::::::::;;;;;;;;;;;-----------------------------------------;,,,:::::::::::;;;;;
:::::::::;;;;;;;;;;;---------------------------------------;:,,::::::::::;;;;;;;
::::::::::;;;;;;;;;;---------------------------------------:,,,:::::::::;;;;;;;;
::::::::::;;;;;;;;;;;----------------------------------===-:,,,::::::::;;;;;;;;;
::;;;;;;::;;;;;;;;;;;-------------------------------+*****=:,:::::::::;;;;;;;;;;
;;;;;;;;;::;;;;;;;;;;;---------------------;;-----=*********-:::::::::;;;;;;;;;;
;;;;;;;;;;::;;;;;;;;;;---------------------------=***********+;:::::;;;;;;;;;;;;
;;;;;;;;;;;:;;;;;;;;;;---------------------------**************;::;;;;;;;;;;;;;:
;;;;;;;;;;;:;;;;;;;;;-;-------------------------=**********?****;;;;;;;;;;;;;;:,
;;;-;;;;;;;;;;;;;;;;;;----------------===+++++****??*************-;;;;;;;;;;;:,:
;;;;;;;;;;;;;;;;;;;;;;;----;------=+**????**********????**********;;;;;;;;;;;:::
;;;;-;;;;-;;;;;;;;;;;;;;---;;---=+++*%%%?+*?****?*???%%%S%?*******=;;;;;;;;;:::;
;-----;;;;--;;;;;;;;;;;;;;;;;--=+++??%%%*++*=+=+++*S%%%%%%%%?*****+;;;;;;;:,::;;
;;-----;;;---;;;;;;;;;;;;;;;;--=+++?%%%%==*+======+%%%%%%%%%%%?****;;;;;;:,,::::
::;--------;;;;;;;;;:;;;;;;;;;-=+=?%%%%?==*+=======*%%%%%%%%%%%%***;;;;;:,,::::;
,:;;;;;---;;;;;;;;::,,,,:;;;;;---*%%%%%*==++===**+++*%%%%%%%%%%%%?+;;;;::,::::;;
,:::;;----;::::;;:,,,,,,,,;;;;--*%%%%%?+===*=+?*++++++?%%%%%%%%%%?-;;;;:,:::;:;;
,,,,:;;--;:,,,,::,,,,,,,,,:;;;-=%%%%S%***+=*===+%SSS%?+*S%%%%%%%%%-;;;;::::;;;--
,,,,,;;;-;:,,,,,,,,,,,,,,,,:;;-+?*%%%%++++=++=+?*%SS%?==?S%**?%%%%*--;;:::;;=+=-
,,,,,:;;;;,,,,,,,,,,,,,,,,,::;;=*+%%%S%SS%+=++===+?%?+==*%++++%%%%%?=;;::;+?%%%%
,,,,,,::;:,,,,,,,,,,,,,,,,,,,,:-==?%%%S+*S+======+**++==*%**++%S%%%%%?*+*%%*=--=
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::;+?%%++*+======+****+=++++++%%SS%%%%%%%*------
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,::-*=--======+*****+=+=+++*S%%%SSS%%%%+;=+++*
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,::,::;=;;;-======+++++====++*%SS%%%%%%%%%%??*+++
,,,::::,,,,,,,:;;:::,,,,,,,,,,,,;=::---;::;-=============+=?%%%%%%%%%S%%%%%%%?**
::;;-;;:,,,,,:;---;;;;::,,,,,,,,-=;;==+;::::-===========*??%%%%%%%SSS%%%%%%%S%%%
-------;;:::;-------;;;::,,,,:::-=;-==??++=-;-==========+*S%%%%%%%%%%%SS%%SS%*++
-----------;---------;;;:::::;=:-=-=*%%%%%?+============++SS%%%%%%%%%%%%%%%%%%%*
;;;;;----------------;;::::::;==+*??%S%%%?;;=============*SS%%%%%%%%%%%%%%%%%%%%
;;;;;;;--------------;;:::;-=+*?%%%%%?%%+?*;;============+SSS%%S%%%%%%%%%%%%%%%%
;;;;;;;----;;---;:::;;;::;+*%%%%%%?+==??-*%?;-==+*++=====+SSSS%SSS%%%%%%%%%%%SSS
;;;;;;;;-;;;;;;:::,,,::;;-*?S%%?+=====*?-+%%%**?S?+=======%SSSS%%SSSSSSSSSSSSS%%
;;;;;;;;--;;;;;::::,,,:;;;+%*=-=======+?++%%%%%%S?======+*?SSSSS%%%SSSSSSSS%%%%%
;;-;;;;;--;;;;;;:;;:::::;;::::;=======+?+=?%%%%%%%;===+*?****?%%SSSSSSSS%%%%%%%%
;;;;;;;;;;;;;;;;;;;;;;::::::;;;-=======+==*%%%%SS%-;+***+====++?%%%%%%%%%%%%%%SS
;;;;;;;;;;;;;;;;;;;;;;;::::::;;-==========+%%%%%%%=-+*+========*SS%%%%%%%%SS%%%S
-;;;;;;;;;;;;;;;;;;;;;;;;:::::;-==========+%%%%%%?=--==+=+==+==+SSSSSSSSSSS%SSSS
---;;;;;;;-;;;;;;;;;;-;;;;:::::;-=========*%%?%%=--===--=======*SSSSSSSSSSSSSSSS
;---;;;;;;;------+***?*==;;:::::-=========*%??*====;:::;=======+%SSSSSSSSSSSSSSS
;----;;;;;;----+++*?%S%%%?=;::::;-=======**?%+====::::::---=====*SSSSSSSSSSSSSSS
;-------;;---++++*?%%?%%%%%*=::::--=====+%%?+-=-;::::::::--======?SSSSSSSSSSSSSS
------------*+++*?%%+-?%%%%%%+;:;--=====*%%=---;:::::::::;-===+==+%SSSSSSSSSSSSS
-----------++++*??%SS%S%%%%%%%+;;;;-=====*+--;;::::::::::;====+===?SSSSSSSSSSSSS
-------;;-=++++??%%SSS%%%%%%%%%=;;;-====----;::;:::::::::-========*SSSSSSSSSSSSS
---------+++++*?%%?*%%%%%%%%%%?-;;:-====---;::;:::::::::;-========+%SSSSSSSSSSSS
--------+++++*?%%%%+?%%%%%%%%%?;;;:-====--;::;;:::::::::-==========?SSSSSSSSSSS%
-------=+++***?%%%SSS%%%%%%%%%%=;;;====+-;:::-:::::::::;-===+======*%%SSSSSS%?**
---=---+*+***?%%%%SSS%%%%%%%%%?=;;;=====;::;--::::::::;-=--=====+==*%%%S%%%S+---
------;+*+**??%%%%SS%%%%%%%%%%*;-;;====;::;--::::::::;--=-======+==+S%SS%SS%----
-------+++**??%%%%%%S%%%%%%%%%+;-:-===;;----;:::::::----=--====++===%%?%S%%*=---
-------=***???%%%%%%%%%%%%%%%%+;-:-==;;----;:::::;;----========++===%?=?%%%*=---
-------=***?%?%%%%%%%%%%%%%%%%?-;;-=;;-----:::::---=---========+====+??%%?*+----
--------***?%%%%%%%%%%%S%%%%%%+-;;=-------;::::---==-==========+====*??*===-----
----=---+*??%%%%%%%%%%%%%%%%%*--:;=---=---::::-=--==-==========+====*==---------
---------**?%%%%%%%%%%%%S%%%%=--:-==--=--;:::;----===================-----==--==
---------+??%%%%%%%%%%%%S%%%?--;;===-==--::::-=--===================++++**+=++++
+==--=====*%%%%%%%%%%%%%%%%S*--;;===-==-::::;-=--===================++++**++=*++
**+=+**+**?%%%%%SS%%%%%?%%%%*+=;-===-=--::::;-=-====================+++++*+*+**+
*???+**+***?%%%%%%%%%%*+S%%%+*+;==+-===:::::-==-====================++++++****?*
???%+**+***?%%%S%%%%%+++?S%?**+===+==-=;---;-=======================**+++***?*?*
???%++*+***??%%S%%%%+++++?%****===+====***+=========================?*+*+***??%*
???%+**+***???+%%%S*++++=+***+*====+===*****?*======================??******??**
????+******???==%%S++++=====+**+===+==+*++**?*+=====+===============???*****?%??
????*******???==?%S%*++========++++++==+++*+*++++================+=+????****????
????**+***????==*?%SS?+===========++++=+==+=+++++++====+===+=====++*???????%????
????******????+=***????*+================++++++++++++=====++=====+?????????????%