⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠛⠢⠤⠤⢤⣀⡀⢀⣠⣶⣲⠀⠈⠉⠉⠑⠒⣦⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⣤⠔⠚⠻⠤⠤⣀⡀⠀⠀⠀⠀⢫⢿⠿⣌⡇⠀⠀⠀⠀⠀⠀⡏⠀⠀⠉⠢⣄⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣴⣾⠟⠁⠀⣄⠀⠀⠀⡀⠀⠉⠳⣄⠀⠀⠀⢿⠀⠘⢳⠀⡀⠀⠀⠀⢸⡇⢀⢠⠃⠀⡨⠗⢉⣟⡁⠀⠀⠀⠉⠉⢲⢤⡀⠀⠀⠀
⠀⢰⠀⠈⠳⣄⢄⠈⢦⡀⠀⠘⣦⠀⠀⢫⣣⠀⠀⢸⠀⡆⠀⣟⡇⠀⠀⢀⡿⠀⣜⢇⡴⠋⠀⠀⣎⠞⠀⠀⠀⠀⠀⣰⣤⠾⠛⠻⢶⡄
⠀⢸⠀⠀⠀⠘⢿⣷⢄⠘⣶⢄⠘⢷⡀⠀⢳⣧⡀⢸⡈⠇⠀⢸⡇⠀⢰⢸⡁⢰⣹⠎⠀⠀⠀⣸⡎⢀⠞⠀⣠⢶⡟⠋⠀⠀⠀⠀⡌⠋
⠀⢸⠀⠀⠀⢀⠈⠻⣷⣝⠮⣳⣄⠀⠻⣆⠘⠸⣧⢀⢿⠀⠀⠈⡄⠀⡞⢸⡇⠀⡟⠀⠀⠀⢰⡟⣴⠃⢠⠞⠀⣼⠀⠀⠰⠀⠀⢀⠂⠀
⠀⠈⢧⠀⠀⠈⠳⣄⠈⠻⣧⣌⠻⣷⡀⠹⣻⡆⠘⡷⠱⣧⠀⣠⣷⣀⣧⣤⣧⣸⠁⠀⠀⢀⣿⡿⢁⣴⠏⠀⢰⠃⠀⢀⠀⠀⢀⠿⢒⠏
⠀⢀⣈⡆⠀⠀⠀⠈⠓⢤⡈⠛⢷⣄⠻⡄⢹⣷⡀⣸⣾⣿⣿⣿⣾⣿⣿⣿⣿⣥⣧⣗⣀⣾⢿⣾⢿⠋⠀⣠⠇⠀⠀⠈⣼⣸⠫⣠⡾⠀
⠞⢁⠀⠹⡀⠀⠀⠀⠀⠀⠙⢦⣀⠙⠢⡹⣄⠹⣷⣟⣿⢿⣼⣿⠿⠙⢛⣛⠷⣾⣧⣿⣿⣷⠋⡡⠋⣠⣪⣿⠀⠀⠀⢰⣿⠃⣦⡞⠁⠀
⣇⠀⠱⣄⢙⡦⡀⠀⠀⠀⠀⠄⣙⠷⣢⣌⣮⡢⣬⣽⣷⣿⣿⣿⡀⠸⢷⡤⢀⣿⣿⣿⣭⣿⠏⠀⢀⣿⡿⣝⣀⢤⣶⣞⣣⡼⠟⠀⠀⠀
⡇⠀⠙⠚⠵⣍⣛⠳⢤⣀⣤⣤⣌⣉⣒⣯⣝⣛⣻⣿⢿⣿⣿⣟⣿⣷⣶⣾⣿⣿⣟⡿⣯⣸⣀⣴⣿⠵⢺⡷⠋⠉⠀⣠⠎⠀⠀⠀⠀⠀
⠹⣄⠀⠀⢠⠀⢉⡳⢆⣨⠽⠖⣦⣤⣬⣭⣭⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣤⣤⣿⣿⣿⡿⠗⠀⢸⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀
⠀⠈⠓⠦⣼⣄⠀⠀⠀⠀⠉⠻⠥⠤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣟⣿⣿⣿⡿⠟⠉⠀⠀⣼⠁⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣶⣾⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣿⣿⣟⣿⣿⣿⣿⡏⠙⡟⣿⢿⣿⠏⠀⠀⠀⠀⣶⠏⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠙⠷⣤⣽⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⢻⡿⠋⡟⠃⣀⡧⠽⠿⠧⠖⢊⣵⣶⡶⠿⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠓⠲⠶⣶⣦⢦⡤⣀⡠⠤⠯⠤⠤⠛⠉⠀⠀⠀⣠⣶⢞⠿⠃⣲⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⡴⠒⠲⣶⠾⡏⡟⠙⣦⠈⠿⡛⢾⢷⣄⣠⣴⣦⣀⣠⣶⠶⠛⡽⢳⠏⣴⣟⡵⠷⢶⡾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠾⣢⡝⢁⣴⣷⠞⣧⣠⢤⡇⠓⠤⠼⣆⠃⠰⡄⠉⠉⢹⣯⣿⣠⠘⢠⢯⢞⣛⡶⢾⣗⣮⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠩⢶⣿⣾⣵⡂⠸⣅⣾⣤⣴⣚⣛⠾⢷⣄⢙⠦⣀⣸⣿⣿⠤⠤⢶⣭⣄⡀⠒⠲⢶⣮⣝⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠋⡾⢻⣄⣤⣿⠟⠁⠀⠙⢦⣀⠀⠈⢋⣱⣾⠋⣡⠆⠀⠀⡄⢳⢹⡿⠲⢄⠹⡮⠑⢯⡣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠯⢅⣿⢞⣼⡷⢳⡀⢸⠀⣷⡷⡄⠀⠘⣇⠈⠠⣝⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡇⠀⡷⠃⠀⢹⠹⣽⣆⠀⠙⠀⠀⠈⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⡇⡼⠷⡾⠃⡏⢧⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢺⡏⣼⡇⠈⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣀⡤⣄⡀⢀⡠⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⠃⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠈⠙⠦⡀⠀⠀⠀⠀
⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀
⠀⠀⠀⡇⠀⠀⠀⠀⠰⣆⠀⢶⢧⢘⠇⠀⠀⠀⡤⠟⠦⡀⠀
⠀⠀⡴⢧⠀⠀⠀⠀⠙⣼⣶⣾⣷⣷⣶⠤⠖⠋⠀⠀⠀⣇⠀
⠀⡎⠀⠈⢧⡀⠸⠧⢼⣿⣿⣿⣿⠃⠙⠦⣄⠀⠀⠀⠀⡏⠀
⢰⠁⠀⠀⠀⠉⠉⣉⣹⠏⠉⠁⠀⠀⠀⠀⠀⠈⡏⠳⡏⠁⠀
⠸⡀⠀⠀⠀⠀⠀⣹⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠈⣆
⠀⢣⡀⠀⠀⠀⠀⢻⠄⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⡇
⠀⠀⠱⣄⡀⠀⠀⣉⣿⠒⠒⠊⠉⠀⠀⠀⠀⣠⣄⣠⠴⠃⠀
⠀⠀⠀⠀⠀⠉⠉⠁⡜⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⣀⠤⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣄⡀⢀⡠⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠈⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⣆⠀⢶⢧⢘⠇⠀⠀⠀⡤⠟⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢧⠀⠀⠀⠀⠙⣼⣶⣾⣷⣷⣶⠤⠖⠋⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠈⢧⡀⠸⠧⢼⣿⣿⣿⣿⠃⠙⠦⣄⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠉⠉⣉⣹⠏⠉⠁⠀⠀⠀⠀⠀⠈⡏⠳⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⣹⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⢻⠄⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⠀⠀⣉⣿⠒⠒⠊⠉⠀⠀⠀⠀⣠⣄⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⡜⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⣀⠤⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⣄⣴⡶⠶⠦⣀⣀⣀⣀⠤⢤⣀⣀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢾⡉⠀⠀⠀⠀⣿⡇⠀⠀⠀⠹⡏⠀⠀⠀⠀⢹⡏⠉⠉⠉⡍⠉⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣇⠀⠘⣷⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠁⠀⠀⣰⡟⠀⠀⠰⠏⠉⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⡴⠿⣄⠈⠳⠈⠿⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⣠⠞⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣠⣶⣶⡶⠚⠉⠉⠉⠓⢲⡦⣼⡇⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⣴⠃⢀⡠⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⡞⠉⠉⠸⠋⠀⠀⠀⠀⠀⢀⣼⡇⠘⣇⠀⠀⠃⠃⣿⠀⠀⠀⠀⠀⣸⠀⠀⣦⠀⠀⠀⣿⠀⠀⠀⣠⡿⠀⠀⠀⠀⣠⠖⠛⠁⠀⠩⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣸⢧⡀⠀⠀⢰⡄⠀⠀⠀⠘⢻⣹⡇⠀⢹⡔⡄⠀⠀⣿⠀⠀⣦⠀⢠⣿⠀⠀⠘⡆⠀⠀⢹⠀⠀⠀⣿⠁⠀⠀⢀⡴⠁⠀⠀⠀⣶⠟⠉⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢹⣿⡈⠱⣄⠀⠀⢻⣶⣆⠀⠀⠈⠙⠇⠀⠀⢻⣹⠀⠀⢿⡁⠀⢸⣆⠘⣿⡄⠀⠀⣿⠀⠀⣿⠀⠀⢰⠇⠀⠀⣠⠎⠀⠀⠀⣠⠞⠁⠀⣸⣧⣤⠶⠾⠿⠒⠲⢦⡀⠀⠀⠀⠀
⢾⠽⠷⣄⡈⠓⠶⠊⠝⢿⣧⡀⠀⠀⠀⠀⠀⠀⢷⠷⠀⢘⣿⣄⠀⢻⣄⠻⡇⠀⠀⡿⠀⢀⡇⠀⠀⣼⠀⠀⣰⠃⠀⠀⣠⠞⠁⠀⠀⣰⠿⠋⠁⠀⠀⠀⠀⢀⣀⣧⡀⠀⠀⠀
⢸⠁⠀⠈⠁⠀⠀⠀⠀⢰⣽⡽⣄⠀⠀⠀⠠⡀⠈⢷⡀⡈⠻⣿⣦⠀⠻⣧⡀⠀⢘⡇⠀⣸⠁⠀⣴⠏⢀⡴⠃⢀⣴⠋⠁⠀⠀⣠⣾⠅⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⢹⡆⠀⠀
⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠳⡽⡞⣷⣄⠀⠀⠻⣆⠀⠳⣝⠦⣈⠻⡷⣄⣘⣧⣴⣾⣷⣶⣿⣀⠶⢉⡴⠋⠀⣰⠟⠁⠀⣠⠶⣻⠟⠁⠀⠀⢀⣠⠀⠀⢀⡴⠶⠶⠒⠉⠹⣆⠀
⠀⠸⡷⣆⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣳⣶⣄⣙⡷⣄⣸⣷⣬⣿⣏⣷⣿⣯⣷⣿⣿⢿⡻⣿⣷⣿⣠⣤⠞⠁⢀⣤⠞⣡⠾⠋⣀⣀⣤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⢹⣿⡆⠀⠉⠳⠦⣀⣀⠀⠀⠈⠑⠻⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣾⣿⣿⣿⣿⣿⢷⣴⣾⠟⠁⠀⠀⣠⣾⠿⠋⠁⠀⠀⢀⢀⠀⡀⠀⠀⠀⠀⠀⢀⣀⡏
⠀⠀⠈⣧⣧⠀⠀⠀⠀⠀⠉⠉⠓⠒⠒⠒⠶⠿⠟⣛⣿⣿⣿⣗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣀⡠⠞⠋⠁⠀⠀⠀⢀⣤⣾⣿⠟⠁⢀⣠⢴⡶⠋⠉⠉⡇
⠀⠀⠀⢹⡟⠀⠈⠓⠒⠶⠶⢤⣄⣀⣤⣴⢶⣖⣶⣦⣹⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠟⠋⠛⢉⣀⠴⠟⠊⠉⠁⠀⠤⠴⡇
⠀⠀⠀⠘⢧⢰⣆⠀⠀⠀⠀⢀⣉⣷⣿⠿⠗⠚⠉⠉⠉⠾⢿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣛⣟⣿⣹⣿⣿⣿⣿⣿⣿⠶⠶⠶⠶⠦⠶⠶⠶⠶⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠳⣌⠀⣠⡴⠊⠉⠉⢀⡴⠂⠀⠤⠖⠀⠀⣠⠟⠁⠉⠉⠳⡟⠿⣽⣾⣿⣿⡷⣽⣿⣾⣿⢿⡿⡝⠲⠶⠶⠶⠶⠦⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠇
⠀⠀⠀⠀⠀⠀⢈⡿⠁⠀⠀⣠⠞⠋⣁⠀⠀⠀⠀⢀⣴⠋⢀⡾⠁⠀⢠⠀⠀⠈⠀⠻⡿⠁⠈⡉⠉⠁⠀⠻⣄⠀⠘⠶⢄⣀⣀⡀⠀⠀⠀⠀⠀⠉⠑⠲⠶⠶⠶⣦⣴⡾⠁⠀
⠀⠀⠀⠀⠀⢰⡿⠀⠀⣠⠞⢁⡴⠊⠁⠀⠀⠒⠊⠉⠁⣠⠏⠀⠀⢠⠏⠀⠀⠀⠀⠀⣇⠀⠀⡇⠀⠀⠀⠀⢹⡄⠀⠀⠀⠉⠉⠉⠉⠒⠶⠄⢀⣀⡀⠀⠀⠀⣠⡿⠋⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣧⡆⠸⠃⠐⠋⡀⢀⣀⠀⠀⠀⢀⣠⠞⠁⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⣿⢹⠀⢷⡀⢀⣦⠀⢰⣷⠈⠻⢷⣠⡤⣀⣠⠴⢶⣄⣀⣈⠙⠛⢫⣴⠟⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠙⠷⢶⣶⠒⠉⠉⠛⠋⠉⠳⠶⠾⠓⠲⡦⣄⣴⢿⡷⠒⠲⠦⢤⣀⣀⢿⠞⠀⠀⠀⢸⠿⠀⣸⠛⠒⠚⠻⠧⠤⠤⠤⠼⠿⠦⠭⠭⠿⠟⠉⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠷⠤⣜⣶⣦⣄⠀⠀⠀⠻⣤⡀⠀⠀⠀⣶⠀⠀⠈⠉⣻⢦⣠⣴⣶⣦⠴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠶⠶⠟⠛⠒⠒⠶⠿⢶⣶⡶⠶⠟⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣟⣓⣲⣤⡄⢀⠀⠀⠀⠀⠀⠀⠀⠐⢻⢿⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣽⡏⠀⠚⠉⠳⣝⢿⣰⠀⠀⠀⠀⠀⠀⠀⠾⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⡿⢿⣄⡀⠀⠀⠀⠋⢳⡽⣶⠀⠀⠀⠀⠀⠀⠠⣿⢿⡒⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡏⠀⠸⣷⠉⢳⣄⠀⠀⠉⢹⣿⣆⠀⠀⠀⠀⠀⠀⢨⣯⣷⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⠏⠀⠀⠀⠘⣇⠀⠙⣷⡀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠹⡽⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣸⡏⡏⠀⠀⠀⠀⠀⠈⣇⠀⠸⣷⠀⠀⠈⣯⣿⠄⠀⠀⠀⠀⠀⠀⣿⡻⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⢀⡇⠀⠀⠀⠀⢹⠀⣶⣿⣦⡀⠀⠘⣞⣇⠀⠀⠀⠀⠀⠀⠀⠻⣹⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⡝⠳⣄⡸⠃⠀⠀⠀⠀⠀⠀⣿⣽⣿⣷⠀⠀⢹⡽⣇⠀⠀⠀⠀⠀⠀⠚⠹⡿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⡇⢀⡈⠙⠲⣄⠀⠀⠀⠀⣰⠟⢙⢻⣿⠃⠀⠰⠿⡻⣄⠀⠀⠀⠀⠀⠀⠈⢻⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡇⠉⣁⡴⠂⠈⠳⣄⣠⠞⠩⣀⣉⣿⡿⠇⠀⠀⠀⠻⡹⣄⠀⠀⠀⠀⠀⠠⠶⣟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⣿⡏⠉⠀⠀⠀⣀⣼⡿⡆⠀⣭⢿⣿⣇⠀⠀⠀⠀⠀⠹⣝⣯⠀⠀⠀⠀⠀⠀⢿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀⠀⠈⠉⢹⠀⠈⢲⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠹⣎⢿⠉⠀⠀⠀⠀⢈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡴⠀⠛⢈⣷⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣮⣷⣀⠀⠀⠀⠀⠸⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⢷⣆⣿⣸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⡏⠀⠀⠀⠀⠰⣏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀