Anya heh face Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢻⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠐⠒⠒⠐⠂⠀⠀⠀⠀⠰⠶⠶⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠶⠶⠤⣄⠠⠤⠒⠒⠺⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⣀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣬
⡅⣈⡓⢴⢮⠠⣴⣥⢄⣀⡀⠀⠀⠀⢰⣤⣹⣦⣄⣀⣀⣠⣤⣤⣀⣀⣤⣤⣶⣿⣿⣇⠀⠀⠀⠀⡀⢀⠚⣯⢯
⠈⠡⠑⡳⢤⢯⣽⣿⣿⣿⣾⡾⠀⣤⣿⣿⣿⣿⣿⣿⡏⣿⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⡄⡈⡺⡾⠣⠁⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠀⠀⢠⣅⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⣤⣶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⢁⣾⡿⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⢁⣼⡿⠁⠀⢀⠂⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡙⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⢀⣾⡟⠁⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⡞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⠙⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣁⣠⣿⡟⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢻⠀⠀⠀⢠⠁⢀⠀⠀⠀⠀⡔⠁⠀⠀⡄⠠⢺⠁⠀⠀⠀⠀⢠⡇⠀⢀⡄⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠰⠀⠀⠀⠀⡈⠀⠈⠀⢀⡠⠈⠀⠁⠀⢸⠗⢁⠆⠀⠀⢀⠠⠊⢠⠆⡠⠆⠈⠀⠠⡀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡆⠀⠀⠀⢀⡇⠀⢠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠐⠂⠀⠀⠀⠛⠁⠀⠀⠀⠀⠢⣷⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡘⠃⠀⠀⠀⢸⠏⠢⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢢⡀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠃⠀⠀⠀⠀⢸⠀⢀⡤⠴⠚⢻⣿⣿⡶⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⣶⣶⣦⡤⣄⠀⠀⡇⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠸⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠠⠄⠉⠉⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠽⠛⠋⠠⠌⠛⠀⣰⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡇⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⠚⢁⡟⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠇⠀⣶⡀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢨⡿⠿⣿⣷⣶⣶⣶⣶⣶⣶⡞⠋⠉⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠐⠀⣧⠱⣄⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡇⠀⠀⠉⠙⠻⠿⠛⠉⢹⣇⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠳⣼⡀⠈⠳⢤⡀⠀⠈⠒⢄⠀⠀⠀⠀⠀⠀⠀⢷⠈⠐⠄⡀⠀⠀⠀⠀⠀⣸⠌⠐⠌⠀⣀⠔⡟⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠉⠓⠶⠤⣤⣈⣴⣤⣀⣀⠀⠀⠈⠀⠀⠀⠈⠑⣠⣴⠊⠁⠀⠀⠀⠈⢣⡾⣿⡇⠀⠀⢀⠠⠔⣁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣧⠈⠁⠒⠁⠀⠀⣠⡾⣯⣿⣷⣄⠀⠀⠀⠀⢸⣿⣦⡉⠈⠉⠁⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⢷⣄⠀⣀⡠⠞⠛⣽⣟⣿⢇⣜⠳⢤⣀⠴⢏⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⠀⠀⢀⣾⣿⣿⡍⢿⣿⣷⣄⠀⠀⠈⢸⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⡤⠀⠀⢛⣛⣛⣛⣃⣛⣛⣛⡛⠧⠤⠤⠇⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣥⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⣇⣼⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢀⠜⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡦⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⡖⠁⠀⠀⠀⡤⠀⠀⢠⠃⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⠉⣣⡀
⣤⣤⣤⣤⣤⣤⡽⠋⠀⠀⠀⢀⠎⠀⠀⠀⠃⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡉⣷⣾
⣿⣿⣿⣿⣿⡜⠁⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡉⣿
⣿⣿⣿⣏⡿⠁⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡌
⣿⣿⡏⣼⠃⠀⠀⠀⠀⡜⠀⠀⣀⠀⠀⠀⠀⠀⢀⠆⢰⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠘⣷
⣿⣟⡹⡇⠀⠀⠀⠀⠀⠁⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠘
⠘⣷⠃⡇⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠀⠘⡆⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢱
⢀⡆⢸⠁⠀⠀⠀⡈⠀⠀⠠⠈⢄⠀⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⠀⠱⡀⠀⠀⠈⠀⠀⠠⠀⠀⠀⢸⡄⠀⢠⡀⠀⠀⠘⡆
⡌⠀⠘⡀⠀⠀⢀⠇⠀⠂⢠⠀⠀⠁⢀⠀⠂⠀⠀⠀⠀⠀⠈⠢⣀⠀⠙⢤⡀⢀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠨⠃⢀⠀⠀⢧
⠀⠀⠀⡇⠀⠀⢨⠔⠁⠀⠈⠂⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠐⠛⠂⠀⠀⠀⠀⠀⠀⠐⠀⠁⠒⠀⠑⠀⣻
⠀⠘⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠆⠘⣄⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠐⠀⢱⠀⠀⠀⠉⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠀⠀⠀⠀⠢⠀⡇
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠈⠄⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠊⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⡟⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢶⣶⣶⣾⣿⣿⣿⣿⣿⡿⠈
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡿⠈⠙⠿⠿⣿⣿⣿⡿⠿⠛⠋⠀⠈⠙⢿⣿⣿⣿⣿⣿⡟⠁
⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠙⢿⣿⣿⠟
⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡎⢷⣶⣤
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⣿⠃
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢷⠃
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢢⡀
⣿⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠉⠂
⣿⡇⠀⠀⠀⠀⡀⠀⢹⡄⠀⠀⠀⠈⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠈⠁
⣿⡇⠀⠀⠀⠀⠃⠀⠘⣯⡀
⣿⡇⠀⠀⢀⣠⠀⠀⠀⢻⣇
⣿⡇⠀⠀⣸⣿⡀⠀⠀⠈⣿⣷⡀⠀⠀⠦⣄⣀⣀⠀⠀⠀⠀⢀⣀⣀⢤
⣿⡇⠀⠀⣿⣿⣿⣷⣿⣷⣿⣿⣿⣷⣄⡀⠀⡈⡟⠉⠉⠉⠉⢻⠄⠀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀
⣀⣀⣤⣤⣤⢄⡰⠖⣠⠞⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣭⠉⢱⣖
⣿⣿⣿⣿⡿⠛⢁⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠈
⣿⣿⣿⠟⠁⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⢻⣿⡟⠂⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠘⣿⠆⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠿⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⢀⡎⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠰⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠃⢡
⠀⣼⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⢇⠀⠀⠀⠀⣇⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢠⠀⠄⢡⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠠
⢰⠋⠀⠀⠀⠀⠀⠀⠁⣰⣆⢸⠀⠀⠄⠀⠀⡅⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠘⠀⠀⠀⠐⠄⠀⠀⢳⠀⠘⠣⡀⠀⠀⡀
⠃⠀⠀⠀⠀⠀⠀⢸⠜⠁⠀⠘⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠐⠐⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠈⠢⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⣶⣶⣶⣶⣶⠂⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢴⣶⣶⣶⣶⡤⠤⣀⡀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⢀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠉⠉⠉⠛⠒⠒⠛⠳⠦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠉⠛⠶⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠶⠚⢫⠏⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠆⠀⣰⠃⢈⣻⡿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⡾⣿⡹⡉⠁⠀⢠⠀⢠⡾⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⠃⢀⣺⠟⡇⠀⠉⠛⠻⠿⠿⠿⠛⠋⠉⠀⣽⠑⣧⡠⠀⠈⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢀⠔⢁⣼⠟⠁⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡂⠈⠑⢥⡠⠋
⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣥⣾⡟⠁⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⡰⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠔⠈⠉⠁⠁⠉⠈⠐⢤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠔⠉⠋⠙⠉⠉⠑⠠⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠤⢀⠾⢰⢂⡀⠀⢀⠀⢻⡿⠀⠀⠀⠀⠀⢀⣠⠀⣀⠀⡀⠀⠸⡏⠀⠀⢠⡎⡴⣀⠀⠀⡀⠙⣿⠏
⠀⠀⠀⠀⠀⠈⠀⠀⢀⢿⢰⠞⠺⠀⠐⠼⠑⠿⠯⣶⠀⡅⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠰⡇⣦⠦⠟⠐⢧⡏⠲⣿⣼⡴⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⣆⠀⠁⠀⠀⠀⠀⠀⠀⡾⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢻⡇⠒⠒⠂⡀⠐⠒⠂⢈⡇⠀⡂
⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⡅⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢰⠔⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠀⢻⣶⡶⠶⠶⢶⣶⠟⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠘⣷⣶⡦⣤⣤⣴⡶⠋⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⣸⠋⠀⣰⡆⢀⠃⠀⣠⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⡿⠁⠀⣶⠀⠹⠁⢀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠘⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠑⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⢀⣠⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⠂⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠙⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⢃⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠸⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⣾⡿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣠⢾⣿⣿⡦⠀⠀⠀⠀⠀⠰⣿⢿⣿⠻⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠁⠀⠁⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠰⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠚⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡚⣶⠦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣶⣌⠙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢉⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⠿⡄
⠀⠀⠀⠀⠀⠀⠀⣴⠿⣴⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⡆⢻⡄
⠀⠀⠀⠀⠀⠀⢰⢏⣶⣿⣿⣯⣉⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠶⡤⢤⡀⠇⠀⢀⡀⠀⠀⠀⠈⠀⣤⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠛⣷
⠀⠀⠀⠀⠀⠀⡞⣻⣿⣿⣿⣦⣬⣄⣀⠀⠀⠀⣀⠉⠀⠀⠀⠀⢀⡃⢠⡆⠈⣅⠸⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⢰⢡⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡀⠀⣀⣀⣀⣬⠥⠤⠀⠀⢹⣤⠧⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣇
⠀⠀⠀⠀⠀⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡆⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠂⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⡄
⠀⠀⠀⠀⣰⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣇
⠀⠀⠀⢠⣏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⢿
⠀⠀⠀⣾⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣟⣬⣿⣿⣿⣿⣿⣤⣾⣤⣿⣿⣿⣿⣧⢸⡆
⠀⠀⢸⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⣿⣿⣿⣿⣿⣿⢿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⡀
⠀⢠⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠻⣇
⠀⣼⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠂⢊⠀⢫⣟⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢻⣿⣷⡔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⢈⣶⣿⣿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣀⣲⡶⠒⠒⠒⠒⠀⠙⢻⠇⠀⠀⠀⠀⠀⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⢠⡀⠀⠀⠀⠀⠹⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⢀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⠀⠀⠀⠀⠀⠹⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⣼⠀⡸⢠⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠨⠛⣹⣿⡆⡄⠀⠁⠇⠀⣰⣿⠃⣀⣀⡀⠀⠇⢸⠇⠀⢰⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⡰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠀⠘⣿⣆⠐⣯⡃⠀⢠⣤⢾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣾⡅⠤⠈⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⣷⣬⠀⢀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠬⠄⢀⡀⠀⠀⠀⠔⠀⢀⣿⣿⢰⠃⠏⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⡀⠠⢤⠀⠀⠀⠀⠀⠀⠀⣠⣿⡷⠃⡸⠀⠀⡀⠈⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢋⠿⠇⠀⠀⠀⠀⠀⢀⣠⣴⠾⠿⠋⠠⠞⣁⠀⣰⠁⢀⠄⠁⠂⠀⠸⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡟⢻⡟⠫⠆⠀⢀⠉⠀⠀⠊⠁⠊⠑⠀⠈⠜⡸⢁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠁⣰⣶⠖⠂⠉⢀⡀⠀⠀⠀⢅⠋⠎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠹⠋⠀⠀⠀⠀⢀⡠⠋⠁⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠔⠋⠀⠀⠀⡀⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠉⠁⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣤⠔⠓⢆⡀⢐⣦⣠⠼⠷⡖⠀⠀⢀⠠⡾⠀⣤⣤⣤⣭⣥⣀⣀⡀⠀⠀⠘⠿⠃
⠶⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⠶⢶⣮⣷⢶⡚⠙⢯⣀⣈⣶⣤⠀⠀⠀⠀⠀⢱⠀⢒⣒⣒⣿⣿⣿⠿⠇
⠀⠈⢻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢻⣧⠟⢾⡉⠉⢳⣄⡈⠀⠀⠠⠄⠀⠀⠀⢴⠚⠛⠩⠉
⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠺⠗⠀⠀⠀⠀⠀⣿⠶⠖⠛⠛⠩⡈⠀⠐⡀⠀⠀⠀⠀⠀⢸⣆⠀⠀⣶⢆⣶⠁⢀⣀⠄
⠀⠀⠀⠀⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠰⠷⠀⠀⠰⡦⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⣠⣧⠀⠀⠀⠀⠀⢸⣿⡆⢰⣿⢸⠇⣴⣿⣿⣿⣦⢠⡀
⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⡀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡭⠋⠀⠀⠀⠀⠀⢸⣿⡇⠀⢿⣾⡄⢿⣿⣿⣿⡿⢰⡿
⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣾⢿⣁⠀⠀⠙⠿⢦⣽⡿⣿⡁⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠀⠀⠀⠈⠓⠛⠋⠁⠀⠀⠀⠀⠀⠀⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠄⠀⠀⠀⠀⢸⣧⠀⠀⣀⣠⣤⣤⣤⣤⣤⡶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠿⠛⢷⣦⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠿⠛⠻⠷⢶⣤⡀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⢹⡇⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⣀⣠⡾⠃⠀⠀⠀⣰⡟⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⠀⠘⣷⡀⠀⠀⠀⠻⣦⣀⣀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠀⠀⠀⠀⠀⣿⠁⠠⣿⣉⣻⡇⠀⠀⣿⣋⣹⡧⠀⢹⡇⠀⠀⠀⠀⠈⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢻⣦⠀⠀⣿⡄⠀⠙⠻⠛⠁⠀⠀⠙⠻⠛⠁⠀⣼⡇⠀⢀⣾⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠸⣷⡀⠀⠀⣤⡀⠀⠀⣠⡄⠀⠀⣰⡿⠀⠀⠀⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡀⠈⠛⠛⠛⠋⠀⣠⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⣿⠓⠀⠉⠛⠿⠶⠶⠶⠾⠟⠋⠀⠐⢻⡿⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠿⠀⠀⠀⣤⣀⠀⠀⠀⢀⣤⡄⠀⠀⠸⠇⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢀⣤⣤⣤⣤⣤⣬⣽⣷⣦⣶⣿⣭⣤⣤⣤⣤⣤⣄⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢸⣿⣦⣼⠿⣧⣤⡿⢿⣤⣼⠿⣧⣤⡿⢿⣤⣾⣿⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣿⡿⠷⠶⠾⠿⠶⠶⠿⠷⠶⠾⠿⠶⠶⠿⣿⡟⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇⠀⠀⢀⣴⡾⢻⣷⠟⠻⣶⣄⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇⣠⡾⠟⠁⠀⢘⣿⣀⠀⠀⠙⠿⣦⡀⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣿⠿⠋⠀⠀⣀⣴⠟⠋⠛⢷⣄⡀⠀⠈⠻⢿⣇⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠺⠛⠁⠀⢀⣤⡾⠛⠁⠀⠀⠀⠀⠙⠻⣦⣄⠀⠀⠙⠻⠂⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⣴⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢷⣤⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣾⣷⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣿⣶⣶⣶⠆





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

⣽⣿⣿⣍
⣿⣿⣿⣛⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢧⡀
⣿⣿⣿⣿⣿⠻⠿⣦⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣼⣿⣮⢻⡤⠦⠶⣤⡤⣤⣀
⣿⣿⣟⠽⠿⡵⢀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣛⡋⢀⣊⡿⣿⡿⢃⢂⡷⣿⠆⠠⡀⠈⠑⠦⡀
⣿⣿⣿⣷⣶⣃⠌⠹⠦⠤⡄⠀⠀⠀⠀⢀⣴⠿⢍⠁⠁⢀⣀⠛⢑⠈⠑⠈⠈⠀⠁⠤⠀⡁⠀⠀⠀⠉⢤⡀
⣭⣭⣥⠈⠙⠛⠳⠶⠄⠀⣀⣤⣶⣶⢿⣟⠁⠀⠀⠈⢠⠟⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⢦⡀
⣿⣿⣿⡝⠂⢀⢀⣦⣴⣿⣿⣿⣿⣿⠟⢸⡄⠀⠄⣰⠃⢠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠁⠀⠀⠀⣸⠁⠐⢶⣤⣀
⣿⣿⣿⣿⣇⣤⠙⣿⣿⣿⣿⣿⣿⡏⠀⠸⢟⡇⣼⠃⡜⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⣽⣦⠀⠿⣿⡶⡟⡆⣄⡀
⣿⣿⣿⣿⣿⣏⡷⡽⣿⣿⣿⣿⣿⣀⠺⠶⠛⣱⢇⣾⠓⣿⠇⠐⡀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢧⠀⢀⠀⢸⣿⡇⢹⠿⣹⣿⣿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⡀⠘⣿⣿⡟⠈⠀⠀⠀⢠⣿⢸⡟⠀⢿⡆⡄⠉⠄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠆⡘⣿⣿⣟⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⠹⠉⠡⢶⢨⣿⠀⠀⢠⠇⠀⣾⣇⣸⣿⣨⣼⣷⣯⢀⡁⠀⠀⢸⡆⠀⢠⡄⠀⠀⠀⢈⡇⠀⠀⠀⠈⣿⣿⣮⡿⠃
⣿⣿⣿⣿⣿⡐⠒⠀⢀⣾⣇⠀⠀⡾⠀⢰⣿⣿⡿⢻⣷⣿⡟⢿⣆⠹⣆⡀⠘⣿⡀⢸⣷⡀⡆⢠⠿⠇⠀⠀⠀⡧⢸⡻⢿⡁
⣿⣿⣿⠉⠻⠷⠊⢉⣾⣿⠀⠀⢀⡇⠀⣼⣟⠋⠀⠀⠙⠛⠁⠀⠈⠳⠽⠿⠤⣿⠷⣼⡍⢻⣧⣎⣸⠃⠀⠀⢰⠃⠈⣇⠈⠇
⣿⣿⣿⣿⠀⠘⠂⣼⣿⡇⠀⡀⢸⡇⢠⣿⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠘⣿⠀⠀⠀⠘⠀⠀⢹⡀⠸
⣿⣿⠿⠋⠀⠀⣸⣟⣿⡶⠀⠃⢸⡇⢸⣇⣤⣶⣿⣷⣶⣬⣀⠀⠀⠀⠀⠀⣰⣿⣶⣷⣶⣶⢦⣄⡸⠀⠀⠀⠀⠀⠀⠸⡇⢸⡀
⠋⠁⠀⠀⠀⢠⡿⣿⣏⡾⣁⠀⠘⣷⢺⣛⣍⣩⢉⠉⢉⠉⠉⠁⣀⠀⠀⠈⠉⡍⢭⣉⠉⠉⠉⠙⠃⠀⡰⠂⠀⠀⠄⠀⡟⢿⡇
⠀⠀⠀⠀⢠⣿⣻⣿⡻⣷⣱⡀⠄⢹⣿⡇⠈⠋⠈⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡼⠃⠀⠀⠈⠀⠀⣧⠈⢳⡄
⣄⣀⠀⠀⣼⣿⣿⣿⣧⣿⣿⣧⠆⠀⢹⣿⠀⠀⠀⠀⣤⣄⣀⣀⣀⣀⣀⣤⣤⣴⠀⠀⠀⠀⢠⣿⠓⠀⢀⠄⠀⠀⠀⠀⣿⠀⠐⣧
⣿⣿⣿⣀⣿⣧⣿⣿⡉⢳⣌⣿⣿⣄⢠⢻⣧⡀⠀⠀⠀⠙⠿⢭⣭⣭⡭⠴⠊⠀⠀⠀⠀⣠⡿⠁⣤⠖⠁⠀⠀⠀⠀⠀⡇⠀⢀⠸⡀
⣿⣿⠻⠋⢻⣿⣿⣿⣷⣎⢀⣇⡬⢛⢷⠁⠹⣿⣷⣶⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣶⣿⢋⡤⠂⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⣿⢰⡇
⣿⣿⣷⣶⢸⣿⣿⣿⣿⣿⣯⣌⡐⠂⠐⠀⠀⢿⣿⣿⣿⣿⡛⠻⠿⠿⠿⠛⣋⣠⠿⣿⣿⠋⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠀⢠⣿⢠⡇
⣿⣿⣯⡇⠈⣿⣿⣿⣟⣿⣿⣿⣧⡄⠀⠀⠀⠀⢿⣿⣿⣭⣻⣾⣵⢅⣴⢾⡛⢀⣰⡿⡋⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡷⣁⣾⣿⢸⠁
⣿⣿⣿⣟⣳⣎⠿⣿⣯⢌⢿⣿⣿⣿⣷⣤⠀⠀⢸⡇⢉⠻⣿⣿⣿⣼⡿⣽⠞⠉⢸⡇⣦⡀⠀⣀⣤⣶⣿⣿⣿⣿⣿⢥⡿⠙⣿⠏

⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⢀⠠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⡀⠀⢀⡀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠢⢤⠀⢠⡠⠀⠀⣠⢀⡲⠶⠔⡉⢅⠉⠀⠀⢭⡉⠐⠦⡀⠀⠀⠀⠀⠨⠤⢉⡀⠄⠠
⠀⠀⠀⠀⠀⠠⠔⠀⠐⠀⠼⡋⠉⢹⠈⠑⠐⣀⠤⣶⣧⠴⠄⠊⢠⡦⢆⡷⠀⠽⠀⠀⡔⣩⣉⢀⡀⠴⣾⠇⣐⠀⠀⠀⠀⠤⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢄⠙⠟⠛⣆⣉⡔⣀⡠⣔⣎⠰⠤⠚⠁⠀⠀⠀⡂⠉⣁⠉⣠⣤⣀⠅⠉⣠⠒⠷⠦⠊⡶⠀⠒⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠘⢆⠤⣄⡀⢌⡨⢵⠀⠌⢰⡀⢠⠰⠞⠁⠠⢁⣄⡘⠿⠈⡃⣠⠈⡛⢲⣋⡴⢣⠇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠐⠈⠔⠂⡀⠰⣄⣈⠐⠀⡤⠃⠤⣈⣧⡀⠧⠭⠀⣀⣤⠃⠀⠉⠉⢉⠔⠁⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⠀⠀⠢⢄⡄⠀⠐⠃⠀⠎⠙⠛⠛⣛⠉⡑⣲⠐⢀⣣⠀⠀⠀⡀⡄⡀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠀⠀⠐⢠⡠⠖⠀⠴⠆⣤⣤⣠⣤⠅⡥⢨⠟⠀⠀⠀⡆⣆⡗⢡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣄⠒⠒⠂⡠⢀⣐⡐⠲⠔⠒⠚⢖⠦⠚⠃⠀⠀⠀⠃⠓⠋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⢀⠀⠀⠀⠀⠀⠈⠒⠀⠄⠀⠀⠀⠀⠀⠐⡀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠍⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⡀⠀⠀⠀⠀⠈⣀⠤⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠐⢉⠁⠀⠀⠀⠈⡒⠰⢂⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠁⠀⠀⠂⠄⢀⠀⠀⠀⡀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⡾⠋⠉⠉⠉⠉⠻⣿⣿⣿⣇⠀⠀⠀⠀⣾⡟⠋⠉⠉⠉⠙⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠈⠻⢿⡿⠛⢿⣿⠟⠛⠻⠿⣿⠀⠀⠀⠀⠀⠠⠴⠄⠠⠦⠤⠀⠸⣿⣿⣿⣧⠀⠀⠀⠻⣤⣤⠀⣤⣤⠀⠀⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠘⠃⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠙⠀⠀⠙⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠘⠃⠀⠘⠃⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣾⣿⣿⣿⣷⣶⣶⡄
⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣆⠀⠀⠀⠀⢸⣿⠿⠛⠛⠛⠛⠛⣿⣿⣷⠀⠀⠀⠀⢈⣿⡿⠟⠛⠛⠛⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣯⣤⡄⠀⣤⣄⠀⢻⣿⣿⠀⠀⠀⠀⠈⢃⣀⣀⠀⣀⣀⠀⢻⣿⣿⠀⠀⠀⠀⠸⣿⠤⠴⠄⠠⠤⡤⠀⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠀⠀⠀⠀⠀⣀⣀⡴⠶⢶⣄⣀⣀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣠⠖⢲⠔⠊⢹⠀⠀⠀⠀⠐⠉⠁
⠀⠀⠀⢀⣀⢀⡜⠁⡰⠃⠀⠀⢸⠀⣠⠀⡰⠊⠉⠲⣄
⣴⡖⠤⠋⠈⠋⠀⠰⡁⠀⠀⠀⣸⡤⠄⠠⢇⡀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡀
⠀⠓⠊⠉⡆⢀⡠⠤⣈⢧⠀⢼⡡⠤⠄⠒⢶⣸⢶⡖⣒⡇⠀⣀⣀⠤⠤⠤⠔⠲⠤⢄⡀⠀⠀⠀⠀⢾⡵⠒⣷⣄⣤⣄⣀⡀
⠀⠀⠀⠀⠉⠁⠀⣀⣈⣯⣷⠀⠉⠓⠒⠚⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡆⠸⡅
⠀⠀⠀⠀⠀⢀⣸⡁⠠⠶⠮⠁⠀⠀⢀⡀⠀⠀⠀⠲⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡔⠛⢤⠀⠀⡟⠲⡁
⠀⠀⠀⠀⠀⠈⢧⢈⣲⠶⠒⢢⠀⠈⠙⡧⣄⠀⠀⠀⠀⠉⠂⠄⢀⡀⠀⠄⠀⢀⣀⡴⠋⠀⠀⠀⠀⠀⠀⢱⠸⠊⠀⠀⠈⡯⣉⣱
⠀⠀⠀⠀⠀⢀⠜⠉⠀⠀⢀⡜⢣⠀⠀⠹⡜⢳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢯⠤⢤⣤⣀⠠⠄⣶⠤⠒⠁⣀⠷⠖⠒⠚⠁
⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠑⢤⣣⣀⡀⠈⠃⠀⠙⠂⠀⠀⣀⣀⣀⣀⡤⢄⣈⢻⠉⠓⠤⢀⠀⡇⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⡝⠀⠙⠢⠤⣄⣈⣉⠉⠉⠉⠁⠀⠀⡆⠀⠀⠉⠳⠝⠂⠀⠃⠉⠛⣢⣄⡀⠀⢀⣀⢠⣤⠤⠒⠉⢉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠒⠂⠃⠀⠀⠀⠀⠀⠀⢀⣠⠒⠉⠀⠤⠋⠁⠀⡠⠊⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠥⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠒⠌⡒⢄⠄⠀⢀⡞⠁⠀⣀⠄⣠⣞⠾⠶⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⡛⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⢀⡤⠞⠛⠂⠤⢾⣴⠊⢉⣼⡶⠋⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⢦⣕⡢⢄⡀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠉⠑⠻⣁⠀⣀⡠⠤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡇⠀⠉⠓⠮⣕⣢⢄⣠⢎⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⡼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢰⠃⠀⠀⢀⣀⠜⣹⠙⠓⠦⠭⣕⣢⣤⣄⣀⣀⣠⡯⠛⠉⡷⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠸⠤⠖⣊⠭⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢩⢹⠀⠀⠀⡰⠁⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠦⠒⣹⠁⠀⠀⣘⠤⠢⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠁⠻⠔⣒⡺⠅⢴⣉⠦⢶⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⠋⠉⠀⠐⢁⠀⠀⠉⠙⠛⠷⢶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣀⡀⣀⣤⡶⠞⠛⠉⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠿⠿⠿⠿⠿⢿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠐⠀⢡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣍⣁⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣦⣔
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⠟⠋⠀⠁⠀⠂⠀⠀⠁⠀⠀⢀⠀⠉⠛⢿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⣿⣿⣿⠟⠁⠀⣜⠐⢶⣶⡄⠀⠀⠰⠿⠛⠀⠇⠀⠀⠀⠈⠛⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣤⣤⣤⡀⠀⠰⣾⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢻⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣠⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠇⠀⠄⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠈⣷⣉⠛⣛⣿⣿⠿⠱⠿⣿⣍⣉⣤⣴⣶⣶⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⠠⠄⢻⣿⣿⡟⠀⠀⠀⠀⠘⣿⣿⣿⠏⢻⣿⣿⣿⣿⡧⠀⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠈⢿⠿⠷⠀⠀⠀⠀⠐⠛⠛⠿⠀⠘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠃⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠠⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠀⠀⠀⠀⠀⠉⠀⠀⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠉⠀⠀⠀⠉⠐⢳⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠰⠁⠀⠀⢈⠂⠀⠀⠀⠀⠴⠒⠀⠀⠀⠒⠀⢳⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠀⠀⠀⠰⠆⠠⠀⠂⠒⠉⢸⠀⠀⠀⠀⠀⢈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⡠⢊⡠⠊⠁⠒⠤⠤⠤⠤⠤⠤⠤⠤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠆⠀⠀⠀⠀⠀⠐⠀⠠⠁⠀⡇⠀⠄⠈⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠬⠙⡆⠀⠀⠀⠠⣤⣤⠤⠴⠋⡼⢄⡀⠀⡀⣀
⠀⠀⠀⠉⠉⠙⠁⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠰⠀⠆⢀⠏⠀⠈⠓⠄
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠕⡦⢀⠀⢀⣠⠆⠀⠀⢸⠀⠀⠀⡰⠈
⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⡌⠀⣠⡈⠓⠢⢸⡀⠀⠰⠁⢱
⠀⠀⠌⠀⠀⠑⡄⠀⠀⠀⠜⠀⠀⠀⠀⠠⣀⠘⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆
⠀⡸⠀⠀⠀⢦⡇⠀⠀⠀⠀⠀⠠⠀⠄⠀⠈⠑⠢⢄⢰⡀⣀⠄⠀⣸
⠀⠁⠀⠀⠀⠤⣙⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢦⠀⠀⣀⣀⡴⠃⠀⢠⠃
⡄⠀⠀⠀⠀⢀⠘⠐⠂⠤⠀⠀⠀⠀⠀⠉⠁⢠⠀⠘⡀⠀⠀⠀⠀⢸⠀⢸
⢷⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⠀⠀⢀⡇⠀⢸
⠀⠳⡇⠀⠀⢸⡤⠤⠤⠤⢀⡀⠀⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⡾⠀⠀⢸
⠀⠀⠹⠄⠠⠒⠀⠐⡄⠀⠀⠈⠑⠢⠤⠤⠤⠜⠀⠦⡀⠀⠀⢠⠇⠤⠚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠁
⠀⣁⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠁⠀⢰⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⡉⠐⠒⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠦⣙⣤⠀⠀⠀⠀⠀⠀⠀⢮⠟⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠶⠶⣦⣫⡶⠋⠀⠀⠀⠀⠈⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠐⠿⣷⣋⡦⠀⠀⠈⢻⢲⣄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⡲⠆⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠸⡾⠀⠑⣄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡔⠁⠀⢀⠄⣢⡆⠀⠀⠀⠀⠀⠀⠘⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣻⡦⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡀⠀⠀⠀⠀⠀⠀⢀⡈⢧⠀⠀⠀⡇
⠀⠀⠀⢀⠊⢠⠃⠰⣷⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣾⡇⠀⠀⠀⢠⣿⡟⠀⢳⠀⠀⡇
⠀⠀⠀⠊⠀⠁⠀⠀⠀⢠⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠋⠀⠀⠀⠀⠘⠛⢁⠀⠀⡇⠀⡇
⠀⠀⠶⢀⡵⠀⠀⠀⠀⢨⣼⠁⠀⠀⠀⠀⠀⠀⢀⣠⣖⣮⣽⣦⣤⣴⣦⣤⣇⣤⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⢄⡀⠸⡄⡇
⠀⢘⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠛⠷⠟⠀⢻⡇
⠀⢨⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠿⣿⣿⠟⠛⠉⠉⠉⠉⠉⠉⠙⠻⣅⣀⠤⢊⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⠁⠀⠀⠀⠰⠀⣠⡇⣸⡃
⠀⣈⡀⠠⠀⠀⠀⣸⡿⢳⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠒⠈⠙⠲⠤⠤⠤⡤⠤⠤⠔⠒⠚⡉⠑⠊⠁⠀⠀⠀⠀⠀⠀⠀⣶⣤⣾⣿⡷⠀⢀⡴⠏⢀⣼⣿⢠⠟⡇
⠀⡍⡄⠰⣴⣆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠐⠀⠀⠉⠀⠀⠀⠀⠂⠀⢀⣀⠀⠀⠀⠀⠀⠈⠉⠉⢀⡤⠖⠋⢀⣤⣿⣿⢿⣿⢘⡇
⠀⠳⡙⡄⠈⢿⣷⣤⡀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⠤⣄⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⣠⣒⠋⠀⢀⣠⣴⣿⣿⣟⣵⣿⡇⢨⡇
⠀⠀⢹⣿⣷⣦⡙⢿⣿⣷⣤⣘⢮⣭⡹⠳⠂⠀⢀⢾⣷⠀⠀⠀⠀⠀⠀⠸⠶⠾⠿⠋⠀⠀⠀⠀⠀⠿⠏⠀⠀⠀⠚⠀⢀⣴⣦⣴⣿⣿⣿⡅⣴⣿⣿⣿⠇⠰⡇
⠀⠀⠈⡄⠙⠿⣿⣜⠿⠿⠀⠙⢻⢞⡂⢀⡀⠀⠻⠞⣡⣶⡦⠀⠀⠀⠀⠀⠄⠀⣀⠠⠤⠤⠀⠀⠀⠀⠀⣀⣀⣠⣤⢰⣿⣿⣿⣿⡿⠟⠃⣼⣿⣿⣿⠏⠀⢸⡇
⠀⠀⠀⠱⡀⠀⠈⠿⢿⣶⣦⡀⠈⠛⢿⡿⣷⠿⠷⣾⣿⣯⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⣛⣯⣤⣴⣾⣿⣿⣿⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠛⠿⣿⣦⣤⣤⣤⣤⣀⣀⡀⠈⠛⢿⣿⣿⣿⣿⣿⠿⠟⠻⠛⢛⣿⣋⣽⣷⣶⣬⣟⣛⣉⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠈⠉⠉⠛⠟⠿⢿⣿⣿⣦⣄⡀⠉⠛⠛⢉⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢦⣴⠀⠀⠀⠀⠉⠈⠛⠿⣿⣷⣦⣤⣶⣿⠿⡿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⣤⣤⣀⠀⠀⠑⠛⠛⢿⣯⣶⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠾⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠤⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏

⠀⠀⠀⠀⠀⢂⠀⢀⠀⠀⠀⠀⠀⠀⢄⠀⣀
⠀⠀⠀⠀⠄⣁⠊⠈⠆⣀⠀⠀⠀⠀⠀⢀⣸⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠠⡂⢀⠘⣸⠃⠆⠀⢀⠀⠁⠙⠉⠈⠙⠙⣿⣥⡀⠀⠒⠀⠄⠀⠀⢀⡀⠔⠛⠀⠀⠠⣀⣀⡀
⠀⠀⠀⠈⢀⡙⣤⡀⠜⠈⠀⠀⠀⢰⠀⠀⠀⠀⠀⢈⣿⡇⠘⢦⡂⡒⢐⠒⠆⠲⡀⡶⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠁⠈⠃⠁⠀⠀⠀⠀⠠⡀⠀⡀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠉⠙⠚⠓⠛⠊⠉⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣻⣄⢀⣠⣿⣿⡟⠀⠀⠀⠒⡀⣀⠀⠀⠀⡇⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠛⠁⠀⠀⠁⠀⠀⠀⡆⠀⢠⠁⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠌⠀⠘⣷⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠙⠈⢀⣤⠃⠀⠀⢇⣀⠀⣀⣀⣦⠆
⠀⠀⠀⠀⡀⠀⠠⡏⠀⠀⠀⠀⢋⡀⠀⠀⠀⠹⣶⣶⣾⡟⣰⠀⠀⠀⠀⠀⣸⠆⠀⠀⠀⠙⠛⠛⠋⠁
⠀⠀⠀⠀⢁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠕⠚⠁⠀⠀⠐⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⢸⠀⢸⠁⠀⠀⠀⠀⡀⠀⣀⣶⠀⠀⡀⣀⢠⡄⢀⠀⠀⠀⠀⣼⠀⠀⣧⡀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠁⢸⠀⠀⠈⠳⠤⡧⡴⠟⠃⠠⠥⠼⡿⠟⠛⠻⣷⡄⠀⠀⣧⠀⠀⠈⠳⢦⣤⣤⡴⠷⠟⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⠀⠀⠀⢸⡇⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠇⠀⠀⠀⣀⠀⠐⢄⠀⠀⠀⣠⣠⣼⠇⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⣤⡼⠋⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠀⢠⠆⠀⠀⠀⢀⠀⠀⣠⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣤⣶⣤⣤⣤⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⡞⠀⠀⠀⠀⡸⠿⠛⠏⣗⢀⡀
⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⡿⠆⠀⠸⣤⣀⠀⠀⠀⠀⢀⣄⣀⣽⡿
⠀⠀⠀⠀⠀⠈⠙⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣀⣀⡤⠤⠤⠤⠤⠤⣄⣀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⢁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⣄⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣨⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠄⠀⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⢀⠖⠉⡉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⡴⠶⠶⠶⠶⠶⠶⣦⣤⣤⣤⣀⡀⠀⠀⠀⠀⢡⠖⠛⢛⢲⣞⠒⢇⠁⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣴⡾⠛⠛⠋⠉⠉⠓⢦⠤⠤⣤⠶⠤⣤⠤⠖⢉⣨⠽⠛⣿⣆⠀⠀⠇⢴⠊⢱⠛⠉⠁⠀⢓⠩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⢀⣀⣈⣷⢖⣚⡷⢀⡔⠒⠛⠉⠀⢀⣴⡿⠋⠀⠀⠀⠘⠑⠎⡳⠖⢟⡠⠳⡄
⠀⠀⠀⠀⣠⣶⠿⠷⠈⠙⢷⣦⣀⡏⠉⠉⠁⠀⠀⠛⠉⠳⣄⣀⣤⡶⠛⢁⠀⣀⠤⠤⢤⡀⠀⠀⠁⢴⠈⢱⡆⣿⠐
⠀⠀⠀⠀⠛⣿⣄⠀⠀⠁⠀⡀⠉⠛⠷⠶⠶⢤⠶⠶⠶⠛⠋⢉⣀⣤⠖⠃⠰⢑⡄⠆⣤⢹⣄⡤⠦⠉⠙⠈⠡⠃
⠀⠀⠀⠀⠐⠈⠛⢷⣤⡀⣎⠉⠉⠀⠀⠀⠠⠀⠠⣄⢀⣠⣴⠿⠋⠁⠈⠀⠀⣈⠃⣄⠿⣸⠃⣠⠰⢤⡘⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⢦⣤⣤⣤⣤⠴⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⢹⠟⠉⢿⢦⡀⠗⣤⡺⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠀⠀⠀⠀⠠⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⣴⣻⡌⡇⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠉⠂⠚⢉⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣽⣿⣿⣷⣿⣶⣶⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉


⠀⢀⣀⠀⢀⡀⠀⢀⣀
⠀⠠⠁⠀⠲⡄⢀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠈⠉⠀⢫⡀⠀⢀⡈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫
⢀⠀⠀⠀⢀⠀⠉⠀⠂⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙
⡎⡀⠀⢀⠂⡀⠀⠀⠀⡌⠀⢸⡀⢠⠤⡴⡂⠉⠀⠀⠂⠀⠒⠀⢒⣀⣾⣷⡶⢠⣐⡀⠀⠀⠀⠀⠀⠂⠐⢻⡂⠀⠀⠀⠀⢧⠙⡄
⢠⠁⠠⠃⢠⠁⠀⠀⢀⠆⠀⠈⠀⣼⢀⠀⢉⠄⣀⠀⠀⣀⡀⣺⡟⡉⢡⢔⠀⠀⠈⡕⠒⢤⡆⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠐
⠇⡰⠃⢠⡇⠀⠊⡁⠀⠀⠀⠀⠀⡄⠘⢀⢸⠂⣿⠴⢋⢹⢭⠉⠉⣡⠱⠛⠓⢤⣤⣄⠠⣠⣶⠀⠃⠀⠀⣴⡈⡀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢡⢀
⠐⠋⣥⡄⠀⠰⠍⠠⢄⠠⠄⠠⢈⣧⠟⣬⡇⠨⠄⠠⢘⣼⠖⣏⠚⣧⡟⢿⡉⢸⡆⡿⠒⢘⣿⣄⠀⡴⠀⠄⡡⠶⠀⠔⠀⡐⠀⠀⠀⠀⠀⡀
⠢⢄⠈⠑⠢⣄⠀⠑⠠⡐⠀⠑⢼⠉⡖⢦⢄⡑⠦⣀⢸⣛⠲⠦⣌⣧⠥⡤⠦⢤⠭⡿⠆⠹⣿⣧⣾⡢⠄⢠⠀⠀⢂⢀⣌⠄⠊⡀⠄⠂⡀
⢸⠀⠹⢢⣄⡀⠙⠢⣀⠈⠐⠄⡆⠀⡇⠸⠨⠘⡑⠠⢙⣎⣂⣁⣚⣛⡛⣶⣯⡴⣊⢀⡤⠤⣭⠿⡏⠀⠰⠈⡄⡈⠌⢋⡠⠐⢀⠠⠂⠁⣀⠴⠞
⠇⢀⠇⡸⢘⡏⢰⠤⠀⠙⠢⢀⠃⢀⠇⡆⠈⠦⠥⠀⠀⠉⠁⠦⠄⣀⠀⠀⠀⠞⣁⣩⠜⠋⠀⠀⠀⠀⠀⠀⠉⢡⠈⢌⡄⠊⢀⣠⠔⠊⢅⠘⡄
⠀⡼⢠⠇⢸⠁⢸⡂⠀⠁⠂⣼⣄⠈⠠⠄⠀⠀⠔⠀⠀⢄⠀⠠⠀⠈⠙⠒⠶⣶⠿⠀⢀⣤⣄⣠⡄⢀⣠⡄⢂⡔⠀⣪⡼⡖⠉⠀⠀⠀⠈⢢⠐
⠀⣇⣞⠀⠛⠀⠈⠁⠐⠂⠈⠉⠉⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⣴⡿⢶⠐⢂⡁⠲⠀⠀⠀⠀⠀⣠⡴⠚⠉⠒⠒⠶⠶⠶⠶⠶⠶⠷
⣸⣿⣟⣉⣙⣑⣀⡒⠒⠂⠒⠒⠖⠖⠀⠈⠳⢦⣄⠀⠀⠀⠀⠀⠠⢀⠑⠦⠖⣊⠴⠊⠁⠀⠀⠀⢀⣠⡖⠛⠁
⣍⣁⣀⣀⣀⣉⣉⣉⣉⣙⣛⣛⠛⠒⠒⠒⠒⠖⠚⠻⠦⣄⡀⠁⠂⢀⠉⠒⠋⣀⡤⠊⠁⣀⣴⠾⠉⢁⣀⣀⣀⣀⣀⣀⡀⠀⠈⠁
⣿⣽⣯⣭⣭⣭⣭⣭⣭⣭⣙⣛⣛⣛⣛⡛⠛⠛⠓⠶⠶⠾⠿⢶⣄⡀⠑⠲⠟⠁⢀⣴⠾⠯⠤⠤⠤⠤⠤⠤⠄⠀⠠⠤⠤⠤⠤⠤⠄⠤
⣿⣭⣽⣯⣭⣯⣭⣭⣭⣩⣍⣩⣉⣉⣉⣛⣛⣛⣛⣛⣓⣶⣶⣶⣮⣿⣶⣤⣤⣾⡛⡒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠤⠄
⣿⣯⡛⣿⣯⣝⣯⣿⢿⣭⡋⠩⠻⣿⣍⠉⠙⠻⣯⣍⡉⠛⢿⣍⠉⠉⠉⠙⠛⠉⠉⠉⣉⡿⠛⠉⠙⣻⡿⠛⠛⠛⣓⡿⠛⠒⢒⡶⠛⠒⠒⣲⡶
⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣖⢹⣿⢷⣌⡠⠐⠉⠻⣶⣄⡙⢷⣄⠀⠀⠀⢀⣤⠾⠋⠀⣀⣴⠟⠉⠀⢀⣤⠞⠉⠀⣠⠶⠋⠀⣀⡴⠛⠁
⣿⣿⣿⣽⣿⣿⣾⣿⣿⣿⣽⡯⣿⢷⣧⡶⣉⠻⣦⣔⠉⡈⠙⢿⣦⡙⢷⣶⡾⠛⠁⢀⣤⠞⠋⠀⢀⣤⠾⠋⠁⣀⣴⠟⠁⢀⣤⠞⠉⠀⠀⣠⠴
⣿⣻⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣯⡻⣿⣿⣦⣗⡬⣻⢷⣤⡀⠀⠙⢻⣦⡉⠀⣀⣴⠟⠁⢀⣤⠾⠋⠁⣠⡴⠟⠉⢀⣠⡶⠋⠁⠀⣠⠴⠋⢁⡴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣮⣿⣷⣤⡷⣝⡿⣦⣄⠀⡈⢿⣿⡟⠁⢀⣴⠟⠁⣀⣴⠟⠉⠀⢀⣴⠟⠁⠀⢀⡴⠋⠁⣠⠔⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠲⣄⠀⠀⠀⢰⣿⡏⠉⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠿⠄⠈⠳⠤⠤⠼⠿⣷⣤⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣖⠶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢁⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⣀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠲⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠖⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⣠⠄⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢸⠇
⠀⠀⠀⢸⠀⣿⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡼
⠀⠀⠀⠈⢧⣹⡀⠱⢄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⡾⠁
⠀⠀⠀⠀⠀⠉⠛⠀⠀⠙⠒⠦⠤⠴⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠶⠶⠶⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠤⠤⣴⣧⣀⠀⠘⠶⠶⠦⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⢀⣤⠶⠟⠋⠀⢉⠀⠶⠶⣾⣅⠀⠈⣉⠙⠳⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⣠⡴⠋⣼⠀⠀⠀⠀⠀⠀⠂⢈⡔⠈⠳⣤⡁⠀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣾⠃⠀⠀⣴⠟⠀⢰⠃⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⢿⣿⡄⠀⢀⡀⢿⣷⣦⢤⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⠛⠁⠀⡅⢀⣾⠃⠰⠀⡏⠀⠀⠀⣀⠀⠀⠘⣯⣠⠀⠀⠀⠀⠀⣿⡿⡆⢘⡏⢻⣿⣿⠸⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠦⠀⠀⢀⡜⠨⣿⠀⠁⢸⠇⠀⠀⢠⠃⡀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠙⣷⡸⡌⡁⠀⢿⣿⡖⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠁⠀⠀⢀⠀⠛⠁⣼⣿⠀⠁⣸⠀⠀⠀⢸⡀⠀⠀⠀⣼⡟⠀⠹⡀⢀⠐⠸⠿⣧⣗⢡⡄⠘⣿⣷⠸⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢿⠇⠀⠀⢀⡌⠀⠀⣼⠏⡗⠀⠀⣿⠀⡀⠀⠀⣿⠸⠀⠀⢸⡇⠀⠀⣷⠈⢀⡀⠘⣿⣯⠀⣀⠀⠸⣿⣧⢹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡄⣾⠀⠇⠀⢀⡟⢀⣿⠀⠀⠿⡆⢇⠀⠀⢹⡄⠀⠀⠘⣿⠀⠀⢻⣇⠘⣷⠀⠿⣿⢀⣿⠊⠃⠹⣿⣧⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⣭⢰⠀⠀⣸⣇⠄⠹⡄⠒⡟⠙⢜⣆⠀⠸⣇⠀⠀⠀⢿⣧⡀⢸⡟⢆⣹⡇⠠⣿⣾⡏⠀⢰⡷⠘⣿⣷
⠀⠀⠀⠀⠀⢀⠀⡀⢀⡟⠀⣿⢰⡀⢀⡿⠈⠀⠀⠈⠢⠄⠀⠀⠙⠢⣄⠘⣷⣜⠂⣼⡈⠻⣤⣷⠈⠹⡿⣰⣿⣿⠇⠀⢸⣟⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠈⠀⢁⡾⠁⠀⣿⡏⠅⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠓⠈⠙⠘⠃⠀⠀⠈⠁⠀⠖⠋⠈⣿⢠⠀⢸⡇⠀⢸⣿⣿⡄
⠀⠀⠀⡀⠀⡀⠀⣸⢷⠀⠀⠸⣧⠀⢸⡇⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⣿⠞⢀⣿⠇⢰⣸⣧⣿⣧
⠀⠀⠀⠙⠀⠀⠀⣿⣾⠀⠀⠀⢿⡦⢼⠇⣀⠤⣶⣶⣶⣶⣶⢠⣄⡀⠀⠀⠀⠀⠀⢀⣄⣲⣿⣿⣿⣟⣀⣀⣤⣧⠆⣸⡟⠐⠚⢿⡿⣟⣿⡀
⠀⠀⠀⠀⠀⠀⠐⣿⢹⡆⠀⠀⠘⣧⢸⡇⠉⠀⠈⠉⠉⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢠⣿⠀⠼⢲⣿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⠀⣿⠈⢷⡄⠀⠀⠸⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣡⡾⠉⠀⢀⣷⣿⡟⣿⣽⣇
⠀⠀⠀⠀⠀⢸⡇⣿⠀⠼⢳⡀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⡁⡀⢀⢲⣿⢿⢩⣿⣿⢿⡀
⠀⠀⠀⠀⠀⢸⡿⣿⠀⢀⢇⣿⣦⡃⠈⢻⡄⠀⠀⠀⠀⠀⠶⣤⠤⠠⠤⠀⠀⠤⠤⠤⠴⠆⠀⠀⠀⠀⣼⢷⡟⢲⣁⡴⣳⣿⠟⢈⣾⣿⣿⢿⡇
⠀⠀⠀⠀⠀⢸⣷⣿⣄⠠⡈⠗⣉⠳⢦⡀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠏⣠⡿⢋⡾⣻⡓⢀⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⢷⡀⠌⢣⡄⠉⠓⠦⠳⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣣⡾⣻⡴⠟⠈⣷⣤⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⡤⣀⠘⢦⡀⠀⠀⠈⢻⣿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣴⣶⣶⣿⣿⣿⣟⡯⣰⠟⠌⠀⣠⣵⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠹⣿⣟⣿⣿⣌⢛⣽⣷⡄⠙⣄⠀⠀⠀⠹⣿⣿⠿⣤⠉⠙⠛⠉⠉⢉⣡⠾⠛⣾⢳⡟⠁⢠⡞⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣟⢻⣷⣍⠀⠙⢷⣼⣷⠀⠀⠀⠹⡋⠠⠈⠑⢠⡀⣠⠶⠋⠁⠀⠀⣿⣾⠇⢠⣿⣴⣾⣿⠿⣿⣿⡟⢻⣿⠟⠿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣄⠈⠛⢷⣾⣿⣛⣿⣷⣦⡀⠀⢷⠀⠀⣀⣼⣿⠻⣦⡀⠂⠀⠀⣧⡿⠀⣼⣿⣿⣿⣦⡄⠉⠁⠰⠿⠁⠀⠐⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⢻⣦⣸⡀⢠⣾⣿⣿⣤⣾⣿⣄⠀⢀⣿⣏⣴⣿⣿⣿⣿⣿⣿⣾⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣴⣿⣿⣿⣿⣿⣿⣿⣿⠘⡎⢻⣇⣾⣿⣻⡟⠁⢻⣌⢻⣷⡋⢿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣧⣤⣭⣿⣧⣿⡇⢠⢬⣿⣿⡿⢧⠼⠷⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⡆⣀⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠖⢂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⡄⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⡿⠿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⠟⠛⢿⣿⠿⠿⣿⣿⠿⠿⠿⠿⠿⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⠉⠁⠈⠋⠁⠺⠛⠁⠸⠛⠉⠀⠈⠋⠉⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⠐⠢⠐⠘⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣄⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠐⠘⠀⠀⠛⠢⠀⠀⠈⠁⠀⠀⠀⠉⢀
⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠄⠀⠀⠀⠘⠘⠀⠈
⠀⠀⠀⠀⠄⠀⣀⣀⠀⠄⠂⠁⠀⠀⠀⠀⠐⠠⢀⡀⢀⠤⠂⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⠀⠠⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠂⠠⠄⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠊⠠⡔⠂⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠒⠀⠀⠀⠀⠀⠢⡄⠈⠀⠀⠀⠀⠀⠗⢀⠀⠀⠆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠥⠄⠁⠂⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡆⠀⠁⡀⠀⠘
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁
⠰⢆⠠⠄⠠⠔⠀⠀⠀⠀⢠⠀⠀⠀⠘⡀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⠲⠂⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂⠀⠐⠒⠂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠦⠀⠀⠀⠀⡀⠀⠀⠀⠈⠐⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠄⠀⠀⠘
⡄⠒⠠⡀⢀⡀⠀⠀⠀⠀⠠⠀⠀⠁⠀⡀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠁
⠃⠀⠀⠈⠀⠉⠀⢀⠁⠀⠠⠂⠀⢀⠂⠉⠁⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢐⡄⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠑⠠⠄⠀⠤⠘⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡈⠣⡙⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠠⠍⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣀⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠐⠣⠄⠀⠀⠀⠀⠀⠐⠀⣀⣑⡎⠀⠀⠀⠀⠀⢀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠉⠙⠒⠄⠁⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⣂⣄⣠⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⡄⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠌⡀
⠀⠀⠀⠉⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠢⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠑⠄⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠂⠀⠀⠁⠀⠘⠐⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑
⠖⢺⡟⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠠⢀⠆⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠐⡁
⠀⠐⠅⠀⡄⠐⠈⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⡀⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢠⡍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠈⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢿⣿⣷⣶⡦⠀⠀⢻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠈⠳⢿⣿⠃⠀⠀⠀⢻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⣥⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿
⣿⣿⡟⠋⠉⠀⠀⠀⠀⠀⠈⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿
⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣿
⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣀⣤⣾
⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿
⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⣤⡴⠶⢛⣩⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣻⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣦⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣦⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢈⠉⣁⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠐⠈⠁⠀⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠂⠀⢄⠔⠒⠢⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠈⠉⠂⢀⠀⠀⠈⠂⠀⣀⡀⣀⠤⠄⠊⠀⠀⠀⠀⡀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄
⣤⡒⠁⠁⠀⠈⠈⠝⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠀⠚⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡎
⠑⠈⠦⣄⣀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠐⠠⠁⠀⠈⠀⠀⡤⠄⠚⢡
⠀⠀⠀⠈⠢⡀⠉⡀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠁
⠀⠀⠀⠀⠀⠈⠁⠙⡄⠀⠀⠙⠄⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠄⠠⠀⠰⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⡀⠀⠈⠐⢲⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⡀⠁⠒⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⢄⠀⠀⠀⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠁⠺⠀⠀⠱⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠑⢄⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢈⠀⢀⣀⣄⠐⠂⠀⠠⢄⠀⢤⠃⠀⠀⠀⠀⠠⠄⠒⠂⠒⡠⠴⢤⡀⠀⠀⠂⠀⠀⠁⠢⡀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⢏⠙⢳⣻⠁⠀⠀⠀⠀⠀⠀⣀⡘⠀⠀⠀⠀⠁⠀⠀⠀⡠⢡⡀⠤⢈⠀⠀⠀⠰⡀⠀⠀⠈⠑⠠⣀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⡋⠀⠄⡀⢐⣠⢜⠤⠉⠀⠀⠀⢄⠀⢠⡄⠀⠀⠁⠀⣀⣀⠚⠁⠀⠀⠀⠈⠐⠢⢀⡀⠀⠀⠈⢛⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠤⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠍⢐⡀⠀⠀⢄⠀⠈⠁⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠁⠁⠀⠄⢛⣒⣐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⢀⢀⡀⠀⠀⠘⠁
⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠉⠀⠀⠈⠉⠉⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⠝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣟⢁⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⢿⣿⣿⣿⢁⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡞⣿⠿⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡍⢻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣣⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⢀⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠹⣌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡾⠟⡙⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣰⠠⠚⠁⠀⠀⠀⡀⢺⠀⢀⠂⠀⠀⠀⠀⠘⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠀⠠⠀⠀⠀⠰⠀⠀⠀⡠⠀⠀⠀⠀⠀⠋⢠⣀⣀⠠⠔⠊⢀⡮⠐⠉⠀⠠⠰⡄⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠇⠀⠇⠀⠀⠀⢸⠆⡘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠄⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⡾⠀⢈⣡⡤⢴⣶⣦⡤⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣄⣀⣀⠀⠀⠷⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠃⠀⠁⠀⠀⠀⠀⡇⠀⠉⠀⠀⠈⠛⠋⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠂⠠⠻⢿⠟⠉⠳⠆⢰⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⡏⠀⢠⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡹⠁⠀⠸⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠀⠀⠃⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠟⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⡄⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢲⣶⣤⣤⣤⣄⣀⣀⣀⣤⠤⠐⠒⠈⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⡙⢄⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢷⣼⠀⠉⠙⠻⠿⡿⠿⠛⣿⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢦⣟⣧⣑⢤⡀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⠡⠀⠀⢀⢴⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⣝⣊⣿⣷⣯⣵⣶⣤⡤⢤⣤⣄⣀⠀⠀⠘⠀⠀⠀⠀⣠⣄⠂⠁⠀⠀⠀⢋⠺⣿⠀⠀⠀⡀⠄⢦⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⣿⣿⣿⣧⠈⠁⠂⠀⠀⣠⣾⣵⢿⢷⡄⠀⠀⠀⢸⣿⣧⣰⣼⣶⣾⣟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣟⣵⣿⣿⣿⣿⣿⣿⡷⡤⠤⠐⠎⢉⣿⣾⠷⣢⣏⠳⠤⠐⠃⣿⣿⣷⣜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢠⣿⣿⣿⡀⣿⣿⣷⡄⠀⠀⢸⣿⣿⣿⣧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⠠⠈⣶⣴⣶⣶⣤⣭⣭⡅⣀⢠⣥⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠣⠀⠀⠀⠀⡠⠒⠉⠁⠀⠈⠀⠠⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠃⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠂⠁⠀⠁⠀⠁⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠙⢦⣀⠀⣀⣀⡴⠂⠀⠀⠀⠀⢁
⠐⡀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠈⠢⢀⠀⠀⠉⠢⠤⢀⣀⠀⠀⠔⡣⠒⠒⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠄⢀⡠⠊⠁⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸
⠰⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢀⠀⠐⠠⠄⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⢄⡀⠀⡠⠂
⠀⠈⠀⠀⢀⡠⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠔⠁⠀⠀⠈⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠠⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢀⣠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⠥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠦⢠⡴⠶⣦⡀⣠⡶⠶⣤⠀⣶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠂⢿⡀⠀⢸⡇⣿⠀⠀⣸⠇⣿⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⠛⠛⠋⠀⠈⠛⠛⠋⠀⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣾⡄⢠⠞⠓⠀⣦⠆⢠⠞⣓⡂⣾⣳⢠⠞⠓⡆⢶⢰⠆⡷⣄⡆⣾⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠋⠘⠉⠙⠈⠓⠚⠀⠋⠓⠈⠓⠚⠁⠉⠑⠈⠓⠚⠁⠙⠚⠀⠃⠘⠃⠛⠚⠁

⠀⢀⣀⣀⡀
⠑⡈⠶⠆⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠖
⠄⢰⠸⠀⢸⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡠⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠚⠁⠀⠀⢄
⠀⠀⡠⠀⠐⢦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠑⠐⠢⠄⣀⠉⠂⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⢀⠀⠀⠀⣀
⠒⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡠⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠢⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⣀⠠⢔⡡⠒⠀⠀⠁⠒⠢⢄⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠇⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠂⠀⠀⣠⠖⠛⠛⠛⠶⣄⠀⠀⡴⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢰⡆⠐⠀⠀⠀⡅⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡄⠘⡀⠈⠂⠄⣀⠀⠀⢀⡠⠔⠁⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠙⠲⢤⡟⠀⢸⠇
⠀⠀⠀⠀⠀⠀⣠⠟⡧⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⣀⣠⠴⢾⣁⣀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⣀⣀⣤⠖⠛⠛⣆
⠀⠀⠀⢀⣶⡈⠉⢻⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠁⣠⠖⠺⣅⣛⢳⡿⣆⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠉⣉⣀⡀⠀⠀⢿
⠀⠀⣤⣏⣤⠟⠉⠙⢷⣄⠀⠀⠀⠉⠓⠶⠤⣤⢶⠀⠀⠀⢠⣧⣯⡞⡿⠛⠙⢯⣹⠏⠀⢹⡀⠀⠀⠀⠀⠀⠈⠳⣦⠀⢠⠞⠉⠈⠉⢳⡄⠈⠳⢤⡀
⠀⠀⠀⠀⢹⡄⠀⠀⠀⠉⠳⢦⣤⣀⣀⣀⡤⠿⣼⠀⠀⠀⠘⡇⠉⠙⢛⠒⠒⣿⠃⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⣠⠞⠀⢺⡀⠀⠀⠀⢸⡇⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠈⠉⠙⠛⢦⠶⡆⠀⠀⠀⠀⠹⡄⠀⠉⠀⡼⠁⠈⠀⠀⠀⠸⡆⠀⠀⠀⠀⡞⠁⠀⠀⠀⠓⠦⠤⠶⠋⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣤⣤⣤⣤⠴⢿⣸⠇⠀⠀⠀⠀⠀⠙⣆⢀⡾⠁⠀⠀⠘⠃⠀⠀⢻⡀⠀⠀⠘⣇⠀⠀⠀⢀⣠⣤⣤⣤⣀⡀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⢀⠀⠀⠀⠈⣧⠀⠀⠀⠈⠛⠒⠛⠉⠀⠀⠀⠀⠈⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⡄⠀⠈⠁⠀⠘⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⢳⣄⠀⠀⠄⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣩⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢯⣭⡇⠀⠀⠈⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⢀⣄⣴⠓⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠻⡄⠀⠀⠀⠀⣴⠂⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠻⣄⣽⢾⣄⣽⣴⠻⣦⢶⡀
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣾⣼⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⢠⡶⠧⠤⠽⢶⡄⠀⠸⡇⢺⠁⠀⠀⠀⣠⣾⠟⠀⠀⢠⡶⠤⣽⣃⣠⠟⡇⣸⣆⠈⢷⡃
⠀⠀⠀⠼⣧⡤⠤⠤⠤⠤⠴⠶⠶⠶⢶⡾⠇⠀⠀⠀⠀⠀⣸⠶⠶⠶⠶⠾⣇⡀⠀⣿⠘⢡⡟⠙⠻⣿⠁⠀⠀⠀⠘⠓⠒⣦⡉⠻⣼⡇⣿⠙⣆⠈⢳⣀
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠈⢳⠀⣩⡤⢬⣷⣄⣈⡿⠀⠀⠀⠀⠀⠰⣶⠿⢷⣤⣬⣷⣧⡼⠿⠖⢛⡏
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⠀⢯⡀⠀⢘⡏⠉⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢰⡟⠛⡻⣿⠳⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢀⡴⠶⣴⣷⣴⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢿⡀⠁⠀⠙⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⣲⣾⠷⠮⡤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⡀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠛⠓⠲⠦⣤⣀
⠀⠀⠀⠀⠀⣰⡯⠶⠓⠶⢭⣉⢿⣩⣽⣇⣀⣀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡤⠶⠖⠒⣖⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣾⣀⡀⢠⡄⠀⠈⠙⠶⣄
⠀⠀⠀⢀⡾⠁⠀⠀⠁⢀⠀⠈⠳⣄⢿⣍⢳⣈⠳⣄⠀⠀⠀⠀⢀⡾⢋⡐⠶⠀⣦⠀⠁⠶⠈⠻⣆⠀⠀⠀⠀⠀⠀⢸⡷⣄⡉⠙⠛⠦⣄⣤⠀⠀⠈⠳⣄
⠀⠀⠀⢸⡇⢀⡀⠀⠀⠈⠀⠠⠄⠈⢧⡈⣿⣩⡗⢹⠀⠀⠀⠀⠸⣇⠛⠀⠀⠀⠠⠴⠀⠀⠚⠃⠸⡇⠀⠀⠀⠀⠀⢸⣧⠉⡷⠀⢘⣂⣀⣙⠳⣴⠞⠀⠘⣧
⠀⠀⠀⠀⠙⢦⡀⠀⠃⠀⠀⠀⠀⠘⠂⢷⠀⣻⣿⢸⠃⠀⠀⠀⠀⣿⣉⣛⣛⡛⠛⢛⣻⣟⣛⢛⣿⣇⠀⠀⠀⠀⠀⢸⠷⠚⢁⡾⣫⣤⢲⢯⣳⣬⠷⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠙⠲⣤⡀⠀⠃⠀⢀⠀⠘⣧⣿⣼⡟⠀⠀⠀⠀⠘⣯⣉⣁⣨⣭⣿⣭⣥⣤⣭⣭⣥⡾⠀⠀⠀⠀⠀⢸⣤⡀⠘⣏⣳⡼⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣈⡀⠀⢻⢡⡞⠀⠀⠀⠀⠀⠀⠘⢯⣉⠀⠀⠀⠉⠛⢶⣄⢀⣴⠃⠀⠀⠀⠀⠀⣿⣠⡽⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠲⠶⠖⠛⠉⠋⠁⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣼⣆⡈⠳⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠁⠔⠈⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾⠟⣹⠟⠁⠀⢠⠎⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠙⣷⡙⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡋⣰⠇⠀⠀⢠⠃⠀⠀⠀⡜⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠈⢿⡌⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢠⡏⠀⠀⢠⠇⠀⠀⠀⣰⡇⠀⠀⢃⠀⠀⢰⡆⠀⠀⢀⠀⠀⢣⠀⠀⠀⢿⡜⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⠀⢀⠏⡀⠀⡤⢠⠇⡇⠀⠀⠈⢆⠀⢈⢡⠀⠀⠸⣆⠀⠈⡆⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⠀⠀⡜⢠⣇⠀⢡⢸⣀⣘⣄⠀⠀⠈⠢⣸⡠⣧⣀⠀⠹⠢⡀⢸⠀⠀⢸⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢸⠀⢠⣇⠆⠘⠢⠼⢿⡉⠀⠈⠓⠤⠤⠤⠼⠓⠀⠈⠉⠉⠑⠈⠚⡇⠀⢸⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠈⡄⢸⠃⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢰⠀⡞⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢡⢸⡤⠶⠾⠿⠿⠿⠷⠶⠭⠀⠀⠀⠀⠀⠭⠷⠿⠿⠿⠿⠶⠦⢸⣰⠁⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢇⠀⠀⠈⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠇⠀⠀⠀⡾⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢸⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⡇⠀⣧
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠈⡆⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠲⢦⣤⠤⠤⠤⣐⣲⠞⠁⠀⠀⠀⠀⡼⠀⠀⠀⠀⡸⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠘⣄⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⢠⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠁⡀⠀⠀⠀⠈⠢⡀⠀⠀⢹⡟⠶⠤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠶⢺⠇⠀⠀⠀⡠⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⡇⣿⡀⠀⠀⠀⠀⠈⠓⠤⠤⠷⠤⢶⡆⠀⠈⠉⠉⠉⠉⠉⠉⠀⢀⣄⣀⣼⢀⣀⠤⠊⠀⠀⠀⠀⣼⣸⠁
⠀⠀⠀⠀⠀⠀⠀⢳⣸⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠈⠉⠁⠀⠀⠀⠀⠀⣠⢞⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠈⠓⠶⢤⣤⣤⣤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣤⣤⣤⡤⠶⠋⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢷⣄⠀⠀⠀⠀⠀⣠⡖⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠁⠉⠉⠉⠉⠉⠊⠀⢀⡇⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢽⣄⠀⠤⠄⠀⣤⡀⠀⠀⡼⠑⠒⠤⠖⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⢍⡓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⣳⡄⢆⣰⣆⡠⠀⣒⡾⠧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡌⢢⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣉⣠⠓⢤⣀⣀⣠⠴⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢱⡀
⠀⠀⠀⠫⡉⠈⠀⠀⠀⠀⢹⡀⠀⠈⠉⠀⠀⠀⠀⠀⡔⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢱⠀⠀⠀⠀⢀⣠⣷⡀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠈⡇⠀⠀⠀⣞⠒⠒⠛⠒⠒⠲⠤⠤⠤⠈⢧⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠳⣄
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠻⠦⠴⠒⠒⠒⠒⠒⠒⠒⠒⠮⠷⣄⡀⠀⠉⠒⠦⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠄⠐⠒⠒⠀⠠⠤⠤⠤⠤⠄⠐⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠔⠒⠊⠉⠉⠉⠉⠒⠒⠲⠤⠤⠤⠖⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠉⠒⢦⣀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡤⢄⠐⠀⡀⠈⣆⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠖⠀⢀⡆⠀⠐⠒⠂⠰⠦⠂⠀⠀⢸⡶⣌⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢺⣀⠐⣯⠂⠾⠋⠀⡀⠲⣀⣤⠀⠀⠀⠀⢀⣼⢡⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠢⠈⠀⠈⢄⠉⢰⡞⠀⡾⢛⠛⠁⢀⣤⣤⣾⡿⠃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠰⠉⠀⠀⣷⣾⣶⣶⣿⣿⡿⠟⠁⠀⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠁⠀⠀⠀⠀⡦⠒⡂⢶⢤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢀⠊⠢⡀⠀⠀⠀⠀⠀⠡⠀⢓⢈⠈⡆⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢀⠀⣀⠔⠉⠁⠂⠀⠀⠑⣀⡀⠰⠋⠢⡘⠀⠣⠁⠀⠀⡌⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠖⠘⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⡂⠀⠀⠀⠀⠅⠀⠢⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⠚⡁⠀⠀⠀⢰⡆⠀⠀⠀⠀⠁⠘⡆⠀⠀⠀⠀⠀⠀⠀⡠⣴⣶⡶⠧⠶⠓⠒⠒⠛⠒⠒⠷⠴⠂⠀⠀⣀
⠀⠀⠀⠀⠀⡠⠃⠀⡇⠀⠀⠀⠜⡄⠀⠀⡲⣤⡀⠀⠺⡄⠀⠈⠀⠀⠀⠀⠁⠒⠻⢧⡤⠤⠠⠀⠀⠀⠀⠀⠀⠀⠤⣤⠶⠞⠃
⠀⠀⠀⠀⡰⠁⠀⠀⢡⠀⠀⡄⠀⢱⠀⠀⠙⣈⠻⣶⣄⡉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡷⡒⢲⠶⠒⠒⡒⠂⠢
⠀⠀⠀⡰⠁⠀⠀⠀⢸⠀⡐⠀⢰⠸⡄⠀⢄⠈⠓⠀⠙⠻⣿⣄⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢱⠈⠢⠌⠀⣰⡇⠀⠀⢁
⠀⠀⠀⡇⠀⠀⠀⠀⠘⠴⠁⠀⠞⠀⠈⡲⢈⠏⢀⠀⠀⠀⠈⢃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢢⡀⠈⡁⠀⠀⠈
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡠⢶⡖⠒⣋⡵⢲⠀⠀⠙⢶⣄⠀⠈⠓⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠀⠀⠀⠀⠐
⠀⢀⡜⣀⣀⣀⠤⣤⠒⠉⠀⠞⢁⡔⠁⠀⡎⠀⠸⡄⠀⠈⠻⢦⣀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠠⡄⠀⠀⠈⢠⡐⠂⠀⠀⠀⡀
⠀⠀⠀⠈⡁⠀⠀⠉⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠊⠈⠑⠒
⠀⠀⠀⠀⠑⠲⠄⠤⠤⠚⠁

⠀⠀⠀⢀⣤⣴⣶⣾⣿⣷⣶⣦⣄⡀
⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⣾⣿⣿⣿⠿⣃⣠⢻⡟⣿⣛⡿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣶⣦⣄
⣿⠟⠉⢶⣿⡟⠁⠈⠃⠘⢿⣿⣷⣝⢿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣿⡇⠀⡼⠋⠀⠀⠀⠀⠀⠀⠉⠻⠿⣷⣿⠇⠀⠀⠀⠀⠀⣰⣿⣿⣿⠿⠿⠿⠿⠿⠿⠛⣿⣿⣿⣿⣿⣧
⣿⠃⠠⠀⠀⠶⠶⢤⡀⠀⠀⠀⠀⠀⠛⢈⡀⠀⠀⠀⠀⠀⣿⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡇
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢄⣀⣾⣿⡆⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⢻⣿⣿⡧
⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢀⡤⠤⠀⠀⠀⠐⠚⠓⠲⠄⠀⠀⠀⢿⣿⠇
⠀⠀⠀⢉⣶⡄⠀⠀⠀⠀⠘⠀⠠⠀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂
⢀⣠⣶⡇⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣤⢄⠤⠰⠆
⣿⣿⣿⣧⠀⠀⣢⣈⡂⠀⢠⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠁⠀⠀⠀⠈⠆⠀⢀⠈
⣿⣿⣿⣿⣄⢐⣛⣘⠷⠀⣠⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡿⠀⠀⠀⠠⠐⠀⠀⢠⡄⠘⢆
⠀⠈⠉⠉⢩⣭⣭⣿⠛⠛⠛⠛⠻⠛⠛⠀⠀⢀⣀⡀⠀⠀⠠⡿⠏⠀⠀⢀⡄⢼⣇⣿⣄⢀⣧⠀⠀⢆
⠀⠀⠀⢠⣾⣿⡟⡟⠀⠀⠀⠀⠀⢀⣀⣤⡄⠈⢹⠿⠛⢿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁
⠀⠀⠀⠘⣿⣿⠇⠀⠀⠀⠀⠀⠀⠁⠠⠅⠂
⠀⠀⠀⠀⠈⠛⡴⠚⢙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣋⣭⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣛⣉⠉⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⣶⣶⣾⠉⠼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣞⣿⠏⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡙⢿⡏⣠⣄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣎⣼⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣿⣿⣿⣿⡿⠛⣻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠠⠀⠀⠀⠐⠐⠀⠀⠈⠀⠀⠀⠄⠀⠀⣂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠒⠛⠟⠿⠷⠶⠀⠀⠀⠀⠀⠒⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠀⠀⠀⠀⠀⠀⠐⢲⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣀⠸⠀⠀⠀⠀⢠⣄⠀⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣧⣸⣿⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢸⣿⣯⣅⠀⠀⠀⠀⠀⠾⢿⣶⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣠⣤⣀⣠⡀⠀⠉⠛⠗⠋⠉⠀⡀⠀⠘⠛⠛⠁⠀⠀⠀⠐⠒⠂⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣶⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢠⢠⣤⣀⣀⣀⡉⣁⡼⡖⣴⠎⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⣩⠔⠚⠓⠢⣍⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⢋⡥⠒⠉⢀⣠⣼⣦⣄⡀⠉⠲⢌⡙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠦⣈⠳⢤⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⠴⢋⡠⠜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠑⠢⣌⡙⠿⣿⣿⣿⡿⠟⢉⠤⠒⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⠉⠒⠤⣉⣡⠔⠊⠁⠀⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⣿⣿⣧⣠⣄⡀⠀⢸⠀⠀⠀⢠⠿⢄⡀⠙⠲⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣿⣿⣿⣿⣿⣿⡄⢸⠀⠀⠀⠀⠳⢄⡈⠑⡶⠊⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣿⣿⣿⣿⣿⠃⢸⠀⠀⠀⠀⠀⣀⣘⠀⢁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⢿⡿⣿⣶⣿⠿⣶⣄⡀⠀⠀⠀⠀⡇⠀⠀⠈⠻⣿⠟⠁⠀⢸⠀⠀⠀⢠⠛⢅⣉⠀⢸⠀⣿
⠀⠀⠀⠀⠀⣀⣴⣾⣿⣛⣻⣿⡿⢟⣿⣻⢿⣿⣟⣿⡿⣷⣤⣀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠓⢄⡉⠀⠸⠀⣿⣶⣤⣄⡀
⠀⠀⠀⢰⣿⡿⣷⣾⣯⣿⣿⣯⡺⠿⠿⠿⢿⣪⣿⣿⣾⣟⣿⣿⢿⡆⠀⠉⠓⠦⣄⡀⠀⠀⢸⠀⠀⠀⢀⣠⣴⣿⣷⣿⣾⣿⣿⠿⠋⠁
⠀⠀⠀⢸⣿⣿⣷⣭⣻⢿⣭⣿⢿⣷⣾⣶⢿⣿⣭⣿⠿⣻⣽⣾⣿⣷⠀⠀⠀⠀⠀⠙⠳⢤⣸⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⢸⡏⠻⣿⣿⣿⣷⣯⣛⠿⣿⣟⣻⡿⢟⣫⣷⣿⣿⡿⠿⠋⣿⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⡿⠟⠋⠁⠈⠁
⠀⠀⠀⠘⠳⣄⡀⠉⠙⠻⢿⣿⣿⣾⡍⣽⣾⣿⡿⠛⠁⠀⠀⣀⣤⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠉⠓⢦⣀⠀⠙⠻⢿⣧⡿⠟⠁⠀⢀⣠⣴⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⢸⠀⣀⣤⣾⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠀⠈⠁⠒⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡘
⠀⠠⣀⡀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⠖⠂⠀⠀⠀⠀⠀⠀⠠⢠⠄⠀⠠⠆
⠀⠀⠈⠀⠀⣠⠴⠉⠀⠀⠀⢀⡀⡈⠀⠠⠤⠒⢀⣰⡾⣍⣉⠜⠀⢂⣻⣄⠄⠀⢀⠀⠀⠀⡀⠘⠁⡠⠀⠀⠀⠀⢀⠀⡀⣣⠋
⠄⠀⠀⠀⠀⠈⢖⢒⣛⣥⡽⠁⠤⠖⠛⠓⠂⠀⣀⠀⠀⠀⠠⠀⠀⠀⠉⠁⠩⠭⠤⠔⠂⠉⠀⠀⠰⡁⠀⢀⡀⡄⠄⠈⣽⠏⠓⡲⠖⠚⠀⠀⢀
⠀⠀⠀⠀⠂⠀⠈⠉⠉⠤⠤⠤⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠐⠀⠀⠨⣟⠓⠂⠈⠀⠀⠀⠆⠀⠨
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠀⠀⡼
⠓⠄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⠤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠐⠀⣠⠞⠁⠠⠄⠀⠀⠁⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣽⡇⠀⠀⠀⠀⠀⢈⠀⠀⠉
⠀⠀⠀⠀⠒⠛⠖⠂⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠸
⠀⠘⡿⡄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠄⠃
⠀⠀⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡊
⢀⣠⠌⢉⢻⡖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠄⠀⠀⠀⠀⠀⠘⠰⢒
⠀⠀⠀⠈⡟⢶⡤⠄⠒⠊⠁⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠔⠀⠀⢂⣤⠖⢃⠀⠀⠁⡨⢶⢀⡄
⠀⠁⠀⠀⠘⣟⠛⠁⠠⠀⠀⠀⠐⠊⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣄⡈⠀⠂⠀⠂⠀⠂⢀⠠⣾⠃⠀⡇⠈⠃
⠀⠁⡆⢀⠀⠘⣟⣀⣐⣃⡀⠀⠀⠀⠀⠀⡇⠀⠀⣀⠄⠤⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⡀⠒⠘⢭⣁⡀⢀⣊⠀⠁⠂⠤⣖⡁⠀⣿⡧⠘⠁
⠂⠐⠁⠀⠀⠀⠘⡝⠋⠀⠀⠀⠰⠀⠀⢀⠇⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣶⡟⠠⢤⠤⣞⡀⠀⢠⡬⠗⢻⠁⠀⣿⡧⠐⠁⠀⠀⢀⣀
⠤⢄⠀⠐⠂⠰⠀⠙⢄⣀⣀⣀⣈⠀⠀⠁⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢻⡀⢠⣤⡀⢸⣯⠉⣷⡀⢸⡻⠐⠁⡀
⠀⢸⠂⢀⠀⢀⠔⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠿⣶⢆⠄⢀⡀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠈⠉⠁⠒⠻⠧⠽⠗⠈⠀⡀⠀⠀⠀⡀⠀⢀⠴⠊⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠏⣰⣿⣿⣿⣿⠟⠉⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣾⣿⠻⡆⠘⢿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣼⣿⣿⣿⡿⠋⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⣺⢱⣿⠿⠁⠀⠁⠀⠀⠻⣿
⣿⣿⣿⣿⣿⡟⣿⣿⣿⡟⢸⣿⣿⣿⣿⣇⣾⣿⣿⣿⠟⢁⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⣠⣴⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠹
⣿⣿⣿⣿⣿⢠⣿⣿⣿⠁⣼⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠞⣿⣿⣿⣿⣿⠁
⣿⣿⣿⡿⣿⣎⣿⣿⣿⣾⣿⡟⣸⣿⣿⣿⣿⠃⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⡟⠻⡟⠀⠀⠀⠀⠀⡀⣤⡀⠠
⣿⣿⣿⢳⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠃⢀⣴⠏⠀⠀⠠⠂⠀⢀⣤⣠⣴⠞⣽⣿⣿⡏⠀⣴⡇⠀⠀⠀⠀⠀⡇⢹⡇
⣿⣿⣿⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡇⢠⡿⠃⠀⠀⠀⣠⣴⣾⣿⣿⣿⣣⣾⣿⣿⣿⠀⣼⣿⣧⣤⠀⠀⠀⠀⠁⢸⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣷⡿⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢼⣿⣿⣿⣿⡆⡀⠀⠀⠀⢸⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡆⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⡿⣧⠃⠀⠀⠀⠸⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣷⣼⣿⣿⣿⡿⢨⣿⢿⣿⣿⣿⣿⣿⡆⢸⣿⣿⣿⣿⣧⠟⠀⠀⠀⡀⠀⠁
⣿⣿⣿⣿⣿⣿⣻⣿⢏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣯⣾⣿⣿⣿⣿⡟⠁⢸⡞⣿⣿⣿⣿⣧⠀⠀⢠⣇
⣿⣿⣟⣿⣿⡇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠸⠿⣿⠟⠀⣿⣿⣿⡿⣿⠛⣿⣧⡄⠀⢇⠈⠻⢿⡇⠀⠀⠀⠘⣿
⣿⣿⣿⣿⣿⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⡟⠀⣠⢿⣿⣿⣿⣿⠀⠈⠻⣷⡀⠘⠆⠀⢸⣷⡀⠘⠀⠀⢻⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠇⠀⡇⠈⣿⣿⣿⢹⡆⠀⠀⠹⣧⠀⠀⠀⢸⣿⣷⡤⠀⠀⠈⣿
⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⠋⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠃⠀⠸⣾⣿⣯⠇⠀⠀⠀⠙⡆⠀⠀⠀⢿⣿⣧⣇⠀⠀⢹⡆
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢰⣿⠃⠀⠀⠀⣰⣶⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣾⠀⠀⠀⠀⢰⡄⠀⠀⠈⢻⡿⣿⡀⠀⢼⠁
⣿⣿⣿⣿⣿⣿⣿⣟⣁⡀⠟⠁⠀⠀⠀⣴⣿⣿⡄⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣇⠀⠀⠀⠀⠳⠀⠀⠀⠘⣇⣿⣧⠀⢸⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣀⣀⣾⣿⡟⢸⣷⠀⠀⠀⠀⡀⠀⠀⢿⣿⣿⣿⣷⣀⠀⣀⣤⣴⣄⣀⣤⣴⣿⣿⠀⠸⡇⠀⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠁⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣧⡀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⠀⠀⢀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⣿⡗⠀⠀⣿⠟⠿⠛⠛⠻⠿⠿⠛⠛⠛⠉⠁⠀⢸⣿⡟⠀⠀⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠀⠀⣸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠹⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⢠⣿⣿⣿⣿
⣿⣿⣿⣿⠸⠛⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣴⣟⠋⠀⠀⠀⠀⠙⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢠⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡄⠀⢹⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠙⠻⢦⣀⣀⣀⣀⣴⣿⣿⡧⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣴⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡷⣰⡌⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠉⠻⣿⡟⠛⠛⠉⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⡟⣿⣿⣿⡿⣟⡿⠛
⣿⣿⣿⣿⣷⡜⣷⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡚⢿⣿⣿⣶⣶⣾⣿⣶⣶⣶⣶⠂⣠⣤⣾⣿⣿⣿⣿⣿⠇⠿⣿⠿⠚⠉
⣿⣿⣿⣿⣿⣿⣞⢷⡌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣽⣾⡖
⣿⣿⣿⣿⣿⣿⣿⣯⠻⣆⠈⠛⠿⣿⡿⣿⣿⣿⣿⣿⣷⣦⣀⣀⣀⣉⣉⣭⣿⣿⣿⣿⣿⠿⠟⠫⠐⠋⠁⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⣷⣦⡀⠀⠉⠲⠬⣝⣛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠀⢁⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⡣⠤⠞⠃⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠨⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⡀
⠀⠀⡠⢀⠀⠃⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⢀⢤⠴⠇⠋⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆
⠀⠀⠀⠈⠃⠒⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⣀⠊⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠃⠈⠲⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⢀⠀⠄⠂⠀⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊
⠀⠀⠀⢀⠀⢀⣀⡤⠀⠠⠖⠂⠈⢁⠀⠤⠀⠒⠒⠚⠁⠀⠀⡀⠀⠀⠀⠀⠐⠐⠀⠈⠈⠃⠉⠉⠉⠉⠙
⠀⡀⠀⠀⠑⠀⠠⠄⠰⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀
⠀⢩⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠄⠒⠈⡉⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠔⠉⢀⡂⠀⢸⠀⠀⢨⠁⠀⢀⠀⣄⡠⠄⡄⠤⠈⠡
⠀⠀⠣⣀⡁⠐⠒⠈⠅⠀⠀⠠⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠈⠁⢂⠘⠈⠀⠐⢸⠀⠀⠸⠀⠀⢈⣉⠀⠈⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠜⠀⠈⠁⠀⢰⠈⠀⠀⠀⠀⠠⠾⠋⠀⠀⠀⠀⠘⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠄⠀⠈⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢁⠉⠁⢀⠀⢀⠀⠐⠀⠀⡆⠀⣥⠤⠀⠀⠀⠀⠀⢀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠀⠀⡀⢤⣤⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠈⡍⠍⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠨⠁⠠⠤⠃⠣⠄⠸⠯
⠀⠀⠀⠀⠀⠀⠀⠀⢲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⠄⢀⣀
⠀⠀⢀⡁⡀⣀⡀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⡁
⠀⢀⠊⠈⠁⠢⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠄⠀⠐⡂⠀⠀⠀⠀⠀⠚⠀⠀⠰⡄⣈⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠐⠇⠊⠀⠀⠀⠠⠀⠄⠀⠒
⠀⠀⠀⣀⠀⠨⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠄
⠀⢰⡄⠀⡶⢢⠀⡄⡄⠀⢸⠀⠀⠀⠀⠀⠀⠾⢁⣈⠷⠆⠀⣀⠁⠀⠀⠀⠀⠀⠰⠒⠠⠒⠃⠀⠐⡀⠀⠀⠀⠀⡄⠀⠀⠀⢀
⠀⠀⠘⠠⠀⠀⠠⠜⠀⠀⠈⠀⠀⠀⡄⠉⡁⠘⠒⣒⡓⣊⠉⠹⡄⠀⠀⠀⠀⠀⠀⢐⠠⠤⣃⡀⠀⠀⠂⠀⠀⠀⠁⡠⠒⠉⠀⠤⠠⠈⠉⠒⣄
⠀⠀⠀⠐⢤⡂⠐⠀⠀⠀⠀⠀⠀⠀⢱⡜⠥⠲⠷⠈⠇⠭⠆⢠⠇⠀⠀⠀⠀⠀⠁⢃⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⡴⠡⠠⠤⠤⠀⠤⠰⠔⠆⢸
⠀⢠⠀⠀⣈⠑⠀⠀⠀⠀⠊⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⣄⠀⠐⠒⠒⠀⢀⡠⠋
⠀⠘⢀⠀⠀⠀⠀⠈⠰⡀⠂⠀⠀⠀⠀⠀⠀⢄⡀⠀⡀⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠉⠙⠛⠛⠋⠁⣀⡤
⠀⠀⠈⠁⠀⠀⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢰⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠀⠀⠀⠀⣤⣤⣆⠀⠀⠀⠀⢀⠀⠀⠙⢿⠀⠀⠀⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢃⠀⠘⠉⠁⠀⠀⠀⠉⠠⠀⠀⡀⠀⠀⠀⠸⡿⠃⠖⠛⠛⠓⠬⠟⢉⡓⠀⢀⠀⠀⣀⣀⠀⠀⠂⠀⠠⣀⣀
⠀⠀⠀⢘⠀⠀⠀⢀⣀⣤⣀⡀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠁⠀⠀⠀⠘⡿⠁⠀⠀⢠⣠⣀⠀⢹⠇
⠀⠀⢀⠀⠀⠀⠀⢾⡿⠛⠙⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠰⠀⠁⣸⣻⡗⠐⠻⣦⢄⠘
⣤⣾⡇⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠨⢂⡆⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⠁⠀⡈⠁⠀⠃⡛⠁⠀⠈⢛⠘
⣿⣿⣷⣤⣤⡄⠀⠈⠧⠀⠀⠾⠃⠀⢠⣿⣤⣌⣸⣇⣄⠀⠀⠀⠀⠀⠈⢉⠙⠛⢻⡇⠀⠁⠀⠀⢠⣀⣀⣀⣀⣠⡄
⣿⣿⣿⣿⣿⣧⠀⣠⠀⣀⢀⣀⣀⣠⣾⣿⣿⣿⣿⣿⡿⠷⢀⣤⡀⠀⠀⢸⣦⣄⠺⣷⣤⣤⡄⠀⠀⠋⠉⣉⠉⢉⣧⠀⣠⡂
⠀⠈⠉⠉⠙⢻⡷⠟⠃⠉⠀⠉⠉⠲⡎⠁⠀⠀⠀⠀⠀⢀⣼⣿⣧⣤⣄⣺⣥⣽⣦⣿⣿⣿⣿⣦⣼⡗⣰⣛⣧⢹⣿⠶⢿⣿⣶
⠀⠀⠀⠀⠾⠋⠐⠠⢀⣀⠀⠀⠀⢀⣽⣄⠀⠀⠀⠀⠀⢀⡿⠿⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡠⡚⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢝⣿⣙⣿⣿⣾⣟⢘⣀⣿⣿⣿⣿⣿⣿⣿⣇⡨⠜⠁
⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠄⠀⠀⠈⠿⠃⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠄⠀⠹⢿⣿⣿⣿⣿⠿⠿⠿⡿⣿⣿⡧
⠀⠀⠀⠀⠸⠻⢉⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡄⠀⢠⣀⡦⠲⢤⡘⠉⢀⡀⠀⠈⠛⠛⠊⠉⠉⠁⢀⢀⣉⠀⡀
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠉⠀⠀⠄⠀⠀⠀⠀⢠⣤⠰⡶⠛⢦⡤⠟⠃⠃⠘⠃⠀⠀⢹⣿⡀⢰⡿⢿⣿⠿⠛⢿⡟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⠙⢀⣓⠀⣀⢄⣀⡴⠶⠶⠀⠀⠀⠘⣿⣿⣿⡇⠀⠇⠀⠀⠘⡄
⠀⠀⠀⢀⠀⠀⠁⠁⠀⠀⠀⢈⣆⠀⠀⠀⢀⣎⠀⠀⠈⠳⣄⣚⣁⣀⣤⣾⠃⠀⠈⠀⣿⣿⣿⠇
⠀⠀⠀⢀⡶⠃⡄⠘⣄⢤⣴⣿⣿⠄⠂⠀⡇⡈⢻⡄⠀⠀⠀⢻⠛⠛⠃⠉⠀⠀⠀⣰⣿⣟⣩⣄⣀⠀⠀⠀⠀⢠⣀⣤⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⡀⠀⣀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠋⠀⠈⠃⠀⠀⠇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠘⡄⠀⠀⠀⣀⣀⣀⠀⠂⠀⡰⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠈⠢⣄⡀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠉⣭⡉⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢿⡟⠈⢆
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠘⠀⠀⠀⢤⡀⢀⠀⢨⡄⠀⠀⢂⢀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡇⠀⠆⢸⠁⠀⠀⠰⣿⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠐⠒⠤⣐⢕⠠⡀
⠀⠀⢠⠠⠤⠤⠤⠤⠤⠄⠀⠐⠒⠀⠒⠒⠒⡄⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⡀⠈⠑⢕⡌⢂
⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢻⣿⣿⣿⡄⠀⠀⠀⠀⠁⠀⠀⠈⢫⠀⠃
⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠉
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢁⠀⠀⢀⣀⠀⠀⠀⠀⠐⠂⠀⠈⠀⠈⠐⠢⠤⣀⡀
⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⢀⠜⠀⠀⣖⡄⠁⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠠⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⢀⣀⡀⣀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠈⢎⠆⠀⠀⠀⠉⠑⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠀⠀⠀⠉

⠀⠀⣠⣤⣼⣿⣿⣿⠶⠀⠀⠀⠐⠁⠐⡄⠀⠀⠀⡠⠤⠶⢶⣦⣀⡀⠀⠀⠀⠶⡄⠀⠀⠀⠀⠠⠞⠀⠠⠞⠁⠀⠴⠋⠀⠄
⠀⠀⠀⠉⠉⠋⡁⠰⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠄⠀⢾⣼⣿⡇⠀⠀⠀⠀⣿⠀⠀⢰⣦⣤⣴⣦⣀⣰⣤⣀⣠⣤⣄⣒⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⠃⠀⠀⠰⠀⡿⠀⠀⠘⠛⠈⠡⠜⠆⠛⠰⠼⠧⠴⠦⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠸⠟⠛⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⠇⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⠀⠀⢀⡀⠀⣠⡆⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠄⢀⠀⠀⠀⠈⠁⠀⠀⢀⣴⡿⠛⠛⠻⠛⠢⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣽⣶⣿⣿⠻⠿⠇⠀⠉⠉⠀⠁
⠀⠀⢀⠐⠄⠀⠀⠀⠀⠂⢈⠄⠀⠀⠀⢀⣽⠏⠁⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁⠀⠀⠀⡄⠀⡀⠀⠘
⠀⠀⣄⠀⠁⠒⠠⠁⠀⠖⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠷⠀⠀⠀⠀⠇⠀⠀⢰⠘⡆
⠀⠀⠓⠺⢿⣈⡢⠞⠁⠀⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⢀⡀⠀⠀⠔⠀⠀⠀⠀⠂⠀⠀⣠⠁⢀⠘⠞⢣⡄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⡷⠈⠀⠀⣤⣶⠆⠀⠀⠀⠀⠀⠀⡅⠀⢠⡇⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣟⣻⣿⣿⣟⣩⠁⠀⠀⠈⢂⠀⠀⠸⣿⣏⠰⡇⠀⠀⠀⠀⠀⢁⠀⠀⡇⡇⢸
⠀⠀⠀⠀⠀⡀⠀⠓⠂⠀⠀⠀⢀⡆⠀⠀⠀⠀⠿⠉⠉⠉⢙⠉⠹⠀⠀⠠⠀⠀⢻⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠇⡁⣹
⠀⠀⠀⠀⠈⢡⡆⢈⠜⢧⠠⠤⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣄⣠⣄⠀⢀⠀⠀⣦⠀⠀⢆⠃⣿
⠀⠀⠀⠀⠀⠉⡤⠋⠀⠈⡆⠒⡀⠀⠀⠀⠀⢀⣠⠄⡐⢻⠷⠢⠀⠀⠀⠀⠀⠀⣸⠀⠀⡟⠛⠋⠉⠻⠀⠈⠙⠂⠀⠀⠀⢸⡃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢰⣮⣿⣷⣽⣭⣦⣴⣿⡤⠀⠆⠀⣠⠋⠀⠀⠷⠂⠀⠀⠀⠀⢀⡀⠀⠀⣀⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣐⠶⢿⣯⣝⡳⣆⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠰⠃⠀⠚⠀⠀⠘⠆⠀⠀⠀⠄⠀⣹⠀⢀⡳⢶⣶⡼⢷
⠀⠀⠀⠀⣀⠀⠀⠛⢷⣦⠋⠟⠁⠛⠁⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡜⠀⣴⠏⠀⠀⠉⠀⠋⠑⠚
⠀⠀⡆⠀⢹⡟⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤⠀⠀⠀⠀⠀⢀⣤⢴⡶⠀⠀⠀⠀⠜⠀⣼⠇⠀⠀⠀⢀⡀
⠀⠀⡇⠀⠈⡱⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠜⠋⠀⠀⠀⠀⠀⠤⣴⠏⠀⠀⠀⠠⢸⢃⠐
⠀⢀⡇⠀⠀⡷⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠈⠀⠀⠄⠀⠀⠀⢀⡴⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⣀⣤⡀⡀⠀⠀⠀⡘⠘⠀⣇⡁⡐
⠀⠸⡇⠀⢀⡘⠂⡁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠟⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠁⠁⠈⢹⡷⠁
⠀⠀⡇⠀⢠⠟⠀⠀⠀⠀⣄⡀⣔⡾⢤⠗⣤⢠⣆⠤⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣶⡄⠄⠆⢂⢐⡐⡀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⣾⡾⠂⠀⠀⢃⣀⠐⣛⣂⢛⡚⣘⣋⢳⠞⠋⠂⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠸⡟⠒⠐⠂⠂⠃⠐⠀⠀⠀⠀⠀⠀⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠐⠲⣌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠘⠇⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡀⠀⠀⠀⠀⠀⢀⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠤⣍⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⡀⠀⢀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⡃⢻⡋⣿⣖⢒⣾⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢿⣩⠿⣧⡼⠻⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣞⢙⣶⡋⣳⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠻⣤⠿⢷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡉⣷⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠁⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⠃⢀⣀⡀⡀⠀⠀⠀⠀⠈⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠃⠀⠀⠀⠉⠑⠦⡀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⢀⣀⡀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⢰⠂⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠈⠁⠲⣄⠀⠀⠈⣄⠀⠀⠈⠁⠸⠄⠀⠀⠀⠀⠀⡈⠀⠘⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⣠⣤⠀⠀⣠⣤⣤⣤⡉⠢⢀⠘⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⣼⢀⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠍⢠⡄⠀⢈⣤⠀⠀⠉⠂⠀⢷⡄⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠁⠘⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠃⠀⠀⠉⠂⠀⠀⠀⠀⢸⣧⠀⠀⠀⠸⠐⠢⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠁⠈⠄⠁⠤⠀⠚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢄⠀⠀⠀⢀⣠⣤⣴⣆⣀⡀⠀⠀⠀⠀⢸⣤⠀⣈⠆⠀⠀⠈⠂⠀⠀⠀⠆⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠠⠂⠠⠄⠀⡀⠂⠀⠈⠙⠻⠯⣭⠍⠿⠛⠋⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⡰⠊⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⢀⠴⠋⠀⢰⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠳⠦⠤⠶⠒⠁⠀⠀⡰⠋⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⢲⣿⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠉⠓⠦⠤⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠰⠒⠁⢠⠋⠀⠈⠸⣖⣤⣤⣤⠠⠄⡐⡅⠀⢀⣠⠂⣠⡊⠀⠀⠈⡉⣲⠶⢄⠀⢰⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠞⡵⠃⠀⠀⢀⡀⠀⠀⠀⠀⠉⠉⢉⠀⠐⡎⠃⢰⡟⠀⣦⠀⡾⢰⠃⠀⠀⠑⢬⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠔⡀⠘⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠀⢰⠀⣤⣼⠁⠀⠁⢸⠃⢀⡔⡀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢿⠁⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢰⣿⡏⡜⠀⡠⡿⢸⣿⡧⠊⠀⣀⣈⢢⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢺⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⡟⢹⢹⡇⣼⢡⠇⡾⡏⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠶⠀⢸⠚⠀⠀⢸⣿⠃⠸⠈⢱⡇⠀⠁⡀⠀⡂⠀⠰⡀⠀⠀⡰⠀⠦
⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠄⢰⡏⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⡏⠀⠆⠀⠀⢧⡀⠋⣀⣾⡗⠄⠀⢈⠉⠉⠀⢀⢰⠂
⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠙⡀⣿⢣⠐⣄⠀⠻⣆⠀⠀⠈⠈
⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡏⠀⡀⠀⠀⠀⠀⠀⠇⠹⡈⡄⠈⢗⠀⠙⠂⠀⠀⡐
⠀⠠⠇⠀⠀⠀⠀⠀⠀⢀⢮⠀⢈⡇⠘⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⡇⠀⠀⠈⠀⢰⠁⠀⠀⠀⠀⡞⠀⠀⠀⠛⠒⠠⠀⠀⠀⠐⠈
⠀⡧⠘⠀⠀⠀⠀⢀⡔⠁⠰⠰⠊⡅⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡏⠀⠀⠀⠀⣽
⠀⠸⡀⠃⠀⠀⡠⠋⠀⠠⠃⡦⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⡇⠀⠀⠀⠀⠃
⠀⠀⠑⣦⠤⠐⠁⢀⡴⠕⠊⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠁⠀⠀⠀⠀⡄
⠀⠀⠀⠈⠒⠶⠒⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⡀
⠀⠀⢠⢴⡧⠖⠚⡛⣷⠀⠀⠀⢀⠌⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢣⠀⢀⠔⣄⠀⢠⣄⣀⡀⠀⠀⣤⢤⡤
⠀⠀⢸⡄⠁⢁⡀⠤⠸⡆⠀⠘⠀⠀⠀⠀⠀⠀⠱⠑⡀⠀⢀⠔⠲⡒⡀⠀⠀⡆⢹⠀⢇⠀⢸⠀⢘⠿⢿⠂⠀⠀⡆⠀⡆
⠀⠀⠀⡧⠠⠀⢒⣂⠁⣳⠀⠰⣄⠀⠀⠀⠀⠀⢀⡴⠁⡠⠃⢄⡄⠈⢶⡀⠀⢁⠸⡆⠀⡕⡅⠀⢀⢳⠁⠀⠀⢀⠇⠀⢇
⠀⠀⠀⢳⡈⠭⢄⡐⠦⠝⠀⠀⠀⠍⠹⠹⠉⠍⡇⠀⣰⢁⣀⡈⠀⠀⣀⢳⠀⠈⢸⠃⠀⠀⡇⠀⠈⢠⠀⠀⢠⡋⠀⠀⠘⢷
⠀⠀⠀⠈⠊⠀⠒⠂⠁⠁⠀⠀⠀⠺⠈⠙⠛⠋⠇⠀⠈⠉⠉⠉⠉⠉⣉⣁⣀⣀⣀⣋⣀⣉⣁⣀⣈⣁⠀⠀⢠⠀⣉⡀⠈⠈
⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⠒⢲⣲⡒⠒⠒⢒⣖⡖⠒⠒⠒⠁⠀⢸⡀⣊⣈⣀⢀
⠀⠀⠀⡔⣡⠔⠂⠁⠐⢌⠢⡀⠀⠀⠀⢀⡔⢈⠉⠈⠉⠒⢄⠀⠀⠀⠸⠘⠉⣠⠖⠒⢦⡉⠙⣮⠀⠀⠀⠀⠨⣀⡀⠀⢀⡰
⠀⠀⠈⠘⠔⠒⡄⣀⢀⣀⢡⠡⠀⠀⠐⠑⠁⡎⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⣿⣶⢠⣴⡇⠀⣿⠀⢀⠄⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⢀⢸⡀⢀⠇⡣⠿⢝⡜⠠⠀⡤⣦⣤⣤⡄⠐⠒⠒⠒⠒⠚⠢⡄⢰⢀⠀⠈⡷⠒⡏⠁⠀⠿⠀⠸⠼⡆⣷⠲⢷⢸⢈⢼⢃
⠀⠀⠀⠢⠋⠅⣀⣈⡒⢛⠔⢁⣀⡈⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠁⠀⠈⠈⠉⢀⠀⠀⠀⠈⠉⠈⠀⠐⠒⠒⠒⠒⠒⠒⠒⠊⠁
⠀⠀⠀⢸⡸⠁⠐⠀⠀⠀⠠⣟⣒⡨⡆⠀⢀⠔⡒⡒⠒⠈⢕⠌⣇⠀⠀⡏⠀⠤⠀⠐⠃⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠌⠃⠹⠁⠀⣤⠰⠀⠃⠆⠃⣀⠴⡍⠀⠀⢳⠛⠉⣽⣟⡇⠀⠀⢹⠓⣛⡛⣛⠛⡛⢛⢛⠻⣻
⠀⠀⠀⢀⣀⣀⢀⡀⠀⠀⠙⠦⠦⠄⠀⠀⠘⠵⣲⣖⡆⠽⠚⠉⠀⠀⠀⢸⡴⣶⠏⢹⠇⠀⠀⢧⠘⣅⠗⠟⠹⡇⠟⠾⢀⢿
⠀⠀⢰⠧⠤⠄⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠛⠀⠀⠀⠂⠁⠀⠈⠈⠀⠀⠀⠀⠁⠘
⠀⠀⢈⡺⠡⠀⠘⡂⠀⠀⢰⢑⡠⠤⠤⠤⠴⠥⠯⠮⠬⠤⠤⠤⠤⡄⡏⡇⠀⢀⠤⠐⣑⣒⣑⡒⠠⠤⣀⡀⠀⠀⢀⣀⡀
⠀⠀⠘⡔⢄⣀⠰⡱⠀⠀⢸⢸⣈⣉⣉⠉⠉⠉⠋⢹⠉⠉⠉⣉⣉⣁⡇⡇⠀⢨⠢⠉⡇⠆⠀⡂⢉⠢⣤⠛⡂⢠⠇⠀⠘⡆
⠀⠀⠀⠈⠑⣲⠉⠀⠀⠀⢸⢰⠒⠒⡆⡇⠀⠀⡀⢸⠀⠀⢸⢰⠀⠀⡆⡇⠀⢸⡤⣀⡃⠂⠀⠃⡈⠴⠙⢄⡀⠀⢙⠉⠉⡅
⠀⠀⠀⠀⠰⠾⠄⠀⠀⠀⠸⡸⠀⠀⡇⠇⠀⠀⣅⡸⠀⠀⠸⡸⠀⠀⠇⠇⠀⠀⠘⠚⠩⠉⠍⠃⠊⠁⠀⠀⠀⠈⢤⣀⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠄⠀⠀⠀⣀⣀⡀⣀⡠⢜⠵⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠠⢄⡢⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠁⠁⠑⠋⠭⠝⠋⠁⠀⠀⠀⢀⠔⡩⠢⠀⠰⠉⠀⠠⢆⠀⠀⠀⠀⠀⠀⠹⣿⣧⠎⠉⠀⠀⠀⠀⠀⠀⡔⢮⣹⠖⢤⡀
⠀⠸⡡⡁⠁⠁⠀⠀⠀⡇⠀⠀⠀⠈⠰⡁⢣⡄⠈⠂⠒⡡⡸⠀⠀⠀⠀⣠⡤⢦⣽⣦⢦⣤⡀⠀⠀⠀⠀⢸⡠⠤⠒⣊⡩⠇
⠀⠀⡱⠕⣂⣀⢀⡠⢮⡀⠀⠀⢄⠘⠂⠘⠒⠃⠀⠀⠁⠚⢁⠄⢠⡤⠼⠿⠿⠿⠿⠿⠿⠿⠷⠤⢤⠀⠀⡗⠐⣋⡉⠥⠔⠃
⠀⠀⠈⠁⠀⠉⠀⠀⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠒⠤⠀⠰⠶⠴⢿⠛⠋⠙⠂⠋⠀⠀⠀⠀⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠈⠣⠆⠀⠀⠀⠀⠠⢀⣰⠒⠛⠃⠀⠀⠀⠔⠒⠀⠚⠀⣠⣄⣬⠀⠀⢂⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣹⠇⠀⠀⠚⣿⠁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠛⠆⢿⣷⠀⠀⢀⡀⠜⠳⢄⣀⡴⠖⠲⣄⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠠⠏⢠⣦⡄⠀⠀⠈⠀⠻⠿⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⣻⣍⣁⠀⠀⠀⠀⢀⣰⡆⠠⠆⠀⠀⠀⣤⣀⡀⠈⠉⠉⠠⣴⠐⠀⠠⣤⡤⠖⠒⠚⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⣠⠀⠀⠀⠀⢀⠀⠉⠈⢡⠉⠀⠀⠀⠀⠈⠁⣾⠤⠀⢠⡖⠛⠉⠛⠀⣁⣀⠀⠀⠁⠈⣷⣶⡏⠀⠀⠀⠀⠀⠙⡷
⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢻⣶⣾⠋⠀⠀⠀⠠⣾⡥⠄⣤⠈⡴⢄⠀⠀⠀⠈⢃⡹⠂⢀⡸⠇⢀⠀⠚⢿⣿⠛⠀⢀⡸⠛⢀⣠⣴⣿⠐⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠋⣸⣆⠀⠀⠀⣤⠃⠘⣷⡘⠧⠀⠀⢦⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠲⣦⡟⠁⠀⠈⠁⣭⣤⣌⣀⠀⠀⠚⡂
⠀⠄⠀⠀⠃⠀⠀⠀⠀⠀⢀⣄⣈⣉⠇⠠⠾⡇⢠⡄⠘⠁⡈⠃⠀⠐⠊⠁⠀⠀⠀⠀⢹⣦⠀⠀⠀⠀⠘⠛⠀⠀⠻⣄⠀⠀⠀⠀⠈⣹⣙⠉⠀⠐⡀⠉⠲⠶
⡀⣀⣰⣆⠀⠀⠀⠀⠁⠈⠿⢧⣄⠀⠀⠀⠀⠀⠀⣻⢾⡇⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠘⠉⠘⢻⠀⠀⠀⠀⠀⢳⣤⡄⣸⣷⠶⠖⠂⠈⠛⢀⣾⣿⣦⠀⠀⢀⡀
⠀⠀⠙⠟⠀⢀⠀⠀⠈⠀⠀⠀⣹⡄⠀⠀⠀⠀⢀⠣⠈⠁⠘⠲⣄⠙⣿⣶⠶⠖⠲⡇⠀⠀⠀⠈⣇⠀⠀⠀⠀⣾⠃⠀⠈⢙⠀⠀⠀⠀⢢⡈⠻⠛⠁⠀⠀⣼⡇
⠀⠀⠀⢦⣤⠾⣳⠴⢀⡀⠀⠀⠉⠁⠂⠐⠆⠀⠀⠐⠀⠀⣀⣼⣿⡄⠀⠀⠀⠀⠀⢰⡿⠰⢀⣠⣄⡄⠂⠐⠂⠛⢃⣉⠀⠈⠁⡟⠀⠀⣤⡄⠸⠻⠇⢠⡔⠉
⠀⠀⠀⠸⡇⠀⠀⠀⠘⠁⠀⡀⠀⠰⠓⠀⠀⣀⢠⠀⢤⣠⡈⢉⣿⣥⡀⠀⡀⠐⣦⠈⠀⠀⠀⢹⣿⣤⡀⠀⠀⢰⣦⠙⢶⡌⠀⠀⠒⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣇⠀⠀⠂⠀⠄⣀⣽⠿⠇⠀⢰⣾⡟⢣⠀⠀⠉⠛⢀⣄⡙⠟⠈⢀⣸⣿⡇⠀⣠⣀⣠⣷⣄⢡⡀⣤⡈⠛⠃⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠉⠛⢇⣸⠀⠀⠀⠀⠸⣿⣇⠀⠀⢿⣿⡄⠀⠀⠉⠹⠾⠿⠛⡛⠛⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢼⠣⢀⣠⣿⣄⡀⠘⠀⠀⠀⠈⡃⢀⠶⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠋⠙⣻⡀⢀⠐⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠄⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠔⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣃⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠂⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⠄⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⢄⡁⠀⡉⠠⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢬⣗⠀⠀⠀⠀⣙⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣤⣿⠀⠀⣀⡴⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠍⠛⠀⠉⠉⠉⠑⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠋⢀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢩⠏⠀⢀⠎⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⠳⡝⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⢣⡏⠀⢠⠆⠀⠀⢸⠀⠀⠇⠀⢀⡄⠀⠀⠀⠱⡀⠀⠹⣎⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⠀⠀⠎⡀⢠⢀⠙⡀⠀⠘⡀⢘⢃⠀⠘⣆⠀⢁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢹⠀⠰⠐⢇⠈⣼⠤⠷⢄⠀⠈⢺⡔⠳⠤⠨⡦⢈⡄⠀⡇⢧
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠸⠀⡷⠁⠀⠉⠉⢀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⢁⢠⠃⠘⡆
⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠆⡧⠴⠶⠿⠷⠶⠯⠀⠀⠀⠀⠺⠷⠶⠶⠶⠦⢸⡜⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⡼⠘⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⡇⢧
⠀⠀⠀⠀⠀⢰⠃⠀⠇⠀⠀⠐⡀⠀⠀⠀⠠⣄⣀⣀⣀⣀⣠⡤⠀⠀⠀⢠⠃⠀⠀⢰⠀⢸⡄
⠀⠀⠀⠀⠀⡞⠀⠀⠈⢄⠀⠀⠰⣀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⢀⠇⠀⠀⢀⠂⠀⠀⡇
⠀⠀⠀⠀⠀⡇⡄⠀⠀⠈⠢⢀⠀⢹⡙⢒⠲⠤⠤⠤⠤⠤⠤⠴⠒⠚⡿⠀⠀⡠⠂⠀⠀⣀⡇
⠀⠀⠀⠀⠀⢳⣻⣄⠀⠀⠀⠀⠁⠈⠉⡹⠃⠀⠀⠀⠀⠀⠀⠀⣞⠒⠓⠒⠈⠀⠀⢀⣴⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⡒⡳⠓⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠃⠅⠚⠙⠣⠴⣶⠪⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢗⠉⠑⠿⠤⣄⢈⢀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠸⠛⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠂⢀⡔⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⣀⠀⠀⠀⠀⠀⠀⡠⠞⠉⡅⠀⣠⠞⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⣁⠀⠀⠀⠈⠙⠒⠤⣀⣀⡾⠀⠀⠀⠈⠙⠁⠀⠀⡠⠕⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⣠⣶⡟⠓⢄⡀⠀⠀⠀⠀⠀⠀⡍⠘⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⢄
⠀⠀⠀⠀⣀⣤⠶⠋⠁⢀⢤⠂⠀⠙⠋⠐⠠⡎⠈⠉⠉⠁⠀⠀⠾⢁⠃⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠈⢦
⢀⣠⠴⠚⠉⠀⠀⠀⠀⠈⢈⠢⡀⠀⠀⠀⠀⠈⢣⡒⠤⣀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠑⢌
⠻⣧⡀⠀⠀⠀⠀⠀⠀⠠⡊⠑⢌⠢⡀⠀⠀⡀⢸⡁⠀⠈⣡⠾⢻⢀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⡇
⠀⣿⠙⠷⣄⠀⠀⢠⠀⠀⠘⣄⠈⠢⠟⣖⠭⠒⢉⣠⣶⣋⡥⠴⡇⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠀⠀⡸⠁
⠀⣿⠀⠀⠈⠻⢦⡈⠘⡄⠀⠈⠢⠖⠉⣀⡤⠞⠋⠁⠀⠀⢀⡄⠈⠒⠤⣀⠀⠀⠀⠀⢀⠀⠀⢀⠞⠁⠐⢀⠞
⠀⣿⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣠⡴⠚⠉⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠉⠀⣰⠟⢴⢡⠄⠈⠧⠚⠁
⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠶⠋⢹⡆⠀⠀⠐⢼⡶⠋⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠋⠁⢀⡸⣃⡠⠎
⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣸⠀⠀⠀⢀⣤⡤⠖⣾⠛⠒⠒⠒⢶⠚⢻⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⢹⡀⠀⠀⠀⠙⡆⠀⣿⠀⠀⠀⠀⢸⡇⠸⣿
⠀⢹⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⡇⠀⠀⠀⡏⢳⡀⠀⠀⠀⠹⡀⣿⠀⠀⠀⠀⢸⡆⠀⣿
⠀⠘⢧⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⢳⡀⠀⠀⠀⢳⡟⠀⠀⠀⠀⢸⠁⠀⣿
⠀⠀⠀⠻⣄⠀⠀⠀⠀⢸⠀⣀⣠⡷⠤⠴⡖⠃⠀⠀⢳⡀⠀⠀⠈⡇⠀⠀⠀⠀⠚⠂⠀⢹⡄
⠀⠀⠀⠀⠈⢷⡀⠀⠀⢸⠋⠉⠀⣀⣀⣀⡗⠀⠀⠀⠀⢳⡀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠸⠷
⠀⠀⠀⠀⠀⠀⠙⣆⠀⢸⡛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⠤⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣰⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣶⣶⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⣤⣤⣄⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣽⣾⣿⣿⣿⣿⡇⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⣿⣿⣿⣿⣞⢦⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⡟⣭⡿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢸⣿⡇⣿⡇⣿⢱⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⢿⠇⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢞⡇⠀⠀⣿⠇
⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇
⣿⣿⣿⣦⡀
⣿⣟⠀⣸⣿⣦⡀
⣿⣿⣿⣿⣿⣿⣿⣦⡀
⣿⣶⣷⣿⣿⣿⣿⣿⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣙⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠇⠀⠈⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠐⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⣀⡀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡐⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢲⣶⡄⠁⠰⡄⠀⠀⠀⠀⠀⠀⡈⠢⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠘⠀⣀⠠⠀⠀⠀⠀⠀⠂⠈⢠⢴⡏⠃⠹⠀⠀⠹⡄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣆⡜⠀⣐⠋⠡⣄⠀⠀⠐⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠃⣴⡎⠇⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠐⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠈⠀⠀⢀⣄⣀⣠⣤⣤⠶⠛⠉⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿
⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⣀⠙⠻⢿
⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿
⣿⣿⡿⠀⡄⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⢀⣿⣿⣿
⣿⣿⠃⣴⡇⠀⠀⠀⠀⠀⣠⣆⠀⠀⠀⠂⠠⠤⢀⠀⡀⠀⠀⣀⣠⣤⡆⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⢠⣿⣿⣧⢀⣾⣿⣿⣿
⣿⣿⢀⣿⣿⡀⠀⠀⠀⠀⣿⣿⣧⡀⠀⠀⠆⠀⠀⠀⢠⣾⣿⣿⣿⣿⣇⣿⣿⡄⠀⠀⠀⠐⢀⡀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣸⣿⣿⣿⣦⣄⣀⠀⠈⠙⣻⣿⡷⠦⣼⣷⣤⣂⣼⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢌⠉⣻⠟⠋⠉⣿⣿⣿⠟⣿⣿⣇⠂⠀⠀⠀⠀⠀⠀⠀⡈⠠⡀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠠⠄⠉⠀⠀⠐⡇⢈⣠⣀⣿⣿⡏⠃⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠄⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⠓⠂⠀⢰⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠘⠢⣀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⠀⠠⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠻⠶⠄⠀⠀⠀⠀⠀⠂⠀⠁⠀⠈⢻⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠙⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⠀⠀⢠⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢸⠀⠀⠘⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠠⢌⡑⠂⢄⠀⠀⠀⣀⣤⣤⣶⣿⣿
⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⣂⡐⠲⠿⣓⡀⢄⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢿⣾⡿⣻⣡⣿⣾⣿⣿⣿⣿⣟⠿⣏⠟⠻⠛⠛
⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠂⠀⠐⠀⠀⣠⠀⠀⠀⠀⠀⠀⢺⢷⡶⠿⠻⢛⠟⣹⣟⢿⡻⣌⢣⠙⣄
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠅⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⢠⣾⣿⣿⢦⠳⡈⢀⢡⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠠⠀⠀⠀⠰⣟⡻⠯⡹⠽⠖⢱⡀⠀⠃⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣐⠀⢀⡀⠄⠀⠀⠀⠂⢀⠀⠀⠀⠀⠒⠉⠉⠁⠂⠀⠀⠀⠀⠀⠚⢸⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣨⣷⣄⡀⢤⡁⠀⠁⠀⠀⢈⢲⣴⣋⠀⣀⠀⢀⠀⣀⠀⣀⣀⣀⣀⡀⡉⠁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⠠⠾⠻⠙⠋⠦⠈⠑⠈⠙⠭⡙⠽⡿⣿⢯⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠈⠑⠀⠀⠀⠀⢸⢦⡀⠀⠑⠤⠑⢌⠛⡾⠁⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⢀⡀⠠⠄⡒⠈⠁⠀⠀⡀⠀⠀⠀⠀⠏⡙⢦⣀⠠⠈⠢⠈⠃⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠌⠷⣶⡾⠁⠀⠀⠀⠔⠀⢞⠀⠀⠀⠀⠁⠈⢻⣷⣄⠐⠄⠐⢃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠋⠀⠀⠀⢀⠄⡠⢁⣾⠀⠀⠀⢸⠀⠀⣛⣷⡍⠳⠄⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⢁⡴⠠⣿⣿⠁⠀⠀⠈⠀⠛⡡⢋⠔⢁⠄⠉⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠁⠄⠀⠒⠀⠈⠒⠀⠉⠑⠀⠀⠀⠀⠘⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠃⠠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠄⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣧⡀⣠⣴⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⢀⣄⣠⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣤⠀⣀⡰⢦
⠀⠀⠀⠀⠀⠀⠃⠀⠀⢠⠗⠲⠔⠂⠳⠀⠀⠀⠀⠀⠀⠀⠀⡐⣇⢠⠉⠄⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠖⠀⣔⠉⢂⣸⡭⡤⠖⢈⠅⣠⠋
⠀⢠⣆⣀⣀⡀⢀⣀⣠⣀⡖⠀⠀⠂⢰⠀⠀⠀⠀⠀⠀⠀⠀⠯⠤⠬⠕⠶⠀⠀⠀⠀⠀⠀⠜⢭⡀⢆⣐⠶⠶⠒⠉⠅⡀⢀⣠⠕⢋⣼⠑
⢤⠃⠊⣿⣓⣀⣐⣛⣓⡿⣅⣀⣄⠀⠘⠀⡄⠀⠀⠀⠀⠀⠀⠃⠐⠒⠀⠘⠀⠀⠀⠀⠀⠘⣆⠘⢷⣺⡔⣺⠃⣠⢒⣓⡩⢿⠓⢚⡭⠊⢠
⠀⠈⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠧⠤⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⡉⠙⠶⠓⠯⠥⠖⠉⠉⣁⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠒⠒⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣴⡿⢦⣠⡞⠁⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⢀⡜⢩⡀
⠀⠀⠀⠈⣾⠛⠀⡀⢉⠀⠘⠈⢦⠀⠀⠀⠀⠀⠀⢀⠀⠀⣶⠤⣞⣉⡉⣱⠞⢫⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡟⠝⠓⠈⠋⠢
⠀⠀⠀⢰⣟⡲⠦⠤⠬⠴⠾⠤⣺⣶⠀⠀⠀⠸⡁⠩⣍⡖⣲⠋⠀⠀⢈⢷⣓⡾⠧⣮⠏⡀⠀⠀⠀⠀⠀⢸⣤⣤⣅⣠⣤⣾⡇
⠀⠀⠀⠀⡏⠈⠉⠉⢹⠉⠉⠉⢹⠁⠀⠀⠀⢠⢹⣫⠍⡾⣻⠦⣤⣐⣫⡾⠤⠓⢔⣩⠿⡇⠀⠀⠀⠀⠀⠀⢛⠛⠛⠛⠛⡋
⠀⠀⠀⠀⣇⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⢸⡷⣝⠚⠚⠤⢞⡚⠋⠉⠗⣲⡶⠊⣩⣾⠃⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠄⡇
⠀⠀⠀⠀⠘⣦⣀⠀⢘⠀⢀⣠⠞⠀⠀⠀⠀⠀⠑⣌⠉⠓⠒⠶⢷⡀⢀⡼⠷⠚⢋⡡⠋⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠹⠄
⠀⠀⠀⠀⠀⠈⠫⠭⠭⠭⠭⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⢦⣽⣯⠴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⡀⣸
⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⣠⠀⢠⣀⣀⡀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋
⠀⠀⣠⠚⢱⠀⠑⣀⣹⣫⣏⣠⣮⢞⣥⡷⠀⠀⠀⠀⠀⠀⠀⣙⡦⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⢰⠃⢁⣌⣵⣷⠛⠁⣤⢸⡍⠁⢺⡀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠁⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠉⠙⠒⢄
⠀⣿⡤⢛⡻⣭⡇⠐⢾⣀⣀⡆⠀⣸⣷⠀⠀⠀⠀⠀⠀⢰⢻⠟⠲⠖⠒⠲⡗⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠀⢈⡀
⠀⣿⢮⣷⣾⣧⣵⣦⣤⣬⣥⣶⣿⡿⣿⠀⠀⠀⠀⠀⠀⢸⠸⡄⠀⠀⠀⢠⠇⡇⠀⠀⠀⠀⢀⣰⠂⠀⠀⠲⠁⠉⠉⠉⠈⠡⠃⠀⠐⣆⡀
⠀⠹⣷⣌⡉⠉⠛⠛⠛⠛⠉⠉⠁⣠⠇⠀⠀⠀⠀⠀⠀⠘⢦⣙⠲⣤⠖⣋⡠⠃⠀⠀⠀⠘⠃⢭⢀⣷⣆⣤⡀⠀⠀⠀⠀⠀⠐⠀⠀⡬
⠀⠀⠈⠳⣍⣙⠒⠂⠀⠀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣉⣯⡁⠀⠀⠀⠀⠀⠀⠀⠸⣌⣛⣙⣛⣁⣀⣀⣀⣀⣈⣋⣀⡠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⠂⡄⠀⠼⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢴⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⡟⣠⡿⠁⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⢿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⣱⡟⠁⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠠⠀⠀⠘⣦⠿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢱⡿⠁⠀⠀⡜⠀⠀⠀⠀⡠⠀⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⢃⠀⠀⠸⣇⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣿⠁⠀⠀⢰⠀⠀⠀⠀⢰⠃⡠⡾⠁⠀⠀⠀⠀⡼⡄⢠⠀⠘⡀⠀⠀⢹⣜⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⢸⠀⡰⢂⣡⣧⢊⠔⠁⠀⣀⣠⢎⠜⢀⣰⡎⢢⠀⡇⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⢄⡔⠡⢢⠀⠀⠀⠘⣤⡷⢋⣵⠌⠙⠍⠁⠉⠁⠞⢁⠞⣩⡍⠛⢿⣷⡇⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⠀⠃⡈⠀⡀⡀⢘⣾⠁⣿⣴⡆⡇⠀⠀⠀⠀⠀⠀⢸⢳⣦⢶⠈⣿⡇⠀⡐⠀⠀⠁⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠆⠰⣀⠀⠁⠀⠐⣄⠸⣿⡀⠻⠜⠳⠃⠀⠀⠀⠀⠀⠀⠘⠞⠋⠟⢀⡿⡇⠰⠁⠀⠀⢀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢠⠀⠀⠀⠀⠀⠀⠈⢦⣃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠁⠁⠀⠰⠀⠈⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢂⡨⠄⠠⠀⠧⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⠀⠀⠀⡇⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠃⠠⠄⠸⠃⠇⠠⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡠⠁⠀⠀⠀⡇⣄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⠀⠀⠒⠁⢀⠆⠀⢡⠐⠄⡀⠀⠀⠑⣄⣀⠐⠀⠀⠂⣀⣀⠀⡴⢊⡏⠀⡰⠁⠀⠀⠀⠀⣷⢻⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠑⠄⠄⠊⠀⠀⢸⣆⠀⠀⠑⢄⠀⠘⡍⠻⡛⠛⠛⢛⠷⢄⣀⢨⠁⠀⠀⠀⠀⠀⠀⡼⠃⡸⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⡌⠀⠐⠢⣀⠀⠁⠀⡇⠀⣬⠖⠶⣅⠀⠈⣿⣿⠀⠀⠀⢀⣠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢱⣦⣀⠇⡁⢀⡖⢠⡌⠳⡄⢼⣿⣤⣴⠊⠩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣏⣠⣀⠏⣁⣀⡂⢤⠀⠈⣿⣿⡟⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⠀⣿⣿⣧⠘⣿⣿⣿⣿⣷⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠒⠒⠀⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣠⣤⣤⣤⣶⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣻⣾⠿⠏⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣾⣛⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⢿⠿⠿⠋⠁⠀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⡛⠿⣦⣌⣉⣉⣉⡉⣻⠉⠉⠉⠉⠉⠁⠀⠀⠀⣉⣈⣀⣀⣠⡤⠤⠖⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣀⡍⠉⠉⠉⠉⠉⠀⠀⠉⠀⠀⠉⠉⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣦⣀⡀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⣿⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢷⣶⠿⠶⢷⢶⣶⠶⠾⣿⠿⠋⠉⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣅⣀⠀⠈⠉⠀⠀⠁⠠⠴⠀⠈⠀⠰⠒⠀⠀⠳⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣠⡤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣾⣦⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠤⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣏⢻⡛⠭⠽⠋⠿⠿⣷⣦⣀⢀⡀⠀⣀⠀⢀⣀⣠⣤⡶⠖⠛⠛⠛⠉⠉⠁⡀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⡶⣦⠄⣀⡀⠈⠉⠉⠙⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣤⣤⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣴⣶⣾⡿⠛⠛⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⡿⢿⡟⠅⠁⠀⠀⠀⠀⠀⠀⣠⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⣿⣿⣿⣿⣿⣭⣟⠉⠉⠙⠙⠀⠀⠀⠀⠀⠛⠋⠉⠉⠀⠀⠀⣁⣤⣄⣤⣤⠶⠛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣛⡳⠦⢙⠻⢿⣿⣿⣿⣿⣾⣾⣶⣶⣶⣶⣦⣶⣶⣶⣾⣷⣾⣿⣿⠿⠛⠋⠡⠀⠐⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⣈⡙⠛⠲⠶⠍⢉⡛⠛⠛⠛⠉⠉⠋⠙⠛⠛⠛⠛⠛⠋⠀⠐⠊⠁⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⢦⣤⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣀⣤⡴⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠋⠁⠀⠀⠀⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢹⡌⢿⡀⠀⠀⠀⢀⡔⠂⠁⠀⠀⠀⠀⠀⠀⢠⡾⣣⠶⣤⡀⠀⠀⠀⠀⠈⠉
⠀⠀⣀⣠⣄⠁⣤⢀⡀⡄⠀⠀⠂⣤⠤⠄⠐⠀⠀⠀⠀⢸⡀⢸⡧⠈⠣⡄⠀⠠⠉⠶⠀⠀⠀⠂⠀⠀⢀⡰⢿⣜⡱⢶⡌⠀⠀⠀⠀⠀⠀⢀⡴⠶⣤
⠀⠀⠉⠉⠉⠀⠸⠼⠃⠇⠀⠀⠺⠶⠒⠺⠛⠁⠀⠀⠀⠘⡗⠈⠡⠀⣰⠇⠀⠀⠤⠼⠒⠈⠤⠀⠀⠀⠚⢵⢏⣺⡷⠋⠀⠀⠀⠀⠀⠂⠀⢺⣗⣤⠟
⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠖⠛⠁⠀⠀⠀⠀⠀⠀⠤⠒⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠰⣦⠙⠃⢰⠀⠀⠀⠛⠿⣿⡿⠛⠁⠀⠀⠀⠃⣀⠈⠓⢄⣤⠀⠀⠀⢠⣿⣿⣿⣿⠷
⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⢞⠁⣽⠆⠉⠀⠀⠀⠀⠛⠇⠀⢀⠴⠋⠀⠀⠀⠄⠀⠀⠀⠀⣀⠀⠀⢠⣴⠏⢠⣤⠀⠙⠃⠀⠀⠀⠀⢠⠙⠃
⠀⠀⠀⠀⡀⠻⠀⠀⠀⠀⠀⠧⠀⢠⣠⡤⠀⠁⠀⠀⠠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠃⠀⠙⣁⣠⣤⣤⣤⡀
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠰⠀⢀⠀⠀⠀⠀⠀⠀⣦⠤⡤⠀⠀⠀⠀⠀⠙⠇⢠⣄⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⢘⠛⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⡉⠀⠀⠈⠀⣀⠀⠐⢦⡀⠀⠛⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣸⡷
⠀⠀⠀⠀⠀⢾⡧⣴⠄⠀⠀⠧⠀⠁⠀⠈⠀⠀⠛⠀⠀⠀⠁⠀⠊⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⡆⠀⠀⠀⠒⠃⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉
⠀⠀⠀⠀⠀⠘⢷⠜⠃⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘⠘⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄
⠀⠀⢀⣴⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⠀⠉⠟⡄⠀⠀⠀⠀⢰⠄⠸⣀⡴⠆⠀⡀⠀⠀⢄⠀⢰⠀⠀⠀⢠⣴⣶⣶⣶⡤⣄⡀
⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⡇⠀⠠⣠⡁⣠⡤⠘⠋⠀⠀⢈⡁⢀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠀⠉⢀⡰⠶⠃
⠀⠀⠆⠀⠀⠀⠏⠀⠀⠀⠀⠀⠈⠀⠀⢸⠁⠀⠀⠀⠀⠈⠁⠈⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⣩⡁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣶⡗⠀⠀⠰⠟⠆⠀⠀⢀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢤⠄⢨⠀⡀⠚⠁⠀⠀⢀⠀⠀⠈⠁⠀⠠⠀⠀⠀⠀⠈⠀⢀⠟⠀⠀⠀⠂⢀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣟⣠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠐⢣⠀⠈⠒⠂⠀⠀⠀⠀⠠⢠⡄⢠⣄⢹⡆
⠀⠀⠀⠀⠀⠀⠠⠘⠃⠀⠀⠀⠀⠟⠋⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠯⠜⠃
⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⢠⣴⠶⠿⣷⣶⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣤⣦⠀⠀⢠⣶⡄⠀⠀⠀⢦⠀⠘⢇⠀⠀⠀⠀⠀⢀⣤⣤⣠⡄⠠⠶
⠀⠱⣤⣤⣄⣄⣰⣤⠆⠀⠀⠙⢿⣤⣿⣦⡼⠇⠀⠀⡀⠀⠋⠉⠃⠀⠀⠀⠠⠀⠀⡀⠀⡀⠀⠰⠀⠀⠸⠃⡰⣿⡟⢶⡏⠓⣤⠀⢿⠯⣍⡼⠇
⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠴⠶⠋⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠂⠈⢱⣏⣠⠼⠷⠞⠁⠀⠀⠀⠈⠁

⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠈⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡤⣤⡤⠀⠀⠀⠀⠘⠿⣿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿⡄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⣄⠀⠀⠀⠀⠀⠐⠀⡈⠀⠀⠀⠀⠀⠀⠀⠻⠷⠀⠀⠀⠀⠀⠆⣠⠆⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢲⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠺⠀⠀⠠⠀⠀⠀⠈⠻⠶⠶⠶⠟⠁
⠀⠀⠀⠀⣀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠐⠂⠠⠀⠀⢑⠀⠀⠀⠀⠀⠀⣀⣀⡠⠄⠀⠀⠀⢠⣬⡑⠢⣄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣾⣦⠄⠀⠄
⠀⠀⠀⠼⠃⣀⠴⢟⠀⠀⠠⠖⠐⠒⠀⠈⠆⠀⠀⠀⢴⣬⣿⣷⣤⡈⠀⠀⠀⠀⠀⣰⡀⠊⣱⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠒⠶⠐⠿
⠀⠀⠀⠲⠀⠙⢄⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣬⠋⣹⡄⢣⡷⠀⠀⠀⠀⠂⠙⠁⡟⠁⠀⠀⠀⠘⠛⢃⣀⡄⢶⡄⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠐⢠⣤⣤⣀⡀⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠐⠖⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⢱⡄⠀⠀⠀⠀⣀⣤⣴⠾⠟⠂⢀⠀⠀⡤⠈⠀⡀⠺⠆⠀⠀⠀⠀⠘⠛⠛⠂⠀⡄⠀⠀⠀⢈⠛⡀⠀⠰⣺⠉⣧
⠀⠀⠀⢶⣿⠿⠃⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⠀⣶⠟⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠦⣿⡶⠀⠀⠀⢄⣠⡀⣀⣸⣀⡲⠐⠀⠀⠀⠀⢀⡀⠀⠀⢑⠊⠁
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠇⠐⠋⠀⠀⠀⠀⣼⡿⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠺⢀⢈

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡛⠏⠙⢻⡋⠉⡟⠛⢻⡙⠏⣙⡿⢛⡋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣿⣿⣷⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣅⠁⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⡴⠊⠁⠀⠀⠀⠀⠀⠀⠈⠙⢶⡲⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣳⠋⠀⠀⠀⡀⠃⠀⠄⠀⠀⠀⠀⠀⠳⡽⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡇⠀⠠⠦⠀⠃⠈⠢⣀⡄⠂⠦⠆⠀⠀⢻⡝⠀⠀⠀⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠤⣤⢤⠄⠀⠀⣤⣤⡤⠀⣾⠀⠀⢁⣠⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠐⢂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠱⡀⠀⠈⠁⠀⠀⠁⠀⠀⣠⡟⠀⠀⠸⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣯⢱⢲⡶⠤⠤⢶⣖⠈⡿⠀⠀⠀⢠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢄⡀⠀⢸⣿⠁⠀⠀⠀⡸⠈⢻⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠘⠋⠀⠀⠂⠀⠁⠀⠑⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣾⣷⣿⣾⣥⣾⣿⣷⣞⣛⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣾⣿⣷⣾⣿⣿⡿⠿⠟⠛⠛⠛⠻⠿⣿⣯⣁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢟⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠻⠋⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠈⣥⡾⣻⣿⠟⠁⠀⠀⡀⠀⠀⠀⡀⠐⠋⠉⠀⠀⠀⠀⠀⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡿⠃⠀⠀⠐⠋⡉⢁⡀⠋⠉⠀⢠⠟⠁⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⢸⣏⠉⠁⠀⠀⠀⠀⠠⠛⠾⠿⠤⠴⠞⠉⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡟⠉⠀⠀⠀⠀⣠⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⢶⣦⠀⠀⠀⠀⠀⠀⠙⠿⣦⣤⣄⡀⠰⠋⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⡄⠫⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⣴⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⣀⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⠒⠚⠋⢰⡋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⠀⠀⠀⠀⠀⠦⣝⣳⡄⠀⠀⠀⠀⠀⠀⠀⣿⡇⠰⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⠶⠶⣬⣓⣶⠋⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⢌⡲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋⠁⠀⠀⠀⠙⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠾⠿⠆⠀⠀⣶⡇⠀⢿⣧⡤⠴⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⣁⠀⠀⠀⠀⢀⣸⣿⡇⠀⠀⠀⢀⣀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠙⠿⠿⠟⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠒⠚⠉⠀⠀⠀⠀⠉⠁⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠶⠄⣤⣤⣴⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠺⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠘⢿⣧⡀⠈⠿⣿⣿⣿⠃
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠻⣿⣦⡀⠀⠟⢹⣦
⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠛⠿⣷⣶⣾⡇⠑⡀
⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠄
⠀⣇⣠⡾⢩⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⣿⣿⠃⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⠑⡀⢸⠑⢄⡀⠀⠠
⠀⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢀⠁⠀⠀⡄⠀⠀⠈⠢⢄⡀
⠀⡟⢷⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢀⠀⠀⢻⠀⠀⠀⠀⠀⣰⠀⠀⢠⠋⠙⢄⠀⠀⠀⡞⠀⡆⠀⠀⠀⠠⣀⣤⣤⣴⣶⣶⣶⠶⢶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠘⠀⠀⠸⠀⠀⠀⠀⠀⢿⠀⢀⠂⠀⠀⠀⢱⠀⠘⠀⠀⠀⠀⠀⠀⣾⠟⠋⡿⠟⠿⢿⠿⠂⣘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣
⡀⡄⠀⠀⠀⠀⠀⠀⠀⠁⣠⠆⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠒⠚⠉⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠁⠀⠀⡆⠀⠀⠀⠀⡄⠀⠁⢀⣀⣠⣤⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⡇⠀⠀⠀⠈⣅⣰⣾⣿⣿⠿⣿⡿⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠁⠀⠀⠀⠀⠐⡿⠋⠘⢂⣠⠤⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠷⠶⠦⣤⢄⣤⣠⣤⣤⣤⣴⣶⣿⡽⠟⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢰⠈⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢀⡼⣟⠛⠛⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠰⠋⠀⠀⠀⠘
⠀⠀⠀⠀⠐⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⣀⣴⡿⠁⠸⡆⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⢀⠔⠁⠀⠀⠀⠄
⠀⠀⠀⣆⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣴⣾⣿⡟⠀⠀⠀⠀⢀⡀⠀⠠⣄⠴⠃⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣷⣤⡀⠀⠐⠀⠀⠀⡠⠐⠁
⠀⠀⠀⠈⠄⠀⠀⠈⠑⢄⡀⠀⠀⠀⠀⠀⠀⠸⢿⣿⡟⠀⠀⠀⡀⠚⠉⠉⠀⠀⠈⠑⠂⠀⠀⠀⠀⠒⣲⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠦⢤⣄⣀⣀⣠⣾⣿⠀⠀⡰⢻⡇⠀⠀⠀⢠⠷⡀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⢋⠆⠀⣰⣿⣿⣦⣤⣶⣿⡆⠑⢄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡏⠨⠀⣼⣿⣿⣿⠇⢸⣿⣿⣿⠀⠀⠀⠀⠟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡆⢀⣤⣬⣉⣙⠂⠘⠻⠿⠿⠇⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣤⠤⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠡⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠂⠀⠀⠂⠠⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⢀⣀⣀⣐⣒⣒⣂⣀⣀⣠⣠⣤⣤⣤⣀⡄⡀⢠⣤⠤⠤⠤⠤⠐⠒⠉
⠀⠀⠀⠀⠠⠔⢶⠀⠀⠀⠈⠙⢲⣤⢀⣀⣀⣤⣤⠶⠛⠉⠉⠁⠀⣀⣴⡀⠀⠀⠀⠀⠐⠉⠈⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⠶⢍⡉⠁⠀⠉⠉⠀⠀⢀⣀⣤⣤⣖⣋⣁⡀⡁⠂⢐⠀⣋⡀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡎⢭⠀⠀⠒⠬⠍⠉⠓⠒⠒⠊⢉⡀⠤⠛⠉⠉⠩⠉⠉⠻⠄⠀⡠⠂⠀⠹⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠲⠀⠀⢀⣠⣴⠶⡋⠉⠉⠉⢀⠀⢀⠀⠀⠀⢀⣀⣀⠰⠀⢠⠀⠀⢀⡀⠄⠁⢀⣀⣀
⠀⠀⠀⠀⠀⢱⠀⠀⠀⢀⠴⢋⠉⠙⠂⠈⠛⠛⠒⠀⠀⠘⠁⠤⠄⠈⠀⠀⠂⠀⠐⠶⠚⠉⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠔⠛⠇⠀⠀⢠⠶⠶⣿⡆⠀⠀⠀⢀⠴⠖⠶⠄⠀⢀⠞⠳⠄⠀⠟⡟⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠑⢾⠀⠀⠀⠐⠤⠐⠁⠀⠀⠈⠃⠐⠂⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


⢀⣀⡀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⡏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡀⣤⠀⠀⠀⠀⠀⡐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢪⢧⡀⡇⠀⠀⠆⠡⠀⠁⢀⠀⠀⠀⠀⠄⠈⠀⠀⢀⠀⠀⠀⠀⠀⢀⠰⣬⠶⣂⡀⠀⠀⠀⠉⠁⡒⣷⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⢀
⠀⠠⠂⡠⢁⠆⢵⠃⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠅⢀⢀⠀⠀⠀⠐⠀⠀⠀⠨⠑⢆⡃⣩⠶⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⢀⡀⢊⠔⠀⢠⠎⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠜⠃⠘⡣⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢸⡤⠁⡤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠈⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠊⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰⣿⢻⡟⠄⠀⠀⠀⠀⠀⠀⠀⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠒⠠⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡄⠀⠀⠀⢰⣦⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠷⠇⠆⠀⠀⠀⢀⠅⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠆⠀⠆⠀⠀⠀⠁⠀⠀⠀⠋⠀⠀⢀⣀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠄⠀⠀⠀⠀⠴⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠺⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠨⠅⠈⠀⠀⠀⠀⠀⠀⢀⢂⡄⠀⠀⠀⢀⠒⠑⢄
⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡁⠀⠀⠀⠀⠀⢦⠆⠀⠀⠀⠐⠛⠀⠐⠀⠀⠀⠀⠐
⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠂⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠔⠀⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⢀⠠⠀⠈⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⢤
⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠀⠈
⠀⠀⠀⠀⡄⠀⠰⢄⠀⠀⠀⠀⠀⠥⠀⠁⠀⠀⠀⢀⡀⠸⡧⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠈⠒⠂⠀⠀⠀⣀⠄⣬⠇
⠀⠀⠀⠰⠇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠤⠿⣿⣻⣶⡝⢪⢁⢄⠀⠀⠀⠀⠀⠀⠈⠑⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⡀⠀⠀⠈⠁⠀⠀⠘⠛⣋⣽⢤⣾⣩⣌⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠸⡣
⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡀⠄⠀⠀⠀⠀⠙⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠓⢄⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⣀⠄⠀⠀⠍⠄⠠⣝⠅⠀⠀⣦⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠷⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠠⠀⢀⡐⡁⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢠⡏⣼⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢂⣄⠀⠀⠀⠀⠀⠀⢀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠆⠀⠀⠋⡋⠉⣠⣤⣴⣿⠃⠸⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡄⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⢿⣷⣾⣿⠿⠟⠃⠀⢸⢇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⠀⠀⠀⡔⠁⠀⠢⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠋⠀⠀⢠⠄⠀⠐⡂⡘⠀⠘⠧⠤⠤⣀⣰⡀⢀⠄⣠⠀⠀⡀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢀⠴⠃⠀⠀⣠⢆⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢺⡀⠀⠜⠁⠀⠐⠀⠀⠀⠀⠰⠀⠁⣰⠇⠘⢰⠁⠀⠀⠀⠀⠙⠂⠠⢶⣎⡁⠀⣤⡶⠿⠿⠉
⠀⠀⠀⠀⠀⢀⠀⠀⢀⣿⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡄⣠⡏⡄⠀⠀⠀⢰⣿⣦⣡⣾⢿⡇⠀⣸⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣿⠀⠀⠀⠤⠀⠀⠀⠀⢠⠀⠰⠃⠉⣷⠁⠙⠁⠀⠀⠀⣿⢏⣿⣿⡿⠀⠙⠔⢁
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⡇⠀⠀⠸⢇⠀⠈⣿⣿⣿⣿⣧⢀⡄
⠀⠀⠀⠀⠀⢀⠔⠟⠁⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠠⠁⠀⠹⣿⡷⠿⠿⠞
⠀⠑⠄⣀⣤⣶⣊⡤⠖⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠒⠀⠀⠀⠀⡜⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠠⠄⠀⠀⠀⠀⡀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠲⠶⠒⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡝
⠀⠀⠀⠀⠀⢀⣠⡶⢃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⢿⣿⡇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠹⠦⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠞⠋⠉⠁
⠀⠂⠀⠀⢰⣆⡤⡆⠀⠀⠀⠀⠀⠊
⢸⠀⠀⠀⢈⡤⡀⡀⡄⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠘⠀⠀⠘⠙⠃⠋⠃⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠞
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠂
⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣶⣦⣤⣴⡎⡱⡄
⠀⠀⠐⡄⠀⠀⠀⠀⢀⠌⠀⠀⣀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣾⣦
⠀⠀⠀⠈⠂⠀⠀⠀⠈⠀⠀⢸⣿⣷⣦⣤⣴⣾⣿⣿⠿⠋⠀⠀⣀⣀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟

⠀⠀⠀⠀⠀⢀⣤⣔⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠉⠀⠀⠀⡀⡀⠛⠀⠠⢀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠒⠦
⠀⠀⠀⢠⡶⠋⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡄⠀⠀⠀⠀⠀⠀⣠⠞⠂⠀⠐⠋⠀⠀⠀⠘⠋⠉⠀⡀⢀⠄⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡜⠁⠀⢀⣤⢀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⡀⠀⠀⢀⠀⠀⠘⡻⠋⠀⠀⡔⠁⠀⣾⠖⠀⠀⠀⠀⣀⡀⠉⠁
⠀⠀⢸⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⣀⠴⠂⠀⠀⠀⠀⡛⠁⠀⣤⣄⣀⠤⠚⠁⠀⠀⢰⡗⠃⠀⠀⠀⠀⠀⠀⠀⠹⠏
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣀⡀⣠⣆⢒⣈⣀⣤⡤⠟⠉⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢸⡅⠀⠀⠀⠀⠀⠘⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢃⠀⠀⠀⠀⠚⠛⠃⠀⠀⠰⡿
⠀⠀⠘⠻⠀⠀⠀⣠⡀⠀⠀⠀⠀⢀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠂⠀⠉⠁⠀⠀⠀⠀⠞⠀⠘⠟⠀⠀⠀⣠⣤⣄⠀⠀⣠⣤⣤⣤⣀⡀⢀⣾⢃⣼⠀⠀⠀⠀⣠⢄⠀⠀⠀⠀⠘⠲⣿⠆⠀⠀⠈⡁⠀⠐⠛⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⣠⣿⠃⠘⠟⠛⠙⠛⠛⠛⣿⡟⣛⡁⣘⠁⣤⢤⢿⣼⠀⠀⠀⠀⠀⠀⠀⠉⠀⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⢸⣿⠇⢀⣶⠲⡄⣼⡏⢹⢠⣿⣏⣼⢣⡿⢸⡧⢊⣔⡙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠈⠛⢂⣾⣯⣠⣧⣿⠣⡼⢻⡟⢿⣄⠘⢓⣉⣙⡉⢸⣇⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⣠⠞⠻⣧⠟⠁⠙⠚⢁⣀⣥⣼⠿⢿⣍⣁⣈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠞⠋⠀⠀⠀⠈⠿⣿⡿⠿⠛⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠞⠃⠀⠈⠉⠁⢀⡀
⠀⠀⠀⠀⣴⠆⠀⠀⣤⠀⠀⠀⠈⠙⠁⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⠒⠓⠚⠃⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⢠⣞⣡⡾⠖⠋⠉⠁⠙⠒⠀⠀⠀⠰⠒
⠀⠀⠀⢠⠀⠘⠷⠀⠀⠀⠀⠀⣴⡶⠂⠀⠀⠉⠉⠀⠄⠀⠀⠀⡠⠂⠛⠁⠀⠐⠟⢃⡀⠀⢀⣤⡀⢀⣀⠀⠀⠀⠀⠘⢿⠁⣠⣤⠀⣀⣘⣀⣀⠠⣤⠀⠀⢤
⠀⠀⠀⣿⣧⠀⠀⣠⣄⢠⣴⣦⠀⠀⢠⣶⣤⠀⠔⢠⠆⠀⠀⣼⠃⠀⢛⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡄⢦⡤⠒⠉⠉⣉⠉⠉⠉⠓⢄⢀⡀⠁
⠀⠀⢰⣿⣿⣿⣧⣤⣤⣄⡀⢤⣄⣀⣀⣀⣀⣀⣴⡏⠀⠀⢀⡴⠀⠀⠙⠁⠀⠀⠀⠤⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⣀⠀⠀⠋⠀⠀⢀⡀⠀⠈⠡
⠀⠀⠘⢿⣽⣋⡉⣉⡉⠹⠿⠟⠛⠉⠉⢉⣽⡷⣀⣶⠀⠀⠈⠳⠀⠀⠦⢀⠐⠀⠈⠀⠁⠚⠉⠀⠀⠒⠀⠀⠀⠀⠀⠀⣏⠀⠀⠋⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠃
⠀⠀⠀⢰⣿⣿⣿⣭⢤⣦⣦⣤⣤⣴⡷⠾⠿⠛⠙⠋⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠉⠀⣀⠀⠀⠤⠀⠀⣀⡀⠀⠀⢸⡇⣴⠄⠀⠀⠀⠀⢴⣆⠀⠀⠀⡀
⠀⠀⠀⢨⣥⣌⣉⠀⠀⣀⣀⣀⢠⠾⠆⠀⣀⣠⣤⣴⡛⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠉⠀⠀⠈⣿⠀⠀⣤⡀⠀⠀⠀⠀⠀⢠⡄⠛
⠀⠀⠀⢉⠾⠋⠉⢭⡉⠉⠋⢹⠟⢻⡏⠉⠉⠉⢈⣉⣤⠀⠀⠀⠀⠀⣀⣀⠀⠰⠖⠒⠒⠀⠉⠈⠁⠀⢠⡀⠀⠀⠀⠀⠘⣇⠀⠈⠁⠀⠶⠂
⠀⠀⢠⣿⡟⢛⡶⠶⣤⣤⣤⣤⡶⠶⠶⠞⠛⢋⡉⠡⢄⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢴⣦⡄⠀⠀⠀⠙⠁⢠⡆⠀⠀⠀⠈⠳⣄⣀⡀⠀⠀⢶⠆⣀⠤⠔⠁
⠀⠀⠀⠨⣷⣿⣿⣄⣀⠺⠷⠀⠀⠀⣀⡀⠀⣛⣣⠴⠊⠀⠀⠀⠀⠀⠀⠈⠁⣀⡀⠀⠀⠉⠁⠀⣀⡀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠉⠃⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣟⠿⠋⠉⠉⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠉⠙⠛⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠋⠁⠀⠀⠀⠐⣿⣏⠀⠀⠀⠽⠿⠁⠀⠀⢠⣴⡀⠀⠀⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⢀⣶⣦⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠉⠙⠁⠀⠀⠀⠀⠈⠺⣻⣷⡄⠀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⠀⠀⠘⠀⢀⣠⣶⠾⠟⠛⠉⠛⠛⠙⠛⠛⠷⣶⣄⡀⠀⠀⠀⢀⣀⣤⣼⣿⣿⠿⠟⠛⢻⣷
⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⣶⡧⠄⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣻⣿⣶⡾⠿⠟⠛⠋⠉⢀⡀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠁⠀⣰⡿⠋⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⠿⠿⠛⠋⠉⢀⠀⣠⣴⣦⡐⣿⡛⠃⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⣸⡟⢁⣀⣠⣤⣴⣶⠿⠿⠛⠛⠉⠁⢀⣀⣤⡄⣶⡀⢹⡇⢿⡇⠹⠃⢹⣧⡶⠄⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⣀⣠⣤⣴⣿⡿⠿⠛⠋⠉⢁⢀⣀⠀⠀⣶⠟⣷⡌⣿⡁⣀⢻⣿⣎⣿⡘⣿⠀⣠⡌⣿⣀⣀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⣀⣀⣼⣿⣷⡾⠿⠛⠛⠉⠁⣀⣠⣄⠀⣶⡀⢻⡇⢿⡆⠀⢻⣇⢬⣧⢹⣿⠛⠈⣿⡘⢿⣧⠹⣧⣼⡇⠘⠛⠛⠀⢀⣀⣠⣤⣿⣷
⠀⢾⣿⠟⠛⠉⠁⠀⣀⡀⣤⡀⢸⣇⢻⡏⠹⣧⢹⣧⠈⣿⡘⣷⠀⠘⣿⡄⢻⡆⢿⣤⣶⠸⠇⠈⠛⠀⢀⣀⣠⣤⣶⣶⣿⣿⡟⠋⠉⠁
⠀⠘⣿⡄⠀⠀⠀⠀⢹⣧⢻⣿⣌⣿⡘⣿⠀⢿⡆⣿⡄⢻⡇⢻⣧⣶⠘⠿⠟⠋⠈⠁⣀⣠⣤⣴⣶⣿⠿⠛⠋⠉⠁⠀⣼⣿⡇
⠀⠀⢹⣷⠀⠀⠀⠀⠀⢿⡎⢿⡙⣿⣇⢹⣧⣼⡷⠘⠻⠟⠃⠀⠉⣀⣀⣤⣴⣶⡿⠿⠛⠛⠉⠁⣽⡟⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠈⣿⡇⠀⠀⠀⠀⠘⣷⠸⡧⠈⠛⠈⠉⠉⣀⣀⣤⣴⣶⡾⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⢠⣾⠟⠀⢀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠉⣀⣀⣤⣤⣶⣾⢿⣿⣟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠟⠁⠀⠐⢾⠿⠀⢀⣼⣿⡏
⠀⠀⠀⠀⣿⣧⣤⣤⣶⣾⣿⡟⠛⠉⠉⠀⠀⠀⠈⠛⠿⣦⣦⣤⣀⣀⣀⣠⣤⣴⣶⠾⠛⠁⠀⡀⠀⠀⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠀⠘⠛⠋⠉⠀⠘⢿⣿⣦⡀⠀⠀⠀⠀⠀⣄⡀⠀⠈⠉⠉⠛⠋⠉⠉⠁⠀⠀⠀⠀⢻⢿⠁⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣮⣦⣄⡀⠀⠉⠟⠇⠀⠀⢀⣦⣖⠀⠀⠀⣱⣿⠄⠀⠀⠀⢀⣠⣶⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣯⣰⣤⣀⡀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⢀⣀⣄⣰⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣾⣿⣶⣶⣶⣤⣤⣶⣶⣾⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠉⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠠⢷⣄⠈⠳⣄⡠⢄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠋⠉⠀⠀⠔⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⢤⣀
⠀⠀⠀⠀⣀⣠⣤⣤⣠⣤⢤⣤⣶⠋⢁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⡀⡀
⠀⠀⠀⠀⢹⣿⣿⣿⠉⢡⡾⠋⠁⠀⢨⢀⡔⠁⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠠⠈⢿⣧⡙⢻⣿⣿⣶⣦⡄
⠀⠀⠀⠀⠈⣿⡿⠀⣰⡟⠁⠀⠀⠀⢠⠊⡅⠀⡀⠄⠀⠌⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢰⠄⡀⠛⢿⣎⠹⣿⣿⡿
⠀⠀⠀⠀⠀⢹⡇⣼⡟⠐⠀⠂⠣⣤⠠⠊⠀⠀⠢⠂⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠈⢿⣆⢸⡿⠁
⠀⠀⠀⠀⠀⠀⣿⢿⠁⠀⠀⠀⡰⠁⠀⠀⠀⠀⢀⡌⡁⠀⠀⠀⠸⠀⠀⠀⠀⢯⠀⠀⠀⠀⣄⠀⠀⠀⢱⠀⠀⠀⠀⠘⣿⣴⠃
⠀⠀⠀⠀⠀⢠⡟⢸⠀⠀⠀⡰⠁⢠⠀⠀⢠⠀⠸⠀⠇⠀⠀⠀⠀⢣⠀⠀⢸⠈⡄⠀⠀⠀⢸⡄⠀⠀⠀⠆⠀⠀⠀⢀⠘⣇
⠀⠀⠀⠀⠀⡾⠁⢸⠀⠀⢀⠃⢀⠟⠀⠀⢸⠀⠃⠀⠈⡀⠀⠀⠀⠀⠡⡀⠸⠀⣘⣄⠀⠀⠀⠏⠢⡀⠀⠀⠀⠀⠀⢨⠀⢹⡀
⠀⠀⠀⠀⢰⠇⠀⢸⠀⠀⠀⢀⠊⠀⠱⡀⠀⡇⠒⠋⠉⠉⠢⡀⠀⠀⠀⠈⠐⡧⠈⠉⠙⠒⠒⠚⠓⠈⠑⠰⠀⠀⠀⡚⠀⠘⡇
⠀⠀⠀⠀⡿⠀⠀⠸⠀⠀⡶⠃⠀⠀⠀⠈⠉⠁⠛⠀⠀⠀⠀⠀⠉⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢿
⠀⠀⠀⣸⠇⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⣇
⠀⠀⢰⠇⠀⠀⠀⠀⠸⠀⡧⠶⠿⠫⠿⠿⠿⠿⠿⠒⣦⣌⠀⠀⠀⠀⠀⠀⠀⢩⣴⠶⠿⠿⠿⠿⠿⠳⠶⣤⡄⡰⠀⠀⠀⠀⠀⢻⡀
⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠸⠈⣧
⠀⢸⠇⠀⠸⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⡆⠀⢹
⠀⡟⠀⠀⠀⠄⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢠⠀⠀⠘
⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠠⣄⠀⢀⣀⣀⣀⣀⣀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡌
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤⠤⠤⠤⠐⠒⠈⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠰
⡏⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⢠⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠈⣟⠲⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠖⡿⠀⠀⠀⠀⠀⠠⠃
⡇⠀⣇⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⢹⡄⠀⠀⠉⠉⠛⠓⠒⠲⠶⠶⠖⠒⠚⠛⠋⠉⠉⠀⠀⢰⠇⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⢠⡄
⢹⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠰⠷⠖⠚⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣼⣀⡠⠤⠚⠁⠀⠀⠀⠀⠀⢀⣾⢠
⠈⣇⢻⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣿⡞
⠀⠘⢮⣧⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠁⠰⠟
⠀⠀⠀⠉⠀⠀⠀⠉⠛⠲⠶⠶⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⠖⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⣀⡀⣀⡀⢀⡀⠀⠀⠀⢀⣀⣀⡀⢀⣀⠀⢀⣀⡀⢀⣀⡀⣀⠀⣀⠀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡯⢸⣿⢸⣿⢸⣷⢸⣇⣸⡇⠀⠀⠀⢸⣿⠀⠀⣸⣿⡇⢸⣿⡇⣼⣿⡇⣿⠀⣿⠀⢻⣇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣌⠁⢸⣿⢸⣿⠈⣿⣿⢀⡀⠀⡀⢸⣿⣀⣀⣿⣿⡇⢸⣿⣿⣿⣿⡇⣿⠀⣿⠀⠸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢠⣌⢻⣧⢸⣿⢸⣿⠀⣻⡏⠀⣹⢮⡁⢸⣿⠀⢨⣿⢻⣷⢸⡟⣿⣿⢿⡇⣿⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣏⢸⡿⢸⣿⠋⠁⠀⣽⡇⠈⠁⠀⠁⢸⣿⠀⢸⣿⠛⣿⣼⡇⢻⡏⣼⡇⣿⠀⣿⡀⠀⣿⡇

⠀⠀⠰⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣤⡀
⠀⠘⢣⠀⠀⠀⠃⢀⡒⠀⠀⠀⠀⠀⣄⣀⠀⣀⠀⢀⠠⠀⡀⠀⠀⠀⣀⣴⣟⣋⢭⢤⡀⠝⢋⣦⣜⠳⠄
⢀⠀⠾⠀⠀⢀⢡⡦⠁⠀⠀⠀⠀⠚⠀⠀⠌⠁⠀⠈⠀⠰⠀⠀⠀⢰⣿⣿⣿⣿⣥⡈⠳⢦⡀⠀⠫⢷⡀⠰⡀
⠂⠀⠀⠀⡄⠈⠘⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⠀⠈⠻⣶⠹⢆⠀⠹⢆⢄⢙⡇⠀⢷
⠀⢠⣦⡾⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣿⣤⣤⣴⡿⡀⠰⢄⢀⣻⣆⠢⣇⠀⠈
⣾⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⠀⠀⠀⢴⠉⣷⠀⢳⢹⣿⣆⠹⠀⠘
⠑⠙⢾⠃⡀⠀⠈⠂⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⢸⡀⣇⡇⣠⠘⣿⢜⠇⡆⢀
⠀⠀⢐⡀⢀⣟⡄⠱⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠀⠀⠀⠐⠶⠖⡿⠒⠒⠒⠛⠃⣻⢿⡌⠁⣝⠄⠀⡷⣷
⠀⠀⢈⡁⠀⡜⣆⠏⣄⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⣤⣷⠄⠀⠀⠀⠀⠻⣈⠟⢷⢈⣇⡀⢰⣟
⠀⠀⠈⢥⠂⠀⠈⠘⠀⢀⠀⠀⢃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠈⠀⠀⠎⠰⡀⢹⠀⠀⠀⢏
⠀⠘⠣⡀⠀⠀⠀⠁⠀⠳⠆⠀⠀⡀⠀⠀⠀⢘⣤⡀⠀⠀⠀⠀⠉⢠⠆⡠⢤⡴⢂⡜⣶⠀⠀⠇⣸⠀⠈⠐⠺⡄⠀⠀⠀⠀⢀⠤⡀
⠀⠀⠐⢋⠆⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠤⠏⠁⣾⡀⠀⠀⣀⠋⣸⣿⢒⣧⣤⣷⡾⢄⠄⠀⠃⠀⢂⡄⠐⢄⣂⡀⠀⠀⠘⠂⠼⠄
⠀⠀⠀⠈⢆⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠃⠁⢰⡾⠁⣴⠟⢿⡟⣼⣄⣈⠠⣀⠆⠀⡄⢳⠀⡀⢟⠣⠀⠙⠒⠂
⠀⠀⠀⠀⠈⠀⠀⣶⠀⠀⠈⣄⠀⠀⠄⠀⠀⠀⡀⠁⢀⠞⠃⠀⣼⠶⣷⡄⣿⣠⠼⣿⠓⢹⡆⢤⠀⠸⡆⠁⠈⠑⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠘⣋⠀⢠⡴⠁⠠⠃⠀⡀⠀⠠⠀⠰⠥⠀⢠⣀⣸⣿⣽⣧⣞⣹⡾⢷⣿⣄⠐⢸⠇⠸⣧⡆⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠘⠁⠀⠞⠀⡀⠁⠀⠀⠀⠀⠀⠀⠐⠁⢠⠀⠁⠠⠘⣯⠿⠛⣿⣿⣽⣯⡶⢿⣿⣼⣿⠊⠇⡀⠡⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⣷⡆⢠⡾⠁⠀⠀⠀⠀⠀⠁⠐⠁⠀⠈⠀⠀⠟⢛⡛⣟⣋⡿⢻⡟⢻⣶⣿⣯⣧⣿⠀⣀⠁⡄⠀⠀⠀⡀⠀⠀⠀⠿
⠀⠀⠀⡨⠁⠠⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄⣌⡀⢻⠛⢤⣼⣷⣿⣩⣧⣼⣿⣬⠀⠿⣦⠀⠠⢠⣄⠀⠀⠀⠀⢀⠖
⠀⠀⠀⠐⠎⠀⣧⡀⠀⠀⠀⠀⠀⠃⠀⡄⠀⠀⠈⠀⠠⠈⢻⠲⢮⣴⡂⠠⢔⣵⣻⣿⡷⣾⣿⠀⠀⠀⠀⠀⠀⠉⠁⠒⠂⠉⠁
⠀⢀⠀⠐⠒⣋⣅⠈⠙⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠡⠀⠲⢟⡤⢽⣶⣿⣋⣭⠃⣵⡇⢿⠀⠀⠀⡀
⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⢀⣄⣳⣾⠟⠛⣿⣷⣶⠿⢳⣿⡆⠺⣿⡇⠀⠀⢤⣶⡆⣦⡛⠰⠀⢃⠒
⠀⠀⣄⠀⠀⠀⠀⠔⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⣻⣧⣄⢳⣾⣿⠷⣿⣶⠏⢰⡿⣄⠈⠁⢸⣷⣤⡉⠃⠉⠐⠆⠉
⠀⠀⢉⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠿⠻⣿⢿⣿⡏⠜⠙⢧⠘⣇⡿⢦⡀⠈
⡀⠀⣬⠛⣤⠧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠆⠾⠁⠀⠁⠀⢠⣬⣧⡉⣶⣿⣿⣵⠾⣥⣶⡌⢠⣿⡇⣀⡽⠦⣀⠀⠀⠀⠲⠄⠀⠖⠦⠤
⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⢀⡔⢠⠀⠃⠀⠀⠁⠀⠀⠀⠀⣾⡁⢿⣷⣝⣟⣿⣟⠒⡟⠹⢧⣼⣿⣿⠿⠖⠛⢻⡻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠄⠂
⠀⢅⠐⠊⢀⠗⠁⠀⠀⢀⣾⣼⡳⠋⠀⠀⠀⠀⠀⠀⠀⢘⡫⠿⠻⣷⣻⣏⣹⡿⣿⡦⢠⣼⣏⣀⠋⡀⠒⠦⣾⠁⠈⠻⣝⢳⠠⢀⡀⠀⠀⠀⠲⠀⠀⢰⠁⡄
⠀⠋⠠⢖⣯⠁⠀⡀⢠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣦⣴⡆⢰⡟⢡⣽⣆⣼⡇⢾⡇⢠⣿⡿⠋⣆⢸⠇⠀⠀⠀⠈⠻⣧⡀⠀⣉⡠⠖⢶⣤⣀
⠀⠀⠀⠀⠃⣀⠰⠦⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠉⠿⠛⢡⣿⣯⣿⡍⠛⠑⢁⡼⠋⢁⢰⡏⣿⠴⠒⠶⠶⠶⠶⢞⣷⣮⣄⡀⠀⠸⡄⠉⠓⢦⣁
⠀⠀⠀⠀⢰⡁⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⣀⣤⣤⣄⠀⢉⡀⠘⣠⣤⣴⠟⠇⢠⣄⣾⣇⢸⡆⠀⠀⠀⣀⡴⢋⢀⢻⡆⠉⠓⠤⣿⣀⣠⠴⠞⠛
⠀⠀⢀⡀⠈⠁⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠘⠀⠀⠀⣹⢿⣹⡟⣷⣀⣡⡏⣴⣦⣄⣿⣿⡏⢼⣦⣹⣄⡤⠞⠉⠠⠀⠾⠀⢿⡀⠀⢠⠼⡿⣄⠀⠀⡀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣄⠀⠀⡟⠠⡟⢁⣿⣋⠙⠿⣯⡄⣼⠋⢸⡷⡀⠹⣿⢳⣄⡀⠄⠀⠀⡾⠀⠈⣳⡴⠉⠀⠁⠈⠑⢤⡇
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⡀⠀⡇⠀⢩⡜⠻⣿⣢⣄⣼⢷⣿⠀⣏⠀⣿⠗⠻⠘⢯⠙⡦⠀⠀⣀⡠⠐⠛⢧⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠓⣚⠛⣽⠋⠙⣿⠛⣦⠟⠚⠻⢦⣄⠀⣾⣞⡴⠾⠛⠛⠳⢄⡀⠈⢧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢠⠀⠀⠀⠈⣿⠞⠹⣃⣼⢅⠀⠙⠣⣄⡙⣿⡏⠀⠋⡉⠀⡰⠆⠀⠀⠀⣉⣳⣬⣷⣄⡀⠤⠀⢒
⠀⠀⠀⠀⠀⠀⠀⢀⣐⠀⣿⠀⠀⠀⠀⠀⠖⠀⠀⠀⢉⠀⠀⢿⡆⠀⠀⠀⢻⡳⣄⢻⡜⠻⣤⣿⠶⠈⢫⠘⠷⠂⠀⢠⠐⠀⢀⣤⣄⠀⠀⠈⠁⠉⠉⠁⠀⡠⠃
⠀⠀⠀⠀⠀⠈⠀⠈⡍⣄⣸⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢄⠀⠀⠀⠵⢜⠳⣿⡆⢸⣇⣠⣀⣄⠋⠀⠂⠀⠉⡠⡄⠀⠀⠙⠆⠀⠀⠀⠀⠀⢀⣶⠁
⠀⠀⠀⠀⠀⠀⢀⠒⠀⣄⡜⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠙⢿⣿⠈⣳⠊⠘⣦⠀⠀⠚⠋⠶⢧⣤⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁

⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠐⠄
⠀⠀⠀⠀⠀⠀⠁⠀⢄⢸⢹⡇⠀⠤⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠄⠸⡘⠁⠀⠀⠀⢸⡭⠚⠁⠀⠐⢠⣤⡬⠂⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⡇⠀⠠⣄⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⡼⠊⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠐⠁⠀⠀⡀⠀⡔⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠀⣀⣁⣀⡀⠀⠀⠀⠐
⠀⠰⣾⣿⣇⠈⡀⠸⢀⠸⠠⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡉⢿⣿⡿⠀⠀⢀⠁
⠀⠀⢿⣿⡏⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠌⣿⠃⠀⠀⠈
⠀⠀⠘⠃⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⡁
⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣀⣀⡀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⢠⠞⠉⣭⣝⢣⠀⠀⠀⠀⠀⠀⠔⢉⣶⣬⠳⡄⠀⠄⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢰⢧⡏⢰⢶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠰⣾⠿⣯⠃⠸⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢸⠘⠀⠀⠘⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢠⠀⠈⠐⠀⢀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠓⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠒⠉
⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⢀⠀⠀⠀⢀⡠⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠘
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁