⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⣀⣴⣷⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⡄⢀⣀⣤⣶⢿⠏⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡟⣿⣿⣿⡟⡇⢸⠀⢿⣿⣿⡟⣰⠃⣾⣿⣿⣿⣿⣿⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡤⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡟⠛⠿⣿⡇⠈⢷⠝⢷⣧⠀⠀⢸⣽⡟⢰⠃⠀⢿⣿⣿⣿⡿⣿⡿⠟⢛⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⢧⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠙⢦⡀⠈⣷⠀⠀⠀⠙⠙⢧⠀⢸⠿⠀⠀⠀⠀⡞⣾⠟⠁⠀⠀⠀⠐⣿⣿⣿⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢳⡀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠓⠒⠒⠒⠶⣼⣆⠀⠀⠀⠀⠈⢧⣿⠀⠀⠀⠀⠀⢻⠏⠀⠀⠀⠀⣠⣶⣽⣁⣀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⢦⣀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⢀⣴⠽⠂⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄⡀⠀⠀⠈⢹⡀⠀⠀⠀⢠⠇⠀⢀⡤⠒⠋⠉⠀⠀⠀⠉⢉⣛⡺⣧⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠉⠓⠦⡀⠀⠀⠀⠙⢦⡀⠀⠐⠛⣲⣶⠶⠤⠤⣄⣀⣀⣀⣀⠀⠀⠀⠉⠂⠀⠀⠀⠃⠀⢀⡴⠋⠔⠚⠓⠒⠒⠲⠦⢤⣀⠀⠀⠀⠀⠉⠙⠻⣦⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢶⢄⡀⠀⠀⠙⣆⠀⢀⡿⠀⢀⡤⠖⠊⠉⠉⢉⡭⠝⠓⠒⠂⢀⣀⡤⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣝⡶⣒⠻⠿⣿⣷⣮⢷⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠙⣦⡀⠀⠈⢣⣟⣤⡶⠋⠀⠀⠀⢀⡔⠉⠀⠀⢀⡤⠞⠉⠀⠀⠀⢀⡆⠀⠀⢀⠐⠒⠦⢤⣀⠀⠀⠀⠀⠀⠉⢻⣷⣤⣤⣙⢿⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡇⠙⣄⠀⠀⢿⡟⠀⠀⠀⠀⣠⡏⠀⠀⣀⣴⡟⠁⠀⠀⠀⠀⠀⡞⠀⠀⠀⠈⢆⠀⠀⠀⠈⠙⠢⡤⣄⣠⢤⣀⡉⠻⣏⠻⣷⣽⡄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⡀⢸⣧⠀⣼⢀⡔⠀⣠⣾⣿⠃⢠⣾⣾⣿⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠘⢦⠈⠙⠛⠿⣿⣿⣧⠘⣇⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣇⠀⣿⣧⣧⡟⣴⡾⠟⢩⡿⠰⣳⠟⠁⣿⡀⠀⠀⠀⢀⢀⠏⢣⠀⡀⠀⠀⠀⢸⡹⢆⢂⠀⠀⠀⠀⢧⠀⠀⠀⡘⠻⡎⠓⣼⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⠀⣿⣿⣿⣿⡏⠀⠀⠀⣇⡼⠃⠀⠀⣿⣇⠀⠀⠀⡌⡾⠀⠸⡄⢷⠀⢀⠀⢸⠿⠮⣷⣄⠀⠀⢳⠘⣧⠀⠀⢾⣄⢸⣶⡿⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣧⢻⣿⣿⣿⠃⠀⠀⣠⣿⣷⣶⣤⣀⠸⣾⠀⢀⢰⣷⠁⠀⠀⢻⡸⡆⢸⠀⣼⠀⠀⠀⠙⢮⠀⠈⣇⠹⣆⡀⠈⢻⣧⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣿⣿⣿⠀⠐⢿⠟⢉⡍⠛⠛⠲⣝⠻⡆⢸⢸⡏⠀⠀⠀⠀⢧⣿⢸⣸⠃⠀⠀⠀⠀⠈⢷⡀⢸⡆⢸⣯⣆⠀⣯⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⣿⣹⠀⠀⣶⠀⠏⠀⣾⣿⣶⡌⢷⠹⣾⣻⠁⠀⠀⠀⠀⢸⣻⣿⣥⡤⠤⠤⣤⡀⠀⠈⣧⣾⡇⣸⣿⣾⡆⣸⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢻⡛⢿⠀⠀⢿⡎⡆⠀⣿⣿⣿⠇⠘⡄⢹⣿⠀⠀⠀⠀⠀⠈⢋⣤⣖⣟⡛⠛⣟⠻⣷⣤⣿⡟⢀⣿⣿⢻⣿⡏⣀⣀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⢿⣦⣤⣌⠛⢽⣦⣀⣉⣁⡤⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢱⣿⣿⣿⣇⢸⡀⠘⠛⠿⠁⣸⣿⣿⡼⠿⣛⡽⠚⠉⢉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣑⣿⣿⣿⡿⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠈⠻⠿⠟⠁⡈⠀⣴⠏⠀⣰⢿⣿⣿⠡⠞⠉⠀⢀⠴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⣿⡿⢋⡼⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣤⣴⣚⣠⡾⠋⠀⢰⣧⠾⠖⣿⠠⠤⠖⠊⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣉⣠⡞⢡⣾⢷⣤⣀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠧⣭⠛⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡖⡆⠸⣿⡟⠀⣸⣿⣿⡆⠀⠙⠢⡀⠀⠀⣀⣋⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠧⠇⢸⣿⡇⣾⣿⣿⣿⣿⠀⠀⠀⢸⣦⣾⠳⠬⣙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣶⣶⣶⣿⣿⠇⣾⢋⣙⣿⣿⡀⣀⣀⡼⣅⢟⢦⣀⣀⡴⡻⠀⠀⠀⠀⣀⡠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣼⣿⠸⣄⣽⣿⣇⢧⣸⠁⠈⠻⣵⣲⣶⣯⠵⠖⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠹⡟⠉⠉⠉⠀⠈⣷⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠙⢿⣯⡉⠁⠀⠀⠀⠀⣠⠜⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣳⣤⣙⣛⠒⠒⠒⠚⠉⢁⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠋⠀⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀