⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠿⠛⠛⠋⠉⠉⠉⠉⠉⠙⠛⠻⠿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀ ⠀⠉⠛⢿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⠀⠀⠀ ⠀ ⠀ ⠀⠈⠻⣿⣷⡄⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣾⣿⣿⣿⡿⣶⣦ ⡀⠀ ⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀
⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣿⣿⣿⣿⣿⣿⣿⣆⠘⣿⠟⣷⣦⣄⡀⠀ ⠀⠀⠘⣿⣷⡀⠀⠀
⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣽⣿⣿⣿⣿⣿⣿⣿⡋⢠⣿⡧⠈⠙⣿⣿⣿⣶⣦⡄⠘⣿⣷⡀⠀
⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣯⣩⣿⣿⣿⣿⣦⠘⣿⣧⠀
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⠉⠉⠛⢻⣿⢿⡿⡿⡿⢿⡟ ⠀⢻⣿⡄
⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⠀⠀⠀⠹⠋⠛⠽⠟⠋⠋⠀ ⠀⢸⣿⡇
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡀⣦⣪⣿⣿⣿⣿⣿⡿⠹⠹⣿⣿⣿⣿⣷⣓⣴⢀⡀⡀⠀⠀⠀⠀⠀⠀ ⠀⢸⣿⡇
⣿⣿⠀⠀⠀⠀⠀⣤⣳⣶⣿⣿⣿⣿⣿⣿⡏⡿⠃⢀⡀⠈⠉⠛⠻⠿⣿⣷⣽⣟⠷⠶⡄⠀⠀⠀⠀⠀⢸⣿⡇
⢸⣿⡆⣤⣄⣻⣷⣿⣿⣿⣿⣿⣿⡿⣿⡁⠇⠁⠀⢻⡞⠉⠀⠀⠀⠀⠀⠙⠻⠿⠟⠛⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣄⣠⣠⡴⠿⣣⣿⡵⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⣿⣿⣿⣿⣴⡞⠋⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀
⠀⠀⠹⣿⣿⣿⣿⣿⣿⢻⠿⣆⠘⠢⡘⠘⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃⠀⠀
⠀⠀⠀⠙⣿⣿⣿⡏⠙⢦⣣⠈⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢀⣄⣀⢀⣦⣀⠀⢀⣠⣾⡿⠃⠀⠀⠀
⠀⠀⠀⠀⠈⠻⣿⣧⡀⠀⠉⠣⡀⠀⠀⣠⣤⣀⣠⣤⣐⣴⣷⣅⣤⣦⣟⣿⢿⣸⣿⣿⣾⣿⣿⠟⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣄⣀⣴⣦⣤⣽⣟⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣿⠰⣦⠀⠀⠀⠀⠀⠀⢀⣼⢡⠾⣷⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⢷⠛⠀⠀⠀⠀⠀⠀⠀⠉⢸⡀⢸⣇⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⢹⡀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣏⢉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡤⠞⠃⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⡟⡈⠉⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣠⠄⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⣷⣤⣀⠀⠀⠀
⠀⠀⠀⠸⣇⠙⢶⢤⣄⣁⣀⣀⣈⣁⣤⠤⠔⢾⣽⠁⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⣄⡿⢿⣿⡆⠀⠀
⠀⠀⠀⠀⠙⠦⣜⠛⠳⣄⡀⢀⡼⠟⠛⠛⣶⠞⠁⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠈⠁⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣭⡟⢒⣒⠒⣋⠁⠀⠀⢸⣅⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⠙⠀⠀⠀⣧⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢩⡇⣠⠿⠛⠧⣄⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀⡀⠀⣀⡀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⢸⠀⠀
⢠⣴⣶⣾⠶⠚⠁⠀⠀⠀⢸⡼⠁⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⢻⣷⡶⠀⠀⡟⠀⠀⠀⠀⠀⣠⣿⣇⡀
⠈⠻⢿⡀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⢠⣿⣆⣨⣿⡆⠀⠀⢻⡄⠀⠀⠀⠀⠈⠛⣹⠁
⠀⢾⠽⠒⠒⠒⠒⠒⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠓⠒⢶⡒⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⣠⠞⠉⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⢹⡀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣳⠶⡞⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⣧⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢠⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡄⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢻⡷⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⣼⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠘⣧⣤⣤⣤⠴⠚⠉⠀⢀⣾⡏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢀⡴⢫⡞⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⡤⠖⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⢀⣤⡤⠟⠓⠒⠲⠤⠤⢿⣄⠀⠀⠀⠀⠀⠀⣏⣩⡤⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣴⣿⣍⣻⠀⠀⠀⠀⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠻⠿⢵⡿⠽⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⡤⢤⡶⢿⡽⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣳⣿⣛⣙⠂⠀