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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠫⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠊⠉⠉⠡⠤⠍⠉⠁⠑⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡠⣤⠤⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⠤⠂⠉⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠠⠐⠉⠁⠐⢄⠀⠄⠂⠈⠉⠃⡀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠐⢄⠔⠁⠀⠀⠈⠢⠀⢸⣇⠁⣀⣄⠀⠀⠀⠏⣀⣤⠀⠀⠀⠈⡀⠀⠀⡇
⢡⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣦⣼⠀⠀⠀⠀⣶⣤⠁⡸⠈⡀⠙⠋⠀⠀⠀⢀⠙⠋⠀⠀⠀⢠⠁⠀⠀⡃
⠀⢣⠀⠀⠀⠀⠃⠀⠀⠀⠀⠉⡱⠄⠀⠀⠀⠈⢁⡶⠃⠀⠈⠂⢠⢤⠄⠊⠖⠑⠠⠀⠤⠐⠁⠀⠀⠀⠁
⠀⠀⠑⢄⡀⠀⠀⠀⠂⠀⠐⠈⠘⠂⠁⠀⠀⣢⠟⠁⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠲⠤⣀⡀⠀⠀⠀⠀⢀⣀⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠒⠂⠉⠉
⣿⡿⠻⠛⢿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⠤⢶⠶⠦⠤⠤⣤⣤⣤⣄⣀
⠀⠀⠀⢀⣴⡾⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⡈⠙⠛⠶⢶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠂⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⡄⠀⣀⠤⠖⠒⠂⠉⠉⠉⠉⠈⠘⠒⠒⠂⠄⠀⠀⢠⠇⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠁⢼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⡤⠚⢩⠤⠄⠈⠉
⠀⠀⠀⠀⠀⢰⠀⠉⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⢠⣴⣒⢩⡷⠋⣀⠀⠘⠒⠀⠀⠀⣞⠑
⠀⠀⠀⠀⠀⠈⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⡡⠦⠀⠉⠀⠉⠉⠅⠀⢼⠉⠂⠀⣈⡁⠀⡤⠤
⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⠀⠀⣠⠔⢁⡀⠀⠑⠒⠀⠀⠠⡎⠓⠀⠀⢀⡉⠄⢸⣄⡀⠀⠓⠄
⠀⠀⠀⠀⠀⠀⢹⠘⡀⠀⠀⠀⠀⢀⠇⠀⣏⡈⠀⠀⣎⠙⠀⢀⣉⣁⡴⠊⢡⡖⠒⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢸⡄⡇⠀⠀⠀⠀⣸⠀⠀⠀⠀⣀⡀⠈⠁⣀⢻⡴⠋⣀⡀⠀⠙⢂⡀⠀⢰⡉⡁⠀⠉
⠀⠀⠀⠀⠀⠀⠘⡆⢳⠀⠀⠀⠀⡇⣼⠉⠀⠸⣄⠁⠀⢸⣉⡽⠁⠘⠧⠀⠀⠰⣇⠉⠀⣠⢤⠀⢠⠒⠆
⠀⠀⠀⠀⠀⠀⠀⢳⢸⡀⠀⠀⠀⢹⡌⠉⠀⠀⠀⢠⠒⠄⢀⡏⠀⠀⠀⢀⢤⠀⠀⢀⣀⠳⠄⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⢸⡘⡷⣤⣀⣀⡀⢳⡀⢠⡎⠃⠈⠒⠀⠘⡆⣏⠉⠀⠻⠤⠀⠀⠧⠀⠀⠰⡉⠁⣀⢤
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡙⠢⠤⠍⣉⣉⣙⣦⡉⠁⠀⠰⣏⠉⢻⡀⠀⠀⠀⠀⣴⠒⠀⠀⢀⣤⡀⠀⠳⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠒⠒⠚⠛⠓⠂⠤⠤⠴⠳⡀⢸⡉⠁⠈⠁⣀⡀⠻⠄⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⣀⠀⠸⣄⠀⠀⠀⣀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠲⠖⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠢⠈⠆⢀⣠⠤⠖⠒⠒⠊⠉⠉⠉⠉⠉⠙⢒⣒⣒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠃⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢮⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣫⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠬⢉⣑⠒⠂⠤⠤⠤⠤⠤⠤⠤⠔⠒⢒⡉⠥⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠈⠉⠉⠀⠀⠐⠒⠀⠉⠉⠉⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⠤⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⣠⢊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⣠⠷⠷⠄⡀⠀⠀⠀⠀⡀⠤⠦⢆⡄⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⢀⡀⠀⠀⠀⢀⡀⠤⠤⢀⡀⠀⠀⠀⢹⠀⡴⠋⠀⠀⠀⠀⠈⠑⣄⡔⠉⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⢰⠇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⠀⠈⠢⣀⠔⠁⠀⠀⠀⠀⠈⠢⡀⠀⣸⠰⠀⣀⣠⣄⠀⠀⠀⠀⡜⢠⣀⣤⠀⠀⠀⠀⠀⢱⠀⠀⠐⢸
⠈⡇⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⣤⣀⣼⠀⠀⠀⠀⠀⣠⣄⣀⢃⠀⡟⠰⠀⠙⠿⠋⠀⠀⠀⠀⢃⠘⠿⠟⠀⠀⠀⠀⠀⡸⠀⠀⠀⢸
⠀⠹⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠻⠿⢛⠀⠀⠀⠀⠀⠹⠿⠏⣈⣼⠁⠀⠱⢄⠀⠀⠀⠀⢀⡰⠉⠢⡀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢸
⠀⠀⠹⣄⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⡠⠋⠢⡀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠉⠐⡖⠎⠁⠀⠉⠁⠈⠁⠒⠒⠊⠁⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠈⠳⣄⠀⢀⠀⠀⠈⠁⠒⠒⠂⠉⠀⠉⠁⠈⠁⠒⠒⢈⣵⠏⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⠤⠤⠤⠤⠤⠴⠚⠋⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⠤⣄⣀⣀⣀⣀⣀⣀⡤⠤⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠁⡐⢂⣠⠤⠒⠚⠉⠉⠉⠉⠉⠉⠉⠉⠩⠭⢕⣒⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⠭⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠒⠠⠍⡉⢐⣒⣒⣒⣒⣒⣒⣂⢉⠡⠄⠒⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠒⠒⠒⠒⠒⠒⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢀⣇⡄⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠔⠉⠉⠁⠀⠢⡀⠀⡠⠒⠀⠉⠉⠛⢄⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠈⠐⠤⡀⢀⠔⠊⠁⠀⠉⠒⢄⠀⠀⣸⢀⠁⠀⢀⠀⠀⠀⠀⠈⠎⠀⢀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢠⠃
⠸⡆⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⢀⡀⠑⡆⠀⠀⠀⠀⢀⡀⠈⠆⠀⡿⢸⠀⢿⣾⠇⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⢨⠀⠀⠀⢸
⠀⢣⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⣿⡿⡇⠀⠀⠀⠀⠸⣷⡿⠘⣰⠃⠀⢄⠀⠀⠀⠀⠀⠀⡠⠣⡀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⢸
⠀⠀⢣⡀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⡐⢣⠀⠀⠀⠀⠀⠀⣠⣶⠋⠀⠀⠀⠑⠂⢤⠤⠔⠊⠐⠖⠈⠂⠄⠠⠤⠒⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠑⢄⡀⠀⠀⠀⠈⠑⠀⠀⠀⠐⠉⠐⠒⠉⠂⠀⠀⠐⢊⡿⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⠤⣄⣀⣀⣀⡤⠴⠖⠋⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⢄⣀⣀⠀⠀⠀⠀⣀⣀⣠⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉


⡿⠿⠿⠿⠿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢻⣿⣿⣿⣿⢿⡿⢿

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠷⠄⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠗
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠉⠈⠉⠉⠉⠉⠉⠉⠉⠀⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡠⠄⠀⠒⠐⠒⠒⠒⠒⠒⠒⠒⠀⠠⢄⠀⠇⡀⡔⠉⠉⡎⠉⢰⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡒⠒⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠒⢚⡴⠯⡀⠈⠒⠊⡁⠀⠈⠢⠤⠜⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡆⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢇⣠⠇⠀⠀⢎⣨⠂⠀⠀⠀⡠⠤⡈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡠⠢⡄⢀⣀⠀⠀⠀⠀⣀⡀⠀⠑⠐⠀⣈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢠⢤⠀⠀⠀⠀⠠⡤⡄⠀⠆⠑⠀⠃⢎⠙⡳⣀⣀⢆⠙⡻⠀⠀⠠⡎⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸⣿⡷⢠⣤⡄⢿⣧⠆⠀⡞⠉⢱⠀⠀⠉⠀⠉⠉⠀⠉⢀⣠⠖⠀⠈⢋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⢖⠊⠀⣰⣶⣦⣤⡴⢶⠟⠋⡇⠀⡊⠉⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣳⣴⠛⢛⡁⠀⠀⠀⢒⡒⠁⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡆⠈⠉⠛⠓⠶⠶⠦⠤⠴⠶⠖⠛⠫⠉⠉⡽⠛⠤⠤⠀⠀⠬⣷⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⡀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡉⠒⠀⠤⠤⠤⠤⠤⠀⠒⢉⡠⠃⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣾⠓⠂⠀⠀⠚⠳⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡤⠶⠶⠶⠶⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠲⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠛⠶⠦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠉⠉⠙⠛⠒⠒⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠔⠒⠒⠒⠒⠛⠛⠉⠉⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢀⣴⠞⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠶⠶⠶⠛⠉⠁⠀⠀⠀⠀⠈⠉⠛⢶⣄⡀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡞⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠙⠛⢶⣤⣀⡀⠀⣀⣠⡴⠛⠉⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠙⠓⠶⠤⣤⣤⡤⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠁⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣦⣤⡶⠿⠶⠶⠶⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⢀⣠⢤⡄⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢠⡾⣻⠿⠉⠀⠀⣀⣤⣄⠀⠀⠀⠘⠧⣤⠟⠀⠀⠀⣀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⣼⡟⠁⠀⠀⠀⠐⢯⣄⣼⠀⠀⠀⠀⣀⣤⣴⠶⠛⣿⠉⠉⠉⠛⠛⠛⠛⠛⠲⠦⣤⣀⣠⠴⢦⡀⠀⠀⠀⠀⣼⡇
⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣠⡴⠟⠋⠀⠳⠶⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⣼⡇⠀⠀⠀⠀⣿
⢀⡾⠃⠀⢀⣀⣀⠀⠀⠉⠀⠀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⣀⣠⡾⠟
⣾⠁⠀⠀⢿⣁⣹⠇⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠸⣥⡴⠟⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠘⣿⠁
⣿⠀⠀⠀⠀⠉⠉⠀⠀⠀⣼⠇⠀⠀⠀⠀⣾⠋⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⣿⠆⢠⡾⣻⡄
⠸⣧⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣮⣀⣸⡆⠀⠀⠀⢀⣿⠀⠈⠛⠋⠁
⠀⠈⠛⠶⢶⣤⣤⣴⢾⣿⣛⢷⣄⠀⠀⠀⠀⠀⠀⠀⢰⡞⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀⣼⠃⣴⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠿⠀⠙⢷⣦⣀⠀⠀⠀⠀⠈⠷⡼⠇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠿⠷⠾⢣⣾⣿⡀
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⣈⢈⣉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠠⠤⠤⠤⠀⠀⠀⠠⠤⠤⠤⠠⢀⡀
⠀⠀⠀⠀⡠⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⣴⣾⣿⣷⣶⣦⣤⣄
⠀⠀⠀⠀⠀⢠⠀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⣠⣿⣿⣿⣀⣸⣿⣿⣿⠿⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠈⡀⢸⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⡶⠶⠛⠉⠙⠛⠛⠛⠛⢿⣿⣿⠿⢿⣿⣿⣤⣼⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠸⡿⠀⠀⠀⢀⡀⠈⠻⣿⣤⣼⣿⡟⠛⣿⣿⡁⣹⣿⡀
⠀⠀⠀⠀⠀⠀⢁⠀⣿⣿⣿⣿⣿⣿⣿⠉⠀⣠⣄⠀⠀⢀⣀⠀⠀⠻⠟⠀⠀⠈⠉⠉⠉⠛⠒⠛⠻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠘⠀⢹⣿⣿⣿⣿⣿⡏⠀⠀⠙⠋⠀⠀⠙⠛⠀⢀⣤⡄⠀⠀⠀⠺⠿⠀⠀⠀⢀⣀⠈⠛⢿⡗
⠀⠀⠀⠀⠀⠀⠀⡆⠸⣿⣿⣿⣿⡿⠁⢀⡀⠀⢀⣤⡀⠀⠀⣀⡀⠉⠀⣤⡄⠀⠀⢠⣄⠀⠀⠻⠟⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⢃⠀⣿⣿⣿⣿⣿⡀⠻⠟⠀⠈⠛⠁⠀⠈⠛⠃⠀⠀⠙⠁⠀⠀⠐⠛⠀⢠⣶⡄⠀⢾⣿⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠸⠀⢻⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠠⣶⡆⠀⠀⢀⡀⠀⢠⣤⡀⠀⢀⣤⡀⠉⢁⡀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠛⠿⠿⢿⣿⣦⠀⠀⢿⠇⠀⠀⢀⡀⠀⠻⠟⠀⠈⠉⠀⠀⠈⠛⠁⠀⠸⠟⠀⣤⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⢀⠀⠰⣿⠆⠀⠀⣠⡀⠀⠀⠙⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠒⠒⠒⠀⠉⠑⠒⠠⠤⠐⠢⡀⠐⠿⠇⠀⠀⣀⡀⠀⠛⠋⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⡀⠀⠀⠻⠃⠀⠀⢀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠐⠒⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠤⢤⣀⣀⣀⣀⣀⣀⣀⣠⠤⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠘⡆⠀⠀⣀⡤⠤⠤⠤⠤⣄⣀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⡋⠻⠗⢀⡀⠠⣿⠗⢄⣇⡗⠦⠤⢤⣀⣀⣀⡤⠤⢚⢇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠘⠛⡤⡄⠓⠃⠀⢠⡦⢀⣿⣹⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢶⠖⠤⣬⡤⠤⠤⢴⣶⠞⢫⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣶⢦⣭⠥⠤⣤⡤⠤⠒⢿⠟⣿⡇⠀⠀⠀⠀⠀⠀⢰⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠿⠟⣲⣤⠤⣍⣁⣀⣀⣠⠴⠻⣿⠀⠀⠀⠀⠀⠀⢸⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠤⣀⣈⣁⣀⣬⠾⠒⣪⣭⠲⡄⣏⡇⠀⠀⠀⠀⢀⣇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠬⠭⠤⠬⠭⠤⠤⠤⠥⠔⠁⠈⠚⠿⠶⠶⠿⠛⠊

⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⢀⡀⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠢
⠀⠀⢰⠟⠀⠁⠀⠠⣄⠀⠀⠀⠀⢶⠄⠀⠀⠀⠀⠔⠃⡀⠈⠁⠀⠀⠉⠁⣠⡠⠂⠀⡀⠋⣀⠀⠀⠀⠀⠠⠄
⠀⠀⡆⠀⠰⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠄⠈⠁⠀⠖⠀⣀⠀⢀⠔⠀⠀⣰⠄⠘⠉⠀⠀⠀⢲⠆
⠀⠀⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢅⣀⣤⣒⣈⡠⠤⠛⠉⠀⠀⠀⠀⢹⡄⠀⠀⠀⢀⡀
⠀⠀⣧⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠀⠉⣁⠀⠀⠛
⠀⠀⠈⠀⠀⠛⠂⠀⠀⠠⠆⠰⠏⠀⠀⣀⣀⠀⢀⣀⣀⡀⠀⢠⠞⡁⠀⠀⠀⣀⠀⠀⠈⠠⣴⡄⠀⠙⠂⡀⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⣡⡿⠀⠛⠉⢉⣛⢻⣿⢫⣡⡇⡤⡼⣷⡀⠀⠀⠀⠀⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢿⡇⣼⠋⣆⣿⣸⣾⣿⡞⢹⡿⠯⠔⡅⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢀⡼⢿⡹⠋⠳⢋⣘⣡⡿⣶⣋⣙⡆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠀⠀⠀⠻⠿⠛⠋⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⡄
⠀⠀⠀⣀⠀⠀⡀⠀⠀⠲⠆⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⢤⠤⠄⠰⠦⡀⠀⠀⠀⠀⠀⣠⢣⣬⠤⠀⠠⠄⠐⠀⢠⠄
⠀⠀⢠⠁⠶⠀⠁⠀⠀⣤⠄⠀⠘⠛⠀⠀⠀⢀⠄⠚⠁⠐⠶⢀⠀⢀⡄⢀⡀⠀⠀⠀⢿⠋⣤⢀⣀⣂⣀⢠⠀⠀⣄
⠀⠀⣻⣦⣀⠰⠆⠲⠖⠀⠰⠖⠆⠀⡜⠀⠀⡏⠀⣭⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⣴⡶⠊⠁⢡⡍⠀⠈⠐⢠⠌
⠀⠀⢿⡿⠛⠛⠛⣶⡿⠷⠶⢖⣶⢛⡄⠀⠐⠅⠀⠀⠀⠀⠀⠀⠸⠂⠀⠤⠀⠀⠀⠀⠻⠁⠰⠆⠀⠀⠀⠒⠂⠀⠠
⠀⠀⢰⡿⣿⡯⠤⣤⣤⣤⡶⠾⠟⠛⠃⡀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠠⠀⢀⣄⠀⠀⣇⠤⠀⠀⠀⠠⡄⠀⠀⡀
⠀⠀⠰⣶⠦⢤⠤⠤⢤⣾⣡⡤⠴⠶⠾⠁⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⠀⠀⠁⠀⠀⠀⢹⡀⠰⠆⠀⡀⠀⠀⠂⠉
⠀⠀⣼⡷⣤⢭⣄⣀⣨⣤⣥⡴⠖⠒⢊⡁⠀⠀⠀⠉⡀⠈⠀⢠⣄⡀⠀⠀⠟⢀⡄⠀⠀⠳⣄⡀⠀⠀⣤⠀⣀⠄
⠀⠀⠘⠷⠿⣦⣝⠛⣀⣀⣀⣀⣸⠧⠖⠁⠀⠀⠀⢀⠉⣠⡀⠀⠉⠀⢠⡀⠀⠘⠀⠀⠀⠀⠀⠀⠙⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠖⠒⠚⠛⠓⠒⠲⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠶⠦⢤⣤⣤⣤⡤⠴⠶⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⣀⣠⣤⣤⣤⣄⣀⡀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⢤⣤⣤⣤⣤⣤⡤⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣠⠖⢚⣍⠉⠉⠉⠩⠏⠛⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⣀⣠⣤⣴⠶⠾⠷⢶⠎⢀⣀⠈⠀⠀⢤⡄⠀⠠⠶⢁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠟⢛⣉⣁⣀⣀⡀⠀⠀⠘⡧⣬⣭⣤⣤⣀⣀⣤⣤⣶⡚⠩⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⡴⢞⠋⠉⠽⠉⢉⣙⠳⣄⢸⠀⠘⠋⠀⠰⠷⠀⠀⠀⢉⣡⠾⣧⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠈⠀⠐⠶⠀⠀⠤⠄⢸⡞⢷⠶⣤⣤⣤⣤⡶⠞⠛⢉⣀⠀⣺⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⡓⠶⣤⣤⣄⣠⣤⣤⡴⠞⠁⠘⠷⣤⣀⣀⡀⠀⢈⣁⣠⣥⠶⢋⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣛⣉⣩⣭⡶⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠟⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⡶⠶⠶⠶⠾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠶⠶⠶⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣴⣶⣶⡿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠉⠙⠻⢿⣶⣄⡀
⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀
⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⣾⡇
⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠾⠛⠉⠀⠀⢿⡇
⢸⣯⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⢀⣀⣤⣶⠾⠟⠋⠁⠀⠀⠀⠀⠀⠀⢸⡇
⢸⡏⠻⢿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣀⣤⣴⡾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⠁⠀⠀⠉⠙⠻⠿⢿⣿⣿⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣾⣿⣶⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⢸⠀⠀⢸⡇⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⠀⠀⣸⣿⠀⢠⣿⠀⠐⣿⣿⣶⣄⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⠀⠀⣿⣿⠀⢸⣿⡄⠀⠀⣿⠛⠛⢸⡇⠀⠀⠀⠀⣶⠀⠀⣀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⠀⠀⣿⢸⡆⣸⢿⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⢠⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⢰⡟⢸⣇⣿⢸⡇⠀⠀⣿⠀⠀⢨⡇⠀⠀⠀⠀⣿⠀⣾⠇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⢸⡇⠀⣿⡟⠀⣷⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⣟⣸⡟⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡀⠘⠃⠀⢿⡇⠀⣿⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⣿⡿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠈⠀⠀⢹⡆⣤⣼⡁⠀⢸⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠈⠁⠻⠿⣿⣷⢸⡇⠀⠀⠀⠀⣿⣿⡀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⣷⣶⣶⣦⠀⣿⢸⣇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠹⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⣿⠀⢿⡄⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠘⣿⡀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣦⣤⣀⣹⣇⣤⣤⣴⣶⡿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⢀⣠⡤⠶⠶⠶⠦⢤⣤⣀⣠⡴⠖⠒⠛⠒⠲⠶⢤⣄⡀⠀⠀⢸⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡿
⢀⡼⠋⣠⣤⣀⠀⠀⣤⣄⠀⣽⠋⠀⣶⣶⣦⠀⠀⠀⠀⢠⣍⠳⢦⣸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡇
⢸⡇⠀⠙⠻⠿⠇⠀⠀⠁⠀⢿⠀⠀⠈⠉⠉⠀⠀⣀⣀⠀⠀⠀⠀⠙⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇
⠈⢷⡀⠀⠀⠀⠀⠀⠶⣶⠀⠘⢧⡀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠇
⠀⣀⣻⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣟⣀⣀⣀⣀⡀⠀⠀⠀⠀⢸⠀⢸
⠈⠻⢯⣭⣉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣩⣭⠽⠋⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠉⠙⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢿⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿

⠀⠀⠀⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢳
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾
⠀⠀⠀⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡤⠶⠖⠚⠉⠉⠙⠒⠲⠦⣄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⢀⡀⠀⢀⣀⡀⠀⠈⠣⣀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠃⠀⠀⠐⠂⠀⠈⠀⠰⡇⠀⢹⠄⠀⠀⠈⢳
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⡼⠀⠀⢀⣠⣀⠀⠀⣀⠀⠉⠛⠉⠀⠀⡀⠀⠸⡆
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠁⠀⠀⣏⠀⢸⠆⠀⠉⠀⠀⠀⢀⣀⠀⠀⠀⠀⢹
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠘⡆⠀⠀⠈⠙⠉⠀⠀⠀⠲⠀⢠⡏⠈⢹⠀⠀⠀⡜
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢹⠀⠀⠀⠤⠀⢠⠤⢤⡀⠀⠀⠙⠒⠋⠀⠀⢸⠃
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⢯⣀⣠⠇⠀⠀⡄⠀⠀⢀⣠⠞
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠤⡤⠴⠚⠋⠉⠁



⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡴⠒⠒⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠐⠢⢀⡀⠀⠀⠀⠉⠁⠀⠀⠉
⢀⣀⠀⠀⡠⣛⣟⡛⠛⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡵⠀⢿⡄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡖⠀⣠⠔⠒⠒⠀⢄
⣹⠟⠀⢰⠷⠶⠭⠥⠤⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠔⠂⠀⣀⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠔⠡⣦⡆⠀⣖⡆⢆
⠁⠀⠀⡌⠘⣴⢃⡁⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠁⠁⠒⠠⡠⡕⠿⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⡠⠈⠀⢸
⠀⠀⠀⡇⢀⣄⠈⢁⣀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡤⠤⣤⠤⠖⠁
⠀⠀⢠⠇⠘⠷⠃⠸⠷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢳
⠀⠀⢸⠀⠈⠓⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈
⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠶⣦⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠉⠁⠁⠲⡀⠀⠀⠀⠀⠀⡠⠀⢀⣀⣀
⠀⠀⣼⢠⣠⠠⢠⢠⡤⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢨⠿⠿⣇⠀⠀⠀⠀⠀⠠⡆⠀⠚⠂⠈⠑⠀⠀⡇⠀⠀⠀⠀⢀⡷⠤⠤⠤⢼
⠀⠀⣿⣀⣁⣉⣈⣀⣁⣼⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠉⠉⠉⢆⠀⠀⠀⣀⣠⢷⣀⡁⠐⠒⠀⠈⡼⠓⠲⠆⠀⠀⣎⣀⣀⣚⣛⣃⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⡶⠆⠄⠰⠈⢣⠀⠀⠉⠁⠀⠀⠈⡟⠒⠂⣏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣤⢨⠈
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⠤⠤⠴⠶⠖⢇⠀⠀⠀⠀⠀⠠⠟⠀⠀⠈⠛⠀⠀⠀⠀⢸⠀⠀⠀⠛⠘⠃
⠀⠀⠀⡠⠉⢁⡀⠀⡉⠉⠒⢄⠀⠀⠀⠀⠀⡇⠀⣖⣢⠀⣶⣓⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢬⡷⠀⣜⡷
⠀⡀⢀⣇⠀⠈⠁⠀⠉⠀⠀⡸⠀⠀⠀⢠⠞⡇⠀⠉⢁⢀⡈⠉⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠄
⠀⠛⠉⠘⠠⢄⡈⠊⠀⢀⡰⠧⣠⡀⠀⠘⠦⣷⡄⠀⠀⠉⠀⠀⠀⢀⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠉⠉⢸⠉⠀⠀⠉⠀⠀⠀⠀⣿⠓⠒⠒⠒⠒⠒⠛⢻⠈⠁⠀⠀⠀⠀⢠⣶⣦⡄⠀⠀⠀⠀⠀⢸⠀⡀⡀⡀⡀
⠀⠀⠀⠀⠈⠋⠀⠀⠈⠓⠂⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⢹⠷⢿⡃⠀⠀⠀⠀⠀⢸⡀⠛⠃⠃⠇⠻⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⡴⠉⠛⠚⢣⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢰⡤⢠⢠⡌⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⢤⣤⣤⣥⣤⡇⠀⠀⠀⠀⠀⠀⢀⠄⢀⠤⠄
⠱⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡄⡠⢊⣀⠀⢀⡀⠱⠀⠠⣾⡆⠀⠀⠀⠀⠀⠀⡇⠀⣶⣢⠀⣴⢦⠀⠘⠀⠀⠀⠀⠀⢠⠀⣴⣂⠀⣴
⠐⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠛⠁⠘⠛⠁⠷⠴⠏⠀⠀⠀⠀⠀⢠⠎⡇⠀⠉⢉⢀⠙⠋⠀⣦⠀⠀⠀⠀⠀⠈⡂⠈⠁⡠⢭
⣷⢤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣈⠀⠙⠃⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠘⢳⣷⠆⠀⠈⠉⠀⠀⠀⣎⣇⠀⠀⠀⢠⣶⠛⠂⠠⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠩⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠓⠒⠒⠶⠶⠶⠶⠏⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⡶⢶⢶⢶⠶⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣾⣷⣷⣶⣶⢶⣶⣶⣶⠶⢶⡶⡶⢶

⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠦⠐⠒⠂⠉⠉⠀⠀⠐⠢⠤⣄⡀
⠀⠀⠀⠀⢀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠈⠒⢆
⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠑⠂⡀
⠀⣴⠁⠀⠀⠀⠊⠀⡀⠀⠀⠠⠠⠔⠀⠀⠀⢀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢰
⢀⠟⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⣀⠠⠐⠊⠁⠀⠉⠉⠓⠂⠤⢀⣀⣀
⡇⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠄⠀⠀⠀⢠⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠢⢀
⢈⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠜⠀⢀⠔⠁⠀⠀⠀⠀⡈⠈⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⢲⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠙⣄
⠈⣄⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠎⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡔⠁⠀⠀⠀⠠⣁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡆⠉⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠳⣌⠂⠄⡀⠀⠀⠀⠀⠀⠀⢀⠈⠈⠀⠀⠤⠤⠤⢀⣀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠀⠀⠙⢆⡀⠀⠉⣉⡠⠜⠋⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠟⠁⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂
⠀⠀⠀⠀⠀⢀⡩⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠤⠔⠀⠀⠀⠀⠀⠀⢀⠊⠁⠀⠀⠀⠀⠈⣀⡠⠀⠀⠀⠀⢰
⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⢦⡑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠈⠀⢀⠀⠀⠀⣇⠀⠂⠤⣀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⢀⠎
⠀⠀⢠⠃⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠷⣄⣀⠀⠉⠂⠄⡀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢠⠟
⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⠈⠉⠉⠢⣀⣀⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⡏⠉⠉⠑⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠹⡀⠀⠀⠀⠀⠀⠙⠒⠤⠴⠒⠊⠉
⠀⠀⠀⠣⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠀⠠⠤⠘⠀⠀⠀⠰⠀⠃⠀⠀⠀⠂⡀
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡸⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡰⠃⠀⠀⢠⠀⡂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠃⠌
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⣈⡠⠤⠐⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠲⢌⣀⡐⠀⠀⠊⠁⢀⡠⠤⠤⠴⠚⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣶⣶⣶⣶⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⣤⠖⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠰⠆⠀⠀⠰⠶⠂⠀⠀⡀⠀⠀⢠⠖
⠀⠀⢀⠋⠀⢀⣀⡀⠀⠻⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠠⠆⠀⢀⠀⠀⠀⠀⠀⢟⠊⠁⠀⠄⠁⣴⠶⠀⠀⠀⣀⡀⠙
⠀⠀⠇⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠒⠀⠀⠀⢀⠋⠀⣰⣆⡠⠔⠉⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⡆⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢤⠤⠶⠭⠤⠴⠖⠊⠁⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣠⣄⠀⠀⢀⣀
⠀⠀⢿⡄⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠂⠀⠀⠀⠀⣀⡀⠀⠘⠁
⠀⠀⠀⠐⠄⠐⠻⠀⠀⠀⠀⡶⠀⡾⠃⠀⠀⣀⣀⠀⠀⢀⣀⣀⠀⠀⢀⡾⢹⠀⠀⠀⠀⣀⠀⠀⠀⠁⢤⣤⡀⠀⠙⠃⢀⠠⠶⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢋⣼⠇⠸⠟⠛⠛⠛⠷⣿⢟⡋⣘⢂⣤⡼⣧⠃⠀⠀⠀⠀⠈⠁⠐⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⣿⡏⢠⡶⢲⢰⡟⢹⣸⣿⣥⠏⣿⣸⣗⡡⣉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠈⢁⣿⡧⣼⢻⣏⡞⠻⠏⢷⣄⡭⠥⢥⡈⠣⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣀⡴⠋⠙⠓⢡⣤⣤⣴⡶⠟⠋⠙⠓⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠴⠆⠀⠚⠛
⠀⠀⠀⢠⠄⠀⢀⡄⠀⠀⠉⠛⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⢠⠤⠲⠒⠒⠈⠛⠢⠀⠀⠀⠀⠀⠀⣠⣏⣶⠶⠂⠈⠑⠂⠀⠁⠀⠶
⠀⠀⢀⡄⠘⠳⠀⠀⠀⠀⠴⠖⠀⠀⠉⠉⠀⠀⠀⠀⡴⢈⡋⠀⠐⠛⢂⠀⠀⣤⠄⢠⡄⠀⠀⠀⠹⠏⢠⡦⣀⣀⣁⣀⣠⠤⠀⠰⡄
⠀⠀⢸⣷⣄⡀⠶⠆⠺⠷⠀⠀⠾⠳⠆⠈⡜⠀⠀⠼⠁⢨⣅⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠖⠋⠀⢠⡄⠀⠀⠉⠂⢴
⠀⠀⢿⡿⠿⠿⠿⠿⣶⣿⠷⠶⠶⢶⣖⠛⣀⠀⠀⢏⠀⢀⡀⢀⠀⠀⠀⠈⠷⠀⠀⡀⠁⠀⠀⠀⠸⠏⠀⠶⠄⠀⠀⠀⠐⠦⠀⠀⠠
⠀⠀⠀⣽⣿⣶⣛⣠⣀⣀⣀⣠⣤⣿⡿⠾⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⡀⠀⠀⣸⢁⡄⠀⠀⠀⢀⣄
⠀⠀⢀⣯⣍⡉⠁⢈⣉⣉⣩⠿⠀⢀⣀⣤⣴⠂⠀⠀⠀⠀⠛⠀⠀⠀⠀⠈⠁⠀⠀⠀⠊⠙⠀⠀⢸⡇⠀⣤⡀⠀⠀⠉⠀⢠⠈⠇
⠀⠀⢈⣼⡋⠉⢿⠉⠉⠹⠏⠿⠉⢉⢉⣉⡥⠄⠀⠀⠀⠠⠄⠀⠰⠊⠀⠂⠉⠀⠁⠀⣄⠀⠀⠀⠀⢻⡀⠈⠁⠐⠂⠀⠀⠀⠀⢀
⠀⠀⠺⣿⣩⣟⠻⢶⣶⠶⠓⠓⠛⠋⣩⡄⢐⡼⠀⠀⠀⠀⠀⡄⠀⠀⠀⠷⠖⠀⠀⠀⠁⢠⠆⠀⠀⠀⠙⠒⠠⣄⣀⡚⠃⠔⠒⠁
⠀⠀⠀⠙⠛⠚⠲⠮⠉⢤⡤⠤⠶⠤⠶⠚⠉⠀⠀⠀⠀⠀⠤⠀⠴⠄⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠂⠉⠡⡖⠦⠀⠈⠉⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠁⢀⣤⡄⢰⣦⠆⠀⣶⢢⠀⠀⢾⣌⡁⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠈⠛⠉⠀⣠⡴⡆⢀⣤⢄⠫⠄⠀⠀⢠⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠒⠀⡐⠀⠈⠋⠀⠈⠑⠉⠀⠀⠀⠒⢁⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⠳⣤⣀⣤⣀⠀⠀⠀⠈⠀⣀⣀⣀⣠⠾⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⠀⠠⠈⠉⠛⠛⠒⠲⡶⠚⠛⠋⠉⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣷⣦⣀⠁⠠⡀⠈⡸⠷⠋⠀⣀⣤⣴⠚⣅⠴⠒⠒⠲⡦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠈⠉⢙⣿⡟⡿⠛⠛⠛⠋⠉⠋⠀⠀⡴⠁⠰⠆⢀⣀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠘⠃⡭⠅⠀⠀⠀⠀⠀⠀⢤⣠⣼⠁⣄⢂⠀⠸⠿⠇⠀⠀⢰⣿⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⠿⡲⣷⢿⣶⠶⠶⢲⢗⢾⠉⠽⣿⠀⠙⢉⠀⠀⠀⠺⠈⠰⠆⠀⠄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⢙⠒⠋⠀⠀⠈⠙⠁⠀⢀⣿⣠⠀⠹⡦⠃⠸⣭⠀⠀⢠⡀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣙⡊⣿⡿⣧⣤⣀⣀⣀⣀⣀⣴⣿⡿⣧⠁⠀⢀⡤⠄⠀⣆⡀⠀⠁⠺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⢲⣿⣿⣧⡀⠉⠛⡙⠁⠈⢉⠉⠉⠠⣄⣂⣹⣧⣀⠀⠙⠇⠀⠀⠀⠀⣶⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠂⠀⠙⠻⠿⣷⣿⣧⣴⣦⣤⣄⣤⣴⣥⣿⣷⣟⢿⡄⠀⠸⠇⠀⠀⠀⠉⠁⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡦⠀⠀⣀⡄⠀⠀⠀⠈⠉⠉⣩⣄⠀⠀⠀⠀⡒⢲⣶⣶⣤⣙⣳⣤⣀⣁⣀⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠼⠿⠇⠀⠀⣀⣩⠁⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠙⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠈⠁⠀⡀⠀⠀⠆⠀⢀⠀⠂⠀⠀⠌⠁⠀⠀⠀⠈⠀⠸⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠁⠀⠠⠀⠀⠠

⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠄⠲⣀⣠⡖⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⡈⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠰⠀⠐⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⢀⣀⣤⡐⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠐
⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠃⠁⠁⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠢⠀⠰⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠁⠠⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⢀⠀⢁⣠⠄⠠⠀⠀⠁⢈⠁⣤⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠈⠉⢀⣠⠤⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠀⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠤⠄
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⠀⠀⠀⢈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡠⠂⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠆⠀⠄⠀⠠⠄⡆⠂⠈⠀⣀⣀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠄⠉⠀⡄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠚⠊⠐⠃⠃
⠀⠈⠀⠃⠀⠀⠐⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠆⠂⠔⠁⠀⡀⠀⠀⠀⡄⠀⢀⣴⠇⠀⠀⠀⠀⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠠⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠆⠀⠠⠤⠀⠙⠀⠀⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⢈⠀⠀⠀⠠⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠆⠀⠀⠂⠘⠋⠉⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠆⠘⠘⠀⠀⠈⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⢠⠃⠀⢠⡄
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢑⠀⠁⠀⠁⠁⠀⠈⠀⠉⠁
⠀⠀⠄⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠉⠀⠒⠂⠀⠀⠤⠤⠄
⠀⡰⠃⠤⠒⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠐⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠤⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠂⠂⠀⠒⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡤⠲⢄⡀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⢄⠀⠃⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣙⡂⠀⣊⣃⣠⠤⡄⠀⠀⠀⠀⠀⠀⠉⠉⠀⢉⠈⠀⠀⠀⡀⠀⠀⠀⠇⠀⣀⢀⠠⠤⠤⠀⢀⡀
⠀⠀⠁⠂⠀⢐⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⡀⠀⠩⡥⠠⠀⠀⡇⠀⠀⠀⠀⠀⢠⢀⠄⠀⠰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂⠒⠀⠀⠉⣤
⠀⠀⠀⠠⡆⠈⠀⠀⠀⠀⠀⠀⠀⠘⠂⠁⠁⠉⠂⠃⠀⠈⡐⠇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⡀⠃⠘⠈⠃⠞
⠐⠀⠀⠠⠀⠀⠠⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣀⣈⣩⣀⣀⡤⠚
⠀⠠⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠄⠀⠀⠀⢀⠀⡀⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⢶⠶⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠋⠁⠀⠀⠈⢡⣶⣿⡎⡳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⣷⣴⣿⣿⡇⠀⠀⠀⠀⠉⠋⠑⠃⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⡀⠀⠀⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠳⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⢳⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⡠⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⠯⠖⠀⠀⡀⣼⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢠⣿⣿⠉⠀⠀⠀⣴⣶⡄⠀⠉⠁⠀⠀⠀⠐⠇⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡄⠀⠉⠁⠀⠀⠀⠀⠨⡝⠁⠀⠀⠀⠀⠀⠀⢀⣴⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⠀⠀⠀⢀⠀⣀⣀⡀⠀⠀⠀⠠⢞⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣦⡀⠀⠀⠚⠸⢿⣿⡿⠀⢐⣣⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠶⠶⠶⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣐⡋⢩⠁⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣯⣄⡉⠉⠀⠐⠀⠀⢀⡀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠊⠑⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠈⠀⠰⠀⠰⠛⢿⣷⡠⡶⠀⠲⡷⠀⠙⠂⣙⠀⠀⠀⠀⠀⣰⠆⣴⡄⠀⠀⠀⠚⠃⠀⠀⢁
⠀⠀⠀⠀⠠⣄⠀⠰⠗⠀⠻⣻⡶⣧⠀⠀⠆⢀⠀⢀⠀⠀⠀⠀⠀⣿⣶⠄⠀⢀⣠⡄⠀⠂⠤⠔⠻⡄
⠀⠀⠀⠀⢀⠈⢳⡠⠕⠦⠤⠈⠻⡻⣶⡆⡀⠈⡀⡾⠀⠀⠀⠀⠀⢻⣳⠐⠀⠄⢨⣥⠀⠀⠲⠄
⠀⠀⠀⠀⢹⣄⠊⣍⢷⣄⠀⣖⡀⠘⢾⣧⡙⣠⠋⠀⠀⠀⠀⠀⠀⠈⢿⣿⠃⠐⠢⠤⠂⣁⡀⢠⠃
⠀⠀⠀⠀⠀⠙⠿⣟⡙⡍⠓⢤⡀⠂⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⠌⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠂⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠤⢰⠃⠴⠄⠀⠀⠀⠋⢀⣴⠺⠌⠁⡙⠋
⠀⠀⠀⠀⢠⠂⠡⠀⠉⠀⠀⠀⠐⣦⡀⠀⠀⠀⠀⠀⠃⠀⣀⡀⣿⡶⠆⠀⠠⣴⠄⣸⠿⡇⠀⠀⠈⢁⣤⡞⠃
⠀⠀⠀⠀⠁⠁⢠⣤⠄⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⡇⠀⠀⢉⡀⣿⣷⣀⡀⠂⢰⣶⣿⠆⠈⠀⣀⣲⣯⠎⢀⡐⠀⡀
⠀⠀⠀⣇⠀⠀⠀⢤⣀⠀⠀⠿⠀⠀⠘⢀⠁⠀⠀⢧⡀⠈⠃⠀⢾⣿⣿⣥⣈⣽⣿⣿⣿⠆⠀⠈⣿⡦⡏⠊⡀⠀⡃
⠀⠀⠀⢿⡈⠀⠙⠀⠀⠀⠀⠀⠁⠐⠀⠸⠀⠀⠀⠈⢳⡆⠀⢀⠑⡿⠻⣿⡿⠋⠉⠉⠙⠇⠠⡄⢽⠁⢠⣦⡁⠠
⠀⠀⠀⠈⢻⣤⠄⠀⠀⠂⣴⡄⠀⠀⠼⡇⠀⠀⠀⠀⠀⠉⠛⠛⠒⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠼⠆⠈⠉
⠀⠀⠀⠀⠈⠻⢦⡤⣶⣀⡀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠆⡀⠀⠀⠙⠛⠉⠀⢀⡀⠀⠀⠀⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠙⠛⠉⠀⢀⣀⠀⠀⠀⠉⠙⠃
⠀⡄⠌⠄⠀⠀⠁⠀⠀⠀⠀⠠⠤⠈⣵⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⡄⠌⠄⠀⠀⠁⠀⠀⠀⠀⠤⠤⠈⣽⠀⠀⠀⠰⡆
⠀⠘⠘⠀⠀⠀⠀⣀⡀⠀⠀⢸⡋⡁⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠀⠀⠀⠀⣀⡀⠀⠀⢸⡋⡁⠐⠆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢶⣀⡈⠁⠀⠀⠀⠋⠑⠛⠀⠀⢀⡀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣀⡈⠁⠀⠀⠀⠋⠑⠛⠀⠀⢄⡀⠐⠂⡀
⠖⠀⠄⠀⠀⠀⠈⠓⠂⠀⠀⠀⠀⡄⠀⠀⠀⠄⠠⠉⠉⠁⠀⠀⠀⠀⠖⠀⠄⠀⠀⠀⠈⠑⠃⠀⠀⠀⢀⡄⠀⠀⠀⡀⠠⠉⠉⠁
⠁⠀⠀⠀⠰⠗⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠁⠀⠀⠀⠰⠗⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠄⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⢀⡋⠀⠀⠀⠡⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠚⠃⢀⠋⠀⠀⠀⠠
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠈⢧⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢣⠈⠈⣀⠞
⠆⠀⠀⡌⢰⠋⠷⢀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⡞⠳⢀⡄⠀⠀⡌⠰⠋⠷⢀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⡞⠓
⠀⠀⠀⠹⣦⣄⡴⠏⠀⠀⡀⠀⠀⠀⣾⣆⠀⠀⠀⠐⠀⠀⠀⠁⠀⢸⠀⠀⠀⠹⣦⣄⡴⠏⠀⠀⠀⠀⠀⠀⣼⣆⠀⠀⠀⠐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⢻⡿
⠀⠀⠀⠤⠀⠀⡀⠀⢰⣶⣴⣾⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⡠⠀⢰⣶⣴⣾⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠤⡀⠀⠈⠙⠛⠉⠀⢀⣀⠀⠀⠀⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠙⠛⠉⢀⢀⣀⠀⠀⠀⠋⠙⠃
⠀⡄⠄⠄⠀⠀⠁⠀⠀⠀⠀⠠⠤⠈⣭⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⡄⠄⠄⠀⠀⠁⠀⠀⠀⠀⠠⠤⠈⣭⠀⠀⠀⠘⡆
⠀⠈⠘⠀⠀⠀⠀⡀⡀⠀⠀⢸⡋⡁⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠀⠀⠀⠀⡀⡀⠀⠀⢸⡟⡁⠰⠆
⠀⠀⠀⠀⠀⢶⣀⡈⠁⠀⠀⠀⠉⠁⠛⠀⠀⢀⡀⠐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣀⡈⠁⠀⠀⠀⠋⠑⠋⠀⠀⢄⡀⠐⠂⡀
⠖⠀⠄⠀⠀⠀⠈⠑⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠉⠉⠁⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠈⠑⠃⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⠉⠉
⠀⠀⠀⠀⠰⠗⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠢⠚⠃⢀⡋⣀⠠⠈⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠢⠛⠃⢀⠋⢀⠀⠈⠠
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠈⢣⠈⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⠀⠀⠀⠀⠧⠈⠀⣀⠞
⠆⠀⠀⡌⢰⠋⠷⢀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⡞⡳⢀⡄⠀⠀⡌⠰⠋⠷⢀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⡞⠳
⠀⠀⠀⠸⣦⣄⡴⠏⠀⠀⠀⠀⠀⠀⣾⣆⠀⠀⠀⠒⠀⠂⠀⠁⠀⢸⠀⠀⠀⠹⣤⣄⡴⠏⠀⠀⠀⠀⠀⠀⣼⣆⠀⠀⠀⠒⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⢹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠂⠀⠀⢻⡿
⠀⠀⠀⠤⠀⠀⣀⠀⢰⣶⣴⣾⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⢀⠀⢰⣶⣴⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡶⠿⠿⠿⠿⠿⠿⠷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⡀
⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⢠⣾⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⣼⡟⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⣼⡟⠀⠀⠀⠀⠰⣿⣿⡇⠀⢠⣶⣦⡀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣿⣀
⢰⣿⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠻⠛⠁⠀⠀⠀⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠿⠿⠛⠛⠛⠛⠛⠻⠿⢷⣶⣦⣄⣀
⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣦⣄
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⡀
⣿⡇⠀⠀⠀⣴⣶⣄⠀⠀⠀⠀⠀⣰⣶⣄⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣄
⢿⡇⠀⠀⠀⠙⠿⠋⠀⠀⠀⠀⠀⠙⠿⠋⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀
⢸⣿⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⢻⣧⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠈⢿⣆⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠙⢿⣆⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠛⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣄⣀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢷⣶⣦⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠉⠉⠉⠉⠙⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣤⣼⡟⠀⠀⠀⠀⢾⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠉⠉⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣤⣀⡀⠀⣾⡏⠀⠀⠀⠀⠈⠻⠿⠷⠀⠀⠀⢀⣶⡄⠀⠈⠙⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣯⣉⣉⣩⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣞⣧⣤⣤⡤⣟⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠉⠉⠉⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢤⣄⣀⣀⣀⣤⠴⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢀⣀⣠⣤⣀⣀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⣠⠶⠋⠉⠁⠀⠀⠀⠉⠙⠳⢤⡀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⡞⠁⠀⢀⣤⠴⠖⠒⠶⢤⣄⠀⠀⠹⣆⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⣸⠁⣴⠋⠀⣠⠞⠉⠀⢠⢦⣠⣦⣀⡌⠙⢦⡀⠈⢳⡀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡟
⠀⠀⠀⠀⠀⢀⡏⡼⠁⣠⠞⢁⣠⣾⣦⣿⣯⣿⣿⣿⡇⣠⣦⠙⢦⠀⠹⡜⡇⠀⠀⣀⣀⣤⣤⠤⠤⠤⠤⠤⠤⡇⠀⣸⣥⣀⣀⡀
⠀⠀⠀⠀⠀⢸⠃⠁⢠⠏⣠⡿⢿⣟⡿⠙⠋⠈⠉⠛⢿⣿⠏⡟⡌⣧⠀⠁⣷⠀⢸⠛⠲⠶⠶⠦⠤⠤⠤⠤⢤⡷⢤⡿⠴⠶⠟⣿
⠀⠀⠀⠀⠀⣼⠀⠀⣿⡤⡷⡰⡿⠈⠀⠀⠀⠀⠀⠀⠀⠛⣾⢟⠷⣸⡆⠀⢸⠀⢸⡇⠀⣶⠀⠀⠀⠀⠀⠀⢰⡇⢸⠇⠀⠀⢀⡇
⠀⠀⠀⠀⠀⣿⠀⢀⣿⠻⢷⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⡿⠟⡇⠀⢸⡇⠘⡇⠀⢻⠀⠀⠀⠀⠀⠀⢸⠃⣾⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⢸⡏⢳⠯⣄⣀⣀⣀⣠⢤⢤⡀⣄⣠⣄⢀⣍⣀⠀⡇⠀⢸⡇⠀⡇⠀⢸⠀⠀⢀⣀⣀⣀⣸⢀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠘⡇⣽⣾⢾⢹⡍⡜⡣⢘⡳⣛⡄⠿⣸⢻⣸⠟⠁⡇⠀⢸⠁⠀⣿⣴⣾⣛⠉⠉⠉⠉⠉⠻⠾⢋⣛⣲⣆⣿
⠀⠀⠀⠀⠀⢸⠀⠀⣷⠈⠙⠉⠈⠛⠃⠈⠋⠉⠁⠉⠉⠁⠈⠁⠀⢸⠇⠀⣸⠀⠀⢹⢻⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⡇⡏
⠀⠀⠀⠀⠀⢸⡆⡀⠘⣆⠘⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⠀⢀⡟⢀⠀⡿⠀⠀⢸⣼⡌⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣇⠹⡄⠹⡄⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢀⡾⠀⡾⢠⡇⠀⠀⢸⡿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹
⠀⠀⠀⠀⠀⠀⢹⡀⠹⡄⠙⣦⠈⠣⣀⠀⠀⠀⢀⡠⠞⠁⢠⠞⠀⡼⠁⣼⠀⠀⠀⠀⡇⡇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠹⣆⠈⠳⣄⡀⠉⠉⠉⠉⠀⣠⠴⠋⢀⡞⠁⢰⠇⠀⠀⠀⠀⣧⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⢷⡀⠀⠙⠓⠲⠶⠚⠋⠁⢀⡴⠋⠀⢠⡏⠀⠀⠀⠀⠀⢿⢹⢿⠀⠀⠀⠀⠀⠀⠀⡦⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠙⠲⣄⣀⠀⠀⢀⣀⡴⠛⠁⠀⢀⡞⠀⠀⠀⠀⠀⠀⢸⢸⣈⠀⠀⠀⠀⠀⠀⠀⣇⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠉⠉⠋⠉⠁⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠸⣇⡈⠉⠉⠛⠛⠛⠛⢛⡏⢁⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⠿⠋⠁⠘⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢯⣥⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⡘⢷⡀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢀⡤⠴⢶⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣴⣿⣿⣶⣤⣹⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣛⠥⠀⠀⠀⠀⠀⠀⠀⠚⠉⠙⠢⠉⠃⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣃⣤⣴⣶⣶⣦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⢀⣀⣠⣤⣤⣴⠞⠲⣶⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⣿⣿⣿⣿⣷⡄⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⡐⠀⠀⢠⣇⣀⢸⣿⣿⣿⣿⠁⠀⢹⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣏⣵⣿⡟⠛⢻⣿⣿⣿⣿⣿⣿⡟⠿⢏⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠿⣿⣷⡀⠸⣮⣝⡿⣿⣿⣿⠟⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠙⠛⠿⢿⣿⣿⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣯⣽⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠏⠉⢉⠛⠛⠻⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠿⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠛⠶⣦⣤⣤⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⡾⠏⠀⣠⣶⠶⣶⣄⠀⠉⠉⠉⠉⠉⠛⠻⢷⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠋⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠈⠑⠛⠿⢯⣝⡛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⣫⠆⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣹⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣴⢟⣵⣶⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⡀
⠀⠀⠀⠀⠀⢀⡟⢡⡾⠉⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀
⠀⠀⠀⠀⢠⠎⣠⡿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣍⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣬⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢷⡄
⠀⠀⠀⣴⢫⠞⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣾⡷⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠻⠿⠿⠃⠀⠀⢀⣾⣿⣿⣿⣦⡀⠀⠀⠀⢹⡘⣿⡆
⠀⢀⣾⠃⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠋⠽⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣼⡿⠀⠀⠀⢸⡇⣿⡇
⠀⣼⡏⠀⠀⠀⣀⡤⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠁⠀⠀⠀⣾⡇⣿⡇
⠀⣿⠀⠀⠸⣿⣿⣇⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣤⡀⠈⠀⠀⠀⠀⠀⠀⠈⠳⢹⣷
⠀⡏⠀⠀⠀⠙⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⢀⠀⠀⠀⠀⠀⣠⣄⢀⠀⢻
⣸⢡⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣧⢸⡆⠀⠀⠀⠀⠀⠀⠀⠘⣯⠹⣿⣿⣿⡟⡞⠀⠀⠀⣠⣾⣿⣿⢸⡆⢸
⣿⣼⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣶⡄⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠰⢯⣼⣿⣿⢸⣇⣿
⢻⡿⣇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣼⣷⠀⠀⠀⠀⠀⠙⠛⢛⣛⡥⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢃⡾⣿⠃
⠈⢿⡙⢷⣄⡰⠶⣄⣀⡙⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣼⡇
⠀⠈⠻⣿⠿⠻⣿⣿⠟⠿⣿⣿⣷⣤⣄⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣾⣿⠃
⠀⠀⠀⠀⠘⢧⠈⠁⠠⡶⠉⠀⠸⠿⢿⣷⠦⠀⣾⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡻⣦⠀⠀⠀⢀⡴⣣⡿⢡⣿⠃
⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢷⡄⢿⣿⣿⣿⢇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣿⠁⠀⢀⣴⣾⠟⣡⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠸⣝⣻⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣟⣛⡯⠅⠀⢠⣿⢟⣵⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠘⣿⡄⠀⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢇⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢰⣟⢀⠀⠀⠀⠀⢰⣿⣿⣤⣿⣧⠀⢀⣀⣠⣴⣤⡾⠷⠾⠿⠛⣋⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⣭⣤⣬⡘⣿⣿⣿⣿⠟⣠⠞⣫⢍⣩⣥⣶⣶⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡿⣿⣭⣴⣶⣮⣤⣭⣉⣥⣤⣶⡾⠿⠿⠿⠛⠉⠁⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣷⣤⣽⣿⣿⣯⣭⣭⣽⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣖⣚⡛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠓⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡧⠤⠤⠬⠴⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⢠⡀⠉⡞⢸⢳⠂⡸⣶⠀⢡⠎⢸⢦⢃⢢⡇⡜⣡⠠⠆⣰⣿⣷⣤⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠁⠀⠀⠀⠈⠁⠀⠈⠁⠈⠁⠁⠉⠙⠘⠘⠃⠛⠒⣠⣿⠁⠀⢙⣻⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠷⠶⠶⠶⠶⠿⠶⠶⠶⣶⣿⠷⠾⠿⠛⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⡴⢶⠾⠛⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⢸⡿⠚⠛⠛⠁⠀⢸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡀⣰⣿⣿⣿⣿⣿⣿⣿⠟⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⢸⣿⣿⡇⣿⣿⣿⣿⣧⣿⣿⡟⣸⣿⣿⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢸⣿⣿⠇⢸⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⣸⣿⣿⠀⠈⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣠⣿⣿⣿⣤⠀⢹⣿⣿⣿⠏⠀⣠⣿⣿⣿⣿⣇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⣿⡿⠀⠘⣿⣿⡟⠀⢠⣿⣿⣿⣿⣿⣿⠀⠀⣾⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠛⠛⠛⠛⠃⠀⠀⠛⠛⠁⠀⠘⠛⠛⠛⠛⠛⠛⠀⠀⣿⠀⠀⠀⢀⣠⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⠴⠶⠶⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣤⣤⣶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⢀⣴⠊⠙⠁⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⡴⠆⠀⠐⠃⠀⠀⠐⠚⠂⡀⢀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⣀
⠀⠀⡘⠁⠀⣠⣄⠀⠈⠛⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⣀⠀⠀⠀⠀⠘⡻⠁⠀⢰⠀⠠⠟⠂⠀⠀⢠⣄⠉⠁
⠀⠀⠇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡦⠊⠁⠀⣀⠠⠁⠀⣾⠦⠐⠉⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⡇⠀⠀⠀⠀⠰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠛⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⢸⠅⠀⠀⠀⠰⠦⠀⠀⢀⡤
⠀⠀⠻⠄⠀⢀⡀⠀⠈⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⢤⣄
⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⠘⠁⠘⠃⠀⢀⣤⢤⡄⠀⣤⣤⣤⣤⣀⣰⣏⡼⢀⠀⠀⣠⢤⠀⠀⠀⠘⠺⢇⠀⠀⣉⠀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢡⣾⠇⢈⣋⠀⣨⢭⢩⣿⠭⡄⣮⣰⢒⢛⢮⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠸⠯⢠⣿⢸⣾⣧⣸⣾⣿⣞⠹⠧⠻⠅⢲⣆⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⣠⠿⣏⠟⠙⠳⢋⣈⣡⡿⢷⣋⣉⣱⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠁⠀⠀⠘⠿⠿⠟⠋⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠠⢤⡀⠀⠀⠀⠀⠀⠀⣠⢇⣈⣁⠠⠤⣄⠀⠂⠀⣀
⠀⠀⠀⠚⢱⡄⠘⠁⠀⠀⣠⡀⠀⠘⠿⠀⠀⠀⠀⢀⠠⠞⠁⠠⡦⠄⠀⠀⣀⠀⠀⠀⠀⠀⠐⣿⠞⣉⠀⠀⠤⠀⠀⣀⠀⠁
⠀⠀⢸⣇⠀⠀⣄⠀⣤⡄⠁⢠⣤⡄⠠⢠⠆⠀⢠⡆⠐⠂⠀⠀⠀⠘⠀⠈⠉⠀⠙⠀⠀⠀⠀⣈⣤⡭⠓⠛⡛⠛⠁⠳⠄⣀⠁
⠀⠀⣾⣿⣿⣦⣦⣤⣠⣤⣄⣄⣀⣤⠤⠏⠀⠀⡰⠀⠈⠉⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⣿⠃⢀⣄⠀⠉⠀⠀⣄⠀⠈
⠀⠀⠙⣻⣦⣤⠆⠈⠛⠉⠀⣀⣰⣿⣧⡾⠀⠀⠈⠀⠈⠀⠈⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⡆⢀
⠀⠀⠈⣛⡛⠛⠉⠙⠛⠛⣻⡏⠉⠁⣀⣀⠤⠀⠀⠀⠀⠲⠀⠀⠀⠀⠘⠀⠈⠁⠠⠒⠂⠀⠀⣷⠉⢀⡀⠀⠀⠙⠀⢀⠀⠆
⠀⠀⢘⡹⠋⢹⣗⠛⠛⡿⢓⡖⠚⠛⠙⣋⡀⠀⠀⠀⠀⣀⠀⠠⠔⠒⠀⠂⠀⠀⢀⡀⠀⠀⠀⠸⡆⠈⠋⠀⠤⠀⠀⠀⠁
⠀⠀⢾⡿⣻⡗⠶⢦⡶⠶⠶⠶⠚⢋⣍⠡⢠⠀⠀⠀⠀⠀⣄⠀⠀⠀⣶⡆⠀⠀⠈⠁⣰⠂⠀⠀⠈⠲⠤⣀⡀⠰⠖⡠⠖
⠀⠀⠀⠛⠻⠷⠶⠍⢁⣀⣠⠤⠤⠤⠿⠚⠁⠀⠀⠀⠀⢄⠀⢤⡄⠀⠀⠀⠀⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⡿⢿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⠙⠿⠛⠋⠉⠉⠀⠈⠉⠛⠻⠃⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡿⠁⠀⠀⣠⣴⣶⣶⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⣡⣶⣤⣼⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢒⠒⠢⠀⠀⣠⣾⣿⢿⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠄⢀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣟⠂⠐⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⠀⠠⣾⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣻⣿⣾⣶⣯⣿⣿⠀⠘⠿⣽⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⣿⣿⣿⣭⣿⣿⠀⠀⠀⢻⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠀⣼⡟⠙⢻⣿⣿⡇⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠤⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠦⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣋⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠉⠉⠒⠒⠢⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠴⠖⠒⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⡴⠖⠛⠛⠛⠒⠒⠶⢤⣄⣀⣤⠶⠚⠛⠛⠓⠲⢤⣀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠞⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣨⠷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠙⠢⣄⣀⣀⣀⣀⡤⠖⠋⠁⠀⠈⠙⠲⠤⠤⠖⠋⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠛⠉⠉⠉⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠶⠋⠁⠀⠀⠀⡠⢤⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠁⠀⢰⠒⣦⠀⠀⠙⢚⣡⣤⡴⠷⠤⠤⠤⢤⣄⣀⠀⣀⣀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠉⠁⣠⠴⠚⠙⠧⠔⠁⠀⠀⠀⠀⠀⠀⠈⠙⠳⢼⡀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣠⠶⡆⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣴⣙⡆⠀⠀⠀⠀⠀⠀⠱⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠁⠀⢠⡏⠀⠀⠀⡖⣲⠀⠀⠀⠀⠈⠉⠀⠀⠀⣠⡄⠀⠀⠀⢹⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⡀⠀⠀⣀⣼⡆⠀⠀⠀⠉⠉⠀⣠⡀⠀⠀⠀⠀⠀⠘⠧⠼⠀⠀⢀⡾⢀⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢹⣷⠙⢶⣄⡀⠀⠀⠈⢧⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⣤⣄⣀⣀⣀⣀⣀⣰⣲⡴⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠟⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠩⠉⠉⠉⢸⣿⠉⠟⠛⠉⢸⣧⢄⠀⠙⠁⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⢸⣿⠀⠀⢀⠀⢸⣿⣴⠯⠄⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣬⣼⣿⣤⣤⣤⣤⣼⣯⣬⣤⣬⣥⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠈⠉⠋⠉⢹⣿⠉⣉⠉⠉⢸⣧⢩⡍⠉⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠂⠀⢸⣿⠔⣿⣃⠀⢹⡟⠉⠀⠊⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣼⣩⣴⣶⣦⣬⣼⣧⣤⣤⣤⠀⠙⠀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠩⠉⠉⠘⠛⠀⠉⠉⠉⠙⠋⠉⢁⠅⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢩⠄⠀⠀⠀⠀⠀⠑⠠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡋⢈⣴⠋⠀⠶⡀⢠⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡸⠁⠀⠀⠀⠀⠐⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡬⠀⠀⠀⠁⠀⠀⠀⠀⣠⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⢶⢇⣠⣴⣿⣿⣶⣶⢞⣁⣤⣴⡶⠒⢻⠁
⠀⠀⠀⠀⢑⠂⢤⣀⡀⢀⡠⠐⢂⠉⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠀⠀⡸
⠀⠀⠀⠀⢸⣧⠀⠀⠈⠉⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢀⣩⠽⠿⠶⠾⠛⠉⠁⢸⣿⣿⣿⣿⣿⣿⡿⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠾⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣝⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣻⠢⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠚⠒⠲⠶⠶⠖⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⡀
⠀⠀⠀⠀⣾⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣧⠀⠀⠀⠀⢀⣀⠀⠀⠀⠉
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠻⣿⡇
⠀⠀⠀⠀⢻⣿⣿⣏⣩⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⠟⢀⣴⣄
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣯⣹⣿⣿⣿⣿⣿⣿⡿⠋⣰⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣠⠚⠉⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠀⣼⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣴⣾⣷⢲⣶⣾⣿⣷⡄
⠀⢀⡴⠚⠁⠀⠀⠀⣼⣿⡄⠀⠀⢀⣄⠀⠙⢦⡀⠀⢸⣿⣿⣿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⡞⠀⢀⡀⠀⠀⠀⠈⠉⠀⠀⠀⠸⣿⡗⠀⠀⠳⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿
⣸⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡏⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠄⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣥⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿
⡇⠀⠀⠀⣴⣦⡀⠀⠀⠈⠁⠀⠀⠀⠀⠰⣿⡷⠀⢸⠁⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿⠁
⢳⡀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠙⠿⣿⣿⣿⣿⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠳⣄⠀⠀⠀⠀⣠⣤⡀⠀⢸⣿⠇⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁
⠀⠀⠈⠢⣄⠀⠀⠘⠛⠀⠀⠀⠀⠀⣀⡠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣰⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⠀⠀⣰⣿⣿⣶⡄⠀⠀⠰⣿⣿⣿⠏⠀⣀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⡀⠀⠙⠿⣿⠟⠀⠀⠀⠀⠀⠉⠋⠀⠘⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠁⢸⠄⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿


⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣻⣟⣿⣟⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡶⢒⢔⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠦⠉⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⠤⠒⠀⠁⠀⠀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠀⠀⠀⠀⠈⠋⠐⠰⢠⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠀⠂⠀⢸⠀⠨⠘⠁⢀⡀⢀⣀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠘
⠀⠀⠀⠀⠀⠲⠒⠩⣩⢉⠆⠀⠀⠀⠀⠘⠀⠠⠀⠙⠉⡐⢕⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠂⢀⡀⢠⠀⢠⡄⠐⠀⡇
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠡⠀⠘⠋⡐⠴⢠⠘⠘⠀⠀⠀⠸⠀⠀⠀⠃⠀⠀⠄⠋
⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⢠⢰⡆⠀⠀⠀⠀⠂⢨⠀⡄⠀⢠⠀⠐⢸⠃⠀⠀⠀⠀⠉⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠈
⠀⠀⠀⠀⠀⠀⠉⠀⢸⠀⠀⠈⠀⠀⢤⠆⢂⡖⢚⡄⡀⠀⢸⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠁⡟⠀⠀⠀⠀⠀⠀⠀⠁⠂⢐⡀⠤⠐⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠠⠀⠂⠉⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⠠⠄⠊⠉⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠐⠥⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⡀⠀⡄⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⡇⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠈⠢⠈⠆⢀⡠⠤⠖⠒⠚⠛⠛⠉⠉⠉⠉⠛⠛⢒⣒⡦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠃⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢮⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡫⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⠍⡑⣒⠂⠤⠤⠤⠤⠤⠤⠤⠐⢒⣊⡉⠤⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠤⠤⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸
⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢀⡰⠷⠷⠤⣀⠀⠀⠀⢀⡠⠴⠦⢦⡆⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠄⣀⠀⠀⠀⢀⠠⠄⠤⠄⡀⠀⠀⠀⢹⠀⡔⠁⠀⠀⠀⠀⠀⠑⢄⠔⠁⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠇
⢸⡀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠈⠑⣄⠔⠁⠀⠀⠀⠀⠈⠲⡀⠀⣼⢰⠀⣠⣠⣦⠀⠀⠀⠀⡌⢠⣤⣦⠀⠀⠀⠀⠀⢐⠀⠀⠀⢸
⠈⣇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣴⣤⣼⠀⠀⠀⠀⠀⣠⣄⣤⢡⢀⡟⠘⡀⠙⠛⠋⠀⠀⠀⠀⢡⠈⠛⠛⠀⠀⠀⠀⠀⡸⠀⠀⠀⢸
⠀⠸⡄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠛⠟⢫⠀⠀⠀⠀⠀⠙⠛⠋⣌⡼⠁⠀⠑⢄⡀⠀⠀⠀⢀⠔⣁⡣⣀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⢸
⠀⠀⠙⢆⠀⠀⠀⠀⠀⠱⢄⠀⠀⠀⠀⢀⠴⣉⡣⣄⠀⠀⠀⠀⣀⢴⡿⠁⠀⠀⠀⠀⠈⢹⡋⠏⠁⠀⠉⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠉⠁⠀⠈⠁⠀⠉⠁⠀⠉⠁⠀⢈⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠉⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠶⠶⠴⠶⠶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠦⠤⢤⣤⣤⣤⡤⠤⠴⠖⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠛⠛⠛⠛⠛⠛⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢠⣶⣶⣶⣆⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠁⠀⠀⠀⠀⠸⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠿⠿⣄
⠀⠀⠀⠀⠀⠀⢀⡞⠿⠃⠀⢀⡴⠀⠀⠀⠀⠉⣠⡾⠁⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣤⣶⣿⣦⠀⠀⠀⠀⠀⠀⠘⠻⠿⠛⠋⠀⠀⠀⠀⠀⣤⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣄⡀⠀⠀⢻⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠉⠉⠉⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠘⠇⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠀⠀⠀⠀⠀⢸⡿⠿⣿⣿⠁⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣤⣦⣤⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣈⠛⠛⠁⠀⠀⠀⠀⠀⠰⠛⠻⠟⠃⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠦⠴⠶⠶⠶⠶⠶⠚⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡶⠟⠛⠋⠉⠉⠁⡄⠀⠀⢦⡈⠉⢹⣧
⠀⠀⠀⠀⢠⣶⠿⠛⣉⠀⠀⠀⢹⠀⠀⠀⣴⠏⠀⠠⠼⠃⠀⣸⣿
⠀⠀⠀⠀⢸⣿⠀⠀⢨⠇⠀⠀⣋⣀⣤⣤⡤⠶⠶⠶⠒⠒⠺⣿⡟
⠀⠀⠀⠀⠸⣿⣦⣀⣤⣴⠾⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠈⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠹⣿⣆
⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠖⠒⠛⠉⠉⠀⠈⠻⣧
⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⢀⣠⡤⠶⠖⠛⠉⠉⠀⠀⠀⢀⣀⠠⣶⠖⣿⢛⡀⠀⠹⣷⡀
⠀⠀⠀⠀⣼⡏⠀⣀⣠⠴⠞⠛⠉⠀⠀⢀⣀⠀⣠⣦⠀⣧⠀⢸⣻⡆⢸⡆⢹⡏⣁⠀⠀⢹⣷⡀
⠀⠀⠀⠀⣿⠀⠛⠉⠀⠀⠠⡄⢰⠾⡄⢾⠉⠁⣇⠈⡇⢹⣀⣨⡏⢳⡀⠇⠘⠋⠁⠀⠀⠀⢿⣇
⠀⠀⠀⢀⣿⠀⣰⠖⠆⢱⠴⢧⢸⡀⣷⠸⣦⡴⠙⠚⠁⠈⠉⠁⠀⢀⡀⠀⡴⢦⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⢸⣿⠀⢿⡀⣀⠈⡇⠸⠄⠙⠁⠀⠀⠀⢀⡀⠀⡤⣄⠀⠀⡏⢳⡼⢡⡟⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⢸⣟⠀⠈⠛⠁⠀⢀⠀⠀⢠⡖⢲⡄⢰⡋⢻⢰⡇⣿⠀⠀⡇⠈⣀⠙⢦⣀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⠋⠻⣤⠟⣼⡆⢻⡘⡇⢸⡆⣇⠹⠶⢶⣿⡈⣯⠓⢦⡽⠂⠀⠀⠀⠈⣿⡄
⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⠀⣰⣎⣰⠇⢷⠈⡇⢷⣸⡇⠻⠴⠖⠋⠉⠙⠁⣀⣀⣀⣤⠤⠆⠀⠀⢻⣇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⠀⣿⠉⠋⠀⠈⠛⠃⠈⠉⢀⣀⣠⡤⠶⠖⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⢹⣧⠀⠀⠀⠘⠶⠟⠀⠀⣀⣠⣤⠶⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠘⣿⡆⠀⠀⢀⣤⠴⠚⠋⠉⠀⠀⣴⠤⠤⠤⣀⣀⣀⠀⠀⠀⠀⠀⣠⣄⣀⡀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⢿⣷⠀⠀⠉⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⡼⡽⠃⣹⠀⠀⠀⠀⣸⠃⠀⠉⣨⢷⢶⡤⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⢯⣿⡟⠁⣴⠃⠀⠀⢀⣼⠁⠀⠀⡸⣡⠏⣠⡇⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠻⣍⠉⣉⣉⣧⣸⠃⣞⠆⠀⢨⣯⣽⣋⣛⣷⢃⡼⠃⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⣈⣁⣀⡀⠀⠀⠘⠷⠦⢤⣤⡧⠞⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠸⠇⠀⣰⠁⠈⠉⣴⢳⠶⡄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⣀⣀⣼⣱⠋⢰⠟⠀⠐⠿⠜⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⢠⠞⠒⠒⠚⠟⠁⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⠀⠀⠀⠀⠘⠛⠓⠒⠒⠖⠺⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣴⣶⣶⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠙⠒⠒⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠔⠒⠒⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠔⠒⠛⠒⠒⠦⠤⠤⠤⠖⠒⠒⠒⠦⣄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠳⣄⠀⠀⠀⠀⠀⣀⠤⠤⠤⣀⠀⠀⣀⠴⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠉⠒⠒⠒⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠁⠀⠈⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⢁⣀⠀⠀⠪⠗⢀⣀⣇⠀⣀⡀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠈⠒⢁⡤⠖⠻⠭⠃⠀⠀⠀⠉⠉⠒⠯⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢠⣒⠄⠀⡴⠉⠀⠀⠀⠀⠀⠠⣏⠇⠀⠀⠄⠀⠈⢶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢰⡃⠀⠀⠯⠇⠀⠀⠀⠀⠀⠀⣔⣢⠀⠀⢸⠐⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠺⣿⠳⣄⡀⠀⠀⢎⡣⠀⠀⠀⠀⠀⠁⢀⡰⠃⠾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⣤⣤⣤⣤⣶⣶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠛⠛⠋⠁⠀⠈⠉⠉⠉⠉⠉⠳⠶⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠁⠀⠀⠀⢀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠠⠄⠀⠂⠀⠀⠀⢄⠀⠀⠀⠀⢀⡠⠠⢄⠀⠀⠘⠒⢦⡀
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠓⠢⡀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠑⠤⠀⣇⠀⠀⠀⠀⠹⣄⡀
⠀⠀⠀⠀⠀⠀⣹⠀⠀⠘⠠⢤⠃⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠐⠀⡀⢀⠀⠀⠀⡀⠀⠁⠁⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠈⠑⠂⠀⠄⠐⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠈⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⣯⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⡃⠀⠀⠈⠑⢤⠄⠀⠀⠀⠀⢘⡇
⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠒⠀⠀⢀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⢺⠃
⠀⠀⠀⠘⢧⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⡀⣾
⠀⠀⠀⠀⠀⢹⡌⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⡏
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣀⡀⠈⠐⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠐⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣄⣀⣀⡀⠀⠀⠈⠑⠀⠀⠂⠁⠀⠀⠈⠁⠀⠀⢀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⠴⢦⣀⠀⠀⣀⡤⢤⣤⠤⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


⠀⠉⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡠⣄⣤⣤⡴⢦⣶⡴⠖⠒⠛⠛⢻⣿⠉⠉⠁⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠀⠀⠤⠄⠒⠒⠈⠉⠁⠀⠈⠁⠀⢰⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠘⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⢀⣀⣿⣧⣀⡤⠤⠤⡾⣿⠷⣚⣛⣫⣭⣽⡆
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣉⣩⡭⠭⢽⣷⠖⠚⠓⠚⠙⣿⡏⠉⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⢈⣧
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠀⠀⣀⣀⣀⣿⡧⠤⣤⠴⠶⠞⣟⣛⣛⣩⣭⡯⣽
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣛⣟⣋⣭⡭⠭⠶⣾⣷⠓⠒⠋⠉⠹⣿⡏⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠉⣿⡇⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣀⣘⡇
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠄⠛⠋⠀⠀⠂⢀⣸⣿⣦⡤⠴⠶⠒⠛⠟⠛⠉⠛⠉⠉⠁
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠐⠀⠉⠈⠚⠅⠀⠂⢀⠀⠄⠀⠀⠀⣈⡦⠄⡀
⠀⠀⠀⠀⠀⠀⠠⠤⠄⠒⠂⠀⠘⠁⢀⠀⠈⠂⡀⠀⠀⠀⠐⠂⠉⣦⣀⢀⡤⠖⠚⠉⠁⠀⠀⠀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢕⠅⢀⡀⣀⣤⠴⠖⠒⠋⠉⠀⠙⠀⡈⠐⠀⡀⠀⠀⠀⣀⠄⠕⠂⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢴⠂⠋⠀⠐⠄⠀⠀⢀⠀⠀⠀⠊⢢⣀⡀⣤⢖⠉⠀⠀⠀⠀⢀⠀⠘⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠘⠀⢀⣀⠤⠐⠐⠀⠉⠑⠢⠀⠀⠂⢀⠄⠀⠀⠀⠀⡠⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⡗⠸⠋⠠⠀⠀⠀⠀⠀⢀⠀⠀⠑⣄⣄⠀⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢄⢀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠠⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⢀⡠⠖⠒⠒⠒⠦⣌⠀⠀⠀⠀⠀⣠⠖⠒⠛⠙⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢀⠤
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⡈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠇⢸⢸
⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠠⡴⠚⠛⠉⠉⠉⠉⠉⠐⢦⠘⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠒⠀⠈
⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠈⠄⠐⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⣁⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣠⡄⠀⠀⠆
⠀⠀⢀⠔⠀⠀⠀⠁⠑⢤⠤⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠁⠁⠀⠐⠠⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢈⠂⠤⠄⠀⠀⠀⠀⠠⠄⠒⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⠂⠀⠀⠄⠤⠤⠠⠀⠐⠘⡀⠀⠀⠡⢀⣀⣀⣀⣀⡀⠆
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠒⠈⡆
⠀⠀⠀⠈⠂⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠁⠀⠂⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠂⠀⠄⢦⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠁⠀⠈⠐⢀⡀⠀⠀⠀⠀⠁⠀⠹⠇⠀⠀⠠⠁⠀⠰⣄
⠀⠀⡐⠀⠀⠀⠀⠀⠀⠉⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠃⢨⠀⠀⠀⢈⠉⠃⡀⠀⠀⠈
⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠠⠰⠁⠀⣨⠉⠉⠀⠰
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠰⡄⠀⠀⢀⠋⠀⠉⠠⠠⠀⠋
⠀⠀⠈⠢⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠑⠤⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠈⢀⠀⠀⡤
⠀⠀⠀⠀⠀⠈⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⣴⣶⣧
⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⣿⣿⣀⠍⠀⡄⣄⡆⠀⡆
⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠂⠀⠀⠐⠀⠀⡰⠋⠀⠀⠑⡄⠀⠒⠋⠈⠀⠀⡤⠁⠈⠁⠠⠁⠀⠀⠀⠁
⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⣀⠗⠀⠀⠈⡀⠀⠘⠇⠈⠐⠀⠃⠐⠒⠀⠀⠢
⠀⠀⠀⠈⠓⠦⣄⣀⣀⠤⠞⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠀⠁⠀⠀⠐⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣄⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠘⠛⠉⠁⠈⠛⢦⡀⠀⠀⠀⠀⠀⣀⡴⠶⣤⠴⠶⠶⠶⣤⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠋⠉⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣤⠴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠐⠃⣠⠞⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠢⡀⠀⠀⠀⠀⠀⠀⣰⣲⡌⣧⠀⠀⠛⠛⠒⠋⠀⠀⠀⠀⢀⡴⠛⣦⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢀⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠋⠀⣼⠃
⠀⠀⠀⠀⠀⢠⡟⠁⠈⢳⠀⠀⠀⠀⠀⠀⣀⣀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠃
⠀⠀⠀⠀⠀⢘⡷⢦⡤⠞⠓⠀⠀⢀⣠⠎⠁⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⣞⡦⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠈⠉⠈⠳⠤⠴⠚⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠀⠠⢤⣤⣴⠖⠾⣏
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⣀⡤⠶⠋⠁⠀⠀⠀⠉⠳⠦⣄
⠀⠀⠀⠀⠸⣦⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⢤⠞⠀⠀⠀⠀⠀⢀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⢠⡏⠀⠀⠈⠉⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⣸⠀⠀⠀⠀⡴⠒⢢⡀⠀⠀⠐⠢⠀⠀⢺⡄
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⢠⡏⠁⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠊⢠⡞⠁⡀⠀⠀⠘⠳⠶⠖⠃⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠘⢳⡆⠀⠀⠀⠈⠳⠤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠋⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⢠⢴⡞⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⠴⠖⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⢠⡀⢠⡀⠀⢸⠁⠀⠀⠀⢹⠔⠒⠊⠉⠀⠀⢀⡴⠛⠲⠶⠶⠚⠓⠒⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢧⡀⠀⠀⠀⠉⠁⠈⠙⠛⢦⣤⣤⠶⠋⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⣤⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⠴⠶⠶⠶⠶⠶⠦⠤⠤⠶⠋⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠾⢧⣤⡼⠇⠀⠀⠀⠀⣠⣤⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉




⣿⡟⣋⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠿⣿⣶⣲⣶⠦⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⠀⠉⢳⣝⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠁⠈⠉⠉⠓⠒⠦⢤⢿⣷⣽⣿⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣾⣿⡿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⢋⣁⣠⢤⣴⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢣⣶⣤⣄⣀⠀⢠⡿⠿⠒⠋⣩⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠈⠃⠘⠻⢿⣿⣿⣿⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢠⠀⡀⠀⠉⣩⣼⠀⠀⠀⠀⠀⠈⠉⠹⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠀⠀⠀⠁⣤⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠉⠀⠠⠄⢰⠇⠑⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣄⣒⡺⠏⠀⢠⣶⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠰⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠛⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠹⢻⣷⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⠃⠀⠀⢀⠇⢰⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⠤⡼⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠣⠤⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡀⠀⠀⠀⠀⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⢸⢸⣿⡿⢟⣀⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣟⡬⠕⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⡿⠿⠿⠿⠿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡄⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⣰⣾⣷⣦⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⢠⣾⡿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠻⣿⣿⡿⠀⠀⢀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣿⡟⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣸⣿⠋⠀⢀⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢰⣿⣿⣿⡄⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠘⢿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠈⠻⠿⠟⠁⠀⠀⣤⠀⠀⠛⠿⠟⠁⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠀⠀⣶⡆⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⢀⣤⣦⣤⠀⠀⠀⢀⡀⠀⠀⢀⣠⣀⠀⠘⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⢸⣿⣿⣿⠇⠀⠀⠙⠁⠀⠀⣿⣿⣿⣷⠀⢸⣿⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠁⠀⠀⣻⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣷⣶⣶⣶⣶⣿⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⣃⣀⣚⣃⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠯⠯⠿⠽⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠐⠂⠀⠀⠀⢲⡆⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣿⡹⡄⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣥⠈⠩⣷⡩⠀⠀⠀⣀⣀⣀⣀⣻⣆⣀⣰⣶⣶⣦⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡫⠽⣝⠲⡬⣧⠀⣶⠿⠟⠛⠛⠋⠉⢻⡍⠁⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⡴⠷⠤⣀⠈⠁⠛⣧⢻⡀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⣴⣤⡀⠀⠀⠀⠐⠉⠓⣶⢿⣧⢷⠀⠀⠀⠀⢀⣀⡀⢻⠄⠀⠤⢖⣂⣒⣠⡄
⠀⠀⠀⠀⠻⣿⣿⣓⠀⠀⡠⠞⠓⠉⠉⠒⢮⣧⣀⢭⠭⠭⣙⣛⣓⣚⡩⠭⠴⠒⠒⠊⠔⢀⣤⡀
⠀⠀⠀⠀⠀⠈⠛⢿⣷⣴⡶⣄⠀⠀⠀⢀⠉⠉⢻⠉⠉⣩⣍⠉⠉⠀⠀⠑⢤⠀⢠⠖⠀⠈⣾⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣮⣅⠀⣀⠠⠀⢈⠠⠃⠀⢿⣯⡳⡄⠀⠀⢂⣠⡇⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣇⠉⠀⠉⠀⠀⠠⠄⠀⠙⢿⣦⠄⢠⣦⢐⣧⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠄⠚⠀⠘⠻⠛⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡉⣻⣿⡝⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣀⠇⠀⠀⠀⠀⠀⢠⣤⣶⣶⣿⣿⣿⣶⣄⢰⣿⠟⠻⠥⠘⠋⠛⠛⠿⣿⣷⣄
⠀⠀⠀⠀⠀⣠⠶⠄⠀⠀⠀⠀⠀⠸⠂⠀⠀⠄⠀⠈⢡⣤⣤⣍⡉⢹⡿⠙⣧⣤⣶⣶⣶⣶⡶⠆⠠⣨⠛⠃
⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠄⠀⠀⠀⠀⠉⠻⣿⣿⣿⠀⠈⢙⠟⠉⠉⠉⠉⠑⠀⠀⠆
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠄⠀⠀⠑⣈⠸⠿⠇⠸⡿⠇⣀⡀⠀⠀⠁⢠⡀⠀⠀⠀⣠⣶⣾⡀
⠀⠀⠀⠀⠀⠀⡀⠈⠳⡄⣶⠰⠏⠀⠀⠀⢸⣄⣴⣦⡴⣤⣶⠇⠬⠀⠀⠀⢠⣿⣹⣭⣷⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⡰⠀⡀⡄⡅⢻⢸⣷⢨⠀⣰⡻⢿⣿⣿⡟⣿⣿⣆⠀⠀⠀⠀⠸⣿⡧⢸⣿⡟⠻⡿⢿⠓⡄
⠀⠀⠀⠀⠀⢠⠀⠘⠁⠀⠀⢸⠈⠆⠀⠛⠿⠷⢾⣿⣿⣿⣿⢻⡆⠀⠀⠀⢀⣾⡶⣼⣿⣧⣶⣦⣜⡉
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣿⡟⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠛⠉⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⢻⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏⠁⠀⠀⠉⠉⠉⠁⠈⠉⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⣁⣠⣤⣶⣦⣶⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢢⡀⠀⠲⣡⡆⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢀⠐⠀⠈⠙⠃⠀⠀⠉⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠀⢠⠀⠀⠀⢀⣀⣴⣳⡀⠀⢼⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠃⣠⣤⣾⣿⣿⣿⣿⣿⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣩⣿⡉⢿⣻⡏⠀⠀⠈⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇⠠⣿⡟⠁⠀⠀⠀⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣠⡾⣻⣿⣿⡁⠀⣿⡏⠀⠀⠀⠀⣹⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⣾⣿⣿⣿⣿⣿⣧⣼⢿⠃⠀⠀⠀⠀⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⣿⣿⣧⣼⣿⣏⣸⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣯⡈⣿⣿⠛⣿⣧⣀⠀⢀⣠⠀⣿⣿⡏⠙⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⣱⣿⣿⣿⣿⣿⣿⣶⣿⡿⣿⣿⣿⣿⠀⢸⣿⣿⣄⣈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⢿⣿⣿⢿⣟⣻⣯⣿⣭⡛⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡋⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣯⣭⣤⣤⣾⣯⣿⣧⣽⣿⣽⣿⣿⣿⣾⣿⣿⣿⣿⣿⡇⠸⠿⠿⣿⣿⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣶⣤⣄⡉⠛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠉⠀⠀⠀⠈⠙⠻⠿⣿⣿⣻⣿⣿⣝⣛⣿⣿⣿⠿⠿⠿⢿⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣓⡶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣉⡉⠉⠛⠻⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⠀⠀⠀⠚⠛⠉⠁⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⡄⠀⠀⢰⣶⡷⠀⠀⠀⠀⠀⠀⠐⣷⡿⡿⢻⡇⠀⠀⡾⠿⠿⠛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣌⢻⣇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⣿⡃⠀⠀⠸⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣿⣷⣿⡆⠀⠀⠀⠀⠀⠀⢹⣿⣿⣽⠀⠀⠀⠀⠹⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠸⡿⠿⠷⠀⠀⠀⠀⡀⢘⣿⣿⠿⠿⠀⠀⠀⠀⠀⠟⠻⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠀⠠⠤⠤⠤⣤⣤⣤⣦⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡉⠀⣠⣶⣿⠇⠀⠀⠀⠭⠉⡁⠀⡐⢀⡑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⢿⣿⠏⠓⡉⢉⠀⡀⠀⠀⠄⠁⠀⠹⠿⠿⠷⣁⣿⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣷⣰⡟⣿⠤⢤⣀⣝⠲⣀⠀⢀⣠⢄⣂⣀⣰⠾⣚⣵⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⡈⠒⠚⠋⢾⠃⣤⣀⠈⠋⣉⣉⣿⣗⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠋⠉⠉⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⣀⢀⣔⣨⣻⠿⠿⠻⢒⡁⠈⠓⠀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠴⠶⣿⠿⠠⢆⣐⠠⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠊⠀⠜⠛⣿⡷⡛⠂⠐⠿⣿⣷⡤⡀⠸⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠸⣻⣆⠠⣤⣴⣤⡀⠘⠿⠿⠃⠠⠘⣿⣧
⠀⠀⠀⠠⡖⠁⠀⣠⠀⣀⣤⣌⠀⠀⣤⣤⣄⠀⢈⠹⣟⣷⡀⡌⣇⠀⠀⠀⠀⠀⠀⠸⣿⣇⣆⡤⢙⡛⢦⡁⠀⠀⠀⠀⠀⢀⠁⠰⡀
⠀⠀⠀⢸⣷⡀⢀⠠⠾⠿⠿⠏⠰⠀⠙⠻⠿⠗⠈⠉⠨⠝⡁⣸⢿⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣄⠹⣿⠆⠹⣆⠰⡈⢷⡄⠹⢿⢀⣧
⠀⠀⠀⠀⢿⣼⡧⣀⠁⠠⡄⢀⡀⣴⣶⣿⡷⠠⠤⣔⣲⣯⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⣄⢀⠃⠧⡘⠛⠣⡀⢰⡿⠋
⠀⠀⠀⠀⠀⠈⠛⠻⠶⢦⣤⣵⣦⣤⣦⣤⡴⠶⠿⠿⠛⠋⠀⢀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢼⣶⣿⣲⣾⠷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠃⣠⣤⡄⠻⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁


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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠈⠳⡄⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠘⢦⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠸⠆⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠒⠦⣀⠀⠀⠀⢹⠀⠀⠀⣠⠤⠤⠤⢤⡀⠀⠀⠀⠀⣠⠤⠤⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠸⡗⠛⠋⠁⠀⠀⠀⠀⠉⠙⠒⠛⠉⠁⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠚⠀⠀⠀⠈⠁⠀⠀⡇⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⣀⡤⠖⠒⠺⡄⠀⠀⡠⠀⠀⠀⠀⠀⢀⡇⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⢀⡼⠁⢀⡄⠀⠀⠹⣄⠈⠀⠀⠐⠋⠀⣠⡞⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⢸⠀⠀⠀⠀⢀⡀⠀⢈⡿⠲⠦⠤⠖⠚⠉⠙⢦⡀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⡏⠀⠀⣄⠀⠀⠀⠀⠀⠀⢳⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠳⡀⠈⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠆⠀⠘⠀⢸⡇⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠈⠓⠶⠤⠤⠾⣇⠀⠐⠂⠀⢀⡀⠀⠀⠀⡼⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠚⠉⠁⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⢉⣠⠶⠳⣄⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣿⡇⠀⠒⠋⠀⠀⠀⠙⠇⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⣤⣤⡈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄
⠀⠀⠀⠀⠀⠀⣀⣴⡞⠉⠠⠞⠉⠀⠈⢷⡀⠹⣧⡀⠀⣀⡀⠀⠀⠀⣀⡼⣿⣦
⠀⠀⠀⣀⡴⠚⠉⠻⣧⡀⠀⠀⠀⠀⠀⠈⠓⠀⠙⣷⣄⠈⠁⣀⡴⠞⣉⣄⠘⢿⣧
⢀⡴⠚⠁⣠⠴⠛⢦⡈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡟⠁⠠⠚⠁⠈⠳⠈⡿⣷⡀
⣿⣄⠀⠈⠀⠀⠀⠀⠻⡌⠻⣆⠀⢤⣤⠀⢀⣠⡴⠛⠉⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣀⣠⠶⠋⣡⠶⠛⢷⡀⠻⣷⡄⠀⡀⠀⠀⠀⠀⢀⠽⠛⠙⠒⠢⣤⣤⣄⣀⡀
⠈⢿⣿⣦⠀⢠⣀⣀⡀⠀⢀⣼⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠁⠀⣀⡤⠚⠁⠀⠀⠀⢀⠀⠀⠀⠀⠉⠉⣉⡶
⠀⠀⢻⣿⣧⡀⠉⣡⠴⢚⣩⣤⣄⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⣴⠋⠁⠀⠙⠶⠀⠀⠀⠈⠻⣤⣴⠖⠛⠋⠁
⠀⠀⠀⠹⣿⣿⡏⠁⠀⠉⠀⠀⠙⠇⠀⢹⣧⠀⠑⠖⠀⣀⡤⠖⠉⠉⠛⠒⠁⠀⠀⠀⠀⠀⣤⣴⡶⠛⢧⡀
⠀⠀⠀⠀⠘⣿⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⣠⠴⢚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠏⠁⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠈⢷⣿⣧⡀⠳⡄⠀⠀⢀⣠⣶⠋⠁⠰⢿⣿⣶⣔⣀⣢⡀⢀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⡿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⢀⡤⠞⠉⢀⣸⣧⣤⣴⡾⠋⠓⠉⠀⡙⠿⢦⣤⡀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣩⣤⠶⠚⠋⠉⠁⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⡌⣿⣆
⠀⠀⠀⠀⠀⠀⠀⡜⠀⢠⣿⣿⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⣠⠔⠊⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠙⣟⣊⣻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠁⡠⠊⠁⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⡠⠊⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⠀⠀⠀⠀⠀⢀⡴⢁⡠⠊⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣿⣷⡀⠀⠀⣠⠊⢀⠞⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣄⠀⠁⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣷⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢻⡄⠀⠀⠀⠀⠀⠀⣀⡴⣻⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣿⡄⠀⠀⠀⣠⣶⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣆⣠⠾⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠶⠛⠶⢶⠤⠖⡾⠖⠚⠋⠛⠒⠶⣤⣀
⠀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠉⠀⠀⠀⠀⠃⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠉⣑⠦⠤⢠⡀
⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠄⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⡴⠋⠀⠀⠠⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠐⠀⠱⡀⠀⠀⠤⠀⠀⠀⠀⠀⠀⣀⠀⠙⢦
⠀⠀⡼⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⠤⡠⠇⠀⠀⠀⠀⢀⠄⠢⢄⠀⠈⠀⠀⠈⠳⣄
⠀⣸⠁⠀⠐⠂⠀⠀⠀⠀⠀⠀⠸⡀⢈⠆⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠠⡀⠀⢀⠇⠀⠀⠀⠀⠀⣸
⢠⡏⠀⠀⠀⠀⠐⠁⢱⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠔⠠⡄⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢻
⣏⠑⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢸⡀
⢻⠀⠁⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠊⠒⢄⠀⠀⠀⠤⠀⠐⠀⠀⢱
⣼⠐⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠘⢀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢧⠀⠁⠀⠀⢀⠀⠜⠀⠄⠀⠀⠀⠀⠀⠁⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⣸
⣸⠀⢄⠀⠀⠀⠁⠀⠀⠀⢀⡀⠀⠀⠀⡄⠁⠀⢱⠀⠀⠁⠀⠀⠀⠀⣀⣀⡀⠀⠂⠀⠀⠀⢀⡠⢤⡼⠁
⢻⡀⠈⡆⠀⠀⠀⠀⠈⠀⠀⠃⠀⠀⠀⠣⣀⣀⠜⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⡀⠀⣇
⠀⢳⡀⡃⠀⠀⠀⠀⠑⠒⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠓⠄⠀⠂⠀⠀⠀⠀⠀⠀⣹⠏
⠀⠈⣇⠐⠄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠹⣄⠀⠈⢂⡀⠉⠀⠁⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠊⠀⠀⢀⣰⡿⠋
⠀⠀⠀⠙⢷⣄⣀⡈⠂⠠⠤⣀⡀⠀⠀⠠⠒⠀⠐⠄⠀⠀⠀⠀⣀⠠⠤⠀⠀⠀⠀⣉⣥⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⠀⠀⠈⠙⣇⡀⠀⠠⠇⠀⢒⣉⣥⡤⠤⠴⠶⠖⠛⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠢⠀⠠⠉⡉⠨⠑⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠀⠀⠀⠁⢀⠀⣀⠁⠠⠖⠀⠰⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⡄⠀⠦⠀⠀⠁⠀⠀⣀⣀⡀⠀⠠⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠈⠓⠤⣀⠀⠈⠀⠀⠀⠉⢉⠐⣁⣤⠊
⠀⠀⠀⠀⣀⠀⠀⠒⣀⠀⢐⠀⡄⠀⡀⠀⠈⠀⠀⡀⠐⠘⢀⢬⠁⠐⠒⠩
⠠⠊⡄⠀⠊⠀⠒⠀⠁⠀⠀⠀⠘⠦⡀⠛⠀⠒⠀⠀⠀⠀⡤⠤⠄⠐⢀⡴⠄
⠃⢰⠃⠀⠀⠐⠀⠀⠰⠖⠉⠀⡍⠘⠈⠙⠒⢀⠀⠤⠤⠀⣤⡤⡀⠈⠁⠠⠘⠒⠚⠉⠭⠙⠋⠑⠢⢄⡀
⢃⠫⠈⠚⠂⢀⠀⠰⠂⠀⠦⠆⣀⠙⣀⠠⠌⠀⠄⠀⢀⠀⠀⡀⠁⣀⡎⠠⠀⠈⠀⣀⠀⠀⠛⠀⠂⠀⠑⢄⡀
⠀⠉⠒⠜⠀⠀⠀⠀⠀⠀⡤⠂⠊⠠⡈⠁⠦⠄⣀⣀⣘⡀⢈⣁⣤⡍⠀⠂⡀⠒⠀⠁⠀⠂⠀⠂⠁⣀⠀⠀⢧
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠚⢣⡠⠀⠑⢂⠀⠀⠥⠠⠁⣄⠀⠀⠇⠠⡄⠀⠀⠀⠠⠄⠀⡀⠠⠄⡀⠀⠄⠐
⠀⠀⠀⠀⠀⢀⠤⠀⠀⡀⠀⠒⠶⠨⢅⣢⠬⠀⢀⣀⣀⣄⣉⡄⠤⠜⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠆⠀⢰⢂
⠀⠀⡀⠊⠙⠀⡀⠈⠀⠀⠀⠀⡀⠀⠚⠃⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠒⢣⠀⠀⠀⠀⠢⠂⠒⡠⢜⡝
⠀⠀⠁⠀⠠⢀⡄⣈⠀⠀⠈⠁⠀⠀⠀⠘⠀⠀⠠⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⠄⢀⣐⣇⣈⣀⡠⠔⠋
⠀⢐⠀⠀⠄⠀⠄⢀⡀⠀⠀⠀⠘⢁⠠⠀⠀⠓⢃⢴
⠀⠈⠦⡋⠀⢠⠀⠀⠀⠠⣒⣢⡄⠀⣀⠀⢀⡈⢱⠎
⠀⠀⠀⠈⠑⠢⢄⣀⡀⠀⠀⠁⠀⠀⠀⣀⠠⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠀⠀⠐⠰⠤⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠝⡤⠀⠖⠀⠀⠀⠀⠀⠀⠈⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠐⠀⠈⠧⠀⢀⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⡄⣴⣴⠂⣠⣦⡄⠀⠀⠀⠀⠀⠀⠿⠏⠁⠀⣠⣴⠖⣧⠈⠁⠉⠀⠒⢯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⢠⠄⠈⠁⢀⣿⡆⢈⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠏⠀⠉⠀⠀⣤⣤⡆⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⢀⠋⠐⠀⠀⢀⠀⠀⢸⣿⣆⢿⡯⠀⣀⠀⢀⣠⣴⣦⠀⠀⣼⠿⣿⡿⠀⢀⡀⠀⠀⠈⠉⠀⣀⣴⡞⡛⠁
⠀⠀⠀⠀⠀⠀⢀⡂⠀⠀⠀⠸⠿⠟⠁⢻⣷⡿⡄⢀⡀⠀⠀⠀⠉⠛⠋⠀⣸⣿⠇⢈⢃⣤⠀⠀⠈⠂⠀⣰⡞⠻⡽⠖⠀⢀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⣀⠀⠘⣿⣧⣿⣨⣈⡀⠘⠒⠀⢸⣿⣭⣿⣿⠷⠞⠂⠁⠀⠀⣈⡉⣹⣿⣢⡞⠁⠀⠀⣈⠁
⠀⠀⠀⠀⠀⠀⣿⡇⠰⡄⠀⣦⡈⠉⠉⠀⣿⣿⣿⣯⡻⠷⠀⣴⠢⢦⣼⣿⣿⣿⣦⣐⡀⠀⠀⠀⠻⢻⣿⡁⡘⣰⠦⡀⠞⠃⠁⠀⡾⠉
⠀⠀⠀⠀⠀⠀⠹⣧⣀⠀⠀⠈⠁⠀⠀⠉⣿⣿⣿⣿⣧⣤⣦⣤⣓⣶⣿⣿⣿⣿⣻⡿⠧⠀⠀⠀⠀⢨⣟⢻⠠⡇⠠⠇⠀⡀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⢀⠀⠂⠀⠐⠲⢛⣩⣿⡿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⢛⣤⠀⠀⠀⠀⠈⣿⣺⠀⠁⡀⣀⠘⠃⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣯⠁⠈⠀⢀⡀⠈⣛⣛⠡⠀⠹⣿⣿⠟⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⢿⡗⠀⣿⡇⠀⠀⣽⣯⣶⡄⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠯⣟⣋⣿⣧⢤⡐⠂⠰⢞⠹⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠹⣷⡀⠀⠛⠨⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠃⠛⠁⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⠟⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠿⠿⠿⠟⠁⠀⢈⣿⡇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⠁⠀⠀⢸⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠋⠀⠀⠀⠼⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠻⠟⠿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡆⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣆⠀⣰⣶⣦⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⡿⢿⣿⠁⠙⣯⣼⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⣶⣿⣷⣾⣿⠻⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣉⣿⣿⡿⣿⠋⠀⣀⣀⣤⢤⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡇⢹⣿⣿⣤⣿⣿⣿⣾⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠁⠈⣿⣿⣿⣯⣿⣿⣿⣿⣿⣧⣼⡿⠛⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡟⠀⢀⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⣿⣶⣿⣙⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡟⠀⢸⣿⣿⡿⢿⣿⣿⣿⣏⣿⣿⣿⡿⢿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣾⣿⣿⣷⣾⣿⣿⣿⣿⣟⠉⣿⣾⣧⠈⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣦⣈⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣂⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠂⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠟⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠏⠀⢹⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣶⣶⣶⣦⣤⣤⣀⣤⣤⣤⣤⣤⣬⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⠿⢿⣿⣿⡿⠿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢫⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⢈⣿⣿⣿⣿⣿⣿⣷⣆⣰⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡟⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡀⠀⠀⢀⣿⣿⣿⡿⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣶⣿⣿⣿⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣮⣍⣛⣛⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣾⣿⣷⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣳⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣭⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠲⣄⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⠛⣁⡤⠶⠚⠉⠁⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⡴⠚⠉⠀⣀⣠⣤⣤⠶⢦⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⢏⢾⣁⡠⢤⣴⡛⠋⢒⡁⠀⠳⣄⠙⢦⣤⢴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣤⠜⠀⣿⠉⣳⣦⠇⢠⡾⠯⠤⠤⣿⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣾⠛⢛⣩⠼⢛⣛⠓⠶⠖⠚⠁⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠉⠉⠀⠀⠛⠚⠁⠀⠀⠀⠀⠀⠀⣴⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠉⠙⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⢀⠌⣷⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⣄⣰⠋⠀⢸⣆⣀⣤⡴⣶⣯⣹⡇⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠹⣇⣴⠃⠈⠉⠉⠀⠀⡿⠓⢻⠃⠀⠀⠀⠀⠀⠀⠉⠛⢷⡀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣹⣧⣄⠀⠀⠀⠀⠀⣷⠐⣼⠀⢾⠓⠀⠈⣿⠀⠀⠀⠀⣷⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡴⠛⠉⣀⡿⣤⡀⠀⠀⢐⣏⣉⣹⣀⣸⠀⠀⠀⡿⠀⠀⠀⣰⡛⠓⢦⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡀⢀⠜⠁⠀⢰⡟⣛⣋⣩⣭⠭⠭⠭⢽⡗⠐⠚⡟⠶⣖⢚⡉⠙⢷⡀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢻⡉⠛⢧⠀⠀⠀⢻⡀⠺⣥⣤⠏⠀⠀⠀⠓⠶⠞⢉⡦⠈⣹⠀⠀⢸⣧⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠙⠶⣌⡳⢄⣀⣠⣟⣒⡒⠒⠒⠒⠒⠒⣒⣒⣉⣭⠴⣾⠃⣀⡴⠟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠛⠛⠶⠦⠭⠭⢭⣭⣭⣭⣭⠭⠥⠴⠶⠒⠛⠉⠁⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡷⢦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⣴⠾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠋⠙⠋⠉⠉⠉⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⢰⡟⣷⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡗⠀⠀⠀⢀⣾⠀⢸⣇⠀⠀⠀⠀⠀⠈⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠈⣿⠀⠀⢻⡆⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠓⠶⠤⠶⠟⢳⠀⢨⣷⣤⣀⣠⣴⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⣄⣀⣀⣤⡞⠀⠈⣷⠀⠉⠉⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠈⠙⢶⣶⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⡿⠿⠃⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⢩⡴⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⢀⠀⠀⠀⠈⠀⠀⠀⣠⠤⠀⠀⠠⢶⠀⣻⠀⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢰⠏⠀⢀⣴⡆⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⣿⠀⠀⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠋⠀⠀⠀⠀⠺⠋⠁⠀⠀⠀⢰⡇⠀⠀⠀⢈⠁⢀⠀⠀⢀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠁⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠁⠀⠀⠀⠹⠿⠀⠀⠸⠀⢱⡀
⠀⠀⠀⠀⠀⢠⠀⠀⠉⠁⠀⠀⠀⠀⣠⣤⠄⠀⠀⠀⡴⠃⠀⠂⢀⡼⠃⠀⠀⠀⠐⢿⣿⣦⠀⠀⠀⠂⠀⠀⠂⠀⣼⡇
⠀⠀⠀⠀⣦⠈⡀⠀⠀⢠⠆⠀⠀⠀⣻⡿⠀⠀⠀⠈⠠⠀⣄⣠⡟⢶⡷⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢺⠏
⠀⠀⠀⠀⠇⢸⡵⡶⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⣴⡿⣿⣶⣄⡐⣂⣀⠀⢀⠁⠠⠒⠀⠀⠀⢠⣴⣿⡿⠋
⠀⠀⠀⠀⢢⣦⡈⠲⠄⠀⠀⠀⠐⠒⠛⣁⣀⣴⣶⣿⣿⣿⣿⠀⠐⣦⣤⣤⠀⠀⠀⠀⢀⣂⣤⣶⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠈⠓⠶⢦⣤⣤⣤⣰⣶⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠻⠿⠾⣿⣿⡷⠶⠾⣟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣄⡀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣩⣭⣤⣤⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⢿⠟⢁⣤⣤⣾⣿⣿⣿⣇⠀⠀⢸⣿⣿⣿⣿⠿⠟⠋⠠⠄⠂
⠀⠀⠀⠀⢀⢀⣀⣤⣤⣦⣴⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠸⠟⣋⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⠟⠋⣠⠄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠉⠁
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠉




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣛⣟⣻⣭⣿⣻⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣭⣿⣭⣿⣯⣭⣯⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⢈⠀⠄⠀⠀⠀⠠⣈⣢⣏⢐⣃⠀⠀⠀⡀⠂⠀⢀⠄⠀⠀⠂⠄⠀⠈⣂⠀⢀⡀⣀⣨⡀⠀⠀⠀⠀⠀⠠⡀⠐
⠀⠀⡄⠁⠀⠀⠀⠂⠀⠀⠭⢚⢹⣿⢟⢔⠀⠀⠂⠀⠀⡠⠉⠀⠀⣦⠀⠠⠃⠂⢘⠀⠠⠀⢿⣿⡿⢀⠁⠀⠀⠉⠀⠈⠁
⠀⠀⠈⠢⠤⠈⠠⠔⠁⠀⠀⠃⠆⠹⠗⠁⠀⠀⠀⠠⠈⠀⠀⠀⠀⠈⠓⠀⠦⠒⠁⠀⠀⠐⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⣤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐
⠀⠀⢀⡕⠠⠂⠀⠀⠀⠀⠴⣿⡟⡩⠄⣠⠀⠀⣚⠽⠚⠩⣖⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢴⣶⡄⠈⡀⠀⠀⠀⣴⣄
⠀⠀⠀⠁⡀⠁⢈⡄⠀⠀⢃⠉⣽⢿⡾⡯⠀⠀⠵⠪⣔⡯⠷⠀⠰⠀⠀⠀⠀⠀⠀⠀⠐⢂⠈⠁⠀⡀⠀⠀⠀⠀⠈⠁⢀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⢀⣠⡲⢄⣀⠀⠀⢀⠀⠀⠙⢲⠄⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⡰⡆⢶⡀
⠀⠀⠀⢨⠇⠂⣉⡏⠀⠀⠀⡤⠀⠀⠀⡀⠀⠀⠀⠘⠿⠏⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢻⡈⡇
⠀⠀⠀⠀⠉⠩⠋⠁⠀⠀⠀⠂⠠⠤⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡤⠴⠶⠶⠶⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠲⠶⠶⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠓⠲⠦⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠉⠉⠉⠉⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣠⠶⠛⠉⠉⠉⠉⠉⠉⠛⠓⠶⠦⣤⣤⣤⠶⠛⠋⠉⠉⠉⠉⠙⠳⢦⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡶⢿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠛⠉⠙⠛⠶⣤⣀⡀⢀⣀⣤⠞⠋⠁⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠙⠓⠦⠤⠤⠤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣤⣤⣴⠿⠶⠶⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⢀⣀⣀⣴⠟⠋⠀⠀⠀⠀⠀⢀⡴⠶⡄⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⢠⣿⡽⠋⠀⠀⢀⡤⠶⡄⠀⠀⠈⠳⠶⠃⢀⣀⣠⣼⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⢀⣼⠋⠀⠀⠀⠀⠘⠶⠶⠃⠀⠀⣀⣤⠶⡟⠉⣹⠇⠀⠀⠀⠀⠈⠉⠉⠙⠓⠦⣴⣞⠳⡄⠀⠀⠀⢀⣿
⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠉⠉⠁⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠉⠻⣧⠀⠀⠀⣸⡇
⢠⡿⠁⠀⣴⠖⢲⡄⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣀⣷⠀⠀⠀⠀⣄⠀⠀⠀⠈⢳⣶⠟⠋
⢸⡇⠀⠀⠙⠓⠛⠀⠀⠀⡾⠀⠀⠀⠀⣠⠖⢶⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣿⠀⢀⣤⣄
⠘⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠙⡇⠀⠀⠀⠀⣿⠀⠻⠷⠛
⠀⠈⠳⢦⣤⣤⣤⣤⣶⣟⢿⣄⠀⠀⠀⠀⠀⠀⠀⣴⠒⣆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⣼⠇⣴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠇⠙⠷⣦⣀⠀⠀⠀⠀⠹⠤⠿⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢦⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⢀⣠⠶⣤⣴⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠻⠿⠛⣾⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣠⠖⡿⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡤⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⡿⠁⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠘⠋⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠔⠀⠀⡠⠊⢀⣄⠀⠀⠀⠀⠀⠰⠦
⠀⠀⣸⠀⠀⠠⡿⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠈⠀⠀⠴⠂⠀⣀⠈⢀⡠⠊⠀⠀⣰⡥⠀⠙⠁⠀⠀⠀⠐⢶⠆
⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠂⠀⢀⡄⠤⠐⠀⣀⣸⠟⠂⠁⠀⠀⠀⠀⢹⡅
⠀⠀⢸⠀⠀⠀⠀⠀⠰⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⠀⠀⠀⠀⠐⠓⠂⠀⠀⢴⠆⠀⠈
⠀⠀⠙⠇⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠑⠀⠉⠁⠀⠀⠀⠐⠋⠀⠛⠁⠀⢀⣠⡤⣄⠀⢀⣤⣤⣤⣄⣀⢠⣟⣡⠇⡀⠀⠀⣠⢤⠀⠀⠀⠀⠓⢿⡆⠀⠀⢈⠀⠤⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢃⣴⡿⠀⠛⠋⠉⣉⣉⠛⣿⣛⣅⢠⡍⣴⢶⠻⣾⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠀⣼⡏⢷⢸⡏⢸⣸⣿⡵⠟⣾⣃⣿⡨⢮⡌⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠉⣸⣿⢦⡟⢻⣉⡟⠹⠏⢻⣦⡬⠵⠶⢄⠘⠳⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣠⠞⠁⠙⠊⣤⣤⣤⣴⣶⠿⠋⠉⠛⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠤⠀⠰⠿⠃
⠀⠀⠀⠀⣠⠀⠀⢀⠀⠀⠀⠚⠷⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠀⠓⠦⠀⠀⠀⠀⠀⠀⠀⢠⠞⣤⣤⠔⠒⠂⠢⠄⠈⠀⢀⡤
⠀⠀⠀⡌⠁⠶⠄⠉⠀⠀⠀⣤⡤⠀⠀⠙⠛⠀⠀⠀⠀⠀⡀⠐⠋⠀⠰⠶⠆⠀⠀⣀⡀⠀⣀⠀⠀⠀⠀⢻⡟⢁⣤⠀⠀⠲⠀⡀⣀⡀⠀⢀
⠀⠀⠰⣿⡄⠀⢀⣄⢀⣴⣤⠀⠀⣴⣦⡄⠠⢢⠆⠀⢀⡾⠀⠚⠂⠀⠀⠀⠘⠃⠀⠉⠁⠀⠙⠀⠀⠀⠀⢀⢱⣬⠕⠊⠉⡉⠉⠉⠑⠢⡀⣀⠃
⠀⠀⢸⣿⣿⣷⣴⣦⣤⣠⣤⣄⣀⣀⣀⣀⣤⡏⠀⠀⢠⠆⠀⠈⠁⠀⠀⠀⠄⠀⣦⡀⠐⠀⠀⠀⠀⠀⠀⢸⡟⠁⣀⡀⠀⠋⠀⠀⣀⡀⠀⠉
⠀⠀⠘⢫⣭⣌⣡⡄⠘⠛⠋⠉⢉⠉⣽⣷⣠⣾⠀⠀⠈⠃⠀⠐⢂⠐⠀⠀⡀⠀⠉⠀⠀⠘⠀⠀⠀⠀⠀⢨⠅⠀⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⢾⠿⢿⠷⠒⠷⠶⠶⠶⡾⠛⠛⠋⠈⠁⣀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠠⠄⠀⠐⠀⢀⠤⢄⠀⠀⢿⡔⠃⠀⠀⠀⠀⠺⠆⠀⠀⢠⡀
⠀⠀⠀⠾⣶⠶⢤⣤⢤⣤⣤⣼⣭⣤⣤⡴⢾⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠄⠤⠀⠀⠀⠀⠀⠀⠘⣇⠀⠺⠆⠀⣀⠀⠀⠀⠓⠈⠁
⠀⠀⢠⣾⣥⣄⣚⡃⠀⠀⣙⣁⣛⣀⣤⡤⠴⠖⠒⠀⠀⠀⠀⠐⠒⠀⠈⠁⠀⣀⠀⠀⠀⠀⢸⠆⠀⡀⠀⠀⠘⢤⡀⠀⠀⠉⢀⣀⠀⠀⣀
⠀⠀⠘⣻⣡⡾⡍⠛⣻⣏⠛⠉⠉⠉⠀⠰⢦⣈⠜⠁⠀⠀⠀⠀⠐⠆⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠾⠁⠀⠀⠀⠀⠉⠀⠲⠤⡤⠉⠐⠋⠁
⠀⠀⠀⠈⠉⠉⠙⠛⠒⠠⠶⠴⠖⠚⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠢⠀⠺⠆⠀⠀⠀⠀⠘⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣶⠶⠶⠶⠶⠾⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠷⠶⠶⠶⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣤⣿⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⣶⣶⣶⠶⠶⠶⠶⠶⠶⠶⠾⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⣹⡟⠉⠉⠛⠿⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣤⡀
⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣷
⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠟⠋⣿⡇
⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠿⠛⠉⠀⠀⠀⠀⣿⡇
⢨⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⢀⣠⣤⣶⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⢸⡇⠙⠿⣿⣿⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⣀⣤⣶⡾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⢸⡇⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣷⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣾⣿⣶⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⠀⠀⢠⣿⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⢸⠀⠀⢸⣿⡄⠀⣸⣇⠀⢸⣿⣷⣦⣄⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠉⣿⠿⢿⠀⣿⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⠀⠀⣼⡿⡇⠀⣿⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⢠⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⠀⠀⣿⠁⣿⢸⡏⢹⡄⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⣾⡏⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣾⠀⠀⣿⠀⣿⣼⡇⢸⡇⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⣸⡿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⣿⠀⢸⣿⠀⢸⡇⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⣧⣿⠃⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠁⠀⠘⡿⠀⠀⣿⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⠠⣶⣿⣄⡀⠀⣿⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⠀⣿⠀⠀⠀⠀⢸⡟⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠈⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣿⡆⢸⡇⢹⣇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⢸⡇⠀⢿⡄⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠙⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠘⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⣄⣀⠀⠀⢹⡇⠀⠀⣀⣀⣤⣴⣶⣿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⣾⣿⠿⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠙⢶⣶⣶⣦⣤⣤⣀⣈⠃⢹⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣧⣾⣌⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠉⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠋⠉⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠃⡄⠀⠀⠀⠀⣠⣤⡤⠀⢤⣤⣤⣀⡀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⣠⣽⣿⡇⠀⣼⣿⣿⣏⣱⠀⠀⠀⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⣹⣿⠁⠀⣿⣿⣏⣿⣿⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢹⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⣠⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⢠⠞⠁⠀⠀⠀⠀⠀⣠⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⡁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣬⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠈⠠⢀⠀⠀⠀⠀⠀⡀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣭⣴⣿⣿⠷⠄⡀⠀⠀⠀⠀⠀⢀⣀⠜⠁
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁⠀⠀⠀⠉⠉⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠉⢹⠿⠿⠿⠿⠿⠿⢿⡟⠛⠛⢻⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⣀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣤⣴⡆⠀⠚⠓⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠒⠒⠒⠄⣀⣀
⠀⠀⠀⣠⠴⠉⠉⠻⣐⠁⡀⢀⣤⠀⠛⠀⣠⠆⠀⠀⢀⣴⠋⠀⠁⠀⠀⡀⠄⡆⠀⣨⡳⡀⠀⠀⠀⡄⣄⠀⠐⠓⠀⠀⠀⠶⠊⠳⠄
⠀⠀⢠⠅⣴⡄⠀⠀⠈⢦⣐⠀⢀⣀⣀⡠⡀⠀⠀⢰⠁⠘⠀⣀⡀⠀⠘⠃⠀⠀⠀⠛⠀⢁⡀⠀⢰⠁⡇⢸⠇⠀⠀⠈⡆⠀⢠⡄⠠⡄
⠀⠀⣼⠀⠈⠁⠀⡈⠀⠈⠉⠓⠲⠷⠋⠀⢠⠀⠀⡰⠀⠀⠘⠿⠗⢀⠀⠀⠀⢀⠀⠀⠀⠀⡇⠀⢸⠀⡗⠀⠀⣤⠀⠙⠀⠀⠀⠀⠀⣈
⠀⠐⡏⠐⠂⠀⣄⠛⠀⠀⣶⠄⠀⠀⣠⡌⠘⠀⠀⢃⠀⠀⣅⠀⠀⠀⠀⠀⠀⠀⠱⢀⡖⠀⠇⠀⠀⠂⠳⣀⠃⠀⣠⡀⠀⠀⢠⣆⢀⠜
⠀⠀⢻⠀⢶⡆⠀⠀⠀⠀⡄⠠⢤⣶⠙⣡⠆⠀⠀⠘⣄⡀⠁⠀⠀⠐⢤⣤⡤⠀⠀⣀⣬⡍⠀⠀⠀⠈⠢⡙⠦⡀⠈⠁⠀⠯⠀⢉⠎
⠀⠀⠈⠳⣤⡄⠀⠀⠀⢶⢀⢀⠀⣡⣾⠏⠀⠀⣀⠀⠈⠙⠶⢄⣀⠀⢀⠀⠀⢒⠨⠝⠊⠀⠀⠀⠀⠀⠀⠉⠂⠽⣧⣷⣶⣶⠴⠋
⠀⠀⠀⠀⠀⠙⠛⠫⠄⣤⣶⠾⠟⠛⠁⠀⢀⠞⠉⠉⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡈⠀⠀⠐⠂⠀⢀⡑⠄⠀⠀⠀⠀⠀⢀⡀⠔⠊⠀⢠⢼⠃⠀⣅⠀⠀⠉⠵⠦⢀
⠀⠀⠀⠠⠬⡛⠈⠀⠀⢀⢜⡍⠉⢡⡌⠉⠙⠓⢆⠀⣀⡀⢈⣁⠈⢀⠀⠀⠀⠀⠉⢠⣶⠀⠀⠀⠸⡊⠀⠀⠉⠀⢀⣀⠱⡈⢂
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠁⢀⡊⠁⠀⠀⣤⡌⢂⠈⠁⠀⡋⠆⠠⠀⠀⠀⣀⠀⠀⠀⠀⣴⠀⠀⢇⠀⠀⠀⣠⠈⠉⠀⢻⠺
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠈⠀⠀⠀⠡⠀⠀⠀⢤⡀⠀⣧⠀⢀⡁⠶⠠⠀⠀⠀⠁⠠⠄⡆⠀⠉⠀⡼⠈⠀⠀⠘⠋⠀⠀⡄⠐⢠⡆
⠀⠀⠀⠀⠀⠀⢻⠀⢀⠀⠀⡾⠀⠀⠀⢠⠀⠀⠁⢀⠟⠂⡙⠁⠀⡀⠀⠀⠘⡃⣀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡨⠁
⠀⠀⠀⣀⠀⠀⠀⢣⡀⠄⠀⠀⠀⠀⢀⢀⣀⡤⠠⠋⠈⠂⠀⠉⠀⠀⠀⠀⠀⠘⢟⠀⢀⠴⠉⠀⣠⡔⢢⠀⢃⢠⣠⠄⠀⠈⠁
⠀⠀⠸⠃⠙⢄⠀⠀⠉⠲⣌⣉⣒⣛⣛⠉⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⢉⠂⡀⠀⡈⠀⢀⣠⡄⠈⣀⢀⡀⠐⠁⡀
⠀⠀⢰⡾⠆⠀⠐⣄⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⡀⠤⡀⠠⠤⡀⢀⠀⠀⠀⠠⠬⠀⠘⣴⠿⠀⣁⠐⠌⠹⠽⠀⠀⠒⡄⠉⠄⠉⠀⢠⡀
⠀⠀⠀⠳⡀⠘⠃⢀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠑⣄⠀⠀⠂⠈⠀⠀⠐⠀⠉⠀⠀⠐⠀⠀⠀⠰⡦⠘⠁⠀⠀⠠
⠀⠀⠀⠀⠈⠉⠉⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⣰⡶⠀⠀⡶⠆⡀⠀⢈⢾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠛⠀⢀⣤⢀⠠⠀⠐⡉⠀⠐⢀⠀⢁⡀⠀⠀⠀⠀⠀⢠⡤⠒⣚⠋⠁⠀⠀⠄⢀⡉⠉⠁⠒⡄
⠀⣠⠒⠐⠒⠦⠤⢀⣀⠀⠀⠀⠀⢇⠀⠀⢀⡂⠠⠐⠈⢀⠀⠀⠐⠉⠀⡄⢀⠀⠀⠀⠀⢘⠁⠀⠀⣀⡀⠀⠀⠀⠀⠉⠁⠀⠀⢈⣡⠄
⠀⢦⡀⠀⠓⠀⠀⢠⠌⠁⢢⡄⠀⠀⠁⠈⠁⠀⠚⠀⠀⠠⠴⣢⠄⠁⠈⢀⡉⠉⠘⡇⠀⠘⡤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⢠⣤⠝⣷⠄
⠀⠀⠑⢄⠢⡀⠀⠀⠀⠀⢀⡀⠈⠢⠀⠀⠀⠉⠀⣠⣼⠀⣀⠛⠁⠀⣀⠈⠋⠀⣼⠁⠀⠈⣧⡶⣬⣍⡉⠉⠉⠉⠉⠁⠰⠆⠀⣀⠤⢲⠇
⠀⠀⠀⠀⠑⢌⡓⠄⡀⠀⠈⠁⠘⠂⠠⡀⠀⠳⣄⡙⠋⠐⠁⠀⠀⠀⠻⠀⣠⠾⠁⠀⠀⢈⠠⡈⠉⠙⢷⡦⠄⢒⣒⣐⡤⠤⠥⠖⠚⡇
⠀⠀⠀⠀⠀⠀⠙⠲⠌⢉⠓⠲⠤⠤⢤⠃⠀⠀⠈⠙⠛⠦⠤⠤⠤⠶⠖⠚⠁⠀⠀⠀⠀⠈⠑⠤⣅⡀⠀⠀⠀⠀⠈⢀⣀⣀⣈⣽⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠉⠉
⣿⣿⠿⢿⡿⠿⠿⣿⠿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣀⠀⠀⣠⣤⣄
⣿⣿⣼⣿⡁⠹⣦⣀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣿⣿⣿⣶⣌⡻⢛⣷⣦⣤⣤⣼⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢀⡀⠈⠲⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡁⠈⠐⠀⠙⠲⣦⣴⠆⢢⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⣠⣤⠶⠒⠈⠁⠀⠀⠀⠀⠀⠀⠂⠀⠈⠒⠁⢀⠠⠖⠃⠈⠈⢯⡌
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠚⠂⠀⠀⠈⠀⣸⡶⣏⣁⣼⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⡤⣶⠈⠀⠀⠀⠀⠀⠉⠀⠀⠠⠀⡀⠀⠈⠀⠈⠁⠉⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⢯⢀⣀⣀⠔⠀⠀⠀⠀⠂⠀⠊
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⣿⣻⣬⣿⣩⠋⠀⠀⠀⠀⠀⠀⠰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢌⡽⠃⣰⡏⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣅⣼⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠟⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣴⠿⣦⣤⣴⠿⣦⣄⣴⣶⣶⣄⣠⣶⠿⣶⣶⣶⣶⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠟⠁⠀⠈⠉⠀⠀⠈⠉⠉⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠁⠀⣹⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⠀⠀⠀⠀⠀⣀⣤⣴⣶⣤⣄⠀⠀⣠⡾⠛⠉⠙⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⠻⠷⠾⠿⠛⠋⠉⠁⠀⠈⠙⠛⠛⠋⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄
⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆
⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇
⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆
⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⣿⠃⠀⠀⠀⠀⠀⠀⢠⡿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢰⣿⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣀⠀⠀⠀⠀⢸⣿
⢸⣿⠀⠀⠀⠀⢰⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⠀⠀⠀⠘⣿
⢸⣿⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡆
⢸⣿⠀⠀⠀⠀⢸⠆⠀⠀⠀⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇
⢸⣿⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇
⢸⡟⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇
⢸⡇⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇
⢸⣷⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇
⢸⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇
⢸⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣤⡄⠀⠀⣤⣤⡀⠀⢠⣤⣤⣄⠀⣤⣴⣶⣤⠀⠀⢀⣤⡶⠿⣷⠀⣠⣶⣤⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇
⢸⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⣼⠉⢻⣦⡾⠉⠙⣇⠀⢹⡃⠈⣿⠀⣿⡀⠀⢿⡇⠀⢸⣯⠀⠀⢾⡿⠋⠀⣽⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇
⢸⣿⠀⠀⠀⠀⠀⣿⠀⠀⣠⡿⠀⠀⠉⠀⠀⠀⣿⡆⢸⣇⠀⣿⡀⣿⡇⢀⣼⡇⠀⢸⣿⠀⠀⠀⠀⢠⣾⡏⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣿⡇
⢸⣿⠀⠀⠀⠀⠀⣿⠀⠀⣿⠃⢀⡄⠀⣤⠀⠀⠘⠇⢸⣿⠀⢻⡇⢿⡇⠀⠻⠿⣶⢸⣿⠀⢠⣦⡀⠈⠻⢿⣤⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⡇
⠈⣿⠀⠀⠀⠀⠀⣿⠀⢸⣿⣀⡈⠡⠖⠈⢁⣀⣀⣀⡀⣿⣄⣹⡇⢻⣇⣀⣀⣀⣿⢸⣿⠀⢸⣿⢿⣦⣀⣨⣿⠇⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠃
⠀⣿⠀⠀⠀⠀⠀⣿⠀⠈⠛⠉⠉⠀⠀⠀⠘⠛⠛⠛⠉⠛⠛⠛⠃⠈⠛⠛⠛⠛⠛⠘⠿⠿⠿⠟⠈⠻⠿⠟⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⢿⡇⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⢸⣿
⠀⢸⡇⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⢸⣿
⠀⢸⡇⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣿
⠀⢸⣇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⣿
⠀⠀⣿⡄⠀⠀⠀⠸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠃⠀⠀⢰⣿
⠀⢀⣿⡇⠀⠀⠀⠀⠀⢸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠁⠀⠀⠀⢸⣿
⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠉⠛⠻⢶⣶⣦⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⡇⠀⠀⠀⠀⠀⠈⠻⣦
⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠿⠿⠿⠿⠿⠶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠿⠿⠿⠿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⠀⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠙⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠇
⠀⠀⠀⠀⠈⠻⢿⣷⣦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠈⠿⠿⠿⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⡿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⠟⠛⠉⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠾⠟⠋⠉⠀⠀⠀⠀⠀⠀⣀⣀⣤⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⠟⠛⠉⠁⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣀⣤⣴⠶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⡶⠾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀
⠀⠀⠀⠀⢀⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⢀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⡶⠟⢿⡇
⣴⣿⣧⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠟⠛⠉⠁⠀⠀⠀⢸⡇
⣿⡇⠀⠈⠉⠉⠙⠛⠛⠛⠿⠿⠶⢶⣶⣦⣤⣤⣀⣀⣀⠀⠙⢷⣄⠀⠀⢀⣠⣤⣴⠾⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠿⢿⣿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⡇⠐⣷⡀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢿⠀⠀⠈⠻⠶⠾⠋⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⢀⡀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣴⣦⣀⣰⡆⠀⠹⢷⣤⣤⡿⠃⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⡇
⣿⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠟⠛⠉⠁⠀⢸⡇
⣿⡏⠉⠙⠛⠛⠿⠿⠶⢶⣶⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣠⣤⡶⠾⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠿⠷⠶⢶⣶⣴⣿⡶⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠿⠦⠤⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠾⠧⠤⠤⢤⣀⡀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⠀⠀⠀⢀⠀⠀⢨⡟⠉⢶⡄⠀⠀⢀⣤⠖⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⣼⠋⠳⣆
⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣡⠤⣤⡀⠀⠀⠀⠀⡾⠉⢻⡄⠘⢷⣤⡾⠛⢦⣴⢋⡤⢤⡀⠀⠀⠀⠀⣴⠋⠙⣶⠀⠻⣦⣴⠟⠳⣄
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⣿⣄⣼⠇⠀⠀⠀⠀⠻⠶⠾⠁⠀⠀⠀⠀⠀⠀⠹⣿⣤⣠⡿⠀⠀⠀⠀⠙⠷⠶⠏⠀⠀⠀⠀⠀⠀⠈⢧⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⢠⣠⡀⠹⡏⠉⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⡀⠀⢀⣠⣄⠈⣷
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡀⠀⠀⠀⠀⠈⠓⠒⠋⠀⠀⢀⣀⠀⠛⠒⠋⠀⠀⢷⣠⡿⠀⢿⠀⠛⠒⠚⠁⠀⠀⣀⡀⠙⠓⠚⠁⠀⠸⣄⣤⠂⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣹⡆⠀⢀⡀⣀⠀⠀⠀⢶⣠⣿⣿⠟⠀⠀⠀⠀⠀⠀⠁⠀⠀⣸⣇⡀⠀⠀⠰⣄⣼⣿⡿⠂⠀⠀⠀⠀⠀⠈⠁⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠀⠀⢿⣤⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣽⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯
⠻⠷⢶⣶⣤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⢿⡀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡁⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠻⠿⠷⠾⣧⣠⡟⠁⠈⢳⡄⠀⠀⢀⡴⠋⠀⠈⠙⣦⠀⣠⡶⢦⡄⢀⣾⠉⠙⣆⠀⠀⠀⣰⠟⠁⠈⠙⢷⡀⢀⡴⠶⣄⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⣤⡾⠃⠀⠀⢸⣇⠀⠀⠀⠀⣿⠀⢻⣤⣤⣿⣾⣧⣠⣴⠟⠀⠀⠀⣿⡀⠀⠀⠀⢸⡇⠸⣧⣠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣍⡀⠀⠀⠀⠀⠻⢷⣤⣤⡾⠋⠀⢀⣩⡿⠋⠙⠻⣯⡁⠀⠀⠀⠀⠘⠷⣦⣤⣴⠟⠀⠀⣈⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣀⣀⣀⣀⣠⣤⣴⠾⠟⠉⠀⠀⠀⠀⠈⠛⠷⢶⣤⣄⣀⣀⣀⣀⣤⣤⡶⠟⠋⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠏⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⢉⣉⣁⣉⣉⣠⣤⣦⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⣉⣉⣈⣉⣁⣠⣴⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⠿⠿⢿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⠿⢿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣄⣀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣄⣀⣁⡀⣙⢛⡛⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣻⣛⣛⣻⣟⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢦⡐⠒⢶⠲⠖⠠⠤⣤⡄⡤⠀⠉⢠⣺⠦⠀⠙⠛⠓⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠻⢿⣶⠟⠀⠀⢐⣠⡄⠀⢀⡞⠀⠠⠀⠀⠀⠀⠀⠡⠖⠐⢶⠄⢩⠃⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠁⠀⠤⠀⠀⠀⣀⠀⠻⠾⢟⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠂⠉⠀⠈⠙⠳⣝⣆
⠀⠀⠀⠀⢀⡤⠶⠁⠀⠈⠜⠀⠀⡀⠀⠀⡈⠀⠀⠀⢱⠀⠀⠁⠀⠐⠀⠀⠀⠀⢰⣿⣯⣷⠀⠀⠀⠀⠤⢸⠈⠑⡄
⢸⣦⣶⣾⣧⡑⡇⠀⠀⠀⠀⠀⣰⡓⣲⡄⠀⠀⠀⢠⠀⠈⠀⠀⠰⠂⢤⡄⠀⠀⠀⠈⠙⠁⠀⠀⠀⠁⠠⠎⠂⠀⠸
⠀⠹⣿⡿⣿⣷⣿⣦⡀⠁⠀⠀⢻⠛⠉⠀⠀⠀⠀⢸⠀⠀⠀⠀⣶⣿⣿⣃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢄⣀⡀⠀⢀⣀
⠀⠀⠈⠛⠿⣿⣻⣿⣧⠀⠂⠄⠀⠀⠀⠀⠈⠀⠁⠈⢳⣄⠀⠀⠀⠀⠄⠀⠀⢠⠀⠀⠀⠀⠀⣠⢴⡆⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠿⠏⠉
⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⡶⢶⣤⣤⣤⣄⠀⠀⠀⣀⠈⢿⣆⡐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⣧⣾⠇⠀⠀⠐⢸⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⢶⣿⣿⣟⡳⠶⢤⣴⣦⣤⣿⣿⣶⣄⡢⣤⣀⠑⠒⠒⠢⠄⠈⠀⢀⢌⣱⡶⠶⢛⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣷⣶⣶⣶⣶⣾⡟⠛⠿⢿⡶⠭⣙⣓⣶⣒⣒⣽⣷⣶⣯⢿⣾⣿⣶⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⢶⣿⣟⡛⢿⡷⢲⣖⣙⡿⠟⢿⣿⣿⠲⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠻⠿⠿⠛⠛⠚⠋⠉





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠒⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠚⠓⠀⠉⠂⠐⠒⠖⠆⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡙⢦⠀⠀⢀⠀⠈⠐⢄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⠉⠁⠀⠊⠀⠉⠉⠉⠁⠀⠐⠀⠚⠓⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠘⠂⡀⢀⣤⣠⣄⠑⠄⡀⢀⣀⣀⡀⠳⢄⠀⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠈⠠⢠⡀⠀⠀⠀⠀⠠⣀⠀⠁⠁⠀⠐⠁⠈⠉⠉⠉⢀⡸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠒⠛⠂⠈⠰⠀⠒⠒⠒⠂⠙⡢⠀⠴⠤⠤⠄⠓⢀⠀⣠⣄⣤⡀⠢⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠲⡀⠀⠀⡀⠀⠐⢦⡀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠁⠰⣄⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠉⠉⠉⠀⠀⢁⠈⠉⠉⠉⠀⠀⠀⠀⠓⠚⠓⠀⠁⠂⠐⠒⠒⠂⠈⢱⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠌⠓⢄⠀⣠⣀⣤⡉⠢⠀⠀⣀⣀⣈⡙⢦⠀⠀⢀⠀⠈⠳⢄⠀⢀⡔⠋
⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠁⠤⣀⠀⠀⠀⠀⠈⢀⡀⠀⠀⠀⠀⠊⡀⠉⠉⠉⠁⢀⠖⠁
⠀⠀⢦⡐⠓⠒⠓⠀⠁⠄⠐⠶⠶⠦⠈⢂⠄⠠⠦⠤⠤⠙⠂⡀⢀⣄⣠⣌⠑⠄⣠⠖⠁
⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠈⠀⢠⡀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠙⢄⠉⠉⠉⠁⠀⠖⠀⠚⠒⠛⠂⠈⠐⠀⠒⠒⠒⠂⠙⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢀⣀⣀⡁⠢⡀⠀⠀⡀⠀⠐⢦⡀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠉⠉⠉⠀⠐⠁⡈⠉⠉⠉⠀⡰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣠⣀⣤⡉⠢⣀⡰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠴⠋

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠈⠒⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⠀⠀⠦⠹⣿⣿⣿⣾⠂⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠰⠿⣿⣿⣿⣯⡀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠄⠀⠀⠈⠁
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⣀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢺⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸⠇
⣿⣿⡇⠀⠀⠀⠀⠀⠀⡀⢤⣤⣼⣿⣤⡀⠀⠀⠀⠀⠈⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠠⣄⠀⣠⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣤⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠿⣿⡿⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣾⡧⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠤⠄⡀⠤⠀⠀⠀⠀⢰⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⢀⠘⢰⡷⠀⠀⠀⠀⠀⠀⠤⠀⠀⣶⣶⣤
⠀⠀⠈⠑⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠦⠙⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠘⠻⣿⣿⣿⣷⡄
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⣤⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠏⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠠⠄
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⣄⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢺⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠃
⣿⣿⠇⠀⠀⠀⠀⠀⠀⡀⢴⣤⣼⣯⣤⡀⠀⠀⠀⠀⠈⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠠⢄⠀⣠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉
⠀⠀⠀⠀⢀⡀⠀⠀⠀⣩⣤⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠻⣿⡿⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣾⡷⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⢠
⣀⣀⣀⣀⡀⣀⢀⣀⣀⢀⣰⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣆⣀⣤⣀⢀⣀⣀⣸⣿⣦⣆⣀⣀⣀⣀⣀⣀⣀⣀⡈⣰⣿⢀⢀⣀⣀⣀⣀⣠⣀⡀⣶⣶⣤⡀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⢉⠔⠒⠒⠲⣄⣀⡬⠽⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⡵⠒⠒⠚⢁⡄⠀⢤⡀⠀⣤⠀⠀⠀⠹⣌⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢀⡼⢀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠈⠀⢀⡄⠀⢈⠙⢦⡹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⠚⠉⠀⠀⠙⠀⠀⠀⠀⠀⠀⠚⠉⠀⠐⠋⠀⠀⡞⠀⠀⡇⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣇⠀⢀⡠⠤⠀⣰⠀⠀⣹⡦⠶⠶⠤⢤⡀⠀⠲⠦⠀⠀⠀⢧⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⠇⠀⠀⠀⠀⠀⠃⠀⡼⠁⠀⠀⠀⠀⠀⠹⡄⠀⡀⠀⠘⠆⠀⠱⡌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡏⠀⠀⠐⠲⠆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡷⠀⠉⠙⠀⡀⠀⠀⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢣⠀⢲⡀⠀⠀⠰⣄⠀⢳⡀⠀⠀⠀⠀⠀⣠⠇⠠⠞⠁⡀⢻⠀⢠⠏⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡇⠀⠀⢠⡇⠀⠀⠀⠀⠙⠦⣤⣤⣤⠴⠋⢠⠀⠀⠞⠁⠈⠀⢸⡄⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠳⢤⡀⠸⠁⠀⠀⠘⡆⠀⣰⠂⠀⠤⡤⠀⠸⠂⠀⠀⠘⠦⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⢹⡄⠀⠲⠄⠀⠀⢀⠁⠀⠀⡀⠀⠀⠀⣰⠀⠀⣀⢀⡴⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣷⢤⣄⣀⠀⠀⠈⠳⡄⠀⠙⢆⠀⠀⠁⠀⣀⣉⣹⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣈⡙⠦⣄⣠⠴⠶⠦⣄⣀⣀⣴⣯⠽⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⠒⠒⠛⠛⠛⠉⠁