⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣦⣶⡾⣾⡾⣿⣟⣿⣟⣿⢾⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⣟⣿⣯⣷⡿⣷⠿⠟⠿⠯⠻⠳⠻⠟⠿⠷⣿⢾⣯⣿⣻⣟⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⢿⣻⣽⣿⣻⡿⣾⢷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠛⠽⣷⢿⣽⣟⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣻⣽⣟⣿⢿⣽⣾⣟⣿⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢯⣿⣽⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣽⢿⢽⣯⣿⣻⡿⣾⣯⣿⣽⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⢿⡷⣿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣾⢿⣻⡾⠋⠀⠀⢈⣿⣽⣟⣿⢾⡷⣟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣟⣿⢷⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣴⢿⣾⢿⠏⠀⠀⠀⠀⠠⣿⡾⣯⣿⣻⣿⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣿⢿⣦⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣾⢿⣻⡽⠁⠀⠀⠀⠀⠀⠐⣿⣻⣿⣽⢿⡾⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣄⣄⣄⣄⣄⣤⢾⣯⣿⣻⣽⣷⡀⠀⠀⠀
⠀⠀⢀⣾⣟⣿⠟⠀⠀⠀⠀⠀⠀⠀⢈⣿⣟⣾⣯⡿⣟⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣻⣟⣿⣯⣿⣻⣟⣷⣟⣯⣿⡾⣷⡀⠀⠀
⠀⠀⣾⣯⣿⡝⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣽⡷⣟⣿⣟⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⢿⣽⣯⣷⡿⣾⣻⣽⠏⠀⠉⢳⣿⣟⣧⠀⠀
⠀⣸⡿⣾⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⢷⣿⣟⣯⡿⣷⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣿⣻⣾⣯⡿⣟⣿⡝⠀⠀⠀⠀⢿⣽⣟⣇⠀
⠠⣟⣿⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣟⣷⡿⣯⣿⣟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣯⣿⣻⣾⢷⣿⢿⡻⠀⠀⠀⠀⠀⠘⣷⣿⣻⠄
⣸⣿⣻⣝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣽⣷⢿⣟⣷⣿⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣯⣿⣽⣾⣿⣻⣟⠃⠀⠀⠀⠀⠀⠀⣻⡷⣿⡇
⣺⣯⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⢷⣿⣻⣯⣿⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡷⣿⣯⣷⡿⣷⢿⣽⡃⠀⠀⠀⠀⠀⠀⠀⢸⣟⣯⣷
⣿⢾⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⢿⣽⣯⣷⡿⣟⣿⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⡆ ⢀⣿⣯⡿⣟⣾⣷⢿⣟⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣽⡷
⢿⣻⣷⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣟⣿⢾⣯⡿⣟⣿⠀⠀⠸⠿⠽⠾⠿⠾⠻⠾⠯⠇ ⣼⣿⢾⣿⣻⡿⣾⢿⣻⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡾⣿
⣽⣿⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣽⣟⣿⢷⣿⢿⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣿⣻⣽⣯⡿⣟⣿⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⡏
⢱⣿⢾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⢷⣟⣿⣻⣽⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣷⣟⣯⣿⡾⣿⣟⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡿⣾⡇
⠘⣾⣿⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣟⣿⣽⢿⣽⣟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣽⣯⣿⣽⣷⢿⣿⣽⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣿⠀
⠀⢹⣾⣟⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣽⢿⡾⣿⣯⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣷⡿⣾⡷⣿⣻⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣽⡇⠀
⠀⠀⢿⡾⣟⣧⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣽⣿⣻⡿⣾⢷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢾⣷⢿⣟⣿⣻⣽⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢾⡗⠀⠀
⠀⠀⠈⢿⣟⣿⣇⡀⠀⠀⠀⠀⠀⠀⠠⣿⢷⣿⣽⣟⣿⣻⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣻⣿⣽⣟⣯⣿⣯⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⢾⡿⠁⠀⠀
⠀⠀⠀⠈⢟⣷⡿⣧⡀⠀⠀⠀⠀⠀⠐⣿⣟⣷⣿⣽⣯⣿⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠙⠈⠃⠋⠉⠃⠙⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣻⣟⡟⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠻⣟⣿⣻⣆⠀⠀⠀⠀⢈⣿⣽⣷⢿⡾⣷⣟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⣰⣿⣯⡿⠏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠙⢽⣯⣿⢷⣄⠀⠀⢠⡿⣷⢿⣻⣟⣯⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣷⠿⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⣟⣿⣾⢿⣻⣿⣻⣟⣯⣿⣽⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⢿⣯⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣿⢾⡿⣿⣽⣯⡿⣯⣿⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢾⣟⣯⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣻⣾⢷⣿⣟⣷⡿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣟⣿⣻⠿⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣟⣯⣿⣟⣿⣻⣷⢷⣶⣶⡶⣶⣶⣶⣾⢾⡿⣟⣿⣻⣽⡿⡾⠏⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠛⠏⠿⠾⢿⣻⡾⣟⣿⡾⡷⢿⠻⠿⠛⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀