⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠠⠀⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠄⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢀⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠿⣿⣇⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣼⣯⣼⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣀⣟⠛⢽⡃⡀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⢰⡀⣞⠋⠁⢀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣼⣦⡠⡛⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡌⠑⢶⣾⡀⢠⣿⣿⣿⣿⣿⣿⠟⢫⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠛⠳⢧⣤⠂⠀⠀⠀⠀⠀⠀⣾⣷⣿⢠⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠛⣿⣿⣿⣷⣤⣸⣿⢧⣼⣿⣿⣿⣯⡁⠀⠀⢨⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⠀⠀⠀⢹⣧⡀⠀⠀⠀⠀⢀⣿⣿⣧⣾⣋⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡖⣿⡟⢏⠋⠠⢺⣿⣿⣿⣿⣷⡄⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣷⡀⠀⠀⢘⣿⡆⠀⠀⢺⣿⠿⠿⠛⢻⣿⣁⠀⠐⣾⣷⣄⡀⣀⣠⣤⣄⣠⣴⣿⣏⢿⡇⢿⣾⣿⣦⣴⣿⣿⣿⣿⣿⡎⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠹⣿⡿⠿⠦⣄⠓⢙⡷⢂⣿⠃⠀⢀⡁⠈⠋⢠⠀⢀⡈⠋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠀⢐⡀⠀⠀⠙⠆⠠⠁⣿⡆⢀⣿⣥⡆⠀⣀⡱⣈⣿⣖⠈⢿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⠉⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡄⠀⠀⠁⠉⣀⢠⣞⣼⠫⣿⡇⠀⣹⣿⣋⠉⠈⠛⠍⢽⣿⠃⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡙⢿⣍⣡⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣾⣿⣿⡷⢂⢀⣠⢤⣤⣽⣛⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠙⠛⠓⠇⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢲⡀⠀⠀⠀⠀⢸⠿⣿⣿⣿⣿⣻⡿⢏⣴⣿⣿⣿⣃⢀⠉⢝⢻⣧⡈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣯⣤⣤⡀⢀⡼⠀⣿⣿⣿⣿⣿⣥⣾⣿⣿⣿⣿⣿⣿⣿⣎⡇⢻⡿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠐⣷⢰⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠾⠿⠅⠀⣴⡟⠁⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⣀⣿⣦⣀⠘⢿⣿⠋⠉⠛⣿⣿⣿⣿⡿⣿⣿⣿⣿⡫⣤⢿⣶⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⡐⢀⠀⠀⢿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣩⡅⣣⠉⣿⣿⣿⠉⠀⠀⣿⣿⣿⡿⠁⠈⠛⠻⣿⣻⣌⡂⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣬⣿
⠀⠈⠁⠀⠘⠃⠀⠀⢻⣿⣿⣿⣿⠻⢿⣿⣿⣿⣿⣿⡿⣟⣿⣿⣿⣿⠇⠀⠈⠙⠿⣷⣄⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣍⠀⠀⠀⠀⠀⣤⣴⣄⣸⣿⣿⣿
⣐⣂⣤⣄⢀⡄⠀⣠⣾⣿⡿⠿⠟⠀⠀⠈⢿⣿⣿⣿⣶⣿⢻⣿⣿⣿⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣦⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿
⣭⣥⣼⣭⡷⣤⡀⠻⠛⢹⣿⣆⠀⠀⠀⠀⠈⠻⢯⣳⡙⣿⣟⢻⡥⠋⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠁⠹⣿⡇⠀⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣾⣛⣛⣃⡌⣿⣿⠀⠀⠀⠀⠀⠀⢸⡿⠃⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⠀⠀⠀⠀⣰⣾⣰⣿⠟⠁⠀⠀⠀⠀⢻⡇⢉⠀⡠⠀⠈⠙⠻⢿⣿
⠿⢿⣿⣛⡙⠩⠽⢭⠽⣟⠛⠿⠀⠀⠀⠀⢀⣴⣾⠃⠀⡀⢹⣇⢀⣀⣴⡒⠋⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡮⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠈⠳⣄⢠⣀⠀⠀⠈
⣴⣿⠷⡿⣧⣷⣦⣬⣴⣀⣄⣤⣀⠀⢀⡀⣿⠟⣉⠖⠋⠉⠀⢿⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠗⠀⠀⠈⠱⢿⡿⣤⡀
⣿⣿⣷⣶⣿⣷⣿⣯⣿⣿⣼⣷⣦⣤⣈⠃⠈⠉⠈⢀⣀⣄⣠⣤⣯⣤⡀⠀⣀⣤⣤⣄⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡦⠀⠀⠀⠀⠀⠁⢺⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⠿⠀⢀⣬⣴⣷⣿⣟⣫⣭⣭⣿⣿⣿⣿⠟⠋⠁⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠄⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠪⣙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠛⠟⠉⠁⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⡿⠿⠻⠿⠿⠷⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⡀
