⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣤⣴⢖⣊⠙⠧⠤⢌⣒⠦⢬⣑⠯⣙⠿⢶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⠶⣉⠉⠒⢌⡑⠢⣉⠑⠢⠤⣈⠙⠢⢄⡙⢦⡉⠒⢷⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣸⣖⡢⣝⠢⢄⡉⣲⣌⣙⣢⣤⡀⣙⣲⣠⣨⢗⣯⢵⣬⣻⣇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⠃⠈⠛⠺⣽⡍⠛⢻⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀
⠀⢠⠯⠒⠒⠒⠒⠪⢃⡀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀
⠀⣯⣦⣤⠶⠶⢦⣄⠀⡇⠀⢣⢀⣴⣚⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀
⠀⡰⢃⠎⣿⣟⡹⠞⠻⠅⠀⢸⡻⣦⡭⢽⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀
⢰⠃⡜⠀⠀⠀⠀⠀⠀⠀⣀⠼⣽⠦⡇⢠⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀
⡏⠀⠀⠐⢦⢀⢀⠀⠀⡜⢇⠿⠬⢯⣣⡴⡃⠀⠀⠀⠀⢀⣀⣀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀
⢈⣿⡝⣟⠯⣰⣿⢯⠾⢷⢎⠳⠷⡘⢝⢇⠉⠉⠉⠉⠉⠉⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠛⢿⣭⣭⣵⣻⢷⢳⢎⢣⢢⢲⢝⣞⢹⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣿⡏⡇⡏⣎⢾⢌⠎⣆⡆⠫⡓⢤⡟⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣮⣇⢡⢣⢷⣹⣫⠘⡞⠺⡀⠲⣒⣺⠃⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡷⡼⢾⢣⠏⢧⢻⣌⢫⣺⡬⠶⠋⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣧⣤⣀⠀⠀⠀⠀
⠀⢿⣤⡏⠁⡿⢸⣨⡭⡟⢹⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⡀
⠀⠀⠉⠒⠓⠛⠚⠁⣴⣷⣡⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⡄⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⢠⠟⠀⢀⣠⣄⡀⠀⣀⣤⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠏⠀⠀⣿⣥⣌⣿⠐⣧⣄⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⢿⣄⠀⣠⢰⣄⠉⣻⠂⣤⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠉⠉⠁⠈⠈⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⣀⣤⣴⢖⣊⠙⠧⠤⢌⣒⠦⢬⣑⠯⣙⠿⢶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⢀⠶⣉⠉⠒⢌⡑⠢⣉⠑⠢⠤⣈⠙⠢⢄⡙⢦⡉⠒⢷⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⣸⣖⡢⣝⠢⢄⡉⣲⣌⣙⣢⣤⡀⣙⣲⣠⣨⢗⣯⢵⣬⣻⣇⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢰⠃⠈⠛⠺⣽⡍⠛⢻⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀ ⠀⢠⠯⠒⠒⠒⠒⠪⢃⡀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀ ⠀⣯⣦⣤⠶⠶⢦⣄⠀⡇⠀⢣⢀⣴⣚⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀ ⠀⡰⢃⠎⣿⣟⡹⠞⠻⠅⠀⢸⡻⣦⡭⢽⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀ ⢰⠃⡜⠀⠀⠀⠀⠀⠀⠀⣀⠼⣽⠦⡇⢠⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀ ⡏⠀⠀⠐⢦⢀⢀⠀⠀⡜⢇⠿⠬⢯⣣⡴⡃⠀⠀⠀⠀⢀⣀⣀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀ ⢈⣿⡝⣟⠯⣰⣿⢯⠾⢷⢎⠳⠷⡘⢝⢇⠉⠉⠉⠉⠉⠉⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠈⠛⢿⣭⣭⣵⣻⢷⢳⢎⢣⢢⢲⢝⣞⢹⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⢀⣿⡏⡇⡏⣎⢾⢌⠎⣆⡆⠫⡓⢤⡟⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⣮⣇⢡⢣⢷⣹⣫⠘⡞⠺⡀⠲⣒⣺⠃⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⡷⡼⢾⢣⠏⢧⢻⣌⢫⣺⡬⠶⠋⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣧⣤⣀⠀⠀⠀⠀ ⠀⢿⣤⡏⠁⡿⢸⣨⡭⡟⢹⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⡀ ⠀⠀⠉⠒⠓⠛⠚⠁⣴⣷⣡⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉ ⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⡄⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⢠⠟⠀⢀⣠⣄⡀⠀⣀⣤⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠏⠀⠀⣿⣥⣌⣿⠐⣧⣄⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⢿⣄⠀⣠⢰⣄⠉⣻⠂⣤⡄⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠉⠉⠁⠈⠈⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀