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