Left hand Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⣤⣦⡄⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⣠⣶⡄
⠀⠀⠀⠀⣠⣤⠀⢻⣿⣿⡄⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⣰⣿⣿⠇⠀⣀⣀
⠀⠀⠀⠀⣿⣿⣇⠈⣿⣿⣷⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⢀⣿⣿⣿⠀⢰⣿⣿⠇
⠀⠀⠀⠀⢹⣿⣿⡄⢻⣿⣿⡇⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⣸⣿⣿⡇⢠⣿⣿⡟
⠀⢰⣿⣆⠀⢿⣿⣿⡜⣿⣿⣿⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣿⣿⢁⣾⣿⡿⠀⠀⢀⣤⡄
⠀⠸⣿⣿⣆⠘⣿⣿⣧⢹⣿⣿⡆⢨⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⡇⣼⣿⣿⠃⠀⣰⣿⣿⠇
⠀⠀⠹⣿⣿⣦⠹⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠀⠀⠀⠠⣿⣿⣷⣄⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⣿⣿⠇⢀⣾⣿⣿⠋
⠀⠀⠀⠹⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⣰⣿⣿⣿⡟⠁⠀⠀⠀⠀⠻⣿⣿⣿⣧⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⡿⠁
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⣠⠖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠓⢦⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⠀⠀⠀⠀⢀⡇⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢷⡀⠀⠀⠀⠀⠀⢸⠀⣀⡀⢸⠀⠀⠀⠀⢸⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⡿⢀⡀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠤⠂⢹⡀⠀⠀⠀⠀⣾⠀⠀⠀⠘⠀⠀⠀⢰⠇⠀⠁⢸⠁⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢳⡀⠀⠀⢀⡇⢀⣀⠀⢸⠀⠀⢀⡞⠀⠀⠀⡸⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⣇⠀⠀⠸⡇⠈⠉⠀⢸⠀⠀⡸⠀⠤⢶⢀⡇⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⠋⠀⠈⣆⠀⠀⡇⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⢀⡼⠁⠄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠸⡄⠀⡇⠀⠀⠀⢸⡀⢰⠁⠀⠀⠀⡌⠀⠀⣠⠎⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⡀⠹⡾⠇⠐⠯⠁⠘⠷⣞⡀⠤⡀⢠⡇⠀⣸⠃⠀⠂⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠘⠉⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠈⠁⠀⠸⣄⣰⠋⠀⠀⢰⠇
⢀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡢⢠⠏
⠻⣄⡀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠦⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏
⠀⠀⠙⢦⡀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⡧⠴⠚⠉⠑⠒⠤⣄⡀⠀⠈⠉⠒⠤⣤⣀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠙⠣⡄⠀⠀⠈⠳⣄⠀⠀⢸⠇⡀⠀⠀⠈⠓⣄⠀⠙⠢⣀⠀⠀⠀⠀⠀⠉⠢⡄⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⠳⢤⠾⠋⠀⠀⠀⠀⠀⠈⢇⠀⠀⠈⠑⠢⣀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠶⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠒⠯⠉⠉⢉⠀⠀⠀⢀⠄⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠴⠶⠖⠒⠒⠛⠉⠁⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⣠⠤⢤⠴⠒⢲⣦⣀⣀⣠⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣤⠶⠿⠯⠤⣀⣀⣀⡀⡀
⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⡏⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠓⠒⠀⠤⡀
⠀⠀⠀⠀⡟⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠢⣤⡀⠀⠀⣀⣀⣀⣀⠀⠀⢀⣀⠀⠀⣀⣀⣀⣀⠀⢀⡜
⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠄⠀⠀⢄⣀⠀⠀⣀⣀⣀⣴⡄⠀⠀⠈⠉⠉⠙⠛⠛⢯⡉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⣇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠙⠛⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢧⡀⠀⠀⠀⠉⠛⠛⠲⣄⣤⡴⠃
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠹⣤⣄⣄⣀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠉⠳⢦⣄⣀⠀⠀⠈⡿
⠀⠀⠀⠀⠀⠈⠳⠶⠒⠒⠒⠋⠀⠀⠉⠙⠛⠳⠶⣦⣤⣄⣀⣛⣦⣀⠀⠀⠀⣽⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠚⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠈⠉⠓⠒⠒⠲⠦⠤⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⠽⠷⠦⠄⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣀⣀⢀⣴⡖⠒⠦⡤⠤⣄
⠀⠀⠀⠀⡀⠤⠐⠒⠒⠚⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢹⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⢄⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⠀⠀⢀⣠⠴⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⡝⠛⠛⠋⠉⠉⠁⠀⠀⢰⣦⣀⡀⢀⠀⠀⣀⡤⠀⠀⠠⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⣀⣠⠾⠛⠛⠣⠉⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣠⠖⠻⠛⠉⠀⠀⠀⢀⡼⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣀⣠⠴⠞⠉⠀⢀⡞⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠏⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⣧⠀⠀⠀⣀⣴⣋⣀⣀⣤⣶⠶⠶⠛⠋⠉⠀⠀⠙⠒⠓⠲⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠛⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠚⠛⠒⠒⠒⠶⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡒⠂⠠⠀⠀⠐⠂⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⡜⠦⡀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⢷⣴⣷⣶⡿⣾⢷⡄⠀⠀⠀⠱⢄⣠⡈⢦⡠⣾⣸⣷⣀⣀⣰⣎⣿⡰⢀⣠⣤⣤⣠⡜
⠀⠀⠀⢠⣾⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣽⡿⢻⠘⡄⠁⠀⠉⠻⢧⣄⣄⣀⣀⠤⢻⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⢸⢁⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣄⢰⣴⣧⣀⣀⠀⡀⣠⡀⡌⢉⢁⡀
⠀⠀⠀⠸⣯⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠏⠚⠁⠀⠈⠉⠉⠉⠉⠉⠛⡿⠛⠁
⠀⠀⠀⠀⢹⣄⡀⠀⠐⢔⠲⠤⢤⣵⣶⢟⣧⣽⣶⣤⣠⣠⡀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡟⠉⠑⠂⠀⠪⢤⣤⠅⠀⢚⡿⡿⠀⠀⠀⠀⠉⠑⠓⢯⣦⣦⣶⠏
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠑⠠⡀⢻⣴⣆⡀⠀⠀⠀⠀⠀⠀⠈⡋⠁
⠀⠀⠀⠀⠀⢧⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣶⣶⣤⣤⠄⣰⠃
⠀⠀⠀⠀⠀⠘⢦⠡⡀⠐⡄⠀⠀⠀⠀⠀⡇⠀⠀⢠⠈⡌⠊⣉⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣾⣮⣄⣀⣖⣀⣔⣰⣐⣠⣮⠤⠬⠄⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡤⠤⠤⠴⠖⠋⠉⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⡀⠀⠀⠀⢀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠛⠁
⠀⠀⠀⠀⠀⣴⠋⠙⠋⠁⠀⡿⠷⠶⠶⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠛⠛⠛⠉⠉⠉⠙⠛⠛⠲⠶⠶⠤⠤⠤⠤⣤⣄⣀⣀
⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢲
⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠉⠙⢷⣤⣄⣤⣤⣴⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠜
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠃⠀⠀⠐⢦⣤⣤⣤⣤⣤⣶⣿⣄⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣼⠉⠉⠙⠳⢦⣄⣀⡀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠾⢧⣀⠀⠂⠀⠈⠉⠛⠓⢦⡤⠶⠋
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢳⣤⣤⣄⣀⡰⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠙⠳⠦⣤⣀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠓⠀⠒⠒⠓⠚⠉⠀⠀⠈⠉⠛⠲⠶⣶⣦⣤⣄⣈⣳⣄⡀⠀⠀⢀⣼⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠳⠞⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠈⠙⠲⠶⢤⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠤⠤⠤⠤⠴⠶⠒⠒⠛⠉⠉⠉⠙⠓⠚⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠦⠴⠿⣇⠀⠉⠛⠉⢳⡀
⠀⠀⠀⠀⢰⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠘⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⡤⣤⣤⣤⣤⣴⣤⣤⣄⣤⣴⠾⠋⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⠈⠀⠀⠀⠄⠀⢀⣼⣶⣤⣤⣤⣠⣤⣤⠶⠀⠀⠀⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣀⣀⣤⠶⠛⠉⠉⢱⡄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣤⠞⠋⠉⠉⠀⠀⠂⠀⣠⠿⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⢀⣠⡤⠶⠛⠁⠀⢠⡗⠀⠀⠀⠀⠀⠀⠰⣀⣀⣤⣤⣼⠃⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢻⡄⠀⠀⠀⣀⣴⣛⣀⣀⣤⣶⣶⠶⠛⠋⠉⠁⠀⠈⠛⠚⠛⠒⠂⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠛⠉⠉⠉⠉⠉⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣶⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠉⠉⠳⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠉⠉⠉⠛⠛⠒⠒⠶⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠭⠿⠶⣦⣤⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⡀⢀⡴⡖⠒⠦⢤⡤⢤
⠀⠀⠀⠀⢀⠤⠴⠒⠒⠒⠛⠛⠋⠉⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢹⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⠔⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⣻⠿⠿⠟⠒⠚⠛⠁⠀⠠⢰⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠿⢝⠓⠒⠶⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⢀⣤⡤⠶⠟⠉⡁⠀⠀⣴⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⠁⠂⠀⠀⠀⠀⢁⣤⠞⠋⠉⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⣀⡤⠖⠛⠉⠁⠀⢀⣴⠏⠀⠀⠀⠀⣀⣀⣨⠤⠶⠒⠲⣿⣁⣀⣀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⣠⡴⠿⠵⠶⠶⠾⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁





⣿⣿⡟⠻⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣠⣴⣾⡟⠀⠀⠀⠀⠀⠘⣿⠷⠶⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠉⠉⠈⠹⣿⣿⡿⠟⠋⠁⠸⣿⣿⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠈⠉⠛⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣌⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠌⠣
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡀⠀⠀⠀⣀⡄⠀⡆⢠
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⠖⠋⠁⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢸⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠁⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⢈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⣿⢀⠇
⠀⠀⠀⠀⠀⠀⠀⢰⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣿⠟
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣰⠇⠀⢸⣿⣿⡀⠀⠀⠀⠀⣀⡠⠤⠶⠶⠶⢤⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⣰⡏⠀⠀⢸⣿⣿⣧⠀⠀⢈⣉⣠⡤⠶⠀⠀⣀⣀⡉⠀⠀⠀⠀⠀⠀⠤⠤⠦⠤⢀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠻⣿⣿⣿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢸⡦⠊⠀⠀⠀⣀⣀⠤⠐⠚⢲⣿⣧⠤⠤⠒⠂⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠙⢿⣿⣦⡤⠄⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⠚⢉⣁⣀⡠⠴⠚⠉⣽⣿⣶⣶⠖⠒⣛⣉⡉⢀⡿⠀⣀⣀⡀⣸⠃
⠀⠀⠀⠀⠀⢿⣷⣄⠀⠀⠀⠀⠀⠻⣿⣿⣶⣶⣶⣤⣶⣶⣶⣿⣿⣿⣷⣦⣄⣀⣀⣠⣤⣴⣶⣿⣟⠻⣿⣭⣉⣉⣉⣥⣴⣿⣛⣋⠉⠁⣰⠃
⠀⠀⠀⠀⠀⠈⠻⣿⣧⣄⠀⠀⠀⠀⠹⣿⣿⣿⡿⠛⠛⠛⠛⠋⠉⠉⠉⠉⠛⠛⣉⣉⣥⣤⣤⣀⠈⠱⡄⠀⠉⠉⠙⠿⣿⣯⣭⣤⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡄⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣄⡾⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣷⣬⣁⡀⠀⠀⠀⠀⠀⠀⣀⣤⣬⣿⣿⣿⣿⣿⣿⣿⡿⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠶⠿⠟⠛⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣾⣿⣿⡆⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⣿⣿⣿⡃⠀⠀⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡄⠀⠀⣿⣿⣿⡇⠀⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⠀⠀⢻⣿⣿⣿⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣴⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⢸⣿⣿⣿⠀⣾⣿⣿⠇⠀⠀⠀⢀⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣤⣸⣿⣿⣿⣀⣿⣿⣿⠀⠀⠀⣰⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢀⣾⣿⣿⠟
⠀⠀⠀⠀⢼⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠓⣆⠀⠀⠀⠀⠀⢠⠞⠛⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠖⢹⡀⠀⠀⠀⠀⠄⠲⠔⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠐⠀⢸⡅⠀⠀⠀⣼⠀⠀⠀⣸⠀⠀⠀⠀⢀⠤⢤⡀
⠀⠀⠀⠀⢠⠞⣷⡄⠀⠀⠀⠀⣿⢀⡀⢸⡇⠀⠀⢠⠇⢀⡀⠀⡇⠀⠀⠀⡰⠫⣤⢼⠇
⠀⠀⠀⠀⠸⡎⠁⢻⠀⠀⠀⠀⡏⠀⠂⢸⡇⠀⠀⡞⠀⠃⠀⢸⠃⠀⢀⡜⠁⠒⠀⡾
⠀⠀⠀⠀⠀⣧⠀⡈⣇⠀⠀⠀⡇⠀⠀⠐⡇⠀⢰⠃⠀⠀⠀⡎⠀⠀⡾⠠⡀⠀⡼⠁
⠀⠀⠀⠀⠀⢸⡀⠀⠘⣇⠀⢸⠃⠀⠀⠀⠛⠤⠟⠀⠀⠀⢸⠃⣠⠎⠀⠑⠀⣰⠁
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⠓⠋⠀⡆⠀⠀⠀⠀⠰⠂⠀⠀⠘⠗⠁⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⢻⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠦⠖⠒⠋⠉⠉⢉⣉⣲⡄
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠿⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠋⠁
⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠋⠉
⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⢤⡀⠀⠀⠀⡸⠀⠀⠀⠀⠈⣆⠀⠀⠀⢀⠤⠒⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠙⢦⡀⠀⡇⠀⠀⠀⠀⠀⢸⡀⢀⠔⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠱⣼⠁⠀⠀⠀⠀⠀⢸⣧⠋⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⡞⠁⠀⠀⠀⠉⠑⠲⢌⣆⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢀⣼⠧⠒⠋⠉⠀⠀⠀⠉⡆
⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⣈⣧⠤⠤⠤⠼⣄⣀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠔⠋⣁⠤⢦⠀⢠⠦⠤⣈⠑⢶⡋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⡔⠁⣴⠋⠀⠀⢸⠀⢸⠀⠀⠈⠑⢄⠙⢦⠀⠀⠀⠀⢀⣠⡶⠛⠁
⠀⠀⢀⡤⠖⠒⠉⠉⠉⠉⠉⠑⠲⠤⠤⡼⠀⡞⠸⡀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠈⢆⠈⣷⠒⠊⠉⠉⠉⠉⠉⠑⠒⠢⢄⡀
⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⠀⠀⢳⡀⠀⡸⠀⠸⣄⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠸⡀⠀⠀⡹⠊⢀⠀⢀⠈⠳⣄⠀⠀⢸⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁
⠀⠀⠀⠈⠉⠒⠲⠶⢤⣤⠴⠖⠒⠊⠉⢳⠀⠳⡤⠊⢀⡴⢻⠀⢸⠷⢄⠈⠳⣶⠏⢠⠏⠉⠑⠒⠦⣤⠤⠶⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⢰⡋⠀⢸⠀⢸⠀⠀⣱⠄⠀⣠⠋⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠲⢄⡈⠙⠚⠀⠈⠒⠉⣁⡤⠊⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⣀⣤⠚⠁⠀⠀⠀⠀⢨⠋⠙⠒⠒⢻⠉⠁⠀⠀⠀⠀⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠑⠤⢄⣠⣤⠤⠶⠛⢻⠃⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠹⡌⠑⠲⠤⠤⣀⡠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢀⣾⠻⡄⠀⠀⠀⠀⠀⢸⡏⢢⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢀⡴⠟⠁⠀⣇⠀⠀⠀⠀⠀⣼⠁⠀⠙⢦⡀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠉⠀⠀⠀⠀⠸⡄⠀⠀⠀⢰⠏⠀⠀⠀⠀⠈⠑⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠖⠒⠛⠛⠓⠲⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⣠⡴⠞⠋⠉⣿⠀⠀⠀⠀⣿⠉⠙⠳⢦⣄⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢻⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⡟⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠸⣇⣴⠞⠁⠀⠀⠀⠀⣠⡴⣿⠀⠀⠀⠀⣿⢦⣄⠀⠀⠀⠀⠈⠳⣦⣸⠇⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣠⠞⠉⠀⣿⠀⠀⠀⠀⣿⠀⠉⠳⣄⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠙⠳⢦⣄⣀⣿⠀⠀⠀⠀⣿⣀⣠⡴⠞⠋⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠦⢤⣤⣤⡤⠴⠶⠖⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⣶⣠⣴⣶⡆
⠀⠀⠀⠀⠀⠀⢀⠠⠠⣀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⢡⠀⠀⡈⠀⠀⢠⣿⣿⣿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠂⠀⠀⢸⣿⣿⡇⣀⡀⠀⠉⠀⠈⣁⣈⠉⢻⣿⠀⠀⠀⠀⢿⣿⠟⠉⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠡⠀⠘⣿⡿⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢄⠀⠀⠀⢸⡏⠀⠤⠄⠀⠀⠀⠀⣤⡀⢈⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠐⣜⠻⡅⠀⠔⠒⠀⠀⠀⠀⠐⠂⠀⢰⠋⡈⠀⠀⠲⢿⠀⣶⡶⠂⠀⠀⠀⣠⡄⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠘⠦⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⡀⠀⠀⠂⢸⠀⠻⠟⠀⠀⠀⠸⢿⠇⠀⢸⠗⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⡨⠤⣄⠀⠀⠰⣶⣶⣾⠇⠀⢀⡾⠛⣃⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠀⠘⠆⢄⡀⠀⠈⢀⡀⠐⠁⠠⠠⠈⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠈⠉⠁⠈⠀⠓⠀⠄⡀⠄⠑⠀⠀⢀⣀⣀⣀⡁⢂⣀⠀⠄⠐⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡂⠄⠀⣽⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⠀⠈⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠶⠀⠈⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⠿⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠐⠒⠂⠒⠀⡤⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠂⢌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠑⠌⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠚⠹⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠤⠄⠀⠀⠀⠀⠤⠄⠐⡎⠀⠀⠀⠀⠀⢸⠻⠿⢿⣿⣿⣿⣿⡿⠿⠿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⡀⠌⠀⠀⠂⣀⠀⠀⡄⠀⠀⠀⠀⠀⠃⡄⠀⣀⠔⠀⠈⠄⡀⠀⢠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠠⡆⠀⠀⠀⠀⠀⠀⢰⠠⠃⠀⠀⠀⠀⠀⠠⠦⢴⠀⠀⠀⠀⠀⠀⡄⢴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠲⠀⠀⠀⠀⠀⡀⠒⢂⠀⠀⠀⠀⠀⠀⠐⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠈⣢⠀⢀⡌⠀⢀⡤⠀⠀⠀⠀⠀⠀⢦⣀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠀⠀⠀⠂⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠐⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠐⠤⢾⠉⠁
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⣴⣦⣤⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⡆⠓⠂
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢰⣦⢀⡞⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠐⡀⠀⠀⣿⣿⣿⣿⡧⠀⠑⠠⣄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠶⣼⣿⣿⣷⡶⠛⠁
⠀⠀⠀⢱⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠻⡃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠚⣿⣿⠟⠓⠂
⠀⠀⠀⡴⠂⠀⢸⣿⣿⣿⣿⠀⠀⢀⠁⢑⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠄⢈⠀⢠⡀⠀⠀⠀⢿
⠀⠐⠉⠁⠀⠀⢸⣿⣿⣿⣯⠀⠀⠐⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡈⠐⠂⠀⡀⣁⢀⡀⠄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⠀⠀⠈⠵⠀⠀⠀⠀⢤⠀⠃⠘⣰
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠤⣴⢆⠻⠰⠵⠆⣙⠳⠖
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠤⠀⠀⠀⠤⠀⠀⠀⠀⠀⢀⡀⠁⠦⠀⠰⡄⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢀⠀⠀⠂⠀⠒⠀⠈⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⢿⡀⣈⠀⢠⣤⣠⡷⡀⠶⠂
⠀⠀⠀⠀⠀⠴⠰⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠁⠈⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣀⠤⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠏⢀⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢉⡀⠐⣀⡀⠀⠈⠁⠀⠠⠄⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡤⠀⣀⠀⠀⠀⢨⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⢄⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢠⣤⣴⣦⣴⣆⣀
⠀⠀⠀⠀⠀⢀⡟⣵⣾⣿⣿⣿⣿⣆⠀⠀⣶⣿⣿⣿⣿⣿⣿⣷⠂⠀⣠⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣀⠀⣀⣼⣿⡿⢿⠿⠟⠻⠟⢻⠀⢰⣿⢋⠉⠛⠻⠛⢿⠋⠀⠀⣻⠃⠄⠉⠛⡛⢻⣿⠂
⠀⠘⠉⠋⣿⣿⡟⡟⠀⠀⠀⠀⠀⢘⠁⢸⡇⢸⡄⠀⣰⡆⢸⠆⠀⠈⡃⢾⠃⠀⢴⡆⢸⠏
⠀⠀⠀⣀⣙⢿⣿⣇⠀⠠⣤⡤⠀⠀⠀⠀⠃⠈⢁⣀⡈⠁⠸⠛⠄
⠀⠀⣼⣿⣿⣦⠈⠙⠀⠠⠀⠤⠀⠀⠀⠀⠀⠀⢀⠉⠁⠀⠀⠀⠀⢀⣤⣀⣠⣄⢀⣄⡀
⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⣸⣿⣿⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⠀⠀⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠛⠛⠿⠿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠃⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠐⠂⠀⠀⠀⠒⠀⠀⠀⠀⠀⠐⠤⠤⡂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣎⡠⠂⠀⠀⠐⢤⠐
⠀⠀⠀⠀⠀⠀⠘⠐⠊⠀⠀⠀⠉⠒⠁⠀⠐⠋


⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⠰⠎⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⣾⣟⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠁⠠⣼⢩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠖⠛⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠆⣀⠀⠲⢦⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡄⡠⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⠜⢁⣤⢄⢁⠘⠳⢷⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡯⠠⠛⡿⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡼⠁⠠⠆⢀⠨⡄⢙⠎⢁⠃⠐⢰⡶⣿⣥⣾⠀⠀⠀⠀⠀⠀⣀⣰⠷⡤⣾⣱⣗⠈⡱⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⢀⡾⠀⢀⠀⠀⠁⠀⢀⡖⢀⡔⠘⠀⢸⠿⠃⠀⠀⠀⢠⣶⣾⡿⣋⣡⣄⠉⢸⣿⣄⢠⣦⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠘⣿⠙⣆⠄⠀⠀⠀⠀⣾⡃⢘⡅⠀⠀⠀⠁⠀⠀⢤⡾⠏⠛⢛⡋⠀⣰⣿⣤⣼⣿⣿⣃⣶⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⡜⠑⢄⡍⠉⠀⠀⠀⠚⣹⡇⣿⡗⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⢈⠍⠘⣿⣿⣿⣿⣿⡿⢟⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠘⡁⠀⠀⢀⡠⠉⠦⠄⠀⠈⣛⣹⡇⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠂⠉⠙⠀⡼⣮⣴⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⢈⣠⡼⣧⠀⠀⠘⠁⠀⠀⢀⣈⣥⡶⣿⣇⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠉⡁⢀⣿⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠸⢴⣿⠀⣿⠀⠀⠀⣀⣤⡀⠈⠀⠀⠉⣿⣿⠘⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠛⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⠀⣾⡃⠀⠀⠀⠀⠀⣹⣿⡇⠀⠀⠀⠀⢽⠉⣧⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢾⠀⣙⠸⠀⠀⠀⠀⣴⣿⡿⠀⠀⠀⠀⠀⣼⠀⢘⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢠⡄⠀⠀⠀⡼⣯⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠎⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⣿⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⢉⣥⠀⠀⢠⡍⠿⢿⡷⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢎⠁⠀⠀⢀⣼⢿⡏⠀⠀⠀⠀⠀⢠⡇⠖⠀⠀⠀⠀⠀⠀⠀⠀⢈⠡⠀⠀⠀⠸⣿⠀⠀⠈⠘⣿⠉⠃⠀⠀⠀⠀⠀⠀⠄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣅⣾⠀⠀⠀⡀⠀⠀⣼⣧⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠄⢠⣾⠿⡆⠀⠀⠀⣿⡤⢦⣴⣴⣶⣤⣶⣤⣴⣶⣤⣷⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⢻⣿⣽⠁⠠⠀⠀⠀⠀⠀⢯⠆⠀⠀⡀⠀⠀⠠⠀⠁⡉⡄⠠⠆⢀⣀⣀⣀⠀⣀⠀⠈⣄⣠⣨⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣻⣿⣿⠀⡄⠀⠐⠁⠑⠀⢸⡁⠶⢀⡀⠠⠀⠀⠀⠀⠀⠂⠀⠀⠀⢶⠴⠤⡀⠈⡆⠀⠙⢻⠟⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢪⡜⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⡼⠶⠄⠀⠠⢂⡄⠂⠀⠀⠈⠀⠰⠀⣿⡆⠀⠰⠀⠁⠀⠀⠛⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢸⣿⣿⠄⠀⠀⠀⠀⠲⠘⡋⠈⠛⠳⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠸⢿⠐⠈⠀⠙⢊⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣾⡿⠎⠀⠀⠀⠀⠀⢂⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠣⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⠿⠇⠒⠃⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠛⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠒⠒⠦⠾⠷⠖⠤⠤⠃⠀⠀⠀⠀⠀⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠴⢁⣨⡰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⣐⠧⣤⠀⠀⠀⠀⠀⡆⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣄
⠀⠀⠀⠀⠀⠀⠄⠀⠘⠀⠈⠀⠀⠘⠁⠀⠀⠀⢡⠀⠈⡄⠀⠀⠀⠀⡀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⠀⠀⠠⠔⠒⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡄⠈⢄⠀⠀⠀⠈⠀⠀⠀⠁⡀⠀⠀⠀⠀⠟⣿⣿⣿⣄⠀⠀⠀⠀⠚⠀⠐⡂⡒⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠙⠢⡀⠢⡀⠀⠀⠈⠁⠂⠀⠂⠀⠀⠀⠀⠐⢻⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢤⠤⠀⠀⠻⠄⠀⠀⠀⠀⠀⠀⠀⡀⡒⠁⠀⠀⠀⠙⠹⣿⣿⣿⣧⣀⠠⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠉⠐⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣄⡀⠀⠠⠔⠀⠀⡀⠀⠀⢱⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡟⠀⠀⢀⣀⠀⠐⣵⣤⣄⡀
⠀⠀⠀⢐⡀⠄⠀⠀⠠⠆⠀⠀⡄⠀⠀⠀⠀⣻⣿⠁⠀⠀⠀⣀⠉⢹⣷⣄⠀⠉⠀⠀⣤⣦⣄⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡄⡤⠚⡿⠢⣄⣿⣿⣿⣿⣦
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠂⢡⠀⠀⠀⠈⠻⡇⠈⠀⠀⠀⠈⠀⠀⢻⣶⣦⠀⡇⠀⣿⡄⣯⠢⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣆⢸⡟⣣⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠣⡁⠀⠀⠀⠀⠤⠆⠈⢻⠀⠀⠐⣄⠀⠀⠀⠀⠈⠡⠀⢀⠀⠘⣾⣿⡆⠘⢦⣿⣷⡹⠇⢡⡀⠀⣀⠀⢹⣿⣿⣿⣿⢿⡿⠧⠌⠁⠘⠟⢻⣭⣩⣍⣁⣴
⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⠐⠀⡸⡂⠏⠀⠉⠁⠀⠀⠰⠆⠰⠛⠀⢦⠀⠙⡟⠻⠆⠀⠀⡛⠇⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣤⡠⡀⠀⠀⠀⣄⣹⣿⣿⣿⡿
⠀⠀⠀⠀⠀⡄⠀⠀⢲⣀⠦⠀⣆⠇⠐⠀⡀⠀⡀⠀⠀⠀⠀⠀⠈⠄⠁⠀⠘⠄⠣⠔⠉⠀⠠⠶⠶⠦⢤⠠⠔⠀⠀⠀⠀⠀⠀⠈⠙⡳⠶⠶⠶⠶⠿⠿⠉
⠀⠀⠀⠀⢠⡃⠂⠀⢸⣏⠀⠑⡜⡆⠀⠀⠁⠀⠐⡄⠀⠀⠀⢀⢌⠖⠂
⠀⠀⠀⠀⠀⡈⡇⠀⢸⣿⣷⡘⡆⠱⡀⠀⢀⢀⠀⠘⡆⠀⠀⠀⠋⣀⡤⠀⠀⠀⠄⠀⣀⡖
⠀⠄⠀⠀⠀⠙⡝⢦⡘⢿⣿⣷⠁⠀⢀⠀⠀⠊⠆⡠⠟⠘⢦⡤⠀⠘⠓⠈⠁⠚⠐⠒⠁
⠀⠀⠀⠀⠀⠀⠳⠀⠈⠙⠛⢃⠀⠀⠈⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⢘⣄⠀⠀⠀⠇⡀⠀⠀⠀⠀⠔⠈⠀⠠⠤⠤⠤⠄⢀⡀⢠⡤⠞⠁
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠈⠀⡋
⣀⣀⣀⣄⣠⣥⣤⣤⣤⣤⣤⣤⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠙⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠾⠿⠿⠿⠿⠷⢶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠁⠀⣀⣀⣤⠀⠀⣤⣀⣀⠀⠈⠙⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⢀⣠⣶⠾⠟⠛⠙⣿⠀⠀⣿⠉⠛⠻⠷⣦⣄⡀⠈⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⣠⣴⠟⠋⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⣰⡿⠁⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠈⢿⣆
⠀⠀⠀⠀⠀⣰⡿⠁⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠈⢿⣆
⠀⠀⠀⠀⢠⣿⠁⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠈⣿⡄
⠀⠀⠀⠀⣸⡏⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⢹⣇
⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⢹⣇⠀⢸⣧⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⣀⣴⠀⠀⣦⡀⠈⠻⢷⣄⠀⠀⠀⠀⠀⠀⣼⡇⠀⣸⡏
⠀⠀⠀⠀⠘⣿⡀⠈⢿⡄⠀⠀⢀⣴⡾⠋⠀⣠⣴⠿⣿⠀⠀⣿⠿⣶⣄⠀⠙⢷⣤⡀⠀⠀⢠⡿⠁⢀⣿⠃
⠀⠀⠀⠀⠀⠹⣷⡀⠘⢿⣄⣴⡿⠋⠀⣠⣾⠟⠁⠀⣿⠀⠀⣿⠀⠈⠻⣷⣄⠀⠙⢿⣦⣠⡿⠃⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠹⣷⡀⠈⠻⠋⠀⣠⡾⠟⠁⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠈⠻⢷⣄⠀⠙⠟⠁⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠺⣿⣄⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⣠⣿⠗⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀⠈⠙⠿⢶⣤⣤⣀⣿⠀⠀⣿⣀⣤⣤⣶⠿⠋⠁⢀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣄⡀⠈⠉⠉⠛⠀⠀⠛⠉⠉⠁⢀⣠⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠷⢶⣶⣶⣶⣶⡶⠾⠿⠛⠋⠁


⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣻⣿⣛⣛⣻⣻

⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣤⣀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡄
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠰⣷⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠰⠧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣦⣀⣀⡀⠀⠀⣀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠉⠛⠉⠙⠉⠉⠋⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⠉⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠊⠉⠀⠉⠑⢦⠰⡁⠀⠀⠀⠀⠠⠛⡄⠀⠀⠀⠀⠀⠀⠀⣶⣄
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣀⢀⠀⠀⠀⠀⢱⢑⠮⠀⠀⠀⡀⢠⠓⡄⠀⠀⠀⠀⠀⠀⢘⣿⣿⡖⠲⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠉⠁⠐⠒⠊⠇⠸⡀⠈⣢⡔⠁⠁⡠⠃⠀⠀⠀⠀⠀⢀⠎⢨⠛⣇⣀⣀⣀⡙⣷
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⡸⠀⠒⠀⠀⢛⠀⠀⠀⢹⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⢰⠊⠂⣠⣀⠸⣿⣿⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣲⣧⣀⠀⠒⣠⠎⠀⠀⠀⠊⣿⣿⡂⠀⠀⣾⣶⠀⠀⠀⢻⠀⠀⣀⠭⡇⠻⣿⣿⡿⢡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠿⣿⡿⠟⠿⠿⢷⠀⠀⣠⠃⣿⡏⠀⠀⢀⣿⡇⠀⠀⠀⠈⠢⣄⠀⠀⣇⣀⣙⣯⠔⠁
⠀⠀⠀⠀⠀⠀⢀⠎⠈⢣⠀⠀⠈⢸⠀⠈⡍⠚⠀⠀⡿⠀⠀⢀⣾⣿⣿⡀⠀⠀⠀⠀⣨⣶⡿⠿
⠀⠀⠀⠀⠀⠀⡌⠀⠀⡘⠀⠀⠀⢸⠀⠀⢡⠀⢀⠌⠀⠀⠀⢸⠉⠛⠋⡅⠀⠀⢀⠼⡏⠁⠀⠘⡀
⠀⠀⠀⠀⠀⡰⠀⠀⢠⠃⠀⠀⠀⢸⠀⠀⠸⡋⠁⠀⠀⠀⠀⠘⣆⣀⣠⠃⢤⡔⠁⠀⠈⡆⠀⠀⢃
⠀⠀⠀⠀⢠⠁⠀⢀⠎⠀⠀⠀⠀⢸⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠙⠛⡇⠀⠈⠀⠀⠀⢠⠃⠀⠀⠸⡀
⠀⠀⠀⢀⠜⠀⢤⠜⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⣀⠤⠊⠀⠀⠀⠀⢣
⠀⠀⠀⠺⢶⡤⠢⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣶⣦⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣀⣀⠠⠬⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣷⣶⣶⣾⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⡏⠸⠿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⡠⠇⢰⣔⣚⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠎⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⢿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣼⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣀
⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠀⠀⣠
⠀⠀⠀⠀⠀⠀⢹⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⢀⠡⠆
⠀⠀⠀⠀⠀⠀⠚⠉⣠⠤⠀⠀⠀⠀⠀⠀⠂⠀⢀⣀⠀⠀⠀⠀⠀⠄⠀⠀⠬⠐⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⢸⣿⠷⣼⣷⡄⠀⡐⡁
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢠⢀⠀⠀⠀⡀⠘⠚⠈⡁⠀⠀⠀⠀⣠⠀⠀⠀⠀⡀⠠⠀⠂⠁⠀⠀⠖⠃⠀⠚⠛⠻⠿⠿⣏⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠺⠅⠀⢠⣄⣀⣴⣦⡀⠀⠀⢉⡁⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠸⠄⢀
⠀⠀⠀⠀⣠⠄⠒⠀⠃⠸⠚⠀⠀⢀⠀⠀⢀⣾⣿⣧⠨⠛⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠑⢀⠀⠀⠚⠁⠀⠁⡴⣶⡏
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠸⡆⠀⢨⠀⠀⠨⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⣧⡿⠇
⠀⠀⠀⠀⠈⢧⠀⠀⠀⢀⠠⠛⠀⢀⡳⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁⠀⡀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⡾⠸⠃⣼⣛⢴
⠀⠀⠀⠀⠀⣈⠷⠒⠉⠀⠀⠀⡀⠉⠁⠀⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠐⠀⠐⠃⠀⠀⣸⠵⠊
⠀⠀⠀⠀⣋⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠰⠀⠘⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠠⠒⠁⢀⣄⡀⠀⡀⠄⠠⡄⢀⣠⣵⡆
⠀⠀⠀⠀⠀⠈⠛⡓⢒⠲⠆⡃⠀⠀⠀⠀⠀⠀⠤⢄⣛⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠈⠉⠉⢠⠀⠀⠈⠀⠃⢀⡀⠀⡈⠱⡏⠙⠉⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⣴⠲⡀⠀⣍⡙⠓⠒⠒⠲⠴⠂⡠⠴⠛⠇⠀⠀⢀⡀⠀⡞⠀⠈⠇⠀⠀⢸⡵⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⡸⠊⢙⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠁⠐⠁⠀⣀⠅⣐⣒⡭⢒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡠⠚⠁⣜⠉⠃⠉⣠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡴⠀⢀⠀⠀⠀⢀⠀⢀⢩⡼⢁⣄⣢⠽⣦⠀⠀⠉⠂⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠋⠲⠶⠓⠁⠀⠀⠀⠐⠛⠉⣾⡭⠊⠁⠈⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⡀⡀⠠⣠⣀⢀⡀⢠⡄⡀⢀⡤⣀⡀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠃⠀⠁⠉⠉⠁⠈⠉⠉⠀⠉⠉⠉⠉⠉⠙⠋⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠶⣄⣀
⠀⠀⠀⠀⠀⠀⠱⡉⠛⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠙⢤⠀⠀⠈⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⢤⣤⣀⣀⡈⠛⣦⠀⠠⣄⠀⠀⠀⡀⠈⣟⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠈⠻⢆⠈⠑⣦⡀⠀⠀⠸⣇⠀⠈⠢⡀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣸⣷⣄⠀⠓⠨⣦⣀⡀⠈⠲⡄
⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠛⠦⣝⣻⣿⣧⢠⡀⠈⠻⣝⣶⣄⠁⠀⠀⠀⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠈⠓⣄⡀⠀⠀⣀⡑⠲⢼⣟⣛⠉⠛⠻⢶⣄⡈⠻⡉⠛⠳⣄⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠐⢦⣈⠁⠀⠀⠀⠀⠀⠉⠓⠶⢭⡿⢿⠦⠀⡀⠈⠙⣶⣤⣄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⣈⡑⠲⠿⠷⣆⣤⣤⣌⣹⣷⠾⡄⠉⠓⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀
⠀⠀⠀⠀⠀⠐⠲⣉⠉⠉⠀⠀⠀⠀⠉⠙⢲⣶⣧⣤⣀⣀⣈⣉⣅⠈⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠐⠴⡀⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠄⡀⠀⠀⢤⣤⣤⣬⠥⠙⠋⠉⠉⠉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠢⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠈⠓⠤⣤⡤⠀⠉⠒⠒⠦⠤⣷⣶⣶⠶⠒⠚⠛⠛⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡉⠳⡄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠈⠙⠛⠒⢶⣯⣥⣤⡤⠤⠾⠟⢿⣀⡀⠈⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠠⢤⣀⠀⠀⠠⠤⠈⣉⣁⣀⣀⣀⡾⠿⠧⠠⠀⠤⠤⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣌⠁⠀⠀⠀⢀⣀⡉⢩⡭⠋⠁⠀⠀⢉⠅⢀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠀⠠⠄⠈⠛⠻⡟⣁⣀⡤⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠒⠉⠉⠹⠟⠛⢋⡀⠀⠺⠃⣀⠀⠀⠂⠀⠀⠒⠴⡄⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠄⣀⣠⠀⠀⠐⠒⠛⠛⠛⠉⣀⣀⠢⠧⢠⡴⠂⠀⣀⠘⡄⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠯⠤⠤⣠⠤⠈⠉⠙⠛⠛⣻⣁⣤⡾⠟⠃⠀⠈⠁⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠥⠤⣠⡤⠒⠚⠛⠛⢛⡻⢄⡀⠐⢀⣘⠤⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠾⠷⠶⠶⣶⠶⠂⠛⣁⡀⢀⣤⠜⢇⣀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣤⡄⠐⠛⠋⣁⣄⣁⠀⢀⣤⠋⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⣠⣥⠄⣈⠀⢀⣀⡤⢀⡽⠋⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠹⠖⠒⠀⠀⢛⣙⠛⠉⢁⣀⣀⣲⠖⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠦⠄⠀⠐⢂⠤⢼⣿⣿⠟⠋⠉⠠⠙⣒⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢀⢰⣶⣶⣶⠞⠙⠩⢙⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠷⠒⠒⣠⠟⠋⠛⣿⣧⣤⣴⡶⠋⠙⣽⢅⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⡄⣀⣈⠕⠛⢻⣏⣤⡶⠊⢠⡾⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠖⠓⠈⠩⠁⢀⡴⡫⠛⢩⣶⣶⣯⣠⡏⢀⣿⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠀⢠⣿⠞⡹⣿⠋⠇⡟⠋⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⣠⠔⠋⠀⡰⠋⠀⢠⠁⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠜⠁⠀⢠⠆⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠁⠈⠀⢀⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠂⠐⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣧⣄⣠⡿⣿⣄⣨⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠙⠋⠁⠈⠛⢙⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⢻⣿⠀⠀⠀⠀⠀⢸⣿⠉⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠈⠋⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣆⠀⠀⠀⠐⠀⠀⠀⣼⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠋⠉⣉⠀⠀⣴⡆⢰⣦⠀⢈⣀⣈⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣶⣿⡿⠿⠿⠏⠀⠀⠻⠿⢿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣿⣿⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠈⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣟⠷⠢⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢳⡑⣬⠑⢪⡃⠛⢷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⢁⢪⡀⣿⡶⠁⢙⢧⣧⡘⣯⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠫⠇⡙⣼⠳⣿⡻⢾⣿⠳⣬⣿⣿⣿⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢠⣷⣰⣿⣾⡷⣏⢳⣜⣿⣿⣿⣿⣿⡛⢷⣽⣷⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠃⠀⡛⣹⡿⢹⣸⣧⣘⣶⡛⡿⣯⣿⣷⡎⣻⢶⣞⢿⣿⣿⣧⡀
⠀⠀⠀⣤⣀⠀⠀⢀⣠⠖⠃⠠⢨⣠⣽⢿⣵⣾⣿⣿⣿⡾⣷⣻⣟⠿⣿⣿⣮⣑⡙⣧⡿⣿⣿⣧⡀
⠀⠀⢐⣫⣙⠛⣛⠅⠄⠞⢁⡜⣸⣾⣿⣼⣿⢿⣿⢻⣜⣷⡽⢷⡹⣷⡜⢿⣿⣿⣧⣈⢻⡙⢿⣿⣵
⠀⠀⢘⣿⣅⡈⢀⡴⣴⣧⣋⣌⣃⡎⣿⣿⡿⡞⣿⡸⣿⣿⣿⡙⣿⣽⣿⣯⣛⣯⠻⣿⣏⣿⣎⣿⡝
⠀⢀⣼⠷⣢⠖⢊⡴⣋⣯⣶⣿⣿⣷⣿⣾⡧⣿⠹⣧⣹⣿⡌⢿⣟⡿⣎⠿⣷⠿⣧⠘⢷⣝⣿⣧⠻⡀
⠀⠈⠁⠨⠶⣤⡞⡜⣿⣿⣿⣿⣿⡟⣿⣻⣿⢹⣧⣿⣷⣝⢿⠮⣻⡷⡝⠧⡈⢳⣜⢷⡌⢿⣿⣎⢧⣡
⠀⢠⠊⢀⣾⢻⣹⣯⣿⣿⡟⢿⡜⣧⢹⡟⢿⣇⠻⡷⢏⡏⠉⠀⠀⠀⠀⠀⠈⠉⠈⠀⠙⢆⢿⣿⡼⡍
⠀⠀⢠⣶⡇⣟⣿⣿⣿⡌⣿⢺⣿⢹⣆⣿⡜⣿⠓⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠛⣿⡇
⠀⠀⠸⠀⣇⣿⣿⣿⣿⣯⠹⡎⢻⣯⡟⠛⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠙⠉⠯⠏⠳⣿⠑⠇⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢀
⠀⡀⠀⠈⠁⠀⣀⠎⠑⠀⠀⠈⠀⠀⠀⠀⠈⠶⠀⠀⠦⡞⣣⠄⠀⢠⣏⣹⠄⣿⣙⠀⣾⣿⡄⣴⠟⠀⣿⣙⠀⠀⢶⣦⣰⣿⠀⢀⡄⠀⠓⠃⠑⠂⠀⡆
⠀⢒⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⢀⣠⣴⡄⠀⠀⠀⠀⠀⠋⠑⠀⠀⠈⠯⠁⠀⠿⠶⠀⠿⢿⠇⠻⠶⠂⠿⠶⠀⠠⠀⠙⣿⠏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸
⠀⠈⠂⡀⠀⠀⠀⠀⠀⡀⠀⢀⡾⠍⠁⠀⣠⠞⠉⡏⠙⠲⡄⠀⠦⠀⠀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠀⣀⠀⠀⢀⣿⡀⠀⠀⠀⢠⡄⠀⠀⠀⠁⠀⠒⡀
⠀⠀⠀⠑⢄⡀⠀⠀⠂⠀⠀⠈⠁⠀⠀⢰⠇⠒⢀⣧⡒⠃⢹⠀⠀⠀⠀⠀⢀⢿⢑⣤⠀⠁⠀⠀⠀⠁⡀⠀⠈⡟⠁
⠀⢀⢀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠶⠀⠈⢧⡴⠋⣿⠙⢲⡼⠀⠀⠀⠐⠉⠙⠈⠀⠁⠀⢀⡀⠀⠱⠇⠀⠀⠀⠂⠀⠀⠀⠵⠎⠀⠈
⠀⢒⡋⡈⠈⠄⠀⠀⠀⠀⢀⣀⣀⠀⢀⣀⡈⠓⠶⠿⠖⠋⠀⠀⠀⣀⣤⣴⣶⣦⣤⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠺⠇⠀⠀⠀⠀⠄⡀⢺⣿⣿⣷⣿⣿⡿⠀⡀⠀⠀⠤⠀⣠⣾⣿⣿⣿⣿⣿⣿⢿⣷⣄⠀⠘⢻⡟⠃⠀⠀⡀⠀⠀⠘⢟⡿
⠀⠀⠀⠀⠀⠚⠀⠀⠀⠈⠀⠙⢿⣿⡿⠋⠀⠈⠀⠀⠀⠀⢰⣿⣿⡏⠋⣯⠋⣍⠏⢸⣟⣿⣇⠀⠀⠀⠀⠀⠸⠿⠄⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⡀
⠀⠀⠢⠤⠄⠀⠀⠀⠃⡀⠀⠀⠀⠛⠀⠀⠀⢀⠘⠀⣤⡂⣿⣟⣿⣷⣿⣷⣷⣿⣶⣾⣿⣿⣟⠀⠋⠀⠉⠉⠉⠉⠉⠉⠃⢠⠀⠂⠀⠀⠀⠀⢐⠀⠀⠖
⠀⠀⢀⣠⡤⣤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⡔⠀⢻⣿⣿⣿⢦⡰⢃⡓⣇⣾⣿⣿⡿⢀⣤⣤⡀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠂
⠀⣠⠏⠁⠀⣼⠀⠀⠙⣆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢻⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⠁⢸⡇⠈⠻⣦⠶⠛⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⡏⠀⠀⢀⣿⡀⠀⠀⢹⠀⡼⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⡿⣿⠿⡿⠟⠉⠀⠀⠀⢻⣄⣴⣿⣄⢀⡼⠃⠀⠀⠁⠀⠀⠀⠐⠂
⠀⢻⡤⠞⠋⣿⠙⠳⣄⡾⠀⠀⠀⠀⠀⠀⠂⠀⠘⠄⠀⠂⠀⠘⠂⢀⣤⣶⡖⠰⢦⡀⠀⠈⠢⡄⠀⠹⣾⣧⠿⠋⠀⠀⠖⠀⠀⠠⠐
⠀⠀⠙⠦⣤⣿⣠⡴⠟⠁⠀⠒⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⢠⣿⣿⢫⠁⠀⣽⡿⡆⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⢀⠀⠀⠀⠁⠠⣤⣴⡶⠶⠦⡔⠁
⠀⠀⠀⠀⡀⠀⠀⢀⢀⡀⠀⠀⠄⠀⠀⠀⠀⠒⠠⠥⠀⠀⣤⠀⠈⣭⣥⡄⠀⣴⣾⣶⡿⠀⣤⠀⠀⠀⣀⡄⠀⠀⠈⠀⠀⠤⣤⣴⣻⣛⣯⡽⢯⡾⠁
⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠸⣧⠀⢮⢿⢟⠀⢻⣿⠏⠀⡴⠇⠀⠀⠉⠟⠣⠀⠀⠀⠀⠀⠈⠉⠚⠑⠋⠉⠀⠋⠀⠘⠁
⠀⠀⠀⠄⢀⢀⢀⠀⠈⠀⠀⠠⠄⠄⠀⠀⠀⠀⠀⢀⠀⠀⠁⣉⣴⢀⠀⠀⠐⠊⠁⠐⣦⡉⠁⠀⠀⠀⠐⠀⠐⣦⡶⣰⢦⠀⣲⡀⢰⡆⡰⣦⠀⣶⣦⡄
⠀⠀⠀⠀⠀⠈⠉⠈⠐⠈⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠘⠛⠟⠘⠃⠀⠸⠟⠻⠃⠙⠛⠄⠻⠻⠄

⡐⠂
⠘⢄⡀
⠀⠈⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⡄
⠀⠀⢘⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆⠀⣠⣶⣶⣶⣦⠀⠀⠀⠀⠼⣿⣿⣿⡃
⠀⠀⠀⡀⠀⠆⠀⠀⢠⣿⣿⣿⣷⡀⠀⠀⠀⠀⠘⣿⣿⣿⠏⠁⠀⠈⠿⣿⣿⣿⠄⠀⠀⠀⢲⣬⣍⡉⣀
⠀⠀⠀⠘⢄⠄⠀⠀⠨⣿⣿⣿⣿⠇⠀⠀⠀⢀⡀⢈⡉⠁⠐⠀⠀⠀⠀⠈⠛⠟⠀⠀⠘⣦⣠⣿⣿⣟⡻⣷⣄
⠀⠀⠀⠀⠀⠣⡁⠀⠣⠉⠛⠉⠉⠀⠀⠀⠀⡀⡘⠘⠂⠁⠛⠛⡛⠛⠲⠦⣴⣤⣤⣤⣤⢼⣿⣿⣿⡿⠼⣿⢿⡄
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠳⠶⠶⠶⠶⠚⢷⣴⠁⠁⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠈⡠⠋⠀⠀⠙⣿⣿⣿⠿⣻⡏
⠀⠀⠀⠀⠀⠀⢸⡍⠀⠀⠀⠀⠀⠀⡰⠀⠁⠀⠀⠀⠀⠀⠀⠀⣼⡟⠳⣦⣄⠀⠀⠀⠀⢘⠀⠸⣿⣿⣴⣿⣈
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⢀⣷⠀⠀⠄⠀⠀⠀⡀⠀⢠⠊⠁⢰⣄⠉⠓⠂⠣⠀⢸⡀⠄⠈⠻⣿⡁⡇
⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⢊⣽⡿⣠⠀⠈⠒⠛⢛⣡⡤⠎⠀⢠⡞⡡⠆⠀⠀⠀⡴⢼⣷⣦⣄⣂⣘⠐⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠩⢼⠀⠘⠷⠶⠾⠛⠉⣁⣨⣧⠜⢋⣈⠒⠂⠀⠀⠤⠄⢸⣿⣿⢿⣿⡿⣿⣷⠄⠋
⠀⠀⠀⠀⠀⠸⠔⣠⣤⣴⣶⣦⣤⣈⠆⢀⠀⢀⡤⠚⡛⣋⣁⣀⢃⡉⣩⣋⣀⣀⣀⣀⣀⣸⣏⡍⠛⡉⢽⣿⡏⠂
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠨⣶⠿⠿⠛⠙⠛⢻⢿⠀⡙⣻⣿⣧⣿⣿⡛⢟⠛⣧⡶⢚⣨⣿⡏⠁
⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⠏⠀⠸⣿⠆⢀⠈⠈⠀⠀⠒⠀⣡⠜⠀⣲⣽⣿⣭⣍⢩⣉⣾⡁⣿⣵⢿⠟⢁⡀⡄
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣶⣄⢰⡇⠀⢘⠀⠀⢀⢸⣧⠈⢡⡆⠀⣍⣿⣹⣿⠛⣿⣿⣽⠧⣿⣽⡿⣟⠋⢸
⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟⢻⡏⠁⡃⠀⠈⠂⠀⠈⣾⣿⡆⠁⠀⠀⠿⣵⣷⣥⡜⢿⣣⡅⢰⣿⡟⠡⢶⣦⠄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠘⣷⡀⣒⠀⠀⢿⡞⠳⠀⡭⣧⣠⠀⠀⠉⢨⠼⢷⠃⠚⠫⣤⣬⠛⠁⠀⠈
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡅⠀⢾⣿⠨⡄⠀⠄⠂⠐⠀⠰⠈⠀⠀⠀⠀⠸⢄⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⢸⣿⠂⠀⠀⠸⣿⢋⡇⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⣈
⠀⠀⠀⠀⠀⠀⢹⠏⠀⠀⠀⢸⣿⠈⠄⠀⠘⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠃⡀⠀⠰⡏⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠀⠀⣀⡁
⠀⠀⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⠹⠂⠀⠀⠠⣈⡷⢰⣶⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⣠⣾⣯⣿⣽⡀
⠀⠀⠀⠀⠀⠀⠸⣷⡄⢲⠀⠀⢰⣶⡎⠛⢧⣾⡇⣸⣷⣣⠤⠀⣶⣶⣦⣀⡄⠀⠠⣤⣿⣶⣟⣻⣿⣿⣹⡧
⠀⠀⠀⠀⠀⠀⠰⠿⠿⠉⠀⠀⠘⢿⣷⣼⡾⠉⠀⠯⠴⠶⠆⠀⠙⠛⢛⡿⠃⠀⠀⠘⠉⠀⠈⠁⠉⠑⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⠿⠻⠿⠛⠿⢿⠿⢻⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡠
⠀⠀⠀⠀⠀⢈⣣⡳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⣜⡁
⠀⠀⠀⠀⠀⠀⠹⣿⣮⡻⢦⣀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⢀⡴⢟⣵⣿⠏
⠀⠀⠀⠀⠀⠀⠐⢦⡙⢯⣑⣩⣿⣿⡟⢶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡶⢿⣿⣿⣍⣊⡽⢋⡴⠂
⠀⠀⠀⠀⠀⠀⠒⠶⢬⡑⠦⠬⣿⣿⣯⠌⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠡⣽⣿⣿⠥⠴⢊⣥⠶⠒
⠀⠀⠀⠀⠀⠀⠙⠳⠦⣍⣙⣓⣛⣿⣿⡽⢗⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡶⣯⣿⣿⣛⣚⣋⡩⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠋⠰⠖⠛⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⠛⠲⠆⠘⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⡋⢝⣿⡿⢟⣿⣿⣿⣷⡀⢀⣾⣿⣿⣿⡻⣿⣿⡋⢙⠒⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⢀⡈⠐⣿⣿⣾⣿⣿⣾⣿⣿⠁⠈⣿⣿⣿⣿⣿⣷⣿⣿⠂⢄⣀⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠉⢸⣿⣿⣿⣿⣿⣿⡿⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⠉⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢾⣿⠏⠃⣿⣿⣿⠛⠀⠀⠀⠀⠛⣿⣿⣿⠘⡹⣿⡷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⡟⡔⢠⢻⢻⣧⠀⠀⠀⠀⠀⠀⣸⡟⡟⣄⢧⢻⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣼⡿⣰⢡⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⢈⠘⠘⠸⣿⣆⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⡴⢱⠃⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢰⢢⡜⢏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠏⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠇⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠨⠋⠛⠛⠛⡒⠲⢄⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⡦⡄⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠐⠀⡀⠀⠄⢹⠀⠈⠀⠀⠀⠀⡨⠰⣀⡄⢀⠀⠀⡇⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⠈⢹⠀⠁⠀⣀⠀⠀⢀⣠⠾⠃⠁⢀⠿⡞⠀⠀⠇⠉⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉⠳⣄⣀⡀⠀⠀⠐⠀⢤⣀⡄⠀⠁⢰⠤⡀⠟⠀⠘⢿⠟⠘⠀⠀⠐⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢒⣩⡩⠭⠉⠉⣋⠴⠂⠀⢌⣛⣃⣞⡀⠐⠒⠉⠀⠀⠀⠠⠄⠀⡠⠊⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠋⠀⢀⣀⠀⢼⠀⠀⢀⠅⠀⠀⠈⠑⠀⠀⣀⠀⠀⢳⣤⣄⠊⠀⢦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡐⠀⢤⡄⠈⠋⠀⢂⠉⠀⠸⣇⠀⠁⠄⠀⣀⠀⣀⣀⣀⠀⠀⠉⢲⠉⠂⠠⠠⠀⠑⣬⣦⡀⡀⠀⢄⣠⡎
⠀⠀⠀⠀⠀⠀⠈⠀⠈⡁⢁⠀⠠⣤⣴⢀⠁⡄⠀⠀⠀⠈⠄⠄⢌⠀⠤⠴⣦⣦⠄⣈⡀⠠⣠⠀⠀⠀⠀⠈⠙⠁⣴⡿⠋
⠀⠀⠀⠀⠀⡀⢁⠀⠉⣥⢠⠑⠦⣤⠡⠎⡄⢰⠀⠀⠐⢄⣶⣾⡇⡇⠀⠀⠈⠈⠓⢀⡙⠰⠉⠂⠶⠞⣠⠜⠁⢪⠁⣀⡀
⠀⠀⠀⠀⠀⡄⠘⢄⡉⠀⠁⣀⠀⠀⢀⣀⣁⡤⠖⢶⠖⡎⠪⠍⠴⠃⠘⠓⠈⠛⠂⢈⠃⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⣸⠀⠀⠀⢀⠀⠀⠈⠗⠉⠀⠀⠀⠀⠀⢸⠀⠘⠦⣀⡀⠀⢄⢀⣀⣡⠀⢸⢄⠀⠀⣰⡄⠀⠀⠀⠀⢀⠆⢀⠀⠀⠳
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠉⠓⠒⠀⠠⠤⢤⡤⠤⠴⡆⠀⠀⡀⠀⠀⠐⠈⠻⡿⠟⠀⠊⠈⠀⣺⣛⡀⢢⠀⠀⠀⠎⠀⠘⡄
⣴⣴⣦⣶⣤⣴⣴⣦⣶⣴⣴⣦⣤⣤⣤⣤⣤⣤⣴⣷⣤⣾⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣷⣷⣶⣴⣶⣶⣾⣴⣶⣴⣧⣴⣴
⣿⣿⣴⣭⣦⣤⣽⣵⣿⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡀⠀⠠⣶⣶⣶⣶⣶⣶⢶⣦⣤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢤⡤⠐⠒⡗⡖⠃⠺⡀
⣀⣠⣤⣴⠿⢿⢟⡿⣿⣮⣿⣿⣿⢻⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠷⣶⣤⢀⡠⠤⣶⣿⡷⠄⠉⣁
⠉⢩⠍⠃⠀⠆⠂⣿⣿⣯⣽⣿⣿⣿⡿⣟⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣗⣲⣿⣡⠀⠙⠶⡬⠙⠁⠠⠐⠱
⣶⡷⡀⠀⠀⢸⣿⣿⣿⣿⢯⣿⣾⡿⢿⣿⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢟⡝⠖⠀⣄⡦⣄⣼⣷⣤⣴⣷
⢥⣆⡥⠀⠀⠰⢼⡿⠿⣿⣶⣿⣿⣷⡮⠭⣯⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠻⠿⢿⠞⣋⣿⣤⠈⠑⣾⣿⣯⠟⣻⣿
⠷⢮⡛⢀⠋⢀⡉⠀⠴⠮⣿⣿⣿⣿⠞⠛⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣬⠉⠁⠄⣀⣉⡟⣻⣠⣄⣤⣍⣳⡟⢩⣽
⡦⣤⣷⠀⠘⠿⣿⣶⠻⠟⠛⣿⣿⣇⠀⠀⢀⣿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⢠⣭⣯⡚⠿⣼⣿⢿⣇⡉⣤⡈⠋
⠐⢛⠑⣛⠊⠤⠋⠀⠃⠀⡀⠀⠀⢀⣊⠀⣼⣿⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣦⣴⣿⡿⠋⠉⠀⣿⣟⣙⡎⠀⠉⠂
⠀⣉⡀⢿⠀⠀⠀⠠⣄⣀⣡⠀⠀⢸⣿⣿⣟⣯⡈⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣼⣿⣿⣿⡿⣀⣀⣄⡃⡁⣻⣷⠀⠀⠄⠐⢠
⣄⡀⠀⣸⠀⠀⠀⠐⣉⠁⠀⠂⠀⣼⠋⠛⣿⣽⡇⣃⠝⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣾⣿⣿⣿⣿⡧⠩⢉⣍⡥⣀⣏⠭⠀⠀⠀⡠⠋
⣿⣿⣿⣿⡄⠀⠀⠀⣼⣧⣀⣤⣾⣿⠟⠀⠋⢛⣻⣷⠐⣶⡗⢻⡆⣶⠃⠀⠀⠀⠀⠀⠀⠐⣶⢲⡷⣳⢚⡦⣿⣟⡁⠀⠷⠀⠘⣉⣿⣿⢼⣿⡷⣢⣼⠷⠦
⣾⢿⣿⠙⣧⠀⠀⣀⣸⣿⡟⡟⠻⠃⣠⣾⣶⢿⡟⠁⠀⠈⠉⠚⠘⠃⠃⠀⠀⠀⠀⠀⠂⠐⠚⠓⠚⠓⠒⠒⢿⣃⡁⣀⡁⠲⠨⣿⣿⣿⣿⣽⡖⠋⢣⣴⡤
⣟⠟⠻⠀⠠⠀⠈⠛⠈⠈⠁⠁⣤⠁⡙⠇⢀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡀⣿⣿⣾⠿⠿⣫⠾⠿⣛⣭⠁⢸⣿⣿
⣷⣈⣫⢏⣡⡀⠄⢦⣤⣄⣄⣐⣟⣄⠠⠏⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠉⢸⠟⠃⢀⣬⠶⣿⣿⣥⣤⠀⠈⠙⢿
⢿⡻⣟⣿⣿⣴⠃⡽⣽⣿⣜⠺⠟⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠛⠯⣟⠈⠕⢻⣿⠀⠀⠀⣘
⠋⢟⣽⣽⢿⣿⣾⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⣤⡶⣶⣿
⣾⣿⡟⡼⢣⡿⢛⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠴⡀⢟⢿⣯
⣾⡇⡼⢁⠾⠀⢺⣿⢯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡶⠞⣤⣧⣿
⣎⣼⢁⡾⡀⠂⡀⠈⠏⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢻⠿⣟⣋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠘⠿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⣷⢰⡶⢀
⠀⠀⣠⣽⣥⡴⠖⠂⢴⣋⣙⣟⣙⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⣾⣖⣦⢠⣞⠹⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣯⣤⠉⠈⢠⣽⣧⡀
⠀⢰⢿⣿⣿⡀⠀⠀⠀⠉⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣟⠉⠀⠉⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣿⣿⣏⡧⢠⣏⣿⣿⣽⠆
⠀⠨⢿⣿⣿⡿⠄⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⣿⣿⡄⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠉⢹⣿⣧
⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⣿⠗⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⣿⡇⠀⠀⠀⣾⠏⣿
⠀⠀⠸⡇⡿⠀⠀⠀⠀⠀⡏⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡏⠀⠀⠀⡟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠘⡇⠀⠀⠀⡟⠀⢹
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⢀⡀⠻⠿⠿⠇⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢃⣄⢈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡔⠀⠀⠀⠀⠀⡠⠠⡟
⠀⠀⢠⣾⣷⡶⠶⠒⠀⠀⣾⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣤⣿⣿⡄⠀⠀⣀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣨⣥⠄⠀⣴⠻⣶⠈⠊
⠀⠠⢯⣿⣿⡇⠀⠀⠀⢠⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⠋⠁⠀⡴⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⣧⠀⠀⠀⠈⢻⣿⡷⣄
⠀⠀⠀⢰⣿⣷⠀⠀⠀⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⢠⣾⣶⢿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡿⠗⠀⠀⠀⣸⣿⡇⠈
⠀⠀⠀⣾⠇⢿⡆⠀⠀⢻⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⠙⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠏⠀⠘⠃⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠛⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠀⠟⠸
⠀⠀⠀⠀⠀⢀⡒⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣴⡌⠋⠋⠁⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠼⠏⣶⠰⡄⠀⠀⠀⣠⡸⠧⠿⠃⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣠⣤⡄⣤⣄⣀⣀⣿⣃
⠀⠀⠀⢰⣾⡟⠛⠁⠀⣺⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁⠀⣠⢶⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠉⠙⢿⡆⠽⠋⠀⠘⣿⣿
⠀⠀⠀⢨⣿⣧⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⢿⣿⡀⠀⠈⣿⣿⡗⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⢿⣿⡿⠇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣇⠀⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢰⡿⣿
⠀⠀⠀⢸⠻⡇⠀⠀⠀⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠈⣿⠀⢹⡏⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠘⡇⠀⠀⠀⠀⠀⠀⡾⠁⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⠀⠈⠀⠘

⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠋⢉⡛⠳⣄
⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⣀⠔⡃⢸⠄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⠿⣿⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⡠⠜⡁⢸⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣽⡇⣿⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠠⠴⠊⠁⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢈⣿⡁⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠠⠖⠋⠀⡟
⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⣸⣿⣇⣿⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠙⠛⠇⢸⠇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠲⣾⣿⣿⣏⢹⡄⠀⠀⠀⠀⠀⢰⡇⠀⠀⠒⠒⢋⠁⣼
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⢾⣿⣿⢸⡇⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢙⡁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢿⣟⡟⠘⠆⠀⠀⠀⠀⢠⠏⠀⠀⠀⢉⣀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣍⣛⠂⣦⠀⠀⠀⠀⢸⠀⠄⠀⣀⣠⡤⠤⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣩⣭⣦⠟⡀⠀⠀⠀⢸⠀⠀⢀⣴⣶⣶⡗⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠛⠷⠟⢻⣽⢸⡇⠀⠀⠀⣮⠀⠀⠉⣡⣤⠴⠆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢠⣽⡎⢷⠀⠀⠀⣿⠀⠀⢀⣀⣠⡴⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⢻⣽⠿⠀⠀⠀⡦⠀⠀⠀⢀⣀⣠⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠠⠤⢙⢦⡇⠀⠀⡇⠀⠀⠘⠛⢉⡁⠀⣿⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣤⣴⣿⣶⡰⣀⢀⠇⠀⢀⣛⢉⣟⣻⡿⠿⠿⢿⣍⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠘⠛⠿⠉⣉⡨⣿⣤⡾⠛⢋⡽⠯⡁⡄⠀⠀⠀⢻⡆⠀⢿⣦⡴⠶⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠷⣾⠿⠛⠉⠀⠀⠀⢹⣆⠀⠀⡿⠀⠀⠈⢸⡇⠀⣾⣿⠁⠀⢀⣀⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⣿⣿⠹⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⡤⠰⢂⣾⠁⢰⣿⡿⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣶⣿⠃⠈⠄⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⢔⣶⡿⠏⠀⣾⣿⠃⠀⢰⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣼⠃⠀⠀⠀⠀⠀⠢⡀⠀⠀⣻⣿⡿⣛⣵⠟⠋⠀⣰⢧⣼⠇⠀⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⠏⠀⠀⠀⠀⠀⢀⢠⣬⣥⣿⣿⣿⣾⡟⠀⠀⠀⢶⢿⣿⠃⠀⠀⣾⣿⣿⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠀⠀⠀⠀⢀⡐⣷⣿⣟⣹⣿⣿⣿⣿⡇⠀⠀⢤⣤⣸⣿⣠⣦⡾⢋⣿⠇⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠐⠈⠙⢿⣿⣿⣿⣿⣿⣿⣯⡧⢀⣘⣻⣟⣿⣁⣡⣽⣷⣿⣿⡷⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢀⠉⠙⠟⠋⣿⣿⣿⣿⣿⣥⣤⣿⣿⣿⣿⣿⠿⠟⠋⠉⠋⠠⣴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⡿⢹⡟⣿⣿⣿⣿⣿⣿⣿⡛⠆⠀⠀⠀⠀⠀⢀⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠁⠸⠃⠟⢻⣿⣏⠀⠀⡱⠀⠀⠀⣠⠖⠀⠀⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠁⠀⠀⢸⢠⣿⣆⠉⠀⠀⡀⣼⠯⠀⠀⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠲⡋⠿⢻⣄⣤⣸⠏⠀⢈⠂⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠃⠀⣿⢹⠆⠈⠀⠀⠀⢸⣷⣿⣇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠙⠘⠆⠀⠀⠠⣴⣿⣿⣏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣉⠀⠠⢘⣧⠞⣤⣶⣰⡆⢸⣿⣿⢏⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠠⣄⡀⠀⠀⠀⠀⠀⠀⡾⠛⠾⠃⣺⣿⢿⡛⣿⣿⣿⣿⣿⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠻⠔⣁⡀⠀⠀⠀⠤⠤⢤⡽⠿⠟⠗⠉⠈⠛⠋⡹⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠺⣥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⠛⠋⠉⢹⣿⡏⠉⠙⠛⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠋⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⣠⣾⣿⡿⢻⣿⡟⢿⣿⣷⣄⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀⠀⠀⣠⣾⣿⡿⠋⠀⢸⣿⡇⠀⠙⢿⣿⣷⣄⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣤⣾⣿⡿⠋⠀⠀⠀⢸⣿⡇⠀⠀⠀⠙⢿⣿⣷⣤⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣏⡀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⣹⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣦⣤⣀⣼⣿⣇⣀⣤⣴⣶⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄

⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣦⡀
⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠙⣿⡀
⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⣸⣧
⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⠀⠀⠀⠀⠀⠀⣴⣿⠃⠀⠀⠀⠘⣿⣇
⠀⠀⠘⣿⣿⣄⠀⠀⠀⢀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⣀
⠀⠀⠀⠹⣿⣿⡆⠀⠀⢾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⣀⣴⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⢀⣾⣿⣿⣿⡆
⠀⠀⠀⠀⢻⣿⣿⣦⣤⣼⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠈⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢠⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣼⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢠⣿⠇⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⠀⣼⠏⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⣿⣷⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢹⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⣿⣿⣿⡿⠀⠀⠀⠉⠁⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣿⣿⠀⠘⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⢿⣿⣿⣿⠀⠀⠘
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣤⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠘⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢻⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣼⣿⠀⢴⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⢸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠀⠀⠈⠻⠇
⠀⠀⡀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠙⠏⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⢸⣿⡋⣀
⣿⣿⠿⠻⡿⠛⠿⢿⡿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠒⠛⠛⠛⠛⠛⠛⠛⢛⣻⠿⠟⠓⠲⠶⠶⠦⢤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⣀⣀⣀⣤⡤⠖⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢠⡞⠋⠉⠉⠉⠀⠀⠀⣼⠿⠶⠶⠦⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠸⣄⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⢻⡍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠛⢷⠀⠀⠀⠀⢸⠋⠀⠉⠉⠉⠙⠛⠛⠛⠲⣶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⣿⡀⠀⠀⠀⣸⠷⠶⠶⠤⢤⣤⣤⣤⣤⡴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢹⡆⠀⠀⢻⣄⣀⠀⠀⠀⠀⠀⠀⢈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⡀⠀⠀⣾⠉⠉⠉⠛⠛⠛⠛⠿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢈⡷⠀⢀⡿⠶⢦⣤⣄⣀⣀⣀⣀⣨⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⣏⠀⠀⢸⣧⣄⡀⠀⠀⠈⠉⠉⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣀⡿⠀⠀⣾⠀⠈⠙⠛⠶⠦⢤⣤⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⣾⠉⠀⠀⣠⠞⠳⢦⣄⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣤⠶⢞⠋⢡⡖⠒⠋⠀⣠⠞⠻⣄⠀⠀⠈⠉⠛⢶⡶⠴⠞
⠀⠀⠀⠀⠀⠀⣿⠶⠟⠛⣋⣀⡴⠚⠛⢧⡀⠀⠈⠳⣤⣀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢹⣄⣠⣞⠉⠻⣷⣄⠀⠀⠙⢶⣄⡀⠀⠈⣿⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠸⡏⣿⠹⣷⣄⠘⣯⠳⢦⣄⣸⡇⠉⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⣇⣿⡀⢹⡌⠳⠾
⠀⠀⠀⠀⠀⠀⠀⢿⡿⠙⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⣀⠀⢰⠈⠉⠉⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣿⣿⣿⡦⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢸⡿⣟⣥⠇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⡾⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣽⠇⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣟⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⡏⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡀⢀⡀⠀⣠⣀⡄⠀⣀⣀⣀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣔⣾⢿⠿⢛⣡⠶⠞⣋⡵⣺⣵⡿⢟⡿⣛⣥⠾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠐⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠛⠉⣁⠀⠘⢉⠀⠖⠊⠥⠾⠛⠳⠾⠻⠞⠋⠁⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠐⠚⠂⠴⠚⠉⠀⠀⠈⠁

⠀⠀⠐⠀⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀
⠀⠀⠀⠀⠠⠀⢤⠀⠀⠀⠀⠛⠀⠛⠀⠘⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘⡄⠀⠀⠀⠘⠂⠐⠃⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴
⠀⠀⢀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠁⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⣁⣀⠤⠁
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠃
⠀⠀⠘⠀⠐⠀⠀⠀⠀⠀⠀⠁⠐⠒⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⣋⠀⠀⠀⠀⠀⡄⠀⠈⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⡀⠀⠀⠐⠀⢀
⠀⢀⠐⠀⠈⠀⠀⠀⠀⠀⡀⠀⣾⠀⠈⢿⠀⠀⠀⠀⠿⢤⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠲⣠⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠋
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠒⠠⣇⠀⠀⠀⠀⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠠⠄
⠀⡠⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠋⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄
⠘⢇⠚⠗⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠐⣧⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠙⡃⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠐⠀⢠⠀⠀⠀⠀⠀⠤⠀⠠⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠉
⠀⢀⠀⠀⠀⠀⠀⠎⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡄⠀⠀⠐⠒⠀⠀⠐⠀⠀⠀⠃⠀⠀⣠⣶⢶⣦⡀
⠀⠘⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠔⠃⣀⢀⢨⠀⠀⠀⠀⢄⢀⠀⠄⠀⠠⣿⡍⣈⣿⡇
⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡘⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡿⠚⣿⠀⠀⠀⠀⠔⠀⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠙⠈⠀⠀⠙⠈⠀⠀⠀⠀⣀⣴⠂⠀⠀⠀⠀⠈⠙⠶⠛⠃⠀⠀⠀⠀⠀⠤⠀⠀⠀⠂⠀⠘⠃⢠⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⡀⠈⠙⠃⠀⠀⠀⡀⠀⠄⠀⠀⢀⡀⠄⠒⠋⠕⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣇⢠⠤⣀⡈⠀⢆⠀⠀⠐⠀⠀⠁⠊⠐⠁⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣰⠚⢢⠀⠀⠀⠀⠀⠀⠀⠸⡎⠀⠀⠁⠀⢸⠀⠀⠀⠈⠃⠀⣤⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⠈⣋⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠖⠚⠛⠋⠉⠉⠙⠛⠓⠒⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣶⣤⣀⡀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣋⣥⣶⣷⡄⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣵⣾⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠿⠿⢛⣋⣭⣤⣶⣾⣿⣷⠀⠀⠀⠀⠀⠀⠘⣆
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡛⠛⠛⠿⠿⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⢹
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠹⣿⠿⠿⠟⢛⣋⣉⣭⣤⣴⣦
⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⢴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⡇⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢻⣿⣿⣿⣿⠿⠿⠿⠛⠛⠋⠁
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣉⣤⣤⣶⣶⣾⣿⣿⣿⣿⡇
⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀
⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠀⠿⠿⠟⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⣸
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠋⠀⠀⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⠤⠤⣤⣤⣤⣤⠤⠤⠶⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢲⡜⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠙⢦⠿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠙⢷⣦⣀⠀⠀⠀⠀⠀⠀⢰⡄⠹⡆⢡⡀
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣀⠀⠀⠀⠀⠙⢿⣷⠀⠀⠀⠀⠀⠸⣷⠀⠛⠛⢿⣄
⠀⢀⣾⡿⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣷⡀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⢻⣧
⠀⠸⣿⠁⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⢻⣷
⠀⠀⢻⡇⠀⠀⠿⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠻⣦⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠈⣷⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠙⣷⣶⠀⠀⠀⠈⠻⢷⣄⠀⠀⠀⠀⠀⢹⣷⡄⠀⠀⠀⠀⢹⣿⠀⠀⠀⢹⡆⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢿⡄⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠘⣷⣦⠀⠀⠀⠀⠻⣷⠀⠀⠀⢸⣿⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⢻⣄⠀⠀⢻⡯⠀⠀⠀⠀⠀⠹⣇⠀⠀⠸⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⠞⠉⠀⠀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠁⠀⠀⠈⣿⣆⠀⠘⣿⠀⠀⠀⠀⠀⠀⢿⡄⠀⢠⣿⠀⠀⠀⠀⠘⡇⠀⠀⠀⣴⠋⠀⠀⠀⡸⠁⠏
⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠈⢻⣤⣄⠀⠀⠀⠘⢿⣧⠀⢿⡄⠀⠀⠀⠀⠀⠘⣷⡀⢸⣿⠀⠀⠀⠀⠀⡇⠀⠀⢰⠁⠀⠀⠀⣼⣧
⣼⡏⢁⠉⢿⣆⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠈⢿⣧⠘⣷⠀⠀⠀⠀⠀⠀⢹⣧⣾⣿⠀⠀⠀⠀⠀⢰⡄⠀⣼⠀⠀⠀⠀⢸⣏
⢻⣧⠘⣷⣄⠙⢷⣀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠈⢿⣆⢿⡇⠀⠀⠀⠀⠀⠀⢿⣿⠟⠀⠀⠀⠀⠀⢸⣧⢀⣿⠀⠀⠀⠀⠈⣿
⠀⠹⣷⡌⠛⠃⠀⠙⢷⣤⡀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⡏
⠀⠀⠈⠻⣆⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠻⣆⠀⠀⠀⠀⠀⠹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣴⠁
⠀⠀⠀⠀⠘⢷⣄⣀⠀⠀⠀⠙⢿⣦⠀⠀⢿⡆⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⢠⡀⠙⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠻⣧⡀⢸⣿⡄⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⢦⡀⢸⠇⠀⠀⠰⠆⠀⢸⠉⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠈⢿⣾⡿⠛⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⢻⣆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠈⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠛⠻⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⢀⣴⡾⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣦⠀⠀⠀⣀⡀⠀⢀⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⣿⣿⣿⣿⣿⣿⡆⢰⣿⣿⣿⣷⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⠀⢿⡿⠿⠿⠿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠘⠿⢿⡿⠛⠛⠋⠻⣿⣿⡁⠀⣠⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣿⡟⠀⠀⠐⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⠀⠀⡈⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⣀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢀⣿⡏⠀⢈⣉⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠿⣿⣿⣿⠃⠀⣼⣶⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠶⢦⣤⣤⠀⣠⣿⣤⣀⣀⣠⣤⣶⣶⡄⠀⠀⠀⠀⢼⣿⡿⠛⣶⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢀⣽⣇⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠸⣿⡿⠈⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠻⠁⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡤⡆⢲⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⠈⠛⠛⣿⡿⣿⢿⣿⠿⠿⠟⢿⠀⠀⠘⠁⢻⠻⣿⣿⣿⣿⣿⣿⣿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠉⠙⠟⠀⠀⠀⠫⠀⠀⠀⠏⠀⠀⠀⠘⠀⠀⠀⠀⡞⠀⠈⠙⠿⠟⠹⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⣾⣏⠀⠀⠀⠀⠀⠀⢀⠀⣄⠀⠀⠸⣧⡀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⢠⡆⠀⢠⣾⡟⠀⠀⢰⣿⣟⣄⠀⠈⣁⣠⣤⣿⣾⠉⣴⢃⣾⣉⣛⡀⣧⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⡀⠀⣀⣿⡀⠾⢛⡇⣀⣀⡻⣃⠀⡄⣽⣿⠻⠘⡟⠆⠙⠛⢻⡾⠉⠿⠟⠋⠛⠉⠉⠛⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠒⠢⠤⠴⠏⡉⠋⠐⢩⠂⠾⠛⠟⠛⠻⠟⠛⠋⠁⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢿⡷⠛⣾⢻⣿⣷⣶⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⠿⠛⠛⠀⢀⣤⣽⣶⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡋⠉⠀⠀⠀⢰⣿⣷⣼⣿⣶⣿⣗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⣏⣿⡀⠀⣿⣿⠛⣛⣻⣯⡉⠈⠛⠙⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⢿⣿⣿⠛⠋⠉⠉⠉⢈⣙⣳⣤⣄⡀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣿⣷⠶⠶⠒⠛⠋⠉⠉⠁⠀⠉⠙⠓⠶⣬⣙⣧⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣣⣀⣀⣀⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⠖⠚⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡿⣿⢻⣿⣟⠛⠛⣿⣿⣿⠿⠿⠛⣫⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣱⣿⡿⢿⣿⣏⣁⣤⠋⠀⠀⣠⡾⢋⠄⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣷⠶⠚⠋⣉⣴⣁⣠⣤⡾⠋⢀⠎⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⣿⡇⣠⡴⠟⢉⡼⠋⣰⠟⠀⢠⠃⡜⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⣿⣏⠀⣴⠟⠁⣼⠏⠀⣠⢃⡜⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢞⢿⣿⣯⣀⣼⡏⠀⡰⢃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠭⣟⣾⢯⣟⣻⢷⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉

⠀⠀⠀⢀⡴⠒⠋⡟⠙⠲⢄⠀⠀⠀⠀⠀⢀⣤⠖⠛⣿⠛⠳⢦⡀⠀⠀⠀⠀⠀⣠⠶⠛⢻⡟⠛⢶⣄
⠀⠀⢠⠏⠀⠀⠀⡇⠀⠀⠀⢳⡀⠀⠀⢀⡞⠁⠀⠀⣿⠀⠀⠀⠹⡆⠀⠀⠀⣾⠁⠀⠀⢸⡇⠀⠀⠙⣧
⠀⠀⢾⠀⠀⠀⢀⣧⡀⠀⠀⠀⡇⠀⠀⢸⠁⠀⠀⢀⣿⣄⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⢀⣼⣇⠀⠀⠀⣻
⠀⠀⠸⡄⢀⠔⠁⡇⠙⢦⡀⣰⠃⠀⠀⠘⣇⢀⡴⠋⣿⠙⢷⡀⢠⠏⠀⠀⠈⢷⣀⣴⠟⢹⡏⠳⣄⢠⡿
⠀⠀⠀⠘⠣⣄⣀⣇⣀⣠⠟⠁⠀⠀⠀⠀⠈⠻⣤⣀⣿⣀⣠⠽⠋⠀⠀⠀⠀⠈⠻⣧⣄⣸⣇⣠⡼⠟
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣦⣤⣀
⠀⠀⢀⣴⠟⠉⠀⣿⠀⠉⠻⣦⠀⠀⠀⠀⣴⡿⠋⠁⣿⡇⠉⠻⣷⡀⠀⠀⠀⣴⡿⠛⠉⣿⡏⠉⠻⣷⣄
⠀⠀⣾⠃⠀⠀⠀⣿⠀⠀⠀⢹⡇⠀⠀⣸⡏⠀⠀⠀⣿⡇⠀⠀⠸⣷⠀⠀⢸⣿⠁⠀⠀⣿⡇⠀⠀⠘⣿⡄
⠀⠀⣿⡀⠀⣠⣾⣿⣦⡀⠀⢸⡟⠀⠀⢻⡇⠀⢀⣴⣿⣷⣄⠀⢠⣿⠀⠀⢸⣿⠀⢀⣴⣿⣿⣦⡀⢀⣿⡇
⠀⠀⠘⢷⣾⠋⠀⣿⠈⠻⣦⡿⠁⠀⠀⠈⢿⣶⡟⠁⣿⡏⠻⣷⣾⠏⠀⠀⠈⢿⣶⡿⠋⣿⡏⠙⣿⣾⠟
⠀⠀⠀⠈⠙⠷⠶⠿⠶⠟⠋⠀⠀⠀⠀⠀⠀⠙⠻⠶⣿⠷⠿⠛⠁⠀⠀⠀⠀⠀⠙⠻⠷⣿⣷⠿⠟⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣛⣙⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⣠⠞⡆⠀⠀⠀⠀⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠉⠉⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣺⠋⠉⢤⡀⡇⠀⢀⡴⠚⠁⠈⠙⠉⠉⡉⠙⠓⢦⡀⠀⣼⠓⠤⢤⣄⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠳⠀⠀⠀⢀⡇⠀⢸⠀⠀⠀⢹⠀⣿⠀⡇⠀⠀⠀⡇⠀⢼⠰⠂⠀⠘⠲⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⢉⡇⠀⢸⠀⠀⠀⠙⠀⠛⠀⡃⠀⠀⠀⡇⠀⣹⡄⠀⠀⢰⠚⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⠛⡟⠉⠀⠀⠈⠳⣄⣀⣀⣀⣀⣀⠁⢀⡤⠞⠁⠀⠸⣄⣀⣠⡏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢾⣀⣀⡷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⠀⠀⠀⠀⠀⣀⣼⠀⣧⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⣀⣀⣀⠀⠀⠹⡄⠀⠀⠀⢀⡤⢤⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⠓⠒⠊⠉⠱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⠃⠀⠈⠳⣄⠀⡇⠀⠀⣠⠋⠀⡼⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⡜⠉⠙⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠣⡀⠀⠀⠈⢻⣇⣠⠚⠁⠀⣰⠃⠀⠀⣀⣀⣀⠀⢠⡏⠀⡞⠀⠀⡰⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠢⣀⠴⠋⠈⠀⠀⣠⠞⠁⠀⠀⠀⠹⣄⠈⠣⣀⣳⡜⠳⢤⡴⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠢⡀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⠉⠀⠀⣰⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡴⠃⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠈⠀⠨⠰⠡⢘⠃⠀⠘⠀⠈⠀⠈⠘⠀⠂⡀⠒⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣆⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡿⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣟⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣯⣾⣿⡿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣵⣿⡿⣯⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⢾⣿⣿⣿⣿⣿⣿⣯⣿⣿⡟
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣫⣾⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣽⣿⣟⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣻⣿⣿⣿⣿⠿⠿⠟
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣛⣽⣾⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠻⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⢇⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⢿⣻⣽⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⡿⣽⣿⣆
⠀⠀⠀⠀⠀⢠⣤⣤⣾⣿⣿⣿⢿⣻⣽⣾⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢯⣿⣿⣿⣿⡿⣿⠆
⠀⠀⠀⠀⠀⢸⣿⣿⣟⣿⣽⣾⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣻⣿⣯⣾⣿⣫⣿⠉
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣾⣿⣿
⠀⠀⠀⠀⢠⣿⣿⣻⣿⣿⣿⣿⣿⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣽⣿⣿⣧⣿⠟
⠀⠀⠀⠀⠘⠋⠈⠉⠛⠛⠛⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣀⣀⣀⣀⣹⡆⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏⠉⠉⠉⠉⣷⠀⠀⠀⠀⠀⠀⣿⠿⠒⠀⢠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⣀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠛⠒⠦⠤⢴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⣿⡄⠀⠀⣸⠇⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⢀⣠⡟⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣀⣤⡶⠾⠛⠛⠛⠙⠛⠶⠦⣤⣀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣱⣿⠟⠿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⡀⠀⢸⣿⠛⠳⠶⣤⣀⣀⣀⡠⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⣿⡀⠙⠛⠛⠁⠀⠀⢠⡿⢿⣯⡁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠘⢿⣤⡀⠀⠀⠀⠀⠘⠁⠀⠈⠻⣶⣤⣤⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠈⣿⡿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠈⠻⣇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⢷⣄⠀⠀⢀⣼⣇⠀⠉⣻⣷⣤⡀⠀⠀⠀⠀⠀⠙⢿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠖⠺⢿⣦⠀⠈⠉⣡⣄⣰⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⢷⡀⣴⣿⣿⣿⣿⣿⠛⢿⣿⡇⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠙⠿⢿⡟⢉⣿⠏⠀⠀⠙⠇⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠘⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⠀⠀⠀⠀⢀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⣀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⡿⠿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠛⠟⠛⠻⣿⠀⠉⠁⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⡀⠀⠀⠀⢸⣇⡗⠀⠀⢀⣀⣴⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠻⠶⠟⠛⠉⠁

⠀⠀⠀⠀⠠⣦⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⢤⡤
⠀⠀⠀⣭⣿⣽⣯⣿⡿⠛⢑⣢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡂⠙⠻⣿⣯⣯⣽⣯⡇
⠀⠀⠀⣯⣿⡟⠋⠀⠀⢠⣿⣭⣿⣼⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣷⣼⣯⣽⣦⠀⠀⠘⠛⣿⣯⠇
⠀⠀⠀⠘⣿⣽⡀⣰⣿⣛⣛⣿⣍⠉⠋⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠋⠋⢉⣽⣟⣛⣻⣧⡀⣸⣽⡟
⠀⠀⠀⠀⠸⣿⣿⡋⢉⣁⣀⣤⠬⠿⠶⢤⣀⠙⢷⡄⢀⠀⠀⣀⠀⣴⠟⣁⣠⠴⠾⠯⢤⣄⣀⣉⠙⣿⣿⡟
⠀⠀⠀⠀⠀⠹⣿⡏⠉⠉⢀⣀⣀⣀⣀⣀⣈⣉⡓⣿⣾⠀⠀⣿⣾⠗⣋⣁⣀⣀⣀⣀⣀⣀⠈⠉⠙⣿⡟
⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⠟⢿⣿⣿⣿⠿⢖⡾⣿⠂⠀⣿⣿⡒⠾⢿⣿⣿⡿⠟⢿⣿⢿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⣿⣯⣭⠜⠁⢀⣴⠏⠀⠹⠀⠀⠛⠀⡙⢦⣄⠀⠙⡬⣽⣿⡿⢿⣯⣧
⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⣉⡤⢶⡤⣶⠟⢁⠌⠀⠈⠀⠀⠉⠀⠀⢄⠙⢷⡦⣴⠦⣌⠙⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠞⢉⡴⠋⣴⠃⢀⠊⠌⠀⠀⠀⠀⠀⠀⠈⡌⢄⠈⢷⡈⢷⡌⠙⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣾⣁⣼⠃⢠⢃⡎⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢆⠈⢷⣀⣹⣶⣿⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⣾⢯⣳⣿⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣟⣯⢿⡾⠝⠋
⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠂⠘⠃⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⣄⠀⠀⠀⠀⠘⢰⡆⠀⢀⡀⢸⣿⣿⢿⣿⣿⢿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆
⣿⠀⠀⠀⠀⢀⠈⠁⠐⣾⣷⢸⣿⣿⣿⣿⣯⣿⣿⣿⡇⣤⢼⣶⣷⣠⣶⣶⡆

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



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⠻⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠠⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠐⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⠁⠀⠉⠊⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡀⠀⠀⣿⣷⡄⢳⣌⡁⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠶⢆⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠁⠀⠀⠀⣀⠤⠀⠀⢄⣤⢀⢀⣿⣿⣿⣿⣿⡧⠀⠄⢐
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢠⠒⠄⠀⣀⠤⠒⠈⠁⠀⠒⠀⠀⠀⠀⠀⡾⡀⠀⢀⡀⠄⠒⠊⠁⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠉⠉⠛⠓⠀⠀⠋⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢀⢀⠀⡶⢷⠀⠻⠀⠀⢰⣦⣤⠾⣿⣦⡀⡀⠄⠠⠄⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣤⠆⠀⠀⠀⠀⡀⠀⢀⠤⠚⢀⡡
⠀⠀⠀⠀⢠⡤⠊⠁⠀⠃⠘⣇⠁⠀⠀⡀⠀⠀⣴⣿⣿⣷⠤⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠊⠓⢄⢠⠔⢨⠁⠀⠁⣰⣮⣿⠃
⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠁⠹⣇⠀⠀⡆⠀⠀⠐⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⣹⡜⣾
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢀⡀⠜⠃⠀⠘⡔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠙⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣠⠀⠀⡀⣾⢀⡇⢸⡟⠁⣔
⠀⠀⠀⠀⠀⠀⢸⡴⠔⠊⠁⠀⠀⢀⠘⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠃⠀⠀⠨⣇⡵⠋
⠀⠀⠀⠀⠴⠊⠁⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢐⠉⠱⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢉⠭⠋⠁⢀⠀⠀⠀⠀⡀⢀⡁⠀⠀⣘⣥
⠀⠀⠀⠀⠉⠑⠲⢦⡤⢤⣄⣀⡘⠀⠀⠀⠀⠀⠀⠀⣀⠀⣻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠤⠤⠤⠰⠚⢁⠀⠀⠀⠋⣹⠀⠸⠀⠀⠠⢖⡼⢛⠟⠯⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢶⠖⠀⠀⠂⠀⠤⠄⣀⡀⠀⠀⠧⢽⣾⣋⣀⣀⡀⢀⡀⠄⢀⣤⡞⠀⠀⠀⠀⠀⠀⡜⠀⠀⢤⠀⠀⠀⢸⢂⠔⠃⠈⠛⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠇⠙⠄⠀⢳⢂⢤⡀⠀⠀⠉⠋⠀⠊⠋⢀⠉⠀⠀⣀⡲⠀⢀⠇⠀⢀⠀⢀⣀⣀⠟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⢀⡤⠖⣠⣤⠲⡍⠱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⢳⡴⠋⢱⠀⠀⠀⠀⠀⠀⠸⠀⢠⣀⣀⡀⠀⠀⠀⣾⠔⠉⠀⣾⡕⠊⠀⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⡄⡀⢘⠀⣠⠀⣀⢀⣶⠂⡤⣍⡯⣛⣦⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠃⠐⠠⠟⠊⠀⠀⠀⠀⠐⠛⠉⢽⣫⠔⠋⠀⠀⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠛⠿⠀⠘⠛⠛⠛⠃⠈⠃⠛⠀⠛⠒⠛⠚⠛⠛⠻⠟⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⢀⡴⠃⣠⠞⠁⠠⠤⢶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⡴⠉⠀⢠⠞⡀⠀⢠⣾⠁⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠋⢀⡤⡶⠟⠁⢀⣠⣿⡿⢊⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⢠⣤⠾⣿⠎⢀⣠⣞⠶⠿⢿⣖⣋⠄⠀⠀⠀⠀⢀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⢀⣡⡖⠋⠃⢀⣠⣶⡿⣋⡥⠖⠊⠀⠀⠐⠓⠒⢠⣄
⠀⠀⠀⠀⠀⢀⡤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠚⢋⣩⣤⢒⣈⣀⣉⣭⣿⣦⠴⠀⠀⠀⠀⢀⡠⠞⠉
⠀⠀⠀⠀⠐⠉⠀⣠⡔⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠎⢐⣚⣉⣉⣉⣉⣥⡤⠔⠊⠁⠀⠀⠈⠉⠑⣒⠦
⠀⠀⠀⠊⠀⠀⢎⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠛⠛⠋⠙⠛⠉⣉⣉⣠⣄⠀⠀⠀⠠⠄⠊⠁
⠀⢠⠀⠀⡰⠛⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠛⠛⠒⠲⢶⣶⣶⠧⠤⠐⠒⠉⠁⢠⡤⠔⠚
⠀⠀⠀⠀⠀⠊⠁⢀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⢤⣛⠛⠷⠶⢤⣴⣮⡷⠒⠊⠉⠁⠀⠘⢧⡀
⠀⠘⣦⡀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄⢴⡛⠯⢤⣤⣤⣬⡄⠀⠀⢀⠀⢠⡄⠒⠛⠉
⠀⠀⠀⠹⣄⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠐⠒⠠⣐⠂⠀⢀⣀⣉⣶⠀⠤⠤⠄⠀⠀⢀⠠⠶⠂
⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠉⠓⢦⣤⣽⣍⣁⣀⡀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⢧⠀⠐⠋⡍⠀⠀⢀⠀⠰⣄⠈⠀⢤⢦⣬⣭⣑⠀⠀⠀⠀⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠲⠄⠀⢻⣬⡉⠰⠶⢦⣤⣤⠤⠀⠁⠀⠀⢲⡀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢠⠄⠀⠄⠈⢻⠷⢾⡶⣤⣄⠀⠀⠈⢻⡈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢱⢂⡀⠉⠁⠲⠤⢤⣤⣀⣨⣿⣬⡁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠪⣀⠀⠉⠐⣉⠓⠶⠆⢈⣀⠤⠄⠈
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠘⢆⡈⢀⡈⠉⠙⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠐⠲⣮⢡⣈⡉⠙⢛⡛⠁⠀⠀⠾⠛⠋⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣒⠋⠀⠉⠙⢻⡿⣿⠦⡴⠆⠄⠠⠴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⡙⠋⠉⠳⣶⣶⣶⣀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⣄⠺⣑⠈⠻⣶⣶⡿⢏⠉⠉⠳⡉⠉⠚⠳
⠀⠀⠀⠀⢀⡀⢤⠘⢦⣈⠻⢶⣼⣏⠉⠰⣤⣤⠦⡉
⠀⠀⠀⣧⡘⣧⡀⠷⣾⡿⣿⣆⠈⠪⣷⡀⠀⠀⠈⠈⠙⠂
⠀⠀⠀⠸⠏⠀⠛⠀⢱⡙⠈⢾⡆⠀⠀⠈⠒⠄⡄
⠀⠀⠀⠀⠀⠀⠰⡀⠀⠱⣄⠀⠈⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠠⡀⢀⠄⠀⠀⠈⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢀⠠⠐⠂⠀⠐⠀⠠⠀⣰⠔⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⢀⣠⣴⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠔⡡⠐⠈⠉⠀⠈⠁⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠲⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⢀⠔⠀⠀⠀⠈⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠛⠻⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⠿⣿⡇⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⢡⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣀⣾⡆⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠟⠋⠀⠈⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠬⣈⢶⡤⣤⠤⠤⠐⠁⠀⠀⠀⠀⢸⢁⡨⠃⠀⠀⠀⠀⠀⠀⠑⠤⠀⠀⢤⠐⢒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠈⢃⢻⣃⠆⠀⠈⠐⠀⠀⠀⠀⠀⡠⠃⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠉⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢰⣦⠒⠢⠄⠀⢀⣀⣀⠠⠂⠀⣘⠀⠑⠀⠁⠀⠀⠆⢀
⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠀⠤⠀⢀⡠⠀⠉⠀⠈⠁⠐⠂⠀⠤⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠦⠀⢂
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠡⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠢⡀⠐⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⡄⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⡨⠼⣀⣀⣀⣀⣀⣀⣀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠀⢸⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠐⠄⡼⡁⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣀⣀⣸⣅⣀⣀⡕⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡇⠉⠁⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠲⠘⠛⠑⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠿⠷⠶⠿⠇

⣀⡀⣀⢀⣀⣀⣀⣀⡀⣀⣀⢀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⡄⢀⣴⣿⣷⣤⡄⠀⠀⠀⠀⠀⣤⣶⣶⣶⣄⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣼⣿⣿⣿⣿⣯⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠟⠛⠛⠛⠓⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⠘⣿⣿⣿⣿⠁⢸⣿⣿⠋⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢹⣿⣿⡏⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣿⣿⡿⠶⢿⣿⣿⣿⡿⠿⢿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠛⠋⠉⠁⢸⣿⡟⣠⣶⣿⣿⠋⠛⠁⠀⠀⢻⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⣿⣿⣦⠿⡿⠿⠃⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⡛⠋⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠟⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠛⠋⠁⠀⠀⢸⣿⣿⡇⠀⠀⠈⠉⠻⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠋⢸⣿⣿⡇⠙⢿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧⠀⠀⢠⣶⣿⣿⡿⠋⠀⠀⢸⣿⣿⡇⠀⠀⠙⢿⣿⣿⣦⡀⠀⠀⣼⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣴⣿⣿⡿⠋⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠙⢿⣿⣿⣦⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣤⣀⡀⠀⠀⢸⣿⣿⡇⠀⠀⢀⣀⣤⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉

⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⣿⣿⣿⣿⣿⠀⠀⠀⢠⣸⡇
⠀⠀⠀⣿⠀⠀⣀⢀⣼⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⠾⣿⣿⣿⣿⣦⣄⠀⣾⣿⡇
⠀⠀⠀⣿⣶⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢀⣿⣿⣿⣿⣇⠀⣀⣄⠀⠀⠀⠀⠀⠀⢸⣿⠃⣠⣿⣿⣿⣿⣿⣿⡇⢿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⢿⠿⠉⠹⣿⣿⡿⠉⢹⡿⠿⣷⣦⣄⡀⠀⠀⢸⣿⢀⣿⣿⣿⣿⣿⣿⣿⣧⠘⣿⡇
⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣠⣾⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣽⣿⣿⣷⡄⢨⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠿⠛⠛⠃⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⢿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⣿⣿⣿⣿⡇⠀⠀⢹⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⡇
⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⡇
⠀⠀⠀⣿⣿⣿⣿⣿⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⡀⣀⣀⣀⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⡀⣀⢸⡇
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠉⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠛⠃⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠋⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣤⣼⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠃⠀⠀⠀⠈⠉⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⠀⠀⠀⠀⠐⠟⣻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠄⠀⠛⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⣴⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠁⠀⠀⠀⠀⠀⠀⣩⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢺⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠳⡄⠐⠂⠀⢀⣤⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⠀⠈⠁⢀⠀⠀⢀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠶⠿⠿⣶⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡋⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠉⠀⠀⠀⠀⠀⠈⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢀⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⣽⣿⡇⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡁⠀⠀⢀⠀⣀⣠⣿⣿⣧⠀⠀⢀⣴⠋⠀⢀⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡇⠀⠀⢀⣀⣀⠈⠉⠙⠿⠗⠒⠛⠛⠃⠀⠀⠀⣀⣀⣀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣰⡿⠃⠀⠀⠙⠛⠿⢿⣿⡷⠶⢶⣿⣿⣀⣀⣀⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⢀⡴⠞⠉⠙⠛⣿⠃⠀⠀⠀⠀⠀⠀⢰⠉⢡⠀⠀⠉⠉⠛⠛⠛⠉⠉⠀⠉⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⢰⣿⠁⢀⠀⠀⠀⢻⣆⠘⣤⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏
⠀⠀⠀⠀⣿⡏⢠⠛⠀⠀⠀⠀⢻⣦⡈⠙⠀⠀⠀⠾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡿
⠀⠀⠀⢀⡞⠀⣼⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣀⠀⠀⠀⠁⠀⠀⠀⠰⣿⡇
⠀⠀⠀⣸⣧⠘⣿⡄⠀⠀⠀⠀⠀⠙⠛⢿⣿⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⣀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⣿⡇⠀⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣷⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⡇⠀⠀⢈⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠘⣷
⠀⠀⠀⢿⣿⠀⠀⠈⢹⣿⣿⡄⠀⠀⢀⠀⠀⠀⠘⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣷⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⠀⠀⠀⢿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠈⢻⣿⣷⡀⠀⣼⠄⠀⠀⠐⢊⡿⣇⠺⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠏⠀⠀⠀⢀⣷
⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⢻⣿⣷⡀⠙⡀⠀⠀⠀⠈⢻⣿⠃⠠⡟⢻⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢀⣿⣿⡀⠀⠀⠀⠀⠛⢻⣿⣿⣄⠀⠄⠀⠀⠀⠀⢿⣇⠀⠐⣿⣩⡿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄⣀⣀⣀⣠⣾⡏⠀⠀⢸⡟⢠⣾⣿⣿⣿⣿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏
⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⢿⣿⣿⣿⡿⠋⠀⠀⠀⠛⠀⣾⣿⣿⡿⠋⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠛⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⢸⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡇
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢸⣿⣿⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⢹⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⠇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⡿
⠀⠀⠀⠀⠀⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⣶⣤⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣤⣴⣦⣾⣿⣿⢇⣀⠀⠀⣰⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣠⣴⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠛⠿⠿⠿⣿⣶⣶⣤⣶⣾⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠀⠀⠍⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣦⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⡝⢧⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠁⠀⠙⠙⢵⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠉⠙⠶⣄⠀⠀⠀⠀⢀⡾⢁⣴⠖⠠⡄⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⣀⠠⠀⠰⣦⠘⣇⠀⠀⠀⢸⡇⢸⠃⠀⠀⠘⠃⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⢀⣤⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⣿⠀⠀⠀⠘⣇⠘⢧⡀⠀⠀⢀⠀⠀⠀⠀⢀⣟⣹⠆
⠀⠀⠀⠀⠀⠀⢸⣅⣸⡅⠀⠀⠀⢶⡀⠀⠀⠀⣰⠏⣰⠇⣀⣀⣠⣤⣼⣷⣄⡉⠛⠛⠉⠀⠀⢀⣴⡿⢿⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠙⣇⠀⠀⠀⠀⠙⠓⠖⠛⢁⣴⠿⣏⣉⣭⠴⠶⠒⠛⣿⠉⠛⠓⠒⠚⠛⠉⠹⠧⢾⡅⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣄⡀⠀⠀⢀⣠⡴⠛⠁⠀⠘⣯⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⡇
⠀⠀⠀⠀⠀⠀⢀⡴⠋⡟⠉⣯⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⡇
⠀⠀⠀⠀⠀⣰⠏⠀⢀⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡗⢦⡀
⠀⠀⠀⠀⢸⠇⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠶⠃
⠀⠀⠀⠀⠘⣦⡀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣏⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⠴⠶⡟
⠀⠀⠀⠀⠀⣠⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠶⠚⢹⡇⢸⡏⠙⠛⠛⠛⠋⠉⢉⣉⡉⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢷⡴⢻⡶⠶⠶⠶⠚⠛⠛⠋⠉⢉⣉⠁⠀⠀⠀⠀⣼⠁⣼⠃⠀⠀⠀⢾⠛⠋⠉⠉⣿⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡴⠶⠚⠛⠋⠉⣹⠀⠀⠀⠀⠀⡿⠀⡿⠀⠀⠀⠀⠘⡇⠀⠀⠀⢸⣄⢀⣀⣠⡤⠞
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢻⡀⢀⣀⣀⣤⡇⠀⣧⣤⣤⣤⣤⡤⠟⠀⠀⠀⠀⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠠⣿⣤⣠⣤⣤⣼⠇⠀⠀⠀⠀⠈⠉⠉⢹⠉⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⠤⠶⠦⣄
⣠⠞⣿⠶⣄⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢻
⠿⠼⠃⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣠⡀⠀⢀⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣁⡿⠀⢻⣉⣻⠟

⠀⠀⠠⠤⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠐⠂⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢒⡀⠀⠀⡄
⠀⠀⠀⠀⠒⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⢀⠔⠂⠉⠉⠑⢦⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡀⡀⠀⠠⣄
⠀⠀⠀⠀⠀⢮⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣏⠉⣹⠻⣄⣀⠷⡄
⠀⠀⠀⠀⠀⠈⠢⢄⡀⠂⣀⠠⠃⠀⠀⢀⡄⢲⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠠⣀⠀⠄⠀⠳⢼⡀⠀⠣⡉⠁⠀⢀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⠀⠀⠀⢐⠡⠋⠈⠀⠀⠀⠀⠁⠰⠀⣄⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⢄⢰⠀⠙⢦⡀⠈⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠑⠒⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⢌⣑⣒⣒⣈⣧
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡒⠂⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠉⠉⠐⠢⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⣀⠤⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⢸⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⡀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⢧⠠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠤⠃⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠒⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⢤⣄⠀⠀⠀⠀⣀⡠⣶⠶⠲⠒⠲⠖⠂⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⡗⠀⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣠⡄⣠⠀⢀⣤⠴⠟⣴⠖⠂⢤⠈⠁⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢶⡦⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢤⡶⠞⠉⠀⠄⠀⢀⠘⠀⠈⠀⠀⠈⢿⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⡀⠀⢠⠆⠀⠀⣼⡃⠀⠈⣿⠥⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⢰⢁⡾⠉⠀⠀⠂⢸⡇⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢸⣀⠀⠁⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣿⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣄⣀⣀⣿⣀⣀⣠⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⡀⢀⣠⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣄⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠋⠉⠛⠛⠛⠻⠛⠛⠛⣉⡉⠀⠀⠉⠉⠙⠛⠛⠛⠻
⠀⠙⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⢈⣿⡟⠳⣿⣿⣿⡤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠉⠁⠁⠀⠈⠁⠐⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣀⣿⣿⡀⠈⠉⠁⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣽⣯⣭⣽⣿⣿⣿⠿⠛⢉⣽⣿⣿⣿⣿⡶⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠿⠋⠁⣠⣖⣭⣶⠿⠟⠋⠉⠀⠐⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠋⠁⠀⣒⣛⠋⠍⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⠗⠂
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠈⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠰⠀⠀⠀⠈⠙⠛⠛⠿⠿⠿⠿⠿⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠐⠉⠉⠉
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠐⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠂⠐⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⢿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⠋⠁⢀⣠⣤⣶⣾⣿⣿⣿⠀⠀⣿⣿⣿⣷⣶⣤⣄⡀⠀⠙⠻⣿⣷⣄⡀
⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⢀⣤⣾⣿⠿⠛⠉⠉⠀⣿⣿⠀⠀⣿⣿⠀⠉⠉⠛⠿⣿⣷⣤⡀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⣠⣾⡿⠋⠀⣠⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣄⠀⠙⢿⣷⣄
⠀⠀⠀⣴⣿⠟⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀⠀⠻⣿⣦
⠀⠀⣼⣿⠏⠀⣠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠹⣿⣧
⠀⣸⣿⠏⠀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠹⣿⣇
⢠⣿⡟⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⢻⣿
⣸⣿⠇⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠸⣿
⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿
⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿
⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⢀⠀⠀⡀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿
⢹⣿⡆⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⢀⣴⣿⠀⠀⣿⣦⡀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⢰⣿
⠘⣿⣷⠀⠘⣿⣷⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⢀⣴⣿⣿⣿⠀⠀⣿⣿⣿⣦⡀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⣾⣿⠃⠀⣾⣿
⠀⠹⣿⣇⠀⠸⣿⣧⡀⠀⣠⣾⣿⠟⠁⢀⣴⣿⠟⠋⣿⣿⠀⠀⣿⣿⠘⠻⣿⣦⡀⠈⠻⢿⣷⣄⠀⠀⣼⣿⠇⠀⣸⣿⠏
⠀⠀⢻⣿⣆⠀⠙⣿⣷⣾⣿⠟⠀⢀⣴⣿⡟⠁⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠈⢻⣿⣦⡀⠀⠻⣿⣷⣾⣿⠋⠀⣰⣿⡟
⠀⠀⠀⠻⣿⣧⡀⠈⠻⠟⠁⢀⣴⣿⡟⠃⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠘⠻⣿⣦⡀⠈⠻⠟⠁⢀⣼⣿⠟
⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠰⣿⣿⣇⡀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⣸⣿⣿⠆⠀⠀⣠⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠈⠛⠿⣿⣷⣦⣤⣀⡀⣿⣿⠀⠀⣿⣿⢀⣀⣠⣴⣾⣿⠿⠛⠁⢀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣄⣀⠀⠉⠛⠻⠿⢿⣿⣿⠀⠀⣿⣿⡿⠿⠟⠛⠉⠀⣀⣠⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣶⣤⣤⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠁
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⠀⣿⣿⣿⡀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⢸⣿⣿⣧⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠘⣿⣿⣿⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⠀⣿⣿⣿⡆⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⣠⣶⣤⡀⠀⠀⠀⠈⣿⣿⣿⣧⢹⣿⣿⣿⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠹⣿⣿⣿⣦⡀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣄⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣴⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉⠉⠉⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢤⣄⠀⠀⠀⠀⠀⠀⠤⢾⠿⢡⡾⣿⣿⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⢯⠶⢻⡇⠈⠓⠶⠤⣄⠀⠀⠀⢀⣴⣾⣇⣼⣧⣠⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠻⠴⠛⣷⡼⠋⠉⠉⠛⠲⣌⢳⡄⣴⠟⠉⠀⣉⠙⠛⠛⠛⣿⣷⠶⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢰⡟⠀⠀⠀⠀⣄⠀⠈⢳⣿⡏⠀⠀⠀⠛⠁⢠⣤⠀⣿⣿⡦⣾⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢠⡾⠁⠀⠺⠂⠀⠉⠀⠀⠀⣿⠀⢠⡏⠀⠀⠀⠀⠁⠀⢻⣿⣤⡼⠟⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢾⡁⠀⠀⢠⡄⠀⠀⢀⡇⠀⣿⣧⡀⠻⢦⣤⡴⠏⠀⠀⠈⢻⡟⠀⢰⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠈⠙⢦⣄⠀⠙⠳⠚⠋⢀⡼⣿⠀⠙⠲⢦⣤⣀⣀⣀⣤⡶⠋⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢀⣴⠿⠷⣶⣶⣖⠛⠋⢀⣧⣤⣤⣤⣼⡆⣸⡭⣋⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣯⠀⠀⠀⠈⠙⠛⠷⢾⣧⣄⣀⣤⡾⠟⠋⠉⠳⠵⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⡴⠏⠀⠀⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣧⣄⣀⣀⣠⡤⠐⠊⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⣶⠛⠩⣿⠿⣦⡀⠀⠀⠀⠀⢀⣾⣏⣀⣀⣀⣀⣤⣤⢤⣀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣰⠊⠀⠈⠻⣦⣀⠀⣠⣿⠋⠙⢯⡉⢉⣵⠟⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠉⠂⠴⠖⠚⠋⠉⠛⣋⣡⣤⡶⠛⠙⠋⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣟⣀⡠⠴⠶⠶⠶⠞⠛⠋⢹⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣈⡿⢦⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⣠⠶⠛⠉⢹⡆⠀⠀⠀⣀⣤⠴⠚⠁⠀⠀⠉⠛⠶⣦⣤⣀⣀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⢀⡴⠋⠁⢠⣤⢤⡴⠻⠦⠤⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦
⠸⡧⠴⠶⠚⠻⠞⠋⠑⠲⢤⣀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣌⢙⣿⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠈⢻⡶⣦⣄⡀⠀⠀⠀⢸⡆⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣿⠉⠛⠳⠶⠾⠇⠀⠈⠛⠿⠶⣶⡶⠶⣶⡶⠶⣿⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣷⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣿⢻⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢦⡿⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠶⢿⣿⡄⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⣄⣀⠁⠿⠻⢄⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠛⠚⣛⣿⣛⣋⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⣄⣀⣹⡄⠀⠀⠀⢺⣅⡀⢀⡴⠖⠛⠉⠈⠉⠉⢀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠷⢤⣤⣤⠤⠶⠚⠋⠁