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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⣀⣤⡀⠀⣟⣷⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⣴⡿⠀⠀⠙⢧⣽⣆⢻⣇⣴⣋⡼⠃⠀⠀⢀⠄⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⢻⡿⠛⢿⣿⢷⣦⣤⣤⣀⣈⣉⣁⠀⢉⣁⣀⣠⡤⠞⠉⢀⣿⠏⠀⡙⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣾⡇⠀⣿⠃⠀⠀⠀⠉⠉⠙⠛⠛⢛⡛⠋⠉⠁⢀⣀⠀⢸⡏⣀⣀⠙⠲
⠀⠀⠀⠀⠀⠀⠠⡴⠚⡇⠀⣿⠇⠀⣿⠜⢹⡇⠀⠀⣴⠞⠋⣵⡿⠁⢠⣶⡰⣻⡿⢀⣿⡗⢁⣼⠇
⠀⠀⠀⠀⠀⠀⠀⡷⠚⠉⠀⣿⡄⠀⠿⠀⣾⡇⠀⣼⢃⡤⠊⣿⠁⣠⣾⡿⠁⡿⠁⡹⠻⢷⡟⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⣄⠘⢷⣄⠀⠀⢿⡤⠞⠉⠁⠀⠀⠙⠊⠁⠙⠃⠀⠙⠊⠀⠀⠈⠳⣤⠴⠃⢤⡀
⠀⠀⠀⠀⠀⠀⠉⠓⣮⣷⣬⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣙⣦⣄⡀
⠀⠀⠀⠀⠀⠀⢠⠴⢡⠟⣡⠟⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣭⠀⠀⠀⠀⣼⣿⣿⠀⢀⣴⣾⣿⠿⣂⣤⣤⣤⣾⣿⡇⠀⠀⠀⠀⣾⣿⣧⠰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⢀⣴⢿⣿⣿⢆⣾⣿⣿⠁⠀⠹⣿⡀⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⡟⢠⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠧⠴⠞⠁⢸⣿⣿⢸⣿⣿⡏⠀⠀⠀⣿⡇⢸⣿⣿⡇⠀⠀⠀⣸⣿⣿⡇⠈⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀⣼⣿⡿⠈⣿⣿⣷⠀⠀⢠⣿⠇⠀⢿⣿⣿⠀⠀⢠⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡁⠀⠉⣻⣿⣿⣅⠀⠘⠻⣿⣷⣤⡾⠟⠀⠀⠈⠻⠿⣷⡶⠟⠀⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣶⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⣾⡏⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣸⡿⠀⢀⠜⠀⢀⣴⠆⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⢀⣿⠇⡠⠊⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⣼⡿⠀⠀⠀⢸⣿⠎⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⣿⡇⠀⠀⡠⣾⡟⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠄⠀⠀⠀⣾⡟⠀⡴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⣿⡇⠐⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⡜⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠐⠘⣿⠃⠀⠀⠀⣿⡇⣠⡄⠀⣠⡖⣆⡀⠠⢺⣆⢺⣷⠀⢸⣿⡎⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⢰⣿⣷⢻⣿⢠⣿⠀⣿⠇⡄⣸⡏⢸⡟⠀⣾⣿⣱⠟⠁⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠌⣿⡇⢸⣿⡜⠻⠚⣿⡰⠀⠻⠃⣾⡇⡸⣿⡿⣿⡄⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢰⠀⠿⠇⠈⠻⠃⠀⠀⠉⠁⠀⠀⠀⢿⣧⠇⣿⡇⢻⣧⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒⣁⠀⠀⠀⠀⠀⡤⠀⠀⢄⠀⠀⠀⠀⠈⠉⠀⠛⠃⠀⠙⠷⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢹⣷⠀⢰⣧⠸⡇⢠⣾⠀⠆⠀⣰⣿⠁⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠘⣿⡄⢸⣿⡀⠈⣿⣿⠀⢻⠀⣾⡟⠀⣿⡇⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⡇⡜⣿⣧⠊⢻⣿⠀⡇⠀⣿⡇⢠⣿⡇⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠃⣸⣿⠀⠈⠛⠚⠀⠀⠘⠓⠃⢻⡇⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠚



⣿⠿⠿⠿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣻⣿⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⣠⡿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢲⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⢰⡿⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⠇
⠀⠀⠀⠀⠀⢠⣶⣿⣿⣾⡏⠀⠀⣿⠃⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣏⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢰⣟⣜⣡⡀⠀⠀⣴⠾⠿⣿⣿⠀⢸⣿⡶⠾⣷⡄⠀⣾⣋⣉⣩⡿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢀⣼⡏⠉⢹⣿⠀⣼⠏⠀⢠⣿⣿⠀⢸⣿⠀⠀⣿⡇⠀⣿⠏⢻⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣆⣀⣴⠟⣿⡇⠀⢸⡟⣠⣿⣄⣠⣿⢻⣿⣤⣾⣿⠀⠀⣿⣦⣾⣿⠀⠀⣿⡇⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠁⠸⠿⠇⠀⠸⠿⠋⠈⠛⠛⠁⠈⠛⠋⠘⠿⠀⠀⠛⠋⠸⠟⠀⠀⠛⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⣶⡆⠀⢀⡴⣶⣦⡄⠈⣷⡆⠀⣰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⠁⢠⡟⠀⠉⢹⡇⢠⣿⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣸⣿⢀⣾⣇⠀⣠⡿⠁⢸⡇⣰⠟⣿⡀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣻⡿⠋⠈⠛⠟⠋⠀⠀⠘⠛⠋⠀⠘⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠤⠂⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠈⠀⠀⠀⠀⠈⡀⠀⠀⠀⣠⠀⡃⢨⣄⠀⠀⢠⡀⠀⢀⠀⠀⠀⡰⠅⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢀⢀⣀⠀⠀⠘⡵⡆⠾⠐⢽⡀⢣⢈⠍⠀⠀⠀⠀⠀⡜⠁⢀⣄⠁⠀⠀⠀⠀⠀⠠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠞⢇⠙⡉⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠹⠀⠒⠀⠀⠀⠠⠠⠄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⢀⠀⡤⠀⠐⠿⠀⠌⠀⠀⠀⠉⡟⠹⡇⢰⠀⠸⡄⢸⡆⢰⢸⢇⡔⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠶⠁⡄⠀⠀⠀⠀⠀⠀⡇⠀⡇⠘⠄⠆⢿⣸⠀⢼⢸⠿⡀⠀⠀⠀⠀⠀⠲⠷⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡁⠀⡁⢀⠀⠉⠀⠈⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡜⢡⡅⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣩⠒⠂⠤⠈⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⢠⣥⡠⠆⢠⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⠃⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠅⠻⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠈⠋⠠⠀⠀⠄⠀⠠⣀⠀⠀⠀⠈⠁⠀⠁⠈⠀⠁⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⣇⠒⠂⠒⠂⠐⠠⠄⢠⠀⠀⠀⠰⢰⠀⢀⠀⠀⢰⡄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠃⠀⠛⡎⠃⠀⡄⠘⠃⠀⠟⠀⢤⡀⠀⠀⠺⠇⡄⠈⢀⡀⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠄⠀⠺⠅⠀⠈⣄⠀⠀⠀⠀⢀⡀⠇⠀⢠⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠀⠀⢀⠈⠀⠀⠀⠐⡊⢃⠀⠀⠀⠀⠀⠀⠈⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠




⣿⠉⢹⡟⠿⠻⠿⡟⠿⡛⠻⠻⠻⠿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⡿⠛⢋⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠘⡆
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣀⠀⣿⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⡇⠀⣿⢀⣤⠀⣠⢴⡄⠀⣿⣠⠎⣿⠇⢻⡔⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠊⣿⢰⣇⡀⣿⡀⠿⠁⢰⡟⠀⢸⣇⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠛⠀⠛⠈⠉⠀⣀⠀⠀⠀⠘⠓⠀⠈⣿⠙⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢸⡆⢰⡟⢯⣀⢰⡿⠀⣾⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⢼⡇⠘⣧⣀⠍⠸⣇⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣴⣿⣦⠀⠙⠛⠛⣰⣷⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠆⠻⠃⠀⠀⠀⢰⣿⣷⡄⠀⡿⠛⠁⠀⠀⠀⠘⣿⣿⣿⣿⡶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⢀⡀⠀⡀⠀⠀⠀⠙⣋⠛⠃⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⣤⡀⠈⣧⠀⠀⠈⢿⡇⠀⠀⠀⣀⠀⠀⠘⠙⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣤⡀⠀⠀⠐⠀⠄⠤⠀⠂⠈⠁⠀⠀⠻⣿⠟⠀⢸⠀⠀⠀⠀⠀⢀⡶⠋⠀⠀⠈⠒⢄⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⣿⣿⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⢰⣶⣦⠀⠘⢧⣀⣤⣄⠀⠀⠈⢳⡄⠀⣏
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⣿⣿⢸⣿⡷⢤⣄⣀⣤⣶⣤⣀⣀⣀⣀⣤⠖⠋⠀⠀⢸⣿⡏⠀⠀⠈⠛⠿⠟⠀⠀⠀⠀⢷⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⠀⠀⠀⠀⠀⣿⠇⢸⣿⠁⠀⠀⠀⠉⠉⢉⣉⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿⠁⣦⠀⠀⠀⠀⠀⢠⡶⠀⠀⢸⡇⢠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠀⠀⠀⠀⠀⣿⠀⢸⣿⡴⠂⠲⣄⢰⣿⡿⠋⢻⣆⣿⡶⠋⠀⠱⣆⢸⡗⢠⡟⠀⠀⢀⣾⣿⣦⠀⠀⠀⣸⢃⡿⠿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⢠⣿⢸⣿⡋⠀⠀⠀⠀⠻⠿⠟⢦⡀⣠⠏⠀⠀⠰⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⣿⡀⣴⠚⠁⢸⣿⣿⡇⠀⠀⣼⣿⢸⣿⠙⢆⡀⠀⠀⠀⠀⠀⣠⠼⠋⠀⠀⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⡀⠀⠀⠿⠀⠘⠟⠀⠀⠀⠿⠃⠳⠀⠤⠞⠛⠻⠃⠀⠀⠻⠟⠘⠿⠀⠀⠉⠀⠀⠐⠒⢋⣷⡆⠀⠀⠀⠸⣿⡿
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⢠⡄⢀⡴⠋⠉⠉⢷⣄⢰⣶⣶⠀⠀⢰⣶⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠻⣿⠟⠁⢀⣴⣶⠟⠁⠀⠀⠀⠀⣿⡿⠀⠀⠀⢸⡇⢸⠇⠀⠀⠀⠘⣿⣸⣿⠏⠀⠀⢸⣿⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠟⠀⢰⣿⠀⠀⠀⠹⣧⠀⠀⢀⣾⡇⠸⡇⠀⠀⠀⢰⣿⢻⣿⠀⠀⠀⣼⣿⠀⠀⠀⣸⣿⠁⠀⠀⠄⠉⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⣂⠀⠀⠀⠀⠀⠀⣤⣧⣄⠰⠀⠀⠈⠙⠚⠉⢸⡇⠀⠙⠦⠀⣠⠿⠃⠀⠻⣤⣤⠼⠛⠿⠀⠀⢠⣿⡏⠀⠀⢀⣤⡀⣀⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢆⠀⢀⣤⣤⣜⢿⣿⣿⠀⠻⠿⠆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣠⣼⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠉⠉⠁⠈⠉⢀⣠⡶⠖⠀⠀⠀⠀⣼⣁⠀⠀⠀⠀⠀⠈⠁⠀⠖⠠⣀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠻⣿⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⣿⠀⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠉⠉⠁⣠⠀⠀⢙⡟⠙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠈⠻⢿⣿⣿⡆⠀⠀⠀⣀⡀⠀⠀⠀⠀⢠⡆⠀⠀⠺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⣰⠟⠀⣴⣿⣿⣿⡀⠀⠉⢉⡄⠀⠻⢿⣿⡿⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠠⠔⠚⠁⠀⠀⠹⣿⢿⡟⠁⠀⠀⣸⣿⡆⠠⣆⠀⠀⠀⠀⠠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠃⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⢠⢤⢀⡀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠰⠀⠀⠠⢀⣠⡾⠒⠒⠒⠢⡀⠀⠠⡀⠀⠸⠀⠀⠀⣀⠤⠖⠒⠒⠲⣄
⠀⠀⠀⠀⢀⣠⢤⠶⠖⠒⠒⠒⠶⠦⠦⢥⣴⣄⣀⢀⡼⠋⠀⣇⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⠀⢸
⡀⠀⠀⢠⠋⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⣨⣾⠁⣰⠏⠉⠁⠀⠈⠲⠤⠤⠞⠁⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠒⠤⠤⠖⠋
⢧⡀⠀⠘⢄⡀⠀⣀⠜⠀⠀⠀⠀⣠⢞⡿⢁⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠉⠒⠢⠤⠬⣭⡤⠤⠤⠔⠒⠉⢠⡾⢁⡾⠋⢹⠀⠀⡰⠊⢹⡶⠀⣰⠖⢉⡆⠀⣼⠟⣩⠗⠀⠀⡠⠒⠒⠂⠒⠠⣄
⠀⠀⠀⢀⡔⠋⠉⠉⠳⡄⠀⠀⣴⠏⠀⣼⠁⢀⡟⠀⣸⠃⢀⡾⠁⣴⠇⠀⣾⠃⣸⢷⡊⠀⠀⢠⠊⠀⠀⠀⠀⠀⢄⡀⠱
⠀⠀⠀⠘⣇⠀⠀⠀⠀⣸⡤⠚⠁⠀⢀⣙⣀⣀⡃⠊⠉⠒⠉⠓⠊⠋⠀⠀⠃⠊⠘⠀⠙⠒⠒⠁⠀⠀⢸⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠈⠁⠒⠒⠉⠁⢣⠀⢀⡞⠁⠀⠀⠀⠀⠉⢉⣰⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⢅⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠀⠈⢧⡀⠀⠀⣀⢀⣴⠟⠁⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠁⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⠀⠀⠉⠉⠉⢡⡞⠁⠀⠀⢀⣼⠏⠀⢀⡀⠀⢀⡀⠀⡀⠀⠀⠀⡀⠀⠀⠀⢄
⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⢠⡟⠀⠀⠀⡠⣻⠏⢠⡞⠙⣿⠒⡾⠁⢰⠏⢀⠔⠁⠀⠀⣨⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣾⠁⠀⢀⠞⣱⠏⠀⣿⠀⡰⠋⣼⠁⣠⡏⡀⠊⠀⠀⠀⠉⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠂⠒⠁⣰⠏⠀⠀⠈⠉⠀⠀⠈⠁⠀⠁⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣀⠀⠀⠀⠀⢠⡼⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠑⢄⠀⡰⠟⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢰⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⡀⠀⣀⣠⠴⠛⠁⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⢀⡠⢄⣀⡠⠤⠤⠐⠀⠤⠄⢀
⠀⠔⢁⡴⡀⠚⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⡴⠁⢠⢊⡅⠀⠀⠴⠖⢛⡭⢝⠓⠀⡰⠁
⢀⡎⡞⡞⢀⠀⠀⠀⠊⠀⠀⠀⠀⠀⣼⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠈⠁
⠈⠘⠇⢿⣄⠑⠤⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⠃⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⠀⡀⠀⢸⠀⠐⡄
⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⢱⠀⣠⠴⠲⡇⠀⠸⠁⠀⠄⠀⡇⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⡾⠀⢀⡏⠰⡇⠀⢸⠃⢠⠃⠀⡘⠀⡸⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡰⠃⠀⠸⠆⠀⠈⠀⠈⠒⠁⠀⠀⠘⡰⢓⠒⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠜⡠⠂⠀⡇
⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⡌⠀⣀⠤⠖⠢⡀⠀⠀⠀⢀⣠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠠⠊⠀⠜⢁⠀⠀⢸⠃⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⣒⣒⣪
⠀⠀⠀⠀⠓⠦⠤⠀⠀⠀⢀⡄⠈⠆⠋⠀⢀⠏⠀⠀⢰⠃⠀⡀⠠⡀⠀⠀⡦⠀⠀⡶
⠀⠀⠀⠀⠤⣀⣀⣀⣤⠴⠋⠀⠀⠀⠀⠀⡞⠀⠀⢀⡟⠀⡎⠀⠀⡷⠀⢀⠇⠀⢸⠃⠀⠀⠀⠀⢠⣄⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⣸⠁⠸⣇⠀⣰⠃⠀⢼⠀⠀⢻⡀⠀⠀⠠⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠐⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠈⠉⠀⠀⠀⠈⠀⠀⠀⠑⠂⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢀⡴⠒⠒⠁⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠀⠤⠔⢊⣡⡄⠸⠦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢴⣶⠾⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠄⣄⠀⠀⠀⠀⢸⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠭⢇⠀⠀⠀⣘⡸⠀⠀⢤⡀⠀⠀⠀⠀⠀⠑⠢⡄⠀⠀⠉⠀⠀⠀⠀⢳⠀⠀⠀⣿⠀⠀⠀⠈⠀⡠⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣆⠀⠀⠋⠀⠀⠀⠀⡹⣀⣀⡰⠺⠆⠀⠀⡋⣢⠀⠀⠠⣲⠀⠀⢸⠿⠀⠠⣻⠀⠀⠀⢀⠊⡰⠁
⠀⠰⡐⡆⠀⠀⢸⣗⣣⡇⠀⠀⢀⢔⡄⠀⠀⠅⡟⠒⠿⠋⠀⠀⠀⡅⣸⠀⠀⢠⡇⠀⠀⢸⡆⠀⢸⠋⠀⠀⠰⢡⠎
⠀⠀⠉⠁⠀⠀⠐⡏⠀⠀⠀⠀⡇⠆⠀⠀⠰⢠⠃⠀⠀⠠⡆⠀⠀⠇⡇⠀⠀⢸⠃⠀⠀⢸⠃⠀⠀⠀⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠅⠂⠀⠠⠤⢤⣱⠀⠀⠀⠀⡆⠀⠀⠢⠆⠀⠀⠀⢤⠁⠀⠀⢘⠀⠀⠀⡾⠀⠀⡔⢁⠜⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢻⠉⠉⠀⠀⠀⡘⡶⢄⡀⣀⠠⠒⣆⣠⠊⢠⡤⠤⣾⠠⣀⣠⢏⣀⣠⡷⠁⠀⠀⠀⢀⠼⢦
⠀⠀⠀⠀⢠⡀⠀⠀⡀⠔⢀⡇⠀⠐⢶⣐⣇⡆⠀⠑⠂⠒⠉⠀⠙⠒⠉⠑⠒⠚⠀⠒⠚⢏⠣⣬⠆⠀⠀⢀⠀⢢⡴⠁
⠀⠀⠀⠀⠀⠈⠙⠚⠚⠉⢠⠓⡄⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⢳⠀⠀⠀⠀⢸⢆⠀⠀⠀⠀⢸⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡞⠀⠀⠀⠀⠸⠺⠀⠀⢀⠄⢲⣄⠀⠀⠹⢢⠀⠀⠛⡇⠀⠀⠀⢘⡄⠀⠀⠀⠀⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡇⠀⠀⠀⠀⢸⡆⠀⠀⣆⠎⠁⠈⠀⠀⠀⠻⠀⠀⡆⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⢹⡀⠀⠘⠀⠀⠀⠀⠀⠀⡇⠀⠐⢦⠀⠀⠀⠀⢀⡇⠀⠈⠳⠶⣚⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢔⡀⣐⡆⠀⠀⠈⡂⠱⣄⢀⣀⣀⡀⠄⣀⠊⠡⡀⠀⠀⠀⢀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡠⡰⠀⠀⠈⠒⠒⠒⠒⠈⠀⠀⠀⠙⠥⠬⠨⠤⠋⢁⠈⠯⠀⠘⠦⣤⣀⠌
⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠔⡰⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠳⠦⠤⠲⠭⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠀⠀⠀⠀⣷⡇⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⢀⡀⠀⣿⠇⠀⠀⠀⠀⠀⠤⠄⣼⠇
⠀⣶⣶⡾⠿⠿⣿⣿⠀⠀⠀⣿⡿⢷⣶⣶⣤⣤⠀⢀⣀⣠⣤⣤⣶⣧⠀⠀⠀⣿⣿⠿⣿⡿⣿⣷⠀⠀⢰⣿⣿⣶⣦⣤⣀
⠀⣿⣿⣶⡆⣿⣿⣿⡆⠀⢸⣿⠇⠺⠿⢩⣿⡇⠀⢻⣿⣿⢉⡹⣿⣿⡄⠀⠀⣿⣿⢠⡹⡇⣿⣿⠀⢀⣾⣿⠁⠟⣋⣽⣿
⠀⣿⣿⣿⣇⣿⣿⣿⡇⠀⣿⣿⣰⣿⡇⣼⣿⠀⠀⠘⣿⣿⢈⣣⣌⣿⣧⠀⠀⣿⣿⣸⣷⣀⣿⣿⠀⣸⣿⣇⣶⡆⢻⣿⡏
⠀⠸⠿⠿⠿⠿⠟⠛⠃⠀⠉⠙⠛⠛⠿⠿⡏⠀⠀⠀⢿⣿⣾⣿⠿⠿⠛⠀⠀⠛⠛⠛⠛⠛⠛⠃⠀⠈⠉⠛⠻⠿⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡆⠀⠀⠀⠀⠀⠀⠀⢰⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣤⣤⣤⣤⠀⢀⣤⣤⣤⣤⣤⣾⡅⠀⢠⣤⣤⣤⣶⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣏⢻⠟⣻⣿⠀⢸⣿⠏⣥⣬⠙⣿⡇⠀⠀⣿⡏⢻⣿⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣾⣿⣿⠀⠘⣿⣆⠻⠿⢃⣿⡇⠀⠀⣿⣷⠸⠿⢃⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠾⢿⣿⡏⠀⠀⣿⣿⣷⡾⠿⠿⠷⠀⠀⢿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣶⣶⠀⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄
⠀⠀⠀⠀⠀⠴⠿⠛⠛⠛⢿⣿⣿⠛⠋⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢀⡞⣿⣿⠃⢀⣤⣀⠀⠀⢠⣶⣤⡀⢠⣤⡄⠀⣤⣶⡄⠀⣤⣤⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣠⠞⠀⣿⣿⢀⣾⣿⡟⠀⢠⣿⡿⠁⣼⣿⣿⠁⢠⣿⣿⠁⣼⣿⡿⠀⣾⣿⢃⡴⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⢰⣿⣇⡞⢸⣿⡇⠀⣸⣿⠃⠀⣻⣿⠇⢀⡞⣿⣿⣸⢹⣿⡇⢠⣿⣯⡞⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⡇⠀⠀⢸⣿⣿⠁⠈⣿⣇⣰⢻⣿⠀⣰⢿⣿⣠⠟⠀⣿⣿⡇⢸⣿⡇⣸⣿⡿⠻⣿⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡾⠟⠀⠀⠀⠘⠿⠏⠀⠀⠛⠛⠁⠈⠛⠛⠁⠘⠛⠋⠀⠀⠛⠿⠃⠀⢻⣇⠙⠿⠃⠀⠘⢿⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡄⠀⢰⣶⣦⢠⡿⣃⣀⠀⠈⢷⡀⠀⣴⣦⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠁⠸⣿⡇⠀⣾⣿⡏⠸⣿⣿⡟⠀⠀⢸⣇⣼⣿⡟⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⣿⡇⡼⣹⣿⠁⢰⣿⣿⠷⠤⠴⡾⠋⢸⣿⠃⢀⣿⡟⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⣿⡏⠀⠈⢿⣿⣆⣀⡴⠁⠀⢿⣿⣤⠞⣿⡇⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣸⣟⣀⣀⣀⣀⣉⣉⣁⡀⠀⠀⠈⠉⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⠚⠛⣻⡟⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⣿⣷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠶⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣧






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡀⠀⡠⢄⠀⠀⠀⡀⡀
⠀⠀⠐⠆⠀⠀⠀⠀⠀⠘⣀⣀⡀⠀⢀⢀⡈⠉⡇⠀⡏⠉⠉⠀⡐⠀⠀⠀⢂⠀⠁⠀⠢⡀⠀⠉⠀⠁⠀⡧⠊⠀⢀⠇⠀⠨⣀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠓⠀⠃⠀⠀⡰⠁⢀⠈⡄⠈⢀⠀⠀⠀⠈⢢⠀⠀⠐⠀⠀⠀⠰⡁⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⢐⠀⢐⠀⠀⠀⡇⠀⠀⠀⠀⠀⡠⠀⡀⠀⠘⠁⠀⠀⠀⠀⠀⠸⠀⠀⢠⡀⠀⠀⠀⢀⠀⡔⢄⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣇⡀⡄⠀⠀⠀⡇⠀⡆⠀⢠⢄⠌⠀⠀⠁⣀⢀⡅⠀⠀⠈⠢⠀⠀⢠⠠⠇⠀⠇⡀⠔⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠉⠁⠀⠈⠉⠁⠀⠀⠀⡠⠤⡀⠀⠉⠉⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠊⠐⠄⡀
⠀⠀⠀⣠⠘⠀⠁⠐⠀⠀⠀⠀⠀⠇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠂⠀⠘⡀⠀⢈⠂⠐⠄⠀⠀⢀⠄⠀⠀⠀⠀⠠⢆⠀⠀⠀⠀⠀⡃⠀⠀⠔⠁
⠀⠀⠀⠀⢂⢀⠀⡠⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠴⠈⡀⢀⠉⢂⢀⢄⠀⠀⠈⠂⠀⠀⠀⠀⠐⠖⠀⠀⠀⠀⠀⠁⠀⠘
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡇⠸⡀⢀⠇⠨⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠁⠀⠈⠂⠀⠀⠀⠀⠀⠈⠀⠐⣤⠀⢀⠠⠊⠐⠊⠀⠀⠀⠀⠀⠀⠊⠀⠀⠁⠀⠈⠂
⠀⠀⠀⡠⠀⠀⠠⡄⠀⠀⠢⡀⠀⠀⠀⡠⠂⠀⠰⣁⠀⠀⠀⠂⠐⢄⠀⢀⠦⠀⠐⠀⠀⠀⣤⠀⠀⠀⠢⡀⠀⠀⠀⡠⠂⠀⠀⡈⠠⠔⢱
⠀⠀⠈⠀⡐⠀⡆⠈⠀⠀⠀⠈⠠⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡠⠊⠀⠀⠘⢄⣉⡖⢠⣊⣠⠃
⠀⠀⠐⢄⠈⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠂⢢⠀⢠⠒⠂⡴⠀⠈⠁⠐⠀⡰⠀⠤⠀⠄⠐⠰⠀⠀⠀⠀⠀⠀⠀⠀⠰⠥⡔⠁⡟⢤⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠡⡀⠀⠢⠁⠀⠔⠀⠀⠒⠢⡀⠀⠑⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠡⠊⠑⠜
⠀⠠⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠞⠀⠈⢄⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠉⠀⢀⠀⠀⡀⣀⠀⢀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠢⡀⠀⠀⠀⠀⡠⠚⡀⠀⠁⠁⠀⡰⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⠄⣩⠀⠀⠀⢠⡆⠒⢢⠀⠀⠀⠀⠀⠐⠀⠘⠀⠀⠀⠁⠒⠂⠁⠀⠀⠈⠂⠀⠀⠂⠀⠀⠀⠀⡀⠢⢀⠁⠈⠉⠉⡆⢩
⠀⠀⠀⢀⠆⠀⠈⠣⣀⣀⡌⠀⠀⠈⢑⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠪⠕⠀⠀⠀⠀⢠⠠⡀⠀⠀⠀⠀⠔⠁⠀⠈⢇⠀⠀⡴⠋⠀⢑
⠀⠀⢀⠰⢀⣀⣀⣰⠡⢒⠀⠄⠒⠒⠡⡤⠉⠀⠀⠀⠀⠀⠀⢠⠂⢀⠔⠢⡠⠐⡄⠀⠈⠈⠀⠀⠀⠆⡈⢡⠤⠀⣠⠈⠠⠙⡀⠀⠀⠀⡇
⠀⠀⠁⠒⠊⠁⠀⠀⡱⠀⠺⡀⠀⠀⡠⠃⠀⠀⠉⠀⠀⠄⠉⠒⠄⢸⠀⠀⡇⡠⠃⠉⠠⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⡦⠉⡄⠉⠁⠙⠣⠤⠕
⠀⠀⠀⠀⠁⡆⠀⣰⠁⠀⠀⠘⡆⢤⠂⠀⠀⠀⠀⠠⠉⠙⢄⠀⠠⠚⠀⠀⠓⠄⠀⡠⠋⢈⠆⠀⠀⠀⠀⠠⣀⢠⠎⠀⠀⠰⡄⠀⢠⠐
⠀⠀⠀⠀⠀⠀⠇⡸⠂⠔⠂⠐⠑⠠⠀⠀⠀⠀⠀⠔⠉⠀⠐⠃⠀⠠⠀⠀⠄⠀⠙⠂⠀⠉⠠⠀⠀⠀⠀⠀⣃⠴⠄⠠⢀⡠⡍⠲⠈
⠀⡖⠀⢠⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠠⢤⠀⠀⡀⠀⠀⣀⠀⠀⡤⠄⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠈⠀⠁⠀⠀⠀⠀⡄⠠⡃⠡⠀⡂⠀⠀⠀⠀⠈⠂⠔⠈⠀⢂⠀⠐⠨⠄⠂⠀⡐⠀⠁⠢⠒⠁⠀⢀⠠⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⡠⠐⢠⡜⢢⠔⠁⡢⠀⠀⠀⠀⠘⠤⠠⠔⢨⠋⠒⠠⠄⠒⠉⡁⠢⠤⠠⠁⠀⠀⠃⠠⠂⢢⠀⠂⠀⠀⠘⠃
⠀⠀⠀⡀⠀⠀⠀⠀⠹⢁⡜⠣⣀⣡⠀⠀⠀⠒⡄⠀⠀⠃⠀⢎⠀⢀⠂⠐⡀⠀⡑⠀⠐⠀⠀⠀⠀⢆⠈⠂⠊⢀⠂⠀⠀⠀⠀⠀⠀⠀⢂⣀⡐
⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠐⠜⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠁⠁⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢨⡏⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣤⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⢨⣿⠀⢰⡄⠀⠀⣰⠖⠀⠀⠀⢀⠀⠀⠀⣿⠂⠀⣾⡇⠀⢲⡀⠀⢸⣟⠀⢰⣦⠀⢠⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣯⠀⣸⡇⠀⢰⣿⠀⣸⡇⠀⢸⣧⠀⢰⡿⠀⠀⣿⡇⣶⠟⠀⠀⢸⡇⠀⢸⣿⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢻⡇⢀⠘⠿⠀⣿⠇⠀⠘⠿⠄⢹⡇⠀⠘⠿⠀⣾⡇⠀⠀⠻⠇⢿⡄⠀⠀⢸⡇⠀⠈⠿⠀⠘⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠈⠙⠀⠀⠈⠇⠀⠀⠀⠀⠀⠈⠀⠀⠘⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠿⢿⡆




⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣆⢀⠀⠀⡁⠀⠆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢀⠙⢻⣿⣿⣧⡤⢈⣀⢀⠀⣀⡴⠠⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⣄⠁⠀⣶⡎⡉⣿⣯⠙⣾⣾⣷⡾⠿⣷⣾⣧⡆⠀⢤⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⣿⣿⣿⣷⣶⠾⣧⢴⠭⣍⣤⡟⢯⠙⣿⣿⣿⠌⣿⣅⣨⢎⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⣫⠛⢿⡟⣎⣷⠈⢆⡹⣅⢀⠱⠊⠉⣈⣝⣉⡴⠿⠐⣾⢸⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠄⠙⠲⢬⡆⢰⠀⠜⢿⣧⣠⡙⢾⡇⣄⣈⡀⣋⢍⢸⣿⣶⣅⣋⣽⣔⠧⠊⠙⣼⠀⠦⣄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠳⢴⠄⠙⠋⣙⠂⠂⠠⣌⣉⣳⠶⣟⣫⣭⣝⣿⠯⣼⣿⣿⣿⠿⠀⠀⠀⠀⠸⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠄⠠⠄⢸⠋⠰⣦⣿⣿⣿⣷⡻⠿⢻⡏⣴⣿⣿⣿⣷⣏⣠⢁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⠀⣄⢀⣀
⠀⠀⠀⠀⠰⠀⠀⣌⠀⠠⢿⣿⣿⣿⡿⠳⠖⢸⡃⢿⣿⣿⣿⣿⡿⢽⣄⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡴⠟⠁⠀⠀⣿⠈⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠈⡀⠈⠀⠛⣔⣂⣉⢻⣜⠙⠾⣉⣤⣂⣰⣿⣿⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣼⠏⠀⠀⣠⠖⠀⠉⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠁⠃⠀⠙⠓⠚⢿⣉⠁⡨⠉⡙⠲⠛⢩⣯⣄⠈⠙⠻⢿⡆⠀⠀⠀⠀⣶⡴⠞⠛⠛⠉⡄⠀⠾⠏⠀⠀⠀⣠⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠀⡉⠉⠁⠚⢿⣧⣤⠞⠃⠀⠀⠀⠉⠀⠀⠀⠀⢄⠻⣤⡀⣀⣰⣟⢁⣙⣉⠀⠀⠁⠘⠋⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⣶⡎⠀⢀⡄⠀⠀⠀⢲⡄⠀⠈⠻⠙⡁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠉⠁⢀⣤⡄⠐⠐⠀⢀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠘⠦⣰⣿⣇⡀⠀⠀⠀⣷⡄⠀⠀⠀⣇⣼⣇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠋⠀⢀⡀⣾⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠂⠀⠘⠋⠀⠀⠀⠀⠹⢿⣿⡲⡆⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣾⠁⠋⠀⠀⠨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⠶⠶⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠳⣄⠀⢀⣀⣀⣀⡤⠒⢉⡁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢖⢢⠤⠚⠋⠉⠀⠀⠈⠉⠒⢯⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡸⠁⠀⠶⠀⠀⠀⠈⠀⠀⠀⠀⢣⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠃⠀⠀⠐⠆⠀⠀⠀⠲⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣛⣒⠆⠀⠀⣖⢒⠆⠠⣖⠤⠄⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡨⢧⡒⠀⠀⣦⡬⠯⣴⠆⠉⠁⣸⠃⠀⢠⠚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠲⢤⣀⣀⣀⣀⣠⠴⠚⠉⢆⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⣇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠚⠓⠒⠒⠚⣋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢶⠄⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⢻⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢽⠃⠀⢀⣺⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⡞⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠀⠀⢼⣁⣞⣩⠾⣀⠼⣟⡃⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⡀⡀⢀⠀⠀⣤⣄⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠫⡞⠫⠞⠧⠎⠀⠀⠛⠋⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠲⡶⠶⠶⠒⠒⠒⠒⠒⡒⢒⠒⠒⠛⢻⢛⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⠋⠛⢹⠆⠀⠀⠸⣙⠋⠋⣉⡷⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⢸⣿⡟⣀⣤⣶⣿⡿⠃⠀⠀⠀⠀⠀⠀⣿⡏⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⣤⣾⣿⣿⡿⠿⠛⠉⢀⠀⠀⣴⡄⣰⡆⢸⣿⠁⣼⡿⠁
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⡿⠿⣿⣿⡟⢹⣿⠇⣀⣄⠀⢀⣼⢟⡅⠀⣿⣿⣿⣃⣾⣿⡞⠋⠂
⠀⠀⠀⠀⢠⣾⣿⡿⠟⠋⠁⠈⠀⣿⣿⠁⣾⡟⣰⣿⡏⢀⣾⡏⣼⢇⣼⣿⢻⣿⡿⣿⠇⣿⣦⣴⠇
⠀⠀⠀⠀⠸⠟⠉⠀⠀⠀⠀⠀⢠⣿⣿⢰⣿⣧⢣⣿⣷⣿⣿⡞⣿⡟⠹⠿⠀⠉⢸⣿⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⣸⣿⡇⣾⣿⡟⢸⣿⠏⠈⠉⠀⠀⠀⠰⠾⣷⡄⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⣿⣿⠁⢿⣿⢇⣴⡆⢀⣤⢶⣦⢸⣷⠀⣶⠌⡿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⣸⣿⠃⣠⣦⠀⣼⣿⢇⣾⡇⠀⣿⣼⡇⣸⣿⢠⠇⣈⣛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠑⢠⣿⡇⢰⣿⣿⢹⣿⣳⣶⠛⠿⡶⠻⣿⠏⠀⢿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⢿⣿⡟⠘⠿⠟⢁⣀⣠⣤⣤⣶⣶⣶⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠃⣼⣿⣷⣶⠾⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠛⠁⠀⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠋⠀⠀⠀⣸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣴⣿⠏
⠀⠀⠀⠀⢀⣼⣿⣿⠏⠀⠀⠀⠀⢠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠟⠛⠋
⠀⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⢠⣾⣿⣿⠃
⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⣠⣿⣿⣿⠏
⠀⠀⠀⢿⣿⣿⡇⠀⠀⢀⣴⣿⣿⡿⠋
⠀⠀⠀⠈⠻⣿⣷⣶⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠤⠷⣾⣈⣉⢉⡁⣿⢥⣅⠀⢸⣧⣤⠁⠀⠀⠀⣋⣚⢛⡂⠀⠀⠀⢀⣛⣛⡛⣀⠀⠀⠀⠀⠀⣿⢾⣦⡄⠀⠀⣼⣾⡿⡇⠀⢠⢶⣤⡄⠀⠀⢴⢳⢶⡼⠂
⠀⠉⠉⠈⢉⣿⣛⡏⠉⠈⠉⠀⢨⡟⣛⡂⠀⠀⠀⠇⢹⠻⠇⠀⠀⠀⠿⠿⠻⠄⠩⣅⠀⠀⠀⠐⣁⣰⣋⣐⢀⠀⣨⣤⣤⡀⠀⢨⣿⣭⠃⠀⢨⣭⠿
⠀⠀⠀⠀⠀⣭⣭⡅⠀⠀⠀⠀⣈⣦⣾⣿⡂⢉⣭⣉⢿⣤⠀⠀⠀⡀⣼⡏⠁⠁⡾⣾⡇⠀⠀⠀⠹⡏⠻⠼⠻⢧⠟⠀⣀⡀⠀⢀⡽⠾⠶⠦⠩⠍⠁
⠀⠀⠀⠀⢰⣰⣶⡆⠀⠀⠀⠀⢈⣱⡯⠄⠁⠀⠀⠒⠚⣧⠀⠀⣈⣷⣏⡱⢤⢶⣦⣼⣳⠀⠀⢘⣛⢿⠄⠀⢲⣤⣆⣶⣸⡆⠀⢸⡃⠿⠽⠄⠀⡹⡭⣁
⠀⠀⠀⠀⢈⣋⢹⡅⠀⠀⠀⠀⢨⢻⣥⡀⠀⠀⠀⠥⠤⠬⠀⢀⣉⣉⡿⠉⠈⠈⠩⣮⢹⣄⡀⢠⡷⠘⠇⠀⠀⠁⢉⣉⢉⠁⠀⢠⣅⣬⡁⠀⠀⠈⠷⢟⠾⣀
⠀⠀⠀⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠁⠀⠀⠀⠀⠉⠈⠉⠁⠈⠉⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠈⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣤⣤⣀⠀⠀⠀⣤⣤⠠⠄⠀⠀⢠⠀⡶⣦⢦⣤⠀⠀⠀⠀⢠⡤⠄⠀⠀⠀⠀⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡈⣉⠀⠀⣨⡭⣬⠄⠀⣤⣧⣤⠀⠀⠠⢦⣤⣤⠀⠀⠀⣦⣿⣴⡆⠀⠀⠐⢺⢷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⠟⢧⡟⠑⠁⠀⠀⣏⡻⠿⠀⠀⠀⠀⣈⠛⣒⡀⠀⣠⣄⣠⠀⠀⠀⢠⣮⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⠛⣿⠁⠀⠀⢠⡛⣇⣥⠀⠀⠀⠀⣀⣒⣚⠀⠀⢼⣿⣽⡆⠀⠀⢀⣦⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣙⣝⠀⠀⠀⠀⠛⢫⡿⢧⣤⣤⣤⣦⡄⠂⠀⠀⠠⠤⡤⡶⡄⣴⣶⣾⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠟⠛⠀⠀⠀⠀⠀⠀⠃⠀⠰⠶⠖⠘⠁⠀⠀⠀⠀⠀⠘⠺⠾⠌⠇⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡴⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⡀⠀⠠⡄⠀⠀⢰⡏⢸⡏⣀⢀⣀⡀⠀⢀⣤⢸⣗⡀⢰⡇⠀⠀⠀⣿⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢺⡇⡸⠇⣏⠾⢽⣧⣿⢻⣏⠼⣿⡁⣼⡇⢾⣼⠃⢿⠀⠂⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠉⠀⠈⠁⠀⠈⠉⠀⠀⠛⠙⠷⠀⠈⠋⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠆⠈⠉⠰⠄⠀⢰⠂⠈⠉⡏⠁⠋⠁⠁⠁⠀⠁⠋⠀⠀⠈⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡦⠀⣀⣠⠀⢀⠀⡀⠀⠀⡇⠀⡄⠒⠀⡄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠧⠠⠄⠀⠄⠌⠀⠀⠄⠀⠀⠀⠀⠄⠄⠥⠀⠀⠀⠦⠄⠀⠠⠄⠠
⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡄⣤⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⡀⢈⠀⠃⠃⠛⠃⠋⡁⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠹⠀⣙⠋⠀⡐⢐⣒⢠⡀⠙⠉⠀⠍⠀⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠒⠒⠁⠀⠀⠀⠀⢠⣟⡅⠀⣼⠋⠀⠀⠈⢆⠈⢓⣵⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠈⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⢀⠄⠹⣟⠠⣀⣿⡎⢰⡖⢰⣧⣰⡆⠈⣴⠂⢠⡖⣿⠀⠀⠠⡀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⡏⠸⣇⠜⠗⠃⣼⡇⣿⠁⠙⠖⠉⠉⢻⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⣴⡿⠛⠻⣷⡸⣿⡄⣸⡿⠀⣿⣟⣛⡃
⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⢿⣧⣀⣠⣿⠃⢹⣷⣿⠁⠀⣿⣟⣛⡃
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⡉⡙⠉⠁⠀⠀⠉⠁⠀⠀⠉⠉⠉⣁
⠀⠀⠀⠀⠀⠀⠲⠾⠂⠿⠿⠷⠃⢻⠟⠿⠻⠞⠰⠹⠺⠙⠏⠿⠇




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢉⣉⡙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢫⣉⠀⠉⠑⢮⡇⠀⠀⢀⣠⠤⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⡞⠁⠀⠰⠙⡳⣮⣀⠀⠙⠦⠚⠉⢀⡀⡠⠄⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡀⠐⠆⣁⠀⠐⠆⠉⡻⢶⣤⣀⡠⠠⠤⣄⡀⠴⢠⠀⢳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠳⣮⣄⠖⢠⣔⡠⠈⠁⠢⠍⢛⠳⠶⣬⣥⣤⣀⣀⣸⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠙⡆⠈⠀⠉⠛⠶⣤⣈⠀⢀⡀⠦⣔⢊⡀⠒⠠⠈⠉⢩⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡘⣄⠀⢬⡉⠁⠐⠉⣛⠷⢶⣬⣀⡙⠒⢈⠀⠁⣠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣀⠣⠢⠬⠘⠆⡘⠲⠀⠩⢙⠛⠓⢦⠾⠥⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣍⠓⠶⣦⣄⣀⣐⠀⠈⠀⣠⠞⠁
⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠤⠴⠖⠚⠉⠁⠀⠈⠳⣄⠀⠘⠋⢉⣹⠖⠛⠶⠤⠤⢤⣤⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠖⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠐⠂⠈⠉⠀⠉⠉⠉⠀⠐⠒⠒⠂⠠⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⢀⠴⠒⠒⢤⡀⠀⡠⠖⠒⠒⠲⢄⠀⠀⠀⠀⠀⠀⢀⠔⠒⠒⢄⡀⠀⡠⠖⠒⠒⠦⡀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⡌⠀⢐⠄⠀⠈⠎⠀⠤⡊⠔⢀⠈⡆⠀⠀⠀⠀⠠⠁⠀⡂⠄⡀⠑⣜⠀⢔⠡⢊⠀⢹⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠰⡀⠀⠄⢊⠔⡀⠔⣊⠔⡊⠅⢠⠁⠀⠀⠀⠀⢸⠀⠨⢔⠩⢂⠀⡠⢐⠡⢂⠕⠀⠐⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠄⢊⠔⡪⢔⠨⠐⡠⠁⠀⠀⠀⠀⠀⠀⢆⠀⠠⢊⠄⡨⢐⠡⢊⠅⠊⡠⠁⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠩⢂⠥⣢⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠁⡨⢂⠥⢊⠕⢊⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠡⠊⢔⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠈⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠃⠌
⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⡀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉⠀⠀⠀⠀⡀⠆⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠒⠒⠒⠒⠀⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⡠⠀⠠⠁⢀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢤⣁⣈⠀⠂⠐⠂⠀⠁⠀⠊⠐⠁⠊⠀⢁⣠⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠒⠐⠊⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⡀⠄⠀⡀⠄⠂⢀⠠⠀⣀⠤⠀⡀⠄⢀⠠⠀⣀⠄⠀⢀⠄⠀⡀⠄⢂⠠⠒⡀⠄⢂⠀⠀⡀⠄⢈⠡⠀⡁⠀⡠⠀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠡⠔⠊⢁⠄⠒⢈⠤⠒⣁⠠⠒⡁⠄⠊⡁⠔⢊⡠⠔⢈⠠⠐⡉⠔⢈⠠⠐⢉⠤⠂⡡⠐⠉⠠⠂⠁⠔⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠉⠀⠂⠁⠀⠈⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⢀⣤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⢠⣿⠃⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣁⡀⠠⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣿⡄⠀⠠⢼⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⢸⣧⠀⠀⠀⣰⡿⠉⠀⠀⠀⠀⢠⡀⠀⠂⢸⣿⠇⠀⠀⣸⡿⠠⠂⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⢠⣿⠃⠀⣼⣷⠀⠀⢸⣿⠈⠀⣾⣿⠀⠀⠀⣿⣧⢰⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⣿⡏⠀⠀⡸⣿⡀⠀⢿⣿⠀⠀⠼⣿⠂⠀⣿⡏⠀⠀⡐⢿⣿⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣀⠔⠀⠈⠁⠀⢿⡇⢀⠌⠀⠈⠉⠀⠸⣿⡠⠂⠀⠉⠀⠀⣿⠇⠀⠀⠀⠈⠁⠀⠻⣧⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢀⡀⠀⠀⢀⣾⠟⠠⣄⠀⠀⣀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⡀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇⠀⠸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⢸⣿⡇⠀⠀⠉⠃⠀⠀⠀⠀⠿⣇⠈⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⡀⠤⢀⠀⡀⠴⠀⢄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣰⡿⠁⠀⢸⡄⠄⠀⠉⠀⠨⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠟⠁⠀⠀⠀⠳⣄⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡴⠛⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⡇⠀⠀⠀⠀⡞⠈⡇⠀⠀⠀⠀⢠⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡴⠃⠀⠀⠀⠀⠳⡴⠃⠀⠀⠀⠀⠈⢷⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣾⣆⠀⠀⡇⠀⠀⢰⣷⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠶⠷⠶⠶⠿⠿⠶⠶⠷⠶⠶⠿⠿⠷⠶⠾⠷⠶⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⣤⡀⠀⣠⣄⠀⠀⢠⠶⢦⡀⠀⡴⣦⠀⢠⠶⡄⠀⣤⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠲⠋⠀⣷⢀⡏⠈⣇⠀⣿⠀⠀⠳⠼⠁⢸⡄⣾⠀⠹⢦⠟⠹⣦⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠛⠁⠀⢻⣠⡇⠀⠀⠀⠀⠀⠈⠳⠏⠀⠀⠀⠀⠀⠀⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⣦⣶⣴⡦⣶⠶⡴⢰⣶⢼⡶⢶⣲⢦⣶⣇⣦⢦⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡈⡀⠁⠈⠁⠀⠀⢀⠀⣀⡀⢀⢀⡀⡀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠁⠈⠀⠈⠁⠀⠀⠀⠁⠈⠈⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠉⠙⠳⢦⡀⠀⠀⣀⡤⠶⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠹⣦⠞⠁⠀⠀⠀⢀⣀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣰⡟⢻⣿⣶⣿⣿⣿⣷⠀⢸⣤⡞⣹⣿⣾⣿⣿⣷⡀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣇⣿⣿⣿⣿⣿⣿⣿⠇⢀⡿⣿⣾⣿⣿⣿⣿⣿⠁⠀⢸⠇⠀⠀⠀⢀⡀
⠀⢀⣤⣤⣤⡀⠀⠀⢳⠀⠀⠘⢿⣿⣿⣿⣿⣿⡿⠋⠀⣸⠃⠈⠻⢿⣿⡿⠛⠁⠀⢀⡾⠀⠀⠀⠛⠉⡟⠷⡄
⠀⠋⠉⣧⠀⠀⠀⠀⠈⢳⡀⠀⠀⠈⠙⠻⠛⠉⠀⢀⣴⠋⠳⣄⡀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⢰⠃
⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠉⠳⢤⣄⣀⣀⣀⣤⠶⠋⠁⠀⠀⠈⠙⠓⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠤⣤⣤⣤⣤⣤⠤⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣤⡤⠖

⠀⠀⠉⠁⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠈⠃⠀⠀⠙⣷
⠀⠀⠀⠀⠈⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⡶⠿⠟⠻⢶
⣠⣤⣤⣴⣶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⣤⣤⣤⣶⠶⠟⠁⠀⢠⣴⠟⠛⡙⠛⢶⣼⠋⠀⠀⠀⠈
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⢠⡿⠁⠐⠂⠀⠀⠠⠉
⠀⠀⠀⠀⠀⣀⣤⡶⠶⢶⣤⣀⣴⠋⠁⠀⠂⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⣴⠏⠀⠀⠂⠀⠈⠻⠃⠀⠀⠀⠀⠠⠄⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣤⣄⣀⠀⠀⡀⢀⣴⠟⠁
⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣤⣿⠃
⠀⠀⠀⠀⠈⠻⣦⣄⣀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣤⡶⠶⢶⣴⡟⠋⠩⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣦⣄⠁⣰⡟⠁⠀⠀⠀⠀⣾⠃⠀⠀⠠⠉⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠟⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⢀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣦⣄⣀⢁⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢛⠛⠻⣦
⠀⢀⣴⠶⠶⢶⣴⠟⠉⠉⠉⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁⠀⠘
⠀⣾⠁⠀⠀⠀⠈⠀⠀⠀⠔⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠉⠙⠷⣦⣠⣴⠾⠛⠛⠷⣦⡀⠀⠀⠀⠀⠻⣆⠁⡀
⠀⢻⣇⠀⠀⠀⠀⠀⠀⠁⣠⡾⠃⠀⠀⠀⠀⠀⠀⢠⡿⢀⠆⠀⠀⠀⠀⠹⠟⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠙⠻⠶⣦⣤⡀⢡⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⣴⠾
⠀⠀⠀⠀⠀⠀⠈⠙⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⠐⠀⠀⠀⠀⠄⣀⣤⡾⠛⠁⠀⠀⢀⣴⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⢀⣴⠟⠋⠀⠀⠀⣴⠟⠛⠻⠁⠀⠀⣿⠂
⠀⢀⣤⣶⣤⡀⠀⠀⠀⣿⠁⠀⠀⠀⢨⡿⠾⠷⢶⣄⠀⠀⠀⠀⠀⠀⠹⢷⠟⠁⠀⠀⠀⠀⠀⣿⡀⠀⠄⢀⠀⣴⠏
⠿⠿⠁⠀⠈⣿⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⡀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠷⢶⣾⠃
⠀⠀⠄⢀⣴⠏⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄
⠶⢶⣦⣾⠃⠀⠀⠀⠀⠀⠀⢹⣧⡶⠶⠶⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠶⣦⣄⣠⡾⠋⠠⠀⠀⠌
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠛⠻⣦⣀⣀⠀⠀⠀⣼⠋⠠⠀⠀⠀⠈⠿⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⠀⠙⠉⠙⢷⡄⢸⡏⠈⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⣴⠟⠋⠉⠉⠛⢷⣄⢀⣠⣴⣶⣤⣄⡀⠀⠀⠀⠈⢿⡄⠀⣀⣀⣀⣾⠃⠘⣿⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣠
⠀⠀⠀⣸⡏⠀⠀⡀⠈⠀⠀⢻⠟⡁⠄⠀⠀⠉⢻⡄⠀⠀⠀⠘⠿⠛⠛⠉⠉⠀⠀⠀⠘⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⢠⡾⠋
⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⣦⣄⠀⣰⠟
⠀⠀⠀⠘⢧⡀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠟
⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⣀⣤⡶⠟⠋⠉⠀⠀⠀⠀⢀⣤⣴⣶⣤⣄⢠⡾⠋⠁⠀⠀⠉⠻⣆⠀⠀⠀⠀⠀⣠⣤⣴⡟⠋
⠀⠀⠀⣀⠀⠀⠀⠀⠀⢻⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠉⠿⠁⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⣾⠋⡀⠈⠀⡀
⣤⣴⠟⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠻⣦⣄⣀⣁⢀
⠈⠁⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠤⣴⡟⣚⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⠀⢤⡀⠀⠀⣰⡟⣰⢿⡾⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢀⡄⣸⢇⣠⣀⠀⠀⠀⣀⣁⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⢡⣾⠵⠋⠀⠀⡎⢀⡾⠀⢠⡟⣸⢀⣴⢋⡞⠀⠀⣠⠞⣱
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢰⡿⠓⢲⡿⠁⠀⠁⠀⠀⠳⢏⠀⣠⣿⠔⢁⣾⠧⠊⠀⠀⣴⠯⠞⠁
⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⢠⡿⠁⢠⡿⣱⣆⣴⢂⡖⣰⠆⣤⢠⡿⣳⢀⣾⠁⣴⢠⡖⣰⠏
⠀⠀⠀⠀⠀⠸⠀⠄⠀⠀⠻⠇⠐⠟⠘⠋⠺⠇⠿⠐⠛⠸⠃⠟⠻⢁⡾⢻⠘⠃⠛⠐⠟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⣶⣦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠤⠀⠆⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠿⠛⠛⠋⠉⠈⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⣍⠀⠀⠀⠀⠀⠀⠀⣦⣰⡆⢸⣇⡌⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠸⡇⢻⠀⠀⠀⢠⣴⣷⢀⣿⡇⢧⠟⢿⢷⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣇⡞⣧⢸⠏⢹⣼⠉⠁⠀⡀⠀⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢹⡇⠘⠇⠀⢠⠖⠛⠲⣄⢹⡀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡌⠥⣀⡸⠀⢧⠀⠀⠀⣸⠀⠑⠒⠋⢀⣤⣄
⠀⠤⠔⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠈⠑⠒⠚⠁⠀⢀⣠⣶⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣶⣾⣿⡿⠿⠟⠿⠛⣟⣿⣟⣛⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⡀⠚⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⠿⠛⠛⠛⠛⠋⠉⠉⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡀⠀⠀⠀⠈⢿⣿⣿⣿⡍⠉⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡄⠀⠀⠀⠘⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠂⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣂⣠⣤⣶⣶⣶⡆⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠤⠤⠤⠤⠖⢒⣛⣛⣉⣭⣭⣭⣿⣿⣿⡇⠀⠀⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣴⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣾⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣄⣀⣠⣤⣤⣤⣤⣤⣤⣤⣠⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⡛⠛⠿⠟⠛⠛⡉⠩⠍⣛⡻⠯⢝⣫⢿⣿⣿⣟⡻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⠪⠓⠢⢍⠉⠉⠩⠤⠔⠂⠀⠥⢒⠈⠁⠈⠁⠊⠙⠙⢚⢗⠀⠺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⡬⠷⡶⠖⡷⠿⢷⠥⠀⠈⠰⠀⠄⠠⠄⠀⠀⠀⠀⠀⠀⠒⠀⠉⠛⠻⠿⣿⡿⢿⣿⡟⠛
⠀⠀⠀⠀⠈⢿⣿⠿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣫⣤⣦⣶⣶⣼⣿⣧⣦⣶⣡⡂⠘⠙⠰⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠐⠁⢀
⠀⠀⠀⠀⠀⡩⠳⠚⠐⠻⢻⣿⣿⣿⣿⣿⣿⣿⣟⣷⣿⣿⡿⠿⠛⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⣶⣤⣈⠙⠻⢶⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⣤⣶⣖⢠⣠⡖⢶⣦⡀⠉⠛⠻⢿⣿⣿⣷⣀⣉⠛⢿⣴⣤⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⠿⠿⣿⣿⣿⣿⠅⠀⠀⠀⠀⢸⣿⢯⢙⣼⣤⣀⣽⣿⡇⠀⠀⠀⠀⠉⠻⢿⣿⣿⣶⣦⡉⠻⢿⣮⣆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⠋⢀⣾⠟⠛⡩⣷⠀⠀⠀⠀⠀⠀⠻⠉⠛⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠐⢹⣿⣿⣿⣿⡆⠄⠙⠻⣯⣢⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠈⠻⠒⠒⠚⠻⠧⣀⠀⠀⠀⠀⠀⠑⢿⣶⣶⣾⠟⠀⠀⠀⠀⠀⠀⠁⠡⠈⠀⠁⠞⠉⠻⠿⡷⠂⠀⠀⠉⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⣲⠦⣄⣈⠃⠶⡠⠄⢀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣦⣤⣀⣁⣀⣀⢀⡀⣀⣀⣀⣠⣤⡴⡾⢯⠣⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠜⠉⢉⡛⠛⠛⠛⠿⠛⠋⠙⠙⠩⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣶⣦⣾⣧⣴⣶⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣼⣿⢿⣿⡏⣵⣟⣿⢫⣽⣿⣿⣇⣠
⠀⠀⠀⠀⠀⠀⠀⢾⣾⣿⣿⣧⡻⣿⣷⣌⣙⣻⣬⣙⣿⣿⣿⣿⣥⡀
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡿⣫⣽⡝⣿⣿⣿⡿⠿⣿⣻⡿⣾⣿⣿⣤⡀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⢁⣿⠳⢟⣫⣼⣿⣿⣿⢻⣇⡿⣻⣿⣿⠏⣾⣷⡦⠀⠀⠀⠀⣰⣿⣿⣿⣿⣦⣤⣄
⠀⠀⠀⠀⢰⣾⣿⣟⢰⡇⢸⣻⡿⢿⡏⢻⢻⠀⡛⡇⣹⢿⢿⠀⣿⣿⣶⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⢠⣾⣿⡇⢸⡇⢾⠸⢸⡟⡇⢸⠘⢠⣹⡃⢫⣘⣘⡃⣿⡿⣿⣿⣿⣷⣦⡘⣿⣿⣿⣿⡿⠉⣸⡇
⠀⠀⠀⠀⠀⣽⣿⣧⢸⣇⣼⣿⣦⣦⣽⣿⣾⢟⣿⡇⢻⣿⠏⣿⣮⣿⣛⣿⣿⡟⣭⣷⣸⣿⣿⣿⡁⠀⣿⣇
⠀⠀⠀⠀⠀⠉⣽⣿⣌⣻⣿⣿⣿⣿⢋⣵⣿⡼⢿⡿⢖⣿⢠⣿⣿⡇⣹⣿⣼⣇⢹⣿⢟⣻⣿⣿⣇⠀⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⡘⢿⢹⣿⢃⡇⠿⢻⢸⡇⣶⡅⣭⣇⢿⠸⡌⢿⣶⣯⣿⣿⣿⣆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⢀⣠⣿⢟⣯⢰⡿⢨⣵⣿⣶⣬⣷⣬⡇⢿⣿⢖⣟⢻⣷⣾⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢠⣤⣴⣶⣶⣿⣿⣿⡿⡿⢃⡾⢸⢇⣿⢡⣦⡿⢛⡟⣻⣬⣿⣜⣿⠘⣟⣿⣿⣿⣿⣻⣿⣇
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡾⡏⢰⢟⢿⡘⣇⢺⠃⣶⡟⣽⢸⣿⣿⣿⣿⣭⣾⣿⣿
⠀⠀⠀⠀⠀⢨⣶⣿⣿⣿⢡⣶⣿⢿⣿⣿⡇⣼⠟⡄⣩⠸⣟⣷⢸⡇⣷⣽⣤⣿⡌⣿⣿⣿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣛⣿⣿⣿⣮⣟⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣋⣸⣵⣿⡿⢛⣿⣿⣿⡏⣾⢿⣿⢻⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢯⣶⣿⣷⣶⣬⣭⣙⣛⠻⠿⠿⠋⠴⠿⢟⣱⣿⣿⣶⣾⠟⣼⡟
⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣮⣯⣿⣿⣿⡿⣯⣭⣭⠝⣃⣐⠶⠿⠿⢿⣿⠿⠿⢛⣡⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣭⣿⣿⣿⡿⠟⠛⠻⢿⣿⣦⣤⣴⣾⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠈⠉⠉⠁⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⣿⣇⠀⠀⢀⣤⡾⢿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⠳⠀⠈⠀⠀⠈⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠀⠀⢀⠠⠄⠀⠀⢤⠄⡀⠀⡀⢄⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢅⡂⣡⠔⢋⢂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⡙⠦⣈⢰⡨
⠀⠀⠀⠀⠀⠀⠀⡠⣫⣞⠥⠚⠑⠈⠀⠀⢀⡶⠛⢻⠄⠀⠀⠀⠀⠀⠁⠊⠓⠬⣳⣝⢄
⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⣿⢸⣷⣼⡇⠀⣸⣿⡄⣿⠀⠀⣿⠀⠀⠀⠀⠀⠈⢻⡇
⠀⠀⠀⡀⠀⢀⠎⡤⡀⠀⠀⠀⠀⣿⢸⡟⣿⡇⢀⣿⢻⣇⣿⣤⡤⣿⣤⠄⠀⠀⠀⢀⢬⠱⡄⠀⢀
⠀⠄⢔⠠⡀⡎⡜⠐⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠻⣤⡆⠀⠀⠀⠀⠀⠀⠀⢂⠂⢣⢳⢀⠄⡢⠠
⠀⠀⢸⢰⢸⢡⡿⢑⣀⣀⢀⡀⣀⢀⡀⣀⢠⡞⣉⣟⢀⣀⡀⠀⣀⣘⢛⣷⡀⣀⣀⣀⡊⢿⡌⡇⣆⡇
⠀⠀⠘⣿⣸⠞⠀⠈⢹⡋⢸⣧⣿⢸⡇⣿⣾⣷⣿⢛⣘⣿⡇⢰⣿⢛⢸⡏⣷⡟⢸⡷⠄⠀⠳⣇⣿⠃
⠀⠀⠀⢸⣿⡀⠀⠀⠘⠃⠘⠃⣿⢘⡋⠛⠙⠃⠛⠛⠉⠛⠃⠀⠙⠛⠛⠃⠛⠃⠘⠛⠃⠀⢀⣿⡇
⠀⠀⠀⠀⢻⡨⡖⠀⠂⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⢀⡶⠛⡷⠀⠀⠀⠀⠀⠀⠐⠀⢲⢅⡟
⠀⠀⠀⢀⢚⢧⢇⡏⠀⠀⠀⢿⡿⠒⡗⣸⠂⣾⣷⢰⣷⣼⢻⣷⡿⢸⣿⠆⠀⠀⠀⠀⢹⡸⡼⡓⡀
⠀⠀⠀⠀⠈⠻⣞⢇⠀⠀⠀⢸⡇⢠⣟⢻⢠⡿⢿⣸⡟⣿⣸⣿⣧⢰⣿⠇⠀⠀⠀⠀⡸⡳⠟⠁
⠀⠀⠀⠀⠐⡸⢞⢫⡧⢬⡢⢒⡀⠀⠀⠸⣧⣶⠘⢧⣴⠀⠀⠀⠘⢦⡶⢀⡒⢔⡥⢸⡕⡳⣇⠆
⠀⠀⠀⠀⠀⠀⠈⠳⣝⢦⡑⢤⣳⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡠⢊⡴⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠭⣦⣌⡑⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢊⣡⣴⠭⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠦⠤⠤⠤⠤⠤⠤⠴⠓⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⠀⠀⠀⠀⠀⢠⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠀⠀⠀⠀⠀⠸⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣰⣿⣿⣦⡄⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣠⣶⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣴⣾⣿⣿⣿⣿⣿⣶⣶⣦⣶⣼⣶⣤⣤⣶⢶⠶⠛⢻⣿⣿⣿⣿⣿⣷⣶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣈⣈⣙⠉⠻⠏⠀⣀⣤⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠿⣿⣿⠿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉⠉⠛⠿⠿⠿⠟⠋⠁⠈⠉⠉⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠈⠙⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⡏⠉⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣿⣿⢛⣛⡀⠀⢰⣶⡆⢸⣿⣷⢰⣶⣶⡆⠀⢸⣿⡇⣾⣿⡇⣠⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⢸⣿⡇⢸⣿⣿⡌⣿⣿⣿⡄⢸⣿⡇⣿⣿⣶⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇⠀⣿⣿⡇⣿⣿⣿⣇⢿⣿⣿⣿⣼⣿⡇⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡇⣿⣿⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⣿⣿⣾⣿⣿⣿⣿⡟⣿⣿⢹⣿⣿⢣⣿⣿⠀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠸⠟⠃⠀⠛⠛⠛⠛⠋⠙⠿⠟⠛⠛⠀⠛⠛⠘⠛⠟⠀⠀⠹⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡄⠀⠀⣾⣿⢇⣴⣿⣿⣷⣄⢰⣿⣷⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⣸⣿⡟⣼⣿⡟⠈⣿⣿⡄⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠀⣿⣿⠃⠀⣿⣿⡇⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⠃⠀⣿⣿⡄⠀⣿⣿⡇⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⢸⣿⣧⣀⣿⣿⠁⣿⣿⡄⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⠙⠿⣿⠿⠃⠀⠘⠿⣿⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⢣⠀⠀⠀⠌⠙⠃⡄⠀⠀⠀⠀⠀⠀⠀⠠⠄⠚⢒⠀⠀⡀⠀⠐⠉⢃
⠀⠀⠀⢠⣀⡀⠀⡄⠓⠂⠋⠈⠈⠈⠈⠒⠒⠄⡀⠀⢀⡤⡀⠀⠀⢀⠱⠆⠊⠁⠁⠁⠉⠑⠐⠢⢀⠀⢀⡀⡀
⠀⠀⠀⡄⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣀⢀⠂⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢀⢀
⠀⠀⠀⣐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣
⡀⠠⢤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠠⠦
⠱⣀⢴⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠠
⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⣠⣴⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣄⠀⠱⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠔⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠛⢻⡆⠀⠀⠀⠀⠀⠀⢰⠟⠛⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⡜⠒⡀
⠀⠡⠠⠼⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣄⠀⡀⠃⠀⠀⠀⠀⠀⠀⠸⣄⠀⣠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣸⠄⠄⠆
⠀⠀⠀⠠⢥⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⡀
⠀⠀⠠⠁⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡋⠀⡐
⠀⠀⠀⠈⠈⠀⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠁
⠀⠀⠀⠀⠀⠠⠈⠙⢢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠶⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⠙⠉⢃
⠀⠀⠀⠀⠀⠀⠁⠂⠃⠈⠂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠑⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠈⠑⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠋⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠨⠓⠦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠖⠓⡀⠈⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠄⠁⡱⠦⢤⠀⣀⠤⠶⠌⠐⠠⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⠌⠉⠐⢀⠠⠂
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⣀⠀⠀⠀⢀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠆⣊⣐⣸⠂⡄⡅⢸⠬⡐⠃⣀⣘⠀⠆⠴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠂⢀⠸⠖⡊⠩⢰⢡⠊⢬⠁⠀⠀⠈⢡⠈⡌⡧⠈⢉⠒⠯⠀⠔⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠐⠁⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠈⠈⠑⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⣿⣿⠋⠛⢹⣿⡏⠘⣿⣿⠀⢀⣿⣿⡀⠀⣿⣿⡄⠀⢹⡇⣿⣿⡇⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣻⠀⠀⢸⣿⣷⣴⣧⣿⠀⢸⣿⣿⡇⠀⣿⣿⣷⠀⢸⡇⣿⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢹⠀⠀⢸⣿⣿⢸⣿⣿⠀⣸⣿⣿⡇⠀⣹⢿⡞⣇⢸⡇⣿⣿⡿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣾⠀⠀⢸⡿⣿⢸⡿⣿⠀⣿⢸⣯⣷⠀⣽⡇⢻⡿⣿⡇⣿⢿⡇⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣗⢾⠀⠀⢸⣧⣿⢸⣯⣿⢠⣿⣴⣿⣿⠀⣿⡇⠈⣧⢽⡇⣿⣾⡇⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⢸⣿⡿⠸⣧⣿⠸⡿⠀⣿⣿⡆⣿⡇⠀⠸⣿⠇⣿⣿⠇⢸⣯⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⢀⣀⠀⠀⣀⡀⠀⢀⣀⠀⠀⢀⠀⠀⢀⣀⡀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡆⡄⠀⠀⠀⣿⢸⠀⣸⠁⡏⣴⣿⣯⣷⡄⣾⡇⠀⣸⣿⣿⠀⠀⠀⣄⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠋⠠⡄⠀⠀⢻⣸⣠⢿⢀⠇⣿⣿⣿⣿⡇⣿⠀⢀⣿⣿⣿⠀⠀⠄⠈⠷⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠀⠀⠀⠀⢉⣁⣾⠎⠀⠻⣿⣿⡿⠃⢻⣆⡜⢸⣿⣿⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⣀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⣿⠁⣿⡏⠁⠉⣿⣿⠉⢸⣿⡉⠀⣿⡏⣻⡄⠀⣿⡇⠀⣿⣇⠈⣇⣾⡋⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡟⠀⣿⡏⠉⠀⡿⣿⠀⢸⣿⠉⠁⣿⡇⣼⡇⢸⣿⣿⠀⣿⣿⣄⡇⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⡇⠀⠀⣿⣿⠀⢸⣿⠀⠀⣿⣧⡿⠃⢸⢹⣿⠀⣿⠹⡿⡇⢀⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⣟⣧⡄⠀⣿⣿⠀⢸⣻⣤⡀⣷⡿⠻⣦⣿⠉⣿⡆⣿⠀⢻⡇⢻⣧⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⡀⢀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠠⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠄⢈⢲⡦⢅⣐⢸⡘⢄⢚⡀⠀⠀⢀⡘⢀⢇⡗⢒⣈⠤⡖⠂⠨⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠢⠆⣍⠩⠹⠤⢓⡃⢘⢒⠁⡍⠉⢹⠀⠆⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠉⠀⠀⠀⠀⠉⠀⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⣠⠤⣄⡀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢴⡷⣼⠕⠁⠀⠀⠀⠠⠀⠈⠑⢲⡿⠐⠀⢠⠞⡁⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡇⢄⠀⢸⡇⢿⡐⣶⠀⡴⢛⡀⣷⠎⡿⢸⡅⡸⢃⣟⠀⣴⢀⡄⣴⢃⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠊⠀⢸⡇⠸⡇⢿⡐⠃⠺⠆⠻⠐⠧⠘⠛⠇⢸⡗⠈⠧⢾⣱⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠼⠁⠀⠀⠀⠀⠀⠀⢠⣸⠂⠀⠀⠀⠀⠸⣇⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢅⡐⠈⠉⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠓⠔⠛⠋⠀⠀⣠⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠉⣄⠀⠰⣄⡀⣸⠧⠖⠒⢶⡇⢀⠀⡄⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣃⡿⠀⣹⠆⣴⢛⠂⣾⠀⢰⡿⢸⡇⠸⣄⢧⡹⣄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣠⠞⠁⠀⠻⠀⠁⠛⠀⢿⡠⠙⠒⠸⡏⢂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠲⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣶⣦⡁⠄⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣄⡀⠘⣤⣀⣽⣿⡇⣤⣤⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠑⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣜⢛⣪⣿⣿⠿⠿⠿⢿⣿⣳⣶⣼⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⣩⣿⣯⡝⠂⠀⠀⠈⠀⠈⠻⣿⠟⣡⣶⣶⣶⣌⢻⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠸⣿⣿⣿⡛⠃⠀⠀⠀⢠⣠⣀⠀⣼⣿⣿⣿⣿⣿⡆⢿⣿⣿⣿⣿⡿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠐⣻⣿⣿⡟⠀⠀⡀⣠⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⠃⣾⣿⣿⣿⣜⡃⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠄⣶⣿⣿⣿⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠟⣃⢴⣿⣿⣿⣿⠯⠚⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣦⣥⣿⣿⣿⣷⡆⠸⣿⣿⣿⣿⣿⣿⣟⠛⠃⢾⢿⣦⠿⣿⣿⢏⡲⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣷⣾⣷⡿⣿⣿⣿⣿⣦⠀⠉⢿⣿⣿⡿⠋⣁⣾⣷⣶⣭⣿⣿⡯⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠁⢠⣾⣿⣿⣿⣿⡿⠯⣙⠂⡄⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠷⡄⠛⠀⠀⢠⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡻⣿⣿⣿⣿⣿⣿⡟⠫⠘⠟⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣠⠿⠛⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠘⠀⠀⠀⡀⠀⠈⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠃⠀⠀⣰⠈⡏⠀⢻⣿⣿⣿⣿⣻⣿⣿⣿⣹⡷⢀⠀⡀⠀⠘⠟⠀⢀⠀⢀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠉⢰⣿⠄⢠⡆⠛⠟⠛⢿⣿⣿⣿⢿⠁⡀⠀⠀⠀⠀⠀⠀⠀⡀⢿⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⢀⣾⣷⡀⠀⣉⣪⣿⡿⠂⠀⠀⠀⠀⠰⠀⠀⢀⠈⠁⠀⣀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠀⠠⠉⠙⠁⠀⠀⠀⠀⠤⠀⠀⠀⠄⠚⣾⣷⣿⣿⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠿⠟⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⢀⠀⠠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣁⠀⣀⠠⠀⢀⡀⢀⠀⣤⢠⡐⠀⠄⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⣤⣠⢤⣖⢿⡏⢹⡇⡰⣧⡘⢶⠉⣴⣄⢀⡶⠠⡴⣰⠈⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣈⠰⢆⠲⣻⣍⣞⠊⣡⣈⣤⣤⣾⣿⡉⢓⡘⢂⣤⣽⣯⣧⢈⡓⢄⣱⠊⠰⡎⠁⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠊⣀⠵⣿⣷⠖⣾⣭⣧⣟⣸⠧⣒⠛⢭⣽⣿⣾⣿⣿⣿⣿⣿⣿⣿⣛⡦⡿⣿⡖⢼⣶⣳⣥⡤⠣⣄⡀
⠀⠀⠀⠀⠀⠀⢀⢀⢎⠃⢩⣿⣟⢻⣿⣿⣿⡿⣶⣿⣿⣿⣿⣿⣻⣿⠟⠋⠉⠙⠻⣿⣿⣿⣷⣯⣼⣧⣸⢪⣿⣿⣿⣶⢯⡀
⠀⠀⠀⠀⠀⠀⠠⢻⣫⣶⢯⢉⡛⢏⣾⣿⣿⣸⣯⣿⣷⣿⣿⣿⣿⡏⠀⠀⠠⡀⠰⠞⣾⣟⡝⡿⣿⣛⣿⣿⣿⣿⣿⣿⢜⢰
⠀⠀⠀⠀⠀⠀⠚⣿⣿⡏⣏⣹⣾⣿⣿⣿⣷⣿⣻⣯⣿⣿⣿⣿⣿⣇⠀⠀⢠⣤⡄⣰⠘⣿⠿⣎⣾⣿⣻⣿⣿⣿⣿⣿⡎⡁⣀
⠀⠀⠀⠀⠀⠘⠂⢿⣿⢻⢦⡟⣫⣽⣿⣿⢿⣿⣿⣿⣿⢿⣿⣻⡿⠿⣷⣴⣶⢭⣠⡟⢀⡿⢿⣟⣧⣮⠚⠻⣾⣿⣿⣿⣇⣴⡀
⠀⠀⠀⠀⠐⠲⠤⣜⡇⡿⢋⣶⣿⣿⣿⣿⣾⣿⣿⣟⣿⣀⡼⢁⣄⣧⡙⣿⣿⣿⣿⠃⣾⣿⣿⣿⣷⠃⠀⣸⣿⣿⣿⣿⣿⡮
⠀⠀⠀⠀⡄⢈⠀⢻⣿⣷⣿⠱⠞⣛⣿⣿⣿⢿⣿⡿⢽⣿⣽⣿⣾⣯⡻⣌⢿⣿⣿⢸⣟⣛⡿⡶⢀⣼⣷⢽⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢠⢾⠂⢻⡿⣫⣛⠿⣿⣿⣿⣿⣿⣷⡟⣱⣿⣿⣿⣿⢯⣙⡷⡹⡎⢿⡵⢸⡿⣿⡿⢡⣾⠿⢿⣼⣿⣽⡿⣿⣿⣀⡁
⠀⠀⠀⠀⠰⠠⢠⠟⣴⢛⡴⠛⣻⣿⢟⣻⣿⢟⣼⣿⣿⣿⣿⣿⣿⣾⣿⣻⢷⣬⣝⡛⣯⠘⢡⡺⠿⠿⣯⣈⣿⣾⣯⣹⣿⡆⡀
⠀⠀⠀⠀⠀⢀⣿⠎⠡⢟⣤⡾⣫⣿⣿⣿⣯⢞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⡾⢿⣽⣦⡿⣽⣿⣿⣏⣻⣿⣿⣿⣿⣿⣠⠂
⠀⠀⠀⠀⠀⠰⣿⠖⣰⡿⢛⢗⣿⢟⣿⣿⣗⡿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⡹⡄⢻⣟⣹⣷⣿⣿⡿⣿⣿⣿⣿⣺⡿⠐
⠀⠀⠀⠀⠀⢀⣿⡴⡿⣊⡴⣛⣻⠛⣿⠿⣛⠿⣿⢿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡟⣵⠿⡄⢙⡉⣅⠉⢻⡇⠘⢿⣿⣟⣼⠐⣎
⠀⠀⠀⠀⠀⠁⣾⣼⣿⣿⠟⣋⣤⣔⣒⣛⣛⢦⣙⣳⣘⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣗⣌⣢⣀⣿⣤⣼⣯⣾⢾⣻⡿⡛⢲⠃
⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠻⠝⠒⠚⠲⠝⠫⡿⠿⠻⠍⠛⠛⠛⠉⠈⠈⠁⠀⠈⠉⠉⠉⠈⠉⠁⠀⠉⠙⠋⠉⠁⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡰⠁⠀⠀⠈⡇⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠈⠃⠀⠀⠀⠀⠸⠃⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡰⡄⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡰⠁⠹⡄⠀⠀⠀⠀⠈⡇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠚⠀⠀⡰⠁⠀⠀⠱⡀⠀⠀⠀⠀⠋⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢠⠃⢰⠛⢄⠀⠱⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⡇⠀⠀⠳⡀⢳⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⡆⢇⠀⠀⠀⢳⠀⡇⣀⣀⣀⣸⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⡤⠒⠉⠉⠙⠒⢤⡀⢈⡷⠋⠉⠀⠈⠉⠓⣌⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡜⠀⡴⢦⠀⠀⠀⠀⠙⠞⠀⠀⠀⠠⣎⣳⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⠀⠉⠁⠀⡀⢠⣀⣇⣰⣁⡆⡄⠀⠈⠀⠀⡜⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⢄⠀⠀⠀⢸⡊⠉⣞⣳⠀⢉⡽⠀⠀⢀⠞⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢌⠳⣄⠀⠀⠏⡲⢪⠥⡖⠻⠈⢀⡴⣡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢮⡑⢦⡀⠀⠈⠀⠀⠀⡠⣪⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢬⡓⢄⡀⣠⢊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⡗⠀⠀⠀⠈⢉⠃⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣠⣿⢇⣼⠃⣿⡀⣾⠀⣿⢸⣧⢸⠙⣴⠏⠀⣴⡟⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⣻⡿⢩⣿⢃⡼⠃⠀⣿⠛⢸⣸⣿⣼⡏⣿⠀⣿⠀⣼⡟⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⣾⣿⠞⣁⡀⠀⠉⠀⠈⠈⠀⢉⣁⠈⣀⠉⢰⣿⡴⢃⣤⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⢸⡿⢡⢞⣿⠃⢠⣶⠻⣤⣦⠀⣼⢃⣾⡿⠀⣿⡟⣵⣫⠾⠃⣸⣿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⣿⡷⠃⣾⠇⣠⣿⢃⡼⣿⠇⣠⡿⠋⣿⣠⣾⣿⣾⣏⠁⣠⠜⠉⣿⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⣰⡟⠀⠀⠛⠀⠀⢿⡾⠋⠻⠋⠀⢿⡴⠃⠀⠀⠉⠉⠘⠟⠀⠛⠛⠁⡴⠚⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⣈⡳⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⡼⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠿⠟⠛⠋⠉⠉⠉⠠⡀⠀⠀⠀⠀⠀⡀⠉⠉⠉⠙⠛⠻⠿⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⢀⠀⠀⠠⠀⠀⠈⡃⠀⡀⠀⠐⠁⢀⡄⠀⠀⠀⠀⠀⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠚⠀⠈⠁⠠⠞⠛⠳⠂⠀⠐⠚⠀⠈⠖⠛⠂⠀⠐⠆⠀⠄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠐⠣⠠⠜⠉⠰⠤⠤⠁⠈⠀⠀⠀⠀⠠






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠚⠉⠉⠉⠉⠓⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠉⠉⠉⠉⠑⠒⠤⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠘⢆⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠉⠓⢴⡁⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠈⠙⠢⣄⠈⡆⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡼⠁⠀⠀⠀⠠⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘
⢠⠏⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⢣⡜⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢀⣌⡉⠛⠻⠂
⡞⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⣴⡟⠻⣿
⡇⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣴⣿⡆⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⣿⣧⣴⣿
⣇⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢸⡀⠀⠀⠀⠀⠀⠀⠈⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠻⢷⣄⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠘⣷⡀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣄⣞⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠘⢷⣄⠀⠀⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠙⠻⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠶⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠒⠉⠀⠙⠻⢶⣤⡀⠀⠀⠀⠀⠀⢀⢔⣿⠇
⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣤⣾⣵⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢗⣦⣀⠀⠀⠀⠀⠀⠀⣀⠤⢺⣧⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢸⣿⠀⠉⠲⢤⣤⠒⠉⠀⠀⠀⢿⡄⠀⠀⢸⡟⠢⠤⠖⠊⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⠁⢸⡿⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⢸⣷⣄⠀⢸⡇⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠼⣿⡿⠷⢸⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠰⠿⣿⡿⠄⢸⡇⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠾⠷⡄⠀⠈⠻⣦⡖⠤⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠸⣀⠀⠈⠲⢄⠀⠀⠀⡠⠼⠧⡄⠀⠀⠀⢀⡤⠿⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢠⠃⠀⠀⠀⠀⠀⠉⠢⡄⠀⢱⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠁⢠⠃⠀⠀⢣⠀⠀⠇⠀⠀⠀⠀⠸⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠰⡤⢃⡤⠒⠒⠋⠀⢸⠀⠀⠀⡠⠔⠒⠁⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠛⠛⠛⠛⠻⠷⣶⣤⡀⠀⠀⠀⢀⣤⣶⡾⠿⠟⠛⠻⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠛⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⠀⠀⠀⠀⡘⣄⠳⣀⣀⠀⠀⠐⠘⣟⣿⣿⠋⠐⠲⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣌⠑⢦⣤⣤⡼⢾⣿⣽⣷⣭⣶⣾⣿⣿⣿⣿⣷⣶⣤⣧⣀⠈⠙⣷⣆⡀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⢤⣬⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡀⠙⢦⡀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢒⣲⣿⣿⣻⣿⣿⠋⢯⣹⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣦⠀⠙⠦⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠹⣿⣿⡇⠀⠘⢻⣿⣿⠟⠺⢷⣿⣿⣿⣿⣿⡟⣿⣿⡿⠻⣿⣮⣳⡀⠈⠂⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠁⢻⣻⣿⠀⠀⠈⣿⣂⠖⠈⠙⢿⣿⣿⣿⡿⠫⢿⡟⠀⠀⢿⢻⣷⡝⠄⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⢰⢿⢿⣧⣄⠀⠘⢿⡀⠀⠀⠀⠏⠉⣉⣋⢠⠟⠀⠀⠀⠈⠀⢙⣿⡄⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠁⠀⠹⣿⣿⣷⣄⠀⠙⠦⣴⡊⡈⠸⡁⢘⠁⠀⠀⠀⠀⠀⢀⣄⢉⡿⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⢹⡗⣿⣿⣿⣷⣶⣤⡀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠊⠉⣿⠛⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢸⣿⠇⠙⣿⢶⡟⠉⠙⣏⡉⠉⠉⠉⠀⠠⠤⠐⠀⠀⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣈⣿⠀⠀⠘⣿⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⢀⣤⡾⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⢿⣦⣿⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠁⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⢻⢷⣦⣀⣀⣤⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢠⡞⠁⡹⠀⠀⢀⡤⠖⠦⣄
⠀⠀⠀⢀⠴⠒⢉⣩⠶⢾⡏⠁⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⢀⣈⠉⠉⣿⠉⢉⣀⠀⠀⣟⠀⠀⠀⠈⡆
⠀⠀⠀⠘⠶⠚⠋⠁⠀⢸⡇⠀⣿⠀⢀⣀⠀⠀⣴⠢⣼⠇⠁⢠⢀⡎⣼⠃⢀⣿⡴⢍⠈⠁⠀⠈⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣰⠋⢸⠃⢸⣇⡼⣿⠀⢠⠿⡿⠀⣿⠀⡜⢻⠁⠈⠳⣄⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠸⡏⠀⣿⠀⡜⠉⠁⠘⠦⠊⠀⠀⠀⠙⠞⠁⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⡤⠴⠦⣄⠈⠚⠁⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⢀⠀⠀⣾⠀⠀⣰⣿⠀⢠⡤⡀⠀⣠⣶⠀⣸⡇
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⡼⠀⢠⡿⠀⣰⠃⢿⠀⢹⡧⢼⠞⢹⡇⢠⠟⣇⠀⠀⠘⢳⡄
⠀⠀⠀⠀⠀⠀⠸⡀⠀⣀⠜⠁⠀⠀⠳⠞⠁⠀⢸⠀⠈⢧⠞⠀⠘⣧⠎⠀⠹⣄⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⡀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠒⠉⠉⠉⠉⠙⠲⢄⣿
⠀⠀⠀⠀⠀⠀⣴⣟⠁⠀⠄⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢶⠄
⠀⠀⠀⠀⠀⣸⣿⠨⠁⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢲⡆⢰⢀⡤⠒⠒⠲⣤
⠀⠀⠀⠀⠀⣿⣿⠀⡀⠀⠀⠀⠠⠄⠐⠒⠒⠒⢲⣶⠀⣿⠀⠸⣿⡄⣾⠃
⠀⠀⠀⠀⠀⢹⣿⣇⠡⢀⠀⠀⠀⠀⠀⠀⡆⢠⣿⡟⠀⣿⠀⠀⣿⠁⢻⣇⡀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠻⣿⣄⡈⠀⢌⢎⡦⠠⢴⢅⣼⠟⠀⠀⠉⠁⠀⠈⠀⠀⠈⠉⠉⠁
⠀⠀⠀⡆⠀⠀⢀⠌⠛⠿⠷⠾⠋⡃⡂⠋⡛⠀⠀⠄⢰⠀⣶⣶⠂⠂
⠀⠀⠀⠰⠀⠀⠈⢀⣤⡠⡤⣦⢁⠠⡀⠡⠃⠀⠀⠀⠘⠀⣿⣿⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄
⠀⠀⠀⠤⡄⣠⠀⣬⣸⠃⠀⠈⢉⢀⠩⠁⢆⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⡇⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⠀⢸⡇
⠀⠀⠀⠀⡌⠡⠁⠀⠀⠀⠀⠀⠺⠆⠉⣦⡾⢠⠬⣅⠀⠀⣿⣿⠀⠀⣿⡏⣿⡆⠀⣸⡄⠀⣿⡆⣿⡏⣿⡇⢸⣇⡔⠁⢰⣯⠐
⠀⠀⣆⠸⠧⢄⠀⠀⠀⠀⠀⠀⠄⠀⠐⠟⠹⠇⠀⠈⠀⠀⣿⣿⠀⠀⣿⡇⣿⣇⣾⠁⠀⠀⣿⡇⣿⡇⣿⡇⢸⡏⣿⡄⠀⠙⣷
⣀⡀⠄⠀⠀⡀⠠⣆⡄⠀⠀⠈⠰⡖⠁⠀⠀⠀⠤⠆⠀⣀⣿⣿⠀⠀⣿⡇⣿⡏⠿⣦⣀⡀⣿⡇⣿⡇⣿⡇⢸⡇⠘⣷⡀⡄⣸
⠀⠀⣠⢀⠈⠐⢰⠂⠀⠀⠀⠀⣦⡂⠀⠐⠁⠀⠐⠀⢀⠄⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀
⠀⠀⢱⠟⡅⠐⠒⠀⠀⠐⣄⣠⠉⠈⢡⣗⠆⣠⣄⢈⣥⠀⠀⡰⢄⣄⠠⠤⠤⠀⢀⠀⣀⣀⣄⢀⣰⣿⣟⣧⣄⣐⣦⣶⡤⠔⠂
⠀⠀⠀⠑⠂⠲⣒⠚⣫⠀⠀⠀⠀⠀⢠⡁⣆⢉⣼⠏⣀⠀⡰⠃⠈⠀⠹⣿⡿⠟⣒⣶⣖⣍⣍⣛⣩⣥⣬⣋⠈⠠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠋⠦⠮⢰⡀⢰⣟⣁⡹⢤⣤⣶⠾⠄⣍⠳⣮⡙⠉⢛⠿⠞⠧⣥⠎⢽⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢼⡦⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣇⠀⠀⡀⠋⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⡀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⠻⣿⣿⣿⣿⠿⠿⠛⣨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣋⣉⡹⢦⣿⡄⠛⠫⣿⣼⠇⠀⣾⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣾⣿⣿⠽⢿⣿⣿⣶⣆⣀⡁⢀⣼⣿⣿⣿⣷⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣧⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠓⢼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣻⣿⣷⡄⠀⠙⠿⣿⣿⣿⣿⣿⢿⣯⣳⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣿⣿⣿⢋⠘⠹⣿⣿⣦⣠⣀⣠⡉⢉⣉⣿⣏⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠻⠟⠀⠠⠄⠛⣹⡿⣿⣿⣿⣷⣿⣿⣟⣦⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣵⣿⡝⠿⣿⣿⣯⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠋⠁⢸⠿⠿⢿⣿⠿⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡃⠀⢤⣨⡄⠰⠛⠁⠀⠈⠠⢀⣠⡀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠎⢈⣿⣿⠶⣶⣿⣷⠀⢄⣀⠀⠀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠲⠀⠀⠀⠂⠀⠀⠀⠈⠁⢿⣿⣿⣷⣤⣬⣿⣿⣾⣿⣿⢧⡄⠀⠀⠈⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⣿⢷⣽⡿⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⠀⢸⣿⣿⣿⣿⣿⢿⠟⠊⠁⠀⠙⠰⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢨⣤⣈⡿⠁⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣼⡽⠃⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠼⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣷⠂⠀⢀⡑⠀⠀⠀⠄⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⡅⡁⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠖⠛⠋⠘⠒⠶⡤⠆⠀⠘⢻⣾⣥⣿⣆⠠⡀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⢀⣐⣄⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣷⣶⣤⣀⡀⠚⠓⠲⣄⠠⣿⣾⣿⣿⣿⣆⠡⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⣤⠀⠀⠀⣤⡄⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠙⠉⠙⠛⠛⠛⢿⣦⡻⠆⠀⠀⠀⢀⣴⣶⣶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠤⠀⠾⠛⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄⣀⠀⠀⢠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⡀⣸⠀⣆⠀⡇⠀⢰⡇⡒⢰⡧⣴⡇⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⣇⢠⡘⡦⣇⣻⣇⣿⣄⣷⣷⣿⠃⣇⣾⢣⣿⣼⣿⡿⣄⡴⣴⡆⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⢌⣿⣧⣿⣿⣼⣽⣿⣿⣿⣿⣿⣿⣾⣿⣷⣿⢟⣿⣿⣾⢋⣿⣯⣾⣣⡞⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠦⣀⣻⣖⣻⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣡⡾⣃⡴⠋⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⣨⣿⣿⣿⣿⣿⣿⡿⠿⠛⢋⠉⠉⣍⣉⡟⢩⢋⠻⡿⣿⣿⠿⣿⣿⣿⣿⣿⣻⣵⣿⣿⣾⣯⡴⢟⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠛⠉⢹⣿⠁⠀⠀⠀⣶⡿⠻⠟⠛⣷⠎⠀⠘⣿⣿⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣫⣴⡯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣿⣿⡿⠋⠀⠀⠀⠀⠸⣿⡛⠀⠀⢘⣿⣧⡀⠀⣠⣿⣖⡁⠀⢿⡿⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣭⣥⣤⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⢿⣧⡠⢔⡩⣻⠿⠷⠾⠿⡃⠀⠀⢀⣾⠁⢀⣠⣤⢤⣶⣴⣾⣿⠿⡿⣍⡛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢚⣋⡙⠻⣤⣄⣀⣀⠀⠀⠀⠀⠀⢀⠻⣿⣧⠞⡅⠀⠀⠀⠀⢳⡌⣠⠞⠁⣀⣢⣤⣶⣿⣿⣯⠛⢫⢔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢩⠟⢿⣶⣶⣿⣯⣭⣤⣌⣙⣿⣶⣦⣤⣤⣤⣶⣿⣷⣾⣿⣿⡿⢿⠛⠓⠀⠈⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⡀⠈⠈⠁⠋⠉⠋⠙⠛⢛⠛⠟⠛⢿⠛⣿⠛⠏⠙⡟⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠁









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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠦⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠴⠤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠙⢦⡀⠀⠀⢀⣴⣋⢀⡴⠚⠛⠲⣄⠀⣠⠖⠛⠛⢶⡄⣈⣷⡀
⠀⠀⠀⢠⡏⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠛⠻⢿⣷⡄⠈⢷⠀⡼⠋⠁⠉⠛⠀⠀⠀⠀⢸⠆⡏⠀⠀⠀⠀⠛⠉⠉⠉⢳⡀
⠀⠀⠀⡿⠀⢰⣿⠟⣋⣉⣀⠀⠀⠀⠀⢠⣾⣷⡆⠙⠟⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⡇⠀⠈⠁⣾⠿⠿⠿⣷⠀⠀⠀⠸⣿⣿⡷⠤⢤⡀⢀⣿⠿⢶⣤⣀⠀⠀⠀⢀⡞⠀⠀⠀⢻⡄⠀⠀⠀⢀⣠⠶⢿
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠹⠏⠀⠀⠀⠘⡏⠓⢦⣤⡞⠀⠀⠀⠀⠀⠻⣄⡴⠞⠉⠀⠀⣿
⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠚⢙⠆⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠠⣦⣄⣀⡀⠀⠀⣀⣠⣴⡄⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠤⠬⠃⠀⠈⠳⣄⠀⠀⠀⠀⣠⠞⢧⡀⠀⠀⠙⢦⣉⠉⠉⠉⣉⡴⠏⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠷⣄⣠⠞⠁⠀⠀⠛⢦⣀⠀⠀⠈⠉⠛⠋⠉⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⠤⡤⠤⠴⠖⠛⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⠤⣤⠤⠴⠶⠚⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠉⠉⠉⠉⠉⡙⠛⠿⣿⣿⠿⠛⠋⠉⠉⣉⣩⣉⣉⡛⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣷⣦⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⣿⣷⣦⢙⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢿⣿⣿⣿
⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡆⠀⠀⠀⠀⡠⠄⡧⣄⡄⢠⢴⢴⡦⣠⣤⣨⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⢰⣿⣿⣿
⣿⣿⣿⣿⣿⡀⠀⠀⠀⠬⠗⠃⠘⠣⠼⠸⠄⠧⠿⠿⠿⠸⠿⠿⣿⣿⣿⣿⣿⡿⠳⠀⠀⠀⢀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢿⣿⡿⠿⠟⠉⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⢀⡆⠀⠀⣠⠞⠁⠀⠀⠀⢀⡔⣿⢳⡄⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⢸⡇⠀⢰⡏⠀⠀⠀⠀⠀⠸⣗⣾⣸⡟⠀⡀⠀⠀⣿⠀⠀⠀⠀⢠⡏⠀⠀⠀⢀⡀⠀⢰⡟
⠀⠀⠀⠀⠀⣾⡇⠀⣿⠁⢠⡄⠀⢀⡴⠀⠈⠉⠁⠀⠀⣿⠀⠀⣿⠀⢠⡀⠀⣾⡁⠀⣄⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠁⠀⣿⡀⢸⡇⠀⣾⠃⢸⡇⠀⣿⡄⢰⡿⠀⢀⣿⢶⡟⠀⠀⣿⠀⠀⣿⠀⢸⡇⠀⣼⡇
⠀⠀⠀⠀⠀⢹⡆⠀⠙⠀⢿⠁⠀⠙⠂⠸⡇⠀⠙⠃⢸⡇⠀⠀⠛⠸⣇⠀⠀⣿⠀⠀⠛⠀⠸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠉⠒⢤⡤⠤⠴⠤⣤⠔⠚⠉⠉⠙⠲⣄
⠀⠀⢀⣠⠴⠒⠒⠦⣤⠋⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠈⢣⠤⠖⠒⠲⢤⣀
⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⣸⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⢹⡀
⠀⢀⡇⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⣀⣠⠤⠒⠉⠀⠀⠀⣇
⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢻
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⣧⠀⠀⠀⠀⠀⠀⢠⡏⠀⠈⠉⠉⠙⠛⠓⠒⠒⠒⠛⠛⠉⠉⠉⠀⠈⣧⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⠤⠶⠖⠚⠉⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⡀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⡟⠛⠋⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⣿⣄⡀⠀⠀⠀⣀⡀⢀⠀⣀⡀⠘⣧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠴⣿⠀⡇⠀⢠⡞⢹⡇⠸⣿⠉⣿⠀⣿⡼⠃⠀⠀⠀⠀⠀⠀⠘⣧⢰⡇⣴⠛⢶⣸⡇⠀⡿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⠀⣿⠀⣇⣠⣾⣀⡿⣧⣀⡇⠀⣿⢀⣿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⣷⣀⡼⠻⣇⡼⣷⡀⢘⡁
⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠉⠁⠈⠉⠀⠈⠉⠁⠀⠈⠉⠁⠈⠁⠀⠀⠀⠀⠐⠀⠘⠀⣸⠇⠈⠉⠁⠀⠀⠀⠈
⠀⠀⢸⡙⠳⣯⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⣤⣀⣠⡴⠏
⠀⠀⣠⠽⠦⣝⡾⠛⠛⠓⢦⡀
⠀⠀⠻⠤⠶⢺⡇⠐⠀⠒⠈⡇
⠀⠀⠀⠀⠀⠀⢧⠀⠘⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⣷⣀
⠀⠀⠀⣰⠞⠉⡟⢧⡴⠋⠁⠀⢠⠞⠋⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⢼⣧⣟⣹⣇⠀⠀⠀⢐⡶⠶⢧⣀⠀⠀⠀⠀⣠⠿⠦⣤⠄⠀⠀⢠⠰⣇⡆
⠀⠀⠀⣇⠘⢀⠀⢸⡇⠀⠀⢠⡏⠀⠃⢀⠀⣽⠀⠀⠀⡾⠛⣆⢰⠇⡟⢸⡆⢹⡟⠒⠛⠀⠀⠘⡏⠰⠀⡌⣧⠀⠀⢀⣼⠃⠆⢠⠘⡷⠀⠀⠈⠻⡟⠀⣼⣡⠄
⠀⠀⠀⠘⢦⣀⣀⣼⣧⣤⡀⢸⠁⠐⠃⣠⠼⠁⠀⠀⠀⣇⠀⢹⣼⠀⣇⢸⡇⠈⡇⠀⡤⢤⠀⠀⠹⣤⡀⢠⣧⣤⣶⡀⣿⠀⠲⠀⣸⠃⠀⢀⠀⢀⡷⠚⠛⠳
⠀⠀⠀⠀⣠⠟⠁⢀⡀⠀⠙⢾⡤⠖⠛⠁⠀⠀⠀⠀⠀⠈⠳⣼⣿⠀⣿⢸⡅⢠⡇⢸⠃⠸⡇⠀⠀⢦⡽⠋⠀⢠⡀⠹⣿⢀⡤⠞⢁⠀⠀⣸⢧⣸⠁
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⡄⢸⡆⠀⠀⢀⡀⢠⣄⣤⣯⣀⡄⠀⢹⠀⣿⠈⣇⢸⠃⡟⠀⠀⡇⠀⠀⢾⠃⠈⠀⠀⠀⡀⢿⠋⠀⢀⡏⢳⣠⡏⠈⣿⡶⡄
⠀⠀⠀⠸⡇⠀⠛⠀⠀⡀⠀⠀⡇⢀⡿⠛⡿⠻⣟⠳⡄⠹⣦⠤⢸⡆⣷⠀⡏⣾⡼⠁⣀⡼⠃⢀⡄⢸⡄⠀⠈⠁⠀⠁⣿⠀⣤⣜⡇⠀⠹⡆⠀⣿⠀⣷
⠀⠀⠀⠀⢳⡀⠀⠀⠀⠁⠀⢸⡇⢹⡇⠀⡇⠀⢹⢰⡇⢠⣧⠀⠘⡇⢹⢸⣧⡟⠛⠉⠁⢨⠟⢋⠙⢓⢷⡈⠀⠀⠆⢠⡇⠀⣧⠈⠳⡄⠀⢻⠀⡟⢠⡟⠋⡷
⠀⠀⠀⠀⠈⢧⠈⠀⠀⢠⠀⣼⠀⠸⡇⠀⢻⠀⡿⡼⢀⡞⠀⠀⠀⣇⣿⢸⣿⠀⠀⠀⠀⢻⣄⠈⠤⠈⠀⣳⣄⠀⠀⣼⠀⠀⠘⢧⡀⢹⠀⣾⢸⢃⡾⣠⠞⠁
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠃⠀⠀⠃⠀⠈⠓⠛⠃⠘⠀⠀⠀⠀⠹⠘⠟⠃⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠉⠙⠂⠐⠃⠀⠀⠀⠀⠙⠊⠀⠃⠛⠚⠚⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣏⠖⣟⣋⣛⣛⣛⣙⣋⣛⣙⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠐⠀⣇⢯⣳⠲⣤⣄⡀⠀⢠⣺⠀⢀⣓⡄⠀⢀⣠⣤⠶⣞⡽⣹⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠠⠄⠀⠀⠈⢷⡛⠳⣴⣌⠻⣆⡇⢡⣂⣐⠍⢲⣰⢟⢡⣦⠞⢛⡾⢁⠂⠀⠠⠄⢀⠀⠀⠄
⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠠⡙⢯⣤⣤⡛⢺⡌⢿⠁⠈⡯⢡⣗⢛⣤⣤⡼⢋⠄⡀⠀⣀⣀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⢀⡈⣙⠐⠂⠰⠀⣱⡜⢦⡀⠀⠀⣠⡤⠀⠉⠉⠈⡝⢉⣀⣤⣄⢌⠁⢀⠉⢀⢤⣄⠀⠀⢀⡴⢣⣎⠀⠆⠰⠆⣋⢁⡀
⠀⠀⠀⠀⠀⠀⠠⠄⠘⠋⠀⠀⠀⠀⣈⡶⠞⠁⠀⣰⠉⡀⡨⢻⣿⣫⣯⣼⡆⢂⠂⠁⠈⠁⠈⠳⢶⣁⠀⠀⠀⠀⠙⠃⠠
⠀⠀⠀⠀⠀⠀⢀⠔⣠⠴⠀⢀⣠⣾⠏⠀⠀⠀⢀⠔⠉⠁⠣⣀⠻⣟⠿⠉⢀⠈⠀⠁⠀⡀⠀⠀⠂⠹⣷⣄⣀⠀⠦⣅⠢⡀
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠊⠸⡿⡟⠀⠀⠀⠀⠀⠓⠈⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⠆⠑⠂⠀⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠉⠈⠁⠐⠂⡤⠄⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⢠⠁⡜⣁⢀⡀⠀⢀⠀⡀⢠⠋⡄⠀⠀⢀⡀⣠⠀⡠⠀⢀⡄⣠⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰⠃⡜⢱⠃⡎⢰⢣⠞⠸⢡⠏⡄⠀⠀⠀⠟⢰⠃⢼⠡⠀⡞⠰⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁



⠀⠀⠀⠀⠀⠀⢀⠀⠠⢤⢰⣷⣧⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⢀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⡆⡠⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠈⠢⠙⠒⠀⠈⠙⢿⣆⠁⠀⠀⠈⠢⣀⡀⡔⠉⣴⣯⣶⣀⠈⢀⠀⡀⠀⠁⠀⠀⠈⣰⡿⠋⠁⠀⠖⠋⠔⠁
⠀⠀⠀⠀⠀⠀⠐⠂⢠⣄⠀⠀⠀⠀⢉⠷⢶⡀⠀⠺⣀⠁⢑⣼⣿⣙⣛⢿⠆⠌⠄⡀⢀⡀⢀⡶⠞⡉⠀⠀⠀⠀⣠⡄⠐⠂
⠀⠀⠀⠀⠈⢁⣩⠰⠆⠰⠀⡹⢢⠞⠁⠀⠀⠙⠓⠀⣀⡀⣀⣣⢈⠉⠛⠋⢂⡀⢈⣀⠀⠚⠋⠀⠀⠈⠳⡜⢏⠀⠆⠰⠄⢍⡈⠁
⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⣡⣞⠛⢛⣥⢼⢃⣼⡀⢀⣷⣘⡧⣬⠛⠛⢳⣌⠀⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠂⠀⠠⢀⡾⣥⡴⠺⢃⣴⠏⠧⣒⠍⠩⣂⢼⠹⣦⡙⠟⢦⣬⢷⡀⠄⠀⠐⠂⠈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⣎⣞⡽⠴⠛⠋⠁⠀⠘⢽⠁⠀⡯⠃⠀⠈⠙⠛⠦⢯⣣⣱⠀⠀⠀⠐⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠒⠒⠦⣄⣀⣀⣀⣀⣀⣀⣀⣠⠶⠒⠒⠒⠶⢤⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⢀⣀⣀⣀⡀
⠀⠀⢀⡴⠛⠉⠁⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠁⠀⠈⠉⠓⢦⡀
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⣽⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡏
⠀⠀⢰⠇⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⢹⡀
⠀⠀⣾⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⣀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠈⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠶⠦⠤⢤⣤⣤⣀⣀⣀⣀⣠⣤⣤⡤⠤⠶⠖⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⣀⣀⣀⣀⣀⣤⠴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣤⣀⣀⣀⡀⠀⢀⣀⣀⣀⡤⠤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣠⡀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣧⣱⠀⣾⠁⡴⠄⣾⠡⠤⠃⠶⠆⠰⠢⠜⠤⠉⣇⠰⡄⠘⣧⠐⣇⡅⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣦⠈⠛⢀⣉⣠⣼⣤⡴⠤⣏⠀⣸⠉⠟⠀⠘⠟⠙⣆⢠⣙⠤⣤⣼⣥⣄⣉⠐⠛⢀⣶⠀⡀
⠀⠀⠀⠀⠀⣤⣀⠀⠺⣀⣈⡡⢼⡷⠚⡋⠉⡤⢠⡎⢠⠞⠁⣐⡉⠁⠌⠀⠀⠠⠊⢉⣆⠈⢲⡀⣷⠀⡤⠉⣙⠓⢾⡥⣈⣁⣀⠇⠀⣠⢤
⠀⠀⠀⡀⢀⠉⢉⣀⡲⠀⢸⠀⢠⡤⠰⠃⠠⠖⠈⠃⠘⠇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⠁⠙⠀⠰⠆⠹⠆⣤⠄⠸⡆⠐⢖⣀⠉⠉⠀⣀
⠀⠀⠀⠉⠉⠃⠀⠉⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⠁⠉
⠀⠀⠀⢼⠿⠿⣿⣿⣿⣿⠛⠿⡿⢸⣿⣷⣾⡆⠐⣿⣿⣿⡿⠀⠀⢀⣾⠿⣿⡇⠀⠀⠘⣿⣿⣧⠀⠀⠀⠸⣿⣿⠐⣿⣿⣿⡗⠀⠀⣰⣷⡗
⠀⠀⠀⠀⠀⠀⣿⠿⢿⣿⠀⠀⠀⠸⣿⠿⣿⡇⠀⣿⠿⣿⣇⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⣿⣿⣿⣇⠀⠀⠐⣿⡏⠀⣿⣿⣿⡇⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⣿⠻⢻⣿⠀⠀⠀⢠⣿⣿⣿⣧⣴⣯⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⣿⣿⣷⣿⡆⠀⢀⣿⡇⠀⣿⣿⣿⣷⣾⣿⣯
⠀⠀⠀⠀⠀⠀⣿⠻⢻⣿⠀⠀⠀⠀⣿⣿⣿⡟⠉⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⣿⣿⡿⠿⣿⡄⠸⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣿⠐⢸⣿⠀⠀⠀⠀⣿⣿⣾⡇⠀⣿⣿⣿⣿⠀⠀⣿⡿⣿⣽⣿⡇⠀⠀⣿⣿⢻⡟⢛⣷⠸⣿⡇⠀⣿⣿⣿⡿⣿⡟⢿⡇
⠀⠀⠀⠀⠀⠀⣿⣴⣾⣿⠀⠀⠀⠀⣿⣿⣿⡇⠀⣿⣿⣯⣿⠀⢸⣿⡇⣿⡛⣿⡇⠀⠀⣿⣿⠈⢿⣿⣿⣿⣿⡇⠀⣿⣿⣿⡇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⠾⢿⣿⠀⠀⠀⠀⣿⣍⣻⡇⠀⣿⣭⣽⣿⠀⣸⣿⠀⢻⣿⣻⣷⠀⠀⣿⣿⠀⠘⣷⡿⣿⣿⡇⠀⣿⣻⣿⡇⠈⣿⣧⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠶⢸⣿⠀⠀⠀⠀⣿⣀⣹⡇⠀⣿⣛⣿⣿⠀⣿⣿⣤⣼⣿⣿⣿⠀⠀⣿⣿⠀⠀⢹⣧⣀⣹⡇⠀⣿⣷⣾⡇⠀⢹⣖⣺⣧
⠀⠀⠀⠀⠀⠀⣿⣴⣶⣿⠀⠀⠀⠘⣿⣿⣿⡇⠀⣿⣙⣻⣿⢀⣿⡿⠉⠉⣿⣿⣿⡆⠀⣿⣿⠀⠀⠀⢻⣦⣿⡇⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠰⣿⠿⡿⣿⠀⠀⠀⠰⡿⠿⠿⠇⠀⠿⠿⠿⡿⠸⠿⠷⠀⠀⣿⣷⣿⣷⠀⣿⣿⠀⠀⠀⠈⠿⠿⠇⠀⣿⣿⠿⡇⠀⢸⣿⣶⣿⣷
⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄
⠀⠀⠀⠀⠀⢰⡜⠂⠀⠀⠀⠀⠀⠀⠀⡶⠶⣦⠀⠀⢠⡶⠶⣶⠀⢀⣠⣶⣶⣦⡀⠀⠀⣴⡦⠀⠀⠀⢠⣶⣶⣶⠀⠀⠀⠀⠀⠀⢀⠃⡆
⠀⠀⠀⠀⠀⠈⣗⢡⣤⠀⠀⠀⠀⠀⠀⣷⠀⣿⠀⠀⣾⡇⠠⣿⢀⣾⣿⣿⣧⣹⣿⣆⠀⣿⡇⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⣤⡌⢾
⠀⠀⠀⢻⠦⣀⣿⠟⠋⠀⠀⠀⠀⠀⠀⣗⠰⡏⠀⢰⣿⡃⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⢰⣿⡇⠀⠀⢀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠙⠾⣿⣀⠴⡟
⠀⠀⠀⠀⠘⠻⡇⠀⡰⠛⡆⠀⠀⠀⠀⢻⡀⣇⣠⡟⣹⠇⣼⠃⢹⣿⣮⣿⣿⣿⣿⣿⠸⣿⡇⠀⠀⣸⢿⣿⣿⣿⡇⠀⠀⢰⠟⢄⠀⢸⠗⠂
⠀⠀⠀⠀⠀⠀⢀⣆⡐⢀⠇⠀⠀⠀⠀⠈⠙⠛⠋⢀⣿⣼⠏⠀⠘⣿⣿⣿⣿⣛⣿⠏⠀⣿⡇⠀⣠⡏⢸⣿⣿⣿⡇⠀⠀⠸⡀⠀⣱⡄
⠀⠀⠀⠀⠀⠀⠈⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠻⠶⠿⠟⠁⠀⠀⠀⠈⠻⢿⣿⠿⠋⠀⠀⠘⠿⠾⠋⠀⠸⡿⠿⠿⠃⠀⠀⠀⠈⠈⢻⠁

⠀⠀⠸⣿⣿⡆⢰⣾⠆⢰⣶⣾⠿⠷⠰⠟⢿⣿⣿⠟⠻⠂⣶⢶⡾⠿⠆⢰⣶⣿⡿⢷⣆⠀⠀⠀⣾⣿⡇⠀⠘⣿⣷⡀⠀⢸⣿⠀⣴⣿⢷⣆
⠀⠀⠀⣿⣿⡇⢸⡿⠀⢸⣿⣿⣄⣀⡀⠀⣼⣿⣿⠀⠀⠀⣿⣻⣧⣀⣀⢸⣿⣿⡇⢸⣿⣇⠀⢰⣿⣿⣿⠀⠀⣿⣿⣧⠀⢸⣟⢼⣿⣇⠸⠟⠃
⠀⠀⠀⢻⣿⡇⣼⡇⠀⢸⣿⣿⠛⠛⠁⠀⢹⡿⣿⠀⠀⠀⣿⣿⡟⠉⠋⢸⣯⣽⡇⢸⣿⣿⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⡆⢸⣷⠘⣿⣿⣦
⠀⠀⠀⢸⣟⣿⣿⠇⠀⢸⣿⣿⠀⠀⠀⠀⢸⣇⣿⠀⠀⠀⣿⣻⡇⠀⠀⢸⣿⣿⡇⢸⣿⡏⠀⣼⡿⣿⣿⡀⠀⣿⠻⣷⣿⣼⣿⠀⠙⢿⣿⣷⡄
⠀⠀⠀⠀⣿⣻⣿⠀⠀⢸⣿⣿⠀⠀⠀⠀⢸⡟⣿⠀⠀⠀⣿⣻⡇⠀⠀⢸⣿⣾⣷⡿⠟⠁⠀⣿⠀⣿⣿⡇⠀⣿⠀⢹⣿⣻⣿⠀⠀⠀⠻⣟⣿
⠀⠀⠀⠀⢿⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⠀⢸⣟⣿⠀⠀⠀⣿⣻⡇⠀⠀⢸⣿⣽⣿⣷⣆⣦⢠⣿⠶⢿⣿⡇⠀⣿⠂⠀⢿⣴⣇⢸⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠸⢿⠇⠀⠀⢸⣿⣿⣷⣶⠂⠀⢾⣿⣿⠆⠀⠀⣿⣼⣷⣶⠖⠸⣿⣾⡇⠈⠻⣿⠸⠿⠀⢸⣿⣿⠀⣿⠇⠀⠈⠿⠟⠀⠻⣿⠶⠿⠃
⠀⠀⠀⣀⢀⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⡀⣄
⠀⠀⠀⠁⠈⣀⣈⠡⠷⠀⠸⡀⠘⠓⠰⡆⠐⠆⢀⡄⢠⠆⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢰⡀⣠⠀⠰⠂⣴⠀⠛⠃⢰⠇⠠⠦⠉⣀⣀⠁⠈
⠀⠀⠀⠀⠀⠚⠉⠀⢶⠉⢉⡝⣺⡷⢤⣅⣀⠛⠘⢇⠈⠦⠀⠩⣁⡤⠂⠀⢀⠐⢄⣨⠟⢀⠾⠀⡿⠀⠛⣀⣩⡤⢾⡓⢩⡉⢉⡦⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠟⢀⣤⠈⣉⠛⢛⠛⠛⠚⣏⠁⢹⡄⣶⠀⢠⣦⣨⠏⠘⣉⠒⠛⢻⡛⠋⣉⠠⣤⠈⠿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠏⡹⠂⢿⡀⠱⠂⢿⣑⠒⡔⠲⠆⠰⠖⣪⠒⣈⡟⠰⠃⢠⡟⠀⡏⠃⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠉⠀⠀⠈⠁

⠀⠀⠀⠀⢿⣿⣿⣿⠇⢸⣿⡇⢸⣿⡇⠀⠀⣿⣿⣿⠀⠀⢸⣿⣧⠀⣿⣿⠀⣿⣿⠀⢠⣿⡟⠀⣿⣿⠿⠀⢻⣿⠀⢸⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⢸⣿⡇⠀⢠⣿⣿⣿⡆⠀⢸⣿⣿⡀⣾⣿⠀⣿⣿⢀⣾⡟⠀⠀⣿⣿⠀⠀⣸⣿⠀⢸⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⢸⣿⡇⠀⢸⣿⠈⣿⡇⠀⢸⣿⣿⣇⢿⣿⠀⣿⣿⣾⡟⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⣷⣾⣿⡇⠀⢸⣿⠀⣿⡇⠀⢸⣿⣿⣿⣾⣿⠀⣿⣿⡟⠀⠀⠀⠀⣿⣿⣶⠀⣽⣿⠀⢸⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡏⢹⣿⡇⠀⣾⡟⠀⣿⣿⠀⢸⣿⡟⣿⣿⣿⠀⣿⣿⣷⡀⠀⠀⠀⣿⣿⠉⠀⣿⣿⠀⢸⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⢸⣿⡇⠀⣿⣷⣦⣿⣿⠀⢸⣿⡇⢿⣿⣿⠀⣿⣿⢻⣷⡀⠀⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇⢸⣿⡇⠀⣀⣀
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⢸⣿⡇⢠⣿⡏⠉⢹⣿⡆⢸⣿⡇⣾⣿⣿⠀⣿⣿⠈⢿⣷⡀⠀⣿⣿⠀⠀⢹⣿⡀⣸⣿⡇⢸⣿⣧⣾⡟⠙⡇
⠀⠀⠀⠀⠀⠸⠿⠇⠀⠸⠿⠇⠸⠿⠇⠸⠿⠁⠀⠸⠿⢧⡼⠿⢷⣿⡿⠿⠀⠿⠿⠀⠈⠿⠷⠀⢿⣿⡆⠀⠈⠻⠿⠿⠋⠀⠸⢿⣿⡿⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⣿⡟⠛⠛⠇⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣠⠏
⠀⢀⣠⣤⣤⣀⡀⠀⠀⣴⡿⠀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣀⣄⡀⠀⢀⣿⡿⠀⠀⣀⣀⠀⠀⣀⠀⠀⣸⣿⡷⠋
⣴⣿⠏⠀⠀⣿⣿⠦⠀⣹⠷⢻⣿⡿⠀⠀⣴⣿⠋⢨⣿⣿⠀⠀⣿⣿⠀⠀⠀⢀⣾⣿⠉⣹⡄⢸⣿⡇⠀⢠⣿⡿⠀⣼⣿⡇⠀⢹⣿⡇
⣿⣿⠀⠀⣸⣿⣿⠀⢠⡟⠀⢸⣿⡇⠀⣼⣿⡇⢀⣾⣿⣿⠀⢠⣿⣿⠀⠀⠀⣸⣿⡏⠉⠉⢠⣾⣿⣅⣤⢾⣿⡇⢠⣿⣿⡇⠀⣸⣿⡇⠀⠀⣼
⠘⠿⣦⠾⠋⣿⣿⣠⡞⠀⠀⢸⣿⣦⡾⠻⣿⣧⠞⠁⢿⣿⣴⠏⠹⣿⣦⣤⡶⠛⣿⣧⣠⡴⠋⣿⣿⡅⠀⠸⣿⣷⠟⠸⣿⣧⣴⠏⢻⣷⣤⡾⠁
⠀⠀⢀⣠⢾⣿⡟⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠁⠀⠀⣿⣿⢻⡀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠈⠁
⠀⣰⠟⠁⣸⣿⣇⣀⣀⣀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⣿⣿⣈⣧⣀⠀⢀⣀⣀⣀⡀
⢸⡇⠀⢀⣿⡟⣿⣿⡿⣿⣿⡄⠀⢸⣿⡇⠀⠀⣿⣿⣿⠅⢀⣾⣿⠿⣿⣦⠀⠀⣴⣿⡿⣿⣷⣿⣿⣿⣿⡿⠀⢸⣿⡿⣿⣿⣷⡀
⠘⣇⣀⣼⠟⠀⣿⣿⠀⠈⣿⣿⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿⡇⠀⢸⣿⡇⢰⣿⡏⠀⢸⣿⡟⢿⣿⡿⠁⠀⢸⣿⡇⠀⢸⣿⡇
⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⠘⣿⣧⠀⠸⠿⠇⠈⣿⣷⠀⠸⠿⠇⢠⣿⣧⠀⠀⢸⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣤⣼⣿⠇⠀⢸⣿⡇⠀⠀⣿⣿⣤⡀⠀⠹⣿⣷⡀⠀⠀⠀⠙⣿⣷⡀⠀⠀⢸⣿⣿⣤⠀⢸⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠛⠻⣿⣦⠀⢸⣿⡇⠀⠀⣿⣿⠛⠀⠀⠀⠙⣿⣷⡄⠀⠀⠀⠘⢿⣿⡄⠀⢸⣿⣿⠛⠀⢸⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⢀⣀⠀⠈⢿⣿⠀⠀⣀⡀⠈⢿⣿⡄⢸⣿⣿⠀⠀⢸⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿⡇⠀⢸⣿⡇⢸⣿⡇⠀⢸⣿⡇⢸⣿⣿⠀⠀⢸⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣄⣤⣿⣿⠀⢸⣿⣇⡄⠀⣿⣿⣤⠀⠘⣿⣧⣄⣼⣿⠃⠈⣿⣷⣄⣼⣿⠇⢸⣿⣯⣀⠀⢸⣿⣇⣠⣾⣿⠇
⣤⣤⣠⣤⣤⣤⣿⣿⣿⣿⣿⣡⣤⣼⣿⣿⣧⣠⣿⣿⣿⣄⣤⣌⣿⣿⣿⣡⣤⣤⣌⣿⣿⣿⣩⣤⣸⣿⣿⣿⣤⣼⣿⣿⣿⣿⣫⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣩⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⢀⡠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠔⠂⠠⢄⡀
⠀⠀⢀⠃⠀⠀⠀⠱⡀⡠⠔⠒⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⡍⢩⣭⡖⠢⣝⢆
⠀⠀⠘⡆⠀⠀⠀⠀⠋⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠈⠢⢉⠠⠊⠀⣹⠂
⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠒⠒⠂⠈⠉
⠀⠀⠀⠀⠈⠢⡄⢀⡀⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠠⠤⠐⠀⠀⠉⠉⠉⠉⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢁⡀⠤⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⢗⡒⠂⢐⣂⣉⡀⠀⠤⠔⠙⣄⡀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⠠⡤⠒⠒⠐⠈⠐⠒⢄⠀⡼
⠀⠀⠀⠑⡶⢇⠀⠀⢀⡠⠤⠒⠂⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠒⠉⠁⠀⠀⢇⡔⠒⠒⠒⠤⠤⠤⠞⢧
⠀⠀⠀⠀⡇⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠂⠈⠉⠁⠒⠠⣡
⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠁⠀⢀⠀⠠⠐⠀⠀⠡
⠀⠀⠀⠀⠀⠁⠒⠀⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢢⠀⠀⡠⠄⠂⠄⡀
⠀⠀⠀⠀⠀⢠⡶⢳⠒⢒⣀⠤⣉⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡠⢣⠊⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠙⢤⠀⠣⣈⠋⣀⠜⠑⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⡔⡠⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠄⠀⢙⠂⠤⠭⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⢇⠀⣀⡤⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠀⠈⠉⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠉⠁⠀⠈⠉⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿
⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿
⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⡄
⡇
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿
⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠶⠖⠛⠛⠋⠉⠉⠉⠁⠀⠀⢠⠄⠀⠀⠀⠒⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠛⣭⠉⠀⣶⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡄
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⢀⠀⢻⡆⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠛⠃⢸⡇⠀⣿⣇⡤⢤⡀⠀⣠⣶⠶⢤⡀⠀⠀⢸⣦⡴⠢⡀⠀⢸⣿⠈⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠸⣧⣀⣀⣀⣤⠄⠀⢸⡇⠀⢿⡟⠀⠀⣿⡀⣿⠁⠀⠀⣿⠀⠀⣼⡏⠀⠀⣿⠀⢸⣧⡴⠟⣷
⠀⠀⠀⠀⠤⠀⠈⠉⠛⠉⠁⣀⠀⢸⡇⠀⢸⡇⠀⠀⢹⡇⢹⡄⠀⣠⡿⣧⠀⣿⡇⠀⠀⣿⠀⣿⣿⡀⠀⠈⠀⠀⠶
⠀⠀⠀⠀⠀⢠⣤⡀⢠⣶⠀⠈⢀⣾⠃⠀⠸⠇⠀⠀⠸⠇⠀⠙⠛⠉⠀⠙⠃⠛⠃⠀⠠⡿⠀⣿⠘⣧
⠀⠀⠀⠀⠀⠈⠛⠁⠀⠙⠶⠾⠛⠁⠀⣀⣀⣤⣤⣴⢶⣶⢶⣶⠖⡛⢷⣶⢶⣶⢶⣦⣤⣤⣀⣀⠀⠘⠷⢤⡶⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⡌⠟⣸⡇⣾⣿⢨⣿⢸⣿⢸⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣙⣠⣿⣷⣬⣥⣾⣿⣮⣥⣮⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠀⠠⠿⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⢴⠄⠀⠀⠀⠈⠉⠉⠛⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⢀⡀⠀⠀⠀⠛⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢄⠀⠀⣠⠖⠲⡄
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⣾⠀⠀⠀⡇⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⣠⠞⠛⠉⠉⠛⠛⠾⣿⠤⠤⠚⠀⣾⠃⠀⠀⠆⠀⠀⠀⠀⠀⢠⠋⠁⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠈
⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⣿⡄⠀⡘⠀⠀⠀⠀⠀⠀⢹⣀⠀⡠⠀⠀⠀⠀⠹⡄⠀⠀⢠⣾⠟⠁⠀⠀⠀⡸
⠀⠀⠈⣇⠀⠀⠠⢄⠀⠀⠀⢹⣿⡀⠀⠀⢻⣇⠈⠀⠀⠀⠀⢠⡾⠓⡄⠈⠉⢁⢤⡀⠀⠀⠀⠈⠢⣴⡿⠃⠀⠀⣀⡠⠊
⠀⠀⠀⠈⠢⠄⠠⠜⠀⠀⠀⠀⢻⣷⡀⠀⢸⣿⠐⠙⣿⠀⢠⣿⠁⠀⢸⠀⡰⢁⣾⣣⠔⣶⡄⢀⣼⠟⣉⣉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠈⣿⡆⠀⣿⡀⢸⡏⠀⣴⠆⡰⢁⣾⠟⠁⢠⣿⢃⣾⡟⠉⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢹⣷⠀⢿⣆⠼⣧⠞⢿⠔⠁⢾⠏⠀⢀⣾⠇⣾⡟⠉⣯⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠈⢿⡀⠀⢿⠇⠈⠉⡠⠤⣄⠀⠀⠀⠀⠀⠀⠘⠿⠺⠿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠀⠈⡆⠀⢸⠇⠀⠈⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠃⠀⠘⠀⠀⠀⠀⠀⠘⢆⠀⢸⡇⠀⣰⡆⠀⣾⠓⠀⠀⠀⠀⠸⣇⠀⠈⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣴⠋⠈⠀⠀⠀⠀⠀⠘⣧⡴⣿⡇⣤⡜⢧⠠⣾⠂⣴⠆⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⣠⡄⠀⠀⠀⢀⢾⡇⣿⠁⡜⠀⣿⡠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠄⢠⡖⡄⢷⣇⡀⠀⢀⠂⢸⡇⠈⠋⠀⠀⠀⠀⠙⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠰⣿⠛⠖⢡⣿⠃⠀⢸⣀⣼⠃⠀⠀⢰⡟⢹⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠻⡤⠃⢸⡇⠀⠀⠀⠉⠁⠀⠀⢀⣿⢃⢦⡾⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠈⠓⠒⠁⠀⣠⣶⠒⠀⣸⣿⢁⣿⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⡀⣿⡇⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⢿⣇⣾⠀⣿⡇⣾⠁⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠙⠓⠹⠦⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠞⠁⠯⠅⠪⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠆⣂⠡⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⣒⠲⠦⢚⣫⣭⠔⠚⢃⣁⠤⠄⠀⠤⠄⡀⠀⠀⠀⠀⠀⠠⠖⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠤⢒⡩⠔⠂⢉⡀⠤⠤⠘⣛⣃⣀⡈⠉⠐⠒⠲⣜⠃⢰⣒⠻⣶⠂⠉⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡡⠚⡡⣒⠥⠐⢂⡩⠤⢴⣲⡚⣏⠙⣌⣱⠬⠟⠒⠀⠀⠉⢳⡄⡈⠑⠦⡈⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡞⡡⠞⠉⢀⡀⣨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠢⡀⠈⠢⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠆⠀⡴⠋⢀⡤⠊⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡚⢆⠀⠈⠢⡀⠘⠄
⠀⠀⠀⠀⠲⠄⢠⣄⠎⢀⡜⠀⢠⢊⡠⠂⠁⠀⢀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣍⢣⠀⠐⢌⢄⠈⡄
⠀⠀⠀⠀⠀⠀⠀⡅⠀⠈⠀⡰⠋⠁⠀⠀⠀⠀⠘⠤⣀⣀⡧⠴⠃⢀⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡂⠷⠀⠈⡆⠱⣰
⠀⠀⠀⠀⠀⠀⡀⣇⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣃⣈⠦⠯⣆⡝⡧⢋⡺⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣉⡇⠀⢓⠂⢻⠢⡀
⠀⠀⠀⢀⣴⠀⠀⠁⠀⡌⢰⠆⡰⠆⠀⠀⠀⡠⠖⠊⠉⢸⠀⠈⠵⠀⠀⡇⣃⡌⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡁⠀⢀⡄⠹⢆⠈⠄
⠀⠀⠀⡎⢸⠀⠀⠀⠘⠀⣎⢎⠴⠀⠀⠀⠘⠤⠤⠂⢀⢴⢢⠀⠀⠀⠐⠁⠁⠣⢔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠸⢣⠀⠄⠉⠺
⠀⠀⠀⡁⡈⣀⢰⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⢸⢼⠖⡆⢀⡀⢀⢀⡀⣿⡄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⡄⠰
⠀⠀⠀⢇⡁⡏⠘⠲⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⢂⠸⠀⠷⠣⠓⠊⠋⣇⡟⠳⠋⢃⢉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡅⠀⠆
⠀⠀⠀⠘⠃⡇⠰⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠙⠀⠀⠀⠀⠀⠀⠀⠁⠐⠤⠊⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠋⠒
⠀⠀⠀⠀⠀⢩⠄⢡⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢖⡂⢠⠂⡄⢰⠴⣀⢷⢚⠀⠂⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠄⣼
⠀⠀⠀⠀⠀⠈⠈⠄⠳⢥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⢉⣋⣉⡁⠀⠀⠀⠸⡠⠋⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⠀⠜⠶⠐⠶
⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠁⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠉⠳⡄⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠡⢟⡍⠀⠀⠤
⠀⠀⠀⠀⠀⢀⢀⣘⡌⠿⠆⠐⠧⠐⠷⢌⡁⠢⡀⠀⠀⠀⠀⠙⠒⠤⠤⠤⠼⢦⠒⠉⣂⠁⣦⠀⠀⠀⠀⠀⠀⠀⡈⠀⡰⡡⠡⡘⠁⢀
⠀⠀⠀⠀⠀⠈⢁⡸⠿⠀⠀⠘⠴⠀⠲⣆⠈⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⣰⢡⠾⠊⠠⠣⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠈⡴⢠⠀⣤⠤⢀⡑⠢⢄⡹⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠔⢪⠃⠀⢀⠔⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣈⡑⠂⢤⣈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡡⢊⣀⠔⣹⣤⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⢀⡀⠉⠒⠤⠄⣀⠀⠀⣀⣀⠙⠟⠴⠚⠯⡒⠕⠈⣡⠟⠨⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⣥⢛⡱⢦⠄⣀⢀⣄⣭⡌⠩⠞⠋⡡⠊⠀⡠⠚⡁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠱⠖⠉⠑⠶⠄⠀⠤⠄⠘⠒⠈⠁⠛⠉

⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⢤⣤⣄⣀⡀
⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⢟⠕⢺⣎⡑⢦⡀⠀⠀⠀⠀⠀⠀⢀⠴⢊⢥⡖⢀⣴⣿⣿⣿⣿⣿⡗⢤⡀
⠀⠀⣴⣿⣿⣿⣿⣿⣿⢟⡕⢩⣧⡳⢝⣴⣿⣿⣦⡀⠀⠀⢀⡔⣡⣖⠷⢋⣴⣿⣿⣿⣿⣿⠟⡩⣜⣧⣝⠦⡀
⢀⣾⣿⣿⣿⣿⣿⢟⢥⣴⡟⣵⢐⣵⣾⣿⣿⣿⣿⡷⡀⢠⢎⢜⠉⢊⣴⣿⣿⣿⣿⣿⠿⠁⢾⣽⡂⠸⣩⣾⣷
⣸⣿⣿⣿⣿⠟⢑⡯⠪⣷⢟⣵⣿⣿⣿⣿⣿⡿⣋⣴⠹⠏⡼⢋⣴⣿⣿⣿⣿⣿⠟⡡⣦⣿⣿⠶⣩⣾⣿⣿⣿
⣿⣿⣿⢟⡵⣱⣹⣍⢎⣵⣿⣿⣿⣿⣿⡿⢫⠰⢟⢏⢦⢪⣴⣿⣿⣿⣿⣿⠟⣣⣎⡟⠚⠝⣩⣾⣿⣿⣿⣿⣿
⣿⠟⢑⣹⢗⡄⢜⣴⣿⣿⣿⣿⣿⡿⣋⣜⡕⣓⠾⢋⣴⣿⣿⣿⣿⣿⡟⠡⣞⣿⠙⠞⣡⣾⣿⣿⣿⣿⣿⢟⣱
⢰⣈⣾⢝⢚⣵⣿⣿⣿⣿⣿⡿⢋⣶⡜⢉⡸⢊⣴⣿⣿⣿⣿⣿⠟⣡⣰⣶⣯⡟⣣⣾⣿⣿⣿⣿⣿⢟⣵⣪⠆
⠈⣇⢛⣴⣿⣿⣿⣿⣿⡿⣋⢼⢑⢍⡾⢋⣼⣿⣿⣿⣿⣿⠟⣡⣺⠾⢿⡞⣣⣾⣿⣿⣿⣿⣿⢟⠵⣸⠻⡧⢼
⠀⠸⣿⣿⣿⣿⣿⡿⢋⣴⣷⢦⡻⢃⣴⣿⣿⣿⣿⣿⠟⣡⠾⡟⢫⡗⣠⣾⣿⣿⣿⣿⣿⢟⢕⣿⣯⢳⢏⣴⠇
⠀⠀⠹⣿⣿⡿⢋⢄⡱⢛⡵⢋⣶⣿⣿⣿⣿⣿⠟⣉⣸⢼⣿⠷⣠⣾⣿⣿⣿⣿⣿⢟⢥⠁⣾⡼⢍⣴⣿⠏
⠀⠀⠀⠙⣏⢶⣿⢏⠶⢋⣴⣿⣿⣿⣿⣿⠟⡡⣾⣽⢯⠺⣩⣾⣿⣿⣿⣿⣿⢟⡕⡱⣯⢿⢛⣵⣿⣿⠋
⠀⠀⠀⠀⠈⢮⠰⢋⣴⣿⣿⣿⣿⣿⠟⡡⣮⠿⣻⠆⣩⣾⣿⣿⣿⣿⣿⢟⣕⣵⡺⣵⢍⣵⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠱⣿⣿⣿⣿⣿⠟⠃⣌⣟⣚⠄⣩⣾⣿⣿⣿⣿⣿⢟⡑⠡⢯⣿⢍⣴⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⣡⣾⣾⣿⡞⣡⣾⣿⣿⣿⣿⣿⢟⠕⣵⣿⡾⢏⣴⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣜⠻⡊⡘⣣⣾⣿⣿⣿⣿⣿⢟⣕⡏⢞⡴⢛⣵⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣥⣾⣿⣿⣿⣿⣿⢟⠵⢹⣻⡞⢎⣵⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⢟⢔⣽⣦⡷⢟⣵⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡱⡞⡎⠝⢏⣴⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣉⣴⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣶⣶⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡶⠟⢋⣉⠁⠀⠀⠀⠀⠀⠀⠈⣉⡙⠻⢶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠛⠻⣶⡾⠛⠛⠻⢷⡄⠀⠀⢠⡾⠟⠛⠛⢷⣶⠟⠛⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⣿⡄⢠⣿⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠻⣷⣄⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⣠⣾⠟⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠈⠻⢷⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⡾⠟⠁⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⣾⠛⠲⠶⠤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣤⠤⠶⠖⠛⣷⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢿⣦⣤⣤⣄⣀⣀⣀⣀⡀⠀⠈⢀⣀⣀⣀⣀⣠⣤⣤⣶⡿⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠛⠋⠉⠉⠙⠛⢿⣿⣿⣿⡿⠿⠃⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠻⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠴⠆⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠉⠓⠶⢤⣤⣄⣠⣤⡤⠶⠚⠉⠀⠀⠀⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠿⠿⠟⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⡀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠈⠙⢶⣀⡴⠖⠶⣾⠋⠉⠛⢶⠟⠉⠛⠋⢁⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⣀⠀⠀⢀⡶⣆⠀⠘⠟⠀⠀⡀⠀⢀⡶⣆⠀⡀⣰⡷⢰⣿⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⣿⠀⠀⢸⡇⢸⠀⠀⠀⠀⢰⡇⠀⠸⡇⡟⢈⠁⣿⠃⣿⠇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⢀⡀⢷⣸⢠⣶⡆⠀⢸⡇⣠⡄⢻⢷⡟⢸⠏⢰⡏⠀⣤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⣼⣧⠈⡿⣻⠋⣀⠀⢸⣥⢻⣧⠏⠸⡧⠻⡤⠛⣧⡼⠋⢀⠼⣋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⠃⢿⣼⠃⠸⠶⠃⠀⠈⢁⠀⠉⠀⢀⣤⡄⠀⠀⠀⠀⠀⠐⠋⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠰⠃⠀⣿⡇⠀⠀⢾⣽⠇⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢠⣶⡄⣴⡆⠀⣴⠀⠀⣹⣇⢀⣠⠞⢿⣄⠀⠀⢀⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢆⠏⣾⡿⢸⡇⣠⣿⢀⡴⠃⣹⡏⠀⢠⡾⣿⡟⠛⠛⠉⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢸⡟⠀⠙⠃⠈⠛⠁⠙⢋⣀⡀⠿⠃⠀⢻⣤⡿⠁⠀⠀⠀⢠⡄⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡉⠁⠀⡀⠀⠀⠀⢠⣶⣆⠠⡟⣠⢹⠀⣰⡆⠀⣴⠄⠀⠀⠀⠀⣾⠃⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡖⠀⠀⣷⠀⢀⡴⠋⠈⣿⢀⣿⡿⢼⠞⣿⢀⡼⣿⠀⠀⠀⢀⣾⠋⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠋⠀⠀⠀⣿⠏⠸⣧⠏⠀⠻⠟⠁⠙⠷⣶⠾⠛⠁⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣤⠄⠀⠀⡼⣿⡇⠀⡀⠀⢀⣴⣄⣠⡄⠀⠀⠀⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣼⠁⣿⠀⢸⠟⠓⠋⠁⠈⠁⠙⠲⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠻⡼⠏⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⣀⣤⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠄⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣇⣇⣿⣸⡗⣿⣁⣀⣀⣤⠖⠃⠀⠀⡀⠀⠰⠋⠀⣀⣠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠻⢾⡟⠙⠉⠛⢁⠈⠙⠋⠉⠀⠀⠀⠀⣾⢙⠀⠀⠊⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣀⠀⢀⣾⡇⣠⣿⠄⡇⠀⠀⢠⢦⡀⢠⡶⡀⣿⣮⡀⢺⣇⠀⠉
⠀⠀⠀⠀⠀⠀⠒⠚⠛⠉⠉⢁⡀⠀⢸⣼⠇⠀⣿⠀⣿⢴⡄⣿⣸⣇⣸⡇⢇⢽⡿⠤⢺⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠻⠴⠹⠆⠳⠋⠋⠈⢁⡁⠀⠀⠀⠀⠛⠃⠀⢰⠋⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⡀⠀⠀⠀⠈⠉⠁⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢦⡁⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠉⠒⠤⠤⣤⠤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡠⠤⠒⠂⠙⠖⢊⠉⠀⠀⠐⠤⡀⠈⠙⠲⣄
⠀⠀⠀⢤⡀⠠⠀⠀⠀⠀⢀⡠⠒⠁⠀⠀⢠⠎⠀⠀⠀⠑⢦⡀⠀⠀⠈⢶⡀⠀⠈⢳⡀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⡸⢗⠋⢉⢢⠀⣠⠊⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⣿⠀⠀⠀⣷⠀⠀⠀⡈⠾⠥⠤⢤⡐
⠀⡔⢰⡃⠈⠆⠀⡧⢠⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠏⠀⠀⠀⡿⠀⡰⢫⠀⢦⡀⠈⢢⠹⣆
⠀⠁⠈⠃⠀⠀⠊⠁⢸⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡾⠁⠀⡇⣿⠀⢸⡇⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⡴⠋⠀⠀⠀⢳⡘⢇⣾⠇⣀⣀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⢀⣀⠴⠒⠦⢤⣀⣀⣠⣴⣯⣤⡴⠞⠋⠀⠀⣀⠤⠐⠒⠛⠂⠁⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⡨⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄

⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠛⠋⠉⠉⠙⠛⠒⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠛⠋⠉⠉⠙⠛⠒⠶⢤⣀
⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⢠⠞⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⢠⡏⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣼⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡏⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣇⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⢿⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠸⡆⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⣀⡤⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⣦⠀⠀⠠⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣉⣙⠛⠿⣟⣿⣧⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡉⠁⠀⠨⣿⡿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣋⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠘⣷⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⢻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⢀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⡾⢿⣯⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣏⣻⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣺⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣬⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠟⠙⣿⣿⣽⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡧⠀⠁⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⠀⠸⣿⢉⣽⣿⣿⣿⣿⡇⢛⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣆⠀⠀⣹⣿⣿⣿⣿⣿⡟⣱⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⠀⠀⣿⣿⣿⣿⣿⡟⣱⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣴⣿⣿⣿⡧⠉⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠻⠁⣴⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⣿⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣃⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⡿⢀

⠀⠀⠀⠀⠀⠀⢠⡄⡰⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⣤
⠀⠀⠀⠀⠀⠀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠤⣤⡤⠄⠀⢠⣤⠀⠀⡄⠀⠀⢠⣤⠀⠀⠀⣤⡄⠀⡀⠀⢠⣤⠀⠀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⡇⠀⠀⣸⣿⡄⠀⠀⣿⣧⠀⠀⠀⢸⣿⠀⡜⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣀⣀⡇⠀⠀⣇⣿⡇⠀⠀⡿⣿⡀⠀⠀⢸⣿⡴⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠈⡇⠀⢠⠁⢻⣧⠀⠀⡇⣿⡇⠀⠀⢸⣿⢳⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⡇⠀⢸⠀⢸⣿⠀⠀⡇⢸⣿⡀⠀⢸⣿⠈⡇⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⡇⠀⡜⠀⠈⣿⡄⠀⡇⠘⣿⡇⠀⢸⣿⠀⠸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠁⠀⠁⠀⠀⠉⠁⠀⠁⠀⠉⠁⠀⠈⠉⠀⠀⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⡉⠉⠉⠉⠉⠉⠁⠀⠘⢿⣿⠿⠃⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⣿⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠠⢻⣿⠀⠀⢸⣿⡇⠀⠀⢠⣿⠏⠐⢦⠀⠀⢠⡀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⠀⢀⠊⣿⡇⠀⠀⣿⡿⠀⠀⠀⠀⠀⢸⣿⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⠀⠌⠀⣿⡇⠀⠀⢻⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⣿⡧⠀⠀⠊⠙⠓⠀⠀⠀⠀⠈⠛⠊⠀⢻⣿⠀⢀⠄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠑⠢⡄⠀⠀⠀⠀⠀⢀⠠⢺⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠔⠁⠀⠀⠀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠘⠃⠘⠤⠤⠤⢤⠴⠥⠄⢼⣿⠆⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠠⠤⠞⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣠⡿⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣑⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⡏⣧⣤
⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣻⣿⣣⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠠⡴⣾⡟⢡⡞⠀⢒⡀⠁⣠⢀⡀⣰⠖⣀⠀⠀⠀⠀⠀⣾⣿⣿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢨⡅⢸
⠀⢰⡟⠀⣿⢸⡇⡾⢸⣣⡿⢹⡃⣿⢶⡀⠀⠀⠀⠀⠀⣿⣯⠁⣾⠃⠞⠟⡽⡟⠻⣿⢡⣟⣿⡇⠀⠀⠀⠀⠀⠀⢀⣄⡀⡀⠀⢀⣈⠳⠞
⠀⠸⠀⠸⠇⣸⢃⡏⣼⠄⣵⢌⡦⠀⠈⠃⠀⠀⠀⠀⠀⣿⣿⢸⡏⣸⢠⣰⣀⣁⡇⣏⡈⢻⣿⡇⠀⠀⠀⠀⠐⢚⡿⢁⡾⣡⢀⡥⡀⢠⣄⡀⣰⢃⠄
⠀⠀⠀⠀⠰⡟⡼⢸⠇⠀⠿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢙⡏⡝⢩⣿⢙⡟⣿⣾⣿⡇⠀⠀⠀⠀⠀⢸⠃⢸⡇⢿⠸⠁⠇⠿⢹⡇⡿⢶⡀
⠀⠀⠀⠀⠀⢐⣁⢤⢥⡦⣴⢲⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡼⣸⣇⣿⣧⣼⣄⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⡞⣴⠂⣰⢆⡖⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠉⠘⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⠃⠛⠀⠙⠈⠃


⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠠⠄⡀
⠶⣶⡿⠋⣴⠂⠀⢀⠀⠀⣀⠀⠀⣰⠆⣀⠀⠀⠀⠀⠠⠴⣾⠏⢡⡏⡀⢀⡀⠀⢠⠀⠀⢠⡆⢀⠀⠀⠀⠀⢀⣠⣾⠋⣰⢃⡀⣠⢄⠀⣤⢈⠀⣰⠆
⢀⡿⠀⣸⠇⣿⢰⡏⣿⢰⡿⣿⢀⣿⣬⠀⠀⠀⠀⠀⠀⢸⡏⢀⡿⢸⡇⣿⢸⢆⣿⢻⠇⣿⣬⡁⠀⠀⠀⠀⠈⢸⡏⢰⡏⢹⡧⡏⢸⢰⡿⣻⠀⣿⣬
⠀⠁⠀⠈⢠⡍⣬⢁⡭⠈⣤⢉⡄⠁⠘⠃⠀⠀⠀⠀⠀⠘⠀⠘⠃⣼⢁⡍⣼⠅⣡⡘⣥⠉⠘⠃⠀⠀⠀⠀⠀⠻⠀⠘⠃⣬⢡⢇⡾⠂⣵⠌⡤⠁⠘
⠀⠀⠀⠀⠺⢧⡇⢿⠁⠸⠿⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⣼⢸⡇⠀⣿⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⡿⢸⡇⠀⠿⠸⡇
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣾⣿⣿⠀⠀⠀⠀⣿⣿⣷⣤⣿⣿⣿⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡄⠀⠘⠛⠻⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡟⠛⠃⠀⢠⡇
⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠋⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⠉⠛⠿⠶⢶⣶⣶⡶⠶⠿⠛⠉⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣄⣀⣀⣤⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠐⠿⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢛⣉⣭⣭⣉⠙⡻⢿⣿⣿⣿⡿⢿⣿⣻⣿⣿⣟⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠙⢿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣷⣿⣿⣿⣿⣷⣾⣾⣶⣶⣶⣾⣶⣿⣿⣾⣷⣿⣷⣶⣶⣷⡆
⠀⢀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣮⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠈⠁⠈⠑⠛⠚⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠐⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠉⢠⣤⣤⣄⣠⣶⣛⣀⣤⣀⣴⣶⣦⣤⣴⣤⡤⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⠻⠿⠿⢿⢿⠿⠿⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠒⠺⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉