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