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