⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠟⠁⢆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢠⠈⡆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⡄⡀⢸⠀⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢲⡇⢧⣧⢢⢸⣀⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⡆⡟⡎⣼⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣟⣶⢋⣴⣿⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣻⣿⣿⣿⣿⣿⣿⠿⣝⣞⣻⢿⣶⣿⣿⠻⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⣿⣭⣟⣻⣽⣯⡾⣿⠝⠙⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣷⣬⣵⣾⣿⣿⣿⣿⡿⠟⠿⢳⢤⠄⣀⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠛⠻⢿⣿⢙⢀⠤⠒⠉⠻⣗⣦⠀⠉⠉⠑⣆⠀
⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡟⠀⠀⠀⠀⣘⢚⡅⠀⠀⠀⠀⠘⣞⣧⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠃⠀⠀⠀⢸⣿⣿⡗⠀⠀⠀⠀⠀⣿⣿⣦⣄⡙⠋⡝
⠀⠀⠀⠀⠀⠀⣾⡷⣿⣿⠟⠁⠀⠀⠀⠀⠈⠁⠉⠁⠀⠀⠀⠀⠀⠹⣿⣿⣿⡧⠀⣟
⠀⠀⠀⠀⠀⣼⣿⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠁⢀⣿
⠀⠀⠀⠀⢰⣿⣿⣿⣧⢃⠄⠀⠀⠀⠀⢀⡌⠀⠀⡀⠀⠀⠀⠀⠀⢀⡼⣿⣿⣆⡀⣿
⠀⠀⠀⠀⣾⣿⣿⣿⡿⣞⠦⡴⣤⢶⣶⢺⣥⣤⡀⠳⣦⣄⣀⣀⡴⣾⣳⣿⣿⣿⣇⣻
⠀⠀⠀⣼⣿⣿⣿⣿⠃⣨⣿⣷⣟⠉⠁⣿⣿⣾⡗⠀⠙⢮⣜⣲⣹⡶⢻⣿⣿⣿⣿⣿
⠀⠀⢰⡿⣿⣿⣿⣷⣄⠘⣷⣯⣹⡗⠀⢸⣿⡟⠁⢀⣤⠾⢼⣻⡷⠀⢸⣿⣿⣿⣿⡇
⠀⠀⢿⣿⡿⣿⣿⣿⣿⣷⢿⠉⠛⠇⠀⠀⡗⠀⠀⣼⠽⡶⢿⠏⣀⣤⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠈⠻⢿⣝⡿⣟⡟⠁⠀⠀⠀⠀⣼⠃⠀⢠⠀⠀⠁⣼⣿⣿⣿⣿⣿⣿⠿⣿⠀
⠀⠀⠀⠀⠀⠀⠉⣱⣏⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠆⠀⠀⢹⢀⠩⣉⣟⣻⣷⠿⠞⠀
⠀⠀⠀⠀⠀⠀⣴⢋⣼⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⢺⡗⠛⠉⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⠟⡬⢸⡇⠀⠀⠀⢠⡟⠃⠀⠀⠀⠀⢀⠔⠁⣧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⢻⡱⢊⡍⡗⢠⠀⠀⣾⠁⠀⠀⠀⠀⢠⠋⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣿⠰⢿⡱⣌⢷⡀⠆⢠⡏⠀⠀⠀⠀⡰⠃⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⠿⠁⡀⠀⠉⠛⢾⣇⡆⣸⡹⢀⠔⣠⠞⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀
⠀⣰⠿⠂⠄⠀⠀⠀⠀⠈⢻⣥⣿⠱⣎⣾⠗⠊⠉⠉⠉⠀⠀⡼⣷⠀⠀⠀⠀⠀⠀⠀
⢀⣿⢂⠄⠠⠀⠀⠀⠀⠀⠈⣧⣛⣷⠟⠁⠀⣀⣤⢤⣤⣤⣤⣙⡼⡀⠀⠀⠀⠀⠀⠀
⣼⣿⡔⠀⠀⠀⠀⠀⠀⠀⠀⣸⡜⠁⠀⠀⠀⢁⣀⡀⠀⠀⠉⠻⣿⡇⠀⠀⠀⠀⠀⠀
⣿⣸⣷⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠞⠛⠉⠉⠙⠻⢶⣄⠰⣙⠇⠀⠀⠀⠀⠀⠀
⣿⡰⢻⣇⠀⠀⠀⠀⠀⠀⣾⠇⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠹⣷⡿⠀⠀⠀⠀⠀⠀⠀
⣿⡍⠈⢿⣷⡄⠀⠀⢠⣾⠃⠀⡰⢣⠃⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀
⢸⣆⠉⠀⠻⠁⠀⢠⣿⠃⠀⢠⠁⡎⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠘⣿⣆⠀⠀⠀⢠⣟⡟⠀⢠⠃⢰⠃⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢹⢿⣧⡀⢠⡟⣸⡇⢀⡎⠀⠘⢄⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⡎⠙⢷⣿⣰⠱⣿⡜⠀⠀⠀⠀⠑⣄⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢹⡀⡴⢿⢿⢣⣿⠀⠀⠀⠀⠀⠀⢸⢣⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢻⢿⡜⢎⡜⣺⣧⡀⠀⠀⠀⠀⣾⣹⣯⣿⣹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢳⣻⠜⣸⡙⡾⣿⣦⣔⣤⣼⣇⣧⣷⢇⡳⡌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⢷⣻⡶⡟⣟⡽⣻⢿⣿⣿⣿⢻⣟⡮⠴⣉⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⣧⠛⡏⠹⡶⢩⣛⣿⣿⣿⡎⢿⡜⣱⠢⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⢇⢹⠀⢱⠀⠉⠛⠿⢿⣿⣎⠧⢥⣋⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⣾⡁⠀⠣⡀⠀⠀⢸⡉⢿⡞⡲⢼⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠈⠢⡀⠨⣇⠘⢿⡵⣫⣞⣿⣵⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⢸⡀⢹⡄⠘⣟⡧⢿⡧⣿⠇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⢤⢶⣣⡞⢠⠏⠰⡀⣿⡿⣟⡼⣽⡃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣏⡇⡞⠀⠁⠹⣿⡳⡇⣿⢺⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠏⢸⡇⠀⠀⠀⢹⡗⣍⡳⣽⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠂⠀⡀⠀⠀⠀⠘⣯⢧⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣥⠀⠀⠀⠀⠀⠀⢹⡾⡹⢿⣇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⡀⠀⠠⠀⠀⢸⣳⣹⣿⠏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠁⠀⠀⠀⠀⢸⣯⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡄⠈⠀⠌⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⢁⠐⢈⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠈⠄⠂⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠌⢂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⠎⡤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡈⢧⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢯⡎⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣛⡕⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⡇⡼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢄⣸⠡⢳⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠡⠇⢏⠰⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠩⡙⣘⠛⣷⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠤⢃⡁⠦⡑⠤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣤⣧⣼⣤⣷⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀