===========--------------=-------------====++++++++++===+**++++==++++++=====++++
===-----=----===-----=----------;;------==========+====-=++++++=====--==========
==---;-----+?%%%%%????*=-=-----==---------==--=======---==+++++===---======-====
-=----;---+%%%????*+====-----+%%??*+====---------====-----=+++++===++++++*+=====
------;;--%%%%SS%+++=+==+--=+%????%%S%%%?%*==-----====----==++++++***++===+**===
------;-++?%S%%%**+===+*-+*+?%%%%%%%%SS%%???%*=-----==---=====+++***?++=++++*+++
----;;;-%S%%S%%?*++*+*?==?+%SSSS%??***?%SS%?***+=======-=++===+++**********+++==
--;;;;-*%SSS#S%*+***?%*=**+SSS%S%%??*+++?%%%?*+??*++=============++++*???****++=
;;;;;-*SSSS##S?+***?%?+=++%#SS%%????**+++?%%%?***%?*++==========++++**????***+++
;;;;;-?SS??S#%**+*?%?*+=+*S%SSS%%%%??**++*???%%?+??**==========++=++****?*****++
;;;;-*%SS%*%S?***?%%%*+=*SSSSSS%%??*?%%??***=?%%*???+--------==--=+++*********++
-;;;+?SSSS?%%%?*%%%%%?+*SSSSS%%%%%?*???%S%?*;-?%%%??*--;;;;;;;;;;-=++==+**+++++=
----*%SSSSSS%%??%%%%?**SSSSSS%%%%?*+*?*%%?+-;;-+%SS%*-=-;:::::::::;-==++++**+++=
;;;;*%SSS##S%%%?S%%%?*SSSSSS%%%%%?%?*?*?+--;;;;-+SS%?+*=;:::::::::::;--=++**+++=
;:;;=%S####SS%%%%%%%?#SSS??%%?*???%%%??*--;;;;;;-%S%??*-;:::::::::;=+++++**+++==
::;;=%S#SSSS%%%%%?**%%SS%?*?%%?*?*?SS?*---;;;;;;-+S%?**-;;;;::::::+***+-;-===---
::;;*%SS%%S#SS%%+++???SSSS%?%S%%?%?S%??***+=----+*S%???+::::::::;-==+*+=-;-=-;;;
::;;*%SSSS##%%?***%%??%%SS%??%%?%%?%%??*?%S%*=-*%*%S%?*-:::::::;-;;;-=+++=-;::::
::;;+%S#S%S%%%?*?SSSS%%%SSS%???%%?%%%S%??SSS%+=%?%S%?*;::::::::;-;:::::;-+++-:::
:::;-?S%+%%%%??%#SSSSSSSS%SSS%%??%%SSSS%SSSS%*-?S#S?*+;:::::::::;;;;;:;--++++-::
:::;;=*=?%%??%SS##SS%%S#SS%%%%????*+?S%****?%?=+?%%**+-:::::::::::::;--==++=--;:
;;;:;=++%%%%%%S###SSSSSSSSS%%%%??*+--+**+-=+??+-++?*+;:::::::::::::::;---=---;::
-;;;+*+%SSS%%S#@##SSSSSSSS%S%S%%??*=---;;;-=*??=--*?=::::::::::::::::::;-;----;:
--;+?**?%%%S#S######SSSSSSSSSS%*?%?*==-----=*??=--*?*-::::::::::::::::::::;;;-;:
=-=?*?***?%SSS########S###SSSSS??%??*+=----=????+=**+-::::;:::::::::::::::::;;::
++??****?%S%SS#######@#SS##S%%%?%%%?*++==---%%SS%-**-:::::+++=-::::::;;-=++*=-::
**?****?%%%%S%#########S%%%%%%??%%%%?*++=---?%%%-;**;:::::-*%?++=+**???%%%%%*=;,
?*****?%%%?%%%S########@#S%%%%%*?%%%??*+=---+*+++-++-;;-=+=+*??%%SSSSS%%??*+=-:,
?****%%%%%%%?S%S#######@#SSS%%%?%%%%??*++=-=*%%%%**?????%S?*++==+*?+-;;:,,::::::
****%%?%%%?+?%%%S#######SS%%%%%%%%%%%??*++=*%??SSSSSSS%%?*?%S%?*+===;:,:::::::;-
***?%??%%?+*%%%%SSS#S###S%%%%%%%%%%%%%??**+=?%?S%+=+--:::-?%%++%?%*==;;;;;;::;--
?*?%%?%S%?+%%%SS?*%SSS##S%%%%%%%%%%%%%%???***%%%=:;-;;:::*%%%=**++*+==---;;;;;--
??????%%%%*?%???*?%%%SSS%%%%%%%%%%%%%%%%%?*+=--+=::-;:::;?%%%?*??+--==----;;;;--
*?????S?SS??*%%%%%%%SSSS%%%?%%%%%%%%%%%%%%?**+++=::;::::*%%%%%%%?*++--=----;;;;;
+?????%%%S%????%%%SSSSS%%%%??%%%%SSS%%%%%%%%%%%?;::;:::-?%%%%%?***=-===------;;;
+??**?%%%%%%%%%%%SSSSS%%%%?????%%%SSSSSS%????*+;:::::::+%%%%%%%%%%?+--=-------;;
*?***%*??%%%%%%SSSS%%%%%%%??????%%%%SSSS=;::::::::::;;-*%%%%%SSSSSS%?+==--------
****?%??**??%%SS%%%%%%%%%????????%%%%%%?;::::::::::;;;-?%%%%SSSSSSS%%?*=-------=
*++?%S%++?%SSS%???????%%%????????%%%%%?+::::;;;;;;;;;;-*%%SSSSSS%%%%%?*-----+++*
?+*?%S%?*%S%%?******???????????????%??*-;;;---;;;;--;;;+%%SSSS%%%????*---=+-*?**
?***+??*%%%?*++====+**???????????????*?=---------=+=----?%SS%%%%?????=---=*+****
%????%%%%?*++==----==+*???????????????S?+==-=====+*++==-*%%%%%%%????+---+*?*****
??*?%S%??*+==--------=+*?????????????#S?%*++++*******+*++%%%%%%%?%%*-==+**??****
%%SS%%?*++==------;--=+*????????????###S*%%?*************%%%%%%%%%?===+++*??***+
%SS%%?*++=------------=+*?????*????S@##SS??*%?******?****?%%%%%%%?+=======*??+=-
SS%%??*+==---;--------==+*????*???%##@#SSS?**%*++++++++++?%%S%%?+=+======+????+=
S%%%%%??*++--;;;-------==+*??**???SS##@#S%%????++++=====+*%%S%?*=-*+++++*??%?%?*
SS%SS%%S%??*+-;;---------=+*******%%S##@#S??*?%?++++++++++%%S%?*=-++++++***?*+?%
SSSSSS%%%%%?%?*=----;------==++*++*?%S##@#S*+*%%?+++++++++%%%%%*=-+*+=====*?*+*?
SSSSSS%%?%%?**?*++---;;;;;----=+=-=+?%S##@@S?+%%%%?*+++++*SSS%%?+=*?+++===*???**
SSSSSSS%%SS%???*?**=-;;;;;;;;;-=---=*%SS#####%?%?%?***++**SSSSS%????***++*%%%?**
SS##SSS%%%????????%*=--;;;;;;;;----=+?%S######%%%%?%?++++*SSSSS%???%**???%%?%%??
##SSSSSSS%?*++*?*???*+--;;;;;;;-----+*%SS#####S?S???%?????%SSSS%%???*????%%%???*
####SSSSSSS%?+==*?+???+--;;;;;------=*?S%######?%??%?%S%??%%SSS%*+-+**????%????+
#####SSS%%??S%*+=+*??%?+--;;--------=+*%%######?%%?%%%%S*?%%%%?*+---****+*%%%%?*
SSSSSSSSSS%*+%%?*+=*%%%?---;---------=*%%S#####S?%%%?%S%%%%%%%%*+---?????**?%??%
##SSS%%?*????**%%%?**%%%*------------=+?%%SS###SSS%SSS#%%S%%%%?*=---+?????*?????
#######S%****%%**?%%?*%%?*------------=*%%SS###SSSSSS#%*S#%%%%?*+---=?????????+*
######SSSS%?**%#S?+*+++%S%+--;--------=??%%SS###SSSS#S%%#%%%%%?*+=---?%%?????*++
##SS##SSSSSSS?*%#S??????%%?*+=---=+==-+*??%SS####SS##S##%%%%%%?*+=---*%%??????*?
###SSS%%%SSS#S%%S#SSSSS#%%??*******???***?%SS####SS####%SS%%%%??+=---+S%%?%??%%%
#####SSSSS##S?%%SSSS#SS##?%%?*++?*%%*%%??%%SS####S####SSS#%%%%%?+=---=S%S??%%?%%
#########SS##S?+*?SSS%%S#S%%%%????S%%%%?%%S##########SSSSS%%%%%?*==---?SSS?%%%%%
######SSSS#S###%**+?S#S%%SSS%%?%%?%SS%%%%%SS#########SSSSS%%%%%?*+=---*SS#%?%%%?
#######SSS%%S###S%?**S##SS%S#S%%SSSSS%?%%%SS####@####SS###%%%%%?*+====*%S#S%?%%%
#SSSSS#####S%%%S##S%??%####SSS%S%SSSSS%S%%#####@####SS##S#%%%%%?*+====*?S#SS%%%%