⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀ ⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⡇⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⢣⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⢀⠀⣀⠜⠀⠀⠀⠀⠞⢀⠂⠀⠀⣄⢀⡀⠄⠀⠀⣀⣀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠬⠾⠷⠛⠒⠓⢺⠒⠒⠒⠲⡄⢾⠊⠇⡉⠀⠐⠒⠚⠋⠁⠀⠀⠈⠑⢺⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠘⣏⡏⠸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⣿⠁⠐⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⢣⠇⠀⠀⠀⠀⠀⣿⠀⢀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠈⠁⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⡠⠛⠀⠀⠀⠀⠘⠘⠧⡀⠀⠀⠀⠀⠀⠀⠜⡄⠀⠀⠀⠀⡇⠀⠀⡠⠃⠀⠘⢇⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣆⠀⢀⣤⡄⠀⠀⠙⠶⢤⣤⣤⠆⠀⠸⠀⠀⠀⠀⢁⠇⠀⠀⠀⡇⠀⠀⠀⠈⠉⢱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠦⠿⡻⢉⣲⠠⠀⠀⣀⡀⢀⢃⠀⢐⠠⢢⠀⠀⠁⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠃⠀⠐⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡤⣾⡗⢡⣶⡞⠈⡆⣸⣼⣿⢈⢘⡉⡀⠀⠀⠀⢂⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠛⢻⠜⠛⣅⢸⢹⠘⠉⠉⠘⠱⡀⠀⠀⠀⠤⠠⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣌⠖⢉⣀⣸⠹⢠⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡒⢆⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣀⠠⢊⡌⠷⣀⢺⡈⠛⠀⢀⣄⢀⠀⣤⡆⠄⠀⠀⠀⠀⠈⠀⢱⣿⠻⠂⠘⣇⡀⠀⠀⠀⠀⠁⠂⠠⠀⠄⠐⠨⠖⣄⠀⠀⣸⠆⢀⡴⢋⡌⠤⢀⣀⠀⠀⠀⠀⠀
⠤⣤⣤⡔⠁⢐⡎⠈⢓⠬⠉⠉⠉⠲⠿⡻⠀⣾⣿⢶⠘⡶⡶⠶⠦⠒⡚⠻⠥⢷⠀⢽⢊⣛⡛⠒⠶⠴⠶⠶⠶⠶⠶⠶⠶⠾⠷⠾⠯⠀⠉⠥⢔⠋⠈⡖⠀⠓⢤⣤
⠀⠀⠀⠚⠲⠌⣑⡄⢁⣤⠶⠒⠒⢲⣶⠃⢀⣿⡟⢻⡖⠛⠛⠛⠗⠒⠫⢀⡘⣮⣖⠈⠩⣐⠤⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠲⢦⢄⠁⣄⡡⠔⠛⠂⠀
⠀⠀⠀⠀⠀⠀⠈⣤⠋⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠜⡳⢀⠀⢈⠀⠀⠀⠀⢹⣇⢋⢀⣼⡾⣉⠉⠁⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠉⠛⢹⠛⠉⢦⡀⢈⢋⠆⢸⠏⠁⢀⡅⠀⣌⠆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠈⣂⠀⠀⠈⠲⢸⢦⠎⠀⠀⠈⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⢦⢀⠀⠀⢸⢀⡥⢀⡀⠀⠀⠀⢀⠀⠂⠃⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⣀⣄⡠⢤⢤⣀⠙⠳⣴⢸⠒⠞⠄⠀⠉⠀⠁⠀⠠⠤⠘⡄⡀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠉⠁⠀⠀⠀⠈⠙⡁⣀⠀⢸⢁⡽⢼⠢⣀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⢸⠀⠀⠀⠉⠳⠳⡀⠀⠀⠀⡇⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢸⣣⠀⠀⠀⢀⣄⠞⠁⢢⠀⠘⢸⠀⠀⠀⠀⠀⣱⢁⠀⠀⠀⡇⠀⠀⠂⠀⢀⠜⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⣿⠂⠀⠀⠈⠁⠀⠀⠀⢡⠐⢸⠀⣄⣠⠖⠉⠘⡻⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢀⣀⢶⠘⠀⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⢄⠀⠀⠀⢰⡣⡀⢹⢸⢸⠎⠀⢠⣴⠖⢛⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⡀⠀⠀⠘⠖⠃⠘⣨⢸⣀⡼⠋⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡐⢂⢀⢆⠀⣨⣷⡞⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠈⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠶⠀⠷⣿⣿⣻⣿⣜⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡴⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣍⢉⣠⠀⠁⠙⠒⠚⠓⠢⣢⠀⠀⠀⠓⠌⠑⠁⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠋⠀⢣⡄⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢠⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⡤⠾⡫⠈⢰⡤⣆⡻⠦⣄⠤⠄⢐⣢⠀⠈⠁⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀