𝕿𝖊𝖈𝖍𝖗𝖎𝖌𝖍𝖙𝖘 Bulletin for Saturday, December 25, 2021
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Generated Sun 26 Dec 02:40:20 GMT 2021
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/2021/12/25/


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

QmbcSKJZiufM2XfxxnB92wQE323m9MsiAMo5YFmwFPu8VN
QmWmC3AgPDFykej4Jpay8nnpKKVGMMEfv8MjggJ9Qw5Lm9
QmNPpiGqWGFQNEECwNKt72LSricBeH2VuxSzamtqTrESHV
QmUNbGTar72DvDUxqyAh6Jdzfvu4JHUqaTFKqWpu9P64bk
QmXFHTXDoB1z4Af36pfwwfrJcsZWK4nkbqqF2EndNyJ9YA
QmcFJuq5PVoQgjijbLaMibeRbgr4h5pFcpzBYWqLap1cNu
QmSBPP9JjsnqHFcHfShHhNsQwTc4eHz66BF5ihuBjaxpKM
QmPrW6GPykdMyQtg8txV2DXmRYjghF1TQPv6CJcwVUx6iy
QmVNVya1kBzPYdBdJekAHEpk3m857qJsnQet4QNxycPonK
QmUQmiHKq1wV1JkS2hwXEf1etD3jM43Afc68V6omxpqaCi
QmbheKKYhpyw3SvZSmHbavdgcv1PSaRtPdaQ746tgUFXbp
QmZqoL722Mk7hDJcSz2oBxydXY7S2G5r4ByHgRdKBw7dUH
QmTZSRzHEP7hwr4tL7ky23HxA9Z3tUBjQrfwYPAieSYsGe
QmRKnWzns1EsWaoxUGG5KEEhiyhSELGYVdfVqSB5Ay4qhc
QmZdVw5kGwFeXoGNexLWZDS8MJxeGKAeup4JpiAnmE946y
QmYdAKSfCu9puR3SWGWwGW19Uzo1jj4L1aYpeGqpC5FCp6
QmZP4LnsTHrznHVdokybFZafDLWJqRPKyvvberG4dVRasQ
QmfL4kPLDVa6CGiB1yXozobGCQ5fTPahv8T7hPstsxvkie
QmP2Y5UBBDqcCEECSknvoPU6WVjL4V93dQMbKj34jYv56N
QmTTqHT1gu1SWzp4ejhsMMHeEr1cnaDHeAQnawx8obkmR5
QmVjqVfy1bLLkjPbYtUagNsq6vfXCToNocEPu1s2Cvo3ab


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

⦿ IRC Proceedings: Friday, December 24, 2021 | Techrights

⦿ [Meme] Lawyers With Their Letters... | Techrights

⦿ Stupid and Legally-Invalid Threats (Physical Letters) | Techrights

⦿ The Car Drives You  -  Part V  -  Part of the Coming War on General Computation | Techrights

 ䷼ Bulletin articles (as HTML) to comment on (requires login):
		http://techrights.org/2021/12/25/irc-log-241221/#comments
		http://techrights.org/2021/12/25/sending-a-letter/#comments
		http://techrights.org/2021/12/25/timewasting-letters/#comments
		http://techrights.org/2021/12/25/war-on-generalised-things/#comments

 ䷞ Followed by Daily Links (assorted news picks curated and categorised):
		http://techrights.org/2021/12/25/rescuezilla-2-3/#comments
		http://techrights.org/2021/12/25/ubuntu-vs-arch-linux/#comments

                    ䷩ 𝚕𝚒𝚗𝚎 57

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

(ℹ) Images, hyperlinks and comments at http://techrights.org/2021/12/25/irc-log-241221/#comments
    Gemini version at gemini://gemini.techrights.org/2021/12/25/irc-log-241221/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 12.25.21⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ IRC_Proceedings:_Friday,_December_24,_2021⠀✐


Posted in IRC_Logs at 2:32 am by Needs Sunlight


Also available via the Gemini protocol at:


    * gemini://gemini.techrights.org/irc-gmi/irc-log-techrights-241221.gmi
    * gemini://gemini.techrights.org/irc-gmi/irc-log-241221.gmi
    * gemini://gemini.techrights.org/irc-gmi/irc-log-social-241221.gmi
    * gemini://gemini.techrights.org/irc-gmi/irc-log-techbytes-241221.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
 QmacZANdVbnvbBKiuTNTRKwhJMUw2DkdSqZ5bRKrVjmk5a #boycottnovell  🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
                                                 (full IRC log
                                                 as HTML)


                                                 IRC log for
                                                 #boycottnovell
 QmUUpWf4xMWtGYu5aUkYbSDMM9ADHH4gwS1i66CFT6iqqE (full IRC log   🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
                                                 as plain/ASCII
                                                 text)


                                                 IRC log for
                                                 #boycottnovell-
 Qmb4ppemFnJ7fezAHRYYr2JhdDoGc9dsVCK7xKv6S1mbCp social          🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
                                                 (full IRC log
                                                 as HTML)


                                                 IRC log for
                                                 #boycottnovell-
 QmehKSybEjnY9VDsD5TpWKMwALR4YgBbHTMUXUCKRkYv83 social          🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
                                                 (full IRC log
                                                 as plain/ASCII
                                                 text)


                                                 IRC log for
 QmNfN8L5DRprpcoLgbvAU7CRQ7M1eS5cA7hgSsaYuWJRHB #techbytes      🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
                                                 (full IRC log
                                                 as HTML)


                                                 IRC log for
                                                 #techbytes
 QmVEzxTasi2u2kshZSkJ2YBCPnZQ2Eqem2WxsDK5FPXvB7 (full IRC log   🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
                                                 as plain/ASCII
                                                 text)


                                                 IRC log for
 QmQT3groARyBQfsMyZvmwSSQaac45tMJpHPrBgXRyjohit #techrights     🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇HTML5 logs⦈
                                                 (full IRC log
                                                 as HTML)


                                                 IRC log for
                                                 #techrights
 QmRH1B9GVpbVJpw2BDa1idbKcAjxWrpX4f377qAcCe9VEs (full IRC log   🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇text logs⦈
                                                 as plain/ASCII
                                                 text)


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇IPFS logo⦈


§ Bulletin for Yesterday⠀➾


Local_copy | CID (IPFS): QmVjqVfy1bLLkjPbYtUagNsq6vfXCToNocEPu1s2Cvo3ab



                    ䷩ 𝚕𝚒𝚗𝚎 184

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

(ℹ) Images, hyperlinks and comments at http://techrights.org/2021/12/25/sending-a-letter/#comments
    Gemini version at gemini://gemini.techrights.org/2021/12/25/sending-a-letter/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 12.25.21⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ [Meme]_Lawyers_With_Their_Letters…⠀✐


Posted in Humour at 6:38 pm by Dr. Roy Schestowitz


                  🄸🄼🄰🄶🄴_🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽_⦇You_sent_a_letter,_eh?⦈_
Disbarring firms that misuse_their_power is always a possibility, albeit it is
                                time-consuming


Summary: Sending a letter is too easy — especially when you acquire a licence
to pointlessly and/or baselessly threaten people


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡀⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠀⣀⣰⣦⣄⣠⣤⣄⢀⣀⢠⢀⣀⣀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⡤⠶⠶⢤⣄⡀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣀⣀⠀⠀⠀⠀⣶⠶⣶⠴⠶⢶⡆⠀⠀
⠀⡀⡮⣿⣉⣿⣀⣶⣸⣰⣚⢸⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠀⣯⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠻⠇⠀⠀⠀⣿⣀⣿⣿⣿⣿⡇⠀⠀
⢸⣿⠛⢉⣉⠁⠋⣭⠉⠈⣍⡈⠈⡉⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡽⡭⠉⠉⠉⠉⠁⠀⠀
⠈⠷⠀⢸⠟⠀⢀⡄⠀⢨⡋⠁⠀⠈⠈⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠃⢿⠀⢸⣿⣸⡇⣿⣷⣾⣿⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠫⢗⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠰⣟⠀⠈⠀⠀⠘⠟⠀⠀⣴⡀⠀⠀⠀⠀⠀⠀⢰⣷⠂⠀⠀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣀⠀⠀⣀⣀⢀⢀⠀⠀⠀⠀⢀⠯⠀⠄⠀⠈⠈⠺⠿⣿⢏⣴⣾⣿⣷⣄⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⣤⠞⠁⠈⠀⠀⠀⠀⢀⣾⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠻⡁⠸⠿⢟⣦⠀⠀⢲⣿⠆⠤⠀⠀⠀⠀⡀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢠⡨⠉⠉⠅⠐⠀⠀⠀⠃⠀⠀⣠⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣾⣷⣦⣶⣾⣷⣤⣤⣼⠉⠀⠀⠀⠀⠀⠀⢧⣀⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡈⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣟⠀⠀⠊⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣄⣀⣀⣀⣤⣀⣴⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⡿⣿⡆⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⢅⣄⣿⠃⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢉⣽⠀⢀⣀⡀⢀⢀⣤⣶⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⡿⠿⠿⠿⠿⠿⠿⠟⠈⠉⠙⢿⣿⣿⡿⠿⠿⠟⣄⣶⣿⣿⣀⣿⣿⣿⣿⠃⠈⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⢲⣮⣿⣿⣿⣿⣿⣿⣏⣿⣿⡀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣷⣶⣶⡇⣀⠀⡀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⡇⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⣿⣿⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⣿⣿⡇⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⡇⣴⣿⣿⣿⣿⣿⣿⣿⡇⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣾⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⡅⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠻⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠀⠀⠛⠛⠛⠛⠛⠛⠛⠁⠘⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠃⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡀⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠀⣀⣰⣦⣄⣠⣤⣄⢀⣀⢠⢀⣀⣀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⡤⠶⠶⢤⣄⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢐⣧⣀⣀⠀⠀⠀⠀⣶⠶⣶⠴⠶⢶⡆⠀⠀
⠀⣀⡮⣿⣉⣿⣀⣶⣸⣰⣚⢸⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠀⣯⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠻⡇⠀⠀⠀⣿⣀⣿⣿⣿⣿⡇⠀⠀
⢰⣿⠛⢉⣉⠁⠫⣭⠉⠈⣍⡈⠈⠉⠉⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣄⣭⡉⠉⠉⠉⠉⠁⠀⠀
⠈⠵⠀⠸⠟⠀⢀⡀⠀⢼⡋⠁⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠃⢿⠀⢸⣿⣸⡇⣿⣷⣾⣿⠀⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠡⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠠⡿⠀⠈⠀⠀⠘⠟⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⢠⣷⠂⠀⠀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀
⡤⢀⠀⠀⣀⣀⢀⢀⠄⠀⠀⠀⢀⠯⠀⠄⠀⠀⠉⠲⠿⣿⢛⣴⣾⣿⣷⣄⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⣤⠞⠁⠈⠈⠀⠀⠀⢀⣾⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠻⡂⠘⠿⠻⡦⠄⠀⢲⣿⠆⠤⠀⠀⠀⠀⡀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢠⡨⠉⠉⠅⠐⠀⠀⠀⠃⠀⠀⣠⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣾⣷⣶⣶⣾⣷⣤⣤⣼⠉⠀⠀⠀⠀⠀⠀⢿⣀⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⢇⠀⠀⠀⠀⠀⠀⡈⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣟⠀⠀⠊⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣄⣀⣀⣀⣤⣀⣴⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⡿⣿⡆⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢅⣄⣻⠃⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣉⢹⠀⢀⣀⡀⠀⢀⣤⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⡿⠿⠿⠿⠿⠿⠿⠟⠈⠉⠙⢿⣿⣿⡿⠿⠿⠟⣴⣾⣿⣿⣀⣿⣿⣿⣿⠃⠉⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⣲⢔⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⡇⣺⣿⣿⣶⣦⣤⣶⡇⣀⢀⡀⣠⣴⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⠁⣾⣿⣿⣿⣿⣿⣿⡇⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣿⣿⣿⣿⡇⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⡇⣴⣿⣿⣿⣿⣿⣿⣿⡇⢬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣾⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⣿⣿⡇⡨⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠀⠀⠛⠛⠛⠛⠛⠛⠛⠁⠘⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣀⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠀⣀⣰⣦⣄⣠⣤⣄⢀⣀⢠⢀⣀⣀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂⣡⡤⠶⠶⢤⣄⡀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣀⣀⠀⠀⠀⠀⣶⠶⣶⠴⠶⢶⡆⠀⠀
⠀⣀⡮⣿⣉⣿⣀⣶⣸⣰⣚⢸⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠀⣏⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠛⠏⠀⠀⠀⣿⣀⣿⣿⣿⣿⡇⠀⠀
⢰⣿⠛⢉⡉⠁⠋⣭⠉⠈⣍⠈⠈⠉⠉⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣝⣭⠉⠉⠉⠉⠁⠀⠀
⠈⠵⠀⠸⠟⠀⢀⡄⠀⢬⡋⠁⠀⠈⠉⠉⠁⠀⠠⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠃⢿⠀⢸⣿⣸⡇⣿⣷⣾⣿⠀⣿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⡹⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠰⡿⠀⠈⠀⠀⠸⠟⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⢀⣾⡂⠀⠀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡤⢀⠀⠀⣀⣀⢀⢀⠀⠀⠀⠀⠀⡧⠀⠄⠀⠀⠚⠾⢿⣿⢋⣵⣾⣿⣷⣄⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⣤⠞⠁⠈⠈⠀⠀⠀⢀⣾⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠻⡂⠸⠿⢿⣦⠄⠀⢲⣿⠦⠤⠀⠀⠀⠀⡀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢢⡬⠉⠉⠅⠐⠀⠀⠀⠃⠀⠀⣠⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣶⣾⣷⣦⣶⣾⣶⣤⣤⣼⠉⠀⠀⠀⠀⠀⠀⢷⣀⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡈⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣏⠀⠀⠊⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣄⣀⣀⣀⣤⣀⣴⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⡿⣿⡆⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢅⣄⣻⠃⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣠⣄⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠉⣹⠀⣀⣀⡀⠀⢀⣤⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣬⣿⠿⠿⠿⠿⠿⠿⠟⠈⠉⠙⢿⣿⣿⡿⠿⠿⠟⣴⣾⣿⣿⣀⣿⣿⣿⣿⠋⠈⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⡲⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣾⣦⣤⣴⡆⢀⢀⡀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⠁⣻⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⣿⣿⡇⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⡇⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣾⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⣿⣿⡇⠼⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠀⠀⠛⠛⠛⠛⠛⠛⠛⠁⠘⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣰⣧⣀⣠⣤⣄⢀⣀⢠⢀⣀⣀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢊⣡⡤⠶⠶⢤⣄⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⣁⣀⠀⠀⠁⠀⣶⠶⣶⠴⠶⢶⡆⠀⠀
⠐⣠⡮⣿⣩⣿⣀⣶⣸⣰⣚⢸⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠐⡧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠙⢿⠁⠀⠀⠀⣿⣀⣿⣿⣿⣿⡇⠀⠀
⢼⡿⠛⣉⡉⠁⢫⠍⠉⢨⣍⠈⠈⣉⣉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣯⡩⠉⠉⠉⠉⠁⠀⠀
⠀⠂⠀⣻⠃⠀⣠⠀⠀⣌⠉⠀⠀⠀⠀⠉⠓⠀⠠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢿⠀⢸⣿⣸⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠫⢮⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠻⠇⠀⠀⠀⠀⠛⠃⠀⢠⣦⠀⠀⠀⢀⠀⠀⣀⣬⣷⠀⠀⡀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⡈⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀
⣴⡄⠀⢀⣀⡀⡄⡠⠀⠀⠀⠀⠸⠄⠐⠄⠀⠈⠙⠿⣿⡿⣫⣷⣿⣿⣦⡀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⣤⠞⠁⠈⠇⠀⠀⠀⣠⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⡇⠀⠻⠟⣻⠆⠀⠐⣾⡿⠠⠆⠀⠀⠀⢀⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣦⠨⠉⠉⠅⠐⠀⠀⠰⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡆⠀⠀⢀⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣶⣶⣾⣿⣶⣤⣴⡏⠁⠀⠀⠀⠀⠀⠘⣇⣠⡀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⢸⡀⠀⠄⠀⠀⠀⢀⠁⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣁⠀⠐⠁⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⣤⣤⣄⣦⣾⣿⣿⣿⣿⣿⡟⢿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⡿⠒⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⣰⣸⡟⠈⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⢀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣀⣀⡟⠀⣀⣀⣀⣀⣠⣴⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⡿⠿⠿⠿⠿⠿⠿⠃⠀⠉⠹⠿⣿⣿⠟⠻⠿⠣⣾⣿⣿⣇⣰⣿⣿⣿⡟⠁⠘⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠆⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢸⣿⡇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⣿⣿⣿⣷⣶⣿⠀⣀⣠⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⡇⣺⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⢠⣾⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢨⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠃⠀⠘⠛⠛⠛⠛⠛⠛⠋⠀⠛⠛⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠃⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣀⣀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣰⣦⣀⣠⣤⣤⢠⣄⢠⢀⣄⣄⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢊⣡⡤⠶⠶⢤⣄⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣤⣀⠀⠉⠁⠀⣶⠶⣶⠴⠶⢶⡆⠀⠀
⢠⣤⡮⣿⣉⣿⣀⣶⣸⣰⣚⢸⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠘⡰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⠿⠀⠀⠀⠀⣿⣀⣿⣿⣿⣿⡇⠀⠀
⢿⠛⢉⣍⠉⠁⠯⠉⠉⣬⡉⠈⠈⣉⣉⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣿⣭⡏⠉⠉⠉⠁⠀⠀
⠘⠀⠘⣟⠀⢀⡄⠀⢰⣏⠀⠀⠀⠀⠈⠉⠀⠈⠠⠄⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠇⢿⠀⢸⣿⣸⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠻⣭⢱⠁⠀⠀⠀⠀⠀
⠀⡀⠘⠿⠀⠀⠀⠀⠈⠛⠀⠀⣸⡆⠀⠀⠀⢀⡀⠀⣠⣿⣷⡀⣀⡀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀
⣤⠀⠀⣀⣀⣀⠀⠄⠀⠀⠀⠀⠿⠀⠲⠀⠈⠈⠛⠻⢿⢿⣿⣾⣿⣿⣦⡀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⣤⠞⠁⢸⠀⠀⠀⢀⣼⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⡄⠘⠛⢛⡷⠂⠀⢰⣿⡗⠴⠀⠀⠀⠀⣠⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠰⡆⠨⠉⠉⠅⠐⠀⠀⠄⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⣴⣦⣴⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣶⣶⣿⣷⣶⣦⣾⡋⠀⢀⠀⠀⠀⠀⠻⣤⣤⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⢇⠀⠀⠀⠀⠀⠀⡐⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡁⠀⠉⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣴⣶⣴⣴⣿⣿⣿⣿⣿⣿⠛⣿⣿⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⠻⠵⠒⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⣶⣼⡏⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣠⣸⡇⢠⣤⣀⣀⣀⣴⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠿⠛⠿⠿⠿⠛⠛⠁⠀⠈⠻⠿⣿⠿⠛⡛⣛⣰⣾⣿⣿⣥⣼⣿⣿⣿⡇⠀⠙⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢗⣶⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣾⣿⣾⠠⣤⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⡇⣺⣿⣿⣿⣿⣿⣿⣿⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⠀⢾⣿⣿⣿⣿⣿⣿⣿⠘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⡿⣰⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠻⠿⠿⠿⠛⠛⠛⠛⠛⠛⠁⠀⠘⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠃⠛⠛⠛⠛⠛⠛⠛⠛⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣀⣀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣰⣦⣀⣠⣤⣤⢠⣀⢠⢠⣤⣄⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢋⣠⡤⠶⠶⢤⣄⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣠⡀⠀⠉⠁⠀⣶⠶⣶⠴⠶⢶⡆⠀⠀
⢰⣤⣮⣿⣩⣿⣀⣶⣸⢰⣚⢸⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠘⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠻⠏⠀⠀⠀⠀⣿⣀⣿⣿⣿⣿⡇⠀⠀
⠿⠋⢩⣉⠉⠁⠯⠉⠉⣬⡉⠈⠈⣉⣩⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣶⣿⡟⢩⠉⠉⠉⠁⠀⠀
⠘⠀⢹⡋⠀⢠⠄⠀⢰⣍⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠇⢿⠀⢸⣿⣼⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣖⡊⠀⠀⠀⠀⠀⠀
⠀⠀⠘⠋⠀⠀⠀⠀⠘⠋⠀⠀⣿⠄⠀⠀⢀⡀⣀⢀⣰⣾⣇⢀⣀⡀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀
⣤⠀⢀⣄⣀⡠⠤⠀⠀⠀⠀⠰⠇⠀⠳⠀⠀⠉⠛⠻⠿⢃⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⣤⠞⠁⠈⠀⠀⠀⢠⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⡄⠙⠛⢛⡷⠀⠀⢺⣿⠓⠖⠀⠀⠀⠀⣄⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢰⡂⠨⠉⠉⠅⠐⠀⢀⠁⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣶⣶⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣶⣾⣿⣷⣶⣶⣿⣁⠀⣀⠀⠀⠀⠀⠣⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⢇⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠁⢀⣀⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣴⣶⣦⣾⣿⣿⣿⣿⣿⡿⠻⣿⡿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡿⠣⣦⣽⠁⠁⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⠄⢻⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢀⣤⣾⠀⣠⣤⣄⣀⣠⢶⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠟⠛⠛⠛⠛⠛⠛⠀⠀⠈⠛⠿⠿⠟⠛⡛⠃⢽⣿⣿⣿⣤⣿⣿⣿⣿⠁⠀⠻⣿⣿⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢓⣲⣤⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⠀⣿⡇⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⡗⢠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⡗⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⠰⣿⣿⣿⣿⣿⣿⣿⡇⢘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⡇⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⡇⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⡿⢨⣿⣿⣿⣿⣿⣿⣿⣿⡧⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣸⣿⣿⣿⣿⣿⣿⡇⢘⣿⣿⣿⣿⣿⣿⣿⣿⠅⣿⣿⣿⣿⣿⣿⣿⣿⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠻⠿⠿⠿⠛⠛⠛⠛⠛⠛⠀⠀⠙⠛⠛⠛⠛⠛⠛⠁⠘⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣀⣀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣰⣤⣄⣰⣤⣤⢠⣀⢠⢠⣤⣄⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢋⣠⡤⠶⠶⢤⣄⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣤⣤⡀⠀⠉⠉⠀⣶⠶⣶⠴⠶⢶⡆⠀⠀
⣠⣤⣮⣿⣉⣿⣀⣶⣸⣰⣚⢸⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠘⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠻⠃⠀⠀⠀⠀⣿⣀⣿⣿⣿⣿⡇⠀⠀
⠙⠋⣩⣉⠉⠁⠏⠉⢩⣮⡉⠈⠨⢍⣉⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣿⣷⣿⣍⠉⠉⠉⠉⠁⠀⠀
⠀⠀⢹⡃⠀⠤⠄⠀⣦⣁⠀⠀⠀⠀⠉⠉⠀⠐⠂⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠃⢿⠀⢸⣿⣸⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠻⡮⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠘⠃⠀⠀⠀⠀⠙⠉⠀⠀⡷⠄⠀⠀⢀⣀⣀⣀⣤⣿⢦⣀⣤⡀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⠀⠀⢀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀
⡄⠀⢀⣤⣀⠄⠄⠀⠀⣀⠀⠸⠃⠀⠃⠀⠀⠑⠛⠻⠿⣫⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⠈⠳⢦⣀⣀⣀⣀⣤⠞⠁⡌⠀⠀⠀⣠⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢑⡀⠙⠛⢻⠗⠀⠀⢻⣿⠂⠆⠀⠀⠀⠀⡄⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢰⡂⠨⠉⠉⠅⠐⠀⠠⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣶⣶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣶⣿⣿⣷⣶⣶⣿⡁⠀⣀⠀⠀⠀⠀⠣⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣇⠀⠂⠀⠀⠀⠀⢁⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡀⠈⠁⣀⣠⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣶⣶⣷⣶⣿⣿⣿⣿⣿⣿⡟⠻⣿⡿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠋⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠲⢦⣿⠁⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣔⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢀⣤⣿⠀⣴⣤⣄⣤⣴⣶⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⠛⠛⠛⠛⠛⠉⠋⠀⠀⠈⠛⠻⠿⠋⠉⣉⡁⢺⣿⣿⣿⣤⣿⣿⣿⣿⠁⠀⠻⣿⡿⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣖⣦⣤⣀⠀⢀⡀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⢀⣿⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⡇⣺⣿⣿⣿⣿⣿⣿⡇⣴⣶⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⡇⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣾⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⠣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠀⠀⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣀⣀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣰⣤⣄⣰⣤⣤⢠⣀⢠⢠⣤⣀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢋⣠⡤⠶⠶⢤⣄⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣤⣤⡀⠀⠈⠡⠀⣶⠶⣶⠴⠶⢶⡆⠀⠀
⢠⣤⣮⣿⣉⣿⣀⣶⣸⣰⣚⢸⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠙⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠛⠁⠀⠀⠀⠀⣿⣀⣿⣽⣿⣿⡇⠀⠀
⠙⠋⣩⡉⠉⠁⠋⠉⢩⣬⠉⠈⠬⢍⣩⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⣿⣷⡯⠭⠍⠉⠉⠉⠁⠀⠀
⠀⠀⢹⡁⠀⠤⠀⠀⣮⡁⠀⠠⡀⠀⠈⠁⠀⠀⠃⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠃⢿⠀⢸⣿⣸⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠟⠟⡟⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠘⠁⠀⠀⠀⠀⠙⠁⠀⢠⡿⠆⠀⠀⢀⣀⣀⣀⣼⣿⠆⢂⣤⡀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⠀⠀⢀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀
⡄⠀⢀⣤⣤⠤⠄⠀⢀⣀⠀⠘⠂⠈⠓⠀⠀⠉⠛⠛⠟⣫⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⠈⠳⢦⣀⣀⣀⣀⣤⠞⠁⡌⠀⠀⠀⣠⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠀⠛⠋⣹⠓⠀⠀⣻⡿⠓⠂⠀⠀⠀⠀⡄⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢰⡂⠨⠉⠉⠅⠐⠀⠠⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⡀⠀⢀⣾⣶⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣷⣿⣿⣷⣶⣶⣏⡀⢀⡀⠀⠀⠀⠀⠣⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣇⠀⠀⠀⠀⠀⠀⢁⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⡀⠈⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣶⣶⣷⣶⣿⣿⣿⣿⣿⣿⡟⠻⣿⡿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠃⠁⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠲⢦⣿⠁⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣀⣠⣿⠀⣴⣤⣄⣤⣤⢾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠛⠛⠛⠉⠉⠀⠀⠈⠛⠻⠿⠋⠉⣭⣃⣶⣿⣿⣿⣴⣿⣿⣿⣿⠀⠀⠻⣿⡿⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣓⣴⣦⢀⡀⠀⣀⡄⠀⠀⠀⠀⠀⣰⣾⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⢠⣿⡇⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⡇⣠⣤⣴⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⡇⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣾⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⢘⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠀⠀⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣠⣀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣰⣤⣄⣰⣤⣠⢠⣄⢠⢠⣤⣀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢋⣠⡤⠶⠶⢤⣄⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣤⣄⣀⠀⠉⠡⠀⣶⠶⣶⠴⠶⢶⡆⠀⠀
⣰⣤⣮⣿⣉⣿⣀⣶⣸⣰⣚⢸⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠀⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠿⠁⠀⠀⠀⠀⣿⣀⣿⣽⣿⣿⡇⠀⠀
⠛⠉⣭⡉⠉⠱⠋⠉⢩⣬⠉⠈⠬⠍⣭⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⣷⣿⠭⠏⠉⠉⠉⠁⠀⠀
⠀⠀⣿⡁⠀⡤⠀⠐⣦⡁⠀⠠⡀⠀⠀⠀⠀⠐⠓⠂⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠇⢿⠀⢸⣿⣼⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⡗⠆⠀⠀⠀⠀⠀⠀
⠀⠀⠘⠁⠀⠀⠀⠈⠉⠁⠀⢰⠷⠄⠀⠀⣀⣄⣀⢀⣾⣯⡧⣤⣤⡀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀
⡆⠀⣠⣤⣄⠢⠄⠀⢀⡀⠀⠞⠂⠈⠃⠀⠀⠉⠛⠛⠛⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⣤⠞⠁⠈⠀⠀⠀⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢉⠈⠛⢉⣟⠃⠀⠈⣿⡟⠒⠃⠀⠀⠀⢰⡄⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⢰⠀⠨⠉⠉⠅⠐⠀⡐⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⢰⣿⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⣶⣾⣏⡀⢀⡀⠀⠀⡀⠘⢣⣤⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣈⣆⠀⠀⠀⠀⠀⠐⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠈⢁⣀⣼⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣷⣶⣿⣿⣿⣿⣿⣿⠏⠻⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠓⠋⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠲⠽⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣦⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣤⣤⣿⠀⣶⣦⣤⣤⡴⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣟⠛⠛⠛⠛⠋⠉⠉⠀⠀⠈⠙⠻⠟⠉⣉⣍⣹⣾⣿⣿⣷⣾⣿⣿⣿⣏⠀⠈⢻⣿⡇⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣧⣴⣤⣄⣀⣠⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⢸⣿⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣿⣿⣿⣿⣿⣿⡇⣾⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠃⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⠡⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠃⠀⠀⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠋⠈⠛⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣤⣀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣰⣤⣀⣰⣤⣤⢠⣀⢠⢠⣤⣄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢃⣠⡤⠶⠶⢤⣄⡀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣶⣶⣤⠀⠀⠈⠀⠀⣶⠶⣶⠴⠶⢶⡆⠀⠀
⣴⣤⡮⣿⣉⣿⣀⣶⣸⣰⣚⢸⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠐⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⡀⠀⣿⣀⣿⣿⣿⣿⡇⠀⠀
⠈⢠⣄⠉⠉⠁⠉⠉⣭⡌⠉⠈⠈⠭⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⡯⠩⠏⠋⠉⠉⠁⠀⠀
⠀⠈⣏⠀⠠⠆⠀⢰⣤⠀⠈⢦⡀⠀⠀⠀⠀⠙⠒⠐⢲⡄⡀⠀⠀⠀⠀⠀⠀⠀⠃⢿⠀⢸⣿⣸⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠿⢿⡛⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠛⠀⠀⠀⠀⠈⠁⠀⠀⣼⠳⣀⠀⠀⢤⢄⣀⣰⣿⣿⣂⣤⣤⡀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⠀⠀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠛⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣤⣤⡀⠀⠀⠀⣀⡀⠀⠋⠀⠉⠃⠀⠀⠈⠉⠛⢛⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⣤⠞⠁⠈⠀⠀⢀⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣀⠈⠉⣩⡟⠁⠀⢸⣿⠃⠛⠀⠀⠀⠀⣼⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⢧⠀⠨⠉⠉⠅⠐⢀⠂⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣼⣷⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣷⣯⣄⣀⣄⠀⠀⡀⠀⠘⢢⣴⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣼⣄⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⢀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⠋⢻⣿⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠚⠉⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠸⠶⡇⠀⠀⠀⠀⠀⠀⢀⣠⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡘⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⣿⣿⣿⣿⣿⣿⣿⠇⣠⣠⣼⡇⢰⣶⣦⣤⣤⠾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠛⠋⠁⡈⣉⡸⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⠈⣿⡿⠃⠀⠨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣷⣖⣦⣤⣠⣄⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡀⣿⡇⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⢰⣮⣴⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡷⣾⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⡟⢀⣻⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠁⠀⠘⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠃⠙⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⣤⣀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣰⣤⣀⣡⣤⣤⢠⣠⢠⢠⣤⣄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢃⣠⡤⠶⠶⢤⣄⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⢶⣴⣄⠀⠀⠁⠀⠀⣶⠶⣶⠴⠶⢶⡇⠀⠀
⣴⣦⡮⣿⣉⣿⣀⣶⣸⣰⣚⣸⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⣀⢠⣤⡄⣿⣀⣿⣿⣿⣿⡇⠀⠀
⠀⢠⣄⠉⠉⠁⠉⠉⣭⡌⠉⠈⠈⠭⢭⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣯⠭⠏⠉⠍⠉⠁⠀⠀
⠀⢘⡅⠀⠰⠂⠀⢣⣤⠀⠘⣧⡀⠀⠀⠀⠉⠛⠋⠂⣲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠇⢿⠀⢸⣿⣸⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⠿⢿⠽⠁⠀⠀⠀⠀⠀⠀
⠀⠈⠃⠀⠀⠀⠀⠈⠁⠀⠀⡿⠻⣀⠀⠀⢴⣦⣤⣴⣿⡿⣐⣴⣦⡀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣴⣤⠀⠀⠂⢀⣀⡀⠈⠋⠀⠙⠃⠀⠀⠈⠉⠻⣫⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⣤⠞⠁⠈⠀⠀⣠⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣀⠈⠉⣩⡟⠀⠀⢹⡿⠃⠋⠀⠀⠀⠀⡾⠄⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⣇⠀⠨⠉⠉⠅⠐⠠⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣾⣿⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣷⣿⣄⣀⣀⠀⠀⣀⠀⠙⣶⣾⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣼⣄⡀⠀⠀⢀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⢀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣟⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠉⢻⡿⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⠛⠁⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠻⢾⡇⠀⠀⠀⠀⠀⠀⢀⣴⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⢇⣠⣄⣾⡇⣴⣷⣦⣴⡶⠿⣿⣿⣿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⡁⠈⠉⠉⠉⠉⠀⠀⠀⠈⠉⠛⠉⠀⣈⢡⣽⣿⣿⣿⣷⣿⣿⣿⣿⣄⠀⠙⣿⡿⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢉⣷⣦⣦⣠⣤⡀⣄⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣿⡇⠀⠀⠈⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⡟⣸⣿⣿⣿⣿⣿⣿⣿⢰⣷⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⠇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⡟⢸⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⡇⠠⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠀⠀⠘⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠃⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢠⣄⣀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣰⣤⣌⢠⣤⣤⢠⣤⢠⢠⣤⣄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⣠⡤⠶⠶⢤⣄⡀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⣷⣦⣄⠀⠀⠑⠀⠀⣶⠶⣶⠴⠶⢶⡆⠀⠀
⣴⣶⡮⣿⣉⣿⣀⣶⣸⢰⣚⣸⢸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢀⣤⣤⣦⡄⣿⣀⣿⣿⣿⣿⡇⠀⠀
⠀⣤⡄⠉⠉⠑⠉⢹⣿⠈⠉⠈⠈⠭⠭⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣷⣿⣋⣟⣭⠉⠉⠁⠀⠀
⠀⢹⡀⠀⠴⠀⠀⣣⡄⠀⠹⣦⡀⠀⠀⠀⠀⠑⠀⠀⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠃⢿⠀⢸⣿⣸⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⡚⠘⠋⠀⠀⠀⠀⠀
⠀⠙⠂⠀⠀⠀⠀⠈⠁⠀⢠⠟⠛⣀⠀⠠⢴⣤⣤⣾⣿⣷⣢⣶⣦⡀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠓⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣦⣄⠒⠐⠀⣀⣀⠀⠘⠁⠈⠙⠁⠀⠀⠈⠉⠛⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢈⠳⢤⣀⣀⣀⣀⣤⠞⠁⠈⠀⠀⣰⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠉⢉⣽⠋⠁⠀⣿⠿⠑⠃⠀⠀⠀⢀⡿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠘⡇⠀⠨⠉⠉⠅⠀⡐⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⢠⣿⣿⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⡀⠀⢀⡀⠀⠓⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣵⣀⡀⠀⠀⣀⡀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣟⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡟⠈⣿⡟⠟⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⠋⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠑⠷⢾⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⣿⣿⣿⣿⣿⣿⡿⣀⣠⣶⣿⠀⣾⣶⣤⣦⡼⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⡀⠀⠀⠉⠙⠉⠀⠀⠀⠈⠉⠋⠁⠀⢤⣶⣿⣿⣿⣿⣶⣿⣿⣿⣿⡀⠀⠙⣿⠇⠀⢸⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣶⣤⣄⣤⡄⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣸⣿⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⡇⣼⣧⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⡇⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⡇⢰⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡯⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣾⣿⣿⣿⣿⣿⣿⠇⣸⣿⣿⣿⣿⣿⣿⣿⣿⠐⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠋⠀⠀⠛⠛⠛⠛⠛⠛⠛⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣤⣄⣀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣁⣰⣤⣀⢠⣤⣤⢠⣤⢠⢠⣤⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⢀⣠⡤⠶⠶⢤⣄⡀⠃⢀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼⢶⣶⣦⡄⠀⠈⠀⠀⠀⣶⠶⣶⠶⠶⢶⡆⠀⠀
⣴⡶⡮⣿⣉⣿⣀⣶⣸⢰⣚⣸⢸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⣀⣤⣤⣴⡄⣿⣀⣿⣿⣿⣿⡇⠀⠀
⢠⣤⠀⠉⠉⠁⠉⣹⡍⠈⠉⠈⠈⠩⠍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣷⣯⣻⣟⠉⠉⠉⠁⠀⠀
⠀⣫⠀⠠⠆⠀⢀⣤⡄⠀⢳⣆⠀⠀⠀⠀⠈⠙⠁⠐⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢿⠀⢸⣿⣸⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣽⡒⠊⠁⠀⠀⠀⠀⠀
⠀⠙⠀⠀⠀⠀⠀⠉⠀⠀⣸⠟⢓⡀⠀⠀⠦⣤⣴⣾⣿⢗⣤⣶⣄⠀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠂⠀⠀⠀⠀⠀⠀⠀
⠀⣤⣤⡄⠀⠀⠀⣀⣀⠀⠋⠀⠈⠛⠀⠀⠀⠉⠉⢙⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⡈⠳⢤⣀⣀⣀⣀⣤⠞⠁⠈⠀⢀⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠉⢁⣝⠁⠀⢈⣿⠏⠒⠀⠀⠀⠀⢰⠇⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⢸⡀⠂⠨⠉⠉⠅⠐⠄⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣤⣀⣀⡀⠀⢀⠀⠈⢳⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣧⣀⠀⠀⠀⡀⠈⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⣀⣠⣼⣿⣿⣿⣿⣿⣿⡿⣋⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠙⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡛⠁⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠹⠷⣿⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣥⡹⣿⣿⠿⠿⣿⣿⣯⣿⡟⠛⢿⣿⣿⣿⣿⣿⣿⠇⢀⣰⣾⡿⢀⣶⣶⣤⣤⠾⢿⣿⣿⡏⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⠀⠀⠈⠛⠋⠉⠀⠀⠀⠈⠉⠉⠀⢀⣵⡀⣿⣿⣿⣿⣾⣿⣿⣿⣯⠀⠈⢻⣿⠃⠀⠱⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡏⣷⣿⣶⣦⣦⣤⣤⡀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣸⡿⠀⠀⠀⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⡇⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⡇⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⠇⣸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⣿⣿⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠃⠀⠀⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠃⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢠⣄⣀⣀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣄⠀⠀⠀⣀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣴⣤⣅⢤⣤⣤⢠⣤⢠⢠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣠⡤⠶⠶⢤⣄⡘⠀⡀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣶⡄⠀⠀⠁⠀⠀⠀⣶⠶⣶⠶⠶⢶⡆⠀⠀
⣶⠴⠮⣿⣉⣿⣀⣶⣸⣰⣚⣸⢸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠛⠁⣠⣴⣤⣴⣶⠇⣿⣙⣿⣿⣿⣿⡇⠀⠀
⣠⡄⠀⠉⠉⠁⢩⣿⠉⠈⠉⠈⠨⣭⢍⡀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⣷⣯⢹⣻⡍⡉⠉⠁⠀⠀
⢸⡁⠀⠴⠀⠀⣣⣄⠀⠛⣧⡀⠀⠀⠀⠈⠉⠉⠈⢳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢿⠀⢸⣿⣼⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⠿⠟⢓⠂⠀⠀⠀⠀
⠘⠃⠀⠀⠀⠀⠉⠁⠀⢠⡟⠻⣄⠀⠀⠴⣶⣤⣾⣿⣿⣧⣶⣦⡀⠀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⢀⣴⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⡷⠈⠀⠀⠀⠀⠀⠀⠀
⢠⣤⣤⠒⠒⠀⢀⣀⠀⠘⠁⠀⠙⠃⠀⠀⠀⠉⠉⣹⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢰⡈⠳⢦⣀⣀⣀⣀⣤⠞⠁⠈⠀⣠⣿⣿⣿⣿⣿⣆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⢉⣹⠋⠀⠀⣹⡿⠑⠂⠀⠀⢀⠀⡿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⢧⠀⢂⠨⠉⠉⠅⡐⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣾⣯⣀⣀⡀⠀⠀⡀⠀⠛⣴⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣼⣆⠀⢀⣀⣀⡐⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⣁⣀⣤⣿⣿⣿⣿⣿⣿⣿⢟⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⣿⡿⠛⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⠋⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠿⢾⡇⠀⠀⠀⠀⠀⠀⣀⣴⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⣿⣿⠟⠻⢿⣿⣿⣿⡟⠛⠻⣿⣿⣿⣿⣿⣿⡿⢀⣠⣶⣾⡇⣾⣷⣦⣶⡴⠿⣿⣿⣿⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣠⣀⠈⠉⠉⠁⠀⠀⠀⠀⠉⠉⠀⢀⣔⣶⣾⣿⣿⣿⣷⣿⣿⣿⣿⠀⠀⠙⣿⡏⠈⢘⢵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣷⣶⣦⣤⣆⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣸⣿⠃⠀⠀⢙⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⣿⣿⣿⣿⣿⣿⡧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⡇⣺⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⢁⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⡟⠰⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡃⣿⣿⣿⣿⣿⣿⣿⣿⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠀⠀⠘⠛⠛⠛⠛⠛⠛⠁⠘⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢠⣄⣀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣰⣤⣖⢠⣤⣤⢠⣤⢠⢠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣠⡤⠶⠶⢤⣄⡘⠀⡀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣷⣶⡤⠠⠈⠉⠀⠀⠀⣶⠶⣶⠶⠶⢶⡆⠀⠀
⣴⠶⠮⣿⣉⣿⣀⣶⣸⣰⣚⣸⢸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠛⠁⣠⣤⣤⣴⣾⡇⣿⣐⣿⣿⣿⣿⡇⠀⠀
⣠⡄⠀⠉⠉⠁⢩⡿⠉⠈⠉⠈⠬⠭⣭⣄⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣷⣿⣻⣿⣏⢉⡉⠁⠂⠀
⢸⡀⠀⠴⠀⠀⣧⡤⠀⠙⣧⡀⠀⠀⠀⠈⠉⠁⠈⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢿⠀⢸⣿⣼⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⡿⣛⠟⠃⠀⠀⠀⠀
⠘⠂⠀⠀⠀⠀⠉⠁⠀⢠⡟⠻⣀⠀⠰⠶⣦⣶⣾⣿⣿⣧⣶⣦⡀⠀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⢀⣤⣦⣄⡀⠀⠀⠀⠀⠀⠠⠀⠲⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠻⠂⠀⠀⠀⠀⠀⠀⠀
⣠⣤⣤⠒⠒⠀⢀⣀⠀⠘⠃⠈⠙⠁⠀⠀⠀⠉⠙⣻⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢰⡈⠳⢦⣀⣀⣀⣀⣤⠞⠁⠈⠀⣠⣿⣿⣿⣿⣿⣆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⢉⣹⠋⠀⠀⣽⡿⠐⠃⠀⠀⠀⢀⡾⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⢧⠀⢂⠨⠉⠉⠅⡐⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⢰⣿⣿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣶⣧⣀⣀⣄⠀⢀⡀⠈⠳⣶⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣼⣆⠐⢀⣀⣀⡐⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠈⣁⣠⣤⣿⣿⣿⣿⣿⣿⣿⢟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡟⠈⣿⡿⠛⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡋⡀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠿⢾⡇⠀⠀⠀⠀⠀⠀⣀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣜⣿⣿⠟⠻⢿⣿⣿⣿⡟⠛⠻⣿⣿⣿⣿⣿⣿⡿⢀⣠⣶⣾⠇⣾⣷⣦⣶⡶⢿⣿⣿⣿⢁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣤⣀⠈⠉⠉⠁⠀⠀⠀⠀⠈⠉⠀⢀⣼⣶⣿⣿⣿⣿⣷⣿⣿⣿⣿⡀⠀⠙⣿⡏⠘⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣷⣦⣦⣢⡆⠀⡀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣰⣿⠃⠀⠈⠹⡷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⡇⣺⣿⣿⣿⣿⣿⣿⡧⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⣿⣿⣿⣿⣿⣿⡧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣿⣿⣿⣿⡇⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⠡⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣸⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠀⠀⠘⠛⠛⠛⠛⠛⠛⠁⠚⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢠⡄⣀⣀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡄⠀⠀⢀⣄⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣰⣮⣇⢠⣤⣤⢠⣤⢠⢠⣄⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡀⢀⣠⡤⠶⠶⢤⣄⡉⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣶⣦⡤⠀⠈⠀⠀⠀⠀⣶⠶⣶⠶⠶⢶⡆⠀⠀
⣴⠐⠯⣿⣉⣿⣀⣶⣸⣰⣚⣸⢸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢋⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠨⠛⠁⣠⣤⣴⣶⣿⡇⣿⣐⣿⣿⣿⣿⡇⠀⠀
⣠⠀⠈⠉⠉⠁⣩⡿⠉⠈⠉⠈⠬⠭⣉⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⣿⣿⣿⣿⣿⣿⣯⣿⡍⠍⢉⠁⠀⠀
⢈⠀⠀⠖⠀⠐⣧⡀⠀⢻⣦⡀⠀⠀⠀⠀⠉⠉⠉⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢿⠀⢸⣿⣼⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⡏⠛⠣⠳⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⠿⠓⡀⠀⠤⠴⣴⣴⣾⣿⣿⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⣠⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣤⣤⠒⠂⠀⣀⣀⠀⠙⠀⠈⠋⠀⠀⠀⠀⠈⢙⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢰⡈⠳⢦⣀⣀⣀⣀⣤⠞⠁⠈⠀⣴⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⢉⣹⠉⠀⠈⣿⠟⠚⠀⠀⠀⠀⢰⡟⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⡀⠀⠈⣇⠀⠂⠨⠉⠉⢅⠐⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⢸⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣷⣿⣥⣀⣀⡀⠀⢀⡀⠈⢳⣦⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣼⡀⢀⣀⣀⣀⢂⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣋⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⣿⡟⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣙⡁⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠙⠷⣾⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣥⣜⣿⡿⠟⠻⣿⣿⣿⡿⠋⠛⠻⣿⣿⣿⣿⣿⣿⠏⢀⣴⣶⣿⠀⣿⣶⣶⣶⠾⢿⣿⣿⡿⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⢠⣷⣶⣿⣿⣿⣿⣾⣿⣿⣿⣿⠀⠀⢹⣿⠇⠈⣽⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣷⣶⣦⣴⡆⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⢸⣿⠀⠀⠘⠻⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣼⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡧⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣿⣿⣿⣿⣿⣿⣿⡇⢼⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⢨⣿⣿⣿⣿⣿⣿⣿⡇⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠋⠀⠀⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠐⠛⠛⠛⠛⠛⠛⠛⠓⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣤⡄⡀⣀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣰⣮⣅⢠⣤⣤⢠⣤⢠⢠⣀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣠⡤⠶⠶⢤⣌⡁⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⢷⣶⣤⡀⠀⠉⠀⠀⠀⠀⣶⠶⣶⠶⠾⢶⡆⠀⠀
⢠⠚⠯⣿⣉⣿⣀⣶⣸⣰⣚⣸⢸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢠⣤⣴⣦⣴⣷⡇⣿⣐⣿⣿⣿⣿⡇⠀⠀
⣀⠀⠈⠉⠉⠁⣹⠝⠉⠈⠉⠈⠪⢭⡉⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⣿⣿⣿⣿⣷⣯⠟⣟⣍⠉⠉⠁⠀⠀
⠁⠀⠠⠆⠀⢀⣀⠀⠈⢳⣆⠀⠀⠀⠀⠀⠉⠉⢀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢿⠀⢸⣿⣼⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡻⣿⣯⣿⠻⠟⠉⠂⠀⠀⠀⠀
⠘⠀⠀⠀⠀⠈⠉⠀⠀⣼⠟⢣⡀⠀⠐⢦⣤⣶⣾⣿⣷⣴⣶⣆⠀⠀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⣠⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠓⣋⠁⠀⠀⠀⠀⠀⠀⠀
⣠⣤⡀⠂⠂⠀⢀⡀⠀⠃⠀⠉⠋⠀⠀⠀⠀⠉⢙⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢲⠈⠳⢦⣀⣀⣀⣀⣤⠞⠁⠈⢀⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⢉⡟⠀⠀⢸⣿⠗⠐⠀⠀⠀⠀⢰⠇⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠘⡇⠀⠂⠨⠉⠉⠅⠐⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⢾⣿⡿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣶⣾⣄⣀⣀⡀⠀⢀⠀⠘⢶⣴⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣽⡀⠀⢀⣀⢀⢂⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⣀⣠⣼⣿⣿⣿⣿⣿⣿⡿⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠙⣿⠟⠏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣋⠁⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠻⠷⣿⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡙⣿⠿⠛⠻⣿⣿⣿⡿⠉⠛⢿⣿⣿⣿⣿⣿⣿⠇⣀⣴⣶⣿⢰⣿⣶⣶⣤⠾⣿⣿⣿⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⣯⣽⣿⣿⣿⣿⣾⣿⣿⣿⣧⠀⠈⢻⣿⠃⠀⡺⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡏⣽⣿⣿⣶⣶⣦⣤⡀⣀⠀⠀⢀⣴⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣾⡟⠀⠀⠘⢚⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⢨⣿⣿⣿⣿⣿⣿⣿⣿⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⣿⣿⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠃⠀⠀⠛⠛⠛⠛⠛⠛⠋⠀⠛⠛⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠃⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣤⠀⣀⣀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣰⣬⣄⢠⣤⣤⢠⣤⢠⢠⣀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣠⡤⠶⠶⢤⣌⡁⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⣶⠶⣶⠶⠶⢶⡆⠀⠀
⠤⠂⠯⣿⣉⣿⣀⣶⣸⣰⣚⢸⢸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠁⢠⣤⣤⣶⣼⣿⡇⣿⣐⣿⣿⣿⣿⡇⠀⠀
⣠⠀⠉⠉⠉⠁⣽⠏⠉⠈⠉⠈⠬⢭⣉⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣿⣿⣷⣿⣿⣿⡭⢍⠉⠁⠀⠀
⠁⠀⠠⠆⠀⢀⣤⠀⠈⣿⡄⠀⠀⠀⠀⠈⠉⠁⢩⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢿⠀⢸⣿⣸⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣟⣿⣽⠹⠛⠋⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⡾⠟⢇⡀⠀⠴⢦⣴⣶⣼⣿⣷⣴⣶⣄⠀⠀⠀⠀⠀⠀⢠⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⣤⣴⣤⣀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣼⠕⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣄⠀⠀⠀⠀⣀⡀⠀⠃⠀⠙⠃⠀⠀⠀⠈⠉⢩⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢷⠈⠳⢦⣀⣀⣀⣀⣤⠞⠁⠈⢀⣾⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⣩⡗⠀⠀⢹⡿⠓⠚⠀⠀⠀⠀⣴⠃⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠸⡄⠀⠂⠨⠉⠉⠥⠐⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢾⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⣶⣧⣄⣀⣀⠀⠀⢀⠀⠙⣦⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣷⠀⠀⢀⣀⡠⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⢀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣟⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢹⣿⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠋⠁⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠻⠷⣧⠀⠀⠀⠀⠀⠀⢀⣤⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣹⣿⠿⠛⠛⣿⣿⣿⡟⠉⠛⢿⣿⣿⣿⣿⣿⣿⡇⣀⣴⣶⡟⢰⣷⣶⣴⣤⠿⣿⣿⣿⡇⠩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣤⣀⡀⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⣠⡖⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⢻⡿⠁⠘⣝⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣿⣿⣿⣿⣴⣦⣶⠀⣀⠀⠀⣀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⣿⡇⠀⠀⠝⣿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⠇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣾⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠠⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠈⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢀⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠃⠀⠘⠛⠛⠛⠛⠛⠛⠃⠐⠛⠛⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠃⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣤⠀⣀⣀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣰⣭⣂⢠⣤⣤⢠⣤⢠⢠⣀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡀⢀⣠⡤⠶⠶⢤⣌⡁⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣶⣤⡄⠈⠁⠀⠀⡀⡀⣶⠶⣶⠶⠶⢶⡆⠀⠀
⠄⠊⠯⣿⣉⣿⣀⣶⣸⣰⣚⣸⢸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢂⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⣠⣴⣤⣶⣴⣿⡇⣿⣐⣿⣿⣿⣿⡇⠄⠀
⣰⠀⠈⠉⠉⠁⣽⡏⠉⠈⠉⠨⠾⢭⣍⡁⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢿⣿⣿⣿⣿⣷⣿⣿⣿⡭⢉⠉⠁⠀⠀
⠁⠀⠤⠆⠀⢀⣄⠀⠘⣿⣄⠀⠀⠀⠈⠉⠉⠉⢹⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢿⠀⢸⣿⣼⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡿⡻⠗⠈⠁⠀⠀⠀
⠘⠀⠀⠀⠀⠈⠉⠀⠀⡿⠿⣇⡀⠀⠴⢶⣶⣶⣾⣿⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⢠⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢟⡟⠁⠀⠀⠀⠀⠀⠀⠀
⣤⣤⠔⠢⠂⠀⣀⡀⠈⠓⠀⠙⠃⠀⠀⠀⠀⠉⣹⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣷⠈⠳⣦⣀⣀⣀⣀⣤⠞⠁⠈⢀⣾⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⣉⡟⠀⠀⢹⡿⠓⠚⠀⠀⢀⠀⣼⠃⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠸⡄⠀⠂⠨⠉⠉⠅⠐⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣾⣿⡷⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⣾⣯⣄⣀⣀⠀⠀⣀⠈⠛⣦⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣧⠀⠀⢀⣀⡠⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⢁⣀⣤⣿⣿⣿⣿⣿⣿⣿⢟⣁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢹⣿⠻⠃⠀⠀⠀⠈⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣋⢁⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠻⠷⣧⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣻⣿⠿⠛⠻⢿⣿⣿⠏⠉⠛⢿⣿⣿⣿⣿⣿⣿⢃⣄⣵⣾⣟⢰⣷⣶⣤⡶⠾⣿⣿⣿⡗⡋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⣻⡿⠂⠙⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⣿⣿⣿⣿⣴⣴⡀⣀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣿⡇⠀⠀⠙⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⡿⢼⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⡇⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⣿⣿⣿⣿⣿⣿⡿⢼⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠈⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢀⣿⣿⣿⣿⣿⣿⡿⠠⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⢿⠿⠿⠿⠿⠿⠛⠛⠛⠃⠀⠐⠛⠛⠛⠛⠛⠛⠃⠐⠛⠛⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠃⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣤⠀⣀⣀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣴⣭⣖⢠⣤⣤⢠⣤⢠⢠⣀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⢀⣠⡤⠶⠶⢤⣌⡁⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⢀⠾⢿⣷⣦⡀⠀⠁⠀⢀⣀⠀⣶⠶⣶⠶⠾⢷⡆⠀⠀
⡄⠉⡯⣿⣉⣿⣀⣶⣸⣰⣚⣸⢸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⢠⣴⣴⣶⣿⣿⡇⣿⣓⣿⣿⣿⣿⡇⠀⠀
⣀⠀⠉⠍⠉⠁⣿⠋⠉⠈⠉⠈⠬⢭⡉⠀⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣷⣿⣭⣿⣍⣭⡍⠁⠀⠀
⠀⠀⠠⠂⠀⢰⣄⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠁⢩⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢿⠀⢸⣿⣼⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣻⣿⡿⢏⢿⠋⠷⠃⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⡾⠛⢧⡀⠀⠄⢴⣶⣶⣿⣿⣯⣶⣷⣆⠀⠀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠹⠅⠁⠀⠀⠀⠀⠀⠀⠀
⣤⣤⠒⠂⠀⠀⣀⡀⠐⠓⠀⠙⠋⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢳⠈⠳⢦⣀⣀⣀⣀⣤⠞⠁⠈⢀⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠄⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀
⠉⣉⠛⠀⠀⢹⡿⠃⠂⠀⠀⠀⠀⣾⠓⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠘⡆⠀⠂⠨⠉⠉⠥⠐⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣷⣯⣄⣀⣄⠀⠀⡀⠀⠛⣶⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣿⣀⣀⣀⣀⡠⣁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⢀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣟⣁⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⡙⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⣁⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⠓⣿⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣹⣿⠿⠛⠛⠻⣿⠿⠋⠉⠉⠻⣿⣿⣿⡿⢿⣿⢇⣤⣶⣶⣿⢸⣿⣶⣶⡶⠻⣿⣿⣿⡏⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠈⣻⣿⠃⠀⠱⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⠏⣿⣿⣿⣿⣿⣶⣿⣆⣀⡀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣿⡟⠀⠀⠛⣿⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⡇⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⡏⣸⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠃⠀⠐⠛⠛⠛⠛⠛⠛⠋⠀⠛⠛⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣤⠄⣀⢀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣴⣯⣗⢠⣤⣤⢠⣤⢠⢠⣀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣠⡤⠶⠶⢤⣌⡁⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⢀⠾⣿⣶⣤⡄⠀⠁⠀⢀⣀⡀⣶⠶⣶⠶⠾⢷⡆⠀⠀
⡄⠉⡯⣿⣉⣿⣀⣶⣸⣰⣚⣸⢸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣀⣤⣶⣶⣾⣿⡇⣿⣲⣿⣿⣿⣿⡇⠀⠀
⡀⠀⠉⠉⠉⠁⣿⠏⠉⡈⠉⠈⠨⠭⣉⢀⡲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣷⣿⣭⣿⡭⣭⠭⠁⠀⠀
⠀⠀⠰⠂⠀⢰⣄⠀⠘⣿⣆⠀⠀⠀⠀⠀⠈⠉⢈⣽⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢿⠀⢸⣿⣼⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣿⣿⡿⡿⠻⠍⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⡿⠛⣷⡀⠀⠂⠦⣶⣶⣿⣿⣿⣶⣷⣦⡀⠀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⣴⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠫⠂⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣄⠒⠒⠀⠀⣀⡀⠐⠃⠀⠙⠋⠀⠀⠀⠀⠈⣩⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢰⡈⠳⢦⣀⣀⣀⣀⣤⠞⠁⠈⢀⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⣀⡗⠀⠀⢹⡿⠃⠂⠀⠀⠀⢀⣾⠛⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠈⣇⠀⠂⠨⠉⠉⠥⠐⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣷⣯⣄⣀⣄⠀⠀⡀⠀⠙⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣀⣀⣀⣀⡠⣁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⢀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣟⣁⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡘⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣍⣁⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⠳⣿⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣹⣿⠿⠉⠛⠛⠿⠿⠋⠀⠉⠻⢿⣿⡿⡿⢿⡿⡇⣀⣴⣶⣿⢼⣿⣷⣶⡦⠾⣿⣿⣿⡇⡠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⣻⣿⠃⠀⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣇⣀⡄⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣿⡏⠀⠀⠑⣷⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⡇⠀⠀⠀⠙⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣅⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⢊⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠃⠀⠀⠛⠛⠛⠛⠛⠛⠋⠀⠛⠛⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢤⠄⣀⢀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀
⠀⢀⣀⣴⣤⣟⢠⣤⣤⢠⣤⢠⢠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⢀⣠⡤⠶⠶⢤⣌⡁⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⢀⠼⣷⣶⣦⡄⠀⠈⠀⢀⡀⡀⣶⠶⣶⠶⠾⢶⡆⠀⠀
⠤⠀⡮⣿⣉⣿⣀⣶⣸⢰⣚⣸⢸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢠⣦⣶⣶⣿⣿⡇⣿⣳⣿⣿⣿⣿⡇⠀⠀
⡀⠀⠈⠉⠉⠁⡿⠏⠉⡈⠉⠈⠈⠩⡉⢀⣒⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣽⣯⣭⡉⠉⠁⠀⠀
⠀⠀⠰⠀⠀⢰⣤⠀⠈⣿⣦⡀⠀⠀⠀⠀⠈⠁⢈⣿⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢿⠀⢸⣿⣸⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣟⢛⠉⠋⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⣻⡀⠀⠂⠶⣶⣶⣿⣿⣯⣷⣿⣦⡀⠀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⣰⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀
⣴⣆⡐⠀⠀⠀⣀⡀⠀⠋⠀⠉⠉⠀⠀⠀⠀⠈⢩⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢰⡈⠳⢦⣀⣀⣀⣀⣤⠞⠁⠈⢀⣼⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠉⣁⡃⠀⠀⢸⡿⠃⠒⠀⠀⠀⠀⣾⠛⠂⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⣇⠀⡢⠨⠉⠉⠥⠐⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣄⣀⣀⠀⠀⡀⠀⠙⢶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣄⣋⣀⣀⡠⣁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⢀⣀⣤⣿⣾⣿⣿⣿⣿⣿⡟⣅⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡈⣿⡟⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣭⣁⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠙⠓⢻⡀⠀⠀⠀⠀⠀⣀⣴⣶⣾⡂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣹⣿⠿⠿⠛⠛⠻⠟⠋⠀⠈⠙⣿⣿⡿⠿⠿⣿⠏⢀⣰⣶⣿⣷⣷⣷⣶⡶⠿⣿⣿⣿⡏⠨⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣞⣶⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠈⣻⣿⠃⠌⣟⢿⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣷⣾⣿⡇⣀⣀⣀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣄⣿⣿⠀⠀⠸⣝⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣾⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⡏⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣿⣿⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⡇⢽⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⠠⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡏⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣿⣿⣿⣿⣿⣿⣿⠁⣻⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠋⠀⠀⠛⠛⠛⠛⠛⠛⠛⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠘⠛⠛⠛⠛⠛⠛⠛⠃⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢤⠄⡀⢀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⢠⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣀⣴⣤⣞⢠⣤⣤⢠⣤⢠⢠⣀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢀⣠⡤⠶⠶⢤⣄⡁⠠⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⢀⠸⠿⣶⣦⠀⠀⠈⠀⢀⣀⡀⣶⣶⣶⠶⠶⢶⡆⠀⠀
⠤⠈⡮⣿⣉⣿⣀⣶⣸⢰⣚⣸⢸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⣤⣶⣾⣾⣿⣇⣿⣚⣿⣿⣿⣿⡇⠀⠀
⡀⠀⠈⠉⠉⠁⢿⠏⢉⡈⠉⠈⠈⠉⠩⣀⣒⣂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠛⢿⣿⣿⣿⣷⣿⣽⣿⣭⠉⠉⠁⠀⠀
⠀⠀⠰⠀⠀⢰⣤⠀⠈⢿⣦⡀⠀⠀⠀⠀⠈⠁⢀⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢿⠀⢸⣿⣸⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⠝⡘⡉⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⡾⠛⢛⣆⠀⠒⠶⢶⣶⣿⣿⣿⣷⣿⣷⡀⠀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⣠⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠐⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣴⣆⡐⠀⠀⠀⣀⡀⠀⠋⠀⠉⠋⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠰⡈⠳⢤⣀⣀⣀⣀⣤⠞⠁⠈⠀⣴⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⣠⡃⠀⠀⢸⡿⠃⠒⠀⠀⠀⠀⣼⠛⠂⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⢳⠀⣦⠨⠉⠉⠥⠐⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣷⣄⣀⣀⠀⠀⠀⠀⠘⢶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣇⣙⣀⣀⣠⣁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⢀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣟⣅⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣈⣻⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⣁⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠙⠛⢺⡆⠀⠀⠀⠀⠀⣀⣴⣶⣶⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣹⣿⡿⠿⠛⠛⠛⠛⠁⠀⠀⠙⠿⣿⡿⠿⠿⣿⣟⢀⣰⣶⣿⣿⣿⣷⣶⡶⠿⣿⣿⣿⡏⠈⣱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣠⢄⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⢻⣿⠇⢈⢺⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⡧⣤⣀⣀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣄⣿⣿⠀⠀⠘⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣼⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⡟⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣻⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⡧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣹⣿⣿⣿⣿⣿⣿⡇⢼⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣾⣿⣿⣿⣿⣿⣿⠁⣺⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠀⠀⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠠⠄⣀⢀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢤⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣰⣥⣛⢠⣤⣤⢠⣤⢠⢠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣠⡤⠶⠶⢤⣄⡙⠠⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡰⠿⣶⣦⠠⠀⠉⠀⢀⣠⣀⣶⡾⣶⠶⠶⢶⡆⠀⠀
⠄⠀⡮⣿⣉⣿⣀⣶⣸⢰⣚⣸⢸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢰⣴⣿⣶⣾⣯⣇⣿⣚⣿⣿⣿⣿⡇⠀⠀
⣀⠀⠈⠉⠉⠁⢿⠏⠉⣈⠉⠈⠈⠉⢍⡀⡐⣐⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⣿⣯⣯⣭⡉⠉⠁⠀⠀
⠀⠀⠰⠂⠀⢐⣤⠀⠈⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⣸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢿⠀⢸⣿⣸⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⠟⢛⠋⠙⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⢻⣧⠀⠀⠒⢶⣶⣾⣿⣿⣷⣿⣷⣄⠀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⢠⣶⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠌⠀⠀⠀⠀⠀⠀⠀⠀
⣴⣤⡀⠀⠀⠀⣀⡀⠀⠋⠀⠈⠉⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣈⠳⢤⣀⣀⣀⣀⣤⠞⠁⠈⠀⣴⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⣠⡃⠀⠀⢸⡿⠁⠚⠀⠀⠀⠀⣼⠛⠃⢀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⢸⡀⢢⠨⠉⠉⢥⠐⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣯⣤⣀⣄⠀⠀⠀⠀⠘⢲⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣘⣀⣀⣠⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⢀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣥⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣀⢹⡿⠙⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣁⣀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠛⢚⡇⠀⠀⠀⠀⠀⣀⣤⣶⣾⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣹⣿⡿⠿⠟⠛⠛⠛⠁⠀⠀⠈⠿⠿⠿⠿⠿⣿⡿⣠⣤⣶⣾⣿⣿⣷⣶⣶⠻⣿⣿⣿⡿⠠⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣾⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⢻⣿⠇⠀⢸⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⡿⣹⣿⣿⣿⣿⣿⣿⣧⣠⣀⣤⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣿⣿⠀⠀⠀⢿⣿⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⡇⣽⣿⣿⣿⣿⣿⣿⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⡷⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣿⣿⣿⣿⣿⣿⣿⡟⣸⣿⣿⣿⣿⣿⣿⣿⡇⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⠇⣾⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⠠⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢸⣿⣿⣿⣿⣿⣿⡏⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⡃⣿⣿⣿⣿⣿⣿⣿⣿⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠀⠀⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠠⠄⡀⢀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⢤⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣴⣥⣟⢢⣤⣤⢠⣤⢠⢠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣠⡤⠶⠶⢤⣄⡙⠠⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣶⣦⡤⠈⠉⠀⢀⣀⣀⣶⠶⣶⠶⠶⢶⡆⠀⠀
⠄⠀⡮⣿⣉⣿⣀⣶⣸⢰⣚⢸⢸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⣼⣿⣶⣾⡿⣷⣿⣛⣿⣿⣿⣿⡇⠀⠀
⣀⠀⠈⠉⠉⠁⣿⠏⢉⣈⠉⠘⠈⠍⠉⡀⣐⣔⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⣿⣯⣯⣯⡝⠍⠁⠀⠀
⠀⠀⠰⠂⠀⢘⣤⠀⠈⢿⣶⣄⠀⠀⠀⠀⠀⠈⠀⣸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢿⠀⢸⣿⣸⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⢟⡻⠉⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⢻⣧⠀⠀⠒⢶⣶⣾⣿⣿⣷⣿⣷⣄⠀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⢠⣶⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣆⠀⠀⠀⢀⣀⡀⠀⠋⠀⠈⠉⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⡈⠳⢦⣀⣀⣀⣀⣤⠞⠁⠈⠀⣰⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⣠⡓⠀⠀⢸⡿⠁⠘⠀⠀⠀⠀⣼⠛⠃⢀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⢸⡀⢢⡨⠉⠉⢥⠐⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⢾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣤⣀⣤⠀⠀⠀⠀⠘⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣈⣁⣄⣠⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣥⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⢻⡿⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣅⣀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠛⢻⡇⠀⠀⠀⠀⠀⣀⣤⣶⣾⢖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣸⣿⡿⠿⠿⠛⠛⠛⠁⠀⠀⠈⠿⠿⠿⠿⠿⢿⡿⠁⣤⣾⣿⣿⣿⣷⣶⣶⠛⣿⣿⣿⡿⢠⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⢐⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠈⣻⣿⠇⠀⣼⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⣿⣿⣿⣿⣿⣿⣿⣠⣄⣠⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣿⣿⠀⠀⠀⢻⣽⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⡗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⡗⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣿⣿⣿⣿⣿⣿⠇⣸⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠀⠀⠚⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠋⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⡤⣀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣀⣴⣥⣞⢣⣤⣤⢠⣤⢠⢠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣠⡤⠶⠶⢤⣄⡘⠠⡄⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⡠⠷⣶⣦⠄⠈⠉⠀⢀⡀⡀⣶⡾⣶⠶⠶⢶⡆⠀⠀
⠤⠀⡮⣿⣉⣿⣀⣶⣸⢰⣚⣸⢸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⣴⣿⣷⣿⣿⡇⣿⣡⣿⣿⣿⣿⡇⠀⠀
⡀⠀⠈⠉⠉⠁⠽⠍⠉⣈⠉⠈⠈⠩⠉⠀⣐⣲⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣷⣿⣯⣽⡭⡝⠉⡁⠀⠀
⠀⠀⠰⠀⠀⢐⣤⠀⠈⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⣸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢿⠀⢸⣿⣸⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢫⠝⠋⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⡿⠛⢛⣷⠀⠀⠒⠶⢶⣶⣿⣿⣽⣾⣷⣄⠀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⢠⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡎⠀⠀⠀⠀⠀⠀⠀⠀
⣴⣤⠀⠀⠀⠀⣀⡀⠀⠉⠀⠉⠉⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢈⠳⢤⣀⣀⣀⣀⣤⠞⠁⠈⠀⣠⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⣀⡋⠀⠀⢸⡿⠁⠐⠀⠀⠀⠀⣴⠛⠃⢀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠘⡆⢢⠨⠉⠉⢅⠐⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣄⣀⣠⠀⠀⢀⠀⠘⣶⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣘⣃⣀⣠⣂⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣀⣤⣿⣷⣿⣿⣿⣿⣿⣿⣫⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⣹⡿⠹⠁⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣍⣀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠛⠚⡇⠀⠀⠀⠀⠀⢀⣤⣶⣶⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣹⣿⡿⠿⠿⠟⠛⠛⠉⠀⠀⠈⠻⠿⠿⠿⠿⢿⣿⣡⣤⣶⣿⣿⣿⣷⣶⣶⠛⣿⣿⣿⣿⠀⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⠠⠀⣀⡀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠈⣻⣿⠏⠀⢙⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⢠⣠⣀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣿⣿⠀⠀⠈⢳⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⣿⣿⣿⣿⣿⣿⣿⡏⣸⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⠀⠀⠀⠀⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⡇⢾⣿⣿⣿⣿⣿⣿⣿⣯⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠁⠀⠘⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠃⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠠⡤⣀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣠⡤⠀⢀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀
⠀⠀⣀⣰⣯⣞⢣⣤⣤⢠⣤⢠⢠⣀⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⢀⣠⡤⠶⠶⢤⣄⡘⢀⡄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⢷⣶⣤⠄⠀⡀⠀⢀⣀⡀⣶⠶⣷⠷⠷⢶⡆⠀⠀
⠄⠈⡮⣿⣉⣿⣀⣶⣸⢰⣚⣸⢸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢠⣬⣶⣶⣿⣿⣿⣿⣋⣿⣿⣿⣿⡇⠀⠀
⡀⠀⠈⠉⠉⠁⣿⠍⠉⣈⡉⠈⠈⠉⢭⡄⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⣿⣿⣿⣷⣯⣽⢯⣯⠋⠋⠁⠀⠀
⠁⠀⠐⠀⠀⢠⣄⠀⠘⣿⣦⣄⠀⠀⠀⠀⠉⠁⠁⣸⣷⣂⠀⠀⠀⠀⠀⠀⠀⠀⠃⢿⠀⢸⣿⣼⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡿⠏⠛⠙⠘⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⡞⠛⣻⣷⠀⠂⠰⢶⣶⣶⣿⣿⣿⣾⣷⣄⠀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⢀⣴⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣄⠀⠀⠀⠀⣀⠀⠀⠃⠀⠙⠋⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢈⠳⢦⣀⣀⣀⣀⣤⠞⠁⠈⠀⣠⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⣀⡏⠀⠀⢸⡿⠃⠓⠀⠀⠀⠀⣼⠛⠓⡀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠈⣇⢂⣨⠉⠉⢅⠔⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣷⣿⣄⣀⣀⠀⠀⠀⠀⠙⣶⣴⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣌⣋⣀⣀⣌⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⢀⣀⣤⣿⣷⣿⣿⣿⣿⣿⣿⢯⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⢹⣿⠻⠃⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣉⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠛⠶⣷⠀⠀⠀⠀⠀⢀⣴⣶⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣜⣿⣿⠿⠿⠿⠛⠛⠉⠀⠀⠈⠻⣿⡿⠿⠿⢿⣿⡃⣠⣶⣾⣿⣾⣷⣴⡶⠿⣿⣿⣿⣿⢡⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣤⣀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⣻⣿⡏⠈⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⢠⣄⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣄⣿⣿⠄⠀⠀⢏⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢨⣿⣿⣿⣿⣿⣿⣿⠐⣿⣿⣿⣿⣿⣿⣿⣿⡟⢼⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢸⣿⣿⣿⣿⣿⣿⡏⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠃⠀⠘⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠠⡤⣀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀
⠀⠀⣀⣰⣧⣝⢡⣤⣤⢠⣤⢠⢠⣀⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣠⡤⠶⠶⢤⣄⡘⢀⡄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⣶⣤⠀⠀⠀⠀⢀⣀⢀⣶⠶⣶⠶⠶⢿⡆⠀⠀
⠀⠈⡮⣿⣉⣿⣀⣶⣸⢰⣚⢸⢸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣴⣾⣶⣿⣿⣿⣿⣉⣿⣿⣿⣿⡇⠀⠀
⡀⠀⠈⠉⠉⠁⠿⠍⠉⢈⡉⠈⠈⠉⠩⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⡷⢛⡭⠻⣏⠙⠉⠁⠀⠀
⠁⠀⠠⠂⠀⢠⣄⠀⠘⢿⣦⣄⠀⠀⠀⠀⠈⠁⠀⣸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢿⠀⢸⣿⣸⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⢏⠈⠂⠂⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠉⠀⠀⡾⠛⣻⣷⠀⠀⠰⢦⣤⣴⣿⢿⣯⣾⣷⣄⠀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⢀⣴⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣄⠀⠀⠀⠀⣀⡀⠀⠃⠀⠙⠋⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢈⠳⢦⣀⣀⣀⣀⣤⠞⠁⠈⠀⣠⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⣀⡇⠀⠀⢸⡿⠃⠓⠀⠀⠀⠀⣰⠛⠃⡀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠈⣇⠂⠨⠉⠉⢅⠔⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣧⠀⣾⣿⡟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣷⣶⣾⣄⣀⣀⠀⠀⠀⠀⠈⢦⡴⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣌⣃⣀⣀⣌⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⢀⣀⣤⣿⣷⣿⣿⣿⣿⣿⣿⢯⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⢹⣿⠻⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣉⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠛⠶⣷⠀⠀⠀⠀⠀⢀⣴⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣘⣿⣿⠿⠿⠿⠛⠛⠉⠀⠀⠈⢻⢿⡿⠿⠿⢿⣿⡁⣠⣶⣾⣿⣶⣷⣶⡶⠻⣿⣿⣿⣿⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣴⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⣻⣿⠏⠈⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⢠⣄⡀⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣿⣿⠂⠀⠀⢹⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⡟⢼⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠈⣿⣿⣿⣿⣿⣿⣿⠰⣿⣿⣿⣿⣿⣿⣿⣿⡏⢾⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢸⣿⣿⣿⣿⣿⣿⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠃⠀⠘⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠠⣤⣀⣀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣤⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣰⣧⣟⢣⣤⣤⢠⣤⢠⢠⣀⣀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣠⡤⠶⠶⢤⣄⡘⢀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣶⣤⠀⠈⠀⠀⢀⣀⣀⣶⠶⣶⠷⠿⢾⡆⠀⠀
⠀⠈⡮⣿⣉⣿⣀⣶⣸⢰⣚⣸⢸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⣴⣿⣶⣿⣿⣿⣿⣘⣿⣿⣿⣿⡇⠀⠀
⡀⠀⠈⠉⠉⠁⣿⠍⠉⣈⠉⠈⠘⠉⡉⡀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣷⣯⣿⣭⣭⠉⠉⠁⠀⠀
⠁⠀⠰⠀⠀⢀⣄⠀⠘⣿⣤⣄⠀⠀⠀⠈⠉⠉⠉⠸⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢿⠀⢸⣿⣸⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⢿⠿⠐⠘⠃⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠿⠿⣿⣷⠀⠀⢠⣶⣤⣴⣮⣷⣣⣶⣶⣄⠀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⠀⣤⣦⣤⡀⠀⠀⠀⠀⠀⠄⠰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣤⠀⠀⠁⠀⣀⠀⠐⠃⠀⠙⠋⠁⠀⠀⠀⠉⠉⣻⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢈⠳⢦⣀⣀⣀⣀⣤⠞⠁⠈⠀⢀⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⣉⠃⠀⠀⢸⣿⠃⠂⠀⠀⢀⠀⣾⠛⠂⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠈⣇⠂⠨⠉⠉⢅⠔⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣧⠀⣾⣿⣿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣶⣶⣯⣀⠀⣀⠀⠀⠀⠀⠛⣶⣴⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣔⣓⣎⣀⢌⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⢀⣀⣠⣿⣷⣿⣿⣿⣿⣿⣿⢟⣑⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⢹⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣋⠁⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠿⠷⣧⠀⠀⠀⠀⠀⢀⣠⣶⣶⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡜⣿⣿⣿⡿⠿⠟⠛⠋⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣃⣠⣼⣾⣿⣶⣦⣤⡴⠾⣿⣿⣿⣿⡇⠨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⣄⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢠⣠⣠⣿⣿⣿⣿⣿⢿⣿⣿⡇⠀⠙⣿⣿⡿⠀⠐⢿⣖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣷⣿⣿⣧⣀⡀⣄⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⣿⣿⡇⠀⠀⠘⠿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠈⣿⣿⣿⣿⣿⣿⣿⠰⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣿⣿⣿⣿⣿⣿⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇⢼⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠃⠀⠘⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠠⣤⣀⣀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢠⣠⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣰⣦⣟⢣⣤⣤⢠⣤⢠⢠⣀⣀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣠⡤⠶⠶⢤⣄⡘⢀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣦⣤⠤⠈⠀⠀⢀⣀⠀⣶⠶⣶⠶⠿⢷⡆⠀⠀
⠠⠈⡮⣿⣉⣿⣀⣶⣸⢰⣚⢸⢸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣴⢋⡁⢀⡀⢀⡀⢈⡙⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⣠⣶⣶⣿⣿⣿⣿⣀⣿⣿⣿⣿⡇⠀⠀
⡀⠀⠈⠉⠉⠁⣿⠉⠉⣈⠉⠈⠈⠉⢉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣾⡇⣾⡇⢸⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣿⣿⣿⣿⠽⢏⣫⠍⠉⠁⠀⠀
⠁⠀⠠⠀⠀⢠⣄⠀⠘⣿⣤⣄⠀⠀⠀⠀⠉⠉⠉⢸⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢿⠀⢸⣿⣼⡇⣿⣷⣾⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠀⠈⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⡿⠿⣿⣷⠀⠀⢤⣤⣤⣴⣿⣿⣺⣶⣶⣄⠀⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇⠈⠀⠀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠠⢶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣄⠀⠀⠀⠀⣀⠀⠐⠁⠀⠙⠋⠀⠀⠀⠀⠉⠉⣻⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⣤⠞⠁⠈⠀⢀⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⣉⠃⠀⠀⢸⡿⠀⠂⠀⠀⠀⠀⣶⠛⠆⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⡀⠀⠈⣇⢢⡨⠉⠉⠅⠔⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣧⠀⣾⣷⣿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣶⣶⣮⣀⢀⣀⠀⠀⠀⠀⠛⣦⣴⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣔⢁⣃⣀⢌⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⢀⣀⣠⣿⣷⣾⣿⣿⣿⣿⣿⢟⣑⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⢹⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡛⠁⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠿⠷⣯⠀⠀⠀⠀⠀⢀⣠⣶⣶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡜⣿⣿⣿⡿⠿⠟⠛⠋⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣃⣠⣶⣾⣿⣶⣦⣤⡴⠿⣿⣿⣿⣿⡇⡈⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣄⢀⢀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣿⣿⣿⣿⣿⢿⣿⣿⡇⠀⠙⣿⣿⣿⠁⠰⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣆⡄⢀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣀⣿⣿⡇⠀⠀⠈⠱⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠅⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣷⣤⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⢉⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠐⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠈⣿⣿⣿⣿⣿⣿⣿⢡⣿⣿⣿⣿⣿⣿⣿⣿⡿⣺⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣿⣿⣿⣿⣿⣿⡟⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠃⠀⠐⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠃⠚⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠠⣤⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣰⣤⣀⢠⣤⣠⢠⣤⢠⢠⣀⣀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢸⢀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⣶⠶⣶⠴⠶⢶⡆⠀⠀
⠀⠀⡮⣿⣈⣿⣀⣦⣸⢰⣚⢸⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣿⣽⣿⣿⡇⠀⠀
⠀⠀⠀⠉⠉⠁⠭⠉⠉⢈⠉⠈⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠘⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠞⠛⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠀⠀⠀⠀⢠⠿⠀⠀⠀⠀⠀⠀⠰⠒⠂⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣧⠀⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣷⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢤⡄⢳⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡄⠀⠀⢀⢈⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⣀⣠⣿⣴⣾⣿⣿⣿⣿⣿⢟⣑⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠹⣿⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡍⠋⠁⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠾⠷⢧⠀⠀⠀⠀⠀⢀⣠⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠘⣿⣿⣿⣿⠿⠟⠛⠋⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣠⣿⣤⣤⣤⡀⠰⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠿⣿⣿⠀⠀⠙⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⠋⣹⣮⣶⣦⣤⣴⣶⠀⠀⠀⠀⠀⣀⣴⣇⣿⣾⣿⣿⣿⣿⣦⣴⣿⣦⣀⡀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⣿⣿⠠⣶⣤⣦⣾⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠈⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣻⣿⣿⠿⠧⠄⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⢀⣀⡀⢀⣤⣤⡀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⣿⣿⣿⠀⠺⠿⠿⠿⣿⣿⠿⠟⠀⠠⡌⠈⠉⠛⢉⣉⣈⣈⣉⣠⣤⣤⣤⣤⣤⢰⣶⣶⣶⣶⡶⣶⡆⠀⠀⠀⣿⣿⠟⢸⣿⣼⣿⡇⠿⠿⣿⣿⠀⠀⠀
⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠟⠛⠿⣿⠟⢛⣁⡙⢋⣉⣉⡉⣭⣍⢡⣤⣤⣴⣶⣶⠈⠁⣶⣶⣷⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⡉⢸⣿⣦⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢲⣶⣾⡿⠀⠀⠀
⠫⣥⡄⣤⡆⣠⣶⣶⣦⢸⣿⣾⣿⡇⠀⢸⣿⢻⣿⢸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⢰⠀⣿⡿⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢿⣿⢀⡀⠀⢻⣿⣤⣼⣿⡇⣿⣷⢸⣿⠀⠀⠀⠀⠀
⡄⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠿⣿⣶⡄⣿⣿⠟⢹⣿⣿⣿⡆⢸⣿⡇⢸⠀⣿⣷⣿⣷⠀⢸⣿⣧⡌⣿⣧⣶⠈⣿⣇⠀⢸⣿⠇⠘⠿⠿⠿⠛⠛⠘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣷⠈⣿⣿⠀⢿⣿⢸⣿⡟⣿⣇⣿⣿⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⣿⠇⠸⠿⠇⣸⡀⠛⠃⠙⠛⢂⣌⣉⣉⣁⣉⠉⠉⢠⣀⣤⣾⣶⣶⣾⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⠇⠻⣿⠀⠾⠿⠿⠿⠶⠟⠛⠛⠁⠚⠂⠈⠉⠉⠀⠀⠀⠀⠐⠀⠀⠀⠒⠒⠒⠚⠛⠛⠛⠃⠐⠛⠛⠛⠛⠛⠛⠛⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⠀⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⢠⣄⣀⣤⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⡀⣤⣤⣤⠤⣤⣤⣤⣤⡤⡤⢀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣠⣀⣀⣠⣤⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⠀⠀⣠⡶⣾⣿⣿⡿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡷⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠼⣴⣶⣞⣻⣿⡟⠗⣏⠋⣷⣶⣶⣿⢶⢶⣆⡀⠀
⣿⣿⣿⣿⣿⣿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⡁⠲⢿⡟⣿⣿⣿⢦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⡿⠻⢿⡏⠻⣿⣷⣿⣿⠄⣿⣈⣿⣽⣿⣿⡇⢤⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣷⣄⢻⣿⣿⣿⣳⡶⣽⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠷⠶⢦⣧⣤⣬⣁⣿⡉⠀⠉⡍⣩⠉⣩⣍⡅⠹⠆
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⡯⠸⣾⣿⣿⣿⣿⣿⣿⣿⠿⢏⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣀⣤⣤⣄⡀⠑⠛⢿⡋⢭⢆⣶⣶⠈⢿⣛⣿⣿⣿⡆⠀
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⠉⢡⣿⢾⣿⡀⢈⣉⣿⣿⣿⣿⣿⠏⠀⠀⠀⠁⠀⠀⠈⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣴⣟⣿⣿⣿⢿⣦⡀⣿⡇⣤⣾⣿⣿⣿⣿⣽⣿⣿⣿⣷⡧
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠰⣿⠿⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠈⢿⣿⣏⣽⡿⠟⣻⣿⣿⣿⣿⣿⣭⣽⣿⣟⣛⣿⡇⣽⣿⣿⣿⣿⣿⣿⠟⠁⠀⠉⠛⠉⠃
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠓⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣠⡌⣳⣧⡤⢀⣤⣤⡀⢠⣼⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠋⠁⠉⠉⠉⠈⠉⠻⣷⣄⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣇⣼⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣤⠀⠀⡀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⡜⠁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢩⠈⠘⠉⡁⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⡦⣶
⠀⠀⠀⠀⠈⣿⡟⠙⢸⣿⣿⣿⣷⣤⣤⣤⠀⣀⢰⡄⢀⣈⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⢙⣿⣿⣷⣿⣿⣿⣿⣿⡀⢰⣶⣧⣷⡶⠟⠿⠻⠙⠙⢿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣏⠁⠒⣳⣼
⠀⠀⠀⠀⠀⣿⣇⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣀⣼⣿⢳⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡿⠛⣸⣿⣿⣿⣿⣿⣿⢛⠻⡷⣿⠋⠉⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠻⠿⠿⣿⡀⠈⠓⠚⠻⣳
⠀⠀⠀⠀⠀⣿⣿⠀⠀⢹⣿⣿⣿⣾⡛⠉⢡⣾⣿⣿⣿⡟⠀⢻⣿⠛⣿⣿⣅⠐⠒⠚⠛⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⠆⣿⣶⣏⡀⠀⠀⠀⠀⠀⠀⠀⢰⠹⣿⣿⣿⣿⣿⡿⠻⠁⠀⠀⠀⠀⠀⠀⠀⠉⠤⣀⣠⡿⠿
⠀⠀⠀⠀⠀⣻⣿⡄⠀⡸⡟⠋⠋⠉⢰⡀⣸⣿⣿⣿⠏⠿⣶⡼⠧⠴⠛⠋⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⢟⠋⠛⠳⣦⣄⡀⠀⠀⠈⢸⠀⣿⣿⣿⣿⡿⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣤⣤⣶⣶
⠀⠀⠀⠀⠀⢻⣿⣯⠘⠀⠀⠀⠀⠀⠸⣍⠉⣩⡿⠋⠀⠀⠄⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⡽⠀⠀⠀⠉⠙⠀⠀⠀⢸⠀⣿⣿⣿⡿⠀⠀⢸⠠⠴⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠋⠀⠟⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⣿⣿⣿⣿⣿⠋⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠻⠻⣿⡇⠀⠀⠘⠀⣀⢂⣀⢀⣀⡀⢀⣤⣤⡉⢻⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⢿⠿⠛⠁⢀⣀⣀⣀⣁⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡎⠀⠀⠀⣿⣿⠟⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠸⠛⠿⣿⡆⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣉⡉⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⢹⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢻⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⢰⣿⣿⣎
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣶⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⢘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⢸⣿⡯⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⡟
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠐⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⠠⣄⣀⣤⣤⣤⣤⣤⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⡀⣤⣤⣤⠤⢤⣤⣤⣤⡤⡤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣀⣀⣠⣤⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⠀⠀⣠⡶⣾⣿⣿⡿⣧⣰⣿⣾⣿⣿⣿⣿⣿⣿⣹⣿⡧⣿⣳⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠾⣰⣶⣖⣛⣿⡟⠓⣌⠏⣶⣶⣶⡷⢶⢶⣆⡀⠀
⣿⣿⣿⣿⣿⣿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⡁⠂⢿⠟⣿⣿⣿⢂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⡿⠻⢿⡏⠿⣾⣷⣿⣧⠄⣿⣀⣿⣽⣿⣿⡇⢤⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣷⣀⢹⣿⣿⣿⣳⡶⣍⣿⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠷⠶⢦⣤⣤⣬⡁⣿⡁⠀⠉⡏⣉⠉⢹⣍⡅⠹⠀
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⢿⡿⣿⣿⣿⣷⡯⠸⣾⣿⣿⣿⣿⣿⣿⣿⠿⢏⠙⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣀⣤⣤⣄⡀⠑⠛⢿⡍⢡⢂⣶⣦⠀⢟⣙⣿⣿⣷⡆⠀
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⠉⢡⣿⢾⣿⡀⢈⣉⣿⣿⣿⣿⣿⠏⠀⠀⠀⠁⠀⠀⠀⠈⠚⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣴⣛⣻⣿⣿⣿⣦⡀⣿⡇⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠰⡿⠿⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠈⢿⣿⣏⣼⡿⠟⢻⣿⣿⣿⣾⣿⣭⣽⣿⣟⣛⣿⣇⣽⣿⣷⣿⣿⣿⣿⠟⠁⠀⠉⠛⠙⠁
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣠⠤⣵⣷⡦⢄⣠⣤⡀⢠⣼⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠋⠀⠉⠈⠉⠉⠉⠻⣷⣄⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣼⣿⠛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⣤⠄⠀⡀⠀⠀
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⢿⣿⣿⣿⣿⣿⣿⣷⡞⠁⢴⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⡏⢩⠈⠈⠉⡁⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣲⣿⡦⣶
⠀⠀⠀⠀⠀⣿⡟⠛⢸⣿⣿⣿⣦⣤⣤⣴⡀⣠⢰⡆⢀⣂⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⢙⣿⣿⣷⣶⣿⣿⣿⣿⠀⢰⣶⣧⣷⡶⠟⠿⠻⠛⠙⢿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⢿⣯⠁⠒⣳⣸
⠀⠀⠀⠀⠀⣿⣇⠀⠀⣿⣿⣿⣿⢾⣿⣿⣿⣿⣿⣠⣾⣿⠰⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡿⠓⣸⣿⣿⣿⣿⣿⣿⢛⠻⡷⣽⠋⠈⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠻⠿⠿⣾⡀⠈⠓⠚⠛⣳
⠀⠀⠀⠀⠀⣿⣿⠀⠀⢹⣿⣿⣿⣿⡛⠉⢠⣾⣿⣿⣿⡟⠀⢻⡿⠋⣿⣿⣅⠰⠒⠚⠛⠋⢠⣿⣿⣿⣿⣿⣿⣿⣿⠆⣿⣶⣏⡀⠀⠀⠀⠀⠀⠀⠀⢠⠹⣿⣿⣿⣿⣿⣿⢿⠉⠀⠀⠀⠀⠀⠀⠀⠉⠤⣀⣠⡾⠿
⠀⠀⠀⠀⠀⣻⣿⡀⠀⠘⡟⠛⠛⠉⣧⡰⢜⣿⣿⣿⠛⢿⣶⡼⠧⠔⠛⠋⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⢟⠋⠛⠳⣦⣄⡀⠀⠀⠀⢸⠀⣿⣿⣿⣿⡿⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣄⣤⣶⣶
⠀⠀⠀⠀⠀⢿⣿⣷⠘⠀⠀⠀⠀⠀⢹⡍⠁⣹⡿⠋⠀⢀⠄⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⡽⠀⠀⠀⠉⠋⡀⠀⠀⢸⠀⣿⣿⣿⡿⠁⠀⢸⠀⠤⠄⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣾⣿⠀⠀⠀⠀⠀⠀⠀⠋⠘⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⣿⣿⣿⣿⣿⠋⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠁⠀⠀⠘⠀⠻⠿⢿⡇⠀⠀⠘⠀⣀⢂⣀⢁⣀⡀⢀⣤⣤⡈⢻⣿⣿
⠀⠀⠀⠀⠀⢨⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡎⠀⠀⠀⣿⣿⠛⢸⣿⣸⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠸⠙⠿⣿⠄⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⡀⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⢘⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢻⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⢰⣿⣿⣆
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢹⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣶⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⢘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⢸⣿⡏⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⡟
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠓
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⠠⣄⣀⣤⣤⣤⣤⣤⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⡀⣤⣠⢤⠤⣤⣤⣤⣤⡤⡤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⡀⣀⣰⣤⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⠀⠀⣠⡶⣾⣿⣿⡿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡧⣿⣷⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠾⣴⣶⣖⡛⣿⡗⠗⣎⠋⣶⣶⣶⡿⢶⢶⡆⡀⠀
⣿⣿⣿⣿⣿⣿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠡⠢⣿⠟⣿⣿⣿⢂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⡿⠻⢿⡋⠿⣾⣷⣿⣷⠄⣿⣀⣿⣽⣿⣿⡇⢤⠀
⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣾⣷⣀⢹⣿⣿⣿⣳⡮⣝⣿⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠞⠷⠶⢦⣥⣤⣬⣁⣿⡉⠀⠉⡍⣩⠉⣩⣍⡅⠹⠀
⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⣿⣿⣿⣷⡯⢸⣾⣿⣿⣿⣿⣿⣿⣿⠿⢏⠙⠻⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣀⣤⣤⣄⡀⠑⠛⣿⡉⢡⢆⣶⣦⠀⢿⣛⣿⣿⣿⡆⠁
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⠉⢡⣿⢾⣿⡀⢈⣉⣿⣿⣿⣿⣿⠏⠀⠀⠀⠁⠀⠀⠀⠈⠚⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣼⣛⣻⣿⣿⣿⣦⡀⣿⣇⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠰⣿⡿⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠈⢿⣿⣏⣽⡿⠟⢻⣿⣿⣿⣾⣿⣭⣽⣿⣿⣛⣿⡇⣽⣿⣿⣿⣿⣿⣿⠟⠁⠀⠉⠙⠙⠂
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠒⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣠⠤⣾⣷⡦⢄⣤⣤⡀⢠⣼⣿⡿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠋⠀⠉⠉⠉⠉⠉⠿⣷⣄⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣼⣿⠛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⣤⠄⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⢿⣿⣿⣿⣿⣿⣿⣷⡞⠁⢴⣿⣯⣻⣿⣿⣷⣿⣿⣿⣿⠏⢡⠈⠈⠉⡁⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣲⣿⡦⣴
⠀⠀⠀⠀⠐⣿⡿⠻⢸⣿⣿⣿⣶⣤⣤⣶⡆⣠⣰⡆⣀⣃⣼⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⢙⣿⣿⣷⣶⣶⣿⣿⣿⠀⢰⣦⣷⣷⡶⠟⠿⠻⠛⠙⢿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⠿⣍⠁⠒⣳⣼
⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⣿⠾⣿⣿⣿⣿⣿⣠⣿⣿⠱⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡿⠛⣸⣿⣿⣿⣿⣿⣿⢛⠻⡷⣿⠋⠉⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠻⠿⠿⣿⡀⠈⠒⠚⠿⣻
⠀⠀⠀⠀⠀⣿⣿⠀⠀⢻⣿⣿⣿⣾⡛⠉⢹⣿⣿⣿⣿⡟⠀⢻⡿⠛⣿⣿⣅⠰⠒⠚⠛⠏⢰⣿⣿⣿⣿⣿⣿⣿⣿⠆⣿⣶⣏⡀⠀⠀⠀⠀⠀⠀⠀⢠⠹⣿⣿⣿⣿⣿⣿⢿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠤⣀⣠⣾⠿
⠀⠀⠀⠀⠀⣿⣿⡀⠀⠘⡟⠛⠋⠁⣧⠰⣾⣿⣿⣿⠹⢻⢶⠼⠯⠔⠛⠋⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⢟⠋⠛⠳⣦⣄⡀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣄⣠⣶⣶
⠀⠀⠀⠀⠀⢿⣿⣷⠀⠀⠀⠀⠀⠀⢛⡍⠁⣹⡿⠁⠀⢠⠀⠐⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⡽⠀⠀⠀⠉⠙⠀⠀⠀⢸⠀⣿⣿⣿⡿⠁⠀⢸⠀⠤⠄⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣽⣿⠀⠀⠀⠀⠀⠀⠀⠃⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⣿⣿⣿⣿⣿⠋⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠁⠀⠀⠘⠀⠻⠿⢿⡇⠀⠀⠘⠀⣀⢀⣀⢁⣀⡀⢀⣤⣤⡈⢻⣿⣿
⠀⠀⠀⠀⠀⢸⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⢿⠿⠛⠁⢀⣀⣀⣀⣁⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡎⠀⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠸⠙⠿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⡀⣤⣄⢠⣤⣤⣴⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⢹⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢻⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⢰⣿⣿⣄
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣶⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⢘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⢸⣿⡏⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣇⣟
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠓
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠄⠀⠀⠀⠀⢀⣤⣤⠀⠀⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⡠⣄⣀⣤⣤⣤⣤⣤⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⡀⣤⣤⣤⠤⢤⣤⣤⣤⡤⡤⢀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣀⡀⣀⣠⣤⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⠀⠀⣠⡶⣾⣿⣿⡿⣿⣰⣿⣾⣿⣿⣿⣿⣿⣿⣹⣿⡽⣿⣯⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠼⣤⣶⡆⡻⣿⡟⠓⣌⠏⣶⣶⣶⣿⠶⢶⣆⡀⠀
⣿⣿⣿⣿⣿⣿⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⡁⠒⢿⡟⣿⢿⣿⢆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⡿⠻⢿⡏⠻⣾⣷⣿⢷⠄⣿⣀⣿⣽⣿⣿⡇⣤⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣷⣀⢻⣿⣿⣿⣳⡶⣽⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠗⠲⢦⣤⣤⣬⡁⣾⡉⠀⠉⡏⣩⠉⣩⣍⡅⠹⠃
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⡏⢸⣾⣿⣿⣿⣿⣿⣿⣿⠿⠏⠙⠻⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣀⣤⣤⣤⡀⠙⠛⣿⡝⢡⢂⣶⣦⠀⢿⣛⣿⣿⣿⡄⠀
⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⠉⢡⣿⢾⣿⡁⢈⣉⣿⣿⣿⣿⣿⠏⠀⠀⠀⠁⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣼⣟⣿⣿⣿⣿⣦⡀⣿⡇⣤⣼⣿⣿⣷⣿⣿⣿⣿⣿⣷⣧
⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠰⢿⡿⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠈⢿⣿⣏⣼⣿⠟⣻⣿⣿⣿⣼⣿⣭⣽⣿⣯⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠉⠙⠛⠂
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣠⠴⢶⣶⡦⢄⣠⣤⡀⢠⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠁⠀⠉⠉⠉⠛⠛⢿⣧⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⡿⠛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⣤⠄⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⠙⣛⢿⣿⣿⣿⣿⣿⣿⣷⡞⠁⢴⣿⣯⣿⣿⣿⣷⣿⣿⣿⣿⠏⢩⠈⠈⠉⡁⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⡦⣴
⠀⠀⠀⠀⠸⣿⡟⠙⢸⣿⣿⣿⣶⣦⣤⣶⣶⣠⣔⡗⣄⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⢙⣽⣿⣷⣶⣶⣿⣿⣿⠀⢰⣶⣷⣷⡶⠿⠟⠿⠛⠙⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⢿⣯⡃⠒⣻⣼
⠀⠀⠀⠀⠀⣿⡇⠀⠈⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣴⣿⣿⠱⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡿⠓⣸⣿⣿⣿⣿⣿⣿⢛⢻⡷⣽⠋⠉⠁⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠻⠿⠿⣿⡀⠉⠒⠶⠿⣻
⠀⠀⠀⠀⠀⣿⣿⠀⠀⢹⣿⣿⣿⣾⠛⠛⢻⣿⣿⣿⣿⡟⠀⢿⡿⠛⣿⣿⣁⠐⠒⠚⠿⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⠌⣿⣶⣏⡀⠀⠀⠀⠀⠀⠀⠀⢠⠹⣿⣿⣿⣿⣿⣿⢿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠤⣀⣠⣾⠿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠘⡟⠛⠋⠉⣄⢢⢽⣿⣿⡿⠹⣿⢶⠼⠧⡔⠛⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⢟⠛⠻⠳⣦⣄⡀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣄⣠⣶⣶
⠀⠀⠀⠀⠀⣿⣿⡇⠈⠀⠀⠀⠀⠀⢟⡉⢁⣽⠟⠁⠀⠢⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⡟⠀⠀⠀⠉⠛⠀⠀⠀⢸⠀⣿⣿⣿⡿⠃⠀⢸⠀⠤⠄⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠈⠁⠸⠋⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠘⣿⣿⣿⣿⣿⠋⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠻⠿⢿⡇⠀⠀⠘⠀⣀⢀⣀⢁⣀⡀⢀⣤⣤⡈⢻⣿⣿
⠀⠀⠀⠀⠀⢼⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠈⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡎⠀⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠛⠛⠿⣿⠄⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⡀⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⢸⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢻⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⢰⣿⣾⣄
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢹⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣶⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⢘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⢸⣿⡿⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⡟
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠓
⢀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣀⣤⣤⣤⣤⣤⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⡀⢤⡠⣤⠤⢤⣤⣤⣤⡤⡤⣀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣠⡀⣀⣠⣤⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⠀⠀⣠⡶⣾⣿⣿⡿⣿⣰⣿⣾⣿⣿⣿⣿⣿⣿⣹⣿⡿⣿⣯⣎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡾⣴⣦⣆⣻⣿⡟⠓⣏⠋⣶⣶⣶⣿⠶⢶⣆⡀⠀
⣿⣿⣿⣿⣿⣿⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⡑⠒⢿⡟⣿⣿⣿⢂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⡿⠻⢿⡏⠫⣼⣷⣿⣷⠀⣿⣀⣿⣽⣿⣿⡇⣤⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣼⣷⣄⢻⣿⣿⣿⣷⡖⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠗⠲⢤⣤⣤⣬⡁⣿⡉⠀⠉⡏⣉⠉⣩⣯⡅⠹⠂
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⡿⣿⣿⣿⣷⡯⢸⣾⣿⣿⣿⣿⣿⣿⣿⠿⠟⡉⠛⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣀⣤⣤⣤⡀⠙⠛⣿⡍⢡⢆⣶⣦⠈⠿⣙⣿⣿⣿⡄⠀
⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⠉⣡⣿⢾⣿⡀⢈⣉⣿⣿⣿⣿⣿⠏⠁⠀⠀⠈⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣼⣟⣿⣿⣿⣿⣦⡀⣿⣇⣤⣾⣿⣿⣾⣿⣿⣿⣾⣿⣷⣧
⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⢿⡿⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢿⣿⣏⣸⡿⠟⣻⣿⣿⣿⣼⣿⣭⣽⣿⣯⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠉⠙⠛⠃
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢰⠴⢶⣾⣦⠄⣠⣄⡀⢠⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠁⠈⠉⠉⠙⠛⠛⣿⣇⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⡿⠛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⣠⠄⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠘⢙⣛⣿⣿⣿⣿⣿⣿⣿⣧⡞⠁⢴⣿⣿⣻⣿⣿⣷⣿⣿⣿⣿⠏⢩⠈⠈⠉⡁⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⡦⣴
⠀⠀⠀⠀⢸⣿⡿⠙⢸⣿⣿⣿⣦⣶⣴⣶⣶⣤⣶⣿⣌⣁⣼⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⠙⣽⣿⣷⣶⣶⣿⣿⣿⠀⢰⣶⣷⣷⡶⠾⠟⠿⠛⠙⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⠿⣞⠃⠒⣳⣸
⠀⠀⠀⠀⠈⣿⡇⠀⠈⣿⣿⣿⣟⠺⢿⣿⣿⣿⣿⣾⣿⣿⠱⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡿⠛⣸⣿⣿⣿⣿⣿⣿⢛⠻⡷⢿⠋⠉⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠻⠿⠿⣿⡀⠉⠓⠺⠿⣻
⠀⠀⠀⠀⠀⣿⣷⠀⠀⢻⣿⣿⣿⣿⠛⠛⢻⣿⣿⣿⣿⡟⠀⢻⡿⠋⣿⣿⣅⠐⠲⠚⠟⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣶⣏⡀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣿⣿⣿⣿⣿⣿⢿⠋⠀⠀⠀⠀⠀⠀⠈⠉⠤⣀⣠⣾⠿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠸⡟⠛⠋⢸⣔⢤⣿⣿⣿⡟⠟⡿⠶⠾⠧⡔⠛⠋⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⢟⠋⠛⠳⣦⣄⡀⠀⠀⠀⢸⠀⣻⣿⣿⣿⣿⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣀⣠⣶⣶
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠘⣯⠉⢈⣽⠟⠀⠀⡢⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⡟⠀⠀⠀⠙⠙⠀⠀⠀⢸⠀⣿⣿⣿⡿⠃⠀⢸⠀⠤⠄⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠘⠁⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠀⠻⠿⢿⡇⠀⠀⠘⠀⣀⢀⣀⢁⣀⡀⢀⣤⣤⡈⢻⣿⣿
⠀⠀⠀⠀⠀⣼⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠈⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡎⠀⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠚⠸⠿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⡀⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⢹⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⢰⣿⣾⣆
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣶⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⢘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⣸⣿⡟⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣟
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠄⠀⠀⠀⠀⠀⣠⡤⠀⠀⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⡠⣄⣀⣤⣤⣤⣤⣤⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⡀⢤⣠⣤⠤⢤⣤⣤⣤⠤⡤⠀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣀⠀⣀⢠⣤⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⠀⠀⣠⡶⣾⣿⣿⡿⣿⣰⣿⣾⣿⣿⣿⣿⣿⣿⣹⣿⡿⣿⣩⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠼⣤⣶⡆⡛⣿⡗⠓⣌⠉⣶⢶⣶⣿⠷⢶⣆⡀⠀
⣿⣿⣿⣿⣿⣿⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⡑⠢⢯⠟⣿⣿⡿⠢⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⠿⠻⢿⡏⠻⣾⣷⣿⣧⠀⣿⣀⣿⣽⣿⣿⡇⢤⠀
⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣼⣷⣀⢹⣿⣿⣿⣷⡶⣍⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⠒⢤⣦⣤⣬⡁⣾⡉⠀⠉⡏⣉⠉⢩⣍⠅⠰⠂
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣻⣿⣿⡿⣿⣿⣾⣷⡯⠹⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⠉⠛⠟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⣤⣤⣤⡀⠉⠛⣿⡍⢡⢂⣶⣦⠀⠏⣙⣻⣿⣿⡆⠀
⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠊⠉⣡⣿⢶⣿⡀⢈⣉⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣿⣿⣿⣿⣻⣦⡀⣿⣇⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⢿⡿⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣏⣽⡟⠟⢻⣿⣿⣿⣼⣿⣭⣿⣿⣯⣿⣿⡇⣿⣿⣷⣿⣿⣿⣿⡟⠉⠀⠉⠙⠛⠃
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢰⠤⢶⣾⣦⠀⢀⣀⠀⢀⣪⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠋⠀⠀⠁⠈⠈⢻⣿⣿⢧⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⡿⠛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⣠⠄⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠘⡙⢈⣙⣿⣿⣿⣿⣿⣿⣿⣧⡞⠁⢴⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⠛⢩⠈⠀⠉⡁⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⡦⣤
⠀⠀⠀⠀⢸⣿⡏⡩⢸⣿⣿⣷⣦⣾⣶⣾⣿⣷⣿⡗⣈⣁⣼⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⢙⣽⣿⣷⣶⣾⣿⣿⣿⠀⢰⣶⣧⣷⡶⠿⠟⠿⠋⠻⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣞⠓⠒⣳⣼
⠀⠀⠀⠀⠘⣿⡄⠀⠸⣿⣿⣿⣿⠚⢛⣿⣿⣿⣿⣿⣿⣿⠡⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡟⠓⣺⣿⣿⣿⣿⣿⣿⢒⠻⡷⣽⠉⠁⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠛⠿⠿⣷⡀⠈⠐⠶⠿⣻
⠀⠀⠀⠀⠀⣿⣧⠀⠀⣿⣿⣿⣿⡿⠛⠛⣿⣿⣿⣿⣿⡟⠀⢻⡿⠛⣻⣿⣅⠰⠶⠚⠟⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⠨⣷⣶⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⢿⠋⠀⠀⠀⠀⠀⠀⠀⠁⠤⡀⣠⣾⠿
⠀⠀⠀⠀⠠⣿⣿⠀⠀⢸⠟⠛⠉⣦⡠⢘⣿⣿⣿⠻⠻⠷⠖⠚⢣⡔⠛⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠘⢟⠋⠛⠷⣦⣄⡀⠀⠀⠀⢀⠀⣻⣿⣿⣿⣿⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣀⣠⣶⣶
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢻⣍⠁⣨⡿⠁⠀⢀⠂⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⡟⠀⠀⠀⠉⠹⠀⠀⠀⢸⠀⣿⣿⣿⡿⠁⠀⢸⠀⠤⠄⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣩⣿⣿⠀⠀⠀⠀⠀⠀⠋⠀⠟⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⣿⣿⣿⣿⡿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠿⠿⢿⡇⠀⠀⠘⠀⣀⢀⣀⢁⣀⡀⢀⣤⣤⡉⢻⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠘⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡎⠀⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠗⠻⠿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⡀⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⢸⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢻⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⢰⣿⣾⣄
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⢘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⣿⣿⡟⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡃⡟
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃
⢀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠄⠀⠀⠀⠀⢀⣤⡤⠀⠀⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣀⣤⣤⣤⣤⣤⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⡀⢤⣤⣤⠤⢤⣤⣤⣤⡤⡤⢀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣀⠀⣀⣰⣤⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⠀⠀⣠⡶⣾⣿⣿⡿⣿⣰⣿⣾⣿⣿⣿⣿⣿⣿⣹⣿⡿⣿⣫⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠾⣴⣶⡆⡻⣿⡟⠓⣌⠋⣶⣶⣶⣿⣷⢶⣆⡀⠀
⣿⣿⣿⣿⣿⣿⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⡑⠒⢿⡟⣿⣿⣿⢦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⠿⠻⢿⡏⠻⣾⣷⣿⣷⠄⣿⣀⣿⣿⣿⣿⡇⢤⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣼⣷⣀⢹⣿⣿⣿⣳⡦⣽⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⠒⢤⣦⣤⣬⣁⣾⡉⠀⠉⡏⣩⠉⣹⣯⡅⠹⠂
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⣿⣿⣿⣷⡯⢸⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⠉⠛⠟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⣤⣤⣤⡀⠉⠛⣿⡝⢡⢂⣶⣦⠈⢯⣻⣻⣿⣿⡆⠀
⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⠉⣡⣿⢾⣿⡀⢈⣉⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣿⣿⣿⣿⣻⣦⡀⣿⡇⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠰⢿⡿⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣏⣽⡿⠟⢻⣿⣿⣿⣼⣿⣭⣿⣿⣯⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠉⠛⠛⠃
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢰⠴⢶⣾⣦⠀⢀⣀⠀⢀⣪⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠋⠀⠀⠁⠈⠈⢻⣿⣿⣷⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⡿⠛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⣠⠄⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠘⡙⢈⣙⣿⣿⣿⣿⣿⣿⣿⣧⡞⠃⢴⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⠛⢩⠈⠈⠉⡁⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⡦⣤
⠀⠀⠀⠀⢸⣿⡟⠿⢹⣿⣿⣷⣦⣾⣶⣾⣿⣷⣿⣷⣈⣁⣼⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⢛⣿⣿⣷⣶⣾⣿⣿⣿⠀⢰⣶⣷⣷⡶⠿⠟⠿⠋⠻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣏⡃⠒⣻⣼
⠀⠀⠀⠀⠨⣿⡄⠀⠸⣿⣿⣿⣿⠚⢛⣿⣿⣿⣿⣿⣿⣿⠱⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡟⠓⣺⣿⣿⣿⣿⣿⣿⢖⠻⡷⡽⠉⠉⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠻⠿⠿⣷⡀⠉⠒⠾⠿⣻
⠀⠀⠀⠀⢰⣿⣧⠀⠀⣿⣿⣿⣿⡿⠛⠛⣿⣿⣿⣿⣿⡟⠀⢻⣿⠋⣿⣿⣅⠰⠶⠺⠿⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⠘⣷⣶⣃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠻⣿⣿⣿⣿⣿⣿⢿⠋⠀⠀⠀⠀⠀⠀⠀⠁⠤⡀⣠⣾⠿
⠀⠀⠀⠀⢰⣿⣿⠀⠀⢸⠟⠛⠉⣦⡰⢎⣿⣿⣿⠹⠿⠷⠖⠚⢣⡖⠛⠋⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠙⢟⠛⠛⠷⣦⣄⡀⠀⠀⠀⢀⠀⣿⣿⣿⣿⣿⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣀⣠⣦⣶
⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⢻⣍⠉⣨⡿⠃⠀⢀⠂⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⡟⠀⠀⠀⠉⠙⠀⠀⠀⢸⠀⣿⣿⣿⡿⠁⠀⢸⠠⠤⠄⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠋⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⣿⣿⣿⣿⡿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠿⠿⢿⡇⠀⠀⠘⠀⣀⢀⣀⢁⣀⡀⢀⣤⣤⡉⢻⣿⣿
⠀⠀⠀⠀⠀⡿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠘⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡎⠀⠀⠀⣿⣿⠛⢸⣿⣸⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠓⠺⠿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⡀⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⢸⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢻⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⢰⣿⣾⣄
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢹⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣶⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⢘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⣿⣿⡿⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⡟
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠀⠀⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⡠⣄⣀⣤⣤⣤⣤⣤⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⡀⢤⡄⡤⠤⠤⣤⣤⣤⡤⡄⣀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣀⠀⣀⢠⣤⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⠀⠀⣠⡶⣾⣿⣿⡿⣷⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⡷⣿⣥⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠼⣴⣴⣆⡛⣿⡗⠓⣎⡋⣶⣶⣶⣿⢶⢶⡆⡀⠀
⣿⣷⣶⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠑⠢⢯⠟⣿⣿⣿⠆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⠟⠻⠿⡏⠻⣾⡷⡿⣷⠀⣿⣀⣿⣽⣿⣿⡇⢤⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣼⣷⣀⢹⣿⣿⣿⣳⡖⣝⡿⢏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠓⠒⢤⣥⣤⣬⡁⣾⡉⠀⠉⡏⣩⠉⣹⣏⠅⠹⠂
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⡿⣿⣿⣿⣷⡯⠸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⡉⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⣤⣤⣤⡀⠉⠙⣿⡉⢩⢂⣴⣦⠈⠟⣙⣿⣿⣿⡆⠀
⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠉⣡⣿⢾⣿⡀⢈⣉⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⠀⠀⠀⠀⠫⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣿⣿⣿⣟⣻⣦⠀⣿⣇⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⢿⠿⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠻⣿⣏⣽⡿⠟⣻⣿⣿⣿⣾⣿⣭⣿⣿⣯⣿⣿⡇⣿⣿⣾⣿⣿⣿⣿⡟⠉⠀⠈⠙⠛⠃
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡾⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢰⠴⢦⣶⣤⠀⢀⡀⠀⠀⢀⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⠟⠁⠈⠁⠁⠈⠈⢿⣿⣿⣧⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡇⣸⡿⠛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⣠⠄⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣀⠀⢙⠉⢈⣙⣿⣿⣿⣿⣿⣿⣿⣧⡜⠁⢴⣿⣯⢿⣿⣿⣯⣿⣿⣿⣿⠓⢫⠈⠀⠉⡁⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢶⣿⡤⣤
⠀⠀⠀⠀⢨⣿⠻⠉⢹⣿⣿⣰⣶⣾⣿⣾⣿⣿⣿⣢⣈⣁⣼⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⠙⣿⣿⣴⣤⣽⣿⣿⣿⠀⢰⣶⣷⣷⠶⠿⠟⠟⠋⠻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣎⠃⠒⣳⣸
⠀⠀⠀⠀⠘⣿⠀⠀⢸⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⠱⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡟⠓⣺⣿⣿⣿⣿⣿⣿⢖⠻⡷⡽⠉⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠻⠿⢿⣷⠀⠉⠲⠾⠿⣫
⠀⠀⠀⠀⢰⣿⡇⠀⠈⣿⣿⣿⡿⠙⠋⢻⣿⣿⣿⣿⣿⡟⠀⢻⡿⠋⣿⣿⣁⠰⠶⠺⠿⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⠘⣷⣶⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⢿⠋⠀⠀⠀⠀⠀⠀⠀⠁⠤⡀⣠⣾⡿
⠀⠀⠀⠀⢸⣿⣇⠀⠀⢹⠟⠋⢣⡘⢴⣿⣿⣿⡏⢟⠿⠗⠒⠚⣧⠴⠛⠋⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠙⢟⠛⠛⠷⣦⣄⡀⠀⠀⠀⠃⠀⣿⣿⣿⣿⣿⠃⢰⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⣀⣀⣦⣶
⠀⠀⠀⠀⠈⣿⣿⠂⠀⠀⠀⠀⠸⣯⠉⢀⣿⠏⠀⠀⡐⠀⠉⠋⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⡟⠀⠀⠀⠈⠙⠀⠀⠀⠠⠀⣿⣿⣿⡿⠁⠀⢸⠀⠤⠄⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣚⣿⡇⠀⠀⠀⠀⠀⠘⠁⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⣿⣿⣿⣿⡿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⠇⠀⠀⠘⠀⣀⢀⣀⢁⣀⡀⢀⣤⣤⡉⢻⣿⣿
⠀⠀⠀⠀⠀⡾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠘⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡎⠀⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠐⠃⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⡀⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⢸⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⣴⣿⣾⡄
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⠸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⢘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⣿⣿⡟⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡃⡟
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠀⠀⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣀⣤⣤⣤⣤⣤⣤⡄⢤⣤⣤⣤⣤⣤⣤⣤⡀⢤⡠⣤⠤⢤⣤⣤⣤⡤⡤⠀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⡀⠀⣀⣠⣤⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⠀⠀⢠⡶⣾⣿⣿⡿⣿⣰⣿⣾⣿⣿⣿⣿⣿⣿⣹⣿⡷⣿⣯⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠺⣤⣶⡆⡛⢿⡟⠓⣌⠏⣶⢶⣶⣷⠶⢶⣆⡀⠀
⣿⣷⣶⣿⣿⣿⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⡑⠰⢯⠟⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⠿⠻⠿⡋⠳⣾⣷⣿⣷⠀⣿⣀⣿⣿⣿⣿⡇⣤⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣽⣷⣀⢹⣿⣿⣯⣷⡦⣉⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠞⠓⠲⢤⣤⣤⣬⡁⣾⡉⠀⠉⡏⣩⠉⢩⣏⠅⠹⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⡿⣿⣿⣿⣷⡏⢹⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⡛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⣤⣤⣄⠀⠉⠙⣿⡙⢡⢂⣴⣦⠈⢿⣛⣿⣿⣿⡆⠀
⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠊⠉⣡⣿⢾⣿⡁⢈⣉⣿⣿⣿⣿⣿⠟⠁⠀⠀⠘⠀⠀⠀⠈⠪⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣿⣿⣿⣟⣻⣦⠀⣿⣇⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⢿⡿⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣏⣹⡿⠟⢻⣿⣿⣿⣾⣯⣭⣿⣿⣟⣿⣿⡇⣿⣿⣶⣿⣿⣿⣿⡟⠉⠀⠈⠙⠛⠃
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡾⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢰⠴⢦⣶⣄⠀⠀⡀⠀⠀⢀⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠋⠀⠉⠀⠁⠈⠙⣿⣿⣿⣧⣄⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡆⣰⡿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⣠⠄⠀⠀⠀⠀
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣀⣈⣟⠋⢈⣙⣿⣿⣿⣿⣿⣿⣿⣧⡞⠁⢴⣿⣯⣿⣿⣿⣭⣿⣿⣿⣿⠓⢫⠈⠀⠉⡁⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢶⣿⡤⣤
⠀⠀⠀⠀⢺⣿⠟⠋⢻⣿⣇⣶⣶⣿⣿⣿⣿⣿⣿⣦⣌⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⠙⣿⣿⣤⣭⣿⣿⣿⣟⠀⢰⣶⣧⣷⣶⠿⠿⠟⠋⠻⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣎⠃⠐⣻⣰
⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⠱⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡟⠓⣺⣿⣿⣿⣿⣿⣿⢖⠻⡷⠽⠉⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠻⠿⢿⣷⠀⠉⠲⠶⠿⣫
⠀⠀⠀⠀⢸⣿⡇⠀⠈⣿⣿⣿⡏⠉⠉⣻⣿⣿⣿⣿⣿⡟⠀⢿⡿⠋⣿⣿⣁⠰⠶⠾⠿⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⠸⣷⣴⣃⠀⠀⠀⠀⠀⠀⠀⠀⡌⢻⣿⣿⣿⣿⣿⣿⢿⠋⠀⠀⠀⠀⠀⠀⠀⠁⠤⡀⣠⣶⡿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⢿⠟⠉⣄⡫⡾⣿⣿⣿⠛⠿⠿⠓⢒⠺⡇⠐⠛⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠉⢟⠋⠙⠷⣦⣄⡀⠀⠀⠀⠇⠀⢿⣿⣿⣿⡿⠃⢰⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⣈⣀⣦⣶
⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⢻⣋⠉⣨⡿⠃⠀⠈⠆⠀⠉⠙⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⡟⠀⠀⠀⠙⠹⠀⠀⠀⠀⠀⣿⣿⣿⡟⠁⠀⢸⠀⠤⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠙⠀⠛⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⣿⣿⣿⣿⡿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⠃⠀⠀⠘⠀⣀⢀⣀⢁⣀⡀⢀⣤⣤⡉⢻⣿⣿
⠀⠀⠀⠀⢀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠘⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡎⠀⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠐⠇⠿⠿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⡀⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⢸⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⣴⣿⣾⡄
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⠸⣿⣧⡄⣿⣧⣶⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⣘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⣿⣿⡿⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡃⡟
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠀⠀⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⡠⣤⣀⣤⣤⣤⣤⣤⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⡀⢤⡤⠤⠤⢤⣤⣤⣤⠤⡄⣀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⢀⠀⣀⢠⣤⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⠀⠀⣠⡶⣼⣿⣿⡿⣷⣰⣿⣾⣿⣿⣿⣿⣿⣿⣹⣿⡧⣿⣫⣎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠼⣤⣤⡆⡻⣿⡗⠗⢍⠋⣶⣶⣶⡿⡶⢶⡆⡀⠀
⣿⣷⣦⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⡑⠂⢿⠟⣿⢿⣿⢂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⠟⠻⠿⡏⠻⣾⣷⡿⣦⠀⣿⣀⣿⣽⣿⣿⡇⢤⠀
⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣼⣷⣀⢹⣿⣿⣯⣳⡦⣉⣿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠷⠶⢤⣥⣤⣬⡁⣾⡉⠀⠉⠍⣩⠉⢹⣏⡅⠹⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣷⡿⡾⣿⣾⣷⡏⠸⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⡛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⣤⣤⣄⠀⠉⠙⣿⡉⢡⢂⣶⣦⠀⢏⣹⣿⣿⣿⡆⠀
⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠉⣡⣿⢾⣿⡀⢈⣉⣿⣿⣿⣿⣿⠟⠁⠀⠀⠐⠀⠀⠀⠈⠪⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣿⣿⣿⣿⣷⣦⠀⣿⡇⣤⣼⣿⣿⣿⣿⣽⣿⣿⣿⣷⣦
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⣿⡿⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠛⣿⣏⣹⡿⠟⣻⣿⣿⣿⣾⣯⣭⣿⣿⣟⣻⣿⡇⣿⣿⣷⣿⣿⣿⣿⡟⠉⠀⠈⠙⠛⠂
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⡾⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡠⠔⣴⣦⣄⠀⠀⠀⠀⠀⠀⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⠀⠁⠈⠀⠙⢹⣿⣿⣿⣧⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⣿⣿⣷⣾⣾⣿⡆⣠⡿⠛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⣠⠄⠀⠀⠀⠀
⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⠀⠀⠀⡀⠀⢀⣀⣹⣟⠉⢈⣉⣿⣿⣿⣿⣿⣿⣿⣧⡞⠁⢴⣿⣯⣿⣿⣿⣽⣿⣿⣿⣿⠃⢫⠈⠀⠉⡁⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⡤⣤
⠀⠀⠀⠀⢻⡿⠋⠏⢻⣿⡄⣶⣶⣿⣿⣿⣿⣿⢿⣦⣌⣁⣼⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⠙⣿⣿⣿⣭⣿⣿⣿⣟⠀⢰⣶⣷⣷⣶⢿⡿⠟⠏⢻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣎⠃⠐⣻⣼
⠀⠀⠀⠀⠘⣧⠀⠀⢸⣿⣿⣿⡟⠉⣿⣿⣿⣿⣿⣿⣿⣿⠱⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡟⠓⣺⣿⣿⣿⣿⣿⣿⢖⠻⡷⡽⠉⠀⠀⠀⠀⠀⠀⠹⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠙⠿⠿⣿⣷⠀⠉⠲⠶⠿⣫
⠀⠀⠀⠀⢸⣿⡀⠀⠸⣿⣿⣿⠇⠉⠙⣿⣿⣿⣿⣿⣿⡟⠀⢿⡿⠛⣿⣿⣁⠰⠶⠾⠿⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⠘⣷⣴⣁⠀⠀⠀⠀⠀⠀⠀⠀⡌⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠁⢤⡀⣀⣶⡿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⢿⠟⢱⣌⠢⣘⣿⣿⡿⠻⡻⠻⢛⣂⢺⡇⠔⠛⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠉⢟⠋⠛⠷⣦⣄⡀⠀⠀⠀⡇⠈⣿⣿⣿⣿⡿⠃⢰⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⣈⣀⣦⣦
⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠈⢿⡉⠁⣸⡿⠁⠀⢹⠀⠈⠉⠁⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⡟⠀⠀⠀⠉⠹⠀⠀⠀⠆⠀⣿⣿⣿⡟⠁⠀⢸⠀⣤⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿
⠀⠀⠀⠀⢈⣻⣿⡄⠀⠀⠀⠀⠈⠋⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⣿⣿⣿⣿⡿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⠁⠀⠀⠘⠀⣀⢀⣀⢀⣀⡀⢀⣤⣤⡉⢻⣿⣿
⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠘⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡎⠀⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠈⠋⠿⢿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣁⣀⡀⣤⣄⢠⣤⢤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⠹⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⣴⣿⣾⡅
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⠸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⢘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⣿⣿⡿⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⡿
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠄⠀⠀⠀⠀⠀⣠⡤⠀⠀⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⡠⣤⣀⣤⣤⣤⣤⣤⣤⡄⢤⣤⣤⣤⣤⣤⣤⣤⡀⣤⡤⢤⠤⢤⣤⣤⣤⡤⡤⣀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⣀⣠⣤⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⠀⠀⣠⡶⣼⣿⣿⡿⣷⣰⣿⣾⣿⣿⣿⣿⣿⣿⣹⣿⡷⣿⣭⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠼⣴⣤⣆⣻⣿⡗⠓⣍⠋⣶⣶⣶⣿⢶⢶⣆⡀⠀
⣿⣷⣦⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⡁⠂⣯⠟⣿⢿⣿⢂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣽⡟⠛⠿⡏⠻⣾⡷⡿⣶⠀⣿⣀⣿⣽⣿⣿⡇⢤⠀
⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣷⣀⢹⣿⣿⣯⣳⡦⣝⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠿⠶⢶⣥⣤⣬⡁⣿⡉⠀⠉⡏⣩⠉⢹⣍⡅⠹⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⡿⣿⣿⣾⣷⡏⢹⣾⣿⣿⣿⣿⣿⣿⣿⠿⢿⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⣤⣤⣄⠀⠈⠙⣿⡉⢡⢂⣶⣦⠀⢏⣙⣿⣿⣿⡆⠀
⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠉⢡⣿⢾⣿⡀⢈⣉⣿⣿⣿⣿⣿⠟⠁⠀⠀⠑⠀⠀⠀⠈⠪⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣿⣿⣿⣟⣳⣦⠀⣿⣇⣤⣼⣿⣿⣿⣿⣽⣿⣿⣿⣷⣧
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⢿⠾⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠛⣿⣏⣽⡿⠟⣻⣿⣿⣿⣾⣯⣭⣿⣿⣟⣿⣿⡇⣿⣿⣷⣿⣿⣿⣿⡟⠉⠀⠈⠙⠛⠂
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣠⠴⣴⣦⡄⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠈⠁⠁⠈⠩⢻⣿⣿⣿⣧⣄⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⣿⣿⣷⣾⣾⣿⡆⢠⣿⠛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣧⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⣠⠄⠀⠀⠀⠀
⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⣀⣹⣟⠋⣈⣙⣿⣿⣿⣿⣿⣿⣿⣧⡞⠁⢴⣿⣯⣿⣿⣿⣽⣿⣿⣿⣿⠓⢫⠈⠀⠉⡁⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢦⣿⡤⣤
⠀⠀⠀⠀⢺⣟⠫⠃⣿⣿⣰⣶⡾⣿⣿⣿⣿⡿⢻⣦⣈⣁⣼⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⢙⣿⣿⣿⣭⣿⣿⣿⡟⠀⢰⣶⣧⣷⣶⢿⠿⠟⠛⢻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣞⠃⠐⣿⣸
⠀⠀⠀⠀⢸⣇⠀⠀⢸⣿⣿⣿⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⠱⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡟⠓⣻⣿⣿⣿⣿⣿⣿⢖⠻⡷⡝⠉⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⠿⠿⣿⣷⠀⠈⠲⠶⠿⣫
⠀⠀⠀⠀⢨⣿⠀⠀⢸⣿⣿⡿⠈⠉⢹⣿⣿⣿⣿⣿⣿⡟⠀⢿⣿⠋⣿⣿⣁⠰⠶⠾⠿⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⠸⣷⣴⣁⠀⠀⠀⠀⠀⠀⠀⠀⡎⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠁⢤⡀⣀⣶⡿
⠀⠀⠀⠀⣾⣿⡆⠀⠀⢿⠋⢧⡙⠶⣻⣿⣿⡟⠟⠿⠟⢛⣘⢾⡇⠐⠛⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠉⢟⠋⠛⠷⣦⣄⡀⠀⠀⠀⡇⢈⣿⣿⣿⣿⡿⠃⢰⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⣉⣀⣦⣶
⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠸⣟⡉⢁⣽⠟⠀⠀⡑⠀⠈⠉⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⡟⠀⠀⠀⠉⠹⠀⠀⠀⡇⠘⣿⣿⣿⡟⠁⠀⢸⠀⣤⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣹⣿⡀⠀⠀⠀⠀⠘⠃⠘⠋⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⣿⣿⣿⣿⡿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⠁⠀⠀⠘⠀⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⢻⣿⣿
⠀⠀⠀⠀⢸⡹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠘⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡎⠀⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠸⠉⠿⠿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣁⣀⡀⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⠻⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⣴⣿⣾⣅
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⠸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⣘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⣿⣿⡟⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⡟
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣤⣤⡄⣤⣤⣤⣤⣤⣤⣤⣤⡀⣤⡠⣤⢤⣤⣤⣤⣤⡤⡤⣀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⣀⢠⣤⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⠀⠀⣠⡶⣾⣿⣿⡿⣷⣰⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣯⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠼⣠⣴⣇⣛⣿⡗⠗⣏⡉⣶⣶⣶⣿⢷⢶⣆⡀⠀
⣿⣷⣦⣿⣿⣿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⡁⠳⣯⠟⣿⢿⣿⢆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⡟⠻⠿⡏⠻⣾⣷⣿⣧⠀⣿⣀⣿⣽⣿⣿⡇⣤⠀
⣽⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣷⣀⢻⣟⣿⣿⣧⡦⣝⣿⣪⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠞⠿⠾⢶⣷⣤⣬⡁⣿⡉⠀⠉⡏⣩⠉⢩⣍⡅⠺⠅
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⣿⣿⣿⣷⡯⠹⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⠛⠻⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⣤⣤⣀⠀⠀⠙⣿⡏⢡⢂⣶⣦⠀⢟⣙⣿⣿⣿⡆⠀
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠉⣡⣿⢾⣽⡀⢈⡉⣿⣿⣿⣿⣿⡟⠁⠀⠀⠑⠀⠀⠀⠈⠫⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣭⣿⣿⣟⣳⣦⠀⣿⡇⣠⣾⣿⣿⣿⣿⣽⣿⣾⣿⣷⣦
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⣿⡷⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣏⣽⡟⠟⣻⣿⣿⣿⣾⣿⣭⣿⣿⣟⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⡟⠉⠀⠉⠙⠛⠃
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡶⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣆⣠⠠⣤⣦⡀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢘⣿⣿⣿⣿⣿⡇⠈⠁⠈⠀⠉⢻⣿⣿⣿⣿⣧⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣶⣾⣿⡆⢠⣿⠛⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⣠⠄⠀⠀⠀⠀
⠀⠀⠀⠐⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣀⣹⣿⣟⠋⢈⣙⢿⣿⣿⣿⣿⣿⣿⣷⡜⠁⢴⣿⣯⣿⣿⣿⣽⣿⣿⣿⣿⠆⢋⠈⠀⠉⡁⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⡤⣤
⠀⠀⠀⠀⢿⡟⠋⠃⣿⣿⣶⣶⣿⣿⣿⣿⣿⡟⣻⣦⣈⣁⣼⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠙⣿⣿⣭⣭⣿⣿⣿⡟⠀⢰⣶⣧⣷⠶⢿⠻⠟⠛⢿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⣟⠃⠐⣿⣤
⠀⠀⠀⠀⢸⡆⠀⠀⢿⣿⣿⣟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠱⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡟⠓⣻⣿⣿⣿⣿⣿⣿⢒⢻⡷⠝⠉⠁⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠙⠿⠿⣿⣷⡀⠉⠲⠾⠿⢫
⠀⠀⠀⠀⢾⣿⠀⠀⢸⣿⣿⠈⠋⢙⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⡿⠟⣿⣿⣁⠰⠶⠾⠿⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠐⠉⢤⡀⣀⣶⡿
⠀⠀⠀⠀⣟⣿⡀⠀⠈⠿⢡⣈⠴⣼⣿⣿⡿⠻⠻⠛⢋⣉⣳⣸⡇⠴⠛⠋⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠘⢟⠋⠟⠳⣦⣀⡀⠀⠀⠀⡇⢈⣿⣿⣿⣿⡿⠃⢰⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣉⣀⣤⣤
⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠈⢿⣋⠉⣨⡿⠁⠀⢈⠂⠀⠉⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⡗⠀⠀⠀⠉⠹⠀⠀⠀⡇⠸⣿⣿⣿⡟⠁⠀⢸⠀⣤⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿
⠀⠀⠀⠀⢘⣿⣿⠀⠀⠀⠀⠀⠛⠀⠟⠁⠀⠀⠈⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠿⠿⣿⠁⠀⠀⠘⠀⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⢻⣿⣿
⠀⠀⠀⠀⣸⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠘⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡴⣶⡶⣶⡎⠀⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠸⠸⠿⠿⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢉⣁⣀⡀⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⠟⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⡿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⣴⣿⣶⣇
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⣘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⣿⣿⡟⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡓⡟
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠓
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣤⣤⡄⢤⣤⣤⣤⣤⣤⣤⣤⡀⣤⡠⡤⢤⣤⣤⣤⣤⡤⡄⣀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⣀⢠⣤⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⠀⠀⣠⡶⣾⣿⣿⡿⣷⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⡷⣿⣯⣯⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠼⣠⣴⡆⣻⣿⡟⠗⣏⡋⣶⣶⣶⣿⠷⢶⣆⡀⠀
⣿⣷⣦⣿⣿⣿⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠁⠠⣯⠟⣿⣿⣿⢆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⡟⠻⢿⡏⠻⣾⣷⣿⣦⠀⣿⣀⣿⣽⣿⣿⡇⢠⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣷⣀⢻⣿⣿⣽⣧⡶⣍⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠿⠿⢶⣥⣤⣬⡁⣿⡉⠀⠉⡏⣩⠉⢹⣍⡅⠹⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⣿⣿⣿⣷⡯⢹⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⠛⠻⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⣤⣤⣄⠀⠀⠙⣿⡋⢡⢂⣶⣦⠀⠟⣙⣿⣿⣿⡆⠀
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠉⣡⣿⢾⣽⡀⢈⣉⣿⣿⣿⣿⣿⡟⠁⠀⠀⠑⠀⠀⠀⠈⠫⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣭⣿⣿⣟⣳⣦⠀⣿⡇⣠⣼⣿⣿⣿⣿⣽⣿⣾⣿⣷⣦
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⣿⡷⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣏⣹⡿⠟⣻⣿⣿⣿⣾⣯⣭⣿⣿⣟⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⡟⠉⠀⠉⠙⠛⠃
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡶⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣠⠠⣤⣦⣀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠈⠁⠁⠈⠛⢿⣿⣿⣿⣿⣇⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⣿⣿⣷⣾⣾⣿⡆⢠⡿⠛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⣠⠄⠀⠀⠀⠀
⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣀⣈⣿⣿⣟⠋⢈⣙⣿⣿⣿⣿⣿⣿⣿⣷⡜⠁⢴⣿⣯⣿⣿⣿⣽⣿⣿⣿⣿⠆⢋⠈⠀⠉⡁⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⢶⣿⡤⣤
⠀⠀⠀⠀⢼⡟⠏⠁⣿⣷⣶⣾⣿⣿⣿⣿⣿⡟⢻⣦⣈⣁⣼⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⢙⣿⣿⣭⣭⣿⣿⣿⡟⠀⢰⣶⣧⣷⠶⢿⠿⠟⠋⢿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢿⣚⠃⠐⢿⣨
⠀⠀⠀⠀⢸⡇⠀⠀⢿⣿⡿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠱⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡟⠓⣻⣿⣿⣿⣿⣿⣿⢒⠻⡷⡝⠉⠁⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⠿⠿⣿⣷⡀⠉⠲⠶⠿⢫
⠀⠀⠀⠀⣼⣿⠀⠀⢸⣿⡧⠘⠛⢻⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⡿⠏⣿⣿⣁⠰⠶⠾⠿⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢿⣿⣿⣿⣿⣿⣿⢿⠋⠀⠀⠀⠀⠀⠀⠁⠁⡤⡀⣀⣶⡿
⠀⠀⠀⠀⣿⣿⡀⠀⠀⡟⢇⡠⠤⣽⣿⣿⡟⠿⠿⠋⢉⣉⢦⢸⡇⠐⠛⠋⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠘⢟⠛⠛⠲⣦⣄⡀⠀⠀⠀⡇⢈⣿⣿⣿⣿⡿⠃⢰⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣉⣀⣤⣴
⠀⠀⠀⠀⠾⣿⣇⠀⠀⠀⠘⣟⡉⢁⣽⡟⠀⠀⠹⠀⠈⠉⠻⣯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⡗⠀⠀⠀⠉⠙⠀⠀⠀⡇⠸⣿⣿⣿⡟⠁⠀⢸⠀⣤⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠈⠇⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⢹⡅⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⣿⣿⣿⣿⡿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠿⠿⣿⠁⠀⠀⠘⠀⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⢻⣿⣿
⠀⠀⠀⠀⢸⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠗⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠘⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡎⠀⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠼⠸⠿⠷⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢉⣀⣀⡀⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⠇⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢻⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⣴⣿⣧⣇
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢹⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⠸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⣘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⣿⣿⡏⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡓⡟
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⣀⣠⣤⣤⣤⣤⣤⡄⢤⣤⣤⣤⣤⣤⣤⣤⡀⢤⡠⡤⠤⣤⣤⣤⣤⡤⡤⠀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⣀⢠⣤⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢀⠀⠀⣠⡶⣾⣿⣿⡿⣿⣰⣿⣾⣿⣿⣿⣿⣿⣿⣹⣿⡷⣿⣭⣬⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠼⣠⣤⡆⡛⣿⡗⠗⣍⡉⣶⣶⣶⡿⢷⢶⡆⠀⠀
⣿⣶⣦⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠁⠠⣮⠏⣿⢿⣿⠆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣿⣿⣿⡟⠛⠿⡏⠻⣼⣷⡾⣶⠀⣿⣀⣿⣽⣿⣿⡇⢠⠀
⣹⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣷⣀⢹⣟⣿⣽⣧⡦⣍⣿⣪⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠾⠿⠶⣶⣥⣤⣬⡁⣞⠉⠀⠉⡏⣩⠉⣩⣍⠅⠰⠀
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⡿⣿⣿⣾⣷⡟⢰⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⠛⠻⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣀⣤⣤⣀⠀⠀⠙⢻⡋⢡⢂⣴⣦⠀⠟⣙⣿⣿⣷⡆⠀
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠉⢡⣿⢾⣽⡁⢀⣉⣿⣿⣿⣿⣿⡟⠁⠀⠀⠑⠀⠀⠀⠈⠫⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣭⣿⣿⣟⣳⣤⠀⢹⣇⣠⣼⣿⣿⣷⣿⣽⣿⣿⣿⣷⣦
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⣾⠶⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠛⣿⣏⣹⡿⠟⢻⣿⣿⣿⣾⣯⣭⣿⣿⣟⣿⣿⡇⣸⣿⣶⣿⣿⣿⣿⡟⠉⠀⠉⠙⠛⠃
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣠⠠⣤⣦⡀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠉⠈⠀⠙⠹⣿⣿⣿⣿⣿⣇⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⣿⣿⣷⣾⣾⣿⡆⢠⡿⠋⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⣀⠄⠀⠀⠀⠀
⠀⠀⠀⠈⢿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣀⣉⣿⣿⣟⠋⢈⣙⢿⣿⣿⣿⣿⣿⣿⣧⡖⠁⢴⣿⣯⣿⣿⣿⣽⣿⣿⣿⣿⠆⢋⠈⠀⠉⡁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣶⣿⡤⣤
⠀⠀⠀⠀⢿⡟⠏⠁⣿⣷⣶⣿⣷⣿⣿⣿⣿⡟⣻⣢⣌⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠙⣽⣿⣭⣭⣿⣿⣿⡟⠀⢰⣶⣥⣷⡶⢿⠿⠟⠛⢿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢿⣟⠃⠐⢿⣬
⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠱⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡟⠓⣻⣿⣿⣿⣿⣿⣿⢒⠻⡷⡝⠉⠁⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠙⠿⠿⣿⣷⡀⠉⠲⠴⠿⢫
⠀⠀⠀⠀⣼⣷⠀⠀⢸⣿⡏⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⡿⠟⣿⣿⣁⠰⠶⠾⠿⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⠘⣷⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠁⠁⡤⡀⣀⣶⡿
⠀⠀⠀⠀⣸⣿⠀⠀⠈⠿⣄⠰⠤⣿⣿⣿⠻⠿⠟⠋⣉⣙⣦⢸⡇⠐⠛⠋⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠈⢟⠉⠛⠳⣦⣀⡀⠀⠀⠀⡇⢈⣿⣿⣿⣿⡿⠃⢰⠀⠀⠀⠀⠀⠀⠀⠐⣶⣇⣈⣁⣤⣤
⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠹⣟⠈⢀⣿⠏⠀⠀⡇⠀⠉⠉⢻⣮⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⡗⠀⠀⠀⠈⠙⠀⠀⠀⡇⢸⣿⣿⣿⡟⠁⠀⢸⢀⣤⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿
⠀⠀⠀⠀⢐⣻⣷⠀⠀⠀⠀⠙⠃⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠿⠿⣿⠁⠀⠀⠘⠀⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⢻⣿⣿
⠀⠀⠀⠀⠈⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠘⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡎⠀⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠐⠸⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⢉⣀⣀⡀⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⠇⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⣴⣿⣶⣇
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢹⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⣘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⣿⣿⡟⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡃⡟
⠀⠀⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⣀⣤⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⡀⢤⣠⠤⢤⣤⣤⣤⣤⡤⡤⣀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⣀⢠⣤⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⠀⠀⣠⡶⣾⣿⣿⡿⣷⣰⣿⣾⣿⣿⣿⣿⣿⣿⣽⣿⡷⣿⣭⣎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠼⣰⣤⡆⡛⣿⡟⠗⣏⡋⣶⣶⣶⣿⢷⢶⡆⡀⠀
⣿⣷⣦⣿⣿⣿⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⡁⠠⣮⠟⣿⢿⣿⢆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣿⣿⣿⠟⠻⠿⡏⠻⣿⣷⣾⣧⠀⣿⣀⣿⣽⣿⣿⡇⣤⠀
⣽⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣷⣀⢻⣿⣿⣿⣧⡦⣝⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠾⠷⠶⣶⣧⣤⣬⡁⣿⡉⠀⠉⡏⣩⠉⣹⣍⠅⠺⠅
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⣿⢿⣿⣷⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⠛⠻⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⣀⣤⣄⣀⠀⠀⠙⢿⡛⢡⢂⣶⣦⠀⢟⣹⣿⣿⣿⡆⠀
⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠉⣡⣿⢾⣽⡁⢈⣉⣿⣿⣿⣿⣿⡟⠁⠀⠀⠑⠀⠀⠀⠈⠫⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣽⣿⣿⣟⣳⣤⠀⢻⣇⣠⣼⣿⣿⣿⣿⣽⣿⣾⣿⣷⣧
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⣿⡷⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠛⣿⣏⣹⡟⠟⣻⣿⣿⣿⣾⣯⣭⣿⣿⣟⣿⣿⡇⢸⣿⣾⣿⣿⣿⣿⡿⠉⠀⠉⠛⠛⠃
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣆⣠⠠⣡⣦⡀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠁⠀⠈⠉⢻⣿⣿⣿⣿⣿⣧⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⣿⣿⣷⣾⣾⣿⡆⢠⣿⠛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⣠⠄⠀⠀⠀⠀
⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣀⣹⣿⣿⣟⠋⢈⣙⣿⣿⣿⣿⣿⣿⣿⣧⡔⠁⢴⣿⣯⣿⣿⣿⣽⣿⣿⣿⣿⠇⢋⠈⠀⠉⡁⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣶⣾⡦⣤
⠀⠀⠀⠀⣿⡟⠝⠃⣿⣷⣶⣿⣾⣿⣿⣿⣿⡟⣻⣦⣌⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠙⣿⣿⣽⣭⣿⣿⣿⡟⠀⢰⣦⣧⣷⡶⢿⠿⠟⠛⢿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢿⣚⠃⠒⢿⣤
⠀⠀⠀⠀⢻⡆⠀⠀⢿⣿⡟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠱⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡟⠓⣻⣿⣿⣿⣿⣿⣿⢒⠻⡷⡝⠉⠁⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠙⠿⠿⣿⣷⡀⠉⠲⠴⢿⢫
⠀⠀⠀⠀⣾⣷⠀⠀⢸⣿⠋⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⡿⠟⣿⣿⣁⠰⠶⠾⠿⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⠸⣷⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠑⠁⣤⡀⣀⣶⣿
⠀⠀⠀⠀⣿⣿⠀⠀⠈⣿⣀⠤⠸⣿⣿⡿⠻⠿⠛⢉⣉⣳⣦⢸⡇⠰⠛⠋⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠘⢟⠋⠛⠲⣦⣄⡀⠀⠀⠀⡇⢈⣿⣿⣿⣿⡿⠁⠲⠀⠀⠀⠀⠀⠀⠀⠐⣶⣥⣈⣁⣤⣤
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢿⣃⠋⣨⡿⠃⠀⠈⠃⠀⠈⠘⢿⣬⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⡗⠀⠀⠀⠉⠹⠀⠀⠀⡇⢸⣿⣿⣿⡟⠁⠠⢸⢀⣠⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠹⠀⠛⠁⠀⠀⠈⠀⠀⠀⠀⠈⡏⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠿⠿⣿⠁⠀⠀⠈⠀⣀⢀⣀⢀⣀⡀⢀⣤⣤⡉⢻⣿⣿
⠀⠀⠀⠀⢼⢼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠃⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠘⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡎⠀⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠛⠸⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⢁⣀⣀⡀⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⠇⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢻⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⣴⣿⣦⡇
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢹⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⡘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⣿⣿⡟⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡃⡛
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣀⣤⣤⣤⣤⣤⣤⡄⢤⣤⣤⣤⣤⣤⣤⣤⡀⣤⡠⡤⠤⣤⣤⣤⣤⡤⡄⢀⡀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⣀⢠⣤⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⠀⠀⣠⡶⣼⣿⣿⡿⣷⣰⣿⣾⣿⣿⣿⣿⣿⣿⣽⣿⡥⣿⣋⣍⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠼⣠⣴⣆⡛⣿⡗⠓⣍⡉⣶⣶⣶⡿⠶⢶⡆⡀⠀
⣿⣷⣦⣿⣿⣿⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠁⠠⣮⠟⣿⢿⣿⢆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣿⣿⣿⠟⠹⠿⡏⠻⣼⣷⣾⣦⠀⣿⣀⣿⣽⣿⣿⡇⢤⠀
⣻⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⢽⣷⣀⢹⣿⣿⣽⣧⡦⣟⡿⢪⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠾⠷⠶⡶⣤⣤⣬⡁⣾⠉⠀⠉⡍⣩⠉⣩⣍⠅⠹⠄
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⡿⣿⢿⣿⣷⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⡛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⣠⣤⣤⣀⠈⠀⠉⢻⡋⢡⢂⣶⣦⠀⠏⣙⣿⣿⣿⡄⠀
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠊⠉⢡⣿⢾⣽⡁⢈⣉⣿⣿⣿⣿⣿⡟⠁⠀⠀⠘⠀⠀⠀⠈⠪⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣭⣿⣿⣟⣳⣤⠀⢙⣇⣤⣼⣿⣿⣿⣿⣽⣿⣿⣿⣷⣦
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠰⣶⠶⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣏⣹⡟⠟⣻⣿⣿⣿⣿⣯⣽⣿⣿⣽⣿⣿⡇⢸⣿⣾⣿⣿⣿⣿⡿⠉⠀⠉⠙⠛⠂
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣆⣠⠄⣡⣤⣀⠀⠀⠀⠀⠀⠀⣿⡟⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠀⠀⠙⠹⣿⣿⣿⣿⣿⣿⣇⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⣿⣿⣷⣶⣾⣿⡆⢠⡿⠛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⣀⠄⠀⠀⠀⠀
⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢙⣿⣿⣿⣟⠋⢈⣉⣿⣿⣿⣿⣿⣿⣿⣷⡔⠁⢤⣿⣯⣿⣿⣿⣽⣿⣿⣿⣿⠇⢋⠈⠀⠉⡁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣶⣾⡦⣤
⠀⠀⠀⠀⢾⡟⠛⠁⣿⣷⣾⣶⣾⣿⣿⣿⣿⡟⢿⣢⣈⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⠉⣽⣿⣤⣭⣿⣿⣿⡟⠀⢰⣦⣤⣷⠶⢿⠻⠟⠛⢿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢿⣞⠃⠐⢿⣤
⠀⠀⠀⠀⢸⡆⠀⠀⣿⣿⡗⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠱⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡞⠓⣹⣿⣿⣿⣿⣿⣿⢒⠻⡷⡝⠉⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠙⠿⠿⣿⣧⠀⠈⠲⠶⢿⢫
⠀⠀⠀⠀⣾⣷⠀⠀⢹⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⣿⠏⣿⣿⣁⠰⠶⠾⠿⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⠘⣷⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢿⣿⣿⣿⣿⣿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠁⠁⣄⡀⢀⣴⣿
⠀⠀⠀⠀⣼⣿⠀⠀⠈⣯⡀⢠⢿⣿⣿⠟⠿⠟⠋⢉⣉⣷⣄⢸⡇⠰⠛⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠈⢟⠋⠛⠳⣦⣄⡀⠀⠀⠀⡇⢈⣿⣿⣿⣿⡿⠁⠓⠀⠀⠀⠀⠀⠀⠀⠐⣶⣇⣈⣁⣥⣤
⠀⠀⠀⠀⣻⣿⡇⠀⠀⠙⡟⡘⠁⣼⡟⠀⠀⠹⠀⠀⠀⠹⣏⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⡗⠀⠀⠀⠈⠙⠀⠀⠀⡇⢸⣿⣿⣿⡟⠁⠀⢸⠠⢤⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿
⠀⠀⠀⠀⢐⢽⣷⠀⠀⠀⠈⠇⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠿⠿⣿⠀⠀⠀⠈⠀⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⢻⣿⣿
⠀⠀⠀⠀⠸⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠛⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠘⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡆⠀⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠐⠸⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡉⢀⣀⣀⡀⣥⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⠇⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢻⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⣴⣿⣦⡇
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢹⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⡘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⣿⣿⡟⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡗⡟
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠴⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⣀⣤⣤⣤⣤⣤⣤⡄⢤⣤⣤⣤⣤⣤⣤⣤⡀⢤⡠⣤⠤⢤⣤⣤⣤⡤⡄⢀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⣀⢠⣤⣀⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⠀⠀⣠⡶⣾⣿⣿⡿⣷⣰⣿⣾⣿⣿⣿⣿⣿⣿⣹⣿⡧⣿⣍⣍⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠼⢠⣴⡆⣻⣿⡗⠓⢏⡋⣶⣶⣶⣿⠶⢶⣆⡀⠀
⣿⣷⣦⣿⣿⣿⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠁⠂⣮⠟⣿⣿⣿⢢⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⠿⠻⠿⡏⠻⣾⣷⡿⣦⠀⣿⣀⣿⣽⣿⣿⡇⢤⠀
⣻⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣷⣀⢻⣿⣿⣽⣣⡦⡽⣿⣪⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠾⠷⠶⡶⣤⣤⣬⡁⣿⡉⠀⠉⡍⣩⠉⢩⣍⠁⠹⠁
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⡿⣿⣿⣿⣷⡏⢰⣾⣿⣿⣿⣿⣿⣿⣿⠟⠻⡛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⣠⣤⣤⣀⠈⠀⠉⢻⡉⢡⢂⣴⣦⠀⠿⣹⣿⣿⣷⡄⠀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⠉⢡⣿⢾⣿⡀⢈⣉⣿⣿⣿⣿⣿⡟⠁⠀⠀⠘⠀⠀⠀⠈⠫⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣭⣿⣿⣟⣳⣤⠀⢹⣇⣤⣼⣿⣿⣿⣿⣽⣿⣿⣿⣷⣦
⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⢿⠷⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠛⣿⣏⣹⡟⠟⢻⣿⣿⣿⣿⣯⣽⣿⣿⣽⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡿⠉⠀⠉⠙⠛⠃
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣆⣠⠄⣡⣤⣀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠀⠀⠙⠹⣿⣿⣿⣿⣿⣿⣇⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⡆⢠⡿⠛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣀⠄⠀⠀⠀⠀
⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢙⣿⣿⣿⣟⠉⢈⣙⣿⣿⣿⣿⣿⣿⣿⣧⡖⠁⢴⣿⣯⣿⣿⣿⣽⣿⣿⣿⣿⠇⢋⠈⠀⠉⡁⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⢶⣾⡦⣤
⠀⠀⠀⠀⢿⡿⡏⠁⣿⣷⣾⣶⣿⣿⣿⣿⣿⡟⢿⣢⣈⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⠉⣽⣿⣤⣽⣿⣿⣿⡏⠀⢰⣦⣤⣷⠶⢿⠻⠟⠛⢿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢿⣟⠃⠐⢿⣤
⠀⠀⠀⠀⢸⡅⠀⠀⢿⣿⡗⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠱⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡟⠓⣻⣿⣿⣿⣿⣿⣿⢒⠻⡷⠝⠉⠁⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠙⠿⢿⣿⣷⡀⠈⠲⠴⠿⢫
⠀⠀⠀⠀⢾⣷⠀⠀⢹⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⡿⠏⣿⢿⣁⠰⠶⠾⠿⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⠘⣷⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢿⣿⣿⣿⣿⣿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠑⠁⣤⡀⣀⣴⣿
⠀⠀⠀⠀⣸⣿⠀⠀⠀⣟⡀⢄⢻⣿⣿⠟⠿⠟⠋⢉⣉⣷⣄⢸⡇⠰⠛⠋⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠙⢟⠋⠛⠳⣦⣄⡀⠀⠀⠀⡇⢈⣿⣿⣿⣿⡿⠁⠓⠀⠀⠀⠀⠀⠀⠀⠐⣶⣧⣈⣁⣥⣤
⠀⠀⠀⠀⢿⣿⡇⠀⠀⠙⡿⡈⢁⣼⡟⠀⠀⠹⠀⠀⠀⠻⣏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⡗⠀⠀⠀⠉⠙⠀⠀⠀⡇⢸⣿⣿⣿⡟⠁⠀⢸⠠⢤⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿
⠀⠀⠀⠀⢱⣿⣷⠀⠀⠀⠈⠇⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠿⠿⣿⠀⠀⠀⠈⠀⣀⢀⣀⢀⣀⡀⢀⣤⣤⡉⢻⣿⣿
⠀⠀⠀⠀⠸⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢟⠛⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠘⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡆⠀⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠘⠸⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡉⢀⣀⣀⡀⣥⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⠇⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢻⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⣴⣿⣦⡇
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢹⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⡘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⣿⣿⡿⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡗⠿
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠴⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡠⣤⣀⣤⣤⣤⣤⣤⣤⡄⢤⣤⣤⣤⣤⣤⣤⣤⡀⣄⡠⡤⢤⣤⣤⣤⣤⡤⡄⢀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⣀⢠⣤⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⠀⠀⣠⡶⣾⣿⣿⡿⣷⣰⣿⣾⣿⣿⣿⣿⣿⣿⣽⣿⡵⣿⣭⣏⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠼⣠⣤⡆⣛⣿⡗⠗⣏⡋⣶⣶⣶⡿⠷⢶⣆⡀⠀
⣿⣶⣦⣿⣿⣿⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠁⠰⣮⠟⣿⢿⣿⠆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣽⣿⣿⡟⠛⠿⡋⠻⣾⣷⣾⣦⠀⣿⣀⣿⣿⣿⣿⡇⢤⠀
⣹⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣷⡀⢹⣟⣿⣿⣣⡦⡝⡿⣪⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠞⠷⠶⢶⣥⣤⣬⡁⣿⡁⠀⠉⡏⣩⠉⢹⣍⡁⠸⠅
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⡿⣿⣿⣿⣷⡟⢹⣾⣿⣿⣿⣿⣿⣿⣿⠿⠻⠏⠛⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣠⣤⣤⣄⠀⠀⠉⢻⡛⢡⢂⣴⣦⠈⠟⣙⣿⣿⣷⡄⠀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠊⠉⣡⣿⣾⣿⡀⢈⣉⣿⣿⣿⣿⣿⡟⠁⠀⠀⠈⠀⠀⠀⠀⠹⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣾⣭⣿⣿⣟⣳⣤⠀⢘⡇⣠⣼⣿⣿⣿⣿⣽⣿⣿⣿⣷⣦
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⣾⠶⢨⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠛⣿⣏⣹⡿⠟⢻⣿⣿⣿⣿⣿⣽⣿⣿⣽⣿⣿⡇⢠⣿⣾⣿⣿⣿⣿⡿⠉⠀⠉⠙⠛⠂
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⠤⢵⣦⡄⠀⠀⠀⠀⠀⠀⣿⡿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⠶⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠀⠈⠫⢿⣿⣿⣿⣿⣿⣿⣗⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡆⢠⡿⠛⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣀⠄⠀⠀⠀⠀
⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⢹⣿⣿⣿⣟⠋⢈⣙⣿⣿⣿⣿⣿⣿⣿⣧⡖⠁⢴⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⠗⢋⠈⠀⠉⡁⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣶⣾⡦⣤
⠀⠀⠀⠀⣺⠟⠏⠁⣿⣷⣷⣶⣾⣿⣿⣿⣿⡟⢿⣢⣈⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠉⣿⣿⣶⣾⣿⣿⣿⡏⠀⣰⣦⣧⣷⠶⠿⠻⠟⠛⢿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢿⣟⠃⠐⢻⣤
⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠡⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡟⠓⣸⣿⣿⣿⣿⣿⡷⢒⠻⡷⠍⠉⠁⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠙⠿⠿⣿⣷⡀⠉⠳⠴⠿⢋
⠀⠀⠀⠀⠼⣷⠀⠀⢸⣿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⡿⠏⣿⣿⣁⠰⠶⠾⠿⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⠸⣷⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠑⠁⣄⡀⢀⣴⣿
⠀⠀⠀⠀⢺⣿⠀⠀⠈⣟⠀⣐⣿⣿⣿⠻⠿⠟⠋⣉⣹⣷⣄⢸⡇⠴⠛⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠈⢟⠉⠛⠲⣦⣄⡀⠀⠀⠀⡇⢸⣿⣿⣿⣿⡟⠁⠇⠀⠀⠀⠀⠀⠀⠀⠐⣶⣇⣈⣉⣥⣤
⠀⠀⠀⠀⣯⣿⡇⠀⠀⠹⣟⠚⢉⣽⠏⠀⠀⡇⠀⠉⠘⢿⡏⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⡗⠀⠀⠀⠉⠙⠀⠀⠀⡇⢸⣿⣿⣿⡟⠀⠐⠰⠠⠤⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⠙⠇⠸⠋⠀⠀⠀⠁⠀⠀⠀⠈⢷⠈⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠿⠿⣿⠀⠀⠀⠀⠀⣀⢀⣀⢁⣀⡀⢀⣤⣤⡈⢻⣿⣿
⠀⠀⠀⠀⠘⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠘⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡆⠀⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠸⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡉⢀⣀⣀⡀⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⠇⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⡿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢻⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⣴⣿⣦⡏
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢹⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⡘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⣿⣿⡿⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠿
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠓
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡠⣤⣀⣠⣤⣤⣤⣤⣤⡄⢤⣤⣤⣤⣤⣤⣤⣤⡀⣄⡠⣤⢤⣤⣤⣤⣤⡤⡄⠀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⣀⢠⣤⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⠀⠀⣠⡶⣾⣿⣿⡿⣧⣰⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣍⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠼⣠⣤⡄⣻⣿⡗⠓⣍⡋⣶⣶⣶⡿⢷⢶⣆⡀⠀
⣿⣶⣦⣿⣿⣿⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠁⠠⡮⠏⣿⢿⣿⠆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣿⣯⣿⡟⠛⠿⡏⠻⣾⣷⣿⣶⠀⣿⣀⣿⣿⣿⣿⡇⢤⠀
⢹⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣷⣀⢹⣿⣿⣿⣧⡦⣝⡿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠷⠶⢶⣥⣤⣬⡁⣟⡉⠀⠉⠏⣩⠉⢩⣍⡅⠸⠄
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣷⡿⢿⣿⣾⣷⡿⢸⣾⣿⣿⣿⣿⣿⣿⣿⠟⠛⢏⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣀⣤⣤⣀⠀⠀⠉⢻⡋⢡⢂⣴⣦⠀⠟⣙⣿⣿⣿⡆⠀
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠉⣡⣿⣾⣿⡀⢈⡉⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠁⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣭⣿⣿⣟⣳⣄⠀⠘⡇⢤⣼⣿⣿⣿⣿⣽⣿⣯⣿⣷⣦
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⢶⠶⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠹⣿⣏⣹⡿⠟⢻⣿⣿⣿⣿⣯⣽⣿⣿⣽⣿⣿⡇⢠⣿⣶⣿⣿⣿⣿⡿⠋⠀⠈⠙⠛⠃
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣧⣰⡦⢹⣦⡄⠀⢀⡀⠀⠀⠀⣿⡿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⢬⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣇⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢠⡽⠛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣀⠄⠀⠀⠀⠀
⠀⠀⠀⠐⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⢻⣿⣿⣿⣟⠋⢈⣙⣿⣿⣿⣿⣿⣿⣿⣧⡖⠁⢴⣿⣯⣿⣿⣿⣾⣿⣿⣿⣿⠗⢛⠈⠐⠉⡁⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣶⣾⡦⣤
⠀⠀⠀⠀⢻⡯⠇⠁⣿⣷⣶⣴⣾⣿⣿⣿⣿⡟⢻⣂⣈⣁⣼⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⠙⣿⣿⣶⣾⣿⣿⣿⡏⠀⣴⣦⣦⣷⠶⠿⠻⠟⠛⢿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢿⣟⠃⠐⢿⣤
⠀⠀⠀⠀⢸⡆⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡟⠓⣺⣿⣿⣿⣿⣿⡷⢔⠻⡷⠍⠉⠁⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠙⠿⠿⣿⣷⡀⠉⠣⠴⢿⢋
⠀⠀⠀⠀⣼⣷⠀⠀⢸⣿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⡿⠏⣻⣿⡁⠴⠶⠾⠿⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⠘⣷⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢿⣿⣿⣿⣿⣿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠑⠑⣄⡀⢀⣴⣿
⠀⠀⠀⠀⣿⣿⠀⠀⠈⣟⠀⣰⣿⣿⣿⠻⠿⠟⠋⣉⣹⣷⣄⢸⡇⠴⠛⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠈⢟⠋⠛⠲⢦⣄⡀⠀⠀⠀⡇⢸⣿⣿⣿⣿⡿⠁⠇⠀⠀⠀⠀⠀⠀⠀⠐⣶⣧⣈⣉⣥⣤
⠀⠀⠀⠀⠿⣿⡇⠀⠀⠻⣗⠚⢉⣿⠇⠀⠈⡇⠀⠉⠹⢿⡏⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⡗⠀⠀⠀⠉⠙⠀⠀⠀⡇⢸⣿⣿⣿⡟⠀⠀⠠⠀⠤⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⢿⣷⠀⠀⠀⠙⠃⠸⠃⠀⠀⠀⠀⠀⠀⠀⠈⣧⠈⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠿⠿⣿⠀⠀⠀⠀⠀⣀⢀⣀⢁⣀⡀⢀⣤⣤⡈⢻⣿⣿
⠀⠀⠀⠀⠨⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠘⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡆⠀⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⡿⣿
⠀⠀⠀⠀⠁⠸⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡉⢀⣀⣀⡈⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⠇⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⡿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⣴⣿⣦⡏
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⡘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⣿⣿⡿⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠻
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⣀⣠⣤⣤⣤⣤⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⡀⣤⡠⡤⠤⣤⣤⣤⣤⡤⡄⢀⣀⡀⠀⠀⡀⠠⠀⠀⠀⠀
⠀⠀⣀⢠⣤⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⠀⠀⣠⡶⣾⣿⣿⡿⣷⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⡭⣿⣍⣈⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠼⣠⣤⡆⡻⣿⡗⠗⣏⠋⣶⣶⣶⡿⠶⢶⣄⡀⠀
⣿⣷⣦⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠁⠀⣯⠟⣿⢿⣿⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣽⣿⣵⠟⠻⢿⡏⠻⣼⣷⣾⣷⠄⣿⣀⣿⣽⣿⣿⡇⣤⠀
⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣷⡀⢻⣿⣿⣿⣧⡶⡍⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠶⣶⣥⣤⣭⡁⣿⡉⠀⠉⡏⣩⠉⢩⣍⡅⠺⠄
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣷⡿⣿⣿⣾⣷⡟⢸⣾⣿⣿⣿⣿⣿⣿⣿⡿⠛⠫⡉⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣀⣤⣄⣀⠈⠈⠉⢻⡙⢡⢂⣴⣦⠈⢏⢛⣿⣿⣿⡆⠀
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠉⣡⣿⣾⣿⠀⢀⣉⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠁⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣾⣭⣿⣿⣟⡷⣄⠀⠚⡇⣠⣼⣿⣿⣶⣿⣽⣿⣿⣿⣷⣦
⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⣿⣶⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣏⣹⡿⠟⢻⣿⣿⣿⣿⣯⣽⣿⣿⣻⣻⣿⡆⢠⣿⣿⣿⣿⣿⣿⡿⠋⠁⠉⠙⠛⠂
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣷⣰⣆⣨⣶⡤⠀⢀⡀⠀⠀⠀⣿⡿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠀⠙⠙⢿⣿⣿⣿⣿⣿⣿⣧⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣴⠛⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣀⠄⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⢻⣿⣿⣿⣟⠋⢈⣙⣿⣿⣿⣿⣿⣿⣿⣧⡔⠁⢴⣿⣯⣻⣿⣿⣾⣿⣿⣿⣿⠗⢛⠉⠀⠉⡁⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣶⣾⡦⣤
⠀⠀⠀⠀⢻⡟⠉⠁⣿⣷⣶⣴⣾⣿⣿⣿⣿⠟⢻⣦⣈⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠛⣫⣿⣶⣾⣿⣿⣿⡏⠀⣴⣶⣤⣷⡶⢿⠿⠟⠛⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢿⣟⠃⠐⢿⣄
⠀⠀⠀⠀⠸⡆⠀⠀⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠡⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡟⠓⣸⣿⣿⣿⣿⣿⡧⢔⢻⡷⠝⠙⠁⠀⠀⠀⠀⠀⠻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⠿⢿⣿⣿⠀⠉⠱⠴⢿⢎
⠀⠀⠀⠀⠘⣷⠀⠀⢸⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⡿⠟⣿⣿⡁⠤⠤⠾⠿⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⢘⣷⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠁⠑⣤⡀⢀⣴⣿
⠀⠀⠀⠀⢻⣿⡀⠀⠀⣿⠀⡐⣿⣿⣿⠻⠿⠻⠋⣉⣻⣷⣤⢸⡇⠴⠛⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠈⣟⠋⠛⠷⣦⣄⡀⠀⠀⠀⡇⢘⣿⣿⣿⣿⡿⠁⠇⠀⠀⠀⠀⠀⠀⠀⠐⣦⣇⣉⣉⣭⣤
⠀⠀⠀⠀⠘⢻⣧⠀⠀⢻⣗⠚⢉⣿⠃⠀⠈⡇⠀⠉⠹⣿⡏⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⡗⠀⠀⠀⠉⠙⠀⠀⠀⡇⢸⣿⣿⣿⡟⠁⠀⠠⠀⣤⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢺⣿⠀⠀⠀⠙⠂⠻⠃⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠹⣿⣿⣿⣿⣿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⠿⠿⣿⠀⠀⠀⠀⠀⣀⢀⣀⢀⣀⡀⢀⣤⣤⡉⢻⣿⣿
⠀⠀⠀⠀⠠⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠃⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠘⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡆⠀⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠘⠸⠿⠿⠄⠀⠀⠀⠀⠀⠀⠀⢀⣈⡉⢀⣀⣀⡈⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⠇⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⡿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢻⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⣴⣿⣦⡇
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⡘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⣿⣿⡟⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⡿
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠃
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⣀⣠⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⡀⣤⡠⣤⠤⣤⣤⣤⣤⡤⡄⡀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⣀⣰⣤⣀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⠀⠀⣠⡶⣼⣿⣿⡿⣷⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⡭⣿⣏⣙⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠼⣠⣤⡆⡻⣿⡗⠗⢎⠋⣶⣶⣶⣿⣶⢶⣄⡀⠀
⣿⣷⣦⣿⣽⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠁⠀⣯⠿⣿⢿⣿⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣽⣿⣿⠿⠻⢿⡛⠿⣼⣷⣾⣷⠀⣿⣀⣿⣽⣿⣿⡇⣄⠀
⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣷⡀⢻⣟⣿⣿⣧⣦⡍⡿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠶⣶⣥⣤⣬⡁⣿⡉⠀⠉⡏⣩⠉⢹⣍⠅⠻⠅
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⡿⣿⢿⣿⣷⡏⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⡛⠻⡉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⣤⣄⣀⠈⠈⠉⢻⡛⢡⢆⣴⣦⠈⢟⣉⣿⣾⣷⡄⠀
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠉⠉⢡⣿⣾⣽⠀⠈⡉⣿⣿⣿⣿⣿⡿⠁⠀⠁⠀⠈⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣾⣯⣿⣿⣟⣷⣄⠀⠘⡇⣠⣼⣿⣿⣾⣿⣿⣿⣿⣿⣷⣦
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⣾⠾⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⣹⡿⠟⢻⣿⣿⣿⣿⣯⣭⣿⣿⣻⣻⣿⡆⢠⣿⣿⣿⣿⣿⣿⡿⠋⠁⠈⠙⠛⠓
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣰⣒⣠⣷⣤⠀⢀⡀⠀⠀⠀⣸⡿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠸⣿⣿⣿⣿⣿⡏⠀⠙⠙⢿⣿⣿⣿⣿⣿⣿⣇⣀⣹⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣴⠛⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣀⠄⠀⠀⠀⠀
⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠈⢻⣿⣿⣿⣟⠛⢈⣙⣿⣻⣿⣿⣿⣿⣿⣧⡜⠃⣴⣿⣯⣻⣿⢿⣾⣿⣿⣿⣿⠦⢟⠉⠀⠋⡁⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣶⣾⡦⣤
⠀⠀⠀⠀⣼⡟⠓⠁⣿⣷⣶⣤⣾⣿⣿⣿⣿⡟⢳⣦⣈⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠙⣻⣿⣶⣾⣿⣿⣿⡏⠀⣶⣶⣧⣷⡶⢿⠿⠿⠟⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⣟⠃⠐⢿⣤
⠀⠀⠀⠀⢸⡇⠀⠀⢻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡟⠓⣺⣿⣿⣿⣿⣿⡇⠐⠻⡷⠛⠋⠀⠀⠀⠀⠀⠀⢻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⠿⢿⣿⣿⡀⠉⠳⠴⠿⢏
⠀⠀⠀⠀⡘⣿⠀⠀⢸⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⡿⠛⣿⣿⠁⠤⠤⠾⠿⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⢸⣷⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠁⠑⣀⡀⢀⣾⣿
⠀⠀⠀⠀⢼⣿⡀⠀⠀⣿⠀⠐⣿⣿⣿⠻⠿⠟⢋⣉⣻⣷⣤⢸⡇⠴⠟⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡯⠀⠉⣟⠋⠛⠲⣦⣄⡀⠀⠀⠀⡇⢘⣿⣿⣿⣿⡿⠁⠇⠀⠀⠀⠀⠀⠀⠀⠐⣦⣇⣉⣉⣡⣤
⠀⠀⠀⠀⠹⣿⣇⠀⠀⢻⡗⠚⢉⣿⠃⠀⠈⠇⠀⠉⠹⣿⡋⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⡇⠀⠀⠀⠉⠹⠀⠀⠀⡇⢸⣿⣿⣿⡟⠁⠀⢠⠀⣀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢨⢻⣿⠀⠀⠀⠙⠀⠻⠃⠀⠀⠈⠀⠀⠀⠀⠙⡇⠈⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠿⠿⣿⠀⠀⠀⠀⠀⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⢻⣿⣿
⠀⠀⠀⠀⢨⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠟⠃⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠘⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡆⠀⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠐⠸⠿⠿⠄⠀⠀⠀⠀⠀⠀⠀⢀⣈⡁⢀⣀⣀⡈⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⠏⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⣴⣿⣦⡇
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⡘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⣿⣿⡟⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠻
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠓⠃
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡠⢤⣀⣠⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⡀⣤⡠⣤⠤⣤⣤⣤⣤⡤⡄⢀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⠀⠀⣀⢠⣤⣀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⠀⠀⣠⡶⣾⣿⣿⡿⣧⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⣥⣿⣯⣍⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠼⣠⣤⡆⡻⣿⡗⠓⢚⠏⣷⣶⣶⣷⢶⣶⣄⡀⠀
⣿⣷⣦⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠁⠀⣯⠯⣿⢿⣿⠆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣽⣿⣿⠿⠻⢿⠛⠿⣼⣷⣾⣷⠀⣿⣀⣿⣽⣿⣿⡇⣤⠀
⣸⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣷⡀⢻⣿⣿⣿⣧⡦⡏⡿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠷⠶⢶⣤⣤⣬⡁⣿⡉⠀⠉⡍⣭⠉⢩⣏⠅⠲⠂
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⡿⣿⣿⣿⣷⡟⢹⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⡛⢫⠉⠻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣀⣤⣤⣀⠀⠀⠉⢻⡛⢡⢂⣴⣦⠀⢟⣙⣿⣾⣿⡄⠀
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠉⠉⢡⣿⣾⣽⡀⠈⡉⣿⣿⣿⣿⣿⣿⠏⠀⠈⠀⠀⠁⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣯⣿⣿⣟⣳⣤⠀⠘⡇⢤⢼⣿⣿⣶⣿⣿⣿⣿⣿⣷⣦
⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⣾⡷⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⢿⣏⣹⣿⠟⢻⣿⣿⣿⣿⣯⣭⣿⣿⣭⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⡿⠋⠀⠈⠙⠛⠃
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣧⣗⣂⣿⣤⠀⠀⠀⠀⠀⠀⢨⡿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⢽⣽⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⣿⣿⣿⣿⣿⡟⠀⠙⢛⢿⣿⣿⣿⣿⣿⣿⣧⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡆⠀⣤⠺⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⠈⢻⣿⣿⣿⣟⠋⢈⣙⣿⣻⣿⣿⣿⣿⣿⣧⡔⠃⢴⣿⣯⣻⣿⠿⣾⣿⣿⣿⣿⠧⢾⠋⠘⠉⡉⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣴⣾⡤⣤
⠀⠀⠀⠀⢻⡟⠅⠃⣿⣷⣶⣤⣶⣾⣿⣿⣿⡟⢳⣦⣨⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠙⣻⣿⣶⣿⣿⣿⣿⡏⠀⣶⣦⣧⣷⡶⢿⠿⠟⠛⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢿⣟⠃⠐⢿⣀
⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡞⠓⣺⣿⣿⣿⣿⣿⡇⡐⢻⡷⠝⠉⠁⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⠿⠿⣿⣿⠀⠉⠣⠴⢿⢏
⠀⠀⠀⠀⣸⣿⠀⠀⠸⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⡿⠏⣿⣿⠁⠤⠤⠾⠿⠃⢴⣿⣿⣿⣿⣿⣿⣿⣷⠸⣷⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠁⠑⣄⡀⢀⣴⣾
⠀⠀⠀⠀⣟⣿⡀⠀⠀⣿⠀⠐⣿⣿⡿⠿⠿⠟⢛⣉⣻⣷⣄⢸⡇⠰⠟⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠉⡟⠙⠛⠷⣦⣄⡀⠀⠀⠀⡇⢘⣿⣿⣿⣿⡿⠁⠓⠀⠀⠀⠀⠀⠀⠀⠀⣤⣇⣉⣉⣩⣤
⠀⠀⠀⠀⠸⢿⣇⠀⠀⢺⡗⠚⢨⡿⠁⠀⢸⠀⠈⠉⠻⣿⡏⢨⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢈⠇⠀⠀⠀⠉⠹⠀⠀⠀⡇⢸⣿⣿⣿⡟⠁⠠⢸⠀⣤⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⢸⣿⠀⠀⠀⠹⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠹⡇⠈⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠿⠿⣿⠀⠀⠀⠈⠀⣀⢀⣀⢀⣀⡀⢀⣤⣤⡉⢻⣿⣿
⠀⠀⠀⠀⢸⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⠛⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠙⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡆⠀⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⡿⣿
⠀⠀⠀⠀⠘⠈⠿⠿⠄⠀⠀⠀⠀⠀⠀⠂⢀⣉⡁⢀⣀⣀⡉⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⠏⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⣴⣿⣦⡏
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⡘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⣿⣿⡟⣷
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠻
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠂
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡠⣄⣀⣠⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⡀⣤⡀⣤⠤⣤⣤⣤⣤⠤⡄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠀⣀⢠⣤⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣠⡶⣾⣿⣿⡿⣧⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⡥⣿⣍⣍⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠼⣠⣤⡆⡻⣿⡗⠗⢈⡏⣶⣶⣶⣿⢶⢶⣄⡀⠀
⣿⣷⣦⣿⣽⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠁⠢⣮⠟⣿⢿⣿⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣽⣿⣷⡿⠻⢿⡛⠻⣼⣷⣾⣷⠀⣿⣁⣿⣽⣿⣿⡇⢤⠀
⢸⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣷⡀⢻⣿⣿⣿⣧⡦⡟⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠷⠶⢶⣥⣤⣬⡁⣿⠉⠀⠉⡏⣩⠉⢩⣍⠅⠻⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠿⣿⣷⡿⣿⢿⣿⣷⣟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣛⠻⡉⠻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣠⣤⣤⣀⠀⠀⠉⢻⡛⢡⢂⣴⣦⠀⢟⣻⣿⣿⣿⡆⠀
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠊⠉⢡⣿⣾⣽⠀⢈⡉⣿⣿⣿⣿⣿⣿⡏⠁⠈⠀⠀⠁⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣭⣿⣿⣟⣳⣦⠀⠘⡇⢠⣼⣿⣿⣶⣿⣽⣿⣿⣿⣷⣦
⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠠⣿⠶⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⢿⣏⣹⡿⠟⢻⣿⣿⣿⣿⣿⣭⣿⣿⣭⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⡿⠋⠁⠉⠹⠛⠂
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣂⣼⣦⡀⠀⠀⠀⠀⠀⠀⡿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⢼⣽⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣟⠀⠙⢛⢿⣿⣿⣿⣿⣿⣿⣇⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡆⠀⣠⠰⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠈⢻⣿⣿⣿⣟⠋⢈⣛⣿⣻⣿⣿⣿⣿⣿⣧⡔⠃⢤⣿⣯⣹⣿⠿⣿⣿⣿⣿⣿⡇⠶⠋⠐⠉⡁⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣴⣾⡦⣤
⠀⠀⠀⠀⢸⡏⠁⠇⢹⣿⣶⣤⣶⣶⣿⣿⣿⠟⢻⣆⣈⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠛⣻⣿⣷⣾⣿⣿⣿⡏⠀⣴⣦⣦⣷⡶⢿⠿⠟⠛⢿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢿⣿⠃⠩⢿⣤
⠀⠀⠀⠀⢰⣇⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⠞⠓⣻⣿⣿⣿⣿⣿⡇⡐⢻⡷⠝⠋⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠙⠿⠿⣿⣿⡀⠉⠢⠤⢿⢏
⠀⠀⠀⠀⡨⣿⠀⠀⠈⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⡿⠟⣻⣿⠁⠤⠤⠾⠿⠃⣴⣿⣿⣿⣿⣿⣿⣿⣷⢸⣷⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠁⠑⣀⡀⢀⣾⣾
⠀⠀⠀⠀⢼⣿⡄⠀⠀⣻⠀⢐⣿⣿⡟⠿⠿⠟⢋⣉⢿⣷⣤⢸⡇⠴⠟⠛⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⡟⠉⠻⠶⣦⣄⡀⠀⠀⠀⡇⢘⣿⣿⣿⣿⡿⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⢤⣧⣉⣉⣩⣤
⠀⠀⠀⠀⢸⣿⣧⠀⠀⢸⡓⠚⣩⡿⠁⠀⢹⠀⠈⠉⢿⣿⡏⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢈⠇⠀⠀⠀⠉⠹⠀⠀⠀⡇⢸⣿⣿⣿⡟⠁⠀⢸⠀⣤⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠨⢹⣿⡀⠀⠀⠹⠀⠟⠁⠀⠀⠂⠀⠀⠀⠀⢻⡦⠈⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠃⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠿⠿⣿⠀⠀⠀⠈⠀⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⢻⣿⣿
⠀⠀⠀⠀⢐⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠿⢟⠛⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠙⢿⠿⠛⠁⢀⣀⣀⣈⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡎⠀⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠈⠈⠿⠿⡄⠀⠀⠀⠀⠀⠀⠂⢀⣉⡁⢀⣀⣀⡉⣤⣄⢠⣤⣤⣤⣶⣶⠀⠁⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⠇⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣦⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⢰⣿⣦⡏
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⡘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⣾⣿⡟⣷
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠿
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠃
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⡤⣀⣤⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⡀⣤⡠⣤⠤⣤⣤⣤⣤⠤⡄⡀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⢀⠀⣀⣠⣤⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⠀⠀⣠⡶⣼⣿⣿⡿⣧⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⡥⣿⣍⣌⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠴⣠⣤⡆⡛⣿⡗⠓⢈⠋⣶⣶⣶⣿⠶⢶⣄⡀⠀
⣿⣷⣦⣿⣽⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠁⠢⢯⠟⣿⢿⣿⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣽⣿⣿⠿⠻⢿⠛⠻⣼⣷⣾⣷⠀⣿⣀⣿⣽⣿⣿⡇⢤⠀
⢺⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣷⡀⢻⣿⣿⣿⣧⡦⡋⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠷⠶⢶⣤⣤⣬⡁⣿⡉⠀⠉⡏⣩⠉⢩⡍⠅⠳⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿⣿⣿⣿⣷⡟⢸⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⠛⠍⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⣤⣤⣀⠀⠀⠉⢻⡛⢡⠆⣴⣦⠈⠟⣛⣿⣿⣿⡄⠁
⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠉⢡⣿⣾⣽⡀⠈⡉⣿⣿⣿⣿⣿⣿⡟⠁⠀⠁⠀⠈⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣴⣿⣿⣿⣟⣳⣦⠀⠘⡇⣤⣼⣿⣿⣶⣿⣽⣿⣿⣿⣧⣦
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⢾⡶⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡇⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⢻⣏⣹⡿⠟⢻⣿⣿⣿⣿⣿⣭⣿⣿⣭⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⡿⠋⠁⠉⠙⠛⠂
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣧⣿⣓⣼⣧⣀⡀⠀⠀⠀⠀⠀⡿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⠀⠙⢛⢿⣿⣿⣿⣿⣿⣿⣧⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡆⠀⣀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠈⢻⣿⣿⣿⣟⠋⢈⣙⣿⣻⣿⣿⣿⣿⣿⣧⡜⠃⢤⣿⣯⣹⣿⠿⣷⣿⣿⣿⣿⣇⡴⠛⠀⠋⡉⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣦⣤⣾⡤⣤
⠀⠀⠀⠀⢸⡿⠀⠉⢹⣿⣦⣤⣶⣶⣿⣿⣿⡟⢳⣦⣈⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠙⣻⣿⣷⣾⣿⣿⣿⡏⠀⣤⣦⣥⣷⣶⢿⡿⡟⠟⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢿⣿⠃⠨⠷⣠
⠀⠀⠀⠀⠈⣇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⠟⠓⣿⣿⣿⣿⣿⣿⡇⡐⢻⡷⠝⠛⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⠿⠿⣿⣿⡀⠈⠢⠤⢿⠟
⠀⠀⠀⠀⢐⣿⡄⠀⠈⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⣿⠛⣻⣿⠁⠤⠤⠾⠿⠃⣴⣿⣿⣿⣿⣿⣿⣿⣷⠸⣷⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠁⠓⣄⡀⢀⣾⣾
⠀⠀⠀⠀⢸⣿⡇⠀⠀⢻⠆⠠⣿⣿⡟⢿⠿⠛⢛⣉⢿⣷⣤⢸⡇⠰⠟⠛⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⡛⠙⠛⠳⣦⣄⡀⠀⠀⠀⡇⢘⣿⣿⣿⣿⡿⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⢦⣧⣈⣉⣩⣤
⠀⠀⠀⠀⢸⣿⣷⠀⠀⢸⠓⠚⢩⡿⠁⠀⠱⠀⠈⠉⢻⣿⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢈⠇⠀⠀⠀⠘⠹⠀⠀⠀⡇⢸⣿⣿⣿⡟⠁⠀⢸⢀⣤⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣼⣿⡄⠀⠀⠹⠀⠟⠀⠀⠀⠁⠀⡀⠀⠀⢻⡆⠈⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠃⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠿⠿⣿⠁⠀⠀⠈⠀⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⢻⣿⣿
⠀⠀⠀⠀⢀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠛⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠙⢿⠿⠛⠁⢀⣀⣀⣈⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡎⠀⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠨⠀⠿⢿⡆⠀⠀⠀⠀⠀⠠⠀⢀⣉⡀⢀⣀⣀⡉⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⠇⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⢰⣿⣦⡇
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⡘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⣼⣿⡿⣷
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⡟
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠃
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⣀⣠⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⡀⣤⡠⡤⠤⣤⣤⣤⣤⠤⡄⢀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣀⠀⣀⢠⣤⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣠⡶⣼⣿⣿⡿⣧⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⣭⣿⣅⣉⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠼⣠⣤⡆⡻⣿⡗⠓⣍⠋⣶⣶⣶⣿⠷⢶⣄⠀⠀
⣿⣷⣦⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠁⠠⢯⠟⣿⢿⣿⡦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣽⣯⣿⠟⠛⢿⡟⠻⣼⣷⡾⣷⠄⣿⣀⣿⣽⣿⣿⡇⢤⠀
⣻⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣽⣷⡀⢹⣿⣿⣿⣧⡮⡙⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠶⢶⣥⣤⣬⡁⣿⡉⠀⠉⡍⣉⠉⢩⣍⠅⠹⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠿⣿⣷⡿⣿⣿⢿⣷⡟⢸⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⠛⢏⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣤⣤⣀⠀⠀⠉⢻⡙⢡⢆⣴⣦⠈⢟⣛⣿⣾⣷⡆⠀
⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠉⢣⣿⣾⣽⡀⠀⡉⣿⣿⣿⣿⣿⣿⣿⠁⠀⠁⠀⠈⠀⠀⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣴⣿⣿⣿⣟⣷⣦⡀⠘⡇⣠⣼⣿⣿⣶⣿⣿⣿⣿⣿⣧⣦
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⣿⡾⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡇⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⢹⣏⣹⡿⠟⢻⣿⣿⣿⣿⣿⣭⣿⣿⣏⣟⣿⣇⢠⣿⣿⣿⣿⣿⣿⡿⠋⠁⠉⠹⠛⠣
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣯⣾⣓⣼⣧⣀⡀⠀⠀⠀⠀⠀⠿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⠙⠟⢿⣿⣿⣿⣿⣿⣿⣇⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡦⠀⢀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠀⠀⠀⠈⢻⣿⣿⣿⣟⠋⢈⣙⣿⣻⣿⣿⣿⣿⣿⣧⡜⠃⣴⣿⣯⣻⣿⠿⣷⣿⣿⣿⣿⣇⣴⠏⠀⠉⡉⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣦⣤⣾⡤⣠
⠀⠀⠀⠀⠸⣿⠃⠁⢸⣿⣦⣤⣶⣶⣿⣿⣿⠟⢻⣦⣨⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠙⣻⣿⣷⣿⣿⣿⣿⡏⠀⣴⣦⣦⣷⣴⢿⡿⡿⠟⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢿⠿⠃⠩⠷⣈
⠀⠀⠀⠀⠈⣇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⠟⠓⣿⣿⣿⣿⣿⣿⡇⡐⢻⡷⠽⠋⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠿⠿⣿⣿⡀⠈⠡⠤⢿⡟
⠀⠀⠀⠀⢰⣻⡆⠀⠈⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⡿⠟⣹⢿⠁⠤⠤⠾⠿⠃⣶⣿⣿⣿⣿⣿⣿⣿⣷⠸⣷⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢿⣿⣿⣿⣿⣿⣿⢿⠋⠀⠀⠀⠀⠀⠀⠁⠘⣀⡀⢀⣺⣾
⠀⠀⠀⠀⢀⣿⡇⠀⠀⢹⠇⣐⣿⣿⡟⢟⠿⠟⣋⣙⢿⣷⣤⢸⡇⠴⠟⠛⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⡛⠉⠛⢳⣦⣄⡀⠀⠀⠀⡇⢈⣿⣿⣿⣿⣿⠋⢰⠀⠀⠀⠀⠀⠀⠀⠀⢦⣇⣈⣉⣩⣡
⠀⠀⠀⠀⠘⢽⣷⠀⠀⢸⡓⠚⢹⡟⠀⠀⡑⠀⠈⠉⢸⣿⡏⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢈⠇⠀⠀⠀⠉⠙⠀⠀⠀⡇⠸⣿⣿⣿⡿⠁⠀⢸⠀⣀⡀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⣸⣿⡆⠀⠀⠹⠀⠟⠀⠀⠀⠁⠀⡀⠀⠀⢻⡆⠈⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠁⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠿⠿⣿⠁⠀⠀⠘⠀⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⢻⣿⣿
⠀⠀⠀⠀⢀⠠⣿⣇⠀⠀⠀⠀⠀⠀⠀⠸⠿⡛⠃⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠙⢿⠿⠛⠁⢀⣀⣀⣈⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡎⠀⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠐⠀⠿⠿⡆⠀⠀⠀⠀⠀⠰⠀⢈⣉⡀⢀⣀⣀⡉⣤⣄⢠⣤⣤⣤⣶⣶⠀⠁⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⠇⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⡿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⢰⣿⣦⡏
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⢘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⣷
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⡇⠹
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠃
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡠⢄⣀⣠⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⡀⣤⡠⣤⠤⣤⡤⣤⣤⠤⡄⡀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣀⠀⣀⢠⣤⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⠀⠀⣠⡶⣼⣿⣿⡿⣯⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⡥⣿⣭⣌⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠼⣠⣤⡆⡿⣿⡗⠗⣈⡇⣶⣶⣶⣿⢶⢶⣆⡀⠀
⣿⣷⣦⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠁⠢⢯⠟⣿⣿⣿⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⠿⠻⠿⡟⠻⣼⣷⣾⣷⠀⣿⣀⣿⣽⣿⣿⡇⢤⠀
⣼⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣷⡀⢻⣿⣿⣿⣧⡦⡙⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠶⢶⣥⣤⣬⡁⣿⡉⠀⠩⡏⣩⠉⢩⡍⠅⠻⠀
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿⣿⢿⣿⣷⡟⢹⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡛⠻⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢁⣤⣄⣀⡀⠀⠉⠻⡛⢡⠆⣴⣦⠈⠟⣛⣻⣿⣷⡄⠀
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠎⠉⢡⣿⣮⣽⡀⠀⡉⣿⣿⣿⣿⣿⣿⣿⠋⠀⠈⠀⠁⠁⠀⠁⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣰⣿⣿⣿⣿⣷⣦⡀⠘⣇⣠⣼⣿⣿⣷⣾⣽⣿⣿⣿⣷⣦
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠠⢾⡾⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠀⠀⣤⠂⠀⠀⠀⠀⠀⠈⠸⣏⣹⡿⠟⢻⣿⣿⣿⣿⣿⣭⣿⣿⣟⣛⣿⣧⢀⣿⣾⣿⣿⣿⣿⣿⠛⠉⠈⠙⠛⠓
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣼⣗⣢⣷⣀⡀⠀⠀⠀⠀⠀⠸⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠉⠙⠟⣿⣿⣿⣿⣿⣿⣿⣧⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠘⢻⣿⣿⣿⣟⠋⢈⣙⣿⣻⣿⣿⣿⣿⣿⣧⡜⠃⣤⣿⣯⣿⣿⡿⣷⣿⣿⣿⣿⣏⣠⠞⠀⠉⡉⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣼⣆⣀
⠀⠀⠀⠀⠘⣿⠛⠀⢸⣿⣦⣤⣶⣶⣿⣿⣿⠟⢻⣦⣨⣁⢸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠙⣻⣿⣷⣶⣿⣿⣿⡟⠀⣤⣤⣧⣷⣴⢾⡿⡟⠟⢻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⠃⠩⠷⣌
⠀⠀⠀⠀⠈⣧⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⠟⠓⣿⣿⣿⣿⣿⣿⡇⡐⢻⡷⠽⠋⠂⠀⠀⠀⠀⠀⠹⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠻⠿⣿⣿⡄⠈⠡⠤⢾⡿
⠀⠀⠀⠀⢈⣿⡆⠀⠀⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⡿⠛⣻⣿⠁⠤⠤⠾⠿⠋⣶⣿⣿⣿⣿⣿⣿⣿⡷⢸⣷⣴⣂⠀⠀⠀⠀⠀⠀⠀⠀⡈⢿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠉⠈⢂⣀⠀⣰⣶
⠀⠀⠀⠀⢘⢿⡇⠀⠀⢹⠇⠀⣿⣿⠛⠿⠟⠛⣋⣙⣿⣷⣤⢸⡇⠴⠿⠛⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⡛⠉⠻⠶⣦⣄⡀⠀⠀⠀⠇⠀⣿⣿⣿⣿⣿⠋⢸⠀⠀⠀⠀⠀⠀⠀⠀⢦⣦⣈⣉⢩⣁
⠀⠀⠀⠀⠨⣻⣷⠀⠀⢸⡓⠚⣹⠟⠀⠀⠇⠀⠉⠙⢻⡿⡏⢨⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢈⠇⠀⠀⠀⠈⠱⠀⠀⠀⠀⠀⣿⣿⣿⡿⠃⠀⢸⠀⣀⡀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢍⣿⡆⠀⠀⠳⠀⠏⠀⠀⠀⠀⢀⠀⠀⠈⣷⣆⠈⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠁⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⡇⠀⠀⠘⠀⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⢻⣿⣿
⠀⠀⠀⠀⠀⠂⣿⣇⠀⠀⠀⠀⠀⠀⠀⠻⠛⡛⠃⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠙⢿⠿⠛⠁⢀⣀⣀⣈⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡎⠀⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠃⠿⠿⡆⠀⠀⠀⠀⠀⠄⠀⢈⣉⡀⢀⣀⣈⡉⣤⣄⢠⣤⣤⣤⣶⣶⠀⠁⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⠿⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⢰⣿⣦⡇
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⢘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⣷
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠹
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠂
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠠⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣀⣠⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⡀⣤⡠⡤⠤⢤⣤⣤⣤⠤⡄⣀⣀⡀⠀⠀⡀⠠⠀⠀⠀⠀
⣀⠀⣀⣠⣤⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⠀⠀⣠⡶⣾⣿⣿⡿⣯⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⣭⣿⣅⣉⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠼⣠⣤⡄⡿⣿⡗⠗⢍⠏⣶⢶⣶⡿⣶⢶⣄⡀⠀
⣿⣷⣦⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠁⠠⠯⠟⣿⣿⣿⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣽⣿⣿⡟⠻⢿⠟⠿⣴⣷⡾⣶⠄⣿⣁⣿⣽⣿⣿⡇⣤⠀
⣹⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣷⡀⢹⣿⣿⣿⣧⡮⡛⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠶⢶⣥⣤⣬⡁⣿⡉⠀⠉⡏⣉⠉⢩⡍⠅⠳⠀
⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠿⣿⣷⡿⣿⣿⣿⣷⡟⢸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡛⠻⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢁⣤⣄⣀⠀⠀⠉⢻⡙⢡⢆⣴⣦⠉⠟⣛⣿⣿⣷⡄⠀
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠉⣡⣿⣮⣽⠀⢀⣉⣿⣿⣿⣿⣿⣿⣿⠋⠀⠊⠀⠁⠁⠀⠁⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣰⣿⣿⣿⣿⣷⣦⡀⠘⣇⣤⣼⣿⣿⣶⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⣿⢾⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠀⠀⣴⠆⠀⠀⠀⠀⠀⠈⠸⣏⣹⡿⠟⢻⣿⣿⣿⣿⣿⣭⣿⣿⣟⣛⣿⣧⢀⣿⣶⣿⣿⣿⣿⣿⠟⠉⠈⠙⠟⠃
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣼⣟⣠⣷⣄⡀⠀⠀⠀⠀⠀⠸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣞⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠉⠙⠟⣿⣿⣿⣿⣿⣿⣿⣇⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠀⠀⠀⠘⢻⣿⣿⣿⣟⠋⢈⣙⣿⣻⣿⣿⣿⣿⣿⣧⡜⠃⣤⣿⣯⣹⣿⡿⣷⣿⣿⣿⣿⣏⣠⠞⠐⠉⡁⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣦⣀
⠀⠀⠀⠀⠘⣿⠛⠉⢸⣿⣦⣤⣶⣶⣿⣿⣿⡟⢛⣢⣈⣁⣸⣿⣟⣿⣿⣿⡿⠋⠀⠀⠀⠉⠙⣻⣿⣷⣶⣿⣿⣿⡟⠀⣤⣤⣥⣷⣶⢾⡿⡟⠟⢻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⠛⠩⠿⣎
⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠡⣿⣿⣿⣿⣿⡇⠀⠀⢀⣀⣀⠞⠓⣾⣿⣿⣿⣿⣿⡇⡐⢻⡷⠽⠛⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠻⠿⣿⣿⡄⠈⠣⠤⣾⡿
⠀⠀⠀⠀⢠⢻⡆⠀⠀⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⣿⠟⣿⣿⠁⠤⠤⠾⠿⠃⣶⣿⣿⣿⣿⣿⣿⣿⡷⢸⣷⣤⣂⠀⠀⠀⠀⠀⠀⠀⠀⡈⢿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠉⠈⢂⣀⠀⣰⣶
⠀⠀⠀⠀⢰⣿⡇⠀⠀⢸⠇⠀⣿⣿⠛⠿⠟⠛⣋⣙⣿⣷⣄⢸⡇⠴⠿⠛⠃⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⡟⠉⠛⠷⣦⣄⡀⠀⠀⠀⠇⠀⣿⣿⣿⣿⣿⠋⢸⠀⠀⠀⠀⠀⠀⠀⠀⢦⣦⣈⠉⢩⣁
⠀⠀⠀⠀⠈⢿⣿⠀⠀⢸⡓⠚⣹⠟⠀⠀⠇⠀⠉⠙⢻⡿⡏⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢈⠇⠀⠀⠀⠉⠑⠀⠀⠀⠀⠀⣿⣿⣿⡿⠃⠀⢸⠀⣀⡀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠳⠀⠏⠀⠀⠈⠀⢀⠀⠀⠈⣷⣆⠈⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠁⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⡇⠀⠀⠘⠀⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⢻⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠻⠛⡛⠃⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠙⢿⠿⠛⠁⢀⣀⣀⣈⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡎⠀⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠻⢿⡆⠀⠀⠀⠀⠀⠄⠀⢈⣉⡀⢀⣀⣈⡉⣤⣄⢠⣤⣤⣴⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⠟⣿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⡿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⢰⣿⣧⡇
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢹⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⢘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⣷
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⢉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠹
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠃
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⢄⣀⣠⣤⣤⣤⣤⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⡀⣤⡠⣤⠤⣤⣤⣤⣤⡤⡤⢀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣀⠀⣀⢠⣤⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⠀⠀⣀⡶⣼⣿⣿⡿⣧⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⣵⣿⣋⣋⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠼⣠⣤⡄⢻⣿⡿⠶⢈⠋⣶⣶⣶⣿⠷⢶⣆⡀⠀
⣿⣿⣦⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠁⠢⣯⠟⣿⢿⣿⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣷⠿⠻⠿⡏⠻⣽⣷⣾⣧⠀⣿⣁⣿⣽⣿⣿⡇⢄⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣷⡀⢹⣿⣿⣿⣣⡦⡛⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⠷⠶⢶⣧⣤⣬⡁⣿⡉⠀⠉⡍⣭⠉⢩⣍⡅⠲⠂
⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⡿⣿⣿⣿⣷⡟⢹⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡛⠻⡉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⣠⣤⣀⡀⠁⠉⠙⡟⢡⠆⣴⣦⠀⣟⣓⣿⣿⣷⡆⠀
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠉⣡⣿⣯⣹⡀⠀⡉⣿⣿⣿⣿⣿⣿⣿⠋⠀⠈⠀⠀⠐⠀⠈⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⡿⢦⡀⠐⣷⣠⣼⣿⣿⣶⣿⣿⣿⣾⣿⣷⣦
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⣾⡿⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠀⠀⢴⡆⠀⠀⠀⠀⠀⠀⠘⣏⣹⡿⠟⢻⣿⣿⣿⣷⣿⣭⣽⣿⣟⣛⣿⣷⠀⣿⣷⣿⣿⣿⣿⣿⠝⠉⠁⠉⠛⠓
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠒⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣼⣟⣢⣷⣄⡀⠀⠀⠀⠀⠀⠸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣗⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⣀⠀⠀⠀⠀
⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⠀⠀⠈⠘⢿⣿⣿⣿⣟⠋⢈⣙⣿⣿⣿⣿⣿⣿⣿⣧⡔⠃⢤⣿⣯⣻⣿⡿⢷⣾⣿⣿⣿⣟⣠⠞⠀⠉⡉⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣤⣴⣦⣀
⠀⠀⠀⠀⠈⣿⠏⠉⢸⣿⣆⣀⣴⣶⣾⣿⣿⡟⢷⣢⣈⣁⣸⣿⣟⣿⣿⣿⡿⠋⠀⠀⠀⠉⠙⣹⣿⣿⣶⣿⣿⣿⡟⠀⣤⣤⣶⣷⣴⢾⡿⡿⠟⠻⣿⣿⡿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⠟⠉⠽⣌
⠀⠀⠀⠀⠀⣿⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠡⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣀⠞⠓⣾⣿⣿⣿⣿⣿⡇⡐⠻⡷⢽⠋⠁⠀⠀⠀⠀⠀⠙⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠻⠿⣿⣿⡇⠈⠣⠤⠼⡧
⠀⠀⠀⠀⢀⢿⡇⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⣿⠟⣿⣿⠁⠤⠤⠾⠿⠃⣶⣿⣿⣿⣿⣿⣿⣿⣷⠸⣷⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠉⠊⢂⣀⠀⣠⣴
⠀⠀⠀⠀⢠⣿⣧⠀⠀⢸⡇⠀⣿⣿⠛⠻⠛⠋⣉⡙⣿⣷⡄⢸⡇⠴⠿⠛⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⣛⠉⠛⠷⣦⣄⡀⠀⠀⠀⢀⠀⣿⣿⣿⣿⣿⠏⢸⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣈⣉⢉⣉
⠀⠀⠀⠀⠀⣻⣿⡄⠀⠸⠓⠚⢹⠏⠀⢈⠃⠀⠈⠁⢿⣿⡏⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢈⠇⠀⠀⠀⠉⠨⠀⠀⠀⢸⠀⣿⣿⣿⣿⠋⠀⠈⠀⣀⡀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡆⣿⣇⠀⠀⠑⠐⠋⠀⠀⠀⠀⣀⠀⠀⠘⣷⠆⠈⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠿⠿⣿⡇⠀⠀⠀⠀⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⢻⣿⣿
⠀⠀⠀⠀⠀⠏⣿⣿⠀⠀⠀⠀⠀⠀⠀⠛⢛⡛⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠙⢿⠿⠛⠁⢀⣀⣀⣈⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡎⠁⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠁⠸⠿⣷⠀⠀⠀⠀⠀⠂⠀⢉⣁⡀⢀⣀⣈⡉⣤⣄⢠⣤⣤⣤⣶⣶⠀⠁⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⠿⢿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢻⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⢰⣿⣧⡇
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⢘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⢷
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠸
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠀
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⢄⣀⣠⣤⣤⣤⣤⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⡀⣤⡤⣤⠤⣤⣤⣤⣤⣤⡤⢀⣀⡀⠀⠀⠀⠠⠀⠀⠀⠀
⣀⠀⣀⢠⣤⣀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣀⡶⣼⣿⣿⡿⣇⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⣥⣿⣋⣊⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠼⢠⣤⡄⠿⣿⡗⠗⢊⠋⣶⣶⣶⣿⠶⢶⣆⡀⠀
⢿⣿⣦⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠑⠒⣯⠟⣿⣿⣿⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣽⣿⣷⠟⠻⢿⡏⠿⣽⣷⡾⣧⠀⣿⣀⣿⣽⣿⣿⣇⢄⠀
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣷⡀⢻⣿⣿⣿⣣⡦⡙⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠶⢶⣥⣤⣬⡁⣾⡉⠀⠉⡍⣩⠉⢩⣍⡅⠺⠄
⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣷⡿⣿⣿⣿⣷⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡛⠻⡉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣠⣤⣀⡀⠁⠈⠙⡝⢩⢆⣴⣦⠀⣟⣓⣿⣿⣷⡆⠀
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠉⣣⣿⣮⣽⡀⠀⡉⣿⣿⣿⣿⣿⣿⣿⠏⠀⠈⠀⠀⠈⠀⠈⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣾⣿⣿⣿⣿⢦⡀⠀⢻⣠⣼⣿⣿⣶⣿⣿⣿⣭⣿⣷⣆
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⣿⠾⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠘⣏⣹⡿⠟⢻⣿⣿⣿⣿⣿⣭⣽⣿⣿⣛⣻⣿⠀⣿⣿⣿⣿⣿⣿⣿⠝⠉⠁⠉⠟⠓
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣼⣟⣢⣿⣄⡀⠀⠀⠀⠀⠀⠸⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠋⠻⡿⣿⣿⣿⣿⣿⣿⣿⣗⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣀⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⣿⢿⣿⣿⠀⠀⠈⠘⢿⣿⣿⣿⣟⠛⢈⣙⣿⣿⣿⣿⣿⣿⣿⣧⡜⠃⢤⣿⣯⣻⣿⡿⢿⣾⣿⣿⣿⣟⣠⠞⠀⠉⡉⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣤⣴⣦⣀
⠀⠀⠀⠀⠀⣿⠛⠊⢸⣿⣆⣀⣤⣤⣾⣿⣿⠟⢻⣢⣈⣁⣸⣿⣟⣿⣿⣿⡿⠋⠀⠀⠀⠉⠛⣻⣿⣿⣶⣿⣿⣿⡟⠀⣤⣤⣦⣷⣴⠾⡿⡿⠿⠻⣿⣿⡿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡏⠹⣿⡟⠉⠽⣆
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠡⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⠞⠓⣾⣿⣿⣿⣿⣿⡇⡐⠿⡷⢽⠋⠁⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠻⠿⣿⣿⣇⠈⠁⠤⣼⡷
⠀⠀⠀⠀⢠⣿⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⣿⠟⣿⣿⡁⠤⠤⠼⠿⠋⣶⣿⣿⣿⣿⣿⣿⣿⣷⠸⣷⣤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠈⠈⢂⣀⠀⣰⣷
⠀⠀⠀⠀⠀⣿⣧⠀⠀⠈⡇⠀⣿⣿⠙⠻⠛⢫⣉⡙⣿⣷⡄⢸⡇⠴⠟⠛⠃⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⣚⠉⠛⠷⢦⣄⡀⠀⠀⠀⢸⠀⣻⣿⣿⣿⣿⠏⠸⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣈⣉⢉⣁
⠀⠀⠀⠀⠀⢿⣿⡆⠀⠀⠗⠚⢹⠏⠀⠨⠁⠀⠀⠁⢿⡿⡏⣸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢈⠇⠀⠀⠀⠈⠸⠀⠀⠀⢸⠀⣿⣿⣿⣿⠋⠀⠀⡆⢀⡀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠰⣿⣧⠀⠀⠀⠀⠋⠀⠀⠀⠀⣀⠀⠀⠘⣷⠆⠈⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠿⠿⣿⡇⠀⠀⠀⠁⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⢻⣿⣿
⠀⠀⠀⠀⠀⡁⢿⣿⠀⠀⠀⠀⠀⠀⠀⠛⢛⠛⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠙⢿⠿⠛⠁⢀⣀⣀⣈⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡎⠁⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠂⠸⠿⣷⠀⠀⠀⠀⠀⠀⠀⢉⣁⡀⢀⣀⣈⡉⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⠿⢻
⠠⣤⡄⣤⡄⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⢰⣿⣧⡌
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⢘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⢳
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠸
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠀
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⢄⣀⣠⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⡀⣤⡠⣤⢤⣤⣤⣤⣤⡤⡤⢀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣀⠀⣀⢠⣤⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⠀⠀⣠⡦⣾⣿⣿⡿⣷⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⣧⣿⣍⣈⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠴⣠⣤⡄⣻⣿⡗⠗⣋⠋⣶⣶⣶⡿⢶⢶⣄⠀⠀
⢿⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠁⠂⣮⠟⣿⢿⣿⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣽⣿⣿⠟⠻⠿⡏⠿⣽⣷⡾⣧⠀⣿⣁⣿⣽⣿⣿⡇⢤⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣷⣀⢻⣿⣿⣿⣣⡦⡙⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⠷⠶⠶⣧⣤⣬⡁⣾⡉⠀⠉⡏⣭⠉⢩⡍⠅⠻⡄
⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣷⡿⣿⢿⣿⣷⣟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣛⠻⡉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣠⣤⣀⡀⠀⠈⠙⡟⢡⠆⣴⣦⠀⢟⣛⣿⣿⣷⡄⠀
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠉⢡⣿⣮⣽⡀⠈⣙⣿⣿⣿⣿⣿⣿⣿⠏⠀⠈⠁⠀⠘⠀⠈⠊⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣾⣿⣿⣿⡿⢦⡀⠀⢻⣤⣼⣿⣿⣶⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠠⢿⠿⢩⣷⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠀⡀⢰⡎⠀⠀⠀⠀⠀⠀⠘⣏⣹⡿⠿⢻⣿⣿⣿⣿⣿⣭⣽⣿⣿⣛⣻⣿⠀⣼⣿⣿⣿⣿⣿⣿⠟⠉⠁⠉⠟⠓
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣼⣿⣢⣿⣄⡀⠀⠀⠀⠀⠀⠸⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠻⡿⣿⣿⣿⣿⣿⣿⣿⣗⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣀⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠈⠘⢿⣿⣿⣿⣟⠋⢈⣙⣿⣿⣿⣿⣿⣿⣿⣧⡐⠃⢤⣿⣯⣿⣿⡿⢿⣾⣿⣿⣿⣟⣠⠞⠑⠋⡉⠼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⡤⣴⣧⢀
⠀⠀⠀⠀⠀⣿⠟⠉⠘⣿⣇⣀⣤⣤⣾⣿⣿⡟⢿⣢⣈⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠛⣻⣿⣿⣿⣿⣿⣿⡟⠀⣤⣦⣧⣷⣴⡾⡿⢿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠹⣿⡟⠉⠽⢆
⠀⠀⠀⠀⠀⢾⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⠟⠛⣾⣿⣿⣿⣿⣿⡇⡐⠻⡷⢽⠛⠁⠀⠀⠀⠀⠀⠘⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⠿⣿⣿⣇⠈⠁⠤⠼⡟
⠀⠀⠀⠀⠀⢾⣷⠀⠀⢻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⣿⠟⣿⣿⡁⠤⠤⠾⠿⠃⣴⣿⣿⣿⣿⣿⣿⣿⣷⠸⣷⣴⣎⠀⠀⠀⠀⠀⠀⠀⠀⢨⠹⣿⣿⣿⣿⣿⣿⡿⡟⠉⠀⠀⠀⠀⠀⠈⠊⢃⣀⠀⣀⣵
⠀⠀⠀⠀⠀⣜⣿⠀⠀⠈⡇⠀⣿⣿⠙⠻⠛⢫⣉⡙⣿⣷⣄⢸⡇⠴⠿⠛⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⣛⠉⠛⠳⣦⣄⡀⠀⠀⠀⢸⠀⣻⣿⣿⣿⣿⡟⠁⡄⠀⠀⠀⠀⠀⠀⠀⢰⣼⣀⣉⡉⣉
⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠗⠚⢹⠇⠀⠨⠁⠀⠈⠁⢿⡟⡏⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠈⠇⠀⠀⠀⠈⠸⠀⠀⠀⢸⠀⣿⣿⣿⣿⠋⠀⠀⡇⢀⡀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠢⢿⣷⠀⠀⠐⠐⠋⠀⠀⠀⢀⣀⠀⠀⠸⣷⡆⠈⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠹⣿⣿⣿⣿⣿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠻⠿⢿⡏⠀⠀⠀⠃⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⢻⣿⣿
⠀⠀⠀⠀⠀⡀⢸⣿⡄⠀⠀⠀⠀⠀⠈⠛⣛⠋⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠙⢿⠿⠛⠁⢀⣀⣀⣈⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡆⠃⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠁⠸⠿⣿⠀⠀⠀⠀⠀⠀⠀⢉⣀⡀⢀⣀⣉⡉⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⠀⠿⢹
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⡿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⢰⣿⣧⡌
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢹⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⢘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⢿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠘
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⢤⣀⣠⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⡀⣤⡠⣤⢤⣤⣤⣤⣤⣤⡤⢀⣀⡀⠀⠀⡀⠠⠀⠀⠀⠀
⣀⠀⣀⢠⣤⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⠀⠀⣠⡶⣼⣿⣿⣿⣷⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⣥⣿⣃⣎⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠼⣠⣤⡎⣻⣿⡗⠗⣋⠋⣶⣶⣶⡿⢷⢶⡄⠀⠀
⣿⣿⣿⣿⣽⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠁⠢⣮⠟⣿⣿⣿⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⠟⠻⠿⡏⠿⣼⣷⡾⣦⠀⣿⣁⣿⣽⣿⣿⡇⢤⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣷⡀⢻⣿⣿⣿⣣⡦⡝⡿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⠷⠶⠶⣧⣤⣬⡁⣾⡉⠀⠉⡏⣩⠉⢩⣍⠅⠻⠄
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣷⡿⣿⢿⣾⣷⡟⢹⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⠻⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣠⣤⣀⡀⠀⠈⠙⣟⢡⢆⣴⣦⠀⢟⣛⣿⣿⣷⡄⠀
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠉⢡⣿⣾⣽⡀⠈⡉⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠁⠀⠘⠀⠈⠊⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣞⣿⣿⣿⡿⢦⣄⠀⢻⣠⣼⣿⣿⣶⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠠⢶⡾⢩⣷⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠀⡀⢰⡎⠀⠀⠀⠀⠀⠀⠙⣏⣹⡿⠟⢻⣿⣿⣿⣿⣿⣭⣽⣿⣿⣛⣻⣿⡀⣼⣾⣿⣿⣿⣿⣿⡟⠋⠁⠉⠛⠓
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣼⣿⣢⣽⣦⣀⠀⠀⠀⠀⠀⠸⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠻⡿⣿⣿⣿⣿⣿⣿⣿⣗⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣶⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⣀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠈⠘⢿⣿⣿⣿⣟⠋⢈⣙⣿⣻⣿⣿⣿⣿⣿⣧⡜⠃⢤⣿⣯⣿⣿⡿⠿⣾⣿⣿⣿⣟⣠⠞⠑⠉⡁⠼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣴⣧⣀
⠀⠀⠀⠀⠀⣿⡟⠉⠈⣿⣇⣀⣠⣤⣾⣿⣿⡟⢷⣢⣈⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠛⣻⣿⣿⣿⣿⣿⣿⡟⠀⣤⣤⣤⣶⣤⡾⣿⢿⠻⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠩⣿⡟⠉⠽⢇
⠀⠀⠀⠀⠀⣸⡅⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠡⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⠟⠓⣾⣿⣿⣿⣿⣿⡇⡐⢻⡷⢽⠟⠁⠀⠀⠀⠀⠀⠈⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠻⢿⣿⣿⣇⠈⠙⠤⠴⣿
⠀⠀⠀⠀⠀⢼⣷⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⣿⠛⣿⣿⠁⠤⠤⠾⠿⠋⣴⣿⣿⣿⣿⣿⣿⣿⣷⢸⣷⣤⣎⡀⠀⠀⠀⠀⠀⠀⠀⢠⠹⣿⣿⣿⣿⣿⣿⡿⡟⠋⠀⠀⠀⠀⠀⠈⠙⠓⣀⡀⢀⣥
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠈⡇⠀⣿⣿⠙⠻⠛⢫⣉⡙⣿⣷⣄⢸⡇⠴⠟⠛⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⡚⠉⠛⠳⢦⣄⡀⠀⠀⠀⢸⠀⣹⣿⣿⣿⣿⡟⠁⡇⠀⠀⠀⠀⠀⠀⠀⠰⣴⣀⣉⡉⣍
⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠗⠚⢹⠇⠀⠨⠁⠀⠀⠁⣿⡿⡏⢈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠈⠇⠀⠀⠀⠈⠹⠀⠀⠀⢸⠀⢿⣿⣿⣿⠏⠀⠀⡇⢀⣀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣻⣷⠀⠀⠀⠀⠋⠀⠀⠀⢀⣀⠀⠀⠸⣷⠆⠈⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠿⢿⡟⠀⠀⠀⠃⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⠻⣿⣿
⠀⠀⠀⠀⠀⠈⢸⣿⡄⠀⠀⠀⠀⠀⠀⠛⣛⡋⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠙⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡴⣶⡶⣶⡆⠃⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠘⠿⣿⠄⠀⠀⠀⠀⠀⠀⢉⣀⡀⢀⣀⣉⡉⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⢀⡿⢸
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⠀⣿⣧⡘
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⢘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⢐⣿⣿⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⠉⡁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠈
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⢤⣀⣠⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⡀⣤⡤⡤⢤⣤⣤⣤⣤⡤⡤⢀⣀⡀⠀⠀⡀⠄⠀⠀⠀⠀
⣀⡀⣀⢠⣤⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⠀⠀⣠⡶⣾⣿⣿⣿⣷⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⣧⣿⣃⣊⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠼⣠⣤⡄⡻⣿⡗⠗⣏⠋⣶⣶⣶⡿⢶⢶⣄⡀⠀
⣿⣿⣿⣿⣿⣿⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠁⠂⣯⠟⣿⣿⣿⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣽⣿⣷⡟⠻⠿⡏⠻⣼⣷⣾⣧⠀⣿⣁⣿⣽⣿⣿⡇⣤⠀
⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣷⡀⢻⣿⣿⣿⣣⡦⡝⡿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⠷⠶⠶⣶⣤⣬⡁⣾⡉⠀⠉⡏⣩⠉⢩⣍⡅⠳⠂
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣷⡿⣿⢿⣿⣷⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⠛⢏⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⣠⣤⣀⣀⠈⠈⠙⢟⢡⢆⣴⣦⠀⢟⣛⣿⣿⣷⡆⠀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠊⠉⢣⣿⣮⣹⡀⠈⡙⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠁⠈⠈⠀⠈⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣞⣿⣿⣿⣿⢶⣄⠀⢺⣤⣾⣿⣿⣶⣿⣿⣿⣯⣿⣷⣆
⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠠⢾⡿⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠀⣀⢀⡞⠀⠀⠀⠀⠀⠀⠉⣏⣹⡿⠟⢻⣿⣿⣿⣿⣾⣯⣭⣿⣿⣛⣻⣿⡆⢸⣿⣿⣿⣿⣿⣿⡿⠋⠁⠈⠛⠛
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣽⣯⣽⣿⣷⡄⠀⠀⠀⠀⠀⠸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣷⣻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠛⢿⢿⣿⣿⣿⣿⣿⣿⣗⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣀⡀⠀⠀⠀
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇⠀⠀⠈⢿⣿⣿⣿⣟⠋⢈⣛⣿⣻⣿⣿⣿⣿⣿⣧⡔⠃⢤⣿⣯⣽⣿⣛⡿⣾⣿⣿⣿⣿⣠⡜⠑⠋⡁⠼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣦⣤⣷⡄
⠀⠀⠀⠀⠀⢿⡿⠏⠀⣿⣧⣀⣤⣤⣴⣿⣿⡟⢿⣢⣈⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠛⣻⣿⣿⣿⣿⣿⣿⡟⠀⣤⣤⣦⣶⣤⡾⣿⢿⠻⠛⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢿⡟⠃⠹⠷
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠡⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡞⠛⣺⣿⣿⣿⣿⣿⡇⡐⢻⡷⢽⠟⠁⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠹⠿⣿⣿⣿⠀⠙⠤⠤⠿
⠀⠀⠀⠀⠀⣾⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⣿⠟⣿⣿⡇⠤⠤⠾⠿⠋⢴⣿⣿⣿⣿⣿⣿⣿⣷⠸⣷⣤⣟⡀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠈⠓⠑⣀⡀⢀⣦
⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⡷⠀⣿⣿⠙⠻⠛⢫⣉⡙⣿⣷⣄⢸⡇⠴⠿⠛⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⣚⠉⠛⠳⢦⣄⡀⠀⠀⠀⢸⠀⢸⣿⣿⣿⣿⡿⠁⡇⠀⠀⠀⠀⠀⠀⠀⠰⣤⣇⣉⡉⣉
⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠗⠚⢹⠏⠀⠨⠂⠀⠈⠁⢿⡿⡏⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠈⠇⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠀⠀⡇⢀⣀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⢸⣿⠀⠀⠐⠀⠋⠀⠀⠀⢀⣀⠀⠀⠘⣷⠆⠈⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠹⣿⣿⣿⣿⣿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⠀⠀⠀⠃⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⠻⣿⣿
⠀⠀⠀⠀⠀⢠⢸⣿⡇⠀⠀⠀⠀⠀⠀⠛⣛⡛⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠘⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡆⠃⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠘⠿⣿⠆⠀⠀⠀⠀⠀⠀⢉⣀⡀⢀⣀⣉⡉⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⢀⡻⢻
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⠀⣿⣷⣸
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢹⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⢘⡛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⠉⡁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠠⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢠⠄⣀⣠⣤⣤⣤⣤⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⡀⣤⡤⢤⠤⣤⣤⣤⣤⡤⡤⠀⣀⡀⠀⠀⡀⠠⠀⠀⠀⠀
⣀⡀⣀⢠⣤⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⠀⠀⣠⡶⣼⣿⣿⡿⣧⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⣵⣿⣋⣊⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠼⣠⣴⡄⠿⣿⡗⠗⢌⠏⣶⣶⣶⣿⢶⢶⣆⠀⠀
⢻⣿⣿⣿⣽⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠁⠂⣯⠟⣿⣿⣿⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣷⠿⠻⢿⡏⠻⣽⣷⣾⣷⠀⣿⣈⣿⣽⣿⣿⡇⢄⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣷⡀⢻⣟⣿⣿⣣⡦⡟⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⠷⠶⠶⣶⣤⣬⡁⣾⡉⠀⠉⠍⣉⠉⢩⣍⡅⠻⠇
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣷⡿⣿⣿⣿⣷⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⠛⢏⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣀⣤⣄⣀⠈⠈⠉⢟⢣⢂⣴⣶⠀⣟⣓⣿⣾⣷⡄⠀
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠉⣣⣿⣾⣽⡀⠀⡉⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠁⠈⠈⠀⠈⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣾⣻⣿⣿⣿⢷⣄⠀⢺⣥⣼⣿⣿⣶⣾⣿⣿⣯⣿⣷⣆
⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠠⣿⡾⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡀⣀⢀⡞⠀⠀⠀⠀⠀⠀⠉⣯⣹⡿⠟⢻⣿⣿⣿⣿⣾⣯⣭⣿⣿⣛⣻⣿⡆⢸⣿⣿⣿⣿⣿⣿⡿⠋⠁⠈⠛⠛
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣽⣯⣽⣿⣷⡄⠀⠀⠀⠀⠀⠸⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠙⢿⢿⣿⣿⣿⣿⣿⣿⣗⣀⣹⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢀⡀⠀⠀⠀
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇⠀⠀⠈⠿⣿⣿⣿⣟⠋⢈⣙⣿⣻⣿⣿⣿⣿⣿⣧⡔⠃⢤⣿⣯⣻⣿⣛⡿⣾⣿⣿⣿⣿⣠⡜⠑⠋⡁⠼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣦⣤⣾⡄
⠀⠀⠀⠀⠀⣼⡟⠙⠀⣿⣧⣀⣠⣤⣴⣿⣿⡟⢷⣢⣈⣁⣸⣿⣟⣿⣿⣿⡿⠋⠀⠀⠀⠉⠙⣻⣿⣿⣿⣿⣿⣿⡟⠀⣠⣤⣦⣶⣤⡾⣿⢿⠻⠛⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢿⡟⠋⠩⠷
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠡⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣀⡞⠓⣺⣿⣿⣿⣿⣿⡇⠀⠻⡷⢽⠟⠑⠀⠀⠀⠀⠀⠈⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠹⣿⣿⣿⣿⡀⠙⠠⠤⠟
⠀⠀⠀⠀⠀⢽⣿⠀⠀⠸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⡿⠟⣿⣿⠁⠤⠤⠾⠿⠋⢴⣿⣿⣿⣿⣿⣿⣿⣿⠸⣷⣤⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠈⠙⠑⣀⡀⢀⣠
⠀⠀⠀⠀⠀⢷⣿⡀⠀⠀⣿⠀⣻⣿⠛⠿⠟⢛⣉⡙⣿⣷⣄⢸⡇⠴⠿⠛⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠉⣛⠉⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠁⡇⠀⠀⠀⠀⠀⠀⠀⠰⣤⣇⣉⠉⣉
⠀⠀⠀⠀⠀⢉⣻⣇⠀⠀⠟⠚⢙⠏⠀⢈⠂⠀⠈⠁⢿⡿⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⡇⠀⠀⠈⠉⠙⠀⠀⠀⠀⠄⢸⣿⣿⣿⡟⠁⠀⠁⠀⣀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⢼⣿⡀⠀⠐⠀⠏⠀⠀⠀⠀⣀⠀⠀⠸⣷⠆⠈⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠙⣿⣿⣿⣿⣿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠸⠿⢿⣿⠀⠀⠀⠃⣀⢀⣀⢀⣀⡀⢀⣤⣤⡀⠹⣿⣿
⠀⠀⠀⠀⠀⢨⢸⣿⣇⠀⠀⠀⠀⠀⠀⠛⢛⡛⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠘⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡴⣶⡶⣶⡆⠃⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠠⠀⠿⣿⠆⠀⠀⠀⠀⠀⠀⢉⣁⡀⢀⣀⣉⡉⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⢀⡽⠟
⠠⣤⡄⣤⡄⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢻⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⠀⣿⣷⣠
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⢘⡛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⠉⡁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠇
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣀⣠⣤⣤⣤⣤⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⡀⣤⡠⣤⠤⣤⣤⣤⣤⡤⡤⢀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣄⡀⣀⣠⣤⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣠⡶⣼⣿⣿⡿⣷⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⣷⣿⣉⣊⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠼⣠⣤⡄⣛⣿⡇⠗⢊⠏⣶⣶⣶⣿⠶⢶⣆⠀⠀
⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠁⠲⣯⠟⣿⢿⣿⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣷⠿⠻⢿⠏⠻⣼⣷⡾⣷⠀⣿⣁⣿⣽⣿⣿⡇⢄⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣷⡀⢻⣿⣿⣿⣧⡮⡝⡿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⠷⠶⠶⣶⣤⣬⡁⣾⠉⠀⠉⡍⣉⠉⢩⣍⠅⠻⠄
⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⡿⣿⢿⣿⣷⣟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⠛⠏⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⣤⣄⣀⠀⠀⠉⢻⢡⢆⣴⣦⠀⢟⢓⣿⣾⣷⡄⠁
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠉⣣⣿⣮⣽⡀⠈⡉⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠁⠀⠈⠀⠈⠊⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣶⣛⣿⣿⣿⠷⣄⠀⢸⣧⣼⣿⣿⣶⣾⣿⣿⣿⣿⣷⣆
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠠⣿⡿⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡀⣀⢀⠞⠀⠀⠀⠀⠀⠀⠉⣯⣹⡿⠿⢻⣿⣿⣿⣿⣿⣯⣭⣿⣿⣛⣛⣿⡇⢸⣯⣿⣿⣿⣿⣿⣿⠟⠉⠈⠙⠛
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣷⣯⣽⣿⣷⡄⠀⠀⠀⠀⠀⠘⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠙⠿⢿⣿⣿⣿⣿⣿⣿⣗⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣷⡄⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⡀⠀⠀⠀
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇⠀⠀⠁⠻⣿⣿⣿⣟⠋⢈⣛⣿⣻⣿⣿⣿⣿⣿⣧⡜⠃⣤⣿⣯⣹⣿⣟⡿⣾⣿⣿⣿⣿⣠⡜⠑⠋⡁⠼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢤⣾⡄
⠀⠀⠀⠀⠀⣻⡟⠁⠀⣿⣷⣀⣠⣤⣴⣿⣿⠟⢷⣢⣈⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠛⣹⣿⣿⣿⣿⣿⣿⡟⠁⣠⣤⣷⣶⣦⡾⣿⢿⠿⠛⢿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⢹⣿⠃⠨⠷
⠀⠀⠀⠀⠀⢸⣇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠡⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡟⠓⣺⣿⣿⣿⣿⣿⡇⢀⢻⡷⣽⠟⠛⠁⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⢿⣿⣿⣿⡀⠙⠢⠤⠷
⠀⠀⠀⠀⠀⠺⣿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⡿⠟⣿⢿⠁⠤⠤⠾⠿⠋⢴⣿⣿⣿⣿⣿⣿⣿⣿⠘⣷⣤⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠈⠙⠙⢀⡀⠀⣠
⠀⠀⠀⠀⠀⣴⣿⡀⠀⠀⣿⠀⣽⣿⠛⠻⠟⠛⣉⡙⣿⣷⣄⢸⡇⠴⠿⠚⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡯⠀⠉⣞⠉⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠁⡇⠀⠀⠀⠀⠀⠀⠀⠰⣦⣇⣈⢉⢉
⠀⠀⠀⠀⠀⠸⣿⣷⠀⠀⠟⠚⢙⠟⠀⢈⠂⠀⠈⠙⢿⡿⡏⣸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⡇⠀⠀⠀⠈⠙⠀⠀⠀⠀⡇⢸⣿⣿⣿⡟⠁⠀⠁⠀⣀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⢻⣿⡀⠀⠐⠀⠏⠀⠀⠀⠀⣀⠀⠀⠸⣷⠦⠈⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠸⠿⢿⣿⠀⠀⠀⠀⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⠹⣿⣿
⠀⠀⠀⠀⠀⠘⢹⣿⣇⠀⠀⠀⠀⠀⠀⠛⢛⡛⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠘⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡆⠋⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⡟
⠀⠀⠀⠀⠀⠈⠈⠿⣿⠆⠀⠀⠀⠀⠀⠀⢉⣁⡀⢀⣀⣈⡉⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⢀⡹⠇
⠠⣤⡄⣤⡄⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⠀⣿⣿⣄
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢹⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⠘⡛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⠉⡁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⢄⣀⣠⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⡀⣤⡠⣤⢤⣤⣤⣤⣤⡤⡤⢀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣤⡀⣀⢠⣤⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣠⡶⣼⣿⣿⡿⣧⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⡵⣿⣋⣊⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠼⣠⣤⡄⡿⣿⡟⠗⢊⠋⣶⢶⣶⣿⢶⢶⣄⡀⠀
⢿⣿⣿⣿⣽⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⠁⠢⢯⠟⣿⢿⣿⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣽⣯⣿⡿⠻⠿⠏⠻⣼⣷⡾⣦⠀⣿⣀⣿⣽⣿⣿⡇⢄⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣽⣷⡀⢳⣿⣿⣿⣧⡦⡙⡿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⠷⠶⠶⢷⣤⣬⡁⣿⡉⠀⠉⡍⣉⠉⢩⡍⡅⠻⠄
⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠿⣿⣿⡿⣿⢿⣾⣷⣟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⠻⠏⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⣤⣄⣀⠀⠀⠉⢻⢣⢆⣴⣦⠀⢟⣛⣿⣿⣷⡄⠀
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠉⢣⣿⣮⣽⡀⠈⡉⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠁⠀⠈⠀⠈⠊⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣴⣟⣿⣿⣿⠷⣄⠀⠘⣧⣼⣿⣿⣶⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠠⣿⡶⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡀⣀⢀⠎⠀⠀⠀⠀⠀⠀⠉⣿⣹⡟⠿⢻⣿⣿⣿⣿⣿⣯⣭⣿⣿⣟⣛⣿⡇⢸⣷⣿⣿⣿⣿⣿⣿⠟⠉⠁⠉⠛
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣷⣿⣹⣿⣶⡄⠀⠀⠀⠀⠀⠈⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠙⠻⢿⣿⣿⣿⣿⣿⣿⣗⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣷⡄⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⡄⠀⠀⠀
⠀⠀⠀⠀⠈⣿⣿⣿⡿⣿⡇⠀⠀⠀⠻⣿⣿⣿⣟⠋⢈⣙⣿⣻⣿⣿⣿⣿⣿⣧⡜⠃⢤⣿⣯⡽⣿⡿⡿⣾⣿⣿⣿⣿⢠⡼⠑⠋⡉⠼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣼⡄
⠀⠀⠀⠀⠀⢽⡟⠫⠀⣿⣷⣄⣠⣤⣴⣿⣿⡟⢳⣢⣈⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠙⣻⣿⣿⣿⣿⣿⣿⡿⠁⢠⣤⣦⣷⣦⡾⣿⢿⠿⠛⢿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢹⡟⠓⠠⠶
⠀⠀⠀⠀⠀⢸⣇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣀⡾⠛⢸⣿⣿⣿⣿⣿⣇⢀⠻⣧⢽⡟⠙⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⢿⢿⣿⣿⡄⠘⠢⠤⠗
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⣿⠟⣻⢿⠁⠤⠤⠼⠿⠏⢱⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣶⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠈⠙⠘⢂⣀⠀⣠
⠀⠀⠀⠀⠀⣸⣿⡄⠀⠀⣻⠀⣸⣿⠛⠛⠟⠛⣋⡙⣿⣷⣄⢸⡇⠴⠿⠚⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠈⢚⠉⠙⠳⢦⣄⡀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣿⡿⠁⡇⠀⠀⠀⠀⠀⠀⠀⠰⣦⣆⣈⠉⢩
⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠟⠒⢙⠟⠀⢈⠂⠀⠉⠙⢿⡿⡏⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⡝⠀⠀⠀⠈⠘⠀⠀⠀⠀⡇⢸⣿⣿⣿⡟⠁⠀⢠⠀⣀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣹⣿⡀⠀⠘⠀⠟⠀⠀⠀⠀⣀⠀⠀⠘⣷⠦⠈⠁⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠘⣿⣿⣿⣿⣿⠏⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠿⢿⣿⠀⠀⠀⠀⣀⢀⣀⢀⣀⡀⢀⣤⣤⡀⠹⣿⣿
⠀⠀⠀⠀⠀⢘⢻⣿⣇⠀⠀⠀⠀⠀⠀⠛⢛⡛⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠘⢿⠿⠛⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡆⠋⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⡏
⠀⠀⠀⠀⠀⠘⠈⠿⣿⡆⠀⠀⠀⠀⠃⠀⢉⣁⡀⢀⣀⣈⡉⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⢀⡼⠇
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢻⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⠀⣿⣿⡄
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⠘⡛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿⡏
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⠉⡁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠠⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⢤⣀⣠⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⡀⣤⡠⡤⢤⣤⡤⣤⣤⡤⡤⢀⣀⡀⠀⠀⡀⠤⠀⠀⠀⠀
⣠⣀⣀⣠⣤⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣀⡦⣼⣿⣿⣿⣷⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⣷⣿⣯⣍⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠼⣠⣤⡌⡻⣿⡟⠗⢘⠋⣶⣶⣶⣿⢷⢶⣆⠀⠀
⢻⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠑⠆⣾⠟⣿⣿⣿⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣽⣯⣿⡟⠻⠿⡟⠻⣾⣷⣾⣷⠀⣿⣀⣿⣽⣿⣿⡇⢤⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣷⡀⢻⣿⣿⣿⣧⡮⡍⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠷⠶⠶⢷⣤⣬⡁⣿⡉⠀⠉⡏⣩⠉⢩⣍⡅⠺⠇
⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⣿⢿⣿⣷⡟⢩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠻⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⣤⣄⣀⠀⠀⠙⢻⢡⠆⣴⣦⠀⢟⣛⣿⣿⣿⡆⠂
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠉⢣⣿⣮⣽⡀⠀⡉⣿⣿⣿⣿⣿⣿⣿⡏⠀⠐⠁⠁⠓⠀⠀⠈⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣴⣟⣿⣿⣿⠷⣤⡀⠘⣧⣼⣿⣿⣷⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⣾⡶⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⣿⣹⡿⠿⢻⣿⣿⣿⣿⣿⣯⣭⣿⣿⣟⣛⣿⣧⢠⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠈⠙
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣷⣟⣺⣾⣦⡀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠉⠻⡿⣿⣿⣿⣿⣿⣿⣷⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡄⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡠⠀⠀⠀
⠀⠀⠀⠀⠈⢿⣿⣿⡿⣿⡧⠀⠀⠀⠹⣿⣿⣿⣟⠋⢈⣙⣿⣻⣿⣿⣿⣿⣿⣧⡔⠃⢤⣿⣯⣿⣿⣿⣿⣾⣿⣿⣿⣿⣁⡴⠑⠉⡁⢼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⡦
⠀⠀⠀⠀⠀⢻⡏⠏⠀⣿⣧⣄⣠⣤⣤⣿⣿⡟⢳⣢⣨⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠛⣻⣿⣿⣶⣾⣿⣿⣿⠃⢠⣦⣦⣶⣦⡾⢿⠿⠿⠛⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠹⣿⠛⠈⠿
⠀⠀⠀⠀⠀⢸⣇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠱⣾⣿⣿⣿⣿⡇⠀⠀⠀⣀⣠⡾⠛⢺⣿⣿⣿⣿⣿⣿⢀⠺⡧⢽⡟⠋⠀⠀⠀⠀⠀⠀⢻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿⣇⠈⠃⠤⠔
⠀⠀⠀⠀⠀⠨⣿⠀⠀⠸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⣿⠟⣿⣿⠁⠠⠤⠼⠿⠟⢰⣿⣿⣿⣿⣿⣿⣿⣿⠄⣿⣦⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢿⣿⣿⣿⣿⣿⣿⢿⠋⠁⠀⠀⠀⠀⠈⠙⠉⢃⣀⠀⣀
⠀⠀⠀⠀⠀⢝⣿⡀⠀⠀⣿⠀⢸⣿⠏⠟⠟⠟⣛⡙⢿⣷⣄⢸⡇⠴⠿⠞⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⢙⠃⠛⠳⢦⣄⡀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⠋⢠⠀⠀⠀⠀⠀⠀⠀⠐⣶⣼⣈⡉⠉
⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠟⠒⢛⡟⠀⠀⠆⠀⠉⠹⢸⡿⡏⣨⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⡹⠀⠀⠀⠈⠉⡀⠀⠀⠀⡇⢸⣿⣿⣿⡟⠁⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢸⣿⡀⠀⠈⠀⠟⠀⠀⠈⠀⢀⠀⠀⠘⣿⠇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠏⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠿⢿⣿⠁⠀⠀⠘⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⠙⣿⡇
⠀⠀⠀⠀⠀⢘⢹⣿⡇⠀⠀⠀⠀⠀⠀⠛⢛⡛⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠻⠿⠛⠋⢀⣀⣀⣀⣁⣀⣤⣤⣤⣤⣤⢠⣴⣶⡴⣶⡶⣶⡆⠛⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠐⠘⠿⣿⠆⠀⠀⠀⠀⠂⠀⢈⣁⡀⢀⣀⣈⡉⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⢀⣼⠏
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢻⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⠀⣿⣿⣄
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣶⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⠘⡛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⠉⡁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠠⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⢤⣀⣠⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⡀⣤⡠⣤⢤⣤⣤⣤⣤⡤⡤⢀⣀⡀⠀⠀⡀⠤⠀⠀⠀⠀
⣠⣀⣀⣠⣤⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣠⡶⣼⣿⣿⣿⣷⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⣵⣿⣏⣈⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠼⣤⣴⡄⡻⣿⡗⠗⢞⠏⣶⢶⣶⣿⠷⢶⣆⠀⠀
⢿⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠑⠦⣯⠟⣿⣿⣿⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣽⣿⣿⠟⠻⠿⠛⠻⣼⣷⣾⣧⠀⣿⣁⣿⣽⣿⣿⡇⢄⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣷⡀⢻⣿⣿⣿⣧⡦⡟⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠷⠶⠶⢶⣤⣬⡉⣿⡉⠀⠩⡏⣩⠉⢩⣍⡅⠲⡂
⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⡿⣿⢿⣿⣷⡟⢩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠻⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⣤⣄⣀⠀⠀⠙⢻⣫⢆⣴⣦⠀⢟⣛⣿⣾⣿⡆⠂
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠊⠉⢣⣿⣾⣽⡀⠈⡙⣿⣿⣿⣿⣿⣿⣿⡏⠀⠐⠁⠁⠓⠀⠁⠈⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣴⣟⣿⣿⣿⠷⣤⡀⠘⣧⣼⣿⣿⣶⣾⣿⣿⣿⣿⣧⣆
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⣾⡶⢩⣿⣿⣷⣿⣿⣿⣿⣿⣿⢿⣿⣿⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⢿⣹⡿⠿⢻⣿⣿⣿⣿⣿⣯⣭⣿⣿⣟⣛⣿⣧⢠⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠈⠛
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣷⣟⣺⣾⣦⡀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠉⠻⡿⣿⣿⣿⣿⣿⣿⣗⣀⣹⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡄⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡠⠀⠀⠀
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡧⠀⠀⠀⠹⣿⣿⣿⣟⠋⢈⣙⣿⣻⣿⣿⣿⣿⣿⣧⡔⠃⢤⣿⣯⣿⣿⣿⣿⣾⣿⣿⣿⣿⣁⡴⠁⠋⡉⢼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⡦
⠀⠀⠀⠀⠀⢻⡏⠏⠀⣿⣧⣄⣠⣤⣤⣿⣿⠟⢳⣢⣌⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠙⣻⣿⣿⣶⣿⣿⣿⣿⠃⢠⣦⣧⣷⣦⡾⢿⠿⠿⠛⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠹⣿⠛⠈⠾
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠳⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡿⠛⢺⣿⣿⣿⣿⣿⣿⢀⠺⡷⢽⡟⠋⠀⠀⠀⠀⠀⠀⢻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠿⢿⣿⣿⣇⠈⠁⠤⠔
⠀⠀⠀⠀⠀⠘⣿⠂⠀⠸⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⣿⠟⣿⣿⠁⠠⠤⠼⠿⠟⢰⣿⣿⣿⣿⣿⣿⣿⣿⠄⢷⣦⣞⡀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢿⣿⣿⣿⣿⣿⣿⢿⠋⠁⠀⠀⠀⠀⠈⠙⠉⢀⣀⠀⣀
⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⡻⠀⢸⣿⠏⢟⠟⠟⣛⡙⢿⣷⣄⢸⡇⠴⠿⠞⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⢙⠃⠙⠳⢦⣄⡀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⠋⢠⠀⠀⠀⠀⠀⠀⠀⠐⣶⣼⣈⡉⠉
⠀⠀⠀⠀⠀⢨⢿⣧⠀⠀⠟⠒⢛⡟⠀⠀⠆⠀⠋⠹⢸⡿⡏⢨⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⡹⠀⠀⠀⠈⠉⡀⠀⠀⠀⡇⢸⣿⣿⣿⡟⠁⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠰⢹⣿⠀⠀⠘⠀⠟⠀⠀⠈⠀⢀⠀⠀⠘⣿⠆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠏⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠿⢿⣿⠁⠀⠀⠘⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⠙⣿⡏
⠀⠀⠀⠀⠀⠘⢺⣿⡇⠀⠀⠀⠀⠀⠀⠛⢛⡛⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠿⠿⠛⠋⢀⣀⣀⣀⣁⣀⣤⣤⣤⣤⣤⢠⣴⣶⡴⣶⡶⣶⡆⠛⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣏
⠀⠀⠀⠀⠀⠐⠈⠿⣿⠆⠀⠀⠀⠀⠂⠀⢈⣁⡀⢀⣀⣈⡉⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⢀⡾⠟
⠠⣤⡄⣤⡄⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⠀⣿⣿⣄
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⠘⡛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⠉⡁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠂
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠠⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⢤⣀⣠⣤⣤⣤⣤⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⡀⣤⣠⢤⢤⣤⣤⣤⣤⣤⡤⢀⣀⡀⠀⠀⡀⠤⠀⠀⠀⠀
⣠⣀⣀⣠⣤⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣀⡶⣼⣿⣿⣿⣯⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⣷⣿⣉⣋⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠼⣤⣤⡆⡻⣿⡗⠓⣋⠋⣶⣶⣶⣿⢶⢶⣄⠀⠀
⢿⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠑⠂⣯⠟⣿⢿⣿⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣽⣿⣷⡟⠻⠿⠋⠻⣿⣷⣿⣷⠀⣿⣁⣿⣿⣿⣿⡇⢄⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣷⡀⢻⣿⣿⣿⣧⡦⡝⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠷⠶⠶⢶⣤⣬⡀⣿⡉⠀⠉⡏⣩⠉⢩⣍⡅⠻⠀
⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⣿⣿⣿⣿⣷⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⣤⣄⣀⠀⠁⠙⢻⣡⢆⣴⣦⠈⢟⡛⣿⣿⣷⡆⠂
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠉⢣⣿⣾⣽⡀⠀⡉⣿⣿⣿⣿⣿⣿⣿⡏⠀⠐⠁⠀⠃⠀⠁⠈⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣴⣛⣿⣿⣿⠷⣦⡀⠘⣧⣼⣿⣿⣶⣿⣿⣿⣾⣿⣧⣆
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠠⣿⡶⢩⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠈⢿⣹⡿⠟⢻⣿⣿⣿⣿⣿⣯⣭⣿⣿⣟⣛⣿⣧⢠⣿⣿⣿⣿⣿⣿⣽⣟⠗⠉⠁⠙
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣧⣟⣺⣾⣦⡀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠉⠻⡿⣿⣿⣿⣿⣿⣿⣧⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡄⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⠄⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠹⣿⣿⣿⣟⠛⢈⣙⣿⣻⣿⣿⣿⣿⣿⣧⡔⠃⢤⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣇⠶⠁⠋⡁⠼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣧
⠀⠀⠀⠀⠀⢺⡟⠃⠁⣿⣧⣄⣀⣤⣤⣾⣿⡟⢻⣢⣈⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠛⣻⣿⣿⣶⣾⣿⣿⣿⠇⢀⣦⣧⣶⣦⡼⢿⠿⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡉⢿⡟⠀⠱
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠡⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡿⠛⢒⣿⣿⣿⣿⣿⣿⠂⢸⡧⢽⡟⠋⠁⠀⠀⠀⠀⠀⠻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⣿⣿⣇⠈⠉⠤⠄
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⣿⠟⣿⣿⠁⠠⠤⠼⠿⠟⢰⣾⣿⣿⣿⣿⣿⣿⣿⡆⢿⣦⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠉⠂⠁⣀⠀⢀
⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⣿⠀⢸⣿⡏⠟⠿⠟⣛⣛⢿⣷⣄⢸⡇⠰⠿⠞⠋⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠁⠈⢹⠋⠋⠳⢦⣄⡀⠀⠀⠀⠀⡇⢘⣿⣿⣿⣿⣿⠋⢸⠀⠀⠀⠀⠀⠀⠀⠀⣶⣼⣀⡉⠉
⠀⠀⠀⠀⠀⢹⣻⣇⠀⠀⠟⠒⢛⡟⠀⠀⠅⠀⠋⠙⢸⡿⡏⢬⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢱⠀⠀⠀⠉⠉⠆⠀⠀⠀⡇⢸⣿⣿⣿⡿⠃⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⢻⣿⠀⠀⠈⠀⠞⠀⠀⠈⠀⢀⠀⠀⠈⣿⠆⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢿⣿⣿⣿⣿⡟⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠿⢿⣿⠁⠀⠀⠘⣀⢀⣀⢀⣀⡀⢀⣤⣤⡉⠙⣿⣇
⠀⠀⠀⠀⠀⢐⢘⣿⣇⠀⠀⠀⠀⠀⠀⠛⢛⡛⠃⠀⠀⢻⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠻⠿⠻⠋⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡆⠻⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠠⠘⠿⣿⠄⠀⠀⠀⠀⠂⠀⢈⣁⡀⢀⣀⣈⡉⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⢀⣾⠏
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⠀⢺⣿⣆
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⠘⡛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⣽⣿⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⠉⢁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣀⣠⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⡀⣤⣠⡤⢤⣤⣤⣤⣤⡤⡄⠀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣠⣀⣀⣠⣤⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣀⡦⣼⣿⣿⣿⣯⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⣵⣿⣋⣈⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠼⣤⣤⡆⡻⣿⡟⠒⣎⠋⣶⣶⣶⣿⠶⢶⣄⠀⠀
⢿⣿⣿⣿⣽⣿⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠑⠂⣮⠟⣿⣿⣿⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣇⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣽⣿⣷⡿⠻⠟⡏⠻⣿⣷⣿⣶⠀⣿⣀⣿⣽⣿⣿⡇⢤⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣷⡀⢹⣿⣿⣿⣧⡮⡙⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠷⠶⠶⢶⣤⣬⡁⣿⡉⠀⠉⡉⣉⠉⢩⣍⡅⢹⡁
⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⡿⢿⣿⣿⣷⡟⢹⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣤⣄⣀⠀⠁⠙⢻⣡⢄⣴⣦⠀⠟⣛⣿⣾⣷⡆⠀
⠀⠁⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠉⢣⣿⣮⣹⡀⠈⡙⣿⣿⣿⣿⣿⣿⣿⡏⠀⠐⠁⠀⠃⠀⠁⠈⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣴⣛⣿⣿⣿⠷⣦⡀⠘⣧⣼⣿⣿⣶⣞⣿⣿⣾⣿⣷⣆
⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠠⣿⡾⢩⣾⣿⣷⣿⣿⣿⣿⣿⣿⢿⣿⣿⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠈⢿⣹⡿⠿⢻⣿⣿⣿⣿⣿⣯⣭⣿⣿⣟⣛⣿⣧⢠⣯⣿⣿⣿⣿⣿⣿⣟⠟⠉⠉⠙
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣧⣟⣺⣾⣦⡀⠀⠀⠀⠀⠀⠈⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠉⠻⡿⣿⣿⣿⣿⣿⣿⣇⣀⣹⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡆⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣵⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢠⠄⠀⠀
⠀⠀⠀⠀⠈⣿⣿⡿⣿⣿⡇⠀⠀⠀⠹⣿⣿⣿⣟⠙⢈⣛⣿⣿⣿⣿⣿⣿⣿⣧⡔⠃⢤⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠴⠁⠋⡉⢼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶
⠀⠀⠀⠀⠀⢻⡟⠁⠀⣿⣧⣄⣀⣤⣤⣾⣿⡟⢟⣢⣈⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠛⣻⣿⣿⣶⣾⣿⣿⣿⡏⢀⣦⣦⣶⣦⡼⢿⠿⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⢿⡟⠀⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠢⣾⣿⣿⣿⣿⡇⠀⠀⠀⣀⣠⣾⠓⢐⣿⣿⣿⣿⣿⣿⠃⠸⡧⣽⡟⠋⠁⠀⠀⠀⠀⠀⠻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠿⣿⣿⣿⣧⠈⠙⠄⠄
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⣿⠟⣿⣿⡅⠠⠤⠼⠿⠟⢡⣾⣿⣿⣿⣿⣿⣿⣿⡇⢿⣦⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠈⠉⠓⠑⣀⡀⠀
⠀⠀⠀⠀⠀⣹⣿⡀⠀⠀⡿⠀⢸⣿⡏⠟⠿⠟⣛⣙⢿⣷⣄⢸⡇⡴⠿⠞⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁⠈⢻⠛⠛⠳⢦⣄⡀⠀⠀⠀⠀⡇⢘⣿⣿⣿⣿⣿⠋⢸⠀⠀⠀⠀⠀⠀⠀⠈⣶⣴⣀⡉⠉
⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠟⠒⢛⡟⠀⠀⠅⠀⠋⠙⢸⣿⡏⣸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢹⠀⠀⠀⠉⠉⠇⠀⠀⠀⡇⠸⣿⣿⣿⡿⠃⠀⢸⠀⣀⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠐⢹⣿⠀⠀⠈⠀⠞⠀⠀⠈⠀⢀⠀⠀⠈⣿⠦⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢿⣿⣿⣿⣿⡟⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠿⢿⣿⠃⠀⠀⠘⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⠹⣿⣿
⠀⠀⠀⠀⠀⠨⢸⣿⡇⠀⠀⠀⠀⠀⠀⠛⢛⡛⠃⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠈⠻⠿⠻⠋⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡆⠻⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠈⠈⠿⣿⠄⠀⠀⠀⠀⠂⠀⢈⣁⡀⢀⣀⣈⡉⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⢠⣿⠟
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⠀⢺⣿⣆
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢹⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⠸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⠘⡛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡖⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⠉⢁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠏
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠠⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣀⣤⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⡀⣤⡤⢤⢤⣤⣤⣤⣤⡤⡄⠀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣠⣀⣀⢠⣤⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣠⡦⣼⣿⣿⣿⣧⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⣷⣿⣏⣈⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠼⣠⣤⡆⡻⣿⡗⠓⣏⠏⣶⣶⣶⣿⠶⢶⣄⠀⠀
⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⠑⠂⣿⠟⣿⢿⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣽⣯⣷⡟⠻⠿⡋⠻⣿⣷⣾⣷⠀⣿⣁⣿⣿⣿⣿⡇⣤⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣷⡀⣹⣿⣿⣿⣧⡦⡝⡿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠷⠶⠶⢶⣤⣬⡁⣿⡉⠀⠉⡏⣉⠉⢩⡍⠅⠳⠅
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿⣿⣿⣿⣷⣟⢩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠻⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣠⣄⣀⡀⠁⠙⢻⣡⠄⣴⣦⠈⢟⢓⣿⣾⣷⡆⠀
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠉⢣⣿⣮⣹⡀⠈⡙⣿⣿⣿⣿⣿⣿⣿⡏⠀⠐⠁⠁⠁⠀⠀⠈⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰⣛⣿⣿⣿⠷⣦⡀⠘⣧⣼⣿⣿⣶⣾⣿⣿⣿⣿⣷⣆
⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⣿⡾⢹⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⢿⣹⡿⠿⢻⣿⣿⣿⣿⣷⣿⣍⣿⣿⣟⣛⣿⣧⢠⣯⣿⣿⣿⣿⣿⣿⣏⡟⠋⠉⠉
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠲⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣧⣟⣺⣾⣦⡀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠉⠻⡿⣿⣿⣿⣿⣿⣿⣇⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣷⡆⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⣮⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⠄⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠹⢿⣿⣿⣟⠛⢈⣙⣿⣻⣿⣿⣿⣿⣿⣧⡔⠃⢤⣿⣯⣿⣿⣿⢿⣾⣿⣿⣿⣿⣇⠴⠃⠋⡉⢼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶
⠀⠀⠀⠀⠀⢿⡟⠃⠀⣿⣧⣄⣠⣤⣤⣾⣿⡟⢳⣢⣈⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠛⣻⣿⣿⣶⣶⣿⣿⣿⡏⢀⣦⣦⣶⣤⡼⢿⠿⡟⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⢿⡟⠃⠀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠡⣾⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⡾⠓⢒⣿⣿⣿⣿⣿⣿⠃⠸⡧⢽⡟⠛⠀⠀⠀⠀⠀⠀⠹⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣷⠀⠉⠤⠄
⠀⠀⠀⠀⠀⢌⣿⠀⠀⢸⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⣿⠟⣻⢿⠁⠠⠤⠼⠿⠟⢱⣾⣿⣿⣿⣿⣿⣿⣿⡆⢷⣦⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⡌⢿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠈⠉⠛⠑⣀⡀⠀
⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⡿⠀⢸⣿⡏⠛⠟⠛⣛⣙⢿⣷⣄⢸⡇⠰⠿⠾⠛⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠁⠈⢻⠛⠛⠳⢦⣄⡀⠀⠀⠀⠀⡇⠈⣿⣿⣿⣿⣿⠋⢸⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⣀⡁⠉
⠀⠀⠀⠀⠀⢘⣻⣇⠀⠀⠟⠒⢋⡟⠀⠀⠅⠀⠋⠉⢸⣿⡏⣼⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢹⠀⠀⠀⠈⠉⠇⠀⠀⠀⡇⠀⣿⣿⣿⡿⠃⠀⢸⠀⣀⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠸⢹⣿⠀⠀⠈⠀⠚⠀⠀⠈⠀⢀⠀⠀⠈⣿⡦⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⠃⠀⠀⠘⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠛⢛⠛⠃⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠈⠻⠿⠻⠋⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡆⠻⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠘⠈⠿⣿⠄⠀⠀⠀⠀⠂⠀⢈⣁⡀⢀⣀⣈⡉⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⢠⣿⠟
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢻⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⠀⢸⣿⣆
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠏⢹⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⠘⡛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⠉⢁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠠⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣀⣠⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⡀⢤⣤⣤⠤⣤⣤⣤⣤⣤⡤⢀⣀⡀⠀⠀⠀⠤⠀⠀⠀⠀
⣨⣀⣀⣠⣤⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣠⡦⣼⣿⣿⣿⣧⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⣶⣿⣫⣉⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠴⣤⣤⡄⡿⣿⡗⠷⢎⠏⣶⣶⣶⣿⠶⣶⣆⠀⠀
⢻⣿⣿⣿⣽⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⠁⠢⣯⠏⣿⣿⣿⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣽⣿⣿⡟⠻⢿⡏⠫⣽⣷⡾⣦⠀⣿⣀⣿⣽⣿⣿⡇⢄⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷⡀⢹⣿⣿⣿⣣⡦⡝⡿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠷⠶⠶⢶⣤⣬⡁⣟⡈⠀⠩⡍⣉⠉⢩⣍⠅⠺⡄
⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⣿⢿⣿⣷⡟⢹⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠻⡉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣠⣄⣀⡀⠁⠉⢻⣣⢆⣤⣦⠀⠟⣛⣿⣿⣿⡄⠂
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠊⠙⢣⣿⣮⣽⠁⠈⡙⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠁⠀⠈⠀⠈⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣰⣛⣿⣿⣿⠿⣦⡀⠘⣧⣼⣿⣿⣶⣾⣿⣿⣿⣿⣷⣆
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠠⣾⡿⢙⣿⣿⣷⣿⣿⣿⣿⣿⣿⢿⣿⣿⠁⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⢿⣹⡿⠿⢻⣿⣿⣿⣿⣷⣿⣍⣿⣿⣟⣛⣿⣧⢀⣿⣿⣿⣿⣿⣿⣿⣯⡼⠋⠉⠉
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠒⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣧⣟⣚⣿⣦⡄⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠉⠻⡿⣿⣿⣿⣿⣿⣿⣗⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡆⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⡄⠀⠀
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠈⠹⢿⣿⣿⣟⠋⢈⣙⣿⣻⣿⣿⣿⣿⣿⣧⡴⠃⢤⣿⣯⣽⣿⣿⠿⣷⣿⣿⣿⣿⣇⡴⠃⠋⡉⢼⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾
⠀⠀⠀⠀⠀⣿⡟⠋⠀⣿⣧⣄⣠⣤⣴⣾⣿⠟⢷⣂⣈⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠛⣻⡿⣿⣶⣶⣿⣿⣿⡏⢀⣤⣧⣶⣤⣼⢿⡿⡟⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢽⡿⠃⠡
⠀⠀⠀⠀⠀⢻⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠢⣾⣿⣿⣿⣿⡇⠀⠀⠀⣀⣀⡾⠛⢙⣿⣿⣿⣿⣿⣿⠀⠸⣷⢿⡟⠉⠁⠀⠀⠀⠀⠀⠹⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⣿⣿⣿⣷⡀⠙⠠⠄
⠀⠀⠀⠀⠀⢺⣿⠀⠀⠸⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⣿⠟⣿⢿⠁⠠⠤⠼⠿⠟⢡⣾⣿⣿⣿⣿⣿⣿⣿⡇⢳⣦⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⡌⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠈⠉⠛⠓⣀⡀⠀
⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⡿⠀⢸⣿⡏⢛⠿⠛⣛⣉⢿⣷⡄⢸⡇⠴⠿⠞⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢻⠛⠙⠳⢦⣤⣀⠀⠀⠀⠀⡇⠈⣿⣿⣿⣿⣿⠏⢸⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣇⣉⠉
⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠟⠒⢛⡟⠀⠀⠅⠀⠉⠉⢸⣿⡏⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢹⠀⠀⠀⠈⠉⠇⠀⠀⠀⡇⠀⣿⣿⣿⡿⠃⠀⢸⠀⣀⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠸⢻⣿⠀⠀⠈⠀⠚⠀⠀⠀⠀⢀⠀⠀⠀⣿⡦⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢿⣿⣿⣿⣿⡟⠁⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠿⢿⣿⡇⠀⠀⠘⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⢻⣿⣿
⠀⠀⠀⠀⠀⢠⢸⣿⡆⠀⠀⠀⠀⠀⠀⠻⢛⠛⠃⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠈⠻⠿⠻⠋⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡆⠻⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠸⠘⠿⣿⠄⠀⠀⠀⠀⠂⠀⢈⣁⡀⢀⣀⣈⡉⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⢀⣿⡿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⠀⢸⣿⣇
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠏⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⠘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⠉⢁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠠⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⢤⣀⣠⣤⣤⣤⣤⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⡀⢤⣤⡤⠤⣤⣤⣤⣤⡤⡤⠀⣀⡀⠀⠀⡀⠤⠀⠀⠀⠀
⣨⣀⣀⢠⣤⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣀⡦⣼⣿⣿⣿⣷⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⡴⣿⣋⣉⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠴⣠⣤⡄⠛⢿⡇⠗⢊⠋⣶⢶⣶⣿⢶⢶⣄⠀⠀
⢿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⠑⠂⣬⠯⣿⣿⣿⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣽⣿⣿⡟⠻⠿⡋⠳⣽⣷⡾⣦⠀⣿⣈⣿⣿⣿⣿⡇⢄⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣽⣷⡀⢹⣿⣿⣿⣣⡦⡙⡿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⠷⠶⠶⢶⣤⣬⡁⣿⡉⠀⠉⡍⢉⠉⢩⣍⠅⠻⠁
⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢿⣿⣷⣟⢹⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡛⠻⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣠⣄⣀⡀⠁⠉⠻⣣⢆⣴⣦⠀⠟⣛⣿⣾⣷⡆⠀
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠉⠉⢣⣿⣮⣹⡁⠈⡙⣿⣿⣿⣿⣿⣿⣿⡟⠁⠐⠁⠀⠈⠀⠈⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣰⣟⣿⣿⣿⠿⣦⡀⠘⣧⣼⣿⣿⣶⣞⣿⣿⣾⣿⡷⣆
⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠠⣾⡾⢉⣾⣿⣷⣿⣿⣿⣿⣿⣿⢿⣿⣿⡃⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⢿⣹⡿⠿⢻⣿⣿⣿⣿⣷⣿⣍⣿⣿⣟⣛⣿⣷⢀⣿⣿⣿⣿⣿⣿⣽⣟⣩⠛⠉⠉
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠲⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣧⣿⣛⣿⣷⣄⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠉⠻⡿⣿⣿⣿⣿⣿⣿⣧⣀⣹⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡆⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡄⡤⠀⠀
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠈⠙⢿⣿⣿⣟⠋⢈⣙⣿⣻⣿⣿⣿⣿⣿⣧⡜⠁⣤⣿⣯⣹⣿⣿⠿⣷⣿⣿⣿⣿⣇⡴⠃⠋⡉⢼⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶
⠀⠀⠀⠀⠀⣿⡟⠎⠀⣿⣧⣄⣠⣤⣴⣾⣿⠟⢳⣢⣈⢁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠛⣻⣿⣿⣶⣶⣿⣿⣿⡏⢀⣤⣧⣶⣦⣼⢿⡿⡟⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⢹⣿⠃⠠
⠀⠀⠀⠀⠀⣸⡅⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠡⣾⣿⣿⣿⣿⡇⠀⠀⠀⣀⣀⣾⠛⢐⣿⣿⣿⣿⣿⣿⠀⠸⡧⢽⡿⠛⠁⠀⠀⠀⠀⠀⠹⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⡄⠉⠠⠄
⠀⠀⠀⠀⠀⣻⣷⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢿⣿⠟⣿⣿⠁⠠⠤⠼⠿⠟⢡⣾⣿⣿⣿⣿⣿⣿⣿⡆⢷⣦⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⡈⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠈⠉⠛⠛⡀⡀⠀
⠀⠀⠀⠀⠀⠽⣿⠀⠀⠀⡯⠀⢸⣿⡏⠛⠟⠛⣋⣉⢿⣷⣄⢸⡇⠴⠿⠞⠛⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁⠈⢻⠛⠉⠳⢦⣤⡀⠀⠀⠀⠀⡇⠈⣿⣿⣿⣿⣿⠏⢸⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⣇⣈⠈
⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠟⠒⢛⡟⠀⠀⠅⠀⠁⠉⢸⣿⡏⢼⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢹⠀⠀⠀⠀⠉⠇⠀⠀⠀⠁⠀⣿⣿⣿⡿⠋⠀⢸⠀⢀⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠸⢸⣿⠀⠀⠈⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠁⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡇⠀⠀⠘⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⢻⣿⣿
⠀⠀⠀⠀⠀⠨⢸⣿⡆⠀⠀⠀⠀⠀⠀⠻⢛⠛⠃⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠈⠻⠿⠻⠛⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡆⠻⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠈⠈⠿⣷⠄⠀⠀⠀⠀⠂⠀⢈⣁⡀⢀⣀⣈⡉⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⢀⣿⠿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢻⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⠀⢸⣿⣇
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⠘⠛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⠉⢁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠗
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠠⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⢤⣀⣠⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⡀⣤⣠⡤⠤⣤⣤⣤⣤⡠⡤⠀⣀⡀⠀⠀⠀⠤⠀⠀⠀⠀
⣤⣀⣀⢠⣤⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣀⡦⣼⣿⣿⣿⣧⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⣴⣿⣋⣉⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠴⣤⣤⡄⡟⢿⡗⠗⢎⠏⣶⢶⣶⣿⠶⢶⣆⠀⠀
⢿⣿⣿⣿⣽⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⠡⠢⡯⠟⣿⣿⣿⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣽⣯⣿⠟⠻⠿⡟⠻⣽⣷⣾⣦⠀⣿⣀⣿⣽⣿⣿⡇⣤⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣽⣷⡀⢹⣿⣿⣿⣧⡦⡟⡿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⠷⠶⠶⢶⣤⣬⡀⣿⡈⠀⠉⡍⣉⠉⢩⣍⡅⠺⠂
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣶⣿⣿⢿⣿⣷⡟⢩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠻⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢁⣠⣄⣀⡀⠁⠉⠻⣣⢂⣴⣦⠀⢟⢓⣿⣿⣷⡆⠀
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠋⠙⢣⣿⣮⣹⡀⠀⡙⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠈⠀⠈⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣰⣟⣿⣿⣿⠿⣦⡀⠘⣧⣼⣿⣿⣶⣞⣻⣿⣬⣿⣷⢆
⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠠⣶⡾⢉⣿⣿⣷⣿⣿⣿⣿⣿⣿⢿⣿⣿⡇⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⢿⣩⡿⠿⢻⣿⣿⣿⣿⣷⣿⣍⣿⣿⣟⣛⣿⣷⢀⣯⣿⣿⣿⣿⣾⣿⣿⣻⠟⠉⠉
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠲⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣧⣿⣚⣿⣷⣄⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠉⠻⡿⣿⣿⣿⣿⣿⣿⣧⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡆⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡄⡠⠀⠀
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠙⢿⣿⣿⣟⠉⢈⣙⣿⣻⣿⣿⣿⣿⣿⣧⡼⠃⢤⣿⣯⣹⣿⣿⠿⣷⣿⣿⣿⣿⣇⡴⠃⠋⡉⠼⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴
⠀⠀⠀⠀⠀⢾⡏⠇⠁⣿⣧⣄⣠⣤⣴⣾⣿⠟⢳⣢⣈⢁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠛⣻⡿⣿⣶⣶⣿⣿⣿⡏⢀⣤⣦⣶⣦⣼⢿⡿⡟⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢹⣿⠓⠀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⣾⣿⣿⣿⣿⡇⠀⠀⠀⣀⣀⡾⠛⢐⣿⣿⣿⣿⣿⣿⠀⠸⣧⣽⡿⠋⠀⠀⠀⠀⠀⠀⠹⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⡄⠉⠁⠄
⠀⠀⠀⠀⠀⣺⣷⠀⠀⢸⣿⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⣿⠟⣿⢿⠁⠠⠤⠼⠿⠿⢡⣾⣿⣿⣿⣿⣿⣿⣿⡇⣷⣦⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⡈⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠈⠉⠛⠚⢀⡀⠀
⠀⠀⠀⠀⠀⢔⣿⠀⠀⠀⡯⠀⢸⣿⡏⢛⠟⠛⣉⣉⢿⣷⣄⢸⡇⠶⠿⠾⠛⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁⠈⢻⠛⠛⠳⢦⣤⡀⠀⠀⠀⠀⠇⠈⣿⣿⣿⣿⣿⠏⢸⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⣆⣈⠈
⠀⠀⠀⠀⠀⢺⢿⡇⠀⠀⠟⠒⢛⡟⠀⠀⠅⠀⠀⠉⠸⣿⠏⣨⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢹⠂⠀⠀⠈⠉⠇⠀⠀⠀⠀⠀⣿⣿⣿⡿⠋⠀⢸⠀⣀⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢹⢸⣿⠀⠀⠈⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠈⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⢿⣿⣿⣿⣿⡿⠁⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡇⠀⠀⠘⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⢻⣿⣿
⠀⠀⠀⠀⠀⢰⢸⣿⡆⠀⠀⠀⠀⠀⠀⠻⠛⠛⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠈⠻⠿⠻⠛⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡆⠻⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠐⠘⠿⣷⠄⠀⠀⠀⠀⠀⠀⢈⣁⡀⢀⣀⣈⡉⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⢀⣿⠟
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢹⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⠀⢸⣿⣇
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠏⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⢸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⠘⢛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⠉⢁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠗
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠠⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣠⣤⣤⣤⣤⣤⡤⣄⣤⣤⣤⣤⣤⣄⣤⡀⣤⠠⣤⢤⣤⣤⣤⣤⡤⡤⠀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀
⣤⣀⣀⣠⣤⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣠⣦⣼⣿⣿⣿⣷⣰⣿⣾⣿⣿⣿⣿⣿⣿⣻⣿⣶⣿⣻⣊⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠼⣠⣤⡌⠻⣿⡖⠗⢛⠏⣶⣶⣶⡷⠶⢶⣆⠀⠀
⣿⣿⣿⣿⣽⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⠑⠢⣮⡟⣿⣿⣿⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣽⣿⣿⡿⠻⢿⠛⠟⣾⣷⡿⣦⠀⣿⣀⣿⣽⣿⣿⡇⢄⠀
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣶⡀⢻⣿⣿⣿⣧⡦⡝⡿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠷⠴⠶⢷⣤⣬⡁⣿⡉⠀⠉⡏⣩⠉⢩⡍⠅⠺⠇
⢹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣾⡿⣿⢿⣿⣿⡟⢹⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣠⣄⣀⡀⠁⠉⠻⣣⢆⣴⣦⠀⣟⢓⣿⣿⣷⡆⠀
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠙⢣⣿⣮⣿⡀⠀⡙⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠁⠀⠈⠀⠈⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣰⣟⣿⣿⣿⢿⣦⡀⠐⣧⣼⣿⣿⣶⣾⣽⣿⣽⡿⣳⠆
⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠠⣶⡿⢩⣷⣿⣷⣿⣿⣿⣿⣿⣿⢿⣿⣿⡇⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⢿⣩⡿⠿⢿⣿⣿⣿⣿⣷⣿⣟⣿⣿⣟⣻⣿⣷⠀⣯⣿⣿⣿⣿⣿⣿⣟⣿⠟⠋⠉
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠲⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣧⣿⣚⣷⣷⣄⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣏⣟⣿⣿⣿⣿⣞⣿⣿⣿⣿⣿⣿⣿⡟⠇⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠉⠻⡻⣿⣿⣿⣿⣿⣿⣗⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⡆⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡆⣠⠀⠀
⠀⠀⠀⠀⠈⣿⣿⡿⣿⣿⡇⠀⠀⠀⠙⢿⣿⣿⣟⠋⢈⣛⣟⣿⣿⣿⣿⣿⣿⣧⡴⠃⢤⣿⣿⡻⣿⣿⠿⣷⣿⣿⣿⣿⣇⡴⠋⠋⡉⢼⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⢾⡏⠈⠀⣿⣧⣄⣠⣴⣴⣾⣿⠟⢻⣢⣈⣁⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠛⣻⣿⣿⣶⣶⣿⣿⣿⡏⢀⣤⣧⣶⣤⣼⢿⡿⡟⠟⢻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢩⣿⡟⠀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⣾⣿⣿⣿⣿⡇⠀⠀⠀⣀⣀⣾⠟⢘⣿⣿⣿⣿⣿⣿⠀⢼⣧⢽⡿⠋⠀⠀⠀⠀⠀⠀⠹⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⡄⠈⠣⠀
⠀⠀⠀⠀⠀⢾⣿⠀⠀⢸⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⣿⠟⣿⣿⠁⠠⠤⠼⠿⠟⢡⣾⣿⣿⣿⣿⣿⣿⣿⡇⢳⣦⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠈⢉⠛⠉⢂⡀⠀
⠀⠀⠀⠀⠀⢪⣿⠀⠀⠀⡷⠀⢸⣿⡏⠛⠟⠛⣉⣉⢿⣷⣄⢸⡇⣴⠿⠾⠛⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁⠈⢫⠛⠙⠓⢦⣄⡀⠀⠀⠀⠀⠆⠈⣿⣿⣿⣿⣿⠏⢸⠀⠀⠀⠀⠀⠀⠀⠈⣷⣦⣆⣈⠈
⠀⠀⠀⠀⠀⠹⣿⡇⠀⠀⠟⠒⢛⡟⠀⠀⠅⠀⠀⠉⠸⣿⡏⣨⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢹⠂⠀⠀⠀⠉⠇⠀⠀⠀⠀⠀⣿⣿⣿⡿⠋⠀⢸⠀⣀⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠁⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡇⠀⠀⠘⣀⢀⣀⢀⣀⡀⢀⣤⣤⡈⢻⣿⣿
⠀⠀⠀⠀⠀⢠⠸⣿⡆⠀⠀⠀⠀⠀⠀⠻⠛⠛⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠈⠻⠿⠻⠟⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢠⣴⣶⡶⣶⡶⣶⡆⠻⠀⠀⣿⣿⠛⢸⣿⣼⣿⡇⠿⠿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠸⠀⠿⣿⠄⠀⠀⠀⠀⠀⠀⢈⣁⡀⢀⣀⣈⡉⣤⣄⢠⣤⣤⣤⣶⣶⠀⠀⣶⣶⣶⠀⠀⢸⣿⡄⢸⣿⡟⠛⢻⣿⡏⠉⢹⣿⠉⢸⣿⣤⡄⣿⣧⣿⡟⠀⠀⠀⣿⣿⠶⢸⣿⡿⣿⡇⢰⣶⢾⡿⢀⣿⡿
⠠⣤⡄⣤⡆⣠⣶⣶⣆⢰⣾⣾⣿⡇⠀⢸⣿⢻⣿⠸⣿⡟⠃⣿⣿⣼⣿⠈⢹⣿⡀⠀⠀⣿⡟⣿⡆⠀⢸⣿⡇⢸⣿⡿⠇⢸⣿⡇⠀⢸⣿⡇⢸⣿⡏⠁⣿⣿⢻⣿⢀⡀⠀⢻⣿⣤⡼⣿⡇⣿⣷⢸⣿⠀⠀⢸⣿⣧
⠀⢻⣷⣿⠇⣿⣿⢹⣿⢸⣿⡏⣿⡇⠀⠘⠻⣿⣶⡄⣿⣿⠟⢸⣿⣿⣿⡆⢸⣿⡇⠀⠀⣿⣷⣿⣷⠀⠸⣿⣧⡄⣿⣧⣴⠀⣿⡇⠀⠸⡿⠇⠘⠿⠿⠟⠛⠛⠘⡛⠸⠇⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⣿⣿⠀⢿⣿⢸⣿⡞⣿⣇⣿⡷⠀⠸⣿⣯⣿⡇⣿⣿⣶⢸⣿⠸⡿⠇⠸⠿⠇⠀⠀⠛⠃⠘⠛⠀⠀⠉⠉⢁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠂⠻⣿⠀⠘⠿⠿⠿⠶⠟⠛⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

                    ䷩ 𝚕𝚒𝚗𝚎 1863

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

(ℹ) Images, hyperlinks and comments at http://techrights.org/2021/12/25/timewasting-letters/#comments
    Gemini version at gemini://gemini.techrights.org/2021/12/25/timewasting-letters/

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 12.25.21⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Stupid_and_Legally-Invalid_Threats_(Physical_Letters)⠀✐


Posted in Deception, Law at 5:58 pm by Dr. Roy Schestowitz


Or: Threatening Letters From Losers, Liars, Perverts and Hypocrites Who
Sabotage_GNU/Linux and Defame_Its_Founders


Part I: Video_download_link | md5sum 3b807595c651995ea2622172293b2998


http://techrights.org/videos/timewasting-letters-part1.webm


Part II: Video_download_link | md5sum 872b7acc595a1b6e595c29d698232970


http://techrights.org/videos/timewasting-letters-part2.webm


Summary: The occupation of lawyers — even outside the realms of patents and
copyrights (so-called ‘IP’ lawyers) — is a net negative; this 2-part video
highlights older and newer examples of that


IN THIS world where “reality hurts” people rush to actions that are senseless
and sometimes legally invalid. We saw this in the case of the EPO and
Microsoft; they like to ‘lawyer up’ their critics. Sometimes they send letters
from lawyers and I’ve lost count of such utterly worthless letters.


“Sometimes they send letters from lawyers and I’ve lost count of such utterly
worthless letters.”This month it took some nerve for a law firm to send a
COVID-19 ‘infection vector’ through the door (in December when we have like
122,000 cases per day), relegating it to a 3-day quarantine before being
opened. This video concerns an American law firm sending_a_ridiculous_letter
for_Michael_Shore. We then deal with a more recent example, which is even more
ridiculous than the first. █


                      🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇Michael Shore⦈
           Screenshot of Michael Shore’s page, archived_by_TechDirt


⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⢰⠐⢰⡄⢸⣿⣽⣽⣿⣿⣕⣿⣿⣼⣿⣇⣿⣻⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠐⠑⠘⠁⢸⣿⣯⣿⣿⣍⣿⣝⣽⣯⣹⣹⣻⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⢏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⡀⠀⢀⢀⠀⠀⡀⠀⣠⠀⠀⡀⡀⣀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠈⠈⠁⠀⠁⠀⠁⠁⠀⠈⠁⠉⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣹⣿⣿⣿⣯⣽⣿⣿⣿⣏⣟⣻⣯⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⣉⡉⣉⡉⣉⡉⣉⡉⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⢿⣿⣿⡿⡿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣇⣀⣀⣀⣀⣀⣁⣉⣉⣁⣈⣁⣉⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢃⣴⣶⣿⡆⠀⠀⢹⣿⣿⣿⣿⣾⣶⣾⣷⣿⣶⣿⣿⣷⣿⣾⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⡂⠒⠒⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡘⣿⢛⡟⠉⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠿⢿⠿⠽⠸⠼⠻⢻⠿⠯⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠘⠏⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣄⢀⣀⡀⣀⠀⣄⣀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⠋⠁⡎⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠉⠀⠀⠀⠀⠹⢖⡤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢨⡴⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
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                    ䷩ 𝚕𝚒𝚗𝚎 1948

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

(ℹ) Images, hyperlinks and comments at http://techrights.org/2021/12/25/war-on-generalised-things/#comments
    Gemini version at gemini://gemini.techrights.org/2021/12/25/war-on-generalised-things/

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Gemini_version_available_♊︎


✐ The_Car_Drives_You_—_Part_V_—_Part_of_the_Coming_War_on_General_Computation⠀✐


Posted in Free/Libre_Software at 3:18 pm by Dr. Roy Schestowitz


                       🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇Speeding Car⦈
                           Speeding towards dystopia


Summary: General-purpose computing and cars which the owner/driver fully
controls are a dying breed that may become extinct unless collective action is
undertaken


THIS will quite likely be the last part of this series (see Part_I, Part_II,
Part_III, and Part_IV), so it is a bit of an outline. We might continue the
series if further findings bubble up to reach the surface.


“Eventually,” as an associate noted, “the supply of old cars will run out and
then even those that have succeeded, until then, in avoiding them will have
their hand forced. The only way out is to influence the supply chain. Another
aspect to cover, one which is as pernicious as the software aspects, is the
push to move all aspects of car use and car maintenance over to subscription
based models.”


“Another aspect to cover, one which is as pernicious as the software aspects,
is the push to move all aspects of car use and car maintenance over to
subscription based models.”
      –Anonymous“Even roadside service subscriptions have replaced the spare
tire + jack. I don’t mean leasing a car or similar to leasing, but the repair
contracts in place of being able to choose where and when to do maintenance.
Also various functionalties, such as the remote start addressed by Louis in his
video. However, whole_cars_are_being_pushed_as_a_subscription_“service”
nowadays. That’s_a_different_matter.”


“You drive us wild, we’ll drive you crazy. You say you wanna go for a spin…”
      –Anonymous“Cory Doctorow, among many others, has also_written_a_lot_about
the_software_in_cars_and_other_consumer_devices.”


See ““The_Coming_War_on_General_Computation”“ (from 28C3 in 2011)


Our associate said that “software showed up in public consciousness, though
briefly” in relation_to_proprietary_software_scandals.


When we started the series we called it “Car Drives You”, but MinceR has
instead suggested (jokingly) “Car Drives You Crazy”.


“You drive us wild, we’ll drive you crazy. You say you wanna go for a spin…”


So said another person in IRC.


We’ve meanwhile learned about how computerised cars are further being exploited
by dealerships and manufacturers.


As Ryan put it about an hour ago: “I got a letter from KIA saying that the
engine might explode and they want to reprogram the ECM. I would tell them I
don’t care if it blows up. Hell, if Scumbucket is in it, I’d be eternally
grateful.”


“Now they even have brakes that catch fire. They don’t call anything a recall.
They say there’s a “Product Improvement Campaign”.”
      –RyanIt’s basically about “[h]ow Hyundai/KIA do the cheapest thing
possible. You can’t repair metal shavings in an engine and defective connecting
rods with a software patch. So they just reset the ECM to display Check Engine
if it detects the condition. Many people with an OLD car just leave the warning
light on and keep driving it, if they’re in a state with no smog test, which is
like, 90% of the country. They extend the engine warranty a bit, but I’ve heard
horror stories about KIA and engine warranties. Like, the dealer will go “We’re
just dropping a used engine in here.” You have no idea if that engine is in
good shape! It could start doing the same thing 10,000 miles later! It’s not a
remanufactured engine where a mechanic has gone through rebuilding it
correctly. It’s just one they took out of a car that went to the junkyard.
They’re extremely disreputable and pretty much flout the law and then dare
their customers and the NHTSA to do anything about it. They basically got a
free pass under Trump, so they never fixed anything right. Now they even have
brakes that catch fire. They don’t call anything a recall. They say there’s a
“Product Improvement Campaign”.”


As it is proprietary, you cannot really check either. Even reverse-engineering
is discouraged if not criminalised.


“Just don’t buy anything from Hyundai or KIA,” Ryan said, “I haven’t seen them
design anything right, and by the time you compare a 2 year old KIA to a 2 year
old Toyota, it’s not much different in price, although statistically you’ll
have way more problems out of the KIA because they always seem to have a lot of
manufacturing defects and they’re so tight that they don’t repair them properly
when they do. To escape the American labor unions, Hyundai and KIA set up their
assembly plants in the deep south, and some of those workers on the assembly
lines only make about $10 or $11 an hour, which is $6-7 an hour less than what
Walmart pays where I live. Most UAW members up north making more reputable
brands make way over $20 an hour. Like $28 was typical a couple years ago. So
KIA is using the deep south states as sweatshop labor compared to that. It’s
how they get the price of the car so cheap and nothing you buy that was made
down there hold up. We bought some furniture that was labeled Made in USA. It
was made in Mississippi, which is the poorest state in the country. The workers
at that plant in 2018 were making $9 an hour. And about 3 months in I sat on
the couch and heard “CRUNCH!”. And the same things happened to our chair and
loveseat. And The Roomplace was sending their warranty people out to our house
constantly to patch it all up again. The Roomplace is about the worst furniture
store ever. They make everything look nice in the showroom, charge too much for
it, and then it falls apart. I got more durable furniture for a small fraction
of the price this time from IKEA and Bob’s Discount Furniture. According to a
Google reviewer last year, they now have a 3 day rule. There’s no fixes or
replacements or warranty anymore.”


The quality is decreasing, but the prices keep going up.


“So KIA is using the deep south states as sweatshop labor compared to that.
It’s how they get the price of the car so cheap and nothing you buy that was
made down there hold up.”
      –Ryan“Scotty Kilmer, a mechanic on YouTube that I like to watch,” Ryan
concluded, “said don’t buy a KIA unless you don’t plan on keeping it very long.
He went over one with (iirc) like 50 or 60,000 miles on the clock (my 2003
Impala has had some work, but closing in on 300,000) and he was going over all
of the stuff on the KIA that was broken. He said they just don’t last. If you
plan to have the car for ten years, don’t get one. They make it real cheap to
buy, but it’s an illusion. On average these days, you buy two cars to last as
long as one would from the 90s, and as long as 3 would from the 80s. The later
the model year, the crappier they made it. The more they cheaped out on the
parts. The more underpaid and disgruntled the labor. Even GM is unbelievably
bad now. They used to go over all of their cars before QA signed off on them.
It was a very tough inspection process, especially for their higher end brands,
like Buick. In fact, mom even bought a 1986 Buick for her third husband and it
ran just fine 30 years later. He’s still driving it I think. The carbon
emissions of a vehicle are not just the amount of fuel it burns for the miles
you get from it. 26% of a car’s LIFETIME CO2 emissions are when it is
assembled. If you have to buy three cars to last as long as one, did you really
save any emissions? When Congress passed Cash 4 Clunkers, they disabled and
crushed cars that were in their prime, and paid for! Then those people got a
car payment. The average fleet fuel economy in the US went up by like, some
abysmal amount. Less than 1%. And all of those cars that took energy and
materials and emissions and lots of water to produce were crushed! As soon as
you start driving a car, the fuel economy starts going down for all kinds of
reasons, from plugged filters nobody ever changes to worn out or fouled spark
plugs. This stuff is cheap to fix, but it will turn any car into a gas guzzler
if you aren’t servicing it. These new Gasoline Direct Inject engines are
amazing, for a little while anyway. On paper, it rolls out of the dealer with
15% more power generated for the same displacement size as a Multi-Port Fuel
Injected engine. But it doesn’t stay that way for long! The EPA has not revised
fuel detergent standards at the federal level since 1996 and if you run fuel
with the bare minimum of detergent in a GDI engine, you will have severe carbon
buildup about 40,000-50,000 miles in and it will be running terribly. And the
solution at that point is of course new plugs and having the mechanic tear
apart the engine to do “walnut blasting” to get the carbon out. That’s not
cheap. If you want to avoid that in a modern engine, you have to pull over at
the expensive gas stations that have the “Top Tier” label. Nobody knows this,
of course, until it’s too late. Murphy is not Top Tier gas and it’s near most
Walmarts and really cheap. I can run my car on it. Don’t try this with a new
one! The new technologies backfire, especially on early adopters. Honda was the
last major automaker in the US to introduce fuel injection. That got along
really bad with US gasoline early on because there were no detergent standards
then at all. It was a disaster for everyone who bought a car like that before
the mid to late 1980s. But Honda has really conservative engineers. VW was the
first one to roll out GDI in the US, and everyone that bought one suffered that
too, because their engine got crapped up and VW had to pay to have mechanics
clean them out. It’s perfectly fine to run a GDI engine but don’t buy bargain
gasoline for it. Hyundai’s THETA II engine was not GDI. In fact, I think it was
a fine engine other than the defective connecting rods. The correct repair is
tear it down and replace those. But the software patch that doesn’t really do
anything is cheap, so they do that. Given every opportunity to make things
right with their customers, Hyundai and KIA just screw the customer harder. KIA
Motors Finance was even in hot_water_for_violating_vehicle_repossession_laws_in
almost_every_state (case_here). There’s one about false credit bureau reporting
and one about violating repo laws. You’ll probably know if the connecting rod
bearings_are_failing_even_if_you_don’t_get_the_patch. Your engine will make a
horrible grinding noise while it’s on. You won’t miss it unless you’re deaf.
The problem is, once the bearings are bad, it won’t run for very long like
that. But whoever has that problem will figure it out. I personally witnessed
two KIAs explode last year within a week of each other. The fire department had
to come to the motel and to Walmart.”


Cars have become a lot more troublesome or troubling than they used to be. I
myself have stopped driving and have not shopped for a car in 10 years, so I
don’t yet know all the nasty bits in the cars to be sold in 2022; the meters/
odometers have become all electronic by now (they won’t last 20 years like the
old generation and ‘repairing’ them would basically mean swapping components).
For a car 20 years old (or more) they would not have those parts manufactured
anymore, which may lead to planned (digital) obsolescence. This, in turn, is
terrible for the environment.


“If your [odometer] stops working you can still drive the car,” Ryan said, “and
sell it, but you’d have to mark “exceeds mechanical limits” for the odometer.
Then it gets a new title saying “WARNING – ODOMETER READING NOT ACCURATE”.
It’ll be a major pain in the ass because age and mileage are how you determine
the value. So you’ll have to sell it to them at a high mileage price or they
won’t take a risk on it.”


There are many other ‘moving parts’ you cannot quite repair, at least not in
the long run. “What happens when you get a single source for parts is exactly
the same thing as prescription drugs,” Ryan noted. “BMW has a monopoly on brake
pads, rotors, and calipers, so a simple brake job that might cost you $199 at a
independent mechanic on a Chevy could run you $600-700 on a BMW. Because he has
to run down and buy the pads from BMW. A used BMW might be cheap to buy, but it
won’t be cheap to keep.”


The “infotainment” bit is not essential for driving, but more critical
components are nowadays becoming computerised and we know that the shelf life
and reliability of electronic devices is far shorter than analogue equivalents.
All the “infotainment” is typically not connected to the rest of the car or
does not pose a security threat although in the case of planes it has been
reported that “infotainment” panels are capable of taking over the whole plane.
I once read and heard that in planes they changed this, so now, as a passenger,
you could leverage “infotainment” to take over the main computer and steer/
crash the plane etc. Not a comforting thought…


“…you can probably get parts for a 1984 Land Cruiser J70…”
      –MinceRMinceR said that in is car “the infotainment thing doesn’t even
seem to be connected to the MID”. That’s about security, never mind economic
and environmental factors, which may merit another part. The shameful thing is
that cars are made to depend on things more complex than they really need to be
and for a car 20 years old they would not have those parts manufactured
anymore.


“You can still get parts for a 2003 Impala,” Ryan said, but that’s a car almost
20 years old.


MinceR said “you can probably get parts for a 1984 Land Cruiser J70…”


“If you’re going to buy an old car, buy one that they built millions of,” Ryan
concluded. “Parts for a 2003 Impala are pretty cheap and they tend to have
multiple sources.” As Ryan noted earlier, more “modern” cars aren’t like this
anymore.


We might do a Part 6 about environmental impact, but if we do, that would take
some time. █


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⠩⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

                    ䷩ 𝚕𝚒𝚗𝚎 2231

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

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 12.25.21⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Links_25/12/2021:_Rescuezilla_2.3_and_Growing_Backlash_Against_COVID-19
Patents⠀✐


Posted in News_Roundup at 5:16 am by Dr. Roy Schestowitz


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇GNOME bluefish⦈


§ Contents⠀➾


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


          o § Audiocasts/Shows⠀➾


                # ⚓ A_First_Look_At_Elementary_OS_“Jolnir”_–_Invidious⠀⇛


                       Elementary OS had a new release a few days ago,
                       version 6.1 which is codenamed “Jolnir”. I’ve
                       always loved the great work that the Elementary
                       team is doing, but the previous version of
                       Elementary had some issues. I’m hoping some of
                       those issues are addressed in this release.


          o § Benchmarks⠀➾


                # ⚓ Linux_5.16_Has_A_Nice_Performance_Gift_For_AMD_Ryzen
                  Laptops_With_Radeon_Graphics_–_Phoronix⠀⇛


                       For those making use of integrated Radeon Vega-
                       based graphics with modern Ryzen laptops at least,
                       the Linux 5.16 kernel is offering some nice
                       performance gains noticed recently as part of the
                       Ryzen 7 PRO 5850U benchmarking with the Lenovo
                       ThinkPad T14s Gen2. Here is a look at the AMD
                       Radeon Graphics performance for that Zen 3 laptop
                       across varying Mesa and Linux kernel versions while
                       then expanding the comparison to multiple devices
                       given the Linux 5.16 performance boost.


          o § Instructionals/Technical⠀➾


                # ⚓ Automatic_login_with_startx_and_systemd⠀⇛


                       If your workstation is using full-disk encryption,
                       you may want to jump directly to your desktop
                       environment after entering the passphrase to
                       decrypt the disk. Many display managers like GDM
                       and LightDM have an autologin feature. However,
                       only GDM can run Xorg with standard user
                       privileges.


                # ⚓ Add_a_lightbox_viewer_for_your_Ghost_blog_in_no_time⠀⇛


                       I have been using Ghost as my primary blogging
                       system for some years now. Although it does not
                       have a rich ecosystem like WordPress or other open-
                       source CMS, it does the job.


                       In Ghost, You will get used to doing everything
                       manually, creating a search functionality using
                       jQuery them Vue, trying out some embedded web
                       widgets for messaging I built with Vue and CouchDB.


                # ⚓ How_to_play_Crab_Game_on_Linux⠀⇛


                       Crab Game is an FPS multiplayer game where players
                       compete through minigames until one player remains.
                       It is based upon the popular Netflix series “Squid
                       Game.” Here’s how to play it on Linux.


                # ⚓ How_to_play_Super_Flight_on_Linux⠀⇛


                       Superflight is a wingsuit adventure game. It was
                       developed and published by GrizzlyGames. The game
                       was released on Windows. However, it is possible to
                       it working on Linux. Here’s how to set it up on
                       your Linux computer.


                # ⚓ How_To_Reverse_a_String_In_Unix_/_Linux_Shell?⠀⇛


                # ⚓ How_to_configure_static_IP_Address_on_FreeBSD⠀⇛


                       his page explains setting up a static IP address on
                       the FreeBSD Unix system using the command-line
                       options.


                # ⚓ How_to_install_the_Brave_Beta_Browser_on_Elementary_OS_6.0
                  –_Invidious⠀⇛


                       In this video, we are looking at how to install the
                       Brave Beta Browser on Elementary OS 6.0.


                # ⚓ How_to_enable_GPU_acceleration_for_your_Linux_applications_
                  (Crostini)_only_on_a_Chromebook⠀⇛


                       Today we are looking at how to enable GPU
                       acceleration for your Linux applications (Crostini)
                       only on a Chromebook. Please follow the video/audio
                       guide as a tutorial where we explain the process
                       step by step and use the commands below.


                # ⚓ How_to_Check_and_Repair_XFS_Filesystem_in_RHEL/Centos_–
                  Unixcop_the_Unix_/_Linux_the_admins_deams⠀⇛


                       Hello, friends. In this post, we will explain a
                       trick that can save a lot of work and even an
                       entire system. So, today you will learn How to
                       Check and Repair XFS Filesystem in RHEL/RockyLinux/
                       CentOS.


                       Before we start, we need to talk a little about
                       XFS.


                       XFS is a high-performance 64-bit journaling file
                       system. Its virtues such as maturity, and debugged
                       code, make it a preferred candidate for use on GNU/
                       Linux servers.


                # ⚓ 2_Ways_to_Install_Homebrew_on_Debian_11_Bullseye_Linux_–
                  Linux_Shout⠀⇛


                       Learn the commands to install Homebrew on Debian 11
                       Bullseye to install and manage various packages via
                       “formulas” and “casks” on your Linux.


                       Homebrew is an open-source project for providing a
                       package manager to macOS because there is no
                       default package manager, unlike Linux systems.
                       However, the Homebrew is also present for Linux,
                       popularly known as Linux brew.


                       The homebrew was started by Max Howell. In 2009 the
                       first version of Homebrew was released. To use it
                       the user either has to be on Linux or Mac. However,
                       using homebrew on Linux doesn’t make sense because
                       all of them almost come with their package manager.


                       Nevertheless, Homebrew does several things, it can
                       be used to install, uninstall and update software.


          o § Desktop Environments/WMs⠀➾


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


                      # ⚓ Krita_5.0_is_Here_As_a_Last-Minute_Christmas_Gift
                        From_KDE⠀⇛


                             Alongside GIMP, Krita is among the most
                             powerful graphics creation applications for
                             Linux. While most well-known for its digital
                             painting features, it is also a capable image
                             editor and piece of animation software. Now,
                             with its brand-new 5.0 release, many of these
                             various aspects of Krita have been improved.


                             Let’s take a closer look!


                             [...]


                             Using a technique called dithering, it is now
                             possible to give the illusion of a smooth
                             gradient in the same bit depth. In simple
                             terms, it means combining the colors of
                             surrounding pixels that the eye interprets as
                             a color out of the range of the bit depth.


          o § Distributions⠀➾


                # ⚓ 11_Most_Popular_Linux_distros_of_the_year_2021-_To_use_in
                  2022⠀⇛


                       Well, the year 2021 is about to end in a few days,
                       of course, let’s welcome 2022 with a warm heart.
                       However, before stepping into it, let’s have a
                       glimpse of the best Linux distributions in 2021,
                       downloaded most so far.


                       Over the past few years, Linux distros have gained
                       quite popularity, and users are now switching to
                       them; as more and more applications are now
                       available for it. Also, due to partly restriction
                       because of the pandemic, many people have been busy
                       learning or relearning the unfamiliar features of
                       Linux; or simply jumping from one distribution to
                       another to test new features.


                       Well, if you are finally thinking to experience
                       some other Linux operating system apart from the
                       one you have right now on your PC or Laptop, then
                       here are some best distributions for 2021 to look
                       forward to in 2022.


                # ⚓ 7_Reasons_Why_I_Use_Pop!_OS_Linux_Distro_as_My_Daily_Driver
                  –_It’s_FOSS⠀⇛


                       Pop!_OS is a popular recommendation for Linux
                       beginners and gamers alike. But, what are some of
                       the reasons that Pop!_OS stands out from other
                       Ubuntu-based distros available? Why should you
                       consider it as a daily driver for your computer?


                       I switched to Pop OS from Ubuntu three years ago
                       and it has been my daily driver since then.


                       Let me point out the benefits of choosing it as the
                       operating system of your choice.


                # § New Releases⠀➾


                      # ⚓ Rescuezilla_2.3_“Swiss_Army_Knife_of_System_Recovery”
                        Is_Here,_Based_on_Ubuntu_21.10⠀⇛


                             Based on Ubuntu 21.10 (Impish Indri) and
                             powered by Linux kernel 5.13, Rescuezilla 2.3
                             is here seven months after Rescuezilla 2.2 to
                             implement several new features, such as basic
                             image verification and a new “Rescue” option
                             to ignore file system inconsistencies and bad
                             sectors.


                             Rescuezilla 2.3 also adds the ability to
                             restore and explore images created by
                             theApart GTK graphical user interface for
                             partclone, improves the image scanning
                             feature to fix incomplete restore image
                             lists, and improves restoring of images
                             created by FSArchiver or the qt-fsarchiver
                             GUI.


                      # ⚓ Rescuezilla_2.3_Released⠀⇛


                             Rescuezilla 2.2 is the latest version for the
                             operating system that is popular for system
                             recovery. Powered by Linux kernel 5.13 and
                             based on Ubuntu 21.10 (Impish Indri).
                             Rescuezilla can create backups that are fully
                             compatible with the Clonezilla tool.


          o § Devices/Embedded⠀➾


                # § Open Hardware/Modding⠀➾


                      # ⚓ Raspberry_Pi_Mega_Cluster_Costs_More_Than_$5,000_to
                        Build_|_Tom’s_Hardware⠀⇛


                             Just when you thought one Raspberry Pi was
                             enough, Skynet Network stepped up with this
                             awesome Raspberry Pi cluster project.
                             According to Skynet Network, it’s dubbed the
                             Raspberry Pi Mega Cluster and costs more than
                             £4,000 (approximately $5,372) to build.


                             The best Raspberry Pi projects typically have
                             at least one Pi but this one has 83. As
                             impressive as that sounds, the original plan
                             was to pack it full of 100 Raspberry Pis.
                             However, network latency had too much impact
                             on performance for that many to be practical.


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


                # ⚓ Cerebro:_Your_Desktop_Search_Assistant_for_Windows,_macOS,
                  and_Linux⠀⇛


                       Searching your local machine for files is no fun,
                       especially when the built-in search tools are not
                       useful to look deeper into file types, locations,
                       and content.


                       But with Cerebro it is different. So, what is
                       Cerebro?


                       Cerebro is a multi-platform desktop app that helps
                       the user to search their machine or the internet
                       with no more than a few clicks.


                       The app is created by Alexandr Subbotin a Russian
                       developer, who also created several useful plugins.
                       Later then, the app attracted many developers who
                       supply it with more plugins and support.


                # ⚓ InvoiceLion:_your_self-hosted_open-source_invoice_manager⠀⇛


                       InvoiceLion is a free, open-source invoicing
                       software for freelancers or anyone who is issuing
                       invoices for customers.


                       Self-hosted means you can download, host, install,
                       and use it on your server.


                       InvoiceLion is created by twin freelance
                       developers, who created it out of their need to
                       automate invoice creation and production.


                # ⚓ 10+_Open-source_Single-Sign_On_(SSO)_Solutions⠀⇛


                       Single Sign-On (SSO) is a login service layer that
                       allows users to sign in to multiple apps with a
                       single login. Using SSO service to improve
                       security, save the user from having to deal with
                       multiple passwords, login sessions, and losing
                       time.


                       In contrast, SSO “Single-Sign-On” is an
                       authentication and authorization process through
                       one login interface, designed to increase security
                       issues, increase employee productivity, and improve
                       customer experience.


                       It is the user first engagement layer for many FIM
                       as IAM services.


                # ⚓ 11_Best_free,_open-source_personal_information_organizer
                  and_personal_CRM_systems.⠀⇛


                       Unlike business-oriented CRM solutions, personal
                       CRM applications are minimal versions of CRM aimed
                       to help individual users organize their contacts.


                       It is challenging to keep track or manually
                       organize your contacts, events, and notes.


                       A Personal CRM system is a minimal CRM app that
                       helps you organize your contact, events, journals,
                       and appointments in one software package.


                       Some personal CRM solutions took it further by
                       managing family members, tasks, chores, shopping
                       lists, and more.


                       Some may disagree with the term CRM “Customer
                       Relationship Management”, and they prefer to call
                       it PRM “Personal Relationship Management”, a
                       solution.


                # ⚓ AMIDE:_Open-source,_free_DICOM_viewer_for_volumetric
                  imaging⠀⇛


                       AMIDE is a feature-rich free, open-source program
                       for viewing, analyzing, and registering volumetric
                       medical imaging data sets.


                       It is one of the oldest open-source, free DICOM
                       software. It provided aid to doctors, and
                       researchers on platforms like Linux and macOS since
                       2003.


                # ⚓ 17_free,_open-source_Bullet_Journal_apps_to_boost_your
                  productivity⠀⇛


                       Bullet Journal is a multipurpose productivity
                       booster journal-taking method, The journal can be
                       anything from a calendar, a goal tracker, a project
                       manager, activity tracker, to a to-do list.


                       Originally created by designer Ryder Carroll,
                       Bullet Journal, or as some may refer to as BuJo or
                       Bujo, became the favorite tool for creative people.


                       In contrast, it is up to you to figure out what and
                       how will you use it.


                # § Web Browsers⠀➾


                      # § Mozilla⠀➾


                            # ⚓ 2021_Retrospection_–_The_Mozilla_Support_Blog⠀⇛


                                   This is probably a moment where we
                                   usually say to ourselves how time flies
                                   so fast. Despite the pandemic and the
                                   uncertainty that lingers this year, I’m
                                   proud and grateful that this community
                                   managed to keep being awesome and
                                   accomplished many things this year.
                                   Through this post, I’d like us to take
                                   a moment to flashback and celebrate
                                   what we’ve accomplished this year.


                # § Programming/Development⠀➾


                      # § Java⠀➾


                            # ⚓ Open-source_software_holds_the_key_to_solving
                              Log4Shell-like_problems⠀⇛


                                   Earlier this month, the existence of a
                                   critical vulnerability in Apache Log4j
                                   2 was revealed and a PoC for it
                                   published. Dubbed Log4Shell, it’s an
                                   issue in a logging library for Java
                                   applications that is widely used across
                                   popular open-source projects and
                                   enterprise-grade back-end applications.
                                   Log4Shell introduced a critical
                                   security risk, scoring 10 out of 10 in
                                   severity.


    * § Leftovers⠀➾


          o ⚓ Blues_Resurrected:_the_Second_Coming_of_T-99_Nelson⠀⇛


                 But on this Independence Day evening at the Waterfront
                 Blues Festival in Portland, Allen wisely chose to tone
                 down (for him anyway) his attire. Even the flamboyant
                 Allen knew there was no way he could compete with the
                 splendor of the Cannonballs’ guest singer for the night,
                 the Jimmy T99 Nelson, the greatest living blues shouter
                 and one of the true progenitors of rock and roll.


                 Midway through the Cannonballs’ rollicking set Nelson
                 strolled onto the stage in a cerulean-colored suit that
                 shimmered so brightly it looked as if it had just been
                 painted by Raphael. Adorned with a captain’s cap and
                 fighter pilot shades, T99 looked like he was ready to
                 rock all night. And damned if he didn’t! Indeed, the 83-
                 year old blues shouter dominated the stage not only with
                 the Cannonballs, but with a host of other top-notch acts,
                 including guitarist Duke Robillard, piano player Marcia
                 Ball and harp master Paul DeLay, stealing the show at
                 what has become one of the nation’s biggest blues
                 festivals.


          o ⚓ Anthropocene_Joyride⠀⇛


                 That evidence was the signature of radioactivity (Carbon-
                 14) from nuclear bomb test fallout incorporated into tree
                 rings for the year 1965 (within October to December) at
                 all latitudes and longitudes. Atmospheric nuclear bomb
                 explosions were set off from 1945 to, primarily, 1963.
                 The peak of such activity was an orgy of bomb tests by
                 the U.S. and USSR in 1962. [1]


                 Prior to 1965 many species had had geographically limited
                 impact on altering environments, like pre-columbian
                 humans in North American making pastures out of forests
                 in Appalachian valleys, herd animals making permanent
                 migration trails, and grazing animals nibbling bush down
                 to grasslands.


          o ⚓ Macedonian_Ramble:_To_Lord_Byron’s_Hellespont⠀⇛


                 This is the eleventh part in a series about a journey
                 across what used to be called Macedonia, which is now
                 divided among Albania, North Macedonia, Greece, and
                 Turkey.


                 About 19:30 the train ended its day at the station in
                 Drama, where the remaining passengers huddled in the cold
                 and then were assigned to one of two replacement buses,
                 depending on their destination. I was on the bus for
                 Alexandroupolis, now a forgotten northern Greek port but
                 once fought over by Greece, Turkey, and Bulgaria.


          o ⚓ The_Best_Albums_of_2021⠀⇛


                 If we all felt lost at times this year, uncertain in the
                 face of tentative hopes and looming unknowns, it was good
                 to know we could call Tyler, the Creator. His lush and
                 multifarious album Call Me If You Get Lost offers
                 comfort, escape, inspiration, and pretty much everything
                 else anyone living in 2021 could use. In music as in the
                 rest of life, the high points of the past year came in
                 smaller numbers and with less impact than we might have
                 expected. Still, they came with impressive persistence in
                 every style of work or genre. Here are my picks for the
                 10 best albums of 2021. (The listing is alphabetical by
                 album title, not a ranking by descending order of
                 quality, though the top project happens to be my top
                 choice of the year, too.)


          o ⚓ “I_Think_of_Myself_as_Attached_to_California:”_Interviewing_Joan
            Didion⠀⇛


                 Q: Your book, Where I Was From (2003), was recently
                 reviewed in The New York Times Book Review. The title was
                 “Giving Up On California.” Have you given up on your
                 native state?


                 A: No, I haven’t. Their title was far broader than any
                 intention I ever had. But people in New York often hope
                 that Californians give up on California.


          o ⚓ The_Best_to_You_Each_Morning⠀⇛


          o ⚓ Holiday_Season_Top_Reading_Recommendations⠀⇛


                 Major book awards and prizes rarely select books
                 addressing corporate crimes and what to do about them.


                 Not surprisingly, you rarely read about these books or
                 see or hear about them on television and radio shows,
                 including PBS and NPR. Corporate funders prefer
                 convenient alternatives such as art, culture, history,
                 and entertainment.


          o ⚓ Why_Christopher_Hitchens_Still_Matters⠀⇛


                 Christopher Hitchens died 10 years ago this month. Many
                 young leftists remember “Hitch,” if at all, as a militant
                 atheist who alternated between debating pastors on the
                 existence of God and defending the war in Iraq.


          o ⚓ Life_Meets_Fiction⠀⇛


                 And it mirrors a striking resemblance to American
                 politics today.


                 I am re-reading “Lord of the Flies” for the first time
                 since high school. It’s a better, more meaningful read as
                 an adult.


          o ⚓ A_Christmas_Story⠀⇛


                 Jenny was awakened about ten o’clock on Christmas Eve by
                 a noise coming from the living room downstairs. She knew
                 that it was her mother and father putting the Christmas
                 gifts under the tree and filling the stockings, so that
                 in the morning they could claim it was all the work of
                 Santa Claus. For a moment she laid in bed listening,
                 huddling under the covers as the street light cast a cool
                 glow through the window, illuminating a bedroom filled
                 with dolls.


          o § Health/Nutrition/Agriculture⠀➾


                # ⚓ In_October,_White_House_Dismissed_Proposal_for_‘Free_Rapid
                  Tests_for_the_Holidays’:_Report⠀⇛


                       As the highly transmissible and now-dominant
                       Omicron variant drives a nationwide surge in
                       coronavirus infections amid a period of widespread
                       air travel, President Joe Biden said earlier this
                       week that he wishes he “had thought about ordering”
                       500 million at-home tests “two months ago.”


                       But new reporting reveals that his administration
                       in October was presented with—and rejected—a plan
                       to scale up rapid testing before the holidays.


                # ⚓ ‘A_Huge_Deal’:_Biden_Rebuffs_Georgia’s_Medicaid_Work
                  Requirement—Last_in_US⠀⇛


                       After the White House on Thursday rejected a
                       proposal to impose work requirements on Medicaid
                       recipients in Georgia—the last state with a federal
                       waiver permitting such restrictions—President Joe
                       Biden received praise for “quietly erasing” one of
                       his predecessor’s “cruelest legacies.”


                       “The announcement from the Centers for Medicare &
                       Medicaid Services, delivered in a 79-page letter to
                       Georgia’s health agency, also reversed a federal
                       waiver allowing the state to charge premiums for
                       the health insurance program for the poor,”
                       according to the New York Times.


          o § Integrity/Availability⠀➾


                # § Proprietary⠀➾


                      # § Security⠀➾


                            # ⚓ Log4j_vulnerability_now_used_to_install_Dridex
                              banking_malware⠀⇛


                                   When executed, the VBS file will check
                                   if the user is part of a Windows domain
                                   by checking various environment
                                   variables. If the user is part of a
                                   domain, the VBS file will download the
                                   Dridex DLL and execute it using
                                   Rundll32.exe, as shown below.


                            # ⚓ Investigation_Shows_Egyptian_Government_Hacked
                              A_Dissident’s_Phone_Twice,_Using_Two_Different
                              Companies’_Malware⠀⇛


                                   Citizen Lab has uncovered more state-
                                   level spying targeting political
                                   opponents and journalists. There’s a
                                   twist to this one, though. One of those
                                   targeted had his phone infected by two
                                   forms of malware produced by two
                                   different companies. And yet another
                                   twist: both companies have their roots
                                   in Israel, which is home to at least 19
                                   entities that develop phone exploits.
                                   Here’s the summary from Citizen Lab:


                            # § Privacy/Surveillance⠀➾


                                  # ⚓ Pushing_Back_on_Police_Surveillance:_2021
                                    Year_in_Review⠀⇛


                                         The San Francisco Board of
                                         Supervisors kicked off the year
                                         by voting unanimously in favor of
                                         special business districts—such
                                         as the Union Square Business
                                         Improvement District
                                         (USBID)—disclosing any new
                                         surveillance plans to the Board.
                                         The Board acted in the wake of an
                                         EFF investigation and lawsuit
                                         that exposed the San Francisco
                                         Police Department’s (SFPD) spying
                                         on last year’s Black-led protests
                                         against police violence. The SFPD
                                         monitored the demonstrations by
                                         using the USBID’s camera network.


                                         EFF welcomes the Board’s small
                                         step toward transparency, but the
                                         city continues to defend the
                                         SFPD’s unlawful surveillance. In
                                         October 2020, EFF sued the SFPD
                                         on behalf of three activists who
                                         helped organize last year’s
                                         protests in the city. This fall,
                                         EFF asked the court to rule that
                                         the SFPD violated the city’s
                                         landmark surveillance technology
                                         ordinance and to prohibit the
                                         SFPD from using the USBID cameras
                                         without prior Board approval.
                                         While the SFPD initially claimed
                                         it did not monitor the camera
                                         feed, an SFPD officer admitted
                                         during a deposition that she
                                         repeatedly looked at the camera
                                         feed during the eight days that
                                         the department had access.


                                         EFF is also in court to protect
                                         your privacy from Automated
                                         License Plate Readers (ALPRs),
                                         which police use to amass large
                                         databases of location and other
                                         sensitive information on millions
                                         of drivers. In October, we filed
                                         a lawsuit on behalf of immigrant
                                         rights activists to stop the
                                         Marin County Sheriff in
                                         California from sharing its ALPR
                                         data with over 400 out-of-state
                                         agencies and 18 federal agencies,
                                         including Immigration and Customs
                                         Enforcement (ICE) and Customs and
                                         Border Protection (CBP), which
                                         violates two state laws.


                                  # ⚓ Edward_Snowden,_Glenn_Greenwald_&_Chris
                                    Hedges_on_NSA_Leaks,_Assange_&_Protecting_a
                                    Free_Internet⠀⇛


                                         NSA whistleblower Edward Snowden
                                         and Pulitzer Prize-winning
                                         journalists Glenn Greenwald and
                                         Chris Hedges discuss mass
                                         surveillance, government secrecy,
                                         internet freedom and U.S.
                                         attempts to extradite and
                                         prosecute WikiLeaks founder
                                         Julian Assange. They spoke
                                         together on a panel moderated by
                                         Amy Goodman at the virtual War on
                                         Terror Film Festival after a
                                         screening of “Citizenfour” — the
                                         Oscar-winning documentary about
                                         Snowden by Laura Poitras.


          o § Defence/Aggression⠀➾


                # ⚓ The_Great_Myth_of_the_”Great“_American_Military⠀⇛


                       Professional sports is a cutthroat business.
                       Succeed and the people running the show reap rich
                       rewards. Fail to meet expectations and you get
                       handed your walking papers. American-style war in
                       the 21st century is quite a different matter.


                # ⚓ Few_Surprises_for_the_Serious_as_January_6_“Commemoration”
                  Nears⠀⇛


                       Thanks in no small part to the investigation being
                       carried out by the US House January 6 Commission,
                       we will continue to learn new details about the
                       specifics of the Capitol Riot that Donald Trump and
                       his cronies instigated in a last-ditch effort to
                       prevent the Congressional certification and
                       inauguration of Joe Biden as the United States’
                       46th president. Still, no serious political
                       observer should have been surprised by the Attack
                       on the Capitol. In May of 2016, the liberal New
                       Yorker commentator Adam Gopnik issued what turned
                       out to be a prophetic warning seven months before
                       Trump’s victory over Hillary Clinton:


                       There is a simple formula for descriptions of
                       Donald Trump: add together a qualification, a
                       hyphen, and the word “fascist” …his personality and
                       his program belong exclusively to the same dark
                       strain of modern politics: an incoherent program of
                       national revenge led by a strongman; a contempt for
                       parliamentary government and procedures; an
                       insistence that the existing, democratically
                       elected government…is in league with evil outsiders
                       and has been secretly trying to undermine the
                       nation; a hysterical militarism designed to no
                       particular end other than the sheer spectacle of
                       strength; an equally hysterical sense of
                       beleaguerment and victimization; and a supposed
                       suspicion of big capitalism entirely reconciled to
                       the worship of wealth and ‘success.’… The idea that
                       it can be bounded in by honest conservatives in a
                       Cabinet or restrained by normal constitutional
                       limits is, to put it mildly, unsupported by history
                       (emphasis added).


                # ⚓ US_and_NATO_Nuclear_Lunacy_Still_Raving⠀⇛


                       Before the Joint Chiefs took it down, Steven
                       Aftergood of the Federation of American Scientists
                       managed to preserve a copy. The manual relies on
                       abstractions and euphemism to depict the
                       unthinkable. It says, “The employment of nuclear
                       weapons could have a significant influence on
                       ground operations.” Of course “employment” means
                       detonation, and “significant influence” means
                       searing fireballs, vaporized victims, blast and
                       shock-wave devastation, demolished hospitals and
                       schools, vast firestorms, and permanent radioactive
                       contamination of water, soil, and the food chain.


                       The manual explains that nuclear attacks create
                       “conditions” without describing them. It says,
                       “Using nuclear weapons could create conditions for
                       decisive results and the restoration of strategic
                       stability.” Then, as if US presidents had never
                       said, “Nuclear war cannot be won and must never be
                       fought,” the report pretends it can and should. “
                       [T]he use of a nuclear weapon will…create
                       conditions that affect how commanders will prevail
                       in conflict.”


                # ⚓ The_Dead_and_the_Living:_Britain’s_Dirty_War_in_Northern
                  Ireland⠀⇛


                       Stephen Travers talked to me of that summer’s night
                       in July 1975 and how everything he saw around him,
                       paradoxically and surreally, seemed more vivid and
                       silent, even momentarily beautiful, as he lay
                       wounded in a field surrounded by the aftermath of
                       violent death in Ulster’s very own heart of
                       darkness.


                       “They all died…I spent about forty-five minutes
                       crawling around reassuring them that everything
                       would be alright…we’d get back to the music…I
                       remember whispering into Fran’s ear…But you don’t
                       accept that type of – it’s… the horror.”


                # ⚓ Republicans_With_Gun-Themed_Xmas_Photos_Are_Usual
                  Recipients_of_Gun_Lobby_Cash⠀⇛


          o § Environment⠀➾


                # ⚓ Climate_Change_Brings_Health_Risks⠀⇛


                       The World Health Organization (WHO) estimates that
                       global warming and rainfall changes cause the loss
                       of at least 150,000 lives every year. This figure
                       could increase significantly if current trends of
                       climate change continue. The WHO also states that
                       the risk of death and disease from climate change
                       will at least double in the next 20 years.
                       Paradoxically, the countries that have least
                       contributed to global warming are the most
                       vulnerable to its negative consequences.


                       Global warming can affect the health of populations
                       both directly and indirectly. Direct effects can
                       result from heat-related deaths or weather-caused
                       disasters such as hurricanes and drought-related
                       wildfires. Indirect effects can result from
                       alterations in complex ecological processes such as
                       changes in the patterns of infectious diseases, in
                       the quantity and quality of domestic food
                       production, and altered potable water supplies.
                       Experts predict that receding waters in the Ganges
                       River could affect the lives of 400 million people.


                # ⚓ Activists_Are_Using_the_“Climate_Necessity_Defense”_in
                  Court_—_and_Winning⠀⇛


                # ⚓ Opinion_|_Climate_Change_Brings_Serious_Health_Risks⠀⇛


                       Much attention has been devoted in recent times to
                       the environmental and economic effects of climate
                       change. Much less attention, however, has been
                       given to the possible effects of climate change,
                       particularly global warming, on the health of the
                       populations, particularly those from the poorest
                       countries. This is a trend that requires prompt
                       attention if the negative effects of climate change
                       on health are to be avoided or minimized. According
                       to some estimates, at least 1 in 6 people worldwide
                       will suffer the consequences of climate change.


                # ⚓ Opinion_|_Climate_Justice_Is_Reproductive_Justice⠀⇛


                       More than ever, people of childbearing age are
                       choosing not to have children. There are a variety
                       of factors that go into this choice — wage
                       stagnation, the housing market, civil unrest,
                       COVID-19. But a common denominator heard among many
                       in their 20s and beyond is grave concern over our
                       climate trajectory. They’re not wrong to be worried
                       about what kind of climate and resources would be
                       available for their potential kids. Climate justice
                       is a crucial part of reproductive justice.


                # § Wildlife/Nature⠀➾


                      # ⚓ What_is_a_Wilderness_Without_Its_Wolves?⠀⇛


                             The Wilderness Act defines Wilderness as a
                             place where the earth and its community of
                             life are untrammeled by humankind, retains
                             its primeval character and where natural
                             conditions are preserved. Simply stated,
                             Wilderness is meant to exist with minimal
                             human interference. Yet within the vast
                             majority of Wilderness areas, the wolf, the
                             apex species with profound ecosystem
                             influence, is now absent—an absence due
                             entirely to the relentless killing by
                             humankind.


                             We need look no farther than Yellowstone
                             National Park to witness the influence wolves
                             have on an ecosystem. The park’s wolves were
                             exterminated by the early 1900s, ostensibly
                             to protect the park’s favored elk herds. What
                             followed was not surprising—an overabundance
                             of elk which led to deleterious impacts to
                             vegetation, particularly lower elevation
                             riparian and willow communities.


                      # ⚓ A_Dark_Day_in_Montana_as_Wildlife_Officials_Sink_to
                        New_Lows⠀⇛


                             This decision is sickening and yet it doesn’t
                             even begin to tell the whole horrific
                             situation that imperiled wolves and grizzly
                             bears have faced all year in Montana. And the
                             stakes are only getting more dangerous as a
                             long, cold winter descends.


                             This year’s start of the wolf trapping season
                             was delayed in parts of western Montana to
                             give threatened grizzly bears more time to
                             safely reach their dens. Despite this,
                             threatened species like grizzlies were not
                             spared from the brutality of indiscriminate
                             trapping.


                      # ⚓ Watch:_Yle_launches_‘Bird_table’_live_stream⠀⇛


                             While the human hustle and bustle of the pre-
                             Christmas rush has subsided, and a Christmas
                             peace descends, a hustle and bustle of a
                             different kind continues in yards and gardens
                             across the country – especially if it
                             contains a bird feeding table.


                             “Finns are very active feeders of birds,”
                             Chief Curator of the Finnish Museum of
                             Natural History Aleksi Lehikoinen told Yle.


                             Although the exact numbers of bird feeders is
                             difficult to gauge, an estimate can be
                             deduced from the amount of bird food imported
                             into Finland. The latest report from the Food
                             Agency is from 2019.


                             “More than ten million kilos of sunflower
                             seeds and more than two million kilos of nuts
                             are imported into Finland every year for the
                             purpose of feeding wildlife,” Lehikoinen
                             said.


                # § Overpopulation⠀➾


                      # ⚓ Do_We_Dare_Stop_Being_Afraid_of_Ourselves?⠀⇛


                             We tell stories, which evolve into myths —
                             and myths are what hold us together. They
                             create the collective entity known as the
                             human race.


                             And myths evolve.


                      # ⚓ There’s_Only_One_Essential_Role_Humans_Have_on_Earth:
                        a_Humbler_Perspective_Could_Save_the_World⠀⇛


                             This is a planet of water in continuous
                             circulation moving through many phases, with
                             each phase intimately linked at every stage.
                             It is the water in the sea, the lakes, the
                             rivers, and the streams. It is the water
                             flowing underground and deep, deep down
                             inside the planet, locked in rock. It is the
                             water in the atmosphere or encased in ice.


                             And it is the water moving through each and
                             every living cell of every plant and animal
                             on the planet.


          o § Finance⠀➾


                # ⚓ Opinion_|_Don’t_Let_the_Deeper_Meaning_of_Christmas_Be_Lost
                  in_Materialism⠀⇛


                       On Saturday, millions of people across the world
                       will celebrate Christmas. Even with COVID-19 still
                       plaguing the world, families will gather; bells
                       will ring; music will be in the air. Each year, I
                       use this column to remind us of the true meaning of
                       Christmas.


                # ⚓ How_Public_Workers_Can_Stop_The_Privatization_of
                  Everything⠀⇛


                       With no VA union members present, and only a few
                       veterans or VA patients in the crowd to present an
                       alternative view, the initial consensus of the
                       group was that public provision of veterans’
                       healthcare was inferior to private sector
                       treatment—a perspective clearly shaped by corporate
                       media coverage of the VA and an ongoing
                       conservative push for its partial privatization.
                       Before the end of the session, participants had a
                       far better understanding of the inter-relatedness
                       of political attacks on public services and jobs,
                       at all levels of government, and the need for
                       greater labor-community solidarity to fend them
                       off. As Cohen and his co-author Allen Mikaelian
                       make clear in The Privatization of Everything: How
                       The Plunder of Public Goods Transformed America and
                       How We Can Fight Back (New Press), any private
                       sector incursion into the public sector is not just
                       somebody else’s problem, it’s an injury to all.


                       In their new book, Cohen and Mikaelian, describe
                       and debunk the vast array of privatization schemes
                       that now litter the landscape, from private prisons
                       and charter schools to for-profit water treatment
                       and trash collection. “Corporations entrusted with
                       public goods have been given the power to lock
                       people up, force them into a parallel justice
                       system, deny them life-saving treatments, place
                       vital information behind paywalls, divert money
                       from poverty programs to wealthy investors, tie the
                       hands of governments, and undermine our
                       constitutional rights.” There are now 2.6 times as
                       many federal contract employees as there are direct
                       employees of federal agencies.


                # ⚓ Opinion_|_What_I’m_Getting_the_American_Oligarchs_This_Year
                  for_Christmas⠀⇛


                       Ho-ho-ho, wait till you hear about the gifts I gave
                       to some of America’s power elites for Christmas.


                # ⚓ Investing_In_Our_Destruction⠀⇛


                # ⚓ Retail_Workers_Are_Organizing_for_Safety_and_Dignified_Pay
                  Amid_Holiday_Rush⠀⇛


                # ⚓ Special_Privilege_in_Admission_to_Elite_U.S._Colleges⠀⇛


                       Consider the fact that, according to a recent
                       study, only about 57 percent of Harvard’s white
                       students were admitted on the basis of academic
                       merit. The other 43 percent received special
                       treatment because their parents had made large
                       donations to Harvard, or they had parents who had
                       graduated from the school (“legacy admissions”) or
                       taught there, or were recruited as athletes.
                       Without this preferential status, some three-
                       quarters of them would have been rejected in the
                       intense competition for admission.


                       Actually, a preferential status for the privileged
                       has long been a tradition in the top ranks of
                       American higher education. During their first
                       decades, elite colleges considered only wealthy
                       white students as worthy of admission. Even by the
                       end of the nineteenth century, their campuses were
                       populated primarily by graduates of private high
                       schools. In the early twentieth century, worried
                       administrators, anxious to limit the number of Jews
                       and other new immigrants, leaned increasingly upon
                       legacy status, family donations, and athletic
                       recruiting.


                # ⚓ Organizers_Are_Building_Solidarity_Through_Holiday_Mutual
                  Aid_Work⠀⇛


                # ⚓ The_Good_News_in_2022_Will_Be_the_Economy⠀⇛


                       In recent weeks, inflation has been front and
                       center in people’s minds as the media have given us
                       endless stories about higher prices for gasoline,
                       milk, and other items. Many have been convinced
                       that inflation will only get higher, outstripping
                       wages and leaving most workers worse off. This is
                       not going to be the case.


                       We now see inflation driven by supply chain
                       problems associated with reopening. This is
                       demonstrated by the fact that we see big jumps in
                       inflation almost everywhere.  The United Kingdom,
                       Germany, Spain, and many other countries have all
                       seen a rise in prices similar to what we see in the
                       United States.


          o § AstroTurf/Lobbying/Politics⠀➾


                # ⚓ Opinion_|_It’s_Not_Too_Late_To_Save_Democracy._Senate
                  Showdown_On_Voting_Rights_Coming_In_January⠀⇛


                       2021 has been a very difficult year for the
                       country, starting with the devastating impact of
                       the coronavirus.


                # ⚓ Report_Spotlights_Massive_GOP_Push_to_‘Hijack_Elections_in
                  This_Country’⠀⇛


                       A detailed analysis published Thursday shines
                       further light on the Republican Party’s sprawling
                       assault on voting rights and the democratic process
                       nationwide, an effort that includes legislation
                       that would “politicize, criminalize, and interfere”
                       with elections.


                       Compiled by the States United Democracy Center,
                       Protect Democracy, and Law Forward, the new report
                       identifies at least 262 bills in 41 states that—if
                       enacted—would “interfere with election
                       administration.” More than 30 such measures have
                       become law in 17 Republican-led states.


                # ⚓ Fighting_the_Power_From_Inside_the_Belly_of_the_Beast:_an
                  Interview_With_Dennis_Kucinich⠀⇛


                       Bernstein:  Welcome Congressman Kucinich.  It is
                       good to talk with you again. Let’s start here. You
                       saw the corporate muscles flexed at the national
                       level as a member of Congress and running for
                       President.  Would you say that what we’ve just
                       seen—some would say we were just Manchined, but
                       what we’ve just seen in terms of Manchin, a right-
                       wing Democrat controlling, essentially the flow of
                       policy in the United States, is this about
                       corporate power?  Is that still the problem?


                       Kucinich:   Well, you know,  that’s one level of
                       analysis, but I’d like to kind of expand the
                       discussion.  What we’re seeing here with the
                       deadlock over Build Back Better is truly a symptom
                       of a nation that is sharply divided among political
                       and ideological lines.  I mean, the basic
                       requirement of a nation is that there be an
                       underlying consensus.  So, unity and purpose and
                       that’s been severely eroded and the consequences
                       are catastrophic, because you have these kinds of
                       divisions that we have right now. This puts the
                       country on a decline.  We have a hyper-partisanship
                       that doesn’t serve any purpose and it’s not just
                       about one party. It’s about both.  They’ve both
                       been sidetracked by campaign financing laws, which
                       make the government an auction to the highest
                       bidder. You cited corporations, but that’s kinda
                       the way it is.


                # ⚓ ‘Not_True’:_Sanders_Counters_Manchin’s_Bogus_Inflation
                  Argument_Against_Build_Back_Better⠀⇛


                       Sen. Bernie Sanders on Thursday countered fellow
                       Sen. Joe Manchin’s recent Fox News appearance with
                       an op-ed on the right-wing outlet’s website aimed
                       at rebutting the West Virginia Democrat’s
                       falsehood-laden talking points against the Build
                       Back Better Act.


                       “Manchin, the Republicans, and corporate America
                       say this bill will add to our national debt and
                       make inflation worse. Not true,” wrote Sanders (I-
                       Vt.), the chair of the Senate Budget Committee.
                       “Unlike the bloated military budget that Manchin
                       recently voted for, which adds $778 billion to the
                       deficit this year alone and costs four times more
                       than the Build Back Better Act over a 10-year
                       period, the White House has said this bill is fully
                       paid for” with tax hikes on the wealthy.


                # ⚓ Sanders_Takes_Aim_at_Manchin’s_Argument_Against_Build_Back
                  Better:_“Not_True”⠀⇛


                # ⚓ 30_Years_Ago,_the_Soviet Union_Collapsed,_But_Socialism_Was
                  Worth_Saving⠀⇛


                       Yet Soviet socialism had more economic successes
                       than failures. The US-USSR comparison used to
                       demonstrate “capitalism good, socialism bad” was
                       apples-to-oranges, and should never have been
                       accepted.


                       British economist Angus Maddison estimates that in
                       1913 the US’ Gross Domestic Product per capita was
                       the highest in the world at $5,301 (in 1990
                       dollars). That of the Russian Empire—the future
                       USSR—was only $1,488–17th in world, less than
                       Mexico. Sixty years later, the US figure had risen
                       to $16,689 and the USSR’s to $6,059, the USSR
                       narrowing the gap by 23%–a considerable achievement
                       considering the extremely different circumstances
                       faced by each nation.


                # ⚓ Trump’s_True_North:_President_for_Life⠀⇛


                       Trump also publicly asked Russia in 2016 for
                       campaign help, a request he soon dropped. This was
                       no secret conspiracy, no hidden collusion. As his
                       son said, they were too stupid for that. Instead,
                       this was an open pitch for intervention. Which
                       doesn’t mean Russians were so cretinous as to
                       comply in any big way. But promoters of the phony
                       Russiagate claptrap mysteriously lost sight of all
                       that, priming the populace for an election fraud
                       narrative and eroding faith in the electoral
                       process.


                       In his hegira from winner to loser, Trump snapped
                       up whatever weapons were at hand along the way, and
                       this disillusionment, caused by the Russiagate
                       fraud, proved most useful. Trump consiglieri like
                       Sidney Powell and Rudy Giuliani turned it on its
                       head, claiming Venezuela and China tampered with
                       voting machines, helping to steal Trump’s crown,
                       while chief of staff Mark Meadows pursued more
                       arcane charges of Italian meddling via space
                       satellites. The Russiagate ruse turned out to be a
                       can of worms.


                # ⚓ Trump_Thinks_He’s_Still_President:_What_Is_the_Evidence?⠀⇛


                       +The Obama-Biden union card check proposal was not
                       on Mr. Trump’s political horizon, nor is it on that
                       of the current occupant in the White House.


                       +The current occupant is ramping up Trump’s
                       unhinged Sino-phobic hallucinations, sanctioning 34
                       Chinese entities for development of “brain-control
                       weaponry.” Not that the Chinese have been angels.
                       In an egregious suppression of freedom of
                       information, the inscrutable Orientals have made it
                       more difficult for US spies to operate in their
                       country.


                # ⚓ Leftist_Presidential_Candidate’s_Landslide_Promises_Clean
                  Sweep_of_Pinochet’s_Fascist_Legacy⠀⇛


                       Now we have a shining example of what needs to be
                       done, in a country that had its own 9-11, but has
                       finally turned things around.


                       Chile, in fact, was the scene of a far more brutal
                       and deadly 9-11 event that occurred on September
                       11, 1973, when the country’s military, under the
                       direction of a fascist military leader named
                       Augusto Pinochet, in a coup backed if not
                       orchestrated by the US under President Richard
                       Nixon and his then National Security Advisor Henry
                       Kissinger, overthrew the democratically elected and
                       hugely popular Marxist President Salvador Allende
                       Gossens, murdering him in the presidential palace
                       and launching a reign of terror that saw thousands
                       of Allende supporters murdered or disappeared.


                # ⚓ The_Greatest_Honor_of_a_Star:_an_Unforgettable_Goal_Against
                  Pinochet⠀⇛


                       An Unforgettable Goal Against Pinochet


                       Among the images that come to us from September 11,
                       1973, the day of the military coup against Salvador
                       Allende, among so many vivid images, one could
                       rightly be of President Allende resisting in a
                       helmet as a last resort, with some loyal militants
                       at the gates of La Moneda palace. This image speaks
                       of a democratic socialist, who by the strength of
                       the ballot box thought he had the power, who is
                       destroyed in the end, defeated with the greatest
                       eloquence of bombs and crime.


                # ⚓ Leaked_files_expose_Syria_psyops_veteran_astroturfing
                  BreadTube_star_to_counter_Covid_restriction_critics⠀⇛


                # ⚓ Democratic_Progress_in_Honduras,_Set_Backs_in_El_Salvador⠀⇛


                       The two countries have taken a 180-degree turn and
                       are now immersed in very different political
                       trends.


                       During the past decade, El Salvador and Honduras
                       went in opposite directions.


                # ⚓ Suyapa_Portillo_Villeda_on_Honduran_Election⠀⇛


                # ⚓ Wisconsin_Republican_Vote_Fraud⠀⇛


                       They should each spend time in a witness stand
                       explaining how and why they tried to bamboozle
                       America. Criminal charges are possible, but no
                       charges have been filed, yet.


                       Throughout the history of America there have been
                       occasions of lowly in-the-gutter personality types
                       that cheat the American public by rigging an
                       election, but in this particular case 10 low-lifes
                       acted in concert to deceive America. They’re
                       considered pillars of the Wisconsin Republican
                       Party.


          o § Civil Rights/Policing⠀➾


                # ⚓ That_Time_the_FBI_Scrutinized_It’s_a_Wonderful_Life_for
                  Communist_Messaging⠀⇛


                       At the Republican National Convention in 2020, one
                       of the speakers compared Donald Trump to the
                       character of George Bailey in the film It’s a
                       Wonderful Life.


                # ⚓ The_Brutal_Legacy_of_India’s_Farm_Laws:_“Repeal_of_Laws
                  Will_Not_Bring_My_Brother_Back’⠀⇛


                # ⚓ ‘Packing_and_Cracking’:_New_Lawsuit_Challenges_South
                  Carolina’s_Racial_Redistricting⠀⇛


                       Alleging that South Carolina’s Republican-
                       controlled Legislature gerrymandered the state’s
                       House district map to intentionally discriminate
                       against Black voters, civil rights groups on
                       Thursday filed a lawsuit against Gov. Henry
                       McMaster and state legislative and elections
                       leaders to challenge the new redistricting law.


                       “The South Carolina Legislature enacted racially
                       gerrymandered and intentionally dilutive state
                       House districts that minimize Black South
                       Carolinian voting power.”


                # ⚓ The_Massachusetts_Flag_Glorifies_the_Violence_Committed_by
                  Colonizers,_Native_Americans_Want_it_Changed⠀⇛


                       It was a little more than 30 years ago when the
                       Massachusetts Turnpike Authority got rid of its
                       offensive logo that featured a pilgrim’s hat with a
                       Native American arrow shot through the middle of
                       it.


                       The state insignia still stands, and like many
                       campaigns to rid schools and towns of demeaning
                       mascots and public statues, the efforts to remove
                       it starts with an accurate version of history. In
                       the case of the Mass Pike logo, it started with a
                       letter-writing campaign by second graders learning
                       about the history of Plymouth Plantation in 1620.


          o § Monopolies⠀➾


                # ⚓ The_Future_is_in_Interoperability_Not_Big_Tech:_2021_in
                  Review⠀⇛


                       We’ve been here before. EFF’s been fighting tech
                       abuses for 30 years, and we’re used to real tech
                       problems giving rise to nonsensical legal
                       “solutions,” that don’t address the problem – or
                       make it worse. There’s been some of that (okay,
                       there’s been a lot of that).


                       But this year, something new happened: lawmakers,
                       technologists, public interest groups, and
                       regulators around the world converged on an idea
                       we’re very fond of around here: interoperability.


                       There’s a burgeoning, global understanding that the
                       internet doesn’t have to be five giant websites,
                       each filled with text from the other four. Sure,
                       tech platforms have “network effects” on their side
                       – meaning that the more they grow, the more useful
                       they are. Every iPhone app is a reason to buy an
                       iPhone; every person who buys an iPhone is a reason
                       to create a new iPhone app. Likewise, every
                       Facebook user is a reason to join Facebook (in
                       order to socialize with them) and every time
                       someone joins Facebook, they become a reason for
                       more people to join.


                # ⚓ ‘Don’t_Cross_the_Picket_Line’:_Apple_Workers_Organize
                  Christmas_Eve_Walkout⠀⇛


                       A group of Apple employees organized a Christmas
                       Eve walkout, demanding better working conditions
                       and calling on customers to not shop in the tech
                       giant’s retail or online stores.


                       “We are Apple,” Apple Together, the group
                       organizing the walkout, tweeted Thursday. “We
                       deserve a respectful workplace. We deserve paid
                       sick time. We deserve protection on the frontlines.
                       We deserve proper mental healthcare.”


                # § Patents⠀➾


                      # ⚓ Senators_Against_Generic_COVID_Vaccines_Have_Received
                        Twice_as_Much_Pharma_Cash⠀⇛


                      # ⚓ In_Six_Weeks_Ahead_of_Christmas,_Rich_Nations_Snagged
                        More_Vaccines_Than_Africa_Got_All_Year⠀⇛


                             The United States, the United Kingdom, and
                             European Union countries secured more
                             coronavirus vaccine doses in a six-week
                             period before Christmas than the entire
                             continent of Africa received in all of 2021,
                             striking inequity that campaigners say is
                             protracting the deadly pandemic and creating
                             conditions for more variants to emerge.


                             An analysis released Friday by the People’s
                             Vaccine Alliance showed that between November
                             11 and December 21, 2021, the E.U., U.K., and
                             U.S. snagged 513 million vaccine doses as
                             they accelerated their booster-shot campaigns
                             in preparation for the holiday season.


                # § Copyrights⠀➾


                      # ⚓ Nintendo_Wins_High_Court_Injunction_to_Block_Access
                        to_Pirated_Switch_ROMs⠀⇛


                             In an effort to restrict access to pirated
                             ROMs illegally made available for its Switch
                             console, Nintendo has obtained a UK High
                             Court injunction against six internet service
                             providers. Targeted against ROM portals with
                             NSW2U and NSWROM branding, the two-year
                             blocking order requires BT, Virgin, Sky,
                             TalkTalk and others to block the sites after
                             they failed to respond to infringement
                             complaints.


                      # ⚓ “Strike_3″_Filed_Over_1,900_Online_Piracy_Lawsuits_in
                        the_U.S._in_2021⠀⇛


                             Strike 3 Holdings is continuing its quest
                             against alleged BitTorrent pirates in U.S.
                             courts. The adult entertainment company is
                             the only prolific ‘copyright troll’ still
                             left in the country and over the past months
                             has filed over 1,900 lawsuits. Most cases
                             have already been closed, with many being
                             settled outside of court.



                    ䷩ 𝚕𝚒𝚗𝚎 3894

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

⠀⌧ █▇▆▅▄▃▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ 12.25.21⠀▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▃▄▅▆▇█ ⌧


Gemini_version_available_♊︎


✐ Links_25/12/2021:_Ruby_3.1_Released_and_Ubuntu_vs_Arch_Linux⠀✐


Posted in News_Roundup at 2:25 pm by Dr. Roy Schestowitz


🄸🄼🄰🄶🄴 🄳🄴🅂🄲🅁🄸🄿🅃🄸🄾🄽 ⦇GNOME bluefish⦈


§ Contents⠀➾


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


          o § Audiocasts/Shows⠀➾


                # ⚓ What_Is_KDE’s_Weak_Side?_–_Kockatoo_Tube⠀⇛


          o § Kernel Space⠀➾


                # ⚓ Etnaviv_DRM_Driver_Will_Now_Work_On_IOMMU-Enabled_Systems_–
                  Phoronix⠀⇛


                       It’s been a while since last having anything major
                       to report on the Etnaviv Linux driver stack
                       reporting open-source support Vivante graphics IP.
                       With Linux 5.17 is an important step forward with
                       the driver now behaving properly for IOMMU-enabled
                       systems.


                       While an IOMMU is important for security reasons
                       and other purposes, the Etnaviv Direct Rendering
                       Manager driver at present hasn’t properly dealt
                       with the DMA configuration of the virtual device
                       and that running into issues with an IOMMU.


                # ⚓ AMD_P-State_v7_Driver_Posted_For_Delivering_Better_Ryzen
                  Efficiency_In_2022_–_Phoronix⠀⇛


                       AMD on Christmas Eve posted their seventh iteration
                       of the AMD P-State Linux driver as their new CPU
                       frequency scaling solution for Zen 2+ to make use
                       of ACPI CPPC for ultimately striving toward optimal
                       power efficiency with a focus on mobile and desktop
                       systems.


                       This driver was developed in cooperation with Valve
                       and has been out for review since September. Now as
                       we end out the calendar year they are up to the
                       seventh revision of this driver and is appearing
                       that it may be ready for mainline soon.


                       The amd-pstate driver has been evolving nicely and
                       now working across more Zen 2 and Zen 3 systems
                       (older processors won’t be supported due to needing
                       ACPI Collaborative Processor Performance Controls,
                       the CPPC support must also be enabled from the
                       system BIOS). This driver can make more informed
                       CPU frequency scaling decisions than the much more
                       basic ACPI CPUFreq driver currently used by Ryzen
                       and EPYC systems.


          o § Applications⠀➾


                # ⚓ Darktable_3.8_Raw_Photo_Editing_Tool_Is_Out_With_a_Lot_of
                  Changes⠀⇛


                       The open-source photo workflow and RAW editing
                       Darktable has been updated to version 3.8. Here’s
                       what’s new!


                       If you’re sick of paying for photo editing software
                       and are looking for an alternative to Adobe
                       Lightroom, then the new Darktable 3.8 may be for
                       you. It’s a free, open-source RAW photo file editor
                       software for photo editing and file management.
                       Darktable has unique features such as virtual
                       lighttable and darkroom clubbed with complete RAW
                       image editing workflow for complex photography
                       projects.


                       The Darktable team has announced its winter feature
                       release just in time for dealing with any holiday
                       photos, bringing Darktable to version 3.8. Here’s a
                       quick update.


                # ⚓ darktable_3.8.0_released_|_darktable⠀⇛


                       We’re proud to announce the new feature release of
                       darktable, 3.8.0!


                # ⚓ Krita_5.0_released!_|_Krita⠀⇛


                       This is a huge release, with a lot of new features
                       and improvements.


          o § Instructionals/Technical⠀➾


                # ⚓ Make_MIDI_loops_on_Linux_with_this_open_source_tool_|
                  Opensource.com⠀⇛


                       There are many great synthesizers for Linux, and
                       like synthesizers in real life, most of them can be
                       controlled using the MIDI protocol. Digital audio
                       workstations like Ardour, Qtractor, and Rosegarden
                       can send and receive MIDI, and they have a plugin
                       structure that permits loading most synthesizers as
                       a component of the application. I’m a fan of
                       modularity, though, and sometimes a big all-in-one
                       audio workstation is more than what I need. Thanks
                       to the combination of JACK, ALSA, and MIDI, it’s
                       entirely possible to open a synth as a stand-alone
                       application and use a stand-alone sequencer to play
                       and record that synth. There aren’t many modular
                       sequencers out there, but I use Seq24.


                # ⚓ How_To_Find_Installation_Date_and_Time_of_Linux_OS_–
                  Unixcop_the_Unix_/_Linux_the_admins_deams⠀⇛


                       Hi guys, In this tutorial, We will illustrate a few
                       ways to find Linux OS’s installation date and time.


                # ⚓ How_to_Build_My_Version_of_Linux_–_Invidious⠀⇛


                # ⚓ How_To_Fix_Lock_/var/lib/apt/lists/lock_Error_[Solved]_|
                  Itsubuntu.com⠀⇛


                       Ubuntu is one of the most popular and widely used
                       Linux-based operating systems. It is often
                       considered the most preferred operating system
                       among first-time users. In this particular post, we
                       are going to show you the steps to solve the issue
                       “Lock /var/lib/apt/lists/lock Error” while using
                       Ubuntu. It is really frustrating sometimes when you
                       are doing something and you are stuck.


                # ⚓ How_To_Install_Discord_on_Fedora_35_–_idroot⠀⇛


                       In this tutorial, we will show you how to install
                       Discord on Fedora 35. For those of you who didn’t
                       know, Discord is an app for voice calling, video
                       calling, and text messaging like WhatsApp and
                       Telegram. It is a boon to several communities,
                       which helps them to expand their projects, reach
                       out to more people, and maintain a community of
                       fans and followers.


                       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 Discord on a Fedora 35.


                # ⚓ How_to_install_Rstudio_Desktop_or_server_in_Debian_11
                  Bullseye_Linux⠀⇛


                       R is an open-source programming language for
                       statistical analysis and graphics whereas RStudio
                       is an IDE for it. Here we will learn the commands
                       to install RStudio along with R Programming
                       language on Debian 11 Bullseye.


                       With RStudio integrated development environment the
                       user can use the GUI to start coding in the R
                       language. This means that the IDE can only work if
                       you have R already installed. Although the IDE can
                       only function alongside R, the programming language
                       can function on its own. Therefore it is not
                       possible to replace one with the other.


                # ⚓ Snowfall_on_your_Linux_Desktop_this_Christmas_–_TREND
                  OCEANS⠀⇛


                       Let us make this Christmas special for you and more
                       of your Linux desktop. Xsnow is a tiny application
                       that lets you experience cold snowfall just like in
                       winter. To be honest, I am feeling cold while
                       staring at my own desktop screen.


                # ⚓ How_to_Install_and_Configure_MariaDB_10.7_on_Fedora_35_–
                  NextGenTips⠀⇛


                       MariaDB Server is one of the most popular open-
                       source relational databases. It’s made by the
                       original developers of MySQL and guaranteed to stay
                       open source. It is part of most cloud offerings and
                       the default in most Linux distributions.


                       It is built upon the values of performance,
                       stability, and openness, and MariaDB Foundation
                       ensures contributions will be accepted on technical
                       merit. Recent new functionality includes advanced
                       clustering with Galera Cluster 4, compatibility
                       features with Oracle Database and Temporal Data
                       Tables, allowing one to query the data as it stood
                       at any point in the past.


                       Welcome to today’s topic on how to install MariaDB
                       10.08 on Fedora 35 distribution.


                # ⚓ How_to_install_the_SuiteCRM_Customer_Resource_Manager_on
                  Ubuntu_Server_–_TechRepublic⠀⇛


                       As your business grows, you’ll probably find
                       yourself needing to manage customer relations. With
                       such a platform available to your business, your
                       staff can better manage their clients, customers,
                       opportunities, leads and much more.


                # ⚓ Flatcar_Container_Linux_on_Raspberry_Pi_|⠀⇛


                       Flatcar Container Linux recently announced the
                       Stable support for ARM64.


                       Perfectly timed! I had a Raspberry Pi 4 lying
                       around and had just ordered a few more to set up a
                       home lab during the holidays. The newer Pis are yet
                       to arrive, so better utilize the time writing a
                       walkthrough on how to use Flatcar Container Linux
                       on your Pis.


          o § Desktop Environments/WMs⠀➾


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


                      # ⚓ This_week_in_KDE:_Samba_printer_browsing_and_more⠀⇛


                             Keep in mind that this blog only covers the
                             tip of the iceberg! Tons of KDE apps whose
                             development I don’t have time to follow
                             aren’t represented here, and I also don’t
                             mention backend refactoring, improved test
                             coverage, and other changes that are
                             generally not user-facing. If you’re hungry
                             for more, check out https://planet.kde.org/,
                             where you can find blog posts by other KDE
                             contributors detailing the work they’re
                             doing.


                      # ⚓ KDE’s_Bug_Fixes_&_Other_Desktop_Enhancements_For
                        Christmas⠀⇛


                      # ⚓ Krita_5.0_improves_resource_management_and_includes
                        storyboards⠀⇛


                             Krita 5.0 You are here to continue the
                             evolution of this popular drawing application
                             that is also part of KDE. Without making much
                             noise, little by little it has been making a
                             niche in a sector that becomes demanding, and
                             it has also done so by being published as
                             free software.


                             Krita 5.0 is a very powerful release, which
                             is good news because it means that the
                             project is alive and well. When it comes to
                             what’s new, we start with the fact that the
                             way the app handles things like brushes,
                             gradients, palettes, and labeling has been
                             completely revamped.


                             Those responsible have explained that before
                             they used a fragile model system to manage
                             the resources, so Krita uses a SQLite
                             database from this version as the nucleus to
                             carry out this task. As a result, a large
                             number of labeling bugs and user interface
                             issues should have been fixed, as well as
                             allowing faster application startup and a 200
                             megabyte reduction in RAM usage.


                      # ⚓ KDE_e.V._is_looking_for_an_event_organiser_|_KDE
                        e.V.⠀⇛


                             KDE e.V., the non-profit organisation
                             supporting the KDE community, is looking for
                             someone to help organise KDE’s event,
                             including our annual global conference
                             Akademy. Please see the call for proposals
                             for more details about this contracting
                             opportunity.


                # § GNOME Desktop/GTK⠀➾


                      # ⚓ World-famous_GTK_themes_for_Linux_–_LinuxH2O⠀⇛


                             This guide is based on the statistics of GTK
                             themes, the themes that are world-famous
                             among Linux users. The list you’re about to
                             go through is a compilation of the 5 most
                             popular themes. These themes work on all the
                             distributions such as Ubuntu, Linux Mint,
                             Fedora, POP OS, Manjaro, Kali Linux, Zorin
                             OS, Debian, etc.


                             Have you ever wondered about the most popular
                             GTK themes for Linux? The themes, that are
                             famous around the world. The themes that
                             people use no matter where they live. Users
                             can be from India, The United States, Russia,
                             Brazil, France, Spain, Japan, or from some
                             other country.


                             So don’t you wanna know what are the famous
                             Linux GTK themes among them!


                             To answer that curiosity, I have written this
                             article. Below you will find the top 5 GTK
                             themes for Linux which are being used by
                             users around the earth. So let’s get started!


          o § Distributions⠀➾


                # ⚓ Merry_Christmas_from_OSMC_–_OSMC⠀⇛


                       The OSMC team would like to wish you a Merry
                       Christmas. We hope you’re enjoying the festive
                       break and watching lots of TV with OSMC.


                       We hope you and your loved ones are safe.


                       While you may be relaxing, we’ve worked hard on
                       delivering a new OSMC update. We’ve kept the
                       changes light to ensure a stable experience during
                       an important break.


                # § Arch Family⠀➾


                      # ⚓ Ubuntu_vs._Arch_Linux:_Which_Linux_Distro_Should_You
                        Choose?⠀⇛


                             Arch Linux and Ubuntu are both top contenders
                             when it comes to being the best Linux distro.
                             Here’s a detailed comparison of both these
                             OSes.


                             Open-source Linux distros are always fighting
                             it out for the top spot. Nonetheless, there
                             is an ongoing discussion about the clear
                             winner, considering that some of the best
                             distros are aiming to reach the pinnacles of
                             success in the world of open-source
                             distributions.


                             Various surveys have rated Arch Linux and
                             Ubuntu as two of the best distros for getting
                             the job done. Nevertheless, it is not
                             suitable to assume why engineers and coders
                             vary in their views about these two distros.


          o § Devices/Embedded⠀➾


                # § Open Hardware/Modding⠀➾


                      # ⚓ [Old]_KiCad_6_is_coming!_A_comprehensive_review_–
                        Tech_Explorations⠀⇛


                             KiCad is arguably the world’s best open-
                             source printed circuit board CAD program.


                      # ⚓ KiCad_6.0_Release_Imminent_As_A_Leading_Open-Source
                        Circuit_Board_CAD_Program_–_Phoronix⠀⇛


                             KiCad 6.0 has been tagged as a major update
                             to this leading open-source printed circuit
                             board design program.


                             KiCad is easily one of the best open-source/
                             free electronic design automation suites.
                             Among many other uses, KiCad has been used
                             for System76 in designing their open hardware
                             products. KiCad has been in development going
                             on 30 years now and with KiCad 6.0 is another
                             big milestone.


                      # ⚓ DIN_SPEC_3105:_Documenting_open_hardware_in_a
                        standardized_manner_–_Market_Research_Telecast⠀⇛


                             When it comes to software, open source is
                             easy to explain: the source code must be
                             published under a free and open license,
                             because everyone has the right to freely use
                             the code, adapt it and distribute it. This is
                             more difficult with hardware: In order to be
                             able to understand, modify and reproduce
                             hardware practically, you usually need parts
                             lists, 3D models, circuit diagrams, circuit
                             board layout plans and so on – depending on
                             the technology and the targeted target group.


                             [...]


                             At the same time, DIN SPEC 3105 is also the
                             first standard that was and was developed
                             under an open license by the open source
                             community together with the German Institute
                             for Standardization (DIN e.V.) is free to
                             everyone.


                      # ⚓ 10_Raspberry_Pi_project_ideas_from_2021_|
                        Opensource.com⠀⇛


                             The Raspberry Pi is one of the most popular
                             brands of single-board computers. Part of the
                             reason for this popularity is the massive
                             number of unique and interesting projects
                             that people have built using Raspberry Pi.
                             Several Opensource.com authors have used
                             Raspberry Pi in various projects and provided
                             instructions so others can recreate their
                             projects. Here are the ten most popular
                             Raspberry Pi articles published by
                             Opensource.com in 2021. I hope they inspire
                             you as you tinker with your own Raspberry Pi.


                # § Mobile Systems/Mobile Applications⠀➾


                      # ⚓ 10_best_mood_tracker_apps_for_Android_–_Android
                        Authority⠀⇛


                      # ⚓ The_best_beer_apps_for_Android_for_hops_fans_–
                        Android_Authority⠀⇛


                      # ⚓ Best_games_to_play_on_your_new_Android_phone_this
                        Christmas_|_T3⠀⇛


                      # ⚓ Tip:_All_Simple_Mobile_Tools_Android_apps_are_free_on
                        Google_Play_currently_–_gHacks_Tech_News⠀⇛


                      # ⚓ 11_first_things_to_do_on_your_new_Android_phone⠀⇛


                      # ⚓ Android_apps:_Install_these_on_your_new_smartphone_–
                        9to5Google⠀⇛


                      # ⚓ 5_Android_apps_you_shouldn’t_miss_this_week_–_Android
                        Apps_Weekly⠀⇛


                      # ⚓ Best_Material_You_widgets_to_try_on_your_Android_12
                        phone_–_9to5Google⠀⇛


                      # ⚓ Samsung_Android_12_update:_These_devices_have_One_UI
                        4.0_–_9to5Google⠀⇛


                      # ⚓ Oppo_Reno_6_Pro_5G_gets_Android_12_ahead_of_New
                        Year⠀⇛


                      # ⚓ LG_Android_11_update_(LG_UX_10)_list_of_eligible
                        devices_&_release_date⠀⇛


                      # ⚓ Samsung_Galaxy_S20_will_get_Android_12-based_One_UI_4
                        stable_update_next,_no_more_betas_–_GSMArena.com_news⠀⇛


                      # ⚓ How_to_Lock_Apps_On_Your_Android_Device_–_SlashGear⠀⇛


                      # ⚓ How_to_Check_Battery_Health_on_Android⠀⇛


                      # ⚓ How_to_set_up_an_Android_phone:_our_guide_to
                        switching_on_your_new_phone_|_TechRadar⠀⇛


                      # ⚓ How_to_read_deleted_WhatsApp_messages_and_recover
                        them_on_Android⠀⇛


                      # ⚓ How_to_take_a_screenshot_on_most_Android_phones⠀⇛


                      # ⚓ Android_Circuit:_Stunning_Galaxy_S22_Ultra_Design,
                        Honor_Magic_V_Revealed,_Best_Mobile_Android_Games⠀⇛


                      # ⚓ The_Nest_Hub_Max_could_soon_become_the_second_Google
                        smart_display_to_run_Fuchsia⠀⇛


                      # ⚓ Vivaldi_Has_Launched_the_First_Android_Automotive_Web
                        Browser⠀⇛


                      # ⚓ Major_Google_Maps_on_Android_Auto_bug_has_finally
                        been_fixed_|_T3⠀⇛


                      # ⚓ “Android_TV_Remote”_is_an_unofficial_remote_app_for
                        your_Android_TV⠀⇛


                      # ⚓ Every_Android_user_warned_to_set_up_this_lifesaving
                        feature_NOW⠀⇛


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


                # ⚓ Cascade_is_a_new_promising_image_editor⠀⇛


                       Cascade is released under GPL-3.0 License.


                # ⚓ Open_source_advent_calendar:_Android_without_Google_–
                  Market_Research_Telecast⠀⇛


                       This is an advent calendar for techies. In the
                       fully commercialized digital world, almost
                       everything belongs to a large Internet corporation.
                       Their software is neither open nor free. As an
                       alternative, there is this small island of the open
                       source world: software whose code is publicly
                       visible and can be independently checked for
                       possible security gaps and backdoors. Software that
                       can be freely used, distributed and improved. Often
                       the drive for work is simply the joy of providing
                       something useful to society.


                       Short portraits of open source projects will be
                       published on heise online from December 1st to
                       December 24th. These are about the functions of the
                       respective software, the pitfalls, the history, the
                       background and the financing. Some projects are
                       backed by an individual, others by a loosely
                       organized community, a tightly managed foundation
                       with full-time employees or a consortium. The work
                       is entirely voluntary, or it is financed through
                       donations, cooperation with Internet companies,
                       government funding or an open source business
                       model. Regardless of whether it is a single
                       application or a complex ecosystem, whether a PC
                       program, app or operating system – the diversity of
                       open source is overwhelming.


                # ⚓ 4_conscious_steps_to_engage_people_in_your_open_source
                  community⠀⇛


                       Getting people involved in your open source project
                       is important for its health and its growth, but
                       telling people about it isn’t enough. You need to
                       take conscious steps to increase involvement and
                       ensure that initial contributions to a project are
                       followed up, tying people in to the project and
                       making them part of the community.


                       If you find this post interesting, you’ll find a
                       lot more about how community and open source are
                       important in my book Trust in Computer Systems and
                       the Cloud, published by Wiley.


                       1 – I’m going to be honest: I wouldn’t know where
                       to start with TikTok. My kids will probably be
                       appalled that I even mentioned it, but hey, why
                       not? The chances are that you, dear reader, are
                       younger and (almost certainly) cooler than I am.


                       2 – I’m guessing the take-up will be a bit lower
                       here.


                       3 – In Real Life. It seems odd to be re-using this
                       term, which had all but disappeared from what I
                       could tell, but which seems to need to re-
                       popularised.


                # ⚓ CryptPad_is_an_open-source,_free_collaborative_web-based
                  office_system⠀⇛


                       CryptPad is released as an open-source under the
                       AGPLv3 License.


                # ⚓ Waistline_is_a_free,_privacy-first_calorie_counter_and
                  weight_tracker⠀⇛


                       Waistline application and source code are licensed
                       under the GPLv3.


                # ⚓ MintHCM_is_an_open-source_Human_Capital_Management_for
                  enterprise⠀⇛


                       Human Capital Management or HCM for short, stands
                       for human resources and employee workflow
                       management for the enterprise.


                       MintHCM is an open-source human resources
                       management system for enterprises.


                       [...]


                       MintHCM is released under terms of the GNU Affero
                       General Public License Version 3.


                # ⚓ Please_use_me_as_a_resource⠀⇛


                       Some of these ideas stick, and many readers have
                       attempted to put them into practice, taking on
                       challenges like starting a business in FOSS or
                       stepping up to be leaders in their communities. It
                       makes me proud to see the difference you’re making,
                       and I’m honored to have inspired many of you.


                # § Programming/Development⠀➾


                      # ⚓ Ruby_3.1.0_Released⠀⇛


                             We are pleased to announce the release of
                             Ruby 3.1.0. Ruby 3.1 keeps compatibility with
                             Ruby 3.0 and also adds many features.


                             [...]


                             Since Ruby 2.6 introduced MJIT in 2018, its
                             performance greatly improved, and finally we
                             achieved Ruby3x3 last year. But even though
                             Optcarrot has shown impressive speedups, the
                             JIT hasn’t benefited real world business
                             applications.


                             Recently Shopify contributed many Ruby
                             improvements to speed up their Rails
                             application. YJIT is an important
                             contribution, and aims to improve the
                             performance of Rails applications.


                             Though MJIT is a method-based JIT compiler
                             and uses an external C compiler, YJIT uses
                             Basic Block Versioning and includes JIT
                             compiler inside it. With Lazy Basic Block
                             Versioning (LBBV) it first compiles the
                             beginning of a method, and incrementally
                             compiles the rest when the type of arguments
                             and variables are dynamically determined. See
                             YJIT: a basic block versioning JIT compiler
                             for CRuby for a detailed introduction.


                             With this technology, YJIT achieves both fast
                             warmup time and performance improvements on
                             most real-world software, up to 22% on
                             railsbench, 39% on liquid-render.


                      # ⚓ Ruby_3.1_Released_With_Speedy,_Experimental_In-
                        Process_JIT_Compiler⠀⇛


                             Ruby 3.1 is out this Christmas day as a big
                             feature release for this programming language
                             while retaining compatibility with Ruby 3.0.


                             Most exciting with Ruby 3.1 is “YJIT” as a
                             new, experimental in-process just-in-time
                             (JIT) Compiler. Ruby 3.1′s new JIT compiler
                             was developed by Spotify for improving the
                             performance of their Ruby on Rails
                             application. YJIT should be very beneficial
                             in improving Rails real-world application
                             performance.


                      # ⚓ The_10_Best_IDEs_and_Code_Editors_for_Linux [Ed:
                        Starting this list with Microsoft proprietary software
                        that spies on everybody is poor taste and
                        misunderstanding of GNU/Linux]⠀⇛


                             A code editor provides a text editing
                             interface to write and edit code. It offers
                             basic editing and formatting features to
                             enhance the readability and management of
                             your code. These apps usually lack advanced
                             features and configurations.


                             A notable benefit of using a code editor is
                             that it is much lighter than an IDE. Many
                             code editors also provide features like
                             compiling, debugging, and support for most
                             programming languages.


                             An IDE is a software development environment
                             that provides tools to develop complex
                             software. It offers various tools like an
                             editor, debugger, and compiler needed in the
                             development cycle.


                             Different IDEs provide different features and
                             configurations suitable for specific
                             projects. IDEs are much more advanced than
                             editors and offer features you won’t find in
                             a code editor.


                      # ⚓ Paul_E._Mc_Kenney:_Parallel_Programming:_December
                        2021_Update⠀⇛


                             It is past time for another release of Is
                             Parallel Programming Hard, And, If So, What
                             Can You Do About It?. But first, what is the
                             difference between an edition and a release?


                             The main difference is the level of
                             validation. For example, during the several
                             months leading up to the second edition, I
                             read the entire book, fixing issues as I
                             found them. So where an edition is a
                             completed work, a release is primarily for
                             the benefit of people who would like to see a
                             recent snapshot of this book, but without
                             having to install and run LaTeX and its
                             dependencies.


                      # ⚓ Upgrade_Node.js_for_your_Ghost⠀⇛


                             When upgrading Node.js, you need to run the
                             update for Node and then also re-install
                             Ghost’s dependencies. This is because Ghost
                             has several binary dependencies which are
                             compiled for the specific Node.js version.
                             Without reinstalling dependencies, Ghost will
                             fail to start with strange error messages.


                      # § Perl/Raku⠀➾


                            # ⚓ Raku_Advent_Calendar:_Day_25_–_Future-proofing
                              the_Raku_Programming_Language⠀⇛


                                   Around this time last year, Jonathan
                                   Worthington was writing their Advent
                                   Post called Reminiscence, refinement,
                                   revolution. Today, yours truly finds
                                   themselves writing a similar blog post
                                   after what can only be called a
                                   peculiar year in the world.


    * § Leftovers⠀➾


          o ⚓ Novelty_Christmas_Songs_You_Definitely_Need_Under_Your_Christmas
            Tree⠀⇛


                 It’s that time of year again, folks! As the weather
                 changes and the calendar pages start falling off rapidly,
                 another holiday season is upon us. In 2019, we embarked
                 on a somewhat ambitious project: The Twelve Things of
                 Tedium. Ernie, Andrew, and I all contributed four items
                 apiece to create a wonderful holiday project of which
                 we’re all quite proud. The interdisciplinary yuletide
                 extravaganza was fun, but was conspicuously absent in
                 2020 (we talked about The Gingerbread Man instead). With
                 2021 wrapping up as yet another unspeakable nightmare, we
                 thought it might be a good time to revisit one of our
                 favorite topics: Christmas novelty songs. So pour
                 yourself a glass of eggnog, gather around the holiday
                 ornamentation of your choice, and crank up the volume as
                 we welcome you, dear readers, to the Tedium Christmas
                 Novelty Song Spectacular.


          o § Health/Nutrition/Agriculture⠀➾


                # ⚓ Eli_Lilly_finds_embarrassment_main_cause_of_underreporting
                  of_symptoms_in_ulcerative_colitis⠀⇛


                       A new CONFIDE (Communicating Needs and Features of
                       IBD Experiences) study commissioned by Eli Lilly
                       reveals that even though bowel urgency ranked
                       second as a symptom of moderate to severe active
                       ulcerative colitis (UC), more than 60% of people
                       surveyed didn’t tell their doctors because they
                       were too embarrassed.


          o § Finance⠀➾


                # ⚓ How_to_Buy_Monero_in_Australia_–_Tech_Guide⠀⇛


                       Monero is an open-source, decentralized
                       cryptocurrency created in April 2014 that focuses
                       on privacy and decentralization. It runs on
                       Windows, macOS, Linux, Android, and FreeBSD.


          o § Civil Rights/Policing⠀➾


                # ⚓ Winter_Session_2021:_A_Digital_Rights_Review⠀⇛


                       The Winter session of Parliament saw frequent
                       disruptions over the suspension of 12 members of
                       Rajya Sabha, demanding the resignation of the
                       Minister over the Lakhimpur Kheri incident and
                       later the Election Laws (Amendment) Bill, 2021. In
                       this post, we take a look at these disruptions and
                       analyse how the Parliament fared with respect to
                       digital rights.


                       A look back


                       The winter session of the Indian Parliament this
                       year was quite extraordinary. Once again, we saw
                       several days washed out, as the protests about the
                       suspension of 12 Rajya Sabha members over their
                       misconduct on the last day of the monsoon session
                       and the Lakhimpur Kheri incident where 8 farmers
                       lost their lives led to multiple adjournments of
                       both the houses.


                       The Winter Session saw 12 Bills, including the Farm
                       Laws Repeal Bill, 2021 and the Election Laws
                       (Amendment) Bill, 2021. The Farm Laws Repeal Bill,
                       2021 was passed on the first day itself in both
                       Houses in less than 10 minutes without discussion.
                       The Election Laws (Amendment) Bill, 2021 was passed
                       within three hours of its introduction in Lok Sabha
                       and within two hours of its introduction in Rajya
                       Sabha. Further disruptions continued, however, with
                       the completion of “essential government business”
                       with the passing of the Election Laws (Amendment)
                       Bill, 2021, the house was adjourned sine die a day
                       ahead of schedule.


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