⠀⠀ ⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡄⠀⡀⠊⠄⠐⠑⢀⠀⢫⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⢄⢸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⢬⠐⢳⠠⣱⠄⢀⠀⢀⣧⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡠⠄⠭⠒⠒⠶⠣⣀⠀⠙⢷⡄⠀⠀⠀⠱⢡⣦⠀⠀⠀⡄⠈⣹⠀⢹⡗⢠⡀⢈⣿⠀⠀⠠⠀⠀⠀⠀⠀⢑⢄⡀⠀⠫⡄⢀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠀⡀⠠⢀⡑⠢⠶⢤⣤⡵⠚⠛⢿⣦⢀⡠⠾⡁⠑⡵⠀⢀⣴⠃⠀⠋⣾⣧⡀⣷⣫⠇⠀⠀⠀⣀⠀⠀⠀⢀⡈⠉⢈⡆⣰⠋⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠈⠁⠀⠉⢳⣶⣅⣀⣀⣀⠀⠀⠻⣿⣍⠉⠑⠾⣇⡀⢘⣷⠂⢀⠄⣛⢷⣶⠋⠀⡎⣠⣲⠚⢈⢦⠀⠀⢠⣃⢠⠈⣵⡇⠠⢆⠀⣤⠔⠁⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣉⡳⠔⠒⠒⡛⠁⣈⣍⠉⠉⠙⢛⣶⣾⣿⣷⡑⠒⣻⠫⣻⣯⣁⣡⠞⢡⣿⠳⡦⢤⣾⡍⠉⣉⠁⠈⢣⠞⣿⠃⢿⡱⠈⢻⡄⣠⣞⣁⡠⠮⠐⠌⠂⠀
⠀⣀⣀⠀⠀⠠⢄ ⠈⡉⠣⡤⠽⠽⡒⠟⢁⠀⠤⣙⣿⣾⣿⡤⠼⠟⠉⢿⣧⡞⠁⠀⢹⡔⠋⠹⢍⣄⣀⣰⢩⡑⢧⣧⡀⡅⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠣⢀⠀⢘⣤⡣⠇⠘⢄⠀⣧⠠⠛⠻⣤⢸⠟⢿⣧⣤⣀⣠⣿⡿⠇⠀⢀⢸⣿⠀⠀⢀⠘⣧⡄⢀⣧⣠⣿⣧⣤⣼⠿⣧⣀⠀⣠⠤⢄⠠⠤⠀⠀⠀⠀
⠀⠀⠀⢀⠀⠀⠐⠈⠀⠉⠁⠀⣈⠔⢤⣀⢉⣤⠤⢤⣀⠘⡕⠚⣲⠽⣿⣿⣍⠙⠀⠀⡈⠙⡟⢯⡀⠀⠀⠙⣿⣋⣩⣼⡟⠁⣼⠟⠁⠀⠀⣀⠝⢦⡀⡀⠅⠐⣂⠄⠀⠀
⠀⠀⠀⠀⢉⠖⠀⠢⢊⣈⠴⠈⠀⠀⠀⠉⠉⠒⠳⣤⡈⠹⣧⡈⠁⠀⠈⣻⣿⣆⠀⢀⠈⠀⠀⠉⠙⡧⣾⣋⣼⡏⣁⣼⣷⠟⠁⠀⣀⣠⣖⣁⠀⠈⠆⠀⣠⣯⠀⠀⠀⠀
⠀⠀⠀⠐⠢⠤⠋⠀⠄⡀⢀⢆⡒⡭⠉⡀⣲⠒⠒⠚⠻⢦⡈⠙⠳⡚⠉⠙⢿⣿⣷⡏⣀⡀⢰⠁⣸⣵⡿⢋⣵⣾⢿⡟⠿⠿⢶⠞⠁⠀⠀⢿⡙⠒⠤⠔⠁⠈⠃⠀⠀
⠀⠀⠀⠀⠠⠤⠀⠀⡤⠈⠁⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠙⠲⣄⠁⣲⠢⣼⢿⣿⣿⣮⣷⠋⢠⣿⣭⣿⣿⡿⠃⠀⠉⠒⠲⠾⢷⡠⣄⠀⡨⠗⣤⠀⡁⠢⡀⠀⠙⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠤⠖⠓⠢⠄⣀⣤⠴⠒⠚⠿⣶⣍⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⣿⢞⡛⡐⢄⠀⠑⠈⢆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡊⠉⠉⠀⠁⠀⠀⠀⠀⠈⠻⣿⣿⡿⠛⢿⣿⣿⡿⠟⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡂⠀⠁⡸⠢⠄⢀⡰⠑⠀⠊⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠃⠀⢀⡿⠿⣇⠀⢸⡇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢱⠂⠄⣀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣡⣄⣻⣿⡿⢱⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢨⠂⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢸⣿⣿⡿⠿⠟⠋⠻⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⢸⣿⠏⣠⣤⣤⣄⡀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣸⣿⣿⣿⣿⣿⣦⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣶⣶⣿⠾⠛⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠉⠈⠀⠉⠉⠛⠉⠁⠈⠙⠻⢷⣦⢶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠛⠙⠛⠋⠁⠀⠀⣴⠒⣿⠓⣭⣶⠁⠀⠀⡄⠀⢠⠀⠀⣴⠀⠀⠀⠀⢠⡆⢰⡇⠀⠀⠀⡄⣴⣿⣦⠀⠀⠀⠈⠉⠉⠒⠒⠒⣦⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⣷⣤⡤⠶⣿⣿⢀⣔⢺⣷⠴⣿⡆⢰⣿⠖⣿⡆⠰⣿⡗⣿⡗⢂⣠⡞⣷⡤⣴⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢃⡠⢹⣿⢸⣿⠘⣿⣆⣿⡇⢻⣿⠀⣿⡇⢸⣿⢀⣿⣯⠀⣿⣇⠋⣸⣿⡀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠋⠁⢿⠧⡟⠁⣘⠟⠁⠻⣡⠿⠋⢹⠟⠁⠻⠋⠈⠟⠙⠿⠟⠁⠙⠟⠉⠀⡹⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⡿⠿⣿⣿⡔⣾⣿⡔⢺⣿⡏⠀⠀⡀⠀⠀⢀⠀⠀⢠⡄⠀⠀⠀⠀⣠⢄⣾⣧⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣿⣿⡇⢸⣿⠀⣤⡞⢻⣿⡟⣿⡇⢸⣿⡗⢺⣿⠈⢻⣿⠣⢴⡎⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⣛⣋⣀⣿⣿⣇⢸⣿⣅⢿⣷⠈⣿⠇⣿⣇⢸⣿⡇⢸⣿⡀⢿⣿⡀⣿⣧⣨⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣋⡀⠉⠟⢋⠈⠟⠁⠈⠟⠉⠈⠛⠉⠀⠀⠟⠋⠱⠟⠁⠙⠟⠁⠸⠋⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀