%%%==%%==%==%%%%%%%%%%%%%%%%%%%%&&&&&&&&&&&&&&&&&&%%?:^.'..,^:;!**)+)+*))++*!~-:!**;*!!!*******!;;!)>%=&%?=%=&&&&&&%%&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&$$$$$$
%=======%%====%%%%%%%%%%%%%%%%%%%%%%%&&&&&&&&&&&%%>)^,.'..,~;!))))*+*)***));-~-;!;:*;;!!!********!!;!))?>+>??==?>?===?=%%%&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&$$$
%==?=%%=%===%%%%==%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%?+)^.''.,^~:!*+>>))***!!**!--:!;:-::;;;**;;!!**))))*;:-----::;;)>>=?*!;!*)>%&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&$$
%=%=%%%===%%%%%%=%%%%%%%%%%%%%%%%%%%%%%%%%%%%===>*,'''.,^-;***;!*))!;*;!*!!:;*;:--::::--:;!!*)))++))*;~^^^,^^^-;!*;~-;;!*)))=&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&$
%==%%%%===%==%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%?=+!,'''',~-;*)*!;:-:!*;!;!**)*)*!;;!:~~~-;;!**))+))))))*!:~~^~----;~^~:;!!*+>>+%%&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
===%%%%%=====%=%%%%%%%%%%%%%%%%%%%%%%%%%%%%=++;^.''.,^~----:;;;----:!;!!)>>+>)**!-~^~-:-:::!!!**)*))*))*:---;;;:-,~~-:;)+>>>++?=%&&%%%&&&&&&&&&&&&&&&&&&&&&&&&&&
===%%%%%===%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%==!:,''',~:;!!!!;-~^~::~-:!!!)>???>>+*;-~^~~~^^~-:;!!!!*))+**!!!;:;:::~^~~--!)+++>+))??%%&&&%%&&&&&&&&&&&&&&&&&&&&&&&&
==%%=%%======%%%%%%%%%%%%%%%%%%%%%%%%%%%%>!-.''',-;!)))+))!*;-~-::-;*)++????>>+);:::-~~~-::;;!!*****!!**!;;;;;!;^^^^~-**))++)))?+>=%&&&&%&&&&&&&&&&&&&&&&&&&&&&&
=============%%%%%%=%%%%%%%%%%%%%%%%%%%?+!-,''.~;*+?====?>>++);;*!!!*++>???>>>+)!!:-~^^^^^~~---::;;;!;;!****)**:~~,~,:;!;***!*)+>)>?=%&&%%&&&&&&&&&&&&&&&&&&&&&&
===========?%%%%%%%%%%%%%%%%%%%%%%%%%%>>!-.`'.~;)>=%&%%==%==??>+)>+++>>???????>>>)*!;;:-~~~~~--::::;;!))+>=?+)*;-~~^,----!*)!)***;>??=&&&%&&&&&&&&&&&&&&&&&&&&&&
%======????==%%%%%%%%%%%%%%%%%%%%%%%=??+;'''.~!>?=%%%%&&%%%&%===>?=??>>?====??>+)!!::-----------::;*)+>>?=?>))**;--^^~^~~**!***!!!+?=>?%&%&&&&&&&&&&&&&&&&&&&&&&
%%=====????===%==%%%%%%%%%%%%%%%%%%=?>>;''''^!>==%%==%&%&&&%&&%%==%=???=?==??>>>+)*!;:----::;;!*)+>>>>????>>+)*!*;;^~^,^^!::!*);:;!+?)+=&&&&&&&&&&&&&&&&&&&&&&&&
%%%=?????>?========%=%%%%%%%%%%%%%%?>?;.'''^*?%&&$&&%%%&%%$$&&&&%%%=?>>????????>>>+)*!;!!*****)))))+>?>>>>+++)*;!!;::~,^^;::**;;;;:!)*!>%&&&&&&&&&&&&&&&&&&&&&&&
%%%====?>>??=========%%%%%%%%%%%%%=?>!,'`'^!?%$$$&&$$&&&&&&$$&&&%==>++>>>>>>+)+>>+++))))!!!!!**)+>+>>>>>>>>>>**;!**!!~^,~::;;;!!;;!!**!?&&&&&&&&&&&&&&&&&&&&&&&&
%%%%==?=>??=?==????==%%%%%%%%%%%%%?>*^'``,;?%&$$$$$$&&$&$$&$&%==?>+))+>>>>>)*!;!+>)!!********))))>>>>+++>++>)*!;!*))*-~^;***;:--;;****;?%%%&&&&&&&&&&&&&&&&&&&&&
%%====????>??????>>?==%%%%%%%%%%%=?*;.'`.->%&$$######$$$$&&%=>+)*;::;*+???>+)!-~:**!;:;!*!!!!!******))++++>**!;;**)++::-!+*:^,^~-;!!!!+=&%%=&&&&&&&&&&&&&&&&&&&&
%======??????=====?===%%%%%%%%%%%?>;-'`'^)=&&$#######$$&%=?+*!:-~^^^^-!?=%=?)*;~^~;*!!!;:;!!*!!!!**!****)*;!!;;!**)>>!!:))*!;;*))*!;;+>??%%%=&&&&&&&&&&&&&&&&&&&
%%====????>?=====%%==%%%%%%%%%%%%?;*.'`.;?%%&&$######$%=>>*;-~~^^,,,^:!!+?==?+;:^,^-;!!!;;;;!****!*))**!;!*!::!!)+>>>+)*+???==??%?);:*+>>?%=%=&&&&&&&&&&&&&&&&&&
%%=====??=>??===%%%%==%%%%%%%%%%=+*:'`'^+?=%=%&$###$%?>))!:-~^^^,,^;;^,^;+?=?>+:-,,~--::;;;;:;;!!!!;;!;**!;:!!!)>++??>>++????=++?>?)::*+>?%===&&&&&&&&&&&&&&&&&&
%%==?===??>>??===%%==%%%%%%%%%%%?**~``.;>>%=>>=&##&=>)*!+;--~^^^^~!^,,,,,-+=%>))-~,^-::---::-:::;;;;!!*;:;!!!*+>+++>=??>+>>>??*)+++);:;)+>=%=%%&&&&&&&&&&&&&&&&&
%%====????))??=======%%%%%%%%%%==*:,`'^*+?=+*!>&$$=+*!;;);--~~^^::^,,,...,:>=%>*!-~,~;*)*;:-:::::;;;:::;!*!!+>>>>>))==?>+)**+)!**++!----:;+>>%%&&&&&&&&&&&&&&&&&
%====????>)+?===?===%%%%%%%%%%%==;-.`'~)+%>;-:?&&%>*!;;+!;--~~~;~~^^^,..'.,*?%%>*:-^^-;)++))*!!!!;!!***)**+?==???>**?==?>+!;!::;!)+*:;----:*+??&&&&&&&&&&&&&&&&&
=%=>????>>?>>???????==%=%%%%%%%=>;~'`':*>=*:^:=&&?=)!!;>*;:--;;~~~~^^^,.'..-+=%=)*:~~~-;)>>>+++))))))))+?=%%=????+*!>==??>)*;-~:*)++!;!:---:)=>=%&&&&&&&&&&&&&&&
%%====>?==??+?=??>>>??==%%%%%%%??;^`'.:*??;-.!%&&=>>)))>+)**!----~~~^^,,...^!?%%?)!:~~-:!!*+?????????=?======???>!!!)====?+!-^~:!))++)*;::::-+%%=%&&&&&&&&&&&&&&
%===%=??===??>?????>?=??%%%%%%%=?;.`',;)>>;~.)%&&=??>+)***!;;:::--~~^^^,,..,-+=%%>**--~:;!!;!)+>+>>>>>??>?????>+*)!!!?===>*-,^^:!*+)+++)*-::-:?=&=&%%&&&&&&&&&&&
%===%???==???++)+>?=???==%%%%%%=)*'`',;)>+:^,+%&&=??>+)))*!!;;::--~~~^^^,...~!?%%=+*!--:!!*!!!**)))++>>>>+>>>)*))*;!!>+>+!;:^,^-!)+))+!*))-;-~>?%=%%&&&&&&&&&&&&
%%%%%=?=?==>+;;*>>>??=?==%%%%%%?>*'`',;*>):,^>%&%=??>++))**!;;::--~~~^^,,''',:+=%%?)*:::*+*!*;;!***)))+>++)**)+)*!*!;+>++)!;:^~:!)>**!;:))!;:,;?%?%%%%%&&&&%&&&&
%%%%%=??=??>+!;:*++==?>==%%%%%%?=:`'',;*+):.-?&&%=??>++))**!;;::--~~~^^,.''''~!>=%=>**:;*+?>***!!******!**))++****)!!=?>>++*;-~-!)**)!!;+)*!:^^)%?&%%%%%%%&&&&&&
%%%%%%=??=??>+)*!>+>=??===%%%%=&=^`'',:!+*-';=&&%=??>>++))*!!;::--~~~^^,.```'^;)?%=?)*;;)>===?)****!***)*))))*)*++);*%%==??+*:--;:!!*!**+)*!;~^:?%%=%%%%&%&&&&&&
%%%%%%=%===??>>))+++?=?===%%%%=%)~''',:!+)-')%&&%=??>>>+))**!;:---~~~^^,.```'^-*>==>)*!;*+=&&%%=?+)))***)))))))>?>!;)%&%%=?+*;~-~;;**!)>***!;-^^+&&=&%&%%%%&&&&&
%%%%%%%====?=?>))+>+?===%%%%%=%?!.'''.-!+)~'>&&&&%=??>>>++)**!;:--~~~^^,.````,-;)?=?))*!!)>%&$$$&%=????>>>+>?==?>*;!>&&&%?>>);:^--!!!!>>*!;!;:~,*%%=&%%%%&%&&&&&
%%%%%%====???=?++>)+?===%%%%%=%+*..''.-;+)^,%$$$&%%==??>>>+))*!!!;!!!***!:^,.,~:!>=?+***!*)?%&$$$$$$&&&%%&&&%==>*;!*?$&&%%=>+);~~:;::!?)*:;!!:~,;=%=%%%%&&&&&&&&
%%%%%=====>>???>>>*>====%%%%%??>~',''.-;++^*####&&%%%===??>>>>>?=%&$$$$$&%%?+)!:;)??+)!*!!)>?=%%&&$&&&&&&%%%=?)*!!)*%&$&%=??>+;~^::--*+!:!***!^,:==%%%&&&%%&&&&&
%%%%%====??>???++>*>??===%%=%?>>..~''.-;++;$@@@@#$$&&%%=====%&&$######$&%=?>>>??++??+***!;*+>?????==????>?>>)*)))+))%&&%%%%=>)*!!:~~-)*!)++))!,,:==%%%%&&&&&&&&&
%%%%%=====>=?==>++*>?===%%%==?>;^.^`'.~:)+!&#@@@@###$$&&%%%&$$$########$&=>)!;;;!+?=>*!!!;;*+?????>+>>++)***))*>=>)+&$&&&%=??>>>;-~~!++>?=>*+!,,!%=%%%%%&&&&&&&&
%%%%%==========>*!>???===%%==+=-^~.`'.~:*)^?##########$&&&&$$$$$###@@@@@@#$%>)*!!)?=>*!!!::!)>>>>?==?++)*)**)+=%=>++%$&%%%%%==>*):;:=====)*)+:,,+=%%%%%&%%%&&&&&
%%%%%%=========>!;???====%===>+-,-.`'.~;**,)#@@@@@####$&%%&$$$$###@@@@@@#$#@#&=>++?=?*;;!:::*>?>+++))*))*)+?=%&&=?+*&$&&&&%%%???))!>===?**+)>,,-%=%%%%%%&%%&&&&&
%%%==%===%=====>:*?==========+*:,;.`'.~;!*^*@@@@@@@@###&===%&$$##@@@@@@@#@&&#@$=?>?=?);;!;:-;>===?????>+)*+&$$&&%=+)$$$&&&&%=%=>?>+>%==>!*))^,^)=%%%%%%&&%&&&&&&
%%%=%%=========*!>>=========?+!:,:.`.,^;;!-*#@@@@@@@@##%>++?=&$$#@@@@#@##$#@@#$%?>?==+;;;!---)=%%%&&=>>>>>))$$$&&?*>$$$$&&&&&%===>==?>+>**;,,^:?=%%%%%&&%%&%&&&&
%%%%%%=======?>:>>>>========>>;-~-'`.,~;:;^-&@@#@@@@@@#>*;:*>=%&&#@##&%&=?%&%=?>+)+??+!::!:--!?%%%%=??%%=%%!=$$&=+;=$$$$$$$&&%&%=%%=?))):-~~-:>==%%%%%%%%%%&&&&&
%%%%%=======?+!)>>>>?======?>>*;-^`'.,-;:-.`,&@#@@@@@@$*:,.:!*)?=&$$&%===%%?):-~^^-)?>*::;!--:+=%%%=?%=?+>%++&&%?!!%$$$$$$&&&&&&&%%=?)++)!;::>%==%%%%%%%%%%&&&&&
%%%%%=====>+)!))>>>>?======?>+)!~.`..^-::-.`.>#@@#@@@@?;~'.^~^,*?%&$$&&%%=?):^.'`'.;+?);-;!:--)?%%%==%?+!;=>+&%?*))%$$$$&&&$&&&&&%==+++*;-;)=%%==%%%%%%%%%%%&&&&
%%%%====)!;;;;*+>>>>?=====?=>>+*,'`..^-::~,`';%#######*-,'...',;)>?=%%=?>)!:~,'```'-*++!;:!;--!>=%%%&%?+!:++)?>>>)>%&&&&&&&&&&&&%%=>!*!:!)?%%%%==%%%%%%%%%%%%%&&
%%%====!*)+**!*)+>>>======?=>>>*,''..^-:-^,`.*%######=;~.'..'',-;**)))*!;:-~^.'``',~;*>>:~:!:-!)+?%%##$%):*)+>??>>>=%&&%&&&&&&&&%?>)!;:;)+%%%%%=?%%%%%%%%%%%%%&&
%%%===>)>>+>+)*)+++?======?=>>+).''.,~-:-^^`~>&$$$$#$)-^`'.''',~-:;;!!;;:-~^^..''.^~-)>);-:!!:!*!!>%$$#$?;!!?=%%%>>%%&%&%&&&%&&%?+**-:~:!>?%%%=?%%%%%%%%%%%%%%&&
%%%===+?=>?>+++>++>=======?=?>>*.'..,~-::~^`;=$$$$$$=;~.'.....^--:::;;:::-~^^,.,,^^~;:*)!-:**;!!;;*%=%&$?!;*%&&&%>>?=%%%%&%%%%=??+!-~:~!)*>%%=?=%%%%%%%%%%%%%%&&
%%%%==+>??>>>++++>?=======?==>>:.'..,~~::~,')%&$$$$%:~^^,,,^^,~-:::::::::--~~^^^^^~;~-;**;;!)*!!!;!=?+>?)!;=#$$&?+++>?=?=%%%=%=?>*:-^--*???+==?%%%%%%%%%%%%%%%%&
%%%====?>?+>>+>)>?========?==>*!.....~~:--.,+&&&&$&;-~,,,^^~~--::::;;;::::----~~~~:^^~;**!!*)**!!*!))*!:;!)#$$&%?+)))+??+>?????>!;::^:)*??=?+?%%%%%%%%%%%%%%%%&&
%=======?+*>+++*>?=========?=?!*.....^~-:-.,>&&&&$+);-..^^^~-:;;;;;;;;;;;::-------^^^-;))*!)***!*;~,,^-!!!&$$&%=>)*!!))++*))+>)::!:~-!>???==?=%%%%%%%%%%%%%%%%%&
%========++++++*)>=========??=**,...',~~::^.)&$$$$>%?)*;-^^,^:!!!!!!!!;;;;;:::::-^~~~-;)+)******!~^---::*%$$$&%?+*;::---::;;!:-:!:~~:*>>==%%%=%%%%%%%%%%%%%%%&&&
%%======?)>>++))*))=========??**^.....~^::-.;%$$$$>&$&%=>!-^^-***!!!!!!!!;!;;;;;~~---;*>??)*!!!!;::--~-*&$$$$&=>)!-^^,,.,^~~~~--~~-:)>>%%%%%%?%%%%%%%%%%%%%%%%&&
%%======>*!-;!***;*>=========>)!~....',^::-,~=$$$$$&$$$##$?*:!***!!!!!!!!!!!!;*:--::!)=$$?**!!!*!;:-:!?$$$$$&%>+!:~^~^^.,~~:-;:!;*:)>=%%%%%%%?%%%%%%%%%%%%%%%%%&
%%======?**)+*:;:;)>==========*!:....'.^~;:^.+&$$######$###&?*;!!!!!!********)!;;;*)>%@@$>*!;!*+++)+?$#$$&&&%?>!:~~~;:,,--;!;;;*)+)?=%%%%%%%==%%%%%%%%%%%%%%%%%&
%%=======?>>>*)**~:)?=========>!!..,..'.^:;-.:=&$$$#$#@#%??)-~--:;;!!!*****))+**)+=$$@@#&+!;;;+%?%&&$##$$&%%=>!:-~--:-.-!*)++);)>=?%===%%%%=?%%%%%%%%%%%%%%%%%%&
%%==========>))*;~~:)=========?**^.,..'',~!;,,+%&&&&&$%&!,..,^~~--:;;!!**)))?++?%#&@@@@$=!;:;!?$=%&&$#$$&&%=+*;;~-!;;-.!!*+??>*>?=%%%=======%%%%%%%%%%%%%%%%%%%&
%%==========>>))^;;*)==========));..,..'.^:!~'-+=??>?!;;~.'''.,^~--:;;!**)+>?%$@$&@@@@#$);;:;!=#%%%&$$$&%%=>)!;-^:*)+-,*>)?==?>==%%%%%%%==%%%%%%%%%%%%%%%%%%%%%%
%%==========?>>*~!)+)==========>>!^.^,.'..~!;..:)!;>&%?+);^```.^~--:;;!!*)+?$@@@$#@@@#$=!;;;;*%#===&$&&&=?>)*;-~^!>=+!.*>>?%%?>=%=======%%%%%%%%%%%%%%%%%%%%%%%&
%============?>*:*??>?=========?+>:,,^,...,:!~.,-~;$##$%=?+!^'',~--:;;!!*)+%$@@#@@@@#@$);;;;;+&@???&&&%=?>)!;-~,;*==?>,-+===%??=%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%&
%=============?)!)??>?==========>>>~.^^,...~;;.',^)$#####$&%?!~^~-::;;!!!*+#@@@@##@&@=)!;;;;;+&#>+>%%==>+*!:-^^~!+=%=?!^~+?==??==%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%&
%=============;;*>???===========??>!,.~^,..,-!^''^)#@@@@@@##$%>!!;;;;!!!**>#@#@@#@##=>:!!!!!*=&$**+>>++)!:--~^-~*?%%%=+*^-)?=?=??%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%&
%============+-*+>>>??==========???+;,,-^...^::'`~?$$#@@@@@@@##$%%>*!!!**)&#&#@@#@>%!-;!!!!*>%&%!!**!;:;;:-~^:!!*%%%%==>)~;++=?=??%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%============):))+>??============???+;,,~^,.,~*;)>)!)>&#@#$&%%&##$&>)***)+&#$%#@@$:~~:;!!!*++%%?;;!:::;--~~~!*++)=%%%===>>~)+=?%=?=%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%============?~)*)>?==============???+-.~~^-*++)))+)?&#&+*!;--:::;)++))))>$#&)=#+:~-:;!!!*)+)%%+:;;~~~~~^^:)))>?==?=%==?==!:+===%=?%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%=============>!))>??===============>>!-:+>>>+>>=%$###=+)))))*****)++++++?$#+:-~~:::;!***))))&&)::;~^~~~-***++?%==%====?>?;:>===%=?=%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%==============?**>??==============>+*)>+++?=&$$##+?#$&%%%=??===?>>>>++++=%+:~-:;;!!*)>)>+*!)%&*-::::;;))*)>+?%%%=%%%%==?=*;=%%%====%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%===============?*+??==========?++*)+)+>?%$$###?!?),%####$$$##$&=?>>>>>+!--:;;;!!*)+>???>)!!>%$!---;;))*)>>?==%%%=%%%%%%=>))%%%%%%==%%%%%%%%%%%%%%%%%%%%%%%%%%%&
%===============?+)?========>+))+)*)+?%&####>:^^,>=~,+$##@@@@#&=?>>>>+*;!!!!!!**)+??==%?)*!;==&!~--!)+++>?==?%%%==%%%%%%%+;?%%%%%%==%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%================?)=====?++*+)))+)?%&$#@#=:...~^,+=?.:&#@@##&=?>>>++*****!!**)+>?=%&$&=?)!;!$%&;~~-;=========%%%%=%%%%%%%)!%%%%%%%=%%%%%%%%%%%%%%%%%%%%%%%%%%%&&
==================+??++)++)+)>=%%$###$&=)~^'''~^^+==;.=#@@$=+)*!!;!*******)+>?==%&###&?>*!;*$=%!~~~-==%==%==%%%%?%%%%%%%=*)?%%%%%%=%%%%%%%%%%%%%%%%%%%%%%%%%%%%&
==================>*?&=+++==&$$###$&===>;^,''.~,~+==?'>###%+!;:;;!!***))+>>?=%&$####@%?+*;;)%==*~~~-+=====%%%%%%?%%%%%%%%%?+?%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%&
==================>;:?%=%&$##@#&%=====?*:^.'',^,->===-*##$%>*;;;;!*)++>?==%%$$@@@@@#@%>)*;:>&%=)~~~~;?==%%%%%%%==%%%%%%%%%==)?=%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
==================#=)*###@@$&%========+*~^'''^,^:?===>-$###=+)***)+>?==%%&##@&@#@@@##=+*!;:>&$>+~~^~~>%%%%%%%%%?%%%%%%%%%%%=?)?=%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
===============???$@%&@#$%===========?)-~,''.^.~*=====!>$##$%=????=%%%&$#@####@#@@#@$?+*!;;>##>?~^^^~;%%%%%%%%=%%%%%%%%%%%%%=>+==%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
===============????%%%=??????=======?>;-~''',,.:>======!)&#$$&%%%%%=&)##@@##$$###@%&#>)!!;!>##+%~^^^^~+%%%%%%%=%%%%%%%%%%%%%%=>>=%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
==============??============?======??;~:..'.,.,*========?*)>>??==?:&&~#@@@@@##$$##$#$+*!!;*=$#+$~^^^^^~?%%%%%%=%%%%%%%%%%%%%%%=>==%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
=============???===================?!-:.,'....:+=================?-&&^@@@@@@$%$&#$@#%)*!!;+%#$*#:^,,,,^:=%%%%%%%%%%%%%%%%%%%%%==?%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
================?=================?)-;,^'..'.^*===================!>#:$@@@@@$%$&#$@@=)*!;;)&##)#*^,,,,,,;=%%%%%%%%%%%%%%%%%%%%%%=%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
===============?=?===============??;:,^''.'',!>===================>;#&>#@@@@$%###@$@?)*;;;)&##+#>^^,,,,,,!=%%%%%%%%%%%%%%%%%%=%%==%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
=============?????==?==?========??*:,~'..''.;>?===================?:$#>&@@@@#$&$$#@#>)!;;:+%##+#%^^,..,,,,!=%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
=============??=????===?=======??+;,~''.''.-)=?==================??:%##%$$####&&$##$>*!;;:>$##)##:,,`'...,,:=%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
===========???===??=?????=======+;^-.'.''.^*?=?===================?;>###%@@@@%?&$$#$+*!;;:&###*$@).```'....,-?%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
===========????==???????=======>!-!,`....^*?==?====================)*$#@+#####$?=#@&>)!;::$###)$@+,'```'.....^+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
=========??=????====?=?=======?*-+-'....,!?===?====================?:&@#&=@@@#$>>#@&+)!;::####)&#=~,'```'.....,;=%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
=======????????=???????=======>:+),'..,.!>==???=??================??:%@@=+@@@>+==##&+)!;;;&###+&@+),,.```'.....,~>%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
========???????????????=====??**?;...^':>==????===================??:=#@#$@@##$&&##&>)*;;;)@##>&@##:,,,''........,*=%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
=======????????????????=====??*>>-',~.^+==?????==================???^>$@@#$######$#$>)*!;!$@##?=@##%^^,,,,........,->%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
=======????????=???========???+>+^.^^.+?=??????==================??:`)&@@@#$@@@@%#@&>)*!;*&###&+##%@)^^,,,.........,^!=%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
======????????=????=??=??=????>+)^,^.->???????==================??!`.!&#@@@?$@@#?$#&>)*!;*&$$##)=#$#$:^^,,,.........,,~)=%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
=======??????=?????=??????????=+)^^^.+>?==????================???:`.*:%#@@@=$@@#&$@&>)*!;)#&$##)>@#$$%~,^,,,.........,,,-+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
========??????????=????===??????)^~,,>?>?=??>================??>~``)>-%$@@@&$@@#$$@&>)*!;!$?$@$>+$@$$$%-^^,,,,,........,,^:>%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
========?=???????==???===?===?=?>-~^,)?????>=============?????).``-&+^%&@@@$=@@#&&@&>)*;:;=+&#$&*>@#%#$%:^^,,,,,........,,,^:+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
======???=??????????????=?===?==?*^-,~>>?>>?============????>-```,%$+^%&@#@@&####$@%+*!;:;=+##$$*)@@=&##$*^^,,,,,,,,,....,.,^~:+=%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
======????????????????????===?????:-~.-*>>??===?=====??????!.```,?$$*^%%##@@@#####@=+*!;:;%)#$$$>)=@$%$#$#?-,^^,,,,,,,,,,,,,,^^~:*?%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
======????????????????????====????>^-.,~!?==????????????>;.`'-!-=$#>+^=%#@#@@######?)*;:-:&*#$%$$!+@@$&$@$@$*~^^^^^,,,,,,,...,,,,^~:)?%%%%%%%%%%%%%%%%%%%%%%%%%%
==========??????????????=??=====?=?*~-.,-;+?=?????????+:'`.:>>)%$##>;-==$#@@@@@####?)!;:-:)+$?+##!+&@#$&#@$$%=>*:-^.,,,,,,,,,,,.,,.,~~:>%%%%%%%%%%%%%%%%%%%%%%%%
=========?=?????????????????=====???;~~.,^-)???????);,``^!>>+?$##@>&-!==&#@#@@@@@##+*!;:-~!&%!;$$?*>#@#%#@@#$#&%>!^,^^^^^^,,,,,,.....~,^:?%%%%%%%%%%%%%%%%%%%%%%
========????????????????????=====???>:~^.~,~!++*:^'``.:+>++>&###@&?>'*??%$##@##@@#$+!;:-~,)??))#&$))>###&##&+*-,^^~~~~~~~^,~^,,.......~,,^)%%%%%%%%%%%%%%%%%%%%%
======??????????????????????=====?????:~^,,'.'````',*>>>>=&####@#)&:`+?>=#&#@$@@#$%);:--~^$+*:+#$%+?;=$&>!~,,^^^~-:::--~~-~^^,,,......^~.,^!%%%%%%%%%%%%%%%%%%%%
=======?????????????????????=?===?????>-,'```.,,```.~*>%#%%##@@@>%+',>++?#%##&@@#$>*;:--~,&?&?=&$?=*!~~^^,^^^~~-;!*!::--::^^^,,,.......-..,^!%%%%%%%%%%%%%%%%%%%
========????????????????????????????>:.`.,^^^~;+)~``'-+&###@@##>>*,`!+)*>#=#$$@@#%);:--~^,&&%+*);~^,^^^^^^~~-:;)=&$?::---~^^^,,,,......,~..,^)%%%%%%%%%%%%%%%%%%
========??????????????????????????+-``.:++;:--!>=?;^``',:+&#$%!~~!`^)**!?=%#%#@##>!:--~~,,:;~^^^^^^^^~~~---;)?%@###?;:--~~~^^,,,,.'''...-..,,~>%%%%%%%%%%%%%%%%%
========????????????????????????>;'`'-?>???!;:*)=%=?:-,```.~:^^-).'!*!;:+?%&>#@&=*;::-~~^^^^~~~~~~~~--::;)=&=?&####);::--~~^^^^,,'```'..-,..,,:%%%%%%%%%%%%%%%%%
=======????????????????????????>^`';>?=?=??>!!!%%&&%?>??+)*!;;:::;!!;;:-=$$$>##==;;::---~~~~~-----::;!)?%#@#$##@##&::::--~~~^^^,.`````..,-..,,^+%%%%%%%%%%%%%%%%
=======????????????????????=??+.`->??=====?==+*$###$%===??>+)))***!!;:--&&&?&##=*;;::::-----:::::;;!>$@&=%##@@@###=~^;:---~~~^^^.````'...:..,,^:%%%%%%%%%%%%%%%%
========?????????????????????+.'!??=======&$$&)&#####&%%%=?>>+)***!;;:-:$$$>$#=>!;;;::::::;;;;;;!!;?@@@@#$$#@@@@#&*,~;::--~~~^^^.``'',*>=%!^.,^~>%%%%%%%%%%%%%%%
=======?????????????????????>.')?==%%%%%%&&$$&&$@@@@@#&%%%==?>+))*!!;:~-==&>&&*;;;;;;;;;!!!!!!!;;:)$$#####=+%@##?)!-:;::---~~~^^,`'.)$#####%:,^~!%%%%%%%%%%%%%%&
=======?????????????????????-')?==%%%&%&$#%=&>%$@@@@@#&%&%%%=?>++)*!;:-^&$$#&?:;;;;;;;;;!!!!;:-::->&%%&%?%=>*%%&%&);!;;::--~~~^^,.,-.^=#####%:^~:%%%%%%%%%%%%%%%
=====??????????????????????)`-?=%%%%&&&$$$$$%=#%#@@$&@$=%%&%==?>++)!;:-^)&#&>:;;;;;;!!!;;;;:::::-^?%==%&?>+?&*=&$>*!!;;::--~~^^^^^;```:&@$&@#+~~:?%%%%%%%%%%%%%%
====???????????????????????~.>=%%&&&&$&$&$###$#$$%?=&&%=%%%%%===>+)*!;:-~>=;;;;;;;:;**!;:--:::---^?@@@@$&&%))?&#=))*!;;;::--~~^^,:%,``^%##?%#=-~:>%%%%%%%%%%%%%%
======????????????????????>`!=%&&&&&$$&$#@@@$$##&$$?+?+%%%%==%%=?+)*!;:--~;!!;;;;;::**;---------~^!#@@@$&&$$$&@$?****!!;::--~~^^^?&-``^?##==#&;~:+%%%%%%%%%%%%%%
======???????????=????????!'?%&&&&&&&&&$@###%&##&?%##%>==%%%=?=%?>)*!!;::;*!!!;;;;;-):::---------~~@$%*+$$&&$@@#))****!;;:--~~^^^#&;'`-&##=>$&!~:)%%%%%%%%%%%%%%
=======?=?????????????????~-%&&&&&&&&&&&###@@#######%>?=======???>+)*!;::***!!;;;;;:!::---------~~,)>>%#?=##@@@>+)))**!;;::--~~^-$?!,'*$##%+$$*~:)%%%%%%%%%%%%%%
========??????????????????.*%&&$&&&&&&%=>$=#@@@@###=>?????????>??>+)*!!;;)**!!!!;;;;--------------~~$@@&?&@@@$?++))))*!!;::--~~^;$%%>>%$@@%>&$*~:)%%%%%%%%%%%%%%
========?????????????????>`>%&$$&&&&%%%=&$$&#@@@@#??>>>>>>>>>>>+?>+))*!;!)***!!!!!;::-------------~^!#%)~?@@+)+>*++))**!;:::-~~^;#$%?$#@@@%=%&;-;+%%%%%%%%%%%%%%
========?????????????????+'=&&$$$&&&%%==?&#&$####=>>>>>++++>>>>+>>>))*!!))****!!*;:::--------------~,%?+-*$)%%?*+>++)**!;;::-~~^-#=???&#@#%%&=--;>%%%%%%%%%%%%%%
=======??????????????????*,%&$$$&&&%===?>$@#%$#$=?>>>+++++++++++)>+))*!!+)******;;;::------~~------~^->)==*$#==*?>>+)**!;;::--~^^&&=&##@##=&&)~:!?%%%%%%%%%%%%%&
=====?????????????==?????!~%&$$$&&%%==??>?#$#?#=>>>++++)))))))++))+))**+))*****!;;;::-----~~~~-----~~,*%?*#$&%=)??>+)**!;;::--~^^+$@$#@@@&=&&:^:*=%%%%%%%%%%%%%&
========?????????????????;-&$$$&&&%==????+?$&$??>>+++))))))*)))))*+))***))**)*!!!;;::----~~~~~~-----~~~+*##&%&%>=?>+))*!!;::---~^^&##@@#&?$&+^~;)=%%%%%%%%%%%%%%
=======?=????????????????;:&$$$&&%%==????>?#&%?>>>++)))*********))*+>>>>?>++***!!;:::----~~~~~~~~-----~~&#&=&$%>=?>++)*!!;:::--~^^-=&$#%=&$%^,-!+%%%%%%%%%%%%%%%
=======??????????????????;;&$$$&%%==???===?#%??>>++))****!!!!*****)*!!!>)))))**!!;;::---~~~~~~~~~--------&>=&$%?=?>>+)**!;;::---~^^~$#$&%$&-,~:*>%%%%%%%%%%%%%%%
========?????????????????;!&$$&&%%=????=%%%==??>>+))***!!!!!!!!!*******+++))))**!;;::---~~~~~~~~~----:::-;>%$$%?%??>+)**!;;;:----~~^~?&&&>~.^:;)?%%%%%%%%%%%%%%&
========?=???????????????;;&$$&&%==???=%&&&%=?>>>+))*!!!;;;;;!!!!*****++>+++)))*!;;:---~~~~~~~~~~---::;;!:>%$#&=%=?>+)**!!;;:::---~~~-:...,^-;!+=%%%%%%%%%%%%%%%
=======??????????=???????!;&$$&%%=?????%$$&%=?>>++)*!!;;;;;;;;;!!*****+>>>>>++)*!;::--~~~~~~~~~~~---::;!!*)%$#&=%=?>+))*!!;;:::------+-~^~-:;!*>=%%%%%%%%%%%%%%&
=======???????????=??????*:&$$&%%=??>??&$$&%=?>>+)*!!;;;:::::;;;!***)*???>>>++)*!;:---~~~~~~~~~~~---:;;!*)+&##%=%=?>++)**!;;:::-----:>----::;*+?%%%%%%%%%%%%%%%%
========?????????????????+:&$$&%%=?>>>?&#$&%?>)++)*!;;:::::::::;!!**)*=????>+;**!;:--~~~~^^^^^~~~~--:;;!*)>&##%%%=?>>+)**!;;:::::-::*!--:::;!*>=%%%%%%%%%%%%%%%&
==========?==????????????>-&$$&%=??>>>?&#$&%?>-*+*!;:::------::;;!*))+====?+'`'!!;:--~~~~^^^^^~~~~~-::;!)+?$##%%%=?>>+))*!;;;:::::::+:::::;!*)?%%%%%%%%%%%%%%%%%
============?==????=??????-&$$&%=??>>>=&##&=>+.!)*;::---------:;;!*))>=%%=?~```!;:--~~~~^^^^^^^^^^~~-:;*)>=$#$%%%=?>>++)*!!;;;;;!+);+*;::;;!)+%%%%%%%%%%%%%%%%%%
==============?????=???=??-&$$&%=??>>?%$##%>+;`.!;::---~~~~~---:;!*)+>=%%%>~'`.!;:-~~~~^^^^^^^^^^^,'`,:!)>=$#$=%%=?>>++)**!!;;;;;!*))>):;;!*)>&%%%%%%%%%%%%%%%%%
===============????=?===??-&$$&%=??>>?&##$?+*'`,;::--~~~~~~~~--::!*)+?%%%=>*'`~!:--~~^^^^^^^^^^^^,````.;)>%$#&=%%=??>++))*!!!;;;;;:+;;;;;!!*+%%%%%%%%%%%%%%%%%%&
================???==?????:&$$&%=???>=$##%+*~`'::--~~~~~^^^^~~~-:!*)+?%&%?>!``;;:-~~~^^^^^^^^^^,,`````~;*>%$#&=%%=??>++))**!!!;;;;:>!!;;!!*)>%%%%%%%%%%%%%%%%%%&
================???===????!%$&%=?????&$##+!-``----~~~~^^^^^^^^^~:;*+>=&&?>)~`,;:--~~^^^^^,^,,,,,'````'-;*>%##%=%==??>++))**!!!;;;;)!;;;;!*)+=?%%%%%%%%%%%%%%%%&&
=======================???)=$&%=????=&##=!~`'---~~~~^^^^^^^^^,..-;*+>%&%))!'`-::-~~~^^^,,,,,,,,,`````,-:*>&#$%?%=??>>>+)))**!!!!!!+;;;!!**)+=?%%%%%%%%%%%%%%%%&&
========================??>>$$%==??=%$#%!-`'---~~~^^^^^^^^,,,```^:*+?%&=*!^`^::--~~^^^,,,,,,,,,.`````^-:*>&#&%=%=??>>++)))***!!!!!)!;!!!*)+?>?&&&%%%%%%%%&&&&&&&