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⠀⠀⠀⠀⠰⠛⠀⠀⠀⠘⠧⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁⠀⠀⠘⠛⠄⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⠀⠐⠦⡀
⠀⠀⠀⡀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠆
⠀⠀⠀⠀⠀⠈⠋⠠⠍⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⢄⠉⠀⡠⠀⠀⠀⠀⢀⡀⠁⠀⠀⠚⠇
⠀⠈⠣⡀⠀⠀⠈⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠⠤⢀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢁⠐⠀⠄⠀⠄⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢁⠐⠀⠀⠀⠰⠂⠁
⠀⠀⠀⠀⠑⢆⠁⠂⠀⠂⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠠⡄
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⠀⠀⠀⠀⢠⠖⠒⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠃⠀⠀⠀⠦⡀⠀⠀⠀⠀⠀⠀⠀⢰⠾⠀⠀⠀⠀⠶⡄
⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⡐⠂⠀⠀⡀⠀⠀⠀⢀⡀⠀⠰⣿⣿⡷⠰⣿⡿⠂⠀⢀⡀
⠀⠀⠀⠑⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁⠂⠀⠊⠀⠀⠀⠀⠈⡄⠀⠀⠈⠩⠔⠁⠂⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠄⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠠⣤⡀
⠀⠀⠐⠺⠶⢦⣤⣀⣀⣠⣤⠶⠞⢛⠀⠀⠀⠀⢠⠐⠋⣁⠀⡎⢉⡓⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠰⠀⠀⠁⡨⠉⢌⠉⠀⠠⠂⠀⠀⠀⠀⠘⠄⠀⠁⢀⠆⠈⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⡖⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠁⠀⠀⠊
⠀⠀⠀⠀⡤⣊⣀⠸⠀⡠⠄⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣤⣀
⠀⠀⠀⢸⢀⠀⠸⠀⠀⠂⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⣀⣠⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠀⠀⠀⠈⠉⠙⠻⢶⣄⡀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠶⠶⠶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⠀⠀⣠⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣤⣀⠀⠀⠀⠀⠀⠀⢻⡆⠀⡠⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⣼⣿⡆⠀⠀⠀⠀⠀⢸⡇⣾⣄⣠⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣨⣿⡿⠃⠀⠀⠀⠀⠀⣼⠇⠘⠧⣼⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢰⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⢀⣀⣤⣤⣄⣀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠈⠻⢷⣤⣄⣀⠀⠀⠀⣀⣀⣤⡾⠟⠁
⠀⢠⣾⡿⠿⣿⡟⠛⠛⠻⢦⡀⠀⠀⠈⠙⠷⣦⣤⣀⣀⣀⣀⣀⣠⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠉⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠙⠻⢶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⣰⠏⠉⠛⠻⠷⢶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⣤⣴⡶⠶⠿⠛⠋⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠈⣿⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⢀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣿⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡛⠛⠛⣿⠛⠛⠋⠙⠋⠙⠙⣿⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣤⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⠿⢷⣶⣤⣤⣤⣤⣴⡶⠿⠿⠛⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢻⠿⠿⢿⠿⡹⠿⡿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣟⣛⣛⣛⣛⣿⣿⣛⣛⣛⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠈⠙⢶⡄⠀⣠⠶⠛⠉⠉⠉⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡠⢤⣄⠀⠀⠀⢹⢀⠠⣤⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⢀⣀⡀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢗⣺⣿⠀⠀⠀⢸⠻⣴⡿⠃⠀⠀⠀⠀⠀⠀⢸⠃⠀⠠⠶⢶⢦⡄
⠀⡴⠛⣏⠉⠃⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣼⠀⠉
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠳⣤⣀⠀⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠉⠳⠦⣤⣤⣤⡤⠶⠋⠁⠀⠀⠀⠀⠉⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⣿⠀⡰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡆⠀⡠⠞⡟⣡⠋⠈⠷⡄⠈⢧⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡴⠁⠈⣧⠊⠀⠀⢯⠁⠀⠀⠀⠀⠀⢸⠃⠘⣦⠶⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣠⠏⠀⠈⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⢦⣼⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠀⠀⠀⠀⢀⡠⠶⠒⠒⠒⠒⠦⣄⠀⠀⣇⠀⠀⠀⢀⡠⠔⠒⠢⣄⠀⠀⠀⠀⠀⠀⢸⡶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠃⡀⡆⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢽⡶⠤⠒⠁⠀⠀⠀⠀⠈⢦⡀⠀⠀⢰⠀⠈⠀⣧
⠀⠀⠀⠀⠀⠀⠀⣀⡘⡄⠀⢣⡙⢤⡎⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⢠⡀⠀⠈⠙⢲⣼⡇⠀⠀⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠙⣏⠁⠀⡱⢭⣻⡇⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⣤⣀⠀⠳⠄⠀⠀⠀⣟⡵⠀⡴⠓⣹
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠍⣲⣶⡇⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⢿⣿⣶⣤⠀⠀⠀⡿⣊⠀⢀⣴⣃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢑⣒⡶⠋⠀⠀⠀⠀⣀⣤⣶⣿⣿⡇⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠈⠙⢿⠇⠀⠀⡿⠁⠴⠋⢁⠎
⠀⠀⠀⠀⠀⢀⣀⣸⠄⠀⠀⠀⡸⠁⠠⠶⠾⠟⠛⠛⠛⠻⣟⠋⠁⠀⠀⠀⠀⣠⣤⣤⣤⡤⣤⡤⠤⢤⡤⠀⢰⡇⠀⡰⢠⠏
⠀⠀⠀⠀⠀⠀⠉⢢⣄⣆⠀⢀⡇⠀⠰⡗⠒⢶⣖⣲⡶⠒⢻⡟⠛⠀⠀⠀⠀⢸⡄⠀⠈⠛⠛⠁⢀⣾⠁⠀⣸⠁⡼⠓⠻⣄
⠀⠀⠀⠀⠀⠀⢰⠋⣁⢹⡄⠘⡇⠀⠀⠱⣄⣀⠉⠉⣁⣠⣾⠁⠀⠀⠀⠀⠀⠸⣯⡓⠦⣤⣤⠶⢻⠏⠀⢀⡏⢸⠃⡤⡄⢹
⠀⠀⠀⠀⠀⠀⣿⢰⠁⠀⣧⠀⢷⠀⠀⠀⠘⠛⠿⠛⢛⡽⠃⢀⣤⠤⠶⠶⣤⡀⠙⠍⠂⠀⠀⠀⠁⠀⠀⢸⠃⣾⠰⠛⡷⣸
⠀⠀⠀⠀⠀⠀⢸⣼⡩⠗⣾⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣟⠀⠀⠀⠀⠀⣹⡄⠀⠀⠀⠀⠀⣰⡄⠀⢸⠀⡿⡀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⢳⡝⢦⠜⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⠚⠋⠀⠀⠀⠀⣠⠞⠁⠀⠀⢿⠼⣳⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣀⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⣻⠁⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠤⠴⠖⢒⣋⣡⣴⣾⡟⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠳⣶⣿⡿⠛⠛⢋⣠⠞⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠈⠛⠶⠶⢒⣋⡥⢠⣄⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠿⣄⡀⠀⠀⠉⢉⣡⡾⠋⠉⠁
⠀⠀⡀⠀⢀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⠛⠛⠛⠋⠁
⣿⣏⢭⣟⡛⣛⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠛⠉⠉⠉⠉⠉⠛⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠟⠛⠛⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⣀⣴⠶⠛⠛⠛⠛⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⣼⠋⠀⣠⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⠀⣼⣇⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢿⣭⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⢉⣿⣿⡄⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣷⣿⣿⣿⠇⢰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⣀⣤⡄⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠋⢠⡿⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⣠⣾⣿⠛⠁⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢀⣴⠟⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠃⠀⠀⠀⠀⠀⠀⠀⣰⡟
⢀⡾⠋⠈⢿⣧⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⣀⠀⠀⣀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁
⠜⠁⠀⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠈⠛⠿⣷⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⡇⠀⠉⠉⠛⢻⡿⠿⠷⠶⠶⢶⣶⣶⣶⡶⠶⢶⡾⠿⠛⠛⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡶⠶⢤⣧⣀⡀⠀⠀⢸⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⢀⣀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠉⠉⠑⠛⠓⠒⠒⠂⠀⠼⠃⠀⠒⠒⠒⠚⠓⢹⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⢀⣠⡶⠾⠛⠛⠛⠛⠛⠻⠶⣦⣴⡟⠉⠀⠀⠀⠈⠙⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⢹⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣶⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠛⠋⠉⠉⠛⠛⠶⣤⡀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣀⣀⣀⠀⠀⣀⣤⣤⣀⠀⠀⢻⡾⠁⢀⣀⣀⡀⠀⣠⣤⣄⡀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⣾⠟⢹⣿⣿⣾⣿⣿⣿⣿⣷⡀⠀⢿⣰⡟⢉⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣿⠖⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣼⠉⠻⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⣼⠀⠀⠀⢀⣤⠶⢶⣶⣄
⠀⢀⣴⢿⣟⠛⠳⠄⠀⠀⠀⢷⡀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⠿⠋⠀⠀⣰⣯⡀⠀⠀⠙⠻⠿⠛⠉⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⣼⠀⠙⠃
⠀⠈⠀⠘⣿⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⢀⣴⠏⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⠀⠀⠀⢀⣀⣤⡶⠛⠁⠀⠀⠀⠈⠙⠛⠶⠶⠶⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⠶⢦⣤⣤⣤⣤⣤⣤⠶⠶⠞⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣄⣀⣀⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠻⠿⣿⣿⣶⣄⠀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠒⠒⠒⠒⠢⢤⣀⠘⢿⡇⠀⠀⠀⣀⡤⠶⠒⠒⠒⠒⠦⣤⡉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡁⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⣇⣨⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢎⢀⡀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠲⢤⣀⠀⠀⠀⣼⠃⠈⠧⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠂⠐⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⢰⡿⠿⣿⡛⠛⠓⢦⠀⠀⠀⠈⠙⠒⠦⠤⠤⠤⠴⠖⠋⠁⠀⠀⠀⠀⠈⠛⠲⠦⠤⠤⠴⠖⠛⠉⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠉⠛⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣤⡶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⡼⠁⠉⠉⠛⠓⠶⣶⠶⠶⠦⠤⠶⠶⠶⣶⠖⠒⠛⠁⠁⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣼⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣸⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⠉⠙⡟⠉⠉⠉⠉⠉⢻⠉⠉⠉⠉⠉⠉⢻⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣿⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣇⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⢀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠾⠧⣤⣤⣤⣤⣤⠴⠷⠞⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠿⠟⠛⠒⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⢿⣷⡀
⠀⠀⠀⢀⡤⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⡀
⠀⠀⠀⢯⣀⣀⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣀⣀⡝
⠀⠀⣀⠀⠈⠉⠀⠈⠁⠀⠀⢀⣤⣶⣶⣷⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣤⣄⡀⠀⠀⠀⠀⠈⠁⡀⠉⠁⠀⣀
⠀⠪⠤⠄⠂⢀⣀⡀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠉⠛⠋⠙⠃⠀⠀⠀⠀⠀⡇⠀⠐⠠⠤⠅
⠀⠀⢀⣴⠾⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠚⠁⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⠙⠓⠶⠤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡤⠤⣄⣀⣀⣀⡀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⣠⠤⠖⠲⠦⢤⡖⠚⠉⠛⢦⣤⡴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⡞⠁⠀⠀⠀⠀⠀⠉⢀⣀⣤⠾⠋⢰⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠶⠒⠚⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠄⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣠⣿⣧⡆⠀⣴⣿⣿⠇⠀⠀⣀⣤⣒⣴⠗
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣒⢦⣽⣿⣿⣿⣿⣧⣾⣿⣿⣏⣤⣶⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠿⢿⣟⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⣠⡾⣿⣿⡿⠟⣉⣀⡠⣄⣀⣀⣠⠴⠚⠉⠉⠉⠙⠲⣤⡙⢦⡈⠉⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢀⡼⣫⣾⣿⠟⡠⠊⠁⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠈⢷⡀⠙⢆⠀⢠⡶⠛⠋⠙⠻⢶⡀
⠀⠀⠀⠀⣾⣽⢿⣿⡏⣸⠁⠀⠀⠀⠀⢀⣶⣶⠀⢹⣴⣶⡀⠀⠀⠀⠀⢸⡇⠀⠀⠳⠋⠀⣴⡟⠳⣦⡀⢻⡀
⠀⠀⠀⢰⡟⠁⣾⣿⠀⡇⠀⠀⠀⠀⠀⠈⠻⠟⠀⢸⠛⠻⠃⠀⠀⢀⣰⡿⠁⠀⠀⠀⠀⠀⠈⠀⠀⠘⡇⢸⡇
⠀⠀⠀⠀⠀⠀⡟⠹⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠙⢒⣲⣶⣾⡿⠯⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠘⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⡀⠀⠀⠀⠀⢀⣤⣾⠷⠒⢻⠉⠁⠀⠀⢳⠀⠀⠀⠸⠈⢳⡀⠀⠐⡖⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠳⢦⣤⠴⢞⠟⢫⠀⠀⠀⢸⡀⠀⠀⠀⠈⣧⣠⣤⣾⣷⣾⣿⡄⠀⢹⡀
⠀⠀⠀⠀⢀⡴⠛⠛⠉⠀⠀⠀⠀⣴⠏⠀⠸⡀⠀⠀⢈⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⣇
⠀⠀⠀⠀⡾⠀⣰⠾⠂⠀⠀⠀⢸⠁⢇⠀⠀⣱⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿
⠀⠀⠀⠀⣧⠀⢿⡀⠀⠀⠀⠀⢸⣄⢈⣷⣾⣿⣿⣿⣿⡿⢻⠟⠉⠉⠉⠛⢿⣿⣿⡿⠛⣿⡿⠀⢸⡆
⠀⠀⠀⠀⠹⣦⠈⠛⠶⠀⢀⣀⠀⢻⣿⣿⣿⣿⣿⣿⡏⠀⠘⠀⠀⡠⠴⢄⡏⠀⠘⣿⡀⢸⠇⠀⣾⡇
⠀⠀⠀⠀⠀⠙⠻⠶⠶⠖⠋⠘⢆⠈⢿⣿⣿⣿⣿⠿⢧⡶⠛⠙⢦⣇⠀⠈⢿⡀⠀⠸⣧⠎⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⢻⣿⠟⠻⡄⠀⢣⠀⠀⠈⢿⡄⠀⠀⢷⡠⠔⣁⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠙⢷⣄⠙⢦⠀⠣⡀⠀⣀⡷⠤⠒⣉⡤⠞⣡⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠙⠚⠛⠊⠉⠉⠀⢀⣈⣩⣵⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠶⠶⠶⠾⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢴⡶⠀⠀⣀⣠⡤⠶⠖⠚⠛⠛⠛⢛⣛⣶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⢁⣴⣫⡴⠞⠋⠉⠀⠀⠀⠀⣀⣤⣖⡛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣩⠽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⢛⡽⢖⣠⡤⣤⣄⡈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢯⣥⠀⠀⠈⡼⠋⠀⠀⠀⠀⠙⠳⣄⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠙⢦⡀⠀⠀⠀⠀⢀⡐⠂
⠀⠀⠀⠀⠀⢀⣹⠀⠀⠀⠀⠀⢴⠚⠉⠀⠀⢀⠞⠛⠀⠀⠸⡇⠀⢀⣤⡀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠙⢦⡀⠀⡤⠤⢬⡀⠀⢌⠉
⠀⠀⠀⠀⢰⢋⣽⠀⠀⠀⠀⠰⣶⠓⠀⠀⠀⠏⠀⠀⣤⠖⠛⠛⠲⣥⣿⡿⠀⠀⠀⠀⣸⠃⠀⢀⣀⣀⡀⠙⣎⠣⣤⡞⠉⠙⣦⡤⣄
⠀⠀⠀⠀⠸⢣⡏⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⡆⠀⠀⠀⣀⡴⠋⠀⠐⠉⠀⠠⡟⠉⠻⣦⠏⠉⠉⠜⢁⣀⣸⠃⡴
⠀⠀⠀⠀⠀⢸⡇⣴⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠈⣧⡀⠀⠀⠈⣹⡟⠒⠚⠛⠉⠀⠀⠀⠀⠀⢀⣠⠟⠙⢲⡈⠀⠀⣠⠞⠉⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠻⠙⣆⠀⠀⠀⣀⣀⡀⢹⠀⠀⠀⠀⠈⠙⠒⠒⢊⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡡⠤⠀⣸⡇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟⠋⢉⣉⣉⣉⠉⠃⠀⠚⢯⣄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢾⡉⠲⢤⡴⢣⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⠛⠷⠶⠋⢁⣴⣄⠀⠀⠈⠙⠒⠦⠤⣤⣤⣤⡤⠴⠖⠛⠉⠙⣆⠀⠈⠣⡀⠙⢧⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣤⣤⠴⠞⠋⠀⠈⠙⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⢩⣷⡀⠀⠀⠀⢀⣧⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢻⡋⠀⠀⠀⠀⠀⢠⠖⠀⠀⠈⠉⠙⠓⠒⠒⢚⡿⠯⠭⢤⣤⣀⣸⠶⠿⠦⠤⠶⠋
⠀⠀⠀⠀⠀⢀⣰⡟⢻⡀⣠⣿⣶⣄⣀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢯⡀⠙⠜⠛⠁⣨⠟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⣾⡋⠑⢂⣠⠼⠉⠓⠶⢤⣄⣀⣀⣀⣀⣀⣀⣠⣤⠴⠞⠁
⠀⠀⠀⠀⡼⠁⠉⠉⠉⠀⠀⠀⠀⢀⡴⢋⡴⠋⠉⠉⢀⡤⠞⠁
⢀⣠⠤⠾⣧⡀⠀⠀⠀⠀⠤⠤⣴⣏⣴⣏⡀⠀⣠⡴⠋
⣟⠀⢠⠞⠧⠽⠛⠲⢤⣤⣄⣴⣋⣤⡤⠴⣶⣫⢿⠉⢳⡄
⢿⣓⣛⣓⣚⣛⣛⣭⠭⠵⠚⠯⢽⣓⣒⠒⠾⠶⠛⣒⣿

⠀⠀⠀⠀⢀⣤⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⢁⠀⠀⠀⠀⠐⠶⠛⠉⠀⢀⠀⠛⠳⠆
⠀⠀⠀⠀⠀⠀⡤⠈⠤⠀⠀⠀⠀⠀⢀⠀⠀⠐⠂⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠐⠰
⠀⠀⠒⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⠀⠀⠀⠴⣄
⠀⠀⠀⠀⠒⠲⣄⠀⠰⢤⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠀⢀⣠⡀⢀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠓⡺⠚⠟⠉⢀⠀⠀⠀⠀⠀⠀⣘⣷⣭⢀⢘⣿⡆
⠀⢀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠅⠀⠀⠐⠈⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠉⠁⠀⠁⠀⠀⠀⠄
⠀⠀⠠⠄⠀⠀⠀⠀⠀⡀⠔⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠈⠦⠀⡀
⠀⠀⠀⠢⣀⠀⡀⠈⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠈⢄⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠠⠴⢤⣄⣀⣀⣀⣠⡤⠶⠖⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠖⠒
⠀⠀⠠⠀⠉⠉⠉⠉⠉⠀⠀⠁⠀⠀⡀⠀⠀⠀⠤⠀⠤⠀⠀⠀⠠⠀⠀⠈⠄⠀⠉⠀⠐⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠈
⠀⠀⠀⣀⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⡤⠒⠈
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⣤⣤⡤⡤⣤⣤

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠋⠁⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠲
⠀⠀⠀⠀⠀⡤⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠃
⠀⠤⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠘⠠⠀⡀⠀⡀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠈⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⠀⠊
⠀⠀⠀⠑⢤⠀⠀⠐⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠠⡀

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠋⠁⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⢀⠋⠁⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠙⠂
⠀⠠⠀⠀⠉⠁⠀⠀⠈⠉⠀⡀⠀⠀⠀⢠⠁⠀⠀⠤⢫⠀⠀⠐⠀⠀⠀⠀⠀⠈⣴⣤⣦⢢⣤⣦
⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠛⠁⠂⠉⠁⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⠀⠀⠁

⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⣀⡴⠆⡠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠞⠃⠀⠀⠘⠳
⠀⠀⠈⠉⠙⠛⠒⠶⠺⠏⠉⠘⠀⠀⠀⠀⠋⠀⠔⠀⡃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⢀⢴⠀⢨⠀⡈⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢤⣀
⠀⠀⠀⢇⠀⠨⠄⠀⠃⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠦⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠙⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⣠⠴⠚⠉⠉⠀⠉⠉⠛⠲⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⢠⡞⣡⠲⣦⡀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡏⠸⣇⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢻⣷⡀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣟⣲⣿⣿⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⣠⣶⠶⠿⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⣼⠁
⠀⢀⡞⠉⢿⡄⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠈⠙⠲⠤⠤⠤⠤⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠈⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠒⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠈⠛⠷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠈⠉⠙⠛⠛⠿⠶⠶⠶⠶⠶⠶⠾⠛⠛⠛⠉⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡖⠒⠦⠤⠤⠤⢤⣤⣀⣀⣀⣀⣀⣀⠀⠤⠤⠤⠤⢲⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣠⡼⠷⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⣠⠾⠛⠉⠉⠉⠉⠉⠛⠛⠶⣏⠁⠀⠀⠀⠀⠉⠳⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠙⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⠶⠶⠞⠋⠁

⣟⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠙⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⣠⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢂⠄⠀⠀⢀⠤⠒⠚⠒⠠⠅⣀⣠⣄⡀⢸⣀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣅⡂⠠⠂⡀
⠀⠀⢀⠠⢀⠀⠀⢀⣌⡀⠄⠉⠀⠀⠀⠈⠓⢤⣤⠤⣴⣶⡞⣽⣥⠁⠈⢶⠀⠐⠂⠀⠀⠀⠀⠀⠀⠙⠁⠀⠻⠏⠀⢀⣠⡀
⠀⠀⠥⠁⠈⡴⣦⠜⠋⠁⠠⠃⠀⠀⠀⠀⠀⠘⠇⠀⠀⠛⠁⠘⠋⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⡠⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉

⠀⠀⠀⢀⠤⠤⠤⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠠⡀
⠀⠀⠐⠁⠀⠀⠀⠈⠀⢑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀
⠀⠀⠀⠀⠀⠁⣤⡆⢠⡤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣨⡅⢀⢘⡧⠠⠴⠃⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⢩⠁
⠀⠀⠀⠀⠀⡀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠠⢀⠀⢰⠶⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⡨
⠀⠒⠀⠀⠀⢡⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢱⠇⠀⠉⠋⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠢⢄⠀⠀⠐⣊⠂⠀⠀⠀⠖⠀⠀⠀⣀⠀⠀⠀⠀⠒⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠄⢤⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠉⢅⣄⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠄⠀⠀⠀⠀⠀⠀⡈⠂
⠀⠀⠀⢀⢦⠀⡄⠈⣤⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠢⣦⠐⢿⠆⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢱
⠀⠀⠀⠁⠀⠀⠓⠄⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⢐⠓
⠀⠄⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⠀⠀⠤⠐⠁
⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠾⠟⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⣴⠦⠀⠀⠀⠛⠆⠀⠀⠀⠀⠀⣀⣀⣴⣇⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢄⠀⠀⠐⡀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠦⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠰⠂⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠟⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⣈⠘⠁⠀⠠
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠒⠂
⠀⠀⠈⠂⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠠⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠈⠅⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠷⠀⠀⠀⢠⣄
⠀⠀⠀⠤⢤⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⡀⠉
⠀⠀⡀⠀⠀⠈⡁⠀⠀⠙⡀⠀⠀⠀⠀⠛⠛⢶⣦⣤⣤⣤⡶⠚⠛⠋⠀⠀⠀⠀⠀⠀⢾⣟⡶⠈⠰⣿⣟⠑
⠀⠀⠀⠀⠀⠀⠐⠰⠀⠀⡀⠀⠀⠠⡄⠈⠀⠀⠀⠔⠉⠢⠀⠀⠔⠁⠀⠀⠀⠤⠀⠐⠛⠿⢋⠔⠜⠛⠋
⠐⡆⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠑⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊
⠀⠈⠂⠄⠀⡀⠀⠀⠠⠐⠁⠀⠀⠀⠀⠨⡁⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠈⠇⢲⠀⠀⠐⠂⠀⠁
⠀⠀⠘⢤⠄⠠⠀⠠⠀⢂⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠘⢀⣄⠀⠀⠀⠀⠀⠀⠀⠈⢼⡄⣀
⠀⠀⠀⠀⠁⠐⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠠⠴⢤⣤⣀⡀⡀⣀⣠⡴⠶⠆⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠶⠲
⠀⠠⡀⠈⠉⡙⠛⡉⠃⠀⡠⠁⠀⠀⠀⠀⠀⠀⢀⡄⠈⢠⠀⠀⠀⢀⡀⠀⠀⠣⠀⠈⠋⠀⠶⠀⠉⠁⠀⠈
⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠠⠀⠄
⠀⠀⠀⢀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂
⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠻⠿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡄⠀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⣽⠟⠉⠉⠉⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠻⢯⣍⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⡾⠃⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠹⣦⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣄⣀⠀⠀⠀⠀⢀⣠⡶⣻⡟⣿⠀⣿⢿⡿⣦⣀⡀⠀⠀⠀⣀⣤⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢯⣍⠛⠛⢿⣿⡏⠁⣴⠟⠀⣿⠀⣿⠘⢷⡀⢩⣿⡟⠛⠛⣩⡿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠙⠷⣦⣬⣭⡴⠞⠁⠀⢀⡿⠀⢻⡀⠈⠻⢦⣭⣥⡴⠟⠉⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠹⣦⡀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠦⠴⠶⠛⠁⣀⣀⣀⣀⣀⣈⠛⠷⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⢯⠀⠀⠀⠀⢹⠋⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣄⠀⠀⠀⠘⡄⠀⠀⠀⡞⠀⠀⠀⣠⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣶⣶⣶⣷⣴⣴⣶⣷⣶⣾⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠉⠉⠉⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠒⠒⠒⠒⠒⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⣀⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⡀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢀⡴⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⡇⠀⢠⣾⣿⣷⡸⡆⠀⠀⠀⢠⡇⣴⣾⣶⡄⠀⢸⡀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⣿⣿⣿⣧⡇⠀⠀⠀⢸⣼⣿⣿⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢧⠀⢻⣿⣿⣿⣿⠃⠀⠀⠀⠈⣿⣿⣿⣿⡿⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠈⢧⣀⠉⢛⡵⠃⠀⠀⠀⠀⠀⠘⢎⡛⠛⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⡶⠤⢤⡤⠤⢤⡤⠤⢴⠶⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⣿⣀⣀⣇⣀⣠⣇⣀⣸⣇⣀⣸⣀⣀⣽⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⢿⠉⠉⡟⠛⠛⡟⠛⢻⡟⠛⢻⠉⠉⣻⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣄⠀⠀⠈⠳⠶⠧⠤⠴⠧⠤⠼⠧⠤⠼⠶⠞⠁⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⠤⠤⠤⠤⠤⠤⠶⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣶⣶⣶⣿⡿⢿⠿⠿⠿⢿⠿⠿⢿⣿⣿⢿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠿⠟⠛⠋⠉⠉⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠈⠋⠙⡟⠻⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣿⣿⠿⣿⣷⣦⣄⠀⠀⠀⠐⠟⢸⡇⠀⠀⠀⡇⠀⢸⡇⠙⢻⣷⡄
⠀⠀⠀⠀⠀⢀⣠⣴⣶⡿⠛⠉⣹⣄⣤⣤⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⣿⠀⠀⠀⠙⣿⣿⣷⡄⠀⠀⠀⠸⠇⠀⠀⠀⡇⠀⢸⡇⠀⠀⢻⣷
⠀⠀⠀⠀⣰⣾⡿⠛⣁⣠⣶⣾⣿⡟⢻⣿⠀⠀⠀⠀⠀⠀⣼⡿⣿⠀⠉⠀⠀⠀⠀⣷⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠃⠀⠀⢸⣿⣇
⠀⠀⠀⠀⢹⣿⣶⡾⠿⠛⠉⠀⣾⡇⢸⡿⠆⠀⠀⠀⠀⢰⣿⡇⣿⡄⠀⠀⠀⠀⠀⢿⠸⠇⠻⣿⣧⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⣿⡟
⠀⠀⢀⣴⣿⣿⡇⠀⠀⠀⠀⠀⠛⠃⠸⡇⠀⠀⠀⠀⠀⢀⣿⡧⣿⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇
⠀⠀⣼⣿⣫⡿⠃⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠈⢻⣿⣦⣄⠀⠀⠀⠀⠀⠀⣠⣾⡟
⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣦⣦⣴⣾⡿⠛⠁
⠀⠀⠸⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣾⠿⠟⢙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡟⢻⡏
⠀⠀⠀⠘⠻⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⠟⣋⡁⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⠀⠀⠀⠀⠸⠇⢸⣇
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁⣁⡀⠀⢉⣁⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠏⢉⣽⣶⣦⠀⠀⠘⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡄⠸⣿⣇⠀⠀⣿⣧⠀⢀⣠⣴⣶⡿⠿⣿⣶⡼⣿⣿⣄⣸⣿⣾⣟⣠⣴⣦
⠀⠀⠀⢀⣶⣶⣶⣶⣤⣄⡀⠀⠀⠀⠈⣿⣧⠀⢹⣿⣆⠀⠙⠻⢿⠿⠋⠉⠀⠀⠀⠀⠉⠁⠈⠉⠛⠛⢹⠟⠛⠛⠛⡇
⠀⠀⠀⠘⣿⣿⣌⠉⠙⢿⢿⣷⣦⣄⡀⠘⣿⣷⢸⡻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠙⣿⣶⡀⢸⠈⡇⠉⠛⢿⣶⣿⡿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠈⢿⣷⣾⡇⠟⠀⠀⠈⠉⢻⡇⠘⠇⠀⢀⣤⣼⣿⣿⣿⣾⣶⣶⣶⣦⣄⣤⣾⡿⠋⠀⢀⣀⣉⣛⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠀⠀⠀⠀⠀⢸⡇⠀⢀⣾⣿⡿⠿⠿⣿⣿⣿⣿⣛⠉⠙⠟⠛⠉⣀⣴⣾⣿⢿⡿⢻⡟⠛⠿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄⠀⠀⠀⠈⠀⠀⠈⠋⠁⠀⠀⠀⠇⠈⠛⠻⢿⣿⣶⣶⣶⣾⡿⠿⠋⠀⠈⡇⠸⠇⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⡉⠉⠉⠁⣀⣀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⡿⢿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠈⢿⣷⣼⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣶⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣌⣤⣀⡤⢾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣶⣤⣿⣿⣿⡿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣷⣄⣘⣿⣿⣿⣿⡟⠛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⡀⢿⣿⣿⣿⣿⣿⡿⠛⠛⢻⣿⣿⣧⣶⣾⡀
⠀⠀⠀⠀⠀⠀⠠⣆⠀⢹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣿⣿⣿⣿⣿⣿⢀⣀⣤⣤⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠶⠟⢻⡄⠉⠛⠻⠿⠿⠿⠿⢟⣿⣿⡿⠋⢻⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢀⣠⣴⣿⣿⠟⠁⣀⡜⠀⠀⠻⣿⣿⣿⡿⢁⣠⣤⣶⡿⠀⠈⣍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣶⣿⣿⣿⠿⠋⢁⡶⢲⣿⣄⠀⠀⠀⠈⠻⢿⣷⣿⣿⣿⣿⠃⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠻⣿⠿⠇⠀⠀⠀⠀⠩⠋⢹⡯⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠟⠛⠛⠛⠛⠛⠿⠷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠻⠄
⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⡀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠟⠛⠋⠉⠉⠛⠛⠻⢶⣦⣀
⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣤⡀⠀⠀⣿⡆⢀⠄⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡤⢤⣿⣿⠀⠀⢸⡷⣿⠀⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⢼⠿⠋⠀⠀⢸⡟⠈⠒⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠀⠀⠀⠀⠙⠻⢷⣦⣤⣄⣀⣀⣀⣀⣠⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣶⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⣤⣴⣦⣤⣀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠻⠿⣷⣤⣀⡀⢀⣀⣤⡾⠟⠋⠉⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠟⠉⠉⠛⢿⣷⣦⣄⣠⣤⣾⡿⠃⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄⠀⠀⠀⠈⠛⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣭⣭⣭⣽⣬⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣶⣶⣾⣿⣷⣾⣶⣷⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⣠⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣠⣤⡀
⠀⠀⠀⠲⠶⠆⠀⠀⠶⠶⠛⠓⠀⠀⠀⠠⢤⠴⠆⠀⠁⠀⠈⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠉⠩⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠂⠐⠂⠃⠸⠀⠓⠀⠤⠀⠀⠀⠀⠀⠘⠃⠀⡀⠘⠋⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠂⠀⠀⠀⠀⠀⠀⠠⠞⠛⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠴⠂⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠰⠶⠄⠀⠀⠴⠶⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠒⠂
⠀⠀⠀⡀⢀⠀⠈⡆⢀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣌⡁⠀⢋⣔⣂⡀⠀⠀⠀⠀⠀⠘⠁⠀⠘⠁
⠀⠀⠀⠐⢌⣁⠤⠓⢄⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⡀⡀⠀⢀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢀⣀⣀⣨⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⠁
⠀⠀⠀⠀⠀⠶⡆⡴⠶⠀⠀⠀⠀⠀⠀⠀⠰⣦⣄⠀⠀⣀⣴⠦⠀⠀⠀⠀⠀⠀⠠⠚⠁⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣭⠁⠈⣭⡄⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠶⠀⠰⠾⠷⡆
⠀⠀⠀⢠⡠⠿⠀⠠⠷⢠⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠠
⠀⠀⠀⠸⣗⢰⣤⣤⠒⣿⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠹⣾⠛⢻⡶⠃⠀⠀⠀⠀⠀⠀⠀⠈⢷⠦⠴⡾⠁⠀⠀⠀⠀⠀⠀⠀⠘⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠲⠀⠉⠉⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⢀⣠⡴⢶⡀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡈⠉⠀⠈⢁⣀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠒⠛⠐⠀⠀⠘⠓⠲⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣴
⠀⠀⠀⠀⠀⣶⣄⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⠀⠀⠀⠐⢎⣀⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢦⣤⡄⠀⠀⠀⠈⣿⣄⠀⠀⠀⠀⠀⢸⠇⢰⠇⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢏⠀⡐⡅⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠀⢸⠀⢸⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⣿⡇⠙⢿⣷⣦⣄⣘⠀⣂⣀⣴⣷⡀⠀⠀⠀⣀⣤⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠉⠛⠿⠿⠿⠿⠛⠉⠻⣿⣿⣿⠿⢿⣿⡟⠀⠀⣀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣀⣠⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠸⡏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠙⢿⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⠖⠛⠉⠉⠉⠓⠶⣤⣿⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠇⠀⢸⣿⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣾⠁⠹⣾⡣⠀⣠⣴⣶⣤⣀⣰⣟⣹⡆⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣀⣀⣤⡾⠟⠓⠒⠶⣄⣹⣯⠝⢦⡀⠀⢀⡇⠀⠀⢻⡗⣾⠋⢰⡖⠛⠉⠉⠉⣿⣷⠀⠀⣧⣀⡀
⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⠉⠻⣿⡟⠁⠀⠀⣀⠀⠀⠀⠙⢿⡯⡈⠻⣄⢸⠃⠀⠀⠀⢻⣻⣄⠸⣇⣤⡄⠀⠀⡿⣿⠖⠒⢻⡀
⠀⠀⢈⡏⠀⡣⠀⠀⠀⠀⠀⢠⣿⣷⢅⣴⠿⠛⠻⢷⡄⠀⠈⣿⣔⠀⠈⠿⠀⠀⣆⠀⠈⢷⡉⠛⠛⠉⠀⢀⣼⢃⣸⣄⣀⣼⡿⠛⠓⠒⠂
⠀⠀⠈⠁⠙⠃⠀⠀⠀⠀⢀⣼⣿⣿⣺⡇⠀⠀⠀⠈⢿⠀⠀⢸⣿⣷⡄⠀⠀⢰⣿⡆⠀⠘⣷⣤⣤⣴⡶⠟⣱⠟⠉⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣧⡀⠀⠀⣠⡞⠀⠀⣼⣷⣿⠻⣦⣀⣸⠀⢻⡶⠾⠿⠉⢉⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⡆⠠⡄
⠀⠀⠀⠀⠐⠚⠛⠛⠛⠉⠉⣹⠿⣿⣿⣾⠉⠛⠛⠁⠀⠀⣰⣿⣿⣿⠄⠘⠛⠋⠀⠀⠀⠀⠀⠐⣤⠷⣖⠒⠒⠛⠦⠀⠀⠀⠈⢠⡤⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⡰⠛⠻⣷⣧⣤⣤⣤⣴⡾⢿⣯⡁⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⡎⠈⢳⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣠⠞⠁⠀⠀⠜⠋⠉⠉⠉⠀⠀⠀⠙⣷⡀⠀⠀⣠⠞⠋⠉⠉⠉⠲⣇⠀⠀⠙⠆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡼⠃⢀⡴⠞⠛⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠒⠈⠦⡄⠀⠀⠀⠀⠀⣿⡼⠁⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣠⣀⡎⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠃⠀⠀⠀⠀⠀⠀⠏

⣿⠛⣿⡟⣿⡟⢻⡟⢻⡟⢻⣿⢻⣿⠛⣿⠛⣿⠛⣿⡟⣿⡟⢻⡟⢻⡟⢻⣿⢻⣿⠛⣿⠛⣿⠛⣿⡟⣿⡟⢻⡟⢻⡟⢻⣿⢛⣿⠛⣿⠛⣿⠛⣿⡟⣿⡟
⣿⠀⣿⡇⣿⡇⢸⡇⢸⡇⢸⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇⢸⡇⢸⡇⢸⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣻⣧⣾⠷⠾⣧⣸⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇
⣿⠀⣿⡇⣿⡇⢸⡇⢸⡇⢸⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇⢸⡇⢸⡇⢸⣿⢸⣿⠀⣿⠀⣿⠀⣿⣿⠟⠁⠀⠀⠀⠀⠈⠑⢼⣿⣤⣿⣄⣿⠀⣿⡇⣿⡇
⣿⠀⣿⡇⣿⡇⢸⡇⢸⡇⢸⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇⢸⡇⢸⡇⢸⣿⢸⣿⠀⣿⣠⣿⠦⠿⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣿⠇⣿⡇
⣿⠀⣿⡇⣿⡇⢸⡇⢸⡇⢸⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇⢸⡇⢸⡇⢸⣿⢸⣿⢠⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇
⣿⠀⣿⡇⣿⡇⢸⡇⢸⡇⢸⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇⢸⡇⢸⡇⢸⣿⢸⣿⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡇⣿⡇
⣿⠀⣿⡇⣿⡇⢸⡇⢸⡇⢸⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇⢸⡇⢸⡇⢸⣿⢸⣿⠀⣿⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇
⣿⠀⣿⡇⣿⡇⢸⡇⢸⡇⢸⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇⢸⡇⢸⡇⢸⣿⢸⣿⠀⣿⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡆⣿⡇
⣿⠀⣿⡇⣿⡇⢸⡇⢸⡇⢸⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇⢸⡇⢸⡇⢸⣿⢸⣿⠀⣿⢋⣀⣀⣤⡆⠀⠀⠀⠀⠀⠀⣠⣄⣀⣀⣀⣠⣴⣾⠋⣿⡇⣿⡇
⣿⠀⣿⡇⣿⡇⢸⡇⢸⡇⢸⣿⢸⣿⢀⠿⡒⠭⣳⣿⡿⢿⣷⣾⣧⣼⣷⡞⠻⠼⣿⣀⣿⠉⣿⠀⣿⡏⣷⡶⢶⡶⢶⡟⢹⣿⢩⣿⠉⣿⠀⣿⠀⣿⡇⣿⡇
⣿⠀⣿⡇⣿⡇⢸⡇⢸⡇⢸⣿⠬⠋⠁⠀⠠⠄⠐⠶⣶⣬⣽⣿⣿⣯⣙⠛⠛⠿⣉⠻⢟⠶⣿⡂⣿⡇⣿⡇⢸⡇⢸⡇⢸⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇
⣿⠀⣿⡇⣿⡇⢸⡇⢸⣧⠜⠁⠀⢀⣠⠶⠞⠛⠛⠛⠛⠿⢿⣿⣛⠧⡙⢍⠓⠤⠀⠁⠀⠀⠀⠙⠻⣇⣿⡇⢸⡇⢸⡇⢸⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇
⣿⠀⣿⡇⣿⡇⢸⡇⣸⠋⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢧⡀⠀⠑⠄⠀⠂⠀⠀⢀⠀⠀⠈⠻⡇⢸⡇⢸⡇⢸⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇
⣿⠀⣿⡇⣿⡇⢸⡿⠁⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠙⣆⠱⡈⠀⠀⠀⠀⠀⠡⡀⠀⢂⢹⣿⡇⢸⡇⢸⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇
⣿⠀⣿⡇⣿⡇⠜⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⢧⡘⣄⠀⠀⠀⠀⠀⠐⡀⠀⢆⢿⡇⢸⡇⢸⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇
⣿⠀⣿⡇⢻⠋⠀⠀⡀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⠶⠾⠶⠦⣜⢦⡀⠹⣎⢳⡀⠀⠀⠀⠀⠀⠀⢲⣾⡇⢸⡇⢸⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇
⣿⠀⣿⡇⠁⠀⢀⠌⢠⣧⡴⠶⠶⢶⣄⠀⠀⠀⠘⠋⠁⣀⣤⣤⣀⠀⠀⠀⠙⣷⢮⣿⣿⡄⢈⡄⠀⠀⢠⠘⣿⡇⢸⡇⢸⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇
⣿⠀⣿⡇⠀⠀⠀⢀⣾⣧⢀⣤⢶⣤⡈⠀⠀⠀⠀⠀⠞⠉⠰⢍⣻⡿⠀⠀⠀⠈⢷⣬⣽⣇⡀⡇⠀⠠⢸⠀⣿⡇⢸⡇⢸⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇
⣿⠀⣿⡁⠀⠀⠀⣾⣿⣿⢻⣁⣘⣢⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠘⠞⢰⡿⣿⡇⠀⠰⠨⢀⡌⢳⣸⡇⢸⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇
⣿⠀⣿⣇⠀⠀⢸⡟⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢿⡇⠀⣿⠀⡌⣇⠀⠹⣇⢸⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇
⣿⠀⣿⡇⣣⡀⠈⢧⢸⠃⠀⠀⠀⠀⢄⡀⠀⣀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⢧⠀⠑⠠⠿⣧⣥⡙⢦⡐⠌⢻⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇
⣿⠃⣿⡇⣿⡏⢦⢾⡼⡇⠀⠀⠀⠀⠀⠙⠁⠀⡉⠀⠁⠀⠀⠀⠠⣀⠀⠀⠔⠁⠀⠀⠀⣿⢷⣤⣀⡀⠀⢄⡈⠂⣱⡄⢠⣿⣸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇
⣿⡇⣿⡇⣿⡇⢸⡾⠃⢿⡀⠀⠀⢀⠌⢀⠜⠂⣀⠈⡩⠡⠄⢠⣤⠌⠁⠀⠀⠀⠀⠀⢰⡟⣼⠻⡿⣿⡶⡄⠙⣆⠈⣿⢸⣿⠇⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇
⣿⡇⣿⡇⣿⡇⠜⠁⢰⠃⡿⣄⠀⠊⠰⠶⠮⠅⠠⠄⠰⠐⢀⠁⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⡇⠀⢳⠈⠃⠈⠀⢹⠀⣰⣿⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇
⣿⡇⣿⡇⣿⡇⠀⠀⠃⡸⢃⢺⡷⣄⠀⠀⠂⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⣠⣴⠟⢹⡇⢃⠠⡀⠳⢤⣤⣿⣼⣰⡿⢻⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇
⣿⡇⣿⡇⣿⡏⡄⠈⠀⢁⣾⣸⣧⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠁⠀⢸⡿⣬⡳⢬⣷⣶⣿⣿⡟⢻⡇⢸⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇
⣿⡇⣿⡇⣿⡇⢸⣦⡰⠀⣿⢏⢻⣷⠶⣾⠷⣄⡀⠀⠀⠀⣀⣤⡶⠚⠉⠀⠀⠀⠀⠀⢸⡇⠈⠙⣳⡟⣿⡏⢹⡇⢸⡇⢸⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇
⣿⡇⣿⡇⣿⡇⢸⡇⢹⡦⢬⣳⣤⣿⣒⣿⠀⣿⠁⣿⡎⣿⡇⢹⣦⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⣿⣿⣿⡇⢸⡇⢸⡇⢸⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇
⣿⣧⣿⣧⣿⣧⣼⣧⣼⣧⣼⣿⣴⣿⣤⣿⣤⣿⣤⣿⣧⣿⣧⣼⣿⣧⠀⠀⠀⠀⠀⢀⣼⣷⣿⣿⣿⣿⣿⣧⣼⣧⣼⣧⣼⣿⣬⣿⣤⣿⣤⣿⣤⣿⣧⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡟⣿⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠁⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡇⠀⠀⣤⡀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠈⣿⣶⣿⣿⠟⠃⠀⠀⠀⠹⣷⣴⣾⠗⠀⠀⢼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣠⣾⡿⢿⣿⣧⡀⠀⠀⠀⢀⣴⠿⣿⣷⣄⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠉⠁⠀⠀⠹⣿⡿⠀⠀⠐⠋⠀⠀⠹⣿⣿⡇⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⢀⣀⠉⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣿⡇⠀⠈⢿⣦⣄⣀⣀⣀⣀⢀⣀⣤⣶⣿⣿⠏⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⠀⠂⠀⠉⠛⠿⢿⣿⣿⣿⣿⠿⣿⢻⡇⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢸⡇⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⠛⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠰⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣦
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⢏⣥⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣵⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠛⠛⠛⠿⣿⣞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠟⠉⠉⠉⠻⣿⣿⣿⣿⣿⢀⣤⣶⣶⣶⣿⣿⣎⢿⣿⡿⠛⠋⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏⣴⣶⣾⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣯⣴⣶⣶⣦⣄⣹⣿⣿⣿⡿⠛⠉⠙⠻⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣄⣹⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣾⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣣⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣍⠛⠻⠿⠿⠿⠛⠋⠀⢰⣿⣿⣼⣿⣿⠟⠋⠁⠀⠀⠀⠙⠻⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣾⣿⡟⣿⣿⠃⢀⣀⣤⣀⡀⠀⠀⠀⢹⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣷⣄⠀⠀⠀⣰⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣷⡀⠀⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣶⣾⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⠿⠟⠋⠀⠀⠀⠙⠻⠿⣿⣿⠿⠟⠉




⡿⠿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⠿⣿⡟⣿⣿

⣿⠋⠙⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠈⡿⡏⡌









⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣩⣩⣭⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣭⣭⣭⣭⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣒⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⠻⡉⠉⣙⡛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⣠⠤⠤⠤⠤⠤⣝⡀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⢰⢄⠀⠀⠀⠀⠀⣀⣠⢶⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢸⣄⠉⠉⠉⠉⠉⠉⣠⡎⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠘⣿⣿⣶⣶⣶⣾⣿⠟⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠈⠻⠿⣿⠿⠟⠁⠀⠀⠀⠀⣼⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣃⠀⠉⠉⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣶⣄⣀⡀⠀⠀⢀⣀⣠⡶⠋⠀⠀⠙⠲⣄⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠉⠉⠉⢉⣩⠟⠁⠈⠙⢦⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠙⢦⡀
⠀⠀⢀⠔⠊⠉⣄⠈⠉⣉⠓⠲⠤⢤⣤⠤⠶⠚⡿⠁⠀⣀⣀⣤⠴⠚⠉⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠈⣷⡀
⡞⠲⠧⠴⣇⣀⣸⢧⡤⣾⣀⡀⢰⡖⠶⠶⠖⠒⠛⠛⠋⠉⠑⠄⠀⠀⠀⠀⠀⣠⡼⠋⢀⡠⠔⠋⠀⠀⠀⢹⡄⠀⠀⠤⠚⠀⢷
⠓⠶⠦⠤⠭⢭⣥⣄⣀⣀⣀⠉⠙⢷⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠱⡆⠭⣗⡶⢫⡀⠀⠀⠀⢀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢸⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣇⠀⠀⠉⠒⣽⡛⡻⠛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⢸⡇⢸⠀⠀⢀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣿⠀⠀⠀⠀⡇⠈⠳⠞⠁⠀⣀⣀⣀⣀⡠⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠈⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢻⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⢸⠇⡼⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⢀⡿⢀⡇⢀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢳⣄⡀⠀⠀⠤⠤⣤⣤⣤⡤⠤⠖⠚⠉⠀⠀⠀⢀⡾⣅⣼⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⣰⠋⠀⠙⠳⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠋⠀⠀⠙⠓⠶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠈⢩⠟⠛⠛⠛⠛⠛⠉⢿⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⣠⠤⣖⠒⠶⠧⠿⣤⠖⠛⠲⣿⠀⠀⡀⠀⠀⢀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣠⣼⠶⠋⠁
⠀⠀⠀⠀⠀⠀⡼⠁⣰⠽⣄⠀⠀⡾⠁⣠⢤⡀⢹⡙⠓⠻⠦⠤⠼⠷⠖⠛⠲⠶⠤⣤⣤⣀⣠⠴⠒⠦⣄
⠀⠀⠀⠀⠀⠀⢧⠀⠙⠋⢙⠗⢾⡇⠐⣇⡼⠃⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢀⡔⢢⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠋⠀⠘⣇⠀⠉⠀⢀⡾⠉⠉⠉⠉⠛⠒⠒⠒⠒⠒⠶⢤⠤⢼⡇⠘⢦⠊⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⣷⣄⣀⣀⣴⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⣶⣶⣾⣿⣷⡴⠛⠉⠁⠀⠀⠀⠈⣳
⠀⠀⠀⠀⠀⠀⠀⣴⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢿⣿⠻⠿⣿⡀⣴⣶⣤⣤⣀⣠⣔⣃⡀
⠀⠀⣠⠖⠒⢂⣾⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠉⣿⣯⣤⣾⣿⣿⣾⣿⣷⣿⡿⠿⣿⣿⣿⣿⣿⡇
⠀⠀⢧⡀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣶⡞⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠦⢿⣿⣿⠟⠉⠉⠻⣿⣿⣿⣶⣶⣿⣿⣤⣬⣿⡤⠖⠒⢲
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠿⣤⣾⣿⣶⠏⢻⣿⡿⢿⣿⣿⣿⣿⣿⣇⡤⠖⠋
⠀⠀⠐⢸⣿⣿⣿⣿⣿⣿⣿⡿⠁⠙⠛⣫⣿⣤⣵⡀⠀⠀⠀⠀⠀⡎⠀⠀⠀⣈⣻⣟⠋⠀⠀⢻⣧⡀⠀⢀⣴⡿⣻⠏
⠀⠀⠀⠈⣿⣿⣿⣿⡿⢟⠏⠀⠀⠀⣾⣿⣿⣿⣿⣿⣦⣄⠤⠤⡀⠈⠉⠉⢉⣉⣀⣼⠇⠀⠀⠀⢻⣿⣿⣿⠏⠀⠻⠤⠤
⠀⠀⠀⠀⢘⠛⠿⢋⡀⠘⣶⣉⣉⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠳⢤⣿⡉⠑⣬⠁⣾⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣤⣶⣿⣿⣷⣶⣤⣤⣄⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⢀⠤⣌⠉⠀⢹⣿⣿⣿⣿⣿⣿⣿⠏⠀⡇⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠿⠿⠿⠿⠟⠋⠁⢤⠴⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣩⣽⣿⣯⣶⣾⣿⣟⣍⣻⣿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢀⡤⠤⣄⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣟⠻⠿⠋⣿⣿⠟⠛⠻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⢀⣠⠴⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣼⣿⣷⣿⣿⣦⠟⠁
⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⠋⠁⠀⣾⣿⣶⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠛⠛⠁⠙⠻⠛⠁
⠀⠀⠀⣰⣿⣿⣿⣿⣿⡟⣻⣄⣴⠉⢸⣿⣿⣿⣿⣿⣿⣿⠛⠛⣦⠀⣤⣄⣀⣀⣤⣤⣶⣿⡇
⠀⣴⠞⠉⠉⢻⣛⣿⣿⣿⣿⣿⣯⣩⣿⣶⣽⣿⣿⠿⣿⣁⣤⠾⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⣠⣤⣤⣤⣤⣄⣀
⠘⣧⣤⡤⠖⠒⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⣠⠤⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⡿⢿⡏⣠⣴⣾⣿⣷⠈⠛⠛⣻⡟⠁⠉⠉⠁⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣞⣥⠖⢿⠿⣿⠿⠉⣿⣷⣶⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠔⠋⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⡿⠗⢿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣷⣾⢡⡶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠶⡟⣻⣦⣿⣧⣙⣲⣯⠶⠠⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠐⠦⣼⣸⢿⣵⠊⠁⠀⠈⣙⢯⣔⡃⠄⠀⠀⠀⠀⣷⢀⡟⠀⡀
⠀⠀⠀⢀⡀⠈⢷⣠⡟⠀⠀⢸⡛⣦⢼⡿⢋⡀⠀⠀⠀⢊⡥⢄⢻⡿⠗⠀⠀⠠⣀⣸⣾⡶⠋
⠀⠀⠀⠀⠉⢓⣬⡟⢻⡄⠀⠐⠒⣨⡿⠁⢾⣿⠀⠀⠀⡋⣶⠀⠘⣷⠀⠀⠀⠀⠈⣿⡼⠋⠛⠒
⠀⠀⠀⠀⠛⠛⢉⡿⠛⢷⢦⡀⠀⣿⡇⠀⠠⠤⠂⠒⠀⠘⠒⠁⠈⣿⡆⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠣⡱⣄⠹⡅⡄⠀⠀⣶⡟⠿⣷⡄⠀⣼⠿⠁⢀⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⠳⣝⢌⡲⣜⡛⠾⠟⢛⣠⡾⢋⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠘⠋⠹⣿⠮⠵⡶⢾⣁⣴⠏⢉⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⠀⠘⢧⣼⠁⠀⠙⠃⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠉⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣀⣀⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⠀⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⣩⢿⣖⠒⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣦⣴⠏⠸⣿⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⣰⡟⠀⠁⠀⠀⠉⢹⡏⠉
⠀⠀⠀⠀⠀⠀⢠⣿⡟⠢⣤⣤⣾⠟⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣰⣶⣄
⠀⠀⠀⠀⠀⠀⠸⠿⠶⠋⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣏⡁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠾⡿

⠀⠀⠀⠀⠀⠀⢠⣴⣠⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣀⣤⠤⠤⠄
⠀⠀⠀⠀⢠⣶⣿⣏⣿⡜⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢡⡞⢁⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠈⠿⢋⣫⣬⣷⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣾⠟⢋⣉⣉⡛⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠛⠿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣴⣤⡀⠀⠸⣿⣿⣏⡜⣁⠤⠄⡉⠳⣌⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⣀⣀⣀⣀⣈⣙⣿⣷⣦⣹⣿⣿⣕⡡⠒⠒⢌⢦⠈⢦⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣶⠿⠿⠛⠛⠛⠛⠙⢙⣿⠟⠛⠛⠛⢿⣿⠿⠿⣶⣾⣦⡱⡀⠑⢄⠙⢿⣦⡀
⠀⠀⠀⠀⣠⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢰⡶⠆⢻⡆⠀⠀⠈⠙⢿⣿⣢⣌⠳⡌⢻⣷
⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠈⠙⢁⣿⠃⠀⠀⠀⠀⠀⠙⢿⣮⢆⢹⢸⣿
⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠶⣿⣟⡁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡞⢸⡿⣇⣀⡄
⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡸⣷⡈⠉
⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⣶⣶⣤⡀⠀⠀⠀⢀⣿⣿⣿⣾⣿⠖⡀
⠀⠀⠹⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⢿⡄⠁⠀⠀⢰⠟⠛⠻⣿⣿⠟⠋
⠀⠀⠀⠀⠉⠛⠻⠿⠷⢶⣶⣶⣶⣶⣶⣶⣶⠶⠿⠗⠀⠀⠀⠀⢀⣼⠋⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⣀⣀⣀⣀⣤⠶⠛⠁⠀⠀⣴⣿⠀⠀⠀⠸⣷⣤⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠉⠉⢿⡇⠀⠀⠀⢀⣤⣾⣿⡏⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⢀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⢿⣶⣶⣾⠟⠋⢁⣾⠃⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣤⣤⣀⣠⣿⣡⡶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠛⢿⣥⣤⣴⠟⠁⣠⡆⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⠛⠛⢿⣿⣿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⢀⣴⡾⠛⠿⠛⠁⠙⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠶⠤⢤⣤⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣿⠃⠀⠀⣀⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠓⠲⠶⠶⠤⠤⣤⣭⣷⣤⣟⣉⣉⣉⣉⣉⣉⣁⣸⣿⣴⡶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢻⣏
⠀⢀⣴⠟⢳⡄⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⣸⠇⠀⢸⡇⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⡼⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⣴⠿⠇⣰⣃⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠿⠶⠞⠛⠛⠻⠿⣶⣶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿⠷⠶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣍⣉⠉⠉⠉⣿⡍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣈⠉⢿⡿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣴⠇
⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⣪⡿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣠⠞⠁⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢠⠞⠁⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⢦⣄⠀⠀⢀⠎⠀⣰⠋⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣾⠀⡼⠁⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣁⣀⣀⡤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡍⠉⠀⠀⣀⡤⠴⠒⠒⠒⠒⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠖⠒⠒⠶⢤⣄⡀⠀⢿⢀⡤⠚⠁⠀⠀⠀⠀⠀⢀⣀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠺⠋⠀⠀⠀⠀⠀⠀⠀⢰⡉⠈⡧⠀⠀⣠⠤⡝⢦⡀
⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠳⡤⠟⠀⠹⡄
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡄⠀⡀⠀⠀⠀⠀⠀⣄⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⡴⠓⡆⠹⡄
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠶⣾⣧⣞⠁⠀⠀⠀⢳⣤⣿⡤⣞⠀⠀⠀⠸⣭⠇⠀⠀⠙⠚⠁⠀⢧
⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢿⣿⢛⡆⠀⠀⠀⡏⢿⣿⠇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⠤⠞⠀⠀⠀⠀⠳⢤⣄⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⡞⠉⠙⢦⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⢳⡀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠈⠧⣀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⡼⠁⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠦⠤⠤⠴⠒⠚⣹⢿⠻⡝⠓⠒⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠘⡇⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⢀⠀⠀⡀⠀⢀⡿⡆⣠⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠸⣷⣄⠀⢳⣰⡇⢠⣸⠁⣷⠇⢸⠁⣤⣴⠆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣝⢦⡀⠢⡈⠲⣿⡙⢶⡀⢻⣇⠀⡏⠀⠙⠀⣿⠀⠁⣾⠞⣰⢁⣤⢔⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠢⣌⡃⠙⢦⡈⡦⠈⢅⡄⠀⢧⠉⢸⠀⠀⡤⠀⠀⢰⠆⠏⢀⠀⠚⠁⣽⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣥⣀⣬⠿⠶⠱⣦⠙⠆⠀⠉⠀⠠⠄⠈⠀⠇⠀⠸⠀⠈⡠⠀⠋⢠⠀⢘⡿⠋⣀⠄
⠀⠀⠀⠀⢤⣀⠀⠀⠒⠮⣿⡌⢍⠁⠠⡀⠈⠀⢢⡀⠁⠀⠀⠄⠀⢀⠀⠂⠀⡀⢀⠁⢠⠀⠁⠀⠠⠴⠯⠗⣶⠃
⠀⠀⠀⠀⠀⠈⠙⠳⠦⠍⠒⠦⣄⠁⠀⠉⠀⠠⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣷⡀⠀⣾⡴⠀⠀⠀⠀⠀⠬⣭⣤⠄
⠀⠀⠀⠀⠀⠉⠙⠛⠒⠲⠀⠠⡀⠀⠀⠙⠂⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠠⣄⣿⠳⠀⠻⠁⠀⠀⠀⠀⠐⢻⠯
⠀⠀⠀⣤⣤⣽⠿⠧⣤⠀⠒⠚⠛⠂⠈⠁⣀⠀⠀⠀⠈⠀⠀⠀⠠⣤⣬⣆⠈⠛⠀⠀⢀⣄⠀⠀⢠⣀⢸⡇
⠀⠀⠀⠀⢀⣀⣠⡤⠀⢉⠿⠒⠀⠠⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢙⣦⠀⠀⠀⠀⠀⠋⠥⡀⠀⠀⠪⢸⡇
⠀⠀⠀⠀⠀⠰⠶⣤⣤⡄⠐⣋⡀⠀⡴⠀⠠⠀⠀⠀⠒⠀⠀⠀⠸⣏⠀⠀⠀⠀⠀⠀⠰⠇⡇⠀⢞⢣⢺⠇
⠀⠀⠀⠀⠀⠐⠋⠉⠐⠚⠉⡉⠁⠊⠀⣠⠀⠀⠀⠄⠀⠀⠀⠀⠀⡞⠀⠀⢀⣀⣀⡀⠈⠒⠃⢀⠬⠛⠛⠛⠻⢶⡞⢻⣦
⠀⠀⠀⠀⠒⠚⣋⣴⡾⠀⠚⠇⠀⠚⠀⡄⠀⠆⠀⠀⠂⠀⠀⠀⠈⠃⠀⠀⠀⠹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠤⠚⢩⣟⣥⡶⢀⣠⣀⣄⡀⡁⠰⠀⠉⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣄⣀⣀⠀⠀⢀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠠⠟⠋⢰⡷⠛⢛⠁⠀⠉⠀⠀⣄⣀⣀⣀⡀⣶⠀⠀⠀⠀⠀⠉⠉⠽⣦⣠⣶⣭⡽⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣴⣤⣤⡼⠶⠏⠈⠉⠉⠁⠙⠳⠶⠶⢶⣦⣀⣠⣤⣾⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠦⢤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄⠀⠀⠀⢀⣤⠶⠚⠛⠛⠛⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠲⣦⡀⠀⠀⠀⠈⣧⣠⠐⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢺⣿⡇⠀⠀⠀⠀⡿⢿⣐⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣤⡶⣶⣦⣄
⠀⠀⣴⣾⡿⠻⠶⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⢸⠇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⣸⠈⠙⠇
⠀⠘⠁⠘⣿⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠙⠷⣤⣀⣀⣀⣀⣀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣤⣤⣤⡴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠷⠶⠶⠞⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣶⣤⣤⡶⠞

⡠⣠⣶⢰⣦⢂⠀⠀⢀⠠⣄⠐⣰⣦⡀⠀⠀⠀⠀⣠⢤⣠⠖⣲⡄⠀⠀⠀⠀⡔⣲⡆⣲⣦
⢧⣈⡥⠑⠤⠜⠀⠀⢌⠛⡃⠠⣙⢋⡇⠀⠀⠀⢰⣃⣺⣃⣈⣿⣸⠄⠀⠀⠀⡇⠛⢡⣌⣸⠇⠀⠀⠀⣠⡄⣰
⠀⠀⣔⣒⡀⠀⠀⠀⠀⠉⠀⠀⠈⢉⡀⣠⠀⢀⣩⣤⣤⣤⠶⠦⠤⠤⠄⠀⠀⠙⠶⢋⣈⣩⠔⠒⠀⠰⠍⠀⣙
⠀⠘⢿⣿⠇⠀⠀⠀⠐⠔⣂⡭⠥⠤⠀⠀⠀⠀⠨⠭⠕⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠒⠀⠀⠀⢠⠤⠤⠄

⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣔⣴⠄⣾⡷⠀⠀⠀⠀⠀⠀⠀⣤⡠⡀
⠐⢾⣬⠇⠀⢰⣐⡇⠀⠀⠀⠀⣾⠰⠟⡠⠿⢊⡆⠀⠀⠀⠀⠀⠸⣆⣡⠂⢘⣲⠁⠀⠀⠀⠀⠀⠶⣿⠇⢸⠄
⠀⡐⢂⣂⠶⠴⠀⠀⠀⠀⠀⠠⢙⣓⣚⡉⠒⣚⣁⡤⠖⠀⠀⠀⠀⢈⠔⢈⠤⠀⠀⠀⠀⠀⠀⠈⠢⠤⠴⠋⣠
⠀⠪⠭⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠈⠁
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢄
⢀⣠⡄⣼⡗⡀⠀⠀⠀⠀⡜⢡⠔⣲⣝⠀⠀⠀⠀⠀⠀⠀⠀⠚⡀⡾⢋⠆⠀⠀⠀⠀⠀⡜⣤⣀⢀⣠⡎
⢸⣙⡱⢤⣴⠃⠀⠀⠀⠀⡜⠟⡠⡙⢋⡆⠀⠀⠀⠀⠀⠀⠈⠉⠰⠤⠞⠀⠀⠀⠀⠀⠀⠳⠽⠋⠯⣫⠇
⠀⠉⣀⣀⣄⠀⠀⠀⠀⠀⠈⠉⢀⣈⣉⠀⠀⠀⠀⠀⠀⠀⠳⣦⣀⣀⣠⠄⠀⠀⠀⠀⠀⣰⣄⣆⡄
⠀⠀⢿⢛⢻⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠈⡻⢟⠁⠀⠀⠀⠀⠀⠀⠘⠹⠻⠻⠇
⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣴⣮⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢚⠉⢑⠰⠻⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡀⠉⠁⠒⠒⠀⠤⠄⠠⣧⠬⠄⠚⡂⡤⢏⡗⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⠄⠐⠂⠒⣿⢸⠀⠀⠠⠒⢧⡟⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡹⣍⢠⠱⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢈⠉⠢⠀⠀⡊⠀⠀⢻⠀⠀⢀⠄⠀⠀⡌⠁⢱⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⣀⠙⣑⠀⣇⠄⢌⠁⡟⠀⠁⠉⢋⡠⠠⠀⢱⢐⡑⢁⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠴⠀⠀⠯⢔⠖⠉⠀⠀⠀⠀⠎⢐⣓⠤⠎⠈⠐⡂⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠔⠹⢀⢀⢠⡎⠀⠀⠀⠀⠀⠀⡄⠑⢄⢑⠀⠀⠘⠈⠢⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠣⡀⠀⠀⠀⢀⠔⠁⡘⠀⢫⠃⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣀⣉⣔⠉⢁⡀⢠⠃⡰⠃⡆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢀⠊⠀⠀⠘⠓⠁⠈⡡⠊⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡑⡈⠢⠔⠁⢰⠀⠀⡔⠁⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⠓⢄⣀⡘⢀⡊⣀⠴⠶⣒⠕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠈⠣⠘⠁⠁⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⡀⠀⠀⠀⡩⣀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠠⡤⢷
⠀⠀⠀⠀⠀⠀⠀⠪⡑⢄⣈⣦⡀⡸⠀⠀⠑⢤⣁⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠁⢘⠻⣅⠀⠀⠀⠈⠀⠉⠒⠦⡤⠌⠡⠤⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣼⡦⡀⠀⠦⢄⠀⢀⠁⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⢕⢝⣶⡙⠀⠉⡟⠛⠂⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠛⠀⠀⠀⢱⣀⣀⣀⣀⣀⣐⣦⣤⣤⣤⠠⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠐⠙⠊⠉⠋⠉⠉⠉⠋⠛⠋⠉⠀⠈⠋⠉⠉⠭⠟⠓⠒⠓⠒⠘⠓⠢⠝⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⠀⡭⢐⠒⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⣀⣀⣀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⢉⡉⠀⠈⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣠⣔⠉⠀⠀⠀⣧⣀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢦⡀⣀⡀⢀⡏⠌⠀⠈⠹⡄⠀⡔⠁⠈⠑⢄⢹⡤⠤⢤⠞⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠁⠀⠀⡹⠀⠀⠀⠀⠀⢻⠸⠀⠀⠀⠀⢸⡀⣇⠀⠀⠀⣶⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣆⠀⠀⠀⠀⡇⠅⠀⠀⠀⣀⣸⡇⠀⠀⠀⣂⣐⡇⢹⠀⠀⠀⡇⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⡇⠀⠀⢠⣾⣏⣿⣧⠀⢠⣾⣯⣽⡇⢸⡀⠀⠀⡇⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢠⡇⡆⢀⣿⣿⣿⡟⠸⡆⣿⣿⣿⣿⠇⡞⢱⠀⠀⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⡠⠊⢰⠸⣜⣿⣿⠟⠳⠀⠹⣿⣿⣿⡟⢰⠃⠀⢱⣄⡁⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡄⠁⢀⠀⠀⢧⠉⠛⠁⠀⠀⠀⠀⠀⠈⠁⣰⠃⠀⢠⠁⠘⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⡀⠀⠀⡆⠀⠈⠳⣄⠀⠘⠋⠛⠲⡄⠠⢾⠁⠀⢀⢬⡀⠀⡄⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⡧⠒⠦⠀⠈⠳⣶⠀⠀⠀⢱⠀⢸⣀⣠⡁⠀⠡⢀⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢸⠀⠀⠀⠀⠀⠐⠁⠒⠳⢤⣦⣴⣾⠀⢸⠁⠀⠄⠀⠀⢸⡇⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠀⡼⠀⠀⠀⠀⠀⢸⡇⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢱⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⢉⠇⣰⠃⠀⠀⠀⠀⠀⢸⡇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡆⠀⠀⠀⠀⠀⠀⡀⣆⣠⠎⣰⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⣇⠀⠀⠀⠀⠀⠈⢧⣉⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠁⠀⠀⠇⢸⠀⠀⠀⠀⠀⠀⠀⢉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠑⣄⠀⠀⡄⢸⠀⠀⠀⠀⠀⠀⠈⠀⡀⠉⠑⢷⣄⠀⠀⠀⠀⠀⢸⡇⣤⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠳⠃⠈⠻⣆⡈⠳⣤⠁⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠘⠁⠁⠀⠀⠀⢬⠀⠠
⠀⠀⠀⠀⠀⠀⢦⠙⠳⠶⣌⡙⠢⠈⢀⠃⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢠⠁⡄⠰⠄⠀⠀⠚⢐⣴
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠐⠉⠀⢸⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢢⡤⢀⣀⠀⠉⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⡷⣄⠀⠀⠀⣀⣀⡀⠀⠀⢀⡇⠀⡼⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉
⠀⠀⢀⡤⣤⣒⢉⠲⡄⠀⠀⠀⠀⢀⡤⠢⡜⠀⢀⡇⠉⢉⠏⡏⠀⠈⠉⠉⠉⣻⠀⢠⠃
⠀⢠⠋⠀⠈⣇⠀⠃⠙⣆⠀⠀⡠⡣⠚⠉⣃⠀⡰⠬⠥⠭⠚⠀⠀⠀⠀⠀⢠⠇⠀⠘
⠀⠆⡀⠀⠰⡇⠀⠀⢣⡘⠒⢊⡔⠁⢀⡴⠃⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⢴
⠘⠊⢣⡃⢆⠘⢤⡀⠀⠉⠉⠁⠀⠀⢈⠟⠒⠒⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡤⠤⠤⣁
⠀⠀⠈⠁⠀⠙⠓⠛⠁⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⠀⡆⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⣏⣀⣀⣀⡇⠀⠀⠘⢦⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣷⣶⣶⣾⣯⣭⣤⣤⣤⣤⣤⣬⣭⣭⣥⣽⣷⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢠⣤⡀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⣀⡀⠀⠀⠀⢀⡤⠾⠀⠀⠀⣀⡀
⠀⠋⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠉⠙⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠰⠀⠀⠀⠀⠆⡀⠀⠀⠄⠀⠠⢶⢶⣤⣀⣠⣀⣠⣤⡶⠟⢻⠉⠀⠃⠀⠀⣀⠙⢠⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⠘
⠰⠀⠸⠇⠀⡇⠛⠀⢀⠂⠀⠀⠀⠀⠀⠉⠉⠩⡁⠈⠁⢀⠌⠀⠀⠇⠀⠀⠿⢠⠙⠁⠀⠌⠀⠀⠀⠀⠀⣤⢀⠀⠀⠀⠈
⠀⠑⠀⠀⠔⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠂⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠂⠀⠉⠀⠀⠀⠈
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠒⠀⠀⠀⠀⠀⠀⠉⠓⠀⠙⢒⠀⠀⠀⢠⠴⠒⠀⠀⠐⠢
⠀⢀⣠⠾⠛⢁⡐⠳⣄⡀⠀⠀⢀⣤⣤⣀⠀⠀⡐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠠⠂⠀⠀⠐⠂
⠐⡏⠁⣤⠄⠨⠄⡤⠀⠈⡃⠀⠀⠀⠀⠉⢻⡆⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠠
⠀⢳⠀⠀⠀⢠⡄⠀⠀⢠⠁⠀⠀⠖⠒⠬⣆⠀⠣⠘⠃⠀⠄⠀⠀⠡⡀⠀⣠⡄⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠈⠁
⠀⠀⠑⠄⠀⠃⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⡄⠀⠀⢀⣀
⠀⠀⠀⠐⠶⠐⠁⠀⠀⠀⠀⠀⠀⠀⢘⠉⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣀⠴⠂⡘⠛⠦⣄⡀
⠀⠀⠀⠀⠀⡁⠀⠀⠀⠂⠀⠀⠀⠠⢡⣶⢦⡄⠁⣴⣶⠆⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠋⠀⠀⠴⠱⠆⠀⠀⡉
⠀⠐⠀⠀⠠⠇⠀⠀⠐⠖⠀⠀⠀⢀⠤⣿⡷⠀⡄⠭⢿⠟⠀⠀⠀⠆⠀⠒⢀⡆⠒⠀⢀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⢀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠂⠈⠂⠀⠐⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆⠀⠀⢀⣄⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⣀
⠀⣰⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠤⠄⡀⠀⠙⢛⡢⠀⠀⠀⢘⠩⠅⢀⠀⠈⠍⢃⠀⠀⠀⠀⠨⠀⠠⠀⠀⠈
⠀⠁⠀⠀⠀⠀⠀⠟⠛⠷⠆⠀⠀⢄⣀⣒⡂⠀⠄⠀⠀⠀⠀⠀⢀⠂⠀⢀⣤⢶⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⣀
⠃⠀⠐⠛⠀⠰⠀⠘⠛⠉⣀⠂⠀⢠⠀⢉⡀⠀⣣⣼⡁⣰⡇⠀⠘⡀⠀⠈⠛⠸⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠈⠣⣙⣋⠼⠓⠉⠉⠚⠀⠀⠀⠻⢄⢀⡴⠋⠲⠤⠚⠁⠀⠀⠀⢦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢠⠴⠾⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⡿⠿⠛⠛⢿⡄⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣸⣿⣿⣤⣴⣶⣿⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢿⡏⠉⣿⣿⠛⠛⠿⠿⣿⣿⡟⣿⡿⠿⠿⣿⠛⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⢀⣾⠃⠹⣇⠀⠀⠙⠒⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢀⣀⣤⣤⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⢀⣴⠾⣿⠉⠉⠁⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀⣀⣀⣀⣀⣠⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⢠⠟⠁⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠶⠖⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠛⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠉⠙⠛⠓⠶⠶⠤⢤⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⣤⢤⣶⡶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡖⠲⠦⠤⢤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⢀⡠⠖⠛⠉⠉⠙⠓⠶⣟⠉⠀⠀⠈⠙⢷⣴⡾⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⣠⡶⠟⠉⠀⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠤⣤⣤⣤⣤⠶⠶⠛⠛⠉⠁⠀⠀⠀⠀⠀⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠀⢀⣟⣉⣏⣋⣋⣉⣫⣙⣉⣙⣉⣏⣋⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣏⣯⣭⣽⣿⣿⣽⣯⣿⣯⣿

⠻⣿⣿⣷⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣷⠆
⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠉⢻⣿⣶⣤⣀⡀⠀⠀⠀⠀⣀⣀⣤⣶⣿⡿⠋⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⣤⢰⣆⠈⠻⣿⣿⣿⠀⢸⠇⠈⡿⠻⣿⣿⣿⣿⣿⣿⡿⢿⠛⢹⡇⠀⢸⣿⣿⡿⠋⢀⡄⣸⡇⣴
⠀⠀⢴⣸⣇⣿⢷⡀⠈⢿⣿⣶⣾⠀⠀⡇⠀⢸⠉⢸⠁⠀⡏⠀⢸⠀⠈⣷⣶⣿⣿⠏⠀⠀⣾⢷⣿⣸⣿
⠀⠀⠘⡏⢻⠉⡀⡄⡄⠈⢿⣿⣿⣦⣤⡇⠀⢸⠀⢸⡆⠀⡇⠀⣸⣤⣴⣿⣿⡿⠁⠀⠀⠀⡏⢸⠇⠋⠸
⠀⠀⠀⠉⠀⢰⢷⡿⠧⠀⠘⣿⣿⣿⣿⣿⣿⣿⣤⣼⣧⣴⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣀⣠⡄⣠⡤
⠀⠀⢀⣦⣦⠈⠀⡀⣴⢠⠀⢹⣿⣿⣿⣿⣿⡿⠿⠿⡿⠿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣠⠟⢿⡾⢋⣴⡞
⠀⠀⠘⡛⡋⠁⣺⢷⣧⣿⡀⠀⣿⣿⡛⡇⢘⡇⠀⢘⡇⠀⡗⠀⢹⡛⣿⣟⠀⠀⠈⠁⠐⢋⠟⣻⢏⢀
⠀⠈⠀⠁⠀⠈⠁⠉⠉⠈⠀⠀⠹⣿⡏⡇⠉⡏⠈⠈⡇⠁⡏⠀⢸⢸⣿⠏⠁⣡⣬⠀⠀⠀⡈⠉
⠀⠀⠀⢰⣾⣷⡇⠀⢰⣧⢠⣷⠀⢻⣿⣧⣀⡇⠀⠀⡇⠀⡇⢀⣼⣿⡟⠀⠸⠻⠟⠃⢰⣷⣷⡇
⠀⠀⠀⠈⠉⠁⠀⣿⢈⣿⢸⣿⣆⠈⢿⡿⣿⣷⣶⣾⣷⣴⣿⣿⣿⣿⠁⠀⠀⢠⠀⡀⠁⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠘⣿⠛⣿⢻⠋⠻⠀⠈⢻⣮⣉⠛⠛⠛⠛⢉⣡⡾⠃⠀⠀⣶⣸⢻⣷
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠛⠀⠀⢀⢠⣄⡄⠉⠛⠿⠿⠿⠿⠟⠋⠀⡀⡀⠈⡏⢹⠻⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣾⡼⡇⠀⠀⠀⠀⠀⣀⠀⠀⡶⣿⠧⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠰⠿⠇⠀⠚⠛⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢀⣤⠶⠶⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⢦⣄⡀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠉⡀⠠⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠄⢈⠁⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠊⠀⡠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⢄⠀⠑⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡐⣴⣿⣶⣠⠀⠀⠀⠀⠀⠀⠀⠀⡄⣶⣿⣦⢁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⢃⣿⣿⣿⡏⠀⠌⠀⠀⠀⠀⠠⠀⢡⣿⣿⣿⡈⠀⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠈⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠁⠀⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡸⠁⠀⡤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠊⠈⢇⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠀⢸⠀⠀⡜⠀⠀⠠⢠⡄⠄⠀⠂⢣⠀⠀⡆⠀⢸⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢹⢧⡀⡆⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢰⢀⡼⠟⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢂⠙⣧⢄⣀⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀⡠⢼⠋⡀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠰⡃⠀⠀⢈⠏⠉⠁⠚⠓⠈⠉⠙⡅⠀⠀⢀⠗⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠈⠀⠄⣸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠀⠁⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠿⠿⠿⠿⣶⣦⣀⠀⢀⣠⣶⡾⠿⠿⠿⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢁⡴⠚⠛⠛⠛⠲⣝⠻⠃⠻⢋⡵⠞⠛⠛⠛⠲⣤⡙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⡀⠀⠀⠈⢷⣠⣠⠏⠀⠀⠀⡀⠀⠀⠈⢳⡀
⠀⠀⢀⣀⣤⠤⠶⠶⠒⠚⠛⣿⠀⠀⠀⣿⣿⡇⠀⠀⢸⡏⣿⠀⠀⠀⣿⣿⡇⠀⠀⢈⡟⠛⠒⠲⠶⠦⢤⣄⣀
⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠈⠉⠀⠀⠀⡼⠁⠸⣆⠀⠀⠈⠉⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦
⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⡤⠞⠁⠀⠀⠙⠶⣄⣀⣀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⢯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⢀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣦⣤⣀⣀⣀⣀⣹⣶⣄⣀⣀⣀⣀⣴⣾⣁⣀⣀⣀⣠⣤⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⢿⠿⠿⠿⡿⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡙⢏⢹⠉⢹⠀⢸⠀⠀⠀⡇⠀⡇⠉⡏⢙⠟⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠉⠒⠼⠤⣼⣀⣤⣀⣇⡠⠧⠔⠋⣁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣄⣀⣀⠀⢀⣀⣀⣤⠴⠞⠉

⠀⣠⣤⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣦⣄
⢰⠟⠁⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⠀⠀⠀⢽⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⠀⠀⠹⣿⣷
⣸⣀⣴⣶⣰⣶⣄⣼⡀⠀⢀⣤⣤⣦⣤⣄⡀⠀⢸⣇⣀⣀⡀⣰⣿⣿⣿⠀⠀⣠⣤⣶⣶⣦⣤⡀⠀⢾⣀⣀⣀⢀⣠⣧⣿
⢻⠀⠀⠈⠀⠀⠀⢸⠁⣰⠟⠁⠀⠀⠉⣻⣿⡀⠨⡏⠈⠉⠀⣿⠈⢹⣿⠃⢰⠋⠁⠀⠀⠸⣿⣷⠀⠐⠀⠈⠉⢸⡉⠹⢿
⠐⡄⢀⣙⣟⡀⠀⡇⠀⣿⣀⣤⡄⢀⣦⣾⣿⡇⠀⡟⡖⠀⠀⣾⣿⣿⡟⠀⣏⣀⣀⠀⢀⣰⣿⣿⠀⠀⠀⢀⣀⣹⣿⣆⡟
⢀⣿⣙⠻⠯⠋⣼⣧⠀⣿⠀⠙⠁⢸⠉⠛⣿⡇⠀⢸⠀⠉⠉⠽⣿⣿⣇⠀⢻⠉⠉⠉⢹⡉⠛⣿⡇⠀⣴⠀⠉⠀⢻⣿⣿
⠛⠻⢻⣿⣿⡿⢻⡟⡇⢨⠀⠀⠀⢾⣷⣸⣿⠇⡖⢻⠀⠠⢤⣴⠟⡿⡿⠂⢿⣱⣁⠀⢾⣿⡞⣿⠁⢰⣿⠀⠐⢾⣿⣿⣿
⠀⠀⢿⠛⠛⠀⠘⠀⠀⢈⣄⠐⠚⠹⣿⣿⣏⠀⠀⠈⠀⠀⠘⠃⢀⠀⠀⠀⡠⣧⡈⠙⠿⣿⣼⣧⡀⠚⣋⣄⠀⠈⢸⣯
⠀⠀⠈⠁⠀⠀⠀⠠⠤⢻⠆⠳⣤⣴⣿⣿⡿⢧⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⠙⢶⣶⣿⣿⣿⡟⠀⠛⠛⠂⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠸⢿⣿⢿⣷⣿⠧⠄⠁⠀⠀⠀⠀⠀⠀⠀⠄⠹⡆⠈⢙⣟⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠒⠒⠒⠒⠒⠶⢤⣄⡀⠀⠀⠀⠈⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⢸⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣁⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣣⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢣⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⣀⣤⣴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⡲⠞⣁⡴⠖⠛⠉⠉⠉⠉⠉⠛⠓⠦⣄⡀⠀⠀⠉⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⢀⣠⠖⠉⠀⠀⢀⣾⡵⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠙⡶⠒⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡴⣻⠁⣴⠷⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠯⢿⡄⠀⢀⡞⠁⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡰⠋⠀⡇⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢸⠃⠀⠀⠀⢀⡿⢄⡀⠀⠀⠀⠀⢀⡠⣄⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠶⠂⢀⡀⠙⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠙⣦⠀⠀⠀⣾⠁⠈⢧⠀⠷
⠀⠀⠀⠀⠀⠀⠀⢠⠖⣉⠭⢍⡀⠀⠀⢿⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠊⠉⠓⠦⡀⢸⠀⠀⠀⠀⠈⠳⣄⠀⣼⠋⠉⢢⣏⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠙⢦⠀⢸⠀⡴⠋⠀⠀⠙⣆⢀⣤⢤⣄⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⡇⠘⣇⠀⠀⣠⣀⣤⣈⡄⣏⠀⢀⡏⠁⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⣷⡀⠀⠸⣇⠤⣇⢧⣠⣤⣤⣄⠀⢾⠀⠀⠘⠃⠀⠠⠴⠖⠛⠛⠛⠿⢿⡷⠀⠙⢦⡜⢡⡏⠀⠈⠁⢉⡒⠋⠀⠀⢀⡜
⠀⢰⡆⢀⡴⠞⠓⢶⠋⠀⢀⠙⠀⠀⣇⠀⢸⠛⠋⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⢸⠁⠈⢳⡖⠓⡆⠘⠃⠀⢠⡴⠋
⡄⠀⠀⢸⠀⠀⢀⣼⡀⠀⢹⡄⢠⠀⠉⠀⡇⠀⠀⢀⠔⠚⠉⠉⠉⣱⠶⠤⢄⣀⡀⢀⣀⡡⠏⠀⠉⠁⠀⠀⠀⣤⠤⠚⢀⡜⠀⠀⠀⠀⢸⠁
⢱⡀⠲⡀⠳⠖⠁⠀⠙⣶⠚⠀⠘⢦⠀⠀⣇⠀⠀⠻⡀⠀⠀⠰⠋⠀⠀⠀⠀⢀⡽⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⣠⠎⠀⠀⠀⠀⠀⡸
⠀⠹⢄⠙⠆⠀⠀⠀⠀⠈⠛⠒⡞⠀⠀⠀⠘⠦⣀⠀⠙⠲⠤⣀⣀⣀⠤⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⣀⠜⠃⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠈⠉⠒⠶⣤⣤⣤⣤⣤⣤⣤⣤⡤⠀⠀⣰⠖⠒⠙⠋⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠒⠀⠚⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⢀⣇⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣠⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⣐⢒⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣌⠙⠲⠦⢤⣀⣠⠤⠤⠖⠛⠉⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⠲⣶⠶⠄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⢤⣀⡀⠀⠀⠀⣘⢻⡀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⡏⠉⠛⠶⣄⡟⠘⡇⣠⡴⠶⠖⠲⣞⣸⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠳⣄⠀⠀⡟⠃⠀⠛⠒⣷⠀⠀⢰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⣤⣤⣬⠦⢤⣄⢸⡇⣸⡳⡄⠀⠀⠀⣷⠶⠚⠛⢛⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠉⣭⡟⠁⠀⣎⠩⠷⠤⣒⡶⠾⠷⠶⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⠢⣤⡀⡀⠀⠀⠀⠀⣨⠇⣀⣠⠔⣫⠃⠀⠀⢸⣇⢠⡞⠁⠀⠀⠀⠀⠀⠀⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢌⣉⣑⣒⣒⣚⣉⣉⠭⠔⠊⠁⠀⠀⠀⠈⢯⡛⠧⣄⣀⠀⠀⠀⢀⣠⠼⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⠉⠉⠉⠁⠀⣤⠖⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣀⣀⣀⣀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢹⠀⠀⠀⠀⢠⣞⣙⡷⠤⠤⠶⠾⠾⠶⠦⠤⣤⣤⣒⠙⢦⠀⠀⠀⢀⡔⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡼⣧⣠⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠛⠿⣄⡀⢀⡜⣀⣬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠀⠀⡴⠚⢶⠀⠀⢀⣠⣀⠀⠀⠀⠠⠊⠉⠇⠀⠀⠀⠈⠙⢿⣎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⠞⠁⠀⠀⠀⠀⠉⠀⠀⠀⠸⣇⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡠⠒⢳⡄⠀⡹⠃⠙⣦
⠀⠀⣀⡀⠀⠀⠀⠐⣧⠚⠁⣠⠴⢤⡀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠞⠀⠚⠁⣠⠖⢷⡒⠋⢻⠆
⠀⣼⠁⢹⣀⠀⠀⡜⠁⠀⠸⣥⣀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡴⠋
⠀⠘⠷⠶⢽⣷⡞⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢻⠀⠀⠀⠀⠀⠀⠀⢀⡤⢦⡌⢷⡀⠀⠀⣠⠶⠶⣄
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⡸⠀⠀⠀⠀⠀⠀⠒⢉⡀⢀⡿⠘⣧⢴⣊⣥⣤⣄⡏
⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣿⣷⡀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⢹⡏⠁⠀⠀⠈
⠀⠀⠀⠠⣯⡇⠀⠰⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⣿⣿⠇⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⠋⢹⠆
⢀⣀⣀⣀⣙⠁⠀⠀⢳⡈⠉⠛⢶⣶⣦⠤⣤⠄⠀⠈⠈⠙⠒⠦⠤⠤⠤⠤⠖⢉⡤⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⣦⠀⠀⠉⣿⢉⣀
⣎⡀⢈⠥⢼⡆⠀⠀⠀⠳⣄⡀⢈⣛⠯⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠋⠀⠀⠐⢿⣀⢹⠆
⠈⠉⠀⢀⠔⡇⠀⠀⠘⠦⣌⣉⡡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠘⠦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡤⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠸⡆⢠⡏⠀⠀⠀⠀⠀⠀⠀⠰⣏⠈⢧⣀⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠜⠀⠀⢠⡀⢹⡶⢤⡀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⡔⠹⡌⠙⠛⠀⠀⢠⡔⠹⣆⢀⡹⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢃⡟⠀⠀⢧⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠳⠤⠿⣆⠀⠀⠀⠀⢹⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⣦⠀⠈⠳⢤⡀⠸⡄
⠀⠀⠀⠀⠀⠀⣦⠰⠽⢦⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣠⠛⣧⠰⠋⠀⠀⠀⢀⡟⠀⠈⠑⢆
⠀⠀⠀⠀⠀⠀⢹⠀⣠⠜⣻⣦⡀⠀⠀⠀⠀⡟⠀⣷⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠦⠴⠃⠀⠀⣠⡞⢥⡀⠸⡆⠀⠀⠀⠀⢸⠀⣤⡀⢰⡈⡇
⠀⠀⠀⠀⠀⣴⠊⠉⢀⡞⠁⠀⢙⡶⣄⡀⠀⠻⠖⠋⠀⠀⠀⠀⢸⣄⣸⠄⠀⠀⠀⠀⠀⠀⣀⣠⠴⠊⠁⢹⡄⠙⡆⢿⠀⠀⠀⠀⢸⡼⠁⠹⣄⡿⡗
⠀⠀⠀⠀⠀⠘⠳⠶⢞⠁⠀⠀⠸⡇⠀⠉⢳⠶⣦⣤⣤⣠⣀⣀⣀⣉⣡⣤⣤⣤⠴⠶⡖⡟⠧⠙⢦⡀⠀⠈⠛⠊⠁⢘⡀⠀⠀⠀⠈⠁⠀⠀⣿⠀⠁
⠀⠀⠀⠀⠀⠀⡞⢀⡎⠀⠀⠀⠀⣷⠀⡴⠋⣼⠁⠀⣰⠁⠜⠁⠹⢡⠀⠘⠒⠂⠀⠀⡇⠙⢶⡄⠀⠙⢦⡀⠀⠀⠀⠘⢧⣄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠇⠈⢶⠲⡄⠀⠀⢻⡘⠷⠤⠟⠀⣰⠃⠀⠀⠀⣇⣸⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁⠀⠀⠀⠙⠳⡶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⠀⣆⢸⣇⡸⠀⠀⠘⣧⠀⠀⢀⡴⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢹⡄
⠀⠀⠀⠀⣄⠹⠦⣬⣷⡼⠟⠀⠀⠀⣽⣂⠀⠈⠳⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣇⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠳⢦⡀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢉⡵⠞⠛⠶⠲⠦⠟⣠⠼⠦⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠇⠀⠀⠤⠍⡁⠶⠂⠴⣤⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⢼⡟⣥⠶⣄⣀⡴⠒⠦⣤⡤⠛⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠭⠔⠒⢦⣀⣠⠔⠒⢠⡀⠀⢀⡨⣷⣄⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠿⠊⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⢧⣽⠙
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶
⣿⣿⣇⣀⣀⣀⣄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠉⠙⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢠⠞⠛⠁⠀⠐⠳⣦⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢉⡤⣦⣄⢠⢤⣤⡈⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢧⠀⠀⢸⡀⠿⣿⣇⠘⢛⡇⠀⠀⠀⣿⣠⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢰⡀⠀⠀⠈⠙⠛⠁⠈⢛⣩⡤⠶⠞⠛⠉⠉⠁⠀⠀⠀⠙⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠖⠀⠀⢴⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡏⠀⠀⠀⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣷⠀⠀⠀⠀⠀⡄⠈⠙⠛⠶⠶⣤⣤⣶⠶⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠘⢧⡀⠀⢀⡼⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢻⠉⠳⢤⣀⣠⣉⣉⡉⠀⠀⢀⣀⣤⡴⠟⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡴⢻⠀⠀⠀⠈⠙⠛⠛⠓⠛⠛⠛⠉⠁⠀⠀⣿⠻⣄
⠀⠀⣀⣀⠀⠀⠀⠀⢀⡴⠋⣡⠶⠋⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⣌⠳⣄
⢀⣀⣑⣬⡑⠖⠋⠉⠉⢠⠞⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠙⢧⡌⠳⣤⡀⠀⠀⠀⢀⡀
⡜⠒⠒⠒⠂⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠙⠣⣌⠹⠶⠦⠴⠡⠗⡲
⠈⣏⣉⡩⠤⠒⠉⠉⡇⠸⣄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⠠⠞⣀⡀⠀⠀⣐⣋⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠼⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠳⠤⠞⠁⠉⠲⣄⣉⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠳⣄⠀⠀⠀⠀⠉⠙⣿⢲⡶⠶⣤⣤⣤⣤⣶⡶⠶⠟⠋⠁⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠹⣦⣤⣀⣀⣀⣀⣹⠀⡇⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣠⡞⠉⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠉⠉⠉⠉⠉⠃⢷⡄⠀⠀⢠⠞⠐⠛⠛⠛⠛⠉⠋⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠶⠶⢦⣤⣤⣤⣴⠇⠀⠀⠘⠦⣤⣤⠤⠴⠶⠶⠶⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⢒⠒⠒⡂⠠⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⠔⢠⠄⠦⠘⠷⠀⠀⠀⠓⠉⣒⣄
⠀⠀⠀⠀⠀⠀⡠⠨⢄⠀⠀⠉⠀⠀⠀⠐⠀⠒⢣⡀⠠⠶⢠⣑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡰⠵⠄⠀⠹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢫⣅⠉⠹⡀⠀⠠⢀⡔⠉⠀⠂⡤⠐⡄⠂⢐⡠
⠀⠀⠀⠀⢀⢁⡌⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢬⣥⠄⠁⣘⡀⠀⠦⢀⣈⠁⠈⢁⠸⠠⠁⠀⠠⡀
⠀⠀⠀⠀⢸⢋⡀⡜⠁⠀⠀⠀⠈⠂⠀⠀⠀⢠⣤⣤⣤⣤⣀⠀⢻⠚⢗⡤⠐⠊⠀⠀⠀⠀⠀⠀⠐⠠⡤⠀⠑⣌⠄
⠀⠀⠀⠀⠀⠪⣥⠃⠈⢀⣠⣤⣴⢞⢠⠖⠒⣀⠉⣋⠭⠭⢽⠉⠘⡧⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠢⠌⠁
⠀⠀⠀⠀⠀⠀⠸⡄⠴⠫⣋⠭⠽⢤⡙⡄⠀⠻⠘⣤⣔⣒⡿⠀⠘⡇⠈⠀⣀⣀⡠⢂⠄⠀⠀⠀⠀⣄⠀⠈⢯⠋⠳⡆
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠘⠥⣄⣒⠾⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢿⣚⠛⣛⡿⠛⠡⠄⣀⠰⢦⣀⡀⠐⢇⢾⣑⢂⠃
⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⢤⣶⡄⢰⣼⢇⠤⠄⣹⠂⡇⠀⣰⢄⣘⣛⡻⢶⣄⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⡀⢀⣄⡜⠁⠶⣦⣤⡼⠋⠀⠀⡘⣇⣾⡗⠀⠉⠉⠁⠀⠀⠀⡄⠩⣦⠄⠈⢢⠉⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠱⡌⢿⠠⠀⠀⠀⠐⠛⠀⠀⠀⢄⣼⣿⣀⡀⡠⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠚⢤⠢⡀⠀⠀⠀⠀⡠⠊⠈⢁⣷⠘⠀⠁⠈⠻⣤⣀⣴⠌⠳⢤⡀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡈⠁⠀⠩⣉⣡⡤⠔⠁⠀⢠⡟⠀⠔⠀⠒⠀⠼⠧⢀⡀⠀⣌⡏⣠⡧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⢀⣀⢀⢀⠜⡃⠀⢂⠠⠄⢀⡀⠀⠀⠈⣂⣸⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠴⠶⠾⠿⠛⠁⠀⢱⠀⠀⠀⠲⢶⠲⠭⠒⠜⣻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⡀⠀⣀⠀⠀⠀⢰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠷⠖⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠿⠿⠿⠿⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⡿⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⢹⡏⠀⢀⡀⠀⠀⠀⡀⠀⣁⡆⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠈⣷⣶⡟⠁⠀⠀⠀⢹⣿⣯⡇⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠘⠋⠈⠁⠀⠀⠀⠀⠁⠀⠉⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠴⢾⣄⣀⠀⠀⠀⢀⣠⣾⢷⡄⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠙⠛⠻⠿⠿⠛⠋⠁⠸⢀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠙⠻⣿⣿⣿⣿⣾⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⡏⣿⠃⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⣰⠶⠀⠀⠀⠀⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⠀⠀⠀⠀⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠂⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠈⠀⠁⠢⡠⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡠⠐⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣠⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠁⢰⠀⠀⢰⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⣬⠈⠁⠀⠀⡆
⠀⢀⣤⠀⠠⢀⠀⢀⠠⠓⠄⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠔⠃⠄⠀⠤⠂⠀⠀⠀⠀⠀⢴⠀⠁⠀⠀⠈⠙⠂⠀⠀⠐
⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠀⠆⣄⡀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⣈⠓⠂⠄⠀⠠⠐⠒⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠒⠈
⠀⠀⠀⠀⠈⠢⡒⠁⠒⠀⠒⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⠉⠈⠁⠁⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠰⠛⠉⠀⠀⠀⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠁⠀⠀⠀⠚⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠃⠀⠀⠀⠀⠻⢦
⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠈⠢⡤⠂⠀⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠐⢄⡀⠀⠂
⠀⠀⠚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠖⢸⠂⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢰⣿⣿⣿⡆⢷⣿⣿⠆⠀⠀⢀
⠀⠀⠀⠐⡄⡀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⢀⠠⠓⠄⣀⡠⠔⠀⠀⠀⠀⠀⢻⠀⠀⠀⠙⠛⢋⠔⠢⢉⡁⠔⠃⠀⠈
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⢈⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠤⣠⠔⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⡀⠐⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠆⠀⠀⠀⢠⣦⡀
⠀⠀⠀⠴⠶⢶⣤⣄⣀⢀⡀⣀⣤⣴⠶⠿⠒⠀⠀⠀⠀⢀⠤⠖⠋⠉⠐⡔⠉⠓⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⢀⠠⠐⠀⠁
⠀⠀⠀⠘⡀⠀⠉⠛⠉⡛⢋⠋⠋⠀⠀⡸⠀⠀⠀⠀⠀⠘⡀⠐⠘⠋⠀⡅⠉⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣤⡄⠈⣤⡄⠀⠀⢡
⠀⠀⠀⠀⠈⠐⠀⠀⠊⠀⠀⠁⠂⠐⠈⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠀⠈⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠰⢄⡀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⣀⠀⢐⠂⠀⢐⠒⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⣠⠮⠧⢤⠘⠂⠠⠞⠒⢲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢏⡢⠤⠘⠒⠒⠘⠓⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠆
⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⣼⣿⣿⣿⢀⣴⣶⣦⡀⠀⣠⣄⠀⠀⣠⣄⠀⢀⣴⣶⣦⡀⣿⣿⣿⣧⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⣿⣿⣿⣿⠸⣿⣿⣿⣧⣼⣿⣿⣇⢸⣿⣿⣧⣸⣿⣿⣿⠇⣿⣿⣿⣿⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡀⠘⣿⣿⣿⠀⢻⣿⣿⣿⢻⣿⣿⡟⢻⣿⣿⡟⣿⣿⣿⡟⠀⣿⣿⣿⠃⢀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⢿⣿⣇⠀⠙⢿⡇⠀⠙⠻⠇⠸⠟⠋⠀⢸⡿⠋⠀⣸⣿⡿⠃⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣶⣄⣠⡄⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⡀⢀⣤⣀⣴⡆⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢸⣿⣿⣿⡇⢰⣿⣿⠀⢀⣴⣿⡄⣾⣶⡄⠀⢘⣛⡗⠘⣿⣿⣿⣿⠘⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡿⠇⢺⣿⣿⣆⢸⣿⣿⣧⣿⣿⣿⢀⣿⣿⣿⠘⢿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠈⠛⠛⠁⠙⠛⠋⠈⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠘⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢳⠀⠀⠀⠀⠀⠀⡈⠒⠀⠀⠀⠀⠀⠀⠀⣿⠐⠂⠀⠀⠀⠀⠀⠀⠐⠏⠆
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠆⠐⠐⠀⠐⠠⠀⠀⠁⠀⠁⠁⠀⠀⠡⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⢀⡀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠈⠁⠀⠆⠀⠄
⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⣀⠀⠀⠀⠀⠀⣀⡔⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠉⣹⠆⠀⠀⠀⢀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⠀⠀⠀⠀⠀⠀⠠⠘⠆⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⠉
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡃⠀⠀⠀⠀⠀⢄⠘⠀⠀⠀⠀⠀⠀⢢⠄⠤⠂⠲⠀⠀⠀⠀⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣠⡀⠨⠃
⠀⠀⠓⠀⠀⠀⠋⠀⠀⠀⠀⠃⡀⠀⣀⡀⠀⠀⠀⢠⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⢀⡀⡀⠅⠀⠀⠀⠀⠁⠀⠀⠝⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠲⡀⠀⠀⠀⠀⠀⠀⡒⢲⡄⠀⠀⡀⣠⡔⠦⡄⣀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠠⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠀⡀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⢀⠀⠠⢤⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠑⠀⣰⠀⠀⠀⠀⡀⢦⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠒⢀⠀⡦
⠀⠀⠀⠙⠀⠈⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⢠⠄⠀⠀⠀⠀⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⡀
⠀⠀⠀⡴⠢⠰⠀⠀⠀⠀⠀⠀⠐⠚⠲⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠂⠀⠀⠬⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠆⠀⠀⠀⠒⠂⠐⠒
⠀⢀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢤⣤⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠈⠠⠀⠄⠀⠌⠁⠀⠀⠐⠀⠐⠀⠀⠂⠀⠚⠁⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⠀⠑⠖⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠢⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⢀⣴⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡄⠀⠀⠀⠀⠀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣶⡀⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⠀⢰⣦⡀⠀⠀⠀⠀⢺⣿⣷⠀⠀⠀⠀⣴⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠻⠿⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⡇⠀⢸⣿⣷⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⣿⣿⣿⡇⠀⠈⣿⣿⠀⠀⠀⠀⠀⢻⣿⣿⠀⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⠃⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡄⠀⠸⣿⣿⣇⠀⣿⣿⣿⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⣿⣿⣿⠀⣿⣿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⢻⣿⣿⡀⢹⣿⣿⡇⢹⣿⣿⡇⠀⠀⢿⣿⣄⣀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀⣸⣿⣿⡇⠀⣿⣿⣿⡈⣿⣿⣿⡀⠀⠈⠛⠛⠻⠿⣷⡄⢰⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠃⠀⠈⠻⢿⠇⠘⣿⣿⣷⣶⣤⣤⣤⣀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡖⢶⡄⠀⢠⣶⣶⣤⡀⠀⠀⢈⣙⠛⠛⠿⠿⢿⣿⣿⣆⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀⠈⣿⠀⢸⣿⡇⠈⢿⡆⢸⣿⡟⠻⣦⠀⠀⣠⣄⠈⠙⠃⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⣀⣀⣀⢸⣿⡇⠀⠸⣷⢸⣿⣧⠀⢸⣧⢰⣿⡟⢧⠀⢀⡀⠀⠻⠿⠃
⠀⠀⠀⠂⠐⠢⣄⠠⠀⠀⠀⠈⢿⣿⡄⠹⣯⠘⣿⣇⠀⠀⣿⡈⣿⣿⠀⠀⣿⡄⢿⣧⡀⠀⢸⣿⡶⣤⣄⡀
⠒⠤⡤⠄⠈⠀⢹⠿⢷⡆⠀⠀⠈⠻⣷⣼⡿⠀⠘⣿⣄⢀⣿⠇⢹⣿⠀⠀⣻⡇⠀⠻⣷⡀⠘⣿⣇⡀⠈
⠀⠀⠀⠀⠀⠀⢀⣀⡈⠛⠒⠀⠤⠄⣀⣀⣀⣀⠀⠈⠛⠛⠋⠀⠀⠻⣧⣀⣿⡇⢀⡀⠹⣷⡀⣿⣿⠉⠃
⠀⠀⠀⠀⠀⠀⠈⠱⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠓⠲⠦⣤⣀⡀⠈⠉⠉⠀⠈⠳⣴⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠑⠂⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠒⠖
⠀⠀⠀⠶⠄⡀⠘⢆⠀⠀⠀⠀⠙⠛⠄⡀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠁⠀⡀⠒⠤⣤⣀⣀⡀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⡀⠀⠀⠑⠀⠀⠀⠙⣄⠀⠀⠈⠉⠉⠉⠒⠶⠄⣀⣀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⠀⢀⠀⠀⠀⠢⠀⠈⠢⢄⠱⠄⣄⠀⠀⠀⠀⣴⣼⣯⠭⡏⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠒⠤⠀⠀⠈⠀⠀⠈⠠⡀⠀⠀⠙⠂⠀⠈⠀⠔⡁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠎⡠⠀⠉⠓⢤⡀⠀⠠⣀⠀⠀⠀⠀⠀⠀⢀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠤⠴⠀⢀⣤⠓⣡⠀⠀⢀⡀⡤⠚⠀⠀⠀⠀⠑⣶⡄⠀⠀⠀⣸⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠿⠛⠣⠤⠀⠠⠤⠤⠄⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠩⠭⠉⠉⠁

⠀⠀⡀⠀⣀⠀⠀⢠⡀⢀⠀⠀⡀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⣀⠀⢀⡀⠀⠀⢀⠀⣀⠀⡀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⢌⠄⠠⠀⣭⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⡀⠀⠰⠶⢀
⠀⠀⠀⠀⠈⠚⡟⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡉⢦⡀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠘⠢⠬⠔⠁⠘⣆⠐⠁
⠀⠀⠀⠀⠀⠀⢣⡀⠀⢈⠆⠀⠀⠀⠀⠀⠀⠀⣿⣦⣌⣑⣒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡆
⠀⠀⠀⠀⠀⠠⠤⠍⠩⠅⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡤⠈⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⣠⠜⡁⢹⣙⢒⠀⠀⠀⠀⠀⠸⠖⠶⠶⣶⠒⢲⡶⣶⣶⡶⠂
⠀⠀⠀⠀⣈⠀⠀⠤⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠃⠈⡏⠀⠀⠀⠀⠀⠀⠀⠈⠁⠊⣹⠀⢸⡏⠒⠋
⠀⠀⠀⠀⣿⣷⣄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠑⠬⠆⠀⠀⠀⠀⠀⠀⠤⠀⢿⠀⢸⡇⣀
⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠜⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀
⠀⠀⠀⠀⡀⣀⠀⢀⠈⠉⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠴⢶⡒⠒⡂⠀⠀⠀⠀⠈⠀⠀⠈⠄⠀⠀⠀⠅
⠀⠀⠀⠀⠙⠋⠀⠚⠛⠀⠀⠀⠀⠀⠀⠀⠐⢶⣒⠆⠉⠈⢳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢠⡀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠦⣄⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠣⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡃⠘⢦⡀⠀⣀⠾⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠇⠐⠀⠈⠉⠀⡄⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠈⠄⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣏⡀⠐⠂⠀⠀⠀⣀⠠⠀⠀⠀⠳⠤⠤⢶⡞⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⣠⠋⢫⠀⠀⢀⣀⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠀⠀⠈⢲⡀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣰⠏⢃⠀⢀⠜⠁⠀⠀⠉⠉⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠡⠀⠀⠀⠀⢀⣃⣀⣀⠀⠀⠀⡰⠁⠀⠀⠑⠁⠀⢀⠁⠀⠈⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⡋⠀⠀⠀⠀⢀⣠⠤⢄⠀⡞⠁⠁⠀⠙⣆⡜⣁⣄⠀⠀⢀⠀⠂⠀⠁⠐⠠⢈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠦⢤⣸⠀⠀⠀⡽⣇⠀⠀⠀⠀⡼⡏⠀⠀⢱⠊⠀⠀⠀⡠⠖⠊⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠙⠒⠊⠀⠈⠓⠤⠤⠚⠁⠑⠤⠤⠊⠑⢤⣀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢀⡤⠤⣄⠀⠀⠀⠀⠀⣀⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣀⡠⠖⠁⠀⠀⠈⠳⣄⢀⡠⠞⠁⠀⠀⠈⢦⠀⠀⡠⠤⢄⡀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠛⠉⠉⠁⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⠋⠁⠀⠀⠉⠒⠶⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠸⠿⠀⠀⠀⠀⠀⠈⠀⠀⠀⣴⡄⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⢶⠆⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠾⠂⠀⠀⠀⠀⠶⠄⠀⠀⠰⠆⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢀⡀⣀⡇⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠸⡁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠁⠙⠓⡲
⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠚⠛⠋⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣽⣵⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣶⣶⣾⣿⣿⣷⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣰⠷⣄⠀⣠⠶⣄⠀⠀⣀
⠀⢀⡤⣾⢻⡏⣷⣀⠀⠀⢀⣀⠀⢸⠉⠳⢤⡏⠀⠈⠛⠁⠀⠹⠞⠋⢹⠀⠀⠀⠀⠀⠀⠀⣀⣴⠳⣄⣤⡀
⠀⠸⣄⡈⠀⠀⣉⣹⠆⠀⢸⡍⠙⠛⠀⣀⣀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠘⠛⢹⠇⠀⠀⠀⢸⣍⠛⠀⠋⠼⠷⡄
⠀⠀⠈⢧⣀⠀⠻⣅⠀⢶⠤⠇⠀⢠⢎⠏⣯⢱⠀⠀⠀⠀⡏⣼⢹⢹⠀⠀⠛⢲⠆⠀⠀⣀⡾⠁⢀⣠⠤⠞⠁
⠀⠀⠀⠀⠉⢧⡀⠀⠙⠾⠇⠀⠀⠘⢾⣦⡯⠞⠀⠀⠀⠀⠑⠿⠾⠋⠀⠀⠴⠯⠶⠶⠛⠁⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⡀⠀⠀⠀⢠⠴⣞⡿⠉⠉⠉⠹⠽⠋⠳⡀⠀⠀⠀⣀⣀⣤⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⡇⠀⠀⣀⣀⡀⣀⣤⣀⠀⠀⡇⠀⠀⢙⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⠀⠀⠀⠳⣄⣞⡁⠀⠉⠁⢀⣈⡦⠞⠁⠀⢘⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠶⢶⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⣀⡀⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⡇⠀⢤⣀⡤⣄⣀⡴⢶⠀⠀⣿⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠉⠀⠈⠉⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡇⠀⢸⠀⠀⠀⠀⠀⢸⠀⢀⠟⠛⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣤⠤⠭⠥⠤⠾⠀⠀⠀⠀⠀⠸⠦⠤⠤⠤⠤⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠁⠀⠀⠀⠉⠙⠳⢤⡀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠒⠒⠶⢤⣌⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⡴⢋⣁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣴⠏⣼⢃⣽⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⠳⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⣹⣿⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣿⣿⡿⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⣸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⣠⣴⡾⠿⠗⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⡼⠋⢹⣧⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠃⠀⠀⠀⠀⠙⠳⠦⢤⣤⣤⠤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠁⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠶⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⠙⠻⠷⢶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⠶⣾⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⣀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⠉⠉⠙⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⢀⣀⣤⣄⣀⣀⣀⣀⡀⠀⢀⣠⠴⠛⠛⠒⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣠⡶⠛⠉⠀⠀⠀⠀⠉⠉⠙⠛⠯⣀⠀⠀⠀⠀⠀⠈⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣄⡀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢦⣄⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⣀⣀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⣰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆
⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⣠⣶⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣶⣄⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⣸⣿⠁⠀⠀⠀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⠀⠀⠀⠈⣿⣇
⠀⠀⠀⣿⣿⠀⠀⠀⠘⠃⠀⣀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⡀⠘⠃⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠚⠛⠻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠻⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡟⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁

⠀⠀⠀⢀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⢸⣿⣇⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠛⠛⠛⢷⣄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡀
⠀⠀⠀⠘⣿⣿⣦⡀⠀⠀⠀⠀⣠⣤⡀⣰⣿⡟⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⠞⠲⣄⠀⠀⠀⢴⠀⢀⣴⣿⣿⡿⠟⠋⠉⠉⠉
⠀⠀⠀⠀⠈⠻⣿⣿⣶⣄⡀⠀⠻⠿⠃⣿⣿⠀⢠⠀⠐⢄⣀⡤⠏⠀⣿⣇⠀⠀⠋⠀⠀⠀⠀⢀⣿⣿⣿⠏
⠀⠀⠀⠀⠐⠂⠈⣻⣿⣿⣿⣷⣶⣶⣾⣿⣿⣦⣤⣄⡀⢀⣴⡄⠀⠀⢹⣿⣦⣤⣴⣶⣶⣶⣶⣾⣿⣿⣏
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡉⠁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠸⠷⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣟⡻⢿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣶⣦⣽⣿⣿⣿⣋⣭⣭⣿⣿⣿⡏⢸⣿⣿⣿⢟⣵⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣤⣶⣿⠿⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⣿⣿⣥⣤⣬⣿⣿⣿⣿⣿⡿⢛⠛⠻⣿⡿⠀⠈⣿⣿⣿⠟⢛⣻⣿⣿⣿⣿⣿⣭⣤⣬⣽⣿⡿⠋⠀⠀⠀⠀⠀⢹⣧
⠀⣰⡟⠉⢳⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠃⢸⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢠⡟
⠀⢿⣧⡀⠀⣾⣿⣿⣿⣿⣿⣟⡿⣿⠿⣟⣿⣿⣿⣿⣿⣿⣤⣴⣾⣿⣿⣿⣿⣿⡿⠿⢿⡿⣛⣿⣿⣿⣿⣿⣿⡇⠀⠀⠑⠲⠖⠋
⠀⠀⠛⠿⢿⢿⣿⣿⣿⡿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡉⠀⠸⣿⣿⣿⣿⣿⣿⣿⡷⠾⠿⢿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠉⠙⠛⠛⠛⠉⢩⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠈⢻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣆⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⣀⠉⢉⡛⠿⣿⣿⣿⣿⡏⠀⠀⢀⣀⣀⣀⠀⢻⡿⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣤⣤⣤⣴⣿⣿⡟⠉⠛⢿⣷⡈⠁⣠⡤⠿⣿⣿⣥⣴⣾⣿⣿⣿⣿⣿⣿⠁⠀⠻⣿⠟
⠀⠀⠀⠀⠀⠀⠰⠆⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢹⣷⠀⣿⡀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠻⢿⣿⣿⣿⠟⠁⠀⡅⠀⠀⠀⢸⡟⠀⠻⣿⣶⣾⣿⣿⡿⢿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣠⣀⠀⠀⠀⠀⠙⠦⠤⠶⠛⠀⠀⠀⠀⢹⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠘⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛


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

⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐
⠀⠀⠀⠘⠀⢩⠁⠀⠀⢩⠀⠘⠀⠀⠀⠀⠚⠀⠭⠀⠀⠀⠭⠀⠞⠀⠀⠀⠀⠳⠀⣬⠀⠀⠀⣥⠀⠓⠀⠀⠀⠀⠃⠈⡍⠀⠀⠈⡍⠀⠃
⠀⠀⠀⠀⢄⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠥
⠀⠀⠀⠀⠀⠀⠒⠐⠐⠂⠂⠀⠀⠀⠀⠀⠀⠀⠂⠰⠐⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠒⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠂⠐⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⠀⣀⡀⠁⠀⣀⠀⠀⠀⠀⠀⠀⢀⠈⣀⠐⠀⠀⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡈⠀⠰⠀⠀⠀⠰⠀⠈⠀⠀⠀⠀⠙⠀⠴⠀⠀⠀⠴⠀⠋⠀⠀⠀⠀⠙⠀⠥⠀⡀⠀⠭⠀⠋⠀⠀⠀⠀⠁⠀⠦⠀⠀⠀⠆⠀⠁
⠀⠀⠀⠀⠂⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠒⠀⠒⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃⠒⠐⠂⠃⠀⠀⠀⠀⠀⠀⠂⠐⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠂⠂⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⢠⠀⠠⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⢠⠀⠠⠀⠀⠀⠄⠈⣠⠀⠀⠀⠀⣄⠀⣀⡀⠀⢀⣀⠀⡄⠀⠀⠀⠀⡄⠀⠤⠀⠀⠠⠤⠀⡄
⠀⠀⠀⢀⠀⠘⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⢀⠀⠒⠀⠀⠀⠒⠀⡁⠀⠀⠀⠀⠈⠀⠛⠀⠀⠈⠛⠀⡀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⠃⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡄⠄⠀⠠⠄⡤⠴⠶⠤⠄⠤⠠⠄⢤⡀⠀⠀⠀⢀⠄⠀⠄⣼⠩⠄⠄⠄⠦⡤⠤⠤⠄⢄⠀⠀⠀⠀⢠⠄⠄⠤⣤⠆⠀⠤⠀⠰⡄⠤⠀⠄
⡇⠀⠀⠀⡜⠋⠓⢔⠶⠲⡶⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢐⡒⠄⠀⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⢀⡀⡀
⠀⠀⠀⠀⢧⣀⣠⠚⣬⣣⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⢲⡃⢄⠑⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠉⠔⠀⠮⠥⣇⡀
⡇⠀⠀⢠⠞⠁⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⣞⠁⢳⡀⠁⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⠀⠀⠀⠀⠀⢀⡽
⡇⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣼⡌⢳⡄⠱⣄⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⢠⠌⠙⠒⠒⠒⠒⠚⠉⢤⠄
⡇⠀⠀⠀⠙⠶⠤⠤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣿⡟⠀⠀⠈⠳⠤⣀⣀⡷⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⢀⡼
⡇⠀⠀⠀⠉⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠤⠤⠤⠐⠋
⠇⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠄⠀⠀⣀
⡆⠀⠀⠰⠞⠋⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢐⠂⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠐⢏⠀⢀⠀⠀⠉⠛
⡇⠀⠀⠀⠸⠇⡀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢐⠩⡆⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢶⢊⣃⢸⣲⢎⣣
⡇⠀⠀⠀⠈⠈⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠠⠀⣄⣀⠀⠀⢤⡤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⢳⡄
⡇⠀⠀⢠⠀⣤⠀⠀⠀⠀⠙⠤⡀⠀⠀⠀⠀⠀⢸⠀⢨⣤⣤⣤⣀⡈⠙⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠴⠚
⡇⠀⠀⠈⣄⠈⠙⠒⠢⠤⠤⠒⠚⠀⠀⠀⠀⠀⢸⠀⠸⢓⠤⠍⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠤⠤⢄
⡇⠀⠀⠀⠈⢕⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠖
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈

⡄⠀⠀⠀⢠⡴⠆⠀⠐⠦⡄⠀⠀⠠⠀⠀⠀⠀⢠⠀⠀⡟⠛⢣⡤⠛⠳⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⡴⣛⠀⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠐⢀⠁⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⢘⣤⣤⡤⣧⣤⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⣤⢱⣆⠈
⡇⠀⠀⠀⠘⠷⠊⠀⠛⢟⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠛⠓⠐⠛⠛⠒⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠚⠙⠛⠐⠂⠂⣄
⡇⠀⠀⠀⡌⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⣠⠏
⡇⠀⠀⠀⢳⣀⠀⠀⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠦⣄⣀⠀⢀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣆⡐⠂⢠⠌⠉⠛⠋⠁
⡇⠀⠀⠀⠀⠉⣉⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠄⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠿⢿
⡇⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣐⡒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠂⠁
⣧⣤⣤⣤⣤⣤⣬⣭⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣤⣤⣤⣤⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡔⠉⠀⠀⠉⢦⠀⠀⢠⠊⠁⠀⠈⠱⡀⠀⠀⡔⠉⠀⠀⠁⢂⠀⠀⢠⠊⠁⠀⠈⠲⡀⠀⠀⡔⠉⠀⠈⠑⢆⠀⠀⠠⠊⠁⠀⠉⠲⡀
⠀⠀⠰⠀⠈⠁⠈⠁⠀⠀⠀⠇⠀⠃⠀⠛⠀⠐⠀⢸⠀⠠⠀⠠⠄⢀⠀⠀⡇⠐⢂⣀⠐⠀⡀⠀⢸⠀⠘⢀⡘⠃⢐⠀⠀⡇⠀⢃⣀⠛⠀⡃
⠀⠀⠀⠑⢌⣛⣛⡡⠊⠀⠀⠈⠮⣙⣛⣋⠔⠁⠀⠀⠳⢄⣒⣒⡠⠊⠀⠀⠘⠤⣈⣀⣁⠔⠁⠀⠀⠣⢄⣉⣉⡠⠊⠀⠀⠘⠤⣈⣈⣁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⠄⠒⠒⠠⡀⠀⠀⠀⡎⢱⠀⠢⢄⠀⠀⠀⢀⠔⠒⠒⠠⡀⠀⠀⠀⡠⠒⠒⠒⢄⠀⠀⠀⢀⠔⠒⠒⠤⡀⠀⠀⠀⡠⠒⠒⠠⢄
⠀⠀⢠⠁⠠⠄⠠⠄⠘⡀⠀⡜⠁⠅⠀⠠⠀⢃⠀⢠⠁⠐⠀⠐⠂⠘⠀⠀⠚⠀⠂⠀⠂⠀⠃⠀⢠⠁⢰⠀⢰⡄⠘⠀⠀⡌⠀⠀⠀⠀⠀⠂
⠀⠀⠈⢄⠀⠀⠲⠀⡰⠀⠀⠡⡀⠐⠛⠂⢀⠆⠀⠈⢄⠺⠿⠿⠇⡰⠀⠀⠡⡘⢿⣿⡿⢃⠃⠀⠈⢄⠆⠀⠀⠶⡐⠀⠀⠣⡰⣄⣀⡰⢀⠂
⠀⠀⠀⠀⠁⠒⠒⠉⠀⠀⠀⠀⠈⠒⠒⠂⠁⠀⠀⠀⠀⠁⠒⠒⠉⠀⠀⠀⠀⠈⠐⠒⠈⠁⠀⠀⠀⠀⠁⠒⠒⠉⠀⠀⠀⠀⠈⠐⠒⠊⠁
⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⢀⠠⠀⢄⡀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⡀⢀⡀⠀⣀
⠀⠀⠀⠜⠁⠀⠀⠈⠢⠀⠀⡐⠁⡀⠀⢀⠘⠄⠀⢀⠎⠀⠀⠀⠈⢂⠀⠀⡔⠁⠀⠀⡈⠱⡀⠀⢠⠊⠀⠀⠀⠉⠆⠀⢰⠉⡇⠀⠀⠀⠑⡄
⠀⠀⠘⡀⠐⢀⡠⠆⣀⠃⠀⢇⢀⣉⣀⣁⠀⡄⠀⢸⠀⠐⣀⣐⠀⢨⠀⠀⡄⢀⣁⢀⣁⠀⡇⠀⢀⠀⣀⣀⣘⡀⡸⠀⠈⠞⠁⢃⣀⡛⠀⡅
⠀⠀⠀⠑⠤⣈⡁⠴⠈⠀⠀⠀⠓⢇⢂⠵⠊⠀⠀⠀⠑⠄⣀⡀⠔⠁⠀⠀⠘⠢⣉⣉⠡⠊⠀⠀⠀⠓⢭⠈⡨⠔⠁⠀⠀⠈⠢⢄⣀⠠⠊
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠈⡱⢪⠀⠀⠉⢢⠀⠀⢠⠊⠁⠀⠉⠢⡀⠀⠀⡔⠉⠀⠀⠑⢆⠀⠀⢠⠊⠁⠀⠉⠢⡀⠀⠀⡔⠉⠀⠀⠑⢄⠀⠀⢠⠊⠁⠀⠉⠢⡀
⠀⠀⠀⠀⠐⢂⡔⠂⠀⠇⠀⡇⠀⠀⠀⠀⠀⡁⠀⢸⠀⢐⣂⣐⠀⢈⠀⠀⡇⠐⠆⠀⠰⠀⡁⠀⢸⠀⡘⠀⠘⡃⢈⠀⠀⡇⠠⠄⣀⠤⠄⡃
⠀⠀⠀⠑⢄⡘⠋⡠⠜⠀⠀⠘⢄⡐⠀⢀⠔⠁⠀⠀⠣⣘⠁⢘⡡⠊⠀⠀⠐⢌⡛⡛⢛⠔⠁⠀⠀⠢⣙⠆⠺⡣⠊⠀⠀⠘⢆⡘⠛⣀⠴⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⡀⠤⠤⢀⠀⠀⠀⠀⢀⠠⠤⠄⡀⠀⠀⠀⠀⡀⠤⠤⣀⠀⠀⠀⠀⢀⠤⠤⢤⣀⠀⠀⠀⢀⡀⠤⠄⣀⠀⠀⠀⠀⣀⠤⠤⢤⡀
⠀⠀⢀⠊⢠⠀⢀⡀⠱⠀⠀⡰⠁⡀⠀⠀⠈⠆⠀⢀⠎⠀⠀⠀⠀⠱⠀⠀⡐⣥⣤⣤⣤⡼⠆⠀⢀⠪⣤⡄⢴⣤⠡⠀⠀⡔⢡⡄⠀⢠⠈⡄
⠀⠀⠘⡀⠈⢤⢤⠁⢠⠁⠀⢣⠀⠩⠤⠍⠀⡌⠀⠘⡀⠀⣤⡄⠀⢠⠀⠀⢃⠉⠉⢀⡉⠁⡄⠀⠘⡀⢈⣀⣈⠁⠠⠀⠀⢃⠀⢰⣶⡄⠀⠄
⠀⠀⠀⠈⠒⠠⠄⠐⠁⠀⠀⠀⠑⠤⠤⠔⠊⠀⠀⠀⠈⠂⠤⠤⠒⠁⠀⠀⠀⠑⠤⠤⠔⠊⠀⠀⠀⠈⠢⠬⠥⠒⠁⠀⠀⠀⠑⠌⠏⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⡔⠊⠉⠉⠑⢄⠀⠀⢀⠖⡉⠉⠉⠢⡀⠀⠀⡐⠊⠉⠉⠑⢄⠀⠀⢠⠒⠉⠹⠟⠻⡁⠀⠀⡔⠈⠉⠉⠑⢄⠀⠀⢠⠒⠉⠉⠉⠢⡀
⠀⠀⢸⠀⠐⠀⡘⠃⠈⠀⠀⡎⠀⠃⠁⠛⠀⠣⠀⢈⠁⡈⠀⠈⣀⠊⠀⠀⡇⠀⠁⠀⠁⠀⡁⠀⢸⠀⠒⠀⠐⠒⠈⠀⠀⡇⠀⠆⠀⠆⠀⡃
⠀⠀⠀⠣⣀⠀⠃⣸⡿⠀⠀⠘⢄⡉⠛⢁⠰⠁⠀⠀⠣⣙⠋⠙⡋⠌⠀⠀⠐⢄⡓⠒⣋⠔⠁⠀⠀⠣⣀⡈⢀⡠⠎⠀⠀⠘⢄⡙⠋⣃⠔⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠻⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠿⠟⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠁⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠉⠀⠀⠈⠉⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠁⠀⠀⠀⠉⠙⠳⣤⡀⠀⠀⣠⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢹⣶⠶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣾⣉⣿⣷⠀⠀⠀⠀⠀⢸⣷⣤⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠙⠣⠟⠋⠀⠀⠀⠀⠀⣾⠙⠢⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⢀⣤⡶⢶⡶⠶⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠋⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠘⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠒⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠤⠤⠤⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠙⢷⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡿⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠁⠀⠀⠙⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⡀⢀⣾⠉⠛⠷⠦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡴⠶⢾⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠃⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣩⡿⠛⠉⠉⣿⡁⠀⠀⣀⣀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠙⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⢀⣀⣀⣿⢀⣠⡤⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣀⡴⠞⠛⠉⠉⠉⠛⠿⣄⡀⠀⠀⠈⠻⡆⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⣠⣤⣀⣀⣀⠀⠈⠁⠀⠀⠀⣀⣿⠞⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⡏⠀⠀⠀⠉⣽⠆⠀⣀⣴⠞⠋⠀⠀⠀⣿⡴⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⡀⢰⣯⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣠⠀⡀⢀⡠⣄⡀⠌⡲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⡴⣋⡀⢈⢀⣤⡈⢳⡐⠀⠀⠀⠀⠀⠀⠀⡤⠖⢂⠀⠀⢀⡀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠿⠟⢹⠸⠿⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠥⠵⠎⠥⠌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⣸⣆⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢸⡀⠸⢣⡄⠷⠀⡸⠁
⠀⠀⠀⠠⢞⡉⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠙⠚⠉⠀⠈⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠒⠋⠙⠒⠊⢁
⠀⠀⠀⠐⣡⠖⠒⢄⠀⠀⣀⣀⣀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⣤⠤⠶⣲⠚⠂
⠀⠀⠀⠀⡇⠀⠀⠈⣷⠋⠀⠀⠈⢣⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⢀⣤⣤⠤⠀⣠⣤⣀
⠀⠀⠀⠀⢧⢠⣶⡄⡇⠀⠠⣿⡆⠈⡇⠀⠈⠀⠀⠠⠖⣂⠀⢀⡠⠄⠀⠀⠀⠀⠀⠀⡀⣀⣤⡆⠀⠘⠿⡿⢋⡔⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠔⠚⢆⠭⠀⠧⠠⠤⠤⠤⣤⡃⠀⠀⠀⠀⠀⡼⠉⣧⠀⡴⢳⠀⠀⠀⠀⠀⡌⣤⢹⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠄⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢃⣴⡿⠐⣧⣾⠀⠀⠀⠀⡀⠉⠧⠞⠑⠃⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⢡⠔⠊⠉⠲⡀
⠉⠐⣦⣀⣀⠈⠁⠁⠀⠀⢀⣀⡀⡄⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠈⣉⣀⠀⠀⠀⣸⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⢿⣆⣈⣸⡷⠀⠀⡇
⠀⠀⠹⡷⠶⠭⢭⣭⣉⣉⣩⠞⠁⠁⠀⠀⠀⠀⠀⣜⠉⢉⣿⣿⠏⠉⠁⠀⠀⠀⢿⡿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠰⡃⠀⠈⣆⣠⠜
⠀⠀⠀⠈⠒⠤⣄⣠⠤⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠟⣁⠔⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠈⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀⣦⠀⠀⡔⠊⠉⠑⡄⠀⠀⠀⠀⠈⣉⠒⠒⣊⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠔⢂⠤⠬⣉⠙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠻⡆⢀⢰⣷⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢏⠀⠀⠈⢧⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠈⢊⠉⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠁⣀⠤⠤⢄⣀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠣⣀⣀⡜⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢤⠒⠭⠒⠉⠉⠉⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢀⣀⣤⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢱⠦⡝⣲⠶⢶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣛⠛⣉⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⣦⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣠⣃⡴⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣴⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠸⣿⣿⣿⣿⡿⢿⣷⣿⣷⣦⣤⣀⣀⣠⠀⠀⠀⣿⣿⣿⡀⠀⢀⠀⢠⣀⢰⠁⠈⠙⠛⠿⠿⠿⢿⣿
⠀⠀⠀⢻⣷⣾⣿⣿⣿⢸⣿⣮⣿⣟⣛⣿⡏⠀⠀⠀⣿⣿⣿⣧⠀⠉⠀⠀⠀⢸⠀⠙⠂⠀⠤⠄⢀⣿⣿
⠀⠀⠀⠀⠻⣿⣭⣬⣽⢸⣿⠿⣿⣿⣿⠟⠀⠀⠀⠀⣿⣿⣿⣿⣧⡀⠴⠛⠓⣸⡀⢄⣀⣀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠻⣷⣶⠿⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⡷⢶⣶⢶⣿⣷⣈⣉⣉⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿