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⠀⠀⠀⠀⢀⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠒⠒⠒⢤⡀
⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠈⠆
⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠠⣇⡀⠀⠀⠀⠀⠀⠀⢀⣰
⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢀⠉⠉⠉⠉⠓⡀⠘⠉⠹⣧⡀
⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⣀⣴⢇⠀⠀⠀⠀⢀⡁⠀⠀⠀⡌⠻⣆⣀
⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⢸⡔⠁⠀⠑⠤⢤⡴⠂⠈⡒⠒⠊⠀⠀⠈⢦⡇
⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡠⠊⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⡜⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠔⠲⠖⠒⠒⠒⠚⠀⠀⠀⠘⠀⠀⠀⠡⢀⠀⠀⠈⠒⠤⠤⠤⠄⠚⠁⠀⠀⡠⠘⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⡰⠢⡄⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠀⠀⠤⠄⠂⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠴⢤⠀⢸⠄⠠⠤⠤⠤⠴⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠦⠔⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⢺⡀⠞⠀⠠⠤⢄⡀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠁⢀⠀⢀⠠⢄⡀⡱⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠀⠀⠀⠀⠀⢀⢃⠔⠁⠀⠀⠀⠉⠒⠲⣆⠠⠤⠤⠤⠤⠤⠤⠄⣰⠖⠒⠉⠀⠀⠀⠀⠢⡘⡄⠀⢏⠀⢠⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠰⣀⠌⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡉⠑⠒⠒⠠⠤⠤⠄⣀⣀⣀⣀⣀⡰⢆⣀⣀⣀⣀⡀⠠⠠⠤⠄⠒⠒⠈⢉⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⠉⠛⡇⠈⠀⠀⠁⠀⠁⠈⠁⢸⠋⠉⢈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠀⠈⠁⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁⠀⠑⠢⢄⡀
⢀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣯⣀⣄⣠⢦⣀⣠⣃⣀⣀⣀⣀⣀⣀⣀⡀⣘⣄⣀⡴⣄⣀⣀⣽⡀⣀⣀⡀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠋⠛⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⣰⡆⠀⢀⠀⠀⠀⢀⡇⢻⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⣤⣀⡀
⠀⠀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⣠⠞⠓⢦⣰⠃⠀⠀⠀⠀⠀⢀⣴⣿⢁⣠⣯⣤⣤⠴⠚⣁⡀⠀⠀⠀⠀⠀⠀⠀⡠⠖⣉⠤⠒⠒⠒⠒⠒⠬⣉⠲⣄
⢠⠎⠁⠀⠀⠀⠉⠛⠲⢤⣠⢤⡿⠀⠀⠀⠃⠀⠀⠀⠀⣀⡴⠞⠁⢉⣉⠀⠀⠀⢿⣠⡴⠶⢬⣄⠀⠀⠀⠀⢠⠎⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢎⠳⡄
⢸⡀⠀⠀⠀⠀⠀⠀⣀⣀⡇⠀⡇⠀⠀⠀⠀⠙⢻⠏⠉⠁⠀⠀⣈⣤⢤⣄⠀⢠⠟⠁⠀⠀⠀⠙⣧⠀⠀⢰⠃⡜⠀⢠⣴⡦⣶⣶⢦⣶⣿⣦⡶⣶⠆⢣⠹
⠈⢷⡀⠀⠀⢠⡞⠉⠉⠉⠻⢤⡇⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⡼⠁⠀⠀⠈⠻⡏⣤⠀⠀⠀⠀⠀⢻⠀⠀⡎⢰⠁⠰⠿⢶⡆⢸⡇⣿⠁⣸⣿⠟⠁⠀⠀⡇
⠀⠈⢷⡀⠀⠸⠻⣄⠀⠀⠀⠀⢻⠀⠀⠀⠉⢳⠀⠀⠀⠀⢸⡇⢠⡀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⣾⠀⠀⡇⢸⠀⠀⠟⠋⠀⠛⠁⠻⠟⠁⠛⠀⠀⠀⠀⡇
⠀⠀⠀⠹⡄⠀⠀⠈⢷⡄⠀⠀⠘⡆⠀⠀⠀⢸⠀⠀⠀⠀⢈⣧⣉⠁⠀⠀⢸⠷⣄⡀⠀⢀⣠⠞⢧⡀⠀⢱⡈⢆⠀⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⡇⡰⢠
⠀⣠⣤⡤⠗⠀⠀⠀⠀⢹⡄⠀⠀⢹⣄⡀⠀⣼⠀⠀⠀⠀⡀⠀⠉⠉⠓⢶⡋⠀⠈⠉⠉⠉⣁⠤⣄⢳⡄⠀⠱⡈⢦⡙⠛⠉⠉⠉⠉⠉⠉⠉⠉⢉⠔⣡⠏
⢸⡇⣠⠶⠶⠶⠶⠶⠖⠋⠀⣠⠞⠋⣉⠙⢷⡏⠀⠀⢠⠿⡁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠁⠀⣸⠂⣷⠀⠀⠘⠢⡉⠲⠤⣀⣀⣀⣀⣀⠤⢒⣩⠞⠁
⠈⢷⡏⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠛⠙⠳⠀⠁⠀⠀⠀⠀⠈⠓⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢀⡿⠀⠀⠀⠀⠈⠙⠒⠦⢤⣤⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⢖⣊⣙⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⢀⡤⣼⡷⠓⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣍⠙⢦⠉⠹⣷⣶⠶⠶⠤⠤⣤⠶⣶⣖⣚⣛⡭⠵⣶⡆⠀⠀⠀⠀⠀⣀⣀⣸⠀⡀⡷⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡏⠈⢷⡈⢧⠀⠹⡍⠓⢦⣀⠀⡿⢆⢹⡍⠁⠀⠀⠀⡟⠛⠛⠛⣉⣉⣉⣉⣉⣉⠀⢿⡵⣾⣃⡈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢻⡈⡇⠀⢹⡄⠀⠈⠻⡇⠸⡄⢣⠀⠀⠀⠀⣟⢹⠉⠉⠁⠀⠀⠀⠀⢹⣤⠼⡇⡇⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢸⡇⢸⠀⠀⢷⠀⠀⠀⢷⠀⢷⠀⢿⠒⠲⠤⣿⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣇⠈⡂⠀⠸⡆⠀⠀⢸⡄⠸⡆⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⢶⣶⣶⠚⠋⢀⣇⣀⣠⣧⠤⠤⠴⢷⠶⠓⠒⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⢻⡆⣿⠉⠉⠁⠀⠀⠀⠀⠀⢀⣀⣸⣀⣀⣀⣠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣷⣌⠙⠁⢸⣶⡶⢶⡖⠛⠛⠉⠉⠉⠉⠙⣧⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⡶⠷⠀⣀⣤⣶⣲⡞⠃⠀⢀⣀⣀⣀⣠⡼⠁⠳⠶⠚⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠋⢻⡀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⡄⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣁⣤⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡃⠀⠀⠀⠀⠀⠀⠨⠭⠟⣻⠋⢓⣂⣠⡼⠦⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡁⠀⠉⢙⣛⠷⢒⡶⠒⠒⠚⠋⠉⠛⠯⢍⣃⣠⠤⠤⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠭⠭⠭⠭⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠛⠛⠛⠛⠛⠛⠓⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⣀⠤⠶⠒⠉⠉⠉⠉⠉⠓⠶⢤⡀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠁⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠱⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠉⠈⢇⣀⣤⣞⣡⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⣠⠾⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣤⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⣸⡇⠀⢠⠀⢸⣧⣠⣿⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⢠⡟⣷⢀⡿⣇⣸⡏⠉⠘⢿⡄⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠾⢿⣤⣴⠶⠋⣠⣿⠟⠁⠹⠟⠀⠀⠀⠈⣷⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣠⣶⡿⠿⠟⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠈⣷⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣿⠀⠀⠀⠀⠉⠀⢀⣠⡤⢤⣤⣉⡻⣦⡀⠀⠀⠀⠀⢴⣤⣴⣾⢿⣋⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣆⡴⠋⠀⠀⠀⠀⠀⢠⡞⠋⠀⠀⠀⠀⠈⠙⢮⡻⠆⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠻⢶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⢹⡏⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣠⣶⣿⡆⠀⠈⢷⠀⠀⠀⠀⠀⣰⠁⠀⣰⣶⣿⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⣿⠻⡆
⠀⠀⠀⠀⠀⠀⢀⣀⣿⣀⠸⡇⠀⢀⡀⠀⠀⠀⢸⡀⠀⠀⠀⢿⣿⣿⣿⡿⠀⢸⠇⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⡿
⠀⠀⠀⣀⡀⣰⠏⠉⠉⠙⢾⣿⠟⠋⠙⢳⡀⠀⠈⠃⠀⠀⠀⠈⠛⠛⠛⠁⠀⡌⠀⠀⠀⠀⠀⠸⣄⠀⠈⠛⠛⠋⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⢈⣃⡼⠃
⣴⠟⠉⠉⠘⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣧⠀⠀⠀⠈⠉⠛⠒⠶⠶⠶⠶⠊⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⠒⠒⠊⠙⠉⠀⠀⠀⠀⠀⠀⠀⣾⠋
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡟⠋⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢀⣼⠋
⣿⠀⠀⠀⠀⣿⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⠇⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠛⠻⣯⠀⠀⠀⠀⣠⡾⠁
⣿⠀⠀⠀⠀⢹⡇⠀⠀⠀⠘⠗⠀⠀⠀⠀⢸⠂⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋
⣿⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟
⢻⡇⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠘⠷⠀⠀⢻⣤⡽⠿⠶⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⡞⠛⠛⠶⣤⡀
⢸⡅⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⣾⠁⠉⠉⠉⠻⣿⣟⣛⣉⣽⠟⠁⠀⢸⡇⠀⠀⠀⠈⠙⢧⡀
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⣉⣉⡉⠀⠀⠀⠀⠘⣿⢶⣄⠀⠀⠀⠀⢻⣆
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⠾⢻⡟⠀⠀⠀⠀⣰⣿⣿⣿⣽⣷⡄⠀⠀⠀⣿⠀⠙⣧⡀⠀⠀⠀⢿⡀
⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠁⠀⢸⡇⠀⠀⠀⠀⢻⡟⣿⠿⣿⣿⡇⠀⠀⠀⢿⠀⠀⢸⡇⠀⠀⠀⣸⠇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⠀⠀⣾⠁⠀⠀⠀⠀⠈⢠⣤⣤⣤⠌⠁⠀⠀⠀⢸⣄⡴⠟⠀⠀⠀⢠⡟
⠀⠙⠷⢦⣤⣤⣤⣤⣴⠴⠶⠿⠿⠟⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠶⠶⠶⠶⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⢘⣧⠀⣴⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⢿⣄⣨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⠀⠀⠀⠀⠀⣿⠛⠛⠛⠶⢶⡀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠉⠛⠛⠒⠂⠴⠿⣤⡀⠀⠀⣨⣷⣤⡴⠶⠒⠒⠛⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣿⠖⠲⣶⠶⠾⠏⠀⠀⠀⠘⠷⠶⢶⡶⠶⠲⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣤⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠖⠒⣚⣋⣉⣉⣉⣉⡙⠓⠒⠶⢤⣄⡀
⠀⣀⣤⡴⠶⡏⠁⢹⠶⢦⡀⠀⠀⠀⠀⣠⠔⠚⠉⠁⠀⠀⠀⠻⣭⠄⠐⠒⠒⠦⣽⠟⠀⠀⠀⠈⠙⠳⢦⣄
⢰⠏⢹⠀⠀⡇⠀⢸⠀⠘⡇⠀⠀⠀⠘⠷⢦⣄⠀⠀⠀⣀⡤⠼⣿⡀⠀⠀⠀⠀⣿⠤⠄⣀⠀⠀⠀⠀⠀⣈⣻⡆
⢸⠀⠘⠀⠀⠃⠀⠈⠀⠀⣇⡴⠛⣩⡷⠀⠀⢹⣦⢖⣉⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣉⡳⢦⣠⡾⠋⠁
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⣏⠀⠀⢰⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠿⣧
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢰⠋⠀⠀⣿⢠⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣭⣤⠾⠛⢡⡄⣿
⣿⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⠀⢸⡆⠀⠀⠙⢿⠀⠀⠛⢂⠈⠉⢩⡉⠉⠉⠉⢉⡍⠉⠀⠲⠏⠀⠈⣷⠟
⢿⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢸⠀⢀⣶⠵⠚⠛⠛⠻⣦⣤⡴⠟⠛⠓⠶⣵⣤⡀⠀⢻
⠈⠻⣦⣀⠀⠘⣷⣄⣀⣠⡤⠖⠋⠙⠓⠒⢶⡏⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⢸⠋⠉⠉⠓⠒⠶⠦⢤⣤⣀⡀
⠀⠀⠈⢿⡉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⢤⣀
⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣷⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣸⣦⡶⠃⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠶⠶⠶⠿⠿⠿⠶⠶⠒⠚⠉⠁⠀⢠⣿⣿⣿⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣦⠀⠀⠀⠀⢠⣴⣶⣦⡀⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠀⠀⠀⠀⠘⢿⣿⠿⠁⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⠋⠉⠛⠿⣿⠆⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡆⠀⠀⠀⠀⢠⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠉
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟⠀⠀⠀⠀⠀⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⠿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠈⠙⠓⢶⣶⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣶⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀

⠀⠀⠀⣠⢞⣿⣿⣿⣭⣭⣭⣝⢦⡀
⠀⣠⢞⣵⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢦⡀
⡞⣵⡿⠿⣿⠿⠿⢿⣿⠿⢿⣿⠿⠿⣷⢹⡆
⡇⣿⠠⢶⣿⣶⢰⣾⠃⣶⡄⣿⠰⡆⢸⣾⡇
⡇⣿⣷⡄⢹⣿⢸⣿⡀⢿⠇⣾⠀⣤⣾⣿⡇⠀⠀⠀⢀⣠⠤⠖⠒⠒⠋⠉⠀⠒⠒⠒⠤⢄⡀
⣧⢿⣦⣤⣾⣿⣼⣿⣷⣤⣶⣿⣶⣿⣿⢻⡇⠀⡠⠖⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠓⢤⡀
⠈⠳⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡵⠋⣠⠊⠀⠀⠀⠀⠀⠠⠟⠛⠀⠀⠀⠀⠈⠛⠳⠀⠀⠀⠀⠙⢦
⠀⠀⠈⠳⣝⡻⢿⣿⣿⣿⣟⣛⡵⠋⠀⢰⠃⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠔⠁⠀⠀⠑⡀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠈⠉⠉⢹⢹⠉⠉⠉⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢘⠀⠀⢠⡆⢸⠀⠀⣤⠀⠀⡇⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠘⢄⠀⠈⠁⣸⡀⠀⠛⠀⢠⠃⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢀⢸⠸⠤⡀⠀⠀⠀⠀⠘⣆⠀⢢⠀⠀⠀⠀⠀⠁⠀⠈⠀⠑⠄⠀⠔⠁⠀⠀⢀⠉⠀⢀⠎⣀⣀⡀
⠀⠀⠀⠀⠀⢀⠅⢈⠁⠀⢸⠀⠀⠀⠀⠀⠈⢧⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢠⠟⠉⢻⠁⠀⢹⣤⣀
⠀⠀⠀⠀⠀⠘⡆⠀⢻⢁⡸⠧⣤⣀⠀⠀⠀⠀⢳⡀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⣸⡄⠀⢸⠀⠀⡜⠀⠈
⠀⠀⠀⠀⠀⠀⠉⢸⢹⠁⠀⠀⠀⠉⠙⠛⠲⠶⠤⣷⠀⠀⠀⠀⠀⠀⠉⠰⢁⡉⠁⠀⠀⣶⠴⠾⠧⣇⠀⠘⠀⠀⠃⠀⡼
⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢀⠀⠧⢄⡠⠋⠘⡦⣀⠀⠹⢄⡀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠈⡇⠀⠀⠑⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⡀⠀⠧⣀⣀⣀⣤⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢄⣀⣀⣀⡠⠴⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒⠒⠉⠳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢩⠀⢀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⡀⡸⠀⠂⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠉⠁⠂⡀
⠀⠀⠀⣰⠋⢹⠀⢠⠞⠉⡆⠀⠀⠀⠀⠀⢸⣀⣀⠤⠤⠒⠒⢊⣀⡚⢥⣒⣄⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⣴⠢⡀⢹⠀⢸⠀⡿⠀⢰⠃⣠⠞⢳⡄⠀⠾⡿⠁⠊⢀⣠⠀⠀⢒⡂⠀⢸⡿⠃⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⢿⠀⢳⢸⠀⢸⣧⡃⠀⡏⣼⠃⢠⡞⠀⠀⠲⢿⡄⠀⢶⢈⠇⠀⠘⠁⠀⠠⣷⠶⣄⠀⠀⠀⠀⢸
⠸⣧⠀⠛⠉⠉⠉⠉⠉⠛⠃⡴⠋⠀⠀⠀⡴⣾⡇⠀⠀⠧⠤⠄⣂⠀⠀⠀⢻⡇⡁⠀⠀⠀⠀⠼
⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠈⢧⡘⣿⠀⠀⢀⣀⣤⣤⣶⡄⠀⢸⡏⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⣿⡀⠀⠀⢀⠴⠒⠚⠛⠒⠚⠛⢶⡄⠀⠀⠉⢹⠀⠀⢸⣿⣿⣿⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠀⠃⠀⠀⠀⢸⡀⠀⠈⠛⠛⣉⠥⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊
⠀⠀⢻⣇⡀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⣤⣄⣀⣤⣤⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡖⡤⠤⠀⣀⡠⠚⠁
⠀⠀⠀⠛⠿⠷⣤⠬⠘⢧⡀⠀⠀⠀⠀⣀⣀⢀⣤⠤⠶⣦⡿⠀⠀⢸⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠳⠒⠛⡛⠉⠉⢻⡇⠀⠀⠈⠳⡟⣱⠞⠁⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡀⠀⣇⠀⠀⢸⡇⠀⠀⠀⠠⢼⠇⠀⠀⠀⠀⢸⡏⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⢺⡇⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣸⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡿⢀⠠⢾⠀⠀⠀⠀⠀⣠⠴⠲⠤⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠹⡄⠀⠦⠤⣤⠔⠞⠁⢠⡛⠒⠊⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠇⠀⠀⠙⠦⢤⣤⣼⡀⠀⠠⢾⣉⣉⣉⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⠀⠀⢀⣼⠃⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠤⣼⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠙⠛⠉⠈⠓⠚⠛⠋⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⢀⡴⠚⠓⠶⣄⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠈⠳⡄⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢀⣀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠠⠞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⡤⠖⠲⢦⣯⣄⣀⣀⣀⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢀⣠⠤⠶⢾⠷⠒⠶⣄
⠀⠀⠀⠀⠀⠀⣜⠤⢄⣀⣀⣁⠀⢈⣉⠉⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⢂⣤⣀⠀⣀⣀⡠⠼⡂
⣤⣤⣤⣤⣤⣤⣬⣽⣶⣶⣶⣤⣤⣤⣬⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣬⣿⣶⣶⣾⣮⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣯⣧⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣷⣶⣾⣿⣿⣿⣿⣷⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⢀⢀⠀⠀⠀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠏⢨⠷⠹⠹⠸⠾⡥⠀⢍⡶⢹⠸⡹⢸⡹⠀⠶⢾⡅⢏⠗⠏⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠾⠿⠿⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠛⠛⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⢀⣼⡿⣷⡄⠀⠀⠀⢀⣴⠟⠁⠀⠀⡄⣶⣀⠀⠙⢷⣄⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⢀⣴⡿⠋⠀⠘⣿⠀⠀⠀⢸⡇⠀⠀⠀⡇⡇⣿⣿⠀⠀⠀⣿⠀⠀⠀⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣴⡿⠋⠀⠀⠀⢀⣿⠃⠀⠀⢸⡇⠀⠀⠀⣷⣿⣿⣿⣴⠆⠀⣿⠀⠀⠀⢿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣆⣴⡿⠋⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠘⣷⡀⠀⠀⢿⣿⣿⣿⠋⠀⣠⡿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⣄⣀⣀⣠⣴⡿⠋⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⣉⣉⡀⣠⡾⠋⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⢻⣿⣿⣿⣦⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⢀⡄⠀⢙⣷⣄
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣾⣿⣿⡿⠹⢸⢸⢸⣿⣿⣿⣧⠀⠀⠀⢰⡏⠀⠀⢰⣾⣿⣿⡇⠋⠀⠹⣆
⠀⠀⠀⠀⠀⢸⣿⣧⡒⢿⡆⣞⢰⡎⡇⣒⣽⣿⠀⠀⠀⢸⣿⣿⣿⣧⠘⠘⠘⠸⠟⣻⣿⣿⠀⠀⠀⢾⠀⠀⠀⢸⣿⣿⣯⣇⣤⠄⠀⣿
⠀⠀⠀⠀⠀⢸⣿⣷⣶⣿⣷⣿⣶⣾⣷⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⣰⣿⣿⡿⠀⠀⠀⢸⡆⠀⢀⣸⣿⣿⣿⣿⠃⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠘⢿⣿⣿⣦⣤⣤⣾⣿⣿⡿⠁⠀⠀⠀⠀⠹⣶⡛⠀⠙⠛⠛⠁⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠶⠶⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡶⠾⣿⠙⠻⠇⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠉⠀⠀⠀⠀⠀⠀⠈⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⣀⠤⠒⠊⠉⠁⠀⠀⠉⠉⠐⠢⢄⡀⠀⣰⠿⠐⠂⠈⠉⠉⠁⠐⠒⠬⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠗⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠛⢦⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⣠⡤⠞⠋⠙⠳⣄⡀⠀⠀⠀⠀⠀⢠⣤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⣀⣀⣀⡈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢀⠀⠀⠙⠳⡄⠀⠀⠀⠀⠀⢀⠖⠉⣁⡀⠈⠙⠲⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣴⠛⣿⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⣴⣿⣿⣄⠀⠀⢹⠀⠀⠀⠀⢠⡏⢀⣾⣿⣧⣀⠀⠀⢹⡄⠀⠀⠀⠀⠀⣿⢳⡄
⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠻⣿⣿⠿⠀⠀⢸⠃⠀⠀⠀⠸⡄⠈⠿⣿⡿⠏⠀⠀⢠⠇⠀⠀⠀⠀⠀⣿⠀⡷
⠀⠀⠀⠀⢀⡶⠛⠉⠳⣼⡶⠖⠒⢦⡀⠀⠀⠘⠒⠤⢤⣀⣀⡀⠀⣀⠚⠀⠀⠀⠀⠀⠳⣄⠀⣀⣀⣀⡀⠠⠎⠀⠀⠀⠀⠀⠀⣋⡼⠃
⣠⠞⠋⠉⢿⠁⠀⠀⠀⣿⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⡿⠀⠀⠀⢸⡀⠀⠀⠀⣿⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢻⠀⠀⠀⠀⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⠆⠀⠀⠀⣰⠏
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⣿⠁⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠞⠋⠉⠁⠙⠃⠀⢀⡼⠋
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡄⠀⠀⠀⢸⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁
⣿⠀⠀⠀⠘⠃⠀⠀⠀⠈⠀⠀⠀⠀⢸⡀⠀⢸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋
⣿⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠸⡇⠀⠸⡏⠙⢳⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠾⠿⣥⣀
⢸⠀⠀⠈⠉⠉⠉⠁⠀⠀⣼⠋⠁⠀⠀⠀⠀⠀⣿⠛⠉⠁⢀⡏⠉⠉⠛⠛⢛⡿⣟⠉⠉⠁⣷⠀⠀⠀⠀⠙⠳⣄
⢸⡄⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⡟⠢⡀⢀⠴⢿⣿⣯⠓⢄⡸⢹⠶⠶⣄⠀⠀⠀⠘⣧
⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠶⠛⠉⡇⠀⠘⠁⠀⢸⣿⣿⠀⠀⠀⠸⡇⠀⠈⢷⡀⠀⠀⠸⡇
⠸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢐⠋⠁⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠙⠂⠀⠀⠀⣇⠀⠀⣘⠃⠀⠀⢰⡇
⠀⢿⡀⠀⠀⠀⠀⠀⢀⣸⣇⣀⣀⣀⣠⣤⡴⠟⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⢸⡴⠞⠁⠀⠀⣠⡟
⠀⠀⠉⠙⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠘⡇⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠿⣿⣿⠿⠁⠀⠀⠀⣿⡀⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⡶⠶⠶⠶⠶⣄⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢠⡇⠀⠀⠀⠀⠹⣆⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⡄⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣫⣄⣀⣠⣼⣧⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣄⣌⣷⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⡠⢔⣢⣭⣷⣶⣶⣮⣬⣔⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⡠⣪⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣕⢄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⢌⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢢⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⡌⣾⣿⠿⢿⣿⠿⠿⠿⢿⣿⠿⠿⣿⣿⠿⠿⢿⣇⠆⠀⠀⠀⢸⣿⡿⠿⢿⡿⠿⠿⠿⣿⡿⠿⢿⣿⡿⠿⠿⣿⡇
⠀⠀⢠⣿⣇⠻⠷⣿⣿⡇⣿⣿⢱⣿⣷⡜⡇⠾⠿⢆⣿⢰⠀⠀⠀⢸⣿⡘⠿⢶⣿⣿⢸⣿⡏⣾⣿⣦⢹⡇⠿⠷⣸⡇
⠀⠀⠈⣿⣝⠿⠗⣸⣿⡇⣿⣿⣜⠿⠟⣱⡇⣺⣿⣿⣿⠘⠀⠀⠀⢸⣯⡻⠿⢂⣿⣿⢸⣿⣧⡻⠿⢋⣼⣇⣿⣿⣿⡇
⠀⠀⠡⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⠇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠡⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⠊⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠈⠪⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠊⠭⠟⠛⠛⠫⠩⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠛⠛⠛⠋⠉⠛⠛⠻⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠁⠀⢀⣀⣀⣤⣤⣤⣤⣄⣀⠀⠀⠈⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⣠⡴⠞⠋⠉⠁⠀⠀⠀⠀⠈⠉⠛⠳⢦⣄⠀⢄⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⡾⢁⣴⣶⣶⣶⣤⡀⠀⣤⡀⠀⢀⣀⠀⠀⣀⣤⣤⡌⢷⡀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣼⠃⣾⡏⠀⠀⠀⢈⣿⠀⣿⡇⠀⢸⣿⠀⣼⡟⠉⠉⠁⠈⣷⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⣿⣇⣠⣤⣴⡿⠃⠀⣿⡇⠀⠘⣿⠀⢿⣦⣄⡀⠀⠀⣸⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⣿⡋⠉⠉⠻⣷⠀⠀⣿⠁⠀⠀⣿⠀⠀⠈⠉⢿⡆⠀⣿⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⢻⡄⣿⣇⣀⣀⣠⡿⠀⠀⢻⣧⣀⣸⡿⠀⣤⣀⣠⣿⠃⢠⡏⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠻⣌⠙⠛⠛⠋⠀⠀⠀⠀⠉⠛⠋⠀⠀⠉⠉⠉⠁⢀⡾⠁⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠈⠙⠲⢦⣤⣀⣀⣀⣀⣀⣤⣤⠴⠚⠉⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⣐⠀⠀⠈⠉⠉⠉⠁⠀⠀⢀⣀⣤⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⠷⢶⣶⣶⠶⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣤⣤⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⣄⣤⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢀⣶⡟⠛⠛⠿⠀⠛⠛⠛⣿⡿⠛⠛⠃⢀⣴⠿⠛⠛⠻⣷⣄⠀⠀⣿⡿⠛⠛⢿⣶⡀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣿⠏⠀⠀⠀⠀⠈⣿⡇⠀⣿⡇⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠉⠛⠿⣶⣄⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣹⣿⠀⣿⣷⣤⣴⡾⠟⠁⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⣿⡇⠀⠀⠘⣿⣆⠀⠀⠀⠀⢀⣿⠇⠀⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠰⣶⣤⣤⣶⠿⠁⠀⠀⠀⣿⡇⠀⠀⠀⠈⠻⣷⣤⣤⣴⠿⠋⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣿⣿⣿⠿⠿⣿⣿⡿⠿⠿⠿⣿⣿⠿⠿⢿⣿⣿⡿⠿⠿⣿⣿⣿⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⠁⣴⣦⠈⣿⣷⡆⠀⣶⣿⠃⢰⣶⡀⢹⣿⡇⢰⣦⡀⢻⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣄⡈⠛⢿⣿⣿⡇⠀⣿⣿⠀⢸⣿⡇⢸⣿⡇⠸⠿⠃⣸⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⠛⣿⣷⠀⢻⣿⡇⠀⣿⣿⠀⢸⣿⠇⢸⣿⡇⢠⣶⣾⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣄⡉⢉⣠⣿⣿⣇⣀⣿⣿⣧⣈⢉⣀⣾⣿⣇⣸⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣜⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠊⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠑⢄⠑⢄
⠀⠀⠀⠀⠀⠀⡠⠊⡠⠊⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠑⢄⠑⢄
⠀⠀⠀⠀⡠⠊⡠⠊⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠑⢄⠑⢄
⠀⠀⡠⠊⡠⠊⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⡤⣄⡏⠀⢹⡞⠉⢣⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠑⢄⠑⢄
⡠⠊⡠⠊⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠑⢄⠑
⡇⢸⠀⢠⠎⠁⠀⠀⠀⠀⠀⠀⠀⡖⠒⣿⠀⢸⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄⠀⡇
⡇⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡿⠀⢸⡇⠀⢸⡇⠀⢸⠀⢠⠖⢲⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⡇⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣷⠀⢸⡇⠀⢸⡇⠀⢸⢀⡇⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⡇⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠋⠀⠈⠁⠀⠈⠁⠀⢸⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⡇⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⡇⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⡇⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⡇⠸⡀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢀⡇
⠙⢦⡙⢦⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⣀⡤⠊⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⢋⡴
⠀⠀⠙⢦⡙⢦⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⢋⡴⠋
⠀⠀⠀⠀⠙⢦⡙⢦⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠙⢦⡙⢦⡀⠉⠦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠔⠃⢀⡴⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣦⣈⣉⣉⣉⣉⣍⣉⣉⣉⣉⣉⣍⣉⣍⣉⣅⣴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⠞⠛⠛⠉⠉⠉⠉⠛⢦⡀
⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⣠⡤⠤⢤⣀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣤⣀⡀
⡞⠁⢀⣀⠀⣀⣀⡀⢀⣀⡀⠀⣀⣀⠀⢹⣀⠔⠋⠀⠀⠀⠀⠀⠙⢦⡀⢀⡴⠋⠁⠀⠀⠀⠀⠙⢦⡀
⡇⠀⣿⣉⠁⠉⡟⠁⣾⠉⢻⠀⣏⣹⡇⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠀⠀⠀⠀⠀⠀⠀⠹⡄
⣷⠀⣈⢻⡇⠀⡇⠀⣿⣀⣼⠀⡟⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⠀⠙⠛⠁⠀⠋⠀⠈⠙⠁⠀⠁⠀⠀⣸⣆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣰⠃
⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠈⠳⢄⣰⠀⠀⠀⠉⠛⠻⠀⠀⠻⠋⠁⠀⠀⠀⣤⡠⠞⠁
⠀⠀⠈⠳⣄⣀⣀⣀⣀⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⢹⡄⠀⠈⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠉⠉⠉⢩⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠒⠀⠠⠤⠀⡆⠴⠆⠀⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⡀⠀⠀⠁⢄⠀⠐⠁⠀⢠⠗⢦⡞⠛⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⡇⠀⢸⠉⠙
⠀⠀⠀⠀⠀⢀⣀⡰⠘⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠠⠄⠂⠀⠀⢀⣀⣀⣸⠀⠀⡇⠀⠘⠀⢠
⠀⠀⠀⠀⠀⢫⣀⡱⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠂⠁⠀⠀⠀⠀⠀⠀⢻⡀⠀⠙⣆⠀⠀⠂⠀⢠⠃
⠀⠀⠀⠀⠀⠈⣤⣉⢶⣁⡠⠌⢉⣒⠢⠄⠀⢀⣀⡀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠈⠙⠄⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠙⠢⢴⠁⠀⠀⠀⠀⠉⠁⠐⠂⠒⠒⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢀⡀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠦⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⣀⡴⠋⠀⠑⢦⣀⠀⠀⠀⠀⣀⡴⠋

⠀⠀⢀⡴⠋⠉⠉⠉⠉⠉⠛⢦⡀
⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⡏⠀⣤⣤⢠⣤⣤⢀⣤⣄⢠⣤⣄⠀⡇
⡇⠀⢷⣤⠀⢻⠀⣾⠀⢹⢸⣧⡿⠀⣷
⡇⠀⢦⡾⠀⠸⠀⠸⠦⠟⠸⠇⠀⠀⣿⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⢀⡴⡟⠈⠀⠀⠀⠂⠀⡀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢠⠄⠀⠀⠐⡀⢀⠁⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠙⠉⠙⠛⠋⠉⠉⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⣼⠉⠙⡏⠀⢸⡤⢄
⠀⠀⠀⠀⠀⢀⢠⡄⠢⠄⠀⠀⠀⠁⠀⠆⠀⡄⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢹⠀⠀⡇⠀⠨
⠀⠀⠀⠀⠀⡔⠀⢄⡀⢸⡀⠀⠀⠀⠀⠀⠀⠇⠠⠈⠁⠑⢄⠀⠀⠀⠀⠀⠀⠀⡠⠂⣴⠚⠓⠺⡀⠀⠁⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠨⣀⡹⡧⠢⡈⠢⢄⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠈⠑⠒⠒⠒⠀⠉⠀⠀⠈⡲⢄⠀⠓⢤⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠌⠀⠀⠀⠄⠀⢀⠀⠂⠰⡀⠀⢠⡀⠀⢀⠀⠰⠁⠉⠀⢼⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠐⠠⢀⠀⠀⠐⠚⠁⠀⠀⡀⢀⠄⠀⠓⠲⠖⠒⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⢄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠂⠀⠀⠃⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠖⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⠉⠉⣟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣿⠀⠀⣿⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠿⠀⠐⠿⣿⣿⣿⣿⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠿⠿⠿⠿⣿⣶⣄⣠⣤⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠁⠀⠀⠀⠉⠳⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⠙⠀⠀⠀⠀⠀⠘⢮⡻⣿⣿⡯⣄⠈⠻⣿⣦⣀⠀⠀⠀⠀⠀⢠⡶⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠳⣦
⠀⠀⠀⠀⢀⣤⣶⣿⣿⡇⠀⠀⠀⡀⠀⠀⠀⠀⠈⠻⢌⡙⠛⠶⣤⡀⠀⠉⠻⢷⡄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⢠⣿⠋⠇⠘⠸⣧⠀⠀⣄⠹⢦⡐⣄⠠⣄⠀⡀⠉⠀⣀⠀⠻⣦⣌⣲⣶⣿⡄⠀⠀⠙⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏
⠀⠀⠀⣿⠋⠀⠀⠀⢤⣈⣳⣦⣌⣛⣶⣿⣿⣿⣾⣷⣿⣶⣦⣤⣝⣶⣌⠻⣿⣿⡿⣷⣄⡀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⣿⣇⠀⠀⠀⠐⣲⣿⣿⡿⠋⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⢷⡌⠻⢧⡀⠙⢿⣦⣁⡤⠤⢆⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣠⡾⠃
⠀⠀⣼⣿⠿⣷⣦⣬⣭⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⠁⠀⠀⡹⣿⡄⠀⠀⠉⠓⠒⠒⠚⠁⠈⠙⠛⠛⠋⠉
⠀⠀⣿⡏⠓⠀⠉⠹⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣮⣧⠀⠀⣷⣿⡇
⠀⠀⣿⣧⡀⠀⢠⣼⣿⡏⣤⠶⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢻⣿⡆⢸⣿⣿⡀
⠀⠀⣿⡟⠿⠿⢿⣿⣿⠋⠀⠀⠀⠀⠈⠙⠻⠂⠀⠀⠀⠀⢠⡶⠾⠟⠛⠉⠙⠳⣼⣿⡇⣿⢏⣿⣿⣆
⠀⠀⠹⣿⣦⣀⣩⣿⡏⢰⣴⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⡀⠀⠀⠀⣿⡇⡏⡸⣻⣿⣿⡆
⠀⠀⠈⣧⡉⢹⡇⢸⡇⠀⠙⢯⣁⣴⢶⡀⠀⠀⠀⠀⠀⠀⠀⠐⣩⣟⡙⠻⣿⡿⠀⣿⠷⡄⢠⢟⡟⣿⡇
⠀⠀⠀⠻⣷⣶⣧⠈⡇⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠗⠚⠁⠀⠀⡿⢠⡷⠋⠊⣰⣿⡷
⠀⠀⠀⠀⠈⣿⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡞⢀⣠⡾⢛⣿⠃
⠀⠀⠀⠀⢀⣿⣯⣍⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⠃⢉⣠⣾⣿⡇
⠀⠀⠀⠀⣸⣿⡏⠇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣤⡿⠟⢋⣽⣿⣿⡄
⠀⠀⠀⠀⢻⣧⠀⢠⣘⣶⠄⠂⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠁⠀⠴⠋⠟⢹⣿⡇
⠀⠀⠀⠀⠀⢻⣦⣀⡤⠃⠀⡀⣨⠔⠀⠃⠒⠿⣿⣿⡷⠦⠠⠀⠀⠀⠀⠀⣼⣷⣦⣄⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⢸⣇⠈⡀⠀⠀⣥⠁⠀⠀⠀⠠⠀⠀⠀⠀⠔⠁⠀⠀⠀⢀⣾⣯⢻⣿⣿⣖⣢⣴⣿⠟
⠀⠀⠀⠀⠀⠘⢿⣾⠁⠀⠀⡏⠀⠀⡀⡀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣴⣿⡳⣝⠷⡌⠙⠛⣩⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢹⡆⠀⠀⠀⠀⢸⠓⢠⡀⠀⠀⠀⠀⢀⡤⠖⠉⠀⣿⣿⣿⡻⠶⠄⠀⠔⣫⡟
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠈⠀⠀⠉⠉⠐⠈⠉⠁⠀⠀⠀⠀⣟⠻⣿⣯⣭⣥⣶⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠤⠉⠉⣩⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⣀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣋⠁
⠀⠀⣀⠤⠔⠚⠉⢸⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠤⣀
⣾⣿⣦⣄⣀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣷⣶⣤⣄⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⣶⣦⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢡⣶⡎⢻⣷⡄
⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⣿⡏⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠈⣿⡇⠀⢻⣿⡄
⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⢹⡇⠀⠀⠘⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⣿⠁⠀⠀⠹⣿⣆
⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⣨⣅⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⢀⣭⡀⠀⠀⠀⠹⣿⣦
⠀⠀⢠⣿⣿⣀⣀⣀⣀⣀⣙⣋⣀⣀⣀⣀⣀⣿⣷⡀⠀⠀⠀⠀⣴⣿⣏⣀⣀⣀⣀⣈⣛⣁⣀⣀⣀⣀⣸⣿⣧⡀
⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⣶⢶⣶⡶⡶⣶⣶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠙⠯⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠈⣢⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠶⢾⣾⢀⠀⣀⣀⣼⠧⠤⠤⠤⣄⡀
⠀⣀⠤⣖⠒⠒⠾⣟⢢⠀⣻⣶⠟⠁⠉⠀⠀⢀⣀⣠⣴⣶⣷⣦⣀
⢰⡃⡞⠁⠀⠀⠱⣿⣷⡏⠏⠀⠀⢠⡄⠀⢄⡀⠀⠈⠉⠉⠉⠛⠻⢷⣦⡀
⠀⠓⢦⠀⠀⠀⣰⠟⠘⢧⣣⣴⣜⣮⣷⣴⣄⣉⠙⣙⣋⣀⡀⠀⠀⠀⠙⠳⡄
⠀⠀⠈⢷⢤⣼⠇⠘⢾⠞⠉⠀⠉⠉⠉⠉⠉⠉⠉⠙⠛⠿⣿⣗⢦⡀⠀⠀⢳
⠀⠀⠀⠀⣸⢃⠘⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⢻⣀⠀⢸⡇
⠀⠀⠀⠀⣿⣆⠛⡿⢀⠴⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣧⣸⡷⠀⣸⠃
⠀⠀⠀⠀⠙⣌⢻⡇⠀⠀⠀⡀⠈⠙⠀⠀⠀⠀⠚⠛⠋⠉⠈⠓⢾⣿⣧⡴⠃
⠀⠀⠀⠀⠀⢸⠙⡇⠙⢿⠛⣿⡗⠀⠀⠀⠀⠀⠀⢰⡶⢶⣶⡤⠘⣿⡟⠁
⠀⠀⠀⠀⠀⠀⡄⡇⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⠻⠒⠋⠀⢰⢙⠿⠄⠀⠀⠀⠀⠀⣠⣴⠶⠶⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠂⠀⠀⠀⠀⢀⣀⣾⠋⠁⠀⠀⠀⠀⠈⠑⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠈⠒⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡇⠀⢠⡶⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⣀⠀⢄⠀⢀⠀⠪⢄⡌⠀⠀⠀⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢨⣷⡀⠀⠀⠈⡘⠻⠿⠟⠂⠀⠉⢄⠀⠈⠆⠀⠀⠀⠈⠣⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣄⠀⠀⠈⠀⠀⠐⠁⠀⠀⢸⡄⠀⡀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⢠⠟⡟⣼⢿⡇⠑⠤⡀⠀⠀⠀⣘⠹⡆⠀⠀⠀⢹⠀⠀⠀⣠⣊⣀⡠⢤⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⣴⠟
⠀⠀⠀⠀⠀⣿⠀⢇⠹⡜⢦⡀⠀⠀⠉⠀⠉⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⠤⠖⠁⠙⠓⠒⠛⠛⠋⠁
⠀⠀⠀⠀⠀⣿⡄⢄⡀⠀⠙⠻⣄
⠀⠀⠀⠀⠀⠘⣿⡓⠉⠛⠶⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⣀⠠⠔⠂⠁⠈⢷⣄⠀⠀⢠⣹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠤⠄
⠉⡃⠂⠐⠀⠒⠤⣾⣿⠇⣠⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢺⠉⠀⠀⡘⠨⠁⡐⡀⠀⡀
⢰⠆⡀⢀⠁⠅⢢⡟⠁⣾⠟⠛⠲⠶⠶⠆⠀⡀⡘⠀⠀⠀⠀⠀⢅⡂⡐⠠⠐⠠⠁⢂⢐⠐⠠⠀⠢
⠨⠦⢐⠠⠈⡀⠀⠛⠦⠿⠐⠠⠡⠁⠌⠄⢀⠐⠀⠄⠀⠀⠀⠀⠈⠰⠠⠁⠄⡒⠨⠠⠈⠀⠄⠁⢀⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣶⣶⣤⣤⣴⣶⣶⣶⣶⣤⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣯⣭⣍⣉⣙⣿⣢⣤⣝⡳⢾⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣩⣭⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠋⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠛⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣶⣦⣄⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠇⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⣻⣷⠀⠀⠀⠀⢺⠏⠀⢺⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⠏⠀⠀⠀⠀⠀⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢰⣦⣾⣿⡿⠸⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣿⣿⡟⠃⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⠏⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⡴⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣯⣴⡿⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠀⠀⠀⢿⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠋⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢻⣿⣿⢿⣿⠀⠀⠀⠺⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠻⠄⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠘⢻⣇⠀⢿⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠠⣄⢠⣶⠿⠀⢲⢶⣾⣄⡀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠛⢻⡆⠈⠀⠀⠀⠀⠀⠣⠀⣶⡀⢰⣶⣶⣦⣤⣄
⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⣌⡀⢀⣀⣀⣤⣝⠋⠀⠀⠀⠀⢀⣴⡾⢯⠀⠀⠁⠀⠀⠀⠎⣿⠄⠀⠀⠀⠀⠀⠀⣾⣆⠁⣸⣿⣿⣿⣿⣿
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣭⣥⣿⣿⣷⣶⣶⣶⣾⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⣿⣏⣰⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣀⣤⢞⣵⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⣱⣾⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣥⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠛⠉⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⣾⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣷⣄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠈⠳⣄
⠀⠀⠀⣠⠞⠁⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠈⠳⣤
⠀⣠⠞⠁⠀⢠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠈⠳⣤
⡾⠁⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠈
⡷⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿
⣷⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⡇⠀⠀⣿⣿⡇⠀⠀⠀⣠⣴⣶⣶⣦⠀⢰⣶⣶⣶⣶⣶⣶⣶⡄⠀⢀⣤⣶⣶⣶⣤⣀⠀⠀⠀⣰⣶⣶⣶⣦⣄⠀⠀⠀⢸⣿⣿
⡇⠀⠀⣿⣿⡇⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣰⡿⠉⠀⠀⠀⠉⢿⣧⠀⠀⣸⡇⠀⠀⠈⢹⣷⠀⠀⢸⣿⣿
⡇⠀⠀⣿⣿⡇⠀⠀⠘⢿⣦⣄⡀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠘⣿⡆⠀⢿⡇⠀⠀⢀⣼⡿⠀⠀⢸⣿⣿
⡇⠀⠀⣿⣿⡇⠀⠀⠀⠀⠈⠙⠻⣷⡄⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⢿⡿⠿⠿⠟⠋⠀⠀⠀⢸⣿⣿
⣇⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⠀⢸⣿⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⢀⣼⡿⠀⠀⣻⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⡏⠀⠀⣿⣿⡇⠀⠀⠺⠷⢶⡶⠾⠛⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠙⠿⢶⣶⠶⠿⠋⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣇⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⢯⡀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⢀
⠀⠙⢦⡀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⢀⣴⠟
⠀⠀⠀⠛⢦⡀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠹⣦⡀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠛⢦⡀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀⠀⠙⢿⣿⣷⣤⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣴⣶⣤⣾⣿⡿⠋⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣦⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣻⣿⣟⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢠⠀⣀⠀⢰⠀⠀⠀⠀⠀⠀⢀⣶⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠔⠺⠷⡄⡼⠗⠒⠄⠀⠀⠀⠐⣾⣯⡟⢿⣧⠆⠀⠀⠀⠀⣀⣀⣀⣨⣿⣀⣀⣀⡀⠀⠀⠀⢰⣿⣥⣴⡆
⠀⠀⠀⠀⠀⠂⣿⡤⠃⠀⠀⠀⠀⠀⢲⣿⠋⠁⠘⢻⣄⡄⠀⠀⠀⠸⣿⣿⠏⠀⢻⣿⡿⠁⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠘⠁⠑⠀⠀⠀⠀⠀⠀⠈⠻⠄⠀⠀⡼⠟⠁⠀⠀⠀⠀⠈⠃⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⢰⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⢠⣦⣀⣴⣶⣦⣄⣴⡆⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⠀⠀⠀⠀⠀⠀⣀⣦⣤⡀
⠀⠀⢠⣤⠾⢤⣿⣤⠿⢤⡄⠀⠀⠀⠀⠀⣿⠟⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠾⣿⡿⠅
⠀⠀⠀⠁⠀⠈⠛⠋⠀⠈⠁⠀⠀⠀⠀⠀⠙⠒⠛⠁⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠒⠒⠀⠀⠀⠀⠀⠘⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣤⣿⠃⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣣⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⢻⠀⠀⠀⠀⣀⢸⣷⣼⡆
⠀⠀⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⢷⠆⠹⣧⠀⠀⠀⠀⠀⠀⠀⣾⢉⡟⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣆⡀
⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣟⣁⣀⣘⣀⣀⣙⣧⡀⠀⠀⠀⠀⠀⠁⠈⡷⠀⠀⠀⠀⠀⠀⣿⣿⢿⢻⣿⡿
⠀⠀⠀⠘⠻⠿⠟⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠀⠀⠛⠁

⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣰⠟⢷⡄⠀⠀⠀⠀⠀⠀⠀⡰⣂⣀⡒⠄⠀⠀⠀⠀⠀⠰⢰⠀⡴
⠀⠀⢀⣀⠀⢒⣿⠖⠀⢀⡀⠀⠀⠀⠀⣼⠏⢸⡌⢿⡄⠀⠀⠀⠀⠀⠀⣿⠛⡿⢻⡶⠀⠀⠀⠀⠀⠀⠂⡈⠠⠔⣁
⠀⠀⠈⠉⠓⢤⣛⡡⠖⠋⠉⠀⠀⢀⣾⠃⠀⢾⡇⠀⢻⣆⠀⠀⠀⠀⠀⠻⣿⣿⣿⠇⠀⠀⠀⠀⠀⣶⡦⠡⢀⡀⠄
⠀⠀⠀⠀⠲⠋⠉⠙⠶⠂⠀⠀⠀⠺⠷⠶⠶⠶⠶⠶⠶⠿⠂⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈

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

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

⠀⠀⠀⢀⡴⢛⣛⣛⣛⣛⣛⣛⣛⠳⣄
⠀⢀⡴⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡳⣆⡀
⣴⢏⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡻⣆
⣷⢸⡏⣉⣙⣏⡉⢉⣉⡟⢉⡉⢻⡏⢉⡉⢻⡇⣿
⣿⢸⣇⠙⠻⣿⣿⢸⣿⠁⣿⣿⠀⡇⠘⠃⣸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⣿⢸⡟⠿⠂⣿⣿⢸⣿⣇⠙⠟⢰⡇⢸⣿⣿⡇⣿⠀⠀⠀⢀⣠⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢄⡀
⣿⡸⣿⣶⣾⣿⣿⣾⣿⣿⣷⣾⣿⣿⣿⣿⡿⢣⡿⠀⣠⠞⠉⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠈⠳⣄
⠈⠙⢮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡴⠋⢀⠞⠁⠀⠀⠀⠀⠀⠀⠞⠛⠁⠀⠀⠀⠀⠀⠉⠙⠳⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠙⢦⡙⠿⠿⠿⠿⠿⠿⠿⢋⡴⠋⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⢄⠔⠁⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠙⠛⠛⢻⠛⡏⠉⠉⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⡄⢸⠀⠀⣠⡄⠀⠈⡄⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠘⠃⢸⠀⠀⠿⠆⠀⢰⠁⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠘⣇⠀⢠⠀⠀⠀⠀⠀⠈⠢⠀⠀⠄⠉⢦⡀⠀⠀⡠⠂⠀⠀⠀⠦⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⢠⠖⠸⠆⠉⠱⡄⠀⠀⠀⠀⠀⠘⣆⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡠⠁⣠⡤⢾⣴⠖⠲⢦
⠀⠀⠀⠀⠀⠀⢼⠓⠐⢄⣀⠂⣿⠀⠀⠀⠀⠀⠀⠈⢧⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠁⠀⡏⠀⠀⡏⠀⠀⢸⠗⠲
⠀⠀⠀⠀⠀⠀⠸⣇⠉⣹⣇⠤⠛⠳⢦⣤⣀⡀⠀⠀⠈⢳⡀⠀⠀⠉⠒⠤⠀⠀⣀⠀⠀⣀⣀⡠⠔⠚⢁⠀⠀⡼⡇⠀⠀⡇⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡇⠀⠀⠀⠀⠀⠉⠉⠛⠛⠶⠶⢷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠅⠀⠀⠀⣶⠖⠚⠳⣾⠀⢀⢃⣀⠀⠃⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⢀⠀⠧⢀⣀⠔⠁⠙⣤⣀⠀⠈⣦⣀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⣸⠃⠈⡧⠀⠀⠈⠃⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣧⠀⠀⠀⠀⣀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠃⠀⠈⠉⠓⠛⠉⠉⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠉⠙⠛⠛⠒⠒⠒⠒⠒⠒⠒⠚⠛⠙⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠲⣖⠒⠒⠒⠒⠒⠒⠒⠒⠒⣲⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣀⣀⡠⣤⡶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠺⣋⠫⠍⠙⠛⢧⡀⣤⠛⠋⠉⠭⢉⠟⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠤⠀⣀⠤⠚⠀⠙⠲⢤⣀⠰⠤⠤⠤⠄⠘⣧

⠀⠀⠀⣠⢞⣿⣿⣭⣭⣭⣭⣝⢦⡀
⠀⣠⢞⣵⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢦⡀
⡞⣵⡿⠿⣿⠿⠿⢿⣿⠿⢿⣿⠿⠿⣷⣹⡆
⡇⣿⠠⢶⣿⣶⢰⣾⠃⣶⡄⣿⠰⡆⢸⣿⡇
⡇⣿⣷⡄⢹⣿⢸⣿⡀⢿⠇⣿⠀⣤⣾⣿⡇⠀⠀⠀⢀⣠⠤⠖⠒⠒⠋⠉⠉⠒⠒⠒⠤⢄⡀
⣇⢿⣦⣤⣾⣿⣼⣿⣷⣤⣶⣿⣶⣿⣿⢫⡇⠀⡠⠖⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠓⢤⡀
⠈⠳⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡵⠋⣠⠊⠀⠀⠀⠀⠀⠠⠟⠛⠀⠀⠀⠀⠈⠛⠳⠀⠀⠀⠀⠙⢦
⠀⠀⠈⠳⣝⡿⢿⢿⣿⣿⣿⣛⡵⠋⠀⢰⠃⠀⠀⠀⠀⠀⢀⠔⠀⠀⠐⢄⠔⠁⠀⠀⠑⡀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠈⠉⠉⢹⢹⠉⠉⠉⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⡆⢸⠀⠀⣤⠀⠀⡇⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠘⢄⠀⠈⠁⣸⡀⠀⠛⠀⢠⠃⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢀⢸⠸⠤⡀⠀⠀⠀⠀⠘⣆⠀⢢⠀⠀⠀⠀⠀⠁⠀⠈⠀⠑⠤⠀⠔⠁⠀⠀⢀⠋⠀⢀⠎⣀⣀⡀
⠀⠀⠀⠀⠀⢀⠅⢈⠁⢀⢸⠀⠀⠀⠀⠀⠈⢧⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢠⠟⠉⢻⠁⠀⢹⣤⣀
⠀⠀⠀⠀⠀⠘⡆⠀⢻⢁⡸⠧⣤⣀⠀⠀⠀⠀⢳⡀⠀⠁⠂⠄⣀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⣸⡄⠀⢸⠀⠀⡜⠀⠈
⠀⠀⠀⠀⠀⠀⠉⢸⠉⠁⠀⠀⠀⠉⠙⠛⠲⠶⠤⣷⡀⠀⠀⠀⠀⠀⠉⠸⢉⡉⡅⠀⠀⣶⠴⠾⠧⣇⠀⠘⠀⠀⠃⠀⡼
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢀⠀⠧⢄⡠⠋⠘⣦⣀⠀⠹⢄⡀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠈⡇⠀⠀⠑⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⡀⠀⢧⣀⣀⣀⣤⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢄⣀⣀⣀⡠⠴⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠑⢄
⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠑⢄
⠀⠀⠀⠀⡠⠊⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠑⢄
⠀⠀⡠⠊⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠑⢄
⡠⠊⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠑
⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⡇⢸⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠉⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠋⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⡇
⡇⢸⣿⣿⣿⡟⠀⠀⣠⣶⣶⣦⠀⠀⢹⣷⣶⣶⣶⠀⠀⢰⣶⣶⣾⣿⡿⠁⠀⣠⣶⣶⣦⠀⠀⠹⣿⣿⣿⡇⠀⠀⣶⣶⣶⣦⡀⠀⠈⣿⣿⣿⡇
⡇⢸⣿⣿⣿⣇⠀⠀⢿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⠇⠀⢠⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣷⠀⠀⣿⣿⣿⡇
⡇⢸⣿⣿⣿⣿⣄⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⢹⣿⣿⡇⠀⠀⠿⠿⠿⠿⠃⠀⢀⣿⣿⣿⡇
⡇⢸⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⡇
⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢻⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⣸⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⡇⢸⣿⣿⣟⠉⠉⢿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⡆⠀⠈⣿⣿⣿⣿⡏⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⡇⢸⣿⣿⣿⡀⠀⠘⠿⢿⠿⠋⠀⠀⣼⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣷⡀⠀⠙⠿⡿⠟⠀⠀⣸⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⡇⢸⣿⣿⣿⣿⣤⡀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣯⠀⠀⣸⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡇⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠑⢄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡠
⠀⠀⠑⢄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡠⠊
⠀⠀⠀⠀⠑⢄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡠⠊
⠀⠀⠀⠀⠀⠀⠑⢄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡠⠊

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⢼⠀⢠⣴⡄⣤⣤⣤⢀⣴⣦⡀⣤⣤⣄⠀⣟
⠀⢸⠀⠹⣦⡄⠀⣿⠀⢸⠃⢸⡇⣿⣤⡿⠀⣿
⠀⢸⠀⢤⣬⡿⠀⣿⠀⠸⣧⣼⠇⣿⠉⠀⠀⣿
⠀⠺⣄⠀⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣠⠟
⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⠀⠀⠀⠄
⠀⠀⠀⠀⠈⠳⣤⣤⣤⣤⣤⣤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠛⠒⣊⠀⠒⢀⠤⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠒⢠⡾⠉⠐⠈⠀⠀⣀⣠⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣿⡒⠒⠒⠒⢚⡦⢄⡀⠀⠀⠀⠀⠀⡴⠒⢲⡞⠛⢦
⠀⠀⠀⠀⠀⠀⢀⣀⣰⠒⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠘⠋⠅⡀⠀⠀⢘⡛⠂⠈⠳⣄⠀⠀⠀⡇⠀⢸⠀⠀⡸⠛⢲
⠀⠀⠀⠀⠀⠀⢹⠤⣼⠀⢀⠠⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⡄⠀⠀⠀⠀⠲⠀⠀⠀⠐⠄⠀⢈⡷⠤⢤⡇⠀⠸⠀⢠⠃⠀⡼
⠀⠀⠀⠀⠀⠀⠈⢣⠬⠶⣇⡴⠠⠬⣉⣐⠂⠤⠤⣀⣀⣠⠇⠀⠀⠀⠀⠀⠀⠾⠂⢠⠲⠃⠀⠀⠄⠀⠘⢦⡀⠀⣇⡀⠀⠐⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠒⠁⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⣿⠀⠀⠹⠀⠈⠂⠤⠤⠐⠁⠑⠂⠒⠈⠀⡠⠁⠀⢹⠀⠀⠙⠂⠀⠀⠘⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠑⣄⡀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⢸⣄⠀⠀⢀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⡍⠑⢲⠀⣠⠐⠂⠁⠀⠀⠀⠀⠀⣸⠉⠉⠉⠉⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠘⠢⣠⠼⢠⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣄⣀⣀⢀⣀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡽⠉⢩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠒⠴⠞⠀⠣⠦⠒⠒⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⠒⠚⠀⠓⠒⠒⠒⠒⠒⠒⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠁⠀⠀⠊⠃⠈

⠀⠀⢀⡴⠋⠉⠉⠉⠉⠉⠙⢦⣄
⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⡏⠀⣠⣄⢠⣤⣤⡀⣠⣄⠀⣤⣤⡀⢸⡇
⡇⠀⣧⣄⠀⢸⡇⢸⡇⢸⡇⣿⣨⡇⢸⡇
⡇⠀⣤⣽⠂⢸⡇⠘⣧⣼⠇⣿⠉⠀⢸⡇
⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠠⡂⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠
⠀⠀⠀⠈⠳⠶⠶⡶⡶⠶⠶⠋⠀⠀⠀⠈⠁⠈⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣶⠀⠀⠀⠀⣶⣶⠁
⠀⠀⠀⠀⠀⠀⠀⡇⣇⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⡤⠈⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢸⡉⡙⠀⠙⡎⠀⠀⠀⠉⠂⠄⣀⡤⠆⠀⠀⠒⣇⠀⠐⠂⠀⡠⣀⠀⠊⠁⢀⡀⢀⣬⡆
⠀⠀⠀⠀⠀⢻⠀⠰⡆⣀⡧⣄⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠑⣄⠠⡏⠉⡏⠀⢸⠤⢄
⠀⠀⠀⠀⠀⠈⠓⡎⠛⠁⠀⠈⠙⠲⢤⠃⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠁⠈⢦⡇⠀⡇⠀⡌⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠢⡀⡎⠀⠀⠀⠢⡀⠠⠀⠀⠀⠀⢀⠴⠁⢠⠀⣞⠉⠙⢧⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⢼⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠈⠑⠄⠈⠑⠄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⢀⠊⠀⢸⠀⠀⠀⠀⠀⠀⢀⣄⠂⠄⠀⠀⡀⠀⠀⠀⠀⠀⢢⠀⠀⢀⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠘⡋⠁⠒⠒⠒⠒⠒⠀⠀⠒⠒⠀⠀⠐⠒⠒⠒⠈⠉⢹⠉⣉⠝⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠤⠤⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠫⢭⣉⡉⠉⠉⠁⠀⠈⠁⢈⣉⡥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡟⠉⢹⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⢿⠀⢸⡧⠤⡤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠀⠀⠉⠄⡀⠔⠃⠙⠤⣀⠠⠁⠀⠈⢦

⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠄⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⡄
⠀⠀⠀⠀⠀⠟⢿⠟⠇⠀⠀⠀⠀⠀⣤⠸⠻⡟⠓⢠⡄⠀⠀⠀⠀⠀⡤⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠓⢄⠈⢣⠀⠀⠂
⠀⠀⠀⠀⣀⣷⡿⣷⣇⡀⠀⠀⠀⠀⠉⠛⣿⣺⠟⠋⠁⠀⠀⠀⠀⠀⠀⠚⠿⠿⠛⠀⠀⠀⠀⢠⣷⣸⡟⣷⠀⠀⠀⠀⠀⠐⠀⡁⠀⠀⡀⠄
⠀⠀⠀⠀⠻⠏⠙⠉⠿⠁⠀⠀⠀⠀⢶⠚⠁⠀⠙⢶⠆⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣾⢿⣷⡷⠀⠀⠀⠀⠀⠀⢀⡁⠂⣐⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⢠⣶⡀⠀⣰⣄⠀⠀⠀⠀⠀⠀⣀⣶⣀
⠀⠀⠀⠀⠴⠿⠄⠼⠧⠄⠀⠀⠀⠀⠀⢙⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠰⣬⡤⠀⠀⠀⠀⠀⠀⠛⠛⠳⡶⠛⠛⠀⠀⠀⠀⠀⠰⣫⠤⢝⠆
⠀⠀⠀⠀⠀⠑⡾⡖⠁⠀⠀⠀⠀⠀⠀⠿⠋⠙⠿⠇⠀⠀⠀⠀⠰⣷⠆⠽⠁⣾⡆⠀⠀⠀⠀⠀⠀⣼⣷⡄⠀⠀⠀⠀⠀⠀⠀⢇⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢰⣶⣦⠀⠀⢀⠀⠀⠀⢀⡉⢦⣤⠊⡀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠰⣽⡷⡾⣿⡇⠀⠀⠀⠀⠀⢠⣴⡿⣷⡄
⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠠⠂⡙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⣇⠈⠁⠀⠀⠀⠀⠀⢾⠙⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢄⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⡀⠈⠉⠀⠀⢀⠀⠀⠀⠀⠈⠓⠒⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣃⣦⡠⠀⠀⠀⠀⠀⠀⢸⣷⡀⢠⡀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣓⣶⡏⠀⠀⠀⠀⠀⠀⢀⣠⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣦
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⠂⠬⠀⠀⠀⠀⠀⠀⠋⣼⠛⠙⠃⠀⠀⠀⠀⠀⠀⠈⣠⣿⠉⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⠉
⠀⠀⠀⣀⣘⣋⣔⣴⣶⣄⡀⠀⠀⢼⣿⣿⣿⣿⣦⣛⠀⠀⠀⠀⠀⠀⠉⠐⠚⠁⠀⠀⠀⠀⠀⠀⠐⠃⢹⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠸⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠙⠓⢶⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣠⠴⢺⠇⠀⠀⠀⠈⠃⢂⠙⡆⠀⠴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⡼⠁⠀⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡇⠀⡇⠀⠀⠀⢓⣨⡀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢡⣀⢳⣴⣿⣶⠋⣡⠥⠄⠀⠰⠃⠰⢋⣴⣥⠀⣠⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢠⡼⠧⠁⠀⠀⢻⣘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡍⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠔⠛⠓⢾⡆⠀⠀⠀⣾⠿⠛⠐⠢⠀⢐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢐⣪⣤⣄⠀⠀⠀⠀⢀⣤⢭⣅⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠠⠀⢀⡀⠈⢒⡆⠀⣘⠉⣀⡀⠈⢃⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠲⢦⣻⢀⡀⠈⠁⠀⢨⡿⠀⣿⡀⠉⠀⠀⡀⠀⡬⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⣤⣀⡟⠀⣵⣂⣀⠒⠉⠃⠀⠘⢿⠒⠒⠋⠉⠀⢿⠁⠤⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠠⢼⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡐⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢸⡄⠀⠀⠀⠀⡀⠰⠂⠀⠀⠀⣀⠀⠀⠀⠀⢰⠇⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣧⠀⠀⠀⡌⢀⣄⠓⠀⠀⣠⡈⠃⠀⠀⠀⢸⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈⠛⠀⠀⠚⠋⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡀⢠⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⠤⣤⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡜⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣧⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠸⠥⠀⠀⠀⢀⣀⠀⠀⠀⠈⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⣆⠀⠶⠶⠶⠶⠄⣰⠟⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⣀⣤⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡞⠀⠀⠀⠀⠸⠇⣤⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈⠳⢤⡀⠀⢀⡤⠚⠁⠀⠀⠁⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⢰⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⠀⠐⠄⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠘
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⣀⣀⣀⣀⣀⣀⣄⣃⣀⣄⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⠟⢿⣿⡿⠋
⣿⣿⡿⠏⠅⣬⡾⠋
⣿⣍⠁⣩⡾⠋
⢿⣿⠟⠋
⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢁⣤⣤⡈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣰⠏⠀⠀⠹⣆⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⣰⠏⠀⣀⣀⠀⠹⣆⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⣼⠃⠀⣸⠋⠙⣇⠀⠘⣧⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡾⠃⠀⠀⢿⠀⠀⡿⠀⠀⠘⢷⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⡾⠁⠀⠀⠀⢸⡆⢰⡇⠀⠀⠀⠈⢷⡈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⢠⡟⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⢻⡄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣠⠟⠀⠀⠀⠀⠀⠀⢸⡋⢙⡇⠀⠀⠀⠀⠀⠀⠻⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠰⡏⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢹⠆⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠙⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠋⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠴⠶⠶⠦⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⢀⣠⠴⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⣄⡀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡠⠞⠉⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣀⡀⠉⠳⢄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⡠⠊⢀⣴⣾⡿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⣷⣤⣀⠑⢄⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠈⣠⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣄⠁⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠈⠁⣀⣀⣠⣇⣀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⣠⣆⣀⡀⠈⠁⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣀⣴⠟⠋⠉⠉⠉⠙⠳⢶⡎⠀⠀⠀⠀⠀⠀⠀⠀⢑⡶⠞⠋⠉⠉⠉⠙⠻⣶⡆⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⣰⠟⠁⠀⠀⠀⢀⣀⠀⠀⠀⠹⡷⡄⠀⠀⠀⠀⢠⢶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣸⡿⠀⠀⠀⠀⣾⣿⡁⢀⡄⠀⠀⢻⠈⠀⠀⠀⠀⠃⡞⠀⠀⢠⣾⣿⠉⢀⠀⠀⠀⠀⢿⡇⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠃⢷⠀⠀⠀⠀⢿⣭⣿⡿⠃⠀⠀⡼⠀⠀⠀⠀⠀⠀⢧⠀⠀⠸⣯⣽⣿⡿⠀⠀⠀⠀⡸⠁⠀⠀⢿
⠀⠀⠀⠀⠀⢀⠖⠓⢺⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠈⠁⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠈⠉⠉⠀⠀⠀⠀⡰⠃⠀⠀⠀⢸⠚⠓⣄
⠀⠀⠀⠀⠀⡞⣤⣄⢸⡀⠀⠀⠀⠀⠀⠑⠦⣄⣀⣀⣀⣀⠴⠊⠀⢀⠴⠂⠀⠀⠠⢄⡀⠢⣑⠤⣀⣀⣀⣀⣀⠴⠊⢀⠀⠀⠀⠀⢸⢀⡤⢸⡄
⠀⠀⠀⠀⠀⣧⠀⣾⠛⡇⠀⠀⠀⠀⠀⠙⠒⠦⠤⠤⠀⠀⠀⠀⢰⡏⠀⡀⠀⠀⢀⠀⢹⠄⠈⠓⠶⢬⣭⣥⡤⠶⠚⠁⠀⠀⠀⠀⡿⢻⡆⢸⠇
⠀⠀⠀⠀⠀⠸⣆⠈⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠙⠲⠖⠋⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢀⡞
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡉⠙⣮⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢣⣿⣿⣿⣿⣿⣇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⢿⠟⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠖⠶⠶⠋⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀⠀⢠⡀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣑⢤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⠶⠶⠖⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠶⠶⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣠⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⡀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠘⢷⡄
⠀⠀⠀⢠⡾⠋⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠙⣦⡀
⠀⢀⣼⠋⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠈⠻⣄
⣴⡟⠁⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠙⢷
⣿⡇⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠈
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣠⣴⡿⠿⡦⠀⢰⣶⣶⣶⣶⣶⣦⠀⠀⣠⣶⡿⠿⣶⣄⡀⠀⠀⣾⡿⠿⠿⢷⣶⣄⠀⠀⢸⡇
⣿⠀⠀⠀⢸⡇⠀⠀⠰⣿⡁⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⣾⡟⠁⠀⠀⠈⢻⣷⡀⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⢸⡇
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠙⠻⢿⣶⣄⠀⠀⠀⠈⣿⡀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⣿⡇⠀⣿⣧⣤⣤⣴⡿⠟⠀⠀⢸⡇
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⣿⣇⠀⠀⠀⠀⣼⡿⠁⢠⣿⠋⠉⠉⠉⠀⠀⠀⠀⢸⠃⠀⠀⢸
⣿⠀⠀⠀⢸⣇⠀⠀⠘⢿⣶⣶⡾⠟⠁⠀⠀⠀⣿⠇⠀⠀⠀⠈⠻⢿⣶⣶⡿⠟⠁⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⢸
⠻⣧⡀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⢀⣼
⠀⠈⠻⣄⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠈⠻⣤⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠷⣄⠀⠀⠀⠀⠈⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠟⠋⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣤⣤⣤⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⡀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⣀⣀⣤⣀⣿⠇

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣼⣿⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠻⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣼⡏⠉⣿⡄⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⢰⣶⡆⠹⣧
⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣾⣿⡇⠀⣿⣿⣆⠹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⢸⣿⡇⠀⠙⣷⡀
⣿⣿⣿⣿⣿⣿⣿⡟⢠⣾⣿⣿⣇⣀⣿⣿⣿⣆⠘⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠈⣿⠁⠀⠀⠘⣷⡀
⣿⣿⣿⣿⣿⣿⠟⢠⣿⣿⣿⣿⡏⠉⣿⣿⣿⣿⣧⠘⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣻⠃⠀⠀⠀⢰⣿⡆⠀⠀⠀⠘⣿⣄
⣿⣿⣿⣿⣿⣏⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⢈⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠠⣿⣥⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣬⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢦⢈⣾⣶⡄⣀⣾⠀⢠⣧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣜⢄⡀⣸⣿⣿⣶⣾⣿⣿⣽⣯⣿⣷⣼⣿⣴⣶⣆⣠⣶⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠢⣌⣿⣶⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡗⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢮⣉⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠋⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣧⣿⡔
⠀⠀⠀⠀⠀⠀⠠⡀⠀⢒⣷⣿⣿⣿⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣷⣴⣴⣣⡆
⠀⠀⠀⠀⠀⠠⣶⣾⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠢⣤⣴⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣶⣦⣄⠀⠀⠀⠀⠹⣿⣿⣿⣷⣤⠦
⠀⠀⠀⠀⢤⣴⣾⣿⣿⡿⠁⠀⠀⢀⣠⣤⣴⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠉⠙⠻⣿⣿⣷⣄⠀⠀⠀⠹⣿⣿⣿⣧⡤
⠀⠀⠀⢈⣻⣿⣿⣿⡿⠁⠀⢀⣴⣿⣿⣿⡿⠛⠋⠉⠉⠓⢤⡀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⡀⠀⠀⢻⣿⣿⣿⣝⠃
⠀⠀⡀⢰⣿⣿⣿⣿⠁⠀⣠⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠁⠀⠀⠘⣿⣿⣿⣽⣆
⠀⠈⠴⣿⣿⣿⣿⠇⠀⣰⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏
⠀⠁⣛⣿⣿⣿⡟⠀⠀⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠿⠛⠛⠷⣦⣄⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣄⡀
⠀⠲⢿⣿⣿⣿⠃⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣤⣾⡿⠿⠿⠿⠿⣷⣄⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⣫⠆
⠀⠐⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠸⣿⣿⣿⠏
⠀⣩⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣿⡇⠀⠀⠀⣠⣦⡀⠀⠀⠀⠈⣿⣦⣦⠀⠀⠀⣿⣿⣿
⠀⠰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⢸⡇⠀⢻⣆⠀⠀⠀⠈⠛⠁⠀⠀⠀⢠⣿⣿⢿⣧⠀⠀⣿⣿⣿
⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⣀⣿⣛⣿⣿⡀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⢸⠃⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⠀⢻⣿⣿⡄
⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣩⠶⠛⠙⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠈⠻⣿⣷⣤⣄⣀⣤⣴⣿⡿⠁⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⣾⣿⣿⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠻⣿⣦⣀⡀⠀⠀⢀⣠⣴⡿⡏⠀⠀⠀⠀⠀⠈⢿⠛⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠟⠋⠀⣿⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⣿⣿⡿⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡞⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣿⠄⠀⠀⠀⠀⠀⠀⣤⠘⣿⣿⣷
⢸⣿⣿⡇⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡾⠏⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⡄
⢸⣿⣿⡇⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠉⣁⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇⢀⡀⡀⣄⢰⡻⢎⢧⢣⢺⣽⣿⣿⣧
⢸⣿⣿⣇⡓⠈⠀⠈⠀⠈⠀⠀⠠⠀⠐⠀⠂⠐⠆⠀⠀⠀⡀⠐⠘⢷⣤⣼⡛⠛⣀⣤⣀⢀⣺⣛⡟⠋⠙⢫⢿⡅⣮⢙⣴⠾⠛⢿⣟⣜⣿⣿⣿⡄
⢸⣿⣿⡏⠁⠀⠨⠀⠁⢈⠀⠉⠠⢀⣡⣬⣄⣀⠂⠈⠀⠀⢐⠀⠈⠀⠛⠻⠹⡏⠋⠛⢻⠟⠛⣋⡴⠷⠿⢬⣈⣙⣥⠟⠁⢀⠠⣀⢻⡇⢻⣿⣿⡇
⢸⣿⣿⡇⠀⠠⠐⢀⡁⠀⠀⠀⣰⠿⠛⠉⠙⠻⣷⣤⣀⣀⣀⣤⡴⠞⠒⠦⣆⣴⠀⠀⢀⣷⠞⣡⠔⠤⣀⠖⠙⣭⠋⠀⠱⡎⠀⠈⢸⣧⠈⣿⣿⣿
⠀⣿⣿⡇⠀⠀⠂⠀⢀⠀⠀⢰⠟⢢⠔⠚⢠⣠⠼⢿⣿⡿⠻⢅⡠⠚⠠⡀⠌⠛⠷⡟⠋⠁⢹⡁⣀⣀⣭⠠⠴⢾⠖⠒⢛⡏⠉⢙⣿⠧⠀⢹⣿⣿⡇
⠀⣿⣿⣿⠠⠤⠀⠄⠈⠀⠀⢾⡀⣺⠒⠀⠤⡧⠤⢤⡿⠤⠤⣼⠦⠔⠒⣿⠒⠒⠒⡟⠊⠉⣹⠇⣀⡠⠛⠦⣴⡛⠳⠔⠋⠘⠒⢋⡟⠀⠰⠘⣿⣿⣷
⠀⢹⣿⣿⡀⠀⠐⡀⠀⢀⠈⡸⣿⡉⠒⠤⠒⠳⢤⣾⣿⣦⣖⠉⠓⠤⠚⣡⡿⠛⠛⠶⣍⡉⠁⠀⠀⠀⠀⣰⠟⠙⢷⣄⠀⢀⣴⡟⠅⠀⠐⠀⣿⣿⣿
⠀⠈⣿⣿⡇⠀⠄⢠⠀⠤⠐⠀⠘⢿⣶⣤⣤⣶⣿⡿⠋⡉⠻⣦⣤⣠⣴⠟⠀⠁⡀⠠⠈⠻⣦⣄⣀⣤⡾⠋⠀⠄⠀⠙⠟⠛⢉⠀⠀⠀⢀⡀⣿⣿⣿
⠀⠀⠹⣿⣿⡀⠰⠂⠠⠗⠀⠁⠀⢀⠌⠉⠉⡉⠉⠠⠀⠀⠀⠀⠉⠉⠄⠀⠀⠀⢀⡠⠴⠦⣬⣉⠉⠅⢀⠀⠀⠀⠁⠀⠂⠐⠈⠒⢀⠐⠂⢠⣿⣿⣿
⠀⠀⠀⢻⣿⣷⡀⠊⠁⠀⠈⣀⠲⢀⠀⠀⠀⣁⠀⠐⠀⠀⠠⡀⠀⠀⠀⠀⠈⠐⠃⡀⠀⠄⠀⠹⢷⡄⠐⠀⠀⠀⠀⠀⠐⠁⠀⢀⠀⠁⣠⣿⣿⣿⠇
⠀⠀⠀⠀⠹⣿⣷⣌⡀⠠⠀⢀⠐⠀⠀⠠⠀⠊⠀⠀⠈⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠶⠄⠉⢸⣿⠀⠂⠀⠀⠈⠀⠈⠁⣀⣨⣴⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠻⣿⣧⣀⡀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⢠⠀⠄⠀⠀⠀⠀⢀⠀⠀⢃⡐⣒⣀⣨⣿⣠⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣤⣄⣈⣁⣀⣠⣀⣤⣤⣤⣶⣤⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⠿⠿⠛⠛⠿⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠻⠿⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦
⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⣀⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⡀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠞⠁⠀⣠⣶⠟⠛⠉⠉⠉⠉⠛⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠋⠉⠉⠉⠙⠻⢷⣦⡀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⣀⣀⣀⡀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⣀⣀⣀⠀⠀⠀⠀⠙⢿⣦⣤
⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⡴⠋⠀⠀⠀⠈⠓⣄⠀⠘⣷⡀⠀⠀⠀⠀⠀⣼⡏⠀⢀⠔⠉⠀⣀⡀⠉⠢⡀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡜⠀⢠⣾⣿⡏⠀⠀⠘⡄⠀⢻⡇⠀⠀⠀⠀⠀⣿⠀⠀⡎⠀⣰⣿⣿⠉⠀⠀⠸⡀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢃⠀⠹⣿⣿⣷⣶⠂⢀⠇⠀⣼⡇⠀⠀⠀⠀⠀⣿⡀⠀⡇⠀⢻⣿⣿⣷⡶⠀⢠⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠈⠢⡀⠉⠛⠋⠁⣠⠎⠀⣰⡿⠀⠀⠀⠀⠀⠀⠹⣧⠀⠘⢆⡀⠉⠉⠉⢀⡠⠋⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠈⠉⠐⠈⠉⠀⢀⣴⠟⠁⠀⠐⠂⠐⠂⠀⠀⠙⣷⣄⠀⠈⠉⠂⠈⠉⠀⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣦⣤⣤⣤⣤⣴⡾⠟⠁⠀⠀⣀⣀⠀⠀⣀⣀⠀⠀⠈⠙⠿⣶⣤⣤⣤⣤⣤⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢊⢔⠤⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠤⡲⠕⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣋⣀⣀⣈⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⣴⣧⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⣼⠿⠼⢧⡹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⡘⣴⣿⣿⣮⢣⡽⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢫⣿⡇⣿⣿⣿⣿⣻⢗⡜⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢡⣿⢻⡟⣿⣿⣿⣿⢹⣿⣿⡜⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡿⣵⡿⠫⠀⣿⢸⣿⣿⡿⣿⠇⠀⠐⣊⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⣡⠈⣁⣴⣿⢿⢸⣿⣿⡇⣿⣆⣀⣀⣵⣎⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⡠⣿⣿⣿⣿⣧⡜⢸⣿⣿⡇⣿⣿⣿⣿⣿⣣⣤⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢏⣾⡵⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇⣧⣿⣿⣿⣿⣿⣿⣵⡹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢋⢾⣿⡿⣿⣿⠟⢩⣿⡿⡘⢿⡿⣣⣿⣿⣷⣿⠏⢹⣿⣝⣧⡽⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢣⣾⣟⣹⣿⣿⡿⠀⠀⢽⡿⢣⣿⣶⡝⣹⡟⠘⣏⣽⣿⣿⣿⣌⣻⡜⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣡⣿⣿⣿⣿⣿⣿⣿⣤⣴⣾⢷⡸⢿⡿⠏⣾⡗⢺⣿⣿⣿⣿⣿⣿⣿⣾⡎⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣥⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣷⣤⣭⣯⣥⣬⣭⣭⣭⣭⣭⣭⣭⣥⣯⣬⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣤⠀⢠⠄⢠⣤⠀⠀⣤⠤⣤⡀⢠⣄⠀⣤⢠⡄⢠⣤⠀⢠⠀⣠⡤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣸⢻⡆⡿⠀⣟⢹⡆⠀⣷⢤⣼⠃⣾⠻⣆⣿⢸⡇⢸⡟⣧⣸⠀⣿⠀⣬⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠿⠃⠸⠋⠉⠻⠀⠇⠀⠻⠄⠿⠀⠹⠿⠸⠇⠸⠇⠈⠿⠀⠻⠦⠼⠇



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠀⠀⠉⠑⠚⠓⠚⠒⠒⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⡠⠖⠀⠀⠈⠉⠉⠉⠉⡉⢉⠉⠉⠉⠉⣍⠫⣍⠫⢝⠫⡛⢷⣄⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠋⠀⡠⢔⠂⠐⠑⠈⠂⠁⠈⠀⠀⠀⠠⣌⠢⣝⠢⣑⠢⡑⠬⡳⢌⡓⢄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠔⠁⠀⢀⠢⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⢝⠢⣉⠲⢝⠪⣑⢬⡲⢌⠲⢌⠲⢷⣌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢡⠔⠁⢀⠠⠌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢬⡑⠦⣙⠮⣑⠦⣑⠮⡑⢌⡂⢝⠢⣉⠢⣙⢢⡘⢆
⠀⠀⠀⠀⠀⠀⡴⢃⠔⠁⠀⢄⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣍⠲⢌⠳⢌⡑⢬⡓⢌⡳⢌⠢⣍⠢⡑⠦⡑⢬⡑⢽⣌⠳⡀
⠀⠀⠀⠀⡠⢊⡔⠁⢀⡀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡓⢬⡑⠦⣉⠢⢍⠢⢍⠲⢍⠲⢕⠮⡑⢬⡑⢬⡒⢌⠲⢌⠳⡘⢦
⠀⠀⣠⠊⡴⠉⠀⢠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠦⣙⠢⡉⠢⣌⠑⢤⡙⠢⡑⠦⣙⠢⣙⠪⡑⢌⡲⢌⠒⢍⠢⣍⠺⢆⠑⣄
⣠⠞⣡⠊⠠⣔⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⠮⣑⠮⡑⢮⡑⢄⡑⠤⡈⠢⣌⠲⢌⠑⢬⡑⢬⡓⢬⡢⢕⠢⣉⠢⡙⠢⣱⣌⠳⣄
⡇⢸⠁⢄⠊⠀⠀⢀⠔⠒⠒⠐⠢⡀⠀⠀⠀⠖⠒⠒⠚⠓⠈⠛⢪⡑⠬⣓⢬⣲⡽⠒⠉⠓⠲⢕⠦⣙⠢⣈⡖⠘⠓⠛⠒⠛⠚⠷⣬⡓⢬⡙⢷⢸⡆
⡇⢸⠀⢐⡀⠀⠀⡏⠀⢰⠒⡄⠀⢸⠀⠀⠀⠓⠒⡆⠀⢰⢶⡒⠾⡈⠒⢌⣓⡏⠀⠀⣀⣀⠀⠀⠱⣌⡑⢮⡇⠀⢀⣀⣤⣀⡀⠀⠘⣮⡑⢬⣻⠈⡇
⡇⢸⠀⣑⠀⠀⠀⣇⠀⠸⣄⡈⠀⠀⠀⠀⠀⢀⡠⡇⠀⢸⡓⢬⡑⢬⡑⢤⡽⠀⠀⡼⢍⡲⣧⠀⠀⢳⢍⡲⡇⠀⢸⡓⢬⡓⣼⠀⠀⣷⢌⠢⣼⠀⡇
⡇⢸⠀⢒⠄⠀⠀⠈⠓⢤⡀⠉⠒⡄⠀⠀⢀⠠⣝⡇⠀⢸⢍⡲⢌⡓⢌⣳⠁⠀⢰⡋⠢⣝⠺⡆⠀⢘⡓⢽⡇⠀⠈⠉⠉⠉⠁⠀⣠⣏⠲⣝⢾⠀⡇
⡇⢸⠀⡢⠀⠀⠀⠀⠀⠀⠈⣆⠀⢰⠀⠀⠬⡑⠦⡇⠀⢸⠲⢍⡢⣍⠲⢼⠀⠀⢸⢮⡓⢬⡙⡇⠀⢰⢍⡲⡇⠀⢰⡶⢶⡲⢖⠺⣙⠦⡑⢬⣻⠀⡇
⡇⢸⠀⠠⠀⠀⣤⠠⡄⠀⠀⡟⠀⢸⢀⠮⣑⠮⣑⡇⠀⢸⣍⠢⢍⠲⢌⣺⠀⠀⢸⡓⠬⡓⢼⠁⠀⣜⠲⣝⡇⠀⢸⢮⡓⢍⠢⣙⠪⡑⢬⡓⡍⠀⡧
⠀⢸⠀⠅⠀⠀⢣⡀⠹⣄⣠⠇⠀⢸⠬⣙⠪⣑⠦⡇⠀⢸⠢⢍⡢⢍⠢⢌⣇⠀⠘⢦⣑⣾⠏⠀⣰⠣⢝⠢⡇⠀⢸⡓⢬⡲⢕⠦⡑⠮⡓⢌⡃⠀⡇
⢠⢸⠀⠅⠀⠀⠀⠑⠤⣀⣀⣀⣠⠯⡑⠪⡑⠮⣑⡇⠀⢸⣉⠢⣍⠒⠬⡂⢍⣦⡀⠀⠀⠀⠀⣠⠯⡑⠦⣙⡇⠀⢸⡉⠲⣍⠢⣕⠪⣲⢌⡲⡇⠀⡇
⢸⢸⡄⠀⠀⠀⠀⠀⠀⠀⣌⠓⢬⡑⠬⡑⢌⡑⠬⡙⠫⣛⠬⡙⠢⡉⠢⣈⠑⠩⢫⣍⡓⠶⡛⠭⡓⠊⠓⠈⠃⠤⡼⠊⡑⢮⡑⢮⡑⢮⡑⢿⣷⢰⡇
⠸⣄⠑⢦⡀⠀⠀⠀⢠⡑⠢⢍⠢⢌⠒⢄⡙⠪⣐⠄⡑⠨⣑⠪⣑⠬⣑⠤⣉⠒⢄⡉⠪⢖⣌⡑⢌⡒⢀⡑⠊⣓⢀⡑⢮⣓⢭⡲⣍⠲⣌⣳⡝⣸⡇
⠀⠈⠳⣄⠈⠢⡀⢈⡒⢌⡑⠢⣍⠒⢌⡒⢌⡒⢤⡑⠬⡑⠤⡉⠢⢍⡊⢕⡪⢑⠦⣈⠒⠤⡉⠻⢦⣌⡢⢈⠓⢬⡑⢌⡢⢌⠳⢌⡻⢝⡤⠋⡰⠋
⠀⠀⠀⠈⠳⢤⡈⢦⣌⡐⠬⣑⠢⢉⠢⢌⠐⠬⡒⠌⡑⠮⣑⠮⣑⠢⢌⠐⠬⣑⠩⣢⢍⠒⢌⡑⠤⡉⠫⡢⢕⠢⣈⠒⢬⡲⢍⠢⡭⠊⣠⠟
⠀⠀⠀⠀⠀⠀⠙⢦⡘⢦⣕⠢⢍⠒⢍⡢⢌⠂⢌⡐⠬⡑⠠⢑⠢⢍⠲⢌⡂⠄⡑⠢⢑⠪⢖⡬⡓⠬⡑⠄⡑⠬⡢⢕⠢⣈⡳⠏⣀⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢮⣖⣌⡒⠬⡒⠍⡒⢌⡐⠌⡑⠦⣉⠢⢍⡒⠬⡑⠢⣑⠢⢉⠢⢌⡒⢕⡢⣑⠨⣒⠬⡑⣬⠞⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢮⡚⠳⣮⢕⠪⢐⠬⣑⠨⣑⠢⢝⠲⢌⡑⠨⢑⠤⣉⠒⠍⣂⢌⡒⠬⡓⠽⣮⣵⠞⢡⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⠷⡂⢍⡛⠶⣭⣒⠅⡒⢍⡐⠬⣑⠨⣑⠢⢍⠊⢝⡢⢄⡑⠪⢕⠮⣱⠎⢁⡶⠋
⠀⠀⠀⠀⢀⠄⡠⢄⠄⣠⠄⠠⠀⠄⠙⢶⣌⣁⣉⣃⣙⣛⢛⠚⠓⠚⠓⠒⠛⠂⠽⠶⠬⠶⠤⠽⠦⠽⠦⠝⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠙⠉⠙⠉⠛⠙⠛⠛⠛⠟⠻⠛⠞⠻⠞⠺⠶⠶⠶⠶⠶⡶⠾⠭⠶⢒⠤⢒⠤⡒⠤⡲⢐⢀⡢⢔⠤⡤⢄⢤⣄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢠⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣤⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⠀⢠⡞⠛⠃⠘⠛⣿⠛⠓⣠⠞⠛⠳⣦⠀⢸⡛⠛⢦⠀⢸⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⠀⠈⠳⢦⣀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢸⡇⢸⣅⣤⠞⠀⢸⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⠀⢀⣀⣀⡿⠀⠀⣿⠀⠀⠻⣄⣀⣀⡾⠁⢸⡅⠀⠀⠀⢸⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⡀⠈⠉⠉⠀⠀⠀⠈⠀⠀⠀⠈⠉⠁⠀⠀⠈⠀⠀⠀⠀⣸⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠙⢿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⠟⠁⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣶⣶⣾⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋


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

⠀⠀⠀⢠⡾⠛⠛⠛⠛⠛⠛⠛⠻⣦⡀
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆
⣿⠀⣾⠛⠇⠛⢿⡟⢃⡾⠛⣷⢸⡟⢻⡆⢸⡇
⣿⠀⠙⠿⣦⠀⢸⡇⢸⡇⠀⣿⠀⣷⠾⠃⠈⡇⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠈⢀⠄
⣿⠀⠶⡶⠟⠀⠸⠇⠀⠻⠶⠟⠀⠇⠀⠀⠀⡧⠀⠈⠁⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠈⠉⠀⠀⠂
⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈
⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠊⠁⠀⠀⠤⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠈⠻⠦⠤⠤⡶⠶⠶⠶⠶⠋⠌⠀⠀⠀⢀⠀⠀⠂⠈⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠆⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⢅⡤⢤⣤⠖⠒⢾
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠂⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡏⠀⠀⢻⠀⠀⢸⠗⠒
⠀⠀⠀⠀⠀⠀⢠⠤⣧⠘⠲⡄⠀⠀⠀⠀⠁⠈⠇⠀⠼⠀⠉⠄⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⢀⠀⢹⠀⠀⡰⠀⠀⠘
⠀⠀⠀⠀⠀⠀⡔⠀⠤⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠈⠀⠘⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢠⠖⠛⠓⢾⡄⠀⠀⠀⠀⠁⠀⡰
⠀⠀⠀⠀⠀⠀⢙⠀⠘⣿⣠⣎⠑⠦⡀⠀⠀⠀⠀⠀⡆⠈⢄⣀⠀⠀⠀⠈⠐⠢⠤⠤⠤⠄⠒⠉⠀⠀⠈⠣⣄⡀⠀⠧⢄⡀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠉⡏⢸⠁⠀⠑⠢⢄⠁⠂⠄⡀⠀⠅⠀⠀⠀⠢⢄⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢐⠄⠀⠊⠀⢹⠀⠀⠀⠁⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠈⠁⠂⠤⢁⠋⠀⠈⠓⠠⣀⠀⠙⠂⠀⠘⢒⠀⡤⠒⠐⠈⠀⢀⠤⠘⠹⣄⠀⠀⢀⣀⡀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢩⠐⠠⢀⠀⡈⠉⠁⢀⠠⠐⡬⠠⠄⠀⠀⠈⢹⠉⠉⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠈⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠒⠀⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⣿⠛⠛⢻⡟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣶⣦⣿⡀⠀⣸⣷⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠿⠟⠛⠁⠀⠙⠛⠿⠿⢿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⣿⠉⣿⣿⡝⠻⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⠈⢿⣿⡆⢘⢏⣠⣤⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⣿⡦⠜⠛⠁⠀⠈⠉⠉⠉⠙⠛⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣬⣽⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠒⢦⡀⠀⠀⠀⠴⠖⠒⢄⠀⠙⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⡖⠦⠈⠀⠀⠀⢐⣶⣦⠀⠀⢠⠟⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⡄⠀⠀⠀⠀⠀⠉⠈⠀⠀⢸⣬⣷⣭⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠐⠛⠂⠁⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⠤⠤⣾⡇⠀⢸⡿⠛⠛⢶⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣟⣡⣠⣿⣟⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠂⠀⠀⢿⡀⠈⣿⣿⡿⣿⡇⠀⠀⠀⠀⢠⣤⣾⣏⣉⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⠀⠠⠀⠈⠱⠀⣿⠀⠀⣸⡇⠀⠀⣠⠔⠾⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠰⡐⠀⠀⠀⠀⠀⠀⢆⢠⠻⣦⡴⠟⣀⡄⠀⢹⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢳⣀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⡅⠀⠈⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠓⢄⠀⠀⠀⠀⢹⠟⠉⢏⠀⠀⠀⠀⠀⣠⣦⣾⣿⣿⣿⣿⣿⣴⠏⠀⠀⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢸⠃⠀⠀⠀⡄⠀⠀⠈⢢⠀⠂⠈⠉⠉⢧⢈⣿⠃⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠹⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢀⡏⠀⠀⠀⣰⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⡾⠁⠀⠀⠀⢀⡤⠶⠚⠛⠛⠳⢦⡀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⢀⡞⠀⠀⠀⢀⡿⠀⠀⠀⠀⢀⠞⡀⠀⠀⠀⠀⠸⡄⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠏
⠀⠀⠀⠀⠀⠀⢠⡟⡆⢀⠞⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠎⠀⢱⡀⠀⠀⠀⠀⠱⡀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢠⡏⡴⠋
⠀⠀⠀⠀⠀⠀⡼⠀⢹⠏⠀⠀⠀⠀⠀⡟⢧⠀⠀⣠⠏⠀⠀⠀⢣⠀⠀⠀⠀⠀⠱⣀⠆⠀⠀⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠃⡰⠃⠀⠀⠀⠀⠀⢀⡇⠈⢆⣰⠃⣀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⠋⠁⠈⡇⠀⣿⠓⠶⠤⠤⠼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢁⠀⢻⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⠁⣴⡄⢹⣷⣤⠀⣤⣿⠏⣠⣤⠈⣿⣿⠀⣤⡈⢻⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣄⠙⠳⣾⣿⣿⠀⣿⣿⠀⣿⣿⠀⣿⣿⠀⠿⠇⢸⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡟⠛⣿⡆⢘⣿⣿⠀⣿⣿⠀⣿⣿⠀⣿⣿⠀⣶⣶⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣄⣉⣡⣼⣿⣿⣠⣿⣿⣧⣈⣁⣴⣿⣿⣠⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣽⢙⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠛⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣒⣦⣤⣀⣠⣤⡤⢤⡀⠀⠀⠀⢀⡤⠖⠚⠃⠀⠀⠀⠀⠀⠈⠉⠙⠓⢦⡀
⠀⠀⠀⠀⢀⣴⣪⢏⢔⡀⠲⣾⣿⣿⣟⣋⠉⢢⠉⢢⡀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇
⠀⠀⠀⣠⣿⠿⢗⠕⢡⠞⠋⠉⠉⠉⠛⠻⢷⡄⢣⡀⠙⣆⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⡴⠋⠀⠔⣡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣷⡄⡜⣦⣈⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⢸⡇⣧⣤⣾⢃⣀⣀⡀⠀⠀⠀⠀⢀⣀⡠⠤⡈⣿⣿⣿⡿⡆⠀⠀⠉⠒⠒⠒⠁⠉⠛⠛⠋⠁
⠀⠈⢧⡹⣿⣏⠈⠀⣈⣉⠁⠀⠀⠀⠉⣁⣀⣀⣀⢸⣿⡟⠁⡇
⠀⠀⠈⣿⣿⣿⠘⠿⣩⣯⠀⠀⠀⠀⠀⢹⡯⠝⠁⣿⣟⣜⡰⠁
⠀⠀⣰⡿⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇
⠀⢸⣿⡵⣣⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⢸⣯⡘⣿⣿⣧⠄⠀⠀⠀⠀⣀⣐⠀⠀⠀⠀⢰⣿⡟⣿⡇
⠀⠀⠛⢿⣮⣿⡇⠀⡐⠀⠀⠈⠛⠋⠁⠀⠀⣠⣿⡿⠁⡹⠁
⠀⠀⠀⠀⠙⣿⡇⠀⠃⢠⡄⠀⠀⠀⠀⣀⢴⣿⣿⠇⣼⠁
⠀⠀⠀⠀⠀⣻⡇⠀⠀⠈⢠⠗⠒⠒⠊⠁⠀⣿⣿⣿⣿⡄
⠀⠀⠀⣴⡿⣫⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣿⡛⡀
⠀⠀⢠⣿⠆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⡇
⠤⠶⢌⢿⣦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣛⢹⣿⡿⣉⠴⠦⠤⣄
⢢⠀⠀⠑⢍⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⠖⠁⠀⠀⡜⠁
⠸⡄⠀⠀⠰⠁⠀⠀⠀⠐⠒⠒⠒⠒⠒⠀⠀⠉⠉⠉⠀⠀⠀⠀⡸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⢿⣿⣿⠻⣿⣿⣿⣿⣆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⡏⠀⠘⢿⡟⢀⣿⣿⡟⢻⣿⣿⡿⠂
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠷⠷⠎⠆⠸⣿⡿⠁⣸⣿⣿⡇
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠯⠽⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢿⣿⣿⡇⠀⠀⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠋
⠁⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠋⠀⠠⠊⠉⠛⠲⠀⠀⠀⢀⠔⠋⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠿⡙⣧⠀⠀⣠⣾⣿⠂⠀⠀⠀⠀⠒⠟⠷⢿⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣵⣾⣥⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡄⠀⠈⣷
⠀⠠⠠⠠⠀⠀⠀⠀⠀⠀⢀⡽⠿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠓⠀⠀⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡶⢾⣿⣇⣀⡽⣻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠳⢿⣷⠀⠀⣿⣦⡤
⠀⢠⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠛⢛⣷⣀⡀⠀⠀⠀⠀⠀⣼⣷⣶⣿⣾⡏⠀⢰⡏⠀⠀⠀⡄⠀⢣⠁⠀⠀⠀⠀⠀⢀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣄⣀⠀⠀⠀⢻⡏⠈⠈⣿⠇⢠⠟⠁⢀⠀⠠⡀⠀⡈⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⠀⠘⠓⠀⡀⠘⣿⣀⣀⣿⢀⠁⡀⠀⠀⠈⠀⠈⠀⡼⢠
⠀⠀⠀⠀⠀⠀⢀⣤⠔⠋⠁⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠛⠒⡄⠉⠉⠁⢸⡄⠀⠀⠀⠀⠀⠀⣠⠖⡁
⠀⠀⠀⢀⣠⠶⠛⠁⠀⠀⠰⣯⠸⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢷⠤⣤⣴⠏⠀⠀⠀⠀⠀⣠⣾⡁
⠀⠀⢡⡾⠁⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⡿⢿⣿⠿⢻⡷⣤⣤⡀⠀⠀⢀⡜⠀⠀⠛⠀⠀⠀⠀⣠⠊⠀⠙⠻⣆⠀⠀⠀⠀⠀⠀⠁
⢁⣴⠞⠃⠀⠀⢀⣀⣀⣀⠀⠀⠀⠈⠁⠀⠀⢻⣆⣼⠁⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⡄⠀⠀⠀⠘⡄⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠁⠠
⢻⡅⠀⠀⣠⠞⠋⠉⠉⠉⠙⠓⠦⡄⠀⠀⠀⠀⢹⠃⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢱⡀⠀⠀⠀⠘⡇
⠀⢻⡄⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⡜⢳⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢳⡀⠀⠀⣼⡇
⠀⠀⠙⣆⢻⡆⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢀⠎⠀⠀⠀⠀⠀⢰⠁⠀⠱⡀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠱⡄⢠⠟⣧
⠀⠀⠀⠈⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⢱⢠⠎⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠹⣄⠀⠀⢸⠻⡄⠀⠀⠀⠀⠀⠹⣞⠀⢹⡄
⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⡸⠋⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠘⣧⢠⠏⠀⣧⠀⠀⠀⠀⠀⠀⠘⢦⠈⠇
⠀⠀⠀⠀⠀⠨⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⣀⠠⠤⠶⣿⠀⢸⡏⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⡿⠀⢸⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠈⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⢡⠀⠈⣌⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⢦⡄⠠⢜⣂⡩⠤⡄⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⣀⣠⠖⠋⢁⠀⠀⠄⠸⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡆⠈⡎⠃⠰⠆⠒⠂⠉⠓⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣆⠀⢳⠀⢠⡄⡜⠀⠀⠀⣀⣀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠘⠂⠈⠀⠀⡵⠃⢠⠀⠀⠠⡴⠣⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠔⠉⠀⢀⣌⡐⣬⢥⠄⠀⡇⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⣠⠂⠘⢆⣷⠃⠀⠀⢡⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⡒⠠⣴⣾⠁⠀⠀⠀⡏⠀⠀⠀⠘⡄⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣤⣨⢥⡾⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⣟⡶⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠋⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣠⠟⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⣉⣁⠈⠛⠛⠛⡏⣉⡉⠡⣼⡀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠈⠉⢙⡇⠀⠀⢀⡏⢘⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠘⠒⠒⠶⠶⠾⠖⠒⠋⠉⡇⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣰⣏⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠇⢻⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⡎⠀⠘⡆⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡼⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠃⠀⠀⠀⢃⠀⠀⢀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠚⠋⠋⠁⠀⠹⠀⠀⠀⠀⡏⠉⠉⠉⠙⠓⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠦⠴⠀⠀⠀⠀⠣⠤⠤⠤⠤⠤⠤⠎


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠟
⠀⠀⠀⠀⣠⣧⡀⢀⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣮⣶⣆
⠀⠀⠀⠘⠉⠈⢡⡌⠉⠉⠃⠀⠀⠀⠀⠀⠀⣶⣤⠀⠰⠓⠬⠤⠄⠀⠀⠀⠀⢀⣤⣸⣽⣿⣿⣼⣠⣄
⠀⠀⠀⠀⠀⠤⠾⠷⠤⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠈⠉⠉⠀⠻⠟


⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⢸⣶⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⢿⡟⢳⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢷⣿⣷⢟⡁
⠀⠀⠀⠀⢸⡿⠁⠈⢿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣳⡶⢶⣛⡇
⠀⠀⠀⠀⠈⠛⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠂⡀⡁⠀⠀⠀⠀⠀⠀⠀⢠⣾⣦⠀⢠⣾⣦
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠶⠘⡉⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⣿⡏⠛⠛
⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣷⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠲⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠈⠉⠉⠉⠉⠉⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢀⡤⣤⠶⢖⡒⠶⢛⠒⠞⣛⡒⠞⣛⠰⠟⡋⠠⢞⡋⠷⣄⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⡶⠯⠕⠂⠉⣄⠠⠉⠒⠈⠁⠂⠈⠁⠒⠨⠁⠀⠬⠀⠒⠨⠍⢓⣤⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠎⢀⣴⠋⠤⢐⡢⠌⣑⠢⠀⣀⠠⠔⠒⠤⣀⠢⠌⣑⠠⠉⠀⣈⠠⢄⣒⠨⠌⢓⣤⠈⠳⡄
⠀⠀⠀⠀⠀⢠⠔⢁⣰⠋⠐⠂⠤⠐⠢⡝⠂⠬⢐⠂⠌⠒⠈⠡⠔⠂⠁⠀⠒⠨⠅⠒⠉⠤⠐⠂⠍⠐⠚⠵⢦⡈⠢⡀
⠀⠀⠀⢀⠔⠁⣠⠛⠅⢀⡀⢤⡐⠦⢙⡀⠤⢐⠠⠄⠂⡀⠤⣐⠀⠄⠀⣀⠤⠄⣀⠠⠄⣒⠠⠄⢂⡀⠤⠒⣈⠽⣦⡈⠢⡀
⠀⢀⡴⠃⣠⡮⢑⠂⠍⠂⠨⠔⢂⠩⠄⠂⣩⠄⢒⠨⠑⢂⡁⠤⠐⠈⠅⠒⣉⠡⠔⢊⣁⠤⢒⠨⣅⡀⠬⠒⢊⡉⡤⠛⣦⠈⠲⣄
⣠⠏⠀⡴⠏⣓⡠⠄⣂⠠⠐⡂⠤⠐⢂⠡⠀⣀⠥⠔⣊⠠⠄⠒⣈⠤⠐⣋⠡⠄⢚⡉⠤⣒⠠⠐⣀⠤⠄⢂⡉⠤⣀⡨⠜⢻⣄⠈⢧
⡇⢰⡖⡈⠍⡐⣢⣥⠶⠎⢡⣖⢈⡅⠶⠞⠓⠲⠞⠓⠲⣖⠈⠥⢐⡶⠛⠒⠶⢮⡕⠊⠡⡴⠾⠃⠲⠶⠛⠴⢦⣉⠔⢒⡩⠕⣗⠀⢸
⡇⠸⡯⢀⡒⢤⡏⠀⣀⣀⠀⠈⡧⢒⣿⣠⣄⠀⢀⣀⣬⡭⠤⣺⠋⠀⢀⣀⠀⠀⢸⡧⠒⣇⠀⠀⢠⡤⢤⡀⠀⠙⣖⠡⢔⣂⡽⠀⢘
⡇⠀⣯⠐⢂⢹⡇⠀⣏⠉⠽⢛⠉⠀⢂⡭⢺⠀⢸⡏⠐⢈⢭⡏⠀⢰⡧⠊⣳⠀⠀⣿⠭⠯⠀⠀⢸⡂⠩⢽⡆⠀⣿⡬⠑⠂⣛
⡇⠀⣻⡀⠤⢚⣇⠀⠙⠱⠶⣥⠀⡂⠤⠖⣻⠀⢸⠗⣀⠄⢺⡇⠀⣼⢂⠠⢾⠀⠀⡧⢐⡗⠀⠀⠸⠧⠔⠛⠀⠀⣯⠤⠒⢃⣭
⡇⠀⣽⠒⢉⠤⠘⠓⠶⢦⡄⠀⡿⠀⠉⠥⢺⠀⢸⡅⢒⠩⢼⡇⠀⣿⠐⡈⣽⠀⠀⣿⠐⡇⠀⠀⣶⠶⠶⠒⠒⠍⠐⢀⠉⠄⣶
⡇⠀⣻⡈⢤⡐⠢⠉⣃⠠⡇⠀⣿⡀⠤⣂⢹⠀⢸⣇⠩⠔⣺⡇⠀⣿⡀⠤⣺⠀⠀⣿⣈⡇⠀⠀⣯⠄⢒⡩⠄⠒⣈⠠⠔⣚⣻
⠃⠀⣿⠈⠥⢐⣞⠙⠷⠼⠃⠀⣾⠂⠭⠖⣺⠀⢸⡗⠪⣍⣺⡇⠀⠻⣎⣡⡜⠀⢀⡯⢐⡇⠀⠀⣗⠈⠤⢒⠊⢍⣒⠊⢡⣖⣺
⠀⠀⣿⠄⣒⡩⠿⣤⣀⠀⠀⣠⡏⣤⡔⠦⢻⠀⢸⡧⠙⣈⡩⢿⣄⠀⠈⠁⠀⢀⡾⣓⠨⡇⠀⢠⡯⢄⡒⠬⢑⡂⠬⢙⡃⢬⠟⠀⣠
⢸⠀⠹⣭⠐⠊⠍⢒⠩⠍⠛⣋⠉⠤⠐⡊⢩⣙⡛⠭⠁⡒⠨⠥⠒⡛⠥⠖⠚⠭⠑⢂⡩⠽⠛⣋⠥⠒⠊⠩⢔⠂⢍⣰⠾⠃⢀⡴⠁
⠈⠳⣄⠈⠻⣕⣂⠩⠄⢒⡈⠥⢄⣒⠢⠍⣃⠩⠤⢒⣀⠭⠔⣒⠨⠅⢒⡠⠤⠒⣉⠡⠄⢒⣈⠥⢔⠺⢅⣐⠬⣙⡶⠋⠀⡠⠋
⠀⠀⠈⠱⡀⠈⠳⣆⡩⠐⠂⠉⠁⠂⢈⡡⠄⠒⠈⠄⠒⣈⡥⠒⠊⡉⠤⠐⠊⢉⡀⠢⠍⡓⠨⢁⡒⢈⡑⢀⣭⠋⠀⢀⠜
⠀⠀⠀⠀⠈⠢⡀⠈⠳⢎⡡⠀⢂⡀⠤⠐⢂⡠⠄⢂⡉⠤⠒⣀⠠⠄⠒⠨⢉⡂⠤⠐⠂⠍⣑⠢⠌⠃⣤⠏⠁⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠈⠱⢄⠈⠱⢏⡄⠐⠂⠍⠀⠢⠭⠐⠢⠬⠀⡂⠩⠀⠒⠨⠐⠂⢈⡐⠤⣁⡢⠬⣳⠞⠁⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠛⢧⠄⣀⠠⠁⣂⠀⠄⢒⡀⠠⣐⡢⢄⣐⠠⣍⠢⠌⣁⠠⣵⡾⠋⢀⡰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠙⠶⠾⠁⠰⠶⠉⠰⠾⠁⠴⠮⠱⠦⢭⠶⣬⠷⠾⠵⠋⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠶⠦⠦⠶⠤⠤⣤⣤⣄⡤⡤⢤⢤⠤⠤⠤⠤⠤⠤⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠑⠮⢈⡴⢃⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢈⠔⡁⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣡⠔⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡡⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡁⢘⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡩⢚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠠⢊⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⡤⠚⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢠⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⢸⡇⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⢸⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢸⡇⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡿⢾⣿⣿⣿⣿⡦⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢈⡁⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢠⡄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⢹⣷⡄
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⢸⡇⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢸⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢸⡇⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡇⢸⣿⣿⣿⣆
⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢈⡁⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣦
⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣴⠋⠉⠉⢀⣒⡒⠶⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⣉⡭⠳⣾⣿⣦⣿⣿⣿⡿⠿⠿⢷⣽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⣡⣤⣶⠟⡻⠟⠛⠛⠛⠛⠻⠻⠿⣿⣿⡿⢾⣽⣟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⠛⠁⠘⣵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⡂⠈⠛⠳⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠁⠼⣏⡏⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢣⢻⡀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣠⣎⡤⠴⠶⢶⣄⠀⠀⠀⠀⠀⡴⠞⠋⠉⠙⠿⣯⠀⠀⠀⠘⢸⡇⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⠘⢸⣿⣥⣴⣶⣷⣶⣮⠀⠀⠀⠀⢠⣶⣿⣿⡿⣿⣷⣿⣦⠀⠀⢸⣦⣧⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⢢⢼⡿⣿⣿⣵⣿⡿⠼⠇⠀⠀⠀⠞⠃⠺⠿⠷⠟⠋⢹⣿⡄⢀⡼⠹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠰⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⢧⠀⢀⠀⠹⢺⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠹⣦⠻⣧⠀⠀⠀⠀⠀⠀⠠⠶⠴⠶⠀⠀⠀⠀⠀⠀⠀⡟⠀⠈⢷⠀⠀⠀⠈⣆⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡏⢧⠘⣧⠀⠀⠀⠀⠀⢀⣠⢤⣤⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡀⠀⠀⠀⣿⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠌⣆⢸⡇⠀⠀⠀⢠⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣿⡀⠀⠀⣇⠀⠀⠰⠁⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡿⡄⣷⠀⠀⠀⠈⢻⣿⣿⣻⡿⠉⠀⠀⠀⠀⣸⠇⠸⠀⠀⠈⠀⢀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⠁⠀⠀⠐⠁⣿⡿⣧⡀⠀⠀⠀⠉⠈⠉⠁⠀⠀⠀⣠⣾⡃⠀⢸⢰⡄⠀⢠⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢹⣿⣦⣄⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣇⠀⣸⡟⠀⠀⡿⠀⡇⠀⠀⡀
⠀⠀⠀⠀⠀⡠⠂⠀⢀⡠⠀⠀⠀⠀⠀⠀⡾⡡⠀⢿⣿⣿⣿⣷⣶⣶⣿⡟⠏⠁⠀⢹⣿⣾⣿⠀⠀⢰⠃⠀⠁
⠀⠀⠠⠂⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⡴⠁⢡⠀⠈⣿⡇⠉⠛⠿⠿⠟⠀⠀⠀⠀⠈⣿⡏⣿⠀⠀⣾
⠀⡐⣠⠀⢈⠄⠁⠀⡀⠀⢀⠀⢀⣠⡾⠁⠀⢀⠳⣦⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠘⢧⠀⢹⡄⠀⠀⠀⠀⢀
⠀⠁⠃⢠⠋⠀⠀⠀⠀⠀⡠⠖⠍⢸⡇⠀⠀⠀⠀⢡⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠂⠙⠹⣧⡀⠀⠀⠀⠀⢠
⠰⠈⢠⠁⠀⠀⢀⡠⠂⠁⠀⠀⠀⢿⢷⡀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠿⡄⠀⠀⠀⠀⠀⠁⠠⣄⠀⠀⠈⠢
⠀⠀⡇⢀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⢸⡟⠉⠁⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠲⡄⠀⠀⠀⠀⠀⡜⠆
⠀⠀⢱⠀⣨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⠀⠘⠖
⠀⠀⠰⡄⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠
⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⣀⡼⠋⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣸
⠀⠀⠀⠀⠘⠆⠀⠀⠀⣀⣠⣤⣶⡛⠵⠋⠁⠀⠀⠀⢀⣁⡒⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠶⠂⡉⠥⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠈⠋
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⣉⡙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⣴⣿⣿⣿⣷⣄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⣾⣿⣿⣿⣿⣿⣿⣆⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢠⣿⣿⣿⣿⠉⠀⠈⢹⣿⣿⣿⣧⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣰⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣷⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣰⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡄⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣦
⠀⠀⠀⠀⠀⠀⠀⣠⠏⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⢧⡀
⠀⠀⠀⠀⠀⠀⣰⠋⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⢳⡀
⠀⠀⠀⠀⠀⣴⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢳⡄
⠀⠀⠀⠀⡼⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠹⡄
⠀⠀⢀⡾⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠹⣆
⠀⢀⡞⢠⣾⣿⠛⣿⡿⠛⣿⡿⠛⣿⡿⠛⣿⣿⣿⠛⠛⠛⠿⣿⠻⣿⣿⣿⠛⡟⢻⡟⢿⣿⣿⠟⣿⣿⠿⠛⠻⢿⣿⣦⠘⣦
⢠⠟⢠⣿⣿⣿⡄⢹⠇⣄⢸⡇⢸⣿⠁⣦⠘⣿⣿⠀⠿⠷⠀⣿⠀⣈⠻⣿⠀⡇⢸⡇⢀⠙⢿⡃⣿⠁⣴⡿⠷⢾⣿⣿⣧⠘⢧
⡟⢠⣿⣿⣿⣿⣷⠀⢰⣿⡀⢀⣿⠃⣠⣤⣄⠹⣿⠀⣶⡄⠺⣿⠀⣿⣷⣄⠀⡇⢸⡇⢸⣷⣄⠀⣿⣄⠙⠿⠆⢸⣿⣿⣿⣧⠘
⢷⡈⢿⣿⣿⣿⣿⣶⣾⣿⣷⣾⣿⣶⣿⣿⣿⣶⣿⣶⣿⣿⣶⣿⣶⣿⣿⣿⣷⣷⣾⣷⣾⣿⣿⣷⣿⣿⣷⣶⣶⣿⣿⣿⣿⠟⣰
⠈⠳⣤⣙⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⢛⣛⣡⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠰⢀⠀⠀⠐⠐⠀⠀⠂⠀⢀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⢈⠁⠆⠀⠀⠀⠀⠀⠈⠢⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⠈⠈⠀⠙⠀⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠁⠉⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠘⠀⢢⠐⠒⢀⢀⠐⠀⠀⠠⡑⡀⢀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢁⠀⠀⠸⣄⠀⠁⠀⠀⠀⠀⠀⢀⠀⠃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠠⠄⠒⠠⠀⠄⠂⠄⡀⠀⠀⠀⠸⠉⠂⠂⠀⠘⠂⠀⠈⠒⠂⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣴⠀⠀⠣⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠉⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠂⠀⠲⠒⠀⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠠⠄⠤⠓⣦⡀⢀⠴⣄⡀⠀⠄⠀⠀⠀⠄⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣀⣔⠃⡀⠠⠤⡀⠀⠉⠣⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢀⠔⠋⠁⠀⠀⠀⠀⠀⢈⠀⡀⠀⠠⠀⠀⠀⠑⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠲⠀⠡⡠⠐⠈⠢⡀⠀⠀⠀⢀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠤⡇⠀⠀⠀⠀⠈⠁⢘⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⢸
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⢀⡀⠠⠀⠀⠀⢈⢃⣦⠐⠌⠃⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀⠀⠀⡐⠀⢢⠀⡔⢦⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⢀⡴⠃⠀⠈⠁⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡡⠄⠂⠀⠀⠀⠀⣀⣀⣂⡄⠵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢀⡀⠀⠠⠌⠓⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠁⠀⠀⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⢐⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⣤⣤⣤⣀⣀⠚⠒⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠤⠤⠤⠤⠤⠤⠤⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⠀⠐⠒⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⡀⡀⠀⠀⣌⠚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠂⡈⠁⢀⡀⠀⠄⠀⠀⢠⠀⠀⡀⠀⠤⠀⠂⠤⡀⢠⠁⠀⠀⠀⢰⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠀⠀⢱⢠⠀⠀⡀⠰⠀⠀⠠⠄⠈⠂⠀⠐⠑⠀⠀⠀⠀⡀⠀⠀⡖⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢀⠀⠀⠸⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢈⣈⣀⡀⠈⠈⠉⠀⢀⡡⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠁⡒⢢⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠀⠀⠀⡀⠉⠑⠀⢸⡂⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠃⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢅⡀⠀⢈⠀⠀⣀⠀⠀⠠⡀⠉⢀⠀⠀⡀⣰⠀⠀⠀⢆⡀⠈⠀⠸⠂⠇
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃
⠀⠀⠀⠀⠀⡐⠂⠀⡄⠀⠀⡀⠀⠀⡆⠀⠀⠀⢀⠔⠊⠙⠂⠀⠀⠀⣆⣐⡒⣒⣒⣒⣒⣶⠄⠀⠀⠀⣄⡤⡀⡀⢀⠄⡄⣠⢀⢄⡀⡀⣀⣀⠀⢸⠰⡇
⠀⠀⠀⠀⠀⢠⠀⠡⡘⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠔⠁⠀⡠⠄⠂⢄⡀⠀⠀⠀⠀⡈⠀⠀⣠⠊⢀⠈⠉⠉⠉⠉⠁⠃⠒⠒⠒⠒⠛⠹⠷⠥⠞⣇
⠀⠀⠀⠀⠀⠀⢇⠀⠈⠀⠀⡀⡀⠀⠀⡀⠀⠁⠀⠀⠌⠀⠀⠜⣀⡀⠀⠀⠑⠀⠀⠀⠔⢀⡴⠊⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠀⣠⠆⣰⠏⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⣀⠠⠀⡠⠀⠈⢁⠄⠕⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠄⡀⠄⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⡀⠠⠀⣀⡐⠋⠀⠀⠀⠀⢀⠤⠂⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠐⠂⠉⠉⠁⡨⠐⡈⠐⡈⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢐⠁⢂⠀⢀⠠⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠂⠀⡂⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⢀⠠⠐⠡⢂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠀⠁⠘⠂⠠⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠈⠐⠀⠀⠀⠀⠁⢀⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢀⠀⢠⠈⡀⠈⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀⢀⠈⡀⢡⠀⠀⡀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⠔⠁⠀⠁⠀⢠⠀⢃⠀⠀⠉⡀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠋⠑⠛⠋⠙⠛⠭⠑⠈⢆⠀⠡⠀⠀⠋⠐⠀⠄⠐⢂⡀⠠⠄⠐⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡉⠀⠀⠀⠀⠈⢆⠀⠀⡼⠁⠀⡀⠨⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠞⠂⠡⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠁⠄⢂⠡⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⠋⠉⠉⠉⠉⠉⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⢒⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠳⣦⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠈⢧⡀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣠⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⢀⡴⠋⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠋⠙⣆⠀⠀⠀⠀⠀⠀⠀⠘⣏⢳⡄⠀⠈⣷
⠀⠀⠀⠀⠀⠀⢰⡟⠀⢠⡾⠁⣴⡇⠀⠀⠀⢀⡖⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠛⠛⠉⠁⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⢻⠀⢻⡆⠀⠸⣇
⠀⠀⠀⠀⠀⢀⣿⠁⢀⡞⠀⡜⣹⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⢸⡆⠀⢿⡀⠀⣿
⠀⠀⠀⠀⠀⣸⡇⠀⠸⠁⣼⠁⣿⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⠛⠓⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⢸⡇⠀⠸⡇⠀⢸⡆
⠀⠀⠀⠀⠀⣿⡇⠀⠀⢠⡇⠀⣿⠀⠀⢀⡟⠀⠀⠀⠀⢀⡤⠖⠚⠛⠲⢦⠀⠀⠀⢀⡤⠖⠛⠛⠲⡄⠀⠀⠀⠀⠸⡆⠀⠀⢸⠃⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⢹⡀⠀⢸⠃⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠈⢧⠀⣰⠋⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⣇⠀⠀⣼⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠃⠀⠈⣧⠀⣼⠀⠀⠀⠘⠁⠀⣠⣴⣶⠤⠀⠀⠈⠀⠃⠀⣴⣶⡶⠂⡀⠀⠃⠀⠀⠀⠀⣿⠀⢠⠇⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⢀⣸⣆⡛⠀⠀⠀⠀⠀⠀⣿⡟⠛⣶⡗⠀⠀⠀⠀⠸⣿⠋⠙⣾⡇⠀⠀⠀⠀⠀⠀⢸⡀⢾⡀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⢠⡟⢁⠈⡇⠀⠀⠀⠀⠀⠀⠙⠳⠴⠟⠁⣀⣠⡀⠀⡀⠙⠓⠚⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⢹⡆⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⢸⡇⠉⢻⡇⠀⠀⠀⠀⠑⠄⢀⣀⣀⡴⣾⠁⠀⠈⠁⠈⠂⠤⠤⠂⠁⠀⠀⠀⠀⠀⠀⢸⡟⠙⢰⠃⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠘⢧⣀⣀⡇⠀⠀⠀⠀⠀⡰⢉⡁⠀⠀⢿⠀⠀⠀⠀⠀⠀⠰⠁⠒⠀⠀⠀⠀⠀⠀⠀⢸⣃⡠⠃⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠉⠉⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢠⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣇⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣴⠆⠀⠀⣼⠃⠀⠀⢸⡟⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⠀⣸⠀⠀⣷⡀⠀⢸⣷⡀⠀⠀⣀
⠀⠀⠀⠀⢸⣿⣀⣠⡼⠃⣠⠀⠀⣼⠇⠀⢠⡇⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⢹⡇⠀⣿⡀⠀⠘⢿⣦⣀⣙⣿⣶⡶⠃
⠀⠀⠀⠀⠘⠿⣿⡧⣴⣾⠟⠀⣴⠏⠀⢀⣾⣡⠃⠈⠻⢦⡀⠀⠀⠀⠀⢀⠙⠖⠃⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠘⣿⣄⠹⣿⣄⡀⠀⠉⠛⢛⡻⠃
⠀⠀⠀⠀⢠⣤⣶⣾⣿⣋⣤⠾⣋⣀⣤⠾⠋⠁⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⠙⠂⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⠀⠈⠛⠷⠮⠿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠀⠀⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⠘⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠀⠀⠀⠀⠀⡔⠉⠉⢢⠀⠀⠀⠀⠀⢻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⣇⣀⣀⣸⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠒⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣄
⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆
⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦
⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣞⠛⠛⠛⠛⢿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠉⠉⠉⠳⣦⠀⠀⠀⠈⠻⣎⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠘⣷⡀⠙⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⢸⣧⣀⣀⣿⣆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡴⠶⠗⠉⠉⠉⠉⠀⠀⠀⠉⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢠⡾⢿⣿⣦⣀⠀⣠⣤⠞⠛⢉⣁⣀⣤⣤⣶⣶⣷⣿⠟⠻⡆⠀⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠻⠿⠛⢉⣁⣤⡶⠾⠟⠛⠋⠉⠀⠀⠀⠀⠈⣷⡀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⢀⣤⡾⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⡼⠋
⠀⠀⠀⠀⠀⠀⢀⣠⠤⣄⠀⠀⢀⣤⢴⣿⣤⣶⣤⣾⡟⠁⠀⠀⠀⠀⠀⢴⡦⠀⠀⠀⣤⡘⠿⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⣞⠁⠀⠈⣷⠀⡞⠀⠀⠙⣥⠀⢻⣍⣵⠶⢀⣀⣀⣀⣀⣀⣀⣠⡶⠶⠟⢻⣶⠶⢶⠶⢿⡇
⠀⠀⠀⠀⠀⠀⢩⡀⠀⠀⠈⣶⠃⠀⠀⠀⢻⡗⠀⠉⠀⠐⠋⣽⠏⠉⠉⠉⠉⠀⠀⠀⠀⠀⠃⠀⠀⠀⢰⡇
⠀⠀⢰⠟⠉⠛⣸⠇⠀⠀⠀⠛⣴⡆⠀⠀⢸⣿⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⢸⡆⠀⠀⢻⣴⡄⠀⠀⠀⣿⡇⠀⠀⠀⢏⣠⡆⢀⣠⣿⠟⠐⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⣠⡌⣱⠀⠀⠀⢿⣇⠀⠀⠀⢹⣧⠀⠀⠀⢸⡿⢛⠉⣾⠏⠀⠀⢸⠀⠀⠀⠠⢤⣤⣤⣤⣤⡤⣾⡿
⣿⠀⠈⢻⣧⠀⠀⠘⣿⡄⠀⠀⠈⠻⠀⠀⠀⠈⢣⡘⣧⣿⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣀⡀
⠁⡄⠀⠀⢿⣆⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣫⣭⠛⠀⠀⢀⣿⣦⣄⠀⠀⠀⠈⠉⣠⣿⡟⠉⠉⢻⣿⠷⣦⣄
⠀⢁⡄⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠁⠀⠀⢀⡘⠛⢄⡈⠛⠛⢳⡾⠛⠁⢸⡇⠀⠀⢸⣿⠀⠀⠙⢷⣄
⠀⠈⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠋⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⢦⣀⣎⣀⣀⢠⣼⠁⠀⠀⢸⣿⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠈⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⢠⠿⢧⣤⡾⠟⢸⣿⠟⢁⡿⠋⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⢻⣦⡀
⠀⠀⠀⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠻⣧⠀⢸⡿⠀⣿⠃⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠹⣷⣄
⠀⠀⢰⡟⠉⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠋⠀⠀⠀⠀⠙⢷⣄⢃⣾⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦
⠀⠀⢸⠇⠀⠀⠙⢳⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀
⠀⠀⢼⠀⠀⠀⠀⠈⢯⡙⠳⠶⢶⠶⠶⣶⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⢸⣇⠀⠀⠀⠀⠘⢷⣤⣄⣤⣤⠾⢋⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠻⣧⣀⠀⠀⠀⠀⠉⠉⠁⣀⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣤⡤
⠀⠀⠀⠀⠈⠛⠿⣶⣶⣶⡶⠶⠟⢉⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣸⡿⠁⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣦⣤⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⠟⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣤⣤⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣀⣀⣤⣤⣤⣴⡶⠶⠿⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠈⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟⠉⠉⠉⠉⠉⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠶⠦⣤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⣀⣀⣀⣠⠶⠶⠖⠛⣿⣷⣄⡀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⣿⡟⠉⠉⠉⠉⠁⠀⠀⠀⢰⣿⠉⠉⠻⣶⣄⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⣰⠟⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⢀⣠⣴⢾⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠛⠿⠿⠷⣶⣦⣤⣤⣤⣤⣤⣤⣴⣿⣷⣴⡾⠿⠟⠛⠉⠁⠀⣿⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣦⣤⣄⣀⠀⠀⠀⠀⠀⢀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣠⣤⣤⣤⣤⣤⣤⣴⣶⣿⠃
⠀⠀⣀⣀⣴⣶⣿⣿⣦⣜⠛⠛⠛⠛⠛⠛⠛⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠙⠛⠛⠛⠉⠉⠉⣤⣴⣿⣿⣶⣤⣄⡀
⢠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣷⡄
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠙⣷⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⡶⢶⣶⣶⣶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⣶⠶⢶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣿⠇
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⡟⠛⠈⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⣠⣀⣀⣠⣤⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠛⠛⠋⠙⠛⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠒⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⡲⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠻⣦⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⠙⢶⣄
⠀⠀⠀⠀⣠⡾⠁⠀⣠⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⠀⠀⠀⠀⠀⠀⠀⢳⣀⠀⠀⠙⣦
⠀⠀⠀⣼⠋⠀⣠⠟⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠾⠛⠉⢳⡀⠀⠀⠀⠀⠀⠀⠈⣏⢳⡄⠀⠘⣧
⠀⠀⣼⠃⠀⣼⠃⣠⡿⠀⠀⠀⢀⡜⠛⠛⠛⠛⠛⠿⠟⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⢻⠀⢹⡄⠀⢹⡆
⠀⢸⡏⠀⢸⠃⣰⢡⡇⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠛⢦⡄⠀⠀⠀⢸⠀⠀⢿⠀⠈⣧
⠀⣿⠃⠀⠇⢠⠇⢸⡇⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢉⣉⡁⠀⠀⠀⠀⠀⢹⡀⠀⠀⢸⠀⠀⢸⡇⠀⣿
⠀⣿⠀⠀⠀⢸⠀⢸⡇⠀⠀⡾⠀⠀⠀⠀⣠⠖⠋⠉⠉⠙⢦⡀⠀⢀⡴⠊⠉⠉⠉⠳⡀⠀⠀⠀⠈⡇⠀⠀⢸⠀⠀⠈⡇⠀⢹
⠀⣿⠀⠀⠀⠘⠀⠀⣧⠀⢠⡇⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠇⢀⠎⠀⢀⣀⡀⠀⠀⢡⠀⠀⠀⠀⣧⠀⠀⡟⠀⠀⢠⡇⠀⢸
⠀⣿⡄⠀⠀⠀⠀⠀⠘⡆⢸⠃⠀⠀⠀⠀⠀⣴⣿⣿⢃⡄⠀⠀⠀⠀⢰⣿⠿⣍⣤⠀⠀⠀⠀⠀⠀⢹⠀⣸⠃⠀⠀⢨⠁⠀⣼
⠀⢹⣷⠀⠀⠀⠀⢠⡴⠛⣾⠀⠀⠀⢀⠀⠀⢿⣇⢀⣿⠇⠀⠀⠀⠀⠸⣧⣀⣼⠏⠀⠀⠀⠀⠀⠀⢸⠒⠛⢦⠀⠀⠀⠀⠀⡿
⠀⠀⢿⣇⠀⠀⠀⡟⠠⢤⣼⠀⠀⠀⠀⠂⠀⠀⠉⠉⢁⣠⠖⠒⠤⠁⢄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣤⢤⢸⠇⠀⠀⠀⢰⡇
⠀⠀⠈⢿⡆⠀⠀⢷⠀⠀⡟⠀⠀⠀⠀⠀⠈⠴⠒⠚⠙⡏⠀⠀⠀⠀⠀⠈⡩⢁⡠⠀⠀⠀⠀⠀⠀⢸⡇⢀⡜⠀⠀⠀⠀⣾
⠀⠀⠀⠘⣿⡄⠀⠈⠓⠶⣧⠀⠀⠀⠀⠀⠊⠐⠂⠀⠀⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠋⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠸⣷⠀⠀⢀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣧⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⡄⠀⠀⣿
⠀⣠⠄⠀⠀⣿⠀⠀⠀⣿⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⢰⠀⠀⣇⠀⠀⢿⣆
⢸⣇⠀⢀⡼⠃⡀⠀⢠⣿⠀⠀⣿⢳⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⡇⠀⣿⠀⠀⠹⣧⣀⠘⠿⣶⣤⡴
⠸⣿⣿⣭⣤⣾⠇⢀⣾⠃⠀⣰⢇⡞⠈⠳⣄⠀⠀⠀⠀⠀⠀⠧⠴⠃⠀⠀⠀⠀⠀⣠⡴⠋⠀⠘⣷⡄⢹⣷⡀⠀⠈⠛⠿⠿⡿⠋
⢠⣤⣴⣾⣟⣁⣠⠞⠁⣀⣴⠿⠋⠀⠀⠀⠈⠓⢦⣀⡀⠀⠈⠒⠂⠀⠀⠀⣀⣤⠞⠉⠀⠀⠀⠀⠈⠻⢶⣽⡿⠷⣶⡶⠶⠋
⣀⣉⣉⣀⣈⣉⣀⣉⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣳⣶⣦⣴⣶⣾⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡼⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠻⣆
⠀⠀⠀⢀⡾⠁⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠹⣦
⠀⠀⢠⡟⢁⡀⠀⣀⠹⣆⠀⠀⠀⠀⠀⠀⣰⠏⢀⣿⢃⠈⢿⡄⠀⠀⠀⠀⠀⢀⡾⠃⡄⣀⢠⠘⣧
⠀⣠⡟⠀⠛⠳⡾⠛⠃⠹⣧⠀⠀⠀⠀⣼⠋⠀⠸⢛⡏⠀⠀⢻⡄⠀⠀⠀⢠⡾⠁⠚⢟⣐⡛⠓⠘⣷⡀
⣰⠏⠀⠀⠀⠺⠿⠀⠀⠀⠘⣧⠀⢀⣾⠃⠀⠀⠀⠹⠃⠀⠀⠀⢻⣆⠀⢠⡟⠁⠀⠀⠐⠛⠂⠀⠀⠈⢷
⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣠⡟⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡉⢷⡀
⠀⠀⠀⣰⠏⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⢀⣾⠃⣴⡄⠹⣆⠀⠀⠀⠀⠀⠀⠀⢠⡟⢰⡇⣎⢷⡀
⠀⠀⣼⠏⢠⣀⣀⡀⠘⢷⡀⠀⠀⠀⠀⢀⡾⠁⠀⢿⡇⠀⠹⣦⠀⠀⠀⠀⠀⣰⡟⢸⣿⣱⡹⠊⢻⡄
⢀⣼⠃⠀⢰⡯⢭⡇⠀⠈⢷⡄⠀⠀⢠⡿⠁⠀⠀⣨⡅⠀⠀⠙⣧⠀⠀⠀⣰⠏⠀⢰⣿⣮⣤⣝⣦⢻⣄
⣾⣃⣀⣀⣁⣀⣀⣈⣁⣀⣈⣿⠄⢠⣿⣀⣀⣀⣀⣈⣁⣀⣀⣀⣸⣧⠀⢼⣏⣀⣀⣈⣉⣁⣉⣋⣉⣃⣹
⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⣴⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡄
⠀⠀⠀⢀⣼⠃⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢻⣄
⠀⠀⢀⡾⠃⣠⣆⠀⢻⡄⠀⠀⠀⠀⠀⠀⣠⡟⢀⣀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠹⣆
⠀⢠⡾⠁⢴⡿⠻⣷⡄⢻⣆⠀⠀⠀⠀⣰⠟⠀⠀⠂⢀⣀⠈⢷⡀⠀⠀⠀⢀⣼⠃⠀⠀⠄⠀⠀⠹⣧
⢠⡟⠁⠀⠘⣇⣀⡟⠁⠀⠹⣆⠀⠀⣴⠏⠠⠤⠄⠾⣾⣿⡿⠎⢷⡄⠀⢀⡾⠁⠀⠁⠀⣿⠀⠀⠀⠘⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠴⠶⠶⠶⠶⠦⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣀⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣀⡀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⢀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⡄⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢀⣤⣶⣄⠈⣧⠀⠀⠀⠀⠀⢠⡏⢀⣠⣤⣀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⣿⣿⣿⣿⣧⢸⠀⠀⠀⠀⠀⢸⢡⣿⣿⣿⣿⣆⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⣿⣿⣿⣿⣿⣸⠀⠀⠀⠀⠀⢹⣸⣿⣿⣿⣿⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠘⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡟⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠈⠛⠟⢋⡾⠁⠀⠀⠀⠀⠀⠀⠹⣝⠿⠿⠟⠁⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⠛⠿⢿⣿⣿⣿⣿⡿⠿⠿⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠁⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠶⠶⠒⠛⠉⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠲⠶⠶⠶⠶⠖⠚⠛⠋⠉