Pet Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣦⡀⠀⣀⣀⣠⣤⣤⣠⣤⣤⣤⣴⢿⣿⣿⣿⣷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠈⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣀⠄⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⣼⠁⠀⢀⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⠀⠀⠘⠀⠀⣷⣾⣿⠈⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠙⠿⠟⠀⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠠⡀⠀⢸⡿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⠛⠛⠉⠀⠀⠀⠀⢀⣀⣤⣤⣀⠘⠀⢸⠃⠙⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠋⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣥⣶⣶⣾⡇⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠻⣿⡗⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⠀⠰⣄⠀⠀⢀⣠⠴⠒⠛⠓⢢⣤⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠆⠀⠀⠀⠀⠸⢻⣿⣧⣤⣶⣶⡿⠛⠉⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠧⢀⡀⠀⠀⠀⠈⠙⠉⠋⠁⢀⣤⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠶⣄⣤⣦⣤⣤⢴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠄⠐⠒⠂⠠⠄⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢡⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⡆⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⡄⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⢀⠃⠀⠀⢸⣯⠆⡠⠤⠤⡀⢯⣿⠀⠀⠀⠇⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠸⠒⠒⠒⠐⠒⢊⣤⣒⣠⣌⠑⠒⠒⠒⠐⠺⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠇⠀⠀⠀⠀⠀⣀⣉⡿⣏⣁⡀⠀⠀⠀⠀⠀⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠫⣀⡨⠁⠀⠀⠀⠀⠀⡰⡖⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠒⠢⠤⢀⣀⣀⣀⣀⠀⠤⠄⠲⡋⢠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣷⠃⠉⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢣⠀⠈⠢⣀⠀⠀⠀⢀⡠⠊⠀⢠⠃⢱⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⡆⠀⠀⠈⠉⢲⠊⠁⠀⠀⠀⡜⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠉⠂⡜⠢⠤⠤⠀⢸⠀⠀⢀⡀⠜⣄⠚⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⣰⣘⣴⡀⢰⠀⣀⠼⢄⡀⢐⠀⢰⣼⣰⣀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠈⠉⠉⠁


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣽⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡀⠀⠀⢀⠤⣀
⠀⠀⠀⠀⠀⢡⠄⠀⢨⠄⠈⢀⡄⡀
⠀⠠⢔⠀⠀⠈⠁⠀⠨⣄⣠⠞⡅⠈⠀⡀
⠀⠈⠑⠀⠤⢔⣮⠽⠀⠈⠁⢸⡇⠀⢀⣽⠛⣶⡀⣤⣤⣤⣀⣀⣠⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠐⠒⠂⠀⠈⢠⡀⠀⢾⠅⠀⠞⠁⢰⣿⡇⠈⣿⡿⠓⢤⡥⡴⢶⠦⠤⠀⠉⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠀⠀⠀⢀⠌⣿⠇⠀⠹⠀⠀⠟⠁⠈⠀⠀⠀⠀⡰⠃⠀⠑⡄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠞⠁⠀⠀⠀⠀⣄⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠸⡆⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣁⣀⡀⠀⠀⠀⠀⠀⠀⠸⢦⣄⣀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢴⣝⠀⠀⠜⡗⠀⠀⠀⠀⠀⠀⠀⠸⡏⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣸⠇⠀⢸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠙⣿⣦⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡹⣿⣷⣄⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠙⡏⠘⠄⠙⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⠁⣠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠐⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢦⡤⠋⠠⡤⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣑⡾⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠓

⠀⠀⣠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⣶⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⢀⡀⠿⠇⣀⡀⠛⠃⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣷
⠀⠹⡿⠻⣿⣿⣿⣿⣷⣶⣿⣿⣷⣿⡁⠀⠀⠀⠀⠀⠀⠸⠇⠀⠈⢿⠟⠀⠀⠀⠀⠀⠀⠙⠻⠷⢦⣄⣀⣀⣀⣀⣠⣤⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠼⠆⢀⡔⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠻⣿⡟⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢻⡿⢹⣿⠀⠀⠙⣿⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠡⣾⠟⠁⠀⠀⠈⣿⠻⣿
⠀⠀⠀⠀⠀⢸⡇⢸⣿⠀⠀⠼⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠿⠄⠀⠀⠀⠀⣿⡀⣿⡀
⠀⠀⠀⠀⠀⠛⠃⠸⠿⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡄⢠⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠹⠸⡈⠠⠹⢸⢘⠸⠰⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠒⣀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠻⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣠⣼⠇⠀⠀⢠⣾⡿⠿⢿⠿⠿⠿⠶⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣦⣶⣶⣶⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣎⠁⠈⠋⢹⠟⢘⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡈⠀⠀⠀⢸⣏⣸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢺⣿⡿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣼⣿⢹⠿⢿⣿⠏⠀⠀⠀⠀⢠⣄⣀⣀⠀⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣿⣏⠁⠀⠀⠁⠀⠀⠀⠀⢢⣶⣿⣿⣿⡇⠀⠻⢿⠿⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡿⠿⠁⠀⠀⠀⢀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⣘⣿⠿⣅⠀⠀⠀⠐⠂⠺⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣶⠂⠀⠀⠀⠀⠀⢀⣉⠁⣀⣤⡗⠀⠀⠀⠀⢀⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⢐⣰⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠌⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣟⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡙⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⢠⡤⠀⢒⣯⣻⣿⣿⣯⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡶⡄
⠀⠀⠀⠀⠸⡀⠂⠈⢟⣿⣽⣿⣿⣿⣿⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠘⡄
⠀⠀⠀⠀⠀⠈⠑⢤⣼⣿⣿⣿⣿⡟⠛⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⠏⠀⠀⠀⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡇⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣋⡉⠛⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣷⣿⡿⠟⠷⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣱⣄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣻⣾⣷⣿⣿⣯⡉⣸⡟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⡇⠀⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⠘⠀⠠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⣋⣩⣭⣦⣦⣄⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣵⠀⢘⡟⠛⠛⠿⠗⠒⠒⠛⠉⠉⠀⠀⠀⠘⢿⣧⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇⠆⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡳⣄⡐⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⢨⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⠏⢻⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣥⣀⡼⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣁⠼⠀⢸⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣦⡠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣄⡜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠋⠙⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠰⣿⠆⠘⢧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⡇⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠛⠉⠀⠀⠓⠀⠀⢀⣠⠞⠃⠀⠀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠁⠀⠀⠀⠀⡴⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠐⠶⢤⣤⡤⠴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄
⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉
⠈⠳⣤⣀⠀⠀⠀⠀⠀⢀⣶⠃⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣿
⠀⠀⠙⠛⠛⢀⣠⣤⠞⠚⡟⠀⠀⠀⠀⠀⠀⠀⢠⡞⣡⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⡀⣴⡿⠹⠃
⠀⠀⠀⠀⠀⠈⠉⠀⠀⢰⠇⠀⠀⠀⠀⠀⢀⣠⡟⠉⠈⠁⣰⡖⠀⠀⢶⣤⣄⠤⣤⣤⡠⠟⠀⠀⠀⠀⢀⡞⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⣠⡴⠞⠋⣹⡄⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⢀⣴⡾⠁⠀⠀⠀⠉⠛⣆⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⢿⡀
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠉⠲⣤⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠸⡇⠀⠉⡳⣤
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠙⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠤⠶⠴⠃⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀⠀⠀⠙⠶⣤⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠱⢦⣤⣀⣠⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⡀⠀⣦⣼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⢀⣤⣴⣶⣤⣄
⠀⠀⠀⠀⠀⢀⣀⣀⣼⠿⣤⣾⡿⢿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠘⣿⣿⡿⠂⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⡀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠻⢝⣲⡤⣤⣤⠼⠉⢹⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠹⡆⠀⠀⡆⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣅⣠⠀⠀⢸⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣴⣧⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⣦⢻⣿⣿⣿⣿⣿⣿⣯⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢿⣟⢛⣱⠋⢈⣿⣿⣿⣿⣿⣿⣿⣾⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⡆⣫⡁⠀⠀⢈⣧⣼⣿⣿⣿⣿⣿⣿⣿⣷⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡋⣦⣈⢿⣦⢶⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢾⣿⠻⣿⣾⡆⠀⢻⣿⣿⣿⣿⠋⡟⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⣿⣆⠘⢻⣷⡀⠀⠹⡝⢿⡇⢰⠁⠈⢿⣿⣿⣿⣿⠟⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⣿⡄⠘⢿⣷⡀⠀⠀⠈⢳⠃⠀⠀⠈⠉⠛⠻⢷⣤⠄⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣦⡀⢸⡇⠀⡀⢀⢸⠀⠀⠀⠉⠀⣴⡾⠶⠧⣤⣝⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡺⠛⣿⡇⠙⢿⠀⢡⣘⣮⡀⠀⠀⢀⡴⠁⠀⢠⣴⣶⣾⣿⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣃⠹⣷⠀⠀⡇⠠⢹⣻⡁⠈⡆⣼⠁⠀⣸⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⣇⣿⡄⠀⡇⠹⣼⣿⣿⣷⣼⣿⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⣿⣿⡇⠀⢈⠘⣿⡯⠙⣿⣿⣿⣇⠀⠸⣿⣿⣿⣿⣿⡿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡛⢿⠇⠀⠘⣤⣿⣇⣄⣈⣻⣿⣿⣦⠀⠈⠻⠿⣿⡞⠁⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡐⢶⡯⢆⣤⣿⠀⠀⡾⢠⢻⣿⣿⣿⣿⣿⣿⠾⣷⣄⠀⠰⢽⣿⣶⣾⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠿⠤⠟⠓⠁⢠⣞⡴⣡⠋⣿⠀⢰⣣⢋⡵⢞⣻⣶⣶⠾⠿⠟⠛⠛⠛⣻⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠋⠀⠘⠛⠛⠷⠾⠛⠧⠀⠀⠒⠒⠒⠒⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⢀⠠⠀⠂⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⢈⠂⠀⠀⠀⠀⠀⠄⠀⠀⠘⠐⠀⠀⠀⡀⠀⠀⠀⢀⠀⢈⠀⠁⠀⠍⠁⠈⠁⡉⠁⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⡐
⠀⠀⢠⠁⠀⠀⠘⠀⠀⠀⠘⠀⠀⠀⠈⡆⠀⠀⠀⠈⠀⣠⠀⠀⠀⡄⠀⠁⠀⠀⠀⠈⠉⡅⠀⡄⢀⣖⠀⢠⠀⠈⠉⠀⠀⠀⠛⡛⠀⠈⠃⠀⠀⠉⠀⠘⡛⠃
⠀⠀⠀⠂⠀⡄⠀⠀⢴⠆⠀⠀⣘⢀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠀⠀⠀⠀⠰⠀⠀⠀⠀⠛⠀⠀⠀⠀⠃⠀⠀⠀⠀⣁⠀⠀⠀⢙⠃⠀⠀⢀⡁
⠀⠀⠀⠀⠈⠉⠢⢠⠀⠀⡄⠄⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⡀⠀⠀⡘⠀⠀⠀⠀⠠⣀⠀⠀⠜⠙⠡⠀⠀⣄⠀⠀⠀⠀⠀⠹⢄⠀⠀⠁⡍⠀⢀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠠⠀⠄⠠⠊⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠒⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠
⠀⠀⠀⠔⡠⠈⠐⠀⠐⠂⠐⠁⢈⠐⡀⠀⠀⠀⠄⠈⠉⠀⠉⠉⠉⠁⠈⠁⠢⡀⠀⠀⠀⠀⠀⠀⠑⣀⢀⡐⠀⠈⡄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠈⠐⠁⠆⠀⠀⠀⠀⠀⠑⡎⠑⠂⠀⠀⠆⠀⠠⠀⣀⠀⠀⠀⣀⠀⠁⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐
⠀⠀⠀⠀⠠⠀⢋⠀⣤⡤⡘⠁⡀⠀⠀⠀⠀⠀⠢⠆⠀⠈⠀⣤⡀⠁⠀⠘⠔⠁⠀⠀⠀⠀⠀⠀⠛⠀⠀⠐⠂⠀⠈⠀⠀⠀⣀⠀⠀⠀⠠⡦⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢡⠆⠀⣩⡀⠈⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠩⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠐⠐⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⢀⠤⠀⠵⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⣀⡀⠄⠊⠀⠀⠀⠀⠀⠀⠑⢄⠀⠂⠀⠂⢀⠄⠁⠀⠀⠀⣿⢀⠀⢀⣀⢄⣀⠀⠀⡀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁⠀⠀⠀⠀⠀⠈⠻⢤⢌⠉⠀⠉⣠⡆⠟⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠠⠂⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⡎⠀⠀⠐⠖⠀⠀⠀⠶⢐⠀⠀⡂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠠⠊⠀⠂⡀⠀⠀⢀⠘⠀⠈⢀⠀⠀⠀⠀⠄⠀⠛⠀⠙⠀⠠
⠀⠀⠐⡀⠀⡄⠀⢀⣄⠀⠀⡬⠀⡐⠀⠀⠀⢀⠀⠀⠀⠲⣄⠀⢠⠖⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠁⢀⡀⠈⠁⠀⠀⢨⠀⠀⠀⢶⡀⢠⡀⠀⢀⡄⠀⣦
⠀⠀⠀⠈⠐⠀⢄⡈⠯⣀⡄⠈⠊⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⣶⠀⠈⠦⠄⠀⠀⠀⠀⠀⢄⠀⡦⢀⡘⢏⣀⠰⠀⢀⠀⠀⠀⠀⠘⢦⠀⠀⣤⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠛⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠂⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⢸⡤⠀⣉⡀⢠⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠄⠀⠀⠀⠀⡐⠈⠘⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠀⠈⠀⠈⢂⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⢖⠀⠀⠀⣲⣀
⠀⠀⡀⠀⠀⠀⠘⠂⣤⠐⠟⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢠⣤⠀⠖⠈⠀⠀⠀⠀⠀⠀⢀⠈⠀⢀⠀⠉⡀
⠀⠀⠉⠰⠀⢤⡄⠀⠲⠆⢀⣦⠀⠐⠈⠀⠀⠀⠀⢀⠀⠀⢀⠸⢃⡀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡅⠀⠀⢀⠃⠀⠀⠀⠀⠀⡎⠀⠀⣛⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠠⠀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠙⠀⠤⠂⠄⠒⠁⠀⠀⠀⠀⠀⠀⠳⠦⣄⡌⣠⠴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⣀⣀⠀⣿⡿⣲⠀⠀⢀⣀⠀⠀⠀⢀⢴⣥⣃⠁⢀⣠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣹⣿⡿⠶⡞⢻⣟⠩⢰⣿⠸⢿⣠⣷⡂⣼⢿⣯⣄⣀⣻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣍⣅⡄⠔⡶⠙⢄⡘⠋⣞⠺⣿⣿⡉⢻⠚⡏⠰⢾⣿⡩⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⠙⠆⠁⠀⠺⠀⡄⢠⢸⣿⣿⢦⡏⡁⢠⣯⠛⣀⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⢌⡒⠾⠘⣷⣿⡟⢾⠳⡇⢸⢋⡐⢛⠛⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠛⣫⣀⣷⣘⠙⣷⠀⠀⢸⢣⡞⣉⡥⡶⢲⡶⣶⣿⣌⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠁⠀⠀⠀⠛⠿⣿⣏⣹⡆⠀⢈⣾⣧⣿⢶⣴⣯⣶⠷⠎⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⠑⠀⠀⠀⠙⢻⣿⢿⡇⠀⠀⣾⣿⣿⡿⢿⣿⣿⡗⠾⢿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠶⠦⠀⠀⠀⠋⠁⠉⠋⠴⣶⡿⠿⠿⣿⣆⡙⠹⠶⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢓⡀⠀⠀⠀⠀⠀⣈⡋⠀⠀⠀⠛⠋⣹⣷⣿⣿⣟⣶⣟⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠑⠢⣄⣀⠀⠀⠀⠉⠉⠓⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣟⠻⣶⣤⣤⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢷⣾⣿⣿⣿⣿⣾⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣱⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⢠⠎⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣸⣴⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⠿⣿⢟⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠘⠁⢈⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢀⣸⣿⡏⣿⣿⡿⢿⣿⣿⣭⣿⣿⢿⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠹⠛⠻⢻⣥⡾⣿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠂⠀⠁⠉⠉⠉⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣈⠁⠀⠀⠀⠓⠀⠀⠄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⠀⠘⠛⡁⠀⠀⠈⠭⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢸⠀⠀⠐⠄⢀⠀⠀⠀⢀⠀⢠⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠜⣄⢣⢀⡠⠤⠤⡤⢴⠋⡠⡞⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠈⠢⡹⠀⠀⠀⡖⡡⠊⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡇⠀⠀⡇⠀⠀⠀⡋⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠰⡀⠀⠱⢄⡠⠔⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢡⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢀⡞⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢀⡆⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠱⡀⠀⠀⢰⠀⠀⠀⢛⠀⠀⠀⠀⠀⡜⢇⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠄⠑⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⢠⠞⠀⠈⢆⠀⠀⠘⡀
⡔⠋⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢣⠀⠀⠘⡄⠀⡰⢻⠀⠀⠀⠈⢣⠀⠀⢃
⡇⢄⠀⠀⠀⠀⠀⢸⠀⠀⠂⠀⠀⡟⠀⠀⠀⠀⠀⠘⡄⠀⠀⢣⠐⠁⣸⠀⠀⠀⠀⠀⢣⠀⠸⡀
⢣⠈⢦⡀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠹⡀⠀⠘⡤⢖⣁⠀⠀⠀⠀⠀⠈⢇⠀⢣
⠀⠑⢤⣈⠉⠛⠉⠉⢳⡄⠀⠀⠢⣇⣀⠀⠀⠀⠀⠀⢀⣱⠀⠀⢸⡁⠀⢙⡆⠀⠀⠀⠀⠀⢣⠈⠓⠤⢄
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠢⠼⠤⠴⠼⠀⠉⠉⠉⠉⠉⠁⠸⣀⢀⠀⢈⢻⠉⠁⠀⠀⠀⠀⠀⠀⠣⣀⣰⣀
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡄⠓⢠⡀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⠈⠉⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⢰⡟⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠄⠀⠀⠀⢱⣤
⠀⠀⠀⠀⠀⠀⠀⠊⠋⠀⠀⠈⠀⠀⣾⣧⠶⠀⠀⠀⠀⠀⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠈⠁⠈⡙⣿⣷⠀⠀⠀⠀⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⡏⠀⣀⣴⠆⣋⡒⢦⣄⣦⣀⡀⠀⣠⣤⢀⣄⢀⣅⣀⠴⠔⠆⣄⠀⠹⣿⡿⠂⠘⠀⠀⠀⠑⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⣼⣿⣴⡿⢿⣍⣿⠿⣿⣿⠿⠿⠿⢿⢿⣿⣿⣿⣿⣖⣿⣷⣮⣡⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣿⣿⣿⣿⡛⠿⠶⠚⠛⠓⠉⠁⠀⠀⠀⠀⠀⠈⠙⢷⠿⠯⢭⣴⠛⣿⣿⣼⡇⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢸⡄⣿⣧⠋⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠚⠟⢻⣿⡇⢸⡟⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣼⡇⢼⣿⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⣾⣿⡇⢸⣧⠀⠀⠀⠀⠀⢀⣄⡾
⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⢠⣸⣿⠇⣿⣿⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠈⠉⠉⠁⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⣽⣿⣇⢰⣿⣶⣤⢸⡀⣸⣾⡿⠁
⠀⠀⠀⠀⠀⠹⣦⣤⡀⠢⠸⠛⢋⣼⣿⣿⢀⢀⡆⠀⠀⠀⠘⠛⡀⣀⣤⣤⣤⣴⣄⡀⡙⠟⠀⠀⠀⠀⡆⠀⣻⣿⣿⣦⡛⠿⠻⠚⠟⣿⠋
⠀⠀⠀⠀⠀⣿⠋⠉⠹⢶⣶⣾⣿⣿⣿⣿⣯⠾⠀⡀⠀⠀⠀⠀⠑⡿⠿⢻⣿⡿⠿⠷⠃⠀⠀⠀⠀⠀⠘⣮⣿⣿⣿⣿⣿⣶⣶⣶⠟⠉⢻
⠀⠀⠄⣴⣾⡇⣀⣠⣿⣿⣿⣿⣥⣨⣿⣿⠋⠀⠌⠀⠐⠑⠚⠂⠀⠙⠿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⡀⠈⣿⣿⣿⡏⣹⣿⣿⣿⣧⡀⡀⢱⣶⣄⠄
⠀⠀⢾⡉⠉⣿⡋⠉⠙⢻⣿⠟⢻⣿⣿⣿⠀⠀⠀⠀⠀⠆⢐⣁⡀⠀⠀⣸⣿⡉⠀⠄⠁⠑⠐⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⢻⣿⡿⠋⠉⢻⣾⠉⠛⡦
⠀⠀⠀⠛⠿⣿⣿⣿⣿⣾⣿⣶⣿⣿⣿⣿⣷⣀⣀⣀⣀⣀⣨⣭⣵⣶⣾⣿⣿⣿⣶⣤⣄⣈⣀⢀⠈⠀⣠⣴⣿⣿⣿⣿⣿⣤⣿⣷⣾⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠁⠈⠉⠉⠉⠉⠈⠈⠀⠀⠈⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠉⠉⠁⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⣠⣤⣴⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡟⠁⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⢰⡿⢿⣿⣿⣿⣿⣿⣿⣿⡄⢹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠃⠸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⡿⠟⠀⠀⠙⢿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢠⣴⣶⡌⣿⣿⣿⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣈⠻⡿⣣⣿⠏⢀⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⢻⣝⣛⣭⠁⣴⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠛⠁⠠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠋⠙⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⠻⠀⠀⠀⠀⣽⢸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠋⠀⡀⠀⠀⣰⣿⠘⠿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣟⠀⠀⠈⣿⣄⣾⣿⡟⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢿⡀⠀⠀⠈⢿⣿⡿⠁⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡃⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣦⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠿⠋⠉⠻⣿⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⣶⣶⣦⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠠⠠⠴⠴⡇⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⢹⣋⣉⣉⣉⣉⣉⣉⣉⣙⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠛⠉⠙⠛⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⣋⡍⢹⣷
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣠⣶⠟⠋⣰⠟⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠰⣿⢡⣬⣭⣙⠛⠿⢶⣦⣤⣀⡀⢀⣤⡶⠟⠛⠛⠛⢿⢶⣾⡟⠉⠀⢀⡾⠁⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠉⠛⢦⡀⠀⠉⣩⡿⠟⠙⢦⠀⠀⠀⠀⡟⠀⠈⠉⠁⢀⣾⣥⣴⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⢀⣠⣽⣷⡄⠀⠀⣠⠖⠈⢧⠀⠀⠀⡇⢀⣳⣄⠀⠀⠡⠏⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠉⠙⠷⠀⠀⢰⠋⣦⣽⣾⡇⠀⠀⣷⣧⣾⠋⠓⠀⠀⠲⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⡤⠀⠀⠀⠀⠈⢉⡼⠁⣀⣀⠈⠳⣄⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡿⠁⠀⠀⠀⠀⣠⠋⠀⢯⣵⣾⡿⠀⠈⠓⣄⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⣼⠁⠀⡀⠀⢩⣯⡀⢀⡄⠀⠈⡇⢀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣤⣤⣀⠻⣄⠀⠙⠒⠋⠀⠙⠋⠀⢀⣴⣷⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣾⣅⡀⠀⠀⠀⠀⠀⠀⣠⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠙⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠁⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢋⡟⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⡀⠀⠘⣏⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⢸⡇⠀⠀⠸⣆⠀⠀⠀⠀⠀⣰⠃⠀⠀⢻⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠘⣇⠀⠀⢸⡎⠳⠀⠀⠀⠚⢱⡇⠀⠀⣸⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⡹⣆⠀⠀⠹⣄⠀⠀⠀⢠⡟⠀⠀⢠⡏⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠁⠘⢧⡀⠀⢈⡛⣦⡞⣋⠀⠀⣠⠏⠀⠉⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠶⠶⠾⢷⣦⣤⣷⣾⣧⣷⣤⣴⠿⠶⠶⠶⠶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠀⠉⣉⠉⠉⠉⠛⠢⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠚⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢐⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⡐⠁⠀⢀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢿⣷⡦
⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣦⡀
⠀⠀⠀⠀⠀⢤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢸⣿⣾⣷
⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⣜⣻⣿⣿⣧⡄⠀⠀⠐⠀⢠⡀
⠀⠀⠀⢀⠠⣿⡿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣰⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⣼⣿⣿⡿⣿⡀⠀⠀⠀⠀⠀⠻⣿
⠀⠀⠀⠄⠉⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣶⣜⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠐⢬
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣤⣤⣄⣀⣀⢠⣀⠀⠀⠀⠀⠈⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⢈⡤⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠛⣿⡿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠈⢿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢀⣤⣾⡼⡟⠂⠀⠁⠀⠀⠀⠀⠀⠀⠐⣆
⠀⠀⠀⠀⠀⢸⣿⡀⣷⠀⠀⠀⠀⢀⠀⠀⠀⠀⠨⣿⣎⠻⣿⣿⣿⡿⠿⠃⡀⠀⠀⠈⠉⠈⠻⣿⣿⣶⣿⣵⣦⣿⣿⡟⠁⢀⠀⠉⠀⠄⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠉⣶⣍⡹⠘⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠑⠈⣀⠀⠀⠀⠈⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡤⠁⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⣀⣀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⠀⠰⠶⠿⢾⡬⠛⠦⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠉⠀⠀⠍⠉⠀⡉⣤⡉⠒
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣌⣹⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡰⢷⠀⠀⠀⠀⣀⠂⢀⣿⡘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠘⠀⡈⠠⠈⣇⣵⡎
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⡄⢠⢀⣿⣿⣿⠀⠐
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣩⠀⣤⣰⣿⣿⠻⢿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⢠⣾⠖⠹⣿⡟⣦
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢛⣿⣶⣿⣿⣿⡏⠀⠀⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣷⡈⡟⠁⣀⣿⣟⡇
⠀⠀⠀⣎⣼⣿⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣾⣿⣿⣿⣿⡿⣱⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣹⣿⣧⠃⠈⣻⣷⣿⡇⢀
⠀⠀⣰⣿⣿⣿⣯⣽⡆⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⣨⣿⣿⣿⡿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⣿⢸⡯⢠⡀⢻⣿⣿⡇⠈
⠀⠐⣿⣿⣿⣿⣿⣿⣷⣶⣧⠀⢀⡀⠐⠢⣄⣤⣴⡶⠋⠀⠀⣽⣿⣯⣿⣛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣟⠁⢺⣧⣼⠟⠋⡧
⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠷⠾⠟⠋⠁⠀⠀⠀⠀⠀⠊⢠⣿⢌⡻⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢌⡀⣾⡏⢟⣧⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣿⠋⠈⠁⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡄⠀⣠⡇⣼⣿⣿⠇⣸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⡏⠅⠰⠀⠀⠀⠳⣎⣴⣇⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⣦⢸⡇⢠⣿⣽⣿⣿⡏⢠⡿⠏⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡀⠀⣤⣷⣩⣿⣿⠀⠒⠀⠀⠀⠀⠀⣍⡇⢀⢩⣼⠃⣸⣿⣿⡿⠛⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠋⣶⠋⠀⠉⠉⢈⣵⠗⣽⠀⠀⢰⡄⠎⣠⡿⠠⠞⡧⠃⠀⠘⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢠⠀⠀⠄⠈⠠⠋⠼⡇⣰⠷⢤⠀⡉⠀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠄⢀⡊⠀⠀⠁⠠⠩⠂⠀⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⡀⠈⠇⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣫⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠁⢼⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠻⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡒⠉⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠈⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠜⠀⢀⡻⠁⠀⠠⠤⠈⠁⠀⠀⠀⠀⠀⠦⠀⠀⠀⠈⢷⡀⠄⠀⠈⠣⡄
⠀⠀⠀⠀⡐⠀⠀⠀⢹⡇⠀⠠⣾⣿⡆⠐⠀⠀⠀⠀⠂⣠⣾⣽⣦⠀⢸⣿⡆⠀⣀⠀⠱⡀
⠀⠀⠀⢀⡇⣀⠀⠀⣿⡇⠀⢀⠈⠉⠄⠀⠀⠀⠀⠀⠈⠃⠈⠉⠀⠠⠄⢻⣧⠀⡏⠀⠀⠀⢀
⠀⠀⠀⣸⡞⠋⠀⠀⢹⡇⠠⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠂⢸⣿⡆⠀⠀⠀⠀⠠⠦⡀
⠀⠀⠀⣿⣿⣦⣀⢀⠘⣇⡖⡀⡀⠀⠀⣰⣶⣶⣶⣿⣦⡀⢀⠀⠀⣀⡉⣾⣻⣷⢿⣶⠀⣰⡦⣴⠆
⠀⠀⠀⢻⣿⣿⢫⣯⡇⣟⢇⠘⣧⠠⠀⠻⢿⣿⣿⣿⠿⠁⠀⠀⣴⠟⣡⣿⣿⣿⢸⣿⣟⠋⣴⠋
⠀⠀⠀⠈⠳⣿⣿⣟⢁⣿⠀⠁⠘⡀⠀⠀⠀⣈⠋⠁⠀⠀⠀⣸⠁⠀⢉⠃⣈⣿⡸⣿⣏⡸⡜
⠀⠀⠀⠀⠀⠛⣿⣏⣼⣥⠗⠀⡀⢿⣦⣴⣶⠿⢿⣶⣶⣤⢖⡹⠋⠀⠁⠁⠸⡻⣇⠿⣿⡶⠃
⠀⠀⠀⠀⠀⠀⠈⠛⠇⠀⠀⠀⠉⠀⠙⢿⡄⠀⠀⠀⣼⣿⠋⠀⠀⠀⠐⠀⢀⣨⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⡀⠀⠀⠈⠿⢶⣶⠾⠟⠁⠀⠀⢀⡀⠰⡄⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡏⠻⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠠⠿⠿⠟⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢷⣀⠀⢀⠀⠀⠀⠀⠀⠄⢨⣅⠰⠄⠀⠀⢄⡀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠙⠿⠦⣷⢲⣤⡶⠾⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠚⠋⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠲⢄⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠦⡄
⠀⠀⠀⠀⢰⠀⠀⠀⡴⠁⠀⠀⠀⠈⢃⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⡀
⠀⠀⠀⠀⢠⠀⢀⡞⠀⠁⣀⡤⠤⠶⠖⠒⠺⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⡇
⠀⠀⠀⠀⠈⠦⠊⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⠞⠛⠛⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢸⡇
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢹⡄⠀⢀⡠⠴⠒⠒⠒⠒⠒⠤⠄⠀⠘⠀⡟
⠀⠀⠀⣀⠄⠈⡼⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠐⣌⠀⠀⠉⠁⠀⠀⠀⠀⠀⣠⣦⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⢰⠁⣠⣤⡀⠀⠀⠀⠀⠀⠀⠧⠞⠀⠐⠀⠀⢀⡠⠴⠒⠒⠓⠲⡄⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⢃⠀⢻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⠇⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠈⠀⠘⡟⠛⠃⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠥⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⢀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⢀⣀⠀⠀⢢⠜⠙⠛⢛⠟⠫⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡐⠃⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⡜⢁⡄⠀⢰⠃⠀⠀⡴⢃⣀⣀⡤⠖⠀⠤⢄⣀⣀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠡⡎⠀⠀⠸⣤⡤⠞⠁⠀⠈⣀⣀⣀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠑⠄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⡤⠤⠼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠳⢄⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠹⢦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠆⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠳⣄⣠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⢶⣶⡆⣶⣶⢶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⢋⣼⠟⠉⣁⣠⣴⠾⠋⠀⠈⠻⢶⣤⣀⡉⠙⢿⣌⢳⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠛⣽⡿⣡⠟⠁⠐⠋⢉⣁⣠⡤⢶⠀⢰⠦⣄⣀⣉⠉⠓⠀⠙⢧⡹⣿⡝⠛⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⢀⣤⣴⣾⠿⠛⠋⠁⠀⢀⣾⠟⡼⠃⠀⠔⠊⠉⠉⠀⠀⠀⠈⠀⠈⠀⠀⠀⠈⠉⠉⠒⠀⠀⠻⡜⢿⣆⠀⠀⠉⠛⠻⢿⣶⣤⣄
⠀⠀⠀⠀⣴⡿⠋⠁⠀⠀⠀⡀⠀⢀⣾⠏⠀⠀⠀⠀⠀⣀⠴⠛⠶⣄⡀⠀⠀⠀⠀⣀⡴⠞⠳⣄⡀⠀⠀⠀⠈⠈⢿⣆⠀⠀⡀⠀⠀⠀⠉⠻⣷⡄
⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⡴⢁⢀⣾⠏⠀⠀⠀⠀⠐⠊⠁⢀⡀⠀⠀⠙⢦⠀⢠⠞⠁⠀⠀⣀⠀⠉⠒⠀⠀⠀⠀⠈⢿⣆⢀⠰⡄⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠸⣿⡀⠀⠀⠀⡼⢡⢎⣾⠃⠀⠀⠀⠀⠀⢀⣠⡶⠟⠻⣦⣄⠀⢸⠀⢸⠀⢀⣤⡾⠛⠷⣦⣀⠀⠀⠀⠀⠀⠀⢻⣆⢦⠹⡄⠀⠀⠀⣸⡟
⠀⠀⠀⠀⢿⡇⢧⠀⣼⢡⡟⣾⠃⠀⠀⠀⠀⠀⠀⣉⡁⠀⠀⠀⢠⠟⢀⠈⠀⠈⢀⠘⢇⠀⠀⠀⠀⣈⠁⠀⠀⠀⠀⠀⠀⢻⡎⣧⠹⡄⢠⠇⣿⠇
⠀⠀⠀⠀⠘⣿⡌⠶⢃⡾⣼⠇⠀⠀⡄⠀⠀⣴⡿⠿⣿⣿⣦⠀⢀⣠⠋⠀⠀⠀⠈⢢⣀⠀⢠⣾⣿⣿⠿⣷⡄⠀⠀⡄⠀⠀⢿⡼⣆⠱⠏⣼⡟
⠀⠀⠀⠀⠀⢸⣷⠀⣾⢻⠏⠀⠀⣰⠇⠀⠀⠸⣧⣄⣠⠈⠙⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠉⢠⣀⣤⡿⠀⠀⠀⢷⡀⠀⠈⣿⢻⡆⢰⣿
⠀⠀⠀⠀⠀⢸⣿⣼⠇⣿⠀⠠⢠⡿⠀⠀⠀⠀⠈⠉⠁⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠉⠉⠀⠀⠀⠀⠸⣧⠠⠀⢸⡇⢿⣼⣿
⠀⠀⠀⠀⠀⢸⣿⡏⠀⢻⣤⢃⣾⠃⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⢔⡤⠖⠒⠀⠐⠒⠦⣄⢀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⢣⣼⠃⠈⣿⣿
⠀⠀⠀⠀⠀⠸⡟⠀⠀⠈⢣⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠃⣰⠃⣠⠖⢋⡉⠁⠉⣉⠓⢦⡀⢳⡀⠃⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠋⠀⠀⠸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢿⣼⠋⢹⣧⡾⠉⢻⣼⠇⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢭⣥⠞⠃⠛⠶⣭⠅⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠠⡀⠀⠀⠀⢀⠃⢰⡇⠀⠀⠀⠀⠙⠳⠆⠖⠛⠁⠀⠀⠀⠀⣷⠀⢃⠀⠀⠀⠀⡠⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⢣⡀⠀⠀⣸⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⡀⠀⠀⣰⢣⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠳⡤⠞⠁⠀⡿⠀⠀⠀⠀⠀⣠⠞⠉⠙⢦⡀⠀⠀⠀⠀⠸⡇⠀⠙⠦⡴⢣⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⣀⣠⠇⠀⠀⡠⢠⡾⢃⠀⠀⠀⢀⠻⣦⠀⡀⠀⠀⣧⣀⡀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣮⣛⠭⠤⠴⣒⣥⣶⢟⡕⠁⠀⠀⠀⠀⠑⣝⢷⣦⣑⡒⠤⠬⢝⣫⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⠿⢟⡫⠗⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⠯⣛⠿⢿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣶⣄⣀⡀⠀⠀⢀⣀⣴⣶⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡷⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣹⣯⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⣡⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠌⢣⢻⡞⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣾⢣⠋⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠈⣎⢿⡸⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣠⣾⡿⣹⣏⡎⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⣠⠏⣸⢇⣽⠏⠙⠉⠉⣉⣭⣭⣽⡟⣿⣭⣭⣍⡉⠉⠙⠉⢾⣅⢿⡈⢧⡀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢀⣤⠞⣡⠾⣿⠟⠁⠀⠴⠚⠋⣁⣠⡀⠈⠀⠈⠁⣠⣀⡉⠛⠲⢤⡀⠙⢷⡻⣦⣙⢦⣄⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⢋⣵⢟⡵⠟⠁⠀⣀⣤⠴⠞⠛⠉⠀⠉⠂⠀⠀⠚⠁⠈⠙⠛⠶⢤⣄⡀⠀⠙⠷⣝⢷⣍⠳⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡟⡕⠋⠀⠀⠀⠚⠉⣠⣤⡤⠶⠶⠦⣄⡀⠀⠀⣀⡤⠶⠶⠦⣤⣄⡈⠙⠂⠀⠀⠈⠓⡝⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠠⣤⣀⠀⠀⠀⠀⡆⠀⠀⡆⠀⠀⠀⢀⣠⡤⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠈⠑⠲⢤⣉⠛⢦⡀⢸⣇⡇⣇⣿⠀⣠⠞⢋⣡⠴⠒⠉⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠶⡿⠿⠿⠿⣮⠳⣄⠙⠏⢾⠀⢸⠎⠟⢁⡴⢫⡾⠿⠿⠿⡷⠶⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⠦⠤⠆⠀⠀⢨⡷⠀⠀⠀⠀⠀⠰⣏⠀⠀⠀⠦⠤⠞⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⢆⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⢀⠆⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⣴⣱⠆⢀⡞⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠘⣆⠀⢶⣱⡄⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⣰⣿⡟⠀⢸⠁⠀⣠⢰⡏⣉⣀⡀⣀⣈⡉⣷⢠⡀⠀⢹⠀⠘⣿⣷⡀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⣿⠃⠀⠀⢠⡞⠁⠀⣽⣏⣨⠟⢿⣈⣿⡅⠀⠙⣦⠀⠀⠀⢻⣿⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⢠⡏⠀⠀⠀⠙⠫⠅⠠⠀⠭⠛⠁⠀⠀⠈⣷⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣾⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⣿⠀⠀⠀⠀⣀⣤⠶⠚⠲⢤⣄⡀⠀⠀⠀⢸⡇⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣝⡻⠄⢀⣴⣿⣏⣀⠀⠀⠀⢀⣈⣿⣷⣄⠀⠼⣛⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠟⠁⠀⠉⠙⠛⠛⠛⠛⠉⠁⠀⠙⠷⠛⠋⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢉⠉⣀⠀⠉⠐⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⢠⡙⣿⠁⣼⠟⣀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄⠉⠁⠃⠄⠹⠀⠋⢈⣷⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢈⣿⣷⠀⠰⢲⢳⠢⠀⢠⣿⢽⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠃⣿⡀⣤⢠⣇⢠⠀⢸⣿⢸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢧⣥⢟⡹⢦⣀⣸⣏⣷⡍⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣽⠃⢲⣿⠿⠊⠑⣫⣾⣿⠇⠘⡆⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⢘⡇⠀⠀⢠⣿⣿⣿⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣾⣷⣆⣴⣿⣿⣿⠁⠀⠀⠀⣾⠁⢠⠀⢸⣄⣤⣤⠀⠀⠀⠻⠦⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⠀⠀⠀⣰⣿⠀⣾⠀⢸⣿⣿⣿⣤⡀⠀⠀⠀⠀⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠄⠀⠀⠙⠻⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⣿⣿⣾⡟⢀⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⣸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢀⣀⡀⠸⣧⣤⣀⠀⢀⣠⣤⣀⣠⣶⠃⠀⢰⣿⣿⣿⡿⠛⠛⠻⢟⡁⠙⠻⡷⣷⡀⠰⠁⢻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⣠⡝⠁⠀⠀⠈⠛⠿⣿⣿⡿⠿⠟⠋⠁⢀⠀⠀⠛⠉⠀⠀⠀⠀⠀⣼⣿⡤⠔⣃⡼⠃⠀⠁⠀⠹⣿⣿⣿⣷
⠀⠀⠀⠀⠀⡰⠋⠁⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⣠⣤⠞⠋⠁⠀⠀⠀⠀⠀⠈⢌⡉⠟⠋⠀⢸⡋⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿
⠀⠀⠄⠒⠉⠁⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⢠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒
⡤⢀⢀⡄⣠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠁⠀⠀⠁
⠓⠛⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢠⠀⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡼⠛⢿⢿⡇⢀
⠀⠀⠀⠀⠀⠀⢠⣤⠀⢠⡘⠿⠷⠀⠀⣾⣿⣿
⠀⠀⠀⢀⣤⣌⠲⠟⠒⠚⠧⣶⠷⠾⣿⣿⣿⣅
⠀⠀⠀⢼⡿⢋⡉⠧⠀⠀⠀⠈⠳⣾⣿⡿⢿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣶⣦⡤⣄
⠀⠀⠀⠘⠃⠈⢁⠀⢰⠀⠀⠀⢀⣿⣿⡄⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⣀⠀⣤⠔⠶⣻⠀⠀⢀⠔⠉⣹⣀⠐⠀⠠⠊⠁⠰⠤⠤⠤⠤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠠⠩⢘⠏⠁⢀⠏⠃⠀⢹⣦⣴⣿⠃⠀⠐⢠⣴⠃⠀⠀⠀⠀⠀⠉⢫⠙⡀⠀⠈⠻⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⢀⠀⠀⢸⡀⠀⠀⢰⡿⠿⠃⠀⠀⠀⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⡇⢃⢀⡀⠢⠹⣿⣿⣿⣿⣝⡂
⠀⠀⠀⠀⠂⠄⡀⠋⠀⠀⠀⢸⣧⡀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠓⠁⠀⢡⡀⡇⠙⠌⠻⢿⣿⣿⡧
⠀⠀⠀⠔⢢⠀⠉⠂⠀⠀⠀⠀⠙⣿⠿⠿⠛⠋⠁⠀⢢⠀⠀⠀⠀⠀⠀⠀⡠⠒⢢⠀⠐⡀⠀⠰⣿⣦
⠀⠀⠀⡀⣰⣧⡀⠀⠀⠀⠀⡆⠀⣿⡆⠀⠀⠀⠀⠀⠀⣁⣀⠀⠀⢀⡠⠞⠁⠀⠀⢆⠀⠬⠛⣿⣿⠛⠁
⠀⠐⣀⣿⠿⠁⠙⢦⣀⣀⢴⣁⡀⠘⣿⣄⢀⣀⡠⠔⠋⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠐⠒⠀⠈⠁
⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⣄⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠉⠊⠉⠁⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣶⣾⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⣿⣿⣶⣶
⠀⠀⠀⠀⠖⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠂⠀⠀⠀⠀⢰⣿⡿⠿⠛⢿⣿⡀⠀⠀⠀⠂⠀⠀⠀⠘⠀⠀⠀⠀⠀⠘⠍⠀⠀⠈⠛⠋⠋
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠘⠟⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣷⣀⣀⣂⣆⠀⠀⠀⠀⠀⠀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠐⠀⠘⠿⠧⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡯⠀⠀⠀⠀⠀⠀⠈⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣧⠀⢀⣾⣿⣆⠀⠀⠀⠀⡞⠃⠌⢁⠀⠀⠀⠰⠟⡿⠃⠀⢰⡏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⡄⠀⠀⠰⢦⣖⣰⠞⠀⠀⠀⠀⠀⠀⣘⣃⡈⠃⠀⠀⠀⣆⠀⠀⠀⢠⠀⠀⠈
⠀⠰⠀⠐⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠸⠇⠀⠂⠀⠀⠘⠟⠛⠛⠃⠀⠀⠀⠀⠙⠙⠀⠀⠀⠀⠀⠀⠘⠽⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡀⠤⡤⠀⢀⠄⠀⠀⡀⠀⣺⣦⣄⡖⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠰⠆⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⠁⠈⠋⠀⠀⠀⠳⠂⠀⠀⠀⠀⠲⠞⠁⠀⠀⢠⣶⣿⠛⣧⣹⣎⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣤⡄⠀⠐⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠹⣋⠠⡆⢀⡈⠀⠀⠀⠀⡸⢧⡏⣤⠀⣧⡛⠀⠀⠀⠀⠀⠀⠀⠄⢠⡄⡇⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⠳⢶⠖⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⢀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢤⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⣤⣦⡦⣣⠔⣶⣄⠀⠀⠈⣷⣿⡿⣿⣿⡇⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠰⣇⣽⠛⢫⣻⡃⠀⠀⠀⠀⣀⠀⠀⢀⠉⢡⡟
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⠿⣿⡿⠄⢠⡾⠯⠙⠂⠀⠀⠟⠋⣤⠝⠛⠃⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠰⣿⣿⠲⢾⣟⣿⠀⠀⠀⠘⠿⠀⣄⢙⣁⠄⠡
⠀⠀⠀⠀⣓⠈⠀⠀⠀⠀⠀⠀⠀⠞⠻⠏⢻⡟⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠐⠀⠉⠁⠀⠀⠀⠀⠀⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠢⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢶⣶⣤⠀⠀⠀⠀⠀⠀⠀⢩⠁⠀⠈⠋⠀⢹⠿⠃⠀⠀⠀⠀⠈⠈⣡⣴⠀⠁⠀⠀⠀⠘⠀⠀⠂⠀⠀⠀⠰⣶⡆⣶
⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠂⣥⣄⢀⡹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣆⠀⢀⠀⠀⠀⠀⠀⠀⡖⠁⠈⡏⠀⢀⠀⠀⠀⠂⠀⠀⠘⠀⠀⠀⠀⠿⣁⠙
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢿⠟⠘⠁⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠲⣴⣼⠬⡵⡃⠀⠀⠀⠀⠀⠀⡇⠀⡂⠈⠀⠠⠀⠀⠀⠐⠐⠗⠈⠀⠀⠀⠀⠀⠉
⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠙⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠟
⠀⠀⠘⠙⠁⢸⡿⣶⣶⡆⠀⠀⢠⢀⠐⣦⡴⡆⠀⠀⠀⣟⣿⡄⣤⣾⡿⠂⠀⠀⢀⣠⣸⣶⣿⣧⡤⠀⠀⠀⠀⠀⠀⠀⣤⠞⠚⢷⣦⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⠘⠃⠀⢐⣿⡿⠀⠸⢦⡟⠀⠀⠀⣼⣻⠇⣟⣿⠀⠀⠀⠀⠀⢩⠿⠛⠛⠛⠄⠀⠀⠀⠀⠀⢀⠀⠿⠛⠓⠚⠂⠀⠀⠀⠀⣰⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣙⢿⡿⠀⠀⠀⠸⣿⠴⣿⢷⣴⡿⠀⠀⠀⠙⠁⣀⠈⠃⠀⠀⠀⠂⠀⠃⠀⣶⡆⠀⠀⠄⠀⠀⠀⠠⠆⢀⣶⢀⣀⢰⣤⢀⡄⠀⠀⠛⢿⠿⠿⡟
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠛⠒⠿⠾⠟⠁⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠘⠀⠀⠼⢄⣀⡠⠀⠀⠀⠀⠀⠓⢺⣿⣬⣋⣼⡷⠜⠁⠀⠀⠀⠘⠦⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠀⠀⠤⣤⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣽⠀⠀⢠⠄⠀⠘⠇⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⡝⠀⠀⠈⠛⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠀⠀⠀⠀⠀⠠⣇⣤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠓⠀⠀⠄⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡗⠲⠛⠦⡀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠙⡄⠀⠘⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⣠⠤⠶⢤⡀⠀⢢⠀⠀⠀⡔⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠉⠀⠀⠀⠀⠹⡆⠈⡄⠀⠀⡇⠀⣠⠀⠸⡇
⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢧⠀⠀⣧⠞⠹⡄⠀⡇
⠀⠀⠀⠀⢸⡷⡄⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢸⠀⠀⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠰⣧⠙⢦⡀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢀⡠⠚⡇⠀⡇⠀⠀⢇⠀⠇
⠀⠀⠀⠀⠀⠈⣷⣄⠈⠓⠦⢄⣸⣄⣀⠀⠀⠀⢀⡤⠚⢁⣠⣖⠉⠉⠙⡇⠀⢧⠀⠀⠈⢣⠘⠂
⠀⠀⠀⠀⠀⠀⠈⠛⠢⠤⠄⠠⠤⠾⣅⣀⣀⣀⣈⣉⣉⡀⣠⡠⠏⠉⠀⠙⢄⠀⡁⣠⠀⠀⠁⠠⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣀⡀⠀⣀⣠⣤⣤⡤⠤⠤⢤⣤⣤⡤⠤⠤⢤⣤⣤⣀⣀⠀⢀⣀⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⢀⣴⠞⠋⠉⠀⠀⠀⢠⣿⡿⠛⠉⠁⠀⠀⠀⠀⢀⡾⠁⠘⣧⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⡄⠀⠀⠀⠉⠙⠳⣤⡀
⠀⠀⠀⠀⣾⠁⠀⠀⣀⣀⡀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⢀⣀⣀⠀⠀⠈⣷
⠀⠀⠀⠀⣿⣠⠶⣟⣹⢯⡞⠉⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠸⣇⠀⠀⣸⠃⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠉⢳⡽⣏⣻⢶⣄⣿
⠀⠀⠀⠀⠟⠁⠀⠙⢧⡟⠀⠀⣠⠞⠋⢉⡉⠀⠀⠀⠀⠀⢀⡿⠀⠀⢿⠀⠀⠀⠀⠀⠀⢉⡉⠙⠳⣄⠀⠀⢻⡹⠋⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠋⣰⠟⠉⣙⢳⡄⠀⠀⢀⡼⠃⠀⠀⠘⣧⠀⠀⠀⢠⠞⣉⠉⠳⣄⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠠⣶⣿⡷⠋⣠⣤⠾⣷⣿⣿⣿⣿⣾⠷⣤⣀⠛⣾⣿⣶⠀⡿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣠⡌⠳⠿⢏⡵⠟⠉⠀⠀⢻⣿⣿⣿⣿⡟⠀⠀⠉⠻⢮⡹⠷⠞⢡⣄⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⠞⢉⣠⣤⠴⢤⡟⠀⠀⠀⠀⠀⠀⠉⢻⡟⠉⠀⠀⠀⠀⠀⠈⢳⡤⠦⣤⣈⠉⠳⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣰⠟⠁⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠻⣆⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣿⠀⢠⡄⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⢠⠀⠀⣿⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣜⣦⠈⣿⣦⣀⡀⠀⣀⣀⣠⣤⠶⠛⠉⠉⠛⠶⣤⣄⣀⣀⠀⢀⣀⣴⣿⠀⣼⣣⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣇⢹⡄⠉⠉⠉⢩⣭⠤⠤⠶⠶⢶⡶⠶⠶⠤⠤⣭⡍⠉⠉⠉⣠⡏⣼⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡙⢦⣄⣀⣻⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣇⣀⣠⡴⢋⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣬⣭⣽⣇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⣯⣭⡥⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⠀⠈⠁⠀⠀⣀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠲⠖⠚⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⢀⣤⡖⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠆⠀⡄⠀⠉⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⢀⣤⣾⠃⠀⠀⠘⠀⠈⠀⠀⣤⠙⣦⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⢁⣼⣿⣟⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣷⡌⣿
⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠔⠋⠙⠛⠛⠛⠻⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⣸⣇
⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣙⣶⣤⣀
⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠈⢿⣦⡁⡀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⠿⠋⡇
⠀⠀⠀⠀⠀⠙⢿⣷⣄⢀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⢀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡆⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣿⠃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣵⣿⣿⣿⣿⣇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣶⣶⣶⣶⢶⣿⣿⣿⣿⣿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠟⢉⣽⣾⡿⠛⠋⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢷⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⡜⠉⣿⠁⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡌⢇⠀⠀⠀⠀⠄⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⢧⠀⠀⠀⠀⠀⠀⠳⡁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⡐⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠰⢱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⣶⣶⣾⣿⣷⣶⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣿⡿⠛⠟⠋⠁⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣶⣶⣶⣦⢹⣿⣿⣿⣿⠿⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⣆⠀⠘⣿⣿⣿⠏⣸⣿⡿⠋⠀⣠⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣤⣽⣤⣴⣿⠟⠁⣀⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⢹⣮⣉⣠⣾⠂⠀⣾⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣹⡇⣹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡇⠘⠀⠀⠀⠀⠀⢠⣿⡇⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⡀⠀⠀⠀⢀⣼⣿⠃⠙⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠰⣿⣄⠀⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢹⣷⠀⠀⠀⠀⠀⠘⢿⣿⣿⡟⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠁⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⠟⠋⠙⠻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⠋⠱⢶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⢀⢀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠘⠀⠉⠙⠉⢻⡄⠀⠷⠀⠀⠑⠀⠀⠐⠀⠀⠀⠁⠀⠈⠀⠉⠉⠁⣶⣦⣤⣤⣴⣶⣤⣤⣤⣤⣤⣶⣤⣶⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⣶⣶⣶⣶⠶⠶⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⣀⡹⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⣻⣾⣿⣿⣿⣿⣿⠃⢠⡄⢤⣀⣦⡀⠀⠙⣿⣶⢄⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣂⣿⣷⣀⣙⣿⣿⣦⣤⣿⣿⡆⢻⣄⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⣸⣿⣿⣿⡿⠿⠿⠿⠚⠛⠉⣉⣩⣿⣿⣿⣿⡟⠋⠉⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣦⠀⠘⣿⣌⠛⠛⠛⠛⠓⢀⣠⣤⣶⣿⣿⣿⣿⡿⠋⣿⣧⠀⢀⣴⣾⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⢸⣿⣿⣷⣶⣿⣿⣿⠿⣻⣿⣿⣿⣿⠟⠁⠀⣿⢸⡀⡛⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣾⣿⣿⣿⣿⣿⠟⢡⣾⣿⣿⣿⣿⠋⠀⠀⠀⣿⠀⣯⣻⡿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣉⣛⣛⣽⠟⠁⣴⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢻⠀⢸⣿⡇⢸⡟⢻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠰⣦⣤⣬⣿⠟⠋⢀⣠⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⡆⠀⣿⠀⠀⠁⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣦⣴⣶⣿⣿⣿⣿⡟⠉⠀⠸⣿⡇⠀⠀⠀⠀⢸⣿⣦⣸⣇⣀⣠⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣿⣿⣄⠀⠀⠀⠈⣿⣿⡟⢻⡁⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠻⣦⡀⠀⠸⣿⣿⣦⡀⠀⠀⣿⣿⡇⠀⠃⠀⡀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠈⠻⣦⣀⠻⣿⣿⣿⣷⣦⣸⣿⣷⡇⣀⣸⣷⣽⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⢸⡀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⠋⠉⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢻⣿⡇⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠈⣿⣷⣄⠀⠀⠀⣿⣆⠀⠀⠘⣿⣿⣿⡃⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠸⣿⣿⣷⣤⣤⣿⣿⣷⡀⠀⢿⣿⣿⣿⡀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣄⣾⣿⣿⠻⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠃⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢦⠀⠀⠀⠀⠀⣰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⠈⡇⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢁⣾⠀⢱⣤⣤⣿⣿⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣾⡇⠀⣿⣿⣿⣿⡻⢷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡇⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣆⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡏⠉⣭⣽⣆⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠉⢹⡿⠃⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠈⢻⣿⣧⡀⠠⣄⡀⠀⠀⠀⠀⢹⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠙⢿⣿⣶⣤⣭⣷⣶⣴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠙⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣯⣍⠉⠛⠉⠉⠁⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣷⡄⠀⠀⠀⢠⣤⣴⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣦⠀⠀⠀⠉⠛⠿⠿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡆⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⢀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⢀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⣏⣱⣾⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡷⠀⢸⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣿⡏⠁⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠻⣟⠦⣄⡈⣀⣀⡽⠛⢿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⣯⠉⠀⠀⡜⢿⣿⣿⣻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠦⠁⠀⠃⠈⢿⣿⣿⣿⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣙⣓⣃⣀⣴⣾⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣛⡛⣿⣿⣿⣿⣿⣿⣿⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣻⣿⢛⡇⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢻⢧⠖⠋⠀⠸⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⠀⢿⣄⠀⣀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠘⣷⣄⣻⣟⣻⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢿⣯⠹⣿⣿⡇⠀⠘⢿⣿⣿⡿⠃⡛⠘⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⣿⣇⠈⢿⣿⡄⠀⠈⢃⠻⣿⢰⠃⠀⠘⣿⣿⣿⣿⠿⠙⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⣿⣄⠈⢻⢿⡆⠀⠀⠀⠸⠆⠀⠀⢀⡀⠈⠉⢛⠿⡶⠠⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣹⣿⣷⡄⣸⣇⠀⡀⢠⡘⡆⠀⠀⠈⠀⢔⡿⠿⠞⠳⠶⣽⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣛⠙⣿⡇⠉⠻⡀⠸⣤⣧⡃⠀⠀⠀⣴⠏⠀⢀⣤⣾⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⣁⠻⣧⠀⠀⡇⡠⡹⣿⣇⠀⠰⣰⡏⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢻⣄⣿⡄⠀⣇⠙⣶⣷⣿⣿⣦⣿⡇⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⣿⣽⡇⠀⢸⠀⣿⣿⠉⢿⣿⣿⣿⡀⠀⠻⣿⣿⣿⣿⣿⡿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⣿⠃⠀⠈⣦⣿⡇⣆⣈⢛⣿⣿⣷⡀⠀⠙⠻⣿⣿⡞⠁⠹⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢶⡧⢆⣥⡿⠀⠀⣾⢁⢟⣿⣯⣿⣿⣿⣿⠷⣿⣦⡀⠀⠠⣽⣿⣶⣾⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢼⣯⡤⠿⠽⠃⢠⡾⣡⢊⡞⣻⠃⢩⣟⡕⢋⡵⢶⣿⣶⣶⡶⠿⠿⠿⠿⠿⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⢾⡶⠿⠀⠘⠿⠷⠿⣦⡿⠿⡍⠀⠤⠤⠤⠴⠴⠶⠟⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡻⣉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⣿⣿⣿⡿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣶⣤⡄⢠⣀⣴⣿⠟⢉⣴⣿⣿⣿⣻⣷⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣁⣉⠉⣉⣠⣾⡿⠿⣄⣠⣿⣿⣿⠋⠙⢻⣿⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⡶⠻⣿⣿⣿⣿⣶⣶⣶⢿⣿⣏⣀⠈⠻⣿⠏⢿⣿⠋⣠⡄⢀⣿⣿⡿⠿⠞⠛⠉⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⣼⣿⡇⢶⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣷⣦⡑⢼⣿⣿⣿⣀⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢠⣿⣿⣷⡈⠻⣿⣿⣿⣿⣿⣿⣿⠘⠉⠉⠁⢳⣈⣿⣆⣿⣿⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⢘⣿⣿⣿⣷⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣸⣿⣿⣿⡿⠀⡿
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃⣸⣿⣿⣿⠿⣿⣾⣿⣿⠏⠉⠛⠿⢿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣅⠻⠁
⠀⠀⠀⠀⠀⣠⣾⣿⠟⢁⣼⣿⣿⠟⠁⠀⠘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⢻⣯⣿⠏⠙⣿⣧⡄
⠀⠀⠀⠀⠀⠉⠉⠀⣸⣿⡟⠋⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠘⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠇
⠀⣠⣤⣤⣤⣠⣬⣻⣲⣶⣤⣤⣤⣤⣄⡀⣀⢠⣤⣤⣤⣹⣷⣤⣤⣤⣶⣌⣿⡄⡀⣀⣀⡀⣄⡀⣴⣿⣧⣠⣶⣶⣶⣦⣦⣦⣦⣠⣤⡀⡀
⠀⠀⠈⠉⠀⠉⠉⠁⠀⠀⠈⠈⠁⠀⠁⠁⠈⠉⠉⠉⠛⠙⠋⠙⠛⠛⠋⠉⠉⠋⠀⠙⠛⠛⠛⠛⠋⠀⠈⠉⠁⠉⠉⠉⠁⠈⠉⠁⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡯⢷⣖⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣴⡶⢽⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⣼⣿⠃⠀⣀⣀⣀⠀⠀⡀⢀⠀⠀⣀⣀⣀⠀⠘⣿⣧⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⢰⣟⢛⣶⣿⣿⣻⣿⣯⢉⡿⢿⡉⣽⣿⣟⣿⣿⣶⡛⣻⡆⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⣾⣷⠋⠉⠀⠀⢀⡤⣠⡞⠁⠈⢳⣄⢤⡀⠀⠀⠉⠙⣾⣷⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⣿⡟⡵⠠⠖⠚⠉⠉⠺⣏⢰⡀⢀⡆⣹⠗⠉⠉⠓⠲⠄⢮⢻⣿⣆⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣿⣽⣿⢏⠞⠀⢀⠤⠴⠖⠒⢄⠘⣧⠇⠸⣼⠃⣠⠒⠲⠦⠤⡀⠀⠳⡹⣿⣯⣿⡀
⠀⠀⠀⠀⠀⠀⠘⡟⡽⣣⠋⠀⠀⠀⣀⣴⠶⢦⡄⠀⢿⠀⠀⡿⠀⢠⡶⠶⣦⣀⠀⠀⠀⠙⣜⢯⢻⠃
⠀⠀⠀⠀⠀⠀⠀⢺⣁⣥⡄⣴⠶⠛⣉⡀⠀⢄⠙⣆⢸⡄⢀⡇⣰⠟⡠⠀⢀⣉⠛⠷⣦⢠⣬⣈⡗
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⡇⢿⠀⢺⡛⠻⣷⡌⢳⣌⠈⠃⠘⠁⣡⡞⢡⣾⠟⢛⡗⠀⡿⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⠉⠁⡼⠋⣀⣽⠆⠀⠀⠰⣯⣀⠙⢧⠈⠉⠀⠀⠀⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⡀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣼⠀⢀⡄⢸⡏⠀⢀⣤⣆⡀⢀⣰⣤⡀⠀⢹⡇⢠⡀⠀⣧⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣤⠞⡄⣸⠇⠀⠸⣾⢉⡇⢸⡉⣷⠇⠀⠸⣇⢠⠳⣤⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⣼⠁⣿⠀⠀⠀⠻⠯⠀⠀⠽⠟⠀⠀⠀⣿⠈⣧⣯⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣸⠃⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⠘⣇⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⢀⡴⠛⠉⠉⠛⢦⡀⠀⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡴⣱⠏⠀⠀⡀⢀⠀⠀⠹⣎⢦⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣔⠜⣱⡿⢦⣄⣾⣄⣠⣷⣠⡴⢿⣎⠣⣪⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠈⠉⠛⠙⠉⠁⠀⠀⠉⠛⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⢟⣹⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣯⡙⢾⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⢿⣮⣿⣿⣿⣿⣿⡿⣫⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⠿⣍⢦⣿⣿⣿⣿⣿⣿⣼⡿⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⡟⢉⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣷⣦⡙⢻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣿⡏⣿⠟⠁⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿⠿⠛⠉⠉⠀⠉⠀⠙⢿⡾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣀⣾⠿⠿⢿⣆⡀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⣶⣦⣤⣤⣶⠟⠁⠀⠀⠀⠻⣷⣦⣦⣤⣲⣾⣿⣿⣿⣿⣿⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠉⢻⣿⢿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣤⣾⣿⣿⣿⣿⣿⣦⠀⢻⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⣷⣹⣿⣏⢿⣿⣿⣿⠃⠀⢿⣿⣿⣿⣿⣿⣿⣿⠀⠈⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠀⠀⠀⠹⢿⣿⣿⣿⠟⠀⠀⠀⢸⣿⣿⣽⣿⣿⡹⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣹⣿⣿⡿⣿⣿⡿⣿⣿⡉⠁⠀⠀⠀⠀⠴⠞⠹⠷⠆⠀⠀⠀⠈⣽⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣽⡿⣿⣿⣿⣧⡀⠀⠀⠀⣀⠔⠚⠷⠦⣀⠀⠀⠀⣠⣿⣿⣷⣭⣛⠻⣿⣿⣟⣹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢫⣾⣿⣿⣿⣿⣿⢷⣶⡙⢤⣤⣤⡀⠐⣾⣿⣶⣾⣿⢽⣿⡝⠻⣿⣶⣿⣼⣿⣿⣝⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡏⠀⢻⣿⣦⠀⠈⠀⠀⠿⣿⣿⢻⣿⣮⣻⣿⣦⢹⣿⣿⣿⣿⣿⣽⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡇⠠⠂⢹⡿⠄⠀⠀⠀⠀⠹⡅⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣾⣿⣿⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡶⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣅⠀⠀⠀⠀⠀⠀⠐⠫⠟⠉⠙⣟⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠈⠁⢠⡃⠀⠀⠳⣍⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⠀⠀⠘⢸⡆⠀⢶⣿⠀⠉⠓⠺⢷⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⢶⣤⣶⣺⣷⣤⠘⣆⣄⠀⠄⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⠀⣤⣾⡉⠙⢋⡙⠻⣩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣛⣿⣦⣀⠉⠻⢿⣿⡆⣀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢿⣿⣿⣶⣾⣿⣿⡏⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣆⠀⠀⠀⠀⢸⣼⣿⣿⣿⣿⣿⣿⣧⠠⠒⠂⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣀⣤⡶⠂⠀⠙⣀⠀⢘⣁⠂⠈⠛⠀⠀⠀⠀⣀⣙⣿⣿⣿⣿⣿⣿⣿⣆⠀⠸⠁⠀⢀⠀⠂⢹⡇
⠀⠀⠀⢠⣾⣿⠟⠀⠀⠀⠈⠛⠃⠈⢋⣤⡀⢸⣶⣶⣀⣤⡼⣿⣿⡿⣿⠿⠟⠙⠿⠋⠀⠀⠐⠚⠛⠛⠳⢾⡇
⠀⠀⠀⢘⣿⡗⠀⠀⠀⠀⠀⠐⠂⢠⠄⠈⠀⠘⣿⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠞⢷⡀
⠀⠀⠀⠀⢻⣿⠆⠀⠀⠀⠀⠀⠛⣤⡀⠀⢀⣈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡐⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣍⡙⠳⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠙⠻⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣦⡙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡖⠲⢽⡍⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠉⠉⠉⣀⠥⠔⣠⡦⠠⣘⠊⠉⠟⢒⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠚⠛⠙⠛⢷⡢⡄⢀⡠⢂⡴⠚⠁⠀⢶⣌⡙⠂⠂⢉⡁⠀⣀⡛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠩⠞⠛⢉⣓⣄⠑⠈⠸⠑⢁⠔⠁⢂⣠⠏⡧⢸⡀⠀⢁⠖⠋⢹⡄⣀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣠⣤⣤⣄⣠⡴⠿⣿⡗⢦⠀⡀⣠⠞⠛⠛⠛⠁⢀⡄⣐⢒⣶⣦⠞⢠⡼⡏⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢰⠃⡼⠉⠀⠀⠒⠼⠓⠒⢤⡈⣷⢸⡇⢸⡇⠀⣠⠀⡴⠞⣿⡯⠁⢉⣁⣀⠀⠉⢀⠳⢦⡈⢷
⠀⠀⠀⠀⠀⠀⠀⠸⣀⢡⠎⠀⢀⣠⡤⠤⠤⢄⠑⢿⠘⡇⠈⡇⠀⠇⠸⠁⣸⢻⠀⠔⠩⢒⣒⣈⣀⠀⠑⠂⠱⠘⣗⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡉⡼⠀⠀⠛⠂⢤⣄⠐⠀⠑⠈⠀⠀⠀⠀⢀⡄⠀⡀⠁⢀⡀⠀⠀⠀⠀⣀⣺⡛⠂⢤⠀⠱⣌⠳⡄
⠀⠀⠀⠀⠀⠀⢀⣦⡴⢃⡇⣀⣠⠖⠉⣠⣤⣤⣼⣧⢠⠐⣦⡄⡜⡿⠟⠻⠊⣿⣿⣾⡖⠊⠉⠀⠀⠁⠀⢠⡄⠠⣈⠑⢦
⠀⢀⣠⡤⠄⠒⠈⠸⠖⠋⠀⠀⢐⠖⠋⠁⠀⠛⠿⢛⣤⢥⠃⠸⡆⠀⠀⠀⠀⠨⠟⣩⣄⡀⠀⢤⠈⡐⢄⠱⡰⡀⢌⢳⡌⡇
⠀⠀⣤⡀⢠⠀⠔⡡⠂⠀⠀⠀⢸⠀⡞⠀⠀⣀⠀⣈⢟⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠉⢳⣆⠂⠘⣆⠀⡇⣇⠀⠸⠁⡇
⠀⠀⠀⢰⢃⡄⠈⢸⡖⠊⡇⠰⠈⠀⢹⠦⠈⠉⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠛⠈⠋⠠⣀⡀⠒⠢⣄
⠀⠀⠀⢸⣺⠿⠀⠈⠀⡄⠻⡀⢀⡸⠂⠀⠀⠀⠈⠉⠀⠀⢀⣠⣤⣤⣤⣄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢱⡆⠀⠀⠂⠤⢀⣀⠢⡌⢳
⠀⠀⠀⠀⠉⢰⢠⠸⢍⡀⠀⠀⠔⠁⢀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠒⠀⠀⢆⠃⠀⠈⢳⡀⢰⠈⠑⠀⠈⠣⠤
⠀⠀⠀⠀⠀⠀⠿⡃⠀⠀⠀⠈⢹⠋⠁⠀⠀⠐⠈⠁⠀⠀⠙⢿⣻⣿⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⠀⡀⡄⠳⢌⣦⠠⣄⣓⡶
⠀⠀⠀⠀⠀⠀⢠⠗⢧⠀⠂⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⢨⠹⡷⢤⣌⡥⢄⢀⡀⡠⠋
⠀⠀⠀⠀⠀⠀⡄⠀⠈⢮⣓⠄⣣⡆⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠄⣶⣶⣦⣦⣀⢀⡀⠀⠑⢄⠀⠀⢠⡏⠀⠘⡄
⠀⠀⠀⠀⠀⠰⠆⢀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⢸⠀⢠⡞⠋⠀⠀⠀⠀⢈⠛⠋⠑⡲⠚⠛⢉⠉⠀⢢⡀⡬
⠀⠀⠀⠀⠀⠎⠠⠃⣔⡥⠖⠈⠀⠀⠈⢧⣠⣀⠀⣄⠼⠞⠁⢱⠀⠀⠀⠀⠀⠘⢀⠔⠖⠁⠀⠠⣈⠳⠐⠂⠆⠑⢄⣀
⠀⠀⠀⠀⠠⠀⠀⢰⡏⡐⠀⠀⢆⠰⡀⢀⡛⠈⠋⠙⠖⠤⣄⣸⡆⠀⠀⠀⠀⢘⠁⡠⠀⠠⡈⠑⠦⡁⠀⡀⠀⠀⠐⠣⡀
⠀⠀⠀⠀⠀⠆⠀⠸⡄⢣⠰⣄⡀⠀⠈⠂⠈⢆⢣⠈⠑⠂⠐⡌⢷⣀⣀⣀⣀⡜⠀⡇⢠⡀⠈⢆⢄⠘⠄⢰⠀⠀⢚⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠁⠀⠉⠑⠀⠀⠀⠂⠌⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠘⠈⠀⠀⠈⠀⠀⠘


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄⠰⠆⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣁⣠⣤⣈⣀⡀⠀⠀⠀⠀⠀⠈⢻⣷⣤⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⢰⣽⣿⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡄⠘⢿⣿⣿⣾⣿⣯⣸⣿⠀⢻⣷⡀⠀⠀⠀⠀⢠⣿⣿⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣠⣿⣿⡇⠀⠀⠀⢠⠀⠹⣿⣿⣿⣿⣧⡀⠰⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡟⠛⣭⣿⠿⠋⠀⠛⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⣿⣿⣿⡄
⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⡋⢶⡀⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠹⣿⣿⣿⣿⣿⣿⣷⡀
⣹⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠁⠀⠀⠀⠀⠀⠀⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠘⢻⣿⣿⣿⣿⣿⣷⡀
⣽⣿⣿⣿⣶⣶⣿⣿⣿⣦⣀⠀⠈⢻⣿⣧⠀⠀⠀⠀⠀⠀⠸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣴⣶⡟⢿⣿⣿⣿⣿⣿⣷⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⠡⡁⠺⣿⣿⣿⢿⣿⣿⡀⠉⠁
⣿⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠙⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣀⠘⣿⣿⣿⣿⣿⣿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢶⠄⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣀⡄⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣬⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣗⣺⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠛⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄
⣿⡿⢿⣂⣽⣿⣷⣤⣌⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⢀⣴⣤⣾⡟⠀⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠆
⣟⣤⣿⣿⣿⣿⣿⣿⣿⣿⣯⡙⣿⣿⣿⣹⣧⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⢀⣴⠟⣻⣿⠟⠀⡀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢦⡀
⣿⡿⡿⠟⢉⢨⣿⣿⣿⣿⡏⠉⠈⢿⣿⣿⣿⡆⠀⠀⠀⠀⠘⣿⣿⠟⢳⣾⡟⢋⣾⣿⡏⠘⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠘⠟
⠋⠀⠀⠀⠈⠙⠻⠟⠋⠁⠀⠀⠀⢸⣿⣿⠟⠛⣶⣦⣤⣶⡿⠛⠁⠀⠈⠁⢰⣿⣿⡿⠃⠀⠿⠳⣌⣿⣿⣿⣿⣿⣿⣿⢿⠟⠉
⣀⣤⣄⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣉⣤⣴⣾⣿⣿⡿⠏⠀⠀⠀⠀⠀⠉⠀⠀⠿⢁⠀⣤⡄⠀⣿⣿⣿⣿⢿⣿⣿⣿⣿⠀⣿
⠉⠙⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⠀⠀⠀⢀⠀⠐⠀⠀⢀⠀⠀⠀⠘⠛⠂⠈⠀⠀⢸⣿⣿⣿⣤⡿⢻⣿⡿⠐⣿⣦⣤
⠀⠀⠀⠀⠀⠉⣻⠋⠛⠻⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⣶⣤⠀⠻⠃⠀⣠⣼⡄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣯⡾⠿⣿⠉
⣷⡄⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⠀⠈⠀⠀⠀⠀⣰⣿⣿⠗⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠁⡀
⣿⣿⡀⠀⠀⠀⣿⣿⡄⠀⣾⣇⠀⠀⠀⠀⠀⠀⠀⢾⠿⠿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠛⠋⠀⠀⠀⠀⣶⠄
⣿⣿⡇⠀⠀⣼⣿⣿⣿⡄⢹⣿⠀⠀⠀⠀⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣿⣿⣷⠀⠀⠛⠛⠛⠛⠟⠀⠁⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠋⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡀⠀⢸⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟
⣷⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣾⡉⢻⣿⣿⠀⠀⠀⠀⠀⢀⣾⣿⣿⣇⠀⣀⡀
⣿⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⣿⣿⣿⡿⠆⠉⠁
⡏⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡆⠀⠀⣤⠀⢻⣿⣿⣿⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣷⠀⣰⣿⡄⢸⣿⣿⣿⣦⠀⠀⠀⠀⢀⣰⣿⣿⡆
⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠀⡀⠀⠀⠀⠀⠀⠈⠁
⣿⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣛⢿⣿⣿⡏⠀⠘⣿⣿⣿⣿⣿⣿⣿⠉⢹⣧⠀⠀⠇⠀⢀⣄
⣿⡇⢠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⠀⢸⣿⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁⠀⠀⠀⠀⠘⠁⣿⣿⣿⣿⡇⣾⠛⣿
⣿⣿⣿⣿⣿⣿⣿⠏⠸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠘⣿⣿⣿⠀⠉⢰⣿
⣿⠿⣿⣿⣿⣿⠃⠀⠀⠀⠉⠁⠀⢀⠀⣶⣄⣀⠀⠀⠀⠀⠈⣿⡟⠛⠀⠀⣼⣿
⣿⢀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣸⣧⣿⣿⣿⠀⠀⣀⠀⠀⠻⠇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠓

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠒⠛⠛⠒⠒⠒⡶⠦⣄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣭⣴⡶⣀⣤⡤⠀⠀⢀⣀⠀⣠⣽⡝⠣⠤⠤⠤⠔⠒⠋⣽⣿⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠽⠿⠋⠋⠉⠋⢁⣰⣿⣿⠯⢾⣿⠿⠛⠛⠀⠀⠀⠀⠀⠀⢀⣛⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣰⣿⣿⣿⡗⣲⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⣿⣄⢻⡿⠟⠿⡥⢠⢁⡤⢂⣠⣤⣤⠀⠀⠀⠀⣤⣶⠞⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⠹⣿⡇⣷⣾⣿⣧⣰⠏⣴⣿⣷⣶⣦⣤⣤⣶⣾⣿⣶⣷⣶⡄⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣿⣆⠀⠀⢀⡀⠛⠁⣿⢻⣿⠿⡏⢘⣻⡿⣇⡬⢘⣏⣻⣿⠟⣿⣿⠏⢨⣷⠀⢁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣦⠀⠨⡗⢀⣼⣿⣻⣦⣤⡀⠀⢀⣴⣮⣍⣛⣛⣛⠻⢿⣿⣿⣿⡿⠙⠷⠤⠇
⠀⠀⠀⠀⠀⠀⢠⣯⡿⣫⣿⣿⣿⢀⣰⣡⣿⡿⠟⠿⠉⠉⠀⠀⢾⢿⣿⠋⠀⠀⠀⠉⠉⠙⠋⠉
⠀⠀⠀⠀⠀⢀⣸⣿⣿⡋⢿⣿⣿⣿⣿⣿⣿⡗⠀⡄⠀⠀⠀⠠⠈⠀⠁
⠀⠀⠀⠀⡰⣿⣿⣿⣿⣿⣷⣯⣷⣙⣻⣿⡟⠀⠀⠀⠀⠀⠀⠀⡄⠆
⠀⠀⢀⡾⠀⣏⠟⠙⢹⣿⣿⣛⡻⠿⣿⣷⣾⣤⣌⡞⣠⡈⡔⠀⢀⡀
⠀⠀⢸⣅⣣⢣⡶⡀⠘⠐⢸⢸⡇⠀⠀⠈⠉⠛⠛⠇⡿⡿⣿⣿⣧⠁
⠀⠀⢸⠏⡠⠏⠠⠁⠀⠀⠀⠈⡹⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠳⡄
⠀⠀⠹⠀⠃⡇⣄⡄⠀⠀⠀⠀⠷⠈⣷⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠁⠁⠘⠇⠀⠀⠀⠀⠀⢀⣿⣸⡏⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠜⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⣿⣿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡺⠑⠤⢲⡞⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡖⠒⢤⣐⡊⠁⠠⠀⠀⠀⠉⢽⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⣤⣽⠁⠞⠀⠀⠀⠀⠀⢠⢜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠁⠝⠛⠛⠛⠋⠀⠀⠀⢠⠄⠐⠂⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠸⡅⡜⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢳⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠼⢶⠤⠔⠋⠙⠁⠀⣇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢀⡘⠀⠀⠈⠙⢦⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣏⣉⠉⠒⠒⣚⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠐⠣⠋⠉⠉⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⠀⠀⠀⠀⠘⠳⢤⡀⠀⠉⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠸⠞⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢄⠀⠀⣦⠀⠀⡆⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⠸⡄⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⢸⠀⠀⡇⣠⠖⠚⠛⠓⠲⢄⡠⠁⠹⢄⠀⠀⠀⢳⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠙⣇⠀⢻⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢳⡀⢠⠏⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠸⡄⢸⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠜⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣋⡄⣀⡬⠽⢧⣸⡆⢀⡵⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢧⡰⠂⠀⠀⢀⣾⠀⢀⠀⠀⣠⣀⡀⠀⠀⢀⣀⣀⣸⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠚⠉⠈⠙⠛⠒⠚⠁⠀⠉⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢛⣛⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⠟⢿⣿⣷⣦⣤⡴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣀⠀⠈⠓⠀⣉⠻⣍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⣤⢀⣤⣶⣶⣶⣿⣣⣾⣿⡧⠀⠀⢸⠃⠀⢹⣆
⠀⠀⠀⠀⠀⠀⣴⣘⠷⢎⠸⠋⠀⠀⠺⢯⣿⡿⣿⣿⣿⣿⠀⡄⠘⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠉⠉⣼⣿⡇⠀⠀⠀⠀⠀⠈⠙⢮⡝⢿⣿⣾⡇⠀⠀⡐⠰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⢰⠀⠀⠀⠀⠀⠈⠢⠙⠿⢁⣤⡴⠃⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⣀⣀⠜⢸⡇⠀⠀⠀⠀⠀⠀⢰⡷⠛⠉⠀⠀⢀⣴⡇⢘⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠁⣩⠟⠁⣀⣨⣧⠤⠔⠒⠉⢀⡔⠁⠀⠀⣠⠤⠞⠀⠠⣴⠺⠿⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠈⠉⠁⠀⣿⠀⠀⠀⢰⡇⣤⣤⠀⠀⠀⢰⠟⠂⠀⠻⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠘⣧⣹⣿⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠈⠉⢻⠀⠰⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⢿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠶⠿⠶⠆⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⢀⣾⣿⠏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⡠⠚⠛⠛⠒⠒⠒⠂⠀⢻⡀⠀⠀⠀⢀⣠⣴⠞⠋⠀⢠⠂⠀⢸⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣴⠾⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢰⠋⠉⡄⠀⠀⢰⠏⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⢰⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢸⡀⠀⠹⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⢸⣇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣼⣧⠀⠀⠹⣆⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⠿⠟⠀⠀⠀⢹⣇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣷⣦⡀⣀⠀⠀⠀⢀⣴⣾⡿⠋⠁⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣷⡀⢀⣾⣟⠉⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠸⠟⠋⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣫⣼⣇⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⢀⣿⣬⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣱⠃⠈⣿⠀⠀⠒⠲⠖⢺⡇⣿⠒⠶⠖⠂⠀⢸⡏⠀⢳⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⣠⣾⠋⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠈⢻⣦⡀⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⢹⡇⣿⠁⠀⠀⠀⠀⠀⠀⠻⣿⣷⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⢠⠀⠃⠃⢠⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡇⠀⠀⠀⠀⢠⠀⠀⠀⢸⢸⠀⢸⢸⠀⠀⠀⢀⠀⠀⠀⠀⠀⡯⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⢀⣤⣬⡳⣄⠠⣘⡿⡆⡾⣟⡠⢀⡴⣫⣤⣄⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡀⠀⠈⣧⣉⠛⠂⣹⠶⠉⠀⠈⠱⢾⡁⠚⢋⣡⡏⠀⠀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣇⠀⠀⠀⠀⠐⡉⣀⠀⠀⠀⠀⠀⢀⡈⡑⠀⠀⠀⠀⠀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⡀⠀⢀⡴⣫⠾⡱⠀⠀⠀⠀⠀⠱⡙⢮⡳⣄⠀⠀⣸⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢳⢣⡟⠀⠁⠸⣡⣤⣄⣀⣀⣤⣤⡹⠀⠁⠘⣧⢳⢳⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣸⠁⠀⠀⠀⢿⡀⢠⡀⣠⠀⣸⠇⠀⠀⠀⢹⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⢷⡚⠀⠘⣲⠇⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡘⠒⢤⢔⣥⢞⣋⣁⣉⣛⢦⣑⢤⠔⠚⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠛⠉⠉⠀⠀⠀⠈⠉⠙⠳⠶⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⢦⡀
⠀⠀⠀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⠀⢩⡀
⠀⠀⣼⣿⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠀⠀⠀⢸⡇
⠀⢠⣿⡇⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠀⠀⠀⠀⠀⣼⣿
⠀⣸⣿⣇⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⢀⡾⣿⣿⡄
⠀⣿⣿⣿⠀⠀⠀⠀⠀⠙⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧
⢰⣿⣿⠙⠧⠀⠀⠀⠀⠀⠈⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿
⠸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿
⠀⢻⣿⠙⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣄⠀⠀⠀⢀⣀⣀⣀⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣟⠉⠉⠉⠀⠀⠉⠙⢿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣿
⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠸⣿⣆⡀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⢀⣦⣼⣿⡏
⠀⠀⠙⣿⡯⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠙⣿⣿⣿
⠀⠀⢠⣯⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣾⣿⣿⡿
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣯⡻⢿⣿⣿⠀⠀⠀⠀⠀⢠⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠁
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣯⣶⢷⡖⣌⢧⣾⣿⠇⠀⠀⠀⠀⡀⢻⣾⣵⠻⣿⣷⣦⡀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⢠⣿⣿⣿⡿⠿⣿⡝⢮⣀⣙⣟⣸⣿⠋⠀⠀⠀⠀⠀⢻⣎⢿⣯⣛⣫⢞⡵⠋⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⡿⠟⠋⠀⠀⠙⢿⣶⣯⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠹⣆⠈⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇
⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⣿⢱⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠉⢉⣵⣿⣮⣙⠉⠁⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⣹⢧⡀⠀⠀⠀⠈⣷⡄⠀⠀⠀⠈⢉⣉⡉⠁⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⢀⣼⠏⣧
⠀⠀⠀⠀⠀⢠⣧⠈⢷⡄⠀⠀⠀⢹⣿⣷⣦⣤⠶⠾⣿⠷⠶⣦⣴⣶⣿⡿⠀⠀⠀⠀⢀⡾⠋⢀⣿⡆
⠀⠀⠀⠀⢠⣿⣿⣧⡀⢳⠀⢸⡀⠈⢻⣿⣿⡿⠀⠀⢿⠀⠀⢸⣿⣿⡿⠁⠀⡴⠀⠀⠸⠁⠀⣸⣿⣷
⠀⠀⠀⠀⣾⣿⣿⡿⠃⠀⠀⠀⠳⣄⠀⠈⠙⡇⠀⠀⠘⠀⠀⢸⡏⠉⠀⣠⡾⠁⠀⠀⠀⠀⢾⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠈⢳⡄⠀⣇⠀⠀⠀⠀⠀⢸⠃⢀⡾⠋⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⡇
⠀⠀⠀⠀⢿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠹⣆⡙⠷⠤⠤⠤⠶⠋⣠⡞⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣤⣀⡀⠀⠀⠀⠈⠛⠷⠶⠶⠶⠶⠟⠋⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣼⣯⣭⣽⣽⣦⣭⣯⣥⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣶⣶⣷⣾⣿⣶⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣽⣤⠎⠀⠀⠀⠀⠀⠀⣀⣤⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣵⣿⠀⠀⠀⢀⣰⣞⠉⠉⣀⣿⡑⠀⣀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⠴⠞⢩⣤⠽⠛⠋⠁⠈⣳⡞⠁⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⠻⡟⠉⣿⣿⡧⠞⣠⠶⠊⠁⠀⠀⠀⠀⢸⠛⣷⡐⠶⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣄⣉⠢⣜⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⢀⣿⠀⢳⡱⡄⢀⣀⣸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡶⠼⣿⣿⣿⣿⣿⣿⡋⠙⠻⠾⠿⠦⠤⠤⠤⠤⠤⢤⣤⣞⣉⣧⣀⣝⡿⢭⣁⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠒⠒⠒⣶⡟⠻⠿⢶⣿⣿⣿⢳⣿⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠛⢼⣏⠠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⣿⣿⣿⢸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣯⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢫⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⡽⠉⣏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⣰⠁⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣾⣯⣋⣞⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⢀⡀⠀⠀⢀⣠⡾⠁⠀⣿⡀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠘⠿⠛⠷⠟⠀⠀⠀⠹⣇⠀⠀⣤⠀⠀⠀⠀⠠⣤⣀⢿⡃
⣘⣏⠛⠋⠈⠁⠀⠀⣴⣿⡇⠀⠀⠀⠀⠀⠀⣿⣷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⣿⡀⠀⠀⠀⠀⠀⠙⢷⣽⣦
⠘⠧⣄⣠⠀⣠⣶⠊⠈⠀⢹⡄⠀⠀⠀⠀⣼⠛⠻⣿⡉⠛⢿⣿⠒⠶⠤⠤⠤⠶⠒⠛⠋⠉⠛⠓⠒⠓⢦⣄⣀⡀⠈⠉⠓⢦
⠀⠀⠙⠛⠛⠁⠁⠀⠀⠀⠀⣷⠀⠀⢀⡾⠁⠀⠀⠀⠻⡄⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡈⠉⣹⢶⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢠⡟⠀⠀⠀⠀⠀⠀⣷⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣩⡏⠀⠙⢁⡼⠻⣄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠶⠎⠀⠀⢹⡀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⡙⢺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣉⣙⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣉⢹⣆
⡶⠒⠲⡶⠶⡶⡶⡶⢶⡶⣶⡶⡶⣿⢿⠿⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠖⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⣀⣀⣀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣳⣿⣿⣿⣿⣷⡄⠀⢱⡜⠒⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠈⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡞⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠜⠸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⡀⠀⢀⠜
⠀⠀⠀⣇⠀⠀⠙⠒⠦⠤⣤⣀⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣷⠀⠙⠿⣿⣿⣿⠿⠓⠒⠒⠔⠚⠚⠁
⠀⠀⠀⠈⠓⠦⣀⣀⠀⢰⣿⣿⣿⣿⣿⣿⣯⠙⠻⠿⠿⠿⠋⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠙⠛⠁⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠔⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⡎⠀⠀⢀⡠⠊⡨⠋⠒⠒⠂⠤⠒⠒⡆⠀⠀⡔⡇
⠀⠀⠀⠀⠀⠀⡇⠀⢠⠊⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣷⠁
⠀⠀⠀⠀⠀⠀⢣⠀⠘⢤⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⢘⢾⣦
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠒⠂⢘⣛⠛⠻⠿⠛⠛⠃⠀⠉⠻⠟⠃⢀⡀⢀⣲⠶⠷⠤
⠀⠀⠾⣿⣾⣾⡶⣾⣿⡇⠀⠀⠀⠀⠀⠄⣵⣐⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣿⣻⣿⠀⣀⣤⣿⣿⠇
⠀⠀⠀⠻⠾⠛⠺⠶⠤⠼⢿⣿⣶⡄⠐⠶⡿⠿⠟⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢻⡟⠛⠛⠂⠙⢿⢻⣧⣴⣰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣈⣛⠲⠂⠀⠀⠀⠀⠀⠀⠀⠿⡻⠋⠈⣻⣿⠆⠀⢺⡆⠀⠀⠀⠀⠀⠀⠐⠂⠷⠖⠒⠀⠀⠈⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢰⡌⣏⠀⠀⢰⠇⠀⠀⢀⣤⣤⣄⡙⣦⡀⠋⠛⠄⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡦⠈⠁⠙⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⡈⠁⠓⠀⠀⠀⠀⠁⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⡀⠀⠛⠉⢱⣦⡀
⠀⠀⠐⠈⠀⠀⢀⣤⣀⢀⠀⠉⠉⠀⢀⣁⣈⣀⠒⠀⠀⠀⠀⠀⠘⣾⡏⢻⣿⡏⣀⠉⣵⡂⠠⠐⠊⠛⠀⠘⠻⣛⠟⠚⠂⠀⠉⠉⣄⣄⠀⢀⣔⠠⠆
⠀⠀⠀⠀⠀⠈⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠈⠁⢁⣘⡟⠈⠓⠐⠛⠉⠀⠀⠀⠀⠀⠁⠈⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⣁⣤⣤⣤⣤⣤⡴⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢉⣴⣟⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⣾⣿⣿⡿⣿⡿⠉
⠀⠀⠀⠀⠀⣀⠄⠒⠒⠉⢀⠀⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡠⠈⡀⣺⡟⠁⠉⠠⠔
⠀⠀⠀⠀⠠⣅⡀⠀⠄⠰⡟⢠⣿⣿⣿⣿⣷⣄⡈⠁⠈⠉⠁⠀⠀⠀⠛⢛⣿⣶⣶⠤⠔⠋⢀⣼⣿⡿⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⢠⣿⣿⣿⡿⡿⠋⢉⣀⣀⠀⠀⠀⠀⠀⢀⣻⣾⣿⠟⠁⠀⠀⠀⠀⢸⠋⠁⠀⠀⠀⣠⠃
⠀⠀⠀⠀⢀⡯⠶⠀⢀⡴⢏⠛⣿⠻⠃⠀⠀⣽⣿⣿⣿⣦⡀⠀⠀⠀⠛⠛⠁⠀⣀⣤⣤⣶⣧⣦⣀⣄⠀⠀⢀⡏
⠀⠀⠀⠀⠐⠄⠤⠖⠋⠀⠘⢦⠉⣱⣄⢀⣨⡛⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠶⢮⡻⣿⣿⡆⠀⠀⠀⠀⣿⣿⡿⣻⣭⣍⠻⢿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⠰⣾⠇⣿⣻⣿⠁⠀⠀⠀⠀⠻⢿⡟⢡⣶⠌⣷⣢⡿⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣶⣾⣿⠏⠁⠀⠀⠀⠀⠀⠀⠈⠐⠲⠥⠞⠋⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣾⣿⣿⣿⣿⣿⠋⠁⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣾⣿⣿⣿⠟⠁⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⠻⢿⣟⡀⠀⠀⢀⠀⠀⠛⣿⢿⣿⡿⠿⠋⠀⠀⠀⠀⣰⡟⠋⡀⣠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⠻⢿⢷⣌⠻⣷⠀⠄⣈⡁⣂⠹⡟⠛⠏⢒⡀⠀⠁⠀⠀⣿⣶⡿⠏⠻⠉⠀⢄
⠀⠀⠀⠀⠀⠀⢠⠖⠛⠻⣦⣀⠀⠀⠀⣿⡆⠈⠀⠒⠀⠀⠶⣂⣿⣦⣐⠒⠰⠄⠆⠀⢠⡿⠋⠀⠀⠀⠀⠀⣸⢆
⠀⠀⠀⠀⠀⠀⢻⣾⣿⡦⠈⠙⠷⣄⡀⠈⠛⠦⣀⡀⣫⣤⠾⠟⠋⠙⠟⠻⣦⣍⣈⣵⠯⠁⠀⠀⠀⣀⣤⠞⠁⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠙⣧⣤⣶⣦⠀⠀⠙⠓⠶⣤⣄⣉⣉⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣁⣤⣤⣶⠞⠋⠉⠉⠀⠰⢛⡿
⠀⠀⠀⠀⠀⠀⠀⢰⡄⠙⠿⣿⣇⠀⠘⢁⣤⣌⠉⠉⠛⠛⠛⠳⠶⠶⠚⠛⠛⠛⠉⠋⠀⠀⠀⠀⠀⢸⣿⣿⡶⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠉⠛⠿⣿⣿⣿⣦⠀⠲⠶⠃⣠⣶⣦⣄⠠⠒⠛⣠⣾⣷⣦⣀⣠⣤⠾⠛⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠈⠉⠛⠛⠶⠶⠿⠿⠿⠿⠿⠶⠶⠾⠿⠛⠛⠋⠉⠁⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣄⠀⡤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠘⣶⡟⣛⣛⡛⣳⣤⠏⣸⣶⡾⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣉⠺⠥⠚⠛⡒⠒⠤⢭⠶⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣆⡀⢿⣿⣿⣿⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁
⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⣿⡿⣉⣿⠿⠿⠿⠿⠿⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣴⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠉⢻⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠃⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡻⢿⡏⠀⠈⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠋⠰⣤⣄⠀⣴⠞⠋⠉⠉⠉⠉⠛⠓⠒⠚⠛⠋⠉⠈⠉⢻⣦⡃⠤⠶⡾⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠘⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠈⢻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⡠⠖⠛⠻⢷⣦⠀⠀⠀⠀⠀⠀⠒⠓⠶⣤⣼⣿⣇⠀⠀⠀⠀⠹⣿⣿⠃
⠀⠀⠀⠀⠰⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣶⠿⣶⡈⣿⠀⠀⠀⠀⠀⣰⡾⢷⣦⡈⢿⣿⣿⡀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⠿⠛⠁⠁⠀⠀⠀⠀⠀⠙⠻⠿⠊⠁⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠆⠀⠀⠀⠀⣠⣤⣴⣦⣤⣄⠈⠙⠓⠲⢶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠘⢿⢙⣿⣿⡏⡿⠂⠀⠀⠀⠀⠀⠙⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⣶⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣀⣀⣀⣠⣴⣿⣿⣷⣤⣄⣀⢠⠴⣶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠛⠛⠛⠋⣹⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣀⣀⣀⣴⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⣤⢴⣦⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠿⠋⠉⠉⠈⠁⠈⠁⠀⠈⠻⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⣐⣠⠼⠷⠀⠈⠑⠢⡀⠀⠀⠀⠀⡠⠐⠒⠒⢲⣦⣤⣀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⣴⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣄⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⢀⣿⡥⠀⠀⠀⣀⡤⣤⣤⣤⡀⠀⠀⠀⣠⡶⢶⣶⣤⣀⠀⠀⠙⣿⡄⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠘⣿⣡⠀⢀⠾⣿⣷⣾⣿⣿⣇⢄⠠⣤⣿⣷⣾⣿⣿⡿⢳⢀⣽⢿⡇⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠛⣿⠟⠋⠀⢽⡟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠈⠛⢻⡥⠈⠉⠉⣤⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⡧⠀⠀⠰⠋⠀⠀⠀⣀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠙⠂⠀⣸⠁⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⢷⣤⡀⠀⠀⠀⠀⢰⣿⣿⣥⣴⣾⣿⣿⠀⠀⠀⠀⠀⢼⡏⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠈⣇⠀⠀⠰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⣛⠋⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⣶⡿⠻⣷⡀⠀⡀⠀⠀⠀⡀⣠⠞⠙⢦⣀⡀⠀⠀⠀⢀⠀⢠⣷⣤⣠⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣘⣶⡖⠚⠉⠀⠀⠀⠀⠁⠙⢳⣤⠤⠮⠤⠞⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠤⡤⠤⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⣴⣿⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠁⠈⠉⢁⣘⣿⣿⣿⡟⠛⠀⠀⠀⠀⢀⡀⢀⣾⡿⠛⠃⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣼⣿⣿⣦⠀⠀⢻⣿⣿⣿⣿⡇⢸⡀⠀⣠⣤⠘⠓⢸⣿⣶⣿⡶⠆⠁⡀
⠀⠀⠀⠀⡠⠐⠀⠈⠀⠀⠀⠀⢀⡀⠀⣀⡀⢀⣿⣿⣿⣿⣄⣀⣀⣿⣿⣿⣿⣧⣾⣄⠀⠙⣿⣷⣤⣤⣿⣿⣷⣶⣶⣗⣀⠄
⠀⠀⢰⣏⠀⣾⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⢀⣈⣻⣿⣿⣿⡟⣿⣥⣶⣿⣿⣿⣿⣿⣥⣤⣀⣿⣿⢰⠀⣿⣿⣯⣯⠰⢀⣴⣲⠖⡃
⠀⠸⠿⠯⣭⠄⠶⢷⡨⠉⠀⠀⠀⠀⠉⠉⠀⠉⢙⡛⠛⠛⠋⡛⡛⠋⠉⠙⠋⠉⠙⠋⠛⢛⠉⡉⠀⠀⠉⠉⠙⢃⠀⠁⠀⠀⠀⠠⠀⣀
⠀⠀⠀⢀⣂⡀⢁⣀⠀⠠⣀⢀⢠⣠⠄⡁⣄⡀⡡⣀⣄⠤⠀⠀⠀⠀⢠⠀⢀⠀⠀⡈⡀⠄⠄⢀⠀⠀⠀⠠⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⡈
⠀⠀⠘⡧⣥⠙⢠⠶⠁⢹⣇⡛⡒⣍⣳⡇⢠⣰⡗⢁⡺⣔⡧⠀⢰⠉⠋⢃⠛⣀⠇⢀⣴⢏⠉⡻⠰⢏⡔⢩⣬⣧⣘⡀⠁⢐⡄⢷⡃⢂⣀
⠀⠀⠠⣿⢸⢡⣞⠞⢠⡈⢻⡄⣰⣍⡏⢸⠊⢹⡇⠪⡽⢙⡿⣶⡆⠙⢰⢢⠟⠟⢠⠟⢙⣶⣋⡁⠜⣿⣌⡹⠄⢰⡴⣄⢠⢼⠁⢻⠿⠉⡉
⠀⠀⠐⣿⢸⡷⣻⠨⣄⣽⢿⡀⣷⣌⡷⠲⡗⢸⣧⣸⡵⠌⣃⢤⣹⡣⡆⣼⢖⠃⢰⡖⠁⠹⢘⢹⣇⣉⣮⠁⢈⡑⣚⣉⠙⣀⢠⡼⠳⢒
⠀⠀⠤⣯⣸⡇⣻⢰⡇⠀⣿⠂⠻⢮⣧⣈⣟⢼⡅⣽⣟⠈⡗⣘⢩⡅⠜⣩⣻⡇⢸⡇⠾⢧⣄⢨⣿⣡⢯⡿⠸⡷⣩⣯⡵⠈⠟⠶⠃⡤
⠀⠀⠠⠿⢤⣀⣹⡈⣁⠃⣿⠐⠶⢓⡷⠼⠇⢠⡇⣺⣁⣘⣇⠐⢱⣍⡎⠉⢹⡃⢜⡃⠶⡫⠅⢼⣯⠘⢡⣿⣸⡟⣩⣟⡛⢘⡟⠸⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠒⠚⠓⠦⠤⢤⣧⣤⣄⣘⡇⠈⠁⢸⣟⠀⠉⠛⢘⡱⠈⠥⠀⢸⣟⠀⣀⣷⣸⡏⣰⠏⠋⢸⡟⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠓⠶⠦⠤⢼⣅⡀⣀⠀⠰⡿⠀⠀⠁⢸⡏⠁⠑⠒⢺⠇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠚⠓⠲⠰⠦⠬⠧⣀⡀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡭⠄⣀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⠀⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠐⠃⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⠳⠂⢠⡀
⠀⠀⠀⠀⠀⠀⠗⢆⠘⠆⢣⣀⠤⣄⣀⠀⠀⠀⡆⡌⠑⠦⠀⠀⡠⠀⠀⠃⠠⡄
⠀⠀⠀⠀⠀⠀⣰⠋⠉⠑⣰⡌⠆⠀⠈⠉⠓⠦⢅⠁⠀⠠⠒⠪⠴⠒⠁⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⡇⢀⡔⠉⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣉⠀⣤⠀⠀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠈⢃⡞⠈⠁⢸⠇⢀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⢄⠘⣆⠀⠘⠆⠀⡀⡀⠠⣄⠀⠀⠀⠀⠀⠀⠀⡠⢞⡭⢭⣉⡓⠲⢖⡂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠀⣸⣧⠔⠋⡁⠀⠀⠀⠀⠀⠀⠘⠦⠘⢦⠀⠀⠐⠁⠸⡔⢬⣀⠑⠒⠒⠶⢴⣋⠐⠁⠐⠒⠦⢭⣉⣩⠴⠂
⠀⠀⠀⠀⠀⡞⠀⠉⢉⠀⠀⠚⠉⠈⢧⠀⠙⠄⠀⢀⣠⠴⠂⠀⠀⠀⠈⠳⣄⠈⢦⡀⢻⡀⠈⣳⠀⠠⡀⠀⠈⠳⣄
⠀⠀⠀⠀⢸⡇⠺⢀⡝⠊⠀⠀⠀⠀⢸⡆⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠱⠀⢏⠉⠁⠀⠀⢹⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠁⠀⠘⠃⠀⠀⠀⠀⠀⠈⠋⠉⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡄⠀⣀⠘⢦⡀⠀⠀⠀⠇⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠁⠤⠤⠤⠄⠀⠀⠀⠀⠀⣸⢠⠇⠀⢹⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⠉⠀⠐⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡾⠀⠀⠀⠁⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⢠⣀⠀⢳⠀⡇⠀⠈⠉⠁⠀⠀⢀⠞⠀⡇⠀⢀⣀⡤⠴⠚⠁⠀⠀⠳⡀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠻⡁⠸⠀⠇⠑⠲⠤⠤⢀⡴⠟⠒⠀⠀⠉⠉⢠⣀⣀⣀⣀⣀⡤⠤⠌⠓⠒⠀⠀⢠⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠈⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡄⢰⠆⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄⠈⠀⢈⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠸⡄⠀⠀⣧⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⣈⠀⠠⢿⣀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠿⠿⠿⢿⣶⣤⡀⠀⠀⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄⠀⣠⡞⠉⠉⠛⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣶⣄⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠀⢰⡄⠀⠀⠈⠙⢦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡾⠋⠉⢉⠉⠛⢷⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠈⠉⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣾⠿⠛⠉⠁⠀⠀⠀⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⡙⠻⣿⣆⠀⣿⣷⡄⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⣰⡿⣧⡉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠹⡆⠘⣿⡀⢹⣿⣿⣄⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⣿⠁⠈⣿⡄⠘⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣸⣷⠀⣿⡇⠀⠙⣿⠿⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⢸⠇⠀⠘⣿⣷⣤⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⢀⣿⠁⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⣿⡀⠀⠀⢿⣿⣿⡿⠁⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⣤⠾⢃⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀⡾⠿⣷⣤⡀⠈⠻⣿⣤⡾⠃⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠙⢧⣴⠶⠒⠲⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠙⢿⣷⣤⣠⡄⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⠿⠿⢿⣷⠀⠀⠀⠀⢘⡏⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠉⠑⠛⠻⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⡀⠀⢀⣰⣿⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⠀⠀⠀⠙⢿⣿⠿⠿⠿⠟⠛⠁⠀⣠⣴⣿⠏⠀⠀⠀⠀⠠⠾⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠛⠿⣶⣤⣤⣄⣠⣤⣤⣴⠾⠿⠗⠶⠾⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠀⠀⠀⠀⠹⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣾⠿⠿⠿⠿⣷⣄⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⠿⢿⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣶⣶⠞
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⣤⣤⣶⣿⣿⡇⠀⠀⢀⡇⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⢠⡇⣆⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠏⢸⠀⠀⠀⢸⣧⠀⠘⣿
⠀⠀⠀⠀⠈⣇⠘⢦⡀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠀⠀⡿⠀⠀⠀⢸⣿⠀⠀⠻⣧⣤⡀
⠀⠀⠀⠀⠀⠹⣄⠀⠙⠶⣤⣤⣿⣿⣧⠀⠀⠀⠀⠀⣸⣿⠿⠟⠛⠛⢦⡇⠀⠀⠀⠘⠿⠛⢦⡀⣄⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣀⣀⣉⣉⣁⣀⢀⠀⠀⢀⣉⣄⡀⠀⣾⣇⠘⣧⡀⠀⠀⣤⠀⢰⠘⣧⣼⣦⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⢿⣿⡿⠿⠛⠙⠛⠿⠿⠟⠻⠿⠛⠿⠿⠿⠿⠷⠟⠻⠟⠀⠈⠉


⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠤⠀⠀⠉⠙⠂⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠠⠤⢄⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⣀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣖
⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡖
⠀⠀⠀⠀⣸⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⣿⣿⣿⡀⠀⠀⢀
⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣠⡐⠀⠀⠀⠀⠀⠀⠀⠰⡉⣝⣾⣿⣿⡟⠀⠀⠀⠀⠹⢶⣤
⠀⠀⠠⠙⠉⢁⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣥⠀⠀⠀⠀⠀⠀⠀⠈⢡⣍⣿⣿⣦⡉⠀⠀⠀⠀⠀⢬⡛
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⣤⣤⣤⡀⠀⠀⠀⢉⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣲⣿⡻⢿⡄⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠠⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣶⣀⠀⠀⠀⠿⠿⣿⣷⣄⢀⡄⠀⣠⣷⡿⠛⠁⢀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠘⠻⣾⣧⠀⠀⠀⠀⠀⠀⠀⢘⡿⢎⠻⢿⢿⡍⠁⠐⠀⠀⠀⠈⠈⣿⣿⣿⣿⣷⣿⠿⠁⠀⠢⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠈⢿⠃⠀⠀⠀⠀⠀⠀⠈⠁⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣙⠋⠿⠛⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠐⢚⣛⡿⠛⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⠀⠀⢁⠀⣠⢶⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⠛⠁⠀⠀⠀⠀⠀⠀⠷⠙⠁⠀⢀⠀⢂⠠⡟⢘⠥
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⢀⠀⣾⣿⡇⠐
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡃⢀⣤⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡌⣰⡧⠻⣿⢳⠀⡄
⠀⠀⠀⡀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⣷⣾⣿⣿⡏⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣧⢹⠃⣰⣿⣿
⠀⠀⢰⣽⣿⡿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡏⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣼⣿⠆⠀⣽⣿⣿
⠀⢰⣿⣿⣿⣷⣿⣄⣴⠀⠀⠄⣀⡀⢀⣀⣠⠄⠁⢻⣿⣿⣿⡅⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢠⣯⡿⢻⣆⢸⣿⢿
⠀⠈⠿⢿⣿⣿⣿⣿⣿⣦⣶⣦⣾⠾⠋⠉⠁⠀⠀⠼⣻⣿⢳⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠃⢠⡟⠿⡄⢈
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣦⡿⠉⠛⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⣅⣾⣾⡇⣸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠟⠃⠆⠀⠀⠱⣦⣾⡀⠀⠀⠀⠀⠀⢤⡆⠀⡇⣷⠀⣿⣿⣿⡿⢀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣣⡄⠴⠿⢽⢿⣃⠉⠀⠀⠀⡀⣰⡇⢄⢵⠏⢸⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠰⠻⢝⡧⢀⣜⡎⠰⠟⢉⠽⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠁⠀⡐⠀⠀⠈⠑⠋⠡⠁⠊⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⡁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡥⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣰⣿⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠚⠉⠉⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⢄⡀⠀⣠⡀⠀⠀⠀⠀⠀⡠⠤⣄⡸⢠⣤⣶⣦⠀⠀⠀⠘⠲⣄⣀⣀⣀⠀⠀⠀⢀⠔⣒⡠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠉⠁⠀⢀⣤⣍⠐⢿⠾⠀⠀⠀⢀⡜⣠⣤⣀⣴⣾⣿⣿⣿⣿⣦⡀⠀⣰⣬⣴⣦⣭⣵⣴⠂⣩⣾⣿⠃⠀⠈⢉⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠔⢀⣀⠀⠀⣴⣦⣉⣭⣰⣾⠷⠀⠈⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡿⢿⣿⣧⣼⡤⡄
⠀⡀⢀⠤⡄⢠⢼⠲⠆⢰⣿⣶⣿⡿⢿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠉⣛⣿⣿
⢀⡇⣤⣀⠈⣁⠈⠳⠿⠟⠫⣿⠀⡷⢯⣶⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣧
⢺⠀⠈⣩⣾⣿⣷⣶⣶⣶⣶⣾⣾⣷⣾⠟⢿⣿⣿⣿⠛⠻⠽⠟⠛⠛⠯⠿⠉⠉⠉⠛⠀⠀⠀⠘⠋⠀⠀⠀⠀⢹⠋⢻⡏⠙⡏⠈⠉⣿⣿⡿⠃⣿⣭
⠳⢄⣶⣿⣿⣿⣇⣿⣶⣶⣶⣿⣿⣿⣿⡀⠀⡁⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠘⠀⠈⡇⠀⡄⠀⢀⣿⣯⣶⣿⣿⣿
⠀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣭⣍⠉⠉⠉⢉⣉⠉⠉⠉⣉⠉⢉⣀⣸⠇⣈⠁⢀⣀⣤⣶⡟⠀⠙⢷⣶⣶⣶⣿⣿⣯⣭⣭⣭⣭⣭⣭⣭⣤
⠀⠀⢀⠀⡄⠀⣀⣠⣴⣿⣿⣿⣿⣿⣭⣭⣭⠤⢄⣠⣠⠠⠌⠉⢉⡙⢻⣿⣿⠁⢠⠈⠿⢹⣿⣿⣿⡇⠀⠀⠈⠛⠛⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿
⠈⠛⠙⠋⠘⢛⠋⠻⠛⠛⠭⠍⠉⠉⠭⠭⠭⢍⣉⣭⣏⣁⣮⣉⣉⣉⣩⣭⡅⠀⢀⡰⠀⠀⡄⠁⠀⠀⠀⠉⠀⠈⠁⠀⠀⠉⠓⠀⠘⠄⠙⣿⣿⣷⣿
⢈⣭⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⣷⣿⡟⠿⢿⠟⡿⠟⢉⡉⢻⣿⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⣸⠀⣿⣿⣿⣿
⢸⣯⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠺⡿⢷⣤⠄⡤⠀⠉⠑⢯⣿⡷⣻⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠸⠟⢠⣿⡆⢸⣿⣵⣿
⢸⣶⡟⣻⣿⡇⠀⠀⠂⢀⣤⣤⣤⣤⡌⢀⡂⣠⡀⠀⠀⠐⠻⣽⣿⠆⣐⣤⣾⣾⣯⣾⣿⣿⡶⠀⣰⣶⣶⢶⣶⢶⣧⣀⣀⣦⢰⠀⠸⣸⣜⡎⢿⡿⣿
⢸⣿⣯⣿⣿⣿⢠⢷⡒⣿⣛⢿⣷⡿⡇⣸⣿⣿⡿⢷⣀⠈⠀⣿⣿⠏⠙⣿⣿⡿⠛⠋⠉⠛⠓⠒⠛⠛⠛⠋⡉⢻⡞⢁⣿⣿⢸⠇⣰⣿⣿⡇⠸⣻⣿
⠀⣩⡽⠛⣻⣿⣼⣿⣷⣿⣿⣿⣿⣿⣇⣘⣿⣿⣷⣦⣾⣷⣤⣾⣿⣶⡄⠾⣻⣧⠀⠀⣐⠀⠀⠀⠀⠀⢀⣤⡉⢨⣵⣾⣿⣧⣾⣾⣿⣿⣿⣷⣾⣿⡿
⢠⠿⣏⣱⣿⣿⣿⣿⣿⡾⣫⣟⡩⢻⣿⣟⣿⡿⣿⣿⢯⣻⣏⠛⠛⠛⡇⣿⣿⣿⠀⣴⠉⡶⣶⡶⠀⢰⣿⣿⣿⣿⡟⢻⢟⣿⢿⢿⣿⢃⣭⣿⣿⡷⠚
⠀⠀⣿⡏⣽⣿⠏⢋⣿⣷⣿⣟⣠⣎⡍⠉⠛⢷⣶⣽⣿⡿⣿⣤⣶⣄⠁⠙⣨⣿⡈⣿⣆⢱⣿⣇⢶⡤⣿⣽⡿⢿⣷⠿⣸⣧⣿⣿⣿⣷⡿⠟⠋
⠀⠀⠟⠳⡏⠉⠀⠈⢡⠿⢛⣽⣛⣁⠀⠐⠂⠀⠹⣟⢻⡿⣶⣿⣯⣛⣷⡶⢽⣿⡧⢸⣿⣤⣿⣿⣼⣧⣸⣿⡶⢿⣶⣿⡯⣷⠾⠻⠟⠁⠾⠿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠧⢬⣿⣿⢿⠋⠀⠈⠀⠈⠹⠿⠟⣿⣿⣽⣿⣿⡿⠃⠼⣿⣿⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣾⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⡀⠀⠀⠀⠀⢠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣂⣀⠂⢉⠀⠀⠈⠉⠛⢿⡖⠁⠀⠀⠀⠉⠣⠁⠐⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡾⢁⠈⡩⠉⠀⠀⠀⠀⠀⠀⢀⠀⠙⠀⡶⠀⡀⠀⠀⠀⠀⡄⡌⢧⡀
⠀⠀⠀⠀⠀⠀⣠⡞⠁⢈⣡⡏⢀⠁⠀⠀⠀⡀⠀⢀⣠⢀⡇⠀⠀⠐⢠⢀⣤⠀⡀⠀⢸⣾⠸⠙⠳⣄
⠀⠀⠀⠀⠀⣼⢹⣿⠀⣿⣿⡇⢸⠀⠀⢀⡾⣿⣶⣿⣧⢿⠀⠀⢀⠀⠀⠻⣿⣽⠦⠄⠈⣿⡀⠀⠀⠻⢷
⠀⠀⠀⠀⠀⠸⣆⣿⠀⠘⣿⡇⢸⡆⠈⠀⠀⠻⠏⣹⡿⠀⠀⠀⢸⡆⠀⠐⠙⢂⣀⠄⠀⣿⠁⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⢹⣷⢸⡗⠀⠘⢦⣄⠀⢾⠇⢀⣠⠴⠚⠋⢀⣀⠰⣟⠃⠀⠀⣿⡆⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠸⣿⢸⡇⠀⠀⠀⠟⠀⠘⢠⠏⠀⢠⢿⡛⠀⢹⡷⠘⡇⠀⠀⣿⣇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⢏⣾⣇⠀⠀⠀⠀⠀⠀⠊⠀⠀⠈⢻⣡⣾⣮⡟⠀⠀⣀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡟⣿⣿⠀⠞⠰⣷⡀⠀⠀⠀⠀⡂⣾⣿⠏⣷⡤⠄⢰⡞⠀⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠸⡿⣆⠀⠀⠻⣷⣄⠀⠤⠄⠶⢾⡟⣾⡿⣧⣡⡼⡄⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⢃⠙⢦⡀⠀⠸⣿⣷⣶⡶⠊⡍⠙⠉⠀⢹⣩⠜⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠈⡆⠀⠈⠳⣄⢀⡁⠙⢷⣷⣍⣒⣀⣀⣠⡏⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠘⡀⠀⠀⢀⠀⠉⠓⠤⣤⣬⡭⠀⠐⠉⠀⠀⠀⢹⡟⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢻⣿⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢀⣼⣿⡄⠀⠀⠀⠀⠀⡀⢸⠇⢸⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢸⣿⠀⠀⠀⠀⠀⠘⣴⢀⠀⠀⠈⠀⠻⣿⠂⠀⠀⠀⠀⠀⠈⠀⠘⢀⣳
⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⢻⣤⡆⠀⠀⠀⠀⣿⣿⢀⡄⠀⠀⠀⠈⠂⠀⠀⠀⠀⢴⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣼⠀⠀⠀⠙⠻⡄⠀⠀⠀⠀⠘⢿⢷⣧⣠⡞⠷⣤⡤⠤⡀⠀⢪⡇⠀⠀⣸⠘⡇
⠀⠀⠀⠀⠀⠀⠀⣾⢻⣿⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠈⠁⠀⠀⠀⠈⠆⠀⠐⠁⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠉⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢦⡀⣀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠙⠷⣾⣦⣐⣦⡀⠀⠀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠛⠛⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠄⡀⣀⡘⡀⠙⠀⠀⠀⠀⠈⠀⠁⠈⠀⢀
⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢠⡤⠤⠖⠀⠑⠀⠀⠀⠀⠩⠠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠱⠂⠓⠀⠀⠀⠂⠀⠑⠄
⠀⠀⠀⢀⠔⠀⠘⠀⠀⠀⠀⠈⡁⠁⠠⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠂⠀⠀⢠⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠛⣦⡀⠀⣶⡄⠀⠁⠐⠠⠀⠀⠀⠄⠀⡀⠐⠀⠀⠀⠂⠉
⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⠉⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠙⢻⣤⣀⡫⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠠⠠⠀⣀⠄⠀⠀⠀⠀⠄⠂⢸⡄⡖⢲⡔⣶⢒⡶⠶⡖⡞⢳⠀⠀⠀⠀⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⠀⢠⠀⡼⠉⢿⠾⣇⠹⠀⠀⠀⠃⠁⠸⢀⠀⠈⠀⠀⠀⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢀⡤⠂⠃⠀⡓⠊⡕⡠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡎⠈⢀⠠⠀⠙⢻⡟⠁⢛⠃⢰⠋⠁⣴⢟⢎⡇⠀⠁⢤⢰⢢⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠂⠈⣀⣸⡇⠀⢿⠰⡿⠦⡀⠻⣍⣉⠄⠀⠀⢀⢟
⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⢠⠀⠀⣿⣉⡽⠉⠒⠢⠤⣧⠀⠤⠀⠀⠒⠀⢰⠄⠈⡙⡀⠀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠸⠀⠀⠈⠉⠀⠀⠁⣆⢄⡇⠀⠀⡀⠈⠀⠀⢐⠣⠈⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⢈⡳⠀⠀⠀⠰⡄⠁⠈⠀⠘⠁⠀⡀⠀⠊⠁⠀⠂⠁⠈⠀⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠆⠊⠀⠀⠀⠀⠒⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣉⡀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⡀⠀⠘⣿⣀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣶⣛⣫⣥⣤⣶⡾⣡⣤⠤⠀⠁⠈⠀⢀⠀⠀⠀⠀⢀⡿⢹⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣤⠀⠘⠡⠭⠭⠵⠶⠶⠦⢤⣴⣟⣿⣶⣤⡀⢀⡴⠋⠀⠀⢀⣀⠉⡰⠉⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠿⢿⣿⣿⣿⡇
⠀⠀⠀⠶⠄⣸⠷⠂⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠙⠻⠷⢏⠀⠀⠀⠀⠀⢠⠞⠀⠀⡴⠁⣲⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣶⣿⠟⠛⠉⠀⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⢠⡿⠇⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⢶⠻⠿⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠙⠳⡄⠀⠀⠀⠀⣿⣿⠁⠳⡀
⠀⠀⠀⢀⣿⡇⠀⠀⠀⣠⡆⠐⢲⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡄⠣⣴⣶⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢸⣿⡟
⠀⠀⠀⡞⢿⣦⣤⣴⡾⠛⣠⡴⠋⠀⠙⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠙⢦⣈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⠀⠀⣻⠀⠀⠈⠙⣧
⠀⠀⣼⣇⣀⣼⢛⡥⠖⠋⠁⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⠬⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠘
⠀⠀⠙⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⣠⠄⠀⠀⢀⣤⣾⡇⠀⠀⠀⠀⠀⠀⠘⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⣂⣠⠴⠖⠋⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠛⠋⠁⣀⣤⠶⠛⠉⠀⠉⠀⠀⠀⠀⠀⢀⣀⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣰⣾⣿⣿⣤⣤⣤⣤⣤⣤⡶⠀⠀⠀⢀⣀⠤⠤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⢹⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢠⣾⡇⠀⢐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠋⠉⠈⠉⠛⠛⠟⠛⠙⣿⣿⡄⠀⢀⠀⠀⠀⠀⠀⡘⠁⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠁⠢⣶⣾⣿⣿⣿⣦⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡖⠶⠀⠀⡀⠈⠋⠀⠀⠀⡤⠀⠀⠀⣠⠀⠀⢋⠁⢂⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⡩⠁⠈⠀⡄⠀⠀⠀⡀⠀⢀⡀⠈⠀⠀⠋⣰⣿⣷⣄⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⢄⠀⠀⠘⢄⠀⠀⢻⣤⣤⡔⠀⢀⠌⠀⢻⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠁⠂⢠⠀⠑⣄⠀⠉⢉⠠⠐⠁⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠬⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢤⠀⠀⠀⠀⠀⢀⠈⡄⠀⢀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠿⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠘⣠⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣤⣤⣤⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠂⣀⣻⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠟⠛⠛⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣨⠀⠀⠀⠀⠀⠀⠀⠄⠻⣿⣯⣟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠁⠀⠀⠠⠀⠀⠰⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠔⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢶⣾⣿⠉⠻⢿⣿⣶⣤⣤⣀⣠⣤⣴⣿⡟⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠋⠉⠁⣄⡈⠂⠀⠀⠤⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣤⣤⣤⣶⣿⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠁⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢿⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠄⠄⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠄⠒⠀⠉⠁⠒⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠹⣿⣧⠀⠀⣀⠠⢆⠀⠀⠀⠀⠀⠰⠂⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠁⠐⠛⠛⠃⠉⠀⠀⠀⠱⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⣄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⡰⠀⠀⠀⠀⠠⠂⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣾⡄
⠘⢿⣦⣤⣄⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⡏⠀⠊⠀⠀⠀⠀⠐⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠑⢄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⣀⣱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣤⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡷⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣤⣴⣤⢖⡖⠒⠲⣶⠟⠯⡘⢏⠱⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⢉⠉⢭⢿⡟⠐⠋⠀⣤⢓⠄⠘⠣⣄⡀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⣠⢔⡀⠀⠀⠀⡌⠁⣸⣾⠁⡀⣄⠁⡂⠀⡾⢷⣾⣤⠁⠀⠀⠌⠓⠦⡀
⠀⠀⠀⠀⠀⠀⣥⠀⡉⢓⡐⣆⠁⠀⢻⣻⢀⠅⢘⠎⢐⣽⣷⣦⢤⣄⡀⠱⣄⠀⢳⣺⡇
⠀⠀⠀⣠⣶⡃⢿⣿⣷⣻⡠⠁⠀⠀⠀⢹⣷⠣⣨⡏⣈⣿⣿⣿⡟⠛⠓⠀⠈⢦⡀⠫⣇
⠀⠀⣴⡿⢧⣔⠠⠹⢻⣿⡷⢀⠀⠀⠀⠀⢻⣷⣏⠑⠈⠉⠿⠣⠴⠾⣢⠈⠠⣀⣈⢀⠙⢦⡀
⠀⠀⣿⣿⣦⠈⡁⠚⠆⣍⠗⠀⠀⠀⠀⠀⢸⣮⠳⠤⠀⠀⠀⠀⠀⠰⠋⡞⣠⡐⢊⣥⢻⣦⣍⠢⡀
⠀⠀⠈⣿⣿⡄⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢻⡄⠱⠄⠈⠳⠀⠀⠀⠀⠀⠀⠀⠦⣜⠛⠒⠪⡕⢌⠢⡀
⠀⠀⠀⠸⣿⣿⣶⣟⠠⡄⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠁⣀⡄⠀⠂⠂⠀⠀⠀⠀⠀⠘⠢⡝⠒⠘⠻⠷⢼⣆
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⢰⠵⡆⢹⣿⠚⠂⠂⠀⠀⠀⠠⣿⣿⠆⠀⠀⠂⠉⣎⣶⣉⡡⠻⢿⣷
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣶⣶⣦⣶⣿⣿⡗⣸⢿⣿⣶⣦⣀⣐⣄⡀⠀⠃⠀⠀⢰⢀⡀⠻⣟⠿⣟⣹⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣻⣿⣿⣿⠿⣿⡟⣠⡿⠀⣀⣼⣿⣿⣭⣉⣛⣻⣦⣉⡘⡓⢾⣍⣷⣤⡿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣝⡙⢛⣿⠷⡿⠛⠉⠐⠉⢉⢤⣟⣿⣿⣯⣿⣿⢿⣏⣻⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣴⣯⠀⠉⠀⠠⢄⡴⢟⣵⣿⣿⠟⡿⠁⠉⠉⠀⠍⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⢿⣿⣿⣃⣖⣒⣛⣍⣾⣿⣿⠟⢃⠀⠀⠀⠔⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠘⠁⢉⣿⣿⣽⢿⡿⢿⠈⠉⡁⠔⡉⠀⠀⠀⠀⠀⠔⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡓⠀⠀⠘⢿⣇⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠠⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠈⠻⡧⠀⠐⠒⠀⠁⠀⠀⠀⡀⣀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠂⠀⠀⠀⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⡀⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣿⣿⣶⣶⣤⣀⣀⣤⡶⠶⠶⢶⣛⣛⣛⣛⣛⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⡟⠛⠛⠳⠶⠟⠁⠀⠉⠀⠛⠙⠿⢿⣟⢁⡤⠒⠋⠉⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣴⠞⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢛⣡⠤⢤⣞⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣻⣦⣀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⢀⣠⣾⡯⠚⠉⠀⣠⣞⣋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⣾⢣⡏⠀⠀⠀⢘⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠙⢦⠀⠀⠀⠀⠀⠐⣿⡿⠟⠁
⠀⠀⠀⠀⢰⣯⡾⠛⠛⢷⣄⣻⠖⠀⠀⢀⡴⣲⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⢨⣿⠃⠀⠀⣯⣾⣿⠟⣿⣷⠀⠀⠀⠀⠀⠀⣾⡿⢿⣿⣷⢳⠀⠀⠀⠀⠀⠀⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠘⢿⣿⣿⡿⠋⣄⡀⠀⠀⠀⠀⢻⣿⣿⣷⢏⡞⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⢀⣤⣶⣶⣤⣄⣤⣿⠀⠀⠀⠀⠀⣤⡄⠺⠉⠑⠾⠙⡆⣠⠶⠿⠤⠭⠭⢍⡁⠀⠀⠀⠀⠀⠀⠀⣠⠼⢿⣄
⠀⢀⣴⠟⢉⣁⢰⣾⣟⠋⠛⢦⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣛⣿⠀⠀⠀⠀⠀⣠⠞⠁⣴⣶⡍⠻⣦⡀
⠀⣼⡃⠀⠺⢿⠇⢻⠏⣾⣷⠈⣇⠀⠀⠀⠀⠀⣀⠀⣾⡋⠉⢉⣻⣧⠘⠀⠀⠀⠀⠉⠙⠢⡄⠀⢠⡇⣾⣷⠻⠿⣰⣾⡿⣷
⠀⣿⣷⠘⣿⡷⢸⣿⣷⣮⡍⠀⣿⡀⠀⠀⠀⠀⠒⠀⠛⢿⣿⣿⠿⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣬⣶⣾⣷⡌⣛⡃⢿⡆
⠀⠈⢿⣦⡄⠰⠿⠿⠛⠛⠁⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⢀⡯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠁⠈⠉⠻⣿⣷⣿⢟⣾⠇
⠀⠀⠀⠙⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⢧⠀⠀⠀⠀⠈⠉⣠⡾⠋
⠀⠀⠀⠀⠀⠈⢿⡌⠙⠀⠀⠀⠀⠀⣯⠸⠇⠲⠶⣤⣄⣠⣤⢀⣀⠀⠀⠠⣤⣹⣶⠒⠚⠋⢰⠏⣿⠃⠀⠀⠀⢀⣸⡟
⠀⠀⠀⠀⠀⠀⠘⣿⠈⠛⠒⠒⠀⠀⠸⡆⠀⠀⠀⠉⠉⠀⠈⠉⠈⠉⠉⠉⠉⠀⠀⠀⠀⢀⡏⠀⠸⠄⠀⠀⠀⣼⠿⠃
⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⢼⠏


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

⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣦⣤⣤⣤⣄⠀⢴⣦⣤⣴⣤⣄⠀⠠⠄⡀⠀⠀⢀
⠘⠛⣷⠟⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠈⠙⡟⠛⠛⠋⠀⠀⠙⡿⠻⣿⠇⠀⠘⠿⠋⠀⠀⠙⠛⠁⠀⠀⠘⠷
⠀⢸⣿⠀⠀⠀⠀⠀⠀⣴⣧⣼⣿⣿⣿⣿⣿⣍⣵⡀⠀⠀⠀⠀⣠⣄⣀⣀⣀⡀⠀⠀⣀⡁
⠀⢸⢹⠀⠀⠀⠀⢀⡞⢻⣿⣿⣿⡟⢻⣿⣿⣿⡟⢳⡄⠀⠀⠀⠀⠙⡿⠿⢿⠀⠀⠀⠈⣿⣦⡀⠀⠀⢰⣾⡷⠀⠀⠈⠶⠀⠀⠀⠐⢦⡄⢸⡇
⠀⠹⠸⠁⠀⠀⠀⣿⡷⠾⣿⡿⣛⡷⢞⣛⢻⣿⠷⢾⣿⠀⠀⠀⢀⠀⣁⢀⢉⡀⣀⢀⢀⡉⠉⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⡇
⠀⠴⠀⡅⠀⠀⠀⣿⡇⠀⣿⣷⡿⣷⢸⢿⣿⣿⠀⢸⣿⡀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⡆⢀⡀⠀⢀⡀⠀⢀⡀⠀⠱⠃⠽⠀⠀⢠⡄
⠄⠀⠀⠁⠀⠀⢀⣿⣿⣿⣿⣟⣿⡟⢸⣿⢿⣿⣿⣿⣿⡂⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠁⠈⠇⠀⠸⠇⠀⠸⠇⠀⠀⠰⠀⠀⠀⠘⠇
⠀⣀⣤⣀⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⢿⠛⣿⣿⣷⣶⣿⣿⣿⣆⠀⣶⣶⣤⣤⣤⣤⣶⣄⠀⢶⣶⣶⣶⣶⣶⣶⣶⡄⠀⣼⣷⣦⣀⣀⣤⣀⠀⠀⠀⢰⣶⡖⣶⣾⣿⠇⣼⣿⣿⣿⣅
⠀⠀⠀⡹⣟⠛⠋⠙⢿⡝⢿⡆⠈⠹⣿⣿⣿⣿⣿⣏⠀⠈⠹⣿⡿⢿⡿⢿⣿⠁⠀⠀⠈⢿⠿⠿⠿⡟⠀⠀⠀⠘⣿⣷⠿⠿⢿⣿⣿⣿⣟⣻⣿
⠀⠀⠀⠑⡄⠀⠀⠀⠃⠇⠀⠀⠀⠀⠸⡇⠀⠈⠹⠏⠀⠀⠀⢸⠁⠀⠀⡸⠻⠀⠀⠀⠀⠸⠂⠀⠀⠟⠀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠈⠻⣿⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡻⠟⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣬⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠻⣷⣤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠯⣍⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠛⣯⣿⣿⣯⣱⣎⡛⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣀⠱⣾⣤⣀⠀⠀⠀⠀⠀⢠⡾⡿⠻⡉⢠⣿⣿⣿⣿⣿⣿⣽⣶⣏⣬⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣌⠉⢉⢩⡁⠈⢰⣤⣿⣿⠁⠀⠁⠾⠁⣿⣿⠉⠦⣈⠉⠛⢿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⡀⠀⠀⢀⣼⣿⣅⠁⢈⠀⠀⠀⠀⣠⣾⣿⣿⣷⣤⡀⠘⠀⠀⢠⡇⠀⠀⠀⠉⣙⠿⠟⠻⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠁⠀⠀⠀⠈⠙⢿⣾⣿⣟⣿⡗⠐⢾⣷⡦⠀⠘⠑⠽⠿⠿⠏⡈⠉⠛⠛⠒⢾⣗⣒⣒⠒⠌
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡇⠀⣠⡶⠂⠀⠀⠀⠀⢻⣿⣿⣾⣟⠀⠀⢶⣾⣿⠆⠀⠀⠀⠀⠀⢻⣦⠀⠀⢀⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠁⠁⡄⠀⠀⠀⣀⠀⠙⠻⠿⣿⣆⠀⠈⠁⡟⠀⠀⠀⠀⠀⠀⢸⡏⠀⠒⢿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⣧⡀⠀⠀⠀⠉⠉⠐⠒⠺⣿⣷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡋⠁⠀⢀⣬⣶⣦⣾⣧⣴⡄⣤⡀⢰⢿⡏⠀⣰⠀⣰⠀⠀⠀⠀⠀⢀⣀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⡀⠀⡸⣿⣿⣿⠟⠻⣿⣿⣿⣷⣸⣾⣷⣏⣹⣾⣿⠀⠀⠀⣠⣶⣿⢏⡴⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⡰⢡⡟⠉⠀⠀⠀⠀⠉⠙⠻⠿⠿⣿⣏⢸⣿⣿⠀⠀⠠⢿⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢀⡜⣱⣿⡏⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠛⠛⢻⡄⠀⠇⠸⣿⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⡇⠸⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣿⠀⠀⠈⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣄⠀⡀⠀⢻⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠘⣻⣦⡀⠈⠙⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢤⣀⣀⣀⣻⣎⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣙⣧⣡⠀⠈⠙⠻⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣯⠇

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

⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢀⠄⠒⡏⢡⡆⠂⠀⠐⡆⢹⡃⠂⢄
⠀⠀⠀⠀⠀⠀⠻⠄⣀⢸⠀⢰⠂⠁⡈⠀⠀⡆⠱⣇⠀⠀⠠⠆⢰⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠔⠏⠀⠠⠄⠠⠄⠤⠄⠀⠀⠄⠠⠀⠠⠤⠀⠄⠧⢠⠈⠀⠀⠙⠵⣆⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠠⠀⠠⠤⠤⠤⠀⠄⠀⡄⠀⠀⡈⠀⠀⠀⠀⠀⠘⠢⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠔⠈
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠐⠀⠀⠐
⠀⠀⠈⠀⠀⠀⢀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢀
⠀⠠⠄⠀⠀⢂⢀⢸⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠤⠀⠀⠀⠀⠎⠀⠀⣀⡀
⠀⠀⢀⠐⠂⠂⣂⠀⢢⠃⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⢴⠆⠀⠀⠀⠀⠀⠘⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠉⠁⠀⠀⠀⠻⠀⠭⠡⠼⠁⠀⣉⡅⢙⠀⣈⠁
⢀⣀⣀⢠⠀⠀⠀⠀⠀⠤⣄⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⡶⠂⠈⠉⠲⣟⣨⠉⣄⠐⠆⠀⠀⣠⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠐⠧⠤⠟⠀⠀⠀⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢊⠋⠀⡭⠘⢃⠀⢳⡏⠡⠈⡈⠁⡏⠀⠀⠁⣈⠀⠋⠛
⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⣤⡄⢀⠀⢰⣤⣀⡄⠀⠈⠁⠀⠀⠀⠀⠧⣄⠀⢶⡀⠀⠀⠀⠀⠀⢸⣙⡄⠘⣧⣭⠀⠀⠀⠧⠄⠠⡀⠅⠠⠄⠸⠄⠀⢄⡁
⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⠀⠙⠋⡀⠀⠘⠋⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⣑⠁⠀⠀⠉⠀⠀⠈⠁⠀⠀⢀⠐⠀⠀⢹⠆⠀⢰⡀⠂⢀⠘⡇
⠀⠀⢤⠏⠀⠐⠗⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠈⠐⢀⢀⠀⠀⢠⣤⠀⡐⠙⡀⠿⣤⠙⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡒
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠈⠀⠄⢀⢠⡄⢺⡇⠐⣿⢂⠁⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠠⡧⠀⠰⣆⡀⠀⢠⡾⠀⠈⣧⠀⣜⠣⢠⣄⠊⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠃⠀⠀⠀⠀⠤⠁⣀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⡄⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⠋⠀⣮⣥⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠈⠉
⣿⣟⠛⣿⠏⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿
⣿⣦⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⣷⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣄⠀⠀⠀⠀⠀⠀⣰⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠸⡆⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⣴⠀⢳⠀⠀⢀⣾⣿⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⣸⡿⠀⣸⣿⣷⣿⣿⣿⡀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣿⡇⢀⣽⣿⣿⣿⣯⡙⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⣼⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣆⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢛⣉⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠈⠉⠀⠈⠙⠛⢻⣾⡇⠱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⢾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⢻⣿⣷⡀⠀⢤⣄⡀⠀⠀⠀⠀⠀⢳⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠙⢿⣿⣷⣦⣬⣝⣳⣦⣤⣴⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠙⠿⣿⣿⣿⣿⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⠉⢙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⢀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠰⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣷⡄⠀⠀⠀⠈⠙⠻⠿⠿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡄⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢀⡾⠁


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