..............................................................................................................................................,::,,,.............................................................................,,...........,.,,,,,,,.....,.....,.
...............................................................................................................................................,,..,............................................................................................,.,......,,.,.....,,
...................................................................................................................................,,,.........................................................................................,........................,,,,.,......
.............................................................................................................................................................................................................................,....................,::,,,,,,.,,......
..................................................................................................................................................................................................................,..........,....................:::,,,.,..,.,.....
...........................................................................................................................................................................................,.....,.................,,,,,,,,.......,,,............,:,,.,....,...,....
.................................................................................................................................................................................................,..,,.,,...................,,,........,,,.......,,,.,,.........,...
........................................................................................................................................................................................................,,..,,,,,,.......,........,,,...,,...........,,..........,..
..............................................................................................................................................................................................................,,,.,,,,..,,,..,,...............,,,,..,,...........,,.
................................................................................................................................................................................................................,,,,,,,,,,,,,,..,,,,,,,,,.....,.....,,...,.........,
....................................................................................................................................................................................,..........,......,..............,..,,,.,,,,,.,,,,..,,,,..,.,,,.....,,..........
....................................................................................................................................................................................::....,...,:::,,,,+:.............,..,....,,,,,...,,..,,,,.,,,,,,,,,:,,,.........
 ...................................................................................................................................................................................,,,.,:,,::::+::::+:,............................,....,,,,,,,.,..,,,,.,,,,.......
..................................................................................................................................................................................,,,::,::,++::+%+%+++,........................................,...,,,,..,.,:,,.....
...................................................................................  ..................................................................................,,,,,,,,,,,,,:::::::,,,,:+++++::...........................................,,,....,..,.......
..............................................................    ........... .............    . ................................................................,,.,,,,,,:,,,,,,,,,,,,,,:+%+:,:+%#%+:,.........................................,,..,........,......
............................................  .................. .................... ...... .................................................................,,.,:,,::::::,,,:+++%%%+++%#%+:+++++%%+::,......................................................,.....
.....................  .....................  . .....  . ....... ...............................................................................................,:+:+%%++%%++:,::+++%++###@#+:+%%++++:::::.....................................................,,...
.....................  ......................  .....  .. ...  ...............  .................................................................................:%++%%%%%%++++:::%##%#@@@@@###++%%++::,,,+:,...,................................................,,..
...................... ....................... ..............................  .......  . .......................  ......... ..... .........................,,,,:%%%%##%%#######%++%#@@@@@@@#@@#####%:,,,,::,,,:,.................................................,,
...................... .................... .. ............................... ......  .. . ....   ..  ... ......    ... .... .....   .....................,,:::+%####@@@@##@@####%%#@@#@@@@@@@@#@####:.,:::::,::...................................................
......................  ...................................................... ............ .   . .  ..  .  ..  ...    .  ..     .   .... .. .............,::+%##%###@@##@##@@@@@#@##@@@@##@#######@###:,:,,::,::...................................................
.............................. ...............................................  .............  ..  .......  . ...   .     ..            . ...  .....   .,,:+%%#############@@@@@@@@@@@@#@###%%#%#%%%###%:++:::+,:,,.....................................,.,.........
...................................................................................... ....... ........ .....  .......... .        ...     ..  .. ..  .,:++%###################@@@@@@@@#@####%%%%######%+%#++:%+,,,....................................,,,,,........
...........................................................     . ............................................ ..... ..  .... .. ..... .  ............,+++%#########%%%%%%#####@@@@#############%#####@#:+#%%::%+,:,........................... .........,,,........
..............................................................................................................  ................. ................. ,::+%########%#%%%%%%%%%###@@@@@@@###########@##@#@@+++%#%::++:, ....................................,,.........
...................................................................................................................................................,::+#########%%%%%%%%%%%###@@#@@@@@@@######@@#@@##@###+%:%#%:,+%:.     ...................... .......,,..........
..................................................................................................................................................,:::%########%%%%%%%%%%%%####@##@@@@#####@@@@@@@@#@@####%++%#+:,++:..               . ................,...........
.................................................................................................................................................,::+%#######%%%%%%%%%%%%%%#######@@@@@###@@@@@@@@@@@@@@###%%###+:+++,,.........        . . .        .....   .......
.....................................,,.,.......................................................................................................,,,+#%######%%%%%%%%%%%%%%%############@##@@@@@@@@@@@@@@####%###%+:++::,.............   .   .. .    .... .......    
...............................................................................................................................................,:,:##%%#####%+%%%%%%%%%%%%##############@@@@@@@@@@@@@@@@##@###@##%%+%+,+.................. . . .. ...... ......... .
...,,..........................................................................................................................................:,,%#++####%%+++%%%%%%%%##################@#@@@@@@@@@@@@@@##@@@@##%%%%+::+............ ..............................
....,...........................,,......,.....................................................................................................,:,:%++###%%%%+++%%%%%%%%####################@##@@@@@@@@@@@@@@@@@@#%%%%++:+,..........................................
....,..........................,.............................................................................................................,,,+%++%###%%+++++%%%%%%%%####################@##@@@@@@@@@@@@@@@@@##%+%%%++::..........................................
.............................................................................................................................................,,:++:%%##%%+++++%%%%%%#%%%#####################@@@@@@@@@@@@@@@@####%%%%#%%+:..................................,,......
..,...................................................................................,......................................................,:+%++%###%++++%+%%%%%%%%%%#####################@@#####@@@@@@@@@##%%%%%%##%%+,................................,,,..,,,.
..,,,........................................................................................................................................::%%+%%##%++++++%%%%%%%#%%%###############################@@@@####%+%+++%##%%+...............................,,....,,,,
...,,,........................,......,.,,....................................................................................................::%+:%%%%%+++++%%%%+%%%%%%%#####################################%%%+:,::+##%%%,........................................
........,.....,......................................,.,....................................................................................,,:%:+%%%%%+++:+%%%+%%%%%%%%%###################################%%%++:,,:+%@%%%+........................................
........,..........,.....................................,..,...........................,..................................................,::+%:++%%%%+++++%%%+%%%%%%%%%##########%######%#%#%%%%#########%%%++::,,,,:##%%%,.......................................
...................,.........................................,.............................................................................,,:%::+%%%%%+++:+%%%%%%%%%%%%%%%##%####%##%#%##%%%%%%%%######%%%%%%++:,,,,,:%###%:.......................................
........,........,,,,.,.....................................................................................................,...,..........:,:%:+%%%%%++++:++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+++:,,,,::#%#%%....... ...............................
...,,,,,,.,,,,,,,,,,,,,,,.....,.....................,......,..........................,..........................................,........,:,++:%%%%%%++++++++%+%%%%%%%%%%%%%#%##%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+++::,::+:+%+%%,......................................
.......,,,,,:,,,,,,,,,,,,,,....................,....................................,,,,,,..............................,......,,,,,,.....:::%,:%#%%%%++++++++%%%%%%%%%%%%%%%#%#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+++:,:::+++%%%+:......................................
,,,,..,,,,,,,,,,,.,,,,.,.......................,,.................................,,,,,,.,..............................,,.,....,,........::++:+%%%%%%+++++++++%%+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%++:::::+++%%++:......................................
..,,,,,,,,........................................................................,,.....................................,...,...........,+,+:::%%%%%++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+::::++++#+++......................................
..............................,......................................,.,,.....................................................,,.........,:,:::+%%%%%++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%++::::+++%%++............................,.........
...........,.........................................................,..........,,..........................,.................,,........,:,:::++%#%%%+++++++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+++::+++:%#%:......................................
...,............,.....,................................................................,.....................................,,,...,,..,,::,::%%###%++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%++++++::+%%+,.....................................
...............,......................................,........................,,.....,,,......................,...........,,,,,.......,,::,+:%%##%%++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%++++++::%%+,.....................................
............................................,...............................,.......,..,,..................................,,,::,.....,,.::,:++###%%++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%++%++::%%+:...............,.....................
........................,.....................,...........................................................................,,,..,.....,,,,::,:++##%%+++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+%+++:%%+:...............,...........,,,.......
.........................,,............,...............................,................................................,............,,,,::::+%%#%%+++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+++:+%+:..........,,,,...........,...,,,.....
.................,.......,.......,....,,..........,...............................,...............,..................................,,.,,::+++%%%++++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+++:+%+:,............,...............,:,.....
...,............,,......,,,,,,,..,,.,,,,...,.,.,..,,...............,.....................................,....................,....,,,,.,,,:+:++%%%+++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%++:+%+:,..................  ................
.......,....,...,....,,,,,,:,,....,.,.....,..,.,,......................................................,,............................,,,,,::::++%%++++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%++:+%+,,..................  ................
...,,.................,,,.,,,,,,..,,,..........................................................................................,.....,,,,:::::++%%++++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%++:+%+::.............. ...........,.........
. ....,...............,,.,,,,,,,..,,,,..,,,...,,....................................,,..,,..................,...........,,....,,.....,,,,:+:::++++++++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%++::++:,..,....................,............
........,....,.......,,..,,,,,,,.,..,...,,,,,........,...................................,..,.................,,...,..,.,,..,..........,:++::+++++++++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+::+::,..,......,,..............,..........
.......,,....,............,:,.,,.,,,,,.,...,,.......,..................................,,.,.,................,,,...,,,,..,..,........,.,+++::+++++++++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+++:++:......,,,....,.,,...,...,............
.................,............,,.,..........,.........,..........................,.......,,,,............,,.,,,,.....,......,.........,::++::+++++++++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+++:+++........,,.,......,,,,,,.............
...............................,,................................,,...............,,.....,............,,..,.,,,,,...,,,...............,:+%+++++++++++++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+++:++:.....,.,.,,,............,............
..,,....,...,.....,..........,,,,,,,,.....................................................................,,,,.,,...,,,..,,,.,,.......::+%+:+%++++++++++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%++++++:............,,,..,..,,,..............
..,..,,,,....................,,,,,,.............,.........,..........,...,.............,,..................,,,,,,..,.,,.,.,.,:,.......::%#%:++++++++++++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+++++++..........,,,,,,,,.....,.............
.....,.....................,,,,.,..............,,,..,,..,.,,..,...,,..,,..............,,,...,,...........,,,,,,.,..,.,.......,,.......++%#++%%+:+++++++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+%%%%%%%%%%%%%%%%%%%%%%%%%%+++++++,............,,.,.,,,...,.......,,,..
.....,......................,,...........,.,,.,,,....,.,.....,....,,..................,,................,.,,....,..,..................++%%:+%%+::++++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+%%+%%%%%%%%%%%%%%%%%%%%%%%%%++++++++........,...............,........,,,,
.........................................,,,,.,,,,,...,,...........................,,,,,:..............,.......,,,,...,.........,::,..::%+::,,,,::+%+++++++++%%%%%%%%#######%##%%%%%%%%%%%%+%%%%%%%%%%%%%%%%%%%%%%%%%%%+++++%+:.....................................
............................,.............,..,,,,,,..,,,,,.,,,...,,,.....,.........,,,,::.....................,,,,,,,..........,+%%#%,,:%#%%+:::,,,,+%%%++++%%%###################%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+++++++%+:...........,....,,.,..,.............,
...........,.....,....,,.,,,,..,,,.,....,,..,,,,,,,....,,...,,,,................,...,................,.......,,,,,,.,..........++++%##%:+::::::::::,:%@@@@@@@@@@@@@@@@@@@@@#########%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%++%++::++++:..................,....,..,.,........
......,........,......,,...,,,,,,,,...........:,.,,.,,,,,.,...,,,......,..,..................................,,,..,...........:%%#%%%###+:::,,,:++::+%@@@@###########@@@@@@@@@@######%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%++++++:+%%%:..........,,.........,....,..........
......,,...,....,..,...,,..,,,,,,.................,.,.....................,.......................,,,,,,,.,,,...............::+####%%%###%+:::::+++%+%@@@##################@@@@@@@@###%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+++++::++%%,...........,.,....,,..,..............
......,,..,,,,,,,....,...,,,,,,,,...,.............,,.........................,..,.................,,.,.,.,.,,,..............::%###%%%%###%+++::::+++++%#@%#######################@@@@###%%%%%%%%%%#%%#########%%%%++++++::++%%:.....................,,..............
.,,,,,,,,..,.,,......,..,,....,,,.,,,,,..........,,,,,,,,..................................................,,,..............,:###%+%#%%%%++++::::++++%+%#%%######@@@@@@@@@@@@#######@@@@##%%%%%##################%+%++:+::++%%,.....,.,,,.,,.,,,,,,,,,,.............
,,,,,.,,,,,,,.,.,,......,,....,,,...,,..........,,,:,,,............,,,...........,.....,,............,,,..........,.........,:###++%##+%%+%%+::::+++++%+#%%###@@@##@@@@@@@#@@@########@@@@@@########@@@@@@@@@@####%%+:+:::++%%,......,,,.,,,.,,,,,,,,,,,...,........
,,,,,,.,,,,,,,,,.,,,.....,.,,,,,:..............,,,.,,.,................,...,.....,.....,.............,,,...,...........,....,+##%+%%##%+%%%%+::::++++++%#%%##@@#@#%#@@@@@#%#@#@########@@@@@@@@@@@@@@@@@@@@@@@@######++::+++%%,..........,.,.,......,,,.............
,,,,,,,,....,,,....,....,,,,,,,::,..............,,................,,.,,,,..,,..,.,,..,.............,.,,,,..........,...,....,%##%+%%%%%+%%%+++::+++++++%#%####%%%###@@@@@#%############@@@@@@@@@@@##########@####@@@##+:::++%%,.....................................
.,,,,,,.,,.,......,....,,,,,,,..,...............................,.,,,::,....,.,......,...............,......................,%##%+%%%%%+%%%+++:::+++++%%#%##%%++%%####@@##%############@######@@@#####################@@#%+:%%,.....................................
......,.,.... .,.,,.....,,......................................,,,,.,,,,,..,,,....................,.,,.,....................%#%%+%%%%++%%%++++::++++++%#%%+++%++%%##################%##%#%%%###@#####@@@@@@###########%%#@#%+,.....................................
 ...  .....,.....,:... ...................,..,.....,,............,..,,,,,,..,,,,,.,,,..............................,,.....,..%#%%++%%%+%+%++++::++++++++#%%+%%%%%+%%#################%#%%%%%%%##@###@@@@@@@@@@@#########+:+%@@%:.................,.........,........
..... ......,.....,..,......,.,,..........,,.,.....,,..............,,,,,,,....,,..,,,,,,........................,,.......,..,+%%+++%%%%#%%++++::++++++++%++%%+%%%%%###########%%%###%###%%%%%%##@###@@@@@@@@@@##@@######%+%+%@@@:.... .....................,........
......,,.........................................,..,,,.............,:,.,,.,,,,,.,...,:,..............,...................,..:%%++%%%%###++++::::+++:+++%+%%%%%%%%%%#############%#%#@##%%%%%%##@###@#@#@@@@@@#%#@@#####%+%%#@@@,.........,..............,..........
.  .,.,,...,,,..................,........................................,.,..........,,...................................,.:%%++%%%%###%+::::::+:::+++%++%%%%%%%%%%%%%%%%#%%###%%#@@#%%%%%%%#@@%###@@##@@@@@#%%#@@####%+%+%@%,... ..,.....,.......................
......,.  ..,......,.....,.....,,..............,.........................,,....................................,............,.%%++%%%####%+:::::::::::++#++%%%%%%%%%%%%%%%%##%#%%%%#@##%%%%%%%%@@#%%######@@@#%%##@@@###%+%%%#+%+,............................. ....
..,,..,,...,..,,,..,,..,,,,..,.............,.,.,,,,............,.......,,,,,,,,,.......................................,......+%%+%%%####++:::::::::::++%+++%%%%%%#####%#%#%%%%%%%#####%%%%%%%%#@#%#######%%#%%%###@@#%%%+%++#+%%+.............................. ...
.,,....,,.,,,,:,,...,.,,..,.,,.............,...............,...........,,,,,,,................................................:%%+%%#####::::::::::::++:+%++%%%%%%%%%%%%%%%%%%%%%#####%++%%%%%%#@#%############%%%%#@##%%%++%%%%#%.........,........................
.......,,,,,.::,,...,.,.....,,...............,.......................,,,,.,,..................................................,%%+%%####%::::::::::::++::%++%%%%%%%%%%%%%%%%%%%%%%###%+++%%%%%+%#######%######%%%%%%###%%++:%%%###..................................
....,..,,,..,,,,,,..,,,,..,,,...........................,..,..................................................,,..............,+%+%%%###+::::::::::::++::+#++%%%%%%%%%%%%%%%%%%%%%##%++++%%%%%+%##%%###%%%%###%%%%%%##%%%++:#%%%##..................................
.......,,,,,,,,,,,...,,,..,,,.,,........................,,.,.....,..,..........................,..............................,:%++%%##%+:::::::::+:::::+++%%%%%%%%%%%%%%%%%%%%%+%#%+++++%%%%%+%##%%%###%%%%%##%%%%%%##%%+++%%%%##..........................,.,.,...
.,..........,:,.,,.,,,,,.,.,.::,...........,..,,..................,,,...........................,..............................:%%+%%##%%:::::::::+::::++++++%##%%%%%%%%%%%++++%%#%++++++%%%%%+%#@%%%%%###%%%%%%%%%%%%%%%++%+%%%%%............................,,,,,.
.,...........,,..,.,,.........,,..........................................,,..........................,........................,+%+%%%##%:::::::::++::::++++++%%%####%%%%%%%%%###%+++++++%%%%++%%@%%%%%%####%%%%%%%%++%%%++%+%%%%%..................................
...,.............,.................................................,,...,.,,.,.....................,...........................,:%%++:+%%::,::::::::::::++++++%%%%%###########%%+++++++++%%%%++%%##%%%+%%%###%%%%%%%++%%++%%+%%%#%........................,..,..,,,.
.,,,,,..........,,,...,,.......................................,........,..,..................,......................,.,......,,:+%%+:+%%:,,:::::::::::::++++++%%%%%%%%%+%+%%%%%+++++++++%%%%%+%%%#%+++%%%%%##%%%%%%++++++#+%%%%#%...........................,.....,
...,,..........,,,,,,,..............................................................................,......................,,,,:::%%+::+%:,,::::::::++:::++++++++%%%%%++++%%%%%%%++++++++%%%%%++%%%##%++++%%%%%%%%%%+++++%%+%%%%%+............................,,,,..
..............,,,,,,.,.......................................................................................,.................,::+%%:+++,,,::::::::+:::::+++++++%%%%%%%+%%%##%%%%+++++++%%%%+++%%%+%##%++++%%%%%%%%+++++%+%%%%%%:..................................
..............,,,..,..................................,...............,......................................................,..,:+%%::++,,::::::::::+::::+++++++%%%%%%%%%%###++%%+++++++%%%%%+++%%%%%%%##%%%%%%%%%+++++%%%%%%%%%,..................................
.......................................,..............,,...................................,,,...................................::+++:++,,:::::::+++:+:::+++++++%%%%%%%%%###%+++++++++++++%%%++++%%%%+%%%%%%%###%%%%%%+:%%%%%%%%.................................,.
............,..........................,,..................................................,..................,.,,................:++::+:,,:::::::+++::::+++++++%%%%%%%%%####+::+++:::++++++%%++++%%#%%+%%+%++%%%%%%++:::+%%%%%%+...................................
............,......,.......................................................................................,..,.,.,...............,,:::::,,:::::::::++::::++++++%%%%%%%%#####+++++++:::++++++%%++++%%#%%%%+++++%%%%%+++::%%%%%%%,...................................
.........,,,,.,,..,,.......................,............................,,..,..................,........,.,,..,,....................,:+::,::::::::::::::::+++++%%%%%%%%##%%%%%%%%%%+++:+++++++%++++++#%%%%++%++%%%%%+++:+%%%%%%+....................................
..........,.....,.,,,,......................................................,,............................,,.,,....,.,..............:++::,,::::::::::::::++++++%%%%%%%##%%%%%%%##@@#%++++++++++++++++%#%%%%%+++++%%%+++:+%%%%%%, .....................,.............
....................................................................,,.....,,,.,,..................,.,........,.....................++::+,:::::::::::++:++++++%%%%%%%###%%%%%%%######%+++%%+++%%%%%%%%#%%%%%%++++%%%+++:+%%%%%+.......,.............,.,...........,,
.....................................................,...............,..,....,........,........,,.,...........,..,.................,::::+,::::::::::++++++++++%%%%%%###%%%##%%%%%%%%##%+%%%+%%%%%##%%%##%%%%%++++++++++++%%%%%,..................,,.,,..............
..............................,......................,...................................,,,.,,,..,..........,.,,..,,...............:::+%,::::::::::++++++++++%%%%%%##%%%%%%%%%%%%%###%+%%%%%%%#####%%###%%%%%%++++++++++%%%%+......................,...............
..........................,....................................................................,.......................,............::++%:::::::::++++++++++++%%%%%%#%%%%%%%%%%%%%%####%%%%%%%#####%%%%##%%%%%%++++++++++%%%%,...........,................  ........
..................................................,...,..,..........................................................................,++%#:::::::::+:++++++++++%%%%%%%%%%%%%%%%%%%%%%####%%#######%%%%%%###%%%%%++++++++:+%%%+.............. ......................,.
...............,,...................,...............,.,...,........................,,.,..............................................:%%#::::::::::+++++++++++%%%%%%%%%%%%%%%%%%%%%%%##########%%%%%%%%%##%%%%%%%++++++:+%%%,........,.,..,........... .........,.,.
..............,,.......,..,,.....,............,.,.....,.....................................................,.............,...........:::,:::::::::+++++++++++%%%%%%%%%%%%%%%%%%%%%%%%########%%%%%%%%%%##%%%%%%+++++++:+%%:....,......,..,....................,....
..............,...................,,......,,.,,...,..,..,................................................. ..,...,,,,...........,.....,::,::::::::++++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%######%%%%%%%%%%%%##%%%%%%++++++:+%+...................,..,,.,............,..
.................................,.......,,,,.,...,...,.....,.....,................................ .........,.,.,,,,,.....,.,,,......:::,::::::::++++++++++++%%%%%%%%%%%%%%%%%%%%%%%#######%%%%%%%%%%%%%#%%%%%%%++++++:++,..................,,,....,.,,.,.,...,....
..................................,,.....,,,................,.....,............................................,.,,,.,.......,,.......::,,:::::::++++++++++++++%%%%%%%%%%%%%%%%+%%%%%%%%###%%%%%%%%%%%%%%##%%%%%%++++++:+:....................,,,.,......,,,,,.,....
...........,..........................,,............................................................ ............,,.......,...........:::,::::::::+++++++++++++%%%%%%%%%%++++++%%+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+++++:+:....................,..,,,,........,.,....
............................,..................................................,.............................,....,,..................::,,::::::::+++++++++++++%%%%%%%%##%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+++++:++....,,................,..,:,,.,..,,,......
...............................................................,.............................,.....,,...........,..,..................::,,::::::::+++++++++++++%%%%%%#################%%%%%%%%%%%%%%%%%%%%%%%%%%%%++++++++......,................,,,,,,,,..,..,.....
.....................,...............,....................,.................................,......,,..,......,.,,.,,.................::,,:::::::+++++++++++++++%%%%%#%++%%%%%%%%%#############%%%%%%%%%%%%%%%%%%%%+++++%+...........,.......,....,....,.,,,,..,....
.....................,,.....................................................................,,................,,,....................,:::::::::::++++++++++++++++%%%%%%%++++%%%%%%%%%%###########%%%%%%%%%%%%%%%%%%++++++:.......,,,,.,........,...............,.. .
.........................................................,,,.................................,...,,.....................,,..,........,::::,::::::++++++++++++++++%%%%%%++++++++%%%%%%%%%%%%%%%%##@###%%%%%%%%%%%%%%++%++%...,..,,,.....................,.....,,.....
...,.....,.......................................................,...........,.,,,,,..,.........................,....,.,.......,.....,:::::::::::++++++++++++++++%%%%%+++++++++++%%%%%%%%%%%%%%%%%%%##%%%%%%%%%%%%%%%%+%,......,,,......,,........,....,,......,...,
...,,..,..,............................,,..,..................,,.........,,..,,..,.....................................,............:,:::::::::::+++++++++++++%%%%%%%++++++++++++%%%%%%%%%%%%+++%%%+%##%%+%%%+%%%%%%%%+..........,.,....,.,..............,,...,..,.,
..........,................................,....,........................,,..,,...,...............,....................,...........,:::::::::::::+++++++++++++%%%%%%%++++++++++++%%%%%%%%%%%%%%%+++++%#%%%+%%+%%%%%%%%,........................,..,..............,.,
............................,.............,...,,..................,.........,,......................,.........................,,,,,%::::::::::::+++++++++++++%%%%%%%%++++++++++++%%%%%%%%%%%%%%%%++++%#%%%%+++%%%%%%%+.,...,,,,.,...,.............,.,.....,..,......
....,.,...........,................,........,.,,..............,,.,,..........,..,...............................,........,::+%%#+:+@::::::::::::++++++++++++%%%%%%++++++++++++++%%%%%%%%%%%%%%%%%+++++%%%%%+++%%%%%%%:.,......,,.,....,........,......,,...........,
.,.,................,...............,...................,,,.,.,..,............,,..,...................................,::++%###+::#@::::::::::::++++++++%%+%%%%%%+++++++++++++++%%%%%%%%%%%%%%%%+++++%%%%%%++%%%%%%%%....,..............,,.....,..,,..,,,,,,,,,..,..
..............................................................,...............,.....................................,:::+%%###+:++#@::::::::::+:+++++++++%+%%%%++++++++%%++++++++%%%%%%%%%%%%%%%++++%%%%%%%++%%%%%%%:..............,.............,.....,,,..,,,,..,,
.................,............................................,.......................,...........................,::+++%%###+:%+%@@::::::::++++++++++++%%+%%%%+++++++%%%%+++++++%%%%%%%%%%%%%%%%++++%%%%%%%%%%%%%%%,.........,.,,,...,,......,,...,.,........,.,,,.
.................,.,.,..,,........................,,.,.........................................................,:+%%+++%####++%++#@#::::::::++++++++++%%%%%%++++++++++%%%%%%++++%%%%%%%%%%%%%%+%%++++%%%%%%%%%%%%%%+.........,.,,,,:,,,.,.........,..........,......
...............,...,.,...................................................................,.......,.,........,::+%%%#%%+%%%%++%%+%#@#::::::::++++++++++%%%%%%+++++++++++++%%%++%+%%%%%%%%%%%%%%+%%++++%%%%%%%%%%%%%%,.........,.,..,..,.,,,.,..,.,,,,,........,......
,..,,....,,.,,......,.................................................... ..............................,,::+++%%%%%%%+%%+++%##%%#@#:++:::::++++++++++%%%%%++++++++++++++%%%%+++%%%%%%%%%%%%%%%++++++%%%%%%%%%%%%%%.......,.....,.........,,.,,.....,..,.,,,..,,...,
,,..,,.,,,,,,,...,..,,......,...............................................,....,,.......,...........,,:+++%%%%%%%%#%++%+%%##%%#@@#+:+:::::++++++++%+%%%%%+++++++++++++++%%%%%%%%%%%%%%%%%%%%%%++++%%%%%%%%%%%%%%:....,,.........,....,,,,....,,..,,.,,,,.,,,,,,,.,
,...,.,,,,,............,...,.....,..................... .......... ..............,,.........,.......,:+++%%%%%%%%%####++%#%%###%#@@#+:+:::::++++++++%%%%%%%%+++++++++++++++++++%++%%%%%%%%%%%%%%%+++%%%%%%%%%%%%%%....,,,,.,,.,.,,,...,,.,.,.,,,.,,,,,.,,,....,,.,.,
..,.,...,,,...............,.......................... ..........................,........,..,,,...:+++%%%%%###%####%##%+%########@@#+++++::++++++++%%%%%%%%%%%%%++++++++++++++++%%%%%%%%%%%%%%%%%+++%%%%%%%%%%%%#:....,,.,,,,,,.,,.,,.,.,,.,,..,,,,.:,,.......,,.,..
.,............... .....................,,,..................................,...,,..,...,,.,...,:++++++%%%%%%######%%%#%#########@@#++++++++++++++++%%%%%%%%%%%%++++++++++++++++%%++%%++%%++%%%%%%+%%%%%%%%%%%%%+......,,,..,,......,,,.....,,,,....,.,..,,,..,,..,,
....,.....,............................... .......................,............,.,,.....,,...:++++%%%++%%%#%%%########%%#########@@#%+++++++++++++++%%%%%%%%%%%%%%%++++++++++++%%++++%%%+++%%%%%%%%%%%%%%%%%%%#%,..........,..,...,,,...,,:.,,,:::.,.......,,,,,..,,
....,........,...,..... ......................... .........,,.....,,......................,:++++++%%%%%%%%%############%#########@##%+++++++++++++++%%%%%%%%%%%%%%%%%++++++++%%%%+%++%%%%%++%%%%%%%%%%%%%%%%%%#:..............,....,...,.,,........,.....,.,,....,,.
....,...,,.,,,,.,,.........................................,,........,........,.........,:::::++%%%%###################%%%%######@##%+++++++++++++++%%%%%%%%%%%%%%%%%%%++%%++%%%%%%++%%%++%%%%%%%%%%%%%%%%%%%#%.........,..,,,.,,,,,.,.,,,,,,,...,,.....,,..,.,,.,,,
.....,,,:,.,,.,,,......,.,................,......... ......,,...,,.............,.......,::+%#####################################@###++++++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%###:.,....,...,....,,..,..,.,,.,,,.,,..,,,,,,,,,,,.:,,,,:
....,,,.,,..,,,..........................,,,,,..,,,..,......,.......................,+%###################################@@@####@@##++++++++++++%%+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%#%%%%###+,,,........,.......,...........,.,.,.,,,,,,,,.,,,..:,,
....,,.,.............,......,,...........,,,,,.,,,,..,,................,...,..,..,+%######################%###############@@@####@@##++++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%####%%#%###::,,...........,.,,,,......,.,.,,.,,,..,,,,,,.,:,,,,.,.
............................,.......,:..,.,......................,,,..,..,.....,++%%################%%################@####@@####@@##%+++++++++++%%%%%%%%%%%%%%%%%%%%%%%########%###%#%%%%%%%#%######%%#####+:::::,...,.,..,..,,,:,,,:,.,.,.,,,,,,.,,.::.,,.,,,,,,,,
...............................,.,,,.,.,....,..,............,..,......,,.....,+++++%##################%##############@######@####@@##%+++++++++++%%%%%%%%%%%%%%%%%%%%%%%###############%%%##########%%#####%++:,::::,,,...,.,....,,:,,,.,,..,,,,,,..,.,..,,...,,.,.,
.................................,.,,,.........,.... .,.,....,.,....,......:+%%++::+%%%%%%%%%%#####################@@########@####@##%+++++++++++%%%%%%%%%%%%%%%%%%%%%%%#############################%####%:::::::,,:::,......,,...,,..,,,.,:,,,,,.,,,,.,,..,.,,,:,.
...........................,............................ ................:%%%%%%%+::++%%###%%%%###########%#####@@@##########@####@##%++++++++++++%%%%%%%%%%%%%%%%%%%%%%%##################################:::::+:::::::::,,,..,.,..,,,.,,.,,,.,...,,...,,.,,,,.,,,,
.......................,..,,.. ......,,..............,. ... .,...,,....:+%%%%%%%+%++:+%%%##%%%%%%###########@@@@@@###########@@###@##%%++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%#######################%#%%####%%%:+%+:::++%%++::,,,,,,,,.,..,::,..,,,....,,,,,,,,,.,,,,,,
.............,..,................,,.,.............,,..,..........,..,:+%%%%%%++++++++:+++%%%%###########@@@@@@@@@@###########@####@@#%%++++++++++%%%%%%%%%%%%%%%%%%%%%%%%%#########################%#####%++#++++%%%%%%++:,,,::,:::,,,,,,,,..,...:,.,,,,..,,,,,.,,,,
.....,.......,,,.................................,.,,.,...........,:%%%%%%+::::++%%%++++%%%####%%%###@@@@@@##################@####@@#%%++++++++++++%%%%%%%%%%%%%%%%%%%%%%%%###########%##########%########+%+##%++%%%+++++%%%+++:::::,,,,,,,,,,.,.,,,.,,..::,,,.,.,,
.....,,,,,,............. .,.,............................,......,:%%%%%+:::::+%%%++%%%%%+%%%%####@@@@####@@####@@############@@####@##%%+++++++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%##################%#####%##+%+++%##%++%%%+%%###%++++++++::::::,,,,..,,,,,,.,,,...,,,,
..,,,.....,.,.,......... ..............,.....,..,.....,...,,,.,:%%%%+::::::+%%+%##@##%%%%%####@@@######@@@#####@@#############@####@##%%%%%%++++++%%%%%%%%%%%%%%%%%%%%%%%######%%#########################%%%+%++%########@@#####%++++%%%%%%%++::::,,,,:...,:,,,.,,,
,,,,,.,,....,..........,.....................,....,,,,,..,,.,:%%%++:::::+%%+%####%%++++%+++++%%##########@####@###############@#####@#%%%%%%%%++++%%%%%%%%%%%%%%%%%%%#%%%##%##############%###########%####+#%+%%%++%###########%##@@@@##@###@@@@###%:,.,,,:,,,.,,..
.,....,,,,,,,,,............................  .  ..........,:++++::,,::+%++%##%++++%%%%%%%%%%+%%%%%%######@#@#@@###############@@####@#%%%%%%%+++%++%%%%%%%%%%%%%%%%%%#################################%####+%%%%++%%+%##########%%%%#######%#####%++#@#:,,,,,:,.,,,,
...........,,..,...................,....................,::::::,,::+%%+%##%+++++%%%%%%%%%%##%%%%########@@@####################@####@##%%%%%%++%%%++%%%%%%%%%%%%%%%%%################################%%####%+##%+++%%%##########%%#########%%%####%+##%#%:,..,:,.,,,
.,:,,.,,...,.,......,...,..........,,.................,++::::::::+%%+%#%+:::++%%%%#####%%################@#####################@@@#@@##%%%%%%%%%%%%%+%%%%%%%%%%%%%%%%###%###########################%%%####%+%##%+++%%%###########%%#########%####%%%#%+%##%:,,,,,.,
,,::,..,,,..,,,.....,..,.,,..,,.,...,,....,........,,+++:::+++++%%+%#%++::++%%##################################################@@##@@#%%%%%%%+%%%%%%%%%%%%%%%%%%%%%%###%###########################%#%#####+%###%%++%############%##########%#####%%##+%%++%#+,..,.
,.,,,,.,,,,,.,,.......,,.,.......,,.,,...,:..,.....:%+::+%%+++%++%#%+++++%######################################################@@###@#%%%%%%%%%%%%%%++%%%%%%%%%%%%#%%%%###########################%%#%%####%+###%%+%%#############%###############%%##%+%%+++##%:..
,,,..,,,,,,,..,,,,.,,,,,..,....,,,,,..,,,.,......:%%+::+%%++%++##%%++++%%#######################################################@@@##@@#%%%%%%%++%%%%%+%%%%%%%%%%%%%%%%%##########################%%%%%%#####+%###%%+%%%###########################%%###%%%#%+###+,,
,,,,,..,,....,,.,.....,,.....,,,,..,...,.......,+#++::%%%++%+##%%%%%++%#%%#####@#################################################@@#####%%%%%%%%%%%%%%%%+%%%%%%%%%%%%%%%%#########################%%#%%%#####%+####%++%%######%#####################%###%%%#%+%###:,
....,,. ...,,,...,.,,..,.....,......,,,.......:##+::+%%%+%%%##%%%%%%%###%%##@#@@##################################################@####@%%%%%%+%%%%%%%%%%%%%%%%%%%%%%%%#%%%%####################%%%+##########+#####%+%%############################%###%%%%#%%###%+
...,..,....,.,.......,.,...,.,......,.,,....,%##+:+%%%++#%###%%%%%%######%%#@@@@@@################################################@@###@#%%%%%%%+%%%%%%%%+++%%%%%%%%%%%%%%%%###################%%%%+##########+%#####%+%############################%####%%#%%+#####
....,.,.,. .....,,:..,......,,,,.,.....,...:###+:%%%%+%#%##%%%%%%%#######%%####@@@################################################@@####@%%%%%+%%%%+%%%%%%%+++%%%%%%%%%%%%%%#%###############%%%%%+%##########%+#####%+%############################%####%%##%+####%
..:,.............,,,,,,,,..,.............,%###++%%%%+%%%#%%####%#########%%####@@@#################################################@@#####%++++%++%%%%+%%%++%%%%%%%%%%%%%%%##%%%#######%%%%%%%%%%%+%###########+######%+%###########################%####%%##%+%###%
,..... ..........,,.,,,,,..,,.....,..,,,:####+:++%%+%%%#%#################%%###@@@##################################################@@###@#%+++%%+%%%%++%%%%%%%++%%%%%%%%%%%%%%%%%%%%#%%%%%%%%%%%%+############%+######%%#################################%##%+%####
.... .....,......,,.....,....,..,.,,:,,+%%%%%+%%#%%%%%#%##%%%%############%%###@@@#################################################@@@####@%+++%%+%+%++++%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%#############+######%+%##################################%%%%####
..... .....,.... .,,...,,............,%#%%++++++%%%%#%+++++%###############%###@@@##################################################@@######%+++++++%+%%++%++%%%%%%%+++++%%%%%%%%%%%%%%%%%%%%%%%%%%#####%#######%%######%%##################################%%%%###%
..............,,.++,,,,,,.,.... ....,+%%%%+%%%%%%%+%%%+%%%#################%####@@@#################################################@########%%++%+++%%%++%%++%%++++++++++%%%%%%%%%%%%%%%%%%%%%%%%######%########+######%+%################################%%#%%####
.. ...... .,,,::::::::,,,.,,,,.,..,,++%%%%%%%%%##%%%%%%####%###############%%##@@@@#################################################@#########%%+%%+++%++++%+%+%%%+++++++++%%%%%%%%%%%%%%%%%%%%%%%######%########%%######%%###################################%%####
....,,....,,,,,::,,,....,.....,,...::%%%%%%%%%###%%##%######%%%#############%###@@@#@@##############################################@##########%%%%%++%+++++++%%%%+++++++%%+%%%%%%%%%%%%%%%%%%%%%%######%########%%######%+%#########################%#########%####
.........,........................,::%%%%%%%%##################%%###########%##@@@#@@@###############################################@########@#%%+%+++++++++++%%%+++++++%%+%%%%%+%%%%%%%%%%%%%%%#######%%########+#######%%###################################%####
...,.......   .... ......... .....,:+%%%%%%%################################%###@@@@@@@##########################################################%%%+++%%+++%++++%+++++++++%%+%%%%%%%%%%%%%%%%%%%#######%%########%%######%%###################################%####
.........     . .... .  ... ... ..::%%%%%%%##################################%#@@@@@@################################################@############%%%+++++++++++++++++++++++%++%%%%%%%%++%%%%%%%########%%#########%#######%%##################################%####
................ ................,:+%%%%%%%################@#################%#@@@@@@@################################################@############%%+++++++++++++++++++++++++%%%%%%%%%++%%%%%%##########%#########%#######%%#######################################
......,....  .. ..........,......::%%%%%%%###################################%#@@@@@@################################################################+++++++++++++++++++++++++++%++++++++%+++%%##########%#########%%#######%#######################################
............ .....,.,.....,.....,:+%%%%%%####################@@################@@@@@@@@################################################@##############%+++++++++++++++++++++++++++++++++++++%%###########%##########%#######%%######################################
.............,,.....,...........::%%%%%%%#####################@@@#############%@@@@@@@@################################################@@###############%++++++++++++++++++++++++++++++++++%%###########%%##########%#######%%######################################
...,...............,....,,.....,:+%%%%%%######################@@@@##############@@@@@@#################################################@##################++++++++++++++++++++++++++++++++%%############%%##########%%#######%######################################
,,..,..,..........,,..,........::%%%%%%%#######################@@###############@@@@@@@####################################################################%+++++++++++++++++++++++++++++%%#############%############%#######%%#####################################
.,,..........,..,,.,.,,...,...,:+%%%%%%########################@@################@@@@@@#######################################################################%+++++++++++++++++++++++++%###############%############%#######%%############################@########
.................:,,....,.,...::%+%%%%%########################@@@###############@@@@@@#########################################################################%+++++++++++++++++++++%%################%############%%#######%############################@########
........,........,:::........,:++%%%%%%##########################@@##############@@@@@@################################################@@##########################%++++++++++++++++%%%#################%############%%#######%%####################################
..............,,,,,:,,,...,..:+%+%%%%%###########################@@@#############@@@@@@@##############################################################################%%++++++++++%%%###################%#############%#######%%##########################@#########
............,...,,,,,.:,.,,,,:++%%#%%############################@@@##############@@@@@@########################################################################################################%#####################%########%##########################@########%
 ....,......,,,.,.::,,::. ..::%+%%%%%############################@@@##############@@@@@@########################################################################################################%######%##############%########%##########################@#########
...........,..,,..,,.,+:,:,.:+++%%%%#############################@@@@@@@@##########@@@@@###############################################################################################################%#######################%##########################@#########
,,,,..,.,,,,,,,:,,,,:,:,,:,,+++%%%%###############################@@@@@@@##########@@@@@@######################################################################################################################################%#########################@##########
,,,,:,:,,...,,.,,,,,::,,,:,:+++%%%%################@###############@@@@@@@#########@@@@@@################################################################################################################################################################@##########
,,::,,,,,,,.,..,,.::,:,,:,,:+++%%%#################################@@@@@@@@#########@@@@@################################################################################################################################################################@##########