
                       Android Studio on Rocky Linux 9. For those of you
                       who didn’t know, Android Studio is the official
                       integrated development environment for Google’s
                       Android operating system, built on JetBrains’
                       IntelliJ IDEA software and designed specifically
                       for Android development. Android Studio is a
                       feature-rich tool to develop mobile applications
                       and user interfaces. It supports a variety of
                       operating systems, including Linux, Windows, and
                       macOS.


                       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 Android Studio on Rocky Linux.
                       9.


          o § Games⠀➾


                # ⚓ Boiling Steam ☛ New_Steam_Games_with_Native_Linux_Clients_–
                  2022-10-19_Edition_–_Boiling_Steam⠀⇛


                       Between 2022-10-12 and 2022-10-19 there were 40 New
                       Steam games released with Native Linux clients. For
                       reference, during the same time, there were 303
                       games released for Windows on Steam, so the Linux
                       versions represent about 13.2 % of total released
                       titles.


                # ⚓ FOSS_Minecraft_Launcher_Destroyed_By_Rogue_Founder [Ed:
                  Minecraft itself is NOT FOSS. Should say, "STOP BECOMING
                  VOLUNTEERS OF MICROSOFT" (rather than what the video cover
                  days); Brodie Robertson knows that Microsoft is not FOSS,
                  GitHub is NOT FOSS but an ATTACK on FOSS, and Minecraft is
                  NOT FOSS either. How many people are fooled by the
                  openwashing propaganda train?]⠀⇛


    * § Distributions and Operating Systems⠀➾


          o ⚓ ZDNet ☛ Want_to_ditch_Windows?_Windowsfx_may_be_the_ideal_Linux
            distribution_for_you_|_ZDNET⠀⇛


                 For many, Windows is the only desktop operating system
                 they’ve ever used. For them, there’s a familiarity that
                 makes dealing with the usual troubles associated with
                 Windows (random restarts for updates, failed drivers, and
                 insecurities) tolerable.


                 But what if I told you there was an alternative? There
                 is, and it’s been around since 1997.


                 That alternative is Linux.


                 Now, before you throw your arms up in the air in
                 frustration, hear me out. Sure, there are tons of Linux
                 distributions available to install for free. Those
                 distributions range from the very basic to the incredibly
                 complex and everything in between.


          o § New Releases⠀➾


                # ⚓ antiX-22_released_–_antiX_Linux⠀⇛


                       antiX-22 is a new release update of our 21 series
                       based on Debian bullseye.


                       As usual we offer the following completely systemd-
                       free and for this particular upgrade – elogind-free
                       flavours for both 32 and 64 bit architecture.
                       Available iso files for sysVinit or runit.


                       antiX-full (c1.4GB) – 4 windows managers – IceWM
                       (default), fluxbox, jwm and herbstluftwm plus full
                       libreoffice suite.


                       antiX-base (c800MB) – 4 windows managers – IceWM
                       (default), fluxbox, jwm and herbstluftwm.


                       antiX-core (c460MB) – no X, but should support most
                       wireless.


                       antiX-net (c180MB)- no X. Just enough to get you
                       connected (wired) and ready to build.


          o § BSD⠀➾


                # ⚓ The Register UK ☛ FreeBSD_comes_to_Amazon’s_lightweight
                  hypervisor_•_The_Register⠀⇛


                       The FreeBSD developer who brought FreeBSD to Amazon
                       EC² has now got it working on the company’s
                       lightweight Firecracker hypervisor.


                       “AWS Community Hero” Colin Percival is the
                       developer of the Tarsnap online backup service and
                       the portsnap tool for updating FreeBSD. It’s thanks
                       to his efforts that FreeBSD is supported on Amazon
                       EC².


                       Now, he has announced that FreeBSD supports another
                       new platform: Amazon’s Firecracker hypervisor. The
                       Reg covered Firecracker when it was announced, and
                       when Amazon cut the price of the Fargate
                       “serverless” platform that Firecracker supports.
                       It’s derived from the lightweight Rust-based crosvm
                       which is part of Google ChromeOS.


          o § SUSE/OpenSUSE⠀➾


                # ⚓ OpenSUSE ☛ Medical_Campaign_uses_GNU_Health,_openSUSE_–
                  openSUSE_News⠀⇛


                       A recent campaign of medical-surgical assistance in
                       Senegal by a Non-Governmental Organization
                       highlights the benefits of using open-source
                       software.


                       A team from Cirugía Solidaria, which is an NGO that
                       provides medical assistance in disadvantaged
                       countries along with other health promotional
                       activities, conducted a campaign from Sept. 23 to
                       Oct. 3 with the Foundation Elizabeth Diouf.


                       The campaign involved several medical professionals
                       carrying out medical assistance while using open-
                       source technologies from GNU Health and the
                       openSUSE Project.


                       The use of GNU Health, which is a Hospital
                       Management Information System, allowed for the
                       multidisciplinary team, which included nurses,
                       surgeons, pediatricians, gynecologist and more, to
                       organize and facilitate daily medical care during
                       the medical assistance campaign. The technology on
                       the team’s devices and the printer that were all
                       running on a local server using openSUSE, and this
                       gave the team maximal opportunity to organize,
                       evaluate and treat patients.


          o § Fedora / Red Hat / IBM⠀➾


                # ⚓ Enterprisers Project ☛ Digital_transformation:_Why
                  observability_is_critical⠀⇛


                       Observability tooling has become critical on the
                       road to digital transformation. As DevOps and
                       cloud, the key enablers of digital transformation,
                       guide us towards ever more federated and
                       distributed processes and systems, incidents become
                       more unpredictable, and observability is our best
                       chance for assuring availability.


                       IT professionals have always dealt with change, but
                       never at the speed of our current digital
                       transformation. The humans of DevOps are being
                       asked to learn and implement new technologies at a
                       pace that often outpaces their current skill level.


                       Here, industry experts Jayne Groll and Helen Beal
                       share their thoughts on why observability is
                       critical for digital transformation.


                # ⚓ Enterprisers Project ☛ 4_ways_remote_leaders_can_build_soft
                  skills⠀⇛


                       Many IT leaders would agree: Building soft skills –
                       communication, empathy, relationship building, and
                       trust, for example – is far more manageable within
                       the walls of the workplace. It makes watercooler
                       catch-ups more probable, reading non-verbal cues
                       more prominent, and developing the right workplace
                       culture more accessible.


                       In today’s hybrid and remote work era, however,
                       honing and mastering these essential skills isn’t
                       so simple. “Many of these interactions just don’t
                       happen anymore,” says Charlie Betzig, managing
                       director at Heller Search Associates, a recruiting
                       firm specializing in senior-level technology
                       executives. “Now you have to pursue opportunities
                       to focus on these skills and be more intentional
                       about them.”


                # ⚓ IT Jungle ☛ IBM_Bolsters_Availability,_Backup_with_Latest
                  TRs [Ed: Fails to disclose the IBM sponsorship; conflict
                  hidden]⠀⇛


                       Nobody wants downtime or data loss, but companies’
                       appetite for both are on the decline as customers
                       increasingly demand 24/7 service. IBM i shops have
                       a lot of solutions to choose from when it comes to
                       ensuring uptime and avoiding data loss, both of
                       which have been enhanced with the latest round of
                       Tech Refreshes (TRs) from IBM.


                # ⚓ IT Jungle ☛ Doing_Disaster_Recovery_Planning_Right_For_Your
                  IBM_Power_Iron⠀⇛


                # ⚓ Fedora Project ☛ Mindshare_Committee_Quarterly_Report-_Q2
                  2022⠀⇛


                       It’s time for the (overdue) Mindshare Committee’s
                       third Quarterly Report. This post covers activities
                       from the Mindshare Committee and related teams for
                       the months of April, May, and June of 2022. We
                       started off strong with getting these reports
                       together, but as time has gone, it’s gotten a bit
                       more difficult to pull them together and out on
                       time. We are addressing the future of these reports
                       and how we can restructure to make it easier to put
                       them together. As always, we welcome feedback on
                       how we can improve these reports in the related
                       Mindshare ticket.


                # ⚓ Red Hat ☛ Open_vSwitch:_The_revalidator_process_explained_|
                  Red_Hat_Developer⠀⇛


                       Open vSwitch (OVS) is a rich implementation of
                       software-defined networking (SDN) L2 switch.


                # ⚓ Red Hat Official ☛ Running_containers_in_cars⠀⇛


                       A little over a year ago, Red Hat announced its
                       intention to collaborate with the automotive
                       industry to help drive the transition to software-
                       defined vehicles (SDVs). In May 2022, this
                       intention became concrete with Red Hat and General
                       Motors announcing their collaboration to help
                       trailblaze SDVs at the edge. Our goal is to produce
                       a base operating system to run all sorts of in-
                       vehicle software for safety-critical use cases as
                       well as non-safety ones.


                # ⚓ Red Hat Official ☛ What’s_next_for_Ansible?_The_path_to
                  end-to-end_automation_across_the_hybrid_cloud⠀⇛


                       IT complexity, whether in your datacenter, in
                       public clouds or at the edge, isn’t going away.


                # ⚓ Red Hat Official ☛ Speed_up_SQL_Server_2022_backups_with
                  RHEL_Logical_Volume_Manager [Ed: Red Hat is shilling
                  Microsoft proprietary software that does not even run on GNU/
                  Linux (it uses Drawbridge)]⠀⇛


                       One of the most powerful—and often
                       underutilized—storage features in Red Hat
                       Enterprise Linux (RHEL) is the ability to take
                       point-in-time consistent snapshots using the Linux
                       Logical Volume Manager (LVM2).


                # ⚓ Red Hat Official ☛ Harmonizing_the_automation_trifecta
                  across_the_hybrid_cloud:_People,_processes_and_platforms⠀⇛


                       There is no doubt that IT automation has proved its
                       worth, with automation technologies now being sewn
                       into the technology fabric of future-fit
                       organizations. It’s an essential component in
                       driving operational efficiency to maximize
                       enterprise potential. Businesses incorporating
                       automation effectively are setting themselves up to
                       excel, and that advantage is vital, no matter the
                       industry.


                # ⚓ Red Hat Official ☛ Automation?_There’s_an_AI_for_That⠀⇛


                       Seems a bit much for your phone’s assistant, right?
                       Even so, imagine if this is the future of
                       enterprise IT, where the command line and even
                       graphical user interfaces are replaced by writing
                       (or saying!) a sentence in plain English. Do this,
                       then that and finally this. That’s it – no
                       complicated syntax required.


                # ⚓ Business Wire ☛ Pandora_FMS_Achieves_Red_Hat_Enterprise
                  Linux_Certification_Boosting_its_International_Projection⠀⇛


          o § Canonical/Ubuntu Family⠀➾


                # ⚓ OMG Ubuntu ☛ Ubuntu_22.10:_What’s_New?_[Video]_–_OMG!
                  Ubuntu!⠀⇛


                       Ubuntu 22.10 is primed, ready to release on October
                       20 and to help ratchet up the excitement I’ve put
                       together a short video to showcase the release’s
                       most exciting new features.


                       Not that much of it of will be a surprise if you
                       read this blog often enough!


                       All told, Ubuntu 22.10 is a fairly modest update to
                       Ubuntu 22.04 LTS. The inclusion of GNOME 43 and the
                       switch to PipeWire as the underlying audio stack
                       are both significant changes, but there’s little
                       else to lure long-term support users away from the
                       luxurious bosom (oo-er) of stability.


                       Not that it’s a total non-event, mind.


                       As mentioned, GNOME 43 and its new pod-like Quick
                       Settings menu is the star attraction (being able to
                       quickly switch wi-fi network from the menu itself
                       was long overdue). Plus, an adaptive Nautilus, and
                       out-of-the-box support for WebP images (meme makers
                       rejoice) are welcome additions.


          o § Devices/Embedded⠀➾


                # ⚓ CNX Software ☛ MaxLinear_unveils_MxL31712_&_MxL31708_Wi-Fi
                  7_chips_for_gateways,_routers_and_access_points⠀⇛


                       
                       Since the WiFi 7 chips are coprocessors there’s no
                       information about software support, but we
                       previously reported that MaxLinear’s AnyWAN URX SoC
                       could run the Yocto-based prplOS (formerly known as
                       prplWRT), RDK-B, or MaxLinear’s UGW (Universal
                       Gateway Software) based on OpenWrt.


                       MaxLinear MxL31712 and MxL31708 Wi-Fi 7 chips are
                       at the pre-production stage with some samples
                       available to selected partners, and mass production
                       is scheduled to start next year. More details can
                       be found on the respective product pages and the
                       press release.


                # ⚓ The DIY Life ☛ I_hope_the_Raspberry_Pi_5_is_this_good_–
                  Khadas_Edge_2_Pro_–_The_DIY_Life⠀⇛


                       Today we’re going to be taking a look at the new
                       Khadas Edge 2. I can tell you upfront that this is
                       one of the fastest SBCs that I’ve tried out, which
                       is made even more impressive by its really compact
                       design and power efficiency.


                       This is an Arm-based SBC (single board computer)
                       that Khadas have recently launched and they’ve sent
                       me their pro version to try out and share with you.
                       They’re currently running an early bird offer until
                       the end of October, so you can get the Edge 2 for
                       $199.


                       Here’s my video unboxing and review, read on for
                       the written review…


          o § Open Hardware/Modding⠀➾


                # ⚓ Purism ☛ Priority_Orders_for_Librem_14_Laptops_(with
                  Standard_Specs)_Available_Now!_–_Purism⠀⇛


                # ⚓ Arduino ☛ Keep_the_air_in_your_workshop_clear_with_this_DIY
                  smart_dust_collection_system_|_Arduino_Blog⠀⇛


                       Workshops get dirty and dusty, especially if you do
                       woodworking. Not only is sawdust annoying, it is
                       also bad for your respiratory system. Dust
                       collection is very important, but outfitting a
                       large shop can quickly become complicated. If you
                       have a big workshop and need to improve your dust
                       collection, then Brumberjack_Woodworks has a design
                       and tutorial for a centralized smart dust
                       collection system that you should check out.


                       This whole system starts with a heavy duty dust
                       collector, which is essentially a huge vacuum
                       cleaner. The exact model isn’t that important, so
                       long as it can handle the workload. Luckily, the
                       key part of this system design is valve-based
                       direction. By only opening the valves you need, you
                       can reduce the necessary capacity of the dust
                       collector.


                # ⚓ Arduino ☛ This_Valorant_fan_made_his_own_functional_Boom
                  Bot_replica_|_Arduino_Blog⠀⇛


                       Valorant is a free-to-play 5v5 first-person shooter
                       game. As in most shooters, players want to avoid
                       getting shot. One way they can prevent incoming
                       fire is to use Boom Bot, which is a little robot
                       that will drive forward and chase enemies before
                       exploding — while the player stays safely hidden
                       out of sight. While he probably won’t be getting
                       into any gunfights, Danny Lum built his own
                       functional replica of the Boom Bot.


                       Boom Bot’s behavior in the game is quite simple.
                       When deployed, it will drive straight forward until
                       it either collides with a wall or detects an
                       opponent. If it runs into a wall, it turns like a
                       Roomba. If it sees a target, it will begin chasing
                       them. Lum was able to recreate that functionality
                       in a conventional two-wheel-drive rover robot. The
                       robot was designed in Solidworks CAD to match in
                       the in-game Boom Bot and then 3D-printed.


          o § Mobile Systems/Mobile Applications⠀➾


                # ⚓ 9to5Google ☛ Galaxy_Z_Fold_2_gets_Android_13_beta⠀⇛


                # ⚓ 9to5Google ☛ Android_13_Go_edition_adds_Material_You,_Play
                  System_Updates⠀⇛


                # ⚓ Android Authority ☛ Your_guide_to_location_services_in
                  Android_12_–_Android_Authority⠀⇛


                # ⚓ Android Authority ☛ What_is_Google_Fast_Pair_and_how_to_use
                  it_–_Android_Authority⠀⇛


                # ⚓ CNX Software ☛ Mini_Pupper_2_–_Raspberry_Pi_4_/_CM4_robot
                  dog_adds_ESP32,_ROS2_support,_servo_feedback_(Crowdfunding)_–
                  CNX_Software⠀⇛


                # ⚓ Online Tech Tips ☛ How_to_Access_the_Clipboard_on_Android
                  Devices⠀⇛


                # ⚓ How_to_Watch_Apple_TV+_on_Any_Android_Phone_or_Tablet
                  Android_::_Gadget_Hacks⠀⇛


                # ⚓ Android Headlines ☛ Your_Brief_Guide_To_Turning_Your
                  Android_Phone_Into_A_Security_Camera⠀⇛


                # ⚓ India Today ☛ Samsung_Galaxy_Watch_5_Pro_review:_Most
                  premium_Android_smartwatch_in_India_right_now_–_India_Today⠀⇛


                # ⚓ 9to5Google ☛ YouTube_Music_playlist_redesign_w/_centralized
                  album_art_rolling_out⠀⇛


                # ⚓ Android Authority ☛ Wallpaper_Wednesday:_Android_wallpapers
                  2022-10-19_–_Android_Authority⠀⇛


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


          o ⚓ OSI Blog ☛ On_the_emerging_landscape_of_open_AI [Ed: OSI runs
            this “HEY HI” series to cover up for Microsoft’s copyright
            violations and mass plagiarism. Microsoft_pays_the_OSI_for_it.]⠀⇛


          o ⚓ Petter Reinholdtsen ☛ Petter_Reinholdtsen:_Managing_and_using
            ONVIF_IP_cameras_with_Linux⠀⇛


                 Recently I have been looking at how to control and
                 collect data from a handful IP cameras using Linux. I
                 both wanted to change their settings and to make their
                 imagery available via a free software service under my
                 control. Here is a summary of the tools I found.


                 First I had to identify the cameras and their protocols.
                 As far as I could tell, they were using some SOAP looking
                 protocol and their internal web server seem to only work
                 with Microsoft Internet Explorer with some proprietary
                 binary plugin, which in these days of course is a
                 security disaster and also made it impossible for me to
                 use the camera web interface. Luckily I discovered that
                 the SOAP looking protocol is actually following the ONVIF
                 specification, which seem to be supported by a lot of IP
                 cameras these days.


                 Once the protocol was identified, I was able to find what
                 appear to be the most popular way to configure ONVIF
                 cameras, the free software Windows tool named ONVIF
                 Device Manager. Lacking any other options at the time, I
                 tried unsuccessfully to get it running using Wine, but
                 was missing a dotnet 40 library and I found no way around
                 it to run it on Linux.


          o ⚓ Medevel ☛ Fractal_Generator:_an_Open_Source_Cross-platform
            Fractal_Art_Generator⠀⇛


                 Fractal art is an amazing never-ending pattern. It is a
                 beautiful loop of colors, structures, and shapes
                 organized in a fancy pattern.
                 Fractal art is a computer generated art that uses
                 algorithms to calculate fractal objects then represent
                 them in a digital image, animation, or video.


                 [...]


                 This software is distributed under the GNU General Public
                 License version 3


          o ⚓ Medevel ☛ Setup_a_Bike_Sharing_Platform_with_the_Open_Source_Bike
            Share_Solution⠀⇛


                 Bike sharing is becoming more popular, for instance,
                 there are several bike sharing startups, and gov
                 sponsored bike and scooter apps and companies, in
                 countries like France, Canada, Turkey, Italy, and
                 Switzerland.


                 Building a bike sharing software from scratch is a
                 challenge, yet it is possible with some effort and time.
                 However, what if you need to setup a bike sharing system
                 for your city, and you do not have enough resources. Then
                 here is your answer: an Open-source bike sharing system.


                 The Open Source Bike Share is the world’s first low-cost
                 and open source bike sharing system. It is built on top
                 of JavaScript, Laravel Framework (PHP), and Vue
                 Framework.


                 The system has been used in shared stands in some
                 universities, some cities around the world.


          o ⚓ Medevel ☛ RadioLogic_is_an_Open_Source_Medical_Imaging_Learning
            Tool_for_Radiologists⠀⇛


                 RadioLogic is a case-based learning and self-assessment
                 tool for the Orthanc Ecosystem for Medical Imaging.


                 Case-based learning (CBL) is an efficient method for
                 radiologist education. RadioLogic is a system to create
                 clinical-cases from real DICOM files and to provide a
                 self-assessment tool to users to view the studies, submit
                 a diagnosis and compare their performance with other
                 peers. The main components are a progressive web
                 application (RadioLogicTutor) optimized for iPAD’s, an
                 Orthanc PACS (RadioLogicArchive) to serve the teaching
                 files and an Orthanc plugin (RadioLogicCreator) to create
                 the clinical teaching cases.


                 [...]


                 The Project is released under the GPL-3.0 License as an
                 open-source project which can be modified, cloned,
                 edited, and redistributed.


          o ⚓ Medevel ☛ GoldenCheetah_is_The_Open-source_App_Every_Serious
            Cyclist_Needs⠀⇛


                 GoldenCheetah is a desktop performance tracker
                 application for cyclists, runners, triathletes, and
                 coaches. It is built to aid the user analyze, track, and
                 optimize their out- and in-door training.


                 The program is available for Windows, Linux, and macOS,
                 free of charge. It is an open-source project released
                 under the GPL v2.


          o § Web Browsers/Web Servers⠀➾


                # § Mozilla⠀➾


                      # ⚓ DebugPoint ☛ Firefox_106_Release_Arrives_with_PDF
                        Annotation,_Wayland_Gestures,_+_More⠀⇛


                             Firefox follows a monthly release cycle, and
                             the new Firefox 106 release brings exciting
                             updates for all of its platforms, including
                             Linux.


                             Firstly, if you are browsing any PDF file
                             over the web using Firefox, now you can
                             annotate it. The preview tab includes two new
                             tools. Text and Draw.


          o § Content Management Systems (CMS)⠀➾


                # ⚓ Medevel ☛ Lavalite_is_a_Laravel_based_Feature-rich_CMS⠀⇛


                       Lavalite is a free open-source CMS that built on
                       top of Laravel framework. It features a user-
                       friendly admin dashboard that supports mobile small
                       screens, informative dashboard, multilingual
                       support, and a clutter-free interface.


                       Originally, Lavalite is built for Laravel
                       developers to build custom CMS system on top of it.
                       It can be used as a base CMS for Real-estate,
                       eCommerce, Classified website, yellow pages, and
                       job listing websites.


                       Developers and even users can easily define a new
                       custom post types, or custom taxonomies without
                       breaking a sweat.


                       Lavalite comes with a rich developer and user-
                       friendly documentation, that guides the newcomers
                       through the installation process, configuration,
                       and server setup.


                # ⚓ Medevel ☛ Payload:_an_Open-source_Headless_CMS_Framework
                  for_Enterprise⠀⇛


                       While Headless CMS solutions are trending, it is
                       not easy to pick the right one for enterprise work.
                       That’s not the case with Payload Headless CMS as it
                       is built for enterprise users.


                       Payload CMS comes with a full support for REST API
                       and GraphQL. It also supports local file storage
                       with a straightforward upload API.


                       With Payload CMS, developers can set custom post
                       types, custom fields, manage the content history,
                       drafts, and revision with ease, and use the block-
                       based layout editor for building eye-catching rich
                       pages.


          o § Education⠀➾


                # ⚓ TecMint ☛ Best_Linux_Open_Source_School_Management
                  Software⠀⇛


                       This tutorial examines some of the best free and
                       open-source school management software for Linux.


                       Running an educational institution in an efficient
                       manner is no mean task. It usually involves
                       handling various aspects such as students’ biodata,
                       grading, timetabling, reporting and other
                       administrative tasks that involve communication
                       between parents, teachers, and the non-teaching
                       staff. As such, having reliable, and efficient
                       school management software is a huge plus.


                       In this guide, we will examine the top 5 free and
                       open-source school management software for Linux.


          o § FSF⠀➾


                # ⚓ FSF ☛ LibrePlanet_2023:_Remember_to_submit_your_session_by
                  November_2⠀⇛


                       LibrePlanet 2023 is the fifteenth edition of the
                       Free Software Foundation’s (FSF) conference on
                       ethical technology and user freedom, which will be
                       held in March 2023, both online and in Boston, MA
                       (exact date and venue TBA). This year’s theme is
                       “Charting the Course.”


                       The free software movement has produced so much
                       more than software used by millions around the
                       world. It also points the way to a freer and
                       autonomous digital existence for all of us: one
                       where it’s the user and not corporate monopolies
                       that control our ways of communicating, how
                       children are educated, and how we monitor
                       information as crucial as our own health.
                       LibrePlanet speakers will show ways of progressing
                       the free software community’s understanding of new
                       opportunities and new threats to the movement.


                       For those of you who haven’t participated in a
                       LibrePlanet conference before: expect a friendly,
                       social, community-focused event with two days of
                       inspiring talks and workshops from prominent people
                       in the free software community, as well as people
                       new to the community. You can choose whether you
                       want to participate online or in person in Boston,
                       MA. LibrePlanet 2023 will offer both.


          o § Programming/Development⠀➾


                # ⚓ Daniel Stenberg ☛ Funded_curl_improvements_|
                  daniel.haxx.se⠀⇛


                       I am happy to announce that curl receives funding
                       from The Sovereign Tech Fund. This funding is
                       directed towards three specific projects that we
                       have identified as interesting and worthwhile to
                       push forward as ways to improve curl and the life
                       of curl users.


                       This “investment” will fund two developers to work
                       on curl over a period of six months: Stefan Eissing
                       and myself. The three projects are explained at
                       some detail below. Of course everyone and anyone is
                       welcome to join in and help out with these
                       projects. Everything will be done in the open, as
                       usual.


                       At the end of the period, we will produce some kind
                       of report or summary of how things turned out.


                       The three projects we are getting funded have been
                       especially created and crafted (by me) to be good
                       solid projects that we really want to see done.
                       This funding is different than many others we have
                       gotten over the years in that we got to decide and
                       plan what we wanted done. These are things that are
                       meant to improve curl as a project and to generally
                       make Internet transfers better and more powerful
                       for a vast amount of users.


                # ⚓ The Register UK ☛ Google_submits_its_pure_Go_container
                  project_to_CNCF_•_The_Register⠀⇛


                       Google has offered its tool for pure Go Kubernetes
                       containers to the Cloud Native Computing Foundation
                       (CNCF) … and may have thought it was a done deal.


                       The CNCF is the Linux Foundation project that acts
                       as the guardian of Kubernetes since Google stepped
                       away.


                # ⚓ Qt ☛ Introducing_Qt_Quick_Effect_Maker⠀⇛


                       I’m super excited to finally talk about the project
                       I have been working with during the past months: Qt
                       Quick Effect Maker (QQEM). This blog post is an
                       introduction into this new tool and more related
                       posts are coming in the near future. So what is it?
                       QQEM is a custom-made tool designed for a single
                       purpose…


                # § Editors⠀➾


                      # ⚓ Linuxiac ☛ 6_Best_Free_Linux_GUI_Code_Editors_for
                        Programming [Ed: The second one is Microsoft and
                        proprietary]⠀⇛


                             This article presents an overview of the six
                             best GUI code editors for Linux that are
                             feature-rich and suitable for every
                             programmer’s needs.


                             A code editor is a must-have tool for each
                             developer. Fortunately, Linux has many of
                             these that are entirely free to use.


                             This list strives to bring you the six best
                             ones based on our own experience with them.
                             But before we begin, we should clarify that
                             we are presenting you with the best GUI
                             (Graphical User Interface) code editors.


                             The terminal-based ones, such as Vim, Micro,
                             Nano, and so on, are outside the scope of
                             this review and will be covered in a separate
                             one.


    * § Leftovers⠀➾


          o § Hardware⠀➾


                # ⚓ Tedium ☛ More_Dongle_History:_Who_Invented_the_Term
                  “Dongle,”_Anyway?⠀⇛


                       Two years ago, I explained the evolution of dongles
                       in the best way I could, how these weird connectors
                       came to be, and why it never seems like our dongle
                       drawer will ever empty. And honestly, when a new
                       dongle emerges, I kind of just want to hand out a
                       prize to the company that made it. And today, I’m
                       handing the flowers to Apple, which just released
                       an unnecessary female-lightning-to-female-USB-
                       C connector because it inexplicably chose not to
                       upgrade the Apple Pencil on the lowest-end iPad,
                       which now uses USB-C. But the recent decision that
                       the European Union made to require everyone to use
                       the same port standard must have Apple out of
                       sorts, hence the weird dongle. But I have to
                       wonder, is this just the dongle’s last hurrah? Will
                       the awkward transition period end these half-
                       solutions for good? Probably not. But given the
                       many shifts in the market in the past two years, I
                       feel like I need to re-assess my thesis: Will
                       dongles eventually go away? Today’s Tedium ponders
                       the dongle some more.


          o § Pseudo-Open Source⠀➾


                # § Openwashing⠀➾


                      # ⚓ TechRepublic ☛ Open_source_is_Disrupting_Video_–
                        Empowering_Developers_|_TechRepublic [Ed: Openwashing
                        by Mac Asay, not a word about the software patents
                        cartel infesting this domain.]⠀⇛


                             After decades of proprietary stovepipes for
                             media players and back-end video
                             infrastructure, a new open source abstraction
                             from Mux may liberate developers.


          o § Security⠀➾


                # ⚓ The Atlantic ☛ A_Museum_Security_Expert_on_How_to_Protect
                  Great_Art_–_The_Atlantic⠀⇛


                       A museum-security expert admits that “it’s pretty
                       darn hard to protect a painting from somebody
                       throwing a can of soup at it.”


                       On Friday, in a bizarre act that immediately went
                       viral, two climate activists covered a 130-ish-
                       year-old Vincent Van Gogh painting with tomato soup
                       at the National Gallery in London. They then
                       proceeded to superglue themselves to the wall
                       beneath the frame. “Are you more concerned about
                       the protection of a painting or the protection of
                       our planet?” one asked. The protesters were later
                       charged criminally.


                       They were concerned about the planet, but also, at
                       least purportedly, about the painting. A
                       spokesperson for the group the protesters are
                       affiliated with, Just Stop Oil, told The New York
                       Times that the group had checked ahead of time to
                       ensure the work was glazed—covered by a thin layer
                       of glass—so that the soup would not damage the art.
                       And glazed it was. Video shows that the orange soup
                       did not seep into the yellow painting but rather
                       rolled and dripped down the front, a barely
                       perceptible layer clearly separating it from the
                       art. The work reportedly suffered no damage, except
                       to its frame.


                       [...]


                       Banks don’t take millions of dollars and put them
                       in plastic bags and hang them on the wall so
                       everybody can walk right up to them.


                # ⚓ LWN ☛ Security_updates_for_Wednesday⠀⇛


                       Security updates have been issued by Debian (bcel,
                       kernel, node-xmldom, and squid), Mageia (chromium-
                       browser-stable, dhcp, dokuwiki, firefox, golang,
                       python-joblib, sos, and unzip), Oracle (nodejs and
                       nodejs:16), Red Hat (firefox, kernel, kernel-rt,
                       nodejs, nodejs:14, and thunderbird), Scientific
                       Linux (firefox and thunderbird), Slackware (git and
                       mozilla), SUSE (amazon-ssm-agent, caasp-release,
                       cri-o, patchinfo, release-notes-caasp, skuba,
                       enlightenment, libreoffice, netty, nodejs12,
                       nodejs14, nodejs16, pngcheck, postgresql-jdbc,
                       python-waitress, rubygem-activesupport-5_1, and
                       tcl), and Ubuntu (frr, git, libksba, and linux-
                       azure-4.15).


                # ⚓ 3_Ways_to_Help_Customers_Defend_Against_Linux-Based
                  Cyberattacks_–_MSSP_Alert⠀⇛


                       Linux operating systems power more than 90% of the
                       world’s public cloud workload, from government web
                       servers to smart manufacturing technologies. But as
                       organizations continue to shift operations to the
                       cloud, cybercriminals are following suit and
                       directing their attention to Linux-based
                       cyberattacks.


                # ⚓ CISA ☛ Oracle_Releases_October_2022_Critical_Patch_Update_|
                  CISA⠀⇛


                       A remote attacker could exploit some of these
                       vulnerabilities to take control of an affected
                       system.


                # ⚓ USCERT ☛ CISA_Updates_Advisory_on_Threat_Actors_Exploiting
                  Multiple_CVEs_Against_Zimbra_Collaboration_Suite [Ed: Way to
                  distract from what's happening to Microsoft Exchange at the
                  moment (Microsoft is not even patching!)]⠀⇛


                       CISA and the Multi-State Information Sharing &
                       Analysis Center (MS-ISAC) have updated joint
                       Cybersecurity Advisory AA22-228A: Threat Actors
                       Exploiting Multiple CVEs Against Zimbra
                       Collaboration Suite, originally released August 16,
                       2022. The advisory has been updated to reference
                       the addition of a new Malware Analysis Report, MAR-
                       10398871.r1.v2.


                # § Privacy/Surveillance⠀➾


                      # ⚓ IT Wire ☛ iTWire_–_Qatar_World_Cup_visitors_cautioned
                        about_taking_mobiles⠀⇛


                             The head of security at the Norwegian
                             Broadcasting Corporation (NRK) has cautioned
                             visitors to the World Cup Football tournament
                             in Qater in November against taking their
                             mobiles when they go over.


                             Øyvind Vasaasen said, after having examined
                             two apps — Ehteraz and Hayya — which all
                             visitors would be asked to download to their
                             mobiles, “It’s not my job to give travel
                             advice, but personally I would never bring my
                             mobile phone on a visit to Qatar.”


                             The NRK is the Norwegian government-owned
                             radio and television public broadcasting
                             company, and the largest media organisation
                             in Norway.


                             Vasaasen said Ehteraz was a COVID-19 tracking
                             app while Hayya was an official World Cup app
                             which would be used to track match tickets
                             and police access to the free Metro in Qatar.


                      # ⚓ Mozilla ☛ A_glossary_of_terms_about_cyberattacks,
                        from_ransomware_to_DDoS⠀⇛


                             If you read news about technology, you’re
                             bound to run into some jargon. Here at
                             Mozilla, we believe that information should
                             be as accessible as possible regardless of
                             your level of expertise. We want to help you
                             approach stories about technology with more
                             curiosity and with a little less head-
                             scratching involved. We’ll break down
                             headline-making topics through a glossary of
                             terms often used to discuss them. Consider it
                             your cheat sheet to all things tech.


                             This month, we’ll give you terms to know
                             about cyberattacks.


          o § Finance⠀➾


                # ⚓ Pune_Rains,_Uncosted_Budgets,_Hearing_Loss_Covid,_Fracking
                  –_Experiences_in_the_community⠀⇛


                       India and the UK have many parallels, it’s mind
                       boggling. Before we get into the nitty-gritties,
                       saw something that would be of some interest to the
                       people here.


                       For those who might not be able to see above,
                       apparently there is place in UK called Tufton
                       Street where there are quite a few organizations
                       that are shadowy and whose finances are not known
                       as to how they are financed. Ms. Truss and quite a
                       few of the people in the cabinet are from the same
                       shadowy organizations. Mr. Kwasi Kwarteng, the
                       just-departed chancellor is and was part of the
                       same group.


                       Now even for me it was a new term to learn and
                       understand what is an uncosted budget is. To make
                       it much more easier I share the example using a
                       common person who goes to the bank for a loan –


                       [...]


                       The Chancellor is very similar to our Finance
                       Minister. Because the UK has constitutional
                       monarchy, I am guessing the terms are slightly
                       different, otherwise the functionality seems to be
                       the same. For two weeks, there was lot of chaos,
                       lot of pension funds lost quite a bit in the market
                       and in the end Mr. Kwasi Kwarteng was ousted out of
                       the job. Incredibly, the same media and newspapers
                       who had praised Mr. Kwarteng just few weeks back as
                       the best Tory budget, they couldn’t wait to bury
                       him. And while I have attempted to simplify what
                       happened, the best explanation of what has happened
                       can be found in an article from the guardian.
                       Speculation is rife in the UK as to who’s ruling
                       atm as the new Chancellor has reversed almost all
                       the policies that Ms. Truss had bought and she is
                       now more or less a figurehead. Mr. Hunt, the new
                       chancellor doesn’t have anybody behind him.
                       Apparently, the gentleman wanted to throw his hat
                       the ring in the Tory leadership contest that was
                       held about a month back and he couldn’t get 20 MP’s
                       to support him. Another thing that is different
                       between UK and India is that in UK by law the PM
                       has to answer questions put up to him or her by the
                       opposition leaders. That is the way accountability
                       is measured there. This is known as PMQ’s or Prime
                       Minister Questions and Answers. One can just go to
                       YouTube or any streaming service and give Liz Truss
                       and PMQ’s and if they are interested of a certain
                       date, give a date and they can see how she answered
                       the questions thrown at her. Unfortunately, all she
                       could do in both times were non-answers. In fact,
                       the Tories seem to be using some of Labor’s
                       policies after they had bad-mouthed the same
                       policies. Politics of right-wing both in the UK and
                       the US seems so out of touch with the people whom
                       they are supposed to protect and administer. An
                       article about cyclists which is sort of half-truth,
                       half irony shows how screwed up the policies are of
                       the RW (right-wing). Now they are questions about
                       the pensions triple-lock. Sadly, it is the working
                       class who would suffer the most, most of the rich
                       have already moved their money abroad several years
                       ago. The Financial Times, did share a video about
                       how things have been unfolding


          o § Civil Rights/Policing⠀➾


                # ⚓ AccessNow ☛ Libyan_parliament_ignores_civil_society’s
                  warning,_solidifies_draconian_cybercrime_law_–_Access_Now⠀⇛


                       Libyan parliament must immediately scrap the
                       dangerous new cybercrime law before it’s used to
                       deny people their fundamental rights. Almost a year
                       after its adoption in October 2021, the Libyan
                       House of Representatives cemented its draconian new
                       cybercrime law, officially publishing this tool for
                       legalised repression on September 27, 2022.


                       As Access Now and other human rights organisations
                       have continually warned, the law will severely
                       undermine freedom of expression, and legitimise
                       authorities’ targeting of journalists, activists,
                       and human rights defenders. The law grants Libyan
                       authorities the power to monitor people’s speech on
                       the internet, and enables the National Information
                       Security and Safety Authority (NISSA) — an
                       administrative and technical governmental authority
                       — to block websites and shut down online services
                       without judicial warrants.


                       “Libyan authorities are systematically constructing
                       a digital hellscape designed to rob people across
                       the country of their privacy and autonomy, and
                       their rights to freedom of expression and
                       assembly,” said Marwa Fatafta, MENA Policy and
                       Advocacy Manager at Access Now. “This dangerous new
                       cybercrime bill was designed to oppress. It must be
                       immediately repealed.”


                # ⚓ Daniel Pocock ☛ Brittany_Higgins_&_Australian_Parliament
                  Rapist:_the_real_verdict⠀⇛


                       The Brittany Higgins trial concluded and the jury
                       is now considering whether to convict the
                       Parliament rapist.


                       The trial revealed that all opportunities to
                       collect forensic evidence have been neglected or
                       obstructed in some way. The public and the jury
                       have been asked to believe. The victim said this on
                       many occasions, she would like people to believe
                       her.


                       The problem with belief is that it is a poor
                       substitute for proof. For hundreds of years, the
                       Catholic Church believed the sun revolves around
                       the Earth. It was only in 1992 that the Church
                       accepted Galileo was right.


                       There is strong scientific proof for the safety and
                       effectiveness of vaccines. Novak Djokovic believes
                       in alternative theories. He was deported and unable
                       to play in the Australian Open.


                       In that sense, it seems odd that Australians are
                       being asked to accept the parliament rapist story
                       on belief without any scientific evidence. Health
                       authorities wouldn’t accept anti-vax conspiracy
                       theories, no matter how strongly some people
                       believe in them.


                       Without the evidence, we can never know if the
                       story was true or false. Nonetheless, the trial
                       revealed other things about political parties and
                       volunteer organizations.


          o § Internet Policy/Net Neutrality⠀➾


                # ⚓ IT Wire ☛ Labor_to_provide_$2.4b_to_NBN_Co_for_1.5m_fibre
                  upgrades⠀⇛


                       The Federal Government says it will provide a
                       US$2.4 billion equity investment over four years to
                       the NBN Co so that an additional 1.5 million
                       premises can be upgraded from copper connections to
                       fibre by 2025. The money will be formally announced
                       in its budget update on 25 October.


                       A joint statement from Prime Minister Anthony
                       Albanese and Communications Minister Michelle
                       Rowland on Thursday — sent to bigger media
                       organisations on Wednesday night — said the
                       investment would benefit more than 660,000 premises
                       in regional areas.


                       Two years ago, the NBN Co announced it would spend
                       $3.5 billion to ensure that about 75% of the homes
                       on the NBN could be upgraded, bringing the total on
                       full-fibre to about eight million. The Labor
                       announcement on Thursday means that total would
                       move closer to 10 million.


                       Just a day prior to that announcement, $700 million
                       was promised for business fibre zones in
                       metropolitan areas and the regions, and $300
                       million to improve regional Internet services.


    * § Gemini* and Gopher⠀➾


          o § Personal⠀➾


                # ⚓ SpellBinding:_TMORSIH_Wordo:_TITAN⠀⇛


          o § Technical⠀➾


                # ⚓ Remote_Work_Blues⠀⇛


                       I live in the American Midwest, but my new job is
                       based out of the east coast. As a result, I now
                       work full-time at home–an arrangement I despised
                       when I was forced to do so at the height of the
                       COVID-19 pandemic.


                       While I’m less put off by the arrangement now that
                       my home office is thousands of miles away, I still
                       find myself subjected to many of the same emotions
                       as before. I get cabin fever quite easily and need
                       to get out of my house regularly. I find it easier
                       to collaborate with coworkers in-person, and I
                       don’t have that opportunity now.


                       [...]


                       It also means that I have fewer opportunities to
                       find interesting things around the city during the
                       day. At my previous job, I would sometimes drive to
                       a local airport during my lunchbreak and watch
                       planes. Sometimes I would just drive around and
                       explore the area. These midday journeys would not
                       only give me a change of pace, but they would help
                       me stay in touch with the goings-on of my town, and
                       I often found plenty of cool things to photograph.


                # ⚓ Speech_to_text_transcription⠀⇛


                       A long time ago, I tried to generate transcriptions
                       of my Halberds & Helmets Podcast using
                       Pocketsphinx. I wasn’t very happy about it.


                # ⚓ Manual_of_Me⠀⇛


                       Yesterday I discovered something that seems like a
                       great idea to do, writing and publishing a manual
                       of myself. This is inspired by the Manual of Me
                       website, but as I don’t want to sign up for any of
                       their things, this will probably just a simple
                       markdown to HTML document.


                # § Internet/Gemini⠀➾


                      # ⚓ My_Month_of_Sanity_&_Not_Posting:_A_Social_Media
                        Power_User_Chooses_to_Touch_Some_Grass⠀⇛


                             One of the things I mentioned when I created
                             this smol blog all those eons ago (and then
                             promptly ignored while I went and sought out
                             yet another hit from the social media feeding
                             tube) was how torn I was about my interaction
                             with online social spaces. I am on Twitter
                             and Instagram every day. Occasionally, a
                             friend will send along a TikTok, and I make
                             the mistake of reinstalling the app and get
                             sucked into a vortex that I am convinced
                             literally steals life like the Six-Fingered
                             Man’s torture machine. In a world
                             increasingly reliant upon online spaces for
                             interpersonal connection, all my social
                             interactions are conducted through text
                             messages, zoom calls, tweets, and
                             shitposting.


                             Part of the therapy work I have taken on is
                             to work on mindfulness, and even before
                             reading Devon’s post, it was apparent that
                             something needed to shift. I used to be a
                             creative person driven to write essays and
                             excitedly explore my interests. When I take
                             the time to reflect upon myself now, I see a
                             pretty empty shell of that former self.


                             [...]


                             You may be asking what ASD and social media
                             have to do with one another. A complex
                             interplay is at play here that touches upon
                             self-worth, self-discovery, mindfulness,
                             social anxiety, obsessive behaviors, and
                             fear. Unfortunately, I do not have a very
                             high opinion of myself from decades of
                             criticism and abuse. This led me to fear that
                             if I remove the frameworks that have worked
                             for me (online spaces), I will be isolated
                             and too fearful of venturing into physical
                             areas to expand my social circle in person.
                             Through years of maskin, I have created a
                             pretty gray world for myself, and I hid my
                             obsessive tendencies away—I honestly don’t
                             think I have had a genuine interest in how I
                             used to in many years. Part of unmasking has
                             been to let go of some of the restrictions I
                             have imposed upon myself, and things like
                             special interests were the subject of much
                             social ridicule. What if, in my quest to live
                             authentically, I open myself up to ridicule
                             (fear again)?


                             [...]


                             I don’t know what this will bring, but I can
                             say that day two feels like quitting smoking.
                             You feel like shit and irritable because your
                             brain keeps cycling back to a thing that you
                             can’t have. It is not very fun! And it is
                             alarming that this effect has been so potent.


                # § Programming⠀➾


                      # ⚓ GitHub_Copilot_is_piracy_but_fascinating⠀⇛


                             GitHub Copilot is a machine learning model
                             sold as a service by GitHub/Microsoft. It
                             works as a super-powerful programmer’s
                             autocomplete. It lets the programmer type a
                             comment like “function to calculate prime
                             numbers” and Copilot will offer a complete
                             implementation or at least a skeleton of an
                             implementation. It can also just see that the
                             programmer is starting to type a prime number
                             calculator, and offer to complete the
                             implementation, without any specific prompt
                             or instruction.


                             [...]


                             Copilot frequently spits out code that can be
                             traced back to the origins, and people have
                             made it generate dozens of lines of
                             copyrighted code verbatim. Whether the code
                             is verbatim, or the ML model has “learned”
                             something, it was trained on code that never
                             authorized this sort of use, and it can lead
                             to unintentional copyright violations.
                             Because the model itself contains those code
                             snippets, it can be argued (very reasonably)
                             that Copilot itself is a copyright violator.


                             This has divided the software development
                             community. Some are horrified with the
                             copyright aspect of it. Others seem to
                             perceive the copyright angle of it as a speed
                             bump in the way of progress, as Copilot is
                             undeniably useful, and clearly points at what
                             the future looks like for software
                             development.


                      # ⚓ The_name_OctoForth_is_already_taken…⠀⇛


                             Damn! I did a google search and nothing came
                             up, so I started using OctoForth… But someone
                             pointed me to a github repo with the same
                             name!


                             So someone has a Rust project with the name
                             OctoForth, and it is a Forth of some sort.


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



                    ䷩ 𝚕𝚒𝚗𝚎 6001

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

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 10.19.22⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Links_19/10/2022:_Microsoft_Laying_Off_Many_Staff,_EasyOS_4.4.2_is_Out⠀✐


Posted in News_Roundup at 5:44 am by Dr. Roy Schestowitz


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇GNOME bluefish⦈


§ Contents⠀➾


    * GNU/Linux
          o Instructionals/Technical
          o Games
    * Distributions_and_Operating_Systems
          o New_Releases
    * Free,_Libre,_and_Open_Source_Software
          o Programming/Development
                # Shell/Bash/Zsh/Ksh
                # Rust
    * Leftovers
          o Science
          o Education
          o Hardware
          o Health/Nutrition/Agriculture
          o Proprietary
          o Security
          o Defence/Aggression
          o Environment
                # Energy
                # Wildlife/Nature
          o Finance
          o AstroTurf/Lobbying/Politics
          o Censorship/Free_Speech
          o Freedom_of_Information_/_Freedom_of_the_Press
          o Civil_Rights/Policing
          o Internet_Policy/Net_Neutrality
          o Digital_Restrictions_(DRM)
          o Monopolies
                # Copyrights
    * Gemini*_and_Gopher
          o Personal
          o Technical
    * § GNU/Linux⠀➾


          o ⚓ FOSS_Weekly_#21_–_$8_RISC-V_computer,_Linux_Tablet,_PostgreSQL
            15,_VirtualBox_7.0,_–_more_|_FOSS_Weekly⠀⇛


                 Here’s everything that happened last week in open source
                 software. Lots of open hardware news from the Pine64
                 team, a new Linux Tablet, and some releases from
                 projects.


          o § Instructionals/Technical⠀➾


                # ⚓ DebugPoint ☛ [Tutorial]_How_to_Install_Viber_Messaging
                  Client_in_All_Linux_Distros⠀⇛


                       iber is a free, secure calling and messaging
                       program for all popular mobile platforms and
                       operating systems.


                       It has a rich set of features such as voice/video
                       calls, text messages with GIFs, stickers, photos,
                       and videos. In addition, Viber features group
                       chats, group calls and disappearing messages.


                       Here’s how to install and use it.


                # ⚓ ID Root ☛ How_To_Install_GPU-Viewer_on_Linux_Mint_21_–
                  idroot⠀⇛


                       In this tutorial, we will show you how to install
                       GPU-Viewer on Linux Mint 21. For those of you who
                       didn’t know, The GPU-Viewer is a free, open-source
                       project to create an easy-to-use interface for
                       glxinfo, vulkaninfo, and clinfo. The program will
                       be developed using Python 3 with GTK3 to display
                       all the crucial details extracted from these
                       programs on one page alongside other valuable tools
                       like grep or AWK.


                       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 GPU-Viewer on Linux Mint 21
                       (Vanessa).


                # ⚓ Linux Capable ☛ How_to_Install_CPU-X_on_Manjaro_21_Linux⠀⇛


                       CPU-X is a free and open-source software designed
                       for GNU/Linux and FreeBSD for system profiling and
                       monitoring application that provides detailed
                       information on the hardware components of a
                       computer system. CPU-X gathers information from the
                       system’s CPU, motherboard, RAM, graphics card, and
                       more, then displays this information in an easy-to-
                       read format. This makes it a valuable tool for
                       troubleshooting hardware issues or simply for
                       getting a better understanding of the inner
                       workings of a computer system.


                       In addition to displaying essential information
                       about hardware components, CPU-X also provides an
                       overview of the system’s overall performance. This
                       can help determine whether or not a particular
                       component bottlenecks a system. Overall, CPU-X is a
                       versatile and helpful application for anyone who
                       needs to gather information about their computer
                       system.


                       The following tutorial will teach you how to
                       install CPU-X on Manjaro 21 Linux. The tutorial
                       will use the command line terminal with the yay AUR
                       helper. Ideally, most users may be using some
                       wrapper for Pacman; for new users, it is essential
                       to install one to keep your packages up-to-date
                       while you learn Arch/Manjaro.


                # ⚓ Linux Nightly ☛ How_to_Install_VMware_Tools_in_Rocky_Linux
                  –_Linux_Nightly⠀⇛


                       When installing Rocky Linux in a VMware virtual
                       machine, you will definitely want to install VMware
                       Tools as well. The VMware Tools software will give
                       you better graphics support in the guest operating
                       system, as well as the ability to drag and drop
                       files between the host and guest system, and
                       configure shared folders.


                # ⚓ Video ☛ How_to_install_Olive_video_editor_on_Linux_Mint_21
                  –_Invidious⠀⇛


                       In this video, we are looking at how to install
                       Olive video editor on Linux Mint 21.


                # ⚓ Linux Nightly ☛ Check_if_File_Exists_in_Bash_Script_–_Linux
                  Nightly⠀⇛


                       In this tutorial, you will learn how to use Bash
                       scripts and if statements to check whether or not a
                       file or directory exists.


                # ⚓ Make Tech Easier ☛ What_Is_Email_Encryption_and_How_Does_It
                  Work?_–_Make_Tech_Easier⠀⇛


                       Email encryption is a technique that disguises an
                       email so that only the sender and intended
                       recipient can access its contents, preventing
                       unauthorized individuals from intercepting the
                       signal and reading the message.


                       Most email doesn’t have this level of protection.
                       Content is sent as regular text and open to data
                       security risks. However, you can step in with
                       proper end-to-end encryption, for instance, to
                       secure your confidential email easily.


                       In this method, email is encrypted on the sender’s
                       system using a public encryption key that scrambles
                       the email and any attachments. The email is then
                       sent to the recipient, who has a private key that
                       can be used to decrypt the message. This technique
                       is so secure that the message cannot be read by any
                       email services, banking institutions, Internet
                       service providers (ISPs), or hackers.


          o § Games⠀➾


                # ⚓ GamingOnLinux ☛ Windows_compatibility_layer_Wine_version
                  7.19_is_out_now⠀⇛


                       Just before the weekend hit, a new development
                       release of the Windows compatibility layer Wine
                       went up with version 7.19. This is part of Steam
                       Play Proton, which allows you to play tons of
                       Windows games on Steam Deck and Linux desktops.
                       Once a year they make a big new stable release, and
                       eventually Proton updates to it too.


                # ⚓ GamingOnLinux ☛ Sweet_point_and_click_dino_adventure_Zid
                  Journey_is_up_on_Kickstarter_with_demo⠀⇛


                       Love your point and click adventures and want to
                       see more? Zid Journey is a new one from the team
                       behind Zniw Adventure. It’s a classic 2D point ‘n
                       click adventure game, developed in a tiny team from
                       Poland. Just like the first game, it will have full
                       Native Linux support.


                # ⚓ GamingOnLinux ☛ EmuDeck_2.0_makes_emulation_on_Steam_Deck
                  easier_than_ever⠀⇛


                       The Steam Deck is a fantastic system for indies,
                       AA, AAA and everything in between but it’s also
                       great for emulation too. EmuDeck is software that
                       can help automate the entire process of setting
                       emulators up, and a huge new release is out now.


                # ⚓ GamingOnLinux ☛ Theme_park_building_sim_Indoorlands_is
                  officially_out_now⠀⇛


                       Indoorlands, a somewhat unique theme park building
                       sim has left Early Access as a finished game ready
                       for you to get clicking around. What’s the
                       different about this one then? Well, as the name
                       hints towards, you’re building everything inside.
                       An indoor amusement park, that you build up using
                       these massive connected rooms. It comes with Native
                       Linux support too.


                # ⚓ GamingOnLinux ☛ LOST_EMBER_should_run_nicer_on_Steam_Deck_/
                  Linux,_as_dev_fixes_video_issues⠀⇛


                       LOST EMBER is an adventure game all about the power
                       of nature. Released back in 2019, it seems recently
                       that went back to it to ensure it works nicely on
                       Steam Deck (and so Linux too).


                # ⚓ GamingOnLinux ☛ Proton_7.0-5_is_coming_soon_improving_Steam
                  Deck_/_Linux_gaming⠀⇛


                       Valve are preparing to launch the latest update to
                       Steam Play Proton, the compatibility layer to run
                       Windows games on Linux and Steam Deck.


                # ⚓ Switch:_Hacker_CTCaer_hints_at_upcoming_Linux_and_Android
                  release_for_new_Nintendo_Switch_models_(2019/Lite/OLED)_–
                  Wololo.net⠀⇛


                       Although new Switch models are harder to hack than
                       the original ones, Switch hardware mods (and
                       therefore, hacks) are a possibility. For these
                       models, hacker CTCaer has shown significant
                       progress on Linux L4T and Android support, with a
                       release ETA of “Soon”.


                       L4T Ubuntu running on Switch OLED


                       L4T (Linux for Tegra) is the package of choice for
                       the Nintendo Switch which runs on the NVidia SoC.
                       The Switchroot team have adapted it for the Switch
                       and coupled with Ubuntu, in particular thanks to
                       work by CTCaer. The team has also ported Android to
                       the console.


                       Until now, L4T Ubuntu for Switch was only
                       compatible with original Switch models, and support
                       of new models have been a “work in progress” for
                       quite some time. It seems this is going to change
                       soon, making the newer Models (2019, Switch Lite,
                       and OLED) even more interesting.


                       The screenshot showcases Ubuntu 18.04.6 LTS (Bionic
                       Beaver) running on the console, but CTCaer also
                       hints that the team are bringing Android and Lakka
                       (a Linux-based distribution specialized in
                       emulation)


    * § Distributions and Operating Systems⠀➾


          o § New Releases⠀➾


                # ⚓ Barry Kauler ☛ EasyOS_Dunfell-series_version_4.4.2
                  released⠀⇛


                       The main reason for pushing out this release, is it
                       has the Android drivers in the kernel, so we can
                       have a got at running Anbox and/or Waydroid.


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


          o § Programming/Development⠀➾


                # ⚓ OpenSource.com ☛ How_to_contribute_to_Hacktoberfest_2022
                  [Ed: "All you need to get started in Hacktoberfest is a
                  GitHub or GitLab account" so you basically need to use
                  proprietary garbage (GitLab's licensing is... complicated)
                  and JS garbage]⠀⇛


                       Hacktoberfest is a month-long celebration run by
                       DigitalOcean to celebrate and give back to open
                       source projects and software. The initiative is
                       open to everyone, and the goal is to encourage
                       everyone in our global community to contribute to
                       open source. In this article, I’ll answer
                       frequently asked questions about how to
                       participate. I’ll also discuss how to contribute to
                       both code and non-code issues.


                # ⚓ Linux Shell Tips ☛ How_to_Install_Latest_Node.js_on
                  AlmaLinux⠀⇛


                       This article guide walks users through the
                       installation and basic usage of Node.js on
                       AlmaLinux 9 and AlmaLinux 8 releases using
                       AppStream and NodeSource Repositories.


                # ⚓ OpenSource.com ☛ Our_open_source_startup_journey_|
                  Opensource.com⠀⇛


                       Before working on ToolJet, I worked with a few
                       enterprise clients as a consultant. Many of these
                       clients were large enough to build and maintain
                       dozens of internal tools. Despite the constant
                       requests from sales, support, and operations teams
                       to add more features and fix the bugs in their
                       internal tools, engineering teams struggled to find
                       the bandwidth to work on the internal utilities.


                       I tried using a few platforms to build and maintain
                       internal tools. Most of these tools were very
                       expensive, and frequently, they didn’t really fit
                       the requirements. We needed modifications, and most
                       utilities didn’t support on-premise hosting.


                       As a Ruby developer, I primarily used ActiveAdmin
                       and RailsAdmin to build internal tools. Both
                       utilities are amazing, but making them work with
                       more than one data source is difficult. I then
                       realized there is a need in the market for a
                       framework that could build user interfaces and
                       connect to multiple data sources. I believe any
                       tool built for developers should be open source.
                       Most of the tools and frameworks that developers
                       use daily result from people from all over the
                       world collaborating in public.


                # ⚓ Dirk Eddelbuettel ☛ Dirk_Eddelbuettel:_RcppSimdJson_0.1.8
                  on_CRAN:_Maintenance⠀⇛


                       RcppSimdJson wraps the fantastic and genuinely
                       impressive simdjson library by Daniel Lemire and
                       collaborators. Via very clever algorithmic
                       engineering to obtain largely branch-free code,
                       coupled with modern C++ and newer compiler
                       instructions, it results in parsing gigabytes of
                       JSON parsed per second which is quite mindboggling.
                       The best-case performance is ‘faster than CPU
                       speed’ as use of parallel SIMD instructions and
                       careful branch avoidance can lead to less than one
                       cpu cycle per byte parsed; see the video of the
                       talk by Daniel Lemire at QCon.


                # ⚓ Dirk Eddelbuettel ☛ RcppSimdJson_0.1.8_on_CRAN:
                  Maintenance⠀⇛


                       The RcppSimdJson package was just updated to
                       release 0.1.8 today.


                       RcppSimdJson wraps the fantastic and genuinely
                       impressive simdjson library by Daniel Lemire and
                       collaborators. Via very clever algorithmic
                       engineering to obtain largely branch-free code,
                       coupled with modern C++ and newer compiler
                       instructions, it results in parsing gigabytes of
                       JSON parsed per second which is quite mindboggling.
                       The best-case performance is ‘faster than CPU
                       speed’ as use of parallel SIMD instructions and
                       careful branch avoidance can lead to less than one
                       cpu cycle per byte parsed; see the video of the
                       talk by Daniel Lemire at QCon.


                # § Shell/Bash/Zsh/Ksh⠀➾


                      # ⚓ Linux Handbook ☛ Pretty_Print_JSON_in_Linux_Command
                        Line⠀⇛


                             While JSON files are preferred for storing
                             data in human-readable form, it is good
                             enough to give headaches when they are
                             minified.


                # § Rust⠀➾


                      # ⚓ CNX Software ☛ Google_KataOS_–_A_secure_OS_for
                        embedded_systems_written_in_Rust_(mostly)⠀⇛


                             Google Research has been working on its own
                             Rust-based operating system called KataOS and
                             designed to secure embedded systems that run
                             Machine Learning (ML) applications.


                             There has been a lot of talk about the Rust
                             programming language in recent times, since
                             it offers about the same level of performance
                             as C programming but helps programmers write
                             more secure code with built-in prevention
                             against buffer overflows for instance. It has
                             gained a lot of traction over the years, and
                             Linux 6.1 will be the first kernel release to
                             include Rust code.


    * § Leftovers⠀➾


          o ⚓ The Nation ☛ J.D._Vance_Declares_Culture_War_on_Tim_Ryan⠀⇛


                 As Democratic Representative Timothy Ryan and venture-
                 capitalist mascot J.D. Vance faced off in Youngstown last
                 night for their final debate before the midterm election,
                 an all-encompassing taking of offense ensued. Each
                 candidate harped on the phoniness, the inauthenticity,
                 and the bought-and-paid-for policy positions of his
                 opponent; each candidate offered carefully presented
                 “commonsense” positions on divisive controversies, and
                 assured the Ohio electorate that he was the candidate
                 most in touch with their struggles and sensibilities.


          o ⚓ Craig Murray ☛ Winners_at_Raindance⠀⇛


                 Nadira’s first feature film as Producer, Winners, won the
                 Audience Award for best film at the Edinburgh
                 International Film Festival to our (frankly surprised)
                 delight. It will play in London at the Raindance Film
                 Festival on Saturday 29 October at 8.30pm, at the Genesis
                 Cinema in Bethnal Green.


          o ⚓ Quillette ☛ Scientology’s_War_on_Psychiatry⠀⇛


                 Hubbard did not launch his career by disparaging
                 psychiatrists as individuals or psychiatry as a
                 profession. In fact, given his early enthusiasm for
                 Freudian therapy, his extensive borrowings from both
                 psychology and psychoanalytic theory, and his desire to
                 contribute to the sciences of mental health more broadly,
                 the virulence of his campaign against psychiatry in his
                 subsequent books and speeches is striking and surprising.
                 The isolated criticisms of psychiatry that can be found
                 in Hubbard’s early work are by contrast almost models of
                 rhetorical constraint.


          o ⚓ Hackaday ☛ Analog_Tank_Driving_Simulator_Patrols_A_Tiny_Physical
            Landscape⠀⇛


                 How do you build a practical tank-driving simulator in
                 the 1970s, when 3D computer-generated graphics are still
                 just a fantasy of the future? If you’re a European tanker
                 school, the solution is to use a large CNC machine to
                 drive a camera around a miniature terrain model (German,
                 translated). In the video after the break, [Tom Scott]
                 takes it for a test drive.


          o ⚓ Hackaday ☛ Whip_Up_Some_Homemade_Artisanal_Flux⠀⇛


                 You don’t think much about the power company until you
                 flip the lights on and they don’t come on. The same can
                 be said of soldering flux. You don’t think much about it,
                 usually, until you try soldering without it. Flux has a
                 cleaning action on metal surfaces that allows for a
                 proper solder joint. The problem is, do you have any idea
                 what’s in the flux you are using? We don’t either.
                 [Catsndogs] has a recipe to make your own flux and then
                 you’ll know.


          o ⚓ Scheerpost ☛ Jail_Telecom_Company_Offered_Sheriffs_Caribbean
            Cruises_and_Florida_Retreats⠀⇛


                 Smart Communications, a for-profit Florida company that
                 sells phone, videochat, and email-like services to
                 prisons and jails, told at least one sheriff’s department
                 that it can live “the resort life” on a trip to Florida.


          o ⚓ The Nation ☛ Gallia_Est_Omnis_Divisa_in_Partes_Tres⠀⇛


                 A pity that she’s now beneath the soil. She rests in
                 peace, where one can never reach her. For otherwise I’d
                 surely take delight In asking that they can my Latin
                 teacher.


          o ⚓ The Nation ☛ Letters_From_the_October_31/November_7,_2022,
            Issue⠀⇛


                 Re “Justice for the People,” by Elie Mystal [September 5/
                 12]: This is an excellent detailing of what Sonia
                 Sotomayor is doing on the Supreme Court. Acting not out
                 of ideology as much as duty, she represents the near
                 future of what needs to be done across the federal
                 branches. Thank you, as well, for reiterating at length
                 the horrors of the shadow docket. Unless it is reined in,
                 the right will use it to try anything. It won’t stop
                 unless more of the public knows about it.Walter Pewen


          o ⚓ Counter Punch ☛ Yassamine_Mather⠀⇛


          o § Science⠀➾


                # ⚓ The Nation ☛ The_Uncategorizable_Bruno_Latour_(1947–2022)⠀⇛


                       Among the most creative and influential scholars of
                       the past 50 years, the protean Bruno Latour was
                       variously described as a sociologist, a
                       philosopher, and an anthropologist. But just as his
                       work questioned the making of categories, he
                       himself transcended categorization. For decades,
                       Latour worked at the Center for the Sociology of
                       Innovation (CSI) at the Paris School of Mines. In
                       2007 he became a dean at the prestigious Paris
                       Institute of Political Sciences (Sciences Po),
                       where he elevated the role of research by
                       establishing its Médialab, a unique experimental
                       space for transdisciplinary work, and integrated
                       environmental concerns—especially climate
                       change—throughout the Sciences Po curriculum.


          o § Education⠀➾


                # ⚓ Times Higher Education ☛ Global_network_formed_to_combat
                  contract_cheating⠀⇛


                       The Global Academic Integrity Network, founded by
                       Quality and Qualifications Ireland (QQI) and
                       Australia’s Tertiary Education Quality and
                       Standards Agency (Teqsa), aims to improve data
                       sharing and collaboration between quality assurance
                       and regulatory bodies.


                       At launch, the United Nations-backed initiative
                       also brings together bodies from France, Hungary,
                       Italy, South Africa, Ukraine, the UK and Zambia.


                # ⚓ Hackaday ☛ If_Only_The_Kids_Knew_About_Pipes⠀⇛


                       Being a parent is hard work. You need to prepare
                       your child to operate in the world. Reading,
                       writing, arithmetic, and how to make good choices
                       are just the beginning. They also ought to know
                       regular expressions and Unix pipes. [Jackdoe] can
                       help with that last one, thanks to their Unix pipe
                       card game.


          o § Hardware⠀➾


                # ⚓ Hackaday ☛ Mining_And_Refining:_Cobalt,_The_Unfortunately
                  Necessary_Metal⠀⇛


                       The story of humankind is largely a tale of
                       conflict, often brought about by the uneven
                       distribution of resources. For as long as we’ve
                       been down out of the trees, and probably
                       considerably before that too, our ancestors have
                       been struggling to get what they need to survive,
                       as often as not at the expense of another, more
                       fortunate tribe. Food, water, land, it doesn’t
                       matter; if They have it and We don’t, chances are
                       good that there’s going to be a fight.


                # ⚓ Hackaday ☛ 3D_Printing_With_A_Drone_Swarm?⠀⇛


                       Even in technical disciplines such as engineering,
                       there is much we can still learn from nature. After
                       all, the endless experimentation and trials of life
                       give rise to some of the most elegant solutions to
                       problems. With that in mind, a large team of
                       researchers took inspiration from the humble (if
                       rather annoying) wasp, specifically its nest-
                       building skills. The idea was to explore 3D
                       printing of structures without the constraints of a
                       framed machine, by mounting an extruder onto a
                       drone.


                # ⚓ Hackaday ☛ An_Oscilloscope_Trigger_For_Vintage_Video
                  Processors⠀⇛


                       Working on retro computers is rarely
                       straightforward, as [ukmaker] recently found out
                       while designing a new display interface. Their
                       oscilloscope was having trouble triggering on the
                       video signal produced by older video circuitry, so
                       they created the Video Trigger for Retrocomputers.


                # ⚓ Hackaday ☛ 3D_Printering:_Managing_Multiple_Printing
                  Profiles⠀⇛


                       I know people who have 3D printers that are little
                       more than appliances. They buy it, they print with
                       it, and they don’t change much of anything. That
                       doesn’t describe me and, I’m guessing, it doesn’t
                       describe you either. This does lead to a problem,
                       though, when it comes to slicers. You have to keep
                       changing profiles and modifying them. It can be
                       hard to keep things straight. For example, if you
                       have profiles for different nozzles, you get to
                       make a choice: keep one profile and edit the parts
                       that change, or keep multiple profiles and any
                       common changes have to be propagated to the other
                       profiles.


          o § Health/Nutrition/Agriculture⠀➾


                # ⚓ NBC ☛ Taking_a_break_from_the_news_can_improve_mental
                  health,_study_finds⠀⇛


                       A constant influx of bad news — pandemic,
                       shootings, inflation, natural disasters, political
                       turmoil — can feel, at best, soul-crushing. Now, a
                       new study from Spain confirms the negative toll
                       constantly being plugged into the news cycle can
                       take.


                       The researchers looked at how people were best able
                       to manage feelings of anxiety and depression at the
                       height of the pandemic, finding that one of the
                       most effective methods was to take breaks from the
                       barrage of bad news.


                # ⚓ TruthOut ☛ The_NFL_Can_Save_Football_and_Preserve_Players’
                  Health_With_This_One_Simple_Fix⠀⇛


                # ⚓ TruthOut ☛ COVID_Has_Become_Indefinite._We_Need_New
                  Practices_to_Keep_Each_Other_Safe.⠀⇛


                # ⚓ Scheerpost ☛ The_Urgent_Reasons_300_Doctors_Are_Calling_for
                  Julian_Assange’s_Release⠀⇛


                       Doctors For Assange wrote to US and UK officials
                       warning of health concerns if Julian Assange stays
                       in prison. He tested positive for COVID.


                # ⚓ Common Dreams ☛ ‘A_Policy_Failure’:_1.3_Million_US_Adults
                  With_Diabetes_Ration_Insulin_Due_to_High_Cost⠀⇛


                       “We have allowed pharmaceutical companies to set
                       the agenda, and that is coming at the cost to our
                       patients.”


                # ⚓ Common Dreams ☛ Analysis_Shows_Toxic_‘Forever_Chemicals’
                  Found_in_Over_80%_of_US_Waterways⠀⇛


                       According to reporting by the Guardian, the new
                       Waterkeeper Alliance analysis “found detectable
                       PFAS levels in 95 out of 114, or 83%, of waterways
                       tested across 34 states and the District of
                       Columbia, and frequently at levels that exceed
                       federal and state limits.”


          o § Proprietary⠀➾


                # ⚓ India Times ☛ Microsoft_continues_layoffs,_sacks_nearly
                  1,000:_Report [Ed: The numbers are wrong. They lies about the
                  extent of the layoffs. See Microsoft_Firing_10%,_Not_1%,_of
                  Staff_(Microsoft_Lies_to_the_Media,_Figures_One_Order_of
                  Magnitude_Off). That does not even include temps and
                  contractors.]⠀⇛


                       Technology giant Microsoft has sacked around 1,000
                       people in a fresh round of layoffs, according to US
                       news website Axios.


                       This is the third round of layoffs at the Big Tech
                       firm since July. Microsoft had fired nearly 1% of
                       its 180,000-strong workforce across offices and
                       product divisions in July as part of a
                       ‘realignment’.


                # ⚓ Axios ☛ Microsoft_becomes_latest_tech_firm_to_cut_staff⠀⇛


                       Details: Microsoft declined to say how many jobs
                       had been cut, but a source said the layoffs
                       numbered under 1,000.


                # ⚓ The Hill ☛ Microsoft_slashes_some_jobs_amid_slowing_revenue
                  growth⠀⇛


                       On Tuesday, shares of Microsoft fell slightly after
                       the layoffs were reported. First quarter results
                       for Microsoft’s next fiscal year is set for release
                       on Oct. 25.


          o § Security⠀➾


                # ⚓ Pen Test Partners ☛ Moto_E20_Readback_Vulnerability_|_Pen
                  Test_Partners⠀⇛


                       The Moto E20 from Lenovo phone uses a Unisoc T606
                       SoC. In common with all Unisoc SoCs, this uses a
                       customised U-boot bootloader as a secondary
                       bootloader which then loads Android. The bootloader
                       can be communicated with using the Android fastboot
                       protocol.


                       Analysis of the bootloader showed that there was an
                       undocumented fastboot oem subcommand, pull, that
                       could read data from the phone (i.e. could upload
                       data from the phone to the fastboot client).


                       [...]


                       As the userdata partition is encrypted, PII cannot
                       be stolen through method 2 by default. It is useful
                       to extract the contents of the bootloader or
                       Android to reverse engineer potential
                       vulnerabilities. The uboot_log and miscdata
                       partitions can contain information useful for
                       potential future attacks.


                       Method 3 can allow cold boot attacks – where the
                       device is booted into the bootloader and the RAM is
                       then examined to look for confidential information
                       from the Android operating system.


                       As a proof of concept, some canary values were set
                       in the Android Settings and Contacts app. The
                       device was booted into bootloader mode. 2 GB of
                       memory was extracted from address 0×80000000 (which
                       usually maps to RAM on ARM cores) and examined. All
                       canaries were found within the memory, including a
                       wireless PSK. Showing it could be possible to
                       compromise user data.


                       Although this was discovered on the version of the
                       firmware described above, the same command has been
                       identified on other versions and software channels
                       for the Moto E20.


                # ⚓ ISTIO-SECURITY-2022-007⠀⇛


                       The Istio control plane, istiod, is vulnerable to a
                       request processing error, allowing a malicious
                       attacker that sends a specially crafted or
                       oversized message, to crash the control plane
                       process. This can be exploited when the Kubernetes
                       validating or mutating webhook service is exposed
                       publicly. This endpoint is served over TLS port
                       15017, but does not require any authentication from
                       an attacker.


                # ⚓ LinuxSecurity ☛ RedHat:_RHSA-2022-7020:01_Important:
                  firefox_security_update⠀⇛


                       An update for firefox is now available for Red Hat
                       Enterprise Linux 9.Red Hat Product Security has
                       rated this update as having a security impactof
                       Important. A Common Vulnerability Scoring System
                       (CVSS) base score,which gives a detailed severity
                       rating, is available for each vulnerabilityfrom the
                       CVE link(s) in the References section.


                # ⚓ IT Wire ☛ iTWire_–_Cisco,_Canberra_uni_to_jointly_address
                  cyber-security_skills_shortage⠀⇛


                       Global networking giant Cisco has tied up with the
                       University of Canberra to address the shortage of
                       cyber security skills, with the focus to be on
                       defence and critical infrastructure.


                       A statement from Cisco said the partnership would
                       expand the National Industry Innovation Network
                       which is focused on how digital technologies can
                       help in Australia’s economic and social challenges.


                       “Cyber security is a major challenge for us at
                       every level, and certainly for industry, government
                       and universities – therefore, it is equally a major
                       source of new jobs and innovation,” University of
                       Canberra vice-chancellor Professor Paddy Nixon
                       said.


                       “There is enormous scope for Australia to improve
                       its resilience against cyber attacks including the
                       use of advanced digital technologies.


                # ⚓ Krebs On Security ☛ How_Card_Skimming_Disproportionally
                  Affects_Those_Most_In_Need⠀⇛


                       When people banking in the United States lose money
                       because their payment card got skimmed at an ATM,
                       gas pump or grocery store checkout terminal, they
                       may face hassles or delays in recovering any lost
                       funds, but they are almost always made whole by
                       their financial institution. Yet, one class of
                       Americans — those receiving food assistance
                       benefits via state-issued prepaid debit cards — are
                       particularly exposed to losses from skimming scams,
                       and usually have little recourse to do anything
                       about it.


          o § Defence/Aggression⠀➾


                # ⚓ Rolling Stone ☛ Feds_Charge_Pentagon_Contractor_With_Lying
                  About_Ties_to_ISIS⠀⇛


                       In a federal criminal complaint filed in Kansas on
                       Monday, the government says Mohammad Rafi Mohammadi
                       communicated with, funded, and, in one case, sought
                       to secure the release of recruiters for ISIS-
                       Khorasan (ISIS-K) both before and after he worked
                       for the U.S. as a linguist in Afghanistan.


                # ⚓ The Hill ☛ French_cement_firm_pleads_guilty_to_funding
                  Islamic_State_in_US_case,_fined_$778M⠀⇛


                       A Justice Department release on Tuesday states that
                       Lafarge S.A., located in Paris, and its subsidiary
                       Lafarge Cement Syria S.A., located in Damascus,
                       pleaded guilty to conspiring to provide support and
                       resources to ISIS and the al-Nusrah Front (ANF),
                       both of which the United States has designated as
                       foreign terrorist organizations.


                       Court documents state that the firms planned to pay
                       ISIS and ANF in exchange for permission to operate
                       a cement plant in Syria from 2013 to 2014.


                # ⚓ Scheerpost ☛ In_Neocolonial_Rant,_EU_Says_Europe_Is
                  ‘Garden’_Superior_To_Rest_of_World’s_Barbaric_‘Jungle’⠀⇛


                       EU foreign-policy chief Josep Borrell showed the
                       West’s neocolonial mentality claiming “the world
                       needs Europe” as a “beacon” and beautiful “garden”
                       to civilize the barbarous “jungle” in the Global
                       South. He also called for regime change to create a
                       “post-Putin Russia.”


                # ⚓ TruthOut ☛ Nuclear_Threats_Escalate_as_NATO_and_Russia_Plan
                  Nuclear_War_Games⠀⇛


                # ⚓ Meduza ☛ Russia_continues_shelling_civilian_infrastructure
                  throughout_Ukraine_—_Meduza⠀⇛


                       Russian forces shelled multiple Ukrainian cities on
                       the morning of October 18. Ukrainian authorities
                       declared an air raid alert for the entire country.


                # ⚓ Meduza ☛ More_Russians_seeking_fake_military_and_law
                  enforcement_IDs_on_dark_web_to_evade_draft_—_Meduza⠀⇛


                       A growing number of Russians have started
                       purchasing falsified law enforcement ID cards and
                       other documents indicating they hold military or
                       law enforcement ranks, Kommersant reported Tuesday.
                       Cybersecurity experts have reportedly observed a
                       flood of document forging services on the dark web
                       and on Telegram since the start of Russia’s
                       mobilization campaign.


                # ⚓ Meduza ☛ Two_Zaporizhzhia_NPP_employees_reportedly_abducted
                  by_Russian_forces_—_Meduza⠀⇛


                       Russian troops have kidnapped two employees of the
                       Zaporizhzhia Nuclear Power Plant, according to
                       Energoatom, Ukraine’s state nuclear power company.


                # ⚓ The Nation ☛ Now_Is_Not_the_Time_to_Playact_Nuclear_War⠀⇛


                       Day after day, the Russian president talks of using
                       nuclear weapons. The American president says it’s
                       been awhile since we “faced the prospect of
                       Armageddon” like this. The UN secretary general
                       says we’re all “just one misunderstanding, one
                       miscalculation away from nuclear annihilation.” A
                       former chairman of the Joint Chiefs of Staff—the
                       highest ranking US military officer—says Vladimir
                       Putin is like “a cornered animal” and therefore “we
                       need to back off that [Armageddon talk]” and
                       urgently engage in real diplomacy before it all
                       goes horribly sideways.


                # ⚓ The Nation ☛ China_and_Russia_Are_Not_Mirror_Images⠀⇛


                       Three months into Russia’s invasion of Ukraine, US
                       Secretary of State Antony Blinken gave a speech
                       claiming that “the foundations of the international
                       order are under serious and sustained challenge.”
                       Yet Blinken passed over the invasion quickly:
                       “Russian President Vladimir Putin poses a clear and
                       present threat…[but] China is the only country with
                       both the intent to reshape the international order
                       and, increasingly, the economic, diplomatic,
                       military, and technological power to do it.”1


                # ⚓ Counter Punch ☛ Oppose_US_and_UN_military_intervention_in
                  Haiti!⠀⇛


                       We strongly condemn the letter submitted on October
                       9th by UN Secretary-General Antonio Guterres to the
                       UN Security Council, proposing the deployment of a
                       foreign, armed occupation force to Haiti. And we
                       denounce theBiden Administration’s drafting of a UN
                       Security Council Resolution calling for the
                       immediate deployment of a foreign “rapid action
                       force” in Haiti, as reported on October 15th.


                # ⚓ Meduza ☛ ‘A_clear_anti-Putin_trend’:_As_the_Russian
                  mobilization_devolves_into_chaos,_people_are,_quite
                  rationally,_blaming_the_president_—_Meduza⠀⇛


                       The Kremlin had never expected mobilization to be
                       “popular” with the Russian people. This is the
                       often-cited reason for Russia’s delay in declaring
                       even a “partial mobilization.” Recent urban focus
                       groups, commissioned by the Kremlin itself, have
                       made clear that practically no one approves of the
                       war in Russia’s larger cities. Only some of the
                       very elderly people, who would never have to go to
                       the front, still approve of the “special
                       operation.” Perhaps even more dismaying to the
                       Russian President’s Office is the the growing
                       “frustration” with Vladimir Putin himself. Meduza’s
                       special correspondent Andrey Pertsev interviewed
                       sources close to the Kremlin about how the
                       government gauges political temperatures around the
                       country, and how it tries to keep the society from
                       reaching a boiling point.


                # ⚓ Meduza ☛ Occupation_officials_announce_transfer_of
                  civilians_to_left_bank_of_Dnipro_River:_‘the_battle_for
                  Kherson_will_begin_soon’_—_Meduza⠀⇛


                       Vladimir Saldo, head of the Russian occupation
                       administration of Kherson region, announced the
                       “organized transfer” of civilians in four
                       municipalities from the right bank to the left bank
                       of the Dnipro River. The measure is necessary, he
                       said, because Ukraine is “assembling forces for a
                       wide-scale offensive” and there is an “immediate
                       danger that the territory will be flooded due to
                       the planned destruction of the dam at the Kakhovka
                       Hydroelectric Power Plant.”


                # ⚓ Meduza ☛ Russia_promises_housing_vouchers_to_Kherson
                  residents_willing_to_leave_region_—_Meduza⠀⇛


                       All current Kherson residents willing to leave
                       their region for Russia will receive housing
                       vouchers from the Russian government, said Marat
                       Khusnullin, the government’s deputy chairman.


                # ⚓ Meduza ☛ Ukraine’s_parliament_declares_‘Chechen_Republic_of
                  Ichkeria’_Russian-occupied_territory_—_Meduza⠀⇛


                       Ukraine’s Verkhovna Rada has declared that “the
                       Chechen Republic of Ichkeria” is a territory
                       “temporarily occupied by the Russian Federation.”
                       287 Rada deputies voted in favor of the resolution
                       that says so, and also condemns “the genocide of
                       the Chechen people.”


                # ⚓ Meduza ☛ The_Su-34_pilots_will_have_‘quite_a_lot_of
                  questions’_to_answer_What_experts_are_saying_about_the_Yeysk
                  bomber_plane_crash_—_Meduza⠀⇛


                       On October 17, a Su-34 bomber plane crashed near a
                       nine-story apartment building in Yeysk, a Russian
                       town in the Krasnodar Region. The plane struck the
                       building and fell apart, reported the local
                       emergency service. Spilled aircraft fuel then
                       caught fire, which spread to the building and soon
                       engulfed it in flames. It took six hours to
                       extinguish the massive fire, which damaged a total
                       of 72 apartments. Rescue teams have now finished
                       searching for people under the rubble. 13 people
                       are reported dead, including three children. Three
                       of the victims died after jumping from the top
                       floors as they tried to escape the fire, said Anna
                       Minkova, the region’s deputy governor. 19 more
                       people have been injured, including five children
                       and two pregnant women. Four of the people who have
                       been hospitalized are still in intensive care. A
                       three-day mourning period has been declared in
                       Yeysk. Meanwhile, experts are beginning to probe
                       what might have caused the crash.


                # ⚓ Meduza ☛ Prosecution_demands_five-year_term_for_Pavel
                  Krisevich,_tried_for_mock_suicide_in_Red_Square_—_Meduza⠀⇛


                       Prosecutors are demanding a five-year prison
                       sentence for Pavel Krisevich, the Russian
                       performance artist who stands accused of
                       ‘hooliganism’ in the Red Square. Pavel Chikov, the
                       founder of Agora, a human-rights advocacy group,
                       reported this on his Telegram channel.


                # ⚓ Common Dreams ☛ Defying_Pentagon_Secrecy,_Reporting_Exposes
                  Retired_US_Generals_on_Saudi_Payroll⠀⇛


                       “Saudi Arabia’s paid advisers have included retired
                       Marine Gen. James L. Jones, a national security
                       adviser to President Barack Obama, and retired Army
                       Gen. Keith Alexander, who led the National Security
                       Agency under Obama and President George W. Bush,”
                       the Post reported, citing more than 4,000 pages
                       documents obtained through Freedom of Information
                       Act (FOIA) filings.


                # ⚓ Democracy Now ☛ How_Kansas_City_Police_Ignored_Warnings_a
                  Killer_Targeted_Black_Women,_Until_One_Escaped⠀⇛


                       Black residents in Kansas City, Missouri, say
                       police ignored their warnings that a murderer was
                       targeting Black women, until one of his captives
                       escaped earlier this month. A 22-year-old Black
                       woman in Excelsior Springs, just outside Kansas
                       City, said she broke free from the basement where a
                       white man held her captive for a month. She also
                       claimed there were more victims, all of them Black
                       women, who were similarly sexually abused. Timothy
                       Haslett Jr. is now in police custody and faces
                       rape, kidnapping and assault charges. Prior to his
                       arrest, the Kansas City Police Department said
                       concerns about a possible serial killer were
                       “completely unfounded” rumors. “This speaks to the
                       violent silencing of Black women specifically, of
                       the Black community at large here in Kansas City,”
                       says Ryan Sorrell, founder of the Black-led
                       independent newspaper Kansas City Defender, which
                       reported on the missing Black women. “This is
                       ongoing. We have cases back in the ’90s where Black
                       women have been murdered,” notes Justice Gatson,
                       executive director of the Black women-led group
                       Reale Justice Network in Kansas City, Missouri.


                # ⚓ Democracy Now ☛ “Rising_Against_Hate”:_Confronting_Anti-
                  China_Rhetoric_by_Politicians_That_Fuels_Anti-Asian_Attacks⠀⇛


                       With midterm elections three weeks away, a new
                       report links reported hate crimes against Asian
                       Americans to anti-China rhetoric used on the
                       campaign trail. This issue is also examined in a
                       new PBS documentary, “Rising Against Asian Hate,”
                       which explores the fight against anti-Asian racism
                       following the Atlanta spa shootings in March 2021,
                       when a white gunman targeted multiple Asian-owned
                       businesses and killed eight people, six of them
                       Asian American women. At the time of the killing
                       spree, hate crimes against Asian Americans had been
                       on the rise after then-President Trump blamed the
                       outbreak of the coronavirus on China, calling it
                       the “kung flu.” “We felt that we had to document
                       this moment,” says executive producer Gina Kim,
                       “and make sure that people recognize that this is
                       an issue that we need to confront as a nation.”


                # ⚓ Democracy Now ☛ 60_Years_After_Cuban_Missile_Crisis,
                  Activists_Demand_World_Leaders_“Defuse_Nuclear_War”⠀⇛


                       As nuclear powers ratchet up tensions around the
                       Ukraine war, the U.S., NATO and Russia are carrying
                       out nuclear war games. Meanwhile, anti-nuclear
                       activists are calling on lawmakers and world
                       leaders to “Defuse Nuclear War.” Their calls come
                       on the 60th anniversary of the Cuban Missile
                       Crisis. “The Kremlin is making nuclear threats that
                       are completely reckless. At the same time, there
                       are things that the U.S. government can and should
                       do that would reduce the chances of nuclear war,”
                       says Norman Solomon, executive director of the
                       Institute for Public Accuracy and co-founder of
                       RootsAction.org. Meanwhile, Democratic leaders “are
                       simply going along with this autopilot heading
                       towards a precipice of nuclear annihilation
                       globally.” His recent piece for Common Dreams is
                       headlined “Don’t Just Worry About Nuclear War—Do
                       Something to Help Prevent It.”


          o § Environment⠀➾


                # ⚓ Barrons ☛ Sweden_Gets_26-year-old_Climate_Minister_In_New
                  Govt⠀⇛


                       Sweden’s coalition government was announced on
                       Friday after Kristersson penned a deal with his
                       partners and the nationalist and anti-immigration
                       Sweden Democrats which pledged to support the
                       government in exchange for policy commitments,
                       especially on immigration and crime.


                # ⚓ Common Dreams ☛ New_Jersey_Sues_5_Oil_Giants,_Industry
                  Lobby_for_Climate_Fraud⠀⇛


                       “The defendants have known for decades that use of
                       fossil fuels is a major cause of climate change.”


                # ⚓ Common Dreams ☛ Activists_Say_Fossil_Fuel_Interests_Have
                  Declared_‘War_on_Africa’s_Sustainable_Future’⠀⇛


                       “The push for investment in fossil fuels is likely
                       to perpetuate the triple injustices of energy,
                       social, and environmental crises.”


                # ⚓ Common Dreams ☛ Opinion_|_We_Can—And_Must—Stop_Carbon
                  Capture_In_Its_Tracks⠀⇛


                       The crises of the past few years have done a number
                       on our economy. From rising rent, to climbing
                       grocery bills; from a deadly pandemic to climate-
                       fueled natural disasters. 


                # § Energy⠀➾


                      # ⚓ The Nation ☛ Electric_Vehicles_Won’t_Save_Us⠀⇛


                             Much of the excitement over the Inflation
                             Reduction Act, which became law this summer,
                             focused on the boost it should give to the
                             sales of electric vehicles. Sadly, though,
                             manufacturing and driving tens of millions of
                             individual electric passenger cars won’t get
                             us far enough down the road to ending
                             greenhouse-gas emissions and stanching the
                             overheating of this planet. Worse yet, the
                             coming global race to electrify the personal
                             vehicle is likely to exacerbate ecological
                             degradation, geopolitical tensions, and
                             military conflict.


                      # ⚓ DeSmog ☛ New_Jersey_Sues_Five_Oil_Companies,_Alleging
                        Decades_of_‘Concealment’_and_‘Public_Deception’_on
                        Climate_Change⠀⇛


                             The lawsuit, filed in the New Jersey Superior
                             Court, states that the companies knew about
                             climate change for decades and actively
                             sought to conceal that information from the
                             public. Instead, they funded PR campaigns
                             aimed at confusing and misleading the public.


                      # ⚓ Common Dreams ☛ Opinion_|_Democrats_Tried_to_Sell
                        Appalachia_Down_the_Pipeline⠀⇛


                             In late September, the now-familiar “debt
                             ceiling showdown” was avoided when Sen. Joe
                             Manchin III, D-W.Va., pulled his permitting
                             reform legislation from the stopgap spending
                             bill because it did not have the votes. Not
                             only was a government shutdown avoided, but
                             this was also a near-miss for anyone who
                             cares about the future of our planet, as the
                             senator’s legislation would have fast-tracked
                             the construction of a new natural gas
                             pipeline through the mountains and rivers of
                             Appalachia.


                # § Wildlife/Nature⠀➾


                      # ⚓ Common Dreams ☛ Western_Wildfires_Cause_Extreme
                        Weather_Up_to_1,500_Miles_Away:_Study⠀⇛


                             Hundreds of thousands of wildfires have
                             burned through nearly seven million acres of
                             land across the West so far this year, but a
                             study released Monday shows the devastating
                             impact the blazes can have across the country
                             as the rising heat causes other extreme
                             weather events as far as 1,500 miles away.


          o § Finance⠀➾


                # ⚓ TruthOut ☛ Congressional_Workers_Get_$10,000_Raises_in
                  First-Ever_Union_Contract⠀⇛


                # ⚓ The Nation ☛ Why_the_Economy_Is_the_Dominant_Issue_for
                  Democrats_In_the_Midterms⠀⇛


                       As early voting gets started in some states,
                       Republicans are focusing their closing argument on
                       the economy, American voters’ biggest concern and
                       the subject of the majority of GOP ad buys. The
                       Democrats’ closing argument, meanwhile, features
                       abortion, the stripping away of women’s right to
                       choose.


                # ⚓ Common Dreams ☛ ‘People_Will_Lose_Their_Jobs’:_Expert
                  Chorus_Against_Fed_Rate_Hikes_Grows⠀⇛


                       With the Fed’s next policy meeting just two weeks
                       away, the Groundwork Collaborative on Tuesday
                       cataloged more than a dozen recent comments from
                       economists, fund managers, and other observers
                       cautioning that aggressive rate hikes are already
                       taking their toll on workers, economic growth, the
                       housing market, and other key areas without
                       tackling the core drivers of inflation—many of
                       which are out of the central bank’s control.


                # ⚓ Common Dreams ☛ Opinion_|_Neoliberalism_Is_a_Failure_During
                  Peacetime—and_Wartime_Too⠀⇛


                       Politically, the G7 and likeminded countries around
                       the world have adopted a war footing to stop
                       Russian aggression. Russian President Vladimir
                       Putin violated the most fundamental principle of
                       international law by launching an unprovoked attack
                       on another member of the United Nations—an
                       institution created explicitly to prevent such
                       aggression. The dangers of appeasement should be
                       obvious. Even a little empathy should make us
                       shudder in horror at the prospect of having to live
                       under Putin’s rule.


                # ⚓ Common Dreams ☛ ‘Believe_Him,’_Say_Critics,_as_McCarthy
                  Signals_GOP_Plan_to_Attack_Social_Security,_Medicare⠀⇛


                       In an interview with Punchbowl News Tuesday, the
                       California Republican outlined plans to use the
                       expected fight over the raising of the debt ceiling
                       next year as leverage to pass several austerity
                       policies and block additional aid for Ukraine, as
                       well as blocking pandemic-related spending.


          o § AstroTurf/Lobbying/Politics⠀➾


                # ⚓ Variety ☛ Spotify_Hires_TikTok_Head_of_Content_Partnerships
                  Bryan_Thoensen_for_Podcast_Team⠀⇛


                       In his new role at Spotify, Thoensen will lead the
                       Talk Creator Content and Partnerships team where he
                       will oversee content strategy and acquisition,
                       business development, partner management and
                       creator community development. He’s also tasked
                       with driving Spotify’s exclusive podcast strategy
                       through strategic initiatives and partnerships.


                       Most recently, Thoensen was head of content
                       partnerships at short-form video app TikTok, where
                       he led TikTok’s relationships with top global
                       public figures and worked with media brands such as
                       the NFL, NBA, Condé Nast and NBC. At TikTok, he was
                       also dedicated to building and supporting diverse
                       creator communities.


                # ⚓ Hollywood Reporter ☛ TikTok’s_Head_of_Content_Partnerships
                  Leaves_for_Spotify⠀⇛


                       TikTok’s former U.S. head of content partnerships,
                       Bryan Thoensen, has joined Spotify to oversee the
                       company’s content strategy and partnerships with
                       individual audio creators, known as the talk
                       creator content and partnerships team.


                       Thoensen will report up to Max Cutler, who was
                       promoted earlier this year to lead Spotify’s
                       partnerships with creators and now holds the title
                       of vp of talk creator content. In his new role at
                       Spotify, Thoensen will also oversee the team’s
                       business development and acquisitions as part of
                       the audio giant’s larger exclusive podcast
                       strategy.


                # ⚓ Hollywood Reporter ☛ Netflix_Will_No_Longer_Share
                  Subscriber_Forecasts_As_the_Company_Sharpens_Revenue_Focus⠀⇛


                       The streamer will still report global and regional
                       membership numbers, but will not provide guidance
                       for them starting in January 2023, when it reports
                       fourth quarter earnings. The company will still
                       provide guidance on revenue, operating income,
                       operating margin, net income and earnings per
                       share.


                # ⚓ TruthOut ☛ Report:_Trump_Considers_Marjorie_Taylor_Greene_a
                  “Great”_Pick_to_Lead_DOJ⠀⇛


                # ⚓ TruthOut ☛ Trump_Billed_Secret_Service_$1.4_Million_at_His
                  Properties_During_Presidency⠀⇛


                # ⚓ TruthOut ☛ GOP_Plan_on_Threatening_Recession_if_They_Can’t
                  Cut_Medicare,_Social_Security⠀⇛


                # ⚓ TruthOut ☛ Expert_Says_Trump_Could_Face_Perjury_Charge_If
                  He_Lies_at_Live_Jan._6_Deposition⠀⇛


                # ⚓ The Nation ☛ Trump_Toadies_Control_the_Republican_Party⠀⇛


                       Amid great fanfare in late September, the
                       Republican Party, under the direction of House
                       minority leader Kevin McCarthy, debuted its pitch
                       to seal a new majority in the next Congress beneath
                       a rubric redolent of the overtures of a bad
                       boyfriend: “A Commitment to America.” Modeled on
                       the landmark 1994 Contract With America pulled
                       together by Newt Gingrich (speaking of bad romantic
                       partners), the 2022 GOP program is mostly a grab
                       bag of harum-scarum cultural resentments papering
                       over a singularly anemic policy wish list.


                # ⚓ The Nation ☛ Who_Is_Red?⠀⇛


                       When some of the people blacklisted during the
                       McCarthy era left the United States to find work in
                       the United Kingdom, they might have thought they
                       had left their troubles behind them. But they were
                       wrong. The FBI passed its files on to MI5, the
                       British Security Service, which seems to have
                       accepted the bureau’s judgments without question.


                # ⚓ Techdirt ☛ FBI_Email:_Whole_Lot_Of_Agents_Think_January_6
                  Capitol_Raiders_Did_Nothing_Wrong⠀⇛


                       Trump threw his unquestioning support behind law
                       enforcement. He did it right after he was elected,
                       promising to elevate cops above the people they
                       served. He claimed the “anti-police atmosphere”
                       police had created themselves was “dangerous” (he
                       didn’t specify to whom, however) and that he, the
                       newly-elected dumpster fire, would get this fixed.


                # ⚓ Techdirt ☛ Turkey_Still_Thinks_It_Hasn’t_Jailed_Enough
                  Journalists,_Add_Prison_Sentences_To_Its_‘Fake_News’_Law⠀⇛


                       Turkey continues to fall back in the freedom field.
                       Or maybe it’s surging ahead in the oppression
                       field! Maybe that’s the list it wants to top.


                # ⚓ The Nation ☛ Georgia_Governor_Brian_Kemp_Plays_the_“Angry
                  Black_Woman”_Card_on_Stacey_Abrams⠀⇛


                       The contrast between Georgia Republican Senate
                       candidate Herschel Walker and the state’s
                       Democratic gubernatorial nominee, Stacey Abrams,
                       could not be more stark. I’m not talking about
                       intelligence or integrity, though there’s that, but
                       the expectations that political pundits and the
                       media generally set for the two candidates in
                       advance of their first debate.


                # ⚓ The Nation ☛ Old_Men_Limiting_Women’s_Choice⠀⇛


                # ⚓ Common Dreams ☛ Opinion_|_This_Supreme_Court_Is_a_Threat_to
                  Asian_American_Communities⠀⇛


                       As we reflect on the disastrous Supreme Court term
                       of 2022, we must acknowledge that this Supreme
                       Court is a threat to the rights of Asian American
                       communities and must be reformed.


                # ⚓ Meduza ☛ Moscow_appellate_court_upholds_opposition
                  politician_Alexey_Navalny’s_nine-year_sentence_—_Meduza⠀⇛


                       A Moscow appellate court has rejected an appeal
                       from jailed Russian opposition politician Alexey
                       Navalny’s lawyers, who sought to have his nine-year
                       prison sentence for embezzlement and contempt of
                       court overturned.


                # ⚓ Meduza ☛ ‘We_don’t_know_how_he_got_to_Belgorod’_Relatives
                  say_alleged_military_training_ground_shooter_disappeared_in
                  Moscow_five_days_before_the_attack_—_Meduza⠀⇛


                       Journalists from Radio Free Europe/Radio Liberty’s
                       Tajik service have identified the family of one of
                       the possible perpetrators of the mass shooting at a
                       military training ground in Russia’s Belgorod
                       region on October 15. The man in question is a 24-
                       year-old named Ehson Aminzoda. The Russian
                       authorities have not officially named him as a
                       suspect, but several days after the incident,
                       Ukrainian blogger Anatoly Shariy published a photo
                       of an official military document listing Aminzoda
                       as a private along with photos of the shooting
                       victims. Shariy didn’t indicate whether Aminzoda
                       was one of the gunmen or one of the victims.


                # ⚓ Meduza ☛ Ukrainian_authorities_warn_of_impending_power,
                  water,_and_heating_outages_as_Russia_targets_infrastructure_—
                  Meduza⠀⇛


                       On Tuesday, Kyrylo Tymoshenko, the deputy head of
                       the Ukrainian President’s Office, told Ukrainians
                       to prepare for power, water, and heating to be cut
                       off throughout the country.


                # ⚓ Common Dreams ☛ Opinion_|_Democrats_Are_Failing_This_Key
                  Rule_of_Communication⠀⇛


                       Republicans are pulling ahead in the polls right
                       now because of the issues of the economy, crime,
                       and homelessness, according to a new New York
                       Times/Siena College poll published just yesterday.
                       More than a year ago, I wrote about this very
                       danger in an article titled, “A Crime Wave Could
                       Take Down the Democrats in 2022″ (among others).


                # ⚓ Common Dreams ☛ ‘Makes_Me_Physically_Ill’:_Footage_Shows
                  DeSantis’_Election_Police_Force_in_Action⠀⇛


                       Through public records requests, the Tampa Bay
                       Times obtained several videos of arrests of people
                       who were accused of voting illegally by DeSantis’
                       new Office of Election Crimes and Security, which
                       was established earlier this year.


                # ⚓ Robert Reich ☛ Why_The_Window_To_Prosecute_Trump_Is
                  Closing⠀⇛


                # ⚓ Telex (Hungary) ☛ Who_said_it?_Hungary’s_Deputy_PM_or
                  Putin?⠀⇛


                # ⚓ Telex (Hungary) ☛ EU_sanctions_depicted_as_actual_bomb_in
                  Hungarian_government’s_anti-sanctions_propaganda⠀⇛


                # ⚓ Telex (Hungary) ☛ Visiting_lecturer_at_Fidesz’_elite
                  education_center_detained_in_London_on_espionage_charges⠀⇛


                # § Misinformation/Disinformation/Propaganda⠀➾


                      # ⚓ Techdirt ☛ Techdirt_Podcast_Episode_334:_Can_You
                        Mandate_Editorial_Transparency?⠀⇛


                             Amidst all the conversation around regulating
                             social media, algorithmic amplification, and
                             disinformation, one idea that tends to get a
                             lot of broad support is mandating editorial
                             transparency. After all, it sounds nice,
                             since transparency is usually a good thing.
                             But in fact, there are huge legal and
                             conceptual problems with mandated
                             transparency. Santa Clara Law’s Eric Goldman
                             has written papers on the constitutionality
                             of the idea and an important Supreme Court
                             case related to this question, and this week
                             he joins the podcast to discuss why mandated
                             transparency isn’t as good (or as
                             constitutional) as many people claim.


          o § Censorship/Free Speech⠀➾


                # ⚓ Al Monitor ☛ US_confirms_Saudi_sentencing_of_American_over
                  tweets⠀⇛


                       The newspaper, quoting the son Ibrahim, said that
                       Almadi, who is 72, was sentenced on October 3 to 16
                       years in prison with a travel ban of another 16
                       years after that.


                       The son told the newspaper that his father had
                       expressed only “mild” opinions with tweets
                       mentioning corruption in Saudi Arabia and the
                       killing of Jamal Khashoggi, the US-based columnist
                       who was dismembered in 2018 after being lured into
                       the kingdom’s Istanbul consulate.


                # ⚓ France24 ☛ US_confirms_Saudi_detention_of_American_over
                  tweets⠀⇛


                       The United States said Tuesday it was raising with
                       Saudi Arabia a prison sentence handed to a US
                       citizen over tweets critical of the kingdom,
                       another source of tension between the historic
                       allies.


                # ⚓ RFI ☛ US_confirms_Saudi_detention_of_American_over_tweets⠀⇛


                       Saudi Arabia has long faced criticism over human
                       rights. Blogger and rights activist Raif Badawi
                       served 10 years in prison through March and was
                       publicly lashed 50 times for charges over content
                       on his website.


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


                # ⚓ IT Wire ☛ iTWire_–_Australia_needs_to_act_urgently_to_free
                  Assange,_says_lawyer⠀⇛


                       WikiLeaks founder and publisher Julian Assange will
                       take his own life if he is extradited to the US,
                       his lawyer Jennifer Robinson says, adding that the
                       situation is very serious and the Australian
                       Government needs to act on it immediately.


                       She told the ABC 730 program on Tuesday: “…the
                       situation is incredibly urgent. Julian’s health is
                       declining in front of our eyes. He has already
                       suffered a mini stroke. He has got COVID right now
                       in prison, which doctors say could be life -
                       threatening for him.”


                       Asked by 730 host Sarah Ferguson if the argument
                       that Assange should be brought back to Australia
                       after his sentencing so he could serve out any term
                       in his own country had any merit, Robinson replied:


                       “The problem with that solution, it is no solution.
                       He would have to go to the US, be convicted,
                       exhaust all appeal processes, and we’re talking
                       about another decade of legal process. He won’t
                       last that long. The medical evidence makes clear he
                       won’t last that long.


                # ⚓ VOA News ☛ Kashmiri_Pulitzer_Winner_Blocked_from_Traveling
                  to_Receive_Award⠀⇛


                       Sanna Irshad Mattoo, a Pulitzer Prize-winning
                       photojournalist from Indian-administered Kashmir,
                       said Tuesday on Twitter that she had been stopped
                       by immigration authorities in New Delhi from
                       boarding a flight to New York to receive her award.


                # ⚓ VOA News ☛ Journalists_in_US_Moving_Closer_to_Improved
                  Legal_Protections⠀⇛


                       The legislation — the Protect Reporters from
                       Exploitative State Spying (PRESS) Act and a federal
                       anti-SLAPP (Strategic Lawsuits Against Public
                       Participation) bill — would, respectively, block
                       federal law enforcement from subpoenaing journalist
                       records and protect media against certain lawsuits.


          o § Civil Rights/Policing⠀➾


                # ⚓ The Economist ☛ Iran’s_protests_spread,_as_a_notorious
                  prison_burns⠀⇛


                       No Iranian prison is as notorious as Evin. For 50
                       years it has been the holding pen, interrogation
                       centre and gallows of the state’s political
                       prisoners. It lies at the base of the Alborz
                       mountains, casting a shadow over the capital below.
                       The fire billowing inside it lit up the city and
                       evoked the torching of the Rex Cinema, back in the
                       summer of 1978, when the death of more than 470
                       people in the building as a result of an arson
                       attack triggered the Islamic revolution that
                       overthrew the shah.


                # ⚓ VOA News ☛ Human_Rights_Groups_Call_for_UN_Probe_of_Iran’s
                  Crackdown_on_Protesters⠀⇛


                       The human rights groups said the evidence suggests
                       a disturbing pattern of security forces
                       deliberately firing live ammunition and metal
                       shells at protesters and bystanders, including
                       children.


                       The statement says more than 200 protesters have
                       been killed during the demonstrations, including at
                       least 23 children, while more than 1,000
                       protesters, activists, journalists and students
                       have been arbitrarily arrested.


                # ⚓ NPR ☛ Worry_grows_for_a_female_Iranian_athlete_who_climbed
                  without_a_hijab⠀⇛


                       A later Instagram post on an account attributed to
                       Rekabi described her not wearing a hijab as
                       “unintentional,” though it wasn’t immediately clear
                       whether she wrote the post or what condition she
                       was in at the time. The Iranian government
                       routinely pressures activists at home and abroad,
                       often airing what rights group describe as coerced
                       confessions on state television.


                       Rekabi left Seoul on a Tuesday morning flight, the
                       Iranian Embassy in South Korea said. The BBC’s
                       Persian service, which has extensive contacts
                       within Iran despite being banned from operating
                       there, quoted an unnamed “informed source” who
                       described Iranian officials as seizing both
                       Rekabi’s mobile phone and passport.


                # ⚓ The Nation ☛ ICE_Is_Subjecting_a_Record_Number_of_Asylum
                  Seekers_to_Electronic_Monitoring⠀⇛


                       When migrants arrive at the southern border and ask
                       for asylum, they currently face two possibilities:
                       Either they’re turned away and shut out of the
                       asylum system under Title 42, the Trump-era public
                       health order Biden has kept in place despite
                       lifting other pandemic restrictions, or they’re
                       admitted into the country and allowed to pursue
                       their cases in immigration court. Those lucky
                       enough to be exempted from Title 42 and admitted
                       into the country—people Mexico won’t accept because
                       of their nationality, including Cubans,
                       Venezuelans, and Ecuadorians—then face a second set
                       of possibilities. Depending on whether they have
                       children with them, family members or other
                       sponsors in the United States, and the availability
                       of beds in detention, as well as other factors,
                       they may be detained in Immigration and Customs
                       Enforcement facilities, subjected to electronic
                       monitoring under ICE’s Alternatives to Detention
                       (ATD) program, or simply told to report to
                       immigration court on their hearing date. The latter
                       option is becoming increasingly rare: An
                       unprecedented number of noncitizens have been
                       enrolled in ATD under the Biden administration.
                       While Biden has reduced the overall number of
                       migrants in ICE detention, his administration has
                       also presided over a significant expansion of
                       electronic monitoring—a scheme that advocates say
                       harms migrants, even as it claims to be more humane
                       than traditional detention.


                # ⚓ ABC ☛ African_Australian_students_are_subjected_to_the_n-
                  word_and_racism_in_the_classroom,_according_to_report_–_ABC
                  News⠀⇛


                       In the message, seen by the ABC, he said, “When
                       slavery comes back, I’m buying her”.


                # ⚓ Scheerpost ☛ Ketanji_Brown_Jackson_Cleverly_Turned_the
                  Right’s_Own_Judicial_Theory_Against_It⠀⇛


                       In her defense of voting rights, Supreme Court
                       Justice Ketanji Brown Jackson brilliantly turned
                       the tables on the right by crafting her own
                       originalist argument to defend taking race into
                       account when drawing district maps.


                # ⚓ TruthOut ☛ Biden_to_Vow_to_Make_Codifying_“Roe”_First
                  Priority_If_Democrats_Win_Congress⠀⇛


                # ⚓ TruthOut ☛ Legal_Experts_Question_How_Judge_Aileen_Cannon
                  Landed_Trump’s_Mar-a-Lago_Case⠀⇛


                # ⚓ Techdirt ☛ DC_Taxpayers_Are_Spending_Millions_To_Rehire_Bad
                  Cops⠀⇛


                       Late last year, a trove of records was obtained by
                       transparency activists Distributed Denial of Secret
                       (DDoS). Those records showed what the Washington DC
                       Metro PD hoped to hide: that the internal
                       disciplinary process was apparently irreparably
                       broken.


                # ⚓ Pro Publica ☛ How_the_FBI_Stumbled_in_the_War_on
                  Cybercrime⠀⇛


                       Retention in the division had been a chronic
                       problem, but in the spring of that year, it became
                       acute. About a dozen young and midcareer cyber
                       agents had given notice or were considering
                       leaving, attracted by more lucrative jobs outside
                       government. As the resignations piled up, Comey
                       received an unsolicited email from Andre McGregor,
                       one of the cyber agents who had quit. In his email,
                       the young agent suggested ways to improve the Cyber
                       Division. Comey routinely broadcast his open-door
                       policy, but senior staff members were nevertheless
                       aghast when they heard an agent with just six
                       years’ experience in the bureau had actually taken
                       him up on it. To their consternation, Comey took
                       McGregor’s email and the other cyber agents’
                       departures seriously. “I want to meet these guys,”
                       he said. He invited the agents to Washington from
                       field offices nationwide for a private lunch. As
                       news of the meeting circulated throughout
                       headquarters, across divisions and into the field,
                       senior staff openly scorned the cyber agents,
                       dubbing them “the 12 Angry Men,” “the Dirty Dozen”
                       or just “these assholes.” To the old-schoolers —
                       including some who had risked their lives in
                       service to the bureau — the cyber agents were
                       spoiled prima donnas, not real FBI.


                # ⚓ Common Dreams ☛ Following_‘Brutal_Union-Busting_Campaign,’
                  Albany_Amazon_Workers_Reject_Unionization⠀⇛


                       “We’re proud of the brave workers in upstate New
                       York who stood up in the face of a vicious anti-
                       union campaign to challenge a trillion-dollar
                       corporation.”


          o § Internet Policy/Net Neutrality⠀➾


                # ⚓ Techdirt ☛ Senator_Sheldon_Whitehouse_Has_Really_Bad_Ideas
                  About_Section_230⠀⇛


                       Over the summer I got a copy of the new book from
                       Lee Bollinger and Geoffrey Stone, two formerly
                       staunch 1st Amendment supporters who have
                       apparently decided to go back on their earlier
                       views, with a collection of essays by a variety of
                       authors about “social media, freedom of speech, and
                       the future of our democracy.” Much of the book is
                       maddening, because there are many essays from very
                       famous people who should know better, but seem more
                       than willing to reject the 1st Amendment because
                       people said bad stuff online.


                # ⚓ Techdirt ☛ 250_Groups_Urge_Senate_To_Stop_Being_Corrupt
                  Jackasses_and_Finally_Confirm_FCC_Nominee_Gigi_Sohn⠀⇛


                       We’ve noted several times how telecom and media
                       giants are running a sleazy smear campaign against
                       Biden FCC nominee Gigi Sohn, in the hopes of
                       keeping the agency in perpetual consumer protection
                       gridlock. The attacks have been carefully seeded
                       across the US press, and accuse the highly popular
                       and qualified candidate of everything from hating
                       police to being an enemy of rural America.


                # ⚓ Techdirt ☛ U.S._Utilities_Are_Poised_To_Disrupt_Telecom
                  Monopolies_Thanks_To_Biden_COVID_Relief_&_Infrastructure
                  Bills⠀⇛


                       As noted previously, I’ve got some mixed feelings
                       on the Biden COVID relief and infrastructure bills’
                       plan to throw more than $50 billion dollars at U.S.
                       broadband. Mostly because we still haven’t mapped
                       U.S. broadband properly (meaning we don’t know
                       where money should be prioritized), and the U.S.
                       has a rich history of failing to police subsidy
                       fraud by the nation’s biggest telecom monopolies.


          o § Digital Restrictions (DRM)⠀➾


                # ⚓ NPR ☛ ‘It’s_the_Great_Pumpkin,_Charlie_Brown’_won’t_be_on
                  TV_this_year._Here’s_how_to_watch⠀⇛


                       In a partnership with the streamer, PBS was allowed
                       to air the specials in 2021. But this year, PBS no
                       longer has the rights, and confirmed that it won’t
                       be airing the Halloween show.


                # ⚓ TechSpot ☛ Report_claims_almost_half_of_systems_are
                  ineligible_for_Windows_11_upgrades⠀⇛


                       In the initial run-up to Windows 11′s launch, much
                       controversy centered on its requirement for TPM 2.0
                       and a relatively recent CPU. Microsoft wants users
                       to have at least an Intel 8th gen or AMD Zen 2
                       processor, but the company is considering allowing
                       some Intel 7th-gen chips and Zen 1s. According to
                       Lansweeper, those requirements are the main
                       roadblock for ineligible systems.


                       The RAM requirement isn’t a problem – 92 percent of
                       surveyed workstations meet it. Conversely, only 57
                       percent of systems meet the CPU requirement, and 64
                       percent have the necessary TPM. Many of those
                       workstations may be compatible with TPM 2.0 but
                       might not have enabled it since some systems must
                       engage the feature manually. While not great, these
                       numbers represent a 12 percent improvement over
                       2021.


          o § Monopolies⠀➾


                # § Copyrights⠀➾


                      # ⚓ Torrent Freak ☛ MPA_Piracy_Investigation_Led_to
                        Property_Linked_to_Other_Crimes_Worth_Billions⠀⇛


                             PrimeWire had outmaneuvered several Hollywood
                             blocking injunctions but a complaint in the
                             U.S. tightened the screws and never let go.
                             An MPA investigation to identify the site’s
                             operator was met with sophisticated
                             countermeasures. It ended at a UK property
                             that has been publicly linked to other crimes
                             worth billions of dollars. And a site called
                             KickassTorrents.


                      # ⚓ Torrent Freak ☛ Megaupload_Pair_Remain_RIAA_&_MPA
                        Lawsuit_Targets_Despite_Guilty_Pleas⠀⇛


                             Earlier this year, former Megaupload
                             executives Mathias Ortmann and Bram van der
                             Kolk signed a plea deal to avoid extradition
                             to the United States. They remain defendants
                             in civil lawsuits filed by the MPA and RIAA
                             but their guilty pleas will undoubtedly
                             change the course of these battles.


    * § Gemini* and Gopher⠀➾


          o § Personal⠀➾


                # ⚓ Feeling_so_depressed.⠀⇛


                       Feeling so depressed.


                       I can’t even handle the smallest problems in my
                       life


          o § Technical⠀➾


                # ⚓ DevTerm_R-01_Review⠀⇛


                       I have been interested in RISC-V for a while. I
                       played a bit with RISC-V cores on FPGA but it is
                       quite hard to find RISC-V cores on general-purpose
                       computers. Fortunately, the company ClockworkPi is
                       making the DevTerm R-01[1], a handheld computer
                       with a RISC-V processor. The DevTerm R1 is a
                       handheld computer featuring an RV64IMAFDCVU single
                       core RISC-V processor.


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


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