⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠓⠒⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢖⣶⡟⠛⠉⢤⣿⠀⠀⣈⣹⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢁⢀⣲⡾⢛⠣⠄⣾⢒⣾⠇⠠⣄⢘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⠈⡄⢨⣧⣴⣇⠲⣡⢆⣻⡟⠀⠀⢀⡄⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣤⣼⣿⠿⣻⠘⢧⡼⣼⣿⠃⠀⣀⡄⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣾⣿⣿⣿⢾⣡⣟⣮⢻⣿⠇⠈⠛⡿⠁⢠⣿⣿⣷⣤⣾⣿⢶⣦⣼⣿⣿⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢛⣿⣻⣳⢏⣯⣻⣭⣿⡟⠀⠀⣯⠁⢀⡿⢿⣿⣿⣿⣿⣧⣀⢸⣿⡟⠛⠻⣤⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⢣⣟⡞⣷⠽⣞⡧⣛⣾⡟⠉⢳⣤⡿⠀⣼⠃⣽⣿⣿⣿⡟⠻⣿⣿⣿⣿⡶⣤⣘⣻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠏⠀⠀⠘⠙⡺⢟⡿⢣⣿⣿⡇⠀⠾⠛⠁⣸⣿⣿⣿⣿⣿⣿⠁⠀⠙⣿⡿⠛⠃⠈⢹⡿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡶⢎⣥⣿⣷⡾⣼⣴⣿⣟⣿⣷⣄⣀⠀⣰⡿⣾⣿⣿⣿⣿⣧⣴⣤⣼⣏⣰⠟⠃⠀⣼⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢴⣸⣿⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡏⠙⢿⣿⣿⡼⢿⣄⣀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣠⣿⣿⡿⠻⢦⣬⣿⣦⡀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⢸⣿⣿⣧⢀⣀⡀⠀⠙⢧⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣦⣤⣉⣿⣟⣿⣇⠹⣦⡀⠀⠙⠆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⠻⢿⣿⣿⣿⣿⣿⣬⣻⣶⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡏⠙⢿⣿⣿⣿⣷⣿⠟⠿⣿⣿⣿⣿⣿⡌⠓⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⠿⠃⢀⡿⠿⠿⠛⠉⠀⠀⠀⠀⠉⠻⣿⣿⣷⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣷⣄⣈⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠉⠀⢿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠸⣿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠏⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣤⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣌⡁⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⡻⠿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠿⠶⣾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡀⢀⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡏⠋⢙⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣏⣿⣄⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠋⢁⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣟⣿⣄⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡐⠠⢀⠀⡀⠄⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⠀⠀⠀⠀⠀⣼⣿⣿⣿⣛⠛⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠣⢈⠀⠒⢀⠁⠢⠐⡀⢂⠡⠀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢗⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣴⣿⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⢈⠂⡀⠄⠂⠡⡈⠔⠀⢌⠰⠀⠌⠠⠑⡀⠢⢀⠁
⠀⠂⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⡄⢠⣿⣿⣿⣿⡏⠉⣿⡏⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠂⠀⢠⠈⡄⠁⢰⠈⡌⠀⢢⢡⠈⠐⢠⠀⠁⠂⡌
⢀⠂⠌⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⣾⣿⣿⣿⣿⣧⣴⣿⠁⠀⠀⠀⠀⠠⢀⠁⠀⡀⠈⠀⠀⡀⢂⠄⡨⠄⠒⠠⠉⠄⠂⡁⢢⠐⠬⡑⢮⡐
⠠⠌⡂⢄⠁⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣾⣾⣿⣿⣿⡟⠛⣿⣿⠃⠀⠀⠐⠂⠁⢂⠀⡈⠔⢀⠐⡈⠐⠠⢁⠊⡔⢈⡁⠆⡁⢎⠰⡁⠦⣉⠒⡍⢦⢩
⠀⠀⠁⠀⠂⠈⠄⢂⠀⠀⠀⠀⢠⣴⣶⣾⣿⣿⣿⣿⣿⣿⡟⠛⠿⣶⣿⣿⡤⠠⢈⠄⡁⢈⠀⠠⠀⡈⠂⠔⡀⠃⡀⡁⠎⢠⢃⠰⡘⣌⠣⡀⠜⡰⢁⡚⠜⣢⠣
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⡀⠀⠉⠻⠿⣷⡌⢒⠰⣀⠈⡀⠐⠄⡑⠂⠌⡐⠠⢁⠌⠢⣌⠲⡑⢢⡑⢌⡱⢌⡱⢌⠲⣁⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡙⠿⣿⣶⣴⠆⠀⣿⡜⢢⠑⡠⠓⠤⢃⡰⢀⠌⡐⢈⠅⡊⠬⡑⠤⠓⡬⢡⠜⡢⢜⢢⡑⢎⡱⡌⢎
⠀⠀⢀⠂⠁⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣬⣿⣶⡀⢀⣿⡏⢆⡑⠦⡙⢌⠒⡤⠉⠲⣱⢊⡜⡱⢢⢍⠲⣉⠖⡡⢎⡱⢊⠶⣩⠞⡰⣍⢎
⠀⠈⠠⠌⢠⡄⠤⡠⢀⠡⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠈⡙⠻⣿⣿⣿⠀⣸⣿⡚⡤⣍⢲⡉⢦⡙⠴⣁⡚⣤⢃⢮⣑⠣⢎⢳⡘⣬⢓⡬⢳⡍⡞⣥⢚⡵⣊⢮