Doctor of medicine Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠼⢤⡙⠈⠶⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠄⠀⢀⠹⡆⠀⠛⠈⡥⠲⠖⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠈⢄⣀⡜⢠⡟⠲⠦⠤⠄⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣇⣠⡴⠟⠼⣷⣄⣀⣀⣀⡼⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠓⠶⣄⠀⢠⣞⠄⠠⡀⡤⠒⢢⡘⡏⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠈⢷⣟⠠⣸⡿⢻⠀⣿⣯⡟⠀⠀⠀⠀⣻⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠈⠱⠟⠋⠉⠁⠀⠀⠀⠁⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡞⠃⠀⠀⠀⠀⠀⢤⣄⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣄⣀⣀⣀⣤⠞⠀⠀⠀⠃⠀⣿⠀⠀⠀⠀⠀⠀⠸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⡿⠋⢁⡀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠘⠻⢦⣤⣤⡴⠟⢁⠀⠀⠀⠀⠀⠀⠀⣿
⣀⣀⣀⣤⠤⠖⠒⠚⠉⠉⢉⡉⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢺⡀⠀⠀⠀⠀⠀⠀⠠⠶⠋⠀⠀⠀⠀⠀⠀⣰⡇
⢻⠁⠀⠀⠀⠀⠀⠀⠒⠂⢤⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⠙⠛⣶⡄
⢸⡀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠙⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⡰⡏⠀⠀⠀⣆⠀⢙⡷⣦⣤⣀⣀⣀⣀⣀⣤⣴⠾⠋⠁⠀⠀⠀⣬⠹⣦
⠀⡇⠀⠀⠀⢔⠉⡠⢀⢈⠀⡁⡀⢠⢸⡇⠀⠀⠀⠀⢀⠞⢱⠁⠀⢠⠖⠋⠀⢸⡗⠉⡀⣩⠟⢯⡏⠉⠡⠿⢤⣤⡄⠀⠀⠀⣷⡄⠈⢷⣄
⠀⣇⠀⠀⠀⢄⡉⠀⠈⠆⠉⡰⠁⢸⠘⡇⠀⠀⠀⡰⠋⣰⡏⠀⠀⠀⠳⠀⠀⢸⡇⠀⣿⠁⠀⢸⡇⠀⠀⢀⣴⠞⠀⠀⠀⠀⣿⠙⢆⠀⠙⢷⣄
⠀⢸⠀⠀⠀⠀⢨⣀⣀⣀⣤⠄⠀⠀⠀⣷⠀⠀⡼⠁⡴⢣⠁⠀⠀⠀⠀⢡⡀⣸⡇⢰⠉⡇⠀⢸⡇⠀⣠⠞⠁⠀⠀⠀⠀⠀⣿⠀⠈⠳⢤⡀⠙⠳⢦⣀
⠀⠸⡄⠀⠀⠀⠀⠨⠤⢤⠤⠃⠀⠀⠀⢻⡤⠚⠃⡼⠁⡌⠀⠀⠀⠀⠀⠀⠁⣿⢀⡏⠀⣿⠀⢸⠃⣼⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣠⠽⠓⠒⠂⣈
⠀⠀⡇⠀⠀⠀⢀⣰⠏⢻⡏⢆⡤⡄⠀⢸⡆⠤⠊⠁⢀⠁⠀⠀⠀⠀⠀⠀⠈⣿⢸⠃⠀⢹⡀⢸⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡾⠋⠀⣠⡴⠚⠉⠁
⠀⠀⢸⠀⠀⠀⣏⢸⡄⠀⡇⢸⠀⣡⠈⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠘⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⢦⡴⠋
⠀⠀⠈⠛⢿⢁⠹⢆⠷⠴⢦⡼⠶⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢰⡆⠀⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢸⡟⣇⠀⠀⠀⡿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣸⠅⢂⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⢦⣀⡼⠧⢾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣄⡀⠀⠀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠓⠒⠲⣶⠶⠾⣀⠋⠙⠛⢻⡇⠸⣥⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡏⠀⠀⠀⢸⠀⢀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⡇⠀⠀⠀⢸⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠈⡇⠘⣧⠀⠀⠀⠈⣇⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠉⠓⠶⠤⣀⣀⣀⣀⣷⣀⣹⡆⠀⣀⣀⣼⡀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⠤⣞⠉⠛⠙⠶⠶⠶⠶⠶⠛⠛⠉⠁⠀⢀⡄⠉⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⣀⣀⣀⣀⣀⣤⣤⠴⠞⠉⠷⠤⠴⠇
⣾⣾⣶⡾⣷⣷⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣿⣷⣾⣶⣶⣶⣿⣷⣶⣿⣿⣿⣷⣾⣷⣶⣶⣶⣶⣶⣶⣶⣷
⣿⣿⣿⣀⣀⣤⣤⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⠒⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⠄⠀⠀⠀⠀⠀⢄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡆⠠⡴⠁⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠁⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣶⠹⣅⠀⠀⠀⠀⠀⢀⡤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠯⢥⣈⣿⡀⠈⠳⣄⠲⣄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⡰⠯⢄⠀⠀⠀⢰⠋⣷⣀⣀⠬⢣⡶⣯⣦⡀
⠀⠀⠀⠀⠀⡜⠀⠀⠀⢳⡀⠀⠈⠲⣼⡏⠑⡄⢸⢠⡿⠉⠻⡑⢠
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢣⠀⠀⠀⠈⢻⢄⠘⣼⣸⢻⠀⠀⠹⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡻⡷⣽⣿⠘⣆⣀⠀⠱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠙⠚⠁⠘⡇⠀⠈⠉⠈⠛⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢣⢇
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⢸⠺⡀⠀⠀⠀⠀⣀⠔⠚⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠈⠓⠦⢄⣀⠀⡇⠀⠀⣀⣀⡼⣅⡑⢦⣤⠴⢊⠁⣀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢩⣷⠊⠁⠀⠠⢤⣠⠿⠋⡵⡛⠍⣺⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠉⠙⠳⠦⣄⡀⠀⢰⣃⡧⠤⠖⢉⡽⠛⣁⣤⠺⣥⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠋⠀⠃⠀⠚⠑⠒⠉⢹⠉⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠋⠉⠑⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⢹⣀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠼⠁⠰⠊⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⡝⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢨⠉⠀⠀⠀⠀⠀⢁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⠀⡴⠒⠂⢀⡜⠛⣦⡀⠀⠀⠀⠰⠂⠅⠀⠀⠀⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⣆⠇⢴⡧⠎⠰⣿⣿⠉⠀⣀⣤⠶⠖⠺⠟⡛⠛⣦⠀⠈⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⣟⣲⡆⠿⠁⠤⠝⠉⠗⠋⠁⠀⠀⠀⠀⠀⠽⢂⡿⠀⠐⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠀⠀⠓⠂⠀⠀⠚⢻⡟⠲⠄⠀⠀⠲⣶⢶⡏⠀⠀⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⢀⡾⠀⠀⠀⠀⠀⢀⡟⠈⡅⡀⢃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⣀⣠⣴⠾⣿⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⡾⠑⠊⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣻⡀⠀⠛⠚⠻⣤⠴⠊⠀⠀⠀⢠⡞⠋⠛⡖⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⢹⢷⡀⠀⠀⡀⠀⠀⠀⠀⢀⣴⠏⠀⠀⣸⠁⠀⠀⠀⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⢸⡆⠙⢦⣬⣄⣀⣀⣤⠶⢻⠏⠀⠀⠈⠛⢛⡶⠃⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠉⡽⢛⠣⡀⡼⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢷⡟⠀⠀⠮⠀⢸⠀⢹⠃⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⡆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡇⠀⢰⠺⠘⠀⠀⢠⣏⡀⠀⠀⠀⡀⠈⠄⡏⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⣨⢸⢀⠀⠀⠀⠈⠙⠶⣄⠀⠡⠀⣼⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⠀⠀⠈⠉⠉⠉⠉⠙⡄⠀⠈⠳⣄⢃⡇⣰⠃⠀⠀⠀⠀⢀⣠⠶⠛⠳⣸⡃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠈⠉⠉⠉⠁⢰⡇⠀⠀⠀⠘⣎⣿⣥⡄⠀⢀⣠⠶⠋⠁⠀⠀⢀⠀⠑⠄⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡠⢷⡤⠴⣄⣀⣤⡤⠾⠶⠒⠚⠛⢹⠋⠀⢰⠋⠉⠁⠀⠀⠀⠀⠀⢠⣄⠉⠲⢄⠐⢆⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠐⠁⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠀⠀⠀⠀⠀⣀⡴⠚⠉⠉⠳⢤⣀⠙⢻⣿
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣧⣀⣀⣹⠂⠒⠒⢚⣿⠁⢀⡄⠠⡀⠠⡄⠙⡏⠉⡟
⠀⠤⠤⢤⣤⣤⢤⣏⠀⠀⠀⠀⢀⣀⣀⣠⣤⠴⠶⠾⠛⠋⠩⠠⠤⠤⠬⠤⠤⡄⠈⢀⣼⠀⢘⡇⢰⡇⠀⡇⢀⡷⠚⠲⠒⠶⠐⠶⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠈⠓⠚⠓⠚⠂⠞⠓⠚⠁⠀⠀⠀⢠⡤⡔⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣶⠒⠒⠒⠒⠒⠒⠒⠛⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢼⣤⣌⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠰⣆⡇⠀⠀⠈⢧⠀⠉⠙⠓⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠡⡄⠤⢀⠀⠀⠀⠀⠀⠀⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠸⡓⠊⠁⠀⠀⢱⡄⠀⠀⠀⠀⠐⡌⢧⠀⠀⠀⠀⠀⢀⡠⠄⠙⢧⡀⠀⠀⠀⢠⡎⠉⠉⠁⠂⢄⡀⠙⢷⣄⠀⠀⠀⣼⣃
⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⣀⠀⠥⠐⠁⠀⠀⠀⠀⠀⠘⡌⢧⠀⠀⠀⣊⡥⠶⠛⠁⠀⠙⢦⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠲⣄⡉⠛⠛⠋⠉⠉⠙⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠰⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢣⠀⠈⠁⢀⣀⣠⡤⠶⠶⠟⠃⠀⠀⠻⣤⣬⡴⣈⣑⡄⠀⠈⠻⢶⣄⣀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠲⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠠⡀⣹⢱⡀⠀⠀⠀⠈⠉⠛⠛⠿⠛⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣂⣤⣤⣔⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠠⢄⠀⠀⠌⠛
⠀⠀⠀⠀⠀⠀⠺⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠴⠟⠀⠀⢲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣶⠖⠒⠒⠒⠈⠙⢿⡃⠐⠒⠒⠒⠀⣤⡀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⡏⠀⠀⣀⠔⠉⢲⠈⠃⠀⠀⠀⠀⠀⣠⡿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠙⠿⠟⠛⠰⣿⡛⠂⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠊⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠂⠀⢀⡘⠛⠛⠓⠀⠀⠀⠀⢀⠀⠀⢺⣶⡾⠋
⠀⠀⠀⠀⠀⠀⣶⡟⠛⠛⠻⣶⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠉⠁⠀⠀⠀⠀⣨⡿⠀⠀⠀⠀⠀⠀⠀⣹⠃⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⣀⣠⣴⡾⠛⠁⠀⠀⠀⠀⢀⡠⠞⠃⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⢸⣧⠄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠛⠛⢻⣧⠀⠀⠀⠀⠠⠤⠚⠉⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠘⣷⠄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠐⠒⠛⠃⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⣿⡆⠒⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡇⠴⠘⣿⣀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠠⠄⢺⡦⠀⠀⣸⠋⠐⢀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣀⣼⣄⠀⣴⣦⠤⠐⠿⠖⢰⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠘⠚⠓⠖⠃⠀⢀⣀⣤⣶⣿⢷⣤⡀
⠀⠀⠈⡁⠈⠿⣯⣥⣤⣤⠤⣶⠖⠉⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣄⣀⣀⣀⣤⣤⣤⡶⠾⠟⠋⠀⣠⠟⠀⠈⠛⠷⣦⣄⡀
⠀⣀⣀⠙⢶⡄⠀⠹⣭⠁⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠹⣍⠉⠉⠉⠉⠉⠀⠀⠀⠀⣀⡴⠋⠀⠀⢀⡀⠀⠀⠉⠛⠷⢶⣄⣀⡀
⣾⡋⠁⠉⠚⠛⢷⣄⢹⡇⢸⠀⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⠑⠂⠠⠤⣤⠤⠤⠐⠚⠉⠀⠀⠀⠀⠈⠻⣷⣦⣤⣀⡀⠀⠈⠉⠙⠛⠳⠦⣤⣤
⠈⠉⠓⠾⠷⠦⣤⡈⢨⡇⢸⠀⡇⠀⢠⡄⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡈⠉⠙⠲⠶⢦⣤⣤⣤⣤⠀⢰
⠀⠀⠀⠀⠀⠀⠈⢷⠀⡇⠸⠀⡇⠀⠈⣧⣤⡀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⣠⡾⠁⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⡇⠀⢠⡟⠉⢷⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⣰⠟⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣬⠛⠛⠛⢻⢠⣾⡁⠀⠸⣇⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⢀⡾⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠈⠐⠻⣷⠉⢳⡄⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠟⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣀⣠⡿⠀⠀⢻⣄⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠙⢶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠘⣧⠀⠀⣠⠶⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠘⡁⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢷⡼⠋⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣠⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⣶⣶⣤⣤⣄⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠋⠉⢉⡿⠛⠛⠛⠛⠛⠛⢿⡿⠛⠻⠿⣿⠟⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⣞⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠉⠷⡄⠀⠀⠀⠀⠈⢷⡄⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢹⡆⠀⠀⠀⠀⠀⠛⣄⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣧⢀⣀⣼⡀⠀⠀⠀⠀⠀⠀⣷⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠉⠉⠉⠙⠿⠓⠒⠚⠒⠙⡛⣩⡄⠀⣈⠉⠀⢻⡄⠀⠀⣠⡴⠞⠁⠀⢀⡀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢦⣀⣌⣀⠀⠀⠀⠈⢁⣀⣤⡶⠛⣧⣀⣼⡷⠔⠛⢁⠀⠈⠁⣠⣾⣇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠻⣿⡛⠛⠛⠓⠊⠉⠉⠈⢣⠀⠀⠈⢀⣠⡾⠋⠀⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⣀⣀⣀⣤⣤⣼⣦⡶⠒⠋⠁
⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠁⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⠀⠀⣠⢴⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠶⢒⣚⡓⠲⣄⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⡞⠀⡞⠁⣠⡿⠋⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⣤⠴⠋⠉⠙⠷⣄⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠤⡀⢨⣠⠼⣁⡤⠬⣏⠁⠀⠀⠀⠀⠀⠀⢀⡇⠀⠱⠤⣄
⠀⢠⠞⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡈⢣⡀⠀⠀⠈⢻⣦⠀⠀⠀⢷⡄⠈⢦⠀⣼⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⠀⠈⢳
⣰⢋⡶⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠱⡄⠀⠀⠀⠙⣷⡄⠀⠀⠙⢦⣠⠇⠘⠶⠶⠶⠋⢀⡀⠀⠀⠀⠸⡇
⢻⡸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⠀⠀⠀⠀⠈⢯⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣌⢢⠀⠀⠀⢷⠀⠐⠀⠀⠀⠀⡄
⠀⠻⢮⡓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠎⡀⣼⠃⠀⠀⠀⠀⠀⠑⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⡸⡆⠀⠀⠘⡄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠙⠲⣌⡙⠲⢤⣄⣀⡀⠀⠀⣀⣠⠴⢏⠀⠀⣟⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠢⣄⠀⠀⠀⠀⠀⠘⡌⡇⢧⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣈⣉⣉⣉⣭⡴⠚⠉⠉⢣⠈⣆⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⢦⡀⠀⠀⠀⠀⢣⢻⡘⡆⠀⠀⢳⡀⠐⠀⠒⠊⢀
⠀⣀⡀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢸⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠳⡀⠀⠀⠀⠸⡎⣇⢱⠀⠀⠀⠙⠓⠒⡄⡞⠁
⠀⡏⠉⢉⠁⠀⠀⠈⠉⠓⠦⣤⡀⠀⠀⠀⠀⠀⠀⢸⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠹⣄⠀⠀⠀⢻⣹⡈⡆⠀⠀⠀⠀⠀⢸⡇
⠀⡇⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⢸⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠘⢦⠀⠀⠈⡎⣧⢳⠀⠀⠀⠀⠀⢸⢳
⠀⡇⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⢤⠀⠀⠀⠳⣀⠙⠲⠤⣤⣤⣤⣤⣤⠄⣀⡀⠀⠘⢆⠀⠀⠀⠣⣠⣀⠹⣼⣮⡆⠀⠀⠀⠀⢸⣾
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⢳⣾⠁⠀⠀⠀⠈⠉⠃⠒⠒⠒⠒⠒⠒⠒⠦⣌⠙⢦⡈⢧⠀⠀⠀⠀⣸⠇⠈⣿⣆⠀⠀⠀⢠⡿⠃
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣟⣿⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠈⢧⠀⣧⢸⣀⣀⡤⠊⠱⠤⢄⠘⠛⠀⢀⣴⡟⠁
⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⠿⠀⠈⠑⠲⣄⠀⠀⠀⠀⢀⡜⡵⢺⡇⠀⠀⠀⡼⢀⡇⠀⠉⠀⡟⠀⠀⣀⣸⠀⠀⣠⡾⠋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡂⠀⠀⠀⠀⠈⠙⠒⠒⠒⢺⠚⢀⣾⠇⣠⠶⠚⠀⠜⠲⢦⠀⠘⢦⡤⠞⠉⠀⠀⢰⢹⠁
⠉⠒⠢⠤⢄⣀⣀⣀⣀⣀⣀⡀⣿⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠿⠃⢸⡃⠀⠀⠀⠀⠀⢈⠇⠀⠀⢀⣀⣀⡀⠀⠈⢯⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⣀⠞⠁⠀⠀⠀⠉⠛⠲⠶⠒⠚⠉⠀⠀⢰⣫⠒⠚⠯⢽⣖⡯⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠐⠰

⠀⣀⣸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢾⡶⠄
⠀⠉⠹⠉⢁⣀⡀⠀⠀⠀⣟⣻⣿⣛⡆⠀⠀⠀⠀⠰⣿⣿⣿⠄⠘⠀⠀⠐⠂⠂⠀⠐⡷⣄⠀⠀⠀⠀⡟⠛⠛⠛⠛⠛⡇⠀⠀⠀⠀⣠⣤⣶⣶⣤
⠀⠀⠀⢠⡿⠟⢿⡀⠀⠀⣯⣽⣿⣭⡇⠀⠀⠀⠀⢠⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⣯⢭⠉⠭⣹
⠀⠀⠀⣾⣧⢀⣼⡇⠀⠀⡷⢺⣿⢲⡇⠀⠀⠀⠀⠘⠛⣛⠛⠂⠀⠀⠠⠀⠀⠀⡄⡠⢤⠀⡸⠀⠀⠐⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠋⠀⠉⠀⠹⠀⠀⣏⣹⣿⣩⡇⠀⠀⠀⠀⢀⣀⣛⣁⡀⠀⠀⠀⠑⢤⠔⠀⠳⢤⠗⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠃⣶⠐
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠷⠾⠿⠶⠇⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠈⠢⠤⠤⠊⢀⣀⣇⣀⠈⠓⠒⠒⠒⠒⠚⠁⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⣰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠉⣿⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠐⣶⣿⣶⠀⠀⢸⣿⠀⡆⢸⠋⡈⢿⠁⣌⣥⡄⣤⠏⣀⢹⠁⡌⣻⣿⡇⢠⣸⠁⣿⠁⡏⢠⢙⡏⠈⣿⠁⡄⢻⡄⢠⣼⣿
⠀⠀⠀⠀⠉⠉⠉⠀⠀⢸⣿⠀⠃⣼⠀⡇⣠⠀⡿⣿⡇⣿⠀⣿⢸⠀⠀⢿⣿⡀⢸⢿⠀⣿⠀⡇⢀⣾⠁⠀⢹⡀⡀⢿⡇⢸⣿⣿⠀⢰⣶⢲⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⣼⣿⣦⣤⣿⣤⣴⣿⣇⣽⣧⣤⣾⣄⣷⣼⣿⣧⣀⣼⣤⣡⣤⣧⣸⣿⣼⣧⣼⣇⣿⣼⣧⣸⣿⡿⠀⠀⣿⢸⣿
⠀⢰⡄⠀⢀⣀⡀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⣿⢸⡏
⠉⢹⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠪⢀⡦⡆⠴⠀⠸⠄⠂⠔⡢⠀⠜⡌⠒⠤⠐⠄⠀⠀⠰⠠⠢⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡏
⠀⢀⣴⣶⣶⣠⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣶⡆⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣤⡄⠀⠀⢀⡀⡠⠤⠤⠤⢤⢀⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣿⣿⣿⡿⠿⢿⣿⣿⣿
⠀⠈⢿⣿⣿⣿⣯⣿⡿⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⢀⢀⣀⣀⣀⣀⣀⢠⣤⣤⠀⠀⠀⠀⣿⣿⣿⠰⣿⠆⣿⣿⣿
⠀⠀⠀⠙⠿⣿⠿⠋⠀⠀⠤⣾⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠜⠓⠚⠒⠒⠚⠛⠛⠛⠸⠿⠟⠀⠀⠀⠀⣿⣿⣿⣷⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡠⠚⣵⣿⣯⢙⣦⠀⠀⠀⠀⣿⡄
⠀⠀⠀⢀⠃⢰⣿⠀⠀⠀⠐⡀⠀⠄⢱⡄⢿⣿⣎⣳⠄⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠈⠀⠘⣿⡀⠀⠀⠀⢡⡀⠀⠀⢹⣾⣿⢶⡋⠊⠀⠀⠀⢠⣿⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⣠⣤⣿⣦⣆⠀⠀⠀⠳⡀⠀⣾⣿⡟⡾⠁⠀⠀⠀⠀⠘⣿⠇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⣼⡟⠉⠉⠉⢻⣷⠀⠀⠀⠱⣸⣯⣿⣾⠁⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⠀⠀⢣
⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘⢧⣾⠃⠀⠀⠀⠠⠂⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡆⠀⠀⠀⠀⢣
⠀⠀⠀⠹⠇⠀⠀⠀⠀⠙⠁⠠⣀⠀⠀⢀⠃⠀⢀⠔⠈⠀⠀⣀⣀⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠁⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠔⠁⠀⠀⠀⠈⠹⣯⣙⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⢣⣰⣯⠍⢀⠟⠀⣀⠀⢲⣶⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⢹⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠊⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠘⣇⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠄⡌⠀⠀⠀⡀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠠⠄⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠐⠁⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠁⠀⠀⣠⣤⢀⡀
⠀⠀⠀⢠⣦⡀⣀⡻⠦⠀⢴⡦⠀⠀⡇⠀⢀⡀⠀⠀⠀⠀⡀⢠⠂⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠵⣬⣤⡀
⠀⠀⠀⠈⠛⠁⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠻⠇⠀⠀⠀⠀⠑⠀⠀⠀⠘⠘⠀⠀⠰⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢀⣖⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣂⣀⡀⠀⠀⠀⣇⠀⠈⠀⠀⠊⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣌⣩⣿⡭⣉⣭⠍⣀⠀⠀⠀⠀⠀⡇⠛⠤⠄⠶⠄⣶⡜⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⢀⠀⢤⡀⠀⠀⠀⠀⠀⡠⣴⣿⣿⣿⣿⣿⣿⡽⠚⠉⠀⠀⠀⠀⠀⢰⠀⡄⠐⢄⠰⣾⡟⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⢀⡀⠀⠈⠙⢦⠀⠘⠛⢋⠙⣉⣙⣋⣁⣄⣡⡠⡤⠤⠤⠀⠀⠀⠀⠀⠈⠆⠀⡠⠋⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠄⠐⣂⠙⠃⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣙⣻⣉⡟⣻⣟⣻⡿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠻⣿⣿⣿⣿⣿⣿⣿⡿⠃⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠈⠛⠻⠿⠿⠛⠋⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣧⠙⣷⣄⠀⠀⠀⠀⠀⢀⣼⠟⢠⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣇⠈⠙⠿⢶⣶⣶⠾⠟⠁⢀⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣿⡇⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠋⠉⠙⢿⣿⡄⠀⠀⣿⡇⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⠀⠛⠀⣸⣿⣿⡄⠀⣿⡇⠀⣼⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡆⣾⣿⣿⣿⣷⡀⣿⡇⣰⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠘⣿⣿⣿⣿⣷⣿⣧⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣌⠻⠿⠟⢋⣼⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠻⠿⠿⠿⠿⠿⠿⠻⠟⠿⠻⠿⠿⠿⠿⠿⠿⠛⠟⠀⠻⠿⠻⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡂⠀⠀⠀⠂⠀⠀⣆⠎⢿⣄⢺⣦⡄⠀⡏⠈⠙⠃⠀⣹⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⡇
⠀⠀⠀⢰⠀⠀⠀⠀⠠⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⣴⣃⡀⠤⢹⣏⣿⡆⠹⣌⠊⠉⡴⠃⡀⠄⠀⠀⠀⠀⠂⢤⠀⠀⡂
⠀⠀⠀⠈⠀⠀⠀⢆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡕⣄⢄⠸⡴⢋⢹⡂⠀⠑⠊⠀⠀⡁⠀⠁⠀⠀⡀⠁⠉⠀⠈⢉⡀
⠀⠀⠀⠐⠀⠀⠀⠈⠒⣖⠐⠁⢠⠆⢠⠒⠁⠈⠒⡄⠀⠀⠸⣇⡴⢎⠀⠀⡟⣶⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠖⢺⡃
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠢⠔⠃⠀⠈⡄⠀⠀⡜⠀⠀⠀⡠⠋⣧⢲⠀⢀⠃⣽⠂⠹⡀⠀⠀⠀⡠⠊⠓⢀⠀⢰⠀⠘⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⣶⠊⠀⠀⠀⢰⠃⢰⡋⢿⡄⣼⡚⠛⠒⢛⢣⠀⠀⠀⢇⡀⠀⣸⠀⢘⠀⠘⠀⢠⡄⠘⣀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠂⠀⠀⡅⠀⠀⢻⡀⠀⠀⢠⡇⠀⣸⠗⠚⣧⣻⠁⠀⢀⡀⠀⡆⠀⠀⠈⠑⠊⠁⠀⠘⠤⠌⠀⠀⠿⠜
⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢸⠀⠰⠛⠁⣱⢇⡉⠠⠤⠴⠴⠄⢸⠀⢐⢂⡌⡀
⠀⠀⠀⠀⠿⠀⠀⠰⡀⠸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠸⣄⡠⠴⠎⠀⢸⠁⠛⠀⠠⡤⠤⠄⠀⠀⢨⡇⡂⠀⠀⡀⠀⠀⠉⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣲⠄⢸⣷⣦⠀⠀⠀⠀⣸⠀⠀⢸⠀⠀⠀⠀⢇⠀⠀⠀⠀⠈⠁⠀⠀⠀⢠⠀⢸⢈⠸
⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⢪⠴⠛⠉⠀⠀⠉⠁⠀⠀⠀⠀⡇⠀⠀⠌⡀⠀⠀⠀⠸⡀⠀⠀⠀⢀⡄⠀⠀⠀⢸⠀⢸⢸⠀⠀⠖⠲
⠀⠀⠀⠀⠀⠀⠠⠀⠔⢀⠁⠀⢀⠀⠤⡀⠀⡖⠄⠐⠀⠀⠀⡇⠀⠀⡗⡇⠀⠀⠀⠀⡇⠀⠀⢀⡞⠳⠀⠀⠠⠜⠀⠠⣘⡶⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⢠⠀⠀⠀⠀⣧⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⠀⠀⣿⢈⡀⠀⠀⠀⣿⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢈⡅
⠀⠀⠀⢀⣤⣤⠀⠀⡈⡇⠃⠀⠀⠀⠀⣀⣄⣆⠀⠀⠀⠀⢀⡇⠀⠀⡟⠀⡇⠀⠀⠀⣿⠀⠀⠀⣀⠀⠀⠀⠀⠀⡘⠀⠀⢠⣂
⠀⠀⠀⠀⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠋⢀⠀⠀⠀⢸⠀⠀⠀⡇⠀⢱⠀⠀⠀⢻⠀⠀⢈⠌⢀⠄⠀⠀⡰⢧
⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⠒⠂⠚⠁⠀⠈⠉⢉⠁⠘⠀⢀⠊⢀⠂⠀⠀⠀⠀⠀⠀⠀⢈⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⡀⠀⠀⠀⠓⠰⣄⠀⠀⠀⠀⠀⠀⡄⠀⠈⠄⠀⠀⠀⢸⠀⠀⠈⠂⢄⡲⠁⠀⠀⠀⠀⠀⠀⠀⠠⠐
⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠸⠓⢀⠑⢄⠀⠀⠀⢠⡇⠀⠀⠸⡆⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣐⣀⡁⠄
⠀⠀⠀⠀⠀⠉⠉⢀⡀⠉⠉⠀⠀⠀⢀⠀⠀⠓⠋⣀⣀⠀⢸⡇⠀⠀⠀⠁⠀⠀⢨⢆⠀⠀⠀⠀⠀⡚⠋⠁⠀⢀⠀⠀⠀⠂⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠢⠀⠀⢠⠏⠀⠀⢸⡇⠀⠀⠀⢸⡀⠀⣼⠛⠀⠀⠀⠰⠄⠁⠀⠀⠀⠣⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⡀⠀⠀⠀⠸⠇⢀⠀⠀⡀⡦⢧⢀⠀⢀⠎⢀⠖⠃⠀⠀⠀⠀⠠⠀⠀⢀⠄⡄⡄⠀⠀⠀⠄⠄⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠜⣟⠁⢬⣆⡀⠀⢰⠊⠙⠢⠊⠁⠑⠀⠀⠀⠀⠈⠠⠀⠠⠀⠀⠀⠰
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⠄⠀⠀⠖⣩⠄⠈⣑⠲⡷⢝⣄⠘⣆⠢⠰⠂⢀⡞⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠽⠆⠂⠀⢀⡏⡲⠿⣄⠈⠳⢄⡴⠋⠀⠌⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠣⢀⣀⢀⡠⠀⠀⡀⠀⣀⡀⢀⣀⠀⠀⠀⠀⣿⠀⢃⡢⠁⠊⢯⣑⠄⠇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠏⠀⠈⠀⠀⠀⠀⡇⠀⠀⠀⠘⣴⡞⠃⠀⠀⢨⢹⡃⢾⡄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠴⢤⠀⠀⠉⠀⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠁⠀⠀⠈⡀⠀⢀⠜⠀⠀⠀⢀⠎⠀⣧⣹⠀⢀⠆⣸⡏⠀⢱⠀⠀⠀⠀⡐⠁⠀⠉⠄⠀⡌⠀⠀⠄⠀⡇⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣶⠁⠀⠀⠀⠀⡎⠀⢸⠀⢻⡄⣼⠫⠋⠉⠉⠙⢣⠀⠀⠀⢃⡀⠀⢀⠇⠀⡇⠀⠈⠀⢀⢇⠀⠉⡀
⠀⠀⠀⠠⠸⠂⠀⠀⠀⠀⠀⠐⠀⠀⢨⠀⠀⠀⢻⡀⠀⠀⠀⡼⠀⠀⣼⠒⠚⣥⣹⠁⠀⠀⢀⠀⠈⡆⠀⠀⠀⠉⠋⠁⠀⠀⠰⠤⠌⠀⠀⠨⠄⠃
⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢠⡇⠀⠾⠚⠀⣴⢇⠉⠁⠤⠤⠄⣦⠀⢳⠀⠀⠂⡀
⠀⠀⠀⠀⢰⡆⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠈⡇⠀⢀⣤⠖⠁⢸⡇⠻⠠⠄⣀⣀⣀⡘⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠂⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠀⢳⣤⣄⠀⠀⠈⠈⠀⡜⠀⠀⢸⡇⠀⠀⠀⠈⡄⠀⠀⠀⠀⡀⠃⠁⠀⠀⠀⠀⠀⢰⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡹⠖⠚⠀⠈⠷⠏⠀⠀⠀⠀⠀⠃⠀⠀⠈⡇⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠉⠁⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⢀⠀⠀⠀⡇⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⠠⠀⠘⢸⢀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⡠⠀⠀⠐⠀⠠⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠃⠀⠀⠀⠀⠘⡄⠀⠀⠰⠋⠈⠀⠀⠀⠀⠀⠀⠀⢦⡜⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣇⠰⠀⠀⠀⠀⠀⡇⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⠀⠀⠀⡟⠀⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⣠⢄⣤⠀⠀⠀⠁⠁⠀⠀⠀⠀⢀⣠⡀⢆⢀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⢣⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⡆
⠀⠀⠀⠀⢸⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠈⠁⠁⠁⠃⠀⠀⠀⠀⠀⡟⠀⠀⠀⡇⠀⠘⡀⠀⠀⠀⡇⠀⠀⠀⠠⢁⠠⠀⠀⠀⡴⠦
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠃⠄⠒⠖⠃⠀⠀⠉⠈⠛⠀⠼⠀⠀⠮⢁⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⢠⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⡀⠀⠀⠀⠀⡆⠀⠀⠀⠬⣀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠌
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠰⣄⠉⠢⡀⠀⠀⠀⠀⢸⠀⠀⠀⠠⡄⠀⠀⢀⠁⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡀⠠⠤⠼⠀⠀⠧⠤⠄⠀⠀⠀⠀⠀⠑⢄⠬⠃⠀⠀⠀⣸⠀⠀⠀⠀⠃⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⢀⠤⠤⠀⠀⠀⠀⢀⠀⣂⣀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢲⠒⠂⠀⣿⠀⠀⠀⠀⢠⠀⠀⠀⣫⡤⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠖⡀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡀⠀⠃⠀⠀⠀⢹⠀⠀⠀⠀⢈⠖⠂⢰⠃⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣨⠀⠀⠀⡔⠘⠃⠠⠀⠀⢠⡸⠓⢗⣀⠀⢠⠊⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠰⡰⠀⠀⠀⠀⠊⠐⠑⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⠀⡀⠂⢀⢀⣠⡠⠬⠥⠤⠤⠥⠤⠤⠤⠤⠄⠤⡤⠀⠀⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠀⠀⠐⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⢀⣀⠀⢸⡆⠀⠠⠀⠂⠀⢉⢉⣉⠁⠀⠀⠄⡁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠞⠓⠂⠊⠚⠂⠘⠁⠈⠀⠀⠀⠔⠑⠲⠴⢤⠈⠁⠂⢂⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⡆⠀⠀⠀⢸⠀⠀⡀⠔⠀⠀⠠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡔⠂⡠⠄⠤⠤⠏⠀⠀⠀⠸⠤⠤⠤⡍⡠⠤⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⡖⠦⠤⠴⣦⡈⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠢⣤⠗⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⠃⢀⡈⠀⣠⠈⢁⠁⠙⣧⠀⠀⠠⡄⠀⠀⠀⢀⣠⣤⠀⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡟⠀⡀⠉⡙⣧⢼⠃⢀⣴⠦⢤⡄⢠⠈⠂⠀⠀⠘⠀⠀⠀⠀⠀⢀⠞⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⢃⠀⠑⣄⠐⢣⠌⠀⣸⡿⠉⠻⣿⡇⠀⠀⡀⠀⠠⠀⠀⠀⠐⠀⢀⠎⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠓⡵⠃⠀⠀⠈⢂⡰⠀⡴⢹⠇⠀⠀⢸⡅⠀⠀⢸⡉⣉⠀⠠⠆⢅⠀⠁⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⢃⠞⠀⠘⡇⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠐⠀⡃⠀⡔⡁⣿
⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠚⠀⠀⠀⠀⠀⠀⢀⠉⠉⠀⠀⡴⢁⡟
⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠦⠤⠤⠴⠥⠤⠤⠤⠤⠤⠦⠤⠴⠤⠤⠤⠄⠤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡗⢄⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠋⠀⡀⠀⠀⠀⠀⠀⣰⠊⠀⠀⠀⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⢀⠗⠂⠠⢛⣳⡄⠀⠀⠐⠪⢀⠀⡀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⣿⣘⢿⣁⢻⡿⣄⡠⠖⠚⠉⠉⢭⡙⡆⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠽⠘⠓⠛⠠⠨⠥⠤⢧⠤⡄⠀⠠⣤⣬⡞⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢀⡄⠀⢀⡾⠀⠀⠀⠀⣰⠁⣁⡐⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⣤⡴⠞⢻⣀⡴⠊⣀⠀⠀⠀⣰⣏⣃⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⢿⡀⠈⠀⠙⠉⠁⠀⢀⡼⠃⠀⡏⠉⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⢰⠙⢦⣠⣀⣀⣠⠴⡿⠀⠀⠚⢒⡶⠀⠀⠀⠈⢱⡄
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢸⢀⠀⠀⡩⢻⠳⣰⠃⠀⢀⡴⠋⠀⠀⠀⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⡞⠀⢀⠀⢀⡤⠄⠀⢈⠟⠀⠀⠃⠀⡄⡍⠀⣠⠋⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢰⠄⠀⠀⠉⠓⢤⡀⠈⠀⣷⠁⡴⠁⠀⠀⠀⠀⠀⢠⡇⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡄⠀⣉⣉⣉⠉⠃⠀⠙⢦⡠⣉⡜⠁⠀⠀⢀⣠⠖⠓⢬⡇⠀⠀⢿
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠇⠀⠀⠀⠀⠀⣄⣀⣀⣠⡷⠟⢲⡄⠤⠒⠋⠀⠀⠀⠤⡈⠀⡀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠐⠝⠋⠙⠒⠚⠉⠉⠁⠀⢠⠃⠀⡎⠀⠀⠀⠀⣀⠴⠚⠦⣈⠑⢬⣾
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣀⣀⡷⠀⠴⠲⡎⠁⡀⢀⠀⡈⢹⠋⡿
⠀⠀⠒⠒⠒⢺⣀⣀⣀⣤⣤⠤⠴⠒⠚⠛⠛⠒⠒⠒⠒⠒⢲⣈⡠⢇⢀⡇⣸⠀⡇⣸⠚⠓⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⢆⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⡤⡄⠀⠁⠈⠉⠉⠈⠁⠀⠀⣀⣞⠻⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡈⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⢠⣰⠉⠉⢹⡉⠛⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢡⡄⠠⡀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⡁⠀⠀⠀⠹⣄⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠃⠀⠀⠀⠘⠂⠀⠀⠀⠐⠘⡄⠀⠀⡀⢄⣂⡄⠳⣄⠀⠀⣼⠀⠀⠀⠀⠦⣈⠳⢤⣄⡴⠧⢤⣄
⠀⠀⠀⠀⠀⠀⠈⣆⠠⠄⠐⠈⠀⠀⠀⠀⠀⠀⠀⠐⡘⡄⠀⠚⠉⢀⣀⣤⣬⠗⠀⠙⢄⣀⣀⡲⡄⠈⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡘⠔⠒⠉⠉⠀⠀⠀⠀⠀⠀⠰⡈⠀⢈⠓⡀⠀⠀⠉⠙⠓⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠉⠉⠉⠉⠒⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠓⢤⡀⠑⠄⠂⣀⣀⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⡄⠀⠶⣍⠳⠶⠾⠿⠛⢉⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡄⠀⣠⣜⠠⢰⡖⠒⢺⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⣀⠀⢝⠁⢀⢀⢀⣞⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣀⣀⣀⣀⣠⠟⠀⠠⡀⠈⠈⢰⠠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡍⢀⠀⢀⡴⢳⡄⠀⠘⣇⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠁⠀⠐⠀⠁⠀⠀⢿⠁
⠀⠀⠘⠌⠓⢤⣀⠀⠀⠀⠀⣠⠖⠶⣄⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷
⠀⣀⢤⡤⠄⢤⡬⠙⡲⠶⠞⡁⠀⠀⠨⠆⠀⠀⠀⠀⣀⠄⠛⠳⢦⣤⣤⣤⣤⣤⡴⢶⠏⠑⢦⡀
⠘⠃⠀⠔⢰⠎⢰⡋⠀⠀⢰⠃⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⣀⡠⠄⠊⢡⡜⢣⡆⠀⡌⠀⠀⢀⠉⠲⣄
⠎⠀⠘⠀⢋⡀⠀⠁⣠⠴⢏⠀⠀⠀⠀⠀⡰⠋⡀⠀⠀⠘⢅⠀⠀⢰⡎⢃⡼⠀⠈⠑⢂⡤⠀⠊⠀⠈⢂
⠀⠦⠀⠈⠀⠀⠀⠰⣇⠀⢸⠀⠀⠀⠀⡰⢁⡾⠀⠀⠀⠀⠈⢦⠀⠘⠀⠸⠃⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠘⡄⠀⢀⠌⢀⢎⠀⡀⣀⠀⠀⠀⠀⠳⡀⢀⡁⠔⠉⠀⠀⠀⠀⠒⡒⢲⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣇⡴⠃⣰⠃⡌⠀⠀⠀⠈⠉⠀⠂⠀⠘⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠙⣠⠞⠁⠰⠀⠀⠁⠐⠀⠤⢀⠀⠀⠀⠇⠀⠀⠀⠀⢀⠀⠀⠀⠀⡘⠀⢐⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠁⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⢸⡤⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠤⣤⣤⣤⣀⣀⣥⣀⣀⣤⣤⣤⣤⣤⡤⠶⡶⠶⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠘⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠒⠾⠧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠤⠐⠀⠀⣠⡄⠀⣱⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣀⣀⡷⢄⠈⠙⠓⠶⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⡀⠀⠀⣀⡤⠖⠉⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⠤⠤⠤⠞
⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⣀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡛⠿⠿⠿⠟⢋⣥⣾⣿⡿⠃⠈⣿⠉⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⢻⣿⣷⣶⣾⣿⣿⡿⠟⠁⠀⠀⣼⠏⠀⠀⠈⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⢻⡿⠋⠀⣀⣿⡿⠿⠟⠛⠁⠀⠀⠀⢀⡼⢋⣠⣤⣤⣈⡛⠙⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠟⣡⢚⡿⢁⣤⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⣰⢟⣴⡿⠋⠉⠉⠙⢿⣷⡄⠀⠉⠛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣹⡿⢡⡞⠁⣸⡷⠛⣿⣿⣿⣿⡿⠻⣦⠀⠀⣠⡞⢡⣿⠋⠀⠀⢀⡄⠀⠀⢹⣿⠀⠀⠀⠀⠈⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⣼⡟⠀⢈⣷⠀⣿⠀⢀⣿⣿⣿⣏⠀⠀⠘⢷⣴⠋⣠⠗⠀⠀⠀⠀⣼⠁⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⣼⡿⠁⢀⣾⠃⠀⣿⠀⣼⣿⣿⣿⣿⠀⠀⢠⡾⠁⣴⠋⠀⠘⣿⡟⠛⠋⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⣠⣤⣤⣀⠀⢻⣇⠀⠀⡇⢰⣿⣿⣿⣿⣿⠀⣰⠟⢀⡾⠁⠀⢀⣴⡿⠃⠀⠀⢀⡜⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢰⣿⣿⣿⣿⡄⠀⢻⣇⠰⡇⣸⣿⣿⣿⣿⠃⣼⠏⢀⡾⠁⠀⣠⡿⠋⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠘⢿⣿⣿⠟⠀⠀⠈⣿⢰⡇⣿⣿⣿⣿⠏⣼⠏⢠⡞⠀⣠⡾⠋⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⣿⡇⣿⣿⣿⡟⣰⡟⢀⡾⠀⣴⠏⠀⠀⣀⡴⠋⠀⣠⡟⠀⠀⠀⣀⣤⠖⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⣿⣿⡿⢰⣿⠀⣸⡇⠜⠁⢀⣠⠾⠋⠀⠀⣰⣿⣀⣠⣴⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⢇⣿⠛⢧⣻⣆⠤⠖⠋⠁⠀⠀⠀⠴⠿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃
⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣾⠇⠀⠀⠀⠙⠳⠿⢂⣤⣴⠶⠶⢟⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠟⢻⣿⠏
⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⠀⠀⠀⢀⣠⣴⣶⣿⣻⣛⣛⣛⣋⣛⠂⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⣼⡟
⠀⢀⣾⣿⠁⠀⠀⢀⡀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣧⣤⣤⣴⣿⣿⣿⣿⣧⣶⡆⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⢠⣿⡟⠁⠀⠀⠀⠈⠙⠳⢶⣾⣿⣿⣿⣿⣿⣿⠟⠁⢀⣀⣤⣤⡤⠤⠬⠭⠭⠍⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⣾⡿⠀⠀⠀⠀⠀⠀⢀⡀⢤⡽⠿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⣿⣧⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁
⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠇
⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠋⢹⣿
⠀⠀⠈⠻⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠐⠶⠶⢶⣶⡶⠶⠶⣶⣶⢿⣿⡿⠋⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠙⠻⢿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠿⠿⠛⠛⠿⢷⣶⣦⣤⣤⣤⣀⣀⣤⣤⣤⠶⠞⠋⢠⣾⠟⠁⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣷⣶⣤⡀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠻⢿⠏⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠈⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⠃⠘⢻⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⠟⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠸⢛⣩⣴⣶⣿⣿⡿⠿⠿⢿⣿⣷⣶⣦⣍⡛⠿⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠻⣏⠀⠀⠠⡾⣿⡿⡦⠀⠀⢴⢿⣿⢷⠄⠀⠀⣹⠟⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢹⡄⢹⡆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⡟⢠⡟⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⢿⡀⠀⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⠄⢀⡿⠁⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠈⣷⠀⠙⠛⠿⠿⠿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠀⣾⠁⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠘⠇⠹⢿⣷⣶⣶⣦⣤⣤⣴⣶⣶⣾⡿⠏⠸⠃⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠠⣌⡙⠻⠿⠿⠿⠿⠟⢋⣡⠄⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣶⣶⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⡀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⠿⡿⠛⠛⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠈⢉⣉⣭⡛⠃⠀⠀⠛⣫⣍⣉⠉⠀⢸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠻⣆⠀⠈⠸⠿⠙⢂⣀⣀⠛⠽⠿⠈⠀⢀⡞⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⡏⡷⠶⠶⠚⠋⠁⠀⠈⠉⠛⠲⠶⠶⡏⣏⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⡀⠀⠀⠀⠀⠀⢀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠙⠒⠤⠔⠚⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠖⣶⠞⠉⠩⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠻⡝⠒⠲⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠋⠉⠀⠀⠀⠀⣇⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢹⡀⠀⠀⠀⠀⠈⠉⠓⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠷⢶⣄⠈⢆⣀⠀⠀⠀⠀⠀⢀⣠⠎⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡄⠘⣇⠉⠉⠉⠉⠉⢁⡟⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠀⠘⡇⠀⠘⡄⠀⠀⠀⢀⡞⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢿⠀⠀⠸⡄⠀⠀⡼⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢸⡄⠀⠀⠹⡄⡼⠁⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠈⡇⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⡇⠀⠀⣠⣶⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⠞⠓⠒⠚⠁⠀⠀⠀⠀⠀⣠⡤⠛⢵⡄⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹
⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣄⠀⡠⡃⠀⠀⠀⠀⠀⠀⠀⡇
⠉⠓⠲⠤⣤⣀⡀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠉⠊⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣀⣠⡤⠴⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠛⠛⠛⠛⠛⠋⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣤⠿⢦⣀⣀⣀⠀⠠⠴⠶⢶⣄⠀⠀⠀⠀⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀⠛⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠛⣀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠘⠋⠀⠀⠘⠛⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⣠⣦⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠈⠀⠀⣠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⢿⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⣤⣶⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠰⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢹⣦⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡼⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠓⢶⡿⠛⠟⠃⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣾⡇⠀⠀⠀⢀⣀⣀⣾⠗⠶⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⢾⣿⠃⢹⡷⠶⠶⠆⠈⠉⠉⣿⠀⠀⠈⢹⡷⢶⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣰⣿⠃⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠈⣿⡟⠉⠱⢦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣘⠋⠀⠀⠀⣿⣿⠀⠘⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠇⠀⣿⡇⠀⠀⠀⢻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡆⠀⣿⡇⠀⠀⣀⠟⠀⠈⣶⡄
⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣿⡇⠀⠀⡿⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⢰⡇⠈⠁⠀⠀⢀⣼⣿⣧⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⣿⠃⠀⣄⠀⠀⠀⠀⠀⠈⢃⡀
⠀⠀⠀⠀⢠⡄⠀⠀⠀⢀⣶⣿⠟⠛⠙⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⣼⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⡸⠃⠀⠀⠀⢸⣿⡷⠾⣿⠶⢿⣿⣿⡀⠀⠀⠀⠀⣿⠀⠀⠀⣘⡁⣀⣿⡏⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡇⠀⢠⡇⠀⣿⣿⠀⠀⣿⠀⢸⡏⣿⣇⠀⠀⠀⠀⣿⠀⠀⣴⠟⠿⠟⢻⣇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀
⠀⠀⠀⠘⠀⠀⢸⡇⢠⣿⠇⠀⠀⣿⠀⢸⡀⢹⣿⠀⠀⠀⠀⣿⠀⢸⡄⠈⠀⠀⠁⢽⡇⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⣼⡄⠀⠀⢸⡇⢸⣿⠀⠀⠀⠿⠀⢸⡇⠈⣿⡇⠀⠀⠀⣿⠀⢸⣥⣾⠆⠸⣧⡌⠁⠀⠀⠙⣤⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠉⠀⠀⠀⢸⣿⢾⣿⠀⠀⠀⠀⠀⣿⡄⠀⢹⣿⠀⠀⠀⣿⠀⣾⠉⠀⠐⠂⠈⠁⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⣾⠀⠀⠀⠀⠸⡇⠸⣿⣄⣀⠀⠀⢠⣿⣀⣠⣿⠏⠀⠀⠀⣿⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠙⠀⠀⠀⠀⠀⡄⠀⠈⠛⣿⣷⠀⣸⣿⡿⠟⠃⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⡿
⣷⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠏⢀
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠛⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠇
⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⡀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠈⠉⠉⠙⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠉⠀⠀⠀⠀⠀⠀⠠⠿
⠀⠀⠀⠀⢱⡄⠀⠀⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠈⢻⡆⢀⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠛⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⣄⠀⠀⠀⢀⡼⠏
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⡀⠉⢳⣶⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣀⣾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⠘⠀⠀⠀⠀⠸⡟⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⡀⠀⣠⠀⠀⠀⠐⠇⠀⡏⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠈⠉⠁⠀⠛⠀⠀⠀⠀⠀⠀⡇⠀⣿⣀⣀⣀⣀⣀⣀⠀⢀⣄⠀⠠⠰⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⢤⣤⣤⢀⣤⣤⣾⡇⠀⠀⠀⠀⠙⣷⣦⣤⠀⠀⠀⠀⠀⢀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⢈⡃⠀⠀⠀⣤⠟⠁⠀⠀⠀⠛⠛⠛⠁⠘⣿
⠀⠀⠀⠀⠀⢀⡼⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠘⠃⣀⠸⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠟
⠀⠀⠀⠀⠰⡏⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠁
⠀⠀⠀⠀⠀⠈⠳⠖⠀⠄⠰⠶⠖⠋⠁⠀⠀⠀⠀⠐⠦⠀⠀⣀⣀⡀⠀⠴⠞⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠙⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⣀⣀⠀⠀⠀⠀⣀⡀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠈⠛⠋⠀⠀⠀⠈⠉⠁⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⡀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⣿⡟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⣀⣀⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣦⣾⣿⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣠⡌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠈⠛⠳⣶⣤⣤⣀⣀⣴⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⠛⠉⠀⢸⣿⣷⢶⣶⣦⡀⠀⠀⠀⠀⣠⡿⠻⣷⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠾⠿⠷⢀⣿⣿⠏⠀⠀⠀⢸⣿⠁⠀⠀⠉⠀⠀⠺⠿⠿⠿⠗⠀⠈⢻⣿⣿⠿⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⠇⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠉⠀⢠⣿⡄⠀⠀⢀⣿⣿⠃⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠘⣿⣿⡆⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡌⠁⠀⠀⠀⠉⢡⡀⠀⠀⣿⣿⠆⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢹⣿⣧⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⠃⠀⠀⠀⠀⠀⢸⡿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢸⣿⣿⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠘⣿⣿⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣧⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣠⣼⣿⣿⣶⣤⡀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣿⣿⡆⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠿⣿⡟⠛⢿⣿⣦⡀
⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁⠀⠀⢻⣿⣷⣀⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣯⣀⣠⣿⣷⣀⣠⣿⣿⡇⢸
⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣼⡿⠛⠿⠛⢣⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠛⠿⠟⠛⠛⢿⣿⣷⣿
⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⡀⠙⣿⣶⠀⠰⠎⢉⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⢸⣿⠀⣠⡀⠀⠀⠀⢿⣿⡃⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⢠⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠹⣿⡟⠃⣀⣀⠀⣰⣿⣧⠀⠀⠀⠀⠀⠀⣾⣿⣿⠃⠀⢀⣎⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠋⠁⠀⠀⠀⠀⠀⠈⠻⡿⢿⣿⣿⠙⢿⡇⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⣸⣿⠀⠀⠀⠀⠀⣿⣿⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠉⢻⣿
⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⣼⡇⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⢠⣿⣿⠀⠀⠀⢴⠀⢀⣾⣿
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣾⣿⡿⠀⠀⣀⣼⣷⣾⣿⠟
⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⡈⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠃⠀⠘⢿⣿⡿⠁
⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣆⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⢸
⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸
⠀⠈⠻⣀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣼⡄
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠈⢡⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠷⠿⠿⠿⠿⠿⠿⠿⠷⠾⣿⡷⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠙⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⢀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣀⣠⣶⣶⠊⠻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⢠⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢷⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⢀⡀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⣀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣤⣤⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠀⢠⣤⣶⣦⣤⣄⠀⠀⣀⠀⣼⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣶⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠒⠀⠰⣾⣿⣿⣶⠶⠶⠶⠶⠖⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⡈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⣀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠿⣿⠁⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⣼⣿⠀⠈⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⣾⣧⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡟⠀⠀⠀⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠃⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⡀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢰⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⣾⣇⠀⠀⠀⠀⠸⡇⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⢻⣇⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠙⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠹⣷⣄⠀⠀⢸⣿⡿⠆⢀⣀⠀⠀⠙⠿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣄⡀⠀⠀⠈⠛⢧⣤⡘⢁⣀⣀⡾⠿⠃⢀⣀⣀⣿⣿⠿⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠉⢱⣶⣤⠀⠀⠀⠙⣿⣿⡟⠉⠀⠠⣶⣿⣿⡿⠋⠁⠀⠈⠛⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⠞⠉⢁⣠⣶⣶⣾⠟⠋⠀⠺⡷⠀⠀⠀⠀⠀⠀⠰⢿⡷⠾⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠉⢀⣀⣀⣀⣠⣷
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣯⣀⣤⣤⣄⣀⣠⣤⣤⣤⣤⣶⣄⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠖⣢⡽⣠⣤⡀⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣫⠞⠁⢾⡿⣿⣿⣿⣿⣷⡄⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣯⣾⠒⠋⢸⣿⣿⣿⣿⣿⣿⣡⡞⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠁⠀⠀⠈⠉⠙⠛⠋⠛⠁⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠈⣿⡇⠀⠠⢤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠃⠉⠀⠀⠀⠴⢶⢯⠙⠂⠀⣴⣶⡶⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠛⠛⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠀⠀⠀⠀⠀⡠⠀⠀⡇⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠏⡇⠀⠀⠀⠀⠀⠑⠦⡤⠧⠀⠀⡜
⠀⢀⣀⣀⠤⠤⠔⠒⠒⠚⠋⣹⣿⢿⡏⢹⠀⢳⡀⠀⠀⠙⠒⠲⠶⠶⠄⢀⠌
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⣿⠁⢸⡇⠈⢧⡀⠙⠲⣄⠀⠀⠘⠓⠀⣠⣾⣤⣀
⢶⠀⠀⠀⢀⡞⣀⡀⠀⠀⢸⢿⠀⠀⠙⠲⢄⣷⠀⠀⠈⠁⠒⠒⠖⢪⠏⢹⣧⠈⠉⠓⠲⠤⣄⣀
⠈⣧⠀⢀⡾⠋⣽⠁⠀⠀⡼⢈⣦⣀⡀⠀⠀⢿⣷⡄⠀⠀⠀⢀⡴⢻⠀⠘⣿⠀⠀⠀⣀⣄⠀⠉⢻⡦⡄
⠀⢸⡆⠀⠀⣸⠃⠀⢠⣾⠎⠁⠈⠻⣿⡄⠀⠈⠻⢽⣆⠀⣠⣾⠐⠃⠀⠀⣿⡆⠰⠚⢁⡟⠀⠀⢸⡇⢳
⠀⠘⡇⠀⢰⠇⠀⠀⠘⣿⡄⠀⠀⠀⢻⣧⠀⠀⠀⠀⠙⣷⠏⣾⠀⠀⠀⠀⣿⡇⠀⠀⣸⠃⠀⠀⢸⠇⠘⡆
⠀⠀⢻⡀⠿⠤⢤⡄⠀⠘⣷⡀⠀⠀⠘⣿⠀⠀⠀⠀⢀⠏⠀⡿⠀⠀⠀⠀⢧⡇⠀⠀⠹⡀⠀⠀⣼⠀⠀⢻⡀
⠀⠀⠈⣷⠀⠀⢀⡇⠀⠀⠸⣇⠀⠀⠀⣿⠀⠀⠀⠀⣼⡜⠀⡇⠀⠀⠀⢀⣿⠀⠀⠀⠀⠈⢧⠀⡏⠀⣠⠜⡇
⠀⠀⠀⢻⠀⠀⢸⡇⠀⠀⠀⢻⡄⠀⢠⡟⠀⠀⠀⢠⡟⠀⠀⡇⠀⠀⢠⢾⡿⢤⠀⠀⠀⠀⢸⣴⢃⠞⠁⠀⢳⡀
⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⠻⣄⣸⠃⠀⣤⡄⣼⠱⡄⠀⡇⠀⠀⢺⣼⣆⣸⠀⠀⠀⠀⠀⣿⢋⡤⠊⠀⠀⢧
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⣶⠦⠤⠞⠉⠉⠉⠉⢠⡇⠀⢸⠀⡇⠀⠀⡬⠷⠶⣋⠤⠄⠒⠒⢺⡷⣫⠶⠋⠀⠀⠈⠣⣀
⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡸⠀⣧⠀⠀⣷⢀⣤⣤⣤⣤⠀⢀⣿⠞⠁⠀⠀⡤⠐⢒⠆⠈⢧⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⠟⢦⡀⠀⠀⠀⠀⢸⠐⠫⢤⠀⡿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⣯⠀⠀⠀⠀⢀⠔⣁⣄⠀⢀⡙⣄
⢀⡀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢀⣰⢌⣶⣄⠀⠀⢸⠀⠉⠁⠀⠁⠀⠀⡿⠀⠀⠀⠀⠀⠀⢸⠉⣿⣦⡀⠀⠟⢊⡴⠁⡼⠡⠉⠛⠳⣄⡀
⠈⠉⠙⠶⣿⣤⣀⣀⣀⠴⠚⠉⠀⠀⠀⠈⠙⠲⢼⡘⠓⠺⠄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡿⠀⠈⣷⣄⠀⠀⡾⠂⡸⠁⣠⠛⠶⠀⠀⠉⠳
⠀⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⡈⠓⠦⣤⣀⡰⠙⢿⣳⢼⠉⠉⠛⠒⠲⠶⠶⠶⠶⠴⠧⢤⣤⣸⣏⣃⠀⣠⣺⠇⡴⠁⠀⢠⠎⠈⡷
⠀⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⣄⣀⠀⣨⠙⢲⣤⠳⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⠇⡜⢳⣄⡰⠃⢀⡜⢡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⣈⣍⣓⣖⣋⣀⣭⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⡽⠁⢻⠎⡲⠛⠊
⠀⠀⠀⠈⠛⠲⠦⢤⣀⣀⣀⣀⣉⣉⣡⣴⣏⡉⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣷⣾⣷⣤⣷⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⢤⣄⡀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣀⣈⣩⣷⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠾⢷⣄⣀⣀⣀⣀⣀⣀⣠⣤⠤⠤⠤⠶⠖⠒⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠶⠒⠀⠉⠉⠀⠒⠒⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣦⣤⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣠⣶⡿⠻⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢉⣽⢷⣥
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢴⣿⣯⡄⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⡀⠀⡀⠀⠈⠀⠀⢠⡆⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢈⣠⠠⢿⣿⣿⡆⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡯⢠⠆⠁⠀⠀⠀⠈⠐⠀⢠⠁⣬⠃⠀⣠⣶⣿⣿⡿⣷⡄⠀⠀⠀⢈⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠁
⠀⠠⠀⠀⠀⠀⠀⠀⢿⡷⡿⠄⠀⠀⠀⠀⠀⠀⠀⢐⢾⡏⠁⢺⣿⣿⣿⡿⠳⣿⣧⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⣀⠀⠤⣾⡇⠀⢸⣿⣿⣿⣿⣿⣿⠟⠀⠀⣰⢿⣿⣿⣿⣿⣿⡇
⠀⠐⠃⠀⠀⠀⠀⠀⠀⢻⡗⠀⢀⣁⣉⣑⠓⣋⠛⠹⠛⠃⠈⠀⠉⠛⠛⣛⡉⠀⠀⠀⣠⠋⣾⠈⣿⣿⣿⣿⣷
⠀⠠⢂⠀⠀⠀⠀⠀⠀⠈⠳⣄⠉⢡⡿⣿⢿⣿⠻⣦⢤⣀⣀⣀⣀⣶⣶⠝⠃⠀⠀⠀⡃⣾⣿⣆⣿⣏⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⢙⠶⠿⠏⠀⢀⡀⠀⢈⣉⠉⣿⠏⠀⠀⠀⠀⠀⠈⠉⢸⣿⣿⠏⠈⠉⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠘⡅⠈⠁⠀⠈⠹⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢘⣏⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⣄⡁⢰⡀⠠⣟⡸⠉⡏⢹⠁⡆⢸⠅⣮⠸⡏⣿⣷
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣏⣄⠈⢿⠒⢇⠭⢙⡇⠷⠠⠄⢶⠠⡷⣴⢧⣿⠏
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣟⡃⠀⠀⠁⠀⠀⠀⠁⠛⠘⠃⠿⠀⢗⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠄⣀⠀⠀⠀⠀⠀⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠒⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠀⠉⠛⠛⠻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠉⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣩⠀⠀⠀⠀⠀⠀⣌⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠃⠀⠀⠉⠛⠙⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣄⣀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⣄⠀⢀⣠⠾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢾⠁⢶⣶⠂⣳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣞⠉⠉⠉⠉⡷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡖⣻⡉⢷⠞⢇⢸⣧⣰⠳⡾⠋⡟⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣿⠙⢷⠞⡖⣶⡾⣰⠳⡾⢹⡟⢶⡀
⠀⠀⠀⠀⠀⠀⠀⣾⢰⠁⠙⡌⠲⣜⣿⣿⣃⠔⠁⢠⠇⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⠁⠹⡌⠢⣘⡌⢱⣣⠞⠁⣸⠁⠀⣷
⠀⠀⠀⠀⠀⠀⢀⡇⠘⣆⠀⠼⠀⠈⠻⡟⠁⠀⢾⣉⠇⠀⢸⡆⠀⠀⠀⠀⠀⠀⢰⡇⠸⣀⠀⡽⠀⠈⢻⠟⠁⠀⢮⣻⠆⠀⢸⡀
⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡟⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠈⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡉⠀⠀⠁⢠⣤⡄⠀⠀⠀⠀⡤⣤⡄⠀⠀⠀⢛⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠀⠻⠿⠃⠀⠀⠀⠀⠻⠿⠃⠐⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⣀⠀⠀⠁⠀⣠⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠳⣶⣶⡾⠋⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⠂⠠⢀⡀⠀⡀⠠⠔⢻⠆
⠀⠈⠁⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⠟⠀⠀⠀⠀⠀⠀⠀⠂⠸⡅⠀⠀⣤
⣆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠀⠀⠀⠀⠀⢂⣀⠙⠁⠀⠀⠀⠈⠀⢀
⠷⠀⠆⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⢉⠐⠂⠰⠖⠉⠆⠀⠃
⠘⢧⠒⠀⢊⡄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⠀⠀⠀⠂
⠀⠈⠀⠀⠉⠲⡀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠛⠁
⠀⠀⠀⠀⠀⠀⠻⢤⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢀⠰⠍⠀⠀⠀⠀⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠔⠠⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡻⠗⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⢀⠂⡀⣀⠀⡀⣀⡀⣀⡤⠖⠉⢡⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠁⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⣀⣀⠀⠄⣀⣀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣤⣤⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣠⠀⠐⠄⢀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⡀⠠⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠺⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠁⠀⠀⠀⢈⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⠻⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠖⠉⠉⠉⠉⠓⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠟⠲⢤⡀⠐⠂⡢⢀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⡀⠀⠤⣝⠲⢤⣤⡾⠿⠛⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢫⣅⠀⣀⣘⠇⢰⣤⠖⠲⡞⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂⠀⠀⠁⠈⠁⠀⣻⠏⠀⠈⠁⡠⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⢀⡿⠀⢀⠀⠈⠒⠘⡏⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛⠓⡛⠉⠀⠀⢠⣷⡀⠀⠀⣇⠑⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠈⠒⠀⠡⠀⠛⠀⠀⢹⡛⠁
⠀⠀⠰⡐⠦⣀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠉⠂⢈⣙⠢⢤⣀⣠⡞⠁⣀⠈⣧⠀⠀⠀⠀⠀⠀⢀⡷⣄⣀⠀⠀⠀⠀⠀⣀⣠⣴⠿⣀
⠠⡞⠉⣉⡩⣭⠏⡤⠒⠉⠁⡜⠀⠀⠀⠉⠀⠀⠀⣠⠴⠊⠁⠀⢸⠉⣻⠟⢛⡿⣹⠋⢁⠃⠀⠈⠓⢦⡀
⡠⠀⢹⣀⣼⠃⠈⠷⠀⢀⣠⠇⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠠⠒⠊⠀⢰⡐⢯⢠⠇⠀⣬⡀⠀⠐⢁⠢⠝⠦⡀
⠁⡁⠀⡀⠀⠂⠀⠀⣾⠋⠉⡄⠀⠀⠀⠀⡠⠋⣴⠀⠀⠀⠀⠱⡀⠀⠘⡇⢠⡟⠀⠀⣀⡤⠛⢠
⠀⠉⠀⠀⠀⠀⠀⠀⠹⡄⠀⡇⠀⠀⠀⡐⠁⡼⠁⠀⠀⠀⠀⠀⠙⣄⠀⡇⢨⢁⡴⠚⠁⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢣⠀⣠⠎⢀⠎⠁⠀⠒⠂⠤⠄⢀⠀⠈⢢⣥⠆⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠸⡞⠁⡰⠋⠠⠀⠀⣀⡀⠀⠀⠀⠘⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⡠⠚⠁⠀⠆⠀⠀⠀⠀⠉⠑⠒⠃⠀⠀⠘⠀⠀⠀⠀⠈⢦⡀⠀⢀⡠⠃⠀⠠⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠦⢤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⡴⠶⢶⠖⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠴⠾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠤⠾⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀⣀⣀⣀⠀⠄⠂⠀⢀⡄⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣰⢄⡉⠙⠲⠦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⢀⣠⠴⠚⠙⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠙⠲⠤⣄⣀⣀⣉⣩⡽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⡀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣀⠀⠠⡀⠀⠀⠀⠀⡀⠀⠀⠀⡤⢤
⠀⡇⡀⠀⡄⠀⣀⠀⠀⣄⡶⠴⠰⠤⠦⠶⠶⡤⠀⠀⠀⠀⠉⠀⡠⠀⠀⠀⠀⠈⠀⠀⠔⠂⠒⠒⠒⠒⠩⠠⠀⠀⡄⢠
⠀⡆⠐⠸⡵⢺⠐⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠉⠈⠉⠉⠀⡐⠀⠂⠀⠀⠀⠀⢀⠤⠄⠀⠀⠄⡀
⠀⠁⠀⠀⢧⠈⠀⠁⠐⠉⠈⠁⠀⠉⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠉⠑⠀⠀⠁⠀⠘⠀⢀⠀⢇⡀
⠀⠀⠀⠠⠇⠀⠀⠀⡀⠀⠅⠀⠀⠀⢈⠀⠀⠀⠀⠀⠒⡀⢆⠄⠀⠀⢀⡠⡤⢧⡀⠀⠀⠀⠀⠈⠀⠐⠅⠀⠈
⠀⠀⠨⢔⠂⠀⠒⠉⠠⠀⠁⠀⠀⠀⠈⠀⠡⠀⡀⠐⠀⢁⡴⠀⢰⠀⣨⠁⠀⠀⢘⠄⠀⢠⣄⠁⠠⠓⠳⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠆⠨⠀⠀⠀⢀⢈⠞⠁⠈⠁⠀⠀⠘⡄⠊⠲⠤⠤⡉⠀⠸⠈⠪⠔⢙⠂⠀⠀⠀⢀⠰⡳⢀
⠀⠄⠀⠀⠠⠄⠀⡀⡀⠀⠠⠀⠍⠀⠀⠀⠀⣁⠀⡠⠀⣄⠀⠀⠀⠈⠓⠤⠥⠤⠔⠋⠉⠀⠀⡄⠠⠀⢰⠂
⠀⠀⠀⠀⢀⠂⠀⠃⠂⠀⢀⠀⠀⠀⠀⠀⠘⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⢀⠀⠀⣈⣔⠀⠲⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠡⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠐⠉⠀⠄⠀⠃⠙⠁⠧⠀⠄⡀⠀⠀⣤⠄
⠀⠀⠁⠀⢀⣀⣀⡀⠀⠀⠈⠨⠀⠀⠀⠀⣠⡄⠀⣠⣶⣶⣄⣤⢤⣤⠀⠐⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠅
⠀⠀⠀⠠⠶⠄⠀⠀⠀⠀⠀⢠⢀⣀⣀⠀⣿⡇⠀⣿⠀⢸⡇⣿⣀⣿⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠂⠒⠀⠀⠀⠀⡈
⠀⠀⣦⢼⢹⠀⠀⣀⡀⠀⠀⠐⡀⢀⠄⠀⣿⠃⠀⣿⠛⠻⡇⣿⠛⣷⠀⠀⢸⠒⠉⠛⠇⠀⠀⢀
⠀⠀⡇⠐⠀⠀⡅⢥⠀⠁⠀⠀⠀⠈⠀⠀⠿⠿⠇⠻⠀⠘⠃⠛⠛⠋⠀⠀⡈⠀⠁⠀⠀⠀⠐⠁⠀⠀⠈⠀⠠
⠠⡀⠃⠀⢀⡠⡄⠈⠀⠀⠀⡦⠤⠤⡄⠀⢰⡟⣿⣟⣻⡟⣻⣿⣿⠀⠀⠀⡇⠐⠁⠂⢀
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠂
⠀⠀⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡄⠀⠀⠀⠀⠁⠂⢀⠀⠀⡅⠀⡰⢀⠠⠀⠀⠴⡀⢀⠀⠀⠀⠀⢀⡀⢀
⠀⢀⠀⢀⠀⣀⠀⠀⠀⠀⠄⠀⠀⠠⠀⡀⡁⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⢠⠀⠀⡱⠄⠀⠀⠩⣣⠀⠀⢀⠀⠘⠁⠀⠘⠁
⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡑⠀⠀⠀⠀⠐⠁⠀⢀
⠀⠀⠀⠀⠀⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠘⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂⡀⢀⡀⠀⠀⠀⢩
⠀⠀⣿⢸⡇⣀⡀⣿⢹⡇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠁⠀⠂⡱⠄
⠀⠀⣿⢻⡏⢩⡇⣿⢸⡇⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢆⢀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⡈⠀⠀⠀⠐
⠀⠀⠛⠘⠃⠺⠇⠘⠛⠁⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠂⠁⠉⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⣀⡉⢛⣤⡽⠭⣍⠛⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣳⠘⢋⢶⣶⣶⡆⠀⠹⢯⠁⢹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠙⠛⣻⠀⠘⠘⠛⠋⠀⠀⠀⢣⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢠⣉⣒⣀⣤⣴⡆⠀⠀⣰⣇⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡙⢦⣤⠶⠋⠀⣠⢾⣿⣿⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠤⢴⡞⠁⢸⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⣀⣤⠤⠔⠆⠳⠶⠍⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠄⠐⣶⣾⠇⠀⢈⣷⠦⣄⡀
⠀⠀⠀⠀⠀⠘⠯⣁⡀⠊⠀⠀⠀⣀⣀⡽⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢁⠄⣠⠞⣿⠏⠀⣠⡟⢹⡄⠀⠉⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠸⡋⠀⡠⠋⠢⣀⠀⠀⠀⠀⠀⣠⠊⠀⣠⢊⢴⠏⣸⣿⣤⣾⠟⠀⡿⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡀⠀⠀⠈⠑⢄⡀⡠⠞⠁⠀⡔⠁⢀⣿⠀⣿⣿⣿⠏⣤⠾⣟⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠙⠆⠀⠀⠀⡆⠀⢸⡇⢰⣿⣿⠏⡼⠡⠀⢸⡦⠀⠳⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⢀⡠⠖⢃⢰⣟⡇⢸⣿⠏⣸⠇⠀⠀⣸⠁⠀⣰⠻⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠤⠒⠉⠀⠀⠘⡛⠿⠃⢸⡟⠠⡿⡄⢀⣰⠏⠀⡴⠃⠀⠘⣄⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢸⡁⠀⠈⠁⠙⠋⢀⡜⠁⠀⠀⠀⡨⠒⠛⠁⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢸⢀⠀⠀⠀⠀⢠⠎⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠉⠀⠀⠀⡰⡇⢀⡴⠛⡁⠀⠀⠀⣀⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠦⠁⠀⠀⠀⠀⡘⠉⠣⣀⣑⠶⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⡆⠀⠀⠀⠀⠀⢃⢰⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡼⢇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠈⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⣦⣄⣀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢷⡶⢶⣶⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠛⠛⠛⠻⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠟⠛⠛⠛⠛⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠤⠤⠤⠤⠴⠆⠀⠀⠠⠤⠤⠄⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠤⡘⠛⢀⡀⠀⠀⠀⠘⢛⣛⣛⣄⠀⠰⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⡀⠀⠀⠠⣶⡆⠀⠴⠒⠆⠀⠈⠿⠇⠀⠀⣀⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠿⠁⠀⢀⣀⣉⠀⠀⠀⠀⠈⠀⠄⠤⠠⠄⠀⠉⠓⠭⢽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠦⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠰⢆⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠀⠀⠀⣀⡀⠤⠤⠀⠀⠀⢦⣤⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠈⢛⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠉⠁⠀⠂⠉⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠈⠛⢶⡦⣤⡤⠶⠟⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣫⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⡴⠃⠀⡟⢳⡀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠜⠈⠇⠀⠀⢹⡄⠙⠢⢀⣀⣀⣄⠀⠈⠀⠇⠀⠀⢸⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠰⢄⡀⠀⠷⠀⠀⡄⠀⢀⡼⢇⠀⡞⠀⠀⠀⠀⣴⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⠋⠀⠀⠀⢰⠀⠠⠊⠀⠀⠘⣄⡇⠸⣤⡜⠁⢸⡞⠁⠀⠀⠒⠀⣽⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⣀⡤⠞⠉⠀⠀⠀⠀⠀⠀⢸⠀⢄⠀⠀⠀⠀⠹⡀⣀⠀⢀⠀⡸⠀⠀⠀⠀⠠⠀⠸⠀⠀⠀⠀⠀⠈⢻⣦⣀
⠀⠀⠀⠊⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠈⢢⡀⠀⠀⠀⢳⡏⠀⠸⣤⠃⠀⠀⠀⢀⣤⡀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠙⡄⠀⠀⠈⣧⠀⢰⡏⠀⠀⠀⠰⠛⠀⠀⠀⠑⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠈⢆⠀⠀⢹⡀⡜⠀⢀⠐⠁⢠⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢶⡄⠀⠀⠀⠱⡀⠈⢷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢁⠀⠀⢀⠀⠀⠀⠀⠀⢸⠃
⠀⣸⠇⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠈⠻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⣷⠸⠀⠀⢸⠀⠀⠀⠀⠀⠀⢲
⢰⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠄⠀⠀⠈⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⢾
⢸⡇⠀⠀⠀⠀⠀⢿⠆⠶⠦⠤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⡠⢦⣀⣀⠼⣇⠀⢀⣀⠀⠀⠀⠀⣼⡶⠃⠀⠀⠀⠀⠈
⠸⠃⠀⠀⠀⠀⣀⣸⡆⠀⠀⠀⠀⠊⡘⠛⠲⢤⣀⠀⠀⠀⠀⣼⣀⣀⣀⣤⣤⡭⣍⡀⢀⠠⠥⣿⣤⣀⣀⡀⢀⡾⠁
⡄⠀⠀⠀⡴⠋⣩⠟⠛⠛⠛⠛⠛⠒⠓⠒⠒⠚⣛⠻⠿⠛⠛⠉⠉⠉⠁⠀⠀⠀⣸⠀⢸⠀⠀⠀⠀⠀⢉⣹⠾⠇⠠
⡧⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢘⣠⡤⠶⠞⠛⠛⠛⠓⠶⢄
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠙⠦⢀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣤⠶⠶⠶⠛⠛⢛⠉⠉⠓⠲⠦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠉⠉⠛⠛⠉⢻⡏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠷⠶⠶⢶⣦⣤⣤⣤⣤⣴⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶
⣿⣛⣛⣛⣛⣙⣛⣃⣜⣉⣻⣋⣉⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣽⣯⣭⣭⣭⣽⣿⣯⣭⣭⣭⣭⣭⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣣⠔⠊⠉⡴⠟⠛⠉⠉⠙⠓⣒⣶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢀⡞⡽⠛⠛⠳⠶⠶⠶⠤⠤⠤⠤⠤⠤⢤⣤⣀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢸⠏⠀⠀⣀⣤⡀⠀⠀⠀⠀⣠⣤⣀⡀⠀⠙⢦⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣮⣤⠴⠿⠯⣭⣅⡀⠀⠀⢀⣨⣭⠽⠿⠶⣤⣌⣇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠐⠂⠀⠀⣿⠶⠶⣾⠀⠀⠐⠂⠀⠀⣿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠱⣿⡄⠀⠠⣿⣧⢄⡟⠀⠀⢹⡀⣾⣯⡄⠀⢀⡏⡴⠇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢏⠙⠶⠶⠿⠷⠟⠀⠀⠀⠀⠙⠾⠿⠶⠶⠛⢀⠇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠼⡄⠀⠀⠁⠈⠀⠠⣀⣀⡠⠀⠀⠀⠀⠀⠀⣸⠤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⡶⠲⢤⡤⠤⢤⠴⠖⢶⢫⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡹⢇⡀⣄⢠⠫⠀⡰⢠⢼⢏⠟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠇⠐⠀⠘⡁⠄⡩⠁⣨⠆⠈⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢠⣤⠎⠀⠀⠀⠁⡀⠡⠊⢀⣁⠀⣄⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠠⣬⠟⠀⠀⠀⠀⠀⡏⠀⠀⡏⠉⢹⠘⢦⡤⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠔⢙⠃⠀⠀⠀⠀⠀⡏⠀⠀⠙⠛⠁⠀⢛⠠⡀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠜⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⠸⡀⠈⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡌⠀⠀⢠⠁⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠱⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠈⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠈⠀⠀⠀⠀⠀⢸⢰⠀⠀⠀⠀⠀⠀⠉⠁⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠧⣄⣀⡀⠀⠀⠀⠀⣸⢸⠀⠀⠀⠀⠀⢀⣀⣠⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⢍⢙⣒⡲⡿⠚⢳⡚⣛⡉⠩⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠈⠀⣀⡧⠀⠸⣀⠀⠀⠀⠀⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠖⠒⠉⠁⠀⠀⠀⠀⠉⠓⠲⠶⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣤⡀⣥⡶⠟⠒⠌⠙⢿⣿⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣵⠈⡜⢿⣿⠟⠀⠈⢣⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠉⢯⣀⠀⠉⢀⠀⠀⠀⡬⢾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡘⢶⣒⣚⣻⠏⠀⢀⣼⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡙⠋⠁⣠⠴⠋⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣣⣑⡖⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣉⣥⣿⠀⠀⣧⣄
⠀⠀⠀⠀⠀⢴⡛⠁⠐⠂⠀⠀⠀⠀⡧⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⡀⣠⢾⣿⠁⢀⣴⢻⡇⠉⠓⠒⢄
⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠛⠊⢹⡋⠀⡠⠣⢄⠀⠀⠀⠀⠀⢀⡔⠁⢀⠎⢰⠏⣾⣿⣤⣿⠋⢸⠃⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢮⠀⠀⠀⠑⢦⡀⢀⠴⠃⠀⡴⠁⣀⣯⢰⣿⣿⡿⢣⡴⣿⠂⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠣⠀⠀⠀⡇⠀⣹⠁⣸⣿⡿⢡⠏⠄⢨⡧⠀⠳⡀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⢀⡠⠖⢃⣸⣿⡇⣿⣿⢃⡿⠀⠀⢰⠇⠀⣰⠳⡄⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠒⠉⠀⠀⠘⡛⠚⠁⣻⠃⠸⣣⢀⣤⠟⠀⡴⠁⠀⠘⢄⠀⡀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣻⠀⠀⠀⠈⠉⢀⡜⠁⠀⠀⣠⠞⠉⠁⠁⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠏⡆⠀⠀⠀⢠⠊⠀⣀⡤⠊⠀⠀⠀⠀⢀⡠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⣀⠀⠀⠀⠐⠣⡤⠞⠉⢄⠀⣀⡤⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠓⠀⠀⠀⠀⢸⠀⠈⠲⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢰⡇⠀⠀⠀⠀⠘⣄⡆⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠘⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣄⡀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠛⠛⠛⠿⠛⠛⠛⠛⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢰⠏⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⣾⡟⠃⢿⣔⠀⠀⠆⠀⠀⠀⠀⠈⣶⣶⣆⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⡛⣼⡿⠿⢸⡎⡌⣀⣥⡀⢀⠀⠀⠀⠀⣲⢹⡿⢿⣧⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⣗⣿⣱⣿⣶⠶⣊⣥⢤⠤⠄⠉⠀⢄⣤⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡽⢿⡻⡀⢈⡽⠞⠛⢻⠒⠒⠴⢤⡤⠎⡥⠍⢀⣿⢽⣯
⠀⠀⠀⠀⠀⠀⠁⠀⣠⡶⠧⢍⡇⠀⢳⣽⠛⠀⠀⣠⣴⣦⣤⣀⣀⣀⡀⠡⢴⡇⠈⣼⡿
⠀⠀⠀⠀⠀⠀⣴⢚⣇⣀⣀⣀⣵⠟⠋⣹⡥⠄⠈⠉⠙⢯⠹⢟⠏⠀⠀⠀⣠⢿⣿⠞
⠀⠀⠀⠀⠀⠀⢟⡀⢿⣰⣥⠤⠟⠀⢀⣿⠀⡀⢻⠀⠀⠸⡆⣊⣤⠴⠶⢮⡁⠀⠉
⠀⠀⠀⠀⠀⠀⠘⣄⣸⠿⠀⠄⠀⠤⠚⢻⡇⢻⣿⠀⠀⠀⡏⠁⠀⠀⠀⠀⠑⢦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⢡⠘⡸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⣋⠞⡀⠰⠶⢢⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠋⢈⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠟⠩⠉⠀⠀⠀⢀⣙⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠉⠀⠰⠀⠀⠀⠀⠀⠈⢇⠀⠙⠟⢻⣷⣄
⠀⠀⠀⠀⠀⠀⣰⡾⠟⠉⠀⠈⠁⠀⠛⢆⡀⠀⠀⠠⠈⠀⠐⣻⣏⣋⣻⣿⣷⣄
⠀⠀⠀⠀⢀⣼⡇⢀⣤⢄⡠⠔⠒⢦⡄⠀⠉⠉⠙⢳⠀⠀⢰⣿⣩⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢀⣏⠠⡙⠁⠀⠀⢠⠃⠬⠄⡏⠀⠀⢀⡴⠊⠀⠀⢺⣿⣷⣄⣬⣾⣿⣿⡄⠁⠢
⠀⠀⠀⣸⠇⡰⠁⠀⠀⠀⠁⠀⣀⣀⠹⣦⠀⣿⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡏⠀⣴⣦⡖
⠀⠀⢀⣿⠘⡇⠀⠈⢀⠀⠀⠊⢁⡠⠇⠻⣆⣿⠀⣤⡄⠀⠀⠈⠙⣾⣿⣿⣿⣿⣿⣿⣭⢯⡀
⠀⠀⠸⣿⠾⣇⠀⠀⢘⠧⡀⠰⠙⠀⠀⠒⠙⢿⣶⣿⡇⠀⣀⡀⠈⠉⣿⣿⣿⣿⣿⢡⣶⣿⣿⡀
⠀⠀⠀⣿⣇⣿⠀⠀⠈⠀⠈⠳⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠘⣿⣿⣦⣴⣁⠀⠔⠀⠉⢁⣹⠷⣬⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠘⣿⣿⣼⣿⣶⣶⣤⡴⠛⣽⣳⡀⣶⣿⣿⣿⣿⣟⣿⣿⣿⣧⣤⣤⣭⣽⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣽⡄⠙⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣬⣙⣿⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠻⣷⠘⣿⡿⠁⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣀⣹⣾⣿⠆⡇⣿⣿⣿⣿⣿⣿⣉⣂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡛⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⢠⡷⣿⣿⣿⣿⣿⣟⣻⣆⢻⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠿⠳⠛⠟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⣿⣿⣿⣿⣟⣿⣿⣿⣦⣙⢿⣯⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⣤⣴⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢿⣵⣿⣿⣿⣿⣿⣧⣽⣿⣽⣧⡹⣷⣿⡆
⠀⠀⠀⠀⢀⡾⠟⠉⠙⠛⠀⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣮⠻⣯⣿⣌⢿⣿
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢰⣿⣿⡿⠁⢀⣾⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⡿⣻⣿⢻⣿⣿⣿⣿⣿⣿⣷⡟⣷⡹⣦⠹⣇
⠀⠀⢠⡆⠀⠀⠀⠀⣀⢠⣿⣿⣿⣇⠀⣿⣿⣿⣿⣿⡇⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠙⢻⣿⣿⣿⡘⢷⣿⡔⢬
⠀⠀⠘⢂⠀⢈⣷⣰⣿⣿⣿⣿⠉⠉⠀⣿⣿⣿⣿⣟⣶⣿⣧⣼⡋⠀⣿⣿⡿⠻⣿⠛⢯⣙⢳⣤⡙⢿⣿⣿⣼⣿⡽⣄⠃
⠀⠀⠆⠈⠈⠉⢻⣿⣿⣿⣿⣿⡀⠀⣼⣿⣿⣿⣷⡟⢻⣿⠛⢿⣷⠀⠛⠋⠀⠀⡿⠀⠀⠙⢳⣽⣿⣌⠙⢿⣇⠹⣷⠹⠰⡄
⠀⢸⠁⢀⠀⠀⣾⡇⢻⣿⣿⣿⣷⡀⢿⣿⣿⣿⣿⡗⣸⣭⣴⣦⣾⠆⠀⠀⠀⠘⡅⠀⠀⠀⠀⢻⣿⣿⡆⠙⢿⡆⣽⣧⠀⢛
⠀⠸⠀⠸⣄⣾⣿⣿⣼⣿⣿⣿⠁⠛⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⡆⠀⠀⠀⠀⡿⠀⠀⠀⠀⢠⡹⣽⣿⡆⢸⣷⠈⣿⡟
⠀⠈⠙⠒⠻⢿⠏⠉⣿⣿⣿⣿⢦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⢸⣿⣀⣄⣴⣾⡧⢄⣥⣤⣶⣷⣾⣿⣿⠇⠠⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠀⠙⠛⠿⠿⠿⣿⣿⣿⣿⣿⡏⢿⣿⣿⣿⠿⠟⠻⠿⠿⣿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠻⠿⢿⡿⠋⠁⠈⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⠀⠔⣛⢻⣶⡄⢰⣿⣛⣳⠀⢸⠗⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠉⠙⠉⠹⠃⠈⠉⠉⠀⠀⣸⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠂⠰⠋⡈⠙⢊⠁⠈⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠤⠤⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⡀⠀⠀⠀⣠⠋⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠻⣧⡽⠿⠶⠒⠂⠁⡀⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠸⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠻⢀⡿⠈⢇⠀⠀⢠⣿⣿⣷⡄⠀⠀⠀⢸⠀⠀⠀⠐⠤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠃⠸⡇⠀⠈⢆⡠⠛⣿⣿⡏⠻⣀⠆⢀⡿⠤⠼⠀⠀⠀⠀⠈⠐⠒⠒⢄
⠀⠀⠀⠀⢀⠐⠉⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠈⢣⡀⣿⣿⣷⠀⡜⢀⣼⢥⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⡞⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠹⣿⣿⣿⡞⢁⡟⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠘⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢘⣇⡠⡀⠀⠹⣿⡟⠀⠸⡇⠀⠀⠀⠘⠇⠀⢠⡇
⠀⠀⠀⣼⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⢴⠿⠋⠀⠘⠄⠠⡏⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⢦⣤⣤⠆⠀⠀⣷⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⢻⣲
⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸
⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢠⡀⠀⣠⡦⣼⢧⡇
⠀⠀⠀⢢⠹⡆⠀⠀⠀⠀⠀⠈⣵⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⡶⡣⠊⡰⡀⠀⠀⠀⢀
⠀⠀⠀⠈⣇⠘⠀⠀⠀⠀⠀⠀⢹⠛⠒⠦⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⢀⠈⡰⠁⠀⠀⠀⠈
⠀⠀⠀⠀⢸⡄⠀⠀⠀⣀⣀⣀⣸⣀⠀⠀⠀⠀⢸⠋⠁⠀⠀⠀⠁⠀⠒⠒⠶⠏⠀⠀⠈⠈⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⣈⣭⣿⡶⢶⣤⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠤⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠂⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣷⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠟⠀⣴⣄⡀⠄⠀⠀⠄⣀⣶⠙⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢻⠁⠸⠀⠛⠁⡘⢉⠹⠀⠙⠁⡘⠈⢹⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⡀⠀⠈⠀⠈⢀⡀⢀⠈⠀⠀⠀⠀⣼⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠉⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆⠀⠀⠀⠀⠀⠠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣭⠁⠀⠀⠀⠀⠀⠀⠉⡽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⢈⣀⣿⣀⢠⠀⠀⠀⠀⠀⠀⠀⢰⠇⢻⡇⠐⠠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢰⣾⡿⠛⡛⠻⣿⣧⠀⠀⠀⠀⠀⣠⡿⠀⢸⡇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⣾⡏⠀⠀⠀⠀⢸⣿⣷⣄⣀⣀⣴⣿⠃⠀⣿⠃⠀⠀⡆⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠏⠀⢰⡏⠀⠀⠀⠃⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⣿⡀⠀⠁⠀⡾⡟⠸⣿⣿⣿⡏⠀⢀⣾⠥⠃⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢡⠁⠀⠀⠸⢀⠀⠀⠹⣿⡟⠀⢪⠀⠀⢠⠀⠀⣰⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡄⠀⡈⠈⠀⢀⠀⠈⢀⠀⠣⠂⠨⠂⠀⠀⠹⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠀⠀⠀⠐⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⣶⠀⠀⠀⢀⠀⠀⣾⣅⠃⠀⠀⠀⣴⣠⣿⡀
⠀⠀⠀⠀⢠⣁⣊⡀⠀⠀⠹⠇⠀⠀⠀⠀⢠⡟⠀⠁⠀⣠⠞⣡⣬⣷⣿⣿⡆⠰
⠀⠀⠀⠀⠈⠛⠉⠉⠓⠶⣦⡀⡀⠀⠀⠀⢸⡗⠀⢀⡾⢡⠞⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡷⣤⠀⠀⢸⠃⠀⣾⠁⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⣰⣟⡀⠀⣹⣦⠘⠛⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠂⠀⣿⣿⣧⣼⠛⠁⠀⠀⠀⠀⣀⣀⣸⣻⣦⡀⢀⣀⣤⣾⣿⣷⡖
⠀⠀⠀⠀⠀⠀⠁⣼⡏⠟⠀⠐⣻⣿⣿⠇⠀⠀⣠⣾⠾⠻⢿⡿⣻⡿⠿⠛⠛⠋⠉⠉⠁⣸⣇⣀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⡄⣾⣿⣿⠋⠀⠀⢰⣿⣿⠿⠛⠛⠉⠉⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣿⣥⠀⡀
⠀⠀⠀⡄⠀⣴⣾⣿⣿⣿⣿⣿⣽⣯⠀⠀⢀⣿⣿⣿⣷⠀⠀⠀⠀⣠⡾⠿⠛⠻⢿⣿⣿⣭⣙⡿⠿⠀⠁
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠏⢹⣿⣷⠧⠀⠈⣿⣿⣿⠗⠀⠀⠀⢠⡏⠀⠀⠀⠀⠈⣺⡟⠻⠿⢿⡷
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣗⠂⣀⣿⡿⡿⠠⠀⠀⠀⡼⠀⠀⢠⣤⣄⠀⢸⡇
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⢷⣿⣿⣿⣿⡟⠁⣰⡶⠓⠒⠀⣰⣇⣠⡾⣟⠀⠉⠳⣼⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣽⣿⣿⠟⠉⠀⠈⠻⢍⣼⡿⠁⠀⠀⢰⣿⣿⣿⠃⠀⠀⠀⠀⢻⡃⠠
⠀⠀⠘⣿⣿⣿⣿⣷⣿⠛⣧⠂⢠⠀⠀⠀⠀⣸⠳⢬⡄⣀⠀⢿⣿⡇⠸⣇⠀⠀⣠⣼⡇
⠀⠀⠀⢹⣿⣿⣿⣿⣿⡞⡯⠀⢋⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⣿⣦⣿⣶⣾⣿⣿⠃
⠀⠀⠀⠘⣿⣿⣿⣿⣿⠂⣧⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠉⠛⠻⠹⡿⣿⣆
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣯⣼⠘⠊⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣾⠃⠳⡄⠀⠀⠀⠠⣹⣿
⠀⠀⠀⢀⡈⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⣡⠖⠋⢹⡵⠀⠀⠱⡄⠀⠀⠀⣺⣿
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣷⠀⠀⠀⠐⠀⠋⠁⠀⠀⣾⠇⠀⠀⠀⢽⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣄⠀⠀⠀⠈⠀⠀⠀⣰⠟⡇⠀⠀⠀⠀⠁⢨⣿⠏⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣿⣦⠀⠀⠀⠀⣴⠃⠀⠘⢀⠀⠀⠠⣰⣾⡟⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⢷⡦⣀⣼⠏⠀⠀⠀⠀⠀⠀⢄⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⡞⣿⣿⠀⠀⠀⠀⠀⢢⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣤⣴⢠⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⣶⣾⡛⢶⡀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⡴⠀⣼⡇⠀⠀⠀⢠⠀⠀⡀⢺⡀⠿⣿⣏⠳⣝⠉⠄⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠆⠀⢿⡇⠀⠀⠀⠀⢃⠀⠀⠀⠻⣮⣿⣯⣵⢏⠄⠀⠀⠀⠀⣿⡇⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⣸⣟⣀⠆⠀⠀⠈⣆⠀⠀⢠⣿⣿⣏⣷⠀⠀⠀⠀⠀⠸⣿⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⢀⣴⡿⠿⠿⢿⣷⡄⠀⠀⠈⢆⠀⣾⣿⣟⣼⠃⠀⠀⠀⠀⠀⢀⣿⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⣼⣿⠀⠀⠀⠀⢹⣿⠀⠀⠀⠈⢢⡯⡿⣼⠃⠀⠀⠀⠀⡀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠃⠀⠀⠀⠡⡀
⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠘⢾⠇⠀⠀⠀⡠⠊⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡜⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠐⠤⣀⠀⠀⠎⠀⢀⠴⠋⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠘⠁⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡈⠀⡴⠋⠀⠀⠀⠀⡀⠸⣷⣬⡿⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢣⣼⣿⠄⢀⡜⠀⢀⣀⡀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠸⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠸⠏⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⡆⡄⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⢤⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠰⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⣠⠀⠀⠙⣦⡄⠀⣠⣄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠁⠘⢛⣄⣋⣁
⠀⠀⠀⠀⠿⠗⠸⠋⠀⠀⠁⠉⠉⠀⠀⠃⠀⢰⣷⠀⠀⠀⠀⠐⢅⠌⠀⠀⢠⠋⠀⠀⠀⡄⠀⢀⡆⠀⠀⠁⠀⠀⠀⠀⠈⠉⠑⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⣤⣤⣤⣤⣤⣀⣀⣁⣀⣀⣀⣀⢀⣠⣀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠞⠀⠀⡠⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠚⠛⠒⠒⠒⠒⠒⠒⠲⠶⠖⠶⢶⡶⠶⠶⠒⠶⠒⠒⠒⠂⠀⠀⢠⠸⣆⠠⠀⣀⠀⣀⢠⠁⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣦⣿⣿⣧⢶⣋⡥⠒⠠⠀⠀⠀⠀⡸⡀⠈⡁⠈⢉⣹⣿⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠠⠆⠐⠲⠄⣀⣀⠀⠀⢠⣲⣾⣿⣾⠿⣿⡟⠍⢳⡋⠁⠀⠀⠀⠀⠀⠀⠀⠃⠐⡄⠈⠦⠙⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠶⠀⢀⣀⠈⠳⠀⠈⠁⠀⠂⠴⠶⠷⠶⠶⠶⠶⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠈⠂⢀⠊⢀⣀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠐⠀⡉⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀⠤⠂⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉






⣿⣿⣿⡿⣿⡿⢿⡿⡿⣿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣾⣷⣾⣶⣿⣶⣾⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉⠁⠀⠀⠉⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⣴⡟⢷⣆⠀⠀⠀⠈⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠻⣦⡾⠏⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣀⣠⣤⣤⣤⣤⣤⣤⣀⣸⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠈⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⢸⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⡀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⢠⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⢁⣿⣧⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣀⣙⣛⣛⣛⣁⣾⣿⣿⣇⢹⡟⢛⣛⣛⣛⣛⣛⣋⡀⠀⠀⠀⢸⣿⣿⣿⣿⣷⢄⣀⡀⢀⣀⢴⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⢿⣿⣿⣿⣿⣿⡆⠁⣿⣿⣿⡿⠋⠉⠉⠁⠀⠀⠀⠉⢉⠉⣉⣩⠝⠀⠀⠀⠀⠀⠈⠯⣉⢉⣉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⢸⣿⣿⣦⣄⡀⠀⠀⢀⣤⣾⣿⡇⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⢸⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⢇⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⢉⣤⣉⠻⣿⣿⣿⣿⣿⣿⣿⡿⢫⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢰⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⡿⢡⣿⠛⠛⠛⠛⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣌⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⡀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠠⢼⡦⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢀⣀⣀⠲⡗⠂⠀⠀⠀⣀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀
⠀⠈⢁⣶⣶⠀⠀⣿⣿⣟⡇⠀⠀⠀⢘⣿⣟⠀⠀⠀⠀⠀⠀⠙⠳⡀⠀⢸⠀⠈⠁⠀⡇⠀⠀⢠⣞⠛⡛⡆
⠀⠀⣼⣇⣼⡇⠀⣿⣿⣿⡇⠀⠀⠀⠻⢻⠻⠀⠀⢀⠀⠀⣀⠤⠀⡇⠀⢸⠀⠀⠀⠀⡇⠀⠀⠈⠁⠁⠀⠁
⠀⠀⠁⠀⢁⠙⠀⣿⣿⣯⡇⠀⠀⠀⣠⣬⣤⡀⠀⠀⠰⡊⠀⢒⠋⢀⠀⢸⣀⣀⣀⣀⡇⠀⠀⠀⠈⠾
⠀⠀⠀⣠⡄⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠈⠀⠐⢻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⢹⡇⠀⠀⣴⠶⢶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠈⠿⠿⠁⠀⣿⢰⢂⠁⡌⠃⣶⣶⢰⠁⡎⡇⢢⣿⡇⣶⡇⣿⢸⠐⣸⠁⢻⠐⢸⡆⢼⣿⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣀⣾⣆⣱⣧⣩⣿⣸⣦⣱⣇⣆⣿⣧⣩⣇⣋⣸⣠⣧⣴⣸⣀⣌⣧⣸⣿⠀⢸⣏⡏
⣀⣦⡀⠠⠄⠀⠀⠈⠉⠉⠉⠉⣉⠉⠉⠉⠉⡉⠉⠉⡉⢉⢉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⣸⠗⣇
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠙⠈⠀⠉⠁⠑⠁⠉⠁⠈⠁⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾
⠀⣠⣶⣤⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡤⠀⠀⣀⣠⣀⣠⣄⣀⠈
⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣿⣿⢻⡻⣿⣿⡇
⠀⠈⠻⣿⣿⠟⠁⢀⣶⣀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠾⠴⠶⠼⠿⠿⠿⣿⠗⠀⠀⢸⣿⣿⣽⣵⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠤
⠀⡏⠅⠀⠀⠀⠀⠀⠠⡴⠤⠦⠦⠦⠴⠰⢤⠀⡀⠀⠀⠈⢀⠠⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠈⠀⠄⠀⠠⠠
⠀⡃⠚⠨⠗⠓⠘⠀⠀⡀⠀⠀⠂⠀⠀⡀⢸⠀⠁⠀⡉⠀⠀⢀⠀⠀⠁⠀⠀⡈⠐⡤⠄
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠄⠀⢀⠀⠀⠀⠂⡀⠂
⠀⠀⢀⠐⠀⠀⠀⡠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠂⠣⠀⠀⠀⢀⠔⠂⠦⢀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠄⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⡀⠎⠀⢠⠈⢥⠀⠀⠀⣉⠄⡀⠢⡄⡀⡊⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠈⢦⡀⠙⠐⠂⠁⣀⡇⠈⠐⠱⠋⢀⠀⠀⠀⠐⠝
⠀⠂⠀⠀⠘⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⢀⠄⠐⠒⢀⠆⠀⠀⠀⠀⠉⠓⠒⠚⠉⠀⠀⠀⠰⠀⠂⠠⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⢀⡀⠀⠀⢤⠀⡠⢐⠀⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⡆⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⠟⢻⡷⣿⠙⣷⠀⠀⠄⠀⠀⠠⠀⠀⠀⢂⢀⠀⠀⠐⠀⢀
⠀⠀⡀⠨⢬⠀⠀⡇⠀⠀⠀⠁⠀⠀⠂⢸⡇⠀⢸⣦⣤⡇⣿⣶⣟⠀⠀⣄⣀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠃⠙⠘⠀⠀⡇⠠⠀⠀⠐⠀⠈⠀⢸⣧⡄⢸⠀⠀⡇⢿⣤⡿⠀⠀⠇⠀⠀⠈⠂⠀⠤⠀⠀⠠⠄
⠀⠀⠄⠀⠀⢈⠁⠅⠈⠀⢀⠀⠀⢀⠀⠀⡴⣶⣶⣶⡶⣶⣶⣶⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠐
⠈⠁⠀⠐⠂⠉⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⢿⣾⡿⣶⣷⢾⢷⢿⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠠⠤⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⢐⠁⠁⢄⠁⠀⠐⡐⡁⠀⠀⠀⠠⠀⠐⠠
⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠉⠐⠂⠀⠀⠀⡤⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠈⠀⠀⠀⢐⡀
⠀⠀⣷⢸⠆⡀⠀⡟⣿⠀⠀⢀⠀⠀⠈⠉⠀⠀⠐⠂⠀⠀⠀⠀⠆⠄⠀⠀⠀⠈⠀⠀⠠⢘⡌⠁⠌⠀⠀⠀⠀⠠
⠀⠀⣿⢻⡜⢹⠆⣷⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠛⠘⠃⠿⠂⠛⠋⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠁⠁⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁
⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣻⣿⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠿⠿⠛⠋⢻⣿⣷⠀⠀⠀⠀⠁⠀⢤⠄⠀⠉⠀⠀⠀⠀⠀⡿⠛⠉⠙⠻⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⣠⣶⣶⣶⣶⣶⣾⡄⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠤⠀⠀⣰⣿⡿⠛⠋⠁⠹⣿⣿⡆⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣼⣷⡄⢠⡿⠋⠀⠀⠀⠀⠀⠈⠛⣿⠀⠀⠀⠸⣿⡏⠉⠉⣹⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡕⠀⠐⡌⣻⣿⡇⢸⣿⣶⡀⠀⠀⠀⠀⢠⣶⣿⠀⠀⠀⠀⢙⡿⠶⠾⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠉⠀⢸⣦⣼⠁⠸⣿⡃⠀⠻⣿⣷⣄⠀⢀⣠⣿⣿⠟⠀⠀⠈⢾⠉⠑⠀⠀⠊⠙⢧⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠃⢀⡠⠈⠛⢹⠄⢰⠏⠻⠀⠀⠀⠀⠀⡀⠱⠀⠀⠀⠀⠀⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢀⠀⠄⠌⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡇⣼⠀⠀⠀⠀⡆⠀⠀⣁⡀⢘⠀⠈⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠐⠀⠀⠀⠸⠗⠀⠀⠐⠐⡇⠀⠀⢻⠃⢸⠀⠀⠀⠀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢰⣶⣶⠀⠀⠀⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠄⠂⠒⠤⠀⠀⠀⠀⠃⠀⠀⠀⠐⠀⠀⡦⠤⠤⠤⠤⠤⠤⠼⠀⢘⡰⠂⠀⠀⠀⡀⣀⠀⠀⠀⠸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠰⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠘⠀⠀⠈⠁⠉⠁⠀⠀⠃⠦
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡤⠤⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⢱⠀⠀⠘⠀⠀⠀⠀⠂⠀⡇⠰⠀⠘⠀⠀⠀⠀⠈⠀⠀⠐⠀⠐⠀⠀⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠀⠈⠛⠛⠛⠛⠛⠒⠒⠒⠒⠁⠘⠒⠒⠒⠒⠃⠙⠛⠛⠛⠋⠀⠈⠛⠛⠛⠛⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣠⠈⢧⣀⣴⣊⡀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⣠⠏⠉⠯⣉⠉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⢤⡤⠀⠹⢄⡉⡇
⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠉⠙⠒⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠶⠌⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠓⣖⠁⠀⢀⣶⢏⠷⡄
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠠⡰⠖⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣿⠔⠦⢤⣶⡿⠛⠁⡰⠉⠐⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠐⢇⠀⠀⠘⡆⠀⠀⠀⠀⠀⣠⢶⠋⡇⠀⠸⡇⢀⣾⠛⠒⠒⠉⡠⠀⢀⠤⠈⠳⣄
⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⢀⡀⠀⡾⣀⣀⣠⠶⠤⠚⠹⡼⢸⠇⠀⠀⣿⣼⠃⠀⠀⠀⡔⠠⣞⣡⡴⡶⡶⣿⡆
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢹⠁⠈⠉⠀⠘⢉⣁⣀⣀⣀⣀⣙⡏⠀⠀⢰⢿⡿⠂⠀⠀⡐⡴⢰⠋⢀⠀⠀⢀⢀⡿
⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢸⠛⠲⢦⣀⣀⣤⠴⠶⠬⣭⣉⢻⡇⢀⣰⢏⡾⠃⠀⠀⠌⡜⢠⡏⠔⢁⡠⠔⠉⣼⡇
⠀⠀⢠⡏⠀⠀⠀⢷⠀⠀⢸⣦⣤⢾⡍⣉⡉⠑⣞⠛⠉⠉⢻⠛⠿⠟⡻⠁⠀⠀⢀⠞⡴⠉⠁⠉⠀⠀⢀⣼⢻⡇
⠀⠀⣼⠀⠀⠀⠀⢸⠀⠀⢸⡏⠀⠈⠻⣶⣄⡀⠈⠒⠲⢤⣿⣀⣀⣰⠁⣀⣀⣰⠧⣼⣅⡀⠀⠀⢀⡴⣿⡧⣾⣀⡀
⠀⢠⠇⠀⠀⠀⠀⢸⡄⠀⠈⡇⠀⠀⠀⠀⠀⠙⠦⠤⢼⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠀⠰⢋⣾⠃⣠⠇⠈⠉⠛⣦
⠀⠸⡧⠄⠀⠭⣤⠰⢇⠐⠂⠙⣄⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⠤⠤⢤⣤⣀⣀⠀⠀⠀⠀⠀⠉⣠⡾⠁⡼⠁⠀⠀⠀⢰⠇
⠀⢸⠃⠀⢒⠒⠛⣏⠈⠳⣄⠀⠈⢦⡀⠀⠀⣀⣀⡤⠾⣁⠀⢀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⡛⠋⢠⣦⣧⣄⣀⠀⢀⡞
⠀⢸⠈⠉⠉⠀⣀⠈⣆⢀⣨⠷⢦⣀⠙⠶⢋⡉⠁⠀⠠⢌⣑⡪⣭⠽⢒⣒⣒⠲⠄⠐⠊⠁⠀⣠⠎⠈⡇⠀⣹⠛⠛⠁
⠀⠀⠀⠀⠀⠉⣉⣀⣉⣐⡁⠀⠀⠈⠑⠲⠤⣼⡷⢤⠀⠀⠀⠀⣀⣀⡀⠀⠠⠤⠤⠄⢀⣀⡘⠁⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢦⣤⣄⣀⣀⢀⠀⠀⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠈⠀⠀⢀⡷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡶⠚⠁⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⡀⠀⢀⣾⡁⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠾⠉⠙⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⣩⣍⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣠⣤⣬⣥⣤⣄⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠉⠉⣉⣉⠀⠀⣘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⡘⠃⡨⢁⠘⢃⡈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠛⠋⠉⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠄⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢡⡧⠀⠀⣬⠄⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠐⠲⠔⠂⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠈⠀⠀⠀⠈⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣪⠕⢯⠑⢂⡔⠒⣹⠲⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡀⠀⣤⣤⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⡿⢧⡸⣤⠘⡏⢻⡟⢛⠇⢠⡼⠨⡏⠒⠦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⡦⢄⡡⢴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢸⠀⠀⡇⠳⡄⢹⣼⣿⡜⢀⡜⢀⣴⣣⡄⠀⠸⡄⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⣀⠀⠀⣨⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠘⠀⣰⠁⠀⠘⢦⣻⣿⡡⠋⠀⠈⠻⠿⠃⠀⠀⡇⠀⠀⠀⠀⠀⣴⣿⣹⣿⣿⣿⣷⣾⡿⠿⠿⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠭⠭⠤⠭⠤⠤⠤⠿⠿⠥⠤⠤⠤⠤⠤⠤⠤⠤⠿⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⣠⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⡿⢀⡇⡇⢸⠁⠀⠉⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣻⡄⠙⢂⠀⣿⠸⣧⡇⢺⠏⡛⠛⠉⠉⠀⠠⠑⡀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⡹⣦⠀⢳⣿⠀⠻⠁⢀⠀⠀⠀⠀⣀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠌⠁⠀⠀⠀⠀⠀⢈⣤⣶⣶⣯⣿⣽⠬⠟⠁⣀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⠷⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⠋⠈⣀⣨⣴⣿⠇⢠⢖⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⠇⣿⣏⡶⣤⣀⠐⣈⣶⡞⣿⣿⣿⠿⠛⠀⢤⣼⣿⣿⣿⡿⡋⠁⠚⠓⣀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡷⣷⡜⣿⣷⣿⣿⣽⣿⣿⣷⣿⠿⠉⠀⠀⣠⣿⠿⠿⠛⣿⣿⣿⣶⣾⣿⣏
⠀⠀⠀⠀⠀⣰⣿⣿⡇⠈⠛⡍⢬⣿⡇⢹⣿⣿⣿⡿⠿⣿⣿⣏⠀⠀⢨⣶⣿⠋⠀⠀⠀⢻⣿⠋⠉⠉⠛⠉
⠀⠀⠀⠀⢠⣿⣿⣿⣿⡞⢾⣳⣾⣿⢁⣿⣿⣿⣿⢛⠐⣹⣿⣿⣷⣴⣿⣿⣧⡄⠻⣆⠀⣾⣿
⠀⠀⠀⠀⢸⣿⣿⠿⣿⣽⣿⡿⠟⣡⠾⠛⡋⠻⠳⢜⣿⡷⠙⠛⢿⣿⣿⡿⠋⠀⠁⠉⠻⣿⡿
⠀⠀⠀⠀⣺⣿⡿⢷⣬⣽⢥⣾⠉⠀⠠⠀⠤⠀⠀⣿⣽⢆⠀⠀⣾⣿⣿⠁⠀⠀⠀⠀⣰⢿⡇
⠀⠀⠀⠀⣿⣿⣷⣿⣿⣚⣚⢿⡀⠀⠀⠂⠀⠀⢀⡏⠘⠿⠲⠂⣽⣿⣏⠀⣠⢰⣦⣤⣿⣿⡇
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣵⣧⣼⢹⡄⠁⠀⠠⠀⠘⠁⠀⠀⠐⢠⣼⣿⣿⡄⠸⣮⡘⠹⠛⣻⠃
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⠙⣝⣨⣦⠈⠉⠀⠀⣀⣩⡵⡶⠚⠉⢸⣿⣿⣿⣦⣼⣿⣷⣿⣿⡀
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡀⠊⠀⠑⠃⠀⢠⢞⡁⠂⠈⠀⢲⠑⣾⣿⣿⡿⣿⡿⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣝⡞⠐⠀⠀⢈⡵⠋⠈⠀⠃⢆⣈⣿⣿⣿⠿⣷⠈⢙⣿⠏⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡟⠐⢂⡴⣋⡀⠔⠀⠀⢔⣥⣾⣿⣿⣿⢇⣿⣻⣶⡦⣤⣬⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣛⡶⠉⢠⡤⢳⣦⢤⡦⣿⣿⢿⣹⣿⠻⣦⣾⣧⡏⢹⣬⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⣿⣿⡑⠉⣶⠚⡁⡅⢿⡟⢸⣽⣿⣦⡘⢿⡿⣶⣶⣿⣿⡿⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣯⣽⡉⠰⢸⣠⠃⢀⣾⣅⣏⣼⣷⣽⣿⣷⣤⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣾⣽⣵⡌⣻⣆⣾⣿⣿⣙⠿⣿⣿⢿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⣻⣇⡟⣿⣿⣾⣿⣿⠁⢄⣹⡎⢹⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣯⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣹⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢉⡉⠙⡅⢀⣠⠖⠉⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⠁⢀⠟⢻⡟⠛⠻⠾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠚⠋⠀⠀⠉⠳⡀⠒⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠈⠛⠋⠀⠀⠉⢦⣼
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢀⠄⠀⠀⠰⣻⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠈⠛⠂⢹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠹⠭⠀⠀⣠⣾⣏⠜⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠠⢲⠿⠋⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢾⡷⠚⠶⢶⣾⡾⠛⠁⣠⠆⠀⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠻⡄⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⣠⢲⠋⣾⠀⠀⢻⡄⢠⡾⠛⠒⠒⠋⢀⠄⠀⣀⠴⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⣞⠀⢀⠀⡴⢤⡤⠊⢱⡸⢠⡏⠀⠀⢸⣧⡾⠁⠀⠀⠀⢀⠂⢤⣚⣁⣠⡴⣶⣾⣷
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢸⡞⠉⠑⠚⠀⠹⠋⠛⠀⠀⠀⠀⠀⠳⡞⠀⠀⠀⣼⣿⣷⠀⠀⠀⢠⢊⠄⣼⠛⠉⠀⠁⠀⠈⣸
⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⡷⢤⣄⡀⠀⠀⢈⣉⣉⣙⣛⠳⠶⣻⡇⠀⠀⣴⢣⣿⠃⠀⠀⡠⢣⠎⣰⠃⡰⠊⠀⣠⠔⢡⣿
⠀⠀⠀⢰⠏⠀⠀⠀⣦⠀⠀⠀⣇⠀⢀⣽⠶⡚⠫⠭⣀⣬⣽⠿⠿⣾⣧⣴⣾⣵⡿⠁⠀⠀⠀⢡⢋⡰⠿⠡⠤⠒⠉⠀⢠⡿⣿
⠀⠀⢀⡟⠀⠀⠀⠀⢻⠀⠀⠀⣿⠿⠛⢾⡡⣒⠂⠀⠙⣄⡀⠀⠀⣼⠈⠉⠉⡰⠁⠀⠀⠀⡴⢁⡎⠀⠀⠀⠀⠀⠀⣰⡿⢁⡏
⠀⠀⣸⠁⠀⠀⠀⠀⢸⠀⠀⠀⣷⠀⠀⠀⠙⠛⠳⢤⡀⠀⠉⢹⢳⡗⠶⠤⠴⠓⠒⠒⠦⠾⠓⢻⡲⢦⡀⠀⠀⡠⢺⡿⠗⣽⠲⢤⣄⡀
⠀⠀⡏⠀⠀⠀⠀⠀⠸⡆⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠙⠒⠒⣮⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠊⣰⡿⠁⣠⠏⠀⠀⠈⢙⡷
⠀⠀⣿⠀⠭⠭⠭⣥⠄⢧⡀⠒⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⠶⠶⠤⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⢁⣴⠟⠀⡼⠁⠀⠀⠀⠀⣸⠁
⠀⢸⡋⠀⠒⠒⠒⠛⣍⠁⠑⢄⡀⠀⠙⣆⠀⠀⠀⢀⠀⣠⠴⢞⡁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⢛⡛⠃⢠⢦⣧⣄⣀⠀⠀⣰⠇
⠀⢸⡇⠛⠉⠉⠀⠀⡈⣆⠀⢀⣹⠦⣄⠈⠳⣤⡚⠉⠋⠀⠀⢀⡉⠒⠬⣝⣒⡖⠒⠒⠲⣒⠠⠤⠒⠊⠁⠀⣠⠏⠈⡇⠀⢹⡟⠲⠋
⠀⠀⠁⠀⠀⠐⠚⠋⠀⠛⠩⠟⠁⠀⠈⠙⠦⢄⣀⢙⣦⣄⡀⠀⠈⠉⠀⠉⠀⠈⠉⠉⠁⢀⣀⡀⠀⠀⠀⠖⠁⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⣁⠀⠀⠀⠀⠀⠈⠉⠉⠈⢳⡀⠀⠀⠀⠉⠉⠉⠉⠀⠀⣀⠀⠀⠈⠉⠐⠂⠀⠀⠀⣿⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⣛⠛⠒⠒⢀⣀⡴⠂⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⠀⠀⠀⠀⠀⣰⠏⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠭⠭⠛⠁⠀⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⣰⠿⣤⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⠃⠀⠀⠉⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡶⠞⠛⠛⠛⠛⠛⠛⠷⠶⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣶⣄⡀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠈⣯⠉⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⡀
⠀⠀⠀⠀⣶⢦⣤⣨⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦
⠀⠀⠀⠀⠘⣦⡈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⣠⡶⠟⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠳⢶⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⡟⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠀⠀⠻⣇⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠀⡿⠀⠀⠘⠋⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠛⠉⠉⠙⠃⠀⠀⠀⠀⣿⠀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⣿⠀⡷⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣠⣦⡄⠀⠀⠀⠀⠀⣿⠀⣸⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣿⣸⡇⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠸⣧⣸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠛⠋⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣷⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⠶⠾⣋⣼⣿⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠹⣿⣭⣭⣭⣭⣤⣶⣾⣋⣸⡇⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠉⠷⣤⣤⡟⠉⣁⣠⡿⠋⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣄⠀⠀⠀⠀⠀⢠⡔⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠀⠀⠀⠀⠀⢹⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣭⣽⣟⠀⠀⠀⠀⠀⢈⣿⣭⣭⠛⢿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⣶⣿⡿⠋⢼⣿⡉⠈⠻⣦⡀⠀⠀⣠⡾⠋⠈⣹⣿⠀⠈⠻⣿⣶⣤⣠⣄⣀⡀
⠀⠀⠀⠀⠀⢀⣤⠞⠛⠛⣿⠋⠉⠉⠉⠁⢀⣼⣿⠁⠀⠘⣿⢷⡀⠀⠈⠻⣤⡼⠋⠀⠀⣰⣿⡟⠀⠀⢠⡟⢿⣯⡈⠉⠉⠉⢻⠟⠛⠛⠳⣤⡀
⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⣿⣿⡀⠀⠀⢿⡘⢷⡀⣠⡾⠛⠳⣦⡀⣼⠏⣸⠇⠀⠀⣾⠇⢸⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⡇⢸⣧⠀⠀⢸⡇⢨⡿⠋⠀⠀⠀⠈⠻⣏⢰⡏⠀⠀⢰⡏⠀⣸⣿⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣧
⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⠇⠈⢿⡀⠀⠀⢻⡏⠻⣤⣀⣀⣀⣠⡾⠋⣿⠁⠀⠀⣸⠇⠀⣿⡟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⣿⣆⠀⢸⡇⠀⠀⠘⣧⠀⣼⠋⠉⠙⢻⡄⢸⠟⠀⠀⢰⡿⠀⢰⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡿⠉⢿⣦⣼⣿⠷⡶⣦⣿⣴⡿⠀⠀⠀⠸⣧⣿⠀⢶⣶⠾⠃⠀⠈⣿⣇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⣿⣶⣿⣛⡿⠷⠶⠶⠛⠛⠛⢿⣛⢯⣿⣿⠁⠀⠀⠀⠀⣿⡇⠀⠀⣿⠇⠀⠀⠀⠙⢿⣷⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣇
⠀⢀⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣏⣿⡆⠀⠀⠀⢠⡿⠀⢠⡾⠃⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡀⠀⠀⣸⠇⣠⡾⠁⢀⣤⡶⠶⠶⣤⣾⡿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡇⠀⢰⡏⣠⡟⠀⣰⡟⠁⠀⠀⠀⠀⠙⣧⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣾⠃⠀⠀⠀⠀⠀⠀⠸⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⣾⣿⠏⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣿⠃⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡏⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠈⠳⣦⣄⣀⣠⣤⠾⠃⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣾⠛⠛⣷⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⢸⢻⣇⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠁⠀⠀⠀⠀⠀⠀⢀⡿⣿⣿⣤⣴⠟⠉⣳⡄⠀⠀⠀⠀⠀⠸⠷⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⢻⡄
⣿⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿⠉⠛⠛⠛⠲⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇
⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⣿⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡆
⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇
⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⡇
⣿⣀⣀⣀⣀⣀⣀⣸⡟⠀⣿⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣸⡇⠀⢸⣇⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠛⠋⠙⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣄⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠊⠉⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⣻⡄⠀⢠⣿⣿⡆⢠⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠙⡀⢸⣷⡀⢸⣿⣿⣷⢸⣷⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢐⣿⠇⢠⣧⠀⣿⣇⠸⣿⣿⣿⠀⣿⣕
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠴⠿⡆⠸⣿⣄⣿⣿⣿⡇⠘⣿⣷⣆⠀⣴⣶⣾⡷⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣰⡀⢻⣿⣿⣿⣿⣇⡂⢌⣉⠣⠀⣿⣿⢛⡧⣾⡎⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⣿⣧⡀⢹⣿⣿⣿⣷⠀⠀⠀⠀⠀⣿⡷⣮⢧⡛⠃⠛⢀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⣿⣷⡈⣿⣿⣿⠿⣷⣤⣤⣶⣶⣿⣶⣧⠸⣏⣩⣴⢿⣿⡟⠿⠄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⡻⣾⣿⣿⣿⣿⣽⣿⡀⠀⡄⢀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢹⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⣀⠀⣀⠠⣽⣊⣿⣿⡟⢿⣮⡛⢯⡀⢠⣟⠻⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠹⣿⣿⠟⠁⠀⢀⡴⠆⠉⣀⣠⣾⣿⣿⣿⢿⣿⡇⠀⠉⠻⣦⡍⠲⢌⢿⣶⣿⡷
⠀⠀⠀⠀⠀⠀⡁⠀⠀⢰⣷⡀⢠⣿⡿⣤⣬⣕⠋⠀⠈⠼⠿⣿⡿⠿⠛⠉⠸⠋⠓⣶⢶⡆⠸⠻⣷⣄⠑⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠈⠿⠁⣿⣿⠈⣇⠀⠀⠀⠀⠘⠙⣿⣄⠀⠁⠀⠂⡇⠸⡇⠀⠀⠘⢿⣷⣄⠰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⣿⡇⢸⠀⠀⠀⠀⠀⠀⡘⣿⠀⠀⠀⠀⠛⠘⠃⠀⠀⠀⠈⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⡀⠀⣴⣿⣷⢸⠀⠀⠀⠀⠀⠀⠁⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠟⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⢦⡛⠘⢇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢡⠀⠈⢢⢹⡄⠈⠁⠸⠒⠒⢒⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣈⠢⠴⠋⣼⢋⠓⠒⠒⠶⢲⠦⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢛⠟⠷⠶⠚⠙⠛⣷⡤⠤⠤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠋⠉⠙⠳⣄⣰⠟⢩⣵⣤⠊⣩⢵⡼⠂⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠘⠳⣬⡼⠋⠱⣜⣿⡾⠅⠀⠀⠀⠼⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣤⠀⠀⠀⠀⠀⢀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⣠⠀⠀⠀⠟⠹⣧⠀⠀⠀⠀⠀⠸⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢶⠾⠛⠁⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠐⢷⣄⣀⣀⣠⡾⠁⠀⠀⠀⠀⠀⠀⢸⠇
⠀⢀⣀⣀⣤⠤⠴⠖⠒⠛⠛⠋⢉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡇⠀⠀⠀⠈⠉⠉⣡⡴⠟⠀⠀⠀⠀⠀⢀⡿
⢿⠉⠉⠁⠀⢀⠤⠄⠤⢄⡀⠀⢨⢸⡇⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠶⢦⡀
⢸⡀⠀⠀⠀⠁⠀⠀⠀⢀⡈⢆⠀⢸⡇⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⢸⠈⠙⡷⣦⣄⣀⣀⣀⣀⣀⣠⣴⠞⠋⠀⠀⠀⢸⠹⣆
⠈⡇⠀⠀⠀⣎⢁⠄⡸⡁⣰⢸⠀⢈⡇⠀⠀⠀⠀⣠⠎⡞⠀⠠⡔⠋⠀⢀⡿⠩⡈⣩⠟⣯⠉⠉⠸⠧⢤⣤⠀⠀⠀⢸⡄⠈⢷⡄
⠀⣧⠀⠀⠀⢄⠀⠀⠑⠀⠠⠃⠀⠀⡇⠀⠀⢀⡜⢁⣼⠁⠀⠀⠘⣆⠀⢸⡇⠀⣿⠁⠀⢸⠀⠀⠀⣠⠞⠁⠀⠀⠀⢸⠙⢦⡀⠙⢦⣄
⠀⢻⠀⠀⠀⠀⢧⣤⣤⡤⠆⠀⠀⠀⣿⠀⢀⡞⢀⡞⡌⠀⠀⠀⠀⠘⡄⢸⡇⣸⠘⡇⠀⣾⠀⢀⡼⠃⠀⠀⠀⠀⠀⢸⠀⠀⠙⢦⡀⠙⠳⢦⣄
⠀⢸⡆⠀⠀⠀⠀⢀⣤⣖⠂⠀⠀⠀⢹⠖⠋⢁⠞⢠⠀⠀⠀⠀⠀⠀⠰⣼⢡⡏⠀⣿⠀⣿⢠⠟⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠴⠚⠛⢀⣀⡬
⠀⠀⠇⠀⠀⠀⡤⡿⠈⡯⠱⡔⢦⠀⢸⡆⠒⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⣿⣸⠁⠀⢹⡀⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣏⠁⢀⡴⠛⠁
⠀⠀⢸⣀⣀⣸⡄⢿⠀⡇⠀⡇⣸⠀⠘⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⠳⠟
⠀⠀⠀⠈⠣⠐⠈⠈⠛⠉⠚⠉⠁⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡿⡇⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⢁⣀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢰⡧⢷⡄⠀⣠⣇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠞⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠶⠋⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠦⠤⢤⣤⣤⣼⠀⠝⠳⠶⢴⠄⢼⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡼⠀⠀⠀⣼⠀⢨⡏⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠰⡇⠀⠀⠀⢻⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢏⠀⢧⠀⠀⠀⠘⡆⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⠉⠓⣦⣄⣀⡀⡀⣸⣀⣸⣆⢀⣀⣀⣿⣀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠴⣏⠉⠙⠛⠓⠒⠒⠚⠛⠉⠉⠁⠀⣠⡍⠉⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠤⣤⣤⣤⣤⣤⡤⠴⠖⠋⠉⠳⠶⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢠⣤⣄⣠⣤⣤⣄⠀⠀⠀⠀⣀⣄⣠⣄
⠀⣷⡀⠀⠀⠀⠀⠀⣸⠄⠀⢠⣿⣿⣿⣿⣿⣿⡆⠀⠀⢿⣿⣿⣇⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⠿⢿⡟
⠀⠘⢿⣶⣄⢠⣶⡾⠏⠀⠀⢸⣿⣿⣿⣿⡏⠉⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⡶⣼⡇
⠀⠀⠀⠘⠛⠛⠋⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡿⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⢿⠀⣹⠃
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣀⡀⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⠃⠀⠀⢀⠰⣾⣿⣷⡦⠠⣄⠀⠀⠸⢷⣦⡀⠒⠐⠐
⠀⠀⣴⣿⣿⣷⣿⣷⡄⠀⠀⠀⠊⣴⡄⠂⠀⠀⠀⣿⣿⣿⣯⠙⠁⠀⠀⠀⠈⠓⠛⣋⡭⢶⣶⣿⡷⠀⠀⠀⠙⢽⣦⡀
⠀⢰⣿⣿⣿⢹⣿⣿⣧⠀⠀⠆⠐⣈⡄⠀⡀⠀⠀⠸⣟⠕⠁⠀⠀⠀⠀⠀⠀⠀⠺⣅⣀⣠⡣⠂⠀⠀⠀⠀⠀⠀⠈⠓⠄
⠀⠈⠋⠉⠀⠀⠉⠉⠋⠀⠀⠈⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢎⠉⣷⠀⠀⠀⠀⠀⠀⠰⣤⣶⣦⡀⠀⠀⠈⢳⣦⣀
⠀⠀⣼⣿⣿⠆⢾⣿⣿⡄⠀⠀⠐⠂⠒⢐⠂⡀⠀⠀⠀⠀⠠⣄⢠⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⡈⢟⣉⣳⡀
⠀⢰⣿⣿⡃⠀⠀⢻⣿⣿⠀⠀⠀⠘⢀⠈⠠⠀⠀⠀⠀⠀⢀⠋⠣⠀⠀⠀⠀⠞⠶⣤⣴⣿⣿⣿⠃⠀⠀⠀⠀⠶⢈⣿⡇
⠀⠀⠙⠛⠁⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠳⠂⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠐⠒⠒⠚⠛⠃
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣤⢀⠀⠀⠀⠀⢀⠀⠀⠤
⠀⠀⠀⢀⣿⠯⣿⡆⠀⠀⠀⠀⢾⣷⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⠀⠀⠀⠀⢠⢸⣿⣸⡆⠀⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠈⣱⣤⣤⡅⠀⠀⠀⠀⠘⠙⠻⣷⣄⡀⠀⠀⠀⢸⡀⠀⢿⣿⡟⠀⣰⡇⠀⢸⣾⣿⣿⡇⢀⣤
⠀⠀⠀⠀⢻⣿⣿⠁⠀⠀⠀⠐⢒⠀⠀⠈⠻⣷⡄⠀⠀⠈⠻⣶⣦⢀⣶⡾⠟⠀⠀⠀⣿⣿⣿⣷⡾⠁⠀⠀⠀⠀⠇
⠀⠀⠐⠶⠿⠿⠿⠷⠆⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠛⠀⠀⠀⠀⠀⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⣠⣤⣴⣐⣄⣀⠀⠀⠀⠀⠀⢰⡆⢰⣶⡆⢸⠀⠀⠀⠀⠀⣸⣿⣿⠶⡀
⠀⠀⠀⣿⣿⡿⣿⣿⣿⣵⠀⠀⠀⠀⣾⣧⠈⣉⠁⣸⠀⠀⠀⠀⠀⠀⢹⢙⣏⣯⠄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠈⠉⠛⠛⠻⠿⠋⠀⠀⠀⠸⠿⠿⠆⠿⠐⠿⠇⠀⠀⠀⠀⠀⠞⠈⠋⢡⠄⠀⠀⠀⠀⠀⠆⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⠶⠉⠙⠛⠉⠉⠲⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣤⣀⠀⠀⠀⢁⠀⡐⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠟⠁⠉⠳⣄⡀⠈⠂⠍⠀⣌⣐⣀⡀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣀⡀⠀⠲⣬⡙⠷⢶⣤⣾⠿⠟⠋⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣉⠀⠀⠀⡈⠻⡀⢀⣀⣤⠤⣤⡞⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠴⠀⠟⠀⠀⢾⣿⠀⢰⠘⠏⢀⣀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢠⡄⠀⠚⠅⠀⢠⣀⡀⣬⢥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⢀⣀⣼⠇⠀⠀⡀⠀⠀⠈⠀⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⢀⡼⣦⠀⠀⠀⢻⡀⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠳⠄⠘⢉⡀⠹⠇⠀⠀⠸⡷⠚⠋
⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠙⢀⠈⠳⢦⡀⠀⠀⠀⠀⠀⣠⡶⠛⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠉⠐⢀⣈⠙⠶⢆⣤⡴⠏⣀⣀⡄⢈⡇⠀⠀⠀⠀⠀⠀⠀⢀⣈⠻⢶⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⣶⠟⠳⣤⡀
⠠⡞⠋⠉⢉⣄⣨⡟⢁⡴⠎⠉⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠊⠁⠀⠀⢸⡅⢹⡟⠟⠙⡿⢋⡛⠁⠀⡟⠀⠀⠀⠙⢶⣄
⠀⡈⠙⢓⠀⢠⡏⠀⠸⣦⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣠⠴⠞⠃⠈⠀⠈⣿⠁⣼⠃⠀⣸⠀⠀⠀⢠⠁⠦⣈⠷⣄
⠧⠀⠒⣩⠙⠛⠀⢠⡤⠀⢀⣴⠞⢋⠀⠀⠀⠀⠀⠀⠀⡰⠋⣀⠀⠀⠀⠀⠸⣆⠀⠀⠀⣾⠃⠈⣦⠏⠀⠀⠉⠁⡰⡦⠈⠈⠁⠀⠀⠈⢡
⠀⢴⠁⠄⠒⠉⠀⠀⠀⠀⢿⠀⠀⢸⡇⠀⠀⠀⠀⢀⡜⠁⣼⠃⠀⠀⠀⠀⠀⠈⢆⠀⠀⠸⠀⠀⡟⠀⠀⣠⡴⠊⠁⠀⠸⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢸⡇⠀⠀⠀⢀⠎⠀⣜⠃⠀⠀⠀⠀⠀⡄⠀⠈⠳⡀⠀⡀⣴⢁⡤⠚⠁⠀⠀⠀⠀⠤⠤⢤⣤⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣳⠀⢀⡴⠃⢠⡞⠘⠀⠰⠒⠂⠤⠤⠀⣀⠀⠀⠹⢆⣷⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢹⡴⠛⠁⣠⠋⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠃⣠⠞⠁⠀⠘⠀⠀⠀⠀⠒⠀⠤⠄⣀⡆⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡸⠀⠀⢰⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠞⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠂⠤⠠⠤⠊⠁⠀⠀⢸⢀⠀⠰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠲⢤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⡶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⠉⠉⢛⡿⠛⠿⠶⠶⠾⠿⠶⠾⠿⠿⠟⠛⠛⠛⠻⠋⠉⠻⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀⠘⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠶⠞⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠳⠶⠿⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠁⠀⢀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⢰⣆⠀⠉⠛⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣀⠉⠉⠁⠀⠈⠉⠉⠁⠀⢀⣤⡴⠞⣧⣤⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠟⠁⠙⠶⣤⡀⠀⠀⠈⠙⠛⠓⠲⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣄⣀⣀⣤⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠶⠖⠛⠋⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣟⣻⣟⢿⡿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣾⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠃⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⣠⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠉⠓⠲⠶⠶⠒⠋⠁⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⢉⣀⣼⣇⡀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣿⠀⠀⠉⠙⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⣴⠟⠉⠉⠉⠻⣦⡀⠈⠳⣄⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⢸⣇⠀⠀⠘⢧⡀⠀⢀⡞⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠹⣤⠏⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣟⢷⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣿⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠛⠛⠿⠂⠀⠀⠀⠾⠶⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣶⠶⢂⣩⢇⣄⣀⠈⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⣫⠞⢁⣴⣿⢷⣿⣟⣽⣷⣦⣌⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣴⢏⣴⢀⡾⢫⣾⣿⣿⣿⣿⣿⣯⡆⣈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠄⠀⠀⠻⠿⠿⣿⣿⡿⣿⠞⠋⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⣿⣻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢸⣿⠀⠀⠀⠶⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠃⠈⠁⠀⠀⠀⠰⢶⡶⣍⠛⠀⠀⣴⣶⡶⠶⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⠿⠟⢂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⠀⠀⠀⠀⠀⠀⠸⣅⠀⢸⣃⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣶⢾⠃⢸⡀⠀⠀⠀⢤⣀⣀⣈⡙⠉⠁⠀⢀⠊
⠀⠤⠴⠖⠒⠛⠋⠉⠉⠁⠀⠀⢠⣾⠟⢹⡇⠈⣇⠀⠳⣄⠀⠀⠀⠈⢉⣉⡙⠛⠀⣠⠃
⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⣿⡍⠀⠸⣇⠀⠙⣆⠀⠈⠳⣆⡀⠀⠀⠉⠁⢀⣴⢿⡶⢤⣄⡀
⢳⡀⠀⠀⠀⢰⢏⣠⡀⠀⠀⠀⣿⢧⠀⠀⠉⠛⠦⣼⣆⠀⠀⠈⠉⠉⠛⠛⠛⣹⠋⢻⣷⠀⠈⠉⠛⠲⠦⣤⣀⡀
⠈⢷⠀⠀⣰⡿⠋⣽⠁⠀⠀⢀⠿⠸⣆⣀⠀⠀⠀⢸⣿⣦⡀⠀⠀⠀⠀⢠⡖⣿⠀⠘⣿⡆⠀⠀⠀⣀⡀⠀⠈⠛⡦⢤
⠀⠘⣇⠀⠉⠀⣸⠃⠀⠀⣴⣷⠚⠉⠉⢻⣷⡄⠀⠀⠹⢯⣳⡄⠀⢀⣴⣏⠴⠃⠀⠀⣿⡇⠀⠴⠛⢁⡏⠀⠀⠀⡇⠸⡇
⠀⠀⣿⠀⠀⢰⠏⠀⠀⠀⣿⣇⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠉⢻⣦⡾⢩⡆⠀⠀⠀⠀⣿⡇⠀⠀⠀⣼⠁⠀⠀⠀⡇⠀⢿
⠀⠀⢸⡄⢀⡟⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⣽⠃⢸⠇⠀⠀⠀⠀⡟⡏⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠸⣇
⠀⠀⠀⣷⡈⠉⠉⠙⡇⠀⠀⠸⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⢃⠀⢸⠀⠀⠀⠀⠀⣷⡇⠀⠀⠀⠙⢢⡀⠀⢸⠁⠀⠀⢻⡀
⠀⠀⠀⢸⡇⠀⠀⢠⡇⠀⠀⠀⢻⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠞⠀⣼⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⢳⠀⡿⠀⣠⠞⠈⡇
⠀⠀⠀⠈⡇⠀⠀⢸⡇⠀⠀⠀⠈⢿⡄⠀⠀⡿⠀⠀⠀⠀⢸⡏⠀⠀⣿⠀⠀⠀⣰⢺⡟⢶⡄⠀⠀⠀⠀⠘⣷⢃⡜⠁⡀⠀⢻⡀
⠀⠀⠀⠀⣧⠀⠀⢸⡇⠀⠀⠀⠀⠈⢻⣄⣸⠃⠀⢠⣤⠀⡟⠱⣄⠀⣿⠀⠀⠀⣷⣾⣇⣠⡇⠀⠀⠀⠀⠀⣿⠏⣠⠞⠁⠀⠈⢷
⠀⠀⠀⠀⣿⠀⠀⠀⠁⠀⠀⣴⣶⣄⣠⠜⠛⠒⠚⠛⠉⢰⠇⠀⢸⡆⣿⠀⠀⢀⣼⣷⣦⣟⣀⣠⡤⠴⠶⢾⣿⢊⣥⠞⠃⠀⠀⠘⢧⣀
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⠇⢻⠀⠀⠸⡇⠀⠀⠀⢀⡀⠀⠀⠀⣿⡵⠛⠁⠀⠀⣀⠤⠤⣤⠙⢷⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⣷⣤⡀⠀⠀⠀⠀⠀⠀⡟⢠⡴⠋⠀⢸⡇⠀⠀⣧⠈⠉⠉⠉⠉⠉⠁⠸⡟⠁⠀⠀⠀⠈⠁⣠⠞⠁⠀⠀⠻⣄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠢⡙⠦⣀⠀⠀⠀⠀⡇⠠⠬⠝⠂⡘⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢰⠿⢤⣀⠀⠀⢀⣞⡥⢒⡟⢀⡴⠲⣼⣧⣀
⠠⠤⣄⣀⢸⡇⠀⠀⠀⠀⠀⠀⣀⠴⠛⠓⠚⠷⣄⣀⠀⡇⣠⣀⣀⡀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⡼⠀⠻⣟⠓⠀⠈⠁⣰⠋⢀⡾⠀⢁⣄⠀⠙⠳⠤⣀
⠀⠀⠀⠙⠿⣿⣶⣤⣤⡤⠞⠉⠀⠀⠀⣀⠀⠀⠀⠉⠛⢧⣌⣉⠙⠂⣀⣀⡀⠸⠇⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢹⣶⣄⠀⠰⠟⠀⣼⠁⢠⠞⠙⠛⣀⢀⠀⠘⢧
⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠛⠲⠦⣤⣠⠾⠉⢻⣹⢸⡏⠉⠉⠙⠛⠓⠒⠶⠶⠶⠶⠾⠷⠤⣤⣤⣧⣉⣀⣀⡔⣻⠇⡴⠃⠀⠀⡰⠃⠈⣻⠀⢠
⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⣄⡀⠀⢀⡍⠛⢦⣤⠿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⠇⣼⠓⣄⢀⡼⠁⢀⡼⢁⡜⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠉⣛⠻⢶⡴⢃⣀⣙⣛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠁⠀⢩⡟⠐⢣⠞⢴⠞⠖⠋
⠀⠀⠀⠀⠶⠤⣤⣀⡀⠀⠀⠀⠈⠙⠛⠛⠛⣉⡿⠛⠋⠉⠉⠉⠉⠉⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣤⣿⡀⣠⣯⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⠛⠛⠛⠛⠋⠉⠙⠲⢦⣄⡀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠛⠙⠳⠦⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠦⣤⣀⡈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠤⠶⠶⠚⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠶⣤⣤⡤⠤⠤⠤⠴⠶⠒⠒⠒⠚⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⢀⠐⠀⠀⠐⡀⢐⡀⠀⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠦⠘⠦⢤⠀⣄⠉⠤⠤⠎⠀⠄⠄⠠⠞⠉⠤⢄⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⢳⠄⠀⢳⡀⠀⡀⠀⠄⠀⠀⠀⠒⠲⠤⣄⡉⠒⣤⠤⠄⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠙⠀⠇⠀⠀⢀⠄⠉⠀⠀⠀⠤⠤⠤⠤⣤⡀⠀⠈⢂⠙⡀⠀⡔⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢃⠀⠀⠀⠰⠃⠀⢀⡔⠀⠀⠀⠀⠀⡀⠃⠛⠀⠐⠁⠞⢁⠀⠞⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢻⣄⢠⡀⠀⢸⣗⠠⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠴⠒⠻⠻⠻⠶⣟⠁⠀⠀⠀⠀⢄⣰⡤⠨⣍⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣡⠜⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢘⣷⣾⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢠⡇⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠀⠼⠿⢿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣏⣿⢢⢄⣰⣣⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣷⣿





⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤
⣿⣿⣤⣴⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣯⢉⣉⡉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠾⣟⣡⣔⠻⠟⠀⠀⠀⠀⡀⠋⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡱⡋⠃⠉⠙⠛⠆⠲⠶⠴⠞⠟⢣⣆⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡷⡟⣛⣳⢶⣄⣈⡰⠞⢿⡭⡉⡂⣿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡟⠻⠌⠁⡀⢹⠁⠀⢀⣂⣀⡐⠁⠉⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⠀⠉⠁⠀⠘⠀⠀⢀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇⡀⠀⠹⣾⣦⠖⠀⠳⠀⠀⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡇⡇⠶⠒⠛⠃⠗⠙⠓⠀⠀⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣧⡻⣷⠀⠀⠘⠛⠉⠀⠀⢀⠆⠀⢀⠈⠉⠛⠓⠶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⡶⢬⠳⢄⣀⠀⣀⣀⠔⡁⣸⠀⠀⣧⠀⠀⠀⠀⠀⠀⠈⠉⠙
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠛⠉⠉⠙⠋⠉⡹⠿⠿⠿⢿⡇⠨⠙⠆⠀⠉⠉⠁⠈⣀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⣠⠤⢒⡍⠀⠋⠀⢀⢈⡳⠥⡬⡗⠲⠶⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⢀⠴⠉⠁⣀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢳⡀⠀⠀⠀⢀⡤⠋⠀⡰⠋⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⢷⠀⠀⠀⢮⡀⠀⠉⠀⡠⠒⠀⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠘⡇⠀⠀⠸⣇⠒⠒⣀⠀⠠⠊⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠢⠀⠀⠀⠂⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠗⠀⠀⠁⠰⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠈⢢⠀⢄⡠⡞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉⠉⠉⠉⠉⠀⠒⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⡰⠓⢄⡀⠀⣀⡀⠀⠠⠄⠰⡀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⣴⣛⠋⠀⠀⠘⣛⠳⠄⠀⠀⠀⠈⢢⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣠⣷⡤⣴⠋⣤⡌⠑⣄⠀⠀⠀⠀⡃⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣣⠃⢣⠀⠙⠁⢀⠏⠉⠉⠒⠰⣦⡤⡃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠺⡁⠀⠀⠉⠒⠒⠁⠀⠀⠀⠀⠀⠱⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠂⠀⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢸⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⡶⢄
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⡄⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠌⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⡰⢁⠀⠀⠀⠀⢰⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠁⠀⠀⡄⠀⠀⡌⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⢠⠁⢸⠀⠀⠀⠀⣸⣇⣀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡈⠀⠀⠆⠃⠀⢰⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠔⠅⠒⡉⠀⠪⡳⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⢠⢰⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣰⠶⠒⡗⠒⢺⢷⢡⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡄⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠐⠀⠀⠀⡅⠀⢴⠁⠀⠀⠀⠀⢸⠀⠦⡄⠀⠀⠀⠀⢸⠀⠀⡀⠋⡀⢈⠉⢦⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠁⠀⠀⠀⡇⠀⠘⠀⠀⠀⢸⠀⠘⠀⠰⢰⠀⠀⠀⠀⠸⠀⠀⠁⠊⠀⠀⢰⠀⡇⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠠⠀⠀⠀⠀⡇⠀⡆⠀⠀⠀⠈⠀⠀⠀⠀⠆⡆⠀⠀⠀⠀⠀⠀⡆⠀⠂⠐⢊⠰⠁⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠈⡃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⢠⠀⠀⠀⠀⠀⢰⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠈⡄
⠀⠐⠀⠀⠀⠀⠀⡇⠐⢆⡀⠀⠀⠀⠃⠇⠀⣀⣼⠂⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠃⠀⠀⠀⠀⠀⡇⠀⠈⠙⠿⡶⠀⠀⠘⠿⠛⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠰
⡘⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣧⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣦⣴⡶⠟⠛⠛⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣽⣿⣿⣿⠋⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢹⣿⣿⣿⡄⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⢿⣿⣿⣧⠀⠀⠀⠀⠀⡎⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⢿⣿⣧⠀⠀⠀⣠⠗⠢⣤⠊⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆⠀⢀⡇⠀⠀⣻⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⡐⢿⡗⠛⠉⠙⠤⠴⠋⠙⠛⠡⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢇⢸⡹⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡏⠸⡀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠃⠀⠀⠀⠀⠸⡝⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣿⡀⠀⠀⠀⠀⢧⠀⠈⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠘⡆⠀⠀⡰⡈⠑⠒⠠⠤⠤⠄⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠤⢼⠙⢦⡀⠀⠀⠸⡀⠀⢐⣥⠀⠀⠀⠀⠀⠀⢠⠒⢲⠉⣄⠔⠁⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⠢⢼⡀⠀⠙⠳⣦⣤⣧⠠⠊⡇⠀⠀⠀⠀⠀⠀⡄⢠⣆⠳⢵⡒⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢀⡀⠉⠐⠒⠀⠤⢬⣏⣷⣥⣀⠀⠀⠀⠀⣠⠇⠈⢸⣦⣤⡿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠁⠐⠀⠀⠀⣘⠫⡯⣻⣟⢕⢄⠀⡔⠙⣤⣶⠟⢁⠼⠣⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⢄⠉⠢⣱⠙⢧⠝⠳⠿⠵⠚⠛⢿⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣷⠀⠀⠀⠑⠲⡌⠓⢾⡄⠘⢦⡀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢘⡀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⢻⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠈⠑⠤⣀⠀⠀⠀⠀⠀⠛⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡄⠀⠀⢀⡼⠄⠀⠀⠀⠀⠀⠄⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠢⠔⠉⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⢀⡟⠒⣲⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠓⠒⠒⠒⠒⠒⠀⢐⣿⣉⣀⡠⠊⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⢸⢀⣀⡏⠏⠲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⢹⠉⠈⠒⠂⢱⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠪⠭⠭⠶⠶⠤⠤⠀⠀⠀⠀⠀⠬⠤⠤⠤⠴⠶⠭⠭⠭⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⠿⠂⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠐⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⡿⠃⠀⠠⠄⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠟⠛⠉⠁⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⣚⠿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠶⠖⠐⠓⠂⠒⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢠⡙⢷⣄⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠠⠐⠀⠀⡠⠀⠀⠄⣦⠄⠷⠆⠄⠴⠖⠉⠀⠴⠀⠀⠀⠀⠀⠀⠀⡑⢶⣦⡐⠀⠄⡀⣀⠀⢀⣀⣀⣀⣨⡆
⠀⠀⠀⠀⢡⠄⠀⠀⣀⣀⠀⢀⣀⠐⣷⠤⠤⢀⣀⡄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣀⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⡄
⠀⠀⠀⠀⣄⠀⠈⠀⠀⣀⣀⢼⣶⣷⣿⣦⣄⢀⢉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣶⣤⣁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢃
⠀⠀⠀⠀⠀⠠⠶⠒⠋⠉⣡⣈⠀⠈⣀⣻⣯⡁⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠷⣯⣙⣓⣀⣀⣈⣁⠀⠀⠀⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣠⢤⢔⠛⠁⠀⠈⠯⢤⠀⠹⣿⡶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⣈⣉⣃⣀⣉⣉⣉⠒⠀⠀⠤⠼⠃
⠀⠀⠀⠀⠀⠈⠁⡠⠛⠁⠀⠀⠀⠐⠰⠀⠀⠈⠃⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣧⡛⠻⣿⣈⡉⠉⠀⠀⠒⢚⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⢉⣉⠉⠉⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣨⣥⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠉⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠫⢭⡙⣆⠀⠀⠈⠷⡒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠱⣄⡼⢃⡟⠙⠒⠦⢤⡀⠀⠙⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢦⣠⡤⠞⠧⣄⣀⠀⠀⠀⠉⠲⣄⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢅⣢⣠⠔⣲⣾⣆⠈⠉⣱⠶⢄⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠛⠛⠳⢌⡙⣇⠈⠿⢿⠏⠀⢠⡇⠀⠀⡯
⣤⣤⡤⠤⠤⠴⠒⠒⠛⠛⠛⠛⡆⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠘⠳⠶⣁⡀⠀⣼⣠⠞⢃⡛⢆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠉⠀⠰⠇⡼
⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢹⠲⠶⠶⠚⠁⠀⠀⣀⠀⠀⠀⠀⠀⠘⡖⠋⠁
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠤⣄⣿⠉⠙⠲⣄⠀⠀⢀⠇
⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⢷⡀⠀⠀⠀⠀⠈⠓⠦⡀⠈⠳⢄⠺⣄⡀
⣠⠴⠒⠛⠋⣱⡆⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⣰⠂⠀⢱⠀⠀⣀⠀⠀⠀⠰⠚⠃⠀⠀⠙⡇⠉⠓⢤⡀
⡕⠀⠒⠛⠉⠉⣹⠆⠀⠀⠀⠀⡇⢣⠀⠀⣠⠃⠀⠀⠀⣿⢙⠗⡦⣤⠤⣞⡁⠀⠀⠀⠀⢹⠀⠀⠀⢳
⠑⣖⠒⠋⠉⠉⣹⠆⠀⠀⠀⠀⣇⡠⠛⣰⠇⠀⠀⠀⠀⡁⢸⡏⢀⠰⣴⡏⠀⠀⠀⠀⠀⢸⠀⠀⡇⠈⡆
⠀⡎⠙⠻⡟⠉⠁⠀⠀⠀⠀⢰⡇⠀⡰⠻⠀⠀⠀⠀⢀⡠⠀⣥⠿⠀⡟⢳⡐⣄⠀⠀⢀⣼⢶⠀⢇⠀⢳
⠀⠈⢳⣟⠥⢤⡔⠒⠒⠿⣍⠉⢁⡞⢡⠇⠀⠀⠀⠀⠻⡀⠀⣿⢸⣼⠃⠀⢹⠞⠙⢲⡟⠁⠀⢇⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠈⠹⠫⢀⡾⠀⠀⠀⠀⠀⠀⠱⣄⢸⡌⡿⠀⡰⠃⠀⠀⠘⣆⠀⠀⠘⡄⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠖⢻⡇⠀⠀⠀⠀⠀⠀⠀⠘⡆⣷⠁⡼⠁⠀⠀⠀⠀⠈⢷⡀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡾⠁⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠊⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⣸⠁⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⣄⣀⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⢠⡉⠉⠉⠛⠓⠒⠒⢲⠶⠶⠶⠶⠶⠶⠒⠚⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠳⣄⠀⠀⠀⠀⠀⢸⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠘⢦⡀⠀⠀⠀⠘⡇⠀⠀⠹⡄
⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣘⣧⣤⣤⣬⣷⠀⠀⠀⠀⢻⡀⠀⠀⠘⡆
⠀⠀⠀⠀⠉⠻⣧⡒⠶⠶⠶⠶⠶⠒⠛⠉⠉⠉⠉⠁⢀⡄⠀⠘⡆⠀⢀⣠⠔⠛⠛⠛⠛⣏
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⢤⣤⣤⣤⣤⣤⠴⠖⢛⣩⣧⡤⠴⠛⠋⠁⠀⠀⠀⣰⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠯⢟⣛⣛⠛⠉⠉⠁⠀⠀⠀⣀⣀⣠⠤⠖⠋⠉⠛⠛⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣉⣉⡁⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣌⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⡄⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⡀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⡀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⣸⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⢿⡆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠃⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⠟⠓⠀⣼⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⢸⡇⠐⠆⠀⠀⣶⠀⠀⠀⣾⣿⠓⠀⠀⢰⡶⠶⢶⠀⢰⣿⠆⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⣷⠀⠀⠀⠀⠸⠇⠀⠀⠈⠁⠀⠀⣀⠾⠁⠀⠘⠇⠀⠀⠀⠀⠀⣰⠿⠃⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡘⠁⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⡄⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠶⠎⠀⠀⠀⢀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⢀⣼⠏⠀⢀⡀⣴⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⡀⠀⠀⠈⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠁⠀⠀⢂⡰⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠙⢿⡦⠤⠒⣬⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠶⠁⡾⠁⠀⠀⠙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠳⡀⠀⠙⣝⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⢁⡴⠃⠀⠀⠀⡆⠀⠑⢤⡀⠀⠀⠀⠀⠀⢀⡜⠀⢸⠀⠀⠀⠡⡄⠀⠈⠱⢷⡄
⠀⠀⠀⠀⠀⠀⢀⡤⠶⠛⠁⠀⠀⣸⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢸⠀⠀⠀⠀⠰⠀⠀⠀⠎⢱⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣏⡇⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⡄⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠄⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢸⠀⠀⠀⡠⠀⠀⠀⠀⠀⣜⠄⠱⣆⡀⠀⠈⢷⡀
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⢻⠀⠀⠀⢸⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠀⠀⠙⢰⠀⠀⠈⣷
⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⡇⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠈⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⢈⠀⠀⠀⠸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡼⢡⡀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠀⣄⠀⠀⢸⡄⢀⣀⡠⠊⠁⢀
⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡘⣃⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡿⠀⠀⠰⠋⢁⡼
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢛⣛⡶⡀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⢠⠀⠷⠀⠀⠀⠔⠋⠀⠀⠀⠀⡀
⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠰⣏⣈⡐⣿⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢰⠏⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣬⣭⡿⠋⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⢰⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⢷⠀⠘⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠈⡇⠀⣄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢸⡄⡏⠀⠀⠂⠀⠆⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⠀⢱⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡄
⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢳
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏
⣿⠀⠀⠀⠀⠀⠀⠀⠠⠒⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉
⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀
⣀⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣄⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣄⣁⣀⣀⣀⣀⣀⣀⣀⣀⣸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠿⠿⠿⠿⠿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿
⣿⣦⣾⣶⣶⣷⣶⣶⣶⣶⣾⣤⣿⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⣀⣠⣤⢀⣀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠷⠛⣩⠶⠒⠛⠻⣿⡿⠛⠉⠩⠭⠙⠻⣾⣿⣿⠗⠀⠐⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠈⠉⠉⠉⠁⠀⠰⠾⠃⠀⣠⠀⠀⣀⣀⣤⠂⠀⣀⣀⠐⠻⣏⣥⠀⢈⣀⠟⠛⠿⠲
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⢀⣴⠖⠒⠛⠛⢷⣆⠀⠀⠀⢸⣷⠤⠀⠈⡉⢣⡀⠙⠛⠁⠀⢼⣿⠴⠾⠟⠛⠀⠀⠀⠙⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣨⠀⣠⠶⠟⠀⠀⣼⡟⠃⠀⠙⢻⣶⢦⣤⣄⣀⡴⠞⢁⣼⠿⣦⠀⢀⡶⠋⢉⣀⣠⣤⠶⢶⡶⠶⠿⠃⠈⠑⢦⣬⣦
⠀⠀⠀⠀⠀⢠⡿⢃⣾⠇⢠⣿⠀⣼⠋⠀⣠⡴⠞⣋⡥⠀⠀⠉⠙⠃⢠⣿⠀⣀⠙⠇⣼⠃⠀⣾⠏⠉⠀⣦⣘⣷⡄⠀⢠⣶⠾⣿⣿⡿⠷⡀
⠀⠀⠀⠀⢀⡬⢀⣿⠁⢀⠜⠁⢠⡗⠀⢸⠋⡴⠞⠛⠦⠀⡀⠀⠶⢀⣸⡇⠀⠉⠁⣼⠁⠀⠸⣿⠀⣰⣴⠀⠙⢸⡗⠀⢸⡇⢰⡆⠀⢷⣄⠹⡄
⠀⠀⠀⠀⠸⣷⠉⠹⣆⠀⠀⢀⠎⠀⠀⢺⡆⠀⠀⠀⠤⢴⣇⠀⠀⣽⡟⠀⠀⠀⡼⠃⢠⠀⠀⣿⣶⣯⣅⣀⣠⣾⣧⣀⣿⣀⢈⣿⠀⠀⠛⣷⣿⡄
⠀⠀⠀⠀⠀⠙⠀⠐⢨⣭⢥⣉⣄⣀⡀⠘⠻⢶⣶⣤⡀⠀⢻⡀⠀⠹⠀⢸⣄⠀⠀⣀⣼⣷⣤⣿⣟⠈⠉⠛⠛⢿⣿⠟⠛⣡⡞⠁⢀⣠⣴⣿⡏⣧
⠀⠀⠀⠀⠀⠀⠀⠒⢀⣁⢀⠈⠀⠈⣉⣱⣄⣄⣉⡀⣀⣀⣿⣿⠖⠂⠲⠛⡟⠟⠿⠋⠀⠀⠉⠙⣿⣤⣤⣤⣠⣀⢁⣠⣶⡏⠀⣰⠟⢏⣙⣿⡇⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣉⠉⠌⢲⣄⣀⣽⠿⠁⠀⠀⣀⣤⣄⡀⠀⠀⠀⣠⠀⠀⣸⡏⠀⣀⣹⣿⣿⠟⠉⢻⣧⣤⣤⡆⢸⣿⣿⣿⢁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠛⠛⠁⣠⣤⣶⠿⠉⠀⠀⠉⠙⠓⣾⡟⠻⠶⣿⣿⠿⠁⠙⠛⠋⠁⣀⣺⣿⡟⣫⣤⣾⣿⣿⣿⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠋⠀⠀⠀⠀⢀⡀⠀⢤⠀⠈⢻⣆⠀⠀⠀⣠⣀⣠⡶⠖⠚⠛⡿⢟⣿⡋⠀⢠⣽⡿⢿⣭⣼⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠇⠀⠀⠠⣴⡛⢇⠀⠀⠳⡄⠀⠙⣷⠀⠀⣸⣿⠟⢧⣤⣶⠟⠁⣌⠉⢧⠀⠈⠉⠁⣆⠠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡴⠿⠃⠈⠳⠄⠐⠒⠲⣂⣺⠛⠚⠋⠀⠡⢶⡇⠀⠀⠘⠻⣦⡈⠛⠶⠲⣎⠙⢷⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣖⢋⣽⢿⣿⣿⣿⣶⣶⣶⣶⣭⣶⣤⣤⣌⣿⡁⣀⣀⣴⢀⣤⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣶⣼⣿⣿⣿⣛⣉⣫⡟⠿⣭⣜⣯⣹⣿⣿⠿⢿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠙⢿⣿⣿⣿⣿⡿⠿⠿⣿⡿⠻⢿⣿⣛⡛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠈⠉⠛⢿⣿⣗⣻⣲⣽⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠃⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢒⣛⣉⣉⣙⠛⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⣀⠴⠊⡀⠀⠀⠀⢳⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⡀⢀⣠⡤⠶⠒⠋⠉⠀⠀⠀⠘⣆⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢔⣡⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢁⠖⡠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢠⢃⡼⢱⡇⠀⣀⣠⣀⡀⠀⠀⠀⠀⠀⣠⣤⠄⠀⣿⡇⠇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠇⡼⠁⢸⠇⠈⠘⠿⠎⠁⠀⡀⠀⠀⠀⠙⠛⠀⠀⡇⡇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢠⠃⡆⠘⡆⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠸⠀⡇⠀⢣⠀⠀⠀⠀⠀⠀⢄⣠⠀⠀⠀⠀⠀⣸⢰⠃⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠈⣆⠀⠀⠀⠀⡀⠀⡀⠀⢄⡀⠀⠀⡏⣼⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢃⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠄⢠⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢀⠈⢆⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡜⢠⠇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡈⢦⠸⡇⠈⠓⠦⠤⠤⠤⠀⠀⢠⡿⢠⣎⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠷⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⡴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡯⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢻⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⡎⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢋⡄⠀⡏⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠸⣆⠈⠓⠦⣄⡀
⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠸⡇⠀⠁⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠤⠤⠴⢾⠁⠀⠀⢀⠓⠈⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⡠⠂⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠄⢀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢄⠁⠀⠀⠀⠈⡆⠀⠀⠀⠈⠃⢀
⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠤⠐⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠈⢧
⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⣇
⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠑⢄⠀⠀⠀⠀⠈⢧⠀⠀⠀⢠⠃⠀⠀⢀⡴⠊⢄⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠙⣦⠀⠀⠐⠤⡀⠀⠀⠈⡆⠀⠀⡎⠀⢀⠴⠋⠀⠀⠀⠣⡀⠀⡏⠀⠀⠀⡇⠀⠀⠀⢸⡆
⢀⡇⠀⠀⠀⢰⡄⠀⠀⠀⠈⢧⡀⢀⡿⠀⠀⠀⠀⠈⠀⠀⡀⠘⡄⢰⢀⠔⠁⠀⠀⠀⠀⠀⠀⠁⣾⡁⠀⠀⠀⡇⣀⠀⠀⠈⣧
⢸⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢰⡌⠀⠀⠀⠀⠀⠀⠀⠀⡀⡬⠁⠈⠀⠠⠆⣷⠃⠀⠀⠀⠻
⡎⠀⠀⠀⠘⡌⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⠏⠄
⣧⣤⣤⣤⣤⣷⣾⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣶⣦⣤⣤⣤⣤
⣿⣭⣭⣭⣽⣧⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣶⣶⣶⣾⣿⣿⣾⣿⣷⣾⣾