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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⠤⠐⠒⢲⣀⣀⣀⡤⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠠⠐⠒⠈⠁⠀⣀⣤⠤⠖⠚⠉⠃⠉⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠒⠀⠉⠀⠀⢀⣀⠠⣤⣐⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⡉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠄⠐⠂⠉⠀⣀⣀⣤⣤⢤⣶⣮⠽⠟⠛⠉⢉⣼⠏⠀⠀⠀⠀⠀⣀⠤⢐⠂⢠⠀⡐⡰⠰⠡⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠤⠆⠒⢒⡀⠨⠍⢒⡈⠌⠓⠈⢉⣀⣤⣤⠴⠶⠒⠛⠉⠀⣀⠠⢐⡨⠕⢂⡩⠐⡢⠐⣱⡡⢁⢂⠂⠱⣇
⠀⠀⠀⠀⠀⠊⠉⣁⠀⠠⠄⠒⢈⣩⡤⢖⣊⡁⠐⠆⠝⠓⠊⠉⠀⠀⠀⠀⢀⡠⣤⠒⠅⠂⢉⠤⠤⣨⡬⡍⠹⣇⢠⢻⢧⠂⢄⡴⢡⢩⡄
⠀⠀⠀⠀⠀⠀⢃⡡⠤⠒⠂⠩⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣒⠨⠑⢈⠅⡄⢨⠂⠆⢎⡔⢧⣺⡷⠂⢙⣧⣾⠿⠎⢄⢃⢂⢂⢷
⠀⠀⠀⠀⠀⠀⠣⠀⡰⢯⢳⠹⣆⠀⠀⠀⠀⠀⠀⠀⣀⢄⠢⠍⣂⣉⠀⡮⡟⡂⢛⡶⠙⢧⣼⠄⠚⢀⣁⡴⠮⠏⠏⡄⠌⢜⠄⠆⠂⢎⢘⡄
⠀⠀⠀⠀⠀⠀⠐⣫⢷⠈⡄⡶⡟⡀⠀⠀⣀⣤⠖⠛⢉⠀⠄⢘⡘⡫⠂⠈⡍⡘⡆⠛⠚⣀⢤⡴⠞⠋⠁⠀⠌⠌⡜⡀⡠⠎⠌⠌⠌⠆⠌⠇
⠀⠀⠀⠀⠀⠀⠐⡙⣟⣧⡈⠀⢻⢃⠀⠠⠀⠠⠄⠠⡄⠠⡫⡀⠑⣸⠀⠀⢈⢉⣤⠢⠙⠉⠁⠀⠀⠀⠀⡌⡘⡐⡐⠐⡐⠜⡈⡐⡈⡘⠈⡸⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠜⢿⣳⡀⠨⡌⡄⠀⠱⣦⠀⠀⣡⠀⠑⠒⡀⠄⣐⠬⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⡜⡐⡐⡠⠠⠑⡘⡐⡐⠐⠠⠠⠁⠐⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⢿⡕⠒⢿⢀⠀⠀⠱⡡⣀⢬⠐⠾⠓⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡰⡱⠠⠡⡡⡐⠠⠠⠡⠡⢡⠡⠡⠱⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⢿⣄⠈⡏⡄⠀⠀⠐⠊⠁⠀⠀⠀⠑⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠡⠡⠡⢡⡱⡐⠰⢁⢡⢢⢂⢃⢁⠢⢁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢎⢻⣆⠹⡔⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢊⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⠡⠡⢡⠡⠱⢠⢃⢂⢂⠂⠆⢂⣂⡂⠥⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢫⣦⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠣⢡⢁⢡⠂⠣⠁⣃⡈⠌⠄⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢻⣻⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣃⠁⠤⠐⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⡿⠀⠀⣀⠠⠄⠐⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡀⢀⠀⢀⠀⡀⠀⡀⢀⠀⢀⠄⡀⠀⡀⠀⡀⢀⠀⢀⠀⡀⠀⠠⠄⡀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠐⠁⠂⠅⠂⠔⠨⠀⠅⠂⠔⠈⠔⠨⠀⠁⠂⠕⠨⠐⠀⠂⠅⠂⠔⠊⠀⠊⠐⠁⠐⠁⠂⠡⠐⠉⠂⠁⠀⠙⠉⠊⠐⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠑⠢⢄⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠒⠤⣈⠑⠢⢄⣀⠖⠁⠑⢄
⠀⠀⡖⠒⠒⠒⠒⠒⠒⠒⠒⢢⠀⠀⠀⡜⠀⠀⠒⠤⣀⠀⠉⠒⠤⣈⠑⠪⢷⣄⠑⢤⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⡞⠀⠈⠒⠤⣀⠀⠉⠒⠤⣀⠀⠉⠒⠀⡜⠳⢄⠙⢦⣀⠤⠔⠚⡄
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠈⠒⠤⣀⠀⠉⠒⠤⣀⠀⠉⠒⠀⡜⠳⣄⠀⠑⢄⠙⠢⡐⡄⠘⡄
⡠⠒⠓⠒⠒⠀⠀⠀⠐⠒⠒⠒⠚⠒⠚⠒⠂⠒⠒⠛⠒⠒⠒⠒⠛⠒⠂⠚⢧⣄⠈⠳⢄⠀⠑⠀⡈⠚⡄⠹⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠳⣄⠀⠁⣠⠞⢣⠀⠸⡄⠱⡀
⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠳⡄⠀⢀⠔⠙⡄⠀⢧⠀⠹⡄⠱⡀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⢀⡔⠁⢣⠀⠘⡄⠈⢧⠀⠱⠀⠱
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢴⠁⠘⡄⠀⢧⠀⠹⠄⠈⠀⣀⠤⠒
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢣⠀⠘⠄⠈⠁⣀⠤⠒⠉
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠘⠄⠀⠁⠠⠔⠊⠁
⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡤⠔⠈
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣤⣤⣬⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠟⠋⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⢿⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠟⠋⠁⠀⣠⣴⣾⡧⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢀⣠⣴⣿⡿⠛⠉⠀⣀⣤⣶⣿⠿⠋⠁⠀⣀⣴⣶⡀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠻⠟⠋⠁⢀⣠⣴⣾⡿⠟⠉⠀⢀⣠⣶⣿⠿⠛⠁⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⢠⡖⠲⠶⠤⣿⡄⠀⠀⠀⣤⣶⣿⠿⠋⠁⠀⣀⣴⣾⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠹⣆⠀⠀⠙⠉⠀⢀⣠⣶⣿⠿⠛⠁⠀⣀⣤⣾⣧⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢰⣼⣆⠀⠀⠀⣾⡿⠟⠋⠁⢀⣠⣴⣿⡿⠛⠉⠀⢀⣤⣶⣿⠿⠃⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⢀⣠⡇⠀⠀⠀⣠⠶⠛⠛⠛⠓⠒⠒⠛⠒⠒⠒⠚⠛⠛⠛⠛⠒⠲⣶⣾⡿⠟⠉⠀⢀⣠⣶⣿⠄⠀⠀⠀⠘⣦⣤⣤⣀⣀⣀
⣤⣤⣤⣼⣽⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⣤⣤⣴⣾⣿⣿⣯⣥⣤⣤⣤⣤⣤⣬⣧⣤⣤⣬⣭⣿⣤⣤⣤
⢻⡄⠀⢸⡟⠀⠀⠀⠀⡏
⠘⣧⠀⢸⡇⠀⠀⠀⢸⡇
⠀⢹⡄⣸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠈⣷⡟⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣼⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢀⡏⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠸⠧⣤⣤⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⢸⠀⠀⠉⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠘⠓⠒⢶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿

⠀⢀⡀⠀⠀⠀⠀⢀⣼⣷⣤⡀
⠀⢸⡿⣷⣤⡀⢀⣾⡟⠀⠉⠻⢷⣤⡀
⠀⢸⠇⠀⠙⠻⣿⣿⣄⠀⠀⠀⠀⠉⠻⢷⣤⡀
⠀⣿⠀⠀⠀⠀⠀⠙⠻⣷⣦⣀⠀⠀⠀⠀⠉⠻⢷⣤⡀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⡀⠀⠀⠀⠀⠉⠛⢷⣤⡀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⠀⠀⠀⠀⠀⠈⣿⡆
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⢹⣿
⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠈⠿⣷⣤⡀
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⡀⠀⠀⠀⠀⠀⠙⠻⣷⣤⡀
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⣀
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣀
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡆⠀⠀⠀⠀⠀⠀⢠⣿
⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠈⠙⠿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠉⠻⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣄⡀⠀⠀⠀⢸⣿⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⣀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⠃
⣶⢶⢶⢶⡶⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣿⢶⡶⣶⣶⡶⠶⣶⢶

⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⣠⣾⣿⡿⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⢿⣿⣷⣄
⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷
⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⢀⣴⠶⠿⠿⠿⠿⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣄⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⣰⡿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣷⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⣿⡇⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⣿⡇⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⣿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠘⠿⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠈⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿
⠀⠀⠙⢿⣿⣷⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣾⣿⡿⠋
⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⢀⣴⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢶⣄
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⣠⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⠾⣯⣍⣉⠙⠛⠶⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣸⣟⠛⠶⣤⣀⡉⠛⠶⢤⣄⡈⠉⠛⠳⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⣿⣼⠯⠝⠛⠶⢬⣍⣛⠳⠶⣬⣍⣛⡓⠶⢦⣤⣍⣙⣛⠳⠶⢦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠈⠉⠛⠻⠶⢤⣤⣀⡀⠀⠉⠛⠲⠶⣤⣍⣙⠛⠳⠶⢤⣭⣉⣙⠛⠲⠦⣤⣍⣙⠛⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠈⠛⣦⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⠦⣤⣌⣉⠛⠳⠶⣤⣄⡉⢷⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⣤⠀⣿⣸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠶⢦⣤⣄⣀⡀⠀⠀⠀⣿⠀⡇⣿⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣶⡀⣿⢸⡇⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⣿⢸⡇⡏⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣹⢸⡇⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠶⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠶⠶⢦⣤⣄⣀⣀⣀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠂⠉⠉⠉⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣦⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢚⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡟⠀⣠⣾⣿⣦⡀⡰⣯⣿⡿⠛⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠐⢇⠀⠈⠻⣿⣿⣿⣟⡿⠋⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠉⠑⠚⠁⠀⠀⠀⠈⠉⠉⠉⠁⠀⠐⠒⠒⠒⠒⠒⠚⣆⠀⠀⠈⠻⠿⠋⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⢀⣤⡶⣶⡟⠛⢛⡟⣻⠛⢻⠀⠀⠀⠀⠀⡏⢀⡄⢸⠀⠀⠀⠀⠀⣀⣀⣀⣁⣀⣉⣉⣈⣉⡀⠉⠁⠀⠂⢰⡿⣥⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⢸⣿⣾⣿⠛⠚⠛⠚⠓⠚⢻⡇⠀⠀⠀⠀⠳⠬⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠾⠮⠾⠵⣵⣾⣋⢨⣿⡄
⠀⠀⠀⠀⠀⣾⣿⢟⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠂⠠⠀⢹⡃⢺⡇
⠀⠀⠀⠀⠀⢹⣿⠎⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡤⠤⠄⢤⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠤⠤⠤⠤⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⡕⢸⡷⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠏⢹⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⡇⠰⠎⢸⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠄⢸⣇⣾
⠀⠀⠀⠀⠀⢸⣿⣰⣼⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠉⠁⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⠀⠀⠁⣸⢏⣿
⠀⠀⠀⠀⠀⠸⣿⣾⣿⡇⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣀⡠⠤⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⣿⢡⡟
⠀⠀⠀⠀⠀⠀⣿⡿⣹⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⢠⠆⢸⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⣤⣤⣤⣄⠄⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⣿⣵⡿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⢸⡧⣺⡇
⠀⠀⠀⠀⠀⠀⢿⡿⡷⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠆⣽⠁
⠀⠀⠀⠀⠀⠀⢸⡷⠇⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡏⢀⡄⢸⠀⠀⠉⠉⠉⣉⣉⣉⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣟⡁⠀⠀⠀⠀⣿⡾⣿
⠀⠀⠀⠀⠀⠀⢸⣧⡞⢼⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠳⠤⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡂⠀⠀⠀⠀⣿⢞⡏
⠀⠀⠀⠀⠀⠀⢸⣯⡾⣻⡇⠄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⢰⣷⣻⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣰⢟⡇⠀⠀⠀⠀⠘⣧⣀⣀⣄⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⠀⢸⡇⣿⠃
⠀⠀⠀⠀⠀⠀⠸⣿⢋⠎⣧⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠈⠉⠈⠉⠁⠉⠙⠉⠉⠉⠉⠙⠋⠙⠛⠛⠛⠛⠛⠓⠁⠀⠀⠀⣼⣷⣿
⠀⠀⠀⠀⠀⠀⠀⣿⢈⣜⠿⠤⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡟⣿
⠀⠀⣀⣀⣀⣀⣀⣿⣾⣿⣟⣵⣟⣔⣑⣴⣿⣽⣫⣫⣟⣝⣉⡟⡩⡫⠫⣫⠟⡕⣩⡵⠣⢪⡿⣯⠟⣽⢟⠝⡩⣩⢯⡽⣩⣿⡿⢋⡼⠫⢊⡿
⣿⡉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠒⠚⠷⠾⠖⠛⠒⠓⠛⠓⠒⠛⠓⠒⠒⠒⢲
⠈⠓⠶⠶⠴⠦⠴⠤⠴⣧⣦⣦⣤⣦⣦⠤⠴⠦⠶⠦⠴⠦⠶⠤⠶⠤⠦⢤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣦⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣠⣀⣄⣄⣀⣀⣀⣀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢰⢀⣀⠀⢀⠀⣀⣀⡀⡀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠒⠒⠒⠒⡄⢹⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⠘⠒⠐⠒⠒⠒⠒⠂⠒⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⠀⠀⠀⠀⢸⢠⣀⣀⣀⣀⣀⣀⣀⣀⣈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠈⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢸⢸⣀⣀⣀⣀⣀⣀⣀⢀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡸⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠠⠠⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⣀⣀⣒⣒⣒⡒⣂⣀⣀⢀⠀⠀⠀⠀⠀⠒⠒⢒⡲⢁⡇⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢨⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠇⠀⠀⠸⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠦⣤⣤⣤⣤⣤⣤⣤⣴⠴⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠁⠉⠉⠀⠀⠀⠈⠀⠉⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣉⡉⠉⡉⠉⠉⠉⠁
⠀⠀⠀⡔⢀⠀⠠⢀⠀⢀⡀⠀⣀⠀⢀⡀⠀⡀⠂⢀⠀⣀⡀⠀⣀⣀⢀⡀⡀⡀⠀⢀⠀⢀⡀⠀⢀⠀⠀⡀⠠⠀⠀⣠⠈⢠⡤⣄⠀⠑⣄
⠀⠀⡼⠀⠦⠤⠤⠬⠤⠦⠡⠴⠌⠠⠄⠥⠴⠭⠥⠤⠥⠤⠬⠤⠬⠥⠬⠭⠴⠤⠤⠤⠅⠰⠨⠄⠀⠠⠀⠸⠄⠣⠀⠈⠓⠓⠲⠬⠦⠤⠊
⠀⠀⠳⠖⠲⠒⠒⠒⠶⠲⠖⠒⠒⠖⠒⠒⠒⠒⠒⠒⠒⠒⠖⠒⠒⠒⠒⠖⠲⠒⠒⠒⠲⠒⠒⠶⠒⠒⠒⠖⠶⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠽⠋⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢊⠕⠉⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠘⡀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⢻⡀⠀⠀⢠⠃⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣴⣠⡀⠳⡀⠀⠸⡀⠀⠀⠀⠀⡠⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⢳⣷⠀⣠⢛⡩⠭⠭⣻⡶⣚⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠙⠤⠥⣰⠞⠁⠈⠉⠒⠦⣀⡀⠀⠀⠀⠀⢀⠎⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣊⡛⠀⠀⠀⠁⠀⠘⣆⠀⠀⠀⠀⠉⠑⠢⢄⡀⢜⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠈⠁⣩⡄⠀⠀⠀⠀⠀⠊⠳⣄⠀⠀⠀⠀⠀⠀⠉⠛⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⡇⠀⠀⠀⠀⠀⠐⠀⠈⠑⠦⣀⠀⢀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⢒⡶⢶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⡜⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠰⢀⡠⢮⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣹⣍⣇⠀⠀⠀⠐⠉⠀⠀⡇⢀⠀⢀⢧⠋⣠⣾⢧⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠌⣑⠲⠤⣄⣴⠤⣤⢿⠋⡇⠾⠞⡄⠉⠀⠸⠒⠒⠾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠉⠲⣤⣈⢣⢀⠗⢧⣧⠀⢀⠟⢆⡀⢚⠉⠉⠲⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠈⠓⢬⡑⠿⣞⠤⠀⡛⣷⠾⡀⢒⣩⣣⠬⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠉⠢⣈⠙⠶⠗⠁⣠⠞⢢⢇⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠈⠳⣄⣠⣞⠙⢖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⡟⠃⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠒⠶⠤⢄⣀⣀⡀⠀⠀⠀⣼⣄⣀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⢀⠔⡡⠎⢉⠜⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⣠⠤⣶⡥⠎⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣤⣤⣼⣁⣀⣀⣀⣠⣎⣀⢠⣾⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠛⠓⠈⠉⠉⠀⠀⠈⠉⠈⠛⠐⠀⠬⠽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⡟⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡙⢻⡆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣾⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⣷⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠿⠿⢿⣶⣶⣦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣠⠤⠤⣤⣀⣀⡉⠉⠙⠛⠿⠿⣷⣶⣶⡤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠀⡟⠀⠀⠀⠀⠉⠉⠙⠓⠒⠶⠤⣤⣀⣉⠉⠛⠚⠿⠿⢷⣶⣦⣤⣄⣀
⠀⠀⠀⣀⣠⠤⠖⠛⠉⠀⢠⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠤⣤⣀⣈⡉⠉⠛⢿⢷
⣴⣾⣋⣉⠀⠀⢀⣠⠴⠒⢻⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡇⠀⡟⣸
⡇⠀⣀⡉⠉⢻⠉⠀⠀⠀⡿⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢸⢃⡇
⣇⠀⡏⢹⠀⢸⠀⠀⠀⢠⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡾⢸⠁
⣿⠀⣿⢸⠀⢸⠀⠀⠀⢸⠃⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⡇⡟
⢿⠀⣿⢸⠀⢸⡆⠀⠀⣾⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣼⢸⠇
⢸⠀⢸⢸⡄⢸⡇⠀⢀⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⡇⡾
⢸⡄⠘⠾⠇⠸⡇⠀⢸⠃⠈⠙⠓⠲⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣼⣸⠇
⠘⡇⠰⣿⡇⠀⡇⠀⠈⠙⣒⡶⢤⣀⡀⠀⠈⠉⠙⠒⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢰⣏⡿
⠀⠙⠲⠦⢤⣤⣧⠴⠞⠋⠁⠀⠀⠀⢉⣽⠶⠦⣤⣀⡀⠀⠈⠉⠛⠲⠦⣤⣀⣀⠀⠀⢀⡏⠀⣼⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠯⣭⣙⡓⣶⣤⣄⣀⡀⠀⠉⠉⠛⠚⠁⢠⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢦⡀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⢹⡟⠓⠶⠤⣤⣀⣼⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣍⠳⠦⣄⣀⡀⠀⠀⠀⠀⢀⣠⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠬⣭⣭⣛⣛⣛⣭⠴⠋

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠒⠒⠒⠒⠒⠀⢄
⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠠⠄⠠⠀⠀⠀⠀⢡
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠂⠖⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠑⠢⠀⢀⣀⡀⠀⠠⠂⠄⠀⠀⠀⠀⠈⠉⠉⠉⠉⠏⠉⠉⠉⠩⠽⠿⠿⠿⠿⣷⠀⠀⠂⠀⡠⠀⠀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠦⠂⠀⠀⢀⢂⠄⢎⣡⠀⠀⠀⠀⠀⢰⡠⠤⠔⠀⠀⠀⣿⠀⠤⠄⠊
⠀⠀⠀⠀⠀⡠⠔⠂⠉⠉⠉⠑⠢⡀⠀⠄⠀⠀⠰⠾⡀⢸⠣⠃⠀⠀⠀⠀⠨⡁⠀⠀⠀⠀⠀⣿⠀⠀⡠⠀⠂⠈⠀⠀⠈⠐⠢⡀
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠉⠀⠀⠀⠈⠄⠀⠀⠀⠀⠢⣙⡓⠂⠄⠀⠀⠀⠀⠠⡳⣴⠣⠄⠀⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠂⠀⠀⡁⠀⠀⠉⠉⡡⠀⢄⠀⠀⠁⠈⠉⠉⠁⠉⠉⠍⡉⠉⠁⢸⠀⠂
⠀⠀⠀⠀⠀⠑⠢⠄⠀⠀⠀⢀⡀⢎⠀⠀⠀⠀⢐⠒⠐⠄⠀⠀⠀⠀⠀⠀⢈⠊⠀⠓⣤⠀⠀⢸⠀⠈⠂⠀⠀⢀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠆⠈⠀⠀⠀⣠⣄⣤⡀⠀⠀⢰⠀⠀⠈⡳⢄⠴⡓⠀⠀⢸⠀⣠⠊⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣂⣀⣀⣀⣒⣠⣬⣦⣤⣴⣤⣤⣤⣬⣦⣶⣤⣥⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠤⠠⠄⠀⠄⢠⠈⡗⡺⣾⣽⣧⣗⣛⣗⠒⠚⠛⠛⢭⡷⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⡤⠬⠤⡅⢭⠸⡅⢯⠿⣿⢹⡿⣤⠀⡤⠕⣀⠱⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠤⠤⠤⠤⠤⠍⠉⠁⠨⢭⣁⣉⣉⣩⠭⠽⠬⢤⢬⠭⣬⣥⣭⣿⣆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠈⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣤⣄⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣁⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣁⣠⣠⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⡟⠋⠉⠁⢠⣉⡉⠉⠀⠈⠉⠛⢳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠁⠁⠀⠀⠀⠈⠿⠿⠀⠀⠀⠀⠀⠀⠀⢈⠙⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠀⠀⠀⠀⠀⢀⠀⠀⠐⠂⠀⠐⠂⠀⢀⠀⠐⣾⣿⠂⠙⢻⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠰⣾⣶⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣂⣀⣀⣤⣤⣬⣿⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣇⣀⣀⣁⣀⣤⣤⣤⣤⠶⠶⠶⠶⠒⠛⠛⣛⡋⢉⣉⣉⡉⣤⣤⣤⣶⣶⠌⡇
⠀⠀⠀⠀⠀⢸⠛⣛⣉⡉⣉⣭⡁⢠⣤⣤⠀⢠⡶⣶⡴⣿⠟⢸⡟⣿⡎⣿⣽⡷⢿⣧⡄⢹⣇⠀⣷
⠀⠀⠀⠀⠀⢸⡆⢻⣿⠁⣿⣿⣿⢸⣷⣿⠆⠘⠿⣶⡄⣿⡿⢸⣿⢠⣄⣿⡿⣷⢸⣿⣀⢸⣿⠀⢿
⠀⠀⠀⠀⠀⠘⡇⠀⣿⡄⢿⣇⣿⣼⣿⠁⠀⠀⢿⣼⡟⢿⠷⠎⠻⠾⠃⠛⠃⠛⣈⣉⣉⣀⣩⣤⣼⠄
⠀⠀⠀⠀⠀⠀⣧⠀⠻⠃⠈⠛⢋⣀⣉⣀⣀⣤⣤⣤⣤⠴⠶⠶⠶⠞⠛⠛⢛⠋⠉⠉⠉⢉⣽⠇
⠀⠀⠀⠀⠀⠀⠻⠶⢾⡟⠛⠛⠋⠉⠉⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠰⣿⡖⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣥⡀⠰⣾⣷⠀⠈⠀⠀⠤⠄⠀⠤⠀⠐⠈⠀⠀⠀⠀⠁⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⡅⠀⠀⠀⠀⠀⠀⠀⢠⣶⡤⠀⠀⠀⠀⠀⣴⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⣤⣄⣀⠀⠀⣀⣀⡋⠃⣀⣀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣵⣽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠿⢶⣽⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⡹⣧⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠉⢦⣄⡀⠀⠀⠀⠀⠀⣹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⠀⢀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠂⠀⣀⣀⠀⠀⠀⠺⣹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⣤⠙⠉⠉⠛⠿⠛⠿⣷⣤⡄⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡜⠁⠀⠀⠀⠀⠀⢸⡃⢸⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⢠⡀⠨⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣠⡾⠋⠘⢯⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠣⡔⠉⠀⠀⠀⠰⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⠄⠀⠀⠀⠘⠇⠀⠴⠋⠀⠀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠠⠀⠀⢳⠞⡀⢪⣀⠀⠀⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠀⢀⡟⠻⠃⠐⠈⢛⣿⠏⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠋⠀⠀⢀⣠⣤⣶⣾⣿⣿⣥⣼⣟⣠⣤⣶⣶⢶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣽⠿⣻⣿⡟⣰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⢿⣿⣿⠿⣯⣛⣭⣿⣟⠻⠭⣽⣾⣷⠟⠁⣰⠛⠟⣾⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⣿⠝⢻⣿⡔⣫⣿⡟⢿⣿⣯⣿⣿⢻⣿⣷⣴⢿⡿⠁⠀⠀⡇⣶⣦⣿⠛⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⣻⣿⣄⣀⣠⡼⢿⣿⣼⢿⣿⣾⣿⣻⢿⣿⢷⡿⣟⣾⠟⠁⠀⠀⠀⡇⣿⢿⡏⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⣟⣽⣿⢿⠿⣿⠿⣿⣯⣛⣿⣿⢾⣏⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⢀⣣⠁⣸⣇⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⡿⠿⠛⠛⠻⠿⢿⣿⣷⡝⣿⡿⠿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣽⠏⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢀⡶⣾⡿⠛⠁⠀⠀⢠⡀⠀⠀⠀⠀⢀⠃⠀⠀⠙⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠐⠄⠀⠀⠀⠀⠀⢀⣾⣁⣠⢀⡔⠦⠘⣧⡀⠀⠀⣠⡿⠀⠀⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠘⠳⣆⢰⡆⠺⠠⣏⠀⠀⢹⡄⠘⠳⠦⠀⠈⢁⣴⢾⣿⡇⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⣠⣼⣶⣯⣬⣽⣆⣀⣉⣉⣀⣀⣤⣤⣴⡶⠶⢶⣿⣷⣾⣿⠗⠠⣤⣤⣴⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣠⣶⣿⣿⣿⡟⣛⣴⣻⣷⣿⣿⣻⠿⠿⢿⣿⣷⣾⣿⣦⣈⢀⣬⣿⣛⢷⣦⣀⣀⠀⠀⠙⠻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣠⣾⣿⣾⣿⣿⣿⣗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣶⣦⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣙⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛⠛⠉⠛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠋⠉⠁



⣄⣀⣀⣀⣀⣤⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠿⠋⠿⢿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣛⢛⣛⣛⣿⣿⣟⡛⢻⡟⢛⠛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡄⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠾⠃⢤⡌⠁⠘⠀⡀⢄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠒⠐⠁⡶⡰⢤⣚⠉⠀⠀⠈⠃⠀⠁⡀⠀⠁⢠⣀⢀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠲⠄⠄⢸⣏⠀⠑⠎⠡⡀⠀⠀⠀⠈⠀⠀⠉⢀⠀⢀⡀⠉⠄⠀⠅⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⡀⠀⢀⣄⢠⡈⠉⠈⣀⠀⠀⠘⠃⠀⡘⠀⠀⢀⠀⠈⠓⢂⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⣠⠀⡀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠸⠠⠀⠀⠀⠀⠀⠐⠀⠀⠙⢋⣀⣼⢶⡀⠀⡲
⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⣤⣀⡟⡳⠀⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠈⠿⣀⡆⠉⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠙⡏⡯⠉⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⣀⠀⠀⠀⠈⠙⣉⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⢘⣎⣤⠀⠀⠀⠀⠁⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠈⠀⠀⡄⠀⠀⢇⡿⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠠⠀⣀⠀⣸⢠⡤
⠀⠀⠀⠀⠀⠀⡀⠀⡄⢀⡄⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢻⡉⠓⡦⢀⠐⠃⠀⢀
⠀⠀⠀⠀⠀⣠⣁⡴⠁⠈⠉⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣰⠐⡠⣄⡀⣀⡈⠗⡎⠠⠍
⠀⠀⠀⠀⠴⠟⠻⢶⡀⠑⠈⠀⠀⠐⠁⠈⠁⠀⡄⠐⠀⠀⠀⠀⠀⠀⠀⢀⣀⡲⣴⡾⠖⡉⢻⣿⣧⢶⢷⣦
⠀⠀⠀⠈⠴⠀⢤⣼⢽⠀⠀⠀⢀⢀⡀⡄⠄⠀⢀⢸⠄⠀⠀⠀⠀⠀⠀⣜⠁⠀⢘⡗⠀⠀⠙⠻⠂⠋⡀⠎⠁
⠀⠀⠀⠀⠀⠀⡝⠁⠈⠀⠁⠤⠬⣈⠄⢀⠔⠀⣮⣼⠦⠀⠀⠀⠀⠀⠀⣝⠉⠉⠉⢳⣴⡆⠀⠉⠀⠰⠗⠂⣉⠁
⠀⠀⠀⠀⡄⠀⠂⠀⠀⢠⠖⠀⠀⠀⠀⢀⠀⠨⢿⣿⠑⠂⠀⠀⠀⠀⠀⠈⢆⠠⣴⢀⠀⠀⠂⠀⠀⠀⠀⠀⡆⠀⣣⡀⠄
⠀⠀⠀⠁⠀⠀⠀⢁⠀⠾⠀⠀⠀⠀⠀⠘⡀⠠⠤⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡥⠀⠀⠀⠀⢀⣰⡃⠙⠷⠇⠶⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠐⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠁⠉⠈⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⡟⡛⢛⡛⣻⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡚⠛⠛⢿⣀⡀⣄⠀⠄⠂⠘⣆⠀⠀⠀⢿⣧⣦⠀⠀⣼⣷⣿⠘⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⡇⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡀⢻⣽⠀⠄⢻⣿⣷⣝⣿⡄⣗⠁⠺⢶⢆⢀⣤⡿⠟⣠⡿⠛⠛⠛⠀⠃⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⡄⠂
⠀⠀⠀⢠⣤⣼⡿⠇⠀⠀⠀⠀⠋⠀⠉⠉⠀⠾⠛⠀⠘⢛⢫⠁⠀⠰⡻⢠⣤⣶⠆⠀⠀⡀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠈⢉⠉⠋⠀⠀⠠⠀⡀⠀⠀⠀⠀⣠⣤⡠⠀⠀⢰⡊⠁⣷⣦⣤⣾⣿⣿⣶⡆⠀⢸⣷⣶⣴⡄⠂⢰⢠⠀⠀⢀⢠⢶
⠀⠀⠀⠀⠠⡄⠠⠀⠰⢮⣻⣦⡀⠀⠀⠙⠛⢷⣦⣴⣿⠇⡈⠈⢫⣏⣿⣿⣿⡿⣿⣅⢀⣻⡏⢉⡐⣶⣻⣛⡀⠀⠀⠒⠁⣀⣀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⣀⣇⡻⠆⠀⣻⡉⠗⡀⠀⠑⠀⢙⠋⡛⠤⣷⣾⣿⣿⣯⣻⣷⣷⣿⣿⣿⣽⡷⠋⠹⠶⠀⡈⠲⣷⣷⣇⣤⢤⠿⠀⠠⠠⡀
⠀⠀⠀⠈⢹⡄⠉⢤⣤⡄⠀⢹⡗⣿⣷⣾⣭⡁⣈⣴⠉⣼⣿⣿⣿⣋⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠸⣧⡾⣿⠟⣿⣿⣿⣭⢿⣽⠂⠏⣾⢃
⠀⠀⠀⠀⠞⠁⣶⠀⠀⠄⠀⢸⡏⠀⣿⣿⣿⡗⠛⠿⢿⣷⣿⡇⣹⣿⣿⣿⣿⣿⣿⣟⣿⣽⣿⢷⠀⣿⡗⠠⠲⣾⣿⣿⠿⢿⠟⣿⣤⢋⢀
⠀⠀⡇⠀⠓⠀⣿⠀⠀⠀⠀⣾⣷⠀⢽⠍⢉⣡⣴⡀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⢸⣿⠂⠀⣠⣿⣿⣿⣠⣾⣿⣭⣭⠋⡅⠀⡀
⠀⠠⡇⠘⣊⠑⣿⡀⣀⣤⣰⣿⣿⠉⣻⣰⣿⣿⡟⣿⡖⣿⣿⣿⣿⣫⢯⣽⣿⣿⣿⣷⣯⣿⣿⣿⣿⣿⣧⣾⣿⣿⣟⣿⣿⣿⣿⡿⠋⠀⠀⠘
⠀⠠⣷⠀⠛⠱⣿⠻⠿⢁⣛⡛⣿⠞⣿⣛⣿⣿⣟⠫⢝⣻⣿⣿⣿⣿⣿⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⢉⣿⣿⡿⢻⠏⣩⣉⡤
⠀⣀⣿⣶⠀⠀⣿⠀⠀⠈⠉⢠⣿⡇⣿⡽⠿⢾⣗⣷⠷⣴⣿⣿⣿⣯⣽⣿⣿⣠⣤⠀⢻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣴⣿⠟⠀⠈⠁
⢀⠈⢻⡿⣱⠄⡽⠀⠀⢀⠀⣷⣿⣷⣿⡇⠰⠸⡗⠀⠀⠀⠰⠿⠟⢿⣶⣾⠿⠟⠛⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠉
⠀⠙⢷⠟⢁⠀⡁⠀⠀⠈⠁⣀⣿⣿⣿⡆⠀⢈⣇⠤⠖⠛⠃⠀⠓⠚⠛⣿⠷⢶⠶⠂⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⡄⢬⡹⠋⠀⠀⠀⠀⠀⢤⣼⣿⢛⠉⠀⠀⠨⣿⠀⠀⠀⠀⠀⠀⠀⠠⢤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⠕⢄
⠁⢙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⢻⡏⠀⠌⠀⠁⠋⠁⠀⠀⠀⢰⣤⣿⠛⠛⠚⢟⠿⠿⣿⣿⡿⢿⣿⡿⠛⠻⢿⣿⣏⠀⡀⠀⠀⠀⠠⠤⢤⣤
⠀⣉⡈⠁⠀⠀⠀⢀⣀⠀⠀⠀⠀⠓⠒⢄⡈⠋⡙⠁⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢐⠒⠀⠀⠀⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣟⣻⣭⠽⠷⠶⠶⠶⠿⠯⣽⣟⡿⢿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⡿⣫⡷⣛⠭⣓⡊⣭⠡⣿⡌⢱⡏⢩⣔⠪⢝⡳⢮⣟⢿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣽⢟⠵⠊⢶⣧⡸⡇⠸⠆⠿⠇⠻⠅⠾⡟⢀⡿⢊⡕⢍⠳⣝⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣫⠞⡕⡁⢿⣷⡌⠋⣁⡤⠖⠒⠒⠛⠒⠒⠦⢄⡈⠠⢞⣡⣶⠕⢌⢷⡽⣿⣿⡀
⠀⠀⠀⠀⣠⣾⣿⣿⣳⢏⠎⠺⣯⣦⠉⡠⠚⠁⢠⣄⠀⢰⢿⠂⠀⣤⡀⠈⠳⣌⠙⣡⠞⠃⢣⢻⡽⣿⣿⣦⡀
⠀⠀⠀⠈⣿⣿⣿⢳⡏⡎⠀⠀⠀⠀⣾⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣭⣤⣤⣤⣬⣧⠈⠀⠀⠀⢣⢻⣹⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⠀⢰⡇⢰⠈⡇⠰⠈⣿⡅⠢⣼⡇⠰⡏⢠⣀⣻⠀⠄⣹⠀⢶⡆⠠⣾⣿⣿⣿⣿⣇
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠀⢸⡇⠸⢀⡇⢰⣾⣿⡛⠦⢈⡇⠰⣇⠸⠋⣹⠀⡆⢹⠀⠾⡇⠀⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠙⣿⣿⡼⣏⢇⠀⠀⠀⠀⢶⠒⣶⢲⡖⢲⡖⣶⠒⡖⢲⡖⣶⠒⣶⢒⡖⠀⠀⠀⠀⡜⣼⣻⣿⡿⠃
⠀⠀⠀⠀⠀⣿⣿⣿⡽⣎⢆⢠⡞⡷⢀⠑⢿⣸⡇⢸⡇⣿⠀⡇⢸⡇⣿⢀⡿⠋⣀⠲⣤⡄⡜⣼⣳⣿⣿⡇
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣝⣦⡫⡉⣼⡿⢃⣄⡉⠓⠾⠧⣿⣤⠧⠼⠗⠛⣁⠰⣌⠻⡷⣤⢊⡾⣳⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣯⡻⣮⡪⢄⢾⡟⣱⠟⣰⠂⣶⡄⣶⠀⣶⣆⠹⡆⢙⡢⢊⡵⣫⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣝⡷⣭⣒⠩⢄⡟⣸⡿⠇⢻⣄⣙⠹⢄⣪⣵⢞⣯⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣯⣿⣓⣶⣶⠶⠶⠶⢶⣶⣛⣯⣵⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠛⠉⠉⠉⠛⠻⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠖⠚⠛⠛⠛⠛⠓⠶⠶⣤⣄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣞⡛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢞⣫⡿⠖⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠒⠶⠶⠦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠓⣾⠋⢰⡾⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣤⠾⢻⡇
⠀⠀⠀⠀⠀⠀⣿⠀⡏⠉⠉⠉⠉⠉⠉⠛⠓⠒⡆⠀⣿⠀⢸⠀⠀⢀⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠉⠉⠉⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠉⣙⣛⣛⡓⠒⠒⠒⠒⠒⠃⢀⡏⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢻⠀⠀⠀⣿⠀⠀⣾⣿⣦
⠀⠀⠀⠀⠀⠀⣿⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⢸⡇⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢰⡇⠀⠀⡿⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸⠃⠀⣿⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠇⠀⢰⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⢀⡏
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡟⠀⠀⣾⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠓⠒⠒⠚⠛⠛⠀⠀⢀⡟⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⢀⡟⠀⣾
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢿⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠁⠀⢸⣇⡴⠇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡖⢲⠀⠀⠀⠀⣸⠃⠀⢀⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⠩⠉⠉⢩⠉⠉⠉⠛⠛⠛⠛⠛⠀⠀⡿⠀⣠⡿⠉
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣿⣀⡴⠋⠈⠉⠉⠉⠛⡟⢛⡿⠛⠓⣷⡒⠶⠶⠶⠶⢶⣦⢤⣤⣤⣤⣤⣴⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠲⠶⠶⠶⠶⠶⠶⠶⠿⠋⠀⠀⠀⠀⣠⠖⠚⠁⢾⣇⡀⠀⠈⠙⠛⠓⠒⠒⠛⠁⠀⣠⡿⠀⠈⠣⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣧⣀⣀⣀⠈⠻⠯⣿⣶⣶⣶⣶⣶⣒⣒⣾⡽⠟⠁⠀⠀⠀⠀⠉⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⡤⠤⡤⢤⡤⠤⣤⢤⣤⣄⣈⣉⣉⣉⡉⠙⠛⠛⠛⠷⠾⠯⢭⣭⣥⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠋⡿⠓⡾⠓⣲⠓⡶⠧⢤⠧⢤⡯⣤⣽⣉⡟⢀⡖⢲⠖⢦⠤⢤⡀⠀⢩⠟⣇⠀⠀⠀⠀⣠⡴⢶⢞⡿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢸⣉⡽⠉⣹⠛⢲⠟⢲⠓⢲⠿⢤⡾⠤⣼⢥⣼⠁⡜⢉⡟⢒⠟⠒⡾⠁⡰⠋⣴⠏⠀⠀⣠⣞⣁⣠⣧⠋⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠤⠤⠬⣉⣉⣉⣙⣛⣒⠛⠒⠛⠲⠾⠤⠴⠧⠴⠃⢸⣉⣽⣉⡟⠓⡾⢀⡜⣡⠞⠁⠀⠀⣰⠋⠀⠀⠀⠈⠉⠲⢄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠓⠒⠶⠶⠶⠯⠭⣭⣭⣉⣓⣒⣒⣒⠒⠒⠦⠤⠤⢭⣭⣉⣠⢋⡼⠃⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠓⠲⠶⠶⠶⠿⠟⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⣴⣲⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠿⠛⠋⢁⣀⣤⠤⠤⠤⠤⠤⠤⣤⣀⡈⠙⠛⠿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⢁⣠⠖⠛⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠛⠲⣄⡈⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠋⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠙⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠏⢀⡴⠃⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠙⢦⡀⠹⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠘⣿⣿⣧
⠀⠀⠀⠀⠀⢰⣿⣿⠇⢀⡏⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢻⡀⠹⣿⣿⡆
⠀⠀⢰⠒⠒⠛⠛⠛⠒⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⠒⠛⠛⠛⠒⠲⡆
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⣤⣤⣄⢠⣤⣤⡀⠀⣤⣤⡄⣤⣤⡄⣤⣤⢤⣤⣤⡀⣤⣤⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⢶⣷⠂⠀⠀⢸⡇⢸⣿⠈⣿⢸⣧⡿⠃⠀⠻⣷⡄⣿⡷⢸⣿⠀⢸⣿⣿⠃⣿⠶⠀⢸⡇⠀⠀⠐⣾⡶⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠛⠛⠋⠘⠃⠀⠀⠀⠛⠛⠃⠛⠛⠂⠛⠛⠚⠛⠙⠃⠛⠛⠃⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠸⠦⠤⣤⣤⣤⠤⢤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡤⠤⣤⣤⣤⠤⠴⠇
⠀⠀⠀⠀⠀⠸⣿⣿⡆⠈⣧⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣼⠁⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣆⠈⠳⣄⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣠⠞⠁⣰⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣄⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⡈⠙⠦⣤⣀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣀⣤⠴⠋⢁⣠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣶⣤⣄⡈⠉⠛⠒⠒⠒⠒⠒⠒⠛⠉⢁⣠⣤⣶⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠿⠟⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣨⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣅
⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⣰⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⡔⢝⠀⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠈⠀⠀⠀⢠⡴⡧⣘⠱⣿⢟⢺⢟⢿⣠⣦⠄⠄⠀⠀⠀⠀⠴⡿⠃
⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⣀⣔⣿⣩⢉⣠⣤⣶⣶⣾⣦⣵⣎⢯⠩⢲⣞⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣾⡉⣴⣿⠿⢫⠛⣦⡏⣫⣙⡻⣿⣮⠛⢉⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣚⣹⡕⣸⣿⣇⡘⣥⢻⡧⡽⢮⠌⡥⢽⣿⣇⣉⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡹⣹⠛⣿⣧⣤⣴⣤⣦⣤⣴⣤⣤⣴⣤⣿⣧⠐⠂⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢉⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⡇⣿⢻⠇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⡿⣿⣿⣿⣸⣿⡏⣿⢸⡻⣿⣿⣿⣿⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣿⣿⣿⣾⡟⡇⣿⣸⣷⣿⣿⣿⣿⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠈⣿⣿⣻⣿⣾⣷⣿⣿⣿⣿⣽⣿⣿⣿⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠉⠙⠉⠉⠙⠛⠋⠉⠛⠋⠛⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠀⡇⢰⠂⣿⠂⣮⠀⡖⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⢷⣄⡀⠀⠀⠀⠀⠀⡈⡆⡇⢸⡀⢿⠀⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⡈⠁⡉⢀⠁⢈⡀⢈⠁⡁⢀⠀⠀⠿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡇⢹⠃⢸⢸⢸⠂⡇⡌⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡙⠀⠻⢸⠀⢠⠘⢸⡄⠇⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔
⠀⠀⠀⠀⠀⢸⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⡀⠀⠀⠀⠀⠀⣷⢰⠀⠀⢹⡄⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣀⣀⣀⡀
⣿⠓⠶⠶⠒⢠⣿⡸⡇⠀⠘⣇⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠑⡙⢦⣌⡀⠀⠙
⢸⡆⠀⠀⠀⢸⠿⠇⢻⡄⠀⢹⣆⠀⠀⠈⢴⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣷⠀⠈⠹⡆
⠀⢧⠀⠀⠀⣸⠃⢄⠀⣿⠂⠀⣿⠀⠀⠀⣸⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠈⡇⠀⠀⣿⠀⢀⡞⠁
⠀⠈⡃⠀⠀⣿⠀⠘⣇⠸⣧⠀⠹⣆⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⡇⠀⠀⣿⠀⠈⢧⠀⡤⣤⣄⣀⣀⣀
⠀⠀⢷⠀⠀⣿⠀⠀⢹⡆⠘⣷⠀⢻⣆⠀⣿⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⢹⠀⠀⠀⡟⣀⠶⠼⣃⡀
⠀⠀⠘⢧⡴⠏⠀⠀⢸⡇⠀⣿⠀⢸⡇⠀⢿⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢸⡇⠀⠀⢰⡂⠀⠀⣇⡽⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣿⠀⠸⣇⠀⢸⠁⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⠃⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣿⢀⠀⠙⡤⠀⠸⠄⠈⢇⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠤⠼⠧⠀⢀⣼⣆⠀⠀⠣⡀
⠀⠀⠀⠀⡀⣀⣤⢾⡏⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠾⣸⡆
⠀⠀⠀⢰⠀⠁⢀⡸⠃⢄⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠒⠲⠦⣤⣀⣀⣀⣀⣀⠀⢼
⠀⠀⠀⡿⠀⠀⠈⠀⠀⠈⠓⠶⣝⡲⠛⠓⠚⠂⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⣀⣀⠝⠋⠉⠀⠀⠀⠘⢧⡀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠐⠊⠁⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣀
⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩
⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋
⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣍⣿⣿⣏⣽⣿⠋⠉⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣩⡟⣝⡟⣝⣿⣿⣿⣿⣿⣏⣿⣗
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡿⡇⣵⣿⣿⣿⣿⣿⣷⣻⡗
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⡿⣛⡿⡻⡟⡻⡟⣻⣟⢻⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⡇⣽⣿⡿⣏⡓⡇⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣷⣿⣷⣵⣿⣷⣷⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡿⣿⣿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⡿⢿⣻⡟⠻⣟⣟⠿⡿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⡟⠻⣇⢱⢀⣇⣰⣾⣿⣧⣸⣿⣀⣿⠇⣸⣿⠛⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⡉⠻⣿⣤⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡿⠉⣽⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣟⣉⣉⣻⣶⣿⣿⣿⡿⣟⣿⣭⣿⣿⣿⣿⣯⣽⣟⣻⢿⣿⣿⣷⣞⣿⣿⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⢠⣿⣿⣿⢯⣉⢻⣽⣿⣿⡿⣫⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⡻⣿⣿⣿⣷⣼⣿⣿⣿⣿⣷⡀
⠀⠀⠀⢠⣿⣿⣿⣻⣤⣬⣿⣿⣿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⡿⠟⠛⡛⣿⣷⡀
⠀⠀⠀⣾⣿⣿⣏⠙⣻⣿⣿⡿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡟⠻⣿⡉⠹⣦⣣⢷⠀⢹⣼⣿⣧
⠀⠀⢸⣿⣿⣿⡿⢿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⠛⣿⡏⢵⣄⢷⡀⢸⡄⢸⣧⠀⠟⢻⣿⡆⠈⣿⣿⣿⡄
⠀⠀⣼⣿⣿⣿⣶⣾⣿⣿⣯⣿⣿⣿⡿⣿⣿⣿⡿⢹⣦⢸⣆⠀⣿⣼⡆⠸⣿⣾⣇⠀⣇⠘⢿⡄⠸⣾⢻⣿⠀⠹⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⠿⣿⠛⡟⠻⣧⠀⢷⠀⢻⣿⣿⠀⠉⠻⣿⡀⢩⡸⣇⠀⢿⣿⢿⡀⠸⡆⠘⡧⢀⣋⣨⣿⣶⣶⣿⣿⡇
⠀⠀⢿⣿⣏⣥⠀⢧⡹⠀⢿⠀⢹⡄⠘⡇⢸⣿⣿⡷⣤⡀⠘⣇⠀⣿⠙⣦⡘⣟⣼⣧⣤⣿⣶⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣾⡄⠘⣿⡀⠸⣇⠀⣷⠀⢻⣿⣿⣿⣿⠘⢿⣰⣿⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⢷⣿⣿⣿⣤⣼⣿⣿⣿⠃
⠀⠀⠀⣿⣿⣿⣧⠀⢻⣧⡀⢿⠀⣿⣂⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿⣿⡿⣿⡟⣿⣿⣿⡿
⠀⠀⠀⠘⣿⣿⣿⣀⣨⣿⣿⣶⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣿⣿⡿⢶⣤⣸⣿⣿⣿⠁
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⢿⣿⣿⣮⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣫⣾⣿⣿⢿⣶⢤⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣧⣾⣿⠿⣿⣿⣷⣯⣽⣻⡿⢿⣿⣿⣿⠿⣿⣛⣯⣷⣿⣿⣿⠟⢵⣦⣩⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣇⣴⣿⠟⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⠛⠻⣷⣤⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣴⣃⣸⣇⠹⣽⠛⣿⡟⢻⡏⢏⠹⡄⢫⢿⣽⣶⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣷⣿⣦⣿⣤⣿⣧⣼⣧⣼⣾⣷⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠻⠋⣿⠹⠘⢻⠏⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⣤⢈⣿⣿⣽⣿⡿⣿⣾⣿⡿⣷⣿⣿⣿⣿⣿⣿⣿⢿⣿⣾⣿⣿⡟⣿⡟⣿⣿⣿⡿⣿⠟⣻⣿⠆⠐⠏⠉⠂
⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣹⣛⣫⣿⣟⣿⣾⢷⣼⣷⣶⣿⣿⡟⣋⣽⣿⣿⣡⢎⣿⣿⣿⡿⣿⣟⣿⣿⣿⣿⠟⠟⣿⣷⡿⣦⡂⣂⡀⡀
⠀⢸⣿⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⢠⡞⢛⣃⣶⣿⢻⣿⣿⣻⣿⣿⣧⣾⣼⣿⣿⣿⣿⣿⣷⣏⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⣷⣼⣿⣿⣧⠈⠇⠈⡁⠌
⠀⠈⣿⠀⠀⠀⠉⠈⢿⣯⢷⠦⣠⡮⠰⣻⣴⡾⢿⡏⣠⡾⣿⣿⣿⣿⣿⣿⣾⣿⡿⠿⠿⣿⣿⢿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣏⡻⣿⣿⡿⠋⠳⣆⡀
⠀⠀⢹⠁⠂⢰⣤⣤⡷⢻⡶⢾⡾⠩⠛⠻⢶⣿⣟⣷⡇⢀⡾⣿⣿⣿⣿⣿⣿⣽⣷⠀⠈⠁⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣿⣿⣭⠊⡹⡿⠱⡀
⠀⠀⠀⠀⡠⣾⠛⢿⢇⠨⣽⣺⢧⠘⠒⣦⣿⣿⣿⣳⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢹⣧⣠⣾⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠏⢿⠏⢠⢷⡏⠰⣇⠄
⠀⠀⢰⣈⣀⢈⠵⠌⠇⠑⢾⣀⠛⠃⠀⢨⡟⠉⣉⣻⡽⠟⠷⠉⠁⠀⠀⠉⢹⣿⠻⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⡟⠈⠁⠠⠁⡌⢧⡴⠈⡀
⢀⠀⢈⣍⣥⣾⠀⠀⠀⠀⢠⠉⠙⠲⢶⣤⣴⠶⡟⣫⡷⠀⠀⡀⠀⠀⢀⢠⡄⣦⣄⡈⢉⡟⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣥⣤⢤⡵⣼⣯⠉⢄⢾
⠀⠳⢦⣽⣿⣿⣶⣷⡆⠀⣭⠭⢹⠿⣿⣿⣿⣾⣿⡃⠩⠆⠈⡠⠤⠄⠴⣼⣻⢧⣿⣿⣿⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣛⣿⣿⡟⠛⠃⣸⣿⠿⠋⡤⠀⠘
⠀⠀⠈⡿⢿⣿⢛⣿⣾⣿⣋⠛⣆⢢⣙⣿⣽⢛⣿⣧⣶⡶⠀⢀⣀⣀⡀⠀⠈⠋⢻⣧⠾⣯⣿⣿⣿⣿⡿⣿⣿⣿⡏⢿⣿⣿⣿⣧⠀⣶⢸⠃⠀⠀⢰⠄⠂
⠀⠀⢘⣧⢨⣿⡿⡿⣿⣿⣛⣷⠈⢿⢵⣿⣿⣏⣿⠛⣅⠄⠴⣾⣿⣛⣣⣤⣤⢤⣴⣮⣾⣿⣿⣿⣿⢿⡞⢛⣽⣿⣿⣿⣿⣧⣿⣷⣺⢿⢻⠀⠀⠀⠈
⠀⢀⠈⣿⠘⣿⠁⢯⣿⣿⡿⡱⠌⠘⠶⢯⣿⣿⣿⣿⠟⣦⣤⣤⣿⣧⣥⠀⠀⡆⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢛⡯⡬⢸
⠀⢴⣤⣽⣤⣿⡀⣼⣿⣿⣿⠇⢀⢀⣠⡃⢩⣿⣿⣿⣦⣷⣼⠀⠉⢀⣉⣀⣸⣷⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢙⣦⣶⠂⢸
⠀⠈⢫⣽⣉⣿⠛⢯⣭⣿⣿⣮⣭⣿⠿⠿⢻⣿⣿⢦⣿⢯⣌⣿⣿⣿⡿⡻⣿⣿⣿⣿⣿⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣼⣿⣿⡧⠘⠁
⠀⠀⠘⣿⣿⣿⡇⠀⡇⢹⣿⡍⠓⣀⡀⠀⢈⡶⡬⡉⠻⠋⢛⣽⢿⣿⣧⣾⣿⣿⣿⡟⣡⣬⡾⠾⠛⠛⠉⠉⢛⣰⣿⣿⣿⣿⣿⣧⣴⠏
⠀⠀⠀⣜⣸⣿⣳⣮⣷⢀⣿⡇⠀⡈⢿⢿⣿⣿⢿⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠈⠁⠉⠒⠲⡖⣛⣋⠛⠁⠉⠉⠙⠙⠿⣿⡋⠉⢀⠈
⠀⠀⠀⠀⠻⣿⣿⡟⠉⠲⡟⠻⠆⠀⠀⠀⠀⠀⠀⠀⠨⠝⠞⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠩⠤⣶⣤⣥⠄⠀⠐⢧⣾⣟⡖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢆⠂⠔⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠋⠀⠀⠀⢀⠀⠀⣀⣀⡄⢤⡀⣅⣿⣤⢴⣦⣬⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣤⣎⣿⣙⣿⣿⡿⢿⣿⣶⣆⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠐⢉⣌⣋⠻⣿⣿⠻⣿⡿⠦⠴⢦⣄⢀⠀⡀⠀⠀⠀⠀⠉⠉⠫⢉⢁⠨⢁⠀⠁⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠀⠈⠀⠀⠀⠀⠀⠶⣲⣟⣳⣦⣻⣿⣟⢃⣀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠚⠂⠀⠀⠉⠉⠉⠻⣷⠆⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⡤⣐⣬⣶⣿⣿⣿⣿⣻⣷⢦⣖⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣬⣷⣿⣿⣿⣿⠱⣿⣿⡿⣟⣻⣿⣿⣛⣛⣿⣾⣽⣮⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣷⢟⣵⣿⡿⣟⣯⣿⣟⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣷⡄⠢
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⣾⢯⣿⡿⣫⣾⣿⣿⡿⠟⠿⣭⣉⣭⡴⢶⣛⣛⣻⣷⣶⣯⣟⡿⣿⣆⢡
⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢿⣏⣿⣟⣽⣿⣿⣟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⣶⣬⡻⣿⣮⣻⣆⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⣿⢻⡟⣽⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣎⢿⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠈⡿⢹⣾⢡⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡎⣿⠿⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⢀⢂⠈⠳⣤⡀⠀⠀⠀⠀⠀⡇⢸⣿⣸⣿⠇⠀⠀⠀⣠⣴⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢸⣿⡇⢿⠀⡆⠀⠀⠀⠀⢀⡠⠄⠘⠃
⠀⠀⠂⠂⠀⠀⠈⠑⠒⠤⢀⢀⣇⢸⠛⠻⣿⡀⠀⠀⠀⣀⣀⣄⣈⣉⠛⠀⠀⠀⠰⠛⠛⠉⠉⠁⣿⣿⡇⡞⢰⠃⠂⠂⠀⠀⠀⢀⡀
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢾⢸⣸⡄⠀⡜⠃⠀⠀⠈⠉⣹⣿⡏⠻⠃⠀⠀⠀⠼⣿⣿⠛⡦⠀⣿⡟⢠⠃⠸⠄⠀⠀⠀⠠⡏⠀⠈⣱
⠀⠌⠌⠄⠄⠄⠠⠠⠀⠀⠌⠌⠎⣿⡇⠐⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣿⢁⢎⣶⢸⠌⠄⢤⠤⢤⠹⠶⠶⠡⠌⠄⠄
⠀⠈⠈⠈⠀⠀⠁⠀⠀⠀⠈⠈⠜⡹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡎⣾⣿⡎⡌⠈⠱⠄⠴⠃⠀⠀⠉⠈⠈
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡙⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠈⠀⠀⠀⠀⠀⣰⣟⢴⣿⣿⣿⣜⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢻⠳⡞⣧⠀⠀⠀⠀⠀⠀⢴⣤⣤⣤⣤⢀⣤⠄⠀⠀⢰⣿⣟⢾⣿⣿⣿⣹⣦⠶
⠀⠠⠡⠁⠀⠀⠀⠀⠀⠀⠀⣡⠅⣁⣹⠀⠡⢹⣷⡄⠀⠀⠀⠀⠀⠙⢿⠈⢿⠟⠋⠀⠀⣠⣿⡯⠿⠆⠙⣿⣿⣷⢿⡞⡁
⠀⠀⠡⠡⠀⠀⠀⠀⠀⠀⢠⠱⣏⠿⣫⡇⠐⡇⣿⠎⠀⢀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢀⡼⡿⣫⣾⣿⣿⣶⢪⣿⣿⡘⣱⠁⠀⠀⠀⠀⠀⠡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡰⣽⣽⣫⡞⣴⢧⡿⠀⠀⠀⠙⠲⣤⣀⣀⡀⠀⢠⣶⠋⣶⡁⢿⣿⡿⣿⠿⣨⠟⢋⣰⠁
⠀⢀⢂⢀⢀⠀⠀⠀⠀⠀⢀⢋⢢⣉⣹⣞⠋⠔⠁⠀⠀⠀⠀⠀⠀⠉⠉⠃⠀⠀⢿⣧⠻⣿⣌⡉⢁⣐⣚⣭⠶⣻⢃⠀⠀⠀⡀⡀⡀⣀⢀⢀
⠀⠂⢂⠂⠀⣂⡠⠤⣤⣶⡖⠲⡎⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠈⣽⢻⣮⣽⠛⣓⣦⠐⠚⠂⢂⠂⠂⠀⠀⠐⠀⠐⠀⠂⠂⠂
⠀⠀⣠⠔⠋⠁⠀⠀⣾⣿⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠀⠀⢈⣿⣿⣿⣿⣶⠂⢤⣄⡀
⢀⡞⠁⠀⠀⠀⠀⠀⣿⣤⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣤⣼⡟⠀⠀⠀⠉⢢⡀
⡜⠀⠀⠀⠀⠀⠀⢠⡛⣿⡟⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⣿⣟⡇⠀⠀⠀⠀⠀⢑
⠃⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⠀⢀⡀⠀⠀⠀⠀⠀⢠⡾⢿⣿⡿⢿⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠐⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠐⠄⠀⠆⠀⠀⠀⠀⣷⣼⣿⣧⣾⠀⠀⠀⠀⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⢸⣏⣿⣏⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠈⠀⠃⠀⠀⠀⠀⢸⣿⣉⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⢿⣿⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠌⠡⠡⠁⠀⠈
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣿⣿⣶⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠳⣾⡀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⢰⣿⣯⣠⣿⣧⣨⣿⣅⣼⣿⣄⣽⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣅⣤⠀⠀⠀⠀⠀⠀⠘⣥⠀⠀⠀⠀⠀⠀⠸⡁
⠀⠀⠀⠀⠀⣾⡛⢻⣿⠛⢻⣿⠛⣿⡟⠛⣿⡟⢛⣿⡧⡀⠀⠀⠀⢀⡔⣻⡟⠛⣿⡟⢃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⡿⢿⣿⡿⢿⣿⡿⣿⣿⠿⣿⣿⢦⡀⣠⢿⣿⡿⢿⣿⡿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡷⠠⠀⠄⠄⠀⠀⠠
⠀⠀⠀⠀⠀⢷⣿⣷⣴⣿⣧⣴⣿⣦⣼⣿⣤⣾⣿⣤⣿⣷⣤⣿⣧⣴⣿⣧⣼⣿⣦⣼⣿⣤⣶⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣷⠀⠁⠈⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⢉⣭⣭⣍⣭⣭⣭⣭⣍⣭⣭⡉⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠶⠶⠶⠶⠶⠶⠶⠶⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⢸⣿⣉⣉⣉⣉⣉⣉⣉⣹⣿⡇⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠶⠶⠶⠶⠶⠶⢀⣀⣀⣈⣁⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣸⣿⣛⣉⣉⣛⣛⣻⡿⠿⠿⠷⠾⢿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢟⣛⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⢀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⡿⠿⣟⣟⣽⣿⣷⣿⠛⣙⣿⣧⢻⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⢀⣀⠠⠠⠒⡀⠈⠈⣤⠦⠐⠸⡄⢸⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣻⣿⣭⣿⡼⠾⣟⣛⣯⣽⣷⣾⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠈⡀⠆⠀⠶⠀⣋⣁⠠⠤⠐⠂⠈⠁⢸⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣻⣭⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠴⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⠤⣤⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠁⠀⠀⠈⠉⠉⠉⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠤⠤⣤⡄⠀⠀⠀⠀⣠⣤⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣤⣦⣤⠀⠀⠀⠀⠀⠀⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠉⠘⠋⠈⠀⠀⠀⠀⠁⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⢠⠀⠀⠀⠄⠈⠣⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⢀⠀⠀⢀⣀⣠⣀⣀⠀⠀⡀⠀⠀⡆⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠂⠹⣿⣛⣛⣿⠟⠀⠸⡀⠀⢀
⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠸⢰⠀⠀⠀⠀⠉⠉⠀⠀⠀⠤⠁⠀⠈⡄⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⣀⣀⣀⣀⢀
⣿⣿⠟⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢷⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⠖⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠲⡄⠀⢸⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡆⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠾⠃⠀⣼⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣒⣀⣒⣀⣀⣀⣠⣏⣀⣤⣤⡤⠴⠶⠖⢺
⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣴⣒⣖⣒⣶⣒⣒⣖⡒⣲⠖⠒⡶⠖⢤⠤⠭⢭⠭⢭⣍⠉⠉⣩⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡾⢁⠾⢤⠾⢤⡾⠥⣤⠯⢬⡽⠭⢭⠿⠭⣽⢯⣉⣽⣋⡲⣾⠋⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⢠⠟⡠⠛⢲⠛⢲⠞⠒⡴⠃⢢⠞⠤⡴⠧⢤⠾⠥⣤⠮⣤⢞⡗⠁⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⢠⠟⠐⠿⠿⠿⠿⠿⠿⠿⠷⠾⠟⠖⠾⢷⣶⣿⣶⣺⣗⣲⡽⠋⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⡿⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠋⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣀⣤⠴⠞⠋⠉
⠀⠀⠀⠀⠙⠛⠛⠛⠒⠒⠲⠶⠶⠶⠶⠶⠦⠤⢤⣤⣤⣤⣄⣀⣀⣀⣀⣿⣤⠴⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⣾⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⣿⡇⠀⣤⡄⠀⣿⣧⣀⣀
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣷⡀⠀⠀⠀⠻⣷⣶⡿⠃⠀⣿⡟⠛⠿⠿⣿⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠋⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⠀⠀⠀⠀⠉⠉⣿⡟
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣴⡿⣿⣦⠈⠛⠋⠀⠀⠀⣀⣤⣶⣄⠀⠀⠀⠀⠀⠀⠀⣿⣏⡀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⣿⣿⠀⣰⣿⠂⣿⡇⢼⡿⠀⠀⠀⢀⣴⣾⠟⠋⣉⣿⡇⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⣿⣶⡄⠀⣶⡆
⠀⠀⠀⠀⠀⣿⣿⠀⠘⢿⣾⡿⠃⢀⣠⣶⣷⣄⠘⣿⣧⣴⣾⠟⠋⣀⣤⡄⠀⠀⠀⠀⣿⣷⣤⣤⣀⡀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣠⣶⣿⣿⡏⠙⣿⡆⠈⠙⠋⣀⣴⣾⡟⠻⣿⡆⠀⠀⠀⣿⣏⠉⠛⠛⠃⠀⣿⡟
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠻⣿⡄⠉⠁⠀⢀⣀⣀⠀⠺⠟⠋⠙⠿⣿⠟⠁⠀⠀⠀⣿⡿⢿⣷⣶⠀⢰⣿⠃
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠙⠛⠀⣠⣶⡿⠟⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣁⣀⠀⠿⠏⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⣀⢰⣿⠃
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠛⢀⣠⣶⡿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠉⠙⠻⢿⣿⡄
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⢛⣿⣷⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣴⡆⠀⣿⣷
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠉⢁⣠⣶⣿⣄⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣶⡿⣿⣇⠘⢿⡆⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠈⠁⠀⠀⢠⣤⡀⠀⠀⣿⡇⠀⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⢀⣠⣿⣿⣄⠀⠀⠀⠀⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠉⠙⠛⠀⠀⠀⠀⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⣿⡿
⠀⠀⠀⠀⠀⢿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡿⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⡶⠛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⢶
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡏⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼⠁⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣹⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⣏⡀
⠀⠀⠀⠀⠀⢀⡼⢋⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡙⢦⡀
⠀⠀⠀⢀⡴⠋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠙⢦⡀
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⢿⠀⡀⠀⠐⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⣤⣄⡀⠀⢀⡀⢸⡇⣷⠀⠀⢸⡀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⢸⣇⠠⠶⠒⠶⣄
⢷⡈⠉⠉⠉⢠⡿⣧⢹⡄⠀⠘⣧⠀⠀⠀⠹⡀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢡⠙⠳⢦⣄
⠸⡇⠀⠀⠀⢸⡇⠛⠀⢻⣄⠀⢹⣧⠀⠀⠀⢸⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡇⠀⠀⣸⠀⠀⠀⢸⠂
⠀⢻⡀⠀⠀⢸⡇⠰⣄⠀⣿⠀⠈⣿⠀⠀⠀⢸⡄⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⡇⠀⠀⢹⡆⠀⢰⠏
⠀⠀⡇⠀⠀⢸⡇⠀⢹⡄⠹⣇⠀⠹⣇⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⣿⠀⠀⢸⡇⠀⠘⢧⢠⡶⢦⣤⣤⣀
⠀⠀⢻⡄⠀⣸⠇⠀⠀⣿⠀⠙⣷⠀⢹⣦⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢸⡇⠀⠀⠘⠃⠀⠀⢸⠉⣠⠶⠞⣂⡀
⠀⠀⠘⢷⣴⠟⠀⠀⠀⣹⠆⢰⣿⠀⢸⡇⠀⢸⣇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⣆⠀⠀⠸⣠⡽⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⡇⠀⠸⣇⠀⢸⡇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⡇⠀⠀⠀⡏⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠳⡄⠀⠙⡆⠀⢧⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⡬⢧⠀⠀⢀⣇⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⢀⣠⢿⠃⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠈⠀⠉⠳⣄⠀⡆⢱⡄
⠀⠀⠀⢀⡈⠐⠋⠁⡾⢀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣸⠂⠀⠀⠊⠁⠀⢱⣤⣜⢦⣤⣤⣤⣤⡤⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠈⠉⢛⡶⠶⠛⠓⠂⠀⢻⡄
⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠤⠖⠲⠋⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀
⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢦
⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟
⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁
⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋
⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⢀⠀⣀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣟⣉⣻⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠦⠼⠋⠁⠀⠀⠉⣠⡶⠆⠀⠀⠀⠀⠀⠀⢀⡀⣴⣠⠂⠠⠔⢀⣾⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣕⣯⢤⣄⠀⠀⠀⠘⠉⣀⠂⠀⡀⠀⠀⠀⡔⠘⣧⡻⠛⠀⠀⠄⠶⠀⣜⢽⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣂⣙⡶⠽⢄⣎⠄⠀⠀⠀⠘⠃⠁⠀⠀⣄⣸⡽⡏⢂⠁⠒⠢⠀⠀⢀⣀⠀⠂⢐⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⠈⡗⠉⠀⠀⠀⠀⠀⠀⠠⢈⠃⡆⣿⢯⡀⠸⢯⡔⠀⠀⢀⠤⣤⠀⡀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⢤⠒⠚⠀⠁⢿⣷⠀⠀⠈⠁⠀⠀⠀⠟⠚⠁⠀⡌⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢴⠀⠠⠈⡉⠀⠀⠀⣀⣀⠀⠀⠀⠀⡬⢠⡞
⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠄⠀⠁⢀⡠⠆⠀⢙⠄⢀⠷⠀⠀⠠⢇⠉
⠀⠀⡠⠖⠁⠀⠀⠀⠄⠀⠀⠀⢂⠀⠤⠊⠉⠉⠣⣀⡀⠀⢀⠃⠀⠀⠀⠈⢂⠀⠉⠁⠀⡀⠔⠣⠜⠀⠀⠄⠀⢀⡀
⠀⠊⠀⠀⠀⠀⠀⠀⠸⠀⠀⠡⠀⢸⣷⡄⠀⠀⡐⠂⠤⠤⠈⡀⣂⠀⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠌⠀⠠⡺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢂⢀⣿⣄⢀⠔⠀⠀⠀⠀⠰⠁⠁⠀⠀⠀⣴⣢⣤⠅⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢾⣿⣿⣷⡀⠀⠀⠀⠐⠃⠀⠀⠀⠀⡠⠛⠛⢯⠁⠀⠸⡀⠀⠀⠠⠄⡎⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠈⢿⣿⣿⠁⠀⠀⠀⢀⠆⠀⡠⠔⠋⠀⠀⠀⢰⠀⠀⠀⠀⠒⠀⠀⠀⠠⠀⠀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠉⠃⠀⠀⠀⢀⡮⠀⠀⠀⠀⠀⠀⢀⡠⠊⢆⠀⠀⢸⠀⢀⡠⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⡠⠔⠉⠀⠀⠀⠣⠀⠘⡰⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡠⢀⢔⣔⠅⢀⠠⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⣾⢋⢮⡞⡥⢊⣥⣒⠤⠐⠀⠨⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠢⡑⠪⡐⢔⢤⡀
⠀⠀⣠⣤⣿⢧⢏⡿⢋⡴⠛⠁⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠠⠂⠀⢱⡀⠈⢆⠘⡄⠙⣄⠘⢄⠱⡝⢆
⠀⢠⡏⢸⡟⠂⡞⣡⠏⠀⠀⢤⡄⠀⠀⢀⣔⡊⠀⠀⠀⠀⠓⠓⠀⠀⠹⣄⠘⡀⠘⡄⠈⢢⡈⡆⢱⢸
⠀⢸⣴⣾⡧⠊⠐⠏⠀⠀⠖⠻⠏⠀⠀⠀⠙⠉⠠⠀⠀⠀⠐⠾⠀⠀⠀⠘⣧⡇⠀⠘⢦⡀⠑⠜⠀⡜
⠀⠸⣽⠄⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⡄⠉⠢⢀⣀⣀⣄
⠀⠀⠈⠇⠈⠂⠷⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠃⡇⠀⠸⡄⠀⠀⠀⠀⠀⢸⣁⠒⠡⠁⠐⠤⠌⠁⠈⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣉⣠⡇⠈⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⣀⠀⠀⡄⣧⡄⡜⡄
⠀⠀⠀⠀⠀⠀⠀⠃⠐⢄⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠐⠀⡄⠀⠠⠖⠀⠀⠀⠀⡇⢠⡏⠀⡸⢀⣿⡇⢠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠇⣾⢁⡜⢁⢎⢾⠀⡇⢸
⠀⠀⠀⢀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⡷⠋⡰⡣⠊⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠐⠂⠉⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣾⡿⢿⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⢛⣋⡭⠥⠶⠖⠒⠒⠒⠒⠲⠶⠤⢭⣙⡛⠿⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⢛⡭⠖⠋⠉⠀⢴⠶⠞⠀⢠⡞⠛⠷⠀⢰⠶⢦⡄⠉⠙⠲⢬⡛⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⡡⠞⠉⠀⢼⣋⣛⡀⢸⡟⠛⣁⠸⣇⣀⡤⢀⡿⢳⡞⠃⢠⡞⠲⠤⠉⠳⢬⡻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢟⡵⠋⣀⣀⠀⠀⠀⣭⣩⡿⠀⢛⣛⣩⡤⠬⠭⢤⣌⣁⡀⠳⢠⣏⠙⠒⠀⠀⣳⣄⠙⢦⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⣡⠎⠀⢾⣁⣽⠀⠀⠀⢀⡥⠖⣋⣭⣴⣶⠶⠶⠶⠶⣶⣦⣭⣙⠲⢬⡛⠀⢀⡴⠋⠈⠁⠀⠹⣌⣻⣷⣤⣴⣶⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣾⠟⡴⢁⣤⣤⣀⠙⢧⠀⣠⠞⣩⣶⠿⠛⠉⢀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣍⠳⣌⠀⣀⣤⣤⣶⡾⠿⠛⠋⠉⠀⠀⢻⣧
⠀⠀⠀⠀⠀⢀⣿⡏⡼⠁⢸⣅⠀⠙⡇⢀⠞⣡⡾⠋⠁⠀⠀⠀⢾⣿⡇⠀⠀⠻⠃⠀⠀⢀⣀⣤⣿⣷⠿⠿⠛⠉⢉⣀⡀⠀⣶⣾⣿⠃⠀⠈⣿⡄
⠀⠀⠀⠀⠀⣼⡟⣸⠁⡤⠀⠉⠛⠛⢡⠏⣼⠟⠁⠀⠰⠷⠀⠀⠈⠁⠀⣀⣠⣤⣶⡶⠿⠟⠛⠉⠁⣀⣤⣤⡀⠀⢻⣿⠋⠀⠀⢹⣿⡄⠀⠀⢹⣷
⠀⠀⠀⠀⢰⣿⢁⡇⢸⠟⠲⠶⡤⢠⠏⣼⡏⠀⠀⠀⢀⣀⣤⣤⣶⠾⠿⠛⠋⠉⠀⠀⣤⣶⣶⡄⠀⢿⣿⣹⣿⠀⠘⣿⣿⠃⠀⠀⢿⣧⠀⠀⠈⣿⡆
⠀⠀⠀⠀⣼⡿⢸⠁⠀⠀⠀⠀⠀⣾⣸⣿⣤⣴⡶⠿⠟⠛⠉⢁⣀⡀⠀⢰⣾⡿⠀⠀⣿⣇⠻⠟⠀⠘⣿⡟⣿⣆⠀⢹⣿⣶⡆⠀⠘⠛⠀⠀⢀⣸⣿
⠀⠀⠀⠀⣿⡇⣼⢀⣀⣠⣤⣶⡾⠿⠛⠋⠉⠁⠀⠀⠀⠀⢸⣿⡻⠿⠀⠈⣿⣷⡦⠀⠸⣿⡄⣾⡇⠀⢹⣿⠼⠿⠀⠀⠉⠀⣀⣠⣤⣶⣶⣿⠟⠛⠉
⠀⢀⣀⣤⣿⣷⠿⠿⠛⠋⠉⣀⣀⡀⠀⢾⣿⢻⣧⠀⠀⠀⠀⠻⠿⣷⣦⠀⠸⣿⣦⣄⠀⠻⠿⠟⠃⠀⠀⢀⣀⣠⣤⣶⡾⠿⠛⠋⠉⣿⢸⣿
⠀⢿⣟⠉⠁⣀⣤⣴⡆⠀⣾⣟⢻⣷⠀⠘⣿⣾⡿⠂⠀⠀⠀⠰⣿⣬⣿⠇⠀⠛⠋⠉⠀⢀⣀⣤⣴⣶⠿⠟⣿⡏⣽⠀⡀⠀⠀⠀⠀⡇⣸⣿
⠀⠘⣿⡀⠀⠛⢻⣿⡀⠀⠸⣿⡜⣿⣇⠀⢻⣿⡀⠀⠀⠀⠀⠀⠈⠉⠁⣀⣠⣤⣶⡶⠿⠟⠛⠉⠁⠀⠀⢠⡿⢡⠇⣼⠧⣤⣀⡀⢸⠇⣿⡇
⠀⠀⢻⣇⠀⠀⠈⣿⣧⠀⠀⢿⣷⣽⡿⠀⠈⠛⠁⠀⢀⣀⣤⣴⣶⡾⠿⠛⠋⠉⠀⠀⠀⢀⣀⡀⠀⠀⣠⣿⢡⡏⣠⡍⡀⠀⠈⢁⡞⣸⣿
⠀⠀⠘⣿⡄⠀⠀⠸⠿⠆⠀⠀⠉⠁⣀⣠⣤⣶⣶⣿⠟⠛⠉⠉⠀⠀⠀⠀⢀⣰⣦⡄⠀⠈⠃⠀⣀⣬⡍⣱⠏⠰⣯⣴⠏⣰⠆⡾⣱⣿⠏
⠀⠀⠀⢻⣧⠀⠀⠀⣀⣠⣤⣶⡾⠿⠛⠛⠉⠳⣌⡻⢷⣤⣀⠀⠀⠻⠃⠀⠀⠹⠟⠃⠀⣀⣤⡶⠗⣠⠞⣠⠶⢶⡄⠈⠲⢏⡾⢱⣿⠟
⠀⠀⠀⠈⣿⣶⠿⠿⣿⣏⠻⣄⠀⠀⠀⠀⠀⢶⣄⡙⠲⣍⣛⠻⠷⣶⣴⣶⣶⣶⣶⠾⠿⣛⣩⠴⢋⣡⣤⡙⢶⣟⠛⠛⣠⠞⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣌⢦⡀⠀⠀⢠⡞⠉⢁⡤⢤⡈⠙⠓⠒⠶⠶⠶⠶⠒⠚⠋⣡⡴⠆⢸⡇⠀⢠⡄⠉⢃⡴⣣⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣍⠳⣄⠋⠀⠀⡟⠀⢀⡟⢰⡟⢛⡆⠀⠀⢰⣟⡻⠀⢿⡴⠖⠀⠙⠶⠞⣡⠖⣩⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⣙⠲⢤⣙⠲⠚⠁⣸⠛⠛⠁⠀⠀⢠⣍⣙⡇⠈⠷⠞⣛⡤⠖⣋⣴⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣬⣙⡓⠲⠤⠤⣤⣄⣀⣀⣀⣭⠭⠤⠶⢚⣋⣥⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢷⣶⣶⣶⣶⣶⣶⣶⣶⡾⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠛⠒⠂⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⢀⠀⠀⠰⠄⠀⠠⠆⠀⢀⡀⠈⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⢀⡀⠀⠉⠀⢀⡀⠀⠀⠀⢀⣀⠈⠁⠀⣠⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠈⠁⢀⠔⠊⠁⠀⠄⠀⠀⠀⠀⠈⠂⢄⠀⠀⢠⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠈⠃⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⠤⣄⣜⣳⠶⢷
⠀⠀⠀⠀⠀⠀⠀⡖⠀⢠⠄⠀⡘⠀⠀⠒⠀⠀⢀⣀⣀⣤⣤⣴⡶⢾⡿⠟⠻⣿⠉⣭⣧⣄⢰⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣠⣄⣤⣶⣶⣾⡿⢛⠻⡏⢉⣽⡏⢰⣶⣧⠘⠃⣿⡄⠶⢾⣿⣈⣿⣷
⠀⠀⠀⠀⠐⣦⠼⠿⢿⡟⣋⡙⣯⢀⡄⢹⣿⣧⣉⡙⢷⠠⠶⢷⣘⢛⣹⣤⣷⠬⠷⣶⠾⠿⡿⠛⠋
⠀⠀⠀⠀⠀⢿⣶⠐⣿⡄⢿⡇⢸⡄⣴⣿⣿⣿⣭⣥⣾⣶⠶⠿⠟⠋⠀⠀⠀⠀⠀⠤⠀⢀⡁
⠀⠀⠀⠀⠀⢸⣿⣇⣿⣿⣦⣴⣿⢿⠿⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠈⠉⠛⡌⠁⠠⠄⠀⠁⠀⡀⠘⠀⠀⠀⠀⡄⠀⠀⡀⠐⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⢀⠀⠀⠂⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠛⠀⠀⡤⠀⠀⠤⠀⠀⠈⠁⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠤⣄⠀⠀⠀⢀⣀⣀⠤⠴⠒

⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡞⢉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⣘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣤⣶⣤⣶⣧⣴⣦⣴⣦⣤⣶⣤⣶⣤⣵⣦⣴⣦⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⢻⣿⣾⣿⣾⣿⣶⣿⣷⡟⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠐⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾⠿⠿⠿⠿⠿⠿⠿⠿⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠂



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠒⠀⠄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠤⠀⠀⠒⠂⠓⠂⠤⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣐⠠⠤⠄⢀⡉⠀⠒⠒⠒⠄⣉⣉⣀⠀⠠⠄⠀⠀⠀⠐⠒⠀⠈⣉⣉⣐⣲⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢅⡀⠤⠤⠀⠐⠂⠀⠉⠁⠀⠒⠀⢤⣀⠉⠠⠴⠆⠒⠀⠀⠀⣈⢉⡀⠠⠤⠀⢀⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣀⠀⠤⠄⠐⠒⠂⠈⢀⠀⢀⠀⠀⠀⠀⠀⠀⠒⠢⠄⡈⢉⠁⠐⠒⠤⠤⠤⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠠⠀⠐⠒⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠲⡔
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠒⠂⠠⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠂⠠⠤⠀⠀⠀⠔⠂⠒⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣋⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⢛⣭⡍⢉⡭⣭⠉⢩⣝⡛⠻⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⢰⣟⣃⠘⣷⠆⢸⡇⣈⠀⣿⣼⠇⣰⡷⠮⢙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣫⡴⣄⠀⠀⢩⣽⡇⣻⣚⣬⣿⣯⣜⣃⡾⢠⣟⡛⠀⣸⠷⠎⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢋⣀⡙⢷⣏⠀⢀⣤⡾⠟⠛⠋⠉⠉⠉⠉⠛⠛⠿⣮⣅⠾⠋⠀⠀⠀⠈⢻⣧⣠⣤⣴⣶⣧
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⢿⣍⡛⣦⢉⣾⠟⠉⠀⠀⢀⣷⡶⠀⣶⡄⠀⠀⠀⠈⢙⣿⣦⣤⡶⠾⠿⠛⠛⣉⣡⣄⠀⢿⡇
⠀⠀⠀⠀⠀⠀⢰⡿⢡⣯⣀⠈⠛⣱⡿⠁⠀⠀⠺⠗⠀⠋⠉⣀⣠⣤⣴⡶⠾⠟⠛⢉⣉⣁⠀⣶⡶⠿⠘⠻⣿⠁⠀⠸⣿
⠀⠀⠀⠀⠀⠀⣾⠇⠈⠀⠈⠉⣱⡿⠀⣀⣀⣠⣤⣶⡶⠿⠛⠛⣉⡉⢀⣴⣶⣦⠘⣿⡛⣿⡆⢻⣷⣴⡆⠀⢿⣇⠀⠀⢿⡇
⠀⠀⠀⠀⠀⢰⣿⠀⡀⣠⣤⣴⡿⠿⠟⠛⠋⠉⠁⣴⣶⣦⠸⣿⠛⠃⢸⣿⠀⠋⠀⢻⣿⢿⣧⠈⣿⣅⣠⡀⠸⣿⠀⠀⠘⣿
⠀⣀⣠⣤⣶⡾⠿⠟⠛⣉⣉⡀⢠⣶⡶⣶⡀⠀⠈⣿⣦⣍⠀⢿⡷⠿⠀⢿⣇⠰⣷⠘⣿⡄⢿⠧⠹⠛⢛⣁⣠⣤⣶⡶⠿⠟⠃
⠀⢻⣏⠁⢠⣶⣶⠷⢸⣿⠙⣿⡄⣿⣦⣿⠇⠀⠀⢀⣉⠻⣷⠘⣿⣤⣶⠈⠻⠿⠋⣀⣉⣤⣤⣶⠶⠿⠛⠋⢹⣿
⠀⠈⣿⡀⠈⠘⣿⡀⠘⣿⡀⢻⣧⠸⣿⠉⠀⠀⠀⠈⠻⠿⠟⢀⣉⣩⣤⣴⡶⠿⠛⠛⠉⢉⣿⢃⡀⠀⠀⠀⣼⡇
⠀⠀⢹⣧⠀⠀⢹⣧⠀⠹⣷⣼⡿⠀⠛⠃⣀⣀⣤⣤⣶⠾⠿⠛⠛⠉⡀⢀⣤⡀⠀⠀⢠⣾⢃⡾⠷⣦⣄⢰⣿
⠀⠀⠈⣿⡄⠀⠈⠛⢀⣀⣨⣥⣴⡶⠿⣿⣟⠋⠉⠀⠀⠀⢴⡆⢨⣿⡷⠀⠉⠀⢀⣴⡟⠡⣿⣴⠆⣀⢠⣿⠃
⠀⠀⠀⢹⣷⣖⡶⠿⠻⣿⡉⠉⠀⠀⠀⣬⣙⠿⣦⣄⣀⡀⠀⠀⠀⠀⢁⣀⣤⡾⠟⠩⣾⣻⣆⡙⠻⣣⡿⠃
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⢿⣄⡀⠀⣰⠟⢉⡶⢦⡉⣛⡛⠿⠿⠿⡛⠛⣋⣭⢰⡟⠓⡈⢻⡌⣻⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣮⣉⠀⣾⣃⡿⢱⣯⣽⠆⠀⢺⣝⡃⢿⡶⠈⠻⣤⢟⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣾⣭⣀⣸⠃⠀⠀⠀⠰⣬⠇⢘⣟⣫⣴⡖⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⢀⣤⡶⣒⣒⣒⣒⣒⣲⣶⣒⣒⣒⣒⣒⣒⣒⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣒⣒⣒⠒⠒⠒⢶⣦
⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⣿⣹
⠀⠀⠀⠀⠀⡏⢰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⢲⡆⣿⢸⡆
⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇
⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇
⠀⠀⠀⠀⠀⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢸⠃
⠀⠀⠀⠀⢠⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢸
⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣼
⠀⠀⠀⠀⢘⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿
⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡿⣿
⠀⠀⠀⠀⠀⣿⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⣿
⠀⠀⠀⠀⠀⣿⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡇⣿
⠀⠀⠀⠀⠀⢹⠀⠻⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠚⠃⠀⣧⡏
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢠⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⠶⠶⠶⠶⠶⠶⢶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠤⠤⠤⠤⠄⡆⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣚⣛⣛⣛⣉⣉⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⣤⣀
⠀⠄⠀⠀⠀⠀⠀⠀⣰⢏⡭⣭⢭⡭⢭⡭⢭⠭⢭⠭⢭⡭⢭⡭⢭⡉⢩⠭⣽⠿⡿⢿⠻⣆⠀⠀⠀⣾⣡⠜⠋⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⡾⢭⠯⢽⠭⢽⠭⢽⠭⢽⡭⢽⡯⠭⡯⠭⡇⠘⣟⣻⣛⣻⣚⣧⠹⣆⠀⠀⠹⣅⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⣰⣏⣈⣓⣛⣒⣛⣒⣚⣒⣚⣒⣒⣓⣚⣓⣒⣓⣒⣛⣀⣱⣤⣷⣤⣷⣤⣇⣹⣆⠀⠀⠘⢧⡀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠻⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡤⠤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠟⠀⠀⠀⠀⠛⠶⠴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠚⢛⠉⡉⢉⡉⠉⠙⡛⠒⠲⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⣉⢠⡄⠶⣘⠇⠙⠥⠍⠈⠁⠙⠌⠃⢛⠰⠖⣤⠈⡙⠲⢄⡀
⠀⠀⠀⠀⠀⡠⠊⢡⡰⢶⠩⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠶⣠⡙⠢⡀
⠀⠀⠀⢠⠞⣀⠺⡪⠂⠁⠀⠀⠰⠀⡂⠶⠴⠀⠐⠄⠆⠶⠆⠂⠠⠆⢆⠀⠀⠀⠁⠪⡰⢆⡘⢦
⠀⠀⢠⠏⢠⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⢡⡄⢧
⠀⠀⡞⠐⠗⠄⠠⢶⡶⢠⡶⡄⣴⠶⡄⠀⢠⡶⣦⢴⡶⠆⣤⢶⣤⣶⢦⣠⡶⠶⠰⣶⠦⠈⡀⠶⠘⡆
⠀⠀⡇⢘⡃⠀⠀⢸⡇⣿⠀⣿⣿⠺⠋⠀⢈⠳⣆⣸⠗⠀⣿⠀⠀⡿⣾⠹⡷⠂⠀⣻⠀⠀⠀⢙⡃⡇
⠀⠀⢧⢈⣅⠃⠀⠘⠃⠘⠻⠋⠛⠀⠀⠀⠘⠳⠋⠛⠓⠂⠙⠟⠛⠃⠙⠚⠛⠚⠀⠛⠀⠀⠃⣨⢱⠇
⠀⠀⠘⣆⠰⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⢆⡞
⠀⠀⠀⠘⢦⡛⣡⡑⠀⡀⠀⠀⠸⠀⠕⠒⠜⠁⠺⠖⠇⠶⠆⠃⠀⠆⠦⠀⠀⠀⡠⠐⢡⡜⣣⠎
⠀⠀⠀⠀⠀⠑⢬⡘⠃⣌⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠀⣤⠺⣪⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣛⠰⠆⣤⢈⣁⢀⠐⠒⠂⠐⠀⠀⡀⣉⢀⡄⠶⣘⡣⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠮⠥⢽⣍⣋⣘⣃⣛⣼⠧⠍⠖⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⢶⡶⠾⠿⠛⠛⢛⠉⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⡶⠶⠿⠿⠛⠛⠋⣉⣉⣁⡀⢠⣶⣶⠆⠿⣿⡟⠃⠀⣿⡄
⠀⠀⠀⠀⠀⠀⡀⣀⣀⣠⣤⣤⣶⠶⠶⠿⠟⠛⠛⢋⣉⡉⢠⣤⣴⣦⢀⣾⠿⣷⡄⣿⡏⢻⣧⠸⣿⣤⣄⠀⢻⣧⠀⠀⢻⡇
⠀⣶⡶⠾⠿⠛⠋⠛⠉⣩⣩⡀⢠⣶⣶⣦⡀⠀⠀⣿⡛⠿⠀⣿⣇⣤⠈⣿⡄⠈⠀⢸⣿⠿⣷⡀⣿⡏⣁⠀⢸⣿⠀⠀⠸⣿
⠀⢸⣧⠀⠶⢿⣿⠓⢸⣟⠹⣿⡈⣿⣄⣿⠇⠀⠀⠙⠿⣷⣄⢸⣏⠉⠀⢻⣇⣘⣿⠈⣿⡄⢿⡇⠹⠿⠛⠃⣀⣉⣁⣤⣤⣿⡄
⠀⠘⣿⠀⠀⢸⣿⠀⢸⣿⠀⣿⡇⢿⣏⠉⠀⠀⠀⠸⣷⣼⡿⠸⠿⠿⠟⠈⢛⣛⣁⣠⣤⣤⣴⣶⡶⠶⠟⠛⠛⠛⠉⠉⠁
⠀⠀⣿⡇⠀⠀⣿⡇⠀⢿⣧⣿⠇⠸⠿⠀⠀⢀⣀⣀⣬⣥⣤⣶⡶⠾⠿⠟⠛⠛⠋⠉⠉⠁
⠀⠀⢸⣧⠀⠀⣙⣁⣀⣤⣤⣴⣶⡶⠾⠿⠟⠛⠛⠉⠉⠉
⠀⠀⠈⠿⠿⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⡴⠒⠛⠋⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣩⣥⣤⣤⣤⣤⣤⣤⣤⣤⠀⢹⡆
⠀⠀⠀⣼⣧⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⡇
⠀⠀⠀⢻⢻⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⢸⢸⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⢸⡏⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠘⡇⣇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⡇⢿⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⢿⢸⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡧⠀⡇
⠀⠀⠀⠀⢸⡘⡇⠈⣧⣤⠤⠤⠤⠤⠶⠶⠖⠒⠒⠒⠛⠛⠛⠛⠉⢉⣉⣉⣉⣉⣉⣀⣀⣇
⠀⠀⠀⠀⠈⣇⣧⣤⠤⠤⠤⠶⠶⠖⠒⠒⠒⠛⠛⠛⠋⠉⠉⠉⠉⠉⢉⣉⣁⣂⣀⣀⡀⠘⣆
⠀⠀⠀⠀⠀⢿⠹⡄⠀⢀⡤⠤⡤⠄⢤⠒⡖⠒⡖⠘⣏⢹⣉⣙⣍⣉⣅⠨⣦⠽⡤⢧⠼⡄⠘⣆
⠀⠀⠀⠀⠀⠘⣆⢳⡀⠈⢧⠄⠺⡖⠚⡖⠙⡉⢹⣉⣹⣄⣵⡤⠴⡤⠼⡆⠲⡒⢳⠚⡗⠙⡄⠘⣇
⠀⠀⠀⠀⠀⠀⠘⣆⢻⡀⠈⢷⠒⠳⠀⠙⡉⠹⡀⢣⠀⢱⣀⣣⡀⢵⡤⠼⡤⢷⠐⢳⠚⡂⠹⡀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠘⣆⢹⡄⠘⣷⠒⢳⠒⠹⡉⢻⣉⣇⣀⣣⣀⣧⣠⣥⠤⠵⠴⠷⠚⠓⠛⠋⠁⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢻⡀⠘⠒⠚⠛⠋⠉⠉⠉⠉⣭⡤⠤⠤⠴⠶⠒⠒⢲⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⣀⡀⢀⣀⣀⡀⠠⢷⡀⠀⠀⠀⠀⠀⠀⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠒⠒⢚⣒⣂⣈⣉⣤⡤⢤⣴⣶⣶⣾⠯⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⡷⠤⣤⣤⣶⣖⣒⣚⣻⣿⣭⠭⠭⠽⠿⠓⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠶⠶⠤⢤⣐⣶⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡭⠿⠛⠒⠀⠈⠁⠀⠈⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⡟⠓⠒⢄⠀⠀⠉⣉⣙⢲⡀⠀⠀⠀⠀⠀⠀⣠⡖⠉⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡎⠀⠠⣄⠈⡋⠉⠁⠀⠀⠀⠀⠀⡎⠁⡀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⡠⢤⡂⠀⢸⠇⠀⡸⢡⠖⠒⠋⠙⠁⠀⠀⠀⠀⠀⠀⠀⠹⣜⢳⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⡿⠂⠀⠘⣆⣸⠀⢠⣇⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢣⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠊⡇⠀⠀⠀⠘⡇⠀⣸⣽⠀⠀⠀⠀⢠⠞⠉⠉⠒⢦⢀⡀⠤⠄⢤⣠⠎⠙⣦⣀⠀⢉⣉⣙⣲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢷⠀⠀⠀⢀⠇⠀⣯⠿⣂⡭⢑⡆⠀⠓⠤⠤⣶⠞⠉⠀⢀⡠⠊⠀⢀⡜⠁⠈⠀⠀⠉⢢⡉
⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠊⠉⣠⠌⠲⣤⡤⠊⠀⠀⠉⠊⣉⣠⠚⠉⠉⠉⠀⠘⣇⣀⡤⠖⠋⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢧
⣠⠤⠒⠂⠉⠁⠀⠀⠀⠀⠀⣠⠞⡹⠀⠀⠀⠀⠀⠀⠀⡎⠁⠈⠉⠉⠓⠒⠒⠒⡆⠀⠀⠀⠀⠀⠀⠘⢫⡭⠭⢑⠆⠀⠀⠀⠀⡇
⣇⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁⠀⡗⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒⠉⠀⠀⠀⠀⡜
⠈⠑⠒⠒⠒⠒⠊⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣔⠲⡖⢇⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢏⠳⡘⠜⣄⠀⠀⠀⠀⢳⢄⣀⠠⡾⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⡌⠀⠈⠿⠳⢤⡀⠘⡀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢀⠀⠀⠀⠀⠀⡀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⢦⠀⠀⣺⡓⠷⠀⢇⡠⣄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣈⡆⠀⠀⢸⡏⣰⠆⣰⠃⠀⠀⠀⠀⠀⠀⣬⢠⠋⠀⠙⣆⠀⠘⣠⠞⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠙⠃⠀⢀⡸⠟⠁⢺⠀⠀⠀⠀⠀⠀⠀⢀⠇⡜⠀⠀⣜⣁⣳⣞⠁⢠⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢫⣹⢀⣀⡤⠔⠚⠁⠀⣠⢶⡅⠀⠀⠀⠀⠀⠀⠀⡜⢸⠁⣠⠴⢊⠉⠁⠈⠓⠃⠀⢤⡉⠦⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⣹⠏⠈⣆⠀⢀⠴⠊⠀⡀⠃⠀⠀⠀⠀⠀⠀⢰⠳⠟⠉⠀⠔⢺⡀⢆⠀⠀⠀⠀⠀⢻⠢⡈⠢⡀⠀⠠⡇⡇⣰⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⠞⡗⠀⠀⠸⠖⠁⢰⠀⠀⢡⠶⢤⠀⠀⠀⠀⠀⠸⠒⠉⠁⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠈⠆⠈⠢⡘⢦⣸⢇⣥⢃⠜⣠
⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⣀⣸⡄⠛⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠘⡏⠙⠈⠁⢙⣊⠴
⠀⠀⠀⠀⠀⠀⢣⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⢿⣀⡠⠖⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠶⠦⢤⣤⣤⣤⣤⣤⡞⣠⠖⡡⠚⠓⠚⠉
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⡗⢲⠶⠦⠶⠖⠒⠒⢛⠇⡎⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠈⠉⡙⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢇⢸⠀⠀⠀⠀⠈⠉⠚⠈⠁⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠠⣸⣄⣇⠀⠀⠀⠀⣀⣀⠴⠁⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡖⢲⣦⠄⠐⠃
⠀⠀⠀⠀⠀⢀⣀⣀⣬⣀⠈⠙⡛⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⣀⣀⣀⣀⣘⡹⡀⢫⢡
⠀⠀⠀⠀⢎⣁⣀⡈⠿⠓⠋⠉⠉⠒⠢⠤⠤⠤⣤⣤⣴⠾⠓⠒⠢⢄⠀⠀⠀⠀⠙⠠⡤⠒⠒⠚⠛⠛⠓⢷⡒⠇⢱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠈⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠦⢤⣤⣤⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠄⠚⠛⠁⠴⢶⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠈⠁⠀⠀⠀⠀⠀⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡐⠊⠀⠀⢠⠔⠀⠀⠀⠀⠒⠒⠄⠀⠀⠉⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠈⢳⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣴⣆⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠘⡃⢤⡀⠀⢸⠀⠀⠀⠀⠀⠀⠻⠏⠀⠀⠹⠟⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠟⠀⢱⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⠀⠀⠀⢸⠂⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣤⣤⠀⠀⣠⠄⠀⠀⠈⢧⡀⠀⠁⠒⠠⠤⠤⠤⠤⠀⠐⠂⠉⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⡴⠖⠁⠀⠀⠀⠀⠀⠙⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⠶⢶⠶⠶⠖⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠙⠛⠛⠛⠉⠛⠋⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⡂⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢣⡈⠙⠳⢤⣄⣀⡀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠤⠁⠀⠀⠀⠀⠇⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠱⣀⠀⠀⠀⠉⠉⠐⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⡤⠊⠀⠀⠀⠀⠀⡀⠀⠟⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡇⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠃⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⡇⠀⠀⠷
⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠘⠣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢾⡄⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⡆⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⠀⠇⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⠀⢰⡆⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠃⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠓⡄
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠃⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰⠤⠇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⢠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣤⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠠
⡼⠓⠒⠒⠲⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠂
⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠛⠉⠛⠛⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣴⣦⣴⣶⣶⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⠶⠶⠶⠶⢶⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠁⠀⠂⠀⠐⠒⠒⠒⠀⠠⠬⠭⠭⠭⠉⠉⠛⢉⣙⣛⠛⠛⠒⠺⠶⢶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠀⢀⠀⠀⠀⠀⡇⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠧⠉⣐⣒⡒⠠⠤⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⡉⠰⢠⠀⠖⢀⠄⠉⡁⠁⣒⠒⠠⠭⠅⢁⣒⠒⠠⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠤⠐⡠⠄⡀⠐⢂⠀⠙⠁⡄⠈⣠⠒⢁⡠⢄⠒⠤⠀⢉⡉⠁⣒⠂⠤⠭⠭⢁⣒⠂⠴⠃⡇⡀
⠀⠀⠀⠀⢀⡠⠒⠉⡀⠀⢁⠀⠈⠠⠄⠀⠡⠀⠈⠀⠒⠈⠔⠂⠄⠈⢡⠐⠈⠀⠠⠆⠀⠤⠈⢀⠀⠀⠆⠀⠤⠀⣀⢥⣟⣉⠇
⠀⠀⠀⠀⢨⣉⠉⠒⠒⠂⠥⠴⢅⣀⡐⠀⠄⠁⠀⠀⠀⠈⠁⠀⠀⠆⢀⡄⠀⣁⠀⠊⠀⠔⠂⡡⠀⠈⣁⠤⠒⣉⠿⠚⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠠⠤⠤⣀⣉⠉⠑⠒⠂⠤⠤⢄⣀⡠⠂⠐⠆⢀⠀⠈⡀⠀⠃⣀⠤⠒⣉⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠦⠤⠤⣀⣉⡉⠐⠒⠒⠤⡤⠒⢉⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠓⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠂⠉⠁⠉⠙⠛⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠠⣄⠀⠀⠀⠀⣀⠄⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⣾⣿⡇⠀⣾⣿⠆⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠤⠐⠂⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠉⠁⠈⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠂⠤⡤⠤⠐⠒⠸⢧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⠀⢀⠗⠀⢀⡁⠐⠲⢄⠀⠀⠀⠀⠀⢰⣄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠄⣀⠀⠀⠑⠦⣀⠀⠀⠀⠀⠠⠊⢻⡟⠀⢲⡌⠠⠟⠈⠁⠀⠢⡀⢳⡄⠀⣀⠖⢀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠑⠬⡕⢠⡀⠀⠀⠙⠢⣄⠐⠁⠀⣾⡇⠀⠀⣧⠇⠀⠀⠀⠀⠀⢰⡀⠐⢘⣥⠔⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⡐⠂⢄⠀⠈⠑⠮⣐⠄⡀⠀⠀⠱⢦⣸⢿⣿⠀⠀⠊⠁⠀⠀⠀⢀⡀⡟⠁⠘⠉⠀⠤⠁
⠀⠀⠀⠀⠀⠀⠠⢀⡞⠀⠀⠱⢀⣀⢱⡀⠀⠀⠈⠑⠬⣣⢄⡀⠀⠉⠛⢯⣈⠑⠀⠀⠀⢀⠰⠦⠴⠄⠂⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠠⢡⠏⠀⠀⡄⢀⠂⡀⢃⡄⡀⠙⢀⡀⠀⠀⠙⠪⡕⡠⠀⠀⠈⠑⠢⣄⢰⡏⠘⢦⡀⠀⠀⠀⡘
⠀⠀⠀⣀⡄⢠⠃⠀⠀⠌⡰⠂⠌⠠⠀⢠⠁⡼⢂⡀⠈⠂⠀⠀⠈⠑⠪⣐⠤⣀⠀⠀⠙⢣⢀⠈⠳⣄⠀⡐
⠀⣀⠁⠈⣶⠃⠀⠀⢎⡐⠁⠀⢐⠀⠠⢃⡾⠁⣎⠄⡀⠁⠀⠀⠀⠄⠀⠀⠑⠦⡁⢆⡀⠈⠀⠑⠢⣌⠋
⡀⠻⡤⠐⠉⠑⠢⣀⠀⠀⠀⢀⠈⠉⠠⠞⡀⠤⠠⠐⠀⢀⠀⢀⡀⢀⠀⠀⠀⠀⠈⠑⠪⣕⠢⢀⠀⠀⠉⠂⠄
⠙⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠀⠁⠀⢀⡐⠁⠀⠀⠀⠀⡸⠀⠈⣁⠏⡠⠢⠀⣉⠠⠄⠀⠈⠑⠢⣐⠠⢀⠀⠀⠐⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠈⠀⡅⠂⣴⠃⡀⠈⣠⠌⡁⢦⠀⣠⠔⢀⠀⠀⠑⠦⣉⠢⣄⠀⠀⠈⠲⢄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠒⢄⡀⠀⠀⠤⠄⠞⠡⢎⡈⠙⠃⠄⠀⠈⡺⠃⠄⡀⢁⠠⠄⡀⠈⠑⠢⣙⠢⢄⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄⡀⠀⠙⠤⣀⠕⠁⢈⠀⢆⡐⠁⠙⠀⣰⠏⠠⢀⣸⠦⠄⡀⠈⠑⢦⠑⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢠⡀⠀⠀⠠⠀⢰⠋⠐⡃⠀⢩⠃⠈⠓⡄⠉⠂⣀⡼⠂⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠦⣀⠀⠈⠂⢠⣁⡞⠠⡊⠑⠂⡠⠁⠌⠀⢀⡴⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠙⠢⣄⠀⠈⠱⢄⡀⠀⠐⡡⠀⠀⢀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠐⠒⠀⠀⣠⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡉⠢⢄⡀⢀⡼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣭⣭⣭⣖⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⠶⠾⠟⠛⠛⠛⠛⠷⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⣴⠾⠛⠉⠁⠀⠆⢰⣆⠈⣶⠋⠈⣷⢀⡶⢠⡍⠙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠿⠀⠙⠆⠠⠟⣸⠇⠛⠁⠀⣴⠂⣉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣀⣤⣶⠶⠾⠟⠛⠻⠿⠶⣶⣤⣀⠀⠆⢘⡧⢠⡶⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⣰⡗⢈⠻⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣌⠰⡏⠑⢀⣁⡈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠻⣦⡀⠀⠀⠀⠟⠠⠾⠃⢀⠙⢷⣄⠀⠀⢀⣴⣤⠀⠀⠀⠀⠀⠀⠙⢷⣌⠁⡟⠉⠉⠀⢻⣆
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⢷⣄⠀⠙⠉⠀⠀⠀⢼⣷⡆⠀⠀⠻⣧⠁⠀⣀⣩⡄⢻⡆
⠀⠀⠀⠀⠀⠀⣬⠁⠀⠀⠀⠀⢀⡀⢠⡀⠀⢀⠀⠄⣸⠏⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢹⣧⠙⣩⣧⣤⠈⣿⡀
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⣿⠀⠀⠙⢷⣄⠀⢰⡟⠀⣠⡾⠓⣄⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⣋⣠⣥⡄⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠶⠶⠦⠴⢸⡏⠀⠀⠀⠀⠙⢷⣄⠀⠾⠋⣷⠚⠋⢀⣄⠙⢷⣄⠀⠰⣿⣿⠀⠀⠀⢸⡇⢋⠙⠂⡀⢸⣇
⠀⠀⠀⠀⠀⢸⡇⢀⣤⣤⣈⢸⣇⠀⠀⠰⣿⣿⠀⠙⠷⣄⠀⠻⣀⠀⢰⡇⢀⠄⠙⢷⣄⠈⠀⠀⠀⠀⢸⡇⣀⣀⣀⡁⢸⡏
⠀⠀⠀⠀⠀⠀⠇⠈⠀⢀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⣸⠗⢁⣴⠟⢀⡙⠳⣄⠀⠀⠀⣼⡇⠈⠉⢉⡁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⠀⠄⠙⣇⠀⠀⠀⢠⣀⡀⠀⠀⠀⠙⢷⣄⠀⠰⡟⠁⡠⢺⡇⠀⡈⠻⣤⣰⡿⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠈⠀⠹⣆⠀⠀⢻⠿⠄⠀⠀⠀⠀⢀⠙⠳⣄⠀⠊⠀⢸⡧⢈⣀⣄⡈⠻⣧⡀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡄⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠐⢿⠿⠀⠈⠳⣄⠀⠉⠀⡿⠋⠀⠀⡦⠈⠻⣦⡀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⢀⡴⠂⡙⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠣⣠⡶⠀⠀⠀⠀⠈⠻⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⠀⠀⡤⠆⠈⠙⠻⠶⣶⣤⣤⣤⣤⣤⣶⠶⠿⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠘⠂⠀⠀⣰⠋⣴⠀⡦⠠⠀⢤⡄⠄⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠇⢸⡄⢰⡟⠂⠀⠀⣇⠀⠀⠀⠀⠀⣀⣤⠾⠛⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠤⢶⣬⢥⣬⣄⣤⣍⡀⢤⠶⠞⠛⠉⠁⠀⠀⠀⠈⠁



⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⡤⡤⠀⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠔⠒⠒⠐⢛⣀⣉⣉⡈⠉⠉⠉⠉⠉⠁⠀⠴⠦⠤⠤⠤⠤⠤⠤⠤⠤⠶⠖⠖⠞⠚⠒⠒⠒⠒⠚⣎⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠒⠀⠉⠉⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡰⠠⢁⠠⠡⡡⡁⡐⢀⢣⣧⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠠⢁⢁⠂⡄⡑⠡⠀⢀⢂⡴⡷⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢠⠃⠂⠂⠌⠘⠡⠡⢁⠂⣼⠂⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⠀⠌⠈⢌⠄⠀⠁⠁⠆⣼⠃⠅⢿⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠆⠜⡌⢀⠂⠆⢀⠀⠌⣰⢃⡮⠠⢻⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡐⡀⠘⢠⠋⠄⡄⠀⠌⣰⠇⡾⢁⠐⢻⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡐⡰⡐⠠⡇⠀⡐⠌⡘⢠⠏⡸⢁⢂⠂⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠐⠐⡐⠁⡝⡐⠐⠀⡐⢠⠏⡰⢃⢂⠂⠲⢻⣗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⠀⡐⢀⠛⠀⠁⠐⠔⢠⡎⣰⢃⢊⢎⢮⠡⢹⣎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠡⠡⠁⠠⠣⠡⠠⠁⠀⢠⡟⣰⢣⢂⠂⢢⢁⠃⢩⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢡⠣⠁⠡⢠⠁⠠⡰⠁⡙⡠⠃⠃⢌⢢⢁⢂⢒⠉⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂⠃⠃⠴⠡⢃⠶⠠⠁⠰⠴⠕⠌⢎⠊⠆⠀⠂⠞⠄⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣖⡂⣰⣆⣀⢀⣉⣉⣉⣉⣀⢀⣀⣀⣀⣒⣒⣒⣒⣒⣒⣒⣞⣒⢒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣾⣾⣟⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⠀⢈⠉⠪⣟⠝⢹⣬⡝⣭⣫⣝⣭⣋⢫⣝⣭⣪⡝⣭⢫⣭⣝⣫⣠⡉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢠⠂⠀⠀⠀⢀⣀⣀⣀⣀⣜⣀⣠⣁⣀⣈⣀⣀⣉⣀⣒⣃⣰⣄⣱⡐⣒⣘⣆⣐⣒⣐⣄⣀⣀⣀⠐⢒⡢⠵⣆⠱⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠎⡰⠁⠀⠀⠀⠀⠀⠀⢀⠈⠀⢀⠃⠀⣠⡀⣀⣀⣀⣀⣀⣇⠀⠀⣢⢐⠒⡀⢣⡐⣤⠚⢢⡀⠀⠀⠀⠀⠈⢆⠲⠈⢧⡙⡄
⠀⠀⠀⠀⠀⠀⢀⠎⡔⠀⠀⠀⠀⠀⠈⣹⠋⣉⣉⣃⡈⣨⡁⠈⠀⣉⠀⠀⢈⡀⠀⠈⡁⠀⠀⠈⣊⠢⡙⠀⡙⣬⣥⠛⣍⠀⠜⢏⠨⡠⡂⠀⢳⣜⡄
⠀⠀⠀⠀⠀⢀⠎⣈⠀⠀⠀⠀⠀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣈⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⣬⡍⠋⠁⠿⡹⢀⡀⢉⣈⡛⣁⣁⣹⡻⡀
⠀⠀⠀⠀⢀⠎⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠥⠍⢭⠁⠉⠈⠀⠀⠙⠬⡑⠭⣋⠉⡋⠧⣉⡋⠙⠛⠦⣳⡀
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠈⠑⠢⣉⠢⢕⠨⣙⠣⡢⢅⠂⢳
⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢍⡢⢅⠂⢅⠨⡀⢃
⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⠀⠚⠢⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠬⣀⣌⡂⢸
⠀⠀⠘⠯⣀⣒⣖⣶⢖⣖⡰⣤⢤⢤⠤⠤⠤⢤⠤⠤⠤⠤⠤⢤⠤⠤⢤⢤⠤⡤⡤⡤⠤⠤⢤⠤⠤⠤⠤⡤⠤⠤⢤⠤⡤⠤⠤⢤⠤⢤⢤⠤⡤⣤⣤⣤⡼⠏
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠘⠁⠙⠋⠊⠘⠉⠛⠋⠋⠚⠋⠛⠋⠚⠙⠛⠛⠉⠛⠛⠛⠛⠛⠘⠑⠃⠛⠉⠚⠑⠛⠛⠋⠋⠉⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⣀⢀⠠⢀⠄⠠⠀⠒⠀⢐⠀⠀⠀⠠⠄⠀⠀⠀⡰⠄⡐⠀⠀⠐⢂⡒⠀⡐⠠⢀⠂⠠⣐⠀⠀⠂⠀⠐⠀⠀⠂⠀⠂⠠⢂⠁⢐⡀⠀⡢⠔⡀⢀
⠀⠀⠀⠀⠀⠁⠀⠁⠐⠈⠀⠀⠉⠐⠀⠀⠊⠀⠂⠀⠒⠁⠒⠁⠐⠊⠀⠊⠁⠂⠁⠐⠈⠀⠂⠁⠐⠈⠀⠂⠀⠔⠊⠀⠊⠁⠐⠀⠔⠊⠁⠂⠁⠐⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠉⠉⠉⠑⠒⠂⠀⠤⠤⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠓⠒⠒⠶⠤⠤⢤⣄⣀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠤⢈⠑⠂⢅⠂⠄⡉⠲⡍⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢠⠘⠀⠈⠐⢨⠐⡢⠄⣉⠖⢩⠙⠲⢎⡑⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⡀⠀⠀⠀⠀⠎
⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⠐⠢⠄⣉⠤⢈⡑⠂⡤⠂⠢⢤⣁⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣼
⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⣁⠘⠠⠐⠋⠤⢈⡰⠆⠀⠀⠀⠠⠗⢀⠄⡀⠀⠀⠀⠀⠀⠀⠤⣀⠠⠇
⠀⠀⠀⠀⠈⠳⢄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⢀⡠⠚⢠⡄⠈⠂⣄⡑⠄⠊⣉⡁⠤⣀⡨⠒⠎⠁⠀⠀⠀⠀⠀⢀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠠⠠⠂⠈⠀⠄⠒⠀⠄⠐⠒⠤⠀⠒⢠⠄⠂⠁⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⡀⠀⢤⠈⠉⠠⠀⠈⡁⠤⠒⠁⠀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⠁⠠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⢀⡄⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⢀⣠⠴⠚⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣈⣈⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣒⣒⣒⣒⣒⣒⣒⣒⣒⣶⣶⣶⣦⣤⣤⣤⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠉⠙⠛⠒⠒⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣼⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢹⡇⢀⣿⡇
⠀⠀⠀⠀⢀⣠⡴⠚⠋⠁⠀⣠⣼⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⣿⠃
⠀⠀⠀⠀⢸⠉⠉⢉⣉⡙⡿⠁⣸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⢹
⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇⡇⠀⢻⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣼⣿
⠀⠀⠀⠀⢸⠀⠀⢸⠀⠃⡇⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡿⡏
⠀⠀⠀⠀⢸⡇⠀⢸⢰⠀⡇⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⣧⡇
⠀⠀⠀⠀⠀⡇⠀⠈⡉⠀⡇⠀⢸⡄⠸⣧⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠀⢸⢿⠇
⠀⠀⠀⠀⠀⣷⠀⠠⣦⠀⡇⠀⢸⣇⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠍⠩⠁⠀⢸⡾
⠀⠀⠀⠀⠀⢻⣀⣀⣀⣠⣧⠴⠋⢉⡏⠉⠉⠉⠉⢉⣉⡽⠭⠭⠭⠭⠭⡿⡟⠛⣟⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣉⣒⣢⣀⡀⣿⡭⠶⠶⠦⠤⢤⢼⠟⠁⠀⠈⠑⠒⠒⠢⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⢽⡿⢛⣿⣻⣏⣩⣭⢭⠭⣭⠿⣿⠟⣒⢒⣒⣒⣶⠶⠦⣤⠀⠀⣀⣼⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠰⠭⠿⢩⣞⣲⣓⣲⠫⣴⠯⡼⠭⡽⠏⡴⣡⣳⣛⡞⡳⢃⢎⡼⠀⣾⠥⣛⡒⠽⡆
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠭⠭⣭⣖⣒⣒⣒⡋⠭⠭⠭⠍⢑⣒⣙⣓⣣⡯⠝⡱⣣⠞⠀⠘⣧⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠓⠒⠒⠲⠶⠶⠤⠽⠖⠁⠀⠀⠀⠻⢷⣶⡶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⡒⠋⠍⠉⠉⠋⠁⠈⠬⠽⠓⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢄⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠁⠀⠈⢓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⢠⣦⡀⠀⠀⠀⠲⣶⣿⣷⡶⠂⠀⠀⠀⣰⣦⠀⠀⠀⠪⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠀⠀⠀⠀⠻⣿⣿⣿⠟⠀⠀⠀⡿⠿⠿⡇⠀⠀⠘⢿⣿⣿⡿⠃⠀⠀⠀⠨⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⠌⠀⠀⢠⡆⠀⠰⠟⠙⠻⠀⠀⢀⣀⠀⠠⠀⠀⣀⠀⠀⠸⠛⠛⠷⠀⠀⣴⡀⠀⠈⢌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢏⠂⠀⠰⣶⣿⣿⢲⠆⠀⠀⡠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⡀⠀⠀⠶⣾⣿⣿⡶⠀⠀⠃⢶
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣼⠟⠻⡇⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⡿⠛⢿⡄⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂
⠀⠀⠀⡄⣶⣶⣦⠀⢠⣴⣶⣄⠀⢠⣶⣶⣦⡄⠀⠀⠀⠀⣠⣶⣦⣄⠀⣴⣶⣶⣶⠀⢀⣤⣶⣦⡀⢠⣶⣶⣶⣄⠀⢴⣶⣶⣶⡄⢠⣶⡴⣶⡆
⠀⠀⠀⠀⢻⡏⠀⠀⣿⡏⠀⣿⡇⢸⡿⣠⣼⡿⠀⠀⠀⠀⢿⣧⣬⣁⠀⣿⣧⣤⠀⠀⣸⡟⠀⠉⠁⢸⣿⣥⣼⣿⠀⣾⣿⣤⠀⠀⠀⠈⣿⡅
⠀⠀⠀⠀⢻⡇⠀⠀⢿⣧⣠⣿⠇⢈⣾⠉⠉⠁⠀⠀⠀⢀⣤⢉⣹⣿⠀⣿⣏⣉⣀⠀⢹⣳⣀⣤⡄⢸⣿⠉⢻⣧⠀⣿⣯⣉⢀⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠙⠃⠀⠀⠈⠉⠛⠉⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠛⠛⠙⠛⠀⠀⠙⠛⠋⠀⠈⠛⠀⠈⠛⠀⠙⠛⠛⠛⠁⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠐⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⢹⣦⣴⡇⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⣿⣶⣿⠁⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀⠀⠐⠻⣿⣿⠿⠂⠀⠀⠈⠒⠄⠀⣀⠀⠀⠀⢀⠀⠠⠔⠈⠀⠀⠀⠾⠿⣿⡿⠷⠀⢀⢄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠘⠃⠀⠰⣶⣤⣶⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⢰⣦⣴⡶⠀⠀⠛⠀⠀⡠⣣⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡠⡀⠀⠀⢼⣿⣿⡿⣦⠀⠀⠀⣿⣷⣾⡇⠀⠀⠠⢾⣿⣿⣿⠄⠀⠀⠠⢪⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⠢⠄⠀⠈⠟⠀⠀⠀⠀⠚⠷⣿⡿⠟⠂⠀⠀⠀⠘⠏⠀⢀⠀⣨⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢮⣖⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠠⣐⣬⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠀⠶⠤⢤⣤⣤⡀⠠⡤⠤⠷⠚⠊⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿