⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠦⣤⡀⠀⣤⠶⠚⠛⠛⠛⠛⠲⠦⣄⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣼⡇⠀⠀⠀⢹⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠉⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣧⠀⠀⢀⣠⣤⠴⢶⡄⢀⡄⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣻⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠀⠀⠀⠀⠉⠻⢏⡀⠀⠀⢸⣿⡾⡇⠀⠀⠀⠀⠀⠀⠀⠉⠛⣶⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠉⠀⣰⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠏⠀⢀⡀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠋⣼⠧⠤⢤⡄⠀⠀⠀⠀⠀⢀⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⣀⣀⠀⠀⣠⣴⡃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢹⢋⣴⡻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢰⣄⡟⠀⢀⣠⡴⢒⣩⠵⠳⠄⠀⠀⢠⠤⠤⠀⠀⠀⠀⣠⠤⣄⠀⠉⠉⠉⠈⠉⠳⣆⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡙⣦⣤⣏⣿⠫⠥⠤⢿⣿⣆⣠⢶⣦⡀⣸⣿⡴⠛⢹⡇⣺⣿⠷⠛⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣩⠟⠲⠶⢲⠟⠁⢠⣈⣣⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠸⣷⣄⣰⣇⡟⠁⠘⢉⣿⣿⡟⣋⣿⡿⠈⠹⠞⠻⢥⡇⠉⠀⠀⠜⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⠀⠀⠀⣾⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⢻⣍⠀⣈⣇⠀⠀⡿⠿⠛⠉⣉⣁⣠⡤⣶⣖⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠄⠀⠀⢀⡼⠛⠻⣿⡀⠀⠀⠻⢦⣀⠀⠰⡉⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠙⢻⡱⣾⣷⢤⣧⠸⣟⠫⠥⠿⠚⠙⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣚⡭⠉⠀⠀⠀⠀⠀⠀⣸⠀⢠⠀⠀⠙⣧⡀⠀⠀⠉⣻⠖⠈⠲⢤⡤⠤⠴⠚⠋⠀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣽⠀⠙⢦⡙⢦⣠⣀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠒⠋⠉⠉⣿⣀⣂⣐⠆⠀⠀⠀⠀⠸⣶⣼⡇⠀⠀⠘⣧⠀⠀⠀⠻⣍⡀⠉⣠⠇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢧⣀⠀⠙⣦⣽⡟⠛⠒⠚⠛⠉⢁⣠⠞⣁⣀⡤⣴⡦⡀⠀⠀⠀⣏⡀⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣛⣸⡟⠻⣟⢲⡶⠒⣶⣾⣽⣿⣾⣩⣄⣰⣿⣿⣿⣤⡀⠀⠈⠛⠻⡏⠀⠀⠀⠀⠀⠀⠀⠄⣸⠃⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⣸⠏⠀⠀⢈⠻⣟⣿⣿⣞⢻⠟⡏⢀⣬⠟⠳⢶⣤⡀⠀⠹⣄⠀⠀⠀⢀⣠⡴⠚⠁⠀⠀⠈⠻⠿⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠎⠸⠿⠚⢻⡉⡿⢀⣈⣀⣀⡉⠭⠿⣿⣿⣥⡀⢀⣀⣹⣽⣄⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣶⠖⠒⠲⠾⠂⢖⡽⠟⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠅⠀⠀⠀⠀⡈⠛⠛⠉⠉⠀⢱⣴⣶⣿⣿⣿⣿⣷⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡠⠄⠠⡌⠓⢦⡀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⣰⡷⣴⣄⡀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠉⠉⠛⢦⡀⢹⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣷⣌⠛⠛⠛⠛⢿⠿⠟⠛⢻⠿⡿⣿⣿⠿⠋⢧⣀⡀⠀⠀⠀⠀⠀⢀⣤⡤⠤⣤⡴⢋⡧⠤⠤⣤⠀⠀⣠⠞⣡⠏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡶⣤⣀⡀⣀⣴⠏⠈⢻⣷⣄⡀⠀⠌⡇⠀⠀⠸⡀⢃⣿⣿⣴⠚⠉⠉⠉⢻⣷⣶⣖⣾⡟⠁⢀⣀⠠⣴⠾⠷⢶⡶⣯⣤⠞⠁⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⡏⠓⠮⢅⡟⠀⠀⠸⢿⣯⣉⣀⠀⢹⠉⠙⠛⣷⢾⣿⣿⣿⡄⣀⠀⠀⠸⣿⣿⣿⣿⣿⡫⠁⣹⣏⣙⢛⡖⠚⣹⣛⣛⠷⠆⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠒⠚⣻⣧⣤⣄⠀⠀⣿⣿⡿⠿⠿⡦⠤⠔⢺⣾⣿⣿⣿⣶⣤⣭⡭⠤⠟⠛⠛⠋⠙⠻⢦⣜⣧⣭⡟⠀⠀⠙⠲⠦⠭⠷⠖⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⢘⣶⣫⣽⣲⣾⡿⣦⡀⣿⣿⠀⠀⢠⠁⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣼⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⢛⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠹⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀