Tater Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠔⠀⢄⡔⠁⠀⠐⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠘⠀⠀⠱⡀⠀⢀⠌⠉⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⣧⠀⠀⠀⠀⢰⠏⠋⠁⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠈⠳⠦⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠰⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠉⠑⠲⣤⣤⣤⣤⣤⣤⣴⣶⣿⡏⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠨⠈⠀⠀⠀⢹⣿⠟⠛⠿⣿⠟⠛⠻⣿⣧⠀⠀⠀⠀⡄⠂⠀⠀⠙⣦⡀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⢀⣨⠟⠃⠀⠀⠀⠐⠊⠀⠀⠀⠈⠳⣤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⡔⡂⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⣼⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠙⠻⠿⠿⣤⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠚⠹⣇⠀⠀⠑⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣀⣤⠤⠶⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠁⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠠⣿⣶⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠉⠙⠛⠛⠒⠒⠒⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣬⣷⡄
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⢄⡄⣼⣯⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠰⠜⠀⢸⠀⠀⠰⠀⠀⠀⢱⣶⠇⣀⡀⢰⠀⠈⠀⠀⠈⠠⣶⠁⣀⣀⠈⣠⠁⠀⠀⠀⡀⠀⠀⢀⠀⢡⠆⠞⠁
⠀⠀⠀⢐⡿⠁⠠⠄⠀⢁⠀⠀⠀⠀⡃⠀⠦⢴⠀⠸⠀⠀⠀⠀⠀⠀⠈⠁⠐⠭⡄⠀⠂⠀⠀⠀⠀⠚⠁⠀⣛⢉⠀⠸⠀⠀⠀⠀⡄⠀⠘⠋⡀⢸
⠀⠀⠀⠀⠆⠀⠐⠶⠀⠈⡆⠀⠀⠀⠌⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⠀⠀⠃⠀⠙⠻⠁⠸
⠀⠀⠀⠀⠀⣀⣀⣀⠠⠴⠁⠀⠀⠔⡈⠲⠶⠔⠛⠁⠀⠀⠀⠀⠀⠀⢀⠈⠦⠤⠶⠒⠃⠀⠀⢀⠀⡰⠑⣀⣀⣀⣠⠴⠃⠀⠀⡤⠒⠦⢤⡤⠶⠋⡀
⠀⠀⠀⠀⠀⠐⠧⠳⠀⠀⠀⠀⠀⡜⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠸⠭⠿⠄⠀⠀⠀⠀⡜⠁⠀⠄⠄⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠙⠀⣠⡆⠆⠀⠐⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠋⠀⠳⠈⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠖⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡁⠀⡁⠀⠀⠀⠀⠀⠀⠀⢦⠀⠲⠌⠀⠀⠀⠀⠀⠀⠀⠀⠐⢴
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡄⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⠘⠴⠰⠀⢂⠖
⠀⠀⠀⠀⠀⠀⠄⠀⡡⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠤⠊⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠙⡿⠭⠁⠀⠀⠀⠀⠀⡴⠃⠀⠈⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣤⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⣠⣤⡤⠶⠞⠛⠉⠉⠉⠉⠁⠀⠈⠀⠀⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⣶⣄⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠉⠁⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⣹⠔⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⢼⠟⢁⡠⠤⠤⠄⡀⠀⣠⠖⠉⠀⠀⠉⠑⢦⡀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⡄⠠⡀⣠⢁⣰⠃⢀⡼⠓⢲⡄
⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠈⢻⠁⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠑⣄⠘⠁⠈⠁⠲⠋⠀⡠⠊
⠀⠀⢰⠏⠉⡁⠀⡁⠈⠂⠀⠀⣾⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⣄⠀⠀⡇⢀⣴⣦⠀⠀⠀⠀⢸⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠊
⠀⠀⢸⣆⢠⣇⣠⣇⣠⣆⠀⠀⣿⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠻⠿⠁⢀⣇⠀⠛⠛⠀⠀⠀⣠⠋⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⢀⣀⡴⠋
⠀⠀⠸⡈⠁⠘⠦⠤⠀⣸⠀⠀⠸⣇⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⢀⡠⠊⠀⠙⠒⠶⠶⠒⠋⠀⢰⡄⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠋
⠀⠀⠀⠙⠦⠀⠀⢤⡞⠁⠀⠀⠀⢹⡆⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠋⠳⠀⠀⠀⠘⣧⣤⣴⣶⣾⠿⠟⠋
⠀⠀⠀⠀⠈⠓⣶⡏⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠰⠟⠳⣦⣤⣤⣀⣀⣀⣀⣤⣤⠶⢚⣫⠟⠁⠀⠀⠀⠀⠀⠀⠘⣏⠉
⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣥⣤⣤⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣶⣶⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣄⣀⣤⣴⡶⠶⠶⠦⠴⠶⢶⣶⣶⠶⠶⠶⠛⠉⣴⣿⠟⠉⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠹⣿⣷⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⢰⢻⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠻⣿⣦⢀⣠⡼⢸⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣧⡀⠀⠀⠀⠀⠀⠙⣿⡟⠀⢁⣿⡇⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⣾⣁⠀⠀⢈⣗⠀⠀⠀⠀⠀⠘⢧⣠⠀⠛⠣⠤⣀⣀⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⣩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣄⣀⣀⣀⣀⣀⣠⣤⣤⣼⡆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣿⣿⡿⠿⣿⠿⢿⠿⣿⡿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣶⣿⣶⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠒⠒⠒⠂⠈⠁⠀⠀⢀⣀⣀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠾⠃⡀⠤⠤⡀⠀⡠⠔⠒⠒⠢⡀⠀⠀⠀⡼⠀⠀⠀⠀⣴⢤⣠⠟⣸⢏⣿⣥
⠀⠀⠀⠀⢤⡀⠀⢹⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⢲⠁⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠘⢦⠉⠀⠋⠞⣩⠞
⣼⠀⣦⣶⢀⣇⠀⢺⠀⠀⠀⠀⠀⡀⠀⠰⣿⠆⠸⠀⢾⡷⠀⠀⡰⠀⠀⠀⢻⠀⠀⠀⠀⠀⢸⡄⠀⢀⡴⠃
⢿⡈⠉⠉⢁⡼⠀⠸⡆⠀⠀⠀⠀⠘⠢⠀⠠⠔⠁⠀⠑⠀⠐⠊⢀⡄⠀⠀⠘⣇⠀⠀⠀⢀⣠⡿⠚⠋
⠀⢹⣧⣼⡯⠀⠀⠀⣽⠀⠀⠀⢰⣄⣀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠈⠀⠀⠀⢹⡶⠶⠞⠛⠁
⠀⠀⠘⣧⠀⠀⠀⠀⣧⡀⠀⠀⠀⠈⠙⠻⠿⣽⣛⣉⣭⠵⠚⠁⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠘⢷⣤⣤⣤⠾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢋⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢦⣤⣤⣴⠶⠶⠶⠶⠷⣶⡶⠶⠶⠛⠋⣴⠋⠀⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠈⢻⣄⠀⣀⠀⡿⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣦⣀⠀⠀⠀⠀⠹⣿⠉⢉⡇⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⠒⠶⠾⠄⠀⠬⣇⠀⠀⠀⠀⠙⠶⡿⠃⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣀⣀⣼⠀⠀⠀⠀⠀⠀⠙⠶⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣰⣶⣄⣀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⢿⣿⠿⣿⣿⠁⡎⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⠹⠿⢛⣡⡞⢡⣌⣿⡿⣋⣭⣭⠍⠀⣤⣴⣶⡟⠀⡶⢸⣿⣆⠀⠿⣿⣿⣿⠟⣋⣵⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⢛⣩⠀⠀⢰⣿⣿⡿⠀⣾⡿⠙⣿⣿⡿⣡⡇⢰⣿⣿⡿⠁⠘⣡⣿⣿⢏⣼⡖⢸⣿⠏⣴⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣏⣴⣿⡏⣸⡇⠸⣿⣿⢃⠀⢸⡇⡀⢻⡟⣱⣿⡇⢸⣿⠏⣴⡆⢸⣿⡿⢃⣾⣿⠀⣿⡏⠜⢛⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠟⣰⣿⡇⠀⣿⡿⢸⣆⠘⢰⣷⣤⠞⢉⢻⡇⢸⡟⣸⣿⠷⣄⣛⣵⡿⢛⠿⠀⠿⢰⡀⠘⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣾⣿⣿⠿⢛⣡⣾⣿⣿⣿⡄⢸⡇⣿⣿⣷⣿⣿⣿⠀⣿⢸⣿⣄⣣⡿⠁⣶⢸⣿⠏⠀⣿⣤⣷⣶⣿⣷⢀⠙⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠶⠾⣿⡿⠿⢿⣿⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿⠀⢈⣾⡿⠛⢿⡇⠀⢋⣼⡏⡀⢸⡿⠁⣿⣿⣿⢣⣾⡿⢂⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠿⠿⢟⣋⣄⠈⡏⣼⣿⡄⢛⡀⠸⡿⣸⣇⠸⢣⣆⠻⢟⣵⣿⠋⣰⣿⡿⢟⣫⣭⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣾⣿⣿⣿⣿⣄⠀⣿⠏⡅⢸⣿⡦⣴⣿⣿⣷⡿⠿⢻⣿⣿⡇⢰⣿⢟⣴⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠏⠁⠹⣦⡸⣎⣡⣾⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣘⣣⣾⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⠃⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠁⠙⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⠃⠀⠀⠀⠀⣹⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠀⠀⠀⢠⣾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠟⠛⠋⠁⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠆⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣀⠀⢀⡀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡞⠁⠀⠀⠀⠹⠁⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠁⢸⡇⠀⠀⠀⠃⢠⣇⡀⢀⠜⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⡀⠀⠀⠀⠳⣤⣀⣀⡤⠋⠀⠉⠁⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠈⠳⢄⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⡄⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣶⣶⣦⣤⣤⣶⡶⠚⠉⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡠⠢⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠐⠒⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠉⠉⢻⡿⠟⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠀⠀⢀⡀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠒⠛⠉⠀⠀⠀⠀⡁⡀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⢀⣠⠞⠋⠁⠀⠀⠀⣿⠀⠀⠀⡠⠂⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠄⠀⠀⠀⢻⡄⠀⠀⠀⢀⣠⠴⠛⠛⠛
⣴⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⠛⠋⠉
⠹⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠐⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠂⠀⠀⠀⠀⠀⠀⠎⠆⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠛⠓⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⡄⠀⠀⠀⠙⢻⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡶⠞⠋⠁⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠙⣆⠀⠀⢠⠏⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⠀⠀⠀⠈⠳⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣤⣤⣤⣤⣤⣀⡀
⣀⣀⣀⣀⣀⣀⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣙⣛⣛⣛⣛⣛⣷⣄⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⠶⠶⠀⢰⣿⣧⠀⠀⢶⡶⣶⡶⠶⢶⣶⡶⠶⠶⢰⣷⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠈⣿⡇⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⢸⣿⡿⠁⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⢠⡏⠀⢻⣷⠀⠀⢠⣿⡇⠀⢸⣿⠀⠀⠀⢸⣿⢃⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣸⣧⣤⣼⣿⠀⠀⢸⣿⡇⠀⢸⣿⣤⣤⠀⢸⣿⣿⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⡇⠀⠈⣿⡆⠀⢸⣿⡇⠀⠘⣿⣀⣀⡀⢸⣿⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣤⠀⠈⠁⠀⠀⠹⠃⠀⠀⠉⠁⠀⠀⠙⠛⠛⠉⠈⣩⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠠⠤⢤⣾⠤⠤⢀⣀⠄⠤⣼⡧⠤⠄⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣄⡙⠣⠀⠀⢸⡏⠀⢀⣿⡈⣧⠀⣿⠀⠀⠀⠐⠋⣀⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⡙⠃⠀⠀⢸⡇⠀⣿⡇⢙⡿⠲⣿⠀⠠⠀⠀⠋⣉⣉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁⠀⣀⣀⣀⡀⠘⠧⠔⠈⠙⠋⠀⠀⠻⠤⠂⣠⡴⠶⢶⣌⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠉⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⠀⠀⠀⠀⠀⠀⣀⣉⠛⠶⢦⣤⣞⣁⣀⣀⣀⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⡤⠤⠂⠀⣾⣉⣉⣙⣶⡞⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠉⠛⠲⠶⠤⣤⣤⣤⡤⠤⠴⠖⠒⠛⠛⣛⡛⠓⠒⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⣤⡞⠉⢳⡤⠤⠖⢋⡁⢀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣀⣤⠆⢀⣤⣤⠀⢸⣧⣿⠀⠀⠀⠀⠈⠙⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⣿⣧⡴⠶⠋⢰⣿⠉⠓⠀⡴⠚⠉⢩⢿⣿⠉⠁⠀⢠⣿⠃⡼⠀⠀⢹⣿⣄⡀⠀⠀⣴⠞⠉⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡰⠚⢁⣤⠞⠛⡿⣿⡇⠀⠀⢀⣿⡇⠀⣾⡇⠀⠀⣰⠋⣾⡏⠀⠀⢠⣾⣿⠞⠁⠀⢠⡟⠀⣼⠃⠀⡾⠁⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠞⠀⠀⣼⠃⣿⣿⠀⠀⣸⣿⣷⢠⡿⣷⠀⡼⠃⠀⣿⡇⠀⣰⠏⠸⣿⠀⠀⣰⠏⠀⢸⣿⠀⣸⢣⡶⣄⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⣀⠞⠀⠀⠀⢀⣼⠃⠀⢸⣿⡀⠀⡏⠘⣿⣼⠃⠉⢛⣵⠶⡄⢿⡇⢠⡏⠀⣀⡙⠷⠞⢁⣤⠤⡸⣿⣠⠏⢿⣧⡿⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⢼⡁⠀⢀⣀⡴⠟⠁⠀⠀⠀⢿⣇⢸⠇⠀⠈⠉⠀⠀⢸⣿⢠⡇⠈⢿⡾⢀⣾⡏⢹⠀⢀⣾⡏⠀⠇⠉⠋⠀⠈⣿⣧⣀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠙⠻⢯⣅⣠⣤⠤⣤⣀⣀⣈⣛⠟⠀⠀⠀⠀⠀⠀⢸⣿⡞⠀⣀⣤⡀⢸⣿⣧⠟⢀⣾⣿⠃⣰⡆⠀⠀⢀⡼⠁⢈⣩⠗⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⢳⠀⠀⠀⣀⣀⡤⠖⢿⣇⢰⠋⠈⢿⣦⢿⣧⠀⡼⠈⣿⣀⡏⢷⣀⣠⠞⠁⣴⡿⠁⠀⢀⣤⠤⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠋⠉⠉⠁⠀⠀⠘⣿⣾⠀⣠⢾⡇⠈⣻⣿⠃⠀⢈⣉⣠⡤⢭⠁⠀⣼⣿⠀⢀⡴⠋⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⣠⠞⠑⢮⢻⡆⢯⡼⠃⢸⠁⠈⠉⠉⠉⠉⠀⠀⢸⠀⠀⢿⣇⣰⠋⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⢀⡴⠁⠀⠀⠀⢳⡉⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⢉⡴⠋⠙⢦⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢠⡞⠁⠀⠀⠈⢳⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠊⠁⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⢨⡇⠀⠀⠀⢠⠴⠒⠁⠀⠀⣠⠏⠀⠀⠀⠀⠀⢸⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣤⣤⡤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⢾⣁⣀⣀⣠⠞⠀⠀⠀⠀⠻⠤⠤⠤⠶⠚⠁⠀⠀⠀⠀⠀⠀⠈⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⠤⠤⠤⠴⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠶⠖⠒⠒⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⢰⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⠋⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠤⠀⠀⠀⠀⠀⠀⣠⣤⡀⢠⠖⠲⡄⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⣎⠁⠀⠛⠤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠐⠀⠀⡼⠁⣀⢳⣏⣤⡀⣷⠀⠀⢰⠀⠀⠈⡷⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⠀⠟⢋⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣄⠀⠀⢀⡀⠀⠀⠀⡇⢸⣿⣿⡽⣿⠇⡟⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣞⠲⠶⣶⠤⠖⠚⠛⠶⢬
⢠⠦⡀⠀⢀⠤⡀⠀⠀⠀⡾⠀⠀⠀⠑⠄⠀⠀⠹⣄⢉⡞⠣⣤⠞⠁⠀⡤⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣠⡾⠋⠉⠁
⠸⡀⢡⠀⡎⢀⠃⠀⠀⠀⢧⠀⠀⢀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣤⣤⣴⠾⠋
⠐⢳⠈⠶⠁⣜⡀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⢈⣣⣤⠀⠀⠀⠂⠀⠀⠀⣻⣀
⠀⣈⡆⢠⠯⠬⠉⣷⠀⠀⠈⣧⠀⠀⠀⠠⠀⠀⠀⠀⢀⣠⣤⣮⣤⣴⣿⣿⣶⣦⣄⡀⠐⢿⣿⣷⡀
⠀⢻⣀⠀⢀⡀⠈⠁⠀⠀⠀⢸⠀⠀⣠⣴⣦⠀⢀⣴⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠟
⠀⠈⠫⢭⣭⡃⠀⠀⠀⠀⠀⠸⣆⠸⣿⣏⣉⣴⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠉⠉⢉⡉⠉⠠⣇
⠀⠀⠀⠀⠉⠛⠛⠶⠶⠶⠤⣤⣿⠀⠙⠛⠛⠛⠋⣉⠁⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⢠⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⡄⠀⠀⠈⠙⠒⠶⠤⠤⠴⠒⠋⠁⠀⠀⠂⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⣀⣀⡤⠤⠖⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⣀⣀⣀⣻⣯⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣤⣤⠤⠶⠶⠶⢿⠋⠉⠉⠁⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠀⠀⢰⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠉⠀⠀⠀⠀⠈⠉⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢿⣿⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠡⠒⠀⠀⠳⡀⠀⠀⠤⠤⠀⠀⠈⢧⡀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠒⠚⠉⠉⠁⣠⠞⠉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠠⠴⢥⢬⣎⣀⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠠⢀⠈⢆⣠⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⡼⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⢀⣠⠦⣀⠀⢀⡖⠃⠀⠀⠀⠀⠀⠀⠀⠲⠆⠈⠉⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⡇⠰⠁⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠐⠊⠉⡁⠈⠀⢰⢯⠲⡀⠀⢠⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⢴⣇⠃⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⡔⠛⠛⠒⠧⣀⢠⡏⢸⠐⠃⠀⠀⠆⢸⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢤⡤⠤⠴⢈⣀⣀⠀⣸⡏⠀⠀⠀⠀⠀⠀⢰⠃
⠀⢰⠁⠀⠀⠀⠀⡏⢸⡇⠸⠸⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣠⠴⡄⠀⣠⡎⠀⠉⠐⠊⠀⠀⠉⠉⠓⢤⡀⠀⠀⠀⢠⠇
⠀⡇⠀⠀⠀⠀⢸⠀⠀⠳⡆⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⡏⠀⠙⠀⠀⠈⢱⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⡆⠀⠀⡞
⢸⠁⠀⠀⠀⠀⡎⠀⠀⡘⠀⠶⢐⠘⡂⠀⣄⠠⠀⠀⠀⢐⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠃
⡞⠀⠀⠀⠀⢰⠃⠀⠀⠱⠴⠖⠁⠉⢇⣐⡤⠶⣤⣤⣤⢼⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡾
⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⡇
⠳⢄⡀⠀⠀⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠃
⠀⠀⠈⠉⠛⠁⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⠤⣄⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠗⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠚⠳⢤⣄⣀⣀⢀⣇⣀⣀⣠⡤⠤⠶⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠿⠛⠻⠷⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⠶⢟⡿⠟⠁⠀⠀⠀⠀⠈⠭⠭⡽⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⢉⣀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⣹⣷⡀
⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⣑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣿
⠀⢰⡇⠀⠀⠀⠀⢀⠄⠂⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⣿⡆
⠀⣾⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⣀⣹⣷
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣿⡆
⢠⣟⡒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⣿
⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢻⣷
⡇⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠈⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠴⢿
⢿⡀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⢸⣇⡸⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⣀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⣰⢿
⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⣰⠋⢸
⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣿
⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢁⣠⠞⢁⣾
⠀⠀⠀⠈⠻⣯⡙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠋⠁⠀⣼⡟
⠀⠀⠀⠀⠀⠈⠻⣶⣄⠈⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣀⠀⠠⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣶⣤⣤⣵⣄⣀⣀⣊⣉⣢⣄⣤⣤⣴⣶⠾⠿⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⣀⠀⠀⣀⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡞⠁⠀⠀⠀⠱⠏⢀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠉⠸⡇⠀⠘⠀⠀⠀⣆⡀⠀⡠⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⣀⠄⠀⠀⠻⣦⣀⣀⣠⠞⠁⠉⠉⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠻⢄⡀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣶⣤⣤⣤⣤⣴⠖⠚⠋⠁⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢠⠄⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠻⣿⠿⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠁⠀⠀⠈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿
⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⠤⠤⠶⠶⠛⠀⠀⠀⠀⢈⢀⠆⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟
⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⢀⣀⡤⣾⠃⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠹⣆⣀⣀⣤⠶⠋
⣴⠚⠉⠀⠀⣿⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠁
⠈⠳⠦⣄⣀⣿⡇⠀⠀⠑⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠰⣾⡿⠟⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⠀⠀⠐⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⠷⠶⠶⠶⠶⠶⠶⠶⠛⠛⠋⠁⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⣿⣶⣶⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣶⣶⣿⣿⣿⣶⣶⣦⡄
⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢈⡒⢄⠀⡀⡠⠊⡡⠂
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠚⠘⣿⠀⠀⠀⠀⡰⠒⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣴⣃⢾⣠⠊
⠀⠀⠀⠀⠀⢀⠀⡆⠀⣰⣴⡛⠀⠀⠀⢶⡅⠀⢠⠄⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⠟⠉⡫⣿⡇
⠀⠀⠀⠀⠀⢸⠀⡇⢰⠟⣉⠙⢶⠤⠤⠀⠙⡶⢠⠘⠁⠀⠀⠀⠀⠀⠀⠀⢀⠞⡴⢋⢰⠈⠳⡈⢣⡀
⠀⠀⠀⠀⠀⢸⠘⡇⠛⠀⣈⠀⢸⣷⣶⡶⢶⡆⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠤⢾⠀⠀⠀⠛⡶⠃
⠀⠀⠀⠀⠀⠀⠭⣀⣄⣸⡇⣀⣸⣥⠤⠤⠬⠧⠧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⣏⡀⠀⠀⣿⠉⠀⠁
⠀⠀⠀⠀⠀⠊⠄⢀⠤⠚⠉⡉⠀⢄⠀⠠⡀⠢⡐⠄⠑⠀⠀⠀⠀⠀⠀⠀⠒⡉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠈⠀⠀⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢠⢂⡖⣄
⠀⠀⠀⠀⠀⠀⠘⠂⡿⡿⢄
⠀⠀⠀⠀⠀⠀⠀⠉⡐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⢰⢧⠺⢀⣤⠤⠤⠤⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡤⣬⣆⡤⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡼⠆⢰⠏⠈⢳⡤⠤⠤⠤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢙⣀⣹⣴⢯⣏⢳
⠀⠀⠀⠀⠀⠀⣀⡗⣆⠸⠀⡷⢸⡌⡗⢸⡇⡧⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠑⣸⣏⣿⢣⢺⣿⣿⣿⣼⢿⣯⠄
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠐⣀⣀⣼⣧⣧⣀⣠⣇⣀⠀⠠⠀⠀⠀⠀⠀⠀⣀⣸⣯⠥⠼⠿⠾⠿⢿⣬⣿⣯⡟
⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠀⢀⠄⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣦⣈⣉⣉⣉⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⠿⢻⣿⣿⣷⣄
⠀⠀⠀⠀⢠⣿⠛⠿⠿⠿⠟⠛⣿⣿⣿⣿⡋⠛⠋⣹⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⢀⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣈⣉⣁⣤⣿⣿⣿⣶⣤⣴⣾⣿⣿⣏⠙⠿⠿⠿⠛⢙⣿⣦
⠀⠀⠀⠸⣿⣄⡉⠛⠛⢉⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣄⣉⣠⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡿⠿⣿⣿⣿⣿⠿⣿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣁⣠⣤⣶⣿⣿⣿⣿⣦⣤⣤⣾⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⢿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⣉⣁⣤⣶⣿⣿⣄⣉⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠻⠿⠿⠟⠋⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣏⡙⠛⠿⠟⠛⢉⣿⣿⠛⠿⠛⣿⣿⣶⣶⣶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣶⣶⣿⣿⣿⣿⣷⣶⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠿⠿⠛⠛⠓⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⣁⣀⡀⠀⠀⠀⠀⠀⠀⢀⡔⢩⣬⠉⢆⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣾⡏⢠⠏⣤⣌⠹⡀⠀⠀⠄⠀⠀⠸⣇⠈⠉⣠⠇⠀⢀⠀⠘⣷⠀⠀⠀⢀⣠⣼⣿⣧
⢰⣟⣻⢯⣟⡁⠀⠀⠀⠀⢸⣿⣁⠘⣄⠉⢁⡴⠃⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠸⡇⠀⠀⣈⡿⠈⢉⠲
⣾⣉⣋⠀⣭⠟⠀⠀⠀⠀⢸⡿⠥⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⢒⡾⡆⠀⠀⠀⣿⣤⠞⠋⣠⡾⠋⠙
⠀⠉⠙⣧⡈⢷⣄⠀⠀⠀⢸⡇⠀⡾⢍⣩⢯⣉⣉⡿⣏⣉⣩⠟⠧⠤⠞⠉⢉⣩⠞⠁⠀⠀⠀⠟⠁⢠⡾⠋
⠀⠀⠀⠈⢳⣄⠙⠳⣦⣀⣸⡇⠀⠉⠒⠶⠤⠤⠤⠤⠤⠭⠤⠤⠤⠶⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠙⢷⣄⡀⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠋⠁⠀⠉⠻⣆⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⡼⠋⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⢿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠲⢿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣤⣤⣿
⣿⣿⠟⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡀⠀⠀⠀⠠⠄⠈⠁⠀⠀⠀⠀⠀⠀⠁⠐⠀⠠⢄⣀⠀⠒⠀⢄
⣀⣀⣀⣀⡀⢀⡀⠤⠤⠤⠤⠄⠀⠀⠁⠒⠒⠒⠒⠒⠒⠐⠒⠒⠂⠙⠛⠲⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠤⠂⠈⠙⠉⠉
⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⣀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠈⠉⠉⠁⠉⠁⠂⠀⠁⠀⠀⠢⠄
⠀⠀⠀⣴⠀⠀⠁⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠄
⠤⠶⠚⠛⠃⠑⠂⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠐⠀⠀⠰⠶⣤⠠⠴⠚⠈⠉⠁⠀⠈⠁⠈⠉⠂⠀⠀⠀⠀⡈⠀⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⠤⠤⠒⠚⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠙⠛⣛⡋⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⣀⠄⠤⠒⠒⠂⠉⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠰⣾⣿⠟⠀⣿⣿⣿⡄⠀⠀⠤⠀⠐⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠁⠠⠀⠛⠛⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠠⢐⣀⣰⠆⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡤⠶⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⡀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠄⠀⠀⠁⢀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠄⠠⠠
⠀⠀⠀⠀⠈⠉⠠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⢀⡀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡀⠀⡀⡀⠈
⠈⠀⢈⠀⠀⠀⠇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠁⠀⠀⠀⠀⠀⠀⠀⣉⡀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⣀⠀⠐⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠠⠤⠀⠠⠀⠰⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡲⠀⠀⠀⢆⡀⣠⠄
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⢠⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠠⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠐⠄⠘⠑⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣉⣀⣈⡉⠤⢴⠀⠀⠀⠀⠀⠈⠉⠁⠈⠀⠀⠀⢀⠀⠀⠀⠇⠀⠠⠄⠀⠀⠤⢀
⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⢰⠀⢂⢀⡄⢲⠐⠀⢀⡇⠀⠀⠀⠀⠆⡐⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⡈⢀⢀⠈
⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠄⠀⠀⡠
⠀⠀⠀⠁⠀⠈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠉⠒⠒⠓⠒⠉⣀

⠀⠀⠀⠀⡠⠶⡄
⠀⡴⠛⣶⠁⢀⡇⣠⠖⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⢋⠙⠛⠛⠛⠛⠳⣦
⠀⡇⠀⢻⠀⢸⡞⠁⢀⠞⠁⠀⠀⠀⠀⢀⣀⣀⣴⣯⣠⣤⠴⠁⡀⣀⠀⠀⠀⠀⠈⠹⣦⡀
⠀⢳⠀⠀⠀⠈⠁⢠⡏⠀⠀⠀⠀⣰⢊⡡⠄⠀⠀⢀⣤⣤⣉⣐⠢⠄⢈⣑⣢⣄⡀⠀⡆⠙⣆
⠀⢸⠁⠀⠀⠒⠒⠾⡄⢤⣤⡀⠀⠹⣆⠀⢀⡔⠋⠁⠀⠀⠀⠀⠉⠑⠒⠂⠂⠒⠚⠓⢇⠀⢸⡄
⠀⠘⡄⠀⠀⠀⠀⠀⣠⣀⠀⢹⠀⠀⠀⢨⠏⠀⠀⣀⣀⣤⣀⡀⢀⡤⠖⠒⠲⢤⡀⠀⠀⠀⢼⠃⢀⡤⠖⠒⠲⢤⡀
⠀⠀⢙⠖⠤⠤⢤⠊⠀⠀⢙⡭⠴⠦⢤⡞⠀⢀⠞⠉⠀⠀⡀⠙⠋⠀⣀⣀⠀⠀⠙⡆⠀⠀⠈⣷⠋⢀⣠⣤⡀⠀⠙⣦
⠀⠀⠈⢢⠤⢤⠞⠀⠀⡴⠋⢀⣤⣄⣰⠃⠀⣾⠀⠀⠠⣿⣿⡧⠀⠸⣿⣿⠇⠀⠀⡟⠀⠀⠀⢻⣠⡟⠁⠀⠈⠀⠀⠘⡆
⠀⠀⠀⡏⠀⠀⡇⠀⢸⠁⠀⠀⠀⢈⡟⠀⠀⠸⣄⠀⠀⣉⣉⡥⡤⣤⣈⣁⠀⣀⠼⠁⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⠀⣸⠁⠀⠀⠀⢨⠗⠋⠁⠀⠀⠀⠀⠀⠉⠛⢧⡀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⣧⠀⠀⣇⠀⠈⢦⠀⠀⢰⠇⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠑⣶⠶⠶⠛⠶⠶⠦⠤⣄⣀
⠀⠀⠀⠘⢆⠀⠈⠓⠤⣈⡓⢦⡟⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢀⣀⣀⣀⣀⡀⠈⠻⡄
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠉⢻⠃⠀⠀⠀⠀⠀⠀⠑⠦⢄⣀⣀⣀⣀⣀⡤⠖⠋⠀⣰⠶⡆⠀⠀⠀⠀⠀⠀⠈⠛⢻⡀⠀⠀⠈⢱⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⣿⠀⠀⠀⠀⠴⠛⢤⣀⡀⠀⠀⠀⠉⠀⠀⣀⣠⠴⠚⠁⠀⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢳⡤⠖⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⡇⠀⠀⠀⠀⠀⣀⠀⠀⠈⠉⠉⠁⠘⠋⠉⠀⠀⠡⠀⢀⡼⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠸⣄⠀⠀⠀⠓⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢻⡦⢀⣠⠀⠀⠀⠀⣀⠀⠀⠀⢠⠏⠁⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠒⠉⠁⣀⡀⠀⠀⠀⠨⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠙⢄⠈⠁⠒⠒⠈⠉⠈⠉⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡔⠁⠀⡰⠋⠉⠙⣦⠀⠰⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⣀⣀⣤⣀⣠⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠉⠛⠉⠀⠀⠀⢰⡋⡇⠀⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢉⡟⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠿⡙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠯⣀⡢⢌⠐⡉⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⡩⠵⠂⡦⢹⠀⣉⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠉⠳⢄⡁⣀⣴⡋⠉⠛⠲⣶⠶⠶⡶⣶⢶⠶⠾⠟⠛⡀⢜⣁⠔⣂⠥⠒⠋⠁⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠙⣅⠀⠀⠑⠤⡀⠀⡱⢖⠖⠀⠋⢢⡀⠀⠒⠙⣢⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢀⠠⢘⠣⠊⢀⣤⠞⢳⡑⢕⢄⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣕⢢⢀⠀⠀⠀⠀⠀⢠⠄⠊⠩⠐⠀⢀⣠⠴⠋⠀⠀⠀⠙⢦⡓⢝⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⡴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣈⡉⠉⠉⠈⠁⢀⣀⣠⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣁⠂⠍⢐⠒⠒⠒⣒⡈⠡⠒⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣤⣤⣤⣤⣤⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠶⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⢠⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣬⣉⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠆⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⣀⢦⣤⡴⠶⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⠦⣤⡴⣀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⣸⡏⠀⠀⢠⣾⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠘⣿⡀
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠀⠙⠳⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠇⠀⢿⡇
⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⠙⠛⠓⠶⠶⠤⣤⣤⣤⣀⣀⣀⣀⣠⣤⣤⣤⡤⠴⠶⠖⡿⠋⠁⠀⠀⠀⠸⣿
⠀⠀⠀⠀⢸⣿⠀⠀⢰⠉⢱⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠈⠒⠊⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡴⠶⡄⠀⠀⠀⢿⡆
⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠘⣄⢀⠇⠀⠀⠀⠘⣿⡄
⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠶⠋⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠸⣷⡀
⠀⣿⠃⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⠀⠀⠀⢹⣧
⢸⣿⠀⠀⠀⠀⢠⠊⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢤⠃⠀⠀⠀⠀⠀⢿
⢸⣿⠀⠀⠀⠀⣾⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⠀⠀⠀⠀⠘⣄⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢰⣿
⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠜⠀⠀⠀⠀⢀⣾⠏
⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢦⡀⠀⠀⠀⠀⠀⠘⠦⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋
⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣶⣤⣤⣤⣀⣀⣀⣀⣠⣤⣤⣤⣶⣶⠿⠿⠛⠋⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠁⠉⢉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⡎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⠿⠋⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣁⠀⢬⡀⠤⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⢀⣀⡄⠠⣄⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡠⠉⠀⡈⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡃⠁⠈⠀⠐⠀⠀⠀⠀⠀⠀⠈⠖⠀⠀⠀⠀⢸⠃⢀⠘⠀⠸⢠
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⣌⠀⠀⢀⠂⠀⠀⡉⠁⠀⠀⠀⡀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠈⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡍⠉⠉⠁⠀⠀⢀⣅⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⠀⠀⠀⠈⡇⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣧⡅⠀⠀⠀⡇
⠀⠀⠀⠀⠐⠶⠶⠤⠤⢠⠦⣄⣀⣀⠀⢀⣀⠔⠀⠀⠀⠀⢨⠀⢀⠹⡀⠀⠀⠀⠉⠉⠉⠉⠉⢀⡜⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣄⣁⣀⠤⠤⠬⠩⠥⠶⠤⠤⠀⠀⢰⣿⣿⣿⣦⣼⠶⠤⢤⣄⣤⣤⠴⠚⠙⠠⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡾⠟⠛⠀⢀⣠⡶⣶⣤⣀⡀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠚⠛⠉⠁⠁⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⢤⠤⠀⠀⠀⠠⠤⣤⣀⣤⣤⡀⠀⠀⠄⠀⠀⠀⠤⠄⠀⢀⣀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣦⣤⣶⣷⣿⠶
⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣟⣽⣭⣒⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⡿⡍
⠀⢀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣷⣿⣘⣨⣿⣊⣻⣽⣍⣡⣤⡄⣠⡶⢾⣷⣀⠀⠀⠨⡯⠃⠀⠀⠀⠀⣀⡀
⠀⢈⣤⣀⣀⡈⣉⠛⢿⣿⠿⣿⣿⣿⡏⣽⡯⣻⣿⣿⡿⣛⣛⣛⣶⠮⠵⢿⣿⠋⢉⣐⣿⣿⡗⠂⠈⠉⠉⠛⠃
⠀⢠⣤⣤⣤⣉⣩⣉⣒⣶⣾⣿⠿⠿⠇⠻⠓⣻⣟⠛⠛⠛⠛⠋⠛⠂⠀⠸⠿⠿⠿⢿⣻⣭⣽⣤⣤⣤⣄⣠⡄
⠀⠘⢟⡛⠋⡉⠉⠉⠀⣤⣴⣶⠀⠀⠀⠀⠀⢹⣿⣥⣤⣴⢤⠶⠒⣶⣤⠖⢠⣤⡄⢠⣽⡿⠛⠿⠿⠟⠛⠛⠃
⠀⢠⣾⣷⣶⣷⣴⣷⣾⣌⠻⣇⡀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡈⣿⣿⡇⣶⠀⣿⣷⡆⢁⣾⣶⣮⣠⣤⣠⣦⡆
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠦⡤⢼⠄⠀⠀⠀⠙⠻⣿⣷⣇⡷⢹⣿⣤⣿⣭⣿⣬⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣄⣀⣀⣹⣿⣀⣤⣾⡄⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣾⣿⣿⣿⣾⣿⣿⣿⣟⣿⣻⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠙⠛⠋⠉⠉⠉⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠛⠛⠉⠉⠉⠉⠛⠛⠛⠒⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⡸⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⣰⠏⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⢠⡿⠀⠀⠘⣇⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣲⠀⠀⠀⢻⡄
⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠸⣧
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⡠⠒⠈⠁⠀⠀⠉⠀⠢⠀⠀⠀⠀⠀⡠⠂⠈⠁⠀⠀⠈⠐⠠⡀⠀⠀⠀⢸⡇
⠀⠀⠀⢰⡿⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠈⣷
⠀⠀⠀⣼⠃⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⢿⡀
⠀⠀⣸⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠸⣇
⠀⢰⡏⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠸⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢹⣆
⠀⣿⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠃⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢻⡄
⢸⡇⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠈⣷
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣄⣀⣀⣤⠴⠋⠀⠀⠀⠀⠀⠈⠳⠦⣄⣀⣀⣤⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢹
⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⡿⠀⠀⠀⠀⣠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⢸⡇⠀⠀⠀⠰⠁⠀⠘⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⣿⠁⠀⠀⠀⢠⡀⠀⢠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠑⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⢄⠀⠀⠀⠀⠀⠀⠀⣠⢦⠀⠀⠀⠀⢰⡟
⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⣇⠀⡸⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢠⡿⠁
⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠋⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠈⠙⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠾⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣻⣷⣶⣶⣶⣶⣶⣿⣟⣛⣋⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣉⣹⣆⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠀⠀⠀⠀⢲⣤⣶⣶⣶⣠⣶⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠆⡆⠀⠀⢴⡾⣿⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡩⠀⠀⠀⠀⠀⠀⠁⠀⠠⠤⢿⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢄⡟⠀⠀⣔⣀⠉⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢀⢀⡿⠁⠈⠀⠁⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢰⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠠⣤⣼⠄⠤⠴⠤⢄⡀⠀⠀⠀⠀⣴⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⣀⠀⠀⠀⠀⠀⠈⠉⢛⣷⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠓⠛⠚⠛⠛⠀⠀⠈⠀⠘⠀⠀⠀⠀⠀⠀⠀⡇
⡖⠈⠭⠭⠭⠭⠭⠭⠥⠤⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⢀⣀⣀⣀⡀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠀⢀⡠⠄⠂⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡔⠀⠉⢀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠆⠀⠀⠀⠀⠀⠀⠀⢰⠃⢰⠀⠀⠀⠁⠀⠀⡄⠰⠀⠀⢃⠀⠀⢠⠀⢰⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢸⠸⠀⢸⠀⠀⠀⠸⠀⠀⡸⠃
⠀⢦⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠈⠆⠀⠀⢀⠀⠀⠀⠀⠀⠘⠘⣾⣾⠾⠬⠽⠴⠄⠚⢊⣉⣀⣐⡀⢲
⠀⠸⣄⣀⣀⠀⢤⠏⠀⠀⠀⠀⠀⠸⡀⠀⠊⠀⠀⠀⠀⡀⠀⠀⢹⡀⠀⠀⠀⠀⠠⡈⢉⣩⡋⢻⣽⣇⣳
⠀⠀⠀⠈⠉⠉⠍⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡷⠀⠀⠀⢀⡓⠠⡟⢙⣿⣀⣻⡭⠇
⠀⠀⠀⠀⠀⢌⡀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠈⠙⠂⠛⢯⢉⡤⠛
⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⡀⠀⠀⠈⢇⠀⣀⠤⠒⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣐⣒⣒⣂⣮⣁⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠒⠒⠠⠤⣀⡀⠀⠀⠀⠀⢀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠒⠒⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠳⠶⠂⠚⠚⠀⠀⠀⣨⡍⢐⢢⣤⠈⠓⠁⡔⠒⠒⠒⠒⠒⡄⠀⠰⡉⣁⠀⠀⢀⠸⠥⠤⠼⢇⡀
⠀⠀⠀⠀⠀⠀⢀⠤⠊⠻⠜⠂⢦⠭⢛⡀⠀⣴⠂⡰⠂⠈⠀⠀⢸⠀⠀⣠⠋⡜⠻⣆⠀⣠⠛⡀⠀⠈⠁⣠⠔⠁⠀⠀⠀⠀⠀⠈⠲⡄
⠀⠀⠀⣀⢔⡎⢡⡤⠤⠐⢒⠶⠫⣁⣤⡀⠀⠀⠉⠀⠀⠺⠀⠀⢘⠀⢰⠧⠤⠼⠾⠧⢴⠧⠤⠶⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⢘⠉⠙⠶⠚⠙⡄⠀⠘⠤⠔⣇⢀⡇⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⠉⠢⡀⠇
⠀⠀⠘⠘⡀⠘⠒⠊⠀⠀⠀⠀⠀⢥⡠⠇⡠⣠⢤⣄⣀⠤⠔⠒⠊⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⠔⠓⠉⠈⠁⠀⡔⠒⠒⠢⡀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠸⣼⣀⠠⠄⠒⠊⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢀⡴⠁⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⣤⣄⠀⠒⠒⠒⠀⠀⠀⠐⠒⠒⠀⢀⣤⡀⠈⠉⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢀⡠⠃⡄
⠀⠀⠀⠀⠀⢸⠀⠈⢉⢍⣑⣒⣒⣒⠒⠀⠈⠉⠉⠀⠀⣇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⣠⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄⠀⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠸⢿⣃⣠⣮⢿⠀⠀⠀⠀⠀⠀⠀⡏⠲⢄⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⡇⢰⣦⣶⢰⢴⠀⣤⡴⠀⢽⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠈⠱⣄⠀⠀⠀⠀⠀⠀⣇⡰⠋⠉⡆⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⢇⠈⠁⠉⠈⠙⠓⠉⠙⠀⣸⠀⡇
⠀⠀⠀⠀⢀⠎⠁⠒⠒⠀⢴⡦⠤⠤⠤⠷⢤⠀⠒⠂⠉⠏⠀⠀⡰⠁⢸⡀⠀⣀⠀⡀⣸⠀⢀⣀⠀⠈⠢⣈⡉⣉⣉⣉⣉⣉⣉⣉⢉⣁⠔⠁
⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠑⢄⠀⣞⡂⢂⡴⠁⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⡜⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⠋⢃⠀⢠⠤⢄⠀⢙⣂⣀⣐⡃⠀⠀⠈⣇⣀⣀⣺⠁⠀⠀⢹⣂⣀⣐⡃
⠀⠀⢸⡀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠑⠁⠀⣈⣉⣉⣉⡁⠀⠀⠀⣏⣉⣉⣹⠀⠀⠀⢨⣉⣉⣉⡇
⠀⠀⠀⠈⠑⠒⠠⢤⣇⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀⠠⠤⠒⠊⠁⠀⠀⠀⠀⠀⠆⠀⠀⠀⢡⠀⠀⡘⠀⠀⠀⠀⡆⠀⠀⡎⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡜⠀⠀⠀⠀⠀⣆⣠⠇⠀⠀⠀⠀⢱⣀⡘⠀⠀⠀⠀⠀⢆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠃⠀⠀⠀⠀⠀⠘⢸⠀⠀⠀⠀⠀⠀⡆⠃⠀⠀⠀⠀⠀⢸⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠊⠉⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⠐⠒⠚⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢠⡄⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠠⠠⣤⡄⠄⠀⠀⠀⠀⢠⠄⠇
⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⢀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣀⣀⣸⣀⡀
⠀⠀⠀⠀⠀⠉⠒⠒⠢⠤⠤⠀⠤⠤⠤⠠⠀⠤⠤⠐⠒⠉⠀⠀⠀⠀⠯⠠⠀⠄⠄⠀⠀⠀⠀⠠⠀⠀⠤⠀⠄⠄⠀⠠⠄⠠⠀⠀⠀⠀⠠⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠟⠻⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠁⠀⢀⣀⣤⣤⣤⡄⠀⡿⠀⠀⢠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷⣦⡀⣤⡄⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⠁⠀⠀⣠⡶⠶⢶⣤⠀⠀⠀⠀⣠⡶⠶⢶⣄⢀⡀⠈⠁⠻⠃⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢠⡄⠀⠀⠀⠰⣿⣶⣶⣶⣿⠇⠀⠀⠸⣿⣶⣶⣶⣿⠾⡇⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠺⠇⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⢠⡆⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⠀⠘⠃⠀⠀⠀⠀⠈⠻⣦⣀⣀⣀⣠⣴⠟⠁⠀⠀⢰⡆⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣄⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣶⣶⣦⣤⣤⣴⣶⣶⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣧⡰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⡧⣛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠛⠋⠁⠀⠊⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣁⡄⠒⢶⡔⢂⡴⠢⠄⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠶⠂⠘⠺⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡆⢀⠀⢐⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⢀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⣘⠁⠀⠀⠀⠸⣁⣠⢏⠀⢃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡄⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣄⠀⣀⠄⠀⠀⡠⠂⠀⠀⢠⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠉⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠉⠉⠁⠀⠀⢀⣇⡀⠀⢀⠀⡇⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⢸⡀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠲⠶⠶⠶⢤⠦⣀⣀⡀⠀⣀⠠⠂⡀⠀⠀⢀⠀⡀⢇⠀⠀⠀⠉⠉⠉⠉⠉⢀⡜⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⣄⣁⣀⡤⠍⠭⠭⠵⠤⠤⠤⠀⢰⣿⣿⣿⣮⣷⠤⣄⣀⣀⣀⡤⠖⠫⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⠽⠿⠇⠀⢀⣠⣬⣀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠶⠶⠶⠚⠫⠋⠂⠉⠙⠛⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⡀⣀⠇

⠀⠀⠀⡶⠛⠛⠉⠉⠑⠢⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⠒⠒⠒⠒⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⣸⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡀⠁⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⣛⣷⡆
⠀⠀⢀⡏⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⡁⠐⠶⣿⣧⣤⣤⣤⣤⣤⡤⠤⠤⠄⣿⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠛⠛⠛⠛⠶⣄
⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠑⢄⠈⢷
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⡆⠘
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⡇⢠
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⢀⣼⢃⡞
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⣿⠀⣠⠀⡄⡄⡄⠀⣤⠀⣿⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣠⣴⠟⢁⠎
⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⣿⠀⠊⠒⠑⠃⠁⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠁⠀⠁
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠛⠷⣦⣤⣄⣀⣀⣠⣤⣴⠶⠛⠁
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠿⠿⠛⠁
⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠸⣧⣀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠘⢿⣷⣤⣤⣄⣀⣠⣤⣴⣾⡿⠃⠀⠀⠀⠈⠙⢷⣄⣀⢀⣀⣠⣤⣶⡶⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠂⠐⠋⠉⠀⠉⠳⣦⣄⡀
⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁
⠀⠀⢀⣀⣤⣴⣶⣛⠉⠉⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿
⠀⣴⠟⠉⠉⠉⠉⠉⠉⠒⠂⠄⢸⣧⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⣀⡤⣀⠀⠉⠙⠒⠶⣤⣀⠀⠀⣿⠀⠀⠙⠻⢶⣤⣀⣤⡴⠚⠋⠉⢹⡇
⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠙⠛⠚⠀⣀⣠⠶⢢⠀⢹⡇⠀⢸⠀⠀⠀⠀⠀⠋⠉⠙⠀⠀⠀⠀⢸⠃
⢸⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠛⢸⣿⠀⠀⠀⢀⡴⠋⠀⣀⣀⣤⣤⠶⠶⠛⠋⠉⠈⠙⠁⠀⢸⡇⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⢸⡇⠀⢰⠆⢠⠀⠀⠀⠀⡎⠀⢸⡟⠀⢀⣴⣿⠶⠿⠛⠋⠉⠁⠀⠀⠀⠀⣰⡏⣠⡇⠀⠀⣼⡇⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠈⣿⡀⠈⠁⠀⠳⠂⠀⠀⢝⠂⠸⡇⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣾⣋⣰⠆⠘⠛⠋⠀⠀⠀⣿⠇⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠞⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⢀⣠⣿⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠉⠓⠦⣄⣀⠀⠀⠀⠀⡇⠀⠀⢻⣾⣛⠂⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠾⠛⠛⠉⠁⠀⠀⠀⠀⠀⠉⠛⠶⠶⠶⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠄⠀⠀⠀⠀⠈⠈⠉⢀⣀⣤⠞⠓⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠆⠈⠛⠉⠀⠴⠶⠶⢀⣀⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣘⣡⣤⠴⠶⠒⠒⠒⠒⠶⠦⢤⣤⣐⠒⠒⢬⡙⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⣩⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣀⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣋⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⡡⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢿⠀⠀⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠛⠶⠀⠀⡆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡀⢀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠛⠉⠙⠳⣄⠀⡇⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡷⠋⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⣠⣀⠀⠀⠀⢹⡆⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠟⠀⠀⠀⠀⠀⠀⠀⢀⡜⢁⡠⠒⠉⠉⠉⠓⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⠀⠀⠀⡇⢸⡆⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⠟⠀⠀⠀⠀⣠⣴⣶⣾⣷⡄⠀⠀⠀⠀⠀⠀⢸⣿⣉⣿⣿⡿⠀⠀⢠⡇⢸⠇⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⡿⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⣿⡿⠿⠿⠛⠒⠒⠾⠃⢰⡆⠀⢸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⢿⣧⣼⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠘⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣀⡽⠟⠋⠉⠁⠁⠀⠀⣤⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣰⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠁⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⣧⢠⡟⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⠔⠊⠁⢸⠀⠀⠀⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⡆⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠠⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠀⠀⠀⠀⠀⡇⠀⢰⠇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠸⡄⠀⠀⢀⣴⣠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⡇⠀⣾⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⡆⠀⠹⡄⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⠀⠀⠀⢀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⢀⡇⠰⠃⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠘⡆⠀⢿⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠉⣀⠤⢤⣄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠙⡄⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⢻⡄⠀⠀⠀⡆⠀⢠⠎⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠸⣄⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡸⠀⢀⠏⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠹⣆⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣠⠃⢰⠏⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠈⢦⡈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠐⣿⣂⡴⠟⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠀⣼⠋⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠘⢷⡀⠀⠀⠀⠈⠛⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡄⠉⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠉⠛⠲⠤⣤⣀⣀⣀⣀⣀⣀⣠⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁




⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣃⣀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢰⣇⠀⣴⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⣿⣄⡀⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡀⠀⢠⣶⣾⣿⣿⡍⡅⠀⠀⠀⢠⡀⣼⣟⣩⣁⡀
⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠀⠀⢿⣿⣿⣿⣧⣗⠀⠺⣿⣄⠀⢹⣿⣟⣋⣠⡐⢀
⠀⠀⣴⡈⢤⠄⠀⠀⠀⠀⠈⠁⠀⠉⠀⢻⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣟⣁⣀⣿⡟⢩⢹⣿⣟⣟⠩⢉⣯⡤⠆
⠀⠶⠿⣭⡤⠶⢖⠉⠉⠉⠁⠈⠃⠋⠉⠙⡛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⠛⠋⠀⠈⠙⠛⠛⠐⠊⠐⠀⠉⠀⣀
⠀⠀⠀⡔⣤⠡⢄⠀⣤⣀⡠⣤⠀⡠⠤⢠⢄⢄⠀⠀⠀⢀⣠⡄⢠⢀⠀⠀⠀⠠⣀⢀⡀⢀⡀⡀⡀⢀⠀⠀⡀⠀⠀⠀⠁
⠀⠀⢹⡶⡌⢱⢛⠀⣿⡌⠉⣢⡽⢇⣦⡏⢀⢜⣮⣆⠀⢇⢁⠡⣉⣬⢠⣴⡧⣌⡅⣚⡤⣡⡶⠗⠠⠄⠀⠵⢹⣅⡠⡄
⠀⠀⣹⣽⣇⣷⢡⠲⣦⡆⢼⡣⡇⣻⡀⣧⠨⠞⢸⡟⠻⣤⡘⢹⣫⡍⣊⡡⢿⢳⣼⠽⢿⠾⠃⢜⡋⠦⠟⠃⢨⣏⣀
⠀⠀⢸⡇⡏⢾⢸⡟⢫⡇⢻⡟⣇⢺⣆⡟⢻⡯⢱⡏⠿⡳⢻⢾⣷⡄⣿⢠⣬⡀⢸⡷⣺⣧⡴⣦⢶⡖⡐⢺⣫⠌⠐
⠀⠀⣹⣏⡉⣺⠘⠒⢸⡇⢤⠮⣯⣼⠇⣷⢸⣋⣸⠎⠑⡟⡴⠛⢻⠂⠿⠠⠾⠉⣸⣏⠟⣻⣐⡗⣻⡿⠆⣶⠡⡆
⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠚⠷⠤⠤⣭⣤⣄⣹⡅⠉⠈⣿⠀⠉⠃⣛⠄⠣⠀⢸⡇⢀⣹⣎⡟⣹⠛⠁⡿⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠛⠒⠒⠦⠿⣀⣀⡀⢸⡇⠀⠀⢸⡏⠈⠐⠋⡟⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠓⠒⠓⠠⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⡾⠿⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡶⠟⠁⢄⡀⠀⠀⠈⣻⣦⡀
⠀⠀⠀⠀⠀⢀⣴⡏⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⢀⡾⠛⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠐⠤⣀⡀
⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣦⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣥⣼⡷⠒⠛⣲⣭⣵⣶⣤⣄⣀
⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⣀⣤⣤⣾⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⢀⣹⣿⠶⠚⠉⠀⠀⠀⢹⣦⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠉⠛⠳⠷⠟⠛⠉⠉⠉⠉⠀⣠⡤⠖⠛⠷⢶⣶⠶⠤⣤⣄⣀⠀⠀⠀⠀⢸⡌⡇⠀⠀⠀⠀⢨⡿⠁⠀⠀⠀⠀⠀⠀⡼⡿⢿⡿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⢀⠘⠀⠀⠀⠀⢠⣬⣷⣆⠀⠀⣸⢇⢣⣤⠀⠀⠀⣼⢹⣾⣥⣀⠀⠀⢀⣀⣠⢯⡞⠁⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢡⣿⡆⠀⢿⠀⠀⢹⣟⣠⣼⣿⡾⠋⠉⠉⠙⠻⣿⣷⣿⠏⣰⡄⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠻⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⣿⠇⠀⢨⡆⠀⠀⣧⣄⣉⣟⠀⡄⠀⡀⣀⣤⣾⡿⢋⣼⠟⣠⡆
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡛⠙⠿⢛⣶⣶⣤⡈⠉⠻⣿⣇⡀⣀⣀⣶⣀⣰⣾⠿⠁⠀⢀⡏⠳⡀⠰⣌⡛⠛⠛⠛⠛⠋⠉⠉⠩⠤⠐⠉⠁⣰⡿
⠀⠀⠀⠀⠀⣠⣴⠿⣿⡀⠀⠀⠀⢠⠉⠙⢻⡄⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢸⠀⠀⠹⣄⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠞⣿⡇
⠀⠀⠀⢀⡾⠋⠀⠀⠉⠁⠀⠀⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠈⠛⠶⠶⢶⣶⣶⣤⣶⣾⣛⣉⣤⣤⣤⣿⠃
⠀⠀⠀⡾⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠃⠀⠀⠀⠀⢤⣄⣀⣀⣀⣠⣤⠶⠞⠛⠛⠛⠛⢿⣶⣤⣄
⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⣀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⡀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⣀⣬⡭⠴⠋
⠀⠀⠀⠙⢷⣄⣀⣀⣰⣾⠿⢤⠥⠾⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣄⣀⣀⣀⣤⣤⣤⣤⣤⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠥⣀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠱⡀⡠⠂⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⢀⠠⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠙⢀⣿⠈⠋⠀⠀⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠂⠐⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠢⣤⣤⠤⠊⠀⠣⢤⣤⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡇
⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⠉⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣬⡇⠀⠀⠀⠀⠀⢀⡤⢤⡀
⠀⢰⠋⠉⢺⡀⠀⢣⡀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢀⡞⠀⢈⠗⠉⠙⡆
⡴⠚⠓⠤⠄⡗⠀⠀⠹⣄⣀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⣀⣠⠏⠀⠀⢚⠠⠤⠚⠓
⠳⣤⣀⢠⠅⠘⠀⠀⠀⠀⠀⠉⠙⠓⠲⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠖⠚⠋⠉⠀⠀⠀⠀⠀⠃⠠⡄⣀⣤
⠸⣁⢀⣀⣀⣀⣠⠜⠉⠉⠛⠛⠒⠒⠶⠤⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠤⠀⠒⠒⠛⠛⠉⠉⠣⣄⣀⣀⣀⣀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢈⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⣀⡀⣀⣀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⠈⡏⢹⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠚⠋⠉⠉⠉⠙⠓⠦⣿⠀⡇⢸⠀⣿⠴⠚⠋⠉⠉⠉⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡧⢾⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⠄
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⢀⡤⠴⠖⠒⠒⠛⠛⠒⠒⠒⠿⠷⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⢠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠖⠒⠚⠛⠷⢦⣤⡀
⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⡏
⠀⣇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠛⠛⠛⢿⠿⣄⡀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠙⣦⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⢀⣀⣄⢉⠓⠒⠲⠶⠖⠚⠉
⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢠⠞⠉⠀⠀⠀⠉⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠉⠙⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⠤⠞⠓⠒⠂⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⣀⠈⢧⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣄⣀⣀⣀⣠⣾⡟⢮⣳⣌⣳⣤⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠙⠻⢿⣿⣷⣤⣀⠸⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢮⣅⣀⠀⠀⠀⢀⡤⣖⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⢀⠖⠀⢰⠀⡇⠁⠀⠀⢸⢰⠁⡉⠉⠂⠐⠉⡉⠑⢦⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢀⠖⠂⢸⢠⡆⠀⠆⠁⠹⠀⡇⠀⢸⠀⡁⠀⠀⠸⠈⠓⢁⡼⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠐⠒⠒⠒⢒⠲⠄⠀⠘⠀⡇⢸⠀⠀⡆⡨⢥⡀⣰⠀⡇⢀⡇⠀⢸⣀⣁⣀⡆⢀⢀⡧⠈⢆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢀⠤⠖⢴⢀⢠⠀⠀⠠⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡇⢰⠱⠀⢴⢸⢸⠠⠀⠀⢀⡠⠄⠒⠀⠈⢹⠀⠀⣀⡀⠀⠀⠂⠀⠀⠀⠐⠒⢠⠀⠔⠈⠉⠑⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⡇⡇⠐⠀⠀⣀⠘⠀⠀⠈⠀⠀⠀⠀⣀⠀⢠⢀⠂⠀⠈⢢⠀⡇⠀⠀⠀⢀⡀⡼⢨⠀⠀⣦⠀⢸⡀⡇
⠀⠀⠀⠀⠀⢠⣉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⠠⢲⠀⠀⠂⠀⠁⠈⠀⢲⠀⠈⠀⠁⠀⠃⠀⡈⠀⠀⢄⢄⠀⠸⠓⠈⠠⡇
⠀⠀⠀⠀⠀⢸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⡇⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⢠⠀⠀⠀⡰⠀⠀⠘⠀⠀⡇⠘⣄⠈⠋⠀⡸⠀⠀⢠
⠀⠀⠀⠀⠀⢸⡧⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠸⠐⠂⠉⠀⣀⠀⢁⣒⠈⠀⠀⠀⡃⠀⠀⠁⠀⠈⠑⠒⠊⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡞⢠⠊⣠⡉⠳⣀⠀⢀⣀⡰⠊⣉⡉⠓⢇⢸⣀⠔⣊⣉⠲⣀⣀⠀⠈⠀⠈⠉⠀⠈⠁⠈⠂⠀⢀⡉⠁⣀⠔⠚⡛⠲⣄⠀⢸
⠀⠀⠀⠀⠀⠉⢻⠸⣅⠜⢠⠏⠉⠉⠉⡇⢸⡁⢸⠆⡸⠈⢇⢸⡀⣨⠇⠸⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡏⠠⡎⠉⡆⢸⠛⠛
⠀⠀⠀⠀⠀⠀⠈⠳⠤⠴⠋⠀⠀⠀⠀⠑⠤⣬⣥⠔⠁⠀⠈⠲⠬⠥⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠉⠋⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡟⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢻⡝⢷⡟⢷⡄
⠀⠀⠀⠀⠀⡠⠾⢿⡋⠉⣿⠛⣷⠦⣤⣀⣀⣀⣀⣠⣤⡶⢻⡟⢓⣺⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣍⡛⢿⠘⠏⢸⣧
⠀⠀⠀⠀⣸⠁⠀⠀⠉⠲⡗⠻⢄⠑⣤⡿⢿⢯⣥⠞⣡⠟⢦⡽⠛⠁⠀⠙⣆⠀⠀⠀⠀⠀⠀⣀⣤⠶⢶⡄⠀⠀⢻⣶⠶⠖⠀⢻⡄
⠀⠀⠀⠀⣿⣄⡀⠀⠀⢀⣐⣤⡴⢟⣫⠴⠛⠦⣝⠿⣥⣤⡋⠀⠀⠀⠀⣠⣿⠀⠀⠀⣠⡴⠛⣉⣤⡴⠾⠃⢰⠟⣩⣥⣄⣀⣤⢾⡁
⠀⠀⠀⠀⠘⠯⣭⣛⣛⣩⣭⠴⠖⠋⠁⠀⠀⠀⠈⠛⠶⢭⣝⣛⣛⣛⣛⡽⠋⠀⠀⣼⠋⣴⠞⠋⠀⠀⠀⠀⠈⠛⠁⠀⠈⠻⣶⠾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⢸⡇⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠙⣆
⠀⠀⠀⠀⠀⢀⡤⠖⠛⢛⣉⣉⡉⠛⠲⠶⠒⢛⣛⣛⠒⠶⢤⡀⠀⠀⠀⠀⠀⢠⣟⣷⣟⣷⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⣿
⠀⠀⠀⠀⠀⠘⣧⠀⡴⠋⠁⠀⠙⢦⠀⢀⡾⠋⠀⠉⠳⣄⢀⡟⠀⠀⠀⣠⡴⢟⡉⠉⢹⡷⠶⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣰⠟
⠀⠀⠀⠀⠀⠀⣼⢸⡇⠀⠀⠀⠀⣸⠀⠸⡇⠀⠀⠀⠀⢹⠘⡇⠀⠀⠀⠻⠶⣿⠁⣾⡀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠹⡄⠳⣄⣀⣀⡴⢃⡜⢧⡹⣄⣀⣀⣠⠟⣰⠁⠀⠀⠀⠀⠀⠛⠞⠃⠛⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠖⠋⠀⠀⠑⠦⠭⡭⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢺⠯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠶⠋⣠⡶⢻⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠒⢦⡞⣩⣤⡀⠉⢳⡄⠀⠀⠠⣞⣿⡯⢤⡤⠶⠖⢻⣥⡴⠞⠁⣼⠃
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢾⣿⣿⣎⣿⡿⠟⢀⣼⠇⠀⠀⠀⠈⠉⠓⣮⡓⠖⠚⠋⠀⠀⣠⡼⠋
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠈⠙⠲⢦⡀⠀⠀⠀⠀⠀⠈⠳⣤⣄⣩⣵⠟⠙⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⠚⠛⠉
⠀⠀⠀⠀⢠⢾⡇⠀⠀⠀⠀⠀⠀⣤⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠹⣄⢳⣄⡀⠀⠀⣀⣴⠟⠛⠁⠀⠀⢠⠞⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠉⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⣿⣿⣄⣀⡀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣄⣄⣀⣀⣴⣿⣿⣿⣿⣿⡷⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠲⣤⣄⠀⢰⣿⣿⣿⡿⠿⠿⠟⢛⡣⢴⢟⡼⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⣾⢁⡴⠲⢿⢸⣇⠀⠀⠀⠈⠳⣌⣙⣛⣓⣀⡵⠋⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠠⡜⣿⠟⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠈⢳⣀⠀⠀⢻⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠖⠛⠋⠀⠀⠀⠀⢀⡴⠒⠲⢦⡀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠠⡟⢧⡶⠓⠆⠙⣆⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠈⠙⢦⡀⠀⢀⣽⣎⢀⠀⠀⠀⢹⡄⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⢀⣠⠟⠀⠀⠈⠛⢻⠸⡆⠀⢀⣼⠁⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠛⠿⠖⠛⠁⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠦⢤⡤⠤⠶⠶⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠄⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣲⠀⠀⠄⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣒⠂⠀⠒⠒
⠀⠀⠸⣍⡴⠂⠀⠀⢀⠐⢚⠫⡍⢉⣤⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡠⣉⢁⡤⠐⠂⠉⡇⠀⠀⠀⠀⠀⠀⣠⣌⠿⣿⣿⣿⡄⠉⢀
⠀⠀⠀⣸⠂⡧⠀⢀⠟⠦⠄⠀⠛⢂⣉⠄⠀⠁⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⡍⢀⣤⣴⡴⠃⡇⠀⠀⠀⠀⠀⠀⠀⠹⢿⡄⠠⡴⠒⠾⠉
⠀⠀⠘⠯⠴⠇⠀⠠⠄⠰⠖⠚⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⣅⡌⠛⢃⣩⣴⠖⠃⠀⠀⠀⠀⠀⠀⠀⡄⠈⠥⠀⠀⠤⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⢐⣊⣁⣋⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⢶⠖⢦⢴⠈⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠴⠖⠒⡊⠉⠉⠉⠓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡒⠶⠶⢒⣂⡴⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣉⠡⢤⡠⠀⠐⢒⣈⠭⠥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⠤⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⣠⡀⠀⠀⠤⠉⠭⠥⠴⠖⠚⠉⠁⠀⠀⠀⣀⠠⣠⡔⢒⠏⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⢤⠤⠤⠄⠀⠀⡀
⠀⠀⢈⡿⣆⡀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠚⣿⠋⠀⡰⢿⣠⠎⠀⠀⠀⠀⠀⢸⠀⠀⠢⢽⡿⠟⢁⣶⠆⠀⡀⠀⠀⡟⠛⢛⠛⣓⣒⣒⡒⢒⠀⠁
⠀⠀⠀⢇⣹⡙⠓⠚⢋⣿⠉⠀⡸⢯⠠⠎⠛⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠰⠋⣴⣸⠿⠋⠀⠀⡇⠀⠀⡿⠀⠸⠛⠙⠛⠋⠃⢀⠀⠸⡄
⠀⠀⠀⠀⠈⠳⠦⠴⠋⠙⠛⠋⠁⢀⣀⣠⡤⠤⠶⠶⠒⠒⠲⠤⣀⠀⠀⠀⢸⠇⠰⣤⡼⢓⣯⣚⣅⠀⢰⡇⠀⢰⠁⠀⣴⡏⠀⠒⠂⠒⠉⢧⡄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣭⣄⡀⠀⠀⢀⣀⣀⣠⣤⡀⠀⠀⠀⠀⣿⡆⠘⢁⣾⣽⠿⠋⠉⠀⢸⡇⠀⣸⠀⢠⠀⠁⠀⠀⠀⠀⠀⠠⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠳⠯⠶⠾⠿⡿⠛⠛⠋⠡⠀⠀⠀⠀⣿⡅⢀⣉⣁⣀⣀⣀⣀⡀⢸⡇⠀⣿⠀⣇⠀⠠⣤⣤⣤⣤⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠧⠀⡄⢖⠉⢈⣓⣂⠀⠐⠀⠀⠀⠀⠈⠀⠚⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⢻⡆⢠⠄⢀⠀⡍⢉⠩⢠⠀⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢱⠈⠂⠀⡈⢈⡀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⡸⠇⠸⠀⡇⠈⡶⠸⠎⠀⠇
⠀⠀⠀⢠⠖⣒⣂⠀⢰⣲⡄⠀⠀⠀⠱⠄⠀⠀⠆⠀⠐⣉⡙⠂⠁⠀⠀⠀⠀⠀⠀⢠⣩⣭⣭⡅⠀⠀⠀⠀⠀⣖⣿⣀⣀⣀⡀
⠀⠀⠀⢈⠉⠉⠉⠉⠉⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠿⢭⠍⠗⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⢠⡎⠤⠀⠀⠀⠄⣼⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⣾⢁⠀⣤⣤⣄⠀⣏⠄⡇⠀⠀⢰⣖⣋⡉⠉⠉⠉⠉⠉⠉⠑⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢠⡄⢻⠀⠀⢰⣿⡿⠿⠿⠿⠿⠯⠍⠽⡗
⠀⠀⡇⢀⠈⠭⠭⣭⣉⠉⠀⡏⠀⠀⢠⣯⡿⠛⢛⣛⣛⠉⠙⣛⠈⠀⠀⠀⠀⣿⠀⠀⠠⠴⠦⠤⡀⢻⢸⠘⡇⠀⢰⣌⣙⣛⠛⠛⠒⠚⠛⠃⢡
⠀⠀⣿⡅⠀⠄⠀⠀⢹⣷⠀⡇⠀⠀⢸⡟⠃⠰⠿⠧⡼⠇⠀⠀⢠⡇⠀⠀⠀⠉⠀⠐⠀⠘⠁⠀⠐⢸⡾⣰⠁⠀⣿⡏⢉⣡⣤⣤⣀⣁⣀⠀⠈⡇
⠀⠀⢻⠘⢷⡶⠶⣒⠋⣿⠀⡇⠀⠀⢸⡇⢉⠙⢛⠛⢛⠛⠉⠁⢸⡇⠀⠀⠀⢇⢠⠀⡧⠀⠶⣀⢀⠀⠻⠃⠀⠀⣿⣰⠋⠤⠤⠄⠠⠤⠄⠙⢦⡇
⠀⠀⢸⠀⢲⡿⠻⣀⡄⣿⠀⡀⠀⠀⠸⡇⠨⠴⠒⠆⠘⠀⠠⢀⢼⡇⠀⠀⠀⢸⣞⠋⠁⠀⠈⠁⢸⠀⢠⠀⠀⠀⣿⡧⠀⠂⠂⠀⠀⠀⢂⠠⢸⡇
⠀⠀⢸⣄⣁⣀⡐⣛⣁⣿⡶⠇⠀⠀⠈⡧⠙⢓⣒⡶⠶⠶⠖⢚⢹⡇⠀⠀⠀⢸⠙⠀⣁⠀⡀⢀⡞⢀⣿⠀⠀⠀⣿⠘⢦⣐⡲⠦⠰⠚⣉⠴⢚
⠀⠀⠀⢧⡈⠉⠉⠩⠉⠅⠠⠁⠀⠀⠐⣇⣀⠙⠻⢯⣭⡽⠂⠀⠸⠃⠀⠀⠀⠀⢷⣆⣀⣋⣉⡁⢀⣠⠏⠀⠀⠀⠈⢧⣄⣈⣉⣉⣉⣉⣠⢄⣻⠆
⠀⠀⠀⠀⠉⠙⠛⠓⠂⠉⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠃⠛⠉⠁⠀⠁⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠀⠀⠀⠀⠀⠀⢀⣠⠦⠤⠬⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠈⠄⠀⠀⣀⠴⠚⠉⠀⠀⠀⠀⣀⠝⠲⢤⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡶⠊⠁⠀⠀⢀⡤⠒⠒⠴⣡⣤⣄⠀⠘⣄⠈⠳⡀
⠀⠀⠀⠀⠀⠀⢀⢾⡀⢀⡴⠋⠀⠀⠀⠀⠀⡏⣴⣦⡄⠀⠸⣿⣟⡠⠤⠧⢵⣄⣱
⠀⠀⠀⢀⡠⠖⠁⢀⣧⣾⣄⡀⠀⠀⠀⠀⠀⢯⠿⠿⠏⢀⠔⠋⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠘⢄⣀⣰⠋⠄⠀⣀⣙⡆⠀⠀⠀⠀⠀⠉⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣄⠀⠀⠀⡔⠒⠢⡄
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⠘⢦⠀⠀⠙⣤⠔⠋⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⣀⣀⡠⠴⠊⠀⡿⢵⠈⣧⠀⠀⣧⠖⠉⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⣀⣑⣤⣥⣤⠜⠁⠀⠀⠀⠀⠀⠠⡞⠓⢦⢌⣉⣉⠀⣀⣀⡀⠚⠣⠈⡀⠘⡆⠀⢏⡖⠉⠀⠉
⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠉⠁⠒⠒⠒⢦⡀⠀⠀⠀⠘⣤⠁⠀⠀⠉⠀⠀⠀⠂⠀⠀⣘⠃⠀⢻⡀⠀⢻⠢⠼⠁⡰
⠀⠀⠀⠀⠀⠀⡇⠀⠀⡾⠲⢄⡀⠀⠀⢀⠇⠀⠀⠀⠀⠘⢷⠀⠔⢦⣀⣀⡠⠴⠒⠋⡘⠀⠀⢸⠛⠶⠿⣅⣐⡨⠁
⠀⠀⠀⠀⠀⠀⣧⠀⢀⡇⠀⠀⠉⠓⠒⠉⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⣀⡠⠞⠀⠀⠀⢸⡃⠀⣀⡼⠋
⢀⣰⠂⠾⡈⢻⠁⠙⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⣾⠉⠋⠁
⢩⡈⠓⢄⡩⠚⠓⠠⠄⣮⡀⣀⡀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⢲⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠉⠦⡀⠀⠀⠀⣀⠼⣁⣀⡴⠃⠀⠀⠀⠀⢠⠞⠁⣠⠔⢈⡠⠔⠂⠘⠳⡄⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠉⠓⠒⠋⠁⠀⠙⢧⡀⠀⠀⠀⣠⢾⠇⡠⠊⠀⠔⠁⠀⠀⠀⠀⠀⡿⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡴⡧⣾⠚⢀⠔⠀⠀⠀⠀⠀⠀⠀⣸⣧⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠡⠃⡔⠁⠀⠀⠀⠀⠀⠀⠀⣰⢿⣉⣶⡦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠃⠸⠦⠤⠤⢤⣀⣀⣀⢀⡴⢛⠦⠭⡴⠁⠀⠈⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄⠠⠀⠀⠀⠀⠀⠀⠀⣩⡏⠔⠁⡠⠎⠀⠀⠀⠀⣀⠟⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⢀⣤⠚⠋⠹⠔⠻⠓⠖⠂⠝⠉⠓⢉⣁⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠶⠚⠉⠉⠙⠛⠒⠲⠶⠶⠶⠖⠒⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣿⡄⠀⠀⢀⣼⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⣹⡿⠟⠿⢦⣾⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣶⣶⣿⠟⠀⠁⠀⢉⡟⠉⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡛⠂
⠀⠀⡆⢸⠀⡆⡀⠀⠀⠚⠛⠛⠛⠛⠉⠙⢻⣿⣿⠋⠀⠀⠀⠀⣞⣀⣀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠒⠂⠤⣄⣠⣼⣿⣿⣧⠀⠀⠒⠒⠶⠛⠀⣿⠀⢸⢿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣹⣿⣿⣿⡀⢰⡄⡦⠀⠀⠀⠉⢠⢜⣿⣿⣿⣯⣼⣿⣿⣿⠟
⠀⣀⣤⡤⠤⠤⢐⣒⣄⡨⣤⠤⠖⢒⣣⣬⣼⡭⢽⠛⠛⣿⣶⡶⠶⣶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣧
⢉⢩⣀⣤⢴⠞⠍⠉⢠⣦⣴⡶⠟⠉⣉⣡⣴⣶⡾⠟⢋⣭⣰⣤⣄⣼⣧⣯⣹⡟⣟⠛⠛⠉⠉⠉⠁
⠛⠛⠉⣀⣤⣤⠼⠚⠛⠥⣤⣴⣶⣿⠛⠙⡋⣠⣦⣼⣿⣿⠏⠛⠋⠉⠉⠀⠀⠀⢳⡝⡄
⣶⡞⡿⠿⠷⠶⠶⠶⢿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣾⣷⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠓⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠔⠒⠒⠒⢶⡇⢠⠖⠉⠁⢠⠋⠉⠑⢖⠲⡄
⠀⠀⠀⠀⠀⠀⣠⠖⢉⡽⠛⠛⢦⡤⠒⠒⠊⠓⠚⢷⡤⠒⠛⠲⣄⡠⠞⠒⠛⠲⣔⠒⠦⣄
⠀⠀⠀⠀⠀⢰⠋⠀⠘⢄⠀⠀⠀⢱⣀⡠⠖⠊⠑⡞⢡⠖⠉⠉⠹⣇⣀⡀⣀⠀⠀⣳⠀⠙⢣
⠀⠀⠀⠀⠀⠈⢦⢄⣀⠀⠉⠒⠴⠤⡿⣁⣀⠀⠀⢸⡇⠀⠀⡔⢉⣀⣀⡀⠤⠓⠉⠁⡀⢀⠞
⠀⠀⠀⠀⠀⠀⢸⠀⠉⠙⣲⢤⣀⡀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⠁⠀⢀⣀⡤⠴⠒⠛⠛⢹
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠁⠀⠉⠉⠒⠶⠶⠶⠦⠶⠒⠒⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⢻
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⣄⣀⣀⣸⠆
⠀⠀⠀⠀⠉⠒⠢⠤⠤⠔⠲⠆⠠⠖⠒⠲⠤⢄⣀⣀⣀⣀⡠⠴⠞⠒⢛⣉⠭⠤⢴⣉⠉⠉⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠙⡆⢀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣀⣀⣀⡤⠞⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠿⠄⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠉⢢⡞⠁⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢛⡏⠀⠀⡄⠀⠀⠀⡇⠐⠀⠀⡸⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⢷⡀⠀⠀⠀⠀⣠⠗⠦⠤⠞⠁⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⢠⡖⠀⠀⠀⠈⠛⠦⣤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠁⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠉⢳⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣄⠀⢸⣇
⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣴⡶⠚⠛⠉⠀⠀⠀⣿⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⡠⡀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢠⣿⠀⠀⠀⠀⢤⡔⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠹⠋⠉⠉⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣄⡀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⡠⢄⠀⠀⠀⣿
⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠄⡠⠃⠀⠀⠸⣧
⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⢰⡏⠀⠀⠀⠠⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢢⠀⠀⠀⠀⢿
⢸⡇⠀⠀⠀⡃⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘
⢸⣷⠀⠀⠀⠣⣀⣀⠎
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⢢⠀⠀⠀⠀⠀⠀⠀⣔⡢⠀⠀⠀⢀⣿⠁
⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠣⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠈⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛
⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⠾⠛⠋⠁
⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣁⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠷⠴⠛⠁⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠁⢀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠠
⠀⠀⠀⢤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠛⠛⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⡀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣀⡀⠠⠴⠀⠒⡁⠀⠀⠤⠤⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠐⠒
⠀⡀⠀⠀⠀⠈⠤⠠⠴⠒⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⡀⠉⠁⠀⠀⠀⣤
⠀⢁⠀⠀⠀⠀⠀⢀⡀⠄⠠⠀⠀⢉⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠔⠈⢀⡂⠀⠀⠀⠀⡅⠀⢀⡀⢀⡤⢠⠠⠀⠁
⠀⠈⠀⠬⠀⠈⠀⠃⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠘⡀⠀⠁⠀⠀⠀⠇⠀⢀⣤⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠐⠛⠁⠀⠀⠄⠀⣙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠂⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠂⠀⠀⠀⢛⠉⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠒⢀⡀⠀⣁⠀⣴
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢨⢀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁
⠀⠀⠁⠈⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠄⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠲⠠⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠘⠆⠘⠃⠂⢂⠄⠀⠀⠀⠀⠀⠤⠄⢙⣐⣒⣉⠠⠤⡆⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁⠀⠀⡀⠀⠀⠀⠀⣀⠤⠀⠀⠠⠤⢀
⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠸⡀⢆⢀⣄⠲⠀⢂⠀⡇⠀⠀⠀⠀⠰⢠⠂⠈⠲⠂⠀⠀⠀⠀⠀⢀⠈⠀⣀⡀⠁⡁⣀⣀⢱
⠀⠀⠀⠀⠛⠄⠀⠀⠀⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⢀⠜
⠀⠀⠀⠀⠁⠀⠈⠠⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠒⠒⠒⠒⠒⠁⣀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠆
⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠄⠤⢤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣤⠎⠩⢞⠀⠅⠉⠒⠠⢬⣃⡉⠉⣹⠋⠐⠤⡀
⠀⠀⠀⠀⣚⠿⠦⠁⠡⠀⠋⠓⢒⠒⠃⠐⠌⠉⡁⠀⣤⡀⠈
⠀⠀⠀⢼⡛⣠⢡⠰⡀⡔⣲⠀⠀⢀⠬⣤⠀⠀⠀⠀⠊⠀⠀⠐
⠀⠀⠀⠈⠁⠁⠉⠷⠃⠂⠠⢸⣆⣁⢠⠰⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡠⠄⣠⠖⢀⡴⠋⣠⠞⠁⣠⠞⠁⣠⠞⠀⠀⡠⠂
⠀⠀⡀⠂⠀⠀⠀⠀⠀⢤⠀⠀⠁⠀⠀⠉⠛⠁⠀⠀⠀⠀⢀⣠⣀⠀⠀⠉⠘⠁⠚⠀⠞⠀⠴⠋⢀⡜⠁⢀⡔⠁⢀⠔⠁⠀⢀⠘⠁⠀⢀
⢀⠀⠀⠠⣀⣠⡄⣤⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⢦⠛⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠜⠀⠀⢠⠂⠀⠀⣠⠋
⠈⠐⠱⣶⠟⠁⠀⠹⠛⣶⠖⠲⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣰⠋
⠀⠀⠠⠿⣧⣄⣀⣾⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⢣⡿⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠂⠀⠉⠉⠉⠛⠃⠦⠤⢯⣄⣀⡀⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦⢀⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠛⠃
⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢀⣤⣶⣿⣠⣴⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠛⠒⠲⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢚⣹⠖⢦⡀
⠀⠀⠀⠀⠀⣀⡞⠁⠀⠀⠀⠀⠀⢀⣠⠟⠉⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠋⣏⢨⡏⠙⡏⡳⢤
⠀⠀⠀⠀⣎⠉⠓⠒⠶⠶⠒⠒⠊⣉⡠⠴⢺⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢀⣀⣸⣅⣉⣏⣉⡇⠀⡇
⠀⠀⠀⢀⣈⣹⠒⠒⠒⠒⠒⠉⠉⠀⣀⣀⣀⣻⣤⡉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣇⣠⠚⠉⠼⣛⣉⠁⠘⣓⣾⣧⠃
⠀⠀⡔⠁⡀⢸⠀⠀⢀⣴⣶⣶⡶⠘⡻⠛⠻⡋⢳⣱⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡴⢡⡠⢸⠇⠀⠀⢸⢉⣤⡑⡜⣡⣌⢻⣎⢝⣆
⠀⠀⡇⢸⠁⡞⠀⠀⠈⢡⠃⠀⣈⣦⡧⣴⣞⣹⠀⢻⣡⠇⠀⠀⠀⠀⠀⠀⠀⠀⢇⣾⣧⡼⠀⠀⠀⠈⠦⡉⣴⠻⣌⣉⠎⢹⣕⣿⠄
⠀⠀⢳⡈⠣⡇⠀⠀⠀⠘⣟⣻⠧⣾⠿⣉⢁⡼⠀⠈⣧⠤⢤⡀⠀⠀⠀⠀⠀⠀⠈⠛⣻⠃⠀⠀⠀⠀⠀⢠⠇⠀⠘⠆⠀⠀⢿⠟
⠀⠀⠀⠈⡹⢇⠀⠀⠀⠀⠈⠒⢲⡇⠀⠘⢏⠀⠀⠀⢸⣴⠖⢸⠀⠀⠀⠀⢀⣠⠔⠊⡇⠀⠀⠀⠀⢠⢄⠬⢲⣤⣖⣂⣀⡀⠸⣆
⠀⠀⡠⠊⠀⡸⠀⠀⠀⠀⠀⢀⡏⠁⠀⠀⠀⣷⣶⣦⣈⡏⡠⠋⠀⠀⠀⢰⢋⡤⠒⢻⠃⠀⠀⠀⠀⠘⣆⠑⠦⠤⠤⠔⡰⠁⠀⣯⡑⣄
⣰⠊⠰⠺⠿⢇⠀⠀⠀⠀⢀⡴⡷⣶⣤⢤⠾⣟⠒⡤⢿⣉⠳⣄⠀⠀⠀⠘⢆⠓⠤⣸⡄⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠚⠁⠀⢠⣅⠗⣸
⠈⠉⢹⣿⡗⣼⠀⠀⠀⠀⠘⢎⣰⢁⣇⢈⡆⣸⠞⣱⣋⣿⡵⠞⠃⠀⠀⠀⠀⡭⢴⣦⣱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣶⠈
⠀⠀⠘⣿⡖⠿⡳⢄⡀⠀⠀⠀⠈⠉⠉⢉⣩⠴⢊⡴⣫⠔⢛⢦⡀⠀⢠⠤⠤⠟⢁⡝⠁⢈⡷⢄⣀⠀⠀⠀⠀⣀⣠⠤⠾⠭⢧⣘⢛⠆
⠀⢀⡞⠢⣍⠓⢌⠢⡈⠉⠓⢲⠒⠚⢻⠉⠀⠀⡎⣼⣁⠔⠁⠀⢱⠀⠀⠁⠒⠊⠁⠀⢀⠏⠐⠠⠤⠭⠙⡏⢹⠥⣄⡒⠒⠒⠢⠬⠗⣄
⠀⢸⠀⠀⠈⠳⡎⢳⢳⠀⣀⡸⠀⠀⠘⠦⢤⣼⣸⣀⣏⣀⣠⠤⠜⠀⠀⠀⠀⠀⠀⠀⠸⣶⡤⣤⣤⣴⡾⠇⠘⠶⠛⠛⠛⠷⠶⠶⠶⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠿⠿⠿⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡶⢿⣿⠟⠉⠀⠀⠀⠀⠒⠈⣉⣉⣻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⠛⠉⢉⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣦
⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢿⣧
⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠀⠒⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⢰⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠒⠤⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⣾⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⣿⡇
⠀⢠⡟⠀⠀⠀⠀⠀⢠⠒⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⣀⠀⠀⣸⣷
⠀⣸⠃⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠈⠐⠒⠀⠁⠀⠀⢀⣈⣿⡆
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⢹⣿
⠀⡿⠢⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠉⠙⢿⣇
⢠⡏⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢹⣷
⣿⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠈⣿
⣿⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠛⢿
⢻⡄⠀⡠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣾
⢸⣧⢰⠁⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⣀⣀⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⡤⠊⠀⠀⠀⠀⠀⠀⠀⣼⢻
⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⢸
⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣿
⠀⠀⠘⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠁⣀⡴⢋⣿
⠀⠀⠀⠈⢿⣿⡳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠀⣼⡿
⠀⠀⠀⠀⠀⠙⢿⣦⡉⠙⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠁⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣶⣄⣀⠒⠒⠒⠤⡀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣶⣶⣾⣦⣤⣤⣧⣤⣬⣷⣴⣶⣶⡾⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⣿⣿⣿⡿⠛⠛⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣶⣿⣿⣿⡅⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢰⠋⠙⣆⢳⡀⡼⢠⣞⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⢸⠀⠀⣠⠸⢤⡤⠏⠈⣷⠃⠸⣭⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠋⠉⠙⠋⠉⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⡔⠲⠀⠀⢀⡆⠀⣸⠏⠀⠀⠀⠀⢀⣼⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⣠⠞⣇⣰⢇⣀⠞⠀⠀⡇⠀⣠⠴⣶⡀⣰⠏⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣩⡟⠁⠀⠈⠁⠀⠀⠀⠀⠀⠳⠚⠹⣄⠜⠀⣟⠀⣠⠞⣅⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠵⠁⠀⡆⠀⡋⠀⡖⢀⣠⠄⢀⡤⣤⡀⣀⣀⣌⠊⠙⠥⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⡟⢻⣄⢠⡿⠤⠼⠃⠻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠠⠇⠀⠇⠀⠙⠂⠳⠤⠖⠀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢶⡟⠲⠀⣋⣙⡆⠰⣾⠓⠆⣼⣉⣙⡇⢻⡖⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⡀⠀⠘⣧⣀⢸⡏⢉⡇⠀⢻⣄⡀⢯⣉⣠⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡟⠋⠀⢀⣠⣄⡀⠀⠉⢉⡁⠀⠀⢀⣀⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣾⠗⠶⢰⡟⠁⠈⢹⡦⠶⣾⠟⠶⠀⣿⡁⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠸⣇⠀⠀⣸⠇⠀⢿⡀⠀⢀⡈⠻⣆⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠙⠛⠛⠉⠀⠀⠈⠛⠛⠘⠷⠾⠋⠀⣼⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣄⠀⣠⡀⠀⠀⠀⠀⢠⣄⠀⣤⠀⠀⠀⠀⠀⣠⡀⢀⣄⠀⠀⠀⠀⠀⣤⠀⢠⡄⠀⠀⠀⠀⠀⣤⠀⣠⡈
⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠨⣤⣤⡅⠀⠀⠀⠀⠀⠈⢠⡌⠁⠀⠀⠀⠀⡀⣭⣤⣬⠄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⡀⠀⢀⠐⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠁
⠀⠀⠀⠀⠛⠀⠙⠁⠀⠀⠀⠀⠈⢋⣤⠛⠀⠀⠀⠀⠀⢙⣡⣬⡋⠀⠀⠀⠀⠀⠙⠁⠘⠃⠀⠀⠀⠀⠀⣻⣤⣝⠁
⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠑⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠷⠀⠺⠆⠀⠀⠀⠀⠰⠆⠀⠶⠀⠀⠀⠀⠀⣺⡂⣐⣗⠀⠀⠀⠀⠀⠿⠄⠰⠇⠀⠀⠀⠀⠀⠾⠀⠾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢄⡀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣦⠀⣴⡀⠀⠀⠀⠀⢠⣄⠀⣤⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⣤⡄⢠⡄⠀⠀⠀⠀⠀⣦⠀⣰⡀
⠀⠀⠀⠀⠀⣤⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⡦
⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣭⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣯⣭⣿⣭⣭⣯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⢀⠀⢠⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⡈⡀⠒⠐⠈⠂⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠤⢀⠉⠀⠀⠄⠊⠁⠃⠢⠀⠀⢦⣀⡠⠀⠴⠈⠀⠀⠀⠀⢌⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠚⠀⠀⡄⠔⠀⠐⠀⠀⠠⠊⣉⠠⠄⠂⠉⠃⠄⠀⠈⠀⢀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⠊⠀⠀⠒⠀⡠⠀⠃⢀⠠⠀⠒⠠⡘⠀⠈⠀⠀⠢⠀⢀⠀⠀⠀⡈⠁⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠄⠀⠁⢀⣀⠒⠀⠀⢀⠀⠀⣠⠋⠠⠀⠄⠐⠂⢀⠘⠀⠂⠌⠀⣠⠂⠈⠆⡀⡀⠀⠚⠣⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠁⠀⠈⠀⠀⣀⠁⠀⡀⠠⠆⠀⠸⡀⡈⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠈⠀⠀⠀⢀⠈⠃⠀⡐⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠁⡀⢀⡀⠁⠀⠈⢀⠀⠀⠀⠈⠀⠀⡕⡂⠁⠀⠀⠀⠀⠀⠆⠐⠒⠀⠑⠀⠉⠀⠘⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⢁⠀⠀⠀⠀⠙⠀⠀⠀⢀⡆⠀⠀⠓⠄⠀⠄⠈⠀⢊⡘⠀⢠⠈⠀⠀⠀⠣⠄⠨⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠁⠀⢀⡀⠀⠀⠀⠠⠊⢡⠠⠀⢡⠠⠀⠀⠠⡴⢀⠘⠄⢘⠁⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢅⡀⠱⣀⠸⠂⠀⠀⠈⠢⢄⠀⠀⠀⢀⡀⠠⠔⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠉⠁⠀⠈⠂⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠛⠀⠀⢁⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⡀⠀⠀⠀⠄⠀⢀⠀⢀⡠⠄⠐
⠀⠀⠀⠀⣀⣁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⢀⠀⠀⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⠀⣠⡀⠀⠀⡀⠄⠀⠅⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⠁⠀⠈⠀⠅⡀⠀⠀⠀⡀⠀⠶⣆⠰⠀⠲⠐⢶
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠉⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⡀⠀⠠⠂⠀⡀⠐⠀⠀⠁⠀⠀⠑⢦⡓⠙⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⢠⠤⢤⠀⠀⠀⠙⠢⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⠠⢀⠁⠐⠁⠀⠀⠸⠦⠼⠀⠀⠀⠀⠀⠈
⠀⠐⠁⠠⠂⠀⠓⠄⠀⢄⠀⠈⠁⠒⠀⠑⠀⠂⠀⠀⠀⠀⠂⠁⠂⠀⠀⠀⠀⠚⠂⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⡠⠀⠄⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⡤⢤⡼⠀⠀⠀⢲⡖⢲⡆
⠀⠈⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠂⠀⠀⠠⠄⠀⠀⣀⠀⠀⠂⠀⠀⠀⣿⣰⡆⠀⠀⠀⢸⢻⣼
⠀⢨⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠂⠦⠴⠂⠀⠀⠄⠤⠄⠤⠄⠀⠀⢐⠈⡅⠀⠀⠀⠀⠀⢻⣏⡅⠀⠀⠀⢸⢸⣷
⠀⢈⡅⡉⠁⠉⠀⠀⠀⠁⠠⠈⠩⠀⡏⢁⠀⠠⠀⠀⠀⠀⠀⣀⡤⠜⠙⠀⡇⠀⠀⠀⠀⠀⢸⡟⡇⠀⠀⠀⢸⡟⢸⣇⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⠄⡐⠠⠐⠀⠀⠀⠀⡇⠀⠀⠀⠀⢒⠤⠉⢙⠀⠀⢀⠀⠀⣇⢀⠠⠀⠀⠀⠸⠑⠛⠒⠒⠀⠈⠉⠉⢉⡀
⠀⠀⡄⣀⠀⠀⠈⠀⠁⠀⢥⡀⢤⠀⡧⠄⠠⠖⠂⠀⠀⠀⠀⠈⠙⠀⠉⠇⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠠⡍⠉⠉⠉⠀⠐⠂⠀⡀⠀⠈⠒⠁
⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⢈⠓⢧⢈⡂⡄⢠⡖⠍⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂
⠀⠩⣁⣨⡇⣨⡶⠈⡀⠒⠂⡠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⡰⠐⡐
⠀⢠⡌⠉⡳⣈⣤⣾⠫⣀⡀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠁⠘⠛⢙⠁
⠀⠀⡀⠈⢻⠷⠌⢶⠶⠘⣠⢅⣠⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⢻⣯⠏⠠⠁
⠀⠀⠉⠀⢲⠀⠀⠀⠁⢈⡋⠡⠀⠄⠀⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣐⣠⡡⣀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢋⡁⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⢀⣀⣀⣀⡀⠀⠀⠀⢀⣈⣉⣩⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠛⠁⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢒⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠁⠞⠉⠙⣳
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠀⠀⠀⠀⠀⠀⠰⡾⠷⠀⠀⠀⠀⠀⠀⠀⣿⢧⡀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢧⠀⠀⠀⣠⡴⢦
⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⠂⠀⠀⠀⠀⠀⠀⢷⣿⠀⠰⠐⠀⡁⠂⠀⠠⡀⠀⠀⠀⠀⠀⠠⢬⣁⠀⠀⠀⠀⠊⠀⣸
⠀⠀⠀⠀⠀⠿⡄⠀⠀⢀⠔⠀⠀⠀⢀⡰⣲⣿⠀⠀⠀⠀⠀⣠⡾⣯⣻⠂⠀⠈⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠏⠉⠙⠚⠃⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠈⠉⢉⢻⣿⠀⠀⠀⠀⢠⣿⣇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢀⠀⠈⠀⢀⣀
⠀⠀⠀⠀⠀⢰⠀⠀⡄⠀⠀⠀⠀⠀⢀⡎⣼⡿⠀⠀⠀⠀⢸⣿⡀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠇⠀⠀⠀⣾⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠟⠁⠀⠀⠀⠀⠈⠻⣷⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠸⡀⠠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠘⠃

⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠛⠛⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⣀⡤⠤⠶⠦⠤⣥⣀⢷
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠤⢄⣀
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⢀⣀⡀⠀⠀⠀⠀⣀⣱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠒⠛⠛⠉⠉⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⠤⠤⠤⠤⠤⠔⠒⠉



⣿⣿⢟⠿⣛⠻⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠛⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠈⠉⠙⠛⠛⢿⠿⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⠀⢰⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢿⠛⠛⠋⠉⠉⠙⠋⠉⢉⠙⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⡿⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢡⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿
⣿⣿⣟⢉⠁⢠⣤⡤⠀⠀⠀⠀⠀⢠⡇⠀⠃⠀⠀⢠⣶⣶⡄⢠⣶⣶⣆⠀⠀⠈⡇⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠿⠋⠀⠀⠀⡀⠀⠀⠀⣼⢢⡄⠀⠀⠠⣘⣿⣿⠣⠬⢿⣿⣃⠄⠀⠀⠀⣦⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⣿⣿⡿⠙⢷⡤⠤⠶⠛⠛⠛⠀⢠⣿⣿⣷⣦⣀⡀⠀⠙⠁⠀⠀⠀⠑⠀⠀⢀⣰⣴⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣤⠂⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣧⣄⣀⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⢀⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣿⣿⣇⣈⠛⠛⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⢿
⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠈⠀⠈⡄⠀⣻
⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣶⣤⣤⣴⣶⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⢱⣸⣿
⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣠⣿⣿⣿
⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠲⢾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠰⢾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣤⣀⣠⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣶⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣀⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⢼⣿⣿⣶⠀⠀⠀⠀⠀⢀⡾⠥⠄
⣼⣿⣧⣌⣴⣆⠖⣶⣶⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⣦⠴⢦⣤⣤⣤⣾⣷⣶⣦⡤⠀⡀⢠⣦⣴⣿⣷⣤⣀⠀⢀⣀⣀⠀⣀⡀⢠⣦⣤⣤
⢻⡿⠿⢿⡿⠿⠲⠿⠿⠧⠔⠀⠀⠀⠀⠿⠀⠀⠉⠉⠉⠙⠻⠿⠉⠉⠉⠉⠉⠀⠈⠀⠀⠀⠀⠉⠉⠉⠙⠋⠛⠛⡙⠉⠉⠉⠈⠉⠛⠛⠛
⡁⠐⣋⠤⠒⠈⢁⠠⠖⠂⡠⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠉⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠀⡠⠀⠁⠀⠀⢠⠀⠘
⠴⠊⢁⡤⠒⠉⡠⠐⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣀⡀⠂⠀⠈⠀⠄⠀⡠⠀⠀⡀⠐⠀⠀⠀⠀⢀⠔⠁⠀⡀⠀⠀⠂⠘⠀⡆
⡠⠖⠁⣠⠔⠀⣠⠀⠀⠐⠠⠄⢈⢁⠂⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠤⠊⣠⠄⠈⢀⠔⠈⠀⠀⠀⠀⢀⠔⠁⠀⠀⡄⠀⠀⠆⠀⡆⠀⡇
⠀⢠⠎⠀⢠⠀⠀⠀⠹⠀⠀⠀⠉⠉⠰⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠂⠁⠀⠂⠁⠀⠀⠀⠋⠀⠀⠀⠊⠀⠀⠀⠀⢰⠃⠀⡇
⠀⠀⠤⠈⠁⣀⠒⠒⠀⠀⠋⠉⢀⣀⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠠⠀⠀⠀⠂⠁⠄⢀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠁
⠆⠙⠂⠀⠤⠈⠁⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠒⠀⢀⠤⠀⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠇⠀⠉⠁⠒⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡴⠛⠁⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⢸⠀⠉⠑⠒⠤⠤⠀⡀⠎⠠⠚⢠⠐⠒⠒⠚⠛⠩⠭⠭⠼⠛⢲⠂
⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⢰⠀⠆⠀⠆⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⢸⡁⠀⠠⠀⠐⠀⠀⢸⠀⠀⠀⡇⠀⢀⠀⢺⡀⠐⠒⠒⠒⠂⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⢸⡇⠐⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠓⠀⠁⢓⠲⠤⠤⠤⣤⣄⣤⠤⠤⠎
⠀⠀⠀⠀⠀⠀⢸⡇⠢⡠⠂⠀⠀⠀⢸⠀⠀⢠⡇⠀⠰⣿⣶⣼⠀⢠⠆⠀⠀⠀⠀⠄⢸⠀⠀⠀⢀⡄⠐⠒⠶⠶⠤⢤⣄⣀⡀
⢀⠀⠀⠀⠀⠒⠂⣇⠀⠀⠀⠀⠘⠀⢸⠀⠀⢸⠃⠐⠷⠿⠿⠇⡄⢮⢀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠸⠀⠀⠀⠀⠈⠀⠐⠂⠈⠉⠛⢲
⢛⠒⠒⠒⠂⠀⠀⣿⢠⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠈⠀⠀⡤⠄⠞⢰⠃⣠⡶⣞⡓⠶⠞⠉⠙⢦⠀⠀⠀⠀⠀⢸⡄
⠀⠙⠒⠒⠀⠀⠀⢹⠀⠀⠉⠉⠉⠋⠀⢰⠀⠀⠀⠀⠀⠀⢀⡇⢰⠀⡴⠀⠈⠀⠀⢨⠾⡋⠀⠈⠻⣄⠀⢀⣀⠀⢳⠀⠀⠀⠀⣿⢇⣀
⠀⠈⡀⠀⠠⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢸⢸⠁⠀⠀⠀⡅⣸⠁⠀⡀⠢⠀⠀⠀⢀⡅⠀⠉⠀⠀⠀⢿⣀⠀⠁⠀⠐⠀⢀⣀⠀⠇⠨⠉⢙⠛⠛⠶⠦
⠀⠀⠑⠤⢤⡤⠤⠀⣀⠶⢠⠀⡄⢀⠀⠀⡎⠀⠀⠀⠀⢡⠇⠀⠀⠈⠒⣢⢀⠂⠀⠀⠀⠀⣀⠀⠀⠀⠙⣧⠀⣄⡀⢳⠀⣀⠀⠉⠐⠒⠂⠼⢃⡴⡫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠐⠨⠀⠂⢱⠃⠀⠀⠀⡾⠀⠀⠀⠀⣎⠀⠀⠒⠶⢶⣤⣄⣉⡀⠀⠀⠀⠘⠇⠈⠁⣀⡇⢀⡉⠁⠀⠀⠀⢰⡯⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⠃⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⠀⠉⠙⠛⠷⢦⣤⡀⠐⡤⠊⠁⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣶⣤⣤⣤⣀⣀⣉⣀⣀⣤⣤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠒⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⣀⠴⠛⠙⢳⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⣆⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⡇⠀⣀⣀⠤⠔⠂⠉⢀⣀⣀⣀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠈⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡍⠉⣉⣠⡤⠔⢲⣶⣿⣯⣅⠀⠈⢳⡀⠀⢀⡠⠤⣄⡀⢠⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⠛⠀⢀⠄⠀⠈⠛⠃⠀⠀⢹⣄⠎⠀⠀⠀⠙⣎⠒⠒⠲⢄⠀⢀⣠⡜
⠀⠀⠀⠀⠀⠀⡤⠚⠉⠑⣼⡻⠁⣒⠛⡄⡏⢈⣍⠑⢦⠀⠀⠀⠀⢻⡈⢦⡀⠀⠀⠘⡆⠀⠀⠀⠉⡇⠀⡇
⠀⠀⠀⠀⠀⣼⠁⠀⢀⠄⡾⡁⠀⣿⠇⡇⣅⠹⠿⠀⢸⠀⠀⠀⠀⠈⢧⠐⠁⠀⢠⠀⡷⠀⠀⠀⢠⠇⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⢎⢀⡇⠱⣄⣀⠜⠉⠙⣖⠲⠒⠁⠀⠀⠀⠀⠀⠈⣧⠀⠒⠃⣰⠇⠀⠀⢀⠎⠀⡜
⠀⠀⠀⠀⠀⢳⠀⠀⠈⣼⠀⠀⢀⠇⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠘⣟⠒⠋⠁⠀⣀⠴⠋⢀⠜
⠀⠀⠀⠀⠀⠈⠳⠤⢴⡇⠀⢠⡋⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⢀⡴⢸⠃⠀⠀⢙⠻⠒⠒⠒⠋⠉⠙⠒⠲⠤⣄⡀⠀⠀⠀⠀⣿⠤⠔⠉
⠀⠀⠀⠀⢀⠔⠉⡀⣾⠀⠀⡔⠁⠀⠀⢀⣀⠀⣠⣀⣰⣄⣱⠤⠽⠂⠀⠀⠀⢸
⠀⠀⠀⢠⠃⡔⠉⠀⢸⠀⣌⣠⣤⡼⢖⠋⠀⠀⠀⠉⡹⢀⠇⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠈⠢⣘⡒⠒⠺⣇⠁⠀⠀⠀⠠⣑⠒⠒⠐⢋⡡⠊⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠉⢉⠓⠘⣦⡀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⡾⠁
⠀⠀⠀⠀⢀⠂⠉⠏⣠⠴⠛⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠾⢍⡀
⠀⠀⠀⠀⠀⠉⢲⡞⠁⠀⠀⠀⠀⠀⠉⠉⠛⡗⠒⠒⢛⡟⠉⠁⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⣳⠀⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠘⠦⣄⣀⣀⣀⣀⣀⣉⣤⠴⠞⠋⠀⠀⣇⠙⠢⢀⡀⠀⠀⠀⠀⠀⠛⡆⣀⣀⠀⠀⡀⣠⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⣀⣀⣀⣤⡤⠴⠃⠈⠉⠀⠀⠁⠙⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠖⠚⠛⠛⠛⠓⠒⠲⠶⠤⢤⣄⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⣠⠴⠛⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆⠈⢧
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈
⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠴⢄⣀⠀⢀⣠⡾
⢸⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⡏
⡻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣢⠀⠀⠀⠀⠀⠀⢀⣴⠏
⢹⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁
⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⡀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⣀⣠⣴⠟⠁
⠀⠈⠳⣄⠀⠀⠀⠀⠠⠤⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⠁
⠀⠀⠀⠀⠙⠲⠤⣄⣈⡛⠊⠀⠀⠀⠀⠀⠙⠒⠒⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⡴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠐⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣁⠀⠀⠐⠀⠀⠠⡀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠐⠠⠁⢀⠐⠁⠉⠀⠀⠰⢤⠄⠐⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃⠉⠀⠀⠀⠀⠀⢄⠠⠄⠈⠀⠀⠐⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠈⠀⣀⠀⠈⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠂⡀⡀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⡀⠁⠀⢆⠄⢀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠠⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠠⠀⠀⠀⡀⠀⠄⢀⠀⠀⠀⠈⠀⠀⠄⠀⠠⣀⢠⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⢀⠄⠠⠀⠀⠀⠄⠁⠀⡘⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠊⠀⠀⠈⠒⠀⠀⠌⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠂⠈⠉⠉⠉⠉⠀⠒⠂⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠀⡀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠑⠤⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢢⣀⠀⠀⠀⠀⠀⠉⠐⠂⠠⠄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⠋⠁⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡠⠃⠀⠉⠒⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⠀⠀⠀⡹⢡⣿⣶⣄⠀⠀⠀⠀⠀⠈⠀⠂⢀⣤⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⢠⡄⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⡰⠁⠘⢿⣿⣿⣿⣶⣤⣀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠔⠀⡰⠁⠀⠀⠀⠙⠻⠿⣿⣿⠟⠀⠀⠸⣿⠿⠋⠉⠙⠻⠿⠃⠀⠀⠀⠀⠀⠘⠆⠈⠉⠉⠀⠢⢀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠀⡰⠁⠀⠀⢀⠔⠂⠉⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠁⠀⠀⡠⠁⠀⠀⠀⠀⠀⢁⠀⠀⢀⠄⠂⠈⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢠⠃⠀⠀⢀⠁⠀⢱⣦⣤⣀⣀⣈⠀⢀⠃⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠎⠀⠀⠀⠘⠑⠠⢈⠛⠛⠋⠀⠨⠀⠋⠀⢤⣀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠘⠀⠀⠀⠈⠁⠀⢐⠃⠠⣀⠀⠈⢿⣿⡿⠃⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡠⠀⠀⠀⡰
⠀⠀⠀⢀⠄⠒⠂⠀⠂⢰⠁⠀⠀⠀⠀⠀⠡⣀⠀⠀⢀⡤⠃⠀⠲⡄⠈⠐⠠⠤⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢀⠔⠁
⠀⠀⠐⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⠉⠀⠀⠀⠀⠹⠄⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠄⠐⠁
⠀⠀⠀⠀⠀⠈⢂⠉⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⢄
⠀⠀⠀⠀⠀⠀⠈⢆⡈⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠃⢠⠚⠈⠁⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⣀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠰⠁⠀⠀⠰⠀⠄⠀⡠⠀⠀⠈⠁⠉⠦⠀⡀⠀⠀⠀⣤⠀⡤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠑⠶⠄⠀⠀⠑
⠀⢀⠔⠂⠁⠀⡠⠂⠄⣧⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠁⠀⠉⠀⠀⠂⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠄⠀⠔⠁⠀⠀⡈⠀⠣⡀⠈⠐⠂⠒⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠈⠃⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠈⠀⠉⠄⠥⡀⠀⠐⢆⠀⠀⠐⠌⠀⠀⠳⣀⠀⠀⠤⡀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢃⠀⢂⠀⡀⠐⠁
⠀⠀⠀⡀⠀⠊⢀⡀⠀⠈⠂⢀⠀⠑⠦⡀⠀⠈⠁⠒⠀⠈⠁⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⣈⡄⠀⢀⠀⡠
⠀⢀⠜⠀⢀⠀⠀⠈⠒⢄⠀⠈⠢⡀⠀⠈⠑⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠊⢁⡠⠀⠈⠀⠉⢣⡈
⠀⠄⠀⠀⠀⠈⢢⠀⠀⠀⠑⢄⠀⠐⡀⠀⠀⠀⠀⠈⠁⠒⢠⠤⢀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠔⠒⠉⠀⠀⠔⠁⠀⢀⠠⠂⠁⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠈⠇⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⡠⠒⠁⠀⠀⣀⠔⠊⠁⠀⠂
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄⠀⢁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠀⠀⠀⢀⠌
⠓⠄⣀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡆⠀⢘⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠂⠀⠃⠀⠈⠐⠂⠁
⠀⠀⠀⠈⠁⠀⠐⠂⠀⠐⠃⠒⠂⠁⠀⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠠⠀⢀⣀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣟⣻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠠⡤⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣯⣿⣿⣿⣿⣿⣛⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠐⠂⠀⠀⠀⠀⠤⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣛⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⣸⣿⢿⣿⣿⣿⣛⣿⠿⣛⣭⣶⣿⣿⣿⣿⣿⡟⣿⣿⣿⣧⠀⠀⠀⠀⣿⣿⣿⣿⠏⠀⡀⠀⠀⠀⠤⠀⣀⠴⠚⠉⠉⠀⠀⠀⠀⢠⡀⠀⠀⠹⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⡿⣣⣾⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠁⠀⠀⠀⢀⡞⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡏⠀⠀⠀⠀⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠻⠿⢿⣿⢸⣿⣿⣏⣿⣿⣿⣿⣿⣼⣿⣿⣿⣾⡟⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣤⣀⣀⢀⡇⠀⠀⠰⠄⠀⠀⠀⠀⠃⠀⠀⠀⠁⣠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠠
⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠄
⠀⠀⢈⡿⠀⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⢻⣿⡄
⠀⠀⠐⢿⢿⣦⣄⡀⠀⠀⡀⠀⠀⠰
⠀⠀⠀⠀⠉⠉⠉⠉⠝⣻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠙⠻⠷⠒⠶⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠤⠤⠶⠶⠶⠖⠒⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠴⠶⠒⠒⠒⠛⠛⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠥⣀⣀⠀⠀⠳⡀⠀⠀⠀⣀⣀⣀⣤⡤⠤⠾⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⡀⡇⠀⠀⠀⠈⠁⠒⢻⠞⠛⠋⠉⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢯⣟⣯⡇⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⢀⡄⡇
⠀⠀⠀⠀⠀⢀⠔⠁⠈⠀⢜⠧⣄⠀⠀⠀⠀⠀⢸⢀⡄⢠⡤⡆⡖⠀⣹⣯⡀⡇
⠀⠀⠀⢀⠞⠁⡔⠀⢠⠀⠀⠑⡈⠙⢦⠀⠀⠀⢸⠀⣿⠛⠇⠇⠛⠛⠉⠁⠀⡇
⠀⠀⠀⡏⠀⠈⠀⠀⠈⠀⠀⠀⠈⠀⠈⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠣⣄⣀⡀⠀⠀⠀⠀⣀⣀⣤⡴⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⡀
⠀⠀⠀⠀⢹⡌⠉⠉⠉⠉⠉⠉⠀⡼⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠻⠿⠓⠤⣀
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠰⠾⠿⣀⠀⢠⣶⣿⠒⢤⣀⡀
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠋⠛⢒⣒⠒⠒⠚⠛⠳⠦⠤⠤⠤⠬⠯⠭⣷⡄
⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⢠⣇⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠙⠿⠷⠀⢰⣾⣷⡀⠀⠀⠀⠙⢷⡄⠀⡇
⠀⠀⠀⠀⠀⣠⣤⠟⠀⠀⠀⢸⣦⡍⠓⠦⣄⡀⢸⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⠀⠀⠛⠿⠧⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢿⣿⣶⣶⣶⣶⣿⣿⠏⠀⠀⠀⠉⠛⠶⠒⠚⠉⠉⢸⠀⠀⠀⠀⢰⣶⣶⠄⠠⣤⣄⠀⠘⠻⠟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠒⠒⠓⠶⠶⠦⠭⠿⠥⢤⣤⣀⣀⣀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣟⠉⠁⠀⠀⠀⠀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠉⠉⠙⠛⠿⣍⣉⠉⠛⠓⠲⠦⣼⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⢀⣠⣤⣤⣤⣀⡈⠉⠛⠛⠛⠛⠛⠛⠉⣽⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⡀⠀⠈⣿⠟⣡⡟⣩⣭⣍⣙⡟⠛⢒⡒⠚⠛⢷⣾⠋⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢠⡟⢩⠔⠶⣌⢻⡀⣼⠏⠀⠿⠾⠋⠀⠀⠉⠀⠀⢿⡛⢦⣄⣈⣿⣾⢋⣭⣭⡙⢦⡀
⠀⠀⠀⠀⠀⢀⡟⠀⠁⠀⠀⣿⢸⣷⡟⠀⠀⠀⠀⣀⡤⠤⣄⠀⠀⣠⠿⠶⢤⡾⠿⣧⢸⡇⠈⠻⣌⣷
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⠋⣸⣿⠇⠀⠀⠀⢰⠋⠀⠀⠈⢳⠀⣇⣀⣀⣀⣳⠀⣿⡀⢳⡀⠀⢹⢸⡇
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢿⣀⣿⣿⠀⠀⠀⠀⢸⠛⣿⣿⡏⠹⡎⡏⢿⣿⠏⢹⠂⢸⣧⣼⠃⠀⠜⢸⠃
⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⣹⣿⡇⠀⠀⠀⠀⠘⣆⠈⠉⠁⣼⠛⢳⣄⠉⢀⡜⠀⠀⢿⣧⣀⣀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⣾⢫⣿⠀⠀⠀⠀⠀⠀⠈⠙⠒⣺⡇⠀⠀⠹⡟⠋⠀⠀⠀⠘⣿⣭⣍⡀
⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⡇⠀⠀⠀⠹⣶⡀⠀⠀⠀⠸⣷⣾⡟⠶⣤⣀
⠀⠀⠀⢀⣤⠾⠋⠁⠀⢀⣿⣿⠀⠀⠀⠀⠀⣀⣠⣤⡼⣧⣀⣇⠀⠀⠀⣰⣳⣇⡀⠀⠀⠀⢿⣿⣧⣤⣀⠙⢳⡀
⠀⠀⢠⣿⠃⣰⠞⠛⠛⠛⢿⡟⠀⠀⢀⡴⠛⣉⠄⠀⡀⠀⢀⠉⠓⠒⠛⠉⠁⠈⠙⣷⢤⡀⢸⣿⠀⢀⣹⠇⢀⡷
⠀⠀⠘⣧⡀⠹⢦⣤⣀⡀⢸⣧⠀⠀⠘⠓⠚⠿⣤⣾⣁⣠⣇⢀⣸⠀⢰⡀⢦⠀⡠⡈⢳⣙⣾⣿⢛⣉⣤⠴⠛⠁
⠀⠀⠀⠈⠛⠶⠤⢤⣬⣉⠻⣿⡄⠀⠀⠀⠀⠀⠙⠶⢭⣅⣉⣉⣉⣳⣾⣧⣼⣧⣽⡿⠲⢻⣿⢟⣿⣷⡶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⡻⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⣴⡿⢿⣉⠉⠙⣿⠉⣿
⠀⠀⠀⠀⣀⣠⡤⣴⠏⠀⠀⡿⣯⣽⣿⣿⣿⢓⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⢿⣿⣿⡶⣿⡓⠲⠒⠚⠁
⠀⠀⢠⣼⣏⠉⠛⠋⠀⠀⡼⢡⣿⣿⢿⣟⣯⠺⣿⣿⣿⣦⠶⠶⣶⣾⠿⡿⢟⡿⢛⡭⢛⡩⠿⢿⣿⣦⡀
⠀⠀⠈⣷⣤⡭⠭⣇⡤⠎⢠⣼⠟⠛⠒⠙⠛⠒⠒⠿⢿⣿⣦⠀⣿⠿⠿⠿⢧⣶⣵⣮⣥⡒⠒⠉⣉⣉⣿⣄
⠀⠀⠀⠈⠛⠞⠛⠉⠀⠀⢾⣟⠳⠶⠦⠤⠤⠤⠶⠒⠋⣉⣿⠀⠿⠶⣶⣤⣤⣤⣤⣬⣭⣉⣭⣭⣭⣭⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠲⠶⠶⠶⠶⠟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣀⡤⠤⠴⠶⠒⠲⠶⠦⠾⢧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠒⠒⠛⠷⢦⣄⡀
⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⣀⣾
⠀⢧⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠉⠁⠀⠉⠻⡓⠦⣄⣀⣀⣀⣠⡤⠊
⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠴⠒⠛⠐⠢⢤⡀
⠀⠀⠀⠀⠈⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⣇⠀⠀⠀⠀⠀⢀⡨⠷⠤⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⣀⠀⠀⠀⠀⠀⠀⠈⢧⡀⡀⠸⡄⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣄⣀⣀⣠⣾⠟⢮⣦⣝⣦⡀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠙⠛⠿⠿⠦⣄⣸⣷⡤⡀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣤⣄⣀⣶⡮⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠂⠉⠉⠉⠉⠉⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⡶⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠁⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠶⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠊⠁⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠈⠐⠒⠒⠒⠒⠒⠒⠒⠒⠈⠉⠀⠀⠀⠀⣀⠤⠖⠚⢧
⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠴⠒⠊⠉⠀⠀⣀⣤⣦⢳⠊⠉⠑⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠉⠉⠀⠉⣹⣿⣿⣿⣿⣷⣦⠀⠀⠀⣠⣴⣾⣿⣿⡿⠀⢣⠠⠀⠀⠀⠙⢄
⠀⠀⠀⠀⢀⠴⠒⠉⠉⠑⠞⠀⠀⠀⠀⠀⠹⠿⠛⠋⠛⠿⠿⠇⠀⠘⠿⠿⠿⠟⠋⠀⠀⠀⢣⠈⠢⡀⠀⠈⠆
⠀⠀⠀⡰⠁⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡴⠊⠉⠑⠢⡀⠀⠀⢣⠀⢱⠀⠀⢰
⠀⠀⠀⡇⠀⠀⢰⠁⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠉⠢⡀⠀⠸⠀⠀⠀⠀⣠⠼⡄⠀⠀⢧⠀⠂⠀⡸
⠀⠀⠈⡁⠀⠀⢀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣸⡄⠰⠶⢶⣾⣿⠁⢀⣼⠀⠀⠈⣇⠀⡠⠃
⠀⠀⠀⢣⠀⠀⠸⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠧⠠⣤⣶⣶⠋⠀⢡⠨⡤⠤⠍⠐⠊⠁⢰⠀⠀⠀⠘⡎
⠀⠀⠀⠈⢦⠀⠀⠙⢀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠈⣙⡩⠤⠒⣫⠄⠱⣄⠀⠀⠀⢀⠎⠀⠀⠀⠀⢻⡤⠤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠑⠦⠤⢼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢍⠀⠀⠀⣰⠏⠀⠀⠈⠳⡒⠂⠁⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣼⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢱⠖⢈⠟⠀⠀⡇
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠐⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠸⣤⠊⠀⠀⡰⠁
⠀⠀⠀⣠⠞⠁⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣟⣀⣀⣀⣒⡄⠀⡇⠀⡠⠊
⠀⣠⠞⠁⠀⣀⠤⠴⢶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠍⢀⡀⠀⠀⢹⢀⠃⠰⢤⡀
⡎⠁⠀⠀⠉⠉⠐⠒⠘⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢉⠝⠓⡺⠒⡤⠶⠶⠚⠉⠢⡀⠀⠈⢳⠖⠊⡼⠀⡀⠀⠱⣀⡀
⠑⠲⠶⠦⠖⣠⡤⠄⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡰⠁⠀⠜⠀⢀⠃⠀⠀⠆⠀⠀⢱⢄⡤⠊⠀⡼⠳⡱⡈⠢⣄⠀⢈⠆
⠀⠀⠀⠀⣯⠔⡐⡠⠀⡼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢅⢀⣜⠀⠀⢸⡀⠀⢀⠶⠤⢔⡣⠊⠀⣠⠞⠀⢀⣣⡱⠶⠊⠉⠁
⠀⠀⠀⠀⢤⠊⡔⠁⣸⣁⠸⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⠃⠑⠲⠁⠄⠒⠉⠀⣠⠞⠁⢀⠔⠉⠀⢈⠴⠤⡀
⠀⠀⠀⠀⠀⣉⡝⠋⠉⠑⠢⡀⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⢀⡔⠁⢀⡴⠊⠀⠀⡠⠔⢦⡀
⠀⠀⠀⢠⣌⡀⠀⠉⠒⠤⡀⠈⠢⡀⠀⠉⠒⠲⠦⠤⠤⢤⡀⠤⠤⠤⠤⡶⠒⠉⠁⠀⠀⠀⢠⠎⠀⡴⠋⠀⠀⡰⠊⠀⠀⠀⢁
⠀⠀⠀⠁⠀⠈⠑⢄⠀⠀⠈⠲⡄⠈⢢⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡎⠀⡜⠦⣀⣠⠎⠀⠀⠀⠀⠀⠈
⠀⠀⢰⠀⠀⠀⠀⠀⠙⢄⣀⠠⠞⢂⠀⢃⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢠⠁⢸⠀⠀⠰⠁
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠈⡆⠈⡄⠀⠀⢀⣠⠜⠀⠀⠀⠀⠀⠀⠈⠒⠒⠶⠤⠼⠤⠼⠤⠤⠴⠀⠠⠤⠤⠤⠖⠒⠉
⠀⠀⠘⠦⠤⣀⣀⣀⣀⣀⣀⡧⠠⠄⠧⠒⠓⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠗⣒⢒⢈⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠁⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢆⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⠀⠀⢀⠜⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⢠
⠀⠀⠀⠀⠀⠠⠄⡆⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡄⢀⠀⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠁⠘⠄⠁⠀⠀⠃⠄⠀⡀⠀⠀⠀⢀⡤⠂⠉⠀⠀⠀⠀⠀⠈⠁⠢
⠀⠀⠀⠀⠀⠃⠄⡀⠀⠀⠀⡀⠄⠈⠁⠀⠀⠀⡎⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠢⠁⠀⠀⢀⡐⠁⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠂⠤⣀⣀⣀⡠⠄⠀⠀⠀⠐⢀⠀⡟⡉⠚⠀⠀⠈⠀⠀⠀⠀⠐⢈⠀⠀⠀⠸⡉⠒⠀⠀⠀⠀⠀⠂⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢀⡁⠀⠀⠀⠀⠀⠲⡢⠂⠀⠀⠀⠀⠑⢔⠀⠀⠀⠀⢀⣄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣄⡀⠁⠀⢀⡄⠀⠀⢤⠀⠀⠂⢒⠀⢀⠀⠀⠈⠉⠈⠀⠀⠀⢀⠀⠒⠂⢀⠄
⠀⠀⠀⠀⠀⠀⡀⠋⠀⠀⠁⢀⠀⠀⠀⠀⠀⡏⣉⠁⠉⠹⠂⠀⠀⢸⠀⠀⠠⠈⠄⠀⠀⠀⡄⡀⠰⠪⠀⠀⠼⠆⠀⠀⠐⠅
⠀⠀⠀⠀⢀⠈⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢐⠄⢀⠤⠀⠀⠀⡀⠀⠀⠀⡀⠄⠀⠠⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠡⣀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠿⠀⠀⠀⢸⠀⢀⢀⢐⠂⠃⡃⠠⠀⠀⠈⠆⠯⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠉⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠂⠀⠀⠀⠂⠀⠀⠠⠄⠀⠀⠒⠒⠂⠂⠒⠒⠒⠒⠒⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠶⠖⠒⠛⠛⠓⠒⠒⠲⠶⠤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠉⠉⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡄⠈⢳⡀
⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⠀⠀⢸
⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣼⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠤⢤⣤⣴⠟⠁
⢸⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏
⣽⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⣾⠆⠀⠀⠀⠀⠀⠀⣠⡾⠏
⢸⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋
⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢆⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠤⢀⣀⣠⣴⠟⠁
⠀⠈⠳⣄⠀⠀⠀⠀⠀⡠⡤⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠙⠲⢤⣀⣀⠑⠒⠊⠀⠀⠀⠀⠀⠑⠲⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠶⠶⠤⣤⠦⢶⣶⠶⠾⠿⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠴⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣠⠊⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠲⠆⠀⠀⠀⠁⡼
⠀⠀⢠⠁⠀⠀⠀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⡆⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⢀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠚⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⣀⠀⠀⣀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀⣐⣯⣿⣶⣦⣤⣋⣉⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣧⣼⣷⣆⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣽⣿⡿⢿⡿⠿⢿⣿⢿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⣿⣿⣿⣿⠀⢀⡀⠀⠀⠀⣀⣤⠐⠋⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣴⣦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣖⣦⣿⣷⣽⡧⡀⢻⣿⡿⣦⣤⣤⣶⣶⣴⣦⣤⡴⡲
⠀⠀⠀⠀⠀⢠⡄⠹⠟⠛⢿⣿⠿⠛⢿⣿⠟⡛⠛⢙⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠿⢿⣶⣿⣿⠿⣿⣶⣦⠼⡄
⠀⠀⠀⠀⠀⢤⢿⡆⡄⡤⠀⠀⣤⣏⠉⣴⣾⣆⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⣿⠟⠡⣶⣾⣿⣀⠀⠛⠙⠛⢹⡷⠀⠂
⠀⠀⠀⠀⠀⠈⠙⠫⡁⢿⠶⠴⣿⣿⣿⠫⢿⣿⢤⣼⣿⣿⣿⣿⣿⣿⣷⣃⡀⣻⣶⣇⣿⣿⣿⣦⣤⢴⣷⠟⢀⣀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣿⣿⣿⠙⡀⠼⠿⣿⣝⣧⣿⣿⣿⣿⣿⣿⣿⣿⠟⣼⠟⠙⠿⣿⣛⣮⣴⣿⣶⣾⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠿⢿⣿⡷⠀⣀⣼⣿⣧⣽⣯⣿⣿⣿⣿⣿⣿⣿⣶⠏⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⡄⠉⠉⠙⠛⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠁

⣿⣿⢛⠻⠛⠻⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⣰⠃⠀⣀⡤⠷⠒⢢⡤⠴⢤⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⡴⠋⠁⠀⠀⠈⠹⣇⠀⢸⠁⠀⠀⠀⡎⠀⠀⠀⠙⣇⠉⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⡴⠊⠉⠉⠲⣤⠖⠋⠉⠉⠉⠉⠙⣷⣶⠚⠉⠉⠑⢄⢠⠖⠋⠉⠉⠛⠯⣍⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠘⢇⠀⠀⠀⠀⠙⡆⠀⣠⠴⠒⠛⠲⡟⢘⡤⠚⠉⠉⠻⣯⠀⢀⣀⡀⠀⠀⠈⢷⡀⠸⠛⡆
⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠙⠲⢤⣀⣀⣻⣹⠁⠀⠀⠀⠀⢳⡞⠀⠀⠀⢀⠔⠉⠉⠉⠀⢉⣳⠤⠒⠉⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠓⠦⢦⣄⠀⠀⠀⠉⠉⠉⠛⠒⠦⠤⢤⣸⣇⣠⣠⣴⠥⠴⠒⠒⠚⠉⠉⠀⢀⣀⣤⣢⡴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠪⢝⡒⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀⣀⣠⡤⠴⠒⠚⠉⠁⠈⠀⡠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠗⠒⠒⠲⠖⠚⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠚⡇
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠴⠢⢻⡀
⠀⠀⠀⠀⠀⠀⢀⡿⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠷⡄
⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣤⣤⣤⣤⣤⡽
⠀⠀⠀⠀⠀⠉⠓⠤⠤⠤⣤⡀⠶⠶⠤⠤⠤⠶⠦⠤⣤⣀⣀⡀⠀⠀⠀⢀⣀⣤⠾⠴⠒⠛⢓⣋⣁⣠⣖⠉⠉⠉⠉⠙⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⢠⠔⠋⠉⠀⠀⠀⠈⠉⠳⣄⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠴⠶⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠆⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⠛⠀⠀⠀⢰⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠒⠤⡀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠹⣀⠞⠁⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⡏⠀⣄⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⢠⢧⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠦⣄⣀⣠⡴⠋⠈⠢⣄⣀⣠⠞⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⠀⠀⢠⡴⠶⣤⠎⠁⢘⡧⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⠇⠀⠀⠸⡆⠀⠈⡆⢠⠞⠀⠀⣠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⡏⠀⠀⠙⡆⠑⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⣠⡴⠞
⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⣀⣀⡤⠴⠋⠀⠀⣠⠶⠤⠴⠚⠁
⠀⠀⠀⠀⣠⠎⠁⢀⡴⠚⠉⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢧⣀⠀⠀⢀⣀⡠⠴⠚⠁
⠀⠀⠀⣰⠃⠀⢰⠋⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠉⠉⠉
⠀⠀⢀⡏⠀⠀⡏⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠒⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⣸⠇⠀⠀⠛⡆⠀⠀⠀⠀⠀⠈⢷⡀⠀⠈⠁⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠏
⢀⡞⠁⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⡼⠀⠀⡴⢿⠳⣄⣀⣨⡗⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⣇⠀⢰⡇⣸⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉
⠈⠙⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣤⣤⣤⣤⣤⣴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⢸⡇⢸⡇⣿⠀⣿⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠉⠉⠉⠙⠛⠶⢾⡇⢸⡇⣿⠀⣿⠶⠞⠛⠉⠉⠉⠉⠉⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⠈⣉⣈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠒⠲⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⡀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⠿⡏⠀⠄⡀⢀⠀⣪⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠇⠀⠀⢠⠂⢀⠟⠀⢐⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠆⠀⠀⠠⠂⠀⠀⠀⣴⢒⢲⠟⡛⢟⠛⣻⠻⣞⠛⡿⢛⠳⣶⡤⢰⣿⣿⣿⣿⣧⠀⠀⠀⠀⢸⠀⡌⠀⠀⠐⢣⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣠⣾⣜⣋⣼⣨⣇⣥⣹⣸⣇⣛⣡⣿⡇⠀⢿⣿⣿⣿⣿⣄⠀⠀⠀⢸⠀⠁⠀⡀⠀⠞⢀⡴⠇
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠀⠁⠀⠀⠈⠻⣿⣿⣿⡟⠃⠀⠒⠚⠒⠐⠒⠓⠀⠉⠉⠀⣦
⠀⠀⠀⠀⠀⠑⠤⢤⣶⣼⣶⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠩⠴⣦⣤⣴⣤⣤⣤⡤⠶⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠤⣀⠐⠒⠀⠀⠀⠀⠀⠤⠶⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠖⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⠅⠀⠀⠚
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠒⠂
⠀⠀⠀⠀⠀⠢⢄⣀⣀⣀⣷⡌⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣻⣟⣿⣻⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣭⣭⣭⣿⣯⣽⣭⣯⣯⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠃⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡔⠉⠀⠐⠆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⠀⢇⠀⠀⢀⠦⠠⠈⠙⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠘⢄⡀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢹⣶⣤⣀⣀⣀⣀⣀⡠⠔⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡿⠿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠘⢄⡀⠁⠀⠘⣿⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢧⣤⡀
⠀⠀⠀⠀⠀⣠⠞⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⠳⣄
⠀⠀⠀⠀⡼⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢧
⠀⠀⠀⢸⠁⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⡆
⠀⢀⣠⠼⠤⣄⡀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢀⣠⠤⠧⣄⡀
⣰⠋⠔⠀⠀⠂⠙⢦⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡴⠋⠐⠀⠀⠢⡙
⡇⠀⠀⠀⠀⠀⢘⢸⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠠⡇⡃
⣇⠀⠀⠀⠀⠀⢸⢸⡀⠀⠘⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⢀⡇⡇
⣿⠀⠀⠀⠀⠀⡰⠋⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢇
⢹⢸⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⡇
⢸⠘⠀⠀⠀⠀⠘⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⠃⠀⠀⠀⠀⠁
⠸⣄⠀⠀⠀⠀⣀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣀⠀⠀⠀⠀⣠
⣶⣾⠿⠿⢿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣿⣿⣿⣷

⠀⠀⠋
⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡠⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⡄⡀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡜⣆⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⡍⠳⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⠿⠿⣣⠀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠠⠊⣀⣸⣀⣠⣀⣀⡀⡰⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠉⠉⢁⣀⡄⢠⣡⣼⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⠏⠁⢨⠛⠂⠘⢫⠈⢁⠀⠀⠀⠀⠀⠀⠐⠁⠁⠀⠀⠀⠈⠉⠀⠀⠉⠇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠘⠀⠷⠲⠘⠀⠰⠀⠀⠀⠀⠀⠀⠑⡤⠀⠀⠀⠀⢸⠂⢀⡶⠀⣿⣀⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣼⠀⠀⠀⠠⡃⠀⠀⠀⠈⠂⠀⠀⠀⠀⡄⣈⠁⠀⢀⠆⠀⠈⣁⠋⠘⠀⢸⣈⡃⠀⢠⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠓⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠂⠈⢑⡻⠓⠒⠁⠀⠀⢸⡁⠀⠀⠀⠘⡇⢱⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣇⠀⠈⠂⠀⠀⠀⠈⠀⠀⣤⢤⠀⢃⠈⠈⡇⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠐⣧⢤⣤⣤⢬⡦⣄⡀⠀⠀⠀⣀⡤⢎⠠⠤⠀⢸⡀⢀⢹⡀⠀⠀⠈⠛⠛⠛⠛⠋⢉⡽⠁⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣊⣀⣨⣭⣩⠿⣭⠧⢤⠤⠀⠀⢰⣾⣿⣿⣿⣽⡶⣤⣄⣀⣀⣀⡤⠖⠯⢄⡀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣸⣿⣿⡽⠿⠿⠁⠀⣀⣴⣯⣄⣀⠀⠀⠀⢙⠓⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠛⠛⠛⠻⠳⠶⠶⠾⠿⠿⠿⠛⠋⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⣤⣴
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⣀⣠⣄⠀⠀⠀⣀⢀⣠⠀⠀⠀⠀⠀⠉⠛⢣⣀⣀⡀⣀⠺⢛⣿
⠤⡼⡿⠿⠧⠴⠶⠤⠶⢶⡦⠶⠾⠿⠿⠛⠛⠁⠢⠈⠻⠻⠿⠻⢿⣿⠿⠿⠾⠿⠀⠀⡄⠀⠀⠀⠰⣻⣿⣿⣴⣿⣷⣿⣿
⢠⠀⠁⢁⠀⠁⠀⡀⠁⠀⠠⢀⠀⠀⠀⠀⠀⠠⠀⢐⠂⠒⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠠⢍⡈⠒⠭⣉⠁⠌
⡀⠀⠸⠀⠄⠀⢄⠈⠢⣀⠀⠀⠀⠀⠄⡀⠀⢄⠀⠀⠀⠂⠀⠤⠀⣀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⢌⡑⠢⢄⡉⠒
⠁⡎⠀⡆⠐⡀⠀⢂⠀⠈⠣⡀⠀⠁⠀⠈⠓⢄⠉⠢⣄⠐⠤⠀⠈⠁⠀⠈⠀⠀⠀⡀⠀⠐⠂⠉⡤⠄⠂⢠⡈⠒⢄⡈⠲
⠀⡧⠀⣧⠀⢁⠀⠀⠳⠀⠤⠈⠂⠀⠀⠈⠂⠀⠑⣀⣈⠁⡀⠀⡀⠀⡀⢀⣀⢀⣀⠀⡤⠈⠉⣁⢀⠀⡏⠀⠀⢢⡀⠙⣆
⠀⠈⡀⠈⠀⠀⠀⠀⢀⠉⢀⠀⠄⠐⠌⠁⠀⠀⠰⠎⠉⠐⠈⠁⠉⠘⠤⢨⣤⠄⢤⢤⠠⣶⣤⠈⢟⣁⣉⠛⣫⣌⠯⡶⢋
⣀⠀⠄⠐⠢⠀⠉⠀⠈⣀⡀⠀⠀⠀⠀⠀⠐⠀⢠⠀⠠⠤⠶⠀⢀⠒⠂⠰⠀⠀⠀⢀⠀⠠⠤⠄⠀⢏⣁⣀⠈⠶⠋⣨⡭
⠈⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⢦⠀⠂⠀⠴⣄⠀⠀⣆⣀⡀⠀⢀⠀⠀⣄⣠⡄⠈⢏⣠⡄⠀⠈⠻⠤
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠒⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢉⣁⣀⠀⠈⠁⠠⠆
⠀⠀⠀⠀⠀⠀⠶⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢁⡤⠤⠀⠀⠀⠠⡶⠚
⠀⠀⠀⢲⢤⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠦⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠈⠀⠀⠀⣀⠤⠀⠀⠀⠀⠠⠖
⣤⣀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠤⠖⠒
⠉⠋⠈⠉⠀⠀⠀⠀⠀⡀⠀⠀⠉⠈⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠋⠈⠀⠀⠀⠀⠠⠤⠖⠒
⠛⠂⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠀⠀⠀⠀⠰⠖⠰⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢠⡤⠤⠄