Attendant Emojis & Symbols

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣯⣉⠁⠈⣉⣹⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠈⠙⢻⡟⠋⠁⠀⠈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠻⠶⣦⣤⣀⣀⣤⣴⠶⠟⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣨⡟⢻⣅⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⣀⣀⣤⡾⠋⠀⠀⠙⠷⣤⣀⣀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⢦⣤⣤⣴⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣿⣀⡀⠀⠀⢀⣀⣿⣦⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠋⠉⠉⠀⠀⠀⢹⡟⠛⠛⢻⡏⠀⠀⠀⠉⠉⠙⠛⠻⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣤⣤⣤⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⡏⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣇⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⢸⡇⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣿⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⢸⡇⠀⠉⠉⠉⠉⠉⠉⠉⠀⢸⡇⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⠛⠒⠲⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢳⡧⠴⠒⠒⠒⠒⠛⠛⠲⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⣼⠀⠀⠀⠀⠀⣀⣀⡤⠤⠬⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡓⠦⣄⣏⣠⠴⠖⢋⣉⣡⠤⠴⡶⠒⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠲⣄⣉⡤⠖⠺⢭⣀⠀⠀⢠⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠈⠁⠀⠀⠀⠀⠈⠙⡆⣼⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⠇⡸⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢻⡁⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⡄⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⢧⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠙⢆⡀⠀⣀⡠⢶⠋⠀⠀⠀⢸⡆⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⢀⣽⠋⠁⠀⢸⣦⡀⠀⠀⢸⠃⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣃⣤⠖⠚⠳⡌⠳⣄⡴⢋⡜⠉⢙⡖⠳⣄⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢱⠇⠀⠀⠀⠈⠲⠿⠖⠋⠀⠀⢸⠀⠀⠈⢧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡟⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠘⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢰⠇⠀⡇⠹⣄⣴⡆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢸⠀⠀⢳⡀⠈⢹⠀⣯⣭⣯⣍⣿⡏⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢸⠀⠀⠀⠳⣄⠻⢄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣼⡀⠀⠀⠀⠈⢦⡀⠙⢦⣀⠀⠀⣿⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣰⠋⡇⠀⠀⠀⠀⠀⠳⣄⣴⣟⣲⣄⣿⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⢠⡟⠒⠓⠒⠒⠒⠒⠒⠒⠋⠀⠉⣁⣠⠏⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⣸⠀⠀⠀⣀⣠⠤⠴⠒⠛⠋⠉⢫⡁⡞⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠗⠚⠋⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⣀⠴⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠠⠀⠐⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢠⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠀⠀⢡⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠂⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣃⠀⢄⡠⠀⣐⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⡷⢦⠤⠚⡂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠁⣄⠀⣄⡀⡀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢉⠀⠀⣆⠀⢒⠀⠄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠁⠘⠂⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠢⠞⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⡀⡀⠀⠀⠠⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣨⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠛⠛⠛⠛⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣄⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠶⠛⠉⠁⠀⠈⠙⠲⣦⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡶⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢴⡇⠠⠶⠒⠲⠶⠀⠀⠀⠀⠀⠀⠶⠖⠒⠶⠄⢸⡦⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⠀⢸⡇⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⢸⡇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢻⡶⣾⡇⠀⠰⠦⠁⠀⠀⢠⡆⢰⡄⠀⠀⠈⠴⠆⠀⢸⣷⢶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠈⠛⢻⡇⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⢸⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⢹⣏⠉⠉⣹⡏⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢈⣷⣄⡀⠀⠀⠀⠉⣛⣛⠉⠀⠀⠀⢀⣠⠞⠁⢀⣠⡴⢺⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠚⠋⠀⠈⠙⠶⣄⣀⠀⠉⠉⠀⣀⣠⠶⠋⠁⣠⠶⠋⠁⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠛⠛⠋⢹⡇⠀⣠⠞⠁⠀⢀⣾⣃⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣧⣤⣤⣤⣤⣼⣧⣞⠁⠀⣠⡾⠋⠉⠉⠉⠉⣉⣭⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠒⠚⠛⠛⠛⠻⣶⣏⠀⠀⠀⠀⠀⠀⠀⠀⣻⣴⣟⣁⣠⣤⡤⠶⣾⣏⣉⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠁⠀⡿⠀⠀⠀⠀⠀⠀⢻⡻⢦⣤⣤⣤⣤⣤⣤⡴⢟⡟⠉⠀⠀⠀⠀⠀⢿⠀⠈⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⣸⠇⠀⠰⣆⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢸⡇⢀⣀⣀⣀⣀⣀⠀⠀⢹⡄⠀⠀⠀⠀⢠⡏⠀⠀⠀⣿⣿⠲⠀⠀⢸⡇⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢸⡇⠸⠧⠤⠤⠤⠽⠀⠀⠀⢷⠀⠀⠀⠀⡾⠁⠀⠀⠈⠃⢸⡧⡄⠀⠘⡇⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠘⢧⣤⣤⣤⣄⣼⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣸⣧⣤⣄⣼⣧⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣧⣠⣤⣄⣤⡼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠛⠛⠋⢉⣭⢿⣿⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⣋⣁⠀⠀⠀⠀⠀⠀⢈⡛⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡾⠋⣁⡴⠞⣩⡴⠖⠀⠀⠀⠀⠀⠀⢨⣉⣻⣿⣿⣶⠄
⠀⠀⠀⠀⣠⣾⣿⣿⠟⠋⣠⠾⢋⣴⠞⢉⣠⡴⠚⠀⠀⠀⠀⠀⠀⢌⣙⢻⣿⣿⠦⣄
⠀⠀⠀⢸⠿⠟⠉⠀⠀⠈⠁⡴⠏⣡⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⢮⣝⣿⣿⣟⢾⡀⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠊⣠⠞⠋⣠⡴⠂⣀⣠⡤⠄⠀⠀⠀⢀⣀⣱⣮⣽⣿⣿⠳⣌⠙⣆
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠜⢁⡴⠛⣁⡴⠞⠋⣁⣤⡴⠶⠛⠻⣿⣿⣍⠉⠉⠛⠛⠿⠾⣧⣹⣇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠠⠞⢁⣠⡶⠟⠋⠁⠀⠀⠀⠀⢹⣿⡝⣦⠀⠀⠀⠀⣄⢳⣽⣿⡄
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢷⡘⣆⠀⣀⢤⡘⣆⣿⣿⡇
⠀⠀⠀⢸⡀⠀⠀⠀⢀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡙⣆⢻⡌⣇⣿⣿⣿⡇
⠀⠀⠀⠈⢣⣤⡴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠙⠻⢾⣷⣿⣿⣿⠿⣿⠃
⠀⠀⠀⠀⠀⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠚⢝⢿⢠⡿
⠀⠀⠀⠀⢀⡾⠷⠛⠙⠛⠃⠀⢀⠀⠀⠀⠀⠀⣴⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠇⢨⣧⣼⡇
⠀⠀⠀⠀⢸⡇⣼⠀⠀⢀⣀⣀⠀⣧⠀⠀⠀⠀⠻⡠⠼⠛⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⣼⠇⢸⡦⠤⣀
⠀⠀⠀⠀⠸⡇⣿⣆⡾⠛⠻⣿⠆⢻⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣤⣿⠀⠀⠘⡏⠀⢀⠙⢦⣾⣷
⠀⠀⠀⠀⠀⢱⣿⣿⣦⠃⠙⠃⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠄⢲⣿⡿⣿⡟
⠀⠀⠀⠀⠀⠀⠙⠿⣿⠀⠀⠀⠀⠣⣄⠠⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠛⠀⠀⠀⣿⣿⣾⣿⢛⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⣳⡄⠀⠀⠀⠀⠀⣤⢹⣿⡿⠿⠛⠁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢴⣾⣿⠿⠟⢋⣩⣥⣾⡟⠛⠁⠀⠀⠀⢀⡼⠃⢸⣧⣤⣴⠶⠛⠉⠉⠉⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⢿⣶⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣰⠟⠃⠀⣸⡟⢃⠤⠤⣄⡀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⠛⠿⠿⠿⠛⠁⠀⠀⠀⢀⣴⠞⠁⠀⢰⣴⣿⠃⢸⡇⠀⠈⢻⣦⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠀⠀⣀⣤⠞⠋⠀⢀⣠⡴⠟⠁⡟⠀⢸⡇⠀⠀⠀⢻⣇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⢴⠛⠉⣀⣠⡴⠞⠋⠁⢀⡄⠀⣇⠀⠀⢿⠀⠀⠀⢸⣿⣦⣀⣀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⠋⠉⠀⠀⣀⣤⠞⠉⠀⢠⣿⠀⠀⠈⢧⡀⠀⠀⣿⠁⣄⠉⠛⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢣⣀⣠⣤⣶⡿⠛⠁⠀⠀⠀⣾⡿⢧⡀⠀⠀⠙⠦⠀⣿⢀⡸⠆⠀⠀⠀⠈⠙⠳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⣸⣿⠿⠛⠁⠀⠀⠀⣠⡴⠛⠁⠀⢈⣷⢄⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣄⠀⢋⣀⣀⣀⣠⣤⠖⠋⠁⠉⠉⠀⠀⢸⠇⠀⠑⢄⡀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⡟⠉⠉⠜⠋⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⣠⠎⠑⢦⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⡾⠋⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡄⠀⠀⠀⠀⠀⠀⢀⡾⠁⣰⠏⠀⠀⣾⡆⠀⠀⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡀⠀⠀⠀⠀⣠⠞⢀⡾⠁⠀⠀⠀⣿⣷⣠⣼⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⣴⠏⣰⠏⠀⠀⠀⠀⢀⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠘⣡⠞⠁⠀⠀⠀⢀⣴⠟⠉⠉⠙⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸


⣶⣶⣶⣦⣤⣤⣴⣦⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣶⣤⣶⣤⣦⣴⣶⣦⣴⣦⣤⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶
⣿⣛⣛⣛⣛⣛⣛⣣⣟⣛⣟⣛⣙⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣭⣭⣭⣭⣭⣿⣯⣭⣭⣭⣭⣭⣽⣯⣭⣭⣭⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⢷⣦⣴⡾⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⢀⣈⣁⡀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⢰⣿⠋⠙⣿⡆⠀⠈⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣤⣀⠀⠘⢿⣦⣴⡿⠁⠀⢀⣤⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⣿⡟⠻⠿⣶⣶⣤⣤⣶⣶⠿⠟⢻⣿⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢸⣿⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⣼⡟⠿⣶⣤⣤⣶⠿⢻⣧⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡟⠁⠀⣠⣾⠿⣷⡀⠀⠉⠉⠀⢀⣾⠿⣷⣄⠀⠈⢻⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣿⣷⡿⣿⡏⠀⠙⣿⣆⠀⠀⣰⡿⠋⠀⢹⣿⢿⣾⣿⣋⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⠉⠀⠀⣿⣧⣀⠀⠈⢻⣧⣼⡟⠁⠀⣀⣼⣿⠀⠀⠉⠛⠻⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠉⢛⣿⡶⠀⠙⠋⠀⢶⣿⡛⠉⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⡀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠙⣿⡆⠀⠀⢀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠻⣷⡀⠀⠀⢀⣾⠟⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠙⣿⡄⢠⣿⠋⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣼⣿⣿⣧⣤⣤⣤⣤⣤⣤⣿⣧⣤⣤⣤⣾⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⢠
⠀⠀⡈⠉⠒⠂⠄⠀⠰⢾⡆⠀⠀⠀⠀⠀⠀⠀⠚⠋⠃⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠠⠀⠀⠀⠀⢀⠮⢶⡶⠆
⠀⠁⠀⠉⠀⠀⠀⢀⣴⣿⣦⡀⠀⢠⠄⠀⠀⠀⠐⢳⣦⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠈⠒⠢⢀⡀⠀⠀⡀
⠀⠀⠀⣀⣴⣦⡴⠟⣿⣿⣿⡇⠀⢰⣳⠦⠆⠀⠀⠈⣿⠀⠀⠀⠀⢀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⣦
⠀⠀⠀⠁⠄⣀⢻⢀⣿⣿⣯⠁⠀⠘⣿⣷⠀⠀⠙⠺⠽⣤⠄⠀⠀⠞⣿⡟⠂⠀⠀⠀⠐⡀⠁⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠈⠈⠙⠛⠛⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⢠⠰⠀⠀⠸⠋
⠀⠀⠀⠠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⢀⣠⣤⢄⠀⠀⠀⣴⣶⣦
⠀⠀⠀⠈⠀⢈⡀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠀⠀⠀⠀⠀⠹⠁⠀⠀⠆⠀⠀⠦⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠍⢉⠟
⠀⠀⢀⡴⠾⢛⣿⣦⡀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⣠⣄⠀⠀⠀⢀⡀⠀⠀⠀⢀⣴⢷⣶⣿⡀
⠀⢠⣾⣿⣶⣿⣿⠛⠳⣾⠆⠀⠀⠀⠀⣼⡗⠀⠀⠀⠀⠀⠀⠴⣿⣷⠀⠀⠉⣉⡀⠀⠀⠀⠀⠀⣰⡶⢿⣿⣤⣶⣿⣧
⢠⣿⠁⢸⣿⣿⣿⡀⠀⠀⠀⢀⣀⣠⠀⢹⣷⠀⠀⠀⠀⠂⢄⡀⣿⣿⠀⠀⢸⣿⣽⡶⠆⠄⠀⠀⠀⠀⢠⣝⣿⣿⠟⠁
⠀⠁⠀⣈⣿⣿⣿⣷⠀⠀⠀⢸⣿⣿⠀⠸⡿⠄⠐⣺⣤⠀⠀⠗⠟⠇⠀⠀⠀⣿⣿⠗⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⡀
⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⢈⡿⣿⠀⠀⠀⠀⡀⠀⢻⡯⠆⠀⣀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁

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

⠀⠀⠠⡄⢤⠀⣦⢠⡄⢰⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡤⢀⡄⣠⠄⡤⢠⠆
⠀⠀⠀⢃⠀⡆⢸⢀⠀⢠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠇⠀⡇⠀⠀⡇⢰
⠀⠀⠀⠸⢠⢀⠆⠜⢠⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⢄⠱⡘⡄⠃⢸
⠀⠀⠀⡃⠁⠊⠊⡔⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠔⠄⠈⡀
⢀⣀⠞⠠⠂⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡈⠢⠑⣄⡀
⠀⢹⢻⡟⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢿⢛⡟⡿
⠀⢘⠇⡝⡌⡎⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢫⢃⢋⢢⡇
⠀⢨⠸⢰⠰⠘⡘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡎⡜⠘⠄⡇
⠀⢸⡔⡎⠀⡇⢧⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⢠⠁⢀⠂⡁
⠀⠈⠀⠸⠁⢢⡨⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠍⡰⠢⠥⠀⠁
⣉⢉⣉⠙⠦⢆⣀⠀⣀⠶⠞⢉⣉⠉⣳⠴⠀⠀⣀⠐⡶⠊⠉⠉⣉⣢⡆⠀⠀⠀⣳⣴⣉⠉⠉⠙⢦⠂⣀⠀⠈⠦⣞⡉⣉⠉⠳⠴⣀⡀⢀⡑⠶⠊⣉⡉⢉
⠘⠛⠰⣄⡠⠀⠘⠛⠀⠒⣄⡠⠘⠛⠐⠀⠒⣄⡨⠛⠂⠂⠀⠒⢄⡿⠀⠀⠐⠀⠒⢿⡠⠀⠀⠀⠀⠛⢇⣠⠀⠐⠀⠛⠓⢤⣠⠀⠀⠛⠃⠐⢤⣠⠀⠛⠃
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠔⠒⠒⢂⠀⡤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠈⠑⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⢀⠤⠒⠊⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠒⠒⠒⢲⡔⠊⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣀⣀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⢀⢀⡤⠖⠂⠀⢈⣉⠩⠅⠀⠀⠰⡄⠀⠀⢀⠤⠒⠀⠀⠀⢖⠉⠓⢤⡎⠀⠀⠀⢀⡠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢜⡄
⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠹⡀⡴⠁⣀⡠⠀⠀⠀⠀⠉⠛⣿⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⢀⠆⠀⠀⠀⢠⡎⠀⠀⢀⣀⡀⠀⠀⠀⠀⣿⣠⡾⠟⠁⠀⠀⠀⠀⡀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠈⠑⢦⡀⠀⣿⡟⡠⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠖⠒⠂⠤⢄⣀⣀⣀⡀⠀⠠⠤⣀⡀⠀⣠⠔⠂⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠈⠳⡶⠲⢾⣶⠦⢄⡀⠀⠉⢉⣩⣥⣤⣿⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠤⣈⠱⠦⣤⣀⣈⣩⡤⠤⣄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠙⠷⣄⣈⠀⢉⠼⣇⠀⠀⢹⠁⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢸⠀⠀⢀⣹⣯⠄⠀⠀⣸⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⠉⣹⠇⡀⠈⠁⠀⠀⣷⣶⣴⣿⣷⣦⣄⣀⡈⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠛⠋⠉⠀⠀⠀⣰⡇⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠉⠙⠓⠀⡷⢄⠀⠀⠀⠀⢠⠃⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠓⠲⢤⣄⡀⠀⠀⠀⠀⢸⠀⢀⣀⣠⡤⠶⣻⣿⠀⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⡔⢺⠀⠀⠉⠉⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⣀⡀⠀⠀⠈⢧⠀⠀⠀⠀⢸⠋⠉⡡⠤⡄⠀⢻⣇⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢃⠐⡄⠐⠒⠶⠀⠀⢸⠓⢦⠀⠀⠀⠀⠀⠤⠶⢄⠀⠀⠀⠀⠀⠼⡆⠀⠀⠀⣸⠀⢰⠁⠀⠆⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸⠀⢫⠁⠂⠀⠀⠀⠈⣇⠘⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⡟⠀⡆⠀⠸⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡆⠘⣏⠑⠒⠒⠀⠀⡿⠀⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⢳⠀⠀⢠⠃⢰⠁⠀⡇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠐⠄⢘⡦⡀⠀⣀⢴⠁⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⡦⠔⠁⠀⣈⠀⠜⠁⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠁⠀⠀⠀⠸⣀⡀⣠⠀⠀⠀⢸⠀⠀⢀⠀⠀⠀⠀⣆⢀⡠⠅⠀⠀⠀⠁⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢹⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢺⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⡄⠀⠇⠀⠀⠀⡖⠒⠢⠀⠀⠀⢠⠤⠤⢤⠀⠀⠀⠀⠖⠂⢲⠀⠀⢠⠁⠀⡆⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠇⠀⢸⠀⠀⠀⠃⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠈⠀⢰⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠸⠀⠀⡆⠀⠀⢠⠀⠀⡅⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠀⠀⡆⠀⢀⠃⠀⡌⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠑⠐⠁⠀⠀⠸⡀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⢆⠀⢠⠃⠀⠀⠑⠊⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠉⠊⠁⠀⠀⠈⠦⠤⠴⠀⠀⠀⠈⠉⠁⠀⢀⣀⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⠤⢤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⠔⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠀⢀⡀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⣤⢤⡤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⢀⣀⠀⠀⡀⡀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠁⢤⣤⣝⣚⡐⢂⡐⢀⡒⢈⠀⠤⠀⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠣⣬⡑⠺⡦⢼⡉⢤⠈⢡⡌⠁⠦⠀⠒⢪⣉⠞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠲⣤⣽⣉⣿⣟⠓⡀⠶⠂⡀⠿⢀⡘⠃⣤⣥⠾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⡛⢷⣬⣯⣭⣿⡋⠛⣀⡘⠛⣠⡈⠉⣤⢤⡞⢒⣉⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⣽⠛⢳⡶⠿⣭⣥⣤⠉⢡⣤⠉⠡⡶⠀⠸⠟⣏⡥⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⡐⢮⣿⡟⣳⣾⠷⠆⠀⠶⠆⠀⠶⠆⣀⠚⢛⣯⣵⢶⠟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣅⡚⢬⣾⣏⣻⣿⡛⠓⣀⡘⠛⣀⡈⠛⣠⣬⡵⢷⣹⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠆⣹⠛⣷⡾⢿⣭⣥⣤⠉⢡⣤⠉⠁⣶⠀⠰⢟⣎⡅⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡐⢯⣽⣟⣳⣶⠿⠆⠀⢶⠆⠀⠶⠆⣀⠚⢓⣯⡡⠖⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢁⡺⢬⣿⣏⣹⣿⡛⠛⣀⡘⠛⣀⡈⠛⣠⣬⡵⢷⣂⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣤⣀⣶⢄⣹⠻⣶⣾⣿⣬⣤⣄⠉⢡⣤⠉⠡⣤⠀⠰⢛⣏⣥⣴⣀⣤⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣾⡏⡇⣾⣿⣽⣿⣷⣾⣿⠂⠀⠲⠂⠀⠲⠂⠀⠚⠋⣽⣿⣷⡾⢋⣷⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⣷⣿⣿⡿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣾⡏⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⣿⠀⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣟⣺⣿⣿⣿⣿⣿⣿⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢻⣿⣿⣿⣿⣿⡏⢀⡟⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡅⣇⢻⡀⠻⣿⣿⣿⣿⣿⣿⣿⠿⢿⢾⣿⣿⣿⣿⡟⢁⡞⣸⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡘⣆⠙⣄⠙⢿⣿⣿⣿⣿⣿⣶⣦⣿⣿⣿⡿⠏⣠⠎⣰⠇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡘⢦⡀⠙⢦⣝⠻⠿⢿⣽⣯⣼⣿⠿⣿⡵⠊⢁⡴⠋⢦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣙⠦⣀⡀⠉⠑⠒⡖⢲⠒⠂⠉⢁⣠⠴⣋⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣳⣮⣍⣛⠛⠛⠃⠘⠛⠛⣛⣩⡴⣞⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⢻⡿⠟⠙⢿⡟⠻⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣿⣿⠀⣿⠒⠒⠠⠤⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠋⠛⠻⠿⣿⣿⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⠏⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣠⣀⢀⡀⡀⠀⠀⠀⠀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠾⢿⣿⣿⠿⢳⣞⣡⣤⣶⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢽⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠛⠩⠿⠿⠿⠿⠟⣛⣛⠛⠿⠿⠿⠿⠍⠙⠛⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣻⡿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣵⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠙⠦⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠖⠦⣄⡤⠴⠖⠁⠀⠀⠀⠂⡨⠏⠙⠟⠢⢤⣀
⠀⠀⠀⠀⠀⢠⡤⣤⡀⠀⠀⢀⣠⠴⠒⠛⠁⠐⠀⠸⠁⠀⠂⢁⡀⠀⠀⠰⡉⠀⡠⠴⠲⠒⠊⠨⢦⡀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⢿⢱⣎⢳⡄⡴⠋⠀⠀⠈⠉⢀⠟⠉⠀⠠⠀⠙⠉⠋⠀⠀⠀⠈⠁⠀⠘⡀⠀⠀⠀⠀⢳⣠⡟⣇⢹⠂
⠀⠀⠀⠀⠀⢸⡞⣯⣧⣿⢰⣦⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠇⠐⡀⢰⣦⡄⠉⢿⣽⢸⡇
⠀⠀⠀⠀⠀⢸⡟⢮⣿⣇⠀⠉⠀⠀⠀⠀⠠⠄⢀⠂⠀⢀⠄⠀⠀⠀⠤⠀⠀⠀⢀⠀⡄⢀⠀⠈⠀⠀⠂⣸⠼⣾⡇
⠀⠀⠀⠀⠀⠀⢷⣤⣽⣍⡙⣮⠥⠄⣀⣰⣀⣀⣁⠄⠖⠺⠐⠄⠀⠀⠀⢤⡂⠀⢸⠀⣠⣘⣋⣠⣦⠶⠾⣡⣬⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣟⠛⣿⡟⠛⢻⣶⣾⣮⣀⣀⡩⠀⠀⠀⠐⠀⠀⠀⣸⡛⣳⣂⣉⣙⣫⣭⣭⠷⠿⠖⣛⣉⡼
⠀⠀⠀⠀⠀⠀⠀⢻⡍⢫⡙⠻⠿⠯⢏⡉⠛⠛⡇⠀⠀⠀⢀⠀⢴⣿⣿⠉⠭⠭⠭⠽⠿⠷⣴⣶⡶⠟⡿⠋⡗
⠀⠀⠀⠀⠀⠀⠀⢸⡗⢾⡛⠷⢶⣆⡚⠛⠛⠛⡇⠀⠀⠀⠀⠀⠀⠉⡗⣒⣲⣂⣴⣶⣶⣬⡥⣤⣶⢿⣶⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⡿⣶⢶⡖⠿⢤⣤⣶⡇⠀⠀⠀⠀⠃⠀⠀⣯⠤⢭⣭⣤⣤⣴⣚⣉⣠⣥⢾⡗⣹
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡳⣽⣦⣌⣹⡷⣦⣤⣽⠆⠀⠀⠀⠀⠀⠀⢠⣗⣒⣚⣛⣙⣩⣤⣟⠉⣭⣐⣿⢆⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⢯⣦⣉⣙⡲⢀⣉⡟⢀⣴⣦⡄⠀⠀⠀⢸⡷⠈⠛⠛⣏⢋⣩⣷⣛⣉⣽⠿⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠢⢧⣙⡛⠯⡬⢛⡃⠘⠻⡿⠀⠀⠂⠀⢸⡕⡬⠵⠾⠟⠋⠡⠕⣛⣫⢾⢛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡶⢦⣍⡙⢻⡍⢛⡇⠀⠀⠀⠀⠀⡀⠀⣏⣩⠷⠿⠾⠝⢛⡫⠗⢋⣴⠷⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⢮⣙⡻⠿⣻⠇⠀⠈⠀⠀⢀⣿⣦⢋⡴⠖⠲⠾⢛⣫⡷⢛⣩⣶⢦⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡷⣮⣛⠭⣝⠢⠦⠤⢥⣤⣖⡦⠍⣛⣛⠾⠴⠛⣛⣫⣛⣩⣵⡿⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⡦⣍⣛⣍⡛⠻⣶⣳⢄⣛⣛⣛⣻⡯⡽⠛⢛⣻⣽⣯⡥⢾⣯⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢸⡙⠨⣛⣦⣉⣛⡿⢿⣍⣩⣉⣩⣍⣭⡔⣛⣉⣭⣾⡯⠖⣉⠛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡙⠳⢿⠿⢮⣙⣛⣛⣂⣤⣭⣥⣤⣍⣉⣹⡭⠶⣗⡴⠊⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣁⠢⠬⢭⢉⢵⣦⡤⣥⡴⡽⠍⠤⠒⠉⣁⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠦⣤⣤⣤⣤⣤⣤⠤⠶⠒⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠘⣆⡀⠠⢀⣒⣒⣻⣦⡀⠀⠢⢄⡀⠠⣬⣀⣁⣀⣀⡠⠤⠔⡢⠐⢀⣼⡄⠀⠀⣀⠠⡶⠒⠣
⠀⠀⠀⠘⣯⡪⣒⢂⣀⣒⣩⢿⣷⣕⡠⢀⠨⠉⣒⣒⣀⣀⣀⠉⠀⡀⣰⣿⠛⠁⠐⠀⠒⠉⢀⠔⠁
⠀⠀⠀⠀⠈⢻⡿⠛⢻⠿⠲⠋⠉⢹⡏⠑⠒⠲⠤⠤⠤⠤⠤⠐⠓⠈⠿⠛⠷⠶⢤⢴⡶⢿⡟
⠀⠀⠀⠀⠀⠘⣏⠀⠈⢀⠄⠐⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⠈⠁⢸⡇
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠆⠐⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⣿⡀⠀⢀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⢣⣤⢨⡇
⠀⠀⠀⠀⠀⢠⣿⠿⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡇
⠀⠀⠀⠀⠀⢈⡿⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠋⡇
⠀⠀⠀⠀⢰⢘⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⡇
⠀⠀⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⡃
⠀⠀⠀⠀⢸⢇⠀⡆⣆⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠺⣸⠀⣷
⠀⠀⠀⠀⠉⠰⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⠿⠿⠿⠿⢿⡿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⢟⣋⠉⠀⠀⠀⠀⠀⠀⡀⠀⠠⣍⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⣾⣿⣿⣿⣏⠀⣈⣭⣤⡟⢧⣤⣤⣀⣀⡀⡀⠈⠁⢽⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⣿⠿⠋⠈⠀⠀⠀⠀⠀⢠⡌⠹⣟⣿⣷⣤⣤⣾⣄⡃⢉⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⡄⠸⠟⢿⣿⣿⣿⣷⣿⣼⣿⡿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠁⠀⣈⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣼⣋⣴⣿⣿⣿⣷⣟⣃⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⣰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⢓⣱⢿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠃⠀⠀⣀⠀⠀⠋⠀⠀⠘⠂⠀⠀⠀⠀⠈⠀⠀⠤⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⡀⢺⣿⠄⠀⣼⠏⠈⠀⠀⠐⠇⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⣠⣴⣿⡟⠉⢻⣿⣿⣿⣿⢿⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢠⣾⣿⣿⣿⣿⣶⣼⣿⣿⣿⣟⣿⣿⣧⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠰⣤⠾⣿⡿⠀⠁⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣾⣿⣿⣿⣃⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢶⣬⡆⣿⣀⣤⣶⣾⣿⣷⣦⠀⢤⣶⣶⣾⣾⣿⣧⣤⣄⣠⡄⡻⣿⣿⣿⣿⣿⣣⣿⢳⣿⠟⣠⣾⣿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣿⣿⣿⣿⣿⣿⣿⠟⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣸⣿⣿⣿⣿⣿⣧⣿⢇⣼⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣾⣿⣿⣿⣿⣯⣶⣄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣏⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢾⣿⡏⢿⣿⡿⠟⣿⣿⣿⣿⡟⢧⣿⣿⣿⣿⣿⣿⣿⣿⠏⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⣤⣤⠀⣤⣿⣿⣿⣿⣷⣴⢃⠀⣉⣉⣙⣛⡉⢁⢂⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣽⣿⣿⣿⣽⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⣿⠇⠘⠛⠃⢿⠿⠃⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡤⠀⠀⠒⠀⠐⠾⣷⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⣠⠀⡀⣿⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣁⣈⣀⣹⢙⡟⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⡋⠉⠿⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⠛⢹⠁⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠸⠆⠀⢠⣤⣤⣤⡄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⠿⠛⢏⠀⠠⠀⠀⠐⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠛⠀⠀⠢⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⣪⢍⠕⢼⡁⠀⠀⠐⢮⡓⢄⠀⠀⣀⠤⠔⢒⣦⢤⣤⣤⡀
⠀⠀⠀⠀⠀⣀⣀⠀⣠⠎⢸⡁⡀⠼⠤⠤⡻⢶⡛⣷⣒⠬⣻⢔⠋⠀⠀⣴⣿⣿⣿⠿⠛⣇⡀
⠀⠀⠀⡎⠉⠀⠀⣈⣭⣟⡛⠛⠛⠓⠒⠒⠯⣉⠁⠀⣀⣙⣎⠁⢳⣠⣾⣿⣿⡯⣒⠒⠒⠁⠈⠉⠩⡇
⠀⠀⠀⡟⠀⣠⢮⣽⣒⣂⡀⠭⠭⢝⣲⠀⠀⣀⣻⣖⡤⡀⠉⢈⡿⡿⢛⡻⣿⣵⣛⢮⠷⣮⣤⠛⡷⣮⣦⡤⣀
⠀⢠⣾⣇⣰⢈⠟⢹⣟⣓⠻⠦⣴⡿⣣⣼⣿⣿⣿⣧⣦⣬⣽⣧⣄⢃⠀⠤⢈⡽⠛⠋⠁⢀⠶⠧⣟⣱⠿⠃⡸
⢰⠋⡇⣨⢣⠊⠀⢸⠀⠀⠉⠑⠒⠬⠭⣙⣛⠻⠿⠿⢿⣿⣿⣿⣿⣬⣶⣿⣀⠠⠷⣦⠔⠁⠀⢀⡼⠖⠒⡏
⠀⠑⠷⠤⡅⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⠉⢆⢀⠴⠋⠀⠀⢀⡇
⠀⢀⠀⠀⠈⢑⠖⣃⠀⠀⠀⠀⠀⠀⠀⣠⡀⣤⡀⡤⣀⣤⣀⢤⠀⠀⢠⢤⢠⣤⢠⢲⢠⣷⢰⣷⠀⠀⢸
⠀⢸⠀⠀⡠⠋⠀⢹⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⡇⡟⣤⣿⢸⡄⠀⢸⣾⢸⢻⢸⢸⢸⢹⢸⢻⠀⠀⣸
⠀⢸⣠⠊⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⢻⡃⢻⡃⣳⡃⣳⠋⣲⠁⠀⢘⣟⢘⣎⢨⣏⢨⢏⡰⢧⠀⠀⡇
⠀⣿⠃⠀⠀⢿⡃⠈⡆⠀⠀⠀⠀⠀⠀⡿⡇⡇⡇⡇⡇⡇⡇⣧⡇⠀⢸⣸⢸⢸⣸⢿⣷⡞⣿⡇⠀⠀⡇
⠀⠸⡄⠀⠀⣸⣧⡤⣻⡄⠀⠀⠀⠀⠀⢧⠇⢧⠇⠷⠇⠧⠇⠧⠇⠀⠸⠼⠸⠜⠙⠊⠙⠃⠙⠁⠀⢸
⠀⠀⠈⢶⡫⠗⠚⠛⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠤⠤⠤⠤⠤⠤⠤⠤⠔⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠒⠒⠂⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⡀⠀⠀⡠⠀⠀⠀⠀⠀⠈⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⡜⢠⠂⠰⠁⠀⠀⠀⠀⠀⢀⠔⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠐⢉⡏⢉⠓⠂⠀⠀⠀⠀⢢⡋⢢⠄⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢈⡁⡌⠉⠒⠀⠀⠀⠀⠀⠸⡏⠡⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠣⡀⠈⠉⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⡠⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣈⠑⠂⠄⣀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠒⢉⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠁⠢⣤⠄⡀⢈⣉⣉⣉⣉⠁⢠⠀⠄⠂⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠬⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⣎⡈⠃⠀⠀⠀⢀⠀⢀⡤⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠢⣀⠉⠑⠒⠒⠒⠒⠈⢁⡠⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠈⠁⠀⠐⠀⠀⠉⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠒⠀⠠⠤⠤⠄⠐⠊⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⠸⣿⡀⢿⣿⡘⣿⣿⣄⠀⠀⣠⣿⣿⠇⣾⣿⡀⢿⣷⡘⣿⣿⣄⠀⠀⣠⣿⣿⢃⣿⡿⢀⣿⣿⣧⡀⠀⠀⣰⣿⡿⢠⣿⡏⢸⣿⢹⣿
⣿⣿⠀⢻⣷⡈⢿⣷⣌⠻⣿⣿⣿⣿⠿⣃⣾⡿⢻⣧⡘⢿⣷⣈⠻⣿⣿⣿⣿⠿⣣⣾⡿⢁⣾⣿⡙⠿⣿⣿⣿⡿⠟⣡⣿⡟⢠⣿⠇⢸⣿
⣿⣿⠀⢸⣿⣷⣄⠙⢿⣷⣶⣤⣤⣶⣾⡿⠛⣡⣾⠿⣷⣄⠛⢿⣷⣶⣤⣤⣶⣾⡿⠛⣠⣾⣟⠻⢿⣶⣦⣤⣴⣶⣿⠿⢋⣴⣿⣿⠀⢸⣿
⣿⡟⠀⣿⣿⠏⢻⣿⣶⣤⣍⣉⣉⣉⣥⣶⡿⠟⠁⠀⠈⠻⢿⣦⣤⣍⣉⣉⣩⣤⣶⡿⠟⠙⠻⣷⣦⣬⣉⣉⣉⣭⣤⣶⣿⠋⢿⣿⡇⠘⣿
⣿⡇⠀⣿⣿⠀⢸⣿⡟⢉⣿⣿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠙⢻⣿⣏⠙⣿⣿⠀⢸⣿⡇⠀⣿
⣿⡇⠀⣿⡏⠀⣿⣿⠁⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⢹⣿⡇⠈⣿⡇⠀⣿
⣿⡇⢰⣿⠃⣸⣿⢁⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⠻⣿⡀⢻⣷⠀⣿
⣿⡇⢸⡟⢠⡿⣡⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡹⣧⠘⣿⠀⣿
⣿⡃⣼⢡⣫⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣮⣣⢹⡄⣻
⣿⠿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡿⢿
⣿⠀⣷⠹⡻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡻⢱⡇⢸
⣿⠀⣿⡆⢻⣌⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢏⣼⠃⣾⡇⢸
⣿⠀⣿⣷⠈⣿⡆⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⣾⡏⢰⣿⡇⢸
⣿⠀⣿⣿⡄⢹⣿⡀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⣸⣿⠁⣼⣿⡇⢸
⣿⠀⣿⣿⡇⢸⣿⣇⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢀⣿⣿⠀⣿⣿⡇⢸
⣿⠀⣿⣿⣇⠀⣿⣿⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⢸⣿⡇⢀⣿⣿⡇⢸
⣿⠀⣿⣿⣿⠀⣿⣿⡆⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⣾⣿⡇⢸⣿⣿⡇⢸
⣿⠀⣿⣿⣿⠀⢻⣿⡇⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⣿⣿⠇⢸⣿⣿⡇⢸
⣿⣶⠿⠿⠿⢶⣿⣿⣷⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣿⣿⣷⣾⣿⣿⣷⣾
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⣄⣈⣑⣋⣀⣤⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠳⢾⡀⢈⠷⠋⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠙⠒⠲⠭⠥⢒⡚⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠲⢤⠤⠤⠒⠋⠈⠒⢾⠀⠀⠀⠀⢀⡤⠖⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠈⠀⠀⠀⠈⠀⢰⠃⠐⡗⠶⣊⡁⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠳⡀⠀⠰⠂⠀⣠⠟⡄⠀⠘⠊⢽⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢸⠒⠒⠒⠺⡁⠀⢸⡀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢺⠉⢳⡀⠀⠀⢠⠏⢹⠛⢇⠀⡔⠉⢁⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠈⠓⡤⠱⡖⣲⠃⡴⠚⠀⠈⣷⠉⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠇⠳⢄⣹⣃⡴⠃⢳⠀⢀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⣄⠀⠀⢹⠁⠀⠀⣸⠀⣸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⢿⠀⠀⢸⠀⠀⠈⣿⠀⡇⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡼⢸⠀⠀⢸⠀⠀⠀⡇⡏⡇⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⠋⠉⠉⠙⠛⠋⠛⠛⠋⠉⠉⠉⠛⠉⠛⠛⠛



⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣻⣿⣿⣻

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




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⡛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣠⡾⠳⡀⠀⠀⠀⢀⡠⣚⢦⡤⡀
⠀⠀⠀⠀⠀⣾⣧⠏⢑⢾⣄⣠⠔⠉⠀⣸⠄⢠⣿⢳⡶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣠⠟⡠⢚⢁⠂⠙⠧⣀⣀⡀⠁⠀⢧⡏⣸⠅⣿⣃⠄⣩⠖⠉⢙⡦⠴⣒⠉⠳⣤⣄⣀⡀
⠀⠀⠀⠸⡗⠒⠚⠋⣉⡭⣷⣺⡥⢠⣿⠢⣠⣠⠇⡯⢾⠉⡠⠊⣠⠴⢚⡩⠔⠚⣩⣄⠀⠙⢶⡉⠍⠻⡄
⢀⣀⠀⠀⡌⠙⠒⣊⣁⣨⠄⢀⢀⡞⢂⢢⢹⡿⠖⠉⡠⠊⡠⠚⠶⠚⠁⠀⡔⠉⠀⠈⠙⠓⠲⠥⢬⡀⡇⡰⠲
⣾⢉⠱⠀⡄⢠⢠⢃⢠⡄⠆⠋⢰⠡⣸⠜⠋⣀⡀⣎⣴⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠁⠳⡅⣽⢡
⠸⡜⣇⠀⢳⠃⢃⡎⢁⠀⢠⡄⠛⡙⠁⠘⢉⢠⡖⠉⢀⡇⢰⠻⣦⣀⣀⣀⡀⢃⣠⠵⠒⠒⠒⠓⠢⡄⠈⢋⣮
⠀⢡⠙⢦⡈⠁⠘⠀⠁⢴⣾⡇⠰⢁⣤⡶⣿⣾⡘⠀⠀⣠⠊⢠⠄⠀⣀⣠⡜⢋⣠⠒⠁⣀⠀⠀⣸⢒⣀⡼⠛
⠀⠀⢹⢤⡀⠀⠀⢀⡠⠿⠷⠧⠶⠥⠞⣛⡛⠃⣾⣆⠀⠁⠀⠉⠀⡟⣏⣧⢁⡻⣮⢣⠀⠉⠀⠀⡏⢉⢀⠆
⠀⠀⠘⣧⠙⠄⠐⠐⣟⠉⢩⣭⠉⠉⢙⣛⡀⠀⣿⢸⡆⠀⡔⠀⡰⠋⢹⣬⠏⠱⣜⢮⣝⣓⣓⡀⠠⡂⡜
⠀⠀⠀⠘⣦⠁⠀⠀⠤⠆⠀⠴⠆⠀⠐⠒⠂⠀⢿⢸⡇⠀⢀⡼⠁⠀⢨⣭⠁⠀⠀⣛⡒⠫⣭⠇⢝⡲
⠀⠀⠀⠀⠹⣆⢐⠀⠩⠥⠀⠩⠭⠄⠀⠭⠄⠀⠬⢿⣇⡴⠯⠭⠀⠀⢘⣒⠀⠀⢀⣒⠀⢠⠄⢠⣱⠁
⠀⠀⠀⠀⠀⢹⡄⠂⠀⠩⠄⠀⠭⠥⠀⠩⠭⠀⠨⠭⠛⠀⠭⠭⠀⠀⢀⣀⠀⠀⠒⠃⠀⠝⢐⣪⡇
⠀⠀⠀⠀⠀⠀⢻⡌⠀⠀⠯⠀⠈⠭⠄⠈⠭⠀⠨⠭⠁⠀⠭⠭⠀⠀⠭⠀⠀⢀⣚⠀⣸⠃⢪⡿
⠀⠀⠀⠀⠀⠀⠈⢿⡔⡂⠈⣧⡀⠩⡭⠀⠭⠥⠈⠭⠅⠀⠭⠭⠀⠀⢭⠂⠀⠸⠁⢀⡞⢨⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡌⠄⠈⢥⠀⢈⡂⠐⣒⠀⢐⠀⠀⣉⣉⠀⢘⣛⠀⠀⡿⠀⣜⢁⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠈⠥⠀⣒⡀⢒⡂⢒⣂⠀⢐⣒⠀⠐⠃⠀⠨⠅⠠⠃⢠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠦⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⡤⠴⠶⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣖⣿⣿⣿⣿⣗⣲⡶⠦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⣯⡷⢿⣛⣭⣽⣿⣿⣿⣿⣿⣿⣿⣶⣬⣉⠛⠲⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣯⣿⢟⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣄⡀⠀⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡾⣫⣾⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠉⠻⣝⣿⣄⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣵⢟⣵⣿⣿⣿⣿⡿⣫⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣽⢫⣾⣿⣿⣿⣿⢏⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣻⣷⡀⢀⠀⠀⢀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣾⢣⣿⣿⣿⣿⣿⣣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⢱⣞⡵⣫⣆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⢏⣿⣿⣿⣿⣿⣱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⢿⣿⣿⣿⣆⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⣾⣿⣿⣿⣿⢧⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣸⣿⣿⣿⣿⡾⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⢳⣿⣿⣿⣿⡏⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣟⣿⣿⣿⣿⣧⢿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣼⣿⣿⣿⣿⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣿⣟⣛⣟⣻⣸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣿⣿⣿⣿⡿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡜⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⣿⣿⣿⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⣷⣿⣲⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢡⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢿⣿⣿⣿⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡟⣸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡜⠀⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⣷⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠉⢁⢀⣠⣤⠭⠿⡿⣿⣿⣿⣿⣿⣿⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⢀⣤⣤⣶⢞⣿⣿⡿⠟⠗⠿⠙⡦⣄⠀⠀⠀⣠⡾⠁⠠⣤⣰⡾⠋⢁⣠⣤⡾⣼⣿⣿⣿⣿⣿⣿⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⢿⣿⢻⣿⡿⠁⢠⣤⣶⣤⣄⡀⠋⢷⡀⣼⠋⢡⣀⣀⡜⠉⢀⣰⣿⣿⡟⢙⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣿⡏⣸⡏⣿⣿⡇⢀⣿⣿⣿⣿⣿⣿⣧⣼⣿⢃⣤⣤⣽⡏⠀⣠⣿⣽⣿⠉⣀⣼⣿⣿⣿⣋⣉⣿⣿⣍⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢠⣿⠀⣿⡇⣿⣿⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⣿⣿⠏⠀⣼⣿⣿⠟⠁⣴⣿⣿⣻⣿⣿⣿⣿⣿⣿⣷⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⢀⣿⡇⣿⣿⣧⠀⢿⣿⣿⣿⣿⣿⣿⣿⢈⣻⣿⣿⣤⣼⣿⣿⠋⢀⣾⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣼⣿⣸⣿⣿⡸⣿⣿⣦⡈⢿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⡇⠀⣼⣿⣿⣾⣴⣿⣿⣿⣿⣿⣿⣿⣿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⣿⣿⣿⣿⣧⠙⣿⣿⣷⣤⠛⢿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣠⣿⣿⣿⣿⡏⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢻⣿⣿⣯⣿⣿⣷⡟⢿⣿⣿⣿⣶⣯⣽⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⣽⡟⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣿⠟⢿⣦⣘⠿⢿⣿⣿⣿⣿⣿⣷⡙⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣔⣼⠟⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣏⠛⢿⣷⣄⡙⠻⢿⣶⣭⣽⣯⣷⠾⠋⠻⣿⣷⡈⢻⣿⣷⡌⢿⣿⣿⣿⣿⣿⣿⣿⡿⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠿⠿⠿⠒⠒⠚⠋⠉⠀⠀⠀⠀⠀⠉⠙⠲⢽⣿⣿⣶⣿⣿⣿⣿⣿⣿⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢆⠀⠀⠀⠀⣠⢖⣦⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡥⠋⣳⣧⣀⡴⠊⠁⢸⡅⢠⣿⢲⠶⣤⣤⣀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢴⣫⣐⡈⠻⢤⣀⣀⠉⠀⠷⡏⣼⢠⣿⡤⣀⠽⠊⢉⡷⠶⣚⠉⠳⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢗⡒⠋⢁⡴⡺⠿⠵⢦⡾⠷⣤⣴⠁⡗⠯⣡⠞⣡⠴⣊⡥⠖⢋⠩⢤⣀⠙⢮⠄⡉⢷
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠆⠍⢋⡥⠤⠶⢀⡄⡾⢙⠆⣻⠟⠋⢁⠞⣡⠞⠑⠋⠁⠀⠼⠁⠀⠀⠩⠉⠓⠒⢦⣸⢰⡋⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡡⡌⠀⣆⡐⡜⣤⠶⠚⠉⢠⣇⡴⢋⣡⠤⢀⡷⢾⡄⢀⡀⠀⠀⠀⠀⡀⠀⡀⢐⣀⣀⡀⠀⠻⣼⢇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠧⢿⡀⠹⠐⢸⠁⠆⢀⣶⠀⡐⠀⢀⣤⣲⢯⠀⠸⢁⡿⠙⠲⠶⠶⠤⣤⠖⢋⠍⠉⠉⠉⣵⢀⠨⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡙⠂⠀⠀⠀⢈⣿⣿⢀⣤⣺⣯⣿⡿⣦⡀⠠⠊⠀⠎⢠⣴⢶⠫⣤⡺⡅⠀⠆⠀⠀⡯⠖⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣝⢦⡀⡠⢾⡦⠤⢼⡥⠴⢶⣶⡂⠀⣿⢷⠀⠀⡀⠀⣼⠼⡼⣦⢦⣻⣜⠤⠤⠤⠘⢀⢂⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⢀⠀⠘⣣⠀⢸⣧⠀⠐⡶⠆⠀⣿⢸⡆⠈⢀⡼⠁⠀⢻⡇⠀⠙⣿⣿⣛⣓⡦⡴⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠤⠀⢒⡀⠐⣛⡀⠀⠘⡀⠀⣹⣾⡇⣰⣿⡄⠀⠀⠿⠆⠀⠀⠭⠀⢈⠁⢐⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⠄⠀⠦⠀⠘⢓⠀⢐⣒⠀⠐⠛⠟⢐⣒⠁⠀⠨⠭⠁⠀⢰⠃⠀⠿⠠⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡈⠀⠨⠤⠀⠶⠄⠀⠒⠀⠀⠤⠀⠀⠭⠅⠀⠭⠭⠀⠀⣒⠀⣼⠃⣺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡠⠀⢹⡄⠘⣟⡀⠸⠧⠀⡭⠀⠈⠩⠅⠀⢨⡃⠀⢸⠁⢀⡖⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡡⠀⠭⠄⠘⠇⠀⣓⠀⣿⠀⠈⣛⡁⠀⣻⡃⠀⡿⠀⣼⢡⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠈⠩⠄⠩⠄⠐⠀⠠⠄⠀⠤⠤⠀⠉⠀⠀⠇⠐⠂⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠲⠶⠶⠤⠤⣤⣤⣤⣤⣤⠴⠶⠶⠶⠒⠛⠋⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠉⠉⠀⢀⡀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡌⠳⣄⠀⠀⠈⠁⠈⠙⡆⠀⠀⠨⠙⢦⠤⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⡐⣄⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠈⢧⠀⠀⡸⠇⠀⠀⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢳⡌⠳⣄⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠹⣄⡀⠀⠀⠈⢷⠚⠁⠒⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠙⠦⡈⠳⣄⠀⠀⠀⠘⢆⠀⠀⠀⠀⠙⡷⡄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⣙⠆⡀⢿⣷⣮⡂⠙⢦⠀⠀⠀⣀⠡⡀⠀⠀⠀⢹⡸⣆⠀⢀⣬⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⠜⠑⠆⠙⠲⠙⠋⠀⢀⡈⣳⡄⢺⣿⣷⣽⡄⢠⣤⣄⢳⡹⣆⢹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⠴⠋⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⠉⠙⢦⡻⣿⠟⠃⢸⣿⣿⣎⠃⠘⠂⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⣠⠞⠁⠀⠀⠀⢀⣠⠖⡻⠛⠁⠀⠒⠀⠀⠀⢀⣠⡤⠶⢂⡀⠻⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⡠⣊⠕⠉⠀⠀⠀⠀⢀⣠⠴⠞⠋⢁⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣈⣴⣆⠀⠀⢀⡴⠊⠀⠁⠀⠀⠀⣀⡴⠚⠉⠀⠀⣠⡶⠋⠁⢤⠀⠀⠀⣀⣀⣤⠤⠶⠞⠛⠋⠉
⠀⠀⠀⠀⠀⢀⣼⣿⡿⠃⣠⠾⣫⣤⠀⠀⠀⢀⣠⠞⠁⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⣠⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠈⠛⠋⠀⠊⣵⣿⣿⠟⠀⣠⠶⠛⠁⠀⠀⠠⠚⠉⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠥⠶⠋⠁⠀⠀⢀⣴⣾⣿⣿⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠯⠴⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⣀⢤⣐⡒⠒⠂⠀⠀⠀⠀⣀⣀⠠⠤⢒⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⡤⠄⠀⣀⠀⠀⢀⠀⠘⠷⣄⡀⠀⡀⠀⠀⠤⡤⠤⠤⠐⡒⠭⢑⣨⣾⣷⡄⣀⡀⠀⠀⠀⠀⠂⠀⠉
⠀⠀⠀⠀⡈⣁⣀⡀⠀⡈⡀⢠⠠⠟⡿⠉⠑⠂⠀⠠⣤⠄⢠⠢⡒⠂⠁⢹⣿⢸⡍⠝⠓⠂⠀⠀⠀⠀⢀⡦⠤⠶⠎⠃
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠁⢀⠈⠂⢄⠘⠋⠁⠀⠀⠀⠀⠀⠀⠁⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡰⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⣠⠤⠶⠖⠒⠒⠒⢚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⢎⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡋⠉⡄⠀⠀⠀⢠⠀⠀⠨
⠀⠀⠀⠀⡈⠀⠀⠌⠤⣡⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠞⠂⢑⠀⠀⠀⠀⠄⠐⠀⠃
⠀⠀⠀⢠⣷⣶⠞⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠋⠉⠀⠀⡀⠀⠀⠡⠀⠀⠀⠐⠄⠁⠈⠄
⠀⠀⠀⢦⡤⣭⢾⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⢀⢠⠰⡀
⠀⠀⠀⠘⠀⣧⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠑⢄⣤⠧⠝⣒⣭⡥
⠀⠀⠀⠀⠀⠈⠀⢃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⣹⠶⣿⣯⠹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠉⠑⠹⠂⠁⠛⠂⠉⠠
⠀⠀⠀⠃⠀⡄⠸⠄⡆⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠋⠀⠀⡇⠀⠀⠀⠌⠀⡄⠀⣶⠀⣀⠄
⠀⠀⢠⣤⢤⡤⣤⢤⢡⡬⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⢀⠤⠜⠶⠶⠦⢤⠬⠤⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠠⠤⠀⡀⢀⠀⢀⢀⣀⢀⣀⣠⠤⠤⠐⠚⠁⠁

⠀⡀⡄⡀⣀⢠⡀⡄⣄⡄⡀⢠⢠⢠⢀⡀⡄⡄⢀⠀⡀⡄⣄⠀⢠⠀⡄⡀⡄⡄⢠⢀⢀⡀⡀⡄⡄⡄⡄⢀⢀⢠⢀⡀⡄⡀⢀⢀⢠⠀⡀⡄⢠⢠⢀⠀⡀
⠀⠇⠃⡇⡇⠈⠀⡇⠛⡇⠃⢸⠘⠸⢸⡇⠃⠓⢸⠀⠃⠇⠁⠀⢸⠀⢱⡇⠃⣗⠸⠀⠘⡇⠇⠁⠀⠇⠀⢸⠘⢸⠈⠁⡇⠇⠈⢸⠘⠀⡇⢻⠘⢸⢸⠐
⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠃⠀⠀⠈⠀⠀⠇⠀⠀⠸
⠠⠖⢶⣄⣠⡶⢛⠿⢦⣄⣤⡴⠟⠋⠙⠷⣤⣠⡤⢶⡟⠓⢦⣤⣠⡤⠶⠚⠋⠉⠛⠶⣤⣤⣤⠶⠛⠛⢻⡦⣤⣤⠴⠶⠛⠳⠶⣤⣀⣀⣀⣠⡤⠶⠶⢦⣀
⠀⠀⠀⠀⡜⠀⠌⠀⠀⠀⠈⠀⠀⢀⠇⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠈⡆⠀⠆⠀⢣⠀⢉⠉⠉⠇⠀⠀⠇⠀⠉
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⢃⠸⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢰⠀⠀⠀⢀⠁⠀⠀⠀⢀⠂⠀⠀⠀⢀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣆⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡈⢆⠱⡀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠸
⠀⡎⠀⠀⢠⠂⠀⠀⠀⡠⠀⠀⠀⠀⠀⢁⣼⠟⠀⠀⠀⠀⠀⠀⠀⡴⡟⣻⣿⣶⡯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠈⢆⠘⢄⠀⠀⠀⠁⡀⠢⠀⢂⠀⠀⡇
⠀⠀⡐⠠⠁⡀⠁⢀⠜⠀⣀⠔⡠⢊⣴⡿⠋⠀⠀⠀⠀⠀⠀⢀⣞⡷⣿⠟⠛⣿⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡈⠣⡈⠀⠀⡀⠀⠐⣄⢀⠈⡆⠀⢃
⢀⣠⡵⢁⠆⣠⠔⢁⡤⢊⠤⢊⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⡇⠋⠀⠀⣬⢻⣟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠘⢆⠠⡘⢆⢢⠱⠘⣼
⢈⡻⢶⣥⣊⠥⢚⣡⣖⣥⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⠄⠀⣠⣾⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠂⡀⠈⠳⡘⣌⢦⢣⣳⡼
⠀⠛⣦⡙⠳⢾⣟⡛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠼⣏⠛⣹⢧⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠢⡀⢙⣮⡾⠟⢉⣠
⠀⠀⡇⠙⢦⣔⣾⠟⠀⠀⠀⠀⠀⠀⠠⣤⡤⠤⠤⠴⠖⣚⣋⣩⡀⣀⣈⠞⢁⣀⡉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⣾⣟⢉⣠⣶⠟⡏
⠀⠃⠇⠀⠀⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠛⠋⠉⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⢧⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠷⢛⠉⡆⡀⠇
⠀⠀⠠⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠀⢣⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢸⠸⠀⡇⠇⠃
⠀⠀⢰⢠⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣯⣿⣷⣄⢸⢸⠀⠀⡀⢀⠀⡀⢀⢀⡀⣀⢀⣀⢀⡀⢀⡿⠁⠖⢢⠀⡇⢸⢰
⠀⠀⠘⢸⢸⢻⣴⣦⣶⣴⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢏⣽⡟⢿⣿⣴⣿⣽⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠃⡜⠀⡧⠀⡇⢸⢸
⠀⠀⠀⢘⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣞⠟⠘⠀⠿⠿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠁⢀⢃⠀⡅⢸⠘
⠀⠀⢠⢸⢸⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢠⣶⣦⢠⣿⣿⡾⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⡀⠂⡘⢸⠀⡇⠸
⠀⠀⢸⠘⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣠⠐⠛⢋⣀⡝⢋⣴⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠰⠠⠀⡇⢸⠀⠇⢰
⠀⠀⠘⠀⡇⡆⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣯⢽⣿⡇⠀⢺⣯⣿⠀⡯⠿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠁⡌⢠⠀⠈⠀⡇⢸
⠀⢰⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣁⣀⢿⣿⡇⠉⢰⣻⣿⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡘⢰⠁⡘⠀⡆⠀⠁⢸
⠀⢸⠀⠀⢃⢰⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣶⠋⠛⣷⠒⢛⣷⣶⣾⣛⣶⣿⣿⣿⣿⣿⣿⣿⣞⣿⠁⡰⠁⡌⢠⠁⠀⡇⠀⠀⠈⡆
⠀⠘⠀⠃⢸⠈⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣤⣦⠀⢀⣤⣀⠀⠀⢁⣀⡈⠀⠅
⢀⣴⣦⠀⣠⣤⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣾⣿⣿⣿⣿⣷⣶⣾⣿⣿⣤⣤
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⡯⣿⣿⡿⡿⠿⠿⡿⠿⠿⠿⠿⠿⠿⠿⠿⡿⣿⢿⣿⣿
⠀⠤⠭⠭⠤⠭⠭⠬⠥⠬⠥⠭⠥⠭⡥⢉⣬⠉⢡⡭⡍⠶⣷⠆⢰⠿⣦⢠⡶⠆⣴⠶⢠⡶⠆⢰⡖⠒⢒⣒⣒⣒⣒⣒⣚⣛⣛⣚⣛⣛⣛⣛⣛⣛⣉⣉⣉
⢀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣒⣒⣚⡃⣼⠿⣧⢺⣀⡄⠀⡿⠀⢸⠿⣏⢸⣟⡃⣙⣻⡎⢛⣳⢮⣭⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣗⣈⣉⣉⣉⣉⣉⣉⣉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠐⢫
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡼
⠀⠀⠀⠀⠀⡜⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⢱⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡾⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣎
⠀⠀⠀⠀⠀⣇⡀⠀⣠⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⠤⠤⠤⠤⠤⠤⢤⣀⡀⠀⢀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⢰⡉⠀⠈⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢉⡆
⠀⠀⠀⠹⣏⣉⣉⣉⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⠏
⠀⠀⠀⠀⢸⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣴⣦⠀⠀⡴⠢⡀⠀⡠⠲⡄⠀⢠⠖⢄⠀⢠⠔⢦⠀⠀⡴⢢⡀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠰⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢸⠀⠀⡟⠚⡇⠀⡟⠒⡇⠀⢸⠒⣻⠀⢸⠓⢻⠀⠀⠓⠚⡀⠀⠀⡇
⠀⠀⠀⠀⢸⡇⠀⠠⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⢹⠀⠀⡧⢤⡇⠀⣧⠤⡇⠀⢸⠤⣾⠀⢸⡤⢼⠀⠀⡧⢴⠁⠀⢸⡇
⠀⠀⠀⠀⠀⡇⠀⠈⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢸⠀⠀⡇⠀⡇⠀⡇⠀⡇⠀⢸⠀⢺⠀⢸⠀⢸⠀⠀⠆⠀⡇⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⣸⠀⠀⡟⠚⡇⠀⡟⠛⡇⠀⢸⠛⢻⠀⢸⠋⢻⠀⠀⡟⠛⡄⠀⢸
⠀⠀⠀⠀⠀⣇⠀⢠⠁⠨⠉⠉⡗⠒⡖⠊⡟⠉⡟⠑⢲⠒⢶⠒⢶⠶⢲⠒⠁⡇⢸⠀⠀⣇⣠⡇⠀⣧⣀⡇⠀⢸⣀⣼⠀⢸⣀⣸⠀⠀⣅⣠⠆⠀⣼
⠀⠀⠀⠀⠀⣸⠀⢰⠓⢚⠀⠀⡟⠚⡇⠀⣟⠒⡇⠀⢸⠒⢻⠀⢸⠓⢻⠀⠀⡗⢺⠀⠀⡇⠀⡇⠀⡇⠀⡇⠀⢸⠀⢸⠀⢸⠀⢸⠀⠀⡇⠈⠀⠀⡇
⠀⠀⠀⠀⠀⢹⠀⢀⡦⢼⠀⠀⡧⠤⡇⠀⣿⠤⡇⠀⢸⠤⣾⠀⢸⠤⢼⠀⠀⡧⢼⠀⠀⡗⠒⡇⠀⡷⠒⡇⠀⢸⠒⢿⠀⢸⠒⢺⠀⠀⠗⠺⡁⠀⡇
⠀⠀⠀⠀⠀⢿⡀⠀⠳⠞⠀⠀⠓⠔⠁⠀⠙⠶⠃⠀⠘⠢⠋⠀⠈⠢⠚⠀⠀⠳⠞⠀⠀⠓⠖⠁⠀⠙⠖⠃⠀⠘⠶⠋⠀⠈⠢⠚⠀⠀⠳⠞⠀⢠⠇
⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢤⣤⣤⣤⣤⠖⠀⠈⠉⠭⠵⠒⠋⣁⣠⠴⠁⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⡰⠂⠀⠀⠔⠚⠉⠁⣀⡤⠶⠂⠈⠙⠲⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⢠⡞⠁⠀⡀⠀⠀⢀⠄⡜⠁⢀⡤⠴⢶⣀⠀⠙⢶⣖⠒⠚⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢶⣏⠀⢀⡞⠀⢀⡴⢁⣾⢀⡴⠋⠀⢀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣼⣿⣴⣿⡗⣠⠎⠀⠾⠛⠋⢀⠀⠀⠀⠙⠲⢤⣄⡀⠈⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣄⣀⠀⠉⠉⠸⠛⠛⠁⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣄⡀⠀⠉⠛⠶⣤⡈⢳⣶⠶⠶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠲⠦⣤⣄⣀⡀⠀⠉⠙⠲⢦⣄⣀⠉⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⠶⠦⠤⣤⣤⣀⣀⣉⣉⡛⢳⣶⣤⣄⣀⣉⠻⣶⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠥⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⡆⠈⠀⠠⠀⠀⠀⠀⠈⠀⠐⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠰⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣷⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⠀⡏⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢳⠈⠓⠷⣦⣤⣤⣠⣤⠶⠝⠊⢱⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⢦⣄⣈⡈⠀⡀⠠⠀⠀⡘⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠁⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠐⠒⠒⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠠⠤⠄⠀⠤⠄⠀⠤⢴⣿⣟⠛⠆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣤⣤⣤⣤⣤⣶⣴⣾⣿⣿⣶⠶⠞⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡴⠙⠃⠙⡟⠋⡏⠛⣿⠙⡟⠝⠻⣿⣁⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠁⠀⠀⠀⡇⠀⠀⠀⠐⠁⠠⠀⠂⠈⠫⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠐⡀⠀⢑⣄⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠠⠀⠀⠀⠃⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢑⡀⠀⠌⢆⠀⠈⢦⡀⠈⡀
⠀⠀⠀⠀⠀⠄⠀⠐⡊⠀⠀⢠⠀⠀⣺⠀⢀⣙⠀⠀⠠⠤⡇⠀⠐⠻⠀⠀⢵⡀⠀⠀⣣⠀⠀⢧⠀⠼⡀
⠀⠀⠀⠀⠠⠀⠀⢠⠁⠀⡀⠇⠀⠀⠆⠀⠈⠨⠀⠀⠀⠀⡯⠀⠈⢨⡁⠀⠀⣇⠀⠀⠘⡆⠈⠈⡇⠀⢱
⠀⠀⠀⠀⠀⢀⠀⡎⠀⠀⢠⠀⠀⠀⠃⠀⠀⢨⠀⠀⠀⣀⣻⠀⠀⠀⡇⠀⠀⣸⠀⠀⡀⣿⠀⠀⢺⠀⢀⡆
⠀⠀⠀⠀⠈⠠⠚⡃⠀⡨⡬⠀⠐⣂⠀⠀⢀⣒⠀⠀⠐⠒⣶⠀⠀⠠⡇⠀⢐⢸⡇⠀⡂⢾⠀⠀⣺⠀⠠⡅
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠣⢀⡀⣈⡇⠀⠀⢸⠀⠀⠈⠀⡏⠀⠐⢐⡇⠀⠀⣽⡁⢀⡉⠟⠀⠒⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠘⠂⠄⠒⠚⠃⠀⠀⠚⠁⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣶⣿⣷⣶⣿⠋⢠⣶⣷⣶⣶⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣁⣾⣿⣿⣿⣿⣿⣿⣋⣶⣿⣿⠏⠁⠀⠒⠲⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢱⣿⣿⣿⣿⣿⣿⣿⣿⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠹⢿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣄⣀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠛⠻⠿⢿⣿⣿⣶⣦⣄⡀⠀⠈⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣈⡉⠛⠻⠿⣿⣿⠀⠙⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣯⣈⣉⣉⠉⠛⠻⠿⣿⣿⣿⣶⣦⣤⡀⠀⣮⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣶⣶⣤⣌⣉⠛⠻⠿⠁⢀⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⢠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⣈⡙⠛⠿⠿⣿⣿⣿⣶⡄⠀⣰⣿⣿⣿⣿⣿⣍⠙
⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣶⣦⣤⣌⡉⠛⠃⠀⣾⣿⣿⣿⣿⡿⢿⡇
⠀⠀⠀⠀⠀⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣄⣉⡉⠛⠻⠿⣿⣿⣷⠀⢰⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣈⣹⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠛⠛⠿⠿⢿⣿⣿⣿⣷⣦⣤⣀⠁⠀⢸⣿⣿⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣄⣉⠙⠛⠿⣿⣿⠃⠘⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣤⣴⣿⣿⣿⣿⣿⣿⣯⣩⣏⣉⡉⠉⠛⠛⠻⠿⣿⣿⣿⣷⣶⣤⣀⠀⠄⣼⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣈⡉⠛⠛⠻⠋⠀⢠⣿⣿⣿⣿⡟⠛⠃
⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣠⣍⡉⠉⠙⠛⠿⢿⣿⣿⣿⣶⣦⠀⠀⣾⣿⣿⣿⣿⣿⣦
⠀⠀⠀⢉⣽⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣷⣶⣶⣤⣄⣈⡉⠉⠃⠀⠀⣿⣿⣿⣿⣧⣈⠁
⠀⢀⣶⣸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⡀⠀⠉⠉⠙⠛⠻⠿⣿⣿⣿⡿⠀⠀⢀⣿⣿⣿⡿⠿⠿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⡆
⠀⠘⢿⡿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠙⠋
⠀⠀⠀⠙⠻⢿⣝⢿⣿⣿⣿⣿⠞⠻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠙⠲⠽⢿⣿⣿⣷⣄⣈⠙⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣷⣦⣍⣙⣛⣻⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠛⠙⠛⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⢿⣿⣿⣿⠇⣴⠾⣶⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣤⢀⢰⣿⣿⡄⠁⢀⣿⡀⠙⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣶⣿⣿⣿⣿⣄⢸⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠹⣿⣟⣦⡉⠸⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠙⣿⠻⢿⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣶⣶⣴⣦⣶⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⣀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⢀⡀⣤⣶⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠷⣶⠓⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠾⠛⠛⠛⠛⠿⠶⠶⠶⣦⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠟⠋⣉⣠⣤⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣀⣉⣉⡉⠛⠛⠛⠷⠶⢶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⢀⣤⡶⠟⠛⣩⣴⠟⠿⣿⡟⠛⠩⣉⠉⠉⠛⠛⠻⠟⠛⢻⣧⣭⣭⣿⡿⣿⣿⡿⣷⣶⣶⣤⣤⣄⣩⣭⣟⢿⡄
⠀⠀⠀⠀⣼⠏⠀⢴⣿⣏⣀⣙⣂⡈⠻⣶⣶⣧⡶⠶⠾⠛⠛⠷⢶⣌⣿⡄⢰⠈⢢⣀⠹⡻⣿⣍⣽⣿⡛⠋⠉⢤⡙⢿⣇
⠀⠀⠀⠀⠹⣧⣀⣀⡀⠉⠉⠉⠛⠛⠻⠶⠿⣿⣶⣶⣔⣀⣒⣒⣈⣉⣻⣿⣼⠶⣦⣄⣤⡶⠿⠿⢭⡉⠙⢦⡀⠈⠁⠀⣿
⠀⠀⠀⠀⠀⠈⠹⣿⠛⠛⠻⠶⠶⢶⣤⣤⣤⣀⣀⣀⠈⠉⠙⠛⠛⠿⠿⢿⣶⣦⣤⣽⡿⢿⡷⣦⡀⠉⠀⠀⠀⢀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠷⠶⠶⣦⣤⣤⣄⣀⣀⣀⣠⣼⣷⠘⣷⠀⠹⠄⠀⢸⡇⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡏⢻⡆⠀⢰⠦⡄⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠀⣿⠀⢿⡄⠀⠀⡆⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⣧⣨⡇⠀⢸⡀⡇⠀⢸⠉⣿⠀⢸⠛⡶⠀⣠⣤⡄⠀⠀⠀⢰⣿⡇⢸⡇⠀⠀⠃⠀⠀⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢰⡶⣦⠀⢨⣭⡁⠀⠘⠶⠿⠀⢸⣤⣇⠀⣿⠀⡇⠀⠀⠀⠸⢾⡇⣿⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣇⣹⠀⢸⡇⣿⠀⢸⡏⢻⠀⢰⠶⣤⠀⢙⡛⠃⠀⠀⠀⢀⣸⣧⢻⣿⠀⡀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⣬⢭⡄⠈⣛⡛⠀⠸⠧⣾⠀⢸⡀⣿⠀⣿⠉⡇⠀⠀⠀⢸⣯⣿⢸⡏⠀⣧⠀⠀⠀⠸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⢸⡀⡇⠀⡏⢹⠀⢰⡶⣦⠀⢀⣉⣉⠀⠻⠶⠇⠀⠀⠀⠘⢻⡇⣸⡇⠀⢸⡆⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠘⠓⠇⠀⢿⣼⠀⢸⣇⣻⠀⢸⡇⢹⠀⢰⠶⣦⠀⠀⠀⢰⡟⣿⣿⣄⣤⣬⣇⣠⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠘⠷⠼⠂⢸⣀⡏⠀⠀⠀⠸⠷⢋⣼⠟⠋⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠳⠶⢶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠶⠶⣦⣤⣤⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡁⠈⡃⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⣿⣿⣿⡟⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠁⠀⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣘⢻⣟⢛⡡⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣝⢦⡤⢤⡴⣯⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠇⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣶⣦⣴⣦⣴⣶⣤⣶⣤⣶⣦⣴⣦⣤⣶⣤⣶⣤⣴⣦⣴⣾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠆⠀⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣺⣿⣚
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠆⠀⠆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣹⣟⣉⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣛⠀⠤⠠⠤⢈⣻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣽⣿⣽⡗⠆
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣆⠀⠀⢀⣸⣻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢃⠻⠿⠿⠟⡡
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣍⠉⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡟⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣇⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠛⠉⠁⠀⠀⡇⠀⠈⠙⠳⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠞⠛⠉⠀⠀⣀⣠⣤⠴⠞⠛⡟⠓⠦⣤⣀⠀⠈⠉⠛⠶⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠋⠉⠀⠀⣀⣠⣤⠶⠞⠛⠉⠉⠀⠀⠀⢰⡇⠀⠀⠀⠉⠛⠲⢤⣄⡀⠀⠈⠉⠛⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠛⠛⠉⠉⠀⠀⣀⣠⣤⠶⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣠⣴⠦⣤⣤⣄⣀⠈⠉⠛⠶⢤⣀⡀⠀⠉⠛⠶⣤⣀⡀
⠀⢀⣀⣤⠴⠚⠉⠁⠀⠀⣀⣀⣤⡴⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠶⠶⣦⣿⡾⠋⠀⠀⠀⠀⠈⠙⣦⡀⠀⠀⠀⠈⠙⠳⢦⣀⠀⠈⠉⢛⡳⢦
⠀⢿⠓⢤⣀⣀⡤⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠉⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠤⢤⣄⣀⣬⠿⠶⠚⠉⢁⣼
⠀⢸⣆⡀⠈⠙⠲⢦⣄⡀⡠⠴⠖⢶⣤⡶⠟⠋⠙⢢⡀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⢶⣤⠒⠋⠁⠀⠀⣀⡤⠞⠋⣿
⠀⠀⣿⠉⠓⠲⢦⣄⠈⠛⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⣀⣀⣀⣤⣀⣀⣀⡀⠀⠀⠙⠶⣤⣄⣀⣠⠴⠖⠛⢿⡁⠀⠀⠀⢻⣀⣤⠴⠖⠋⠁⠀⠀⢠⡇
⠀⠀⢸⡆⠀⠀⠀⠈⢙⠶⣤⣀⠉⠛⠷⣦⣀⠀⠀⠀⠀⠀⠙⠯⣍⣉⣉⣉⠉⠉⠉⠙⠛⢷⣤⠤⠖⠋⠉⠀⠀⠀⠀⠘⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⣷⠀⠀⠀⠀⢸⣠⡇⠈⠛⠦⣄⡈⠙⠻⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣦⠸⡇⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⢹⡆⠀⠀⠀⠈⣟⣇⠀⠀⠀⠀⠙⠲⣄⣀⠈⠉⠛⠷⣦⣄⡀⣀⡤⠖⠋⠉⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡿⠀⣿⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠈⣧⠀⠀⠀⠀⠘⢮⣗⣦⣄⡀⠀⠀⠀⠉⣻⠶⢦⣄⡀⠉⣿⠁⣀⡤⠖⠚⠉⢉⡇⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⣿⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣷⣦⣤⡴⠋⠀⠀⠀⠉⠓⣿⠋⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡟⠀⡄⠀⠀⠀⣿⠀⠀⠁⠀⢸⡇⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣸⠃⣸⠁⠀⠀⠀⣿⠀⢀⣀⣀⣼⡇⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠀⠋⠀⠀⠀⠀⣿⠾⠛⠋⠉⠉⠀⠀⠀⣀⣴⡟
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠃⠀⠁⠀⠀⢀⣀⡛⠀⠀⠀⠀⢀⣠⣴⠿⠛⠉
⠀⠀⠀⠀⠀⠈⠧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣤⣤⡶⠶⠟⠛⠉⠀⢀⣠⡴⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣇⣠⡴⠞⠉⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⡀⠀⣿⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠿⠶⠞⠋⠉

⣿⣿⡿⣿⣿⢿⡿⣿⡿⣿⢿⡿⣿⣿⡿⣿⣿⣿⣿⢿⡿⣿⣿⡿⣿⢿⣿⢿⣿⢿⡿⣿⣿⣿⡿⣿⣿⢿⣿⢿⡿⣿⣿⣿⡿
⣿⣿⣷⣿⣿⣿⣿⣿⣿⡻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣟⣻⣾⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠜⣻⣿⣿⣿⣿⣿⣿⣻⠿⠽⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣏⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⢻⣿⣿
⣿⡏⡟⣞⡏⡾⣸⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡏⣧⢹⡎⡿⡞⡇⣿
⣿⣿⣧⢿⢳⢧⣿⢇⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢹⠏⡏⠛⡇⠋⢻⠻⢹⡟⡟⡿⣿⣿⣿⣿⣿⣿⣿⣷⡸⡌⣇⣿⡇⣷⣿
⣿⣿⣹⣾⡟⣾⢏⣾⣿⣿⣿⣿⣿⣿⢻⡟⡇⢸⠸⠀⡇⠀⠇⠀⠸⠀⢸⠀⠁⡇⣿⢿⣿⣿⣿⣿⣿⣿⣧⣳⣹⢸⢿⢻⣿
⣿⣿⣿⡿⣼⣿⣿⣿⣿⣿⣿⣿⣿⠋⢸⠃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠇⠋⢸⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣸⣿
⣿⣿⣽⣳⣿⣿⣿⣿⣿⣿⣿⡟⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡇⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣻⣿⡿⣿
⣿⣿⡇⣷⣿⣿⣿⣿⣿⣿⣿⣧⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣷⣿
⣿⣿⣿⣇⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠰⠠⠄⠦⠰⠤⠆⠄⠆⠤⠴⠠⠴⠀⠄⠦⠦⠸⠸⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣤⣙⠻⠿⠿⠟⠋⣤⣙⠻⢿⣿⠿⠟⠁⠀⠙⠻⠿⠿⠿⢋⣥⣦⢙⠻⠿⠿⠟⣋⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣷⣾⣿⢰⣿⣿⣿⣶⣶⡖⠀⠀⠀⠀⠀⣷⣶⣾⣿⣿⣿⠸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⣾⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣇⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⡿⢃⣾⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣦⠹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⡻⠟⣋⣴⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣬⡙⠿⠁
⠀⠀⠀⠀⠀⣰⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠
⠀⠀⠀⠀⠀⣿⣧⠸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⢡⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣇⢻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢃⣾⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⡄⢶⣶⣶⣶⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⠆⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠿⠿⠛⣡⣾⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣙⠻⠿⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⠂
⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⢠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⢣⡱⡞⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡙⠄⠀⡠⠔⠉⢉⣀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⢀⡀⠀⠀⠀⢀⡠⠊⠁⠀⣈⣆⣯⠀⣠⣶⠗⠉⠱⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠁⢠⢃⢾⠱⣀⠠⣚⢣⠀⢀⠴⠊⠀⠑⠤⡙⠓⠤⠤⠀⠠⠬⠆⠒⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡌⠰⢁⠎⠀⠳⡈⠢⣀⠡⠻⡁⠀⠀⠀⠀⠀⠀⠁⠒⠂⠀⠀⠒⠂⠉⠁⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡜⢠⠁⠎⠀⠀⠀⠈⠲⢄⡐⠤⡘⣄⠀⠀⣀⠀⠀⠠⠠⢄⠀⠀⡠⠒⠢⢄⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠊⠀⣂⡸⠄⠒⠉⢃⠀⣀⡀⠬⠷⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠈⠢⣀⠷⡀
⠀⠀⠀⣠⠄⠠⠤⢒⡡⢊⠥⠔⠊⠁⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠈⠢⡀
⠀⠀⠀⠏⠀⠀⠈⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⣘⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠃⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⡘⠒⠄⣀⠂⣀⠠⠤⠐⠀⠲⠀⢂⠀⠀⠀⢀⡠⠤⠦⡄⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠤⢤⡀⡀⠀⢸⠀⡠⠤⠀⠀⠒⠂⠀⠀⠠⠄⣀
⠐⡤⠄⠤⠦⢼⠀⠀⠀⠀⠀⠆⠀⢨⡄⠄⠤⠼⠵⡄⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⡴⠡⠤⠠⠤⢯⠀⠒⢄⠀⠀⠀⠀⠀⠀⠈⠱⢲⣄⡀
⡠⠇⠀⠀⠠⡌⠀⠀⠀⠀⠀⠰⡀⡰⠀⠀⠀⠀⠀⠈⢆⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠎⠀⠀⠀⠀⠀⠀⢂⠀⠀⢣⠀⠀⠀⠀⠀⡰⣱⣗⣝⢝⢓
⠓⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠈⠗⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠨⠀⠆⠀⠀⠀⠀⠀⠀⠀⠓⠢⣸⠀⠀⠀⠀⡔⢰⢏⢾⣢⢃⠊
⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⡌⢀⣀⠀⣀⣀⠀⣀⡀⢀⣈⣄⠀⠈⢢⠀⠀⠀⠀⠀⠀⣀⣤⣧⣀⢀⣀⡀⢀⣀⠀⣀⣀⠀⠉⠢⢄⡌⣰⢣⡿⣱⢃⠂
⠀⠀⠀⡶⠒⠀⠒⠀⢀⠀⠀⡐⠁⠀⠀⡇⡇⠅⢸⠀⣀⠸⠀⠈⢦⡀⠀⠱⣄⣠⠔⡖⣋⢠⠀⡳⢾⢸⡰⡇⡇⠀⡅⡇⠘⠀⠀⠀⠀⠉⢳⢿⣼⢣⠂
⠀⠀⠀⠐⣄⠀⠀⠀⠀⠑⣴⠁⠀⠀⠀⡅⡇⢹⢸⠀⣧⢰⠀⠀⠂⠐⠤⠆⣉⢘⡩⡇⠖⠸⠀⠽⢸⢸⠀⡇⡅⠀⡃⡇⢘⠀⠀⠀⠀⠀⠘⠀⠈⠁
⠀⠀⠀⠀⠈⡗⠂⠀⠒⠉⠀⣒⡁⢈⢒⠁⢑⣋⠀⣓⡋⢈⢒⠁⢙⣊⠀⣑⡋⢈⣓⡁⢘⣊⠀⣑⡋⠈⣒⡁⢉⣚⠀⣑⡊⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠀⠀⠈⠀⡇⡇⠈⢸⠁⢉⢸⠐⠆⡥⢬⠀⣇⡿⠀⠀⡇⠀⠈⠀⠃⢸⢸⠀⡇⠁⠀⠁⠃⠈⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⡇⠀⢸⠀⠈⢸⠀⠀⠆⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⢸⢸⠀⡇⡄⠀⡀⡆⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠂⡁⢈⠂⠁⠑⡊⠈⠁⡃⢈⠐⠁⢙⢘⠀⡃⡃⠈⡀⡁⢘⢈⠀⠑⠈⠈⡂⡁⢉⢈⠁⡑⡊⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠁⠈⠀⠇⡇⠈⢸⠁⢐⠸⠀⠀⠁⠈⠀⠁⠈⠈⠀⡇⠀⢨⠀⡇⢸⢸⠀⡇⡇⠀⠁⡇⢈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⡇⡇⠀⢸⠀⠀⢰⠀⠀⠄⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⢾⢸⠀⠇⡇⠀⠁⡇⢰⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⣀⡃⢈⣐⠁⢑⣊⠈⣃⡀⢈⣐⠁⢙⣈⠀⣃⡋⢈⡂⡁⢘⣈⠀⣃⡈⠈⣐⠁⢙⣈⠁⣁⡊⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠀⠁⠈⠀⠇⡇⠈⢸⠁⠀⢸⠀⠀⠅⠈⠀⠁⠀⠈⠀⡇⠂⠨⠀⠁⢸⠸⢆⡃⠏⠀⠀⡇⠸⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸⠀⠀⠀⠀⡆⡇⠀⢸⠀⠀⢸⠀⠀⡀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠀⢸⢸⠈⡇⠃⠀⡆⡇⢠⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠈⠂⠃⠈⠒⠁⠑⠊⠈⠓⠂⠘⠒⠁⠑⠘⠀⠃⠊⠈⠂⠁⠑⠘⠀⠑⠊⠈⠒⠃⠑⠒⠁⠓⠊⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠔⠁
⣿⣿⡿⠛⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⡶⠺⠿⠛⠟⠓⡲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠫⠃⠁⠀⠀⣀⣠⣤⣤⣈⡲⡈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡡⠂⠀⠀⣠⠶⠋⠉⠀⠀⠀⠈⠙⢶⡄⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⠊⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠡⠃⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠣⠃⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠻⢶⡴⠿⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠱⢺⠋⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⠁⡞⠀⢠⣎⠉⠙⢦⠈⢷⡀⠀⠀⣀⣤⠤⠶⠶⢼⠇⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢼⠀⣇⠀⢸⠀⠱⡀⠀⢣⠈⣧⠴⠋⠁⢀⣠⠤⠔⣮⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⢸⠀⠘⡄⠀⢱⠀⢀⠖⠁⢀⡤⠚⠁⠀⠀⡔⠀⠀⠁⢢⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⡆⠈⢆⠀⠱⡀⠈⡴⠃⢀⡴⠁⠀⣀⠔⠂⠉⠀⣀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠸⡄⠈⢆⠀⠈⢺⠁⢠⠋⠀⢠⠊⠀⡠⢔⡩⠠⠄⢑⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣙⢦⠈⠣⡀⢂⢀⠇⠀⢰⠃⠀⠌⡐⠁⠀⢀⣤⢂⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣬⣽⣾⡀⠀⢸⠀⠀⡀⠑⠤⡔⣇⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣤⣤⠬⠽⠖⠛⠁

⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠠⡄⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠉⠉⠐⠄⠀⠀⢀⣀⣀⣀⣀⣤⣄
⠀⠀⠀⠀⠀⣞⠁⠀⠀⢱⡄⠠⡎⢀⠊⠢⡀⠀⠀⣀⠀⢀⠃⠀⠀⠀⠀⣤⣘⣒⠩⠭⠍⠀⠲⣖⠐⡏
⠀⠀⠀⣴⠒⣎⠲⠤⣀⡐⣣⣢⢞⣦⠘⣧⠈⠲⡎⣴⣿⣾⠀⠀⠀⠀⠀⠈⡏⢸⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠈⠉⠁⠀⠀⠈⢮⣊⡵⣿⣽⡷⡄⠀⢠⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⣧⠸⠀⠀⠄⠀⠀⣘⣀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⢮⢞⢡⡾⣟⣤⠀⠱⣄⠣⠀⠀⠀⠀⠀⠀⢀⡽⠤⠷⠄⠛⢋⣉⠭⢤⣔⣃⡀
⠀⠀⢀⠀⠀⠀⡀⠀⢴⡦⠀⠀⠑⣥⢞⣫⡾⣣⡀⠘⠀⢣⡀⠀⠀⢀⣀⠀⠁⠉⠉⠀⠀⠀⠀⠀⠀⢿⣿⣿
⠀⠄⠃⠀⠀⠀⢀⠴⠐⠂⠂⠉⡂⠈⢯⣕⣾⣯⣷⡀⠀⠈⢏⠐⠒⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿
⠣⠀⢀⠀⠀⠀⠐⠢⣄⠀⠀⢠⡃⠒⡘⢻⣫⣵⡾⢯⠀⣛⠄⠀⣀⠪⣆⠼⠂⢤⣠⣄⠰⠾⡟⠻⠿⠟⠛⠋
⠀⠀⠈⠚⠢⡀⠀⢀⣜⠿⡅⡍⠙⠼⠁⠒⢻⣷⣾⠋⠁⠀⡠⠔⠣⡻⡀⠩⠄⢈⠛⣿⣧⡀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢨⠓⠂⠒⢴⠀⣿⣿⢋⠥⠒⠁⠀⢀⣀⣅⡱⢰⡈⣷⠀⠘⡿⠁⠀⠀⠀⠀⣀⣐⡄
⠀⠀⠀⠀⡠⠔⠂⠀⠉⠀⠁⠀⠀⠀⢨⣀⠟⠋⠁⢀⣠⠔⠋⠁⠀⢀⣀⠈⠁⠀⠀⢠⡅⡇⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⡜⠀⠀⠀⠀⢀⠀⠀⠀⠀⡴⠉⠀⣀⠤⠒⠥⣀⠀⠀⢀⣾⠟⠿⣽⣆⠀⠀⠈⠉⠀⠀⠀⠀⢰⣤⡀⠛
⠀⠀⠀⡀⢀⡄⠀⡇⠀⠙⠀⠀⠀⠘⢦⣍⣀⠀⠀⠀⡀⠑⠀⠈⢿⡗⢠⣿⡜⠁⠢⡀⠀⠀⠀⠀⠀⣾⣿⡿
⠀⠀⠀⢣⠸⣿⠀⠈⠒⠊⠀⠀⢀⠀⣀⣨⣿⣿⣿⡶⢤⣄⡀⠀⠀⠉⠛⠙⠂⠀⠃⠴⠑⢄⠀⠀⢀⠛⠋⠁
⠀⠀⠀⠘⠦⡙⠁⠀⠀⠀⠀⠐⠁⠀⣿⣿⠿⣿⣿⣀⠀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢵⣄⠊⠀⣀
⠀⠀⠀⠀⠀⠈⠁⠒⠒⠂⢁⠤⠐⠊⢁⡀⠀⠘⢿⣿⣿⣿⣿⣧⣤⣴⠖⠈⠁⠈⠠⢄⠀⠀⡒⠒⠵⢆⠀⠱⠄⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⡌⠀⠀⠈⣆⠈⢿⣿⣿⣿⠟⠋⡤⠒⠒⠂⠀⣶⡶⠁⠶⣅⠀⠀⢀⠇⠀⠀⠘⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠐⠄⣀⡀⠍⠀⠈⢿⠛⢁⣀⣀⡈⢆⡎⠀⠀⠉⠀⠀⠐⠤⠈⡉⠁⠀⠀⠄⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⢀⠤⠄⠀⠀⢂⠁⠒⠒⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⢑⣁⣤⣴⡾⠀⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣿⣟⣛⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣹⣛⣝⣛⣟⣽⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠞⠛⠻⢶⣤⡀⠀⣠⡶⠞⠛⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠈⠛⠟⠉⠀⠀⠀⠀⡀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠸⣟⠻⠷⣶⡶⠾⠿⢶⣶⠶⠟⢻⡟⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢿⣤⣠⣿⠀⠀⠀⠀⢹⣆⣠⡿⠁⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⡀⠀⠀⠀⠈⠙⢿⣄⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⠻⠶⢶⣤⣤⣀⣉⠛⠛⢋⣀⣠⣤⣶⠶⠟⠛⠋⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠉⠁⣠⣤⡀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠾⠋⠙⢷⣤⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡃⠈⠙⠛⠻⠶⠶⠶⠾⠛⠛⠉⠀⠀⠀⠀⠀⠈⠙⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⠀⠀⠀⠺⠂⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⢀⣴⠏⠀⠀⣤⣤⡀⠀⢀⣠⣴⠟⠋⠁⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠘⢿⡌⠛⠿⠛⠉⣀⡄⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⠀⠀⠀⠀⠈⢿⣄⣠⣴⠿⣿⠃⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠹⣧⡀⠀⠀⠀⠐⠷⠶⠆⠀⠀⠀⠀⣰⡟⠹⣧⠀⠀⠀⠀⠀⠙⠉⠀⣼⠏⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠘⢷⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠙⠷⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢸⡟⠻⠶⠶⠶⠶⠞⠛⣏⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣀⣀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣀⣀⠀⠸⣧⠀⠀⠀⠀⠀⠀⣀⣤⣤⡴⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣼⡷⠾⣿⠉⠉⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⠉⢹⡿⢶⣿⡀⠀⠀⠀⣴⡾⠋⠁⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⣿⡀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⣿⠁⠀⠈⢿⡄⠀⣸⣏⣀⣀⠀⢀⣴⠾⠛⠁
⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⢹⣧⣄⣀⠀⠹⣧⣤⣤⣤⣤⣴⡟⠀⣀⣠⣤⡿⠀⠀⠀⠈⣿⣠⡿⠉⠉⠛⠛⣿⠁
⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠉⢹⡗⠀⠘⣷⡀⠀⣼⠏⠀⠸⣟⠉⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣷⠀⣿⣄⠀⠀⠈⢷⣾⠋⠀⠀⢀⣿⠆⢸⡆⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⣾⠃⠀⠀⠙⠷⣦⡀⢸⡇⢀⣠⡾⠛⠁⠀⠀⢿⡄⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠛⢿⣿⠟⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⣾⠁⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣾⣿⡿⠆⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢾⣿⣿⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⢻⡆⠀⣾⠃⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⣸⡇⠀⠀⠀⠀⢀⡟⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⢸⡇⢀⣿⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⣿⠁⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠘⣷⠈⣿⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⣿⠀⠀⠀⠀⢀⣿⠁⢸⣇⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⣇⠀⣿⡀⠹⣧⣀⠀⣀⣴⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠀⠒⠒⠀⠀⠀⠀⠒⠒⠀⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠉⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⣀⠠⠐⠒⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠤⡀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⢠⠂⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠐⡄
⠀⠀⠀⠀⠀⢀⠃⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠐⡀
⠀⠀⠀⠀⠀⡈⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢀
⠀⠀⠀⠀⢠⠁⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⡄
⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢠⠀⡆⠠⠐⠉⠀⠱⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠎⠀⠉⠂⠄⢰⠀⡈
⠀⠀⠀⠀⠈⡄⢸⠑⡀⠀⠀⠀⠐⡄⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢠⠂⠀⠀⠀⢀⠊⡇⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠈⢄⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡠⠁⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠈⡀⠄⠀⠈⢆⠀⠀⠀⠈⢂⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⡐⠁⠀⠀⠀⡰⠁⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠇⠈⠄⠀⠈⢢⠀⠀⠀⠀⠣⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⡜⠀⠀⠀⠀⡔⠁⠀⠠⠁⠸
⠀⠀⠀⠀⠀⠀⠘⡄⠈⢆⠀⠀⠡⡀⠀⠀⠀⠱⡈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠎⠀⠀⠀⢀⠌⠀⠀⠠⠁⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⢆⠀⠀⠱⡀⠀⠀⠀⠐⡌⢢⠀⠀⠀⠀⠀⠀⠀⠀⡔⢠⠂⠀⠀⠀⢀⠊⠀⠀⡰⠁⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠢⠀⠀⠘⠄⠀⠀⠀⠘⢄⠡⡀⠀⠀⠀⠀⢀⠌⠠⠁⠀⠀⠀⠠⠂⠀⢀⠔⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠑⡀⠀⠈⢆⠀⠀⠀⠈⢂⢡⠀⠀⠀⠀⡌⡐⠁⠀⠀⠀⡰⠁⠀⢀⠊⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢈⢆⠀⠀⢢⠀⠀⠀⢀⠏⠀⠀⠀⠀⠹⡀⠀⠀⠀⡔⠀⠀⣰⠁⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠒⠤⠵⠐⠊⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠮⠤⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠾⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠳⠶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⢭⣥⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣜⣣⣯⢪⣕⣭⠿⣭⢪⡵⢯⠷⣭⢪⡽⣯⡷⣭⢺⡽⢿⣷⣣⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡵⢮⡇⠀⢴⡄⠀⢠⡤⠀⠀⠀⠀⠀⢠⡆⠀⠀⣤⠀⢸⣯⢮⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠼⢯⣝⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣽⠿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡬⣼⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣭⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠰⢈⡿⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢸⣿⡇⠶⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⢛⣡⣵⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣮⣅⡛⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠴⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠦⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⢸⠀⠀⡄⠀⠠⠀⠀⡐⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠑⠄⠀⠀⢀⠀⠈⠄⠀⠐⠀⠀⠀⠀⢸⠀⢀⠀⢀⠂⠀⠠⠀⠀⠐⠀⠀⡀⠁⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠠⡀⠀⠂⠀⠈⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠄⠀⡠⠊⠀⠀⠀⠊⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠑⠀⠀⠀⢀⠀⠠⡀⠀⠀⠀⣀⢠⠀⡤⢠⠄⠀⠠⢄⠀⠀⠀⢀⠄⠀⢀⠄⠀⠀⢀⠄⠈⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠈⠐⠠⢀⠀⠀⠁⠠⠀⠀⠈⠢⡀⠑⠀⠀⠀⠘⠃⠃⠈⣀⣁⣀⣀⣀⣀⢀⠑⠄⠐⠁⠀⠔⠁⠀⠀⠀⠁⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠑⠠⡀⠈⠀⠀⠀⠰⠖⢛⡉⢀⣭⣭⣥⣬⣭⣽⡀⠳⠖⠀⠀⠀⢀⠄⠈⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠄⡀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠸⢛⣉⡁⢠⣬⣤⣤⣤⣤⣯⠀⡛⢏⠀⠠⠂⠀⠀⠀⠁⠀⠠⠀⠂⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠂⠀⠀⠈⠀⠠⠀⠀⣛⣋⣩⠀⢠⣤⣤⠤⣤⣤⣭⠐⡙⢃⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⣩⣤⠀⢴⡶⠶⠶⢶⣦⣬⢀⣙⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣥⡤⠀⢴⣶⠶⠶⢶⣶⣦⢈⣙⡃⠀⠀⠀⠀⠀⠀⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠍⠀⠀⣥⣤⠦⠀⠶⠶⠶⠶⠶⢶⡄⢨⣍⠄⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⢀⣤⠦⠶⠀⠶⠶⠶⠶⠶⠶⠆⢠⡭⠀⠀⠈⡀⠀⠀⠀⠄⢀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠈⢄⣆⢷⢸⡄⣧⢸⡆⢀⠀⠀⣀⠄⠀⠀⠂⠄⡁⠀⠀⠄⡀⠀⠁⠐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⡠⠐⠈⡠⠂⡀⠀⠙⢫⠃⠛⠘⠣⢟⡄⢐⠊⠀⠀⠀⠀⡁⠀⠀⠑⠠⢀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠠⠀⠀⢀⠔⠁⡀⠈⠀⠀⠀⠘⠀⢲⣖⠀⢸⣥⡌⢀⠀⠀⠈⢂⠈⠀⡀⠐⠄⠀⠈⠀⠄⡀⠀⠀⠁⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⡠⠂⠁⠀⠀⠀⠔⠀⡔⠀⠀⠑⡀⢀⡣⠐⠋⠀⠀⠂⠀⠢⠀⠑⡀⠈⠂⠀⠀⠂⠀⠀⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠔⠊⠀⢀⠔⠀⢀⠈⠀⡐⠀⠠⠀⠐⢉⡉⡗⠈⡁⠀⠀⠀⠄⠀⠡⠀⠈⢀⠀⠀⠂⠀⠀⠑⠠⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⢀⠔⠁⠀⡀⠁⠀⡜⠀⠀⠀⠀⠘⣿⠇⠃⠀⢁⠀⠀⠂⠈⠄⠀⠐⡀⠀⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠃⠛⢀⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡇⢀⣸⠀⠀⠀⠄⠀⠀⠀⠀⠀⢄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣹⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠓⢦⠀⢀⡴⠚⠋⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠈⣹⣋⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠔⠁⡌⠙⠲⠤⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣆⠀⣇⠀⡆⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠏⠀⣿⠀⠹⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠈⠀⠀⠁⢠⠞⠉⠀⠈⠙⠄⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⢀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⢀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠙⢦⡀
⠀⠀⠀⠀⣠⠋⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠹⣄
⠀⠀⠀⡼⠁⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠈⠳⡀
⠀⢀⡞⠀⢀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠙
⢀⠈⠀⢠⠃⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢀
⠁⠀⣰⠃⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠔⠃
⢀⡴⠁⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⢀⠀⡇
⠘⢦⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣫⠴⣋⠉⣷⠹⡎⣸⠁
⠀⠀⠉⠓⠤⢀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⢀⡠⢖⡺⠭⣉⠓⢄⡀⣰⢋⡜⠁⠀⠈⢷⣇⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⣋⠥⠒⠁⠀⠀⠈⢇⣀⣉⠵⠋
⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⣉⣉⣁⡀⠀⠉⠉⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⣷⣛⣋⣉⣉⣉⣿⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠉⢉⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⠤⣄⠀⠀⣠⠤⣄⠀⠀⣠⠤⣄⠀⠀⣠⠤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡇⠀⣿⠀⠀⡇⠀⢹⠀⠀⡏⠀⢹⠀⠀⡏⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣇⠀⣿⠀⠀⣇⠀⣼⠀⠀⣇⠀⣸⠀⠀⣇⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢛⣁⠀⠀⠙⢛⣃⠀⠀⠙⣛⡋⠀⠀⢙⣛⠛⢧⠞⣛⡓⢦⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣾⠉⣷⠀⢰⡏⠹⡇⠀⣼⠉⢹⠀⠀⡏⠉⣧⠀⢰⠋⢹⡆⢻⡀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣿⠀⣿⠀⢸⡇⢸⡇⠀⣿⠀⢸⠀⠀⡇⠀⣿⠀⢸⡀⢸⡇⠘⡇⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠶⠋⠀⠈⠳⠞⠁⠀⠘⠲⠛⠀⠀⠛⠾⠃⠀⠈⠳⠞⠁⠀⡇⠀⠀⢰⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⡴⠲⣆⠀⣰⠚⣦⠀⢠⡖⢶⡀⢀⡶⢳⡄⠀⣴⠒⣆⠀⣰⣿⠀⠀⡿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡇⠀⣿⠀⣿⠀⣿⠀⢸⡇⢸⡇⢸⡇⢸⡇⠀⣿⠀⣿⠀⣿⠘⢶⠴⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠻⠴⠃⠀⠹⠦⠟⠀⠘⠧⠞⠁⠈⠳⠼⠃⠀⠻⠴⠏⠀⠘⠦⠟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡴⢦⡀⢀⡴⢦⡀⢀⡴⢦⡀⢀⡴⢦⡀⢀⡴⢦⡀⢀⡴⢦⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⠷⠼⠃⠈⠧⠼⠃⠈⠧⠼⠁⠈⠧⠼⠁⠘⠧⠾⠁⠘⠧⠾⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⣤⡄⠀⣠⣤⡀⠀⣠⣤⠀⢀⣤⣄⠀⢀⣤⣄⠀⢠⣤⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣿⠀⣿⠀⣿⠀⡇⠀⡇⢸⡇⢸⡇⢸⠀⢸⠀⣿⠀⣿⠀⡷⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠹⣤⠟⠀⠻⡴⠇⠀⠷⡼⠃⠘⢧⡾⠀⠸⢦⠟⠀⠻⣤⠏⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠉⠙⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣋⣀⡀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠉⠉⠉⠙⠓⢦⣄⠀⠀⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣤⡀⢸⡧⣤⠄⣀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠈⠛⢿⣱⣟⡀⠀⠙⢦⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣄⡀⠀⠀⣠⠋⠀⠈⠑⠦⣾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠈⠙⢲⡔⠁⠀⠀⠀⠀⠀⠀⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢦⣀⡀⠀⢏⡀⠀⣀⠴⣳⡄⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠉⠓⠦⣍⡉⠀⣠⠋⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠦⣄⡀⠀⠀⠀⠀⠉⢻⡇⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠓⠦⢄⣀⣀⡜⣇⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠢⣄⡀⠀⠀⠀⢙⠁⠁⠘⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⢮⠏⠙⠒⡶⠋⠀⠀⠀⠀⠀⠠⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⠏⠀⠀⣼⠁⠀⠀⠀⠐⠀⡀⠠⣖⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⢤⣀⣼⠟⠻⢶⡄⠀⠒⠚⢳⠀⠹⡆⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢢⡴⠟⠋⠀⠀⠀⢿⡄⠀⠀⠀⢣⠀⠙⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣌⢿⣄⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⢿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠉⢩⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠒⠋⢉⣀⡤⠴⠖⠋⣻⠙⠲⢤⣀⠉⠛⠲⢤⣄⡀
⠀⠀⠀⠀⢀⣠⠤⠴⠒⠚⠉⢁⣀⡤⠴⠒⠋⠉⠀⠀⠀⠀⢀⡇⣠⣤⣄⣈⡉⠓⠢⢤⣀⠉⠓⠦⣄⡀
⠀⣤⠒⠊⢁⣀⡤⠤⠖⠚⠉⠉⠀⠀⠀⠀⢀⣠⠴⠒⠚⠓⢺⠟⠁⠀⠀⠀⠙⢦⣀⣀⡈⠙⠢⣄⣈⡩⠗
⠀⣇⡉⠚⠯⣄⣀⠤⠴⣤⡴⠖⠲⣄⣠⠖⠋⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢀⣀⣠⢬⡭⠟⠋⠉⣀⠤⢺
⠀⢹⡈⠉⠲⢤⡙⠲⢤⣄⡀⠀⠀⠈⠣⣤⠤⠤⠤⠤⠤⣤⣈⠑⣲⠤⠖⠊⠙⡇⠀⠀⢷⠔⠚⠉⠀⠀⢸
⠀⠈⣇⠀⠀⢰⣿⠑⠢⢌⡙⠶⣤⣀⡀⠈⠉⠉⠁⠀⣀⢴⡛⠉⠀⠀⠀⠀⠀⢱⠀⢠⢸⠀⠀⠀⠀⠀⡟
⠀⠀⢹⠀⠀⠀⢿⣄⡀⠀⠉⠲⢤⣈⡙⠳⢦⡤⠒⢉⣠⠴⡇⠀⠀⠀⡀⠀⠀⢸⠀⣸⢸⡆⠀⠀⠀⢠⡇
⠀⠀⠸⡆⠀⠀⠀⠙⠛⠷⣤⣀⡴⠁⠉⠓⠲⡗⠋⠁⠀⢰⡇⠀⠀⡸⢀⠀⠀⢸⠀⠋⠀⡇⠀⠀⠀⡼
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⢀⠇⡌⠀⠀⢸⣀⡤⠴⠇⠀⠀⢠⠇
⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣾⠀⠀⠘⠀⠀⠀⢀⡼⠁⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠈⠓⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡿⠀⣠⠴⠒⠒⠋⢉⣀⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠛⠋⠁⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⠀⠀⡇⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠧⠖⠚⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣆
⠀⠀⠀⠀⠀⢀⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣇
⠀⠀⠀⠀⠀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡆
⠀⠀⠀⠀⢰⠟⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⢷
⠀⠀⠀⠀⡈⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⡄
⠀⠀⠀⠀⣧⡄⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣤⡇
⠀⠀⠀⠀⠃⢠⠸⠀⠀⠀⢀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⠀⠀⠀⠸⠀⠀⠁
⠀⠀⠀⠀⢸⠈⡀⢀⡖⠚⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠒⣆⠀⡈⢰
⠀⠀⠀⠀⠀⠆⠡⢹⠁⠀⢻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠂⠀⢹⠡⠁⡆
⠀⠀⠀⠀⠀⠐⣤⠿⠀⠀⠘⢿⣿⣿⣿⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢿⣿⣿⣿⠇⠀⠀⠸⢦⡔
⠀⠀⠀⠀⠀⠀⣠⢘⡇⠀⠀⠘⣿⣿⣿⣇⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⣿⣿⣿⡟⠀⠀⠀⣞⢠⡁
⠀⠀⠀⠀⠀⠀⢸⡏⠑⡀⠂⠀⢽⣿⣿⣿⣦⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣾⣿⣿⣿⠅⡀⠀⡐⠉⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣦⡳⣜⣷⣬⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣯⣴⣟⡼⣇⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠛⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⡇⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⢸⣿⣿⣿⣿⣿⡇⣿⣿
⣿⣿⡇⣿⣿⡟⣿⣿⣿⣼⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⣾⣯⣽⣟⣛⣛⣛⣛⣿⣯⣽⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⡇⣿⣿⡇⣿⣿
⣿⣿⣇⣿⣿⣧⣿⣿⡏⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⢿⣷⣾⣭⣭⣭⣛⣻⣭⣭⣽⣶⣿⢿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣟⣿⣿⣧⣿⣿⣧⣿⣿
⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣯⣿⣿⣿⠃⠉⠻⢿⣿⣶⣯⣭⣝⣛⣛⣛⣛⣯⣭⣷⣾⣿⡿⠛⠁⢻⣿⣿⣯⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿
⣿⣿⣿⣿⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠉⠁⠀⠀⠀⠀⢻⣿⣿⣼⣿⣿⣿⣿⣿⡾⣿⣿⣿⣿
⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿
⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣻⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿
⣏⡛⠻⠻⠿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠿⠻⠛⣉
⣿⣿⣷⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣿⣿
⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⢛⡛⢟⡛⠛⡟⣛⡿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⠀⠀⢀⢄
⢠⣄⡰⠂⣈⡥⠂⠀⠶⣟⣀⢄⡾⠛⣁⣀⣤⡀
⡀⠀⠀⡩⠟⠿⠿⠷⠶⠾⠷⠾⠟⠋⠉⠙⣋⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠏
⠈⠉⠐⠠⠤⢄⢀⣀⡀⠀⠀⠀⣀⡀⠀⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠠⠤⠀⠠⣠⠤⠄
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢿⡏⠀⠀⠀⡠⠔⠶⠁⠐⠐⠈⠀⠀⠒⠻⣷⣬⡁⢀⣀⡉⢈⣺⡷⢂⢤⡀
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⢿⢢⠀⠀⠈⠁⠀⠀⠀⠐⠀⠀⠀⠀⠘⠂⠌⠉⠄⠄⠀⠀⠀⠈⠙⡎⠡⠆
⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡋⠀⠀⠀⠀⠀⠲⠆⢤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⣐⡀⣬⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⣽⡆⠀⠀⠀⠀⠀⢙⡂⠲⠤⠀⠀⠀⠀⢈⣉⣉⢨⣭⠍⠬⠤⠄⣒⣚
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣿⡇⠀⠀⠀⠀⠀⠀⠭⢘⡒⠀⠀⠀⠀⠠⠥⠤⠐⢒⡂⣒⣛⢉⣭⠥
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣯⡅⠀⠀⠀⠀⠀⠀⢑⡠⢭⠀⠀⠀⠀⢘⣛⣃⢨⠭⠄⠤⠶⠐⣒⡂
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠰⠶⠆⠐⠒⠀⠛⠁⠈⠉
⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣭⣿⣯
⠀⠀⠈⠀⠠⠀⠒⠀⢤⠴⠶⢾⣟⠹⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⠀⠀⠀⠀⠀⠀⣔⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣮⣟⡵⠃⠀⠀⢀⡀⠈⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠲⠀⣀⢀⡠⠄⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣸⠆⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡴⠛⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠧⠤⠀⠀⠀⡀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣻⡿⣿⣷⣄⠀⠀⣠⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢡⢟⠃⠈⣼⣿⣿⣿⣿⣷⣌⣃⠘⠶⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣿⣁⢛⣛⡤⢴⣿⣿⣿⢯⣻⣿⣭⣛⠻⠿⠾⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⡀⠉⣉⣠⡴⣻⣿⣿⢏⡜⣿⣿⣿⣿⣻⠷⢶⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠟⢋⣵⣿⡿⣹⠏⡼⣿⣿⣾⣿⡟⣿⣏⠳⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢉⡾⣿⠿⠟⠋⡟⢡⡿⢸⡿⠿⠀⠁⢹⣯⠈⢦⡀⣼⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⡞⢹⠇⠀⠀⢰⠥⣾⠇⠘⠳⠐⡀⢨⠈⠉⣇⠀⢳⡈⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢈⣿⠿⢲⡞⠛⠟⣴⣿⠀⢀⡀⢀⠁⠀⠄⠐⣸⡄⠀⠹⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠋⠹⠉⠀⣿⡟⠀⢜⠁⠘⠂⠀⢁⣀⡥⢻⡀⠀⠘⣄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡼⠀⠀⢰⣿⡇⠀⠘⠁⠐⠀⢀⢐⠀⠀⠈⢧⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣼⢸⣇⠀⠰⠆⠀⡄⠀⠘⠀⠓⠀⠸⣄⠀⠀⢨⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢸⠈⣿⠀⠀⢲⠀⡀⠀⢀⠀⠀⡄⠀⢹⠀⢀⣸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⣃⣿⣧⢀⠈⣀⠁⠀⠈⠀⠀⣀⠀⠸⣷⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠸⣿⣿⣿⣦⣴⣿⣧⠀⠐⠂⠀⠈⠀⠀⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⡾⠀⠀⠀⠚⠀⠃⠀⠀⠀⠀⠀⠀⠉⠙⡿⢿⠟⠁⠀⠀⠀⠀⠀⠀⠸⡙⠁
⠀⠀⠀⠀⠀⠀⣀⡖⡯⡘⣿⠃⠀⠠⡇⠈⠋⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢷
⠀⠀⠀⠀⢠⡾⢛⡰⢖⣳⠎⢀⡀⡁⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⣠⣼⣧⠀⠀⠀⠀⠀⠀⢱⣄⣄⡀
⠀⠀⠀⢤⠁⠀⠨⠅⠉⠀⡰⠾⠆⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢰⣄⣤⣤⣄⣤⣀⣤⣤⣦⣿⣾⣿⣿⣿⣿⣆⠀⠀⠀⢠⣠⣿⠟⠀⠀⢠
⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⢯⣿⣸⣗⣩⣿⣯⢻⣿⡿⠿⣟⡻⠾⠛⠋⠁⢀
⠀⠀⠀⠀⠁⠂⢀⠀⠀⠀⠀⢀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣉⣿⣿⣿⣻⣿⣽⡼⢿⡟⠻⠏⠙⠃⠀⠀⠀⠸⡥⡤⠤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠲⠤⠀⠠⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣈⣤⣶⣾⣿⠁⢀⠠⡑⠌⠂⠀⠁⠐⢷⠀⠐⣀⣤⣶⣤⡠⠖⠀⠈⡁⠒
⠀⠀⠀⠀⠀⠀⠠⢢⡾⠿⣛⣭⣴⠄⣸⣆⠀⠁⢰⠀⠀⠀⠀⠈⢀⡸⠿⠻⠛⠈⠀⠀⡀⠔⠈⠀⡤
⠀⠀⢀⠀⠀⠀⣰⣶⡾⠟⠛⠭⣶⣿⢿⣿⢁⣢⡴⠌⠄⡂⣡⢾⠟⠁⢀⠀⠀⠀⠀⠀⠀⣠⡤
⠀⠀⣇⠀⠈⠛⢿⠋⠶⠒⢂⠀⠔⣈⠤⠀⡤⡀⠖⠂⠀⠂⠈⣃⡉⠥⠴⠒⠂⣀⠤⠐⠈
⠀⠀⠌⠢⡀⠀⠀⠑⢢⡄⣨⣸⠰⣃⡐⢋⡠⠆⠈⠉⡉⠀⠤⢋⣀⠠⠴⠆⡭⢀⡌⠁⢑⡀⠰⠂
⠀⠀⠀⣀⠈⢦⠀⠀⠀⠈⠣⡀⠑⢤⡤⠀⢒⣀⠈⠀⡤⠶⠚⠋⠀⢠⢒⣡⣤⣤⠋⠀⠻⠰⠓⠄
⠀⠀⠀⠀⠀⠀⣬⣦⡀⡀⡠⠈⢢⠀⡔⠉⠈⠈⢱⠀⠀⠀⠀⠀⠀⡀⡘⠉⠉⠍⠑⢋⣀⣰⣤⣾⠇
⠀⠀⠀⠀⠀⣶⣿⠋⠉⢪⣀⠀⠀⢢⡑⠒⠉⢙⡁⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠘⠝⢿⠃⠁⡀
⠀⠀⠀⠀⢾⡏⠐⡤⠒⠈⠉⠑⢄⠀⠈⠉⠫⣶⣦⠀⡃⢐⠀⠀⠀⠀⠀⢀⣀⠄⠃⠁⣀⣶⡀
⠀⠀⠀⠀⠀⣴⡟⠀⠁⠀⠈⠀⡰⢁⣄⠀⠀⠌⠁⡄⣠⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠛⢿⠇
⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠃⢰⣿⡏⠀⢢⢠⠀⠈⣇⠀⠀⠀⠀⢠⡀⠲⡀⢙⡆⠀⣸⣧⠄
⠀⠀⠀⠐⠉⠃⣌⠢⣀⠀⠀⠀⣀⠦⡙⠛⣁⡜⠁⠠⠐⢍⠀⠀⠃⠃⢀⠀⠀⠚⠁⠒⠂⢹⣿⠄
⠀⠀⠀⠠⠀⠀⠀⠑⢤⡀⠨⣿⣿⡈⠀⡰⠉⠉⡄⢢⠀⠘⢆⠢⠀⠉⠉⠀⡠⠋⢢⡤⠴⠿⠋
⠀⠀⠀⠀⠈⠐⠄⣀⣈⣇⠀⠈⠻⣿⠠⠉⣒⠂⠈⢡⢀⣤⡀⠙⠒⠤⠐⠊⢀⣠⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⡒⡀⠀⠈⠁⠒⠈⠇⠒⠁⠁⠀⠉⠓⠒⠀⠒⠒⠉⡀⠻⣋⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀⡠⢀⠀⡄⠀⠀⣀⣄⠢⠀⠀⠀⠢⠀⠀⢠⠄⠻⠿⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠀⠐⠊⢲⡾⠁⠻⠇⠢⠠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⢤⣤⢤⡔⢲⡖⠲⣖⠺⣿⠒⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⣠⠊⠀⠈⣷⠀⣿⠀⣿⠀⣿⠀⢿⡀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢠⠃⠀⠀⠀⠛⠀⠙⠀⠉⠀⠈⠀⢀⣀⣀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣠⣼⣤⣤⠶⠶⠶⠶⠶⠀⠀⠀⠛⠛⠋⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⢈⣄⣀⣀⣠⣤⣤⣤⠄⠀⠀⠶⠶⠶⠾⠛⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠛⠉⡏⠉⠉⠁⠀⣀⣀⡀⠀⠀⢠⣤⣤⣤⣤⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠶⠶⠿⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⣀⣀⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠤⢤⣠⣤⣾⡶⠶⠶⠶⠶⠟⠓⠀⠀⠈⠋⠉⠉⠉⠉⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⣶⣾⣿⢀⣸⣀⣠⣤⣤⣤⣤⣦⠀⠀⠀⠶⠶⠟⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⢈⣿⣿⡿⠋⠉⡍⠉⠀⣀⣀⣀⣀⡀⠀⠀⣤⣤⣤⡶⠶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠛⢻⣿⠶⠿⡟⠛⠛⠛⠉⠉⠉⠀⠀⠀⢀⣀⣀⣀⣠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣷⣤⣤⣷⠶⠶⠶⠞⠛⠛⠃⠀⠀⠈⠉⠉⠉⠉⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⢀⣀⣰⣤⣤⣤⣤⣴⠶⠦⠀⠀⠐⠛⠛⠛⠛⠛⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣯⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣯⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⢤⣀⣀⣀⣉⣲⣤⣤⠤⠤⠤⢤⠤⡶⢲⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⣯⠀⠀⣀⣀⣀⠠⠤⠤⠄⠒⢒⣉⠔⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⡀⠛⢿⣭⡤⠄⠒⣒⣒⣈⠩⠭⠥⡔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⠉⡏⠉⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣇⡠⠤⠤⠒⠒⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣸⣀⡀⠤⠤⠒⠒⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡏⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⣇⣀⠤⠤⠐⠒⠒⠉⠁


⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣼⣿⣼⣼⣧⣿⣤⣼⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣶⣾⣷⣾⣿⣷⣶⣶⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠠⠜⠉⢆⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⠚⠒⠈⡐⡆⢄⡀⠀⡐⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⡤⠔⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠉⡠⠔⢀⡀⠀⠀⠛⢶⢇⠀⠀⠀⠀⠈⠄⠀⠈⠂⠔⠉⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢄⣀⡠⠤⠤⠤⠇⠚⠁⠀⢠⡞⠉⠉⠁⠀⠀⢂⠁⠀⠀⠀⠀⠀⠀⠀⣄⡐⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣔⠩⡒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⡰⢨⠀⠀⠀⠀⠀⠀⢠⣉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⠚⠋⠴⠒⠲⠦⠤⣄⣀⣀⣐⣹⠄⣀⠀⠀⠀⡀⠎⠂⠉⡄⠀⠀⠀⠀⠀⣬⣹⣀⣀⡀⠤⠤⠤⠀⠒⠂⠈⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠈⠉⠙⠛⠋⠋⠉⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢉⢑⠒⠢⠤⠤⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⠒⠈⡿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⠬⠀⠀⡃⠀⠆⠀⢸⠈⠉⠉⠉⠷⢒⡒⠀⠀⣀⠐⠪⡏⠀⢘⠁⠈⠇⠀⢰⠊⠀⠀⠀⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠀⠀⠈⠉⠁⠀⠘⠀⠀⠀⠀⠆⠀⠉⠀⠀⠀⠑⠤⠆⠀⠘⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢿⠀⠀⣏⣰⡆⠀⢠⠤⠤⠀⠀⣀⣀⣀⠀⠀⣀⣀⢀⡀⠀⢠⠤⠀⡄⠀⢰⣉⠽⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠸⠀⠀⠇⠈⠗⠀⢨⠀⠀⠀⠀⡀⠀⠘⠀⠀⡁⠀⠀⠀⠀⢀⠀⠀⠁⠀⢰⠀⠘⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠢⠀⠀⠃⠀⠈⠀⠀⠁⠁⠂⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⢠⠤⢤⠀⠀⣀⣀⣀⠀⠀⣀⣀⣀⡀⠀⢠⠤⠤⠄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⡁⠀⢸⠀⠘⠀⠀⠗⠀⢠⠀⠀⡅⠤⠤⡄⠀⠸⠒⠒⠃⠀⢠⠤⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠉⠃⠀⠈⠀⠀⠀⠀⠃⠀⠠⠀⠀⠇⠀⠠⠆⠀⠀⠀⠀⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠐⠂⠀⢠⠤⣄⠀⠀⣀⣀⣀⠀⠀⣀⠀⠀⠀⠀⢠⠤⠤⠄⠀⢰⠈⢹⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⡇⠀⢨⠀⠈⠀⠀⡁⠀⠈⠀⠀⠗⠀⠠⡄⠀⠘⠀⠀⠀⠀⢸⢀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⠐⠃⠀⠈⠀⠀⠀⠀⠓⠠⠨⠀⠀⢤⡤⡤⠄⠀⠘⠀⠀⠁⠀⠈⠉⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢰⡒⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢰⠒⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡝⠆⠀⠸⠀⠘⠀⠀⠆⠀⢴⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠃⠀⢨⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠴⠀⠀⢤⣀⣰⠀⠀⡍⠀⠰⠇⠀⠰⠤⠤⠃⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠠⣄⠀⠀⠀⢀⡀⢀⠀⠀⣀⣀⣀⡀⠀⠀⠤⠤⡄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠈⠉⠀⠀⠈⠀⣻⠀⠀⡏⠀⠈⡁⠀⠠⠄⠀⠁⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⠤⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠄⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⢀⡠⠊⢀⠔⢀⡴⠁⡀⠠⠤⣉⣒⠢⠤⠤⠤⠤⠤⠄⢀⣀⡠⠤⠒⢀⠹⣄⠐⢦⡈⠢⣄
⠀⠀⠀⠠⠔⠋⣠⠖⢁⡴⠋⠀⠀⠈⠓⠤⣀⡈⠉⠉⠉⠉⠉⠉⠉⠁⢀⣠⠤⠚⠁⠀⠈⠳⣄⠙⢦⡈⠓⠤
⠀⠀⠀⣀⣀⣭⠴⠚⢹⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠒⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⢸⠙⠲⠬⣅⣀⣀
⠀⠈⠉⠁⠀⠀⡀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⡀⠀⠀⠉⠉
⠀⠀⠀⢀⠀⢠⠇⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢳⠀⢠
⠀⠀⠀⢸⠀⡞⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⣆⠘
⠀⠀⠀⡆⣼⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢼⡄⡇
⠀⢀⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣄⠂
⢰⢛⢋⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢉⠛⢳
⢸⢸⠘⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⢀⢸
⢸⢸⠀⡄⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢸⢀
⢸⠀⠀⡇⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⡃⠀⢘
⠀⠀⠀⡇⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡇
⠀⠀⠀⠃⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣯⣽⣻⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠻⣿⣿⣿⣷⡄
⠀⠀⠀⢠⣿⢯⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡽⣿⡄
⠀⠀⢠⣿⢏⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡹⣿⡄
⠀⠀⣾⣿⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿⣷
⠀⢸⣿⣹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣏⣿⡇
⠀⣿⡇⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣷
⢀⣿⣇⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣸⣿
⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⢹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠈⠻⣿⠿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣸⠿⣿⠟
⠀⠀⢻⣷⣼⡗⠀⠀⢀⠀⣴⣿⣿⣛⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣛⣿⣿⣦⠀⡀⠀⠀⢾⣧⣾⡟
⠀⠀⢀⣿⣿⠃⣴⠾⠋⣸⣿⣭⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣭⣿⣇⠙⠷⣦⠘⣿⣿⠁
⠀⠀⠘⠟⡥⢰⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⡆⢬⡻⠃
⠀⠀⢠⣾⡇⣾⣿⣿⢸⣿⣉⣿⣿⣿⡏⣽⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢻⣿⣿⣿⣯⣿⡗⣿⣿⣿⢸⣿⣄
⠀⢠⣿⣿⡇⣿⣿⣿⢸⣿⠛⣿⣿⣿⢰⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⠿⠛⣿⣿⣿⠟⣿⣇⣿⣿⣿⢸⣿⣿⣆
⢠⣿⣿⣿⡇⣿⣿⣿⠘⣿⣞⣿⣿⣿⢸⡟⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢻⡇⣿⣿⣿⣿⣿⠇⣿⣿⣿⢸⣿⣿⣿
⢸⣿⡿⣿⡇⣿⣻⢻⡆⣿⣯⣿⣿⣿⢸⣷⣌⣿⡟⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢻⣿⣡⣾⣇⣿⣿⣿⣽⣿⢰⡿⡿⣿⢸⣿⢿⣿
⠸⣿⣷⣿⡇⢿⡿⡞⣧⢹⣿⣽⣿⣿⡇⣿⣿⣿⣷⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣾⣿⣿⣿⢸⣿⣿⣯⣿⡏⣸⢷⢿⡿⢸⣿⣾⣿
⠀⣿⣿⣿⣷⢸⣷⣧⣿⡘⣿⣿⣿⣿⣷⢹⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⡟⣾⣿⣿⣿⣿⢃⣿⣼⣾⡇⣾⣿⣿⣿
⠀⢹⣿⣿⣿⡈⣿⣿⣿⣧⢻⣿⣿⣿⣿⡜⣿⠟⣿⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⠻⣿⢷⣿⣿⣿⣿⡟⣸⣿⣿⣿⢡⣿⣿⣿⡏
⠀⠈⣿⣿⣿⣧⢹⣿⣿⣿⡌⣿⣿⣿⣿⣧⢻⣴⣿⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣿⣦⡿⣼⣿⣿⢿⣿⢣⣿⣿⣿⡏⣼⣿⣿⣿⠁
⠀⠀⠸⣿⣿⣿⡄⣿⣿⣿⣷⡸⣿⣿⣿⣿⡞⣿⣿⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣿⣿⢳⣿⣿⣿⣿⢏⣾⣿⣿⣿⢰⣿⣿⣿⠇
⠀⠀⠀⠘⢿⣿⣷⠸⣿⣿⣿⣧⡹⣿⣿⣿⣿⣮⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣵⣿⣿⣿⣿⠏⣼⣿⣿⣿⠇⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠈⠻⢧⠹⣿⣿⣿⣷⡹⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣿⣿⠏⡼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣬⡻⢿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⡿⢟⣥⣾⣿⣿⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⣠⣾⡿⠋⣠⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢀⣿⡇⢠⣿⣿⡇⢠⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⢿⣿⠟⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣾⣿⣶⣶⣶⣶⣤⣤⣈⠉⠛⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⣀⣨⣿⣅⣀⣈⣉⠉⠙⠛⠿⣿⣶⣾⣿⣿⠟⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠛⣿⡟⠛⠻⠿⠿⣿⣷⣶⣤⣀⣹⣿⣿⣿⣶⣄⡈⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣾⣿⣷⣶⣶⣶⣤⣤⣀⠉⠛⠿⣿⣿⣿⣅⠉⠻⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣄⣠⣿⣯⣀⣈⣉⠉⠛⠻⠿⣿⣷⣾⣿⣿⡟⠿⣿⣦⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢻⣿⠛⠛⠿⠿⣿⣿⣶⣤⣄⣈⣿⣿⣿⣷⣦⣀⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣶⣶⣶⣤⣄⣈⠉⠛⠻⣿⣿⣿⣯⡈⠙⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣬⣿⣅⣀⣉⠉⠛⠛⠿⣿⣷⣶⣿⣿⡿⠛⠿⣷⣤⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢩⣿⠛⢻⣿⣿⠿⣿⣷⣶⣦⣄⣈⣹⣿⣿⣿⣶⣤⡈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⣿⣿⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠉⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡘⣿⡀⢿⣿⡇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣦⣙⣳⣄⠙⠿⣿⣿⣿⣿⣿⣿⣟⣝⣙⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠶⢦⣽⣿⣿⣿⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡗⢰⣿⣿⣿⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⣩⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣇⣻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣙⣛⣫⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠘⡆⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⢻⠀⢀⣀⣴⣡⣀⣠⡄⠀⠀⠀⠘⢶⠶⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠤⡤⠤⠤⠄⢰⠀⠀⠀⠈⢷⠀⠀⠀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣠⣀⢀⠎⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⢸⢁⣼⣿⡿⠀⠀⠀⠀⡾⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠘⢺⣿⡿⠁⠀⠀⠀⢸⠇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⢀⣤⡀⠀⣠⠆⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢿⠀⢄⣀⣀⣀⣀⣀⣀⢠⣇⣾⣿⣿⠇⠴⠃⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⢸⡆⢤⣤⣭⣩⣭⣭⣤⡌⣿⡿⠟⠁⠀⠀⠀⢀⠜⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⣿⣿⣿⣿⣿⣿⢁⡄⣠⣾⣿⡇⣠⡴⠋⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢻⣿⣿⣿⣿⣿⠸⣴⣿⡿⠿⠛⠁⠀⠀⢀⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣉⣿⣾⡿⠃⣀⡴⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⣿⣿⣿⣿⣿⣧⠙⠉⠭⠒⠋⣡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠻⠿⠛⠛⠛⠛⠀⠃⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠛⣷⠛⠛⠉⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠘⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡟⠩⠹⣟⣻⣛⣟⣟⣿⣿⣻⣻⣟⣿⣟⣽⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣠⣄⣀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⠿⠻⢿⣿⣿⠿⠛⢿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⡋⠙⣿⡏⠀⠀⠹⡏⠀⠀⠈⢿⡇⠀⠀⠀⣿⡇⠀⠀⠀⣿⠃⠀⠀⠈⣿⠁⠀⠈⣿⠋⠉⣿⣿⣿⣿⣿
⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣶⣶⣶⣷⣤⣤⣤⣾⣧⣤⣤⣤⣿⣧⣤⣤⣤⣿⣦⣤⣤⣴⣿⣶⣶⣶⣿⣿⣿⣿⡿⡟⠋⠁
⠀⠀⠀⠀⠈⠉⢿⠁⠈⠹⣿⡿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⡟⠉⠉⡟
⠀⠀⠀⠀⠀⠀⠘⣆⠀⣰⣿⣧⣤⣼⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣤⣼⣿⣇⠀⣠⠇
⠀⠀⠀⠀⠀⠀⣤⣿⡿⠻⣿⡿⠉⠉⢻⡏⣿⡇⢹⣿⠀⠀⠘⣿⡇⠀⠀⢸⡟⢹⣿⠉⣿⠋⠉⢻⣿⠿⢿⣿⣤⠆
⠀⠀⠀⠀⠀⠀⠋⢹⡇⠀⠸⣿⠀⠀⢸⡇⢸⡇⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⡇⢸⣿⢠⡿⠀⠀⣼⡇⠀⢸⣏⡹
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⣶⣿⣧⣀⢈⣿⣾⣷⣤⣿⣀⠀⣸⣿⣇⠀⢀⣸⣧⣼⣿⣼⡇⣀⣴⣿⣷⣴⣾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⣿⣿⡛⢻⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⠛⣿⣿⠁⠀⡝
⠀⠀⠀⠀⠀⠀⠀⢠⣈⣷⡾⣿⣿⠛⠻⣿⠻⣿⠏⣿⠁⠀⠈⣿⠃⠀⠀⣿⠟⣿⡟⢻⡿⠛⢻⣿⠷⣾⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⠀⠀⢻⠀⠀⢿⠀⣿⡄⢸⡇⠀⠀⣿⠀⠀⠀⣿⠀⣿⠀⢸⠇⠀⠸⠁⠀⢹⠻
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⠀⠀⢸⠀⠀⢸⡄⣿⡇⣸⡇⠀⠀⣿⡀⠀⠀⣿⠀⣿⠀⣼⠀⠀⡄⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⠖⣾⣧⣀⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣄⣴⡿⠒⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠿⠿⡿⠿⠿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠿⠛⠿⠛⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠃⠀⠀⠀⠀⠀⠈⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣿⣶⣶⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠈⠙⠛⠿⠛⠛⠁⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠙⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠙⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠈⣿⡿⠋⠉⠉⠉⠉⠛⢿⣿⠿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣠⣿⣆⠀⠀⠀⠀⠀⢀⣸⣿⣦⡄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠛⠋⠹⣿⠏⠛⢷⣄⠀⠀⣤⡾⠋⠹⣿⠃⠀⠛⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⢰⣷⣀⣤⣤⣌⠻⠿⠿⠋⣡⣤⣄⣀⣿⡆⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⣾⣿⠿⣿⣿⣿⣿⡄⠀⠀⠀⣿⣿⣿⣿⡿⣿⡆⠀⠀⠀⠉⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣿⣷⣤⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⢀⣀⣀⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣴⡏⠁⠀⠀⠀⠀⠀⣰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠿⢿⣿⣿⠿⠟⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⢀⣴⠿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⢳⣦⡀⠀⠀⠀⠀⠈⠹⣶⡄
⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠋⠀⠀⠙⢿⣤⡀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⣶⠋⠁⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⢰⣟⠁⠀⠀⠀⠀⠀⠉⠿⣆⡀⠀⠀⠀⠀⠻⣧
⠀⠀⣸⠟⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⡀⢠⣶⣤⣾⣷
⣀⣾⣿⣀⣤⣄⠀⢀⡼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡽⠿⠃⠀⠀⠀⠈
⣿⠃⠀⠈⠉⠉⠰⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⣀⣴⠾⠿⣧⡀⠀⠀⠀⠀⠀⣀⣶⠟⠉⠀⠀⠀⠀⠀⣠⡾
⠛⣶⡄⠀⠀⠀⠀⠀⠘⠻⣦⡄⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀⠀⠀⠀⠀⢀⣀⡀⠀⢠⣾⣿⣿⣆⡀⠈⠿⣄⠀⢀⣴⡾⠛⠀⠀⠀⠀⠀⢀⣴⡞⠋
⠀⠀⠙⠿⣤⣄⠀⠀⠀⠀⠀⠙⠿⣦⡀⠀⣤⠿⠃⢀⣬⣿⣿⣄⠀⠀⠀⠸⠿⠃⠀⠀⠙⢿⣿⡿⢿⣄⠀⠻⣿⣿⡋⠀⠀⢀⣠⣤⠼⠿⠃
⠀⠀⠀⠀⠈⠉⠳⣶⣄⣀⡀⠀⠀⠘⣿⣾⠋⠀⢰⣿⣿⣿⠟⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠀⠈⢻⣦⡀⣿⣿⣿⣶⡶⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢠⣴⣾⣿⣷⣤⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⣹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣶⣶⣤⣀⣀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢀⣀⣤⣶⣶⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣤⣼⣿⣿⣷⣤⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠛⠛⠛⠿⢿⣿⣿⡿⠿⠟⠛⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡀⠀⣿⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⡇⠀⢻⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⣾⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⠀⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡇⠀⠀⠀⢻⠇⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠘⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⢘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢟⣀⣼⡇⠀⠀⠀⠀⣼⣯⡀⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢛⣤⣾⣿⣿⡇⠀⠀⠀⠀⢿⣿⣿⣤⡟⢳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⡆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠉⠁⠈⠛⠛⠉⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⡿⢿⣿⢿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣶⣶⣶⡶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁⠀⣀⣤⠶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⣠⣶⠟⠋⠀⠀⠀⣁⣀⣠⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⢀⣤⡾⠋⠅⣀⡤⠖⠛⠉⠁⠀⠀⠀⠈⠉⠉⠛⠓⠶⢤⣀⡀⠀⠀⠀⠀⠌⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣴⣿⡟⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀⠀⠀⡀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⣠⡾⣏⡿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣷⣶⣿⣿⣄
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢠⡾⢋⡙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⠿⠿⡿⡟⠻⣆
⠀⠀⠀⠀⣠⠋⠀⠀⠀⣴⣿⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠉⠀⢹⡆
⠀⠀⠀⣰⠇⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⢻⡀
⠀⠀⢠⠋⠀⠀⠀⣸⡟⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⡀⣀⣀⡀⠘⣧
⠀⠀⡎⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣝⡀⣻⡄
⠀⢸⠁⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠉⠉⠛⡉⢻⡄
⠀⡎⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠅⠀⠀⠀⠁⠘⡇
⠀⠃⣀⡀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⢐⠀⣇
⢸⠛⠛⢍⣉⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠀⠤⠄⣰⣇⣀⠀⠀⠀⠀⢷
⠘⠀⠀⠌⣾⡏⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⡀⠀⡀⢺
⡇⠐⠙⠒⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢿⣆⠀⢸
⠇⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣤⡾⠻⣷⡀⠀⠀⠀⠰⡣⠀⢿⣥⠈
⢰⠀⠀⠀⣿⡀⠀⠀⢀⣠⠴⣟⠙⠶⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡁⠀⢠⣾⡟⠀⠀⢦⡷⠀⠀⠀⠀⣱⠀⢿⣷
⠸⠀⠀⠀⣿⡆⠀⠀⣾⣟⠠⠈⠀⠐⢽⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⣸⡿⡆⠀⠈⢻⠅⡀⠀⠀⠀⢘⠀⣿⣿⡀
⠀⡆⠀⠀⢹⣇⠀⣾⣿⡟⠗⣄⢳⠀⠀⠉⠳⡆⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣰⠀⢰⣿⡿⠂⠀⠀⣹⠃⠁⠀⠀⠀⢸⠁⣼⣿
⠀⠹⠀⠀⠸⣿⡄⠹⣿⣞⠦⡀⢁⣠⠀⠁⠀⠘⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠄⠀⢸⣿⣧⠀⢨⣰⠏⣰⠀⠀⠀⠐⡏⢨⡿⣿⣋
⠀⠀⢣⠀⠀⢺⣿⡀⢹⡿⣦⡙⠀⠉⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⣼⣿⣿⣑⣶⠟⠨⠀⠀⠀⠀⣴⢀⡾⢠⣟⣿
⠀⠀⠀⠙⣷⣷⣿⣷⡀⠓⠉⣷⣌⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⢸⡀⠀⠈⢿⣿⣿⡿⠛⠀⠀⠀⠀⠠⣴⠃⡜⢡⣿⣽⠇
⠀⠀⠀⠀⠹⣿⡀⠘⢷⡄⠙⠦⣍⢦⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⣄⠀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣼⢣⢎⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠘⢷⡀⠨⣿⣄⠈⠣⠗⢂⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⢀⣄⣞⠵⠓⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠳⡀⠈⢿⣧⡀⠀⣈⣂⠀⠀⠀⠀⠀⡄⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣄⣀⣀⣠⠴⠟⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⡛⣿⣄⠘⢿⣦⠀⠀⠀⠀⢱⣀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡙⢯⣿⣧⡈⢻⣇⡀⠀⢀⠘⣧⣶⣆⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠻⢷⣄⢻⣿⡷⣾⡞⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠁⢻⡎⠛⠓⠛⠚⠁

⠀⠀⠀⠀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⣤⣄
⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡟⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿
⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⣼⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⡄
⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⡇
⠀⠀⣠⣼⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⣠⠈⢩⣿⣿⣯⡄⣄⠀⠀⠀⠀⣠⠈⢁⣿⣿⣿⣥⣄⠀⠀⠀⠀⢠⣴⣷⣿⣿⣿⣧⣄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⡇⡇⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢻⠀⠸⣿⣿⣿⡇⡇⠀⠀⠀⠀⢸⠀⠸⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⡏
⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⡀⠀⣿⣿⣿⢡⠇⠀⠀⠀⠀⠸⡄⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠈⢿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢷⣤⣿⣿⣿⠞⠀⠀⠀⠀⠀⠀⠷⣤⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠳⣤⣤⣤⣤⡼


⠀⠀⠀⢸⣿⣿⣿⠛⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⡆⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⡟⡆⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⡆
⠀⠀⠀⢸⣿⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠠⠜⠋⠉⠙⠙⠣⠀⡀⠀⠀⢀⣠⣜⣋⣉⣉⣛⣣⣀⡀⠀⠀⢀⣀⣬⣿⣿⣿⣷⣥⡄⡀⠀⠀⠀⠀⣨⣽⣿⣿⣿⣅
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠿⣿⣿⠿⣿⣿⠁⠀⠀⢸⠀⠀⣀⡀⠀⣀⠀⠀⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠇⠀⢰⣶⣤⣴⣶⡆⠀⠀⠀⠀⠀⣿⣿⡀⠈⠁⢀⣿⣿⠀⠀⠀⢸⡆⠀⢿⣿⣿⡿⠁⠀⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⠻⣿⣿⠟⠀⠀⠀⠀⠀⠀⢿⣿⣿⣶⣶⣿⣿⣿⠀⠀⠀⢸⡇⠀⠀⠙⠋⠀⠀⢰⡿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⠿⠃⠀⠀⠀⢸⣿⣄⣀⣀⣀⣀⣠⣾⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⢂⢀⣠⣤⣤⡤⣤⣤⣤⣤⣤⣤⣈⣁⣐⠂⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⣀⠔⢊⣥⣴⡶⠭⠗⣩⢻⣿⣿⡶⠀⠀⠀⠐⢛⣿⣟⠿⡏⢻⣻⣶⣤⣌⡑⢄
⠀⠀⠀⠀⠀⢠⠞⢁⣴⡿⢿⡍⣓⡩⢝⠁⠻⣿⡗⠈⢉⡒⠀⠈⠿⣻⣿⠆⡙⠦⢉⣉⡉⣹⣿⣄⠑⢄
⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠛⠓⢲⡶⠁⠀⠀⠙⠻⠂⠀⠀⠀⣀⠊⠛⠁⠀⠈⠑⣦⣤⣤⣴⣾⡿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⠇⠁⢸⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡷⣾⢻⠂⠷⡇
⠀⠀⠀⠀⠀⡀⠘⣿⢠⠀⢸⠃⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠇⠺⢸⠀⠀⡇
⠀⠀⠀⠀⠀⡇⢰⣿⢸⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠀⠀⡇
⠀⠀⠀⠀⠀⡇⢸⡏⢸⠀⡌⢠⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠘⠀⡆⡇
⠀⠀⠀⠀⠀⡇⢸⡇⣼⢠⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⠀⡄⢡⡇
⠀⠀⠀⠀⠀⠀⢸⡇⡞⢸⢀⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠸⠀⣇⢸⡇
⠀⠀⠀⠀⠀⡇⢸⣧⠇⡎⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡇⠇⣇⡇
⠀⠀⠀⠀⠀⡇⢸⣿⣴⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣾⠇
⠀⠀⠀⠀⠀⠃⢹⡿⠻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡋⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⡂
⠀⠀⠀⠀⠀⠀⢸⡏⡅⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⢀⠀⠀⡈⠀⣼⡇
⠀⠀⠀⠀⠀⠀⢸⡇⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠄⠸⠇⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣇⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠈⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠀⡀⠀⠑⢦⣀⣀⣀⣀⣤⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠈⠢⡀⠀⠙⢦⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠹⡶⣄⠀⠘⣆⠀⠈⣷⡄⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⣱⣿⣷⠈⢧⡀⣸⣆⠀⠘⡿⡄⣼⣇⠀⠀⠀⠀⠀⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⡴⠊⠁⢀⣩⠴⠂⠳⢿⣿⣧⢰⣿⡙⠿⣿⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢋⡴⠋⠀⣠⣾⠟⠁⢀⡤⠖⠂⣩⠞⠛⠿⣇⣀⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⡋⠀⡠⢞⠝⠁⣠⠖⠉⣀⣴⠊⢁⣠⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠈⠿⠿⠛⠐⢫⣶⣿⡧⠞⠁⠀⠘⠛⠓⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⣠⣴⢶⢿⣿⣿⡟⣦⣄
⢀⣾⣍⡑⠋⠃⠠⢤⣵⠘⢸⣿⣶⣄
⠸⠏⠒⠒⠰⠖⠽⣷⣾⠀⢸⣿⣿⡿⣗
⡧⠉⣭⡬⡭⠤⣼⣿⠃⠀⣾⣿⣻⣿⣿⠀⠐
⠨⢀⢄⡁⢌⣮⣗⠏⠰⣰⣿⣗⣥⣾⠇⣄⡐⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠐⡴⣿⣿⡾⠋⣄⣿⣻⣽⣿⣾⣿⣄⠊⠀⠀⢀⡠⠀⠀⣀⠀⠀⠀⠀⠐⠈⠑
⠀⠀⠀⡠⠛⠒⠛⠉⠉⠉⠉⠁⠔⠁⠉⠉⠛⠙⠛⠒⠶⠎⠁⣑⡓⠀⠀⠀⠀⡘⠓⠂⠀⠛⢀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠋⠙⠛⠳⠿⠞⠉⠙⠛⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠙⠀⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠍⣒⠂⠄⠀⠠⢀⢀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠀⠀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⢠⠎⠘⠿⢟⠝⡠⠀⠛⠟⢫⠏⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠙⠿⢫⠀⡰⠃⠀⠀⢠⢫⠞⠁⠀⠀⡰⣫⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠈⠑⢦⠀⡴⠋⠀⠀⡴⣣⠞⠀⠀⠀⠀⠀⠁⠀⠀⢀⡾⠋⠀⠙⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢤⠈⠀⠀⢀⣾⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠃⢸⡀⠀⠀⢸⡆⠀⣠⡾⠃⠈⠉⢀⡤⠊⠤⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠈⣇⠀⣀⣀⢳⣴⠟⠀⠀⢀⡴⠋⠀⠀⠀⠀⡀⢠⣄⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣻⠀⠀⠀⠘⣦⣿⣿⣧⡳⠀⢠⡶⠋⠀⠀⠀⢀⡴⠋⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⢹⡆⠀⠀⠀⠈⠉⠻⠿⠃⠀⠉⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢻⡄⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠘⣦⠀⠀⠀⠙⣦⠠⣾⣷⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢧⡀⠀⠀⠈⠃⠹⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⣠⠞⢁⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠈⠳⣄⢰⣿⣷⡐⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⡞⠁⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢦⡀⢰⣾⣧⡹⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⣠⠏⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠙⠲⠝⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠛⠁⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢶⣶⣿⣿⣿⣿⣷⡶⠆⠀⠀⠀⠀⠀⣀⣠⣤⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⢀⣼⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠒⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠈⠁⠚⠛⠛⠓
⠀⠀⠀⠀⠀⠘⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀
⠀⠀⠀⠀⠀⠴⣆⠀⠀⠀⠁⣠⣶⡀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⢀⣶⣄⢀⣠⣦⡀
⠀⠀⠀⠀⠀⠀⠹⣿⣶⣤⣾⣿⠋⠀⠀⠀⠀⠀⠀⠸⣿⣷⣶⣶⣿⡟⠁⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⡿⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣤⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢐⡅⠀⢀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠀⠀⠀⠀⠀⠘⣿⡟⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠂⠒⠠⠤⣄⠒⠂⠠⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠈⠉⠀⠀⠀⠀⡠⠒⠒⠒⠠⠁⠂⢤⣀⠤⠀⠌⠉⠗⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡨⠠⠠⠤⠤⠀⠤⠶⡩⢍⡉⠉⠁⠀⠐⠒⡩⠐⠒⠒⠲⠤⠦⠤⠬⣇
⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⢠⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠒⠒⠀⠀⠀⠀⠂⠚⠃⠀⠀⠐⠺⡀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡧⠤⣤⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠔⠁⠉⠁⠉⠙⠉⠉⠉⠋⠈⠉⠉⠛⠉⠉⠉⠉⠈⠓⢄⣀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⢤⡤⠤⠤⡤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠊⠉⠙⠉⠉⠙⠉⠙⠁⠁⠛⠈⠉⠉⠙⠋⠉⠛⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⣄⠄⠠⠀⠤⠤⠠⣠⠤⠤⠤⠀⠤⠤⠠⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠉⠉⠛⠉⠉⠉⠋⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢀⠤⠤⠤⠤⠤⠤⠤⢤⡤⠤⠤⡤⠤⢤⡤⠤⠤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠥⠤⠤⠼⠫⠤⠤⠤⠤⠤⠤⠤⠯⠯⠤⠤⠼⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠄⠀⠠⠴⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣾⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣁⠀⡙⣻⡷⢒⡶⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠈⠉⠀⠁⠁⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠻⣷⣤⣀⣤⣄⣀⣤⣤⣶⣤⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⣿⣦⣉⣻⣿⣿⣿⣿⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⢻⡇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⠇⠸⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠦⣀⠲⣮⣿⣿⠀⠀⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠻⣄⠀⠀⠀⠀⣠⣴⠷⢬⣙⣿⣿⣿⣦⠠⣇⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢷⣄⠀⣴⣿⣧⡀⠀⠙⠋⡟⠿⣿⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣻⣷⣦⣤⣈⡑⠪⢽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝⣿⢻⡟⢻⡟⢻⣿⣿⣿⣷⣦⣝⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣸⣇⣾⠁⠘⠇⣠⠂⠘⢉⣿⣭⡻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢠⡟⣽⠇⠠⠇⠀⠉⠀⢂⠸⠀⠉⢳⣌⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⡟⢰⠏⠀⣠⣤⣴⣶⣦⣤⣇⡀⠈⡈⠙⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⢀⠞⢁⡟⠀⣴⡿⣿⣿⣿⡿⢿⣿⣿⣦⢡⠀⢹⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⣠⠌⠀⣤⠀⣼⣿⡅⢸⣿⣿⠀⠀⣿⣿⣿⣯⢳⡘⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢃⡴⠊⠀⠀⠐⠁⠀⣿⣿⣧⠀⢻⣿⡀⢠⣿⣿⣿⣿⡇⢱⣼⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢡⡞⠁⠀⠀⠀⠇⠀⢠⣿⣿⣿⡆⠈⢿⡇⢸⣿⡘⣿⣿⣿⡌⠘⣿⣿⣿⡙⣿⡙⠻⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠘⡃⠀⠀⠀⡼⠀⠀⣼⣿⣿⣿⣿⠀⢸⡇⣸⡿⣿⢀⢿⣿⣷⠀⠐⢻⣿⣷⣜⣿⣄⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⡗⠀⢀⡼⠃⠀⣼⣿⣿⣿⣿⡿⠷⢾⣷⣿⡇⣧⠸⢸⡟⣿⡄⠀⠈⣿⣿⢿⠿⣿⣆⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠇⠀⠎⠀⢠⡞⣱⣿⣿⣿⣿⠇⠀⣸⣷⣿⡇⣿⠀⠚⣇⢿⣷⡑⠀⣿⡇⠈⢇⡈⢿⣷⡀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠨⠀⠀⢼⣧⣿⣿⣿⣿⣿⠀⠀⣹⣷⣿⡇⢿⡰⠀⣿⢸⡏⣿⣄⠹⣷⡀⠀⠹⡌⢿⣿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢠⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⡿⠘⢾⡇⠀⣿⣺⡇⡀⢹⣦⣿⣯⡁⠠⡘⡇⢹⣿⠏⠹⣷⡀
⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠆⠀⠀⢸⣠⡿⢳⣿⣿⣿⣿⣿⣿⡀⠀⣿⣿⠃⡀⢸⡇⢀⣿⠸⣇⠱⡀⢿⠀⠙⢿⣦⡘⠈⢻⣿⡇⠀⢻⣇
⠀⠀⠀⠀⠀⢠⣼⡀⠀⠀⠀⠀⠀⠀⢘⣿⣧⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⡿⠀⠠⢸⡇⢸⣿⠀⢻⡄⠁⠈⡄⠀⠀⠛⣷⡀⠈⢿⣷⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⢀⣴⠾⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄⢹⣧⠀⢀⡿⠀⢸⡏⢠⠈⣿⠀⠀⢡⠀⠀⠀⢸⣷⡄⢸⣿⢃⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⣶⣿⣅⣰⣿⣿⣿⡇⣿⣿⣿⣿⣿⡿⣦⣸⡏⢀⠚⡇⠀⢸⡇⠨⡄⣿⡆⠀⠈⣄⣀⡀⠀⣻⠇⢀⣷⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣶⣿⣿⢸⣿⣿⠃⣿⠙⣧⡌⠀⠀⠀⢠⣷⣤⣴⣿⣿⣄⠀⢿⡿⠿⠟⠉⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠉⠹⢿⣿⣿⠿⣏⢸⣧⠀⢻⣷⣤⣤⣤⣼⡿⠟⠛⠿⠿⣿⢶⠾⠃⠀⠀⠀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢻⡿⠁⢀⣨⡏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣈⣤⣉⣧⣤⣌⣠⣽⣌⣠⣿⣋⣶⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿