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