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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣄⣤⣤⣶⣶⣾⣿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣶⣶⣾⣿⠿⠿⠟⠛⠛⣋⡉⠉⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣾⣿⡿⠿⠿⠛⢛⣋⠉⢩⣥⠀⣶⡶⠾⠇⣿⣿⠉⠁⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣶⣶⣖⣳⡿⠿⠿⠟⠛⠛⣉⣉⣥⣄⢰⡶⢶⡿⠛⠸⣿⡄⠘⣿⡇⢿⣧⣠⣤⢸⣿⡶⠾⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⣀⣀⣤⣤⣴⣶⣶⣿⡿⠿⠟⠛⢛⣋⣉⡉⢩⣤⣤⣶⠶⠆⠀⠀⠸⣿⡇⠙⠛⠁⠀⢸⣿⠀⠀⣿⣇⠀⢿⣷⢸⣿⡉⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⢸⣿⡟⠛⠉⠉⠉⢠⣴⠀⢰⣿⠀⣿⡏⠙⣿⡆⠁⢸⣿⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄⠀⠘⣿⡇⠀⢸⣿⣀⣼⡿⠀⣿⠇⠀⠀⠛⠃⢀⣀⣀⣤⣤⣤⣶⣿⣿
⠀⠈⣿⣇⠀⠀⠀⠀⠸⣿⣆⣸⣿⡆⣿⣇⠀⢻⣷⠀⠀⣿⡇⠀⠀⠀⠀⢰⣶⣄⣼⡿⠀⠀⠿⠇⠀⠀⢉⣋⣉⣥⣤⣤⣴⣴⣶⣶⠿⠟⠛⠛⠛⠋⠉⠉
⠀⠀⢻⣿⠀⠀⠀⠀⠀⣿⣏⠉⢿⣇⢸⣿⡀⣸⣿⠀⠀⢻⡷⠀⠀⠀⠀⢀⣉⣉⣭⣤⣤⣶⣶⣾⣿⠿⠿⠿⠛⠛⠉⠉⠉
⠀⠀⠸⣿⡆⠀⠀⠀⠀⢸⣿⠀⠸⠿⠀⠙⢛⣉⣁⣠⣤⣤⣦⣶⣶⣿⠿⠿⠿⠛⠛⠋⠉⠉⠁
⠀⠀⠀⣿⣧⠀⣀⣀⣠⣤⣤⣶⣶⣶⣿⡿⠿⠿⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠹⡿⠿⠿⠟⠛⠛⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠙⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠈⠻⣦⣀
⠘⣦⠀⠀⠢⡀⠀⠈⠻⢷⣤⣀
⠀⠘⣧⠀⠀⠙⢦⡀⠀⠀⠙⠿⣕⠦⡀
⠀⠀⢻⣆⠀⠀⠀⠻⣆⡀⠀⠀⠈⠳⣎⢳⡀
⠀⠀⠘⣿⣆⠀⠀⠀⠙⢟⠦⣀⠀⠀⠈⡇⢹
⠀⠀⠀⢻⡝⣆⠀⠀⠀⠈⢧⠈⠙⡓⠛⣡⠞
⠀⠀⠀⠀⠻⣌⢷⣄⣀⢀⡸⢠⠦⡉
⠀⠀⠀⠀⠀⠙⠷⣌⣉⣉⡤⠋⠀⠙⢦⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠑⣌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⠀⠀⠀⠀⢀⣤⠶⠟⠛⠛⠛⠛⣿⡟⠻⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡑⢄⡴⠋⠀⠀⠀⠀⠀⠀⢸⠈⠣⡀⠀⠀⠈⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠔⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣄⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠐⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⡀⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⢠⣿⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⠑⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠁⠈⠑⢤⡀⠀⠈⣆⠀⠀⣀⣸⠋⠀⢢⡀⢀⠀⢀⣠⢴⢋⡾⠁⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⢙⢧⠀⠀⠀⠀⠀⠈⠁⠢⡎⠀⠀⠠⠟⣡⣿⣋⠤⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⣀⣀⣤⡀⠀⠀⠀⠀⣠⠼⠂⠛⠛⠀⢻⣉⣁⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣠⢤⠄⡀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣽⠘⠳⠒⠒⣶⡒⢫⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢆⠹⡄⠀⠀⠀⠀⠀⠀⣸⠃⠀⡜⠁⠀⠀⡤⠔⠋⠈⢦⣹⣤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡠⢇⢸⡟⠻⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣸⠠⣹⠞⣉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⢸⡇⠀⠀⠀⠀⠀⣠⡴⢿⡄⠀⠀⠀⢀⡼⠀⠀⠀⢰⠇⠀⠙⣿⠐⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⣴⠞⡡⠤⠘⡙⠶⠶⠖⠋⠀⠀⠀⣰⢯⣀⣀⡴⠃⠱⣄⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠳⣤⣤⣾⠁⠀⠳⠤⠗⠈⠂⠄⠀⠀⠠⣴⠟⠁⠀⠈⠉⠀⠀⠀⠈⠳⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⠤⢼⣇⠔⠛⠄⠀⠀⠀⠀⠀⠤⠒⢹⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠢⣄⠀⠀⢀⠠⠐⠁⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢠⠟⠻⡄⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⡀⠀⢀⡇⠀⠀⢻⡀⠀⠀⠰⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⢀⢈⠇⠀⠀⠀⠳⢄⣀⣀⣈⣉⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠘⢦⡀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠹⡢⢄⣈⠏⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣕⢤⣀⠀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠉⠉⢹⠏⠙⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡐⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢰⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠐⢦⠀⠀⠀⠘⠁⡐⠀⣠⢖⡷⠁⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠤⢦⠀⠀⠠⠒⠈⠁⢹⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠐⡄⠀⠀⠀⠀⢰⠀⡨⠑⠂⠶⠂⠉
⠀⠀⣤⠄⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼⡾⠄⠀⠀⠀⡇⠀⠀⠀⠏⠀⢘⢛⡟⢢
⠀⠀⡸⠐⢚⣾⣷⠛⣶⣶⠂⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡇⠀⢀⡤⠚⠳⠄⡠⠜⠀⢀⣎⣀⡝⢘⢆
⠀⠀⠳⢚⣚⠛⠛⠒⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣑⣰⡏⣀⠐⠀⠀⠀⠀⡾⠁⠀⠁⠀⠀⠡⡑⢄
⠀⠀⠀⢨⠁⣷⠒⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢄⡀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠈⠁⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠊⠙⡄⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠚⠀⠀⠈⠂⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢀⣀⠀⢰⣄⠀⠀⠈⡄⠀⠀⣀⡀⢀⣾⠂⢀⣀⣀⠀⠀⠀⢀⣀⣀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⣾⡿⠛⢷⡈⣿⣆⠘⡄⠃⡔⠉⠀⠀⣸⠇⣴⣿⠋⠙⣧⠀⣰⣿⠋⠙⣧⠀⠘⡇⠀⣠⡶⠾⠓⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⣿⡟⠀⠀⢸⡇⢹⣿⠀⡇⢈⠔⣉⣁⢀⡟⣸⣿⠃⠀⠀⡿⣰⣿⠃⠀⠀⣿⠀⢠⡇⢸⡟⠀⢀⣠⣤⣤⣄⡙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢸⣿⠁⠀⠀⣾⠀⣼⠇⠀⡃⠐⠉⢀⠀⠘⠃⣿⡟⠀⠀⢸⡇⣿⡟⠀⠀⢰⡏⠀⡾⠁⢸⡇⢠⡟⠁⠀⠀⠈⠙⢷⡙⢧
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢨⠀⣾⣿⠀⠀⣰⠇⡰⠋⠀⠔⢀⠔⠡⠀⠠⢀⢰⣿⡇⠀⢀⡟⢠⣿⡇⠀⢀⡟⢀⡾⠁⠀⠈⠁⢸⣇⠀⠀⠀⠀⠀⠈⣷⠈⠁
⠀⠀⠀⠀⠀⠀⡀⠘⢠⣼⠀⢿⣿⠀⣰⠃⣠⣴⣦⡀⠀⠘⢀⣾⡿⣷⠸⠈⣿⡇⢠⠞⠀⠘⣿⡇⢠⠞⠰⠋⢀⣠⣤⣤⡀⠀⠙⠛⠛⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⡀⡰⠀⢸⠀⢸⣿⠞⢡⡞⠁⢨⣿⡇⠀⠀⣾⣿⢀⡿⠀⠀⣿⣷⠃⠀⠀⠀⣿⣿⠋⠀⠀⢠⣿⣿⣿⠈⣷⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠐⠡⣶⠆⠀⠈⣿⡆⡾⠀⠀⣾⡿⠀⠀⢰⣿⡿⠋⠀⠀⢠⠿⣿⡀⠀⠀⢀⡿⣿⡀⠀⠀⢸⣿⡍⢿⣦⣿⡀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣤⡀⣀⠀⠀⠸⣿⡇⡀⠀⣿⡇⠀⢠⡟⢿⣧⠀⠀⣠⢟⣤⡹⣧⡀⢀⡾⠁⢻⣧⠀⠀⣸⠿⣿⣦⡾⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢿⠀⢿⣷⣦⣀⠈⠀⢈⠃⠹⣧⣤⠟⠀⠈⠛⠿⠛⢁⣾⠋⠻⣮⠉⣩⣀⣿⠀⠻⣷⣴⠟⠀⠀⠀⠀⠀⣀⣀⣤⡄⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠘⣿⣿⣿⣿⣶⣶⣤⣤⣄⣀⣀⣘⠃⠸⡿⠂⠙⠁⠘⠂⠶⠾⠏⠛⠋⠀⣀⣀⣡⣤⣤⣤⣶⣶⣿⣿⣿⠟⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣀⡀⣸⡟⠘⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣄⡈⠁⠘⢿⣿⣿⣿⠛⢻⡿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⠿⣿⠛⣿⣻⡟⢩⣿⡇⣶⣿⡟⠀⠀⠻⠿⠣⠿⠛⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠈⠻⢿⡿⠀⣿⠃⣼⠃⣼⣿⡇⢰⣿⣿⣿⡏⢿⣾⣿⡆⢹⣿⠀⣿⣿⡇⢸⣿⡇⢸⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⡇⢰⡷⢠⡇⠀⣿⣿⡇⢸⣿⣿⣿⣧⡀⠙⣿⡇⢸⣿⡄⢻⣿⣿⠘⣿⣿⣈⣿⣿⣿⣿⣿⣿⣶⣶⠄⢀⡼⠃
⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣧⣼⣿⣄⣻⣿⣇⣸⣿⣿⣿⣯⣿⣠⣿⣧⣼⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⣡⡴⠋
⠀⠀⠀⠀⠠⣤⡈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣉⡙⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠉⠉⠉⠀⠀⠀⢰⡦⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠷⠶⣶⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⡤⠴⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠈⠘⠀⠀⠀⠐⠀⠂⠘⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠁⠠⠀⠀⠀⠀⠂⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⢢⣄⠈⠳⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢄⠀⠑⠁⠲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠭⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠒⠲⠒⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠹⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢭⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠁⠀⠀⠀⠀⠀⢰⠁⠀⠀⢀⣀⡜⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠘⠂⠀⠀⠁⢰⠡⠔⢹⠋⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠈⢹⠀⡐⢁⢀⡀⠜⠉
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢠⢆⠀⠀⠀⡆⠐⠁⢰⠈⠐⠶⡶⡄
⠀⠀⠇⠐⠂⠓⢲⣶⠴⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⠀⠀⡠⠔⠧⣀⠠⠂⠀⣜⢈⠃⠲⡀
⠀⠀⠓⣀⡒⠃⠚⠛⠂⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣂⡞⠀⠐⠀⠀⠀⢰⠋⠀⠁⠀⠈⠪⡢⡀
⠀⠀⠀⢌⠘⠆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢠⡄⠀⠈⡆⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠁⠈⢄⠀⠃⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⠉⠩⢕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⡄⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠚⠛⠉⢳⡀⠀⢠⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⣀⣱⡤⠶⠚⠋⠉⣁⡠⠤⠐⠒⠳⡄⠀⠙⣆⠈⠎⡢⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠋⠉⣀⠠⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⢧⡀⠀⠉⠀⠀⠪⡱⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡔⠁⠀⢀⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠈⢳⡀⠀⠀⠀⠀⢡⠁
⠀⠀⠀⢨⠀⠀⠀⠁⠀⠀⡠⠁⠀⣠⠊⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢷⠀⠀⠀⢳⡀⠀⠀⠀⢸
⠀⠀⠀⠈⠀⠀⠈⠢⠀⠀⠁⠀⡰⠃⣀⡀⠀⢰⠃⠀⠀⢀⡠⠔⠒⠈⠉⠉⢳⣀⠤⠒⠉⢀⡠⠊⠀⠀⠀⢀⣷⠀⠀⠀⢸⢰
⠀⠀⠀⠀⠀⠀⠀⡠⠃⡠⠀⡴⠁⠀⠉⠙⣆⢸⣄⣀⠀⣸⢤⣤⣤⡤⠤⢖⡽⢃⣠⠴⠚⠁⠀⠀⠀⣀⠔⠉⠘⡇⠀⠀⠈⠢⡓⠄⢀
⠀⠀⠀⠀⢀⠈⡔⢀⠎⠀⡼⢡⠶⣂⠔⠚⢺⣿⣀⣀⣉⣉⣑⣒⡲⠶⠿⠓⠊⠁⠀⠀⠀⠀⣠⠴⠊⠁⠀⠀⣠⠇⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⡀⠀⠆⠀⠀⡰⠁⡇⢶⡇⠀⣾⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠉⠉⠳⣤⣬⣯⡄⠀⠀⠀⠀⠀⠀⣀⣀⠤⠔⠚⠉⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⣠⠴⠚⠉⠉⠉⠓⠒⠒⠒⠒⠒⠒⠶⠖⣶⣶⡊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢹⠈⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄
⡔⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⡋⣠⠏⠀⠃⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢋⡠⠃
⠑⠤⣑⠢⢄⣀⡀⠀⠀⠒⠒⠒⠚⠋⠉⠀⢉⣉⣠⠆⠀⠀⣀⣀⣤⠤⠖⠚⢉⣡⠴⠚⠉
⠀⠀⠀⠉⠒⠠⠭⢉⣛⡓⠒⠒⠒⠒⠒⠒⠒⠛⠓⢛⣉⣉⣉⠤⠤⠖⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⣀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠰⠟⠁⠀⠀⠀⠀⠉⠓⠲⢤⣀⣿⡇⠀⠀⠀⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣴⢿⡟⠙⣦⠀⠀⠀⠀⣸⣿⠓⠦⠤⣀⣀⠤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣠⡿⠃⠸⣧⠀⠸⣇⠀⣀⡼⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⣿⠃⠀⠀⠙⠳⠶⣿⠛⠉⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⣸⡿⠀⣾⡏⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⢀⣿⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⣿⠇⢸⣿⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣸⣿⠀⠘⣿⡀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣿⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠉⠉⠉⠉⣿⡟⠀⠀⢿⡇⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⢿⡇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠘⢿⣄⡀⣀⣴⠏⠀⠀⠀⠀⠀⠈⢿⣄⡼⠁
⠀⠀⠀⠀⢰⡏⢱⡄⠀⠀⠀⠁⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⣀⣤⢤⣄⠀⠀⢀⣤⠖⠋⢳⡄
⠀⠀⠀⠀⢸⣷⡀⡇⠀⠀⣠⠤⠶⠲⣿⠖⠶⠤⠤⠤⠤⠴⠂⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠈⡇⣰⡿⠁⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠙⢻⣧⣀⠘⠁⠀⠀⢰⣿⠀⠀⠀⢀⣄⠀⠀⠀⢠⣿⠀⠀⢀⣿⠁⠀⠀⠀⣼⢣⡿⠁⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⢸⠃⠙⢷⣄⡀⣰⠋⣿⠀⠀⠀⢸⣿⠀⠀⠀⣾⡏⠀⠀⣸⣟⢀⣀⡤⠞⠁⣾⡇⠀⢀⡴⠋⣀⣠⡤⠤⠤⣤⡀
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢙⣿⡅⠀⣿⡆⠀⠀⢸⣿⠀⠀⣸⣿⡇⠀⠀⣿⡏⠉⠁⠀⠀⣀⣿⠥⣖⡭⠖⠋⠁⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⣼⠁⠀⢀⡴⠁⠈⣿⡆⢿⡇⠀⠀⣼⢿⣄⡴⠃⢸⡇⠀⢀⣿⡗⠲⢟⡉⠉⢩⣿⠯⣅⡀
⠀⠀⠀⠀⢰⠇⠀⣰⠋⠀⠀⠀⢹⡇⠘⣿⠀⢀⡟⠀⠀⠀⠀⠈⢿⣀⡞⢻⡇⠀⠀⠙⣦⠀⣿⠀⠀⠙⣆
⠀⠀⠀⢠⡟⠀⢰⠃⠀⠀⠀⢀⣿⠇⠀⠘⢧⡼⠃⠀⠀⠀⠀⠀⠈⠉⠀⠈⣿⡀⠀⠀⢸⠀⢻⡆⠀⠀⢸⡄
⠀⠀⢀⡾⠀⠀⠸⣄⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠋⠀⠀⠻⢆⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⢻⡟⡛⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠟⠻⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠠⠄⠀⠀⠰⡀⡆
⣿⡏⢴⢤⣤⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠐⠈⡠⠀⠀⠀⠀⠘⠂
⣿⣷⣈⠀⢀⣡⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠁⠐⠢⠀⠀⢉⣄⠔⡄⠀⠀⠀⠀⡀
⣿⣿⠛⠉⠉⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠤⠇⠤⠋⠡⣌⠄⠀⠀⠀⠀⢈
⣿⡇⠾⠺⠾⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣢⣀⣀⡀⠀⠀⠐⠢⠀⣀⠀⡸⠀⠀⢀⠀⡠⠁
⣿⣿⣦⣤⣤⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⢤⣄⠠⢬⡑⠀⠀⠀⠀⡠⠔⠁⠀⢀⠂⡠⠁
⣿⡟⣁⣀⣀⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠄⠨⡤⠈⢀⠀⠈⠣⡀⢀⣀⠃⡐⠁
⣿⣇⠛⠚⠘⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠑⠢⡀⠈⣢⢶⣜⠂
⣿⣿⠷⠒⠲⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣫⡀⠀⠙⠢⣀⠀⠈⠉⠉⢸⠉
⣿⡏⣀⣤⣤⡀⢿⠀⠀⠀⠀⠀⠀⠏⠐⣀⠀⠀⠐⠁⡠⠶⡀⠀⢒⡲⡃⠉⠰⢒⠖⠁
⣿⣷⣀⠀⢈⣠⣿⠀⠀⠀⠀⠀⠀⢸⠦⠫⡀⠀⠸⣀⠁⠀⠈⠹⡁⠀⠀⣤⢣⠎
⣿⡿⠋⠉⠉⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⠀⠐⠩⠗⣄⠀⠀⠘⣄⢤⢢⢇
⣿⡇⠦⠟⠺⠃⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡠⠞⠛⢃⣋⢻⠃
⣿⣿⣦⣤⣤⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢳⢯⠧⢓⣟⠎
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣵⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⣀⣠
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢿⣿⣿⣿⢀⣀⣄⡀
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠉⠉⠉⠀⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄⠀⠀⣠⠄⠀⠀⠀⠀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⠀⢰⣇⠃⡴⢷⡀⣼⢁⠂⣼⢋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⢾⠁⠀⣱⢎⠁⣿⠂⢸⡧⠂⢀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢈⡟⠀⢸⡄⠰⡟⠁⡰⣷⠀⢼⡇⢀⣿⠀⡤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠞⠁⠀⣈⣳⠄⠉⠋⠀⠈⣁⣈⣷⠂⠈⢉⢸⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠂⠉⡡⠤⠤⣓⢊⣉⡭⡡⢴⠖⠀⠉⢱⠤⢬⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢀⡇⠀⠀⢸⢾⡿⡇⣿⠸⡇⠀⠀⡸⠀⠠⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠣⢄⣀⣘⡈⠥⠬⠑⠒⠹⣋⣉⡁⠤⠤⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠲⣦⠴⢶⣶⠶⢰⣶⠀⣶⡆⣶⡶⠶⠖⣶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣍⠀⢸⣿⠀⢸⣿⠀⣽⡇⢸⣧⣤⠀⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡈⣿⠆⢸⣿⠀⠘⣿⣀⣿⠃⢸⡇⠀⠀⣿⡀

⣿⠛⠛⠙⠋⠛⠛⠛⠛⠋⠻⣟⠉⠛⠋⠉⠙⠛⠋⠉⠉⠉⠉⠉⠛⠉⠉⠉⠛⣛⠻⡟⠉⠛⠛⠛⠛⢻⡟⠛⠋⠛⣿⡟⠉⠛⠛⣿⣏⠛⠋
⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢹⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣧⠀⠀⠀⢱⡟⣆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⢲⣴⠶⠾⠟⠛⠛⠛⠛⠛⠷⠶⢦⣴⠃⠀⠀⢸⠇⠀⠀⠀⢰⠹⡇⠀⠀⠀⣿⢻⡀⠀⠀⠀⣧⢸⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⠀⠀⠀⠀⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠴⢾⣄⠀⠀⠀⢸⠀⣧⠀⠀⠀⣿⢸⣇⠀⠀⠀⣼⠀⣇
⣿⠀⠀⠀⠀⠀⢀⣠⠾⠋⠈⠳⠄⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠸⡆⠹⣦⠀⢠⣿⠀⢿⡆⠀⢀⣿⠀⣿
⣇⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⢹⡇⠀⠀⢿⡀⠙⠷⣾⠏⠀⠈⠻⣶⡿⠋⠀⡿
⡿⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢸⣷⠀⠀⠈⠻⣤⡀⠀⠀⠀⠀⣀⣀⣀⣠⡞⠁
⡏⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⡿⢢⡈⠙⡻⢿⣇⠀⣿⠿⡉⠁
⡇⠀⢀⣿⠁⠀⠀⡴⠛⣷⠀⢠⣶⣦⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠃⠀⢹⡄⡇⢀⠙⣶⠟⢀⣧
⣷⠀⢸⣿⠀⠀⠀⣧⠀⠻⣦⣻⡄⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⣠⠟⢸⡆⠱⡈⢷⣿⠶⠋⢨⡇
⣷⠀⢸⣇⠀⠀⠀⠘⣧⣀⠬⠿⢟⡀⢘⣿⠀⠀⠀⠴⢂⣀⣀⠀⠀⠀⠀⣶⠀⠀⠴⠞⠛⠇⢸⣧⣠⠻⡄⣸⠃⣠⠏⢸⣿⡖⠒⢹⡆
⡇⠀⠸⣿⠀⠀⠀⠀⣼⠿⠒⠶⣤⡗⢨⣿⠀⠀⠀⠰⠟⠛⠛⠻⠀⠀⢀⡼⠟⠒⣦⠀⠀⠀⠀⠈⠻⣾⢡⣿⠟⠁⠀⠀⡏⠘⢻⠋
⣟⠀⠀⢿⡆⠀⠀⠀⣿⠀⠶⣤⣼⠂⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢻⡿⠁⠀⠀⠀⠀⣇⠀⠸⡀
⣿⡷⣦⣬⣿⣄⠀⠀⠘⢣⡄⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡰⢻⡀⠀⡇
⣿⡅⠀⠉⠉⠛⠛⠛⠛⢲⡇⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⡠⢿⣄⡀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⢀⣾⠁⠀⠀⢀⡾⠁⢸⡇⠀⠇
⣏⢻⡐⢶⠤⣀⣀⡀⠀⣼⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠀⠸⣿⢿⣿⣿⣷⠛⠋⠉⠁⠀⠀⢠⡾⠃⠀⠀⣠⠞⠀⠀⠸⣇⠀⢸⡄
⡟⠈⢣⠀⠀⠩⡄⠉⢐⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠹⢁⡏⠀⠀⠀⠀⢀⣴⠟⠀⠀⢀⣴⠋⠀⠀⠀⠀⣿⠀⠘⡇
⣏⠀⠈⠳⣄⠀⠉⠓⢺⣿⠀⠀⠀⠀⠀⠀⡾⢧⣄⣀⠀⠀⠀⠀⠙⠶⠴⠞⠁⠀⣀⣠⡶⠟⠁⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⣿⡀⠀⢳
⡿⠀⠀⠀⠈⠓⢦⣤⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⠀⠀⠀⠀⠀⠀⠒⠚⠉⠁⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸
⣧⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡀
⣧⠀⠀⢀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⣇
⣷⠀⢀⣾⠃⠀⠀⠈⠉⠛⠷⣦⣄⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⡠⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠃
⣿⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣤⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⡜⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⣿⠀⢾⠁⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢻⣤⣽⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢃⡞⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣧⠀⢹⡄⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢾⠁⢸⠍⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢸⣤⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⣿⠀⢸⠟⠛⠳⣄⠀⠀⠀⠀⢻⠀⠀⢸⠷⢿⡀⠀⠀⠈⠙⠛⠲⠶⠶⠶⠖⠛⢹⡆⢀⣿⠁⣿⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡟⢠⠾⠆⠀⠀⠹⡆⠀⠀⠀⢸⡆⠀⣿⠀⠀⠙⠳⠦⣤⣄⣀⣀⡀⢀⠀⢀⡴⢿⣤⡾⠃⢠⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣄
⡿⣾⠒⠀⠀⠀⠀⠙⠦⠀⠀⣸⡷⢦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⡴⠋⠀⠘⢿⡀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⡇⢹⡀⠀⠈⠳⠄⠀⠀⠀⢰⡟⠀⠀⠙⠦⣤⣀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠻⣏
⣧⠈⠳⠤⣤⡀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠉⠛⠶⠶⣾⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇
⣧⠀⠀⠀⠈⠛⢶⣤⣤⣶⠟⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣼⠃
⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⠃⠀⠀⠀⢸⡏⠉⠉⠉⠉⠻⡏⠉⠉⠉⠉⢹⣿⣿⣿⡏⠉⠉⠉⠉⠹⣿⠛⠉⠉⠉⠻⣿⠉⠉⠉⢹⣿⠉⠉⠉⢹⡏⠉⠉⠉⠉⠹⣿
⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⣿⣿⠀⢀⡄⠀⢸⣿⠀⠀⡟⠀⠀⣿⡇⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⣿⠀⠀⢸⠀⠀⣷⠀⠀⣿⠀⠀⠀⠈⡇⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⣿
⣿⣿⠀⠈⠃⠀⢸⣿⠀⠀⠀⠀⣠⣿⡷⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇⠀⠀⠀⠀⢠⣿
⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⡀⠀⣿⣿⣷⠀⢸⣿⣿⣿⣿⣿⣿⠀⢠⣦⠀⠀⣿⡆⠀⢿⠀⠀⣿⡀⠀⣧⠀⠀⢀⡇⠀⢸⣇⠀⢠⣦⠀⠈⣿
⣿⣿⠀⣾⣿⠀⠈⣿⠀⠰⣧⠀⢸⣿⣿⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠉⠀⢠⣿⡇⠀⠀⠀⢠⣿⡇⢰⣿⠀⠀⢸⡇⠀⢸⣿⠀⠈⠁⠀⢸⣿
⣿⣷⣶⣿⣿⣷⣶⣿⣶⣾⣿⣶⣶⣿⣿⣶⣼⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣦⣤⣴⣿⣿⣷⣾⣿⣷⣶⣾⣷⣤⣾⣿⣶⣶⣶⣶⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣤⠤⠤⠤⣶⣖⣒⣶⣾⣿⣿⣭⣿⣯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⣠⣤⡤⢤⣴⣶⣶⣒⣶⣾⣿⣭⣽⣿⣿⣶⣶⣿⣿⣿⣿⡿⠿⠿⠻⣿⣿⣿⣿⣿⣿⣿⢹
⠀⠀⢀⣀⣀⣀⣠⣤⣤⠤⢤⣀⣐⣖⣒⣾⣿⣽⣯⣭⣷⣶⣶⣾⣿⣿⣿⣿⠿⠿⣿⠿⠛⠛⠛⢻⠋⢻⡏⠉⣿⠀⢀⣀⣼⡇⠀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢸⢹⣿⣶⣶⣾⣿⣿⣿⡿⢿⣿⠿⢿⡿⠛⠛⠻⣿⠉⠉⠉⣁⣿⣿⣿⠀⢠⡄⢘⣷⣦⠀⢶⣿⡀⠸⣷⠀⢻⡄⠘⠿⢿⡇⠀⠛⠛⣿⣿⣿⣿⣿⣿⣿⡯⡇
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⠀⢸⡇⠀⣷⠀⢸⣿⡇⠀⣿⣿⣿⣿⡄⠈⠛⢿⣿⣿⠀⢸⣿⡇⠀⣿⠀⢸⡇⠀⢠⣼⣿⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣧⣷
⠀⠀⣇⣿⣿⣿⣿⣿⣿⣿⡇⠈⠛⠀⠸⣇⠀⣿⡆⠈⣿⣧⠀⢻⣿⣿⣿⡿⢷⣄⠀⢻⣿⠀⢸⣿⣇⠀⠿⠇⢸⣇⠀⢻⣿⣿⠀⢸⣿⣿⣿⣿⣟⢿⣿⣿⣿⢻
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣶⡆⠀⣿⠀⢸⡇⠀⣿⣿⠀⢸⣿⣿⣿⣇⡀⠉⢀⣼⣿⣇⣀⣿⣿⣦⣤⣤⣾⣿⣶⣾⣿⣿⣿⡿⠿⠿⣿⣿⣟⣛⣿⣛⠯⠼
⠀⠀⢸⢻⣿⣿⣿⣿⣿⣿⣷⠀⢸⣇⣀⣿⣦⣄⣠⣴⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⡿⠿⢿⣟⣿⣛⣛⣿⡿⠿⠭⠿⠿⠛⠒⠒⠓⠋⠉⠉⠉⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣟⣛⡛⠫⠯⠭⠽⠿⠗⠒⠒⠛⠛⠋⠉⠉⠉
⠀⠀⠀⢯⠭⠭⠯⠿⠐⠂⠀⠛⠋⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠉⠛⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⡆⠀⠀⠀⣿⡆⢠⣾⠿⠿⣷⡜⠿⢿⣿⠿⠿⠂⠀⣠⡾⠿⠿⠿⠗⠾⠿⣿⡿⠿⢿⣦⠀⠀⠀⣾⡆⣴⡿⠿⠿⠿⢶⣾⠿⠿⠿⠷
⣿⣇⣀⣀⣀⣿⡇⣿⡇⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⢿⣇⣀⣀⣀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⣿⡇⢿⣧⣤⣤⡀⠸⣷⣤⣤⣄
⣿⡏⠉⠙⠛⣿⡇⣿⡇⠀⠀⢸⣿⠀⢀⣿⠀⠀⠀⠀⠈⠙⠛⠛⠛⢿⡆⠀⣿⡇⠀⢸⣿⠀⠀⠀⣿⡇⣴⡿⠿⠿⠃⢠⣾⠿⠿⠏
⣿⠇⠀⠀⠀⢿⣧⣽⣷⣤⣴⡿⠃⠀⠠⣿⠀⠀⠀⢀⣤⣤⣤⣤⣤⣾⠇⠀⣿⡇⠀⠘⢿⣦⣤⣴⡿⠁⣿⡇⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠈⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⢀⡈⠁⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣤⣆⣀⣀⡀⢀⡔⠳⣀⡀⢠⠀⠀⠀⠀⠈⠙⣦⢀⡘⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⢶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⣠⣿⣿⠁⠀⢿⣿⣷⣤⡴⠤⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⣤⣿⣤⡤⣿⡇⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⣶⠁⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⢠⡴⠂⠀⠂⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠶⠀⠀⠀⠀⠀⣴⣶⣤⣤⠾⠃⠀⢠⣿⣷⡇⠀⠀⠸⣷⣦⣦⣴⠟⠁⢠⣿⣿⠷⠶⠀⣼⣶⡦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⢙⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⡈⠀⠁⠀⠉⠀⠘⠛⠀⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣧⣼⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⠤⠚⠉⠉⠸⠫⡝⠃⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠻⣿⣿⣿⣿⡇⠀⠀⢰⠀⢿⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⢨⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣿⠟⠹⢿⡇⠀⠀⢸⠀⡈⠣⡀⠀⠀⠀⡹⢿⣄⠀⢀⣴⠗⣠⣻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠘⠶⠒⠞⠁⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠁⠰⠿⣃⣩⠁⡐⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣶⣤⣤⠀⠀⠀⠀⠙⢦⡀⠀⣀⠀⠀⣀⠀⠀⠈⠀⠀⣨⠛⠁⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⢿⣿⣿⣿⡇⠀⠀⠋⠀⠀⠉⠁⠉⠃⠀⠀⠀⠀⢀⡤⠔⠉⠀⠀⠀⠐⠙⢻⠗⠂⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠈⠙⢿⣿⡇⠀⠀⠆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢷⣎⠃⠀⢀⣼⠟⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠷⠴⠿⠟⠁⠀⠀⠀⠀⡀⠀⣠⠆⠀⠀⠀⠂⠀⠀⠀⢠⠀⠁⠀⣠⡾⠃⠀⠀⠀⢀⠊⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣄⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⣀⢀⣤⠧⠰⠄⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡆⠀⢀⡤⠚⠁⠀⠀⠀⠀⠟⠀⠀⠀⢀⣄⠀⡀⠘⠏⠀⠀⠀⠀⠠⣬⠄⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⠿⠓⢆⣽⣿⡧⠐⠈⠁⢀⠀⣀⢤⠔⠀⠀⠀⠀⢀⡈⢁⡀⠀⢀⠀⠀⠙⠛⠁⠀⠈⠒⣸⡗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⠀⣿⣾⡿⠿⠁⠂⣠⡶⠀⢠⠁⠀⠀⠀⠀⠀⢀⣀⣉⣀⣠⡄⡤⠀⠀⠀⡀⠀⠀⠀⠈⣿⠑⣦⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣀⠀⠀⠠⠀⠹⠂⠀⠀⠆⠀⠀⠃⠈⠉⠉⣉⡁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠁⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⣿⣿⡖⠳⢶⣤⡀⠀⠀⠀⢀⣀⣸⢄⠀⣀⠀⢈⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢰⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡇⠀⣹⣻⣿⣧⠋⠀⡂⢸⢧⠄⢻⣥⠘⠁⣾⣤⡁⠀⣼⠁⠀⠀⣄⠀⢀⠓⠄⠀⢰⡜⠂⠀⠂⡸⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⠆⣴⣏⣿⣿⡀⢄⠀⣡⡎⢠⡤⠈⢹⠄⣀⠀⣉⡀⢻⡜⢳⡞⡠⠃⢘⣯⣅⢐⣴⠄⡴⠇⠀⠀⡃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠑⣄⣀⡀⠀⠀⠀⠊⠁⢤⠄⠀⢸⡄⠈⡉⠨⣉⠙⠋⣹⢿⠋⠡⣤⣸⣧⡙⣁⢸⣦⡾⠀⠞⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⢘⣇⢀⣀⠳⠄⢀⠀⠀⠀⠐⠂⠀⠀⠤⠀⠉⠦⠠⠀⡦⠀⣀⢈⣤⣿⣷⡙⢊⡁⢀⡴⣶⢿⡶⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠠⠔⠊⡉⠋⠉⣹⠂⠈⠀⠀⠀⠀⠀⠀⠈⢳⠷⠴⠚⠃⠀⠧⠀⠉⢈⣈⠙⡍⠁⠉⠈⠛⠤⠏⠀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠚⠉⠑⠒⠒⠀⠙⠒⠐⠂⠺⠇⠀⠀⠧⠀⠀⠀⠐⠀⠀⠉⢰⣞⠉⠀⠐⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤