⣴⢖⡶⣲⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀ .⣶⣶⣿⢾⣒⣦⠀
⣿⣿⡾⣫⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⣨⣷⣌⠳⣝⣿⡄
⣿⡿⣿⣿⣿⣻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢠⣾⣟⣿⣿⣿⢾⣿⠆
⠀⠀⠈⠹⣿⣿⣟⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢷⣀⣀⣀⠀⢀⣀⣀⣶⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣻⣿⠟⠁⠀⠀⠀
⠀⠀⠀⠀⠉⠿⣿⣿⣖⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣯⣸⣿⡋⠉⠀⠈⠙⣯⣏⣏⢿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣿⢿⣿⡿⠉⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠛⢿⣿⠶⢚⣻⣦⡀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡷⠂⠀⠀⠰⣿⣿⣾⣮⢧⡀⠀⠀⠀⠀⠀⠀⣰⣟⡛⠻⢿⡿⠏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣭⣭⣿⣿⣦⡀⠀⠀⣠⣾⣿⣿⡯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣧⣙⣦⡀⠀⢀⣴⣿⣿⣿⢭⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣟⣯⣭⣽⣽⣶⣾⡟⡿⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡷⣿⣶⣮⣽⣯⣟⣻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⣿⣿⣿⡿⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢿⣿⢿⣿⣷⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠿⢿⣧⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠀⣴⣿⠿⢷⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣀⣀⠀⠀⣀⣠⣞⣷⡿⠿⠟⢦⠈⠳⣜⣢⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣞⠞⠁⣠⠾⠿⢿⣮⣳⢤⣀⠀⠀⣀⣀⡀⠀⠀⠀⠀
⠀⠀⠀⢹⡿⣿⢖⣻⣽⣾⣿⡏⠀⠀⠀⠈⠳⣄⠈⠳⣝⢄⠈⠳⣄⠀⠀⠀⠀⠀⢀⡼⠃⢀⣞⠷⠉⢀⡞⠁⠀⠀⠀⠸⣿⣷⣾⢭⣓⣿⣿⣿⠀⠀⠀⠀
⠀⠀⠀⠘⢓⣿⣶⣾⣿⠁⠈⠀⠀⠀⠀⠀⠀⠈⢳⡄⠈⠶⡳⡄⠈⢳⡄⠀⠀⣴⠋⢀⣴⡿⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠉⠘⢻⣷⣮⣿⣟⠉⠀⠀⠀⠀
⠀⠀⠀⠀⢸⡿⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⢮⠣⡄⠙⣦⠚⠀⣴⣿⠋⠀⣠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠘⡿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠑⣬⠞⠁⣠⣞⠟⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡶⠋⢀⢴⠵⠁⣠⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⡴⡛⠉⢀⡴⠋⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⡰⠋⠂⢀⣴⣏⠈⢢⣣⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣠⡞⠂⠀⣠⠞⠀⠙⢦⡀⠙⢭⣂⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⢀⢔⠗⠁⣠⠞⠁⠀⠀⠀⠀⠙⢦⡀⠑⣗⣄⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢀⣴⠛⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠛⠳⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⡀⠂⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠺⣂⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠣⣀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠠⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⡼⠁⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢄⡀⠳⡄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣾⡵⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣿⣄⠀⠀⠀⠀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀