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⠀⠀⠀⠀⠀⠀⡞⠉⠉⠀⠀⠀⠀⠈⠉⠙⠒⡶⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡦⣏⠀⢰⠇
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡿⢠⠏
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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⣄⡀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠙⢦⡀⠀⠀⠀⠀⠀⠾⣽⠂⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
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⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠈⠙⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢀⣟⠉⠛⠒⢒⣲⠤⠀⠀⠀⠀⠀⠀⠀⢲⠛⠉⠉
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠈⠳⡀⣀⠀⠀⡞⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠟⠉⠙⣦⠿⡄⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠈⠹⣄⠀⠀⠀⣇⠀⠸⡇⠳⣄
⠀⠀⠀⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢻⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣁⣀⣀⡤⠞⠻⠤⠴⠳⣄⠀⠀⠀⠀⠀⠀⢀⡼⠻⡄⠀⢸⡄⠀⡇⠀⢼⣄⡀
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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠋⠁⠀⠈⠉⠈⠁⠀⠀⠈⠉

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⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠁⠈⠙⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋⠁⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠙⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣄⠀⣀⣠⣤⣶⣶⣶⣶⣶⣶⣶⣦⣤⣄⣀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
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⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠟⣋⣩⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢿⣧⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣱⣦⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⣀⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠠⠄⣀⠀⠀⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⣠⠤⢂⡥⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⠀⠀⠀⠀⠀⠐⠬⡑⠢⢄⡈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⢀⠔⠉⠀⠀⠀⠀⠀⠀⠐⠤⠖⠋⠛⠶⠖⠀⠀⠀⠀⠀⠀⠈⠳⣄⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣





⣿⠿⠿⠿⡿⢿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣛⣛⣿⣿⣟⣿⣛⣿⣿⣿

⠀⠀⠀⠀⠀⣴⢿⠛⠳⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⠶⠶⣶⣦
⠀⠀⠀⠀⢰⡏⠀⠳⡄⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠁⠀⠀⢀⠞⠁⢹⣇
⠀⠀⠀⠀⢸⡇⠀⠀⠘⣆⠀⠀⠀⠙⠳⣄⢀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⡔⠋⠀⠀⠀⠀⠀⣴⠋⠀⠀⠈⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣦⠀⠀⠤⠖⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⢤⣀⢀⡼⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠈⣷⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⢹⡄⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠱⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡟
⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢠⡞⠋⠁⠉⠙⢦⠀⠀⠀⢠⣤⡄⠀⠀⠀⡴⠞⠛⠛⠷⣄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⠞⠉⠀⠉⢻
⠤⣤⣄⣀⣀⢸⣇⠀⠀⠀⢿⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⣻⠀⠀⠀⠘⠁⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢸⡇⠀⣀⣀⣾⣁⣀⡀⠀⠀⣸
⠀⠀⠀⠈⠉⠙⣿⠦⣄⠀⠀⡀⠀⠀⠀⠀⠀⠑⡦⠤⠖⠉⠳⠤⢤⡔⠀⠀⠀⡀⡀⠈⠁⠀⠀⠠⠶⣾⠟⠉⠉⡿⠁⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇⢀⡠⠤⠤⠤⣰⠇⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⡿⠀⠀⢸⡇⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⣻⣦⠖⠁⠀⠠⠀⠀⠀⠀⠀⠹⡋⠀⠀⠀⠀⡸⠀⠀⠀⠐⠀⠀⠀⠀⠂⢤⣤⣾⠁⠀⠀⢸⡇⠀⠀⠀⡼⠁
⠀⠀⠀⠀⣀⡴⠚⠉⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠙⠳⣦⣀⢸⡇⠀⠀⢸⠃
⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⠴⠖⠛⠉⠀⠀⠀⠀⠀⠙⠻⣇⠀⠀⣾
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⣸⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⢷⡀⠀⢀⡴⠋⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠙⢲⠃⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢸⡷⠖⠋⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡰⠀⡄⠀⠀⣼⠁⡀⠀⡀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⣠⣾⣤⠤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠴⠧⠤⠖⠻⢤⣅⣸⣀⣠⡴⠿⣀⣀⣀⣀⣀⣒⣒⣛⡽⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠢
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠋⠉⠉⠀⢀⡇
⠀⠀⠀⠀⠀⢠⠂⢀⢳⠢⡀⠀⠀⡔⠉⠑⠄⠀⠀⢀⣀⡀⠀⢈⠅
⠀⠀⠀⠀⠀⡏⠀⢠⡑⡇⠈⢢⡌⠀⠀⡠⠂⠀⠂⠁⠀⢸⠄⠋
⠀⠀⠀⠀⢰⢗⠀⠀⣑⠤⠄⠀⠑⠤⠀⠦⢐⠉⠀⠀⣀⠤⠄
⠀⠀⠀⠀⠈⣀⠖⣉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠊⠁⢀⠄⠂⠀⠀⡆
⡤⡀⠀⠠⠊⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⠷⣂⠀⠀⠀⡇
⢃⠉⠀⠀⢴⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⢨⠊⠁⠀⠀⣸⡃
⠀⠀⠀⠀⢠⠰⣿⡄⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠰⡀⠀⢀⠔⡑
⠀⢢⠀⣒⡦⡙⠛⠃⠀⠀⠀⠀⠔⢋⣽⣿⣿⠂⠀⠀⠀⠈⠂⠄⠄⡀
⠀⠀⢳⡒⠚⠻⡆⠀⡠⠀⠀⠀⠠⠤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⢘⣲⠄
⠀⠀⠀⠑⢄⠀⠐⠒⠦⡄⠀⠼⢅⡒⠬⢍⠀⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠈⠓⢄⡐⠀⠀⠀⠀⠀⠉⠢⡀⠁⠀⠀⢀⡠⠔⠉
⠀⠀⠀⠀⠰⡶⠀⢌⡐⠀⠀⢀⡴⠊⠁⠉⠲⡐⠈⠁
⠀⠀⠀⠀⡘⠀⠀⠀⠈⠁⠀⡞
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⢀
⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⢀⠔⠁⠋⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠈⡌⠢⠀⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠂⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⡈⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠰⢀⡀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⣊⠄⡀⠀⠀⢠⠀⠀⠀⠘⡁⠀⠑⠄⠀⠀⡠⠃⠀⠀⢀⣀⣀⠠⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠦⠆⠊⠀⠀⠀⠀⠘⡄⠤⠠⠜⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠘⣇
⢠⡶⠒⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠈⠀⠀⢻
⣿⠀⠀⠄⢄⠙⢷⡀⠀⠀⢀⣀⣤⠴⠶⠒⠛⠛⠛⠛⠲⠦⣄⠀⠀⠀⠀⢸⠁
⣿⠀⠈⠀⠀⠐⠄⠹⣦⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⣾
⢻⠀⠈⠀⠀⢀⡴⠋⠀⠀⠐⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠶⠶⠄⠀⠹⣴⡏
⠘⣇⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠙⣆
⠀⢹⡆⣸⠁⢀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣶⣷⡀⡈⠳⠀⠀⠀⢹⡆
⠀⠀⢻⡟⠀⠈⠁⢀⣠⡄⠠⢄⠀⠀⠀⠀⠀⠀⠀⡻⢿⡿⠃⢠⠀⠀⠀⠀⢿
⠀⠀⠀⣏⠀⠀⠀⠺⣿⣷⣴⡆⢡⠀⠀⠀⠀⠀⠀⠈⠒⠐⠒⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢻⠀⠀⠀⠐⣌⠛⠋⣀⠎⠀⠀⣴⣪⣿⠆⠐⢀⠀⠀⠀⡀⠀⠀⢀⡼⠁
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠉⠉⠀⡠⠂⠀⠈⠉⣇⠀⠀⠌⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢀⠊⠀⠈⠢⠤⠞⠁⠀⠀⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠻⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡠⠤⠲⡋⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣻⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⢸⠃⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⢷⡀⠀⠀⠀⢹⡄⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⢿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⢸⠇⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣧⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡀⠀⣠⠟⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡀⠀⠀⠈⡇⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢸⣥⠾⠋⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣄⣀⡇⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡟⠀⢀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠀⢠⠖⠉⠁⠀⢠⠇⠀⠀⠀⠀⣞⠀⠀⠀⠀⣰⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⢦⣤⣤⣤⣼⣀⢀⠀⣀⣀⣿⣤⣤⠴⠞⠁
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⠀⠀⠀⢀⠸⡇⢀⡀⠀⠰⠀⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⠇⠃⠟⠀⠀⠀⠀⠈⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⡴⣶⣦⠀⠀⠀⠀⢠⣞⣿⡆⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠷⠆⠀⠀⠻⣿⡿⠁⠀⠀⡀⠘⢿⠿⠃⠀⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠳⢀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⡠⠚
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠐⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠊
⠀⠀⠀⠁⠆⠕⠀⠀⠀⠀⠀⠠⡄⠈⠢⢀⡀⠀⠀⢀⡀⠐⠁⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠠⠀⢉⢁⠈⠀⠀⠀⠀⠀⠐⠀⠀⠐⣀⠯⠇⢀⠇⢀⡀⠈⠃⠀⠀⠀⠀⠀⠀⠠⠀⢀⣜⠄
⠀⠀⠀⠀⠈⣹⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡴⠃⠉
⠀⠀⠀⠀⠀⠭⣀⠈⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⡄⠀⠠⣀⠀⠂⠀⠼⠀⠈
⠀⠀⠀⠀⠐⠀⠾⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠈⠉⠣⠀⡜⠂⠐
⠀⠀⠀⠀⠀⠐⢆⠆⡎⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠂⠀⠀⡤⠤⡀⠂⠀⠂
⠀⠀⠀⠀⢠⠊⡀⠉⠸⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠾⠁⠀⠀⡰⠀⠀⢈⡀⠀⠄
⠀⠀⠀⠈⠐⢀⡄⠀⣀⠖⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠃⠀⡀⠄⠗⠂⠤⢌⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⠀⠀⠀⡀⠀⠀⠀⠄⠠⠀⠀⠀⢠⠀⠀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠈⠃⠂⠊⠀⠀⠈⠒⠒⠁⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⣸⡷⣄⠀⠀⡀⠀⠀⠀⣠⣬⡁⠀⣀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢩⣿⡿⠟⢻⣏⠹⣾⡋⢿⣼⣆⣸⠿⣯⣄⣘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣶⠶⠀⢛⠉⠢⠉⣹⣝⢻⡯⠈⣋⣇⠙⢻⡗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢁⡀⠨⠄⡇⢾⣿⣿⣽⢰⠀⡿⢠⢶⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠾⣋⢹⡼⠓⣆⠈⢉⣇⡟⣘⣤⢤⣥⣴⣖⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡉⠀⠀⠉⢛⣿⣯⣹⡀⠀⢽⣧⣿⣦⣾⣶⠷⣙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠉⠀⠀⠈⣹⡿⢿⡇⡀⣸⣿⣿⡟⢿⣿⣓⣻⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢭⡄⠀⠀⠀⠀⠈⠿⠛⠙⠻⠿⢦⣬⣭⣿⢟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢀⠀⠀⠀⠐⠓⠢⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠛⢶⣤⣄⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⣏⣴⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠁⢨⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⣾⢿⣼⣿⡟⣻⣿⣷⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀⣀
⠀⠀⢲⠖⢉⣹⠆⠀⠀⠘⡝⠃⠑⢎⡏⠀⢐⠉⠍⠛⠋⠍⣢⠀⠀⠀⠠⡀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠙⡄⠈⠀⠀⠀⠀⠉⢈⡱⠆⠀⠀⠀⡶⠤⠤⠴⡆
⠀⠀⡇⢠⠃⠀⠠⡄⢟⣤⠁⠎⠱⠸⡄⠀⢀⠤⠈⡗⢪⠀⠁⠀⠀⢀⠖⠑⠀⠛⠃⠐⠃⢧⠀⠀⠀⠀⠁⠀⠛⢀⡀⠛⠀⠈⡆⠀⠀⠀⠀⣆⠀⠀⠂⡠
⠀⠀⠡⡈⠂⠠⢛⡄⠘⡇⠰⠃⠒⡄⢃⠀⠀⠑⢿⡇⢨⠀⠀⠀⠀⡀⠀⠰⠄⠀⠀⠆⠀⠀⡆⠀⠀⠀⠑⣄⠀⠀⠁⠀⢀⡜⠀⠀⠀⠀⠀⠸⠓⠒⠙⣀⠖⣡⣆
⠀⠀⠀⠈⠁⠈⠁⠀⠀⠃⠂⠀⠀⠘⠋⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⠀⠀⠈⠁⠒⠈⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡿⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠐⠒⠒⢀⡦⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠚⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠤⠇
⠀⠀⠀⠀⡀⢀⡄⠀⠀⠀⣠⠴⣤⡀⣬⣽⡀⠀⠀⠀⡶⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⠀⠀⢀⡠⣤⠆
⠀⠀⠀⠀⡏⠁⢙⡄⠀⠀⠘⠁⠘⡀⠀⠚⠃⠒⠒⠆⠁⠈⠀⠀⠀⠀⠀⠀⢀⠠⢄⡀⠀⠀⠀⠀⠀⠹⡀⠀⠀⢀⡔⠁⠀⢀⣴⠂⠀⠀⠀⢀⣄⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠠⠛⢀⠸⠇⠀⠀⠀⠀⠀⣧⣴⠂⠀⠑⠿⠒⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠱⣀⡰⠏⠉⠁⠐⢻⠁⠀⠀⠀⠀⡘⠀⠉⠈⠁⠈⢹⠁
⠀⠀⣀⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣗⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣴⢲⡖⢲⣼⠀⠀⠀⠀⠀⠃⠒⠒⠀⠤⡤⡜
⠀⠀⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠠⠤⠤⠿⠀⠀⠀⠀⠀⠺⠄⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠀⠀⠀⢺⡆⠀⠘⡇⠀⠀⣇
⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠐⠂⠓⠠⠐⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠋⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠶⣀⡀⠀⠠⢾⠀⠀⠀⠀⡜⠉⠀⠑⡄⠀⠀⠀⣀⠄⡀⠀⠀⠀⠀⢠⡀⠀⣠
⠀⢀⡐⠃⠈⠙⡆⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⣦⠠⠠⣦⠸⡀⠀⠀⢸⠀⠟⠀⠀⡤⠀⠀⠠⣵⠊⠁⠀⠀⠀⠀⢸⠀⠈
⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⡋⣠⡀⠀⠀⠀⢀⡔⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠑⠰⠒⢺⠃⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⣺⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⣠⢄⠀⠀⢀⣀⡈⠻⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀⡇⠀⠀⠀⠀⢀⠀⠀⠁⠒⢄⣠⡞⠀⠀⠀⠀⠀⠀⠀⣙⢱⡋
⠀⠀⠀⠀⢀⣠⣤⣠⣄⣀⡀⠀⠀⠀⢇⠀⠻⠿⠃⠀⠻⡿⠃⢀⠃⠀⠀⠀⠈⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⣤⡀⠀⠀⠋⠀⠀⠀⠀⠀⢤⡀⢀⠋⠈⢰
⠀⠀⠀⠀⠧⢤⣤⣭⣤⣤⠼⠃⠀⠀⠀⠑⠄⣀⣈⣁⣀⢀⠤⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠛⠃⠠⠔⠃⠀⠀⠀⠀⠀⠈⢗⣼⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡧⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡁⢙⡶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⢸
⠀⠀⠀⠀⡠⠸⡇⠀⣙⡧⣀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀⠀⠀⣶⠄⠀⠲⣰⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⡠⠄⠂⠂⢄⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⡸
⠀⠀⠀⠰⡄⣈⠛⠒⠚⣁⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⣠⡘⡄⠀⠀⠀⠀⠈⠃⣀⠀⢀⠄⠊⠀⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠉⠑⠂⠀⠐⠒⠁⠀⠀⠀⠀⢀⡀⠤⠀⠀⠀⠀⠀⠀⡊⠈⠁⠀⠈⠀⢹⠆⠀⠀⠀⠀⠀⠈⣴⠁⠀⠀⠀⢀⠜⠀⢀⠼⠲⣖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠀⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠁⠁⠁⠀⠀⡀⡇
⠀⠀⢰⠓⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⡀⠘⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠛⠀⠀⠀⣭⠃
⠀⠀⠘⠂⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠀⣠⠔⠒⢱⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡽⢄⣠⡄
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢄⣼⣁⣠⠴⠋⣀⡀⢤⣿⠄⠀⠀⠀⠀⠀⠀⣄⢀⡄⠀⠀⠀⠀⠀⠀⢠⡆⡞⣡⢀⢀⣌⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⢲⡄⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠰⢻⠣⡝⢸⠇⠀⠀⠀⠀⠀⠀⠈⠉⠈⠃⠀⠀⠀⠀⠀⠀⠀⢸⠣⠤⠤⠤⠤⠤⢼⠄
⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠘⢍⠉⣹⠃⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⣧⡿⣄⣠⠾⣄⣀⠀⠀⠀⠀⠀⠁⠐⠀⠊⠀⠀⠀⠀⠀⠀⠸⣸⠀⠶⠀⠀⠀⠲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡯⠙⢷⣤⡀⠀⠀⠀⠀⣠⣼⡿⠄⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⢀⣠⣴⣿⠻⡿⣿⢿⣿⣷⣶⣌⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠈⠉⠁⣀⡀⠀⠀⠀⢉⣩⣭⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠹⠿⠿⠀⠀⠀⢾⠿⠋⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⢻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠟⠀⢠⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠠⠀⠀⠉⠓⢶⣈⣁⣼⣛⣩⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢙⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⡿⢿⣿⣿⣿⠿⣿⣷⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣷⢻⣿⣿⣯⣷⣽⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣻⣶⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡰⣟⠏⠁
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠻⠃⠀⠀⠀⠀⠀⠔
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠐
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣠⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠉⠉⠙⠻⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀
⠀⣴⣿⣿⣝⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠙⠻⠿⠿⢆⠀⠀⠀⢹⡄⡄
⠀⢻⡿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡆⠀⠀⠀⠉⠃⢀
⢰⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣴⣧⣀⡀⠀⢀⣀⣴⡤⠤
⠈⠉⠛⠛⠛⠛⠛⠟⠛⠛⠂⠠⠤⠤⠤⠄⠰⠖⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣆⠀⠀⠀⠀⢠⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆⣤⣤⣀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⣽⣿⡏⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠁⠀⠹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠁⠀⠀⠁⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣆⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢀⣼⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⢿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠋⠀⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⣿⡏⠙⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡟⠀⠀⠛⠛⠛⠉⠀⠀⠀⠀⠸⠿⠿⠿⠇⠀⣿⣿⣿⡿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠋⠀⣠⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⢀⣾⠿⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⣶⡀
⠀⠀⠀⠀⢸⡏⠐⠢⣌⠛⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⢉⡤⠂⣸⣇
⠀⠀⠀⠀⣾⡇⠀⠀⠭⢧⡀⠈⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⢀⡴⢋⡀⠀⢹⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠐⣒⣙⣦⡀⠀⠉⠻⣦⣄⣀⣤⣤⣤⣴⣶⣶⢶⣶⣦⣤⣤⣤⣄⣀⣠⡾⠛⠁⠀⣰⣋⣩⠥⠄⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠊⣉⣉⡛⣄⠀⠀⠀⠙⠉⠉⠈⢿⣿⠀⠀⢸⣿⡿⠀⣼⠋⠉⠙⠉⠀⠀⢀⣞⣻⣥⡤⠤⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⣥⣤⡍⠉⠋⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢸⣿⠁⢰⠏⠀⠀⠀⠀⠀⠀⠋⠈⢙⣤⣴⠆⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠘⠇⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⣀⣤⣴⣶⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠴⢦⣤⣄⡀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⣼⠇⠀⠀⠀⢠⡞⣩⣿⣿⣿⠀⠀⣹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣇⠀⢀⣿⣿⡙⢦⡀⠀⠀⠀⢻⣇
⠀⠀⠀⢰⣿⣀⣠⡤⠀⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⡇⠀⢤⣄⣈⣿
⠀⠀⠀⢸⣿⣿⠋⠀⠀⣧⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢿⣿⣿⣿⣿⣿⣿⣿⡏⢠⡇⠀⠀⠻⣿⣿⡆
⠀⠀⠀⢸⣿⣅⣀⠀⠀⠘⣦⡈⠻⣿⣿⣿⣿⡿⢋⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠙⢿⣿⣿⣿⡿⠏⢀⡞⠁⠀⢀⣀⣸⣿⡇
⠀⠀⠀⠘⣿⣿⡿⠿⠃⠀⠈⠙⠶⠤⣬⣭⡤⠶⠋⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠈⠓⠦⢤⣭⣤⠴⠞⠋⠀⠀⠘⠿⢿⣿⣿
⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⢀⣀⡤⠤⠿⣷⣞⠋⠉⠉⠛⢛⣛⠉⠙⠒⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠐⠒⢂⢉⠉⠉⠉⠉⠉⠉⣽⡿⠗⠤⠤⣀⣀
⠈⠁⠀⠀⠀⠀⣈⣿⣷⣶⡉⠁⠈⠉⠀⣀⠄⠀⠀⠀⠀⠙⠶⠶⠋⠈⠳⠶⠞⠀⠀⠀⠀⠀⠠⡈⠈⠀⠈⠉⣃⣶⣾⣏⢀⡀⠀⠀⠀⠀⠉
⠀⠀⠠⠔⠮⠓⠉⠀⠈⣙⡿⢷⣦⣤⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣤⣴⡾⣟⠉⠀⠀⠈⠙⠡⠂⠀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣼⠏⠙⠻⠿⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣴⡶⠿⠛⠛⢿⡄⠈⠑⢦⡀⠀⠀⠀⠀⢰⡟⠁⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⣼⣧⣄⣀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠙⠄⠀⠀⠀⠈⣿⡀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢹⣇⠀⠀⠀⠀⠀⢀⡿⠁⠈⠛⢻⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⣿⡀⠀⠀⠀⣠⡾⣿⣆⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠄⠀⠀⠀⠀⣿⠃⠀⠀⢸⣇⢀⣤⣾⠋⠀⠙⢿⣷⠟
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣷⠀⠀⠀⠀⠐⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠞⠀⠀⠀⠀⢠⡿⠀⠀⠀⢸⣿⠋⢻⣿⣧⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢿⡀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢸⡏⠀⣀⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠸⣇⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢀⣿⠀⠀⠀⢀⣾⠷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⢀⣿⡀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣸⣇⡀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣹⣇⣺⡀⢸⡀⠀⣹⣦⣤⣤⣤⣤⣤⣤⣤⣾⡀⠀⢰⠆⢸⣇⣿⣈⣿⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣠⠠⠠⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠆⠰⠆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⣀⠀⢀⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠈⠐⠉⠀⠈⠋⡀
⠀⠀⠐⣧⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠀⠀⠀⠁⠁⠒⠦⠴⢤⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠐⢿⣾⣿⣿⣿⣿⣿⣿⣾⣿⡶⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢰⣤⢬⣉⡅⡉⠽⠉⠿⠯⠶⠷⠳⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠘⠻⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠤⠽⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠄⠀⠀⢤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢄⠀⠁⡀⢠
⠀⠀⠀⠐⢪⣷⣻⠯⡑⠀⠳⠟⣓⡃⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠈⠹⠧⠃⡁⠤⠚⡤
⠀⠀⠀⠀⢸⡋⠚⠧⢵⠒⢲⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡜⣒⠙⠉⣀⢐⠸⡆⣀⡚⡀
⠀⠀⠀⠀⠀⣷⠋⠠⢈⣵⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⡟⠻⡶⢄⠻⣯⠑⣾⡟⢘⡟⣾
⠀⠀⠀⠀⠀⠘⣆⠀⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⢛⢯⠟⣾⣠⣿⣯⣽⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠁⠚⠾⠉⠒

⠀⢀⡀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠤⠄⡀⠀⠀⠀⢀⡀⠀⠀⠀⡀⣠⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢠⠄
⠀⠘⠀⠀⠀⠀⠀⠈⢄⠀⠰⠁⢀⠀⠉⠆⠀⠀⠌⠠⠊⠠⠀⠟⠊⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠈⠀⠡⣈⢀⡄⠐⠰⣲
⠀⠀⠐⠀⠐⠀⠀⢉⠣⠀⠀⠣⡀⠈⠀⠈⠆⠈⢠⠴⡀⠊⠀⠀⠀⠂⠀⠠⠀⠄⠀⠁⠀⠀⡠⠁⠀⠒⠄⠀⠉⠉⠀⢐
⠀⢀⠀⣋⡀⠀⠀⠀⠈⣀⢀⢀⠈⠒⠈⠄⢀⡴⠀⡯⢚⠁⠀⠀⢠⠐⠂⡐⠠⢀⢀⠀⢀⠜⠀⢣⢄⣀⡀⠀⠀⠀⠤
⠀⠘⠀⠀⡆⠀⡠⠊⡁⠀⠃⠂⢌⡑⢄⠀⠸⠋⠀⢸⡵⠀⠀⢸⠃⠀⢀⡅⡀⠘⠄⠐⠁⢀⠀⠠⣀⡀⠀⡀⠠
⠀⢠⠖⢦⠀⢄⣠⠒⡂⢜⣊⠼⠓⢊⠁⢠⠒⠒⠤⡀⠃⠀⠀⠀⠁⠨⠉⠁⠀⠀⠀⠀⠠⠂⢤⠀⠠⠀⠀⠁⠀⠈
⠀⠀⡆⣈⢂⠀⠀⠀⠀⠒⠒⠒⠈⠁⠀⡌⠀⠀⢰⠀⠰⢠⠀⠀⠂⠆⠀⢀⣴⣶⢻⣟⣳⡆⠀⡀⠀⢀⠀⢀⡰⠉⡘
⠀⠀⡄⠂⢠⠀⠀⠀⢖⠒⠀⠠⠀⢲⠆⠑⠀⠲⠈⠀⠀⠁⠃⠀⠀⠀⠀⢀⢀⠽⠉⠉⠉⠉⠀⠀⡀⠐⠀⠀⠉⠉
⠀⠀⠈⠉⠁⠔⢄⠀⡄⠉⠀⠒⠀⠁⢀⠀⣴⡢⡄⠀⠠⠀⠀⠀⢀⠖⡆⠀⠀⠀⠀⠍⠀⠁⠀⡀⠃⠀⠀⡔⠉⢆⠁⢠
⠀⠦⠀⠀⠈⠀⠀⠀⠀⢀⠀⢈⠈⠡⠏⠓⢉⠠⠣⢶⠠⢅⢤⡤⠀⠀⠀⡀⠈⣁⠀⠈⠈⠀⠂⠀⠀⠀⠨⠀⡠⠊⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠐⠀⠀⠂⠈⢀⣐⣃⡠⠤⠗⢉⠀⠀⠀⠀⠰⠀⠀⡀⠼⢑⠀⠉⡂⠲⢀⠑⢀⡱
⠀⠀⠐⠀⠘⢀⠀⠀⠀⠀⠀⠤⠀⠀⠘⠀⠈⠉⣤⠀⠀⢘⠭⠁⣀⠀⠐⣁⠀⠀⠀⠁⠀⠀⠀⠀⠀⢈⠀⠀⠀⠎
⠀⠐⠂⠀⠀⡠⠄⠀⠀⠊⢟⠀⠀⢠⠶⠛⠉⠐⢹⣶⠄⠈⡠⠀⠀⠀⠀⠈⠀⢠⠄⠀⠀⠀⠀⠀⠀⣡⠀⠀⠘⠀⠂⠄
⠀⠰⠠⠍⢀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⢈⠠⠒⠀⠀⠀⠀⠉⠉⠉⠈⠐⠡⠄⢀⣀⠀⠂
⠀⢠⠀⠀⡀⠄⠈⡆⡕⠀⠈⠀⠐⠀⠀⠠⠒⠃⠀⠀⠀⢰⠀⠀⠀⠌⠀⠀⠄⠠⠏⠀⡀⠀⢀⠠⠀⠀⠙⠇
⠀⠀⠀⡀⢀⡄⠀⢡⠁⢀⡀⠌⠀⡄⠂⠂⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡰⠒⠀⠀⠒⢅⡀⢀⠰
⠀⠠⠐⠊⠸⠈⠆⠈⢀⠡⠀⠀⠦⠠⢖⠀⠀⠀⠈⠁⠀⠈⠀⠐⠀⡄⠀⢠⠂⠉⠀⠀⠀⠈⠀⢀⠜⡀⠀⠀⢻⠧⠂⠁
⠀⠀⠠⠨⠄⠀⠂⠀⠀⠒⠈⠀⠀⠀⠀⠄⠀⠀⠀⠠⠒⠃⠀⠀⣀⠀⢌⢀⠢⠤⠬⠔⡀⠤⠔⢊⠈⠃⡀⠈
⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠈⠠⠂⠀⣀⡀⡔⠒⠒⡆⠀⢢⠀⠐⠀⠀⠁⠀⠈⠀⠠⡈⡀⢄⠀⠠⠀⠈⠀⠀⠀⠀⠐⠁
⠀⢀⡀⠐⠀⠀⠀⠀⠂⠄⠁⠀⠀⠘⠄⠂⢘⣀⣸⠀⠀⡀⠀⠐⠄⢤⠆⠀⠀⠈⠚⠁⠡⠊⠈⠠⠁⠀⠤⠀⠐⠀⠒⢤
⠀⠀⣠⡶⢀⡶⠀⢀⠀⠃⡀⠂⠈⠠⠀⠀⡀⠄⠠⣀⠀⠀⠘⠀⢀⠀⠀⠀⠃⠀⠠⣀⣠⣤⠄⠀⠀⠀⢄⠀⠀⠀⠀⠔
⠀⢸⣆⠅⠊⠀⡀⠁⠀⡂⠀⠀⡀⢊⠀⠌⡠⠊⢉⡉⠠⠀⠈⠠⠊⠀⠀⠀⠁⠀⢀⠜⠈⠡⡀⠀⠀⠀⡈⠀⠈⠁⠁⢠
⠀⠀⠁⠂⠀⠀⠃⠀⠀⠈⠀⠉⠀⠈⠀⠋⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠘⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠓⡄⠀⠀⢠⠚⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⢀⠆⠀⠀⢣⡀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⠤⠤⣀⠀⠀⠉⠉⠉⢆⠀⠀⡰⠉⠉⠉⠀⠀⣀⠤⠤⠤⠤⠤⠄⣀
⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⣀⣀⣀⡀⠀⠉⠢⡀⠀⠀⠘⡄⢠⠃⠀⠀⢀⠔⠉⠀⣀⣀⣀⡀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⡰⠁⠀⠀⡠⠒⠉⠀⠀⠀⠀⠉⠒⢄⠈⢢⠀⠀⢡⡂⠀⠀⡔⠁⡠⠒⠉⠀⠀⠀⠀⠉⠒⢄⠀⠀⠈⢆
⠀⠀⢰⠁⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠗⠊⠉⠉⠑⢾⣀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠈⡆
⠀⠀⠄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢀⡀⢀⡀⢀⡀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡆
⠀⠀⠈⢆⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠄⣀⣀⣀⣀⠠⠤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⡰⠁
⠀⠀⠀⠈⠢⡀⠀⠙⠢⢄⣀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⡠⠔⠋⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠈⠑⡲⠂⢀⠜⠁⠀⠀⠀⠀⠀⠸⠉⠐⠒⠒⠒⠒⠂⢉⠃⠀⠀⠀⠀⠀⠈⠳⡀⠐⢖⠊⠁
⠀⠀⠀⠀⠀⠀⡔⠁⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣀⣀⣀⣀⠠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠈⢣
⠀⠀⠀⠀⠀⢰⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⠀⠀⠀⠀⠀⠀⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⡆
⠀⠀⠀⠀⠀⠘⡄⠀⠈⢆⠀⠀⠀⠀⠀⣀⠔⠁⡰⢗⠒⠒⠒⠒⡺⢆⠈⠢⣀⠀⠀⠀⠀⠀⡰⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠑⠒⠒⠒⠉⠀⡠⠎⠀⠘⣄⣀⣀⣠⠃⠀⠳⢄⠀⠉⠒⠒⠒⠊⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⣀⣀⣀⠤⠒⠉⠀⠀⠀⠀⠘⢄⡠⠃⠀⠀⠀⠀⠉⠒⠤⢀⣀⣀⠤⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠈⠀⠀⠦⠀⠀⢘⣺⣄⡠⢀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠄⣠⡁⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠀⠋⠀⠀⠐⠀⠀⠀⠀⠀⢀⠘⠛⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇⢀⠀⠀⡆⠀⠠⠘⢥⠽⠀⠀⠀⡀
⠀⡀⠀⠐⠀⢠⠀⠒⠈⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠠⠄⠀⠀⠀⠀⢀⡠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠰⣷⢄⠀⠀⠂⠀⠁⠀⢠⡇
⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣊⢄⠀⠀⠀⠐⢜⢃⡀⢀⣀⣐⣤⡤⢊
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠑⠈⠉⣓⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠈⠀⠀⠀⠈⠆⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠄⠀⠀⢀⡀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡀⢊⠀⠀⠀⠀⡊
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢈⡠⠀⠀⠀⠉⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠁⠀⠀⠀⠀⠀⠐⠊⠀⠀⠐⡀⠀⠀⡰⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠠⡀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡉⢉⡒⡄⠀⠀⠀⠐⠠⠀⠀⠀⠀⣄⡠⠉
⠀⠀⠀⠀⠀⠤⢤⢤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠠⡄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠺⠀⠀⠈⢶⠀⠀⠈⢱⠂⠐⠒⠒⠋⠁⠄⠀⡇
⠀⠀⠀⠀⠀⢠⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠢⣄⡀⠐⢆⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠈⠃⠀⠘⠀⠀⡠⡧⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠈⠉⠘⣆⡀⠀⢀⠀⠲⡄⠀⠐⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⠄⠼⠀⠀⠀⠀⠀⠀⠀⢌⠢⣄⠐⠀⢀⡴⠤⢛⢽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠛⠐⠒⠂⠈⡼⠀⠀⠀⠀⡀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⣀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⣠⠀⠑⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣠⠞⠁
⠀⠃⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠛⠫⠄⠀⠀⠁⠀⠐⠤⢀⣀⣀⡠⠴⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠘⠀⠀⠘⠋⠀⠠⠀⠐⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⣰⠀⠀⠀⠀⣠⠄
⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⣿⣯⣿⠇⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣜⠿⠿⡿⠀⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠋⠁⠠⣤⠀⠸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⢹⣦⣜⣛⣛⣿⣿⣿⣿⣿⣿⡿⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣴⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠤⢀⣀⣻⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠙⢋⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⡄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣿⣿⣿⣿⣿⣦⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠺⠿⠃⠀⠄⠀⠀⠀⢠⣿⣄⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣇⢾⣿⡟⠁⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⣾⣿⣿⣿⡄⠀⠀⡀⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡀⢀⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠐⠾⣿⣿⣿⣿⣿⡏⠉⠉⢻⣿⠀⠀⠃⢀⣿⡿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠿⣿⣿⣶⣤⣦⣿⣆⠀⠀⢸⣿⣧⣄⣠⣴⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠻⠟⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠛⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠦⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠠⣶⡄⠀⠀⣧⠶⠛⠛⠉⠉⠉⠙⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⢀⣀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⣀⣀⣀⣀⣀⣠⣴⡾⠛⡅⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠘⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⢿⢻⠛⡟⢻⢻⠛⠙⢹⡇⢀⣷⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⡶⡟⣿⢷⣤⡾⣿⠀⡇⢸⠘⡀⡇⢸⣈⣀⣠⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⣴⢿⠉⡏⢻⡇⣧⢸⢸⡈⡇⢸⣀⣷⡾⠟⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⠶⠾⠿⠟⠛⠁⠀⠀⠀⠘⠃⠀⠀⢠⣴⣿⣾⡄⢇⢸⣧⣼⣼⣤⣇⣧⡾⠛⠁
⠀⠀⢠⡶⠋⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⢸⣿⣧⣼⣼⣿⠁⠈⠙⠛⠋
⠀⢰⡟⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠙⢿⡜⢿⡏⠀⠈⠉
⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⣠⢾⣷⠟⠋
⠀⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⣼⣷⢸⣿
⠀⢸⣷⢹⠛⡿⢿⡿⡷⢶⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣼⡿
⠀⠀⠻⣿⣄⣧⠘⡇⡇⢸⢼⣧⠀⠀⠀⠻⠇⠀⠀⠀⠀⢀⣾⣿⠋
⠀⠀⠀⠈⠙⠛⠿⠿⠿⠾⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠻⡿⣶⣤⣤⣀⣤⣴⡿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⡇⠀⡏⢹⠉⢻⠈⡇⢣⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣷⣬⣦⣼⣤⡷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢄⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀
⠀⠀⠀⠀⠀⠀⢠⡄⠀⠰⣀⡸⠀⠀⠀⠀⠀⠀⠀⢎⡀⡱⠀⢀⣄
⠀⠀⠀⠀⠀⠀⡃⠈⡄⠀⠖⠲⠀⠀⢀⠴⡄⠀⠀⢠⠀⠄⠀⡌⠀⡂
⠀⠀⠀⠀⠀⠀⡭⢭⠀⠀⠀⢸⠀⠀⠨⣀⠸⠀⠀⠐⠀⠀⠀⢨⢭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⢰⠀⢦⠀⠀⠸⠀⠀⠀⠐⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠄⠒⠊⠉⠑⢄
⠀⠀⠀⠀⠀⢀⡄⠠⠒⠒⠀⢸⠉⠉⢹⠀⢘⠉⠉⠙⠀⠒⠂⢆⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠰⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⢸⠀⠸⠀⠀⠈⠉⠁⠀⢐⠀⡇⠈⠢⡀⠀⠀⢀⣤⣤⣒⡀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠷⢀⠀⠐⠚⠀⠀⠀⠀⠀⠀⢸⠀⣨⠀⠀⠀⠀⠀⠀⠘⠒⠃⢀⠴⡁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠒⠂⠀⠐⠒⠒⠂⡞⠅
⠀⠀⠀⡂⠀⠁⠒⠀⠤⠤⠀⣀⣀⣀⣈⣉⣁⣀⣀⣀⡀⠠⠤⠄⠐⠊⠁⠀⡇⠀⠀⠰⠆⠀⠀⠀⠤⠤⠄⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⡆
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠐⠒⡏⠄
⠀⠀⠀⣉⠑⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⢈⠀⠀⠀⢐⠒⠂⠀⠠⠤⠤⠤⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡆
⠀⠀⠀⠈⠉⠒⠠⠤⢀⣈⠉⠈⠉⠀⠀⠀⠀⠀⠀⠉⠈⢉⣀⠠⠤⠒⠈⠨⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠈⠉⠁⡏
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠠⠤⠄⠄⠀⢀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠆
⠀⠀⠀⠈⠒⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠐⠒⠒⠂⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠂⠒⠒⠒⠒⠒⠒⠂⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⣄⠤⢀⡤⠠⢄⡠⠤⣄⣀⡀
⠀⠀⠀⠀⠀⢀⠔⠉⡠⠔⢒⡠⠶⠺⣦⠴⠦⣴⠳⠶⢄⠀⠩⡢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢠⠊⡠⠊⠀⠠⠊⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠑⣄⠈⠢⡱⡀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠤⡀
⠀⠀⠀⢠⠃⡔⠁⠀⢀⠃⠀⠀⠀⢰⠀⠀⠀⠀⢸⠀⠀⠀⠈⠆⠀⠑⡘⠄⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄
⠀⠀⢀⠆⡜⠀⠀⠀⡘⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⠀⠀⠐⡈⠆⠀⠀⠰⡁⠑⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⢁⠇
⠀⠀⠎⡜⠀⠀⠀⢠⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠃⠀⠀⠐⡈⠄⠀⠀⠋⢲⡐⠢⠤⠬⣉⣉⠈⠉⠀⠁⣉⣀⣀⡉⣁⠠⠀⢐⡴⠙⠆
⠀⠈⡶⠀⠀⠀⠀⠆⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠘⡀⠀⠀⠰⡆⠀⠀⠁⠦⠌⣁⠐⠂⠀⠄⠀⠠⠨⠤⠀⠀⠠⠀⠀⠒⣈⣁⣰⠋
⠀⢰⠁⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⢁⠀⠀⠀⠈⠒⠤⡉⢈⡁⠀⠐⠒⠀⠀⠀⠈⠉⢉⣀⣁⠤⠒⠁
⠀⠈⠒⠀⠄⠀⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠰⠂⠀⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠂⠉⠂⠤⢀⣀⠠⠔⠊⠁⠒⠒⠒⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢹⢀⠼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⢸⠈⣆⢙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡳⣆⠀⠀⠀⡰⡃⡇⠀⠀⠀⠀⣼⢰⡿⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠞⣹⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡉⢣⡀⢈⢱⢁⠀⠀⠀⠀⠀⠒⢸⡇⢘⡃⠀⠀⠀⠀⠀⠀⢀⠞⣴⠇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣃⢿⡄⠳⡏⣿⢨⠀⠀⠀⠀⠀⡇⣸⡇⢨⠀⠀⠀⠀⠀⠀⢀⡞⢰⡟⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡘⣿⡄⢖⣿⢸⠀⠀⠀⠀⠀⠆⣿⡇⠾⠀⠀⠀⠀⠀⠀⡌⢡⣿⢃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣸⣿⣸⡟⡶⢶⠶⣛⣶⠾⣿⣿⣷⠛⠶⠦⣤⣀⠀⡰⢣⣼⢇⣮⠠⠔⠖⠒⠺⢟⣻⠖⠛⠃
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣿⠿⠋⠐⣃⠨⣭⠉⠃⢹⠍⢯⣉⣠⠀⠀⠈⠙⢷⣽⣿⣾⡶⠶⢿⣛⢫⠔⠋⠁
⠀⠀⠀⠀⠀⣵⢦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠈⠻⣦⡾⠶⠋⠀⠈⠉⠁⠀⠀⠀⠀⢀⡙⢿⣯⣴⢶⣎⠉
⠀⠀⠀⠀⠀⠘⣇⡝⢦⠀⠀⠀⢠⣾⠇⠙⠀⠀⢀⣤⣶⢤⣄⠀⠀⠀⠀⠀⠀⣀⡴⢾⣿⣟⣳⡄⠀⠀⠉⢸⠋⢨⡑⢍⠳⣄
⠀⠀⠀⠀⠀⠀⢡⡹⣦⠁⡄⠀⣾⠡⣤⡇⠀⢠⡿⠛⠛⠳⣤⡇⠀⠀⠀⠀⠀⣧⡞⠋⠁⠈⠙⣿⡀⠀⠀⢹⡄⣄⠀⣂⢧⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠱⣛⣿⡹⡾⢏⣠⡟⠀⠀⠛⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠈⠀⢀⣀⣀⠀⠈⠀⠀⠀⠈⣿⢹⠰⡇⢸⣆⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠄⠀⣀⠀⠀⠀⠙⣿⣿⠃⠈⠛⠁⠀⠀⢀⠖⠋⠛⢦⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠳⣄⠀⠀⠀⠀⠙⣜⣃⡴⠋⢻⡆⢻⡄⠀⠀⢀⣠⠤⠖⠛⣋⠙
⠀⠀⠀⠰⣄⣤⣀⢠⣴⠏⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠈⠉⠐⠋⠀⠹⣆⢻⣦⠚⣫⣦⠞⢀⡴⠃
⠀⠀⠀⠀⠈⠺⢝⠻⣿⠀⠀⠰⢓⡆⠀⢀⠏⠀⣀⣵⣥⠀⢱⠀⠀⠀⠀⠀⠸⡇⠀⠀⢈⣬⣧⠀⢻⠀⠀⠀⠀⠀⣠⢶⡖⠄⠀⢻⣯⣴⡿⠟⣁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣺⠰⣧⠀⠸⠀⠰⣏⣠⣿⠆⡌⠀⠀⠀⠀⠀⠸⠄⠀⢀⣿⣩⣿⠃⠘⡆⠀⠀⠀⠐⣏⢈⡀⣴⠀⢸⣿⣭⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢿⢠⡟⠀⠀⠀⠀⣿⢻⡇⠀⠇⠀⠀⠀⠀⠀⠀⡄⠀⠈⠸⡟⣿⡀⠀⡇⠀⠀⠀⠀⠀⢯⠞⠁⠀⢸⡏⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠁⠀⠀⢠⠀⣦⣿⣿⢃⡻⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⣄⣿⣿⠃⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⡝⠳⠀⠀⠈⣄⠘⠿⢟⡾⠁⠀⠀⠴⠒⠤⠀⠀⠈⢦⠀⠹⢿⡿⢁⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⠾⠆⠀⠀⠀⠓⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣠⠴⠋⠀⠀⠀⣤⠤⣄⠀⠀⠀⠀⠀⣸⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢸⠋⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢦⣄⠀⣸⠛⠀⢸⠂⠀⠀⠸⢀⡿⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⡀⠀⠀⠈⠀⠸⣿⡟⠓⠤⣀⡀⠀⠀⠀⠀⢀⣀⡤⢴⣶⣶⠀⠀⣿⠀⣷⢠⢃⣾⠃⠀⠀⢀⡾⢁⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠁⢸⠐⠀⠀⠀⠹⣿⣦⣤⣀⣉⣙⠛⠛⠉⢁⣀⣀⣼⣿⠟⠀⠀⠁⠀⠈⠛⠋⠁⠀⠀⠀⣾⣛⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡘⢷⣦⠀⠀⠀⠹⣿⣿⡿⠿⠻⣿⠿⠿⠛⠛⢛⠛⠋⠀⠀⠀⠀⢠⣾⣿⡆⠀⠀⠀⣼⣿⠹⠿⠁⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⠀⠀⠀⠀⠙⢧⡀⠀⠈⠀⠀⣶⣿⣿⣿⠇⠀⠀⠀⠀⠀⠈⠻⠛⠁⠀⠀⠊⣀⣤⢴⣶⣦⡀⢀⣴⣿⡙⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⡿⣷⠄⠀⠀⠀⠀⠙⠲⢤⣀⡀⢹⣿⣿⣿⠀⢀⣤⣤⣄⠀⣶⣾⣿⣿⠆⣠⣾⣿⡃⢸⣿⣿⣿⠸⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠱⠏⠜⠛⢦⡄⠀⣀⣤⣤⣤⣤⡀⠀⢸⣿⣿⣿⣠⣿⡿⠟⠃⠀⢻⣿⣿⡟⠀⣿⣿⣿⠷⣿⡿⠟⠁⢀⣭⣿⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⢶⣶⣤⡀⠀⢠⣾⣿⠇⠈⣿⣿⣿⡄⢸⣿⣿⣿⢻⣿⣿⣆⠀⠀⢸⣿⡿⠿⠀⣛⠻⣿⣆⠀⠀⣴⣶⡆⠻⠿⠷⠶⠛⠁
⠀⢀⣴⡶⠻⣿⣿⣦⠀⣴⣿⣿⠁⠀⣿⣿⣿⡄⣿⣿⣿⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⣸⣿⣿⣿⣷⡄⣿⣿⡅⣾⣿⣿⠇⠙⠛⠛⠃⢙⣋⣁⠀⣠⡶⣶⣦
⣰⣿⣿⠇⢀⣿⣿⡿⢰⣿⣿⣿⠀⠀⣿⣿⣿⠃⣿⣿⣿⡀⢀⣿⣿⠟⠀⣿⣿⡿⠿⠃⠙⠛⠛⢉⡀⠀⠀⠀⢸⣿⣿⡴⣶⣶⣦⠸⣿⣿⡿⢸⣿⣷⣾⣯⡀
⣿⣿⣿⡀⠀⠛⠋⡁⠈⢿⣿⣿⡀⢠⣿⡿⠋⠀⠈⠛⠿⠿⠟⠛⠁⠀⣀⢀⣠⣤⡀⠀⠀⣴⣿⠋⢿⣿⣆⠀⢸⣿⣿⠀⢸⣿⣿⠀⣿⣿⣇⢀⣽⡿⢿⣿⡇
⠹⣿⣿⣧⣄⣠⡼⠃⠀⠀⠉⠛⠛⠋⠁⠀⣀⣠⣤⣠⣶⣶⣤⠀⢾⣿⣿⠏⠙⣿⣿⡆⢀⣽⣷⠞⢿⣿⣿⠀⣸⣿⣿⣀⣼⣿⠟⠰⣿⣿⠿⠈⠛⠛⠚⠉
⠀⠀⠉⠉⣉⣥⣄⣀⠀⣠⣶⡟⢻⣿⣦⠀⢻⣿⣿⠁⠘⣿⣿⡆⢸⣿⣿⡇⢀⣿⣿⣇⢸⣿⣿⣠⣾⣿⠿⠃⠻⠛⠋⠙⠋⠁
⠀⠀⣠⣾⡏⢨⣿⣿⠂⣹⣿⡶⢾⣿⣿⡀⢸⣿⣿⡄⢸⣿⣿⣧⢼⣿⣿⠇⠸⠟⠛⠋⠀⠉⠉⠁
⠀⠀⣿⣿⡇⠘⠛⢉⢰⣿⣿⣄⣸⣿⣿⠇⠺⠿⠟⠃⠈⠋⠉⠁
⣀⣀⣙⣻⣷⣄⣐⣋⣀⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠿⢻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣬⣤⣤⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡍⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠑⠛⡇⠀⠀⠀⢀⣀⠠⠀⠀⠀⠀⠀⠐⠶⣦⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠀⠈⣿⡤⠔⠀⣈⠁⠀⠀⠀⠀⠀⠆⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠠⢶⣾⣯⠙⠿⠿⠒⠢⢤⣤⣽⣿⣿⣾⣛⣿⡦⢀⣼⣿⡆⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣶⣂⠈⢀⣩⣿⣿⣿⣿⣿⡍⠙⢿⣿⣿⣿⣇⡔⠀⢀⠠⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⡟⢻⣿⣿⣿⠏⠀⢀⣾⣿⣿⣿⣿⣿⡟⢛⠤⣀⣄⣈⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣅⣈⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⠻⠿⠟⠀⠀⠘⢿⣿⣿⣿⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠪⠻⣧⠠⠀⠀⠀⠠⢆⣈⣿⣿⣿⣷⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡹⠄⠀⡜⢢⠀⠀⠀⠙⠻⢩⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⠄⠀⠘⠁⠀⠀⠀⣀⣴⣶⣾⣿⣿⣿⡃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠠⠾⣿⣿⣿⣿⠀⠁⠘⣷⣶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⣀⣠⣿⣿⡿⠇⠀⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⣿⣿⣿⠿⠿⣿⣿⣿⣿⠁⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠙⡄⠀⠿⣷⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⣿⡏⡂⠀⠀⠈⠻⢇⡀⠘⢀⡿⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣟⡏⠢⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠑⢤⣼⠇⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⡳⣄⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢊⣿⣿⣿⠷⢶⣦⣀⠀⠀⠀⠀⠀⠹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢱⡇⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣇⣾⠟⠋⠁⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⢢⡙⢷⣄⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠙⢷⡄⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠻⢾⡿⠿⢿⣿⣟⢿⣿⣿⣿⡟⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠳⢍⠙⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⡿⢿⣿⣿⡟⠿⣿⡿⢿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⠞⠁⠀⠀⠙⠻⣿⣷⣦⣍⣙⡒⠾⢭⣤⣈⡙⠲⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⣝⡢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⠋⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣬⣍⣙⠓⠲⢮⣭⣳⢮⣷⣵⡝⣿⣿⣿⣿⣿⣿⣿⣿⣦⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠚⠭⢷⣒⣶⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣟⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠛⠓⠂⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣶⠈⠀⠹⣿⣿⣿⣿⣿⢧⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠰⠿⠋⢁⠀⣀⣀⣀⡀⢀⡠⠤⠒⠉⠁⠙⣿⣿⣿⣿⣧⡐⢀⣿⣿⣿⣿⣿⣷⡁⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢰⡶⠿⠗⠀⣠⠊⠉⠀⠉⠻⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠸⡟⠘⣿⡞⣿⣾⣿⣿⣿⣿⣿⣿⣷⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣷⠬⢄⣀⢠⣀⡇⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣀⡼⡟⢺⢋⠳⡀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⠀⠀⢃⠀⡇⠀⠙⠢⢄⣀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣆⠀⠈⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⡇⡿⣸⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣷⢠⠉⠉⠑⡄⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⡇⠁⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣏⠄⠀⠀⠈⢆⢈⡤⠴⠒⠊⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⡟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠘⡟⣦⠀⠀⠀⠀⠡⡄⠐⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡏⢿⣿⣿⣿⣿⡟⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢱⠘⣷⡀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⣠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠇⢸⠻⣿⣿⡿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡇⠈⢿⣄⠀⠀⠀⠀⠈⠂⠀⠒⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠈⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠸⡇⠹⣆⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⢠⣞⣣⠶⣤⠀⠀⠀⠀⠀⠈
⠀⢀⣤⣤⡄⠐⡄⣤⢰⠀⠀⠁⠶⠀⠂⣈⠀⠀⠀⠀⣧⠀⣷⠀⠙⡆⠀⠐⠉⢃⠀⡀⠈⠀⠀⠀⣴⣻⢞⡽⣻⠎⠀⠀⠀⠀⠀⠀⣤⠚⣶
⠀⠀⠀⠀⠀⠀⠓⠃⠘⠀⠀⠈⠛⠋⠛⠉⠀⠀⠀⠀⢹⣁⠀⢂⡼⠃⠀⠀⠒⠊⠉⠀⡂⠀⠀⠈⠐⠫⣼⠟⠁⠀⠀⠀⠀⠀⠀⠈⠿⠵⠋
⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣤⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⠀⠀⠀⢀⣤⣤⣠⣦⣤⠀⠀⠀⣀⠐⠄⣈⠀⠀⠀⠀⠀⢠⣄⣠⣄⣤
⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⢀⠄⢠⡒⠦⠀⠀⠀⢠⣤⠙⠃⠀⡼⠂⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⣴⠆⠀⠑⢿⡄⠀⠀⠘⠛⢿⣿⠋⠁
⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⣤⠐⠂⠛⠁⠠⠀⠀⠀⢀⠈⠀⠔⠊⠀⠀⠀⠀⠁⠀⠀⡀⠀⣦⠀⠠⠟⠃⠐⠛⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠈⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠘⠛⠛⠛⠛⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠐⣶⠀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡠⠚⠀⠁⠒⠀⠀⠀⢤⡀⠈⠹⠇⢤⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⣀⡀⠀⠈⠿⠀⠀⠀⠀⠀⠠⠁⠀⠀⡙⣷
⠀⠀⠀⠀⠠⣶⣤⣤⠀⠀⢾⡀⠁⠀⠀⠀⠰⠇⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣤⠀⠀⠀⠠⠞⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠁
⠀⠀⠀⠀⠀⠿⣮⠗⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠘⠘⠃
⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⢠⠦⠀⠀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠀⠉⠛⡂⠀⠀⠀⠀⢰⠀⢃⣠⠖⠀⡀⠀⠀⠄⠀⠰⠀⠀⠀⣴⣶⣶⣶⡶⢤⣀
⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⡣⠀⠠⢤⡄⣤⠖⠉⠀⠀⠀⣅⢠⠄⠀⣀⡀⠀⠀⠀⠳⢄⠀⠀⡴⠖⠋
⠀⠀⠂⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⠹⠁⠀⠀⠀⠀⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠡⡄⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢺⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠠⢾⡿⠁⠀⢀⠋⠁⡀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣰⠀⡆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠐⣄⠁⠠⣀⡀⠀⠀⠀⠀⠀⠀⠐⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠰⠀⠀⠐⣿⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠈⠁⠀⠀⠀⠀⢀⠈⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠻⣴⣷⢄⡇
⠀⠀⠀⠀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠄⠀⠉⠑⠀⠄⠀⠀⠀⣀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⠀⠀⠀⢀⠀⠰⡇⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⣀
⠀⢆⣀⣀⠀⠀⣀⣠⠀⠀⠸⣿⣍⣶⣍⣻⡀⠀⠀⠀⠰⠒⠆⠀⠀⠀⢀⠉⠉⠀⠀⠉⠉⡀⠀⢀⡜⠁⣴⣴⣀⡶⢤⡀⢠⣾⣛⢛⣿
⠀⠀⠙⠛⠛⠛⠛⠃⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⢶⣤⣄⣤⠶⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠠⠀⠃⣜⢁⣸⡧⡴⠃⠀⠁⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣶
⠀⠀⠀⠘⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⢤⣤⠀⠀⠀⠀⠘⢿⡿⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⡆⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡂⡄⠀⠀⠀⠀⠀⠂⢠⠁⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠄⣀⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠙⠀⠀⠠⠀⠀⠀⠘⠳⠶⠶⠟⠉
⠀⠀⠀⠀⠄⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⢀⠒⠀⢀⡇⠀⠀⠀⠀⠀⠤⢤⠐⠂⠀⠀⠀⣶⣦⠙⢦⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠾⠷⠂⠐⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠉⢠⡜⠛⣅⠀⠀⠉⠉⠉⠀⠀⠃⠀⠀⠀⣷⡾⡿⠻⣆⠀⠀⠀⠀⠀⣾⢀⡀⠟⠀⠀⠀⣤⣤⠀⠀⢀⠀⠀⠀⢀⠸⠋⠙⠉⠙⠁
⠀⠀⠀⠉⠀⠉⠂⠉⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠳⠊⠐⠓⠀⠛⠁⠀⠀⠠⣅⠈⢨⠅⠀⠀⠀⠀⠉⠠⣦⠆⠻⠃⠀⠀⠘⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠓⠂
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢃⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⠀⠀⠀⢀⠤⠀⠀⣤⠀⠀⠀⠀⠀⣤⣤⡀⠀⡀⠀⠀⠀⠿⡇⠀⢀⣠⠤⢄
⠀⠀⠀⣠⣶⣶⠀⠀⠀⠀⠠⠀⠀⠘⠇⠀⠀⠀⣴⠾⠟⠉⠁⠀⠐⠀⠀⠁⠀⣘⣦⡄⠀⠀⠀⠀⠀⠀⢠⡀⣠⠡⠀⠀⠀⠁⠐⠀⠀⠿⠁⠟
⠀⠀⠀⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠠⠟⠁⠀⠀⢈⣿⡷⠒⠾⠟⠂⠀⠀⠀⠀⠀⡞⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⣶⣶⣶⣶⣶⣦⣶⣴⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠻⠛⠛⣿⠋⢻⠛⢛⡛⠟⣛⣿⢛⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣿⣿⣿⣾⣿⣶⣾⣿⣷⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡔⠒⠛⡹⢉⠽⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠋⣉⠤⠔⠊⠉⢣⠚⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⢡⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⢦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⣫⣽⠿⡆⢹⡟⣻⣶⣭⣓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣺⣵⠾⢻⣿⣿⣿⣿⣉⣻⣿⣿⣿⣯⡻⣮⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢫⣾⣿⣵⣶⣾⣿⠿⢛⠂⢹⡙⠯⣙⠻⢿⣿⣾⣿⣌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣱⣿⣿⡿⠟⠋⠉⠀⠀⠁⠀⠀⠂⠀⠈⠣⡀⠉⠻⢿⣿⡎⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣿⢟⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢢⡙⢿⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢹⡄⠁⠋⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠈⣿⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⣌⡙⠒⠤⠴⣁⣀⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡇⢀⣀⣸⠧⠟⠃⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠐⠬⢭⣒⠲⠦⠤⢤⣬⣍⣉⣉⣉⣉⣉⣉⣉⣉⣭⣭⣥⠤⠤⠶⠒⠋⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⢍⣀⠀⠡⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢊⡩⢄⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⣿⣿⣷⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⡾⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⣿⣿⡿⠁⣶⢻⣿⣿⣿⣿⢛⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⣹⡇⠀⣿⣿⠟⣻⣿⠇⢾⣿⠃⣾⣿⢿⣿⣿⣿⣟⡟⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣿⣧⠀⢿⡿⠀⣿⣟⠀⣿⡇⠰⡟⢀⣿⠋⣾⣿⣿⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣿⣿⣷⣾⣿⣶⣾⣿⠐⣿⡇⢿⣿⣶⣿⣶⣿⣿⡏⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⣿⡏⢹⣻⡟⢿⣿⡷⢼⣿⡶⠿⡿⢿⢫⣿⣿⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠸⣿⣇⢸⣿⣧⠸⣷⡇⢘⣿⡇⢠⣿⡮⢸⣿⠇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣶⣾⣿⣦⣿⣧⣼⣿⣧⣷⣾⣶⣿⡿⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢹⣿⣿⡄⢹⢿⣿⠍⢸⣇⠰⣿⢰⣿⣿⠇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡘⣿⣿⡇⢸⡏⣿⠀⢼⣿⡀⠋⣾⣏⡿⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢻⣿⣷⠘⢧⣿⢰⣾⣿⣷⢤⣿⣹⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠘⣯⣿⣴⣿⣧⣸⣇⣿⣬⣾⢧⡟⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣍⣛⡛⠿⠿⣿⣿⡿⠿⠟⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠒⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠀⠁⠈⠀⠀⠃⠀⠀⠀⡌⠁⠉⠀⠈⠁⠈⡆⠀⠀⢠⠁⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⡠⠈⠀⠁⢄
⠀⠀⠀⠀⢸⠀⠉⢀⡀⠉⠀⡄⠀⠀⠀⠇⠀⠉⡀⡈⠁⠀⡆⠀⠀⠸⠀⠈⢁⢀⠉⠀⢸⠀⠀⠀⢠⠡⠄⠀⠠⠌⠄
⠀⠀⢀⠀⠀⠀⠀⡠⢄⠀⠀⠃⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⢀⠂⠐⠂⠀⠸⠀⠀⠒⠀⠀⠄
⠀⠀⠀⠀⠀⡄⢠⠁⠀⠆⠀⠀⠀⠀⠀⢸⠀⢸⠀⠸⠀⢠⠀⢀⡀⠀⠀⢃⠀⠀⢀⠃⠀⠀⠀⠀⠀⠱⠀⠀⠀⡌⠀⠆
⠀⠀⠀⠀⠀⠡⠜⠀⠀⠠⠜⠀⠀⠀⠀⠀⠦⠆⠢⠇⠃⠌⠀⠀⠀⠀⠀⠀⠢⠠⠊⠀⠀⠀⠀⠀⠀⠀⠣⠀⠜


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣽⣭⣴⣭⣿⣵⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⣀⠀⣿⣿⣇⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⣴⠂⠀⠠⠈⠀⣼⣿⡟⢋⣁⡈⠻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠙⣏⠀⠀⠆⠀⠀⢈⣭⡖⠚⠿⠃⢨⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣄⠀⠀⢸⣿⠃⢸⡿⠀⠀⠃⠀⠀⠙⢿⡿⢣⣶⣾⡝⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣋⠉⠙⣆⠀⠈⢿⠂⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⡂⠜⠀⠀⠀⠀⠘⠄⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢺⣿⣿⣿⣠⣶⣦⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠛⢄⣁⠈⠹⢿⣿⠇⠀⠀⠀⠀⢸⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⡧⠀⠾⣿⡗⠀⣻⣦⣄⠀⠀⠀⠀⡍⠀⠀⠀⠋⠙⢋⣈⣹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣆⡔⢀⣀⣤⣿⣿⡟⠀⠀⠀⢠⠉⠀⠀⠚⠀⠲⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⢿⣿⣿⡷⠀⠉⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⡜⠀⢠⠆⠀⠀⣼⣷⣿⣷⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡠⠃⣴⣶⣤⣹⠿⠿⢻⡥⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⠄⠀⢾⣿⣿⡿⠈⢣⣄⣌⠀⠙⣻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠐⠀⠀⢉⣩⡖⠒⠺⠿⣷⠀⢉⣽⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠰⣿⣿⣿⣆⣴⢀⣀⣰⢼⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡀⠀⠀⠀⡀⠂⠘⠿⠟⠉⢼⣿⣿⣿⡇⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠔⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⠈⠛⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⢿⣶⣄⠀⡘⡏⠀⠀⠁⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣄⣀⣈⡀⠙⠛⢠⡿⠁⠀⠀⢀⠖⠉⠛⠛⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠛⠿⠷⠒⠐⢮⠃⡄⠀⠀⠉⠀⠐⢶⣶⡄⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡥⠀⠀⠀⠀⠀⠀⠈⠻⣆⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢆⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢳⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠳⣄⠀⠙⠳⠤⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⠿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⣀⣤⣖⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣷⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣿⣌⣿⣯⣟⣹⣬⣿⣷⣿⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣿⣿⣿⣿⠿⠛⠻⠻⠿⠿⠿⠿⣿⣿⣿⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡧⢶⣶⣦⣀⣀⣠⣶⣶⣖⣀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣈⣻⣿⣿⡏⣒⡀⣿⣿⣿⣿⣿⣿⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡹⠟⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⠦⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⡇⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⣅⠀⡌⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⣍⣁⠲⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣆⠀⠀⠀⠈⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢰⣿⣿⣦⣀⠀⣰⣟⣄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠉⠉⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⣥⠤⠞⠓⠒⠒⠚⠓⠢⢄⣙⢢⠀⠀⠀⠀⡠⠐⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣫⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢃⢀⡴⠋⠀⢀⡠⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡈⠋⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⠤⡄⣤⠠⠤⠒⠊⠁⠀⡼⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠋⠀⠀⡀⠠⠄⠀⠀⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢛⠻⢷⣄⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⡀⠐⠄⣴⠋⣰⠊⠉⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⠃⠀⠀⢹⣇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⣸⠇⠀⢇⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠘⢦⡀⢀⠄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠤⠬⠷⣖⠊⢸⠀⠀⠀⠀⠀⠀⠀⣸⠀⢀⠠⠄⠐⠂⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⡸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠑⠄
⢀⡤⠤⠀⠶⣀⠠⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⣀⣀⣀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠑⠠⡰⠆⠠⠤⣄
⣏⠀⠀⠀⠀⢸⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⢸⠀⠀⠀⠀⢈
⠈⠁⠀⠀⠐⠊⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠈⠒⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠢⢍⠒⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⡰⢲⠶⢿⢷⡠⢌⠁⠀⠀⠀⠀⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠴⣒⡁⠒⠈⠙⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢰⠃⠀⠇⡎⠎⠞⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢞⣠⡿⢻⠛⢷⠚⣙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠒⠚⠀⠗⠁⣀⡠⠤⠐⢒⡈⠀⠀⠀⠘⠢⢀⡀⠀⠀⠈⢸⣰⠸⡀⠀⡇⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⠤⠒⢊⠩⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠍⣐⠢⠄⣉⠀⠉⠊⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠨⠙⠒⠤⠴⠋
⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣽⣯⣭⣬⣭⣬⣭⣭⣥⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣾⣿⣶⣾⣿⣶⣶⣶⣾⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⠀⢀⡀⡀⠀⢀⣄⢄⠀⣀⠀⢀⣄⣠⣀⣀⠀⢀⡄⢠
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠳⣻⠛⠉⠟⠙⠖⠛⠃⠀⠃⠈⠒⠋⠋⠀⠛⠀⠳⠊⠷⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⡤⢦⡀⠀⠀⣠⠄⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⣠⢎⡴⣰⢃⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠚⠳⠚⠚⠛⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡛⠘⢏⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠡⠀⠈⠙⠢⢤⣀⠰⠜⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢱⣄⠀⠀⠀⠙⣷⣾⠇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠄⠀⣠⡆⢀⠤⠀⣐⣈⡀⠐⢿⡇⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠰⠖⠉⠒⠓⠳⢰⠋⠉⣩⡶⠛⠁⠙⢦⠀⠀⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⠃⠀⠀⠀⠀⢨⣷⣾⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠤⠤⠞⠃⠀⣼⠀⠀⠀⠀⠀⡸⠃⢸⡥⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠞⢃⣀⣀⣤⠶⠛⠁⠀⣠⣵⢀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⣴⠖⠲⠤⠔⠒⠒⠒⠐⠂⣀⡤⠖⠛⠉⠉⠁⠀⢀⣀⣠⣤⣿⣷⣟⡻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠉⠢⣄⡀⠀⠀⠀⣠⣴⠟⠁⠀⠀⢀⣀⣤⣴⣾⡿⢟⡭⠏⠹⡋⣋⢱⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠈⠙⠛⠛⠛⠻⠻⠟⠗⠖⠻⠟⠻⠋⠉⠁⢀⠈⢰⣞⣴⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠹⡦⣤⣄⣠⡄⠀⣀⠀⣀⢀⣀⣀⣁⣘⢠⣂⣠⣤⣼⣾⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢣⢹⡄⢹⡟⠈⢿⠛⣿⠛⢻⡟⢻⠟⢻⡟⢻⡏⢩⣇⣸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⡎⣧⠀⣿⠀⢸⡀⢿⠀⣸⡁⢸⠃⢼⠀⢺⠇⢺⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢠⣸⡆⢿⠂⠸⡇⢸⢢⣼⡇⢸⠂⣿⡀⣿⢓⣿⣿⣿⢿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠸⣿⣇⢺⣧⠀⣷⢾⡧⣿⣧⣿⢲⣿⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣶⣷⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⣿⡸⣿⣿⡧⣿⡎⣿⣾⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣧⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⣿⣿⣿⣧⣿⣿⣷⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣶⠧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣾⣿⣿⣿⣯⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠾⢟⡛⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠿⠟⠻⠛⠛⠙⠉⠉⠉⠁



⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⠶⠶⠶⢶⣶⡶⠶⠶⢶

⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⢠⠄⡀⢠⠄⢠⣤⡄⢤
⠀⠀⠈⠶⢆⠸⣇⠠⢨⡇⢸⠶⠀⠰⠶⠀⠀⡇
⠀⠀⠘⠂⠋⡀⠘⠋⠋⠀⠸⠀⢂⠘⠀⠀⠀⠃⡀
⠀⠀⠈⢸⠁⢱⡎⠀⢣⠘⠀⢱⠂⢸⠁⠈⣅⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠰⠄⠈⠣⠤⠊⠐⠤⠜⠀⠸⠆⠀⠧⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣀⣀⣀⣼⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠻⢿⣟⠛⠋⠉⢹⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣠⡤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⢸⡀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢸⣿⣷⢠⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⣿⣿⡀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⣀⣴⣿⣿⣿⣿⣧⣤⡐⣶⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣿⣿⣿⣿⣿⣿⡟⣡⣿⠃⣸⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢄⣸
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⣿⣿⡛⢌⡛⢿⣿⣿⣯⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣽⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡸⣽⣿⣿⣿⡟⠟⢿⣼⣷⡷⢻⣾⡟⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⠀⠄⠛
⠀⠀⠀⠀⠀⢀⡶⠁⣸⣿⣿⣿⣿⣄⠀⢻⣿⣷⣿⣿⡇⢸⠀⡆⢠⠀⡀⠀⠀⠰⣄⣀⡀⠀⠄⠀⢠⣀⣴⠃⠀⠀⠀⠀⠀⣰⠏⢸⡀
⠀⠀⠀⠀⠀⣿⢊⡉⣩⣿⣿⣿⣿⣇⣴⣿⣿⣿⣿⣿⡇⢸⡄⠀⢸⣿⣧⠀⠀⢠⣾⣿⣿⡟⢂⡀⢻⡟⠁⠀⠀⠀⠀⠀⣰⠃⠀⢸⡇
⠀⠀⠀⠀⠘⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⣸⣿⣧⣤⣀⠉⢿⣿⡿⠉⠀⠀⢛⡿⢀⡀⠀⠀⠀⢰⠃⠀⠀⠘⠃
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢸⡇⢠⣿⡿⣿⣧⠝⠂⣿⣿⡟⢦⠄⠚⠳⠐⠀⢰⡄⠀⣰⠃⠀⠀⢚⠳⡄⠛
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⢸⣿⣷⣿⡟⠀⠀⣿⠿⠿⠰⡧⠶⠤⡾⢷⣾⣀⣰⠋⠀⠀⠀⠰⠆⠀⠳
⢆⣨⠆⠀⣸⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⣿⣿⣿⠀⢸⡇⢸⣿⣧⣿⠆⠀⢰⣿⠀⠀⠀⡆⠀⠀⣿⠸⡏⢩⠏⠀⠀⠀⠀⠀⣾⠂
⠈⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣼⣿⣨⣿⣹⣟⠀⠀⣾⣇⡤⢺⠄⡉⡀⢀⣿⠀⢻⠈⠲⠐⣷⠀⠀⠀⣿⢀⡤⢤
⠀⠧⠀⣿⣿⣿⣿⣿⣿⣿⡏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⡀⠀⢰⣿⣯⡍⢸⣶⣶⡏⢿⣿⠏⢀⠀⠀⠀⠉⠀⠀⠀⣿
⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⠶⢸⣿⣿⣿⣿⣦⣴⣦⣽⣿⣿⣿⣿⣿⡁⠈⢳⡞⣿⣟⣀⣸⠛⣻⣏⠘⣿⠀⡌⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠷⠆⢻⠇⣿⣯⣭⣤⣤⣿⣿⡄⠀⣸⣧⠀⠀⠀⠀⠀⣀⡀⡇
⠀⠀⠀⠀⠀⠈⠉⠛⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡏⠀⠀⠀⠀⠹⣆⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⠀⣾⡇⠀⠰⠿⠟⠁⠇
⠀⠀⠀⠀⢿⠀⠀⠀⢸⣸⣿⣿⢯⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⣀⣀⣀⣠⣤⣿⣿⣿⣿⣿⠇⠀⡄⠉⠀⠀⠀⠀⠀⢀⣠⢤⣀⠀⣀
⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⡟⣼⣿⣿⡏⣿⣿⣿⣿⠀⢸⡇⠀⠀⠉⠉⠉⠛⠛⢿⣿⣿⣿⡟⠁⢠⡆⠀⠀⠀⠀⢴⣦⣤⣉⡁⠿⢾⣿⣄
⠀⣠⢀⣤⣼⡧⠖⠌⠉⣿⣿⣳⣿⡏⠀⠀⣿⣿⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡁⠀⠸⠗⠒⠊⠉⠉⠋⠛⠙⠿⠛⠉⠀⠉⠛
⣿⣿⣷⢋⠍⠀⠀⠀⢸⣿⣿⣿⡿⠁⠀⢐⣿⣿⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃⢸⡦⢘⡀
⠛⠛⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⢹⣿⣿⡿⠀⢸⡧⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣼⣷⣬⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⣿⠀⣠⡟⢿⣿⣿⡇⠀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣿⣿⣴⡟⠀⢸⣿⣿⡇⢠⣿⣿⠀⢰⣶⣦⣤⣄⠀⠀⢀⣿⠀⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⣿⣿⣃⣤⣴⣿⣿⣷⣾⣿⢿⡄⠸⠿⠿⢿⣿⣿⣿⣿⣿⣺⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡽⢿⢏⣹⣿⣍⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⡙⠀⣿⣿⡃⢀
⣿⣿⣶⣶⣶⣶⣦⣤⣤⣤⣀⣙⣿⡭⡆⠰⢿⣿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠐⠀⠐⠆⠀⠀⠀⢐⠁⠀⠈⣡⠕⠀⠀⠀⠸⢏⣙⡆⠀⠀⠀⠀⠀⢠⡆⠤⠀⠀⢤⠀⠀⠀⢀⡠⠜⠿⢦
⠀⠀⣈⠙⠓⣾⣛⣷⠀⢰⣿⡿⢿⣂⣸⡄⠀⠀⡀⠀⠀⠀⣸⣄⠀⠀⠠⣷⣦⣽⣷⢆⣰⡆⠀⠀⠀⣾⡟⠒⠚⠈⢰
⠀⠀⠦⡀⢘⣫⡭⠶⠀⠈⣿⠂⠂⠈⠉⠀⠀⠀⠰⢸⢸⣿⢯⡇⠀⠃⠀⠀⠹⠿⠀⠀⠻⠆⢄⡤⠀⢸⡷⣦⡄⠀⣼
⠀⠀⠁⠀⠬⠷⠚⠉⠀⠀⠛⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠙⠊⠁⠀⠐⠀⠀⢀⣀⡀⠀⢠⣤⣤⡤⠀⢸⣄⣈⡁⠀⣈
⠀⠀⠀⠀⢀⢀⠀⠤⢀⡢⠀⠀⠀⢠⣾⣛⣛⡿⠞⠀⠀⠀⠀⠀⠀⠀⠤⡅⠈⠧⠺⠖⠶⠶⠶⠖⠀⠈⠉⠉⠉
⠀⠀⠀⠀⢸⣿⣟⡶⣶⢤⣌⡀⠀⠀⠐⠀⠤⢍⣙⣷⠀⠀⠠⠔⠀⠀⣤⣶⣢⠀⠀⠀⠀⠀⢀⣚⣁⠚⠻⢶⢂⣰⠄
⠀⠀⠀⠀⠀⠷⣿⣿⣻⣏⣫⡟⠀⠀⠯⣉⡓⠢⢤⣙⠀⠀⠁⠀⠀⠀⡠⣾⡏⠀⠀⠀⣄⡀⣉⣉⠉⢀⢠⠼⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠀⠀⠀⠀⠈⠑⠺⠗⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠈⠙⠂⠐
⣶⣶⡶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠉⠓⢄⠀⠀⠀⠀⡰⠚⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⢸⡃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣀⣤⡎⠀⠀⠀⠀⢳⣄⣀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠔⠒⠒⠒⠒⠒⠢⠤⣄⡀⠀⠀⠀⠀⠀⠱⡄⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⣠⠤⠔⠒⠒⠒⠒⠒⠢⠤⣄⡀
⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⢳⠀⢀⡏⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⣀⠴⠒⠉⠉⠉⠉⠉⠑⠲⢄⡀⠈⠳⡀⠀⠀⠈⡆⢸⠀⠀⠀⢀⠜⠁⠀⣠⠖⠊⠉⠉⠉⠉⠙⠒⠦⣀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⢀⡎⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠹⡄⢀⣠⠧⠤⣄⡀⢠⠏⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⢱⡀
⠀⠀⠀⡼⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣠⠟⠁⠀⠀⠀⠀⠉⠻⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢧
⠀⠀⠀⡇⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣀⠀⠀⠀⠀⣀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢸
⠀⠀⠀⣇⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢯⡽⠀⠉⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⣸
⠀⠀⠀⢹⡀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢀⡇
⠀⠀⠀⠀⢣⡀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠒⠲⠤⠤⠤⠤⠤⠤⠖⠒⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠑⠦⢄⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠊⠁⠀⠀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣤⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠲⠦⠤⠤⠤⠤⠴⠖⠒⡛⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⣤⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠒⠤⠤⠤⠤⠤⠤⠒⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢷⠀⠀⠀⠀⠀⠀⠀⠀⡿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⢨⣷⠲⠤⠤⠤⠤⠖⣾⡅⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠘⠦⣀⣀⣀⣀⡠⠴⠚⠁⢀⡴⠃⠸⡄⠀⠀⠀⠀⢀⠇⠈⢦⡀⠈⠓⠦⣄⣀⡀⣀⣀⠴⠋⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠱⡦⠤⠤⢴⠎⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⣤⣀⣀⣤⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠙⣄⡤⠋⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⣀⣀⣤⠤⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠈⠉⠉⠀⠀⠁⠀⠒⠂⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠃⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣀⣀⣀⣠⡶⠾⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣉⣀⣀⣀⣠⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠉⠉⠝⢻⡇⠀⠀⠀⠀⠀⠀⣄⣀⡀⠀⠀⠙⠛⢶⣄⠀⠀⠈⠑⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠰⡇⠀⠀⠠⠙⠿⠷⢶⣄⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣤⠶⠃⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠷⠛⠉⠀⠀⠀⠀⠀⠀⠈⠛⠾⠶⠶⣶⣶⡶⠶⠶⠟⠛⠉⠉⠀⠀⠀⠀⢀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⠁⠀⠁⢰⣿
⠀⠀⠀⠀⠀⠀⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏⠀⠀⠀⠀⠀⠤⡦
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣖⣤⣤⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠒⠦⠀⠀⠀⡄⠀⠀⢀⡎⠀⠁⢀⡖⠀⠀⠀⠀⢸⠃
⠀⠀⢸⣶⣤⡀⠀⠀⠀⠀⠀⠉⠛⠒⠳⠶⢢⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠄⠀⠉⠀⠀⠀⣀⡤⠖⠉⠀⢀⣤⢸⡂⠀⠀⡘⠋⠀⠀⠀⠐⣿
⠀⢀⠉⠉⠛⠛⠽⠆⣶⣶⠀⢸⡗⠒⠛⡿⠟⠁⠀⠀⠀⢀⣀⣀⠈⠉⠑⠂⠀⠀⠀⢀⣀⠦⠚⠉⠀⠀⠀⠀⢰⠃⠉⠀⡻⢣⡞⠁⠀⠀⠀⠀⠜⠁
⠀⠈⠻⢿⣶⣤⡄⢀⣀⠀⠀⣾⠂⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠈⠉⠀⠀⣀⡠⠴⠂⠉⠀⠀⠀⠀⠀⠀⠀⢀⣸⡀⠀⠀⢁⡌⠁⠀⠀⠀⢶⣾
⠀⠀⠀⠀⠀⠉⠁⠉⠛⠛⠀⠁⠀⠀⠀⠀⠒⠲⠀⠀⠀⠀⢀⠠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠛⣇⡄⣴⠉⠀⠀⠀⠀⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⣀⣤⡤⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢀⣿⣤⠍⠀⠀⢀⣠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣀⠄⠂⠈⠀⠉⠉⠓⠶⠤⣀⠀⠀⠀⠀⠀⠀⡾⣿⡁⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⠀⠛⠒⠒⠚⠛⠋
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⡄⡀⠀⠀⠀⠸⣿⣿⣿⢷⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠤⠦⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠉⠛⠿⣿⣮⣭⡙⠳⢦⣤⠔⠒⠀⠀⢠⣼⠃⠀⠀⠀⠀⠀⠠⡶⠋⠉⠋⠉⠓⣆
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⠀⠉⠙⠟⢿⣷⣀⣠⡦⠔⠋⠉⠀⠀⠀⠀⠠⠎⠈⠐⢦⣄⣀⣀⠀⠀⣸⠷⣦⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠰⠛⠙⢸⡟⢛⣏⠀⠀⠘⡇
⠀⠀⠀⠑⣤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠁⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡄⣠⠀⠀⠀⠀⠈⡿⠁⠀⠀⢰⠇
⠀⠀⠀⠀⠈⠙⠿⠿⢿⣏⣀⣚⢛⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣤⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⢻⡄⠀⠀⠀⡀⠑⢤⣴⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠲⠿⠛⠋⠁

⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣉⣈⣀⡀⠈⠛⡉⠀⢀⡹⠿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣯⣓⣒⠋⠀⠠⠁⠀⠀⢬⠶⠤⠉⠉⢉⢉⡁⠁⠠⣾⣿⠟⠁⠀⠲⣿⢿⣧
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠶⠶⠋⠀⠀⠀⠀⠀⠀⣬⢛⡋⠀⠀⠀⠙⠋⠉⠈⠛⠂⢀⣀⣬⣽⠟⢛⣿
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢉⡿⣆⣀⣒⣆⣀⣭⣿⡛⢠⣤⣤⣀⡐⢤⣁⡂⠠⣀⣀⣀⣀⣈⣉⣩⣿⣿⣿
⠀⠀⠘⠈⠀⠀⠉⠉⣉⠉⠁⢈⣀⠙⠛⠉⠀⠈⢿⡷⠋⠋⠻⡿⢿⠍⠁⠩⠯⠤⣭⣤⣤⠭⠀⢈⣉⣁⠐⠈⢹⠻⠿⢿
⠀⠀⠀⠀⠀⠀⠀⢠⣁⣬⢌⣵⡶⣰⠀⣼⣿⣿⠟⣱⠉⠉⠉⣹⣿⣷⣲⣤⡆⠀⣶⣾⣧⣤⣤⣤⣭⣟⠋⢙
⠀⠀⠀⠀⠀⠀⠀⣇⡿⣣⣿⡟⣰⣿⢠⣿⡿⣡⣾⣿⠀⢰⢰⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⠰⣿
⠀⠀⣠⣤⠤⠤⢤⠈⡵⢿⠟⠼⠿⠇⠸⠯⠾⠿⠿⠿⢀⣸⢀⢿⣿⣿⣿⡿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠚⠀⣤⠴⡾⠅⠂⠙⠫⣭⣭⣶⣶⡖⠂⠀⠀⢰⣿⣿⠸⠈⣿⣿⣿⠁⠀⠀⠉⣿⣿⡟⠉⢹⣿⠇⠀⠀⡯⠄⠀⠄
⠀⠀⠀⠰⠁⠀⡂⡄⡍⠛⢁⣀⡊⠉⣉⡀⠀⣠⢴⡄⠙⢻⠀⠀⢻⣿⡏⠀⠀⠀⠀⠉⠉⠁⠀⣠⣤⡄⢠⡆⡦⠀⠀⠂
⠀⠀⠀⠰⣯⠿⡅⡅⡃⣂⠙⠾⡃⠀⣡⣶⣦⣌⠛⢁⣤⣼⡀⠀⠀⡌⠀⠀⢠⣾⣿⡧⠀⢠⣤⣬⡙⠇⠈⢣⠀⠀⠀⣉
⠀⠀⠀⢰⠂⠠⠴⠓⠐⠖⢒⣚⣛⣺⡿⠛⠛⣿⣦⡾⣟⣛⣛⣒⣒⣋⣛⣛⣓⣛⣛⣙⣳⣿⠛⠻⣿⣆⠁⢄⠆
⠀⠀⠀⠈⢹⣿⣿⣶⣾⠿⠿⠿⠛⢿⣧⣈⣀⣿⡟⠉⠉⠉⠉⠉⠙⢿⣿⣿⣿⢉⣛⣛⣛⣿⣄⣤⣿⡋
⠀⠀⠀⠀⣀⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠷
⠀⠀⠀⠈⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣡⣦⣄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢰⡦⠀⢀⣠⡾⠛⠛⠒⠒⠒⠚⡛⣦⣄⠲⢄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠛⠒⠿⢶⡐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣿⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢹⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣠⠂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀
⠀⠀⠀⠀⠀⢠⠚⠁⡄⠸⣇⠀⡀⠀⠀⠀⠀⠀⡠⠀⠐⠋⠀⠘⠁⠚⠿⢿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠸⡄
⠀⠀⠀⠀⠀⠈⠄⡄⠰⠀⢛⣼⠀⠀⠀⠤⠂⠉⠀⢀⡀⢀⣀⣐⣄⣠⠀⠀⠈⠉⠻⠷⠀⠀⠀⢀⣀⣀⣀⡄⠀⠇
⠀⠀⠀⠀⠀⠀⠈⠾⣦⡳⡀⡟⠀⠀⠀⠀⢀⢴⠀⡀⠙⠿⢿⣿⣿⣽⣝⣢⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⠙⠿
⠀⠀⠀⠀⠀⠀⢀⠠⠟⠿⣿⣿⠀⣴⠋⠀⡂⠀⠈⠉⠲⣶⣤⡝⠋⠃⠀⠹⠗⠒⠢⡀⠀⣀⣤⡶⢦⣤⣞
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠹⣇⢸⠇⣴⡹⠁⠄⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⣇⣀⣉⡿⠛⠐⣼⣿⠛⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣠⠎⠉⠉⢿⠀⠀⠀⠉⣤⣷⠀⠈
⠀⠀⠀⠀⠀⠰⡀⢆⠀⠀⢀⠀⡋⢸⠈⠁⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣀⡴⠞⠡⠄⠀⠀⠠⠀⠀⠀⠀⣠⢏
⠀⠀⠀⠀⠀⠀⠙⢦⣥⣄⣈⢳⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠞⠁⢸⠂
⠀⠀⠀⠀⠀⠰⠁⠈⠀⠈⠉⢿⡏⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢿⡆⢸⣦⡀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠁⠶⠐⠆⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⣿⡟⠀⢑⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠑⢆⡀⠀⠀⡀⣿⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢴⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡦⣤⣰⡿⠀⠀⠀⡀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢀⡀⢁⠬⠁⠌⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡹⣁⢰⣧⡞⠂⢤⠀⠈⠀⠀⠀⠀⠐⠖⠆⢸⡷⠋⣠⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢯⡀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢲⣄⡈⠳⣏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢷⠀⠀⢰⡇⠀⠀⠀⠀⠀⣾⢉⢻⣶⠿⠀⣀⣷
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⠲⢿⡇⡀⠘⡟⠀⠀⠀⠀⠀⢹⣧⣾⠁⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢋⡒⣾⣿⣀⣶⡇⠀⠀⠀⠀⠀⠀⡉⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏⠻⠿⠋⡕⠀⠀⠀⠀⢀⠐⠉⡒⠖⡲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡶⢲⣆⠁⠑⣄⠀⠀⣾⡠⢠⣇⠨⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠋⠆⠀⢻⡆⠀⣿⢃⣾⠃⢸⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠰⡸⡄⠈⠟⠀⢹⣾⡏⢠⡏⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⣀⢷⡁⠀⢸⠀⠈⣽⡁⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠸⠏⠿⡀⢸⠀⠀⠙⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣙⣛⣛⣹⣯⣩⣉⣿⣋⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣭⣽⣭⣽⣿⣯⣭⣭⣭⣭⣿⣭⣭⣯⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⠀⠀⠀⠀⠰⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠉⠐⠢⣀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⢀⠀⠀⠀⠱⣀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⠂⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠐⠦⠀⠀⢦⠀⠀⠀⠸⠀⠀⠀⠀⢨⢀⡈⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⢛⠀⠐⠛⠛⠓⠄⠐⣄⠀⠀⠀⠀⠆⠀⠀⠀⠀⠚⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠙⠂⠀⠀⠀⢸⡆⠀⠀⢈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⠀⠀⠤⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠰⡀⠀⢠⣴⡶⣿⣦⣤⡀⠀⠀⢸⡆⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⡐⠐⠀⠀⠀⠀⠙⠁⠈⠁⠀⡠⠂⢘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⣤⡤⠄⠈⠀⡠⠊⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢠⡔⢒⣿⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢑⠲⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⢈⢄⡄⠁⠀⠈⠓⠒⠲⠌⠤⠵⠀⠰⠎⣰⠇⡀⢀⡄⡝⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠊⠀⠀⠀⠃⠀⠀⠀⠀⠐⠆⠀⠀⣀⡏⠀⠋⡼⠈⠀⠀⠀⢘⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢰⣿⣱⠀⡰⠇⣿⡇⠀⢀⣘⢂⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠏⠏⣏⡼⢸⠱⡋⢀⠀⠀⠉⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⣿⠁⡀⠀⢇⠀⡄⢠⣦⠀⠐⠀⠀⠂⢈⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡏⠀⠀⠀⠈⠑⠰⡿⠁⡀⠀⣆⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡇⢣⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠰⠁⡀⠀⠀⠀⢀⠉⠇⣂⠘⠂⠉⠂
⠀⠞⠀⠀⠀⠀⠀⠐⠣⡀⠇⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠀⠘⠀⠀⠀⢰⠁⠀⠀⢀⠂⠀⠀⠈⠉
⠀⠃⠁⠀⠀⡠⠊⢀⠂⠅⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⠀⠀⠀⠘
⠀⠈⢦⠀⠔⠀⣠⠠⠊⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠁⠂⠉⠀⠀⠀⠘⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠈
⣿⣿⣿⣿⣛⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣟⣛⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡔⣀⣀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠁⢀⠀⠀⠀⢀⡠⢊⣀⡠⠂⡀⠀⠀⣚⡫⣇⣤⣠⣄⠨⠄⠀⣃⡀⠀⢀⣀⣀⣀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⢦⡐⠖⡉⠛⣷⠀⠀⠻⣅⠈⠩⢅⠈⠓⠲⠌⠉⠀⠈⠙⠀⣞⠁⠀⠂⠀⢍⠲⠭⠉⠉⠉⠩⢏⠂⠋⡇
⠀⠀⠀⠀⠀⢀⠉⠀⠉⠐⠒⠁⡀⠀⠘⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠠⠃⣀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠠
⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡅
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠨⠻⠿⠂
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠄⢁⠀⢘⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⠈⠀⠠⣀⠀⠀⢢⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠬⢥⠴⠤⠤⠤⠮⠤⠤⠴⡿⠄⠀⣶⣾⡆⠀⠀⠤⠤⢒⠒⡶⢐⠿⣄⠀⠀⠙⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠉⠉⠀⡐⠀⠀⠀⠠⠔⠉⠐⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐

⠀⠀⠀⠀⠀⠀⠈⠁⠠⠤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⢠⠀⠀⠀⠀⣴⡀⠀⢰⡄⠀⠀⣸⣇⣾⡇⠀⠀⠀⢠⣄⢀⣶⣿⣧⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀
⠀⠀⢸⡄⠀⠀⠀⣿⣇⢰⣾⣇⠀⣶⣿⣿⣿⣷⢰⡇⠀⢸⣿⣸⣿⣿⣿⣼⣷⠀⢠⣿⣸⣷⠀⠀⣾⡀⣠⠀⢀⣾⠃
⠀⠀⢸⣧⠀⠀⢠⣿⣿⣿⣿⡇⠀⣿⣿⠙⢻⣿⣾⣧⠀⣼⣿⣿⢿⠏⠘⣿⡿⠀⠘⣿⣿⣿⠀⠀⣿⣿⣇⠀⣼⣿
⠀⠀⢸⣿⡆⠀⢾⠙⣿⣿⡇⠁⠄⢸⡇⠀⠈⣿⡏⠹⠏⠸⣿⡟⠀⠀⠀⠹⠀⢀⠀⢹⣿⠁⠀⠸⣿⣿⣿⣆⣿⡇
⠀⠀⠘⣿⣧⠀⠘⠀⢿⣿⠀⠀⠀⠀⠁⠀⠀⠿⠀⠀⠂⠀⣿⠇⠀⢠⠀⠀⠀⠸⠀⠈⠃⠀⠀⡆⣿⡏⠸⣿⣿⠃
⠀⠀⠀⣿⡿⡇⠀⡅⠸⡿⠠⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡄⠘⣿
⠀⠀⠀⢹⠇⠀⠀⡄⠀⠀⠠⠀⢀⣀⣤⣄⣙⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠶⠀⠀⠀⣀⠀⢄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣾⠋⠉⠀⠉⠛⠀⠀⡙⣷⡀⠈⢆⣀⣀⠂⢀⣴⣿⣿⣿⣷⣶⣄⠀⢉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠳⣿⠘⣿⣶⡸⣿⡏⠀⣸⣿⡟⠋⠉⣿⣿⡿⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⢠⣄⠀⠠⢰⢿⣤⣿⠅⣿⣿⣶⣿⣿⣿⣏⣿⠿⠎⣰⣿⢟⣷⣿⣿⡗
⠀⠀⠀⠀⢀⠀⣤⣶⣧⠈⣤⣀⣠⣿⣄⠀⢨⡿⡏⠁⣸⣿⣿⡟⠁⠀⠙⣿⣿⣶⣦⣴⣶⣿⠟⠉⠉
⠀⠀⠀⠀⠘⣷⣮⣿⣮⣷⣌⠻⠛⠛⢨⡆⠘⠷⣀⣴⣿⠿⠟⠁⠀⠀⣤⠘⠙⠿⢿⢿⠹⣯
⠀⠀⠀⠀⠀⠘⠛⠻⢿⣼⣿⣿⣶⣦⣬⣤⣶⠞⢻⡇⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠐⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⢻⣿⣿⣿⡞⠋⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢹⣦⣀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢈⠉⢋⠁⠀⠀⠀⠀⠈⠀⠀⣦⢸⣦⢺⠀⣶⣶⡀⠉⠛⢦⡀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⣶⡶⠀⠉⠀⠀⠀⠀⠀⠘⠃⢸⠆⢀⠙⣦⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠄⠁⠀⠀⠀⠀⠰⡀⠛⠀⠀⠀⠠⡀⠒⢀⣀⠀⠀⠀⠀⠘⠂⠘⣧⠀⠀⢻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠆⠉⠀⠀⠈⠙⠀⠀⠁⣸⣿⢀⠀⠀⠀⠀⢰⡀⣽⣶⣶⣿⣿⡟
⠀⠀⠀⠀⠀⢀⣷⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠐⠀⢸⡉⣽⠛⠛⠂⠀⡰⡏⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠱⣴⣿⣀⣀⣠⡀⠀⡀⢀⠀⠉⠀⠀⠀⠘⢷⡏⢠⣤⠰⣞⠃⠀⢠⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⣿⣿⣿⣿⣷⣄⠙⠘⣇⢧⡰⣤⣦⡀⠛⠋⣿⠃⢰⣄⠀⣠⣿⣿⣿⡿⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢸⣿⡀⢰⠀⠀⠙⣿⣿⣿⣿⣿⣷⣤⡁⠘⠃⠁⠀⠘⠀⠚⠁⠀⢈⣡⡾⣿⣿⣿⣿⡅⠀⢰⡆⠀⢸⣿
⠀⠀⠀⠀⠀⢾⣿⡇⢸⣇⠀⠀⢨⠙⠻⢿⣿⣿⣿⣿⣷⡆⣤⣤⣤⣤⣤⣶⣾⣿⣿⡇⣿⣿⣯⣸⣿⣁⣾⣧⠀⣿⣿
⠀⠀⠀⠀⠀⢸⣿⡇⣼⣿⠀⠀⣼⡇⠀⢠⣹⡿⣿⣿⣿⣿⡜⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣯⣙⣼⠟⠛⢻⣿⠃⢸⣿
⠀⠀⠀⠀⠀⠈⣿⠁⢹⣿⠀⠰⣿⣷⠀⣼⣿⣿⠟⣼⣻⣿⡿⢦⡿⡿⣿⣿⡿⢩⣿⢸⣬⠿⣽⣿⠃⠂⠸⣿⠐⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠻⠀⠀⡿⠀⠀⢻⣿⠀⣿⡏⢻⣿⢶⣟⣿⣝⠿⣿⢷⣿⡙⢡⣶⣦⣬⡍⠀⠀⢻⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠁⠈⠁⠀⠉⠙⠛⠇⠓⠈⠿⠟⡷⠿⠏⠻⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠒⠂⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠘⠃⢀⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠁⠀⢀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠋⠀⠀⡀⠈⠁⠀⠡
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠘⠂⠀⣤⡀⠈⠁⠀⢀⣀⠀⡗⢄
⠀⠀⠀⠀⠀⢠⠎⠀⠑⠄⣀⣀⣀⣀⡀⠀⠄⣀⡀⣉⠰⣡⢀⠦⡀
⠀⠀⠀⠀⠰⠁⢄⠐⠃⠀⠀⠀⠒⠀⠀⠠⠤⠀⠀⠀⢀⠤⢊⣀⠈⢆
⠀⠀⠀⠀⡆⠀⢀⠉⠒⠀⠤⠀⠀⠀⠀⠤⠄⠀⠒⠈⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⣄⠈⠁⠀⠀⠋⠀⠀⠀⠐⠒⠀⠀⠘⠁⠀⠉⢁⡀⡴⠁
⠀⠀⠀⠀⠀⠸⠈⠉⠀⠀⠉⠉⠀⠉⠉⠀⠉⠁⠀⠉⠁⠉⠀⢀⠃
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠩⠶⠦⠀⠀⠀⠀⠤⠶⠭⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠢⠴⠃⠀⠀⣀⢄⢶⠋⢉⠔⠖⢄⣀⡀⠤⠐⠈⠉⢡
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡤⠠⠤⠄⠀⢀⠔⢠⢂⡆⢀⠊⠘⠀⠱⠱⡤⠄⠒⠁⠀⠉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣀⣀⠧⠀⠀⠀⠤⡜⠀⠂⢻⠃⣬⡀⠿⠆⠀⠀⡇⠠⠤⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡀⢄⢰⠀⠸⠈⣻⠏⢀⠀⡜⡰⠁⠒⠠⢄⣀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⠀⠈⢀⡠⠘⢢⡀⠀⠄⠰⡀⠈⣎⠔⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠒⠁⠀⠀⠀⠈⠑⠚⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠀⠀⠚⣭⣶⣿⣷⡄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢩⣴⣾⣿⣷⡀⠈⣿⣿⣿⡇⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⣛⢉⣩⣭⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⢛⠻⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⣰⣿⣤⠀⢀⣠⣿⣿⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠟⠛⢛⣉⣉⣭⣬⣽⣿⣦⣀⣀⣀⣴⣿⣿⡿⣣⣾⣿⡿⢋⣿⣟⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣻⣟⣵⣿⣿⡟⢠⣿⣿⣿⡏⢳
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣬⡙⢿⣿⣿⣿⣿⠿⠛⢛⣉⣉⣉⣩⣭⣥⣤⣶⣿⣿⣿⣿⣿⠟⢁⣿⣿⣿⣿⣷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣦⡙⢿⣿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⢛⣉⣥⣴⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⢀⣶⣦⣬⡙⠿⣿⣿⣿⣿⣦⡹⣿⠿⠛⠻⠛⠛⠛⢛⣋⣥⣶⣿⣿⣿⣿⣿⣿⣷⣤⣬⣉⠁⣁⣴⣿⣿⣿⡆
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣶⣌⠻⢿⣿⣿⣷⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⣛⣉⠽⣟⣫⣴⣾⣿⠋⣾⣿⣿⡇⣴⣄
⠀⠀⢀⣴⣦⣄⣈⡙⠻⢿⣿⣿⣿⣦⡙⢿⣿⣟⠻⠛⠻⠿⠿⠿⢛⣫⣴⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠟⢸⣿⣿⡟⢠⣿⣿
⠀⢠⣾⣿⣿⣿⣿⣿⣶⣤⡙⠿⣿⡿⢻⣾⣷⡾⢿⣿⣿⣿⣿⡿⢿⣿⣿⡿⠟⣛⣯⣥⣤⣴⣷⣿⣟⣋⣠⣴⣿⣿⣿⢁⣿⣿⡿
⢀⣙⣙⠛⠛⠛⢿⣿⣿⣿⢿⣷⣾⡆⣷⣿⣿⣴⣷⣿⠛⣿⣿⣿⣎⢻⣿⣿⣦⠀⠉⠉⠉⠉⠛⠛⠻⣿⣿⣿⣿⠟⢡⣾⣿⣿⠇
⣿⣿⡿⠿⠛⢛⣛⠟⣽⢃⣿⣿⣿⡇⢹⣿⣿⣷⣹⣿⡇⢹⣿⣿⣿⣆⠻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣙⡻⢧⣴⣿⣿⡿⠋
⠋⣡⣴⣾⣿⣿⢋⣾⡇⢸⣿⣿⣿⡇⢸⣿⡟⢿⣿⣿⣿⡈⢿⣿⣿⣿⣇⢹⣿⣿⡇⠀⠀⠀⠀⠠⡄⠘⠛⠁
⣿⣿⣿⣿⠟⣵⣿⣿⡇⢸⣿⣿⣿⠇⣾⣿⣇⠸⣿⣿⣿⣷⠈⣿⣿⠿⠟⠀⠉
⣿⣿⠟⢡⣾⣿⣿⡿⢀⣿⣿⡿⢣⣾⣿⣿⣿⠀⠿⠿⠟⠛⠁
⠟⢁⣴⣿⣿⡿⠏⣴⣿⣿⣿⠇⠘⠟⠛⠉⠁

⠀⢀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⡀⠀⠀⠀⣀
⠀⠑⠤⠃⠀⠀⡖⠐⡄⠀⠀⠀⠀⢔⠉⣣⠀⠀⠀⢀⣠⠤⠖⠻⣄⠘⠤⠃⠀⠘⠒⠊⠀⠀⠀⠀⠐⢍⠆⠠⡞⠣⠁⢸⠀⠀⢇⣈⠆
⠀⠀⠀⢠⠒⢢⢉⠨⡀⠀⠀⠀⠀⠈⠉⣁⡤⠖⠊⢉⢤⠀⠁⠀⠚⣄⠀⣸⠏⠓⡶⠚⠉⠉⠉⠓⢦⡀⠀⠀⠙⢦⣠⠼⠦⣄
⠀⠀⠀⠀⠑⠁⠰⠤⠜⠀⢀⣠⠴⠚⠉⠀⣤⢄⠻⠣⠉⠁⠀⠐⠀⠉⣇⢳⣆⣈⣇⠀⠀⠀⠀⠀⢀⡏⠩⣽⠃⣏⠤⠐⣂⡘⡆
⠀⢪⣉⠆⠀⢀⣠⠤⠖⠋⠁⡤⠀⣖⣢⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⠉⠀⠈⠓⠒⠂⠐⠒⠛⠦⣴⠎⠀⢷⢊⡡⠄⠀⢻⡀
⠀⢀⣠⠶⠛⡩⠀⣀⣦⠫⡹⠑⠒⠀⢀⡤⠚⠉⣿⠤⠶⣲⠀⠀⠀⣠⠴⠋⠀⣀⣀⠀⠀⠀⠀⣀⣠⣤⣀⠀⠀⠘⣇⠴⠚⠉⠉⣧
⠰⢯⠐⠀⠈⠧⠤⠈⠀⠀⡐⠒⣢⢩⡇⠀⠀⣰⣧⠤⠞⣁⡤⠖⠋⠀⢀⣀⠀⢇⣨⠃⢠⢶⠎⠁⠀⠀⠀⠙⣆⠀⠹⣤⢒⣉⣥⡟
⠀⠙⣆⡀⠀⠀⠀⠀⢀⠀⠼⠦⠔⠘⠖⠒⠊⣀⡤⠖⠋⠁⢄⣱⠀⠀⠣⠤⠃⠀⠀⠀⠘⣎⢦⡀⠀⠀⠀⠀⢸⣄⠀⠉⣽⢀⠀⣙⡆
⠀⠀⠘⣧⠀⢂⠀⠀⠀⠈⠁⠈⢁⣠⠴⠚⠋⢡⠤⣄⡀⢀⠤⠤⢤⣀⣀⣀⠀⠀⢨⣈⠇⠹⡑⠪⠅⢒⣒⣒⣪⠞⠀⠀⠹⢴⣤⠿
⠀⠀⠀⠙⣧⠀⠁⢄⣀⡤⠴⠚⠉⠀⠀⠀⢸⣉⠀⠀⠉⡧⠒⠒⢊⠔⣒⣋⡩⢷⡤⠤⢤⠀⠉⡶⢦⠤⠖⠋⢀⣠⣀⡀⠀⠀⢀⠠⡄
⠀⠀⠀⠀⠙⠧⠚⠋⠁⠀⠰⣉⡇⠀⠀⠀⢀⡙⠒⠒⠛⣷⣒⠒⠩⠔⠁⢀⡠⣾⠀⢀⣛⣶⢸⠁⡎⠀⠀⢠⡏⡎⠉⠉⠉⠲⡄⠙⠁
⠀⠀⢎⣈⡆⠀⡀⠀⣀⣠⡤⠔⡲⢚⢷⠞⠁⠛⢲⠀⠠⠒⢌⠉⠙⠒⠓⠒⠶⠋⠙⠒⠾⢁⡏⣸⠀⠀⠀⡏⢠⢣⠀⠀⠀⠀⢸
⠀⠀⠀⢀⠜⣌⣶⢋⠏⡔⠀⡰⢡⠊⡰⣧⣤⠤⠿⠀⠘⢒⣃⣀⣀⣀⡤⠶⡆⠀⢎⢹⠀⡸⢠⠇⠀⣀⠴⢳⡀⢢⢣⡀⠀⠀⡸
⠀⠀⢸⡏⠉⠉⠹⡌⡘⠀⢰⢁⣃⡴⠴⠋⠀⠀⢀⡴⠊⠉⠀⠀⠈⢣⠀⢙⡇⠀⠀⠀⢠⠃⡾⠒⣉⠤⠒⠉⠳⠤⣵⣭⣒⣺⠇
⠀⠀⠸⠶⣎⡴⠚⠙⠓⠒⠉⠉⠀⠀⠀⣤⠒⠒⣿⠀⠀⠀⠀⠀⢀⠜⠒⠮⠟⣀⡤⠖⡟⢸⠓⠉⠀⠀⡀⠀⡂⢓⡄⠀⠀⠀⢀⡀
⠀⠀⢀⠤⡀⠀⠀⠀⠀⡔⠒⡄⢎⢙⠄⢼⣡⡀⢈⡦⠤⠤⠴⢚⣁⠀⠀⡖⣋⠥⠒⢻⠃⡇⠀⠀⠀⡏⢈⡆⠉⢉⣀⠀⠀⠀⢏⣩⠆
⠀⠀⠱⢤⠟⠀⠀⠀⠀⠑⠚⠁⠀⠁⠀⠀⠘⠖⠋⠀⠀⠀⠀⠫⣠⠇⠀⠉⠀⠀⠀⠈⠋⠀⠀⠀⠀⠈⠉⠀⠀⢧⣀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠿⠅⠀⠀⠀⠀⠠⣿⣾⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡤⠀⠀⠀⡿⠋⠻⣷⣄⣠⠤⢼⣿⡦⠤⣄⣠⣾⠟⠋⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢿⣌⣓⠂⢉⡻⠀⠘⣿⠀⠀⢟⡉⠐⣒⣡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⠞⠛⠁⠀⠀⠉⠀⠀⠉⠛⠳⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠛⠀⠀⡞⠿⠄⠀⠀⠀⢠⠿⢱⠀⠀⠛⠛⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠷⣦⡤⠤⠄⠀⠑⢐⠀⠀⠀⠀⠀⠒⠂⠀⠠⠤⢤⣶⡾⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡀⢀⡤⠂⠀⠁⠐⠢⣎⡀⠀⠀⠀⢀⣣⠔⠂⠈⠁⠒⢄⠀⢀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠈⠀⠀⠠⡄⠈⠀⠘⢿⣿⣿⣿⡿⠃⠀⠀⡠⠄⠀⠀⡇⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠘⡄⠀⠀⠘⡄⠀⠀⠀⠀⣿⡀⠀⠀⠀⡰⠁⠀⠀⣠⠃⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⢦⡀⠀⠈⢲⠀⠐⠊⠀⠉⠀⠀⡍⠀⠀⢀⡴⠕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢼⡀⠀⠀⠀⠀⠀⢠⠧⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣒⠒⠒⠒⣂⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠁⠀⠉⠻⠿⠏

⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⣤⡄⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⡤⡄
⠀⠀⠀⠈⠳⠄⣀⡀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠁⠀⠀⠀⠀⠈⠳⠄⣀⡀⣀⡴⠂⠀⠀⠀⠀⠀⠈⠈⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠸⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⡿⠇⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⢽⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⢿⠟
⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆
⠻⠋⠀⠀⠀⠀⢀⡠⠐⢊⣱⡀⠀⠀⠀⠀⣀⡾⠤⣄⠀⠀⠻⠋⠀⠀⠀⠀⠀⣠⠖⠈⣱⡀⠀⠀⠀⢀⣠⡾⠤⣄
⠀⠀⠀⠀⠀⢀⡛⠀⠤⠊⠉⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢀⠠⠂⠉⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠾

⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⣤⡄⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⡤⡄
⠀⠀⠀⠈⠳⠄⣀⡀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠈⠳⠄⣀⡀⣀⡤⠂⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠸⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⡿⠇⠀⠀⠘⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣤⣄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟
⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆
⠻⠋⠀⠀⠀⠀⠀⡠⠴⣚⣵⡀⠀⠀⠀⢀⣠⡾⠤⣄⠀⠀⠳⠏⠀⠀⠀⠀⠀⣠⠔⢘⣱⡀⠀⠀⠀⢀⣠⡾⠤⣄
⠀⠀⠀⠀⠀⢀⠛⡁⠐⠂⠉⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠁⠤⠊⠉⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡌⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⢀⡹⠁⠄⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠠⠀⠈⡟⠁⡀⠐⡤⠀⠠⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⣉⠀⡌⠁⠀⢳⡥⡞⠀⡥⢠⠆⠀⠈⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⢤⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠠⡱⣇⣅⣂⢴⠡⣀⢄⣠⠄⠐⢈⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠘⠀⠀⠀⠀⠅⣠⠖⠉⠀⠀⢹⠟⠉⠉⠙⡄⢈⠋⠠⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⠁⡀⢀⣡⡄⠀⠀⡞⠁⠀⠀⠀⢀⠃⠀⠀⠀⠀⠙⡅⠰⠂⠀⠠⠀⠀⢤⢀⠀⠄⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡲⠛⠁⠀⠀⣰⠷⠓⠂⢄⡀⠀⢸⠀⠀⡀⢠⣀⠴⠁⠀⠁⢂⠀⣰⠰⠟⠓⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠃⡠⠤⠀⡜⠁⠀⠀⠀⠀⣤⢖⠟⠙⠛⠫⡙⢥⣀⠀⠀⠀⠘⡌⠉⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢀⡄⠄⠀⠐⢉⠁⠀⡀⠀⠀⢸⣷⠁⡠⠀⡀⠀⠀⢂⢳⠠⣀⣠⠴⢧⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⠘⠓⠚⠀⠀⠀⢼⣀⠀⠐⠂⠈⠐⢀⣼⣿⠚⠁⠀⠀⡼⠀⡐⠈⠐⠲⢤⣂⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⢀⡲⢆⠀⠀⠀⠎⢿⣶⡤⣄⢠⣤⡾⠏⠀⠈⠐⠒⣹⠰⣂⡀⢀⠂⠀⢢⠀⠙⠉⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⣤⣄⣆⢴⠁⠀⢀⡥⣖⠀⠀⠀⠀⠩⠋⠙⠈⠀⠀⠀⢹⠒⠊⠪⠼⣆⣲⠋⣘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠠⠜⠃⠚⢆⠀⠈⠀⠈⢄⡀⢀⣠⠂⠀⠀⠀⢀⣀⣠⣼⠳⣷⣞⠁⠈⠇⠋⠱⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠈⡄⠀⠁⠀⠄⠐⢩⡆⠠⠀⠐⠂⠀⠈⠁⠁⢠⡖⠒⢲⠉⢽⠅⡛⠀⣿⢫⠩⠁⠶⠀⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠀⠀⡤⠈⠀⢀⠣⣌⡾⢀⠤⢀⡠⡀⠀⠑⣷⡖⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠐⠀⢀⡀⠀⠁⠀⠛⢻⠀⠀⠀⠀⠇⠒⠀⠀⣳⣀⠠⠀⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠈⠛⠂⠀⠀⠀⠐⠀⠀⠀⣆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃

⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠙⠛⠛⠁⠀⢀⠀⠀⠀⠀⠘⠑⠛⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠘⠙⠛
⠀⡀⣀⢂⠀⠀⠀⠁⠀⠀⠀⠀⢠⣠⠄⢙⡇⠀⠀⠀⠐⣦⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢂⠀⠀⠈⠁⠀⠀⠀⠀⢠⣀⠄⢙⡇⠀⠀⠀⠐⣶
⠀⠱⠂⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢚⠈⢡⡄⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢚⠈⢡⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠈⠓⠇⠀⠀⠀⠻⠲⡰⠦⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠓⠇⠀⠀⠀⠻⠲⡰⠶⠀⠀⠀⠀⠀⡀
⣀⠀⠀⠀⠀⠀⠙⠶⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠃⠢⠤⠴⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⠶⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠦⠤⠴
⠉⠀⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠂⠈⠁⠀⢀⡀⠠⠀⠀⠀⠉⠀⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠐⠂⠈⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠂⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⢤⠤⠤⣄⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠁⡁⠀⠀⠀⠀⠀⠀⢐⠀⠀⡀⠀⢀⢤⠤⠤⣄
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠂⠋⠁⠀⡋⢠⠂⠂⠀⠆⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠉⠁⠀⡋⢠⠒⠂⠀⠆
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⠀⠀⠀⢣⡀⠀⢁⡰⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⠀⠀⠀⢣⡀⠀⢁⡠⠃
⡆⠀⠀⢰⠁⡞⠛⡆⢀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⢠⢚⠶⢀⡆⠀⠀⢰⠁⡞⠛⡆⢀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⢰⠾⠶
⠀⠀⠀⠸⣦⣀⢀⣡⠟⠀⠀⠀⠀⠀⠀⢀⣶⡄⠀⠀⠀⠠⠀⠐⠀⠈⠁⠀⢸⠁⠀⠀⠸⣦⣀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠠⠀⠂⠀⠈⠁
⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠰⣶⡆⠀⠀⠘⣿⣧⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠘⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠰⣶⡆⠀⠀⠀⣿⣧⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣠⡄⠀⠀⠉⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣠⡄⠀⠀⠙⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠉⠁⠀⠐⠀⠀⠸⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢠⠤⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠀⠀⠘⢿⣿⡿⠇⠀⠀⠀⠀⠀⠀⣤⢤⣼⠁
⠀⠀⠀⡀⠀⠈⠱⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠡⠆⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⢀⣀
⠀⢂⡒⡐⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣡⠰⠇⠀⠀⠀⡀⠟⠀⠀⠀⠀⠀⠀⠀⠀⢂⡒⡐⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣡⡰⠆⠀⠀⠀⠀⠟
⠀⠀⠀⠑⠀⠀⠀⠠⣄⡄⠀⠀⠀⡯⣑⠀⣘⠃⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠀⠀⠀⢀⣄⡄⠀⠀⠀⣯⣁⠀⣘⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠐⢦⣀⡀⠁⠀⠀⠀⠈⠈⠘⠁⠀⠀⠤⡄⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⡀⠁⠀⠀⠀⠈⠈⠊⠉⠀⠀⠤⡄⠀⠃⢀
⠶⠀⠄⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⢀⠄⠀⠀⠃⡀⠠⠍⠉⠁⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠉⠓⠃⠀⠀⠀⠀⢀⠄⠀⠀⠀⡀⠠⠉⠉⠁
⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈
⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⣀⠤⠆⠀⡴⠉⣉⡉⠃⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⣀⠤⠇⠀⡴⠉⢉⡉⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠀⠀⠀⡅⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠀⠄⠀⡅
⡆⠀⠀⢀⠄⣠⣤⡀⠀⠀⠀⠀⠠⢶⡽⠒⠀⠀⠀⠀⠘⠀⠤⠰⠊⠀⣠⣀⠀⠂⠀⠀⢀⠄⣠⣤⡀⠀⠀⠀⠀⠠⠶⠽⠖⠀⠀⠀⠀⠈⠀⠤⠰⠊⠀⣠⣀
⠃⠀⠀⢸⡀⠃⠀⠃⣴⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⣀⠀⠸⡄⠉⢰⠇⠀⠀⢸⡀⠃⠀⢃⣴⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⡀⠀⠸⡔⠉
⠀⠀⠀⠈⠻⠶⠴⠞⠁⠀⢀⣀⡀⠀⠀⢰⣿⡇⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠻⠶⠴⠞⠁⠀⢀⣀⡀⠀⠀⠀⣿⡇⠀⠀⠀⠨⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⢿⠟
⠀⠀⠀⠤⠄⠀⢀⣀⠀⢸⣶⣶⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠂⠀⠀⠀⠀⠠⠄⠀⠀⡀⠀⢰⣶⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢈⠀⠀⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠈⠲⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠀⠈⢢⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⣠⠀⠈⠹⠀⠤⠒⠀⠀⢰⣤⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠣⡀⢻⡀⠞⠁⡀⠐⣤⣤⡀⠀⠀⢀⣾⠰⣇
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠢⢁⠾⠁⢺⢨⠉⣀⣴⣶⣿⠏⢸⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠿⣿⠿⠟⠛⠉⠀⢈⣀⡈⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⡀⠀⠀⠀⠀⠈⠀⠰⠘⢹⠈⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠒⠀⠀⠀⢀⠀⠐⠁⡐⠈⠂⠀⠀⢈
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠂⠉⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⡐⢩⠀⠀⠀⢸⠀⠀⢀⠀⠀⠘⡀⠀⢀⠀⠀⠀⢐⠶⣿⡛⠊⠁⠈⠉⠉⠉⠁⠂⠁⠀⣀⣄
⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠈⡀⠀⢬⠳⣦⡀⠹⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⣄⢀⣀⡀⠀⢀⡀⡐⠋⠉
⠀⠀⠀⠌⠀⠀⠀⡇⢀⠀⠀⢁⠀⡀⠑⠌⠙⢷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⢈⡀⢀⡾
⠀⠀⢸⠀⠀⠀⠀⠃⠂⠀⠆⠀⢑⠌⠖⠀⠀⠀⠈⢃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢄⠈⣁
⠀⠀⠘⠀⠀⠀⠀⠀⡠⢲⡖⢚⡭⢲⡀⠀⠑⢦⣀⠀⠐⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⡀
⠀⠰⠡⡤⠄⠒⠖⠁⠀⠋⡠⠋⠀⠈⠀⠘⠀⠀⠉⠳⠦⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⠌⡁
⠀⠀⠀⠣⣀⡀⠀⢀⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⡀⣠⣿⡿⢰⣿⡇⠀⢠⣿⣷⠀⢠⣿⣿⡀⢀⣾⣿⡇⢀⣾⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟⠁⢸⣿⡇⠀⢸⣿⡇⢀⣾⣿⣿⣧⣾⣿⣿⡇⢸⣿⡿⢃⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⣇⢀⣾⣿⠃⣼⣿⠏⣿⣿⡿⢻⣿⡇⠘⣫⣶⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠻⣿⣿⡿⠃⠰⣿⡟⠀⠘⠛⠁⠘⣿⡗⢈⣭⣉⣉⣉⠉⠁
⠀⠀⠀⠀⠀⠀⣶⣿⣶⣄⠛⠋⢰⣷⣆⠀⣠⣶⡄⣤⣦⠀⠀⣴⣦⠀⠀⢠⣤⡄⠀⠙⠻⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⢓⣀⠀⢻⣿⣶⣿⡟⠁⣿⣿⠀⠀⣿⣿⠀⢠⣿⣿⣧⠀⣠⣿⣷
⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⠟⠀⠀⣿⣿⡏⠀⠀⣿⣿⠀⢸⣿⡟⢀⣿⣿⣿⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⠿⢋⣴⣿⠀⣿⣿⡇⠀⠀⢿⣿⣤⣿⣿⢁⣾⣿⠃⢸⣿⡿⠋⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠹⠿⠁⠀⠀⠈⠻⠿⠟⠁⠘⠿⠃⠀⠀⠉⠁⠀⢹⣿⠇


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣭⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⡐⠐⢶⠄⠦⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⠈⠡⡗⢤⢳⠄⣤⣱⠉⢨⠕⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠃⢹⣲⠇⠊⠀⠀⣼⣷⣦⣚⣙⡁⠀⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧⡖⣸⣤⡾⠎⠀⠀⠀⠙⢯⣟⡟⣷⣿⣜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠝⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰⠀⠨⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠒⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢺⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⠈⠉⠁⣶⠆⠀⠀⣶⡆⠁⢸⣼⣏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠠⢀⠄⠛⠳⣤⠴⡋⠗⢀⣼⡿⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢹⠗⡄⠇⠀⠀⠘⠀⠀⠁⠀⠘⢿⣷⣜⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠁⡄⢸⠀⠀⠀⠀⠀⢰⠀⡄⣎⣿⣧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣏⠀⢄⠀⣸⠀⠀⠀⠀⠀⠸⡁⠀⠏⣿⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠎⢈⠁⠀⠄⠀⠀⢀⠀⠉⠀⢲⣿⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡔⠛⠃⠀⠂⠀⠀⠐⠀⠄⠀⠀⠘⣯⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠣⢀⣀⣀⣀⠀⠠⠤⠔⠓⠒⠒⠐⠒⠂⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⡿⢿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠋⠁⠀⠀⠀⠀⠉⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠋⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁⠀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⢀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠘⢿⣷⡀
⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣶⣶⣤⣬⣿⣿⣦⡀
⠀⠀⠀⠀⢠⣿⡟⠀⠀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠿⣿⣦⡀
⠀⠀⠀⢠⣿⠏⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⣿⡟⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⣿⣧⣄⣀⣀⣀⣤⣤⣶⣶⣿⠿⠿⠛⠛⢻⣿⡏⠉⠉⠉⠉⠛⠛⢻⣿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⠇
⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⣠⣿⡟⢀⣄⡀⠀⣀⣀⠀⠀⢿⣷⠈⠙⠻⠿⣿⣷⣶⣶⣶⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣾⣿⣿⣷⣼⣿⢿⣿⣶⣼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⠋⠻⠟⠃⢸⣿⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⣀⣀⣀⣤⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠈⠐⠄⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⢀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠀⡐⠀⠀⠀⠀⠀⠀⠐⡀⢀⠀⠀⢠⠀⡈⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⢀⠀⡐⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠈⠀⢰⣧⣷⠀⠀⢰⣮⣷⠀⠘⠀⠀⠀⠀⢼⣥⡧⠀⠀⣠⢠⡐⠄⠀⠀⠀⠀⠈⠀⢰⣧⣶⠀⠀⢠⣴⣦⠁⠰
⠀⠀⠀⠠⠶⠉⠉⠘⠛⠈⠉⠱⠦⠀⠀⠀⠀⠶⠂⠉⠁⠶⠆⠻⠿⡃⠀⠀⠀⠀⠀⠀⠠⠦⠙⠉⠰⠖⠘⠛⢫⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⡄⡆⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢒⣫⡀⠀⣜⡒⠌⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢁⡀
⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⣰⣦⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠠⠟⠁⣠⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⡇⠀⢘⠀⣸⣿⠐⣸⡗⠀⠀⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡀⠻⠏⣸⡟⠀⠀⣀⠀⠀⠒⠶⢶⡆
⠀⠀⠀⠀⠀⠐⠚⢶⡄⠀⠀⠀⢠⣦⠀⠰⠙⢷⣄⠀⠘⣷⠀⡘⠓⠀⠀⢻⠀⠀⢴⡟⠁⠰⣦
⠀⠀⠀⠀⠁⠀⠀⢸⣿⠀⠀⠀⣾⡏⠀⣀⢂⡈⠿⠆⠀⠈⠉⠠⠀⠂⠐⠂⢄⠀⠹⠄⠄⠀⢿⡆
⠀⠀⠀⠀⣆⠀⠀⣾⡇⠀⠀⣶⣽⡀⠀⢻⡌⠛⠂⠀⠀⢠⠂⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠈⠛⠂⣿⡇⠀⠀⠙⣿⠉⠀⠈⢧⡀⠀⠀⠀⠈⠓⠚⠛⠁⠀⠠⠂⠀⠀⠀⠀⠀⣿⠃
⠀⠀⢰⣶⣦⡄⠀⠘⠧⠠⠀⠀⢹⡇⠀⠀⠀⠉⡀⠀⠀⠀⠄⠀⠐⠀⠠⠀⠀⡄⠀⠀⠀⣼⠏
⠀⠀⢀⣠⣤⠄⠀⠀⠀⠀⠀⡀⠸⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⣠⣴⠇⠀⢀⡼⠋
⠀⠀⠸⡿⠃⣴⡞⠀⠀⠐⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⡈⣁⣠⠴⠋
⠀⠀⠀⠀⠀⠉⠁⠀⠁⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢼⣶⣿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢳⠁⣀⡀⡱⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢹⡧⣟⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⣷⠋⠀⣀⣀⣀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⡚⢛⡛⠻⠟⠛⢓⠔⢻⠀⢸⣿⣿⠿⢟⢃⣤⣾⣿⣿⣿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠰⣇⠎⡠⡀⢃⡄⠀⠈⠀⠈⣃⠈⠅⠀⠀⠠⡏⠁⠀⠀⣤⣴⣶⡶⠷⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠂⠀⡀⠐⢻⠀⠏⠦⠬⠤⠿⠧⠶⠉⠁⣦⡏⠦⠤⠦⣠⣤⣥⣤⣄⠐⣨⠒⠠⠦⢠⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⠑⠀⣠⣶⣿⣿⣿⣷⣦⠀⠀⢀⣀⣀⡷⣧⣀⠔⡿⠚⣿⢄⠀⠊⡗⠐⠒⠀⠠⣼⠐⣅⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣿⣿⠀⣠⣵⣿⣿⣿⣷⡜⣿⣴⣾⣿⣿⣿⣿⢩⠏⢰⠁⠀⠈⣿⡂⣤⢣⠶⠿⠄⠀⣿⣏⢉⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⡛⠛⠉⠻⢿⣿⣿⡿⣿⠀⠀⠁⢸⣿⣿⠿⠘⢄⣿⡀⠀⠀⣼⣇⣏⣨⠃⠒⠒⢰⡿⠕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢤⣀⣀⠀⠘⠒⠒⠒⠛⠛⠀⠀⠀⠀⠈⠉⠉⠉⢉⣀⣈⠩⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣺⣿⣿⡏⢻⠿⣶⣶⡖⣶⣖⣶⡾⢿⢻⣿⣷⣿⣶⣶⠶⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⡟⢹⠀⠈⠉⢓⢺⡀⠀⢹⣯⣿⣿⣃⡒⠠⢸⡇⠀⠉⠙⢏⣾⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡠⠚⠓⢺⡒⠦⠠⠾⠎⠉⠁⠈⠀⣌⡇⠀⠉⣉⣉⣑⠒⠠⠸⠟⠛⠿⠋⢩⣉⣉⣭⣽⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠇⠀⠉⢘⠛⢛⣀⠀⠀⠄⢀⣀⡈⠓⢛⡿⣿⣿⣿⣿⠛⢷⣶⠛⠻⠿⣿⠿⠟⢻⣷⣾⠧⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠠⡀⠀⠀⢨⡾⠥⣀⣀⠀⠀⢀⣀⣨⡆⣸⣼⣿⣿⣿⣿⣷⠈⣿⡃⠀⠀⠠⠀⣀⣇⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠭⢍⣛⣉⠓⠤⢀⣀⠈⢉⣁⣀⡽⠿⠛⢿⣿⣿⣿⣿⣿⣾⣋⡈⣁⣂⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒⠒⠲⠶⠶⠶⣦⣤⠦⠤⠉⠑⠒⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠭⠽⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀⡾⠵⠥⠤⠴⢲⡎⢳⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡛⠿⣤⣤⣤⣤⣤⣀⠢⠭⢛⣩⣿⣷⣦⣄⣀⡀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠂⠀⠐⠒⠒⠛⠛⠛⠋⠉

⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⠟⠀⣾⣿⠃⠀⠈⠂⡀⠀⠀⠀⠉⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠈⢻⣿⣿⣿⣿⡷
⠀⠀⢸⣿⣿⣿⣿⠟⠀⣼⣿⣿⣶⣤⣤⣯⢄⠀⠀⢀⡀⢘⣃⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠃⠈⣿⣿⣿⣿⣧
⠀⠀⢸⣿⣿⣿⡏⡀⠘⠻⣿⣿⣿⣿⣿⣿⣦⠄⠀⠀⡷⠿⠟⠀⠀⠀⠀⠀⠀⢃⣤⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⣿⣿⣿⣏
⠀⠀⢸⣿⣿⣿⢇⠁⠀⠀⠈⣽⣿⡏⠁⠉⢻⣾⠀⠸⠂⠀⡀⠀⠀⠀⠀⠀⢀⣨⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡎⣿⣿⣿⣿
⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⣀⣿⣿⡇⠀⠀⠀⠘⡇⠀⠀⠀⢿⡇⠀⠀⠀⠀⢠⣴⠟⠛⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡸⣿⣿⡟
⠀⠀⢸⣿⡟⠁⡀⠀⠀⠀⣿⣿⣿⡁⠀⡀⠀⠀⣧⠀⠀⠘⡞⠒⠒⠀⠀⠀⣿⡏⢰⠀⡀⠀⠀⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡄⠀⢳⣿⣿⡷
⠀⠀⢸⣿⣴⣦⣿⣿⣦⣄⣸⣿⣿⡇⠀⠁⠀⠀⠃⠀⠀⡼⠁⠀⠀⠀⠀⣀⣽⣷⡆⠂⣷⣤⠂⢣⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠘⣿⢹⣷
⠀⠀⢸⣿⣿⠛⠻⡿⠟⠙⢿⣿⡿⠟⣤⡀⠀⠰⣶⣶⡞⠀⠀⢴⠞⠀⠹⣿⠟⢿⣧⢤⣿⣿⡤⣼⣀⡄⠀⠀⠅⠀⠀⠀⠀⠀⡀⡄⠀⡀⢿⣿⡟
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠔⠀⢹⢿⠆⠀⠹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⣿⣿⡧⣼⣿⠁⠀⣤⠀⠀⣴⡂⣀⡀⡈⠀⢀⡀⢸⣿⣿
⠀⠀⢠⡀⠀⠀⠀⠀⠐⠀⠀⣸⢂⡠⠐⠾⠆⠰⢶⣿⣷⣾⣆⠀⠀⠀⠀⠀⠀⢸⣯⡴⠛⠛⠣⠤⠈⠀⠐⠂⠈⠉⠀⠈⠀⠉⢛⣭⣿⣤⣼⣿⡿
⠀⠀⢨⣭⣤⣤⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⢩⡍⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣤⣤⡤⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⣩⠁
⠀⠀⢸⣿⣿⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⢸⣿⣟⡉⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⢸⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡄⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⠁⠀⢀⡀⢠⡇⢿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⢸
⠀⠀⠀⠳⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣬⢁⡰⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈
⠀⠀⢀⠀⠂⠀⠀⠀⢤⣤⣴⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⣿⡟⠛⡾⡇⣷⡄⠀⢀⣀⣴⣶⣶⣦⣤⡴⠃⣠⠚
⠀⠀⢸⡇⢠⠁⠀⠀⣼⣿⡍⠋⠉⠑⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣤⠆⢴⡇⡟⠈⠂⠐⣿⣿⣿⡿⣩⡶⠲⢯
⠀⠀⠘⣿⣄⡀⠙⠛⢿⡛⢻⡆⢰⡀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⢀⠘⡇⠁⠀⠀⠲⢾⣶⣍⣿⣿⢅
⠀⠀⠀⠈⠀⠁⠀⢀⣸⣇⠘⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠈⠡⠾⠇⠀⠀⠀⠀⣩⣿⣿⣿⡋⠈
⠀⠀⠰⠀⢀⣤⣶⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⢀⣠⣴⠾⢿⣿⠈⠋⠁
⠀⠀⢀⣤⣾⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡿⠀⠀⠀⠀⣿⣿⣿⣿⣄⡙
⠀⠀⢸⣿⣿⣿⡿⣿⣶⣿⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠁⠀⠀⣴⠆⣿⣿⣿⣿⣟⣓⣠
⠀⠀⢸⣿⣿⣿⠃⠉⣿⢿⣿⠏⠀⠘⠚⠁⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠈⣴⣿⡆⣻⣿⡿⠁⢉⢿⣧⠚⠁⠀⠀⠲⠒⠒⠒⠤
⠀⠀⢠⣷⣶⡆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶
⠀⠀⢸⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠙⣿⣿⣿⣿⣿⣿

⠀⠀⠸⢫⣶⡟⢀⡀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⡇⣾⡇⠀⠘⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠘⠃⠀⠀⢀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⡼⣿⣿⣿⣿⣿
⠀⠀⢰⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢠⣬⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⣻⣿⣿⣿
⠀⠀⢸⣿⡏⣼⡏⢀⠀⠀⡀⠀⠀⠀⣈⡀⠀⠀⠀⠠⠘⣤⣄⠀⢸⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢻⣿
⠀⠀⢸⣿⣾⣿⠀⡾⠀⣴⡇⠀⣴⡈⣙⣻⡆⠀⠀⠀⢀⠉⢟⠀⣸⣿⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⡀⢻
⠀⠀⢸⣿⣿⣿⣶⣥⣾⣿⣷⣤⣸⣿⣿⡉⠀⠀⠀⠉⠀⢐⣿⣎⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠋
⠀⠀⠀⠹⠿⢿⣿⣿⣿⣿⣿⠿⢿⣿⣿⠏⣇⠀⠀⠠⠀⠂⠻⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⣿⣿⣷⣿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢧
⠀⠀⠀⠀⠀⢀⣤⣾⡟⠋⠄⠐⢛⣿⡿⣆⢉⡀⠀⠀⠀⣀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⡟⠃⠐⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠖⠒⠈⠉⠁⢀⣈⣢⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⡶⠿⠋⣉⣭⠶⠛⣋⣴⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡏⠀⠈⠉⠛⢛⠟⠛⠭⢠⡄
⠀⠀⠀⠀⠀⢀⣀⠤⠴⠒⣢⣀⠀⠀⣾⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⡟⠓⠒⠦⢤⣄⡀
⠀⠀⢀⠴⠊⠉⠀⢔⣮⡅⠻⢮⣿⣿⣿⡶⢤⣤⣇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣌⣬⣵⢦⠄⠀⠉⠑⠦⡀
⠀⠀⠀⠀⠀⣶⣾⣿⣭⠵⠶⠛⠋⣉⣉⣿⠿⠿⢿⣯⣯⣷⣒⣶⣧⣄⣀⣀⣿⠛⠉⠁
⠀⠀⠈⠢⢄⡀⠀⠈⠐⠀⠀⢴⣿⠿⠖⠒⠛⣩⣽⣛⡥⠴⠟⠛⠙⠛⠓⠒⠋⠀⠀⠀⠀⠀⣀⡠⠖⠁
⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⠤⣤⣀⣀⣀⣀⣀⠈⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡤⠤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣳⠘⠁⠀⠀⠠⠂⢠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⠀⠀⠀⠑⠀⠀⠀⣷⣶⣾⡓⠀⠀⢡
⠀⠀⡀⠄⠀⠀⠛⢀⡌⠀⡴⢁⠔⠉⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠴⢂⡀⠀⠴⠖⠈⠈⢹⡿⠏
⠀⠀⠀⠀⣘⣿⣀⠀⠀⠈⠐⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⣸⠃
⠀⠀⠀⠀⣿⠟⠻⡁⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠈⡇⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⣐⣿⣧
⡀⠀⢀⣌⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠙
⠿⣿⣿⣍⠀⠀⠘
⠀⢨⡿⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠆⠤⣄⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠑⠲⣀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣈⡂⢇
⠀⣴⣄⢀⡠⠄⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈
⠉⢹⣿⣿⠉⠀⢤⠀⠀⠀⣀⠔⢁⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⠀⠀⠀⣶⣤⣴⠯
⣠⡾⠋⠁⠀⠀⠐⠁⢀⡔⠡⠠⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠙⣿⣿⠆
⠁⠀⠀⠀⠀⠂⣺⣶⣾⡋⠁⠉⢖⠒⠂⠐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠉⠁⠀⢹⡟⠁⠀⠀⡀⠀⢀⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⣶⣶⡆⢀⡀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⢤⣀⠀⠀⡔⠋⠱⠤⢶⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢠⠒⣄⠀⠀⢀⠚⠠⠂⢀⠤⣄⠓⠛⢢⡑⢄⣆⠀⡀⡱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠑⣄⠀⡴⢶⠲⡒⠒⡖⡚⢲⠲⣆⣠⠙⠚⡗
⠀⠱⡤⡑⠼⠟⢍⡑⣄⡀⠘⠂⠊⠀⢤⡠⣵⠋⠁⢌⣀⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢠⣍⣘⢃⠀⢣⢸⠀⢰⢀⠀⡇⣿⣁⠀⠘⢆
⢀⠞⠀⠀⠀⠀⣠⠞⠳⠥⠀⠀⠀⢇⣠⠇⢸⡧⣵⠀⢠⠒⡾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠀⢀⡀⢹⣤⠂⢸⢀⠀⢸⡄⢠⢇⡿⢹⣱⠒⠋
⢇⠀⢀⡴⢾⡉⠙⠢⣄⣔⡇⠈⡂⢀⠤⣤⠋⠧⣈⣁⡜⠀⠀⠀⠀⢀⡀⠤⡤⠤⠤⢤⣤⣤⣀⡀⠀⠈⠓⢻⠃⣧⠜⠋⠈⠒⠞⠚⠒⠛⠛⠉⠉⠀⠀⠉
⠀⠉⠉⠀⠀⠑⠤⠴⠛⠾⣌⣉⣠⡧⠞⠁⠀⠀⠀⠀⠀⠀⠀⢠⡎⠑⠠⠀⣌⡶⡷⠷⢶⡀⠤⣍⣣⠀⠀⠘⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⡒⠲⠤⠤⡶⠤⠤⠼⠖⠚⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢁⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠉⠙⠲⢄⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢸⠀⠀⠀⡇⠀⠀⢀⡄⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠒⡩⠤⢕⢤
⠀⠀⠀⠀⠀⡰⠲⡋⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠈⠐⠚⠠⠤⠤⠧⠤⠤⠚⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢡⠀⡠⢀⠑⠤⠚⠀⣣
⠀⠀⠀⠀⠀⣧⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠒⠁⠀⠢⠴⠃⢀⣀⠸⣄⡆
⠀⠀⠀⠀⠀⢏⠳⣄⠈⠓⠤⣀⡀⠀⠀⢀⣰⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠐⠒⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠠⡄⠀⠐⢄⠸⣄⡨⠃⡠⠁
⠀⠀⠀⠀⠀⠘⢦⠀⠑⠢⠤⣀⣉⡉⢉⢉⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁⠀⠀⠀⠀⣀⡴⠂⠈⠉⠑⠢⠤⣀⠀⠀⠈⠣⣆⠀⠁⡁⠄⣀⣰⠻⠥⣀
⠀⠀⠀⠀⠀⠀⠀⢹⠦⣀⡀⠀⠈⣉⡡⠟⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠂⠁⠀⠒⠀⠤⣠⠀⠈⠁⠀⠀⠀⠀⣀⠄⠊⠉⡄⠀⠀⠀⠀⠈⠈⡑⢟⡉⢠⠻⡖⠊
⠀⠀⠀⠀⠀⠀⠀⡼⠀⡰⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣎⣁⡘⠊⢇⡀⠀⠀⠀⠂⠀⠀⠀⠉⠓⢂⠔⠊⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠉⠁⠈⠙⡀⠸⡄
⠀⠀⠀⠀⠀⠀⣰⠃⡰⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⠀⠀⠈⠉⢀⠿⣓⣦⣀⠀⠀⢀⣤⣀⠂⠀⢸⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠱⡀
⠀⠀⠀⠀⠀⣰⠃⣰⠃⠀⠀⠀⠀⠀⠀⠀⣞⡋⠤⡀⠀⠀⠀⠀⠀⠔⠅⠈⠀⠀⠀⢉⡹⠓⡄⠉⡆⠀⣸⠗⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠱⣄⢑
⠀⠀⠀⠀⢰⠃⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠙⠲⢬⠑⠒⠒⢮⠒⠉⠓⢦⠒⠒⠲⣤⠐⠀⢠⠀⡾⠊⠀⠀⠀⠀⠀⣠⠔⠛⠉⡡⢈⠩⠛⢦⡀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠘⠲⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣆⠠⣀⣔⠠⠤⢄⣍⣀⠀⣀⣸⠀⡠⠞⠉⠁⠀⠀⠀⠀⢠⠞⠁⠀⠀⢀⣀⣀⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⢏⠠⡀⢀⡴⢋⢀⢔⢹⡏⠩⢑⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠑⠚⡟⠀⠰⡉⢁⣼⢅⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠤⣼⠗⠀⠈⠲⡀⠀⠀⠀⠀⠀⠀⠈⢷⡤⢀⣷⣀⠴⡻⠉⠈⠎⣆⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⠚⠉⠙⠢⡈⢆⠀⠈⣷⡖⠀⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣄⣙⣺⣤⡤⠴⡟⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⠐⠚⠉⠁⢄⠙⢆⠀⠀⠀⢹⠈⣄⡴⠙⠛⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⢆
⠀⠀⠀⠀⡀⠠⣀⡀⠀⠀⠔⠉⢍⡉⠲⣌⠢⠀⠀⠀⠑⠘⣀⣠⠤⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠳⡄
⠀⠀⠀⠐⡀⠀⠸⡈⠑⠚⠒⢀⡀⠱⠀⠈⢆⣷⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠘⢆
⠀⠀⠀⠀⡝⠉⠀⡨⠛⢳⠔⠣⠤⠤⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠘
⠀⠀⠀⠀⢇⠀⣸⠽⠛⠉

⠀⠀⠀⠀⠀⠀⢀⠖⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⣰⠲⡄⠀⠀⠀⠀⠀⠀⠀⢀⣞⡅⠀⠀⠀⠀⠀⠀⠀⡴⠂⠒⢦
⠀⠀⠀⠀⠀⠀⢮⣀⣄⣄⡼⠀⠀⠀⠀⠀⠀⡰⠣⠭⢽⡀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠦⡀⠀⠀⠀⠀⢀⠷⠔⠒⢺⡄
⠀⠀⠀⠀⠀⠀⠈⢟⠢⡱⠁⠀⠀⠀⠀⠀⠀⣗⢒⠒⢒⡇⠀⠀⠀⠀⠪⣀⢤⣀⡠⣀⡼⠀⠀⠀⠀⠸⡀⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠃⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⢸⠁⠀⠀⠀⠀⠀⠸⣴⣤⣦⠇⠀⠀⠀⠀⠀⠀⠈⢹⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⢠⢾⣱⠦⡀⠀⠀⠀⠀⠀⠀⢀⡖
⠀⠀⠀⠀⠀⢀⡤⠒⠒⠒⠢⡀⠀⠀⠀⠀⢀⡠⠚⣧⣼⣦⠀⠀⠀⠀⠀⡰⠁⠉⠉⡖⠁⠀⠀⠀⠀⠀⡴⠷⠞⢄
⠀⠀⠀⠀⢰⣫⢤⣀⠤⣄⡠⢌⣲⠀⠀⠀⣏⣩⣭⣤⣤⣤⡇⠀⠀⠀⠀⠳⡒⠒⢲⠞⠀⠀⠀⠀⢀⣠⣏⣙⣉⣹⣄⡀
⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠊⠀⠀⠀⠀⠯⠭⠭⠭⠤⠴⠃⠀⠀⠀⠀⠀⢹⢻⠏⠀⠀⠀⠀⠀⠈⠑⠒⣢⣖⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠽⠦⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⠤⠶⠭⡉⠑⣄⠀⠀⠀⠀⢰⠣⢢⢔⢮⢣⠀⠀⠀⠀⠀⡔⠉⠀⠈⠱⡄⠀⠀⠀⠀⢸⣿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⡏⡞⡉⠳⡈⡆⢸⠀⠀⠀⠀⠸⡰⣐⣾⣮⡾⠀⠀⠀⠀⢰⠳⡖⢶⢲⡖⣷⠀⠀⠀⠀⢸⣿⣿⡛⢻⡇
⠀⠀⠀⠀⠀⠱⣱⢌⣒⣁⠷⠊⠀⠀⠀⠀⠀⠉⢲⣿⠃⠀⠀⠀⠀⠀⠈⣇⣧⢸⢸⢧⡏⠀⠀⠀⠀⢸⠋⠛⠻⠿⣯
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡔⠋⠀⢙⠲⡀⠀⠀⠀⠀⠀⠀⢀⣠⡓⢤⠀⠀⠀⠀⠀⠀⡞⡁⡙⡆⠀⠀⠀⠀⠀⠀⣀⣄⣤⣀
⠀⠀⠀⠀⠀⡜⠡⠆⠀⠈⠀⢳⠀⠀⠀⠀⠀⡴⡏⢣⠈⡞⠁⠀⠀⠀⠀⡖⠚⠈⠘⠒⡆⠀⠀⠀⣰⡎⢱⡀⢀⠎⢑⡆
⠀⠀⠀⠀⠀⢸⣀⠀⠀⠙⢀⠎⠀⠀⠀⠐⣮⠣⣸⡤⠋⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⢏⠀⠀⠀⠀⠙⢌⡏⠙⠋⢹⣡⠃
⠀⠀⠀⠀⠀⠀⠈⠙⣾⠉⠁⠀⠀⠀⠀⠀⠈⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠯⠤⠋⠳⠬⠇

⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⡒⠒⢄
⠀⠀⠀⢠⠒⡉⢀⠠⠈⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔
⠀⠀⠀⡃⠀⠀⠠⠀⠈⠀⠄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⣰⢦
⠀⠀⠀⠰⠀⠂⠀⠀⠐⠀⠄⢠⠀⢰⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⢇⠀⠉⢀⣇
⠀⠀⠀⠀⢣⠀⢰⠉⠂⠚⠀⠀⠃⠔⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠉⠉⠁⠈⠣⡀
⠀⠀⠀⠀⠀⢆⡸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⣀⡴⡆⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢔⠞⠝⠀⠣⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⢠⠊⠠⠃⡌⠀⠀⠣⡈⠁⠒⢽⢄⡠⡀⣀⡀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⣀⣤⠁⠀⠃⠀⡇⠀⠀⠀⠘⡄⠀⠀⠣⢕⠁⡇⠓⢖⠊⡦⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠈⠩⠀⡸⠆⣀⢃⠀⠐⣄⠀⠀⢀⣄⣀⠴⣗⢤⠭⠑⢆⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠬⠉⡾⠥⢓⣄⡠⠢⣉⠔⠫⣉⠜⢅⠔⢀⣀⢴⡤⠵⢤⣠⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠂⢹⠒⠆⠨⠇⠀⢠⠀⠀⡄⠀⠰⠀⢠⡡⠔⠈⠉⠀⠀⠀⠉⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⡆⠘⡀⠸⠀⠈⡀⠀⠄⠀⡄⢀⠊⠀⢀⠀⢠⠦⣔⡑⠤⡀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣘⠀⢀⡀⢸⡤⠓⠠⣇⠀⠃⠀⠀⠀⠇⡆⠠⠀⡀⠱⡋⠀⠀⠀⡈⠃⠀⢀⡀⢐
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠙⠀⠈⠂⠀⠁⠀⠀⢸⡀⠐⠀⠀⡐⠀⣧⠲⠠⢄⠀⠈⠙⠉⠁⠀⠀⡠⠁⢈⠇
⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⢀⠇⣀⠀⣀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠓⠊⠑⠤⠊⠉⠈⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠠⠤⠄⠒⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢔⣠⣾⣿⣿⣿⣿⣿⣧⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⣁⣥⠖⠛⠛⠛⠿⣿⣿⣿⣿⣿⣷⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡗⣒⣂⡀⠀⠀⠉⠙⠛⢿⣿⣿⣿⣧⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢋⣹⣿⣿⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣆⣀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡄⠀⠀⠀⠀⣀⡀⠀⠀⠸⣿⣿⣿⣿⣆⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⢰⣯⣥⣾⠃⠀⢀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡿⣿⣿⣿⣿⣧⡀⠀⠈⠈⠉⠀⢠⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠹⣿⣿⣿⣿⣿⣷⡶⠆⠀⠐⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣹⡄⣈⣛⣛⣛⣛⠻⢫⡄⠀⠼⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠟⠉⠙⣿⣿⣿⣿⣿⡄⠀⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣆⠘⢿⣿⣿⣿⣿⣿⣧⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠑⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆⠘⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡽⣿⣿⣿⣿⣿⣆⠈⣻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⣾⠟⠛⠟⠚⠛⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠁⠊⠁⠀⠀⠀⠀⠺⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠫⠛⠁⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡄⠀⠀⠀⠀⠙⠛⠛⠉⠉⠉⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⢀⠠⠀⠓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋⠉⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠈⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠁⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀⠀⠀⠀⢀⣦⡀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠠⠀⠂⠀⠈⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡇⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⢿⣿⣿⡿⠋⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠠⠟⠋⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠄⠀⠀⠀⠀⠐⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡆⠀⠀⠀⢠⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⠐⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀⠀⠀⢀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⢠⡟
⠀⡦⠶⠾⠤⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⡘⠁⠀⠀⢠⣌⠆⠀⠁⠀⠀⠀⠖⠈⠙⠉⠉⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⠀⢠⡈⠀⣀⠁⠀⠀⠀⠀⠈⠂⠀⠈⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢃⠀⠀⠀⠀⠀⠀⠄⠃

⠀⠀⠀⠀⠠
⠀⠀⠀⠈⢀⠀⠀⠀⠀⢰⠉⠁⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠤⠖⣆⠀⠀⠀⠑⠀⢂⠐⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠸⠄⠀⠀⠀⠀⠈⠄⠀⠠⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠠⠆⠈⣁⣈⡉⠂⠀⠀⠐⣶⡀⢱⣮⠄⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠐⠈⠠⢻⠍⠀⠀⠀⠀⠂⠀⠨⠙⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣊⣄⡤⠄
⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⣸⠀⠀⠀⠀⠀⠀⠠⢀⢀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⡀⠀⠀⠀⢀⡴⠆⠄⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠈⠀⢀⣀⣤⣀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠊⢁⠀⡀⠁⡄
⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠊⠀⠀⠀⠉⠰

⠀⠀⢀⢀⡀
⠀⠀⠸⣿⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡶
⠀⢐⠀⢫⠀⢀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠛⠁⠀⠀⢠⠀⠀⠀⢠⡀
⠀⠸⠀⠸⠀⠸⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠃⠀⠀⠀⠀⠀⣀⠀⠀⠶⠞
⠀⠀⠄⠘⠀⠀⠀⠈⠁⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⢤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⠀⠀⣴⠛⠛⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠔⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠤⢾⠆⠀⣿⠤⠄⠀⠀⢸⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠸⠁⠀⢷⣤⣤⡄⠀⠸⣤⣤⡤⠀⠀⢧⣤⣤⠀⠀⠘⢦⣤⠜⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⣤⢤⣄⡈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠐⠊⣉⢤⣄⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣞⠀⠀⠀⠙⠳⣄⠈⠳⣄⠀⢀⣀⣀⣀⣀⡀⢀⡤⠚⢉⡤⠖⠋⠁⠀⠈⡣⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠧⠀⠀⠀⠀⠀⣠⠗⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠀⠰⣉⠀⠀⠀⠀⠀⢀⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠰⡆⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⡼⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠲⣀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⣸⠃⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡤⠝⠦⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⠴⠃⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⠦⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠊⠉⠀⠒⠒⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣻⣽⣯⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠠
⠀⠀⠦⢄⢀⠘⠂⠀⠀⠐⠀⣈⡭⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣤⢐⣢⡄⠀⠀⠀⠐⠀⢠
⢀⠀⠀⠀⣙⡀⠈⠀⠀⠀⠂⠀⠀⠀⠃⠀⠈⠀⠁⠀⠀⠀⠀⢚⡩⣥⡶⡇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠒⣒⡀
⠙⠛⠤⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠀⠐⠀⠀⢀⠁⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣉⠀⠐⠤⢐⡀⢠⡆⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡿⠀⠀⠀⠀⠀⠂⡤⠀⠀⠀⠀⠉⠛⠓⠲⠾⠭⠦⠀⠀⠘⠤⢄⣀⣀⡇
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⢀⢀
⠀⠀⠘⠀⠀⢈⡁⠀⠩⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣟⣿⣿⡂⠀⢦
⠀⠀⠁⠀⠸⠋⠙⢛⠛⢤⠀⠀⢂⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠘⠛⠉⠁⠀⠀⠀⢸
⠀⠀⠀⡠⢤⣮⣆⣠⠤⡎⠀⢀⠁⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠃⠀⠀⢃
⠀⠄⠀⡇⠀⠀⠀⠈⢉⢲⠄⠀⠒⠠⠰⠖⠁⠀⠀⠀⠀⣠⠀⡀⠀⠄⠀⢀⠀⠀⠀⠐⡀⢸⠀⠀⠒⠀⠈⠆
⠀⠀⠀⣷⣀⣀⣀⡀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠢⢌⣉⠙⠰⠂⠦⢤⡀⠀⠀⠉⢺⠀⠀⠀⠀⠀⠀⣀
⠀⠠⣀⡍⠠⠄⣀⣉⡙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣂⠱⠂⠀⠉⢕⣒⡒⠡⢬⠀⠀⠀⠀⠱⠶⠶⠶⠒⠛⠉
⠀⠀⠀⠈⠉⠛⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠴⠬⣄⣉⣡⡏

⠀⢀⠔⠀⠉⠉⠑⠒⢄⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠁⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠈⠉⠀⠒⢄⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠉⠐⠢⡀
⡰⠁⢰⡱⠀⠀⢎⡡⠀⢣⠀⠀⠀⢀⠊⠀⠀⡦⠀⢠⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢠⠃⠠⣤⠄⠀⢤⡤⠀⠱⡀⠀⠀⠀⡜⠀⡞⠒⡄⢰⠒⠢⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢈⠀⡀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⢰⠀⠀⠈⠉⠀⠀⠉⠁
⢡⠐⣇⠀⠀⠀⠀⡠⠓⡐⠀⠀⠀⠘⡄⠣⣀⣤⣤⣤⣄⡔⢀⠇⠀⠀⠀⠀⠀⠀⠸⡀⠀⢤⣤⠄⠀⠀⠀⢠⠃⠀⠀⠀⢆⠀⢀⣴⣿⣿⣆
⠀⢹⣿⣟⠂⠐⢈⠠⠊⠀⠀⠀⠀⠀⠈⠢⢈⣙⠛⠛⣉⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣘⠚⠀⢀⡠⠔⠁⠀⠀⠀⠀⠀⠓⢌⡙⠟⣿⣉⠠⠊
⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⡠⠂⠁⠀⠀⠈⠑⢢⡀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠉⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠉⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⡔⠊⠁⠀⠀⠉⠑⢄
⡜⠀⠠⠤⠀⠀⠠⠄⠀⠱⠀⠀⠀⢠⠃⠀⠒⠂⠀⠀⠒⠀⠈⡆⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠃⠀⠪⠀⠀⠘⡄⠀⠀⠀⠎⠀⠐⠃⠀⠀⠚
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠸
⠰⡀⠢⡀⠀⢰⣿⡦⢂⡜⠀⠀⠀⠈⢆⢰⣄⡀⠀⠀⡠⠆⡠⠃⠀⠀⠀⠀⠀⠀⠘⢄⠀⠓⠤⠒⠤⠒⠀⡰⠁⠀⠀⠀⢣⡀⢐⡶⡖⠒⠒⠂⢀
⠀⠈⠂⠄⣉⣉⠡⠔⠊⠀⠀⠀⠀⠀⠀⢿⡿⠍⣉⡁⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⢀⣀⡀⠤⠊⠀⠀⠀⠀⠀⠀⠑⠢⠝⣁⣀⠠⠔⠁

⠀⠀⢀⡀⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠤⢀⡀
⢀⠔⠁⠀⠀⠀⠀⠈⠑⡄⠀⠀⠀⠀⡠⠊⡀⡀⠀⢀⢀⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⢀⠊⠁⠀⠀⠀⠀⠈⠢
⡌⠀⠀⠁⠀⠀⠈⣡⠀⠘⡀⠀⠀⢰⠁⠀⠓⠂⠀⠀⠒⠁⠀⠇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠀⠀⠀⠀⠀⠈⡆⠀⠀⢀⠃⠀⠐⠒⠀⠀⠒⠂
⠇⢀⡀⢀⣀⠀⠀⠀⣀⢀⠃⠀⠀⢠⠀⢀⡀⣀⣀⣀⣀⣀⠀⡄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡀⠀⠀⠀⡆⠀⠀⠈⡀
⠘⢄⠑⠾⣿⣀⡠⠖⢡⠎⠀⠀⠀⠀⠣⡈⠢⢼⣿⣧⠤⢋⡔⠁⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠠⠛⠄⠀⢀⠜⠀⠀⠀⠀⠑⡄⠀⠈⠻⠟⠁⠀⡠
⠀⠀⠑⠒⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠈⠒⠂⠉⠁⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⠈⠑⠂⠠⠀⠐⠊



⠀⠀⠀⠀⠀⢦⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⣷⠃⠀⡇⠀⣶⠀⡟⠉⡗⠉⡇⠀⡏⠉⡖⠉⡇⠸⡄⣰⠀⠀⠀⡞⠉⢶⠀⡗⠉⠀⠀⢸⠈⢧⣸⠀⣰⠉⠹⡄⢹⠉
⠀⠀⠀⠀⠀⠀⠻⠀⠀⠳⠤⠟⠀⠇⠀⠇⠀⠃⠀⠇⠀⠇⠀⠇⠀⢹⠃⠀⠀⠀⠳⠤⠞⠀⠃⠀⠀⠀⠸⠀⠀⠙⠀⠘⠦⠜⠁⠘⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂⣀⣀⣀⡀⠀⣀⣠⣔⢂⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣻⣿⣿⣿⣿⣿⣿⢿⣿⡤⢐⡀⣀⠀⠀⠀⠄⠈⢷⡀⠈⢮⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣧⣼⣿⢸⣿⠶⠟⠛⠋⠀⠀⠀⠀⠀⠀⠙⢷⣾⣧⣠⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣚⣿⡿⣻⣿⣧⡈⢷⣿⣷⣔⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣽⣸⡿⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣿⠿⠉⠛⢿⣿⣾⢿⠉⠋⠛⠿⠿⠤⠀⠀⠀⠀⠀⠑⢹⣿⣿⣿⣿⣿⡿⠛⠛⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡎⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢛⣹⣿⣿⣧⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣀⣠⣾⣷⡿⢟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢄⠀⠀⠀⠀⠀⠀⠀⠈⣉⣽⠿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⡀⣀⣀⠀⠀⠈⠀⣈⡴⣞⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⢿⡏⠛⠑⣲⠚⠻⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢈⡘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰
⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡄
⠀⠀⠁⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⡄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⣿⣼⡇⢸⣿⣧⠙⣿⠃⢸⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⢹⡇⢸⣯⣿⠀⣿⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠛⠘⠃⠙⠁⠛⠁⠋⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⠀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⣤⡴⠆⣶⢸⡆⢿⣟⠀⣿⢹⡇⢹⣿⡏⢸⡏⣿⢸⣷⣾⢰⡶⠆⠀⢀⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣇⠠⠈⣿⠀⢹⣿⡇⢸⡏⠀⣿⢿⡇⠀⣿⠁⢸⣇⣿⢸⣿⣿⣸⠋⠀⠀⠛⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠨⡇⠀⠙⠶⠀⠻⠃⠘⠛⠃⠛⠈⠋⠀⠙⠀⠈⠙⠋⠘⠋⠃⠛⠃⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣐⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠦⠒⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠀⣖⠀⠀⠀⠀⣀⠀⠀⡀⢐⠀⠥⠒⠉⠀⠀⢀⡀⢠⠜⢲⡀

⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠊⠀⢴⣾⣿⣿⡄⠈⠂⠙⠛⠀⠀⠐⠀⠀⠀⠀⠎⠀⠀⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠋⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣐⣀⣀⠀⠀⠀⠀⠀⠀⠋⡿⠃⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⣼
⠀⡠⠒⠃⠀⠒⢀⠀⠀⠀⠀⠀⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢄⢀⣾⣟
⠞⠀⠀⠀⠀⠀⠈⠁⡄⠀⠀⠀⠉⠉⠤⠤⠤⠄⢀⡴⠒⠒⠒⠠⡀⠀⠀⠈⠃⡀⠀⠀⠀⠄⠠⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢀⣀⣶⣿⡟⢉
⠂⠒⠂⠐⠄⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⣄⡀⢈⢢⠤⢤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣷⣷⣿⣿⣷⣾⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠐⠒⠄⣀⠀⠀⠀⠀⠀⠀⠀⠉⠩⢯⣿⣿⢿⣿⣿⣯
⠷⠄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘⠈⠃⠀⠀⠀⡠⠒⠀⠀⠐⢉⣿⡿⠄⠉⠏⠛
⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢵⣤⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠰⢄⡀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣿⣆⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣶⠇
⠀⠀⠀⠀⠈⡟⠒⠠⠀⠀⠀⠣⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠉⠉⠙⠧⠀⢀⠀⠀⠀⠒⠤⢤⣿⣿⣿⡇⠀⣿⣇⣠⣴⢅⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠀⠠⣤⡼⣿⣿⣿⣸⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢀⢈⡄⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠠⠀⠀⠀⠑⠀⢀⡎⠀⠄⠺⠃
⠀⠚⠀⢤⡀⠀⠀⠀⠀⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⢔⣾⠿⠛⢿⡇⠐⠦⠀⠀⢀⣀⡀⠀⠀⠀⠃⠀⠀⠤⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀
⠀⠈⠐⠛⢧⡀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠄⣀⣀⣀⡀⠤⢰⠞⠋⡠⠀⡀⠀⠙⣄⠀⠀⠀⠀⠀⠉⠁⠀⠊⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠁⠀⠀⠀⠀⠀⠀⠀⠸⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠖⠒⠻⣷⢶⣷⠀⡀⠀⠀⠀⠀⠀⠀⠄
⠔⠋⠀⠀⠀⠀⠀⠀⠀⠂⠣⣄⣀⡄⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠄⠀⠀⠀⠀⠀⠁⠉⠉⠉⠚⢆⠀⠀⠀⠀⢀⠀⠀⠀⣠
⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠉⠘⠈⠁⠁⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠀⢸⣇⣸⣀⣠⣾⠿
⠀⠀⠀⢀⡇⠉⠉⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢱⣄⣆⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⣽⣧⣾⣿⣿⡿⠿⠃
⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠀⢿⣿⣆⡀⠀⡜⠁⠀⠀⢀⣴⣞⡥⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢿⡏⡁⠀⠀⠀⠐
⠀⠀⠀⡀⠠⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠀⠀⢤⣀⠀⠀⠀⠀⢻⣿⣷⣀⣠⡄⠀⣤⣿⣿⠟⢀⡀⠀⠀⠀⠀⠢⡄⢀⠍⠀⠂⠉⠷⣁⠂⠤
⠒⠀⡉⠁⠀⠈⠀⠀⠐⠄⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠈⠙⢿⣿⡛⠃⢰⣿⣿⣩⠆⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠉⠓⠒
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣿⣿⣶⣶⣉⡄⠓⢦⣹⣿⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠁⠀⠀⠀⠒⠋⠉⠉⠈⠞⠛⠙⠋⢻⡿⢿⠅⣠⣆⠋⣼⣿⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣤⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠈⣿⣥⠽⠬⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠙⠻⢶⣴⠿⢿⣄⠀⠀⠁⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⣶⣦⣶⡞⠁⠀⠀⢈⣷⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠖⠋⠉⠻⢿⣧⣬⣁⣠⣼⣿⠁⢀⣀⣸⡃
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠴⠉⠩⠭⢌⢻⣿⣿⡿⢿⡉⢉⣿⣃
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠒⠖⠃⠀⠀⠀⠀⠀⠁⣗⣀⣶⡆⢛⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⣸⠂⠀⠀⠀⠀⠀⠠⣿⠅⣿⠇⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠤⠤⠄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⣿⣆⠀⠀⠀⠀⠀⢀⣽⣸⡟⢠⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣠⣤⠴⢒⣾⡷⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣤⡙⣿⣿⣤⣤⣀⣴⣿⣿⣿⣵⠿⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⠋⠁⠒⠻⡿⠿⢆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⠛⣿⣿⣿⣿⣿⣿⠿⠛⠑⠉⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⣄⡀⠀⣾⠀⠀⢸⠀⠀⠀⠀⣇⠀⠀⠀⡗⠒⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣗⢶⡖⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⣴⣷⡄⢛⣿⣿⣿⣖⣦⣼⣀⡀⠀⠀⢹⠀⠀⠀⣥⣾⡗⠆⠀⠉⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠻⠿⢟⠛⠉⣉⣤⣶⣛⣻⡿⣿⠿⢿⣿⣿⣶⣦⣤⡏⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠎⣸⢀⣆⠠⣤⣀⡀⠀⠀⠀⠀⠀⠈⠒⠰⠤⢄⣀⣀⠙⠉⠁⠀⠐⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⣀⣠⠤⠄⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⣴⣯⣀⢨⣦⢈⣁⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣠⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⢿⣖⠲⣶⡤⣀⡈⠛⠛⣩⣟⣛⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠺⠾⢇⣀⣿⣶⡙⢦⠛⣿⡶⢆⠉⠉⠁⠀⡈⠳⣄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠄⠁⠀⠀⠀⠀⠀⠀⠙⢀⣬⣏⣆⣀⡠⠥⣦⠷⣦⡄⢠⠀⠀⢽⡇⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⣁⣤⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣨⣟⣁⡙⣿⣳⢶⣿⢻⠘⠀⠀⢸⡇⠠⡅⠁⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠘⠿⠿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣣⣭⣿⠛⠿⢾⣿⣦⣭⡿⢀⡁⣰⣿⣿⣿⣧⣌⠁⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⣂⣾⣿⣿⡶⠶⣞⣿⠷⠉⢙⣲⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠈⠀⠸⠿⣓⡲⠶⠶⠾⠟⠻⠉⠉⢠⣶⡿⠿⠷⠞⠛⠉⠉⠉⠉⠉⠉⠉⠁⠀⢀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠄⣀⣀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠔⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠐⠒⠒⠂⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠠⣒⣺⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠁⠘⡦⠀⠦⣤⣔⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡆⠛⠫⣿⢹⣼⣿⠅⠀⠀⠀⠀⠀⡓⠀⠘⢦⣀⡄⢫⠙⠒⠶⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⣐⣀⣦⣤⣈⣡⢿⣬⡀⠀⠀⠀⠀⠃⢐⠛⠘⣿⠋⠙⣳⣤⣿⡥⠐⠚⡆⠀⠀⠀⢤⣀⣀⠔⢞⠀⠁⠀⢙⡆
⠀⠀⠀⠀⠀⠩⡀⠁⠀⠁⡡⠄⠀⠁⠀⠀⠀⠈⠓⠤⢆⣴⣛⠶⠖⠃⠉⠀⠀⠀⠀⡇⠀⠀⢠⢲⠊⡁⠐⠀⣠⡀⢢⣼⡷⣀
⠀⠀⠀⠀⢺⡀⣀⡍⣠⣄⣁⣽⡀⠁⠀⠀⠸⠤⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣨⣛⡁⠀⠈⠋⠀⢁⣈⣤⡴⠿⠂
⠀⠀⠀⠀⠀⣏⠉⠁⠉⠀⠉⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠋⠀⠀⠀⢠⠀⠀⠉⠉⠁⠀⠈⠀⠀⢀⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣣⣤⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠉⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢰⠀⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⣶⣄⣀⣀⣀⣀⣤⣼
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢛⣛⡛⠻⢟⣛⠕
⠀⠀⠀⠀⠀⠀⠀⣥⠀⠁⠀⠀⠀⠀⠀⢀⠆⠀⠀⢀⠀⠀⢠⠀⠀⠀⡀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠢⣌⢣⣀⢸⡀⢠⣴⠁⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⢙⠀⠀⠉⠉⠁⠉⠉⠙⠧⣀⡠⠊⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⢀⡀⠀⠤⠒⠂⠐⠀⠠⠤⢄⠀⠈⠢⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠲⢒⢶⣧⠀⠀⠀⡀
⠀⠀⠀⠠⠀⠀⠀⠠⡄⠀⢀⡄⢠⡄⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠄⣥⣮⣽⣿⠿⢀⠞⠋⠔⠡
⠀⠀⠀⡆⠠⠀⠚⠄⣸⣦⣾⢤⠖⢀⠆⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠂⠀⣄⡈⠛⠛⠋⠉⢉⡉⠙⣁⣔⣀⣀⣀⡀
⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠀⠉⠙⠋⠁⠉⠉
⠀⠀⠀⠂⠉⠢⠖⠐⠂⠄⠒⠀⠤⣄⠏⡴⠀⠀⠀⠐⠒⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣀⣀⠀⢀⣠⡄
⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⢀⠀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣀⠀⠀⠀⠈⠛⠿⠛⠺⠽⠿⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠂⠒⠐⠃⠐⠒⠒⠂⠐⠒