Playfulness Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠁⠒⠠⠄⠸⡄⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠊⠉⠉⠉⠁⠒⠲⢄⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣄⠀⠻⣄⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠈⡳⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠉⠓⠤⡴⠁⡴⠁⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢂⠀⢀⣀⠀⠀⠀⠀⠘⠣⡀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⡠⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢿⡎⠔⣵⢶⡇⡀⠀⠀⠀⠀⠈⠢⣀⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠘⠠⣄⠀⠁⠂⠤⣀⠀⢀⡤⠊⠀⠒⠊⠰⡀⠛⡻⢁⡅⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⣸
⠀⠂⠒⠒⠒⠠⢀⠀⣀⡾⠋⠀⠀⠀⠀⠀⠀⠈⠉⢀⠎⣵⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣿⠀⠀⠀⠀⠀⠀⡏
⠀⠀⡔⠀⠀⠀⠀⠚⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⣰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠞⣁⠔⠁⠀⠉⠐⠒⠣⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⢻⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠙⠦⣤⣀⣀⣠⠎⠀⠀⠀⠀⣀⡤⠚⣩⣌⡉⣹⠶⠦⢤⣀⡀⠀⠀⠀⠈⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢸⠀⠀⠘⣆⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠉⠂⠠⠄⠔⠂⡹⠁⢠⠂⠀⠣⢄⣀⣠⠔⠊⠁⠉⠓⠲⢤⡴⠋⠀⠀⠀⠀⠀⠀⢀⣀⡤⡾⠉⠀⠀⣸⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠋⠉⢀⠔⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠉⠳⣆⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⣀⡼⠀⠀⠀⠀⠀⠀⡇⠀⠀⠳⣄⣀⣢⠀⠀⠀⠀⢸⠀⠀⠔⠉⠉⠁⠀⢲⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠴⠃⠀⠀⡜⠁⠈⠙⡇⠀⠀⠀⢸⠀⠀⠙⠧⠤⠼⠆⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠇
⠀⠀⠀⠀⢠⠋⠁⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⣀⣀⣀⡤⠤⠤⠷⠶⠦⠤⠿⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠃⠀⠀⢸
⠀⠀⠀⠀⠈⠒⠧⠴⣅⡴⣒⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣠⠀⢀⣄⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣀⣠⠂⠀⢰⠒⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠆⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠀⠃⠀⠀⠀⠊⢀⢀⠄⠰⡆⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠐⠛⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠢⠂⠀⡀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⣠⠔⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⣀⡠⠀⠀⠀⠀⠂⠀⠀⠀⠀⡌⠀⠀⠁⠀⠀⠀⠀⢀⡴⠇
⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠃⠐⠧⠒⠒⠈⠀⠈⠒⠂⠚⠀⠄⠄⠐⠈⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠁⠀⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⠤⠀⠤⠄⠠⠤⠤⠄⠀⣀⡀⠀⠀⠀⠀⣀⡠⠔⠀⠀⠀⠀⠀⠉⠓⢄⡀⡀⠀⠀⠀⠀⠴⢠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠻⣿⣿⣿⣿⡇⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢿⣿⡇⠀⠀⠀⢘⡔⠚⠛⢃⠸⠓⣟⠟⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡇⠀⠀⠀⠘⠙⠛⠛⠛⠛⠂⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠶⠀⠈⠣⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦⣌⣁⣴⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⠉⠚⢱⣾⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣄⣀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⣴⣀⣄⣄⠀⠀⠀⢿⢿⠁⠀⠀⠀⠻⠿⠃
⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣧⠘⠛⠀⠀⠀⢰⣶⣾⣶⣶⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣧⡀⠘⠂⠀⠐⠂
⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠉⠉⠋⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⣿⣿⣿⡿⠿⡫⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣶⣀⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⠹⠍⠋⡃
⠀⠀⠀⠀⠻⣿⣇⢻⡀⣷⣼⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⡿⣟⣿⢿⣏⣻⠛⠛⢿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣼⣀⡔⠁
⠀⠀⠀⠀⠀⣽⠿⠿⠿⠛⠻⣿⣿⣿⣿⣿⡃⠘⣿⢯⣯⠿⣅⠿⣿⣮⣯⣯⣳⠒⠈⠘⠿⠟⠛⣹⣿⡋⠉⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠸⣷⣶⣶⣷⣶⣿⣿⣿⣿⣽⡉⠲⠈⠁⠙⠃⡀⡉⢘⠃⠒⠎⢳⣾⣿⣶⣾⣾⣿⣿⣿⣿⣿⣷⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⢶⠆⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⡀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⢀⡀⠀⠐⠀⡇⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠆⠀⢻⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠘⠀⠀⠒⠊⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡄⣠⡀⠀⠀⠀⠀⡀⠑⠲⠴⠒⠁⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁⠀⠁⠈⠐⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⢀⠄⠀⢄⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠉⠢⢄⣀⣀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⡀⠀⠀⠄⠁⢠⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠀⡐⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⡄⠀⠀⠀⠀⠡⣬⠀⠸⡉⠀⡜⠀⠀⠀⠀⠀⠀⡠⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢀⡜
⠳⠄⠀⠜⠁⠐⠀⡅⠉⠉⠈⠀⠀⠀⠀⠀⡠⠊⡜⠓⠂⠤⢤⠊⠉⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢠⡄⠠⠤⠔⠈⣠⣾⣤⣀⡀⠀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠈⢦⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠤⠤⠐⠋⠁⠙⠁⡐⠀⢹⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⡄⠀⠀⠀⠑⢤⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠀⠘⢦⣀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⣀⠵⠀⠀⢀⡤⠊⣀⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠂⠤⠤⠤⠄⣀⡨⠀⢨⠁⠒⠚⡄⠱⡤⠤⠄⠒⠊⡡⠐⠂⠉⠀⠀⢀⠔⠋⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠄⢀⠀⢠⠤⠤⠀⣠⠖⠒⠚⠒⠒⠢⢼⡄⠱⡤⣄⣄⣀⡟⠉⠉⠒⠒⠞⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢾⣀⣼⣀⡸⠀⠀⠀⡇⠀⡄⠀⠀⡀⠀⣄⠈⠂⠘⢄⠀⠸⣀⡆⠀⡄⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⣆⣀⣀⡧⠤⠇⠈⠒⠢⠚⠀⠀⠀⠈⠉⠙⠒⠚⠁
⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⡴⣂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⠀⠈⠈⠸⡷⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⣀⣤⣤⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⢰⡿⠀⢀⣾⡿⠋⠁⠀⠀⠀⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠑⠧⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⢠⣿⠃⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠘⠐⠲⠞⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠆⠁⠀⠀⠀⠀⠀⢠⣾⠏⠀⢀⣾⠇⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⢀⣾⠏⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⢀⣾⠏⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠃⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⣠⣿⠃⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣅⠀⣴⡿⠁⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣁⣨⣿⠟⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠀⠀⣿⡇⠀⠀⠀⠘⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⡿⠛⠁⠀⠀⣠⣶⡿⠋⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⣨⡷⠀⢀⣴⣾⠟⠉⠀⠀⢀⣿⠟⠉⠉⠙⣦⠀⠀⠀⠀⠀⠀⣴⣿⣷⣦⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣡⣴⠟⠁⢀⣾⡟⠁⠀⠀⠀⠀⠈⣿⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠻⣿⣿⣿⠇⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠁⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⡾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⡿⢷⡄⠀⠀⠀⣸⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠘⣿⣿⣿⣿⠀⠀⣴⣾⣶⣤⠀⠀⠘⢿⣿⣿⠏⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠹⣆⠀⠀⠈⠙⠋⠁⠀⠀⢻⣿⣿⣿⠇⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣀⣤⣾⠿⠋⠀⠀⠀⠀⠀⠀⠨⢄⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⣦⣤⣤⣤⣤⣴⣶⡾⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢡⠖⠒⠢⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢹⡀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠊⠀⠀⠈⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠠⠂⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠊⠉⠉⠉⠉⠒⢄⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠁⠀⠀⠀⠉⠳⢤⡁⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣀⣀⠀⠀⠈⠀⠀⠀⢀⠋⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢠⠀⠀⠀⡌⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠚⠤⡀⢠⠁⢀⠟⠒⠲⠤⠤⣀⣀⡣⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢞⠓⠀⠀⠰⡄⠀⠉⠰⢏⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⣰⠁⠀⢙⡆⠀⠀⠀⠀⠠⢲⡆⢨⡴⠊⠘⡄⠃⠀⠀⠀⠀⢣⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⡏
⠀⠀⠀⢀⡸⠄⠀⠘⠄⣀⠀⢀⠀⠓⠈⠐⡙⢾⠅⢰⠀⡆⠀⠀⠀⠀⠀⠧⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡃⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⢏⠀⢀⣄⡀⠀⢀⠜⠁⠀⠀⠀⠀⠘⠦⠔⠃⡸⠑⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡇⠀⠀⠀⠀⠀⡞
⠀⢠⠀⠈⠉⠁⠒⠰⢟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠈⡇⠀⠀⠀⠀⠀⡇
⠀⡆⠸⠂⠀⠀⢀⡀⠀⡗⠀⠀⠀⠀⠀⠀⠀⢀⠔⡀⠈⠐⠢⠄⣀⠀⠀⣠⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⣧⠀⠀⠀⠀⠀⡇
⠀⠣⡎⠀⠀⢀⠋⠂⠀⡇⠀⠀⠀⠀⠀⠀⡠⠊⠴⢦⣤⣀⡀⠀⠀⠉⠘⠃⠀⠀⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⠂⠀⠀⠸⡄⠀⠀⠀⠀⢻⡀
⠀⠀⠈⠑⠲⠼⣦⣤⡴⠁⠀⠀⢀⣠⡴⠊⠀⢠⠐⠂⠁⠀⠈⠁⢲⠀⠀⠀⡶⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⢸⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠛⢩⠁⠀⠀⠀⠁⠉⠉⢸⠀⠀⠀⢸⡀⠀⠀⢱⠀⠀⠀⠀⠀⢀⣠⠴⠖⠋⠉⣣⠀⠀⢸⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢣⠀⠀⡘⠀⣄⠀⠀⠙⠒⠚⠁⠀⠀⢸⣠⠞⠉⠀⠀⢀⡤⠊⠀⣀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡠⠎⠀⢠⠇⠀⠈⣱⠒⠦⢴⡀⠀⠀⠀⢸⠁⠀⢀⠊⡉⠉⡀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⡏
⠀⠀⠀⠀⠀⢀⡠⠤⠠⠤⠤⠤⠄⠀⠀⠤⢤⠤⠖⠋⠀⠀⢰⠃⠀⠀⠀⠇⠀⠀⠀⢸⠀⠀⠘⠒⢦⠤⢇⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⠦⠾⠀⠀⣸⠁
⠀⠀⠀⠀⢠⠎⢠⠀⡀⠀⠀⠀⠀⣀⠤⠤⠤⠤⠤⠤⠤⠤⠿⠀⠀⠒⠒⠚⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⢺⣀⡃⢠⡇⠀⡷⡶⢋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠾⣀⣠⣇⠀⢀⡝
⠀⠀⠀⠀⠀⠀⠉⠙⠓⠚⠹⣴⠋⠀⡞⠀⠀⠸⡏⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠦⠤⠤⠾⠃
⣿⣷⣶⣶⣶⣾⢷⣾⣷⣷⣿⣶⣶⣿⣶⣶⣶⣿⣷⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣷⣾⢷⠾⠷⠿⠷⢾⣿⠿⠿⠿⠿⠿⣿⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⣦⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣸⣿⡿⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠇⣰⣿⣿⣧⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢡⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⣏⣶⣿⣿⣿⣿⡷⠃
⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⣠⠴⢒⣋⣭⣥⣴⣦⣬⣕⠾⣾⣿⣿⣿⠟⠁
⠀⣾⣿⣷⣮⣭⣭⣁⡒⠤⢀⠀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢠⠞⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣀⣀⡠⠠⠀⣄⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡰⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠶⣾⣿⣿⡿⠿⠟⠿⣿⣿⣿⣿⣿⡿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⡜⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠈⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⢿⣪⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣾⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣯⣶⣿⠟⠀⢰⣿⣿⣿⣾⣿⣿⣇⢿⡿⡗⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣧⠀⠆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣭⡆⠀⠘⣿⢿⣏⡽⣿⣿⣿⣯⣁⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢱⠇⢿⣿⣿⣿⢿⣿⠘⡄
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠖⠋⠀⠀⠀⠙⣦⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⠇⡼⠀⡞⣿⣿⣿⠈⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠋⠹⠿⠛⠀⢀⣀⣀⡀⠀⠘⠛⠉⠙⣿⣿⣯⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⢠⠃⠀⠸⡿⠙⣿⡀⠀⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡀⠀⢀⠆⢰⣿⣛⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣰⠎⠀⠀⠀⠹⡄⠈⠃⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠙⢛⡛⠛⢀⡠⠃⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡴⠋⠀⠀⠀⠀⠀⠈⠲⢤⡀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡘⠻⡉⠻⠝⡇⠠⣾⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⢟⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢳⡀⣀⠜⣶⡘⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡿⢋⡸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⡼⣿⣿⣿⣿⣿⣿⣸⣿⣿⢟⠕⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡤⢴⣶⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣇⡩⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠐⠀⠀⢣
⠀⠀⠀⡄⠀⠀⠀⠂⠀⠀⠀⠐⢟⣻⣿⠿⣟⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠴⣄⣀⣰⣤⡼⠃
⠀⠀⢨⡁⠀⠀⠀⢀⡠⠄⢒⠨⠀⠈⠀⠀⠀⠈⠀⠛⠋⣉⣁⣋⣙⠋⠵⠃
⠀⠀⠀⠈⠀⠒⠐⠋⠀⣆⠀⠀⠀⠀⠀⡠⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠧⣤⣦⡰⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣿⣥⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣿⣶⣾⣿⣯⣿⣷⣾⣿⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠄

⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠈⠻⣆⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠀⠠⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠁⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡠⠄⠒⠒⠒⠢⣄⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢋⠑⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠊⡀⠑⠄⠀⠀⠀⠘⡀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠀⠀⠀⠀⠁⠢⡞
⠀⠀⠀⠀⠀⠀⠠⠊⠱⡀⠀⠢⠀⠀⢀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡠⢔⠋⠉⠉⠉⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡠⢄⠁⠀⠀⠀⠀⠀⠘⠓⠠⡄⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠐⢦⡀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⢀⠄
⠀⠎⠀⠀⠐⠀⠀⠀⠀⠀⠛⠁⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢠⠄
⠀⢆⣣⣀⠴⠀⠀⠀⠀⠀⠀⠀⢀⡄⣰⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⠁
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣠⠎⢠⡿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠈⠢⠤⣀⡠⠤⠴⡊⢁⠔⠻⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠔⠀⡠⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢸⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⡸⡇⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠂⠀⠀⠀⠀⠀⠀⠄⠀⠀⢆⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⡰⢡⠃⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⣀⡴⢋⠔⠁⠀⠰⢁⡜⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠠⠤⠒⠉⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⠀⠱⡖⠊⠁⢮⣀⡄⢀⣠⠗⠈⠀⠀⢀⠎
⠀⢀⠆⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠤⠒⠓⠀⠀⠀⠀⢱⠀⠀⠀⠀⠉⠡⡅⡀⠀⠀⢀⡜
⠀⠈⠮⣀⣀⣀⣀⡀⠀⠤⠤⠐⠒⠒⠒⠒⡲⠛⠉⠀⠀⠀⠀⠀⠀⣀⣠⡤⠞⠀⠀⠀⠀⠀⠀⠀⠑⠂⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣀⣀⠀⣠⣀⡤⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠈⠉⠉⠀⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⠤⠖⠒⠒⠤⣄⡀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⣄⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⢀⠠⠒⠁⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠈⢀⡎⢸⡀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠤⣀⠀⡜⠀⢠
⠀⠀⠀⠁⣀⠤⠤⠒⠲⠖⠤⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢧⠀⠀⠀⢣⠀⠀⠀⠀⠀⣀⠔⠊⠀⠀⠀⠀⠀⠀⠙⢧⡴⠋
⠀⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⢀⡈⠂⠀⠁⠀⠀⠈⠳⣄⣀⣠⠇⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠊⠈⡆⠋⠈⢳⡀⠀⢣⠀⠀⠀⠀⠉⠁⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⠇⢰⣦⠉⢷⡆⠀⣇⠀⠀⢂⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠈⠦⠤⠊⠀⠳⣄⣠⠇⠀⠈⢦⠳⢄⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⣰⠋⠀⠀⠉⠉⠀⠀⠙⡆⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⠀⢑⠀⠀⢰⠟⠀⣠⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⡀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠓⠲⠤⠧⠖⠓⠚⠁⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢰⢧⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣄⣀⠀⠀⣸⢦⣀⠀⣀⣀⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡏⠈⣧⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡏⠙⢥⣤⣭⠟⠻⡁⠱⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⡇⠀⠈⢣⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣴⢋⠇⢀⡔⠁⠀⢠⠏⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⡘⢤⣠⠞⠀⠀⠀⠀⠀⣠⠴⠋⢉⠁⢠⡏⠀⠀⢀⡏⠀⢠⠏
⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋⠀⠀⡗⠦⠤⠤⠘⠦⣀⠀⠀⠀⢲⡖⠉⠣⢧⢤⣇⡤⢞⠔⠒⠒⠚⠀⢀⡞
⢠⠊⠁⠀⠉⠉⠛⠛⠋⠉⠉⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⣀⣨⠷⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⢯⣰⠀⠀⡆⠀⢻
⣇⣰⠃⢠⠀⠀⡀⠀⢀⣀⣀⡤⠖⢛⡭⠖⠒⠒⠚⠛⠉⠁⠀⠀⣀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠘⠓⠒⠓⠶⠚
⠈⠉⠳⠾⣤⣤⠿⠉⠉⠉⠀⠀⠀⡟⢰⠂⢠⠆⠀⢠⡶⠒⠛⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣻⣶⣾⣦⣴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠈⠉⠉⠉⠁⠒⠦⣄
⠀⣀⡀⠠⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠉⠲⣄⠀⠀⠀⠀⡠⠊⡠⠔⠒⠒⠒⠒⠤⣀⠀⠀⢈⡇
⡞⠁⠀⠀⠀⠀⠉⠲⢄⣀⠠⠤⠒⠒⠒⠤⣠⠋⠀⠀⠀⠀⠀⠈⠣⡀⠔⣋⠔⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉
⠑⠲⠴⠤⠤⣄⣀⠠⠤⠄⠒⠒⠉⠀⠑⢢⠀⠀⠀⠠⠀⠀⠓⠀⠀⠀⠺⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠐⠃⠀⠛⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⣀⣤⠤⠤⠤⠤⢤⣀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⡴⠋⠀⠀⠀⠀⠀⢀⠀⢈⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⡀⠀⠳⡀⠰⣄⣀⣤⣴⠫⠔⠉⠀⠀⠀⡼
⠀⢀⣀⣴⠛⢳⡤⣄⠀⠀⠀⠀⡜⢠⠃⢣⠀⠀⠀⠉⠉⠉⢻⡄⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢻⡀⠙⠀⠘⠀⢈⠇⠀⠀⠀⢇⠘⠂⣐⢧⡀⠀⠀⠀⠀⡘⠈⠲⠤⣤⡤⠖⠁⠀⠀⠀⠀⠀⠀⠀⢴⠉⢀⠀⠉⠢⡀
⠀⠀⠙⡄⢀⣀⢢⡏⠀⠀⠀⠀⠀⠉⠉⠀⠀⠙⠢⠤⠤⠚⡅⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠻⣆⣠⣀⡹
⠀⠀⠀⠹⡄⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⢰⠏
⠀⠀⠀⠀⠙⢦⡀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠙⠳⢤⣄⡉⠓⠶⠤⣤⣀⣀⣀⣀⣀⡜⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⢀⣀⡤⠞⠋⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠶⠤⣤⣤⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣉⣉⣀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢉⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⣿⣤⠶⠶⠛⠛⠛⠶⢦⡀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠒⠦⣄⡹⣄⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣠⠞⢁⡾
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠉⠉⠓⠂⠈⠐⠤⣀⡀⣀⠠⠔⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠖⠛⠳⠤⠤⠤⠄⠒⠉⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⣀⣀⣀⠀⠤⢄⣀⡀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠄⠀⠀⠀⠈⠉⠉⠉⠛⠛⠻⢶⡒⠒⠒⠒⠒⠋⠁
⠀⠀⠀⠀⣠⠴⠒⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣈⣹⠏⠛⠛⠛⠛⠉⠙⠧⠤⠤⠶⠶⠶⠶⠶⠦⡄⠀⢀⠀⢀⢹
⠀⠀⠀⠀⠧⣴⣠⡀⢰⠆⠒⠒⠚⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠼⠥⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢀⡴⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠰⠤⣀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠀⠀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠐⠊⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠘⣄⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠤⠄⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠈⠉⠓⢦⡶⠒⠓⠒⠂⠀⠁⠀⠋⣀⢤⠀⠀⠀⠘⣤⣤⣤⣤⡤⠤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢀⡀⠀⢈⠠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠁⠀⠀⡴⠛⠂⠘⠀⠀⣤⣬⡁
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠐⠀⠀⠀⠠⠀⠀⠀⠀⠙
⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢠⠏⢸⠀⠀⠀⠀⠀⢀⢠⠀⢀⣄⠀⠀⣀⡀
⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢀⠰⠋⠀⣴⠀⠀⠐⡻⠛⠊⠀⠀⠈⠁⠀⠔⠻⢁⡀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⣼⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⣲⣾⣿⡿⠿⠿⢿⣷⡄
⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣷⣤⣤⣾⣿⣿
⠹⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢲⠞⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠘⠿⣿⣿⣿⣿⣷⣄⡀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠈⠀⠀⢢⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠛⠋
⠀⠀⠀⠈⠙⠻⠿⢿⣿⠿⠋⠁⠀⢹⣿⣿⣶⣶⣶⣶⣶⡿⢻⣧⠀⠀⠀⠡⡀⠀⠀⠀⠈⠑⠒⢀
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠃⠀⠙⢷⡀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠈⠰⡆⠦⠀⣀⡄⠀⠀⠀⠀⣀⠔⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⡞⠀⢀⣴⣿⣿⣿⣶⣄⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠑⠠⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⢸⠁⠀⢾⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣶⣿⣿⣿⣶⡄⠀⠀⠉⠢⢄⠀⠀⠀⠀⠈⠐⠂⠼⠷⢦⡀
⠀⠀⠀⠀⠀⢸⡆⠀⠈⠿⣿⣿⣿⡿⠋⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠁⠒⠄⣀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣰⠖⠢⡀
⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠂
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠐⣆⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠼⠿⡶⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠾⠿⠁⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡘⠀⠀⢄⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠘⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠈⠆⠐⠀⠀⠀⠀⠁⠀⠀⠂⠠⢀⠀⠀⢀⠀⠤⣀
⠀⠀⠀⠀⠈⠓⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑⠀⠑⠄⠙⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⡈⢢⡀⠐⣀⢸⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣴⣶⣿⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⣀⣤⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡌⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⣹⡇
⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣦⡿
⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⢸⣿⣿⣿⣿⣿⣿⠉⠉⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣤⠖⠊⠉⠁⠀⠈⠉⠉⠒⠶⣤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢇⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⢀⠇⠀⠀⠀⣿⣙⠄⢀⡠⢤⡀⠠⣋⣿⠀⠀⠀⢸⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢸⠤⠤⢤⣀⣉⡩⠔⣉⠤⠄⣈⠳⠭⣩⠤⠤⠤⠬⡇⠀⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⡇⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⢸⠤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠢⡴⠞⠙⡶⠖⠂⠀⠀⠀⠀⠀⢸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠙⠦⠔⠃⠀⠀⠀⠀⠀⢀⡴⢻⠁⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠑⠒⠤⠤⠤⠄⠤⠤⠤⠤⠔⠒⠉⡇⠀⣎⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢹⡎⠀⢉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢹⠀⠀⠑⢄⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⡇⠙⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢇⠀⠀⠀⠉⠒⢦⠔⠒⠉⠀⠀⠀⣸⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠦⠤⢘⣆⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣠⡃⠠⠴⠥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡀⢀⠀⡎⡀⠉⠉⠉⠀⢸⠀⠀⠈⢑⠋⢁⢹⠀⡀⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠼⠤⠟⠧⢄⣧⠤⠴⠚⠑⠢⠤⠼⡤⠼⠞⠤⠧⠜⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⠿⠿⠿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠀⠀⠀⠀⢀⣤⣴⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⠀⠀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠻⢷⡶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣙⣷⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⢀⣴⣾⠟⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠉⠁⠀⠀⠀⠀⠀⠀⠘⠻⣷⣄
⠀⠀⠀⣀⣴⣿⠟⠁⣀⣤⣶⣾⣿⡷⠀⠀⠀⢀⣼⠏⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠙⢷⡄⠀⠀⢰⣿⣷⣤⣀⡀⠈⠻⣷⣦⡀
⢀⣴⣾⣿⣿⣷⡿⠿⠛⠋⠉⠀⣿⡇⠀⠀⠀⣼⣇⣀⡀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣀⠈⣿⠀⠀⢸⣿⡍⠉⠛⠿⣷⣶⣬⣽⣿⣷⣤⡀
⣿⣿⣟⡋⠉⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⣿⣿⣏⣽⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣩⡇⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿
⠀⠙⠛⠿⣿⣶⣶⣤⣤⣤⣤⣾⣿⠁⠀⠀⠀⠙⠿⢿⣿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⡷⠟⠀⠀⠀⣿⣧⣀⣀⡀⢀⣀⣠⣤⣶⣾⣿⠿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠋⠙⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣯⣍⣙⢲⡄⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠈⠛⢿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⣀⣀⣀⣤⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⡀⠀⠀⠀⠀⠈⣽⡿⠛⠻⠿⠿⠿⠿⠿⢿⣿⣥⣬⣿⠿⠿⠿⠟⠛⠛⢿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⢿⣧⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠈⢿⣷⣤⣀⣀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠈⠛⠻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⡀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⡀⠀⢀⣤⣶⡿⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⣶⣶⣶⣶⣶⣶⣾⠿⠟⠁⠛⠿⣷⣶⣶⣶⣶⣶⣶⣿⠿⠿⠛⠋
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⡀⡀⠀⠐⠀⠀⠀⠀⢄⣠⠀⠀⠀⠀⠀⠈⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠠
⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠆⠂⠀⠐⠄⣀⠀⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠠⠀⠀⢀⠀⠘⠖⠀⠀⠀⠰⠐⡂⠐
⠀⠀⠀⠀⠀⠀⠤⡌⠀⠀⠀⠀⠀⠀⠷⠀⠀⢠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢈⡄⠀⠐⠈⠒⠉
⠀⠀⠀⠢⠂⠀⠀⡄⣀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⡠⠀⠀⠤⠩⠀⠀⠀⢄⠀⠀⠀⠀⡠⠀⡀⠀⠁⠀⠀⠀⠀⠠⠈⢁
⠀⠠⡀⠀⠀⠀⢀⠀⢈⠁⠀⠀⠀⠈⠀⣤⠄⠀⠘⠤⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠀⠀⠂⠈⠀⣀⣁⠈⠀⠀⠘⠁
⠹⠆⠀⠀⠐⠀⣁⣠⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢨⣤⠠⣄⡖⠢⠀⣀⠀⡀⠀⠈⠹⢆⡀⠀⠀⡀⡐⠂⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡦⢀⠄⠀⠀⠀⠀⠙⠁⠐⠀⠀⠆⠈⠀⠁⠀⠀⠀⠠⠴⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⡂
⠀⠀⠀⠀⠁⠀⢀⣄⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠳⡂⠀⠠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠚⠛⢄⡀⣀⠀⠛
⠀⢘⢀⠀⠀⡼⡏⠁⠀⠀⡀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠄⡀⠐⠂⠀⡓⢠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠦⠄
⠀⠦⠀⠡⠔⠀⠸⠀⠀⠀⠀⠀⠀⢦⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⠀⠈⠁⠗⠀⢦⡀
⠀⠠⣄⠓⠀⣆⠸⢧⢀⣀⣀⣀⠸⠏⠁⠀⠀⠀⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⠢⠄⠐⠿⠀⠀⢠⠀⠀⠀⠀⠰⡦⠃
⠀⠀⠈⠁⠈⠁⠀⠤⠴⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠴⠆⠀⠰⠦⠆⠈⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣰⢁⡁⠀⣰⣆⠀⢰⡖⢀⡀⣀⠀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠛⠉⠙⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⢤⡀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⡀
⠀⠀⠀⠀⢀⠂⠀⠀⢠⠞⠀⠀⠀⠀⠙⡀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢳
⠀⠀⠀⠀⢸⠀⠀⣰⠋⠀⢀⣀⣠⡤⠤⠴⠤⠧⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢸⡇
⠀⠀⠀⠀⠈⢦⡴⠃⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣤⡴⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⢸
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⣀⣤⠤⠤⠤⠤⣄⡀⠀⠀⡸⠀⡟
⠀⠀⠀⠀⠀⠠⢣⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⣧⠠⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⢀⡽⠀⠀⠹⠞⠁⠀⠀⠀⠀⠀⣀⡀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠟⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣛⡟⠀⢰⠁⠀⠀⠀⣀⣠⣤⣤⣄⡀⠀⠸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⣸⠀⢸⣿⣷⣄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠒⠉⠁⠀⠀⠀⠀⡷⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠘⢄⠘⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⣀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⠀⠀⠀⢘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠌⢳⣤⣄⣀⣀⣤⡴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⢀⠤⠤⠀⠀⣠⠊⠀⠈⣩⠏⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠚⠈⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⡏⢠⠂⠀⢰⠃⠀⢀⡴⠃⠤⠤⠖⠋⠉⠉⠒⠲⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⡇⠀⠀⡈⠳⠶⠋⠀⠀⠀⠠⠤⠤⠤⢤⣀⡀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠈⠢⠀⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⡡⠴⠒⠒⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠘⠢⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠈⠳⢦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠉⠉⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢠⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⡀⠀⢀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⠦⣀⡤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠄⠶⢤⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⣶⡞⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠉⠢⡄
⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⣤⡜⠀⠀⠀⣛⣃⠂⠀⠀⠀⠀⠀⠀⠀⠙⢉⣀⡀⠀⠀⢹⣦⠑⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⣻⠇⠀⠀⢼⣿⣿⡆⡐⠀⠀⠀⠀⠀⢁⡠⣿⣧⣽⠦⠀⠸⣿⣷⠀⠀⡄⠀⠘⢆
⠀⠀⠀⠀⢨⡇⢀⠀⠀⢸⣿⡀⠘⢀⠀⠉⠡⠄⠀⠀⠀⠀⠀⠀⠘⠇⠈⠉⠁⠀⢠⠅⢻⣿⡀⠰⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣮⡴⠟⠀⠀⠈⣿⠃⢀⡚⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠤⢀⣿⣧⠀⠀⠀⠀⠀⠠⣘⣄
⠀⠀⠀⢰⣿⣦⡀⡀⠀⠀⢿⡆⣄⠁⠀⠀⠀⠀⢠⣀⣀⣀⣤⣶⣄⠀⠀⠀⠀⠀⠀⠲⣠⡿⣿⡄⣦⣀⠀⢀⣤⡀⣬⡃
⠀⠀⠀⢸⣟⣿⣿⣧⣦⣦⢸⡇⡇⠊⣕⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠆⠈⡂⠀⢸⣾⡆⣿⣷⣿⡇⣿⢿⣦⣼⡟⣸⠿⠁
⠀⠀⠀⠀⣿⡟⣿⢱⣷⣾⢸⣧⠣⡄⢻⡇⠂⠀⠉⠹⢿⣿⣿⠿⠋⠀⠀⠀⣸⠟⢁⠼⣿⠿⣿⣷⢹⣿⣿⠄⡴⢏
⠀⠀⠀⠀⠀⠹⣿⣿⣿⡁⢸⣿⠀⠀⠀⢡⡀⠀⠀⠀⢠⣈⡠⠀⠀⠀⠀⢠⡏⢀⢀⣀⡁⠠⣹⣿⠸⣿⣫⡠⢣⠁
⠀⠀⠀⠀⠀⠀⠹⢿⣿⣡⣏⣴⡗⠀⢀⠸⣿⢦⣴⣦⠾⢿⣿⣶⣶⣤⡴⣋⠞⠁⠀⠀⠀⠈⢟⢿⡇⢿⣻⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢺⠉⠁⠀⠀⠘⠓⠈⠻⣿⡄⠀⠀⠀⠀⣨⣿⡞⠁⠀⠀⠀⠐⠄⠀⠀⣑⣿⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⠀⠀⠀⠀⠀⠀⠸⣷⣤⣤⣤⣴⡿⠋⠀⠀⢀⠀⠀⢠⡀⠀⢹⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣷⡀⠂⠀⠀⠀⠀⠈⠉⠋⠋⠉⠀⠀⠀⠀⠀⢿⡂⠀⣡⣤⣼⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⠃⠘⠧⠾⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⠁⠀⠋⠛⠋⠈⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠘⢷⣀⡀⠀⣄⡀⠀⠀⠀⠀⠢⠀⣢⡄⠘⠂⠂⠀⠀⠤⢀⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠦⣳⡖⣶⣤⣿⠾⠗⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⣤⣤⠀⠀⣠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⢀⣀⠀⠀⠀⣀⣰⣾⢿⣷⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢸⡟⢻⣷⣴⣄⣻⣿⣆⠀⢠⣿⠛⣿⡄⣿⣿⠀⠀⣀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⣿⣿⢿⣿⣿⣿⣿⡿⣿⣦⣾⡏⠀⣿⣇⣿⡿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣷⣿⡇⠈⣿⡇⢻⣷⠀⠘⣿⠇⠀⠀⢹⣿⡿⠿⠇⠀⠰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⣿⠿⢷⣤⣴⣶⣬⡟⢃⣶⡿⠃⠀⢸⣿⠀⠀⢸⣿⣤⣾⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣏⣀⣾⡟⠉⠉⢻⣿⣾⡟⠀⠀⠀⠀⣤⣄⣀⣾⡿⢿⣿⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠋⠋⢿⣧⣀⣀⣼⣿⠻⣿⣄⣀⣀⣤⣨⣿⣿⣿⣥⡄⣈⣉⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡉⠀⢠⣤⣽⣿⣟⣿⣥⣶⣾⠛⠛⣿⣯⣿⠟⢿⣿⠛⣋⣿⡛⠛⡏
⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠀⢸⣿⣇⣹⣿⢿⣿⣀⣀⠀⢀⣿⡋⣿⡇⢸⣿⣾⣟⠋⡀⠀⠁
⠀⠀⠀⠀⠀⠐⣿⣇⠀⠀⢀⢹⣿⣿⣿⣁⠸⣿⡟⠉⠀⢸⣿⠿⣿⣧⢸⣿⠘⣿⣿⠛⢀⠀⠤
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⣿⡇⠙⢿⡷⣿⣷⣿⠿⣿⡯⠀⢚⣿⣼⣿⠀⠘⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣆⡀⢀⣀⣀⣀⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣍⣉⠁⠀⠀⠀⠀⠀⠀⠀⠉⣉⣭⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⡿⠈⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⢧⡄
⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢀⡼⠁⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⢠⡟⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⢠⡟⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⣿⠁⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⢀⡇⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⠴⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⣼⣿⣄⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠈⠉⠉⠛⠲⢦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠶⣦⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣦⡀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠑⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠤⣀⡉⠉⠓⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠉⠲⣤⠤⠤⣌⡑⠢⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⣼
⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢀⣤⣤⠀⠀⠀⠛⠄⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡆⢠⡟
⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣾⠁
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⣾⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣼⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⣼⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⡟⠰⣀⠔⢋⣤⣶⣿⣷⣶⣶⣶⣶⣦⣤⡄⢸⡄⠀⠀⠀⠀⢘⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⢷⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⢳⡈⢳⡀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣉⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠒⠐⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠐⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠃
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⡀⢃⡀⠀⠀⠀⠀⠀⠀⠀⡄⠠
⠀⠀⠀⠀⠀⠀⣰⠀⠀⢰⠀⠀⠀⠀⠀⠀⣀⠾⠋⠁⠀⠠⠀⢐⣠⠄⠐⠈⠡⠒⠠⠤⠀⠉⠙⠻⣗⠁⠀⠀⠀⠀⠀⡡⠀⠁
⠀⠀⠀⠀⠀⠀⡍⠾⠃⢸⡆⠀⠀⢀⡠⠞⠁⠀⠀⠀⠈⠀⠠⠻⠁⠀⠀⠀⠀⠀⢩⡰⠊⠀⠀⠀⠈⠀⠀⠀⠀⢀⡚⠁
⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⣻⣦⡄⠈⠀⠀⠀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠠⠀⠀⠀⠀⠀⠠⡈⠁⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡿⠁⠀⠀⠀⠀⡀⠀⠠⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠻⢫⠆⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠈⢐⣬⣵⣦⣽⣿⣤⡇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⣀⢖⣨⣥⣭⣒⢶
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⣼⣿⣿⣿⣿⡷⣷⣋⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⠐⡱⢣⣿⣿⣿⣿⣿⣶⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠐⢌⣿⣿⣷⣿⣷⠿⠛⠀⡠⠤⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠱⢾⣿⣾⣿⣿⣿⢛⡆⠂
⠀⠀⠀⠀⠀⠀⠀⠆⣐⠂⠀⠀⠀⠠⣙⠛⠻⠁⠁⠀⠄⠁⠀⣀⣴⣿⠶⠶⣾⣶⣄⡈⠑⢤⠀⠀⠐⢫⣭⢥⣞⡋⠀⠀⡀⡇
⠀⠀⠀⠀⠀⠀⢸⣶⡷⠄⠀⠊⢀⡰⠜⠋⠀⠀⠄⠀⠀⠀⣼⠟⣥⣶⣄⣠⢠⣄⠉⢻⡄⠀⠐⠦⡀⠀⠹⢟⡩⠀⠀⠀⠁⠇
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠯⠈⠀⠀⠐⠁⠀⠀⠀⠀⠙⣾⣧⣿⡟⣿⠻⣿⣦⣼⠁⠀⠀⠀⠘⢢⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣙⠃⣿⣄⣌⣤⠏⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⢀⢰⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⢟⣻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠁⠀⠰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠱⠷⣖⠈⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣘⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠤⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣼⣿⣿⣿⣿⣷⣧⣀⠀⠀⠀⠀⠀⠀⠀⡠⠉⢤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⣘⠢⣀⠀⠀⠀⣴⡾⠛⠛⠛⠛⠛⣻⣿⡟⠋⠛⠛⢿⣿⡶⠢⢀⣆⡈⡹⣺⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠙⠶⣽⣷⢺⣿⡏⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠙⣆⡘⣋⢉⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢬⡍⣋⣽⣇⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠈⡿⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣏⡻⣦⣄⡀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢠⠟⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣭⣋⠛⠳⠶⡾⢿⣶⡶⢦⣶⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠳⠾⠶⠶⠖⠛⠊⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⡿⢿⠿⠿⠿⠿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⡿⠻⣿⣿⢿⠿⠿⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡬⢻⡶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠈⢼⠇⣾⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡦⢼⠆
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⠀⠀⠀⠀⠀⣤⣤⣜⠀⠀⠀⠀⢃⣤⣤⠀⠀⠀⠀⠀⢶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⣸⣿⠟⠁⠀⠀⠀⠀⠀⠈⠻⣿⡆⠀⠀⣰⣿⣿⣧⣤⡀⠀⠀⠀⠀⠉⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠀⠀⣿⣧⡻⡿⢛⡃⠀⠀⠀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠙⣿⣷⡄⠀⠀⠀⠀⢀⣼⣿⡏⠀⠀⠀⠙⣿⣷⣾⠚⠃⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⠐⣻⣿⠿⠷⠀⠀⠀⠀⠘⢿⣿⡀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⢘⣿⣿⠟⠾⠆⠀⠀⠀⠀⠈⢿⣧⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⢠⣤⣤⣀⠀⠀⡀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠈⢋⣴⠸⣿⣿⡟⠀⠀⢧⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠘⢦⣀⡀⠀⠀⠀⠀⠉⣵⣇⣬⣴⠃⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣤⣀⣀⣈⣉⣸⡿⠋⠀⠀⠀⠀⠘⢶⣤⣀⣀⣀⣀⣤⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡏⠉⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠻⡇⠙⠀⢰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⠏⠠⠒⠈⠀⠀⠀⣿⣏⠀⠔⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠋⠄⠀⠀⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢁⠀⣀⠀⡠⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠂⠀⠀⠀⡐⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⡀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢄⢈⢀⡁⠀⠨⣒⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⣉⣁⣧⡀⠀⠈⠘⠢⠄⡀⠀⠀⠀⠀⠐⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠼⡗⠀⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠉⠀⠀⠀⠃⠀⠀⠀⠀⠆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠄⠀⠐⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠓⡄⠀⠲⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠁⠠⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡠⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⣶⠖⠒⠛⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⡞⠋⠉⠉⠓⠦⣄⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣁⣠⠞⠀⠀⠀⠀⠀⠀⠈⠙⠲⣾⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢰⡇⠀⠀⠀⢀⡴⠶⠶⢤⣀⠀⢀⣀⣀⣀⣀⡟⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⡾⠀⠀⠀⢰⡏⠀⠀⣴⣚⠉⠉⠻⢧⡀⠀⠈⠉⠳⣄⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⠀⠀⣼⠁⠀⠀⠀⠈⢷⡀⢸⡇⠈⠙⢦⠀⠈⢷⠀⠀⠀⠀⠘⡆⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠹⡄⠀⠀⠀⠀⠈⢷⠀⠓⠦⣤⠞⠀⠀⣸⠀⠀⠀⠀⢀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡟⠲⢤⣀⡀⠀⠈⣧⣀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⡞⢀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣘⡇⠀⠀⠈⠉⠛⢲⡇⠈⠙⠛⠋⠉⠀⠀⠀⠀⠀⣸⠋⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⠀⠀⢀⣠⠶⠚⠁⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢈⣩⠷⠒⠒⠋⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠈⠙⣳⠖⠛⠻⣍⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⢻⠀⠀⢠⡟⠀⠀⠀⠈⠉⠓⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⡟⢸⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⣾⠃⣀⡤⠶⠒⠛⢛⣋⣩⡽⢿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣷⣸⣾⢗⣋⡤⠴⠖⠚⠋⠉⢁⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⡿⢟⣉⣠⣤⡤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠾⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣛⣙⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠆⠁⠀⠀⠰⡍⠙⠆⠄⡀
⠀⠀⠀⠀⠀⠀⠠⣗⠀⠀⢰⠃⠀⠀⣴⠍⠠⡀
⠀⠀⣒⡈⠀⠀⠀⠀⠀⠀⠘⣆⣠⠾⢫⡆⠀⠈⢠⡀
⠀⠀⠉⠂⢀⣤⠤⣺⣫⠿⠀⠈⠁⠀⢹⠁⠀⠀⣀⡿⠙⣶⡀⢤⣄⣀⣀⣀⣀⣀⣀⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠠⢦⠶⠁⠀⠈⢀⠂⠀⠠⣼⠂⠀⢰⠋⠁⣠⣿⣇⠈⢻⣿⣿⢯⣍⣉⣉⣡⣭⣤⣈⡉⠉⠉⠁⠂⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⢻⠀⠀⠁⠀⠀⠙⣿⡏⠀⠘⡟⠃⠀⣸⡿⠉⠳⠙⠃⠀⠀⠀⢀⡄⠀⠐⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⡜⠀⡿⠃⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⢸⠠⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠶⠾⠋⠀⠀⠀⠀⠀⠀⢢⡀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠀⢹⣇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⣀⡂⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣢⡙⠄⠀⢀⡼⡥⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⡟⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⢙⣦⠀⠀⢰⡧⠤⠤⠄⠀⠀⠀⠀⠀⠀⠈⢷⣀⠙⢿⣷⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣼⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠸⣿⣿⣦⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡝⣿⣿⢷⡄⠐⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⣿⠀⠑⡀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢈⠁⢠⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡇⠀⠀⠃⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠁⠎⢀⡈⡀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠧⣴⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⠉⢴⣭⣍⣡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠛⠋⠀⢦⢠⣍⠂

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠏⠀⠀⣡⠀⢠⡀⠀⠀⠀⠀⣠⣶⣿⣤⣤⣴⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠄⠀⠀⠋⠀⠈⠀⠀⠀⠀⠘⡉⠛⠿⢛⡛⠛⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠰⣦⣤⣤⡤⠀⠀⠀⠀⠀⠸⣿⡀⠀⢸⣷⡀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠳⡦⠄⠀⠀⢀⣤⣶⣤⣤⣭⣶⣄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠯⠀⠠⡆⢲⣤⣾⣿⣿⣿⡏⠁⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢄⣧⡀⠀⠀⠛⠋⡟⠉⠸⠿⠇⠀⠀⠈⠟⠁⠀⠀⠀⠀⠀⠀⣤⣶⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⣀⣀⣿⣇⣾⣇⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⡴⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⠿⠗
⠀⢀⣠⠖⠋⠀⠸⣿⣿⡟⠛⠛⠛⢻⣿⣯⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⢻⣇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇⡀⠀⠀⢸⣿⣿⣀⡀⠀⠀⠀⠀⠀⠚⠛⠻⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⡿⠃⠀⢼⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡖⠛⠛⠲⢄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠃⠀⠀⠀⠈⠣⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠉⢀⠴⠒⠒⠢⢄⠀⠁⠀⠺⡀⠀⠀⠀⠉⢢⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⡤⠋⠀⠀⠀⠀⠀⠁⠀⣰⠶⢸⠀⠰⢶⡄⠀⠙⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠠⡦⠀⠀⡶⠰⠄⡀⠀⠀⣰⣿⣿⣟⠛⠛⠛⢿⣷
⠀⠀⢀⠀⠀⠀⢀⡔⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠐⠀⠀⠒⠀⠐⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠀⠀⡴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⣼⠁⡇⠀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋⠃
⠀⢀⠀⠠⡇⠀⢷⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡄
⠀⠀⣄⠀⢧⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⠇⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠓⢤⣀⠙⢦⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠻⠀⠀⢠⣔⡉⠉⠑⠒⠢⠤⠄⢀⣀⣀⣀⡠⠔⠂⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⣲⠏⠁⠀⠀⢻⣿⣿⣿⣿⣶⣾⡿⠀⠘⢄⠀⠀⠀⠐⠺⢿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠿⠏⠓⠒⠶⠤⠤⠤⢦⠀⠀⡼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠀⠀⠀⠀⠄⣀⠂⠄⣸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡝⠂⠉⠀⠀⠀⠀⠈⠀⠐⠠⣄⠒⣌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠘⠚⢄⣀⠀⠀⠀⠀⠀⠀⠀⠢⡐⢂⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠒⠒⠒⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠂⠐⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⣠⠖⠛⠛⠛⠓⢦⡀
⠀⠀⢀⣠⠤⠤⠤⢤⣀⠀⠀⠀⠀⣴⡛⠓⠈⠳⢦⣀⣰⠃⠀⣠⠔⠒⢤⠀⠙⡆
⢠⡞⠉⠀⣀⣀⠀⠀⠈⢧⣠⠶⠋⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⢧⣀⣠⠆⠀⠀⡿
⣿⠀⢰⠋⠀⠈⢱⡀⠀⠈⠁⢀⣀⣀⠀⠀⠀⠀⠀⠲⢿⣿⣶⡄⠀⠀⠀⣀⠼⠁
⢻⡀⠀⠳⠤⠤⠞⠁⠀⠀⣴⣿⠿⠋⠀⠀⠀⠀⠀⢀⡤⠦⣄⠀⠀⠀⠈⠛⠶⣤
⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢦⣄⠀⠀⠀⠀⢸⣋⣦⠈⢧⠀⠀⠀⠀⠀⠺⣄
⠀⠀⠀⢉⡟⠁⠀⠀⠀⠀⠀⡏⠀⣤⣿⣧⠀⠀⠀⠸⣿⣿⠃⣸⠀⠀⠀⠀⠀⠐⢿
⠀⠀⠀⠈⣿⡁⠀⠀⠀⠀⠀⢧⠀⠻⣿⣿⠀⡠⠤⠀⠘⠳⠖⠛⠋⠀⠀⠀⠀⠀⣈⡷⠶⠶⠚⠓⢦⣀⡀
⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠴⠞⠓⠤⠞⢁⣤⠒⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠠⣼⠀⠀⢀⡤⢄⠀⠀⢹
⠀⠀⠀⠈⢧⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣠⡶⠋⡧⠀⢫⣀⡸⠀⢀⡼
⠀⠀⠀⠀⠀⠳⠶⢦⡀⠀⣀⠤⠄⠠⢀⠀⠀⠀⠉⢻⠀⣄⠀⠀⠀⠠⣀⣼⢋⣠⠞⠁⠀⠀⠀⠀⢺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣹⠟⠁⠀⠀⠀⠀⠱⣄⡀⠐⠾⠟⢀⡀⠄⠀⠀⠈⢻⠉⠁⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠄⠀⢀⡀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢠⠎⠀⢀⠔⠃⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⡆⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢰⠇⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⢽⠓⠛⠉
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⣮⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠁⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠈⢀⠀⠀⠀⠀⠸⠗⠀⠀⠀⢀⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠑⠤⣀⡀⠀⢀⣀⡠⠔⠋⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠂⢸⡿⠛⢳⡖⠂⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠠⠄⠘⠷⢤⡀⢀⡼⠇⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⡀⢀⣽⣏⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣤⣤⢴⠧⠞⠁⠈⠳⠤⠴⣤⣠⡤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠟⠏⠙⠉⠀⠀⠀⠈⢋⡀⠀⠀⠀⠀⠸⡿⠛⠻⠗⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⠛⠋⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢐⣞⣟⣿⣶⡄⠀⠀⠀⠃⠀⠀⠀⠀⠘⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢛⡿⢙⣿⡀⢿⠆⠀⠀⠀⠀⢐⡀⠉⢀⠈⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠆⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠡⡞⢹⡗⠈⠀⠀⠀⠀⠀⢸⡷⠄⠀⢳⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡟⣻⠋⠐⣰⠀⣠⡼⠃⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢸⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣤⣿⠟⠉⠀⠁⢀⠘⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡴⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡷⠀⠀⠀⠀⠀⢸⠝⣿⡆
⠀⠀⠀⠀⢰⣄⠈⣿⣿⠀⠀⠀⠀⢼⣆⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⢁⣴⣢⡶⠂⣀⠀⣴⣷⡄⠀⠐⣧⣀⣀⡤⣤⣄⡈⠀⠏⠸⡀
⠀⠀⠀⠀⠀⡇⣼⣿⣿⣀⡀⠀⠀⠀⠹⢷⠆⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢀⣠⣃⣤⣾⡿⠙⣿⣤⠔⠛⣿⠿⠛⠓⠛⠿⢿⣟⠀⠁⠈⢂⡶⠀⠀⠁
⠀⠀⠀⠀⠀⢁⠸⣿⣿⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠋⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣀⠈⣷⣀⡀⠂
⠀⠀⠀⠀⠀⠀⠙⡿⢿⣿⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⠀⢾⣿⣿⣿⡀
⠀⠀⠀⠀⢀⠀⠦⠚⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⣿⣿⣇⠪⢿⣿⣿⣿⡇
⠀⠰⣄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠉⠀⠸⣿⣿⣿⡾
⠀⠰⠦⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⡇⠀⠀⠀⠀⢀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⣀⢀⡀⠀⠀⠂⢸⣿⣿⡇
⠀⢀⣆⠰⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣼⣆⠠⠑⠂⠀⠀⡰⠋⠀⠉⠉⠻⠻⣷⢶⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣴⣴⣾⣿⣿⣿⣿⣥⣦⣀⣀⣠⣿⣿⣿⡄
⠀⠀⠙⠳⠄⣀⠀⠀⠀⠀⠀⢀⡄⠀⠺⠇⡉⠀⡀⠀⠀⠈⠁⠀⢀⣤⠶⢴⡶⠷⢢⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣭⣁⠀⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⣼⣟⣧⠀⣾⠿⣤⡁⠀⠀⠀⣤⣴⣿⣶⣾⣿⣿⣶⣧⣌⠑⠀⠀⠀⠀⢰⢿⣿⣿⣿⣿⣻⣟⣿⣿⣷⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢰⠏⠉⠀⠀⢩⣄⣠⡟⠀⠀⠀⠈⠉⢻⠟⠸⣿⣿⠿⠛⠋⠀⠀⠀⠀⠀⠊⢢⣼⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠀⠀⠀⢸⣿⣿⡁⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⠀⠀⢹⢿⡿⣿⢿⡟⠋⣰⢠⣿⣿⣿⣯⣿⠃
⠀⠀⠀⠀⣷⠄⡀⠀⠀⢣⣀⣀⣤⣶⡇⠸⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⣇⣸⡿⣿⡿⠟⠁
⠀⠀⠀⠀⠙⠷⣿⣦⣤⣄⡿⠉⣿⣿⣷⠘⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⢇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⣧⠴⣿⣿⣿⠃⣿⣟⣧⠐⠆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣶⣤⣀⣴⣾⡛⢁⠘⠀⠀⠀⠀⠀⠀⢀⣶⡾⣿⣿⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠘⢋⠉⠠⣤⣽⣿⣿⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣟⢻⣿⣏⠖⠀⠀⠀⠀⠀⠀⠀⣍⢽⣿⡿⢜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠐⠛⠛⠛⠉⢩⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⣉⡉⠓⠀⠄⠀⠘⠃⠀⠒⢸⣿⣾⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠂⠀⠀⠀⠐⠀⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⢤⣮⡉⠀⠀⣠⣧⠀⠀⠀⠈⠀⠀⠀⠈⣮⣛⣛⡳⣾⣿⣽⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡽⠊⠁⠀⠀⠀⠀⠈⠀⠀⠈⠙⢿⠿⣿⣿⣿⡿⣿⣿⣧⠀⠀⠀⣀⠰⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠁⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠛⠁⢀⠋⠉⠉⠀⠀⢺⣏⠀⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣐⣫⣶⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⢈⣠⣾⣿⣟⣩⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠁⠈⠛⠳⢶⣶⣶⣶⣶⣿⣿⡿⣿⣷⣧⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⢤⠀⠀⠀⠀⠀⠀⣼⣉⣿⣿⡿⣿⠷⠶⣤⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡞⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠀⢻⣴⡴⣳⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡛⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢄⠀⠀⠀⠀⠀⡠⠐⠒⠀⠀⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠊⣄⡘⠃⣨⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠳⢴⣾⠃⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣸⠂⠄⠁⠈⠉⠁⠀⠉⠀⠄
⠀⠀⠀⠀⠀⣀⠤⠐⠁⠀⠀⣠⣀⡀⣮⡷⠀⠀⢧⣐⡠⡁⢠⣠⣤⡀⢀⣤⣔⡜⡁⠀⢈⠀⠀⠀⠀⠀⠀⢀⡘⠋⠑⠠
⠀⠀⠀⠀⠀⡇⠚⣷⣆⢠⢤⡌⣋⣵⣨⡻⡞⢠⣿⣅⡁⢳⡄⠉⠱⣤⣤⡍⠹⢼⡾⠀⣼⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢰⣀⣨⣟⡎⢛⣵⠳⢶⣿⣀⣢⠎⠈⠏⠀⠀⣹⠓⣶⣔⣛⣲⡶⠞⠛⠲⠏⠐⠀⢀⡀⠀⠀⡜⠃⡁⠐⠁⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣬⣭⣤⣠⣡⠞⠉⠀⢀⡏⠠⠀⡏⠀⢌⢠⠉⣡⣈⠠⢀⠁⡸⣦⢴⡆⢣⣾⣥⣤⡶⠁⠀⡄⠄⠓⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠶⠤⢒⣘⠾⢈⠀⢀⠔⠀⠠⠋⡨⠢⢼⡀⠀⠾⡐⠀⠈⡓⣶⣦⣶⠖⠋⡂⠀⠠⠀⠲⠀⠄⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠚⠒⠈⠉⠉⠉⠉⠉⠉⠃⡠⢀⡄⠀⠠⠔⢤⡒⢠⡔⠀⠀⡠⠼⢋⣀⣀⣒⣐⣿⠀⠀⠀⠀⣇⠢⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⢼⠀⠀⠀⠀⠀⢁⠴⠀⢀⠔⠁⢀⠀⠀⠀⠈⠉⡻⠀⠀⢀⢎⠀⢠⠄⡏
⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠠⠆⢈⡁⠀⡠⠆⠀⠀⣤⢀⢜⠠⠄⠀⠂⠀⠀⢀⡜⠀⠀⢠⠀⠀⠁⡈⠔⢃
⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⠀⡀⢀⡇⠀⢀⠔⡡⢘⣹
⠀⢀⣀⣀⡀⠀⣀⠀⢤⣤⣄⣀⣀⢀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⣠⢚⠝⡩⠞⢡⠟
⠀⣀⠄⠀⣀⣀⠀⠀⣤⣤⡤⠠⠌⢀⣈⣛⣿⠦⣤⡀⠌⠀⢀⡴⡄⠲⠤⣤⣤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠚⢤⣡⣪⠤⠊
⢰⠕⠊⠉⠁⠀⠀⠀⠀⠀⠁⠉⠃⠈⠀⠒⠛⠶⢢⣥⣦⡙⠖⠇⠤⠑⠂⠀⠈⠀⠉⠉⠉⠁⣀⠀⠀⠠⠔⢁⠀⠈⠾⠈⠁
⢨⣮⣐⣒⣂⣀⣀⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣄⣀⡀⠀⠀⢫⣀⠤⡔⠁
⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣙⣛⣒⣒⣂⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⢉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠒⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⠀⠀⠀⢀⡈⠆⠐⠒⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⢀⠔⡡⠊⠀⠀⠀⠀⠀⠀⠀⠑⡆⠁⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⡠⠔⠁⡐⠁⠀⠀⠀⠀⢀⠀⢀⠀⠀⠘⡒⡀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⣠⢎⡀⠀⠀⠀⢀⠔⠁⠀⠀⠑⡄⠀⠀⠰⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢱⣿⣟⡀⠃⢀⠊⠀⣠⣄⢤⡀⠐⠀⠀⠀⠡⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠛⠻⠃⠀⠀⠀⢸⣿⣯⡄⡏⠀⡇⠀⠀⠀⠀⠑⠠⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢾⣍⣉⠉⠙⣦⠀⠉⠛⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠻⣿⡿⠿⠛⠀⠤⠀⠤⠔⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠈⠉⠀⠀⠀⠉⠀⠀⡀⠔⣮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡂⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⡶⠀⠂⠒⠂⠀⡈⠅⠂⡩⠂⠁⠂⠄⡀⠀⠀⠀⠀⠀⠀⠐⠀⠈⡔⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢛⡖⠂⠀⠀⢁⠠⠐⠁⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠁⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠻⢶⡒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⢰⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⡇⠀⠀⠠⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡘⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⡴⢇⠀⠀⠀⢣⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠁⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢸⠀⠀⠀⢸⡄⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠸⠀⠀⠀⠘⠃⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠄⠀⠀⢀⠔⠁⠀⠀⠀⣘⠤⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠣⠰⣥⠂⡄⠀⢀⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠂⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢿⠏⠩⠁⠀⠀⠀⠀⠤⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⢰⢠⠄⡄⠀⠂⠈⣄⡉⠉⠫⠿⠫⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡠⠄⠰⢏⣟⣯⣒⣂⣀⡀⠘⠃⠤⢠⣤⡤⠘⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠒⠊⠉⢠⠤⣞⣠⢟⣎⣞⣿⣶⡶⢄⠂⣠⠀⢀⠀⠀⠌⡀⠹⢎⣳⡽⢂⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠄⢀⣀⠐⠄⠿⠟⣻⡿⡿⡿⢿⣞⢴⣏⠀⢣⢀⡒⣆⢠⢦⠁⢀⠀⢁⡵⠛⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⢠⣾⣽⣿⣷⡁⢪⢼⡟⣙⣗⣯⣿⣄⠛⢀⡀⢾⠘⠢⠌⠻⣃⠂⢉⠟⠒⣚⢁⢀⣠⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣭⣾⣼⣿⢾⡿⠿⣷⡽⢃⢋⡟⣉⡤⢻⠧⠈⢳⣄⣠⡀⣥⡄⠁⠀⠀⣁⣠⢿⠁⢐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣮⡹⡾⣿⣾⣏⣯⠸⣿⣣⢁⣸⡟⠡⠆⠀⠲⡀⢸⠛⠾⣛⠚⣷⢀⡀⢦⠘⣧⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢿⡿⣷⣿⣷⡿⣳⣾⢋⡔⢁⣀⠤⠒⠒⠒⠶⣭⣿⠀⣃⣼⣙⣿⠟⢦⣈⡿⠇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣭⢷⣺⢹⠟⣷⣿⡷⣯⣴⣿⠁⠀⠀⠀⠀⠀⠘⣷⣦⣮⡘⠺⣏⡔⡐⣾⣛⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡭⢽⣋⠿⣬⣻⠿⣿⣽⡿⠀⠒⠀⠀⠀⠀⠀⢻⢋⣢⡑⢾⡋⠞⢻⠯⣁⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣵⣿⢿⣿⣿⠾⣾⢽⡼⣿⡏⠁⠀⠁⠀⠀⠀⠀⢀⣿⡘⠷⣻⠮⢯⡟⢛⠒⣅⣐⣋⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⣟⣿⣟⣛⡾⢺⣿⠇⠀⠀⠀⠀⠀⠀⣀⣼⡿⠉⣻⠆⡜⣦⣌⡿⢉⠀⢿⣧⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣤⠀⠀⠟⡷⣸⣿⠀⠀⠀⠀⠀⠀⠀⠙⠋⢀⣼⡅⢾⣷⢄⣿⡷⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⠿⣾⡇⢆⣱⢸⣿⠀⠀⠀⠀⠀⡠⠂⠀⠐⠿⢿⣿⣾⣻⡫⠫⡧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⢹⢟⢻⡄⠀⠀⠀⠈⠀⠀⠀⠀⠘⣷⡿⣿⣧⠧⢘⠴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢿⠸⣄⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⣏⣔⣁⣄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⣉⣻⡟⠦⣄⣀⠀⠀⠀⢀⣷⡀⣀⠔⢹⢹⡤⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⢄⡿⣉⣙⣿⣙⡿⢋⣻⣿⠤⠧⢜⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢭⡟⠁⣠⠸⢿⡮⣌⠈⠑⣦⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡼⠁⠄⣮⡀⠁⠳⢄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠀⠁⠀⠉⠒⠒⠚⠛

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

⠨⣕⠢⠁⠀⠀⠀⠀⠀⠀⠀⣀⠨⣑⠬⠁⠀⠀⠀⠀⠀⠀⠀⡀⢭⡓⠌⠀⠀⠀⠀⠀⠀⠀⢀⡀⢍⡢⠍
⠍⠀⠀⠀⠀⠀⠀⠀⢀⠠⣕⠢⠅⠀⠀⠀⠀⠀⠀⠀⣀⠨⣑⠬⠑⠀⠀⠀⠀⠀⠀⠀⡀⠬⡒⠌
⠀⠀⠀⠀⠀⢀⡀⢄⡲⠍⠂⠀⠁⠀⠀⠀⠀⢀⠠⢄⠢⠑⠂⠀⠀⠀⠀⠀⠀⢀⠠⣐⠮⠑⠀⠀⠀⠀⠀⠀⠀⡀⠤
⠀⠀⠀⡀⠤⡒⠬⡒⠀⠁⠀⠀⠀⠀⠀⠀⢄⡲⠍⡒⠀⠁⠀⠀⠀⠀⢀⠠⢔⠢⢑⠂⠈⠀⠀⠀⠀⠀⢀⠠⣐⠮⢑
⠀⠤⣐⠪⠑⠊⠁⠀⠀⠀⠀⠀⠀⠄⡐⠬⡒⠈⠀⠀⠀⠀⠀⠀⠠⢄⡲⠍⠂⠁⠀⠀⠀⠀⠀⠀⠤⢐⠢⢕⠂⠁
⠢⢕⠂⠁⠀⠀⠀⠀⠀⠀⠀⣐⠪⡑⠈⠁⠀⠀⠀⠀⠀⠀⠄⡐⠬⡒⠈⠀⠀⠀⠀⠀⠀⠀⢀⡒⢌⠂⠉
⠭⠀⠀⠀⠀⠀⠀⠀⢒⠢⢕⠂⠥⠀⠀⠀⠀⠀⠀⠀⣂⠨⣑⠨⠅⠀⠀⠀⠀⠀⠀⠀⡂⠬⡓⠈⠄
⠀⠀⠀⠀⠀⢀⡀⢌⡲⠍⠂⠀⠀⠀⠀⠀⠀⢀⠠⢍⠢⠁⠂⠀⠀⠀⢀⡀⢀⣀⠠⣑⠮⠁⠀⠀⣀⠀⣀⡀⣀⣀⢬
⠀⠀⠀⡀⠤⡐⠬⠐⠂⠀⠀⠀⠀⠀⢀⡀⢄⡲⠍⠒⠀⠀⠀⠀⠀⠀⣇⠈⣍⠠⠽⠾⠿⠿⠿⣿⣿⣿⣿⠯⣕⠮⣷
⠀⠬⣐⠪⠑⠈⠀⠀⠀⠀⠀⠀⠀⠬⡐⠬⠒⠈⠀⠀⠀⠀⠀⡀⠠⢅⡒⠌⠒⠚⠒⠐⠒⠒⠒⡛⠻⢝⠺⠝⠚⠛⠋
⠢⠝⠂⠁⠀⠀⠀⠀⠀⠀⠠⣐⠪⠑⠀⠁⠀⠀⠀⠀⠀⠀⠠⡐⠬⠓⠈⠀⠀⠀⠀⠀⠀⠀⢄⡒⠍⠂⠈
⠉⠀⠀⠀⠀⠀⠀⠀⢄⠢⢝⠂⠁⠀⠀⠀⠀⠀⠀⠀⣐⠢⣑⠈⠁⠀⠀⠀⠀⠀⠀⠀⡐⠬⡓⠈
⠄⠀⠀⠀⠀⢀⡒⢌⣒⠉⠀⠀⠀⠀⠀⠀⠀⢀⠢⢝⡂⠁⠀⠀⠀⠀⠀⠀⠀⣐⠪⣑⡊⠁⠀⠀⠀⠀⠀⠀⠀⣐⠬

⣿⣆⠀⠀⠀⡀⣀⣴⣆⣤⢠⡀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⢀⢠⣄⠀⢀⡀
⣿⣿⡄⠀⠀⡀⢀⣤⣿⣯⣶⣷⣾⣿⣿⣿⣿⣿⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣤⡀
⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡿⠆⠀⠀⠀⠀⠀⠀⢰⠶⠆⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⣤⣤⣀⠀⣾⢶⠆⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠟⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡟⠻⣿⣿⣿⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣦⣍⠀⣼⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⡿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⡗⠷⠻⠋⣽⣏⠀⢀⡂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣷⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⠖⣀⠀⣶⣿⣿⣷⣿⣿⣦⡈⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠃⠸⣿⣿⣿⣵⣿⣿⣿⣧⡒
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣯⠔⡁⠀⡀⣽⣿⣿⣿⠿⣻⣿⠋
⣿⣿⣿⡿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⡿⢋⡾⣿⣿⣿⣿⣿⣿⢺⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⡟⣿⠿⡉⠀⠁⣳⣿⠿⠿⢁⣀⡚⠛
⣿⣿⡟⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠈⠀⠉⠀⠀⠀⠈⠛⢿⣿⣿⣿⣧⣡⣡⣆⣅⠈⠆⣿⣿⣶⣿⣼⠙⠃
⣿⣿⡀⠀⠀⠀⠀⠈⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⠿⠟⠡⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣾⣧⣧⣶⣾⣿⠿⠿⡇
⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⡀⠀⠈⠑⠲⠤⠀⠀⠀⠀⠀⠉⠉⠉
⣿⣿⡟⣀⣤⣤⣤⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣸⣿⣿⣿⡶⠤⠀⠀⣀⡀
⣿⢛⣼⣿⣿⣿⣿⣧⠀⠀⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⣰⣿⣿⣿⣿⣷⠀⢀⣾⣿⣿⣦

⢀⡀⠀⣴⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⣆⠀⢀
⢨⡇⠀⣿⡄⢀⡀⠀⣿⡇⠀⠀⠀⣶⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⣶⠀⠀⠀⢸⣿⠀⢀⡀⢠⣿⠀⢸
⢸⡇⠀⣿⡇⢀⠁⠀⣿⡇⣤⡄⠀⠺⠀⣽⣇⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣸⣯⠀⠟⠀⢠⣤⢘⣿⠀⠈⡀⢸⣿⠀⢸
⠀⡃⠀⣿⡇⠀⠀⢠⣿⠁⣿⡇⠀⢽⠀⠛⠛⠙⠋⠁⠀⠀⠀⠀⠈⠙⠋⠛⠛⢀⡧⠀⢸⣿⠈⣿⡆⠀⠀⢸⣿⠀⢘
⠀⢄⠀⣿⡇⠀⠀⢸⣿⠀⣿⡇⠀⠊⠉⠑⠈⠋⠋⠉⠀⠁⠀⠀⠙⠙⠋⠁⠋⠋⠘⠂⢸⣿⠀⣿⡇⠀⠀⢸⣿⠀⠠
⢠⡈⠀⣿⡇⠀⠀⢸⣿⠀⣿⣷⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣾⣿⠀⣿⡇⠀⠀⢸⣿⠀⢁
⠀⡇⢀⣿⡇⢠⡄⢸⣿⣦⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣦⠀⠀⣴⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣴⣿⡇⢠⡄⢸⣿⡀⢸
⠸⡅⢸⣿⠄⣿⣷⣈⡉⠉⠉⠉⠉⠉⠉⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠉⠉⠉⠉⠉⢉⣁⣼⣿⠐⣿⡇⢨
⠀⡅⢸⣿⡇⡛⠛⠛⢉⣀⣀⣀⣀⣀⣀⣀⣁⣈⣁⠀⠀⠀⠀⠀⠀⣈⣁⣈⣀⣀⣀⣀⣀⣀⣀⣉⠛⡛⢛⢰⣿⡇⢸
⢀⡇⠸⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠇⢸
⠀⠑⠆⠒⠲⠴⠢⠴⠒⠒⠦⠈⠑⠀⠂⠒⠶⠶⠶⠲⠆⠀⠀⠰⠖⠶⠶⠶⠒⠐⠀⠊⠁⠴⠒⠒⠶⠶⠢⠒⠒⠰⠊
⠀⡠⠆⠤⠤⠲⠖⠲⠤⠤⠖⢀⡠⠀⠀⠠⠶⠶⠶⠴⠆⠀⠀⠰⠦⠶⠶⠶⠤⠠⠀⢄⡀⠲⠤⠤⠖⠢⠖⠦⠤⠰⢆
⠈⡅⢰⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⡆⢨
⠀⡃⢸⣿⡏⣥⣤⣤⣈⠉⠉⠉⠉⠉⠉⠉⡉⢉⡉⠀⠀⠀⠀⠀⠀⣉⡉⢉⠉⠉⠉⠉⠉⠉⠉⣁⣤⣤⣬⠨⣿⡇⢘
⢐⡃⢸⣿⠂⣿⡟⢉⣁⣀⣀⣀⣀⣀⣀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣀⣀⣀⣀⣀⣀⣉⡉⢻⡿⠐⣿⡇⢈
⠀⡇⠈⣿⡇⠘⠃⢸⣿⠟⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠟⠀⠀⠛⠿⠿⠿⠻⠟⠛⠛⠛⠛⠛⠻⣿⡇⠈⠃⢸⣿⠁⢸
⠘⢁⠀⣿⡇⠀⠀⢸⣿⠀⣿⡿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⢿⣿⠀⣿⡇⠀⠀⢸⣿⠀⡈
⠀⠊⠀⣿⡇⠀⠂⢸⣿⠀⣿⡇⠠⢄⣠⡀⢀⣄⣤⡄⠀⠀⠀⠀⣀⣤⣠⡀⣀⣄⡠⠄⢸⣿⠀⣿⡇⠀⠀⢸⣿⠀⠐
⠀⡄⠀⣿⡇⠀⠀⠘⣿⡀⣿⡇⠀⠻⠀⣤⣤⣠⣄⡀⠀⠀⠀⠀⢀⣠⣄⣤⣤⠀⡗⠀⢸⣿⢀⣿⠃⠀⠀⢸⣿⠀⢨
⢸⡇⠀⣿⡇⢈⡀⠀⣿⡃⠛⠃⠀⣴⠀⣻⡏⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⢹⣟⠀⣦⠀⠘⠛⢨⣿⠀⢀⠁⢸⣿⠀⢸
⢘⡇⠀⣿⠃⠈⠁⠀⣿⡇⠀⠀⠀⠿⠀⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠿⠀⠀⠀⢸⣿⠀⠈⠁⠘⣿⠀⢸
⠈⠁⠀⠻⠀⠀⠀⠀⠛⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠟⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠶⠒⠂⠒⠀⠒⠢⢤⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠖⠒⣤⣦⡤⠂⠀⠄⠚⠉⠉⣈⡆⠀⠀⠀⠀⠀⠀⠀⢀⣭⣐⢄⠀⠠⣦⣤⣶⡶⠦
⠀⠀⠀⠀⠀⡴⠋⠀⠤⡄⢸⡟⠁⠈⠀⢀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢛⠺⢦⡈⢻⡿⢀⡀⠀⣄
⠀⠀⠀⠀⢰⠁⠴⠀⠀⠀⡜⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⣿⣗⠀⠉⠀⠁⠀⠈⠢⣸⡄
⠀⠀⠀⠀⣿⢸⣶⡦⠀⣞⠀⡤⠒⠲⣴⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣿⣿⠛⠳⢆⠰⣠⣄⣀⠈⢡
⠀⠀⠀⠀⡇⠘⠛⠁⣠⣧⣾⣴⣿⡿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠧⣾⣿⠿⣿⣷⣼⣄⡟⠙⠿⠇⢸
⠀⠀⠀⠀⠙⠒⠢⠾⣿⣿⣿⣿⠿⣶⠿⠟⠁⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠈⠻⣿⢟⣿⣿⡿⣿⠦⠴⠗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡻⣿⣿⡾⠋⠀⠀⠀⠀⣰⢟⣭⣶⣿⣇⣩⠻⡆⠀⠀⠀⠀⠘⢿⣿⡟⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣧⠘⣿⠃⠀⠀⠀⠀⢠⣇⣾⣿⡟⣿⣿⣿⣷⣿⡄⠀⠀⠀⠀⠈⢻⡆⠀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡀⠀⠀⠀⠀⠀⠀⠘⣿⣛⠛⣹⣿⢿⣽⠟⣻⠀⠀⠀⠀⠀⠀⠘⡅⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣼⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣟⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⡇⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣸⡏⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⣿⡿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⢧⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⣿⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣯⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠻⠟⠿⠿⣻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⣿⢁⠀⠀⠀⠀⠀⠈⠁⡡⢿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠄⠀⠤⠔⠂⠐⠉






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢿⡿⡿⢿⣿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣟⣛⣟⣛⣿⣟⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠀⠀⣤⡴⠶⠶⠛⠛⠛⠛⠛⠿⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠶⠶⠶⠶⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠳⣤⣤⣤⡴⠶⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠈⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠉⠙⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣼⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⣿⣶⣶⠶⠶⠶⠤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⢠⡞⢉⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⢀⡠⠤⠴⠶⠶⠶⠶⠶⠶⢤⣤⣀⠀⠀⠀⠀⠙⢧⡉⠀⢋⣀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠉⠻⣆⠀⠀⠀⠈⢷⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢰⡟⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⡈⢱⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⡟⠁
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⣄⠘⣤⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⠏
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⠈⠉⠈⢳⣤⣴⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢠⡟
⢻⡄⠀⠀⠀⢠⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⠀⠀⠀⠀⠀⠀⠻⢯⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠂
⠈⢷⡄⠀⠀⠈⢧⡀⠡⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⠔⠉⠀⠙⢳⣄⠀⢀⡏⠀⠀⠀⠀⢀⣴⡿⠛⠁
⠀⠀⠙⢦⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠊⠉⠀⠀⠀⠀⠀⠀⠹⡆⢸⡇⠀⠀⠀⣠⡿⠉
⠀⠀⠀⠀⠉⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠓⠀⠐⠶⢤⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢹⣤⠶⡄⠀⠀⠀⠀⢹⠀⡇⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⡴⠋⠉⠀⠟⢁⡾⠁⣀⠀⠀⠀⠈⡄⢃⠀⠀⣾
⠀⠀⠀⠀⡞⠛⠟⠉⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣼⠁⠀⠀⠀⠀⡾⠷⠛⣩⠇⠀⠀⠀⠃⠈⠀⠀⡟
⠀⠀⢀⣸⠃⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⣤⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠠⠴⠟⢳⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢰⡇⠙⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⠀⣀⣠⠴⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠶⠚⠋⠁⠀⠀⠀⠀⢀⡴⠋⠉⢷⡄
⠀⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⢳⡈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠠⠀⠀⠁
⠀⢿⡀⠀⡀⠀⠀⠀⠈⠀⠀⠈⠷⡀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁
⠀⠘⣧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠙⠶⣄⣀⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠈⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠿⠙⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢛⣿⣿⠟⢛⣿⣿⠛⠻⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣷⣤⣤⡶⠖⠚⠛⠓⠒⠛⠛⠛⠓⠟⠹⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⡟⠀⠀⠀⣀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣏⣻⣿⣿⣿⣿⠀⠀⠀⠃⠀⠀⣾⣮⣿⡆⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠻⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣾⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⣿⠘⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠟⢹⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠒⠒⢦⡄⠀⢠⡇⠀⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⢻⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⣿⣿⡿⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠛⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⡇⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣡⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠒⠤⣀⣀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠹⢶⣶⣎⣉⣠⣤⣴⡶⠖⠊⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⣤⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠉⠀⠀⣄⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⢀⡀⠀⠀⡀⠀⠀⠀⣠⣶⡿⠟⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠒⠛⠛⠲⠿⠛⠹⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⠉⠢⡀⠀⠀⠀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡀⠀⢸⢀⡀⢀⣀⠜⢹⡀⠀⠀⠀⠀⠀⠀⠀⠔⢢
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠈⢆⠸⡀⠀⠀⠀⠀⢨⠇⠀⠀⠈⠁⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡤⠤⠓⠒⠒⠻⢤⣱⡀⠀⠀⠀⡎⠀⠀⠀⠂⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡠⠊⣁⡀⠀⡠⠔⠒⠒⠲⣌⠳⡄⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⡠⠴⡶⠦⢄
⠀⠀⠀⠀⢠⠎⠀⠀⡠⠤⡊⠀⠀⠀⠀⠀⠈⣆⠹⡄⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⢀⠏⢀⠀⡜⠀⣠⠃⠀⠀⠀⠀⠘⡀⢸⠀⢻⠈⠀⠈⠓⠦⡤⠤⠤⣖⠋⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢣⠀⠀⠀⠸
⠀⣀⠤⣸⠀⠁⠀⢙⠋⠁⠀⠀⠀⠀⠀⠀⡇⢸⠀⢸⠀⠀⠀⠀⠀⡱⠂⢰⣉⣄⣀⣀⣀⠀⠀⠀⡰⠃⠀⠀⠀⠣⡀
⡞⠀⠤⣸⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⢁⡟⢀⡟⠀⠀⠀⠀⡈⢀⡴⠋⠀⠀⠀⠀⠀⠉⠲⣜⠁⠀⠀⠀⠀⠀⠈⠒⡺
⢧⠀⠀⠈⢧⠀⠀⠈⢧⠀⠈⠲⠤⠤⠖⢡⠞⢀⠞⡠⠂⠀⠀⠂⡼⣉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⣀⡴⠊⠁
⠈⠳⠤⠤⠞⠳⣄⠀⠀⠑⠢⠤⠤⠴⢚⣡⠖⠉⠌⠀⠀⠠⢀⠜⡁⠁⠀⠀⠈⠐⠠⡀⠀⠀⠀⠀⠀⢳⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠶⡶⠒⠛⠉⠀⠀⢠⠀⠀⡐⠠⠠⠊⠠⠂⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠓⢄⡀⠀⠀⠀⠘⣀⠔⡐⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⢀⡤⠒⠋⠁⠀⠀⠀⠈⡏⠑⠚⠀⠁⠴⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⢰⠋⠀⠀⠒⠠⢄⠀⠀⣸⠃⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⡴⠋
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢡⣰⠋⠀⠀⠀⠀⠀⠀⠀⢙⣤⡀⠀⠀⠀⠀⠀⣠⠞⢁⣠⠔⠋
⠀⠀⠀⠀⠀⠈⠓⠲⠶⠶⠒⠋⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠉⠓⠶⣤⠶⠿⠖⠚⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠐⠚⠉⠀⠀⠀⠀⠀⠀⣿⠈⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠂⠉⠉⠐⠢⣀⠀⠀⠀⠀⠀⡼⠡⠆⠀⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠱⡄⢀⣠⠞⠀⠀⠐⡀⠀⡇⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠶⠞⠛⠉⠀⠀⠀⣀⠖⠁⠀⠙⠓⠒⠲⠤⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⡶⣾⣿⡿⣿⣿⣷⢶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⠿⣛⣛⣙⣶⣴⣧⣷⡾⣿⣿⣶⣮⣼⣦⣶⣁⣛⣻⠿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⢻⣟⣥⣞⢿⣿⣛⣯⢩⣤⡦⠚⣷⣟⣻⣷⠓⣰⣤⡭⢽⣛⣿⡿⣳⣬⣳⡝⠻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⣫⣟⡿⢛⡩⢷⣿⡭⠾⡇⠠⠥⠤⠂⠘⠁⠈⠃⠀⠠⠤⠄⠸⠷⢭⣽⡾⢍⡛⠿⣻⣝⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣫⣾⣿⣛⡔⡺⠟⢃⡄⠈⠂⣀⢀⣴⠛⣿⢙⡖⢠⡄⢲⡍⣿⠛⣦⡀⣀⠐⠁⢀⡘⠻⡓⢨⣛⣿⣷⣝⢿⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣾⠟⣎⣴⣷⣻⢏⡷⠿⠀⢈⣥⣾⠹⡖⢺⡧⠘⠂⠉⠉⠁⢸⡇⠀⠉⠉⠐⠃⢼⡗⢲⠎⣷⣮⡛⢀⠿⢾⡹⣏⣾⣦⣱⠿⣷⡄
⠀⠀⠀⠀⢀⣴⣿⠷⣾⡷⣣⣞⠙⠛⠤⠀⣀⠞⣤⠛⠋⠁⠀⢙⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⡏⠀⠈⠈⠛⣴⠻⣀⠄⠠⠛⠉⣳⣜⢿⣶⠾⣿⣦⡀
⠀⠀⠀⢀⣾⡿⣣⢾⢯⡜⠛⢁⠜⣥⠲⣆⠹⢧⡀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢠⠠⠄⡄⠀⠀⠀⡘⠁⠀⠀⠀⠀⢀⡼⠏⣰⠒⣤⢡⡌⠛⢡⡽⣷⣜⢿⣷⡀
⠀⠀⢀⣿⡿⢰⣿⣿⢫⡿⢄⠀⣞⠛⡷⠁⠀⠀⠻⣆⠀⠀⠀⠰⠌⠇⠁⠀⠈⢀⡀⠁⠁⠈⠸⠡⠆⠀⠀⠀⣸⠿⠀⠀⠀⢶⠛⣳⠁⠠⢿⡝⣿⣿⡆⢿⣿⡀
⠀⠀⣾⢿⣁⣿⢣⠷⠿⡀⠠⢤⣍⠋⠀⠀⠀⠀⠀⢠⠳⡉⠀⠀⠀⢺⡆⢀⡄⢹⡇⢠⡀⢠⡇⠀⠀⠀⢉⠞⡄⠀⠀⠀⠀⠀⠙⣩⡤⠄⢠⠿⠾⡼⣿⣈⣽⣷
⠀⣸⣿⣤⣿⢃⣶⠆⡤⣠⠴⣮⠉⠓⢶⣤⡀⢀⡀⠈⠁⠘⣶⡀⠲⡄⣻⣄⣷⠼⠧⣾⣠⣟⢠⠖⢀⣶⠅⠈⠁⢀⡀⠀⣤⡶⠚⠉⣰⠶⣄⢦⠰⣶⡘⣿⣤⣿⣇
⢠⣿⡏⣸⣯⣌⡉⠲⢱⠿⠶⠆⠀⠀⠀⠁⢙⠓⠍⣀⡀⠰⣄⢹⡶⠛⠁⣀⣰⣤⣤⣄⣀⠈⠛⠶⣏⣠⠆⢀⣀⠨⠘⡋⠀⠀⠀⠀⠐⠶⠻⡞⠒⠉⣣⣽⣇⢸⣿
⣸⣿⢹⣷⣷⢯⡗⠂⠈⠟⠒⠀⠀⠀⠀⠀⠠⠁⠀⠙⠻⢦⡼⠛⢀⣴⣞⢻⣯⢼⡧⣽⡟⣷⣦⡠⠙⢧⡴⠟⠋⠀⠈⠀⠀⠀⠀⠀⠀⠓⠻⠁⠐⢾⠽⣶⣿⡏⣿
⣿⡟⢸⣯⣿⣼⠇⠀⢉⠛⠛⠒⢶⣶⠠⡼⠁⢀⣀⣈⠓⠶⠇⢰⣿⣧⣟⠿⣛⣯⣥⣟⠿⣻⣼⢿⡆⠼⡷⠚⢁⣀⡀⠈⢡⠄⢲⡶⠒⠛⠛⡉⠀⠼⣧⣟⣼⡇⢿
⣿⡷⣼⡏⣬⡄⢠⢰⣏⣙⠀⠀⠀⠀⠀⠐⠂⠀⠉⣉⢙⡇⠀⣾⠷⣾⡏⣼⣿⠟⠫⣿⣧⢹⣷⠾⣿⠀⢸⡋⣉⠉⠀⠐⠒⠀⠀⠀⠀⠀⡋⣹⣆⡄⢠⣥⢹⣧⢾
⣿⣧⢿⡇⠻⠃⠸⠸⣏⣸⠀⠀⠀⠀⠀⡠⠄⢀⣀⣉⣩⡏⠉⢿⡶⣿⣇⢻⣯⣬⢴⣿⡟⣼⣿⢶⡿⠉⣹⣍⣉⣀⡀⠀⢄⠀⠀⠀⠀⠀⣏⣹⠇⠖⠘⠟⢸⡿⣼
⣿⣧⢸⣟⣻⢳⡖⠠⣘⣥⣤⠤⠶⠖⠒⢃⡉⠀⠋⢩⣤⢿⡆⠹⣷⡟⣯⡶⣽⣛⣛⣯⢾⣽⢻⣾⠇⢰⡿⢤⡍⠙⠀⢉⡚⠒⠶⠶⠤⣤⣤⡁⠀⢲⣞⣟⣿⡇⣼
⢻⣿⣸⡿⠿⣞⣧⠄⠀⣦⣤⠀⠀⠀⠀⠀⠀⡀⠀⣀⣤⠞⢳⡤⠈⠿⣿⣹⣟⢺⡟⣻⣏⣿⠟⠁⣤⡞⠳⣤⣀⠀⢀⠂⠀⠀⠀⠀⠀⣤⣴⡀⠠⣼⣣⠿⢿⣇⣿
⠘⣿⣇⢿⣟⢋⣁⠰⡸⣦⡴⠄⠀⠀⠀⠀⣈⡅⠂⠉⠁⠰⠋⣹⠦⣤⡊⠉⠹⠛⠛⠏⠉⢑⣤⠼⣏⠙⠆⠈⠉⠐⢬⣁⠀⠀⠀⠀⠠⢶⣴⢇⠆⣈⡙⣻⡟⢸⣿
⠀⢹⣿⠻⣿⡌⡿⠆⠡⠙⠶⠛⣀⡠⠾⠟⠀⠈⠁⢄⠀⠀⡾⠃⢴⠋⣹⠛⡷⢲⡖⢾⠛⣏⠙⡦⠘⢷⠀⠀⡠⠈⠁⠀⠻⠷⣄⣀⢛⠶⠋⠜⠰⢿⢡⣯⠗⣿⡏
⠀⠀⢿⡾⡛⣿⡼⡶⣶⠁⠀⠚⣋⣁⠀⠀⠀⠀⠀⠘⡴⢃⡀⠀⠀⢼⠇⠈⠃⢸⡇⠘⠃⠸⡇⠀⠀⢀⡘⢤⠃⠀⠀⠀⠀⠀⣈⣙⠃⠀⠈⣶⢶⢧⣿⢙⢶⡿
⠀⠀⠈⣿⣷⢸⣿⣿⣜⣷⠂⠀⣯⣠⠧⡀⠀⠀⣴⡮⠀⠁⠀⠡⣂⠆⣀⣀⢀⠈⠁⠀⢀⠀⠰⡰⡌⠀⠈⠀⢵⣦⠀⠀⢀⠴⣄⣽⠀⠐⣾⣣⣿⣿⡇⣾⣿⠃
⠀⠀⠀⠘⢿⣷⡟⢿⣞⠣⣤⠈⢮⡻⠴⠏⣠⡞⠁⠀⠀⠀⠀⠀⡜⠉⠁⠀⠘⠲⠖⠃⠀⠈⠁⢡⠀⠀⠀⠀⠀⠈⢳⣄⠹⠦⠟⡵⠁⣤⠼⣳⡿⢻⣾⡿⠃
⠀⠀⠀⠀⠈⠻⣿⡦⢿⣧⡙⠟⢀⣀⠀⠀⠋⢤⠖⢤⡀⠀⠀⣸⠋⠀⠀⠀⠀⢲⡆⠀⠀⠀⠀⠘⡇⠀⠀⢀⡤⠲⣤⠙⠀⠐⣀⡀⡻⢋⣼⡽⢔⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣤⢏⢻⡾⣹⣝⡳⣶⠁⣌⡳⢯⣠⡏⢰⡗⢀⣀⣀⠀⠀⢸⡅⠀⠀⣀⣀⡄⢺⡆⢹⣄⡿⢞⡁⠈⣶⢞⣫⣏⢷⡟⡻⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡝⢿⣿⣭⡧⢤⣆⡉⠃⢠⠄⠉⠀⠻⣤⣿⣸⠧⢸⠇⠼⣅⣽⣤⠟⢁⠐⠠⡀⠘⢉⣰⡄⢼⣭⣿⡿⢫⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣮⣝⣿⣶⢭⣓⡾⣟⡳⣶⡄⡐⠒⠚⠁⢠⠀⠀⡄⠀⠓⠒⠂⢠⡶⣞⣻⢷⣚⣭⣴⣿⣫⣵⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⣜⣭⡛⢿⣿⡿⣭⣗⣚⠿⠗⢠⡿⣯⣽⢿⡄⠾⠿⣓⣺⣭⢿⣿⡾⢛⡩⣧⣴⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣮⣭⣭⢍⡿⢿⡿⠿⠯⣿⣿⡽⠿⢻⡿⠿⡫⣭⣭⣷⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠶⠷⣿⣿⣶⣶⣿⣿⠶⠶⠿⠟⠛⠛⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠿⡿⠿⢿⣿⠿⠿⠿⠿⢿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠉⠉⠉⠉⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣩⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡁⠔⠺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⡤⠊⠀⠀⠁⠀⠀⠀⠀⠀⢠⣤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡰⠎⠀⡀⠀⠌⠀⠀⠀⠀⢀⠄⠂⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⡘⠀⢀⣀⣀⠛⠢⢀⠀⠀⢰⠁⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠀⡐⠀⠀⠀⢀⠋⠉⠒⠤⣉⠀⠈⢄⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠁⠀⠃⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠒⠉⠀⠀⠒⠒⠀⠀⠀⠀⠒⠂⠠⠄⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡏⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉⠀⠀⠀⠐⠀⠄⣀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⢸⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠸⠀⢀⠠⢒⡈⠀⢒⣠⡀⠀⣠⣀⠀⣀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⢦⡦⠊⢀⠤⠐⠈⠉⠁⠀⠀⠉⠙⠃⠾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⡀⣀⣼⣁⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⠟⠙⠛⠿⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠁⠢⡑⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⣿⣿⣿⣿⣿⣷⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⡠⠤⠅⡚⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠁⣠⠀⠠⠤⠄⠐⠂⠉⠁⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⢀⣿⣿
⣿⣿⣿⣿⠋⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢡⠀⢀⡸⠀⠀⠄⠂⠢⢄⠀⠀⣸⡀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⣸⣿⣿
⣿⣿⣿⣏⠠⠤⠀⠒⠲⡄⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⢀⣸⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⢠⣿⣿⣿
⣿⡿⠋⠁⠀⠀⠀⠀⢀⠜⢆⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣤⣿⣿⣿⣿
⡏⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠑⠢⣀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿
⣷⣄⣀⣀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⡀⠤⠤⠤⠤⠤⠀⠒⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠂⠀⠀⠀⠃⠤⣀⠀⠀⢠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⣿⣿
⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠡⠀⠀⠀⠀⠀⠉⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠈⠙
⣿⣿⣿⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠰
⣿⣿⠏⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⢀
⣿⡏⠀⠀⠀⠂⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣰⣿⣶⣤⣤⣤⣿⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠍⠓⠒⠀⠤⡤⠤⠀⠒⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣀⠀⠀⠀⠀⡠⠊⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⡂⠉⠀⠀⠀⢂⡀⠀⠀⠀⠀⢀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⣠⣤⠀⠀⢀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⢰⣾⢿⣷⣤⣾⡟⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⡿⢿⣇⣠⡀⣿⣿⡄⠀⢀⣾⡟⣿⡇⢹⣿⡏⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿⠿⠷⣿⣷⣿⣿⣿⣿⣿⣿⣿⡄⣼⡿⠀⣿⣧⢸⣿⠁⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣼⣿⠋⢹⣿⡜⣿⣏⠁⠹⣿⡟⠁⠀⢸⣿⣼⣿⣆⠀⢀⡓⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣿⣿⣿⣤⢈⣉⣁⣹⣿⠆⣠⣿⡇⠀⢰⣾⡋⠉⠉⣶⡆⣠⣴⣶⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠋⢉⣿⡿⠛⠻⣿⣦⣼⡿⠁⠀⢀⠈⣉⡃⠀⢸⣿⣿⣿⣏⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⢾⣿⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠻⣷⣶⣿⠟⠈⠛⠻⠿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠐⢻⡿⠇⠀⠀⠻⣷⣶⣶⡿⠋⣈⣻⣷⣶⣶⡿⢾⣿⣿⣿⣿⣿⢰⣶⣶⣿⠟
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠗⠀⠀⣿⣿⠿⣿⣷⣿⡿⠛⠛⠀⠀⣼⡿⣿⡇⢸⣿⣀⣴⣾⠗⠀⠸
⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⢀⣿⣿⣶⣿⠟⢻⣷⣴⡶⠀⢠⣿⣦⣼⣧⢸⣿⡙⣿⣤⣤⡄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢻⣿⣦⣀⣤⣤⣸⣿⠻⣷⣦⡸⣿⣅⣀⣀⣸⣿⠛⢻⣿⣸⣿⡇⠘⣿⣏⣠⣤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠛⠹⠿⠀⠈⠻⠟⠿⠿⠛⠛⠿⠗⠀⠸⠿⠿⠿⠇⠀⠸⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢠⣶⣶⣶⣶⣶⣶⣶⣴⣦⣴⣶⣶⠄
⠀⠀⡇⣿⢸⣼⣧⣷⣾⣿⣿⡇⣿⢸
⠀⢠⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈
⠀⢸⣇⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⡄
⣶⡘⠛⠛⠛⠛⠛⠛⠛⠛⠿⠛⠛⠛⠃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣄⡀
⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡖⣣⡾⢻⣿⣿⣿⣿⣿⡝⣦⢲⣾⣿⣿
⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣦⣾⣿⣿⣿⡿⠻⠿⣿⣿⢾⡇⠙⣿⣿
⣿⣿⣿⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⠈⣿⣿⣿⡇⠈⢹⣿⣿⣿⠉⠉⣿⣿⣿⡿⣿⣿⣿⣿⢡⣴⣶⡌⣿⣾⣟⣦⣽⣿
⣿⣿⣿⠀⠀⣿⣿⣿⡇⠀⠸⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⢠⣿⣿⣿⣏⣼⣿⣿⡇⣾⡧⠘⢸⣿⣿
⣿⣿⣿⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⣷⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⠀⠀⣿⠟⢁⡈⢻⣿⣿⣬⣍⣟⠛⣻⠟⡇⠀⢸⣿⣿
⣿⣿⣿⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⠿⠀⠀⠁⣰⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠡⡇⠀⢸⣿⣿⡄
⣿⣿⣿⠀⠀⣿⣿⣿⣧⠀⠀⠿⠛⠛⠛⠀⠀⡙⢉⡉⢁⣤⣄⣤⣴⣖⣸⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⣷⠀⢸⣿⣿⡇
⣿⣿⣿⠀⠀⣿⠟⠋⠁⠀⠀⠐⠲⣄⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⢸⣿⣿⣿⠶⣶⣤⣤
⣿⣿⣿⠀⣴⣯⣴⣶⣤⣤⣄⡀⠀⠈⠻⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⠀⠀⠀⠈
⣿⣿⣿⣾⣿⣿⣿⡿⠿⠿⠿⠿⠦⡄⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⢏⣤⡈⠉⢉⣙⡀⠀⠀⡀
⣿⣿⣿⣿⣿⣿⣷⣶⣷⣾⣦⣤⣀⣈⣀⡀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣻⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⡸
⣿⣿⣿⣿⣿⣿⣿⣿⡛⣿⣟⠁⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢻⣿⣿⣷
⢿⣿⣿⣿⣿⣯⣛⠛⣿⣙⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣀⣤⣤⣶⣶⣶⣶⣿⣿⣿⣿
⣿⣷⣦⣭⣭⣿⣿⣷⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠟
⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠻⣿⣿⣟⠛⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡴⠶⠶⡿⠿⠋⠻⠿⠚⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⢷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠆⣀⣴⠧⠉⠉⠓⠢⡀⠀⠀⠀⠀⠀⣀⠤⠔⠒⢒⣦⣄⣀⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⢀⣼⣟⣁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⣳⣶⡀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⣼⡿⡋⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⠀⠀⠀⠈⢻⣿⡀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⢀⣿⡿⠁⠀⠀⣠⣾⣟⣹⣿⣿⣷⠀⠀⠀⠀⣿⣯⣹⣿⣿⣿⣦⡀⠀⢤⣿⡗⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢸⣷⣶⣤⡴⠙⣿⣿⣿⣿⣿⡟⠒⠒⠘⠚⠟⠿⠿⣿⣿⡿⠃⢿⣴⢿⠙⠇⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠈⢹⠟⠉⠁⠀⣺⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡂⠀⠀⢀⡶⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡀⠀⠀⠀⢸⠆⠀⠀⠚⠁⠀⠀⠀⢀⣤⣶⣶⠶⣶⣦⣄⠀⠀⠀⠀⠉⠀⢀⣿⠁⠀⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⠀⠀⠀⠺⣦⣤⠄⠀⠀⠀⠀⠀⣿⣿⣿⣶⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⢼⡏⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣤⣤⣶⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠀⠈⠹⣆⠀⠲⣄⣀⡤⠾⠞⠁⠀⠀⠙⠛⠶⣶⣀⠀⣀⠞⢀⡼⠉⠛⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠳⣄⣀⣀⡀⠀⠀⠀⣀⣠⡴⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡄⢀
⠀⠀⠀⠀⠀⠴⡽⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⢂⠀⠀⠀⠀⠀⠀⠁⠲⡴
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⡴⣄⠀⢀⡼⠂⠀⠀⠐⠌⠀⡆⠀⠀⠈⠣⠈⠙⠁
⠀⠀⠰⠄⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⢠⠄⣿⡿⠛⡴⢇⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠰⣤⡇⠀⠀⠀⠀⠀⠈⠀⠃⠀⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⢁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠁⠁⠀⠠⡇⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠀⠀⠀⠀⣄
⠀⠀⠀⣞⠡⡄⠀⠀⠤⢀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⢄⠒⣆⠀⠀⠀⠀⠀⠀⠐⠌⠊⠁⡀⠀⠀⠨⡁⠼⠀⠀⠀⠀⠀⠐⠋⢶⠤⠀⠀⠀⠀⠄
⠀⠀⠀⠈⠠⡁⠀⠰⠀⠈⠑⣃⡔⠀⠀⠈⠀⠀⠀⠐⠂⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⣿⠀⠀⠀⠀⠰
⠀⠛⠞⠂⠀⠀⠀⠐⠂⠀⠈⣀⢀⠀⠀⢀⠀⠀⡌⠀⠀⠀⡄⠈⢦⠀⢀⠀⠀⠀⠀⠀⠀⢢⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠁⠀⠀⠀⠀⠂⠀⠀⠀⢠⠀⠀⠀⠀⠀⣏⠀⠠⠆⢠⣮⣿⠁⠤⠠⠀⠽⠷⠀⠀⠀⠤⠀⠄⡀⠀⠀⠄⠀⠀⠀⠠⣉⡄⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⢠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⢂
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⣉⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠐⠃⠀⠀⢤⠀⠀⠀⠀⠀⠀⢰⡀⢀⣤⡄⠀⢀⡔⠀⠀⠀⠀⠃⠀⠀⢦⠀⠈⠀⠀⠳⠈
⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠂⠐⠀⠀⢠⠀⠀⠀⠀⠀⠀⣀⡜⣿⡷⢛⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠘⣖⡶⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁⠀⠈⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠠⠂⠘⠦⠴⡄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠁⠈⠀⠓⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠠⡆⠠
⠀⠀⠀⠔⢃⠁⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢾⡇⠀⠀⠁⠀⠄
⠀⠀⠀⠂⠀⠠⠀⠀⢘⠀⠠⡤⠆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠
⠀⠀⠊⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠓⠀⠦⠀⠀⠉⠀⠀⠀⠀⠀⠀⠆⠠⡬⠏⠀⠀⠀⠈⠉⠂⠀⠀⠀⠈⠀⠀⠀⠀⠘⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⣀⡀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠀⠀⢼⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠹⣿⣿⣿⣿⣿⣆⠀⢀⣬⡄⠀⠀⠀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⡤⠀⠻⠻⣿⣿⣿⣿⣀⢿⡿⠃⠀⠀⠀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠘⢪⡿⣿⣿⣿⣿⣯⣀⡀⠀⠀⠀⠀⠸⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠅⣿⣿⣿⣿⣿⣿⣿⣿⡧⣤⠋⣸⣿⣿⣿⣿⣿⣿⣷⣿⣯⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠘⢿⣿⣿⣿⣿⣿⡿⣻⣷⡄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⣤⣈⣙⡿⣿⣿⠟⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣠⣤⣦⣄⡀
⠀⠀⠀⠀⠰⢶⣿⣿⣿⣿⣿⣿⡿⣿⣹⣡⣾⣿⣿⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠈⠉⣠⠞⠁⠈⠙⠉⠟⠚⠋⠈⠋⠙⠛⠛⠛⠉⠉⠃⠈⠉⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⠋⠙⠛⠉⠙⠋⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢹⣯⡏⡉
⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠀⣠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⡸⠏⣨⡿
⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠐⠀⠒⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠃⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠓⠏⠀⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠡⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⣀⠀⠀⠀⠀⠀⠠⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢉⣡⡁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠿⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠘⠀⠈⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⠀⣀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⠉⠀⠀⠀⠀⠈⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⡀⠀⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠂⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠺⣟⡛⠛⢲⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠙⢾⡓⠿⠉⠋⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣧⢀⣤⢤⣤⢬⣟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡿⠦⠉⠃⠉⠀⠹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢈⣛⣶⣤⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢸⡏⠉⠛⠿⣍⡉⠉⠛⠛⢩⣭⡭⠿⣿⣿⠳⢶⡶⠶⠶⠒⣶⣤⠴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢀⡀⠀⠀⢸⡇⠀⠀⣤⣈⡙⢄⠀⠀⠀⠬⣽⣿⡗⠀⠀⠀⢀⣤⡶⠛⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠜⠉⠀⠀⠀⠀⢷⠀⠀⠞⠀⠀⠀⠀⠀⠀⠈⣿⡋⠉⠀⠀⠀⠘⠁⢀⣠⡤⠀⢠⡟
⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⡟
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢻⣻⣿⡹⠀⠀⠀⠀⠀⢸⣿⣿⣳⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⣿⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠉⠉⠁⠀⡀⠀⢀⠀⠠⠉⠉⠁⠀⠀⢀⣴⠏
⠀⠀⠀⠀⢰⡇⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⠘⠀⣀⣉⣶⣏⣀⠐⠀⠂⢀⣠⠴⣿⠋
⠀⠀⠀⢀⣾⠇⠀⠀⠀⣸⣿⣦⡀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠉⠓⢦⣄⣀⠀⣀⡤⠴⠒⠋⠉⠀⠀⠘⢧⡀
⠀⢀⡤⠞⠁⢀⡀⠀⢰⡟⠀⠈⠻⣦⡀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⣴⢋⡤⠖⢠⡏⠀⢀⡿⠀⠀⠀⠀⢸⣿⡋⣀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠈⠙⠷⠤⠴⠛⠛⠛⠁⠀⠀⣀⡴⠋⠁⠘⢛⣷⡄⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠀⠀⠀⠀⢀⣰⣿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠉⠀⣢⡀⠀⣨⣿⠿⠒⠛⠋⠛⠓⠶⠶⠶⠶⠚⠛⠋⠉⠛⠻⣦⡀⠀⢀⡐⢿⣿⠑⠀⠈⠛⢦⣄⡀
⠀⢀⡤⠶⠖⠚⠛⠉⠁⠀⠀⠀⢀⣾⠿⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣿⡟⠃⠀⠀⠀⣶⣄⣈⠙⠳⠶⣤⣀⡀
⠀⠿⠧⣤⠖⠀⠀⣠⡆⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣤⣼⡋⠀⠁⢀⠀⢠⣄⡀⠉⠙⢶
⠀⠀⠀⠻⠤⠴⠚⠁⠹⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢾⡅⠈⣳⣄⢈⡿⠶⠖⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣯⣥⣬⣽⣧⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⡴⠶⠛⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠀⠉⠙⠛⢶⣼⣿⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠈⢳⣴⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢷⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢾⣯⠀⠀⢀⠔⠈⠉⠉⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠎⠀⠀⠈⠛⠉⠀⣀⡚⠒⠂⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠀⠀⠀⠀⣠⠎⠓⠀⠈⠉⠀⠉⠁⠀⠌⡠⠚⢃⡠⠔⠂⠉⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⡄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢊⠜⠰⠞⠁⠀⠀⠀⠀⠀⠀⠴⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⡇⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠠⠶⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⡀⠀⠀⢳⣶⣦⡘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⢴⡀⠀⢀⡧⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠐⢷⠾⢲⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠀⢄⠁⠀⠠⣗⢟⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠠⠌⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠈⠁⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠘⣏⠽⠇⠀⠀⠀⠀⠀⠀⣴⠀⠉⠉⢡⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠐⢀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠓⠂⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢦⡿⠶⠒⢢⠀⠀⠳⠶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠠⣖⠛⠢⢄⣁⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠉⠉⠉⠷⠀⠀⠠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠐⠾⡄⠀⠀⠹⡀⠉⠉⠉⠉⠉⠉⠀⡰⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⡀⠀⠀⠀⠀⠐⢻⢁⡠⠤⠤⡙⠚⠛⠉⠉⠙⠒⠚⠁⣠⠀⢤⡀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠳⢌⠀⠠⠔⢊⠵⠃⡇⢀⠀⠠⢹⠀⠀⠀⠀⠀⠀⠀⣼⠑⠀⠂⢹⡆⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠬⢥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⡴⠁⠀⠀⠑⢤⣈⣡⠎⠀⠀⠀⠀⠀⠀⠀⠙⣤⣀⣉⠜⠁⠀⠀⠑⢦⡀⠀⠀⠀⠀⡀⠀⠈⠁⠀⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡦⠤⠤⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠑⣦⣄⣀⣐⠄⠀⠀⠀⠀⠐⠥⣹⡆
⠀⠀⠀⠀⣠⣤⠆⠰⣤⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢻⣄⣤⠶⣤⡀⠀⠰⣨⠗
⠀⠀⠀⠾⠛⠁⠒⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⢶⣄⠀⠀⠀⠉⠛⠛⠁
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣀⣀⣀⡀
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠉⠛⣦
⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠉⠉⠀⠀⠀⠈⣿
⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢀⣴⠶⠶⠶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠈⠻⣤⡀⠀⠀⠀⠀⠀⢸⢀⡶⠛⠋⠀⠀⠀⠀⠀⠉⠛⠛⠲⠶⠆⠶⠶⠚⠛⠁⠘⢷⣄⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⣀⡀⣀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡸⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣦⣀⠀⠀⢀⣀⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣞⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣾⣷⣤⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣬⣉⣻⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣯⣕⣲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⡿⣿⣿⡏⠈⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣶⣾⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⠁⢠⢡⣿⡿⢻⣿⣿⡿⣿⣯⣽⣿⣿⣿⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣟⡘⠃⢉⣀⠀⠐⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠟⠋⠉⠀⢀⡀⠘⠁
⠀⠀⠀⠀⠀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠹⢿⠿⢿⣿⣯⣄⣤⣾⣿⣿⣿⣿⣿⡿⠿⣿⣿⠻⡶⠄⠢⣀⣶⡀⠀⡀
⠀⠀⢠⣠⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡻⢿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠉⠙⣷⣦⣿⣿⣷⣾⣄⣀⣠⣶
⣄⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣷⣬⡁⣮⢻⣿⣿⣿⣿⣿⣿⣿⣵⣿⡟⠋⠛⢻⣿⣿⣷⣦⣄⣤⣴⣿⢿⣿⣿⣿⣿⣿⣟⣋
⠉⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣏⣷⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⡿⡀⠀⢶⣼⠹⣟⡟⣻⣿⣟⣋⡀⢦⣼⣿⣿⣿⣿⣿⣯⣤
⠤⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠁⣿⠟⠹⡏⠀⠀⣌⡿⠿⠟⣽⣻⣿⣿⣿⣿⣶⡒
⠀⠀⠉⠉⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⢿⣿⣿⡁⣶⣧⣶⣆⣴⣶⣦⣤⠀⠀⢀⢩⣿⣿⣿⡿⠿⠟⠛⠁
⠀⠀⠠⠚⠉⠉⣾⡿⠿⣿⣿⣋⣙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠊⠉⠀⠀⡼⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠛⠋⠁⣺⠿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢰⠋⠹⠿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣏⡙⠛⠛⠛⠋⠉⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠙⣆⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣸⣀⠀⠀⠀⠀⠀⡰⣻⡳⣀⠳⣦⣄⠀⢀⣠⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡖⠊⠁⠈⠉⠓⠒⠒⠚⠣⢿⢫⣴⣵⠘⠀⠋⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠈⡗⠪⠒⠛⠋⠀⢀⠀⠀⢠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⢄⣀⣀⡀⢤⣾⠇⠀⠀⠀⠀⠀⠈⡗⠒⠁⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⠒⠒⣒⠮⢡⠎⠀⠀⢀⡰⠃⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⠀⠛⡄⠀⠀⠀⠀⠀⢠⠃⠀⠀⡐⠑⣂⡥⠚⠒⢚⡿⠀⠀⠀⠀⣿
⠀⠀⠀⡄⠰⢆⠀⢸⣀⡀⠀⢠⠃⡤⠤⢀⡀⠀⠈⠒⠒⢊⠠⠒⠉⠉⡧⠖⣋⠀⠀⠀⠀⠀⡿
⠀⠀⡜⠻⠁⠘⠋⠉⠈⠉⠑⡇⢘⠄⠀⠀⢳⠀⢀⠔⠊⠁⣠⠔⢊⠼⠚⠉⡀⢱⠀⠀⠀⠀⡇
⠀⡜⠀⠀⠀⠀⠀⠀⠀⠈⠑⣭⢞⣀⡠⠁⣸⠐⣕⡀⢰⡟⡡⠚⠁⠀⣠⠜⣱⠃⠀⠀⠀⢰⠃
⣤⠇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⡤⠞⠁⠀⠀⠉⢹⣞⢁⠄⢠⠚⠁⠔⠁⠀⠀⠀⠀⡞
⡍⠇⠀⠀⠀⠀⠀⡴⢋⠀⢀⣴⠋⢳⣀⡀⠀⠀⢀⡤⠋⠈⠘⢲⠋⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠇⠀⠀⠀⠀⠀⠀⠈⠛⡤⠄⠈⠆⢸⠀⠙⣶⠊⠁⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠸⡀⠀⠀⠀⢀⠀⡸⠳⠇⠀⠀⢸⠂⠀⠀⡽⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠱⣄⠀⠀⠘⡗⠁⠀⠀⠀⠀⠀⠀⢀⡴⠃⣀⡠⠔⠊⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠈⠓⢤⣀⡀⠀⠀⠀⠈⠑⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⡀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⢀⠀⠒⠀⠒⠒⠒⠒⠀⠐⠒⠂⠀⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠒⠒⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡄⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⢀⣶⣶⣶⣆⠀⠀⣠⣶⣶⣶⣄⠀⠀⢀⣀⠀⠀⠀⣠⣤⡄⠀⠙⠛⠁⠰⣦⣤⡄⠀⠀⠀⠸
⠀⡀⠀⠀⢿⣿⣿⣿⣿⠃⢰⢿⣿⣿⣿⠿⠀⠀⠿⠿⠃⠀⠐⠿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠘⠧⠆⠀⠀⠄
⠀⡇⢀⣀⡑⠌⢛⡉⠞⠁⠀⠹⠏⢋⠿⠈⠀⠀⠀⠁⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⡀
⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠇
⠀⠀⠈⠙⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠇
⠀⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣶⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠸⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣴⡆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠛⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⢿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⠯⠴⡾⠇
⠀⣠⣤⣀⡀⢀⡀⣀⠀⣀⠀⠀⠀⢀⢀⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⣿⣿⣿⡇⠀⠀⢀⣄⣠⠶⠴⢦⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⠀⡤⠤⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠘⠋⠀⠀⠀⠚⢹⣏⣀⣠⠀⠀⢠⣦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⢀⠈⠉⠀⠀⠀⢀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠁
⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠘⠁⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠂⠈⠀⠀⠠⠀⠀⢰⣿⠇
⠀⡀⢀⣠⡀⠔⠀⠄⢠⣶⣶⣶⣦⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢦⡴⠂⠀⠐⠠⠀⠀⢄
⠀⠀⣴⣿⣶⡀⠈⢳⠀⠀⠀⢼⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢳⡆
⠀⠀⣿⣿⣿⣿⣶⡇⢠⡄⠀⠘⠛⢉⣴⡆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁⠘⠃⠐⠀⠂⢀⣾⠇
⠀⠀⠈⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⣄⠑⢀⡞⠁⡔
⠀⠀⠀⠀⠛⢻⡿⠀⠀⠀⡤⢤⠄⠢⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠐⠆⠀⠀⠀⠀⠀⣄⡀⠄
⠀⠀⢰⠀⠀⢽⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣔⣂⡀⠀⠀⡀⠀⠈⠁
⠀⠀⠈⠀⠀⣾⠋⢀⠀⠀⢀⡀⢀⡀⣠⣄⣀⣠⣄⣄⣠⣠⣄⣠⣀⣄⠀⣀⠀⠙⠋⢀⠀⠈
⠀⠀⠀⠀⣴⠛⢷⣄⡀⡀⣸⡇⠈⣥⣬⣿⣟⣿⣿⠋⣿⣭⣿⣿⣿⢛⣤⡁⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠜⠁⠀⠀⠈⠉⠻⠋⠁⢀⠿⠘⠿⠻⠻⠗⡰⠿⢗⠛⠲⠿⠛⠿⠃⠀⠀⢠⡀⠀⠀⣀⣶⣿
⠀⠀⠀⢠⡄⠀⠀⠀⠀⠈⠋⠀⠈⠀⠀⠀⠑⠋⠈⠙⠃⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠿⠛⠋⠉⠁
⠀⠀⠀⠈⠁⠰⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠔⢢⠀⠀⠀⠀⠈⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⡀
⠀⠀⠀⠀⠀⠀⢠⠔⢸⠃⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠻⠆
⠀⠀⠀⠀⠀⠗⠀⠀⣀⡀⢠⠀⠐⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⡤⠀⢆
⠀⠀⢄⡀⠀⠀⠀⠰⠿⠃⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡑⠀⠀⣰⠃
⠀⠀⠀⠙⠓⠶⠶⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠶⠦⠀⠠⠶⠶⠶⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣞⣉⠀⢀⣶⣾⣦⣤⣄⣴⣤⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣾⣷⣆⣶⠶⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣿⣿⡿⠋⠀⠀⠂⠀⠙⠟⠋⠀⠀⠀⢹⣿⣿⠀⠴⠀⠂⠀⠀⢹⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢩⣿⣿⣿⣿⡁⠏⠀⢀⠴⠀⠒⠁⠀⠀⣀⠀⠀⠀⠈⠑⢄⠀⠀⠇⠀⢸⣿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣿⣿⠿⠟⠋⠀⠀⠀⠂⢀⠀⣁⢠⠀⡍⠰⢇⡴⠛⠉⠞⢣⠶⠀⠀⢰⠘⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⡆⠉⠀⠀⢠⡝⣄⠸⠀⠀⠀⠀⡌⠀⢀⠀⠘⠐⠈⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠁⠀⠀⠀⠆⠀⠀⠀⢘⠁⠀⠀⢀⡜⢡⡼⠀⠀⠲⢤⡞⠀⠀⠀⠌⠀⢤⠀⠈⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⡀⠀⠀⠠⠈⠆⠀⠀⣏⡀⣹⠀⡀⠀⠀⠀⠁⠀⠀⠤⢀⡆⠀⠀⢰⠏⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠁⠀⠀⠈⠸⠀⠀⠶⠆⠸⠿⠷⠄⡀⠰⠆⠀⠀⠀⠀⠈⠀⠂⠀⠸⠂⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⢸⣰⠄⢀⡁⠀⣄⡀⠠⠄⠠⠖⠋⠙⠁⠀⠀⠄⠀⣀⠄⠂⢀⠀⠀⠀⠀⡼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⡀⠀⠀⢠⠞⠉⠁⠀⠀⠈⣇⠐⠖⠀⠀⠲⠰⠖⠂⣀⠤⠐⠌⠙⠀⠀⣼⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠠⠀⠀⠊⠀⠀⢀⡄⠀⠀⢁⠀⠀⠀⢺⠀⠀⢀⡜⠁⠄⠀⠁⠀⠳⣄⠀⠀⠀⢦⡀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⡀⠀⠀⠀⡀⠀⢻⠁⠀⠀⠉⠡⡀⠤⠀⠑⢄⡀⠀⠉⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⢀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠇⠀⠀⠀⠉⠑⠺⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⢀⣴⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣶⣴⣸⣷⡄⠀⠀⠸⢶⣇⡀⠄⠀⠀⠀⠀⠀⢀⣈⣠⣝⣦⣤⣼⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣶⣀⠀⣀⢸⣻⣿⣷⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣧⣄⡀⠉⠻⠿⣿⣿⣿⣿⢿⣯⠙⠻⠿⢿⣿⣿⡿⠁⠉⠋⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠙⢿⣿⠚⠁⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⢠⢪⡝⡆⠀⠀⠀⠀⠀⠀⡠⣒⢶⣖⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⣖⢦
⠀⠀⠀⠀⠀⢠⢎⣾⣷⡱⣄⠀⠀⠀⠀⣸⢼⣿⡵⣿⣷⢱⠀⠀⠀⠀⠀⠀⠀⢀⡮⣽⣫⣿⢿⡽
⠀⠀⠀⠀⢀⣟⠟⠛⠛⠻⣽⡀⠀⠀⠀⠸⣼⠽⣻⣷⣟⢾⠀⠀⠀⠀⠀⠀⡴⣯⢗⢾⣿⡿⣮⢢
⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⠈⠑⠒⢮⡻⣮⣒⢤⠀⠀⠀⢸⢼⢹⠉⠓⡯⣷⡖⠋
⠀⠀⠀⠀⠸⡿⣄⠀⠘⣣⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣟⣗⣪⢯⠃⠀⠀⠘⢞⠯⣗⣺⡵⡣⠃
⠀⠀⠀⠀⠀⠈⠚⠛⠛⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⠀⠀⠉⠓⠒⠉
⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⢰⣫⣶⣶⣿⣦⡀⠀⠀⠀⠀⠀⡞⡷⢺⣭⣭⣭⣭⡖⢮⢣⠀⠀⠀⡰⡊⣷⣦⣴⣯⡷⣄
⠀⠀⠀⠘⣝⢿⣤⢛⣳⣕⢄⠀⠀⠀⠀⢱⣹⣿⢿⣿⣿⡿⣿⡏⡏⠀⠀⢸⣽⣿⠃⠀⠀⠙⣿⡾⡄
⠀⠀⠀⠀⠈⠓⢽⣮⡛⣭⣳⡵⡄⠀⠀⢸⣹⣿⠊⠰⠎⢱⣿⢧⠇⠀⠀⢸⣹⣿⡀⠀⠀⠀⢸⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢿⡮⡯⣯⠇⠀⠀⠀⠑⢽⣣⡠⢄⣞⠽⠋⠀⠀⠀⠀⠑⢽⣳⠄⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⣄
⠀⠀⠀⠀⠀⠀⠀⡧⣿⣿⣮⢢⠀⠀⠀⠀⠀⡠⡖⠺⠉⣫⢣⠀⠀⠀⠀⠀⢀⡔⣭⠾⠖⠿⢿⠂
⠀⠀⠀⢰⢒⣒⣒⣃⣿⣿⣿⡏⡇⠀⠀⠀⢀⡇⣶⣾⣿⣿⣏⣇⠀⠀⠀⠀⡼⣻⣫⣾⣏⢳⣽⡄
⠀⠀⠀⠈⢇⢿⣿⣿⣿⣿⡿⣳⠃⠀⠀⢠⢯⠾⣿⣿⣿⣏⡠⣸⠄⠀⠀⠀⠙⢢⢻⣿⣿⣿⡿⡁
⠀⠀⠀⠀⠈⡷⡭⡟⢻⢯⢾⠁⠀⠀⠀⠘⣞⢄⢈⡿⣟⠽⠊⠁⠀⠀⠀⠀⠀⠀⠣⡻⣿⣿⣷⣳
⠀⠀⠀⠀⠀⠙⠭⠵⠪⠭⠞⠀⠀⠀⠀⠀⠈⠙⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⠿⠵⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣤⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠘⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠙⠂⡁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⠀⠘⡀⠀⠀⠀⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡰⠀⡃⠑⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢄⡄⠈⠎⠡⠄⠀⠀⠀⠀⠀⠀⠀⡉⠓⠂⣤⠀⠀⡤⠤⢤⠀⠀⢀⡀⠀⠀⠀⢀⠌⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠨⡀⠄⢠⠺⠒⣄⠀⠀⡰⠁⠀⠀⠀⡁⣈⡿⣄⡁⠱⢄⣤⣏⢀⣹⠇⠀⠲⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡃⢰⠴⠈⠀⠑⠊⠀⠀⠀⠀⠠⡄⠤⢡⠀⠉⢀⡄⣘⡁⢀⡉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⢠⠀⣠⢶⠀⠀⠀⠀⠀⠀⠀⠂⠳⠐⠺⣀⢢⡏⠀⠈⢻⠀⢸⡄⠓⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠁⣰⠀⢀⢇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠙⣹⢃⠙⠦⠴⢿⡇⢀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⠃⡐⣄⠠⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⣟⣀⣀⡄⢀⡼⠄⣀⠛⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⡀⢆⠃⠀⠀⠀⠀⠀⠠⢰⠆⠠⠹⠈⠁⣿⠀⠉⣦⠀⠼⠀⠰⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣿⣤⡀⠀⠀⠀⢄⠀⠀⠀⠆⠐⠹⡊⠱⡈⠓⣹⣵⡔⠢⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠐⡐⠆⠁⠉⠈⠀⢰⣿⣿⣿⣀⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡟⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡻⣿⣿⣾⣀⠀⢄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠁⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣾⣿⣿⣿⡆⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠏⠁⠀⠀⠁⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⠠⠀⠀⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⢜⢿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡀⠀⠈⠀⠉⠛⠿⠿⠿⠿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠟⣿⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣆⠀⠀⠀⠀⠀⢠⣤⣌⠁⠀⠠⠍⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠒⠂⠀⠀⠻⣿⡿⠀⠰⠄⠰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣤⣔⠚⠋⠉⠀⠙⠲⢶⣤⡀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣆⠙⢿⣷⡶⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣼⠀⠀⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠹⣿⣿⣿⣿⡇⠀⠀⠀⢀⣿⣶⣾⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⠀⠀⠀⢸⣿⡿⠋⠀⢻⡀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⡿⠋⢰⠀⠀⠈⡇
⠀⠀⠀⣰⠟⠛⠷⠞⠛⠉⠉⠉⠙⠳⠀⠀⠀⠀⠀⢸⡏⠉⠙⣿⠀⠀⠀⡇⠀⠀⢃
⣀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⢻⣧⠀⠀⢠⠀⠀⠀⠢⡀⠀⠀⠀⠀⠠⢄
⠋⠁⠀⠀⠀⠀⠀⠒⠦⣤⣀⣀⣀⠀⠀⠀⣀⣠⡾⠃⠀⠀⠀⠀⠻⣦⠀⠀⠉⠉⠳⣄⠈⠐⠂⠤⠶⠒⠛

⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠒⠛⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠯
⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡔⢀⣀⣰⠃
⠀⣠⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢀
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠠
⠀⠀⠀⠀⠀⡄⠀⡀⠚⠛⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠄⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠐⠀⠄⠀⠀⠀⠀⠀⢀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠉⠀⠀⠓⠀⠀⢠
⠄⡀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣀⠀⠀⠀⠀⠀⠀⠠⡌⠂⡀⢄⠈⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢲⡀⠀⠀⡤⠀⠠⡄⠀⠀⠀⠀⢠⠀⣤⠿⠤⡀⢠⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐
⡀⡀⠀⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠁⠀⠀⠀⠀⠀⠀⠠
⠇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇

⠤⠤⢀⠀⢀⣀⠀⠀⠠⠦⠄⡀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⠀⠀⠀⠀⠠⠒⠒⠢⠀⠀⢀⠓⠤⠒⠲⢤⣀⠀⠀⢀⡀
⠀⢀⠀⠀⠉⠈⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠉⠀⠀⠤⢄⠀⠈⠁⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡀⠀⡀⠀⠀⠀⠀⠀⠈⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⡀⠀⠀⠀⠈⠐⠀⠈⢀⡀
⠀⠀⠡⠄⠀⠀⠀⠀⠠⠇⠀⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠑⠀⠀⠀⠀⠐⠒⠀⠒⠃⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠑⢀⣠⠶⣶⣿⢿⣿⣿⡟⢶⣶⠶⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⢀⣠⣶⣿⣿⣦⣭⣿⣶⠙⣿⣿⣿⣽⣧⣶⠾⡿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⣄⠀⠁⠀⠙⢀⣹⣿⣿⣿⣿⣿⣿⣿⣿⡷⢶⣾⡤⡆⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⢋⣠⡴⠋⠁⠀⠈⢃⣰⣦⣉⠻⣧⣌⣹⣿⣿⣿⣿⣷⢶⡽⢿⣁⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠈⢠⡴⠟⣁⡀⠀⠀⠀⢴⣾⣿⣿⡟⢷⣾⣿⣿⢿⠛⠻⣿⡵⠛⣴⠟⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠠⠊⢀⡤⠚⠁⡀⠞⠁⠀⠀⠀⠀⠀⠈⢻⣏⣃⠘⣿⠊⢛⠿⢳⣧⠁⣘⣛⣭⣶⣬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⡠⠂⢉⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣷⣤⣼⣧⣾⣟⠛⣻⠋⠉⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠊⠷⠾⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠙⠟⣿⡄⣿⠛⠤⠷⢄⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡤⠴⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠈⢿⠟⠀⢀⡘⠃⣀⠀⡰⢆⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠐⣂⣀⢬⣤⣤⡄⣤⣤⣶⡊⣿⣿⡄⠀⠀⠘⠇⠀⣄⠀⠀⠀⠀⠀⠀⠀⠁⠀⣩⣤⣄⡀⣰⣿⣿⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⣡⣿⣿⣿⣸⣿⣿⡿⣿⣿⣿⠇⣿⣿⣿⠀⠀⠀⠐⣧⠘⢷⣄⠀⠓⠄⠀⠀⣀⡼⠋⡷⣉⣿⣿⣿⡏⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣤⣿⢁⣿⣼⡏⢻⣿⠃⣿⣿⡟⠀⣿⣿⣿⠀⠀⠀⡀⠘⣷⣎⠻⣦⡴⠘⠛⠀⠈⠁⣠⣾⡿⣿⣿⡟⠰
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣏⣿⠿⠾⣿⣿⣿⣬⣯⡀⣿⠿⠦⡀⣿⡿⠛⠀⠀⡀⢿⣿⢿⣿⡅⠈⠁⢰⡟⢒⣴⣿⣿⡿⢡⣿⣿⢣⠇⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣇⣀⠶⣿⣿⣿⢿⣟⠁⣙⠣⠄⢻⣿⣿⡄⠀⡀⠙⠄⠃⠀⠙⠃⠀⢀⣾⣿⣿⣿⣿⣿⠁⢘⣿⣷⡆
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⣾⣭⢠⣿⣿⣧⣼⠛⢷⣿⡖⢀⣾⣸⣿⣷⣾⠛⠂⠀⠀⢀⣀⣤⡾⣿⣿⣿⣿⣿⣿⣧⠀⣿⠟⢹⡷
⠀⠀⠀⠀⠀⠀⣿⣿⣧⣿⣿⣧⣿⣿⠿⣿⡿⢷⡶⠿⠻⣻⡛⢿⣿⣾⣇⠀⢀⣤⣶⣿⣿⣿⣿⠈⠻⢿⣿⣿⣿⣿⡾⢃⣶⡞⠋
⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣯⡿⠾⣿⢀⠀⢑⣆⡀⢒⣖⣳⣦⣤⣙⣯⣾⣿⣿⡿⣿⣿⣿⣿⣟⠀⠀⠐⠞⠿⡿⢋⡤⡰⠟⠃
⠀⠀⠀⠀⠀⠸⣯⣿⣿⣿⣉⣼⣶⢧⣤⣴⣿⣿⣿⣨⣽⣿⣿⣿⣿⣿⣿⣿⣿⡝⠛⠈⠛⠿⣧⠀⠀⠀⠀⠀⠐⠟⢣
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⠾⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣦⣭⣙⣻⣷⣤⣴⣶⣦⡿⢀⣤⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⠛⣻⢇⠈⣿⣿⣿⣿⡟⢧⡀⠈⠀⢿⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠘⠶⣄⠈⠈⠻⡿⣿⡇⠀⠀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢈⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⣶⣶⣶⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠙⠛⠛⠲⠶⠾⠵⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠉⠀⠀⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⡠⢤⡤⠚⠉⠀⠈⠉⠐⠢⣀⠀⠀⣀⠀⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⡠⠞⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠁⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⣀⠴⠊⢀⠌⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⢻⠢⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠉⠀⠀⣠⡊⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠙⢄⠀⠀⡄⢡⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣱⣿⡷⠈⢢⠀⢠⠃⠀⣀⡀⡀⠀⠘⡆⠀⡇⠀⢢⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠹⢿⣿⠆⠈⢀⠃⠀⣼⣿⡷⠈⢻⠀⢰⠀⡇⠀⠀⠑⠢⢀⡀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⣤⠶⠦⠤⢼⣀⠀⠙⢿⣿⠧⠊⠀⢸⠀⠇⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢿⣶⣦⣤⣤⣼⡇⠀⠀⠀⠀⠀⢠⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠙⡿⠿⠛⠉⠈⠀⠐⠒⠒⠊⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⢀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠈⠉⠉⠙⠉⠉⠁⠀⠀⣀⠤⢺⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠶⡤⠀⠤⠀⠐⠂⠂⣉⠬⠖⢊⠔⠊⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⢣⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠷⠤⠠⠤⠒⠊⠁⢀⠤⠊⠁⠀⠀⠀⠀⠀⠉⠓⠤⣀⠀⠀⠀⠀⠀⢃⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼⣀⠀⡀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠘⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢰⡹⠀⠀⠀⢱⡀⠀⠀⠀⠀⣀⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⡜⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣤⠾⢳⠀⠀⠀⠀⢇⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠀⠀⠀⠀⢸⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠀⠐⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢠⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢉⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⣎⣀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣎⣀⣠⢃⠄⠀⠀⠀⢀⡜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠤⢇⡠⠔⠃⠙⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠃


⣀⣀⣀⠀⡀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻
⢀⣾⠿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣴⣶⣤⣼⣿⡏⠉⢻⣿⣿⣿⣿⣿⣿⣷⣦⣴⣦⡀
⣼⠏⣶⠀⠀⠀⢀⣀⣀⣠⣴⡷⣿⣿⡿⠟⠙⢿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠏⢸⣿⡄⠀⠀⠉⠙⠛⠟⠻⣦⣼⣿⣿⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠟⣷⣀⡀⢀
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠙⠛⣿⣿⣯⣌⣭⣬⣅⣤⣤⣿⣿⡿⠛⠋⠁
⣠⡿⠝⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣽⠿⣷⡆⠀⠀⠀⠀⠀⣿⣸⣿⡟⠙⣿⡏⠙⢿⣿⣿⣧⣠⠶⣿⢤⣾⣿⣿⣿⣿⡿⠛⠿⣟⢿⣿⡇⠀⠀⠀⠀⢠⡶
⢇⠀⢀⠀⠀⠀⠀⠀⠀⢿⠛⠿⣿⣶⣴⣿⣶⣟⣿⣫⡿⠷⢶⣿⣞⣃⣾⣿⣟⣿⣶⣰⣶⣯⡗⣿⠃⠀⠀⠀⠀⠻⣷⡄
⣤⠀⣸⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣟⣿⡗⠿⠉⠛⣿⣿⣧⣐⣮⣿⣿⠿⢛⣡⣶⡽⠟⢫⣿⡛⣻⠀⠀⠀⠀⠀⠀⠙⣿⣄
⠇⢀⣸⠆⠀⠀⠀⠀⠀⠀⢺⣿⡽⡛⣹⢌⠳⣶⣶⣾⣿⣿⣿⡿⢛⣿⣷⣿⣿⣉⣜⣿⣟⣿⣣⢹⣆⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣼⣿⠀⠀⠀⠀⠀⢀⣤⣨⣷⣰⡷⣿⣿⠟⢛⣉⡉⢾⣿⣿⣿⣷⠎⠉⠙⠿⠿⣶⣿⣶⣿⣶⣸⣏⡀⠀⠀⠀⠀⠀⠀⠙
⢠⣿⣿⢠⣾⡇⠀⠀⠀⢽⣿⣯⣤⣤⡿⡇⠀⠩⢏⣀⣤⣼⣿⣿⣿⣿⣿⡆⢦⣤⠽⡿⠿⣟⣿⣿⣿⠏
⣼⣿⣇⣿⣿⠃⠀⠀⠀⠀⠀⠙⠳⣽⣿⣟⢿⣧⣼⣿⣿⣿⣿⡟⠛⠋⠉⢀⠐⡂⣤⣤⣶⢾⠟⡁⠀⠀⠀⠀⠀⠀⠀⢦
⣿⣿⣿⣿⡿⠀⠀⠀⠀⢰⣷⠀⠀⠀⣿⣿⡟⣿⣿⣿⣿⣿⣯⣤⣤⣴⡿⠻⣿⢟⣽⣿⠗⠀⠿⡿⠀⠀⠀⠀⠀⠀⢀⣸⡇
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠈⠁⠰⠂⠀⣿⠈⠓⣿⣟⣿⡏⣀⢿⣿⠩⣿⣴⣶⣋⣾⣿⡇⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⣿⣿⣿⡿⠀⠀⠀⠀⠀⠐⠛⠀⠀⠆⣿⠀⣴⣿⣿⣿⣿⣿⣸⠀⣠⣿⣿⣿⡟⠉⢸⣇⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⠚⢁⣬⡘⠛⣿⣿⣿⣿⣿⣿⠀⣿⣿⡿⠿⠏⠀⢸⡟⢻⡿⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⡇⢀⣤⡀⠀⠀⠀⠀⠂⠀⠸⣿⠃⠀⢻⣿⣿⠀⢻⣿⣤⣿⠟⠁⠀⠀⠀⢸⡇⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽
⣿⣿⣿⣷⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⢸⣿⣿⠀⢸⡿⡿⢉⣷⣶⡎⠁⠀⠸⠁⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠸⠿⠿⠀⠸⡇⠀⠀⠋⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⣊⡵⢋⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⡖⢛⡋⠉⠙⠛⣩⠴⢚⣡⡴⠋⠁⣀⣤⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣩⠖⣫⠵⢋⠁⣀⠀⠀⠀⠘⠋⠥⠶⠋⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⢫⠞⢁⠞⠁⡴⢋⡞⠁⠀⠀⠀⠀⠀⠀⠀⡀⢦⡀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡽⢱⠃⠠⠏⠀⠘⠁⣼⠁⣰⠀⠀⠀⠀⠀⠀⠀⠙⠢⣍⣀⡘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢰⠃⠀⠀⠀⠀⠀⠀⠀⠹⡄⢿⠀⣿⢠⡀⠀⠀⠀⠀⠀⠀⠈⠁⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢷⡌⠷⠙⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠒⣮⣿
⠀⠀⠀⠀⢀⣠⣤⠤⡤⣴⠋⡹⠸⡇⠀⠠⣶⣲⡄⠀⠀⠀⠀⠀⠀⢠⣤⢤⠀⠀⠀⠀⠀⠀⣧⠙⣱⠏
⠀⠀⠀⠀⡞⠠⣈⣦⣤⣼⡴⠗⢺⡇⠀⠀⠈⠁⠀⠀⠀⠄⢠⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢠⡟⠋⠁
⠀⠀⠀⠀⣧⡀⠈⠀⠀⢀⣠⠔⢻⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢀⡔⠉⠛⠀⠀⠘⣏⣁⣄⣀⣹⣦⣀⡀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢀⣠⡴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⢀⡞⢉⣧⣤⡀
⠀⢀⡟⠀⠀⠀⠀⠀⠀⣿⣍⠀⢩⡁⠀⠀⠈⠉⠛⠛⠉⠉⠉⠙⠛⠛⠛⠛⠛⠉⠉⣹⠶⠶⠶⣤⡀⠀⠀⠀⠀⠀⠙⡎⠹⠉⠀⠉⣡⣜⣁
⠀⢸⠁⠀⠀⠀⠀⠀⡴⠃⠈⢧⠀⣧⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⢀⣀⣹⣦⠀⠀⠀⢀⣰⡇⠀⢰⡆⠀⠀⢠⣼
⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⢹⣦⠖⠋⠉⠉⠀⠀⠈⣀⣠⠶⠦⠞
⠀⠀⠈⠳⣤⣄⣀⣀⣀⣀⣠⠏⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠈⠉⠉⠉⡏⠋⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⠶⢤⣀⡀⠀⢀⣀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠿⠷⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣈⢷
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣅⣀⣀⣀⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⠀⠉⠛⢧⣄
⠀⠀⠀⠀⣠⠶⠟⠋⠉⠁⠀⠀⠈⠉⠛⠦⣄⠀⠀⠀⢤⠀⠀⠀⠀⣠⠞⠁⠀⢀⣀⣤⣤⣀⠀⠙⢷⡀
⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢸⡇⠀⠀⠀⣀⣀⣀⡴⠋⠁⠀⠀⠉⠻⣦⠈⢿⡀
⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠚⠀⠀⣠⠞⠉⠀⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣇
⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢶⣤⠒⠚⠻⣤⣶⠺⠛⣆⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢸⡇⠀⣿
⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡙⣧⠀⢰⡟⠀⠀⠀⠈⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⡾⠁⢠⡏
⠀⠈⠛⠶⢶⡄⠀⠀⠀⣠⣞⣷⡄⢷⣼⡆⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⡞⠁⣠⡟
⠀⠀⠀⠀⠀⠙⢦⣀⣰⣓⠟⠁⠀⠀⠘⢧⡀⠘⢷⣄⠀⠀⠀⠀⣿⠲⠶⢶⡾⠿⠤⠴⠞⠋⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⠀⠀⠀⠀⣠⠀⠙⢳⡀⠉⠳⢦⣤⣴⠏⠀⠀⠀⠙⠷⣶⣶⡶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⡉⠀⠀⢠⡇

⠀⢠⠄⠀⣴⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⢀
⠀⢸⣿⢃⣿⣿⣇⣀⢠⣌⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡞⠁
⠀⢸⡏⢸⣿⣿⣟⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠘⠻⣀⠀⣈⣿⣿⣯⢹⣶⣶⡆⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⡄⠀⠰⠆⠐⠀⠀⠀⡀⠁
⠀⢺⣷⡌⣿⣿⡘⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⡟⣀⠻⢿⡇⠀⠀⠀⠀⣠⡄⠀⠈⣠⣀⠀⢀⡀⠀⠐⣰⣷⠾⡗⠀⡹⢂
⠀⠸⢏⡳⠉⢁⣿⡁⠒⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠾⠛⠁⢀⡟⠻⢿⠛⠻⠦⠉⠀⠀⠀⡀⠀⠀⠀⠈⣿⣏⣀⣩⣵⡶⢦⠉⠉⠐⢤⣤⣾⣧⣀
⠀⠀⣼⠃⠀⢸⣿⡽⢂⣤⣶⣶⡶⢶⠟⢻⠋⠁⣤⠀⢰⡀⣾⣷⠀⠘⡆⠀⠀⠀⣦⣤⠀⣉⡀⠀⠀⢠⡺⠀⠉⠀⠀⢸⡁⡀⢠⣶⡁⠈⠻⣿⡟⢀⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⡅⣾⣾⣇⣸⡇⣿⠀⢰⣿⡆⠀⢃⢹⣿⡀⠀⢹⡄⠀⠈⡿⢁⣾⡟⢃⠀⡀⣰⡏⠀⠀⠀⠀⠰⠆⢿⡌⢿⣷⣦⣄⣀⡅⠀⠁
⠀⢀⣴⡄⢠⣿⣿⣍⡷⣾⣯⡿⣼⠛⣽⣿⡟⢠⡟⡇⠀⢸⠀⣿⠁⠀⣸⡇⠀⠀⠀⣾⡏⢠⠟⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠙⠻⠿⣿⠁
⠀⢸⣿⣿⣿⣽⣿⡿⣿⣿⣿⠳⢂⣱⡿⣿⠁⠟⠀⡇⠀⠸⡇⣿⡆⠘⣿⡿⠃⣇⠀⢙⠀⡉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠛⠁⢀
⠀⢸⣿⣻⣿⣾⣿⡷⣿⡥⣤⣄⡈⠉⠀⠀⠀⣀⣤⣤⠀⠀⡗⣿⣶⣰⣿⣧⠘⠟⢀⡤⠶⠃⠀⣶⠀⢶⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⢸⣿⣿⣿⡿⣿⢷⡉⠀⠸⠟⠁⠀⠀⠀⠀⠙⠿⠃⠀⢀⣇⣾⣿⣿⣿⠇⠀⠃⢈⠁⠘⣠⠆⣼⠀⠘⠇⠛⣿⡄⠀⠀⠀⠀⢀⣤⣤⡄⣀⢠⣿⡟⢸
⠀⢸⣿⣿⣿⣼⣿⣾⣿⣄⣠⡄⣤⣶⣁⣰⣦⣄⣠⡆⠀⣸⢿⣯⣿⣿⡏⠀⠈⢇⠀⠀⠀⣥⢶⣾⣦⣤⡀⠀⣼⣿⣿⣿⣷⣾⣶⣿⣿⣿⠀⠈⠟⠀⢫
⠀⢸⡏⢹⣿⣿⣿⣟⣿⣿⣿⣿⣏⣿⣿⣿⡿⣽⣿⡆⢰⣿⣿⣿⣿⣿⠀⢠⠀⣆⠀⠀⠸⠁⠀⣽⣷⡌⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢀⠀⠀⡴
⠀⠸⣧⠀⢿⣿⣿⣿⣯⡿⠋⢿⣿⣿⣻⣏⣿⠿⣿⢧⣿⣿⣿⣿⣿⠇⢠⣧⣼⡟⠀⠘⠂⠀⠀⢿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠁⠀⣠
⠀⢰⠋⠁⠈⣿⣿⠋⠀⢀⡤⠀⠈⠛⠻⠻⠏⠀⠠⣶⠗⡌⢻⣿⡏⣠⠟⠛⢿⣷⠀⠀⠀⠀⠀⠻⠿⠟⢿⣿⣿⣿⠿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠜⣠⠆
⠀⠈⠀⠀⠀⠛⠛⠁⠀⠘⢷⣄⡀⠈⣞⠋⢀⣀⣴⠟⠀⠀⠛⠿⠃⠀⠀⠀⠀⠉⠀⠀⠀⠳⠀⠀⠘⠣⡴⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⢠⡄⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⢀⠀⠀⠈⠉⠀⡀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣤⡂⠀⠀⠀⣤⡀⠀⣠⣶⡶⠀⠀⢠⣴⠋⠀⠀⠀⠊
⠀⠰⠀⠀⠘⢃⠅⠀⠐⠀⠀⠀⢀⡠⠄⠠⠀⠀⠀⠈⡰⠂⠀⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣇⣶⢢⡾⠿⣷⠄⠈⠁⠀⠀⠀⠉⠀⢀⡀⠀⠀⠐
⠀⢠⠀⠀⠀⢦⣰⣾⣷⣶⣶⣷⣀⠀⠀⠂⢠⣿⣶⣷⣾⣿⣶⣰⡆⠀⠀⢀⡘⡇⠀⠀⠀⢸⣿⣿⣿⣇⠘⠓⠀⠉⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⣠⡀
⠀⢠⣴⣤⣀⠈⣿⣿⣿⡟⢉⠛⠙⠧⠶⠞⠈⢻⣿⣿⣶⣾⣿⣿⡇⣀⣴⡇⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠛⢷⣤⣀
⠀⢸⣿⠛⠻⠟⠛⠿⠛⠳⠾⠀⡀⠀⠀⠀⠀⠘⠟⠛⠛⠛⠛⠛⠛⠛⠿⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⢛⣥⣦⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⡹⠏⢀⡀
⠀⢸⣿⠀⠀⠀⠈⠀⠙⠆⠇⠀⠀⠀⠀⠀⠀⡀⠂⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⣠⠟⠁⠁⠠⠀⠀⠀⠀⠀⠀⠀⡀⠈
⠀⢸⣿⠀⠀⠀⠀⠰⠄⠀⠀⠀⢠⠀⠀⠀⢀⠃⠀⠀⢀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠛⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠂
⠀⢸⣿⠀⠀⠀⠀⠂⠘⣆⠀⠀⠀⢀⡰⢄⠄⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⡀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⣿⠀⠀⠀⠀⠀⠀⠈⢸⣷⣇⠰⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾
⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⢠⡴⣶⡇⠀⣈⣤⠉⡀⠀⠀⠀⠀⠀⢀⣼⡇⠀⣠⣾⡼⣄⡀⠀⡄⠀⠀⠆⣽⡁⠂⠀⡀⢀⠀⠀⠀⠀⡤⠉⢉⡰
⠀⠸⣿⠶⣄⡈⠰⣌⠄⠙⠻⣆⠰⠺⢦⡾⠓⠀⣿⠋⠁⠠⡀⠀⣠⠈⢤⣿⣿⢧⣼⣿⣿⣿⣭⣱⣤⠀⠀⠀⠀⠈⠀⢤⣾⣷⣿⢠⣾⢀⢘⠃⠰
⠀⢘⣧⣎⡀⠑⢒⣷⣶⢦⢰⡉⠂⠀⠀⠀⠀⠀⣨⠂⢰⡖⢱⠚⠁⣨⣾⠟⡅⢺⣿⣿⣿⣿⣿⡏⠀⠘⠻⠾⠶⠀⠈⢸⣇⠈⣿⢿⣿⢿⣽⣧
⠀⢸⠟⠛⠷⠴⢋⠌⠛⠃⣿⣦⣂⣤⢤⣤⢤⣺⡅⠀⠙⠁⢌⡛⠘⠋⢁⠀⠀⢺⣿⣿⣿⣿⡯⠄⠀⠀⠀⠂⠀⠀⠀⠾⣿⡄⣱⠎⣁⣴⠟⣋⡄
⠀⢰⣾⣦⠖⣰⠋⢠⣤⣾⠋⠋⠉⠉⠉⠀⠉⠈⠁⢸⣷⣦⣀⢻⡜⢳⠣⣀⣤⣿⣿⣿⠿⢋⣤⣄⠆⠀⠈⠀⠀⠀⠀⠀⢹⣿⣷⣾⠟⣀⠸⠋⠀⠀⡀
⠀⢸⣿⠘⢿⣭⡽⠻⠟⢿⡸⣧⡀⠀⠀⠀⠀⢠⡄⢸⡟⠛⢛⢧⡼⠏⠐⠀⠘⠿⣯⣿⠑⠋⠉⣉⠀⠫⣔⢀⠀⠀⠀⠀⠀⠉⠙⢿⣿⠟⠀⠀⣰⡟⡀
⠀⢸⣿⠀⢦⠔⡈⢷⣾⡿⢃⣿⣿⠶⠤⠀⢸⣿⣀⣾⣿⡶⠃⠄⠀⢀⡀⠀⠤⣀⠿⠋⠀⠀⠀⠀⣀⣴⣿⣾⢳⡌⠃⠀⠀⠀⢀⣼⡃⢰⣶⠻⣿⣿⣷
⠀⢸⣿⠀⢔⣨⡽⢲⣿⣷⠻⣬⢿⣷⣶⣾⣿⡏⠾⢸⣿⣿⠓⢶⣤⣄⣀⠬⢁⡜⠂⠀⠀⠰⣐⡗⠛⢡⣿⠟⠀⠻⠂⠀⣀⡝⠛⠓⠂⠚⠛⠃⡼⠟⠉
⠀⠸⣿⡟⠛⡟⣡⣾⣿⣿⣷⣀⡌⢿⡿⣿⡟⣤⣰⡿⣿⣿⣷⣜⠋⠉⣿⣷⣬⣿⠞⠛⠛⣛⣍⠁⠀⠈⠻⠠⠲⣶⡄⠙⠉⠀⠀⠀⣀⣤⣴⡞⠀⣠⡖
⠀⢠⣿⣁⢀⣿⣿⣿⣏⡁⡌⠻⢷⣮⣷⣞⠶⠿⠋⡤⠦⢿⣿⣿⣧⣰⣾⣿⣿⡇⠀⠀⠘⢿⡥⠀⠀⠀⡠⢁⠙⠈⡟⠀⠀⠀⠀⣰⣿⣿⣿⣡⡾⢋⣤
⠀⢸⣿⣿⣿⣿⣿⣿⡅⣶⡀⠀⠀⠀⠐⠀⠀⠀⠀⢰⡆⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⡅⠀⡰⠀⠁⠀⣀⠶⠿⠿⠛⠉⣿⢣⣶⣿⣿
⠀⢸⣿⣾⣿⣿⣿⠏⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⡄⢻⡿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠛⠛⠋⠀⠀⢐⣠⣾⣷⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣦⣽⣷⣦⣄⣀⡀⠀⢀⣀⣠⣴⣾⣿⣶⣿⠻⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⣤⠶⣦⡑⠶⣦⠚⠓⠆⠀⢀⣴⠟⠋⠁⠀⠈⣿⣿⣿
⠀⢸⣿⣿⣿⣿⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢰⣿⣿⣿⠿⠿⢿⣿⣿⣿⣶⣌⣀⡀⢼⡇⠠⣿⠀⠄⠀⢰⣿⠏⠀⠀⠀⢀⣶⣾⣿⡿
⠀⢸⣿⣿⡿⣿⡄⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣾⡏⣿⡧⢠⠓⣺⣿⣿⣿⣿⢇⡌⠹⠄⠳⠀⠷⠀⠂⢠⣿⡏⠠⣀⠀⠀⢾⡿⠿⠿
⠀⠸⣿⣿⠇⠿⠇⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠋⠀⠀⠛⠃⠛⠁⠉⠉⠉⠉⠉⠉⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢠⠔⠁⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠐⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠉⠂⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠂⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠂⠀⠂⠁⠈⢀⠔⠊⠁⠀⠉⠙⠢⠈⢦⠁⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡔⠁⠀⣀⠀⢣⠀⡆⠘⠐⠃⡠⠑⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠐⠒⠀⠀⠘⠀⠁⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⡈⠐⠂⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⡀⠀⠀⡀⠈⠐⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⢰⠀⢈⠈⡀⠀⠀⣳⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣄⠀⠀⠀⢀⠠⠚⡄⢠⣤⣀⣀⠄⢠⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡀⠈⠈⠀⠀⠐⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠀⠐⠀⠀⠂⢢⠀⢨⠑⡀⠀⠐⢄⡀⠈⠢⡀⠈⠂⠐⠲⢤⢄
⠀⠀⠀⠀⢀⣤⡀⠒⠀⢄⣀⡀⠄⠀⠈⠀⠀⣀⠈⠀⠀⣨⠀⠀⠠⠑⠄⠀⡰⢀⠀⠈⠆⠀⡏⠅
⠀⠀⢀⡾⡭⠋⢀⠔⠁⠀⢀⠔⠀⠄⢀⢀⡀⠀⠉⠉⠁⠘⣦⠀⠀⢠⠂⠀⠄⠀⢀⠀⡀⠀⢃⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠃⠀⠀⠂⠀⠀⠦⠐⠁⠀⢁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠈⢑⡀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⡀
⠀⠀⠀⠀⢰⣧⠆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁
⠀⠀⠀⠀⢻⢸⣏⠀⠙⠁⠀⠀⠀⠀⠹⠗⠀⠀⣀⡀⠐⠀⠀⠀⠐
⠀⠀⠀⠀⠈⠉⠀⠠⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣟⠃⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⠀⠀⠀⣀⣾⣏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⢩⢠⠋⣼⡆⠸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣷⣴⣾⣿⡻⢋⡀⠀⠤⢄⡀⠀⣆⡀⠠⡀⢠⣧⣰⣿⣨⡷⢙⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣿⠀⠄⠀⠀⣀⠠⠄⠀⠉⢨⣻⠗⣿⠛⢙⣿⣿⢥⣤⣶⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣭⣈⠓⠀⠐⠉⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠉⠻⣫⣶⡶⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡷⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠅⠄