Rental Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣧⣤⣤⣾⣿⣦⣤⣤⣤⣤⣤⣤⣶⣿⠟⠁⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⢸⣿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣾⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⢿⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⣴⠶⠶⠶⠶⠶⣶⠀⠀⢰⡶⠶⠶⠶⠶⣶⠀⠀⠀⠀⣿⠀⠀⠀⣶⠶⠶⠶⠶⢶⡆⠀⠀⣶⠶⠶⠶⠶⢶⡆⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⢸⡆⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⢸⡅⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠙⠛⠛⠛⠛⠛⠋⠀⠀⠘⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⣿⠀⠀⠀⠛⠛⠛⠛⠛⠛⠁⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⣸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⣦⣤⣤⣾⣿⠀⢸⡟⠛⢷⡄⠀⣿⡟⠛⠛⠀⢸⣷⣄⠀⠀⢸⣿⠈⠛⢻⡟⠛⠀⣿⣿⣤⣤⣤⣴⣿⣤⣤⣤⣤⣼⣧⣤⣤⣼⣿⡇
⠀⠀⠀⠀⠈⠉⠉⠉⠉⢻⣿⠀⢸⣧⣤⡾⠃⠀⣿⡷⠶⠶⠀⢸⡏⠹⣧⡀⢸⣿⠀⠀⢸⡇⠀⠀⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡏⠻⣦⡀⠀⣿⡇⠀⠀⠀⢸⡇⠀⠈⠻⣾⣿⠀⠀⢸⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠘⠃⠀⠙⠓⠀⠛⠛⠛⠛⠀⠘⠃⠀⠀⠀⠈⠛⠀⠀⠘⠃⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋




⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢶⠶⡶⠶⢶⣶⠶⠶⠶⡶⢶⡶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤⢰⣶⢰⣿⠛⠃⣿⣟⣋⢸⣿⣉⡁⠀⠀⠔⡹⠀⠀⠀⢸⣿⢿⣦⢰⣶⣦⠀⣤⡀
⢀⣀⢠⣤⢰⡟⢻⣷⢸⣷⢸⣿⠸⠿⣿⡇⣿⣟⡋⢈⡉⣿⡇⠀⠊⡜⢲⠤⡿⠀⢸⣿⢿⣏⢸⣿⣦⠀⣿⣇⣽⡀⠶⣦⣤
⢸⣿⣴⣿⢸⡇⢸⡟⢸⣿⣼⡿⠰⠶⠿⠇⠛⠛⠛⠘⠛⠋⠁⠈⠀⠑⠉⠘⠷⠂⠸⠿⠸⡿⢸⣿⣤⠀⣿⢻⣿⡇⠀⣿⡏
⢸⣿⢹⣿⠘⠿⠚⠃⠈⠉⠁⠀⣀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠨⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠉⠀⠻⠇⠀⣿⡇
⠸⢟⡈⣉⠀⠀⠀⠀⠀⠀⠀⣸⡏⢠⡝⠀⢀⡀⠀⡃⠀⠀⠀⣀⡄⠀⠀⠀⠀⡀⠀⠀⠘⢰⡅⠀⠀⠸⠀⢠⣿⡿⠁⣭⠁
⢀⣿⠘⣯⣶⡐⠂⠀⠀⠃⠀⠻⣿⡟⣷⣄⠈⠀⠀⠉⣀⠈⢴⡉⡇⠂⠀⠀⠀⠂⢰⠀⠃⣾⣷⠀⠀⠀⠀⣾⣿⡇⠃⢻⣷
⢸⣿⣤⣸⣿⣿⣰⣠⣴⣶⣀⣄⣸⣿⣿⣧⣀⣀⣘⣁⣘⣀⣸⣇⣄⣈⣼⣿⣄⣁⣸⣄⣸⣿⣿⣄⣗⣸⣇⣿⣼⣇⣷⣸⣶
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉







⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠤⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⠻⢦⣄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠙⢷⣌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⠢
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀⠁⢄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⠀⠀⠀⣤⣤⣤⠄⣠⣴⣤⣄⠀⣴⣶⣶⣦⡄⠀⠀⠀⠀⣿⣿⠈⠳⣄⠑⢄
⠀⠀⠀⠀⠠⠾⠿⠿⢿⠀⠀⠀⣿⣤⣤⢰⣿⠁⠀⣹⡆⣽⣥⣤⣾⠇⠀⠀⠀⠀⣿⣿⠀⠀⠈⠑⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠘⢿⣄⣠⣿⠃⣿⡇⠙⣿⣄⠀⠀⠀⠀⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠀⠀⠀⠀⠉⠉⠀⠀⠉⠁⠀⠈⠉⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣤⣤⣤⡄⢠⣤⣤⣤⠀⣤⡀⠀⣠⢀⣀⣀⣀⣀⡀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣤⣼⠟⢸⣯⣤⣤⠀⣿⣿⡄⣿⠀⠉⢹⡟⠉⠁⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⢻⣆⢸⣏⣈⣉⡀⣿⠈⢿⣿⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠈⠉⠉⠉⠁⠛⠀⠈⠛⠀⠀⠸⠇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠲⠶⠤⠤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠓⠒⠲⠶⠶⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⢿⣿⡿⠿⢿⣿⢿⣿⢿⡿⠿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠘⢂⣿⠀⣛⣻⡇⢈⠻⢸⣿⡆⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣼⣦⣿⣤⣭⣽⣧⣼⣷⣼⣿⣧⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣿⡏⠉⠉⠉⠉⢹⣿⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⢿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡿⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⢉⣿⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿⣿⡉⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⠄⠦⡄⠆⠀⣰⡀⠀⠀⣰⡀⣰

⠀⠀⢠⣤⣤⣤⣤⡀⠀⢀⣤⣤⣤⣤⣄⠀⠀⣠⣤⣤⣤⣤⡀⠀⣤⣤⣄⠀⠀⣠⣤⣤⠀⢀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⢸⣿⣍⣉⣿⠟⠀⢸⣿⣯⡍⣿⣿⠀⢸⣿⣿⠉⣿⣿⡇⠀⣿⣿⣿⣦⣼⣿⣿⣿⠀⢸⣿⣿⣭⣭⡅⠀⠀⠀⠀⠀⠀⠈⠙⣦
⠀⠀⢸⣿⡏⠙⣿⣦⠀⣸⣿⣯⣠⣿⣿⠀⠘⣿⣿⣤⣿⣿⡇⠀⣿⡏⠹⣿⡟⢹⣿⣿⠀⢀⣭⣭⣽⣿⡟⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⢠⡆⠀⢠⣶
⠀⠀⠈⠉⠁⠀⢉⡡⠊⢁⡉⠉⢩⡍⠁⠀⠀⠉⠉⠉⠉⠉⠀⠀⠉⠁⠀⠈⠀⠀⠉⠉⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⠙⣦⣤⠀⠀⢠⡄⣻⣿
⠀⠀⣠⡤⣀⡠⠋⣀⠜⠉⡿⠖⢩⡟⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢠⣤⠼⠂⠉⠉⠤⠀⢛⣹⣿⣿
⠀⠀⠀⢨⠟⡛⠀⠀⠑⠀⠀⣀⣈⣭⣭⣅⠀⠀⣤⣤⣤⣀⣄⠀⢀⣤⣄⠀⢠⣤⠀⢀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠤⠚⠀⠀⠀⠀⣤⣶⣾⣿⣿⣿⣿
⠀⠀⣰⠋⢠⠇⠀⠀⠀⠀⠀⣿⣯⣈⣿⡿⠁⠀⣿⣿⣅⣁⣁⠀⢸⣿⣿⣷⣼⣿⠀⠀⠀⢿⣿⠁⠀⠀⠀⢀⠀⠀⠀⣀⠄⣪⡆⠀⣿⣿⣿⣿⣿⣿⣿
⠀⢰⣇⡠⠋⠀⠀⠀⠀⠀⠀⣿⣿⠉⣿⣷⠀⠀⣿⣿⣉⣉⣁⠀⢸⣿⠙⢿⣿⣿⠂⠀⠀⣼⣿⠀⠀⠀⢠⣈⣢⣄⣀⡁⠚⠋⠀⠀⢿⣿⣿⣽⣿⣿⣿
⢀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣀⣈⣉⣀⣀⣉⣉⣉⣉⣉⣀⣈⣉⣀⣀⣉⣉⣀⣀⣀⣉⣉⣀⣀⣀⣘⣛⣿⣿⣛⣁⣀⣀⣀⣀⣈⣿⣿⣿⣿⣿⣿
⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠟⠿









⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢤⠠⠤⠀⠄⠀⢠⠠⠠⠀⠀⠠⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢓⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⢀⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡅⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⡇⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿⣿⣿⡇⣿⣿⠀⣿⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⠀⣿⣿⣿⣷⢸⣿⡇⢸⣿⣿⣿⡆⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿⣿⣿⡇⣿⣿⠀⣿⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⠀⣿⣿⣿⣿⢸⣿⡇⢸⣿⣿⣿⡇⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⠇⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⡇⢸⣿⣿⣿⡇⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⠋⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡙⣿⣿⡇⢸⣿⣿⣿⡇⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⡟⢛⠛⢿⣿⠛⣛⣛⣻⡟⢿⣿⣿⣿⢻⣟⡛⠛⣛⣿⣿⡇⣿⣿⣇⠘⠛⠛⠛⠃⣿⣿⣿⣿⡇
⠀⠀⠀⠘⠻⠿⠟⢸⣿⣿⡇⠿⠗⣸⣿⠀⠛⠛⣿⡇⢠⡙⢿⣿⢸⣿⣿⢸⣿⣿⣿⡇⠻⠻⠿⠻⠛⠛⠿⠟⠟⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣴⡘⢿⣿⠀⣿⣿⣿⡇⢸⣿⣦⡉⢸⣿⣿⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣾⣿⣦⣿⣦⣤⣤⣾⣧⣾⣿⣿⣿⣾⣿⣿⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢟⣽⣾⣷⣿⣻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢟⣭⣾⣿⣿⣿⣿⣿⣿⣷⣿⣳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢟⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⢭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣤⠀⢸⡆⠀⣀⠀⢸⡆⠀⠹⡇⠀⣷⣤⠀⢀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡿⠀⢸⡇⠀⠙⣻⣿⡇⠀⠀⠁⠀⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡄⠀⣾⡇⠀⣾⠛⢿⡇⠀⣦⠀⠀⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣸⣷⡀⣸⣧⣀⣀⣀⣼⣇⣠⣿⣧⢀⣿⣿⣄⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣴⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠉⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢰⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⣿
⠀⠀⠀⠀⠈⣿⣿⣿⡟⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡆⠀⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⢻⣟⡟⡟⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⠀⢠⢀⠤⡄⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣾⣷⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⠀⠈⠈⠙⠁⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⠀⠆⣽⠐⣺⡇⡙⡇⣷⠀⣾⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⠀⢸⣭⠇⢸⡭⢸⢷⣸⡏⢹⠉⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣤⣷⣽⣬⣽⣧⣿⣤⣿⣦⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠿⠀⠘⠊⠓⠚⠶⠘⠂⠛⠃⠘⠂⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⢿⡶⠶⠴⠶⢶⣴⣦⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤
⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⠆⠀⣠⡶⠾⠷⢶⣄⠀⠀⣶⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⡄⢰⡏⠀⠀⠀⠀⢹⡇⠀⣿⣀⣸⡟⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⣧⡀⠀⠀⢀⣼⠇⠀⣿⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠈⠛⠶⠶⠟⠁⠀⠀⠿⠀⠘⠷⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⢀⣀⣀⣀⠀⢀⡀⠀⠀⠀⢀⡀⢀⣀⣀⣀⣀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠉⢹⡧⠀⢸⡏⠉⠉⠀⢸⣿⣦⡀⠀⢸⡇⠈⠉⣿⠉⠁⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⢶⡟⠁⠀⢸⡟⠛⠛⠀⢸⡇⠈⠻⣦⣸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠈⢻⣆⠀⢸⣧⣤⣤⠀⢸⡇⠀⠀⠘⢿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠈⠉⠉⠉⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⢹⣿⠉⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⠿⣿⡿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠁⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠁⣤⣄⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠟⠀⠀⠛⠋⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣟⡃⢻⡿⠀⠀⠀⢶⣶⢰⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿⣿⢿⣿⠀⠀⠀⣶⠚⠐⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠉⠁⠈⠉⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⢀⣶⣶⣶⣶⡀⠀⣴⣶⢰⣶⢰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⠉⣿⣿⠉⠃⠀⣭⣥⢨⣭⢨⣭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠈⣿⣿⠀⠀⠀⣿⣧⢸⣿⢸⣿⡆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣀⣀⣨⣿⣿⣀⣀⣀⣀⣥⣤⣬⣤⣭⣤⣼⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⠿⢿⣿⣿⣿⡟⠛⠛⠛⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏⠁⣤⣀⣙⣿⠏⠀⣠⣄⠀⠹⣿⠀⠀⣿⣿⣿⡇⠀⢰⣦⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣄⣀⠈⠉⢿⠀⠀⣿⣿⠀⠀⣿⠀⠀⣿⣿⣿⡇⠀⢸⡿⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣄⠀⠙⠃⣀⣾⣧⣀⣈⢁⣀⣼⣿⣀⣀⣀⣀⣠⣧⣀⣤⣤⣤⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠯⠐⠒⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠤⠤⠤⠴⠒⠒⠒⣺⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸⡿⠷⠴⠒⠒⠒⠚⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿⡿⠁
⠀⠀⠀⠀⠰⡚⠉⠉⠉⠹⡆⠀⠀⢸⡇⠀⢀⣀⣀⣀⣤⣤⣤⣤⣶⣶⠶⠶⠿⠿⠛⣟⢛⡛⣋⠉
⠀⠀⠀⠀⠀⣧⣤⣤⣶⣶⡇⠀⠀⢸⡿⣿⢛⠛⠉⠽⠏⠉⠀⠀⠒⠂⠀⠈⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣿⣿⠸⠀⠀⠀⠀⣤⣤⡤⣠⣤⣄⢀⣤⣤⣄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⡟⣿⠀⠀⠀⠀⢀⣿⠤⢰⡿⢀⣿⢸⣧⡴⠋⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣿⠀⠀⠀⠀⠘⠇⠀⠈⠳⠞⠃⠺⠉⠓⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⣿⠈⠀⣰⡞⢻⣆⣶⠟⢓⣶⡆⣾⠖⢾⡟⠃⣼⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⣿⣿⠀⠀⣿⢿⡋⢱⣟⣛⢸⡟⣿⡟⠀⣾⠃⠀⡿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⡄⠀⠉⠀⠉⠈⠉⠉⠈⠁⠈⠁⠀⠉⠀⠈⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢹⢿⡇⣇⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⡇⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⣷⣶⡇⣧⢸⣿⣧⣾⣅⣴⡀
⠀⠀⠀⠀⠀⣀⣤⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣄⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡙⢛⠛⠻⠶⠶⠦⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠀⢰⣤⣦⣤⣀⣈⣉⠉⡙⠛⠛⠳⠶⠶⠶⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢀⠀⢀⠀⣿⣿⣍⣹⣿⡿⣿⣷⣾⣿⣿⣷⡀⠁⣆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⠨⠀⠒⢸⣿⣛⠙⣿⣟⣸⣿⣿⣿⣷⣾⠿⢃⠀⠀⠰⢸⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣰⣿⣿⣷⣍⣤⣤⣭⣿⣛⠁⢿⠏⠻⡷⠀⠃⠈⡰⠠⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣦⡀⠀⠀⠀⠀⣸⠇⠀⢠⣿⣿⣾⡿⣿⣿⣯⣹⣿⣿⣷⣿⠿⣿⣿⣷⣦⣶⡶⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⢀⣤⣶⣿⣿⣿⡟⠉⠻⣿⣿⣦⡀⠀⠀⣿⠀⠇⠸⣿⠻⣿⣶⣿⣿⣭⣿⡿⣿⣿⡟⢰⣿⡟⢀⣿⡿⡁⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⠻⣿⣿⣦⡈⢿⣿⣷⣶⣶⣤⣤⣤⣉⣉⡁⡙⠃⠛⠿⢀⠿⡿⡁⣼⣿⠃⡁⠈⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣀⣀⠀⠀⣴⣤⣄⡈⠻⣿⣿⣦⡀⠀⠉⠉⠉⠛⠛⠛⠿⢿⣿⣿⣿⣶⣶⣦⣤⣤⣀⣁⡀⠠⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⢟⣴⡿⠿⠟⠀⠀⠉⠙⡛⠿⣦⠈⠻⣿⣿⣦⡀⠀⠀⠀⢀⠤⡤⠾⠿⠿⣄⠈⠉⠉⠙⠛⠛⠿⠿⢿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠟⠁⠙⢉⠖⠛⠳⡀⠀⡴⠋⠉⠳⡀⠀⠀⠘⢿⣿⣷⣄⠀⠀⣏⢈⡧⠴⠤⠄⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⡇⠀⠀⠀⣟⣠⣤⠀⣷⠀⡇⣶⣦⠀⡇⠀⠀⠀⠀⣽⣿⡿⠃⢀⡜⢿⡧⠤⠤⠀⣰⠃
⠀⠀⠀⡴⠒⣆⠀⢾⣿⣿⠿⠛⠉⠁⠀⣧⠀⠀⠀⢻⣿⣿⣀⡟⠀⢹⣿⠿⠤⠇⠀⠀⠀⠀⡷⠶⠒⠛⠉⣹⣼⣶⣶⣦⠴⠃
⠀⠀⣀⣇⣀⡟⠀⠀⠀⢸⡀⠀⠀⠀⠀⢸⠀⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢸⣇⣠⠴⠖⠛⠉⠀⣿⣿⠇
⠠⡞⠁⠀⢨⡏⢶⣄⣀⣨⡧⠀⠀⠀⠀⢸⡄⠀⠚⡗⠤⢤⣀⣀⡠⠤⠴⢊⡟⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢿⡙⠉⣁⣹⣟⣟⣈⣀⣀⡠⠄⠀⠀⠘⡇⠀⠀⢻⣶⣤⣤⣤⣴⣶⣾⠋⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⣾⣿⡏
⠀⠀⠻⢭⡭⠴⠋⠈⠉⠁⠈⡇⠀⠀⠀⠀⡇⠀⠀⠀⠻⣿⣿⣿⣿⠿⠿⣄⡀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠓⠷⠾⠶⠖⠋⠀⠀⠀⠀⠀⣷⣦⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣀⣶⣿⣶⣿⡇⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣶⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣸⣀⣀⣠⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⡶⠁⠙⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠁⣀⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠐⠿⠟⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⢰⣿⡇⣾⣷⠀⠀⠀⣀⣀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⢠⣶⡆⣶⣶⠀⠀⠀⣩⣹⢈⣉⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠿⠇⠛⠿⠀⠀⠀⠛⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⣀⣀⣀⣀⠀⠀⢀⣀⢀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣼⢿⣿⣿⢿⡆⠀⠿⠿⠸⠿⠟⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⢸⣿⢓⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⢸⣿⣿⠀⠀⠀⠻⠿⠸⠿⠖⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⠟⠛⠛⠛⠛⣿⠛⠛⢿⣿⠛⠙⡟⠉⠉⠉⠉⠉⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⢰⣶⠀⠀⣿⠀⠀⠶⠶⠶⣿⡄⠀⠀⠹⠄⠀⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⠀⢠⡄⠀⠺⣿⡄⠀⢶⣶⡶⣿⡇⠀⣷⡀⠀⠀⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣀⣸⣿⣆⣀⣹⣇⣤⣤⣤⣤⣼⣧⣤⣿⣿⣦⣤⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⢸⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣤⣤⣤⣄⡀⠀⠀⣤⣤⣤⣤⣤⠀⢰⣄⠀⠀⠀⠀⠀⣤⡄⢠⣤⣤⣤⣤⡄⠀⠈⠻⢿⣷⣄
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⢹⣷⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⣷⣄⠀⠀⠀⣿⠁⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⡀
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣧⣴⡾⠏⠀⠀⣿⡿⠿⠿⠗⠀⢸⣿⠈⠻⣷⣄⠀⣿⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣶
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡏⠻⣧⡀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠈⠻⣶⣿⡁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠁
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠿⠇⠀⠙⠿⠄⠀⠿⠷⠶⠶⠶⠀⠸⠿⠀⠀⠀⠀⠈⠿⠅⠀⠀⠸⠇⠀⠀⠀⠀⢀⣴⣿⠟⠋
⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣠⣀⣀⣀⣠⣨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠃



⡿⠿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⢉⣩⡶⠛⣁⠤⢄⡉⠻⢮⣍⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣠⠶⠛⠁⠀⠀⡇⠀⠀⡇⠀⠀⠉⠻⢦⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠉⠒⠊⠁⠀⠀⠀⠀⠀⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⡅⠀⡖⠚⠒⠒⢲⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⠒⡆⠀⡍⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⡇⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠳⠤⠤⠤⠞⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠤⠴⠃⠀⣧⣾⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡤⠦⠦⠴⢤⠀⣠⠤⠴⠴⢤⡀⢠⠤⠤⠤⠤⡄⠀⣿⣿⣿⣦⣉⢻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣤⣇⠀⡇⠀⠀⠀⢸⠀⣿⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⡇⢠⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⣋⣻⣿⣿⣷⢧⣀⣀⣀⡼⠀⣿⠀⠀⠀⢸⡇⠸⣄⣀⣀⣠⠇⠘⢿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣝⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⢊⢚⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣟⡛⣡⣿⣿⡿⠶⠶⠶⠶⠶⠶⠷⠶⠤⠶⠶⠷⠶⠦⠶⠶⢴⡀⠋⠉
⠀⠀⠀⠀⠀⢀⣀⣹⣿⣿⣿⣿⣿⣏⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡷⠁
⠀⠀⠀⠀⠀⠘⠻⠿⡿⣿⣿⣯⡿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠉⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⣷⠀⢀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀
⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀
⠀⠀⠀⠀⣠⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀
⠀⠀⠀⠸⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠙⢷⣄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⠳⣤⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣾⠛⠛⠛⣷⠀⣿⠛⠛⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡄⣿⠀⠀⠀⠀⠀⢈⣻⣮⡴⠾⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⣿⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⢰⡏⣀⣠⣴⠶⠟⠛⠉⢠⣤⠆⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣠⡴⠾⠛⠉⢁⣠⠤⢰⣦⣹⡄⠹⡄⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣴⠶⠶⠶⣦⠀⣶⠶⠶⢶⣦⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⡀⠐⣞⣻⡆⢷⣛⡄⠳⠈⠃⣀⣠⣤⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⣿⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣷⠀⠹⠌⠙⠈⣁⣤⡴⠶⢛⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠿⠶⠶⠶⠟⠀⠿⠶⠶⠾⠟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣧⣤⠶⠞⣛⠉⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⡾⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠁⠈⠀⠁⠀⠈⠀⠁⠀⠀⠂⠀⠈⠀⠀⠀⠐⠀⠁⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠛⠉⠀⠀⠙⠛⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠁⠀⣶⠀⠀⣶⣄⣼⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣄⠀⠙⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⠀⠀⡀⠀⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠛⠻⢿⠀⠀⠿⠀⢀⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣦⣄⣀⠀⠀⣀⣤⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣄⣀⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢐⡆⠰⢆⢲⣖⠊⠴⣢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠐⠀⢨⡏⠂⠀⠐⠀⢹⣨⡷⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢵⡀⠐⢀⠀⢁⠀⡈⠀⢀⡱⠓⠁⠁⠪⢆
⠀⠀⠀⠀⠀⡠⠔⠒⠒⠂⠄⣘⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⡐⣥⣼⣷⢨⢬⡣⠸⣇⠀⠂⠀⢀⡕⢣
⠀⠀⠀⢠⠊⡠⠂⠉⡀⠀⠀⠈⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢏⠌⠸⠺⡼⠶⠚⢸⣦⡇⡠⡪⠊⠉⠈
⠀⠀⠀⡆⠐⠀⠀⢏⣈⠂⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠟⠛⠚⠛⠚⠢⣮⢏⢴⠎
⠀⠀⠀⢃⠘⡀⠀⠢⡤⠃⢀⠄⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⠁⡠⠊⠉⠉⠉⠢⡈⢻⠁
⠀⠀⠀⠈⠢⡈⠒⠄⠀⠔⢉⢠⠁⠉⠉⠉⡩⠭⠭⠭⠭⡉⠉⢉⢟⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠑⢄
⠀⠀⢰⠉⠅⠭⠩⠩⠉⠉⠉⠉⠉⢢⠤⠤⠭⠬⠭⠬⠭⠥⠤⡎⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠊⠛⠻⠛⠛⠋⠉⠉⠉⠉⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢆
⠀⠀⢱⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠈⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠒⠒⠒⠂⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠒⠂⠛⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⣽⣉⣉⣋⡦⢰⠒⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠏⠋⠑⠛⠀⢸⣀⡠⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⢀⡠⠄⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⡇⠀⠠⠔⠁⠀⠰⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣐⣀⣀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠥⠚⢧⠀⢀⠗⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠂⠈⠓⡴⠡⠤⠀⠓⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠠⠀⡀⠀⡀⡀⠀⠀⠀⠀⠀⠀⡜⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠀⠀⠀⠀⣄⢋⣰⠈⢢⠠⠤⠤⠤⠼⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢀⠨⠈⠂⠀⠀⡀⡀⠀⠈⠁⠆⡈⢸⣀⡀⠀⠀⠀⠀⡎⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡘⠂⠀⠀⠀⠀⠈⠧⡟⢱⠒⡦⢄⡡⠂⠀⠀⠉⠉⠁⠐⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣧⡠⠊⠀⠀⢀
⠀⠀⠀⠀⠀⢀⢤⠖⡫⠉⠉⡐⠠⢍⡁⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⡠⠊
⠀⠀⠀⡠⠚⡱⠃⠀⠀⠀⡠⠃⢠⢄⡌⢝⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠇⠀⢀⣠⠴⡁⠑⡄⢀⡠⠖⠉
⠀⠀⡜⠀⠘⠂⠤⢀⣠⠌⠀⣐⠣⠆⣠⠋⠲⡑⢨⠃⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⢇⠀⠘⣄⠜⠉
⠀⢰⠁⠀⠀⠀⠀⠀⠀⠉⠒⠠⢉⡒⠦⢌⠔⢀⠃⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⡆⠀⠀⠀⠑⢄⡇⠀⠀⠃
⠀⠙⠒⠠⠤⠤⠤⠀⠀⠀⣀⣀⡀⠈⠑⠢⣤⠎⠀⠀⠰⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠢⠄⣀⢤⣀⠀⡀⠀⢀⡁⠤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠤⠤⠤⠄⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠄⠀⠀⠀⠀⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⡤⠤⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⠤⠴⠶⠖⠒⠛⠛⠉⠉⠉⠉⠀⠀⠈⢧⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⠉⢹⡏⠁⠀⠀⠀⠀⢀⡴⠖⠛⠛⠛⠲⣄⠀⠀⠀⠀⠀⠀⠉⠓⠚⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⣠⠟⠀⠀⠀⠀⠀⣰⠏⠀⢀⣼⣷⣤⠀⠈⢳⡄⠀⠀⣠⣤⡀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠁⠀⢀⣀⠀⠀⠀⡏⠀⠀⢿⣯⣤⣀⠀⠀⠀⣷⠀⠐⣇⣀⡽⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⡇⠀⠠⡏⠉⣳⠀⠀⣧⠀⠀⠀⠉⠙⣿⡷⠀⠀⣿⠀⠀⠈⠉⠀⠀⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠉⠀⠀⣇⠀⠀⠙⢛⡥⠖⠒⢻⣆⠀⠘⠛⢿⡟⠁⠀⣰⠃⠀⠀⠀⠀⠀⡴⠋⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢿⠤⣤⠞⠉⠀⠀⠀⣸⠋⠛⠦⣤⣤⣤⠴⠚⠁⠀⠀⠀⠀⢀⣸⣇⣀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⣀⣠⡼⠞⠁⠀⠀⢀⣠⠞⢁⣀⣀⣀⣤⡤⠤⠴⠶⠖⠒⠛⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠛⠉⠉⠀⠀⠀⠀⢀⡴⠛⠋⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⣤⢾⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡴⢦⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣉⣩⣴⡏
⠀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠳⠶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠘⢧⣤⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠛⣛⣛⣛⣛⣛⣛⣃⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⣿⡿⠿⠿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⣿⣟⣛⣛⣛⣛⣛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⡷⠶⠶⠶⠶⢾⣿⡇⠀⠀⠀⠀⠈⣿⣿⠀⣿⣯⣭⣭⣭⣭⣭⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⣟⠛⠛⠛⠛⢻⣿⡇⠀⠀⠀⠀⠀⠸⣿⠀⣿⣯⣤⣤⣿⣿⢛⣛⡃
⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⣿⣟⣛⣛⣻⣿⠿⠿⠇⠀⠀⠀⠀⠀⠀⢻⡀⢿⣿⣿⣿⣿⡿⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⡇⢸⠟⠀⠀⠀⠀⠀⠀⠀⠈⣷⣶⣶⠶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡏⣩⠭⠭⠭⠭⠭⠭⠭⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣀⣀⣀⡀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⡇⣿⢰⠒⠒⠒⠒⠒⠒⠒⠒⡆⠀⠀⠀⠀⢠⠖⠚⠁⡇⢈⣉⣉⣉⣉⣁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⣿⢸⠀⠐⠒⠒⠒⠒⠂⠀⡇⠀⠀⠀⠀⠈⣇⠀⠀⡇⠠⠤⠤⠤⠤⠤⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⣿⢸⠀⠒⠒⠒⠒⠒⠒⠀⡇⠀⠀⠀⠀⠀⠸⡄⠀⡇⠀⠖⠒⠀⠀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⣿⢸⠀⠘⠙⠛⠋⠛⠃⠀⡇⠀⠀⠀⠀⠀⠀⢳⠀⣇⣀⣀⣀⣀⣀⡧⠊
⠀⠀⠀⠀⠀⠀⠀⠃⣿⢸⠀⠉⠉⠉⠁⠀⡖⣶⠇⠀⠀⠀⠀⠀⠀⠈⣇⠀⣀⣀⠤⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠋⠉⠀⠀⣉⠉⠉⠉⠛⠛⠛⠛⠛⠳⢶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉⣉⣩⡽⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠛⠋⠉⠀⠀⠀⠀⠠⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢛⣡⡴⠶⠻⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣿⣶⣶⠿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⣴⣿⣷⠾⠿⠷⠶⠶⢶⡶⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣻⣿⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⡀⣀⣴⣿⣿⣵⣶⣶⣶⣶⣾⡿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠀⢀⠈⠙⠻⣿⣶⣤⣀⠀⢄⡀⠀⠉⠙⠋⠁⠀⠀⠀⣠⣶⣿⣾⣥⠶⢍⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⢶⣿⠛⠛⠛⢛⣛⣛⣛⣛⣛⡛⢿⣟⡶⣌⡁⠀⠀⠀⠀⢀⣠⣾⣿⠿⠛⢉⡰⢜⠃⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠾⣧⣤⣬⣤⣀⣀⣀⠒⠀⠒⠤⠻⠶⣹⣦⣄⢀⣴⡟⠯⢑⢠⡔⠊⠁⠈⠀⣀⡭⣷⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣠⣤⣤⠠⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⡭⣿⣁⣤⡶⠿⠃⣀⣤⡶⢾⡛⠉⠀⠼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡍⠻⡶⢦⣌⣉⠛⠛⠛⠶⢶⣦⣤⣤⣀⣒⣰⣟⣉⣤⣴⣶⣟⡋⠁⠀⢀⣹⣶⣤⣤⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠶⣿⡀⠈⠄⠈⠉⠛⠷⢦⣤⣀⠀⠈⠙⠛⠿⠿⣿⡿⣿⡿⠛⢉⣠⣴⠾⠟⣛⣫⡭⠴⠶⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢷⡈⢿⡀⠀⠀⠀⠀⠀⣀⣀⠉⠛⠶⣤⣀⠀⠀⠀⢉⣠⣴⣾⣿⣿⣷⠾⣿⣿⣶⣦⣀⠘⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⢳⡈⠛⠛⠛⠛⠛⠙⠛⠛⠛⠷⠶⣤⣍⣛⣶⠶⢛⣽⠟⠋⣉⣤⣤⡒⠀⣀⣤⢼⡏⠙⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣞⣁⣀⣀⡐⠒⠤⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⣤⣼⣿⣿⡿⠷⠛⢋⠛⢉⣠⣴⠾⢻⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣏⣉⣉⠉⠛⠓⠶⢦⣤⣀⡀⠀⠀⠀⠀⠀⢀⣴⠏⠉⢹⡇⡄⠀⢀⣴⠶⠟⠋⠉⠀⠀⠘⡿⠛⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡈⠉⠙⠓⠶⢶⣭⣍⣛⠶⣦⣄⣴⣿⣧⣴⣶⣾⡧⠶⠚⠉⠀⠒⠒⠒⠀⠀⠀⢠⣷⠀⠀⢵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡹⠛⠛⠛⠓⠶⠦⣤⣌⣉⡛⠛⠿⢷⣿⡃⢄⡸⢦⣑⣲⠿⠿⡛⠛⠛⠛⠻⠶⣦⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣷⣶⣤⣀⠀⠀⠀⠙⣯⠙⠛⠳⣶⣟⡉⠛⠿⢿⣶⡵⠞⡩⠐⢉⠄⠂⠀⢀⣠⠿⣍⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣿⣿⣿⡍⠛⣷⣤⣄⠘⠓⠛⠛⠉⠉⠙⠷⣦⣤⣀⠀⠀⠀⠊⣁⣤⣶⣿⣯⣉⣠⠬⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⣠⠟⠙⠛⢷⣤⣾⣿⣿⣶⣤⡀⠀⢀⣤⠾⠿⠬⣿⣷⣶⣶⣿⡿⠟⣋⡩⢬⡐⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⣀⣀⣀⠉⠛⠻⠿⣷⣾⣿⢾⣏⣁⣀⣠⣴⡾⠜⠛⣉⠵⢞⣋⣽⣾⣃⡴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠉⣻⣿⢿⣶⡶⢦⣤⣤⡴⠟⠉⠛⣻⣗⠤⠔⠒⢊⣽⠿⢋⣵⣯⣼⠻⡶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⢸⠁⠈⣷⡈⠙⠛⠷⢦⣤⣀⢖⣚⣿⡍⠑⣲⢛⣻⠶⣞⣛⣭⠴⠾⠃⠘⣆⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠉⠲⢤⣀⢻⣤⣴⠶⠋⠉⠉⠹⠷⢮⣿⣿⣭⠷⠞⠛⣉⣥⠄⠂⠀⠀⠀⢹⡆⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠐⠦⢄⡈⠛⠿⣤⣀⠀⠀⣰⡷⠲⣤⡉⠉⠀⠀⠐⠉⠁⢀⣀⣤⡶⢾⣛⣛⣡⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣶⣤⣀⠀⠉⠓⠀⠀⠉⠛⣶⣟⠀⠀⡏⢻⣄⠀⣀⣤⡶⠿⠟⠛⠋⠉⠉⠉⢨⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣌⡝⠳⢦⣄⡀⠀⠀⢰⣿⠙⣿⠸⣧⠀⠹⠿⠋⠁⠀⣠⡀⠀⠀⠀⠀⠒⠚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠈⠱⠤⠉⠛⠷⣦⣼⣿⣿⠃⠀⣹⠆⠀⠀⠈⣉⣡⣤⠴⠖⢊⡽⠋⠁⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⣿⠃⠀⠀⠀⠀⠀⣀⡀⠀⢁⡀⠀⠀⣿⡤⠴⣚⠋⠉⠁⢀⣠⠞⠋⠠⣦⡴⠴⣿
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣶⣤⣍⣛⠶⢦⣤⣀⠀⠀⠈⠉⠙⢶⢧⣤⣾⢁⠤⠄⠉⠓⠾⠋⠁⠀⠉⠉⢀⣰⣤⣿⠃
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢻⡿⠿⠿⣶⣮⣭⣻⣶⣄⣀⠀⠉⠑⣺⣏⢀⡀⠒⣀⡠⠤⠔⣒⣠⠶⢟⡛⠉⠁⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣷⣦⣄⡀⠀⠀⠀⠀⠙⠻⢿⣶⣦⣬⣿⣠⣄⣠⣤⠴⠖⠋⠡⠔⠊⣁⡤⠐⠀⠀⢀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⣶⣤⣀⠀⠀⠀⠈⣙⢛⣿⠛⠛⣯⠳⣄⠀⠀⠀⠀⠊⠁⢀⣠⠤⠚⠋⠁
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣦⣤⣈⣉⢻⣀⣤⣬⢷⡌⣳⡀⠀⠀⠔⠚⠉
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠾⠛⠋⠐⠉⠉⠁⠛⠊
⠀⢀

⠀⠀⠀⠀⢠⣾⣿⣿⠀⠀⠀⠀⠀⢸⣶⣶⣾⠀⠀⠀⠀⠀⠀⠀⢠⡜⢛⠛⠀⠀⠀⠀⠀⢸⡗⣿⢺⡇⠀⠀⠀⠀⠀⠀⢀⡞⠉⢻
⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣟⠀⠀⠀⠀⠀⠀⢀⣿⡧⠄⠀⠀⠀⠀⠀⠀⢘⡋⣿⣙⠃⠀⠀⠀⠀⠀⢀⣿⡗⠓⢺
⠀⠀⠀⣨⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⠀⠀⠀⠀⠀⣠⣿⣿⡿⠛⠀⠀⠀⠀⠀⠈⠱⣿⣿⡇⠀⠀⠀⠀⠀⣨⣿⣷⡶⠚
⠀⠀⠀⠙⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠅⠀⠀⠀⠀⠀⠙⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠿⠄⠀⠀⠀⠀⠀⠙⠻⠟

⠀⠀⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡀⠀⢀⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠿⠿⣿⣯⣽⠿⠿⠃⠀⠀⠿⠿⣿⣿⣽⠿⠿⠃⠀⠀⠿⣿⣿⣿⣿⣿⠿⠃⠀⠘⠿⠟⡍⣉⡩⠻⠿⠃⠀⠀⠿⠟⠍⣉⡩⠻⠿⠃
⠀⠀⠀⠀⣿⣿⣿⠀⡶⠀⠀⠀⠀⠀⣿⣿⣿⠀⣦⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣏⠉⢱⠀⡶⠀⠀⠀⠀⠀⣊⠉⢱⢀⣆
⠀⠀⠀⠀⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢄
⠀⠀⠀⠀⢸⣿⡇⢜⢦⠀⠀⠀⢀⠞⠛⢷⣦⣄⠀⠀⠀⠀⠀⢸⣶⣶⣶⡇⠀⠀⠀⠀⠀⢸⠛⠛⠛⡏⠀⠀⠀⠀⠀⠀⣤⣴⣦⣾⣧
⠀⠀⠀⠰⢁⣀⠀⣇⠀⠀⠀⠀⡜⠠⠸⡋⠉⠈⡄⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠘⢤⣀⡠⠃⠀⠀⠀⠰⢶⣤⣠⢣⠀⠀⠸⡀⠀⠀⠀⠘⢻⣿⡿⠃⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠇⠀⠀⠀⣶⣿⣿⣿⣯⣤⣤⡀
⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠷⠶⠛⠃⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠸⠶⠿⠷⠇⠀⠀⠀⠀⠿⠛⠛⠛⠻⠿⠿

⠀⠀⠀⠀⣤⣿⣤⣀⣤⡀⠀⠀⠀⠀⣤⣿⣦⡀⠀⠀⠀⠀⠀⠀⣤⣿⣤⡀⠀⠀⠀⠀⠀⠀⣤⣿⣤⣀⡀⠀⠀⠀⠀⠀⣤⣿⣤⣠⡀
⠀⠀⠀⠀⣇⣀⣸⠉⠄⢣⠀⠀⠀⠀⣇⣀⣸⣿⣿⡇⠀⠀⠀⠀⣇⣀⣨⣿⣷⠄⠀⠀⠀⠀⣇⣀⡰⠛⡑⢄⠀⠀⠀⠀⣇⣀⣹⣿⣷⡀
⠀⠀⠀⠀⣿⣿⣿⢗⢂⣀⡆⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⠍⠉⠀⠀⠀⠀⠀⣿⣿⣿⠆⣤⡄⠀⠀⠀⠀⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠻⠿⠟⠈⠋⠉⠀⠀⠀⠀⠻⠿⠟⠈⠉⠀⠀⠀⠀⠀⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⠈⠁⠀⠀⠀⠀⠻⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⡉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⣀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⢰⡏⠉⠉⠉⠉⠉⣷
⡏⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠓⠒⠒⠒⢺⡇⠀⠀⠀⠀⠀⡿⠒⠒⠒⠒⠒⠒⠒⢶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢶⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣻
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⢤⠀⢀⡤⠴⠶⠤⣄⠀⠀⡤⠤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢲⠒⠚⢠⠋⢀⠖⠒⢆⠈⢣⠀⡇⢰⡒⡆⠈⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢈⣉⣽⢸⠀⢸⠀⠀⢸⠀⢸⠀⡇⠀⣉⠁⣜⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⠀⠀⠈⢦⡀⠑⠒⠃⣠⠏⠀⡇⢸⠉⢆⠈⢦⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠉⠉⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⡖⠒⢒⠒⠲⣄⠀⡖⠒⠒⠒⣲⢰⠒⠒⢲⡀⢰⠓⢲⢰⠒⠒⠒⠒⢒⡆⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⢯⡽⠀⡸⠀⡇⠀⠯⠭⡅⢸⠀⣶⡀⢳⣸⠀⢸⠈⠉⡇⠀⡏⠉⠁⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⣤⡀⠸⡅⠀⡇⠀⡖⠒⠃⢸⠀⡏⢱⡀⠻⠄⢸⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⣧⣀⡇⠙⣄⣙⣆⣇⣀⣉⣉⡽⢸⣀⡇⠀⠱⣀⣀⣼⠀⠀⢧⣀⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠸⣇⣀⣀⣀⣀⣀⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣤⣤⠀⣠⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠰⠶⠶⠶⠆⠠⠶⠶⠄⠀⠀⠀⠀⠀⢀⡴⠖⠛⡛⠳⢦⣄⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⣈⣁⡄⠀⠙⣧⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⠤⠤⠤⠤⠴⠤⠦⠤⠤⠴⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠻⠦⠤⣤⠀⠀⢹⠆⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡄⠀⣤⠀⢠⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⢷⠹⣄⠀⠐⠒⡒⠋⠀⢀⡿⠀⠀⠀⢸⡇⠛
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣌⠳⣝⠶⣤⣈⣁⣤⠴⢋⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣇⣙⣛⣛⣋⣉⣉⣉⣉⣉⣙⣙⣛⣋⣸⡇⠀⠀⣤⣤⡄⠀⠀⠈⠳⣬⣛⡲⠶⠶⠶⠚⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⠀⠀⠚⠛⠂⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢏⡴⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⢷⡘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣡⠟⣀⣀⣈⠉⠉⣁⣀⡈⠉⠉⣁⣀⡉⠉⠉⣀⣀⡉⠉⢉⣀⣀⡉⠉⢉⣀⣀⠉⠉⢉⣀⣀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠉⠉⠉⠀⠀⠉⠉⠉⠀⠀⠉⠉⢁⣀⣈⣉⣉⣁⣀⣈⠉⠉⠁⠀⠈⠉⠉⠀⠀⠈⠉⠉⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣏⣉⣉⣉⣉⣉⣉⣙⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠖⠀⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⠐⠛⠛





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠒⠂⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠦⠤⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠑⠄⠀⠀⠈⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠉⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠁⠀⠄⠀⠀⠊⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠐⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠰⢷⠀⠀⠀⠀⠀⠀⠙⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢠⡂⠀⠡⠂⠀⠄⠀⠀⠀⠀⠠⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⡁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠔⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠰⠀⠀⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⡁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀⡀⠀⠄⠞⠂⡑⢄⠀⠀⠀⠀⠔⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠊⠈⡀⢀⠀⠀⠀⠀⠘⢄⠁⠀⠀⠀⠈⠢⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⢴⡀⠀⠀⠀⠀⠀⠀⠀⠄⠰⡄⠀⠄⠈⠕⠀⠀⠀⠀⠀⣬⡤⠴⠓⠀⡐⠀⠀⠠⠑⠀⠀⠠⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠢⡑⢄⠂⠀⠀⢈⠢⡀⠢⡑⣌⠢⣀⠘⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣦⠀⠊⠁⢈⣣⡚⠀⠀⢂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⢀⠁⠁⢀⠀⠋⠜⠂⠈⠑⠶⠈⢢⡄⠀⡀⠤⣖⡱⠐⠀⠀⢀⠀⠰⡣⠀⠀⠨⠓⠁⠀⢀⢘⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠁⢌⠢⢃⠂⢄⠈⠀⠀⠒⠲⠦⣤⣤⠖⠋⠁⠰⠂⠁⠘⠛⠛⠃⠁⠀⠀⠁⠘⠁⡤⣌⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢡⣌⠓⡀⠄⠐⣊⣤⠗⠋⡙⠦⢰⢤⠂⡿⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⢀⡆⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⠀⠜⢋⡡⠶⡛⠁⠀⠀⣚⣅⢀⡉⡆⠰⡏⠀⠀⠀⠉⠉⠉⠉⠙⢻⠀⢸⠀⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢁⠡⠂⠀⠀⠀⠢⡙⢯⠀⠈⠁⠀⠀⠀⢠⣘⣇⠀⠀⣠⣤⡀⠀⠀⠀⠌⠀⢸⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠢⣑⠄⠀⠀⠀⠰⡏⠀⠀⣹⣿⠁⠀⢙⣷⣤⣤⣤⣤⡞⢠⠍⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠊⠁⠈⠑⠒⠛

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡿⠿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣿⣧⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣤⣴⡿⠋⢀⣤⣤⣄⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣽⡿⠋⢀⣴⡿⠋⠙⢿⣷⡀⠈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠀⣠⣿⠟⢁⣀⣀⣀⠙⢿⣦⡀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⣠⣾⠟⠁⣴⡿⠛⠛⠻⣷⡄⠙⣿⣦⡀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⢀⣾⡿⠃⠀⢸⣿⡀⠀⠀⠀⣹⡷⠀⠈⠻⣷⣄⠈⠻⣿⣄
⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢀⣴⡿⠋⠀⠀⠀⠀⠻⣷⣤⣤⣶⡿⠃⠀⠀⠀⠈⠻⣷⣄⠈⠻⣷⣄
⠀⠀⠀⢸⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣏⢀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⡀⢘⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠛⢹⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠙⠿⠿⠋
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠾⠆⠀⠾⠆⠰⡿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣠⣴⣶⣾⢶⣶⣶⣄⡀⠀⠀⠀⠀⢸⣿
⠀⢀⣤⣤⣤⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣾⠟⠉⠀⠀⠀⠀⠈⠻⣿⣆⠀⠀⠀⢸⣿
⢠⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⢸⣿
⢸⣿⠀⢸⣿⠿⣿⣆⠀⢸⣿⠿⠂⢸⣿⣿⢸⣿⠀⠸⠿⣿⡿⠗⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿
⢸⣿⠀⢸⣿⣤⣾⡟⠀⢸⣿⣤⡄⢸⣿⣿⣾⣿⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿
⢸⣿⠀⢸⣿⠉⢹⣷⠀⢸⣿⢉⠀⢸⣿⢻⣿⣿⠀⠀⠀⣿⡇⠀⠀⣿⣇⢀⣀⣀⣀⣀⣀⣀⡀⣿⣇⢀⣀⣸⣿
⢸⣿⠀⠘⠛⠀⠘⠟⠀⠘⠟⠛⠃⠘⠛⠈⠛⠛⠀⠀⠀⠻⠃⠀⠀⣿⡿⠛⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠟
⠈⢿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡿⠃
⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢽⡏
⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇
⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣸⡇
⠀⠀⠀⠘⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠂⠀⠀⠀⢒⠤⢹⡶⡖⠒⠒⠒⠀⠂⠀⠀⠐⠒⢒⡊⢳
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢸⡦⡸⢹⡄⣀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠇⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡧⠄⠀⠉⠁⠀⠘⣡⣿⡞⠃⠀⠀⠀⠀⠀⠈⠀⠈⢉⡀⠀⠀⠇⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠄⠀⠀⠀⠀⠀⣽⡏⢡⡀⠀⠀⠀⠀⠀⡤⠀⠀⠻⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⣀⠀⡈⠀⠀⠻⢣⣸⡇⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣯⠘⠋⠁⠀⠀⠀⢀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⢘⡫⠀⠀⠀⠀⠐⠂⠎⠸⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀⠀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣯⠀⡀⠈⢉⣭⣿⣶⠆⡖⠰⣦⠴⠆⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡤⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢿⣧⣿⣿⣯⣿⣟⢚⠸⡱⠿⢿⣻⣖⠀⠀⠀⠀⡀⠀⠀⠐⠛⠆⣳⣾⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣯⣭⣭⣟⣻⣿⠏⠁⣀⡀⠙⠛⠳⠶⠶⣤⣤⣄⣀⣀⠚⠛⠉⠀⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒⠤⠤⣈⣀⣐⠶⠯⢥⡀⠉⢟⣛⠿⢿⣶⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠙⠛⠛⠙⠒⠚⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠙⠻⡿⠿⣿⢿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣈⡄⠨⠄⠀⠿⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣆⣀⣀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⠛⠿⠿⣿⡿⠃⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⠿⠀⢀⠀⠀⠀⠀⠂⠘⠿⠿⣿⣿⢿⠶⠎⠙⠿⠿⠹⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠭⠉⠋⠽⣶⡖⠒⠞⠠⡈⠀⠀⠀⠀⣀⡀⣀⣀⠐⠤⠤⠴⠲⠤⠄⠀⠠⠼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠔⠒⠒⠉⠉⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠚⠚⠉⠉

⠀⠀⠀⠀⢀⣀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠈⠉⠁⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠦⠀⠄⠂⠀⠀⠀⠁⠀⠔⠃⠀⠀⠀⠀⠀⡤⠂⠤⡄
⠀⠀⠀⢀⣯⣤⣶⣶⣄⣄⠀⠀⠀⠈⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢈⢠⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠆⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⠠⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠘⠛⠟⠁⠀⠸⡆⠀⢀⡤⢀⣀⣀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠊⣉⠥⠐⠛⠁⠀⠈⠈⠩⠉⠀⠂⠀⢼⣆⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⠁⠀⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⢸⣿⠀⠀⣠⠶⠦
⠀⢀⠈⡠⠂⠀⠀⠀⠀⠀⢸⣿⠀⠀⣴⠀⠀⠀⠄⠀⠀⡀⢀⣶⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠘⠁⠔⠀⠀⠀⠀⠀⣸⠏⠀⠀⢼⣒⠲⠞⢒⣠⣈⡀⣸⡏⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⠓⠀⠀⡆⡂
⠀⠀⠀⠰⠖⠀⠒⠀⠀⠞⠁⠀⠀⠀⠀⠉⠛⠛⠛⠻⠿⠶⠞⠁⠀⠀⠚⠓⠚⠓⠀⠐⠐⠛⠓⠛⠂⠀⠀⠀⠀⠀⠄⠀⢀⢯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢡⠋⠔⠁⠀⠀⠠⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⡀⣠⢀⡀⢀⣠⣤⣤⣶⣶⡆⢾⠁
⠀⠀⠠⢀⠀⠀⠀⠀⠀⠄⠀⠠⢁⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡟⠋⡽⠋⠹⠈⠈⢣⡼⠁⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⣀⡀⠂⠀⠀⠤⣤⣤⣤⢃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠊⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡶⠦⠄⠀⠀⠀⡀⠠⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡟⢸
⠀⠀⠀⠀⠀⢀⠴⠶⢾⠲⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠐⠀⠄⠈⠆
⠀⠀⠀⠀⠠⠀⠀⠀⠘⠄⠐⠓⠀⠂⠐⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠤⠤⠖⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⡅
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊⠀⠀⠀⠀⠠⠤⡄⣤⣤⣤⣠⣀⣤⣠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣀⠀⢀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⣄⣀⣄⣀⢀⣀⣀⣀⣠⡀⣼⠀⡓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢸⠀⡇⠀⡉⢉⠉⢉⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⣀⠀⠁⠀⠀⠀⢸⠀⠱⠤⠤⠤⠤⠤⠤⠠⠤⠤⠤⠤⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠈⠑⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠦⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠳⠚⠁⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⢤⡀⠀⠵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡷⠀⠄⠘⠦⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒⠒⢒⡒⠒⠒⠒⠒⢺⠀⠀⠀⠀⠀⠀⠸⣆⠀⠘⠒⠁⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⢒⠒⠒⣒⠒⣐⢒⠒⡒⡒⠒⡒⠒⢲⠀⠀⠀⡠⠶⠉⠋⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠋⠀⠀⢠⠟⣡⠌⠻⡄⠀⠀⠙⡆⢸⠀⠀⣜⠃⠀⢰⠛⠄⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⠀⠀⢸⡀⢈⡦⢀⡇⠀⠀⠀⡇⢸⠀⠀⢏⡀⠀⢀⣙⠆⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠳⠤⠤⠤⠷⠦⠴⠾⠤⠤⠤⠖⠁⢸⠀⠀⠈⠣⣤⠀⡁⢀⣤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠁






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⣛⣛⣛⠛⣿⣿⣿⣿⢛⣛⣛⡛⢻⣿⣿⡇
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⣿⣿⣿⠀⣿⣿⣿⠀⣿⣿⣿⣿⢸⣿⣿⡇⢸⣿⣿⡇⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣤⣉⣍⣩⣠⣿⣿⣿⣷⣌⣩⣉⣡⣼⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⡏⣭⣭⣭⡉⡇⣿⣿⣿⠉⣭⣭⣭⠉⣿⣿⣿⣿⢩⣭⣭⡍⢹⣿⣿⡇⣿⢩⣭⣭⣭⢹⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⠇⡇⣿⣿⣿⠀⣿⣿⣿⠀⣿⣿⣿⡿⢸⣿⣿⡇⢸⣿⣿⡇⣿⢸⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣧⣤⣤⣤⣴⡇⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⡇⣿⣦⣤⣤⣤⣼⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⡇⣶⣶⣶⡄⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⢰⣶⣶⣶⢸⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⡇⠿⠿⠿⠇⡇⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡇⣿⠸⠿⠿⠿⢸⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣶⣾⡇⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇⣿⣷⣶⣶⣶⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠛⠛⠛⢿⡇⣿⣿⣿⠛⠛⠛⢻⡇⢸⣿⣿⡇⣿⡟⠛⠛⠛⣿⣿⡇⣿⡿⠛⠛⠛⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣐⡄
⠀⠀⠀⢠⣦⡄⠀⠀⠀⠀⠀⠋⠄⠀⠀⠀⠐⠒⠀⠀⠱⠀⢀⠀⠸⣹⣿⢳⡆⠀⣾⣹⣿⡿⢿⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⣤⣦⣤⣦
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢰⣀⣶⠀⠀⠀⢀⣀⣸⣀⣀⠀⠘⠤⠄⢻⠀⠀⠀⠀⠿⠯⠥⠥⠼⠃⠀⠀⢻⡯⣝⠛⠀⠀⠀⠷⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠰⢀⠀⠀⠀⠀⠀⣿⣦⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⢠⡶⢶⠀⠀⠀⣀⣤⣤⡄⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠉⣿⣿⣿⣶⡆⠀⠠⣿⢟⣷⡿⠃⠀⠠⠒⠶⠶⠖⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠻⠿⠿⠀⠀⠀⠀⠿⠦⠤⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠙⠟⠁
⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣿⣿⣿⡿⠀⠀⠀⠰⠶⣾⣶⣶⠀⠀⠰⠶⣾⣂⣀⠀⠀⠀⡀⢸⣦⠀⠀⠀⢀⣶⣿⡏⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⢺⡿⢷⡦⠀⠀⠀⠰⠴⠿⠿⠿⠀⠀⠀⠀⢹⠉⠁⠀⠀⠀⢙⣍⣙⠃⠀⠀⠠⠤⠤⠤⠿⠧⠀⣰⣷⣾⣧⣤⡀⠀⢾⣦⡎⠱⠿⠆
⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠋
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢾⡥⠙⠀⠀⠀⠀⣴⣶⣶⡄⠀⠀⢰⣦⣀⣠⣞⠀⠀⢠⣀⣂⡄⠀⠀⠀⠀⣠⢀⣀⣀⠀⠀⠀⣬⠀⢀⡃⠀⠀⠀⠀⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠈⠁⠈⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⠋⠛⠛⠋⠀⠀⠀⠀⠙⠈⠁

⠀⠀⠀⢠⠞⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⠀⠀⠀⠀⢀⡄⢀⣀⣀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⢀⣶⣠⡄⠀⠀⠀⡶⢶⠶⢤⡀
⠀⠀⠤⠀⠀⠙⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠘⠛⠛⠉⠛⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠛⠟⠀⠀⠀⢀⣷⣾⡶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠒⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠠⠀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⢀⣉⣍⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠨⢉⢉⣹⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢘⠆⠀⠀⠆⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠉⠀⣀⠒⠈⢌⡐⠂⠤⢀⣀⠠⣤⠼⠉⠠⠶⠏⣤⣀⡀⢐⠢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠉⠉⠅⢘⡂⠸⠁⠀⠈⠉⠁⠀⠀⠀⠀⢀⣀⣤⣠⠀⠐⢊⠓⠟⠇⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠂⠀⠐⠢⠖⠚⡇⠀⠀⠀⠀⠀⠀⢀⠂⣅⡉⠉⢙⠃⠀⠀⠐⠛⠛⠉⠲⠦⠤⠀⡠⠀⠀⠀⢲⡀
⠀⠀⠀⠀⠀⠖⠒⠒⠠⠴⠀⡄⠸⠂⠀⠀⠀⠰⠤⠜⢃⣀⡀⠄⠤⣄⢰⢒⣋⡕⣒⠷⠶⣣⣅⣒⡦⠤⡁⠀⠉⠓⢈⡉⢄⠔⢒⠉⠉⡅
⠀⠀⠀⠀⠀⠃⠤⠀⠠⠼⠁⡅⠰⠦⣀⣀⣀⣀⣀⡠⢞⠈⠉⢈⠢⣿⠸⠀⠒⠨⠍⠛⠁⠀⠬⣁⡈⠉⠀⠍⠐⠽⢒⠉⢉⠩⢀⠀⠀⡇⡀⣠⢀
⠀⠀⠀⠀⠐⠣⠤⢤⡗⠟⠋⠭⣝⠄⠀⠀⠀⠀⠀⢀⡼⢄⠀⠀⡥⣴⠶⢄⠤⣠⣤⣀⣤⣄⣤⣄⡀⠀⠀⢀⣤⠤⠶⠎⣒⣩⡅⠖⠓⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠧⣀⣐⣀⣠⣆⣀⠀⠀⠀⣙⣰⠞⠓⠲⠒⢚⢭⢆⢲⠂⠀⠀⠀⠀⠈⠙⠙⠻⠽⠀⡠⠤⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠁⢪⡁⠒⠀⠐⡒⠂⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠐⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢀⡀
⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠀⠀⢀⣠⠔⠙⠓⠛⠀⠀⠀⠰⠅
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁⠀⠄⠀⠀⠀⣀⣴⣋⠀⠈
⠀⠀⠀⠀⠀⠂⠈⠀⠀⠈⠁⠒⠀⢤⢀⢀⠀⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⠈⠀⠀⡠⢔⡾⠋⠁⡠⢔⡫
⠀⠀⠀⠀⠀⠀⡀⠄⠒⠤⢄⣀⠀⠀⠀⠀⠁⠉⠉⠋⡉⠀⠠⠀⠀⠀⣀⠠⠂⣩⠔⢁⡠⢒⠍⢚⠵⣺
⠀⠀⠀⠀⠀⠀⠀⠐⢠⣄⢀⢀⠉⠒⠠⠤⣀⡀⠀⠀⠀⠀⣀⡠⠔⢊⡠⠖⡁⢔⡪⠔⠉⢀⡤⠖⠋⠩
⠀⠀⠀⠀⠀⠀⠤⣀⣀⣀⠀⠈⠈⠙⠒⠒⠤⠤⣍⣑⣒⡋⢁⡠⢖⡡⣐⠥⠒⠁⣀⠴⠚⠉⠀⣠⡠⡇
⠀⠀⠀⠀⠀⠀⠗⠀⠒⠂⠈⠭⠅⣒⣀⠤⠄⣀⠀⠀⠈⢈⠤⣒⠅⣊⢄⣨⠵⠊⣁⣠⢤⡺⠟⣋⣶⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠈⠩⠀⠚⠩⠭⠕⠊⣔⡯⠑⠉⢥⢴⣛⡽⠚⡹⢖⡯⠗⣻⠃
⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠁⢀⣨⠴⢞⠍⠀⣀⠥⠚⡡⠔⡋⢸⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢺⣷⡾⣫⠕⠋⠄⠒⠉⣠⡔⠬⠐⠁⢠⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠁⠐⠆⣭⠝⠊⠁⠂⢀⣠⠶⠛⠁⢀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠦⣄⣀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠄⠂⠀⢀⠤⠚⠉⡀⢀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠢⠄⣀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠠⠄⠀⠈⠀⠠⣐⡼⠞⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⠄⣰⠿⠭⠄⠀⠀⠀⠀⠒⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠵⠋⠀⠒⠒⠒⠒⠦⠄⣄

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⡀
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠈⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣰⣶⣦⡀⠠⠄
⠰⠶⠲⠤⠤⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⡖⠐⠒⠂⠈⠉⠉
⠀⠤⠀⠀⠀⠀⠀⠠⠤⠶⠦⠄⠀⠀⠀⠀⠀⠀⠀⠢⠽⠿⠟⠁⠀⠀⠀⠀⠀⠠⠤⠮⠤⠄⠀⠀⠀⠀⠀⠀⠤⡤⢤
⣤⣤⣤⣤⣄⣀⣀⡀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢀⣀⡀⢀⠀⠀⣤⣤⡄⠀⣤⣤⣤⣤
⣿⣿⣿⣿⣶⣿⣿⣷⣶⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⣿⠄⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⣤⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠹⣷⣤⣾⢿⠯⠿⠛⣿⣯⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⡄⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⠀⡒⡺⠟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠴⣚⠶⠿⢦⣶⣮⣽⣿⣿⣿⣿⣿⣛⢛⣿⣶⣦⣌⣛⢷⠶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⢤⣯⣛⣻⠿⣿⣟⣛⡛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠐⠒⠶⠒⠶⢶⣦⣴⡶⢦⣍⠉⢿⣿⣿⣿⡿⣛⢛⠉⣙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣛⣫⣾⣻⣿⣿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠤⠥⢤⣄⣿⣷⣮⡥⢶⣶⣤⣬⣹⣿⣿⣿⠷⠖⠂⠙⢿⣿⣷⣶⣮⣝⡛⠿⢿⣿⣧⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣀⣀⣀⣤⣤⣍⣙⣛⣛⡛⠛⠀⠀⠀⣊⣉⣙⡛⠛⢛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢀⣤⣴⣦⣤⡄⠀⠀⢐⠒⠒⠂⣀⡀⠀⠀⠀⢀⣰⣾⣾⡆⠀⠀⠀⠠⢤⠤⠀⠀⠀⠀⠀⣠⣾⣧⣠⠂
⠀⠘⠿⠿⠿⠿⠇⠀⠀⠘⠀⠀⠣⠀⠀⠀⠀⠻⡿⠛⠻⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣆⠀⠀⠀⠀⠄⣦⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣾
⠀⠀⢸⣿⣤⣄⠀⠀⠀⠀⠸⠿⠿⠿⠄⠀⠀⠀⠀⠚⠁⠟⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢰⣾⣶⣾⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢠⣾⣿⣄⠀⠀⠀⠀⠠⢨⡂⠀⠀⠀⠀⢸⡇⣶⣿⣿⠀⠀⠀⠀⠀⣠⡺⠟⠁
⠀⠀⢹⣿⣾⡿⠀⠀⠀⠀⢠⢏⠷⡄⠀⠀⠀⠈⣇⣻⣿⡟⠀⠀⠀⣠⣦⠟⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢠⣤⡄⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⡏⡇⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣀⣤⣶⡦
⠀⠀⣾⣿⣿⣿⡄⠀⠀⠈⠉⠉⠉⠈⠁⠀⠀⠀⠀⡇⢥⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠿⡻⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⠉⣨⣴⣿⣿⣿⣿⣿⡟⢿⣿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣷⣶⣼⣾⣷⡟⢩⣿⣿⣿⣛⢛⣭⣿⠎⣉⡠⢧⡃⠉⢻⡆
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠿⣟⣟⣭⣭⣽⢻⣬⣷⣾⣿⣿⣿⣿⢿⣿⣿⣿⣭⠤⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠞⠁⢁⣬⣶⣿⣿⣿⣿⣿⢿⣯⢿⣿⢿⣿⣿⣦⡥⢿⣿⢿⣿⣿⣿⡿⠒⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⡟⣼⣿⣿⢿⣹⣿⣟⣻⣿⢹⢿⣿⣿⣿⣿⢸⢻⣿⣿⣷
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠷⣼⣿⣿⣿⣆⣻⣿⣿⣖⢿⡜⠘⠏⢹⣿⣿⣌⡀⠀⠹⠛
⠀⠀⠀⠀⠀⣼⠿⠟⠏⠈⢠⣿⣿⡝⠯⡧⢹⣿⣯⡓⠸⣷⠀⠀⠏⢿⢿⡓⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⡿⡝⠆⠀⢸⣿⣦⡉⠀⣿⢀⣤⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠑⠈⠀⠀⠀⠻⠇⣩⣶⣿⡝⢿⣿⡿⣶⣤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⡿⣫⠵⢿⣹⡇⠀⠀⠹⡄⢹⠷⠄⠀⣀⣙⠿⣔⣲⣤⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠛⠛⠛⠛⣿⣷⣾⣖⣲⣤⣄⣿⡗⠛⠛⠓⠂⢲⣶⡼⠶⢦⣸⣷⣶⡿⠿⠿⠿⠛⠛⠛⠻
⠀⠀⠀⠀⠛⠉⠛⠒⠒⠒⠚⣁⣤⣤⠤⠤⠌⠈⠉⠒⠒⣿⡇⠉⢁⣷⡆⢠⠈⣡⠈⣋⣉⣣⠤⠴⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⡾⢸⢻⢃⣮⢞⢥⠀⠠⠤⠭⣭⣿⣿⣯⣈⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣬⡏⢀⡟⣯⣈⣬⡃⢻⣾⣯⣀⠀⠐⣾⠿⠿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠒⠊⠉⢛⡽⠋⠋⠀⠸⠘⠹⠉⠳⠈⠙⠛⠿⠿⢽⣿⠿⠿⠓⠒⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⣀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣵⣿⣿⣿⣿⢿⣾⣙⣻⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣽⣿⣿⣿⢿⣵⣿⣿⣾⣿⣿⣿⣭⣭⣝⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⢀⣀⣤⠄⠀⠀⠀⠀⠈⠿⠋⠈⠋⠁⠉⠉⠛⠛⠋⠉⠉⠉⠛⠛⠻⠿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣏⠀⠀⣀⣠⣤⣶⠿⠛⣉⣠⣤⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣭⣴⣶⡿⠟⢋⣁⣤⣴⡶⠆
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣶⡿⠿⠛⠋⠁
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣟⣵⣿⣏⣿⣿⣿⣿⣿⣏⣀⣤⣤⣶⡶⠶⠂
⠀⠀⠀⠀⠨⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⢻⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁







⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣤⣀⡀⠀⠀⠀⣿⡄⠀⢀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣷⡀⠀⠈⠉⠈⠁⠄⡀⠀⠀⠀⡼⠁⠀⠀⢀⣤⠞⠉⠉⠉⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⢟⡛⣷⣄⡀⠀⠀⠀⠈⣓⣤⣠⣇⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣍⣹⣿⣿⠟⠉⣻⣤⣄⠲⠿⠛⠉⠉⠙⢦⣴⡟⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⡀⠃⢬⠙⠻⠸⣯⣿⣿⣶⣤⠀⠀⠀⠀⠸⣏⠙⠢⢀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⠦⠀⠀⠀⢀⡀⠈⠉⠛⢿⣶⣦⣤⡀⠀⣭⡄⠀⠀⠙⣦⣀⣰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣟⣿⣿⣯⣥⣶⡿⠦⡀⠀⠁⠠⣄⠀⠀⠈⠙⢫⣿⣷⢾⣧⣀⡀⠀⠸⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⣿⣿⣾⣿⢿⣿⣿⣿⣯⣦⣀⠀⠈⠁⠀⣀⠀⠀⠈⠛⣿⡟⢛⣽⣶⠀⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣴⡿⠛⣴⣿⣿⣿⣿⣿⢻⣷⣤⡀⠀⠀⠙⠂⠀⠀⠀⠻⣾⣿⣶⡿⣯⣥⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡿⣶⢿⢏⣽⢷⣿⢿⣿⣿⣿⣋⣴⣄⡀⠀⠀⠀⠠⢄⡘⢛⣤⣜⣿⣿⠏
⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠈⣿⣛⢿⣻⣶⡜⢿⣿⣿⣿⣿⡿⣿⣿⡿⣷⡮⢱⣤⢀⡠⠀⠈⠂⢽⣿⡿⠃⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⣻⣿⢿⣿⣿⣦⣼⣿⣿⣿⢿⣟⣿⣿⣿⣟⢻⣿⣾⢛⣽⣶⠄⠀⠀⠀⠀⠀⡀
⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠠⣟⣿⣿⣿⣿⡿⢿⣿⣷⣾⡿⡟⠃⢙⣷⣿⣯⣾⡟⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣰
⠃⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰⣿⡿⢶⡿⠋⣴⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣤⣙⣿⣿⡉⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⡜⠡
⠇⠀⠀⠰⠀⠀⠀⠀⠀⠀⣼⠉⣻⣿⣻⣿⡿⠟⣰⣿⣿⣟⢻⣽⣿⣷⣿⡿⠿⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠠⠦
⠂⠀⠀⠀⠀⢠⡀⠀⠀⢀⣿⣶⣾⡟⠋⣽⣿⣆⠻⠿⣿⣿⠿⠛⠋⠁⠀⢠⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⣠⢄
⡄⠀⠀⠀⠀⠈⡃⠀⠀⣬⣿⣿⣿⣾⣿⡷⢿⣿⡶⠚⡃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡀⠈
⠁⠀⠀⠀⠀⠐⠀⠀⣐⣿⣿⣿⣿⣽⣿⣿⠟⢿⣿⠃⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠄⠁⡄⡀
⠇⠀⠀⠀⠀⠀⠀⣠⣿⣿⣹⣿⣿⣿⡟⠁⠀⠈⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⡿⠋⠀⠀⠀⣴⣦⠀⠀⠀⢀⣀⣤⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠃⠀⠀⠀⠀⢀⣠⣄⠀⠀⢉⠛⡁⣀⣀⠀⠀⢸⣞⣠⣴⣾⣬⡽⠿⠁⠠⠔⠲⠀⡤⣤⢄⣆⣠⣀⣀⣀⣀⣀⣀⣠⣆⣀⣀⣀⣀

⠀⠀⠀⠀⠀⢀⡀⡠⠐⣲⣷⣿⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣯⠭⠁
⠀⠀⠀⠀⡴⠛⣻⣟⣿⣡⢾⠏⠚⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣷⠏⣯⡉⠀⠀⣠⣶⡒⣂⣤⣤⡀
⠀⠀⠀⠀⠀⢸⠟⠹⠓⡹⠋⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠁⡙⢀⠀⠀⠋⠘⢷⠈⠋⠗
⠀⠀⠀⠀⢀⠎⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠄⡀⠀⠑⠠⡀
⠀⠀⠀⡠⠃⠀⡠⠋⠀⠤⠀⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡈⠑⠀⡀
⠀⠰⠞⢁⣴⠟⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠗
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⢿⠿⡿⡿⣿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠒⠀⠀⠀⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⢉⡀⢉⣉⣁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠞⡶⢄⡶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡤⢏⠉⣠⠖⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢠⡤⢉⠤⡴⡩⠏⣀⠠⡾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡕⢠⠤⠤⢠⠄⢐⣈⠃⡀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⢀⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣤⣄⡈⠋⠴⠨⠛⠐⠒⠏⠠⣄⢎⡟⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡶⣧⡤⠃⡒⢠⠔⢲⣠⡦⠜⠒⡒⠲⠎⠁⠀⠀⠀⠀⠀⠀⠀⡖⠤⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡜⠁⠀⠀⠉⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣴⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⠂⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣘⣷⣤⣧⣾⡻⠋⠉⠙⠛⠲⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣣⠅⠼⠘⠞⠒⠝⠸⠞⢀⠖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣬⣭⣻⣿⣿⠵⠾⠿⠿⠒⠶⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⢾⡿⢛⡿⣯⣛⠟⠻⣶⣤⡈⡂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠁⠞⣵⣿⡆⣇⠙⣦⡀⠈⠙⠆⠙⠅⠀⠀⠀⠀⣠⣼⣷⡛⣚⠷⠦⠀⢀⣳⣠⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠋⡄⣤⢻⣿⡄⠹⢶⣽⣿⣄⠀⠀⠀⠐⡀⠀⠐⣣⢞⢿⣷⣬⠈⢳⢤⣴⠞⣻⣿⠛⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢡⠻⠘⣿⡇⠘⠀⢹⡷⠙⠐⠆⠀⠀⠀⠀⣸⠧⠈⠈⣿⣟⢷⡀⢸⡏⢺⡏⢌⠓⣆⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠙⠀⠀⢠⡀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠸⣿⡁⠀⠀⠀⠁⠂⠰⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠦⣄⡀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠈⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠠⡀⠀⠈⠙⢦⣀⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠤⠬⠯⠿⠿⠤⠴⠖⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠻⣆⣤⡀⠀⠀⠹⣿⡄⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢈⣉⡀⣠⣤⠴⠚⠋⠀⠀⢠⢴⠖⠈⠀⠀⠤⠐⠂⠙⢿⣯⢦⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣤⡤⠶⠶⠶⣶⠾⠿⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠈⠐⣴⣤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⣯⣷⡀⠀⠈⠳
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡤⠤⢤⠤⣤⡤⠂⢀⡤⠖⠲⠿⠟⠛⠋⢀⡾⠀⠀⠀⠀⠀⢀⣤⠖⠒⠈⠁⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠶⢞⣋⣩⣥⣤⡤⣶⢶⡶⠶⢶⣦⣤⣄⣀⣀⣀⣠⣤⡤⡶⠚⠋⠀⠀⣀⣀⠤⠚⢉⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠠⠤⠤⠤⠤⠴⠶⠖⠒⠒⠂⠉⠉⠉⠉⠀⣀⣀⡤⠨⠀⠀⠴⠒⠊⠁⢀⡀⠄⠚⠁⠈⠀⠀⢀⣜⠛⣋⣲⣆⣤⡤⠟⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⢉⣉⠥⠀⠀⠀⠀⣀⠠⠄⠀⠲⠉⠀⠀⠀⠀⠀⣠⡖⣥⣦⡾⢸⡞⠈⢙⡓⠢⠄⠀⠀⠀⠀⠀⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠒⠒⠒⠒⠚⠉⠀⣠⠴⠒⠋⠉⠀⠐⠂⠀⠠⠂⠀⢀⠤⠐⠛⠱⠋⠽⠇⠀⠙⠋⠙⠛⠑⠃⠀⠀⠀⠀⠈⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣦⡀⠀⠀⠀⠀⢰⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⢟⡻⢿⣿⣷⣄⡀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⣫⣶⣿⣿⣶⣝⠻⣿⣿⣦⣜⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⣋⣵⣿⣿⣿⢟⡻⣿⣿⣿⣮⣙⢿⣿⣷⣮⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⢟⣥⣾⣿⣿⣿⡿⣣⣾⣷⣜⢿⣿⣿⣿⣷⣬⡻⢿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠟⣩⣶⣿⣿⣿⣿⣿⡏⣼⣿⡿⢿⣿⣧⢹⣿⣿⣿⣿⣿⣶⣍⠻⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠋⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡏⣶⣶⣽⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣿⣿⣿⣿⡇⣿⣷⣤⣬⢹⣿⢸⣿⣿⣿⣿⡿⣋⣭⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡦⣹⣿⣿⣿⣿⡇⣿⣷⣍⣩⣾⣿⢸⣿⣿⡿⠿⣃⡹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣱⣶⣦⣭⣭⣭⣁⣛⠛⣛⣍⣙⣛⣘⣋⣥⣶⣿⣿⣷⡙⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣱⣿⣿⣿⣿⣿⡟⣫⣴⣿⠿⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠜⢿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⡿⠱⢿⣿⣿⣿⣿⣿⣿⣶⣥⣶⣿⣿⣷⣶⣭⡛⠿⣿⣿⠟⣫⣵⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⠙⣡⣶⡎⣛⣛⢿⣿⣿⣿⣿⣿⡻⢿⣿⣦⡐⢶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢡⣿⡿⣫⣾⣿⠟⠸⣿⣿⣿⣿⠿⣿⣷⣬⡛⠟⣸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⣡⣾⡿⠸⣿⣿⢿⣷⣮⡙⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⢾⠿⠘⠿⣷⠌⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠉⠙⢿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⣠⠶⠒⠒⠲⢤⡀⠀⠀⣀⣤⣄⡀⢘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⠒⠋⠁⠀⠀⠉⠛⠋⠙⠻⣿⣿⣿⣄⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⢤⣀⡞⠙⢦⡀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⡴⠋⠉⠙⠳⣄⡼⠁⠀⠀⠀⠙⠓⠲⡄⠉⠙⠲⣤⠀⠀⠰⠿⣿⣿⡿⠟⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠐⢦⠴⠃⠀⠀⡴⠋⠁⣀⣠⣤⣤⣤⠄⠀⠀⣿⠀⠀⠀⠈⢳⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⡀⠀⠀⣠⠞⠁⣠⠞⠉⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⢧⣠⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿
⠿⣍⣛⣉⣁⡴⠊⠀⣀⣀⣀⣀⣀⣤⠖⠋⠁⠀⡴⠚⠲⣄⠀⢀⣽⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣿⡿⠟
⠀⢀⣨⣭⣤⣤⠴⠚⠉⠀⠀⠀⠀⠈⠙⢦⡀⠈⢧⣀⡀⣈⣩⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⣀⡴⠒⠒⠦⣀⠀⠈⠳⢤⣀⣀
⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠙⢄⣀⣨⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⠀⠀⠈⠳⠦⣤⣤⠤⣴
⠀⢀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠿⠟⠛⠛⠶⠶⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣻⣿⣿⣿⣿⣷⣶⣤⣤⣶⣿⣿
⠀⢸⣷⣶⣶⣶⠾⠟⣋⣡⣴⣶⣶⣿⣷⣶⣶⣤⣄⡈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣿⣿⠿⠛⢉⣡⣤⣤⣤⣤⣤⣌⣉⠙
⠀⠀⠉⣉⣥⣤⣶⣿⣿⣿⣿⣿⣿⣿⢯⣭⣭⣤⣬⣥⣤⣤⣄⡉⠛⠻⠿⠿⠿⠟⠛⠉⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠘⠟⠛⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⡿⠿⠿⠿⠿⢿⣿⠷⠶⠛⣣⣴⣿⣿⣿⣿⣿⣿⣦⠤⠤⠶⠿⠿⠟
⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⣿⣶⣶⣤⣄⣀⠀⠙⠷⣶⣶⣶⣶⣶⣶⠶⠾⠿⠿⠿⠿⣿⠿⠛⣉⣤⣤⣶⣶⣶⣶⣤
⠀⠀⠀⠀⠙⢿⣿⣿⡿⠟⠛⠛⠻⣿⣿⣿⣿⣷⣦⣄⠀⠰⣶⣶⣶⣶⣶⣶⣶⠶⠟⣁⣤⣶⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢳⣴⠶⠶⠶⢶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣬⣭⣭⣭⣭⣤⣶⣿⣿⣿⣿⣿⣿⠟⢛⣉⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣦⣤⣤⣀⡉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠛⠳⠦⠄⠉⠛⠻⠿⠿⢿⣿⣿⠿⠿⠟⢛⣉⣤⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⣶⣾⣿⣷⣦⣀⠛⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣤⢠⡄⢀⠊⠀⠐⠦⡀⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣈⠉⠙⠓⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠤⠀⠀⠛⡀⢠⠇⠀⠀⠂⠂⠼⠇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⠀⠀⠀⠁⠀⠈⠉⠙⠓⠲⠶⢤⣄⣀⡉⠉⠛⠒⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⠀⠀⠀⠀⠀⠀⠀⠀⢹⠳⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠤⠤
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⡠⠁⠀⠐⠂⠀⠀⠀⠀⠒⠀⠀⠀⠀⠱⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡠⢤
⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢸⡤⣤⢀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⡈⠛⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠋⠔⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠈⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠐⠀⢹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠠⠀⡀⠀⠀⠓⠐⠁⠀⠀⠀⠀⠀⠒⡌⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⣴⠟⢻⣧⢀⣼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠙⠷⠾⣿⡾⠃⢠⡞⢻⡗⣴⠟⠷⣰⠟⣿⠇⣺⠟⢃⡾⢠⡞⢻⡆⣸⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⠟⠁⠀⠘⠛⠛⠓⠙⠓⠒⠛⠛⠛⠒⠛⠓⠘⠓⠚⠛⠛⠑⠋⠘⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠄⠠⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡂⠀⠀⣀⠊⠀⠈⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⠁⠀⠁⠀⠀⠇⠓⠠⠄⠠
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⢠⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠦⠀⠀⠰⠀⠀⠀⠒⠀⢤⡄
⠀⠀⠀⠀⠀⣀⣉⡂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠟⠀⠀⠉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠈⠇
⠀⠀⠀⠘⠿⠟⠛⠃⠐⠿⡷⠆⠀⠀⠁⠀⠀⡀⠀⠀⠀⢀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⣴⣶⣄⡀⠀⠀⢀⣠⣶⣦⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⢀
⠀⠀⠈⠛⠿⠋⠀⠀⠈⠿⠟⠁⠠⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠸⣿⠟⠃⣠⣈
⠀⠀⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠲⢠⡶⠀⡰⠠⡀⠀⠀⠠⠼⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⡀⢸⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠀⠐⠚⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢠⠃⠀⠠⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⢻⣿⠛⢻⣿⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠤⢾⣿⠤⢼⣿⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⣧⣀⣼⣿⣀⣸⣿⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⡷⠶⢾⠶⢾⡇⠀⠀⠀⠀⠀⠀⢸⣇⠀⣸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣯⣭⣽⣍⣽⡇⠀⣀⣴⣄⠀⠀⢸⡏⠉⢹⣿⠉⢹⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⣷⠶⢾⠶⢾⡇⣴⣿⣿⣿⣷⡄⢸⡟⠚⢻⣿⠛⢺⣿⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⣯⣭⣽⣭⣽⡇⣿⣿⣿⣿⣿⡇⢸⣷⣶⣾⣿⣶⣾⣿⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣿⣿⠁⢸⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠈⠉⠉⠈⠉⠀⠈⠈⠀⠀⠉⠁⠈⠈⠉⠉⠉⠉⠉⠁⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⣄⣀⡀⠀⠀⠀⠀⣀⣀⣠⣤⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁




⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣤⢤⢤⠤⠤⣄⢀⡤⠤⠤⠤⠤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⠚⠚⠚⠒⠒⣿⢸⣛⣛⣛⣛⣛⣛⣛⣛⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣤⣄⠀⠀⣿⢸⢐⠶⡶⡶⠂⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠤⠘⠧⣌⠠⠄⣹⠘⠒⠖⠒⠶⢒⣲⣒⠲⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠶⠟⠀⠀⣿⠀⠀⠀⢀⡼⠋⣭⡙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣠⣀⣄⣿⠀⠀⠀⠸⣇⠀⣽⡆⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⠤⠿⠧⠤⠤⠟⠀⠀⠀⠀⠙⠳⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⡿⠿⣿⣿⣿⡿⠿⢿⣛⡇⠀⣰⠞⣛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣏⢺⡇⡷⠀⠀⣿⡇⢸⡇⠀⣻⡅⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣙⣶⣶⣾⣿⣿⣷⣶⣾⣉⡇⠀⠻⣦⣭⣥⠾⠁





⠀⠀⠀⠀⠀⠀⠀⢰⣖⡆⣴⡄⢦⡴⢰⣄⣶⢰⣒⢰⣆⣴⢲⡖⠀⣦⣠⡆⣶⡒⢲⡖⢰⣀⣆⡔⢲⣰⠖⢦
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠘⠉⠙⠘⠁⠘⠉⠘⠘⠒⠘⠈⠋⠈⠁⠀⠃⠁⠃⠛⠒⠈⠃⠘⠀⠃⠑⠚⠘⠓⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡰⠫⠍⡒⠤⣀
⠀⠀⠀⠀⡰⠑⠀⢠⣀⠁⠀⠉⠒⠤⣀
⠀⠀⠀⡰⠁⠀⢠⡏⠻⠿⣶⣤⣀⠀⠀⠉⠒⠤⣀
⠀⠀⡰⠑⠀⢠⣿⡃⠀⠀⠈⠉⢿⣓⡄⠀⠀⠁⠂⠍⡒⠤⣀
⠀⠠⠀⠀⢠⡿⠷⠹⠦⣄⡀⠀⣼⡟⠃⠀⠀⠀⢀⣤⣀⠀⠀⠁⠀⠄
⠰⠀⠀⢀⡻⣏⠀⠀⠀⠈⠐⠂⠚⠃⠀⠀⣀⣄⢻⢿⣿⡇⠀⠀⠀⠁⠂⠅⡢⢄⡀
⠡⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠾⡿⠛⠁⢸⣿⡇⠀⠀⠀⠀⠀⣄⣀⠁⠀⠕⡢⢄⡀
⠀⠀⠁⠒⠤⡀⠀⠀⠀⠀⠀⢀⣴⣶⡍⠿⢳⣦⣀⣨⣿⡇⠀⠀⠀⠀⣼⡿⠏⠀⠀⣤⣀⠁⠊⠕⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠙⢻⣿⡇⠀⠀⠀⢸⣾⠏⠀⠀⠰⣿⡿⢿⣶⣤⡀⠁⠊⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⣀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠤⣄⠉⠀⠀⠐⣿⡟⠀⠀⠈⠙⣿⣷⡀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢐⣄⡟⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠄⠂⠀⠀⠈⠀⠀⢀⠂⣌⡀⠀⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⠳⢗⣦⣠⣶⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⠀⠀⠀⠈⠉⠉⠀⠀⠀⠐⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣠⡀⣤⢀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢠⣤⡄⣶⡆⣿⢸⣿⣉⡁⣿⣭⡜⣿⠀⠀⠀⠀⠡⠃⢀⠀⢸⣿⣻⡇⣿⡷⠆⣶⡄⣄
⠀⣤⠀⢀⠀⡌⢹⡇⣿⡇⣿⠀⣉⣿⡇⣿⣧⡄⠄⠈⠀⠀⠀⣁⠗⣹⡀⢸⣿⢻⡆⣿⡟⠃⣿⣿⠎⠀⠈⡀⠂
⠀⣿⡄⠀⠀⠇⠸⠃⠻⠟⠛⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠈⠁⠛⠛⠇⠿⠁⠀⠀⠀⢁
⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡻
⠀⢰⡇⣮⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠻⠤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣴⠔⠊⣷⡄
⠀⣼⣧⣸⣿⣿⣰⣠⣤⣄⣀⣀⣤⣤⣄⣀⣀⣂⣀⣃⣸⣇⣄⣃⣾⣆⣀⣠⣄⣀⣤⣄⣀⣀⣦⣿⣼⣸⣇⣷⣿
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠂
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⣸⢆⠀⠀⠀⢠⠉⡀⠀⠀⠀⠀⠀⢠⣾⢵⣤⣤⡤⡴⠾⣆⠀⠀⠀⠀⠀⠀⠰⣦⣄⣀⣠⣧⡀
⠀⠀⠀⠀⠀⠀⠁⠀⢂⠀⢀⡄⠀⣷⠀⠀⠀⠀⢠⢞⣵⡿⣿⣧⣄⣇⣀⣤⠀⠀⠀⠀⠀⠰⠀⠀⠽⠯⠅⠀⢹⡆
⠀⠀⠀⠀⠀⠸⣄⣀⣿⣀⣼⣡⣼⡟⠀⠀⠀⠀⢁⣾⡇⠁⢿⡎⠀⢻⠉⢸⡆⠀⠀⠀⠀⠘⣶⣄⣉⣽⣯⣉⡹⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠙⠂⠸⠀⠀⠚⠐⠾⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀
⠀⠀⠀⠀⠀⢠⡋⠛⠁⠃⡜⠁⠀⠑⡄⠀⠀⠀⠀⢰⡇⣽⣿⣷⠻⢟⣣⡀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠘⢧⣄⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠙⠁⣠⡉⢡⣶⣾⡻⠃⠀⠀⠀⠀⠀⠻⣾⣟⠛⣻⣿⣿⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠊⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠇⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⣄⣾⣶⠀⠀⠀⠀⠀⢀⣴⣿⣷⣴⣾⡆⠀⠀⠀⢰⣾⣆⣤⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠙⢿⣿⣿⣿⠀⠀⠀⢀⣴⣿⡿⠋⠻⣿⣿⣅⠀⠀⠀⢘⣿⣿⣿⠟⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠙⢿⣿⣷⣄⣴⣿⡿⠋⠀⠀⠀⠈⠻⣿⣷⣄⣴⣿⣿⠟⠁⠀⠀⠀⠈⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⣟⣻⡃⠀⢸⣿⣿⣿⠟⢹⡆⠀⢘⡇⣟⠀⠀⣾⠙⢿⣿⣿⣿⠀⠀⣻⣛⡇⠀⠀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠛⠛⠃⠀⢸⣿⣿⣿⠀⢸⡇⠀⠘⠛⠛⠀⠀⣿⠀⢸⣿⣿⣿⠀⠀⠛⠛⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡀⣀⣀⡀⢀⢸⠿⢟⣿⢶⡼⣇⣀⡀⣀⣀⢀⡀⡿⣴⢾⣟⠿⢿⣀⣀⣀⢀⣀⣀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒

⠀⠀⠀⠀⠀⠀⢐⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠶⠶⠶⢤⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⣀⣠⣄⣀⣀⡀⠈⠙⣦⠡⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡇
⠀⠀⠀⠀⠀⠀⠐⠀⢀⠄⡀⠀⢂⣠⣠⣤⣐⣲⣤⣤⣤⣤⣀⣀⣘⣷⣿⢉⣀⠀⠈⠩⠙⣳⣶⠋⠄⢂⣠⣤⠒⣠⣐⡠⠠⠀⠄⣒⣄⣂⢠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢐⠪⣢⠶⠶⠶⠿⠛⣫⣩⠟⠛⣲⣶⡿⠟⠉⠉⠭⠿⠿⠍⠩⠭⠭⠿⠿⠿⠿⣿⡿⠟⠓⣲⡮⠽⠟⢛⣛⣛⣛⠛⠫⢉⢳⡄
⠀⠀⠀⠀⠀⠀⢈⠰⣿⡀⡀⢀⡠⠞⠁⠀⠀⢸⡏⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⡏⠀⠀⢎⠀⠀⠀⡀⠈⡆⣯⠀⡗⠀⡄
⠀⠀⠀⠀⠀⠀⠠⠁⣿⠀⡴⠁⠀⠀⠀⠇⠀⠘⢣⠀⠀⠀⢈⡄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⡇⠀⢀⠀⠙⢆⡒⠋⠀⢷⡇⢰⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠰⡅⢻⠜⠁⠀⠀⠀⠀⠛⠛⠛⡟⠀⠀⠀⢀⠈⠁⠁⡰⠖⠀⢀⠀⠀⠀⠀⠀⠟⠉⠉⢺⡇⠀⠀⠀⠀⠀⠙⢄⣀⡜⣇⠀⣧⠀⡇
⠀⠀⠀⠀⠀⠀⠐⠠⢺⠀⡀⠂⠁⠂⠘⡀⠀⡠⠇⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠘⢄⠀⡠⠛⠑⠀⠀⠀⠀⠠⡐⠦⠀⠇⢻⠀⣿⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠂⣿⠀⠁⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠐⡝⠀⠀⠀⢸⠀⣿⠀⡄
⠀⠀⠀⠀⠀⠀⢠⡀⣿⠀⠀⠁⣤⠆⠀⣼⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠒⠚⠈⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠁⣿⠀⠀⠈⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⡀⠀⠀⣿⠠⠁
⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠄⠠⠤⠄⡀⢨⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢨⡀⠙⠊⠀⠂⠄⡀⡇⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠀⣿⠀⠁⠀⠀⠂⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠅⣁⣀⠌⠙⢺⠃⠃⡆⠀⡟⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⢸⣄⠀⠀⠀⠠⡀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠆⠀⠀⠀⠰⠇⠀⠀⠓⠠⠀⠈⠁⠒⢿⠆⢀⠀⡇⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠁⣯⠀⠣⡀⠀⣠⠶⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠠⠀⠀⠀⡀⠀⠀⡀⠀⢡⠀⠀⠀⠀⢀⢸⠀⡇⠈⡇⠀⠁
⠀⠀⠀⠀⠀⠀⢸⡀⣿⢀⣰⠈⢦⡀⠀⠸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣠⡄⢣⡘⠆⠄⠈⡇⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠻⠶⠶⠶⣶⡿⠶⠶⣤⢤⣼⠷⠤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡏⠀⠀⠀⡠⠂⢐⡦⠀⣠⣿⡶⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠻⠁⠦⠄⠐⠒⠂⠉⠀⠘⠆⠠⠁⠀⠈⠉⠉⠉⠩⠉⠙⠛⠛⠛⠛⠙⠉⠝⠓⠲⠶⠟⠛⠓⠿⠤⣤⣤⣤⡼⠋⠂⠈⠗⠀⡇
⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠤⠀⠀⠠⠬⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣄⠀⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠻⣿⣷⣄⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁⠐⠆⠲⠀⠙⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⢈⣃⣭⡤⠴⠶⠾⠿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢰⠀⠐⠄⠀⠀⡆⠀⡀⠐⠐⠄⠀⢰


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣯⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣴⣿⣿⡿⠷⢦⣕⠀⣀⣿⣿⢿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣼⠇⠈⡿⠀⠀⠘⠆⢰⠟⢡⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠀⠰⡀⠀⠀⠈⠀⢸⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⣤⣿⡾⠦⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⢸⠀⠀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠟⠉⠓⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣼⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⢀⣀⣄⣀⣻⣤⣴⣶⣿⣿⡿⣿⣷⣯⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣲⣦⣶⣿⣷⣾⣿⣾⣿⣽⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠁⠉⠲⠿⠿⠦⠬⠭⡍⢉⣉⣉⠉⠙⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠔⠒⠀⠀⢀⣠⣤⡴⠂⠄⠀⠀⢀⣀⡠⠭⠄⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡀⠁⠠⠤⠀⣀⣐⣲⣦⣤⣤⣤⣤⣄⣀⣤⣤⣤⣤⣤⣶⠶⠿⠛⠁⣠⢄⠀⠀⠀⠪⠹⠭
⠀⠀⠀⠀⣀⣀⢀⣀⣀⡠⠭⠵⠶⠶⢚⢛⡛⠛⠽⠟⠛⠋⠉⠉⠉⠉⠈⠉⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⣳⣷⠶⢦⠀⠀⠐⠄
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⢤⣤⣖⣲⡶⡺⠿⠿⠟⠛⠋⠀⢀⡞⠀⠀⠀⠀⠄⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠾⠛⢛⣉⣁⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡴⠞⢫⡀⠀⠀⠀⢀⡠⠔⠋⡀
⠀⠀⠐⠒⢀⣉⣉⣉⣙⣉⣉⣭⠤⠤⠶⠟⠛⠛⠋⠉⠉⠈⠃⠛⣉⣀⢀⣙⣩⠽⠛⠛⠛⠛⢛⣫⠭⠉⠓⠋⠉⠒⠻⠧⠒⣒⠉⠁⠀⢀⣀⣱⠄
⠉⠁⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠴⠖⠒⡋⠉⠅⠐⠊⠉⠀⠀⢉⣠⠴⠒⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⢸⣶⣶⢷⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠒⠀⠀⠀⢀⣠⡤⠤⠒⢈⠄⠙⠂⠀⠀⠀⠀⠀⠀⢀⡤⢮⣱⣎⡿⢟⣓⣿⠉⢀⣉⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠒⠒⠒⠒⠛⠛⠉⠁⠀⣠⡤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠊⠀⠐⠀⠂⠉⠰⠏⣥⠼⠃⠀⠉⠉⠁⠘⢋⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⡏⠉⣩⣭⣭⣭⣭⡉⠉⠉⠉⠉⣩⣭⣭⣭⣭⡉⠉⣿⡏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⣿⡇
⠀⠀⠀⠀⠀⣾⣿⠿⠿⣿⣿⡿⠿⢿⣿⣿⡇⠀⣿⣀⣀⣀⣸⡇⠀⠀⠀⠀⣿⣀⣀⣀⣸⡇⠀⣿⣿⢿⠿⢿⣿⢿⣿⣿⡿⣿⣷
⠀⠀⠀⠀⠀⣿⣿⠀⣀⣀⣀⣀⣀⡀⠀⣿⡇⠀⣈⣉⣉⣉⣉⡁⠀⠀⠀⠀⣉⣉⣉⣉⣉⡀⠀⣿⣿⠀⢀⣀⣀⣀⣀⡀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⣿⠉⠉⠉⢹⡇⠀⣿⡇⠀⣿⠉⠉⠉⢹⡇⠀⠀⠀⠀⣿⠉⠉⠉⢹⡇⠀⣿⣿⠀⢸⡏⠉⠉⠉⡇⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⣿⣀⣀⣀⣸⡇⠀⣿⡇⠀⣿⣀⣀⣀⣸⡇⠀⠀⠀⠀⣿⣀⣀⣀⣸⡇⠀⣿⣿⠀⢸⣇⣀⣀⣀⡇⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⢉⣉⣉⣉⣉⡁⠀⣿⡇⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⣿⣿⠀⢈⣉⣉⣉⣉⡁⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⣿⠉⠉⠉⢹⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⡏⠉⠉⠉⡇⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⣿⠀⠀⠀⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⡇⠀⠀⠀⡇⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠙⠛⠛⠛⠛⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠈⠛⠛⠛⠛⠃⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠛⠛⠛⠛⠀⠀⣿⡇⠀⠛⠛⠛⠛⠀⢸⡇⠀⠀⢸⡇⠀⠛⠛⠛⠛⠀⣿⣿⠀⠀⠛⠛⠛⠛⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢋⣉⡭⠷⡄⠀⠀⠀⠀⢠⡏⠗⡦⣄⣀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣉⡡⠴⢚⡉⠀⠀⠀⢹⠀⠀⠀⠀⡼⠛⠲⢤⣈⠈⠟⢲⢤⣀⡀⠈⠙⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣉⡤⠖⢋⡥⠖⣚⣨⠵⠚⠛⣆⢸⠀⠀⠀⢠⠇⢀⠀⠀⠈⠙⠓⠦⣌⡁⠿⢳⠦⣄⣀⠀⠉⢣
⠀⠀⠀⠀⠀⠀⠀⡞⢀⡜⠁⢀⣠⠤⠒⠋⠁⠀⠀⠀⠀⣿⢸⠀⠀⠀⢾⡀⠈⠁⠲⠄⣀⠀⠀⠀⠉⠛⠲⢤⣁⠛⠹⠖⢾
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠉⠓⠦⢄⣀⠈⠁⠲⠄⣀⠀⠀⠀⠈⠉⠓⠦⣼
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢰⡇⠀⠀⣠⠖⢋⣉⣉⠙⠲⠄⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⣿⣨⠀⡀⢠⠞⣡⠚⣉⣤⣍⠙⠂⠀⢺⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠤⣄⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡇⡏⡼⠁⡴⠁⡠⠄⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⡋
⠀⠀⠀⠀⠀⠀⠀⣧⣽⠘⡇⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⢼⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⣚⣩⠷⠒⣿⡉⣿
⠀⠀⠀⠀⠀⠀⠀⡿⢿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⣀⣠⠤⠖⠚⣉⣡⠤⠖⠋⠁⠀⢀⣀⣻⡧⠿⢤
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⢀⣠⠴⢺⡇⢸⢸⠀⠀⣴⣋⣁⣠⣤⣖⣋⣭⡤⠤⠴⠖⠚⠋⠉⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡆⡇⠀⢀⣠⢴⢚⣭⠰⣼⠀⡇⢸⢸⠀⠀⡇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⢚⣹
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⡇⠀⣿⣾⢉⣿⣯⡇⢹⢸⡇⢸⢸⠀⠀⡗⠶⠖⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠉⢹
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⡇⠀⣿⣿⠉⠸⠇⢃⣨⠼⠃⢸⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⢺
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⡇⠀⡏⠋⣀⠴⠚⠉⠀⠀⠀⣼⢸⠀⠀⡷⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⢾
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⡇⠀⠛⠉⠁⠀⠀⠀⠀⣠⠞⠁⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡃⡇⠀⠀⠀⢀⣠⠤⠖⠋⠀⢀⡴⠋⠀⠀⣟⠒⠒⠚⠉⠉⠀⠀⠀⠀⣀⣀⣠⠤⠴⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡀⠙⠒⠚⠋⠉⠀⠀⣀⡤⠞⠉⠀⠀⠀⠀⠙⠶⠶⠒⠒⠒⠚⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠾⣇⡀⣀⣀⡤⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠛⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡂⠤⠒⢒⣠⢶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⠀⠀⠋⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⣈⠔⢊⠭⠴⠒⣻⣿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⡴⣿⣯⠔⠀⠭⠄⣐⡟⠋⠙⠀⣠⠀⠀⠀⠀⠀⠀⠠⣄⡀
⠀⠀⠀⠀⡀⠐⠒⠒⠂⠀⠠⠄⠤⠔⠈⠙⠋⠁⠸⣽⣿⣷⣾⣿⠉⠷⠄⠀⡐⠁⠀⣡⠀⠀⠀⡄⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣽⠿⡛⠿⣻⢿⡊⠀⠀⠐⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⡀⠀⠈⠘⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡁⣌⣦⣽⣿⡁⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣀⢠⣽⡿⢿⣿⣯⡥⠀⠀⠀⠀⢻⡉⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡰⣌⣿⠋⠀⠈⣿⣿⣾⠄⠀⠀⠀⠘⠀⠀⠀⣤⣄⡀⢠⣼⣤⠄
⠀⢠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣀⡻⣇⡿⠁⠀⠀⠀⠸⣿⣿⠀⢁⣤⣀⠀⠀⠀⠘⠛⠉⠙⠛⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡿⠇⠈⠀⠀⠀⠀⠀⠀⠈⠉⠙⢪⡿⣽⣧⡀⠀⠀⠈⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⣷⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠒⠒⠲⠂⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣃⡴⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⣢⣥⣀⣀⣀⣀⣠⣤⣤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⡙⠛⠛⠛⠛⠛⠻⠿⠿⢿⣿⣍⠉⠉⠁⠉⠉⠁⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⡠⠔⠊⠉⠉⠉⠁⢒⣤⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⠀⢠⣎⣀⠀⠀⠀⠀⢀⠔⠉⠀⢛⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡄⠀⢝⠂⠀⠀⠈⠙⢲⡀⢠⠀⠀⠀⠀⢸⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⡉⠉⠋⠀⠀⠀⠀⠀⠀⢱⣇⣀⡀⠀⠀⠀⡇⠀⠀⠆⠀⠀⠀⠀⠀⢀⠀⡀⡰⣖⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠀⠀⠀⠀⠆⡄⠀⠀⠀⠀⠀⣸⠋⠑⢄⠀⡇⢀⠀⡾⠀⠀⠀⠀⠀⠀⠠⣀⡳⠜⠀⡏⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠈⢀⡀⠀⠀⠀⣰⠃⠀⠀⠈⠷⠋⡠⠈⠴⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣴⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⢤⠠⠒⣓⡄⢠⢀⠤⢡⠒⠢⢴⠃⠀⡠⠀⢀⡠⠊⡱⠷⢆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠈⠖⢥⠀⠀⢠⠃⠀⠀⠀⠑⡏⣀⠴⢋⠔⠙⠦⠤⠊⢀⡀⠀⠀⠀⢀⠔⢲⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢽⡖⠨⡷⠔⠂⠄⠀⠀⡾⠀⠀⠀⢀⣠⢼⡥⢚⡁⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠸⡠⠎⢳⣀⣠⠗⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠃⠠⡀⠔⠀⠐⢦⣀⣇⠀⡠⠞⠉⠀⢀⡟⡉⢁⡀⠀⠀⠒⠀⠀⠀⠢⡀
⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢀⡿⠀⠄⠉⠋⠀⠑⠀⡇⠀⣏⠀⠀⠀⠀⣼⠁⡀⠄⠩⡒⠀⠒⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠒
⠀⢀⡗⡁⠀⠀⠀⣀⣂⣀⣃⣤⣼⢥⣿⣿⣱⠁⣠⣨⣁⡖⡀⣉⣞⣀⣑⣤⡠⠾⠷⠏⠘⠚⣢⠈⠂⠀⢄⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣳⡜⢻⡄
⠀⡌⠁⠁⢠⠀⣀⡤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠙⡖⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠉⠀⠉⠂
⠀⡃⠀⠘⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠇
⠀⠿⠲⢶⠃⠀⠀⣠⠔⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡏
⡴⠛⠈⠚⠲⡀⢸⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⣠⣇⣀
⡇⠀⠀⠀⠀⡇⠀⠀⢠⠁⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⡆⡴⠁⠀⠀⠈⡷⠀⠀⠀⠀⠀⢈⠀⠀⠃
⠘⠠⠄⣀⠴⣧⠃⠀⠘⢀⡀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠠⠀⡠⠀⢁⢧⠀⠀⠀⡰⠃⠰⠄⠉⢹⡂⠘⠆⠁⢀
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠙⡤⠤⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠄⠀⠀⠀⠀⠀⠈⡏⠍⠉⣉⢱⠀⡔⠀⠀⠠⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠈⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣾⣿⣿⣷⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢳⣰⡏⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠋⠄⠸⠿⠿⠿⠛⠙⠛⠛⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣆⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⠿⡃⢸⣿⣿⣿⠿⠿⠿⠿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡈⠈⣤⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢻⣴⡇⢠⣤⣤⣤⣤⣤⣤⣤⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠂⠁⢂⠘⠛⠛⠛⠉⠉⠉⠉⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢧⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢜⣛⠅⢸⣻⣿⣿⠛⠛⠛⠛⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⣾⡿⣿⡿⣻⠟⢿⣿⣽⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣶⡿⠿⢟⠛⢉⢥⣴⣻⡿⠻⢽⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢡⣿⡟⢡⣲⣾⣿⣮⣿⠟⠉⠁⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣠⣾⣟⡥⢙⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⣀⢻⣿⣿⣿⣿⣀⣤⡶⡾⠒⠄⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣧⡄
⠀⠀⠀⠀⠀⠀⡠⠂⣀⣤⣿⣿⠿⠀⢐⣾⣿⣿⡿⠟⠟⠉⢁⣀⣴⣀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠐⠉⠀⠀⠛⠉⢋⡀⣄⣶⣾⣿⣿⢯⠙⠐⠊⠉⠛⠿⢿⣿⣦⣀⣀⣠⣴⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢺⣿⣧⣧⣾⣿⣿⣿⣿⣿⡻⠆⠀⠀⣠⣴⣄⣻⣿⣿⣿⣿⠟⠋⠀⠋⠙⢻⠟⡁
⠀⠀⠀⠀⠀⣠⣶⣾⣿⣛⣛⣿⣯⣭⣿⠉⢂⠁⠀⣀⣸⣿⣿⣿⣿⣿⡿⠛⠂⠀⠀⢀⡤⣴⣾⠿
⠀⠀⠀⠀⠀⢻⡿⠿⢿⣿⣿⣿⡿⣯⡉⠉⠀⠀⢺⣿⣿⣿⣿⣿⣏⠉⠀⠀⠀⠀⠀⠀⠀⠈⡁
⠀⠀⠀⢠⣶⣦⣄⡀⠀⠅⠑⠀⠀⢀⣠⡀⣀⣦⣾⣿⣿⡟⠎⠙⠻⠷⠀⠀⠀⠀⢀⣤⣶⡿⠃
⠀⠀⠀⣿⣿⣿⣿⣷⣢⣦⣿⣼⣷⣾⣿⣷⣿⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⣀⣚⣫⠭⠁
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠉⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢐⣒⡒⠒⡇⠀⠀⠀⠀⠀⠻⠴⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢶⣄⠀⠀⢙⣛⠃⠀⠀⠀⠀⠠⣶⡆⢰⠐⣻⠋⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡼⠁⠀⠛⢯⠲⣄⠀⠀⠠⠤⠀⠀⠠⠀⡗⠀⣰⠃⠀⠈⢷⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢵⡼⠁⣶⡖⡀⠹⣣⣸⡂⠚⠀⠀⠐⠃⠀⠀⠀⣰⠇⢰⣶⢒⠈⢷⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⡰⠆⢀⣼⠃⠀⠃⠃⠃⠀⠘⣽⣤⣤⣀⡀⠀⡀⢀⣀⣶⠏⠀⠐⠐⠘⠀⠈⢿⣄⠀⢴
⠀⠀⠀⠀⠀⠀⠀⢂⢀⢞⠀⠀⢸⠀⢸⣿⡇⠸⠟⠂⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⣸⣿⢀⣸⣿⣇⠀⡟⠀⢈⡄⠀⢃
⠀⠀⠀⠘⠀⠀⠀⠀⠮⠶⠀⠀⢸⠀⡏⠉⠉⠉⠉⣿⡄⠀⠙⡟⠀⠀⠿⢽⠀⠀⢀⣈⣉⡏⠉⠁⢹⣧⡆⠀⠐⢃⠀⠀⢂⠔⠄
⠀⠀⠀⠀⠠⠄⠈⠙⠮⠅⣀⣈⡹⠀⠉⠛⠙⠙⠙⠫⠀⠀⠀⠈⠉⠉⣉⠉⢁⣀⡀⠀⠁⢹⢒⠓⢚⠅⡇⠀⠀⢀⠀⡄⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠐⠂⡂⠄⠁⠿⢉⠀⠀⢀⢤⡦⡄⠠⡄⠀⣤⣤⠀⠀⠀⠈⠒⠉⠁⠀⠀⢨⣄⢠⣤⠀⠂⡀⡠⠀⢂⠀⢀
⠀⠀⠀⠀⠀⠀⠄⢀⠀⣤⠖⠀⠈⡆⠀⠀⠀⠁⠀⠈⠁⠀⠉⠛⠂⠘⠀⠘⠂⠦⠈⠉⠀⣘⠛⠉⠛⣸⡷⡍⢳⡿⡀⡃⠨⠴⠹⠃
⠀⠀⠀⠀⠀⠁⠐⠀⠀⠆⠀⠀⠄⢸⡀⠀⠈⡁⢀⠀⠀⠀⠉⠢⡁⣀⠀⠀⢠⣤⠠⠤⢤⠶⠀⠛⠊⠉⠐⠟⠐⣟⡛⠻⠷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠂⠈⠉⠙⠠⠭⠦⠦⠬⠏⠉⠀⠏⠉⠉⠥⠤⠥⠠⠀⠀⠀⠀⠀⠈⡂

⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠁⠒⠚⠃⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠱⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⣤⠴⣤⡁⠂⢄⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠁⠀⡤⢾⣦⡰⡟⠻⡉⠉⠙⢦⢌⠢⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡕⠁⠈⠁⠈⠛⠒⣻⢏⠎⣂⢰⠀⠀⣼⠀⠑⡈⢂⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠒⡄⠀⠀⠀⠀⠎⢀⠇⠯⡀⠘⠔⠁⠀⠀⠀⠈⡄⢂⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡔⠀⠀⠀⠀⠀⠉⠀⠖⠺⠀⠀⠀⠀⠀⠸⣮⠿⡀⡀⢠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠆⢀⠀⠀⢄⠀⠀⢀⣀⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⢐⠵⠾⡇⠁
⠀⠀⠀⠀⡀⠨⠰⠉⠉⠉⣷⠨⠀⡀⠈⠓⢄⣇⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⡄⡀⢀
⠀⠀⠀⠎⠇⠀⠀⠀⠀⠀⠀⠀⠀⡉⡄⠀⠀⠀⠙⠶⠒⠂⠀⢀⠀⠀⠀⠀⠀⠑⠂⢲⢁⠁⠈
⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⡀⢀⠀⠁⠀⠀⠀⠀⠠⡀⠀⠀⠀⡄⠵⠤⠤⠤⠤⠤⡧⠾⠴⠓⠲⢟⣚⡛⢍⠉⠙⠛⠛⢋⠆
⠀⠀⢰⠀⠘⠠⡷⠀⠀⠀⠰⡧⠈⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡏⠈⢻⡆⠂⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠘⣄⡀⠀⠀⠀⠀⠀⠄⠁⠰⢢⢀⠀⠀⠀⠀⠰⠀⠀⡠⢄⠙⠒⠚⠁⠃⠀⠀⠀⠀⡔
⠀⠀⡄⠀⡠⠐⠒⠒⠒⠒⠒⠒⠠⠀⠀⡷⢤⡁⡐⠂⠀⠒⠀⠒⣘⡜⡠⠀⠡⠄⠀⠬⠅⠀⠀⠈⢆⠀⠀⠠⠀⠀⠀⠀⠀⡠⠂
⣆⠀⠀⠀⠃⠤⠤⠤⠤⠤⠤⠤⠄⡆⠀⠂⢀⠧⠀⢀⡀⠀⠀⡀⠠⠖⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⢀⠦⠀⡖⠐⠀⠂⠈⠀⠠
⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠰⠠⠀⣠⠑⠶⣷⣴⡀⠀⠀⠀⠀⠀⡔
⠣⣘⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⡠⠂⠐⠒⠓⠒⠒⠓⠒⠀⢀⠀⠀⠀⠀⠀⢀⡤⠊⠈⠁⠐⠃⠀⡀
⣀⣈⣄⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣌⣀⣀⣁⣀⣀⣀⣀⣀⣈⣀⣎⣶⣶⣶⣦⣬⣤⣦⣤⣖⣒⣒⣚⣉⣉⣉⣉⣉⣉⣉⣉
⣿⣿⣋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠢
⠀⠀⠀⠀⢘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⠀⠀⠀⠪⣷⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠤⣄⠀⠀⠀⢀⡀
⠀⠘⠷⣄⠂⠈⢽⣇⡕⠏⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠳⠲⠄
⠀⢠⣄⠙⢻⡀⠱⢃⣴⣤⣧⣦⣶⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⡀⣀⡀
⠀⠀⠨⢧⠈⢥⣠⢀⣌⡀⠀⠈⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠑⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈
⠀⠀⠀⠈⣣⠜⣉⡁⠙⠛⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡜⢲⠟⠛⠷⢧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⡀⡆⠃⢀⢠⠇⡄⢠⠃⡄⢀⠄⠀⠀⠀⠀⠂⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡰⣌⢆⠣⢱⣠⣘⣘⡰⠡⢃⠎⡰⢋⠀⠀⣀⠀⠀⠀⢀⠀⡀⠀⢀
⠀⠀⠀⠠⠔⠀⢀⠀⠈⣇⠀⠀⠀⠐⠦⣄⡐⠦⣀⠦⡑⢌⠾⠋⠉⠀⠀⠀⠉⠓⢮⢊⠴⣋⠴⠊⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⡀⠀⢈⠳⠀⡿⠒⠒⠒⠠⠤⣀⣉⠓⠬⡑⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠵⢚⣁
⠀⠀⠀⠀⠁⠀⠈⠂⢧⢈⡉⠓⠶⠆⠀⠠⠬⢭⡑⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠭⠤⠒⠒⠉⠁
⠀⠀⠀⠛⠀⠀⠀⠨⠶⢦⡉⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠈
⠀⠀⠐⠂⠀⠈

⠀⠀⡀⡀
⣤⣤⣉⣄⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠖⠛⠋⠉⠙⠛⠛⠒⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢠⠀⠀⢠⠀⠀⠐⠸⣇
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⡶⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡘⠀⠀⡌⠀⠀⢠⠀⣿
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⡇⠰⠀⠠⠐⠲⠾⠄⠀⠀⢀⡴⠁⢀⡴⠧⢄⣠⡟⠀⣿
⠀⠀⠀⠀⠀⢸⠀⠀⣀⠀⢯⡇⠀⠀⡠⠮⠔⠒⠒⠉⠉⠉⠉⠉⠁⠀⠈⠉⡅⣸⠇
⠀⠀⠀⠀⠀⠸⡄⡘⠀⠀⠠⠇⠐⠮⠖⠲⠦⡄⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⡿⠁
⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⢠⠀⡄⢠⢠⠄⠒⠒⠒⠀⠤⠀⠀⠀⠀⡈⢰⡇
⠀⠀⠀⢀⣤⠞⠉⠉⠁⢲⣦⡊⠘⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠇⡰⠡⢁⢾⠇
⠀⠀⠀⣾⠁⠀⣀⠔⠊⠁⠀⠀⢳⠦⠄⣀⣀⡐⠒⠠⠐⠊⠁⠀⠀⢀⣴⠋
⠀⠀⠀⣇⡠⠞⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠫⢥⣟⠀⠴⠂⠘⢻⠁⠈⠙⠒⢦⡀
⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠈⢷
⢰⠋⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⠀⠀⠘⣧
⠸⣆⠀⠀⠀⠀⠀⢰⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠾⣇⡀⠀⠘⢣⡀
⠀⠈⢳⣄⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢨⠁⠀⠀⢄⢹⡄
⠀⣰⠋⠀⠁⡄⠀⠀⠀⠙⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠄⠀⠀⠀⠘⢧⡀
⣰⠃⠀⠀⢠⠁⠀⠀⠀⡰⠊⠉⠐⠒⠂⠀⠀⠠⠤⠀⠀⠀⠄⠠⠤⠴⠃⠀⠐⠒⠀⠀⠀⠀⠈⣧
⡟⠀⠀⠀⢰⠀⡄⠀⡀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣷⠀⠀⠀⠀⠑⠓⠃⠃⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠈⠙⠒⠦⣤⠀⢀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⡀⣀⠤⠋⠉⠈⠉⠉⠉⠉⠉⢡⠆⠀⠀⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⢰⣤⣤⣤⣶⡇⠀⠀⠉⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢻⡄⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢷⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠈⠉⠉⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠉⠓⣦⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⣀⣥⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠻⣤⣤⡴⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⣶⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠐⠋⠀⠀⠀⠀⠈⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣊⠉⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠉⠀⠀⠀⠀⣀⡀⠀⠐⠁⣀⣈⡠⡤⠶⠥⠚⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢉⣹⣋⣀⣀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢰⡏⡠⠊⠉⠀⡼⠁⠀⢈⠆⡷⠖⠒⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠧⡠⢤⠞⠙⠓⠒⠛⠚⠧⢤⡀⠒⠠⠀⠒⠠⡁⢶⣀⡂⠤⠬⠍⢰⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠒⢏⣀⡀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠈⠛⣄⢀⠀⠀⡀⠍⠀⠀⡀⠀⠀⣭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣀⠀⠀⠀⠰⠁⠀⠀⠀⠃⠀⠉⣹⡖⢒⡒⢠⣖⢚⡛⠲⠦⢄⣂⢭⣀⣒⠢⠭⡅⠹⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠤⣀⠀⠀⠀⠀⠀⠀⢰⠦⠬⠃⠀⢀⠇⠀⠀⠀⠈⠀⢀⡴⢉⠁⣿⡏⠹⠼⠁⢹⣶⣤⣤⡈⠁⠀⠀⠀⠀⠈⡦⠤⠄⣀⡀
⢹⡿⣭⡉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠒⢢⣤⠀⠀⠈⠒⠒⠣⢄⡜⠀⠀⠀⠀⡷⠖⠉⠀⢸⠀⣿⡇⠀⠀⠀⣸⣿⠀⢸⡿⠀⠀⠀⣤⣀⠀⠇⠀⠀⠀⠙⡇
⠈⣿⠷⣿⣷⣦⢄⡀⣀⣀⣠⢤⣴⠶⢮⣛⠯⣻⠆⠀⠀⠀⠀⠀⢰⠁⢀⠠⡴⠶⠁⠀⠀⢠⡎⢸⣿⣧⣤⣶⣶⣿⣿⠀⣼⠇⡄⢸⠙⠛⠈⣠⠀⠀⠀⠀⢀⠁
⠀⢻⢔⠽⣿⣝⠑⢫⣉⡢⠼⠓⠂⠉⠀⠀⠀⢘⡇⠀⠀⠀⠀⢀⠃⠀⢀⠜⠁⠀⠀⠀⠶⣡⠁⣾⣿⣿⣿⣿⣿⣿⣿⢀⣿⢠⢠⠺⡄⠀⠀⢙⣤⠀⠀⠀⡜
⠀⢸⡗⠦⣦⢝⣳⠜⢢⡄⠀⠠⢄⡀⠠⠤⣀⣻⠀⠀⠀⢀⣀⡎⢀⠔⠁⠀⠀⠀⠀⠀⣠⡞⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⡇⠆⡌⠀⠓⢒⣋⡅⠀⠀⠀⢠⠃
⠀⠀⣇⡈⠀⠝⢿⣛⡴⢶⠛⠻⢟⣛⠛⠋⢭⡿⠀⠀⠀⢹⠟⠴⣿⠄⠀⠀⠀⠀⠘⠊⣰⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⠀⣇⣀⣀⣈⡅⢿⠀⠀⣠⠃
⠀⠀⣯⢻⢮⣔⣽⣯⠑⠒⠛⠉⠀⠀⠈⠉⢻⡇⠀⢀⠤⢿⠂⢴⠁⢀⡀⠀⠀⠀⠀⢠⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡷⠛⠁
⠀⠀⣟⠡⡳⢊⣛⡲⠤⠄⠒⠢⠤⠬⣤⣀⣿⡇⢀⡟⡧⡇⠀⣿⠀⡸⠀⠉⠒⠂⠤⠧⢈⡻⠿⢿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠈⣉⣉⠁⠈⠁
⠀⢈⡷⣮⣮⢥⡿⠿⠛⠛⡛⠛⠒⠒⠶⠴⣿⠀⡸⣤⡽⣤⡤⢭⡶⠥⠤⠤⠔⠊⠙⢦⣀⠙⠶⢦⣬⣉⣙⡛⣿⡿⠿⠾⢄⠀⡠⠊⠀⠀⠉⠒⠠⠄⣀
⠀⢸⢟⠗⡻⣷⠮⣍⣓⠒⠤⠉⠉⣉⣈⠀⣿⡀⠁⣈⣵⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣲⣆⡀⢋⠉⠁⠀⠀⠀⠀⠀⠫⡀⠀⢀⡀⣀⠔⠓⠂⠬⣀
⠀⣿⡭⡫⢺⠶⢿⡶⠾⠟⠛⠓⣂⣀⠌⠭⢿⣇⠼⠿⣦⣞⠳⣄⣀⣀⣀⡤⠤⠴⠂⣺⡋⠠⢚⡅⠈⠉⢣⡀⠀⠀⠀⠀⠀⠀⠙⢶⡁⠀⠉⠑⠒⠦⠤⣄⣉⠒⢠
⠀⠀⠀⠀⠻⠛⠲⠜⠛⢛⣛⡉⠥⠖⠓⠚⠋⠉⠑⢤⡀⠙⠯⣢⣄⣉⣑⣦⣤⣀⡴⣍⠂⠉⠉⠉⢁⠟⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⣠⡚⠕⠊
⠀⠀⠀⠀⠀⠀⠈⢫⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⡀⠀⠉⠁⠀⠀⠀⠀⠀⣉⡵⢂⡴⠁⠀⠀⠰⠀⠑⢦⣀⣀⣠⡤⠀⠂⠂⠉⡀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⣀⠀⠀⠀⣀⣀⡠⠴⢒⠩⠒⠙⠢⣀⣠⣤⠴⢖⡒⠉⢀⠔⠉⠉⠙⠲⣖⠃⠀⠀⢀⡨⣖⠉⠈⠁⠐⠛⠁⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠁⠙⢏⠉⢉⡡⠐⠛⠀⡀⠀⠀⠀⠈⢁⡠⠔⠚⠉⠛⠁⠀⠀⠀⣠⠊⠈⠲⡤⠚⠁⠀⠀⠈⣁⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠳⣤⠔⠊⠉⠁⠒⠒⠤⠄⣀⠀⠀⡔⠣⡀⠀⠀⢀⣀⠠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠙⠂⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⢀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⡀
⠀⠀⠀⠀⢸⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⣽⠛⠛⠛⠛⠛⣿⠀⠀⣾⡛⠛⠛⠛⠛⣿⠀⠀⠀⠀⣿⠀⠀⢰⣟⠛⠛⠛⠛⢻⡇⠀⠀⣿⠛⠛⠛⠛⢻⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⠃⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡆⠀⠀⣿⣄⣀⣀⣀⣠⣿⠀⠀⣿⣅⣀⣀⣀⣀⣿⠀⠀⠀⠀⣿⠀⠀⢸⣯⣀⣀⣀⣀⣸⡇⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⣷⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣷⣶⣷⣿⣿⣿⣿⣿⣿⣷⣾⣾⣿⣶⣶⣶⣷⣿⣿⣿⣿⣾⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⡠⠀⠀⠐⠄⠀⠀⠀⠀⠂⠀⠀⠂
⠀⡤⢊⠄⠀⠀⠀⡀⠠⠀⠀⠀⡡
⠀⠄⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠰⡄⠀⡆⠄
⠀⠠⠀⠀⠀⠀⠀⠈⠘⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢰⣿⣿⣿⣿⣿⡇⡄⠀⢀⣶⣿⣿⣿⣿⣿
⠀⠀⢹⣿⣿⣿⣿⣿⣷⠁⠀⠀⢿⣿⣿⣿⣿⣿⡾
⠀⠀⠀⠙⠿⠿⠿⠛⠁⠀⠀⠀⠈⠛⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡾⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠿⠟⠃⠢⡤⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢀⡠⠀⠂⠉⠐⠐⠀⠉⠛⠻⠿⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⠀⠂⢐⣶⡶⠶⣤⣤⣄⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⢬⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠀⢀⡖⠈⠉⣍⢳⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣄⢁⡀⠤⡀
⠀⠀⠀⠐⠀⠉⠀⠀⠀⠀⠺⠶⠶⠀⠠⠘⠀⠈⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⠐⠐⠂⠀⠀⠀⠀⠀⠀⠀⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢈⣄⡀⠀⠀⠀⠠⠆⠠⠀⠄⣀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣓⢾⡧⠀⠀⠀⠀⣺⣿⣿⣿⡿⢿⣿⣿⡀⠀⠀⠀⡀⣀⣀⣭⣭⡏
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⢠⠤⠤⠤⡇⠀⠀⠀⠁⠭⢾⣷⠶⠷⠛⣛⢹⠶⠠⠀⠉⠉⢛⣭⡶⠯⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢔⣀⡀⠈⠧⠀⠢⠤⠀⠀⠀⠤⢄⣀⣶⣶⣿⡃⠀⢈⠀⠤⠒⣒⠚⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⠍⠉⢅⣲⣇⠀⠀⠀⠀⠩⣔⣟⣻⣯⠽⣾⣿⣤⠀⣪⢠⡀⢠⠄⠬⠔⠓⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠒⠛⠛⠛⠛⠛⠚⠰⠆⢰⣿⣿⣿⣛⠛⠉⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠤⡤⠒⠒⠒⠒⠂⠐⡖⠐⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠁⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠖⠂⢩⠶⠿⠙⠉⠀⠀⠘⠟⠛⠙⠉⠀⠒⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⢂⡀⡄⠈⠀⠤⠤⢤⡤⠤⠤⣤⢤⡀⡆⠈⠀⣉⠁⠐⠂⠳⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢁⡄⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠑⠈⠀⠰⢀⠀⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⣀⠀⠀⠀⠀⠀⠀⢠⠖⠀⡠⠀⠀⠀⠃⠈⠀⠉⠈⠀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠊⠀⠁⠀⠀⢀⠠⠓⠖⠁⡀⠉⠀⠀⠀⠀⠀⢀⡀⠤⢀⡤⠶⡶⢿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠊⠀⠀⠀⠀⢀⠄⠈⠀⣠⠞⠁⠀⠂⠀⠀⠀⠙⢦⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠃⠀⠀⠀⠀⡰⠁⠀⠀⡜⠁⠀⠀⠌⠀⠀⠀⠈⠀⠀⢣⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠂⠀⡀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠑⠂⠀⠘⠀⠀⠀⠰⠄⢀⣀⡀⣀⠀⠀⠀⡇⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠰⢷⡴⠻⡿⠿⠦⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠐⠦⠤⠀⣀⣀⠁⠀⠀⠈⠁⠐⠃⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢳⠼⣦⡀⣿⠉⢹⣿⠗⠀⠢⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠏⠘⣡⠆⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣈⠙⠉⠘⣿⣷⣼⡋⠈⠋⢀⠀⠀⠀⠀⠀⢇⠀⠀⠀⡤⠀⠀⠀⠀⠀⢀⠇⠀⡾⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠰⠀⠙⠶⠶⠶⠛⠿⠿⠷⠦⢀⠀⠁⡀⠀⠀⣤⠀⠑⠀⠞⡈⠀⠀⠀⡀⠀⢘⡄⣪⠃⡜⠁⢙⠃⢆⠅
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠂⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢠⣌⣉⡀⢈⠀⠀⠈⠀⠀⠀⠀⠑⡄⢋⠜⢁⡞⠁⡠⢞⡴⠁
⠀⠀⠀⠀⠀⠀⢀⠂⠐⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠁⠀⠀⠀⠀⠐⠶⣀⠀⢞⣐⡭⠞⢟⣤⠞⠁⠘
⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⠂⠀⠀⠀⠀⡀⠠⢼⢝⠳⢯⡐⠦⣞⣪⠦
⠀⠀⠀⠀⠀⠀⠐⢀⡃⠐⠀⢲⡎⠉⠉⣶⡎⠉⠙⣦⣀⣤⣤⣤⣤⣤⠧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡬⠑⠂⠈⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠙⠒⠊⠉⠙⠒⠺⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠀⠀⠀⠀⠀⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠑⠀⠠⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠈⢦⠁⢈⣀⠀⠄⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠈⠁⠈⠀⠀⠀⠂⢨⠀⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣬⠐⠀⠐⠀⠠⣢⢠⢁⠀⠀⠢⢠⡾⠀⠉⠠⠿⠇⠄⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠾⢮⡁⠁⠈⠉⠹⣘⠉⣀⡄⠀⠀⠆⠐⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⢀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠐⠁⠐⠋⠓⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣠⡇⢀⠶⣾⣤⣤⡿⡦⠀⡀⣾⣷⣴
⠀⠀⠀⠀⢀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠉⠉⠀⠀⠁⠘⠚⠀⠀⠀⠀⠙⠀⠈⠁⠈⠋
⠀⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠂
⠀⠀⠀⠂
⣀⣀⣀⣐⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣐⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣴⣿⢿⡿⠷⢦⡄⠀⣨⣿⡿⢿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠇⠀⡿⠀⠀⠘⠀⠸⠋⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠐⡀⠀⠀⠀⠀⡀⠀⠠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⣶⣿⡟⠂⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠞⠀⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣧⠀⣆⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⢦⣄⢠⣤⣤⣄⣿⢶⣴⠟⣽⣿⣿⣿⣿⣶⣿⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣽⣦⠾⠿⠿⠿⡿⣿⠷⣿⣿⠿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠒⠒⠒⠂⠀⠤⠄⠀⢉⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠒⠒⠃⠁⠀⣀⣤⡤⠒⠚⠐⠂⠀⣀⣤⠤⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⣤⣤⣤⣭⣽⣶⣶⠶⠶⠶⠤⠶⠶⠶⠶⠾⠖⠛⠋⠁⠀⠰⢀⣀⡀⠀⠑⠘⠂
⠀⠀⠀⠀⠀⠀⠤⠤⠴⠖⠒⠚⠛⠉⠭⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣄⣤⡽⠟⠉⡧⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⠶⠶⠶⠒⠒⠚⠒⠒⠛⠛⠛⠉⠤⠰⠖⠉⠉⠉⠉⠁⠀⠀⣀⡤⠃⠀⠀⠀⠀⠀⢀⡠⠚
⠀⠀⠀⠀⠀⠀⠤⢤⣤⣀⣠⠤⠶⠶⠛⣛⣩⣤⣤⣶⠶⠶⡶⢶⠿⠶⠖⢾⡶⣦⣤⣤⣤⣠⣤⣠⡤⠴⢶⡾⣏⣁⣲⡄⠀⣀⠠⠄⠂⠁⠠
⠀⠠⠀⡤⠠⠤⠤⠤⠴⠖⠒⠒⠒⠂⠉⠉⠁⠁⠀⠀⠀⣀⣀⡬⠤⠄⣒⡤⠖⠒⠉⠀⠀⣀⠠⠒⠋⠁⠀⠀⠀⠈⠉⠉⠀⣶⡀⣀⡀⣤⡤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠒⠚⢋⡉⠁⠀⠀⠀⠀⠀⢀⡀⠀⠠⡖⠋⠁⠀⠀⠀⠀⠀⠀⡠⢀⢠⣠⣀⠸⣿⠏⠿⠷⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡄⣀⣀⣀⣀⣠⡤⠶⠛⠉⠀⢀⣠⡤⠖⠐⠉⠁⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠤⠊⡴⠾⢷⡛⠒⠾⡧⠀⣲⡾⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠁⠙⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣽⣭⣯⣯⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣄⣆⣴⡾⠷⠢⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣬⣝⣿⣿⡷⡶⠶⠶⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⢏⡽⡛⣝⠙⠷⣦⠠⡀⠀⠀⠀⠀⢠⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢃⠀⣿⡇⢱⣘⣷⡀⠀⠁⠀⠀⠀⠠⣸⣿⣏⠀⢍⠁⣠⢾⣶⠧⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠇⢹⡗⠀⠈⣿⠙⠠⠀⠀⠀⠀⡾⠉⠘⣿⣧⡈⢸⠳⡟⢹⡆⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⡈⠣⣄⠀⠀⠀⠀⠀⠀⠀⠉⢿⡅⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠙⠦⣄⡀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢄⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠈⠙⠦⣀⠀⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠓⡂⣂⣀⣲⢄⠀⠀⠀⠀⠀⠀⠈⠑⠲⢤⡀⠀⠀⠈⠑⢦⡀⠀⠀⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⢻⠛⠚⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠈⠳⣄⡀⠀⠘⢦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢦⡀⡀⠀⠙⣶⠄⢴⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⠲⣀⠹⣷⣄⢀⡈⠈⣲⣄⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠆⠀⠀⠀⠀⢀⣠⡴⠞⠊⠉⢉⡽⠥⣑⣌⠿⣹⡽⣟⣠⣶⠉⣿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣤⣴⠾⠉⠀⡄⠀⣴⠖⡀⡄⢠⣆⠹⣿⣇⣤⡀⠘⢙⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢋⣭⢏⣴⣾⢐⡋⢡⢰⣷⡷⡿⠿⢿⠿⠿⠿⠿⢷⣄⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠟⠛⠿⠿⠷⠿⠷⠿⠋⠃⠀⠀⠀⠀⠀⠑⠀⠐⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⣀⣀⣀⣀⡐⠂
⠀⠀⠀⠀⡠⣤⡤⠤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣬⣈⠬⠄⠁⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠉⠉⠉⠉⠈⡇
⠀⠀⠀⠀⡇⣿⡇⠀⣠⣄⡋⠀⢂⠀⠀⠀⠀⠀⠀⠀⢰⣀⣂⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⢨⢐⢒⢒⢒⢒⢊⠠⠀⠀⠀⠀⢀⣈⣀⡠⠀⠀⠀⣠⠀⠀⠇
⠀⠀⠀⠀⡇⣿⣿⡈⠷⠊⠑⠄⠀⠣⠀⠀⠀⠀⠀⠀⠰⠒⠒⠒⠂⠐⠲⢰⡆⠀⠀⠀⠀⢸⠨⢎⡨⢌⡨⢌⠀⠀⠀⠀⠀⡆⠀⢾⣵⠔⣁⣈⡀⠇
⠀⠀⠀⠀⡇⣿⣿⡇⡄⢑⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣀⣀⢸⡇⠀⠀⠀⠀⢸⢨⣣⢠⡣⢘⡣⢐⠀⠀⠀⢀⣥⣤⣤⣶⣾⣿⣿⣶⣧⣤⡀
⠀⠀⠀⠀⡇⣿⣿⠇⠃⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣐⠆⣂⠆⢆⡒⢸⡇⠀⠀⠀⠀⢸⠐⡱⠔⡁⠄⠡⠈⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘⠤⠒⡠⡃⠢⠔⢸⠇⠀⠀⠀⠀⠘⢴⢲⣶⠒⠒⡦⠄⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠊⡒⠉⠒⠉⠑⡀⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠭⠤⠂⠀⠀⠀⠀⠀⠀⢸⣭⣭⣭⣭⣭⣷
⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⢁⣀⣸⣀⡀
⠀⠀⠀⠀⠀⣾⣒⣛⠃⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠓⠠⣤⠄⠃
⠀⠀⠀⠀⢨⣝⠩⣫⢡⡢⢙⣥⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣄⣰⣖⣲⢰⡦⣑⡀⠀⠀⠀⠀⠀⠀⠠⡏⠉⠉⡹⠋⣹⡂
⠀⠀⠀⠀⠀⣆⣀⣀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠄⠀⢽⢹⣿⡆⠉⠀⣯⠀⠠⠀⠀⠀⠁⠀⢀⣠⡀⡠⢊⣼⡶⠀⠀⠀⠀⠀⠀⣇⠠⠊⡠⠊⢸⡁
⠀⠀⠀⠀⠀⠥⠤⠤⠤⠥⠤⠄⡠⣪⣾⠗⢄⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡹⢊⣴⣿⠋⠀⠀⠠⠀⠀⠀⠀⡇⡠⠊⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠠⡴⠀⡠⣪⣾⣿⠁⠀⡠⠃⠀⠀⠀⠀⢸⣿⠿⠇⠀⡇⠀⠀⠀⠀⠀⠖⠖⣪⣴⡿⣫⠇⠀⢀⠔⠁⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠰⢀⣀⣀⣁⣀⣘⠿⡻⠁⡠⠊⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠐⠀⠛⢌⠁⢀⠔⠁⠀⠀⠀⠀⠀⠀⠇⠀⢀⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠃⠀⠐⠁

⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣿⣿⣷⣦
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⡀⠀⣀⣀⡀
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢷⣀⣸⣿⣇⣀⣀⣀⣀⣀⣀⣠⢄
⠀⠀⢘⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⢻⣿⠟⠁⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠑⣄
⠀⢰⣿⣿⣿⣿⣿⣿⠁⠐⠀⠀⠒⢸⣿⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠈⠣⣀⣴⣦
⠀⢻⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣼⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡄
⠀⠀⠛⠿⣿⣿⣿⣿⣿⣦⣤⣤⣾⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⣸⠟⠋⠁⠈⢆⠀⠈⠻⢷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡏⣿⣿⡄
⠀⠀⠀⢀⠅⠀⠀⢲⡆⠀⠀⠀⢀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣭⣭⠁⣭⣭⠁
⠀⠀⠀⢈⠀⠀⠰⠀⠈⠳⣄⠠⠞⣠⣴⣶⣦⣄⣐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⣿⠀⠦⠤
⠀⠀⠀⠠⠇⠀⢰⣄⠀⠀⢁⣤⣾⣿⣿⣿⡿⠿⠿⠶⠆⠀⠀⠀⠀⠠⠶⠲⠶⠶⠶⠶⠖⠒⠒⠂⡇⠀⠉⠉⠁
⣀⣤⣴⣶⡄⠀⠀⢀⣀⣶⣿⣿⣿⣿⣛⢉⠁⠀⠀⠀⣀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠉⠃⠀⣷⣶⣶⣤⣤⣄⡀
⣹⣿⣿⣿⣿⣿⠟⠋⠁⠀⢠⠀⠀⢹⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣿⣤⣤⣀⣀⣾⣤⣤⣬⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⡿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⢀⣀⢀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠤⠀⢐⣧
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣀⣴⣿⣿⣿⣿⡿⠋⠙⢦⡀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡟⠻⢿
⢸⠙⠛⠿⣿⠟⠁⠾⠓⢀⡉⠁⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⣼⣿⣿⣿⣿⠟⢀⣠
⠾⣀⠀⣸⡇⣴⠄⣾⣷⠘⠃⠀⠀⠀⠀⠙⢿⡆⠀⢠⣤⡀⠀⢸⡿⠋⠀⠀⠀⣾⠀⠈⢹⡏⠀⢸⣿⣠
⠀⠈⠙⠺⣇⣬⠤⠿⠛⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠻⠦⠤⣼⣧⡴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⣀⣀⣀⣀⣠⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣀
⢀⣾⣿⣿⣿⠟⠁⠀⠙⣇⠀⠀⠀⠀⢀⣤⠶⣤⡅⣤⠁⣤⢨⣤⠶⣤⡀⠀⠀⠀⠀⠰⣿⣿⣿⡿⠃⠙
⠈⣭⠉⣭⠅⠀⢠⣤⠀⠀⠀⠀⠀⠀⠉⣥⠀⣌⠁⢠⣤⠀⠈⣥⠀⣌⠉⠀⠀⠀⠀⢀⣿⣿⣿⠀⣶⡆
⠤⠤⠤⠤⠤⠤⠾⠿⠦⠤⠀⠀⠀⠠⠤⠤⠤⠤⠤⠼⠿⠦⠤⠤⠤⠤⠤⠄⠀⠀⠀⠈⠙⠛⠿⠶⠿⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⣀⡀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠉⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⢠⡀
⠀⠀⠀⣠⠞⠁⢀⣾⣿⣾⡇⠀⠀⠀⠀⠀⠘⠛⠛⠛⠀⠛⠀⠙⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠈⠻⣿⡇
⠀⠀⠀⣿⠒⠀⢻⠛⣿⠛⡟⠀⠀⣠⣾⣿⣿⠟⢢⠀⠸⠆⠿⢰⣿⠟⢢⠀⠀⢀⣴⠟⠁⠀⠀⠀⠈⠻
⢶⠚⢿⠿⣿⠿⣿⡿⠛⣿⡇⠀⠀⠉⠉⠉⠁⡄⣤⠁⢀⣤⡄⠈⠁⡄⣤⠁⠀⠀⠀⡀⢀⣼⣿⣷⡆⢀
⣼⣀⣼⣤⣿⣤⣿⡇⠀⣿⣇⠀⢀⣀⣀⣀⣀⣁⣀⣀⣸⣿⣇⣀⣀⣁⣈⣀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿