Hook Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠉⠁⠀⠀⠈⠉⠛⢿⣿⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⣾⣿⡏⠉⠉⠙⣿⣿⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⠰⣿⣿⣇⠀⢀⣰⣿⡟⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠟⠁⠀⠀⢀⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣤⣤⣤⠈⠛⠿⠋⠁⠀⠀⠀⠀⠿⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⢀⣤⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠐⠿⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⣿
⢀⣿⡇⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⣿
⢸⣿⠁⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⣾⣿⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⡄⠀⠀⠀⠀⢸⣿⡿⠛⠛⠻⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠁⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⣿⡆⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠸⣿⣇⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣸⣿
⠀⢻⣿⡆⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿⡿
⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⠀⢶⠄⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠈⢿⣿⡀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠘⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⣶⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⡿⠿⠉⠻⠿⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⢀⣀⣀⣀⣀⣀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣴⣧⠻⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣿⣿⣾⣿⠃⠀⠀⠀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠟⠁⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣼⡿⠟⠛⠋⠁⠀⠀⠀⠀⢯⠀⣠⡔⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢲⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡆⠰⠒⠀⢸⣀⠀⠠⠄⢀⣀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠘⠁⠈⠀⠀⠀⠀⠈⠈⠀⠈⠜⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠏⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠓⠀⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⡟⠀⠀⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣾⣷⣾⣿⣶⣶⣄⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠩⠂⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠣⠈⠁⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠰⠆⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⣿⣿⣿⡿⣻⣿⣻⣿⣿⣧⠘⣞⠁⠉⢉⡙⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⢃⣼⣿⣿⣿⣮⣿⣏⢛⣿⣿⣿⡿⣧⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⣾⣿⣿⣻⣿⣟⣿⣿⢬⣿⣿⠿⠠⠋⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⡀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣀⣾⣿⣿⣿⣯⣽⣿⣴⣿⣌⣿⣿⢰⠇⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⠞⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣤⣿⣯⡙⠳⠀⠀⠀⠀⠀⠀⠀⣠⣾⣾⣿⣿⣿⣿⣿⣷⠼⣯⢳⡟⢹⣴⣿⡆⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣿⣿⣿⣿⣿⣷⣤⣀⡀⢀⣠⣶⣾⣿⣿⣿⣿⣿⢿⣿⣿⡿⣶⣾⣿⣧⣼⣿⣿⣿⣧⡀⠀⠆⠰⠄⠀⠀⠀⠀⠀⢠⣿⠷⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣿⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢟⣿⣿⣿⣏⢻⣶⣄⡙⢿⣧⡤⢿⡹⢿⣿⣶⣤⣀⣠⠀⢀⣀⣠⡄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⢫⣾⣿⠟⣛⣿⡍⣿⣿⣿⣯⡁⠢⣼⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣾⣿⣿⣿⣿⠉⣳⣽⣍⠻⡇⠸⠏⣿⡿⠿⢿⣿⡅⢸⣿⣿⣿⣧⡰⠙⣿⠉⠾⣿⣿⠿⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡄⣿⣿⣿⣧⣿⣆⣄⣿⢃⠀⣠⡝⣿⣿⠻⠿⠿⣿⣧⡀⢡⢀⣰⣿⠏⣗⣸⣿⣿⣋⣩⣽⣟⡳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⡉⣹⣿⣿⣿⣿⣾⢂⣙⣷⠏⠀⠀⠀⠀⠈⣿⣷⣮⡎⠁⠑⠞⣠⣿⣿⣿⠏⠀⠀⠉⢻⡎⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣷⠏⠉⣿⡭⠿⣭⣟⣟⣥⣤⣾⣷⡦⠀⠀⣿⣿⣿⣷⡄⠀⡤⠻⣿⡿⠁⠀⠀⠀⠀⠈⣷⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢈⣅⡿⠈⠀⣴⣿⡁⠀⢈⡍⢉⡿⣿⠻⣿⠻⣿⠞⣿⣿⣿⣿⣷⣤⣶⡆⣿⠁⠀⠀⠀⠀⠀⢠⡇⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⣤⣿⣿⢿⡀⣀⠟⠀⣰⣶⣶⣤⣤⣴⣯⣴⠾⢶⡿⠶⣿⣿⣿⣿⣿⣿⣁⠈⣉⠉⠀⠀⠀⠀⡠⠋⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠴⠿⣿⣿⣿⣿⣵⣄⢠⣾⣃⢉⣹⠯⠞⠥⢸⣾⠇⡄⠟⢹⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⢀⡴⢶⣿⣷⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣩⢿⡏⣟⣸⠃⠀⠰⠀⡾⢁⡔⢽⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣾⣿⣿⣿⣿⠋⠀⠚⢷⣤⣧⣈⣠⡶⠂⢠⡏⢀⡿⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡾⣏⣀⣀⣤⠖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢈⣴⣿⣿⣿⣿⣿⣧⣤⣤⣄⣀⠈⠉⠁⢀⡤⠟⠀⡸⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣉⣽⡛⢋⣱⡿⣧⡀⠤⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⣾⢫⣿⣿⣿⣿⣿⣿⣷⣾⠿⠻⠛⠶⢾⣿⠃⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠹⣿⣿⣛⠿⠁⠘⠛⣻⡄⠹⢶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⢇⡻⣿⡿⣿⣾⡿⠛⠛⠁⠰⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡿⣿⣿⡷⣿⢿⣷⢀⡘⣿⢧⣴⠆⠀⢁⠤⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⡇⣾⣼⣿⠁⡤⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣻⣿⣽⡿⣿⣺⣿⣿⣿⣿⣸⠛⣥⢨⡁⢠⠄⠀⢀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣿⣿⣿⠠⠀⠀⠀⠀⠀⠀⢀⡾⢁⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠘⠣⠾⠤⠤⠾⠛⠛⣫⣍⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠹⣿⣿⠀⠀⠀⠀⠂⡀⣠⢟⡥⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣻⡗⣿⡆⣠⣴⣶⣿⣿⡿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣧⣠⣀⣀⢛⣷⡿⢻⡅⣼⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣿⠿⠃⣨⣵⣿⣻⣿⣿⡿⣴⠺⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡷⠹⠿⠟⣿⡇⠘⢻⡄⠛⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠋⣀⣼⣿⣿⣿⣿⣿⣽⣿⣿⠿⡥⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⠁⠠⠤⡎⠀⠀⠈⡘⠀⠀⠁⠸⣿⣿⣿⣿⣿⣿⣿⣿⢺⣎⡁⡆⢹⣿⣿⣿⣿⠛⡫⣾⣃⠀⡞⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠛⢁⣴⣶⣤⣤⣀⠀⠀⠀⠀⠀⢀⠀⠹⣿⠹⣿⡿⠿⣻⣷⡟⣽⣿⢶⣬⣙⠷⡿⠛⠖⠃⠀⠀⣀⣥⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⡈⠸⠋⠘⠁⠉⢱⠆⡀⢀⠀⠈⠁⡞⣤⣾⣿⣤⣦⣟⢿⣶⣿⣿⠼⠋⠙⠛⢜⣃⣰⣤⣶⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⡌⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⡟⢫⡿⠝⣿⣿⣿⡆⠻⣿⣿⠀⠘⢛⡛⣦⣿⡿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣠⣴⣶⣾⣶⣶⣦⡀⣄⣤⡦⠀⠘⢣⣄⣴⣶⣿⡷⠀⠉⠛⡶⠶⣾⡽⡿⠓⡄⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠟⠊⠛⠻⠿⠊⠉⠋⠑⡟⠃⠀⠐⠶⠟⠋⠏⠉⠋⠀⠀⠀⠀⠈⠀⠁⠐⠟⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢰⠞⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⣥⡍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⠈⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣭⣽⡅⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣶⡐⢶⣾⣾⣿⣿⡿⢫⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣭⣉⣉⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢈⣿⡃⢀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣄⣀⡀
⠀⠀⢀⣿⡹⣯⡀⠀⠀⠀⠀⠀⣤⣤⣼⣷⣿⣂⠀⠀⠀⠀⠀⣿⣴⣂⣀⡦⡄⠀⠀⠀⣴⣿⣿⠿⢻⣿⠶⠀⠀⠀⠀⢰⣿⣿⣿⢿⡀
⠀⠀⠘⠿⠶⠿⠃⠀⠀⠀⠀⠀⠛⠛⠷⠶⠋⠁⠀⠀⠀⠀⠤⠿⠿⠽⠿⠧⠤⠀⠀⠀⠀⠙⠿⠾⠿⠋⠀⠀⠀⠀⠀⠸⠿⠿⠿⠾⠁

⠀⠀⢠⣶⣶⣶⡀⠀⠀⠀⠀⠀⢀⣠⠦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠴⠶⡄⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖
⠀⠀⢸⣿⣿⣷⢲⡄⠀⠀⠀⠀⠾⢀⣞⠹⣿⠓⠀⠀⠀⠰⣟⣿⠀⠀⢸⣏⡷⠀⠀⠀⠀⣶⣒⣚⣛⣓⡂⠀⠀⠀⠀⠀⣰⣟⢽⣦
⠀⠀⠈⠙⠛⠙⠚⠁⠀⠀⠀⠀⠦⠌⠏⠠⠟⠁⠀⠀⠀⠀⠈⠁⠳⠶⠂⠉⠀⠀⠀⠀⠀⠀⠻⠿⠿⠯⠇⠀⠀⠀⠀⠀⠙⠿⠾⠋

⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡠⠀⠠⡀
⠀⠀⢾⣿⣾⣿⡧⠀⠀⠀⠀⠀⢰⣿⣿⣤⣤⠀⠀⠀⠀⠀⢀⡴⣤⡐⢀⠇⠀⠀⠀⠀⠀⣿⣿⣽⣭⣿⡛⠋⠀⠀⠀⢰⠀⣺⠀⢨
⠀⠀⠈⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠙⠿⠟⠋⠀⠀⠀⠀⠀⠘⠯⠿⠃⠁⠀⠀⠀⠀⠘⠛⠛⠒⠚⠓⠛⠁⠀⠀⠀⠀⠀⠂⠤⠄⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⡤⠴⠒⠒⠒⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⡏⠙⠉⠀⠀⠀⠀⣶⢲⠏⢹⠛⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡇⠀⠀⠀⠀⠀⠈⠉⢉⣙⣿⠢⣄⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠉⠓⠒⠒⡻⠛⠉⠉⠉⠉⠑⢮⣦⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣀⠤⠤⣄⡀⠀⢱⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⡞⢁⡴⡶⠄⠹⡄⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣇⠀⣭⣯⠇⢠⠇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⠀⠀⠈⠳⠤⠥⠴⠋⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢠⣀⣤⣠⢀⠀⣠⠄⢠⢄⣄⡤⣠⡄
⠀⠀⠀⠀⠀⠀⢠⣤⡀⣠⣀⡀⣀⢄⣄⣀⢠⠀⣈⠄⣀⣀⢠⢠⣠⣄⣀⡠⣠⠀⣀⢀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢰⣶⣶⣷⣶⣾⣶⣶⡆⠀⠀⠀⠠⣤⢀⣹⣿⣅⠠⣤⠀⠀⠀⠀⠀⣦⣤⣤⣀⣯⠗
⠀⠀⠀⠀⢸⣿⣿⣏⡲⢽⣿⣿⡇⠀⠀⠀⣾⣿⣾⣏⡩⣿⣷⣿⣦⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⢸⣿⣿⣷⣭⣼⣿⣿⡇⠀⠀⠀⠛⠛⣿⣷⣵⣿⣿⠛⠋⠀⠀⠈⠙⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠈⠉⠛⠉⠉⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣀⣤⣶⣳⣦⣤⣀⠀⠀⠀⠀⠀⣀⣈⡋⢀⣄⠀⠀⠀⠀⠀⠀⢀⣀⢠⣦⢘⡟
⠀⠀⠀⠀⠀⣿⠀⣿⢸⡇⢸⡇⠀⠀⠀⠀⠸⣿⣸⡇⠘⠚⠁⠀⠀⠀⠀⠀⢘⣟⢰⣷⢸⣿
⠀⠀⠀⠀⠀⣿⠀⣿⢸⡇⢸⡇⠀⠀⠀⠀⢶⣌⡉⠤⣶⣦⣀⡀⠀⠀⠀⠀⢸⣿⢸⣿⢸⣿
⠀⠀⠀⠀⣴⣿⣶⣿⣾⣷⣾⣷⡄⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠇⠀⠀⠀⠀⠾⠿⠾⠿⠾⠷⠆

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣽⣭⣭⣽⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣐⣳⣧⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⢉⡱⠀⠀⠀⠀⡀⠀⠦⠄⠠⢒⣼⠒⢐⢚⣺⠀⣻⡃
⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠂⠈⢉⠥⠤⠄⠒⣒⢲⡂⢎⢫⣻⡦⠭⢯⠟⡖⠋⠉⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⢀⡐⠀⡄⠀⠂⠀⣀⠀⠀⠀⠀⠓⠂⠋⠉⠉⠉⠁⠹⠀⠚⠚⠁⠀⠀⠖⠀⠀⠐⠨⢍⣀⠀⠀⢄
⠀⠀⠀⠤⢶⡤⠀⠀⠀⣠⠐⣲⣾⡞⠨⢵⣶⠅⠐⣖⣈⢥⡀⠶⢁⠁⠀⠀⢀⠀⠀⠀⢈⡐⢁⡢⣀⠀⠀⢀⠀⢀⠀⠀⠀⠀⣈⣤⠴⠧
⠀⠀⠑⠤⡐⠀⠐⠀⢄⠙⠒⢋⣜⣿⣀⢘⣻⠤⢀⣰⣏⡿⣥⡀⠀⠁⢂⡐⠀⠌⣃⢠⠐⠀⠦⢄⠠⠑⠤⣏⣀⣈⣟⣶⡞⠉
⠀⠀⠀⠀⠈⠁⠠⡀⠀⢢⠄⠀⣠⠤⢉⣉⠩⢅⠀⠒⠾⣟⠉⠤⠴⠂⢀⡌⠀⢠⠈⠃⠄⢀⣀⡀⣰⡄⠴⠄⠵⢤⠶⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⢡⣒⡶⠃⠀⠐⠁⠄⢬⡼⠿⣾⠷⣀⠀⠀⢉⡁⠀⡀⢀⣀⡀⣀⣠⣠⣜⣠⠒⠡⢶⡄⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⡛⠉⠁⠀⠈⣿⣦⠟⠋⠛⠛⠋⠉⠀⠉⠁⠈⠀⠁⠀⠀⠌⠠⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣢⣀⡰⠆⠴⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠴⠖⠚⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⠂
⠀⠀⠀⠀⢰⣶⡶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⢤⡀⣠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡄
⠀⠀⠀⠀⢸⣿⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣽⣿⡇
⠀⠀⠀⠀⢀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠈⢠⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣷⣶⣤⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⢀⠁⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣯⣿⣿⣿⣿⣽⣿⣷⣿⡿⠃
⠀⠀⠀⠀⡀⠒⡒⢀⡀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢿⠿⢿⣿⡿⠛⠻⠛⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟
⠀⠰⠖⠞⠦⠢⠔⠀⠶⠂⠷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣄⠀⠀⠀⠀⠀⠀⠀⠀⠺⠘⠃⠂⠃⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣄⣍⠠⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⡸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢦⠀⠀⢄⠀⠀⡇⠀⡇⠀⡇⠀⢰⠀⠀⡞⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠐⣆⠈⢆⠀⠸⡀⠀⡇⠀⡇⢀⡇⠀⡇⠀⡼⠀⢠⠂⠀⡠⠋
⠀⠀⠀⠀⠠⡀⠀⠀⣄⠈⢣⡀⠈⢆⠘⡆⠀⢇⠀⢧⠀⡇⢸⠀⢸⠀⣰⠁⣰⠃⢀⡜⠁⢀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠢⣄⠈⠣⡀⠑⣄⠈⢆⠘⡄⠸⡀⢸⠀⡇⣸⠀⡇⢠⠃⡰⠃⣠⠊⢀⠔⠁⢀⡴⠊
⠀⢀⡀⠀⠀⠤⡀⠈⠳⢄⠈⢢⡈⢢⡈⢧⠹⡀⢇⢸⠀⡇⡇⢸⢀⠏⡰⠁⡴⢁⡴⠋⣀⠔⠉⢀⡤⠂
⠀⠀⠉⠒⠤⣀⠈⠑⢤⡀⠑⢤⡙⢦⠑⣌⢧⠱⠘⠈⠀⠁⠃⠃⠞⡰⢡⠎⡠⠋⡠⠞⢁⡤⠚⠁⣀⠤⠒⠉
⠀⠀⠠⠤⣀⡀⠉⠒⠤⣈⠓⢤⡙⠦⡑⠈⣠⠔⠚⠉⠉⠉⠉⠑⠲⢄⡁⠊⡴⢊⡤⠚⣁⡤⠖⠋⢁⣀⡤⠄
⠤⢤⣀⣀⠀⠉⠙⠒⠤⢄⣙⠲⢬⠃⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠘⡡⠖⣋⡡⠤⠒⠊⢉⣀⣀⠤⠤
⠀⢀⣀⣀⣉⠉⠑⠒⠒⠤⠬⢍⠃⢰⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⢐⡩⠥⠴⠒⠚⠉⠉⢀⣀⣀⡀
⠀⠀⠀⠀⠉⠉⠉⢉⣉⣑⣒⣲⣤⣿⣤⣤⣤⣿⣿⣿⣾⣶⣷⣴⣶⣤⣶⣦⣼⣿⣒⣒⡚⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⢄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢸⢸⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣒⣚⠚⢒⣪⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⣿⣿⣿⡇⡔⠉⠉⠉⠉⠉⠉⠉⠉⠉⡭⠛⠉⠛⠭⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢱⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠉⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⠤⢀⣀⣀⣈⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢓⠁⠀⠀⠀⠉⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡃⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⣄⡀⠀⢀⣴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠘⠒⣶⠒⡒⢲⠒⡖⢲⠒⣖⢲⡇⡇⠀⠀⠀⠀⠀⠀⢀⠜⠁⢣⢨⣛⡁⡆⠈⠢⠀⠀⠀⠀⠀⣔⣖⡤⡾⣸
⠀⠀⠀⢱⠆⠲⠐⠆⠰⠀⠆⠰⡀⠇⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢩⠉⡏⠀⠀⠀⡇⠀⣴⡎⠉⠹⠞⠉⠫⠏⠉⢳
⠀⠀⠀⠀⢯⡌⢥⠩⡌⢥⠨⡄⢡⡄⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣘⠀⡃⠀⠀⠀⡇⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠘⡖⠐⠂⠒⠐⠂⠒⠐⡇⠣⣀⣀⣀⣀⣀⣀⣸⣤⣤⣤⣬⣶⣥⣤⣤⣤⣇⣠⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠓⠤⠤⢤⢤⠤⠤⠤⠤⠤⠤⠤⡤⠤⠤⠤⡤⠤⠤⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⢠⡔⡲⡢⡤⠤⠤⠤⠤⠤⡤⣚⡸⡀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡇⠀⠀⣜⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠮⣒⡣⠋⠉⠉⠉⠉⠉⢇⣣⣋⠇⠀⠀⠀⠰⣍⣉⣉⣉⣉⣉⣉⣉⠇⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠄⠒⠂⠀⠒⠒⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⢄⣀⣀⠉⠁⠀⠀⠀⠀⠀⣈⣉⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢙⠦⣈⠛⢦⠀⠀⢀⠴⣶⠶⢤⣀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠑⠁⡎⣸⣿⣿⣆⠙⣿⣞⡾⣥⣿⣍⣉⣙⣒⠲⢤⣤⣄⢂⡤⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⡆⢫⠙⣿⣟⠛⢋⢿⣝⢿⣿⣶⣌⠙⢷⣶⣖⠚⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃⣹⢿⣿⣷⠘⢧⡀⠛⠃⣸⡼⣯⡼⡟⢻⣿⣿⣦⣮⡻⣿⣃⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⣱⣾⣃⣏⣻⣿⣇⢀⣹⣧⠀⢡⠄⡏⠃⣿⠇⠟⢉⣿⣿⣿⣮⡻⣤⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡀⠀⠴⠶⠶⠦⢬⡻⠿⢿⣶⡶⠚⡈⠀⣼⣿⡿⠄⠀⠉⢹⣿⣷⣽⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢾⣤⣒⠢⡀⢀⡀⢄⠒⠤⠄⡴⢺⣿⠟⠋⠀⢀⣴⢻⡿⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣌⡑⠛⠦⠕⠒⠊⠀⠀⠈⠓⠂⠒⢫⡍⣨⢁⣾⢻⣿⣯⢢
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣷⢱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢈⡏⣼⢿⣿⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡞⢆⠰⣤⣄⣀⠀⠀⠀⠀⠀⣉⣾⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⢧⡀⠳⣜⠿⣮⠙⢶⣄⠀⠀⢈⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠋⣻⣷⣦⡈⣆⣼⣿⣷⣬⡛⠿⣄⢼⣿⣷⡄⠈⣺⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⠿⢿⣿⣿⢿⣿⣿⣿⣾⡿⡟⠿⣹⣷⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠄⡀⢀⡀⠀⠀⠛⣦⣹⣿⣷⠻⣿⣿⡈⠁⠉⢀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢶⣷⠀⠀⣿⣿⣿⣿⣯⠈⢉⣟⣿⡶⠒⢹⠻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⢀⣮⣿⠀⣷⣿⣿⣿⢿⡟⢰⠿⣫⣟⣡⣤⣼⣶⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⠏⣼⣿⢇⣿⡟⣼⡠⠴⣺⢿⣻⣿⣿⣿⠻⣿⣟⠀⠙⠆
⠀⠀⠀⠐⢶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶⡶⠶⠀⠀⠀⣠⣴⣶⣿⣿⡿⠋⣰⡿⢋⣾⠟⡼⠉⠀⢘⣵⢿⣿⣿⠿⠏⠙⠈⠛⣿⡄
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⢂⣼⡿⣛⣽⣾⣿⠁⡈⠉⣴⡿⠃⠈⠀⠀⣰⠟⣵⠿⠟⠉⠀⠀⠀⠀⠘⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠫⣾⣿⠿⠋⠛⠛⠁⠸⠋⠀⠀⠀⠀⠈⠁⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⡿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⢿⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣀⣄⠀⣤⣀⠀⠀⠀⢰⣿⣿⣿⣿⣻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣾⣿⣿⣇⢿⣿⣧⠀⠀⢸⣿⣿⣿⣿⣿⠋⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣦⣴⣤⡼⠿⠟⠛⠛⠀⠈⠛⠀⠀⠀⠀⠀⠐⠐⠀⠠⠄⢠⠀⠀⡠⠀⢀
⠀⠀⠀⠀⣾⣿⣸⣿⣉⠘⣿⡟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠈⠛⣹⣿⣿⣀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⢀⣰⣦⣤⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠙⠙⡀
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣷⣤⣀⠀⠀⠀⡙⠉⠉⠛⠻⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠃⠒⢦⠀⣀⣠⣿⣀⣀⣀⡀⠀⢀⠤⠤⠀⢴⣶⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⢡⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠻⢿⣿⡿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡁⠀⢼⣏⡀⠀⠀⠸⠻⠿⠿⢿⠋⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣷⠺⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⢰⣿⡄⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⣿⣿⣶⠶⠿⠿⠿⠶⣾⣿⣿⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡾⠛⣁⣤⣴⣶⣶⣶⣦⣤⡉⠙⢶⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣠⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣤⣶⣿⣿⣿⣤⣽⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⣀⣛⣛⣛⣂⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠶⠾⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣉⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⠍⠉⠉⠉⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣯⣭⣭⣽⣯⣬⣽⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣿⣿⣿⣯⣿⣿⣭⣿⣯⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣄⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣻⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣻⣾⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣻⣾⣿⣿⡿⠿⠿⢃⣀⣀⣀⣀⣤⠴⠶⡄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣾⣿⢟⣩⣴⣶⣶⣾⣿⡿⣻⣿⡟⠡⡜⢿⣼⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣶⣤⣀⠸⣿⣿⠟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣀⢷⢹⢸⡿⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣮⡉⣸⣿⡿⣿⣿⣿⣿⣿⣿⡟⠛⠀⡟⠀⡟⡾⡇⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣇⢹⣿⡇⣿⣿⣧⣈⣉⡙⢿⣷⣶⣤⣀⡀⠀⢙⣦⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⣛⣛⣻⣽⡔⢮⣅⣈⣛⣛⣫⣥⣾⣷⣮⣝⠻⢿⣿⡿⠿⠛⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠛⠿⡿⣦⣝⡛⠿⠟⠛⢋⣛⣿⣿⡿⠿⠿⠟⠁⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠈⠉⠉⢉⣴⠟⠉⠉⠀⠙⠃⠀⠀⠀⠐⠳⣶⣶⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⣼⡿⠿⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⣿⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣶⣤⣤⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡤⣤⣤⢀⣤⣤⣤⣤⢠⣤⡄⣤⣤⡤⣤⣤⢠⣤⣄⢠⣤⡄⢀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡌⠁⢸⣿⠘⢿⣷⣬⣉⠀⣿⣦⣼⣿⠀⣿⡇⢸⣿⣿⣦⣿⠀⣿⡏⢈⣛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⣾⣿⠼⡿⢶⣿⠿⢰⣿⡆⢼⣿⠤⣿⣧⢼⣿⠌⠿⣿⠀⠻⣿⡶⢿⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣷⣶⣦⣄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠟⠋⠉⠉⠀⠀⠉⠉⠙⠻⢿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠉⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠺⠿⠿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠀⠀⢀⣴⣶⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⠁⢸⡇⠙⣿⡷⠀⠀⠀⠀⠀⠀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣷⣼⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⡿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠀⠀⠀⠀⠀⢰⣿⣧⠀⢸⡇⠀⣿⣿⠇⠀⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣼⣧⣶⣿⠟⠀⠀⠀⠀⢀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡀⠀⠀⠀⠉⢹⡏⠉⠀⠀⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⠉
⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⡿⠿⢿⣿⣷⣦⣄⣀
⣴⣿⠛⠿⢿⣿⣶⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣶⣤⣶⣤⣀
⢻⣿⣤⣀⠀⠀⠈⠉⠙⠛⠿⠿⣿⣷⣶⣦⣤⣿⡟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢋⡛⢿
⠀⠉⠛⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣷⣶⣶⣶⣶⣶⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠈⢁⣾
⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⡿⠿⠿⣿⣿⣿⠃⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣶⣤⣤⣶⣶⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠻⣿⣶⣤⣾⡿⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣥⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⢿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠉⠀⠈⠉⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⣿⣿⣧⣴⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠛⠛⠛⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢿⣿⣿⡇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣦⣤⣀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠋⠙⠁⢠⣿⣿⣿⡟⠀⠀⠀⣀⣤⣴⣶⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⣄⠀⠙⢿⣿⣿⣿⣄⣸⣿⣿⣿⡿⠋⠀⣠⣶⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⠙⠛⠛⠛⠛⠋⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⡀⢀⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⣿⡇⢸⣿⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡉⢉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡇⢸⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⠄

⣤⣀⡀
⣿⣿⣿⣿⣶⣶⣤⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⠀⠀⠀⠀⠀⢀⣤⣤⣶⣾⣷⣦⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣰⣿⣿⠿⠟⣿⣿⣿⣿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠙⠋⠉⠀⠀⠈⠻⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⡏⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡄⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⣀⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣾⡧⠀⠀⠀⠀⠀⠀⠀⢠⣷⣶⣆⠈⠁⣀⣠⣤⣶⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⢒⢤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⠉⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⠄⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⡿⢤⡀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢼⣆⣀⠀⣀⣤⣶⣿⣟⣛⣿⣿⣿⣿⣿⡟⢿⣷⣶⡅⠀⠀⠀⠀⢀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⢀⣼⣷⡘⢷⣽⡿⠿⠿⣛⡋⡉⠛⠛⠻⣾⣿⠏⣴⣿⣶⣄⠀⠀⠓⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣞⠉⠀⣴⣿⣿⣿⡿⠈⠁⠀⠀⠀⠀⠀⠈⣉⣉⣉⣡⣄⣜⣷⣿⣿⣿⣷⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⡁⢈⣧⣾⣷⣾⣿⡯⣄⡀⠀⠀⣠⠴⡒⡋⠉⠉⠀⠈⠠⠀⠀⣉⣻⣿⠿⢿⡿⢿⡿⣿⡿⢟⣻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢍⣉⣿⣿⡿⠋⠀⠀⠉⣽⠟⠉⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⠀⠀⠞⠐⣛⠛⠈⢀⡼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡅⢠⠀⣱⠟⠇⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣅⠀⠠⣄⡈⣿⣦⣾⣥⣇⣤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢫⠀⣒⠿⡈⠠⠂⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠙⣛⠶⣄⠈⠁⠉⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡷⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢦⣄⡠⠀⠀⠀⣤⠀⠀⣰⡦⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡷⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠰⢖⣛⣿⡦⠖⢻⣿⣿⣿⠦⠂⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⡿⠉⣹⠃⠀⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣄⠀⠀⠀⠀⠀⡀⢰⢖⡓⢲⠶⠚⠁⠀⢰⠇⣰⠏⢇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠙⣿⣿⣿⣷⣀⠀⡀⡠⠚⠙⡟⡇⡿⠀⠀⠀⠀⢸⢞⣅⣴⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⢻⣿⣿⣿⣷⣾⣄⡜⡇⢀⣿⠇⠠⣀⠀⣀⡄⠀⢻⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡀⠀⡱⢄⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣆⠀⢹⠃⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⡐⠀⡄⡌⠙⢻⣶⣯⣿⠿⢿⣿⡿⠿⠿⣿⣿⣿⠿⠛⡿⠲⠁⠀⡠⡀⠡⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢳⠮⠥⡘⠃⠀⠀⠀⠙⢿⣷⣆⠈⠚⢿⣿⣿⣿⡿⠁⠀⡰⠃⠀⠀⠀⠈⠛⣀⠭⢶⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠐⣤⠀⠀⠀⠀⠙⢻⡄⠀⠈⠹⣿⣿⠀⠐⡏⠀⠀⠀⠀⢀⡔⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣥⠀⠀⠀⠀⠀⠀⠿⠒⠦⣤⣿⢟⠁⠀⣧⠀⠀⠀⠀⢤⡁⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠻⣄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⡀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⠀⢀⡀⠠⡆⠀⠀⠀⠈⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣴⣿⣸⣧⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⠀⠡
⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡙⣿⣄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣆⢻⣷⡘⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠹⡷⠈⠉⢀⢐⠂⠀⠀⠀⠀⢀⠀⠺⠇⠀⠀⠂⠀⠀⠀⠈⠉⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⢈⠛⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠒
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡔⠀⠀⠀⠀⠆⠀⠀⠀⠺⢠⣠⠔⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⣐⠂⠁⠐⠚⡁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠑⠂⠂⠐⠒⠁⠀⠀⢀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠂⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⢀⣠⣴⣶⡿⠟⠋⠁⠀⠀⠀⠀⠒⠠⢀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⠀⡀
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠳⠜⠜⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣠⣴⣾⣿⡆⠀⠀⠀⠀⠡⠀⠀⠀⠀⠈⡀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈
⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠲⠒⠂⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠄⠀⠠⠶⠀⠀⠀⠀⠀⡀⠀⣠
⡿⠋⢁⡄⠀⠙⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⢠⠜⠀⠀⠀⠐⣠⣴⣿⣿
⠁⠀⠺⡤⠀⠀⢸⣿⣿⣿⡿⠀⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⡄⠀⠀⠤⠄⠀⣼⣿⡿⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠒⠀⠀⠀⠀⠈⣿⣿⣿⣿
⣿⣶⣤⣤⣴⣾⡿⠋⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠈⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠠⠄⠀⠀⠀⢻⣿⣿⣿
⣿⣿⣿⣿⠟⠁⠈⠀⠁⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⡐⠠⠈⠀⠀⠀⠀⠀⠸⣿⣿⣿
⣿⡿⢋⠀⠀⠀⠑⠀⡀⠀⡠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢈⠀⠀⢀⠄⠀⠀⠀⠀⠀⢿⣿⣿
⠁⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠈⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠲⠄⠀⠀⠄⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢄⣀
⡠⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢨⠖⠐⠀⠁⢀⠤⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠰⠀⠐⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲
⠀⠀⠀⠁⠀⠀⠀⡀⠀⣄⠀⠈⠀⠀⠀⢀⠞⠋⠀⠀⠀⠀⠀⠀⢓⡆⠀⠀⠀⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡖⠀⠘⡅⠀⠀⠀⠀⠀⢉⠂⠐⠀⢹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠋⠀⠉⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠒⠢⡁
⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⢇⠀⠁⠿⢤⡀⣀⠠⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣦⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠹
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠄⣀⠫⠀⠀⡀⣀⡴⡃⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⣠⡾⣿⢻⡿⢷⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠛
⣇⠀⠀⠐⠀⠀⠀⠀⣀⣤⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣀⣿⢸⣷⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘
⠀⠄⠀⠀⠀⠀⢠⣿⣯⢿⣿⣷⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣾⣿⣶⣄⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠈⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⢠⣤⣿⣤⣤⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣿⣅⣿⣀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠈⠉⣿⢙⣷⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠛⠛⣿⠛⠃⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠉⠀⢀⣠⣴⠟⠁⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⠶⠶⣦⣤⣀⠀⠀⠉⠛⠛⣿⠛⠛⠉⢀⣴⠟⠋⠉⠉⠉⠙⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠈⠙⢷⡄⠀⠀⠀⣿⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⣿⠀⢠⣿⣤⡄⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⡀⠀⠀⠐⠻⣾⣷⡀⠀⣿⣴⠾⠿⢷⣤⣤⣤⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠟⠛⠋⠉⠛⠿⣿⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢶⣶⣶⣶⣶⣶⣶⠿⣶⠶⣶⣶⣶⡶⢶⢶⣤⡀
⠀⠀⢠⣶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⢸⣿⠀⠀⠀⣠⣤⡀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⡃
⠀⠀⠸⡿⠀⠀⢸⡏⠈⣿⠆⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⢰⣧⠀⠀⠈⠛⠛⠋⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⢷⣤⣤⣤⣤⣤⣾⡏
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢹⣧⣠⡶⠾⠷⣦⡄
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⡾⠟⠉⠀⠀⠀⠈⣿⡄
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⠛⠶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠷⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗⠾⠿⠿⠿⠛⠛⠋⠁
⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⠔⢡⠔⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠎⡐⢁⠞⡠⣊⢔⡁⢒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣰⣘⣃⣃⣊⣐⣡⣊⣊⠕⡰⠲⠕⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⣀⣴⣲⠟⢹⣿⡿⡪⠋⠛⣽⣽⡝⡟⠹⣷⣷⣯⣀⣠⡾⠪⣂⢐⢌⢂⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⢀⣤⠔⠛⠿⠱⠑⡀⡢⠙⣿⡇⡄⠜⣾⣿⣟⣄⣜⣿⣿⣻⡟⠊⠻⣿⣷⡶⢵⣯⡇⠀⠀⠀⠀⠀⠀⣠⣶⣫⡂
⢀⣠⠖⣁⠣⠖⠇⠀⠀⠀⠀⠍⠉⠎⢅⢄⠌⢜⣿⣞⢀⠈⠜⣿⣽⡟⠔⠴⣽⣽⣟⣔⢜⣿⣿⢁⣦⣿⣷⣘⡮⣿⣥⠂⢀
⠫⣁⠑⢬⡁⣧⢠⠆⠀⠂⡙⠂⠃⣀⢂⡎⠊⠊⠻⠏⠔⠐⡠⣻⣿⠏⡊⡀⣾⡿⣫⠎⠨⠛⠋⠉⣉⢉⢩⣉⡡⣿⣆⠡
⢀⠈⣢⢤⠳⡈⠜⣠⠤⠈⠀⡠⢠⡾⠛⠇⠀⠀⡐⠄⡸⡈⢀⡿⠋⠈⢁⠁⠙⠈⢄⢠⣢⣥⢾⣉⠀⠀⠀⠀⠙⠯⣍⠒⡠
⠀⠀⠈⠁⠉⠐⠲⠤⢤⡄⠀⠐⠰⡋⠅⠲⡀⠄⡠⢑⠀⠀⠎⡈⣖⣱⣕⡤⠴⠚⠻⠛⠈⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢺⣷⠖⣚⣒⠛⠒⠓⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣲⡽⡟⢌⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠂⠀⠈⠀⠠⠔⠊⠁
⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢵⡄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⡄⠀⠀⡆⠰⠀⢀⣀⠄⠀⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⢁⡀⠀⠧⣈⡀⠬⠃⠀⠂⠓⠘⠂⠑⠐⠠⣙⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠘⣄⡠⠀⠀⠀⠀⠀⡀⢆⠐⠀⠀⠀⡀⠀⡄⠀⢠⠕⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⠸⡀⠀⢸⠠⡄⢼⡀⠰⠤⣇⠀⡸⢠⠉⠁⠤⠀⠀⡄⢀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⡀⠀⢧⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⢦⣀⠀⠀⢸
⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠐⠤⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠣⠀⠀⠀⠀⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠻⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⡀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠿⠿⠁⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠁⣤⣤⣤⣤⣤⠀⢹⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⠀⢹⣿⣿⣿⠿⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⠀⢸⣿⣿⠁⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⠀⢸⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⠀⢠⣤⣴⣿⣿⣿⣧⡄⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⣠⠒⠛⢐⣤⣀⠀⠀⠀⢠⣴⣾⣿⣿⣶⡤⠀⠀⠀⠀⢀⣤⠤⢤⠠⣤⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢻⣾⡿⣷⣿⠃
⠀⠀⠀⢾⣹⡇⠀⣿⣧⣿⠆⠀⠀⠀⢹⠀⢸⠀⢸⠀⠀⠀⠀⢰⣿⣷⣬⣿⣠⡾⠀⠀⠀⠀⢰⠾⣧⡿⢷⠀⠀⠀⠀⠀⢸⣯⡿⣯⣿
⠀⠀⠀⠀⠙⠄⣄⠬⠋⠉⠀⠀⠀⢀⣾⣄⣾⣄⣼⣄⠀⠀⠀⠘⠟⠛⠻⠻⠋⠀⠀⠀⠀⠘⠛⠒⠛⠓⠚⠛⠂⠀⠀⠀⠺⣿⣿⣿⣿⠆

⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣀⣴⠹⡍⢩⠏⢦⡀⠀⠀⠀⢾⢉⢹⡄⢤⡄⠀⠀⠀⡠⠤⠤⠤⠤⠤⠤⡀⠀⠀⠈⣃⣜⢉⢟⡖⢤⠀⠀⠀⠀⢠⣤⠯⢧⣀
⠀⠀⢺⡇⣻⡇⣿⣿⠀⢸⡧⠀⠀⠀⢨⣧⣯⣵⣼⡄⠀⠀⠀⠑⣺⣿⣿⠀⣿⡒⠁⠀⠀⢺⠋⠀⢹⠺⠤⡎⠀⠀⠀⢶⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠉⠻⣰⣣⣞⣆⠞⠀⠀⠀⠀⢸⣿⣿⣿⣽⡇⠀⠀⠀⠀⠛⠛⠛⠀⠙⠃⠀⠀⠀⠈⠒⠐⠻⣄⡛⣤⠀⠀⠀⠀⠙⣿⠿⢿⡿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣤⣤⣤⣤⣶⣶⣄⠀⠀⠀⠀⠀⣸⣷⡆⠀⠀⠀⠀⠀⠀⢠⡿⠛⠿⣦⠀⠀⠀⠀⢠⠻⡈⢹⠐⢒⡆⠀⠀⠀⠀⣴⣶⣶⣶⣶⣤
⠀⠀⠀⣯⣽⣿⢺⢺⠾⡷⡇⠀⠀⠀⠂⠙⠛⡉⠛⠶⠄⠀⠀⠀⢘⠗⠀⠐⢟⠀⠀⠀⠀⠙⠒⠃⢸⠀⣎⣘⡄⠀⠀⠀⠘⢿⣿⣿⡿⠁
⠀⠀⠀⠛⠛⠛⠛⠿⠿⠋⠀⠀⠀⣠⣶⣶⡄⠂⠀⠀⠀⠀⠀⠠⣿⣿⣶⣾⣿⡧⠀⠀⠀⠀⠀⣀⣸⣀⡈⠁⠀⠀⠀⠀⠀⠀⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠻⠛⠻⠂⠀⠀⠀⠀⡇⢴⡦⠀⣿⣻⣿⡇⠀⠀⢠⢝⢿⣿⡿⣆⡀⠀⠀⠀⢀⣀⣴⣒⣲⣆⣀⠀⠀⠀⠀⠀⠀⠀⣤⢰⣷
⠀⠀⠀⠀⠀⠘⠂⠊⠀⠀⠀⠀⠀⣥⣤⣤⡄⠛⠛⠛⡁⠀⠀⠇⠀⠑⠀⠛⢩⡁⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣦⢸⡇⢿⣼⡿
⠀⠀⠀⠠⣾⣿⣾⣼⣿⡆⠀⠀⠀⣿⣯⣸⡇⠐⠟⠀⡇⠀⠀⠈⠢⢤⡄⢴⣿⣿⠂⠀⠀⠀⠿⠿⠷⠿⠿⠇⠀⠀⠀⠀⠉⠈⠁⢿⣾⠟
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⡀
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠈⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣰⣶⣦⡀⠠⠄
⠰⠶⠲⠤⠤⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⡖⠐⠒⠂⠈⠉⠉
⠀⠤⠀⠀⠀⠀⠀⠠⠤⠶⠦⠄⠀⠀⠀⠀⠀⠀⠀⠢⠽⠿⠟⠁⠀⠀⠀⠀⠀⠠⠤⠮⠤⠄⠀⠀⠀⠀⠀⠀⠤⡤⢤
⣤⣤⣤⣤⣄⣀⣀⡀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢀⣀⡀⢀⠀⠀⣤⣤⡄⠀⣤⣤⣤⣤
⣿⣿⣿⣿⣶⣿⣿⣷⣶⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⣿⠄⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⣤⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠹⣷⣤⣾⢿⠯⠿⠛⣿⣯⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⡄⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⠀⡒⡺⠟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠴⣚⠶⠿⢦⣶⣮⣽⣿⣿⣿⣿⣿⣛⢛⣿⣶⣦⣌⣛⢷⠶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⢤⣯⣛⣻⠿⣿⣟⣛⡛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠐⠒⠶⠒⠶⢶⣦⣴⡶⢦⣍⠉⢿⣿⣿⣿⡿⣛⢛⠉⣙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣛⣫⣾⣻⣿⣿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠤⠥⢤⣄⣿⣷⣮⡥⢶⣶⣤⣬⣹⣿⣿⣿⠷⠖⠂⠙⢿⣿⣷⣶⣮⣝⡛⠿⢿⣿⣧⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣀⣀⣀⣤⣤⣍⣙⣛⣛⡛⠛⠀⠀⠀⣊⣉⣙⡛⠛⢛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⡿⢿⣻⣛⣓⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠻⠿⢿⣿⡿⠿⠿⢟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠠⠶⢤⡀⠀⢀⣀⣀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⠀⠓⠂⢻⠀⠤⠤⠀⢁⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢽⠀⢀⠰⠆⠠⠀⠀⠀⡌⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠒⡂⠤⠈⡠⢊⠵⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠻⣷⡄⠀⠉⠀⢰⣡⢋⠊⣄⠙⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣶⣿⣿⣿⣿⢻⠀⢩⡛⠓⠒⢂⡿⢿⠾⢻⠟⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠻⡿⠟⠻⠿⠛⠋⢉⡆⢸⡏⢸⡀⠀⠑⣄⡠⠼⣷⡀⢠⠂⠀⣰⢿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠀⢧⠀⠀⠀⠀⠀⡼⠀⠈⠁⠀⡇⢀⡴⣿⣿⡄⢻⣿⡀⠀⠀⠈⢆⠀⠈⠉⠙⠲⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⣦⠀⢸⡀⠀⠀⠀⠸⠁⠀⠀⠀⠀⢳⠈⣴⣿⣷⡀⢸⣿⣷⠀⠤⢂⡌⠱⡀⠀⠀⡔⠀⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠧⡐⠦⣄⠀⡇⠀⣌⠉⠉⠉⠛⠃⠀⠀⢸⣴⣿⣿⣿⣇⣿⣿⣟⡃⢀⣾⠀⠀⠑⣄⠘⢀⣴⡇
⠀⠀⠀⠀⠀⠀⣴⠛⠧⣀⡀⠀⠈⢳⣷⡀⢿⣆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣷⠀⠀⠘⡆⠀⠀⠈⢲⡟⣹⣇
⠀⠀⠀⠀⠀⢰⡟⢆⠀⠀⠻⢯⡛⣯⣽⠇⠈⢻⡄⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠁⠘⢿⣿⣷⠀⠀⠑⠀⠄⠀⠀⢻⡁⠈
⠀⠀⠀⠀⢀⣿⠇⠈⠑⠢⠤⠤⣙⣰⣿⠀⠀⣸⡿⡄⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠘⢿⣿⣧⣀⠀⠀⠀⠀⠀⠀⡳⢬⡳⡀
⠀⠀⠀⡠⣫⠵⣚⣛⡛⠛⠟⠞⣿⣿⣿⠀⠀⣿⠁⠹⡄⠀⠀⠀⠀⠘⣿⣿⠁⠀⠀⠀⠈⢿⣿⣿⢧⡀⠀⠀⠀⠀⢱⠈⠇⣷
⠀⠀⢸⣿⡿⠏⠁⠘⠙⠳⣄⣴⣿⣷⡛⡀⢰⡏⠀⠀⠱⡄⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢈⣿⣘⣧⡙⠂⠀⠀⠀⠀⢇⢀⡏⣠
⠀⠀⢸⠉⡜⠀⢀⡤⠆⠀⠈⡙⢿⡿⠛⢥⢾⢧⡀⠀⠀⠹⡀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢀⣾⣿⣟⠉⢿⣤⡀⠀⠀⠀⠈⠚⣴⠃
⠿⠿⠿⣦⣀⣴⠋⠀⡠⠔⣫⣾⡏⠐⠦⡀⠓⢟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠿⢾⣯⣛⣷⣄⢀⣠⡾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠙⠿⠷⣶⣦⣼⣄⣏⣻⡅⡙⠶⢁⠐⠠⠜⣿⠿⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣈⡹⠿⠆⠉⠳⣝⣃⡀⠈⠗⠒⠙⠇⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⡀⠀⠀⠀⡀⣀⡤⣤⣀⠠⠴⠶⠖⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠼⠶⠖⠁⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⠀⢀⢀⣀⠤⠴⠞⠓⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠠⡄⠀⠀⠀⠀⡏⠀⠀⢠⡆⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖
⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢻⡄⠀⠀⠀⡇⠀⠀⣾⠁⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠘⢦⡀⠈⢧⡀⢀⡀⢳⡀⢶⠀⡇⠺⢰⠇⡔⠀⡴⠃⠀⠀⣠⠞⠃⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀⠀⠀⠙⢦⡀⡑⣄⠱⡄⢣⠸⠀⡇⠆⡌⡼⢁⡞⢠⠜⣠⠞⠁⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠢⡙⠢⡈⠣⡀⠀⢀⡀⠇⡀⠈⣠⠋⠔⡡⠊⢁⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⢦⣤⣀⠀⢬⡑⠬⡂⠀⠠⠂⣠⠤⠄⠢⠄⡁⠐⠄⢀⡠⠖⣉⠤⠒⢀⣠⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠶⠶⠦⣤⣤⣀⣀⣀⠀⠈⢉⡒⠈⠑⢈⠟⢠⠊⠀⠀⠀⠀⠀⠈⠳⡔⢅⠀⠪⠄⠒⠊⠁⢀⣀⣠⡀⣀⣀⣀⣴⠖⠶⠤⠤⠤⠶⠶⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠈⠉⠉⠉⠑⠒⠒⠤⠌⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠂⠀⠐⠒⠊⠉⢉⡝⠒⠛⠁⠈⠉
⠠⡤⠶⠒⠒⠒⠚⠋⠉⠀⠀⠀⠉⠉⠳⢄⣀⡉⠉⠉⠀⠐⠀⢇⠀⠀⠀⠀⠀⠀⢀⣀⡧⠬⠥⠤⠬⠭⣱⠖⠋⢀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⠤⣄⣠⠤⠸⠓⠒⠒⠒⠒⠚⠉⠁⠀⠀⠀⢀⣠⠼⠃⣀⠖⠋⠀⠀⠈⠉⢁⣠⠤⠤⠀⠀⠒⠀⠉
⠀⠀⠀⠀⠀⠐⠂⠁⠁⠐⢦⡀⣀⡀⠀⠀⠀⠤⡀⠀⡀⢠⣳⣖⡒⠒⠠⠤⠤⠤⠤⠤⣴⠞⠉⢉⡀⠐⠬⠈⠀⠴⡆⡀⠾⣉⣉⠁
⠀⠀⢀⣄⠀⣤⠄⢤⠄⠀⣠⠀⠤⠤⠬⠭⠥⠴⠦⠤⠤⠤⠴⠶⠿⠖⠂⠀⠀⠀⣤⠼⢃⢰⣀⠉⡤⡀⣀⠠⠀⣠⠔⢁⡽⠄⠈⢫⣉⣙⡤⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢠⡼⣃⣤⠾⠷⡼⠖⠛⠶⢅⣀⣀⣀⣜⣠⣤⣅⣐⣀⣈⣉⣄⣁⢇⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡖⠒⣒⠲⠤⠶⠻⢛⣛⣋⢀⣤⣄⠤⠤⠤⠤⠤⠶⠤⠭⠭⠁⠀⠀⠀⠉⠉⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⣀⣴⠶⠒⠒⠚⠛⠚⠛⠛⠒⣾⣉⡉⠀⠀⠀⠈⣡⣤⣵⣐⠒⣒⣒⣃⡀⠀⠀⠀⠀⠀⠤⠤⠤⠤⡤⠤⠤⠔⠀⠀⠦⠤⠤⠤

⠀⠀⠀⠀⠀⢰⡏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⢹⡆
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⢰⣱⠀⠀⢀⠤⠒⠉⢹⠉⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⢸⣹⠀⡔⠁⠀⠀⠀⢸⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⢸⣹⣸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠸⠼⢿⠀⠀⠀⢀⡰⡞⠀⠀⠀⠀⠀⢸⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠘⣄⡠⠖⠁⡰⠁⠀⠀⠀⠀⢀⡜⠀⡔⠒⠚⡦⡀⠀⠀⣀⠴⠦⡀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠈⠢⣀⢠⠃⠀⠀⠀⢀⡠⠊⠀⠀⣧⠀⠠⡇⣈⣖⣉⡖⠒⠒⡎⠲⣄⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠋⠁⠀⠀⠀⠀⣧⠤⢤⡇⡄⠀⢼⡗⠒⠒⡇⠤⠬⢤⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⡇⠖⠒⢺⣟⠉⠉⡇⣗⣒⣺⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⡇⡋⣉⣹⣏⢉⢉⡇⡒⡒⣺⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠁⠈⠉⠉⠀⠉⠉⠁⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠸⣇⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⣀⣀⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠠⡀⠀⠑⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡟⠉⣶⣄⣻⣯⣿⠿⢟⣻⡦⠀⠀⠰⡆⠀⠄⠀⢨⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⡇⠀⢿⣿⣿⣏⠀⢸⣽⣽⠇⠀⠀⠀⠀⠀⠸⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡰⣿⣿⣦⣀⣬⣙⣛⣷⠿⠋⠁⠀⠀⠀⠀⠀⠀⠐⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⠙⣻⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠻⢿⣿⣿⠟⠛⠋⣀⣤⠞⠋⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⣀⣤⠤⠴⢶⣯⣵⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢪⣿⣿⣿⣷⡕⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠰⣿⣿⣿⣿⣿⡟⣻⠷⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡹⣿⣿⣿⣿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⢁⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠖⠙⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⠀⠀⡌⠀⠹⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠐⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣷⣆⣶⣶⣶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠻⣧⠺⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⣿⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠒⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⢠⢠⡀⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠢⡾⠁⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⣴⡶⡀⠙⡄⠂
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠷⠏⠀⠁⠀⠀⢀⣨⣴⢙⣿⡿⣿⣟⣿⣾⢿⣿⡇⠀⠑⠳⠽⠇⢲⡿⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠙⢯⠠⠑⢶⣶⣬⠭⣤⣥⣭⣤⣼⣿⣿⣣⣬⣛⢿⣿⣇⣐⣀⣖⣚⣿⣷⣾⣷⣿⣿⣦⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠶⣛⣽⣒⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡭⢙⣮⣡⣌⣿⣿⣿⣿⡯⠿⢿⣏⡷⠝⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠉⠁⠀⠈⠉⠋⠉⠋⠉⠚⠋⠉⠙⠛⠋⠋⠉⠁⠉⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣣⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠃⠀⠀⣀⣤⣤⣾⠿⠛⠉⢀⣤⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠘⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠂⠀⠀⠀⠀⠀⢀⣠⣶⣶⣸⣰⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠻⣿⣿⡉⢹⣿⣇⣀⡀
⠀⠀⠀⠀⣠⣴⡄⡆⣰⣦⡀⠀⠀⠀⠀⣀⣤⣶⢨⡍⣁⠀⠀⠀⠀⠀⢸⣿⠟⢻⣿⣿⣿⣀⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣌⣻⠿⠿⠿⠇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⢀⣾⡿⢻⣿⣾⣾⣿⣷⣀⠀⠀⠀⣼⡿⠀⢸⣿⣿⣿⡿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠰⣿⣭⣼⣿⣦⣴⣿⠂
⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⠀⠀⣸⣿⠁⣾⣿⣿⡿⠈⠛⠿⠿⠗⠀⠈⠁⠀⢸⣿⣿⠁⠀⢸⣿⢠⣤⠀⠀⠀⠀⢴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⡆
⠀⠀⠀⣾⡟⣼⣿⣿⣿⡇⣿⡇⠀⠈⠁⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠸⣷⣶⣿⣿⣷⣶⣿⢂⣀⣿⣿⣿⣿⠛⠉⠻⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠈⠀⣿⣿⢿⣿⡇⠈⠀⠀⠀⠀⢸⣿⣿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠃⣀⣰⣿⣿⠋⠉⠻⣿⣿⣼⡿⣿⣿⣿⣷⣀⠀⣠⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⣸⣿⡏⠈⢿⣿⡄⠀⠀⠀⠀⠀⠰⣿⠟⠁⠀⠉⠻⣿⣷⣤⠀⢴⡿⡿⠋⢵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⣿⣿⠀⣿⣷⠀⠀⠀⠀⢠⣿⡟⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠿⣿⣿⡿⠿⣷⠆⠀⠀⢹⣿⡿⣿⣿⣿⢿⣿⡁⠈⠁
⣶⣶⣶⣶⣶⣿⣷⣶⣿⣿⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣿⣶⣾⣿⣶⣶⣿⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⡿⣿⣿⣿⡿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣟⢱⣿⣿⣿⣧⡀⠉⠉⠀⠀⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠟⣿⡿⠀⠈⣿⡿⢀⣀⣀⡀⣿⣿⣿⣰⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠰⣿⠇⢸⣿⣿⣇⠛⠋⠁⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⣾⣿⣷⡸⠛⠉⢸⠁⢀⣠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠿⠋⢹⡇⠀⣀⡘⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠸⠟⠉⣿⠀⢀⣘⢳⣾⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⢹⡇⠀⢀⠛⣶⣿⣧⢸⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢳⣾⣿⠀⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠃⠘⠟⠿⠀⠻⠿⠗⠘⠿⠛⠃⠻⠿⠟





⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡾⠛⠉⠉⠉⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠤⣾⣏⣠⣤⣤⡄⠀⠀⠀⣀⠹⣷⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⣽⣿⣷⣾⣿⣷⣶⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⡿⠟⠶⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠺⠿⠿⠿⠿⠿⣿⡟⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⠿⠟⠉⠉⠉⠉⠛⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣷⣄⠀⠀⠀⠀⠀⠀⣠⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠶⠦⠴⠶⠿⠟⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⠀⠀⠀⠉⠉⠉⠙⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡂⠀⢐⣾⠂⠀⠀⠀⠀⢀⠀⠀⠠⠆⠀⢀⣼⣻⡀⠟⣃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⣿⣷⣼⣯⣿⣷⣶⣼⣦⣤⣶⣶⣴⣧⣴⣶⣿⣿⣿⣷⣶⣿⣿⣿⣤⣤⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠀⠁⠈⠉⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣀⣠⠊⠙⢂⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣤⣤⣀⣀⣀⡀⠈⠉⠉⠀⠀⠈⠀⠀⠀⠀⠠⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣉⣷⣶⣄⣀⠀⠀⠀⠀⠴⠀⠀⠠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⢻⣿⣿⣿⣶⡶⢤⡤⠤⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠚⠿⠿⠿⠿⠟⠷


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⠀⠀⠈⡀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⠅⠀⠀⣠⠂⠀⠀⠠⠎⠀⠈⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠐⠁⠀⠀⠁⠀⡠⠀⠀⠀⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠄⠀⠀⠀⠂⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢄⡤⠀⡀⠈⠀⠀⠀⠀⠐⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠄⠀⠀⠀⠀⠀⠀⠀⠐⠁⠒⠀⠀⡀
⠠⡀⠀⠀⠀⠐⠈⠁⠀⠄⢠⠀⠀⠀⡄⢰⣉⣃⣘⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠈⠀⠀⠀⢄⠀⠀⠀⡄⣶⣹⣁⣛
⠀⠀⠀⢀⠤⢠⡄⠀⠀⠀⠈⠃⠀⠀⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡃⠀⢠⠤⢠⠄⠀⠀⠀⠈⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠂
⠉⠀⠀⠌⠀⠼⠀⢠⡄⠀⠂⠀⠀⢀⠀⠀⠀⡀⠂⠀⠀⠀⠰⠃⠉⠁⠀⠀⠌⠠⠼⠀⢠⠄⠐⠂⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⡀⠀⠀⠀⠀⠄⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠒⣀⠀⢀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢢⡀⠀⢠⠄
⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠂⠰⠀⠊⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠀⠐⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠂⡀⠄⠀⠀⠀⠀⢀⡐⠒⠢⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠠⢂⠐⠀⠂⠀⠀⠀⠀⠀⢀⠐⠒⠂⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠄⢂
⠀⠠⠠⠞⠀⠀⠀⠈⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠉⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⠈
⡀⠀⠀⠀⠐⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠁⠀⠀⠀⠂⠀⠀⠄
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠍⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠟
⠀⠀⠀⠀⠉⠁⠐⠂⠀⢀⡀⠀⠀⠀⡀⢠⢰⡄⢲⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠁⠔⠀⠀⢀⡀⠀⠀⠀⠀⣠⢰⠄⡦
⠈⢢⠀⠀⡀⠀⡀⠀⠀⠀⠘⠆⠀⠀⠃⠜⠃⠒⠒⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀⠀⠀⠀⠀⠀⠚⠄⠀⠀⠃⠟⠐⠒⠂
⠒⠊⠀⠠⠉⢩⠁⠠⠔⠀⠀⠀⠀⠀⠀⠠⠂⠀⠄⠀⠀⠀⢀⡂⠒⠒⠀⠀⠠⠉⢨⠁⠠⠐⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⣠
⠀⠀⠀⠉⠈⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠁⠀⠀⠀⠀⢂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠁⠀⠀⠰⠐⠈⣄⠃⠀⠘⠁⠀⡀⢀⠀⡄⠀⠀⠀⠀⠁⢁⠀⠀⠈⠁⠀⠀⠔⠀⠀⡌⠁⠀⠈⠀⠄⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠉⠙⢿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⣠⠶⠒⠒⠲⢤⡀⠀⠀⣀⣤⣄⡀⢘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⠒⠋⠁⠀⠀⠉⠛⠋⠙⠻⣿⣿⣿⣄⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⢤⣀⡞⠙⢦⡀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⡴⠋⠉⠙⠳⣄⡼⠁⠀⠀⠀⠙⠓⠲⡄⠉⠙⠲⣤⠀⠀⠰⠿⣿⣿⡿⠟⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠐⢦⠴⠃⠀⠀⡴⠋⠁⣀⣠⣤⣤⣤⠄⠀⠀⣿⠀⠀⠀⠈⢳⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⡀⠀⠀⣠⠞⠁⣠⠞⠉⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⢧⣠⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿
⠿⣍⣛⣉⣁⡴⠊⠀⣀⣀⣀⣀⣀⣤⠖⠋⠁⠀⡴⠚⠲⣄⠀⢀⣽⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣿⡿⠟
⠀⢀⣨⣭⣤⣤⠴⠚⠉⠀⠀⠀⠀⠈⠙⢦⡀⠈⢧⣀⡀⣈⣩⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⣀⡴⠒⠒⠦⣀⠀⠈⠳⢤⣀⣀
⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠙⢄⣀⣨⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⠀⠀⠈⠳⠦⣤⣤⠤⣴
⠀⢀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠿⠟⠛⠛⠶⠶⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣻⣿⣿⣿⣿⣷⣶⣤⣤⣶⣿⣿
⠀⢸⣷⣶⣶⣶⠾⠟⣋⣡⣴⣶⣶⣿⣷⣶⣶⣤⣄⡈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣿⣿⠿⠛⢉⣡⣤⣤⣤⣤⣤⣌⣉⠙
⠀⠀⠉⣉⣥⣤⣶⣿⣿⣿⣿⣿⣿⣿⢯⣭⣭⣤⣬⣥⣤⣤⣄⡉⠛⠻⠿⠿⠿⠟⠛⠉⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠘⠟⠛⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⡿⠿⠿⠿⠿⢿⣿⠷⠶⠛⣣⣴⣿⣿⣿⣿⣿⣿⣦⠤⠤⠶⠿⠿⠟
⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⣿⣶⣶⣤⣄⣀⠀⠙⠷⣶⣶⣶⣶⣶⣶⠶⠾⠿⠿⠿⠿⣿⠿⠛⣉⣤⣤⣶⣶⣶⣶⣤
⠀⠀⠀⠀⠙⢿⣿⣿⡿⠟⠛⠛⠻⣿⣿⣿⣿⣷⣦⣄⠀⠰⣶⣶⣶⣶⣶⣶⣶⠶⠟⣁⣤⣶⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢳⣴⠶⠶⠶⢶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣬⣭⣭⣭⣭⣤⣶⣿⣿⣿⣿⣿⣿⠟⢛⣉⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣦⣤⣤⣀⡉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠛⠳⠦⠄⠉⠛⠻⠿⠿⢿⣿⣿⠿⠿⠟⢛⣉⣤⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⣶⣾⣿⣷⣦⣀⠛⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⣆⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣟⠿⣿⡿⠿⠿⢯⣭⣿⣿⣾⣿⣮⣜⡿⣆⠀⢉⣦⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣒⠚⠓⠈⢀⣀⣀⡼⢥⣿⣿⠼⠋⠛⠻⢿⣼⡄⠀⢀⣧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⢠⣾⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠒⠒⠒⠢⣤⣤⠰⢿⣿⡦⠌⠛⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣤⣤⣤⣤⡾⠙⠏⢀⣼⠅⠀⠀⠒⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⣀⡔⢿⣿⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠀⠀⠉⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⢀⣼⣿⠟⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠛⢿⣿⣿⠟⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠋⠀⠀⠀⢸⣿⡿⠿⠿⠿⠿⢿⣿⡇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠋⠉⠙⠋⠉⢻⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢹⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣶⣶⣶⣶⣾⣶⣦⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣷⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢼⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣿⣿⠀⢸⣿⣷⣶⣶⣶⣶⣾⣿⡇⠀⣿⣿⣶⣶⣶⣶⣶⣿⣿⠀⠀⣿⣷⣶⣶⣶⣶⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁



⣿⣿⡿⠿⣿⡿⢿⢿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣴⣿⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠠⡞⠡⡼⣹⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⣛⡛⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⣽⢿⡍⣹⣿⣷⣸⣆
⠀⠀⠀⠀⠀⡠⢚⠙⢿⣿⡇⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢟⡻⣿⡄⠀⠀⠀⠀⠀⠀⢸⣇⠉⠈⢿⣪⠟⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠃⣚⡠⣼⣿⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡲⢍⣿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⣔⠲⢦⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠒⠠⠤⠞⢡⣤⠀⠀⠀⠀⠀⠀⢰⡶⠒⡲⣒⣒⡲⡒⠲⣶⠀⠀⠀⠀⠀⢀⣾⠃⢸⣿⡦⠄⠛⠉
⠀⠀⠀⠀⠀⢠⣅⠐⣿⡇⣶⡆⢸⣿⠀⠀⠀⠀⠀⠀⢸⠀⠶⡹⣯⢜⡷⡱⠆⢸⠀⠀⠀⠀⠀⠘⠛⠃⠉⠈⡆
⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠸⠷⠤⣱⡯⠭⠷⠥⠴⠿⠀⠀⠀⠀⠀⠀⠀⡔⢪⠖⣩⡒⣴⢢
⠀⠀⠀⠀⠀⠈⠩⠤⠉⠡⠉⠡⠌⠩⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡺⣄⡞⣠⣟⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⢰⣄⡤⠭⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⡀⠽⣋⡀
⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⢞⣵⣶⣄⠁⣠⠟⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⢶⣿⣦⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠦⡝⢻⣿⣿⠃⠀⠀⠀⠀⣠⡞⠁⡘⠷⡧⣯⠞⠁⠻⣧⡄⠀⠀⠀⠀⣿⣿⣿⣿⡀⣫⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣶⣿⣿⠃⠀⠀⠀⠀⠀⠈⠳⣄⣁⣠⠞⡅⣿⣦⣴⡿⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠱⠚⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠛⡁⢈⣙⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⡘⠃⠘⠻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠿⠇⠸⠀⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣶⣆⣰⣾⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⠿⠋⠀⣀⣀⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣦⣀⠀⠀⠀⠈⢹⡏⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢸⡇⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢸⣇⣠⣼⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠟⠛⠷⣦⣼⡟⠉⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣆
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⢸⣿⣿⡟⠉⠉⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠐⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⢸⣿⣿⣷⣤⣴⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣉⣉⣉⣉⣉⣉⣉⡙⠛⠟⠿⠟⠿⠋
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠐⣶⣶⣶⣆⠀⠀⢀⣀⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠿⠟⠛⢋⣠⣶⣿⣿⣿⣧
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⣷⣶⣯⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣶⣶⡆⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣖⠀⢻⣿⠀⢰⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣆⠘⡇⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⢿⣿⣿⣿⣧⣰⣿⣿⣿⣿⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠈⢿⣿⣿⣟⣹⣿⣿⣿⠃⠈⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣄⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⡇⠀⠀⠀⣀⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢈⣽⣶⣀⣴⣾⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣻⠿⣻⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⢻⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠹⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⠀⠀⣴⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠀⠀⠻⣿⡿⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⢀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⡏⠉⠉⠉⠉⠉⠁⠀⠈⠀⠒⠤⢤⣄⠤⠄⣐⠒⠀⢨⡈⠍⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠫⡳⡄⠀⠈⠆⢠
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⡇⠀⠤⠴⠤⣀⠀⠀⠀⠀⠣⡤⠤⠤⣤⢄⠀⠈⠻⡀⢀⢞⠘
⠀⠀⠀⠀⢸⡿⠿⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠉⠒⢤⣀⡀⠈⠢⡤⠨⠟⣛⣢⣄⠈⢣⠬⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣈⡛⠓⠦⠯⠹⠾⠷⠞⠀⢉⠕⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢍⣒⠨⢿⣍⣡⠶⠒⠀⢲⠂⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠭⢐⡒⠂⡍⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠐⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢆⣀⣘⣄⡔⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠍⡙⠃⠀⠀⠋⡩⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⣱⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠎⢀⡠⠤⢤⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡎⠀⡏⠀⢎⡀⢹⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠳⣄⣒⣁⡼⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢱⡀⠀⠈⠈⠁⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠚⠉⠒⠒⠒⠒⠒⠊



⣿⣿⣿⣿⣛⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣿⣿⣟⣻⣻⣻

⠀⠠⠤⠤⠤⡤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣄⣾⣿⣿⣇⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠿⠛⡿⠿⠿⠿⠿⠟⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠺⠀⠐⠉⠘⠛⣿⣿⣿⡟⠙⠁⠀⠠⠀⠀⠀⠀⠀⠀⡹⠯⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠒⠒⠂⠐⠀⠂⠀⠀⠀⠀⠒⠀⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⡎⢠⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠠⠀⠀⠀⠀⢀⠀⠀⠐⠀⣀⠤⠀⠀⢀⠀⠀⠀⠘⠄⠀⡀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠾⠀⠉⠠⠀⠘⠓⠁⠀⠀⡐⠂⠾⡐⢄⡀⠀⢠⡘⠀⠄⢼⣷⡤⠅⢷
⠀⠀⠀⠀⠀⠠⠄⢰⡀⠀⠂⠠⠀⠀⠀⠀⣰⡿⣣⣾⣧⣁⣈⠀⣇⠉⠀⠀⠀⡀⠱⡄⠈
⠀⠀⡆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⢀⠀⠈⠆⠠⠄
⠀⠀⢳⡀⠀⠈⠛⠿⠿⠷⠾⠲⠶⠶⠶⢶⣶⣒⣛⣛⣛⣛⣛⣉⠉⣿⠀⡸⠃⠀⠀⠀⠀⠂
⠙⠛⣉⣉⣐⣛⣛⣛⣋⣁⠀⠀⠈⠛⠿⠯⠭⢥⣤⠀⠀⠀⠀⠀⠀⢹⡗⠰⠀⠀⠀⢱⠀⠀⠁⠀⠀⣠⡀
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣦⣬⣭⠉⠁⠀⡀⢠⠀⠀⠀⠬⣤⣴⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠘⡉⠀⡀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣧⠀⠀⠀⣿⡇⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠃⢰⡀⠀⠀⠛⠁⠀⠀⢰⡀⠐⠠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣄⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡼⠁⠠⠄⠀⠀⠀⠀⠀⠀⠁⠐⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡓⠀⠀⠀⠄⠈⠛⠉⠓⠄⠀⠠⠀⠀⠐⠊⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⠽⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⣀⣀⣤⣤⣶⣦⣀⣀⣀⣓⣠⣤⣤⣤⡀⠀⠀⠶⠿⠿⠛⠛⠉⣉⣀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠿⠿⠿⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆⠀⠉⠉⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣯⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣴⣿⣿⡿⠷⢦⣕⠀⣀⣿⣿⢿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣼⠇⠈⡿⠀⠀⠘⠆⢰⠟⢡⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠀⠰⡀⠀⠀⠈⠀⢸⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⣤⣿⡾⠦⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⢸⠀⠀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠟⠉⠓⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣼⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⢀⣀⣄⣀⣻⣤⣴⣶⣿⣿⡿⣿⣷⣯⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣲⣦⣶⣿⣷⣾⣿⣾⣿⣽⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠁⠉⠲⠿⠿⠦⠬⠭⡍⢉⣉⣉⠉⠙⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠔⠒⠀⠀⢀⣠⣤⡴⠂⠄⠀⠀⢀⣀⡠⠭⠄⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡀⠁⠠⠤⠀⣀⣐⣲⣦⣤⣤⣤⣤⣄⣀⣤⣤⣤⣤⣤⣶⠶⠿⠛⠁⣠⢄⠀⠀⠀⠪⠹⠭
⠀⠀⠀⠀⣀⣀⢀⣀⣀⡠⠭⠵⠶⠶⢚⢛⡛⠛⠽⠟⠛⠋⠉⠉⠉⠉⠈⠉⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⣳⣷⠶⢦⠀⠀⠐⠄
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⢤⣤⣖⣲⡶⡺⠿⠿⠟⠛⠋⠀⢀⡞⠀⠀⠀⠀⠄⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠾⠛⢛⣉⣁⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡴⠞⢫⡀⠀⠀⠀⢀⡠⠔⠋⡀
⠀⠀⠐⠒⢀⣉⣉⣉⣙⣉⣉⣭⠤⠤⠶⠟⠛⠛⠋⠉⠉⠈⠃⠛⣉⣀⢀⣙⣩⠽⠛⠛⠛⠛⢛⣫⠭⠉⠓⠋⠉⠒⠻⠧⠒⣒⠉⠁⠀⢀⣀⣱⠄
⠉⠁⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠴⠖⠒⡋⠉⠅⠐⠊⠉⠀⠀⢉⣠⠴⠒⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⢸⣶⣶⢷⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠒⠀⠀⠀⢀⣠⡤⠤⠒⢈⠄⠙⠂⠀⠀⠀⠀⠀⠀⢀⡤⢮⣱⣎⡿⢟⣓⣿⠉⢀⣉⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠒⠒⠒⠒⠛⠛⠉⠁⠀⣠⡤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠊⠀⠐⠀⠂⠉⠰⠏⣥⠼⠃⠀⠉⠉⠁⠘⢋⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡔⣤⣐⢦⣄⡰⣥⣄⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⠠⠤⠄⠐⠊⠀⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣣⠶⣿⡦⠾⣷⠶⣿⣤⡽⠞⠛⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠂⠁⠀⠆⠀⠀⠀⣁⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⢠⡿⠀⣼⠃⣰⡟⠀⠀⢀⣀⠀⠈⢙⣟⣂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠁⠀⠀⠀⠂⠀⠀⠀⠉⠀⠈⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⣼⠇⣼⠇⣰⡟⠀⠀⠀⢟⣿⡇⠀⠀⢳⣍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢀⠀⠀⠀⠀⠀⠀⠆⣺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣿⣿⢴⣿⣴⡿⣶⡿⠐⢀⣀⠓⠀⡀⠀⠀⢀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠡⠇⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⣿⣵⣿⣶⣿⣴⣎⡍⠉⠙⢦⠈⠄⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⠀⣤⠤⡤⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢿⣿⣽⢿⠿⣿⡇⠻⠿⠻⣿⣸⡇⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⢀⠄⢠⣠⣨⡜⠂⠑⠀⠀⠀⠀⠀⠀⢠⣻⣟⡯⣿⣿⣿⠏⠀⢻⣿⣦⣀⣘⣩⠟⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠈⡄⠈⠀⠇⢀⠐⠀⡄⠀⣀⠀⠀⠀⠒⢿⡿⠿⠿⣿⠃⠀⠀⣈⠉⠃⡄⠄⠀⣸⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢀⠀⠊⠀⠀⠀⠀⡇⢠⠈⠀⢠⡆⠀⠀⠘⢉⠍⠉⠈⠱⠴⠅⠉⠉⢈⣭⠍⣭⡍⠉⠈⠉⠉⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠸⠀⠀⠀⠙⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠠⠢⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠸⠀⠀⠀⠀⢠⢃⠀⠠⠀⡀⠀⠀⠀⠀⢠⣎⢊⣄⠈⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⡄⠆⠀⠀⠨⠀⠆⠐⢠⡄⠀⢠⢠⠀⠀⡀⠀⠀⠀⡄⣄⠀⢠⠀⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠰⡄⠀⠀⠀⠀⠀⠀⠁⠠⡀⠀⠀⡄⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⠂⠡⡂⠀⠀⠀⢘⠀⠀⢂⠀⣰⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⡇⠀⠀⠀⢈⡀⠀⠀⠀⡆⠀⠀⠀⠤⠀⠀⠤⠀⠄⠀⠀⠀⠠⠠⠀⠤⠀⠆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠁⠡⠀⠀⠀⠁⠀⠨⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠄⠀⠀⠃⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⠀⠒⠒⠂⠒⠀⠂⠒⠐⠂⠐⠒⠒⠒⠀⠓⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠁⣉⢀⡉⠀⠈⡀⢈⠈⠉⢈⠀⠆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢄⠀⠀⠀⠀⢀⣆⣀⡀⠔⢠⣀⠀⠀⣀⠀⡀⠀⢀⡀⢀⢀⠀⣀⠀⡇⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠔⠀⠀⠀⠀⠘⠂⠀⠤⠤⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠠


⣾⣿⣿⣶⣶⡷⢿⣾⣿⣿⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣶⣿⢷⡿⠿⡿⢶⣶⣿⡿⠶⠶⢶⢿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠂⠀⠐⢲⣦⡀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⠀⣀⣀⣤⡖⠂⠀⠑⣲⣆⣀⢀⣀⣀⣀⡀
⠀⠀⣯⣉⣉⣛⠛⠿⡿⠿⢛⣛⣛⣛⣻⣯⣟⣻⣛⣯⣿⡷⠀⢠⣯⣉⣉⣛⠛⠿⡿⠿⣛⣟⣛⣛⣻⣭⣟⣛⣻⣽⣿⡇
⠀⠀⣭⣭⣭⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢠⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠛⠛⠋⠉⠉⠉⢛⠛⠛⠛⠙⠛⢛⡉⠛⠛⠛⠛⠛⠃⠀⠈⠛⠛⠉⠉⠉⠙⣟⠛⠛⠋⠉⠉⢙⠉⠛⠛⠙⠛⠛⠃
⠀⠀⠀⠀⠀⣦⠀⣠⣾⠀⠀⠀⠀⠀⢸⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣠⣿⠀⠀⠀⠀⠀⣿⣄⢰⡇
⠀⠀⠀⠀⠀⣹⡞⣫⣾⣇⠠⠤⢤⡀⢸⣯⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢞⣡⣿⣄⠤⠤⣤⡄⢿⣯⣻⣧
⠀⠀⠀⢀⠔⢻⣾⣿⠧⠀⢴⡶⠾⠯⠤⢿⣿⣿⠳⣄⠀⠀⠀⠀⠀⡠⠞⣿⣾⡿⠅⠀⠤⡴⠯⠭⠌⢿⣿⣏⠳⣄
⠀⠀⠖⠁⣴⣿⡿⠁⠀⠀⠀⠠⠤⠤⠾⠯⣿⣿⣷⡎⠳⡄⠀⠠⠊⢠⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠻⢯⣿⣿⣷⡌⢳⡄
⠀⠀⣴⣿⡿⣿⡉⠉⠉⣉⢉⡉⢉⢉⠉⣉⣛⣻⣿⣿⣇⡈⠀⠀⣶⣿⣟⣟⡉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣻⣿⣿⣧⡁
⠀⠀⣿⣟⣼⠩⠤⠤⠄⠀⢀⣀⣀⣀⣀⡈⣈⣸⣿⣿⣿⣷⠀⠀⣿⢏⡾⠥⢤⡤⠤⠄⢀⣀⡀⣤⡀⠀⣤⣼⡿⣿⣿⡇
⠀⠀⣟⡾⣉⣀⣀⣀⣀⠀⠀⠐⠀⠀⢀⣤⣄⡍⢭⣿⣿⡿⠀⠠⢏⡟⠁⣀⡀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣤⠭⢭⣽⣿⡇

⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠐⠀⠨⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀⣠⣀⣤
⠀⠀⠀⠲⠀⠁⠀⠀⠀⠈⡔⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣼⠃⠀⠀⣠⣤⡄
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢣⠐⠠⠤⠄⠠⠰⡿⡿⢿⠿⡿⠿⠿⢿⡟⠀⠀⠀⠸⠳⣿⡒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡂⠀⠀⠀⠁⠁⠉⡉⢉⢉⠩⠉⠉⢹⣇⠀⠀⠀⠀⠁⠁⠀⠀⠀⢤⣄⡀
⠀⠀⠀⠀⠠⠄⠤⠀⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠀⠀⠀⠘⡧⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⢠⡄⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⢖⠑⠛⣿⠀⠀⠀⣸⠀⠀⠀⣀⠀⢰⡏
⠀⠀⠀⠀⠀⠘⢿⡀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣐⣾⠀⠀⠀⢸⡆⠀⠚⠺⢯⡿
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠠⠐⠃⠀⠉⠀⠀⠀⠀⠀⠉⠈⠉⠁⠁⠀⠀⠀⠀⠀⢚⡇
⠀⠀⠀⠀⠀⠒⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠿⠿⠿⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣽⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡺⣿⣿⣿⣿⣿⣿⠿⠟⣂⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⢟⣛⣭⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⣠⣾⣿⣿⡿⠟⣫⣵⣾⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⣴⣿⣿⡿⣋⣴⣾⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢀⣀⠀⠀⠀⠀⠀⣠⠞
⢀⣾⣿⣿⢏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⣿⣿⣿⣿⠋⢡⣭⠻⣿⣿⣿⣿⣿⠟⣡⣾⣟⣹⣇⣠⣤⡶⠞⠃
⣸⣿⣿⢇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⡰⣾⣿⢣⣿⣿⣿⠟⣡⡾⢋⣤⠏⠉⠉⠉
⣿⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣇⠻⠿⠟⠁
⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡟⢋⣉⣤⣴⣾⣆⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠘⣿⡇⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⠟⣡⣶⣄⡉⠙⠛⠛⠛⠛⠉⠁
⠀⠈⠳⠸⣿⣿⣿⣿⣿⣿⣧⣙⠛⠛⣁⡼⠟⠋⠁
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⢶⣶⣾⣿⡿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣻⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡷⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠄
⠀⠀⠀⠀⠀⠀⠀⡄⢤⡐⣮⢱⣆⡲⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣘⠹⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠰⡠⠀⠀⠈⠱⢾⠿⠉⠯⡉⠁⠀⠸⡄⠀⠀⠀⠄⠀⠀⠋⠀⢉⡇⠈⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠃⠀⠀⠀⠀⣀⠘⣷⠀⠀⠀⠠⠆⠁⠀⢀⠀⠆⠀⠀⠀⠀⣼⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠄⠟⠁
⠀⠀⠀⠀⠄⠐⠀⢿⡀⢂⠉⠀⠀⠐⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⣠⠃⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡍⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⢤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠐⠂⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢡⠏
⠀⠀⡸⠀⠀⢹⠀⡁⢱⠀⣿⠁⢸⣿⡆⠀⠀⠀⠀⢀⢊⣾⣷⢢⣤⡤⠔⠀⠄⠄⠔⠀⠀⠀⠀⠀⠀⢀⠀⣄⠀⠈⠀⡠⠋
⠀⠀⢻⡀⠀⠘⠀⠐⠋⠀⠷⠖⠘⠀⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢵⣢⡀⢀⠜⠁
⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⢄⣠⠋
⠀⠀⠀⠀⠈⠙⠛⠶⠤⠤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⣠⢄⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠅⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠚⠂⡐⠋
⠀⠀⠀⠀⠀⠀⢀⡀⠠⠠⡀⠀⣖⠀⠹⠆⠀⠀⠀⠀⠀⠁⠀⠀⠈⣏⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠁⢠⡖⠆⢱⣀⡉⣡⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢿⣶⣦
⠀⠀⠀⠀⠀⠈⠣⣌⠁⣀⠞⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠉⠳⠒⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣄⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠘⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠟⢱⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠁⠀⣸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣸⠇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⢻⣿⣿⣦
⠀⠀⠀⠀⠀⢀⣾⣿⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⡀
⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣤⡀
⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠟⠁
⠀⠀⠀⠀⣿⠃⠀⢠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⢀⣄⠀⢁⣤⣄⡀⠀⣤⡀⢠⣄⠀⣤⡀⢠⣄⠈⢩⡍⠁⢀⣤⣀⡀
⠀⠀⠀⢰⡟⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠋⠀⣿⠀⣿⠀⠘⠇⠀⣿⠀⢸⡇⠀⣿⠁⠸⣿⡄⢸⡅⢠⡟⠀⠘⠇
⠀⠀⠀⠸⡇⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣴⠀⠀⣿⠀⢿⣦⣀⠀⠀⣿⣀⣸⡇⠀⣿⠀⢀⡻⣇⢸⠇⣾⡷⣆⣀⣀
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠛⠀⠀⣿⠀⠈⠙⢻⣷⠘⣿⠉⢹⡇⠀⣿⠀⠰⡆⢻⣼⡅⢿⡇⠀⢸⡏
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⣆⠀⠀⣿⠀⣿⠀⢸⡇⠀⣿⠀⢰⡇⠘⣿⠀⠸⣇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣃⡀⠀⢐⣟⣀⠛⣳⣖⠁⢀⣿⣆⢸⣷⡀⣿⣇⣸⣧⡀⢹⡀⠀⠙⣳⣞⢿
⣿⣿⡿⠿⣿⡿⠿⠿⡿⣿⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣾⣶⣾⣶⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⡆⠈⠛⢿⠟⠡⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡀⠀⢸⣇⠀⢹⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢺⣿⡆⠀⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢸⣿⣷⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⡆⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢸⣿⣿⣿⡇⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢿⣿⣿⣿⠘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠋⣀⣀⣀⣀⣀⣀⣙⣛⣁⣀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⣾⡿⣿⡿⣻⠟⢿⣿⣽⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣶⡿⠿⢟⠛⢉⢥⣴⣻⡿⠻⢽⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢡⣿⡟⢡⣲⣾⣿⣮⣿⠟⠉⠁⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣠⣾⣟⡥⢙⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⣀⢻⣿⣿⣿⣿⣀⣤⡶⡾⠒⠄⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣧⡄
⠀⠀⠀⠀⠀⠀⡠⠂⣀⣤⣿⣿⠿⠀⢐⣾⣿⣿⡿⠟⠟⠉⢁⣀⣴⣀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠐⠉⠀⠀⠛⠉⢋⡀⣄⣶⣾⣿⣿⢯⠙⠐⠊⠉⠛⠿⢿⣿⣦⣀⣀⣠⣴⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢺⣿⣧⣧⣾⣿⣿⣿⣿⣿⡻⠆⠀⠀⣠⣴⣄⣻⣿⣿⣿⣿⠟⠋⠀⠋⠙⢻⠟⡁
⠀⠀⠀⠀⠀⣠⣶⣾⣿⣛⣛⣿⣯⣭⣿⠉⢂⠁⠀⣀⣸⣿⣿⣿⣿⣿⡿⠛⠂⠀⠀⢀⡤⣴⣾⠿
⠀⠀⠀⠀⠀⢻⡿⠿⢿⣿⣿⣿⡿⣯⡉⠉⠀⠀⢺⣿⣿⣿⣿⣿⣏⠉⠀⠀⠀⠀⠀⠀⠀⠈⡁
⠀⠀⠀⢠⣶⣦⣄⡀⠀⠅⠑⠀⠀⢀⣠⡀⣀⣦⣾⣿⣿⡟⠎⠙⠻⠷⠀⠀⠀⠀⢀⣤⣶⡿⠃
⠀⠀⠀⣿⣿⣿⣿⣷⣢⣦⣿⣼⣷⣾⣿⣷⣿⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⣀⣚⣫⠭⠁
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠉⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠦⠕⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠻⣿⣿⣿⣦⡀⠀⠀⠀⣾⣿⡟⣡⣙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠋⢠⣤⡈⢻⣿⣿⣷⡀⠀⠀⣿⣿⡷⢦⠌⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣦⣀⣉⠙⢻⣿⣿⣿⡇⠀⠀⠙⢿⣿⣦⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡀⠙⠿⠃⣠⣿⣿⣿⠁⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣷⣠⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠛⠻⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⢀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⣄⣀⠀⠉⠁⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠻⠿⠿⠿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢹⣆⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⡾⠀⠀⠀⠛⠛⠛⠋⠉⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⠟⢷⡀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣠⡞⠁⠀⠀⠙⢦⣌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠋⠀⠀⠀⠀⠀⠀⠙⠋⡀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢿⡻⣆⢻⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡙⢷⣌⣷⡘⢿⣷⠘⣧⡾⢻
⠀⠀⠀⠀⠀⢰⣦⡀⣿⣇⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣽⣦⠉⠛⠁⠈⠉⠀⠛⢃⡟
⠀⠀⠀⢠⣦⣌⣿⡻⣿⢿⡿⣿⣴⢻⡆⠀⠀⠀⠀⠀⠈⠻⢦⣤⡀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠹⢮⡛⠃⠀⠀⠀⠙⢣⡞⠀⢀⣀⠀⣤⡀⢠⣄⠀⠈⢿⣄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢠⡏⠀⠀⢸⡝⣿⡇⣿⣿⢹⠀⠀⠀⣿⠀⠀⠀⠀⠸⣧
⠀⠀⠀⢀⣤⡀⣼⢧⢀⡶⣦⢸⡇⠀⣶⠾⣷⠈⠃⠘⠋⢾⡾⢻⠆⣿⠀⠀⢠⣦⡀⣿⡆⢀⣤⡄
⠀⠀⡶⣼⡏⣷⣿⢸⣿⠃⡿⢸⠁⠀⠈⠳⣄⠀⠀⠀⠀⠀⣰⠏⠀⣿⢀⠀⢸⡌⣷⡟⣿⣸⢃⣇⣴⡦
⠀⠀⢷⡘⣿⠘⠟⠈⠛⢸⣧⡿⣦⠀⠀⠀⠹⡆⠀⠀⠀⣴⠃⠀⠀⣿⣏⠻⣾⡇⠻⠃⣿⡟⢸⡟⣹⠇
⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢁⣴⡃⠀⠀⠀⢠⡇⠀⠀⠀⣿⠀⠀⠀⣿⠙⢷⡀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⢀⡇⠀⠀⠀⢠⡴⠋⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢿⡀⠈⠷⣄⣀⠀⠀⠀⢰⡏
⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⢸⠃⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢹⡄⠀⠀⠈⣷
⠀⠀⢀⣿⠀⠀⠀⠀⢸⡇⠀⣼⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠸⣇⠀⠀⠀⠘⣧⠀⠀⠀⢸⣇
⠀⠀⣸⠃⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢻⡆⠀⠀⠀⢻
⣿⣿⠟⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠟⠛⠿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⠉⠀⠀⠀⠀⠀⠙⠻⢿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⠉⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠀⠀⠀⠀⠉⠻⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⢻⣿⡟⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⢻⣿⡟⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⢛⣿⡟⠛⠛⠛⠛⣿⡟⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣾⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣾⣿⡇⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣾⣿⡇⠀⠀⠀⠀⠸⣿⣷⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣶⢦⣀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣄⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⣿⣟⣷⣼⣿⣿⣿⡿⢟⣯⣽⡟⠟⡴⢋⣴⣷⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣛⢷⣶⣿⣿⣿⣿⡿⠋⠑⢋⡽⢛⡅⣠⢿⠟⡉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠻⠿⣿⡘⣿⡇⡄⠀⠀⠀⢉⡞⠟⢸⣴⣷⣦⣿⡤⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣿⣿⠀⣘⣶⣶⣝⠻⡧⢳⣄⣀⢀⣼⣤⡄⣨⡭⠭⣂⣀⡨⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡄⠀⠽⣿⣿⠀⢷⣆⠛⠛⠛⢛⡭⠀⢰⣶⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣤⣤⣶⣾⣷⣤⠙⠿⠶⣟⣫⣴⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠩⢭⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⣠⠴⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⣾⣣⣸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢹⣿⣿⡿⠃⠀⠀⠀⠀⢀⣾⣿⢿⣿⣿⡏⠉⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠈⢿⣿⣧⠀⠀⠀⠀⠀⠈⣿⡇⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣄⠈⠻⣿⣦⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⡇⠀⠈⢻⣷⣄⠀⠀⠰⠿⣧⣀⠀⠀⠀⢀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⡇⠀⠀⠀⢿⣿⣷⣦⡀⠀⠀⠉⢿⣿⡟⠋⣉⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⡇⠀⣦⡄⠈⠿⠟⠙⠻⣷⣤⣤⣾⣟⠳⣌⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣧⠀⣿⡇⠀⠀⠀⠀⠀⠀⠙⢿⢻⡝⢳⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢷⡙⢦⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢺⡐⠭⣧⠇⠀⠆
⠀⠀⠀⠀⠀⠀⠀⣸⣥⣤⠖⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢀⣀⡀⠀⠀⠀⠒⡂
⠀⠀⠀⠀⠀⠀⠠⣿⣿⣷⢶⣄⠀⠀⡀⠀⢖⡲⠀⡊⠁⡆⠈⠉⠁⠀⣼⣤⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⡝⠛⣏⣹⠄⠀⣠⡄⠉⠈⠁⠀⠀⠀⠀⠀⡟⠻⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠈⢡⡀⠀⢽⠆⠀⢀⢠⠀⠀⠀⡀⡔⠃⠀⠁
⠀⠀⠀⠀⠀⡀⠖⣀⣀⠀⢠⠖⠒⢄⠀⢿⡀⠀⠀⠀⠀⠄⠂⠀⠠⢇⣹⠃⠀⠄⠒⢌⠂⠀⢀⣼⠃
⠀⠀⠀⠀⣼⣿⠉⠉⠁⠀⠹⣀⣀⠜⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠸⡀⠀⡠⢀⢠⣾⡿
⠀⠀⢀⣴⠟⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠉⠀⠀⢨⡿⣿⡀
⠀⠰⠟⠁⠠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡖⡆⠄⠀⣤⢢⠰⡂⢰⢲⠂⢠⠖⡄⡖⠆⢸⠀⠀⠀⠀⡉⠔⠢⡔⠒⣤⡖⡄⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠁⠈⠁⠈⠊⠈⠊⠀⠈⠀⠈⠐⠁⠁⠀⠈⠁⠀⠀⠀⠁⠈⠊⠈⠊⠀⠁⠁⠑⠁

⠀⠀⠀⠀⠀⠀⡀⠐⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣺⣿⣗⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠤⣤⣼⣷⠄⠀⠀⠀⠀⠀⢠⣶⢳⣦⠀⢶⣦
⠀⠀⠀⠀⠀⠀⠿⠀⠿⠘⠟⠀⠀⠀⠀⠀⠀⠀⣾⣾⢿⡻⣫⣷⢻⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣶⠄⠈⠁⠀⠀⠀⠀⠀⠀⠈⠿⠼⠏⠀⠀⢸⣧
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⣝⢫⣽⡵⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⣽⠆⠀⡦⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⠿⠿⠇
⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠻⣦⡀⠀⠀⠀⠀⠀⠰⣿⠶⠶⠶⠶⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⣠⣤⣤⣤⣼⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⢿⣽⠿⣅⣯⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⣼⣧⣘⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣾⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠻⠟⠺⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡗⢶⣦⡀
⠀⠀⠀⠀⣿⠛⠋⢸⡷⠙⠛⢿⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣥⣿⣿⠁⠀⠀⠀⠀⠀⠀⠲⣦⡄⣴⡾⢿⣶⠀⠀⠀⠀⠀⠀⠀⠖⠛⠋⣵⠂⠈⠛⠂
⠀⠀⠀⠀⠁⠀⠀⠈⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠟⠻⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠐⠒⠲⠒⠛⠓⠒⠂⠆

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣠⡀
⠀⠀⠀⠀⠀⠀⢠⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣠⣴⣶⣄⠘⠁⠀⠀⠀⠀⠀⠀⣶⢀⢀⣿⣉⡇⠀⡀⡆⠀⠀⠀⠀⠀⠀⠰⠗⠮⣭⠧⠾⠄
⠀⠀⠀⠀⠀⠀⣾⡇⣿⣿⢶⢢⠀⠀⠀⠀⠀⠀⠤⢠⣿⡟⠿⣿⣷⠠⠤⠀⠀⠀⠀⠀⣿⣙⣹⡿⠿⠿⡿⡷⣷⠀⠀⠀⠀⠀⠀⠸⢷⣻⣿⣿⣾⡃
⠀⠀⠀⠀⣀⣤⣿⣧⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠤⢼⣿⠶⠷⠶⡶⠞⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⣿⣿⣿
⠀⠀⠀⠀⣧⣼⠿⡿⢿⢿⣠⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⣾⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⡼⣿⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡭⣭⢽⡀
⠀⠀⠀⠀⢀⣴⣧⡒⢿⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣾⣋⠀⠀⠀⠀⠀⠀⠀⠀⣴⣲⣄⣎⣀⣇⢰⣲⣄⠀⠀⠀⠀⠀⠀⠀⠈⢮⣶⡵⠋
⠀⠀⠀⢠⣾⣿⣿⡇⣠⣴⣇⡻⡀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠟⠛⣷⠀⠀⠀⠀⠀⠀⢠⣿⣿⢿⣾⣿⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣶⡔⢂⢶⣶⡄
⠀⠀⠀⠘⠟⠝⠶⠭⢿⣿⣿⣯⠏⠀⠀⠀⠀⠀⠀⢸⣷⠏⠈⢿⣿⠀⠀⠀⠀⠀⠀⠘⠛⠛⢸⣿⣿⣿⡏⠛⠛⠀⠀⠀⠀⠀⠀⢸⣿⣿⣍⣾⣾⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⣿⣶⣶⣾⣷⡀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠟⢋⡄⠀⠀⠀⠀⢲⡴⠋⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⣠⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠘⢃⠀⠿⢿⣿⣿⣦⡾⠁⠀⠀⢠⡶⠋⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠻⣿⣿⡥⢾⠀⢠⣶⡇
⠀⠀⠀⠀⠠⠀⠀⠸⠀⠀⠀⠀⠀⠈⠳⠀⠘⠟⠁⣿⠇⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⡀⠈⠙⠻⢿⣧⠀⠀⠀⠀⠀⢠⠀⠀⠉⠉⠙⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⣿⣷
⠀⠀⠀⠀⣰⠿⠁⠀⠈⠉⠀⠀⠀⠀⠈⠈⠓⢀⡀⠤⠤⠄⠻⠀⠀⠀⠸⠹⡤⠀⠀⡐⠀⠔⠈⡇⠀⠀⠀⠀⠀⠲⠈⠀⢀⣀⡀⢻⡆
⠀⠀⠀⢠⠏⠈⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢰⡆⢛⠶⠄⠀⠀⠀⠀⢀⠑⠀⠀⣶⣄⠀⠁⡠⠂⠀⠀⠀⠀⠀⠀⠠⣍⠡⣦⡁⣀
⠀⠀⠂⠈⢢⡈⠛⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠰⠋⣠⡎⠀⠀⠀⠘⢿⡀⡠⠖⣁⡠⣴⠊⠀⠀⠀⠀⠀⠀⠙⢶⣾⣿⣶⣶⠟
⠀⠀⠐⠗⠤⣅⣀⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠋⠀⠀⠀⠀⠀⠀⠀⠻⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠉⠛
⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣽⣯⣯⣭⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣽⣿

⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⣴⣿⡿⡟⠃⠀⠀⠀⠀⠀⠀⣰⣿⡿⠛⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⢿⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠍⠿⠿
⠀⠀⠀⠀⢀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢀⠛⢻⡇⠀⠀⠀⠀⠀⣀⣀⡀⣠⣤⣀
⠀⠀⠀⢠⣾⣿⣿⣿⣦⠤⠀⠀⠀⠀⠀⣼⣿⣿⣿⡆⢺⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠿⠶⠿⠗⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣀⣀⣴⣶⣾⣿⡆
⠀⠀⠀⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠟
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈
⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠟⠈⠿⠿⠿⠋

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡇⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⡀⣀⡠⠄⠠⠄⠀⠀⠀⠀⠀⠒⠈⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠈⠁⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠂⠄⢀⡀⠀⢀⠀⡆⠀⠀⠀⠂⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠀⠀⠀⡀⠀⣤⠠⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣴⣿⢿⡿⠷⢦⡄⠀⣨⣿⡿⢿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠇⠀⡿⠀⠀⠘⠀⠸⠋⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠐⡀⠀⠀⠀⠀⡀⠀⠠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⣶⣿⡟⠂⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠞⠀⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣧⠀⣆⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⢦⣄⢠⣤⣤⣄⣿⢶⣴⠟⣽⣿⣿⣿⣿⣶⣿⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣽⣦⠾⠿⠿⠿⡿⣿⠷⣿⣿⠿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠒⠒⠒⠂⠀⠤⠄⠀⢉⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠒⠒⠃⠁⠀⣀⣤⡤⠒⠚⠐⠂⠀⣀⣤⠤⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⣤⣤⣤⣭⣽⣶⣶⠶⠶⠶⠤⠶⠶⠶⠶⠾⠖⠛⠋⠁⠀⠰⢀⣀⡀⠀⠑⠘⠂
⠀⠀⠀⠀⠀⠀⠤⠤⠴⠖⠒⠚⠛⠉⠭⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣄⣤⡽⠟⠉⡧⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⠶⠶⠶⠒⠒⠚⠒⠒⠛⠛⠛⠉⠤⠰⠖⠉⠉⠉⠉⠁⠀⠀⣀⡤⠃⠀⠀⠀⠀⠀⢀⡠⠚
⠀⠀⠀⠀⠀⠀⠤⢤⣤⣀⣠⠤⠶⠶⠛⣛⣩⣤⣤⣶⠶⠶⡶⢶⠿⠶⠖⢾⡶⣦⣤⣤⣤⣠⣤⣠⡤⠴⢶⡾⣏⣁⣲⡄⠀⣀⠠⠄⠂⠁⠠
⠀⠠⠀⡤⠠⠤⠤⠤⠴⠖⠒⠒⠒⠂⠉⠉⠁⠁⠀⠀⠀⣀⣀⡬⠤⠄⣒⡤⠖⠒⠉⠀⠀⣀⠠⠒⠋⠁⠀⠀⠀⠈⠉⠉⠀⣶⡀⣀⡀⣤⡤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠒⠚⢋⡉⠁⠀⠀⠀⠀⠀⢀⡀⠀⠠⡖⠋⠁⠀⠀⠀⠀⠀⠀⡠⢀⢠⣠⣀⠸⣿⠏⠿⠷⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡄⣀⣀⣀⣀⣠⡤⠶⠛⠉⠀⢀⣠⡤⠖⠐⠉⠁⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠤⠊⡴⠾⢷⡛⠒⠾⡧⠀⣲⡾⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠁⠙⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣽⣭⣯⣯⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⡀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⣲⣦⣄⡀⠀⠀⠙⢲⡄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⣻⠿⢿⣛⠋⠁⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣤⣬⣵⡆⠀⠀⠀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠙⠻⣄⠀⠀⠀⠀⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⣀⣼⠇⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠛⢯⡁⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⢀⣿⠂⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠸⣯⣕⣶⣤⣀⣀⣈⣳⣿⣅⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠉⠻⡍⠉⠁⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠉⢻⡗⢦⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣶⡾⠷⠋⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣹⣄⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⡀⢈⣏⢱⡄⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⡟⠛⠋⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠈⠀⠀⠀⢀⣀⠀⠀⠀⣠⠶⠶⣦⣝⣦⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠻⣦⣽⣶⣄⠘⠦⣄⠀⠈⠉⠁⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠷⠄⠀⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣶⣶⣤⣤⣤⣴⣶⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⢿⠟⠻⠿⣿⠟⣿⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⡇⠀⠀⢠⡄⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠘⣆⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⢆⠀⠈⠳⡀⠀⠀⠘⣆⠀⡄⢸⡇⡆⢰⠇⡤⠀⣠⠋⠀⠀⢀⡴⠆⠀⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠑⢄⠀⠙⢆⠘⣆⠘⡄⡇⢸⠀⠃⡏⡼⢁⡴⢁⡔⢀⡴⠋⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⢠⡑⢌⠠⠣⡈⠂⠁⠀⢸⠀⠀⠐⢡⠎⡠⢋⠔⠋⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀⠀⠀⢉⠲⢌⠢⠀⠀⢈⠀⠂⣁⣈⣈⠀⠐⢁⠈⠀⢁⡤⠞⢉⡠⠀⠀⣀⣠⠴⠖
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠈⠉⠙⠲⠤⣉⠂⠈⢂⠔⢁⠔⠉⠀⠀⠀⠈⠐⢄⠠⡒⠁⠐⢊⠁⠤⠒⠉⠁⠀⠀⠀⠀⠀⢀⣠⣄⣀⡀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠒⠶⠦⠬⠅⣀⣀⡀⠂⢠⠇⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠈⣀⠀⠤⠴⠖⣒⠿⠥⠖⠛⠓⠋⠀⠀⠉⠉⠉⠁⠀⠈⠉⠁
⠀⣀⣀⣀⣀⣄⣤⠴⠒⠒⠒⠒⠒⠶⢤⡀⠀⠀⠒⠀⠀⠀⠉⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⣀⣀⡒⢀⣠⠼⠃
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⡀⠀⠀⠈⠀⢣⣀⣤⡤⣄⣀⡤⠴⠚⠉⠁⠉⠉⠀⢠⡞⠁⣀⠴⠀⠐⠂⠖⠂⠀⠀⠀⠀⠀⢠⡄⠠⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠙⠒⠪⣉⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠚⢉⠰⠲⠁⠀⣀⠀⢀⡤⠴⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠒⠂⠠⠤⠄⢈⡓⠁⠐⠀⠓⠻⣲⣭⣀⠉⠉⠉⠉⠉⢩⠟⢁⠀⠀⠀⠀⠉⠀⠀⠈⠓⡂⢬⠖⠲⣤⠤⢤
⠀⠀⠐⠒⠀⠛⠃⠈⠃⠀⠐⠂⠈⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣰⡟⠉⡴⡀⢴⣐⣋⡂⠆⠀⠀⠐⢉⠀⠞⠁⠀⠐⠳⠶⠎⠡⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠚⣁⣀⣜⣫⡶⠛⠉⠙⠋⠁⠉⠉⠳⠶⠶⠶⠦⠖⠛⠒⠲⠶⠶⠶⠶⠶⠧⠿⢦
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⢀⡠⠤⠴⠶⠶⠚⠭⠭⠍⠭⠟⠉⠉⢰⡄⠒⠐⠒⠒⠊⠉⠐⠒⠊⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠰⠾⠋⠉⠉⠁⠀⠀⠀⠀⠉⠹⠦⢤⠀⠀⠀⠀⠐⠶⠞⠺⠭⠥⠥⠄⠠⠀⠀⠀⠀⠀⠀⠘⠛⠓⠒⠒⠒⠛⠋⠀⠀⠙⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡗⣭⢛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣰⡏⣼⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⣿⢰⣟⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣷⣾⣷⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡟⠛⠀⠀⠀⢀⣀⡿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇⠀⠀⣶⣶⠞⠀⠀⠀⢤⣤⡴⠀⠀⣤⡄⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠹⣆⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇⠀⠀⣿⡟⠀⢀⣤⡀⠘⣿⠁⢠⡄⢸⡇⣠⣼⠇⣄⡏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⣿⣿⣿⡇⠀⠀⣿⡇⠀⢸⣿⣷⣾⣿⠀⣿⣿⣿⡇⠙⢻⡇⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⠀⣿⡿⢿⡇⣶⣾⠻⡆⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⣿⡇⠀⠸⣿⡏⠈⣿⠀⠙⠃⣸⡇⠉⠸⠆⠐⠃
⠀⠀⣠⣴⣶⣶⣿⡇⠀⠀⠀⣤⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠸⣿⡿⠃⠀⠠⣿⣿⡀⠀⠀⠀⠈⠛⠃⠀⠀⠉⠁
⠀⠾⠿⠛⠛⠉⠉⣿⣷⣶⣾⣿⡇⠀⠀⠀⠀⠈⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛



⠀⡁⣉

⠀⠀⠀⠀⢀⡠⠖⠚⠉⠉⠉⠙⠒⠦⣀
⠀⠀⠀⡴⠋⢀⠴⠚⠉⢉⣩⠙⠢⡄⠈⢣
⠀⠀⡼⠀⣰⠃⠀⠀⢸⡉⠉⠂⠀⠘⡆⠀⣇
⠀⠀⡇⠀⡇⠀⠀⠀⠀⠱⡄⠀⠀⢀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢒⣚⣋⣉⣉⣉⣛⡒⠲⢤⡀
⠀⠀⢣⠀⠹⣄⠀⢳⠖⠚⠁⠀⣠⠎⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⡎⠠⡞⠭⣴⣀⣠⠖⠛⢳⠾⢙⡆⠘
⠀⠀⠈⢷⣄⡈⠙⠒⠲⠖⠒⠋⢀⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣈⡙⠒⠒⠒⠒⠒⠒⠚⣉⣤⣾
⠀⠀⠀⠀⠙⠿⣷⣶⣶⣶⣶⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢀⣠⠤⠤⠴⠶⠶⠤⠤⢤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢩⠔⣒⡏⠉⣩⠭⠿⣗⣲⢤⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡈⠺⠭⢿⣟⣁⣀⣤⠥⠴⠚⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣶⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⣀⣀⣀⠉⠙⠿⣿⣿⣿
⠀⢀⣠⠤⠖⠒⣒⣒⡒⠒⠦⠤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⡴⠋⠁⠀⠀⠉⠲⣄⠈⢿⣿
⡔⠁⡔⣊⣩⠉⢉⡽⠛⢯⡭⠖⡄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⠀⣠⠄⠀⢀⣠⣀⠈⢆⠈⢿
⣷⣄⡙⠒⠮⠭⠭⠤⠤⠤⠖⠚⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⠈⠹⣄⣠⠎⠀⣸⠶⢸⠀⢸
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠣⡀⠀⠀⠀⠀⠁⢀⡞⠀⡸
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠉⠉⠉⠛⠋⠉⠉⠉⠀⠘⢿⣷⣄⠉⠒⠤⠤⠤⠔⠋⠀⡴⠃
⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣤⣤⣤⣤⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠀⠈⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠃⠀⠀⠀⠀⠀⠀⢻⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠆⠀⠀⠉⠛⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣀⠀⠀⢠⣤⣤⣀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠉⠻⣦⠀⣸⠃⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠘⣧⡿⠀⠀⠀⠀⣠⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣤⡀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⢀⣴⣿⡿⠟⠛⠉⠙⠛⠿⣿⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⢀⣀⣠⣾⣿⣏⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠘⠛⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⢀⣀⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠘⠛⠻⣿⣿⡟⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣄⣀⣀⣀⣀⣤⣶⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⠿⠿⠋⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁


⣿⠿⠿⠿⡿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⡿⣿⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠉⠀⠘⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠟⠁⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠟⠋⠁⠀⠀⣠⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠉⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠘⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀⠀⠀⠈⠻⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣦⣄⣀⠈⠛⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣦⡀⢈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⡎⢁⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⡘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣵⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⠿⢿⣷⣦⣀⠀⠀⠀⠀⣠⣿⣿⢿⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣄⠀⠀⠉⠻⢿⣿⣿⣶⣿⣿⢿⣽⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣟⠋⠀⠀⠠⢶⣿⣩⣿⣿⡟⣿⣟⣿⡿⠿⢿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⠻⣿⣿⠁⠀⠀⠀⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⣿⣿⣿⡏⣿⣷⣿⣿⣶⣦⣤⡸⣦⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⣿⣷⣿⣿⢿⣿⡏⠙⠻⣿⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠏⠀⢸⣿⣷⣾⣿⣿⡇⠀⠀⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣄⡀⠉⠛⠋⠉⠀⢀⣰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⠀⠀⢰⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣤⣄⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢿⣿⡇⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠸⣿⣄
⠀⠀⢀⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⢿⣿⡀
⠀⠀⢸⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠘⣿⣧
⠀⢀⣾⡟⠙⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⢸⣿⣆
⠀⢸⣿⠁⠀⠈⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠈⢿⣿⡄
⠀⣾⡿⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠘⣿⣧⡀
⢠⣿⠇⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠁⢸⣿⣇
⢸⣿⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⣿⣿⡄
⣿⣿⠀⠀⠀⣰⣶⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⡀⠹⣿⣧
⣿⣿⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⣿⣿
⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⣸⢻⣿
⣿⡇⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢨⠸⣿
⣿⣿⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⡄⣿
⢿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣽⠛⣿
⠸⣿⡇⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣠⣿⢠⣿
⠀⢿⣿⡀⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠏⣬⣿⣿⣿
⠀⠘⣿⣷⡀⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢏⣿⣿⣦⣿⣿
⠀⠀⠘⣿⣷⡀⠀⠀⠀⠘⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢏⣹⣿⣧⣾⣿⠇
⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⠀⠙⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡿⣣⣿⣯⣹⣿⣿⠃
⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠀⠀⠀⠙⠻⠿⣿⣷⣶⣤⣤⣤⣄⣀⣠⣤⣤⣴⣾⣿⣿⡿⢿⣭⣿⢿⣣⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠀⠀⠀⠠⠈⠙⠛⠛⠿⢿⣿⣿⣿⠿⠿⠿⠛⠻⣥⣴⣾⣽⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣶⣤⣉⡀⠐⠂⠀⠆⠀⡤⠈⣉⣵⣶⣶⣟⡻⠷⣎⣭⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣴⣦⣴⣶⣶⣿⣿⣿⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⢢⠀⠀⢄⣔⣤⢰⠂⠴⢦⢂⢠⣒⡄⡴⠆⢸⠄⠀⠀⠀⠈⡠⠒⢤⠒⢄⣴⢢⠠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠈⠈⠀⠉⠀⠡⠀⠐⠁⠀⠈⠨⠈⠢⠀⠁⠀⠈⠃⠀⠀⠀⠀⠈⠤⠉⠠⠀⠉⠈⠐⠌

⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⢶⣶⣶
⠀⠀⠀⠀⠀⠀⣴⣾⣯⡃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣨⣿⠁⠀⠀⠀⠀⠀⠀⠰⣿⡦⢻⣿⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⣤⣜⣿⣷⢘⣿⣟⣤⡀
⠀⠀⠀⠀⠀⠀⠛⣿⡟⡃⣹⣇⠀⠀⠀⠀⠀⠀⠀⡴⣶⣿⠈⡏⢹⣷⡶⡄⠀⠀⠀⠀⠀⠙⠻⠟⠻⣶⡋⢁⡾⠗⠀⠀⠀⠀⠀⠛⣿⢽⣿⣿⣿⢹⡟⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣟⠆⣻⡿⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⠀⡇⢸⣿⠃⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡜⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⠿⠿⠿⠿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠛⠙⠋⠋⠛⠙⠋⠙⠛⠙⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣄
⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⢿⡆⣀⠀⠀⠀⠀⠀⠀⢀⣴⣤⡀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠐⠒⠶⣆⣀⣠⣤⣴⣳⠄
⠀⠀⠀⠀⢀⣟⠁⢿⣿⣿⣿⠇⢹⣇⠀⠀⠀⠀⠀⠀⠛⠛⣠⣤⡘⠛⠃⠀⠀⠀⠀⠀⢀⣀⣤⡟⠛⠈⢠⣾⡏⠀⠀⠀⠀⠀⠀⣸⣿⣻⣀⣄⣸⣻⣿
⠀⠀⠀⠀⠯⠈⠇⠘⠿⡿⠟⠀⠇⠨⠇⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⢰⣷⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣮⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣥⣿⣤⡀⠀⠀⠀⠀⠀⠀⠈⠋⢙⡟⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⢶⣶⡶⣶⣶⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⡦⠀⠰⡶⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢠⡄
⠀⠀⠀⠀⠀⣀⣤⣤⣄⢠⡄⢠⣄⠀⠀⠀⠀⠀⠀⢸⣿⣼⣯⢿⢣⣿⣿⠀⠀⠀⠀⠀⠀⠘⢿⣛⡿⠃⠀⣴⢶⠀⠀⠀⠀⠀⠀⣰⣿⡿⣻⣵⣶⠒⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⡇⠀⠀⠀⠀⠀⠸⠿⠿⠿⠾⠾⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⣉⡁⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣧⣽⣿⢿⣹⡿⣘⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠾⠀⠰⠗⠀⠺⠆⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡄⠀⠻⠟⠀⠀⠀⠀⠀⠀⠻⣿⢟⣿⡟⢯⣾⠏
⠀⠀⠀⠀⠀⠷⠿⠿⠟⠙⠿⠿⠿⠇⠀⠀⠀⠀⠀⠸⠿⠗⠾⠿⠒⠿⠿⠀⠀⠀⠀⠀⠀⠸⠶⠾⠶⠗⠀⠺⠷⠀⠀⠀⠀⠀⠀⠀⠸⠚⠛⠛⠛⠁


⠀⠀⠀⠀⠀⠀⠀⣰⡿⠻⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣶⢺⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠃⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⢿⣿⢬⣿⠂⠀⠀⠀⠀⠀⠀⠀⣰⢖⣧⣾⢻⣦⡓⢶⡀⠀⠀⠀⠀⠀⠀⠸⣿⣍⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⢛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣬⣟⠛⣩⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣝⠿⠨⣟⣵⣿⡇⠀⠀⠀⠀⠀⣤⣀⠈⠉⢁⣠⣤⣄⡀⠀⠀⠀⠀⠀⠉⠀⢿⣿⣨⣿⠇⠈⠁
⠀⠀⠀⠀⠀⠀⣼⣿⣆⢦⣿⣿⡄⠀⠀⠀⠀⠀⠀⣿⢿⣯⣾⣿⣮⣿⣿⡇⠀⠀⠀⠀⠀⠈⠛⢿⣜⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠖⠀⣿⣿⡇⠐⠄
⠀⠀⠀⠀⠀⠀⠙⠋⠙⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠵⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⡂⠀⠀⠭⢖⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠅⠀⠀⡭⢵⡂⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⣅⠀⠀⣈⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠀⠀⣒⡫⠅⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⣖⣚⠁⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠍⠀⠀⠭⣒⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣁⠀⠀⠉⠑⠂⠀⠄
⠀⠀⠀⠂⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⠀⣭⠭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡂⣘⢐⡂⡂⠐
⡀⠐⠀⡁⠐⠀⠐⠀⡁⠀⠀⠀⢰⢀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡒⠡⠀⠄⢈⣉⡁⡀⣂⢀⣀⢀⣀⣠⠨⠅⠄⠆⣐⢀⡀⡆⢀⣀⡀⣀⣄⢀⣀⣀
⠀⢀⠀⠀⠈⡁⢀⠀⠄⠄⡂⠭⠨⠐⠀⠠⠀⠀⠀⠀⠀⠀⠨⠀⠄⣒⢀⠀⡃⢒⠈⠈⠁⠛⠙⠁⠈⢤⣺⡥⡬⠅⠦⢽⠸⠇⠯⢿⣙⡆⣿⢿⣧⣼⣭
⠀⠠⠀⠂⠀⡀⢀⠀⠀⡀⠂⠅⠨⠀⢀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠅⠀⢐⠀⡂⠸⠨⠀⠃⠒⠀⠀⠀⠀⠙⠚⠃⠀⠉⠉⠠⠄⡭⣸⢻⡇⣿⣧⣌⣹⡿
⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠅⠁⠈⠨⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠅⠁⠨⠀⠅⠠⠘⠈⡇⠤⠂⠀⠐⠀⠐⠂⠀⡄⠛⢽⠠⡄⠛⠚⠻⠃⠛⠓⠞⠛⠓
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠄⠄⣀⢘⠈⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠁⢠⠀⠁⠈⢀⠀⡀⠄⠀⠀⠀⠀⠀⠐⠂⡀⠂⣀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠐⠂⠀⠀⠁⡁⠂⣀⠐⢀⠈⠠⠀⠀⠀⠀⠀⠀⠐⠂⡀⠂⠀⠀⡀⠀⢀⠀⠂⠁⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⡀⢀⠐⢀

⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢾⣿⣼⣿⣻⣟⣩⣿⣾⣀⢴⡿⣿⠯⢀⣩⡐⢺⠦⣿⠙⣽⣾⢻⣟⣹⡯⣿⢻⠤⠦⢀⣶⣂⠤⣤⣤⣄⠀⠃⣶⣴⡄
⠀⠀⠀⠀⠉⠉⠁⠉⠉⠁⠉⠉⠀⠀⣩⣿⣿⡿⠻⠙⢷⣄⡙⠀⣇⣼⡞⣯⣿⣷⣿⣨⣧⡏⠈⠁⠝⡘⣯⡿⢹⠋⢰⢞⢼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡟⢋⣠⣤⣦⣿⣿⣿⣀⠈⠭⠵⠯⠌⠸⠟⠟⠇⢤⡤⠙⣿⣿⠇⠀⣴⣗⠼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣫⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣧⠀⢸⡧⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣯⣿⣻⣿⣿⢟⣻⣿⣿⣷⣯⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣼⡇
⠀⠀⠀⠀⢀⡀⣴⢿⣿⣿⣿⣿⣿⣿⡛⠩⡽⠎⠈⠓⠊⠁⠉⠙⢼⡽⠃⢘⣿⡯⣵⠉⢻⣿⣷⣄⡀⠀⠀⠀⠀⠉⠛⣻⡷⠇
⠀⠀⢰⡾⣏⡿⢋⣿⣿⡟⣿⣿⣿⠁⠂⠀⠀⢀⣀⠀⠀⠀⠀⢠⡶⠁⠀⠈⠉⠁⠀⠀⠈⠴⠬⠻⣧⡀⠀⠀⠘⠳⠾⠙⠇
⠀⢠⢤⣷⡽⣤⣿⣿⣿⣴⣿⡿⠉⠀⠀⠀⣀⠜⢿⣦⠙⢷⣤⣸⣿⠳⠤⠀⠀⠀⣀⡔⠀⣀⣤⠴⠒⣬⣶
⠀⢹⡄⢿⣷⣿⣿⣿⣿⡿⣿⡇⠀⠀⠘⠋⣁⡈⠀⠙⠃⠘⠻⠿⠌⠳⣄⠈⠶⠬⠄⡁⠈⠙⢁⣴⢋⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠈⣿⣷⠙⣿⣻⣿⣿⣿⣿⣷⣤⠶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣤⣤⣴⣽⣿⣦⣤⣤⣬⣷⣤⣥⣈⣙⣳⢦⡆⠀⠀⠀⣿⣿⠋
⠀⢀⣿⣿⠀⠛⢿⣿⠻⣿⣿⣾⣧⢀⣿⣿⣿⣿⣿⡿⣻⣾⣿⣿⣿⣿⠿⣻⣯⣥⣼⡿⣿⣦⢈⠉⢭⡿⠎⠀⠀⠀⠀⠈⠉
⠀⠈⣽⣿⣧⢀⣠⠙⣷⡌⠛⠚⢿⣿⣿⡇⣾⣿⣿⠷⠾⣫⣾⣿⣿⣿⣿⠿⠋⢉⣿⣿⠇⠀⠀⠀⠄
⠀⠀⠸⣻⣿⣦⡈⢁⣙⠻⣦⣀⠈⠉⢻⡇⢸⣿⣯⡿⠟⣵⣿⡿⢃⣴⣶⣶⠏⢉⠛⠉
⠀⠀⠀⠹⣿⠿⠿⢦⣝⣄⠀⠉⠳⣀⣠⣇⢸⣿⠏⢀⣠⣷⣻⠿⠋⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠸⠟⠙⠋⢂⠰⡶⠾⣿⡧⢿⡇⠀⣰⣶⣦⣤⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠵⣮⠓⠉⠁⢾⡇⠀⠀⠀⠤⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠄⠛⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠁⡀⢀⣀⢠⣀⠠
⠀⠀⠀⠀⠈⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢀⡀⢘⠂
⠀⠈⠄⠠⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⢀⡀⣀⠠⠤⠐⠒⠒⠒⠦⠤⣀
⠀⠀⠀⠀⠀⠀⣀⡀⣠⣔⣀⡂⠐⢀⠰⠤⠀⠚⠛⠉⢁⡐⠢⡀⠀⠀⣀⠤⠶⠒⠙⠒⠚⠓⠒⠲⠤⠤⣀
⠀⠀⠄⠂⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠁⠀⠀⢁⣀⠥⠤⣁⣉⡐⠒⠲⠀⡀⣀⡠⠤⠄⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⣀⡠⠤⠤⠑⠐⠒⠚⠃⠀⠈⠑⣄⣀⠤⠉⠁⠠⢍⡀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⢀⣀⡀⠠⠄⠀⣀⡀⡀⣈⡆⡴⠒⠉⠉⠀⢄⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠠⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠈⢦⡠⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⣤⡀⠁⠀⠀⢏⣀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣶⢻⣶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣧⣿⣿⣿⣿⠋⠴⣤⣤⡤⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⣿⣿⣿⣿⣥⡛⣦⡞⢡⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⣟⣿⣷⣄⣠⣾⣿⣿⣿⣿⣻⣷⣿⣿⣿⣇⡻⣡⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⣿⣿⣿⣿⣟⠫⠍⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣁⠽⠿⣿⣿⣏⡠⠖⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⡿⢿⣿⣿⡏⣰⣿⣟⠙⠿⠋⠙⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⡿⢩⡶⠝⣫⡍⠃⠸⠁⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⠿⣷⡨⣷⠀⠉⢠⡆⢸⠀⢸⡇⠀⠀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣤⣶⣤⡂⠀⣤⣤⣦⣴⣄⣾⡇⠀⢰⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢝⡃⢁⣾⣿⣿⣿⣾⠟⡁⠀⣾⣿⣧⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠺⠉⠛⠛⢉⠠⠈⠀⠀⠿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⢯⠄⡀⠀⠂⢠⡄⠂⠁⠀⠀⠀⠀⠀⠈⠙⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⣿⣿⣿⣿⣿⣿⠒⡫⠈⠁⢀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⡑⠀⣤⣀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⡌⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⢱⡇⢿⣿⡿⠟⠉⠀⢀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣸⣷⡈⠁⠀⠀⣠⣾⡷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⡿⠓⢶⣶⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄⣸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⡄⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡀⠀⠀⠀⠀⢀⣄⢠⡀
⠀⠈⠻⢬⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⢆⣀⠀⠀⠀⠀⣀⠀⢀⡀⠉⠃⠀⠀⠀⠀⠉⠀⠀⠀⢀⠄⠀⠀⣏⣩⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠢⣄⡄⢐⠀⠀⠀⠀⠀⠉⠀⠀⠀⡤⣰⡃⡿⠁⢠⣤⠆⠀⠀⠀⣠⣼⣲⣆⡀⠀⠁⠀⠈
⠀⠀⡠⠒⢰⠄⢨⣤⡾⠛⠃⠞⠃⢄⠀⠀⠀⠀⢀⠠⣀⠀⢽⠟⠁⠀⠀⠀⠀⡀⠔⠚⠉⠈⠉⠻⣷⢀⣦⠀⢰⡐⡄
⠀⢠⣠⠏⠈⠀⠘⠔⠀⠀⠀⢤⣄⡀⠈⠲⡄⠀⠀⠓⠂⣴⠿⢀⠀⠀⠀⡔⠀⢀⣠⣤⣀⠀⠀⠀⠀⠻⢦⠃⠈⠛⠃
⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢠⢾⣿⠛⠀⠀⠈⠀⡀⠀⠈⠁⣨⠟⠁⠀⢈⣀⣾⣽⣋⣉⣁⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠈⠻⢿⣿⠶⡀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠘⢻⣸⡓⠓⣚⠀⠀⠀⠀⠀⠀⠀⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣌⣷⢿⡇⠀⠀⠀⠆⠀⠀⠀⠀⠀⣇⠀⠀⠛⠭⠭⠼⠃⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⠟⠀⠀⠀⠠⠃⢀⡀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠀⠀⠈⠛⠲⠇⠀⠀⠀⠙⠢⠄⣀⣀⠤⠄⠀⠀⠤⠐⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠉⠉⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⡄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛
⠀⠀⠀⣞⠱⠆⠀⠀⠻⣥⠀⠀⠀⡠⠀⠀⡞⠀⠘⠀⠀⠀⠀⠀⡁⠀⠀⢀⡀⠀⠀⢢⡄⠀⠀⣄⢄⣀⣀⡀
⠀⠀⠀⠙⢄⣀⣀⠀⠀⠀⠀⠀⠀⠁⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠸⡇⠀⠀⠀⢿⠀⠀⢿⣆⠌⠿⠂⠋
⠀⠀⠀⠀⠀⠉⢹⠆⠀⠀⠀⠀⠀⡴⠋⠀⡀⠀⠀⠀⣠⡤⣰⡄⡇⠀⠀⠀⡇⠀⠀⠀⠘⠷⣄⣀⠘⠓⠂⠀⢰⣦
⠀⠀⠀⢸⣡⠄⠃⠀⢀⡤⠀⣠⠞⠀⠆⠀⠿⠀⡇⠀⠷⠖⠉⠀⡇⠀⠀⠀⡇⠀⠀⢠⡀⠀⠀⠉⠓⢦⣀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⣤⠀⠀⠀⡍⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠈⠃⠀⠀⠐⠀⠀⠉⠳⠶⡦
⠀⠀⠀⠀⠀⣤⠞⠁⠐⡆⠀⠀⣿⠀⠀⢠⡁⠀⡇⠀⠻⣙⡀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⡆⠀⠂⠀⠀⢳⡄
⠀⠀⠀⢀⠔⠁⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸⡇⠀⡇⠀⠀⠈⠀⠀⡇⠀⠆⢀⡆⠀⡄⢰⠀⠀⠀⠀⡇⠀⠀⠀⠂⠀⠱
⠀⠀⠀⣉⠠⠤⠧⠤⠶⠧⠴⠷⠛⠒⠖⠾⠷⠖⠇⠀⠀⠀⠀⠀⠛⠛⠉⠋⠙⠛⠛⠛⠛⠒⠛⠒⠓⠂⠀⠄⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣠⣤⠀⠀⠀⠀⠀⠚⠶⢤⡀⠀⠀⣀⠠⡄⠀⠘⣃⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⢸⣷⠀⢸⣤⣴⣶⡆⠠⣤⢐⣏⠁⠀⠀⠋⣰⠃⠀⡰⠣⠁⠀⠀⠤⠤⠚⠃
⠀⠀⠀⠀⠀⣄⠳⠊⢁⡐⣛⠒⢕⣫⣾⢿⡏⠀⣼⠏⠉⠿⠀⠀⠀⠙⠉⠀⠀⠈⠉⠀⢀⡜⠁⠀⢀⠀⢦⠀⠀⠀⢀
⠀⢉⣈⣐⣒⡏⠀⠐⣟⣛⡂⠈⣿⠅⠀⠜⢉⣷⠶⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⡄⠰⡏⠁⣸⠂⠀⠀⢠⡀
⠀⠘⢂⣀⠀⣿⣀⣀⡿⠖⢀⠀⠹⡄⠋⣀⠈⠛⠐⠉⠁⠀⣀⣈⠀⠀⠀⡀⠀⠀⡉⠁⠀⠀⠈⠂⠤⠴⠋⠀⣠⡎⠈⠁
⠀⠐⠊⠁⠀⡴⢿⣧⢿⡶⠀⠀⠑⢠⡎⠀⣀⠀⠐⢀⣠⡞⢹⢿⣐⢰⡻⠏⡙⠿⣆⠀⠀⠀⢌⠀⠀⢠⠴⢚⢧⠠⣄⠈
⠀⠀⠀⠲⡄⣀⡢⠥⠀⠀⢀⣚⠫⠓⠛⠉⠀⠀⠠⠄⠈⠫⠛⢲⣬⣿⣏⣈⣧⠴⠀⠀⠝⠓⠀⠀⠀⢐⠒⠀⠾⠀⠉
⠀⠀⠀⠀⠈⠉⠛⠒⠒⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠿⠶⢧⣿⠖⠖⡂⠢⠀⠀⠀⠀⠀⠀⠈
⣠⣠⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡛⣛⣿⢉⡛⣛⡛⣻⣟⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠘⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢠⡤⠀⠤⣤⣤⣤⡤⡤⠀⠀⠀⠀⢀⣠⣤⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡴⠉⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⡠⢄⡀⢀⡴⢋⠉⢉⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⠤⣀⣀⡜⢁⠂⠉⠋⠀⢸⠀⢸⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡰⠋⡆⠀⢩⠀⠸⠄⠀⡇⠀⢸⠀⠘⣠⠴⠒⢒⠒⠲⢄⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡜⠁⠀⡖⠚⠒⡆⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢻⠀⠀⠀⠧⠤⢄⡀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠘⡄⠀⠀⡄⠀⠀⡇⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠘⠦⣀⠉⠹⠉⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁




⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢳⣄⠀⠀⠀⠀⠀⠀⢀⡤⢚⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣿⣾⣾⣿⣿⣿⣶⣤⣤⣾⣍⣴⠋⣀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡱
⠀⠀⢀⣠⣤⣶⡾⢿⣿⢿⣿⠿⠿⣿⡿⡿⢿⢿⡟⣻⣿⡿⣿⢿⢿⡿⣿⣿⣷⣤⣄⣀⣀⠀⠀⠀⣰⠋⣰⠁
⠰⢾⠋⠩⣽⣿⠃⠀⠉⢷⣶⣶⠶⠾⠿⠷⠚⠳⢶⣷⡌⠀⠁⠉⠚⠁⠉⠁⠋⠙⠛⠛⢛⣿⣾⣾⣿⣿⠃
⠲⢦⣧⡶⠈⠁⠀⢀⡄⣾⣸⣄⣀⠄⠒⠒⠒⣈⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠟⠋⠋⠉⠉⠙⣿⠿⢇
⠀⠀⠈⠉⠛⠲⢦⣼⣿⣟⣻⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣤⣶⣶⣶⣾⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣟⠿⡿⠛⠛⠛⠛⠋⠉⠉⠉⠙⠯⣉⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢬⣦⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡈⠙⠻⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠉⠻⢿⣶⣶⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⢿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⡿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⡿⠿⢿⣶⣦⣄
⠀⠀⠀⠀⠈⢿⣧⠉⠛⠻⢿⣶⣄⡀⠀⠀⠀⢀⣤⣶⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠈⠙⠻⣿⣦⣀
⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠉⠻⢿⣦⣴⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⢀⣠⣴⡿⠟⠛⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⢀⣿⣟⣠⣴⣾⡿⠟⠋⠀⠀⠀⠀⠈⠙⠿⢿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⢾⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⡿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⣀⣤⣾⡿⠛⠛⠛⠿⠿⠿⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⣶⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠘⣄⡰⠙⣄⣤⢀⡿⠀⣠⠴⡇
⠀⠀⠀⡀⠀⠀⢠⡄⣽⠑⠾⠀⠈⠃⠀⠙⠈⠃⠰⠞⠁⠐⠇⢀⣠
⠀⠀⠀⠙⢟⠃⠶⠆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠚⢩⠃
⠀⠀⠠⣤⠦⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⣠⠶
⠀⣀⣀⠨⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠏⣀⣀
⠈⠙⠒⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠔⠒⠊⠉
⠀⣠⣔⣋⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠉⠦⠄
⠀⠀⠀⠀⣼⣁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣘⣆
⠀⠀⠀⠈⠁⠀⣯⠴⠆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠹⡄⠁⠀⠀⠀⠀⠀⡴⠈⠀⣠⠛⢩⣶⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣇⠞⠻⢀⡜⢧⡞⣆⡞⠳⣄⣯⠙⠶⣄⡘⢆⠀⠀⠀⢀⣼⡗⠀⠀⢿⠀⠛⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠟⠀⠀⠀⠈⠀⠀⠀⠙⠀⠀⠀⠙⠺⣦⠀⠀⣼⠏⠀⠈⠁⠀⠀⠠⡀⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⢠⡿⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⠀⢸⡇⢀⡖⠂⠀⠀⠐⣒⣚⡇⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠚⠁⠀⠀⠀⣾⢡⡀⠀⠀⠀⠀⠀⠀⠈⣡⠀⢡⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣾⣇⠀⣠⠀⠀⣀⠀⠀⣿⡀⠈⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣆⠹⣤⣤⠋⠀⣴⢟⣧⠀⠈⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣠⠖⠟⠛⠛⣡⣠⠆⣀⣀⣴⡿⠋⠘⣿⣷⣄⠀⠙⣿⡳⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠌⣾⠛⠄⠀⠀⠈⠃⠉⠀⠉⠛⣯⠠⠶⣄⠈⠉⠻⠷⣦⡈⠹⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠼⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣽⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠱⢦⡀⠀⠀⠀⢠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢀⣤⣹⡌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡶⡄⢀⠀⠀⠀⣿⣿⠟⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⠃⠀⡈⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⣀⡀⠀⠀⣦⣀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣠⡤⠀⢀⣴⣿⡄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠻⣿⣿⡄⠀⠀⢿⠇⠀⠀⠀⠉⣿⠨⠲⠕⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠀⠘⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡗⠀⠀⠀⢧⠀⠀⠀⠀⣿⡄⠀⡢⠀⡀⢅⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠈⢣⠀⠀⠀⠸⡇⠂⠁⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇⠀⠀⠀⠀⡏⠀⢦⡄⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠫⠄⠀⠀⠀⠇⢠⠈⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢀⠆⡀⢀⠀⠀⠀⢠⡄⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠣⣈⣀⣴⠆⠀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠉⠑⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⢀⣿⣿⡄⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣶⣶⣤⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⣤⣼⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠐⠀⠈⠀⢸⠀⠀⠀⠀⣤⠀⠸⣿⣿⣿⣿⠋⢀⣾⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡀⠀⠀⠀⠀⣾⣿⣿⠿⢠⠀⠀⢀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠈⣄⠀⠟⢻⣿⡿⣿⣿⠁⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃⠀⠀⠀⠀⠙⠿⠿⠖⠛⠆⠀⠿⢿⠁
⠀⠀⠀⠀⠀⠂⠀⠀⡘⠀⠀⠀⠀⠀⠀⠈⠓⠀⠈⠛⠀⢻⡏⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠻⣿⣿⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠛⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣾⣿⣷⡄
⠀⠂⠀⠀⠀⣠⣾⠁⠀⠐⠿⠀⠀⠀⠀⠀⠂⠘⠦⠁⠀⢀⣀⣀⠹⣇⠀⠀⠀⢰⢲⣀⠁⠀⠀⡀⠀⡠⠘⡇⠀⠀⠀⠀⠀⡠⣄⠐⠀⠀⠀⠉⢻⣷
⠀⢀⠀⠀⣸⠯⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢈⡉⢡⣤⡈⠀⠀⠀⠀⠀⡙⠤⠄⠈⢀⣀⠀⠀⠇⠀⡀⠀⠀⠀⠀⠈⠀⠤⠐⠛⢛⣀⠙
⠀⠀⠄⠀⣏⠀⢠⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠟⣠⡎⠉⡰⠀⠀⠀⢠⣤⠔⠀⢀⣸⠏⠃⠀⣠⠖⠀⠀⠀⠀⠀⠀⣤⢀⣀⣿⣈⡙⢳⡔⠆
⠀⡀⠈⣆⠈⠳⠂⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠈⢁⡠⠾⠇⠀⠀⠀⠀⠙⢧⣠⢉⣥⠤⠒⠿⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⡻⣿⣿⡟⠛⠁
⠀⠀⠀⠛⠙⠲⠤⣤⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⣀⣀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠
⠀⠀⢰⠈⠉⠉⠉⢉⡍⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠂⠀⠀⠀⡖⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠁⠀⠀⠐⠈
⠀⠀⢸⠀⠀⠀⠀⠊⠘⠆⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⠀⠀⠀⠃⠈⠠⠤⠤⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠤⠤⠤⠬⠄⠀⠤⠀⠀⠶
⠀⠀⢸⢀⡌⠉⠉⠉⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⠇⡄
⠀⠀⢸⠀⡆⢀⠴⠀⠦⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⠤⢀⠀⠀⠀⣤⣀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠆⠘⢆⡀⠀⢀⠶⡀
⠀⠀⠸⠀⠀⠁⠀⠉⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣈⣉⣉⣁⣀⣉⣤⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠀⠀⠀⠉⠁⠀⠈⠂
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠟⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠒⠒⠒⠒⠒⡒⡂⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠁⠀⠀⠀⠁⠈⠁⠘⠓⢠
⠀⠀⠀⢅⢹⠉⠛⠉⠀⠀⢸⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡈⠉⠉⠁⠀⠀⢸⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠂⠀⠀⢠⠆⠀⡼
⠀⠀⠀⠓⠀⠀⠐⠒⠒⠒⢚⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠄⠀⡄⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⣀⣸⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠈⠉⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠉⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠐⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠑⠠⢁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡡⠒⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⡐⢄⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⢀⠠⢀⠤⣠⣶⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢷⣄⠀⠀⠀⠀⣀⡀⠼⡧⢐⡒⠈⠂⠈⠀⠁⠉⠉⠛⠦⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡒⡀⡼⠛⠓⠚⠉⠉⠉⠔⠁⡣⢌⠙⣌⣐⣄⠐⣠⠀⠖⠀⠰⢓⡩⣕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢏⢀⠺⣄⠫⠀⠀⠀⡀⠄⠦⣀⡀⡩⠚⠈⡉⠝⠀⣀⡢⠅⣒⣩⢥⡶⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣄⢳⠠⡹⣇⡠⠶⠦⢀⣀⡤⣤⡮⠍⠭⢗⡶⠮⠽⡆⠒⠏⣊⣥⣶⣶⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠱⢤⡈⠺⢟⢺⡯⠭⠶⠒⣒⡋⠡⣈⡥⢔⣨⣕⣶⡿⢟⣿⡿⣿⣯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠈⠀⠙⢦⠉⣶⣦⣤⣖⣺⠭⣟⣫⣽⣻⣝⣫⣾⠟⢋⣥⣶⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡂⢴⡆⠩⣭⣔⡒⠊⣩⣽⣯⣵⣾⣿⣾⣥⣴⡾⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⢗⣿⠂⠑⣲⣾⣭⣥⡿⡷⠿⠟⠛⢋⠁⡀⢀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠨⠀⠀⡈⠉⠩⢈⠁⠈⠀⠀⠐⠠⡂⢀⠁⡨⢀⠅⡈⢐⠁⠁⡀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈