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