Old lady Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠆⠔⠉⢳⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠃⠀⠀⠀⠀⠀⣤⠠⠄⠠⠀⣀⠀⡀⡀
⠀⠀⠀⠀⠀⢠⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠠⠤⠄⢀⣄
⠀⠀⠀⠀⠀⢨⠄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣃⣀⣀⠀⡀⠀⠀⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠣⡄⠀⠀⠀⠤⢔⠒⠊⠀⠀⠀⠀⠀⠀⠡⠈⠉⠀⢀⡀⠈⠄
⠀⠀⠀⠀⠀⠀⠉⠁⠒⠀⠀⢠⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⣨⡵⢿⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⠄⠊⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢣⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⢀⠶⠉⠀⠈⠀⠘⢄⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠈⡁⠣⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⢠⠞⠀⠂⠀⠀⠀⠀⠐⠀⠀⠈⡇⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⢠⡁
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠘⢄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠠⢄⡀⣀⣀⣀⠴⣇
⠀⠀⠀⢨⡓⣦⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢴⠆⠀⠀⠈⢳⡀
⠀⠀⠀⢸⣀⡟⠙⡿⠒⠒⠀⠁⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⢹
⠀⠀⠀⡞⢰⠀⢰⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠦⣄⣀⣀⣀⣠⣮⠀⠀⠀⠀⠈⣡⠈⠀⡀⠀⠀⠠⡄
⠀⠀⣸⠁⠨⠀⡟⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⣾⣄⠀⠀⠀⠀⠀⠀⠨⠁
⠀⠀⣿⠀⠇⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠁
⢀⠤⢇⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢳
⠌⠓⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⡡⠄⠊⠁⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸
⠀⠨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢾
⠀⠀⠈⠉⠒⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠐⠀⢀⡠⠤⠊
⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⣿⠒⠒⠢⠐⠤⠤⠤⠠⠤⠤⣤⣶⣤⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⡎⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⣠⠴⠂⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠐⠦⣄⡀
⠀⣠⠖⠉⠁⢀⠀⠄⠂⠁⢀⣠⠞⠁⠀⠀⠀⠀⠈⠢⣄⡀⠈⠐⠂⠤⠀⠈⠈⠳⣄
⣞⠁⠀⠀⢀⣀⣠⠤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⣄⣀⣀⣀⣈⠇

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠀⠀⠉⠉⠀⠁⠐⠠⢄
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠤⡀
⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⢀⠊⢀⠤⢄⠀⠀⠈⢂
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⡈⠀⢁⡀⠀⠡⣴⡀⠀⢂
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠐⠀⣴⢥⠶⡤⡤⠦⡵⠦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦⡀⣀⠀⠀⠀⠈⢂⠙⡀⠘⢄⡠⠀⢠⢃⠎
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠽⣊⠋⠄⠀⠀⠀⠀⠈⠚⡒⠒⠂⢖⠋⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠄⣁⣸⡄⠀⢀⣴⣾⡟⠬⠐⠂⠅⢻⣷⢦⡀
⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⣿⣷⣿⢀⣾⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣳⡀
⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢿⢿⣿⣿⢿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⣧⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠤⠄⠠⠤⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠿⠿⠿⠇⠀⠀⠀⠀⢸⠿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠊⠉⠁⠘⠛⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⣀⣀⣀⣀⣀⣀⣀⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔⠛⠃⠉⠉⠉⠉⠉⠉⠉⠉⠁⠙⠛⠢⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠀⠀⠀⠀⠀⠀⢠⣴⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⢶⡈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⠐⠁⠀⠀⠀⢀⣴⠏⠀⠀⠀⢳⣄⠑⣤⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣠⠂⠀⠀⠀⡀⣰⡟⠁⠀⠀⠀⠀⠀⠙⢧⡀⠉⠲⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣴⣀⣀⡠⠔⣃⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠈⠉⣉⣤⠞⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠳⢦⣄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠶⠄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⠁⠀⠀⡀⠀⠀⣿⠇⠀⠀⢀⡈⠀⠀⠀⠀⠀⠘⡿⠗⠀⠀⠀⠀⠈⢷⣼⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠛⢷⡀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡼⠛⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠊⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⡰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⣤⡾⠛⠧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⢹⡷⠶⠒⠚⠛⠛⠛⠒⠒⠉⠛⣹⠇⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣾⠋⠀⠀⠀⠀⢿⢦⣈⠀⠠⠀⠀⠈⣉⡤⠾⡟⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢾⠀⠀⠀⠀⠀⢻⠀⠉⠙⠛⠛⠛⠛⠉⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣦⣾⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠰⠀⢰⡆⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠇⣸⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢻⣄⠆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢀⡉⠀⠀⠀⠀⠰⣄⠀⣰⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣼⠟⢹⡇⠀⠀⠀⠀⠚⠛⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⣹⠀⢸⡇⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⣠⣤⣤⣀⣿⣁⡀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⢿⣀⣀⠴⣄⣉⣿⡀⢸⡇⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠠⣾⡇⡀⠀⠙⠿⢋⠁⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠻⣧⠞⠉⠉⠉⠱⣾⠃⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣆⢁⡄⠀⣰⡏⢀⣀⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⢀⣴⠇⣼⠶⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣎⣩⡄⡿⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⠁⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠴⠟⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇⢀⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⡀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⣴⠀⠀⠀⠀⠻⠿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢁⡴⠎⠀⠀⠀⢀⣀⣤⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣣⣴⠛⣤⣀⠴⠶⠞⠋⠉⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⢹⠛⠛⠓⠒⠒⠶⠶⠶⠖⠲⠶⠛⠛⠛⠋⠉⠀⠁⠀⠀⢀⣤⠴⠖⢛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠘⣦⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⡶⣊⣁⣀⢠⡶⠖⠋⠉⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣿⠀⠈⠉⠙⠛⠛⠛⢿⠛⠛⠋⠀⠈⠁⠀⠈⡇⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⣀⣠⡄⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⣀⡅⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣤⣀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢠⡿⢦⣤⣄⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⢰⡄⠀⠀⠙⠷⣤⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠒⠀⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⠀⠀⠀⠀⠠⢀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠎⠀⠀⢠⠄⠀⠀⠀⠣⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠠⡀⠒⠁⠈⠢⢄⣀⣀⡇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⡀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠿⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡌⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠑⣢⣤⣀⣀⣀⣀⣠⡴⠞⠉⠀⢀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⠌⡉⢻⣄⠀⣨⡿⠀⢄⠀⢠⠊⠉⠁⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣠⠎⠀⠁⣸⠀⠉⠀⢢⡀⠸⡀⢈⠀⠀⠀⠠⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠘⠣⢤⣤⣞⠁⠀⡄⠀⠀⠉⠓⠛⢀⣀⠀⠀⠀⢇⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢠⢞⡳⠁⠀⠠⠖⠲⣍⠁⠈⠁⠀⠀⡸⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠖⠤⣀⠯⠼⠁⠀⣤⣀⠀⠀⣸⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠇⠠⠒⡂⠒⠛⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠁⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⣠⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣈⡑⠲⠤⣀⣀⠀⢀⠀⠀⠀⠀⣀⣀⡀⠤⠤⠔⠒⣩⣀⣠⠾⠋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⢦⣤⣤⠀⢨⠈⣉⡁⣀⠤⢄⠠⡴⠓⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡁⠀⢨⢀⠐⠰⣿⡶⠛⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⠛⠒⢻⠙⠶⠖⢻⡃⠀⠤⠂⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠟⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠟⡁⡶⢂⣠⡀⠀⣹⢶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⣿⣿⠎⣰⠠⠁⠈⠘⠖⠘⠉⠈⠋⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢟⣵⣾⣿⠻⢿⣿⣷⣶⡀⠀⠀⣤⣴⠀⣦⡉⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⣸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡁⠈⣹⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢷⣾⣿⡿⠀⠀⢀⣀⠀⢠⠀⣀⣀⡀⠀⠀⠈⣿⠀⠀⣼⡁
⠀⠀⠀⢠⠆⡀⠀⢠⠒⡆⠀⠀⠀⠀⠘⡷⠾⣿⣿⣧⠀⣩⣽⣿⡀⠀⣾⣿⣿⡷⣄⠀⠀⡟⣷⡀⠐⣻
⠀⠀⠀⢸⠀⡅⠀⠨⠀⠇⠀⠀⠀⠀⠈⢳⠄⢙⣿⣟⠉⠀⠀⣹⠇⠀⠻⠉⠀⠀⠀⠀⠁⡏⢁⣿⡶⠋
⠀⢇⠠⠘⠀⢤⠀⡀⢸⠀⠀⣀⠀⠀⠀⠈⠳⢿⣿⡇⠀⠀⣠⣇⣀⠀⣀⠀⠀⠀⠀⠀⠀⣧⢾⡟⠁
⠀⠘⡀⠀⣆⠀⢠⡁⠨⡤⠊⢠⡀⠐⠀⠀⠀⠸⣿⡇⠀⠀⣏⡀⠈⠀⠀⣀⡸⣄⠀⠀⠀⣃⡼⠁
⠀⠀⠐⣄⠈⠀⠀⠀⢀⣀⡖⠁⠀⡱⢺⠀⠀⠀⠙⢧⠀⠐⠃⠙⠷⠶⠞⠋⠀⠏⠀⠀⡰⠋⠁
⠀⠀⠀⢸⡆⠀⠀⠐⢉⠎⠹⡤⠒⠃⡸⠀⠀⠀⠀⣾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠙⢷⡀
⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⢀⡼⠁⠀⠀⢀⣼⠙⢻⣿⣦⡑⠠⠄⠠⠒⣠⠶⠋⡿⠁⠌⢧
⠀⠀⠀⠈⢷⡀⠀⠀⠸⠀⠀⢠⠞⢀⣀⡤⢶⡿⠁⡀⠂⣽⣧⡉⠀⠀⠀⠉⠀⣀⣾⠅⠁⠠⠈⢿⢦⣤⣀⣀⣀
⠀⠀⠀⠀⠼⠃⠀⠀⠀⠀⠴⠟⠊⡉⠂⣀⣾⡄⠢⡴⠾⡿⣿⣿⣦⡀⠀⣠⠞⣫⣽⠦⣌⣠⡈⢸⣯⠀⠈⠀⠈⡙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠂⡀⠂⢀⠂⢀⠀⢀⢈⡿⠻⠻⣞⠡⠈⠨⠿⣄⠈⠀⠍⠁⠌⠁⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⠐⢀⠐⢀⠐⣰⣯⣠⠈⠀⠹⡆⠄⠀⣄⣩⣦⠁⠄⠁⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠐⠀⠀⠀⠈⠉⠝⠁⠄⢻⠄⠋⠍⠁⠄⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠁⠄⠱⠿⠉⠆⠁⠀⠀⠀⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣶⣦⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠉⢉⡈⡙⢦⡉⠉⠹⣿⠹⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠃⢀⣴⣿⣿⣿⡦⠙⠻⣿⣿⡖⣻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠇⣴⣿⡟⠛⢛⣳⣄⡀⠀⢾⣿⣗⠵⡛⣿⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡁⢿⣾⣿⡥⠴⢶⣿⡿⠿⠟⢠⣾⡏⠀⢿⡿⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠙⢯⣵⣾⡿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠟⠒⢲⣿⣿⢛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣦⣽⣿⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣠⣀⢸⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣛⣿⡄⠰⠋⣉⣛⠃⡀⠀⢀⢉⣭⣄⡉⠸⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣊⠻⣏⠀⠀⠾⠿⣿⠓⡙⠀⠿⠛⠛⠛⠋⠀⡟⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢼⣿⣿⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⣇⠀⢀⡀⠤⠀⠒⠳⠞⠋⠀⠄⠀⢸⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠘⠀⠈⠙⠫⠭⠽⡻⠉⠀⢈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢦⣤⡀⠀⠘⠒⠛⠁⢀⣠⣯⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣇⢻⣄⠈⠙⢦⣤⣤⣤⣤⣾⣿⠏⠀⣿⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⠻⣏⠘⠈⠀⠛⢷⣄⡀⠀⠉⠉⢁⡾⠋⠀⠈⠻⡏⣿⠀⣾⣹⡟⣦⣤⣀⣀
⠀⠀⠀⠀⠀⣀⣠⡴⣿⡅⣿⢹⡄⢻⡄⠀⠀⠀⠀⠈⠙⡷⣒⢖⣻⡀⠀⠀⠀⠀⠀⣿⠀⢣⡟⡆⣿⣿⣶⣽⠳⣆
⠀⠀⠀⣴⣟⢩⣴⣶⡏⡇⢹⡌⣧⠈⢿⣄⠀⠀⠀⣀⣤⣧⣓⣄⣡⣷⣦⣄⡀⠀⢸⡇⠀⢸⠃⡇⣿⣿⣿⣽⣿⡞⣷⡄
⠀⢀⣾⣇⢸⢸⣿⣿⣿⢱⠀⠀⠘⠄⠈⢿⣷⣶⣾⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠁⠀⠘⠀⡇⣿⣿⣿⡏⣿⡇⣿⣿⣆
⠀⢸⣿⣿⡌⡇⣿⣿⣿⡎⡆⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⢻⣿⣿⡟⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⡇⣿⣿⣿⣷⣿⡇⣿⡿⣿
⠀⢸⣿⣿⣷⢹⣸⣿⣿⣷⢳⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⢸⣿⣿⡇⡏⣿⣿⣿⡿⠁⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⡇⣿⣾⣿
⠀⣸⡟⣿⣿⡎⣧⢿⣿⣿⡎⡇⠀⠀⠀⠀⠀⠀⠙⢿⣧⣸⣿⣿⣇⣧⣿⠿⠋⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⡟⣿⡇⣿⣿⣿
⢸⣿⡘⠻⢿⣿⣸⡜⣿⣿⣿⡸⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢹⣿⢱⡿⢩⣿
⣨⡿⠿⠿⢶⣿⡇⡇⣿⣿⣿⣧⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣿⣿⣿⣼⡿⠚⠁⢛⣿
⣿⣿⣤⣹⣿⣿⣷⢱⠹⣿⣿⣿⣷⡹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣣⣿⣿⣿⢸⣿⣇⢀⣴⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢋⠀⠀⠀⠀⠀⠀⠈⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⣄⣠⣤⣶⠀⠀⠰⠆⢸⠅⠚⠂⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠁⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠊⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣀⠀⠀⠀⠀⠀⢀⣤⣭⠐⠒⠂⠒⠀⠀⠀⢤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠛⠛⣃⠀⠀⣴⣇⠀⠀⠀⠂⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠂⠉⣿⣤⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠃⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠋⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢁⠠⢾⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠁⠀⠀⠠⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡇⠀⢠⣼⣷⡆⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣸⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠱⡌⠙⠃⡀⠀⠀⠀⠀⡇⠀⠠⣬⣏⠁⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆⣀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢡⡀⠀⠀⠈⢳⡄⣀⢽⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠉⠉⢡⣄⣀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢣⣤⣤⣄⠊⠁⠀⠀⠀⠀⠛⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣐⠈⠀⠀⡤⠄⠀⠐⠒⠀⠁⡀⠈⠒⠪⠀⠀⠸⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⢙⠗⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠘⠂⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠤⡀⡀⡙⠂⢀⠄⣀⡄⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠠⠂⠀⠀⠉⠀⠐⠊⠠⢄⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠠⠂⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⡇
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣤⣤⠀⠀⠀⠀⠀⣠⣤⣶
⠀⠈⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠉⠀⠈⠁⠀⠀⠀⠀⠀⢉⠀⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⣀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡁⠈⠀⡀⠀⠀⠄⠈⠁⠀⠀⠀⠄⠉⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠖⠀⠀⠁⠀⠀⠐⠀⠉⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠏⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠉⢑⡒⠦⣄⡀⠤⠤⡀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⣀⣈⣀⣀⠀⠀⠙⢆⠀⠈⢆⠀⢀⠔⢩⠀⠀⠈⠙⠢⡀
⠀⠀⠀⠀⠀⠀⡰⠋⠠⠀⢀⡠⢄⣀⠀⠀⠀⠀⢈⢀⠀⢈⠐⠁⢠⠃⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⡇⠰⠀⡦⣀⠉⠡⠌⠉⠳⢦⣄⣘⡧⠶⢏⠀⠀⠀⢀⠄⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠳⡠⠰⠃⣁⡀⢠⠚⡇⠲⠊⠀⢹⣿⡄⠣⡀⠀⠀⢀⣀⡤⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⢁⣰⠅⠿⠋⢣⡭⠜⠀⠀⡜⣿⣿⣦⣈⣒⣂⣦⢱⡤⢀⣀⡀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠑⠤⠺⠄⠀⡠⠊⣵⠸⣿⣿⣿⣿⣿⣿⠀⡄⠈⠑⠢⣸⣕⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⣤⣤⠠⠂⠠⠜⠉⢆⠈⠻⣿⣿⣿⠏⡸⡂⢀⠀⠁⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡘⢿⠀⢢⠀⠀⠀⠀⠃⣀⠀⠀⢀⠔⢁⠌⠁⢤⠀⠀⢠⠔⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢳⣄⡑⢂⠀⠰⡀⠀⢀⣤⠎⠀⢠⠀⠀⠀⠃⢀⠄⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⠀⠈⢳⠚⠛⠅⠀⠎⠀⠀⠞⠀⠘⠘⠢⡄⠨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠠⠀⣀⡐⠀⢀⠄⠓⢤⡾⣀⠔⡀⠀⠀⠣⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢢⠤⠔⣡⠇⠀⠀⠀⢀⠞⠀⠀⡔⠠⠤⢄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣠⠴⠁⠀⠀⠀⠠⠃⠀⣠⠞⠀⠀⠀⠀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠄⠀⢀⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠊⠉⠀⢄⠀⠀⠀⠘⠤⠄⠀⢀⠄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝⠁⠀⠀⢈⠀⠀⠀⠀⣀⠔⠀⠀⠰⡉⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⣤⣤⣤⣤⣀⡀⠒⣋⡁⣤⣤⣄⡀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⠿⠿⠿⠻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢚⠛⠛⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⢀⣄⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠶⠄⠐⠁⠀⢀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣆⠀⠈⣀⡠⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠗⢄⣀⣀⠔⠉⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣦⣾⣿⣿⣦⡄⠀⠘⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠏⠀⠘⣿⣿⠀⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠻⠀⠀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⡠⠒⠠⠤⢾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠐⢿⠛⠟⠁⠀⡿⠿⠟⠛⠀⠀⠀⠀⠀⠀⠻⠿⠿⡇⢈⣤⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⣾⡆⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢹⡜
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡗⠛⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣠⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⢇⠀⠀⠀⠀⠀⠀⡠⢴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡂⠒⠈⡇⠀⠈⣄⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⢿⣷⣤⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣼⡇⠀⠀⠀⠙⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠚⠉⠉⠒⠒⠉⠓⢦⠤⠴⠤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⡀
⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠹⣄
⠀⠀⠀⠀⠀⡼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢣
⠀⠀⠀⠀⢞⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡀
⠀⠀⠀⠀⣸⠂⠀⠀⠀⡤⠴⠋⠉⠉⠀⠀⠈⠉⢧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠰⡇⠀⠀⣠⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⢳⠀⣞⠁⠐⠲⠄⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠘⣆⣿⠔⠛⡓⢦⣀⣤⣀⡤⠶⠦⣄⠀⠀⠀⠀⠘⣦⠀⣀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⢿⡀⢸⡇⣸⠃⠀⢸⢰⣿⠀⢸⠗⠒⠒⠒⠂⢸⢞⡥⢳⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⢻⠙⠒⢚⠁⠀⠀⠈⠳⠧⠶⠋⠀⠀⠀⠀⢀⣿⣿⠇⣼⠀⠀⡷
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝⣋⡼⠃⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠘⣇⠀⢻⡛⠛⠛⢒⡶⠀⠀⠀⠀⠀⠀⢀⠎⠀⠁⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠉⣒⡚⠉⠀⠀⠀⠀⠀⢀⡴⢻⠀⢀⡤⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⣀⡤⠖⠉⠀⢸⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠴⡖⠋⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⡞⠁⠀⠀⠀⠀⠀⠀⢀⡴⠛⠒⠦⠤⣀⣀
⠀⠀⠀⠀⣠⠴⡖⠋⠉⠀⢀⡞⠓⢤⣀⣀⣀⣀⡤⢶⠟⠀⠀⠀⠀⠀⠀⠈⠉⢓⡶⠦⢤⣀
⠀⠀⣰⠋⠀⡸⠀⠀⠀⠀⡾⠀⠀⠀⠈⠉⠉⠁⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠈⠳⣄
⠀⣼⠁⠀⢠⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠈⢆
⢰⠇⠀⠀⡎⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢧
⢸⠀⠀⣸⠁⠀⠀⠀⠀⠀⣧⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡆
⢸⠀⢀⡏⠀⠀⠀⠀⠀⠀⢸⡄⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⣸⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢳⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⡿⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⡇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢧⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠈⢧
⣷⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣤⣤⣬⣷⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣈⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡤⠤⠶⠶⠶⠶⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠛⣩⠽⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠉⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣀⣀⣤⠴⠊⢈⡓⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⣼⠏⣠⡼⠖⠀⠀⠀⠀⠀⠹⣷⠶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⢸⣿⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣤⣄⣀⣤⣤⠴⠊⣀⣠⠄⠀⢀⣴⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠘⣿⣾⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠙⠛⠷⠶⠚⣛⣩⢤⣠⡴⠟⠁⢀⠀⠀⠀⠀⠘⢿⣧
⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠠⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣉⣥⢴⣫⡴⠋⠀⠀⠀⠀⠀⠀⡹⡆
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠙⣋⣉⣙⠉⠀⢀⣤⠶⠖⠛⠂⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠘⠿⣭⣭⡿⣶⠄⠀⠀⠀⢰⣏⣭⠭⠭⠿⠷⣦⡀⠀⠀⠀⠘⠿⠇⣠⡴⠋⠀⣀⠒⠛⠓⠀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⡖⢿⡉⣭⣿⣿⠀⣖⣒⣲⣤⠖⠛⢯⣍⣛⠲⣄⠀⢀⣀⣠⠤⠼⣟⣓⣺⠶⢋⣭⡝⢶⡀⠀⢸⣾⠇
⠀⠀⠀⠀⠀⠠⡇⠸⠟⠉⠿⠟⢸⠁⠀⠀⢹⠐⠛⢹⣿⠟⠛⢈⡽⢖⣒⠛⠛⠉⠉⠉⠀⠀⣾⡄⠀⢸⣇⣀⠸⣿⣄
⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣤⡴⡋⠀⠀⠀⠘⢶⣆⠓⠒⢂⣠⡾⠃⠐⠎⠁⠀⠀⠀⠀⠀⢸⣁⠈⢀⡾⠋⢉⠃⢠⣽
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⡾⠁⠀⠀⠀⡀⠀⠉⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⣰⡟⢤⣀⡾⠀⠲⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡠⠙⠒⠀⠓⠶⠛⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⠞⠋⡀⠸⣇⢀⠀⣳⠏
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠰⠁⢀⣠⣄⣀⣠⣤⣄⡀⠉⠳⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠰⣌⣛⣡⡤⠟⣼⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠙⠿⣿⡿⠟⠋⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣿⠂⠭⣥⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠠⠴⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠟⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⠁⠀⠀⣀⡸⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⠳⠶⠖⠋⠁⠀⠀⢀⣀⣀⣀⣤⡴⠊⣀⣤⠾⠋⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠿⢦⣤⣀⣀⠀⠀⠀⠉⢉⣁⣤⡴⠾⠋⠁⠀⠀⠀⠀⠀⣈⡿⠶⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠘⢲⠋⠁⠀⠀⢈⣿⠿⠿⣟⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢠⡜⣋⡀
⠀⠀⠀⠀⠀⡤⠶⠶⠖⣿⠀⡞⠀⠀⠀⠀⣼⠃⠀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣰⠟⠃⠀⠈⠙⠳⢶⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠻⣄⣀⡤⠀⣿⠀⢰⣿⡿⠀⡀⠀⠀⠀⠀⠀⠀⣀⡼⠋⣰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣧⣤⡴⠴⠋⠑⠿⡿⠟⠳⠴⣮⡝⣲⠶⠶⠶⣛⢋⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠋⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠛⠋⠻⣿⠟⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡄⠰⠶⠶⠾⠶⠶⠶⠶⠶⠦⣠⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢀⡀⠀⠀⠀⠀⠀⢠⡾⠃⠀⢷⡄⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⡤⠉⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠈⠳⣆⠈⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢠⡜⠀⠀⠀⠀⡀⣰⡾⠉⠀⠀⠀⠀⠀⠀⠘⢷⡀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣱⡌⠀⠀⢠⡔⢃⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⢉⣠⠾⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⢦⣀⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡾⠋⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠛⠲⣤⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⡀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⡸⠃⠀⠀⠀⣀⠀⣤⣤⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠙⣇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⡿⠃⠀⠀⠀⠂⠀⠀⠿⠋⠀⠀⠀⣴⠄⠀⠀⠀⠀⠀⠀⢻⠏⠀⠀⠀⠀⠀⠀⠙⣿⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢀⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣆⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠊⠁⠀⠀⠈⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⢀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⢧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠛⠻⣶⣦⣤⠤⠾⠿⡿⠷⠶⠦⢘⡻⠛⣻⠀⠀⠀⠀⠙⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏⠉⠀⠀⠀⢿⣦⡀⠂⠀⣀⣀⡀⠠⠖⠈⣠⣾⡏⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠉⣿⠀⠀⠀⠀⠀⢸⡏⠙⠳⣤⣤⣤⣤⣤⠖⠛⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁⠀⠀⣿⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠘⣿⡟⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢀⠀⠀⡘⠃⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠈⣄⠀⣿⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⣠⡇⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠘⢷⣰⠂⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠠⢄⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣬⣿⠋⣻⠀⠀⠀⠀⠀⠐⠶⢾⠗⠀⠀⠀⣀⣀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⣧⠀⠀⠀⠀⢀⣀⣀⡀⢸⣷⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣆⠀⠀⣀⡀⠴⢾⡇⠀⣿⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣴⡟⠉⠉⠙⢿⡿⠶⠃⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠹⣶⣞⣩⠟⠛⠻⢷⣀⣻⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣿⣧⠐⠀⠀⠈⣡⡾⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠘⢿⠁⠀⠀⠀⠀⣹⡋⣀⢀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣤⣟⣀⠴⠿⠃⣴⡤⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⣤⣤⡾⠏⠁⡿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠶⠾⠃⣏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡅⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠛⠃⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠷⠴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⢀⡠⠀⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⢁⡴⠋⠁⠀⠀⢀⣤⣤⠶⠟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣷⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣏⣠⡶⡶⠞⠛⠋⠁⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⣿⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠻⠟⠛⠛⠉⣉⠀⠀⠀⠀⠀⣀⣀⣴⠶⠛⠋⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⣦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⢎⣁⣠⣤⣴⡶⠞⠉⠉⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⡇⠀⠀⠀⠈⡏⠀⠉⠉⠙⠛⠛⠛⠛⣿⠛⠛⠛⠁⠀⠉⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠸⡇⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡀⠀⣀⣀⣀⣀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡌⠃⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠉⠉⠉⠙⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠰⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠢⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⢸⠋⠙⠻⢷⣤⡀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣀⡀⠀⠈⣻⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⢛⣛⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠛⠁⠈⠙⠛⢶⣬⢳⡌⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⠀⠠⡀⠐⠄⡄⣿⣇⡇⠀⡿⠘⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠬⣾⣗⣤⡼⠆⢧⣰⡿⠟⠠⠞⠁⠀⠀⠀⠉⠳⣆
⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⢀⡀⠄⠀⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⢀⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡄
⠀⠀⠀⠰⠉⠀⠀⠀⠀⣇⣀⣀⡀⠀⠀⠀⠀⠀⣀⣤⣤⢤⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⢀⣨⣉⠣⠀⠀⠀⠊⣉⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⠁⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⣇⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡂⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⢦⣄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠐⠦⡀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢀⠀⠀⠈⠀⠌⠡⠈⠁⠀⠀⠀⠀⠀⡠⠀⠀⠸⠀⠀⠀⠀⠀⠀⣽⠇
⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢹⣆⣀⡀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠐⠲⢤⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠙⢶⡀⠀⠀⢸⠀⣄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠤⣥⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠃⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠈⠉⠉⠉⠙⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠙⢷⣤⣤⣤⣤⣤⣴⣶⠾⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⣀⣤⡶⠖⠋
⠀⠀⠀⠀⠀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⠿⠋⠁
⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠿⠿⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠟⠋
⢢⣀⡀⠀⠀⠀⠀⢀⣀⣀⣠⣴⣾⣿⣿⠟⠋
⠀⠙⠻⢿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠉
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⡶⠶⠦⢶⣴⣶⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⣶⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣠⣽⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣴⣶⣶⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠉⠉⠠⣼⣿⣿⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣀⡀⢀⣶⣄⠀⣀⡀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⡀⠀⢀⣂⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⣿⣿⣶⣿⣦⢰⡇⠀⣿⣂⣀⣾⣿⣿⢿⣿⡏⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣭⣿⣿⣿⣿⣿⣄⣸⣿⣿⣿⣿⣻⣿⡛⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⠛⠙⠛⠻⢿⢿⣝⢛⢿⣿⣿⣿⣿⣿⣿⣛⡉⣿⣿⣿⠟⣻⠯⠼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢡⡀⠀⠀⠀⠀⠀⠂⠐⠒⠛⠉⠻⣿⣿⣿⣯⡀⢾⡛⠿⠂⠃⠰⠃⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢩⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⣿⡏⠀⠘⢷⡀⠀⠀⠀⠀⢸⠟⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣰⣾⣿⣧⡀⠀⣌⠘⣧⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡟⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠹⢆⣀⠄⠀⡀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡴⣿⢀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⠻⠿⣿⣿⡿⣿⣿⣾⡀⠀⠨⠳⠘⡰⠁⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡏⠀⢸⡿⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠺⣿⣿⡇⠸⣿⣿⣷⡄⠀⠈⣢⣀⠀⠀⣼⣿⣽⡿⠿⣿⠿⣈⣋⣀⣀⠙⠻⢾⡇⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠻⣿⣿⣧⠀⢨⣿⡗⠀⣸⣯⡾⠋⣠⣤⠷⢞⣛⣭⣉⣙⠻⣄⢸⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⡔⢻⣿⣷⣆⠸⣿⣧⣴⣿⣿⣿⣿⣿⡵⠟⠋⠉⠉⢹⣿⡆⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣦⣿⣿⣿⣘⣿⣿⠹⣿⣿⣿⠿⢋⣀⣤⣴⠖⠲⢿⣿⣇⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣯⣩⣴⣿⠟⡟⢱⠆⡀⢀⢹⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⠻⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣿⣇⣼⣿⣼⣿⣧⢨⣿⠟⣿⣿⣿⣿⣿⣿⣿⣶⣄
⢀⡤⣠⢖⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⡀
⠈⡰⢁⠈⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⠁⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠰⠀⢣⠎⠜⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⡳
⠀⢣⠁⢊⠜⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣧⣹⣿⡿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡏⢫⡁
⠀⠀⢦⢊⠞⣴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠻⢿⣿⣽⠛⡟⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠾⠎⠊⠁
⠀⠀⠘⣆⢚⡜⡵⣻⣿⣿⣿⣿⣿⣿⣿⣿⡐⢹⣿⣦⡀⠀⠀⠀⢀⡼⡏⠄⠀⠀⡆⠙⢻⣄⠀⠈⠁⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢀⠐
⠀⠀⠀⠈⢺⡜⡔⡵⡻⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⢿⣿⣦⡀⣰⣿⠃⠁⠀⠀⢠⠁⠀⠘⣿⣆⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡁
⠀⠀⠀⠀⠀⠉⠸⢜⢞⢜⣿⣿⣿⣿⣿⣿⣿⡀⢻⠘⠟⣿⣿⠏⠈⠀⠀⠀⠀⠸⠀⠀⠀⢈⠿⠿⢿⣿⣿⣟⣿⣿⣿⣿⢿⣿⡿⡿⡯⠛⠪⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠙⢿⣿⠿⡿⢹⡇⠀⠀⠀⢹⠇⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⡿⠮⠻⠝⠝⠛⠇⠙⢌⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⢿⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢰⠀⣰⣟⠧⠢⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⠙⠂⠂⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⣦⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⠋⠁⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠡⣶⣶⣦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣏⣿⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠘⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣸⡟⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⢹⣿⣿⣿⣆⠀⠀⠀⠀⣾⣩⣿⢻⣷⢿⣏⣻⣷⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⢿⣿⣿⣿⣦⠀⠀⣾⢿⣏⣿⣿⣿⢾⣿⣿⣽⡷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠸⣿⣿⣿⣿⡆⠸⣻⡾⣿⣽⣟⣻⡾⣿⣧⡿⢾⣟⣿⣄⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⢿⣿⣿⣿⣿⠈⣿⣿⢿⣷⣿⣿⣷⣿⣯⠿⣾⣻⣿⣻⣯⣾⢷⡾⠃⠀⠀⠐⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠸⣿⣿⣿⣿⣇⣻⣷⠿⠟⠺⠯⣿⣾⣷⣶⣶⣾⣯⣭⣤⣍⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⣿⣿⣿⣿⡏⢹⣿⡿⢿⣶⣶⣤⣈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠉⠉⠉⠉⠛⠓⠒⠋
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠈⣟⣷⣿⣷⣿⣿⣿⣷⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢻⣿⣿⣿⡄⠛⠛⠻⠿⠿⠿⠿⠿⠿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣇⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣧⣤⠶⠻⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣿⡇⠈⠉⣿⡏⠙⠛⠛⠛⠛⣿⡉⠉⢿⣧⠀⠀⠀⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠸⡇⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⢰⠇⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⢧⠀⠙⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⡞⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠈⢿⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠾⠋
⠀⠀⠀⠀⠐⠁⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣴⣶⣾⣿⣟⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠉⠉⠉⠛⠛⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠛⠀⠐⠒⠀⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡈⠙⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠉⠀⠀⠀⠀⠀⢃⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⣠⠊⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀⢀
⠀⠀⠁⡀⢉⡀⢰⠁⠀⠀⠀⠀⠀⠀⢠⡦⠀⠈⠢⡀⠀⠈⠂⠁⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠠⠶⠶⠶⠶⠶⠶⠶⠦
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⡞⡀⠀⠆⡀⢡⠀⠀⠀⠀⠀⠀⠸⠀⠀⠠⠀⢤⡠⠂⠐⠢⡄⢀⡄⠨⡅
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡄⠀⠁⠀⢃⠀⠀⠈⡄⢀⠃⠀⠀⠀⠀⠈⡇⢄⠀⠀⢸⡏⠀⢀⡔⠀⠀⠀⠀⠦⠄
⡇⠀⢀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠀⠀⠁⠀⠈⠢⡀⠀⠀⡂⠀⠀⠀⠀⠀⠀⡇⠈⠓⠒⠋⠈⠉⢁⠄
⠃⠀⠀⠀⠸⢇⣀⠀⠀⠀⠀⠀⢸⠁⠀⠠⠀⠐⠀⠀⠀⠀⠈⢆⠀⢹⠀⠀⠀⠀⠀⢀⠗⢀⣀⣠⡗⠀⠀⠆
⠀⡀⠐⠒⠰⠋⠉⠉⠛⢯⠉⡉⠁⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⢸⠀⢨⠤⠀⢀⣀⠀⡜⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢄⡀⠀⢀⠠⠀⠀⠀⢸⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠀⠈
⠀⠀⢄⣀⣀⠠⠂⢀⠠⢊⠈⠁⠀⠀⠀⠀⠀⠀⠖⠂⠀⠀⠠⢀⣀⣀
⠀⠀⡞⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⡀
⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠀⢀
⠀⠈⠑⠒⠲⢤⠤⢄⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⠠⢤⠀⠀⠀⣀⣃
⠀⠀⠀⠀⠀⠘⠀⢸⣿⠀⢸⣿⠀⠈⣿⡏⠁⠈⣿⠇⡀⠈⣿⠁⠇
⠀⠀⠀⠀⠀⠈⠀⣾⡇⠀⠘⠛⠀⠀⣿⡇⠁⠸⣿⡄⠀⠐⠻⡆
⠀⠄⠀⠀⠀⠂⠠⠿⡇⠀⢸⣦⣄⡀⠙⠄⠀⡘⠛⠃⠘⠒⠀⠀⠀⠈
⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣥⡀⠂⠄⠀⠀⠀⠐⠿⠃⠀⡈⠄
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠈⠉⠀⠀⠀⠂⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⢲⠂⠲⡴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⢓⠾⢾⠄⢙⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠠⠄⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢴⢂⠏⣛⠛⣿⣆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣢⣤⡶
⠀⠀⠀⣠⣄⠈⢀⠀⠀⡈⠛⢠⠙⠴⠁⠛⢾⣿⣗⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⡟⢀⣽⣿⡯
⠀⠀⠀⠺⠿⡇⠐⠐⡺⣻⣆⠞⣶⡀⡉⠰⠊⢙⠿⢷⡶⡤⣦⠀⣀⢀⠀⡀⡀⠀⠀⢀⡀⢀⣀⣾⣯⣽⣷⣾⢿⡿⠊
⠀⠀⠀⠀⠀⠀⢠⠀⡀⠈⢻⣷⠿⡿⠃⣤⡶⣴⣢⣤⣤⣽⣘⣉⣻⡟⢠⣷⣻⣶⢦⣾⣷⡟⠻⠿⠏⢽⡛⡵⢢⠄
⠀⠀⠀⡀⠀⠀⠀⠂⠹⡞⠸⠓⣦⣷⠾⠿⢿⣟⢹⡍⢉⡉⠉⠉⠩⢥⣿⣇⣻⣿⣿⣿⡏⡞⡤⣠⠘⣿⣗⡆⢞⠁
⠀⠀⠈⣋⢁⡀⠀⠀⣈⡀⠠⠤⢀⣠⠄⠈⠙⠉⠁⠑⠀⠈⠉⠃⠀⣨⢉⠙⢿⣛⠙⡛⠓⢒⠒⡛⣚⠛⣛⣓⣂⣀⣀
⠀⠀⠀⠈⠋⠛⠓⠀⠈⠉⠉⠀⠘⠋⠛⠛⠛⠛⠀⠛⠓⠒⠐⠂⠈⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⠛⣛⣛⣛⣻⠿⢉⡙⠋⢨⣉⠻⣦⡀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣼⠟⣉⣩⣬⣤⣼⠿⣿⣿⣿⣾⣿⣷⣮⣿⣷⣌⠻⢛⣉⣭⠋⠻⠶⣦⡄
⠀⠀⠀⠀⠀⠀⣾⠛⢛⣩⣭⣶⣿⣟⣛⣯⣵⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡟⢥⣾⡃
⠀⠀⠀⠀⠀⠀⢙⣷⠮⣉⣽⣿⣿⣿⣿⣿⣿⣟⣛⠻⠯⣙⣻⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣦⣀⣀
⠀⠀⠀⠀⠀⣠⡟⢁⣾⣿⣿⣿⣿⣿⡿⠛⢓⣂⣩⣿⣿⣿⣿⣿⣿⣥⣽⣿⣿⣿⣿⣯⣭⣙⣻⣿⣿⣿⣮⣉⠙⢷⡄
⠀⠀⠀⠀⢀⡿⣠⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⡿⠟⠻⣿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠄⢸⡇
⠀⠀⠀⠀⣼⡇⠋⣿⣿⣿⢛⣿⣿⣿⣿⣿⣟⣴⢢⣾⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢿⡇
⠀⠀⠀⠀⠙⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⡛⢫⣼⠇
⠀⠀⠀⠀⠀⣿⡸⣿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣭⣻⣿⣿⣿⣿⣿⣏⢻⣤⡿
⠀⠀⠀⠀⠀⠘⢷⣤⢸⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⢀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡌⡷⠿⠿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⠛⠿⠏⠻⣿⠋⡀⠈⢴⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⠀⠸⡇⠈⠯⠉⠙⠛⠿⠿⠯⠭⠟⠋⠛⠛⠎⠉⠁⢀⡤⠀⠀⠹⠀⣠⢎⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡜⡄⠀⠀⠁⠈⠓⠶⣤⣄⣀⣀⣀⠀⠀⠀⢀⣀⣀⣤⣶⣏⠀⠀⠀⢰⡘⡇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡜⡄⠢⡄⠀⠀⠘⠋⠙⠛⠛⠉⠀⠀⠀⠈⠉⠉⠛⠃⠈⠁⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢇⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢛⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠸⡀⠀⠀⠀⠀⠀⢠⣦⣀⢀⣒⣈⣽⣶⠀⠀⠀⠀⠀⡴⢵⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡫⠞⠳⡀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠿⣿⣿⠀⠀⠀⢀⠌⠑⠢⣝⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠜⠁⠀⠀⠐⢀⠀⠀⠀⡿⢻⣧⣀⣁⣠⣿⡌⠀⣠⡔⠁⠀⠀⠀⠈⢮⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠌⠀⠀⠀⠀⠀⠀⢙⠲⠤⠔⠈⠛⠛⢛⣛⠩⡰⠟⠁⢀⠀⠀⠀⠀⠀⠀⠋⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⡈⠑⠂⠤⠄⠊⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡰⡄⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠌⠒⠀⠠⠀⠂⠄⠀⠀⠀⠀⢘⠀⠀⠀⠀⢠⡞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢘⠳⢤⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠶⡎⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠈⢉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⡇⡻⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡞⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢃⠀⠀⡇⠈⠒⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢀⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠶⠶⢶⣶⣤⣤⣤⣶⡶⠶⠾⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠀⠀⠀⠒⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡠⡠⠴⠆⠠⠀⢀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠸⠠⠀⠀⠀⠠⠠⠈⠆⠇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢓⠀⠐⠀⠀⠐⠀⠀⡏⠂⠀⠀⠀⠀⠀⢀⠊⢀⡀⡀⠀⠀⠈⢢
⠀⠀⠀⢀⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⠀⠔⡀⠀⣐⢄⢀⠗⠁⠀⠀⠀⠀⠀⠎⠀⠃⠀⠈⢂⢀⠀⠀⢢
⠀⠀⠀⠸⠜⢦⠢⡜⠂⠀⠀⠀⠀⠀⠀⠀⠀⣣⣀⣁⠀⣁⣠⡎⠀⠀⠀⠀⠀⠀⡜⠀⣸⠀⣀⠀⢈⣉⠣⣀⡀⢁
⠀⠀⠀⠈⢎⠒⢄⠀⢠⠀⠀⠀⠀⠀⡀⠤⢾⣿⣶⣄⢀⣴⣾⣿⠦⠄⡀⠀⠀⠀⢇⠘⣭⠑⠠⠛⠃⠀⠋⡧⢁⠜
⠀⠀⠀⠀⠀⠢⢄⢀⣸⣄⠀⠀⣴⠉⠀⠀⠀⠹⣿⣿⣾⣿⣯⢿⡧⡔⡆⢱⣄⠀⠀⠁⢬⣢⣀⠑⠊⣀⣔⠤⠊
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⢸⣿⡆⠀⠀⠀⠀⠈⠻⣿⠟⠉⢌⠂⠄⠰⣸⣿⡄⠀⢀⣀⡤⢏⢀⣀⡹⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⢴⣿⣿⣇⣴⣿⢿⡏⠐⠚⠒⠈⢹⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣽⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣽⣿⣷⣿⢽⢿⡇⠀⠀⠀⠀⠀⡯⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢀⠀⠀⠀⠀⠀⠐⠀⢈⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⡀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡴⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠠⠴⠂⢀⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⡰⢊⠔⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡔⠀⠈⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣇⠎⡠⠁⠰⠀⠄⠀⠔⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠑⠂⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡸⢃⠜⠀⠀⠀⠑⠠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡠⠄⠋⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⡀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⡰⢁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⡑⠀⠁⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠠⠀⠀⠀⢀⠄⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠉⠀⠀⢈⡤⠀⠀⣀⠠⠄⠠⣌⣅⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡠⠊⡄⠀⠀⠀⠐⢀⠔⢉⡴⠊⠁⠀⠀⠀⠀⠀⠀⣀⡀⠈⠒⢄⡀⠀⠀⠀⠃⠸⠄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡰⠁⢰⠀⠀⣀⠄⡠⠃⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢌⣢⣄⠐⠠⠄⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠃⠀⠀⠀⠤⢁⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⣀⣀⠐⣂⠓⠒⠲⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⠃⠀⠀⠀⠀⠁⡼⠀⠀⠀⠀⠀⠤⠂⢀⠠⠤⢀⣀⣀⠁⠉⠉⠉⠉⠉⠉⢹⠉⡋⠙⠓⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡏⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢀⠔⠊⠓⠒⠤⠤⢤⣤⣀⣈⣉⣉⠁⠀⠒⢺⠀⡇⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡴⠃⠄⢀⣀⣀⣀⠒⢤⡀⠀⠀⠈⠉⠉⠉⠉⢹⠀⠁⠀⠀⠠⡆⠒⠋⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⠈⠀⠀⠀⠀⡀⢸⠀⠀⠀⡼⠁⠀⠉⠀⠀⡠⠖⠉⣀⡉⠂⢄⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠈⠁⠉
⠀⢀⣀⡀⡤⠤⠀⡔⠒⠁⠀⠀⠀⠀⠀⢠⠘⡀⠀⢠⠃⠀⠀⠀⠀⠎⢠⠞⠉⠈⡏⢳⢦⠑⡄⠀⠀⠀⠈⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡾⡉⠄⠄⣨⣠⡤⠾⠶⢶⡦⢤⣀⠀⠀⢰⡀⢃⠀⢸⠀⠀⠀⠀⢸⠀⠇⠀⠀⢀⡇⢸⠈⢧⠰⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⣩
⠑⠪⡷⠛⠉⠀⠀⢀⠀⠀⠉⢦⠉⠙⠲⠤⣧⠘⣇⠸⡆⠀⠀⠃⠈⠧⡠⠀⢀⠞⢠⠇⠀⣸⠀⡆⠀⠀⠀⢰⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢋⡴⠛⠁
⠀⠀⠻⣄⡒⠆⠄⣀⢀⣀⡃⢠⠀⠀⠀⠠⠊⢇⠸⣄⠱⡀⠀⢠⠤⠤⠴⠚⢁⡴⠋⠀⢀⡟⢰⠀⠀⠀⠀⠈⢇⠈⢦⡀⠀⠀⡸⢱⠀⠀⣠⠴⢋⣡⠞⠋
⠀⠀⠀⠀⢹⣉⣉⡐⠒⠚⠛⠻⠭⠭⠭⠤⠤⠬⢧⡈⢦⡙⢖⠚⠢⠖⠒⠊⠉⠀⠀⣠⠞⢠⠃⠀⠀⠀⠀⠀⠈⠳⢄⠉⠒⢊⣁⡼⠔⣋⡥⠖⠋
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠒⠂⠒⠒⠂⠛⢦⡙⠲⣅⡒⠠⠤⠤⠤⠖⠋⣡⠔⠁⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠛⣋⡥⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣍⡓⠲⠶⢶⣎⣭⣤⣤⠤⠤⠤⠶⠒⢚⣉⣡⡤⠴⠚⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣲⣶⣶⣦⣴⣶⣶⣶⣖⣚⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⡛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠘⡻⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠙⠂⠸⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⢄⢀⠀⠀⠐⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠅⠉⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡐⠂⠁⠀⠈
⠀⠀⠀⡠⠀⢀⡠⠛⠉⠉⠉⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠐⡍⠀⢀⠇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣀⢠⠀⢀⡀⡴
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⠊⠁⠀⠀⠒⠈
⠀⠀⠀⠘⡀⢄⠐⠂⠀⠘⠡⢤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠐⢀⠘
⠀⠀⠀⠀⠈⠠⣀⡄⠓⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠒⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠐⠊⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠶⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠂⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠘⠍⠡⠁⠘
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐
⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⠈⠡⠀⠀⡄⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣰⠁⡆⠑⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡠⡊⡀⠂⠃⠰⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⠘⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣾⣿⠛⠛⢻⠛⠛⠛⠛⠲⢶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠃⠀⢸⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠙⠛⠿⠟⠛⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⠟⠳⣄
⠀⠀⠀⠀⠀⣴⣿⣿⣿⠋⠀⣠⣤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⢭⣭⡁⠀⠀⠀⠘⢷⡄
⠀⠀⠀⢀⣼⠟⠛⠋⠀⢀⡞⠁⣠⣄⡀⠙⢷⡀⠀⠀⠀⣴⣶⣄⢸⢀⣴⣶⡄⠀⠀⠀⣰⠛⠁⣀⣤⡀⠙⣆⠀⠀⠀⠀⢻⣄
⠀⠀⢀⡾⠁⠀⠀⠀⠀⠈⢧⡀⠀⢀⡷⠀⠀⢹⣄⣀⣀⣉⡛⠃⢸⠀⠛⣋⣁⣀⣀⡼⠁⠀⠰⣇⠀⠀⣠⠏⠀⠀⠀⠀⠀⠹⣆
⠀⠀⣼⠃⠀⢀⣀⣀⠀⠀⠀⠉⠛⠉⠀⠀⣠⠞⠉⠁⠀⠈⠙⠳⣾⡴⠛⠉⠀⠀⠉⠙⢦⡀⠀⠈⠙⢋⣡⣀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⣸⠇⠀⣴⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⣸⠧⢄⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⠤⢿⡀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⡀
⢀⡿⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠓⣦⡤⢀⠀⣀⡠⣤⡖⠋⠉⠀⠀⢸⡇⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⣠⣶⣿⣇
⣸⡇⠀⠀⠈⠻⠿⠟⠋⠀⣠⡴⠶⠛⠋⠹⣆⠀⠀⠀⠀⠈⠁⠀⣨⡁⠀⠉⠀⠀⠀⠀⢀⡾⠉⠛⠳⠶⣤⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⢧⣀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠈⠳⢤⣄⣀⣠⡤⠞⠁⠙⠦⣤⣀⣀⣤⠴⠋⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠈⢙⣻⢿
⢿⣄⠉⠙⠓⠶⠦⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡤⠶⠖⠚⠉⢁⣼
⠀⠙⠳⠶⣤⣤⣀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣠⣤⡴⠶⠛⠁
⠀⠀⠀⡰⠋⠀⢉⡽⢷⣄⠀⠀⣴⠒⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢲⠄⠀⢀⣴⠿⣍⠁⠈⢳⡄
⠀⠀⢰⡇⠀⢠⡟⠀⠀⠉⠛⠶⣬⣭⣉⣙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣉⣩⣭⡴⠞⠋⠁⠀⠘⣧⠀⠀⣷
⠀⠀⠀⢷⣄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠳⢦⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⠶⠟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣰⠇
⠀⠀⠀⠀⠈⡇⠀⠀⠀⢀⣀⣀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠉⠉⠉⢹⠉⠉⠉⠁⠀⠀⠀⠀⣀⣀⣀⠀⢀⣀⣀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠹⣄⠀⢰⠁⠀⠀⠹⠁⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠹⠁⠀⠀⢱⠀⣀⡼⠁
⣤⣤⣤⣤⣤⣤⣬⣽⣿⣷⣶⣶⣾⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣾⣯⣿⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣾⣶⣶⣶⣿⣿⣭⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣬⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠒⠠⠂⠠⠀⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⡔⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣷⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⣹⡦⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⣟⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀⢀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠿⡷⠌⠁⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠴⠟⢙⣻⣿⣿⣿⣿⡶⡗⠈⣽⣿⣿⣿⣷⣶⣦⣄⡙⣆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠶⠿⠿⢿⣿⣽⣿⠿⠀⠀⣽⣿⣿⣃⣛⣿⣿⡟⠛⡙⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠂⠿⠛⠋⠉⠀⠀⠀⢀⣿⡿⠃⠀⠀⠉⠀⠀⠀⣧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠈⣿⣷⣦⡀⠀⠀⠀⠀⠀⣿⡆⠸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⣼⡏⣠⣀⡀⣀⣠⣴⣿⡇⠀⠀⠀⠀⣴⣿⡇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⣰⣾⣿⣋⠀⠀⢹⣿⢿⣾⢿⣿⡖⠒⣶⣾⣟⡿⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣗⠀⣟⣿⣛⣀⣚⣗⣻⣯⣸⣤⣦⣬⣿⣷⠀⠉⠉⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⡆⢹⡿⡿⠟⣛⠛⠛⠻⠟⠓⣿⣿⢿⣿⣦⣄⡀⠀⢰⣿⣿⣿⣿⣶⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣬⣸⣇⠘⡟⣧⠀⠀⠓⠲⠶⠒⠚⠋⠁⡀⠿⣽⣿⡆⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣷⣿⣤⡀⠟⠀⡀⠀⠀⠀⠀⠀⠛⠁⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣴⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣀⠀⣰⠀⠀⣀⣱⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⡿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣝⠛⢿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋
⠀⠈⢿⡿⠋⠀⠈⠉⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠑⠀⠨⢿⣿⠿⠃⣰⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠤⣠⡤⢀⣾⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣴⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡀⠀⠀⠀⠄⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢀⡀
⠀⣲⣄⢤⡄⠀⠀⠀⠀⠀⠀⠂⠹⣄⠀⠁⢻⡄⠀⠀⠀⠀⢀⠩⢄⡀⠀⠀⠀⢠⠀⠀⠀⢈⣄⠀⠀⣦⠈⣿⡀⢒⠀⣤⡶⠶⢦⣤⡀
⡆⣿⣿⢸⡇⣥⠀⢀⡄⣰⡀⣸⣆⣿⠀⠀⢘⡇⠀⠀⠀⢀⣸⡇⣹⣞⣿⣦⠶⠾⠷⠒⠒⠘⠿⠶⢦⣿⡇⣼⣇⡄⣷⢼⣧⣄⡀⣿⡅
⠀⢿⣟⠼⠛⠛⠛⠻⠿⢿⣿⣿⣿⡟⢀⣼⣿⣿⣤⢁⣴⡼⠿⢟⣾⡿⡭⠐⠀⠈⠉⠀⠀⠀⠀⠁⠀⠙⠻⣿⠿⣿⣿⣸⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⢿⣿⣿⣿⡟⣉⠴⠚⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⣍⠉⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⣿⠀⠇⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿⢟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡀⠀⠀⠘⢸⠀⢠⠾⠟⠩⠄⠤⢤⣌⠹⢦⡀⣀⡀⠀⠀⣠⣴⠚⠉⠁⠀⠉⠑⠢⠀⠘⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣝⠢⢄⡆⢘⢀⣏⠀⢀⡪⢀⡦⣆⠔⠳⠀⢷⣤⣤⣤⣾⡟⠁⢀⣴⣩⣧⡽⠇⡄⠈⣆⣀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣏⠀⠥⡘⣻⣿⠀⠘⢁⠻⣾⣿⢏⢹⡇⢸⠟⠛⠛⣿⡃⠀⢘⡀⠼⣭⣿⣎⡁⠀⣟⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠛⠦⠤⢾⢿⣧⠀⠲⣄⣸⣀⣴⡾⢃⠈⠀⠀⠀⣘⣧⡀⠘⢄⡀⠛⠉⡤⢁⣤⠉⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⠿⡷⡦⣤⣈⣀⣡⠴⠿⢿⣿⣿⣿⡿⠿⠷⢦⣀⣈⡉⣀⣠⣴⣷⣶⣶⣿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠉⠀⠈⠉⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⠔⠊⠉⠉⠉⠉⠐⠢⡄⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢰⠃⠀⠀⢀⡴⡀⠀⠀⠀⠈⢆⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡜⠀⢐⠒⠁⠀⠈⠢⣀⣀⣀⣸⡀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠖⠃⠀⠠⠀⠀⠀⡈⠁⠀⠉⠁⠀⢣⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠰⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠕⠁⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢈⣵⣦⠤⠤⠀⣤⣤⠤⠶⠛⠉⠀⢀⣀⣀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣠⠁⡌⢹⢆⡀⠀⣼⢷⠅⢆⠀⠀⡰⠉⠀⠉⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⡰⠃⠀⠁⢸⠀⢸⠉⠀⢇⠀⠈⡄⠀⠇⠀⠀⠀⠈⡄⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠠⢎⣀⢀⣀⠴⠋⠀⢸⠀⠀⠈⠓⠦⠵⠄⢸⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠉⠁⠐⣢⣄⡸⠀⠀⢀⣀⣤⠀⠒⠚⠛⠆⠀⠀⠀⡇⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⡾⢡⡞⠀⠀⠀⠀⠀⠈⠛⡄⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠑⠢⡤⠓⠚⠅⡀⠑⡦⡤⠂⣀⣼⣇⣀⣠⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠘⠤⠔⠊⡑⠈⠈⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠓⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⢄⡀⠀⠀⠀⠀⠀⠃⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⣀⠀⣀⡾⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⣀⡈⠑⠢⠤⣀⣀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠤⠐⠒⢉⣅⣀⣠⡴⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⣤⣤⣄⣀⣸⠀⠀⡉⠉⢉⠉⢁⡠⡤⣀⢀⡴⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠻⠀⢠⠁⠀⣉⣀⣿⣧⡰⠾⠤⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣉⣁⡶⣀⣸⡽⢆⣀⣠⡟⡏⠀⠀⠀⡈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠀⠀⢀⡇⠀⠈⠁⠘⠧⣤⣭⣭⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠁


⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣦⣤⣬⣴⣿⣶⣤⣼⣧⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣷⣾⣿⣿⣿⣷⣶⣶⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡘⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣷⣌⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⠀⠈⠙⠿⣿⣿⣿⣿⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣶⡍⠙⠉⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠈⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿

⣿⣿⠛⢿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠶⠾⠿⠛⠛⠛⠛⠛⠿⠷⠶⠶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠷⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢶⣦⣄⡀
⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷⣦⣄
⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⡀
⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄
⠀⠀⣾⠇⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⣸⡏⠀⠀⠴⠿⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⢀⣿⠀⠀⠀⠀⡠⠚⢉⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠛⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⢸⡏⠀⠀⠀⡜⠀⣰⡿⠻⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⢿⣄⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⢸⡇⠀⠀⢰⠁⢀⣿⣇⣠⣿⣿⣿⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡀⢸⣿⡀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⢸⡇⠀⠀⠸⡀⠈⣿⣿⣿⣿⣿⡿⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⢿⣿⣿⣿⣿⣿⠃⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣷⠀⠀⠀⢱⡀⠹⣿⣿⣿⡿⠃⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠘⣿⣿⣿⣿⡟⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢘⡏⠀⠀⠀⠈⢿⣦⣌⣉⣉⣠⣴⡿⠋⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠻⣷⣄⡈⠛⠛⣋⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣾⠁⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀
⠈⢿⣄⣠⣤⣴⠶⢾⣿⡛⠛⠛⠉⢉⣭⡉⠉⠉⠉⠁⠀⠀⠀⣠⡿⣿⡍⠉⢉⣉⡙⠛⠛⣷⣶⣦⣄⡀
⠀⠈⠙⢿⣄⠀⢀⣾⠛⢿⣆⠀⢀⣿⠋⠻⣦⣾⠟⢿⣦⣀⣾⠋⠀⠈⢿⣶⠟⠉⠷⡴⠟⠁⠹⣆⣉⠻⠿⣦⣄⣀⣀
⠀⠀⠀⠀⠻⣧⣼⡇⠀⠀⢻⣆⢸⡿⠀⠀⠈⠁⠀⠀⠙⠛⠁⠀⠀⠀⠀⣀⡀⢀⡤⣄⣠⢤⣀⣀⣠⣤⣤⠤⠿⠟⠛⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠻⠿⠀⠀⠀⠀⠙⠿⠇⠀⠀⠀⠀⠀⣀⣀⠀⢀⡾⠻⣦⣾⢋⣹⣿⣴⡶⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣼⡟⠙⢷⣿⣁⣤⣼⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⣿⠉⠻⣿⣟⣠⣴⠾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢰⣿⣦⠀⢀⣿⠃⠀⠉⢀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⢿⡇⠹⣧⣾⢃⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⢸⡇⢀⣽⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠹⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⠶⠦⣤⣤⣤⣤⣀⣀⣀⣤⣤⣤⣤⠶⠶⠛⠋⠉
⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣼⣧⣭⣧⣷⣶⣼⣼⣭⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⣶⣶⣾⣿⣶⣶⣶⣷⣶⣶⣾

⠀⡶⠻⠷⡆⠀⣰⠾⣿⣦⠀⠀⠀⠀⡶⠿⠷⡄⠀⠀⠀⣰⠿⣷⣆⠀⠀⠀⠀⡶⠿⠷⡆⠀⠀⠀⠀⠀⣰⠿⣷⣆⠀⠀⠀⠀⡶⠻⠷⡄⠀⢰⠾⣿⣦
⠀⠀⠀⠀⠁⠀⠋⠀⠐⠙⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠁⠂⠀⠉⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠃⠂⠐⠉⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⠀⠐⠙
⢀⣤⣶⣦⡄⠀⠀⣄⣠⡄⠀⠀⠀⢀⣤⣶⣦⡄⠀⠀⠀⠠⣆⣠⠄⠀⠀⣀⢀⣤⣦⣦⡄⠀⡀⠀⠀⠀⠠⡄⢠⠀⠀⠀⠀⢀⡤⣦⢦⡄⠀⠀⣄⡰⡄
⢾⣿⣿⣿⣿⠀⠀⢿⡿⠃⠀⠀⠙⠟⢻⣿⣿⡿⠄⠀⠀⢐⣿⡿⠂⠀⠀⠙⠿⢻⣿⣿⠻⠟⠁⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠚⢿⣷⣾⠛⠀⠀⣴⣦⡄
⠀⢿⣧⣾⠀⠀⠀⣷⣾⡇⠀⠀⠀⠀⢿⣧⣿⠀⠀⠀⠀⢸⣷⣾⠄⠀⠀⠀⠀⢾⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⣿⣿⡇
⠀⢸⣿⣿⠀⠀⠀⠿⠿⠃⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠘⠿⠿⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠘⠿⠿⠁⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠿⠿⠃
⠀⠸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠘⠋⠋⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠘⠋⠛⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠘⠋⠋⠀⠀⠀⠐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠻⠀⠀⠀⠀⡞⠛⠿⡇⠀⠀⠀⠀⠠⠏⠉⠙⡆⠀⠀⠀⠀⠀⡞⠛⠿⡆⠀⠀⠀⠀⠸⠋⠉⢱⠀⠀⡟⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⣴⣾⣶⡄⠀⢠⣶⣷⣦⠀⢰⣴⣾⣿⣷⣴⠇⠀⠀⢰⣧⣴⠀⠓⠀⠀⣄⣠⣾⣿⣷⣆⣠⡀⠀⠀⠀⢰⣤⣶⠀⠀⠀⠀⢶⣴⣿⣿⣶⠀⠀⢰⣤⣵
⣰⣿⣿⣿⣷⡀⠻⣿⣿⡟⠀⠈⠉⣿⣿⣏⠉⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠈⠋⣿⣿⣿⠉⠉⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠈⢩⣿⣿⣿⠀⠀⢸⣿⣿
⠈⢻⣏⣿⠋⠀⠀⣼⣫⡄⠀⠀⠀⣿⡿⡏⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡿⢻⠀⠀⠀⠀⠀⠘⣿⣿⡟⠁⠀⢸⣿⣿
⠀⢸⣿⣿⠀⠀⠀⠉⠋⠁⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠈⠙⠉
⠀⢈⣿⣏⠀⠀⠀⢀⡀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⢀⡀
⣶⣶⣶⣶⣶⣶⣶⣾⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣴⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷
⣿⣿⣭⣮⣯⣴⣭⣽⣮⣾⣯⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤
⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠡⠀⠀⠀⠀⠀⠀⠬⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣰⣦⠀⠀⠤⠃⠀⠀⠀⠀⡩⢆⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠙⠁⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀
⠀⠀⠀⠀⠢⡀⠀⠄⠀⡀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡠
⠰⠀⠁⠀⠒⠁⠀⠠⡆⠁⠄⠀⠀⠀⢋⠖⡒⢢⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⡀
⠐⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠐⠘⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢉⡀⡀
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⠛⢛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠁⠀⠀⢀⡀⠀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠠⠈⣠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡠⠀⠠⠊⠉⢁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠊⠀⠐⠀⠀⠃⠘⠟⠀⠀⠀⠀⠀⠀⠈⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠠⣄⠀⠆⠐⠦⣄⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠤⠠⠤⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠰⠀⠀⠀⠐⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠂⠁⠈⠐⠀⠰⠄⠀⠠⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⣸⠃⠀⠀⠀⢸⠂⠀⠀⠀⡄⠐⢂⡄⠀⠱⣄⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⢰⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⡇⢠⠋⠀⠀⠀⠈⢘
⠀⠀⠀⠇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠃⠀⢷⠀⠀⠀⠀⠈⠃
⠀⠀⠸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⠀⠀⠀⢣⠀⡌⠀⠀⠀⢰⠆
⠀⢠⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡏⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠐⢀
⠀⢸⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⠀⡏⢀⠀⠀⠰⠃⠀⠈⣦
⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⢀⠀⠀⠀⣧⠁⠀⠀⡆⠀⠀⠀⢸⠸⠀⠆⡀
⠀⡏⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠸⠀⠀⠶⣁⠀⠀⡸⠀⠀⠀⠀⠈
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜⠁
⢸⠇⠀⠀⠀⢀⣀⣀⡀⢀⠀⠙⠆⠀⠀⢠⠋⡀⠈⠁⡀
⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣏⣤⠋⡃⠀⢰⡀⠉⠐⢂⠄⢀
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢀⡸⣤⠊⠁⠀⠀⠀⠈⠁⠒⠁⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠
⡧⣤⠀⠀⠀⠄⠀⣀⠀⣀⠀⣀⠀⢀⣀⣸⠂⡎⠀⠦⠤⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠓⠶
⠀⡾⠀⠀⠀⠀⠀⠋⠀⠀⠀⠸⠀⠀⠀⠁⠰⠇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⢠⡃⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡆⠀⠀⠀⣤⠀⡆⠀⠀⠀⠀⢠⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠘⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⢠⠇⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠐⠊⠉⠁⠀⠀⠀⢀⠀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡘⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⢠⡇⠀⣄⠀⠀⠀⠀⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢀⣾
⢠⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣄⠁⣀⠀⠀⢸⡄⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⣩⠆
⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡟⠀⠋⠀⠀⠀⣣⠀⠀⣸⠀⠀⠀⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡴⠋⢁
⠀⡆⠀⠀⠀⠀⠀⢯⡀⠀⠀⠀⠀⠀⠇⢸⠀⠀⠀⠀⢸⠀⠐⠇⢰⠀⠁⠀⠉⠉⠀⠐⠒⠒⠒⠀⠈⠉⠁⠀⡇⠀⡇⡀⠀⠠⡄
⠀⣿⠀⠀⠀⠈⠁⢀⣴⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡸⡆⠀⣠⡀⠀⠂⠀⠀⠄⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⠀⠂⠂
⠀⢻⡀⠀⣀⡴⠚⠉⠀⠈⠀⠂⢠⡀⠀⡁⣠⡴⠋⠀⠀⠀⠀⠀⠀⠈⠁⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠘⠋
⠀⠘⠷⠾⠉⠐⢲⠤⢤⡤⠀⠀⡄⠉⠈⡟⠓⠀⢩⣽⠃⠒⠖⠒⠒⠒⠒⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⡇
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣃⣀⣀⣀⣀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣶⢀⣧⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠟⠛⠿⢷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⢀⢤⠀⠀⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠃⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⠛⠀⠀⠘⠀⢸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡼⣿⣿⣷⠀⠀⠀⠀⢻⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⢀⡼⡇⠀⠛⠁⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠯⢈⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⢠⣤⣶⠾⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠃⠀⠉⣁⣀⣤⡞⠓⢤⣀⠀⠀⢀⠴⠖⠘⣿
⠀⠀⠀⠀⢠⡀⡀⡠⠤⢈⡀⢉⠉⠻⠘⠿⠋⠁⠀⠀⠀⢹⣟⠷⠋⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⢶⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠚⣁⣘⣶⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⣠⡜⠁⠀⠀⠀⠀⠀⣿⣹⡇⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠠⠀⠉⠉⣉⣉⡀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⣸⣟⠉⣤⠀⠀⣰⠊⡱⠆⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠁⠀⠀⠀⠐⢠⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣇⠀⢿⣀⠶⠙⠦⣽⠀⠀⠀⠘⢫⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣰⡿⣻⣿⠋⠐⠀⠀⠀⠀⠀⠀⣎⣻⢷⣾⣇⣠⣤⣿⠁⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⢿⠀⠀⠀⣼⣿⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠺⠝⠛⠋⠉⠀⠀⠀⠳⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⢻⣿⢿⡟⠀⠀⠀⣸⣿⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⢠⠀⠀⠀⠠⣄⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⢨⣿⣿⡀⠀⠀⢀⣹⡟⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⡄⠀⠀⠀⠀⠀⠀⠀⠰⡀⠈⠀⠀⢀⠀⠘⣇
⠀⠂⠀⠀⠀⠀⣾⣿⡟⢡⡀⣀⣿⣉⢰⣿⢤⠀⡤⠀⠀⢘⣃⠀⣰⠄⢐⢿⠙⣧⡤⡄⠀⢀⣆⠀⢃⠀⢱⡀⠀⢀⡸⢆⠀⢹⣧⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠠⣿⣿⠁⠀⣸⡿⠁⠀⣿⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠋⡀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⢳⠀⠀⠁⠀⠃⠀⢹⡆
⠀⠀⠀⠀⠀⢸⣿⡿⠂⢠⡿⠁⠀⢰⠏⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢀⣼⣿⠇⢀⣾⡟⡾⢠⡿⢠⠇⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠈⣿⣿⠀⣼⡿⢹⠃⣿⡁⠊⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⢠⣿⣿⣶⣿⠃⡎⣰⡟⠳⡇⠀⠘⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⢀⠀⠀⢸⣇⠀⠀⠀⢠⡀⠀⠤
⠀⠀⠀⠀⠀⣿⣿⣿⡿⣼⠃⣿⠀⠀⠇⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠻⢛⣻⣛⠋⢸⡇⠀⠀⠀⣼⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⡇⠀⠀⢰⣦⣀⠀⠀⢘⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢢⣾⡿⣿⡿⠀⣾⣁⠀⠀⢠⡟⠁⠀⢀⣮⣄⠀⠀⠀⠀⠀⠀⣼⡇⠉⡶⣦⣀⡀⠀⠀⣤⠹⡀⠁⠒⠀⣀⡄⢀⡠⣄⠀⠙
⠀⠀⠀⣼⠨⣿⣏⠻⣿⣿⣟⣿⣆⡀⠈⢗⣤⠴⣿⠁⢸⠗⢤⣄⣀⠀⢀⣿⣿⠀⢣⠈⣏⠻⣿⡂⢻⡄⢳⡀⠀⠀⠈⡿⠇⠀⠀⠀⠀⠀⢻
⠀⠀⠀⣼⣧⡙⣿⣷⣿⣿⣿⣿⡟⢡⣴⣿⡏⢰⡿⠀⢸⠆⠀⣿⠉⠙⠛⢻⣿⠀⢸⡄⢹⠀⢏⠹⣌⣷⠈⣧⠀⠀⠀⢐⠂
⠀⠀⠀⢿⣿⣷⣦⣉⣙⣛⣋⣥⢀⣿⡇⣿⠃⢸⣇⠀⢸⠀⠀⣾⠀⠀⠀⢹⢹⡇⠈⣇⠀⠀⠘⠆⠉⢻⣇⠸⡄⢀⠀⢈
⠀⠀⠀⣼⣿⣿⣿⣿⣿⠟⢻⡟⢸⣿⢰⣿⠀⢸⡟⠂⢸⠄⠀⢸⡀⠀⡀⢸⠀⣇⠀⢻⡄⢰⡀⠑⠀⢻⣿⠀⢳⠀⡆⠸⡆⠀⠇
⠀⠀⠀⣿⣿⣿⠿⢱⡟⠀⣿⠁⢸⡟⢸⡿⠀⣿⡃⠀⢀⡇⠀⢸⣇⠀⠀⠀⠀⣿⠄⢿⣿⠈⡇⠀⠀⠀⢿⡁⠈⠀⠙
⠀⠀⠸⣿⣿⣤⠀⠀⠀⢸⠃⢀⡞⠀⡾⠃⠀⣿⣶⠀⠸⠿⠀⠀⠉⠀⢸⠀⠠⣧⠄⠸⠅⠀⢃⠀⠀⣀⣸⣧⣀⣤⣄⣠⣤⣤⣠⣤⣶⣶⡶⠁
⠀⠀⠀⣿⣿⣿⣿⣿⣶⣾⣴⣦⣤⣤⣀⣠⣾⣁⣀⣀⣦⣀⣀⣴⣶⣶⣆⣠⣴⣧⣤⣴⣶⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡉⠁
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠟⠻⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⣊⣭⡆⣖⠻⠋⠙⠟⣲⢰⣭⣐⠈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣧⠀⢻⣾⣷⠴⢾⣞⢾⡿⣳⠷⠦⣾⣷⡟⠀⣼⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡙⢷⡀⠈⢷⡴⠋⣿⣕⣪⣿⠙⢦⡾⠁⢀⡾⢋⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⣹⣿⡄⠈⡇⣰⣟⣒⣒⣻⣆⢸⠁⢠⣿⣏⠻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡾⠁⢘⡇⣰⠟⠋⠀⠉⠉⠀⠙⠻⣆⢸⡁⠈⢷⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡇⠀⣾⡷⠣⣒⣲⣄⠀⠀⣠⣶⣒⠜⢾⣷⠀⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⣧⡀⠙⣦⠘⠓⡴⢿⡇⢸⠿⢦⠚⠃⣴⠋⢀⣼⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣴⡛⣷⠀⢸⡇⠀⣀⡼⠀⠀⢧⣀⠀⢸⡇⠀⣾⢛⣦⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣠⡷⠟⠛⠀⣸⠃⠸⠋⣁⢱⡎⣈⠙⠃⠸⣇⠀⠛⠻⢾⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⣀⠀⡀⣰⢿⣆⠀⠿⣒⠚⢓⣒⠿⠀⣰⡿⣆⢀⠀⣀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⣵⡿⣽⣇⠈⢿⣦⣀⠉⠀⠀⠉⣀⣴⡿⠀⣸⣯⢿⣮⣀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠋⣹⠙⣆⠈⢧⠉⠙⠓⠚⠋⠉⡼⠁⣰⠋⣏⠙⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⢶⠃⠀⢘⡆⠸⡇⠀⠀⠀⠀⢸⠃⢸⡃⠀⠘⠶⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⣿⣿⣷⡼⠃⠀⠀⠀⠀⠘⢧⣾⣿⣿⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡩⢍⠢⡀⠀⠀⠀⠀⠀⠀⡠⣪⢭⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⠔⠀⠘⢄⠀⠀⠀⢀⠎⠀⠣⠬⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠦⠤⠤⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠈⠉⠀⠀⢀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠄⢘⠁⠄⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡶⠃⠐⢄⣀⣀⣠⠊⠢⢄⡠⠜⠀⠈⠳⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢡⠎⠀⠀⠀⠀⣤⣿⠛⠉⠛⣷⣦⡀⠀⠀⠀⠘⢄⣑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡄⡔⠁⠀⠀⠀⠀⢸⣿⣿⣿⣶⣿⣿⣿⡷⠀⠀⠀⠀⠀⠹⡇⠈⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⡘⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠰⢰⣿⣷⡄
⠀⣠⠒⡢⠤⢄⠎⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠿⠟⢆⠤⠤⠒⢢⡀
⣸⡁⠙⡆⠀⢨⣀⡀⠤⡠⣆⠀⠀⠀⠀⡀⠀⠀⠀⠉⠁⠀⠀⠉⠁⠀⠀⠄⡀⠀⠀⠀⢀⠧⡀⢄⣀⣸⠀⠂⡆⠀⣱
⠳⠤⠭⠠⣤⡀⠀⠀⠀⠀⠈⢦⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠⠊⠀⠀⠀⠀⠀⣠⡤⡨⠥⠴
⠀⠀⠀⢀⠃⠈⠉⢙⠧⣀⠀⠀⠉⠒⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠖⠋⠀⠀⢀⡠⢟⠉⠉⠀⢃
⠀⠀⠀⢸⠀⢀⣤⡾⢄⡀⠉⠒⠢⠄⣀⣀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⢀⣀⡀⠤⠒⠊⠁⣀⠼⡄⠀⠀⢸
⠀⠀⠀⢸⠀⢾⣿⠧⡀⠉⠒⠠⢄⣀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣀⣀⠤⠒⠊⠀⡠⢇⠀⠀⢸
⠀⠀⠀⢸⠀⠀⢹⣄⠈⠑⠢⢄⡀⠀⠈⠉⠁⠐⠒⠒⠒⠒⠒⠒⠒⠒⠂⠉⠉⠀⠀⣀⠤⠒⠉⢀⡼⠠⣿⣿
⠀⠀⠀⠸⠀⠀⢤⡀⠑⠢⢄⡀⠀⠉⠁⠒⠂⠀⠤⠤⠤⠤⠤⠤⠤⠄⠀⠒⠂⠉⠁⠀⣀⠤⠊⠁⣠⠀⠙⠛
⠀⠀⠀⠀⢷⣦⢠⠈⠒⢄⡀⠈⠉⠐⠂⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠒⠈⠉⢀⣠⠔⠊⢰⡀⠀⠇
⠀⠀⠀⠀⠘⣿⠏⡗⠤⣀⠈⠉⠒⠢⠤⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠒⠊⠁⢀⡠⠖⣿⡇⡜
⠀⠀⠀⠀⠀⠙⢄⠈⢦⡀⠉⠒⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠐⠊⠁⣠⠞⢋⠜
⠀⠀⠀⠀⠀⠀⠈⠓⢶⣿⠲⢤⣄⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣀⣤⠴⠚⣁⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠬⣹⠛⠛⢶⠶⠶⠶⠖⠒⠒⠶⠶⠶⢶⠞⠛⢻⠩⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠉⢹⠀⠀⢸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠑⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠐⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠂⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⢀⠀⠀⠀⠒⠀⠀⠀⠀⠂⠀⠀⠀⠔⠡⠀⠀⠀⠀⡠⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⠠⠄⠀⠄⠀⠀⠀⠀⠀⠀⢠⠋⠔⠀⠀⠁⠀⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⣀⡠⠞⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⡀⠀⠄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠁⠀⠀⠀⠀⡤⢋⡤⠒⠊⠉⠀⠉⠉⠉⣐⠂⢄⡀⠀⠀⠀⠀⡦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠁⡸⠀⠂⡀⠠⢊⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⣢⣄⠰⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡰⠁⠀⠀⠠⢁⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣒⣓⣒⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⢷⠁⠀⠀⠀⠂⡜⠀⠀⠀⠀⠐⢁⡀⠐⠀⠬⠁⠀⡀⢀⣀⣀⡀⠀⢸⠉⠉⠉⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢐⡏⠀⠀⠀⠀⠀⠇⠀⠀⠀⡠⠊⠀⡈⢉⡁⠒⠒⠲⠤⠤⠤⠤⠤⣅⠸⠀⠀⠀⣀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢪⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠠⠒⠂⢐⠢⠬⠦⢀⠀⠀⠀⠀⠀⢰⠀⡀⠀⠀⠣⠤⠔⠃
⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠀⠀⠀⢘⠀⠀⠀⢰⠁⠀⠀⠀⡔⢁⠔⠒⠲⢦⡀⢄⠀⠀⠀⠸⠀⡇
⢠⠖⠋⠡⢁⣐⣬⣶⣤⣤⣄⡀⠀⠀⡄⢃⠀⣾⠀⠀⠀⢸⠀⠃⠀⠀⡀⢸⠘⡆⠆⠀⠀⠀⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣫
⠈⠒⣵⠚⠁⠀⠀⠀⠀⠉⢆⠉⠒⠦⢴⠘⣇⢹⠀⠀⠀⠈⠦⠄⢀⡴⢁⠆⠀⡇⢸⠀⠀⠀⢇⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⣡⠞⠁
⠀⠀⠘⠷⣞⣰⣀⢆⠤⡅⣰⢀⣀⡀⠊⢣⡘⣄⢣⡀⠐⠤⠄⠒⣁⠴⠋⠀⢠⠃⠄⠀⠀⠀⠈⢆⠑⢄⠀⢀⠎⠱⢀⡠⠖⣩⠴⠋
⠀⠀⠀⠀⠘⠤⠤⢭⣍⣉⣁⣙⣛⣛⣛⣒⢳⣌⠢⣍⠉⡉⠉⠉⠀⠀⣀⠔⢁⠎⠀⠀⠀⠀⠀⠀⠑⠢⣬⡥⠴⢛⡩⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⡑⠦⢭⣉⣉⣭⠤⠚⠁⠀⢀⣀⣀⡤⠤⠖⢚⣩⠥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⢭⣍⣉⣉⣉⣭⡥⠤⠴⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣿⣿⣷⣶⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠋⡉⡉⠙⢯⡉⠈⠙⣿⡏⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠁⢀⣤⣾⣿⣾⣦⡙⠻⣾⣾⣧⢼⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⣿⠇⣴⣾⡿⠟⠿⢭⣅⠀⠀⢌⣿⡿⢊⠽⢿⣿⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠹⣧⣾⣿⣋⣡⣤⠾⣿⣿⡷⠄⣼⣯⠁⢱⣾⣾⣿⣿⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣖⣭⣽⣿⠁⠀⠈⠙⠓⠶⠖⠒⠛⠻⢶⡦⢭⣴⣾⣿⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡤⣙⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⠀⣠⠶⠶⠶⡆⠀⠀⠀⠠⠖⠛⠛⠦⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠳⣾⡟⠀⠀⣠⣶⣶⣤⠲⠀⢰⣠⣿⣿⡷⠆⢸⣯⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⢶⣼⣇⠀⠀⠉⠉⠛⢠⠃⠂⢸⡀⠀⠀⠀⠀⢘⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⢻⡀⠀⠀⠀⠀⡀⢀⣄⣠⣧⠄⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⣧⠀⢰⠒⠫⢀⣠⠌⠭⣤⠤⠁⠀⢠⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠈⠀⠀⠀⢉⣛⣋⡵⠀⠀⢀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣿⠓⢶⣄⡀⠀⠀⠀⠀⢀⣶⣿⡆⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⢿⣇⠹⣦⡀⠈⠛⠶⣶⣶⠶⠿⣿⠟⠀⣸⣿⣷⡶⠶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⡿⣏⢹⡇⠈⠈⠀⠙⠻⢶⣄⡀⠀⠀⢀⡾⠋⠀⠀⠈⠻⢹⡟⢀⡏⣾⢻⣷⣶⠤⣤⣀
⠀⠀⠀⠀⠀⣠⣤⠶⢿⣿⢸⡇⣿⠀⢿⡄⠀⠀⠀⠀⠀⠈⣽⢫⠯⠝⣆⠀⠀⠀⠀⠀⢸⡇⠈⢹⢿⣿⣿⣿⣿⣦⢛⢷⡀
⠀⠀⠀⣴⢿⢡⣴⣾⣿⢸⠀⢷⠸⡆⠈⣿⣄⠀⠀⠀⣠⡴⣷⣜⣒⣠⣿⣷⣦⣄⡀⠀⣼⠃⠀⢸⢸⡇⣿⣿⣿⣸⣼⡏⢻⣦
⠀⢀⣾⣧⠸⡎⣿⣿⣿⡇⡆⠀⠀⠹⠀⠈⢿⣷⣶⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠘⠈⡇⣿⣿⣿⡏⣿⡟⣼⣿⣷
⠀⢸⣿⣿⡆⢷⢻⣿⣿⣷⢹⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⡇⣿⡿⢿
⠀⢸⣿⣿⣿⡸⡎⣿⣿⣿⡎⡇⠀⠀⠀⠀⠀⠈⢻⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⡇⢻⣷⣿
⠀⣸⡿⣿⣿⣧⢷⠹⣿⣿⣷⢹⡀⠀⠀⠀⠀⠀⠀⠹⣿⣧⣸⣿⣿⣿⣟⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⣇⣿⣿⣿⡟⣿⣿⢸⣿⣿
⢰⣿⠁⠻⣿⣿⡜⡇⢿⣿⣿⣇⢧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣿⣿⣿⡇⣿⣇⣾⢏⣿
⢨⣿⠿⢶⣦⣿⣷⣿⢸⣿⣿⣿⡌⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣿⣿⣿⣧⣿⠸⠋⠸⢿
⣿⣷⣀⡳⣾⣿⣿⡸⡜⣿⣿⣿⣿⣎⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⣾⣿⣿⣿⡟⣿⠀⢠⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⢂⣈⣉⣉⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢊⠤⠂⠋⠀⠀⠀⠀⠈⠢⡄⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣡⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⠤⠛⠋⠉⠈⠉⢉⡐⠢⠀⠀⠈⠆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⠀⠀⠀⣑⠄⠀⠎⠉⠙⢳⠀⠀⠀⠸⠏
⠀⠀⠀⠀⠀⠀⠀⢀⠴⢴⠇⠀⠢⠘⣛⣁⠼⠂⠘⢌⠗⠀⠤⠆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠣⣀⣈⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠖⠤⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠀⢃⠀⡴⠁⠀⠀⠀⠀⠈⢰⠀⠀⢀⠃⠀⠀⠸⢖⠉⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⢨⠴⠦⠀⢀⡄⠀⠀⠒⠛⣤⠐⠁⠀⠀⠀⢰⡏⡸⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⢀⡔⠦⡀⠀⠀⠀⠀⠢⠄⣄⠀⠀⣜⢀
⠀⠀⠀⣠⡤⠒⠒⠋⠓⠢⠤⣤⢴⡊⠁⠐⠠⠘⠓⠂⠤⠤⠔⠊⠁⡠⢊⠝⠉⠉⠉⠙⢲⣄
⠀⢀⡾⠉⡇⠀⠀⠀⠀⠀⠀⠈⢢⠹⡁⠳⣄⣀⣀⡄⠀⢀⡤⠲⠏⣠⠋⠀⠀⠀⠀⠀⣸⠈⡆
⠀⡾⠁⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠳⡘⠾⠁⠉⡅⠀⠀⢈⠶⢁⠞⠁⠀⠀⠀⠀⠀⠀⡇⠀⢸⠄
⢸⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⢦⠀⡇⢀⡴⢂⠴⠉⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡀
⡎⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠰⠗⣡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠁
⡇⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⠶⠞⠷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢰
⡇⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣀⡴⠒⢉⡤⢤⡔⢲⠖⢢⣙⢶⡤⠄⣀⠀⠀⠀⠀⣼⠀⠀⠀⢸⠁
⡇⠀⠀⠀⠀⠇⢀⡀⢀⡠⠊⠀⢑⠴⣉⡠⣾⠀⢿⠀⢸⠉⣦⢈⣑⣏⠀⠀⠀⢠⡄⠀⠀⠀⡇
⡇⠀⠀⣠⡴⠛⠀⠈⠉⠱⣄⠀⠳⣴⠁⢀⠿⠤⢿⣀⠾⣤⣿⡀⠉⡄⠈⠑⢤⣤⣠⠀⠀⢰⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠟⠊⢃⠀⢸⡄⠀⠀⠀⠀⠀⠀⣷⠀⢸⠀⠀⠀⠋⠙⠀⢀⡎
⢳⣀⣀⣀⡤⠤⠖⠚⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⢸⠢⡀⠀⠀⠀⢀⡾
⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠙⠶⣤⡴⠛⡇
⠀⠀⠀⢸⠀⠀⠀⢀⡴⠞⠛⠉⠉⠉⠉⠛⢳⡄⠀⠀⠀⠀⠀⣿⠀⢸⣀⣤⣤⣄⣀⣠⠁
⠀⠀⠀⣿⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠄⠀⠤⠤⢄⣿⠀⢸⡇⠀⠀⠀⠀⠉⣳
⠀⠀⠀⢹⡖⢸⠃⠀⠀⢦⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣼⠃⠀⠀⠀⠀⢠⡟
⠀⠀⠀⢸⡇⠀⢠⠇⠀⠀⠙⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠂⣿⠀⣸⠀⠀⠀⠀⠀⠉⢸
⠀⠀⠀⠸⡇⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢰⠀⣿⠀⢻⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⢻⡄⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⠀⣿⠀⣾⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠙⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠶⠼⣿⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⢸⣯⣭⣭⠉
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⣿⠀⢿⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⠀⣿⠀⣿⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣾⠀⣿⠀⣿⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⡟⠀⣿⠀⣿⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⣀⣀⣠⣄⣀⣀⣀⣿⢰⡇⠀⣿⣀⣽⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⣸⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⡷⠺⠁⠀⠈⡟⠛⠛⠛⠛⠉⠛⠛⠓⠲⠶⣤⣀
⠀⠀⠀⠀⠀⢠⡎⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣀⢸⣧⠀⠀⠀⢸⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢀⣀⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡴⢋⣘⡙⠖⠒⠶⠶⢜⣛⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢄⡙⣶⠋⢰⠋⣤⡌⡆⠀⢀⣠⣤⣉⠀⡜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣼⡃⠀⠈⠲⢤⠔⠁⠠⡏⠠⠶⣸⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠈⣧⣄⣀⣀⣭⡟⠓⠀⠙⠓⠚⠁⠀⠀⢸⠛⠓⠒⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⠃⠀⡇⠀⠀⠀⠀⠳⣄⣀⠀⣀⣴⡖⠤⢄⡬⠤⣄⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⢀⣤⣤⣤⡴⠉⠀⣀⣠⣤⣽⣦⣤⣤⣄⣠⣤⣭⣭⣭⣵⣬⣵⠏⠀⠀⣨⠧⠤⠤⢤⡀⠀⠈⢧⡀
⠀⠀⠀⢸⠀⠀⡠⠇⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠠⡇⠀⠀⠀⢸⡇⠀⠀⠀⢳⡀
⠀⠀⠀⢸⠀⠘⠶⣎⣀⠦⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣇⣰⠶⠋⠀⠀⠀⠀⡇⠀⠀⠀⠀⢳
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⡇
⠀⠀⠀⣿⠀⢸⠀⢠⡄⢀⡄⣾⣉⡄⠀⡿⠀⠀⢠⠖⠂⠀⡤⠲⡄⣸⢦⡼⡇⢰⣋⠁⠀⠀⡇⠀⠀⠀⠀⠀⢷
⠀⠀⠀⣿⠀⠈⣧⡏⢧⡼⠁⠿⣤⠄⠀⠷⠔⠀⠸⠤⠤⠀⠙⠚⠁⠛⠀⠀⠃⠸⠧⠤⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠛⠛⠶⣤⣀⣇⠀⣀⠀⠀⠀⡟
⠀⠀⠀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⡏⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠤⠤⠤⠶⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣅⡄⣤⡄⠠⠜⣧⣴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣁⣫⢟⡞⣿⣤⣤⡿⠚⢋⣋⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣤⣶⣾⣿⣿⣿⣞⢱⣿⢸⡇⣟⣈⡉⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣸⣃⣀⣴⣏⣿⣿⣿⣿⣿⢿⢻⣿⣼⣿⢛⢳⣬⡧⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⢿⣋⣼⣿⣿⡿⠿⢿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠙⢻⡿⠯⣿⣿⣦⣯⣿⣿⣿⢿⣷⢾⣿⣿⣿⣿⣿⣿⣿⣧⣹⣿⢿⣿⢋⠐⢆
⠀⠀⠀⠀⠀⠀⠈⣀⣴⣿⣷⣾⠟⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡅⠬⢿⣧⡽⣿⣷⣿⡅⣠⠤⠄
⠀⠀⠀⠀⠒⠻⠿⢿⡿⢟⠿⣏⣽⣿⡿⠟⡽⣻⣿⣿⣿⣿⣿⣿⣏⣿⣼⣟⣵⡷⣿⢉⡋⢀⠤⠄
⠀⠀⠀⠀⠀⠀⣀⢀⣤⣮⣽⣿⣿⣿⣿⣿⣛⡛⢿⢿⠉⠉⢤⣿⣾⣿⠟⠋⣹⣶⣾⢘⡕⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠈⠛⢿⠿⠟⠓⠚⠁⠀⠻⣾⣿⣷⣏⠄⠆⣞⣛⠌⣿⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠼⠧⠮⠰⢻⣷⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢪⠭⡐⠄⠀⠀⠀⠀⠀⢀⠔⡩⢍⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠐⠁⠈⢂⠀⠀⠀⢠⠊⠀⠃⠡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠧⠤⠤⢅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠈⠀⠀⠀⠄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⠀⠂⢨⠀⠂⠀⢠⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⡚⠁⠀⠲⠤⣄⣴⣡⣑⡠⠔⠃⠀⠈⠻⡒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢡⠊⠀⠀⠀⠀⣴⣿⣍⡀⣈⣿⣷⡄⠀⠀⠀⠈⢦⣍⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡷⡐⠁⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠹⡀⣸⣦
⠀⠀⣀⠀⠀⠀⡰⠿⠟⢡⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⢁⢻⣿⣷⡀⠀⠀⢀⡀
⢀⠼⢠⠀⠉⠎⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠨⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠉⠁⠎⠈⠀⠘⢆
⡞⠂⠤⠁⠀⠓⠐⠂⠐⠋⢆⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠎⠓⠈⠒⠐⠃⠀⠁⠄⢚
⠈⠉⠉⠙⠗⠢⠄⢤⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠁⠀⠀⢠⠀⠤⠒⠟⠉⠉⠉
⠀⠀⠀⢸⠀⠀⣀⡎⠉⠒⠤⣀⡀⠈⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠉⠉⠀⣀⡠⠔⠊⠉⡆⠀⠀⢸
⠀⠀⠀⠀⠀⣾⣿⠀⠑⠦⢄⡀⠈⠉⠐⠒⠂⠠⠤⠤⠤⠤⠤⠤⠀⠒⠒⠈⠉⠀⣀⠤⠔⠉⢸
⠀⠀⠀⠀⠀⠙⣏⠉⠢⢄⣀⠈⠉⠐⠒⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠐⠒⠈⠉⢀⣀⠔⠊⢉⡄⣴⣶
⠀⠀⠀⢸⠀⠀⡏⠑⠦⣀⠀⠉⠒⠒⠤⠤⠀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠔⠒⠊⠁⠀⡠⠖⠉⡇⠛⢿
⠀⠀⠀⠈⣦⡄⠏⠲⢄⡀⠈⠒⠢⠤⠄⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠤⠤⠐⠀⠀⣀⠴⠊⡇⠀⡌
⠀⠀⠀⠀⢻⣿⢹⠦⣀⠈⠑⠢⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠒⠉⢀⡠⢾⡿⢠⠃
⠀⠀⠀⠀⠀⠫⡀⠱⡀⠙⠒⠤⢀⣀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢀⣀⠠⠔⠊⠁⡰⠛⡡⠃
⠀⠀⠀⠀⠀⠀⠙⠶⣾⡷⢤⣄⡀⠈⠉⠉⠉⠀⠐⠀⠒⠒⠂⠀⠈⠉⠉⠉⠀⣀⣤⠴⠋⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠬⢉⡟⠒⠶⠶⠶⠒⠒⠒⠒⠒⠲⠶⠶⠖⠚⣏⠩⠄⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡇⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠉⢢
⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠂⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠾⠈⢿⣇⡀⠀⠀⠴⠋⠀⠀⠰⠆⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⠃⠀⠀⠀⡼⠁⠀⠀⣀⠀⠀⠀⢠⣄⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⡶⠀⣠⣤⡝⠋⠂⠙⠯⠴⠂⢸⣇⣨⡇⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⡏⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⠤⠀⠰⣮⠀⠀⠀⠀⡤⠄⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⡀⠙⠦⣀⠀⠀⠲⢤⣄⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣿⡀⠀⠀⠸⣿⣦⡀⠈⠙⡲⠀⣸⠏⠉⠛⠶⠦⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠁⠀⠀⠈⢷⠀⠀⠀⣿⣿⣿⣷⣶⠃⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⠀⠐⠀⠀⠄⠀⠈⣷⣄⠀⠙⢦⣤⠟⠁⢠⡏
⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠠⠀⢠⣆⠀⠀⠀⠀⠰⠉⠻⣆⠀⠀⠀⠐⢧⡈⢧
⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢠⠀⠀⠈⣷⣄⡀⠀⣠⠇⠈⣧⣀
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠠⡄⢃⠀⠀⠀⠈⠀⢀⠎⠀⠀⠉⠉⠁⠀⠀⠈⡎⢢⡀
⠀⠀⠀⠀⢰⠁⢀⠁⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠌⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠱⡄
⠀⠀⠀⠀⡘⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⡤⠞⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠹⡄
⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠘⡄
⠀⠀⠀⠀⠇⠀⡇⠀⠀⠀⠀⠠⠄⠐⠂⢢⠒⠒⠛⠀⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡓⣄⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠷⠂⠀⠀⠀⠀⠀⠀⠀⢸⢄⣀⢀⣀⣀⣤⡀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⡷⠈⢧⠀⠸⡀
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠖⠈⠀⠀⣿⠉⠀⠀⢀⠇⠀⣰⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⡆⠀⢱⡀
⠀⠀⠀⢸⠀⠀⣀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀⡄⠢⠆⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠈⡧
⠀⠀⠀⠈⠳⠋⠁⠀⠐⠒⠒⠒⠒⠂⠀⠀⠀⢸⠇⠀⠀⠀⠙⠁⠀⢀⣀⣀⣀⣀⣤⡤⠂⠀⣿⠀⠸⠀⢀⣣⣀⠀⣀⡀
⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠠⠄⠀⠀⠀⣇⢰⠛⠛⠋⠁⠀⠀⠈⢙⣆⢀⣀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⢆⢹⠀⢰⠀⢸⢰⠼⢁⣈⠁
⠀⠀⠀⠀⢸⡀⢀⣀⣀⣀⣀⣀⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠘⠙⡀⠻⠖⠉⠉⠀⠀⠉⠈
⠀⠀⠀⠀⠈⠉⠀⠉⠉⠉⠉⠉⠉⠉⠙⠓⣯⣀⣀⣀⣀⣀⣤⣤⣤⡤⠤⠤⠶⠶⠶⠶⢶⠶⠞⠀⠀⠀⡇⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢸⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠘⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⡄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⣤⡴⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⣠⠄⠀⢿⡀⠀⠀⠀⡇⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠁⠀⠀⠘⣇⠀⠀⠀⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⣀⣡⡀⠀⠀⠤⠤⠤⠤⠄⠀⠀⣀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⢦⡄⢀⡧⠠⣇
⢀⡤⠐⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⢻⣄⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠋⠉⠙⠣⣤⠤⠤⠴
⠉⠙⠛⠒⠒⠒⠂⠂⠒⠒⠒⠒⠛⠛⠋⠉⠁⠈⠉⠉⠉⠛⠶⠦⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⢿⠿⣿⠿⣿⣿⡿⠿⠿⠛⠛⠉⠉⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠐⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠀⠀⠀⣴⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⣿⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⣿⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⠀⠀⠀⠙⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠤⠀⠀⣤⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⠀⠀⠀⣿⠀⠈⡇⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⣿⠀⠀⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠈⠀⣿⠀⠀⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠋⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠻⢶⣶⣤⣤⣀⣀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠾⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠈⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⡿⠟⣻⣿⣿⣿⢿⣿⣿⣿⣿⣿⣧⡀⡅⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠔⣫⡾⠋⠁⠀⠀⠈⠛⠛⢿⣿⣿⣇⢌⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣩⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣷⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⣀⠠⣶⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⢠⣾⣿⣿⣿⣿⣥⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⢰⣿⣯⣿⣾⣿⠟⠛⠋⠛⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢠⣿⠋⣾⣿⣿⣇⠀⠐⠿⠿⣿⣶⡌⣿⣷⣶⣶⣶⠶⠶⠿⠶⢶⣶⣦⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⢸⣿⣿⠿⣿⣆⠀⠀⠀⠈⠉⣱⡿⠁⢹⡿⣴⡞⣿⣶⣦⡄⠈⠉⣽⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠆⠚⣈⣿⡟⠀⠀⠉⠛⠶⠶⠶⠛⠉⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⢀⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠴⠖⠋⢩⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⠀⠀⠀⠀⠀⠹⣿⠒⠲⠤⢤⣄⣀⠀⠀⠀⠀⠀⢠⡿⠡⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣆⠀⠀⠀⠀⠀⠈⠀⠀⠻⠟⠋⠁⠀⠀⠀⠀⣠⣿⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠇⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⣿⣿⡟⡩⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⡛⠋⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠎⠐⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠰⣿⡟⠀⠄⠀⠙⠂⠀⠀⢀⣀⠤⠞⣿⣿⣿⣿⠃⣰⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠐⣹⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢟⣄⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⡔⢁⣾⣿⣿⣆
⠀⠀⠀⠀⠀⢀⠶⠂⠜⠀⣰⣄⠀⠀⠀⠀⠀⠀⣸⣿⣫⡾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠕⢀⣾⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⣰⣿⠛⠀⠀⠀⠀⠀⣴⣿⣷⣟⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣿⣿⣿⠿⠛⢁⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡠⢚⡼⠃⢀⠂⠀⠀⣠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡿⠿⠛⠁⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠠⠐⣰⠏⠀⠀⠁⢀⣤⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡯⠊⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⠀⢀⣴⣿⣿⣿⣿⠿⠋⠅⠋⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠙⢿⣿⣿⣿⣠⣿⣿⣿⣿⠟⢁⠐⠀⠀⠀⣀⡶⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠟⢛⡋⠀⠀⠀⠀⢠⣼⣿⣾⣿⣿⣿⣿⠿⠿⠻⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿⠟⠉⢠⠐⠉⠀⠀⢀⠠⠖⠻⣿⠿⣿⣿⠿⠛⠁

⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣶⣶⣤⣤⣀⡀
⢠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠉⠉⠛⢻⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⣷⣼⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⡿⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡉⠉⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⢻⣿⣿⣿⡇⠀⠀⠈⠉⠙⣿⠋⠉⠉⣿⣧⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠻⣧⣼⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣆⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠻⠷⣶⣤⣤⣤⣤⣶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣶⣶⣶⣶⠶⠿⠛⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⡿⠿⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠻⠶⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠁
⠀⠀⠀⠀⠀⠀⣾⣶⠀⠀⣠⣴⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⢿⣿⣴⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣧⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢀⡀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠿⢦⡀⠀⢸⡇⠀⢀⡴⠿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠞⠃⢠⡾⢷⡄⠘⠓⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣯⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⣀⡈⣛⠉⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢠⣍⣙⠻⢶⣟⣉⣹⣆⠀⣤⣤⡄⢀⣤⠾⣟⢻⡷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⡁⠀⠀⣼⡋⠛⠶⠟⠉⠉⠉⠀⣧⣠⡇⢸⡷⣤⠘⠋⢠⡶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⢶⣄⠈⠻⣦⡀⣴⠶⠶⠶⠶⠾⠷⠶⠾⠷⠿⠶⠶⠾⠷⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣸⠋⠳⣆⠈⣷⠻⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡶⠟
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡟⠀⠀⣿⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢾⣀⣸⠃⠀⠀⢿⡀⣹⠀⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠙⠉⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠰⠟⠛⣷⠀⠀⠀⢸⡇⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠘⢧⣤⣤⣤⣤⣬⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠆⠀⣀⣀⣀⣀⣀⣀⣀⡀⢠⡴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⣴⠶⠛⢶⡌⢉⣉⣉⣉⣉⣉⡁⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⣿⠀⣥⣬⣭⣭⣭⡅⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⣿⠀⣤⣤⣤⣤⣤⡄⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣿⠶⡤⠾⣿⡐⠶⠶⠶⠶⠶⠆⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⠾⠋⠙⢷⣠⡾⠉⠛⠶⣤⡐⠶⠶⠆⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠈⡋⠀⠀⠀⠀⠈⣇⠘⠛⠃⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⣟⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡟⢠⣤⣤⣄⢹⡿⠁




⡟⠉⣻⣛⣻⣛⣛⣟⣻⣛⣻⣛⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣛⣛⣛⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠐⠊⠉⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠻⢷⡄⠀⠛⣶⣦⣀⠈⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⡶⡄⢠⣿⣷⣶⣿⣿⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣚⣵⣾⣿⣷⢿⣿⠿⡇⠀⣰⣿⣿⣿⣿⣿⣿⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⡿⢏⢻⣷⣤⣿⣴⣇⡀⠟⠉⠀⠈⠀⠈⢿⣷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢾⣅⣤⣤⣾⣾⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣀⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢠⣟⠋⣉⢱⡿⣸⣿⣿⡿⠛⢿⣎⣙⡷⠀⢸⣭⣿⣟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠉⠉⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠾⠿⣿⣿⣿⣿⣿⣿⣿⡿⡆⠀⠀⠀⠀⠶⣾⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣾⣿⣷⣄⡀⠀⠶⣿⣿⣿⣿⣿⣿⣩⡻⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣯⡿⣿⣶⣤⣤⣿⣿⣿⣿⣿⣷⣤⣄⣈⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⣤⣈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣦⣈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⣽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠚⠉⠉⣭⣵⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⢀⡙⢶⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⢠⡀⡟⠹⠴⠃⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⣼⠉⡀⠀⠀⢀⠤⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠤⢧⡀⠀⠀⠀⢰⡞⠉⡇⠈⠑⢲⣧⠖⠋⢉⠟⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠘⠁⠀⠀⠀⣸⡆⠀⠱⣀⠖⠉⠙⣦⠴⡿⠒⠋⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢘⣑⣧⠀⠀⠈⠓⠦⠤⡾⠀⠉⢳⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣰⡯⠴⠶⡄⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢠⡀⠀⡀⠀⠀⣇⣇⣰⡠⡄⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⠳⣄⢳⡀⣄⠀⣀⠀⡰⠃⠀⠄⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠘⠶⠙⣏⣿⡿⠛⠁⣀⡠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⢸⢢⣴⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢤⣀⡀⠀⣠⠞⠉⢠⠄⠀⣮⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠉⢿⣅⡀⣰⣁⠀⠀⢃⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡤⣴⣫⢾⡄⠁⢠⠷⣦⣼⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣞⣉⡻⣿⣏⣤⣀⠎⡰⡋⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣉⣓⡒⠃⠀⡠⠺⠔⢊⣏⣀⡃⡇⠸⡜⢆⡰⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣒⡭⠽⠄⠓⢆⣘⣽⠒⠦⠾⠭⠽⡝⠀⠀⢸⠊⢡⣠⠗⣄
⠀⠀⠀⠀⠀⠀⣠⢖⠝⢋⣤⠔⣻⣷⣠⣾⡿⠛⢛⣿⡿⠋⠉⠁⠀⠀⠈⢻⡙⣄⣱⠊
⠀⠀⠀⠆⡄⡰⣱⠋⠦⢴⣹⡝⠻⣿⡿⠋⠀⣠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⢧⡇⠀⢠⠞⠉⠀⠀⠀⣤⡠⢬⣟⣿⣁⣀
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⡠⣞⣙⠒⠚⠯⠿⢻⠍⠉⢙⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠚⠩⠭⠭⠭⠭⠭⠝⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣀⣠⣉⣹⣿⣿⣿⣿⣿⡿⠛⠿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠉⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠈⠈⠑⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠛⠋⢁⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠐⣶⠋⠉⠻⡆⠀⠀⠀⡾⠋⠉⢻⡖⠀⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⢀⢁⡀⠀⠀⠀⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⡀⠀⠀⠈⠛⠁⠀⠀⣀⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡖⢶⠲⡖⢶⣿⣿⣿⣿⣿⡟⠋⠛⠚⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣴⠚⣤⣿⣿⣿⣿⣿⣿⣁⡤⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⢿⣿⣿⡿⠿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⣿⢿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠔⠛⠛⠋⠙⠓⠒⠲⠦⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⠟⠛⠉⠉⠉⠉⠉⠛⠶⠶⠦⣄⡀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠂⢉⡰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣨⡇⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠚⠉⢢⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⢀⣈⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⣱⠋⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠁⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠘⢰⣦⢤⢈⣁⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣒⠉⠉⠒⣄⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⢸⢧⣀⣠⠁⠀⠀⠀⠐⠦⡿⠁⠀⠀⠉⣹⠒⣶⡖⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠱⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⡅⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢠⡟⠣⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡇⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⡾⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⡀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡇⠀⠀⠀⠀⠋⠀⠀⠀⢠⡾⠁⠀⡠⠉⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠋⠛⠓⠚⠛⠉⢁
⠀⠀⠀⠀⠀⡔⢠⠀⠀⠀⠸⡀⠀⠀⠀⠘⢤⣠⡴⠟⠁⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⢠⠆⠈⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠸⠀⡇⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢦⣄⣀⣀⠀⠀⠀⣀⣀⣀⣤⡤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠒⠒⠊⡉⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠇
⠀⠀⠀⠀⢅⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠲⠉⠀⠀⠀⢠⢤⡎⠛⢨⠀⢷⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⡄
⠀⠀⠀⠀⢸⠀⢸⡄⠀⠀⠀⠀⠀⣀⠴⠊⠀⡠⡐⠇⢶⠖⠹⠃⢨⠀⣀⡜⢂⠘⢢⣈⠉⠒⠤⢀⡀⠀⠀⢀⡰⠃⠀⣾⡇
⠀⠀⠀⠀⠈⣇⠈⠻⠦⠤⠴⠂⠋⠀⢀⡴⠋⡴⢁⣴⠎⢀⡇⢀⣍⣼⢩⡀⠀⠉⠋⠉⠛⠦⣄⣀⠁⠀⠀⠉⠁⣀⡼⠈⡇
⠀⠀⠀⠀⠀⠎⠳⣄⠀⠀⠀⢀⣠⠞⠉⠀⢰⠀⢼⠈⠓⠙⠢⠚⠀⢸⠈⡇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠖⠛⠉⠀⠀⡇
⠀⠀⠀⠀⠀⣦⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠸⠤⠿⠀⠀⠀⠀⠀⠀⣼⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠿⢿⡶⢄⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⢀⠄⢀⠔⠒⠘⣦⠀⠢⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⡇⠀⠀⠢⡀⠀⠀⠀⠀⡿⢸⡇⠀⠀⢀⠔⠁⠀⢸⠆⠀⠊⠉⡆⠀⠀⠀⠀⠀⢸⡀
⡏⠉⠉⠉⠉⠷⠀⠀⢀⡠⠖⢪⠁⠀⢘⡉⢦⠀⠀⠈⠐⠄⣀⠀⡿⢸⣇⠠⠒⠡⠀⠀⣠⡿⠀⠀⠀⡄⢃⠀⠀⠀⠀⠀⠀⠷⠐⠒⠒
⡇⠀⠀⠀⠖⠂⠉⠉⠻⢤⠀⡀⠀⠀⢸⡅⠠⠷⢤⡀⠀⠀⠀⠈⣿⢸⡇⠈⠁⠀⣠⣴⣉⡇⠀⠀⢀⡇⠘⠁⠒⣦⠤⠤⠤⠼⠄
⢙⢛⠛⠻⠛⠛⠛⠛⠓⠊⠙⡇⠀⠀⣼⠃⠀⠒⠊⠛⠳⠶⠦⠤⣿⢸⡷⠶⠚⠛⠁⠖⢾⡇⠀⠀⠸⡷⠶⠶⠿⠶⠶⡶⢶⠶⣶⣶⠶
⡄⡈⠀⠘⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡇⠀⠀⠀⠀⢰⠐⠀⠀⠀⡜
⡇⡇⠀⢰⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⠸⣇⡀⠀⠀⠀⠀⠀⠛⠀⠀⠈⡇⠀⠀⠀⠀⡄⡆⠀⠀⠀⡇⡇
⡇⡇⠀⢸⠀⠀⠀⠀⠀⠠⣇⡀⠀⠘⣓⠄⠀⠀⠀⠀⠀⠀⢸⠀⠙⠚⠁⠁⠀⠀⠀⠀⠠⣎⣀⣀⣘⡷⠀⠀⠀⠀⡇⠇⠀⠀⠀⡇⡇
⡇⡇⠀⠈⠃⠀⠀⠀⠀⢠⠊⠉⠉⠉⢡⣄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠁⠀⠀⡀⢱⠀⠀⠀⠀⠑⠂⠀⠀⠀⠃⡇
⠁⠁⠀⠀⠀⠀⠀⠀⠀⡦⠠⣇⠀⠀⠀⠌⠙⠲⢦⣤⣤⣀⠸⣤⣀⣠⣤⠇⠀⣠⡞⠁⠀⠀⠀⣾⠠⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠉⠳⢦⣄⡀⠀⠊⠁⠀⠀⠈⠙⠢⣄⠀⢀⡴⠛⠉⠁⠂⡀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⣄⡀⠀⠀⠀⠀⠘⣦⠏⠀⠀⠀⠀⠀⣨⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠖⠛⠛⢳⣦⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⡡⠀⠀⠀⠀⠀⠀⠠⠃⠀⠃⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠂⣠⢶⡀⠀⠀⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡴⠊⠀⠀⢀⡠⠊⣀⡴⣿⣠⣾⡵⠞⠁⠀⢷⡰⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢀⣠⣴⣯⠴⠛⠉⠀⠟⠋⠁⠀⠀⠀⠀⠈⢿⣆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠻⠛⠉⣩⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠆⠘⡟⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⣀⣙⡳⠖⢀⡆⠀⠠⠴⠚⢁⡀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⡟⠀⣼⠀⠀⠀⠀⣿⣿⡇⠀⠀⣿⢸⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠘⠢⠴⠂⢠⡏⠀⠀⠀⠀⠬⡫⠔⠀⠀⣷⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠙⠓⠂⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠃⠉⠉⠉⠭⠍⠉⠉⠈⠁⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣶⠦⡤⠤⠤⠤⠶⣾⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⣶⠞⢹⠁⠀⢣⠀⠀⠀⠀⢸⡏⢳⡦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠉⠉⠀⠀⡟⠀⠈⣧⠀⠀⠀⠀⠀⠀⣼⠀⠀⣧⠀⠀⠉⠉⠛⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠐⣄⠀⠀⠀⠘⡇⠀⠀⠈⢧⡀⠀⠀⢀⡾⠃⠀⠀⡟⠀⠀⠀⣼⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠘⢆⠀⠀⠀⢿⣤⠶⡆⠀⠙⢦⡴⠋⠀⣶⠶⡼⠃⠀⠀⢀⠃⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠦⣤⣸⣀⣤⠤⠟⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣟⡀⠀⠀⠀⠀⠀⠀⣏⠉⠉⠉⠉⠉⡇⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣄⡉⠓⠦⣄⣀⣦⠀⢿⡀⠀⠀⠀⣀⡇⠀⠀⠘⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⡤⠖⠋⢁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠳⠶⢦⣬⣿⠀⠈⠙⠓⠖⠛⠁⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣤⡴⠶⠛⢻⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⣀⣸⣯⣀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠒⠶⠿⠦⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣇⣇⣀⣀⣤⣤⠶⠞⠛⠉⠉⠁⠀⠀⠀⠀⢸⡇
⠤⠤⠤⠤⠤⠤⠤⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⠶⢤⣽⣿⠿⠟⠛⠛⠛⠛⠳⠾⣿⣷⡄⠀⠀⠀⠀⠀⢀⡿⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⣹⠀⠀⠀⠀⠀⠀⠀⠠⢬⡻⣦⡀⣀⣀⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⡦⠶⠶⠿⠿⠶⣦⣤⣄⣀⡀⢤⣝⡲⣝⢾⣿⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢁⡤⢔⣤⣤⣤⣤⣽⣖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⣣⡾⠟⠉⠉⠉⠉⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⠀⠀⠀⠀⠀⢀⠀⠀⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡋⢿⠀⠀⠒⠀⡀⠀⠛⠀⢻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⢇⠀⠀⠀⠀⠤⠄⠄⠀⠀⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠓⠒⠚⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣀⡠⣤⠤⢠⠎⠰⠒⠤⠤⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⢿⣎⠁⠙⠻⣿⣟⠉⠀⠀⣧⢠⡐⣤⡶⠶⠚⠓⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠁⠀⠀⠀⣿⣆⣀⣴⠇⠈⢷⠦⡾⣿⡎⣿⣿⣄⠙⣛⠛⠛⢒⣂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢤⡘⣿⣦⠀⡠⢸⡏⡻⢿⠖⠀⠐⠄⠪⠳⢿⣾⣿⡗⠀⠈⠩⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢙⡎⢿⣇⠈⠂⣷⠀⠀⠀⠀⠂⠀⠀⠀⢀⣿⡿⠃⠀⠀⠀⢺⣿⣿⣇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠈⣿⣇⠀⢸⡖⡖⠒⢚⠋⡋⢉⢩⣻⠋⠀⠀⠀⠀⠶⣿⣿⣿⡧⠠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢄⠀⠀⠀⣿⣿⣧⣸⠀⠀⣄⢆⣦⣆⣴⣸⡇⠀⠀⢀⠤⣒⣿⡿⠁⠀⠀⠐⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢨⡲⢄⡀⠀⢻⡯⢳⣿⣴⣿⣿⣾⣿⣿⣿⣿⡧⠤⡤⢄⣩⠽⠿⣷⣄⡠⣤⠅⢀⣼⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⡌⠀⢀⡤⣾⣷⡾⠿⠿⠟⠛⠛⠙⠋⡉⠀⠀⠀⠠⠄⣹⣿⣴⣿⣿⣦⡖⣺⣿⢃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢽⣂⡀⡰⠟⠉⠀⠀⠀⠀⠀⠀⢷⠰⡄⣱⠀⠀⠀⠀⠀⠀⠀⠈⢽⣿⠏⣰⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⣿⠀⠁⠀⡀⠀⠠⢔⠠⣀⠄⠀⢀⣼⣾⣿⣦⣄⠀⢀⣀⢀⣀⡶⣣⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡘⠀⠀⠀⠂⢐⠨⢔⣪⣴⡦⣞⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣤⣠⣬⣭⣭⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠿⢿⡿⠿⠿⠟⠛⠛⠛⠛⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠦⠬⠉⠉⠉⠐⠒⠤⡀
⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⢀⠿⢸⠙⠻⢶⡒⠀⠀⠀⠻⡒⠠⠄⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢤⢄⡀
⠀⠀⠀⠀⠀⠸⢀⠃⠀⠀⠀⠀⠉⠐⠂⠀⠘⠂⠀⠀⠃⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣀⠀⠀⠀⡈⣆
⠀⠀⠀⠀⠀⢠⠎⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⢸⡆⠀⠀⠀⠀⠀⡤⠊⠁⠀⠀⣠⠀⠀⠀⠀⠈⠂⢄
⠀⠀⠀⠀⢀⣸⠀⢀⠀⠀⡀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⢹⣄⡀⠀⠀⠀⠎⠀⠀⠀⢀⡼⠁⠢⡀⢀⠀⠀⠀⠈⢂
⠀⠀⠀⠐⠁⢸⠀⠀⠀⠹⠿⠀⠀⠀⠀⠀⠨⠿⠀⠀⠀⢈⠀⢱⠀⠀⢘⠀⠀⠀⡴⠋⠀⠀⠀⠙⢆⡁⠀⠀⠀⠈⡄
⠀⠀⠀⠈⢄⠀⡄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠞⢀⠌⠀⠀⡏⢁⠤⠊⠀⠀⠀⠀⠀⠀⠀⠙⠲⠄⡀⠀⡇
⠀⠀⠀⠀⠀⠉⢣⠀⠀⠀⠀⡆⠈⠉⠀⠰⠀⠀⠀⠀⢠⠋⠁⠀⠀⠀⠇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠁⠀⠉⠁⠀⠀⠀⠀⣠⡂⠀⠀⠀⠀⡔⢻⠁⠀⠀⠙⠁⠀⠐⠀⠀⠀⠛⠁⠀⠀⢙⠓⢢
⠀⠀⠀⠀⠀⠀⠀⠀⢨⢶⠄⠠⠤⠀⠀⠤⠠⠀⣾⡻⠉⠒⢄⠀⠀⠣⠤⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡬⠤⠞
⠀⠀⠀⠀⠀⠀⠀⣴⠁⢘⡆⠀⠀⠘⠂⠀⠀⣨⡷⠁⠀⠀⣠⠓⢄⠀⠀⠈⠢⣀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣠⠊
⠀⠀⠀⠀⠀⠀⡔⢨⠀⠀⠝⣄⠀⠀⠀⢀⣔⠟⠀⠀⠀⢀⠁⠀⠀⠣⡀⠀⠀⡠⠝⢲⣦⠤⠶⠦⢐⢖⡟⠀⠉⠢⢄
⠀⠀⠀⠀⠀⠰⠀⠘⠀⠀⠈⠢⡙⠚⡫⠕⠁⠀⠀⠀⠀⠸⠂⠀⠀⠀⢱⡔⢻⠀⠀⠈⡗⠩⠀⠩⠔⠊⡇⠀⠀⠀⠀⡱⢄
⠀⠀⠀⠀⠀⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢧⢸⠆⠀⠀⢁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠰⠀⠀⢣
⠀⠀⠀⠀⢸⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠒⣦⣂⠀⠈⡘⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠐⡇⠀⠀⠀⠣
⠀⠀⠀⢀⠜⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠀⠝⠀⠀⢠⡇⠀⠀⠀⢸⠀⠀⠀⠀⠈⡀⠀⠀⠀⠸⠀⠀⠀⢸
⠀⠀⢠⡯⠒⢢⣙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠒⢤⣠⠃⠄⠀⠀⠀⡜⠀⠉⠉⠀⠀⢃⠀⠀⡀⣀⣧⠀⠀⢰⡀
⠀⠀⠮⢦⡂⡎⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡉⠀⠉⡷⠖⠁⠀⠃⠀⠀⡔⠀⠀⠀⠀⠀⠀⠈⢢⣾⡀⠈⢻⡄⠀⡔
⠀⠀⠀⠀⠉⠁⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⢶⠃⠀⠀⠀⠰⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡶⢹⡟⢸
⠀⠀⠀⠀⠀⢸⠁⢠⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠁⠸
⠀⠀⠀⠀⠀⠘⢰⢸⠤⠤⠤⠀⠉⣉⣉⣉⣉⣉⡉⠡⠤⠤⠀⠒⡆⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡠⠂
⠀⠀⠀⠀⠀⠀⡎⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹
⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⢀⠄⢲⡆⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠇⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⡁⠘⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠁⠢⠤⣦⠀⠀⠠⡆
⠀⠀⠀⠀⠀⠀⢸⢿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⡄⠀⠀⠀⠀⠀⠸⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠇
⠀⠀⠀⠀⠀⠀⠘⢸⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠀⠤⠀⢠⠀⠠⠀⠀⢲⠈⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠈⠀⠀⠀⠀⢃⠀⠀⠀⠀⠄⠀⠀⠀⠘⡆⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣇⡠⠤⠬⣁⠀⡇⠀⠀⠀⠀⠀⢶⠚⠛⠒⢶⠀⠀⠀⠀⠈⣇⠀⠀⠈⡁⠀⠀⠀⠀⠑⢤⣄⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣈⣉⣉⣉⡹⠇⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠰⡉⠑⢦⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡀⠀⠙⠻⣷⣶⠆⠀⠀⠀⡾⠟⠁⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⡠⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢉⣗⠚⠙⠁⠀⠘⠉⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠸⡟⠀⠠⠀⠀⠀⠀⠛⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⠠⠔⠁⠀⠀⠀⠀⢀⡠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠴⠒⠒⠒⠒⠒⠒⠋⠁⠀⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠯⠤⢄⣤⣀⡀⠀⠠⠤⠤⠒⠋⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠴⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣀⣀⣀⣀⡤⠖⠋⠁⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣭⣵⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⢿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⣿⡸⣿⣿⣿⣿⢿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣯⣾⣿⣿⡿⣻⣿⣿⣿⠏⢠⢧⣿⠁⠀⢁⣁⣿⠀⠀⠘⣄⠀⠙⠧⡀⠉⠫⣯⡉⠉⠹⠿⣦⡀
⠀⠀⠀⠀⠀⠘⢟⠛⢛⠛⢿⡛⠛⠛⠿⢿⠟⠛⠻⢿⣿⠿⡿⢿⠿⠿⡿⢿⣿⠿⠿⠿⢾⡶⠿⠷⣿⣶⡶⢾⣿⣶⡶⠶⢶⡿⡦
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢀⠙⢆⠠⡀⢈⢣⡀⠀⠀⢳⡀⠀⠈⠀⠀⡇⢸⢹⠆⠀⠀⠜⠁⠀⣠⢟⡡⢖⣡⠖⠁⠀⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡈⠙⠻⢷⣀⠀⠀⡓⠀⠘⠀⠀⡇⠀⡌⠀⠀⣠⡎⢔⣩⣊⠥⠚⠁⠀⢀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡑⠈⠠⠀⠀⠙⠳⣤⡘⠄⠀⡁⠀⣧⠀⣀⣀⣴⡏⡶⡿⠋⠁⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⠣⠀⣷⢄⣿⣴⣿⠟⠛⠁⠐⠁⠀⠀⠀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢉⡻⢿⡏⠁⠀⠀⡐⠀⠀⡀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠀⠈⢄⢰⡼⢤⢷⣿⡄⠀⡜⠀⣠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡈⢾⣧⣼⢦⡝⣿⣬⡴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠿⣟⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⠿⢛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⣊⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠓⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡨⠇⠒⠉⠒⢄⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠀⠄⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠓⠒⠒⠦⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⡀⠀⢀⡶⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⢠⠊⠀⠀⢀⠶⠀⢠⠄⣀⣀⣛⡓⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠀⣀⠀⠀⠀⠀⠀⠀⠈⠲⠏⣀⠤⠤⠬
⠀⠀⠀⣀⣀⠀⠀⠀⡠⠊⠀⢀⠤⢤⣄⢺⡀⠀⠀⠘⠷⢦⠆⠘⠷⠂⠀⠉⠉⠉⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⢀⡴⠋⠁⠀⠈⢑⠎⠀⠀⢰⠃⠀⠀⠈⠳⡉⠓⠒⠒⠒⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠈⠉
⡾⠀⠀⠛⠠⢀⠃⠀⠀⠀⠘⠷⢤⣄⣀⣆⠙⢀⠀⠀⠀⢿⢦⣀⡀⠀⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠃⠀⠀⠀⠀⢀⣠⠴⠂⠁
⠹⢤⣀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⡌⠠⠆⠀⠀⠑⠌⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣥⣴⣯⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⣀⡀⠈⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠲⠶⣤⣤⠤⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⠻⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢄⡀⠀⠈⠉⠒⠀⠀⠀⠀⣀⠔⠊⠁⠢⢄⡀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠉⠑⢢⣀⢀⠠⠴⠒⠉⠀⠀⠀⠀⡈⠀⠈⠉⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⠤⣤⣄⡄⢀⠀⠀⠀⢠⣴⣊⠁⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢅⣀⣀⣉⠀⠤⣉⡉⠓⠚⠋⠛⣿⠦⡀⠀⡖⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠉⠉⠉⠉⠑⠲⢤⠖⠉⠲⢼⣼⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⡏⠉⡟⠀⠙⠛⠂⠒⢲⡞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣦⣼⣆⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⢶⣾⠢⣤⣄⣀⣀⢀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣀⣀⣈⣻⣾⣿⠥⠴⣟⣿⠋⣙⣳⣶⣒⣒⣿⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠒⠒⠚⠋⠉⠀⠀⠉⠒⠬⢙⣛⠫⠄⣀⣀⣹⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠚⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢰⢄⢤⡆⠶⠠⠄⢠⠤⡤⠴⠄⠀⠀⠀⠆⠒⠐⠀⠂⠆⠒⠐

⢌⡟⠙⠹⠳⠤⠏⢹⣷⣞⣧⠀⠀⠀⠀⠀⠀⢀⡉⠁⠈⠹⠷⢦⣰⣠⡺⢿⡔⣷⢬⣜⠠⢤⣤⢶⣿⣿⣿
⠀⠀⡀⡂⠴⡀⠀⢀⢉⡙⠻⠀⠀⠀⠀⠀⠀⡄⠄⠀⠀⣘⣃⣞⣫⣉⠲⠄⠉⢻⠿⣿⣿⣾⡿⠒⢻⣿⢓
⡄⠀⣈⣴⣿⣿⣝⣿⣿⣿⢿⡦⡀⠀⠀⢲⣶⣶⡾⠗⢚⣯⣤⣹⣿⣮⣥⠂⠁⡄⠈⠛⢿⣿⣏⣵⣼⢯⡏
⠀⠀⣟⣻⡇⣽⣵⣫⣥⣾⣸⣯⡧⡄⠀⠀⣻⣿⢿⣧⣢⣿⣿⠟⣿⣿⣿⣿⣷⣳⣄⠀⠀⠉⠙⠛⠛⠟⠛
⠀⠠⢿⣏⡙⠏⠉⠉⠉⠙⠉⠉⠉⠉⠁⠀⠀⣿⣿⣿⣿⣿⣧⣤⣭⣿⣿⣿⣿⣿⣿⣷⣦⣴⣤⣄⡀⣀⣠
⠀⠀⠀⠈⠁⣤⠀⣤⢀⠀⢰⠑⠀⠀⠀⢄⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⡃⡀⠀⢠⠍⠁⢘⡗⡄⠀⡼⡓⠀⠀⢠⡤⣦⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢉⢁⣿⡟
⡑⡝⠀⠈⠀⠀⢀⣇⠹⡄⠀⢹⠀⠀⠀⠣⠽⣷⢀⠀⠀⠀⠈⠻⡿⡿⣿⣻⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⢷
⠇⡃⠀⠀⠀⣦⠜⣻⣷⡭⡟⢹⡂⠀⠀⣸⣾⣟⣀⠀⠀⠀⠀⠀⠘⡆⠼⠅⡂⠻⣗⣏⠿⠛⣻⢟⣿⣑⣠
⣿⡇⢀⡦⠀⢷⠘⣛⣿⣿⠃⢠⡽⠈⠄⠎⢻⡏⠡⡴⠀⠀⠀⠀⠀⢹⡂⠴⠀⠴⣏⠤⠄⠀⢀⠦⣿⣿⡫
⠄⢑⣸⠀⠀⠘⢿⣿⣿⡟⠀⢸⠉⢀⠀⢀⣾⣶⣠⡇⠀⠀⠀⠀⠀⢸⠟⠚⠀⠒⢉⡃⡛⠒⢫⣠⣾⡿⢝
⡦⢙⣍⡁⢠⣠⠘⠉⠅⢀⣴⣏⠀⠂⣈⣿⠻⣿⣿⡃⠀⠀⠀⠀⠀⡏⠃⢂⢨⠤⠄⠠⠀⠀⡄⠀⡀
⣶⣾⣷⢛⣏⣑⠾⠒⡹⠉⢹⣇⠿⢨⢿⡙⢦⣝⠻⠃⠀⠀⠀⠀⢠⠃⢄⠈⠿⡞⠲⢢⠔
⣷⣾⣿⣺⣷⣧⡄⠛⣇⠠⠙⠳⣾⣿⠈⡗⢠⣽⡟⠀⠀⠀⠀⢠⢇⠆⢎⢣⠐⣦⣄⣠⡄⠑⠢⣀⠉
⡉⢙⡿⠋⣨⣚⣷⢤⠌⠀⣀⣼⣵⣜⠇⣿⡌⢷⠧⠀⠀⠀⠀⠌⣼⡀⢻⡌⣦⡘⢿⣿⣿⣵⣤⡀⠈⠲⢤
⡿⠟⢿⡿⠿⠻⣿⣴⣾⢿⠀⠈⢹⣿⣦⣳⠱⣀⡗⠀⠀⠤⠀⠙⠁⠂⠀⠀⠀⠙⠮⡙⢍⠉⠛⠟⠢⣤⡀
⠄⠐⠋⠀⠀⠀⠀⠀⠹⠖⠀⣸⡌⠑⢶⠉⠓⠼⣽⣄⡀⠐⠀⣀⢴⡂⠀⠀⠀⠦⠀⠀⢀⡀⣂⠈⠉⠈⠛

⠀⠀⣏⣙⡿⢿⣿⡎⠛⠋⠉⠠⣄⠀⢀⡀⠿⣟⡛⣿⣿⣿⡿⠝⠛⠛⠓⠠⣟⣛⡻⠷⠶⠤⠤⠉⠉⠉⠉⠈⠉⠛⠗⠆
⠀⠀⠛⠛⢫⡙⢿⡤⢄⠀⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠀⠀⠀⠀⠀⠀⠉⢍⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠳⣷⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠶⠄⣠⣀⡐⠲⠦⠐⠂⠀⢀⣄⣀⣀⠀⠀⠙⠁⠉
⠀⠀⡀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠈⠀⠂⠠⣄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⡄
⠀⠀⠀⠀⣀⡉⠉⠻⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠸⠃⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⣙⣿⣿⣿⣦⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⣿⡿⠟⣉⡾⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣶⣾⣿⣼⣿⣿⣿⣦
⠀⠀⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⡀
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶
⠀⠀⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⣷⡠⠄⠀⠀⠀⠒⠂⠘⠤⠒⠀⢠⣼⣿⡯
⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣠⡀⢀⣴⠀⣾⣿⡃
⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣻⣿⣧⡉
⠀⠀⣷⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⣿⣴⣶⣾⣶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢢⣶⣶⠶⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⣿⣆⣄⣀⠀⠀⢙⠂⠒⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠙⠛⢿⣿⡧⠀⠘⠛⠂⠀⠀⠀⠀⠤⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠉⠿⣿⣿⣻⣿⣿
⠀⠀⡆⢀⣀⠰⠶⡄⠀⠀⠀⠀⠀⠐⢶⣶⢦⣄⣠⣤⣍⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠛⠋⠉⣠⡄⠀⠀⠀⣾⣿
⠀⠀⠿⠂⠴⣤⡀⠈⠘⠛⠓⠀⠀⠀⠀⢉⣲⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⣿⣿⢿⣿⣷⣾⣿⣿⣶⡞⠛⠛⢻⣿
⠀⠀⠲⠀⠀⠛⠛⢷⡶⢶⣤⡄⠲⠶⠶⠿⣷⣮⣿⣿⡿⠷⣭⠼⠟⠛⠻⠿⣿⣾⣿⣿⣷⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠐⢳⣶⣾⣦⣿⡷⠶⠄⢤⣤⣤⣴⣶⣶⡏⠉⠀⠀⠀⠐⠺⠛⠿⠿⢿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣀⡉⠉⠹⠿⣶⣅⡀⠀⢠⣤⡄⠠⢶⣾⣍⡉⠉⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⠛⠛⠋⠉⠉⠉⠉⠉⠉⠛⠉⠋⠉⠉⠛⠛⠛⠛⠉⠘⠋⠛⠛⠚⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⠙⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠿⠛⠛⠛⠛⠛⠛⠛⠛⠿⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠉⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⢀⣴⡶⠟⠛⠉⠉⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠐⠟⠀⠀⠙⠿⣦⡀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⣼⣏⣀⣀⣠⣴⡿⠂⠀⣤⡀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠁⠀⣠⣾⠟⢿⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⠿⠿⠿⠛⠋⠁⠀⠀⠙⢻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣤⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣶⣄⠈⠉⢀⣴⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⡀⠀⠙⣷⣶⡟⠁⠀⣹⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠉⠈⢿⣴⡾⠋⠉⠻⣦⣾⠏⠈⠉⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠁⠀⢀⣀⠀⠈⠁⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⣠⣾⠟⠁⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠈⠻⠋⠁⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⠶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⣀⠀⠀⠀⠀⢀⣠⣦⣀⣠⣤⣄⣤⣄⡴⠶⠆⠴⠰⠂⠀⣀⠂⠀⠄⠀⡀⠀⠠⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠈⣃⠀⣠⣾⣿⣿⣿⡟⠿⠆⠙⢹⠿⠂⠀⢁⠠⣀⢠⣰⣿⣷⣤⣶⠄⠀⠀⠈⠂⠀⠀⠉⠀⠀⠀⠀⠀⢠⡆⠀⠀⠈
⠀⢸⡲⠄⠐⠿⠿⢿⣿⣿⣟⠓⠆⠀⣣⣤⣤⣶⡴⠞⠛⢻⣿⠟⢛⣿⣿⠳⠤⠤⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣌⣷⣄⡀
⠀⢸⠂⣴⣀⠀⠶⣿⣿⣿⣿⣶⣶⣾⠛⠋⠉⢿⣥⡄⠀⠨⣿⣯⣾⣿⡭⠤⢶⣦⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡿⠏⠁⣹⣷⡶⡅⠄⠀⠀⠄
⠀⠸⡀⣿⣿⣿⣶⣿⣿⣿⣿⣻⡟⢁⢁⡴⡆⠀⣿⣥⠄⣠⣿⣿⡿⠋⢨⣵⣾⣿⣷⣼⣇⢀⣠⡀⠀⡀⠀⠀⠂⠠⢾⣭⢭⡙⢷⠀⣆⠀⠀⠀⠁⠠
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣿⡟⠃⠙⢦⣿⣿⣿⣿⣿⠟⠄⠘⠉⠉⠃⢀⣼⡟⣿⣟⠋⢁⠄⠀⣉⠀⠀⡄⢉⡳⠊⠌⢸⣦⠏⢛⡗⠂
⠀⢰⠀⣿⣿⣿⣿⡟⠦⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⡟⠛⡏⠀⠀⠀⠀⠀⠀⠿⠛⠷⠬⠉⠋⢠⡦⠀⠀⣟⢁⡈⢾⡏⠀⠀⠀⢹⠀⠁⢰⡥
⠀⢸⠀⡿⢩⡉⣿⡷⣾⣿⣿⣿⣿⡟⠠⡀⠀⣿⣿⣿⣷⢼⢃⠌⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠄⣍⠈⠳⠾⢢⡤⠠⠔⢨⣄⠀⠃
⠀⢸⠀⣇⣸⡴⠟⣠⣿⣯⡹⢸⣿⡇⢰⡵⣷⣿⣿⣟⠉⠘⣿⡴⢁⡄⠀⢀⣀⡄⠀⡾⢶⡖⠀⡜⢰⠄⠀⡏⣠⡌⢀⠀⠁⡀⢀⠀⠈⣴⠃
⠀⢸⠀⣿⡄⣴⣾⣿⠛⣿⣿⣛⠛⠛⢷⣹⡟⣿⣿⢣⢄⢨⣿⣦⣿⣧⠀⡸⢀⣿⣼⡇⣾⣏⠀⢐⠁⢣⣸⣿⠙⠀⢸⠀⠀⢿⡬⠧⠤⣿
⠀⢸⠀⣿⣿⡈⢿⣿⣷⣜⣻⠿⠀⠀⠉⣿⣩⠼⠛⠟⠀⠛⠻⠛⠛⠃⢠⠁⠺⢿⣿⠟⠛⠻⠁⠛⢠⠚⠉⢙⣷⣶⡌⠃⠀⢁⠀⢠⠄⡄
⠀⢈⠀⣿⡻⠁⣠⣿⣿⣿⣿⡇⠀⢄⣣⣽⠇⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⢀⡆⢀⡿⠀⢀⠶⠦⢐⡁⠀⠔⠋⢸⣿⣿⠷⠶⢋⡀⣀⢠⡇⠀⡀
⠀⢸⠀⡏⠉⢻⡿⣿⣿⣿⣿⡷⠀⢺⣿⠏⡀⡄⠀⠀⡀⠀⢠⡴⠛⢠⣂⣈⡇⢈⡃⣀⣀⣆⡀⠀⢠⣀⣀⣀⣸⣿⠀⠀⠀⠘⣿⡿⣮⣇⠀⠀⠂
⠀⢰⠄⡇⠀⣾⠁⢹⡿⣿⣿⡗⠐⣊⠀⠀⠐⡇⢠⠀⠐⠒⣾⡦⠄⠊⠉⠙⠛⠚⠛⠛⠁⠀⠀⠀⠈⠈⠁⢩⣿⡇⠀⠁⠀⠀⡟⠓⠘⠋⢠⡀
⠀⢸⠀⣿⠂⢿⣦⣾⣿⣿⣿⣿⣁⣿⣧⡀⢿⡰⣠⠘⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠈⣇⢀⠀⠀⠀⣧⠐⢧⡄⠀⠉
⠀⢸⠀⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⡧⢠⢢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡎⠀⠀⣟⢸⠀⠀⠀⣻⣶⣦⣽⣿⢗⠉
⠀⠍⠀⣻⣿⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠀⠄⡼⢛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠟⠙⡂⠀⢠⣿⣿⣿⣿⣿⣿⣷
⠀⣾⠀⣿⣿⣏⡛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⠞⢗⣫⣆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣀⡀⠚⠃⢀⠉⠉⠁⠏
⠠⠏⣠⠟⠉⠉⠁⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣯⣰⣿⣿⣯⣍⢃⠃⠀⠄⢠⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⠠⡇⠀⠀⣿⠙⠠⠀⠏⢈⡀⠀⣁
⠀⣤⠕⠢⠅⠄⠐⠄⢐⠄⠀⠀⠀⠉⢛⢿⣿⣿⣿⣿⣿⣻⡏⡑⢲⠀⠠⠉⡉⠉⠉⠉⠙⠛⠛⠿⡯⠉⣋⡸⠧⢤⣀⡟⠀⠀⠁⣨⣏⣁⣤⣬⡤
⠀⠘⠳⠶⢆⠖⢂⠤⠰⠢⡀⠀⡄⡁⡁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⡁⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⢀⣈⢉⣁⣰⣿⣿⣿⣿⣿⣿⡙
⠀⠛⠓⠀⠀⠈⠙⠋⠈⠉⠀⠘⠓⠓⠀⠀⠁⠚⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠦⠀⠀⠁⠒⠓⠚⠛⠛⠛⠛⠛⠒⠛⠙⠛⠛⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠼⠛⠀⠀⠀⠀⠀⠀⠀⠃⠐⠀⢦⠉⠂⠀⠀⠀⠀⠀⢍⡉⠀⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⣠⠇
⠀⠀⠒⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣀⡀⠀⡀
⠀⠈⠐⠂⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠤⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠀⠀⠂
⠀⠀⣀⠀⠛⣴⣶⠃⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠲⠤
⠀⠀⡃⠀⣚⠋⠱⠦⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⢱⠀⠁⠐⠷⡀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣦⠀⠀⠈⠀⠀⣀⣀⠀⠁⠀⠦⠀⠀⠀⠀⡠⠤
⠀⠀⠈⠁⠀⠀⠘⠱⣦⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⡟⢢⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣤⠀⣀⣾⣿⣿⡆⠀⠀⠀⠀⣸⣿⣷⣤
⠀⠀⠈⠁⠀⠀⢠⣷⣿⣇⠀⠀⣼⣿⣿⣿⣿⣿⢁⠿⡞⠙⠢⡀⠀⣠⣾⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢰⣿⣿⡿⠁
⠀⢐⣀⣤⣴⣾⣿⣿⣿⣿⣧⣸⣿⡿⠋⣽⣿⡇⣈⣰⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠻⣿⣿⣿⡇⠀⠀⢠⣿⣿⣿⣷⠀⠀⠀⠀⠀⡄
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢰⣿⣿⢁⠁⢾⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠈⠙⠛⠃⠀⠀⣾⣿⣿⣿⣿⣷⡄⠀⢀⣾⠄
⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⣿⣿⡏⠀⠀⡆⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠟⠏⠀⢀⣞⣿⡆
⠀⠀⠻⠿⠿⠿⠿⠿⠁⠀⠀⠄⢀⣀⣉⡙⠃⢹⡖⡃⡀⠀⠛⠛⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠀⠀⠀⢀⣾⡙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠀⠀⢀⡀⠀⠀⠄⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⠀⠀⢠⠛⡎⣷⣿⡇
⠀⠀⠀⠄⢀⡀⠀⠀⠀⠠⠂⠀⣊⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠇⠀⠀⣰⠁⢠⣹⣿⣿⡇
⠀⠀⠀⣀⡨⠷⠀⠚⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⢸⣿⣿⡿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠀⠀⣠⠛⢤⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠾⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠁⠀⠀⠀⠀⠈⠈⣝⣷⣄⡀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⣴⡓⢢⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣿⠀⠀⢀⠀⠆⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣿⣿⣷⣦⣄⡀⠀⣾⣿⣿⡇⠀⡸⡟⣿⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠉⠍⠋⠐⢢⡜⠧⠀⠀⠰⠶⢀⣀⠴⠟⢻⠉⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⡰⠁⢱⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡏⠀⢰⣠⣾⣇⣘⠃⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸⡄⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣰⣇⣥⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠿⠀⣾⣿⣿⣿⣿⣷⣶⣶⣶⣤⣤⣤⣤⣤⣶⠐⠚⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣀⢠⣿⡤⢿⣿⣿⣿⣿⡇⣼⣿⠿⠿⢿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣠⣾⣿⣛⡛⢀⣤⣾⣸⣿⣿⡇⣽⣿⣿⠀⢈⣿⣿⣿⣿⣿⡟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠉⠉⠉⠁⢰⣿⣿⣿⣿⣿⡇⡏⢿⡯⠀⣙⣿⣿⣿⣿⣿⠁⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⡇⠚⠁⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠛⢿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠟⠻⠿⠀⠃⠒⠃⠀⠛⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠻⠿⠿⠿⠛⠛⠞⠛⠿⠿⠻⠷⠟⠚⠻⠛⠃

⠀⠀⠀⣠⡷⠆⠀⠀⠀⠀⣶⠀⣰⡆⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠰⣦⣄⠀⠀⠀⠀⣲⠀⢀⣠⡄
⠀⠀⣸⣿⠙⠄⠀⠀⠀⢼⣿⡯⣿⡟⠆⠀⠀⠀⢞⣿⣇⠀⠀⠀⠀⠠⠛⣿⡇⠀⠀⢰⣿⠳⣿⠟⡄
⠀⠀⢹⣿⠀⠀⠀⠀⠀⠘⣿⠁⢿⡇⠀⠀⠀⠀⡾⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⢹⠀⣿

⠀⠀⠀⣽⣇⠀⠀⠀⠀⣼⣇⢀⣤⡗⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⢀⣟⣠⣽⣅⠀⠀⠀⢷⣄⡀⣹⡄
⠀⠀⢰⣿⣯⠃⠀⠀⢀⣿⠋⣿⣏⠱⠀⠀⠀⠘⢹⡿⠑⠀⠀⠀⢻⣿⠳⣿⡏⠃⠀⠀⠏⢻⡟⢿⣷
⠀⠀⠸⠿⠙⠂⠀⠀⠸⠙⠃⠻⠇⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠘⠟⠀⠿⠃⠀⠀⠀⠀⠸⠇⠸⠸
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢀⡀⠀⢀⡄⠀⠀⠀⣀⠀⠀⢀⡀
⠀⠀⢀⣴⡿⡁⠀⠀⠀⠀⣀⡶⢀⣾⣆⠀⠀⢀⣴⣷⣄⠀⠀⢀⣼⣧⣀⣴⣧⡄⠀⢠⣾⣄⣴⣾⡃
⠀⠀⢸⣿⠀⠁⠀⠀⢀⡼⠛⠋⠁⢸⢻⠀⠀⠈⢸⣿⠌⠀⠀⠁⢸⡟⠈⢸⣿⠈⠀⢸⢻⢸⣿⡇⠁
⠀⠀⠈⠛⠀⠀⠀⠀⠈⠉⠀⠀⠀⠘⠘⠀⠀⠀⠈⠉⠀⠀⠀⠀⠘⠃⠀⠘⠉⠀⠀⠘⠘⠀⠛
⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⣤⠀⢴⠄⠀⠀⠀⢠⣶⡀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣴
⠀⠀⢰⣿⡶⡄⠀⠀⠀⣴⣿⣞⣿⡷⠄⠀⠀⢀⠞⣿⣆⠀⠀⢸⣿⠦⢸⣧⣄⠀⢀⢼⣿⠦⡴⣿⡦⡀
⠀⠀⠀⠀⢸⢧⡀⠀⠀⠘⣿⠀⣿⡇⠀⠀⠀⠀⠀⣿⡟⠀⠀⢸⢻⠀⠘⠉⢙⣦⠀⠈⣿⠀⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢆⣀⣶⣴⣬⠶⢏⡨⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣯⠀⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣾⣧⣿⣇⣿⣿⣿⣿⣿⡿⡉⢉⡉⢡⡄⠀⠀⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⢀⡆⠰⣾⡇⣿⡿⣿⡄⣿⣿⡿⡿⠷⢿⡬⠘⠸⠀⠀⠀⠀⠀⡯⠠⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠂⣾⠁⢿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡁⠀⠦⠰⠀⢠⠀⠜⡧⢀⣿⡂
⠀⠀⠀⠀⠀⠀⢸⡇⠉⡯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⣤⣶⡄⠀⠀⠀⡏⢸⣿⣿⣤
⠀⠀⠀⠀⠀⢰⣸⡁⠠⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣰⣿⣿⣧⠀⠀⠴⡗⢺⣿⣿⣿
⠀⠀⠀⠀⢀⠀⢿⣿⣀⣃⣸⣿⣿⣿⣿⣿⡟⣟⡿⣿⣿⣹⣿⣿⣿⠿⠿⠧⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠃⣻⣿⣿⣯⣿⣿⣧⣾⣿⣿⣿⣿⣿⡿⢿⠿⡷⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣶⣿⣿⣿⣿⣿⡏⢀⣾⣿⣿⢻⠿⠿⠯⠽⢻⣿⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣴⣿⣿⣿⣿⣿⡟⠀⣼⣿⣿⣟⣼⣧⡀⠲⣤⢠⢤⡈⠉⠁⠀⠀⠀⠈⠛⢻⣿⣿⣿⣼⠄
⠀⣌⣿⣿⣿⣿⣿⡿⠀⣺⣿⣿⣿⣿⣿⣻⣷⡄⠹⣿⣆⢳⣌⠂⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣄
⢠⣿⣿⣿⣿⣿⣿⣡⣾⣿⣿⣿⡍⢱⣼⣿⣿⣻⣆⠈⢿⣴⣿⣦⠀⡀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣆⣀
⠘⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⠋⣿⣧⣀⣉⣉⣉⣁⠈⢂⠀⣀⣀⣀⣀⣹⣿⣿⣿⣿⡆
⠀⠀⠀⣈⣙⠛⠛⠿⠿⠿⠿⢿⣿⣭⣭⠉⢊⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠈⠀⠐⠂⠀⡀⠀⠀⠀⠉⠁⠉⣙⣛⣿⣍⣉⠉⣭⣭⡉⠁⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠶⠊⠙⠿⣿⣿⢿⠂⠛⠛⠃⠀⠀⠀⠀⠐⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠉⠉⠉⠀⠀⠂⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⣀⡀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠈⠁⠀⠀⢀⣀⣀⡀⠀⠁⠪⠀⠀⠀⠀⠉⠢⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡤⣒⣚⠉⠁⠒⠀⠤⠀⢀⠁⠈⠑⠂⠄⣀⠀⠀⠃⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠀⠀⠀⠀⠀⠉⠀⠐⠠⢀⡀⠈⠐⠠⢀⠀⠀⠉⠂⠬⡀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠹⡦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⡀⠈⠒⢄⡀⠀⠈⠐⠤⡀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⡀⠹⣄⠁⠂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⢈⠐⠤⡀⠀⠈⠑⠄⡀⢂
⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⢀⠃⠀⠈⢦⡀⠀⠈⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠥⠢⣄⣀⣈⠊⠄
⠀⠀⠀⠀⢀⠎⠀⠀⠀⢀⠌⠀⠀⠀⡠⠙⠲⣄⡀⠀⠀⠈⠐⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠛⢉⠊⠄
⠀⠀⠀⢀⠎⠀⠀⠀⡠⠊⠀⠀⢀⠔⠁⠀⠀⠀⠙⠡⢐⡀⠠⠀⡀⠀⠈⠁⠐⠒⠀⠀⠀⣠⠤⠴⠒⠀⣩⡵⠟⠐⠉⠅⠈⠢⡀⠀⡀
⠀⢀⠠⠂⠀⠀⡠⠊⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⠁⠤⠀⠀⠀⠬⠠⠠⠔⡂⠛⠂⠀⠀⠐⠂⠉⡰
⠐⠁⣤⡤⠒⠁⣀⠠⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢲⠙⠅⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠏⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⠀⠰⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢨⠐⠀⠀⠒⠉
⠀⠀⠌⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢡
⠀⠀⢄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠜
⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠄⣀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⣀⠀⠄⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⣊⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢔⣩⣤⣬⣛⠒⢀⣈⠳⡀
⠀⠀⠀⠀⢀⡤⠚⠒⠳⠏⠐⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⣶⡷⠞⠋⣟⡝⡗⡄⠓⡲⠘⢆
⠀⠀⡀⠀⠱⡩⣄⠀⡮⡵⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣣⠁⣄⡀⠀⠀⠘⣼⡘⣾⠈⣓⡶⠈⡆
⠀⡐⠁⠀⠀⢇⠟⠀⢉⡅⠀⢰⡲⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢤⠈⠱⠆⠀⠦⠬⠛⢻⠀⣩⠭⠆⢃
⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⡳⢙⡳⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡈⢾⡀⢄⠀⠀⢀⢀⠔⣄⡤⢩⠉⡂⡀
⣀⡇⢠⣒⠆⠀⠀⠀⠀⡀⠀⠀⣠⢣⠉⠀⠀⠀⠀⠀⡠⠒⣲⣶⣿⣯⢿⣛⢿⣮⣑⠂⠁⠈⠓⠒⣉⡹⡖⠲⠃
⠳⢛⡖⠶⡳⠰⠤⠤⠔⠁⡠⠂⢻⠁⠀⠀⠀⠀⠀⢾⣶⣞⠻⠣⢴⠦⢵⠒⠾⢷⣙⠛⣛⣭⠟⢯⠘⣽⡇
⠀⠘⠒⠲⣓⡦⠤⠔⠂⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠈⢹⠒⠒⠶⣖⡒⠊⠁⡀⠉⣿⣿⢧⡜⠀⢻⣷⣉⡆
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢸⣶⣾⣿⠀⠀⠀⠙⠻⢷⠁⠀⠀⡇⠛⠉⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠱⣄⣀⣀⣀⢤⣎⣀⡤⡤⣤⡀⠀⠀⢸⣷⠿⠛⠀⠀⠀⠀⠀⣼⣠⣴⡶⡇⠀⠀⠀⠀⠀⣘⣄
⠀⠀⠀⠀⠀⠀⢸⡿⠤⠔⣿⣇⣀⠵⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⢿⢟⣽⣿⡗⢲⣖⣶⠝⡿⠯⠝⢦
⠀⠀⠀⠀⠀⠀⢸⣀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢟⣉⠀⠉⠉⡉⢉⣉⣀⠤⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢒⡁⠈⠀⣀⠀⠉⠀⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠄⠔⠁⠀⠀⠀⠀⠑⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠠⢃⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠡⠀⣀⠠⠀⠀⠘⠄⠘⠃⠀⠀⣀⢪⡴⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣂⡀⠷⠘⣀⠀⠀⠀⢾⠀⣆⠀⠀⠀⢠⡙⢶⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠶⢆⠸⡆⠀⠀⠈⢦⠈⠀⠀⠀⠀⠁⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠨⡇⠀⠀⠈⠑⠆⠀⠀⠀⢠⡀⠰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⠀⠈⠀⠀⢠⠌⡄⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠀⠀⠀⠀⠆⣠⡄⡡⣘⠁⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⠀⠀⠀⠀⡀⠀⢸⠒⠉⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢢⡄⠠⠚⠀⠀⠀⠁⠀⠀⠀⠀⠠⠔⠋⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠀⠀⠀⡒⢌⠁⠀⢀⠄⠈⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡡⠚⢆⡠⠊⠀⠀⠈⠀⠀⢀⠌⡠⠊⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢐⣡⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡟⠉⣉⡁⠀⢐⣒⡃⠀⠠⠤⣤⢚⡦⠐⣂⢼⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠇⠀⠀⠀⠀⠁⠀⠀⠈⠙⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⡀⠀⢀⡀⢀⣀⣀⣤⣤⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣿⠿⠸⠛⠛⠛⠃⠛⠛⠛⠉⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣴⣶⣶⣶⡾⠟⠀⠀⠀⠀⠀⣀⣀⣀⣀⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠀⠀⠀⠂⠒
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⢀⣀⣰⣿⣭⡛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢿⣅⠀⠀⠈⠹⣆⠀⠀⠀⣾⠋⠉⡉⠀⠈⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡶⢶⡿⠀⠀⠀⠀⣿⠀⠀⠀⢻⡶⠛⠛⠳⠶⣤⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠼⣿⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠉⠀⠀⠀⠈⣷⣄⡀⠀⣠⡾⠃⠀⠀⠀⠘⣷⡀⠀⠀⣠⡟⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⣿⢈⣿⠟⠉⢿⡀⠀⠀⠀⠀⠈⠛⣿⠟⠋⠀⢠⡄⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣰⣿⠀⠀⠀⠀⠀⠀⢻⡿⠁⣀⠀⠘⣧⠀⠀⠀⠀⠀⣼⠏⠈⢷⡀⠈⢿⡄⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢹⡆⠀⠀⠀⣰⣏⠀⠀⠈⣷⠀⠘⣧⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡄⠀⢿⡄⠀⣴⠏⢻⡄⠀⠀⣿⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⠉⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣷⠀⠈⣷⡾⠃⣠⣾⡇⠀⠀⣿⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⠳⢾⡇⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⢸⣶⠶⢻⣧⡾⠋⢽⡀⠀⢰⡟⠳⣾⡏⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢰⡆⠀⠀⠀⣿⠀⠀⠿⣤⡼⠏⠀⠀⢸⡟⠛⠛⣷⠶⠛⢿⠛⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣿⣀⣀⣀⣀⣀⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢀⣀⣸⣇⣀⠀⢀⣿⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⣿⠉⣿⠉⠉⣹⠏⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣼⣇⣿⣄⣀⣉⣉⣉⣉⣉⣿⣌⣀⣉⣹⣧⣀⣀⣀⣀⣰⣟⣛⣉⣀⣿⣿⣛⣉⣀⣿⣄⣀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⡴⠛⠈⠨⠥⠦⠦⠮⠤⠍⠈⠳⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⡀⡄⠆⠀⠰⠒⠂⠂⠐⠒⠂⠆⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣿⡇⠀⠀⠀⡴⠫⢤⣤⡀⠀⢀⣠⡤⠴⡄⠀⠀⠀⠀⣿⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢦⣿⡇⠀⣰⣿⡆⠀⠉⠉⠁⠀⠈⠉⠉⠀⢘⣆⠀⠀⠘⣿⡖⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠙⢋⠙⠛⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣸⣿⠀⠀⢱⣿⣧⠀⠀⠀⣀⣀⡀⠀⠀⢀⣿⡎⠀⠀⠀⣿⣇⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡶⣰⣿⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢾⡇⠀⠀⢰⣿⢟⣧⢰⠟⠍⠉⠩⠛⣆⣾⣼⣗⠀⠀⠀⢸⡿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⡴⣻⢫⣿⣷⢶⣏⣶⡶⣠⣶⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⡿⣿⡙⢿⢆⣬⢿⢧⣰⡟⢋⣿⢘⠷⢦⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡿⠁⠛⠻⠟⠷⠛⠟⠛⠛⢿⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢡⡘⡠⡂⢻⣿⡿⣷⣷⢿⣼⣿⢿⠁⡦⠀⢁⣃⡀
⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠾⠚⠲⠞⠿⠾⢲⣻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣤⡄⣻⠁⠀⢠⡟⣿⣿⣏⣿⣿⡯⢻⡄⠱⠀⣮⠳⡝⠀⢀
⢴⡖⢂⣐⡈⢉⣉⠀⣀⠀⠈⢁⣈⠉⣉⣉⠉⣉⣀⡉⢉⡀⢀⣀⠒⣶⠄⠀⣼⠇⢸⣿⡯⠁⠀⢸⡤⢸⣽⡟⣽⠟⣿⡁⢹⢪⠁⣀⠀⡽⠆⢻⡇⢸⣷⡀
⢸⣿⣟⠛⢻⣿⢻⣷⣿⡇⣼⣿⢻⣷⣿⣿⡷⣿⣿⣽⣿⣷⣿⣿⣯⣿⠀⠈⣿⠀⣟⠜⠀⠸⠂⡈⢧⡀⢧⠀⠱⣇⡑⡴⢀⡈⡀⡈⠂⠀⡄⠞⡇⢸⣫⡇
⢸⣏⣻⣿⣟⣻⣿⣋⣿⣿⣏⣿⣿⣋⣿⣛⣷⣿⣻⣏⣳⣏⣻⡿⣿⣿⠀⠈⠿⠀⣛⠀⠀⠀⠀⠃⠸⠀⣾⣏⣀⢹⣮⡆⠀⡀⠐⠀⠀⠠⠁⠘⢿⠀⠏⠁
⠀⠙⠈⢹⡇⠀⠀⡯⣿⡷⢰⣯⣆⢰⡙⣻⣇⡿⣟⠀⠀⠀⣿⠀⠀⠉⠀⠀⠀⠘⠇⠀⠈⠠⠄⠀⠀⠀⠹⢏⣿⣚⣛⣏⠀⠃⠈⠀⠀⠆⠀⣤⢸⠀⡆
⠄⠀⠀⢸⡇⠀⠀⠿⠇⠀⠷⠚⢺⠎⠾⠯⠞⠯⠷⠀⠀⠀⣿⠀⠀⢠⠀⡀⡀⠀⣀⠀⠀⡀⠀⠀⠀⠀⣾⡟⡞⣿⣃⣿⡀⠰⠀⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠁⢀⠀⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⢸⢸⡆⢉⣀⡁⣰⠀⡇⠀⠀⠀⠀⢢⢌⠇⣱⡨⠽⠀⠰⠌⠀⠀⠀⡄⠀⣰⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠿⠼⡿⠿⠾⠿⠿⠷⠿⠸⠹⠿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣧⣶⣼⣿⠆⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⢀
⠀⠀⠀⠀⡄⡀⠀⠀⠀⣘⡀⠀⠀⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢰⠋⣼⣿⣿⡻⠃⠘⠄⠀⠀⠀⠀⢐⠐⣂⠂⣀⠀⡀⢂⠒⠒⡒⡐⠐⣀⠂⠒⣚⢺⢦
⠀⠀⠀⠀⠈⠛⠛⠛⠀⠻⠙⠃⠘⠛⠛⠁⠙⠛⠛⠃⠘⠋⠀⠀⠀⢸⣜⠛⣧⣿⡏⠀⠀⠀⠀⠀⠀⠀⠛⠉⠛⠛⠛⢀⠋⠛⠋⢛⡛⠛⠛⠛⠛⠂⠛⣸⣴
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠖⠳⠴⠖⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠂⣀⠀⠀⠀⠄⠀⠢⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠐⠊⠉⠀⠐⡠⡀⠀⠀⠀⢀⠠⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠤⠤⠐⠂⠀⠈⠒⠤⣁⡂⠄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠄⢆⣒⣒⠰⡂⢸⠯⢓⠫⠝⠉⠢⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠏⡆⠀⠀⠀⠨⠄⠤⠀⠀⠀⠀⠀⣴⣥⠉⠉⠉⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⡠⠊⠱⡀⠀⠊⠉⠉⠉⠉⠁⠀⠀⡰⠁⠀⠓⠤⣄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠉⠀⢀⠠⠛⠲⢤⡄⣀⣀⣀⠤⠐⠊⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠉⠀⢉⣠⠀⠀⠀⠀⠉⢓⣄⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠑⠦⠀⠀⠀⠀⠀⡀⠐⠉⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠠⠤⠤⠄⠒⠈⠁⠐⠂⠀⠈⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠒⠒⠒⠒⠂⠐⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠓⠙⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣭⠭⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠭⠕

⡕⣿⡘⠿⠿⣛⣾⣿⣶⣭⣭⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣛⣻⠟⠭⠭⠉⠉⠉⠉⠉⠛⠿⠛⡛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿
⡄⠸⢿⡿⣻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡻⡻⠽⣒⣭⣭⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⡙⠹⢿⣿⣿⣿⣿
⣝⣢⣴⣿⣿⣿⣿⡿⣿⣻⣿⣿⣿⣿⣿⣿⠟⢽⡻⠕⣪⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣌⡻⢿⣿
⣿⡿⣿⣿⣿⣿⡿⣽⢷⡿⣽⣿⣿⣿⢿⢖⣋⣵⣶⣿⣿⣿⣿⣿⠿⠟⠋⠉⠁⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿
⡟⣼⣿⣿⣿⡿⣽⡯⡾⣹⣿⢿⡯⣳⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡾⣱⣿⣿⣿⣿⣏⡾⣡⡯⣸⣴⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢰⡿⣫⣾⣿⣾⣽⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣷⣦⣤⣀⠀⠀⠀⠀⣀⣠⣴⣾⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢷⣯⢞⠋⣊⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⡀⠀⢉⡛⢿⣿⣿⣦⣄⣀⣴⣿⣿⡿⢋⡁⠀⢤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣉⣥⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣶⣶⠿⣿⣿⣿⣿⠟⠛⢻⣿⣿⣿⡟⢻⣷⡿⢻⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠭⠿⠽⠛⢻⣿⡿⠀⠀⠘⢿⠹⡛⠻⠯⠍⠛⠋⠉⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠁⠀⠀⠀⠀⢠⣾⣿⣧⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠙⠻⣿⣿⣿⣿⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠀⠀⠀⢲⣤⣄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣏⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠙⠻⠿⣶⣿⣿⣿⣿⣇⠀⠀⠀⠀⣠⣾⣟⠉⠀⠀⠀⠀⠀⢀⠉⠙⣷⣤⡀⠀⠀⠀⢀⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⣤⣤⣾⣿⡿⠿⣿⣿⣦⣤⣤⣴⠿⠿⠿⠛⢿⣿⣶⣶⣶⣿⣿⣿⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠯⢹⣿⣿⣿⣿⣿⣿⣿⣯⣤⣶⣤⣄⣠⣤⣦⣤⣤⣾⣿⣿⣿⣿⣿⣿⡟⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⣿⣿⡽⣿⣿⣿⣿⣿⣿⣿⣦⣉⣉⠛⢋⣉⣡⣴⣾⠟⠁⣻⣿⣿⣿⠏⣰⣯⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣏⣾⣧⡼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣾⣿⡿⠋⢀⣴⣿⣿⣿⢃⣀⣟⣻⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢲⠊⠙⣶⣿⣷⣾⣿⣿⣿⣿⣿⣿⡿⠛⠛⠋⠉⠁⠀⣠⣾⣿⣿⣿⣿⣾⣿⣿⡟⣛⣓⣒⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣼⢷⢻⢟⠟⣻⢿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣟⠿⣿⢿⡛⠻⣽⠏⢋⡷
⠀⠀⠀⠀⠀⠀⠘⠲⠸⠆⣺⢻⣡⣾⢿⣏⣒⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣽⡿⣿⣦⢛⣗⢨⣆⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣰⣿⣿⣿⢻⣟⠛⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢉⣿⣿⣿⣿⣧⠚⠛⣿⣻⣿⣿⣿⣿⡿⠿⠿⢶⣶⣤⣤⣀
⠀⠀⠀⠀⢀⣀⣤⣶⣿⡏⣉⣻⣼⣿⠹⡍⣲⠿⣿⣿⣿⣇⠘⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⣿⣆⠋⡽⢽⣿⠿⠟⢉⣿⣦⣄⣀⣠⣼⠿⠿
⣤⠴⠶⠟⠛⠋⠉⠉⠛⣬⢥⣤⣸⣬⣧⣿⣿⣦⣿⣿⣿⣿⡄⠸⣿⣿⣿⡟⢀⡿⣿⡏⢻⣿⣁⠿⣿⡧⣴⣼⣦⡽⢫⡿⠟⠉⠁
⣀⣀⣀⠀⠒⢶⣶⣶⣿⣿⡕⡒⠷⣏⡜⠋⣠⡾⣿⡃⢸⣿⡷⠀⢹⣿⣿⠃⠸⠀⠙⠧⣠⡞⠻⣦⣠⢻⣾⡻⠶⢊⣼⡿⢶⣦⣄⠀⣾⡿⠃
⢰⢿⣿⣿⣿⣿⣿⣿⡿⠙⡇⠭⠽⢿⣠⡾⠟⠁⠸⣿⠹⠋⠀⠀⢸⣿⣿⡄⠀⠀⠀⣸⠟⠀⠀⠙⢿⣄⡼⢿⠋⠅⢻⣿⠀⡈⣁⣼⣿⠃