⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⡀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠔⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠹⡗⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⣠⢿⠀⢸⠙⢦⡀⠀⢢⡀⠀⠀⠀⡘⢆⠉⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⡴⠁⠈⣇⡜⠀⠀⠙⢦⡀⢻⢦⡀⠀⠱⣦⡀⠀⠀⠈⢷⠒⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⡠⡿⢀⡞⠀⠀⠀⠸⠇⠀⠀⠀⠀⠙⢮⣧⠱⣄⠀⠙⣍⢢⡀⠀⠈⢧⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⠀⠀⠀⠀⢠⢣⠃⠀⢀⡴⠋⣰⢁⡞⠀⡄⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢣⣢⡘⣆⠙⣄⡄⠘⣇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⣀⠀⣠⠔⠂⠀⢀⣿⠃⢠⡞⠀⢠⣣⠃⢀⠔⠁⠀⢀⡇⡞⢰⢳⠃⣰⡇⠀⡄⠀⠀⠀⢀⠀⠀⠀⠀⠹⡄⠹⣿⣮⢣⡈⠻⡀⠘⣆⠀⠀⠀⠀
⠀⠀⠀⠀⢀⠴⠚⠁⠀⠀⢨⠍⠀⠀⠀⠀⠸⣷⣰⣿⡤⢶⣿⠇⣰⠋⣠⠞⢀⣼⣰⠃⣾⣿⢀⡇⡇⠀⣿⠀⠀⠀⠘⣆⠀⠀⠀⠀⢻⡀⠘⡿⣿⢷⠀⢳⡀⠘⣆⠀⠀⠀
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡟⠁⣰⠋⣸⢰⢃⢾⣏⡟⠁⠀⣿⣴⢿⣿⣸⠀⢳⠀⣿⡆⢀⡀⠀⠸⣦⢀⠀⠀⣧⣷⡀⢹⠘⣾⡆⠀⢣⠙⡌⢧⡀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢻⢀⡤⠖⠋⣹⣯⠏⣼⡏⠀⠀⣀⣼⣿⣾⣿⡿⢀⣈⣷⣼⠽⣆⢳⡄⠀⣏⢿⣦⠀⢸⡏⢧⠘⡇⠹⡇⢀⠀⢣⠸⡦⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⣠⡶⠛⠹⣟⣠⣵⡵⠚⠉⣀⣿⠿⠗⣚⣯⣩⠤⠤⠽⠗⢚⣻⣿⡄⣿⣭⣻⣆⠀⡏⢹⡆⡇⠀⠃⠸⡆⣤⣧⢣⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⠁⠀⠠⠖⠫⢁⣴⣷⣶⣯⣓⣒⣋⣉⣥⠤⠤⠖⡖⠚⠉⠉⠀⠘⢧⡟⠛⠓⢿⢀⣏⡿⣷⡇⠀⠀⠀⣻⣼⣿⣼⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⡠⠤⠒⠊⢹⡯⡉⠙⣏⣁⣸⠁⢰⠇⠀⠀⢰⠃⠀⠀⠀⠀⠘⣽⣿⢷⣶⣬⣿⢉⣧⣿⡇⢸⣶⠀⣿⠈⠻⣿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢧⠈⠓⣬⡻⣇⣠⡟⠓⠒⠒⠋⠀⠀⠀⠀⠀⠸⣩⣿⣿⡟⢻⣿⠾⣻⠃⣷⢸⡿⣆⢿⠀⠀⣿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⡀⢠⠃⠀⠀⠸⡄⡄⠀⠈⢧⠀⠘⣟⢿⣯⠀⠀⠀⠀⡰⠋⡇⡄⠀⠀⠀⠀⠉⠙⠒⠋⠁⣼⣡⠀⣾⣾⠇⠘⣾⡄⠀⠏⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⡞⠀⠀⠀⢱⣸⡄⠀⠀⠳⣄⠘⣆⠈⠓⠀⠀⠀⢇⣤⣿⡇⠀⠀⠀⠀⠀⠀⣠⠴⢚⣉⣠⡴⢻⡏⠀⠀⠘⠿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⢠⠀⠀⠀⣷⣿⠢⡀⠀⠹⣷⢼⣦⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠐⠛⢉⣿⠋⢁⡼⠃⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡇⢸⣇⢀⠀⢧⢻⣦⢈⣷⣄⠙⣆⠉⠓⠀⠐⠒⠲⠄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣣⠴⠋⠀⢠⡀⢻⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⢿⣸⣆⢸⡀⢹⣧⢿⢯⣳⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠿⠋⠁⠀⡆⠀⡾⡇⠘⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠈⣟⣿⣷⡇⢸⠈⢿⡷⡙⢿⣧⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⢠⡇⠀⡇⣷⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⣿⠇⢹⠳⣘⡇⠈⠃⠈⢦⡙⠀⠀⠀⢀⣠⣴⣾⣿⠀⠀⢀⣇⠀⢰⡀⠀⢸⣧⢸⣽⠸⣆⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠈⢻⠄⠀⠀⠀⠈⠓⠒⠚⠉⣸⠃⠹⡟⣧⠀⢸⠉⢧⠘⣧⠀⣿⢹⢸⠇⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠉⠒⢤⡼⠁⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠈⠳⣼⡲⠈⢧⡏⣧⠟⣟⣿⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠋⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠈⡃⠀⠸⠃⠘⠀⠉⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠸⡄⠀⠀⠀⠀⠀⠀⣟⣀⣀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⡼⠟⠛⠙⠛⠯⣗⡢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡇⠀⠀⠀⠀⠀⠀⡇⢢⡙⡝⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣍⡒⠤⠤⠤⠐⠒⠒⠈⠉⠳⢄⠙⣄⠀⠀⠀⠀⠀⢧⠀⡇⠘⡜⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡏⠒⣦⠤⣀⣀⠀⠀⠀⠀⠀⠑⢌⢆⠀⠀⠀⠀⠘⣧⡇⠀⠘⡬⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠀⢸⡀⠀⢫⠉⠓⢲⣦⣄⣀⠀⠙⢆⠀⢀⡠⠴⡾⠁⠀⠀⠀⠈⠳⣝⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡟⠀⡄⢉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⢧⡇⠀⢧⠀⠀⢯⠻⣿⣿⣶⣌⣆⠀⠀⡼⢁⣀⠀⠀⠀⠀⠀⠀⠉⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⡸⣠⠋⠙⢎⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠘⣷⠀⠈⢧⠀⠈⢧⠀⠙⠻⢷⣿⣦⣴⣿⡿⠋⠉⡗⠲⡤⠖⠒⠛⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠁⠁⠀⠀⡸⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠈⢧⠀⠈⢧⠀⠀⢀⣼⣿⣿⣿⣦⡀⠀⡇⠀⣇⠀⠀⠀⢻⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀nako: