Big what Emojis & Symbols

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⢠⣿⣷⡄⠀⠀⠀⠀⠀⠰⡿⠻⠿⡷⠀⠀⠀⠀⠀⢠⣄⣀⣀⣴⣦⠀⠀⠀⠀⠀⣠⠖⢲⣄⠀⠀⠀⠀⠀⠀⠀⡔⡒⢦
⠀⠀⠀⢸⠑⢋⡇⠀⠀⠀⠀⠀⠀⣇⠀⣀⡇⠀⠀⠀⠀⠀⠋⢸⡿⠿⣿⠈⠀⠀⠀⠀⢸⣁⠶⡧⢰⡇⠀⠀⠀⠀⣠⣴⢷⣶⢧⣄
⠀⠀⠀⡿⢿⡯⢿⠀⠀⠀⠀⠀⢸⠈⠻⠋⣽⠀⠀⠀⠀⠀⠘⠶⣤⣤⠴⠛⠀⠀⠀⠀⠀⠙⠴⠶⠊⠀⠀⠀⠀⠀⠘⠿⠟⢧⣧⡼
⠀⠀⠀⠉⠈⠁⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⢀⣤⡀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⢻⡿⢿⡟⠀⠀⠀⠀⠀⢘⣓⠀⠀⣚⡂⠀⠀⠀⠀⢀⣀⣠⣤⣶⠄
⠀⠀⣶⣮⡿⣷⣿⣆⠀⠀⠀⠀⣠⣤⣿⣿⣿⡄⠀⠀⠀⠀⠻⣿⠓⠚⣿⠗⠀⠀⠀⠀⣸⣞⣀⣀⣷⣇⠀⠀⠀⠀⠈⣿⣿⡿⠿⠆
⠀⠀⠀⠉⠁⠉⠉⠀⠀⠀⠀⠀⠛⠛⠛⠈⠛⠁⠀⠀⠀⠀⢸⣈⡷⢾⣁⡇⠀⠀⠀⠈⣿⡿⠋⠙⢿⣿⠀⠀⠀⠀⠀⠈⠛⠿⠂


⠀⠀⣤⣀⣀⣀⣀⣤⠀⠀⠀⠀⢰⠒⠚⠓⠒⡆⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⢠⣄⣷⣀⡁⠀⠀⠀⠀⣾⡟⠛⠛⢻⣷
⠀⠀⣿⠼⠾⠦⠤⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠿⠁⠀⠀⠀⠀⠀⠈⠰⢿⡿⣯⠁⠀⠀⠀⠀⣿⡷⠶⠶⢾⣿
⠀⠀⠛⠙⠋⠙⠋⠛⠀⠀⠀⠀⠈⢽⠯⠽⡯⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠍⠀⠀⠀⠀⠀⣿⣧⣤⣤⣼⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣴⣶⣾⡇⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠾⠂⠀⠀⠀⠀⠀⢠⠐⠐⠰⢄⠀⠀⠀⠀⠀⠀⠀⡀⠇⢈⠉
⠀⠀⠀⠀⣿⢀⣼⡇⠀⠀⠀⢸⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣷⠄⠀⠀⠀⠀⠁⣤⣄⣀⣴⡅⠀⠀⠀⠀⠀⣼⣛⢳⡯
⠀⠀⠰⢿⠟⠘⠛⠁⠀⠀⠀⠀⠑⠤⢀⣈⡉⠽⠁⠀⠀⠀⠀⠻⡿⢿⠟⠀⠀⠀⠀⠀⠀⠉⠻⠟⠋⠀⠀⠀⠀⠀⠓⠮⣛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢤⣤⡀⠀⠀⠀⠠
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢋⡂⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠈⢻⢆⠀⠀⠀⠀⠀⠃⠀⠀⠀⠙⡵⡄⠀⠀⠀⠀⠀⠈⣝⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢣⢂⠀⠀⠀⠀⠀⢀⠀⠀⠀⢃⠃⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⢸⢃
⠀⠀⠀⠀⢃⢂⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠤⠤⠄⠠⣄
⠀⠀⠀⠀⠈⢧⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⢄⠈⡶⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠠⠀⢸⡇
⠀⠀⠀⠀⠀⠈⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢱⣃⠀⠀⠀⠀⠀⠘⡃⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠈⠀⠐⠘⡿⠀⠀⠀⠀⠀⡀⠐⠀⠲⢦⣄⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⠀⠀⠤⣿⠀⠀⠀⢠⠀⠸⣧⡠⠤⣤⣄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠁⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠰⠶⡆⠀⠀⠐⠄⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⡇
⠀⠀⡀⠠⡄⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠐⢻⡇⠀⠀⠀⠂⠀⠀⠀⢀⡿⠁⠀⠀⠀⠘⠁
⠀⠀⠀⠐⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠤⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠐⠄⠀⠐⠉⠁⠀⠀⠀⠀⠂
⠀⢀⣀⡀⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⡘⠱⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⡀⣯⣣⣄⡀⠀⠀⠀⠀⢀⣀⡤⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠒⠒⠒⠂⠀⠤⠶⠶⠶⠒⠒⠒⠚⠛⠛⠉⠉⠛⠓⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢰⢋⣳⡀⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⡀⠀⠀⠀⠀⠸⡈⠉⠉⠉⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⢈⣉⠙⠳⣄⠀⠀⠈⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠉⠉⠙⠆⠈⠀⠀⠀⡀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⡔⠈⠙⠓⠄⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⢀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⢰⠃⠤⢤⡀⠀⢠⢀⣠⣤⠼⠀⣤⢴⠋⠁⡇⠀⠀⢀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⢸⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⣀⠀⣾⠠⠤⢤⠀⠀⡨⠋⢀⡄⣠⣤⠶⠋⠀⠸⠧⠤⠖⠋⠙⠓⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠃⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⡆
⣤⠶⠒⣋⡁⠄⡇⠰⠶⢀⡤⠊⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡟⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⠁
⠉⠉⠉⠁⠀⠀⡇⣠⠖⢉⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠁⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⣏⢙⣛⡉⠁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢀⡟⠹⣄
⠀⠀⠀⠀⠀⠀⣿⠤⠤⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣸⠃⠀⢸
⠀⠀⠀⠀⠀⠀⢹⡒⠒⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣀⣀⣰⠃⠀⠀⢠⡏⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠈⣧⠉⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣾⠀⠀⣧⠀⠀⠀⠀⣼⠁⠉⡉⠛⠋⢥⣄⡟⠀⠀⢀⡞⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢧⡀⠀⠈⠙⡄⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡟⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⢹⠀⠀⢠⢸⡇⠀⠀⡞⠀⠀⠀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠈⠀⣀⡴⠁⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢀⡇⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⣼⠀⠀⣾⠸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠸⡇⠀⠀⠀⣇⠀⠀⠀⢸⠀⢀⡏⠀⠀⣿⠀⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⡇⠎⠀⠀⠀⣿⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠘⠆⠀⠀⢸⡆⠀⠀⠀⠀⠙⢞⣓⣠⠇⣼⣁⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⣄⡤⣄⠀⢘⠄⠀⢸⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⠛⠒⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠊⠉⠁⠀⢀⣾⣀⣀⡀⢀⣤⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠤⠶⠒⠒⠒⠚⠛⠛⠙⠉⠉⠙⢻⣆⡴⠶⠤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣴⡞⠛⠉⠁⢀⡀⠀⠄⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠘⠋⠀⠀⠠⣄⣀⠀⠉⠙⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠁⠀⡀⠉⠁⠀⠀⠉⠀⠀⢀⡀⠤⠄⠀⠐⠀⠀⠒⠒⢤⡀⠀⠠⠀⠀⠀⠀⠈⠙⢄⠀⢲⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⡠⠄⠀⠁⠀⠀⣀⠀⠀⠀⡰⠊⠁⠀⣠⠶⠒⣒⠒⠢⡄⠀⠀⡷⠀⠀⠀⢠⠆⠀⠀⠀⠜⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⢸⡃⠀⠀⠠⠀⠘⣄⠀⠈⠀⠀⠸⡅⠀⠀⠰⡇⠀⠀⠀⠀⢀⣉⣉⠡⠔⠀⠀⡰⠃⠀⠐⠒⠁⠀⠀⠀⠶⡤⢤⣀⢀⣠⢾⡇
⠀⠀⠀⠀⠀⠈⣟⠲⢄⡀⠀⠀⠈⠑⠀⠀⠁⠀⠈⠀⠀⠀⠈⠒⠢⠤⠄⠀⠀⠀⠤⠤⠒⠉⠀⠀⠀⠀⣀⡠⠄⠀⠀⠀⠘⣆⢀⠏⠀⣸⠁
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠉⠓⠦⣄⣀⡠⠖⠋⡟⠀⠀⠀⠀⠤⠤⠀⠀⣀⣀⡀⠀⢀⠀⠀⠤⠤⠰⠒⠋⠁⠀⢀⣀⡤⠶⠚⣋⡜⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠹⡀⠀⢸⠗⠤⠤⠤⣤⣤⣀⣀⣀⣀⣀⣀⣤⠏⣦⣤⠤⠤⠤⠖⠒⠚⠉⠉⠀⠀⠀⠀⣾⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠘⠀⢃⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⡄⠀⡇⠘⣄⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠁⠀⡇⠀⠹⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠇⠀⠀⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠃⠀⠀⠀⡇⠀⠀⠀⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣼⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⡀⠀⡆⠀⢀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡇⠀⠀⢸⠀⠀⢀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠀⢡⠀⠸⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⡄⠀⠀⠀⠀⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠃⠀⠀⠇⠀⠀⠀⠀⢰⠀⠀⠀⡄⠀⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠈⠀⠀⠆⠀⠈⠀⠀⠀⡇⠀⠀⢸⠀⠀⠈⠀⠠⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡄⠀⠀⠇⠀⠀⠈⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⣠⠚⠉⠙⠲⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⡇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⣀⣀
⠀⢀⣀⣴⠃⠀⠀⠀⠀⠀⡠⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠦⣠⠛⠉⠀⠈⢷⡀
⢰⠋⠀⠀⠀⠀⡔⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠈⠙⠢⣀⠀⠀⠀⠀⠀⣧⣀
⣸⣀⡠⠤⠤⠚⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠘⣦⡀⠀⡀⠀⠈⠉⢳⡀
⣸⠃⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠹⢄⠀⠀⠀⠀⠀⠈⠛⢿⡓⠶⣆⠀⢸⡇
⢯⣀⠀⠀⡰⠛⠲⠖⢺⠟⠋⠀⠀⠀⢀⡴⣇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠒⠲⣟⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠈⢻⡛⠛⠻⡄⠀⠀⠈⣧⠀⠘⠶⠞⠳
⠀⠉⠙⠛⠁⠀⠀⠀⢿⣤⣀⣀⠀⣠⠞⠀⢿⣀⡀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢹⣦⡀⠀⡀⠀⠀⠀⢱⣤⣄⠀⠀⣀⡇⠀⠘⣇⣀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠁⠀⠀⠀⠉⢹⠀⠀⠀⠀⡜⠛⠋⢷⠀⠀⠀⢠⣿⡁⢀⡟⢦⣀⣠⡼⠀⠈⠛⠋⠉⠀⠀⠀⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣶⣶⣶⣚⣁⣀⣀⣘⣳⣶⣶⣟⣈⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠀⠈⣿⡟⠉⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠈⢿⢸⣿⣎⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣧⣴⣶⣾⣦⠀⠀⠢⢀
⠀⠀⠀⣠⣾⡏⣀⣠⣴⣶⡾⠁⠀⠈⠳⣤⡌⣆⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣟⣻⣿⣿⣿⣿⣿⣿⣶⣦⣀⠁⢠
⠀⠠⣾⣿⣿⣾⡟⠁⠙⣿⡀⠀⠀⠀⠀⠙⣿⡿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣰⡾⣻⣿⣿⡟⠛⣿⣿⣿⣿⣿⣿⣷⣄⠀⡀
⠀⠀⠈⠛⠛⠉⠓⠚⠛⠋⠀⠀⠀⠀⠀⠀⠘⣦⡬⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣴⢿⣾⣛⠛⠛⠛⠋⣛⠋⢩⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣵⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢠⣿⡏⣾⣿⣿⣿⠀⢰⣮⣿⣄⠘⣿⣏⣿⣿⣿⣿⣏⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠄⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠿⠀⣿⣿⣿⣿⣶⣧⠄⡀⠈⠓⡏⠙⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣦⠙⠻⢿⣿⣿⣿⡇⠀⠀⠀⠀⡆⠈⣿⣿⣿⣿⣿⣿⣦⣀⡘⠂⠀⠀⣿⣿⣿⣿⠟
⠀⡆⢀⣤⣤⣤⣴⡿⡠⠀⢀⣄⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠉⠻⢿⡇⠀⠀⠀⢠⠀⣄⣿⣿⣿⣽⠋⠉⠁⢤⣄⠀⠀⡘
⠀⠀⠈⠉⠁⢀⠄⠊⠀⣤⡞⠛⠇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠠⠀⠀⣀⡼⢸⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⢀⠀⡤⠔⠚⠉⠀⠀⡀⠀⢀⡄⢰⡇⣿⠁⠀⠀⠀⠀⠀⠘⣠⣜⡿⣇⣾⣿⣿⣿⣿⠋⠀⠀⢀⣤⡶⣿⢧⣶⡀
⠀⠀⠀⠀⣿⠾⣁⣀⣀⣀⣤⣾⣤⣶⡿⠀⣼⢧⣟⣀⣀⣤⣤⡤⠶⢿⣭⢀⣝⢮⣟⠿⣿⣏⠁⠀⠀⢀⣿⣿⡿⣫⣿⣿⣿⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠉⠉⣉⣿⣿⡿⠉⠀⢁⣰⠟⣛⣩⣭⣅⣀⣠⣴⣆⣠⡴⢋⣾⢸⣿⣿⣮⣝⡳⣄⡀⣸⣟⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠶⣤⣀⣀
⠀⠀⠀⠀⢀⣠⣴⣾⡋⠀⠀⠀⢀⣴⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⢡⣿⣿⣼⣿⡿⠿⠿⠿⠎⠉⠈⠉⠙⠻⢿⣿⣿⡿⠿⠛⠛⢿⣿⣷⣦⣄⡉⠙⠦
⢶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣴⣶⣾⣿⣿⣿⡿⣿⡟⠛⠛⠛⠿⠛⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣼⣿⣷⣷
⠈⠻⣄⡈⠿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠱⣿⣿⣿⣿
⠀⠀⠈⠙⠲⠦⣀⡈⢭⣭⡿⣋⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢹⣿⣿⣽
⣀⣀⣀⣀⣀⣀⣀⣈⣀⡌⠙⠻⠿⠿⠿⣿⣷⡻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⢠⣭⣀⠠⠆⠀⢤⣮⡙⣷⡿⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⣦⣀⡄⠀⡇⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⢀⠞⣡⡾⢿⣿⣿⢷⣦⣄⣻⡛⢘⠃⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠘⣿⣷⣾⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢿⢀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⠁⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠘⣿⣿⠀⠀⠠
⠀⠀⠀⠀⠀⣸⣿⠞⠋⣀⣤⣿⣿⣿⣯⣻⣿⣿⣷⣀⡤⠤⠤⠴⠛⠒⣺⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢠⠃
⠀⠀⠀⠀⠴⠛⢁⣴⡿⣛⣿⣿⡿⠏⠁⠁⠀⠈⠉⠀⠀⠀⠀⠀⣠⡞⠉⠉⠉⠁⠂⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⣿⣀⠃⠀⠐
⠀⠀⠀⠸⣶⣾⡟⣡⣾⠿⣋⡍⠀⣤⢆⣠⣤⣤⣤⣤⣤⣤⡶⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⣿⡇⡴⠀⠀⡐⠋
⠀⠀⠀⠀⡟⣡⡿⢋⣥⣾⡿⠀⣼⣳⣿⣿⣿⣿⣿⣿⣿⣿⢱⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠰⢦⣄⡈⠀⢀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠘⣿⣤⣄⣴⣾⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⢀⠃⣿⣿⣿⢿⣿⠿⣿⢻⠇⣾⣿⣿⣿⡦⡄⠀⣀⣀⠀⠀⠀⠀⠀⠉⠁⠂⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⢸⡿⡿⢛⣻⣿⣿
⠀⠀⠀⠘⠛⢯⣙⠻⣿⡟⣿⠈⠀⠉⠈⠉⠀⠈⠀⠈⠀⠀⢿⣿⣿⢿⣼⡇⠀⢹⡟⠻⢶⣤⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⢸⣧⢱⣿⡿⠋⣉
⠀⠀⠀⠀⠀⠀⠈⠙⢮⣛⣻⠤⠶⠶⠶⠦⠤⠤⣤⣀⣀⣀⣈⠛⣡⣿⣿⣿⣶⠘⠁⠀⠀⠉⠓⡄⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⣸⣿⠘⣋⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⣆⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣾⠟⢛⣛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠛⠿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢀⣸⣿⣿⣧⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠉⠛⠛⢁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⠾⠛⠉⢠⡟⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣴⣶⣿⠿⠋⢁⣠⣴⣾⡟⠀⠀⢀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⢀⣤⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠿⠟⠋⠉⢀⣠⣾⡿⣿⣿⣿⡿⠁⠀⠀⠀⢄⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣠⣤⣴⣾⣿⢟⣡⣾⣿⣿⣿⣿⣄⠀⠀⠀⣽⡄⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⣷⣄⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⢛⣫⣴⣿⣿⣿⣿⡿⢋⡴⣿⣷⣄⠀⠘⢀⣼⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣀⣬⣍⡳⢄⠀⠀⠀⡀⠠⠀⠒⠒⠂⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⡾⠃⠀⠀⠙⣎⢃⣠⣮⣀⠀⠐⠀⠀⠀⠀⠀⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⣿⢀⠀⠀⠀⠀⠏⣯⣤⢟⣁⢤⠒⠀⢀⠀⠀⠀⢸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⢿⠀⣠⣀⣠⣮⠞⠁⢁⡜⠃⠛⣡⣆⠀⡁⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠉⠈⠐⣄⠀⠈⠣⣨⣿⡍⠀⠀⢀⡄⢈⠃⢰⢸⡍⢿⣿⣧⣤⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡠⠀⣶⣼⣿⣶⡶⠽⠟⠡⠀⠀⠈⢀⡘⡀⢀⢿⣰⠀⠉⠐⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣧⣀⠀⠋⠉⠀⠀⢀⣠⣴⠀⣴⣶⡿⣇⡧⣈⣷⡙⠒⠀⠒⠂⢻⡐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠁⣠⣴⣶⣶⡿⣿⣿⠏⡼⣫⠚⡅⠠⡤⠴⡘⢻⡦⠐⡒⠒⣦⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡸⠫⢿⣿⠏⢹⡿⠃⣴⣿⢥⣿⣶⡿⠛⢲⣷⣸⣁⡆⠀⠀⡇⣰⡗
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⣾⠁⠀⠀⣁⢀⡜⢀⣼⣟⡞⣿⣬⡻⣇⠀⠘⣽⣿⡿⠧⡀⣸⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢿⣆⢠⣾⠖⠋⢰⠟⠉⢨⣺⠻⠻⣿⠉⠉⢁⡀⢙⣷⠀⠀⠑⡈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠘⣿⣿⠋⢠⠖⠁⠀⠀⣸⡿⠀⠀⠸⠀⡔⠉⠀⠈⢉⣳⡀⠀⣸⣂⣦
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡆⣿⡏⠀⣼⠀⠀⣠⠼⠋⠀⢡⡄⢸⡮⡇⡔⠉⢻⣕⣛⣡⡴⠃⠘⣾⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣂⠀⠔⣰⠟⡁⠈⢿⡀⠀⠁⠀⠀⠀⠈⣇⢸⡇⢣⠐⠀⠀⠉⠉⠉⠀⠀⠀⠥⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⢌⢧⣧⠀⠀⠀⠀⠀⢸⡿⢨⠃⠀⠏⠀⠀⠀⠀⠀⠀⠀⠠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢷⠁⠀⠀⠀⠀⣿⠃⡜⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠞⠀⠀⠀⢀⡿⢁⠌⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠊⠀⠀⠀⠀⣷⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠀⠀⠀⠀⠀⡏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠠⣅
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣠⣤⡤⠾⠷⠶⠶⠚⠛⠛⠒⠒⠒⠒⠶⠶⠦⠤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠟⠛⠉⡽⠆⢀⠀⠀⠉⠁⠀⠰⠀⠐⠒⠀⠀⠶⠄⢀⠀⠰⠀⠀⠈⠉⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣠⠴⢛⣭⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠰⠦⠀⠀⠛⠂⠀⠘⠃⠙⠳⣦⡀
⠀⠀⠀⠀⢰⡿⠋⠁⠀⠀⠀⠀⠀⠾⠂⠀⠀⠛⠁⠀⠀⢠⡄⠀⠈⠂⠀⠀⠀⠘⠅⠀⠀⠶⠀⠀⠀⠤⠀⠀⢀⠄⠀⠈⠓⠀⠙⣦
⠀⠀⠀⣰⠏⢰⠃⠀⠠⠖⠀⢠⡤⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠛⠈⢳⡀
⠀⠀⢰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠑⠀⠐⠀⠔⠒⠀⠈⠑⠒⠀⠁⠀⠁⠀⠠⠀⠊⠢⠀⠀⠀⠀⠐⠒⠀⠀⠀⠰⠦⠀⠀⢳⡀
⠀⠀⣿⡟⣼⠉⠀⡀⠀⠀⠀⠄⢊⠐⢊⣠⣤⣄⠀⣄⢀⢤⣀⣄⣆⣀⣄⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⠐⠂⠀⠀⠀⠀⠀⠸⡇
⠀⠀⣿⣾⣿⣧⣾⡉⣯⣤⣄⣀⣿⣿⣼⣿⣿⣿⣿⣿⣿⡾⠿⠿⠻⠿⠿⠿⠟⠿⠿⠿⠶⡶⠶⢤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⢹⣿⣿⣽⣥⣿⣿⣿⣿⣿⣿⠿⠛⠛⣋⣭⡥⢤⣤⡤⠾⢦⠶⠖⠛⠓⠶⠤⣤⡴⠛⠙⠛⠒⠶⠶⠻⠧⠭⣭⣉⠋⠛⠛⠛⠛⠛⠻⣄
⠀⠀⠀⠙⣿⣿⠿⢿⣛⣋⣉⣁⡀⠀⣠⠞⠉⠀⢴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⠀⠀⠀⠀⠀⠘⠀⠀⠉⠓⢾⡟⠛⠳⣼⣷⣮⠷⠆
⠀⠤⠶⢿⣥⣾⡿⠋⢋⡋⠉⠙⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢀⣠⠉⠸⡇
⠀⠀⠀⠸⣿⡿⣃⡀⠈⠁⠀⠘⠂⠀⣠⡶⠾⣷⣖⣚⣠⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣤⣤⣄⠀⠀⠀⠀⠀⠦⠀⠈⠀⠀⢠⡇
⠀⠀⠀⠀⠙⢿⣿⣷⡆⢠⢶⡼⠷⠾⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠙⣷⣾⣿⡄⢠⣿⣿⣟⣉⣈⠉⠉⢙⣿⡀⢠⣤⣼⠿⢷⠶⠶⣶⡖⠋
⠀⠀⠀⠀⠀⢸⣿⣿⣇⡜⠀⣀⣀⣀⣀⣠⣤⣤⡤⠤⠴⠶⠶⠖⠒⠒⠛⠛⠿⠿⠾⠛⠋⠀⠈⠉⠉⠉⠙⠛⠓⠿⠤⠤⢤⣄⣀⣀⣴⡏
⠀⠀⠀⣠⣶⣿⣿⠟⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡄
⠀⢀⣾⣿⣿⣿⣿⣷⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⢠⠀⡴⠖⠀⠀⠀⠀⢀⣤⡴⠞⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⠷⣤⠶⣦⡀⠀⠀⠀⣤⣽⣦
⠀⢸⡏⠍⠒⢲⠖⠂⠀⠉⠙⠛⢷⣦⣄⠀⠀⠀⠸⡤⠃⠀⠀⠀⣠⡾⠟⠁⢀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠤⡀⠹⣧⠀⡾⢷⠇⠈⢻
⠀⠈⢷⣶⣶⡿⠿⠀⠀⠦⠀⠀⠀⠈⠛⢿⣦⣄⣀⠀⠀⠀⣠⡶⠏⠀⠀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠙⠷⠃⠀⠤⢀⡼
⠀⠀⠀⣽⣿⣷⣶⣦⣶⣤⣤⣴⣤⣤⣤⣤⣬⣿⣿⣿⡄⣸⣯⣤⣄⣀⣤⡽⣶⣤⣴⣶⠶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣷⣤⣤⣤⣴⠞⠁
⠀⠀⠸⣿⢿⣷⡄⢠⠟⠻⠿⠋⠁⠈⢿⠉⠉⣿⢿⣿⣿⡿⠉⠹⡟⠉⠁⣤⣄⠀⣸⠃⠀⣿⠀⢈⡟⠙⠳⠮⠴⠋⠁⢿⡇⠈⠉⣾⠛⢿⣿
⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠘⠳⠞⠀⠀⠀⠀⠀⠀⠀⢷⡀⣸⠁⠙⠋⠁⠀⠀⠺⠗⠋⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⣰⡇⠀⢀⣿
⠀⠀⠀⠀⢸⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢰⡿
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠾⠃
⠀⠄⠀⠀⠀⠀⠉⠛⠿⢿⣿⣷⣺⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠶⠶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⡤⠴⠶⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⠁⠀⠒⠾⠃⠐⠓⠂⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣦⡄⠀⠀⠀⠀⢀⡀⣀⡿⣧⡀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣀⡀⠀⠀⢠⠾⢿⣿⣿⣿⠉⠀⠀⠀⠀⠈⠉⣹⣿⣎⡷⣄⠀⠀⠀⣠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣴⠶⠚⠋⢹⣿⣅⣺⡄⠀⣤⣤⠀⣴⠾⣿⡟⠻⢌⠛⠒⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠿⠛⣽⣤⡤⣄⣀⣀⣠⡜⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠁⠉⠽⠛⠛⢻⣅⠀⠀⠀⠀⠠⣤⣄⠀⠀⠀⠀⣠⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠛⠿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠂⠀⠀⠈⠄⠐⠂⠀⡀⠀⣿⡿⢠⠀⠀⠰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠀⠀⠙⠷⡀⠀⠀⠀⠀⠀⣀⣉⠀⠒⠀⠀⠀⣈⡀⠀⢆⠀⢴⡋⠚⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⢤⡆⣤⣤⠀⠀⢠⡤⠀⠁⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠘⠉⠀⠀⠈⠐⠀⠁⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡏⠰⠦⠈⠀⠀⠀⡀⢈⠄⠀⣻⡀⠀⠀⠈⠀⠀⠀⠀⣸⠂⠠⣬⣁⠀⢠⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣶⠀⠿⢶⡶⠖⠀⠀⠀⡁⠀⠀⠀⠹⣷⣄⠀⠀⣦⡌⠀⣼⡟⠁⡄⢀⢸⡀⠀⢶⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣠⠴⠦⠄⡀⠀⠘⠁⠀⠀⠀⠔⠻⣿⣆⢶⠽⣷⣰⡿⢃⠀⢡⠈⣿⣧⡀⠀⠀⠀⢤⣤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⡠⠀⢀⡳⠰⠂⠀⠀⢸⠇⠀⠀⠹⣿⠆⠀⢸⣿⡓⢸⣧⠀⠿⣠⡟⠁⠀⠀⣴⣾⣿⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣶⡿⠇⠀⠀⠀⢀⡯⠀⠀⢀⣴⡿⠻⠶⠛⠁⠀⠈⣿⠀⠀⠁⠀⠀⠀⠐⠿⡿⠿⠷⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠁⠀⠀⠀⠀⠀⡇⠠⣶⡀⠁⠀⠀⠀⠀⡀⣤⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠹⢄⠛⠃⢠⠀⠀⠀⠀⠘⡷⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠀⠈⠀⠀⠀⢀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢠⠀⠀⠀⠀⠀⠄⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣠⠀⠀⢹⠄⠀⠀⠐⢠⣷⠀⠀⠀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣧⣤⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁

⣿⣿⡿⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⢹⣧⠙⢧⣹⣷⡙⢦⡙⠻⢶⣶⣦⣿⣿⣿⣿⣾⣿⠋⣴⣿⣿⠿⣋⣵⣿⣴⣿⡿⣃⣴⣯⡾⠃⣠⣾⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠻⣷⡀⠙⣿⣿⣿⣿⣷⣦⣀⠉⠻⣿⡌⢻⣿⣿⣼⣿⣿⣷⢟⣭⣽⣿⣿⣿⣿⣿⣿⢋⣠⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡀⠻⣿⡄⢸⡿⣠⣈⡉⠛⢿⣷⣄⣿⣇⣿⣿⣿⣿⣿⣿⣵⣿⣿⠿⠛⢉⣡⡄⣿⣧⣿⡿⠋⣰⡿⢱⣿⣹⣿
⣿⣿⣿⣿⢿⣷⡄⢹⣷⣾⠇⡿⣬⡝⠳⣄⠈⠻⡿⠟⠛⠛⠛⠛⠛⠿⢿⠟⠁⣠⠞⣩⡣⣿⢸⣿⠏⣠⣾⠟⠁⣾⢣⣿⣿
⣿⣿⢿⡿⣧⡻⣿⣆⢿⣿⢸⡇⡏⢯⣽⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡡⣽⣿⡿⢸⢸⣿⣾⠟⠁⣠⣾⣷⣿⣿⣿
⣿⣿⡎⣿⡌⢻⣾⣿⣿⣿⣾⣇⠿⢫⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣿⠿⣸⣼⣿⣃⣤⣾⣿⡿⣋⣽⣿⣿
⣿⣯⣿⣾⣿⣦⡙⢿⣿⣿⣿⠿⠾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⢿⣿⣿⣿⣿⣿⣵⣾⡿⣫⣿⣿
⣿⣭⠛⠻⢿⣿⣟⠳⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⢟⣿⣿⣯⣾⣿⣿⢿
⣿⠶⢖⣒⣶⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⡞⠋⢹⡶⣄⠀⠀⠀⠀⠀⣤⢾⡏⠙⢧⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣶⣶⣾
⣿⣿⣿⣿⣿⡿⠿⠒⠒⠉⠀⠀⠀⠀⠀⣇⠀⣾⡇⠘⡇⠀⠀⠀⢸⠁⢸⡇⠀⡸⠀⠀⠀⠀⠀⠀⢒⠚⠿⣿⣿⡿⢿⣿⣾
⣿⢋⣴⣿⣿⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠦⣼⣃⡴⠃⠀⠀⠀⠘⢦⣸⣧⠜⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣗⠮⣽
⣿⣟⣥⣾⡿⢃⣠⠴⣶⣶⡶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠊⠯⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⣶⡶⢤⣄⠙⣿⣿⣿⣷⣾
⣿⣻⣿⣿⣷⠟⢡⡾⠋⠁⢀⣼⣿⠉⠛⠶⣤⣀⠀⠀⠀⢀⣦⠀⠀⠀⠀⣀⣤⠶⠛⠹⣿⣧⠀⠈⠙⢷⡈⠻⣼⣿⣯⣿⣿
⣿⣿⡟⢻⣿⢠⡟⠀⠀⠠⠚⢁⣾⠀⠀⠀⠀⠉⠉⠛⠻⣽⡿⣭⠛⠛⠉⠉⠀⠀⠀⢠⣧⠈⠓⠄⠀⠀⠑⠀⣿⣿⣿⡟⣸
⣿⣿⡇⠘⣿⠈⠀⠀⠀⠀⠀⢈⣼⣆⠀⠀⠀⠀⠀⠀⣰⠃⣧⠹⡄⠀⠀⠀⠀⠀⠀⣷⣧⡀⠀⠀⠀⠀⠀⢀⣿⣿⢻⣿⣿
⣿⣿⣿⡄⠹⣇⢸⣄⠀⠤⠶⢛⣽⣿⢀⠀⠀⠀⠀⣰⠃⡼⡟⢦⠹⡄⠀⠀⠀⠀⡀⣿⣯⠛⠒⠀⠀⣰⠆⣼⠃⠃⣼⣿⣿
⣿⣿⣿⣿⣦⡈⠈⢿⡓⢒⣶⡿⠋⢻⣿⢀⠀⠀⢰⢃⡞⠁⡇⠈⢧⡹⡄⠀⠀⡀⣷⡏⠙⢿⣖⡒⣛⠟⠐⠁⣀⣼⣿⣿⣿
⣿⢿⣿⣟⠻⠿⠿⣿⣿⣿⡿⠶⣦⡈⢿⡎⠀⠀⣯⡎⠀⠀⡇⠠⠀⢳⣿⠀⠀⢱⡿⢀⣴⠶⢻⣿⣿⣿⠿⠿⠟⣿⣿⠿⣿
⣿⣷⡙⢿⣆⢠⣾⣿⣿⡿⣿⣷⡌⣷⢸⣧⡀⠀⣿⡇⠀⠄⠛⠀⠀⣼⣿⠀⢀⣾⢁⡿⣡⣾⣿⢿⣿⣿⣷⡄⣸⡿⢡⣾⣿
⣿⣿⣷⠈⢿⣿⣿⣇⣿⣧⣿⡟⣿⣾⣾⡇⠙⢆⢹⣿⠀⠀⠦⠀⠀⣿⡇⡴⠃⢻⣾⣷⣿⢿⣿⣾⡏⣸⣿⣿⡟⠀⣿⣿⣿
⣿⣿⣿⡄⠸⣿⡨⢻⣿⠟⣿⣧⣿⣿⡿⠁⠀⠈⣾⣿⡄⠆⠀⠀⢸⣿⡿⠁⠀⠸⣿⣿⣯⣼⣿⠻⣿⡟⢸⣿⠁⢸⣿⣿⢹
⣿⠹⣿⣧⡀⢿⣿⣾⢛⡀⣫⣭⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⠇⠀⠀⠀⢸⣿⣿⣭⡍⣪⣻⣷⣿⠟⢀⣾⣿⠏⣸
⣿⣷⣾⣿⣷⣦⣭⣛⣛⡿⢿⣿⣿⣿⣧⠀⠀⠀⠈⣿⣇⠀⠀⠀⣿⣿⠀⠀⠀⠀⣾⣿⣿⣿⠿⢟⣛⣩⣥⣴⣿⣿⣷⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠈⢠⠀⣀⣀⠀⡈⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠴⣾⠿⣿⣗⢠⡀⠀⠀⠀⠓⠂⠀⠁⠀⠀⠄⡈⢩⡏⢉⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⣴⣧⣄⡠⣙⣷⢿⡇⢸⠃⠉⠆⣰⣿⣽⣦⢻⠃⠀⢸⠃⠈⠁⠈⢠⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⢿⣻⣿⣿⣻⣿⠀⠀⣆⠀⢰⣿⣿⣿⣿⡄⠀⠀⢻⡇⢀⠀⡀⠐⣿⠄⠀⠀⠒⢴
⠀⠀⠀⠀⠀⠛⠂⠙⠀⠀⠀⠈⠛⠿⣿⣿⣷⣀⢀⣿⡄⣼⣿⢿⣿⣿⡇⠀⠀⠀⢻⣦⣷⡜⢖⡿⠃⠈⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⠀⠘⣿⣼⣷⣾⠛⠓⢿⣿⡼⣿⣿⡇⠀⠀⠀⠸⠛⠋⠛⠿⠁⠈⠠⠌⠁⠀⠈
⠀⠀⠀⠀⠀⢠⡄⠀⢾⣿⢁⡈⠛⠟⠛⣿⡏⢹⡟⠀⠀⢸⣿⡇⢻⣿⡇⠀⠠⠀⠀⠀⠀⠀⠀⠥⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⡀⠀⠀⠀⠀⠿⡏⠸⠃⠀⠀⣬⣽⣷⠿⠿⠇⠀⠀⠀⠀⠀⠀⠘⠀⠬⠀⠠⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣛⣭⣍⣻⣋⣿⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠒⠲⢤⠒⠉⠉⠉⠉⠉⠉⠓⠒⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠞⠁⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀
⠀⠀⠀⡠⠀⠀⠠⢀⠔⠉⠀⠀⠀⠀⠃⠀⠀⠀⠀⠤⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⡇⠀⠀⠀⠀⠀⠀⠊⣩⠀⢠⢣⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⡇⠀⠀⠀⠀⠀⠀⠚⠛⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⢣⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡀⠄⠃⠀⢀⠀⠀⢀⠂⠀⠀⠀⡾
⠈⣆⠀⠀⠀⢀⢾⠀⠀⠀⠀⢀⠈⢰⠃⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠘⢆⠀⡠⢁⡼⡀⠀⠀⠠⠀⡠⠃⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠨⠋⣴⡋⠀⣇⠀⠤⠔⢺⠁⠀⠀⠀⠈⠣⣄⡀⠀⣀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠉⠉⠉⡆⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠄⠀⠀⢱⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣿
⠀⠀⢀⠤⣀⢀⠤⠚⠁⠀⠀⠀⡸⠳⡀⠀⠙⢲⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠑
⠀⠀⢇⣄⣄⠀⠀⠀⠀⠀⢀⡴⠁⠀⠹⡀⠀⠈⡇⠀⠀⠀⠀⢿⠙⠒⠒⠒⠛⢻⠉⠉⡇⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠈⠢⣄⣐⣀⡠⠤⠒⠁⠀⠀⠀⠀⢳⣰⣀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⠤⣾⠀⢀⠀⠀⣄⢀⣠⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠐⢲⠉⠟⠳⣌⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⣘⣿⣶⠆⠀⢹⠁⠀⠀⠀⠀⠀⢀⣤⣴⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠄⠀⢊⢋⣿⠀⠠⠘⡇⠀⠀⠀⠀⠰⡷⠕⠲⣜⡳⢭⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢀⣸⠈⠛⠀⢀⠂⠱⡀⠀⠀⠀⢩⠳⠤⢘⠇⠈⢎⢷⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⣦⢴⣽⡏⠀⠀⠀⢸⡆⠀⠐⢄⠀⠀⠀⠉⠛⠁⠀⠀⠐⡼⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⡀⠀⣼⣿⣿⠋⠀⠀⠀⠀⢸⣯⠀⠘⢮⠢⢀⠀⠀⠀⠀⠀⠀⢀⣇⢟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠏⣀⣴⡔⠀⣠⣿⣿⣧⠀⠈⣷⣦⣈⠐⠢⡀⠀⠀⡜⢪⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡿⣡⣾⣿⡿⢃⠔⣿⣛⠭⢝⣦⣀⠈⠻⣿⣢⡄⣸⠀⣰⣩⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣩⣾⠿⡿⠁⡰⠧⣪⣾⡁⠀⣘⢿⣦⣥⡄⠈⣷⠏⣟⡾⠚⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣁⠜⢡⠖⠋⢀⣠⣴⣾⣿⣿⣧⢀⢿⠃⢙⠃⢙⢷⡯⣿⣩⢦⣿⣷⣶⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡙⢁⣤⠾⣿⣿⣫⣶⢊⣿⣿⢻⡃⢢⠋⠀⣇⡞⢸⠋⠉⠙⠻⢙⣿⡗⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢺⠿⣤⣾⢿⢊⣞⣅⠟⠡⣾⠏⢬⣇⣀⣦⠶⠾⠶⣟⣿⠀⠀⠀⠀⠀⠈⢷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢿⣞⡼⣿⠣⢳⢢⣔⡥⢾⢻⣿⣾⡖⢻⣋⡎⣏⢞⣆⠀⠀⠀⠀⠀⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡛⠙⢹⡇⠶⠨⣏⣩⢶⠟⢹⣛⠇⠘⠿⠔⠙⠫⣷⣱⠦⢀⠀⠀⡼⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣳⢶⣿⠹⠽⠯⣀⣠⠇⣼⣩⢆⣀⡀⡀⣀⣀⠀⠈⠛⢶⠑⢤⣇⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⢹⣿⣷⣄⡃⠋⣃⠔⣯⢗⣩⡴⣾⠤⠔⠤⢬⠿⡆⠀⠉⢻⢟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⣆⠾⢻⡯⣉⣁⣶⢶⣿⣹⡾⢉⣁⣉⢓⠦⢀⠀⠀⠀⠀⠀⠘⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣘⡿⡇⣿⣦⣖⣿⣽⡯⠊⠀⢙⣻⡆⢿⣖⡩⢤⣄⣐⡀⠄⣑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠷⣾⣺⠽⠞⠉⠀⠀⢰⠡⢀⡿⣸⠓⠻⠿⠾⠾⠯⠿⠶⢌⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠲⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣻⡈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠫⣔⡽⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢠⠂⠠⠐⡂⠀⠐⠒⠐⠠⢄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠛⠀⠀⠀⠧⠀⠀⠈⠄⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣇⠀⢀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⠨⡀⣁⣉⡈⠀⠀⠀⡆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⠎⢘⠀⠀⠀⢠⠀⠀⠀⠀⢠⡇⢣⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡄⠀⠀⠀⠀⡌⠘⢎⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡰⢇⠀⠀⠀⡜⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠙⠚⠁⠀⠗⠤⠊⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⠀⡇⠀⠀⠀⡆⠀⢀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄⠀⢸⠀⢸⠀⠀⡇⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠄⠀⠀⠀⠠⡠⠐⠀⢸⠁⠀⠀⠳⣸⠀⢸⠀⢠⡇⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⡇⡠⠊⠉⠙⡆
⠀⠀⠀⢆⠈⠀⠐⠄⠀⡎⠀⢸⠀⠀⠀⠀⢹⠀⡾⠀⢸⠃⠀⠀⢠⡇⠀⢸⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⡿
⠀⠢⠀⢠⡀⠀⠀⠀⣠⠃⠀⢸⠀⠀⠀⠀⢸⠀⡇⠀⢸⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⢇⠀⠀⠀⢀⡇
⡀⠄⠀⠀⡝⠀⠒⠉⠀⡀⠀⠘⣇⠀⠀⠀⢸⠀⡇⠀⢸⠀⠀⠀⠈⡇⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⡸
⠃⠀⠀⡜⠀⠀⠀⠀⠀⢠⠀⠀⠹⡄⠀⠀⠸⠀⡇⠀⢸⠀⠀⠀⠀⡇⠀⢾⠀⠀⠀⠀⠀⢸⠀⠀⢠⠃⠀⠀⠀⠀⠶⢄⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢹⡀⠀⠀⠀⣇⠀⢸⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠈⠀⢀⠏⠀⠀⠀⠀⠀⠀⣸
⠠⠀⣀⠀⠀⠀⠀⠀⠀⢠⠔⢰⠤⠤⠧⠤⡀⠀⢻⠀⠸⡄⠀⠀⠀⡇⠀⢸⡀⠀⠀⢀⠤⠤⠤⠯⢅⡀⠒⠠⠤⠤⠖⢁⡜
⠘⢇⡈⠀⠀⠀⠀⠂⠀⠁⣠⢣⠀⡄⢠⣀⡷⡖⠚⠀⠀⠛⢲⢲⠖⠃⠀⠈⠉⢧⠴⠯⣴⠀⡆⠀⣌⠇⠈⠉⠉⠉⠉⠁
⠀⠀⠈⠙⠢⠄⠢⠀⠂⠀⠉⠙⠒⠓⠚⠁⠀⠓⠾⣤⣄⣰⠴⠛⠶⣇⣴⣠⠧⠞⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢠⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡄⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠈⠻⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡆⢶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠃⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⠿⣿⡿⠛⠿⠟⠁⠀⠀⠀⠉⠻⢿⡟⢿⣿⣿⣿⣿⣿⣿⣿⡿⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⠃⠀⣡⣤⣄⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⠻⣿⣿⣿⣿⣿⣿⠁⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⣿⣿⠀⠘⣿⣿⣿⠇⠀⠀⠀⠀⠀⠘⣿⣿⡿⠀⠀⣿⣿⣿⣿⣿⡇⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢸⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣄⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣾⣿⣿⣿⣿⣿⠤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣴⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⡿⣿⡃⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠘⣿⣿⣿⣿⣿⣿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠘⢷⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣤⠟⠀⠙⠻⢿⣿
⢿⣿⣿⣿⣿⣿⠟⠉⠉⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⢸⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠋⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⢸⡿⠿⢿⠏⡌⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠿⠿⠄⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⢀⣆⣰⠁⠀⠀⠀⠀⠀⠀⠀⢡⢹⡄⠀⠀⠀⠀⠀⢸⣿⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇
⠀⠀⢼⣉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡋⣷⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⢻⡿⠛⠳⠶⣶⢶⡶⠶⠶⠶⣶⠾⠛⠛⠛⠛⠉⠁
⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⣿⢸⡇⠀⠀⠀⡷
⠀⠀⠀⠛⠶⠶⢦⣤⣤⣤⣤⣤⡤⠤⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡇⠘⡇⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡯⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⣷⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣼⡇⠀⣿⣀⣤⣼⠇
⠀⠀⠀⠀⣀⣤⣼⣧⣴⣷⣴⣧⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣼⣿⣿⣿⣦⣤⣿⣿⣿⣿⣦⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣿⣛⠲⣄
⠀⠀⠀⠀⡴⢋⠩⢉⠍⡉⠍⠙⢦⠀⠀⠀⠀⠀⣠⣖⣛⣛⣛⣳⡀⠀⠀⠀⠀⠀⢀⡇⠘⡇⠀⠀⠀⠀⠀⢠⠖⣫⡥⢾⢳⣗⣯⣙⣮⡇
⠀⠀⠀⢸⡧⠬⠤⠤⠤⠥⠤⠤⢼⡇⠀⠀⠀⠀⢧⡤⠤⠤⠤⢤⡟⠀⠀⠀⠀⠀⣼⣉⠉⢻⡀⠀⠀⠀⢠⠏⡾⡥⣌⢸⢸⠿⣚⣭⢶⡀
⠀⠀⠀⢸⡗⠚⠒⣖⣛⣲⠒⠓⢺⠅⠀⠀⠀⠀⠸⡇⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣏⢈⡇⢸⡇⠀⠀⠀⢸⠸⡇⢓⡛⠾⠖⣫⣅⢸⢸⡇
⠀⠀⠀⠀⢿⣙⣯⡉⠉⢉⣽⣋⡿⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢹⠉⠀⢸⠀⠀⠀⠀⠘⣇⠳⣜⠿⢊⣤⣙⣡⢏⡼
⠀⠀⠀⠀⠀⠉⠉⠙⠶⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⢻⣀⣀⣰⡇⠀⠀⠀⠀⠀⠀⣟⠀⠀⢸⡄⠀⠀⠀⠀⠈⠳⢬⣛⣛⣚⣋⡵⠚
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⣠⣖⢳⡏⣿⢓⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠻⣟⢦⡀
⠀⠀⠀⢸⠷⠾⢦⣷⣯⠼⠶⢻⠂⠀⠀⠀⠀⠀⣠⣤⣭⣼⣤⣤⠀⠀⠀⠀⠀⢀⡴⠋⡁⠂⠐⢈⠙⠦⡄⠀⠀⠀⠀⢀⡞⡁⢈⠀⡉⢺⡇
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢸⠋⣳⠀⠀⠀⠘⢶⠒⢶⠖⠒⣶⠒⡶⠃⠀⠀⠀⠀⣸⡀⠄⠠⢀⡠⠏
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣀⣀⣻⣀⣀⣀⣀⣿⣛⣁⠀⠀⠀⠀⠘⡇⢸⡆⢀⡏⢰⠇⠀⠀⢀⣠⡴⣋⡽⠶⠚⠋
⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠼⠁⠀⠀⠀⠀⠻⣄⣀⣀⣀⣀⣀⣀⣠⠽⠁⠀⠀⠀⠀⠹⢤⣧⣼⡧⠟⠀⠀⠀⠈⠳⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣙⣦⡀
⠀⠀⠀⠀⠀⢀⣀⡤⠞⢧⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠙⢦⠀⠀⠀⢠⣯⣤⣬⣷⣬⣷⡀
⠀⠀⠀⠀⣰⠯⠤⠤⠤⠬⢿⡀⠀⠀⠀⢰⠏⠁⠀⠟⠁⠀⠿⠁⠀⠉⢧⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠓⠺⡆⠀⠀⠈⢷⠛⢓⣺⠷⡾⠁
⠀⠀⠀⠀⣷⠶⣤⠴⢤⡴⢶⡏⠀⠀⠀⣞⣶⡒⣲⣒⣒⣖⢒⣖⣒⣲⣺⡆⠀⠀⠀⠀⢀⣴⠋⠉⠳⢤⣠⠇⠀⠀⠀⠸⡖⢋⣥⢶⠇
⠀⠀⠀⠀⢹⡀⢸⠀⢸⠁⣸⠃⠀⠀⠀⢿⠲⠭⣽⠮⠥⣬⠭⢾⡭⠵⣻⠇⠀⠀⣀⣴⡏⠉⠳⡄⠀⠀⡽⠀⠀⠀⠀⠀⢻⣉⡤⡞
⠀⠀⠀⠀⠈⣧⣸⣆⣸⣀⡏⠀⠀⠀⠀⠈⠉⠙⠛⠒⠒⠒⠒⠚⠛⠉⠉⠀⠀⠈⠳⢤⣽⣄⣀⡽⠶⠚⠁⠀⠀⠀⠀⠀⠈⣇⣸⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⢶⣤⣄⠀⠀⠀⠀⢀⣤⡀⠀⢀⣤⡀⠀⠀⠀⠀⢠⡤⠤⣤⠀⣰⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢺⠶⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠛⢧⡼⣷⣦⢷⡄⠀⠀⣠⠏⢠⠙⢶⠏⢠⠙⢦⠀⠀⠀⠈⣿⠉⡏⠘⣷⠶⡿⠀⠀⠀⠀⠀⠀⢀⡴⣳⢛⡞⠁⠙⣦
⠀⠀⠀⠀⢀⡴⠋⠀⠀⣸⠦⢾⣹⠏⠀⠀⠀⡇⠀⡼⠃⢸⡇⡼⠃⢸⡇⠀⠀⠀⡿⠀⣧⡞⠃⠀⠛⡆⠀⠀⢀⣠⢖⣫⣞⡵⠋⠀⠀⢠⡟
⠀⠀⠀⣰⠋⠀⠀⢀⡴⠋⣠⠞⠁⠀⠀⠀⠀⠻⣄⠓⢀⡼⡄⠓⢀⡾⠀⠀⠀⣾⠁⠀⠈⡇⠀⠀⠀⡇⠀⠀⢸⡖⣻⠖⠋⠀⠀⢀⡴⠋
⠀⠀⠀⠘⢦⣀⡴⠋⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠘⠶⠋⠀⠘⠶⠋⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠙⢧⡀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠙⠷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠳⠶⠒⠶⠃⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠐⡆⢀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡀⢠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠑⠒⠰⠀⠄⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊⠁⠉⠱⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠠⠤⠀⠀⠐⠀⠀⠀⠀⠀⠄⠐⠢⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠄⢀⠀⠆
⠀⢀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⢂⠘⠢⠊⠠⠀⠀⠁⠲⢆⡶⠜⠛⠁⠀⠀⠀⠀⠀⠀⠒⠀⢨⠀⡀⠂⠀⠀⢀⡀
⠀⠨⣀⠀⣀⠀⠠⠄⠒⠀⠀⠀⠀⣀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠃⠀⢒⡤⠈⠁⠀⡀⡀⠀⠀⠀⠀⠀⢀⡘⠡⣏⠟⠉⣢⠀⠃⠁⠈⠊
⠀⠀⢉⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠀⠁⠀⢀⠈⠁⢂⠘⠀⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠀⠀⡄
⠀⠀⠈⠁⠀⠀⠀⠀⡘⠡⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠶⠊⠃⠄⠀⠁⠀⠀⠀⠀⠐⠀⠁⠀⠀⠄⠀⠀⠀⠀⠅
⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠈⢀⣀⣀⣤⣤⣤⣀⣀⣀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⡠⠚⠁⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢒⠀⠀⡊⢀⠀⠀⠀⣰⣾⣿⣿⣏⣉⣉⣙⣿⣿⣿⣿⣦⠀⠀⠀⠁⠀⢄⠀⠡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃⠐⠀⠀⠀⠀⠀⠀⠂
⠀⠈⠅⠀⠀⠀⠠⠀⢰⣿⢩⠛⡏⣯⢩⣿⡩⠍⠛⠙⠉⢿⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⢠⠀⢠⣠⡤⠠⣄⠀⠠⠤
⠀⠀⢂⢰⠀⠀⠀⠀⠈⣿⢰⣀⠷⢹⢸⣿⢰⡀⠀⠄⠘⣸⠇⠀⠀⠀⠀⠁⠀⠀⡀⠘⠀⠀⠠⠀⠀⠀⠀⠁⠀⢂⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⡀⠀⠀⠔⢀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣷⣿⠾⠟⠋⠀⡄⠀⠀⠈⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠄⠀⢀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⢠⠠⠰⠄⠘⠀⠀⠀⠀⡆⠀⠀⠈⠀⠀⠀⠀⠄⡢⠂⠉⠁⠂⠀⠀⠐⠐⠄⠀⠀⠁⠉⠉⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠄⡀⠀⠀⡐⠀⠀⠀⠐⠢⠀⠀⠄⠀⠀⡀⠀⠀⠑⠞⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠁⠀⠀⠀⠈⠄
⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠃⠀⠀⠃⠠⣀⢀⠼⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠠⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠐⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⢀⠀⠠⠤⢀⢀⠀⠀⠀⠀⠐⢀⡀⢀⠀⠀⠈⠁⠀⠀⠀⠐⠀⠀⠅⠀⠀⠀⢠
⠀⠉⠀⠔⠀⠂⠈⠈⠠⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⡄⠀⠀⠀⠁⠀⠈⠑⢧⠋⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠂⠀⠀⠰⡀⠀⠀⢀⠄⠀⡀⠀⠀⠀⠀⢀⠀⡈⠁⢀⢁⢄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠈⡄⠐⠈⠀⠀⠀⠁⠠⠀⠑⠢⠄⠐⠀⠀⢠⢰⠚⢣⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⣀⡀⠀⠐⠃⠀⠊⠡
⠀⣀⠀⢂⠠⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠀⠀⠐⠀⢀⠀⠉⠑⠰⠼⠓⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠢⢀⠐⠀⠀⠁⠄⠀⠁⠀⠀⠖
⠀⠀⠀⠈⠀⠀⠤⠄⠐⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠉⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠲⢦⡀⣠⠴⠒⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⢰⠛⣆⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢀⡞⢳
⠀⠀⠀⠀⠀⠀⢸⠀⢻⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢸⠃⢸
⠀⠀⠀⠀⠀⠀⢸⠀⣸⠦⣄⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⢀⡤⢾⡀⢸
⠀⠀⠀⠀⠀⠀⠸⡄⠹⣄⠈⠳⣄⠀⠀⠀⠉⠲⣄⠀⢀⡴⠚⠁⠀⠀⢀⡴⠋⢀⡼⠁⡼
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⢳⡀⠈⠙⠲⡄⠀⠀⠈⠓⠋⠀⠀⠀⡴⠚⠉⠀⣰⠋⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠙⣦⠀⠀⠙⢆⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⠞⠁⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠁⠀⠀⠈⡇⠀⠀⠀⠀⡏⠀⠀⠀⠉⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠇⠀⠀⠀⠀⠇⠀⠀⠀⠛




⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⠶⡄
⠀⠀⠀⢀⣤⠞⠈⠀⠀⠉⠈⢀⠀⠀⠀⠀⠀⠀⠂⠀⡇⡏⡇⢤⣌⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⢀⠀⣀⡀⠀⢀⠀⠀⠈⠳⠀⠀⠀⠀⠀⢡⡤⡏⠏⠇⠀⠉⠳⡦⠀⠀⠀⢠⣤⡵⠄⠀⠩⣅⠈⠃
⠀⠀⠀⠸⢸⣿⡇⠀⢸⣈⣥⣄⠀⠀⠀⠀⠀⠀⢰⣶⡏⠉⠀⠀⠈⠘⠁⠀⠀⠀⠘⠛⠁⠀⠀⠀⢚⣤⠄⠲⠆
⠀⠀⠀⠀⢸⠉⠇⠀⢠⡟⡇⠀⠀⠀⠀⠀⠀⠀⠈⠳⣃⣃⣐⢆⡀⠀⠀⠀⠀⠀⠀⢷⣦⣤⠾⠀⠘⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠉


⣶⣶⣶⣶⢶⣶⣶⢶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢤⡤⠤⠄⢀⣀⠀⡸⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠣⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠀⠀⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠂⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠈⡉⠉⠉⠁⠀⠀⠒⠂⠐⠈⠉⠉⠁⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠢⠤⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠃⠘⠉⠭⠽⠟⠛⠛⠛⠋⠈⠉⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠚⠛⠻⠛⠛⠓⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣠⠤⣄
⠀⠀⠀⠀⠀⢀⡴⢚⠁⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠈⠙⢦⡀⠀⢀⣠⡤⠖⠛⣉⠥⠄⠠⠜⠻⣦⡀
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣞⠉⠀⠀⠀⠁⠀⠔⠊⠉⠴⠊⠩⡝⢳⣄⡀
⠠⢶⠤⣰⣇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⢣⡺⡤⣿⣌⠙⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣿⣠⢼⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡲⠤⣄⣀
⡠⢀⠙⠺⠻⣷⣶⡤⠤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣩⡾⡃⣀⣚⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⣠⣤⣽⣦⣀⠤⢶⢺
⠀⠐⠣⡅⣙⢄⣋⣭⣛⣓⣶⠶⢶⣤⡤⣤⡴⠶⠶⠶⠶⠚⡟⠛⠶⠿⢱⣚⡏⠩⠈⠓⢄⡀⠀⠀⠀⠀⣀⣤⣤⣶⡶⢿⢿⠿⠿⠟⠛⠉⠉⢀⣀⣼⠿
⠀⠀⣾⣷⡔⠀⢨⠃⠂⠘⠑⠷⢶⣧⣲⡖⠃⢂⡵⠷⠒⠛⣓⣲⡒⠄⡽⣿⡀⠀⠀⠀⠀⠙⠢⣀⣴⡟⠛⢿⡟⠀⣠⣲⣷⣲⠶⢶⣒⠚⠿⢍⡳⢤⣤
⠀⠀⡟⠬⡙⠗⠦⣤⣀⠀⠀⣀⣀⠀⠀⣙⡉⠀⢀⠭⠒⣈⠡⠖⠴⠼⢂⣿⡁⢠⢠⡔⠀⠀⢠⣾⡇⠘⡳⡖⠀⡴⣱⡁⢸⡌⠉⢳⢦⣍⡒⠤⣉⣲⠖
⠀⢀⠛⣦⡘⠀⢀⡈⠉⠛⠖⠀⣀⣀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⢀⣠⠎⢱⠙⢦⡸⢃⠄⠀⣾⣿⡇⠀⣷⡉⢙⡴⣝⢧⡀⣷⠀⠈⣶⣄⣹⣲⣇⡰
⠀⠐⡀⠈⠀⠩⠴⠿⠧⠤⠤⢤⣁⣉⣥⣌⡱⣶⡴⠭⢄⣒⣂⠭⠔⠋⠀⡸⠀⠀⠙⢼⠃⢠⡟⣿⣿⠀⠸⣿⡋⠵⢾⣭⣏⣻⡖⠐⡧⣄⣈⣉⣏⣹
⠀⠀⠑⢄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠙⢮⡇⠈⢻⠇⠀⢳⡉⠲⣤⣄⠤⣞⠛⢟⠿⣓⠊
⠀⠀⠀⠀⠈⠉⠒⠛⠛⠉⠁⠲⠶⠶⠶⠦⠤⠤⠄⠀⠐⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠈⢿⠭⡛⡘⣤⡉⢉⠇⠈⠑⢤⣙⡚⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⣈⣆⠱⣸⣌⣄⠉⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢣⢩⢻⢿⣦⠀⠀⠀⠀⠀⠀⠀⠴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡻⠎⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠆⠀⡀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⠉⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠳⣄⠀⠀⠀⠀⠀⣤⢠⡄⠀⠀⠀⠀⠉⠐⠁
⠀⠀⠀⠀⠀⠾⠿⠿⢿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠈⠣⡀⠀⣤⠀⣶⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⣄⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠛⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⠄⠖⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣀⢠⣶⠛⡆⠖⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠸⡿⢷⡈⠀⠀⠀⠀⠀⠀⠀⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⣷⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡏⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⡿⡿⠿⠿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢉⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢉⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣃⢸⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠓⡏⠀⠀⠀⠀⢀⡠⢄⢴⠊⠉⢱⡀⡏⠉⠀⣁⢀⠖⠤⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠰⡁⠀⠈⢎⡆⠀⠀⢣⠡⠀⠀⢠⡌⠀⠀⡰⡰⠁⠉⢢
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣼⠀⣧⣤⣤⣤⣀⣱⡀⢺⠁⠘⡀⠀⠀⣾⠀⠀⢸⠁⠀⢠⠛⠳⡄⢀⠎
⠀⠀⠀⢰⡉⢖⠉⠉⠀⠀⠀⠘⠤⠃⠀⠀⠀⠉⢁⡣⢉⣇⠀⢇⠀⠀⢹⠀⠀⡍⠀⠀⠸⠀⡸⠀⡎
⠀⠀⠀⢸⠪⠭⢉⣛⣒⣛⠛⠚⠶⠚⠛⣛⣚⣛⡉⠭⠑⡟⡀⢸⠀⠀⢸⠀⠀⠇⠀⠀⡇⠀⠇⣼
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠧⣀⡇⠀⠸⠀⢸⠀⠀⠀⣃⡴⠊⠸
⠀⠀⠀⠀⡇⠀⠀⠀⣼⢷⡀⠀⠀⠀⢠⡾⣦⠀⠀⠀⢸⡀⠀⠈⠉⠒⠓⠒⠚⠒⠒⠉⠁⠀⠀⡇
⠀⠀⠀⠀⢡⠀⠀⠈⠁⠀⢁⣴⣶⣦⡉⠀⠈⠁⠀⠀⡏⡇⠀⠞⠋⠳⠀⠀⠀⠀⠞⠉⠳⠀⢰
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠘⣿⡟⣻⠁⠀⠀⠀⠀⢠⠄⢻⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠛⠁⢀⣀⠀⣀⠀⢼⢀⣸⡀⠀⠀⠀⠀⠻⡿⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⡴⠋⠒⡝⡤⠖⢀⣉⣛⠫⠽⠿⣓⠒⠶⣶⣶⣶⣶⣤⣬⡿⣍⡓⠛⣉⣐⡶⣚⠭⠭⠓⢄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡃⠀⢀⡴⠛⢒⡒⢛⡒⠢⠤⠤⠒⢬⠲⣦⣄⣢⣶⠖⠥⠦⠶⠶⠛⠛⠛⠚⠲⢄⠀⠀⢸
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠉⢲⠋⠀⠀⠰⠃⠈⠁⠺⡄⠘⣀⠘⢣⡄⠰⠂⣰⠂⡤⢸⡇⣨⡃⠘⠁⠀⠀⠀⠱⡖⠁
⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⠀⠀⠀⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠉⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠆⠤⠤⠤⢀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⡀⠤⠤⠄⠒⠒⠊⠁


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

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠁⠀⠀⠀⠀⠀⠀⡠⠆⠤⡀⠀⠀⠀⠀⠀⣀⢤⡤⢠⣆⣠⠤⡄
⠀⠀⠀⠀⠀⡔⠋⠀⠄⠀⠀⠀⠉⢢⠀⠀⠀⠀⠀⠀⣠⣖⣛⣒⣲⣀⡀⠀⠀⠀⣔⡉⢉⣎⣑⡄⠀⠀⠀⠀⣨⠞⢀⡎⠲⣁⡞⢻⣉⠆
⠀⠀⠀⠀⢸⡤⠤⠤⠤⢤⠤⢤⠤⢤⡇⠀⠀⠀⠀⠨⣅⣀⣀⣀⡀⣠⠇⠀⠀⠀⢳⣾⣛⡳⠶⡇⠀⠀⠀⠀⠹⡓⠾⠽⠤⠿⠚⠉⢩⠃
⠀⠀⠀⠀⠀⣿⣉⣩⣭⣭⠽⢭⣙⣲⠎⠀⠀⠀⠀⠀⢡⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢟⣉⠷⡾⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠘⢤⣀⣀⣙⣇⢛⡧⠇⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⣍⣴⠃⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠤⠤⠞⠀⠀⠀⠀⠀⠀⠀⠸⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠒⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⡠⠊⡡⢤⠤⠤⣌⣱⣠⠖⢆⠀⠀⠀⠀⣠⠊⠉⠈⠙⢦⠀⠀⠀⡔⠱⠆⢠⡍⠰⠗⠄⠀⠀⠀⠀⣀⠤⢄⡀⠀⢀⣀⢀
⠀⠀⠀⠀⡜⢰⠛⣲⣁⢀⠴⣿⠿⠃⢱⠸⠀⠀⠀⠀⡇⠀⡀⠁⡁⢈⠆⠀⠀⢳⠲⡖⠦⡲⠖⠲⠇⠀⠀⢐⢉⡿⠿⣶⠾⡽⢷⣶⢾⡛⡂
⠀⠀⠀⠀⢇⢸⠈⠔⠁⠉⢁⡞⢻⠾⠍⠉⠀⠀⠀⠀⠃⡀⠀⠀⠄⡜⠀⠀⠀⠀⡇⢱⠀⡃⡸⢸⠀⠀⠀⠀⢻⠀⠂⠀⠉⠀⠉⠀⡁⢹⠁
⠀⠀⠀⠀⠘⣌⠻⣅⣠⣱⣀⡰⠃⡼⠀⠀⠀⠀⠀⠀⠀⠑⢴⠒⡼⠁⠀⠀⠀⠀⠑⠠⢄⠵⠒⠃⠀⠀⠀⠀⠘⠦⣈⣀⣀⣀⣀⡠⠔⠃
⠀⠀⠀⠀⠀⠈⠑⠢⠤⠭⠤⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠺⡭⣭⠎⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂⢰⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣞⣋⣹⠀⠀⠀⠀⠀⢀⡀⠀⢀⡰⢄
⠀⠀⠀⠀⠀⢠⣾⠥⣄⡴⢮⡱⢄⠀⠀⠀⠀⠀⠀⡴⠶⠶⠷⠾⠷⠖⣦⢤⠀⠀⠀⢀⡼⢀⠟⠁⠈⠦⢤⠀⠀⢠⠧⡌⠛⢧⡀⡆⠙⢦
⠀⠀⠀⠀⢀⡏⡰⠋⠀⠉⠑⠂⠺⡓⡆⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⣰⡿⠵⠃⠀⠀⢩⠀⠘⡄⠀⠀⠀⢸⠀⠀⡇⠀⡟⠀⠀⢃⢱⠀⠀⡇
⠀⠀⠀⠀⢸⡴⠃⠀⠀⡀⠀⡀⠂⠙⡗⠀⠀⠀⠰⣚⠓⠒⠈⢉⣉⣉⣉⠏⠀⠀⠀⢺⠀⠀⡇⠀⠀⠀⠈⡇⠀⠑⠤⢝⡂⣠⢧⣈⡴⠊
⠀⠀⠀⠀⠸⡤⠤⠤⠄⠒⠒⠒⠒⠒⠃⠀⠀⠀⠀⠈⠉⠉⢀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠒⠒⢧⠠⠤⠄⠀⠁⠀⠀⠀⠀⠙⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠉
⠀⠀⠀⠀⠀⠀⢀⡠⠤⡔⠒⢤⣀⡀⠀⠀⠀⠀⠀⠀⣿⡿⡇⠀⠀⠀⠀⠀⠀⢰⠒⠟⣻⣓⡆
⠀⠀⠀⠀⠀⡔⠉⠀⠀⠘⣄⠘⡄⠈⢢⠀⠀⠀⠀⠀⡟⡇⣣⠀⠀⠀⠀⠀⠀⢘⢁⠿⡵⠉⢹⠀⠀⠀⠀⡴⠒⢤⠔⠒⢲⠔⠲⡤⢲⢦
⠀⠀⠀⠀⡰⠻⣄⠀⠀⠀⢈⡏⠉⠓⠋⠀⠀⠀⠀⢜⣉⣉⠈⠑⡆⠀⠀⠀⠀⢸⠀⡜⠀⣀⢸⠀⠀⠀⢀⣇⣀⣁⠤⠤⣁⡠⠤⢤⠤⣴⡄
⠀⠀⠀⠀⡟⠦⣈⡱⠦⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢈⠆⡜⠀⠀⠀⠀⠀⠘⡞⠠⠀⠀⠀⡇⠀⠀⢮⣵⣒⡬⣉⡡⢔⣂⠭⠭⡭⢧⠜
⠀⠀⠀⠀⠸⡀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠉⠀⢇⠀⠀⠀⠀⠀⠀⡃⠀⠰⠀⠄⢱⠀⠀⠀⠲⠤⠒⠒⠒⠒⠒⠊⠉⠉⠉
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣄⣠⡨⠇⠀⠀⠀⠀⠀⠧⠤⠒⠒⠒⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠄⡡⢊⠄⠦⠓⡀⠀⠀⠀⠔⠀⠉⡄⡁⠀⠀⠒⠲⢀⢀⢀⠀⡒⠒⢒⣒⠒⠀⠀⢀⡤⠄⠀⢀⡀⠀⡄⢀⣀⠀⠠⡄⠀⢠⠴⠒⠢⠀⡀⡀
⠀⠀⠀⢀⣀⡀⠂⠈⠐⠭⠀⠀⢁⠠⢀⡄⠐⢦⠼⢚⡋⠕⠀⠀⡗⠄⠜⠣⠠⠛⠄⠨⠣⠀⠸⠀⠀⢨⡇⠀⠀⢹⡧⠀⡗⠈⠁⠈⠀⠀⠀⠀⡆⣲⡀
⠀⠀⠠⡋⠀⢈⠀⠠⠀⠀⠐⠚⢡⢐⠁⠀⠀⠀⠉⠀⠀⠒⠀⠀⠈⡉⠉⠉⠉⠉⠈⠉⢉⠉⠁⠀⠀⠀⠓⠀⠉⣁⣄⠀⠀⠀⠁⠀⠀⠀⣄⠀⠑⡀⢥⠀⠉
⠀⠀⠀⡈⠒⢋⡀⠐⠑⡄⢠⠛⢡⣸⣇⡀⠀⠀⠀⠀⣤⠒⠒⠁⠀⠀⠀⠂⠂⠂⠀⠀⢀⢀⡰⡕⠆⠀⠀⠞⠋⠁⣈⡄⠀⠀⠠⠤⠄⠤⢼⠀⠀⠢⠔
⠀⠀⠡⡄⠀⠉⠀⡀⠀⠊⢈⡟⠁⠀⠐⠉⠑⠀⠀⠀⡄⠈⠁⠀⠊⠉⠉⠙⡉⠋⠉⠡⠀⠀⠄⠇⠈⠀⠀⠘⡞⣝⡬⡅⠀⠀⢈⠀⠁⠀⣸⠀⡀⠀⠁⣌⠶⡀
⠀⠀⠀⠐⠄⠘⠁⣀⠀⠀⠈⠈⠀⠀⠀⠀⠂⢃⠀⠀⣍⠐⢀⠀⢰⡄⠘⢀⠀⠰⠎⠀⠀⠀⡀⠃⠀⠀⠀⠀⢉⠀⠀⡀⠀⠀⠈⠐⠐⠂⠐⠀⡤⢆⠻⠄⠎⡅
⠀⠀⠀⠢⢆⠐⠋⠁⣀⡀⠀⠀⢠⠀⢰⠴⢚⡀⢂⠀⠁⠀⠀⠎⠀⠁⠀⠀⠀⠐⠀⠀⠀⠠⠀⡇⠀⠀⡀⠀⠀⢲⡛⠝⠀⠀⠀⣶⡄⠀⠀⠀⠐⠨⠀⠤⠊
⠀⠀⡄⠀⣈⣔⠾⠛⠉⠑⠈⡀⠀⠀⠖⡿⣂⢁⠈⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⡃⠀⠀⢠⠀⠀⠀⠀⣠⣆⡴⡆⠈⢣⡈⠓⠠⢠⣄⠀⢶⣦
⠀⠀⣇⠘⠉⠀⠀⡀⢀⡀⠀⠠⠀⠠⠀⢼⣀⡥⠀⢠⠀⢣⠘⠀⡀⠀⠐⠀⠀⠠⠀⢀⡠⠄⠀⡇⠐⢀⠈⣠⣾⢆⣤⣫⡻⢻⠃⡆⠈⢳⡀⠀⢀⣠⢤⣌⡉
⠀⠀⢻⣤⠀⡈⢿⠿⠡⠙⡄⠀⢃⠀⠡⠀⠄⠂⠁⠀⢁⠀⠂⠀⠀⠀⠂⠀⠒⠐⠀⠈⠐⠒⠀⠈⠉⠀⣀⠈⠁⡆⠀⠀⡅⠈⠞⠁⠀⠀⢀⣀⠃⠈⠀⠌⠹⣄⡀
⠀⠀⢸⣿⡄⠑⡀⠀⠁⠩⠅⠰⡀⠀⠀⠀⢀⡀⠄⠐⠀⠀⠀⢀⠀⠀⠀⠚⠋⢉⠉⠁⠀⠀⠀⠈⠉⠁⠀⠀⢀⣝⣛⣻⡇⠀⠀⠀⣃⠑⡄⢻⡞⡐⡰⣀⢰⡟
⠀⠀⠈⢿⣧⠀⠐⢈⠠⢄⠦⠂⣡⠈⠂⠉⠁⡀⣤⡄⠀⠀⠀⠀⢰⠇⠑⣰⡆⣶⢀⢄⠀⡂⣄⡰⡶⠄⠀⠀⠈⠉⠉⠙⠁⠀⢰⠚⠁⡉⠀⠀⠙⠬⣉⠡⠊
⠀⠀⠀⠀⠂⠀⡄⠈⢈⠄⠂⣉⣀⡄⠀⢰⡁⠀⢈⠿⠀⠀⠀⠐⢀⡐⢀⠀⣄⣄⣨⠀⡀⠀⡀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃⠀⠘⠀⣘⠁⠆⠀⠀⣠⠼⣄
⠀⠀⡀⠈⠀⡠⠁⠀⣈⠲⠚⠉⢁⢄⠀⠀⠙⠀⠁⠀⠰⠣⡄⠀⠤⠶⠂⢀⣄⣄⠀⣠⣀⣠⢠⣦⠈⠀⣠⢤⡀⠀⢀⡤⠸⡍⠀⠀⣦⣀⠼⠀⠀⠀⠀⠂⠀⠂
⠀⠀⠈⡉⠁⡀⠀⠂⡠⠈⢂⠐⠀⠈⠀⠀⠀⡘⠉⠂⠀⠇⠉⡀⠀⠀⠓⠀⠉⠓⠈⠉⠀⠀⠋⠀⠀⠸⣗⣊⡕⠀⠰⠿⠟⠇⠀⠀⠀⡶⡀⠀⡐⠁⠀⠒⠛⠠
⠀⠀⠸⠁⣚⠀⠈⢀⠔⠂⠀⠑⠒⠀⠀⢀⠀⡀⠀⠈⡀⠨⠔⠊⠄⠀⢰⠾⠒⠀⠀⢠⡖⡐⠄⠀⢀⣀⡈⠁⠀⣞⠐⣶⣖⢐⣇⠀⢰⠓⠃⠀⡀⠈⠉⠉⠉⢰
⠀⠀⠀⠤⡝⠀⠈⠃⢀⡖⠁⠀⠄⢠⠀⠇⠀⢷⣒⠀⡇⠀⠀⠀⣠⡀⠁⡀⠀⠃⠀⢠⡀⠀⠃⠀⣯⠬⣹⠆⢘⠛⣃⠄⢴⣞⢸⠀⠈⠀⢰⠀⠈⠒⠒⢒⠚⠁
⠀⠀⡼⠲⡂⠀⠀⠀⠉⡀⠀⠀⠈⠉⢀⠄⠀⠘⡀⠀⠇⠀⠀⠋⠁⠀⠀⠐⢄⡸⠀⠃⠀⢹⠂⠀⠉⠛⠋⠀⢠⣘⠿⢧⣴⣄⣬⠀⠀⠐⢪⠀⠀⠀⠀⣘⣲⡀
⠀⠀⢀⣩⠰⣆⠀⠔⠉⠁⠐⠉⠉⠀⠰⠞⠃⠀⠐⠚⠀⠀⠀⠠⠤⣀⡀⠀⠀⠀⠈⠛⠒⠎⣀⣀⡄⠀⠀⠀⠀⠉⠉⣉⣉⣍⣤⣤⣬⣿⠟⢲⡆⠀⢸⣵⣦⣿
⠀⢸⡿⠿⠃⠁⣠⠀⢔⠀⠡⢶⠄⠀⣀⠀⣀⠀⢸⠋⢆⠀⠀⠀⠀⠀⠬⢹⠂⠙⠁⠀⠀⢀⣀⣀⣦⣠⠈⠈⠀⠉⠛⠋⠙⡞⠌⢉⣉⣁⣀⣛⡁⠀⢸⡀⠁⣻
⠀⠀⠁⠌⠀⠀⠘⢅⠀⠈⠉⢀⠔⠅⠀⠈⠐⠊⣺⡄⢺⡀⠀⠀⠀⠈⡡⠞⠁⠉⠉⠀⠀⢀⣀⠀⠀⠀⡠⠐⠐⡐⣢⢄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠃⠀⠉
⠀⠀⠀⠈⠀⠀⠀⠈⡢⠀⠀⢢⢁⣤⢠⠆⠴⠘⡋⣷⢸⡇⠀⠢⠐⠈⠀⣀⡤⢴⡾⠎⠉⠈⡾⡀⠀⠐⠠⠐⠂⠈⠉⠀⡇⠀⢐⠒⠀⢀⢉⣉⣀⣀⡀⠀⠀⢀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢥⠀⠁⠊⡀⠀⠀⡀⢔⠐⠀⠙⠂⠠⠰⡂⣩⡀⠇⠀⢈⣰⠢⠀⠀⢡⡇⠀⠀⠋⠐⠀⠒⡄⠳⠁⠀⠀⠑⣀⣀⡀⠀⠚⠋⠁⠤⠤⠌
⠀⠀⠀⠀⠀⠈⢀⣀⠀⠀⠀⠀⠀⠀⠁⢁⡀⠁⢀⠀⠀⢰⢰⡄⢛⠧⠆⢀⠀⡀⠀⠀⠀⠀⠚⠁⠀⠀⠈⠂⠰⠊⠐⠁⠀⠀⡀⠀⠀⠐⠒⠂⠐⠀⠀⠉⢳
⠀⠀⠀⡐⢁⠺⡄⢸⠶⠌⠀⠀⠀⠐⠀⠌⠀⠀⠀⠀⠀⢸⠈⡇⠘⠂⠃⠉⠀⠀⠀⠀⠀⠠⢤⡄⠠⠐⣀⠁⠄⠒⠆⢠⡤⠀⢶⠀⠀⠒⠂⠀⠘⠂⠀⠀⢫
⠀⠀⢰⡀⣏⠢⠱⠪⢺⠇⡤⠀⠀⠀⠄⣈⠤⠠⠄⠂⠀⠀⠁⠁⣠⠖⠋⠋⠉⠳⡠⠀⠀⠀⠈⠀⠐⣀⠈⠠⠊⠀⠀⠘⡇⢤⠘⡄⠠⠰⠀⠀⢰⠃⠀⠀⠋⡆
⠀⠀⠘⠑⠌⠀⠓⠒⠈⠄⠁⠀⡀⠀⠀⠀⠤⡤⣴⣎⣉⡆⠀⠐⣅⡀⠀⠀⢀⡼⠁⠠⠀⠀⠀⠀⠀⠐⠂⠂⠂⡆⠀⠀⣷⠀⠀⡇⠀⠀⠒⢲⠊⠃⠈⠹⠁⠃
⠀⠀⠀⠑⠠⠀⠀⠀⠠⠐⠁⠀⠁⠀⠀⠀⠈⠉⠙⠋⠴⠃⠀⠀⠈⠉⠓⠚⠁⠀⠀⠰⡖⠲⠶⠶⠤⠼⠧⠌⠈⠁⠀⠀⠸⠍⠒⠈⠀⠦⠄⠁⠤⠤⠖⠂⠀⠁

⠀⠀⠀⠀⢀⡴⠞⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣧⣀
⠀⠀⠀⠀⣾⠟⠿⠿⢏⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠛⠿⢿⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣠⡞⢠⡦⠀⠀⠀⠀⠀⠀⠀⢘⣛⣭⡶⢶⣭⣛⡃
⠀⠀⠀⠀⢻⡄⠉⣀⠈⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⠁⠀⠉⢀⡟⠀⠀⠀⠀⠀⠀⠀⢰⡏⣠⠞⠁⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠰⣆⢸⣿⡇
⠀⠀⠀⠀⠸⠿⢦⣉⡤⠾⠧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣌⣛⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠶⠶⠿⠿⠇



⠀⠀⠀⠀⠀⢀⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢠⣆
⠀⠀⠀⠀⠀⢸⡏⠛⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⢦⣤⢾⢀⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣿⡛⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠟⠙⣣⠄
⠀⠀⠀⠀⠀⣸⡇⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠺⣿⣿⣿⠊⠃⠀⠀⠀⠀⠀⠀⢸⣯⡀⣿⠈⢙⣻⢤⣦⠀⠀⠀⠀⠀⠀⠀⢀⣬⣭⣥⣶⣶⡿⠟
⠀⠀⠀⠀⣿⣿⣷⣤⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡛⠛⣻⣤⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠘⣿⡆⣻⡧⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢾⣿⠿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⢬⣭⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⣶⣾⡷⠤⢾⣷⡦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⣶⢹⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡼⣿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢣⣦⡄
⠀⠀⠀⠀⣴⠟⢧⣀⡾⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡏⣛⠛⠛⠟⠃⠀⠀⠀⠀⠀⠀⣀⣤⠴⣦⡘⠿⠿⠆
⠀⠀⠀⠀⠙⠶⠞⠉⠷⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠹⠷⠾⠟⠋



⠀⠀⠀⠀⢀⣴⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡟⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠴⠶⠖⣦
⠀⠀⠀⠀⠉⢵⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⢻⡆⣄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⢹⣷⣦⠀⠀⠀⠀⠀⠀⠀⢀⢸⡇⣴⠛⠀⢿
⠀⠀⠀⠀⠀⠀⠙⠿⣟⣶⡀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣼⡇⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⢶⣿⢁⡿⠀⠀⠀⠀⠀⠀⠀⢸⡿⣇⠙⠒⢃⣸⡆
⠀⠀⠀⠀⠰⠚⢳⡀⢉⡛⠗⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠇⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣀⡼⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⣿⡿⠟⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠉⠀⠉⠀⠀⠀⣠⠤⠀⠀⠐⠠⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠐⠒⠤⡄⠴⡴⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠞⠣⠒⠗⠒⠀⠀⠀⢄⢠⠤⠄⠤⣤⠲⠈⠀⡠⠶⠈⠑⠀⠤⣥⡔⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣁⣂⣀⠁⠀⠈⠀⠀⠀⠀⠀⢂⠤⣎⣁⠠⠄
⠀⠣⠀⠐⠠⠄⠀⠀⠀⠤⠀⠀⠀⢠⠞⠀⠀⠀⢎⡀⠀⠀⠀⠀⠀⠀⠁⢀⡠⠐⠁⠀⢠⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠤⠄⠊⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠓⠦⢀⡀⠀⡠⠖⡩⠄⠀⠀⠀⢸⡄⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠐⠈⠀⠈⠀⠀⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠭⠔⠉⠀⠀⠀⠀⠀⠘⡇⡞⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠠⠄⢠⡄⠀⠒⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⢀⡀⠀⠀⠀⠄⠠⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⠀⠀⠈⠀⢠⠘⣠⠀⠅⠤⠀⠃
⠀⠀⠀⠒⠂⠀⠀⠀⠀⠠⢀⡌⠀⠈⠀⠸⠁⠐⠀⠀⢀⠁⠉⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠠⠴⠄⠊⠀⢀⠘⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠠⠃⠀⡜⠚⠀⠁⠂⠀⠀⢄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠄⠀⣴⠄⢀⡀⠙⠂⡀⢀⡀⠀⠀⠐⠧⡄⠣⡀⠀⠀⠀⠀⠀⠛⡠⠄⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⡀⠀⠰⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠄⢠⣄⣀⠀⠀⠀⢀⠀⠐⠠⠤⠀⠉⠈⠒⠈⠁⠀⠀⠀⠀⠀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠈⠠⢌⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄⠀⠀⡐⠁⠀⠀⠀⠁⠀⠀⠀⢰⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠄⠈⠀⢀⠀⠒⠂⠉
⠀⠀⠀⠀⠀⢆⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⠀⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠨⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⢀⠀⡀
⠰⢦⡀⠴⠄⠦⠔⠃⠀⠀⣀⠤⢠⡄⡶⠠⢄⠀⡆⠀⠀⠀⡀⠀⡝⠤⢀⡀⢠⡄⢤⣤⠀⣀⡠⠄⠀⡢⠄⠠⠄⠀⠀⣀⠤⠀⢨⣅⡰
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠐⢠⠀⡥⠀⣶⠀⠀⠀⠀⠀⠀⠀⠁⢀⢈⣛⡀⠀⢠⠀⠀⠘⠡⡤⠀⠁⠀⠕⠀⢀⡀⣤⠀⠀⠂⠁⠁
⠀⠀⠀⠀⠀⠀⠠⠄⠆⠀⠀⠐⠈⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⢩⠁⠀⣠⡀⠑⠓⠐⠆⠀⠀⠀⠀⠈⠉⠀⠉⠀⠂
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠀⣀⣀⣈⡀⠴⠗⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠤⠀⠀⠀⠠⠀⠀⠃
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⢬⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠁⠀⠈⠁⠉⠀⠦⠤⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠷⠀⠀⠐⡶⠀⡀⠀⠀⣀⡀⠀⠁⠀⠄⠀⠺⢷⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠔⠀⠀⠠⠤⠀⡀⠠
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠌⠓⠀⠀⠁⠀⠒⠀⠀⠀⠑⠠⠌⠂⠉⠐⠂⠉⠀⠀⠀⠀⠔⠀⠑⠒⠂⢀⣁⡀⠀⠈⠁⡠⠀⠀⠀⠄
⠀⣤⣄⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊⠆⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⡀⠀⠀⠀⢈⠄⠠⠐
⡇⠀⠀⠁⢄⠘⠦⣄⢠⡄⠔⠀⠘⣰⠀⠀⠀⣀⣦⠄⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠡⠀⠀⣤⠀⠀⠀⢠⠁⠀⢠⡏⠀⠀⠀⠀⠀⡘⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠄⡀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⡀⠘⢇⣀⠀⣀⠀⠔⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⢋⡤⠀⠀⠀⠀⠐⢀⠁⠀⠀⠠⠒⠈⠉⠉⠁⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠀⠀⠀⠇⠀⠀⠀⠀⢀⠄⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠤⡀⠀⢀⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⣦⠀⠀⠀⢰⠀⠀⠃⢄⣀⡀⢀⣀⡰⠞⠀⡈⠀⠀⠀⠀⠀⢀⠀⠘⠀⠀⠀⠊⠔⣈⠉⠋⣀⠲⠀⠀⡄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⣊⠤⠖⠶⠤⡑⠀⠀⠡⠀⠀⠀⠀⠄⠀⠀⡐⠀⠀⣀⡀⣀⣀⠈⠁⠈⠣⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠊⠀⠀⠁⠀⠀⠀⠀⠀⠑⠀⠀⠀⠐⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡧⠼⣀⠀⠀⠀⠀⠀⠆⠀⠴⠀⠀⢀⠤
⠀⠀⠀⠀⠓⠈⠜⢉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⣰⠁⠀⠀⠀⠀⢠⠈⠁⠜⠂⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⢶⠀⠃⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣀⣠⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⣈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⢂⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⢀⡜⠹⠋⠟⡆⠀⠀⢠⣉⣀⣀⠀⠀⠀⢀⣀⣀⣀⢁⠀⠀⠀⠀⠀⣠⣴⠟⠉⠀⠀⠄⠀⠀⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⣢⡈⢂⠀⠀⢀⠃⠀⠠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠅⠀⠀⠀⡴⠟⠁⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⣀⡴⠊⠀⢁⠐⢎⡠⠚⠀⢀⠈⡇⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣅⠀⠀⠀⠀⠠⠀⠀⡄⠀⠀⡀⠐⠀⡸
⠀⡠⢔⣂⠍⠁⢀⡤⠆⠈⡔⠀⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⣧⠀⠁⠀⠄⠀⠀⣀⡀⠀⡄⠀⠀⠀⠀⢼⠿⣶⣶⠤⣀⣤⢀⣀⣀⢠⡤⣔⠚⡄
⠘⠐⣈⡄⠖⡀⠩⡀⢀⠀⡃⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢻⡤⠤⠤⠤⠤⠒⠁⠀⢀⠁⠀⠀⠀⠀⣾⢦⠥⢌⣾⠛⠛⠀⠀⣢⡀⠀⠨⠼⣧⠄
⡸⠋⠀⠰⠈⠁⠀⠀⠀⠊⠀⠀⠀⠈⠀⡰⠁⠀⠀⠀⠀⠀⢸⠀⢀⣀⡤⠄⠀⠤⠖⠩⠀⠀⠀⠀⣀⠧⢈⠉⠣⠜⠓⠨⠭⣽⣚⡤⠎⠔⡨⠟
⢫⣄⠀⢀⠠⣀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠈⡁⠈⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⠘⠠⣄⠄⡀⢈⣁⡀⡠⠞⠡⠤⠄⠂⠀⠀⢠⠐
⠀⠁⠒⠒⠑⠢⠭⣉⣉⡀⠤⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣂⣀⣀⣀⣀⣀⣰⠁⠀⠀⠀⠀⠀⠈⠒⠳⠭⠤⣄⡀⣀⡀⣀⠀⠤⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢀
⠀⠀⠀⣀⠜⠁⠀⠀⡀⠢⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣗⡈⠉⠀⠀⣞⣽⣷⡻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⠀⠀⠗⠀⡄
⢀⠔⢪⡀⠀⠀⠆⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡑⠦⠌⠭⠉⠉⠩⠍⠄⠀⠉⡀⠀⠀⠀⠀⠀⠀⢠⠆⠁⢀⠄⠀⠀⢀⠠⠗
⠘⠀⠗⡹⢄⠀⠀⠀⠀⢀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠊⠀⠠⠄⡌
⠀⠀⠀⠀⠈⡆⠀⣴⠃⠉⠀⠁⠠⣀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡀⠄⠂⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠢⠀⠂⠀⢀⠀⠀⢀⠄
⠸⡦⡀⠀⢠⣧⠞⠁⠈⠲⢄⡀⠀⠠⡑⠄⡀⠀⠀⠀⠀⡟⠉⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢀⡈⠇⣀⠂⠆
⠀⠘⡊⢗⢍⠡⠀⠀⠀⠀⠀⠈⠢⣄⠔⢀⣼⠀⠀⠀⠀⣻⡀⠀⠀⠠⠐⠊⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠡⢔⠉⡰
⠀⠀⠑⢀⠙⠫⡂⠀⠀⠀⠀⠠⠀⡠⣔⡝⠁⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠤⢉⠃
⠀⠀⠀⠀⠁⠠⡀⠀⠀⡂⠤⢐⠬⠊⠜⠀⠀⠀⠀⠀⠀⠸⡓⠀⠀⠀⡀⠄⠂⠁⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡁⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⢠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⢠⠇⢻⡀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⢀⡏⠀⠀⢧⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⡼⠀⠀⠀⠈⣇⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢰⠃⠀⠀⠀⠀⢹⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠘⣇⣀⣀⠀⠀⣀⣀⣸⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⡏⠀⠈⡇⣼⠁⠀⢹⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢸⠀⠀⠀⡇⣿⠀⠀⢸⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣼⠀⠀⢀⡅⢨⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠏⠀⠀⢸⠃⢸⡄⠀⠀⠧⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⣼⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⣠⠖⠦⣄⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣠⠴⠦⡄
⠀⠀⠀⡰⠃⠀⠀⠈⠙⠧⣄⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⢀⡠⠞⠋⠀⠀⠀⠙⣄
⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠘⢦⡀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⣀⡼
⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠙⠫⣄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠟⠋⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⠔⠂⠈⠉⠀⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣴⠶⣋⢛⢦⡀
⠀⡃⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠃⠘⠀⠠⠄⡀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠸⠪⠝⣃⡥⢚⡡⠖⠛⢇
⠀⡉⠀⡀⠉⠉⠀⠀⣀⣉⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠘⢈⡀⢈⣉⡩⠤⠒⠉⠀⠀⠀⡜
⠘⠒⠢⠂⠄⠐⠐⠤⠠⢶⠂⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠍⣉⣀⣐⠒⢒⣒⣒⡾⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠂⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠐⠀⠐⠀⠀⢨⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⡠⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡔⢪⠔⠒⡶⡄
⠀⠀⠀⠀⡔⠒⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⠐⢉⡉⠀⠁⢑⣻⡇
⠀⠀⠀⢀⠗⢐⠢⡀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡙⠃⠀⠁⡥⠜⠉⠤⠌⢣
⠀⠀⢀⠞⠐⠀⠇⠈⠆⠈⢇⠀⠀⠀⠀⠀⠀⠐⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠒⠒⠒⠒⠒⠒⢺⢁⡤
⠀⢀⠞⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠠⠀⠀⠀⠀⠀⠄⠐⠐
⢀⠚⠁⢪⣹⡠⠮⠒⠛⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠆⠀⠀⠀⣸
⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠈⠒⠋
⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⣠⠴⠊⠽⢣⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⢄⡀
⢀⣴⣋⣀⣀⠀⠠⢥⠤⠤⠤⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢐⡀⠥⠭⠤⢐⠙⢦⡀
⠈⡤⠤⠄⠤⣀⡠⠔⠢⠤⠴⡇⠀⠀⠀⠀⠀⠀⡠⠔⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠖⠁⠀⠀⠀⠀⠀⠈⢺⡇
⠀⠧⠀⢀⣀⣀⣀⣀⣀⣀⠠⠇⠀⠀⠀⠀⠐⣊⣀⣀⠀⠠⠀⠀⠀⠀⠤⢝⠢⠀⠀⠀⠀⠀⠈⠁⠀⠠⠤⠤⠤⠤⠤⠄⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠉⠀⠐
⠀⠀⠀⠀⢀⠆⡐⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣯⡀
⠀⣠⠀⢀⣀⡀⠐⢒⡒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⡄
⠀⠘⡉⠁⠀⠒⠒⠂⠀⣻⠯⣱⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⢀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⢀⡀⠱⣤⢀⣀⣀⡀⣴⢉⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠒⠒⠁⠀⡠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠈⠉⠀⠀⠀⢁⠆⢀
⠘⠠⣤⣤⡤⠤⠤⠤⠤⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⠂⡀⠈⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠀⠀⠀⠀⠠⠈⠰⠋
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢀⡀⠀⠠⠅⠀⠀⡠⠂⠈⠢⡈⠁⠠⠄⠄⠀⠂⠀⢀⠴⠀⠐⠛⠻⠀⠸⠇⠄⠄⢄⠀⠀⠊⠅⠀⠀⡠⠊⠉⠠⢀⠇⠘⡄⢀⠀⠀⠀⢀⠴⠀⠐⠛⠳⠁⠸⠔⠠
⠀⠀⠈⠀⠐⠀⠂⠁⠀⠀⢀⠜⠀⠀⠖⠀⠨⠀⠠⡀⠀⠀⡠⠊⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠂⠑⠄⠀⢀⠘⠀⠀⠖⠂⠀⠀⠀⠀⠀⠀⠠⠊⠀⡀
⠺⠂⠳⣀⠀⠈⠈⠒⠚⠀⠁⠀⠀⠀⠠⠀⠠⠃⠀⠈⠠⠈⠀⡔⠓⡜⢂⠀⣾⠃⠺⢆⠽⣠⠀⠈⠈⠒⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠔⠂⠔⢂⠀⣺
⠀⠘⠂⠁⠀⠀⡠⠐⠀⠁⠐⠠⠀⠀⠀⠀⠈⠀⠀⠀⡀⢀⠈⠲⢪⡖⡛⠀⢙⠒⠂⠛⠛⠏⠀⠀⠠⠐⠀⠀⠐⠢⠀⠀⠀⠀⠌⠀⠀⠀⢀⠀⠈⠶⢢⠔⡛⠀⢛
⠀⠀⠀⠀⠀⠀⡶⠤⣂⢴⡤⢴⡆⠀⡏⠀⠄⢩⠂⠀⠀⠆⠐⠈⡧⢾⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⡶⠄⣂⢠⡆⢶⡆⠀⡍⠀⠈⢩⠂⢰⠀⠀⠀⠈⡳⢮⠃⠀⠈
⠀⠀⠉⠀⠀⠀⠉⠉⠷⠘⣣⠍⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠆⠀⠐⠂⣀⢀⡀⠁⠀⠀⠉⠀⠀⠀⠉⠉⠳⡜⣫⠝⠁⠀⢠⠀⠀⡀⠀⠀⠑⠀⠆⠀⠐⠂⢀⢀⡀
⠀⢀⠊⠀⠈⢢⠀⠀⠀⠈⠁⠀⠀⠀⠸⠄⠀⠄⠀⠀⢀⣲⠦⠀⠀⡁⠀⠀⡀⣥⠀⠠⠈⠀⠈⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⣀⣶⠦⡀⠰⡉⠀⢀⡀
⠀⢨⠀⠀⡠⠂⢀⢀⣑⠄⢀⣴⠤⠂⠀⠀⠀⣦⣡⡾⠟⠁⠀⡀⠀⠀⠂⠀⠀⡀⠀⢀⡀⠀⡀⠂⠀⢂⡐⠄⢀⡕⠴⠠⠀⠀⠀⣤⣱⡾⠟⠋⠀⠀⠀⠁
⡄⢢⠘⠈⠀⠀⢈⠠⡔⠲⠵⠷⣄⠒⠀⠀⠈⢜⠁⠀⠀⠠⠊⠀⠀⠀⠒⠐⠀⠀⠄⠡⠚⠈⠀⠀⠄⠀⡴⠲⠾⠶⣄⠂⠀⠀⠀⢎⠀⠀⠀⠠⠊⠀⠀⠈⠂
⢀⡁⠀⣀⠀⠀⠈⢐⠂⠠⠄⢂⠄⢀⠀⢠⣂⠀⠀⠀⠀⠀⠘⠀⠇⡈⣄⠀⠀⠀⢀⡀⠀⣀⠀⠀⠐⢐⠂⠠⠄⢂⠄⠀⠀⣘⣀⢀⠀⠀⠀⠀⠸⠁⠇⡘⡀
⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⡯⠇⠀⠀⢀⠀⡰⠂⠺⠀⠀⠠⠀⠌⠢⠰⠁⠄⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⣠⡿⠇⠀⠀⢀⠀⡯⠒⠲⡀
⠀⠃⢀⠀⠀⠀⣠⠄⠒⠀⠲⠀⠐⠂⠁⠂⠈⠁⠐⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⢀⠀⠀⢀⠀⠃⠀⢠⢄⡒⣀⢶⣄⠐⠂⠑⠐⠈⠁⠐
⠀⠂⠁⠐⠊⠀⠃⠀⠀⠀⠀⠀⠆⠀⠀⡀⠀⠀⡀⡀⡀⠀⠄⢀⡀⠀⠀⢀⠀⠀⠀⠒⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⠀⡀⠀⠄⠀⡄
⠀⠀⠀⠀⠀⠀⡆⣀⠠⠀⠒⣀⠁⠀⠦⢷⠀⠘⡉⠁⡙⠀⠀⠀⠉⠀⠀⠉⠀⣀⠀⠀⠀⠀⠀⠀⣂⣀⠠⠀⠒⡀⠁⠀⢨⠷⠄⠘⠃⠉⠛⠀⠀⠀⠉⠀⠀⠉
⠉⠀⠉⢉⡄⠀⠀⠀⠀⢀⠨⠀⠁⢀⣀⡀⠄⠀⠠⠀⠀⡀⠴⣀⢦⠀⢀⡁⠉⠀⠉⠀⠋⢉⡆⠀⠀⠀⠀⠀⡤⠀⡃⠀⢌⡄⣄⠀⠃⠀⠂⠠⠴⣐⢦⠀⢀⡀⠉
⢉⣑⠈⢛⠄⠀⠀⡠⠊⠈⠐⡈⠃⠀⠖⠀⢀⠂⠀⣀⠖⠐⠒⠋⠺⠀⠰⡃⣀⣄⢢⣄⠨⢋⠖⠀⠀⡀⠊⠈⠀⢀⠃⠐⡔⠀⠀⠀⠀⠀⠔⠐⠒⠉⡳⠁⠘⡏⣑
⠀⠀⠀⠀⠀⠀⠪⠀⠀⠀⢀⠘⠀⠀⠔⠀⠰⠀⠨⡀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢋⠐⠀⠀⢀⠜⠀⠀⠆⠂⢀⠀⠈⡀⠀⠀⡠⠈
⠿⣆⡵⣠⠀⠀⠈⠃⠈⠃⠀⠀⠀⠀⠂⠀⠀⠃⠀⠈⠂⠊⠀⡐⡁⣜⢂⠀⣷⢃⡿⢂⠵⣠⠀⠀⠈⠁⠈⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠈⠂⠂⢀⡜⡁⣄⢂⠀⣾
⠒⠛⠈⠁⠀⠀⡀⠂⠁⠀⠁⠢⠀⠀⢀⣀⣃⣀⠀⠀⠤⠀⡀⠒⠢⢔⡚⠁⠙⠐⠓⠛⠋⠇⠀⠀⠀⠀⠉⠉⠁⠢⠀⠀⢀⣀⣂⣀⠀⠀⠤⠄⡈⠖⠢⢔⡺⠀⢋
⠀⠀⢀⠀⠀⠀⡷⢒⣒⢒⡒⢶⡇⠐⡅⠀⠀⢨⠂⢀⠀⠆⠀⠈⡗⠺⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⣶⢒⣂⢒⡒⢶⡆⠐⠍⠀⠀⢨⠂⢰⠀⠀⢐⠈⡷⢮⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠌⠶⡜⣣⠍⠀⠀⢀⠀⠀⠈⠀⠀⠁⠒⠒⠀⠐⢂⠀⠀⢀⠀⠀⠀⠀⣀⡀⠀⠙⠌⠷⠜⣯⠈⠁⠀⠀⠀⠀⡀⠀⠀⠁⠒⠂⠀⠐⠂⠀⠀⢀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠁⠀⠀⠀⠰⠀⠀⠆⠀⠀⣐⣶⠶⡄⢠⣁⡀⣀⡀⡤⠀⠠⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⣶⠶⡀⠰⣁⠀⣀⣀
⠀⢠⠀⠀⡀⠂⢀⢀⡠⠄⠠⡔⠴⠂⠀⠀⠀⣤⣦⡿⠟⠁⠀⠈⠀⠀⠂⠀⠀⡀⠀⢠⠀⠀⡠⠂⠀⢀⡈⠄⠨⡔⢤⠢⠀⠀⢠⣔⣤⡿⠟⠁⠀⠄⠀⠁
⡔⢢⠊⠀⠀⠀⠀⠀⠌⠒⣋⠶⡴⠒⠀⠀⠈⢊⠀⠀⢀⠄⠊⢀⡀⢈⠁⠂⠉⠀⣔⠠⠈⠁⠀⠀⠁⠀⡌⠒⣟⠧⢴⠒⠀⠀⠀⢎⠀⠀⠀⠠⠊⣀⡀⢈⠁⠂⠉
⢀⡁⠀⣀⠀⠀⠈⠜⠂⠰⠄⢀⠄⠀⠀⢠⡇⠀⠉⠁⠀⠀⠘⠒⠃⡌⣄⠀⠀⠀⢀⡁⠀⣀⠀⠀⠀⠈⠊⠡⠤⢈⠄⠀⠀⢘⡄⠀⠉⠉⠀⠀⠘⠒⠂⡬⣄
⠁⠈⠂⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣨⠯⠇⠀⠀⢀⠀⡸⠂⠸⠀⠀⠠⠁⡀⠃⠰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠨⡀⠀⠀⣨⡿⠅⠀⠀⢠⠀⡯⠂⠒
⠀⠁⠀⠀⠀⠄⣠⢀⡔⠭⣳⠄⠘⠄⣉⣀⢈⡁⠜⠀⠀⠀⠁⠀⠀⠀⠀⢘⠀⠀⠀⠀⢀⠀⠃⠄⣠⢆⡔⢭⣳⢄⠘⠆⣉⣀⣈⡁⠜⠀⠀⠀⠁
⠀⠂⠀⠐⠀⠀⠁⠂⠀⠀⠀⠐⠇⠀⠀⡀⠂⠀⣀⣀⡀⠀⠄⢠⡀⠀⠀⢰⠀⠀⠀⠂⠀⠑⠀⠀⠡⠀⠀⠀⠀⠈⡆⠀⠀⠠⠂⠀⣀⣀⣀⠀⢀⢠⡀
⠀⣀⠀⠀⠀⠀⣎⠀⠄⠀⠀⢈⠀⠀⠲⢿⠀⠘⠅⠀⠜⠀⠀⠀⠀⠀⠀⢀⠀⡀⡀⣀⠀⠀⠀⠀⣂⠀⠄⠐⠀⣀⠀⠀⢰⡷⠀⠸⠁⠀⠘⠀⠀⠀⠀⠀⠈⢀
⣁⣀⣉⣈⡀⠀⠀⠀⠀⠀⡈⣀⣁⢀⣀⣀⣄⠀⠠⠠⠀⣀⡒⣑⢢⡀⢀⣀⣁⣀⣉⣀⣃⣀⡂⠀⠀⠀⠀⢀⡤⣀⣃⢀⣌⣄⣀⠀⠃⠀⠂⣀⠐⣑⣦⡀⢀⣀⣁
⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢨⠀⠀⢠⣆⣧⣴⣶⣶⣷⣿⣿⠿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⢄⣄⠀⠀⠀⢀⣄⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠆
⠀⠀⠀⠀⣠⣿⢿⣿⡿⠛⢻⣿⣿⠳⣾⣿⠀⠓⠒⠀⠀⠀⠀⣠⣼⣿⣿⣤⣀⠘⠋⠛⠿⡧⡄⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⡀⠀⣿⣿⢸⡟⠁⠄⢻⣿⣷⡄⠼⠻⣿⣶⣨⡅⠀⠀⠀⠙⢿⣿⡿⠛⡋⠀⠀⠀⠀⠀⢈⣀⡄⠀⠀⠀⢺⣿⣷⠄
⠀⠀⢃⣤⣿⣿⡧⠀⠀⠀⢹⣿⠋⠀⠀⠀⡈⢻⠑⡆⠀⠀⢠⣴⣾⣿⣿⣦⢹⡆⠀⠀⠀⠐⠀⠈⣿⢦⠀⠀⡜⠈⠈
⠀⢀⡆⢿⣿⡟⠀⣗⠀⠀⢸⣿⠀⣄⠀⠠⢷⡄⢠⡇⠀⢰⣿⣯⣾⣿⣿⣿⣎⡇⢰⡄⠀⠠⢶⠀⣿⡿⠂⢸⠁
⡀⠋⢇⢸⣿⣷⣄⠛⡀⢀⣼⣿⠃⠁⠀⠀⠀⠘⣿⣇⣴⣾⢿⣿⣿⡿⣿⣿⣿⣿⠟⠁⠀⠀⠈⠀⣿⣷⣴⠎
⡟⠀⣀⢸⠿⣿⣿⡙⠢⢭⣴⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⠹⠺⠈⠿⣷⣿⣿⡛⡄⠇⠀⠀⠀⢀⣶⣿⣿⠏⠀⠀⢀⠡⠀⡄
⠀⠀⠋⠈⢄⣠⣶⡇⢸⡂⠄⣿⠀⠀⠀⠀⠀⡀⠈⣸⣿⠀⠀⠇⣀⣿⣏⠙⣻⣟⠙⠛⠀⠀⠉⢯⢻⡁⠀⠀⠀⠀⠶⢰⠆
⣀⠀⢀⣖⢺⠒⢺⣷⢸⣧⣩⣿⡄⠀⠀⠀⠀⢀⠀⣽⡏⠀⠀⠂⢹⢿⡏⠀⠐⠻⡖⠇⠈⢸⠀⠀⣄⡗⣾⡂⠀⢰⢰⠈
⡬⣴⣄⡖⣶⣠⣤⡟⢸⢸⣿⢻⡇⠀⠀⠀⠀⠠⠀⢠⡇⠀⠀⠀⢰⠘⣿⠯⣙⣷⡇⠰⠠⣤⣸⠄⢺⠃⠃⡇⢠⡌⠸
⢓⠚⠻⣟⣿⠟⠋⠀⣶⢸⣿⠸⣦⠀⠀⠀⠀⢐⠀⢸⡅⠀⠀⠀⡸⢨⣟⠠⡟⡧⠄⢈⣰⠟⠋⢆⡘⠧⢰⣿⣿⠁⠘
⠀⠀⠀⠐⠠⢄⠀⠀⢿⢸⡟⠘⣳⠄⣀⡀⠀⠀⠀⣼⠆⠀⠀⠀⣡⢸⣿⣿⣟⢂⣠⣾⠙⠌⡃⠀⢃⢠⢼⠿⢿⡀
⠀⠀⢻⡀⠂⢘⡧⠔⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠈⠸⡟⡯⣟⣛⡯⢭⣾⣿⣷⣾⣿⣯⢿⣿⣼⣷⣶⡿⢦
⠠⠰⣿⠤⡀⠉⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣙⣽⣿⠧⠶⣿⣿⣏⣽⣻⡽⣟⣿⣿⠿⠿⠝⠓
⠀⠘⠂⠀⠀⠈⢓⡤⠬⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢐⡄⠅⠈⠉⠉⠙⠲⠮
⣾⣾⣶⣦⣤⣤⣶⣾⣿⣶⣷⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⢐⡀⠐⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡡⠐⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠐⡆⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⢠⠀⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀⢨⠒⢢⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠘⠚⡃⣘⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠰⡀⡆⠀⠀⠀⠀⠀⠀⠀⠘⠒⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢛⠀⡏⠆
⠀⠀⠀⠀⠀⠐⠁⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠃⠈⠀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⢄⡀
⠀⠀⠀⠀⣠⢤⣀⠀⠀⣠⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠐⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢁⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⡄⣉⠑⢋⠁⢀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢁⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠨⠀⡏⢉⡹
⠀⠀⠀⠀⢠⠒⢍⠀⠸⠬⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠶⠖⢲⠮⠁⠀⠀⠀⠀⠀⠀⠀⠔⢁⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠂⡁⠐
⠀⠀⠀⠀⠀⠀⠈⢀⠰⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠈⡂⠀⠀⡊⠀⠀⠀⠀⠀⠀⠀⠀⠤⠉⡰⠒⠠⠀⠈⠀⠀⠀⠀⠀⠀⠘⠐⠨⠒⠄⠁⠠⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀
⠀⠀⠀⠀⠀⢠⣄⣀⣀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣰⡂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢪⣂⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄⡀
⠀⠀⠀⠀⠀⢸⣰⣇⠉⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠤⣾⠤⢤⡀⠘⡄
⠀⠀⠀⠶⣤⡾⡟⢺⠗⠚⣲⠶⡋⠀⠀⠀⠀⠀⠀⣚⡀⠀⠴⠆⡈⠩⠀⠀⠀⠀⠀⠀⠥⠆⠀⠄⡖⠒⣒⠀⠀⠀⠀⠀⠸⠀⠀⠹⠆⣴⡤⠾⠅
⠀⠀⠀⣃⣿⣗⣇⠘⠢⣀⣽⡾⠧⠀⠀⠀⠀⠀⠀⠁⠴⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠬⠄⠀⠬⠀⠨⠥⠨⠅
⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢮⠀⠀⠨⠅
⠀⠀⠀⠒⠒⠒⠂⠘⠀⢐⠒⠒⠒⠀⠀⠀⠀⠀⠠⠈⠣⣊⠀⠀⢀⡖⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⣆⣛⣛⣟⡰
⠀⠀⠀⢈⠂⡑⠂⠐⠀⠑⠊⠚⠀⠀⠀⠀⠀⢠⠃⠀⣐⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⠀⠄⠀⠀⠀⠀⠀⡴⠓⠀⠀⠐⠋⠀⠁⢀
⠀⠀⠀⠀⠀⠈⠐⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠈⠀⠀⠀⠀⠀⠊⠀⠐⠀⠀⠐⠂⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⢴⣶⣶⡦⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣻⡭⠷⠚⠛⠉⠉⠉⠉⠉⠉⠛⠓⠾⢯⣟⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⡽⠞⠉⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠉⠳⢯⣳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠉⠀⠀⢀⣤⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣤⡀⠀⠀⠉⠳⣝⢦⡀
⠀⠀⠀⠀⠀⠀⣠⢿⡞⠁⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠈⢳⡿⣄
⠀⠀⠀⠀⠀⣴⣳⠏⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⣀⣿⠉⠉⠹⣇⡀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠹⣞⣦
⠀⠀⠀⠀⣼⣳⠃⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⢠⠖⠋⠁⠀⠀⠀⠀⠀⠉⠳⢦⡀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠘⣞⣧
⠀⠀⠀⢰⢧⠏⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⢀⡴⠒⠒⠦⣄⠀⣠⠾⠃⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢹⡼⡆
⠀⠀⠀⡿⣾⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⣄⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣷⢿
⠀⠀⢀⣇⡇⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠈⠙⠓⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⣸⡀
⠀⠀⢸⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⡇
⠀⠀⠈⡏⡇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⡀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⢹⠁
⠀⠀⠀⣷⣿⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢦⡀⠀⠀⠀⠀⢹⡄⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣿⣾
⠀⠀⠀⠸⡞⣆⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢾⡁⠀⠀⠙⠳⠦⠤⠴⠛⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣸⢳⠇
⠀⠀⠀⠀⢻⡽⡄⠀⠀⠻⡄⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⢠⠟⠀⠀⢠⢯⡟
⠀⠀⠀⠀⠀⠻⡽⣆⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣀⣀⣰⡟⠉⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣰⢯⠟
⠀⠀⠀⠀⠀⠀⠙⣾⢧⡀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⢀⡼⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢦⣀⠀⠀⠈⠛⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠛⠁⠀⠀⣀⡴⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢯⣳⢦⣀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⣀⡴⣞⡽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⣽⣳⡦⢤⣤⣀⣀⣀⣀⣀⣀⣤⡤⢶⣚⣯⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⠺⠿⠿⠗⠒⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠔⠒⡄
⡴⠒⢤⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠈⠑⠚⠁
⠙⠒⠉⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⣀⣀⣀⡀⢸⠀⠀⠀⠀⠀⡰⠊⠁⠀⠀⠀⠀⠀⠙⢆⢀⡀
⠀⠀⠀⠀⠀⣀⡷⠖⠛⠉⠉⠉⠉⠉⠙⠓⠦⣄⡀⣸⠁⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠉⠉⠓⢦⡀
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆
⠀⣸⠁⠀⠀⠀⠀⣰⣷⡀⠀⠀⠀⣰⣷⡀⠀⠀⠀⠀⠈⣇⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⣿⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⣟⣰⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠉⠉⢹⣿⣷⣶⢤⣄⣀⡠⠖⠋
⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⢸⣿⣿⣿⠀⠈⠙⣷⣄
⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⡷⣄
⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⠤⢤⡤⠶⠖⠛⠉⠀⢸⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⡇⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⣼⣿⣿⡿⠀⠀⢰⣿⣿⡇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⢰⣿⣿⠇⠀⠀⢠⣿⣿⣿⠇⠀⠀⣸⣿⣿⠀⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⣰⣿⣿⠟⠀⠀⠀⣾⣿⣿⡟⠀⠀⢠⣿⣿⠇⠀⢀⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡋⠀⠀⢀⣼⣿⣿⡿⠁⠀⢀⣾⣿⠟⢀⡴⠟⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢾⣿⣿⣟⣀⣀⣠⣾⡿⠿⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣀⣀
⠀⠀⣠⣾⣿⣿⡀⠀⢰⠋⢍⡩⢳⡄⠀⠀⠀⠀⣼⣧⠀⠀⢰⡿⠿⢿⡄⠀⠀⣾⠟⠻⢿⠀⠀⠀⣾⡍⣿⡇
⠀⠀⢸⡽⢯⡗⠀⠀⠸⣄⠈⠀⣸⠃⠀⢼⢴⡟⢿⡿⠀⠀⠨⠐⡀⢀⠃⠀⠀⣵⠀⢈⢸⡀⠀⠀⢨⡟⣯⡀
⠀⠀⠈⠁⠈⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠀⠁⠈⠁⠀⠀⠀⠈⠀⠁⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠈⠁⠈⠁
⠀⢀⣤⡀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠠⡦⠤⠤⡆⠀⠀⡄⠀⡄⠀⠀⠀⠀⢀⡤⢦⣀⠀⠀⢀⠴⠶⠲⢦
⠀⠀⡿⡧⠤⡧⡬⠀⢐⡥⣧⡀⠀⠀⠀⠀⡇⣀⡀⡇⠀⠀⡇⠀⡟⠀⢸⠀⠘⣿⣾⢿⣮⠃⠀⢸⡾⢃⣦⢻⡇
⠀⠀⠓⠇⠀⠓⠓⠀⠐⠓⠇⠙⠦⠀⠀⠰⠃⠉⠀⠇⠀⠀⠧⠤⠷⠤⠼⠀⠀⠿⠇⠸⠿⠀⠀⠈⠓⠾⠿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⢀⢀⣤⣄
⠀⠘⣶⣾⣶⡷⠆⠀⢀⣴⣶⣾⡏⠁⠀⣀⣀⣀⣄⡀⠀⠀⢰⢶⡠⢆⡀⠀⠘⠿⣅⣠⢹⠟⠀⠀⣿⣿⣿⡿
⠀⠀⠁⠤⠿⠄⠀⠀⠀⠯⠤⠬⠧⠀⠀⠻⠟⠛⠻⠟⠃⠀⠕⠏⠀⠜⠜⠀⠀⠐⠌⡡⠜⠀⠀⠀⠈⠃⠋

⠀⠀⡖⢛⡛⠓⡆⠀⣤⠐⢲⠖⠢⡄⠀⢰⠴⠦⠠⣶⠀⠀⢾⡷⠀⣿⡇⠀⢰⣶⣶⣶⡶⠆⠀⢀⣠⣔⣼⣴
⠀⠀⡧⡤⡥⡴⡃⠀⣿⠀⢸⠀⠀⡇⠀⢸⠬⠭⠩⢽⠀⠀⣠⣄⠀⣤⡀⠀⢸⣿⣿⣿⡷⣃⠀⠠⢤⡤⢼⠤
⠀⠈⠛⠛⠛⠛⠁⠀⠙⠛⠛⠛⠛⠃⠀⠘⠀⠀⠀⠈⠀⠀⠈⠁⠈⠙⠁⠀⠈⠉⠉⠉⠉⠀⠀⠈⠙⠂⠘⠘
⠀⠀⣠⣶⣶⣦⠀⠀⢠⣦⣴⣤⣦⠄⠀⢴⣄⠀⣠⣦⠀⠀⠀⡄⡆⣠⠀⠀⠀⣤⣄⠀⢠⠄⠀⠀⡶⠶⠶⡶
⠀⠀⡇⠴⡆⠀⡇⠀⢸⣿⠁⠸⣿⠂⠀⠀⣙⣷⣟⠁⠀⢀⡀⢳⣤⣥⠋⠀⠰⡌⢄⡴⣃⡄⠀⠀⣇⣀⣀⡇
⠀⠀⠧⠤⠧⠤⠃⠀⠘⠿⠻⠟⠿⠁⠀⠼⠋⠁⠙⠡⠀⠀⠈⠻⠇⠟⠀⠀⠀⠩⠿⠤⠶⠃⠀⠀⠉⠻⠹⠁
⠀⠀⢠⡀⢀⣠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣲⣉⣉⣰⠀⠀⢈⢉⣉⣩⣩⡄⠀⠏⣿⢹⢹⢙⠀⠀⠀⠀⢸⢷⡀⠀⠀⣠⢦⣠⢦⡀⠀⠀⢀⣾⣦
⠀⠀⢸⣿⣿⣿⠀⠀⠘⠸⠻⠻⠿⠇⠀⣄⣁⣨⣸⣸⠀⠀⢠⣾⣿⠀⠀⠀⢸⠉⠉⠉⠉⣳⠀⠀⣸⣿⣿⣆
⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⠥⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠤⢴⠀⡦⠤⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠂⡀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⢀⠰⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠨⠉⠉⠁⠒⣒⣒⣒⠒⠀⠉⠉⠁⠀⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⢀⡼⢷⡀⠀⠀⠀⢀⡼⢷⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⢁⣴⣶⣦⡈⠁⠈⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠀⠙⠀⠐⠈⠉⠁⠀⠤⠀⠀⠀⠀⠀⠘⣿⡟⢻⠃⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢄⠁⠀⠁⣀⠀⠀⠡⠀⠀⠀⢀⡌⠀⠀⡈⠀⠁⠀⠄⠈⠓⠃⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉⠂⠀⠀⢢⠀⠀⠘⠁⠀⢠⠒⢠⠀⢀⠠⠐⠀⠠⠀⠂⠀⠀⠀⠈⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠘⠀⠀⢸⡆⠀⠀⠀⠀⠆⠀⠀⢀⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢠⣶⡄⠀⠀⢠⣶⡄⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⠄⣀⠀⠀⠀⠀⠀⠀⢀⡀⠜⠀⠁⠀⠀⠀⠀⠈⠛⢡⣶⣶⡌⠛⠁⠀⠀⠀⠀⠀⠘⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣈⠉⠀⠈⠁⢁⣀⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⢏⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠃⢀⡀⠀⠋⠈⠉⠀⣐⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠂⠁⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠄⠀⠀⠀⠀⠄⠱⠄⠀⠀⠀⠀⠀⠠⣤⠒⠉⠉⠉⠁⠀⠀⣀⠀⠄⢺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⢀⠃⠀⠀⠑⠚⠉⠁⠀⠀⠉⠀⠙⠀⠘⠋⠁⠀⠀⠀⢰⡆⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠈⠉⠁⠨⠍⠁⠉⠉⠉⠉⠉⡉⠉⢉⠉⠉⠉⠉⠉⠻⠄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠈⠐⠀⠀⠂⠀⠉⠉⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠒⠂⠈⠁⠀⠀⠀⠀⠀⠁⠀⠒⠒⠒⠂⠀⠒⠒⠒⠒⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠛⠉⠙⠛⠒⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⢿⣿⣿⠟⠋⠉⠉⠉⠉⠳⣶⣾⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠻⢿⣀⡀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⣄⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⡄⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠉⠒⠒⠲⠶⠒⠒⠉⠀⠀⠀⠀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠋⠉⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⣠⣎⣉⣳⡀⠀⠀⠀⠀⡤⡤⠒⡶⣠⠒⡦⡀
⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠉⢆⠀⠀⠀⠀⢠⠮⠭⠭⠭⠽⠄⠀⠀⠀⣸⠥⠄⠬⠦⠨⣆⠀⠀⠀⣜⠁⣼⡙⣧⢎⢫⣾
⠀⠀⠀⢠⡗⡒⡒⢖⠠⣒⡲⣶⠾⡀⠀⠀⠀⠘⡎⠉⠉⠉⢹⠁⠀⠀⠀⠙⣟⢻⡭⡛⢹⠁⠀⠀⠀⢹⠙⠓⠓⠚⠚⢉⡇
⠀⠀⠀⠀⢹⠒⠾⡉⣘⢿⠓⠞⡟⠀⠀⠀⠀⠀⢣⠀⠀⠀⡌⠀⠀⠀⠀⠀⠘⡟⢤⣺⠃⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠉⠁⠘⠳⣰⠞⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⡁⡝⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠄⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢀⡤⢒⣂⡉⣐⠲⡄⣠⢤⠀⠀⠀⠀⠀⣀⠤⢤⠒⠢⡄⠤⡀⠀⠀⢀⡠⠔⠒⠒⠠⢄⠀⠀⠀⠀⠀⢀⠤⠒⠢⢄
⠀⠀⠀⢠⢋⠖⠧⠎⢀⡠⡫⠞⠹⡆⠇⠀⠀⢠⠞⠁⠀⠀⠣⡀⢣⣀⣈⠆⠀⣎⣀⣀⣀⣀⣀⣠⡕⠀⠀⠀⢠⠋⠀⠀⠀⠀⢣
⠀⠀⠀⢸⠈⠆⠏⠋⠻⠒⡿⢭⢅⠙⠂⠀⠀⡜⠣⡀⠀⠀⢀⡝⠁⠀⠀⠀⠀⠈⡄⢠⠈⡄⠰⢠⠁⠀⠀⠀⠸⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⢆⠳⢖⢐⢢⣀⡰⢋⡞⠀⠀⠀⠀⡗⢄⣈⡶⠒⠁⠀⠀⠀⠀⠀⠀⠀⢡⣘⠀⢃⣃⠎⠀⠀⠀⠀⡼⢑⣤⠴⠊
⠀⠀⠀⠀⠀⠑⠢⠤⠭⠤⠒⠋⠀⠀⠀⠀⠀⢇⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠸⢭⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠠⠄⠀⠀⠀⠀⠀⠀⠀⣔⣒⣢⢠⡆⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⣴⢖⣂⠴⡒⣒⣄⠀⠀⠀⠀⠀⡤⠤⠤⠧⠧⠤⣄⣀⡀⠀⠀⠀⢸⠀⣇⠾⠼⠄⠀⠀⠀⢀⡔⡢⣄⡠⢚⠦⡀
⠀⠀⠀⠀⡎⡡⠒⠉⠒⠒⠒⢝⢦⠀⠀⠀⠀⢧⠀⠀⠀⠀⢀⣯⣗⠅⠀⠀⠀⡸⠀⡧⠋⠈⠣⡄⠀⢠⠊⠀⢱⠈⢳⠸⡄⠈⠆
⠀⠀⠀⢸⡴⠁⠀⠀⠀⠀⠀⠈⢿⠂⠀⠠⡆⠀⠷⠤⠄⠖⠋⠃⡲⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⠀⠸⡀⢀⠇⠀⢸⠀⡇⠀⡄
⠀⠀⠀⠸⠤⠤⠤⠀⠒⠀⠀⠤⠊⠀⠀⠀⠈⠑⠒⠒⠒⠒⠂⠉⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⠀⠀⠑⢤⣃⠴⠛⢄⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠣⠤⠤⠄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣄⡀⠀⠀⠀⠀⠠⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣀
⠀⠀⠀⠀⠀⠀⣀⠀⣞⣵⣵⠗⠺⡄⠀⠀⠀⢠⠃⠘⡀⠀⠀⠀⠀⠀⠀⣠⠊⠱⢼⢌⢷⠀⠀⠀⣠⣀⡠⠤⣄⡀⢀⣀⣀⡀
⠀⠀⠀⢰⣊⠭⣔⡺⣽⠝⠁⠀⠀⡇⠀⠀⠀⡞⣚⠓⠙⡆⠀⠀⠀⠀⣴⠥⠔⠚⠉⢉⡏⠀⠀⢸⠀⠘⠀⣀⠈⠀⠀⠋⠀⢹
⠀⠀⠀⢎⣪⠛⠊⠉⠀⠀⠀⣠⠎⠀⠀⠀⠀⠹⠀⡇⢰⠃⠀⠀⡠⠊⠀⠀⠀⠀⡠⠋⠀⠀⠀⣞⡋⢛⡓⠒⡞⠉⣖⡛⢳⣋⡢
⠀⠀⠀⠈⠡⡀⠀⢀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠸⠋⠁⠸⡄⠀⠀⠣⢄⡀⠀⠀⡔⠁⠀⠀⠀⠀⠘⠬⠅⠈⠫⠬⠍⠤⠬⠥⠜
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⡄⠠⠃⠀⠀⠀⠀⠈⠢⠞
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠦⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣀⢤⣑⠠⠄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸⠁⠈⠀⠇⠀⠀⠀⢀⠄⠂⠙⠳⢮⠉⠀⢙⠀⠀⠀⠀⢨⠁⡀⠱
⠀⠀⠀⠐⠠⢤⣬⡠⠀⠀⢦⠁⠚⠠⠀⠀⢀⣅⣀⣠⢴⡂⠀⠀⠀⠀⠂⠀⠀⢀⠴⡇⢐⠦⢀
⠀⠀⠈⠰⡀⢘⠿⠳⠬⠂⠈⠍⡄⢀⠔⠀⠀⢿⠙⠁⠐⠃⡸⠁⠠⠊⠀⠀⠀⠆⢠⠁⣶⠠⢠⡅⢀⠐⠤
⠀⠀⠀⠘⣡⡀⡠⠃⠀⣀⠥⠒⢐⢮⡁⡠⠂⢝⣄⠀⠂⣠⠁⠀⢀⠀⠄⠀⠄⡀⡨⢾⣁⡤⢏⢁⡈⠘⢁⡄
⠀⢀⠃⠀⠁⠈⠑⠀⡆⠄⣇⠀⡀⡀⢯⡠⢫⣽⢦⡁⢁⠜⡓⠀⡤⡂⠈⠣⠐⠈⠁⠀⡈⠀⢺⣀⡶⢄⠀⠁
⠀⠀⠀⠀⠀⠈⠉⠐⠁⠀⢉⡖⠭⢴⠁⠑⢄⣽⣶⣬⣉⡒⠂⡼⣡⢀⠀⢠⢦⡅⠀⠨⠤⠄⠀⠉⠀⡀⢃
⠀⠀⠀⡀⠀⠀⠀⡀⠀⡐⢁⣜⠄⠀⠀⠀⠀⠁⡀⠀⠀⠀⡙⠀⣳⠌⠐⠣⡀⠀⠀⠀⢀⠓⠀⠀⠀⡉⠁
⠀⠀⢠⠌⠁⠀⠈⢄⢠⢡⠋⠀⠀⠀⠁⢢⠀⠀⡍⠢⡀⠰⢀⣠⠁⠠⢄⢆⠑⡀⠀⠀⠌⠀⠀⠀⠀⢁
⠀⠀⢀⠀⠀⠃⠀⠀⡆⣆⠀⢀⡄⠀⢰⡄⠀⠐⢧⣄⣨⠣⠌⠝⠃⠀⠈⠈⠳⠐⡀⠐⠤⡠⣀⡀⢀⠀⢻⠖
⠀⠀⠀⠄⠈⠈⠀⠉⠸⠁⠀⠀⠀⠀⡀⠈⠒⢄⠔⠁⡸⢄⡀⠠⠤⠂⡄⠀⢀⡶⣴⠀⠀⠆⠀⠁⠎
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠁⠚⣉⣩⠂⠂⢀⢌⠀⣔⠩⠚⠛⠒⢄⠀⠁⠰⠈⠋⠀⠓⠃
⠀⠀⠀⠀⠁⠀⠒⠀⠀⠀⠀⠑⠠⢄⣀⡀⠀⠀⠈⠀⠒⠌⣹⠁⠀⠀⢀⠠⠂⠀⠀⠀⠀⠀⠀⠁⠀⢄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠉⢀⡀⠆⠠⠭⠔⠲⠀⠀⢀⡑⠒⠓⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠀⠀⠃

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠁⠀⠁⠀⠀⢀⠠⠤⠀⠀⠀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⢀⠤⡀⢤⠰⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠦⠤⠐⠀⠈⠀⠠⠀⡀⣤⠀⠤⢤⡌⠛⠂⠀⠤⠆⠉⠉⠁⠠⠤⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠂⠀⠀⠀⠀⠀⣀⣡⡒⠦⠤
⠀⠂⠀⠀⠢⠄⠀⠀⠀⠄⠀⠀⠀⡀⠞⠀⠀⠀⠆⠀⠀⠀⠈⠀⠀⠀⠀⢀⡠⠘⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁
⠀⠀⠀⠐⠀⠀⠀⢀⣀⠀⠠⠄⠊⠁⠀⠀⠀⠀⠀⠁⠒⠦⢀⡀⢀⡤⢊⠡⠀⠀⠀⠀⢠⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠔⠈⠀⠀⠀⠂⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠥⠖⠁⠀⠀⠀⠀⠀⠸⡄⠃⠀⠄⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⡤⠄⠤⠀⢤⠄⠀⢂⠀⡰
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠉⠀⠀⠀⡆⠀⠀⠄⠂⢀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠐⠄⠃⡤⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠌⠀⢀⠀⠘⠀⠐⠄⠀⠀⠁⠀⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⡀⠀⠀⠀⠀⠀⠀⡐⠁⢠⡕⠁⠀⠀⠀⠀⢠⡄⠀⠀⠀⠇
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⣀⠂⢀⠀⠙⠂⠀⡀⠀⠀⠀⠑⢧⠘⠀⠀⠀⠀⠀⠀⠐⢁⠤⠀⠀⠘⠀⠠⢀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠄⣀⠀⠈⠀⠀⠀⠀⠢⠤⠀⠈⠀⠂⠈⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠈⠠⢌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠁⠀⡄⠀⠊⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠦⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⢀⠀⣀⢀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠘⠲⡀⠶⠄⠦⠐⠃⠀⠀⡀⠀⢠⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢀⠁⠤⠄⠀⠀⠀⠤⠄⠀⢀⡀⠀⢀⠠⠐⠠⠀⠀⠀⠠⠄⠀⠀⢀⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⣁⡀⠠⠄⠄⠀⠈⠠⠄⠈⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⠀⠠⢤⠄⠙⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⠀⠀⠠⠤⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⢰⠀⠀⠨⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠈⠉⠉⠀⠦⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠐⠖⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠛⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠄⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠐⠈⠁⠀⠂⠀⠀⠀⠀⠐⠒⠈⣀⣁⠀⠀⠀⡁⠀⠀⠀⠀⠃
⢀⠤⢄⡀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠂⢀⠀⠀⢀⠀⠀⠐⠢⠀⠀⠀⠀⠀⡄
⡇⠀⠀⠈⢂⠘⠢⠄⠀⠠⠒⠀⠈⡠⠀⠀⢀⡄⠆⠆⠀⠙⣂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠰⠀⠂⠀⠀⠀⠀⠁⠂
⠀⢀⠀⠀⠀⠱⡀⠀⠄⠒⠀⠀⢰⠁⠀⢰⠑⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⢂⠈⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠊
⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣀⠀⢆⣀⣀⡀⠀⠂⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠥⠄⠀⠀⠀⠀⡀⠀⠀⠀⠄⠐⠈⠉⠉⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠁⢀⠀⠀⠀⠠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠓⢠⠄⠀⠀⣤⠚⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠐⠢⠄⡈⠣⡙⠃⠀⠘
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠔⠒⣉⡐⠒⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⣠⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⣤
⠀⠂⠀⠀⠀⡀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡀⠁⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠶⠾⡄⠀⠁⠀⢂⠐⠃⠐⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠑⠀⠈⣉⠲⠀⠀⠀⠀⠀⠀⢀⡉⠀⠭⠭⣝⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠲⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣑⣤⣤⡤⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠿⠿⣯⣿⣿⣿⣽⠿⠿⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣻⣷⣿⣘⣓⣾⣿⣿⣿⣷⣚⣃⣿⣾⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢿⣿⣿⣷⣶⣾⣿⣿⡿⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣳⠇⢿⣿⣯⣿⣿⣿⣽⣿⡿⠘⣎⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄⠀⢉⣿⣿⣿⣿⣿⡏⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⣸⣿⣿⣿⣿⣿⣿⢀⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠛⠲⣄⠀⠀⠀⠀⠀⢸⠟⠶⢄
⠀⠀⠀⠔⣲⢿⣿⣿⡍⠀⠀⣀⠘⠓⠾⣿⡆⠀⠈⢢⣤⣾⡀
⠀⢀⠂⡼⠁⢿⣿⣿⣿⡄⠈⠉⠀⠹⠆⣿⠟⠃⠁⠻⠿⣿⠃
⠀⠈⡄⡇⠀⠘⢿⣿⣿⣷⣤⣤⣤⣴⠖⠁⠀⠀⠀⠐⠶⠯⣙⣖⡶⣤⣤⠤⠤⢄
⠀⠀⠈⠳⣀⡤⠚⠉⠛⠿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠙⠀⠀⣀⠀⠙⢷⠖⠤⣤⣤⡞⣁
⠀⠀⠀⠀⡿⢅⠀⠐⠂⢠⡾⢉⣿⣇⠀⠀⢀⣀⡄⣀⠀⠀⠀⠀⠈⠀⠛⠓⠶⠄⠀⠈⢳⣦⠔⠋⢿⣿⣠⡤⠄
⠀⠀⠀⠀⡇⠀⠙⠲⣶⠷⠤⣿⣿⣿⣷⣄⠀⠈⢉⠪⢳⣦⣤⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⢀⣿⢟⠇⠀⠈⡀
⠀⠀⠀⢀⣿⡀⣀⡾⣧⣦⣦⣮⣿⣿⣯⣿⣷⣶⣴⣿⣿⣿⣿⣼⣷⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠒⢆⡼⠁
⠀⠀⠀⠛⠻⡿⠟⠛⠛⠛⠛⠻⣿⢿⣟⠛⠛⠾⠿⢿⣿⣿⣿⣟⣻⣿⢻⣿⣻⣿⢿⠵⡿⢹⣯⢾⣽⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠈⠀⠀⠀⠈⠈⠙⠟⠯⠿⣿⡟⣶⡿⣾⣿⣾⣟⣹⡟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢩⠿⠟⠛⠯⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⣤⢀⡀⣶⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣶⣾⣿⣿⣿⣿⣶⣾⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣯⡀⠉⠛⣿⣿⣿⡿⠛⠉⣀⣽⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡷⠂⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠐⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣼⣿⠏⢀⣤⣤⣤⣐⢄⠀⠀⠀⠀⡠⣀⣤⣤⣤⡀⠹⣿⣧⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⢠⡿⠋⠭⣽⣶⣿⣿⡾⠛⠋⠉⠛⠻⢷⣷⣤⣤⣾⡾⠟⠛⠉⠙⠛⢷⣿⣿⣶⣯⠭⠙⣿⡄
⠀⠀⠀⠀⠀⢸⡇⠘⠻⣿⣿⣿⠏⠀⢀⣤⡶⢶⣤⡀⠙⣿⣿⠋⢀⣤⡶⢶⣤⡀⠀⠹⣿⣿⣿⠟⠃⢸⡇
⠀⠀⠀⠀⠀⠘⣇⠀⡞⢛⣿⣿⠀⢠⡟⣡⡀⣳⣦⢹⡀⢸⡇⢀⡟⣅⢘⣷⣎⢻⡄⠀⣿⣿⡛⢳⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠙⣦⣇⠈⢾⣿⣆⠘⣇⠻⣟⣿⡿⢡⠁⡸⢇⠈⡆⢿⢻⣿⡟⣸⠇⣰⣿⡷⠁⣸⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣤⣿⣿⠷⡌⠳⠬⠭⢖⣡⠾⠗⠺⠷⣌⡲⠭⠥⠖⢁⠼⣿⣿⣤⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣧⣄⣀⣀⡤⠋⠄⣀⠀⠀⣀⠠⠙⢦⣀⣀⣠⣴⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡟⠀⠀⠀⠉⠃⠘⠉⠀⠀⠀⢹⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⢄⡀⠀⠀⠀⠀⠀⠀⢀⡠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡉⠓⠒⠒⠒⠒⠚⢉⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣤⣤⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠹⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤
⠀⠀⠀⠀⠀⠰⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠂
⠀⠀⠀⠀⢀⣴⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀
⠀⢠⡄⣠⡟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢷⡄⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⡟⣿⣿⡿⣧⠀⣤⠀⠀⠀⠀⠀⠘⢶⣿⡷⠛⢶⣿⣷⠟
⢀⣼⠟⠋⠀⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⠀⠉⣮⣿⣏⡀⠀⠀⠀⠀⠀⢠⣾⣷⠋⢸⣿⣿⣷⠈⢳⣿⣆⠀⠀⠀⠀⠀⢨⣿⡇⠀⠀⣿⡇
⢸⣿⡀⠀⣰⡿⠉⢿⡆⠀⠀⠀⠀⠀⠀⢠⡿⠉⢿⡄⠀⣹⠿⣍⠃⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⢹⡇⠀⢸⣿⠆⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⡇
⠉⠀⠃⠀⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠈⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡴⠛⢛⢲⡀⠀⠀⢀⣀
⠀⣀⢀⣀⡀⠀⠀⠀⠺⡿⠃⠀⠀⠀⠀⠠⡄⡢⢰⡄⠀⠀⠸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⠀⢿⡟⠀⠀⠀⠀⠸⡇⠈⠁⢠⡇⠀⠀⠸⣿⠃
⠀⢉⠘⠋⠑⠀⢳⣶⣿⣿⣿⣦⠀⠀⠀⠀⠀⠉⠈⠀⠻⣶⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣾⣿⣿⣧⠀⠀⠀⠀⠉⠓⠒⠋⠘⠻⠟⢿⣿⣿⡄
⢀⣿⡁⠀⣸⣇⠀⠀⣿⣿⣟⠁⠀⠀⠀⢀⣿⡀⢀⣿⡀⠀⢸⣿⡟⠉⠀⠀⢀⣤⡀⠀⢠⣤⠀⠀⢰⣿⣿⡹⠇⠀⠀⢀⣿⡀⢀⣿⣃⠀⠀⢸⣿⡏⠿
⢾⣿⡿⠼⣿⡿⠆⠘⣿⢿⡟⠀⠀⠀⠀⢿⣿⠷⢾⣿⡿⠀⢸⣿⡇⠀⠀⠀⢠⣿⣤⡀⢨⣯⣤⡀⢻⡿⣿⠃⠀⠀⠀⠾⣿⡿⠾⣿⣿⠀⠀⣾⢿⡇
⠈⠿⠇⠀⠿⠇⠀⠀⠿⠸⠇⠀⠀⠀⠀⠸⠿⠀⠈⠿⠇⠀⠺⠈⠷⠀⠀⠀⠘⠿⠛⠦⠸⠿⠻⠆⠸⠇⠿⠀⠀⠀⠀⠰⠿⠇⠀⠿⠇⠀⠀⠿⠸⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠛
⢀⣿⠃⢠⡄⠀⠀⠀⣸⡟⠀⡀⠀⠀⠀⠀⢘⣿⠃⠀⠀⣀⠀⠀⢘⣿⠃⠀⠀⠀⠀⠀⢰⡇⣀⡐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⠀⣶⣷⡆
⣾⣿⠻⠟⠀⠀⠀⢸⣿⡟⣿⢃⣀⠀⠀⠀⣿⣿⢻⣿⣶⡆⠀⠀⣿⣟⢳⣾⣶⠆⠀⠀⢸⡗⢿⠇⣾⠇⢿⡿⣿⠿⡧⠀⠀⠀⠀⠈⠀⣼⣿⣷⠝⠛⠁
⣽⡟⢻⡆⠀⠀⠀⢘⣿⣻⡇⢈⡇⠀⠀⠀⣿⣟⣿⠀⢿⠀⠀⠀⣿⣟⡿⠀⡇⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠻⡇⠀⠀⠒⢒⣛⣓⠶⣚⣛⡒⠒
⠉⠁⠀⠁⠀⠀⠀⠈⠉⠉⠁⠈⠉⠀⠀⠀⠉⠉⠉⠀⠉⠁⠀⠀⠉⠉⠁⠈⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠀⠈⠉⠁⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣶⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃
⠀⠰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢺⡏⢉⠉⢹⠇⠀⠀⠀⠀⣾⡆⢸⣁⡠⠀⠀⢹⠀⠀⠀⠀⠰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠰⣿⠆⠀⠀⢾⡇
⢠⣿⡷⣤⡀⣠⣤⠀⠀⠀⠀⠀⣠⣾⣿⡾⡶⠇⠀⢸⠀⠀⠀⢀⣼⣿⡟⢫⣉⣀⣀⣀⣸⠀⠀⠀⢠⣿⡷⣄⣠⣤⠀⠀⠀⠀⠀⣴⣿⣷⡀⣰⣿⣿⡆
⣾⣿⠀⠈⣥⠘⡷⣤⡀⠀⠀⠀⠉⢸⣿⡼⣧⣤⣤⣼⠆⠀⠀⠘⠁⣿⡇⠈⠉⠉⠉⠉⠉⠀⠀⠀⣿⣿⠀⣾⣿⠛⢀⠀⡀⠀⠸⢻⣿⡟⠷⢻⣿⣿⠻
⣿⠹⡇⠠⠸⣬⡇⡆⠀⠀⠀⠀⠀⢸⢿⡇⢸⠁⠘⡇⠀⠀⠀⠀⢰⡟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠹⡆⠉⠈⠰⢿⠿⡿⠂⠀⢸⡿⡇⠀⠈⡿⣿

⠀⠀⠀⠀⣤⠃⠘⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⣴⠃⠐⠀⢠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⠀⠝⢱⠀⠼⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠍⠱⠀⠼⢉⡀
⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡀⠀⠀⠀⠀⡴⠋⡏⡩⠦⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠐⠀⠀⠀⡴⠫⡎
⠙⢣⡴⠀⠀⠔⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠆⠅⠜⢃⡶⠀⠀⠔⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠄
⠀⠀⠀⠀⠀⠐⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠰⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠰⠤⠒
⡄⠀⢀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⠀⠐⢷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠐⢷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠖⡼⣿⡆⢊⠀⠀⠀⠌⠄⢉⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣬⣖⣾⣻⡆⠊⠀⠀⠀⠌⠆⢙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⠀⠀⠀⠀⠶⢋⠘⠃⢦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡤⠀⠀⠀⠀⠶⠛⠘⠇⢦⡴⠂
⠀⠀⢀⣀⠀⠀⠈⠀⠀⠃⠀⠀⠨⣁⠀⡤⠈⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣀⠀⠀⠈⠀⠀⠃⠀⠀⠈⣀⠂⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢉⠀⠁⠀⠀⠀⠀⠀⠀⠚⠀⠈⠀⠀⡠⠦⡴⠐⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⡁⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⢀⡠⠢⡤⠀⠀⠀⠀⠆
⠀⠀⠈⠛⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠖⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠆⠀⠀⠉⠛⠃
⠀⠀⠀⠀⢀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂
⠀⠀⠀⢀⠉⡰⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡰⠉⠆
⠀⠀⠀⠀⡶⠃⠘⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠃⠠⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠜⠹⠀⠼⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠱⠀⠍⠹⠀⠨⢡⡀
⠦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠐⠀⠀⠀⣴⠁⡏⠩⡦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠐⠀⠀⠀⡰⠉⡏
⠙⡃⡶⠀⠀⠔⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠔⠁⠜⠃⡴⠀⠀⠔⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆
⠀⠀⠀⠀⠀⠐⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠐⡤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠒
⡀⠀⢀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠐⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠐⠷⠆
⣗⠺⣻⡆⢂⠀⠀⠀⠌⠂⢉⠿⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡷⣊⣖⢾⣹⡖⠊⠀⠀⠀⠈⠂⢉⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣤⠀⠀⠀⠀⠖⢋⠘⠃⢦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⡴⠀⠀⠀⠀⠷⢛⠘⠃⢢⠴
⠀⠀⢀⣀⠀⠀⠀⠀⠀⠃⠀⠀⠈⣁⠐⠤⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣀⠀⠀⠈⠀⠀⠃⠀⠀⠈⣁⠂⠤⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢉⠀⠁⠀⠀⠀⠀⠀⠀⠚⠀⠉⠀⠀⡰⠦⠶⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠁⠀⠀⠀⠀⠀⠀⠚⠀⠉⠀⠀⡤⠲⡴⠐⠀⠀⠀⠂
⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠆⠈⠀⠉⠛⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠆⠀⠀⠉⠛⠃⠀⠀⢀
⠀⠀⠀⠀⢀⠀⡠⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠅⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠈⠉⠰⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠉⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠃⠀⠊⠀⠀⠈⠀⠐⠀⠓⠀⠀⠀⠂⠐⠀⠀⠀⠐⠈⠒⠒⠀⠁⠁⠘⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠻⣿⣿⣿⣿⢿⣿⣻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡯⠅⠀⠀⠀⠭⢍⠙⣿⣿⣿⣿⣿⡷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⢠⠀⡀⠀⠀⢀⠘⣿⢿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⢁⠜⠀⠈⠒⠒⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠠⢄⡀⠀⠀⢶⠆⠀⣤⣾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⡀⠀⣴⣿⡶⣰⡿⠉⣿⣻⢟⠟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣠⠿⠿⣿⡇⣿⠱⠆⡇⢰⣿⢘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡞⣱⣿⣿⣎⣿⣷⣿⢮⣵⣎⣃⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢱⣿⣿⣿⢏⣴⡆⣹⢠⡍⣹⢟⡻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣯⣿⡜⠿⠿⣃⡸⠟⠰⢣⢏⣼⢣⢟⣵⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⢸⠿⢋⣿⠟⡛⢿⣿⣿⣿⣶⣿⡸⢛⣄⣛⣵⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⢀⣾⣿⠃⣾⣿⠀⣿⢠⣿⣿⠛⣿⠛⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣟⣸⣿⣿⠘⣿⠏⣸⡇⣼⣯⡿⢸⡇⠸⠿⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⢛⣻⣷⠶⣾⣿⣧⣙⢛⣡⣿⢁⣆⢹⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡹⣿⡇⢾⣾⢏⣿⣿⢛⣛⣿⣾⣿⣦⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠻⢇⣿⢰⣶⡿⣸⣿⡟⣘⣿⣿⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣮⣽⣇⣛⣻⣇⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠤⠄⠄⠀⡐⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠰⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠐⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠉⠉⠙⠓⠒⠒⠀⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠛⠉⠙⠒⠲⢤⡀⠀⠀⠀⢀⣸⠤⠔⢢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⠖⠋⠁⠀⠠⠖⠡⠟⡷⣄⡀
⠀⠀⠀⠀⠀⠐⡒⡾⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠰⡄⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⡈⠹⠿⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⢟⣹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⣉⣲⣄⣀⢤⠄
⠀⠀⠀⠀⠀⠀⢁⣮⠆⢽⡍⢿⣖⢶⠶⢶⠒⣒⣚⣹⠉⠞⣳⡽⠉⠈⠢⣀⠀⣀⣤⣴⡶⢛⣟⢛⡛⣉⣁⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠸⢛⠦⢄⣀⠀⠈⠉⠙⡓⠀⣉⠀⠘⠋⡥⢎⡟⢠⢀⠀⠀⣽⠉⢤⡏⠀⡎⢻⠛⢯⣄⡈⢙⣭⠗
⠀⠀⠀⠀⠀⠀⠘⠦⢀⣠⡌⠛⠁⠤⢄⢠⣈⣁⡀⠀⡀⣠⠎⠁⠣⣌⠄⢸⣿⡄⢸⣬⣖⡽⠄⣇⢀⠦⠽⢾⡜
⠀⠀⠀⠀⠀⠀⠐⢄⢀⠈⠉⠁⠈⠀⠈⠁⠉⠉⠉⠁⠈⢀⠄⠁⠀⠈⠢⣥⠙⣧⠈⣟⠬⣉⣁⡟⢴⣟⠓⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠀⠐⠒⠒⠒⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢂⠙⡓⡁⢶⠨⠇⠈⠂⠭⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠱⠘⠏⠇

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

⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣀⣀
⠀⠀⣠⣾⣿⣿⡀⠀⢰⠋⢍⡩⢳⡄⠀⠀⠀⠀⣼⣧⠀⠀⢰⡿⠿⢿⡄⠀⠀⣾⠟⠻⢿⠀⠀⠀⣾⡍⣿⡇
⠀⠀⢸⡿⢯⡇⠀⠀⠸⣄⠈⠀⣸⠃⠀⢼⢴⡟⢿⡿⠀⠀⠨⠐⡀⢀⠃⠀⠀⣵⠀⢈⢸⡀⠀⠀⢨⡟⣯⡀
⠀⠀⠈⠁⠈⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠀⠁⠈⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠈⠁⠈⠁
⠀⢀⣤⡀⠀⠀⠀⠀⢀⣴⡀⠀⠀⠀⠀⠠⡦⠤⠤⡆⠀⠀⡄⠀⡄⠀⠀⠀⠀⢀⣤⢦⣀⠀⠀⢀⠴⠶⠲⢦
⠀⠀⡿⡧⢤⡧⡼⠀⠘⡤⣧⡀⠀⠀⠀⠀⡇⣀⡀⡇⠀⠀⡇⠀⡟⠀⢸⠀⠘⢿⣿⢿⣮⠃⠀⢸⡾⢃⣦⢻⡇
⠀⠀⠓⠇⠀⠓⠓⠀⠘⠓⠇⠹⠦⠀⠀⠰⠃⠉⠀⠇⠀⠠⠧⠤⠷⠤⠼⠀⠀⠾⠇⠸⠿⠀⠀⠈⠓⠾⠿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⢀⢀⣠⣄
⠀⠘⣶⣾⣶⡿⠆⠀⢀⣴⣶⣾⡏⠁⠀⣀⣀⣀⣀⡀⠀⠀⠰⢶⡠⢇⡀⠀⠘⡟⣅⣠⢹⠟⠀⠀⣿⣿⣿⡿
⠀⠀⠉⠴⠿⠄⠀⠀⠀⠯⠤⠬⠷⠀⠀⠻⠟⠛⠻⠟⠃⠀⠵⠏⠀⠜⠜⠀⠀⠑⠌⠡⠜⠀⠀⠀⠈⠃⠋

⠀⠀⡖⢛⡛⠓⡆⠀⣤⠰⠲⠖⠆⡄⠀⢰⠶⡦⠠⣦⠀⠀⢾⡷⠀⣿⡇⠀⢰⣶⣶⣶⡶⠆⠀⢐⣢⣴⣼⣼
⠀⠀⡧⣤⡥⡴⡇⠀⣿⠀⢸⠀⠀⡇⠀⢸⠬⠭⠩⢽⠀⠀⣠⣄⠀⣤⡀⠀⢸⣿⣿⣿⣷⣃⠀⠠⢴⡤⢼⠤
⠀⠈⠛⠛⠛⠛⠁⠀⠙⠛⠛⠛⠛⠃⠀⠘⠀⠀⠀⠈⠀⠀⠈⠁⠈⠙⠃⠀⠈⠉⠉⠉⠉⠀⠀⠈⠁⠀⠘⠈
⠀⠀⣠⣶⣶⣦⠀⠀⢠⣦⣴⣤⣦⠄⠀⢴⣄⠀⣠⣦⠀⠀⠀⡄⡆⣠⠀⠀⠀⣤⣄⠀⢠⠄⠀⠀⡶⠶⠶⡖
⠀⠀⡇⠰⡆⠀⡇⠀⢸⣿⠁⠸⣿⠂⠀⠀⣙⣷⣟⠁⠀⢀⡀⢳⣤⣥⠋⠀⠰⡌⢤⡴⣃⡄⠀⠀⣇⣀⣀⡇
⠀⠀⠧⠤⠧⠤⠃⠀⠘⠿⠻⠟⠿⠁⠀⠼⠋⠁⠙⠡⠀⠀⠈⠻⠇⠟⠀⠀⠀⠩⠻⠤⠶⠃⠀⠀⠉⠻⠻⠁
⠀⠀⢠⡀⢀⣠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣲⣉⣉⣲⠀⠀⢈⢉⣉⣩⣭⡅⠀⡋⣿⢹⢹⠹⠀⠀⠀⠀⢸⢷⡀⠀⠀⣠⢦⣠⢦⡀⠀⠀⢀⣾⣦
⠀⠀⢸⣿⣿⣿⠀⠀⠘⠸⠻⠻⠿⠇⠀⣇⣁⣈⣸⣸⠀⠀⢠⣾⣿⠀⠀⠀⢸⠉⠉⠉⠉⣳⠀⠀⣸⣿⣿⣆
⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠠⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠘
⠀⠀⠀⠀⠂⡀⠘⠓⢁⠀⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠉⠰⠀⠀⠠
⠀⠀⠀⠀⠘⠓⠐⠘⠈⠁⠉⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣈⠠
⠀⠀⠀⠀⠐⠂⠀⡀⠀⡂⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢦⠀⠀⠰
⠀⠀⠀⠀⠀⠙⠄⣦⠜⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠘⠉⠉⠀⠰⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠄⠀⢠⠀⢠⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠈⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠱⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠠⠀⠀⠂⠤
⠀⠀⢀⠹⡀⠀⡄⠀⠀⠆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠈⠁⠁⠀⡀⠂⠚⠢⣶⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⣀⠀⠀⣠
⠀⠀⠀⠀⠀⢠⠱⠀⠐⠉⠛⠁⠰⢇⠀⢸⡂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢨⠂⠀⢁⣀⡄
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠂⠤⠔⠂⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠘⠁⠀⠀⠈⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢂⡉⣈⠀⠀⠐⢠⣤⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠊⠑⠉⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠒⠀⠂⠂⠀⠀⠘⠀⠀⠀⠁⠀⠘⠀⠀⠀⠠⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠖⠦⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠋⠀⠀⠰
⠀⠀⠀⠀⠀⣀⠄⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠙⠀⠀⠈⠁⠀⠀⠄⠀⠀⢀⠀⠠⡀⠀⠀⠀⠀⠠⣰⡄
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚
⠀⠀⠀⠀⠀⠈⠃⠀⠐⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠂
⠀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠈⠀⠂⠀⠀⠀⠀⡀⠈⠀⠀⡕⢈⠀⠀⠀⠀⠀⡲⠤⣤⡀⢀⣀⠀⠀⠀⢀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠔⠀⠀⠀⠀⠉⢀⡂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠑⠀⠈
⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⡩⢣⠀⣤⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠓⣀⢦⠠⠤⡏⠇⠰⠰⡅⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠐⢆⡰⢌⡄⠹⡤⠿⠲⠁⣎⡹⠆⠡⠀⢂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⣄⢀⡸⠒⠎⠘⡀⡠⠞⢢⠀⠨⠁⠈⡀⢂⠈⠎⠖⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⢹⠇⣤⠋⢄⠀⠑⢻⠲⠉⢇⠀⠁⠡⠐⠀⣃⣸⠞⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⣶⠀⠀⠀⠀⠀⣀⡇⠀⠈⢆⠈⡆⠀⠈⠓⠋⢇⢃⠘⡆⠀⠀⣣⡶⠟⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣰⠃⠀⠀⠀⠀⢸⡗⠶⠦⢤⣬⣤⣽⣄⣦⣤⣥⣸⣬⣤⡴⣶⠞⠉⠀⠀⠀⠀⠀⡇
⠀⠀⣠⡤⠔⠒⠂⠀⣈⡧⡤⠓⠒⠤⢤⡀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣠⣤⣤⣤⣤⣧⣤⣀⣀
⠀⣼⣏⠀⠀⠀⣠⣾⡿⠀⠀⠀⠀⠀⣀⢟⡆⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠛⠉⠉⠁⢀⠀⠀⠉⠉⠉⠛⠻⢷⣦⣤⡀
⠀⡷⠬⣍⣛⣒⠚⠛⠒⠒⠒⣒⣚⡉⠥⠚⡇⠀⢸⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⢉⡁⡄⠀⠀⠠⠀⠂⠀⠉⠊⠁⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣄
⠀⡏⠀⠀⠈⠉⠉⠉⠉⠉⠁⠁⠀⠀⠈⠀⡇⠀⠸⡆⠀⠀⣀⣴⠾⠋⠁⡠⢰⠄⠉⠀⠂⢂⣠⠤⠖⠒⡈⠁⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠈⢿⣷
⢠⡇⠀⠀⠀⠀⠀⠠⠀⠀⠄⢀⡀⠀⠀⢀⡇⠀⠀⣇⣠⡾⠋⠁⠀⠀⢀⣈⣀⣤⠤⣤⠴⠿⠤⠤⣤⣉⣀⠀⢴⡴⠀⠉⠉⠠⡠⠆⠀⠀⠀⠈⣿
⢸⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡇⢀⣴⠟⠁⠀⢀⠀⠐⠊⠉⠁⣠⠔⠋⠀⠀⡄⠀⣀⠀⠈⠉⠛⠲⣄⠀⣆⠜⠀⠀⠀⠀⠀⠀⣸⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⡀⢸⣷⠟⠁⠀⠀⠺⠈⢀⡠⠂⣤⠞⠁⠀⢀⡀⠉⠀⠠⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⣼⡿⠃
⢸⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠄⡆⠀⣠⠞⠁⠐⠁⠐⠒⠁⢠⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⢻⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣰⡟⠀⠀⠀⠀⠀⠀⠋⠀⡴⠃⠀⠀⠸⠥⠛⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⢀⣤⡞
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⠉⣀⣤⠞⠋⣠⡇
⢸⡇⠀⠀⠁⠀⠀⠀⠀⡜⠀⠀⢨⣿⣦⣄⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠿⠛⠉⣀⣤⠶⠛⠉⢀⣤⠞⠉⡷⢤⡀
⣿⡇⠀⠀⠀⠀⠀⢠⠎⢀⡀⠔⠉⠳⣍⡛⠿⢿⣶⣶⣴⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⡿⠿⠟⠛⢉⣁⣤⠴⠞⠋⠉⠀⣀⣤⠾⠋⠁⣀⡼⠃⠀⣿
⣿⠇⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠸⣇⠙⠓⠦⢤⣈⣉⡉⠉⠉⠉⠉⠉⢉⣉⣁⣠⣤⠴⠖⠚⠋⠁⠀⢀⣠⣤⠶⠛⠉⠀⣀⣴⠞⠋⠀⠀⢠⣿
⣿⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⢦⣄⣀⠀⠉⠉⠉⠛⠛⠛⠛⠉⠉⠉⠀⠀⢀⣀⣤⡴⠶⠛⠋⠉⢀⣀⣤⠶⠛⠉⠀⠀⠀⢀⣴⡿⠁
⣿⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢦⣌⠉⠙⠛⠓⠒⠲⠶⠶⠶⠖⠒⠛⠛⠋⠉⠁⢀⣀⣤⡴⠶⠛⠋⠁⠀⠀⠀⠀⠀⢀⣤⡿⠋
⣿⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠓⠲⠶⠤⢤⣤⣤⣤⣤⠤⠴⠶⠶⠞⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠁
⠘⣿⣦⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⢛⣻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠿⠛⠁
⠀⢿⣝⡛⠻⠷⠶⠿⠿⠟⢛⣋⣩⡴⠖⠛⠁⠀⠉⠛⠷⠶⣤⣤⣤⣄⣀⣀⣀⣠⣤⣤⣤⣴⠶⠶⠟⠛⠋⠁
⣤⣤⣬⣭⣿⣿⣷⣾⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣭⣏⣼⣉⣯⣩⣿⣹⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠊⠓⠁⠀⠘⠃⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠸⠀⠀⠘⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠠⠀⢀⠀⠀⡄⠘⠔⠀⠀⡠⠒⠐⠂⠄⠀⣀⠀⠀⠀⡀⠀⠠⢒⠄⠀⠀⠀⠀⠀⠚
⠀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠇⠀⡄⠀⠀⠀⠰⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠑⠀⠂⠂⠀⡀⠀⠀⠠
⠀⠁⠀⠀⣠⢂⢶⡀⠀⠀⠀⠀⠀⠀⠈⠐⠔⠀⠁⠀⠀⠄⠈⢀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠈
⠀⠀⠀⠀⠅⠬⠋⠀⠀⠀⠀⠄⢀⡀⠠⡀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠁⡂
⠀⠀⢀⡄⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠚⠀⠂⠀⢠⠉⡄⠠⠀⠀⠀⠀⠂⠁⠠⠀⠐⣢⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠠⢴⠰⡄⢀⡐⠀⠀⠈⠀⠀⠀⠀⠀⡀⢤⡶⠄⠘⢊⣁⠠
⠀⠀⠀⠂⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠁⠉⠀⠀⡀⠤⢀⠀⠀⣼⡟⠁⠀⠀⠀⠈⠀⠠⡴⢡⢿⡄⠈⠂
⠀⠀⠀⠀⠐⠀⠧⠰⢺⠀⠀⢀⡀⠀⠐⠀⠀⠀⡊⠀⠀⠀⢁⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠒⠒⠁⠀⠀⣀
⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⣮⣹⣲⣲⠀⠀⠱⢄⣀⡠⠜⠀⠀⠄⠃⠀⣯⣩⡣⢭⠀⠀⠀⡀⠄
⠀⠢⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣿⡨⠟⠀⠀⠄⠀⠀⠀⣀⠰⢆⠠⡀⠠⠀⠀⠀⠀⠀⠀⠀⢨⠁⢀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣇⠊⠀⠀⠀⠀⠀⡀⠀⠃⠀⠈⠀⠃⠀⠀⡄⡐⡠⠀⠀⢁⠈⠀⡌
⠀⠀⠠⠀⠀⢰⠃⠀⠀⠀⠀⠀⠉⠀⠀⠐⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡆⠀⠄⠐⠀⠘⠒⠓⡇
⠀⡄⠀⠐⠤⠤⢴⠂⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⢀
⡐⢀⡂⠀⢠⣀⣼⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠒⠀⠐⠂⠂⠀⢬⡣⣁⣸⠀⠀⠐⠀⡂⠀⠀⠐⠂⠀⣘
⣿⡫⠁⠀⠀⢉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⡻⣥⡆⠀⠀⠅⠀⠀⠀⠀⠀⠀⠄⠸
⠎⠋⢀⣴⠿⠝⠛⠁⠀⠀⢀⠀⠀⢀⣾⣦⡄⠀⠀⢨⢀⠠⠶⠀⠀⠉⠊⠀⣀⣀
⣄⠀⠀⠉⠀⠀⠀⠀⠀⠐⠀⡂⠀⣾⣿⡿⠟⠁⠀⠈⠠⠄⠊⠀⠀⠀⠰⠎⢲⡻⠄⠀⠀⠠⠠⠀⠀⠠
⠄⠀⠀⠆⠀⠀⠀⠀⣏⡀⠐⡆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⢄⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠐⠂⠀⠀⠀⠀⠤⠀⠀⠀⠉⠀⠨⠁⠸⠆⠀⠀⠀⠰⠂⠠⠀⠀⠡⠀⠄⠀⠀⠀⠔⠂

⠛⠃⠠⠀⠀⠈⠈⠈⠂⠁⠀⠁⠀⠀⠂⢡⠄⢠⠄⠂⠀⠀⢀⠂⡀⡀⢀⠒⡀⠀⠀⠀⠀⠠⠘⠛⠄⠀⠀⡅⢘
⠂⠀⠀⠀⠀⠀⠀⡔⠀⢢⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⢘⣴⠁⣁⠘⣤⠃⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣉⣛⡉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠁⠇⢘⠉⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠢⡌⢱⡄⢰
⢸⠁⠁⠀⠀⠀⠀⠤⡀⠤⠀⠀⠀⠀⠈⠉⠐⠢⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⠄⠳⠎⠽⠇⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠸⣀⢿⣰⠀⠀⠀⠀⠄⢀⠂⠀⠀⠀⠀⠀⠀⠀⢲⢀⢖
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⢠⠤⣆⠉⡄⠀⠀⡀
⢠⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣔⣤⡈⢡⠀⠀⠀⡰⢀⣀⡰⠄⠀⠀⠈⢰⡸⣀⠆⣆⠁⠀⠀⠁⠈
⠈⠆⠠⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠈⠤⠜⠀⡈⠀⠘⠀⠀⠀⣇⠘⣉⢧⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈⠀⠀⠀⠀⠈⠁⡄⠈⠀⠀⠀⠀⣖⠀⠀⣀⡀
⠑⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠠⣤⠀⠀⠀⠀⢀⡔
⠀⠀⡑⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⢠⢀⣈⡦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⣶⠀⡗⠀⠀⠀⢊⡀
⠀⠀⠀⠀⠀⢸⣀⠇⠒⢰⠀⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰⠉⣶⢁
⠀⠀⠀⠀⠀⠰⠿⠤⣾⠤⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠚⠙⠚⠀⠀⠀⢀⠀⠦⢀⠤⠂⠄
⢀⠤⠀⠀⠀⠀⠀⠚⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⡄⡀
⠘⣀⡘⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠂⠐⠀⠐⡀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠣⠀⠐⠁⠀⠀⠀⠁⡤
⠈⠀⠀⠀⠀⠀⡎⠰⠶⡎⢡⠀⠀⠀⠀⠈⠆⠚⠀⠀⠀⠀⠀⠐⠂⡅⠐⠀⠀⠀⠀⠀⠁
⠘⠂⠐⠀⠀⠀⠶⠄⠒⠹⠏⠀⠀⠀⠁⠐⠀⠐⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⡄
⠀⠠⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠇⠂⠆⠀⠀⡠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠛⠻⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⠟⠿⠿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⢠⡄⠹⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⠀⠸⠷⠂⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠉⠁⠀⢻⣿⣿⣿⡇⢸⣿⣿⣿⣿⠀⠰⠶⠆⢘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣤⣾⣿⣿⣧⣤⣿⣿⣿⡇⢸⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠇⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠸⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠋⣁⣤⣉⣹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣷⣄⠉⠛⢉⣹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣠⣤⣤⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⡏⠓⠒⠯⠝⢻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠆⠀⠀⠹⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⢉⡐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣾⣿⣥⣤⡀⡥⠀⠀⠀⠀⠀⠀⣐⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢰⠞⡃⠠⡠⢹⣦⣾⣟⣽⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⣻⣿⣿⣻⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣯⠙⢋⡿⠣⣮⣉⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣿⣿⣿⣏⡉⠙⠁⠀⠀⠒⠦⠈⠉⠉⠉⠿⣿⣿⣿⣿⡟⣶⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠛⠻⠛⠲⠶⠶⣄⣲⣀⡀⡒⠒⠺⠽⠛⠛⢋⣼⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣛⣉⣤⣶⣶⣮⠙⠉⠉⠉⠉⠉⠉⢉⣭⣶⣶⣤⣀⣾⣿⣿⠏⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡿⣭⣙⡻⣿⣿⣿⣷⡄⠀⠀⢀⣴⣿⢿⡟⣛⣹⣿⣿⣿⣿⢾⣾⡏⠎⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⠆⣈⡴⣋⠿⡝⢁⣙⠳⢶⡟⡇⠈⠀⢽⠋⡛⠞⢻⠁⢀⣰⡗⣵⠐⣯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⡔⢡⠗⡂⣨⣍⠳⢮⠀⣀⣚⠋⠘⠀⠀⠀⢡⣴⣿⣉⣸⡏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣏⡟⠳⠓⠜⠉⠉⠐⠚⠛⠿⢆⠍⣀⣎⣹⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓⠒⠒⠶⠤⠤⠤⠄⠒⠂⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠙⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠙⠋⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⣦⣴⡿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⡀⠀⣸⣏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠋⠀⠙⣿⣿⠆⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣇⡼⣿⣿⡇⠀⣶⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⡇⠀⠀⢀⢸⢃⠀⠀⢸⣿⠙⣿⣿⣿⣿
⠀⠀⠀⣾⣿⣿⣞⣿⣿⣿⢹⢃⣿⣿⣿⣴⣿⣶⠀⠀⠀⢸⣿⣿⣿⠁⠀⠈⢳⠀⠀⢸⡆⡞⠀⠀⠼⠋⠀⠙⣿⣿⣿
⠀⠀⢠⣿⣿⣿⡄⣿⣿⣿⠀⠀⣿⣿⢉⣿⣿⡿⡀⠀⠀⢸⣿⣿⡟⠀⠀⠀⢻⠀⠀⢸⣿⣿⣀⢠⣷⣶⠀⣶⣿⣿⣿
⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⠀⠀⢿⡿⠀⢸⣿⡇⠀⠀⠀⢸⣿⣿⣿⡗⣤⣽⣽⠀⡄⢨⣽⣿⣤⢸⣿⡇⠀⣿⣿⣿⣿
⠀⠀⠀⠸⠃⠇⠀⠟⠉⠳⠀⠀⠞⠳⠀⠘⠛⠇⠀⠀⠀⢸⣿⣿⣿⣧⣿⣿⣿⣤⣷⣼⣿⣿⣵⣌⣿⣧⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢰⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠂⠀⠀⢀⠀⠀⢀⣀⡀⠀⠀⢀⠀⠤⡄⠀⠀⠀⠀⡄
⠀⢀⡼⠟⠛⠿⣦⠀⣤⡶⡶⢶⠶⣦⠀⠆⠁⠄⠀⠠⢴⣟⣻⡦⠤⠀⢠⠀⠀⢀⣴⣶⣆⠀⡄⡈⠁⢀⠀⠀⠀⢀
⠀⠸⢷⠾⡿⢿⡿⠀⢿⠀⠀⠸⢀⢸⠀⠀⢠⠨⡀⠀⠄⢀⠀⠠⠃⠀⠉⠀⠀⠸⣶⠷⡾⠀⠀⠀⠀⠘⠈⠩⠄⠘⡀
⠀⢀⢿⣶⣛⣾⠇⠁⢸⣤⣤⣤⣄⣼⡀⠀⡄⠀⠇⠀⠀⠠⠆⠀⡄⣀⠠⠀⠀⢠⣽⣿⣥⡀⠀⠀⠀⠀⢄⠨
⠀⠀⠀⠙⠛⢋⠀⠀⣼⠂⡇⢀⠀⢸⡇⠐⢂⠀⠑⢂⠀⠛⠀⢀⠀⠉⡤⠀⠠⡍⠉⠉⠉⠁⡇⠀⠀⠂⠀⠈⣄
⠀⠀⠀⠅⠀⠠⡁⠀⠘⠛⠛⠉⠉⠉⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠁⢀⠀⠀⢀⠂⠀⠀⢀⠀⠁⠂
⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⢀⠤⠠⠀⢀⡅⠰⠂⠀⠓⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠐⠀⠠⠐⠀⠀⠐⠈⠀⢀⠉⠀⠁⠀⠀⠀⡰⠐⠁⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠅⠀⠀⠀⠀⡆⡐⠈⢀⡶⠒⠢
⠀⠠⠀⠀⠀⡠⢖⠆⠀⠀⡴⠀⢐⣿⣦⣄⠠⡀⠀⣠⢀⣀⠰⠀⠀⠀⠈⠛⡀⡄⠀⠀⠀⠀⠀⢀⡀⠁⠄⣈⡅⠀⢀
⠀⠀⡠⠤⢄⠈⢊⠄⠠⡀⠘⠀⣿⠉⠈⠹⡧⠈⢁⠀⢷⣗⠄⢣⠀⠀⠐⠀⠐⠀⠀⠼⠁⣀⢀⠀⠀⠀⠀⡀⠀⡉
⠀⠐⡄⠀⢀⠄⡁⠊⣀⠇⠀⠀⢡⠀⠀⠀⡇⠘⡮⠒⡀⠀⠀⠐⠀⠀⠀⠀⠀⠁⣀⡀⠘⠀⣾⢸⡇⣼⡇⣷⡇⠈
⠀⠀⢀⢁⢀⣀⠈⡍⣀⢀⠀⠀⠀⠠⠠⡂⠀⠀⠘⢀⣠⣴⡦⠀⠀⢧⣯⣾⡂⠀⠠⠀⠀⡀⠈⢀⠁⠁⠠⡁⡀⠰⣧
⠀⠉⠸⠻⠼⠹⠆⠇⠛⠈⠀⠴⣿⣧⣀⣼⣿⠆⠐⣿⣿⣿⠟⠀⠀⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠺⠿⢷⠖⡄⠀⠀⠀⠊⠀⠸⣿⣩⣿⠃⠢⠀⠈⠉⠁⠀⠀⠀⡀⠀⣀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⢀⡀⠀⠀⢀⠄⠃⠀⣔⠀⠀⠐⠙⠛⠛⠀⠀⠀⣢⠀⠀⢠⠒⢉⠁⠀⠂⠀⠀⠍⠒⡄⠐⠤⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠐⣂⠉⠐⢀⠀⠀⠠⠀⡈⠈⠂⠖⠠⠄⠂⠀⠈⠀⢁⠀⢄⣀⡀⢠⠘⣀⡠⠆⡀⠈⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢈⣠⣤⡈⠀⡀⠁⠀⠈⠀⠁⠈⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⢀⡀⠤⠀⠀⠀⡔⠲⠒⠂⠂
⠀⡦⣭⡭⠭⣭⡄⡆⠀⠀⠒⠀⠤⠤⠀⠌⢝⡠⠀⡀⠀⢨⠄⢲⡀⠀⠀⠀⠀⡠⠀⠈⠁⠀⠀⠄⠀⠀⠁
⠀⢹⡁⠀⠀⢈⡓⠀⠀⠠⠀⠀⠀⢠⠀⠈⠀⠀⠚⠽⠀⠠⠁⢁⡀⠀⠀⠀⠚⠂⠀⠂⠈⠀⡤⣀⡀⠀⠀⢀⣀⡀⠆
⢀⠊⠑⡄⢠⠊⠑⠄⠀⠀⣴⣶⠶⠀⠀⠀⠄⠈⠀⢈⠀⠀⠀⢿⡽⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠉⠡⠀⠀⠹⠧⠃
⠀⢠⣤⣀⣀⣶⡄⠀⠀⢀⠈⠁⢠⠀⠀⠸⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠇⡩⠄⠀⠀⠀⠀⠀⠀⠈⠂⠀⠲
⠀⠛⠻⠿⠿⠿⠛⠀⠀⠀⠉⠀⠁⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠈⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⣿⡏⢩⣩⣉⣩⣍⣭⣍⠉⣿⡇⠀⠀⠀⠀⣿⡟⢉⣩⣉⣩⣩⣭⣍⠉⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⡇⠀⠘⣿⣿⠀⣿⡇⠀⠀⠀⠀⣿⡇⢸⣿⠁⢠⠀⣻⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠇⠰⠀⣿⣿⠀⣿⡇⠀⠀⠀⠀⣿⡇⢸⣿⠀⢠⠀⢻⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣄⣴⣄⣽⣿⠀⣿⡇⠀⠀⠀⠀⣿⡇⢸⣿⣄⣈⣠⣾⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣷⣤⣤⣤⣤⣤⣤⣤⣴⣿⡇⠀⠀⠀⠀⣿⣷⣤⣤⣤⣤⣤⣤⣤⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢩⣭⣭⣭⣭⣭⣭⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠃⢠⠀⢹⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⢸⠛⢻⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣦⣬⣤⣾⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠆⣀⣠⣀⡀⢠⣄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠁⠙⠉⠑⠈⠈⠙⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣯⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠟⠻⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠒⠛⢻⠀⠀⠀⣿⠉⠉⢹⡏⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣤⣀⣀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⡏⠉⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⣀⣀⡀⠀⠀⠈⠛⠛⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡟⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⠸⡿⠿⣿⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⣿⠶⠶⠾⠷⠶⢶⣿⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⣀⣤⠤⠞⠀⠀⠀⣿⠶⠶⢶⡶⠶⠶⣿⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣶⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠷⠒⠛⠉⠉⠀⢀⣤⣦⣤⣄⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠀⣿⠀⠀⠸⡇⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢹⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠅⠀⠀⠀⣿⣀⣀⣠⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣤⣶⣦⡛⠛⠃⠙⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⡄⢴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠛⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠄⠘⠛⠛⠀⠓⢰⣿⠂⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣦⠀⠀⢸⣿⠀⠀⠀⢠⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠉⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒

⠀⠀⠀⠀⠰⠀⠂⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⣁⣀⣀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠈⠁⠂⠜
⠀⠀⠀⠀⠠⣀⡀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⡆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠠⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢀⠈⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢀⠂⢀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠠⡀⠀⠀⠀⠀⠀⠀⠄⣀⢀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠂⠁⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠑⣄⠎⠌⠐⠬⡂⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃⢡⠀⡀⠠⠄⠀⣚⠠⠀⠀⠀⠀⠀⠀⠀⠈⠺⢀⡀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠊⠋⡒⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠐⠦⠒⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⣐⠀⠀⠀⠈⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠁⠂⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡂
⠀⠀⠀⡠⠠⢄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡁⠢⡀
⠀⠀⠀⠡⡠⢢⠊⠁⠂⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⢀⣂⠀⠀⠀⠀⠀⠀⠀⡠⠊⠐⠌⠀⠀⠈⡆
⠀⠀⠀⠀⠑⠣⡀⢄⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢍⠁⠐⠞⠀⠀⠀⠀⠀⠀⣰⠐⠈⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠈⠐⢕⠠⠠⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠀⠀⠀⠀⠀⠀⠸⠦⡑⢄⣀⡠⠔⠁⠀⠀⠀⠀⠀⡔⠒⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠄⠛⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡈⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠈⠢⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠐⠈⡤⡀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⡀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡤⢊⠌⢃⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⢐⠄⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠀⠀⠀⠀⠀⢀⠧⡀⠀⠄⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠓⢠⣑⠖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠐⠂⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠠⠬⠁⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⠀⠀⠑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢈⡀⠀⢀⠂
⠀⠀⠀⠀⠀⡠⠊⠐⠂⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⢰⠂⢠⠭⠁
⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠮⢀⡁⠀⠀⠀⠀⠀⠥⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣀⠀⠀⠀⠀⡄⠐⠀⠀⠀⠀⠐⠀⠂⠀⠀⣀⠊⠀⢀⡈⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀⢀⢀⠀⠀⠀⠄⠈⠀⠀⠀⠀⢀⡄⠀⠀⠀⢄⠀⠉⠒⠠⢁⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠁⠀⠀⡠⠀⢠⠠⠴⠤⠆⠥⡙⢦⡢⡔⠁⠀⠀⠂⠀⠀⠿⡢⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠆⡀⠀⠁⢈⡄⠙⠁⠢⠜⠀⠈⠀⠀⠀⠀⠀⠀⡸⣄⣿⠈⠢⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠆⡂⠄⠰⠏⠀⠠⠂⠀⢀⣈⡦⠀⠀⠀⠀⠀⠀⠈⠁⠈⣢⡄⢧⠀⢤⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠀⠐⠠⠀⠀⠘⢀⠀⡀⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⡌⠃⠀⡠
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠓⢰⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢀⠀⢹⡇⠉⠀⠀⠄⢤⣰⠁⠀⠀⠀⠐⠀⡁⠀⠈⠀⢀⠄⠀⠀⠰⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠰⡈⠰⡀⣀⠀⠁⡠⠊⢀⠕⠀⠊⠀⠀⢀⠔⠁⠀⠠⠂⢂
⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠄⢡⠀⠀⠱⢄⢉⠀⠤⠈⡠⠀⠅⠀⠠⠂⠀⠐⠁⡠⠂⠀⠀⠔⠀⠑⠢⠀⠀⠀⡤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠉⠀⢣⠀⠂⠀⠀⡑⠢⠖⠆⠐⠂⠒⠮⢄⢔⠀⡊⠀⠀⠀⠀⡀⠀⠀⠸⠀⢢⢬⠀⢇⡂
⠀⠀⠀⢠⢤⡔⢂⠀⡀⠀⠁⠂⠠⡱⡈⠀⠁⠀⠀⠐⠇⠐⣂⠀⠀⠀⠀⠉⢡⢠⣤⣄⢤⣤⣤⣮⣥⣤⢎⣥⣝⣬⣤⣤⣤⣄
⠀⠀⠀⠐⠀⠹⢌⢄⡈⡄⠀⠀⢈⠈⠌⡂⠐⠀⡂⠀⢉⠲⡈⢅⠀⠀⠀⢀⠚⢸⡦⠦⠦⢢⠶⠤⠒⠾⠿⠿⢿⣇⣿⠏⢻⠇
⠀⠀⠀⠀⠑⢄⡀⠈⠛⢴⣄⠀⠰⣀⠀⠈⠠⠁⡂⠤⣔⠀⠆⠀⠀⠀⠀⡜⠀⠘⢻⡵⠚⠉⠀⠀⠗⠐⡂⢁⠟⠋⠁⣠⠋
⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⠈⠉⠑⠄⠀⠀⠀⠁⠈⠙⢃⠒⠠⠠⠒⠉⠀⠀⠀⠀⠀⣠⠒⠓⠀⠋⠈⠁⠀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢀⣠⡰⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠤⠀⡀⠀⠀⠀⢁⠀⠀⠀⠠⡨⢀⡀⠀⡥⠤⠁⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠢⠀⠈⠀⠀⠀⠨⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠪⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⡼⠙⣄⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⢤⣀⣿⣿⣦⠁⠀⠸⣾⣿⣿⢠⠔⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠈⠙⣿⣿⡇⠀⢠⣿⣿⡟⠀⠀⣸⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣄⠀⠘⣿⠧⠤⠾⢿⡟⠀⢀⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠛⠻⠿⣿⣷⠊⠀⠀⠀⠀⠀⠙⢾⣿⣿⠿⠋⠐⠒⢲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠤⠀⣀⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠈⣇⣀⣀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⣇⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⣿⣿⣿⣷⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⠟⠉⠀⣀⣼⣦⣀⠀⠀⠀⣀⣤⣯⡀⠀⠉⠉⢩⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣀⠀⣼⣿⣿⡟⠁⢹⣿⡏⠁⠹⣿⣿⣶⣤⢄⣠⠽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢸⣿⡿⡟⠀⠀⣾⣿⣿⡀⠀⠙⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢧⠀⡠⠸⣿⣿⠳⣄⠀⡇⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁⠀⠙⠏⠀⠀⠙⠃





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡄⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄⠀⠀⠀⠈⣿⣿⣿⡀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⠀⠀⠀⣿⣿⣿⣇⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⠀⠘⣿⣿⣿⡇⠀⣿⣿⣿⡇⠀⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⡀⣹⣿⣿⣧⣼⣿⣿⣿⣇⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠟⠿⢿⣿⣿⣿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠋⠀⣀⣴⠟⠁⢀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡋⢀⣴⣾⡟⠁⢀⣴⠋⠀⣤⠀⢀⣾⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠋⠀⣰⣿⠃⢀⣼⡏⠀⣼⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡁⣠⣾⡿⠃⢠⣾⣿⠃⢰⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣴⣿⣿⣿⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠐⠒⠶⠖⠲⠆⠀⠀⠰⠒⠦⠶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢸⡍⣷⠀⠀⢀⡄⣶⣤⣠⣴⠀⣶⣄⡰⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣇⠀⠀⣽⠀⠀⠘⠃⠁⡉⡉⠀⢀⡀⠁⠀⠀⡆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠐⠋⢀⡇⠸⠀⡖⠀⠏⡖⡸⠂⣾⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠀⠛⠀⠂⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠰⠒⠛
⠀⠀⠀⠀⠀⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡠⠒⡰⠄⢢⣎⣁⠀⠐⠚⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠶⠶⠒⣒⡒⠲⢤⡀⠀⡤⠀⠀⠀⠀⠀⠀⢀⣠⠖⠻⡄
⡟⠓⠺⣄⠀⢀⣇⣀⣹⡄⠀⢸⣄⠀⠀⠂⠀⠀⠀⠀⢸⣁⠀⠀⠀⠀⠀⢀⠀⠈⢑⡦⢷⡀⣌⠐⢣⡄⢀⣀⡰⠟⢫⣾⠟⠁
⠇⠀⠀⠀⠉⠉⠀⠀⠀⣹⢿⣧⡎⠀⠀⠀⠀⠀⠀⠀⢸⡂⡈⠀⠀⡐⢀⡀⣈⢀⣀⣤⣾⠁⠉⠀⠛⠓⢀⣤⠶⠛⠉
⢰⠀⠀⠀⠄⠀⠀⠀⡐⠉⠀⢧⠀⠀⢐⠀⠀⠀⠀⢴⠉⠻⠽⠿⣭⠿⣯⠽⠛⣛⡻⢟⡟⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀
⠸⠀⠀⠀⠀⠀⠀⠐⢀⣠⠴⠚⠀⠀⠀⠉⠉⠁⠀⠈⠲⢍⣀⣒⣐⣒⣀⡨⠭⠄⠒⠉⠀⠀⠰⠀⠀⠀⠀⠈⠀⠈⠁⡆⠀⠀⡀⠀⠀⠨⢆
⠀⠉⠙⠛⠛⠒⠚⠞⠋⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⢀⡀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠀⠀⠱⢂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡨⠓⠀⠀⠀⡈⠓⢦⡀⠀⠀⠘⣆⠀⠀⠀⠀⣼⠃⠀⠀⠀⣤⣶⠶⣤⣤
⠀⠀⠀⡀⠀⢀⡄⠀⠀⠀⠀⠰⢄⣨⠁⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠈⠢⡀⠙⢆⠀⠀⠁⣆⠀⠀⠀⡇⠀⠀⠀⠀⠁⢀⠠⠐⡄
⠀⠀⠀⠀⠐⠳⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠤⠀⠀⠀⢀⠎⠀⠀⠐⠦⠀⠀⡀⢀⣨⡦⠠⠇⠀⢀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠀⡁⢄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠄⠀⠀⢀⠀⣠⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡀⢂⣤⠾⠛⠁⠐⠃⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠄⡆
⠀⠀⠀⠀⠀⠀⠄⢙⣣⡄⠀⠀⠈⠀⠉⠀⠀⠀⠀⢸⣀⣴⠟⠋⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠘⠶⠤⠀⠀⠀⠒⠃⠀⠀⠀⠀⠀⠘
⠀⢀⠄⠒⠢⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡤⠒⠛⠛⠉⠉⠉⠓⠦⡀⠀⠀⠀⠀⡤⢀⢄⡀
⠀⡀⢰⠀⠀⢧⠀⠀⠀⠀⣠⣄⣴⣄⠐⢃⣀⡀⠀⠀⠀⢠⠏⠀⠀⠈⠀⠀⠀⠀⢀⣠⢼⠀⠀⠤⣜⠷⠻⣇⢱⠈⡁⠀⠀⠀⠐⠀⠀⠈⠠
⠀⠄⣸⠀⠀⢸⠀⠀⢠⣾⠏⡠⠊⠀⠀⠀⠀⠉⢶⠀⠀⡟⠀⠤⠀⠀⠀⣠⣤⣶⠿⢳⣻⡀⠀⠀⣿⠄⠀⠟⣸⣄⡿⠃
⠀⡇⣇⠀⠀⣼⠀⠀⢼⡫⠜⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠳⣤⡤⠶⠊⣙⢋⠽⣷⢦⣎⡥⢴⠋⠀⡋⠘⠚⠷⠛⢛⡇⠀⠀⠀⠀⠀⡀⠠⠤
⠀⢶⢸⠀⠀⡇⠀⠀⠀⠉⠓⠲⠤⠤⠤⠤⠤⠀⠠⠃⠀⠀⠈⣼⣄⣸⣭⣴⡶⠿⠋⠃⣱⠎⠀⠀⢸⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⠃⠀⠄
⠀⢸⡌⠀⢰⠇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⢀⣀⣈⣉⣀⠤⠒⠊⠁⠀⠀⠀⠀⠃⠀⠀⠠⠓⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠳⣤⠟⠀⠀⠀⠀⠀⠀⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠂⠐⠚⠀⠀⠀⢀⠀⠐⡀
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠟⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣠⡌⠢⡀
⠀⠀⠀⢸⠀⠀⠀⠐⠂⠀⠀⠐⠖⠒⠲⡤⠠⢤⡶⠀⠀⢀⠤⣠⠶⠶⠚⠋⠉⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡽⡆⢀⣬⡦
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠀⠂⠀⠀⠁⢨⠀⠀⠀⠀⠀⣀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠘⠦⠂⣀⣒⠀⠉⠝⠃⠚⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵⣤⣤⣤⣮⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⠤⣽⣋⣛⣛⣛⣛⣋⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣄
⠀⠀⠀⠀⢀⡤⠶⢒⠒⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠠⢻⡀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠈⣧
⠀⠀⠀⢰⢋⡀⢂⠌⢄⠂⠈⢧⠀⠀⠀⠀⠀⠠⣶⣒⣖⣻⣖⣖⣶⡄⠀⠀⠀⠀⠀⠀⣼⠂⣄⢀⡀⢠⢼⡀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⡼
⠀⠀⠀⢸⠄⠁⢇⣀⣠⠆⠁⣿⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢠⡄⣹⠀⠀⠀⠀⠀⠀⠀⠈⣏⠩⢋⠹⣺⠏⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⡴⠁
⠀⠀⠀⠈⢧⣠⠂⠌⠑⢂⡴⠃⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡔⡈⢆⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡴⠃
⠀⠀⠀⠀⠀⠉⠛⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⣺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀
⠀⠀⢀⡖⢋⠩⠅⠠⢀⡠⢉⢳⡄⠀⠀⠀⠀⠀⠀⢀⣼⣙⣲⣀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⣶⣏⣿⣽⣿⣿⢶⠄
⠀⠀⠀⢷⣡⢢⠋⢃⣞⠵⢆⡟⠁⠀⠀⠀⠀⠀⢸⡉⠤⢀⠤⢭⡇⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣂⣀⣐⣀⣧⠀⠀⠀⠀⠀⠀⣿⣿⢹⡏⡟⣯⡾⠃
⠀⠀⠀⠈⢧⡁⢫⠈⣬⣢⡟⠀⠀⠀⠀⠀⠀⠀⢼⣀⡀⣈⡆⢀⡿⠀⠀⠀⠀⠀⠀⢸⣭⣭⣉⣭⣭⣉⣭⡇⠀⠀⠀⠀⠀⠀⢿⠛⠺⠗⠛⢿⡇
⠀⠀⠀⠀⠈⢿⠛⣤⣩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣉⣁⣈⠾⠁⠀⠀⠀⠀⠀⠐⠫⡍⠹⠭⠛⠍⠿⢉⡟⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠳⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠈⠧⠤⠤⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⠀⠀⠀⠀⠀⠀⢀⢤⣤⢖⣦⡴⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⣀⣀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠛⠛⠉⠀⠀⠀⠀⠀⠀⡏⢸⣺⢼⠟⡇⠸⡆⠀⠀⠀⠀⢸⢯⠻⣿⣿⣳⡀
⠀⠀⢠⣖⠟⠙⠉⠉⠉⠙⠻⣦⡀⠀⠀⠀⠀⠀⠐⠈⣿⣿⡀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⣉⢺⣾⠇⠀⡇⠀⠀⠀⠀⢸⡆⠳⡹⣿⢷⣷⡄
⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⢶⠟⠯⠭⠭⠭⠍⢿⠂⠀⠀⠀⠀⠀⢸⠀⢘⣷⡞⢋⡇⠀⡇⠀⠀⠀⠀⢸⡇⠀⠙⣝⢿⣿⣷⡀
⠀⠸⣧⣀⣀⣀⣀⣀⡀⢀⢀⣀⣸⠇⠀⠀⠀⢸⣤⣀⣀⣀⢀⣀⣼⠀⠀⠀⠀⠀⠀⠸⡄⠈⣟⠱⢲⡇⢀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠈⢪⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⡟⠻⠟⠿⠿⠛⡇⠀⠀⠀⠀⠀⠀⠀⢧⠀⣿⡾⢛⣇⣸⠁⠀⠀⠀⠀⠈⢧⣀⣀⣀⣀⣱⣻⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢠⠖⠲⣄⡖⠛⢦⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠛⠳⡄
⠀⠀⠀⠀⢸⠀⠀⠘⡀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⣒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢶⡦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⡇
⠀⠀⠀⠀⡟⠓⠐⠒⠑⠒⢛⡆⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢀⣴⡯⠒⠉⠉⠁⠑⢽⣷⡀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠠⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣿⡔⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⢸⠆⠀⠰⡇
⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⣀⣴⡯⠟⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠈⢦⣀⡴⠃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠉⠻⠭⣉⢉⣩⠿⠛⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠈⠛⠚⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠲⠴⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣤⣤⣤⣤⠟⢲⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣞⠉⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢧⡀⠀⣠⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣚⣉⣙⡆⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠉⡍⠁⠳⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⣣⠼⣿⠖⠋⠙⠲⣞⡯⢬⣷⠀⠀⠀⠀⢀⡖⠋⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢸⠀⠀⠀⠀⡿⠀⠈⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠙⠲⣄⠀⠀⠀⠀⠀⡠⠚⠉⠉⠉⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⡇⠀⠀⠀⠀⠀⢻⠳⣄⠀⣠⢾⣁⡀⠀⠀⠀⠀⠀⣀⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⣼⠁⠀⠀⠀⠀⠀⢸⣦⣈⠟⠁⣠⢻⡇⠀⠀⠀⠀⠀⡟⢆⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠖⠁⡘⠀⠀⠀⠀⠀⠀⠘⡇⠙⠲⠚⠁⠸⠗⠦⠤⠤⠴⠶⡇⠈⠓⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⣛⡛⠛⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⢸⠛⢛⣛⠛⢳⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣒⡇⠀⡇⠀⠀⠉⠏⠉⢹⢹⡏⠉⣟⠁⢸⠀⢏⣈⠀⢸⠀⡞⢱⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠿⢤⡥⠤⠤⠧⠤⠀⣾⣙⣆⠸⠀⡀⠀⡇⠀⠸⣤⠤⠥⠤⠾⣄⡇⢸⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⣶⠀⠀⡇⢀⣻⡆⢀⣯⣉⣉⣨⣇⠀⠙⠛⠁⠀⠀⢸⠀⣶⣆⠀⣿⠀⠈⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠛⠀⢀⡇⠘⠒⢂⡿⠤⠤⠤⠤⠬⢧⡀⠀⠀⠀⠀⣸⠀⠷⠜⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢉⡽⠦⠤⠤⠤⠤⠤⠴⢧⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠶⠶⠶⠶⠶⠞⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⡏⠉⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼⠉
⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡒⠂⢰⠐⢢⠀⡆⠒⡖⠀⡶⡀⢰⠒⡦⢰⠀⠀⡖⠒⠀⢠⡖⠂⠒⡖⠂⡖⢲⠄⡔⠒⢦⢰⣠⠖⢰⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡀⢠⣀⡸⠁⡇⠀⡇⣸⠒⢧⢸⣉⡷⢸⣀⡀⣏⣁⠀⢀⣈⡷⠀⡇⠀⡏⢹⡈⢆⣀⡼⢸⠙⢆⢸⣉⡀
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡆⠐⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⣀⣤⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣂⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⣱⠿⠛⠂⠀⠀⢠⣤⣤⣤⣤⣤⣄⠀⠀⠀⡖⠂⠀⠀⠀⠀⠀⠀⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣈⣀⣉⡉⣉⣉⣀⢀⣉⡀⣙⠉⠉⠙⠿⠛⠛⠰⠀⠀⢠⡁⢀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣹⣉⣳⣅⣉⣉⣿⢀⣉⣿⡉⡉⢹⡇⠀⠀⠀⠀⡤⠀⠘⠃⠘⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⡷⢦⠀⢀⡢⠁⢖⣤⣤⣦⡁⢩⠍⠈⡏⠀⠁⠈⠉⠉⠉⠉⠉⠃⠀⠀⠀⠤⠒⢊⠀⠀⢀⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠉⣠⢀⠀⠀⣈⣸⡟⠐⢰⣿⢻⠀⢁⢧⣶⣶⣶⣶⣶⣶⣶⣀⡀⠀⠀⠀⣋⠛⠉⡍⣫⠀⠴⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠛⣁⠀⢈⡀⡉⠚⠃⠤⠒⢋⡈⠉⠛⢺⡍⠈⠈⠛⠛⠛⠛⠛⠇⡀⠀⠀⠀⠀⠀⠀⠙⠆⢲⠄⠠⡤⠀⣼⡇
⠀⠀⠀⣋⡀⠀⠉⢀⡈⠃⠀⠀⠀⠼⣀⡀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⠀⠀⠀⠠⢂⠀⡀⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠭⡤⠀⠁⡀⡈⠂⡀⠀⠀⠙⠉⡒⢠⠤⠄⣲⣠⠤⠀⠀⠀⠂⠚⠒⠚⠻⠏⣼⣵⡷⠶⠾⠦⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠖⠀⠶⠾⡄⢳⠳⠄⠹⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡤⠀⡀⠀⡰⢠⠴⠦⠆⠅⠈⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢀⣀⠄⠀⠈⠀⠈⢢⠀⠀⠀⠀⠀⠀⡄⡀⠀⡀⢀⣶⣥⢢⡴⡦⠤⠶⠘⠁⠛⠁
⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⡠⠐⠉⠀⠈⠐⠒⠂⠀⠀⠚⠀⠀⠀⠀⡀⠬⠐⠘⠊⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠄⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⡌⠀⠂⠀⠀⠆⠤⠀⠀⠀⠀⠤⠄⣀⠀⠀⠀⠨⠀⠱⠀⠒
⠀⠀⠀⠀⠀⢰⢀⠼⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤
⠀⠀⠀⠀⢡⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠌
⠀⠀⠀⠀⠘⠀⠀⢀⠄⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠃
⠀⠀⠀⠀⢀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠈⠶⠀⠀⣀⣀⠀⢀⡀⠀⠀⠀⠀⠀⡀⠀⣀⣀⡀⠀⠶
⠀⠀⠀⠀⠀⢦⡀⠀⠙⠀⢄⡀⠉⠄⠀⠀⠀⠈⠀⣀⠀⠌⠁⠀⡐⢀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⡔⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⣄⣘⡁⣴⢗⣰⢛⠀⠀⣀⣴⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⢀⣑⠦⠐⡀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠉⢡⠋⠀⠀⠀⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠠⠄⢄⡀⣄⢀⠄⠈⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣀⡀⠄⠉⠈⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⠁⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠅⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠛⠛


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢣⢫⢻⢿⣦⠀⠀⠀⠀⠀⠀⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠎⠀⠈⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⡀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⠉⠀⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠳⡄⠀⠀⠀⠀⠀⣤⢠⡄⠀⠀⠀⠀⠈⠐⠃
⠀⠀⠀⠀⠀⠼⠿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠈⠢⡀⠀⣤⠀⣤⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⣤⣀⠀⠁⠀⠀⣾⣿⣇⠀⠀⠀⠀⠀⠀⠛⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠜⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣀⢀⣶⠶⠄⠴⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢸⣿⢷⡉⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣷⠀⠀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⣿⣿⣿⣿⣿⢿⡿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠉⠙⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠦⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡉⢀⣾⣷⡀⣽⣿⣏
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⢰⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣧⣷⣿⣾⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣯⣠⣾⡇⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣯⢿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡟⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⠟⢺⣿⣿⣾⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣦⠈⠻⠿⠋⢠⣾⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⠿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⣠⣄⣀⠘⢿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⢸⣿⣿⣿⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣟⣃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣼⣟⣿⣻⣥⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈

⠀⠀⠀⠀⣴⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⢀⣀⠀⠀⠀⠀⠀⠀⡾⢂⣤⡀⠀⠀⢸⠋⣷⢠⡆⣠⠒⣆⠀⠀⠀⢈⠰⣿⢊⡁
⠀⠀⠀⠀⣙⣚⣁⢠⣤⠀⠀⠀⠀⣼⣧⣀⣤⢤⣄⠀⠀⠘⢧⣿⣴⡾⠋⠀⠀⠀⠀⠀⣾⡷⠼⠦⠋⠀⠀⠀⠹⠷⢿⡿⣿⡖⠋⠀⠀⠀⠈⢦⣶⡔
⠀⠀⠀⢠⡟⢻⠻⡆⠁⠀⠀⠀⣸⢯⡄⣿⡇⠸⣿⠂⠀⠀⢀⣼⠿⣅⠀⠀⠀⠀⠀⢀⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠐⠋⠀⠋⠀⠛⠀⠀⠀⠚⠉⠘⠃⠈⠛⠛⠋⠀⠀⠀⠚⠁⠀⠛⠀⠀⠀⠀⠀⠚⠁⠙⠂⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠸⠙
⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⢀⣶⣿⠟⠻⣷⣆⠀⠀⠠⡀⠀⣴⡶⠂⠁⠀⠀⠀⢠⣶⡖⠢⣶⣦⠀⠀⠀⢠⡦⠀⣀⣀⠀⢶⡄⠀⠀⢠⡄⣴⢲⡄⣰⠀⠀⠀⣤⠄⢀⢄⠀⢠⡄
⠀⠀⢸⣿⣯⠀⣹⣿⣿⠀⠀⢀⠂⢈⣿⣋⡀⢀⠂⠀⠀⠨⣽⣛⣛⣛⡭⠀⠀⠀⣇⡀⠸⣏⣻⠇⢀⡁⠀⠀⣶⣶⣾⣶⣶⡆⠀⠀⠠⣦⣀⢙⢬⡁⣠⣶
⠀⠀⠈⢿⣷⣤⣤⣿⠟⠀⠀⠈⠠⠊⠉⠉⠑⠌⠁⠀⠀⢘⣒⣶⣷⣖⣺⠀⠀⠀⠞⠓⠒⠚⠁⠒⠛⠃⠀⠀⣿⣿⣿⣿⣧⡟⠀⠀⠀⠂⠒⠒⠒⠐⠒⠂
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⣤⠶⢤⡀⠀⠀⠀⠀⣀⣀⣤⣄⣀⣀⠀⠀⠀⠀⢠⣄⣀⣤
⠀⠀⢸⠀⢠⡖⣦⠀⢹⠀⠀⠀⢸⡇⣴⡖⢸⡇⠀⠀⠀⣰⠏⢡⡶⠌⢛⣀⠀⠀⠀⣸⣃⣀⣈⣗⡀⠀⠀⠀⣿⡟⢩⡍⠻⣿⠀⠀⠀⣶⠞⣡⡶⠉⢶⡆
⠀⠀⢸⣄⣈⣑⣁⣀⡽⠀⠀⠀⢸⡇⠿⠶⢸⡇⠀⠀⠀⠿⡀⣽⣯⡄⢨⡏⠀⠀⠀⢿⣿⠉⣽⣿⠁⠀⠀⠀⢻⣆⠿⠅⣰⡟⠀⠀⠀⣼⡄⣿⣭⡄⣸⡄
⠀⠀⠲⠶⠶⠶⠶⠶⠶⠂⠀⠀⠘⠿⠿⠿⠿⠃⠀⠀⠀⠀⠛⠶⠴⠶⠋⠀⠀⠀⠀⠿⠿⠤⠽⠿⠆⠀⠀⠀⠀⠙⠿⠿⠋⠀⠀⠀⠀⠀⠹⠶⠲⠾⠁

⠀⠀⠀⠀⢠⣾⡓⠀⠀⠀⠀⠀⠐⠀⣟⠰⠂⣄⠀⠀⠀⠀⢀⠀⣿⣆⡀⠀⠀⠀⠀⢰⢊⣥⠍⢂⠀⠀⠀⠀⠀⢀⡴⢺⡷⢦⡀⠀⠀⠀⣠⠖⢛⡳⢦⡀
⠀⠀⠀⠀⢸⣯⣤⢀⣀⠀⠀⠀⡙⢱⢼⣋⠕⠰⠄⠀⠀⠀⡿⠻⢋⠻⡇⠀⠀⠀⠀⠰⣬⠯⢄⠜⠀⠀⠀⠀⠀⢸⡄⣿⣃⢨⡇⠀⠀⠸⠇⠀⣿⠁⠈⡇
⠀⠀⠀⠻⠷⢼⠴⠿⠏⠀⠀⠀⠘⠃⢸⡟⠉⠃⠀⠀⠀⠀⠁⢼⡯⠁⠁⠀⠀⠀⠀⠀⠈⣄⡈⠀⠀⠀⠀⠀⣠⣾⠟⠓⠚⠋⠀⠀⠀⣠⣾⠏⣉⣡⡼⠃
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠉⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈
⠀⠀⢠⣶⣿⣽⣽⣷⣤⠀⠀⠀⠀⢀⣬⣧⡀⠀⠀⠀⠀⠀⣈⣧⣧⣴⣆⠀⠀⠀⠀⣴⣿⠛⣿⣷⡀⠀⠀⠀⣴⣾⡿⢚⣶⣦⠀⠀⠀⣶⣿⣶⣴⣿⣿⡆
⠀⠀⠀⣷⢸⡇⣿⢰⡇⠀⠀⠀⢀⣾⡏⢩⣷⡀⠀⠀⠀⣼⣿⣿⣿⣿⣴⣦⠀⠀⠀⢻⣿⣐⣻⣿⣇⠀⠀⢸⣿⣿⠄⠼⣿⣿⠇⠀⠀⢿⣿⡏⢐⣿⣿⠇
⠀⠀⢠⣿⣼⣧⣿⣼⣧⠀⠀⠀⢸⣿⣥⣬⣿⡇⠀⠀⠀⢹⣿⡿⠿⣿⡟⠉⠀⠀⠀⠀⣿⣿⣿⠿⠃⠀⠀⠀⡿⠿⠶⠶⠟⠋⠀⠀⠀⠀⠙⢧⣤⠿⠃
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣤⡤⠀⠀⠀⣤⡄
⠀⠀⢀⣼⡗⡒⢤⣶⣆⡀⠀⠀⠀⠀⠺⣄⡿⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⣾⠋⣷⠙⣿⣆⠀⠀⠀⠀⢀⡀⠀⢾⣿⡷⠄⠀⠀⠋⢫⣶⠲⣮⠋⠃
⠀⠀⢸⣿⣦⣧⣾⣿⡟⠀⠀⢰⣾⣿⣷⣶⣦⡶⠆⠀⠀⣿⣿⣿⣯⣽⣻⣆⠀⠀⠉⠒⠋⢠⣼⣿⣦⠀⠀⢸⡏⣿⡆⣰⡿⠃⠀⠀⠀⠀⡸⣇⣘⡿
⠀⠀⠈⠻⠟⠛⠉⠉⠀⠀⠀⠈⠉⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠟⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠈⠛⠿⠞⠋⠁⠀⠀⠀⠀⣿⡏⠀⠀⠈⣿⠇

⠀⠀⠀⠀⠀⢀⣄⣀⠠⠤⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⣆⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⡶⢦⣤
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠀⠀⠀⣠⡾⠃⣀⠎⠀⠉⠢⢅⡠⠀⠀⠀⠀⠀⠀⢀⠔⠒⠈⠉⠁⠀⠀⠈⣽⢧
⠀⠀⠰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢹⡃⠀⠀⠀⡠⠚⠂⢀⠠⠇⠐⠀⠚⠀⠠⠉⠒⢄⠀⠀⠀⠀⢸⠒⠶⠿⣿⣿⣿⣿⡿⠿⢺
⠀⠀⡆⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢀⠀⡜⠐⢤⢀⠇⢀⣀⠀⠘⠃⠃⠀⠑⠂⠀⠉⠢⣀⠀⠀⡧⢄⠀⠀⣄⣤⡄⡄⠀⡈
⠀⠀⠇⢷⣄⠀⣀⡀⡀⠀⢀⣀⠄⢠⣤⣟⣿⢈⠀⠘⠦⡀⠌⣒⣤⡁⠤⢄⡀⢉⣀⣀⠀⠓⠒⠀⠎⡃⠀⡇⠀⣩⠃⠚⠋⠠⠥⠤⡇
⠀⠀⠀⠘⡯⢵⡶⠦⠬⡭⠵⠔⠚⣛⠾⢋⠆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠒⠂⠤⠭⠤⠭⠭⠖⠁⠀⢠⢸⣟⢴⣀⣀⣀⣼⡀⡇
⠀⠀⠀⠀⠈⠢⢍⣈⡉⣉⣀⣉⣡⠤⠔⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⢦⠤⣀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠘⠢⠬⣈⡛⠋⠛⠿⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡘⠠⠱⣆⣹⢕⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢠⠁⠀⠁⠐⡌⢳⠼⠏⢶⢄⠀⠀⠀⠀⠀⠀⠑⠠⢄⣀⣀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⡀⢠⣴⡷⣾⣿⣿⡶⣦⠀⠀⠀⠀⠀⠀⠀⢂⠎⡄⢀⣀⣐⠈⠚⠢⢀⡼⡜⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⢠
⠀⠀⠀⣴⣦⡛⣿⣼⣿⣷⢿⢿⣿⢓⢻⣾⠀⠀⠀⠀⠀⢠⠃⢠⠊⠁⠐⠂⠙⢲⠓⠋⠉⠣⡇⠀⠀⠀⠀⠀⣀⠤⢽⣊⠿⣶⣋⣿⠧⠤⢌⢢
⠀⠀⠀⢘⣨⣷⢹⢿⢿⣿⣿⣿⣷⠞⡞⢫⠀⠀⠀⠀⢀⠞⠈⡇⢏⣗⣼⢦⠃⡦⢇⡀⠤⡜⠀⠀⢀⡠⣖⡩⢿⡺⠽⠞⠉⠁⠀⠁⠀⠀⠀⢻
⠀⠀⠀⣹⡉⠉⢾⢯⣟⡎⢿⣹⣿⠞⠀⢠⠀⠀⠀⢀⢼⡀⠀⢱⡚⠛⠋⠙⠄⠃⡏⢀⠞⠀⠀⢠⠟⠉⠀⠈⠏⠁⠀⠀⠀⠀⠀⠀⢀⡀⠂⠌
⠀⠀⠀⠘⢧⠀⠈⡆⠈⠉⠉⠉⠀⠠⡀⢈⠀⠀⠀⠈⢦⠹⢮⡀⠈⠂⠭⠤⠴⢊⠀⡜⠀⠀⠀⢿⡈⠀⠀⡐⢀⢀⢀⠀⠀⠄⣒⠭⠓⠈⠁
⠀⠀⠀⠀⠀⢣⠀⠘⡄⠸⡀⠸⣆⣀⠇⢸⠀⠀⠀⠀⠀⠑⠦⡙⠲⢤⣀⠀⠉⢁⠜⠀⠀⠀⠀⠈⠫⢀⣉⠀⡈⠤⠐⠂⠉⠁
⠀⠀⠀⠀⠀⠀⢻⡄⠘⡄⠉⠈⠀⣀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠈⠁⠒⠬⠍⣑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⡰⠆
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠘⣄⡠⠅⠒⠊⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⣚⠓⠙⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⣢⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢒⡤⡓⢒⡂⠰⢎⣁⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢙⣈⣖⣶⡏⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠓⠒⢀⣉⣁⠤⣚⡶
⠀⠀⠀⠀⠀⠀⢀⠠⠤⠒⠛⠻⢟⢳⢦⢄⢀⡀⠀⠀⢨⣇⣞⣻⠿⠮⠄⠒⠒⠛⠢⡀⢀⠀⠀⠀⠀⠀⠀⢀⡌⠉⠉⠁⠤⠤⠐⠬⠔⢳⡄
⠀⠀⠀⠀⠀⣔⡅⠀⠀⠀⠀⠀⠀⡱⡄⠙⢵⡕⡄⠀⠠⡑⢄⠀⠀⠀⢀⠀⠀⠀⠐⠈⠓⢕⠦⠀⠀⠀⠀⠀⠳⠶⢒⣒⠀⠐⠒⢂⡉⢴⢡
⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⠀⠀⠀⠈⠹⠁⠀⢻⣌⠀⠀⠈⠂⠙⠢⡀⠀⠀⠀⠀⠄⢀⡔⣩⣵⢦⡀⠀⠀⠀⢠⡐⠀⠀⠨⠉⠏⠐⠐⣨⠈
⠀⠀⠀⠀⠐⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠇⢉⠠⣗⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⣾⡷⢪⠐⢸⡟⠀⠀⠀⠀⢨⠉⣩⠄⣤⢠⡉⡍
⠀⠀⠀⠀⠀⠀⠀⠙⠷⠦⢤⡤⢀⣐⡀⠑⣤⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⡋⢂⣀⣴⢟⠜⠀⠀⠀⠀⡆⡄⣙⡖⢙⡖⢑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠊⠁⠀⠀⠀⠀⠀⠀⠁⠡⠥⠬⠧⠬⠥⠂

⠀⠀⠀⠀⠀⣀⣀⡤⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⡖⠭⠒⢀⡍⡁⣈⡠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠃⠢⠀⠒
⠀⠀⣀⠁⠀⠀⠀⠈⠀⠀⠀⢀⡰⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⡐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠂⠀⠖⠀⠀⠊
⠀⠀⠁⠀⠀⠀⠒⠒⠒⢒⣈⣩⠴⠚⡅⠀⠀⠀⠀⠀⠀⡠⠞⠍⠀⠄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠤⠐⠒⠒⢀⣀⡠⠤⠖
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⡀⠠⠔⢻⠀⠀⠀⠀⠀⠀⣼⠁⠀⠐⠀⢠⠊⠁⠀⠀⠀⣠⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⠀⢀⣠⠄
⠀⠀⢸⠄⠀⡀⠀⠀⠀⠀⢀⣀⡤⢾⠀⠀⠀⠀⠀⠀⠇⠀⠀⢂⠘⠀⠀⠠⠤⠐⢊⠁⠁⡔⠊⡥⠀⠀⠀⠀⠀⠀⢩⠝⣀⢋⠼⡟⢣⠉⡄
⠀⠀⠘⠀⠄⠀⠈⠉⠉⠭⠀⢀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠔⠁⠀⠞⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⢼⣤⣦⣼⣤⢧⢿⠁
⠀⠀⠀⡇⠀⠀⡀⠄⠐⠀⠀⠙⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⢜⣕⢕⡵⡡⡏
⠀⠀⠀⠘⠀⠀⠀⣀⣈⣀⡠⠴⡺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡯⡪⣾⢮⡇
⠀⠀⠀⠀⠀⠀⠒⠀⠠⠄⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⢠⡆⠀⠀⠀⠀⠀⠀⣀⠄⠘⠀⠀⡚⢀⡀⠀⠃⢦⡐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠞⠁⠀⠀⢀⣠⠀⠀⠀⠀⣀⢄⠈⢱⠚⠇⢸⠁⠞⠂⡜⠓⢀⠀⠀⠀⠀⠀⣀⠄⠊⠀⠀⠐
⠀⠀⡠⠖⠂⠒⠉⠈⠁⣄⣀⣠⠞⠁⠘⠉⠀⠀⠀⠀⠁⠈⠃⣰⠸⠀⠟⠲⠀⣼⠇⠀⠈⠀⠀⠀⠀⡸⠤⠠⠄⠤⢀⡀
⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡆⠀⠂⠅⠧⠼⠿⠏⠀⠀⠃⠀⠀⠀⠀⠀⣺⣁⣊⣐⣂⠐⠋⠻⠐⠠⢀⠤⠀⠈⠆
⡌⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⠌⢀⡘⡈⠀⠀⠀⠀⢄⣁⣠⠎
⠇⠀⠀⠀⠀⠀⠀⡀⠀⢈⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢨⠆⠀⢀⡈⠉⠀⠒⠃⠐⠂⠐⠀⠜
⠀⠀⠀⠀⠀⣀⣴⡡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢀⡀⠀⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠐⠀⠤⠌⠀⠉⢐⡋⣘⠁⢀⠀⢀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⣀
⠀⠀⢀⣠⠖⠊⠉⠄⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⢸⠁⠄⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢧⡀⠀⣀⡀
⠀⣠⠋⠀⡈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⠀⠂⠀⠀⡄⠀⡄⠀⠠⠀⠀⢀⡀⠀⠀⠀⢸⠁⠀⠈⠃⠪⡁⢟⣠⣄
⠀⣇⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⣡⠞⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡒⢠⠉⠂⡠⠀⠀⡄⠀⠘⠀⠀⠀⠀⠈⠓⠀⠤⠘⣤⣗⣲⣄
⠈⢙⡁⠶⠄⠤⣤⠤⠤⡬⢄⠒⠚⠋⠀⡰⠁⠀⠀⠀⠀⠀⠀⠠⡔⠀⠀⡁⠂⡠⠊⠀⢀⡰⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⡢⡑
⠀⠈⡀⠀⠉⠀⠐⠈⠀⠐⣁⠤⠚⢀⡁⠴⢃⠀⠀⠀⠀⠀⠀⠀⠀⠄⢈⠕⠊⠀⢀⡴⠋⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞
⠀⠘⠀⠘⠀⠤⠶⠒⠒⠒⠀⠚⠩⠉⠂⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢠⣪⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠂⣒⣨⠔⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠓⠾⡆⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠶⠶⠶⡆⠀⠘⢲⡶⡶⠶⠛⢮⢀⡀⠀⠀⢰⠶⠶⠶⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠭⣿⠀⡇⠀⠀⠀⢁⡹⢤⡤⢾⡀⠁⠀⠀⢸⠀⡞⠙⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣽⣀⣚⣋⣀⣧⣀⡀⠀⠈⠀⠸⠂⠀⠉⠀⣀⣀⣸⣀⣈⣉⣀⣽⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⡄⠀⡯⠀⠲⡆⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⠀⢸⡅⢰⣦⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⠇⠀⡧⠀⠯⠄⠀⡇⠀⠀⢀⡴⢄⡀⠀⠀⡇⠸⠛⠇⢸⡇⠸⠽⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⠞⠁⠀⠀⠙⠷⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⢀⣀⡴⣄⣀⠀⠈⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣥⠤⠤⠤⠤⠤⠤⠤⠤⢤⡖⠋⠀⠀⢀⡿⣬⠷⣯⠾⣄⠀⠀⠀⠙⢦⡤⠤⠤⠤⠤⠤⠤⠤⢤⡽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⣀⣀⠀⠀⢸⠀⠀⠀⠀⠈⣷⣛⡶⣏⣳⠉⠀⠀⠀⠀⢸⡇⠀⢀⣀⣀⣀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣧⣬⣧⣼⠀⠀⢸⠀⠀⠀⠀⠀⠈⠉⠓⠋⠉⠀⠀⠀⠀⠀⢸⡇⠀⢸⣤⣿⣤⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠧⠴⠧⠼⠀⠀⢸⠀⠀⣠⠤⣤⢤⡄⠀⠀⢠⠤⣤⠤⡄⠀⢸⡇⠀⠸⠤⠿⠤⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⣀⣀⠀⠀⢸⠀⠀⣿⠶⡷⠴⡇⠀⠀⢸⠶⣿⠶⡇⠀⢸⡇⠀⢀⣀⣀⣀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣯⣩⣏⣹⠀⠀⢸⠀⠀⠛⠒⠛⠚⠃⠀⠀⠘⠒⠛⠒⠃⠀⢸⡇⠀⢸⣉⣿⣉⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢧⣬⣧⣼⠀⠀⢸⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⢸⡇⠀⢸⣤⣿⣤⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⡇⠀⠀⢙⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡇⠀⠀⣹⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣧⣤⣤⣿⣤⣤⣤⣧⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣼⣧⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡂

⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣇⣀⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠁⢰⠃⠑⣆⢰⠈⢻⠉⢠⢷⠀⢸⢈⡆⢰⠀⠀⣎⡁⠀⠠⣎⠁⠈⣟⠁⢺⠈⡆⡴⠁⠑⡄⣆⡴⠀⣞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⢸⣀⣠⠏⢸⠀⢸⢀⡞⠚⣇⢸⣀⡇⢸⣀⡀⣇⣀⠀⢀⣀⡱⠀⣏⠀⣸⠉⣆⠳⣀⣠⠃⡏⠱⡄⣏⣀
⣿⣿⡟⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⢄⡀⠛⠛⠛⠓⠃⠀⠀⠃⠑⠒⠒⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡇⠸⠤⣧⠀⠀⡆⠘⢵⠶⠟⠸⠿⠆⠗⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠟⢠⠀⡇⢀⠄⣤⡀⣰⢠⡄⡀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠈⠀⠃⠃⠇⠸⠣⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠤⠾⠁
⠀⠀⠀⠀⢠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⣀
⠀⠔⢂⠶⠘⣆⣁⡀⠔⠒⡄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⠶⠖⣒⡒⠒⠤⡀⠀⠤⠒⠀⠀⠀⠀⠀⣠⠖⢻⡀
⡏⠙⠳⣄⣀⣭⠤⣽⣄⣀⡼⡄⠀⠂⠀⠀⠀⠀⣯⠀⡀⠀⠀⠀⠀⠀⠈⢶⠼⡄⢌⢑⣦⠄⠀⠠⠟⣩⠾⠛⠁
⠇⠀⠀⠀⠀⠀⠀⠀⡜⢻⠾⠃⠀⠀⠀⢀⠀⢀⣧⣠⣀⣀⣌⣠⣄⣦⣤⣴⣾⠇⠀⠄⡉⣁⠴⠚⠉
⢰⢰⠀⠀⠀⠀⠀⠈⠀⢀⡇⠀⠀⠄⠀⠜⠀⢧⡈⢙⣛⣚⣛⡓⠋⠍⢝⡪⠟⠀⠀⠀⠀⠠⠤⠤⠤⠀⠀⠀⠀⠀⠋⠢⡀
⠘⠤⢤⣤⣤⣠⣦⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠚⠃⠉⠁⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⢠⠆⠀⠀⠁⠰⡐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠯⠄⠉⠙⠲⣄⠀⠀⠀⢹⡀⠀⠀⠀⢠⠏⠀⠀⠁⢀⣤⣀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢄⣀⡒⠁⠀⠂⠀⠀⠀⠀⡠⠁⠂⠀⠀⠀⠢⡀⠑⢄⠀⠀⠃⡀⠀⠀⠘⠀⠀⠀⠈⠍⠉⡉⣝
⠀⠀⠀⠀⠲⠅⠀⠀⠀⠀⠀⠉⠁⠀⠀⢄⠀⠀⠀⣰⠁⠀⠀⠆⠀⠀⠀⣈⣦⠄⠆⠀⠀⠁⠀⠆⠀⠀⠀⠀⠀⠀⠐⠄⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠡⠀⠀⢀⢀⣤⣶⠟⠋⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⠃⠀⣠⡴⠚⠉⠀⠘⠀⠀⠀⠀⠀⣸⠀⠀⠀⠠⠀⡆⠀⠀⠀⠀⢰⣼
⠀⠀⠀⠀⠀⠠⠠⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠺⠶⠛⠁⢀⣠⣤⡦⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠁
⠀⠔⢉⠉⢳⠀⠀⠀⠀⠀⢀⡀⢀⡀⡀⠀⠀⠀⠀⣠⠞⠙⠠⠁⠉⠉⠀⠉⢱⡀⠀⣀⣠⣁⣉⣡⡀⡀⠀⠀⠸⣽⣋⣦⣄
⠀⠀⢆⠀⢸⡄⠀⠀⣴⠟⢪⠟⠈⠐⠒⠣⣄⠀⢠⠏⠀⠀⠀⠀⠀⣀⣥⣾⣽⠀⠀⢉⣏⠉⠹⡟⡇⣧⠄⠀⠀⠈⠉⠀⠈
⠀⢰⡃⠀⢸⠃⠀⣺⣛⠔⠁⠀⠀⠀⠀⠀⠈⡀⠸⣄⣀⣤⠔⢲⠟⣿⣥⡴⣚⣡⡆⠀⡇⠢⢴⡽⠿⣏⠀⠀⠀⠀⠀⠀⢀
⠀⡇⡅⠀⢸⠀⠀⠈⠓⠢⠤⠤⠀⣀⠀⠀⠤⠁⠀⠈⣭⣄⣀⣋⣭⣾⡟⠋⢱⡜⠀⠀⢇⠀⠀⠀⢰⠃⠀⠀⠀⠀⠈⠁⡠
⠀⢹⠁⠀⡞⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢅⣀⣉⣉⡠⠴⠒⠉⠀⠀⠀⠘⠀⠀⠀⡟⠀⠀⠀⠀⠀⠉⠉
⠀⠈⠣⡜⠁⠀⠀⠀⠀⠀⠁⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠂⡀
⠀⠀⠀⡇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡀⣀⣤⣴⠟⠛⠉⣻⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⢶⠈⠢⡀
⠀⠀⠀⡁⠀⠀⠈⠁⠀⠀⡙⠉⠉⠛⠉⠑⡉⠀⠠⠐⠗⠛⠉⠉⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⡿⣼⢄⣶⠟
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁⠀⠀⠀⠆⣀⣀⡀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⡖⣾⢻⣻⢹⡟⡗⡖⣦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⣿⣎⠧⠗⣛⣋⠉⠭⠭⠉⣙⣓⠣⢾⣾⣹⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢎⣳⡽⢊⠥⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠬⡑⢯⣞⡵⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢮⣙⡾⡡⠊⠁⠀⠀⠀⠀⠀⠀⡏⠈⡇⠀⠀⠀⠀⠀⠈⠑⢌⢶⣋⡵⡀
⠀⠀⠀⠀⠀⠀⠀⢠⢯⣳⠏⡔⠁⠀⠀⠀⠀⠀⣠⡴⠒⠁⠀⠁⠐⠂⠄⠀⠀⠀⠀⠈⢢⠹⣞⡽⡄
⠀⠀⠀⠀⠀⠀⢀⣟⣓⠏⡌⠀⠀⠀⠀⠀⠀⢰⠋⠀⠔⠉⠀⠉⠙⠚⠀⠀⠀⠀⠀⠀⠀⢡⠹⣚⣽⡀
⠀⠀⠀⠀⠀⠀⣸⠤⡽⢸⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢷⠶⣇
⠀⠀⠀⠀⠀⠀⣽⣽⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠈⠒⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠭⢭
⠀⠀⠀⠀⠀⠀⣗⣒⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⣄⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠘⢸⣟⣟
⠀⠀⠀⠀⠀⠀⢹⠶⢷⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⣞⠒⡏
⠀⠀⠀⠀⠀⠀⠈⣟⡭⣆⢃⠀⠀⠀⠀⠀⠀⢰⠦⠄⣀⣀⣀⡴⠃⢀⠌⠀⠀⠀⠀⠀⠀⡘⣰⢭⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣞⡽⢆⠣⡀⠀⠀⠀⠀⠙⠢⠤⢄⡀⠀⠤⠔⠉⠀⠀⠀⠀⠀⢀⠜⣰⢯⣳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢖⣩⠷⡑⢄⡀⠀⠀⠀⠀⠀⢸⣇⢀⠀⠀⠀⠀⠀⠀⢀⡠⢊⡼⣍⡳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢞⡽⣳⢌⡒⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢒⡡⣞⢯⡱⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⣏⡿⡷⢢⢭⣍⣀⣒⣒⣀⣩⣭⢴⢺⡹⣿⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠻⠼⢼⣽⣇⣯⣇⡿⠼⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡘⣀⠁⠐⠲⠦⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡖⠒⠀⠈⠁⠙⠛⠿⠿⠿⠯⠝⠉⠉⠁⠀⣤⣶
⠀⠀⠀⠰⣄⠀⠀⠂⠀⠀⠀⠀⠀⡄⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠖⠒⠒⠂⣀⠠⠤⠄⠀⢤⠤⢤⡤⠴⠶⢛⡛
⠀⠀⠀⠀⠻⡄⠀⠀⠀⢸⡀⠀⠀⠣⠀⢠⠀⠀⣦⡎⠀⠀⣺⠀⠀⠀⠀⠀⠀⠀⡀⠁
⡀⢀⠀⠀⠀⠂⠀⢠⠀⠀⠁⠀⠀⠀⡄⠀⡄⠀⠈⢠⠄⠀⠋⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠌⡀⠀⠀⠀⠀⡀⠼⣇⠀⠀⡇⠀⠰⠃⡄⠃⠀⠰⣧⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈
⠠⠟⠀⠀⢀⣀⣠⣀⣉⣀⣀⣇⠀⠀⠀⠁⠈⢀⠠⠄⠤⠤⠤⠔⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣑
⢸⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠂⠀⢀⠀⠀⠒⠂⠀⠀⠀⠒⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿
⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⡀⠎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡆
⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⢀⢹⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⡀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠋⠉⠉⠉⢉⠉⠛⠙⣻⠦⠐⡒⠀⠁⠀⠀⠀⢀⠀⠀⠐⣻
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠐⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠀⠀⡀⠀⠀⠀⣀⡄⢀⠖⢀⡀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠈⣾
⠀⠀⢔⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⣄⠀⡀⠀⢀⣀⣠⣀⠀⠀⢀⠀⠄⢀⡈⡉⠀⠀⠐⠂⠂⠀⣀⡀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⢸⠚⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⠀⠀⠀⠖⠙⠆⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠠⠤⠶⠦⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠛⠳⡶⠲⠶⠓⠔⠂⢀⠀⠀⠖⠒⠂⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⠀⠀⠀⡐⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁

⣿⣿⣿⣿⣿⣿⣿⣦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣤⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠙⢿⣿⣿⣄⠀⠹⣿⣿⣿⡿⠛⠋⠉
⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠙⢿⣿⣆⠀⠈⢿⣿⣇
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡏⠙⠻⣿⣷⣄⠀⠙⢿⣦⠀⠀⠻⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⣄⠈⠙⠻⣷⡀⠀⠙⠇⠀⠀⠙⢿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⢿⣿⣷⣤⡀⠈⠙⠆⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣄⡀⠉⠙⠛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣇⠀⠰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡆⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠉⠀⠀⠀⠉⠛⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁
⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣿⣿⣦⣄⣀⣀⣀⣠⣴⣿⣿⣇
⣿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠛⠉⠛⠿⠿⠿⠿⠿⠛⠉⠛⢿⣿⣆
⢿⣿⡄⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⠀⢀⣴⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧
⠈⢿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣥⣤⣤⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⢠⣿⡿⠋⠉⠉⠉⠙⠛⠛⠛⠛⠛⠻⠿⠋⠀⠀⠀⠀⢀⣠⣴⣾⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠙⢿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠿⠛⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠛⠛⠋⠉⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠿⢃⣠⠴⣳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⡶⣫⣥⠊⠙⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣽⣾⡿⠋⠠⡄⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣯⣿⠄⠀⠈⢁⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠛⠛⠋⢀⡀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢁⡼⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣚⣽⠛⣏⠀⠀⢀⣴⢃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢺⡈⠙⠓⠒⣦⡐⢒⠿⣛⣩⡝⢾⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣧⠘⡄⠀⠈⣿⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⡀⠀⠉⠀⠀⠀⢟⠋⠀⣸⡇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡆⢱⡀⢰⠇⣸⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠐⠈⡀⢆⡀⠀⠀⣉⠘⠋⠳⢤
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣧⠀⣿⣶⠀⠄⠀⠀⠀⠀⠀⣰⣦⣄⣹⣶⢶⣦⠀⠀⠀⠀⢸⣿⠀⠀⠀⡀⠈⠀⠀⠀⠻⢶⠀⠀⣸⠇
⠀⠀⠀⢀⣾⣿⣿⣿⡟⣿⣿⣿⣿⣿⠀⣿⣿⠀⡆⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠩⡀⠀⠀⠀⢸⣿⠀⠀⠀⠸⠆⠀⠀⠀⠐⣤⣤⣮⡽
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⡟⠀⠃⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣸⡏⠀⠀⠀⠈⠀⢐⡁⠀⠀⠈⣿⣅
⠀⠀⢸⣿⣿⣿⢿⣿⣿⣿⣿⣿⡿⠃⡰⢋⡴⠃⠀⠀⠀⠀⠀⠀⠀⠒⠒⣤⠄⠘⠛⠁⠀⢸⠀⡟⠀⠀⢀⣠⣴⠎⡿⢧⠀⠀⣸⠇⠈⠓⣦
⠀⠀⠘⣿⣿⣿⣿⣟⠿⠿⢟⡉⠀⢀⣴⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⢸⠞⠀⠀⣴⡟⠀⠁⣼⠃⠘⠀⣰⠃⠀⠀⢀⣼
⠀⠀⠀⠘⢿⣿⣿⣿⣦⠄⠀⢠⣤⠛⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⡴⠃⠀⠀⢸⣿⡇⠀⢠⣿⢀⣴⡿⠁⠀⢀⣴⣿⣿
⠀⠀⠀⣼⣿⣟⣿⡟⠉⠑⠄⠀⠀⢣⠈⣻⣏⠀⠀⢤⣀⠀⠀⠀⠠⣄⣘⣛⣇⡄⠒⢸⣇⠀⠀⠀⣿⣿⠋⠀⢸⣿⣿⠟⠁⠀⠀⣸⣿⣿⣿
⠀⠀⣼⣿⣿⣿⠇⠐⢀⡴⠊⠀⠀⠘⠘⠿⣧⣥⣄⢀⠉⠛⢿⣿⠿⢛⣷⣤⣶⣶⣿⣿⢻⡄⠀⢠⣿⣇⠀⠀⢸⣿⠃⠀⠀⣠⣼⣿⣿⣿⣿
⠀⢠⣿⣿⣿⡟⠀⠀⠀⠑⡀⠀⡰⢄⠁⢀⣿⣿⣿⣦⡹⠇⠘⣩⣿⣽⡋⣿⣛⣛⠻⣿⣧⠙⣠⣿⣿⣿⣆⠀⢸⣷⣦⣴⣾⣿⣿⣿⣿⣿⣿
⠀⠘⣿⣿⣿⠇⠀⠀⠀⠀⠀⠬⠁⠎⠁⠀⢉⣻⢿⣿⣿⡟⠷⣿⣿⣿⣷⣿⣭⠁⠘⢱⣿⡽⠏⠋⠙⠙⣛⠛⢶⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣟⠐⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣿⠁⠈⠁⠀⠀⠀⠙⠛⠛⠛⠁⠀⠑⠶⠤⣭⣿⣛⣍⣀⣨⣤⠤⢈⣀⣠⠇⠀⠈⠹⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣦⣄⡀⠀⠀⣲⣬⣿⣷⣬⣿⣿⣗⠠⠦⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠒⠂⠀⢒⣈⠭⠼⢛⣃⡸⠷⡄⠘⠻⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠉⠁⡿⠀⠀⠀⠀⣀⠀⢉⣠⡴⠖⠀⠤⠀⡀⠀⠀⢤⣿⡿⠿⠖⠚⠋⠉⢲⡀⠀⠘⠲⠲⣿⡿
⢀⣀⡿⠿⠟⠛⠋⠉⠁⠀⠀⠀⣀⡀⠀⠤⠤⠀⡗⠀⠀⠀⢉⣠⠾⠛⠉⠀⠀⠀⠀⠀⠀⠈⡏⠉⠉⠁⠐⠤⣄⡀⠀⠈⡇⠀⠢⢄⡀⠈⠛
⠀⠀⠀⠀⠠⠀⠀⠀⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠻⣐⡶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠳⢴⡇⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣸⠁⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠠⠐⠀⠁⠀⠁⠀⠀⠀⠑⢤⡀
⡀⢀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠜⠛⢉⡀⠀⠀⠀⠀⠀⢀⢀⡀⢀⡀⡀⠈⢀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿