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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣤⠖⠂⠛⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⡰⠁⠀⠀⠒⣒⠤⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡌⠉⠢⢄⣠⠤⡀⠁⠀⠀⠀⠙⠒⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢠⡀⢄⣀⠉⠖⠀⠘⠋⡣⠀⠀⠀⠸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢷⡈⠀⡇⠀⢀⠀⢸⠀⡀⠰⣄⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠬⢐⠉⠀⡜⠀⣿⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣟⣾⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⠦⠟⢶⠤⠔⠚⣷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢷⡙⡼⡄⣢⢶⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⢖⡟⡙⠀⠀⠀⠘⣶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡑⠀⠀⡯⢨⠀⠀⠀⠀⠀⠀⢀⠖⠁⣠⠦⠾⢤⣝⣢⠤⠔⠺⠃⡜⠑⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠘⠀⢸⠀⠀⠀⠀⠀⣠⠃⠀⡜⠁⠀⢠⣯⡯⠏⠀⣰⠁⣰⠁⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⢀⡨⢆⠀⠀⠀⠰⠃⣤⣶⡇⠀⠠⣿⣿⣇⠀⢠⢥⡴⢋⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠀⠀⢣⠀⢰⠁⠀⠘⣿⣷⠶⠉⠙⢿⣿⣄⠎⣠⢳⠃⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⢳⠁⠀⠀⠰⠹⣿⡀⠀⠀⠈⢿⣿⡜⣡⠁⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠈⡆⠀⠀⡄⠀⢹⣇⠀⠀⠀⠈⢿⣷⠃⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠈⠀⣰⡇⠀⠀⣿⡄⠀⠀⠀⠘⠏⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⣰⠋⠅⠀⢸⠈⢿⡄⠀⠀⠀⠀⠀⠀⢀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡤⠞⠁⢰⠁⠀⣘⠀⠈⠟⠢⣄⡀⣀⣀⡴⠋⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡟⠉⠁⠀⠒⡇⠉⠛⠋⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⣀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣀⣤⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠙⠾⠇⠀⠀⠀⠀⠣⢤⠴⠶⣿⠿⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠃⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠐⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠃⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡎⢰⣿⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠘⠀⠻⠿⠇⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⣀⣀⠀⠀⢠⣤⣦⣄⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠈⠅⠘⢀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⡇⠀⣿⣿⣿⣿⡆⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢢⠄⢀⠈⣌⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⠛⠋⠁⠀⠈⠙⠛⠋⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⣸⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⡀⠀⣾⠆⠸⠆⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⢀⠄⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠣⠒⠤⠢⣤⠦⠴⠋⠒⠚⠁⠀⠀⠀⢀⡄⢰⠙⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠠⡀⢀⠤⠄⠒⠃⠈⠉⣉⡉⠵⠀⣎⡀⠁⣠⠄⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠣⡄⢀⡀⠴⠄⠊⠑⠂⠈⣁⠈⠆⠘⠀⠀⠰⢤⠀⠳⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢁⠌⠁⠀⠀⠀⠀⠠⣀⢀⣀⠠⡀⠀⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⡄⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⢀⡁⢀⡬⠭⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡒⠤⠃⠀⠀⠀⠀⠀⠀⣈⣉⣉⠉⢀⣃⣎⣀⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠒⠘⢠⡠⠀⠀⢠⠋⣠⣤⡀⠉⠒⠉⣡⣤⣤⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠜⠀⠊⠈⠀⠀⠀⠸⡄⢿⣿⣿⠇⠀⢾⣿⡿⠟⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠾⡓⠊⠑⠒⠰⢶⡲⢚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢀⠀⠈⠀⠀⠀⠀⠀⠘⠂⢤⡲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢼⡠⢄⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢔⣊⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢂⡠⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢤⠘⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠘⣐⢄⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢴⣒⠡⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⣀⢀⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⠤⠔
⠀⠀⠀⠀⠀⠀⠘⠑⠁⣆⠆⡒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡁⠰⣠⠈⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⢻⢹⣯⣣⣀⣴⣞⣿⣿⡿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠿⠛⠛⠯⣿⣿⣿⣿⠿⠛⠛⠿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠹⡿⠁⠀⠀⠀⠀⠀⠙⣿⡷
⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⠶⡄
⠀⠀⠀⠀⠀⠈⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⠀⠀⠼⠦⠤⠤⡶⠄⠀⡤⠒⡄⢸⠀⠀⢸⠀⠀⡔⣩⠃⠙⡄⠀⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⡀⠀⠀⡇⠀⡼⠔⠊⠀⠈⠀⠀⢸⠀⢸⠞⠁⠀⠀⢰⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⡇⠀⠰⡀⠀⠀⠀⠀⠀⢸⠀⠘⡄⠀⠀⠀⡼⠀⠈⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠒⠊⠀⠀⠣⠕⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⢶⣄⢀⣠⣴⣶⡾⠿⣦⡀
⠀⠀⠀⠀⠀⢀⣾⠏⠉⠉⠙⠛⠿⣷⣦⣤⣶⣶⣶⣶⣶⣶⣾⠋⠀⠀⠀⠀⠹⣦⠛⠉⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⣼⠃⠀⠀⣴⠶⢶⡄⠻⢶⣤⡀⢀⣀⣀⣛⡀
⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⣄⡛⠁⠀⠀⠀⢺⣏⠉⠉⠉⠙⣷
⠀⠀⠀⠀⠀⠈⣵⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣀⣀⣹⡧⠀⠀⠀⠀⣈⠙⣻⡄⠀⠀⣿
⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠈⠻⠶⠶⣦⠙⠿⠛⠀⢀⣼⠃
⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣤⣴⢾⣷
⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⣠⣤⣤⣴⣿⣦⣤⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠶⠶⠿⠿⠿⠿⠟
⠀⠀⠀⠀⢶⡆⠀⠀⠀⠀⠀⢼⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠀⠀⠀⠀⢀⣀⣿⡇
⠀⠀⣠⣤⣾⣿⡶⠂⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢠⣶⣶⣶⡀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⣹⡿⠛⠛⠓
⠀⠀⠀⠀⠀⠘⣷⣄⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣴⣿⣁
⠀⠀⠀⢀⣤⡶⠟⠋⢵⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠎⠉⠙⠻⠦
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠛⠷⣶⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠻⠿⠿⠿⠿⠿⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠛⠷⣄⠀⣀⣤⣶⣶⡿⣷⣦
⠀⠀⠀⠀⠀⠀⣼⡟⠉⠉⠙⠛⠿⢷⣶⣤⣤⣴⣶⣶⣶⣶⣶⣾⠋⠀⠀⠀⠀⠘⣷⡙⠋⠁⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⣼⠇⠀⠀⣠⣶⢶⣄⢸⣧⣤⣀⠀⠀⢀⡘⠿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣄⠸⠋⠀⠀⠈⠡⣾⠛⠛⠛⠛⢻⣆
⠀⠀⠀⠀⠀⠘⢋⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠈⢙⣿⠀⠀⠀⠀⠀⡛⠻⣦⠀⠀⢀⡿
⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁⢶⣤⣤⣤⡘⢿⣶⠟⠀⠀⣼⠃
⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⢀⣠⣾⡇
⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⢸⣿
⠀⠀⠀⣀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣦⣤⣤⣤
⠘⠛⠛⠛⢛⡛⠛⠃⠀⠀⠀⠀⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⡀⠀⠀⠀⠈⠉⢉⣽⠉
⠀⠀⠀⠀⢸⣿⣀⣀⠀⠀⠀⠀⠿⠿⠃⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠘⠿⠿⠀⠀⠀⠀⠀⠴⣾⣿⣦⣤⣄
⠀⠀⠰⠟⠛⠻⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⢀⣹⣷⡶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠿⢿⣦⣄⡀
⠀⠀⠀⢠⡶⠟⠉⠀⠘⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠁⠀⠈⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣶⣶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⣶⡾⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡶⠀⢀⣤⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⢿⣿⠀⣴⡿⠁⠀⣸⡟
⠀⠀⠀⠀⠀⣀⣤⠴⠦⣤⡀⠀⠀⠀⣀⣀⠀⢻⠀⡼⠋⠀⠀⠀⠀⠙⣿⣿⠥⠴⠞⠋
⠀⠀⠀⢀⣾⡟⠁⠀⠀⠘⣿⡄⠀⠀⢹⣿⡇⠘⠀⠠⠶⠶⠦⠀⠀⣸⣿⠃⣠⡞⠉⡇⠀⠀⣀⡀
⠀⠀⠀⣾⣿⠃⠀⠶⣶⡄⣿⣿⠀⠀⢸⣿⡇⠀⢀⣴⡟⠙⣷⣄⢀⣿⡟⢰⣿⠁⢰⢧⢞⣭⠉⢻⣷⡄
⠀⠀⠀⢻⣿⡄⠀⠀⢸⡇⣿⣿⡄⠀⢸⣿⡇⠀⣾⣿⠁⢀⣿⡿⢸⣿⡇⣿⣿⣰⠋⢸⣿⣿⠀⣈⣿⣿
⠀⠀⠀⠀⠙⠳⠤⠤⠚⣤⣿⣿⣷⣶⣾⣿⣷⠀⣿⣿⠖⠛⠋⠁⣼⣿⠁⣿⡟⠁⠀⠘⣿⣿⠉⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⠃⠉⣿⣿⡇⠀⠹⣿⣆⠀⢀⡼⢻⣿⡇⢻⣿⠀⠀⡼⢻⣿⡄⢀⣿⠏
⠀⠀⠀⠀⠀⢠⣿⠿⠆⠀⣿⣿⠀⠀⢹⣿⣇⠀⠀⠈⠙⠛⠉⠀⠀⢿⣿⣄⠛⠓⠊⠀⠀⠈⠙⠛⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡿⢏⡀⠀⠀⠉⠁⠀⣀⠀⠀⠀⠀⠀⣀⣀⠈⠁
⠀⠀⠀⠀⠀⠀⠙⠲⠶⠟⣸⣿⣿⣿⣷⣄⢰⣿⣿⣿⣿⣦⢠⣾⡿⢿⣿⠀⠀⠀⠀⠀⠀⣾⣷⠀⢀⣾⡷
⠀⠀⠀⠀⠀⢠⣀⣴⡀⠀⣿⣿⡇⢈⣿⣿⢸⣿⡇⠀⣿⣿⡿⣿⣷⣦⠀⠀⢀⣀⣀⣀⡀⣿⣿⣴⣿⡿⠁
⠀⠀⠀⠀⠀⢨⣿⣿⠟⠀⣿⣿⣷⣿⡿⠋⢸⣿⣷⣾⣿⠟⣾⣿⠟⠁⠀⠀⠛⠛⠛⠛⠋⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿⣿⡏⠁⠀⠀⢸⣿⡿⣿⣷⡄⠻⣿⣷⣶⣾⡷⠀⠀⠀⠀⠀⣿⣿⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣌⡉⠀⠀⠀⠀⠈⠉⠁⠀⠙⠁⠀⠈⠉⠉⠉⣀⣀⣀⣀⣤⣄⣈⣉⡀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣤⣄⣀⣀⣀⣀⣠⣤⡤⠶⠖⠛⠋⠉⠉⠀⠀⣀⣀⣀⣀⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣀⡤⠤⠖⠒⠋⠉⠉⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⣒⣒⣂⡂⠌⠉⠭⠭⠩⠉⠭⣍⣉⣉⣀⡒⠢⠤⢀⣀
⠀⠀⠀⠀⠀⣀⠤⢒⡊⠭⠒⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠢⢍⡑⠢⢄
⠀⠀⢀⡴⢫⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡱⣄
⠀⢠⠎⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠹⡌⢂
⢀⠁⡆⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢸
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠇⣠⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⡠⠖⢀⣄⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⣀⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⡄
⢇⢠⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠈⠅⠀⣤⣤⣴⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⢳⡇⠀⠀⢛⠋⠀⠀⠀⠀⢰⢇
⠘⡎⡄⠀⠀⠀⠀⠀⠀⠀⠐⣀⣰⡄⣄⠈⠁⣓⢄⢀⣀⣈⡇⠀⠀⣘⣷⢤⡀⠉⢀⡹⠀⣵⣤⡖⠀⠀⢠⢎⡞
⠀⠸⡐⢄⠀⠀⠀⠚⠋⠀⠘⠛⠃⠈⠓⠛⠛⠁⠀⠙⠛⠋⠈⠒⠛⠋⠀⠑⠚⠛⠋⠀⠀⠀⠀⠀⣀⠔⡡⠊
⠀⠀⠈⢦⡓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣲⣤⣶⣾⡶⠚
⠀⠀⠀⠀⠈⠲⠍⣒⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⢤⣬⣭⡭⠭⠭⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠉⠥⣣⠀⠀⠀⠀⠀⠀⠸⡰⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡑⡀⠀⠀⠀⠀⠈⣇⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡢⢀⠀⠀⠀⠜⢦⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢭⣒⠤⢤⢤⣙⠮⠓⢤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⢀⣀⠀⠀⠀⠀⠀⠀⠟⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢻⣿⠀⠀⠀⣴⡄⠀⠀⠀⠀⢠⣿⡟⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣤⣤
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢸⣿⠀⠀⢠⣿⠁⠀⢀⣀⠀⢸⣿⠃⠀⢸⣿⠁⠀⣤⡶⡄⠀⡠⠂⠀⠀⠉⠉
⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⢸⣿⠀⣀⣼⣿⠀⣰⡿⢉⡇⢸⣿⠀⠀⣿⣿⠀⣸⡟⠑⢺⠋
⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⢸⣿⠚⠁⢸⡇⠀⣿⡗⠋⠀⣸⣿⡀⠀⣿⡏⠀⢿⣧⢀⡞
⠀⠀⠀⢸⠁⠀⠀⠀⢀⠔⢹⣿⠀⠀⢸⣿⠀⢸⠿⠤⠚⠁⢿⣧⣰⣿⡇⠀⢸⠛⠋
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣸⡟⠀⠀⠈⠛⠖⠃⠀⠀⠀⠀⠀⠈⠀⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢾⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡆⠀⣾⣿⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⣿⢰⣿⡿⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⢸⣿⣿⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣀⠏⢀⣤⡀⠀⠀⣴⣿⠟⢻⡆⢸⣿⣟⢀⡇⢸⣿⣿⠀⠇⠀⠀⠀⣴⠶⣤⡀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⠀⡜⢸⡇⠀⢸⣿⡟⠀⣰⠇⢸⣿⣇⡜⠀⢸⣿⣿⠠⠀⠀⢰⣷⠙⠶⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⢰⡇⢸⣷⠀⢸⣿⣷⠒⠁⠀⠘⣿⣿⡀⠀⠀⣿⣿⡇⠀⠀⣸⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣀⡴⠊⠸⣿⣿⣿⠀⢸⣿⠀⢀⡿⣿⣆⣀⣀⡴⢻⣿⣧⡀⠀⣸⣿⣷⣄⣠⠟⣿⣦⣠⣼⡟
⠀⠀⠀⠀⠈⠁⠀⠀⠀⢻⣿⣿⡆⠈⢿⣦⠎⠀⠈⠙⠛⠉⠀⠀⠙⠿⠿⠛⠁⠈⠛⠛⠁⠀⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠶⠒⠚⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈






⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢂⠖⠁⣉⡉⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠠⠣⡀⠀⡀⠒⠑⠀⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⡀⠀⢻⠉⠰⠒⡀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢷⠈⠨⠀⠀⠀⠚⢀⠀⣄⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠣⡀⠐⠆⢀⡆⣸⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡲⣽⡗⡀⢀⠀⠀⠀⠀⠀⠀⠀⠦⠔⢛⣹⡇⠂⠉⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡽⡜⠃⢰⠙⠀⠀⠀⠀⠀⢀⠤⠒⡘⣝⣑⡄⣀⡀⢸⠋⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠠⠂⢸⠀⠀⠀⠀⢠⠊⢀⠊⠁⢠⣿⣿⠀⢠⠂⡘⠀⠈⠑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⢀⡸⠄⠀⠀⠠⠃⣤⣮⠀⠠⣿⣿⡀⢀⢇⠴⡃⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠈⠆⢠⠁⠀⠹⣿⡲⠉⠙⢿⣧⠎⡴⡓⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠇⠀⠀⠀⢻⡇⠀⠀⠈⢿⣏⡘⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠘⠀⢠⠇⠀⣿⡄⠀⠀⠘⡿⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢠⠟⠀⠀⠏⢷⡀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠔⠁⠰⠀⢘⡀⠘⠟⢤⣀⣀⣠⡴⠋⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢸⠉⠁⠀⠹⠀⠉⠁⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⣄⣸⠀⠀⠀⠀⠀⢀⣀⣤⣤⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠉⠀⠀⠀⠀⠉⠉⠉⢻⠉⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢸⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠⢀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠆⢼⣿⠄⠀⠀⠀⠀⠐⠄⠀⠈⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠙⣦⠀⠀⣴⠒⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⡟⠀⣼⠃⠀⣸⠃⠀⡓⠛⢹⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣰⠃⣸⠇⠀⢠⠏⠀⠀⢧⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡰⠃⢀⣿⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⡤⢤⠀⢸⣿⠞⠁⠀⢸⡷⠚⠁⠀⢠⡖⢁⣀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣿⣿⡇⠀⣠⠊⣷⠀⢰⣟⡠⠎⠀⢸⣿⠀⠀⠀⣼⡇⠀⠀⢠⣿⠐⣏⢸⡧⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⣿⡇⢰⠃⢰⡟⠀⣿⠏⠀⠀⢀⠿⣿⠀⠀⢰⣿⡇⠀⠀⣼⡇⠀⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⣿⣷⠇⠀⢸⡇⢀⢿⡄⠀⢀⠎⠀⢿⡀⣠⠃⠸⣇⢀⡼⢹⣧⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠄⠀⢸⣧⠎⠀⠑⠐⠁⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡤⢿⣄⠀⠀⠀⠀⣠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⡂⠀⠀⠀⠀⠀⢀⡞⠁⢈⡇⠀⠋⠿⠮⡿⡃⠀⢀⡞⠁⢀⡇⠀⣠⠞⠁⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠈⠄⣀⠀⢀⡾⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⢀⡜⠀⣼⠇⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⢙⡀⠘⠃⠙⠁⣼⠇⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡠⠊⠀⣸⡏⠀⠀⡴⠃⠀⠀⠀⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠃⠘⠃⠀⠀⠀⢀⣿⠞⠀⠀⠀⢀⡴⠉⢀⠀⠀⠀⣿⡏⠀⠀⢠⣿⣃⡴⠊⠀⠀⣠⢠⡁⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠿⠀⠀⠀⠀⣼⡏⠀⠀⠀⢀⣼⡇⠀⣿⠀⠀⢠⣿⡇⠀⠀⢸⣿⠁⠀⠀⠀⢀⡄⠀⠀⠗⠀⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⣧⡤⠄⠒⠉⣿⠀⣸⢿⡆⠀⡎⢿⡇⠀⠀⣸⣿⠀⠀⠀⠀⡞⠀⠀⠀⠘⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢸⣧⢳⠀⠀⠀⢿⣤⠏⢸⡇⡼⠀⠸⣷⠀⢠⠟⣿⡀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢿⣸⠇⠀⠀⠀⠁⠀⠈⠛⠁⠀⠀⠙⠲⠋⠀⠙⠷⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣤⣄⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠿⠻⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣅⠂⠀⠉⠁⠀⠀⠸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠹⠇⠀⠀⢀⣤⡋⠀⣻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⣤⠀⠀⠀⠛⠁⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⢠⣄⣀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡀⠁⠉⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⠂⠛⠁⠀⣾⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⡞⠳⣮⡀⠀⣀⣠⠾⠛⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠈⠻⣦⡼⠟⢛⣏⠀⠀⣀⡤⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⢰⠷⡾⠈⠉⠉⠁⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢻⡀⠀⠀⠀⠀⠀⠀⣿⢸⠇⠀⠀⠀⠀⠀⣴⡀⠀⠀⢿⠀⠀⠀⠀⣀⠀⠎⡿⢻⣧⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢷⣄⠻⢼⠀⠀⠀⠀⢰⠇⡈⠀⠀⠀⠀⠀⠘⠋⠀⢀⣀⠜⠷⠀⠀⠀⡏⡶⠀⠑⢋⠼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠙⠷⢸⡇⠀⠀⠀⣸⢳⡇⠀⠀⠀⠀⠀⠀⢀⣂⣉⡴⠶⣇⠀⠀⠀⡇⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣰⢻⡇⠀⠀⠀⣿⣼⠃⠀⠀⠀⠀⠀⠀⠈⠹⣏⠀⠀⢻⡆⢀⡼⠃⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⣰⠇⢸⡇⠀⠀⠀⣧⣼⠀⠀⠀⠀⠀⠀⢠⡟⠀⠹⡄⠀⠀⡼⠋⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣿⠀⢸⡇⠀⠀⢰⡇⣿⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢻⣄⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢰⡇⢀⡴⣧⣤⡀⣸⢀⡇⠀⠀⠀⠀⣀⣘⠁⠀⠀⠀⠀⠹⣄⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⣴⠏⢠⡿⢷⠀⡟⠛⠛⢲⡿⢏⠉⢃⠹⣧⡀⠀⠀⠀⠀⠙⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣴⠟⠁⢰⡟⠀⡟⠐⠇⠀⠀⠘⠃⠀⠀⠈⠃⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢀⣾⠏⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠸⡀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆
⠀⠀⠀⠀⠀⠀⠛⠂⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢴⠀⠀⢀⠒⢂⠀⣠⠒⢀⠀⠖⢠⢀⢆⠀⠀⠀⠀⢠⠒⢂⠀⠀⠀⠀⡠⠐⠂⢠⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠸⠀⠀⣠⠀⢻⠀⡏⠠⡎⠉⠋⠁⢸⠀⠸⠀⠀⠀⠘⠂⢸⠀⠀⠀⠌⠁⠔⠉⠉⢣⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⡡⠤⣄⡀⢸⠀⣯⠀⢨⠀⠨⠀⠸⠀⠸⠀⠀⠀⢠⠀⣸⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⢸⠀⠆⠀⣃⢀⣀⡀⠀⠄⠀⣄⡀⣀⠸⣈⠉⠀⣀⡀⢃⠀⠡⢄⣠⠊⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠚⠀⠀⠘⠠⠊⠀⠑⠢⠠⠠⠴⠂⠀⠓⠠⠤⠴⠞⠀⠓⠤⠤⠤⠊⠀⠑⠒⠒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡴⠘⣀⣀⣑⠀⢘⠁⡗⠀⠀⠠⠭⢑⠀⣼⠙⡈⣱⡀⠀⠀⠔⠟⠈⠆⠀⡾⠑⡉⢦⠀⠀⢀⠞⠁⠙⠆⠀⠎⢁⣩⣥⣉⠄⢸⢃⣈⣍⢑⠢⡀
⠀⠀⠀⣆⠘⠦⡀⠀⠀⢨⠁⠀⠀⠀⠀⠀⢸⠀⠂⠀⣿⠀⢥⠀⡌⠁⣴⠀⠰⠀⠁⢰⡧⠀⢇⠀⠏⢈⡗⠈⡇⠀⠀⠘⠄⣀⣀⠀⣹⠀⣇⣀⡽⠀⡅

⠀⠀⠀⠈⠑⠢⡀⠑⡄⢀⠀⣈⠀⠀⢀⠀⢸⠀⡀⠀⡘⡀⠨⣦⠁⢠⠃⠀⢀⠀⡃⠰⠫⡀⠀⣜⡀⠘⢁⠀⡀⠀⡂⢀⠄⠶⠞⠀⢁⠀⡀⢀⠄⢎
⠀⠀⢰⠲⠆⠤⠌⢀⠇⠘⡁⠘⠠⠤⠪⢀⠆⠀⡯⠈⡆⠸⡈⠀⢠⠃⠸⠠⠉⠀⠂⠘⠀⢃⠁⠤⢡⠁⣈⠀⡇⠈⡁⠘⠤⠤⠠⡀⠈⠀⠇⠀⢓⠀⢥
⠀⠁⠀⠉⠒⠀⠒⠁⠀⠀⠈⠒⠢⠀⠒⠁⠀⠀⠙⠒⠁⠀⠑⠒⠁⠀⠐⠒⠂⠀⠂⠐⠀⠀⠑⠘⠁⠀⠐⠀⠃⠀⠑⠒⠒⠀⠒⠁⠙⠂⠃⠀⠈⠒⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣦⣤⣀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⡿⢿⣿⣿⣿⡿⢾⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣟⢠⠀⠐⠛⠉⠉⠀⢸⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣶⠀⠀⠀⠀⠀⠆⠀⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠁⠀⠀⠀⠸⠿⠀⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠈⠃⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡈⠻⠳⠖⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠻⠷⣤⣄⣀⣤⠤⠚⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⡧⠀⠀⠀⠀⢿⣿⣿⡿⠳⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⣿⡉⠻⣦⣀⣀⣤⠶⠛⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠙⠳⠶⢛⠉⣽⣇⣀⣀⡤⠖⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠁⠀⠀⠀⠀⠀⠀⠀⡾⢻⠇⠀⠁⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⢠⡄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⢻⣄⠀⠀⠀⠀⠀⠀⢸⡇⣾⠀⠀⠀⠀⠀⢀⣴⡀⠀⠀⠹⡀⠀⠀⠀⢀⡀⢀⢋⡿⢻⡵⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⣄⠻⢴⡇⠀⠀⠀⠀⡿⠀⠇⠀⠀⠀⠀⠀⠈⠋⠀⢀⣀⠠⠷⠄⠀⠀⢸⢹⡎⠈⠑⢣⠼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠙⠷⢀⡇⠀⠀⠀⢀⡟⣾⠀⠀⠀⠀⠀⠀⠀⠀⠒⣉⣡⠶⣾⠀⠀⠀⢸⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢠⢿⡇⠀⠀⠀⢸⣧⣿⠀⠀⠀⠀⠀⠀⠀⠘⠻⣏⠁⠀⠸⣧⠀⣠⡞⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⢠⡟⢸⡇⠀⠀⠀⢸⣁⡇⠀⠀⠀⠀⠀⠀⢀⡶⠀⠹⡆⠀⠀⣩⠞⠁⠀⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⣾⢹⡇⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢻⡄⠀⠁⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⢠⡟⠀⢈⣧⡀⠀⠀⡟⢸⠃⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠻⣄⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⣼⢃⡼⠋⣽⣯⡓⢲⠷⠾⣦⣤⣤⡄⠶⡒⢷⣄⠀⠀⠀⠀⠀⠹⣦⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⣿⠟⢁⣴⠟⢸⠃⣼⠀⠀⠀⢿⠏⠑⠀⠱⠈⠻⢧⡀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣰⡿⠁⠀⠘⠏⠀⡾⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⡾⠃⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⢇⡀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠈⠛⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠤⠤⣀⡀
⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⡜⠀⠀⠀⣿⠀⠀⢰⡇⠀⢠⡖⠒⠶⠆⠀⣾⠀⠀⠀⢰⠀⠀⠀⢀⡴⠒⢦⡀⠀⠀⠀⠀⢷⡀
⠀⠀⣸⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⣇⣀⣀⡀⠀⢹⠀⠀⠀⢸⠀⠀⠀⢸⠁⠀⠀⢻⡀⠀⠀⠀⠘⣇
⠀⠀⡇⠀⠀⠀⠀⣿⠒⠒⢺⡇⠀⢰⡏⠉⠉⠁⠀⢸⠀⠀⠀⢸⡀⠀⠀⢸⡀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠘⣇⠀⠀⣸⠇⠀⠀⠀⢸⢿
⠀⠀⡇⠀⠀⠀⠀⠛⠀⠀⠘⠃⠀⠀⠓⠒⠚⠀⠀⠘⠒⠒⠂⠘⠓⠒⠒⠂⠈⠓⠚⠋⠀⠀⠀⠀⡜⡞
⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣰⠃
⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⡰⠃
⠀⠀⠀⠀⠀⠙⢶⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⢊⡰⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠚⠺⡄⠀⠀⢠⠚⣖⣒⣒⣒⣒⣒⣒⣒⣘⣙⣍⡭⠴⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡔⢣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠴⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠤⠦⠴⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠤⠤⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⢠⠞⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⣰⠃⠀⠀⠀⢰⡆⠀⠀⠀⢰⡆⠀⠀⢠⣦⣤⣤⣤⡄⠀⠀⣶⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⣤⡶⣤⣀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⢠⠇⠄⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⠃⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⡞⠈⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⢸⣧⣀⣤⣤⡄⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢀⡇⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⢻⡇⠀⠀⢸⡏⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⢸⣟⡆
⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢹⡆⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⢸⣉⡇
⠀⠀⠸⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⢸⣇⣀⣀⣀⡀⠀⠀⣿⣀⣀⣀⡀⠀⣷⣀⣀⣀⣀⡈⢻⣄⣀⣠⡾⠃⠀⠀⠀⠀⠀⣼⢸⠁
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠂⢠⠇⡟
⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣸⠃
⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡴⠃⡰⠃
⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣀⠴⠋⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣲⡦⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠖⠛⣉⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠒⠶⠼⣆⠀⠀⠀⠀⡰⠻⣛⢒⠒⠲⠒⢒⠒⠖⠒⠒⢒⠛⣋⣋⣉⣡⣤⠦⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢠⠞⢉⡶⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠋⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣤⡶⠄⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡆⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡷⠖⠒⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢪⢀⡇⣀⣀⣄⡈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡞⠠⣔⠀⡀⢲⠊⠎⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢰⡅⠀⠈⠀⠱⠀⠀⠜⡆⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⣦⠀⠀⠤⠤⠔⠀⡌⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠘⢦⣀⣀⣀⣀⡔⠀⣆⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠣⠄⢸⠉⠉⠉⠀⢤⠠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⡖⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠴⡸⠀⠀⠀⠀⠸⣐⠒⡦⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⣀⠆⡌⣘⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠌⠳⣄⡀⢀⢀⣠⠎⢠⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠌⠀⡏⠀⠀⠘⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠅⢀⣄⠈⠉⢹⣿⠆⡄⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠘⡄⠀⠱⡀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⡁⠘⠛⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢠⠃⠀⠀⢁⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣀⠤⡟⠀⠀⢠⠃⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣻⠀⠀⢠⠀⠀⢰⣶⡄⠀⡌⠀⠀⠀⣤⠴⠊⠀⠀⢱⠀⡠⠁⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡟⠀⠀⠈⠀⠀⠀⠈⠀⠀⡇⠀⠀⠀⢩⠀⠀⠀⠀⠀⡷⠁⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠠⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡀⠀⠀⠀⠘⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠃⠀⠀⠀⠇⠿⠗⠀⠀⢸⡇⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⠀⠀⢀⠀⡆⠀⠀⠀⠀⣾⠀⠀⠠⡀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢸⠀⢀⠌⠀⢡⡀⠤⠠⠶⢿⠀⠀⠀⠑⢄⡀⠈⣄⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢘⡠⠊⠀⠀⢸⡄⠠⠀⠀⠘⠂⠀⠀⠀⠀⠙⠶⣼⣦⡤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣶⠉⠀⠀⠀⠀⠘⡇⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⣇⠈⢺⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠉⠀⣀⣀⣀⣀⣀⣿⠀⠀⡀⠀⢸⣀⣀⣀⣀⣀⣀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠙⢻⠉⠉⠉⠉⠉⠁⠀⠀⡇⠀⠀⠁⠀⠀⠉⠉⠻⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡶⣶⣤⣄⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡴⡿⢹⣿⣕⡄⢀⣴⣟⣿⢿⣿⡟⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⠿⠿⣝⣧⣿⣷⣿⡿⣿⡿⠭⠿⣿⣽⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠉⠪⣿⢿⠟⠁⠀⠀⠀⠀⠉⠺⣽⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠛⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣟⠆⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠘⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣆⠀⠀⠀⠀⡇⠀⠀⠀⡜⠀⠀⠀⠀⡀⠀⠀⡇⠀⠀⢸⠀⠀⠀⣠⠴⡄⠤⡀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠂⠀⠀⠉⡏⠉⠉⠉⡟⠁⠀⡴⠋⢉⡇⠀⡇⠀⠀⢸⠀⠀⡰⢁⡔⠁⠀⠸⡄⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⢰⠗⠚⠉⠀⠀⡇⠀⠀⢸⠀⠀⡗⠉⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠘⣆⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢧⠀⠀⠀⠀⢀⡇⠀⠀⠁⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠂⠀⠀⠀⠇⠀⠀⠘⠀⠀⠀⠙⠒⠤⠴⠊⠀⠀⣎⣱⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠣⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠙⣦⠀⠀⣴⠒⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⡟⠀⣼⠃⠀⣸⠃⠀⡓⠛⢹⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣰⠃⣸⠇⠀⢠⠏⠀⠀⢧⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡰⠃⢀⣿⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⡤⢤⠀⢸⣿⠞⠁⠀⢸⡷⠚⠁⠀⢠⡖⢁⣀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣿⣿⡇⠀⣠⠊⣷⠀⢰⣟⡠⠎⠀⢸⣿⠀⠀⠀⣼⡇⠀⠀⢠⣿⠐⣏⢸⡧⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⣿⡇⢰⠃⢰⡟⠀⣿⠏⠀⠀⢀⠿⣿⠀⠀⢰⣿⡇⠀⠀⣼⡇⠀⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⣿⣷⠇⠀⢸⡇⢀⢿⡄⠀⢀⠎⠀⢿⡀⣠⠃⠸⣇⢀⡼⢹⣧⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠄⠀⢸⣧⠎⠀⠑⠐⠁⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡤⢿⣄⠀⠀⠀⠀⣠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⡂⠀⠀⠀⠀⠀⢀⡞⠁⢈⡇⠀⠋⠿⠮⡿⡃⠀⢀⡞⠁⢀⡇⠀⣠⠞⠁⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠈⠄⣀⠀⢀⡾⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⢀⡜⠀⣼⠇⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⢙⡀⠘⠃⠙⠁⣼⠇⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡠⠊⠀⣸⡏⠀⠀⡴⠃⠀⠀⠀⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠃⠘⠃⠀⠀⠀⢀⣿⠞⠀⠀⠀⢀⡴⠉⢀⠀⠀⠀⣿⡏⠀⠀⢠⣿⣃⡴⠊⠀⠀⣠⢠⡁⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠿⠀⠀⠀⠀⣼⡏⠀⠀⠀⢀⣼⡇⠀⣿⠀⠀⢠⣿⡇⠀⠀⢸⣿⠁⠀⠀⠀⢀⡄⠀⠀⠗⠀⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⣧⡤⠄⠒⠉⣿⠀⣸⢿⡆⠀⡎⢿⡇⠀⠀⣸⣿⠀⠀⠀⠀⡞⠀⠀⠀⠘⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢸⣧⢳⠀⠀⠀⢿⣤⠏⢸⡇⡼⠀⠸⣷⠀⢠⠟⣿⡀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢿⣸⠇⠀⠀⠀⠁⠀⠈⠛⠁⠀⠀⠙⠲⠋⠀⠙⠷⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣷⡀⠀⣿⣿⡏⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠸⡇⢰⣿⣿⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⡇⢸⣿⣿⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡼⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⣿⣿⣿⠀⢸⡇⢸⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣸⠁⢀⡾⣧⠀⠀⠀⣴⣿⡿⠋⠙⣷⠀⣿⣿⣿⠀⡾⠀⢸⣿⣿⡇⢀⠃⠀⠀⢀⠀⣾⠛⠻⣦⣀⣠⣾⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⡼⠁⣿⡄⠀⢸⣿⣿⡇⠀⢠⡟⠀⣿⣿⣿⣰⠃⠀⢸⣿⣿⡇⠌⠀⠀⠀⣿⣧⠙⠷⣶⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢠⡇⠀⣿⡇⠀⢸⣿⣿⡧⠴⠋⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⡗⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⣿⣿⣿⣼⠁⠀⣿⡇⠀⠈⣿⣿⣧⠀⠀⠀⠀⢨⣿⣿⣧⠀⠀⠀⢹⣿⣿⣆⠀⠀⣸⣿⣿⡀⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠠⡴⠞⠁⠀⠀⢻⣿⣿⣿⡀⠀⣿⣿⠀⢠⠏⠻⢿⣷⣶⣶⠾⠋⠘⢿⣿⣷⣤⣤⠞⠻⣿⣿⣷⠾⠃⠘⢿⣿⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇⠀⠘⠿⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠴⠶⠖⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠴⢒⣒⣒⣒⣒⣉⣉⣉⡉⣉⣉⣉⣉⣉⣉⣀⠒⠂⠤⢄⣀
⠀⠀⠀⠀⠀⠀⣀⡠⠔⣒⡩⠥⠒⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠤⢉⡑⠲⠤⡀
⠀⠀⠀⢀⡔⢊⡥⠖⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣌⠳⡀
⠀⠀⡰⢃⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡘⢦
⠀⠜⢁⠈⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠄⠀⠠⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⡁⠀⠀⠀⢱⠘
⠐⢀⠇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⡄⠘⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⢁⣠⡀⠀⡤⠚⠁⠴⠦⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⡠⣄⠈⠢⢀⣠⣠⠀⠀⠀⠀⠀⣠
⢃⢰⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⡗⠀⠠⣶⣶⡾⠿⠀⠀⢄⡀⠀⠀⢀⣀⠀⠀⠈⢙⡞⢀⠀⠈⡍⠁⠀⠀⠀⠀⢀⡇
⠈⡇⢇⠀⠀⠀⠀⠀⠀⠀⡀⠀⣆⣀⣧⠡⣀⡀⠀⣳⢆⡠⣀⣀⣻⠠⣀⣀⣊⣷⠦⣀⡀⣀⣠⠇⠠⣖⣬⠆⠀⠀⢠⢎⡸
⠀⠘⢆⠢⠀⠀⠀⠐⠛⠋⠀⠘⠛⠋⠀⠙⠒⠚⠋⠁⠀⠉⠚⠚⠀⠑⠺⠓⠋⠈⠙⠒⠛⠋⠁⠀⠀⠀⠀⠀⣀⠔⣃⠖⠁
⠀⠀⠈⢦⡑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣐⣢⣤⣶⣾⡷⠞⠁
⠀⠀⠀⠀⠉⠲⢌⡒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣈⣙⣚⣛⣙⡉⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠪⠭⠵⣄⣐⣒⡐⡄⠀⠀⠀⠀⠀⠀⠀⡏⣩⠗⠒⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡝⢄⠀⠀⠀⠀⠀⠀⢵⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⡢⡀⠀⠀⠀⠀⠈⢇⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢬⡒⠤⣀⠀⠈⠈⠳⣌⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢍⣑⣒⠒⠞⢛⢈⣱⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢠⠄⠀⠀⡠⠒⡄⠀⡠⠆⡀⠠⠆⣄⢀⢄⠢⠀⠀⠀⢀⠔⢦⠀⠀⠀⠀⢀⡤⠐⠀⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠇⠀⠀⡁⠐⡇⠀⡇⢠⡋⠒⠂⠁⢸⠀⢺⠀⠀⠀⠸⠆⠨⡄⠀⠀⠠⠉⢠⠊⠉⠓⡀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⣈⣉⣉⣁⠀⡇⠀⣇⠀⢉⡉⣱⠀⢸⠀⢸⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⡄⠀⠀⠀⠁⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⠀⠀⡍⠀⡇⠀⡁⠰⡁⢀⣀⠀⠀⠀⠀⣀⢀⡀⠀⠇⠋⡀⠀⠀⢠⠀⠑⡄⠀⡠⠃⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠇⠀⠀⠯⠠⠇⠀⠧⠌⢁⣈⡩⠇⠀⠦⢀⣉⣀⡽⠀⠫⣀⣀⣈⡭⠀⠑⠶⠤⠤⠤⠜⠁

⠀⠀⠀⢀⠴⠆⠀⠒⡄⠀⠖⢲⠀⠀⠀⢐⡒⠆⠀⡔⠖⠒⢄⠀⠀⠀⣤⢖⠒⡄⠀⣴⠢⠒⢄⠀⠀⠀⡠⠖⠐⢆⠀⢠⠒⠐⢒⡀⢂⠀⡴⠀⠐⠒⠦⢀
⠀⠀⠀⡉⠰⢎⠉⠑⠁⠈⠦⠈⠀⠀⠀⠈⠀⡆⠀⠃⠴⡆⠙⡆⠀⡰⠀⣠⠀⣀⠀⡇⢀⣏⠉⣇⠀⡰⠑⣦⠔⢶⠀⠄⠁⠎⠉⠉⠉⠀⡧⢘⡎⠉⡦⡀⡃
⠀⠀⠀⠙⠢⣈⠁⠢⡀⠀⠀⢰⠀⠀⠀⠀⠀⡇⠀⠇⠈⠇⠀⢹⡠⠁⢀⠏⠀⠀⠀⠦⢨⡟⡀⠘⣠⠁⠈⠟⠀⢘⠀⠀⠀⢈⣈⣉⡇⠀⡇⠈⠁⠉⠁⡔⠁
⠀⡀⢀⣀⣀⠀⢱⠀⢸⠀⡆⠘⢄⠀⢀⠂⠠⠅⠀⡆⢸⠈⠆⠸⠁⠐⡎⢰⢀⠺⠀⡇⠠⡆⢇⠀⠉⠆⡜⢸⠂⢺⠀⠔⠀⣆⠀⠀⠀⠀⡇⢨⠉⠑⡅⠘⡄
⠀⠀⠘⠢⢤⠉⢁⠄⠁⠀⠑⠄⣀⣉⣁⠵⠊⠀⠀⢧⡰⠀⠘⠆⠀⠞⠀⠠⣈⠀⠀⠄⢠⠅⠈⠢⠨⠔⠁⠸⠀⠸⠀⠰⠀⣈⡉⢁⡩⠀⢇⠀⠂⠀⠀⡀⡸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣤⣤⡶⠔⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠦⠤⠶⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡉⢩⠂⣀⣀⡀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⠓⢉⠤⠈⠩⡥⠁⡞⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣜⠀⠀⠃⠀⡆⠉⠀⡱⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠫⡀⠀⠀⣀⣀⠠⠀⡔⠁⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢳⡀⠀⠀⠁⢀⡰⢱⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠢⣀⠀⢿⠛⠚⠚⠉⡀⢀⡿⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣽⠀⠀⠀⠀⠿⠨⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢠⠃⠏⢌⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠉⠀⠳⡁⠀⠀⠀⠀⢈⡲⢈⠁⠀⠉⠑⢲⡀⠀⠀⠀⠀⠀⠀⢀⠇⢱⠃⠘⠐⠏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠠⠀⠈⠑⠒⠒⢶⣯⡀⠈⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⡎⠀⠸⡀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠈⠀⣾⡷⠀⠀⠘⠛⠁⠇⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠸⠀⠀⠘⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠀⠀⠀⡰⠁⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⢠⡇⠀⠀⡄⠀⠀⣀⣀⠀⠀⠸⠀⠀⠀⠀⢀⡠⠖⠋⢧⠀⠀⡰⠁⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡇⠀⠀⡇⠀⠀⠼⠿⠁⠀⡇⠀⠀⠀⢿⠋⠀⠀⠀⠈⣆⡐⠁⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⠁⠀⠀⢀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠘⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⢸⠀⣠⡀⠀⠀⢸⠃⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡌⠀⠀⠀⠈⠀⠙⠁⠀⠀⣾⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⡇⠀⠀⡄⠀⡇⠀⠀⠀⠀⣿⠀⠀⠐⢄⠀⠀⢣⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⡇⠀⡔⠀⠀⣷⠒⠂⠈⠛⢻⠀⠀⠀⠀⠱⣄⡀⢣⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣀⡥⠊⠀⠀⠀⢿⠀⡇⠀⠀⠘⠀⠀⠀⠀⠀⠈⠙⠻⢽⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡆⡇⢀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⡟⠀⠀⠀⠀⠀⠘⡇⠈⢹⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢈⣀⣀⣀⣤⣤⣄⣠⡇⠀⠈⠀⠀⢸⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠉⡏⠀⠀⠀⠀⠉⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈⠙⢿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠗⠀⣠⡶⠒⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀⡀⠀⠀⠀⢿⣧⢠⣾⠏⠀⣠⡿⠁
⠀⠀⠀⠀⢀⣤⠖⠒⠲⣦⡀⠀⠀⢠⣤⡀⢸⡆⠘⠃⠀⠀⠀⠀⢹⣿⠟⠒⣚⣉
⠀⠀⠀⣰⣿⠃⠀⠀⠀⢹⣷⡀⠀⠘⣿⣿⠀⠀⠐⣛⣛⡓⠀⠀⣾⡟⢠⣾⠁⢸⠀⣀⣤⣤⣀
⠀⠀⠀⣿⣿⠀⠀⠛⢿⢸⣿⡇⠀⠀⣿⡿⠀⢠⣾⠏⠈⣿⣆⢸⣿⠃⣿⡏⢠⠟⣾⣶⡆⠘⣿⣧
⠀⠀⠀⠹⣿⡄⠀⠀⡼⢸⣿⣇⡀⢰⣿⡇⠀⣿⣿⣀⣰⣿⠇⣼⣿⢀⣿⡧⠋⠀⣿⣿⣤⠤⢿⣿⡆
⠀⠀⠀⠀⠈⠉⠉⠉⠰⢻⣿⡿⠿⢿⣿⡿⠃⢻⣿⡉⠉⠀⢠⣿⣿⠈⣿⡇⠀⠀⣽⣿⡄⠀⣸⣿⠁
⠀⠀⠀⠀⠀⢀⣤⣄⠀⢸⣿⡇⠀⢸⣿⡇⠀⠈⠻⢷⣤⠴⠋⠸⣿⣆⠻⣧⣀⠜⠁⠻⢷⣤⠿⠋
⠀⠀⠀⠀⠀⣿⠋⠉⠀⣸⣿⠃⠀⠈⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠗
⠀⠀⠀⠀⠀⠙⢄⣀⣠⡟⣽⣶⣶⣤⡀⠀⣴⣶⣶⣤⡀⢀⣤⣶⣶⣄⠀⠀⠀⠀⠀⢠⣤⡀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⣿⣟⠉⢿⣿⣼⣿⡏⠙⢿⣿⣾⣿⣏⡉⠉⠀⠀⠀⠀⠀⢸⣿⣇⣴⣿⠏
⠀⠀⠀⠀⠀⢹⣿⣿⠦⢸⣿⣿⣤⣿⡿⢹⣿⣧⣤⣿⡿⣼⣿⠿⠃⠀⠀⢶⣿⣿⡷⢸⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠉⠉⠋⠀⢸⣿⣿⠛⠉⠀⢸⣿⣿⣿⣯⡀⢿⣿⣤⣤⣴⡄⠀⠀⠀⠀⢸⣿⡏⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣌⠛⠋⠀⠀⠀⠈⠛⠃⠈⠛⠃⠀⠙⠛⠛⠋⠀⠀⢀⣀⣀⣘⠛⠁⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣄⣀⣀⠀⣀⣀⣀⣀⣤⡤⠶⠖⠚⠋⠉⠉⠉⠉⠉⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠉⠉⠁⠀⣀⣀⣠⠤⠴⠒⠚⠉⠉⠉⠉⠉⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠋⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⣊⠃⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣠⣴⣶⣄⠀⢀⣼⣿⣿⣷⡄⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⢉⢀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠘⣿⣿⣿⣿⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⣄⢀⡦⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠉⠉⠀⠀⠀⠈⠉⠛⠉⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢈⡨⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢀⠀⠀⢰⣶⠀⢲⡆⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⠄⠊⢀⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⢇⡰⣄⡡⣀⣀⣄⣠⠖⠲⠦⠖⠃⠀⠀⠀⠀⢀⠔⢌⠲⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⣀⣀⠠⠏⠢⠠⠶⠂⠀⢄⢀⠄⠀⠀⠂⠉⢀⡉⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠐⠊⠀⠙⡂⢔⣉⡁⠤⠀⠠⠀⢰⠒⠒⠈⠁⠈⠓⡆⣠⠔⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣉⡣⣀⠔⠂⠉⠉⠁⠀⢀⠀⠠⠖⠈⠀⠈⠉⠉⣉⣉⠵⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡠⠆⠀⠀⠀⠀⠀⠀⠐⠥⠀⠠⠤⠒⡄⠀⣎⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠒⠀⣉⠀⢀⡤⠤⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⣒⡬⠃⠀⠀⠀⠀⠀⠀⠀⠈⣚⡃⠉⠉⢨⢁⡞⢀⣀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢅⡠⠤⠱⢀⠀⠀⠀⠀⢠⠞⠉⣀⠈⠉⠓⠮⠛⠋⣁⣀⡀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡆⢀⡒⠨⠉⠀⠀⠀⢧⠀⣿⣿⣿⣦⡀⠀⣴⣿⣿⣿⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣍⣛⣟⣋⣤⣀⡙⠛⠛⣁⣀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠙⢙⠆⠀⠀⠀⠉⠉⢩⠙⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢦⡑⠦⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⣆⠲⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢔⣉⠕⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⣀⠄⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢄⡈⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠮⢭⡄⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡴⠽⢃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠄⠈⠀⠒⢪⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠉⠁⠀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠄⡌⠑⣒⠒⣒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣑⠒⢚⢍⠁⣀⢢
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠸⠜⠀⢤⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠆⠘⠒⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⡀⣄⣀⡀⣠⣰⡢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣝⢿⣷⣿⡅⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡻⣷⣽⣿⣿⣮⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣘⠛⠻⢿⣿⣿⣿⣿⣿⣮⣻⣿⣿⣿⣿⣩⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡀⠀⠀⣿⡟⠃⠀⠀⠻⣿⣿⣯⡤⠻⣷⡀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡳⣦⣴⣀⣴⢛⣛⡳⣦⡙⣿⡀⢱⣄⡽⣿⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣟⣼⣧⠘⠻⢃⣿⣿⢹⣷⠛⠛⡄⠘⠆⠳⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⠁⠙⠛⠛⢻⣿⣿⣴⣿⣧⡤⠛⠠⠀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣤⣄⣄⣠⣬⣵⣾⣻⣿⣿⡗⠀⠱⡀⠅⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣶⣭⣿⣿⣿⣿⣭⣭⡿⣿⣷⠀⢠⠽⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⢿⣿⣿⣿⣷⣿⣈⢉⢈⣦⣷⣾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡙⢦⣈⠛⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠈⠙⠛⠻⠿⠇⠀⠂⠀⠀⠺⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠎⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡎⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢧⠀⠀⡀⠄⠚⠓⠒⠒⠂⢄⠀⠀⠀⠀⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠑⠪⠄⠀⠘⠷⣦⡀⠀⠀⢑⣤⡀⠀⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠙⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠒⠠⠀⠁⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠁⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠒⠒⠂⢠⣽⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⢀⠄⠤⠀⠁⢊⠄⠊⠁⠉⠉⠈⡺⠃⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠉⠂⢄⠀⠀⠀⠀⣸⡦⠀⡀
⠇⢀⣠⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⡟⠓⠠⢀⠑⢄
⠀⠉⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠐⢄⠠
⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠈⠄⠂
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠔⠀⠻⡄
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠈⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣒⣤⣄⣀⣀⣀⣀⣈⣐⣂⣀⣤⣀⣀⣀⣂⣁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠉⠢⢀⠀⠀⠀⠀⢀⠠⠐⠂⠁⠁⠀⠀⠀⠀⠉⠐⠠⢄⠀⠀⠀⢀⡠⠊⠀⢡⡀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠂⠄⡀⠑⠢⡠⠊⠁⠀⠀⡠⠐⠈⠀⠀⠀⠈⠐⢄⠀⠀⠈⠂⢔⠁⢀⠄⠊⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡰⠊⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠈⠂⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⠑⠈⢀⠀⠀⠀⢠⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠰⠀⠀⠀⠐⠀⠂⠀⠐⢧
⠀⠀⠀⠠⠐⠠⡀⠀⠀⠀⠀⡘⠤⢃⠀⠁⠃⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠰⢀⢁⠉⡀⠀⠀⠀⢀⠔⠀⢢
⠀⠀⠀⢁⠀⠀⠈⠄⠀⠀⠀⠃⠀⠅⠅⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠀⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠔⠀⠇⠀⠀⢀⠃⠀⠀⠘⢀⠀⡀
⠐⠁⠈⠐⢦⡀⠀⢰⠀⠀⢀⡤⠐⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠑⠠⢄⠀⠀⢸⠀⠀⡴⠋⠀⠀⠈
⢇⠀⠀⠀⠀⠙⠀⠈⠀⠈⠁⡠⠂⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠢⡀⠈⠁⠁⠀⠈⠀⠀⠀⠀⡠
⢀⠁⠖⠂⠤⠄⠀⠀⢀⠔⠊⠠⠀⡈⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠈⡀⢀⠁⠑⠤⡀⢀⣀⠒⠐⠛⠁⠂
⢀⠀⠀⣀⣠⠂⠀⠈⠑⠠⡀⠈⡤⠁⠰⠀⠀⠀⠀⠀⠘⠁⠒⠠⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠘⠀⠐⡌⠀⡠⠐⠉⠀⠀⢱⠦⠄⣀⡠
⠀⠈⠀⠀⠄⠂⠄⣀⡀⠀⠈⠒⠁⠀⡀⠀⠀⠀⠀⠀⠃⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠊⠀⢀⣀⣠⠴⠡⠂
⠀⠀⠀⠀⢀⠂⠉⠉⠉⠉⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠐⡀⠀⠀⢀⡠⠔⠀⡄⠰⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⢠⠀⠒⢤⡀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠉⢁⠋⠀⠀⡸⡀⠀⡁⢀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢘⠀⢀⢆⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠢⠠⠔⠀⠀⠁⠀⠈⠀⢀⠉⠒⠠⠄⠀⢀⣀⠀⢀⣀⣀⠀⠀⠤⠰⡊⠀⢁⠀⠀⠈⠁⠀⠑⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⡀⠐⠤⣴⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣈⣠⠐⢂⡒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠁⠀⠀⠀⠀⣀⡀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠁⠀⠀⠀⠈⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠢⠀⣀⡀⠌

⢀⣤⠀⣔⣃⡠⠤⠤⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡀⠀⠀⠀⡴⢦
⠘⢬⡎⠉⢀⡄⡀⠀⠔⠩⢍⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠙⣦⠎⢱⣠⣄⠀⠀⠀⠀⠀⠀⠀⡎⠀⢱⠀⠀⢰⠇⢸
⠀⠀⡀⢠⢬⡀⠁⣀⣃⣈⣁⣀⠀⠀⠀⠀⠀⢀⡖⠢⠴⠃⠀⠋⠀⠘⠁⢸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠉⠉⠀⠈⡆
⠀⠀⡇⣷⠀⠙⠫⠿⡼⣇⠧⠟⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⢠⡎⠰⠰⠆⠰⣂⣒⡏
⠀⠀⡇⠙⢄⣀⡀⣲⡐⡤⣂⠔⣆⠀⠀⠀⠀⣘⠰⣍⣱⣀⠝⠀⠯⠮⠝⢺⡴⡄⠀⠀⠀⠀⢸⠘⢞⣣⠛⣤⢣⠟⣜⣧
⠰⣆⡇⠀⠀⠀⠉⠉⠉⠉⠁⠀⣇⣴⠆⠀⠀⡏⠐⠲⠿⠶⠷⠮⠷⠾⠽⠞⠒⠛⠳⡄⠀⠀⠸⡄⠀⠀⠀⣀⣁⣁⣀⡠⠁
⠀⠉⠑⠤⣀⡀⠀⠀⠀⠀⣀⠴⠁⠀⠀⠀⠀⣧⡤⢤⡀⣀⣀⡀⢀⣀⣀⢀⣀⢀⣀⡝⠀⠀⠤⢵⠀⠀⠈⠉⠉⠀⢠⠅
⠀⠀⠀⠀⠀⠡⡌⠉⡽⠏⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠚⠀⠀⠣⠏⠉⠙⠼⠙⠞⠉⠃⠀⠀⠀⠀⠣⠜⠀⠀⠈⠻⠍
⠀⠀⠀⠀⣀⡄⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡐⡄
⠀⠀⠀⡼⠁⣁⠐⢁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢡⠍⢲⠼⠶⠞⠁⢠
⠀⠀⢀⠗⠀⠉⠀⠈⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⡀⠀⠀⠀⠑⠶⣾⢁⣄⢀⡄⢀⣀⡙⡽⠄
⠀⢠⠃⡤⢤⡦⡄⢰⠱⢋⠎⡆⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⡞⠁⠀⠀⠀⠈⢢⡀⠀⠀⠀⢸⠈⠉⠉⠙⠛⠣⠯⢼
⠀⡎⢀⡉⣉⠉⣁⡀⣩⣁⢄⡇⠀⠀⠀⠀⠀⢰⢻⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠇⠘⠚⠤⠿⠤⠿⠴⠯⠜⢳⠀⠀⠀⠀⠀⠈⡎⢧⡀⢀⠇⠀⠘⡗⠀⠀⠀⠘⠦⡠⠤⠃⠀⠀⠀⠀⠀⠀⢰⠃
⡰⢋⣀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢉⠆⠀⠀⠀⠀⠙⢄⡉⠁⠀⣠⠇⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠑⠉⠈⢲⠄⣀⣀⡀⢠⡀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⡉⡦
⠀⠀⠀⠘⠜⠁⠀⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠌⠙⡖⢢⠤⠤⠀⠒⠒⠛⢄⠜
⠀⠀⡀⠀⢀⣴⣧⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠉⠉⠀⠀⢀⡄⡀⢠⠀⠤⠤⠤⠤⠤⣄⣄⣠
⠀⠈⢯⠒⢁⡤⡄⠀⢏⣂⠶⣀⡠⢤⠎⢳⡀⠀⠀⢀⠔⠒⢤⠖⢲⠎⣀⠉⢱⡀⠀⠀⠘⠋⠸⢠⡪⠏⣱⠀⡖⠶⠢⢸⠈
⠀⠀⠀⡇⠘⠤⠂⠀⣠⢲⡶⠭⠧⠚⠒⠁⢸⠀⠀⢺⡘⣃⠴⡄⠘⢄⣉⣠⠜⠁⠀⠀⠀⠀⡇⢀⢩⡭⣄⢤⢈⣭⡥⠚
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⡞⣷⢢⡜⡏⠀⠀⠀⠀⠀⢀⠴⠃⠀⠉⠑⠉⠋⢹⣉
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠀⠀⠀⠀⡰⠋⠉⠉⢹⠃⠀⠀⠀⠀⠀⠷⢹⠀⠀⠀⠀⠀⠀⢸⠳⡇
⠀⠀⠀⠈⢙⢲⢲⠀⡎⠈⢳⢸⠉⡏⡟⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⡀⠀⠀⣀⡼
⠀⠀⠀⠀⠸⡍⠸⠀⠃⠀⠸⠎⠀⠥⠀⠀⠀⠀⠀⠀⠀⠀⠱⣲⠖⢜⡄⠀⠀⠀⠀⠀⠀⠀⠀⠵⠉⠉⠹⡍
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢠⡄⢀⢄⢠⡀⣠⡀⠀⠀⠀⠀⣀⣤⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣀⣀
⠀⠀⠀⣠⡾⠛⠉⠛⢷⡀⠀⠀⠀⠐⠷⣤⠶⠿⢿⣧⡀⠀⠀⠀⢀⣈⣿⣿⡾⣾⣿⠯⢄⠀⠀⠀⢰⣟⠀⣦⠀⣹⠇⠀⠀⠀⠀⠀⣀⣠⠞⢿⢻⡿⢷⣶⠃
⠀⠀⣰⡟⠘⠓⠘⠓⣈⢿⠀⠀⠀⠀⢸⡇⢰⣧⡄⣻⣿⠀⠀⠀⠸⣖⣽⡇⠶⢌⣿⣯⢎⠀⠀⠀⣼⠿⠶⠶⠞⢋⢠⣄⠀⢰⡿⣿⢉⣿⡾⣾⣉⡷⣾⠗⢶⣤⣀
⢾⣇⣿⢸⣷⣿⣾⣿⡿⢸⣇⡶⠀⠀⠘⣿⣦⣤⣴⣿⣿⡇⠀⢀⡳⣾⢛⠛⠷⢛⡛⣿⣏⡤⠀⢸⡇⠀⢀⣀⣀⠙⣿⠉⠀⠀⠀⠀⢸⣿⣿⣿⣽⣿⡟⠀⠐⠏⠿
⠀⢸⣿⣦⣬⣤⣥⣤⣴⣿⡏⠀⠀⠀⠀⣿⣿⣿⠠⠆⢹⡇⠀⠈⢻⣿⢐⣄⢀⡈⣁⣿⣿⠁⠀⠈⠻⣿⡿⠿⣿⡟⠁⠀⠀⠀⠀⠀⣿⠟⠉⠉⠻⣿⡇
⠀⠘⠿⢿⡿⢿⡿⣿⠿⣿⠇⠀⠀⠀⣰⣿⣬⣿⣷⣶⣿⣷⡀⠀⢸⡿⣿⠿⡿⣿⡟⣿⣿⠀⠀⠀⠀⢸⡿⠿⢿⡇⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠈⠁⠈⠀⠉⠀⠉⠁⠀⠀⠐⠋⠙⠋⠙⠉⠙⠁⠈⠁⠀⠘⠃⠟⠘⠃⠙⠁⠋⠘⠀⠀⠀⠒⠛⠀⠀⠀⠛⠦⠀⠀⠀⠐⠚⠃⠀⠀⠀⠀⠀⠹⠶⠄
⠀⢀⣠⣤⣄⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡄
⠀⢺⣿⣿⠿⡆⠀⢠⡾⢿⣿⣿⡄⠀⠀⢼⣗⣠⣤⣤⡽⠛⠀⠀⠀⠀⠀⢸⣧⣀⠀⠀⠀⠀⣀⣿⠀⠀⠀⣰⠛⠛⠛⠦⣄⠀⠀⠀⠐⣧⣤⣶⣶⣦⣼⠇
⠀⠀⠘⠁⢀⣽⠶⠾⣄⠀⠉⠁⠀⠀⠀⢀⣾⢟⠛⡛⠿⣦⠀⠀⠀⠀⠀⠀⠈⢻⣿⢿⣿⡿⠛⠋⠀⠀⠀⡟⠐⠂⠰⢠⡏⠀⠀⠀⠀⢸⣿⣐⣂⡿⣿⡀
⠀⠀⠀⠀⣾⠁⢴⡄⢘⣧⠀⠀⠀⠀⢠⣿⣧⣈⠁⢉⣠⣿⣧⠀⠀⠀⠀⠀⠷⣾⠰⠆⢸⣤⠇⠀⠀⠀⢸⣷⣤⣤⣀⣼⠁⠀⠀⠀⠀⠘⡷⢩⢻⣖⣿⣧
⠀⠀⠀⠀⠹⢷⣤⣤⡾⠃⠀⠀⠀⢠⣿⣿⣿⣿⣟⣛⣿⣿⣿⡆⠀⠀⠀⣠⣴⡿⠛⠺⣿⣦⡀⠀⠀⠀⡌⠀⣿⢹⡏⢹⠀⠀⠀⠸⣇⣠⡟⡲⣎⢍⣿⣿⢀⣠⠇
⠀⠀⠀⠀⠀⡴⠁⠙⠦⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⠇⢸⣷⣿⣷⡇⠸⠃⠀⠀⣰⣀⢠⡇⢸⡇⢸⠀⠀⠀⠀⠀⠸⣷⣶⡿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢸⠟⠛⠻⣇⠀⠀⠀⠀⠓⠀⢸⡇⢸⠀⠹⠷⠀⠀⠀⠀⣶⣴⣤⣴⣿⣿⡟
⠀⠀⠀⠀⢀⣼⠿⠻⢷⣶⠃⠀⠀⠀⠠⠏⠩⠏⠹⠏⠹⠍⠳⠄⠀⠀⠀⠀⠛⠀⠀⠀⠹⠀⠀⠀⠀⠀⣀⣤⠇⠸⣤⡀⠀⠀⠀⠀⠀⠈⣛⣿⠿⣟⠋
⠀⠀⠀⠀⢾⡇⠸⡶⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⢀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⢈⣿⠶⢶⡾⠋⠀⠀⠀⠀⠀⠀⠸⣏⣀⣀⡾⠃⠀⠀⠀⠀⠀⢰⢀⣠⣤⣾⣄⠀⠀⠀⠀⣆⠉⢀⣠⣤⣤⣀⣀⡤⠋⢀⠄⢄⠀⠠⣉⠌⠀⣥⣂⠅
⠀⠀⠀⠀⣿⡁⠶⠄⣿⠀⠀⠀⢠⡆⠀⣠⣾⠟⠛⢿⡗⠀⣤⠀⠀⠀⠀⠀⠿⣟⢡⠙⣯⣂⠀⣀⠀⠈⢻⣿⠉⠆⠆⢹⣿⠀⠀⠘⠈⠺⣄⢰⡇⢠⡞⠁
⠀⠀⠀⠀⢨⡿⠒⢾⡇⠀⠀⢀⡘⢷⣴⣿⣯⠐⠷⢀⣷⡴⠋⢀⠀⠠⣤⠘⣵⣿⠶⢻⢿⣿⡴⠋⠀⠀⡜⠛⡶⠶⢶⠾⢏⠀⠀⠀⠀⠀⢸⣾⣧⣿⡀
⠀⠀⠀⠀⢸⣧⣛⣼⠇⠀⠀⠘⢿⣄⣼⣿⠿⣷⣶⣿⠿⣷⡶⠿⠁⠀⠉⠉⣿⡇⠃⠀⡄⣿⡇⠀⠀⢘⡁⠀⡁⠀⠈⠀⢨⡆⠀⠀⠀⠀⢀⣯⣐⣨⣧⡀
⠀⢤⣶⡂⠹⣏⣊⣿⠐⣶⣦⠄⠀⠀⠹⣿⣄⡈⠈⣁⣼⣿⠇⠀⠀⠀⠀⢠⣿⣤⣥⣤⣴⣾⡇⠀⠀⠈⠁⠐⡇⠀⢾⠀⠸⠃⠀⠀⣴⠟⠛⣿⠟⠉⠉⠙⢷⡄
⠀⠈⢀⣀⡜⠉⠉⠙⣆⣀⠉⠀⠀⠀⠀⠙⠿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⣾⠟⣿⠟⣿⡟⢻⡇⠀⠀⠀⢀⣰⣧⠀⢸⣧⣀⠀⠀⠀⠻⡀⠀⢿⡀⠀⠀⠀⡸⠃
⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠈⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀
⠀⠀⠀⣹⣥⣤⣤⣼⡋⠀⠀⠀⠀⠀⠀⢀⣄⢀⣴⣧⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣀⣀⡀⠀⠀⢠⣤⠤⠒⠒⠒⡒⠶⣷⠀⠀⠀⠀⠀⢉⣤⣤⣤⣤⡍
⢶⣄⣾⣿⡿⠿⢿⣿⢷⣤⠦⠀⠀⠀⣠⣈⣿⡿⢿⡿⢿⣿⣀⠀⠀⠀⢸⡏⢍⣩⣭⣩⠏⣿⠀⠀⢸⣧⣀⠻⠀⠘⣛⣠⡿⠀⠀⠀⠀⣰⡟⠩⠄⠦⠙⣿⣆
⠀⣾⡱⢨⣇⠻⢢⣇⣂⣿⠀⠀⠀⠀⢙⣿⡏⠊⢵⣫⢕⢿⡟⠀⠀⠀⠈⡇⢾⣿⣿⣿⠀⡏⠀⠀⠀⠉⠙⠋⠻⡟⠉⠁⠀⠀⠀⠀⠀⣿⣇⠒⠀⠀⣠⣿⣿
⠀⠙⢿⡿⠿⢻⣿⣿⣿⣿⠀⠀⠀⠛⣿⣿⣮⣃⣞⠭⣪⣾⣿⠃⠀⠀⠀⢧⣈⣉⣩⣬⡴⠇⠀⠀⠀⢠⣤⡄⠀⡇⠀⠺⡟⠀⠀⢠⡞⠻⣿⣿⣿⣿⣿⣿⠏⠙⠂
⠀⠰⣶⣾⣷⣿⣿⣿⡿⠉⢢⣀⠀⠀⣿⣿⣿⣿⣿⠿⢟⣿⣿⠀⠀⠀⠀⠀⢸⣍⣥⣼⠀⠀⠀⠀⠀⠈⠉⣄⢠⣿⣄⠴⠁⠀⠀⠀⠀⠀⠈⢻⠟⣿⠛⣧
⠀⠀⣈⡹⠛⠛⢋⡁⠀⠀⠁⠉⠀⢠⣿⣿⣿⣶⣤⣶⣿⣿⣿⣆⠀⠀⠀⠀⠈⣿⠛⢻⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⠀⠘⡄
⠀⠀⠈⠁⠀⠀⠘⠁⠀⠀⠀⠀⠀⠺⠟⠛⠛⠿⠿⠛⠻⠿⠛⠛⠃⠀⠀⠀⠾⠉⠿⠳⠛⠂⠀⠀⠀⠀⠀⣠⠛⠛⠋⢦⡀⠀⠀⠀⠀⠀⠶⠗⢀⣸⣄⠀⠿⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠶⢶⡶⢶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⢠⡞⠁⠀⢸⡇⠀⠙⢷⡀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠀⠀⢠⡟⠀⠀⠀⢸⡇⠀⠀⠈⢷⡀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠛⠶⣤⣀⣾⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠇⢀⣀⣤⡤⠶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢸⡏⠙⠃⠀⠀⢈⣀⣤⡴⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠸⠀⣀⣀⡀⠀⠀⠉⠛⠓⠶⢤⣤⣀⣀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⠀⠀⣠⠶⠛⠉⠉⠉⠛⠳⣄⠀⠀⠀⢸⡇⠈⠉⠛⠳⠶⠞⠛⠉⠁⠀⡧
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠀⣼⠋⠀⣀⠀⢀⡀⠤⠀⠈⢧⠀⠀⣸⠷⠤⢤⣤⢤⣤⠤⠤⠤⠤⠴⠟
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢠⡇⠀⠀⣿⣤⣼⡇⣷⠀⠀⢸⡄⢠⡇⠀⣠⠟⠁⠀⠉⠛⠒⠒⠶⠄⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⣷⠀⠀⠿⠀⠸⠇⠿⠀⠀⣼⠃⢸⡇⢀⣿⠀⠚⠃⠀⠘⠛⠀⣶⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠶⠘⢷⣄⠀⠀⠀⠀⠀⣀⣀⣙⣦⡄⣷⢸⡟⣧⡀⠶⣤⠶⢀⡼⢷⡀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠈⠛⠲⠶⠖⠛⠉⠉⠉⠁⢠⡟⢸⣇⡈⠛⠶⠶⠶⠛⢁⠘⡇⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⡀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢸⡇⠙⢦⣤⣀⣤⡴⠋⢰⡇⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠶⠀⣰⢿⡇⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢨⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠻⢦⣼⠃⠀⠀⠀⠀⠀⠀⠀⠘⣧⡴⠟⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠖⠛⠛⠛⠛⠛⠛⠲⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠴⢴⠟⠁⠀⢀⡤⠖⠋⠉⠀⠉⠀⠀⠉⠉⠛⠲⢤⡀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⢰⣦⣴⣾⣭⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠹⣦
⠀⠀⠀⢀⡴⠛⠳⣦⡞⠉⠙⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⣠⠞⠋⠉⠀⠀⠀⠀⠀⠈⠉⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠘⣆
⢠⡴⠖⢾⡇⠀⠀⢸⡇⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣻⣆
⣿⠀⠀⠘⣧⠀⠀⠀⣷⠀⠀⠀⢷⣤⡤⣤⡀⠀⠀⠀⢸⠇⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⢀⣾⠀⢀⡀⢰⣦⢀⣿⠹⣆
⢸⡆⠀⠀⢻⡄⠀⠀⢹⡆⠀⠀⠘⡇⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣞⣁⣀⣠⠞⢻⣀⡾⣧⣸⡏⠛⢻⡆⢹⡄
⠈⣧⠀⠀⠈⣧⠀⠀⠈⣃⠀⠀⠀⢹⠀⠈⡅⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⡾⠛⠁⠀⠀⠀⠀⠀⠉⢩⣍⡁⠒⠛⠉⠀⠙⠋⠀⠀⠀⣿⠈⣷
⠀⢻⡀⠀⠀⠿⠀⠀⠀⠈⠀⠀⠀⠈⣧⠀⢹⡆⠀⢘⡇⣀⡴⠟⠁⠀⠀⠠⠶⢟⣛⣻⣭⣍⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢸⡆⢿
⠀⠘⡇⠀⠀⢀⣀⣤⣤⠶⠶⣶⠟⠀⠀⠀⠀⢻⡞⢻⡟⠉⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣙⠻⣦⣄⠀⠀⠀⠀⢸⡇⢸
⠀⠀⢿⡀⠀⠉⠉⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⡾⠀⠀⢀⣴⣿⣿⡇⠀⠸⡆⠀⠀⠀⠀⢀⡴⠋⢁⡀⠀⠉⠙⢦⠉⠀⠀⠀⠀⣸⠀⣿
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢸⣧⣸⡇⠀⠀⠀⠀⠀⠀⣇⠀⠀⠸⣿⣿⣿⣿⣾⠀⡇⠀⠀⠀⠀⡟⢀⣾⣿⣿⡇⠀⠀⠈⣷⠀⠀⠀⠀⠛⢶⣏
⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣸⠇⢹⡇⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣙⠛⠿⠛⢁⡼⠁⠀⠀⠀⠘⡇⠸⣿⣿⣿⣿⡟⠀⠀⣿⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠈⣆⠀⠀⠀⠀⢀⣀⣽⡶⠶⠒⢲⣏⠁⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠈⣤⣈⣛⣛⣋⣀⣀⡴⠃⠀⠀⠀⠀⠀⣾⢀
⠀⠀⠀⠀⠘⠳⠶⠚⠻⣏⠉⠀⠀⠀⠀⠀⠙⢷⡀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⢡⣤⡀⠀⠀⠀⠀⣰⣷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠙⢦⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠦⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⠙⢦⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣠⡴⠞⠋⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⡀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⣤⣀⣀⠀⠙⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⠶⢼⡟⠛⠷⣶⠦⢤⣤⣤⡤⣶⡶⠖⢻⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠳⠦⣤⠴⠞⠋⠀⠀⠸⡏⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⣴⣶⣤⡀⠀⠀⠀⣿⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢶⢾⣿⣿⣿⣿⠀⠀⠀⢹⡆⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠈⠉⠁⠘⠋⠉⠀⠀⠀⠸⡇⠀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣾⣇⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠛⠛⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⣿⡄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣧⣤⣀⣀⣀⢀⡼⠋⠉⠉⠈⢻⣄⣀⣀⣀⣀⣬⣿⣤⣠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠉⠉⢹⠂⠀⠀⠀⢸⡉⠉⠉⠉⠀⠀⠀⠘⡯⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣤⣤⣤⣄⣀⣀⡿⠀⠀⠀⠀⠈⣧⣀⣠⣤⣤⣤⣤⡴⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢰⡏⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⢧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠶⠛⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠚⠷⢶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢯⠲⣄⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣀⡤⠴⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⠳⣩⢷⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⠞⢧⣹⠞⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⢃⣀⣙⣲⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣖⣋⡀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡞⠋⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠹⡄⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⢠⢤⡖⢲⢀⣀⡀⠀⠀⢹⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡏⠀⠀⠀⠀⠀⢀⣄⡀⡀
⠀⠀⠀⠀⠀⠀⢰⠾⡌⠳⠘⢋⣨⠇⠀⠀⠘⡇⠀⠻⣤⣤⠴⠶⠒⠲⠦⣴⠃⠀⢸⡇⠀⠀⠀⠀⡖⠻⠸⠋⣿⡄
⠀⠀⠀⠀⠀⠀⠈⠲⠶⣄⠀⠈⢧⡀⠀⠀⢀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⢀⣽⠆⠀⡴⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠈⠳⢤⡞⠀⠀⣠⣤⢤⡤⢤⡤⠴⣞⠉⡹⠉⠓⡄⠈⠷⠖⠋⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⢠⠎⠀⠳⠞⠳⠞⠑⠚⠉⠉⠁⠀⠀⢸⠀⢀⣤⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⠀⠀⠀⢀⡤⢤⣀⡖⠓⠲⡄⠀⠀⡸⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⢇⠀⢠⡇⠀⠀⠘⠁⠀⢀⣹⠀⢠⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⢦⡀⡷⢄⡠⢦⡖⢲⣏⣈⠶⠋⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠉⠓⠶⠓⠋⠉⠉⠉⠁⠀⠀⠀⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣀⣀⣀⣀⣀⣠⠤⠖⠒⠦⣄⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡼⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⢹⣄⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣀⠲⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠧⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢤⣤⡼⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣢⠂⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠒⢦⠀⠀⣼⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⡇⠀⢸⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢀⡞⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢡⠀⠘⡇⠄⠀⠀⠸⡀⠀⠀⠀⠀⢸⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠘⡄⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⣼⠀⠀⠀⠀⡇
⠀⣠⠖⠶⣄⠀⠀⠀⠈⡇⠀⠀⠀⢣⠀⢳⠀⠀⠀⠀⢻⠀⠀⠀⠀⡏⠐⠀⠀⠀⡇
⠀⡇⠀⠀⠘⡆⠀⠀⠀⡇⠀⠀⠀⠘⡆⢸⢀⠀⠀⠀⠈⡇⠀⠀⢀⠃⠀⠀⠀⠀⡇
⠀⢳⠀⠀⠀⠹⡀⠀⠀⢳⠀⠀⠀⠀⢳⢸⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⠀⠀⠀⢸⠃
⠀⠈⡇⢀⠀⠀⢱⠀⠀⠈⡀⠁⠀⠀⠘⡏⡀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⢸
⠀⠀⠘⡄⠀⠀⠀⢣⡀⠀⡇⠀⠀⠀⠀⡇⢃⠀⠀⠀⠀⡇⠀⢠⠇⠀⠀⠀⠀⡄
⠀⠀⠀⠹⡀⠀⠀⠀⠧⠀⢹⠀⠀⠀⠀⢣⣼⠂⠀⠀⠀⠿⠤⠜⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠹⡀⠀⠀⠘⡄⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠂⠉⠉⠉⠲⡀
⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠿⠶⠋⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊
⡀⣀⡀⣀⡀⣀⡀⢀⡀⣿⣄⣄⣀⣀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⢢⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠁⠀⠈⠉⠉⠉⢙⠳⠆⠀⠀⠀⠀⣤⠞⠁
⠃⠀⠀⠐⠀⠂⠐⠀⠁⢺⠋⠀⠀⠀⠀⢲⡆⡠⠈⠀⠀⠀⠀⠀⣿
⠒⠚⠒⠛⠒⠒⠚⠐⠁⣿⠓⠓⠊⠒⠙⠘⠓⠒⠚⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⢀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣄⠀⠀⡀⠀⠀⣠⡦⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⣰⠏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣾⠁⠸⣆⡼⢳⣤⠞⠁⢹⠀⡀⢠⠞⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢦⡀
⠀⠀⠀⠀⣰⠏⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠸⣯⠀⠀⣿⠃⠀⠟⠀⠀⣾⠾⣷⡎⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢻⡄
⠀⠀⠀⠀⡟⠀⠀⠀⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⢉⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠻⠁⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠈⠹⡄
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⢀⣠⡴⠶⠛⠛⠉⠉⠁⠀⠀⠀⠀⠈⠉⠙⠛⠶⠶⣤⣼⡁⠀⠀⠀⠀⠀⠀⣠⡴⠞⠃⠀⠀⠀⣇
⠀⠀⠀⠀⣿⠀⠈⠀⠀⠀⠠⠀⠀⣠⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢦⣄⣤⠶⠋⠁⠀⠈⠀⠀⠀⠀⡽
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢀⠀⠀⠀⠐⠀⢰⠇
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣄⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⡶⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⡤⢴⡶⠂⠘⠛⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠠⠀⠀⢹⠉⠉⠙⣿⠛⠛⠛⣿⠟⠛⠛⢻⡟⠛⠉⢹⣏⠉⠉⠙⣿⡀⠀⠀⡇⠀⠀⠀⣉⠀⢈⠀⣿
⠀⠀⠀⠀⠀⠀⠈⡆⠐⠀⠐⠀⠀⢸⣄⣀⡴⠛⠶⠶⠶⠛⠦⣤⣤⡾⠻⣤⣀⣴⠟⠲⠶⠚⠁⠙⠓⠛⠁⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⢻⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢧⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢸⡇⠀⠀⠀⠉⠙⠛⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢤⡤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠀⠀⠀⠀⠀⠉⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠙⣦
⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢧
⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢳⡀
⢠⡟⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠳⡄
⡜⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹
⡇⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⢸⡀
⢳⡀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠙⢦⣀⠀⠀⣠⠞⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠈⠉⠉⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣯⠙⢦⣀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠉⠳⠦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⠙⠢⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠉⠛⠲⠶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠶⠖⠛⠉⠁⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠉⠉⠉⠉⠉⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⣀⣀⣤⠴⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⠶⠶⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠒⠛⠒⠲⢦⣄⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⣤⣾⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣀⣤⣄⣀⣠⣤⠶⠛⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠛⠳⢴⣧⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠛⠙⠻⢿⣷⡶⠋⠉⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣭⣩⣉⣭⣄⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⠤⠬⠭⠭⠤⠤⠤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⡆⠀⠀⠀⠀⠀⠀⣶⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⢀⡀⠀⠀⣿⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⢠⡄⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠻⣄⣠⡇⠀⠀⢸⡇⠀⠀⣿⣄⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣴⡇⠀⠀⢸⡇⠀⠀⢻⣤⡼⠃
⠀⠂⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢸⡇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣼⣧⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠾⠧⠶⠞⠁

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤⠄⠀⠀⠠⢀⠀⠀⠀⢀⠄⠀⠀⣀⠀⠀⠒⢄
⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⢼⣿⣿⠃⠀⠀⡐
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⣉⠥⠥⠒⠉⠁
⠀⠀⠀⠐⠀⠀⠀⣾⡇⣠⣿⠏⠶⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡄⠀⠀⠀⣽⣿⠛⣿⣰⡆⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠇⠀⠀⠀⢻⡇⢀⣿⢻⡇⠀⠀⡔
⠀⠀⠀⠐⠀⠀⠀⠘⠃⠈⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⣴⡆⠀⣶⣦⠀⠀⢀⡇⣸⠁⡀⠀⠀⠀⢰⣂
⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⡏⣷⣴⣆⣸⠃⠉⢾⡟⢸⣾⣇⣿⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠤⠤⠤⠄⡆⠀⠀⠀⠀⠀⣿⡏⢻⡇⠇⠟⠉⠙⠟⠁⠘⠃⠘⠋⠉⣿⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠇
⠀⠀⠀⠀⠀⠀⢸⣿⠾⣿⢰⡆⣰⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⡄⠀⣠⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢷⠾⠷⠋⢡⣾⠛⠻⠆⠀⠀⠀⠀⠀⣼⣧⡾⣡⡀⣿⢿⠇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⠀⠀⠀⠀⠀⠀⠀⠀⣿⢁⡏⣏⣠⢯⡾⣠⠧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡀⢀⠏⠳⣄⠀⡰⢆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠒⢄⣘⣀⣱⣮⣤⣦⣬⣎⣥⣘⡠⠞⢹⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⢒⣾⠼⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠳⢼⡤⠒⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣮⢔⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢧⡎⠤⠴⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠐⠒⢄⠀⠀⠀⠀⡇⠀⠀⠻⣎⣰⡡⡆
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⣤⠀⠀⡇⠀⠀⢰⡇⠀⠀⢠⡌⢳⣠⣃⠴⠀⠀⠀⠀⢀⡠⢴⡄
⠀⠀⠀⠀⠀⠀⢸⠃⢀⡔⠂⠐⠢⡀⠀⠀⠀⠀⠀⠣⡀⠀⠀⢀⠜⠁⢀⡴⠫⠵⠀⠀⠀⢰⣶⠙⢾⣼⠔⣆⣄⢠⠬⣧⡟⡇
⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⣴⠄⠀⡇⠀⠀⠀⠀⠀⠀⠈⠙⠉⢁⣠⠶⢉⠀⠀⠀⠀⠀⠀⠀⢩⣤⡛⣙⡳⢯⣬⣮⣴⠞⡅⡃
⠀⠀⠀⠀⠀⠀⢸⡀⠈⠣⠄⠠⠔⠁⠀⠀⠀⠀⠀⣀⣠⢴⠚⠋⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠛⢡⠛⣡⣷⣤⠁⢻⡃⠀⡇
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⣀⣀⡤⡴⢖⢛⠉⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⢈⣛⡋⣤⡝⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠉⠉⠙⢭⠁⠑⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠿⠃⠈⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣠⠞⣰⠶⠒⠲⠦⣤⡀⠀⢀⣀⣀⣤⠤⢤⡀⠀⡞⠁⠀⢈⡇⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⡼⠁⠀⠀⠀⠀⢈⡇⣼⠋⠉⠀⠀⠀⢀⡷⠀⢹⡀⠠⣯⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣎⣒⣛⣀⡿⠀⠀⠀⠀⣶⣋⣡⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠉⠀⠀⠉⠀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⢂⠀⠔⠂⠀⠐⠒⠁⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠎⢐⠢⢀⡀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢸⡀⠀⠀⠀⠰⢄⡀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢤⠀⠀⡂⡄⠀⠀⠀⠀⠔⠀⠁⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠌⠚⠐⠠⡷⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠡⡟⠀⠂⡤⢤⠃⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠀⠀⠉⠐⠒⠃⠀⢅⢀⠀⠀⣸⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠦⠤⡬⠃⠀⠀⠀⢀⠀⠀⠒⠒⠒⠒⠒⠒⠂⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠣⡄⠂⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠠⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠂⠁⠈⢂
⠀⠀⠀⠀⠀⠎⠈⠁⢀⠀⠀⠀⠡⠀⠀⠀⠈⠐⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠠⠐⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡴⢧⠀⢰⢦⡀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⡠⢰⡯⡔⠁⠀⠀⠠⠹⣂⠀⠀⠀⠀⠀⠀⡆⠀⠁
⠀⠀⠀⠀⢠⠰⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠔⠁⢀⡩⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠀⢃⠀⠀⠀⠀⣀⠤⠐⠂⠈⡠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠑⠄⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢀⠀⠀⢆⠀⠀⠐⠀⠀⣠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠠
⠀⠀⠀⠀⠈⠀⠀⠀⠢⠤⠓⠂⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠠⡀⠀⠀⠀⣀⣀⣀⣀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⠉⠂⠈⠁⠀⠈⠙⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⠲⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⢠⠑⠄⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠌⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠴⠶⠦⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣽⣶⣄⣀
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠉⠉⡶⢤⣀⡰⠋⠀⠀⠀⠙⢯⡻⣦⣀⡀⣀⣀⣀⡀
⠀⠀⠀⢠⠞⣠⠖⠋⠉⠉⠉⠙⠓⠶⣄⡀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣳⡌⢻⡙⢯⠀⠉⠉⠻⣦⣄⡀
⠀⠀⢠⢏⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⡀⢧⠈⢣⠀⠀⠀⣷⠀⠙⠳⣄
⠀⠀⣾⡟⠉⠙⡆⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣧⠸⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⢷⡀
⠀⢠⢟⠀⠀⢠⣿⣿⣿⣿⣿⣷⠀⠀⡠⡄⠐⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠞⠉⣿⠀⠂⠀⠀⠀⠀⠇⢀⠀⠀⠀⠀⠙⣦⡀
⠀⡜⠈⠒⠒⣿⣿⣿⣿⣿⣿⣿⡇⠁⠈⠈⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢩⠏⠀⠀⣻⡇⠀⡆⠀⠀⠀⠀⠸⠀⠀⠀⠠⡀⢩⠳⡄
⠀⡇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣾⠀⣸⠁⠐⠁⣀⠴⠧⣄⠀⠀⠀⠁⠸⠀⢻
⠀⢷⠀⠀⠀⣿⣿⣿⣿⣿⣉⣿⠁⠀⠀⠀⠀⢣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢀⡟⣰⠟⠋⠉⠉⠁⠀⠀⠈⣯⠁⠀⡇⠀⠀⢸⡆
⠀⠸⡆⠀⠀⠘⠿⣿⣿⣿⠟⠡⢳⣿⡆⣠⣧⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⡧⠀⠀⢸⠃
⠀⠀⢻⣆⠀⡐⠀⠀⠀⠀⠀⠀⠈⢉⣴⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⡇⢠⠂⡞
⠀⠀⠀⠻⣝⠦⣄⣀⣀⣀⣠⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠐⠀⠀⡼⠁
⠀⠀⠀⠀⠈⢷⣄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢁⡴⠋⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠰⠀⠀⠀⠀⠀⠀⢀⣧⣀⠤⣀⠀⠀⣠⡴⠋⢀⡔⠋⠁⠈⠉⠛⠲⠶⢶⣖⣚⠉⠁⠀⢔⣠⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⣀⡀⠀⢀⣀⣀⣤⡤⠿⠛⠁⠀⢀⡎⠀⣴⡯⢤⣄⣀⠤⠀⢀⣀⣤⣤⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣸⠀⣸⠁⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣌⠳⢤⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠰⡀⢀⠂⠀⠀⠰⠀⠀⠀⠂⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⣵⠁⢸⠀⠀⢸⠀⠀⠀⢇⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⠀⠀⠀⠀⣀
⠀⠁⠠⠇⠀⠀⠀⠟⠀⠘⠀⠀⢸⠀⠀⡇⢸⠁⢸⠀⠀⠀⢸⠀⠃⢀⠆⠀⡀⠀⠀⢀⠄⠀⠀⠀⠀⠈⠃
⠀⠀⠐⡡⠄⠀⠀⠀⡀⠀⡆⠀⢸⡀⠀⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⠀⢠⠀⠰⢦⠈⠀⠀⢀⣀
⠀⠀⠀⠈⠀⠀⠀⠐⡅⠀⠿⠀⢸⠁⠀⡇⠀⠀⠘⠀⠀⠀⢸⠀⠀⠈⠀⡂⠀⠀⢼⠀⠀⠀⡀
⠀⠀⠀⢽⡀⠀⠀⠀⠸⠀⠀⠀⠜⠀⢀⠇⠀⡀⢀⠀⠀⠀⠘⠀⠀⠀⢠⠁⠰⡀⡀⠀⠀⠀⠃
⠀⠀⠀⠈⠪⠒⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠚⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠁⠀⠀⠀⢠⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⡀⠀⠂⠋⡀⠀⠀⠐⠂⠀⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠈⠠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠤⠀⢀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣿
⠀⠀⠀⠀⣠⣾⡿⢻⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣿
⠀⠀⠀⣰⣿⣟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣿
⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠈⠁⢈⣍⡙⢻⣿⣿⣿⡇⣿
⠀⠀⠘⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠈⣿⣿⣿⠃⣿
⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣜⢹⡇⠀⠀⠀⠀⠀⠀⠀⢸⠙⣷⠀⠿⣿⡟⠀⣿⠀⠀⠀⣀⣀⣀⣀⣀⣀⢀
⠀⠀⠀⠈⣿⠿⠻⢿⣿⡇⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⢈⣸⠇⠀⢸⡇⣠⠞⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⡇⠈⠙⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⢸⠋⠀⠀⠈⠛⠁
⠀⠀⠀⠀⢙⣦⣶⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠈⠉⠀⠀⠀⣀⠀⠀⢸⣷⣶⣦⣤⣤⣀
⠀⠀⣠⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣤⣤⣤⢤⡤⠴⠚⠁⠀⠀⣾⣿⣿⣿⠃⠀⣽
⠀⣼⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠹⣍⠛⠛⣿⠀⠀⠀⠀⠀⣸⣿⣿⠛⠳⣤⠴⢧⡀
⢺⣿⣿⣿⣿⣿⣿⣿⠏⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠁⠌⠀⠀⠀⠀⣠⡾⠛⢿⡇⠰⢤⡏⢀⠞⢻⡆
⠙⠿⢿⣿⡿⠿⠛⠁⠀⠀⠀⠙⣳⡦⣤⣀⣀⣀⣀⣀⣀⣀⣤⠴⠺⣧⡀⠀⠈⣷⠀⠈⠙⠻⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠁⠀⠀⠀⠉⠛⠻⠿⠿⠛⠁⠀⢀⠀⢙⣦⡞⠁⠀⢠⠖⠚⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠹⣆⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠟⠀⠀⠀⣰⠏
⠀⠀⣀⣤⣤⣠⡤⠞⠁⠀⢀⡴⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢤⣤⠞⠁
⠀⠸⣇⠀⠀⠀⠀⠀⠀⡴⠋⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠸⣟⣛⣻⢿⡄⠀⢹⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⠉⠀⠀⠙⠛⠋⠀⠀⢻⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣶⡖⠀⠀⠀⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⠚⠛⢻⡇⡟⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⠀⡇⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⡼⢧⣤⣤⣼⠀⣷⣤⣤⣴⢧⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣉⣙⣃⣛⣁⣀⣀⣸⣆⣇⣀⣀⣈⣛⣘⣋⣉⣳⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣷⣶⣶⣶⣶⣴⣶⣶⣾⣋⣷⣶⣶⣶⣶⣶⣶⣶⣾⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠼⠆⠀⠀⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⡀⠀⠀⠀⠈⢣⡀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠱⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡗⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⢴⣾⣛⠛⣲⠶⢦⡤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⣴⠁⢀⡈⢻⡃⢴⣆⣱⠀⠈⣉⣉⠓⠒⠶⠤⣤⣀⣀⣀⣀⡤⠎⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⡀⠤⢷⠀⠀⠀⠻⣄⡘⣷⡟⠙⠾⠛⢁⡴⠋⠁⠉⠛⢦⠀⠀⠀⠈⠉⠉⠁⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⡠⠄⠦⡀⠱⣄⠀⠀⠀⠀⠈⣧⠀⠀⠀⠈⠽⢋⣀⠀⣀⡀⢾⡁⣾⣏⡆⠀⢸⡇⠀⠀⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠀⢀⡤⠖⢾⠁⠀⠀⠘⢄⠈⠛⠒⠚⠛⠉⢹⡆⠀⠀⠰⡅⠈⢉⣀⣉⡁⠸⢧⡈⠉⠀⢀⡼⡁⠀⢀⣠⡖⠋⠁
⠀⠀⣰⠋⠀⠀⢈⡳⠶⠖⠀⠈⠳⣄⠀⠀⠀⠀⠀⡇⢰⠂⠀⣠⠞⠉⠀⠀⠈⣷⣄⠙⠛⠛⠛⠊⢁⣴⠋⠀⠈⠉⠉⢀⡴⠁⢀⠔⠊⠉⠙⠂
⠀⠀⡏⠀⣀⡀⠀⢻⠀⠀⢀⡀⠀⠈⡆⠀⠀⠀⠀⡇⡼⢠⣾⣥⣤⣤⣴⣤⣴⣿⡈⢦⠀⠀⠀⠀⡼⠉⠛⠲⠶⠶⠒⠉⠀⢰⠃⠀⢀⠀⢁⡞
⠀⠀⠣⡜⠉⣟⣠⢞⣀⡀⢨⡟⢦⠀⢸⠀⠀⠀⠀⣷⣦⡿⠿⡿⠿⣿⣿⣿⣿⣿⣿⣿⡇⠆⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠈⠙⠉⠙⢦
⠀⠀⠀⠀⢰⠁⠙⠺⡄⠗⠋⠀⢸⠀⢸⠀⠀⠀⢰⣿⡿⠇⠂⠀⠠⠤⢤⣀⣀⣿⠋⠻⣿⣸⠀⣸⠣⡀⠀⠀⠀⠀⢀⡴⠃⠀⣠⠄⣀⡴⠃⠀⠀⡆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⡞⠀⠀⢧⠀⠀⢸⠿⠀⠀⠀⠀⠐⠶⠦⠤⠈⠉⠓⠢⣌⣿⢀⡿⣄⠈⠙⠒⠒⠋⠁⠀⢀⣴⡅⠄⠻⣄⣀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠙⠒⡿⠀⠀⢀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⣿⠸⡇⠉⠳⠦⣤⣤⣤⠤⠖⠋⠀⠙⠦⣤⡤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣠⡾⠁⠀⣴⠋⠈⡇⠀⠀⣰⠋⠙⢷⣄⠀⠰⠇⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠛⠉⢀⣠⠞⠁⠀⠀⡇⠀⢀⡏⠀⠀⠀⠙⢦⡀⠀⠐⣄⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⣠⡴⠚⠋⠁⠀⠀⠀⢸⠀⠀⢸⠃⠀⠀⠀⠀⠀⢻⡄⠀⠘⣦⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⠙⡄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⢹⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢐⡇⠘⡆⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢻⡄⠀⢳⡀
⠀⠀⠀⣀⡤⠴⠶⠤⠤⣤⡾⠀⠀⣰⣅⣀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣧⠀⠀⠀⠀⠀⢸⡀⠀⢸⡃⠀⠀⠹⣄⠀⠙⠦⠤⠠⠒⠒⠒⠒⠦⢤⡀
⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⣠⡤⢄⣀⣼⠄⠀⠀⢸⣀⣀⡀⣀⡤⠼⠧⠀⠘⢧⣄⣀⢀⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡗
⠀⠀⠳⣤⣤⡽⢦⣀⣀⠀⣀⣴⣄⣀⣸⣤⠞⠀⣞⠁⠀⢀⡀⠀⠀⠀⠀⠀⢤⡀⠉⠉⣳⠀⢰⡄⠀⠀⠀⠀⢍⣳⡀⢀⣤⡀⠀⣎⣀⡼⠓⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠈⠁⠀⠀⠀⠹⠦⡴⠞⠳⣄⣀⣀⠂⠀⢈⡷⠶⠚⠛⠶⠺⡄⠀⠀⢀⠀⢀⡟⠈⠉⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠛⠒⠚⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢲⡀⢀⡤⠖⠊⠉⠉⠉⠒⠢⣄⠀⡠⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠉⠉⠀⠀⠀⣀⣀⣀⣀⣀⠀⠉⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀⠔⠊⠉⠀⠀⠀⠀⠀⠙⣆⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢰⠃⣀⣀⠀⠀⢀⠔⠋⠉⠳⡜⡆⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡇⡎⣀⠀⢱⠀⢸⡀⠲⠂⢀⠇⢹⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢰⠀⠑⠬⠤⠾⣄⢀⢝⠒⢒⣁⣀⠈⡇⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢸⠀⢀⢤⠤⢴⢥⢤⡪⠴⣅⠼⠈⢷⡇⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡜⣴⠁⠀⠑⠃⠀⠀⠀⠀⠀⣀⣀⣾⢣⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⣼⠃⠹⣗⢄⣀⢄⢀⠴⠒⠒⣦⢗⡵⠁⢸⠀⠀⠀⠙⢤⣀⣀⣴⣧⡀
⠀⠀⠀⢀⣦⣀⡠⠞⠁⠀⢀⡼⢹⠀⠀⠈⠑⠪⠽⠿⠶⠾⠭⠒⠁⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⢹⠅⠀⠀⠀⢀⡤⠊⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠑⠤⣀⣀⡠⠔⠋
⠀⠀⠀⠈⠑⠒⠒⠋⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⢰⢾⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠈⠦⣀⠀⠀⣠⠊⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡾⠏⠀⠀⠻⠶⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠾⠀⠀⠀⡄⠀⢀⠀⡄⢀⠀⣀⡄⠀⠀⣷⣣⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣤⣀⣤⣼⠈⠛⠈⠁⠈⠙⠁⢧⣤⣶⠟⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣽⡿⠋⢀⣀⠀⠀⠀⠀⢀⣀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢡⣾⠁⠀⠉⢢⢠⣾⠁⠀⠉⠂⠀⠀⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠻⣿⣄⢰⡷⡸⢹⡿⣄⢾⡆⡜⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⠗⣤⣝⣓⣒⣋⣁⣀⣙⣓⣒⣫⣤⣆⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡀⣿⣿⣦⣿⣦⣿⣼⣿⣼⣷⣾⣿⡿⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⢻⣿⣧⠻⣽⠻⡿⢿⡿⢿⣿⡿⣿⣛⠟⠁⠀⢀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⣸⣿⣿⠗⠬⠉⠛⠚⠓⠒⠛⠛⠉⠁⠀⠀⠀⢸⡇⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⢨⣿⡆⠀⠀⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⢘⡁
⠀⠀⠀⠀⠀⠀⠀⠸⢻⣷⣠⣤⣸⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢟⣴⣀⠎⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠯⠀⠙⣸⡿⠿⠿⣷⡀⠀⠀⠀⠀⠀⠀⣠⠄⠰⢌⠃⠠⠿⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢹⢿⣿⣶⣶⣴⣷⣿⣿⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡽⠷⠄⠀⡂⠀⠈⠉⠉⠉⠀⠈⣿⡧⠄⠰⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣿⣶⣷⣾⣿⣶⣷⣶⣶⡶⢶⢶⣶⣾⣿⣿⣿⣿⣿⣷⣦⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠀⠃⠀⠀⠀⠀⠠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣤⣀⠀⠀⠀⢀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⣿⠀⠀⣶⠒⠒⠂⠀⣶⠄⠀⠀⠀⣶⠀⠀⠀⠀⢠⡶⠛⣦
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⢀⣴⣿⠟⢹⡿⠀⠀⠀⠀⠀⢸⡟⠋⠙⣿⠀⠀⣿⠛⠋⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⣴⣿⠏⠁⠀⣼⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⢀⣿⠁
⠀⠀⢸⣿⡁⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠃⠀⠀⠉⠀⠀⠙⠛⠛⠁⠀⠙⠛⠛⠁⠀⠙⠛⠋⠁⠀⠀⠙⠛⠁⠀⠀⠀⣤⢶⣦
⠀⠀⠈⠉⠛⠛⠛⠛⣿⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⣿⣧⣤⡀
⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢹⣿⡄⢠⣤⠀⠀⣾⣿⠀⣠⡀⢰⡿⣿⡇⠀⠀⠀⠀⢠⡄⢰⡿⣿⡇⠀⠀⠀⠀⠀⣴⡶⣶⡀⢀⣿⠁⣿⣷
⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⢸⣿⠇⣾⡟⠀⢰⣿⣿⠀⣿⡇⣿⠁⣿⣇⣾⣷⠀⠀⣾⡇⣾⠁⣿⡇⣾⣷⠀⠀⣸⣿⢁⣿⠃⢸⣿⠀⣿⣿
⠀⠀⠀⢀⣾⠋⠀⠀⠀⢀⣾⡟⠀⣿⡇⢀⣿⢻⣿⠀⣿⣷⡏⠀⣿⣿⠹⣿⠀⣴⢿⣷⡟⠀⣿⣿⠹⣿⡀⣰⣿⣷⠾⠃⠀⣼⡏⠀⣿⣿
⠀⠀⢀⣾⠃⠀⠀⠀⢀⣾⡟⠀⠀⢻⣷⣾⠃⠸⣿⠀⢸⣿⡇⠀⣿⡏⠀⠛⠟⠁⢸⣿⡇⠀⣿⡟⠀⠙⠟⠁⢻⣿⠀⠀⢠⡿⠁⠀⣿⣿
⠀⢀⣾⠃⠀⠀⠀⣠⣿⠏⢀⡀⠀⠀⠈⠀⠀⠀⢻⣄⠸⡿⠀⠀⠙⠁⠀⠀⠀⠀⠘⠿⠁⠀⠙⠃⠀⠀⠀⠀⠀⠛⠷⠶⠛⠁⠀⠀⢸⣿
⠀⢸⡏⠀⠀⣠⣾⠟⠁⠀⠀⠙⠦⣤⣤⠤⢶⡀⠀⠉⠀⠀⢀⡴⠦⣄⣠⡴⠛⢦⡀⠀⢀⣤⣀⢀⣠⠏⠙⠛⠳⣄⣀⣀⣤⢶⡀⠀⠀⠛⠃
⠀⠘⠿⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠲⡄⠀⠀⠀⢀⡖⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⢠⠃⠀⠀⠀⠀⢑⡆⡞⢀⣀⣤⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⣒⣒⣒⣚⠃⠑⠒⠒⠚⠉⠉⠀⠉⠉⠀⣀⣠⣤⣀⡀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⠖⠉⠁⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣀⣀⠉⠳⡄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣸⠁⠀⠀⠀⡤⢶⣤⡀⠀⠈⣧⠀⠀⠀⠀⢀⡏⠀⠀⣾⣉⣿⣷⠀⢻⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢻⠀⠀⠀⠸⣷⣾⣿⡇⠀⠀⣿⠀⠀⢀⡀⠘⣇⠀⠀⠛⠿⠿⠋⢀⡟⠀⡿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⢆⠀⠀⠀⠉⠛⠉⠀⠀⣠⠇⠀⢰⡏⠉⢢⡈⠢⣄⣀⣀⣀⡴⠋⢀⡼⠁⢸⠿⢷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠓⠦⣤⣄⣠⣤⠴⠞⠁⠀⠀⠀⢧⠶⠗⠃⣠⡄⠀⠁⠀⣠⠶⠋⠀⢠⠏⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣄⠀⠀⠀⠀⠀⣤⠤⣤⣤⠤⡤⢶⢋⢿⡇⠀⢠⡞⠁⠀⣠⠴⠁⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠘⣧⢉⡀⠒⢈⡁⢌⡿⠀⠀⠀⠙⠛⠉⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠀⡀⠀⠀⠀⠀⠀⠈⠻⠴⠧⠾⠼⠋⠀⠀⠀⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⡴⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⡞⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠻⠋⡇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣤⣼⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡛⠶⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠚⢉⡇⠀⢀⡤⠦⣄⠀⠀⠀⠀⠀⠀⠀⣠⢄⠀⢿⠉⠳⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠖⠋⠀⠀⢸⠀⠀⡴⠋⠳⡄⠀⡇⠀⠳⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠞⠁⠀⠀⠙⠚⠁

⠀⠀⠀⠀⠀⣀⣠⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡒⢒⢀
⠀⠀⠀⢠⠖⠉⠀⠈⠀⠉⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⡶⡆⠀⠀⠀⡋⠗⠀⠀⠀⠀⠀⠀⠎⣡⠸⡆⠀⠀⠀⣼⡷⡀
⠀⠀⢰⠇⠀⣤⠀⡄⠀⡄⠀⠈⣢⠀⠀⠀⠀⠀⠀⠀⠈⢙⠆⠀⠀⢻⠀⠀⠀⠀⠀⠀⠈⠣⠭⠐⢇⠀⠀⠑⠌⠨⠘
⠀⠀⠧⠀⠀⠀⢀⠀⢀⠀⠀⠀⢙⡀⠀⠀⠀⠀⠀⠀⢠⠊⠔⢠⡄⢂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠣⡀⠀⠁⠆
⠀⠀⠳⠀⠀⠀⠈⠙⠉⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⢀⠀⠢⠀⠠⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠙⠉⠈⠣
⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⣐⣀⡀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢀⠀⢀⣳⠧⣀
⠀⠀⠀⠀⠓⣤⣤⠤⣤⣴⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡘⠉⠳⠋⠀⠀⠀⢈⣂
⠀⠀⠀⠀⠐⠛⠛⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣔⣃⠄⠰⣀⡓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠈⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⡄⠂
⠀⠀⠀⠀⠹⠟⠆⠀⠀⠸⠽⠅⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⡄⠀⢀⠀⠈⢢⣀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠺⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣠⠄⢀⠀⡁⠀⠈⠀⡀⠀⣤⠙⡄⠀⠀⠀⠀⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⡂⠀⠀⠀⠘⠀⠸⡀⠀⣶⣷⣾⣾⣶⠇⠀⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠰⠳⠀⠒⠒⠒⠒⠀⢀⠀⢋⠀⠀⠀⠰⣤⠇⠰⣄⠈⠘⠀⠃⠋⢠⠖⢽⣤⡀⠀⠀⠀⠀⢀⠀⠀⠄⠂⠐⢄⢀⣄
⠀⠀⠀⠒⠀⠀⠀⠐⠒⠊⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⡖⡆⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠁⠤⢀⣤⠀⠈⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡇⡇⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠂⠀⠄⠀⠠⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠺⠥⠤⠃⠑⠂⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠
⠀⠀⠀⠀⠀⢠⡀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⡄⣀⠀⠀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡵⠖⠂⠀⠲⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠓⣀
⠀⢀⠀⠀⠰⠀⣉⣀⣀⣬⠀⢡⠀⠴⣄⠀⠀⠀⣴⢤⠊⣶⣷⣀⣾⣿⢷⢣⣸⠗⠀⠀⠀⠀⢰⠀⠀⠶⠀⠀⠰⡂⠀⠈⡆
⠀⢪⠠⢈⠁⠀⠙⠿⠟⠋⠀⠀⡀⠂⠋⠀⠀⠀⠀⢾⠀⠉⡉⣀⢳⡿⠋⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢃⠀⢿⢿⣿⡗⢀⠎⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠳⣄⠀⠀⡠⢞⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠡⠀⠤⠀⡄⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⠠⠀⠑⠊⠀⡈⣄⠀⠀⠀⠀⠀⠀⢆⣀⡀⡀⣀⢀⣀⠀⣀⣀⠠
⠀⠀⠀⠀⠘⠀⠀⠃⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠒⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⢀⠀⠀⢀⡼⠓
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⡄⣤⡈⠑
⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠈⢈⡁⣉⠉⣁⠖⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠟⣁⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠰⣿⣻⠷⢞⣻⡿⠇
⠀⠀⠀⠀⠀⠈⢻⣦⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠐⡫⠊
⠀⠀⠀⠀⠀⣼⠁⣶⠘⣇⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠮⠖⠀⢘⢁
⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠦⣀⣠⠊⠣⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠃⠀⠀⠀⠀⠀⠀⠉⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡠⠤⢀⣀⣀⠠⠤⠢⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢰⠁⠀⠄⡀⠀⢀⠀⠀⢱⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠙⣮⠀⠀⣴⠀⠀⠀⡦⠀⠸⡌⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣽⠀⠀⠀⠀⠀⢄⠀⠀⢀⡧⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⣤⣤⣤⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠈⠉⠁⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⢹⢟⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠑⠒⠉⢸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢄⡜⠀⠀⠈⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠐⠒⠋⠉⠢⣀⣀⡠⠔⠉⠀⠈⠉⠑⢲⠤⡀⠀⠀⠀⠀⠀⠳⡀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠘⢦⠀⠀⠀⠀⠀⠸⠉⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢣⡀⠀⠀⢀⠃⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠱⡀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠣⡀⠀⠀⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠱⣄⠇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⠀⢀⡠⠖⠙⠒⠋⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠈⢦⡀⠀⠀⣠⡇⠀⠀⠀⡇
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⡤⠚⠁⢀⡀⠔⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠻⡁⢀⠁⠀⠀⠀⡇
⠀⠀⠀⢀⣀⡱⠉⠑⠴⠊⢁⡠⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⣼⠀⠀⠀⢰
⠀⢚⡉⠴⡆⠀⠀⠀⣸⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⠤⣀⡠⠃
⠛⣉⣥⡯⣵⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠈⠉⠠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣈⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠓⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⢲⡀⠀⣿⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⢹⠀⠀⠀⠘⡆⠀⠀⠀⠀⢠⠏⠁⠉⢦
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢰⠀⠈⡔⠀⠀⠀⢣⠀⠀⠀⠀⠸⠀⠀⠀⢸
⠀⠀⣀⡀⠀⠀⠀⠈⡆⠀⠀⠀⡆⠀⡇⠀⠀⠀⢸⡀⠀⠀⠀⡠⠀⠀⠀⢸
⠠⡎⠉⠙⣆⠀⠀⠀⡇⠀⠀⠀⢰⠀⠁⠀⠀⠀⠈⡇⠀⠀⠀⡇⠀⠀⠀⢸
⠈⡇⠀⠀⠘⡆⠀⠀⢹⠀⠀⠀⠀⡇⠇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢸
⠀⠸⠀⠀⠀⠙⡀⠀⠈⡆⠀⠀⠀⢰⢠⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡜
⠀⠀⢣⠀⠀⠀⠱⡀⠀⢇⠀⠀⠀⠈⡌⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠇
⠀⠀⠀⢧⠀⠀⠀⢡⠀⢸⡀⠀⠀⠀⣇⣷⠀⠀⠀⠸⡄⡼⠀⠀⠀⠀⢠
⠀⠀⠀⠈⢆⠀⠀⠀⢄⠈⣧⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠘⡆⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠉⠉⠉⢦
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢼⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⡠⠊⠀⠀⠀⠀⠀⢀⡞⠉
⠀⠀⠀⠀⠀⠀⠇⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠉⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠲⠒⢒⢒⣒⣒⢋⡉⣉⡉⡉⡙⣛⣙⡉⣉⠉⠓⠒⠒⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⢒⣋⡭⠴⠔⠒⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠒⠒⠦⠤⣌⡉⠒⠢⢤⣀
⠀⠀⠀⠀⠀⢀⠴⠚⣉⡤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⣈⠑⢦⡀
⠀⠀⠀⢠⠞⢀⠴⠋⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢇⠙⣆
⠀⠀⡴⠁⡴⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠈⢳⡉⢇
⠀⡜⠀⡔⠀⠀⠀⠀⠀⢀⣀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠒⠂⠀⡠⡖⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠉⡀⠀⠀⠀⠀⠧⠈
⠰⠀⡼⠀⠀⠀⠀⠀⠀⠊⠘⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⡃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣠
⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇
⡇⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠠⠖⠁⣀⣄⠀⠘⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡤⠊⠀⣀⠀⠑⢄⠰⠰⡀⢈⠀⠀⠀⠀⠀⠀⡅
⡴⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢯⠇⠀⢸⠁⠀⢀⣈⣉⣀⣆⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⡌⠀⠀⠞⢯⣷⠀⠀⡔⠾⢿⢋⠀⠀⠀⠀⠀⢠⡇
⠰⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠀⠀⢸⠠⠀⠘⠻⠗⢚⠋⡃⡀⠀⠣⢤⡀⠀⠀⠒⢲⡅⡀⠀⠃⠼⠃⠰⡤⢁⢴⠁⠠⡁⠀⠀⠀⠀⡼⢡
⠀⢱⡘⡆⠀⠀⠀⠀⠀⢀⣀⣸⠀⠀⢠⣤⡼⢧⡱⠤⣄⣀⣩⠞⢣⡳⢤⠤⡽⣇⠣⣤⣤⣥⢿⣙⠢⢤⣀⣤⡴⠃⠈⠮⡵⠾⠁⠀⠀⢀⠞⢤⠝
⠀⠀⢳⡘⢆⠀⠀⠀⠀⠋⠉⠁⠀⠈⠉⠉⠀⠀⠉⠉⠘⠉⠁⠀⠀⠉⠉⠉⠀⠈⠑⠒⠊⠁⠀⠈⠉⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⡔⢃⡰⠂
⠀⠀⠀⠹⣄⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⣒⣠⣤⣶⣿⣾⠶⠋
⠀⠀⠀⠀⠈⠓⢤⡑⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠲⠶⠒⠺⠛⠿⠟⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠬⣍⣓⠒⠤⠤⠤⢤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⣒⣋⣻⠭⠯⠟⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠒⠒⣮⢧⠀⠀⠀⠀⠀⠀⠀⢁⢸⡐⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠓⡀⠀⠀⠀⠀⠀⠀⡘⣇⠞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠢⡀⠀⠀⠀⠀⠑⡘⣄⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢌⡓⠤⣀⠀⠀⠁⠈⠣⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠦⣍⠓⠲⠶⠦⠌⠻⠀⢛⣲⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠁⠀⠀⢉⠙⠂⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠀⠀⠄⠀⠀⠠⠀⠀⠱⡤⠶⠚⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡴⠒⠂⠉⠀⠲⠴⠇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠠⠀⣠⣤⣤⣠⢀⣐⣦⡶⠚⠻⢦⠜⣿⣶⠾⣿⠷⠞⠛⣛⣀⢂⣤⢠⣀⣤⡴⠊⠀⠀⠀⠀⠀⠀⠀⠞⠀⢰⣧⣶⣀⣠⣶⣆⡸⢃⠀⠀⢀⡤⠰⡄
⠀⠀⣠⣄⣠⣼⡿⠏⠀⡉⣭⣦⠑⠋⠀⢀⣤⣄⢀⡈⢻⡌⡁⠘⠓⠛⢩⣿⢈⣩⡼⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⢿⣿⣿⣿⣿⣿⣿⣷⡘⡀⠀⡇⠀⠀⣇
⠒⠢⣹⣿⣦⡾⢿⣗⣩⡅⠛⢁⣀⡰⡄⠉⣛⡁⢾⣿⣿⣅⢉⣉⣼⠞⣴⠿⢏⠹⠀⣤⠀⠀⠀⠀⠀⡴⣯⢱⣠⠀⡇⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠹⡄⠀⠈
⠶⠟⢹⣇⣹⣷⠾⠏⣉⡅⢄⠿⠿⢢⣶⡞⠻⣷⣿⡟⠛⢼⠇⠈⢻⢀⣶⠺⠦⠵⢶⣿⡄⣼⣿⣦⠀⢉⣶⢴⡷⠊⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⠄⢠⡀
⣷⠏⠀⠙⣋⣻⣴⡜⢿⠿⡘⢳⠄⠉⠻⣷⠚⣿⠉⢁⣈⣶⠸⣿⣦⠀⠿⡆⢠⡄⠚⢁⠀⣉⣤⣿⣏⠋⣉⢹⡇⠀⢀⠎⠸⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⢸⠟⣛⠻
⣡⣼⠿⠟⢙⣿⠿⠿⣴⣿⣤⡤⠐⠦⠄⠙⢦⡈⠈⠿⠛⠉⣦⠄⢠⣤⣶⡎⣤⠀⣆⡉⠩⠁⢽⢠⣾⡆⠽⣶⣇⠲⠈⠀⠀⠟⢡⣿⡿⢿⣿⠉⠈⠀⠀⣾⣿⣿⠟
⠿⠿⢷⣬⡿⠁⢠⣤⡄⢤⣥⣤⣶⣶⣶⡾⠛⢿⠆⠀⠤⠈⢻⡗⠸⣿⢃⣤⣮⡁⣟⣥⠀⠀⡀⠈⢋⢀⣤⡶⠚⡆⠦⡀⠀⠀⠈⠙⠃⢸⡿⠃⠀⠀⣡⣿⠏⠰⢶
⢳⠆⠀⠉⠐⢂⠛⠟⣁⡼⢿⡟⠉⠙⠿⣇⣀⣿⠶⠀⠀⠀⠉⢱⡜⠁⠈⠉⠙⠡⣼⠿⣧⡀⠀⠀⠀⠻⡿⠃⠀⠃⠀⢨⠀⠤⣀⠀⠀⠀⠀⢀⠠⠊⠙⣧⣴⣿⣶
⣤⢀⣠⡴⠞⠙⣿⣦⡁⣠⣄⢠⣠⣀⣐⡛⠻⡏⠀⠀⣀⣀⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⠀⠀⠀⣤⠄⠀⠀⠀⠀⠁⠀⣠⣾⣛⣻⣶⣎⠁⠀⠀⡀⠘⠩⢋⣽
⣿⣦⣤⡀⢋⣴⣿⡟⠁⣼⣿⡈⠉⠣⠀⠀⣴⣧⣤⣶⣴⣶⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠲⡀⢠⡷⣎⣠⠀⣄⠀⣀⣼⣿⣿⣿⣿⣿⣿⡄⠀⠰⠁⠀⠀⣠⣾
⠀⠙⣿⡏⢨⣿⠉⠁⠃⠻⣿⡏⠠⡀⠀⣴⣿⣿⣿⡿⠿⠥⠴⠀⠀⠀⠀⠀⠀⠀⣀⣰⢾⣟⠀⠘⡇⣠⣴⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣴⣯⣍⠉
⣠⡴⠊⣥⡀⠁⢀⡀⣀⣀⣨⣿⣤⣿⣾⣿⡿⠁⠀⠀⠀⠐⠂⠁⠀⢀⢐⡀⢴⣾⣿⣿⣟⣭⣤⣶⣿⣿⡿⠟⢁⢀⡠⠄⠀⠉⠛⢿⣿⣿⣿⣿⣦⣤⡾⣿⣿⣿⠇
⠁⣠⣌⢹⣿⣶⡈⠙⠛⣉⣤⡀⠐⢿⣯⠀⣰⡀⠀⠠⠤⣤⣄⡈⡻⢿⣿⣿⣾⣿⣿⣼⣿⣿⣿⣿⡿⠋⠀⢀⠘⠈⠧⠤⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠇⠻⠿⠛
⣷⡟⠉⠙⢿⣇⣤⣼⡿⠋⠉⠙⠗⠚⠷⢿⣿⣷⠀⠀⠀⣿⠃⡞⠀⢯⠈⠘⣫⡿⢿⣿⣿⣿⣿⡟⠀⠀⢠⠄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢿⣷⣦⣄⣀
⣽⠇⠀⠀⠀⢀⢤⡈⠁⢠⣾⣷⡈⣿⠶⣿⠟⠛⠷⡀⣠⢷⠀⠙⠄⠀⠐⠺⠿⠿⠀⣹⣿⣿⣿⡇⠀⠀⠈⠒⠀⠀⠀⠀⠀⠀⡀⠀⢰⣄⠀⢰⣦⡀⢙⣿⣿⣿⣿
⠋⡀⣀⣠⣶⠟⠈⢀⣄⠹⣿⠟⠀⠀⠀⣰⡕⣶⣶⣿⣧⠈⠓⠤⠤⠤⡀⠀⠀⢀⣴⣿⣿⣿⣿⡇⢰⣷⣄⠀⣀⣀⠀⠀⣶⣿⣿⣦⣸⣿⣤⣼⣿⣿⠿⢿⣿⣿⡿
⢈⣿⣬⡀⠠⠶⠈⠉⠙⢦⠀⢀⡀⢺⣿⣿⡇⢈⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠈⢿⣿⣠⣿⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⠆⠘⢿⡋
⠻⣿⣿⠿⠂⠸⣷⣶⣶⣶⢠⣀⠁⣀⠉⢉⣀⣃⣴⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠙⠿⠿⠛⠛⠛⣙⡆⠀⠻⣿⣿⣿⡟⢻⡟⢿⣿⣿⠉⠃⠻⠟⠀⠀⠀⠀⠀⣿⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠰⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀
⠀⠀⠀⠀⠛⢤⣶⣿⣿⣦⣝⠁⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢸⡵⠀⠘⠛⠀⠺⡗
⠀⠀⠀⠀⠀⠀⡩⢙⠹⠉⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠛⢀⠀⠀⠺⢵⠂⠀⠀⢀⣀⠀⠀⠀⢻⠗
⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⡀⠈⡈⠀⡀⠄⠀⣤⡄⠀⠀⠀⠘⠋⠄⠀⠀⢰⣤
⠀⠀⠀⠊⠀⠁⠀⢐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⡀⠁⠉⠁⠀⠀⠀⠉⠄⠀⠀⠠⠄⠐⠁⠀⠁
⠀⠀⠁⠈⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠠⡀⠀⠁⠀⠂⠃⠀⠀⠄⠀⠀⠀⠀⠀⢸⡇⢰⡇
⠀⠀⠀⠀⣀⣀⡈⠀⠀⢄⣀⣠⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠠⠾⠇⠸⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠿⠿⠿⠿⠇⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠃⠰⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠺⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠂⡀⠀⠀⢐⣂⡂⠀⠀⠀⡄⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢂⠀⠀⠀⢸⣦⣀⠀⠀⠀⠀⠀⢈⣁⣶⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠈⠀⠀⠀⠘⣿⡿⠀⠀⠀⠀⠀⢺⣿⠟⠀⠀⢀⠀⠈⠀⠀⠀⠆⠀⢀
⠀⠀⢠⠛⠀⠀⠀⠈⠁⠀⠀⠀⠘⡆⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠰⠀⠀⠒⠒⠂⠀⠀⠃
⠀⠀⢼⠀⢂⠀⠐⠠⠐⠊⠁⠀⠠⡇⠀⠀⠀⠀⠈⢀⣀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠐⠂⢲⠂⠈⠑⠂
⠀⠀⠀⠀⠀⠁⣐⠒⠀⣒⡀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠈⠁⠀⠀⡠⠁⠀⠀⠀⠀⠀⠚⠂⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣒⠂⠐⢂⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠊⠈⠉⠛⠁⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠚⠉⠀⠀⣠⣼⠀⠖⢶⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠗⠀⠀⠀⣠⣜⠁⠈⢿⠀⠈⣆⠀⠀⠀⠈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⡖⠆⢠⠾⠥⠤⠇⠀⠈⠑⠣⠸⡄⠀⡀⠀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⣎⠀⢰⠇⠰⡿⠛⡆⠀⠀⠀⣶⡿⡟⣄⠑⡄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣖⢢⣸⢹⡀⠀⠀⠀⠈⠛⢲⠋⠀⠀⠉⠈⠁⠴⠲⠆⠉⠉⠁⠉⢳⠾⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣃⠘⠂⠙⠸⡇⠀⠀⠀⠀⠀⠀⠳⣦⡀⠀⠐⡖⠒⠒⠂⡶⠂⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⠁⠠⠀⠀⠙⢁⠜⠁⠀⠀⠀⠀⠈⠙⢦⣄⠈⠁⠀⠈⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠀⠀⠰⠋⠀⠀⠀⠀⠀⢀⣀⣀⣠⣬⡟⠻⢿⣟⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢱⠀⠀⠀⢀⡴⠋⠁⠀⠹⣷⠤⣀⣀⣀⣬⠟⠛⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⣦⣴⣶⠋⠀⢀⡀⠀⠀⠀⠉⠓⠛⠉⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠇⠈⢻⣧⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⢀⣀⣠⢿⣦⠋⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢶⠀⣀⢤⣼⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠓⢺⣫⠔⠋⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⡄⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢻⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡹⠛⠐⠒⢲⠖⡖⠛⠉⢋⠉⠙⡏⠹⡄⢀⡸⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠸⠀⡇⠀⠀⠀⠱⢤⡇⠀⣷⠉⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠲⣄⠀⠀⠀⠀⠀⢅⢃⣹⡆⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠘⡄⢿⡝⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡏⣇⠀⠀⠀⠀⠀⠱⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢤⠤⣀⣠⣤⣰⠃⠹⣤⣤⣤⣤⠤⠤⢷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⢸⠀⠀⠀⠀⠀⢧⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡾⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠄⡀⠴⡇⠀⠀⠀⠀⠀⠀⣧⠤⠴⠚⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠾⢯⡉⠦⠤⠤⡆⠀⠀⠀⠀⠀⠀⠙⡒⠤⠖⣉⡽⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠚⠙⢌⣢⠥⠴⠋⠀⣴⠟⡇⠀⠀⠀⠀⠀⠀⢸⣗⡌⢪⠘⠧⣔⡹⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⡢⠤⠤⢤⡐⠒⡒⣙⣉⣉⠇⠀⠀⠀⠀⠀⠀⠈⠓⠻⠶⣦⣄⣀⠀⣀⣠⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠒⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⠤⠤⠶⠶⠶⠦⠶⠶⠤⠤⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀
⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡇⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠘⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⠤⣄⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠤⠶⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟

⣶⣶⠶⡶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢰⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⠷⣀⠀⠀⢀⡠⢼⡋⣱⢖⡛⠛⠉⠉⠉⠉⠉⠉⠉⢛⡷⣏⢹⠷⢤⡀⠀⠀⣀⠾⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⡰⢩⠗⡞⠁⠀⠀⠻⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠟⠀⠀⠈⢶⠺⡙⡆⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡾⢸⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠒⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣧⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢀⣴⣦⡀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⠻⠟⠁⠀⢸⠀⠀⠀⠀⠀⠀⠠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⣀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡇⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠂⠀⠀⠀⠀⠀⡏⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠒⠒⠉⢣⠎⠀⠀⠀⠀⠀⠀⡇⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⢷⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡠⣼⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠞⠘⠦⣄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠓⠒⠲⡶⠶⠦⢤⠤⠤⠤⠤⡤⠤⠤⢶⠔⠒⠒⠋⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣼⠀⠀⠀⠀⣇⣀⣀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠶⠾⠂⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⡋⠀⠀⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠡⠃⣩⠈⠉⡥⠒⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣆⠀⠈⠀⠁⠁⠀⡆⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠰⡄⠀⠂⠄⠂⢀⡁⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠱⣦⣤⣤⠴⠁⢹⡤⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢒⡃⠀⠀⠀⡖⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡐⠹⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠉⠠⠀⠀⠀⠀⢁⠆⠅⠉⠑⠒⣄⠀⠀⠀⠀⠀⢀⠋⡶⠁⠁⠛⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢈⠐⠒⢪⣧⠰⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢀⠀⠡⡀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠈⠛⠁⠀⠈⠁⡄⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡈⠀⠀⢁⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠤⡞⠀⠀⡐⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇⠀⠀⠀⠀⠰⣿⠂⢠⠀⠀⠀⡶⠊⠁⠀⠸⡄⠜⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡆⠀⠀⠀⠀⢸⠀⠀⠀⢁⠀⠀⠀⢀⠏⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡇⠀⠀⠃⢠⡄⠀⠀⣿⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢸⠀⠁⠀⠀⡏⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢸⠀⢀⠂⢨⡀⠀⠠⠼⡇⠀⠀⠑⢄⠀⢁⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⡤⠁⠀⠈⡇⠀⠀⠀⢠⠀⠀⠀⠀⠑⠾⡵⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⠉⠀⠀⠀⠀⣧⢐⡀⠀⠸⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⠃⠀⠀⠀⠀⢸⠀⠉⠀⠀⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠴⠖⠒⠶⠾⠀⠘⠀⠀⠓⠒⠒⠲⠶⣦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⣿⣿⣻⣻⣛⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣻⣿⣟⣟⣿⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⣀⣀⣴⠦⠄⠀⠀⠀⠀⠠⢤⣄⡀⠀⡔⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠔⠊⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⠢⡀⠀⡌
⠀⠀⠋⠀⠀⡀⠀⠀⠀⠔⠄⠠⠂⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⢲
⠃⠀⢉⠀⠉⠜⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠢⠁⠆⠂
⠀⠀⠓⠊⠑⠅⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠐⠀⠀⠀⠁⠢⢹
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⠀⠀⠈⠐⠠⡄⠀⢀⣀⠀⣀⣀⡀⠠⠔⠊⠁⠀⠀⠀⠀⡀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⢁⠊⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⣀⢄⢀⠠⠄⠀⠀⡀⠀⠀⠀⠘⡀⢀⣀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠅⣀⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⠤⠒⠊⠁⠀⠀⠀⡄⠀⢀⠀⠡⡀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⢢⡀
⠀⠀⠀⠑⠐⠒⠀⠒⠂⠒⠠⠤⠄⠀⠤⠐⠂⠀⠤⠤⠍⠁⣀⡀⠀⠠⠤⠐⠒⠀⠒⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠶⠶⠖⠓⠒⠲⠄⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⡛⠉⠠⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠛⠷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠘⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠋⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠱⠀⠀⠀⠀⠀⡀⠀⠈⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡋⠅⠀⠀⠰⡁⠀⠀⠀⠀⠀⠀⠀⢰⡄⠃⠀⣴⠁⠀⠀⠈⠃⡀⠀⠀⢻⡆⠀⠝⢻⣦
⠀⠀⠀⠀⠀⢀⣼⡏⠀⠀⠀⢠⡟⣠⣀⠀⠰⠀⠀⠀⢀⣄⣿⡜⢲⣿⣦⣄⣄⣦⣆⣹⣴⣦⣿⡇⠐⠀⠈⢻⣆
⠀⠀⠀⠀⢀⣼⡋⠁⠀⠀⠂⢸⣿⡟⠟⠻⠱⡜⠛⢲⠏⢻⡿⠁⠈⣿⡉⠋⠉⢙⠛⠁⠀⠀⣾⡿⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⣼⠏⠁⠀⠀⠀⠀⠀⠿⣧⡀⠀⠀⠇⠀⠀⣠⣾⠃⢰⠀⢨⢷⣄⠀⠈⠀⠀⣀⣼⠫⠀⠀⠀⡀⠀⠘⣯
⠀⠀⠀⢠⡿⠉⠀⠀⠀⠀⠀⠀⠀⠃⠹⠷⣶⣷⣶⡚⠋⠀⢀⠀⠀⠀⠸⠏⠻⣻⣶⠿⡏⠁⠀⠀⠀⠀⠀⠀⠐⢻⡇
⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠃⠉⠃⠀⠀⠘⠃⠀⠀⠀⠀⠀⠉⠉⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⢼⣧
⠀⠀⢰⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣠⠁⢀⣾⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢀⡀⠀⢨⣿⡀
⠀⠀⣼⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⣶⣿⣦⣿⣿⣿⣿⣿⣿⣿⣤⣄⣠⢂⡀⠀⠀⠀⠀⠀⠈⠁⠀⠠⣿⡇
⠀⢸⡏⠀⠀⠚⠀⠀⠀⠀⠀⣠⣠⠀⢿⣾⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠉⢉⣝⡻⢿⡿⣃⡀⠀⠀⠀⠀⠀⡀⠀⠀⢈⣷
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡷⣼⠛⢣⠈⠹⡍⡛⠻⠿⢿⣷⠿⠛⠛⠹⠇⠘⠃⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄
⠀⣻⣇⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⡇⢀⡄⠠⠀⠀⠀⠠⠃⠀⢠⠉⠀⡀⠀⡄⠀⠀⠀⣿⡟⠄⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇
⠀⣿⠏⠐⠀⠀⠀⠀⢠⠀⠘⣽⣿⠡⠀⠀⠀⠀⠆⠀⡀⠀⢀⠋⠠⠀⠀⠀⠁⠀⠀⠀⣿⡳⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿⡏
⠈⣿⢠⡿⠃⠀⠀⠀⠀⠀⠢⣽⣿⡷⠀⠀⠗⠈⠀⠀⠀⣰⣶⢶⣆⣀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠘⠀⢠⠀⠋⢼⡧
⠘⣿⡋⠑⠀⠀⠀⠀⠀⠣⠀⢻⣿⠗⠀⠀⠀⠀⠀⠠⣼⣿⡿⠿⢿⣿⣷⣀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⠰⠀⠆⠀⠌⣹⡇
⢀⣿⣷⠀⠐⠀⠀⠀⠀⠀⠀⢼⣿⣧⠄⠀⠀⠀⠀⣇⣾⠃⠀⠀⠀⠙⣿⣏⠄⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠸⢮⡟
⠀⢹⣿⡁⢀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠐⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠹⣷⡖⠀⠀⠀⢸⣷⡀⠀⠀⢠⠀⢀⠀⠈⢰⢻⣏
⠀⢸⣿⡗⢫⠀⠀⠀⠀⠀⠀⠘⠿⣿⠀⠀⠀⡀⣿⣿⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⢸⡟⡅⠀⠠⠀⠀⠀⠀⢈⠃⣹⡟
⠀⢸⣿⣧⢊⡙⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠠⠁⣹⡏⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡷⠀⠀⠀⠀⢀⠀⢀⡏⢹⢸⡇
⠀⠸⣿⣿⣼⡗⣦⢣⣘⣄⠀⢀⣾⣿⡇⠀⠀⣴⣼⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠈⢸⡗⣟⣣⢰⠀⣠⠔⣯⢻⣶⣿⡇
⠀⠀⣿⣿⣧⣼⣿⣷⣾⣶⣶⡿⠛⠛⠿⢃⠀⠐⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⣿⣿⣯⣉⣤⣇⣾⣿⣾⣿⣾⠃
⠀⠀⠉⠻⣿⡟⠉⠙⠉⠈⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⢠⠜⠁⠉⠛⠛⠛⠉⠛⠛⠻⣿⠏
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢈⣿⡄⠀⠀⠀⠀⠀⠀⠀⢹⡇⠇⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠐⠃⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣧⠀⢀⠀⠀⡀⠀⢀⡀⠀⣶⣳⣾⣿⠂⠀⠀⠀⠀⠀⠀⠀⢸⣷⡖⠐⣤⠀⠀⠠⠃⠀⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠹⣄⢈⡀⠀⠀⠀⠈⣃⢀⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡾⢿⡶⣿⣿⣾⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣏⣣⣤⣤⣆⣀⣄⣀⣼⠁
⠀⠀⠀⠀⣀⡴⠛⠉⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠋⠉⠁⠀⠀⠀⠀⢠⡞⠛⠋⠉⠈⠋⠉⠉⠙⠏⠻⡏⠛⠻⡀
⠀⠀⠐⠊⠁⠀⠊⠀⠰⠂⠀⡤⠀⠀⠀⢀⣤⠴⠓⢆⠀⠀⠲⠀⠀⢸⠁⠀⢸⣦⡀⠀⠀⠀⠀⡀⠀⡀⠀⣄⠉⠢⡀
⢀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⢀⠜⠋⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢻⣆⠀⢿⡀⠙⠳⣤⣤⠈⠛⢦⡈⠢⡈⠠⡀⠹⡄
⢀⠋⠀⡀⣴⠊⠀⢠⡄⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢲⡄⠐⠂⠀⠀⠀⠻⢦⣌⢙⡆⠀⠀⠉⠛⢶⡄⠙⠆⠐⠀⢠⡄⣷⡀
⠀⠐⠉⠁⣁⢀⠴⢋⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⢧⡀⢰⣄⣤⠰⣧⣸⠇
⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡼⠤⠛⠀⠃

⠀⠀⠀⠀⠀⣀⡠⠤⠶⠖⠒⠲⠶⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⢄⣀⣀
⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣄
⠀⢠⠏⢀⣀⠀⢀⠀⣀⣀⣀⢠⣄⠀⣀⠤⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠒⠂⡤⢆⠴⠌⠏⠀⢀⣟⠐⣧⠀⠈⢳
⠀⡎⠀⢾⠀⡄⢇⡽⣀⠿⡼⢫⠇⠘⢸⡁⢀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠃⠀⠀⠀⣀⠀⡄⢀⠀⠀⠀⠀⠀⢸
⠀⢡⡀⠈⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠜⠉⠃⠓⠘⠒⠐⠂⠀⠀⡜
⠀⠀⠑⡄⢀⠄⠀⠀⠀⠀⠀⠠⠤⠄⠐⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠢⠤⠤⠤⠤⠄⠠⠀⠤⡄⠀⠎
⠀⠀⠀⠀⢸⠀⠀⠀⢀⣠⢶⡶⠖⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣄⠀⢀⠇⡰
⠀⠀⠀⠀⠈⠃⠀⠰⠋⢀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠟⠛⠛⢻⣿⡇⠉
⠀⠀⠀⠀⠀⠀⠀⢲⡸⢿⠀⠀⠠⠂⢀⠃⠀⠀⠀⠀⠖⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠄⠀⠀⠀⠀⠀⠀⢠⡄⠀⠤⠀⠀⢿⠟⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣎⠁⠀⠀⠀⠬⠂⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⢸⠀⠀⠀⠀⣠⣞⡎⡙⠉⠀⠃⠀⠀⡴⠃
⠀⠀⠀⠀⠀⢀⣀⠘⡟⠉⢑⣄⣉⡛⢁⠃⠀⠀⠀⠀⠈⠀⠠⡀⠀⡀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡅⠀⢸⠀⠀⠀⠀⣫⠚⣏⢦⠀⠤⠂⢀⣼⠃⡶⠄⡀
⠀⠀⢀⣤⡶⠋⢀⠤⠁⡠⠾⣿⣿⠏⠑⠀⠀⠀⠀⠀⢀⡆⠀⠀⠐⠃⠀⠰⠄⠀⠀⠊⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢹⡀⣇⠆⣢⢤⠖⡿⢇⡸⠁⠀⠀⠙⣽
⠀⣰⣿⠋⠀⡔⠁⠀⢀⡗⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡎⠀⠀⠀⠀⠀⡼⠈⠍⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⢹
⠀⣿⡧⠀⠸⢤⢀⢀⠞⠀⠀⠀⠀⠀⢂⡴⠏⠀⢠⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⣸⢃⡼⢳⠇⠀⢸⠇⠀⠀⠀⠀⡄⠀⠀⢸
⠀⠘⣧⡀⢀⡶⣾⠋⠀⠀⠀⠀⢀⡴⣃⣐⣁⣘⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⢀⡀⡟⠁⠀⠸⡇⠀⠸⡆⠀⠀⢀⡰⠇⠀⠀⢨⡆
⣀⣀⣈⣉⣉⣁⣉⣁⢀⣀⠀⠀⢀⣀⣈⡀⣈⣁⣁⣀⣀⣉⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⣰⣧⠒⠚⠛⣟⢄⠀⠀⠀⠀⠀⡸
⠀⣇⡅⠀⡇⠀⠀⠀⢢⠀⠀⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠎⣁⡰⠟⠚⠲⡴⢊⠙⢿⣠⡤⠖⢲⠋⡇
⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⡾⠛⠻⣤⡖⠄⠀⠈⢆⠀⠀⠀⠱⡀⢸⠇⠃
⠀⣿⠃⢴⣿⠀⠀⠀⢰⣿⣿⣷⣦⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⣿⣇⣠⠤⠞⠂⠀⠀⠀⡧⠞⢸
⠀⣿⠀⠀⡟⠀⠀⠀⢸⣿⡏⠉⠙⢷⠀⠀⠀⠀⠙⢦⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠉⢩⠁⠀⢃⠀⠀⠀⠀⡗⢰⢸
⠀⣿⡇⠀⡇⠀⠀⠀⣸⠿⠷⠤⣄⠈⠳⡀⠀⢀⠔⡉⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠘⡄⠀⠈⡀⣀⡀⠀⠁⠸⢸
⠀⣿⡇⢀⣟⠉⣙⡖⠟⠀⠀⠀⠻⠃⠀⠈⠲⢃⡂⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠛⠋⠀⠈⣣⠀⠘⠟⠻⠐⡆⠁⠀⠼
⠀⠉⠁⠈⠐⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠣⠀⠴⠃⠁⠀⠐⠉⠀⠀⠂

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

⠀⠀⠀⠀⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⡆
⠀⠀⠀⠀⡇⠀⡄⢠⠀⠤⠠⡄⢠⠀⡄⠠⠄⢠⠀⡄⠠⠄⢠⠀⡄⠠⠄⢠⠀⡄⢠⠄⠤⠀⡄⠠⠄⢤⠀⡄⢠⠀⡤⠀⡄⢠⠀⠤⠀⡄⠀⡇
⠀⠀⠀⠀⡇⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡇
⠀⠀⠀⠀⡇⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⡇
⠀⠀⠀⠀⡇⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⡇
⠀⠀⠀⠀⡇⠀⡁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇
⠀⠀⠀⠀⡇⠀⡄⠀⠀⠀⠀⡇⠀⡆⠀⠀⡀⡀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡄⢀⠃⠀⢀⠀⠶⠀⠂⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠤⠀⡇
⠀⠀⠀⠀⡇⠀⠄⠀⠀⠀⢸⠉⠉⠃⠞⢱⠃⡟⣸⠸⢃⠇⠏⠐⡃⠀⠀⠀⠀⡟⠀⡇⠀⠘⠀⡆⢸⠀⠄⠊⠝⡰⢹⠇⡎⠐⠀⠀⠀⠖⠀⡇
⠀⠀⠀⠀⡇⠐⠂⠀⠀⠀⠘⠀⠀⠀⠀⠀⢨⠃⠁⣞⠈⠀⠀⢸⠁⠀⠀⠀⠀⠋⠐⠁⠁⠈⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠓⠀⡇
⠀⠀⠀⠀⡇⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠊⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⡇
⠀⢀⡆⠀⡇⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⡇⠀⣄
⢠⣾⣿⠀⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠆⠿⠂⠿⠇⠺⠇⠖⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⡇⢸⣿⣦
⢾⣿⣯⣴⡿⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠸⣷⣬⣿⣿
⡌⣾⡿⠟⣧⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⣤⡟⠿⣿⡎
⡏⠀⣴⣾⣿⢋⠃⢀⣤⣾⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣸⣦⡄⠀⣉⢻⣿⣶⡄⠈
⢀⠘⠛⢛⡷⡿⡗⢁⡀⠈⠛⠛⢿⠀⠀⠀⡀⠀⣴⡄⢠⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣦⠀⣴⡄⠀⠀⠀⠀⢰⠟⠓⠋⠀⣀⠁⡾⣷⣟⠛⠛⢀
⠁⠀⠀⣁⣀⢠⣴⡾⠟⢦⡀⠀⢈⣀⡀⣴⣿⢰⣿⣷⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣿⣷⢸⣷⡄⣀⣈⠀⠀⣠⠞⠿⣶⣤⢀⣀⡁
⣴⢋⡏⠁⢘⣴⣿⠀⢀⣠⣿⣶⠈⢿⡇⣿⣿⠀⠛⠃⢟⣡⣤⣤⡀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣙⠃⠻⠃⢸⣿⡇⣿⠏⢐⣾⣧⣀⠀⢸⢷⣜⠀⠉⣏⠲
⠙⠻⣇⣠⡾⣯⢛⡷⠫⣽⣿⣘⢆⠀⠀⠈⠛⢀⡼⢂⣻⣟⡛⠉⠉⠀⠀⠀⠀⣄⡈⢉⣿⣿⣿⣃⠺⣄⠘⠋⠀⠁⢀⠞⣼⣿⡩⠳⣟⢫⡿⣦⣀⡿⠛
⡿⠃⠻⣯⣉⠖⣿⢋⠑⢾⡏⠀⠀⢧⠀⠀⣤⢮⣾⡿⠻⢿⠿⠲⡄⠀⠀⠀⣀⢻⣷⣼⣟⣿⠿⢿⠿⠊⢦⡄⠀⢠⠇⢄⠈⣿⢔⢋⢺⡗⢆⣹⡿⠃⠻
⣿⡄⠀⢈⡿⠛⣿⣶⣕⣸⣧⠴⢦⣌⣦⣘⣿⣶⠛⠀⠀⣸⣀⢀⡸⣄⡀⠀⢻⣧⢹⢟⣛⡁⠀⠀⠀⢲⣾⣟⣠⡶⣩⡶⢤⣿⣐⣥⣾⡟⠻⣟⠀⠀⣼
⡛⢿⡖⢹⣇⣠⣤⠋⠀⠉⢱⣶⠴⠛⢿⣍⢫⣭⣽⣿⡆⠘⡝⢿⣔⡮⠜⣦⡈⢻⠴⠿⠋⠀⢶⣶⣚⡏⡩⢋⣽⠟⠳⢴⣶⠉⠁⠈⢣⣤⣀⣿⠑⣾⠟
⠀⣠⣷⣿⠻⣿⣠⣧⣤⣶⣿⣷⣧⣤⣘⣿⣷⡿⣿⣿⠷⠀⣅⣁⣙⣷⣬⠟⠓⢖⠻⣏⡙⣲⠾⢿⣯⡟⠷⣧⣏⠀⢰⣷⣿⣷⣦⣴⣧⣸⡿⢻⡷⣧⡀
⣶⡯⡾⡡⡲⢾⠁⠀⣭⣿⣯⣿⣿⣻⣿⣿⣿⣥⣏⠁⢀⡀⠨⣉⡟⢡⣧⣠⣤⣾⠋⠙⠧⣵⣷⠿⠟⠊⠰⣬⣧⣀⣸⣿⣯⣿⣯⡅⠀⣹⠦⡢⣹⡮⣷
⣸⣿⣫⣬⣵⣿⣀⢀⣈⡽⣿⠿⠂⢙⣴⣿⢿⣶⣶⡎⢉⣿⣶⠿⣏⠀⢼⣿⣟⣿⡿⣀⡴⣿⣿⠏⠀⠀⠀⠻⡅⠘⠁⠺⢿⡿⣍⣀⢀⣸⣷⣭⣮⣹⣿
⠙⠛⠿⣫⣳⣿⣿⣍⣹⡁⠁⡀⣀⡼⠉⠁⡾⠛⠉⠀⢠⣎⣁⣠⣽⣆⣀⣹⠹⠿⠛⠉⢳⣟⠁⠀⠀⣤⡀⠀⠈⠹⣄⡀⠈⠁⢹⣉⣽⣿⣷⣹⡽⠟⠛
⡿⢿⠿⡿⠿⢟⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣟⣛⣿⣿⣻⣿⣟⣿⣟⣻⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠁⣶⡈⣿⣿⣻⣷⢾⢿⡿⠻⣫⣿⠋⣩⠻⣿⡿⢋⣍⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣇⢸⣿⢃⣿⣷⣾⣿⣿⣿⣫⣾⣿⡏⢰⣿⡇⣿⡇⣼⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠸⡏⣼⣿⣿⣿⣿⣿⣿⣵⣿⣿⡇⢸⣿⠀⣿⠀⣿⡟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠀⢰⠿⢿⣿⣿⣿⣿⠿⢿⣿⣿⡇⢸⠇⣸⣿⠀⣿⠁⣾⣿⣿⡿⢿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣷
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣯⣤⠀⢀⣴⣾⣿⣿⣿⠁⣴⣿⡿⣿⡇⠘⢠⣿⣿⠀⠃⣸⣿⣿⣿⣷⣼⣷⠉⣛⣯⣴⣿⣿⣿⣿⣿
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⢸⣿⡏⢸⣿⠟⢀⣿⠇⢠⣿⣿⡿⠀⢰⣿⣿⡿⢛⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣿⢸⣿⡇⢸⣥⣴⣾⠏⠀⢸⣿⣿⠇⠀⣾⣿⣿⠁⢸⣿⣿⣿⠇⣼⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣟⢸⣿⣿⡇⢸⣿⡇⢈⠻⠟⣡⣴⣧⡸⡿⢋⣴⣆⠻⡿⣁⣶⡘⢿⡿⢋⣼⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣍⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠿⡿⣿⣿⡿⠿⠿⠿⠟⠛⢻⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁

⠀⠀⠀⠀⢀⡤⠖⠀⢀⠔⢛⡠⠄⠖
⠀⠀⠀⢠⠊⠀⢀⡰⠁⠀⠁⠀⠈⠉⢉⣙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠐⠒⠀⠐⠒⠒⠤⢄⡀
⠀⠀⠀⠆⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⡀
⠀⠀⢰⠀⠀⠀⠀⠀⣀⠃⢀⣠⠐⠄⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⡤⠚⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠮
⠀⠀⢸⠀⠀⠀⠆⠀⠀⠀⡎⠐⠊⠀⠠⠁⠠⢀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡁⠀⠀⠀⠀⠀⠠⡉⠀⠚⠋⠙⠓⠃⠀⠀⡖
⠑⠢⢄⠀⠀⠀⠄⠀⠀⣺⠡⢾⠇⠦⠴⠺⡇⠈⡙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⠔⠁⠀⢄⠀⡄⠀⢐⡐⢢⠀⠀⠀⠀⠀⠀⠀⠆⠀⠦
⠀⢠⠜⠀⠒⠀⡏⠀⠀⠀⠑⠀⠀⠀⣀⡴⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣷⠀⠀⠀⠌⠤⣃⠘⢌⣁⠌⠀⣤⠄⠀⠀⠀⠀⠐⠀⠐
⠀⡇⠣⠀⠈⠐⠂⠀⠀⠀⠀⠀⡠⠠⠄⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢻⠀⠀⠀⠀⠸⣅⡀⠀⠀⠀⠰⠁⢡⡀⠀⠀⠀⠀⢸⠚
⠀⠈⠒⣎⠀⠀⠀⠀⠀⠀⠀⢸⢿⣶⠆⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣠⡾⡇⠀⠀⠀⠀⡜
⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠰⡬⠋⠁⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠮⣤⡾⠋⣠⠅⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⣶⠲⡶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⢤⣤⣤⡤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⠶⠠⠌⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⡠⠐⢠⠒⠁⣀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠂⡉⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢡⠀⠀⠀⢀⡘⠀⠠⠊⠀⠐⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠
⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠓⠄⢂⠀⠀⠀⠷⠀⠀⠀⠀⠳⠈⠓⠈⠀⠀⠄⠈⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣤⠀⠐⠀⠈⢄⣀⡤⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⢆⠀⠀⢄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢀⠜
⠀⠀⠀⠀⡀⠀⠀⠶⠀⠀⠈⠀⢰⢠⣤⠄⢳⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠊⡰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⢀⡴⠁
⠀⠀⠀⠀⡇⠀⢀⡀⠤⠐⠀⠀⠸⠀⠰⠘⠦⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄⠈⠁⠐⠒⠒⠀⠈⠉⠫⣀⠀⡁
⠀⠀⠀⠀⢨⠉⠀⠒⠒⠒⠒⠀⠀⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄⢤⠀⠀⢸⠀⠀⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢰⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠀⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠐⠚
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣙⠀⠀⠀⢇⣀⣀⠼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠤⠣⢄⠀⠀⠀⠀⢘⠓⡆
⠀⠀⠀⠀⢠⠟⠒⠚⠛⠥⢀⡠⠂⠘⢍⠄⠀⠀⡙⠋⠈⠙⢦⠀⠀⠀⠀⠀⣄⠀⠀⠄⠀⠀⢀⡀⣤⠀⠀⢸⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⢧⡈⣀⡀⠀⠀⡘⠀⠀⠀⠈⡟⠠⠤⠖⣀⣀⠀⡠⣇⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁⠀⠀⠈⠈⠉⠉⠉⠉⠁
⠀⠀⣀⠀⡉⠈⠈⣉⣁⣈⣁⣀⣀⣀⣐⣇⣀⣀⣈⣁⠀⣀⠐⡈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⡀⠀⠀⡀⠀⠀⡀⣀⣀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠞⠛⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⢩⠀⣀⣀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⡜⠩⡄⢉⢹⠉⠟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠅⠀⠀⠐⠀⠀⢺⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢱⡀⠈⠐⠈⢀⢃⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⢻⠶⠶⠒⠃⢘⡷⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⡽⠀⠀⠀⠸⠀⢠⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠎⠏⣨⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⡎⠣⣀⣀⣀⣠⠋⡄⠀⠀⠈⢳⡀⠀⠀⠀⠀⠠⠀⡏⠀⠀⣡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⡇⢤⣄⠀⠘⠿⢂⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢰⠀⠐⡄⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢀⠂⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢰⡆⠀⠀⠀⠀⣠⡀⠀⡀⠀⠀⢀⣠⠔⠊⢣⠀⢀⠂⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠇⠀⠠⠀⠀⠙⠁⠀⡇⠀⠀⠙⠀⠀⠀⠀⡧⠃⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⠀⠀⢠⠀⢀⠀⠀⢸⠃⠀⠀⠈⡀⠀⠀⠀⠁⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⠀⡀⠛⠀⠀⢸⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡆⠀⢀⠀⡇⠀⠀⢀⣿⠀⠀⠢⡀⠀⢂⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡆⡠⠃⠀⢻⠉⠀⠈⠹⠀⠀⠀⠈⠢⣄⣦⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡔⠋⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡷⠀⠀⠀⠀⠘⡇⠙⡄⠀⡇⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣠⡤⠤⠤⣤⠇⠀⠇⠀⠱⠤⠤⠤⣤⣤⣀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣤⣤⡴⠶⠶⠤⠤⠤⠤⠤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠯⣄⡀⠀⠈⠉⠙⠛⠒⠶⠦⢤⣤⣀⣈⡉⠙⠛⠻⠿⢭⣝⣓⡒⠒⠶⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢄⡀⠉⠙⢓⣶⡤⠤⠤⠤⠤⠤⠤⠤⠬⠭⢭⣛⣲⣶⣤⣤⣉⡉⠛⠓⠶⠶⣤⣬⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡀⠉⢳⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠶⢦⣤⣄⣀⡉⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠉⠙⣻⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⡀⠀⠀⠀⠹⡤⣬⣉⣙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠙⠲⡇⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⠀⠀⠀⠀⢱⡤⣄⣉⣻
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠙⠓⠶⡇⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢀⣿⠀⠀⠀⠀⠈⡦⣄⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠲⠦⣤⡇⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣾⣶⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣷⠤⣄⣹
⠀⠀⠀⠀⠀⠀⠀⠀⡷⠶⢤⣼⠂⠀⠀⠀⠀⢹⣿⣿⣀⣤⣶⣶⡄⠀⢀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⠶⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⡷⢤⣄⣸⡀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣿⣿⡇⠀⢸⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⠦⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⣧⣄⣀⣉⡇⠀⠀⠀⠀⠘⣿⣿⠁⠀⠈⣿⡇⠀⢸⣿⡏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣸⠒⠶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣀⡉⠉⣇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢹⡇⠀⢸⣿⠁⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⡟⠐⠒⠪⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠉⠉⠛⣆⠀⠀⠀⠀⢹⡏⠀⠀⠀⠘⠃⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠓⠶⠦⣤⣤⣀⣉⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠉⠛⠒⠾⢷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠓⠶⠦⢤⣄⣀⣀⡉⠉⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠓⠲⠦⣤⣄⣉⡙⠛⠻⠶⣶⣶⣶⠶⠦⠤⠤⠤⠤⠤⣤⣶⡒⠒⠛⠛⠒⠶⠶⢤⣤⣄⣀⣉⠉⠉⠛⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠤⠤⣍⣉⣛⡒⠶⠤⣤⣉⡙⠓⠲⠦⢤⣄⣀⡀⠈⠉⠙⠓⠶⠦⣤⣤⠖⢤⡀⠉⠉⠙⠛⠲⠶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠷⠶⠦⠤⠬⣭⣭⣲⡶⢤⠤⠤⠔⠋⠀⠀⠙⣟⠒⠦⢤⣄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡶⠤⣄⣀⠈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣀⠉⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠈⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⠉⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠜⠐⢄⡀⠀⠀⠀⠀⢀⡠⠄⠂⠀⠉⠁⠀⠈⠉⠀⠐⠢⠄⡀⠀⠀⠀⢀⡠⠊⠀⢢⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠚⠂⠄⡀⠈⠒⢄⡠⠂⠁⠀⠀⢀⠔⠂⠈⠉⠉⠀⠐⠤⡀⠀⠈⠑⠠⡔⠁⢀⠤⠂⠁⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⠊⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠈⠲⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠰⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠂⢠⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⡀⠠⡀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠁⠀⠀⠀⡄⠀⠀⠀⠣⡀⠄⠀⠨⡆
⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⣏⣹⠀⠰⠐⠄⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠄⠄⠈⠠⠻⠀⠀⠀⠀⠀⠀⠤⢄
⠀⠀⠀⠈⠀⠀⠑⠄⠀⠀⠀⢸⠀⠠⠐⡆⠉⠀⠀⠀⠀⠀⠁⠂⠄⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠑⠁⢈⣹⠀⡇⠀⠀⠀⡔⠀⠀⠀⠇
⢀⠔⠂⠌⣆⠀⠀⠘⡀⠀⠀⠐⢀⠠⠈⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢄⡀⠁⠀⠀⢰⠀⠀⢀⡼⠐⠁⠐
⡆⠀⠀⠀⠈⠳⡀⠀⠇⠀⠄⠊⢁⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠄⡈⠑⠢⠄⠘⠀⠠⠊
⠑⠄⣀⡀⠀⠀⠀⠀⠀⠀⣀⠔⡁⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⢂⠀⠈⠆⢄⠀⠀⠀⠀⢀⣀⣠⡤⢂
⠐⠈⠉⠉⠉⠁⠤⠄⢤⠊⠀⠀⠁⢠⠁⠘⠀⠀⠀⠀⠈⢢⠄⣀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠁⠀⠀⠀⠀⠀⠈⠆⢰⠀⠀⢉⠦⠀⠒⢠⡀
⠐⠄⠠⠄⠚⢇⠀⠀⠀⠈⠐⢄⡸⠃⠀⡄⠀⠀⠀⠀⠀⡀⠀⠀⠘⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⢆⣀⠄⠁⠀⠀⠀⡠⢛⠀⠒⠒
⠀⠀⠀⠀⠐⢂⠓⠠⠤⢤⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠤⠴⠶⠭⢀⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠡⣀⣀⣀⡤⠊⠁⠰⠀⠠⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠱⠀⢱⠀⠈⠲⢤⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⢠⠫⣀⡀⠃⠘⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠀⠠⢀⠠⠣⡀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠐⠁⠀⠀⠀⠀⠂⠀⠐⠀⠁⠐⠀⠤⠀⠀⠀⠀⠀⠀⠀⠤⠐⠂⢫⠀⠀⠃⠀⠀⠀⠀⠀⠈⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀⠄⠰⠦⣤⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⣧⠴⠀⠀⣐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⡀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢀⡈⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠑⠢⠀⣀⡀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣘⣿⡾⣯⣳⣔⣄⡞⣿⡶⣿⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⠀⠀⠀⠀⠂⢠⣀⣹⣿⣾⣿⣿⣷⣿⣿⣿⣹⡭⠿⠿⠷⣯⡀⠀⠠⠀⠀⠰⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠿⠋⠀⠰⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠈⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⣠⠞⠉⠁⠁⠉⠀⠈⠛⠛⠙⠉⢻⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣴⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠏⠀⠀⠀⣀⡤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣄⣀⠀⠀⠀⠈⢯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠉⠳⡄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠘⣇⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣼⠁⠀⠀⢀⣠⣤⣄⡀⠀⠀⠸⡆⠀⠀⣰⠃⠀⠀⢀⣤⣤⣄⡀⠀⠀⠈⣇⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠐⣿⣿⣿⠟⠻⡆⠀⠀⣷⠀⠀⣟⠀⠀⠔⣿⣿⣿⠛⠻⡄⠀⠀⢸⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡇⠀⠀⢧⠀⠀⠀⢿⣿⣿⣶⣴⡟⠀⢀⡏⠀⠀⢻⠀⠀⡄⢿⣿⣿⣦⣾⡇⠀⠀⡾⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡇⠀⠀⠈⢧⡀⠈⠢⢙⡛⠛⠟⠁⣠⠞⠀⠀⠀⠈⢳⡀⠈⠢⢙⢛⠻⠋⠀⢀⡜⠁⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⠀⠀⠉⠳⠤⠤⠤⠤⠴⠚⠁⠀⠐⠛⠂⠀⠀⠉⠓⠦⢤⡤⠤⠴⠚⠉⠀⠀⠀⠀⣸⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣿⣿⠀⠀⠀⠀⠀⠀⠰⡟⣶⣶⠒⠒⠒⣶⣶⣶⣶⡖⠒⠒⣻⣿⣏⣛⣆⠀⠀⠀⠀⠀⠀⢀⣿⠃⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣤⣾⣿⣿⣿⣿⣷⣤⣴⣿⡿⢿⣿⡟⠀⠀⠀⠀⠀⠀⣼⣏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣽⣿⣦⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⡏⠀⠙⣿⣿⣿⣿⣟⣠⠴⠛⠀⠀⠀⠀⠀⠀⣰⣿⣁⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢷⡿⠿⠙⣶⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣼⠋⠿⢿⡾⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣔⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠈⠁⠙⠻⢻⣿⣶⣤⡄⣲⣤⣠⣄⣄⣀⣀⣤⣬⣴⣶⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⡿⠻⣿⠛⠟⠛⠿⣿⠻⠟⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⣶⠾⠃⠈⢻⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠹⢶⣶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣼⣿⣾⣁⣀⣀⣀⣼⣶⣶⣶⣶⣶⣶⣶⣾⣇⣀⣀⣀⣸⣮⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠓⠲⡄⠀⠀⠀⢰⠇⠀⠀⠗⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢺⠀⠀⠀⡼⠀⠀⢰⡇⠀⣰⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡄⠀⢠⡇⠀⠀⣸⠁⢠⠃⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⣼⠁⠀⠀⡟⠀⡎⠀⠀⢰⠁⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢣⠀⡏⠀⠀⢰⠇⣼⠀⠀⢠⠇⠀⢀⡴⠁⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣸⠇⠀⠀⡼⣴⠃⠀⢀⡟⠀⣠⠋⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠀⠀⠀⠛⠃⠀⠀⡼⢀⡞⠁⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠈⠑⢦⡀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠈⢧⣸⡇⠀⠀⠀⠈⠙⠛⠲⠦⣀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠙⠓⠶⣤⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠈⠙⡆⠀⣠⠞⠋⠁⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠶⠦⠄⠀⠰⣄⡀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠈⠙⠛⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣄⣀⣀⣀⣀⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⣀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠟⠛⠛⠉




⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣹⣿⣻⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⣤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⣿⣾⡿⣿⣿⣿⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣘⢿⣿⣿⣿⣽⣧⣿⣿⡿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠻⠶⣿⣿⠻⡿⣿⣿⣷⣾⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠀⠀⠀⠄⠀⠀⠀⠀⠒⠀⠛⠟⠁⢸⠁⠀⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠠⠀⠱⠂⠀⠄⢀⡀⠀⠀⠀⠀⠀⠰⢠⠀⠀⡿⠀⠀⠀⠀⠀⣀⣠⣤⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⡁⢠⣶⠘⠛⠏⣀⠀⡀⠀⠀⠙⡿⢦⠀⡇⠀⢀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠘⢈⡇⣺⣿⣯⠉⣴⠉⠧⠀⠀⠀⠀⠀⠀⠀⣷⣾⠋⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠷⣌⠹⠄⠐⠃⠀⠀⠀⠀⠀⠀⣿⡷⠎⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠈⠊⢀⠰⠋⠀⡀⠀⠀⠀⣤⠀⠀⢠⣄⣿⠂⠀⡀⠸⠿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠂⠀⠀⠀⠀⠀⣿⣧⣄⠈⢿⣿⢠⠞⢀⣀⠀⠈⠙⢻⠛⠉⠛⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠠⢀⣀⣀⢉⡙⢷⣶⠀⡿⠂⠀⢿⣿⡿⣄⠀⣸⠀⠀⠀⠀⠀⣿⣛⣧
⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⢸⡇⠀⠰⡆⠻⣝⣹⣿⠃⠈⠛⠄⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⢀⠜⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢸⡇⠀⠂⠉⠊⠟⠳⠛⠀⠀⢀⡀⡇⠀⡀⢀⣠⡀⣸⢶⠃⠘⠃⢀⣼⣶⣵⣤⡀
⠀⠀⠀⠀⣷⡄⢀⡀⠀⠀⢸⡇⠀⠀⠀⠀⢀⠀⢁⠀⠈⡿⠃⡇⠀⡃⢠⣽⣿⡏⣤⡀⠀⢀⢸⠿⣿⣿⣿⣿⣆
⠀⠀⠀⠈⣻⣿⣧⣿⣷⡀⠸⣅⠀⠠⢾⠀⢼⠀⣀⢁⣤⡤⢀⠏⠀⠄⠸⢃⠺⣤⣼⡧⢠⠇⢀⡑⢻⠾⣯⣿⠇
⠀⠀⠀⠀⠨⡻⢿⣜⠿⣃⠘⠇⠀⠀⠀⢀⠠⠋⢇⣼⣿⣯⣷⠄⠀⠀⠀⠘⠀⢸⠅⢰⡎⢶⠺⠄⠈⠛⠋⠁
⠀⠀⠀⠀⠀⢌⣵⣬⣧⣽⣦⢷⡄⢀⣾⣿⡓⢿⡿⣿⣿⣷⠟⠀⢠⠀⠀⣿⠀⣼⠀⠸⢠⠃⢠⠀⣠
⠀⠀⠐⠀⢰⣟⣻⣿⣿⣶⣟⣱⣿⢻⣿⠟⢀⡿⠷⣿⣿⣿⡇⣠⣾⡇⠀⢻⠀⠀⣼⣿⣓⣦⣸⣿⣇⣴⢃⣤⡄
⠀⠀⠀⣇⠺⣿⡿⢿⣿⣷⣾⠿⠾⣻⣿⠄⢠⣷⣾⣄⢡⠻⢿⣿⣿⡇⠀⢸⣀⣾⠿⣷⡄⠻⣿⡿⢯⣿⣏⣞⣡⡤
⠀⠀⠀⠈⠟⠴⡿⢻⣄⡹⣸⡟⠙⠟⠁⠐⣿⣿⣿⡿⠘⡀⠀⡿⢿⣅⣠⢸⠉⡇⢰⠿⣿⡄⣠⣴⣟⣻⣽⣯⣥⡤⠄
⠀⠀⠀⠀⠀⠉⠑⠋⠘⡁⠉⣿⠀⠀⣄⠀⠈⠙⣻⣀⣀⣧⢤⣷⣼⣤⣽⠾⠶⢷⣮⠰⢒⡿⣿⣿⣤⣧⡠⡉⠐
⠀⡓⠐⢦⣜⣿⣷⣤⣄⡇⠀⣏⣾⠀⢻⣷⡀⣼⣿⢟⣽⣾⣿⣿⡏⡿⣿⣿⣛⣿⣿⣷⣻⠇⠋⡏⢻⠙⠛⠃
⠀⡇⠀⠀⠈⠉⠛⠛⠛⠛⠀⣿⠇⠀⠾⣿⣧⠟⣧⣿⡿⢿⠁⢸⣴⡇⡿⠿⣯⣭⢿⡟⠎⠀⢀⣴⣾⡿⣿⠿⣷⠆
⠀⠃

⠀⠂⠀⠀⠀⠀⠀⠃⠐⠃⠘⠃⠘⠁⠘⠀⠀⠉⠐⠇⠐⠃⠀⠀⠒⠃⠻⠀⠓⠀⠡⠀⠀⠇⠀⠓⠀⠃
⠀⠂
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⢲⡶⠒⠒⠛⠻⠷⠻⠿⠟⠛⠛⠛⠛⠁⠹⠗⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣆
⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡄⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸⡃⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠠⣿⣀⣀⣀⣸⡟⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡄
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣽⠀⠈⠉⠉⡏⠀⠀⣿⡀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠁⠀⠀⠈⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠲⣶⣤⣤⣤⣤⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠉⠙⠉⠛⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⡀⣀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠀⠀⢀⣤⡦⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣃⣠⡴⠴⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠾⠟⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣾⢏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠈⢻⣇
⠀⠀⠀⠀⠀⠀⣸⡏⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⣿⢁⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣷⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⣿⠀⣿⣶⣖⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠒⣂⠀⢹⣿⡰⣣⡦⠀⣼⡇
⠀⠀⠀⠀⠀⠀⣿⠀⢻⠟⠛⠉⣀⣀⣈⢙⠒⠒⠒⠒⡠⠤⠞⣁⡴⢂⡀⣉⣁⠰⢾⡗⣿⣿⠀⣿⠇
⠀⠀⠀⠀⠀⣴⠻⡆⣼⢃⣶⣿⣿⣿⣿⣿⣿⣦⡯⠁⠂⣤⣭⣵⣾⣿⣿⣿⣿⣷⡀⢿⣶⡏⢠⡟⢛⣦
⠀⠀⠀⠀⢰⣿⡄⢻⣿⢿⣿⡿⢛⣉⡙⢿⣿⣿⠅⢤⣠⡹⣿⣿⡿⢛⠛⡳⣝⣿⣷⡘⣿⠁⡼⣠⣿⣹⡆
⠀⠀⠀⠀⣼⠹⣷⡼⢟⣿⣾⣧⣈⣛⣃⣼⡿⣿⡆⣤⡸⣿⠟⢿⣄⣘⡻⣃⣸⣿⢿⢇⡘⠆⢱⣏⠋⣿⠃
⠀⠀⠀⠀⣿⢰⣿⡀⠐⠿⡿⠿⠿⠧⢯⣿⡇⡜⣸⣿⣿⣜⢷⣷⣞⠛⢩⢭⠴⢚⣣⠞⠀⢠⡟⣿⡇⡿
⠀⠀⠀⠀⠹⣎⠻⣧⣀⣀⣀⠄⠉⠉⠁⠉⢀⣾⣿⣿⣿⣿⣧⢻⣟⣆⣈⢉⡙⣉⢩⣤⣶⡿⢱⠟⢱⡇
⠀⠀⠀⠀⠀⠙⣧⠆⢙⢻⣿⢀⡀⢎⠄⠀⠘⠛⠟⢡⠻⠿⠟⠘⠁⣄⣠⣾⡇⣿⣿⣿⡟⡇⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠹⣦⣼⡆⡿⣾⣷⡏⡆⡄⠀⢀⠀⢀⠀⠀⠀⠀⠀⠈⢻⡏⠘⣿⠋⡿⣴⣷⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⢳⣹⣿⡴⠷⣷⠷⣾⠷⡾⢤⡴⢾⡾⣦⣴⣼⡹⢰⣿⣸⢿⡇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⢹⣿⣷⣤⣿⡀⣸⡀⣇⣸⣠⣼⣄⣴⣤⣿⢹⣴⣿⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠀⢟⠛⠿⠿⢿⠿⡿⢿⣿⠟⣿⡟⢻⡿⠟⣨⠟⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠰⡌⠻⣶⢧⣼⣠⡄⣰⣿⣀⡦⠵⢊⣴⡞⢁⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡉⠰⡉⠀⠲⣦⣬⣥⣤⣤⡶⣶⡟⠁⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡿⣦⡀⠚⣄⠀⠉⢩⣤⣴⡆⠈⢀⣴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢾⡜⣠⣶⢰⢠⣤⣠⢴⢾⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠛⣷⠇⠸⡿⠞⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠤⠄⠀⠀⣰⡷⠀⠀⠀⢠⣄⠀⠀⢠⣾⠿⠛⢿⡄⠀⠀⠀⠂⠀⠀⢸⣿⠗⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⠀⠸⠏⠀⠀⠈⠀⠃⠐⠃⠁⠀⠀⠀⠑⡀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣴⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠿⠿⢿⣷⠀⠀⠀⠲⠀⠀⠀⠋⠭⠩⡛⠇
⠀⠀⠀⠋⠉⠉⠀⠀⠀⣾⣽⡆⠀⠀⠀⠒⠚⠒⠀⠀⣾⣇⠂⠀⠈⢸⣿⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⢠⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠉⣁⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡛⠻⣿⣷⠀⠀⠀⡆⠀⢠⡾⠿⣿⡆
⠀⠀⠀⠐⠤⠢⠀⠀⢾⣿⠁⠀⠀⢈⣿⡄⠀⠠⠠⠂⠀⠀⠐⠋⠙⠛⢠⣿⣿⠀⠀⠀⠁⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢂⠟⠀⢀⢉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠠⠛⠄⣀⣀⠁⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⢤⢀⠙⠀⠙⢱⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⣆⠀⠄⡠⠀⡅⢤⣸⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣬⠦⡄⠁⠴⡥⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡻⠇⢡⠴⠀⠀⠀⠀⠀⠀⣈⢥⢴⠞⡃⠀⠀⢳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⡄⢠⠀⠀⠀⠀⠠⠊⠠⠚⠚⣶⣿⠒⢈⠏⡘⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⠀⢠⡀⠀⠀⡰⢁⣠⠁⠀⣼⣿⠉⠀⢎⣰⠃⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠁⠑⡀⡰⠁⠘⣿⡧⠖⠙⣿⣇⡘⡸⡖⠁⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠰⠁⠀⢀⠸⣿⠁⠀⠈⣿⣎⡔⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠁⠀⡼⠀⢹⡆⠀⠀⠘⡿⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⡴⠃⠀⡘⢻⡄⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠊⠀⠆⠀⣇⣀⠛⢢⣤⣀⣤⠞⠁⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢠⡇⠀⠀⠈⠀⠀⠀⠀⠀⢀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠲⢼⠇⠀⠀⠀⢇⣤⣴⣶⢾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢰⣿⡇⠀⠀⠀⠀⠠⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁