⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⣤⣤⣤⣄⡀⣀⡤⣖⡛⠛⡻⠟⠳⠦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠁⢀⣦⠏⣹⣿⣝⢍⡽⢁⠊⢁⣡⣿⣣⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⠁⡔⠁⣾⠃⠀⡇⠁⢸⣿⡇⡞⠀⣏⡟⣿⣎⢷⣵⢂⠀⠠⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠈⠘⠀⢠⡏⠀⠀⡇⡆⠉⣿⣃⡴⣉⠿⠀⠈⣿⣦⢿⣿⡆⠀⢠⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼⡇⠀⠀⡗⡟⠛⠿⡿⠻⠛⠧⠄⠀⠹⣿⣿⣿⡀⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠀⣿⣧⠀⠀⢙⠃⠀⠀⢆⠀⠀⠀⠀⠀⠰⣿⣿⣿⣦⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡿⣿⡀⠀⢸⠀⠀⠀⢘⣆⠀⠀⠀⠀⢼⣿⣿⡟⣿⠆⠀⡀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⡀⠇⡿⡇⠀⢸⣴⠖⣒⡛⣸⣯⡹⣧⠀⣸⣷⣻⣱⡇⠂⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢠⢧⣇⣿⢠⠇⢷⠀⢠⣯⣴⣿⡿⡏⣿⡷⣿⣷⣿⠇⣿⡇⡅⢶⠀⠄⢸⣔⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⡦⣺⣾⡿⣿⢿⠀⢸⠀⢸⣿⣿⡿⣿⣿⠙⠛⢡⣿⠃⠀⠙⠇⣇⠸⡀⠘⡼⣏⢿⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⢀⣾⣟⢽⡫⠚⠁⠀⢳⢼⠀⢸⡟⣯⣻⠛⢸⢀⠀⢸⢿⡀⠀⠀⠠⣹⣄⠑⠄⠘⣽⣆⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡃⣴⣿⠿⠋⢁⠀⠀⠀⠀⢸⢸⠀⢸⣷⢙⡏⠀⡏⠈⠆⠘⣿⣇⠀⠀⠀⢹⣿⣷⡌⢦⣈⢿⣧⡀⠉⢧⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⢩⢎⣾⡻⠃⠀⠀⠀⠃⠀⠀⠀⣼⢸⠀⣮⣿⡞⣇⠀⠻⡴⠁⢀⠄⢸⠀⠀⠀⢸⢹⣿⣮⣌⡣⢕⠙⡗⢄⠈⣷⡀⠀⠀⢠⡆⠀
⠀⠀⠀⠀⠀⠀⣀⡤⠴⣚⣩⡤⢔⣿⡿⡱⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⡿⠃⠀⣿⢳⡀⠘⢦⡎⠀⢸⠀⠀⠀⢸⠈⣿⣞⣷⡹⣷⠱⣿⢷⣿⣶⠧⢴⡶⠋⠀⠀
⠀⠀⠀⢠⣴⣿⠓⢛⣉⢥⣶⣿⡷⠯⠊⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡇⣼⠎⠀⠀⡼⡽⠀⣷⡄⠀⠉⠒⣼⡇⠀⠀⢈⠀⠸⣝⢯⢷⡙⣧⠈⢢⡈⠻⣿⠀⠀⠀⠀⠀
⠀⢀⣴⣻⣸⡾⠛⠛⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⢠⠟⠁⠀⠀⡼⠁⡇⠀⠘⡟⡄⠀⠀⠈⡇⠀⠀⢸⠀⠀⠘⡿⢇⠁⢹⡆⠀⢑⢤⡨⢂⠀⠀⠀⠀
⡴⣫⡿⣿⢧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡆⠀⠀⠀⠀⢀⠞⠁⠠⢿⠀⢀⡇⠈⢧⡀⠀⠀⠀⠀⡞⠀⠀⠀⢣⠈⠣⣊⢳⢀⠘⣯⡙⠿⣗⣤⣄⠄
⢉⢉⡇⣾⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠔⠉⠁⠸⡀⠀⣠⠔⡏⠀⠀⠈⡈⡀⠸⡇⠀⠘⡏⠓⢦⣀⣠⠇⠀⠀⠀⠈⢄⢡⠋⣮⠘⡌⣞⢿⡄⢻⣮⢧⠀
⡿⠘⣧⢿⣏⡄⠀⠀⠀⠀⠀⢣⣀⣀⡤⠤⠒⠙⢦⡀⠀⠀⠀⠙⠚⢹⢀⡇⠀⠀⠀⣇⣧⠀⡇⠀⠀⣅⢠⡀⠈⠁⠀⠀⠀⠀⠀⠈⢾⠀⣿⠂⡇⢸⣯⡿⢾⣿⡆⢣
⣰⡆⡿⡼⡌⣟⠁⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⣿⠀⢇⠀⠀⢸⠉⡼⣰⡇⠀⠀⢹⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡏⣠⠇⢁⣿⣼⣺⢽⣷⡤
⠙⡷⡡⢸⣧⠸⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢠⣧⠜⡀⠀⣸⠚⠚⠉⡇⠀⠀⡿⡄⠀⠀⠀⠠⣀⢄⠀⠓⣄⡴⢋⣼⢿⣿⠘⣿⡽⡇⠘⣿⢧
⠀⠣⣑⡿⢦⡳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠈⢢⠔⠒⠊⣿⠸⠻⢄⣇⠀⠀⠀⡇⠀⠀⣇⣈⠦⠒⠒⠈⠉⠉⠉⠀⠚⠛⠭⣺⣻⡭⣺⣹⡇⡿⣆⡼⣞
⠀⢀⡜⠁⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠈⡇⠀⠀⢸⡷⢧⣐⡇⠀⠀⢀⡧⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣫⢺⡷⠙⣇⠘⣿⣜⡧⣈
⡀⡏⠀⣀⣀⠤⢄⡀⠉⢷⣀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⢸⡀⠠⠞⡇⠀⣙⡧⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠴⠶⠛⠋⢡⣟⢞⡻⢿⣿⣽⣾⣤⡀
⠯⣴⣾⠋⢀⣀⡀⠬⠴⠒⠿⢦⡀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⢿⠛⠉⣧⠞⠁⠀⢀⣄⣀⣀⠀⠤⠄⣀⣀⣀⡀⠀⠀⠀⣀⣀⣠⣤⣶⣯⡕⠋⠀⠀⢉⣛⣛⣞⣟
⡀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠈⢆⠀⠀⢀⡼⠃⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠤⠤⠴⠾⠟⠄⠙⠢⣄⠰⠛⠛⠛⠋⠉
⣬⠤⠤⠤⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠓⠶⠚⠉⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠒⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠲⠦⠤⠤⠤⠴⠒⠒⠒⠚⠋⠉⠉⠉⠉⠉⠑⠒⠲⠤⣀⡀⠀⡘⣿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛⠛⠀⠀