⠀⠀⠀⠀ ⣀⣯⣾⣿⡿⢟⣿⠛⠉⠩⠁⠀⠀⡟⠁⠀⣀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⠟⠁⣐⡮⠁⠀⡐⠀⠀⠀⠀⢰⣅⠈⠀⠒⠄⡀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠋⠀⠀⠀⡬⠁⠀⡐⠀⠀⠀⠀⠀⠀⠟⢂⠀⠀⠀⠈⠂⡀⠀⠀⠀⠈⠙⢿⣿⣻⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠏⠀⠀⠀⠠⡙⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠐⡈⠑⡀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢗⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⠁⠀⢀⠈⠀⢠⠁⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⢄⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠦⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠂⠀⠀⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢂⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣗⡀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣨⠁⠀⠀⠄⠀⠀⢀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡰⠀⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡘⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣣⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⢃⠀⠀⠀⠀⠀⠸⢆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢁⢀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠋⡀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡈⠀⠀⠀⠀⠀⠀⠰⢰⣿⡆⠀⠀⠀⠀⠀⠀⠄⢂⠀⠐⡀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠈⢡⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢨⠀⠀⢀⠁⠀⠀⠀⠀⢀⠀⠆⣿⣿⣧⠀⠠⠀⠐⠀⠀⠘⡀⠠⠀⠐⣀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠈⡄⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠀⠀⠀⠀⠀⠸⠰⢸⣿⣿⣿⣆⠀⢂⠀⠀⠀⠀⠐⡀⠐⠀⠐⠑⠀⠀⠈⠄⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢁⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡊⠀⠀⡀⠀⠀⠀⠀⠀⢐⢳⣾⣿⣿⣿⣿⡌⠌⣆⠀⠀⠀⠀⠐⠀⠈⠂⠈⢀⠢⡀⠀⢂⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡅⠀⠀⡇⠀⡀⠀⠀⠀⢰⣧⣿⣿⣿⣿⣿⣿⣄⠚⢆⠀⠀⠀⠀⠈⢄⠀⠑⠀⠁⡀⠀⠀⠂⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡆⠀⠀⡆⡆⡔⠄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⠀⠃⢸⢀⡇⠀⠀⠠⠈⣻⣿⣿⣿⣿⣿⣿⣿⣦⠙⡧⡀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⡀⢀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⠆⠀⠀⡇⢰⠀⢳⠁⢷⠗⡀⠀
⠀⠀⠀⠀⠀⠀⢹⢰⠀⠈⣸⢰⠀⠀⠀⠒⡹⣿⣿⣿⣿⣿⣿⢿⣿⣷⣜⣮⣦⠀⠀⠀⠀⠈⠊⠀⠀⠀⢄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢤⢸⠀⠀⠀⡇⢸⢰⣾⢠⠆⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠸⢸⠆⠀⢡⢀⠃⠀⠀⢀⠱⡹⣿⣿⣿⣿⣯⣿⣼⣹⣻⣮⣿⣿⡦⡀⠐⠄⠀⠈⠢⡢⢀⠀⠠⡣⡀⠀⠀⠀⠀⠀⠀⠀⡿⢸⠀⠀⠀⡇⠸⢘⠙⣾⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⣾⣯⣶⠀⢸⠎⠄⡀⠈⡷⣼⣌⢻⣿⣿⣿⠿⢻⠩⠁⠀⠈⠑⢖⠠⠔⢄⠀⢌⠐⠨⠢⠁⠣⠂⠰⢠⣄⠀⠀⠀⠀⢠⡙⡈⠀⠀⠀⣌⢀⢸⠀⠏⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢹⣁⠱⡇⢸⢰⡘⢇⢄⢡⠈⢿⣷⠟⠋⠀⠀⠀⢅⠀⠀⠀⠀⠀⠁⠀⠀⠈⠂⡝⡰⢠⠄⣐⡆⠀⢀⠀⠈⠀⠂⢠⢲⢡⠃⠀⠀⣰⠂⣸⡆⠘⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢩⣆⢼⢡⠈⠂⠱⢕⠌⡅⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠊⠔⣣⣀⣀⣀⠸⠀⣠⢏⡲⡉⠀⠠⢠⡏⢠⣗⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢯⣛⣥⠀⠀⠂⠈⠩⢊⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣵⠉⠁⠀⠄⡆⡊⣜⡚⡔⠀⢠⡵⣿⢠⠯⠂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡎⠙⢓⠄⠀⢆⠀⠑⠱⢄⠀⠀⠀⠁⢃⡀⠀⠉⠀⠀⠀⠀⠀⠀⢀⢐⠚⠹⠀⡀⣨⢰⡀⢨⡣⡞⣠⡰⠉⢀⣭⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠤⠢⠽⠦⢈⠑⢦⢀⠳⡅⡠⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡠⣐⠁⡂⢀⢃⣔⣔⣇⢛⢣⡜⣷⠮⡧⠃⠈⢐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡆⠀⢠⠈⠑⠊⣾⣺⢳⣿⠙⢲⣄⡀⠀⢀⣀⣤⡔⢻⡔⠃⠁⣀⡔⢹⡏⢹⣿⠁⠙⣷⣟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡄⣴⠾⠟⣴⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠟⠃⣿⣿⡅⠘⠈⠉⠉⠁⠀⠀⠀⠃⠛⠃⢸⣷⣼⡏⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣠⠤⢓⣵⣾⣿⣷⣄⠈⢘⠲⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣶⣶⣶⣤⣤⣤⣤⣶⣶⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⡤⠓⠁⣰⡿⣽⢾⣟⣿⣿⣧⡄⡘⠢⡟⣧⣠⡀⠀⠀⢀⡀⣀⣴⣁⠟⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠟⠁⢀⣾⢯⣟⣽⣻⣞⡿⣿⣿⣿⣮⣳⣜⣻⣿⣷⠒⠊⠉⠁⣠⣴⣶⣦⣤⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣤⣠⠀
⠀⡠⢟⣞⣯⣞⣷⣻⣾⣿⣿⣿⣿⣿⣿⣾⣷⣿⢿⣵⣤⣶⣿⣿⡿⣿⣿⣿⣿⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡴⣠⠀⢀⣠⠠⠒⠒⠒⠒⠋⣰⣿⡿
⢀⠝⣸⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡐⣉⠯⣄⠀⠀⠀⠀⠀⣴⣿⢯⣿⢷
⢈⢠⣿⣽⣻⣿⢿⣯⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣫⡽⣷⠀⢀⣠⣾⡿⣯⣿⣾
⠀⣼⣷⣻⣿⣾⣿⣯⣿⣯⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡽⣧⠛⠿⣿⣽⣿⣿⣿⣿
⢠⣿⡽⣟⣷⣿⣿⣿⣷⣿⡿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣟⣯⣿⣿⣿