⠀⠀⠀⠀⠀⠀⠀⢀⡀⠸⡆⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠘⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⣼⢃⢀⡴⡷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⡙⠳⢿⡄⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠙⢷⣄⠀⢀⣀⣤⡴⢶⡶⠒⠒⠶⣮⣀⡀⠀⢀⡼⠃⡿⠋⢠⡇⢠⠇⣠⢴⠟⣠⡴⢞⡿⠃⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠐⡄⠀⠀⠀⠀⠀⠀⠙⣄⠀⠻⣄⣀⠀⠀⣀⡀⠀⠉⠳⣦⡀⠀⠀⢀⣽⠟⠋⢉⣴⣾⣿⣿⣿⣿⣿⣶⣮⣿⡳⢾⣅⡀⠇⠀⢸⠇⣼⠞⢡⣯⣾⠟⣴⡯⠖⠚⣻⠿⢾⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢷⠀⠀⠲⣖⠢⠤⣤⣌⣂⠀⠙⠿⣄⡀⠀⠙⠳⣤⡀⠈⠻⣦⣰⠟⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣌⠙⠳⣦⣄⣠⡇⠀⠀⣱⠏⠀⠁⢀⣠⣞⡁⠀⡼⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠸⡆⠀⠀⠘⢦⡀⠀⠀⠉⠙⢶⣤⣘⡷⣤⣀⡀⠈⠙⠆⢀⣼⠋⢀⣼⣿⣿⡿⠿⠟⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠉⠛⢷⣤⣐⠃⠀⠀⠠⠛⣻⠟⠁⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢿⠀⠀⠀⠀⣹⣶⣤⣄⣀⣀⡏⢻⢧⡴⢋⣩⣛⣳⡦⡾⠁⠰⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣦⡡⡀⠈⠛⠷⣦⣀⢀⡞⠱⡇⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠘⣧⠀⢠⣤⣬⣿⣦⣄⠈⠉⣿⣤⠞⣴⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠈⠙⠒⠦⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣧⠙⣄⠀⠀⠈⠛⢿⣦⣀⡧⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⡆⠀⠙⠶⣄⡀⠈⠓⠄⢸⡏⢸⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠈⠻⢷⣄⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢿⡀⠀⠠⣼⣿⠟⠒⠂⣼⠀⢸⣿⣿⣿⠏⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠙⢿⣇⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⣷⠀⠀⠀⢿⣿⠳⢠⡇⠀⣿⣿⣿⡏⠀⠀⠀⠀⡇⠀⢧⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣷⣇⠀⠀⠻⢿⣾⠁⠀⣿⣿⣿⠀⠀⠀⠀⠀⣧⡄⠘⡆⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⡟⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⣿⢳⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢹⣟⣆⠀⢸⠇⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⢸⡌⣇⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⣾⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠘⣇⠘⣆⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠸⡄⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠹⡆⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢹⣦⡀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠹⡄⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢿⠙⢦⡀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣇⠀⠙⣆⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⣇⠀⠙⠲⣄⡀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠸⣇⠀⠘⢧⡈⠳⡀⠀⠀⠀⠀⠀⠀⠈⠃⢹⡄⠀⠀⠈⠙⠦⣄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣀⣸⢦⠀⠀⠙⢦⡙⠆⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠘⣶⣄⡀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠉⢿⣍⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠞⠋⠁⠈⢳⡀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢠⢞⣭⣽⣿⣿⣦⣽⣯⣿⣦⣼⡿⣧⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠉⠓⠄⠀⠀⠀⠀⠀⠀⢹⡆⠀⣀⣤⣿⣿⠟⠛⠛⠛⣿⢿⣿⣿⣿⣄⠙⠂⠀⠀⠀⢰⣧⣀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠘⣇⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⢀⣐⣦⣽⣶⣄⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⢻⡀⢾⣿⡿⠿⣽⠳⣤⣀⣸⡇⠙⣿⣿⡇⠀⠀⠀⠀⠀⠈⣷⠈⠙⠳⠶⣶⣭⣿⣦⣀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⣾⠀⢿⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀⠀⢧⠈⠓⠾⣧⡌⠁⣷⠚⠷⢦⣾⣋⣸⠇⢰⠏⡏⣧⠀⠀⠀⠀⠀⢀⣸⣦⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⡿⡇⠀⢠⡇⠀⢸⡆⠀⠀⠀⠀⠀⢠⡏⣷⣴⣾⣿⡿⠋⢹⣦⡷⣄⠀⠹⣎⠙⠿⠉⠉⠉⠛⠛⠂⠀⠀⠈⠀⠀⠸⣆⠀⠈⢻⡀⠀⠀⠀⠀⡟⢿⠀⠀⠀⠀⠀⠀⠙⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⠿⠶⠟⣿⣾⠁⠀⣸⣷⠀⠀⠀⠀⠀⠸⠃⢹⣿⣿⣟⠹⣏⠹⣿⡀⢈⡿⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠛⠀⠀⠀⠀⢠⡇⢸⡆⠀⠀⠀⠀⢀⡟⠚⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣰⠏⢹⡆⠀⠀⠀⠀⠀⠀⠀⢿⡿⢿⣆⠙⢦⡀⠙⣿⡿⠋⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⣰⡏⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠉⠁⠈⠙⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⡇⠀⠀⠀⠀⢸⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣴⢟⡇⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⡇⠀⠀⠀⠀⢸⣿⣷⡏⢻⣶⣄⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⡾⠃⢸⣇⡄⢀⣾⢿⣆⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⢠⡿⠀⠀⠀⠀⢸⣿⣿⣷⡈⢿⣿⣿⣦⣀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⣸⣿⣇⣾⡇⠀⢻⣆⠀⠀⠀⠀⠀⠀⢘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⡿⣁⡴⠋⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣼⣿⣿⣿⣿⣷⣄⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠛⣽⡿⢻⡇⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⢻⣿⡶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡸⠃⠀⠀⠀⢀⣠⠞⠋⢀⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⣇⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⢻⣧⠀⠙⠻⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠈⠉⠛⠳⢶⣦⣤⣤⣀⣀⣀⣤⠞⠋⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⣿⣿⣿⣿⣿⣿⣿⡟⣸⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠉⠛⠓⠒⠶⠶⠿⠿⢿⣿⣿⠿⣿⣦⣄⡀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢸⣿⣿⣿⣿⣿⣼⣿⠣⣿⣿⡇