⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠝⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⡿⠀⠀⢸⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢇⠀⠀⠹⣿⣿⣤⣤⣶⣶⣛⡋⠉⠉⣿⣿⡇⠀⠀⠘⢀⠀⠘⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⠀⠘⢆⠀⠀⢻⣿⣿⣿⣿⣿⡉⠛⠛⠻⠿⠿⣧⣤⣀⣞⢹⡆⠈⣿⣿⣷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠠⣤⣌⡂⠀⠈⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⢯⣿⣿⡏⠙⠻⠶⣤⣤⣄⣀⣠⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡿⠀⠀⢻⡈⠙⠲⣄⣸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠐⢄⠀⠈⠓⢄⡀⠀⢶⣶⣭⣽⣷⣄⠀⠀⠀⢀⣽⠟⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⣤⣤⣷⣄⠀⠈⠻⣿⡿⠿⠻⠛⢻⡆⠀⢆⠀⠀⠱⡀⠀⠀⠙⠢⡀⠈⠻⣿⣏⠹⣀⢶⣞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠘⢦⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠰⡄⠀⠀⠈⢆⠀⠀⠀⠈⠢⡀⠈⠻⣶⣌⢳⡙⠳⣆⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⡀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢳⠀⠀⠀⠀⢂⠀⠀⠀⠀⠈⢦⡀⠈⠻⣦⡹⡄⠉⠳⣦⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣻⣤⣼⣦⣶⣴⣶⣶⣶⠀⠀⠀⠀⠀⠑⠀⠀⠘⣷⡼⡄⠀⠘⠳⣦⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣿⣿⣿⠕⢛⣿⣷⣶⣶⣤⡷⠄⠀⠀⠀⠘⠉⢉⣿⣎⠘⡌⠀⠀⠙⠙⠃⣄⠀⠀⠀⠀⠀⠠⡀⠈⢻⣿⠀⠀⠀⠈⠙⢷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⡇⣿⡇⢠⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣌⢦⣀⣤⣤⣤⣘⡆⠀⠀⠀⠀⠀⠹⡄⠀⠻⣇⠀⠀⣀⡀⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢳⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠂⠀⠀⠘⣆⠀⢿⡄⠀⠙⢿⡛⠛⠉
⠀⠀⠀⠀⠀⢰⣶⣄⡀⠀⠀⠀⠘⣧⣤⣿⣦⣿⣿⣿⣿⣿⣿⠟⠻⣿⡂⠀⠀⠀⠀⠐⢻⡏⢩⡄⠹⣿⣿⣿⣿⣿⣿⣿⣶⡶⠶⠂⠀⠀⠸⣆⠸⣧⠀⡀⠈⣷⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣷⠘⡇⢀⠀⠀⠀⠀⢸⡇⠘⠇⢠⣿⣿⣿⣿⣿⣿⡍⠁⠀⠀⠀⠀⡄⠀⣿⡄⣿⠀⠸⣦⢹⡇⠀
⠀⠀⠀⢰⣶⣶⠶⠖⠀⠀⠀⣤⣶⣾⣿⣿⣿⠿⣿⣿⣿⣿⣿⣆⣘⣠⠗⠋⠈⠀⠀⠀⠀⢛⢛⠟⡝⠉⠋⠁⢿⣿⠁⠀⡄⠀⠀⠀⠀⣧⠀⣻⣧⣿⠀⠀⠹⣿⡇⠀
⠀⠀⠀⠈⠁⠀⢀⣠⡄⠀⠀⠘⢿⣿⣿⣷⣄⠉⠛⠛⠟⠟⡽⠿⠉⠰⣶⣿⡿⠃⠀⠀⠀⠀⠀⠈⠐⠀⠀⠈⠸⣿⡆⠀⣷⠀⠀⠀⠀⣿⠀⣾⣿⡟⠀⠀⠀⢿⡇⠀
⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠈⠙⢿⣿⣿⠗⠈⠀⠈⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⢀⣿⣧⠀⣿⠀⠀⠀⠀⣿⢀⣿⠿⢁⠀⠀⠀⢸⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠲⠤⠒⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣠⣿⣿⠛⢢⣿⠀⠀⠀⣰⣇⣼⠇⠀⢸⠀⠀⠀⣸⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣹⣿⣥⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠏⠁⣿⣠⣾⡿⠀⠀⣰⣿⡿⠣⠀⠀⢸⣷⠀⠀⣿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⢁⡴⠀⠉⠉⠁⠉⢻⣿⠶⠶⣶⢦⣤⣴⣶⡶⠖⢀⣾⣯⣭⣙⠻⣧⡄⣿⠟⢹⡇⢀⣵⠿⠟⡷⠁⠀⠀⣿⢻⡆⣸⡏⠀⠀
⠀⠀⠀⠀⠀⣀⣤⣤⠶⠾⡋⢉⣀⣤⠴⠋⠈⠀⠀⠀⠀⠀⢻⣇⢀⠏⠈⡹⠻⠿⠛⢟⠛⠛⠉⢇⠀⢘⢀⣾⠃⢁⡴⣿⣵⠟⠁⣠⠞⠀⠀⠀⢠⡟⢸⣷⡟⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠛⠛⠛⢛⣿⠟⠋⢀⡤⣒⣥⣤⣤⣤⣄⣀⣈⣿⠎⡄⠀⢧⠀⠀⠀⢸⠀⠀⠀⡞⠀⡸⠾⣧⣴⠋⠘⠋⠁⣠⠞⠁⠀⠀⠀⢠⡟⠁⠾⠋⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣔⣫⡾⠟⠉⠀⠀⠉⠉⠙⢻⣇⠊⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡇⠀⠑⠀⣸⣿⣴⣤⣴⣮⣡⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣤⡾⠁⠀⡴⢫⡿⠋⠀⠀⠀⠀⠄⠀⠀⠀⠀⠙⠷⣤⣀⡇⠀⠀⠀⢸⠀⠀⢀⣇⣠⣼⡾⠋⠉⠀⠀⠀⠉⠙⠻⢷⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡟⠁⢀⡼⢡⣿⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠙⠙⠛⠓⠂⠘⠛⠋⠉⠉⠉⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⠁⠀⡼⠑⢸⡇⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⡇⠀⣴⡇⠀⣾⡇⠃⠀⠘⢷⣀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢰⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣇⣸⡿⡇⠀⣿⡇⠀⠀⠀⠈⢻⡄⠀⢀⠔⠀⠀⠀⢀⣄⣀⡀⠀⠀⠀⠀⠀⢀⣀⡈⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⣷⣿⠀⢻⡄⣿⡅⠀⠀⠀⣰⡟⠁⠔⠁⠀⠀⠀⢰⡏⠀⠉⠙⢶⣄⣠⠶⠛⠉⠉⠉⣷⠀⠀⠀⠀⠀⠀⣴⡋⠀⠀⠀⠀⢸⡇⠀⣀⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠻⣿⠂⠀⠀⣼⠏⢀⠊⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⡳⠝⠃⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⢸⡇⠀⠸⣿⣶⣄⢸⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⣼⠋⠀⠆⠀⠀⠀⠀⢠⠁⠀⠰⠹⣦⣀⢝⡀⠀⠀⢀⣤⢞⡅⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⣾⠁⠀⠀⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⣿⡀⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢬⡻⣍⢇⣠⠶⣋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣿⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣿⣿⣿⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢿⡄⠀⠀⠀⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣀⣀⣀⣀⣠⣄⣠⣧⣤⣤⣤⣶⣾⣾⣿⣿⡏⠀⠈⠀⢻⡄⠀⠀⠘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⢻⡏⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡇⠀⠀⠀⠀⣻⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡄⣿⠃⠀⠀⠈⠉⠛⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉⠈⣿⠁⠀⠀⠀⠀⠀⡿⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢡⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⠀⠀⠈⣧⠀⠀⢠⣿⣿⣿⣿⣿