⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡯⣿⣾⢿⢷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣝⣿⣷⡇⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⢿⡁⢸⠟⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣷⣯⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣿⢯⣿⡿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡃⡌⣻⡏⡀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣺⣿⣿⡅⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡿⡿⣷⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡷⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⣴⣻⣳⡄⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠉⣳⢿⣷⡏⠉⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣄⡀⠀⠀⢀⡴⢶⣦⡔⢲⣤⡶⠾⣷⣾⣿⣣⣹⣿⣶⡿⠶⣦⣤⢖⣤⣴⠶⣄⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠳⣤⡴⠟⠉⠉⠑⠉⠛⠟⠉⠳⠋⣿⣿⣮⣻⣿⡟⠻⠛⠉⠛⠛⠙⠝⠋⠙⢿⣍⣯⠛⠓⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣩⠿⡽⣥⢏⡉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠃⠀⣹⣇⠀⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠫⢶⣦⣜⣟⣿⣡⣶⠶⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⢿⣿⣷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⠀⠉⢙⣿⣽⣋⠁⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠒⢄⠀⢀⠞⠁⠀⠈⣿⣿⣿⣦⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠈⠀⣠⣾⣿⣿⣿⡍⠑⢮⡁⠘⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣾⣷⣧⣄⡀⠀⠀⢹⣿⣿⣿⣷⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⠀⢀⡴⠋⠉⢻⣿⣯⡛⠁⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⢿⣿⣿⣿⣷⡀⠀
⠀⠀⠀⠀⠀⠀⠀⡸⠀⡸⠀⡼⠀⠀⠐⣿⡗⣟⣷⠀⠀⢀⠇⠀⢤⣶⣤⡈⢦⡀⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⣿⣿⣿⣿⣇⠀
⠀⠀⠀⠀⠀⠀⢰⣇⢠⠇⠀⡇⠀⠀⠀⢙⣾⣽⠁⠀⢠⠞⠀⣼⣿⣿⣿⣿⡄⠹⣄⣈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⣿⣿⣿⣿⣿⠀
⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⢠⣇⣀⠀⣰⣿⢿⣿⣷⠖⠋⠀⢀⣿⣿⣿⣿⣿⣿⠀⣿⡏⢀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠖⣛⣹⠛⠛⠛⠋⠀⠀⠂⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⠘⣇⠀⠁⠠⠤⢅⡀⠉⣉⣛⣋⡉⢀⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠻⣿⣿⣿⠇⡈⣄⠀⠀⠀⠀⠈⠑⠾⠿⣿⣿⣿⣿⡟⣿⣿⣿⠀
⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠛⠛⠁⠀⡈⠛⡅⠀⡇⢸⣟⠂⠀⠀⠀⠀⠀⠀⠀⠋⠛⢋⣼⣿⣿⣿⠀
⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠐⡏⠀⠀⠀⠀⡇⠀⠀⣧⠀⢢⠀⢱⠸⣿⣯⣠⠀⠀⠀⠀⠀⣀⣴⣠⣺⣿⣿⣿⣿⠀
⠀⠀⠀⠀⢠⠃⢀⣀⠤⠐⢉⣩⣉⠁⠀⠀⠀⠀⡜⢀⣄⣀⠦⡘⠛⢂⠜⠛⢧⠞⠳⢘⡀⢹⣿⣷⠀⠀⢀⡀⣊⡁⠈⠙⣿⣿⣿⣿⣿⠀
⠀⠀⠀⠀⣸⠐⢁⣴⢶⣶⣿⣿⣿⣿⣄⠉⢆⠀⣑⡜⠉⣟⣀⣤⣀⣤⠦⠤⢼⣅⣨⢸⢸⣿⣿⣿⣇⣠⣤⣿⣥⣾⣷⣶⣌⠉⠙⢻⡇⠀
⠀⠀⠀⠀⠙⠷⣾⠣⣿⣿⡿⢿⣿⢻⣿⣷⡜⠀⡛⢿⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠑⠙⣶⠙⣿⣿⣏⣿⠟⣿⣿⣿⣿⣿⣧⠀⢸⡇⠀
⠀⠀⠀⠀⠀⠀⠀⢳⣿⣿⡆⠘⣿⢨⣿⣿⡗⢰⠿⠶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢉⣹⣿⣿⡿⢰⣿⡇⣼⣿⣿⡿⠀⢴⠅⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠘⣿⠘⣿⣿⣿⠸⡀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠖⢊⡝⡿⠃⢸⣿⠀⣿⣿⣿⣷⡆⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢸⣿⠀⣿⣷⢻⣻⣾⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠋⠁⠀⣿⡇⢠⣿⣿⣿⡿⠀⣇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢨⣿⡇⣽⣿⠟⣷⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢸⣿⣿⣿⠇⢀⡎⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⣿⡏⣿⡿⠷⠻⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣾⣿⣿⠟⠀⣿⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣼⡆⢀⡙⢷⣿⣷⡞⣾⣻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠐⣿⡟⠁⠀⣼⡿⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠘⢾⣿⣯⠗⢿⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⣿⣷⣾⣿⡿⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠂⠀⠀⠹⢷⣿⠿⣿⣧⡤⢶⢲⣦⣤⠤⢤⡠⢄⣀⣀⣤⢤⡦⡴⡆⠀⣿⡃⠄⢀⣻⣿⣿⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣖⠀⠀⠀⢀⠉⡀⢈⠀⢻⡉⢹⠁⡷⠀⠸⠀⠘⡀⢻⡄⢸⣇⣿⡅⢀⣿⠋⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣆⠀⠀⠈⠐⠄⠘⠉⠉⢻⠞⠒⣾⠲⡴⠒⠾⠓⢺⠙⠉⡿⣿⣶⣾⡿⠂⠀⠐⣿⡿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢆⠀⠀⠀⠀⡀⠀⠀⠘⠀⠀⠈⠀⠇⠀⢘⠀⠚⠀⠀⠇⢻⣿⣿⡧⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⠁⠐⠶⠆⠐⠀⠀⠀⠒⣀⣠⣆⣀⠀⢤⣠⣾⡿⠉⠀⠀⣀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⢿⣿⣿⡿⠿⡿⣯⣶⠆⢠⠌⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣕⠶⣪⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣟⢪⣹⡟⣿⠓⠛⠒⠒⠖⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣏⢳⢣⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡞⣾⢾⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡅⡟⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠰⣿⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡷⣗⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀