⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣀⣤⣤⡀⠀⣤⣾⣿⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣾⣆⣴⡶⣶⣆⠀⠀⣴⡄⠀⢠⣾⠿⠻⢷⣤⡄⠀⣿⣿⡿⠋⢻⣿⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣿⠋⠀⠈⣿⡆⠀⣿⡇⠀⢿⣧⡀⠀⣰⣿⡇⠀⣿⣿⠃⠀⢸⣿⡆⠀⠸⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢹⣿⠀⠀⠀⣿⡇⠀⢻⡧⠀⠀⠛⠿⠿⠟⣿⡇⠀⠛⠛⠀⠀⠈⠛⠃⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⢀⣀⠀⠀⣀⣸⣿⣀⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁⠀⣠⡄⢀⣤⣤⡀⠀⠀⢈⡁⠀⠀⣠⣶⣶⣶⣀⡀⠀⣿⣧⣾⠿⢻⣷⠀⠙⢻⣿⠛⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠋⠙⣿⡄⠀⢸⡇⠀⢸⣿⠁⠀⢈⣿⡟⠀⢿⣿⠏⠀⢸⣿⡆⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣿⣿⣟⣛⣿⣿⣷⣤⣸⣧⠀⠘⢿⣶⣴⣿⣿⣧⠀⢸⣿⠀⠀⠀⢿⠇⠀⠀⠹⠿⠿⠏⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢲⡢⡀⠈⠉⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣊⣥⣿⡿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⢸⣿⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢩⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡿⣟⣸⡆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠙⠛⠿⠿⣯⣷⠀⠀⠀⠀⠠⣤⣤⡀⠀⠀⠯⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣀⣼⡿⠿⣿⢉⣽⣷⣿⣿⣷⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠀⠀⠀⢒⣠⣤⣝⣿⣧⣄⡀⠀⠀⠀⣼⣁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣿⣿⢃⣤⣿⣾⣿⣿⣿⣿⣿⣆⠨⣹⢿⣿⣿⣿⣿⣿⡇⠈⠃⠀⠀⠈⠿⠟⠛⠋⠙⠯⣿⣿⣷⣦⣄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⢿⣻⣽⣿⣿⣿⣶⣻⠈⠻⢿⣿⣿⣭⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢱⠹⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⢷⣶⢺⣿⠿⠛⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣸⠀⢹⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠾⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⢀⠠⠔⠋⠉⠉⠀⠀⠈⠉⠁⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣀⠄⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀
⢠⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⣀⠄⠒⠒⠈⠉⠓⠒⠣⣄⠀⠀⠀⠀⠀⠀
⠸⡀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠈⠀⠀⡴⢫⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀
⠀⠀⠠⡀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣦⣬⣻⣿⣿⣿⣿⣿⣿⣿⣷⣇⡀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠞⠀⠂⢸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀
⠀⠀⠀⠠⠀⠀⠀⠹⣿⣿⣿⣿⢿⣿⣿⠻⠿⣿⡕⣍⠛⢻⢿⣿⣿⣿⣿⣿⠿⠷⠚⠁⠀⣠⡋⠀⠀⠀⠈⠂⠀⣀⠳⠒⢒⣂⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣤⡉⠉⠢⠄⣈⠾⠚⠁⠀⠀⠉⠉⠉⠀⠀⢀⡤⠤⠄⠃⠀⠣⡀⠀⢤⠛⠘⢟⠃⠱⢀⡰⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠢⡀⠀⠹⣿⣿⣿⣿⣿⣿⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠐⠀⠈⠢⢈⡂⠀⢀⢀⣼⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡈⢻⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠀⡰⠁⡠⠀⠈⠀⠀⡀⠀⠑⢄⢲⡋⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠏⠀⠥⠎⠀⠀⠀⠀⡀⢀⠐⠀⠀⠁⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠀⠀⠀⠀⠀⡀⠀⠀⠐⡂⢀⠆⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠄⢸⡧⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣭⠷⠸⡧⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠞⠛⠛⠛⠛⠲⢦⣄⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡰⠋⠁⠀⠀⠀⠀⠲⠂⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀
⠀⠀⠀⠀⠀⡼⠀⠀⠀⢠⠶⠛⡓⠀⠀⠈⠙⠲⣄⠀⠀⠙⢦⠀⠀
⠀⠀⠀⢀⣰⡁⡀⢀⡀⠀⢀⠾⠀⠀⠀⠀⠀⣦⢈⠁⠀⠀⣈⣇⣀
⠀⠀⠀⣾⣼⣿⠁⡞⠀⢠⡎⢴⠄⣆⠀⠀⢀⡇⠿⣞⣇⡀⢿⣿⣸
⠀⢀⡤⣾⣿⡟⢠⠋⢹⠉⠀⡌⠓⠋⠂⠤⠚⢀⡄⠈⣛⡇⣸⣿⣿
⠀⠘⣷⣽⣿⣷⡞⠀⠘⠲⡌⠃⠀⠀⣀⣀⡀⠘⠃⢀⡇⣡⣿⣿⣧
⠀⠀⠀⢿⣮⣽⠀⠀⠀⠀⡅⠀⠀⢨⠀⠀⡹⠀⠀⠸⡞⢻⣭⣵⠾
⠀⠀⠀⢸⠉⠛⢳⠀⠀⠀⢇⠀⠀⣀⣉⣉⣁⡀⢀⣤⡇⣠⠞⢸⠀
⠀⠀⠀⡎⠀⠀⡞⠀⢀⠤⠚⠉⠉⠉⠉⠉⠉⠉⠛⢾⠛⠁⠀⠸⡀
⠀⠀⢰⠃⠀⠀⡇⠀⢯⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢧