Prisoner Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⢿⣶⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡁⣶⡞⠛⠿⢋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠙⠀⠈⠀⢀⢠⠞⠉⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣲⣤⡤⠄⣒⣂⣥⣦⣤⣷⣶⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠿⢿⡿⠿⠿⡟⠛⠛⠛⠛⠛⣯⣀⣀⣈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣶⣶⣾⣴⣶⣶⣷⣶⣿⣿⣿⡿⢿⡛⡛⠛⠛⢆
⠀⠀⠀⠀⠀⠀⠀⢠⢋⣩⣀⡞⠉⠉⠉⠉⡧⠀⠀⠀⠀⠀⣠⣏⣀⣀⡀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⢿⡿⠿⠿⠿⢿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⣷⣶⣶⣶⣶⣾⣧⣤⣤⣤⣾⣶⣶⣶⣾⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠈⡦⠴⢯⣉⡉⠉⠉⢹⣯⡏⠉⠉⣲⣫⣵⣯⣭⣼⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠉⠉⠉⠉⠈⣍⣓⣵⣔⢋⣉⣉⣉⣉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣼⣶⣶⣶⣶⣶⣶⣶⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠿⠿⠿⠿⡏⠉⠉⠉⠉⠉⡹⠋⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣤⣶⣶⣿⠿⠿⠿⠿⣿⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⣘⣦⣀⣀⣠⠃⠀⠀⣸⣿⡁⢀⣽⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡛⠻⡟⠛⠻⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣾⣟⣶⡷⢤⣀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⢠⡾⠏⠀⠀⠀⠉⠉⠁⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⠀⠀⢲⠀⠀⠸⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⠀⠀⢰⠄
⠀⠀⠀⠘⠒⠀⠈⠁⠠⠀⡆⠀⢰⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⠀⠀⡄⢀⠀⠉⠀⠠⠼
⠀⠀⠀⠀⠀⠀⣀⣠⣄⢀⣷⣀⣾⠤⠋⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣨⣄⢠⡇⠀⣄⡀
⠀⠀⠐⡆⠈⠻⠟⠉⠉⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠙⣿⠖⠂⠔
⠀⠀⠀⠹⣤⣀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠙⠓⢦⣤⣤⣠⡤⢞⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠺⣤⣄⣀⣀⣠⡴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠠⢤⡀⠉⠀⠀⢦⢔⣤⢤⣄⡂⢒⣂⣀⡀⢄⡄⠉⠘⠉⢀⣈⠙⢛⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢨⠆⠀⠀⠀⢨⠑⠉⠀⠈⠀⠀⠊⠀⠊⢻⠀⠀⠀⠀⢬⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠤⠇⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇

⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠈⠹⢿⣟⣿⣟⠛⠛⢿⠓⣰⣶⣶⣃⣀⡀⠀⣠⡒⠀⠈⠀⠀⠀⠀⠚⠛⠉⠀⢤⣤⣄⣀⡀⠀⠀⢂⣀⡀⣐⠂⠀⠀⢀⠤
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠈⠉⠁⢤⠀⠀⠙⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠈⠉⠉⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⢿⣿⣿⣿⣿⡿⠃⠀⠀⣔⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⠤⠗⠶⠶⠦⠐⠒⠶⠿⣷⣤⣄⣸⣿⣿⣿⣿⣧⣴⣶⣾⠋⠐⠓⠠⠤⠦⢀⠄⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣷⡀⠀⠀⢠⠤⠤⠤⠤⠶⠀⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠲⠀⠠⠄⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⣆⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠐⠁⠙⠓⠈⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡈⠉⠛⠙⠂⠂⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠹⠋⠉⢉⠉⠙⣿⣿⣿⣿⣷⣶⣶⣤⣤⠾⠿⠿⠿⠿⠿⠿⠿⠯⡿⠿⡿⢻⣿⣿⣿⣦⣤⣤⣤⣶⣶⠿⠿⠟⢹⣭⠉⠭⠉⠉
⠀⠀⣴⣶⣶⣶⣦⣤⣴⣴⣤⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣦⣤⣤⣴⣿⣷⣶⣶⣶⣷⣶⣶⣶⣾⣷⣶⣤⣤⣤
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟

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

⠀⠀⡆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠒⠺⡗⠒⠒⠒⠒⠒⠒⢺⠓⠒⠒⠒⠒⠒⠒⢺⠒⠒⠒⠒⠒⠒⠒⠒⢺⡗⠒⠒⠒⠒⠒⠒⠒⡟⠒⠒⠒⠒⠒⠒⠒⣿⠒
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠄⠐⠆⢤⣀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⣀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠿⠃⠘⠟⠀⠀⢸⡏⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠚⠀⠀⠀⠀⠀⠀⢸⠒⢄⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠦⡄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢀⡇⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠙⡏⠉⠉⠉⠉⠉⠉⢹⠋⠉⠉⠉⠉⠉⠉⢻⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⡟⠉⠉⠉⠉⠉⠉⠉⣿⠉
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⣀⣀⣇⣀⣀⣀⣀⣀⣀⣸⣄⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣈⣉⣀⣸⣇⣀⣃⣀⣀⣀⣀⣀⣧⣀⣀⣀⣀⣀⣀⣀⣻⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠟⠻⠿⠿⠿⠟⠛⠟⠻⠟⠛⠛⠛⠻⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⡄⠄⣤⠀⠀⢠⠄⠀⢠⡄⠀⠠⣤⠀⠀⣤⠠⠀⣤⠀⠀⢠⡄⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⢸⡆⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⢸⡇⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣇⣀⣯⣀⣀⣸⣄⣀⣠⣇⣠⣀⣧⣀⣀⣬⣀⣀⣼⣀⣀⣸⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⢸⡇⠀⣿⠀⠀⣼⣆⠀⠀⣿⣿⣿⡇⠀⢀⣾⡀⠀⢸⠀⠀⢸⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢸⡇⠀⣿⠀⢀⣿⠇⠀⠀⣿⣿⣿⡇⠀⠈⢿⣇⠀⢸⠀⠀⢸⡇⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⢸⡇⠀⣿⠀⣼⣿⣀⣤⣶⣿⣿⣿⣿⣷⣦⡼⣿⡄⢸⠀⠀⢸⡇⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠀⠀⢸⡇⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⠇⠀⣇⠻⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣶⣿⠀⠀⢸⠀⣻⣿⣿⣿⣿⣿⣿⡿⢯⠉⠀⢸⠀⠀⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⠀⣿⣿⣿⣿⣿⣿⣿⣧⣘⠀⠀⣸⠀⠀⢸⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⢹⣿⣿⣿⣿⣿⣿⣿⣿⡛⠀⠀⢹⠀⠀⢸⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢸⠀⠀⢸⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⠸⠀⠀⢸⠀⠀⢸⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⢸⡇⠀⣿⠀⠀⢸⣿⣿⣿⣿⠛⣿⣿⣿⣿⣟⠀⠀⢻⠀⠀⢸⡇⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⣿⠀⠀⢸⣿⣿⣿⣿⠀⢻⣿⣿⣿⣿⠀⠀⢸⠀⠀⢸⡇⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⣿⠀⠀⢸⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⠀⠀⢸⠀⠀⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⢸⡇⠀⣷⠀⠀⢸⣿⣿⣿⡇⠀⠸⣿⣿⣿⡧⠀⠀⣼⠀⠀⢸⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⣿⠀⠀⢸⣿⣿⣿⡁⠀⠀⣿⣿⣿⡇⠀⠀⢸⠀⠀⢸⡟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⢸⡇⠀⣿⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⠀⠀⢸⡇⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⠧⠤⠿⠤⠤⣼⣿⣿⣿⠧⠤⠤⢿⣿⣿⡷⠤⠤⠾⠤⠤⣼⠇⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠋⢈⠟⠋⠀⢹⣿⣿⣿⣿⠋⠉⠉⣿⣿⣿⣿⣏⠉⠉⢣⡉⠉⠙⠳⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣉⣀⣠⣴⣁⣀⣀⣴⣿⣿⣿⣿⣿⣄⣀⣀⣽⣿⣿⣿⣾⣄⣀⣀⣹⣆⣀⣀⣈⣽⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⡿⢟⣿⠟⠋⠀⠀⣠⠜⠁⠀⠀⠀⡼⠃⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⡿⡄⠀⠀⠈⢳⡄⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠉⠚⠋⠁⠀⠀⠀⠘⠁⠀⠀⠀⠀⠚⠁⠐⠛⠛⠛⠛⠿⠁⠀⠀⠸⠿⠿⠿⠿⠿⠛⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠙⠋⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢰⠀⠀⠀⢴⠀⠀⢠⡄⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⢸⡇⠀⠀⢸⠀⠀⠀⢹⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⢸⡇⠀⠀⢸⠀⠀⠀⢼⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⢿⠀⠀⠀⣿⠀⠀⠘⡄
⠀⠀⠀⠀⠀⢠⠀⠀⡇⡇⠀⡇⢸⡇⠀⠀⢸⠀⠀⠀⡝⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⢻⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢠⢀⠀⡇⡇⢸⡇⢸⡇⠀⠀⢸⠀⠀⠀⣷⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⢸⡀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡈⣸⠀⡇⡇⢸⡇⢸⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⢹⠀⠀⡇⢸⡇⢸⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⠃⠀⠀⠀⡇⠀⠀⢸⡂⠁⠘⠀⠈⡇⠀⠀⢸⠀⠀⠀⣇
⠀⠀⠀⠀⠀⡇⡆⢸⠀⡇⢸⡇⢸⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡆⠀⠀⠀⡇⠀⠀⢸⠇⠀⠀⠀⠠⡇⠀⠀⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡇⡇⢸⠀⡇⢸⡇⢸⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠘⠃⠀⠀⠀⠀⡁⠀⠀⢸⡄⠀⠀⢹
⠀⠀⠀⠀⢀⠇⡇⢸⠀⡇⢸⠀⣸⠀⠀⠀⣿⠀⠀⠀⡿⠀⠀⢸⡇⠀⠀⢰⡇⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⢸⠃⡇⣸⠀⡇⣿⠀⣿⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠇⣿⠀⡇⣿⠀⣿⠀⠀⠠⣿⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⣿⢸⡇⣿⠀⣿⠀⠈⠀⣿⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⣇⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⣏⢸⡇⣿⠀⡿⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠈⢸⠀⡟⢸⠃⡏⠀⡇⠀⠀⢠⡇⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠘⡇⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⡇⢸⢀⡇⢸⡇⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⡄
⠀⠀⠀⠀⡆⠀⠀⡇⢸⢸⡇⢸⡇⠀⠀⢸⡇⠀⠀⢀⡇⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⡇⡀⠀⡇⢸⢸⡇⢸⠇⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠃⠇⢸⡇⣼⢸⠁⢸⠀⠀⠀⢸⡇⠀⠀⢸⠇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠁⠛⠘⠀⣸⠀⠀⠀⣸⡃⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⠿⠀⠀⠀⠸⠃⠀⠀⠀⠐⠇⠀⠀⠀⠇⠀⠀⠈⠇
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠂⠀⠀⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣆⠀⢀⣤⣶⣶⣶⣶⣤⡀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡟⠛⠉⠉⠛⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠚⠛⠛⣿⣧⠀⠀⠉⠛⠛⠛⠋⠁⠀⠀⣼⣿⠛⠛⠓⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠧⠤⠤⠤⠴⠶⠾⢿⣷⣄⡀⠀⠀⠀⠀⢀⣤⣾⡿⠧⠶⠶⠶⠶⠦⢼⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣿⣿⣿⣿⣿⣿⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣦⣀
⠀⠀⠀⠀⠀⠀⡜⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢣
⠀⠀⠀⠀⠀⡜⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢣
⠀⠀⠀⠀⢰⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⡆
⠀⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡇
⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇
⠀⠀⠀⠀⠸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠩⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠶⠿⢛⣀⣤⣾⣿⠟⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡅⠀⠀⢸⣿⣿⣿⡇⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⣿⣿⣿⣧⠀⠀⣿⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣀⣠⣿⣧
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠉⠉⠉⠁⠀⠐⠘⠛⠛⠉⠙⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠙⠃⠀⠀⣤⣴⣶⠶⠄⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠑⡀⠀⠀⠏⠁⠀⠑⡄⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢠⣠⠀⠀⢄⠀⠁⢀⠆⠀⣸⣏⡤⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠈⡇⢉⣈⠶⠦⠀⠀⠈⠈⠁⣀⣀⣩⣽⡷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⣧⣯⣭⣿⣦⣶⣶⣾⣻⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⣹⣿⣿⣍⠻⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⢻⣷⠀⣿⣍⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⡿⣿⡇⢸⣿⣆⣿⣿⣆⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢸⡇⣿⡇⠀⠿⣿⠘⠷⣿⣦⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⣼⢸⡇⣿⣿⠀⢺⣿⡇⠄⢨⣿⣧⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢼⣿⠸⡇⣿⣿⠀⢸⣿⡇⡇⢀⣷⢿⣇⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠙⢀⡇⣿⣿⠀⢸⣿⡟⣿⠞⢡⣾⠏⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢸⡇⣿⣿⠀⢸⣿⣇⢹⣷⡿⢃⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⣿⣿⠀⠀⣿⣿⠀⣿⡖⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⡇⠀⣹⣿⡤⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢻⣿⣿⣿⡍⣿⣷⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢸⣿⣿⢹⣇⢹⣿⡎⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⣿⣿⠀⣿⡄⣿⣷⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢠⣿⣿⡆⢿⣇⢹⣿⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢨⣿⣿⡇⢸⣿⠸⣿⣧⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢸⣿⣿⡇⢀⣿⠀⣿⣿⢼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣋⣸⣿⣿⡷⠈⠋⣉⣉⣴⣴⣾⡷⣫⢵⠂⢠⣤⡀⠀⠀⢀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣼⣭⣭⣽⣽⣿⣿⡿⠐⠞⠉⣱⣿⣿⣿⣟⠳⣷⣶⣷⣟⣗⣶⣴⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠟⠛⠛⠃⠐⠻⠿⠿⠿⠿⠛⠛⠀⠀⠁⠀⠉⠁⠉⠉⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣛⣿⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠈⡇⠀⠀⠀⠈⡏⠀⠀⠀⠀⡏⠀⠀⠀⠀⣯⠀⠀⠀⠀⣿⠀⠀⠀⠀⢹
⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⢀⣷⣶⣿⣿⣷⣧⣄⠀⠀⠀⡟⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸
⠖⠒⠒⠒⢺⡗⠒⠒⠒⢺⡗⠒⠒⠒⠲⡗⠒⠒⢲⣿⣿⣿⣿⣿⣿⣿⣿⡖⠒⠒⡿⠒⠒⠒⠒⣿⠒⠒⠒⠒⢾⠒⠒⠒⠒
⠀⠀⠀⠀⢸⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸
⣀⣀⣀⣀⣸⣅⣀⣀⣀⣸⣇⣀⣀⣀⣀⣇⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣿⣀⣀⣀⣀⣿⣀⣀⣀⣀⣸⣀⣀⣀⣀
⠁⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠈⡏⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠈⣏⠀⠈⠈⠀⣿⠀⠀⠀⠀⢹⠁
⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣏⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸
⠶⠶⠶⠶⢾⡷⠶⠶⠶⢾⡷⠶⠶⠶⠶⡷⢶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⠶⡿⠶⠶⠶⠶⣿⠶⠶⠶⠶⢾⠶⠶⠶⠶
⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⣿⠀⠀⠀⠀⢸
⡀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⣿⠀⠀⠀⠀⣸⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣋⣛⣉⣛⣟⣻⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⠶⠶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣥⣤⠀⢰⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢠⠶⣶⡶⠲⢶⠶⢦⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠉⠉⠙⠛⠋⠉⠉⢿⣿⣷⡀⠀⠀⠀⠀⠀⢸⠀⠏⠇⠀⠸⠀⠀⠀⡄⠀⠸⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⡴⠦⠀⠀⠤⣤⡀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡄⠀⠀⠀⣀⠈⠉⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢺⡇⠀⠀⠁⠀⠀⠈⠛⠀⠀⠀⠀⠀⣸⣿⣿⣿⣤⡀⠀⠀⠀⡇⠸⡄⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠸⢤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⣠⡤⠬⢻⠀⠀⠀⡇⠸⣇⠀⠀⠀⠀⠀⠀⠀⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⣀⠞⢀⣴⣾⣟⣛⣛⣒⡳⠷⠶⢶⡦⢤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠙⠋⠛⠻⢶⣄⠀⠀⠀⠀⠀⠀⠉⠉⡿⠀⠀⠈⢧⣀⣀⠀⠈⠉⠉⠙⠛⠛⠒⠲⡖⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠐⠒⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠶⠶⠶⠦⣴⠧⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣦⣤⣤⣠⣤⣤⣤⣾⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠚⡛⠉⢙⠛⠛⠛⠛⠋⠛⠛⣿⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⣿⣷⣾⣏⣉⠉⠛⢛⣦⣤⣄⣿⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡜⢡⠟⠻⠿⠿⠿⢿⣿⣿⣿⣿⢿⡿⢿⡛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⢸⠁⢸⣦⣤⣄⣀⣀⠀⠀⠀⠀⠀⢀⡇⠈⢷⡘⡆
⠀⠀⠀⣀⡠⠤⠔⠒⠒⠚⠛⠛⠛⠻⢍⣉⣁⣠⣶⣿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣀⣈⠇⠙⢦⣤⣀⡀
⢰⡚⠛⠒⠒⠲⠶⠤⠤⠤⠤⢀⣀⣀⣀⣀⣀⣇⠈⢛⣿⣶⣿⣿⣧⡀⢹⣿⡏⢹⣿⣩⠏⣔⣉⣀⣀⣀⡀⠲⠯⣝⢶⠄⠀⠀⠀⠀⠀⠠⠤⣀
⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡿⠿⣿⡏⠉⢩⡿⠿⢿⠛⠛⠷⠾⠟⠁⠀⠀⠁⣀⢀⠉⠉⠉⠉⠉⠁⣀⣀⣀⠀⠠⠤⠒⠊⢻
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣤⡇⠀⣼⣷⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣯⠁⠀⠀⠀⠀⠀⠀⠀⡟
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣿⡿⢿⠇⠀⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠈⠛⠓⠶⠶⠶⠶⠶⠶⠦⣴⠋⠀⠀⠀⠇⠏⡟⣻⠶⢶⠶⣾⡧⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⢤⣄⣀⣀⣀⠞⠁⠀⠀⠃⠉⠁⠀⠀⠘⡗⠶⠶⠦⠤⠤⠤⠤⢤⣤⣀⣀⣀⣀⡀⢸⣀⣠⡤⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⣀⡠⠦⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠌⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣷
⠀⠘⠛⣿⣿⢿⣿⡿⠯⠿⣿⠯⠍⣩⢋⡽⠋⠀⠀⠀⠀⡼⠯⠉⠹⠀⠀⠀⠀⠠⠿⠯⠋⠉⣉⣉⡉⡉⠉⠁⠈⠭⠍⠁⠉⠉⠏⠉⠨⠟⠁⡀
⠀⠀⠀⠒⠛⠛⠛⠛⠓⠒⠚⠛⠛⠛⣿⣿⡟⠛⠛⠒⠚⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠉⠉⠉⠉⢽⣶⣴⠐⠒⠛⠛⠈⠐⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠠⠤⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠄⠼⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠛⢀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣧⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⣯⠀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠙⠙⠻⠛⠓⠚⠛⠛⠛⢿⡛⠶⠤⠐⠀⠀⠀⠀⠈⠲⠿⠛⣉⡁⠈⠛⠃⠒⠾⠶⠦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣶⠄⢨⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⠲⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠶⠶⠒⠀⣈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠙⠓⠶⠶⠦⠄
⠀⣤⡤⠤⠤⠤⠤⢤⣤⣤⣴⣤⣤⣤⣀⣀⣀⢀⣰⣯⣤⣤⣤⣤⣤⣤⣤⣤⢤⣤⡤⣤⣤⣤⣬⣯⣀⡀⠀⡀⣀⣀⣀⣤⣤⣤⠤⣤⡤⠀⠠⠄
⠀⠀⢀⠀⠐⣚⠀⠀⢛⠿⠛⠛⢿⣿⣿⣿⣿⣟⣋⣉⣩⣍⣉⣉⣉⣈⣉⣑⡛⠉⠘⠃⠾⠛⠿⢛⣿⣛⡛⠛⠛⢻⣟⣉⣉⣉⣀⣿⣋⡀⠉
⠀⠈⠈⠀⠀⠁⠀⠀⠀⠀⠁⠀⠈⠉⣿⣿⡏⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣦⣤⣤⣦⣤⣤⣶⣤⣤⣤⣤⣤⣴⣦⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠿⠿⠿⠿⠿⠿⢿⡿⠿⠿⠿⠿⠿⠿⢿⡿⠿⠿⠿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣀⢸⡇⢀⣀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣴⣿⣿⡇⢸⡇⣹⣿⣿⣦⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⢠⣿⣿⣿⣿⡇⢸⡇⢾⣿⣿⣿⣿⡆⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⡇⢸⡇⢸⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢹⡇⣿⣿⣿⣿⣿⡇⢸⡇⢸⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⢹⣿⣿⣿⣿⡇⢸⡇⢼⣿⣿⣿⣿⡏⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⢿⣿⣿⣇⢸⡇⣻⣿⣿⡿⠋⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢼⡇⠀⠀⠀⠉⠛⠗⢸⡇⠺⠛⠉⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣴⣶⣶⣶⡆⢸⡇⣶⣶⣶⣶⣶⡖⢸⡇⣶⣶⣶⣶⣶⣶⢸⡇⢰⣶⣶⣶⣄⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⢾⣿⣿⣿⣿⡇⢸⡇⠉⠉⠉⠉⠉⠉⢸⡇⠉⠉⠉⠉⠉⠉⢸⡇⢸⣿⣿⣿⣿⡷⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠉⠻⢿⣿⡇⢸⡇⣤⣤⣤⣤⣤⣤⢸⡇⣠⣤⣤⣤⣤⣤⢸⡇⢸⣿⡿⠟⠉⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠂⣿⡇⣶⣄⠀⠀⠉⠃⢸⡇⠻⠿⠿⠿⠿⠇⢸⡇⠹⠿⠿⠿⠿⠟⢸⡇⠘⠉⠀⢀⣠⣶⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⣠⣄⠀⣿⡇⠻⣿⣿⡶⠀⠀⢸⡇⣀⣀⣀⣀⣀⡀⢸⡇⢀⣀⣀⣀⣀⣀⢸⡇⠀⠀⣶⣿⣿⠟⢸⣿⠀⣠⡀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠘⢿⣿⡇⣿⡇⢀⣈⠙⠀⢠⣤⣾⣇⢻⣿⣿⣿⣿⡏⢸⡇⢻⣿⣿⣿⣿⠟⣸⣷⣤⡄⠈⢋⣁⡀⢸⣿⢸⣿⡿⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⢠⣤⡀⠀⠈⠃⣿⡇⣿⣿⡇⠀⢸⣿⣿⣿⣦⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣾⣿⣿⣿⡇⠀⢸⣿⣿⢸⣿⠈⠁⠀⢀⣤⠀⠠⣿⣿
⠀⠀⠀⣿⣿⠀⠸⢿⣿⣷⣤⡀⣿⡇⣿⣿⡇⠀⢸⣿⣿⣿⡿⢰⣿⣿⣿⣗⢸⡇⣾⣿⣿⣿⠸⣿⣿⣿⣿⠀⠀⢸⣿⣿⢸⣿⢀⣤⣾⣿⡿⠇⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠉⠻⢿⡇⣿⡇⣿⣿⡇⠀⠸⣿⣿⡟⠁⠉⠉⠉⠉⠁⢸⡇⠈⠉⠉⠉⠁⠙⢿⣿⣿⠀⠀⢸⣿⣿⢸⣿⢸⡿⠟⠉⠀⠀⢀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠁⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⣤⣤⣤⣤⣤⣄⢸⡇⣠⣤⣤⣤⣤⣤⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⠈⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠛⠛⠛⠛⠛⠃⢸⡇⠚⠛⠛⠛⠛⠛⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⣀⣀⣀⣀⣀⡀⢸⡇⢀⣀⣀⣀⣀⣀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠛⠛⠛⠛⠛⠋⢸⡇⠙⠛⠛⠛⠛⠋⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠐⠀⠐⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⢩⣿⣿⣿⡏⢹⣿⣿⣿⠋⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣾⣿⣿⣿⣷⠌⠁⠀⢤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠛⠙⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠈⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣹⣠⠤⠶⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠦⠤⣈⡏⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⣤⠤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠒⠒⠒⠒⠢⠤⣄⡀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠈⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢉⡗⠒⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢤⣤⡀⠀⣄⣀⠿⠿⠾⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠿⠿⢿⣁⡼⠁⢀⣤⡤⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠛⡇⠀⡟⣽⣤⣤⣤⣠⣀⣉⣒⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣚⣉⣀⣠⣤⣤⣤⡏⣇⠀⣸⠓⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡯⠿⠿⠷⠈⠛⠁
⠀⠀⠀⠀⢀⣤⣤⣶⣦⣤⣄⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠁⠀⠀⠀⢀⣀⣀⡀
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⢳⣄⠀⠀⠀⢻⡯⢿⣿⣿⣷⡮⣿⣿⣴⣾⣿⢷⣖⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⠇
⠀⢠⣾⢾⡿⣿⣿⡿⡟⣿⣿⢻⣿⣧⠀⠀⠈⠉⠊⠛⠛⠛⠉⠛⠛⠛⠿⠿⠟⠓⠿⠿⠛⠻⠿⠿⠿⠟⠚⠻⠿⠛⠛⠋⠛⠛⠛⠟⠋
⠀⢸⣿⣷⣯⣾⣭⣽⣾⣿⣿⣭⣿⣿⢤⣀⣠⡤⢼⠁⠀⠀⠘⠛⠒⠀⠀⠀⠲⠀⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠉⠉⠑⢺⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣄
⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲
⠀⠀⠀⠀⠀⠀⠉⠉⠉⡟⠉⠉⠉⠉⠉⠉⢹⠉⠉⠉⠉⠉⠉⣿⠉⠉⠉⠉⠉⠉⠉⡏⠉⠉⠉⠉⠉⠉⣾⠉⠉⠉⠉⠉⠉⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⣿⣿⣶⣤⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⠛⠛⠉⠙⠛⠛⡟⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⣿⡀⢀⣀⣀⡀⢀⣷⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣿⣧⡛⠛⠛⢛⣼⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡟⡇⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠠⠴⣶⣿⢻⣿⣿⣿⣿⡟⣉⣷⠦⢄⡀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⡰⢻⣆⡀⣈⣽⠛⠻⠿⠿⠟⠛⠉⣧⢀⣤⡟⢆⠀⠀⢿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⡟⢦⡉⠛⠻⣿⣿⣶⣶⣶⣶⣿⠿⡿⠛⢉⡴⢻⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠙⣿⣶⣾⣤⣤⣤⣤⣤⣤⣴⣿⣿⠋⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢠⣀⠈⢹⠉⠛⠛⠛⠛⠉⠉⣟⣠⡄⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⠿⢿⣿⣶⣶⣤⣤⣶⣶⣶⣿⠿⡇⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⣦⣤⣼⣉⡉⠉⠉⢉⣁⣀⣿⣴⡇⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⡉⠙⢻⠛⠿⠿⠿⠿⠛⠛⣿⢉⡇⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣸⣀⣃⣀⣸⣻⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⢟⣇⣀⣘⣀⣀⣸⣀⣀⣀⣀⣀⣀⣇⣀⣀
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨

⠀⠀⠀⢀⡄⠀⠀⡀
⠀⠀⠀⣻⡇⠀⢀⣥⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢄⠀⢄
⠀⠀⣸⣽⣷⠊⠁⡸⠤⠀⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠑⠦⠀⠶⣀⣇⡀
⠀⠀⠩⣼⣿⣷⣄⠀⡡⠒⢐⡩⣄⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⢹⣀⣴⣆⠹⢎⡇
⠀⠀⠀⠙⣟⣦⢿⢮⣀⣞⠻⠆⣛⠉⠸⠏⣹⡶⠶⠄⠀⠠⠀⣤⠄⣤⣰⣣⢤⣤⡴⣿⠿⡃⠙⡦⢾⡅
⠀⠀⠀⠀⠘⣿⣏⣹⣏⡝⠀⠀⠀⠑⢆⠀⠀⠀⠀⠀⠀⠠⠀⠙⠃⠉⡀⠿⠾⢷⡀⠈⢦⡸⠚⠁⢈⣙
⠀⠀⠀⠀⠀⠀⠸⠳⢄⡀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠈⠤⣤⠞⠁⠀⠀⡌⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹

⣶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠿⠿⢿⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠂⠒⠛⠛⠋⠉⠁

⣿⣿⣿⣷⣦⣀⣀⣀⣀⣀⣀⣀⣀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣶⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣷⣶⣄⣀⣀⣀⣀⣀⣀⣀⣀⣴⣾⣿⣿
⠉⠉⢉⣽⣿⣿⣿⠿⠿⠿⠿⠿⣿⣽⣿⣯⠉⠉⠉⠉⠁⢩⣿⣿⣿⡿⠿⠿⠿⠿⢿⣯⣿⣿⡍⠉⠉⠉⠉⠉⣽⣿⣿⣿⠿⠿⠿⠿⠿⢿⣭⣻⣿⡉⠉
⠀⠀⢸⣿⡿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⠂⠀⠀⠀⠀⢿⣿⢿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⠐⣿⣿⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⣀⣀⡠⢛⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣀⣀⣀⣀⠨⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣀⣀⣀⣀⣀⢝⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀
⠶⠶⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠶⠶⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠷⠶⠾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⢀⣀⣀⣀⣠⣖⡭⠛⠚⠛⠯⣝⣢⣀⣀⣀⣀⣠⣚⡽⠒⠛⠛⠛⢯⣝⣦⡀⣀⢀⣴⡮⠛⠛⠛⠻⢭⣲⣄⠀⠀⢠⣞⠟⠛⠉⠉⢛⣽⣦⣀⣀⣀⡀
⠀⠐⠻⠿⠿⠿⠿⣿⢧⡀⠀⠀⢀⡸⢿⡯⠿⠿⠿⢿⡿⣇⣀⡀⠀⣀⡸⠠⣠⠘⠀⠻⡿⢇⡀⠀⠀⠀⣻⢿⡿⠿⠽⢿⡿⡆⠀⠀⢀⣘⠿⡿⠻⠿⠿⠯⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠺⠿⠽⠽⠖⠋⠀⠀⠀⠀⠀⠙⠒⠚⠻⠿⠿⠒⠛⠁⠀⠀⠀⠈⠓⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠙⠛⠛⠛⠛⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦

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

⠀⠀⠀⢹⣄⢲⡲⢡⡇⠀⢠⠘⣴⠢⢼⣠⠂⠀⢧⡙⣏⡙⢣⣸⠀⠀⡇⢸⢨⡇⢸⠀⠀⣣⢰⡌⡴⠀⠀⢂⣱⠞⠲⣀⡇
⠀⠀⠀⡄⡟⢦⣽⡎⠃⠀⠘⢳⣌⣵⣎⠹⡀⠀⡏⢻⣿⣽⡏⢱⡀⠀⡃⠸⣼⠃⢰⠀⠀⢹⠦⠵⠈⠀⠀⡤⣻⣤⣼⡏
⠀⠀⠀⢷⡘⣤⣭⢃⡆⠀⢸⣴⠿⢍⠙⢶⠁⠀⡷⣬⡰⣇⡠⡞⠀⠀⣳⠄⠐⢀⣿⠀⢸⣈⣤⡬⢠⠃⠀⡀⢏⡉⣍⠁
⠀⠀⠀⡎⣷⡌⢢⠙⡇⠀⢃⠈⣷⠺⢷⢈⠂⠀⢁⠹⡟⡟⠇⢸⠀⠀⢱⠀⢸⣿⡏⠀⠀⡋⢸⣴⡏⠀⠀⢱⢾⡀⢠⢖⠁
⠀⠀⠀⡇⢣⡙⠚⡇⡀⠀⠘⢣⣌⣱⣄⠹⠀⠀⡞⢶⣩⣷⡞⠣⠀⠀⣼⣆⢰⡌⠣⠀⠀⢇⢰⠬⡼⠀⠀⠇⡟⠿⠟⠃⡄
⠀⠀⠀⡸⣦⡹⣙⢰⠇⠀⡆⢸⠀⠀⢸⢀⠀⠀⢢⡈⢻⣅⠳⡼⠀⠀⠁⢿⣿⡁⠶⠀⠰⢹⡶⢷⠑⡀⠀⢦⣀⠊⢑⣠⠇
⠀⠀⠀⡁⣟⢳⣾⡆⡇⠀⣧⢸⠀⠀⠸⠉⠀⠀⡄⣹⠷⠾⡗⢛⠀⠠⡀⠈⢸⠃⠀⠀⢰⡘⡤⠤⢠⠂⠀⠐⣽⣶⣾⡏
⠀⠀⠀⢻⡈⠦⠤⢠⡇⠀⡇⢸⠀⠀⢠⠀⠀⠀⡼⣦⡑⢢⡬⢻⠀⠀⡇⠀⢸⠀⠈⠀⠀⠉⠸⢶⣯⠀⠀⡆⠡⠔⠣⠁⡀
⠀⠀⠀⡊⠿⣧⣵⡌⡇⠀⢇⠸⣄⠀⢼⣠⠀⠀⢧⡘⣏⡉⢳⣸⠀⠀⡇⣰⢸⠀⢀⠀⠀⣠⠰⡞⠇⠀⠀⠈⢓⣀⣠⡿⡁
⠀⠀⠀⢃⡹⣌⣙⢁⡇⠀⡜⢳⣬⣹⣄⢹⠀⠀⡏⢻⣷⣼⡏⢹⡄⠀⡇⠙⣼⠀⣿⠀⢠⢹⠇⠺⠈⠀⠀⡄⠏⣻⡋⠇⡄
⠀⠀⠀⡜⣷⡝⢮⡙⡇⠀⢰⡈⠳⢤⠙⣾⠀⠀⡵⣬⡰⣧⡠⢞⠀⠀⠷⠀⡰⢠⡿⠀⠘⣌⠆⡤⢀⠆⠀⠑⣬⠁⢨⡤⠃
⠀⠀⠀⡇⢣⠹⠿⡇⡗⠀⢆⠘⣗⠚⢷⣨⠀⠀⢄⠹⣟⡟⠇⣸⠀⠀⢳⠀⢿⣿⠃⠀⠀⡍⢸⢷⣯⠀⠀⠀⡼⢿⠾⡇⡄
⠀⠀⠀⢸⣆⢳⡒⢱⡇⠀⠜⠳⣌⣱⣄⠹⡀⠀⡞⢷⣽⣷⡞⠃⠀⠀⣞⡄⢠⠈⢆⠀⠀⣡⢰⡆⡴⠀⠀⢂⣰⠊⠳⣀
⠀⠀⠀⡄⣟⢦⣼⡎⠃⠀⠀⠀⠀⠀⠈⠀⠁⠀⢳⡀⢫⣅⠱⡴⠀⢸⠈⣿⢾⠃⣸⠀⠀⢹⠆⠿⠀⠀⠀⡤⣹⣤⣴⡟
⠀⠀⠀⢳⡈⣤⢬⢁⡇⠀⢀⠀⠀⠀⢸⠈⠀⠀⣄⣽⠷⠾⡗⢻⠀⢸⠀⢾⢸⠀⢼⠀⢸⣌⠄⠸⢠⠃⠀⡀⢋⡝⢍
⠀⠀⠀⡎⣷⡌⢲⠙⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⡜⣦⣀⣢⡼⢻⠀⢸⡀⢸⢸⠀⢸⠀⠀⡉⢸⣶⡏⠀⠀⢙⢾⡀⣰⣟⠁
⠀⠀⠀⣇⢣⡙⠛⡇⡀⠀⢠⡈⢶⠂⠤⣀⠄⠀⢧⡘⣏⡉⣓⣸⠀⢸⠁⢸⢸⠀⣌⠀⠀⢧⠸⠈⡬⠀⠀⠇⠟⠻⠛⠃⡆
⠀⠀⠀⡼⣦⡹⣝⢰⡇⠀⠰⢓⣤⣹⣦⠸⡄⠀⡏⢻⣷⣼⡇⢹⡄⢸⡇⢸⣸⠀⣹⠀⠀⢹⡶⢿⠁⠀⠀⢦⣄⠈⢙⣴⠃
⠀⠀⠀⡁⢏⢳⣾⡆⡇⠀⢘⣌⠙⢤⡙⢶⠁⠀⡽⣬⡐⣧⡨⢟⠀⠸⣷⠂⡤⢠⡟⠀⠸⡈⠄⠤⢰⠆⠀⠐⣽⣶⣾⡏
⠀⠀⠀⢹⡈⠢⠴⢠⡇⠀⢨⠈⣿⠒⢿⣀⠆⠀⢇⠹⣟⡟⠇⣸⠀⠀⢱⠀⢿⣿⡇⠀⠀⡉⠸⢷⣯⠀⠀⣇⠡⠜⠡⢀⡄
⠀⠀⠀⡆⠻⢧⣵⡌⡇⠀⠘⠱⣤⣙⣆⠙⡀⠀⡞⢷⣭⣷⡞⠣⠀⠀⣾⣄⠸⠄⢳⠀⠀⣠⠰⡞⢏⠀⠀⠈⢻⣠⣰⡋⡀
⠀⠀⠀⢇⡹⣌⣉⢁⡇⠀⠀⠀⠀⠀⢀⣀⡁⠀⢳⣈⢺⢄⠱⡾⠀⠀⡏⢿⣿⠀⣴⠀⢠⢹⡗⠛⠈⠀⠀⡀⠋⣹⣋⠇⡄
⠀⠀⠀⡞⣷⣝⢶⡙⡇⠀⢀⣴⠀⠀⢸⠉⠀⠀⣄⣽⠿⠾⡗⣙⠀⠀⡆⢸⠇⠀⢫⠀⠘⣌⠆⡴⢀⠆⠀⠐⣮⡁⠨⡴⠃
⠀⠀⢀⣇⣃⠙⢿⡇⣇⠀⡈⢸⡀⠀⢀⢀⡀⢀⡞⣶⣁⣲⣜⣫⡀⠀⣇⣸⣰⣀⣸⡀⠀⣍⣸⣷⣟⠀⠀⣀⣾⣿⣿⣇⣄
⣿⣿⠛⢿⠛⡟⢿⠟⣿⢟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠶⠒⠒⠒⠒⠒⠒⠶⠶⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠉⠉⠉⠙⠓⠦⣄⡀⠀⠀⠀⣀⣤⠴⠖⠛⠋⠉⠛⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⠶⠶⣟⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⡾⠋⠙⠛⠛⠋⠛⢷⠶⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⣀⣤⠴⠞⠋⠉⠁⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠈⠉⠛⠲⠦⣄⣀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⣧
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡄
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠶⠶⠶⠶⠶⠖⠒⠛⠋⠉


⣀⣀⣀⠀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢠⢒⡭⣑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢚⠭⣐⢤⡀
⠀⠀⠀⠀⠀⠸⡘⣄⢈⣧⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⣄⢈⣧⠱
⠀⠀⠀⠀⠀⠀⠙⠮⣙⣢⡙⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣙⣢⡙⢄⣀⣀⠰⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⡑⡦⣝⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡑⡦⠕⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡹⣄⡜⢆⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡹⣄⡔⢆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠭⠦⡑⢤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠭⠦⡑⢤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢢⡽⠲⡝⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢢⡽⠲⡝⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡳⢄⡸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡳⢄⡸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⣋⡙⠋⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⡶⠟⢻⣇⣀⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣼⠋⠀⠀⠀⢹⡟⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠔⠛⢦⡀⢀⡴⠋⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣀⡼⠛⣉⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⢲⠖⠁⠀⣠⡾⠛⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⣵⠾⣿⡁⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣴⠏⠁⠀⠘⢻⡟⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢐⠧⢤⡀⢀⣴⠟⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠁⠀⣠⠗⠋⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⣤⠀⠀⠀⣠⠾⢷⠶⠟⠋
⠀⠀⠀⠀⠀⠀⣠⠖⣩⣶⣶⡎⠁⠀⣠⠞⠁
⠀⠀⠀⠀⢀⡴⢡⡞⠉⠀⠘⢷⣶⠞⣿
⠀⠀⠀⠀⣼⠁⡿⠤⣄⠀⣠⡾⢃⣼⠏
⠀⠀⠀⠀⠟⠃⠀⢀⡼⠞⢋⣴⠟⠁
⠀⢀⠔⠁⠀⣀⣴⣿⣶⠾⠛⠁
⣴⠁⠀⣠⡾⠋
⠹⣷⠞⠉
⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠿⠿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⢀⣠⣤⣀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠠⣿⠁⠀⠙⢷⣄⠈⠳⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠙⢧⡀⠀⣀⣹⣦⣄⣈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠈⠙⢿⣏⣉⣀⣉⡙⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠙⠻⠟⠉⣻⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⣦⣤⣤⡾⢻⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡏⠀⠀⠀⣾⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⣧⣤⣤⣠⡿⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⡏⠈⠉⠉⠻⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠈⠻⠾⠿⣦⡀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠳⣶⡶⡶⠿⣧⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠘⣧⠀⠀⢹⣇⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢹⣆⣀⣀⣿⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⠟⠉⠉⠉⣹⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⣠⡾⠿⠟⠋⠀⣠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⣰⡿⣦⣤⣤⣴⠾⠉⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⣸⠇⢠⡿⠀⠀⠀⢀⡿⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢁⣀⢀⠉⠛⠷⣾⡇⠀⠀⠀⣾⠃⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣶⡛⠉⠉⣳⣶⣄⠀⠙⢷⣄⣼⠇⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠙⢿⡛⠛⢿⣇⠘⣿⣦⣤⣼⠏⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠹⣦⣌⠙⢿⣆⢹⣧⣄⣀⣀⣀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⣟⠙⠷⣤⣿⡄⣿⣍⣉⣿⠟⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⣿⠀⠀⠈⠙⣧⢸⣏⣉⣁⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣻⣦⣤⣤⣀⣻⡄⣿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠟⠋⢀⣀⡈⠉⢻⡇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣠⡾⠛⠻⠿⠿⠛⢁⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⣴⠞⠉⠻⢶⣤⣤⡶⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣴⣿⣥⡶⢶⣴⣟⠁⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⣯⣤⣤⣤⣄⠙⢷⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡀⠀⢀⢀⣠⣿⣦⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⡟⠛⠁⢨⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⢰⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡿⠛⠛⠛⠿⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠈⢿⣦⣤⣄⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣦⣀⠉⠉⢹⣧⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⡟⠛⠛⠛⣿⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣇⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⣆⣀⣴⠟⠀⣸⡇
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠉⠉⠁⣀⣴⠟
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠶⠶⠞⠛⠁
⠀⠀⠇
⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⢤⡔⠲⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠙⣧⠀⠘⡆⠀⠙⣄⡑⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠙⢿⡄⠀⢿⡄⠀⢿⡄⢠⣿⠿⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⢶⣄⠀⠣⠄⠀⣀⣤⣾⣩⠿⠿⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣿⣧⣤⣾⣷⣾⠟⠉⠉⠀⠀⠈⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⢹⠿⢿⡟⠇⠀⠐⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠼⣶⣄⣀⣀⣴⣦⣄⣴⢶⣿⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⣷⡀⢰⣦⠉⠌⠓⣈⠙⠛⠃⢘⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣙⣦⣿⣧⣤⡀⢉⠲⠄⢀⡞⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡔⣻⣼⢷⣦⣙⣻⢿⣷⣤⣾⡇⢸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⣿⡇⢙⠣⠉⠻⣿⣉⠡⡙⢻⣿⣇⠸⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠦⣹⣶⣤⣉⣻⣿⣿⣤⡏⢹⣷⣿⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⡿⠸⣿⡋⠛⠿⠿⣭⣼⡷⣾⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣄⢂⣤⣉⡙⢲⣴⣿⠟⠋⡙⣯⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠸⢷⣧⣠⣽⠋⠉⠉⠳⠁⡠⢺⣋⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣁⣛⣽⣿⣿⡿⡷⣺⣶⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⣛⡀⠤⠿⠿⠟⣹⣿⠙⠙⢿⣆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣕⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠤⠤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠁⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⠿⠋⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⡼⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⢰⠃⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⡎⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⡀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢿⣿
⠀⠀⢸⡇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢻
⠀⠀⠀⡇⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢸
⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠈⠻⢿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⡆
⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⠿⠿⠛⠋⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⢀⣴⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄
⠀⠀⠀⠀⠀⣰⠏⠀⡀⠀⠈⢳
⠀⠀⠀⠀⠀⡟⠀⢨⡟⢃⠀⠀⣇
⠀⠀⠀⠀⠸⣧⠀⢸⡷⢾⣾⡀⢸
⠀⠀⠀⠀⠀⣿⡄⠈⢷⣄⡈⠻⣾
⠀⠀⠀⠀⠀⡟⢻⣄⠀⠀⣹⣦⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢣⠀⠸⣟⢉⡭⢹⡇⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠨
⠀⠀⠀⠀⠀⠈⢦⡀⣸⠃⠀⣺⠃⠀⣿⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠑⡀⠀⠀⠀⠀⠀⢰⠏⡐
⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠸⡟⢀⣰⣿⡄⠑⢄⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⢠⣟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠹⣷⣁⣬⠿⠶⠟⠷⠶⢦⣤⡀⠀⠀⠀⠀⠈⠧⢠⡆⠀⡾⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠈⠛⠶⠖⣾⠞⠶⠶⣤⡀⠙⢶⡶⠤⠤⢄⡀⠀⠁⠀⡀⠀⠀⠴⠞⠛⠋⠉⠁⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⣀⣠⣎⡔⠋⢻⡄⠀⢱⠄⣀⡎⢹⠆⠀⠘⢋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣥⣉⠨⣇⠀⠘⢯⡀⠊⠀⠈⠙⠋⠀⠀⢸⠋⡿⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢧⡀⠈⠛⠦⣄⡀⢀⡀⠀⠆⠈⠛⠧⣄⡀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣾⣁⡵⠀⠀⣀⣀⠀⠀⢠⠊⢳⡄⠀⢹⠨⠝⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⠠⡇⢸⠋⠒⢇⠀⠙⣇⣰⡸⢹⠛⠛⠙⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣒⣶⡶⠶⠛⠃⠀⠛⠁⠀⠁⠀⠉⠓⠲⠶⠾⢧⣀⠈⠻⢶⣦⣴⣮⡶⢦⣄⡀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⣤⡀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣤⣤⣤⣶⠞⢋⣿⣆⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⢿⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠸⣷⡏⠀⣹⡆⠀⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡀⠀⠀⠀⢸⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠀⢠⡿⠅⢀⡟⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠐⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢸⣧⠐⣋⣹⡄⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠻⣏⠉⠉⠙⣦⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣦⡉⠛⢷⡀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡿⣶⣾⡇⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢡⣸⡇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⠉⠁⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠴⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠒⠒⠚⠛⠉⢉⣻⠶⠶⠲⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠛⠉⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙⣄⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠘⣆⣠⣤⡦⢴⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⣴⡿⠋⠉⢙⡎⣇⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣀⣠⡤⠖⢺⣿⠀⠀⣠⠞⡼⠃⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠒⠒⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⣞⣠⠔⠋⠉⠀⠀⠀⠘⣮⣛⣩⠵⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⣄⡀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⠈⣧
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡄⠘⡆
⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣷
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⢸⡀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⢧⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠈⢧⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠶⠦⠤⠤⠤⠤⠤⠤⠴⠖⠚⠉⠁

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

⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠚⠀⠀⠀⠀⠀⢠⠂⠀⠠⢤⣤⣄⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⣠⣠⠄⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣻⡟⠸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢠⣿⢧⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⣠⠆⠀⠀⠀⡴⢛⡇⠀⠀⠀⣾⠀⡦⠂⠑⣾⢏⡞
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠉⠁⠀⠀⠀⠊⢁⣼⠃⠀⠀⠀⣿⣶⠀⢠⣾⣿⣾⡅
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣩⣇⠀⠀⠀⠐⠀⣴⡟⠁⠀⠀⠀⠀⠙⠿⣇⠘⣿⠿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⢉⣿⣿⠄⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠡⡀⠹⣿⣳⠀⠀⠀⠀⠀⣠⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣴⣿⣿⣴⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠱⣴⡉⠉⠓⠀⠀⠴⢯⣠⣤⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⣠⣾⣿⠿⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⢘⣿⣷⣶⣦⣤⣤⣀⣀⣨⡍⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣤⣾⣦⣾⡟⢻⣧⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠋⠀⠈⢻⣿⣿⣏⠀⠀⠈⢿⣄⣆
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠙⠻⠃⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣬⠓⣄⠀⡴⠻⣿⣧⡀
⠀⠀⠀⠀⢸⠀⣾⠟⢤⡀⠀⠀⠀⣡⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣬⢻⣿⡀⠹⣿⡇
⠀⠀⠀⠀⠀⢿⠋⠀⣼⣷⣠⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣇⠀⠛
⠀⠀⠀⠀⣀⠀⠀⣼⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⠘⣯⡈⠢⡀
⠀⠀⠀⢸⣟⠀⣰⣿⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡿⣿⣦⠈
⠀⠀⠀⡿⢿⣿⡿⢿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⢯⠉⠀⠈⢻⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠖⠒⠒⠒⣤⣤⠤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠁⠀⢀⡤⠴⠒⠉⠉⠉⠀⠈⠙⠲⠤⣉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⠀⢀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣬⣻⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⣀⣤⣴⡾⠟⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢳⣴⣶⠿⠟⠋⠁⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⢀⡞⠁⢠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⢠⠏⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⣀⠤⠤⠤⠖⠒⠒⠢⠤⠤⢤⣰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⢠⠏⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⢀⡏⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⣸⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣆⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⡧⠖⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⢠⠃⠀⠀⢯⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠦⠤⠤⠤⠤⠔⠻⡛⡄
⠀⠀⢸⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠙⡇
⠀⠀⠘⡄⠀⠀⠀⠘⣾⣄⣠⡤⠤⠖⠒⠉⠁⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢹⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠈⣇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠘⣆⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠘⣆⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢦⡀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠈⢦⣠⣾⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⢀⣠⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢅⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡏⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠈⠉⠓⠒⠦⣴⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠈⠻⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠛⠷⠤⠤⠴⠒⠒⠒⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⠴⢦⣤⣤⣄⡀⠀⠀⢀⣀⡀⠀⠀⣀⣤⣤⣴⡶⠶⠶⢶⣤⣤⣀
⠀⠀⠀⣠⣤⣶⣶⣾⣿⠻⢷⣶⣾⣿⣭⣶⣶⣶⣿⣷⣶⣶⣾⣿⣿⣿⣿⠛⠿⠿⠿⠿⣿⣿⣿⣿⠿⢿⣿⣿⣶⣍⢳⡄
⠀⠀⣼⣟⣭⣷⣶⠶⢿⣶⠶⣶⢶⣦⣍⣿⣷⠀⠀⠀⣿⢋⣽⣶⣟⣿⣛⣿⣻⣿⣿⣶⡎⣿⣷⠀⠀⠀⠀⠀⢹⣿⣿⣷
⠀⠀⠘⢿⣿⣻⣿⣿⣿⣿⣿⣿⣜⢿⣿⣾⣯⣤⣶⣤⣿⣿⡿⢿⣽⣿⣿⣿⣿⣿⣟⢿⣿⣿⣷⣶⣶⣶⣶⣿⡿⣿⣿⠃
⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠙⠷⠾⣯⣭⣽⣯⣿⡿⠿⠖⠛⠉⠉⠉⠉⠉⠉⠙⠛⠿⠿⢿⣿⣯⣿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠰⣦⣄⣀⣠⣤⣤⡀⠀⠀⣰⠿⣶⣎⣠⣀⠀⠀⣠⡀⠙⣠⣄⡀⣀⠀⠀⣾⠛⠛⣛⢿⣦⣠⣴⡾⠛⢳⡄⠀⢸
⠀⠀⠀⠈⠙⠻⣭⠧⢸⣿⣿⣿⢿⣷⣾⠟⠿⢿⡛⠀⠀⠀⠀⠀⠉⡩⣿⣶⠾⣯⣽⣤⣿⣿⠿⣿⣿⣶⣠⣼⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠘⠷⣦⣼⠿⠉⠀⠉⠁⠀⢠⠧⠀⠀⠀⠀⠀⠀⠀⠈⠿⠁⠀⠀⠈⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠆⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒⠛⠒⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠤⠠⠀⠀⠠⡄⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⢀⢠⢰⡼⠚⠉⠉⠙⠢⡀⠐⢥⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⢠⢐⡨⠞⠁⠀⠀⠀⠀⠀⠀⠹⡀⠠⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢀⢐⣨⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠘⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⢲⠎⠀⡜⣵⣯⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢔⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢁⢀⠄⡘⡀⠀⢸⣯⣦⠀⠙⡆⠀⠀⠀⠀⢀⡤⠞⢁⢠⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠄⢎⡥⠞⠉⢷⠲⣀⠈⢻⡮⠁⠀⢸⠀⢀⡠⠖⠁⢤⣰⠜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠌⣨⠞⠁⠀⠀⠀⠈⢣⣙⢷⣼⠇⠀⠀⣿⢏⢡⠠⣐⠼⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠴⠞⠚⠓⠤⢄⡀⠀⠀⠀⠀⢈⠽⠁⠀⡀⣼⣼⡼⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⣰⢰⣶⡶⣦⣙⣦⣀⡤⠞⠁⢀⠠⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⡀⠄⣪⡷⡻⡳⠤⠬⠿⠛⠋⢁⢀⡀⡜⣊⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠄⢒⡥⠞⠙⠧⣇⣡⢱⢠⢰⠰⠐⣊⡬⠵⠊⠁
⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⢠⠰⣨⠔⠉⠀⠀⠀⠀⠀⠉⠙⠓⡶⣾⡛⡏
⠀⠀⠀⠀⠀⡴⠁⠀⢀⡜⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠻⡲⠃
⠀⠀⠀⠀⢰⠁⠀⢀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⡐⣣⠞⠁
⠀⠀⠀⠀⡇⠀⠀⠼⠐⣇⠀⠀⠀⠀⠀⠀⣠⠖⠉⡀⢆⡴⠊⠁
⠀⠀⠀⠀⢳⠒⣒⡂⠀⠈⠢⢤⣀⡤⠖⠊⠁⡠⢘⡴⠋
⠀⠀⠀⠀⠀⠳⣄⠒⡤⢀⡀⡀⠀⠀⢠⠐⣢⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠥⡮⢝⣁⠥⠖⠋⡁⠀⠀⠀⡀⠀⡀⠀⡀⢀⡀⢀⡀⢀⡀⢀⡀⢀⡀⢀⡤⢀⠀⡀⠀⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠂⠵⢊⡴⢊⠴⢊⡴⢊⠔⣣⠔⣉⠔⢋⠴⣊⠕⣫⠔⣩⠖⣩⠔⣩⠖⣩⠞⣡⠖⡡⠞⣡⢖⡥⢊⠔⣫⠞⡵⣫⠞⡵⢋⠞⡵⣣⠞⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠁⠀⠁⠀⠁⠀⠁⠀⠁⠀⠀⠀⠁⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⣒⣙⠇⠀⠀⠀⠀⠀⠀⠀⠀⢦⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⡦⣄⠀⠀⠀⢀⣠⢴⣖⡀
⠀⠀⠀⠈⢻⣿⣶⣤⣤⣶⡿⢷⣦⣄⣀⣀⣤⣾⣃⠙⢿⣾⣤⣶⣦⣄⠀⠀⣴⡶⠄⠀⠰⣶⣦⣤⣠⣄⠀⠀⢺⣝⣉⡉⠀⣤⣛⣻⣷⣶⣾⡿⠚⠉⠈⢻⡆
⠀⠀⠀⠀⠀⠈⠙⠓⢯⣭⡷⠰⣴⣶⣿⣟⡟⢿⣾⡷⣩⡿⠻⠯⣭⣛⡳⠀⠀⠀⠀⠀⠀⠀⠀⣠⠽⢷⣾⣿⣿⣭⣼⣥⣤⣽⣿⡿⣿⣿⢿⣿⡷⢄⣤⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣄⣈⣉⣿⡟⠋⠐⠶⠞⠋⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣫⣿⠃⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠘⠷⣵⡾⠽⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⢀⡀⠀⢀⣤⣤⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣶⡿⠃⠀⠘⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡽⣤⠚⠉⢣⢀⡄⠀⠀⠀⠀⠀⠀⢠⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣤⣠⡷⠒⠁⣪⠎⠀⠀⠀⠀⠀⠀⠙⢻⢷⣆⡉⣷⣄⣂⡶⠐⠒⠒⠒⠤⣤⣄⣀
⠀⠀⠀⣀⣠⣤⣐⢒⣒⠶⣴⠤⠤⣀⣀⠀⢠⡞⠁⣴⣋⣈⣀⡠⠐⠈⡀⠀⠀⠀⠀⠀⠀⣎⣔⣺⣀⠣⢧⣽⣠⣿⣡⡾⢿⡏⠉⡓⡌⣿⣷⢽⡓⠒⣠⣤⡀
⠀⠀⠀⠀⠉⠉⠉⣿⠻⡅⢸⢳⠶⣤⣍⢲⢾⠇⣾⠉⡋⢩⠟⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠉⠙⠛⠓⠶⠤⣍⣩⣤⣾⣽⣶⣾⣿⣿⣧⣼⠋⣽⠟⠻⠷
⠀⠀⠐⠒⠾⠷⠶⠿⠦⢬⣭⣧⡀⠛⠛⣚⣋⣉⣠⠴⠟⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣛⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠃


⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣶⣶⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠛⠀⠀⠒⠈⠁⠀⠈⠀⠀⠸⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠒⠒⠒⠲⠶⠶⠢⠲⠄⠀⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢤⣤⣤⠤⢤⣄⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠠⠤⡤⠤⣤⣤⣤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠤⠤⣤⣤⠄⠀⠀⢀⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡀⠀⠀⠀⢀⣀⢀⣶⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⣷⣤⣀⣈⠙⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⢍⣀⣄⣀⣈⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢡⣄⣀⣀⣄⣄⣀⣀⡀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠠⡀⣀⣠⣀⣠⣄⣀⣀⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠉⠉⠉⠉⠉⠉⠁⠁⠙⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠚⠂⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠐⠲⠶⠶⠶⠖⠀⠀⢀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠠⣾⣿⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣶⣤⣤⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠯⠙⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡿⢋⣡⣤⣤⣤⡀⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⢁⠀⡀⢀⠀⣀⣀⠀⡀⢻⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠈⠈⠉⠉⠉⠉⠉⠉⠉⠉⣷⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⠀⠘⠒⠛⠛⠛⠓⠒⠀⠀⠀⢉⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠞⣶⠀⠘⠒⠓⠚⠋⠘⠛⠀⠀⢸⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡼
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⣿⣿⡏⠻⠿⠟⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣹
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻
⠀⠀⠀⠀⠀⠀⣿⣿⣇⣴⣶⣦⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢷
⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣸
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻
⠀⠀⠀⠀⠀⠀⣿⣿⡏⠻⠿⠟⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢷
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡾
⠀⠀⠀⠀⠀⠀⣿⣿⣇⣴⣶⣦⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼
⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢷
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡾
⠀⠀⠀⠀⠀⠀⣿⣿⡏⠻⠿⠟⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣇⣴⣶⣦⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣹
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⢼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠉⠋⣿⣿⣿⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⠶⠶⠶⠶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣶⠦⠤⠴⢂⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠉⠀⠀⠀⢀⣠⣴⣾⣿⣿⣧⣤⣤⣴⠶⠖⠒⠒⠂
⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠖⢻⣟⣻⣶⣶⣦⣿⣿⣷⣶⣶⡶⠛⠉⢉⣙⡛⠛⠛⠋⠉⠀⠒⢻⣿⡟⠛⠛⠉⣉⣁⣀⡀⠀⢀⣀
⠀⠀⠀⢀⣤⠞⡫⠏⢱⡤⠛⠭⣁⡣⠬⠥⠪⠦⢝⡿⣿⡏⠠⠚⠛⠿⣿⣿⠟⠛⠒⠶⠤⠶⠾⠿⣟⣓⣁⣀⣉⣿⣉⣉⡭⠉
⠀⠀⣠⠟⢡⢎⣅⡵⡫⢈⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣤⣤⣾⣿⡷⠾⠛⣋⣠⡴⠾⣛⣉⠉⠉⠈⠉⡉⠉⠁
⠀⣼⠋⡐⠇⢆⠞⠖⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠁⠀⠀⣀⣤⣶⡿⠛⣩⡶⠛⢁⣴⣶⣿⣛⣩⠴⠒⠂
⢰⠇⢰⠇⠆⡞⠊⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠿⣿⣤⣾⠏⣠⡾⠿⠿⠿⣛⣩⡤⠤
⣿⠀⢼⠘⢸⠙⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢀⣀⣤⠿⠋⣡⣾⣿⣷⣶⠶⠛⠉
⣿⠀⡃⢘⣾⢈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣀⣀⣀⣤⣾⣿⣿⣏⣁⣀⡤⠒⠁
⢿⡀⢯⢈⢹⢈⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠿⠿⠿⠿⠿⠛⠋⠁
⠘⣧⠸⡀⡙⡎⣄⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⠀⠀⠀⣠⡴⠖⠒
⠀⠘⣧⣓⡀⡹⡄⢌⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣀⠀⠀⣠⣾⡋⣀⣤⠶⠛⠉⠁
⠀⠀⠈⠻⣷⣀⠜⢦⡢⢙⠦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠉⠛⠿⠿⠿⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠈⠛⠶⣷⣙⡪⢅⣦⢛⡲⢢⣄⣀⣤⠶⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⣚⡓⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⠖⠉⠁⠉⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢐⡁⠀⠀⢀⡜⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⡽⠒⢚⣁⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⣢⠊⠁⣠⠞⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡴⢫⠔⠉⠈⠳⢶⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢃⠀⠀⠀⠀⠀⠀⣇⠮⡀⣀⡠⠔⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣀⠀⠈⠳⡀⠘⠀⡠⠔⠒⡲⠊⢁⣴⣉⣥⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠀⠀⠀⢮⣀⢠⡞⢀⡴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢙⡤⠈⠀⠀⡙⠋⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠔⠁⠀⡠⢤⡈⠱⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⢋⡷⠒⠊⢀⡠⠋⠀⢄⠀⠙⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⡤⠊⢀⡴⠋⠉⠒⠉⠁⠀⢰⠀⠈⢧⠀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⠂⣏⣠⡔⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣨⡀⠀⢀⡴⡱⠀⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⣠⣓⣒⡩⠜⠁
⠀⠀⠀⠀⠀⠀⡠⢔⣩⠎⢀⢴⠁
⠀⠀⠀⠀⠀⡜⡴⠁⠈⢊⠇⡼
⠀⠀⠀⠀⠀⡄⠓⠤⠒⣁⠜⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣴⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠋⠉⢉⣭⠟⠋⠉⠉⠉⠙⠛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣇⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣠⠤⠤⠄⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠛⠿⢶⣦⣤⣤⣤⣾⣯⣤⣤⠶⠶⠟⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣦⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠉⠀⣽⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣿⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣀⣤⣴⣶⠿⠿⠟⠛⢛⣿⠟⠛⠛⠛⠛⠛⠿⠿⠿⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣷⠿⠛⠉⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠄⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣠⣶⡿⠿⠿⣿⡿⠿⠿⠿⢷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⢀⣶⠟⠉⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠉⠛⢿⣥⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⢄⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄⡀⣴⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣦⡀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣈⣝⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣟⣦⣄⣀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⣤⣶⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣿⣷⣐⣿⣶⣦⣥⣿⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣥⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⠽⡿⢟⣿⠿⣿⣿⣿⣿⣿⣿⡿⠿⢿⠿⠿⡿⠻⢿⠛⠛⠋⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⢀⣿⠃⠠⣄⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⢸⣿⠀⠀⡄⢉⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⢸⡇⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠤⠤⠤⣀⣀⣀
⠀⠀⢸⣿⠀⢺⣟⠲⢀⣤⡀⠀⠨⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⣐⠒⠶⠄⠀⠀⠀⠀⠀⢰⡿⠒⠒⠓⡲⣌⢳⡤⣍⡒⠦⣄
⠀⠀⠀⣿⡄⢸⣿⣭⡀⠉⠄⡀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠰⠙⠂⠀⠀⠀⠀⠀⠀⣸⡧⠤⠤⠤⠵⣌⠛⠅⡀⣉⠳⡌⢣
⠀⠀⠀⠸⣷⡘⣯⡿⣯⣒⠄⢁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠓⠛⠛⠂⠉⠉⠉⠒⠮⡟⢠⣷⢿
⠀⠀⠀⠀⠹⣧⠈⠿⣲⣿⣛⣦⣄⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⡇⢸
⠀⠀⠀⠀⠀⠹⣷⡀⠙⢹⣿⣿⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⢨⣇⣸⢱⡌
⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠙⣿⣿⣿⡿⣅⣿⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣇⡞⠀⠏⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠙⠙⠙⠻⠿⢿⡶⠻⠿⣖⡶⠴⠶⠓⠛⠀⣀⣴⠾⠋⠀⠀⢀⡤⠶⠚⠛⣉⣉⣉⣉⣉⡝⠛⠚⠶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣄⣀⡀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡶⠟⠋⠁⠀⠀⣠⠞⠁⣠⡴⠞⠋⠉⠉⠉⠉⠉⠉⠛⠶⣤⡈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⣰⠃⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⢻⡆⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣾⡁⠀⢀⡠⠤⠶⠖⠲⠶⠾⢧⣀⠀⠀⠀⣿⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣧⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣼⠇⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣄⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁⢀⡼⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⣼⡇⠀⠀⠀⠀⠀⢀⠀⠠⠖⠚⠉⢀⣠⠔⠋⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⣇⡇⠀⠀⠀⠀⠀⢸⠶⠶⠖⠒⠋⠉⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣠⠤⠖⠊⠁

⠀⠀⠀⠀⢀⡤⠒⠒⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠒⢲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⢤⡀
⠀⠀⠀⡴⢡⠔⠊⠉⠓⢄⠀⠈⠳⡄⠀⠀⠀⠀⢀⠾⣇⠀⠀⠀⠀⣼⠀⠈⢣⡀⠀⠀⠀⠀⠀⢀⡤⠚⠁⣠⣴⣤⠐⠂⠀⢹
⠀⠀⢸⢠⠃⠀⠀⠀⠀⢸⡀⠀⠀⢱⠀⠀⠀⠀⡞⠀⠈⢷⡀⠀⠀⣿⠀⠀⠀⣇⠀⠀⠀⠀⢠⠋⠀⣴⣿⠟⠛⠈⠀⠀⠀⣸
⠀⠀⢸⢸⠀⠀⠀⠀⠀⠸⡇⠀⠀⣸⠀⠀⠀⠀⣇⠀⠀⠈⡇⠀⠀⠹⣇⠀⠀⡟⠀⠀⠀⢠⠇⢀⠔⠉⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠈⢯⣇⠀⠀⠀⠀⠀⠳⣄⣠⠃⠀⠀⠀⠀⠘⣆⠀⢀⡇⠀⠀⠀⠈⢲⡴⠁⠀⠀⠀⢸⢠⠊⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠙⠳⢄⣀⣀⣠⠤⠚⠁⠀⠀⠀⠀⠀⠀⠈⠓⠾⣅⣀⣀⣠⠴⠊⠀⠀⠀⠀⠀⠘⠧⣄⣀⣀⣀⡤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⢀⡴⠿⠥⠤⠤⣽⡉⡶⣄⠀⠀⠀⠀⠀⠀⢠⠖⢹⠿⠻⡏⠉⠙⢢⡀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠀⠌⠁⠍⠳⢦⡀
⠀⠀⢠⠏⠀⠀⣀⠤⠖⠁⡸⢣⡜⣆⠀⠀⠀⠀⢰⣁⣴⠁⠀⢀⣴⣤⣤⠊⠛⡄⠀⠀⠀⠀⡼⠁⠀⠀⠠⠀⠀⠀⠀⠀⢱⡀
⠀⠀⢸⡠⠚⠉⠀⠀⠀⡴⠁⢸⠙⣼⠀⠀⠀⠀⣿⣿⣿⠗⠂⠹⣿⣿⣿⠀⠀⣷⠀⠀⠀⠀⡇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠸⡀⠀⠀⠀⣠⠎⠀⠀⡄⠀⡟⠀⠀⠀⠀⢳⠉⠫⡀⠀⢀⠎⠀⠀⢱⣾⠇⠀⠀⠀⠀⢇⠀⠀⠀⠀⠠⠀⢀⠀⠀⢨⠃
⠀⠀⠀⠙⢄⡠⠚⠁⠀⠀⢠⣣⠞⠁⠀⠀⠀⠀⠀⠳⣄⣸⣿⣿⣦⠠⠄⡺⠏⠀⠀⠀⠀⠀⠈⠣⣄⠀⠂⠀⠄⠀⢀⡵⠋
⠀⠀⠀⠀⠀⠉⠒⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠛⠉

⠀⠀⡠⠐⠐⠠⠀⠀⠀⠀⡀⠐⠂⠤⡀⠀⠀⠀⢀⠠⠀⠀⢠⠀⠀⢀⠤⠐⢲⠤⠀⠀⠀⠀⠠⠶⠆⢀⠀⠀⠀⢀⠤⠒⠢⢄
⠀⠈⠀⠀⠀⠀⠀⠀⠀⡌⠑⡀⠘⡀⠀⠀⠀⡀⠡⠶⠋⠀⢀⠀⠀⡀⠀⠀⠀⠑⢡⠀⠀⣶⡆⠰⣷⠎⠁⠀⠀⠃⠀⠀⠀⠀⠂
⠀⠘⢄⠀⠀⢡⠔⠀⠀⠀⢀⡇⠀⢑⠄⠀⠀⡄⠀⠀⢀⠀⠂⠀⠀⠡⡀⠀⠀⡀⠈⠀⠀⠉⣠⣤⡀⡸⠃⠀⠀⠢⡀⠀⠀⡠⠁
⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⣠⠖⠉⠑⢄
⠀⠀⠀⢠⡇⢀⡖⠄⠀⠆
⠀⠀⠀⢸⡇⢸⡗⠷⣆⢸
⠀⠀⠀⠀⣷⡀⠙⠶⣌⢿⡀
⠀⠀⠀⠀⡀⠙⡖⢒⢻⡆⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢁
⠀⠀⠀⠀⠐⣄⡼⠁⣰⠃⢸⡧⡀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠈⢷⠀⢻⣤⠞⣷⣈⣦⣀⣀⠀⠀⠀⠀⠳⣄⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠙⠾⣭⢥⡤⣤⣉⠛⢦⣤⣀⣀⠈⠘⠂⠉⠀⠀⡤⠶⠖⠒⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⣀⣤⣠⠋⣷⠀⢱⠄⣴⢹⠆⠀⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⢺⡄⠘⢦⡀⠀⠈⠉⠀⠐⣥⡟⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣩⢓⡄⠁⢀⠀⠀⢉⠷⡄⠙⣦⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⢀⠀⣏⡟⠐⢆⠀⢿⣠⠜⠞⠛⠛⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠴⠛⠛⠉⠀⠉⢀⠀⠀⠈⠁⠚⠚⠳⣄⡙⠒⠶⠗⢛⣶⣄⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢸⠄⠀⠀⠀⠀⠀⠉⠙⠻⡛⢿⣤⠚⣧⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢸⣁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢨⡇⢠⡏⠀⡜⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠸⣇⡭⠼⣆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡙⠦⣄⠈⢷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⢶⣼⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⣼⠃⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣀⡠⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠐⠂⠀⠀⠒⠀⢤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠃⠀⠤⢀⣀⣤⣤⣦⣀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣧⡀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠁⠁⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠄⠊⠀⠀⠉⠙⠛⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠰⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠘⢿⣿⣿⡿⠟⠂⠄⠀⠀⢀⣴⣿⣿⣿⡿⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠹⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢤⣤⣀⠀⠀⠀⠀⢀⠸⠛⢛⠩⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠖⠈⠀⠒⠈




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

⠀⠀⠀⠀⡀⠀⠀⢧⣉⡂⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⣠⠤⠤⢐⡶⠖⠒⠒⠠⠤⣄⣀
⠀⠀⠰⣆⡈⠲⢤⣪⠗⢒⡀⠙⠲⢄⡀⠀⠀⢀⣀⣠⠔⠒⠋⢉⣀⣀⡤⢄⡙⢿⡶⠛⢿⡇⣠⠲⣶⠤⠄⡀⠈⠻⣦⣄
⠀⠀⠀⠈⠛⢷⣶⣬⣍⡉⠀⣑⢦⣀⠩⣿⣅⡋⣽⣇⢠⠒⣿⡇⠀⢸⣻⡈⠣⠾⠷⠒⠚⠉⢡⣺⢟⡀⠀⡰⠋⠢⠈⣿⣏⠑⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠈⠙⢿⡶⠛⠢⢼⠏⠱⣸⣿⠤⠼⠛⠋⢐⡫⠯⠭⠾⣛⠶⠶⠒⠒⠒⠚⠉⠙⠂⠔⠦⣭⣇⢆⠸⣴⡯⠋⠉⠙⡷⠖⣛⣉⣉⣀⣐⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠴⣾⠷⠤⠶⠶⠶⠟⠛⠙⠛⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣄⠉⠒⠦⢼⣀⣩⡴⢲⣶⠒⠒⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⣀⣈⣁⣀⡾⠛⠠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠾⠖⠀⠜

⠀⠀⠀⠀⠀⢠⡀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⠀⢀⡀
⠀⠀⢤⡀⠠⣠⡿⠅⠉⠒⢄⡀⠀⠀⠀⣀⡠⠔⠒⠂⣈⣀⣉⢻⣧⠶⢾⠅⡤⣤⡄⣈⠉⠲⣤⡀
⠀⠀⠀⠙⠷⣮⣭⡍⢉⡰⢤⡈⢿⣍⠉⣿⢠⠖⣿⡁⠸⣿⠘⠲⠛⠐⠚⢉⣰⠿⣀⢀⠞⠐⠈⣿⣍⣒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠙⠿⡉⢐⣿⠈⠺⣟⣒⣋⣅⣬⢟⡑⠻⠛⠓⠒⠉⠉⠉⠉⠁⠒⠒⢾⣞⣌⠲⢟⡀⠈⡝⡋⠭⠤⠤⠬⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠙⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠤⣈⠙⠒⠞⠋⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠿⠦⠤⠀⠹

⠀⠀⠀⢤⣀⡀⢀⡤⢒⣻⣾⣷⣶⣦⣤⣀
⠀⠀⠀⢿⣶⣯⣭⣟⣻⡏⠙⠛⠻⠿⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⠙⣿⣿⣿⠃⠀⠀⠀⠀⢀⣞⣻⠿⣆⣀⣀⡀⠀⡴⣪⣵⣶⣾⣿⣶⣿⣯⣶⣤⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣶⣤⣄⣀⣨⢿⣿⣿⣿⣷⣾⣿⣯⣭⣟⣫⠀⠀⠀⠀⠈⢙⣿⣿⣷⣀⣀⡀⠀⢠⣞⣵⣾⠿⠿⠿⠿⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠛⠁⠀⠀⠉⠉⠹⣿⣿⣏⠀⠀⠀⠀⠀⢺⣾⣶⣶⣿⣿⣿⣿⣭⣭⣭⣷⠀⠀⠀⠀⠀⣼⣭⣭⣭⣯⣭⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣯⣿⡿⠋⠈⠉⠉⠉⠹⣿⣿⣧⣤⣤⣤⣤⣤⢼⣻⣿⠟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠟⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣆⠀⢸⡇⠀⢠⣄
⠀⠀⠀⣰⠤⣀⣀⡔⣪⣭⣿⣿⣷⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⠀⠰⠛⠁
⠀⠀⠀⠿⣿⣷⣾⣭⣟⡏⠉⠉⠛⠛⠿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢀⡤⢖⣦⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠈⣿⣿⣟⡁⠀⠀⠀⠀⢠⣾⣟⣛⣧⢤⣀⣀⣀⣜⣵⣾⠟⠛⠀⠀⠰⢾⣿⣦⡀⠀⠀⠀⠀⠀⢀⡠⠤⣀⣒⣒⣂⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣷⣶⣤⣤⣤⣛⣿⡿⠿⢿⣿⣷⣾⣿⣯⣷⠀⠀⠀⠀⠀⢀⠿⠿⢷⣀⣀⣀⣀⣠⣯⣶⡿⠛⠛⠛⠛⠻⢿⣿⣿⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⠛⠋⠀⠀⠀⠀⠀⠹⣿⣿⣧⣀⣀⡀⠀⠀⢹⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⡷⠀⠀⠀⠀⠀⢾⣿⣾⣿⣷⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣾⠟⠁⠀⠀⠀⠀⠙⣿⣿⣷⣦⣤⣤⣤⣴⣞⣽⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠚⠋⠉⠉⠉⣡⠤⠛⠛⠉⠛⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖⠉⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠱⡀⠀⣀⣀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⣠⣿⠟⠛⢮⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⣀⡤⢴⣿⠁⠀⣠⢎⡞⠉⠉⠉⠉⠉⠉⠛⠒⠒⠢⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⣰⣡⠴⠚⠉⠀⠀⠘⣯⣒⣪⠵⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠤⣀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⢳
⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⣼⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢿⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠘⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⢻⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⢳⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠳⡄⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢦⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢤⣄⣀⣀⣀⣀⣀⣀⣀⣤⠤⠖⠚⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠾⠛⠛⠛⠛⠷⠶⣤⣴⡶⠾⠛⠛⠛⠛⠳⠶⣦⣤⠶⠾⠛⠛⠛⠛⠷⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⣀⣤⣤⣴⣦⣤⣤⣄⠀⠙⢷⣤⣤⣤⣶⣤⣤⡾⠋⠀⣀⣤⣤⣴⣶⣤⣤⣄⡀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡟⠉⠀⠀⠀⠀⢠⡿⠉⢿⡄⠈⣿⠀⠀⠀⠀⣿⠁⢠⡾⠉⢿⡀⠀⠀⠀⠀⠉⢻⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠘⢷⣄⡀⠀⠀⠀⠈⣿⠀⠘⣧⣄⣿⠀⠀⠀⠀⣿⣀⣼⠃⠀⣾⠁⠀⠀⠀⠀⣠⡾⠃⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠉⠛⠛⠛⠛⠛⠛⢷⣄⠀⠉⠛⠛⠛⠛⠛⠛⠉⠀⣠⡾⠛⠛⠛⠛⠛⠛⠉⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⢶⣤⣤⡴⠶⠾⠛⠛⠷⠶⢶⣤⣤⣴⠶⠾⠛⠛⠷⠶⢶⣤⣤⡴⠶⠾⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢹⣇⠀⢀⡿⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆⠀⠉⠀⠈⠁⠀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⠶⠶⠶⠶⢦⣤⣤⡶⠶⣦⠀⠀⣴⠶⠶⣤⣤⡴⠶⠶⠶⠶⠶⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⣀⣠⣤⣤⣤⣤⣄⣀⠈⠙⣷⣤⡿⠀⠀⢿⣤⣴⠛⠁⡀⢠⢤⣤⠤⢤⣤⣄⡈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢠⡾⠋⠁⠀⠀⠀⢠⡟⠙⣷⡀⠘⣿⠀⠀⠀⠀⣾⠃⢀⣾⠃⢶⠆⠀⠀⠀⠉⠙⢿⠆⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⣧⣀⠀⠀⠀⠀⠘⣧⠀⠸⣧⣀⣿⠀⠀⠀⠀⢿⣀⣼⠏⠀⣿⠃⠀⠀⠀⠀⣀⣾⠃⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠈⠙⠛⠛⠿⠿⠛⠛⢧⣄⠈⠙⠛⠛⠿⠿⠛⠛⠋⠁⣀⣼⠛⠛⠿⠿⠛⠛⠋⠁⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⢶⣤⣤⣤⣤⡴⠾⠛⠛⠶⢶⣤⣤⣤⣤⡴⠶⠟⠛⠷⢶⣤⣤⣤⣤⡴⠶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⣿⡅⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢹⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⣿⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡻⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣴⠊⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠘⣿⠸⠗⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣣⠏⢧⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢲⡏⢉⡀⡠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⢸⠃⠘⢋⡇
⠁⠀⠀⠀⠐⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄⠚⠘⠋⠀⠀⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⢿⠀⠔⠶⡄⢀⠐
⠀⠀⠀⠄⢀⠀⠛⣄⠈⠂⢀⠀⠀⢀⣠⣤⡴⠂⠀⠀⠀⠀⠀⣠⡶⣲⠟⠁⠀⠙⠟⠃⠈⡆⠠⣰⠆
⠀⠀⠀⡀⠐⠀⢀⡾⠀⠀⠀⠀⠀⠉⠁⡜⠀⠀⡤⡠⣄⠀⠘⠛⠛⠁⠀⢰⠃⠓⡄⠀⠀⣘⣦⣀⠐
⠀⠀⠀⠀⠀⠎⠀⠀⠠⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⢘⡆⠀⠀⠀⠀⠀⠀⠀⡀⠃⡆⠉⠉⠁
⣧⣄⡀⠀⠀⠀⡀⢐⣀⣠⣤⣤⣄⡀⠀⢣⠀⠀⠂⠠⠏⠀⠀⣀⠠⡤⢤⣄⡀⠀⠐
⠶⠯⠿⠛⠛⠉⠀⠀⠀⠀⠉⠙⠳⣮⣑⠦⣧⣄⠀⠀⢀⠀⠀⠀⠉⠙⠋⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣾⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢛⣛⣛⣛⡛⢷⣤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⢟⣿⢻⣏⣩⡿⢻⣾⣛⣷⣚⣛⣻⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣛⣭⣽⠿⣿⣻⠷⠻⠭⠿⠟⠳⠯⠭⣽⣿⣽⠛⠛⣷⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣯⣄⡿⢻⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠯⣝⣺⣣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣼⣷⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣋⣿⣿⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⠛⢁⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣉⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⢻⡜⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⠟⠷⣿⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣧⡿⣦⡤⢫⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⣤⣤⣀⡀⠀⣀⣀⣀⣀⣀⠀⢠⡶⣚⣿⣿⡿⢫⣾⠟⠷⠋
⠀⠀⣠⣴⣟⡛⢶⣀⣯⣾⡉⢛⣶⣮⣿⣶⣿⣾⣿⣿⣋⣷⣿⣾⢏⣯⢘⣿⡿⢹⡇
⣴⢟⣯⠾⣿⣿⣾⣿⣿⣾⣇⣘⣷⣾⣭⣭⣴⣯⣀⣹⣯⣭⡷⢾⣿⣯⠿⠟⠛⠉
⣿⣾⣥⣄⣽⣿⣿⠏⠙⠛⠷⠶⠾⠟⠁⠀⠙⠿⠿⠿⠷⠞⠁⠀⠉
⠛⢿⣿⣿⣋⡉
⢴⣛⣿⣻⣼⢻⡄
⠈⣿⣿⠋⢸⣎⢿⡀
⠀⠸⣿⣿⣛⣿⡾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠙⠛⣛⣛⣭⣤⣶⣶⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⢰⣾⣿⣿⣿⠿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠈⠉⠉⣰⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠳⣿⣷⣄⠀⠀⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⣿⣿⣧⢸⣿⣿⢀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡿⠋⠀⠈⠻⢿⣿⣿⡟⣼⣿⠟⠁
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⣰⣿⣿⠏⠘⠉⠁
⠀⠀⠀⠀⣿⣿⡟⠉⠉⠉⠉⠉⠉⣭⣉⠻⣿⢟⣵⣾⣿⠟⠁
⠀⠀⠀⠀⣿⣿⣿⣿⣶⡀⠀⠀⠀⢻⣿⣷⣾⣿⡿⠛⠁
⠀⠀⢠⣇⢿⣿⡗⣿⣿⡇⣀⡀⠀⠀⠉⣉⣭⣵⡖
⠀⢠⣿⣿⠈⠛⣽⣿⡿⣼⣿⣿⣿⣿⣿⣿⠿⠋
⢀⣾⣿⢇⣤⣴⣿⣿⠃
⣼⣿⣿⣿⣿⣿⣿⠇
⢿⣿⣿⣽⡎⣿⣿
⠀⠉⣿⣿⡃⣿⣿⡀
⠀⠀⣿⣿⡇⣿⣿⡇
⠀⠀⢿⣿⡇⢻⣿⡇
⠀⠀⢸⣿⣇⣸⣿⡇
⠀⠀⠀⢻⣿⣿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣤⣤⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣶⣦⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⠿⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣛⠛⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⣾⣶⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢛⣋⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⣭⣉⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⣿⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣥⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠿⠿⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣴⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢛⣋⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣬⣉⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠻⣿⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⣶⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣷⣦⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠿⠋⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣙⠛⢻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢠⡾⢿⣿⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⢉⣥⣼⣧⡀⠀⠀⠀⠀⣀⣾⣷⣦⣍⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠿⢋⣿⣤⣀⣠⣴⣿⣍⠻⢿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣜⣿⣿⠏⣿⣯⡘⣿⣿⣷⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣥⣘⣿⣿⣧⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⣦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⣾⡗⠛⢶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⡟⣶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉





⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀
⣿⠿⠿⠿⠿⠟⠻⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣟⣛⣛⣿⣿⣛⣛⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠛⠉⣩⡾⠟⠁⠀⠀⠀⠀⠀⠉⠉⣻⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠉⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⣠⡞⠋⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⣰⡟⢿⣄⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧
⠀⢠⡿⠀⠈⢿⣆⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠈⣿⡆
⠀⣾⡇⠀⠀⠀⢿⡆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠸⣿
⠀⣿⠀⠀⠀⠀⠈⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⠀⠀⣿
⠀⣿⣄⠀⠀⠀⠀⢹⡇⠀⠀⠸⣷⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⡙⠷⣦⣀⠀⢸⣧⠀⠀⠀⢿⡄⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⣀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢻⡇⠀⠀⠙⠛⠾⣿⣤⣤⣄⣸⣧⣤⣤⣤⣤⣤⣽⣷⡶⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠘⣿⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠹⣏⠉⠉⠁⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠃
⠀⠀⠹⣷⡀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠈⠛⠷⢶⣤⣤⣀⣀⣀⣠⣤⣴⡶⠿⠛⠉⣼⠏
⠀⠀⠀⠙⣷⡀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠈⢿⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣤⣀⣀⠀⠀⠀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣤⣄⣀⡀⠀⠀⠀⠀⢀⣉⣙⣿⣿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡶⠒⡊⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠚⣀⡜⢙⣁⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⢧⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡎⢠⡯⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢩⠁⠆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣏⠳⠀⠀⠀⠀⠀⠀⠀⠌⢸⠅⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠠⠗⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡄⠘⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⠀⣀⣁⠂⠀⠀⠀⠀⠀⠀⠀⢸⠹⡛⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠔⠊⡠⠿⣿⡏⠐⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢓⡆⢡⣄⠀⠀⠀⡀⠀⠀⡀
⢤⠖⠒⠒⠒⡐⠛⠲⠆⠂⠃⢰⠏⣏⠠⠊⠁⠀⠀⢂⣴⣄⡀⠀⠀⠀⠀⠀⠠⣈⠳⢟⣠⠼⢂⡈⠔⠖⢲⣄⣳⠲⢤⡀
⠠⠴⢋⡝⠙⢻⡋⠶⠆⠀⡈⠁⠖⠁⠀⠀⠀⢀⠦⠶⠐⢆⡙⢢⠀⠀⠀⠀⠀⠀⠑⠢⠉⠐⠃⡁⠠⠰⣄⡉⠛⠲⠇⠴⠶
⠀⠀⠁⠀⠉⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⢲⠉⢅⢄⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠐⠂⠈⠀⠀⠉⠒⠢⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣝⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠦⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠖⠠⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣤⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿