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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⡿⠿⣿⣿⡔⣾⣿⡔⢺⣿⡏⠀⠀⡀⠀⠀⢀⠀⠀⢠⡄⠀⠀⠀⠀⣠⢄⣾⣧⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣿⣿⡇⢸⣿⠀⣤⡞⢻⣿⡟⣿⡇⢸⣿⡗⢺⣿⠈⢻⣿⠣⢴⡎⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⣛⣋⣀⣿⣿⣇⢸⣿⣅⢿⣷⠈⣿⠇⣿⣇⢸⣿⡇⢸⣿⡀⢿⣿⡀⣿⣧⣨⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣋⡀⠉⠟⢋⠈⠟⠁⠈⠟⠉⠈⠛⠉⠀⠀⠟⠋⠱⠟⠁⠙⠟⠁⠸⠋⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀