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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣤⣠⣄
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⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⢛⢷⡍⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠛⢻⠛⣿⢻⡟⣻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣿⢸⣇⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠂⠉⢀⠈⠸⠇⢾⣿⣿⣷
⠀⠀⠀⠀⣀⣀⣀⣿⣀⣀⣠⣇⣸⣿⣀⣀⣸⣇⣀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⡷⠀⣀⣀
⠀⠀⠀⢠⡏⠉⠉⢿⣭⣍⠉⠉⢹⣿⣍⠉⠉⠉⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⣩⣍⠉⠉⠉⣩⣭⠉⠉⠉⢹⣿⡇⣿⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠛⠋⠀⠀⠀⠛⠛⠀⠀⠀⣿⣿⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣴⡟⠛⠀⠀⠀⣿⠛⠀⠀⠀⢸⣿⡇⣿⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣿⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⡇⣿⣿⡀
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⣇⠀⠀⠀⢀⣿⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠻⣦⣀⣠⣼⠇⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿⡧⣿⣿⡇
⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣷
⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⢿⣿⣷⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⡻⣿⣿
⠀⠀⠸⠷⠶⠤⣤⣤⣤⣤⣄⣀⣀⣀⣀⡀⠀⠀⣿⣯⣿⣿⣿⠿⠀⠀⠀⠀⠀⠐⠷⠶⠤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⠀⠀⢸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⣤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⣖⣒⣦⣄⣠⢯⠟⠋⠉⠉⠙⠺⣟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⠞⠉⠀⠀⠉⢹⣿⣯⠀⠀⠀⠀⠀⠀⠘⣮⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡏⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⢸⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⠀⠀⣀⣤⠤⣀⣸⣿⢸⣼⡄⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡴⠚⠁⢸⠀⠀⠉⠉⠺⣿⣧⣀⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢠⠶⣒⡋⠉⢹⣿⠀⠀⠀⢸⠀⠀⠘⢿⣶⣿⣇⠀⠉⠙⠲⠤⣤⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠓⢾⣿⡀⠀⠀⢸⠀⠀⠀⠀⠈⣿⡿⠿⣷⣦⣤⣀⣀⠈⠙⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢸⣿⠉⠑⠦⣼⡀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠉⠉⠛⠃⠀⣀⣠⠤⠞⢻
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠙⠲⢤⣀⣿⠇⠀⠀⠀⠀⠀⢀⡴⢿⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠳⢤⣀⠀⢀⡴⠋⠀⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠀⠀⠈⢙⡋⠀⠀⠀⢻⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢾⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⢸⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⡇
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⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡸⠠⠤⠤⠤⣤⡇
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡼⠁⠀⠀⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡼⠁⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠀⠀⠀⠀⢸⠀⡜⠁⡠⠚⠁
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⢸⣞⠴⠋
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁
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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣞⣀⣀⣀⣈⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠍⠉⠉⠉⠉⠍⠉⢹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⣀⣠⣋⣇⣈⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠇⠘⠁⠈⠉⠋⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⠒⠓⠒⠒⠒⠒⠒⠚⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠛⠛⠛⠛⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⠆⢘⣛⠁⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠈⢛⡋⠐⣷⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⣴⣿⣿⣷⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣾⣿⣿⣷⠈⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠹⢿⣿⠿⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠻⣿⣿⠟⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢀⡤⣤⣄⡈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⣈⣷⡀⡯⠀⠀⠀⠙⠳⢤⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⡀⠰⡒⠲⠤⣍⣉⠛⠶⣄⠀⠀⠀⠀⠙⢦⡙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢄⠀⠀⠙⣟⠦⣌⠛⢦⡀⠀⠀⠀⢳⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢦⡀⠘⢦⠈⢷⣦⡙⢦⡀⠀⠀⢷⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢦⡈⠳⣌⢧⡙⢦⡙⢷⡄⠈⢷⡈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣬⣷⣄⣹⣄⣘⣷⡙⢦⠹⡄⢻⡄⠈⣧⠸⡇⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠓⠿⠶⣿⣤⡼⠶⠷⠶⠶⠶⠟⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⡇⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⢀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⡀⡿⣀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⢀⣤⠞⠁⠀⠈⠛⢦⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠋⠀⠀⠀⠀⠀⠀⠀⠙⣃⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠷⠶⠶⠶⢶⣶⣶⣶⣾⣿⣤⣤⣴⣶⣶⠶⠶⠿⠿⠛⠛⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⡖⢲⣤⣼⣤⣶⣤⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣽⣿⡟⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣶⣾⣷⣶⣿⢻⣿⣿⣿⠀⡀⣘⠁⡂⠀⠀⠀⠀⢸⣿⣿⣿⠋⠉⠉⠉⠁⡄⠀⠀⠀⣿⣿⢿⣿⢿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⡌⣿⣿⣿⡀⣿⣿⣿⣿⡀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣷⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⢀⣿⣿⣿⣿⣿⡇⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⢻⠀⠀⠀⠏⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠸⠿⠿⠿⠿⠿⠧⠚⠛⠛⠻⠿⠿⠿⠿⠇⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⣸⣧⣤⣤⣄⣀⣀⣀⣀⣹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣲⡊⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢠⠁⠙⡀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠃⠀⠀⢡⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠈⡄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢢⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡰⠁⢣⠸
⠀⠀⠀⠀⢰⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠎⠠⠁⠀⠀⡆⡇
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⠷⠶⢶⣶⢶⣶⣶⣶⣶⣴⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⣤⢿⣿⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣸⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⣿⣿⣷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣶⣴⣿⣿⣿⣿⣬⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⢠⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡆
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣽⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠈⠉⠙⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣙⣛⣻⣿⣟⣛⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⢀⡤⣲⣺⡯⢯⣛⣑⡛⣲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⢀⡤⠤⠤⢤⣉⠛⠷⣉⠀⠀⠀⠀⠈⠉⠉⠓⢮⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⠞⠁⠀⠀⠀⢰⠌⠽⢦⡈⠳⣄⠀⠀⠀⠀⠀⠀⠈⢳⡸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡎⠀⠀⠀⠀⠀⣿⠰⡇⠀⠱⣄⠈⣢⠀⠀⠀⠀⠀⠀⠈⣅⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⡤⠄⠒⠺⣀⠀⠉⢻⡆⠀⠀⠘⡄⠐⡂⠀⠀⠀⠀⠀⠀⢹⡌⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠉⡀⠀⡇⡀⣀⠀⢃⠀⠀⠀⣅⠀⠀⠀⡾⠀⢹⠀⠀⡀⢀⡀⠀⠀⡇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⡅⠀⡆⢹⣏⡴⠘⠄⣠⠔⠻⠷⣤⠆⠀⡇⠀⡆⠘⠉⠁⠈⠭⢒⡓⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠣⡀⢀⠀⠇⠀⡇⠘⣿⣿⠆⣸⠟⠳⣾⣐⢰⡠⠤⡅⠀⣿⡇⠀⠀⠀⠀⠀⣸⢹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠉⠻⢤⣠⠠⡇⠰⠁⢿⣷⡯⠤⡶⣿⣦⣹⠀⢀⡇⠀⡧⣱⠀⠀⠀⠀⢰⢭⡾⠚⠊⠋⢕⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠹⠐⣷⡆⢠⡩⢸⡗⠚⠋⠁⠈⠉⠛⠲⠇⢸⡏⢹⣹⠉⠋⠍⣛⠛⠳⣤⡀⠀⣾⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠑⠀⠀⣟⠈⠚⠧⣼⡁⠶⠆⠀⠀⠀⠰⢶⠀⢘⠉⢿⠇⠈⠀⠀⠈⢑⡦⠤⢽⣶⠃⣰⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣒⣸⠉⠢⠤⣀⠉⠒⠰⢶⣎⣙⡀⡋⠀⢞⣃⢀⣳⣄⠄⣀⣤⣦⣸⠊⠥⠛⠋⠉⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢉⡓⠓⠮⣒⣲⠑⡇⡈⣻⠛⠉⠉⠉⠀⠁⠀⢀⡀⠀⢠⣐⠶⠆⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢔⡠⢄⡀⠩⠱⣦⡼⠀⠒⠈⠉⠉⠉⠉⠀⠴⠊⠁⢀⠄⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⡀⠀⠀⠀⢂⠄⢊⠔⢁⠄⡠⠀⡀⡠⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡡⠊⡡⠊⡠⠊⡠⠪⠊⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⢊⠔⠉⡠⢊⠔⡡⠊⢀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠔⢉⠔⡡⠊⢀⠔⡡⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠅⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔⢁⠔⡡⠊⡀⠔⢁⠔⢀⠆⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠅⢀⠀⠀⠀⡠⠖⡡⠊⡠⠚⠁⡤⢃⠔⡡⠂⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠄⠂⣉⠤⠂⠀⠈⠀⢊⡠⠊⢀⡠⢊⠔⢁⠀⢀⠊⠀⠘⣖
⠀⠀⠀⠀⠀⠀⢀⡸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠀⣉⠤⠒⣁⠀⢀⠔⠂⡠⠒⣡⠖⢁⠔⢁⠔⢁⠈⠔⠀⣖⡈⠨⠅⢒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠓⠤⣀⠠⠄⠀⠀⠀⠀⠀⢀⠀⠤⠄⢒⣊⡡⠄⢒⡉⠀⠀⡠⠔⢉⠤⠊⡠⠀⡁⠄⣁⠖⡡⢊⠐⠅⢸⣒⠀⠉⠠⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠦⣄⡀⠀⠀⠠⠄⠒⠂⠉⣁⠠⠔⢊⡡⠔⠀⢈⠠⠂⡁⠔⢁⠤⠊⡠⠀⡡⠊⡴⠃⠀⢂⣈⣇⡂⠩⠄⠐⢂⡈⠉⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠶⢤⣀⡂⠈⠁⠤⠒⣉⠥⢒⠀⠂⠁⠔⠈⡠⠚⠁⠔⠁⣀⣉⡀⠤⠒⠚⣉⠉⠅⠀⢀⡀⠨⠀⢂⣀⠠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠦⠤⢬⡥⢬⣤⣠⡴⣀⣤⠵⢒⠪⠙⢉⠡⠄⠒⣈⠁⠄⢐⠀⠁⠀⠂⠈⠥⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣄⡄⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠟⠉⠁⠈⠉⢪⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡟⡤⢤⡀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⠤⠤⢤⣤⢞⣽⣶⣃⣉⡑⢿⣦⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⢰⣯⠏⠀⠀⠉⠉⠉⠉⢻⣿⠒⠒⠛⠲⠶⠤⠤⢤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢿⣷⡷⡶⠾⠶⢶⣶⣶⣿⣯⣾⣿⣧⣼⣧⣤⣦⣴⣿⣄⣰⣤⡦⣤⠀⣄⡀⣴⠶⢟⢟⡻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢻⣋⡿⠃⣈⠀⠈⠫⠟⣿⢁⠀⠀⠀⣠⡤⣄⠉⠉⢹⣏⣿⣟⠛⠛⠛⣿⠿⣿⣧⠀⡫⢊⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣏⣻⠀⠀⠸⡭⠀⠀⠈⠸⡐⢔⡹⠀⠀⣜⡿⠩⠃⠐⠀⠘⢿⣢⠞⣿⠀⢐⠵⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡍⣷⠀⠂⢠⣴⠮⠏⢶⣄⠂⣠⡄⢄⠈⠁⠄⠁⠀⠈⠀⠀⢠⡀⢀⡀⠀⠀⢸⣿⡇⠀⡪⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣌⣱⣟⠀⢀⣿⠁⣶⢥⣤⠙⣷⠝⠶⢠⠃⠀⠀⡤⠤⠄⠀⢰⢡⢂⢲⢰⠈⠀⣠⡿⣿⠀⢒⠕⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡍⠉⠉⠉⠉⢉⣽⡿⠿⢿⣿⡛⠛⠛⠛⠛⠛⠓⠒⠶⠾⠾⠾⣧⡄⠐⡈⠠⢵⠏⠀⠈⠹⡯⢼⡇⠀⡪⠚⣧
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣚⣫⣀⣀⢀⠰⣠⣿⠃⠰⠆⡀⢸⣷⡀⠰⠆⠀⠀⠀⠀⢀⣠⡴⢚⠙⡇⠀⠈⠉⠁⡀⠀⣠⣠⡿⠘⣿⠀⠐⡣⢻⡆
⠀⠀⠀⠀⠀⠀⢸⠏⠉⠉⠙⠛⠛⠛⠛⢿⡷⠶⠶⠶⠷⢿⣼⣷⣤⣼⣴⣤⣤⣤⣼⣆⠀⠑⡀⣻⠀⡀⠀⠀⠀⠀⡻⣿⣿⠄⢹⡇⠨⢆⠜⣷
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠀⠀⢼⠇⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⡈⢘⡄⣴⡮⠰⡄⠠⠈⠈⣯⢁⠀⣿⠀⠡⠊⢻⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⣔⠅⣿⡹⠳⢗⡟⠀⠀⠀⣿⢄⡀⢹⡆⠈⡀⠍⣿⡀
⠀⠀⠀⠀⠀⢀⣿⠖⠶⣤⡤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠄⡀⢿⠈⠉⠉⡄⠀⠀⠀⣿⠔⠠⠈⣇⠀⢤⠴⠘⣇
⠀⠀⠀⠀⠀⢸⡇⡸⢎⠀⠙⣤⠊⠉⠢⣠⠋⠙⢍⡩⠟⠛⡍⢉⠝⠛⡍⢉⠝⢛⡛⣻⣧⠀⠊⢀⠹⣀⡄⡁⠀⢠⠤⢤⣹⡈⠀⠀⢻⠀⠠⠊⠄⢹⡆
⠀⠀⠀⠀⠀⢸⣏⠀⠀⡱⣞⠀⠑⣤⠊⠉⠢⣠⠞⠑⣄⢀⡜⠣⡀⢀⠜⠣⡀⣠⠟⢽⣿⡄⠐⡁⢨⡿⠻⠃⠀⢹⡶⣶⢿⡇⠀⠀⣼⠷⠶⢷⣿⣿⠟
⠀⠀⠀⠀⠀⣸⡏⢣⡜⠀⠈⣢⡞⠀⠳⣠⠖⠉⢦⢀⡼⠣⡀⢀⡜⠣⡀⡠⠚⢦⠀⣸⣿⣧⠈⡪⠀⡧⠀⠀⠀⠀⡈⠡⠘⡇⠀⣸⠟⣨⣶⠿⠋
⠀⠀⠀⠀⠀⣿⡇⠁⠈⣦⠚⠁⠙⣤⠞⠉⠢⣀⡴⠛⢄⢀⠜⢯⠀⡠⠛⢤⠀⡠⠛⢼⣿⣿⡀⠰⠁⢿⣞⣕⡄⠀⠀⣴⠲⣗⣸⣿⡿⠛⠁
⠀⠀⠀⠀⠀⣿⠈⣢⡞⠁⠑⣤⠞⠁⠱⣄⠔⠉⢦⣀⠔⠳⡄⣠⠛⢄⠀⡰⠛⢄⢀⢼⣿⢾⣇⣀⡦⢸⠿⠿⠤⠥⠼⠿⢷⣿⠟⠉
⠀⠀⠀⠀⠀⡿⢎⠀⠈⢢⡔⠉⠣⣀⠔⠉⢢⣀⠔⠓⢄⡠⠞⢣⡀⡠⠻⡀⠀⡴⠻⣼⣯⣿⡏⡛⢻⣿⠄
⠀⠀⠀⠀⠘⠷⠤⣵⣶⣥⣌⣤⣔⣉⣢⣀⡔⠙⢄⡠⠊⠳⡀⡠⠛⢄⠀⡨⠫⡀⢀⢼⣿⡿⣈⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠚⠷⠶⠤⣼⣧⣀⣀⣿⡀⠀⡹⢇⢸⣿⣹⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠓⠶⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⡟⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠁⠈⠙⣿⠀⠀⠀⠀⣀⣀⣀⣀⣸⣼⣤⣤⣯⣧⠤⢤⣤⡀
⠀⠀⠀⣤⣤⠤⠤⢤⡿⢧⣤⣤⣤⣿⣤⣤⣤⣼⣿⣿⣛⣷⣤⣿⠤⠶⠶⠗⠚⢋⡉⢹⣀⣴⠚⠳⡄
⠀⠀⢠⡿⠿⠤⠤⢤⣧⣤⣤⣤⣄⣼⣀⣠⣿⣽⣿⠋⠁⠈⢻⡇⠀⠀⢀⣤⣤⣼⠉⡿⠦⢿⣇⠀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠁⠀⠀⠀⢠⣿⠒⠲⠶⠾⡿⠖⠒⠒⠺⡗⠒⠒⠚⣻⢷⠾⣄⡇⠀⠀⣿⠉⠛⠦⣄⡀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣼⣻⣇⣀⣀⣀⣇⣀⣀⣀⣠⣇⣀⣀⣀⣿⣿⡄⠈⠛⠲⢤⡟⠀⠀⢠⡤⠽⢷⡆
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠉⠋⠉⠉⠁⠘⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠋⠓⢦⡟⡇⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⢸⡇
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⢸⡇
⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠓⢦⣀⠀⠀⠀⠀⡇⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠳⢤⡀⠀⣇⣧⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡇⠀⠀⠀⠀⠀⠉⠳⠟⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀⣤⢤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⡙⡵⠛⠻⣿⡙⢎⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢀⢱⠀⠀⠀⢣⢷⠀⠏⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⢼⠀⠀⠀⠈⣸⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⢰⢈⠀⠀⠀⢀⣿⡀⢈⢠⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠴⠒⠀⠉⠉⠉⠉⠉⠉⢹⡼⠀⠉⠀⠀⠈⠁⠈⣮⠃⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⢸⡁⣸⠀⠀⡇⠀⠀⠀⠀⠀⠀⢾⣿⠄⠀⠀⠀⠀⠀⠺⢿⠗⠀⠀⠀⠀⠐⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢻⠋⢽⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠔⠠⡀⠀⠀⠀⠀⠀⠰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣾⢊⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢊⢈⠔⡠⡀⠀⠀⠀⢊⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠨⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠊⢊⠄⠔⠀⠀⠀⡥⡀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣪⠂⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠕⢼⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⡐⡄
⠀⠀⠀⠀⠀⠀⠀⡯⠐⣽⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⡈⡐⠄⢔⠆
⠀⠀⠀⠀⠀⠀⠀⡧⠂⣽⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⠎⢈⢀⠌⠔⠈⣰⠢⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡡⠠⠀⡀⠠⠪⠢⢂⠌⡼⠡⡡⠆⢰
⠀⠀⠀⠀⠀⠀⢰⡅⡂⢺⠀⠀⢀⠄⠀⠀⠀⠀⠀⢊⠄⠔⠐⠁⡠⡔⠐⡑⡑⠡⠂⡊⠡⡐⠌⢄
⠀⠀⠀⠀⠀⠀⢸⢗⠉⣺⠀⢘⠑⡁⡁⡠⠢⢀⠄⡀⡠⢀⠂⠊⢊⢊⢊⠄⡰⠑⡁⠀⠐⡀⠄⢈⠐
⠀⠀⠀⠀⠀⠀⢸⠕⠨⢺⠀⢠⠨⡪⢊⠔⡰⠡⢊⠌⠔⡑⡑⡱⡡⠢⢊⠜⠜⠔⡐⠀⠐⠀⠌⠀⠊
⠀⠀⠀⠀⠀⠀⠸⢵⣾⠿⡆⢸⠐⡔⡡⢊⠔⡁⡡⢂⢊⢎⢌⡔⡕⡡⢪⢂⢈⠄⡄⡰⠄⢄⢂⠠⡥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠛⠖⠒⠒⠒⠲⠆⠁⠃⠤⠬⠬⠬⠐⠓⠓⠲⠒⠚⠚⠒⠐⠒⠒⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⡀⢀⠄⠀⢀⠀⡠⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⡀⢀⠀⣀⠄⡀⢐⠁⠀⢀⠩⢀⠁⠠⡈⠀⡁⡊⠔⡡⢂⠄⡨⢊⠔⡨⢀⠕⡠⢊⠔⡩⢂⠅⡠⠨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁⠈⠀⠁⠈⠀⠁⠈⠀⠁⠈⠀⠁⠐⠁⠊⠀⠁⠀⠀⠁⠈⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⢀⣀⠀⣴⠋⠁⠈⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⡿⡛⠛⢲⣴⢿⠿⠷⣶⡀⠤⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⡟⠻⣅⣲⡶⠿⢤⣀⢨⣿⣷⠂⠠⠄⢸⣷⠀⠂⠀⠉⠑⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⡖⠠⠤⢤⣤⣶⢾⣿⡤⠤⠄⣿⡃⠘⠀⠀⢻⡍⢩⡉⠒⠠⠄⣠⡟⠉⠁⠒⠠⡞⡍⠉⠉⡇
⠀⠀⠀⠀⢀⡤⠐⠋⠀⢠⡿⠁⠠⡌⠙⣷⠀⠤⠿⣇⠉⠁⠐⣺⠧⠄⢀⡀⣴⠒⠋⢹⠀⠀⠀⠀⣇⢱⠀⠀⢹
⠀⠀⠀⠀⡇⠀⠀⠈⠉⢹⡗⠠⠄⠀⣀⣿⠀⠘⠀⠀⠀⢉⣐⣿⡄⠀⠀⠀⣿⡀⠀⢸⡇⠀⠀⠀⢹⠀⣇⠀⠈⣇
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢠⡏⠉⠐⠒⠠⢿⡄⠀⠤⣧⠀⠀⠀⢿⣇⠀⠀⣇⠀⠀⠀⠈⡇⠘⡄⠀⢸⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⢻⠀⠀⠀⢸⢸⠀⠀⢸⠀⠀⠀⠀⢹⠀⢡⠀⠀⢧
⠀⠀⠀⡌⠀⠀⠀⠀⣩⣿⢿⣿⣿⡦⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⢸⡀⠀⠀⢸⡅⡇⠀⠘⡇⠀⠀⠀⠈⡇⠀⢆⠀⠸⡆
⠀⠀⠀⡇⠀⠀⠀⠀⢱⣿⣇⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡀⠀⠘⡇⠀⠀⠘⡇⢱⠀⠀⢻⠀⠀⠀⠀⢹⠀⠘⡀⠀⢻⡀
⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⣿⣆⠀⠰⠀⠀⠀⠀⠀⠀⣟⠀⡇⠀⠀⡇⠀⠀⠀⡇⠘⡄⠀⠈⣇⠀⠀⠀⠈⡇⢠⠛⣻⣶⠗
⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠈⢿⣷⣄⣀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⢻⠀⠀⠀⣷⠀⣣⣀⣀⢺⡉⠉⠒⠒⠺⠗⠋⠁
⠀⠀⠀⣇⣀⠀⠀⠀⠈⠃⠀⠸⣿⣿⣿⡷⠀⠀⠀⠀⡇⠀⢱⠀⠀⠸⣆⣀⠀⢹⡰⢋⡭⠽⠛⠁
⠀⠀⠀⠀⠉⠉⠛⠒⠦⠤⣄⣀⠀⠉⠉⠁⠀⠀⠀⠀⡇⢀⡜⠦⠤⣀⣧⠈⠙⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠤⠤⣀⡀⠀⣧⢊⣠⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣧⠀⠀⢀⣤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠆⠀⠉⠑⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⣠⠆⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠐⢀⡋⠋⠀⠀⠀⠀⠀⡎⠀⠀⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⣗⠲⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⡴⠁⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⣿⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣷⠁⠀⠀⠀⠀⡀⣀⣀⡤⢀⠀⠀⠿⡄⠀⢠⠀⠀⠀⠀⠀⠄⠀⢰⠘⠀⠿⠅⣤⠀⣀⣀
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠈⡇⣿⠀⠀⠈⠀⠀⠀⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠛⠀⣧⡟
⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡆⠀⠀⠀⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃
⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠖⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠁⠀⠀⠀⠀⠀⠇⢨⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠜⠀⠀⠀⠀⠀⠀⠀⠄⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⠒⠂⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡆⠀⣀
⠀⠀⠀⠀⢆⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠈⢆⠀⠀⠀⢀⣀⡀⠤⠤⠛⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠿⠋⠁
⠀⠀⠀⠀⢸⣀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠉⠁
⠀⠀⠀⠀⠈⠋⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠛⠿⣿⠿⠿⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡄⠙⠒⡄⠀⠀⠛⠀⠠⠂⣀⠀⢀⡀
⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢼⣅⠀⠀⠀⠀⠀⠀⣇⠀⠀⠱⠀⠀⠀⡄⠀⠀⠀⠀⠀⣉⣴⠚⠛⠛
⠀⡥⠂⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠃⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠘⢈⠀⠀⡏⠀⠀⠀⠐⣿⣿⡟
⠀⡄⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⣇
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣣⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢤⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⡄
⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⡀⠀⠀⢠⣄⡄
⠀⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⣄⡾⠋⠁
⠘⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⢀⣤⠶⠛⠃⠈⠙⠳⣝⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠴⠟⢱⡦⣄⠀⠀⠀⠀⠈⢷⡀⠙⢓⣀⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠛⣦⠀⠀⢰⡇⣀⢰⡶⠞⠛⠛⠷⡞⣣⣦⣼⠛⠋⠈⣧⢉⣉⣭⣽⣧⣀⠀⠀⠰⡏⢉⣷⢾⠙⠓⡆
⠀⠀⠀⠀⠀⣰⠏⢀⣤⠘⣧⢀⢸⡏⢩⡟⠉⣻⠆⠀⢀⡿⣫⣄⠉⠳⣄⡀⣿⠈⠁⠀⠀⠀⠈⠛⣦⡴⢣⡏⢀⡼⣱⠋⠁
⠀⠀⠀⠀⣰⠃⠀⣄⣉⣠⣼⠏⢸⡿⠏⠀⣴⠋⢀⡴⠋⠀⠙⠚⣠⠶⠚⠃⣿⢀⠀⠀⠀⠀⠀⠀⠈⣿⣻⠟⠉⠀⢻⡄
⠀⠀⢀⣾⣁⡆⠀⠀⢀⡼⠃⢀⢸⡧⣤⡾⠁⣴⠏⠀⢰⡋⣳⠸⣇⡀⠀⠀⣿⠈⣷⠦⣄⡀⠀⠀⢀⣿⠁⠀⠀⠀⠀⡇
⠀⠀⠀⢸⡏⠁⠀⢀⡾⠳⢦⣿⢸⡇⠀⣠⠞⠁⠫⠽⠀⠉⠁⠀⣿⠉⠛⠂⣿⠰⠋⠀⠈⢩⡟⠒⠚⢻⡀⠀⠀⢀⣼⠃
⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⢸⡏⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⣿⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡿⢷⣤⣶⡄
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡇⢸⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇
⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇
⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⡇
⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⣷
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⡿⠀⠀⣿
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣇⠀⠀⣿
⠀⠀⠀⢸⣧⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡾⠛⣛⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠓⠲⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠷⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⢋⡩⠤⠤⣤⣍⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢁⡞⠁⠀⠀⠀⠀⠻⣷⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢀⠏⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡈⡄⠀⠀⠀⢀⣀⣀⣀⣼⣇⠄⣌⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⠶⠾⠿⠿⢧⢿⠛⠛⠛⠛⠋⠉⢩⠯⠝⣼⠉⠉⠉⢿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⢲⡻⠾⢈⠄⠀⠀⠀⠀⠈⡿⠊⠡⠂⠀⠀⠈⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣈⣇⠀⠀⠀⠀⡏⣂⢁⠀⠀⠀⠀⠀⠀⢿⢒⠈⠁⠀⠀⠀⢸⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⡄⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡴⠇⠀⠀⠀⠀⢧⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⣼⡟⢿⡇⠀⠀⠀⢹⠶⠷⠀⠀⠀⠀⠀⠀⠀⣿⠀⠆⠀⠀⠀⠀⠸⡆⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⢲⡟⠁⠸⣿⠀⠀⠀⠸⣆⡜⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠛⡄⡄⡄⢸⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⢀⠇⡌⠀⠀⢼⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⢸⡴⠁⣀⣀⣸⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⠃⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⢀⣀⣀⡇⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⢹⡇⢰⣿⣿⣿⣿⣿⣾⣿⣀⣀⣠⣤⣤⣤⣶⣶⣾⣿⣿⠿⠿⠿⠿⠿⠯⠛⠿⠟⠉⠀⠀⢀⣼⣿⠟
⠀⠀⠀⠀⠸⡇⠘⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⡉⢥⠠⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠿⠋
⠀⠀⠀⠀⠀⠻⣄⠀⠀⠂⠙⠛⠻⠿⠿⠛⠋⠉⡉⠔⣊⠥⠚⠋⠀⠀⠘⠛⠻⠿⠿⠶⠶⠾⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀⣀⣀⣀⣀⡠⣀⡽⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠁
⣤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣸⣧⣭⣯⣤⣵⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠓⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣶⣤⣤⣿⣇⣀⣀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣸⣿⣤⣤⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠻⠶⡶⡆⢰⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢤⣀⠀⠀⠀⠀⠀⢰⡟⣿⠟⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⣀⣸⠟⠁⠀⠀⠀⠀⠈⢻⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠄⣐⠒⠲⢤⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣠⠴⡖⠒⠮⡕⣄⣠⠼⠶⠶⠊⠉⠛⠛⠀⠉⠁⠀⠘⠛⠋⠩⠿⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡞⠀⢸⡀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠸⢰⠀⣠⠜⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣧
⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⢀⠀⠈⠉⠛⢺⠖⠁⠂⣀⣀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⢠⡮⠊⢩⠋⠀⠀⠈⠉⠙⡶⠊⠉⠉⠀⠉⠉⠉⠓⠲⠤⣔⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻
⠀⠀⠀⠀⢀⢳⠁⢀⠇⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡇
⠀⠀⠀⠀⡜⠃⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢲⡆⢀⣱⠇
⠀⠀⠀⠀⠁⠀⠀⠏⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⠄⠀⠀⠀⠀⢶⣋⠭⠿⣿⠶⠒⡋⢀⡭⢿
⠀⠀⠀⠀⡄⠆⠀⢠⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⠴⠖⠒⠒⠒⢦⡀⠀⠀⠀⠀⠀⠈⠲⣴⣲⣆⢀⣤⣤⣀⣰⡯⣷⠝⠊⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠸⠀⠀⠀⠉⠉⠉⠁⠀⠀⠃⠈⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡀⡀⠀⡆⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢀⣤⠀⠀⠀⡿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⢤⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠇⢡⠀⢣⠀⠀⠀⠀⠀⡇⠀⠀⠈⢣⡤⠄⢁⣀⣀⣴⠃⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⢐⠋⠁⠀⠀⠀⢸⡇⠀⠀⠀⣸
⠀⠀⠀⠀⠀⢸⠘⡆⠘⡆⠀⠀⠀⠀⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⢦⠀⣀⡠⠴⢺⡀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⡇⢷⠀⠸⡄⠀⠀⠀⢷⠀⠀⠀⠘⠦⣄⣀⣠⡤⠞⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⡇⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⢰⣸⡀⠀⠘⢆⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠴⠞⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⢇⠳⡀⠀⠀⠱⢄⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡉⠢⢄⣀⣀⣉⣉⠽⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠊⠉⠉⠁⠈⠉⠓⠦⢤⣄⣀⣀⣀⣀⣠⠼⠓⠒⠒⠒⠒⠋⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢀⠀⠀⡀⠀⡰⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⣠⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢠⣠⢄⡔⢀⡔⡤⠀⡾⢃⢄⡀⡨⢁⡴⢃⡄⡀⠀⠀⠀⠀⠁⠀⠈⠀⢀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠷⠒⠛⠋⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⣠⠇⠈⠣⠚⠂⠙⠚⠓⠈⠃⠙⠈⠉⠓⠉⠐⠉⣻⠗⠁⠀⠀⠀⠄⠀⠀⠄⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠀⠀⠀⠀⠀⣀⠠⣤⡀⠀⠀⢀⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠈⠁⠀⢶⡄⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠂⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠈⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠄⠀⠀⠀⢠⣤⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⣄⠀⣤⢀⣀
⠀⠀⠀⠈⣾⠀⠀⠀⣀⣀⠀⠀⢀⡴⠟⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⢴⡋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⢀⣿⡄⠀⠀⢈⠉⠉⡠⠄⡐⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠱⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣀⣀⣴⠶⣶⠶⠶⠂⠀⠀⠀⠀⠀⠀⠀⠄⣤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠘⠉⠁⠀⠸⠗⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⠟⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⢀⣀⣀⡀⠀⠐⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⡆⠀⠀⠄⠀⠀⠀⠀⠐⠆⠀⠀⣹⣆⠀⠀⠀⠀⠀⠀⠠⠁⠀⠠⡤⢄⠀⠀⠉⡏⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠟⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠢⠜⠀⠀⢐⡇⠀⠀⠀⠀⠀⠀⠀⢀⠈⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠰⠄⢀⠎⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠢⡄⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣲⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡀
⠀⠀⠀⠀⠂⢀⠎⠀⠀⠿⠛⠙⠿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⣄⣤⡶⠀⠀⠀⠀⢀⣀⠀⠀⢀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⡎⠀⠀⢀⠀⠀⢀⣀⡀⠀⠀⠠⠆⠀⠀⠀⠘⠂⠀⠰⠖⠀⢤⣀⣀⣤⡄⠀⠀⠀⠀⡸⠋⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠸⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣀⣀⣂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣰⣿⣿⣷⠦⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠁⠉⠉⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣫⠀⠈⠉⠀⠀⠀⠀⠀⠀⠠⣤⣤⠄
⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⡤⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠛⡇⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢨⣷⠾⡤⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡇⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⠀⢉⣠⠶⠀⡿⠻⠄⠲⠛⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠓⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠈⠈⠀⠀⠀⠀⠀⠈⠁⠀⠃⢠⡆⠀⠀⠀⡇⡀⢋⠀⢀⠀⠀⠀⠀⠀⠙⠏⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⡙⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢉
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠘⠇⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠴⠶⠒⠒⠒⠶⠶⠤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠀⢀⡠⠤⠴⠶⠖⠒⠒⠶⠶⠤⢤⣀⠀⠈⠙⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⠀⠀⠙⠲⠤⠤⣤⣤⣤⣤⣤⡤⠤⠴⠚⠀⠀⢀⡠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣨⠟⠒⣶⠤⠤⣤⣤⣀⣀⣀⣠⣤⡤⠤⢴⡶⠊⢉⡴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠖⢺⡟⠁⣠⠞⠁⣠⠞⠁⣠⠖⠁⢀⠔⠁⢀⡴⠋⢀⡴⠋⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⡠⠴⠚⢧⣞⠁⣠⠞⠁⢠⠜⠁⢀⠔⠁⢀⡴⠋⢀⡴⠋⢀⣴⡯⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⣀⠀⠀⠙⠒⠲⠦⠬⠽⠷⠶⠶⠧⠤⠴⣥⣤⣴⣯⢤⡴⠯⠶⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠉⠙⣲⡖⠦⠤⠤⣤⡤⠤⠤⠤⣶⡖⠒⠋⠉⣠⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡀⠀⠓⠃⠀⡀⠀⠓⠃⠀⣀⠀⠓⠃⠀⣀⠀⠓⢻⡷⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣼⡽⠀⠀⠀⠸⠽⠀⠀⠀⠸⠽⠂⠀⠀⠸⠽⢀⣠⠞⠧⣄⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢲⠶⣤⡴⠾⠦⠤⠤⠴⠾⠶⠾⠯⠥⠴⠖⠒⠋⠀⢀⣠⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠉⠿⠶⠖⢶⡴⠤⠤⠤⠤⠤⡴⠶⠲⠶⠿⠉⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣿⠂⠀⠀⠐⣏⣹⠂⠀⠀⠐⣿⣹⠇⠀⠀⠀⢻⣉⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣍⣀⣀⣤⡀⠁⠈⠀⣀⣤⣀⠉⠈⠀⢀⣴⣀⣈⣠⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠲⠦⠤⠽⠶⠯⠤⠴⠖⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢀⣠⣴⣿⣿⡀⠀⠀⠀⢸⡁⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠻⣿⢷⣷⣿⣿⠓⠀⠀⠀⠀⢀⣴⣶⣶⣄⠀⠀⠰⣿⣿
⠀⠀⠙⢿⣿⣿⣿⡧⠀⠀⠮⠘⡇⢸⠀⢰⢀⠄⡄⢠⢀⠄⢰⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢀⡟⠁
⠀⠀⠀⠀⠛⠻⣿⡗⠀⠀⠀⠀⡇⠀⠑⡈⢉⠁⠃⠺⠈⠂⡸⡀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣯⠀⠀⠀⠀⠀⠙⠿⡿⠿⠃⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠁⡇⠀⡇⡜⢀⠀⡀⡀⢸⠁⠀⠀⠀⠀⠀⣀⢿⣿⡻⣿⡿⢄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⡇⢸⠀⠇⠘⠐⠻⠄⢯⡮⠀⠀⠀⠀⠀⠙⠓⠶⠷⠶⠷⠛⠀⠀⢈⠇⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⢀⡀⠀⠀⠀⠀⢹⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢀⣀⣤⠤⠆⠀⠀⣠⠄⠒⠈⣁⣤⣤⣤⣄⡀⠀⠀⠁
⠀⠀⠀⠘⡟⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡶⠟⠛⠉⠀⠀⠀⠀⠀⠃⠀⠀⣾⣿⣿⣿⣿⣿⣷
⠀⠀⣤⠘⣷⠸⢀⠀⠀⡟⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠃
⠀⢠⣸⡄⣿⢸⣿⢀⢴⡁⣀⠀⠰⣿⣿⡟⠟⠋⠉⠀⠀⣤⣶⡤⣾⣷⢠⣶⣤⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠀⠀⢸⣷⣾⡗
⠀⠀⣿⣇⢿⣼⣿⡙⡜⢻⣠⠇⠀⠀⠁⠀⠀⠀⣴⣿⣧⣽⣿⣿⣿⣿⣿⣷⣯⣼⣿⣦⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠠⠀⠀⠀⠻⠏
⠀⠀⠘⣿⣸⣿⣟⣧⣧⡎⠀⠀⠀⠀⠀⠀⠀⣀⣙⣿⣿⣤⣍⠻⣿⢹⡝⣿⢏⣿⣿⣋⣀⠀⠀⠀⣀⡀⠀⢠
⠀⠀⠀⢿⣿⣿⣿⣿⣿⠁⠀⠀⢀⡈⠑⠀⠀⣿⣿⣿⣿⣿⣿⡇⣿⢸⣷⡈⣾⣿⣿⣿⣿⠀⢠⠞⠉⠉⣷⠀⢀⣠⣤⡈⠀⠀⠀⠀⠁
⠀⠀⠀⢸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠹⣶⠀⠀⠈⣩⣿⣿⡉⣋⣵⣿⢸⣿⣇⣿⣿⣿⣍⠉⠀⣟⠀⠀⠀⣾⢠⡟⠁⠀⠹⡆
⠀⠀⠀⠘⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣘⠟⠀⠀⠻⣿⡟⣻⢿⣿⣿⣿⣿⣿⣟⢻⣿⠟⠀⠀⢻⣄⠀⣰⡇⣾⠀⠀⠀⣸⠇⠀⢀⣰⠆
⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⢠⣤⠻⠃⠉⠀⠀⠀⠀⠀⠀⠛⠿⠃⢿⡿⠚⠿⠛⠀⠀⠀⠀⠀⠀⠉⢻⣿⣰⣿⣤⣤⡾⠋⠀⢠⣿⣿⠃
⠀⠀⠀⠀⠛⠛⠛⠛⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⢸⢿⡿⠋⠈⠁⠀⠀⢠⣿⣿⡏
⠀⠀⠀⠀⣴⣶⣶⣦⠀⠀⠙⣶⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣶⣶⡄⠀⠀⡶⡶⠀⠀⠀⠀⠀⢀⣿⣿⠟
⠀⠀⠀⠀⣿⣿⣿⣿⠄⠀⠀⠀⠁⠀⢂⠀⠀⠰⡶⢸⡆⠀⢀⡅⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⠁⠀⠀⠀⠀⢀⣾⣿⡏
⠀⠀⠀⣠⣾⣿⣿⣷⣤⠀⠀⠈⠁⠀⠈⢣⠘⡀⠠⠈⡇⠀⢸⠁⠀⠀⠀⠙⠿⣿⣿⡿⠟⠀⠀⢸⠇⠀⠀⠀⠀⢀⣾⣿⡟
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠣⠐⠀⠀⠃⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣡⣴⣾⣿⣿⣶⣦⣀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣴⠀⠀⢀⢀⠀⠚⠀⠀⡏⡀⢀⠀⠀⠐⠂
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢻⠀⠉⣿⢿⠀⣇⣾⢸⠡⡇⢿⠱⡏
⠀⠀⠀⠙⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡴⠶⠶⠶⠶⢶⣤⣶⣻⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡛⢶⣏⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⢷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠿⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠉⠙⠛⠳⠶⠶⠦⣤⣤⣤⣄⣀⠀⠀⣾⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⣤⡏⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⡷⠦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠲⠶⠶⠀⠀⣸⠀⢠⡟⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠇⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣾⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡟⠀⠀⢸⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⡇⠀⠀⣼⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⢦⣤⣀⡀⠀⠀⠀⠀⢸⠁⠀⢸⡇⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠶⠶⢾⣤⣤⡾⠷⠞⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢀⣄⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⢄⡀⠀⢸⠁⠉⠢⣄⠀⢀⡞⠈⠣⣎⠀⠈⠙⠒⠤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠙⠦⣏⠀⠀⠀⠈⠓⢾⣀⠀⠀⠈⠑⠦⡀⠀⠀⠀⡠⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⡁⠢⣌⠓⠦⣄⡀⠀⠀⣸⠇⠀⠀⠀⣠⠋⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⠑⠤⡑⠢⠀⡽⠀⠀⣿⠁⠀⠀⢰⠃⠀⠀⡟⠢⠤⠤⡤⠤⠴⠒⠉⠢⡀
⠀⠀⠀⠀⠀⢸⣯⣉⠉⠉⠙⢦⡑⠢⢌⠓⠀⣿⢃⠀⠘⣆⠀⠀⢸⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⣤
⠀⠀⠀⠀⠀⠨⣿⣿⢿⣶⣤⣀⠉⠲⢤⣙⠂⠙⣷⡀⠀⠈⠳⣅⠈⢧⠀⠀⢻⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⢥⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢞⠿⣿⣿⣦⣄⠈⠙⠶⢬⣙⣦⣄⣀⠈⠳⢤⣓⢦⣀⠑⢤⣀⡀⣀⠀⠀⣀⣠⣤⣶⠿⣛⣛⣻⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⢟⢝⣿⡻⣦⡀⢀⣠⠬⢭⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣛⣿⣿⣟⣻⣿⣿⣟⣛⣯⡭⠭⠭⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢭⡻⣮⣻⣟⡮⠭⢟⢛⣛⣿⣿⣯⣽⣿⣿⣯⡭⠭⠽⠓⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢗⡶⠶⠟⠛⠋⠉⠉⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣟⣝⣻⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⡏⢉⣹⣾⢋⡻⣦⡶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢾⣧⣴⡿⠻⠶⠟⠛⠷⠾⢿⣔⣶⡿⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢻⣷⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢦⣷⡿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⡶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡩⡍⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣥⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣔⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡟⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢠⠹⣏⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⠏⠆⣿
⠀⠀⠀⠀⠀⠀⢰⡏⠄⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣼⠏⠀⠰⢹⡆
⠀⠀⠀⠀⠀⠀⣼⠃⠄⠀⠀⠈⢷⣍⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣢⡾⠁⠀⠀⠀⠜⣧
⠀⠀⠀⠀⠀⠀⣿⠐⠀⠀⠀⠀⠀⠙⠷⣤⡂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣤⠾⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣮⣄⣂⡀⠄⣶⣿⣿⣷⠤⢀⣂⣨⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠠⢹⡇
⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣿⣿⣿⣿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣿⠁
⠀⠀⠀⠀⠀⠀⠸⣇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠙⣧⡑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢐⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⣧⣤⣥⣤⣄⣐⣀⣀⣀⣒⣀⣐⣀⣀⣁⣀⣈⣤⣬⣥⡴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⣋⣉⣤⣤⣤⣬⣍⣛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢋⣡⡶⠛⠉⠉⠀⠀⠀⠀⠈⠙⢳⣌⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢋⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢃⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢡⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢁⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠷⠾⠷⠶⠶⠶⠶⠤⠤⠤⣤⣤⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⣾⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠶⣤⡿⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⠇⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠙⣻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣱⠿⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠸⣷⣦⣤⣤⣴⠏⠀⠀⢀⣀⣀⣀⡀⠀⠀⢀⡾⣱⡏⢀⡾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠀⡿⠀⠀⠀⢀⡾⠉⠈⠉⢉⡿⠀⣼⢣⣿⣄⣾⢣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣭⣶⣶⣶⣖⣢⣤⢤⡿⣀⢸⣇⠀⠀⠀⠾⠧⣤⣤⣤⡾⢁⣼⠇⠸⢧⣿⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠉⠉⠉⣩⣿⣇⣀⣼⡿⠿⠿⣿⣿⣾⣾⣿⣿⠿⢿⡏⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠐⣿⣬⣿⣿⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⢀⡼⢋⠘⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣰⡟⠁⣩⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⣿⣍⡒⠦⢄⣀⡀⠈⠛⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⡀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡛⠓⠶⢯⣭⣟⣒⣒⣒⣒⣲⡶⠤⠤⠤⢤⣤⣴⠟⠀⢀⣼⠃⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣤⣀⣀⡉⠉⠉⠉⠉⠉⠛⠛⠛⣻⣭⣥⣤⡴⠟⠁⣠⡟⠁
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠳⠶⠶⠶⠶⠶⠶⠦⢤⣤⡴⠶⠶⠛⠉
⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⡠⢀⠄⢠⢎⢀⣀⢠⢌⡀⠀⠀⠀⠀⠀⠀⣀⠀⠑
⠀⠀⠀⠀⠘⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣤⡜⠈⠁⠁⠉⠁⠉⠈⠈⠉⠀⠈⠝⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠠⠆⠘⠃⢠⡀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁⠁⠀⠐
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀
⠀⠀⢠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⡀⡄⣀
⠀⠀⢸⡇⠀⠠⠤⠀⠴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠺⠁⠀⠀⠀⠀⠀⠄⠀⠤
⠀⠀⠘⠃⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠃⠀⠀⠀⢀⣀⣤⢤⠤⠄⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠁⠀⠋⠀⠀⠀⠰⠆
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⢸⡄⠀⠀⠀⠀⠀⠀⡤⡄⠈⢻⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠈⠀⠠⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣴⡄
⠀⠀⠀⢠⠃⠀⠋⠉⠃⠀⠀⠀⠀⠀⢀⠀⣀⡀⠀⠀⠂⠀⠀⠀⣤⠶⠃⠀⠀⠀⠆⠀⠀⠀⠀⢀⣀
⠀⠀⢀⠁⠀⠀⠀⣀⣈⠀⠀⠁⠀⠀⠀⠀⠀⢀⣽⣿⡯⠀⠀⠀⠁⠀⠀⢀⣀⡀⢀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⢳⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢶⠦⠄⢁⡀⠀⠀⠀⠀⠀⠈
⠀⠀⢸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠄⠀⠀⠠⠖⠀⢋⠁⠉⢁⠀⡤⠀⠀⠀⠀⢴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⣴⠃⡇⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐

⠀⣀⠄

⠀⠀⠀⠈⠀⠀⠀⣸⠂⠀⠀⠉⠁⠀⠀⢠⠀⠀⢸⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⡿⠀⠀⣤⡀⠀⠀⠀⠃⡀
⠀⠀⠀⣆⠀⠀⠀⣿⠀⠀⠀⣀⡀⠀⠀⢹⠀⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠃⠀⡇⠀⠀⣿⣿⠇⠀⠀⠸⠟⠇
⠀⢀⣼⣿⣦⠀⠀⠙⠀⠀⢰⠟⠃⠀⠀⠿⠀⠀⠘⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⡇⠀⠀⠛⠛
⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⠇⠀⠀⠓⠂⠀⠀⠀⠀⠈⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠶⠟⠀⠠⠆


⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⡇⠀⢻⡆⠀⡀⠀⠀⠀⠴⣤⠀⢠⣶⣆⠀⠀⠀⠀⠀⠀⠀⠈⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢳
⡇⠀⢸⡇⠀⢿⣿⠀⠀⣄⠌⠀⢸⣿⠿⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠂⠀⠈⠑
⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠛⠈⢄⠀⠀⢰⠀⠀⠀⡆⢀⣴⣶⣶⡒⢲⡄⠀⠛⢿⡶⠚⠁
⡇⠀⠈⠀⠀⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠚⠀⠀⢸⠀⠀⠀⡇⠀⢿⣿⣿⣿⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⢰
⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡉⠁⠀⠀⢸⠀⠀⠀⡇⠀⠳⠿⠿⠿⠿⠃⠀⠀⢸⣷⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⠍⠲⡲⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡭⣿⣝⢟⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡠⠤⣄⡀⢸⢸⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢟⣛⢿⢿⡧⡇⣿⣿⢸⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢸⠀⢸⢸⣿⣿⣿⣿⣿⣿⡏⢹⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣷⣿⡇⡏⠀⠀⠀⠀⠀⠀⢰⡞⣿⣿⣿⣿
⠀⠀⠀⢰⣶⣾⣿⣶⣾⣿⣿⢹⣿⣿⣿⣿⣷⠘⡀⠀⠀⠀⢸⣿⣿⣿⡏⠉⠁⠁⠉⠁⠁⠀⡆⠀⠀⠀⠀⢸⣧⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠀⡇⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣿⢹⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡀⢿⣿⣿⣿⣿⠀⢣⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⣿⣾⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡆⢸⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⠿⠿⠿⠿⠧⠞⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣀⣀⣀⣀⣛⣯⣿⣿⣿
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠠⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠔⠒⠈⠉⠉⠐⠒⠢⡙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠒⠓⠓⠒⠀⠤⠤⠤⠤⠤⠤⠼⠼⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢸⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠜⡙⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡆⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠃⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⠞⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠑⢄⠑⠤⣀⣀⡀⠀⠀⣀⠔⠊⡠⠖⠁⠀⠀⠀⡇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠉⠒⠾⣭⠭⠭⠥⠔⠂⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⠚⠉⠀⠀⠉⠉⠒⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⡆⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄⠀⠀⠀⠀⠀⢠⢡⠒⡝⣼
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⡈⠇⠀⡇⡿
⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⡟⠀⠀⠁⡇
⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠀⡇⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⡇⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢱⠸⡈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡑⠤⣉⠢⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢸⠀⡄⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⢍⣒⡒⠤⢈⣁⡒⠒⠠⠄⣀⣀⡠⠎⢰⠀⠘⠀⠇⠀⠀⣠⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠋⠥⠶⠔⣒⡒⠒⠁⠀⠀⢇⠈⠢⢒⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠚⠒⠒⠉

⠀⠀⠀⠀⠛⠛⠛⠓⠂
⠀⠀⠀⠀⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠈
⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⠊⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⣄
⠀⠀⠀⠀⡇⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢧⠴⠋⠀⠀⠐⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠲⢦⢤⣄⣀⠀⣸⠁
⢠⠴⠒⠒⠒⠒⠚⠛⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠈⣆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⢄
⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂
⡇⠀⠀⠀⡟⠆⠀⠤⠤⠀⠀⠐⠁⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⡇⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠂⠤⠤⢤⣀⠀⢀⠀⠀⢀
⡇⠀⠀⢠⣧⡀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠁
⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⡠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠤⢰⡏⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡠
⢀⠀⠀⢸⡟⠦⠤⣒⣒⣨⡥⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠔⠀⠀⠀⠀⠀⠉⠒⠲⠤⠔⠁⠸⣀⠀⠄⠀⠀⠀⡆⢧
⠀⠀⠀⠀⠙⠿⠛⠉⠉⠉⠀⣠⡤⠶⣦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠈⠉⠒⠒⠲⠾⣅⡼
⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⠿⠴⠟⢶⠗⠀⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⢉⣉⣉⣇⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢤⢄⠀⡀⢢⣀⣠⣿⠤⠤⠤⠒⢳⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⢳⣡⠸⡀⠀⠻⠀⠀⠀⠐⠂⠈⠀⠈⡇⠀⠀⠀⢰⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠈⣇⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣀⡠⠤⠄⠒⠀⠁⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⡴⠚⣉⡿⣏⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⡾⠛⠉⠙⢦⠀⠈⢻⡀⢸⠃⠀⢸⡀⠀⢳
⠀⠀⠀⠀⠀⣠⠞⢉⠞⠁⠀⠀⢀⣠⣼⠶⠶⠞⠛⠋⠉⠉⠉⠉⠉⠙⠛⠲⢤⣀
⠀⠀⠀⢠⠞⠁⣰⠋⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⣰⠋⢀⡾⠁⠀⠀⣼⡷⠛⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⢰⠇⠀⣾⠁⠀⠀⣸⠏⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⡏⠀⢸⠇⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⢸⠃⢀⣿⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⣾⠀⢠⡇⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⡏⠀⢸⡇⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⡇⠀⠈⡇⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠁⠀⠀⠀⠀⣀⡤⠤⡄⠀⠀⠀⢀⡏⢷
⡇⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡄⠀⠀⠀⠀⠐⡇⠀⠀⢿⠤⠤⠴⠞⠁⠸⡇
⣷⠀⠀⣧⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠦⣤⣠⡤⠖⢷⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻
⢹⠀⠀⢿⠀⠀⡿⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢸⠀⠀⠀⠀⠀⠀⠀⠘⠶⠒⠚⠀⣀⣠⣤⠤⢤⡘⡇
⠘⡇⠀⢸⡇⢰⡇⢸⠁⠈⠉⠉⠀⠀⠀⠀⡞⢸⡀⠀⢀⣀⣀⣠⣤⣤⠤⠶⠖⠛⠉⠀⠀⠀⠀⣧⢷
⠀⢷⠀⠈⣇⣼⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢸⡄
⠀⠘⡆⠀⢽⡏⠈⡟⠛⠛⠛⠛⠒⠶⠶⣤⣼⢻⡇⠀⢸⡇⠀⠀⢀⣐⣠⣤⠴⠶⠶⠒⠛⠛⠉⣇⢸⡇
⠀⠀⢹⡀⢸⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⣼⣾⢸⡇⠀⣾⠙⢶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡇
⠀⠀⠀⢧⣾⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡇⣿⢸⡇⠀⣿⠀⣸⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⠴⠶⢛⡾
⠀⠀⠀⠈⣿⠀⠸⣦⣀⣀⠀⠀⠀⠀⢀⡇⣹⢸⠃⠀⠙⠦⢯⡤⠶⠞⠛⠉⠉⠁⠀⣀⣀⡤⠶⠋
⠀⠀⠀⠀⠹⣄⡀⠀⠉⠉⠙⠛⠛⠒⠚⠓⠋⡾⠀⠀⠀⠀⢀⣀⣠⣤⠴⠖⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠤⣤⣤⣤⣤⣾⡥⠴⠖⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀⣤⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⡙⡵⠛⠻⣿⡙⢮⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢱⠀⠀⠀⢣⢷⠀⠎⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⢸⢸⠀⠀⠀⠈⣸⠀⢠⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⢸⢀⠀⠀⠀⢀⣻⡄⢈⢰⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⢤⣤⠴⠒⢀⠈⠉⠉⠉⠉⠉⢹⡼⠈⠉⠁⠀⠀⠀⠈⣨⠃⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠉⣸⠀⠀⡇⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠺⢿⠗⠀⠀⠀⠀⢐⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡊⢽⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠐⠀⡀⠀⠀⠀⠀⠀⢰⠁⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠂⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⢄⠔⡠⡀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠪⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠃⢊⢀⠔⠀⠀⠀⠕⡁⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢪⠂⠇
⠀⠀⠀⠀⠀⠀⠀⣿⠐⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⡐⡄
⠀⠀⠀⠀⠀⠀⠀⡯⠐⢽⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡈⡐⡄⣔
⠀⠀⠀⠀⠀⠀⠀⡧⠀⣹⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢂⠎⠈⢀⢌⠔⠌⡴⠣
⠀⠀⠀⠀⠀⠀⠀⡇⠂⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡑⡡⠡⠀⡀⠠⠪⠀⢊⠎⠸⠀⡐⠇⠰
⠀⠀⠀⠀⠀⠀⢰⡇⠨⣺⠀⠀⢀⠄⠀⠀⠀⠀⠀⠊⠌⠜⠐⠁⡠⠔⡑⡑⠑⠡⠂⡊⠠⠀⠌⢀
⠀⠀⠀⠀⠀⠀⢸⢇⠄⣺⠀⢘⠑⡁⡁⡠⠠⢀⠄⡀⡠⠀⠊⢊⢊⢊⠊⠌⡠⡑⡁⠀⠐⡀⠄⢀
⠀⠀⠀⠀⠀⠀⢸⢕⠬⣺⠀⢸⠨⡪⢊⠔⡰⠡⢊⠈⠔⡑⡑⡑⡡⠢⢂⠄⠘⠔⡐⡀⠐⠀⠈⢀⠊
⠀⠀⠀⠀⠀⠀⠸⢵⣾⠿⣆⢸⠐⡔⡡⢊⠔⡁⡡⢂⢊⢌⢜⡔⡑⡡⢪⢊⢎⠌⡄⡰⠀⠄⢂⠠⡧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠛⠒⠒⠒⠒⠂⠀⠁⠣⠤⠬⠬⠌⠔⠃⠃⠐⠒⠒⠒⠒⠒⠒⠒⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⡀⢀⠄⠀⢀⠀⡠⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠈⢂⠄⡠⢐⠁⡀⢄⠈⢀⠅⠠⡈⠀⡁⡊⠔⡡⢂⠄⡨⢂⠔⡀⢀⠔⡠⢂⠔⡠⢀⠁⡀⠪⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠐⠁⠈⠀⠁⠈⠀⠁⠈⠀⠁⠈⠀⠁⠀⠁⠊⠀⠁⠈⠀⠁⠈⠀⠀⠀⠁

⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣿⣤⣶⡄⠀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣿⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣟⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⡟⠛⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣇⣠⣄⠀⠀⠠⠄⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⡏⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡗⢿⡿⠃⠀⠰⠦⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠴⠦⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⡀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡷⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡧⣾⣿⠄⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⢸⡿
⠀⠀⠀⠀⣿⣧⣶⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⡇⠈⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⣇⣴⣶⡀⠀⠈⠁⠈⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⡇⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢠⠀⠀⣿⣇⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⡏⠙⠛⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣼⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠠⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠄⠀⠀⣿⠇
⠀⠐⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⠀⢠⠀⠀⠙⠋⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⢠

⠀⠀⠀⠀⡤⡄⠐⠀⠂⠀⠀⠄⠠⠠⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠹⢀
⠀⠀⠀⠈⠀⢹⠀⠂⠐⠀⠆⠀⠀⠄⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠒⠀⠂⠰⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠐⠬⡔⢀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠂⢨⠂⠀⠠⠖⠄⠀⠂⠀⠀⠀⠀⠀⠀⠐⠠⠈⠀⠀⠀⠀⠀⠠⡇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀⡦⠤⠶⠄⠀⠂⠀⠀⠀⠀⠀⢀⠀⢀⠀⠠
⠀⠀⠀⣁⠉⠠⡠⠀⠐⠀⠐⠀⠀⠄⠀⠂⣀⠀⠐⠀⠀⢠⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⢠⠔
⠀⠀⠀⠙⡁⡄⠄⠀⠀⠀⡀⠀⠈⠄⠀⠈⠁⠀⡁⠈⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠙⠴⠀⡀
⠀⠀⠀⣤⠀⠁⠈⠀⠀⠀⠑⠂⠀⠄⠀⠀⠠⠱⡓⢤⠀⠀⢀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢠⠀⣠⡤
⠀⠀⠀⠈⠃⠀⢀⠀⠀⠀⠀⠄⠀⠀⠠⡄⠠⠀⠑⢎⠢⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠄⠀⠉⠉⠀⠄⢀⣣⠀⢄⠀⠀⠈⠀⠀⠀⠀⠀⣠⡄⡂
⠀⠀⣠⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠊⣀⠜⠀⠀⠠⠀⠀⠀⢀⠉⠠⢤⢻⠋⠀⢀⡀⢲
⠀⠀⠀⠒⢤⠀⠀⠇⠀⠀⠀⡁⠀⠀⠀⠁⠀⠀⠀⠀⠰⠭⠀⠀⠅⠀⠀⠀⠂⠀⠀⢸⠅⠀⠀⠁
⠀⠀⠂⠀⠀⢨⠀⠀⠀⠀⠰⠷⠤⠀⠀⢁⡀⠀⠀⠀⠀⠀⠀⢀⠐⠀⢀⣁⠡⠀⠀⠺⡄⡠⣄⣀⠀⠄
⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠐⠘⠀⠁⠁⠉⠀⠁⠀⠀⠀⠀⠀⠘⠙⠒⠠⠁⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣧⣉⣛⣫⢼⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠈⠷⣤⣀⣤⠞⠀⠈⢳⠀⠀⠀⠀⠀⠄⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢸⠀⢤⡀⠀⢀⡀⠀⠀⢀⠀⣸⠒⠂⠀⠁⠀⠀⠀⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠊⡤⡌⢣⠀⢸⡉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⠀⠀⠀⠀⢀⣄⣐⡁⠀⠈⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⡇⠸⡿⣾⠀⠀⣀⡀⡀⣀⡀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠉⠛⢿⡠⠁⠀⣀
⠀⠀⠀⠀⣀⡀⠀⢀⠐⣠⠞⠁⠸⣿⣿⠀⠀⠒⠀⠀⠁⠑⠀⠀⢸⠁⠀⠀⠀⠀⠀⣀⣤⣼⣷⣿⣿⣿⠷
⠀⠀⣞⠓⢢⠈⣁⡒⠆⢳⠀⠀⠀⠈⢻⠀⠀⠀⠈⢉⣁⠀⠀⢀⠼⢖⢒⣒⣛⠛⠿⣟⣩⠅⠐⣒⣐⣚
⠀⠀⠬⣁⣈⣠⠃⠉⠻⠈⣄⡀⠀⠀⠘⢆⡀⠀⠈⠉⠀⠀⢠⢋⢆⠀⡻⠿⠿⠚⠃⠀⡷⠾⠿⠿⠟⠁
⠀⣀⠄⢊⠁⠢⡀⠀⠀⠃⢱⠈⠙⠲⢤⣉⢫⠙⠓⠒⠒⢲⡯⢎⡲⡠⠄⢄⡲⠈⢁⡶⢹⣤⣤⡀
⠐⠁⠂⠄⢠⠁⠙⡄⠀⠈⡈⣇⡀⠀⢀⠀⣥⢀⡴⠛⡗⠀⣏⡏⣤⠖⡆⡈⢶⢼⣿⣿⡾⠿⠿⠃
⢀⠘⠀⠒⠀⠊⡐⣹⣄⠀⠁⡎⠀⣠⡶⣳⣌⠋⠁⠀⢛⡉⢻⣷⠶⣚⣩⠅⡾⣾⣿⣿⠃⢀⡀⣶⡶
⠀⠙⢤⠀⠐⣪⠰⡿⢣⣧⣶⣷⣾⣿⡿⠋⠀⠀⠒⠲⡜⠀⡏⠙⠻⠿⣿⣿⢿⣿⣿⣁⣴⡅⠨
⠀⠀⠀⠻⡚⠊⠁⠀⢺⣿⣿⡟⠁⠀⠈⣀⣂⣠⡿⡂⡇⠀⡟⢒⡤⢔⡼⠚⠙⠋⠄⡧⠾⠃
⠀⠀⠀⠰⣿⡦⠤⠖⠛⠻⢿⠷⣗⣂⣠⠯⠙⣽⣄⠈⣇⠠⢛⣥⠾⠋⠉⠒⠦⢄⣀⣼
⠀⠀⠀⠀⠀⠀⠀⣠⣄⡘⣀⣀⡠⠿⣄⣠⣶⠏⡹⣫⡵⠊⡏⠁⠀⠀⠁⢠⣀⣼⣏⡴
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡗⠀⠉⠳⠤⢖⣡⣲⠟⠋⢹⠀⠀⣧⡀⠀⣶⠴⠞⠛⠛⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠠⡀⢠⡟⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠉⠀⠈⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠉⠀⠀⣀⠤⠖⠊⠁⢀⠀⠀⠈⠑⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠚⠉⠁⠀⢀⣠⠤⠖⠉⠀⣀⠀⠀⠚⠉⢀⣠⠄⠀⠀⠀⠉⠓⠤⣀⡀
⠀⠀⠀⠀⠀⠀⣀⡤⠖⠘⠉⠀⠀⠀⣀⡤⠖⠋⠁⢀⣀⠤⠖⠋⠁⣀⡤⠴⠊⢉⣀⠤⠖⠃⠀⢀⡄⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⣻⠦⣄⡀⠀⠀⠀⠈⠉⠁⣀⡤⠖⠋⠉⢀⣠⠴⠚⠋⠁⠀⠴⠚⠉⠀⢀⡄⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⣽⣓⠦⣍⠲⢤⣀⠀⠀⠀⠀⠀⢀⠴⠚⠉⣀⠀⠰⠚⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⢰⠶⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⢄⣀
⠀⠀⠀⠀⠀⢚⣭⡑⠦⣝⠲⢬⡓⠦⡀⠀⠀⠀⠀⠀⠛⠉⠀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣦⡀
⠀⠀⠀⠀⠀⣹⠶⢬⡓⠦⣝⠲⢌⡓⢌⡓⠦⣀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⢀⠞⠉⠉⠉⠉⠉⠉⠀⢀⡠⠖⠋⣀⡤⠖⠀⠀⠀⠀⣀⡤⠖⣋⡷⠄
⠀⠀⠀⠀⠀⢼⣛⠲⢌⡓⠦⣙⠢⣍⠲⢌⡓⢬⡑⠢⢄⡀⠀⠀⠀⠙⠲⢄⣀⡜⠀⠀⠀⠀⢀⣠⠴⠚⣁⡤⠒⠋⠁⠀⠀⢀⡤⠖⣋⡥⠖⣋⠥⣟⡁
⠀⠀⠀⠀⠀⣺⡭⣙⠢⢌⡓⠬⣙⠢⣝⠢⣍⠲⢌⡑⠢⣉⡓⢤⡀⠀⠀⠀⠈⠙⠲⠤⠔⠊⠉⠠⠒⠋⠀⠀⠀⢀⡠⠔⢊⡡⠖⣋⠥⠒⣉⠤⠒⣯⠄
⠀⠀⠀⠀⠀⣽⠖⠦⣙⠢⢍⡓⠬⡑⠦⣑⠦⣍⠲⢬⡑⠦⣍⠲⠬⣑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢊⡡⠔⢊⡡⠖⣋⠤⠒⣉⠤⠒⣩⡷
⠀⠀⠀⠀⠀⠾⣋⡒⠬⣑⠢⢍⠒⢌⡓⢬⡑⠦⣙⠲⢌⡑⠦⣍⠒⠤⣙⠢⢌⡓⠦⣄⠀⢀⡠⠔⢊⡥⠔⢊⡡⠖⣋⡡⠖⣋⠤⠒⣩⠤⢚⡉⠴⢞⡁
⠀⠀⠀⠀⠀⣛⠤⣌⡲⠬⣑⠢⣍⠲⢍⠲⢌⡓⠤⣉⠲⢌⡑⠦⣉⠒⠬⣙⠢⢌⡑⠦⣉⠥⠔⣋⠥⠖⣊⡡⠔⣋⠥⠖⣉⠤⠒⣉⠴⢂⡩⠄⢚⡻⠄
⠀⠀⠀⠀⠀⢬⡓⠢⣍⠒⢬⡑⠦⣙⠢⣍⠲⢌⡓⠤⣉⠢⢌⡑⠦⣍⡒⠤⣑⠢⢌⡑⠤⠒⣉⠤⠒⣉⠥⠖⣉⠤⠒⣉⠤⠒⣉⠴⢊⡡⠔⢊⣩⣟⠂
⠀⠀⠀⠀⠀⢲⣍⡑⠦⣉⠲⢌⡑⠬⡑⠦⣙⠢⢍⡒⠤⣙⠢⢌⡑⠦⣍⠒⠤⣑⠢⢌⣓⣉⠤⠒⣉⠤⠒⣉⠤⠒⣉⠤⢚⡩⠴⢚⡡⠔⣊⡡⠴⣯⡀
⠀⠀⠀⠀⠀⣹⠦⢌⡓⠤⣉⠢⢌⡓⢬⡑⠤⣙⠢⢌⡓⠬⣑⠢⢌⡑⠦⣍⠒⠤⣑⠢⠬⠤⢒⣉⠤⢒⣉⠤⢒⣩⠴⢒⡩⠔⢊⡡⠔⣋⠥⠔⣋⡷
⠀⠀⠀⠀⠀⢼⣛⠢⢌⡑⠦⣙⠢⢍⠲⢌⡓⠬⡑⠢⢌⡓⠬⣙⠢⢌⡑⠦⣍⠒⠤⣑⠒⢊⡩⠴⢚⡡⠴⢂⡩⠴⢚⡡⠔⢊⡡⠔⣋⠤⠒⣉⠤⣟
⠀⠀⠀⠀⠀⢚⡭⣙⠢⢌⡑⠤⣙⠢⣝⠢⢍⡒⢬⡑⠢⢍⡒⠤⣙⠢⢌⡑⠦⣉⠒⠬⣉⡡⠔⢊⡡⠔⢊⡡⠔⢊⡡⠔⣊⡡⠔⣋⠤⢒⣉⠤⠒⣫⠄
⠀⠀⠀⠀⠀⣭⠖⠤⣙⠢⢌⡓⢬⡑⠦⣑⠢⣍⠲⢌⣑⠦⣍⠒⠬⣙⠢⢌⡑⠦⣍⠒⠤⠔⢊⡡⠔⢊⡡⠔⢊⡡⠔⣊⡥⠖⣋⠥⢒⡉⠤⢒⣉⡷⠂
⠀⠀⠀⠀⠀⢾⣋⡒⢬⣑⠢⢍⡒⢬⡓⢬⣑⠦⣉⠲⢬⡑⠦⣍⠒⠬⣙⠲⢌⡓⠦⣉⠒⣋⡥⠔⣊⡡⠔⣊⡡⠖⣋⡥⠖⣉⡥⢒⣩⠴⢚⡩⠴⣯⡁
⠀⠀⠀⠀⠀⣻⠤⣍⡳⢬⣙⠢⣍⠲⢍⡒⢬⡑⠮⣙⠲⢌⡓⠦⣉⡒⢬⣙⠢⢍⡓⠦⣉⡥⠖⣋⠥⠖⣋⡥⠖⣋⡥⠒⣉⠤⢒⣩⠴⢊⡩⠴⢚⡣⠄
⠀⠀⠀⠀⠀⣽⡓⠢⣍⠒⢬⣑⠦⣙⠢⣍⠲⢌⡓⠤⣙⠲⢌⡓⠦⣍⡓⠬⣙⠢⢍⡓⠦⠒⣋⠥⠒⣋⡥⠖⣋⠥⢒⣉⠤⢚⣩⠴⢚⡩⠴⣚⣩⣟⠂
⠀⠀⠀⠀⠀⠀⠉⠓⠦⣙⡲⢌⡑⠬⡑⠦⣙⠢⣍⡓⠤⣙⠢⢍⡓⠦⣍⡓⢬⣙⠲⢍⣛⣉⠥⢒⣉⠤⢒⣉⠥⢚⣉⠴⢚⣩⠴⢚⡩⠔⣊⡩⠴⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢍⡓⢌⡓⠬⣙⠢⣍⡓⢬⣙⠲⢍⡓⠦⣍⡓⢬⣙⠢⣭⠴⢚⣩⠵⢚⣩⠤⢚⣩⠴⢚⣩⠴⢚⡩⠖⣋⡥⠖⣛⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢌⡓⢬⣙⠦⣍⡓⢬⣙⠢⢍⡓⠦⣍⠒⢬⣙⠲⢚⣩⠴⢚⣩⠴⢚⣩⠴⢚⣩⠴⣚⡩⠔⣋⡥⠖⣋⡥⠟⠂
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢬⡓⠢⣍⡳⢬⣙⠢⢍⡓⠦⣉⡓⢬⣙⡩⠴⢚⣩⠴⢚⣩⠴⢚⡩⠴⢋⡡⠔⣋⡥⠖⠋⠁
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣍⠒⢬⣙⠢⢌⡓⠦⣉⠲⠤⠔⣊⡩⠔⢚⡩⠔⢚⡡⠔⣊⡥⠔⠋⠁
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢬⣑⠢⢍⡒⠦⣙⠒⢊⡡⠔⢊⡡⠔⢊⡡⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢌⡓⠤⣉⡥⠔⢊⡡⠖⠊⠁
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠖⠊⠁
⠀⢰
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⣂⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡡⠊⠁⠀⠉⢆⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠈⠉⠉⠳⣍⠉⠒⣄⠀⠀⢸⠀⡇⠀⠀⠀⠀⢨⠀⠇⠀⠀⢀⠔⠚⣩⢝⡉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⣠⠖⠉⠀⠈⠆⠀⠘⡆⠀⢀⣃⣧⣀⣀⣀⣀⡎⡜⠀⢀⡔⠁⠀⡔⠁⠀⠈⢢⡀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢀⠜⠁⠀⠀⠀⠀⢸⠤⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⢎⡀⠀⡜⠀⠀⠀⠀⠀⢱⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢀⠊⠀⠀⠀⠀⣠⠚⠁⠀⠀⢀⡠⠤⠐⠒⠒⠂⠤⣀⡀⠀⠀⠀⠉⢲⡁⠀⠀⠀⠀⠀⠀⠇⠀⠀⢰
⠀⠀⠀⠀⠀⢰⠁⠀⠀⢀⠎⠀⠀⠀⠀⡔⠁⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠸⠀⠀⠈⡄
⠀⠀⠀⠀⠀⡆⠀⠀⠀⡜⠀⠀⠀⠀⡜⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⡆⠀⠀⠃
⠀⠀⠀⠀⢠⠁⠀⠀⢠⠁⠀⠀⠀⢰⠇⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠸⡆⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠸⠀⠀⠀⡌⠀⠀⠀⠀⡞⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢻⠀⠀⠀⠀⠀⢡⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⢠⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡆⠀⠀⠀⠀⢰⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⡌⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢲⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠒⠒⠒⠒⠒⠒⠒⠒⠀⠈⠉⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⡄⠀⠀⠀⢠⠁⠀⠀⣀⠠⠤⠤⠤⠀⠀⠒⠒⠒⠒⠒⠂⠀⠀⠤⠤⠤⠤⠄⢀⣀⡀⠀⠀⡆⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⡇⠀⢀⠴⠾⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠗⠒⡄⠀⡇⠀⠀⠸
⠀⠀⠀⠀⡇⠀⠀⡇⠀⡜⠀⡆⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⠀⠁⠀⡇⠀⠀⡆
⠀⠀⠀⠀⠃⠀⠀⠃⠀⡇⠀⡃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠄⢰⠀⠀⢰⠁
⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇⠀⠁⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡇⠀⢸⠀⠇⡘⠀⠀⡸
⠀⠀⠀⠀⠈⡄⠀⠘⡄⡇⠀⠀⠀⢸⠀⠉⠉⠈⠉⠉⠁⠀⠀⠐⡖⠒⠒⢐⡆⠈⠉⠉⠉⠉⠉⠉⠀⠀⠃⠀⢸⠀⣷⠁⠀⢠⠃
⠀⠀⠀⠀⠀⢃⠀⠀⠱⡇⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⣠⠃⠀⢀⠏
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠱⣄⡇⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠘⠀⢀⡰⠋
⠀⠀⠀⠀⠀⠀⠈⠣⣄⠀⠀⢃⠀⠀⠀⠉⠒⠒⠂⠤⠤⠤⠤⠤⠤⠤⠤⠠⠤⠤⠤⠤⠤⠤⠄⠒⠊⠁⠀⠀⡦⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠀⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⡜⣁⡴⠋⠙⢦⣌⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⣤⣤⣬⡙⢳⡏⠀⠀⠀⠈⢯⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣠⠞⠁⠀⠀⠛⣆⢳⠀⠀⠀⠀⠸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢬⣧⣤⣄⣀⣸⣐⣿⠀⡇⠀⠀⠀⠀⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⡟⠐⣾⣿⣿⢸⣻⣿⣿⣏⡆⢷⣤⣄⣀⠀⣽⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢯⣯⣝⣻⠁⢽⣿⣿⣿⡏⣿⣻⣿⣿⡇⢸⡿⣿⣻⣯⣿⣦⢧⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠖⢃⡠⠚⣿⠀⣲⠿⡿⣝⣛⡿⣿⣿⣟⡄⢸⣿⣯⣿⣟⣿⣽⢿⣷⣿⣿⣗⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡖⠉⣣⡔⡏⢀⣮⣀⠼⠃⢈⢿⠓⢤⣍⡇⢸⣿⣿⣿⣿⣾⣿⣯⢻⡿⣿⡿⠷⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠐⠌⠀⢰⣿⢰⡟⢁⡠⠛⠄⢀⢶⠂⠈⡇⢸⣷⠮⢭⡛⡟⠓⠚⠒⠛⠉⠁⠄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⡰⡖⠈⢘⡿⠓⠉⢄⡠⠚⢅⠀⢠⢀⡧⣨⢀⠴⡊⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠫⣀⡽⡞⠁⢈⣤⠚⠉⢃⡠⠜⠣⠀⢸⣷⠞⢧⣠⠜⠻⠀⢀⡜⠀⠀⠀⢠⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠂⠁⣀⠼⢞⠁⢈⢦⠔⠉⠁⣀⠄⠻⡁⣈⢶⠊⠑⣤⠾⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠴⡎⠑⣀⠌⢲⠁⠀⣱⡔⠊⠑⣄⠴⠛⠅⢀⡱⢞⠈⣰⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡱⡤⠰⡏⠈⣀⡠⠳⠉⢀⡨⢦⠊⠈⣢⡀⠘⠫⣀⠼⠻⣀⡀⠀⣼⠀⠀⠀⠀⡕⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣨⢦⠚⠋⠀⣣⡄⠘⢇⢀⣨⢷⡈⠀⣀⣤⠊⠑⣤⠖⠉⡇⠀⡡⠀⠀⠀⠀⠙⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢷⡞⠻⡀⠙⢦⠚⠣⣀⡠⠴⣎⢀⡠⠷⡁⢀⠳⣌⠃⠀⢡⠂⠀⠀⠠⠎⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠦⠠⣷⡊⠙⣦⡤⠊⢣⣀⠔⠚⣣⡔⠋⣻⡀⠀⠆⠀⠀⠀⠈⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠂⠵⣖⣉⡥⠖⣋⡠⠜⠻⣀⣅⠀⠐⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠿⣤⣔⡊⢉⣻⣄⠠⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡍⠀⠈⠛⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣇⠃⠀⠀⠀⠀⠈⠙⢿⣶⣶⣶⠶⣖⣛⣛⣛⣛⠛⢿⡿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣾⡘⠉⠑⡀⠀⠀⣀⡀⢀⡇⠀⣀⠀⡸⠀⠀⡼⠸⠀⢰⠀⠀⠈⡙⠛⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣅⣀⡀⠈⠢⣀⠅⠀⢰⠀⠀⠀⠀⠀⠀⢸⠀⣇⠎⡝⠐⠒⠒⠁⠀⢠⠁⢸⣛⢿⡷
⠀⠀⠀⠀⠀⠀⣼⡟⠂⢤⣈⡆⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⡠⢃⠔⡄⢀⠁⠀⡔⠐⢢⠀⠀⠙⠙⢆⣸⠇
⠀⠀⠀⠀⠀⢀⣿⠦⡀⠀⠈⠀⡰⠑⢂⢀⠃⠀⠀⠀⠀⠑⢄⠓⠒⠁⠘⠀⡌⠀⠀⠸⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠣⡀⠀⠠⢃⠦⠈⢸⡂⠀⠀⠀⠀⠀⠀⠣⠀⠀⡆⠘⠄⣢⡔⠐⠁⠰⢖⠢⣰⡇
⠀⠀⠀⠀⠀⠙⠿⣵⣤⡉⠀⠁⠘⠃⠀⣆⣑⢄⡠⠊⠑⠄⠀⢘⠀⢠⠁⠀⠠⠔⢢⠀⠀⡀⢨⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⡤⡀⠀⠀⠯⠜⠀⠀⠀⢠⠁⢠⠃⠀⠸⠀⠀⠱⡀⠘⠤⠊⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣼⣀⢀⠀⠠⠤⠤⢷⣧⣤⣤⡦⣀⠀⡠⠃⠀⣆⡰⠀⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠙⠛⠾⣷⣶⣤⣤⡖⠀⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠙⠻⢿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⢶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠐⠀⠀⢀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⡀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠤⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⢿⣿⣿⣟⣿⣿⣿⣿⡿⠏⠡⠙⢿⣿⣿⢿⠿⠿⠟⠂⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣷⣟⣿⣿⣿⣿⣶⣿⡿⠛⠀⠈⠀⠀⣼⣶⣦⠰⠀⣁⡡⠀⠻⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢂⡚⠿⣯⣿⣿⣿⣿⡿⠛⢛⣓⣿⣿⣦⠀⠀⣿⣭⣭⣬⣶⣶⣶⣿⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⡇⠀⠀⠐⣊⠙⢧⡐⠄⡈⠠⡄⠉⡷⣍⠉⠛⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⣾⣿⣏⠉⠃⣔⠁⣲⣀⠹⠀⣸⣦⡀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⠁⠀⠈⠉⢉⣴⣶⣔⡀⠱⢀⡘⢀⣼⣿⣿⣿⣶⣦⣄⠀⠉⠙⠛⠿⠿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠃⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣦⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠈⠉⠓⠂⠤⢄⣀
⠀⣀⣀⠤⠤⠒⠒⠉⠉⠀⠀⣀⠤⠒⠉⠉⠻⠿⠿⠿⠿⠿⠿⣿⢿⠿⠿⠿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠷⠤⣀
⠈⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠶⢦⣤⢠⠀⠼⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀
⠀⠀⠀⢀⡠⠔⡊⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠘⠲⢦⢌⡧⣄⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀
⢠⠴⠚⠁⠀⢀⣠⣤⡀⠀⠀⢀⡀⠀⢠⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈
⠀⠀⠀⣠⡞⠉⠀⠀⣸⣷⡶⢟⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠓⢤⣄⡀⠀⠰⠆⠈
⠀⠀⠀⠻⣷⣤⡤⣾⣿⣷⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠓⠤⣄⡀
⠀⠐⠀⠀⠀⢠⡾⠋⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠠⠀⠸⣷⣤⡤⠖⠋⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⢈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠀⠀⠀⠨⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣯⣿⣿⣽⣯⣭⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣽⣿⣭⣭⣭⣽

⠀⠀⠀⡀⠀⡀⢀⠀⢀⡀⢀⠀⢀⠀⠀⣀⣤⣤⣄⡀⢀⣀⣀⣀⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠐⣧⠀⣧⠀⡇⠀⠁⢸⠀⢸⠀⡼⠋⠀⠀⠀⠉⢉⠁⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⢠⠀⠀⠀⡔⠙
⠀⠀⢸⠀⠀⠙⣠⠇⢸⠄⣸⠀⡌⠀⠀⡀⠀⠀⠀⠁⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⢀⢀⠀⠀⠀⠀⠀⡏⠉⢹
⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⢀⠀⠠⡀⠀⠀⠀⣄⠀⠀⣄⠀⠈⠀⠀⢀⡀⣀⠀⡀⠀⣀⢀⣀⠔⡁⠀⠀⠀⡉⠂⠀⠰⠋⠉⠉⠴
⠀⢰⠇⠀⢈⣀⠀⢀⠀⠀⠈⠢⠀⠁⠺⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠻⠄⠿⠵⠛⠖⠋⠈⠇⠀⠁⠂⠁⠃⠩⠀⠈⠀⡀⢀⠀⠀⠉
⠀⠸⠀⠨⠉⠁⠀⠈⠨⠅⠂⠂⠐⠀⠠⠄⠀⠤⠄⠠⠠⠀⠤⠀⠀⡂⠀⠀⠀⠂⠂⠐⠐⢁⡀⠀⠲⠤⠖⠀⣠⠙⠀⢰⡂⠀⡀⢠
⠀⠀⠀⠀⠀⢙⢚⡁⠂⠀⠀⠀⠀⠀⠄⠀⡀⢰⠆⠰⠇⡀⢀⡤⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠅⠉⠀⣁⡔⠂⠀⠇⠀⠉⢉
⠀⠀⠀⠂⠀⠘⠐⠀⠂⠈⠀⠄⠀⠀⠀⢠⡄⠀⣀⡀⠀⠁⠈⠀⠀⣤⣤⣤⣬⣤⣥⡄⠀⠐⠄⠀⠂⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠁⢀⡸⡄⠐⢈⣆⡠⠀⠀⠠⠠⠀⠀⣹⠀⠿⠆⡤⠤⣄⣠⠤⣀⢠⡤⠤⢤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠄⠀⠄⠰⠀⠀⠀⢰
⠀⠀⢀⠀⢨⡀⠀⠀⡄⠀⠀⠀⠀⠠⠀⠀⣿⠀⠀⠀⡇⣤⡈⠁⣤⢸⡆⢹⠀⣿⣶⢰⡟⠁⣠⢶⣄⣴⣦⠀⠀⠀⠀⠀⠀⠀⠒⠘
⠀⠀⠁⠀⠰⠂⠀⠀⠂⠀⠀⢃⣀⣀⢈⡇⠀⢷⠀⠀⡇⣼⡇⠸⡿⢸⠇⢸⠀⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⢸⡁⢀⠀⠸⠀⠁⠰⡀⠀⢸⠀⡂⢸⠀⠀⠀⣿⡀⣇⣩⠴⣤⣠⠞⢰⣏⣀⣹⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀
⠀⠀⣀⡉⢠⠉⢀⢈⢀⠉⠀⠉⠉⠉⢈⡀⣀⣀⣬⢥⣄⣀⣀⣛⣠⣤⣀⠀⣀⣀⣈⣋⣀⣀⡀⣀⣈⣀⡀⠁⠁⡀⠀⠀⠀⠀⠨⠀⠨
⠀⠒⠈⠤⠤⠤⢹⡀⠰⠂⠀⡄⠐⠀⢯⢹⡏⢠⢀⣤⡀⠙⣿⠀⠀⣶⡌⠛⣥⡙⢠⣶⣿⢸⡇⡇⠲⢾⡇⠀⠀⠀⠠⢀⠀⠀⠠⡀
⠀⠀⣀⣀⣀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠈⢧⠁⣿⢸⣿⠃⠀⣿⠀⠀⡉⢶⣧⣟⢿⡄⣭⣿⢸⡇⡇⣿⠙⠃⠐⠀⠀⠂⠘⠂⡝⢽⡇
⠀⠀⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣄⣉⣴⡀⠛⢀⡀⣿⡈⣇⠛⢀⡄⣛⣿⣘⣶⣇⣻⠠⠮⠿⠧⠀⣠⠀⢳⡿⠾⠟
⠀⠀⢠⡴⠀⠀⠀⡀⡀⠀⠸⣿⣿⡇⠀⠘⠛⠃⠈⠉⠀⠙⠛⠉⠛⠻⠉⠛⡟⠋⢉⡩⢉⠉⠉⠙⠙⠀⢰⣀⠀⠰⠇⠀⠀⠀⠠⣀⠄
⠀⠀⠀⠁⠀⠀⠐⠒⠒⠀⠀⠀⡀⠀⠀⠀⠘⠈⠉⠉⠁⠀⡀⠀⠀⠀⠀⠀⠁⠀⢬⠀⡠⠃⢀⣀⠀⠀⠀⠟⠀⢶⣶⣶⡖⠀⠀⡯
⠀⠀⢁⢨⡤⠀⠄⠤⠤⠤⢬⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠓⠆⠀⣀⢱⠎⠀⠀⠁⣤⣾⣿⣿⣿⣿⠀⠀⠀⠈⠸⠏⡁⡄⠀⠟
⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⠀⡂⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡠⠀⢀⠁⠂⠈⠁⡀⠀⠠⡄
⠀⠀⠀⠀⠀⠟⡁⠀⠉⠃⠀⠃⠀⠀⡀⠀⢀⠀⠀⠀⠀⠰⠀⠀⠈⠀⠀⠀⠀⠮⠆⠭⠂⠀⠀⠂⠈⡇⠀⠁⠐⠀⠀⠄⠤⠈⠀⠁⡀
⠀⠀⠀⢠⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠨⡅⠈⠯⠀⠀⠀⠀⠀⠀⠀⢰⡂
⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠄⠀⢰⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⡟⠀⠀⠀⠀⠀⠀⠓⠀⠀⠴⠛⠂⠀⠛⠳⠆⠀⠁
⠀⠀⠐⠐⠃⠈⠀⠀⠀⠉⢙⠠⠄⠀⠸⠃⠀⡀⠐⠀⠰⠀⠀⠀⠸⠈⠀⠀⠀⠀⡇⢰⡋⢸⠀⢸⡅⣶⠚⠛⢲⠰⣤⠂⡖⠛⠋⢷
⠀⠀⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠉⠁⠀⠘⠁⠀⠀⠀⠈⠀⠉⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢒⣩⣭⢭⣽⣗⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣡⠞⢉⠔⣊⠥⠤⠤⠬⣭⡓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣰⠃⡴⣡⠊⠁⠀⠀⠀⠀⢸⡼⣧⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡏⢰⢱⠃⠀⠀⠀⠀⠀⠀⠀⡧⡇⢣⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡇⡌⡌⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣈⡮⣇⣀⣀⠠⣤⡀
⠀⠀⢀⣀⣤⢤⠤⠤⠤⡒⡒⡒⡛⣳⠇⣷⢈⢋⢍⠍⡭⡩⡩⡩⣡⣢⢄⡇⣗⡔⡶⠾⠿⣇⣀
⠀⠀⡌⢠⢍⢍⠩⣩⣿⢮⣪⢊⣼⢿⣗⣗⣕⣵⣣⣪⣪⣮⣾⣼⡿⡽⠵⡇⣿⠾⠷⠶⠒⠒⢒⡛⢳
⠀⠀⠇⠪⡐⢕⡟⢋⢉⢉⠉⠉⠉⣭⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣾⡷⡄⠀⠀⠀⠈⠅⢸
⠀⠀⠀⡑⠮⡲⡇⠈⠀⠀⠀⠀⠀⠺⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠽⠎⠀⠀⠀⠀⠀⠉⢸
⠀⢰⠀⠪⡑⢌⡇⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⠀⠀⠀⠀⠀⠀⠈
⠀⢸⠀⡑⢌⠢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⢘⠮⡢⣙⠡⡦⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⡇⠐⢕⢔⢾⢘⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⡇⣌⢄⠢⣽⠡⡢⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠄⡇
⠀⡁⢎⠢⡑⣾⠀⠀⠑⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⢎⢎⠞⡅
⠀⢅⠡⡑⢬⢻⢀⠀⠀⢊⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠪⢣⢪⢣⢪⢎⡇
⢰⠠⡑⢌⡃⢸⠰⡡⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡔⡕⡱⡱⡱⠡⢡⢣⢊⡇
⠸⢐⢌⢪⠁⠀⢈⠜⡠⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡠⠪⢂⢊⠎⠌⡔⡰⡱⡡⡣⡣⠣⣇⠀⣀⡀
⡄⠢⡡⡏⠀⡀⠠⢊⠔⡡⢂⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⢔⠌⠔⡑⡱⡡⢃⢊⠌⡜⡔⡕⡑⡑⡑⡱⣿⠤⠤⠌⠉⢐⣂⣐⣂
⢷⡚⠛⢣⡄⡇⠔⡡⢊⠔⡡⢢⢄⠄⡀⢀⢀⢔⠕⡑⡡⢣⢊⢜⠔⡔⡡⢣⢊⠎⢜⠜⠜⠜⠜⠴⠎⣿⣉⣉⣀⣀⣀⡐⠒
⠀⠈⠓⠦⣵⡇⠐⠱⠡⠊⠔⠱⠋⠊⠐⢑⣁⣊⣈⣈⠐⠡⠈⠭⠬⠄⠁⠒⠒⠒⠒⠒⠒⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⢀⣴⠒⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣄
⠀⠀⠀⠀⠀⡀⠤⠥⠤⠬⣙⡆⠀⠀⠀⠀⠀⠀⣾⡇⢀⠩⠐⠶⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠞⠉⠰⡄
⠀⠀⠀⠀⡆⠀⡀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⣰⡿⠀⠃⢀⡴⠒⣈⣉⢲⠀⠀⠀⠀⢠⡇⢀⠀⠤⠀⣠⣽⣷
⠀⠀⠀⣠⣄⣀⣀⣀⠀⠀⠀⢦⣇⠀⠀⠀⢰⠉⠀⢠⠀⣸⠀⠀⢈⣃⣸⠀⠀⠀⠀⢸⠀⢰⠖⠒⠐⠂⡟⢿⡄
⠀⠀⠀⢩⡖⠀⡠⠠⠄⠀⠀⠈⡟⡀⠀⠀⣿⣀⠈⠛⠂⡇⢠⠀⢠⠂⢀⢴⠀⠀⠀⠛⢠⢀⡄⠀⢠⠀⡅⠀⣿
⠀⠀⣏⣸⡇⠀⠑⠄⡀⠄⠀⠀⣇⠁⠀⠀⠉⠛⠀⠠⡆⠀⣄⡶⠂⠀⠙⢺⠀⠀⠀⠀⢀⢂⡀⠀⠠⠄⣃⣨⡽
⠀⠀⠈⠈⠃⠠⠀⠠⠀⠀⠒⠂⠉⠀⠀⠀⠀⠈⠓⠤⠇⣄⣏⠀⢈⡤⠴⠊⠀⠀⠀⠙⠉⠈⠀⠤⠤⠬⠥⠞⠁

⠀⠀⠀⠀⠀⣀⡤⠶⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣒⣢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣄⣀⡀
⠀⠀⠀⢠⠏⢩⠖⠒⠲⣿⠿⣦⠀⠀⠀⠀⠀⢠⠎⠕⣒⣒⣒⠫⡳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐
⠀⠀⠀⢸⠀⡇⠀⣀⠠⠉⡀⠬⣇⠀⠀⠀⠀⡘⢘⡏⠀⠀⠀⡿⡕⡇⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠈⣀
⠀⠀⠀⠨⡇⡇⡿⠄⠀⠀⠀⠀⡆⠀⠀⠀⠀⣧⡞⠁⠀⠀⠀⠀⢹⢿⠀⠀⠀⠀⢀⡿⠃⠀⠀⢸⢸⢀
⠀⠀⠀⢀⡇⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠀⡀⠀⠀⠀⢠⡈⠺⡄⠀⠀⠀⢸⡇⠀⠀⠀⣸⢼⠈
⠀⠀⠀⠺⢻⡧⣇⠀⠀⠀⠀⠀⢠⠀⠀⠀⡼⠂⠐⠇⠀⠀⠀⠈⡇⠀⢱⠀⠀⠀⠸⡇⠀⠀⠀⠉⢸⠀⠠⠔⣎
⠀⠀⠀⠀⠀⢸⠸⠟⢀⣤⣤⢷⣾⠀⠀⠀⠇⡀⠀⠀⠀⠀⠠⠜⠁⠀⠸⠂⠀⠀⠀⢻⠆⠀⠀⠀⠀⠀⢀⡠⠛
⠀⠀⠀⠣⠤⢾⠀⠀⠀⠩⠛⠋⠁⠀⠀⠀⠘⠓⠶⠶⠤⠴⠶⠶⠖⠛⠀⠀⠀⠀⠀⠚⠁⠀⠠⠷⠒⠊⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡏⣿⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠒⠈⠉⠀⢸⠀⠈⠉⠒⠢⢄⣀
⠀⠀⠀⠀⠀⠀⢀⡀⠤⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠄⣀
⠀⠀⠀⠀⡔⠡⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠱⠆
⠀⠀⠀⠰⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠘⡀
⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢀⠀⣀⠴⢄⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠑⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⡅
⠀⠀⠀⠀⠀⡇⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢧⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁⠀⠀⠇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⠀⠀⡘⠀⡏⢆⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⢀⡸
⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⢠⠁⠀⡇⠘⡄⢀⠔⠊⠀⠀⠀⠀⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠈⠑⠃⠀⠀⡇⠀⠈⠁⠀⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⡇⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⡧⠔⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣶⠶⢶⡄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠰⠞⠛⠒⠛⠒⠲⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢠⣿⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠛⠚⠓⠓⠚⠂⠚⠛⠛⠳⠖⠓⠶⠦⠤⠦⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢸⣿⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠒⠛⠛⠚⠛⠛⠛⠋⠙⠉⠁⠀⠀⠚⠲⠶⠚⠒⠒⠒⠖⠒⠶⠲⠶⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠚⠛⠂⠚⠛⠛⠙⠓⠛⠙⠚⠛⠚⠛⠛⠒⠒⠀⠒⠛⠓⠒⠒⠲⠆⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠤⠤⠤⠴⠤⠤⠤⠤⢤⡤⠤⠤⠤⠄⠀⣤⠤⠤⠤⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠉⠉⠛⠛⠛⠛⠛⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⠛⢻⡇
⠀⠅⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣼⡇⠀⠀⠈⣿⡦⣾⠃
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣾⠟⠁⠀⠀⠀⢸⣷⡟
⠀⠀⠀⢸⣇⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⠀⠀⠀⠀⠀⢀⣯⣴⠟⠁⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠁
⠀⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠋⣩⠞⠫⣍⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢰⠃⠀⠀⠘⡇⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣿⣤⣤⣿⣤⣤⣴⣶⣿⣶⣶⣾⣷⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣿⣿⣿⣿⣿⣿⣿⣿⣺⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣟⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⣰⣭⣜⣦⠀⠀⠀⠀⢀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠻⡀⠰⠠⠀⠀⠀⠀⠐⠊⠄⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠖⠀⡉⢉⠉⠁⠘⠂⠶⡀⢀⠊⠀⠀⠈⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠕⠀⠀⠀⠂⠠⢀⠀⠀⠀⢀⠀⠈⠢⡀⠀⠀⠀⠠⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣷⡄⠀⠘⡄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⡀⠁⠀⠁⢻⠄⠀⠐⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠊⠉⠀⠓⠐⠠⠄⠠⠀⢀⠈⠗⠀⠠⠀⠈⢁⠀⠀⢁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⡀⠀⠀⠇⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡎⠰⡆⠁⠀⠀⠰⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⢙⡇⠈⠀⠠⠉⠀⠁⡀⠀⢀⠊⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⠀⠀⠁⢸⡀⠈⢀⠏
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡐⠂⠊⠀⠀⢠⣖⠀⠀⠀⢸⣷⠀⣜
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡔⠂⠀⣐⢫⡘⢣⠈⠹⡿⡘⠈⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⣀⣀⣀⠀⠀⠔⠀⠀⠀⠀⠘⢫⡰⠂⣐⢓⢷⡁
⠀⠀⠀⠀⠀⠀⠀⠐⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣝⡠⠒⠠⢍⡽⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⢮⡷⢒⡤⢮⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⡠⠀⠀⠉⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢸⡇⠙⢦⡀⠀⠀⠀⢠⠟⣁⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠸⣇⣀⣀⣙⣦⠀⠀⣿⠞⠙⠾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠛⠛⠛⠛⠃⠀⠛⠛⠛⠛⠃⠀⠀⠈⠉⠉⠉⢹⠀⠀⢻⣤⣤⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠰⠶⠶⠶⠶⠄⠀⢸⠀⠀⣾⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⡀⢀⠀⠀⠀⠀⢸⠀⢀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡆⠀⢸⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⡞⠛⡦⠀⠀⠀⣀⠀⠀⠀⠀⢸⡇⠀⢸⠀⢸⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠙⠛⠁⠀⢀⡴⠋⢧⡀⠀⠀⢸⡇⠀⢸⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢀⡀⠀⠀⢠⠞⠁⠀⠈⢳⡄⠀⢸⡇⠀⢸⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⣠⠞⠙⣦⣴⠋⠀⠀⠀⠳⡄⠹⣄⢸⡇⠀⢸⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣷⠋⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠘⣾⡇⠀⢸⠀⠈⣇⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⠀⠀⢻⡀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠋⠀⠀⠈⠳⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢀⠀⡀⣀⡀⢀⢆⢀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠈⠈⠊⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢀⣀
⠀⠀⠀⠀⠀⢀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠀⠀⢀⠠⠔⠙⠛⠛⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⣁⠀⠄⠀⠀⠀⣀⣴⣋⡀⠈
⠀⠀⠀⠀⠀⠂⠘⠀⠀⠈⠅⠒⠠⢤⢀⣀⠀⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⠈⠀⠀⡠⢔⠾⠋⠀⡠⢔⡪
⠀⠀⠀⠀⠀⠀⡀⠄⠒⠤⢄⢀⠀⠀⠀⠈⠉⠉⠉⠋⡉⡀⠀⠀⠀⠀⣀⠠⠂⣩⠔⢁⡠⢐⠭⡊⠕⣺
⠀⠀⠀⠀⠀⠀⠀⠐⢠⣄⢀⣀⠉⠒⠠⠤⣀⡀⠀⠀⠀⠁⣀⡠⠐⠋⡠⠖⡁⠔⡪⠕⠈⢀⡤⠖⠋⠩
⠀⠀⠀⠀⠀⠀⠤⣀⣀⠀⠀⠈⠈⠑⠒⠒⠤⡤⣍⣑⣒⠋⢁⡠⢖⡡⣐⠭⠒⠍⣀⠴⠚⠉⠀⣀⡤⡇
⠀⠀⠀⠀⠀⠀⠇⠀⠒⠂⠨⠭⠅⣒⣀⠤⠄⣀⡀⠀⠈⢈⠠⣚⠅⣊⢔⣢⠵⠋⢀⡠⢤⣸⢟⣋⣴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠂⠈⠉⠀⠚⠩⠬⠕⠨⣔⡯⠛⢉⣡⢴⣻⡽⠞⡩⢖⡯⠗⣻⠁
⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡟⠋⢀⣨⠵⢞⠭⠒⢂⠵⠊⣹⠔⡋⢸⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢾⣷⡾⣫⠕⠊⠀⠒⠉⣠⡴⠮⠒⠡⣰⣺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠁⠀⠆⣿⠝⠊⠡⠂⢀⣠⠴⠛⠁⠀⣠⡴⠞⠉⠀⠀⠠⠤⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡤⢄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄⠃⠀⢀⠤⠊⠁⡁⢀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠢⠄⣀⠀⠀⠁⠀⠂⣀⠀⠀⠀⠀⠠⠤⠐⠈⠀⠀⣐⡼⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠤⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⠿⠭⠄⠀⠀⠀⠐⠒⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⡷⠛⠀⠒⠒⠒⠒⠠⠤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⢠⢒⣐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢠⠕⢿⠓⠚⠑⡀⠀⠀⠠⠂⢉⣥⠛⠓⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣀⠀⠿⠸⠨⠇⢀⡀⢁⠠⠂⠀⡠⠀⠀⠑⡀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠋⠙⠒⠒⠀⠤⠀⡀⠀⠈⠙⠣⣀⠰⠀⠀⠀⠀⠐⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠓⠊⠁⠀⠀⠈⠁⠂⠤⡀⠀⠀⠠⠀⠀⠀⠱⡀⠀⠀⠀⠀⠃⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⣠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠢⡀⠀⣸⣶⡀⠀⠘⡄⠀⠀⠀⠘⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⣀⡀⠐⡀⠈⠈⡷⡀⠀⠘⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠚⠂⠉⠃⠠⠀⡀⠀⣀⢀⠀⠉⠷⠀⠐⠀⠀⠙⠃⠀⠀⢀⠀⠀⠀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠽⠀⠀⠀⠀⠆⠀⠀⠩⠀⠀⠈⡀⠀⠀⢠⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡆⠀⠀⠃⠀⠀⢸⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠃⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡤⠀⡀⠀⠀⠀⠈⣄⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢆⠀⠀⡄⠀⠀⠀⠀⠀⠀⣤⡄⠀⡄⠀⠸⣇⡈⡕⢰⠀⠀⠀⣇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⣀⠀⠀⢀⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⢿⡀⠇⠀⡔⠀⠀⣈⢢⠀⠀⡈⠀⣵
⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠈⠀⠀⠂⠅⢀⠀⠀⠄⠀⠀⠀⠀⠀⠈⢘⠀⠀⠀⡄⠀⠀⠀⢸⣇⢀⠀⣠⣃
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠐⠂⠠⢈⡀⠂⠊⠁⠀⠀⣤⣀⠀⠀⠀⢨⣿⠏⢰⡏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡰⠆⠀⠀⡗⣇⠔⢖⡖⠲⣶⡆⠖⠐⠦⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠑⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠷⡏⢢⠌⠑⠆⣼⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⣇⡏⠃⡤⡋⡐⢒⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢠⠳⣌⣀⠈⠁⠟⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠁⠲⢀⡀⠄⢀⠀⠀⠀⣀⠄⠀⠇⠈⠛⣭⣴⣞⣩⠏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠌⠁⠠⠄⠒⠈⠉

⠀⠀⠀⠀⠀⢠⣶⡿⠿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⣿⡏⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣤⣀⠉⠉⠙⠻⢶⣤
⠀⠀⠀⠀⠀⠻⣿⣦⣤⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣿⠙⠻⢿⣿⣷⣶⣶⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⣿⢷⣄⣀⣀⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠈⠉⠉⣽⣿⣃⠀⠀⠀⠀⣰
⠀⠀⠀⢠⣶⡿⠿⣷⣆⣠⣽⣷⣉⠻⣯⣁⣀⣀⣀⡉⠙⠛⠶⣤⡀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⡄⠀⣰⣿
⠀⠀⠀⣿⡏⠀⠀⠈⣿⡏⠉⠉⠻⣦⡈⠙⢿⣭⡉⠉⠉⠉⠉⠉⠙⠃⠀⢸⣷⣶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⣼⡿⠀⢰⣿⣿
⠀⠀⠀⠻⣷⣤⣤⣾⡿⠁⠀⠀⠀⠈⠻⣦⣄⠈⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠿⠿⣿⣿⣶⣶⣾⡟⠁⢠⣿⡿⣹
⠀⠀⠀⠀⠈⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣍⣛⣶⣄⡀⠀⠀⠀⢀⣀⣠⣤⣤⢀⣆⠀⠀⠉⠛⠋⠀⠀⣾⣿⢃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣩⣭⣴⣶⣾⠿⠟⠛⠋⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⣸⣿⡟⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⡿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢠⣿⡿⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⣾⣿⢃⣿⡏
⠀⠀⠀⠀⢠⣶⣶⣿⠿⠿⠛⠋⠉⠀⢀⡀⠀⠀⠀⠀⢀⣠⠴⣀⣠⣀⣈⠁⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⢰⣿⣯⣾⡟
⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠄⠀⣴⢟⣵⠟⣋⣍⡈⠙⣷⡆⠀⠶⠶⠶⠀⠀⢹⣇⠀⠀⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣼⡏⠾⠁⣼⠇⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⡄⢰⣿⣿⡟
⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠹⠷⠶⠞⠋⣠⣾⡟⠀⢶⣦⣤⡀⠀⠀⠀⠈⣿⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⣠⡴⠂⠀⠙⢿⣷⣶⣶⣶⡿⠟⠉⠀⠀⠀⠉⠛⠃⠀⠀⠀⢰⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣠⠀⠈⠉⠁⠀⠀⠀⠀⢀⣠⣴⣶⠾⠟⠛⠓⢺⣿⡿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢀⣠⣶⡿⠛⠉⠀⠀⣀⣀⣤⣤⣿⣽⣽⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣴⣾⠿⢛⣁⣤⣴⣶⠿⠟⠛⠋⠉⠉⠀⠈⠻⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⣠⣴⡿⠟⠛⠋⠉⠉⢉⣩⣥⡶⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡾⠛⠁⠀⠀⣀⣤⠶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢀⣤⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠛⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣀⣀⣤⣁⠚⠛⠛⠛⠳⠾⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠚⠉⠉⠉⠉⠉⠉⠉⠱⠂⠀⠀⠀⠀⠉⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢀⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠢⢄⡀⢀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠳⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⡇⠀⠀⠀⠀⠀⣲⠂⠈⠉⠉⠛⣮⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡸⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣡⠃⠀⠀⠀⠀⠀⡏⠃⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠒⠀⠤⠤⠤⠶⠗⠇⠀⠀⠀⠀⢀⣼⠃⠀⢀⣴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⣦⣤⣄⣀⣀⣀⣀⣠⣤⣤⣶⣿⣥⣶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣀⡀⠠⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⢐⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠐⢷
⠀⠀⠀⠀⠚⡀⢳⠀⡄⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡀⠀⠀⠀⠠⠠⡀⣀⠀⠠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢀⠈⠀⠀⠀⠐⠆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⠠⢠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠈⠐⢁
⠀⠀⠀⢀⠀⢐⡀⠄⠐⣀⠄⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⢀⠐⢠⠈⠀⠀⠀⠀⠀⢨⠑⠀⠆⠒⠀⠀⠒⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠺⠐⠐⠀⠊⠁⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠉⠁⠀⠀⠀⠀⠁⠀⠸⠓⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⣐⢐⠂⠀⠀⠀⠀⠀⠸⠒⠐⠙⠃⠀⠀⠀⠀⠀⠀⠐⢀⠀⢀⠀⢠⠀⠀⠂⡈⢂
⠀⠀⠀⠸⡄⠁⠂⡤⠀⠀⠂⠀⠃⠀⠀⠄⠈⠐⠀⡀⠀⠛⠀⠐⠀⡆⠀⠀⡐⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠆⡠⠂
⠀⠀⠀⠐⠋⠑⡀⠄⠀⠀⠀⢀⠀⠀⠈⠄⠀⠀⠉⠀⠀⡁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠠⠐⠈⠣⠄
⠀⠀⠀⢀⡬⠉⠁⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡀⣄⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠂⠀⠀⠀⣠⡀
⠀⠀⠀⠈⠲⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⠈⠢⡌⢅⠂⠀⢠⠀⠀⠀⠀⠀⠀⠃⠃⠀⠀⠊⠀⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠉⠁⠀⠀⠡⠐⠆⠀⡀⠐⢄⠡⣀⠂⠀⢀⠰⠀⠀⠀⠀⠀⡀⠀⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠁⠈⠀⠀⠀⢀⡡⠐⢺⠀⠈⠀⠐⠅⠀⠀⠀⠤⠄⠀⠁⣛⡧⠐
⠀⠀⠐⠂⠀⠀⠀⠸⡄⠀⠀⠀⠀⡂⢰⠀⠀⠀⠀⠀⠀⠀⠔⠊⣀⡀⠀⢘⠘⠀⠀⠠⠂⠀⠘⠚⢸⡇⠀⠂⠤⠀⠈
⠀⠀⠀⠈⠐⠰⣀⠀⠃⠀⠁⠀⢀⡄⠀⠀⠀⢀⠀⠀⡀⠀⠀⡈⠓⠃⠀⠀⡂⠀⠀⠊⠈⠀⠀⠀⢸⠂⠀⠀⠀⠀⠇
⠀⠀⠐⠀⠀⠀⢤⠀⠀⠀⠀⠀⠳⠷⠤⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⢀⡅⠌⠂⠀⠒⠛⠦⣀⣢⡀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⠀⠐⠈⠁⠉⠈⠁⠉⠀⠃⠀⠀⠀⠀⠀⠀⠘⠉⠐⠢⠌⠀⠜

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠰⠿⠤⠴⠿⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⢲⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠲⠶⠤⣝⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⡤⠴⠂⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠿⠡⠄⠸⣷⠠
⠀⠀⠀⠀⠀⢰⠁⠀⠀⣀⣀⣀⣀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⡸⠀⢀⣤⠶⠒⠛⢉⠓⣆⠀⠀⠀⠀⠀⠀⡞⠀⢀⣀⢀⣀⡀⣈⣽⣿⣇
⠀⠀⠀⠀⠀⠐⠀⠎⠁⠀⠀⠀⠀⠉⢲⡸⡇⠀⠀⠀⠀⠀⡘⠟⠁⠀⡅⠀⢸⠁⠀⠀⠻⠟⠀⣿⠀⠀⠀⠀⠀⢀⡇⠰⢩⣀⡀⠠⠤⠍⢱⢿⡟
⠀⠀⠀⠀⠾⡟⡞⠂⠒⠒⠒⠒⠀⠀⠀⢳⣷⠀⠀⠀⠀⢠⡇⠀⢤⣸⠆⠀⡿⠀⠀⠠⠠⢒⡒⠛⡄⠀⠀⠀⠀⣤⡇⡇⠸⠃⠀⠀⠀⠀⢸⠀⠛⣧
⠀⠀⠀⡈⠀⣿⠃⠀⡰⠒⠒⠊⠀⠀⠀⠘⣇⣁⠀⠀⠀⢸⣷⢸⠀⠈⠉⡇⠇⢠⠀⠈⠳⡍⠀⣤⢳⠀⠀⠀⠀⡏⡇⣧⢸⡶⠀⠀⠼⠀⢸⠀⡀⣿
⠀⠀⠀⡇⣭⣿⠀⠀⠑⢤⡀⢀⠄⠀⠀⠀⣷⠈⠂⠀⠀⠈⠘⡟⠄⢀⣴⠇⠀⣸⢸⡶⠂⠀⠉⠹⣼⠀⠀⠀⠰⠀⡆⢸⢈⡀⠐⠂⣠⡄⢻⡀⡏⣽
⠀⠀⠀⠑⠙⢧⡀⡀⠀⢀⣈⡁⢀⣀⣀⠴⠛⠂⠀⠀⠀⠀⠀⠳⣆⣀⣸⠀⠀⢻⠋⠀⣈⠀⢀⣀⠼⠀⠀⠀⠀⠳⠗⠚⠘⠁⠁⠀⠀⠒⠛⢉⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠈⠓⠛⠂⠈⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀
⠀⠀⠀⠀⠀⣴⠚⢛⣡⡤⠤⢬⣿⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣩⠭⠭⢭⣙⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠙⢉⡉⢤
⠀⠀⠀⠀⠀⡇⠀⡏⠀⠔⠈⠉⠵⠉⠙⠻⡄⠀⠀⠀⠀⠀⠀⡼⢡⣧⠖⠒⠒⠒⢢⡝⣌⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⡆⠐⠀⠠⠤
⠀⠀⠀⠀⠀⡇⢸⠀⣸⢠⠄⠂⠀⣆⠁⠒⢧⠀⠀⠀⠀⠀⢀⠇⢀⡟⠀⠀⠂⠁⠸⠻⡌⢿⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⡃⠀⠀⣀⣠⡤⠄
⠀⠀⠀⠀⠀⣧⢸⠀⣿⡇⠀⠀⠀⠀⡀⢸⡇⠀⠀⠀⠀⠀⢾⣤⡟⠀⠀⠀⠀⠀⠀⠀⢿⠿⡄⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⡇⣿⢠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣿⢸⠀⡇⠀⠀⠀⠀⠀⢁⠀⠁⠀⠀⠀⠀⠀⣀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⣇⣽⠘⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣰⣿⣸⠀⡇⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡤⠀⠀⠀⠀⠀⢻⠀⢰⢱⡀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢸⠏⣿⠀⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠉⠃⡾⢛⢧⣄⡀⠂⠀⠀⢀⠀⢰⡇⠀⠀⠀⣶⠛⠀⠀⠃⠀⠀⠀⠀⠀⢸⠀⠈⠀⣿⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠈⠀⣿⠀⠔⠊⠉⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠀⡟⠇⢈⣤⣠⡼⠖⣿⠇⠀⠀⠀⢻⢀⠀⠀⠀⠀⠀⠀⠤⠴⠋⠀⠀⠀⡿⠀⠀⠀⠀⠀⢹⡚⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠛
⠀⠀⠀⠘⢦⡀⢀⣯⠀⠀⠀⠈⢉⣫⠵⠛⠉⠀⠀⠀⠀⠈⠿⣦⣤⣤⣄⣀⣤⣤⣤⣤⣤⠶⠂⠀⠀⠀⠀⠀⠠⢾⠧⠀⣀⣀⣴⣤⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠈⠉⠉⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣟⣻⣻⣹⣻⣻⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣽⣯⣿⣿⣯⣽⣿⣯⣭⣯⣽⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣴⣶⡶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣲⣿⣿⣶⣤⣴⡿⠋⠉⠉⠉⠙⢿⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⠟⠁⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⢳⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡏⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⢀⡤⢶⠒⠦⢼⣿⣸⣽⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⣿⠞⠉⠀⢸⠀⠀⢠⣄⢹⣿⠲⠤⣄⣀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣾⠛⠤⣄⢸⣿⠀⠀⠀⢸⠀⠀⠈⠛⢿⣿⣤⣄⡀⠈⠉⠒⠿⢇⣀
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠙⣿⠦⢄⠀⢸⠀⠀⠀⠀⢸⣿⠉⠛⠛⠿⠶⢦⣤⠀⠈⠉⠒⣤⣄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠰⠿⠀⠀⠉⠙⠢⢄⣀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠉⠁⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢺⣿⣀⠀⠀⠀⠀⡰⠊⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠈⠙⠲⡤⠊⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⠦⠤⠤⢤⣬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⠃⠀⠀⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣰⠃⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⡇⣰⠃⡠⠚⠁
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⡀⣷⠵⠊
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⠒⠊⠉⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠔⠒⠉⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⡄
⠀⠀⠀⡤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠉⠁⠀⠀⠀⠘⡄
⠀⠀⠀⠙⢦⠀⠀⠀⠀⠤⠔⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠒⠉⠉⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⣀⡠⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⣠⡔⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠉⠁⠀⠀⠀⡇⠀⠀⠀⣀⠔⠋⢀⣽⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⡠⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡤⠒⣹⢷⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠚⠁⢀⡤⠚⡣⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠊⠉⠀⠀⠀⢠⢶⡡⠔⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠐⠓⠉⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⡠⠔⠚⠉

⠀⠀⠀⢀⡤⠴⠶⢞⡛⠻⠷⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣶⣆⠀⠀⠀⠀⡖⢶⣄⡀
⠀⠀⡜⠁⠀⠀⣎⡀⠀⠑⡄⣠⠄⢍⠻⣦⡀⠀⠀⠀⠀⢠⡟⠛⠳⡏⠀⢀⣠⣞⣳⠞⠀⠈⣤⡀⢰⡆⣴⠃⢀⠀⢀⣠⡀
⠀⠀⠳⣤⣤⣤⣤⡉⠳⠞⢱⡇⠀⢠⠀⠈⢿⡄⠀⠀⠀⢸⡇⠒⢢⠀⢀⣾⠁⢈⠇⢀⣀⠀⢿⡇⠸⣶⠃⣴⠟⣱⠿⠛⠁
⠀⣀⠤⠂⢀⡠⢛⡁⠀⠀⡠⠽⡛⠋⠀⠀⣸⡇⠀⠀⠀⣾⠅⢀⠃⠀⣼⠃⢠⠎⠀⠘⠿⣧⣈⣷⣾⣿⣿⣿⡟⢁⣀
⢠⡁⠀⠀⣿⠀⠀⠈⣱⠀⣆⣀⣸⠀⢀⣴⡿⠁⠀⠀⢰⡇⠀⡞⠀⣼⠇⡠⠃⠀⠀⠀⠲⠾⢿⣿⣿⣿⣿⣿⣿⣏⡉
⠈⢷⣦⣄⣈⡓⣓⣚⣋⣀⣬⣭⣵⣾⠿⠛⠁⠀⠀⠀⣼⠃⣸⠁⣼⠏⡼⠁⠀⠀⠀⠘⠛⠒⢶⣿⣿⣿⣿⣿⢿⣏⠛⠃
⠀⠀⠈⠉⠑⠛⠛⠛⠛⠛⠛⠉⠉⢀⣀⣀⠀⠀⠀⢠⣿⣠⠃⣰⢏⡞⠀⠀⠀⠀⠀⠀⢀⣠⠞⢻⡟⠻⣿⡅⠀⢹⡇
⠀⠀⠀⠀⢀⣀⣠⡤⠴⠶⠲⠚⠋⠋⢿⡅⠐⠀⠀⣸⢁⡏⣰⣯⠏⠀⠀⠀⠀⠀⠀⠰⠟⠁⢠⡾⠁⠀⣿⣿⡄⠈⠁
⠀⣀⠔⠘⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠽⠚⠀⠀⠻⠞⠰⠷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠈⠿
⠘⠛⠀⠀⠷⠖⠒⠒⠛⢉⣉⣁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣤⣤⣀⠂⠤⣄⠀⠀⢀⣴⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣞⢩⡭⣍⢻⣆⠀⠀⠀⠀⠀⡞⠁⠰⣿⣿⡿⠿⠃⠀⣸⡇⠀⠙⠿⠿⠟⠋⠀⠀⣾⠋⢽
⠀⠀⠀⣠⡶⢋⠳⡄⢹⠈⣧⣼⢀⡿⠀⠀⠀⠀⠀⢿⠲⠶⠶⠶⠶⠶⠖⠛⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣼⡇
⠀⠀⠀⣿⢰⡉⢣⡘⢦⡇⠀⣖⠋⠀⠀⠀⠀⠀⠀⠈⣒⣤⣤⠤⠴⠒⠓⣾⠿⠶⠶⣤⡀⠀⠀⠀⢰⣿⣛⠒⢞⡩⢿
⠀⠀⠀⠻⣦⣝⣯⢤⠀⣿⠀⢹⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠣⣄⣀⣠⡴⠏⠀⠈⠀⠀⣷⠀⠀⠀⢸⡧⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣾⡇⠸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⣀⣀⠠⠤⠤⠤⠰⡄⢸⠀⠀⠀⢸⣇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⣠⡄⠀⠀⠈⢷⢸⣿⢆⠀⠀⠀⠀⣷⠀⠀⠉⠉⡁⢠⡀⢠⢠⣄⠤⠀⡇⢸⡆⠀⠀⢸⡧⠥⠀⠀⠀⢸
⠀⠰⣿⡞⣿⣿⣷⡦⠀⠀⠸⣘⡟⢦⣵⣄⠀⠀⢹⡆⠀⡆⠀⡇⠸⠱⣼⢸⠉⢆⠀⡇⠸⡇⠀⠀⢸⡯⠀⠀⠀⠀⢸
⠀⠀⣸⢾⣿⣿⣿⣥⡀⠀⠀⣿⡇⠀⠈⠻⢦⡀⠸⣧⠂⢁⣀⣀⣀⡄⠠⠤⠄⠒⠒⢃⣀⡇⠀⠀⢸⡯⠅⠀⠀⠀⠈⡄
⠐⣶⠴⣾⣿⣿⣿⣯⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣤⣀⣤⣤⣤⣶⠶⠶⠾⠛⠛⠋⠉⠀⠀⠀⡾⠛⠓⠒⠒⠒⢛⢻⡆
⠃⠘⢰⣿⣿⣿⣿⣿⣅⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠸⣻⣿⣿⣿⣿⠟⡏⠁⠀⠀⠀⠀⠀⠀⣰⢒⠒⣛⣛⣋⣉⣀⣀⣀⣀⣀⣀⡀⣀⠀⠀⠤⠤⢤⣀⡀
⠀⠀⠟⡟⢸⡟⣿⣧⠀⠘⠀⠀⠀⠀⠀⢿⣉⣀⣀⡀⢸⣀⣤⣤⣤⣤⣄⣄⣄⣤⣤⣤⣤⣤⣤⣄⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⢉⡴⢛⣿⠢⣍⡻⢶⣤⡀
⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⠀⣩⠞⠁⣿⠦⣄⠉⠲⢬⡙⠳⣦⣀
⠀⢀⣠⡴⠛⠁⣠⣤⣤⣤⣤⡀⠁⢀⠔⢻⠦⣌⠑⠤⡀⠀⠀⠀⠙⠻⢦⣄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⠴⠶⢶⣶⡀
⠀⢹⡇⠀⢠⡾⢋⣥⣤⣄⠈⢻⡆⠁⠀⣸⡗⢌⡙⠢⠀⠀⠀⠀⠀⠈⠓⢍⡛⣿⠒⣛⠛⠋⠉⠉⠉⠀⠀⠀⠀⢀⠔⣻⡏⢷⡄
⠀⠸⡇⠀⣾⠁⢸⣇⢀⣽⠆⢸⡇⢠⠊⢸⣧⣀⠉⠒⠄⠀⠀⠀⠀⠐⠦⣄⠉⣿⠞⠁⣀⠀⠀⠈⠻⣦⣴⠾⠃⠎⣠⣿⠹⡀⠹⣦
⠀⠀⣿⠀⢹⡆⠀⠉⠋⠁⠀⣾⠁⠀⠀⢸⣧⣈⠙⠂⠀⠀⠀⠀⠀⠐⢄⠈⠳⣽⣠⢞⡅⠀⠀⠀⡶⠛⠻⣦⠀⠞⢁⡿⡆⠁⠀⠈⢷⡀
⠀⠀⢿⡀⠈⢿⡄⠀⠀⠀⣼⠃⠀⠀⠀⠀⠁⠀⠉⠂⠀⣀⣀⡀⠀⠀⣄⠑⢄⢸⡷⠋⠀⣠⡄⠀⠀⠀⠀⠀⢀⠔⢩⡇⠘⡄⠀⠀⠀⠻⡆
⠀⠀⢸⡇⠀⠀⠻⣦⡀⣸⠏⠀⠀⠀⠀⠀⠀⢀⣤⠀⠉⠉⠉⠀⠰⣤⡀⠑⢄⢹⡇⠀⢰⠟⠀⠀⠀⠀⠀⠀⢋⠴⢻⢷⠀⢳⠀⠀⠀⢸⠃
⠀⠀⠈⣷⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠐⠟⠁⠀⠀⠀⠀⠀⠀⠈⠗⢀⠀⢹⡇⠀⣀⠀⠀⠀⠀⠀⠀⠐⣫⠔⣿⠀⢇⠀⠀⠀⢀⡟
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠚⢁⣤⡟⢧⠀⠀⠀⠀⢸⠇
⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠀⠀⠀⠀⢠⡶⠟⠛⠛⠷⣦⠀⠈⠿⠄⠀⠈⠁⠀⠶⣤⣀⡀⠘⠁⢸⣷⠘⠄⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢰⡀⠀⠀⣿⠁⣴⠛⠻⣧⢸⡇⠀⠀⠀⢶⣄⠀⠀⠀⠀⠉⠁⢰⡄⣼⣇⠁⠀⠀⠀⠀⡿
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⣸⡟⢦⡀⢹⡄⠹⢦⡾⠋⣸⠇⠀⣼⠀⠀⠙⠷⠀⠀⠀⠀⠀⢠⡿⣿⠈⠆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⢀⣀⣀⠀⠀⢀⣴⠂⣠⠞⢹⣟⠢⣙⠈⢷⡀⠀⠀⣴⠏⠀⢆⣿⡴⠁⠀⠀⠀⠚⠶⠄⠐⠟⢹⠇⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⢸⡆⣾⠋⢉⣷⡶⠛⠁⠀⣡⢖⡽⣿⠳⣌⠓⠀⠻⣦⣼⠃⠀⢄⠈⣿⠀⢀⠄⣠⠂⠀⠀⠀⣀⣤⡾⢆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠙⠛⠋⠁⠀⠀⠀⣸⠕⢋⣠⣿⣖⠄⠀⠀⠀⠀⠀⠀⢀⡈⠣⣿⠞⢁⡼⠁⠀⠀⠀⢚⡡⣾⠳⠈⠆⠀⠀⠀⠀⡿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣤⡴⠞⠛⠉⠀⠉⠛⠶⢦⣤⣀⡀⠀⠀⠱⡄⣿⣠⠞⠀⠀⠀⠀⠐⣋⣴⣿⡀⠱⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⣿⢀⣠⠴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣄⣈⣿⣁⣀⣤⣤⠶⠞⠛⠉⠻⣄⢣⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠹⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡗

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⣤⣶⣖⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⡌⡖⠉⠙⡷⡌⢪⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⠀⠀⠀⢰⣣⠀⢫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⢸⣾⠀⢾⡃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡄⠤⠤⠤⠤⠿⠦⠰⢼⠀⠀⠀⠒⠚⠒⠂⣗⠒⠒⠒⠒⠒⢢
⠀⠀⠀⠀⠀⠀⢠⠀⢨⠁⠀⡄⠁⠀⠀⠀⠀⢰⣿⡄⠀⠀⠀⠀⠀⣤⣶⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢸⡷⢺⠀⠀⠂⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⢗⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠢⠊⢄⢀⠀⠀⠀⢐⠀⡄
⠀⠀⠀⠀⠀⠀⢸⣃⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠔⠁⡡⡠⠀⠀⠀⠁⠀⡂
⠀⠀⠀⠀⠀⠀⢸⢃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠺⠀⠅
⠀⠀⠀⠀⠀⠀⢸⠅⢸⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⢸⡂⣹⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⢐⠄⡘⡀
⠀⠀⠀⠀⠀⠀⣼⠀⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠔⠀⠤⢈⢈⢂⡞⠀⡂
⠀⠀⠀⠀⠀⠀⡼⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢊⢈⠄⠈⠠⠂⠠⡠⠠⢂⠞⠀⡬⠁
⠀⠀⠀⠀⠀⠀⠏⠢⠸⠀⠀⢔⠀⠀⢀⠀⠀⠁⠁⠀⡠⡀⠘⠁⠀⠐⠀⠑⠈⠀⠂⠀⠐⣀
⠀⠀⠀⠀⠀⠀⡗⠀⢰⠀⠐⢐⢕⠕⠕⡡⠢⠊⢊⢈⠀⠀⢂⢀⠀⠄⠄⠀⠈⠀⠀⠀⠘⠜
⠀⠀⠀⠀⠀⠀⣗⣤⣾⡅⢰⠨⡡⠂⢊⠔⠐⡐⡁⡡⡡⡡⡠⠀⠊⠊⢂⢂⢂⠀⡁⠄⢠⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠹⠾⠤⠤⠥⠥⠬⠬⠬⠌⠘⠘⠴⠵⠥⠥⠥⠥⠥⠮⠤⠤⠬⠀⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⡀⢀⠄⠀⡀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠁⢁⠀⡀⠀⠀⡈⠀⠀⠨⠀⠀⠀⠀⡀⢐⠁⡪⢀⠕⡨⠂⠐⠁⠈⢀⠑⠠⢊⠐⠁⡨⢂⢀⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠈⠀⠁⠈