⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀ ⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀ ⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀ ⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀ ⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀ ⢰⡿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇ ⠘⠃⣿⣿⣿⣿⣿⡿⠿⠓⢚⣟⠛⠛⠻⡏⠉⠉⣽⠛⠛⠿⣿⢿⡿⢿⣿⣿⣿⣿⣿ ⠀⠀⣿⣿⣿⣟⠋⢧⠀⠀⠈⠁⠀⢀⡼⠃⠀⠀⠙⣄⠀⠈⠋⠀⠁⣠⠋⢻⣿⣿⣿ ⠀⠀⡜⠀⠙⢿⠀⠀⠑⠲⠤⠤⠒⠉⠀⠀⢸⡇⠀⠈⠑⠦⠤⠤⠖⠁⠀⣼⡿⠛⡏ ⠀⠀⠘⢄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⠇ ⠀⠀⠀⠀⠉⠁⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠄⠀⠀⠀⠀⠀⠀⢠⠖⠒⠚⠁⠀ ⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⣀⣬⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣧⡀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀ ⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀ ⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀ ⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀ ⠀⠀⢸⠀⠉⠙⡟⢿⣿⣿⣿⣿⣿⣿⣿⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⢹⠀⠀ ⠀⠀⢸⡀⢀⡼⠁⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⡔⡀⡼⠀⠀ ⠀⠀⠀⠉⠉⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠉⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠀⢹⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⠀⠻⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⢍⠿⠉⢱⠀⠀⠀⡇⠀⢰⣋⣐⢅⡅⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠑⣦⡜⠀⠀⠀⢹⡒⠁⠀⡄⠙⡶⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠷⠴⠖⠋⠀⠀⠀⠀⠀⠙⠲⠤⠵⠔⠁⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠹⣄⠀⠀⠀⠀⠀⠀⠀ ⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟⢏⠉⠃⠀⠀⠀⠀⠀⠀ ⠤⠤⠤⣠⡤⢄⣀⣀⡠⠤⠤⠤⠤⠤⠤⠤⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⢤⠤⠤⡤⢤⣤⣤⣥⠬⢷⡄⠀⠀⠀⠀⠀⠀ ⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠿⢳⡀⠀⠀⠀⠀⠀⠀⠀ ⠖⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠶⠶⠴⠶⠶⠶⠶⢶⠶⠶⠶⠶⠶⣶⣶⣶⣗⢶⡶⠶⠶⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⡶⣖⠛⠂⠀⠀⠀⠀⠀⠀ ⡀⣀⣀⢀⢀⡀⠀⠀⠀⣀⣀⣀⣀⣀⡀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣬⡿⠍⠉⣯⣍⡁⢉⡉⠛⠀⡉⠉⠉⡛⠒⠒⣺⣇⣀⣀⣀⣀⣀⣋⣘⣦⡀⠀⠀⠀⠀⠀⠀ ⣛⣛⣛⣛⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣹⣿⣛⡋⣩⣤⣴⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣦⣷⣷⣤⣝⠓⠦⣧⣼⣿⣭⣿⣭⣭⡁⠀⠀⠀⠀⠀⠀ ⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⣭⣟⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣺⣷⠶⠶⠿⠶⢷⡀⠀⠀⠀⠀⠀⠀ ⠒⠒⠒⠒⠒⠊⠙⠒⠚⠉⠉⠉⠉⠙⠒⠂⠒⠒⠒⠒⠚⠉⠉⠓⠒⣋⣫⠽⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢯⣮⡷⡠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⣿⣿⣿⡿⣿⠟⣹⡟⢹⡟⠻⣿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡴⣿⠛⡈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠋⠋⠀⠃⠀⠉⠀⠈⠀⠀⠛⠈⠋⠀⢻⡿⢿⣿⣿⣿⣿⣿⣿⣿⡃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠣⢵⢾⠿⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡟⣠⣴⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⣿⣿⣿⣿⣿⣯⡁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣠⡖⠋⠉⠁⠘⠧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠉⣀⣀⣀⣈⠹⣦⠀⠀⠀⠻⠟⠛⠛⠻⢷⣆⠀⠀⢿⣿⣿⣿⣿⣯⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⢀⡸⠋⠀⣠⠞⠉⠉⠉⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢼⣅⡟⢀⣨⣧⠈⠀⠀⢠⡞⣩⡍⠒⠦⡄⠉⠂⠀⢸⣿⣿⣿⣿⡏⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢀⠞⣁⡤⠚⢁⡤⠶⠒⠒⠒⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠉⠩⠟⠁⣸⠀⠀⠘⢿⡒⠲⠶⣶⠇⠀⠀⠀⢸⢟⣩⢿⠉⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⡴⢋⡞⠁⢠⡴⠊⠀⠀⢀⣀⡤⠦⠬⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠠⠶⡤⠀⠀⠀⠀⢠⢾⠋⢸⠁⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⡇⠘⠀⢀⡆⠁⠀⠀⡠⠞⠉⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⣧⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡴⢢⣾⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⢹⢠⠀⠘⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⢈⡴⠋⣿⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠈⢸⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢰⡖⠦⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢿⠀⠀⡧⢸⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠘⠳⠤⠾⠇⠀⠀⠀⠀⠀⠀⣠⡞⣹⢱⢹⠀⠀⣗⢸⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⣠⠞⢱⠃⢸⣼⢸⠀⠀⡇⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠁⡆⠀⠀⠀⠀⠀⠀⢀⢾⠶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁⠀⡞⣠⡟⡟⣾⠀⠀⡇⢸⡇⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣄⢹⠀⠀⠀⢀⣠⠔⣺⠽⠚⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⣿⡏⠉⠉⠛⠉⠉⠀⠀⠀⠀⢠⣣⠛⡠⣷⠙⣧⣀⡇⢸⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣈⠛⠖⠒⣋⡥⠖⠋⠁⠀⠀⠀⣇⣀⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⣸⠙⣝⢦⣀⠀⠀⠀⠀⣀⡤⠖⢋⠤⠊⢸⡟⠀⢹⡿⣿⢾⣄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣏⣄⠀⠀⠉⠉⠙⠳⠦⠤⣤⠖⠛⠛⣻⠛⠁⠀⢰⡇⠀⠈⢦⡈⢓⣶⢖⡉⢁⠤⠊⠁⠀⠀⢸⠁⠀⠀⢷⠈⢦⠈⠙⠻⢄⡀⠀⠀⠀⠀⠀⠀⠀ ⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠈⣧⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⡞⡇⠀⠀⠀⢙⣿⣘⣯⢿⡁⠀⠀⠀⠀⠀⡞⠀⠀⠀⠸⡆⠈⠀⠀⠀⠀⠉⠙⠒⠲⠶⠤⢤⣄ ⠙⠳⠶⣤⣀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠸⡄⣸⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⢀⡇⣇⠀⠀⢤⡞⠀⠀⠀⢰⠹⣄⠀⠀⠀⠀⡇⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸ ⠀⠀⠀⠀⠙⠙⠶⣤⡀⠀⠀⡏⠀⠀⠀⡇⡏⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⢸⠠⢹⠀⣰⠓⡇⠀⠀⠀⣾⠁⠘⢦⡀⠀⢸⠃⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃ ⠀⠀⠀⠀⠀⠀⠀⠈⠋⠳⢦⡇⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠙⠒⢤⣀⠀⣀⡀⢾⣠⢾⣴⠋⠀⡿⣄⣠⣴⡧⠔⠒⠺⣧⣀⢾⡀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡆⠀⠀⢰⠃⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠋⠀⣿⢣⠈⠁⠑⢤⢧⡇⡇⢸⠀⠀⠀⠀⣸⡏⣾⠃⢀⣠⠤⠴⠖⠛⠀⠀⠀⠀⠀⠀⡇⠀⢀⡏⠀⠀ ⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠈⠻⣇⡀⠀⠀⠀⠀⠀⢠⠖⠋⠀⠀⠀⠀⣧⠈⣧⠀⠀⣸⠂⣿⠁⢸⠀⢀⡠⠟⠁⢱⣿⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡼⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠈⠓⠤⡀⠀⠀⠀⠀⠀⠀⢹⣄⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠈⢧⣀⡗⠀⠈⠀⠸⠀⠈⠀⠀⢀⡞⡇⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⡇⢠⡇⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢹⠁⠀⠀⠀⡆⠀⠀⠀⢀⡞⠀⡇⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠘⢳⣄⢸⡀⠀⠀⠀⠀⠀⣧⠀⠀⠀⡞⠀⠀⠀⠀⡇⠀⠀⢠⡞⠀⢠⠇⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⣇⡏⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠈⠉⠻⣇⠀⠀⠀⠀⠀⣿⠀⠀⢀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣰