Sugary Emojis & Symbols

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

⣠⠚⠉⠓⢦⡀
⣇⠀⠀⠀⠀⠙⢄
⠈⢤⡀⠀⠀⠀⠀⠳⡀
⠀⠀⠙⢦⡀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡖⠛⠛⠉⠑⢦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠘⢦⡴⠒⠊⠙⠶⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡼⠲⣄⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉⠳⣄
⠀⠀⠀⠀⠀⠀⢠⠐⠉⠀⠀⠀⠀⠈⣦⣀⣳⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⢀⣀⣠⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⣤
⠀⡞⠉⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀
⠀⡞⠃⠀⠀⠑⠤⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠒⠉⠀⠀⠈
⠀⠑⡶⠠⢄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠐⠒⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚
⠀⢰⠇⠀⠀⠀⠀⠀⠐⠒⠀⠠
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠐⣇⠈⠁⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡟
⠀⠀⠀⠀⠈⠓⠶⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣓⣒⣲⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣖⣒⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡴⠋⠉⠀⠀⠀⠉⠙⠒⠢⠤⣄⡀⠀⠹⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠏⠁⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠉⠓⠒⠶⣤⡀
⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣄⣀
⠀⠀⠀⣠⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀
⠀⠀⠀⢧⡀⠉⠒⠀⠀⠤⠀⢀⣀⠀⠀⠀⠀⠀⠈⢀⣁⣀⣀⣀⣀⣀⣀⣀⡀⠠⠤⠤⠤⠴⠒⠉⣹⠆
⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⢠⡇⠀⠀⠀⣀⡤⣄⠀⢀⣀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣀⠀⠀⢀⡠⠤⠤⡀⠀⠀⠀⠀⠘⡇
⠀⠀⢸⠃⠀⠀⡜⠁⡔⠊⠀⢸⢸⠀⢸⢡⢠⠊⡠⠔⠁⠀⠀⡏⠈⡇⠀⡇⢀⣄⡀⠘⡀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠱⢤⣓⡒⠢⣼⠸⠀⠀⢸⡏⢸⠁⠀⠀⠀⢸⢱⡇⢹⠀⢇⠀⠓⢁⡰⠁⠀⠀⠀⢠⡇
⠀⠀⢸⡆⠀⠀⠀⠀⠀⢘⡆⢹⠀⠧⠃⡼⡇⠸⡀⡯⠉⣑⣼⠈⢣⠀⠇⢸⠀⣄⠘⡄⠀⠀⠀⠀⣸⠁
⠀⠀⠀⢷⡀⠀⢠⡖⠒⣉⡠⠛⣄⢀⡠⠃⠙⢄⣈⣴⣀⡇⡆⢠⠀⢣⠸⡘⠀⡏⡆⠘⡆⠀⠀⣰⠃
⠀⠀⠀⠀⠳⡄⠀⠉⠉⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠈⠒⠃⠓⠁⠈⠋⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠳⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣠⣤⣴⠏



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⡀⡀⠀⠀⠀⢀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠚⠛⠉⠉⠙⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠉⢧⡀⠀⠀⠀⠀⠀⠀⠀⠿⠆⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⠀⠀⢀⣻⠴⠶⠶⠒⠒⣦⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⢀⣄⠀⠀⠀⢸⡧⡀⢀⡤⠖⠉⡟⠀⠀⠀⠀⠀⣘⣦⡴⠛⠓⢦⡴⠚⠁⠀⠀⠀⠀⢀⣀⡤⠶⠚⠉
⠀⠀⠀⠈⢋⣀⣀⣀⣬⣧⣉⡷⠖⠒⠒⠓⢲⣤⡖⠉⠉⠁⠈⢦⢀⡴⠋⠀⠀⠀⢀⣠⠴⠚⠉⠁
⠀⠀⠀⠀⣿⠉⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢸⡄⡟⣆⠀⠀⢀⡼⠃⠀⠀⣠⣴⠚⠉
⠀⠀⠀⠀⣿⠀⠀⠐⠀⠸⡄⠷⠀⠀⠀⠀⢸⡇⢱⠘⢆⡰⠋⠀⣠⠔⠋⠀⢸⠀⠀⠀⢀⡀
⠀⣀⣠⡤⠿⠒⢶⠒⢲⠚⠛⠋⠉⠉⠹⡗⢶⡻⠛⠒⠚⠳⢴⣏⣀⣀⣀⣠⠼⠤⢤⠀⠙⠃
⠀⢿⠀⠀⠀⠀⠘⡆⠀⢧⠀⠀⠀⠀⠀⣇⠘⡇⠀⠀⠀⠀⢸⠉⠀⠀⠀⢹⠀⠀⠘⡇
⣠⡼⠦⠤⠤⠤⠤⠿⠴⠼⠶⠶⠶⠦⠤⠾⠤⠷⠤⠤⠤⠤⠾⠤⠤⠤⠤⠼⠤⠤⠤⠷⠶⠶⢶⡄
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⢤⡤⠴⠂⠀⠀⠀⠀⠀⠐⠲⠦⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠂⠲⠥⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⣶⢤⣀
⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⠇⠀⠀⠀⠀⠀⠀⢀⡴⢃⣀⣠⡤⠶⣦⣤⡏⠀⠀⠀⠀⢸⠀⠈⡇
⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⢸⠁⠀⠀⠀⢸⠀⠉⢳⡀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢼⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠚⣻⠀⠀⠀⠀⢸⡄⠀⠄⠀⠸⡄⠀⢸⣇⣀⣀⣸⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠲⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠖⠚⠛⠁⠀⠀⠀⢠⣼⠓⠒⣶⣖⠚⠓⠒⣏⣹⠏⠉⠀⠉⣩⢤
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣯⣤⣬⣷⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭
⣿⣿⣄⣭⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣏⣁⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⠷⢶⣄⠀⠀⢀⠀⢹⢿⣿⢇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠈⠀⠉⠙⠳⠶⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢂⣀⣬⡷⠾⡋⠁⠟⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⣠⠀⠀⠀⢤⠀⠀⠀⠉⠛⠒⠶⠚⠲⠶⢶⢲⣾⣺⢗⡪⣽⡿⠚⡂⠀⢀⣴⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡀⠉⠛⠛⠻⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠹⠻⠻⠁⠐⡿⠙⠙⠁⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣦⣀⡠⣄⡀⠈⠛⠻⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠻⣂⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢟⠛⠉⠛⠷⣷⡤⣤⣄⡀⠀⠀⠀⣀⣴⣖⣳⠤⠀⠀⠀⠀⠐⠷⣶⣢⣤⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠉⠛⢿⠛⠛⣛⠷⠶⠿⠿⠟⠛⠛⠿⣟⣻⡿⠛⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡛⠁⠀⠀⠀⠀⢋⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠣⡈⠿⠛⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⡖⠄⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⠸⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡐⠀⠀⠀⠀⠀⠀⠐⠂⠀⠂⠙⠓⣦⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡭⣽⣯⡍⠁⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⣰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⡶⠞⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠩⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡍⠑⠿⡷⡀
⠀⠀⠀⠀⠀⠀⠸⣵⣀⣀⣤⣦⡄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣌⣻⣷⣶⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠋⠧⡀⢠⣄⣀⣀⣀⣀⣀⣀⣠⠈⠉⠬⣅⣀⣠⣤⣠⣤⣤⣠⣼⡿⠿⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠤⡤⢤⠤⡤⠤⠤⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡘⠀⠃⠘⠀⠃⠘⠀⠃⠘⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠉⠉⠋⠙⠉⠉⠉⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠄⠤⠒⣤⣤⣒⣒⣦⣤⡒⣲⡇⠀⢀⣀⣀⡀⠤⠤⠐⢒⣒⣒⡒⠒⢊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⢆⠀⠒⠊⠉⠉⠀⠀⠀⠀⠁⠀⡇
⠀⠀⠀⠀⡴⣂⠉⠉⠒⣤⠔⠒⠒⣲⢆⠀⠀⠀⠀⠣⡀⠀⠀⣀⡀⣀⢀⡄⢸⣾⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⢰⠀⠀⠙⠎⣿⣁⣀⣠⠞⣁⠈⢇⠀⡀⠀⠙⠁⠓⢃⠛⡏⠛⠉⠀⢸⣿⠀⠀⢸⠀⠀⠧⣀⠀⡀⣤⣀⣶⢰⠀⢸
⠀⠀⠀⠀⠓⡤⠜⠋⠉⠒⡢⡄⢣⠀⢁⠜⠀⣇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣘⡏⠀⠀⢰⠀⠀⠊⠁⠓⠃⣉⡇⠉⠀⠀⢸
⠀⠀⠀⠀⢸⠁⠒⠲⡔⢉⠀⢸⠤⠷⠋⠀⠀⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⡇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠈⣄⠀⠀⢃⠀⢁⡜⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⡇⠀⣀⣀⣀⣀⣀⣀⠀⠠⠄⠀⡇
⠀⠀⠀⠀⠀⠈⠉⠒⠸⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⡇⡠⠤⠤⠤⠄⠒⠒⠒⠈⠉⠓⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⡚⢦⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡰⠚⠙⠒⠐⢦⡀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠉⢀⠀⠑⡄⢣
⠀⠀⠀⠀⠀⠀⣰⣋⣀⣀⣀⣀⣀⣀⣹⣄⠀⠀⣠⠔⡋⠀⠀⢰⡜⠋⠀⣀⠜⠚
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡝⡅⠀⠀⠇⠁⢀⡪⠔⠊⠁
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠁⠈⠀⠀⠀⠀⡇⠀⠳⡘⢆⡠⠴⠒⠁⠀⠀⡀⠴⠔⠒⠢⣀⠀⠀⠀⠀⢀⣀⣀⠀⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠘⣄⣀⣀⠀⢀⡀⢀⡰⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢸⡙⠒⠒⠒⠒⠚⠃⢐⣀⠭⠤⠤⠀⠒⠒⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⠤⠔⠒⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⡞⠁⠽⠿⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃⣀⣠⠴⠛⠛⠳⡄⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⣵⣋⣉⡝⠁⠀⠀⠀⣸⣥⣤⡧⠤⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⢀⣠⣾⠿⠛⢉⣤⣤⡤⡔⠛⠓⠒⠓⠶⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠉⠀⣀⠴⠛⠁⢀⣠⠞⣿⣚⣯⡀⠧⡀⠀⠀⠀⢀⡧⣤⡼⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⢀⡤⠖⠋⠁⢀⣠⠖⡏⠀⠀⡇⡇⠀⠀⠀⢸⣄⣀⣀⣸⣇⠘⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⢀⣠⡴⠚⠉⠀⢀⡤⡞⠉⠉⠉⢱⣤⣴⣇⡟⠉⠉⠙⠛⣆⠀⠀⠀⢸⣉⡩⠭⠛⠛⡶⢤⡀
⠀⠀⠀⠀⠀⡠⠞⠉⣀⡤⠖⠋⠁⠀⣀⡴⢚⡇⣠⠦⣄⣠⠤⠾⣭⣤⣸⠀⠀⠀⠀⡏⠉⠉⠉⠉⢹⡿⠤⠤⣄⠀⡇⠀⠙⣦
⠀⠀⠀⠀⣮⣤⠴⠚⠉⠀⠀⣀⠴⠊⢷⡀⠈⢷⠃⠀⠀⠈⠉⢲⠀⠈⠉⠉⠙⠛⡻⣇⣀⣀⣀⣀⣸⠛⠛⠓⠚⢳⠟⠀⢀⡟
⠀⠀⠀⠀⠘⣆⠀⠀⣠⠴⠋⠁⠀⠀⠀⠙⠶⣇⣀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠋⠀⠉⠉⠉⠉⡏⠀⠀⠀⢀⣸⡤⠞⠋
⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣶⣶⣧⣤⣤⣤⣄⣀⣸⣇⣤⣤⣤⣤⣤⣿⣶⠶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠚⠚⠛⠛⠛⠚⠛⠚⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠛⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⣶⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⢧⣤⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠻⣧⠉⣹⠟⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠈⠉⠛⠛⠷⠶⠶⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠶⠶⠟⠛⠋⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡷⢶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡴⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠉⠉⠙⠛⠛⠷⠶⠶⠶⠶⠶⠶⠶⠶⠾⠟⠛⠛⠋⠉⠉⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠴⡆⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⣶⠤⣤⣤⣀⣀⣀⣀⣤⣤⡤⠴⠶⠒⠛⠉⠉⣁⣀⡀⠀⢻⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⣀⣄⡀⣀⠀⣀⢀⣤⣤⡀⢀⣶⡄⠸⣯⣼⡇⠀⠘⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⢀⡻⢦⡅⣿⠀⣿⢸⡅⠰⣦⢸⡷⠻⡆⠿⠀⠙⠂⠀⣿⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡀⠈⠓⠚⠃⠙⠶⠛⠀⠙⠛⠉⠈⠀⣀⣀⣠⡤⠶⠒⠛⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠀⠀⠀⠈⠉⠛⠒⠒⠶⠶⠶⠶⠖⠒⠒⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠷⠶⠶⢦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⡶⠶⠶⠞⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁

⡠⠊⠉⠓⢄
⣇⠀⠀⠀⠀⠳⣄
⠘⢦⡀⠀⠀⠀⠈⢢⡀
⠀⠀⠉⠢⡀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠳⡀⠀⠀⣀⣀⣀⣀⣠⡴⠾⠁⠀⠀⠀⠙⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠹⡞⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠤⢄⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠔⠒⠚⠉⠳⡄⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄
⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠘⠒⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡄
⠀⠀⠀⠀⢀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⢢⡀
⠀⡴⠒⠒⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢱
⠀⡷⠄⠀⠐⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠊⠀⠀⠛
⠈⠧⡤⣀⠀⠀⠀⠉⠉⠉⠑⠒⠒⠠⠄⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠄⠒⠒⠂⠉⠉⠉⠀⠀⠀⠀⠄⢾
⠀⢰⠃⠀⠈⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠸⣄⠈⠁⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠈⠙⠒⠦⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠉⠳⢤
⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢸⠂
⡰⠋⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠚⠙⠉⠀⠀⠀⣠⠶⠖⠲⠚⠲⡄
⣧⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣿
⠘⢦⡀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠦⢤⡀⠀⠀⢰⠇
⠀⠀⠙⢦⡀⠀⠂⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⣇⣀⣤⣀⡟
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡞⠉⠉⠉⠀⠉⠷⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⢳⣤⠴⠒⠋⠉⠳⠞⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⠦⣄⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠛⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠉⠉⠀⠀⠙⢦⡀⠘⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠎⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣄
⠀⠀⠀⠀⠀⠀⣠⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠘⠧⣄
⠀⠀⣀⣤⡤⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠲⢦
⠀⣾⠁⠀⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣰
⠀⡼⠋⠀⠀⠈⠑⠤⢄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠋⠀⠀⠀⠈
⠀⠻⢦⡤⣀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠒⠒⠲⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠒⠒⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠔⢺
⠀⠀⡾⠀⠀⠈⠁⠒⠀⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸
⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⠤⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠐⠁⡆⠀⠀⠀⠀⠀⠀⡾
⠀⠘⣇⠀⠀⠀⠀⠀⠀⡇⠀⣎⠉⠀⠀⡏⠓⠀⡖⡆⠀⡠⠒⢂⣉⣐⡆⠀⠀⡠⠄⠂⢦⠀⠀⢠⠞⢉⡀⠀⠈⣆⠀⡇⢀⠃⠀⠀⠀⠀⠀⣸⠃
⠀⠀⢻⡄⠀⠀⠀⠀⠀⠙⠒⢬⡉⢢⠀⠀⠂⢠⠁⡇⠚⠀⡴⠉⠀⠉⠀⠀⢀⠇⢀⣄⠘⡀⠀⠘⠀⠸⣉⠇⣀⠄⠀⣇⡜⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⣠⠅⢰⠀⡆⠰⠞⢠⠃⡇⠀⣇⢰⡆⠀⠤⠄⡸⠀⢘⠁⠀⢃⠀⠀⠀⣠⡀⠘⠣⡀⢀⣈⠁⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⢳⡀⠀⠀⠀⠾⠤⠴⠖⠃⠀⠳⠤⠤⠎⠀⠘⢤⣈⣩⡀⠀⡗⢊⡇⠀⠏⢣⠀⠘⡀⠀⣀⣀⠰⣄⡴⠋⠘⠊⠀⠀⢀⡴⠏
⠀⠀⠀⠀⢰⠛⠒⠤⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠉⠀⠀⠑⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⡀⠴⡿
⠀⠀⠀⠀⠘⢧⣄⡀⠀⠀⠈⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠦⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣛⣛⣓⣲⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣒⣚⣋⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠋⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠿⠿⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣀⡀⠀⢈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⢠⣾⡿⠁⠀⣼⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⢠⣿⠟⠁⣠⣾⠟⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠈⠁⢀⣴⡿⢁⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⣴⡿⠋⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠛⠛⠛⠛⠿⢷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⣠⡄⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠸⠗⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠈⠁⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠋⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠸⠗⠀⠀⠀⠙⠻⢿⣦⣄⡀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠉⠀⠰⠆⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣤⡀⠀⡀⠀⠈⠙⠿⣶⣤⣿⠃⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⣠⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠋⠀⠀⠀⠾⠎⠛⢿⣦⡀⠀⠀⣠⣿⠏
⠀⠀⠀⣀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⢠⣤⠀⠀⠠⣦⠀⠀⠀⢀⡀⠀⠀⣤⡄⠙⢿⣦⣼⡟⠁
⣰⣾⣿⣻⡿⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⢠⣄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠋⠀⢀⣤⠀⣤⡄⠹⣿⣿⣶
⣿⡏⠙⠻⠿⣶⣦⣄⣀⡀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣤⠀⠀⠀⢠⣦⠀⠈⠋⠀⠀⠀⠀⠉⣀⣬⣶⡿⠟⠉⣿⡆
⣿⡇⠀⠀⠀⠀⠈⠙⠛⠻⠿⣶⣶⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⡶⠿⠛⠋⠁⠀⠀⠀⢻⣇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⢀⣤⣾⢿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⠟⠋⠁⢸⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢰⣶⣶⡾⠿⠟⠛⠉⠁⠀⠀⠀⠀⣾⡇
⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠃
⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⠿⢛⣿⡿⠁
⠀⠀⠀⠀⠙⠿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠻⠿⠿⠛⠛⠛⠋⠉⢁⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠾⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠈⠻⣶⠶⠶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠩⣝⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⡀⠀⢀⣀⡀⢀⡀⠀⢀⠀⠀⢀⣀⡀⠀⢀⣄⠀⠀⣀⠀⠈⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢸⠁⠰⣿⡉⠋⢸⣿⠀⣿⡇⢰⡿⠙⠃⠀⣼⣿⡄⠀⣿⣷⡄⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⠀⠀⠙⣷⡄⢸⣷⠀⣿⠃⢸⡇⢺⣶⠀⣿⣾⣧⠀⣿⣿⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠈⠀⠻⠶⠿⠁⠀⠻⡾⠏⠀⠸⣷⡿⠁⢸⡿⠀⢿⠆⣿⡿⣧⡀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠁⠈⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡦⠓⠀⠠⢤⣤⣤⣤⠀⠀⠀⠀⠀⣤⡤⠤⠶⠒⠋⠁⠀⣼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣋⣀⣀⣬⣬⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣁⣁⣀⡈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠛⠛⠛⠲⢦⣄⣀⡀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⢴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠛⠳⣤⣤⡀⠀⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠲⢦⡀
⠀⠀⠀⠀⢀⣤⡞⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⣀
⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠛⢦⡀
⢠⠞⠻⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠀⠀⠀⠀⡄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠷⣦
⠈⢷⣤⠀⠈⠉⠙⠒⠂⠤⠤⠤⠄⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠤⠤⠤⠤⠤⠴⠖⠒⠒⠚⠉⠁⢀⣼
⠀⣼⠃⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣼⠀⠀⠀⠀⣠⠖⠋⢙⡧⠀⢰⠒⡆⠀⢠⢤⠀⠀⣠⠖⠒⢲⠀⠀⠀⢀⡴⠶⡄⠀⠀⡴⠊⠉⠉⠉⠳⡀⢀⠀⡀⠀⠀⢸
⣿⠀⠀⠀⢠⡇⠀⡞⠁⠀⠀⢸⠀⣿⠀⢸⠘⡄⡼⠁⡴⠚⠉⠀⠀⠀⢸⠁⠀⢹⠀⠀⡇⠀⡶⢦⡀⠀⣇⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠣⣄⣙⣛⠒⠢⣼⡀⢿⠀⢸⠀⣷⡇⢸⠃⠀⠀⠀⠀⠀⣼⢸⣇⠈⡇⠀⣇⠀⠙⠚⠀⣠⠏⠀⠀⠀⠀⠀⢸
⢻⠀⠀⠀⠀⠀⠀⠀⠉⢳⠀⢸⡇⢸⣀⡏⢰⣿⡀⢸⠀⢴⡒⠒⠲⡄⡇⠘⠿⠀⢹⠀⢹⠀⢀⠀⠸⡁⠀⠀⠀⠀⠀⠀⣾
⢸⡄⠀⠀⠀⠀⢀⣀⣠⠼⠀⣸⡇⠀⠉⠀⡸⠈⣇⠈⠓⠖⡁⠠⡖⢻⠇⠀⡜⢧⠈⣇⢸⠀⢺⢧⠀⠱⡄⠀⠀⠀⠀⢰⠇
⠀⢳⡄⠀⠀⢸⣉⣀⣠⡤⠞⠁⠱⣄⣠⠴⠁⠀⠘⠳⠤⠖⠧⠤⠇⢸⠀⣼⠀⠈⣆⢸⡄⡇⢸⠈⢇⠀⣹⠀⠀⠀⣠⠏
⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠘⠚⠁⠉⠉⠀⠈⠉⠀⠀⢀⡴⠋
⠀⠀⠀⢺⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠋
⣤⣤⣤⣤⣬⣽⣿⣿⣿⣿⣿⣿⣽⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⢠⠠⡄⠤⠠⡄⢤⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣃⣘⣀⣃⣈⣀⣃⣙⣀⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣒⣒⡚⡰⣶⣮⣭⣷⣶⢞⡑⣇⠀⢠⠤⠤⠤⠒⣒⣂⣉⣭⣭⣉⠉⢹⠃
⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠈⠣⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⢠⠛⠤⣀⣀⣙⠖⠊⠉⢩⠟⢆⠀⠀⠀⢘⣢⡄⢠⢠⣄⡤⡤⡆⠈⣿⠀⠀⢸⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⡀⠀⢸⣡⣇⠒⠐⢶⠁⢠⠈⡆⠠⠀⠈⠀⠈⠉⢌⠗⠉⠁⠀⠀⣿⠀⠀⢸⠀⠀⣓⠤⣀⡤⣶⠰⡷⠫⠀⡇
⠀⠀⠀⠀⢹⣚⡉⠀⢀⡭⠛⡄⢧⢀⡜⠁⢸⠤⠤⠤⠄⠀⠀⡀⠤⠤⠤⠬⠿⠀⠀⠘⠀⠀⠁⠀⠉⠰⠴⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢸⠀⠈⠙⡇⠦⡄⣿⠘⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠦⣄⡀⣇⣀⠔⠁⠀⠀⠀⠀⠀⠢⠤⢤⡄⠀⣀⣀⡀⢠⡤⠔⠁⠀⠀⠀⡇⠀⠤⠤⠄⠀⠒⢒⣂⡈⣅⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠧⠞⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠈⢣⠙⣆
⠀⠀⠀⠀⠀⠀⢀⠔⠃⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⡠⠔⠋⠀⢀⣴⠄⠀⢣⣸
⠀⠀⠀⠀⠀⠀⡏⠈⠂⠘⠀⠀⠐⠀⠛⡆⢠⠔⠉⠘⢠⠤⠞⡳⢀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠂⠀⠀⠀⠀⢰⠀⢺⡘⢄⠀⢀⡠⠴⠊⠁⠀⢀⣀⡠⣀
⠀⠀⠀⠀⠀⠀⢡⠀⠀⠈⠀⠀⠀⠀⡸⠀⠀⠑⠜⠋⠉⠀⠀⠀⢠⠮⠥⢀⣀⣈⣑⣄⣀⠤⠒⢒⣂⣈⣉⣉⠩⠭⠝
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠐⠒⠒⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⣀⣀⡤⠒⠉⠉

⠀⠀⠀⠀⠀⠀⢀⢠⠤⠤⢤⣄⡀⢐⠋⠒⠐⢆⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡔⢱⠤⣀⡴⡁⠈⠀⠒⡦⠉⠙⢎⣢⠇⠠⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠆⣰⠚⢍⢍⠙⡉⠉⡝⣧⡷⠉⡝
⠀⡹⠐⠉⠉⠐⣼⡦⠀⠉⢠⠀⢺⢻⣅⠄⠒⢊⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠇⠘⠒⢆⡀⢸⡜⠀⣇⠀⡁⣿⡦⣀⠸
⡎⠀⢀⣤⠒⠪⢀⣁⠤⠀⠄⢁⢁⡼⢏⠙⢀⠇⠉⠀⠀⠀⣀⡀⠀⡀⢀⣀⡀⠀⠀⠸⠴⡆⣜⠷⠎⠢⠼⠴⠤⠿⠔⠓⠃⠘⠃
⠈⠐⠉⠀⠱⠤⠜⠣⢇⣒⣱⡥⠚⠀⠀⠉⠉⠀⠀⢀⡒⠃⠄⢈⡶⡶⠭⣄⠬⣙⡆⠀⠘⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠑⠒⠂⠀⠖⠀⠒⠗⠊⢻
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠀⡇⠀⠀⡇⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⡠⠤⣀
⠀⠀⠀⠀⢀⢠⠔⠁⠀⠀⠀⠀⠉⢢⠀⠀⠀⠀⠀⠈⠠⢀⡇⣀⣀⣇⣀⣀⡧⠔⠃⠀⠀⠀⠀⠀⠀⠀⢀⠞⠅⠀⠀⢰⣉⡈⠣
⠀⠀⠀⠀⣀⠈⢆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠠⠂⠠⣈⠆⠀⠀⢆⠃
⠀⠀⠀⠀⢘⠢⡀⠑⠤⣀⠀⠀⠀⣠⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠂⠄⣀⡀⠀⠀⠀⠀⠀⠀⢣⠠⡀⠀⠠⠀⣀⠹⢀⠁
⠀⠀⠀⠀⠈⢦⡈⠑⠂⠤⢈⣉⣉⡤⠊⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⣠⠖⠂⠈⠁⠒⢢⡄⠀⠀⠑⠆⡈⣉⢀⣤⠴⡃⢤⡀
⠀⠀⠀⠀⠀⠀⡝⠒⡤⠤⠠⠴⠊⠀⠀⠀⠀⠀⢀⠤⢦⣊⠀⠀⠈⠀⠱⠂⠀⣀⣀⠀⢀⠄⠂⠁⢰⠀⠀⠀⠀⠀⢠⠩⠤⡰⠹⡉
⠀⠀⠀⠀⠀⢠⠃⡰⠁⠀⠀⠀⠀⠀⠀⠀⣀⠄⠓⠒⠂⠬⢲⡤⡀⠀⠀⠀⢀⠀⡠⠊⡄⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠱⡀
⠀⠀⠀⠀⢠⠏⡰⠁⠀⠀⠀⠀⠀⣠⠖⢉⠀⠀⠀⠀⢀⠔⠀⠊⠉⠑⠒⡶⡉⠓⡄⠀⡧⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠱
⠀⠀⠀⢠⠏⢠⠃⠀⠀⠀⠀⠀⠀⢈⠑⠢⢅⠂⠠⢴⠐⠓⢢⠤⠀⠴⡄⠀⠉⠀⡶⠈⠀⠀⠀⢀⡠⠖⠋⠉⠙⠙⠢⡄⠀⠀⠀⠈⠉
⠀⠀⠀⠈⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢸⠄⠄⣠⠀⠀⢼⡠⠄⠤⣇⠠⠂⠉⠁⠀⠀⠀⣠⠋⠀⠀⢀⡤⢤⣀⣀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢐⣀⡜⢁⡤⠄⣻⠀⠊⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢰⣧⠈⠦⡀⠀⠀⠀⠀⠀⠈⣆⠀⠸⡀⢀⡦⠔⢋⢆⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠚⢍⠡⠀⠉⣧⡤⠀⡆⠀⠀⠀⠀⠀⠀⠈⠣⣌⡙⢧⣀⣀⣼⠞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡤⠒⠘⠃⢉⠢⠀⠀⠀⢃⢁⡠⠛⠳⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠠⡈⢆
⠀⠀⠀⠠⠠⣤⣀⠀⡔⠉⠭⡉⠢⡐⡀⠀⠀⣂⣣⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠳⡀
⠀⠀⠀⢠⢄⡘⢄⣉⡉⡠⠀⠁⠀⣸⠼⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⡆
⠀⠀⠀⣼⠀⢠⢦⠤⠋⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢶⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣴⣤⣠⡀⠀⠀⠀⠙⢿⣿⣿⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠽⠛⠒⠧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠻⠆⠀⢐⣦⣽⠟⢫⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⠀⠀⢀⡀⠀⠉⠓⠶⠦⣤⣴⢤⣤⣤⣤⠤⠶⠔⣾⣿⣿⠍⠀⢁⣤⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡊⠛⠶⠶⢶⣥⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠃⠛⠻⠿⠾⠀⢰⡟⠙⠛⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⡀⣠⣄⠈⠙⠻⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡶⣢⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠻⠟⠙⠻⢿⣦⣤⣤⣀⠀⠀⠀⣀⣴⣞⣦⣄⠀⠀⠀⠐⢷⣶⣠⣤⣶⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⡀⠀⠀⠈⠉⠉⠋⠉⠉⠹⢿⢛⣛⡟⠳⠛⠛⠛⠛⠛⢻⣿⣿⣟⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠸⡘⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⡅⠀⠀⠀⠀⠀⠀⠈⠣⠸⠟⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣗⠀⠀⠀⠀⠀⠇⠃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢠⣰⣦⣶⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⡴⢴⣿⣀⡄⠀⠀⠤⠀⠠⠄⠙⠉⠉⠿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣚⣻⣗⡂⠀⠀⠀⠐⠒⠦⠐⠦⠠⢴⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠾⠓⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⡉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠹⠿⣿⣇
⠀⠀⠀⠀⠀⠀⣤⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣆⠰⣿⣀⣍⠉⠂
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠷⢶⠳⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣜⠃⠀⢀⠀⣀⢠⠀⣰⣿⣿⠿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉⠁⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠉⠉⠉⠉⠉⠙⠛⠒⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢖⡿⠗⠒⠒⠲⠦⣄⣀⠀⠀⣀⣀⡤⠤⠦⣤⡈⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡞⠀⠀⠀⣀⠀⠀⠀⠈⠙⠉⠉⠀⣀⣦⠀⠀⠳⡄⠀⠀⠙⢦⣀⡠⠤⢤⣤⢄⣀
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣼⠁⠀⠀⠲⡾⠇⠀⠀⢄⡰⠀⠀⠀⠙⠋⠀⠀⠀⠙⢦⣀⡔⢾⠀⠶⠦⠘⠁⢀⣤⠍⢓⣄
⠀⠀⠀⠀⠀⣼⠟⠛⠉⠉⠁⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⡔⠛⡋⠠⢶⡀⢜⠛⠀⠛⢓⣀⡛⣁⠑⡄
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈⠉⠉⣠⠴⢒⣲⡿⠿⠿⢭⣝⡲⢄⠀⠘⠃⡸⠀⠈⠙⠁⠺⠇⠈⠙⠐⠛⠁⠉⠀⠛⠀⠘⡀
⠀⠀⠀⢰⢿⠀⠀⠀⠀⠐⠶⠶⠀⢀⡾⢁⡴⠋⠁⠀⠀⠀⠀⠀⠉⠺⣧⠀⢴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⡿⢸⡆⠀⠀⠀⠀⠀⠀⠀⡞⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠜⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⢷⠀⠀⠀⠀⢲⡄⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⢸⠀⠀⢀⣀⡀⠉⠘⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⢷⠀⣼⠀⠀⠉⠉⠀⣀⣀⠘⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣲⢿⠙⡀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠔⠊
⠀⠀⠀⠸⡄⣿⠀⠀⠀⠀⠀⠈⠉⠁⠈⠻⢷⣄⡀⠀⠀⠀⠀⢀⣠⠞⠁⠸⡄⠉⠉⠀⠈⠒⠂⢦⠀⠀⢀⡜⠉⠳⠄⠀⠜⠁⡘
⠀⠀⠀⠀⢳⡸⡄⠀⠀⠀⠀⠘⢦⡀⠀⢀⠀⠈⠉⠛⠓⠛⠋⠉⠀⢠⣤⣦⡇⠀⢀⠀⢠⣶⡦⠈⠓⢐⠊⠀⣶⣶⠀⠀⠁⢀⠃
⠀⠀⠀⠀⠀⢳⡹⣄⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣠⡄⠀⠀⠘⢷⡀⠀⢰⠏⢿⠀⠘⡀⠀⠉⠁⠀⠈⠉⠀⠀⠉⠁⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠙⣎⠙⠲⠤⣤⣀⠀⠀⠀⠀⠘⠉⠀⠐⢦⡀⢀⠀⢀⡞⠀⠘⡇⠀⡃⠀⠀⡇⠀⠀⡆⠀⠀⠀⠀⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⢨⠟⠉⠙⠾⠤⡔⠒⠟⠉⠉⢣⠀⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠙⢦⣀⠀⠀⠀⣀⣼⡀⠀⠀⠀⠀⣽⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠯⢭⣍⣉⣁⣀⣹⣦⣤⡤⠔⠛⠲⠤⠤⠞⠉⠉⠁⠀⠒⠒⠀⠈⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⣤⣀⠀⠀⠀⢀⣴⠞⠛⣆
⠀⠀⢀⣀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠙⢷⡀⠀⢾⠁⠀⠀⣻
⣠⠾⠋⠙⡇⠀⣼⠃⠀⠀⠀⠀⡠⢒⡆⢘⡇⠀⠈⠳⠚⠋⠉
⢧⠀⣀⣀⡿⠀⢻⡄⠀⠀⠀⠀⠢⣀⣀⣼⠃⣠⠶⠶⠶⣤⡀
⠈⠛⠉⠉⠀⠀⠈⠻⣦⡀⠀⠀⠀⠈⠉⠛⠛⢻⠀⣀⢀⠈⠹⣦⡀
⠀⢀⣴⠞⠛⠛⠷⣦⡈⠛⠶⢤⣤⡴⠶⠒⠲⢦⣦⡁⢈⡃⠀⠘⣷⡀⠀⠀⠀⠀⠀⣀
⢀⡾⠁⠀⠀⠀⣀⣈⢻⡄⢠⡞⠁⠀⠀⢀⣀⠀⠸⡮⣉⠀⠀⠀⠘⣷⠀⠀⢀⡴⠟⠉⣷
⢸⡇⠀⠀⠀⢈⠰⠞⣸⠇⣾⠀⠀⠀⠀⣇⠚⠀⢠⣇⠈⢢⠀⠀⠀⢹⡇⠀⢸⡇⠀⣀⣹⠄
⠘⣧⠀⠀⠀⠈⠓⠾⣯⣄⣹⣇⠀⠀⠀⠈⠑⠰⠿⠼⢦⣬⣆⣀⡀⢸⡇⠀⠀⠙⠋⠉⠉
⠀⠈⢷⣄⠀⠀⠀⠀⠀⠈⠉⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄
⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠈⠐⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⣶⡶⠒⠒⠢⡌⠀⠀⠀⢰⢋⠉⠡⣀⠀⠀⠁⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⢤⡜⣄⠀⠀⠀⠉⠀⠠⠉⠑⢄⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠳⠀⠁⠈⡳⠦⠀⠀⠘⠢⡀⠀⠀⢳⡀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⢢⠀⠀⣧⠀⣸⡟⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠶⠶⠶⠶⠶⠶⠶⣤⡀⢣⡀⢸⠀⣿⠛⠉⠉⠉⠙⠛⠶⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⠿⠇⠃⠈⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠙⢷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠈⠙⠲⠦⠤⠤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⣤⠴⠶⠋⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠀⠀⠚⠛⠛⠓⠒⠒⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢠⠞⠁⣠⠴⠃⠀⢸⠁⢻⠀⢠⡞⠆⠀⣠⠞⠉⣀⣽⠀⠀⠀⠀⡞⠉⠹⡄⠀⠰⠏⠀⠀⠀⠀⠈⣧⠀⠀⠐⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⢸⣁⡀⠀⠀⢸⠀⢸⠀⢸⠇⣿⢰⠇⢠⡞⠁⠀⠀⠀⠀⠀⡇⣀⠀⢧⠀⠀⡇⠀⣿⢑⡆⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⠢⠤⣭⣉⠉⠑⣼⡂⢸⠀⢰⠀⣿⣾⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⣿⡄⠸⡄⠀⣇⠀⠈⠉⠀⣠⠟⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⢸⡇⠘⣆⡼⠀⣿⣿⠀⢿⠀⢾⡉⠉⠑⢆⡸⠀⠓⡃⠀⢷⠀⣿⠀⢀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣀⣤⠞⠀⣸⡇⠀⠀⠀⢠⠇⢸⡄⠈⠳⠚⡁⠀⡞⠉⡇⠀⢠⠛⡄⠘⡄⢸⠀⢿⢳⡀⠘⣆⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⣏⣁⣀⣠⡤⠞⠁⠸⣄⣀⡴⠋⠀⠀⠙⠶⠤⠾⢧⣤⠇⢠⡃⢠⠇⠀⠹⡀⢳⠘⡆⢸⠈⢧⠀⣸⡆⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⠀⠱⠞⠀⠛⠋⠀⠈⠛⠉⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣻⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣁⣀⣀⣀⣀⣀
⣿⣿⣿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢰⠄⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⡀⠁⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⠀⠙⡆⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⢛⣩⠴⠚⠋⠉⠓⢄⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣖⠿⣋⠁⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⡩⠊⠡⡶⢈⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡵⠋⢺⣆⡀⣠⢻⠃
⠀⠀⠀⢠⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡱⠋⠀⠀⠀⠉⠉⢏⠇
⠀⠀⠙⠄⢁⠌⢧⣀⣰⣶⢰⣶⢰⣶⣴⣦⣴⣦⣠⡄⣀⣠⢋⠞⠀⠀⠀⡄⠀⠀⡾⡸
⠉⢛⣃⠀⣏⠀⠀⢈⣙⣛⣛⠛⠛⠛⠿⠿⠿⣗⣛⡷⢿⣧⣿⢦⣴⣄⡀⠀⠀⠀⡆⡟
⠀⠉⣡⡄⠈⢠⠞⠉⠹⣿⣿⣿⣿⣶⣦⣅⡃⢴⡆⠈⢳⡒⠮⢝⡻⠾⣟⣶⡆⣀⡇⡇
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠈⠈⠉⠛⢿⢿⣯⡲⣄⣀⣾⢃⠄⠀⠉⠑⠮⣙⠿⣿⣷⣿⠀⢴⣄⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⢟⣦⡳⡀⠀⠀⠀⡄⠀⠀⠈⠑⢬⡛⢿⡿⠚⠷⠷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⣿⡜⣆⠀⠀⠀⠀⠀⠙⢗⡀⠙⢦⡑⢴⡿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡜⡎⠠⡄⡥⠀⠀⠁⠈⠀⠠⡑⣌⠻⣟⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠸⡀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠈⢆⠹⣯⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⡇⢣⢇⠀⠀⠀⠠⠃⠀⠀⠔⡄⠈⢆⠹⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡷⣎⢋⣱⠸⠀⠠⠀⠀⠀⢀⠄⡀⢈⡤⠀⠸⡄⣷⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠏⢛⣻⡄⡇⢠⣷⠀⠀⠑⠊⠀⢿⢿⠂⠀⡇⢸⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠇⠈⠟⠄⠀⠈⠉⠀⠈⡓⠂⠀⡇⢸⣛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣽⡇⢸⠀⠀⠀⠀⠠⣄⠀⠀⠁⠀⢀⠇⢼⣛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣇⢸⠀⠦⠀⠀⠀⠁⠀⠀⡰⠀⡼⢀⣯⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡀⢇⠀⠠⠀⠀⢀⡠⠀⠀⡴⢁⣾⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣷⣌⠳⣄⡀⠀⠀⣀⡤⠊⣠⡾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡿⣳⣤⣍⣉⣉⣤⢴⣏⠙⢟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠹⠟⠺⠏⠿⠋⠋⢰⣯⣯⠁

⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿
⠘⠿⣿⣿⣦⡀⠀⠀⠀⣠⣾⣿⣿⠟⠁⠻⢿⣿⣷⣄⠀⠀⠀⢀⣴⣿⣿⡿⠋⠘⠿⣿⣿⣦⡀⠀⠀⠀⣠⣾⣿⣿⠟⠁⠻⢿⣿⣷⣄⠀⠀⠀⢀⣴⣿⣿⡿
⢠⣶⣿⣿⠟⠁⠀⠀⠀⠙⢿⣿⣿⣦⡀⣰⣾⣿⡿⠋⠀⠀⠀⠈⠻⣿⣿⣷⣄⢀⣶⣿⣿⠟⠁⠀⠀⠀⠙⢿⣿⣿⣦⡀⣰⣾⣿⡿⠋⠀⠀⠀⠈⠻⣿⣿⣷
⣿⣿⠟⠁⠀⢀⣴⣶⣄⠀⠀⠙⢿⣿⣿⣿⡿⠋⠀⠀⣠⣶⣦⡀⠀⠈⠻⣿⣿⣿⣿⠟⠁⠀⢀⣴⣶⣄⠀⠀⠙⢿⣿⣿⣿⡿⠋⠀⠀⣠⣶⣤⡀⠀⠈⠻⣿
⠟⠁⠀⢀⣴⣿⣿⣿⣿⣷⣄⠀⠀⠙⠿⠋⠀⠀⣠⣾⣿⣿⣿⣿⣦⡀⠀⠈⠻⠟⠁⠀⢀⣴⣿⣿⣿⣿⣷⣄⠀⠀⠙⠿⠋⠀⠀⣠⣾⣿⣿⣿⣿⣦⡀⠀⠈
⠀⢀⣴⣿⣿⣿⡿⠻⣿⣿⣿⣷⣄⠀⠀⠀⣠⣾⣿⣿⣿⠟⢿⣿⣿⣿⣦⡀⠀⠀⠀⣴⣿⣿⣿⡿⠻⣿⣿⣿⣷⣄⠀⠀⠀⢠⣴⣿⣿⣿⠟⢿⣿⣿⣿⣦⡀
⣠⣿⣿⣿⣿⠋⠀⠀⠈⢻⣿⣿⣿⣷⢄⣼⣿⣿⣿⡟⠁⠀⠀⠙⣿⣿⣿⣿⡦⣤⣿⣿⣿⣿⠋⠀⠀⠈⢻⣿⣿⣿⣷⢄⣼⣿⣿⣿⡟⠁⠀⠀⠙⣿⣿⣿⣿
⠀⠙⣿⣿⣿⣷⣄⢀⣴⣿⣿⣿⠟⠁⠀⠘⢻⣿⣿⣿⣦⡀⣠⣾⣿⣿⡿⠋⠀⠀⠙⣿⣿⣿⣷⣄⢀⣴⣿⣿⣿⠟⠁⠀⠘⢻⣿⣿⣿⣦⡀⣠⣾⣿⣿⡿⠋
⣷⣄⠀⠙⢿⣿⣿⣿⣿⣿⠟⠁⢀⣴⣿⣦⡀⠈⠻⣿⣿⣿⣿⣿⡿⠋⠀⣠⣾⣷⣄⠈⠙⢿⣿⣿⣿⣿⣿⠟⠁⢀⣴⣿⣦⡀⠘⠻⣿⣿⣿⣿⣿⡿⠋⠀⣠
⣿⣿⣷⣄⠀⠙⢿⣿⠟⠃⢀⣴⣿⣿⣿⣿⣿⣦⡀⠈⠻⣿⡿⠛⠀⣠⣾⣿⣿⣿⣿⣷⣄⠀⠙⢿⣿⠟⠋⢀⣴⣿⣿⣿⣿⣿⣦⡀⠈⠻⣿⡿⠛⠁⣠⣾⣿
⠹⣿⣿⣿⣷⣄⠀⠉⢀⣴⣿⣿⣿⡿⠉⢿⣿⣿⣿⣦⡀⠈⠁⣠⣾⣿⣿⣿⠏⠹⣿⣿⣿⣷⣄⠀⠉⢀⣴⣿⣿⣿⡿⠙⢿⣿⣿⣿⣦⡀⠈⠁⣠⣾⣿⣿⣿
⠀⢈⣹⣿⣿⣿⡷⠰⣿⣿⣿⣿⣉⠀⠀⠀⣉⣿⣿⣿⣿⠆⢾⣿⣿⣿⣏⡁⠀⠀⢈⣻⣿⣿⣿⡷⠰⣿⣿⣿⣿⣉⠀⠀⠀⣩⣿⣿⣿⣿⠆⢾⣿⣿⣿⣏⡀
⣴⣿⣿⣿⡿⠋⢀⡀⠈⠻⣿⣿⣿⣷⣀⣾⣿⣿⣿⠟⠁⣀⠀⠙⢿⣿⣿⣿⣆⣰⣿⣿⣿⡿⠋⢀⡀⠈⠻⣿⣿⣿⣷⣀⣾⣿⣿⣿⠟⠁⣀⠀⠙⢿⣿⣿⣿
⣿⣿⡿⠋⢀⣤⣾⣿⣦⡀⠈⠻⣿⣿⣿⣿⣿⠟⠁⣠⣴⣿⣷⣄⠀⠙⢿⣿⣿⣿⣿⡿⠋⠀⣤⣿⣿⣦⡀⠈⠻⣿⣿⣿⣿⣿⠟⠁⢠⣴⣿⣷⣄⠀⠙⢿⣿
⡿⠋⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀⠈⠻⣿⠟⠁⢀⣴⣿⣿⣿⣿⣿⣷⣤⠀⠙⢿⡿⠋⠀⣰⣾⣿⣿⣿⣿⣿⣦⣀⠈⠻⣿⠟⠁⢀⣴⣿⣿⣿⣿⣿⣷⣄⠀⠙
⠀⣠⣾⣿⣿⡿⠋⠈⠻⣿⣿⣿⣧⡀⠀⢀⣴⣿⣿⣿⠟⠁⠙⢿⣿⣿⣷⣄⠀⠀⣠⣾⣿⣿⡿⠋⠈⠻⣿⣿⣿⣷⡄⠀⢀⣴⣿⣿⣿⠟⠁⠙⢿⣿⣿⣿⣄
⠺⣿⣿⣿⣿⣄⠀⠀⢀⣼⣿⣿⣿⠿⠓⢿⣿⣿⣿⣧⡀⠀⠀⣠⣿⣿⣿⡿⠟⠺⣿⣿⣿⣿⣄⠀⠀⢀⣼⣿⣿⣿⡟⠓⢿⣿⣿⣿⣧⡀⠀⠀⣠⣿⣿⣿⣿
⠀⠈⠻⣿⣿⣿⣷⣴⣿⣿⣿⡿⠃⠀⠀⠀⠙⢿⣿⣿⣿⣦⣾⣿⣿⣿⠟⠁⠀⠀⠈⠻⣿⣿⣿⣷⣴⣿⣿⣿⡿⠋⠀⠀⠀⠙⢿⣿⣿⣿⣦⣾⣿⣿⣿⠟⠁
⣷⣄⠀⠈⠻⣿⣿⣿⣿⡿⠋⠀⠀⣠⣾⣤⡀⠀⠙⢿⣿⣿⣿⣿⠟⠁⠀⢀⣴⣧⣄⠀⠈⠻⣿⣿⣿⣿⡿⠋⠀⠀⣠⣾⣆⡀⠀⠙⢿⣿⣿⣿⣿⠟⠁⠀⢀
⣿⣿⣷⣄⠀⠈⠛⠿⠋⠀⠀⣠⣾⣿⣿⣿⣿⣆⠀⠀⠉⠿⠟⠁⠀⢀⣴⣿⣿⣿⣿⣷⣀⠀⠈⠹⠿⠋⠀⠀⣠⣾⣿⣿⣿⣿⣦⡀⠀⠉⠻⠟⠁⠀⢀⣴⣿
⠈⠻⣿⣿⣷⡀⠀⠀⠀⣠⣾⣿⡿⠟⠁⠙⢿⣿⣷⣄⠀⠀⠀⢀⣴⣿⣿⠿⠋⠈⠻⣿⣿⣦⡀⠀⠀⠀⣠⣾⣿⣿⠟⠀⠙⢿⣿⣷⣆⠀⠀⠀⢀⣴⣿⣿⡿
⣠⣾⣿⣿⠟⠁⠀⠀⠀⠙⢿⣿⣿⣦⣀⣴⣿⣿⡿⠋⠀⠀⠀⠈⠻⣿⣿⣷⣄⣠⣾⣿⣿⠟⠁⠀⠀⠀⠙⢿⣿⣿⣦⣀⣴⣿⣿⡿⠋⠀⠀⠀⠈⠻⣿⣿⣷
⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿
⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣄⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡄
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⡿⢋⡁⡄⢀⡈⡝⢿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿⣿⣦⣀⣀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣿⣤⣀⣀⣀⣀⣤⣿⠀⠀⠀⠀⠀⠀⠀⣸⠋⠄⢀⠤⠈⡃⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠘⠉⠉⠉⠉⠉⠋⠉⠙⠋⠛⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣾⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣒⢾⣦⣀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡄
⠀⠀⠀⠀⣰⣶⣆⠀⠀⠀⣠⡴⠶⠶⠂⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣟⠿⣽⠟⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⣇⣀
⠀⠀⠀⣼⣿⣿⣿⣧⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⡉⠈⠙⠻⠆⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣁⣁⣈⣉⣤⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⣄⠠⡀⠀⠀⠀⠀⠀⠀⠀⢈⣛⣿⣴⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣠⠷⠂⠀⢀⡀⣀
⠀⠀⠀⣀⣀⣀⣰⣁⣄⣸⣀⣀⣀⡀⠀⠀⠀⠀⠘⠟⠱⣷⠄⢰⡐⣦⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋⠀⠙⣿⣿⣷⠀⠀⠀⠀⠀⢀⣼⣿⠟⠀⢠⠼⠿⠟⠉
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢹⡆⠘⢷⡜⠟⠀⠀⠀⠀⠀⠀⢻⠛⠛⠦⣀⠴⠛⠋⡟⠀⠀⠀⠀⠀⠙⠃⠀⣶⣤⣤⣤⡀
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣀⣀⣀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠖⠛⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠒⠂⠀⠀⠀⠀⣼⠛⣷⣼⣿⣶⡄⠹⣆⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿
⠀⠀⠀⣤⣤⣤⣤⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⡧⠀⣿⠀⠀⠀⠀⠀⠀⠛⠿⠿⢛⣡⣴⡶⢶⠂⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡉⠻⠿⠛⢷⣼⠃⠀⠀⠀⠀⠀⠀⠿⠛⣋⣉⣱⣤⠽⠖⠂⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢈⣿⣀⣀⣀⠀⠀⢠⡤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠭⡝⡁⣠⡾⠃⠀⠈⠉⠉⠉⠉⠛⠀⠀⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠂⠥⠚⠁⠂⠠⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢲⣡⡀⠀⠀⠀⠀⠠⡀⠀⣉⠖⠒⠲⠶⡖⠒⠒⣦⠀⠀⠀⠀⠐⠒⢍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠣⢬⠯⢄⠜⠀⡠⠒⠒⠢⣤⠊⣡⠖⠋⠉⢳⠈⠀⠀⣭⡀⠀⠀⠀⠀⠘⢦⡙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢙⠚⠒⢌⠢⢐⡁⠀⠀⠰⢎⠟⠲⠀⠀⢀⠏⠀⠀⢤⣌⠰⣀⡠⠖⠋⠙⣆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢄⠀⢀⡀⠀⠀⠀⠀⢧⡜⠀⠀⡨⢦⠭⡉⠐⠒⠂⢻⡀⠠⠤⠒⠧⠀⢀⡴⢃⠜⠃⠨⠚⠛⢉⠞⠁
⠀⠐⡆⠀⠀⠀⢐⠃⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠣⣀⠈⠀⠀⠀⠀⠁⠀⠀⢀⡠⠖⢣⠔⠁⡰⠂⠀⠐⠊⡡⠢⡤⢄
⠀⠀⠘⠀⠉⠁⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠤⠤⠒⠊⣁⠤⠒⣁⠴⢋⠿⠳⠀⠀⠼⢱⡜⠒⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢠⡄⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡝⠒⠂⠤⠄⢀⣀⡀⠤⠤⠒⠉⠁⢠⢋⠟⠀⠀⠈⠧⠾⠥⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣽⣻⠰⢾⢺⣆⡶⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣶⣄⠀⠀⠀⠀⠀⠀⢀⡜⠁⠐⠚⠁
⠀⠀⠀⡀⠀⠀⠀⢠⠘⣣⠖⢌⢻⡜⣶⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡀⡻⢿⣷⣄⠀⠀⠀⡠⠊
⠀⠀⠀⠇⠀⠀⠀⠀⣾⡙⠙⠛⢩⡭⠄⠀⡇⠀⠀⠀⠀⠀⠀⢠⠊⢀⣠⠊⠊⠀⠀⠙⡿⢧⣠⠞
⠀⠀⠀⢁⠀⠀⠀⠀⠉⠈⠄⠙⢢⠀⠀⠀⡗⠢⢄⡀⠀⠀⠀⠰⣀⡘⠙⠃⠀⢀⠴⡟⠳⢄⠁⠀⠀⠀⢀⢀⡀⢀⠎⠸⠧⠀⠀⠀⡀⠀⡴⢊⢝
⠀⠀⠀⠈⠂⠄⢠⡐⠒⠙⠀⠉⠡⠋⠠⠴⠃⠀⠀⠈⠓⠤⣀⠀⠈⠣⣀⡀⣀⣀⡜⠉⣷⠘⡦⠀⠀⢀⢴⡢⡘⠫⡌⠀⠀⠀⠀⠀⡠⠂⢔⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠐⠂⢤⠀⡄⣤⣄⡀⡀⠀⠀⠉⠒⠤⣀⡀⠤⠘⣗⠉⢒⣶⠁⡰⢐⡥⠊⠀⠈⠀⠌⠲⡐⠒⡲⡊⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠁⠒⠂⠦⠤⠤⠤⠜⠿⢒⠇⠀⠀⠀⠀⠈⠉⠲⢥⡈⠉⣩⠙⡡⡞⠀⠀⠀⠀⠀⠀⠀⠑⠌⠻⣮⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠳⢍⡀⠈⢆⡀⠤⣄⢀⣀⣀⡠⢒⡳⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⢲⡊⠐⠈⠀⣀⠔⡩⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠉⢀⡠⠌⠀⠀⢴⣊⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠊⠁⣀⠠⠐⠂⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠁⠀⡀⠤⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠊⠉⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣿⣾⣷⣿⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠟⠛⠋⣉⣁⣀⡀⢀⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠰⢲⡖⠀⢠⠀⠀⠀⠀⠀⠀⠐⠐⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣔⢾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠰⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⢀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠀⠀⠀⠂⠀⠀⠀⢀⣀⣩⠓⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⡐⠦⣀⠤⠄⠀⠀⣀⠜⡰⠠⠃⡀⠋⠹⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠛⢁⣼⣿⣿⡿⠛⠁
⠀⠀⠀⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠃




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⠘⣟⠛⣿⢛⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⣹⣯⣯⣿⣼⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣿⣦⣨⣿⡿⠴⠿⠶⢧⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⠷⠟⠛⠛⠉⢙⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠟⠛⠉⢉⣀⣀⣀⣀⣀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⠉⠀⠀⣤⣤⣶⣦⣤⣴⣿⣿⣿⣿⣿⣿⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣿⣿⣿⣿⣿⣿⣯⣿⣿⡟⠛⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡉⡿
⠀⠀⠀⠀⠀⢠⣶⣞⣿⣽⣿⣿⣿⣿⣿⣿⣿⣏⠁⢹⠇⠘⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⢿⠇⠩⠉⠁
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⢃⠚⡸⠿⣷⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⠿⣿⣿⣿⣿⡿⣷⣻⣿⣿⣿⣿⡀
⠀⣀⣴⣾⣿⡿⠟⠁⠀⠀⠉⣡⢯⣤⡘⠾⣷⣶⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠾⣿⣯⣿⣿⡴⠊⠑⣡⣿⣿⣿⠟⣻⣿⣿⣿⣿⠇
⠿⠿⣿⠿⠯⠒⠈⠁⠀⠠⠖⠛⠊⠉⣠⣾⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠟⠛⠛⠋⠁⠀⠀⠉⠉⠉⠉⠀⠾⡿⠿⡟⠉
⠀⡀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠁
⠀⢆⣀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢦⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣤⣀⡀⠀⠀⠀⠀⠙⢻⣿⣿⣶⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠙⡿⠤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠻⣆⠀⠀⢰⣐⣿⡿⢯⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠈⠉⠳⠤⣤⣀⣀⣀⣀⣀⣀⠀⣀⣠⣀⠠⣴⣶⣾⠛⠉⠀⣈⣩⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣊⠷⣤⣤⣤⣜⠃⠀⠀⠀⠀⠈⠉⠉⠉⠉⠩⠝⠿⢗⢲⣼⣿⣯⣯⠀⢠⡾⣯⡿⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣄⠀⠀⡈⠙⠻⢶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⢉⠁⠀⠘⢁⣀⡀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡷⢶⣽⣦⡀⠀⠈⠉⠉⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⣸⣵⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡅⠋⠀⠀⠘⠛⠛⠛⠻⠶⠤⠤⠤⠶⢿⣿⣿⣿⣦⣤⣀⣤⣬⣽⣶⣿⣿⣿⣿⣿⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣺⡿⠀⠀⠀⠀⠰⠠⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠆⠶⠀⠀⠀⠀⠀⠀⠀⠹⡛⢻⣽⣮⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⣆⣁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠂⠈⠀⠀⠀⠀⠀⠀⠈⠌⡀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠈⠈⠈⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⣠⡈⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠶⣋⣹⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡛⣋⣿⣦⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣿⡯⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣺⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣘⡛⢿⣷⡀
⠀⠀⠀⠀⠀⠀⢶⣆⡀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡄⠈⠁⠀⠀⠀⠀⠀⠀⣼⣿⣇⣹⣿⣤⣤⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠗⠻⡄⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠩⢤⣍⣀⣀⣤⣀⡤⣄⣀⣾⡿⠿⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠉⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢡⠈⣇⠈⠀⠁⠀⠈⡈⠌⠀⠉⢡⠈⣇⠈⠀⠁⠀⠈⡈⠌⠄⠁⢡⠈⣇⠈⠀⠁⠀⠈⡈⠈⠄⠉⢡⠈⣇⠈⠀⠁⠀⠈⡈⠈
⠸⣆⠘⠀⠀⠀⠀⠀⠈⠰⡠⠀⠸⡆⠘⠀⠀⠀⠀⠀⠈⠰⣠⠀⠸⣆⠘⠀⠀⠀⠀⠀⠈⠰⣠⠀⠸⣆⠘⠀⠀⠀⠀⠀⠈⠰⡠
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠁⣰⠀⡤⠀⢠⠠⢃⠀⠀⠀⠀⠁⣰⠁⡆⠀⠠⠡⢃⠂⠀⠀⠀⠁⣼⠀⡄⠀⢠⠡⢃⠀⠀⠀⠀⠃⣰⠀⡄⠀⢠⠡⢡⠂
⠀⠀⠀⠃⣰⠇⠀⣔⠇⠁⠀⠀⠀⠀⠀⠃⣰⠇⠀⡔⠂⠁⠀⠀⠀⠀⠀⠃⣰⠃⠀⡔⠇⠁⠀⠀⠀⠀⠀⠃⣰⠃⠀⣄⠇⠁
⠀⠀⠀⠀⠈⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠃
⠀⠀⠀⠀⢠⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⡀
⠀⠀⢀⢠⠙⠀⢰⠏⢠⠀⠀⠀⠀⠀⢀⢰⠛⠀⢰⠏⢠⠀⠀⠀⠀⠀⢀⢐⠝⠀⢰⠇⢠⠀⠀⠀⠀⠀⢀⢀⠝⠀⢰⠇⢠
⠀⢀⢃⣂⠃⣀⣘⢀⡏⢠⠀⡀⠀⣀⢁⢂⢃⣀⣘⢀⡏⣀⠀⡀⠀⢀⢁⣂⠃⣀⣘⢀⣏⢀⠀⡀⠀⣀⢃⢂⢂⣀⣘⢀⡏⢀
⢹⡄⠁⣸⠁⡍⠉⢠⠡⢡⣡⠈⢹⣄⠉⣸⠁⡏⠉⢠⠡⢃⣡⠈⠹⣄⠁⣼⠁⡌⠉⢠⠉⢁⣡⠈⢹⣄⠁⣸⠁⡍⠉⢠⠩⢁⣡
⠈⠋⠄⣇⢰⠇⠀⣔⠣⡏⠹⡆⠀⠋⠆⢇⣰⠇⠀⣄⠣⡏⠸⡆⠀⠎⠄⣇⢰⠇⠀⣔⠣⡎⠸⡆⠀⠎⠆⣇⢰⠃⠀⣄⠃⡏⢹
⣀⠘⡀⡈⡙⢀⡸⠋⡀⢹⡀⢃⣀⠐⡐⡈⡙⢀⣸⠋⣀⢻⡀⣃⢀⠐⡐⡈⡙⢀⣸⠋⡀⢹⡀⣃⢀⠐⡐⡈⣙⢀⣸⠋⡄⢹⡀
⢡⠈⣇⠈⣠⡏⠁⣌⡈⠌⡄⠁⢩⠈⣇⠈⣠⡏⠁⣌⡈⠌⠄⠁⢩⠈⣧⠈⣠⡏⠁⣌⡈⠌⠄⠉⢡⠈⣧⠘⣠⡏⠁⣌⡈⠌⠄
⠸⡆⣸⢢⠘⠀⢰⠏⢱⠰⡰⠀⠸⡆⣸⢢⠙⠀⢰⠏⢱⠐⣰⠀⠸⡆⣸⢲⠝⠀⢰⠇⢱⠐⣰⠀⠸⣆⣸⢰⠝⠀⢰⠇⢱⠐⣠
⠀⢋⢁⣂⠃⣀⣚⢀⡏⢀⠙⣆⠀⢋⢡⣀⠃⣀⡘⢀⡟⢀⠙⣇⡀⢋⢁⣂⢂⣀⣚⢀⡏⢀⠙⣇⡀⢋⢁⢂⠃⣀⣘⢀⡏⢀⠙
⠈⠠⠁⣸⠁⡍⠉⠩⠡⢁⠉⠀⠈⠀⠁⣸⠁⡏⠉⢠⠡⢁⠉⠀⠈⠀⠁⣼⠁⡏⠙⠙⠋⢫⠟⠉⠙⢽⡯⣿⢣⣿⡿⣹⣿⣿⡏
⠀⠀⠀⠃⢰⠇⠀⣔⠇⠁⠀⠀⠀⠀⠀⠃⣰⠇⠀⣔⠁⠁⠀⠀⠀⠀⠀⠃⢸⡇⢁⣬⠥⠉⠀⠀⠀⠀⠈⠉⣸⡿⢷⣿⠿⠋
⠀⠀⠀⠀⠈⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠃
⠀⠀⠀⠀⢀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡀
⠀⠀⢀⢠⠙⠀⢰⠏⢠⠀⠀⠀⠀⠀⢀⢰⠝⠀⢰⠏⢠⠀⠀⠀⠀⠀⢀⢰⠘⠀⢰⠏⢠⠀⠀⠀⠀⠀⢀⢰⠝⠀⢰⠇⢠
⠀⠀⢁⠂⠃⠀⠘⠀⡏⢀⠀⠀⠀⢠⢡⢂⠃⠀⠙⠀⠟⢠⠀⠀⠀⠠⢡⠂⠃⠀⡸⠀⠏⢀⠀⠀⠀⠀⢡⠂⠃⠀⠘⠀⡏⢀
⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠊⠆⡀⠀⠀⠀⠀⠀⡄⠸⡆⠀⠏⠆⠀⠀⠀⠀⠀⠀⡄⠹⡆⠀⢋⠆⡀⠀⠀⠀⠀⠀⡄⠸⡆⠈⠊⠆⡀⠀⠀⠀⠀⠀⡄⠸
⣂⠐⡀⡘⡀⠀⢀⠀⡀⢹⡀⢃⢀⠐⡰⡈⡀⠀⢀⠀⡀⢹⡀⢃⢀⠐⡐⡈⡀⠀⢀⠀⡀⢹⡀⢃⢀⠐⡐⡈⡀⠀⢀⠀⡀⢹⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢰⣾⡇⠀⣰⣶⣆⢰⣿⣿⠀⣰⣄⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⣾⣿⣷⢠⣾⣿⣿⡆⣿⣿⣿⢸⣿⣿⡀⣿⣿⣀⡀⣿⢿⡇⢰⣾⣶
⠀⠀⠀⠀⣀⣿⣿⡇⢰⣶⣶⡆⣿⣿⣿⢸⣿⣿⣿⣧⣿⣿⣿⣸⣿⣿⣷⣿⣿⣿⡇⣿⣿⡇⢸⣿⣿⡆⠀⣠⣤⡀
⠀⠀⠀⢸⣿⣿⣿⡇⣿⣿⣿⣷⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⡇⣾⣿⣿⣷⣼⣿⣿⣇⣀⡀
⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠀⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⢀⡀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⣿⣿⠀⣾⣿⣧⣤⠀⣿⣿⣿⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⢠⣿⣿⡇⣾⣿⣶⣿⣿⡄⣿⣿⣿⣿⡆⣿⣿⣿⣰⣿⣿⡇⢀⣀⠀⣿⣿⣿
⠀⠀⠀⠀⣤⣤⣤⢸⣿⣿⣾⣿⣿⣇⣿⣿⣿⣿⣿⣱⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⡇⣿⣿⣧⣿⣿⣿⣠⣶⣤
⠀⠀⣶⣦⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣶⡆
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠎⢁⣿⠄⣀⣀⣀⣠⡄⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⠢⢤⡞⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠄⠌⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣒⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢍⣖⠋⠀⠀⠀⠀⠀⡀⠀⣠⠄⠒⣤⠀⠀⠘⠀⠀⠀⠀⠐⢎⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠒⠓⠀⠀⢀⠀⠀⠰⢿⠎⠆⠀⢀⠏⠀⠠⣌⠀⠀⠀⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⢠⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠀⠐⠀⢹⡀⠐⠒⠓⢀⡠⢊⠔⠁⠀⠀⠉
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢉⠗⢁⣴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⢀⡀⠀⠀⠀⠀⢐⣈⠤⠚⢡⢮⠉⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠶⢩⡯⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⣄⠀⠉⠀⠈⠉⠀⡠⠂⠠⠗⠁
⠀⠀⠀⡄⠀⠀⠀⠁⠀⠂⠓⣸⣷⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠻⣷⣄⠀⠀⢀⠔
⠀⠀⠀⡁⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠈⠙⠢⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢠⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢜⠀⠁⠀⠄⣉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠉
⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠒⠄⡈⠒⠒⠒⣾⠤⠟⡆⠀⣀⢄⡠⠀⠠⠀⢤⠀⣀⣀⠄⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃⠀⠤⠈⢀⣉⣉⠷⠤⡄⠀⠀⠀⠑⠢⢄⡈⠲⢖⠛⣠⠇⠈⠀⠀⠀⠀⠀⠑⢤⣂⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠐⠙⢉⠀⠀⢀⠠⡀⠀⢀⣀⢄⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠅⠂⠀⠈⠀⠠⠊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠈⠁⠀⠀⠀⠀⠸⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠊⠁⡀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⠁⢀⠠⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣧⡀⠀⢀⡴⣥⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠚⠋⠉⠁⠀⠀⠀⠈⠀⠈⠀⠀⠀⠈⠉⠛⠲⢤⣄⡀⣠⠞⠁⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠋⣠⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠛⠲⠶⢤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣡⣾⠿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⡴⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣧⡈⠁⢠⠀⠀⢠⡄⠀⢀⡴⠒⢦⠀⡞⠉⢱⠀⠀⣿⠉⠁⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣩⠷⠸⣆⣀⡼⣄⣀⠸⣄⣤⣾⠀⠓⠔⠛⠲⠀⠙⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣄⣀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⠀⠀⠀⣀⣠⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢷⣶⣶⣶⣶⣶⣶⣶⣶⣶⠾⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣒⡾⠟⠓⠒⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢸⠁⠀⠀⠀⢀⡤⠖⠛⠉⠙⠒⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⠀⠀⠀⠀⢸⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠸⡄⠀⠀⠀⠀⠳⣄⠀⡟⢀⣀⣠⣤⣄⡀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⢦⡀⠀⠀⠀⠈⣳⣿⢯⡁⠀⠀⠀⠈⠳⣄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠉⠙⠛⠛⠉⠁⣸⠀⠈⢦⠀⠀⠀⠀⠘⡆⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠈⣇⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠤⣄⣀⣠⠤⠞⠁⠀⠀⠀⠀⡏⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⢳⣄⡀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⠔⠉⠀⠉⠒⠦⠤⣤⣤⣶⡯⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡴⠋
⠀⠀⠀⢀⡴⠋⠀⢀⡴⠋
⠀⠀⢴⡋⠀⢀⡴⠋
⠀⠀⠀⠙⠶⠋

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

⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⢀⠴⡄⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⢰⣟⣽⣿⣢⡀⠀⠀⠀⣾⣿⣿⡄⠀⠀⠐⣿⣿⡄⠀⠀⢀⡈⠀⠘⠀⠀⢰⡟⣇⠉⣧
⠀⠀⠀⠀⠌⠁⠌⣿⡕⠁⠀⠀⢸⣿⣟⣻⣿⠀⠀⠘⠛⣏⢳⠀⠀⠎⠀⡔⠂⠀⠀⢸⣭⣙⡄⠞⣠
⠀⠀⠀⠀⠚⠀⠠⢼⣷⡖⠀⠀⠘⣿⣾⣷⠟⠀⠀⣀⣤⣿⣾⣇⠀⠄⠀⠀⡀⠟⠀⠀⠳⢬⣿⡀⡏
⠀⠀⠀⠀⢀⠀⠀⢸⣿⢻⠀⠀⠐⢿⣿⠏⠀⠀⠀⡿⣿⣿⣛⢽⠀⠀⠄⠀⠁⠀⠀⠀⢘⣦⣽⡇⢀⡄
⠀⠀⠀⠀⠈⠀⠀⣼⢥⣾⠀⠀⠀⠀⣿⠂⠀⠀⠀⡆⢸⡟⡂⢸⠀⠀⠀⠀⠐⠀⠂⠀⠀⠠⢙⡇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣎⣀⡤⠀⠀⠀⠀⡿⠀⠀⠀⠀⡵⠜⢉⡐⠁⠀⠀⠀⣠⣀⠐⠁⠀⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠰⠿⠍⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠹⣾⣿⠛⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠐⢿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠀⠀⠀⠄⠀⠀⠈
⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠂⠀⢠⠛⠀⠀⠀⠀⡀
⢠⡀⠀⠀⠀⠀⠀⠀⠑⠀⠠⠀⠘⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠎⡀⢠⣿⠗⢀⠀⠀⢀⠀⡴⢀⠀⠰⡆⠂⢀⣡⠀⣀⣰⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⣀⠀⡀⢠⠌⣴⠀⠀⠀⠐⠀⢚⠁⠀⣀⠀⢀⡾⣸⢳⣌⠈⣧⣦⠀⢡⣯⣾⢁⡤⣅⣾⢻⣴⠆⣿⣰⣿⣿⣦⣰⡟
⠀⠀⡆⠀⢀⠀⠀⡄⠀⠁⢰⠠⠘⠁⠐⠃⢸⠺⠀⣄⠀⠀⣇⣴⣼⣶⡖⣧⣼⣿⠃⣿⣿⣿⡿⣿⡿⣶⣸⣿⡟⣺⣧⣿⣿⢸⣿⣸⣿⡇⣿⣿⣿⣿⢹
⠀⠀⠀⠀⠈⠂⠈⠃⠀⠃⢐⣇⠀⠀⠆⠸⣈⠐⠁⡃⠄⠰⡇⢸⣿⢉⡧⢹⣽⣷⢸⡟⡿⠿⠁⣿⢸⣿⠘⠻⠋⠻⠛⢾⠿⠘⠇⠚⠹⠇⠿⠏⠻⠛⠾
⠂⠀⣀⢆⠀⢠⠀⠒⠄⠐⣘⠀⣧⡀⢀⣇⡏⢸⡆⢻⡂⡆⡏⢹⠛⠚⠙⠈⠉⠁⠈⠐⠁⢀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡂⠀⡀⠀⠀⠤⡀⠀⠀⢀⠀⣀
⡰⠀⡄⠸⠀⠈⢄⠀⡦⠼⡿⠀⡼⡦⠀⣿⢠⣎⡏⠹⠁⣿⡅⠊⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠈⠀⠐⠒⠀⠄⠀⠀⠈⡠⠀⠈
⣠⢸⣷⠀⡀⢐⠨⠀⠀⠀⠳⠤⠃⠀⠀⢉⠈⠉⠉⠙⠈⠻⠷⠀⠒⠀⠀⠀⠀⠀⠁⠠⢄⠐⠒⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⣀
⡋⢸⡏⡀⡇⠘⣀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⣀⠀⠀⠀⠀⠠⠀⠈⠑⠒⣄⠈⠁⠀⠄⠀⠐⠀⠃⠀⠀⠀⠀⠘⠀⠐⠀⠂⠀⠀⠀⠀⠀⢀⡀
⣃⣘⣃⣄⣃⣀⣈⣀⣀⣉⣄⣈⣀⣀⣀⣀⣀⣀⣈⣑⣂⣐⣒⣀⣀⣀⣰⣓⣂⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣠⣤⡶⠾⠟⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠶⠖⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣴⡾⠛⠉⠀⠀⠀⢀⠴⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⢰⠄⠰⠃⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣴⠟⠁⠀⠀⠀⣠⢴⠛⣰⠃⠀⢀⡆⠀⠀⠀⡀⠂⠀⠀⣄⠀⠀⠀⠀⠈⠐⠛⠁⠀⠠⠂⠀⠈⢻⣆⠀⠀⠀⠀⠸⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⢀⡴⠋⠀⢸⡇⠡⠀⠀⢀⢁⢰⡧⠂⠀⡜⠓⠠⡌⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠙⣧⡀⠀⠠⠀⣀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠐⠀⣠⡞⠀⠀⠀⠀⡍⣦⠀⡠⠉⠂⢬⣄⠠⠀⠄⡀⣠⠀⠘⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡂⠁⠝⣷⠀⠀⠀⠉⣦⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠦⣄⢰⠟⠁⠀⠀⠀⠀⠊⡠⠊⠁⢠⣤⠛⠉⢷⣀⠚⠢⠿⠀⣴⠃⠐⠌⠀⠀⠂⠀⠀⢀⠊⠀⠀⠂⠁⠈⡹⠆⠓⡄⠶⠿⠀⡀
⠀⠀⠐⠒⠂⠀⠀⡀⢸⡇⠀⢀⡇⠘⠀⡀⠀⠀⢀⡴⢋⣠⡷⠖⣾⢀⠙⠦⣈⣁⠴⠇⢶⠀⠡⡴⠖⠀⡼⠂⠀⠀⡀⢀⡀⠉⠀⠒⠤⡾⡄⠓⡴⠓⠂⢐⡠⠃
⠀⠀⠀⠀⠀⢈⡄⠀⣾⠇⠀⡾⠀⣾⢀⡡⢂⠔⠀⠀⢀⠒⣤⠀⠈⡻⠖⠁⣤⠁⠀⠀⠀⠛⠠⠢⢴⠖⠁⠊⡠⠆⢱⠊⠀⡀⠘⠓⡀⠋⢡⡖⠉⡄⡏⢛⡃
⠀⠀⠀⠀⠆⠘⠀⠀⢹⠀⢸⡇⠀⢈⠋⣠⡁⠴⢄⣀⣹⠛⠠⠤⠀⡱⠆⠈⣤⠀⠀⠀⠀⡘⠁⡤⣄⡁⠤⡈⠡⠄⠀⠀⠀⠑⠴⠊⠐⠛⣇⡑⠚⠠⢈⡛⣅
⠀⠀⠀⠰⠀⠂⠀⠀⢸⡄⢸⡁⢀⠀⡴⠋⠉⠀⢴⠞⠁⠉⠒⠒⠂⢷⡤⠄⠀⢵⠀⡶⠉⠁⡱⢤⠒⠐⠒⠒⠁⠀⠀⠀⠀⠀⠣⠨⢄⢹⢀⡉⢐⢂⡨⠿⡤⠐
⠀⠀⠀⠀⠀⠃⠀⠀⢸⣇⠈⢧⡎⢰⢀⢀⠠⠖⠁⠀⠀⠀⠀⠀⠀⠁⠁⢰⠆⠀⡄⠠⡤⠹⢆⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠲⠀⢎⣉⠴⣌⠀⡴⠆⠉
⠀⠀⠀⠰⠀⠀⠠⠀⢈⣿⠀⣼⢦⡟⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠇⠀⠴⠀⠈⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠒⠈⠈⣶⡈⠂⠛
⠀⠀⠀⠀⠁⠀⠀⠰⠎⠹⣧⠇⠰⠉⡄⡄⠀⠐⡐⣀⣓⣂⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠄⠀⠈⣚⣀⣁⣀⡐⠀⢀⠀⢀⡄⠀⣸⠃⠙⣌⢮⠆
⠀⠀⠀⠀⠀⠘⠋⢸⠧⠄⠃⡜⠚⡃⣣⢀⣴⡾⠛⠋⠉⠉⠛⠛⠶⣄⠠⠀⠀⠀⠀⠀⠀⠠⠀⣠⡶⠟⠛⠉⠉⠉⠉⠙⠳⡄⠀⢼⠁⢠⣿⣀⠂⠿⡎
⠀⠀⠀⠀⠀⠀⠢⣤⠂⠐⢠⠛⢺⠉⣧⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⣋⠀⠀⠀⠀⠀⢰⠄⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠸⣀⣾⠋⠉⡦⠀⠇⡇
⠀⠀⠀⠀⢠⠊⣠⠈⢀⠶⠠⣀⣼⠁⢳⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠖⠁⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣈⡿⢇⠀⠀⠀⡰⠀⠆
⠀⠀⠀⠀⠈⣰⠄⢠⡽⠠⠀⢉⣠⠅⡇⣇⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠣⠍⠀⠀⠀⠀⠀⠈⠃⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡂⣷⠃⠂⠄⠁⠤⡪⠄
⠀⠀⠀⠀⠈⠛⢶⠋⣥⠀⠆⢀⡿⢘⠓⠘⠻⠶⢶⣤⣤⣶⠾⢻⠁⠂⢃⠔⢀⠠⠉⠀⡀⠐⠀⠛⠀⢙⡛⠶⠶⠶⠶⠶⠛⠁⠀⢀⠹⣎⢄⠀⠀⠀⡖⠉⠀⠐
⠀⠀⡠⠊⠀⡘⠿⢇⠙⠒⢀⣿⠃⢈⠐⢈⣨⡀⠐⡀⠐⠤⠀⡀⣈⡌⠁⠀⢈⠖⠛⠶⣦⠀⠀⢐⠶⠆⠬⣠⢄⣒⣀⠀⡀⠠⣴⠦⠤⢿⣆⠀⢄⡀⠀⢀⠄⢀
⠀⡜⠁⠀⢀⢹⠀⠀⠇⠀⡾⠁⠠⠊⡀⠀⠀⠈⠂⠒⠉⠐⠿⠄⠀⠊⠀⢠⡟⠀⠀⠀⠘⣧⠀⠁⠐⠁⠁⠀⠀⠀⠀⠤⠔⠉⠀⢤⣄⣀⣿⡅⠂⠤⠈⠠⠐⠁
⠀⠿⠦⡴⢾⠀⠀⠀⠀⣾⡧⢊⡶⢀⡧⣤⡰⠂⠀⢠⡖⠒⠀⢄⡀⠘⢀⡾⠀⠀⠀⠀⠀⠸⣆⠀⠀⠃⠀⢀⠁⠀⠀⠀⢉⡰⠖⠋⠉⠉⢹⡇
⠀⠀⠀⠠⣾⢀⠀⠀⠀⠛⣷⡅⠚⠉⠀⠀⠀⠉⠁⠜⠀⠂⠀⠀⠀⠀⡾⠁⠀⢀⣠⣄⡀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣤⠴⢦⣿⠃
⠀⠀⡀⠀⠙⠘⡀⠀⠀⢠⠘⢷⣄⠀⠁⠞⠀⡤⠊⠀⠀⢀⠘⠄⣐⠂⡉⠓⠛⠋⠉⠉⠛⠛⠛⢁⠀⠠⣀⠒⢀⠀⠀⠀⠈⠀⠶⢀⣤⡾⠃
⠀⠘⢧⠀⠀⢃⠹⡄⠂⣀⢆⠀⠙⠳⠶⢶⣤⣤⣤⣤⣰⣡⡆⣸⣯⣥⣇⠀⠀⡀⠀⢀⡀⠀⠀⣨⣄⢲⡿⢺⣤⡧⣤⠴⢶⡶⠚⠛⠉
⠀⠀⠘⢷⡀⠀⢂⠙⡀⠀⠈⠣⡀⠀⠀⠘⡟⠡⢿⡍⠉⢻⣯⣭⣝⣡⣄⣘⣁⣤⣉⣡⣤⣁⣡⣤⣄⣥⣤⣾⠁⠀⣼⠦⣸⡇
⠀⠀⠈⠀⢡⠀⠀⠳⢌⢆⠀⠀⠀⢄⠀⠀⡿⢢⣸⡇⠀⢸⡇⡇⠈⡇⠀⠙⠁⠀⢸⠁⠀⠉⠁⠀⡏⠀⠀⢸⠀⠀⣿⠠⢻⠁
⠀⠀⠀⠀⠀⠄⠀⠀⢈⠇⡱⠄⠀⠀⢡⠀⢱⠎⣸⡇⠀⠘⡷⢷⠤⡷⣤⡴⣦⠤⢼⡤⢤⡶⢤⡴⠿⠲⡿⡇⠀⢀⣿⡀⣿
⠀⠀⠀⠀⠀⢰⠀⠀⡞⠀⠐⡌⢷⡶⢈⠀⢸⡶⢉⣷⠀⠀⣧⣼⣀⣄⢸⣄⣀⣀⣀⣀⣀⣅⣼⣤⣤⣤⣧⡇⠀⢸⣇⡰⡟
⠀⠀⠀⠀⠀⢠⠀⠘⡄⢀⡶⠛⠀⡃⠈⠆⠸⣇⡌⢉⡷⠾⡛⠚⠟⠉⠉⠩⢭⠭⣭⡭⣭⢩⢥⡖⡀⠒⠒⢛⠚⢯⣀⣤⡇
⠀⠀⠀⠀⠀⠈⠖⠀⢧⡎⡇⠀⢸⠀⠀⡼⠀⢯⣀⠀⠡⠖⠁⠉⠀⢠⡐⠎⡣⣼⠋⠙⢶⣩⢞⡰⠡⠍⠼⠹⠉⠯⠍⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠘⠀⢀⠃⠀⠘⢷⣼⣐⡀⢙⣀⣉⠤⠖⢈⣭⣹⢤⠆⠛⠶⢋⣹⠛⣲⠶⢤⠶⢶⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⡠⠂⠀⠀⠀⠀⠉⠻⣯⣷⠶⡶⢶⡛⠻⠤⠘⠁⢴⣄⡀⠀⠈⠉⠙⢒⣾⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣀⠀⠀⠀⠲⠀⣶⠈⠴⠀⢀⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠒⠛⠓⠒⠋⠉⠁
⡇
⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠉⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠟⣧⠀⠀⠀⠀⠀⠹⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣦⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣧⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⢧⡀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣵⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢷⡀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡄⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣽⣆⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⣦⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣧⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣷⡀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⢀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢟⣦⠀⢠⠔⠋⠀⠈⠀⠘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡟⠃⢀⡤⠄⠀⠀⠀⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣼⠏⢀⡀⠀⠀⠠⣤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⠟⣤⡿⢇⢠⠎⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⡀⣸⠃⠸⣿⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠹⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢸⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⡇⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠁⠀⠀⠀⠀⠈⠉⠛⠋⠉⠶⠶⢤⣄⣼⡇⠀⠺⠀⠘⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣠⠶⠆⠙⠉⠋⠉⠉⠁⠰⠶⠶⠦⣀⣀⡀⠀⡿⠉⠀⠤⠀⣀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡿⠃⠀⠆⠀⢠⠀⠀⢻⡤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⡀⠈⠍⠓⢤⣀⠀⠒⠂⠠⠄⠐⠒⠒⠒⠂⠉⠀⢀⡾⠀⠀⠈⠀⢀⠈⢡⡈⠙⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠙⢦⡄⢡⡀⠈⠑⢢⣤⡀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠀⠀⠀⡷⠀⢸⡀⠘⠷⣤⡀⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⠈⠁⠾⣦⡀⠀⠈⠛⣉⠰⠆⢀⣀⠈⠉⠀⠀⠀⠀⠐⠁⠀⢸⠀⠀⠀⠀⠈⠑⢦⡀⠈⣷
⠀⠀⠀⠀⠀⠀⢀⠉⠀⣀⡀⠀⠀⠀⠀⠀⠈⠁⠶⢄⣈⠳⠄⡀⠀⠈⠙⠿⢂⡤⠶⠤⠀⢰⣏⣀⣈⣀⠀⠀⠀⠀⢙⡶⢿
⠀⠀⠀⠀⠀⠀⠸⠀⢰⣉⣙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠶⣄⣀⡀⠀⠀⠒⠤⠄⣀⠀⠀⠀⠀⠙⠋⠠⠤⢄⣠
⠀⠀⠀⠀⠀⠠⡇⠀⡄⠻⡌⠀⠈⠙⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠒⠀⠠⠤⠴⠶⢦⣤⡄⠀⠠⣤⠤⡏
⠀⠀⠀⠀⠀⡄⠀⣀⢣⣄⡿⠸⠀⡆⢠⢤⡉⠑⠒⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠃⠀⠙⢦⡀⠀⢧⡞⢠⡏⠀⢠⠿⠀⢠⣄⠀⠉⠉⠐⠶⠦⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠦⣀⠀⢠⣤⢲⡼⣶⠀⣼⡜⠃⠀⢀⠘⠦⣀⣤⠀⡰⠆⠀⣈⡉⠉⣉⠉⠘⡇⠀⠀⢸
⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠈⠛⠦⢄⣀⠀⠛⠐⠏⡀⠀⠀⢸⡦⠖⣹⣧⠀⠙⣦⠀⣭⠀⣸⠤⠀⣠⠁⠀⠀⡆
⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠤⠤⢀⣀⡀⠈⠀⠸⠇⠙⠠⡤⠏⠀⠟⠀⢽⡀⢀⡟⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠒⠒⠂⠀⠠⢤⣤⡤⠀⠀⠀⠚⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠲⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠤⠤⠄⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠤⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠉

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢰⠇⢠⠀⠀⠀⠀⠀⡀⠎⠂⠀⡾⢀⡄⠀⠀⠀⠀⢀⠰⠙⠀⢰⠃⣠⠀⠀⠀⠀⠀⡀⠊⠃⠀⡾⢀⡄⠀⠀⠀⠀⠀⠀⠑
⡈⣀⣅⣠⣀⣀⡀⣘⣰⣘⣀⣀⢁⣸⣀⡀⣀⣀⢀⣂⣆⣂⣀⡈⣀⣏⣠⢀⣀⡀⣀⣀⣐⣀⣀⢁⣸⣁⡄⣀⣀⢀⣁⣂⣀
⠀⠈⡌⠈⠀⠀⣆⠸⡆⠁⠀⠀⠀⢡⠁⠁⠀⢰⡀⢳⠈⠀⠀⠀⠈⡈⠌⠀⠀⣆⠘⡆⠀⠀⠀⠀⢠⠡⠁⠀⢠⡀⢷
⠀⠀⠀⠸⣣⠀⢹⡄⠁⠀⠀⠀⠀⠀⠀⢆⡆⠈⣧⠈⠀⠀⠀⠀⠀⠀⠰⣣⠀⢹⡄⠉⠀⠀⠀⠀⠀⠀⢏⡆⠈⣧⠈⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠶⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡠
⢸⠃⡴⠀⠀⠀⠀⢀⠄⠀⠁⠀⡟⢠⠆⠀⠀⠀⠀⠠⠐⡈⠀⢸⠃⣴⠀⠀⠀⠀⠀⠄⢐⠁⠀⡟⢠⠆⠀⠀⠀⠀⡠⠘⡊
⠄⠤⢧⢼⠤⠤⠤⢬⠬⡬⠤⠤⠡⡼⡤⡤⠤⠤⠠⡥⢥⠥⠤⠄⠤⢧⢤⠠⠤⠄⢤⠼⠬⠤⠤⠠⠼⡤⡦⠤⠤⠠⡥⠥⠥
⠀⠀⠆⢈⠀⠀⣧⠘⣆⠀⠀⠀⠀⠰⠀⠀⠀⢸⡆⢹⠀⠀⠀⠀⠀⠆⠂⠀⠀⣧⠈⣇⠀⠀⠀⠀⠠⠀⣀⠀⢸⡄⢻⡀
⠀⠀⠀⠘⢽⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠫⡆⠀⠷⠀⠀⠀⠀⠀⠀⠀⠘⢴⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠣⡆⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢀⡆⠀⠀⠀⠀⠀⠀⠀⡠⡛⠀⣰⠀⠀⠀⠀⠀⠀⠀⢀⢴⠃⢀⡆⠀⠀⠀⠀⠀⠀⠀⠠⡞⠀⣴⠀⠀⠀⠀⠀⠀⠀⢀⢔
⢸⠀⡞⠀⠀⠀⠀⠠⢂⢉⠀⠀⡏⢰⠃⡀⠀⠀⠀⡜⡈⡁⠀⠸⠁⡾⠀⠀⠀⠀⢀⢂⢁⠀⠀⠇⢰⠃⠀⠀⠀⠀⠄⡈⡀
⠂⠲⠳⠲⠐⠒⡂⢲⡒⠖⠒⠒⠐⠞⠖⠖⠒⢒⠐⣖⠲⠒⠒⠒⠲⠳⠲⠐⠒⡆⢲⡚⠖⠒⠒⠐⠚⠖⠓⠒⢲⠐⣖⠓⠒
⠀⠀⠃⢀⠀⠀⢳⠈⠇⠀⠀⠀⠀⠘⠀⡀⠀⠸⡆⠸⠀⠀⠀⠀⠀⠃⢁⠄⠀⢷⠈⠧⠀⠀⠀⠀⠘⠈⠠⠀⠘⡆⠹
⠀⠀⠀⠈⠺⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠛⠀⠀⠀⠀⠀⠀⠀⠈⠺⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢠⡇⢀⠀⠀⠀⠀⠀⠀⠔⠍⠀⣼⠀⡀⠀⠀⠀⠀⠀⢠⠺⠁⢠⠇⢀⠀⠀⠀⠀⠀⠀⠔⠍⠀⡼⠀⡀⠀⠀⠀⠀⠀⢀⠪
⠸⢀⡏⠀⠀⠀⠀⠐⠡⠰⠀⠀⠇⢸⠁⡀⠀⠀⠀⠎⠄⠄⠀⠸⠀⡏⢀⠀⠀⠀⠰⢡⠠⠀⠀⠇⢸⠃⡀⠀⠀⠀⠊⠄⠄
⠁⠉⠙⠙⠈⠉⡅⢹⡉⠋⠉⠉⠀⢉⠋⠁⠉⢁⠈⣯⠙⠉⠉⠁⠙⠙⡙⠈⠉⡀⠩⡉⠉⠉⠉⠈⢉⢋⢇⠉⢩⠈⣋⠈⠉
⠀⠀⠁⢰⢀⠀⢻⡀⠃⠀⠀⠀⠀⠈⠀⡴⡀⠈⣇⠘⠂⠀⠀⠀⠀⠁⢠⢆⠀⢻⡀⠃⠀⠀⠀⠀⠈⠀⠖⡀⠘⣇⠘
⠀⠀⠀⠀⠙⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⢀⣠⡤⠴⠒⠒⠛⠛⠛⠛⠛⠒⠒⠶⠤⣄⡀⠀⠀⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢀⣠⠴⠚⠉⠀⢀⣀⣤⠤⠴⠶⠶⠶⠦⠤⣤⣀⡀⠀⠉⠙⠦⣄⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⣀⠴⠋⠀⢀⡠⠖⠋⠉⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠉⠛⠢⣄⡀⠀⠙⢦⡀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⢀⠞⠁⠀⣠⠖⠋⠀⠀⣀⠤⠖⠋⠉⠉⠀⠀⠀⠉⠉⠙⠲⢤⣀⠀⠀⠙⢦⡀⠀⠙⣆⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⡰⠋⠀⢠⠞⠁⠀⢀⡴⠊⠁⢀⣀⡤⠴⠒⠒⠒⠒⠒⠦⢤⣀⠀⠈⠳⣄⠀⠀⠙⣆⠀⠈⢧⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⡼⠁⠀⣰⠋⠀⠀⣠⠋⠀⣠⠔⠋⠀⢀⣀⡤⠤⠤⠤⢤⣀⠀⠈⠙⢦⡀⠈⢣⡀⠀⠈⢦⠀⠈⢧⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣼⠁⠀⣰⠃⠀⠀⡼⠁⢀⡜⠁⠀⡠⠚⠉⠀⣀⣀⣀⣀⡀⠈⠙⠦⡀⠀⠙⣆⠀⢳⡀⠀⠈⣆⠀⠘⡆⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⡟⠀⢠⠇⠀⢰⠇⠀⠀⣼⠁⢀⠎⠀⢠⠞⠀⢀⡴⠋⠉⠀⠀⠀⠉⠲⡄⠀⠙⣆⠀⠘⣆⠀⢷⠀⠀⢸⡄⠀⢻⠀⠀⢧
⠀⠀⠀⠀⠀⠀⢸⢁⣠⠼⠶⠒⣚⡲⠦⣴⠇⠀⡞⠀⢀⡏⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠘⡆⠀⢸⡄⠘⣆⣠⠴⠗⠒⠾⠦⣄⣸⡄
⠀⠀⠀⠀⠀⢀⡿⠋⠀⠀⠀⠀⠀⠈⠓⠈⠳⣼⠷⠶⢼⡁⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢀⡽⠶⠤⣧⠞⠉⠺⠋⠈⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠲⠒⢦⠀⠀⢹⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡏⠀⠀⠴⠳⠖⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⡖⠋⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⠳⠶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⠞⠁
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⡤⠤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⢤⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠙⠶⠤⠤⠶⠒⢦⣀⣀⢀⣀⡭⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢭⣀⣀⣀⣀⡴⠒⠦⠤⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠰⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠒⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢧⠠⠀⠀⠀⠀⠀⠀⠀⠀⠍⠉⠙⠁⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣶⠦⠤⣄⠐⠑⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⢀⠑⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⢎⣷⣄⡀⠀⠅⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠆⠀⠀⡀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠖⠑⣆⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⠀⠀⢀⣉⢁⠈⠁⠀⠐⠈⠀⠀⢀⠄⣠⡴⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⠀⠀⠀⠀⠐⡏⠀⠰⠀⠀⠀⠀⠰⠀⠛⠚⣯⠴⠿⠷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠄⠢⠄⣀⣀⠺⢾⠋⢀⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⣬⣵⣏⢻⠀⢀⠀⠀⠈⠐⣶⣿⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠻⣿⢿⣿⣿⣾⡏⠉⠀⠀⠀⠀⠘⢟⠃⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⢄⡼⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡀

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⣼⣦⣦⣘⢦⣣⠾⠒⢰⣯⠀⠀⠀⠀⢀⣦⣾⣿⣿⣿⣦⣆⠀⠀⠀⠀⠀⠀⣰⣴⣿⣿⣿⣷⣴⡂⠀⠀⠀⠀⠀⠀⣸⣦⣴⣾⣿⣿⣷⣦⣴⣇
⠀⠀⢻⠇⠠⡴⢺⠷⣍⠛⠻⣟⠀⠀⠀⠀⠈⡟⢿⣿⣿⣿⠟⡏⠀⠀⠀⠀⠀⠀⢹⠻⣿⣿⣿⡿⠻⡃⠀⠀⠀⠀⠀⠀⢻⡟⢿⣿⣿⣿⣿⡿⢻⡟
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠉



⠀⠀⠀⠀⠀⢐⣶⣷⡂⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣷⣶⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡖⣶⢲⣄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⡿⠦⢷⡷⣏⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠻⢿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⣱⠟⣦⡝⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⢴⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠻⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠰⣤⣴⣶⣤⡤
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣀⡀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣷⣾⣿⣿⣿⣿⣿⣾⣿⡀⠀⠀⠀⠀⠀⢸⡷⠬⢷⣳⣛⣸⣇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⠏⠹⠿⠿⠿⠿⠟⠛⢿⠀⠀⠀⠀⠀⠀⠸⠓⠚⣱⠹⡎⠙⠇

⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⣰⡿⠋⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡆
⠀⠀⠀⠀⣿⠆⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠖⠒⠒⠒⠒⠶⠦⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠁⢀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣦⣤⣌⣉⠓⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⣀⣤⣶⣿⠿⠟⠛⠛⠉⠉⠉⠉⠉⠉⠙⠛⠛⠿⢿⣷⣦⣌⡙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣴⣾⡿⠛⠉⢀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⡈⠙⠻⣿⣶⣄⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣴⣿⠟⠁⢀⣠⣶⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⣷⣤⡀⠙⢿⣷⣄⠀⠙⢶⡀
⠀⠀⠀⠀⠀⠀⢀⡼⠃⢠⣾⡟⠁⠀⣰⣿⠟⠁⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣤⣄⣀⠀⠀⠙⢿⣦⡀⠙⢿⣷⡀⠀⠙⣦
⠀⠀⠀⠀⠀⢠⡞⠀⢠⣿⠟⠀⢠⣾⠟⠁⠀⢀⣠⣴⡾⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠉⠙⠻⣷⣄⠀⠀⠹⣷⡄⠈⢻⣿⡄⠀⠈⢷⡀
⠀⠀⠀⠀⢀⡞⠀⢀⣿⡟⠀⢠⣿⠏⠀⢀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠻⣷⡄⠀⠘⣿⡄⠀⠻⣿⡄⠀⠀⢷⡀
⠀⠀⠀⠀⣼⠁⠀⣸⣿⠁⢀⣿⠏⠀⣠⣿⠋⠀⠀⠀⢀⣠⡤⠶⠛⠛⠉⠉⠉⠉⠉⠙⠳⢦⣄⠀⠙⣿⡄⠀⠘⣿⡀⠀⢻⣿⡀⠀⠈⣧
⠀⠀⠀⢠⡇⠀⠀⣿⡏⠀⣼⡏⠀⣰⡿⠁⠀⢀⣠⠞⠋⢁⣀⣤⣤⠤⠤⠤⠤⠤⢤⣤⣄⡀⠙⢷⡀⠘⣿⡄⠀⠹⣷⠀⠈⣿⣇⠀⠀⠸⡆
⠀⠀⠀⢸⠁⠀⢸⣿⠃⢠⣿⠀⢰⣿⠁⠀⣠⠟⢁⡤⠞⠋⠁⣀⣠⣤⣤⣤⣄⣀⡀⠀⠀⠉⠳⡄⢳⡀⠹⣷⠀⠀⢻⣇⠀⢸⣿⡀⠀⠀⣷
⠀⠀⠀⣿⠀⠀⢸⣿⠀⣸⡇⠀⣿⠇⠀⣰⠃⣰⠋⠀⢀⡴⠚⠉⠀⣀⣀⣀⣀⠉⠙⠳⣄⠀⠀⠹⡆⢷⠀⢻⣧⠀⠘⣿⠀⠈⣿⡇⠀⠀⢸⡀
⠀⠀⠀⡇⠀⠀⣾⣿⠀⣿⠁⢸⣿⠀⢰⠇⣸⠃⠀⢠⠟⠀⢀⡴⠛⠉⠁⠉⠉⠳⣦⠀⠈⢧⠀⠀⢻⠸⡇⠈⣿⡄⠀⢿⡇⠀⢻⣿⠀⠀⠸⡇
⠀⠀⢠⡇⠀⠀⣿⡟⢠⣿⠀⣾⡇⠀⣾⢠⡇⠀⢀⡟⠀⢀⡞⠁⠀⣀⣤⣤⣀⠀⠈⢷⠀⠘⡇⠀⠸⡇⣷⠀⢹⣇⠀⢸⣷⠀⢸⣿⡀⠀⠀⡇
⠀⠀⢸⡇⠀⠀⣿⡇⢸⣿⠀⣿⡇⢠⡇⢸⠁⠀⣸⠁⠀⡼⠀⣰⠛⠁⠀⠀⠉⢳⡄⠈⣇⠀⣿⠀⠀⣧⢸⠀⠸⣿⠀⠈⣿⠀⠘⣿⡇⠀⠀⣧
⠀⠀⢸⠀⠀⠀⣿⡇⢸⡇⠀⣿⠀⢸⡇⣸⠀⠀⡿⠀⢠⡇⣸⠃⢀⡴⠶⠲⢦⡀⢻⠀⢹⠀⢸⠀⠀⣿⢸⡆⠀⣿⡀⠀⣿⡆⠀⣿⡇⠀⠀⣿
⠀⠀⢸⠀⠀⠀⣿⡇⢸⡇⢸⣿⠀⢸⠁⣿⠀⠀⡇⠀⢸⠁⡟⢠⡟⡴⠛⠳⡄⢷⠘⡇⢸⡇⢸⡄⠀⣿⢸⡇⠀⣿⡇⠀⣿⡇⠀⣿⡇⠀⠀⣿
⠀⠀⣾⠀⠀⠀⣿⡇⢸⡇⢸⣿⠀⢸⠀⠿⠀⠀⡇⠀⣼⠀⠇⠸⠸⠇⠀⠀⠇⠿⠀⣧⢸⡇⢸⡇⠀⣿⢸⡇⠀⣿⡇⠀⣿⡇⠀⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠿⠇⠸⠇⠈⠛⠀⠈⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠃⠈⠁⠀⠉⠈⠃⠀⠙⠁⠀⠛⠃⠀⠙⠃⠀⠀⠉

⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠉⠙⣶⣶⣶⣶⣶⣶⣶⣶⣟⠉⠉⠉⢱⣶⣶⣶⣶⣶⣶⣶⣾⡋⠉⠉⠙⣶⣶⣶⣶⣶⣶⣶⣶⣎⠉⠉⠉⢳⣶⣶⣶⣶⣶⣶⣶⣾⡋
⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡁
⣿⣿⠉⠉⠁⠈⠉⠉⠉⠉⢹⣿⣿⣿⡏⠉⠉⠀⠈⠉⠉⠉⠉⣿⣿⣿⣿⠉⠉⠁⠈⠉⠉⠉⠉⢹⣿⣿⣿⡏⠉⠉⠀⠈⠉⠉⠉⠉⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠛⠻⣤⣤⣤⣤⣤⣤⣤⣤⡾⠛⠛⠛⢧⣤⣤⣤⣤⣤⣤⣤⣤⠟⠛⠛⠻⣤⣤⣤⣤⣤⣤⣤⣤⡼⠛⠛⠛⢧⣤⣤⣤⣤⣤⣤⣤⣴⠟
⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣶⣾⠛⠛⠉⠉⠉⠙⠛⠛⢻⣶⣶⣶⡟⠛⠋⠉⠉⠉⠛⠛⠛⣷⣶⣶⣾⠛⠛⠉⠉⠉⠙⠛⠛⢻⣶⣶⣶⡟⠛⠋⠉⠉⠉⠛⠛⠛⣷
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠿⢿⣀⣀⣀⣀⣀⣀⣀⣀⣸⠿⠿⠿⣇⣀⣀⣀⣀⣀⣀⣀⣀⡿⠿⠿⢿⣀⣀⣀⣀⣀⣀⣀⣀⣸⠿⠿⠿⣇⣀⣀⣀⣀⣀⣀⣀⣠⡿
⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠆
⣤⣼⠿⠟⠛⠛⠛⠻⠿⠟⢳⣤⣤⣤⡾⠿⠛⠛⠛⠛⠿⠿⠛⣦⣤⣤⣴⠿⠟⠛⠛⠛⠻⠿⠟⢳⣤⣤⣤⡾⠿⠟⠛⠛⠛⠿⠿⠛⣦
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡅
⣀⣠⣿⣿⠿⠿⢿⣿⣿⡿⣧⣀⣀⣀⣼⣿⡿⠿⠿⣿⣿⣿⢿⣀⣀⣀⣠⣿⡿⠿⠿⣿⣿⣿⡿⣇⣀⣀⣀⣼⣿⠿⠿⠿⢿⣿⣿⢿⣄
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿

⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠙⠻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⠀⡄⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠘⣮⣿⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⡠⠋⠐⢻⢞⠁⢀⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠟⡇⠀⠱⡙⡿⣎⢘⠀⠹⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣦⡀⢰⣐⣔⣹⣆⡨⣤⡴⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢁⢀⢈⠿⡏⢩⠻⣿⡗⣯⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣤⣥⣠⣾⡎⡇⡄⣆⣹⡟⢷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣿⣯⣬⡯⡁⢈⠰⣿⣿⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣻⡭⣿⡿⢧⡰⢿⡆⡟⡟⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣙⢪⣤⣸⣰⢾⣼⣿⡇⡞⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠀⠉⠉⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠟⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣶⣿⣿⠻⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡐⠃⠀⠀⢀⣠⣴⣾⣿⢿⡦⣹⣿⣿⣿⣿⡶⠄⠦⠀⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢟⠀⠀⣹⠟⢻⡇⠙⡛⠋⠋⠁⠀⠓⢲⣗⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡿⠯⠀⠈⠀⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠙⣹⣷⣌⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢻⡟⠀⠀⠘⠉⠀⠀⠀⠀⣀⠀⠀⢃⡀⠐⣾⣿⣄⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡱⠈⠁⠀⠒⠁⠀⠀⠀⢀⡄⠹⠶⠀⠘⠟⠀⠉⠛⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⣀⣠⣴⣶⣄⠀⢸⡷⠀⣤⣶⣤⡀⠀⠀⠀⠀⠀⡇⠀⠐⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡦⠀⣰⣿⣿⣿⣿⣿⣿⡆⠀⠁⠈⣿⣧⣽⢟⠝⣃⡄⠀⢰⡇⣀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠳⠀⠹⣿⣿⣿⣿⣿⣿⠃⢀⡀⠈⣿⠶⣷⣿⣿⣿⡟⢄⣼⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠈⠛⠻⣿⣿⠋⢀⣾⣷⡀⠘⣿⠧⠿⠛⠋⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⡁⠈⠀⠰⠄⠀⠀⠈⢠⢨⣿⣿⡟⡀⠉⠀⠀⠐⠀⡄⠹⣿⣿⣭⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⢻⣔⡠⠄⢤⠄⠰⠁⠘⣎⠿⢿⣵⠇⠐⣦⠠⡄⠠⢐⣢⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢸⠹⡿⠾⢿⠛⣶⣦⡄⠀⠀⠀⠐⠀⠀⠀⠀⢠⣶⣶⢿⣿⣿⣿⣍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⡥⣶⣶⣿⡾⣻⡇⠹⡿⠷⠤⡤⠤⠠⠦⠤⢠⢶⣿⣿⠏⠸⣿⡿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⢫⣿⡦⠃⠝⡇⠀⠐⠀⠀⠀⢤⢤⡤⣤⡤⠶⣔⡟⠆⢠⣈⣉⣉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡟⠁⠙⠁⣀⠘⠃⠀⢀⠙⠒⠶⢴⢶⣶⣶⠷⠟⢛⡁⠀⣾⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠀⠁⣀⡁⠀⠁⠀⠀⠀⠀⠣⢀⣴⣦⡲⠿⠆⠀⠀⣼⣿⣿⣿⡛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣥⠒⠠⠴⠦⣄⡀⠀⢸⣿⡿⠇⠀⢀⣠⣾⠿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢌⠙⣿⣶⢿⣷⡖⣿⣿⣿⣻⣦⠈⠛⠋⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⢀⣦⣾⠶⠀⠩⢿⡟⠛⠙⠛⠛⠤⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠛⠈⠉⠀⠀⡀⢠⡇⠀⠀⢀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠙⠇⢀⠘⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠉

⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠁⠀⠩⡉⠀⠀⠩⢉⠓⠀⠀⠐⠄⠄⠀⢀⠉⢁⠀⠛⠱⠀⢠⠈⠃⠀⠀⠈⠁⠈⠋⠁
⠀⠒⠀⠀⠂⠀⠐⠀⠀⠂⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢬⣠⡀⠀⠀⠀⠐⠀⡀⠀⢀⠀⠠⠅⢀⡈
⠀⠀⠀⠀⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⠨⠤⠄⠀⠃⠀⠃⠀⠈⠁⠐⠂⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠇⠀⠀⠀⠠⠀⠀⠁
⠀⢐⠀⠂⠂⠀⠀⠃⢀⠀⠀⢀⠀⠀⠠⠢⠀⠀⠈⠉⠀⠀⠀⠀⠐⠀⢂⠂⠀⠈⠀⠀⢠⠄⠀⠀⠀⠁⠀⠀⠀⢤⡑⢦⠦⠅⢐⡈
⠀⠀⠀⠀⠄⠀⢁⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⢀⠀⢀⡁⠀⠠⠤⠤⣭⡌⠒⠁⠀⠀⠀⡄⢠⠀⠀⠀⠁⠐⠀⠀⠀⢀⡀⠀⠢⠀⡁
⠀⠰⠀⠁⠀⠠⠀⠁⡄⠁⠠⠐⠀⠠⠔⠚⠀⠀⠀⠤⠄⡀⠀⠀⠀⢄⠀⠀⠈⠄⠀⠀⠀⠀⠁⠈⠀⠃⠀⠀⢐⡀⠘⠓⠈⠁⠐⠆
⠆⠀⠀⢀⢀⠀⠀⠁⠀⠀⠀⠀⡀⠀⡠⠀⠀⠀⠀⠀⠤⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⣉⡉⠁⠀⠑⠀⠠⠀⠀⠀⣀⠀⢈⡅⠀⢀
⠃⠀⠀⢀⠬⠂⠀⠀⠀⠀⠠⢄⠀⠀⢀⠄⠒⠀⠠⠐⠒⠲⠀⠠⠀⣄⠀⠀⠐⢙⠀⠀⠁⠁⡆⠈⠀⢤⠀⠀⠀⠀⠚⠀⠈⠀⠀⠎
⠇⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⣀⣄⠀⠲⠋⠁⠀⢀⠉⠵⠀⠐⢆⠈⡀⠀⠠⠸⠀⠂⠒⠂⠧⠀⠤⠄⠀⠀⠀⠀⣠⠀⠆⠀⠄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠒⠒⢀⡁⠢⠀⠁⠀⠀⡀⠀⢰⠀⠀⠐⠀⠀⡀⠈⠀⠁⠀⠠⠀⠀⣀⠠⠀⢠
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠀⢀⡤⠂⠀⠀⠀⠀⠘⢦⡄⠀⠰⡀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠃⠀⠠⠀⠀⠀⠠⠀⠆⠠
⠀⠀⠀⠃⠄⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠰⠈⠀⠀⠀⢀⠀⠀⠐⠂⠀⡀⠒⠂⠀⠀⠀⠒⠀⠐⠐
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠀⠐
⠀⠀⠀⡆⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠈⠐⠀⠐⠀⠀⠀⠀⠀⠉⠀⠈⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠺⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡜⠠⠀⠀⠀⠈⠀⠂⠆⠀⠀⡄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄
⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠂⠀⠄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠐
⠀⠀⠀⠈⠀⠈⠐⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡤⠀⠀⠀⠀⠀⠀⠀⡂⠀⢀⠀⢀
⣀⠁⠀⠀⡠⠀⠀⢉⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠈⠁⢀⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀
⡉⠀⡀⠠⠀⠀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠈⠁⠀⠈⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠠⠀⡆
⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⡌⠀⠀⡇⠀⠀⠀⠀⠀⢰⠃⡄⣸⡇⠀⠀⡆⢠⠀⠈⠅⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠠⡄⢠⡆⢰⡆⠰
⠀⠁⠀⠀⠃⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠁⢀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠛⠀⠙⠁⠘⠀⠈⠁
⣿⣿⠟⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠉⠀⠐⠀⠀⠀⡸

⠀⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣿⣿⣿⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠈⠛⢻⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⡟⠛⠁
⠀⠀⠀⢸⣿⠀⣤⣤⣤⣤⣤⣤⡄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢘⣿⡇
⠀⠀⠀⣼⣿⠀⣲⠒⠒⠀⠐⠀⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⠛⠙⠛⠋⢻⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⡷
⠀⠄⠀⢿⣿⠀⣙⣉⣁⣀⢀⡅⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⢸⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⠟⠉⠉⠉⠙⠻⣿⣿⡇⢸⣿⣿
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠁⢉⠉⠉⠉⠉⠉⠉⠉⠉⡉⠁⢸⣿⣿⣿⣀⣀⣀⣀⣸⣿⣿⡇⠘⣿⡇⢀⡄⠀⠀⠀⡀⢸⣿⡇⢸⣿⡿⠤⠤
⠀⠀⠀⢸⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠨⢸⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⠿⠿⣿⠉⠙⢻⠿⢿⡇⠘⣿⡆⠀⠀⠀⡀⠀⡀⢈⣿⡇⢸⣿⡇
⠀⠀⠀⢸⣿⠀⢸⣿⡇⠀⠀⣹⣿⣿⠀⢠⣼⣿⣿⣿⣿⡏⠉⠀⢸⣿⣿⠀⠆⢿⠄⠀⢺⠐⠸⡇⠀⣿⠀⣤⠐⠀⠀⣠⠰⠟⣿⡇⢸⣿⡇
⠀⠀⠀⣸⣿⠀⠘⣻⡛⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⠿⠿⣷⣶⠀⢸⣿⠘⠉⠀⠘⠀⠉⢲⡴⣆⡇⠀⠃⢲⡶⢶⡆⠀⠀⠐⠒⡆⡇⢸⣿⣇
⠀⠀⠀⣿⣿⠀⠀⠙⠁⣿⠉⠉⣿⣿⠀⢀⠀⢸⣿⣆⣆⠻⠛⠀⢸⣿⢰⣾⢷⣆⣀⣀⡆⢷⢹⡇⠀⡛⢸⣷⣸⣇⡐⣀⣄⣀⡇⡇⢸⣿⡇
⠀⠀⠀⢸⣿⠀⢰⡖⠲⣿⣶⣶⣿⣿⠀⢰⣶⣾⣿⣷⣶⣶⣶⠀⢸⣿⠈⠉⣽⢩⣥⣦⠐⡖⠉⡅⠀⣷⣶⣶⣮⢉⣉⣴⣶⠀⡆⡇⢸⣿⡇
⠀⠀⠀⢸⣿⡀⢸⣷⣶⡆⢀⠘⣿⣿⠀⢸⣿⠛⠛⠛⠛⣿⣿⠀⢸⣟⢀⠐⠻⠞⠛⣛⢚⠿⠆⣶⣴⡿⢿⠿⢟⣀⡈⠀⠈⠀⠒⡇⢸⣿⡇
⠉⠉⠉⣿⣿⠁⠈⠉⠩⢡⣌⠀⠙⣿⠀⢸⣿⣀⣀⣐⣀⣿⣿⠀⠘⣿⠀⢉⣷⣸⣟⣿⣿⡗⠐⢻⣿⡇⣸⣿⣽⣷⠀⠈⠙⠛⣦⡇⢸⣿⡧⠐⠒
⠀⠀⠀⣿⣿⠀⠀⣤⣄⣸⣿⣷⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⠿⠀⠀⠀⢠⠤⢸⣿⡇
⠀⠀⠀⣿⣿⠀⢸⣿⣿⣭⣟⣻⣿⣿⠂⡸⣫⣶⣶⣶⣮⡹⣿⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⠀⠀⢠⡿⠒⠺⣿⡇
⠀⠀⠀⣿⣿⠀⢀⣀⣀⠁⠉⠉⠙⠋⠺⣧⣿⡄⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⣿⣸⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⣠⠞⠀⠀⠩⢋⠳⣆⠀⢀⠀⠀⠹⠋⠐⢠⠖⠛⠉⠈⠙⠳⣄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣿⣿⣿⣿⡆
⠀⠀⢰⠇⠰⠀⢰⣃⠀⠀⠘⣇⠀⠀⢄⣠⣠⣴⣏⠀⢤⡦⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⢸⡀⠀⠀⠈⠃⠈⠀⠀⡿⠀⠀⢼⠄⡟⢬⡃⠀⠸⡏⠺⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡀
⠀⠀⠈⢷⣀⠀⠀⠀⠀⣀⡼⠃⠀⠀⠚⢀⠈⠉⢻⣄⠀⠀⠈⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠉⠀⠐⠒⠛⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠉⠛⠦⠶⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠘⠛⠛⠛

⣿⣷⣿⣟⣿⣿⣯⡛⠛⠿⠟⢻⣿⣿⣯⣽⣧⣶⠶⠒⠺⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣷⡀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣭⣭⠿⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⠴⠨⣿⣿⣿⣟⣦⡑⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣀⣠⡬⠉⣙⡿⣯⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢖⡎⠋⠉⠐⠒⣒⢛⣛⣫⣥⣄⣡⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣿⢿⣯⣴⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⠛⠣⢞⣤⡄⢀⣼⣏⡽⠿⠗⠌⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⡿
⣿⣿⣿⣿⣫⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⢤⠤⠶⢶⡖⡋⠀⠀⠀⡀⠀⠤⠒⢋⠿⠟⣻⣇⡔⠀⠀⠀⠀⠀⢀⠀⢹⣿⠿⠿⠻⠟⠩⠔⠋⠁
⣿⣿⣿⣿⣽⣛⣿⡛⠁⠒⠀⠠⠶⠒⣢⣶⠟⠛⢋⣀⣤⣦⣦⣶⡾⢋⡠⠴⠃⠀⠀⣠⣴⡇⠀⣰⣿⠟⣴⢆⣠⠞⢁⣊⡥⠔⠉⠕
⣾⣿⣿⣿⣿⣿⣿⣤⣤⠄⠀⢀⣐⣚⣩⣤⣴⣾⣿⣿⣿⣽⣿⣵⣾⣋⣴⣶⡴⣶⣿⣿⣿⣿⣷⣿⣿⣼⣷⣿⠧⠀⣙⠥
⣿⣿⣿⣿⣷⠿⠻⣿⡟⢀⣴⣿⣿⣿⣿⡅⠠⠼⣿⣿⣿⢿⣿⢿⣿⢟⣿⣿⣄⣽⣿⣿⣿⣿⣿⣿⣿⣿⠟⡣⠶⠛⠁
⣯⣤⣽⣏⣤⣶⣿⣿⣧⣿⡿⣟⣿⣿⣿⠇⢒⣨⣿⣽⣿⣿⣷⣾⡿⢿⣿⣿⡿⣾⣿⣿⣿⣿⡿⢿⡿⠽⠛⡒
⣿⣿⣿⣿⣿⣿⣿⣿⡽⢫⣽⣿⣿⣿⠏⡰⠿⠍⣿⣿⣶⠟⠉⠉⣠⣾⣿⣿⣴⣿⣿⣿⣿⣿⣽⣶⣦⣄⠀⠤
⣿⡿⠿⠟⣿⠟⠛⣁⣴⣿⣿⣿⡿⠃⠀⠀⠀⠘⠉⡩⠤⢄⣴⣿⣿⣿⠟⣛⣿⣿⣿⣿⣿⣹⣿⣿⠾⢦⠀⢤⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠤⠴⠚⠛
⣿⣿⣿⣍⣁⣴⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣯⢳⡶⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠉⠀⠀⠀⠀⢀⡤⠔⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣠⠄⠀⠀⠀⠀⠀⣈⠃⠀⣀⣴⠿⠟⢛⣷⣿⣎⣻⣿⣿⣿⣿⣿⠃⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠐⠒⠛⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⠀⠀⠀⠀⢤⠔⠋⠉⠀⠀⠐⠋⠽⢛⣽⣷⣾⠟⢻⡿⢏⡰⠃⢻⣷⣶⣤⡀⠀⠀⠀⠀⠈⢙⣛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠉⣸⡿⣧⣊⣤⡴⠛⠁⠈⣹⣿⡓⠂⠀⠀⠀⠠⢤⣥⣉⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⣿⣿⡾⠟⠗⠚⠏⠉⠿⢧⠾⣫⣾⣿⣿⠇⠀⠀⠈⠑⠲⣶⡛⠋⠁⠀⢀⣠⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠋⠉⠁⠠⠀⠀⣤⣶⣗⣤⣶⡞⣁⣩⣿⠁⠀⠀⠀⠀⠀⠀⢈⣹⠶⠻⠋⠉⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⣀⣤⣶⣿⣷⠊⠰⠶⠶⠤⢀⡴⢋⣽⣾⣿⡿⠿⣿⣩⡿⠻⣧⡄⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠐⠢
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⢛⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⣰⣿⣽⠿⣿⣿⣶⢿⣿⣿⣴⠇⠀⠀⣠⣾⠛⠁⠀⠀⠀⢀⡀⠤⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣛⣭⣭⣥⣈⣈⣛⣛⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⣺⣿⣿⣏⣽⣿⣿⠃⣰⣿⡿⢿⡃⢀⣾⣟⣡⣴⣒⣥⡤⠄⠒⣒⣀⡀
⣿⣿⣿⣿⣿⣿⣟⣯⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠋⣀⣴⡶⠚⢀⣤⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣯⣤⣴⣿⣿⣿⣿⣿⣭⣥⠖⠒⠛⣹⡯⣿⣿⣻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣻⣿⣿⣿⣷⡟⣡⣾⣿⠏⠀⣠⣾⣿⣿⣿⣿⢿⣿⣿⠿⣿⡿⠟⢛⣿⣿⣿⣿⠿⠟⢿⣿⣿⠟⠀⠀⡁⠐⠛⢉⣩
⣿⣿⣿⣿⣿⣿⣿⠿⣿⣋⣥⣶⣿⠿⢛⣽⣿⣿⣿⣿⣿⣿⣿⡉⣠⣾⣿⣿⣿⣿⣿⡟⠉⡜⠁⢠⠋⢀⣴⣾⣿⠟⠋⠀⠀⠀⠀⠉⠛⢿⣤⣀⠁⠀⠊⣽⡿
⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣼⣿⣿⣿⣿⣿⣿⣿⡇⠈⠀⠀⣧⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢥
⣿⣿⣟⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣇⣀⣴⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⢀⣠⠶⣶⠂⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⢏⣼⣿⣿⡟⣿⣿⣿⣿⣿⣿⠁⠀⠀⢀⡴⢿⣷⡄⠼⣒⣛⢒⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⠁⣰⣟⢘⣷⣿⢻⣷⣿⣿⣿⣿⣿⣿⠀⢠⣿⣞⣵⣿⢟⣤⣾⢿⠟⢻⡿⣲⡄⠤⣄⠤⣀
⣿⣿⣿⣿⣿⡿⠛⡿⢿⣿⣿⣿⣿⣱⣟⢫⣿⣿⣿⠀⠿⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⡿⠿⠟⠋⢡⣬⠿⣮⣁⣬⢃⣿⣵⢦⣓⡀⠺⠀⠀⢧⣀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠒⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⢠⡄⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢏⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢶⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠉⠂⠀⠀⠄⠓⠀⠂⠐⠒⠂⠖⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠈⠉⠣⠀⠈⡹⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀
⠰⡆⠀⣀⠂⠰⡀⠀⣀⣤⢦⠠⠔⢲⣄⣀⢀⠀⠀⢀⠠⠄⠸⠀⠀⠂⠀⠀⠀⡇⠀⠧⠀⠔⣩⢬⠑⡀⠀⠀⠈
⠀⠘⠛⠂⠀⠀⠀⠰⠁⠛⠛⠛⠒⠛⠛⠷⠁⠀⠀⠈⠀⠀⠀⠀⠀⠈⠢⠀⠰⠁⠀⠀⠀⢀⡁⢀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢲⠆⠒⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠁⠂⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠒⣮⡀⠢
⠀⠀⡤⢀⣀⡀⢰⢰⣤⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢠⠄⠀⠀⠀⠀⠀⠐⠂⠀⢀⠀⠀⡀⡆⠀⠘⢸⠈
⠀⠀⠀⠀⠀⠀⠙⠚⠋⠉⠉⠙⠛⠋⠂⠀⠀⠀⠀⠀⠀⠀⣀⠄⡴⠂⢍⠁⢀⠀⠧⠤⠂⢀⠇⢲⠀⠙⢦⠀⠇⠁⠀⡀⡄
⠀⠀⠀⠀⠀⠀⣀⡀⠲⣜⠚⠴⠟⣻⡀⠀⠀⠀⠀⠀⠀⡴⡄⢃⡡⠠⠮⣹⠾⢿⣗⠰⢍⠁⠀⠀⠀⠀⠈⠱⠀⠀⠀⠁⠁⢀
⠀⠀⡀⠀⠀⠀⡏⢫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠁⠀⠂⠉⠁⠡⠄⠰⢈⢠⠄⠀⠀⣁⣀⣀⠈⡀⠀⠈⠀⡀⠘
⠀⠀⡇⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠘⠇⢴⠢⠠⣭⠆⢨⣐⣢⡀⠀⠈⠈⠂⠈⡽⢋⣻⠛⠀⠀⠆⠀⠀⠁⡂
⠀⠀⠣⠤⢄⣀⣀⠀⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⣀⡡⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠠⢀




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠂⠐⠄⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠐

⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⢹⣿⣿⣿⡆⠀⢸⣿⣿⣿⡇⠀⠈⣿⣿⣿⣧⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⡆⠀⢸⣿⣿⣿⡇⠀⠈⣿⣿⣿⣧⠀⠀⣹
⠉⠉⠀⠀⢸⣿⣿⣿⡇⠀⠈⠉⠉⠉⠁⠀⠀⣿⣿⣿⡟⠀⠀⠉⠉⠉⠉⠀⠀⢸⣿⣿⣿⠇⠀⠈⠉⠉⠉⠁⠀⠀⣿⣿⣿⡏⠀⠀⠈
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⣿⣿⠀⠀⢸⣿⣿⣿⡆⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⣇⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⡆⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⣧⠀⠀⣿
⣿⣿⠀⠀⠘⠉⠋⠙⠁⠀⢸⣿⣿⣿⡇⠀⠀⠋⠙⠉⠋⠀⠀⣿⣿⣿⣿⠀⠀⠘⠉⠋⠛⠁⠀⢸⣿⣿⣿⡇⠀⠀⠋⠙⠙⠋⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⢲⣶⣶⣶⡄⠀⢸⣿⣿⣿⡇⠀⠐⣶⣶⣶⡆⠀⠀⣿⣿⣿⣿⠀⠀⢰⣶⣶⣶⡄⠀⢸⣿⣿⣿⡇⠀⠐⣶⣶⣶⡆⠀⠀⣿
⠛⠛⠀⠀⢸⣿⣿⣿⡇⠀⠘⠛⠛⠛⠃⠀⠀⣿⣿⣿⡟⠀⠀⠛⠛⠛⠛⠀⠀⢸⣿⣿⣿⡇⠀⠘⠛⠛⠛⠃⠀⠀⣿⣿⣿⡿⠀⠀⠛
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⣶⣦⠀⠀⢸⣿⣿⣿⡄⠀⢰⣦⣶⣶⡆⠀⠀⣿⣿⣿⣇⠀⠀⣶⣶⣶⣶⠀⠀⢸⣿⣿⣿⡄⠀⢰⣶⣶⣶⡆⠀⠀⣿⣿⣿⣇⠀⠀⣶
⣿⣿⠀⠀⠸⠟⠟⠿⠃⠀⢸⣿⣿⣿⡇⠀⠀⠿⠻⠻⠟⠀⠀⣿⣿⣿⣿⠀⠀⠸⠟⠟⠿⠃⠀⢸⣿⣿⣿⡇⠀⠀⠿⠻⠻⠟⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⢤⣤⣤⣤⡀⠀⢸⣿⣿⣿⡇⠀⢠⣤⣤⣤⡄⠀⠀⣿⣿⣿⣿⠀⠀⢠⣤⣤⣤⠀⠀⢸⣿⣿⣿⡇⠀⠠⣤⣤⣤⡄⠀⠀⣿
⠿⠿⠀⠀⢸⣿⣿⣿⡇⠀⠸⠿⠿⠿⠇⠀⠀⣿⣿⣿⡗⠀⠀⠿⠿⠿⠿⠀⠀⢸⣿⣿⣿⡇⠀⠸⠿⠿⠿⠇⠀⠀⣿⣿⣿⡷⠀⠀⠿
⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⣤⣤⠀⠀⢸⣿⣿⣿⡀⠀⢠⣤⣤⣤⡄⠀⠀⣿⣿⣿⡇⠀⠀⣤⣤⣤⣤⠀⠀⢸⣿⣿⣿⡀⠀⢠⣤⣤⣤⡄⠀⠀⣿⣿⣿⡇⠀⠀⣤
⣿⣿⠀⠀⢸⡿⡿⣿⠇⠀⢸⣿⣿⣿⡇⠀⠀⣿⢿⢿⡿⠀⠀⣿⣿⣿⣿⠀⠀⢸⡿⡿⣿⠇⠀⢸⣿⣿⣿⡇⠀⠀⣿⢿⢿⡿⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⢀⣀⣀⣀⠀⠀⢸⣿⣿⣿⡇⠀⢀⣀⣀⣀⡀⠀⠀⣿⣿⣿⣿⠀⠀⢀⣀⣀⣀⠀⠀⢸⣿⣿⣿⡇⠀⠀⣀⣀⣀⡀⠀⠀⢿
⣿⣿⠀⠀⢸⣿⣿⣿⡆⠀⢸⣿⣿⣿⡇⠀⠈⣿⣿⣿⣧⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⡧⠀⠀⣽
⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏
⣀⣀⠀⠀⢸⣿⣿⣿⠀⠀⢀⢀⡀⣀⡀⠀⠀⣿⣿⣿⡇⠀⠀⣀⢀⢀⣀⠀⠀⢸⣿⣿⣿⠀⠀⢀⢀⣀⣀⡀⠀⠀⣿⣿⣿⡇⠀⠀⣀
⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⣷⠀⠀⣿
⣿⣿⠀⠀⠈⠉⠉⠉⠀⠀⢸⣿⣿⣿⡇⠀⠀⠉⠉⠉⠁⠀⠀⣿⣿⣿⣿⠀⠀⠈⠉⠉⠉⠀⠀⢸⣿⣿⣿⡇⠀⠀⠉⠉⠉⠁⠀⠀⣿
⡿⢿⢶⢷⢶⠶⣶⠶⢶⣶⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣷⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⢾⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⡄⠀⠀⠀⢀⠀⠀⠀⠀⢀⠎
⠃⠀⢀⠔⠁⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⢀⠖
⠀⠞⠁⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠂⠄
⠀⠀⠀⠀⣀⠔⠁⠀⠀⢀⠄⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⢉⡠⠤⠤⠤⣉⡑⠢⠀⠀⠀⢀
⠀⠀⠀⠙⡁⠀⠀⢀⡴⠁⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⠔⠋⠀⠀⠀⣀⡀⠀⠉⠢⡀⠑⡀⠀⠁
⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢠⠋⠀⡠⠀⠁⠀⡀⠀⠁⠄⠀⢸⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⠈⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡆⠀⠸⠁⠀⣰⠋⠁⠀⠀⠘⠀⢸⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⠀⠀⢀⠔⠁⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⣇⠤⣀⣀⡤⠂⢠⠆⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⠔⠁⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠘⡄⠀⠈⠑⠢⠤⠔⠚⠁⠀⠀⣠⠁⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠐⠃⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⣀⠔⠁⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠑⢤⡀⠀⠁⠒⠂⠀⠒⠈⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠁⠀⠀⠀⢀⠔⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠈⠑⠒⠤⠤⠤⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⡴⠁⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⢣⠀⠀⠀⠀⢀⠞⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠑⠀⠀⢀⠊⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⠢⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠈⠂


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠈⠚⠭⣉⡒⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠭⣍⣒⠢⠤⢀⡀⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣦⡀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⣒⡋⠭⠥⠬⠻⣦⣄⡀⠀⢀⣠⠃⠠⠭⠭⢿⡲⢦⡄
⠀⠀⠀⠀⣀⣀⣀⠀⠀⢿⣅⣏⣀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⢀⣜⣸⠇⠀⠀⠀⣀⣀
⠀⠀⡰⢪⠕⠒⠻⠟⡧⠀⠀⠉⠉⠉⠉⠛⠛⠓⠒⠒⠒⠚⠛⠛⠛⠛⠋⠉⠉⠁⠀⣄⢴⠒⠏⠓⠒⢝⢦
⠀⠀⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⡎⡇
⠀⠀⠙⣜⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⡸⣱⠃
⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣪⠞⠁
⠀⠀⠀⠀⠀⢷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⢭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠤⠀⡀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠉⢉⣉⣀⣠⠤⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢑⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣈⡙⠁⠀⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡋⢳⡚⠁⠀⠀⠀⠀⠀⠀⢀⡤⠊⣡⠴⠒⢢⡘⠂⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠟⠀⠀⠀⠀⠀⠀⠀⢠⡝⡤⢫⠀⠀⠀⣸⠁⠀⢀⡈⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⢀⣀⡨⢵⠃⠀⠀⣀⣴⠃⠀⠀⡼⢋⡦⠈⢚⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠒⠊⠀⠀⠀⠘⠒⠀⠖⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠈⠀⠀⠀⢀⣴⠊⡠⠊⣀⠀⠁⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠤⠤⠤⠤⠀⠀⠀⠒⠒⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠴⠊⣁⠼⠋⣠⢾⣥⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠀⠀⠐⣊⣡⠴⠋⢡⢎⡉⠁⠀⢩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠜⢶⡐⣇⢂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠒⠒⠘⠉⠉⣀⠀⢀⣣⠞⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠵⠜⢺⣩⢳⡞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠼⢾⠽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⠃⠈⠁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⠞⡡⣪⠗
⠀⠀⠀⠀⠀⠈⠐⠢⠼⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠈⠣⣄⣀⣀⣤⡪⢑⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡪⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠲⠀⠂⢲⡀⡢⢠⢤⣤⢀⠀⠀⠀⠉⠒⠤⣀⡠⠐⢘⣏⠉⢐⣧⠂⢰⢂⡴⠂⠀⠀⠀⠢⡙⢢⡑⢀⢖⠕⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠉⠑⠒⠒⠶⠶⠀⠐⠛⠓⡿⠀⠀⠀⠀⠀⠀⠙⠒⢄⡀⠉⡩⣈⡡⠞⠁⠀⠀⠀⠀⠀⠀⠈⠐⠍⠳⡵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢅⡀⠈⠀⢀⠠⣆⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠊⠁⢀⠜⠀⠀⢠⡔⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠈⠀⠀⠀⠒⠈⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠀⠀⢀⡠⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠉⠀⠀⣀⠤⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠐⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠀⠆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠢⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠁
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢈⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢃
⠄⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠠⠀⣠
⠐⠀⠈⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠂⠀⠀⠀⠂⠃⠀⠀⠀⢀⠂⠈⠀⠀⠀⡀⠀⠠
⠀⡀⢀⠀⠀⠀⠀⠘⠀⡌⠑⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⡀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠁⠀⠀⠀⠒⠀⠀⠀⠀⢀⡤⠀⠐⡦
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠐⠈⠅
⡆⠀⠀⠈⠀⠀⠀⠊⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠄⠊⠀⠀⠀⠀⠀⠀⠈⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⢀⠀⠀⢀⠄⠀⡀⠀⠀⠘⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⡀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠁⠁⠠⠈⠠⠘
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⡄⣠⠀⣰⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⠀⠀⠀⠀⠀⣽⣯⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⣧⣾⣶⣷⣿⣶⣿⣤⡂⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⡆⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⣠⡻⣟⢏⠃⠃⠇⡟⡸⢻⣿⣥⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣀⡾⠋⠉⠉⠙⢦⡀⠀⠀⠀⠀⠀⠒⢶⣝⡳⠌⠀⠀⠀⠀⠀⠑⣭⣾⣿⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠏⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠐⡲⣶⠦⠍⠀⠀⠀⠀⠀⠀⠀⢸⠿⢿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣠⣤⣾⡷⠖⡀⠀⠀⠀⠀⠀⠀⡒⢭⣿⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⢠⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣟⡵⢀⠀⠀⠀⣄⢾⣾⣿⠿⣍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠄⠈⠗⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⣺⠟⣵⡯⣾⣸⣞⣮⣿⣿⣯⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡿⠟⠀⠀⡿⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠠⠋⠀⠏⠈⢹⠛⣏⠻⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣩⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢰⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠈⠉⣿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢯⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠈⢳⣄⠉⠙⠷⣦⠤⢄⣀⣀⣀⣀⠀⢰
⠀⠀⠀⠀⠙⠻⢖⠤⠄⣀⣈⣉⣛⣩⠟⠁⠀⠀⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢀⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣤⡶⢋⣽⠟⠀⠀⠐⣄⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠥⠝⢛⣉⣵⣾⠟⠁⠀⠀⠀⠀⠈⠙⠉⠉⠉⠾⠴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣖⢲⣶⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣹⣿⡟⣻⣿⡧⢸⣿⣿⣿⣿⣎⣨⣑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣿⣿⣬⣿⣷⣿⣯⡄⢺⣿⣿⠛⣿⣽⣷⣾⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢷⣯⣭⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⣿⣿⣿⣿⣿⣿⢿⣻⣿⡿⣿⣿⢟⣿⣿⣿⣿⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢤⣔⣺⣽⣿⣿⣿⣿⣿⣿⣿⣯⣽⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣭⣶⡾⠟⢿⣿⣿⡟⠙⢻⣿⣿⣿⣿⡿⢿⣎⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣥⣾⣿⣯⣽⠿⠂⠐⠿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⠏⣩⣴⣦⣤⢀⡀⠀⠀⠀⣀⣘⣫⡛⠻⣿⣿⣿⣿⣿⢿⣿⣥⣶⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠀⠹⠧⣤⡌⢩⠇⠀⠀⢨⢩⣉⣙⣿⠿⣿⣿⣻⣽⣿⣾⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡄⠀⠀⣼⣇⣤⡀⠀⠠⠀⡰⠚⠉⠉⣀⣿⣿⣿⣿⡵⣿⣼⢄⡣⢛⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⣿⣿⣿⣧⠀⠉⠙⠉⠉⠛⠂⠀⢷⠀⣀⣸⣿⣿⣿⡿⣿⣿⣿⣿⣾⣷⣧⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢹⣿⣿⣾⣿⣻⣯⠛⠓⠄⣠⣤⣦⣤⡀⠀⣺⣎⢷⣿⣿⣿⢿⣿⣿⢿⣿⢻⣿⣿⣿⡿⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢟⣁⣭⣿⣿⡄⠀⢸⣿⣶⣶⣾⣿⡦⠉⠁⠀⠈⠙⣿⣿⣧⣿⣌⣿⣯⣿⣿⣿⣿⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⣿⣿⣿⣿⣿⡄⠀⠙⠿⠶⠟⠋⠀⠀⢀⡠⠤⠶⠿⢿⣿⣿⣿⢿⡿⠟⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⢻⣆⠀⠠⠆⠀⠀⢀⡴⠋⣠⣤⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣳⣿⣿⣷⣶⣦⣴⣶⡾⠁⢰⣿⣿⣿⣿⢻⣿⣛⣹⠿⠿⠿⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⡈⠙⠻⣿⠃⠀⠛⠻⠿⠿⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⠩⣶⣿⣵⢟⣫⣷⠀⣸⠃⠀⠀⠲⣶⣶⣦⣤⣌⢻⣿⠟⣹⣷⣦⣄⠀⠀⠛⠣⠊⡀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⣦⢸⣱⣿⣿⣿⠐⠁⠀⠀⠑⢤⣀⠙⢦⠍⠙⠳⣽⣿⣿⣿⣹⠋⣳⠀⠀⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠟⠈⠻⣿⠟⠁⠀⠀⠀⢤⣤⠀⢘⡧⠀⠂⠀⠀⣾⣿⣿⣿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⢀⣾⡾⢡⠊⢡⣖⠀⠀⠀⠀⠙⠯⢍⣻⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠊⠁⠀⠀⡀⢀⡴⣻⡿⠁⠑⠦⠽⠄⠕⠄⠀⠀⠀⠀⠀⠀⠉⠛⣄⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⡀⠀⠀⣠⣶⣤⣾⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⡄⣠⢴⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣎⣹⡣⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠋⠉⠉⠁⠀⠀⠀⠈⠙⠒⠚⠛⠙⠛⠛⠶⡶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⢀⠀⠀⠰⠆⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠃⠀⠉⠀⠀⢠⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢹⡄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠐⠤⡈⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠳⣄⠁⢀⠀⠄⠒⠺⡗⠓⠓⠲⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣧⢸
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠠⠊⠀⡈⢷⣠⠴⠒⠊⠉⠉⠙⠓⢦⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⢺⢸
⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⠀⠀⠀⠉⠀⢀⠀⠀⠀⠁⠀⢀⡾⠁⡰⠊⠁⠀⠀⠀⠀⠀⠀⢀⣼⣥⠤⠀⠀⠰⠀⠀⠀⠀⠉⠒⢦⣞⠑⠘⠪⣿
⠀⠀⠀⠀⠀⠀⣸⠃⠀⢸⠀⠀⠀⠃⠀⠁⢀⡀⠀⠀⠀⢸⣻⣾⣁⣀⠀⠀⠀⣀⣤⠴⠚⠉⠁⠀⠀⠀⣤⠀⠀⠠⠀⠀⠀⠀⣰⠏⢷⡈⣰⠇
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⠆⠀⢠⠀⠀⠀⠈⣠⠤⠖⠛⠉⠉⠉⠉⠉⠉⡷⠋⠁⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⣷⠏
⠀⠀⠀⠀⠀⠠⡇⡇⠀⠀⠀⠀⠀⠀⢀⠴⠋⠁⢀⡀⠀⠀⠀⠤⠂⠀⡸⠁⠀⠀⠈⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠁⠀⠴⠀⠀⠁⠀⠀⠀⠀⠀⢠⠃⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⣠⠄⠐⠀⠀⠀⠠⡆⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⢹⡄⡄⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠅⠀⢀⠄⠦⠈⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢳⡜⠾⠈⠛⢄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡤⠒⠊⠉⡅⡠⢰⢢⠀⢠⢣⠈⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣌⠀⠀⡌⢉⠉⣙⡛⠛⡛⡁⠃⢁⢄⢉⠀⠀⡿⢠⣵⠲⠀⢼⣱⠳⠯⡞⠌⠈⢀⣠⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠲⠣⡡⠬⢬⢠⡡⣩⣾⡶⡁⡄⠈⠞⠿⠇⠊⠏⠇⠀⣠⣥⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⣄⣀⣀⣀⠂⠁⠁⠈⣁⣀⣽⢄⣠⣤⡤⠴⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⢄⣴⣶⣿⣿⣿⣿⣿⡿⣶⣆⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠊⠁⢠⣴⡿⠟⠁⠈⣽⣿⠙⢅⠡⠈⠻⢿⣦⡄⠀⠑⠦⢄
⠀⠀⠀⠀⠀⢀⡔⠊⡄⠀⠒⢲⡿⠛⠀⡠⠂⣾⣿⣥⠀⠀⢠⠐⠄⠀⠛⢿⡖⠒⠂⠐⠑⢢⡀
⠀⠀⠀⠀⣴⣿⣷⢴⠈⠁⠀⠘⠂⡞⢁⣴⡶⣿⣿⣷⠀⠀⢰⢶⣶⡌⢓⠀⠃⠀⠀⠀⡦⣿⣿⣦⡀
⠀⠀⢠⢎⢸⣿⠀⡇⠀⣀⠀⠀⠀⠘⢄⣙⡡⠋⠛⣻⣄⠀⠑⢌⣋⡠⠃⠀⠀⠀⣀⠀⢸⢸⣿⠇⡵⡀
⠀⢠⠃⠈⢦⣉⡔⠛⣎⠀⡱⠖⢦⠀⠀⢀⡀⠆⠀⣿⣿⠀⠠⠄⡀⠀⠀⡴⠒⢯⠀⣱⠛⢲⣉⡴⠃⠘⡄
⠀⣥⣾⡗⠀⡞⠓⠶⣿⣶⣿⠴⠾⡄⠠⠎⠁⣀⡀⢸⣿⢠⣄⠈⠁⠄⢠⠷⠶⣾⣦⣿⠶⠊⠣⠀⢻⣷⣤
⢰⡿⢫⠆⠁⢑⡴⠲⣿⢻⡷⠦⡤⣎⣀⣀⠘⢻⣟⢿⣻⢟⣛⠁⡀⣀⣱⣤⠔⣾⡟⢿⠗⢦⡌⠉⠲⡝⣿
⣜⣷⡆⡠⠂⠈⠢⠴⡁⢀⠷⠠⠞⠿⠿⠿⠟⢻⣧⣿⡿⢺⡿⠻⠿⠿⠿⠣⢀⠼⡁⢈⠦⠔⠁⠐⢄⢰⣾
⣿⡿⢉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⡀⠀⠀⠐⢿⠟⢻⡟⠘⡿⠃⠀⠀⢀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠈⡉⢿
⠟⣿⣿⡷⠈⠀⠆⢴⣶⣄⣾⣿⡆⣀⣤⡀⡀⠀⠀⣾⣿⠀⠀⢀⢀⣤⣄⢰⣿⣧⣀⣶⡦⠰⠀⠁⢾⣿⣿
⠰⣹⠇⢀⢠⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣀⣂⡀⠀⠿⠿⠀⢀⣘⣀⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣄⡀⠸⡏
⠀⢻⠀⢠⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢣⡄⠀⠀⢠⠌⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⡄⠀⡟
⠀⠀⢢⠜⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡄⢀⡀⠠⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠣⡔
⠀⠀⠸⠰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣰⡶⣣⣌⢦⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆⠇
⠀⠀⡆⠠⠅⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⡞⣰⣿⣿⣆⢛⠛⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣧⠠⠄⢰
⠀⠀⢷⣶⠂⠉⣷⠸⣿⡟⢻⣿⣿⠙⠿⠎⠘⣰⣿⣿⣿⣿⡎⠃⠉⠿⠟⣻⣿⡟⢻⣿⡇⣼⠁⠐⣶⡾
⠀⠀⠈⢷⣾⣷⡆⣴⠉⢠⠀⠉⠉⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⠉⠉⠀⡄⠁⣦⢰⣾⣷⡾⠁
⠀⠀⠀⠀⠙⠿⠻⠿⢶⣿⣴⣗⣀⠀⠀⠀⠀⠻⢿⡿⢿⡿⠏⠀⠀⠀⠀⣄⣺⣦⣿⡦⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠘⠀⠀⠀⣿⣿⣟⠁⠀⠀⠀⠀⠁⠈⠀⠈⠐⠈⠀⠀⠀⠀⠈⣻⣿⣿⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⣥⡔⠀⢻⣿⠛⠺⠴⣶⣦⡦⣶⣰⠲⡖⡆⣶⣴⣰⣦⡦⠗⠚⣿⡟⠀⢢⣬
⠀⠀⠀⠀⠀⢠⣭⡝⠀⠀⢻⡤⣀⠀⠀⠀⠀⠉⠀⠘⠁⠀⠉⠀⠀⠀⠀⣀⢠⡟⠁⠀⢫⣭⡄
⠀⠀⠀⠀⠀⠠⢶⠾⡅⠀⠀⢢⡀⠀⠐⠠⠀⠤⠀⠤⠄⠀⠤⠀⠄⠒⠈⢀⡖⠀⠀⢨⠳⡶⠄
⠀⠀⠀⠀⠀⠀⠾⢟⣒⣰⠶⠀⠈⠒⠠⢄⣀⣀⢀⣀⠄⣀⣀⣀⡠⠄⠂⠁⠀⠶⣆⣞⡻⠶
⠀⠀⠀⠀⠀⠀⠈⢋⣱⡇⡄⢰⡶⠀⠀⠀⣠⣄⢀⣶⣷⣄⢠⣤⠀⠀⠀⢶⡆⣠⢸⣮⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠺⣋⣽⢠⡀⠻⠏⠛⢋⣬⠛⠛⣯⡙⠛⠻⠟⢀⡄⣯⣙⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢿⢫⣵⢲⡤⡀⢍⡠⢄⡩⢀⢤⡖⣮⡝⠷⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠻⠹⠧⠸⠇⠸⠇⠻⠏⠿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠂⠀⠀⠐⠀⠂⠀⠀⠀⠂⠀⠀⠂⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠦⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣦⡤⠀⢀⠀⢀⠀⠶⠀⠁⠀⡜⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠲⠦⢄⣙⣢⣶⣠⡀⡇⠰⠀⠀⣀⣴⠆⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣧⠄⡠⠚⢙⡏⠀⠀⠠⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠻⣿⠀⡗⠖⠈⠉⡄⠀⠐⡆⠀⠀⠀⣰⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⣉⡉⠉⣛⣳⣦⣀⠀⠹⢸⣿⣄⠀⠀⢰⣶⣶⡿⠂⠉⠛⠉⠁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠹⢦⣼⠁⡠⢴⣈⣿⠟⠁⠀⢢⣀⣨⡁⢀⣀⠀⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠙⠞⠉⢻⡛⢲⣦⣼⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠩⠑⣤⣀⣿⢋⠩⢠⡏⠟⠁⢀⣠⠤⠴⠦⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠸⡖⢀⣠⣿⣿⣿⡇⡀⠙⢣⡴⠚⠋⠚⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠠⠀⢀⣀⡀⠁⢙⣿⡿⢻⣧⣗⣛⣉⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⢠⠈⢹⡟⠉⠋⢹⡁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠤⣾⠲⡾⢇⠀⠀⢸⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡇⢸⡀⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢺⡟⠟⢸⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⢸⡃⠸⠘⠏⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢏⠘⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠘⠀⢇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⣿⣶⣿⣧⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡇⣿⠈⣽⠇⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠃⢿⠀⣼
⠀⠀⠀⣠⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⢸⠀⣿⠀⡀
⠀⠀⠀⠻⡄⢁⡄⣄⢠⢠⡄⡄⣄⠠⣄⠀⢸⡇⠀⣠⢠⢀⡀⣄⢠⢄⠀⠀⢸⡅⠀⣸⣶⣿⣄⡇
⠀⠀⠀⢴⢹⠾⡇⣇⢼⢸⡿⡇⣇⠀⢻⠀⠸⡇⡠⣏⢸⢸⡇⡇⢾⠀⠀⠀⠸⡷⠾⢻⠋⢿⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣸⡇⣇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠛⠉⠙⠁

⣀⢘⣿⣿⣿⠀⠀⠀⣿⣿⣿⡃⣀⣀⢘⣿⣿⣿⠀⠀⠀⣿⣿⣿⡃⣀⡀⢘⣿⣿⣿⠄⠀⠀⣿⣿⣿⡃⣀⡀⢘⣿⣿⣿⠀⠀⠀⣿⣿⣿⡃
⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠛⠛⠛⠛

⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿
⠿⠻⣿⣿⣿⠀⠀⠀⣿⣿⣿⠟⠿⠿⠻⣿⣿⣿⠀⠀⠐⣿⣿⣿⠟⠿⠿⠿⣿⣿⣿⠀⠀⠀⣿⣿⣿⠟⠿⠿⢻⣿⣿⣿⠀⠀⠀⣿⣿⣿⠟
⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⢰⣿⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⠇⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⡇
⣦⣴⣿⣿⣿⠀⠀⠀⣿⣿⣿⣧⣶⣦⣴⣿⣿⣿⠀⠀⠀⣿⣿⣿⣦⣶⣤⣬⣿⣿⣿⠄⠀⠀⣿⣿⣿⣦⣶⣦⣤⣿⣿⣿⠀⠀⠀⣿⣿⣿⣦
⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿

⣤⣤⣦⣤⣤⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⠀⠀⠀⣤⣤⣤⣤
⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿
⠉⢨⣿⣿⣿⠀⠀⠀⣿⣿⣿⡅⠉⠉⠈⣿⣿⣿⠀⠀⠀⣿⣿⣿⡇⠉⠉⢨⣿⣿⣿⠀⠀⠀⣿⣿⣿⡇⠉⠉⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⡅
⣀⢘⣿⣿⣿⠀⠀⠀⣿⣿⣿⡃⣀⣀⢐⣿⣿⣿⠀⠀⠀⣿⣿⣿⡃⣀⡀⢘⣿⣿⣿⠀⠀⠀⣿⣿⣿⣁⣀⣀⢘⣿⣿⣿⠀⠀⠀⣿⣿⣿⡃
⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠀⠀⠀⠛⠛⠻⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠟⠛⠻⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠛⠛⠻⠻⠛⠛⠛⠛⠛⠛⠀⠀⠀⠛⠛⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿
⠿⠻⣿⣿⣿⠀⠀⠀⣿⣿⣿⠟⠻⠿⠿⣿⣿⣿⠀⠀⠐⣿⣿⣿⠟⠻⠿⠻⣿⣿⣿⠀⠀⠀⣿⣿⣿⠟⠻⠿⠻⣿⣿⣿⠀⠀⠀⣿⣿⣿⠟
⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⢰⣿⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⠇⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⠇
⣶⣴⣿⣿⣿⠀⠀⠀⣿⣿⣿⣶⣶⣶⣴⣿⣿⣿⠀⠀⠀⣿⣿⣿⣦⣶⣶⣴⣿⣿⣿⠄⠀⠀⣿⣿⣿⣶⣶⣶⣴⣿⣿⣿⠀⠀⠀⣿⣿⣿⣦
⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿

⣤⣤⣤⣤⣤⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⣤⣤⣤⣤
⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿
⠉⢈⣿⣿⣿⠀⠀⠀⣿⣿⣿⡅⠉⠉⢨⣿⣿⣿⠀⠀⠀⣿⣿⣿⡅⠈⠉⢈⣿⣿⣿⠀⠀⠀⣿⣿⣿⡅⠉⠉⢨⣿⣿⣿⠀⠀⠀⣿⣿⣿⡅

⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠉⠉⠉⠉⠛⢷⣦⣤
⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠷⣤⣀
⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⢷⡶
⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀
⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢿⣲⣤⣄⡀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠇⠀⠃⠀⠈⠃⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠛⠁⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠈⠙⠒⠯⣿⣖⣒
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠒⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡃⠀⢴⠀⣸⠁⢀⠏⢸⡄
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢌⣈⣀⣁⣀⡠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢃⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡄⠀⢀⡤⠒⠒⠀⠈⣙⠒⢤⡚⠔⠉
⣄⣠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡴⣃⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⢤⡄⣠⢤⡠⡄⣤⣠⠄⡖⢲⠀⠀⠀⠀⡀⡇⠀⡞⠀⢿⠈⣇⠈⡹⠃⢨⠇
⠀⠀⠈⠁⠐⠪⠥⢐⣂⣠⠤⣄⣎⡜⠀⡓⢢⡄⣆⣸⠺⣅⣿⢸⡸⠇⠏⠘⠇⠛⠐⠁⠙⠁⠉⠀⠁⠈⠀⠀⠀⢀⠄⠓⠐⠻⠦⣈⡀⢁⣀⠁⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢿⡿⠧⣀⠉⠉⠀⠈⠀⠀⠑⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢰⠊⠈⠉⠬⠍⠉⠭⠤⠀⠂⠀⠐⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠰
⠻⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠉⠙⠛⠿⢶⣤⣄⣀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢶⣖⣄⣀⡀
⣿⣿⣿⡿⣿⡿⢿⢿⡿⡿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣷⣦⣶⣷⣶⣿⣶⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣴⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⡀⣀⠀⠀⠀⠀⠀⠀⢰⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡉⠉⠙⠛⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢰⡶⠶⠶⢶⣦⠀⠀⠀⢀⡞⣿⠹⡄⠀⣸⢻⠳⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣿⡷⡄⢀⡞⣿⠣⡀⠀⡼⠁⣿⠀⠙⣴⠃⢸⡀⠑⣄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠎⢸⡇⢨⠿⣀⣿⢀⣨⠾⠅⣀⣿⣀⠜⠉⠳⣼⡇⠀⠈⠳⢄⡀⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣧⢀⡴⠃⠀⢸⡗⠁⠀⠀⣿⠉⠀⠀⠀⠈⣿⠀⠀⠀⠀⢸⡗⢤⡀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⣀⣤⣶⣽⣧⡶⠶⠶⢿⠿⠿⠟⠛⠛⣿⠿⠿⠷⠶⢾⣷⣤⣍⣓⠢⢤⣿
⠀⠀⠀⠀⠀⣤⣤⣸⣿⣟⣋⣉⣡⣸⣧⣤⣤⣤⣾⣶⣾⣿⣿⣿⣿⣶⣦⣤⣤⣤⣧⣀⣉⣙⣿⣶⣿⣤⣤
⠀⠀⠀⠀⠀⢸⣿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⣸⣿⣇⠀⠀⠀⠀⠀⢰⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠙⣿⣿⡀⠀⡤⠀⠀⠘⢧⠰⠿⠛⠋⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⣻⣿⡇⠘⠀⠀⠈⠀⢸⠆⡄⡔⢰⢀⣆⠎⡇⠘⡄⣼⢠⡎⣠⢂⠄⡆⡶⣦⠀⢰⣿⣿⡀
⠀⠀⠀⠀⠀⢨⣿⢿⣿⠀⠈⠀⠀⣀⣥⣦⣁⢁⢉⢸⠈⠀⠉⠀⠃⠙⠈⠃⠈⠊⠁⠀⠁⠉⠀⢸⣿⣽⡇
⠀⠀⠀⠀⠀⢸⣟⠺⣿⠀⠀⠀⠀⠀⠀⠀⣹⠉⠉⠘⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⡆⠀⠀⠀⣿⡇⢺⡇
⠀⠀⠀⢠⣴⣶⡏⢺⣿⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣀⢀⡀⢸⠀⠀⠀⢀⢀⠀⢠⡃⠁⠀⠀⠀⣿⣿⢨⣷⣦⡄
⠀⠀⠀⠀⠈⢩⡷⠙⣿⠀⠀⠀⠀⠀⠀⠰⠃⠀⢸⠊⠘⣇⢾⢸⠃⡇⠐⡇⣿⠸⡇⠀⠀⠀⠀⣿⠛⢰⡏⠁
⠀⠀⠀⠀⠀⢸⣯⣩⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⣹⡧
⠀⠀⠀⠀⠀⠐⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⠇
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⡀⠀⢀⢀⠀⢀⠀⢀⡀⠀⢀⡀⠀⠀⢀⠀⠀⢀⠀⠀⡀⢀⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⣴⣿⣿⠀⠑⠄⠀⡇⠀⢸⢸⡀⢸⠀⠀⡆⠀⢘⡀⠀⠀⢸⠀⠀⠈⢄⠀⡃⢸⠸⡀⢻⣿⣷
⠀⠀⠀⠀⠀⢹⣿⠇⠀⠀⠀⠀⠁⠀⠈⠀⠁⠈⠀⠈⠁⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠀⠁⠈⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⢛⡿⠿⠷⣖⡒⠒⠶⠦⠤⣄⣀⣀⣠⡤⠤⠶⠒⠒⣲⠶⠿⠿⣟⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢺⣿⢳⢦⣙⠳⢤⣄⣰⣿⢿⡿⣿⣧⣀⡤⠖⣋⡥⢞⢹⡿⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠈⠉⠛⠛⠛⠛⠉⠈⠛⠛⠛⠛⠋⠁⠀⠈⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠿⠛⠛⠛⠿⠶⢶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠛⠉⠀⠀⠀⠀⢀⣀⡀⠀⠀⠈⠙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢁⣠⠴⠒⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠲⠿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⡀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢳⡈⠓⠶⠶⠒⠒⠛⠉⠉⠉⠉⠉⠉⠉⠛⠓⠲⠦⢤⣀⡀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡾⢷⡀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⣰⡟⣻⠀⠻⣄⠀⠀⠀⠙⢦⡀⠀⠀⢠⡖⠋⠉⠉⠉⠉⠉⠛⠒⠶⢤⣄⡀⠀⠀⠀⠈⠳⢮⣻⣆
⠀⠀⠀⠀⠀⣰⡟⣰⠃⠀⠀⠈⠳⣄⠀⠀⠀⠙⢦⣄⠸⣄⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⠙⠲⣤⡀⠀⠀⠀⠙⢿⣧
⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⢀⣠⣤⡈⠳⣤⡀⠀⠀⠈⠓⠦⣍⡉⠉⠉⠉⠉⠉⠙⠲⢤⣀⠀⠀⠀⠙⢦⣄⠀⠀⠀⢹⣧
⠀⠀⠀⢰⡿⠃⠀⠀⢀⡴⠋⠀⠀⢹⠄⠀⠙⠳⢦⣄⡀⠀⠀⠉⠓⠶⢤⣄⣀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠘⢧⡀⠀⠀⢻⣆
⠀⠀⢠⣿⠁⠀⠀⢠⠞⠁⠀⠀⣠⠏⠀⠀⠀⣠⡤⢬⡉⠓⠲⠦⠤⣤⣀⣀⠉⠙⠲⣄⡀⠀⠀⠙⢦⡀⠀⠀⠹⣆⠀⠀⢿⡄
⠀⠀⣾⠇⠀⠀⢠⠏⠀⠀⢀⡞⠁⠀⠀⣠⠞⠁⢀⡼⠃⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄⠀⠙⢦⡀⠀⠀⠳⡄⠀⠀⠘⣆⠀⠘⣷
⠀⢰⡿⠀⠀⢠⡏⠀⠀⢠⡟⠀⠀⢀⡼⠁⢀⡴⠋⠀⠀⢀⣠⠤⠶⠦⠤⣄⠀⠀⠈⠳⣄⠀⢳⡄⠀⠀⠹⡄⠀⠀⠘⡆⠀⢻⡇
⠀⣾⠇⠀⠀⣼⠀⠀⠀⡾⠀⠀⠀⡼⠁⠀⡞⠁⠀⠀⡴⠋⠀⠀⠀⠀⠀⠈⠙⣆⠀⠀⠸⣆⠀⢻⡀⠀⠀⢹⡀⠀⠀⢻⠀⠘⣿
⢠⣿⠀⠀⠀⡇⠀⠀⢰⡇⠀⠀⢠⡇⠀⢸⠇⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢻⠀⠈⡇⠀⠀⠈⡇⠀⠀⠸⡇⠀⣿
⢸⡇⠀⠀⢰⡇⠀⠀⢸⠀⠀⠀⢸⡇⠀⢸⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⡗⠀⠀⠀⣿⠀⠀⠀⡇⠀⢸
⣿⡇⠀⠀⠘⡇⠀⠀⢸⡀⠀⠀⠘⡇⠀⠘⣇⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⢀⡟⠀⢠⡇⠀⠀⠀⡟⠀⠀⠀⡇⠀⢸
⣿⡇⠀⠀⠀⣇⠀⠀⠘⡇⠀⠀⠀⢳⡀⠀⠹⣆⠀⠀⠈⠳⢦⣄⣀⣀⣠⡴⠛⠀⠀⢀⡾⠁⠀⡼⠁⠀⠀⢠⡇⠀⠀⢸⠇⠀⠈
⣿⣿⠀⠀⠀⢹⡀⠀⠀⢻⡀⠀⠀⠈⢳⡀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⡴⠋⠀⠀⡴⠃⠀⠀⠀⡼⠀⠀⠀⡾
⣿⡘⡇⠀⠀⠈⢧⠀⠀⠀⢳⡀⠀⠀⠀⠙⢦⡀⠀⠈⠙⠲⠦⢤⣤⣤⠤⠶⠚⠉⠀⠀⣠⠞⠁⠀⠀⠀⡼⠃⠀⠀⣸⠃⠀⠀⢀
⢻⡇⢹⡄⠀⠀⠈⢧⠀⠀⠀⠳⣄⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠁⠀⠀⠀⢀⡞⠁⠀⠀⣰⠋⠀⠀⢀⣾
⢸⣧⠀⠻⡄⠀⠀⠈⢳⡄⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⠒⠛⠋⠉⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡼⠃⠀⠀⢀⡞⣹
⠀⣿⡄⠀⠙⣆⠀⠀⠀⠙⢦⡀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠀⠀⢀⣠⠟⠀⠀⠀⣠⠞⢀⣿
⠀⠸⣷⠀⠀⠈⠳⣄⠀⠀⠀⠙⠦⣄⠀⠀⠀⠈⠙⠓⠶⠤⢤⣤⣤⣤⣤⠤⠶⠒⠋⠉⠀⠀⠀⣀⡴⠛⠁⠀⠀⢀⡴⠃⠀⣼⡏
⠀⠀⠹⣧⠀⠀⠀⠉⠳⣄⠀⠀⠀⠈⠛⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⢀⡴⠋⠀⠀⣰⡟
⠀⠀⠀⠹⣷⣄⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠦⠤⠤⠤⠤⠶⠶⠚⠋⠉⠀⠀⠀⠀⢀⣠⠞⠋⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠉⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠈⠻⣧⣙⠶⣄⡀⠀⠀⠀⠀⠈⠉⠓⠒⠶⠦⠤⠤⠤⠤⠤⠴⠶⠖⠚⠋⠉⠀⠀⠀⠀⠀⣠⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⡙⠳⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⣫⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⣀⡉⠉⠓⠒⠶⠤⠤⠤⣤⣤⣤⡤⠤⠤⠶⠖⠚⠋⣉⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⣦⣤⣤⣤⣀⣀⣀⣀⣠⣤⣤⣴⡶⠾⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠉⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠁⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⢀⠀⠀⢻⣿⣦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣼⣿⠇⠀⠀⠉⠛⠻⠿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣸⣿⣿⡿⠃⣀⣴⣷⠄⠀⠀⢀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣭⣴⣶⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⠋⢁⣴⣿⡿⠛⠉⠀⠀⠈⣿⣿⡿⠟⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋⠀⠀⠀⠻⠟⠉⠀⠀⠀⠀⠀⢰⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⢀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣿⣿⣿⣇⠀⠀⠀⣀⣤⣾⣿⠟⠁⠀⣠⣾⣿⠟⠁
⠀⣀⣤⣶⣿⡿⠿⠛⣛⣽⣿⣿⣿⣿⣷⣾⣿⡿⠟⠋⣀⣤⣴⣿⡿⠋⠁
⣾⣿⡟⠋⠁⠀⠀⣾⣿⠟⠋⣽⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠙⠿⣿⣿⣶⣶⣦⣌⠁⢀⣼⣿⠏⣿⣿
⠀⠀⠀⠀⠉⠉⠛⣿⣷⠈⠛⠁⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡟⠁
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣼⣿⣿⣿⣿⠟⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠙⢿⣿⡿⠋⠀⠀⠈⠻⣿⣿⣿⣿⣦
⣦⣼⣿⣿⣿⣿⠟⠁⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿
⣿⣿⣿⣿⠟⠁⠀⠀⢀⣰⣿⣿⣿⣿⣿⡿⠋⠀⠻⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠙⢿⣿
⣿⣿⠟⠁⡀⠀⠀⠰⣾⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣷⠄⠀⢀⣴⣿⣿⣿⣷⣄⠀⠀⠙
⠉⠁⣠⣾⣿⣦⡀⠀⠙⢿⣿⣿⣿⣿⣿⣦⡀⠀⢠⣴⣿⣿⣿⣿⣿⠟⠁⢀⣶⣿⣿⡿⠋⢿⣿⣿⣷⣄
⣠⣾⣿⣿⣿⣿⣿⣦⡀⠀⠙⢿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠟⠁⠀⠘⢿⣿⣿⣯⡀⠀⢀⣽⣿⣿⣿⠇
⣿⣿⣿⠟⠙⢿⣿⣿⣿⣦⡀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣠⣤⡀⠀⠙⢿⣿⣿⣶⣿⣿⣿⠟⢁⣰
⣿⣯⡁⠀⠀⠀⣹⣿⣿⣿⣿⠆⠀⠀⠙⢻⣿⣿⣿⠟⠁⠀⣠⣾⣿⣿⣷⣦⡀⠀⠙⢿⣿⡿⠋⠀⠀⢾⣿
⣿⣿⣿⣦⣠⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠙⠛⠁⠀⣠⣾⣿⣿⣿⢿⣿⣿⣿⣆⠀⠀⠉⠀⠀⠀⠀⠀⠙
⠙⢿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠁⠀⠙⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢀⣤
⣦⡀⠙⢿⡿⠟⠁⠀⠀⠀⠀⢀⣴⣿⣤⠀⠀⠀⠈⠻⣿⣿⣿⣦⡀⢀⣴⣾⣿⣿⡿⠋⠀⠀⠀⢀⣴⣿⣿
⣿⣿⣦⡀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣤⡀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢀⣴⣿⣿⣿⠿
⢻⣿⣿⣿⣦⡀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠈⠿⣿⣿⡿⠋⠀⠀⠀⠀⣴⣿⣿⣿⡟⠁
⣿⣿⣿⡿⠃⣠⣾⣿⣿⣿⣿⡿⠋⠈⠻⣿⣿⣿⣿⣷⣄⠀⠀⠀⠘⠋⢀⣴⣀⠀⠀⠀⠈⠻⣿⣿⣿⣦⣀
⣿⠟⠉⠀⠺⣿⣿⣿⣿⣿⣍⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣷⠄⠀⢀⣴⣿⣿⣿⣶⣄⠀⠀⠀⠈⠻⣿⣿⣿
⠁⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⣄⢀⣴⣿⣿⣿⣿⣿⠏⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠈⠻⣿
⣾⣷⣄⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⣠⣼⣿⣿⣿⡿⠋⠀⠙⢿⣿⣿⣿⣦⡀⠀⠀⠀⢀
⣿⣿⣿⣷⣄⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠋⠀⠀⠈⠻⣿⣿⣿⣿⣄⠀⠀⠀⣠⣿⣿⣿⣿⡟⠀⢀⣰⣿
⡿⢿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠘⢻⡿⠋⠀⢀⣄⠀⠀⠀⠈⠻⣿⣿⣿⣷⣤⣾⣿⣿⣿⠟⠃⢀⣴⣿⣿⣿
⠀⠀⠙⢿⣿⣿⣿⢷⡄⠀⠀⠀⠀⠀⠀⣴⡿⣿⡷⣤⠀⠀⠀⠈⠻⣿⣿⣿⡿⣿⠟⠁⠀⣴⣿⣿⣿⣿⠿
⠀⣀⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠸⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⢠⠛⠀⠀⠀⠀⠀⠀⠐
⠀⠃⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠
⡄⢀⢤⠀⡴⠒⠚⠿⠟⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠆
⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⠆⠀⠂⠁⠈⠁⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣦⠀⠀⠀⠐⠀⠀⡄
⠀⠀⠀⠀⣄⠤⢠⠔⢀⡆⠀⠀⠀⠀⠀⠸⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈
⠀⠐⠀⠐⢪⡄⠠⠤⠀⠀⠀⠀⠀⠀⠀⢀⢈⡀⠀⠀⠀⡀⣀⠀⣄⢠⠐⡆⢴⢰⠔⣼⠲⠈⠆⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣄⠀⢀
⢀⠀⠀⠀⠠⠓⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒⠼⠆⠇⠳⣸⠘⠚⠈⠁⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⡴⢦⡄⠠⣤⡜⣧⣶⠀⣿⣹⡆⢶
⠐⡔⢢⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠘⠷⠴⠃⠘⠓⠃⠈⠉⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⠆⠠⠰⠘⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣮⣵⣶⣴⣦⣴⣯⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⣶⣛⡀⠘⠛⠒⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠿⠒⠈⠉⠉⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⢩⡀⠀⣦⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣌⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠙⠒⠒⠒⢶⠛⠉⠙⠐⠒⠒⠒⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⢢⡴⠋⠙⢆⠀⠈⢹⠂⠀⠀⠀⠀⠀⠀⠀⠀⣤⢈
⣤⠀⠐⠀⡤⢢⡤⠤⠤⠤⡤⣄⠀⡀⠤⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⠂⠤⠴⠁⠀⠀⠈⡄⠀⢎⠀⢀⠔⢉⡑⢢⠀⠀⠀⠈⢏
⠈⣦⠽⣏⣅⣀⣛⣉⣹⣻⣿⣿⣾⣆⣴⣿⣿⣫⠆⠀⠀⠘⠣⢀⣀⡈⠑⠒⠒⢇⠀⠀⣠⠁⠀⠀⠀⠀⠘⠋⠉⢀⠀⠀⠀⠀⢸⡄
⠀⢀⠀⠀⠀⠀⠀⠀⢁⣠⣀⣀⣀⣀⣀⣀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⠀⠀⠉⠉⠁⣤⡄⠀⠀⠀⡶⠤⠔⠋⠀⠀⠀⣀⠜
⠀⢸⠀⠀⠀⠀⠀⠐⣽⣿⣻⣿⣽⣿⣶⣛⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⡆⠆⠀⠀⠀⡗⠀⠒⠒⠂⡩⢛⣣⣴⣒⣀⡠⠀⣀
⠀⢸⠀⠀⠀⠀⠀⡆⡬⢼⣿⣤⣭⣭⠬⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⡸⠀⢸⠀⠀⢠⠁⠀⠀⠀⠀⣧⠋⠀⠋⡇⠀⢈⢷⡦⡈
⠀⠸⠦⠄⣀⣀⣀⡃⡧⣜⢣⣷⣤⣿⢴⡆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⢀⡴⠁⠀⡸⠀⠀⢈⡑⠚⢕⢄⣀⢛⠀⢸⠀⠃⠀⠘⢸⠁⠁
⠀⠀⠀⠀⠀⠀⢨⠛⠛⠉⠛⠋⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⡴⠒⢤⡁⠀⢀⠀⠧⠤⠖⠁⡼⠁⣺⠀⠈⢾⠀⠘⢸⠀⠀⡆⡈
⠀⠀⠀⠀⠀⠀⢨⣀⣴⢦⠖⣿⣬⡿⢟⣶⠀⠀⠀⠀⠀⠀⠀⣀⣢⠀⠄⢣⠀⣶⡋⢿⣿⣿⡶⣰⣍⠋⠀⠀⠀⠀⠀⠈⡕⡆⡈⠀⠀⠃⠇⠀⠇
⠀⠀⠀⠀⠀⠀⢨⢻⠊⠉⠉⠁⢀⠀⡽⠛⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠚⠧⡀⠠⠼⠷⣖⣋⣩⢤⠒⠇⣀⡄⠀⠐⠂⠀⠘⡼⠇⠀⠘⢸⠀⢰
⠀⠀⠀⠀⠀⠀⡈⠀⠙⠈⠁⠠⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⣔⣂⣩⣥⣔⡶⣒⠦⠤⣄⡀⠀⠈⠑⠥⢣⠔⡶⢠⠵⢶⢾⠉⠈⢆⡇⡆⠀⡄⢰
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠠⠤⠄⠒⠃⠀⠀⠀⠀⠀⠀⠀⠘⢍⠛⠂⠋⠹⠉⠻⠽⠰⣼⠂⠀⠀⠀⠀⠙⠲⠧⠔⠚⠂⠧⠀⡈⣣⡅⠀⡆⡌
⠀⠀⠉⠑⠒⠴⠧⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠀⠤⠤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⢤⡀⠀⠀⣠⠖⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⠀⢠⠞⠁⠀⠀⣀⡙⣦⠀⢧⣠⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢀⣈⡇⢻⡄⠀⠀⠨⡀⢠⡏⣠⠴⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣀⣤⣀⠻⢦⡀⠀⠀⣉⣉⠁⠠⠠⡈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢈⠳⣄⢩⡟⠋⠉⠀⠙⣷⡒⠁⠀⢻⡄⠀⣀⡴⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢁⣃⡏⣿⠀⠀⠠⡘⢀⣿⠈⠆⠀⠈⣧⠀⢯⣀⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠈⠙⠓⠾⢆⠀⠀⠀⠉⠉⠓⠚⠒⠠⢿⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⠀⠁⠀⣄⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡿⠃⠀⠺⠀⠀⠐⠥⠈⠢⢀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢈⠙⠁⡠⠀⠀⠰⡀⠀⠣⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣀⣀⣀⡀⠈⠂⠀⠁⢰⣏⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣉⣹⣦⠡⠀⠘⠀⠀⠀⠀⠈⠛⠛⠶⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠓⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⡀⠀⠀⠄⠄⠤⠤⠤⠒⠒⠉⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢀⠤⠤⡄⠀⡠⡄⠀⣀⠀⠀⣀⠤⣀⠀⠀⠀⣀⡀⠀⠀⡠⠔⠒⢤⡀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡇⢰⠊⠁⠀⠅⡇⠀⡍⠀⡜⢡⠔⠋⠀⠀⢸⠀⢸⠀⢸⠀⠤⠄⠀⢡⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠣⠬⢍⡁⢲⡄⠃⠀⠆⣿⠀⡏⠀⠀⠀⠀⡀⢿⠀⢇⢸⡀⠈⠁⣠⠊⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⡱⠀⡇⠘⠞⢠⠻⡀⢠⡸⠍⢉⠥⠇⠈⣄⠘⡀⡇⢰⡀⠡⡀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⢴⣂⣉⡠⠖⠱⣀⣠⠞⠀⠳⠤⠴⢆⡤⢸⢀⡄⠈⡄⢧⠃⢸⠡⡀⣱⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠃⠈⠉⠀⠉⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠦⠤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠟

⠀⠄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠈⠉⠁⠀⠓⠒⠂⠀⠀⠀⠀⠄⠀⠀⢁⠀⠀⠀⠁⢀⠀⠀⠀⠒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠤⠄
⠀⠦⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢩⡀⠀⢺⠅⠄⣽⠋⠀⠀⠤⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠁⠔⠠⠀⠀⠤
⠀⠀⠀⠀⠘⠃⠀⠀⠈⣤⣀⠀⠛⣀⠀⠒⠁⠀⠀⠀⠀⠈⠀⠘⢧⣀⠀⠁⣀⡀⠀⠀⠂⠀⡄⠀⠀⠐⡀⠂⠀⠀⠉⠁⠀⠠⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠘⠁⠀⡀⠀⢠⡆⠀⠀⠐⠚⢑⠲⠀⠀⠀⣤⠈⣉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠂⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⢄⠠⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⠀⢠⣽⠀⠀⠚⠃⠈⠛⠀⠀⠀⠉⠀⠀⠀⠠⣦⠀⠀⠀⠀⢀⡆⠀⠴⠦⣄⢠⡀⠀⠀⠀⠀⠆⠀⠀⠀⠘⡀⠂⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠐⠀⠀⠀⢀⠀⠀⢻⠆⠀⠀⠀⠀⠘⡇⢡⠀⠉⠀⠀⠀⠀⠈⠡⡐⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠽⠧⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠀⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠃⠀⠀⠆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠂⠤⠄⠂
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡴⠀⢰⣬⠇⠈⠁⠉⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠕⡀⢀⡀⡐⠒⠀⠀⠈⢁⡀⠑⠀⠀⠢⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⣆⢞⠇⠀⠁⡤⠀⣀⡈⠛⠁⠃
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢈⠀⡀⠀⠀⠂⠀⠀⡁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠠⡀⠀⠈⠴⠂⠀⠀⡨⠀⠨⠦⠄
⣀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⢀⠀⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠈⠀⠀⠀⠀⠀⠄⠀⠤⠀⠀⠀⠀⠐⠀⢀⡀⠀⡀⠈⢢⠀⠀⠉⢀⠀⠤⠀⠂⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡅⠐⠀⠈⠀⠗⠀⠀⠆⠀⢠⠀⠀⠂⠀⠀⠀⠂⠀⠠⠀⡀⠠⠆⠀⠀⢂⠁
⠀⠀⠀⠀⠂⠀⠀⠑⠀⠀⠀⠀⠠⠼⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⢈⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑⠅
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⣀⠀⠀⠠⠀⠀⠈⣀⠁⠀⢀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠠⡘⠉
⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠁⠀⠀⠀⠢⠀⠰⣬⠁⠀⠀⠈⠂⠀⠀⠈⠀⠈⠀⠀⢀⠀⢀
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢠⠖⠀⠲⢏⣀⠐⠃⢀⡉⠁⠀⠛⠈⠀⠈⢘⠃⠀⢠⢀⢌⠀⢀⡁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠈⠀⠀⠀⠁⠀⠙⠃⠠⡄⠀⠀⠀⠈⠠⠁⣠⠀⠀⢠⠄⠈⠀⢀⠀⣸⡄⠀⠐⠘⠆⠙⠂⠀⠔⠒⠄⢉⠀⠀⠰⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠁⢀⡄⠀⠐⣈⠀⠘⠓⠠⣔⠀⡀⠀⠀⠁⠈⠃
⠀⠀⠀⠀⠀⠄⠀⢀⠔⠁⠀⠂⠀⠀⢀⠀⠀⠐⠂⡀⠀⠀⢀⠀⠀⠀⠶⠄⠠⠀⠀⠁⠠⠆⠀⠀⠀⠀⠀⠀⠐⣀⠒
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠀⠈⠀⠒⢀⠡⠐⠡⡌⠀⡴⠂⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⡄⠀⠀⠁⠀⠘⠈⠀⣼⠀⠈⠁⠀⠀⡜⠀⠈⡈⠀⠀⠀⡐⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⡄⠠⠄⢀⠀⢀⡀⠀⠀⠁⠀⠓⠆⠀⠠⠼⠀⠀⠀⣰⠀⢬⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⡃
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣼⠀⠀⠀⡿⠀⡽⠙⣛⡭⠀⠰⠂⠀⠀⠀⢀⠀⠀⠀⠈⠁⠀⠑⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠔⠁⢌⢀⢀⡉⠁⠀⠓⠺⠯⠁⠀⣀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⡀⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⢴⡃⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠾⠘⠀⠈⢃⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⠃⠀⠀⠠⠆⠀⠀⠀⠰⠄⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠁⠀⠀⠀⡀
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣤⣶⣶⣤⣶⠲⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡆⢠⣄⡀⣀⣀⢘⣿⣿⣿⣿⣿⣿⣿⣶⡾⢿⣿⣗⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢥⠺⣿⡿⢻⣿⣿⣿⠉⣿⠿⢿⣄⣸⡏⠻⣧⡁⠙⡟⠛⠳
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣼⣧⣿⣿⣾⣿⣿⣿⠶⠋⠠⣶⠜⠻⢷⣦⠼⣧⡀⢁⠀⠀⢠
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡆⠀⠀⠈⠑⢦⡄⠸⠟⢾⣿⡞⣇⡀⢸
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⠈⠀⠀⠀⠀⠀⢸⡧⠀⠀⣸⣿⣵⠏⠁⠌
⠀⠀⣰⣿⢟⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣹⠗⠀⠀⠀⠀⠀⠸⡿⠀⠀⣘⣉⡅⠀⢀⣤⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠙⠋⠁⣿⡋⣀⣠⣤⣶⣶⡶⣞⣁⣀⣴⣯⣿⣧⠔⠀⠉⠁
⠀⠀⠀⠀⠈⠁⣉⠻⠍⠁⢰⣾⣿⠀⠀⠠⣤⣶⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⡿⠟⢫⡟⠁
⠀⠀⠀⠀⠀⠂⡹⠀⠀⠀⣈⣿⣅⡀⣴⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣁⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⣿⣿⣿⣿⡿⢍⣴⡟⠛⠛⠿⠶⢿⣿⣿⠓⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠛⠛⠉⣠⣾⡏⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣄⡀⡀⣀⣀⣀⠀⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢠⣿⡇⡇⢻⣿⣿⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⠀⠀⠀⠀⠀⠀⠀⠈⡿⠸⡿⡇⠇⢼⠿⠿⠀⠷⠁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⣀⣀⣀⣀⣀⡀⣀⣠⣤⠀⣀⣀⣠⢀⣀⣄⢠⡄
⠀⠀⠀⠀⠀⠀⢠⣤⣴⣿⠟⠁

⠀⠀⠀⣠⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠤⢤⣄⠀⠀⠀⠀⠀⠤⣄
⠀⠀⠀⢸⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⠀⢀⡤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⢀⡏⠀⠀⠀⠀⠀⠄⠀⠀⢠⡄⠁⢰⣿⣽⣷⣄⠀⡀⠠⠴⢦⣤⣾⣯⠿⠶⠶⣤⣶⣷⣶⣄⠀⢈⠢⣄
⠀⠀⠜⠀⠀⠠⡷⠚⠃⠄⡠⠄⠀⠻⠶⢿⡟⠙⢈⡽⠯⠴⠶⠶⠶⣿⣿⣷⣶⣄⣀⣬⣧⣌⣿⣿⡄⠀⠀⠈
⠀⣼⠃⠀⣄⣀⣧⠀⠀⠀⠁⠀⣀⠦⣽⣆⠙⣶⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣦⠌⠁⡀⠙⣿⡟⠁
⠀⢁⠀⢸⣿⠟⠁⠀⡄⢠⠖⠛⠜⠃⠈⡠⠋⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠙⢆⡹⣿⡦⣼⣯⡀⠀⠀⢲
⠀⢸⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⢀⠄⠁⠄⠀⠀⠀⠀⠈⢣⣹⣿⣿⣿⣽⣧⣤⣈
⠀⣾⠀⠀⠹⣽⠀⠀⣠⡴⠛⡞⠁⠀⠣⡀⢤⣴⠲⢤⣴⣎⠀⡀⣠⠀⠀⠀⠀⠀⣠⣼⣿⣿⣩⣿⣿⣿⣿⣿
⠀⢹⢀⣾⠟⠉⠛⠐⠃⠀⣠⡠⠀⠀⣠⠏⠙⠿⡇⠈⠛⠋⠀⠀⣨⠀⡇⠀⠀⠀⣾⣿⣿⡣⣿⣿⣿⠁⢻⣿⡆
⠀⢸⣾⣿⠆⢀⠀⣀⣠⡾⠋⠀⠀⠚⠃⠀⠀⢣⢙⡠⢤⠆⠀⢠⡟⢤⡇⠀⠀⣰⣿⣿⣟⣼⣿⣿⠋⠳⠾⢿⡇
⠀⣼⣿⣯⢠⣿⣶⡿⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⡿⢟⠃⣀⠔⠉⠀⢀⣋⠀⣰⣿⣿⣿⣿⣿⣿⣿⣛⢶⡄⢸⡇
⠀⢹⣿⣿⡿⠃⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠠⠿⠧⠴⠜⡳⠋⠉⢳⣿⣿⣿⣿⣿⣿⣿⣿⣟⢦⣵⣾⡆
⠀⢸⣿⢣⡇⢶⠀⢻⠓⠚⠀⢀⡠⠤⠄⠀⠴⠊⠀⠀⠀⠀⢠⠎⠀⠀⠀⣸⣿⣿⣿⣿⣿⠿⣿⠻⢌⡀⢹⣿⠁
⠀⢹⡿⢻⡷⠐⠄⣀⡧⡀⠞⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⣄⣴⣿⣿⣿⣿⣿⣿⠈⢿⣷⣶⣿⢾⣿⡇
⠀⢸⡀⢸⠁⠀⠆⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣉⣧⣨⣿⠿⡋⠒⣿⡇
⠀⢸⣇⢸⠂⠀⠐⠀⠣⣘⣤⠢⠤⠤⡤⠄⠀⠀⣀⠤⢰⣤⢌⣷⣿⣿⣿⣿⣿⣿⣿⠃⠙⠧⠿⣿⡆⢸⣿⣿⡇
⠀⢸⣷⠈⢧⠀⠀⢀⠀⠸⣆⠙⠒⣶⠈⠐⠠⠤⠴⠒⠉⠀⢀⠏⢠⣿⣿⣿⣿⣿⣏⠈⠀⠀⠀⡘⡟⢶⣼⣿⡇
⠀⢸⣿⠀⠘⢦⡀⠀⠡⠄⢾⣤⣜⡸⠀⠀⠀⠀⠀⠀⠀⢠⡊⢠⣿⣿⣿⣿⣿⣿⠂⠙⢦⡀⣠⠃⢿⣿⣿⣿⡇
⠀⢸⣿⣷⣀⠸⠥⢶⡶⠟⡛⣿⣯⣴⡄⠀⠀⠀⠀⠀⠀⢈⣉⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠈⠁⠀⢸⠿⣿⣿⡇
⠀⢸⡟⣿⣏⠀⠀⠸⣆⠈⠛⡾⠉⠁⠐⠀⠀⠀⠀⠀⠀⠀⣙⠿⢿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⣸⣾⣿⣿⡇
⠀⢸⣧⣿⣇⣠⡄⡰⠉⣆⣼⣿⡂⠀⠸⣧⣤⣴⣶⠒⣺⡉⠀⠒⠋⠁⠀⠉⣏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⡁
⠀⢸⢩⣟⢹⣏⣿⠃⠀⢹⣿⣻⣷⠀⣠⠟⠃⢹⣊⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡟⢿⢿⣿⡇
⠀⢸⣻⣿⣾⣽⡏⠀⠀⠈⠛⠛⠉⠘⠃⠢⠄⣀⡈⢘⡂⠀⣀⡀⠀⣤⣀⣄⣼⠀⠀⠀⠀⠀⠀⣼⢷⣯⣾⣠⣄
⠀⢨⣙⠶⠿⠿⢿⣤⣤⠀⠀⠀⠠⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⢰⣧⢡⡞⣽⣏⡅
⠀⢈⠙⠛⠛⠺⢿⠤⣿⣿⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⣠⣯⠃⣾⢋⡀⠄
⠀⠈⠁⠀⢀⣀⣀⠀⠈⠽⣧⡈⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⡶⣿⣏⠳⣞⣙⡻⠚⣤⣨⡄⠊⠃
⠀⠠⠀⣀⡤⣩⠝⠛⠛⠒⠀⡁⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⢟⣭⠴⠛⠛⢻⣟⠉⣀⠀⠑⡶⠇
⠀⠰⠖⣁⠤⠛⠴⠀⠑⠀⠀⡇⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⣿⢿⠃⠀⠀⠀⠈⣿⣿⡯⢷⠀⢱⡆
⠀⢰⠒⡉⠒⠏⢶⡆⢠⠄⠀⢳⠀⠀⠀⠀⠀⣸⠡⡇⠀⠀⠀⠀⠀⠀⢠⣷⣼⣀⣀⣠⣤⡌⣿⡤⠷⠾⣷⢛⡁
⠀⢸⡞⠛⢀⡀⠚⠛⠃⣀⡀⢈⡀⠀⠀⠀⢠⡃⠄⣿⡀⠀⠀⠀⠀⠀⣾⡒⠚⣿⣿⣟⢳⢁⣹⣆⠈⣀⠘⣟⠁
⠀⠘⣷⣶⣬⠤⡄⠀⠀⠀⠭⠬⠀⠠⠄⠀⠩⠩⢯⣩⠥⠤⠤⠀⠄⠨⠀⠠⠀⠀⣤⠤⢴⠌⠹⠳⠠⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⣵⠋⠀⠀⣈⣉⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢟⣵⣿⣿⡿⠶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣇⠀⢠⠞⠉⠀⠉⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⡿⣱⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢳⣀⠀⠘⢦⣿⡴⠖⠶⢤⡀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣞⢿⣿⣧⣙⠓⢛⣭⣽⡻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠉⠛⠒⠛⣻⠙⢦⠀⠀⢹⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣭⣭⣵⡆⣶⡝⣿⣿⣆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⣀⣠⠴⠃⠀⢸⠇⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⣫⣾⣿⡗⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢉⡿⢦⣄⣀⣀⣤⣏⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠎⠛⠿⢿⡿⠟⠵⠟⠋
⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡴⠋⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⣠⠞⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⣠⠞⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⣾⣁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠁
⠀⠀⠈⠉






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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣋⣴⣶⣶⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠞⠁
⣠⣤⣤⠤⠤⠖⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⣵⢾⣿⣿⣿⣿⣿⠽⠓⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠀⠀⠀⠀⠈⠛⠭⡿⢟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡸
⣤⣤⠀⠀⠉⠑⠒⠄⠤⣀⡀⠀⠀⠀⠀⠈⣉⡹⠹⠿⠃⠀⠀⢀⠀⠀⠠⠉⠈⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢤⠞⠁
⣿⣿⠀⢀⣀⣀⣀⣀⣀⣀⡈⠉⠙⢢⡀⡚⢀⣀⢆⠀⠀⠀⠘⡟⣭⠿⠀⠀⠀⠀⠀⡈⢇⡀⠀⡀⠀⠀⠀⡀⠀⠁⡜
⣿⣿⠀⠀⠈⠀⠀⠁⠀⣠⠈⠁⠀⣈⡡⠄⢀⠈⠓⠛⠱⠂⠤⣙⠛⢀⢀⣤⡖⠀⢀⣼⣶⢶⠀⠀⠀⠀⣠⠟⠚⠉
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠁⡠⠗⠊⠁⠀⠀⠈⠙⠈⠉⠐⠘⣽⣓⡋⠈⠉⠀⠉⠳⢶⠚⠉⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⢀⡰⠀⠀⠀⠀⠀⠱⢄
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⠈⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⠷⠀⠀⠀⠀⠀⠈⡀
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⢁⠔⠃⠀⠀⠀⠀⠀⣠⠴⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⠀⠨⠁⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⢀⡠⠊
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠉⠁⠀⠀⠀⠀⠀⠀⠐⠀⡄⢀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⡀⠠⠊
⠛⠛⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡠⠒⠁

⠀⠤⠤⣄⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠉⢠⢀⡀⠄⣟⠊⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⢠⡕⢤⡀⠀⣦⣀⡈⠂⢸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢸⣿⣥⠈⣠⣜⣿⣌⡀⣷⠀⠀⠈⠀⠀⠐⠐⠀⠀⠀⡄⠀⠀⢺⣿⢙⣀⣤⣴⡶⣫⣆
⢠⣿⣿⣷⣬⣿⣿⣿⣿⣏⠀⠀⠀⣤⣬⠿⠿⠐⣠⣎⠁⠠⣶⣾⣿⣿⣿⠿⠋⠘⣍⣿⣆
⢨⣿⣻⣿⣻⣻⣿⣿⣿⣿⣭⣶⣷⣬⣭⣙⣷⣿⣿⣿⣿⡻⣿⣿⣿⠟⠁⠀⠀⠀⠛⢹⣿⣧⠀⠀⠠⡄
⠀⡟⢻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣵⣶⣾⣿⣿⣿⠀⠠⠀⡀⠀⠀⠈⠻⢿⣧⡼⡄⡧⢠⠀⠠⡄
⠀⡟⡋⣹⣿⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣟⣻⣿⣭⣭⣷⣾⡀⠠⠤⡇⣂⡄⢐⣒⠲⠞⣽⠈⢷⢘⣶⠾⣯⣤
⠀⣿⣶⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢶⣿⣿⣿⣛⣷⣯⣭⣥⣴⡮⠭⣭⣃⣀⡓⠂⠬⠅⣯⣽⣿⠿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⢻⣿⣿⣿⣿⣿⣿⡿⣿⣜⠘⣿⣿⣿⣟⢻⢯⢷⠖⢲⣿⣿⣿
⠀⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⢸⣿⣿⣹⣿⣿⣿⣿⠿⣿⠀⣶⣶⣶⢁⢸⢶⣶⠀⣰⣶⡟⢻
⠀⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣟⣿⣿⡇⣿⣿⣿⣿⢸⣿⣿⣹⣿⣿⢟⡫⡕⣻⢀⣿⣿⣿⣆⢰⣾⣿⡆⢹⣿⣧⢼
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⣿⣤⣭⣁⢎⣉⣀⣉⣭⣡⣼⣂⣥⢹
⠀⣿⣿⣿⠿⠿⠉⠉⢈⣉⣻⢿⣿⣿⣭⠉⠉⠉⠙⢛⠛⢶⡦⡮⣭⣽⡛⣟⣺⣻⡲⠯⡭⠩⠍⡉⠁⠋⠟⠙⠷⠾⠺
⠀⣷⣾⡿⠿⢿⢿⣖⣾⣿⣿⣿⣟⣿⣽⡛⠒⠀⠈⢒⢐⡂⢐⠒⣒⡂⠆⠤⠍⠭⠩⠭⠭⠍⢉⡭⣯⣹⣯⣉⠽⠃⡸
⠀⠿⣿⣦⡶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⢘⠁⣤⣝⠥⢷⡤⠀⠀⠈⠀⠠⠅⠤⠖⠂⠢⠦⠭⠌⠃⠨⠤⠄