Nine Emojis & Symbols

⠀⠀⠀⢀⣴⣶⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣻⣷⣶⣄
⠀⠀⢀⣾⣿⡿⠛⠛⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣍⠙⠛⠋⣹⣿⣧
⠀⠀⠸⣿⣿⣇⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⣴⠸⣿⡅⠀⢹⣿⣿⡀⠀⠀⠀⠀⢐⡟⣀⣀⣤⠀⣠⠴⡄⠀⠀⠀⠀⠀⣿⣽⡿⠀⠀⠀⢿⣯⣿
⠀⠀⠀⠹⢿⣿⣷⣴⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⣿⡄⠙⢿⣦⢸⣿⣿⡇⣼⡟⣻⡇⣿⢃⣿⠁⣿⢸⡯⠜⠁⠀⠀⠀⠀⠀⠙⢿⣧⣼⢶⣦⣼⡿⣷
⠀⠀⠀⠀⠀⠈⣰⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣾⡿⣸⣿⣿⢀⣿⠀⣿⣴⣿⣼⡏⠘⠿⠞⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⢹⣿⡟
⠀⠀⠀⠀⠀⠔⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣰⣿⡿⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣾⣿⢾⣿⠁⣿⣏⣿
⠀⠀⠀⢠⡄⣤⡄⣄⣄⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣦⣴⣶⣦⣼⣿⠇
⠀⠀⠀⢸⢻⡟⡇⡿⡇⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⠟⠛⠁
⠀⠀⠀⠘⠈⠉⠁⠁⠋⠙⠃



⠀⠀⠀⠀⠀⣀⠠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⢀⣀
⠀⠀⢀⣶⣾⣿⣿⣷⣶⣝⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣿⣋⣽⣟⠛⣿⣆⡀
⠀⠀⣾⢳⣿⡛⠁⠉⢽⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⢠⡄⠀⠠⠀⢐⡂⣀⢠⣇⢰⡆⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣾⡿⣶⣿⠟⠾⡏⣿⣊⢻⡷
⠀⠀⢻⢸⣿⡡⣀⠠⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢸⡷⠀⢸⣿⢸⡇⣿⢸⢿⣼⡇⣿⠚⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⡅⠀⠀⠈⣿⢟⢸⣿
⠀⠁⠈⠧⡻⢿⣿⣿⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⣿⠏⠀⢸⡟⣿⡇⣿⢸⠀⣿⡇⣿⡴⠆⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⣆⠀⠀⢀⣿⡟⣿⡏
⠀⠂⠀⢀⣄⣀⣀⣠⣼⣿⡟⡐⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢀⡤⠀⢸⡇⠻⠇⠛⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⢀⡙⠛⠀⢀⣾⠟⣿⠟
⠀⠀⠀⢧⠻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠷⢢⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣼⣿⢸⠄⡀⢤⡾⢫⡾⠋
⠀⠀⠀⣤⠈⣅⣤⡄⢠⣠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠘⠘⢢⢴⣭⠾⠉
⠀⠀⠀⠿⠻⠿⠿⠟⠿⢿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠟⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⡆⠀⠀⠀⠀⣰⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⠿⠃⠀⠀⠀⠀⠻⠿⠿⢿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⣄⡀⠀⠀⠀⠀⢀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠈⠳⡄⢀⡴⠋⠀⠀⠀⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⡾⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣴⡦⠀⠀⡇⠀⠀⠀⠀⠀⢾⠆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣄⠀⠀⠀⠀⠀⠀⣼⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠉⠈⢷⣄⠀⠀⠀⠀⠀⢀⣴⠋⠀⠙⠷⣤⣤⣤⡴⠟⠁⠀⠈⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⣤⡄⠀⠀⠙⠻⠶⠶⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠂⠀⠀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠈⠙⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠓⠚⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠛⠉⠁⠀⠉⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⠀⠀⠀⣀⣤⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⢀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠶⢦⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⢀⡾⠋⠛⠋
⢀⣤⣶⣶⣦⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⠀⠀⠀⠀⣠⡴⠋
⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣷⣶⣶⣶⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣤⣤⣤⣤⠶⠛⠉
⠘⢿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⡿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⣀⡀
⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⡿⠟⠋⠁⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠈⠉⠙⠛⠷⢶⣤⣄⡀
⠀⠀⠛⠿⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠉⠙⠻⠿⣿⣶⣦⣤⣀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣋⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⣤⣭⣤⡴⢶⡶⠶⢶⣤⣬⣤⣐⣀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡴⣺⣿⣾⠿⠛⢉⣡⣤⣶⣾⣿⣿⣶⣶⣾⣿⣯⣽⣛⡿⢿⣿⣿⣿⡠⣄
⠀⠀⠀⠀⠀⠀⠔⠀⢀⡴⣫⣾⠟⠋⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣻⣿⣿⣿⡻⣄
⠀⠀⠀⠀⠀⢀⠈⣴⢯⣾⠟⠁⣠⣾⣯⣿⣿⣿⣿⣿⡿⠟⠋⠉⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⣝⣿⣿⣟⣧⡀
⠀⠀⠀⠀⠀⠀⡜⣷⡿⠃⢠⣾⣿⢫⣿⣿⣿⡿⣻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣯⣷⡀
⠀⠀⢀⢁⠀⡼⣽⡿⠁⣰⣿⠟⡁⣾⣿⣿⡿⣱⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠄⠀⢸⢳⣿⠃⢠⡿⡟⠀⢸⣿⣿⣿⢇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡘⣿⣿⣿⣿⣇⣿⣧⠂
⠀⢠⢠⠀⡇⣿⡏⠀⠞⣿⠀⡠⢸⣿⣿⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠸⡟⣿⣿⡏⣿⣿⡜
⠀⡀⠀⣰⢻⣿⠀⢰⣧⠇⠒⠀⢸⣿⣿⡟⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⢙⣿⣧⡄
⠀⠁⠃⠻⢾⠁⠀⠸⠃⠀⠀⠀⣾⣿⣿⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠀⠸⣻⣿⣿⡟⣿⣿
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡀⣶⣶⣧⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇⣿⢿⠐
⠀⠀⠀⠀⠀⣿⡄⠀⠀⢄⠡⣱⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣧⣯⣼⣤
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⢆⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣟⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⢦⠻⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡏⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠰⡍⢿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠀⠀⠀⣿⣿⣿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣦⠀⠀⢲⠼⢿⣿⡿⣿⣿⣿⣷⣤⣔⣂⣀⣀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣿⣿⣿⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠈⢶⣽⡿⣿⣮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢠⣿⣿⡇⣸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡙⢿⣿⣿⣿⣶⣽⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⠇⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⢼⣿⣿⠀⠀⠀⣿⣿⣿⣰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠿⠛⠛⠛⠋⠉⠁⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠀⠀⣻⣿⡿⠛⢛⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⣴⣶⣾⢇⣤⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢁⣼⣿⣿⠏⣾⣿⡟
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣟⣿⣿⣿⢿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣇⣾⣿⣿⠏⣼⣿⡟
⠀⠀⠀⠀⢀⣾⣿⣫⣾⣿⡿⠿⣿⣷⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⢸⣿⣏⣿⣿⠁⠀⠀⣹⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠸⣿⣏⣿⣿⣆⠀⠀⣿⣿⣿⣿⣧⣄⣀⣀⣀⣠⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠙⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢟⣫⣹⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣶⣾⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⢛⣛⡛⠛⢛⣛⡛⠋⠉⠉⠁
⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣦⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⠻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣇⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢋⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡴⠶⡶⠶⠶⠶⢶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⡉⠉⠁⣄⣀⣨⣦⠤⠷⠶⠶⠦⣭⣿⣳⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⡋⠁⠐⢄⠀⣈⡶⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡁⠀⠘⢦⣀⣤⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠈⠦⡀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢅⠀⠀⠠⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠑⢄⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⣶⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠢⡀⠀⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⢹⡆⠢⢝⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⢲⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢸⡗⠢⢄⡈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠁⠢⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠗⢦⡀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠲⢄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣾⠑⢤⡙⢦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣸⠏⠡⡀⠙⢤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣀⣠⣾⠋⠱⡀⠑⣄⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠑⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣥⣄⢳⡀⢳⣀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠄⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣜⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠉⠛⠲⠦⣤⣤⣀⣀⣀⣀⣀⣠⣤⣶⠾⡋⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠟⡻⠉⠉⠻⣦⡀⠀⠀⠀⠀⠉⠉⠉⢈⡟⢀⠈⢦⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⡡⠃⠀⡜⢠⠁⠀⠀⣀⣨⣿⣦⡀⠀⠀⠀⠀⣠⢾⠁⡀⢆⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠳⣦⣀⣠⣧⠶⠞⠛⠉⠁⠀⠈⠳⣤⡤⠶⠏⠙⢨⠀⠀⢈⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠶⠶⠶⠶⠶⠶⠚⠛⠉⠁

⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠂⠄⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣐⡀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠁⠀⠀⠀⠀⠀⠀⠀⠠⢂⠀⠁
⠀⠀⢀⡴⠚⠈⡇⠀⠀⠀⠀⣠⠊⠉⠋⠙⠛⡄⠀⠀⠀⠀⠀⣏⠈⠑⠛⠙⡽⠃⠀⠀⠀⠀⠀⡜⠑⠁⢺⠀⠀⠀⠀⠀⠀⢸⠁⢉⣻⣋⡑
⠀⢸⢁⣠⠀⠂⡇⠀⠀⠀⠀⠑⠦⠊⢙⠄⠀⡇⠀⠀⠀⠀⠀⠀⡠⠋⠀⣜⠁⠀⠀⠀⠀⢀⡜⢀⠜⡁⢸⠀⠀⠀⠀⠀⠀⢸⠀⠧⢄⡀
⠀⠈⠁⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠞⠀⡰⠃⠀⠀⠀⠀⠀⠈⠦⠤⡄⠀⠱⡄⠀⠀⢀⠞⠀⠘⠚⠁⠘⠲⡄⠀⠀⠀⠀⠘⠒⢢⡀⠈⢦
⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⡔⠁⠀⠘⠓⠒⡆⠀⠀⠀⠀⡤⠤⠴⠃⠀⢠⠇⠀⠀⠘⠒⠒⠒⢲⠀⢰⠊⠁⠀⠀⠀⠀⡴⠤⠞⠁⠀⡼
⠀⠀⠀⠸⠦⠤⠇⠀⠀⠀⠀⠀⠧⠤⠤⠤⠤⠴⠇⠀⠀⠀⠀⠧⠤⠤⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠸⠤⠼⠀⠀⠀⠀⠀⠈⠲⠤⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡀⠈⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠄⡀⢦⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢤⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢀⡨⠴⠶⠔⢲⠀⠀⠀⠀⡔⠒⠒⠴⢗⣂⠷⡄⠀⠀⠀⠀⡠⢤⢧⣴⡀⠀⠀⠀⠀⠀⡠⠶⡠⢗⡰⣄⠀⠀⠀⠀⠀⢀⠤⠶⢷⡨⣄
⠀⢠⠋⠀⣠⣒⡒⠋⠀⠀⠀⠀⠧⠤⠤⣤⠀⠀⣀⠇⠀⠀⠀⡎⠀⣔⢐⠄⢸⠄⠀⠀⠀⢸⠁⢠⠊⠑⡄⢘⡆⠀⠀⠀⡴⠁⡠⠒⠒⢄⠈⢧
⠀⡎⠀⠀⡀⣀⠉⢢⠀⠀⠀⠀⠀⢀⡞⠁⣠⠞⠁⠀⠀⠀⠀⣱⠄⠈⠉⠠⢎⠀⠀⠀⠀⠸⡀⠈⠒⠚⠁⠉⡇⠀⠀⢰⠇⢰⠃⠀⠀⠘⡄⢸⡀
⠀⡇⠀⢈⠀⢈⠆⠈⡇⠀⠀⠀⢀⠎⠀⣰⠃⠀⠀⠀⠀⠀⡼⠁⢠⠒⠐⡆⠀⢣⠀⠀⠀⠀⠉⠒⣲⠆⠀⢰⠃⠀⠀⠸⡆⠈⢧⣀⣀⡼⠀⢸⠁
⠀⠙⢄⠀⠉⠉⢀⡼⠁⠀⠀⠀⣾⠀⢀⡇⠀⠀⠀⠀⠀⠀⠹⣄⠈⠑⠊⠁⣠⠏⠀⠀⠀⠰⡏⠉⠀⢀⠔⠃⠀⠀⠀⠀⠱⣄⡀⠉⠁⢀⣠⠏
⠀⠀⠀⠉⠙⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠠⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⡀
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⡀⡄⠀⠀⠀⠀⠀⣀⣀⣐⠂⠀⠀⠀⠀⠀⡀⡀⡀⣐⡀⠀⠀⠀⠀⠀⢀⢂⠌⠀⠀⠀⠀⠀⠀⡀⡢⡈⡀
⠀⢀⠴⠋⠁⡇⠀⠀⠀⢠⠊⣀⣉⠉⢱⠀⠀⠀⠀⠸⠄⢠⠉⢨⠎⠀⠀⠀⠀⢠⠋⡁⠘⠀⠀⠀⠀⠀⢸⠀⡭⠿⠠
⠀⠡⠔⡆⠀⡇⠀⠀⠀⠀⠉⠁⡸⠀⡸⠀⠀⠀⠀⠀⢔⠁⠀⠣⡄⠀⠀⠀⡰⠁⡜⡁⠀⣀⠀⠀⠀⠀⢸⣀⡑⠒⢄
⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⣠⠞⠀⠴⠥⣄⠀⠀⠀⢀⣈⣈⠇⠀⢸⠀⠀⠰⠥⠤⠤⡄⠀⡴⠃⠀⠀⠀⢀⣀⡸⠀⢈⠆
⠀⠀⠀⠧⠠⠇⠀⠀⠀⠀⠧⠤⠤⠤⠤⠜⠀⠀⠀⠸⢄⣀⡠⠔⠃⠀⠀⠀⠀⠀⠀⠧⠠⠀⠀⠀⠀⠀⠧⢄⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⠤⠦⠔⡆⠀⠀⠀⡠⠤⠤⠴⡆⡡⡄⠀⠀⠀⢀⣬⣦⣥⡀⠀⠀⠀⠀⡠⢄⠴⡨⣄⠀⠀⠀⠀⢀⡤⠶⣦⣅
⠀⡰⠁⢠⣖⣒⠃⠀⠀⠀⠧⠤⢤⠄⠀⣀⠇⠀⠀⢰⠁⢔⣐⠀⠱⠀⠀⠀⢸⠀⡔⠁⡆⢨⡆⠀⠀⢠⠋⡠⠒⠲⡄⠱⡀
⠀⡇⠀⡠⠠⡈⢣⠀⠀⠀⠀⣠⠋⢀⠞⠁⠀⠀⠀⡠⠓⢀⣀⠐⢅⠀⠀⠀⠸⣄⡈⠉⠁⠀⡇⠀⠀⡏⠀⡇⠀⠀⡱⠀⡇
⠀⣇⠀⠦⠤⠃⢸⠀⠀⠀⢠⠃⢀⡏⠀⠀⠀⠀⠀⡇⠀⢇⣨⠇⢀⡇⠀⠀⢀⡤⠭⠋⠀⡼⠁⠀⠀⢳⡀⠑⠤⠔⠁⢠⠃
⠀⠈⠒⠠⠤⠖⠁⠀⠀⠀⠘⠒⠚⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⠔⠊⠀⠀⠀⠈⠦⠄⠐⠈⠀⠀⠀⠀⠀⠙⠢⠤⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⣴⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠈⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣄⣤⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣄⣠⣼⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡿⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⡿⠟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣶⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠛⠻⠷⠶⠶⠶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⠶⠶⣤⣤⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⠀⠀⠀⠀⠀⠈⠙⠻⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣦⠀⠀⠀⠀⢸⡇
⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠘⣿
⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠖⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⣸⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⣀⣀⣈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⣿⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡋⠉⣈⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⣿⡄⠸⡄⠀⠀⠀⠀⠀⠀⠀⠰⣤⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⡶⠿
⢹⡇⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠻⢾
⠈⣿⡀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠹⣧⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠹⣧⡀⠀⠀⠀⠉⠛⠶⣤⣤⣄⣠⣇⣀⣀⣇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠙⢷⣄⠀⠀⠀⠊⢠⡄⠀⠉⢹⡏⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣶⣤⣤⣤⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣀⣤⠀⠀⠀⠀⠀⠰⠤⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⡤⠤⠤⠤⠤⠶⠶⠶⠒⠒⢺⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣴⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⡄⢀⠀⠠⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡐⠿⢀⣄⠀⠐⠈⠁⠴⠈⢉⠐⠀⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢿⣿⣿⠇⠀⠈⠉⠀⠀⠀⠀⠰⣿⠆⠀⠂⠘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⣭⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢛⣀⣄⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⣿⣿⡿⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡄⢾⣿⣿⡇⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠌⠉⢴⡆⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⣈⠙⠋⠀⣶⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠈⡀⠰⠆⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠰⠄⣰⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠛⠁⢿⣿⠆⢀⠛⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠰⠷⠀⠙⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢃⣤⣤⡀⢾⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⣀⠙⠻⠛⣁⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢸⣿⣿⡟⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢛⡁⣶⡆⠛⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⠈⠀⢠⡌⢡⣶⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠶⢠⣾⣿⣶⠐⠿⠃⠀⣠⣤⡀⠉⠠⢈⣤⣄⠂⠀⠄⣩⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠻⠿⠛⠀⠃⢀⠀⢿⣿⠇⠀⠀⣿⣿⣿⠇⠛⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⠌⠉⢡⠀⢰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡃⠛⠂⠠⠌⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠀⠐⣀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠂⠀⢁⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣶⣿⣶⠀⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢤⠄⡈⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡘⠃⠰⠆⣴⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠃⠶⢀⡅⠶⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⠿⠻⠿⠛⠿⢿⠿⢻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠛⠋⠉⠙⠛⠒⠲⠤⣄⣰⠋⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣸⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣠⠖⠋⢉⣀⣤⠤⠤⠬⠭⢷⠀⠀⠀⠀⠀⠀⡞⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⣇⢠⡞⠉⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢰⠃⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠘⢾⣇⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⡼⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⢀⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣀⣀⣀⣀⣀⣀⣀⡤⣤⠀⠀⠀⠀⢸⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⠭⢭⣭⣥⣤⡤⣯⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⢧⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠸⡆⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⡰⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣉⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠺⠿⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⡙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⡴⠶⠶⣦⣄⡀⠀⠀⠀⣠⡾⠋⠉⠉⠉⠙⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠁⠀⠀⠀⠀⠉⠻⣦⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⣠⣶⢦⢸⡇⠀⠀⠀⠀⢀⣴⡶⢦⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⣿⣧⣼⢾⡇⠀⠀⠀⠀⢸⣿⣧⣼⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠻⠿⠟⠸⣇⠀⠀⠀⠀⠘⢿⣷⡿⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⣴⠿⣄⠀⠀⠀⠀⠀⠉⠀⠀⢠⡾⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠙⠷⣤⣀⣀⣀⣴⠾⠋⠀⠙⢧⣀⠀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠈⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⠛⠳⠞⠛⠋⠁⠀⠀⠠⣶⣶⣦⣄⠉⠻⣦⡀
⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⠙⠷⠀⠈⠻⣦
⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⢿⡁⢸⣿⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣶⣤⣤⣤⣤⣤⣤⣤⣴⡶⠾⠟⢋⠁⠀⠀⠱⢈⣿⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⢻⣦⠀⠀⠈⢦⠀⠀⠀⢹⡿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⢇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠶⠶⠶⢶⣦⣤⡀⠀⠀⠘⣷⡀⠀⠈⠆⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠉⠻⣷⡄⠀⠸⣧⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⢻⡇⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢈⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠰⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣄⣀⣀⣠⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⣶⣤⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⣴⡶⠾⠛⠋⠉⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⣿⠿⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⢠⣿⠀⠀⠈⠙⠛⠿⢶⣦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋
⠀⠀⠀⠀⠉⠻⢷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣴⠶⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠀⠒⠂⢀⣀⣀⣀⣀⣄⣀⣀⠐⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⣨⣥⡶⢶⣛⣻⣿⣯⣭⣭⣭⣬⣭⣭⣽⣿⣿⣛⣶⣶⣮⣄⠂⠡⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⠀⠂⣠⡶⢾⣻⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣛⡷⣤⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⡴⢟⣿⣿⣿⠟⠛⠉⢀⣠⣴⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣷⣮⣭⣛⡻⠿⣿⣿⣿⣿⣅⢢⣄⠀⡀⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⣠⡾⣯⣶⣿⠟⠉⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣟⡿⣿⣿⣷⣿⢷⣅
⠀⠀⠀⠀⠀⠀⠀⠂⠴⣡⡾⢯⣾⡿⠛⠀⣠⣴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡻⣿⣿⣷⣿⣷⡡⠈
⠀⠀⠀⠀⠀⡀⢀⠂⣰⣟⣼⣿⠟⠁⣠⣾⣿⢿⣿⣿⣿⣿⣿⡿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢿⣿⣿⣮⣻⣎⠄
⠀⠀⠀⠀⡰⠀⠂⣰⠋⣿⣿⠃⠀⣴⠿⣿⡏⣼⣿⣿⣿⣿⠟⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣯⢿⣿⣿⡽⣿⣿⣷⣿⡔
⠀⠀⠀⠀⢁⠀⣸⢯⣿⣿⠃⢀⣾⣿⠟⡀⢰⣿⣿⣿⣿⠏⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣷⢿⣇⠘
⠀⠀⢀⠀⡈⢠⠿⣼⣿⠃⠀⣾⣯⠟⠠⠀⣾⣿⣿⣿⡟⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⣿⣿⡏⣿⣿⣠⠃
⠀⠀⡈⠀⠁⣼⢰⣿⡏⠀⣸⣿⠍⡣⠀⠰⣿⣿⣿⣿⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⡟⢸⣿⣇⡰
⠀⠀⠇⠰⢀⣇⣿⡿⠁⠀⣩⢼⠇⡀⠚⠀⣿⣿⣿⣿⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⠘⣇⣸⣿⣿⣧⡿⣿⣿⡇⢠
⠀⢀⢀⠀⣼⢹⣿⠇⠀⢸⣿⡎⠀⠁⠀⢰⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⢻⣿⣿⣿⣿⡁⣿⣿⣧⠇
⠀⠈⠘⠀⠿⢸⡏⠀⠀⠼⠋⠀⠀⠀⠀⢸⣿⣿⣿⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠸⣟⣿⣿⣿⡿⢹⣿⣿⠀⡄
⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⣨⣭⣭⡍⣤⣬⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⢿⣿⣿⣿⣷⢸⣿⣿⠰⡇
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠠⡄⢆⣿⣿⣿⣿⣿⣿⡇⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⢸⠇⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⡆⠐⣼⢿⣿⣿⣿⣿⣿⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠀⠀⢸⣿⣿⣿⣿⣾⣿⣿⣶
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠙⣆⢻⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠸⣄⠻⣼⣿⣿⣿⣿⣿⣿⡆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠻⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⣬⡑⣿⢿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣻⣿⣿⣿⣿⣿⡟⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⣃⠜⣿⣿⣿⣿⣿⣿⣿⣿⣄⠂⠀⢀⣀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢸⣿⣿⣿⠀⣽⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⢽⡚⣝⣿⣿⣿⠻⣿⣿⣿⣿⣿⣶⣥⣤⣤⣶⣶⣶⣾⣿⣿⣷⡿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⣾⣿⣿⡟⢠⣿⢿⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠻⣯⣭⡟⣿⣷⣍⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢠⣿⣿⣿⡇⢸⣿⣾⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣄⠘⠻⣿⣿⢻⣿⣿⣮⣽⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢸⣿⣿⣿⠃⣼⡟⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⣿⣿⣿⡇⠀⠀⠀⣾⣿⣿⡿⢠⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠀⠀⢨⣿⣿⣿⡇⠀⠀⢠⣿⣿⣿⣷⣾⣿⣿⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⣼⣿⣿⣿⣿⣿⢿⣿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠈⣿⣿⣿⠏⠛⢛⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⢀⣬⣴⣾⣿⢠⣄⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇⢠⣾⣿⣿⡿⢁⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣶⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟⢠⣿⣿⣿⣿⢣⣼⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⣿⣽⣿⣿⣿⡛⢿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣧⣿⣿⣿⣿⢃⣾⣿⣿⡁
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣫⣾⣿⣿⡿⠿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⢠⣿⣿⢿⣿⣿⡟⠁⠀⠀⢘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⣿⣿⡾⣿⣿⣇⠀⠀⠀⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⢻⣿⣷⣿⣿⣿⣦⡀⠀⢹⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⢿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣋⣄⣻⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣭⣭⣽⣶⣾⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⣸⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠛⠻⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣬⣙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⠛⠳⢶⣄⠀⠀⢀⡴⠋⠁⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠈⢷⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⣀⡀⢀⡿⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣼⣿⢹⣼⡇⠀⠀⠀⠀⣼⣿⢙⡆⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢹⣿⣽⢻⡇⠀⠀⠀⠀⢻⣿⣿⡟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠿⣆⠀⠀⠀⠀⠀⠉⠁⢀⣿⡀⠀⠀⠀⠀⠛⠋⠀⠀⣰⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠁⠀⠙⢷⣄⡀⠀⠀⣀⣴⠟⠉⠻⣄⠀⠀⠀⠀⠀⢀⡼⠋⠀⠈⠙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠀⠀⠀⠀⠈⠻⠶⠶⠶⠞⠋⠀⠀⠀⣤⣤⣄⡈⠛⢷⣄
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠛⠆⠀⠙⣷⡀
⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⡋⢸⣿⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⢶⣦⣤⣤⣤⣤⣤⣤⣶⠶⠟⠛⡉⠀⠀⠱⣈⣿⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠱⠀⠀⠀⢹⡟⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⢡⡀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⠛⠛⠛⠻⠷⣦⣄⠀⠀⠹⣧⠀⠀⠐⣀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⢻⡆⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠈⣿⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⠀⠀⠀⠀⣀⣤⡾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠁
⠀⠹⣧
⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⠶⢶⣦⣤⣤⣤⣤⣤⣶⡶⠶⠿⠛⠋⠉⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⢸⡏⠙⠻⠶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠉⠙⠛⠿⠶⢶⣦⣤⣤⣤⣤⣤⣤⡶⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁
⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉
⠀⠀⠀⠀⠉⠻⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠷⠶⣶⣦⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣤⣤⣤⣴⡶⠾⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⣙⡏⠀⠀⠀⠀⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⡉⠉⢙⡇⠀⠀⠀⠀⢧⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⠀⠀⠀⠀⢀⡤⢒⡚⠉⠙⠯⣙⠋⠉⠙⡇⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢣⠀⠀⢀⠔⠁⠰⣇⡸⢠⣞⢲⠀⠉⠉⠈⠑⢄⠀⠀⠀⡸⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⢦⣠⠋⠀⠀⣠⠔⠒⠢⡉⠉⠀⠀⠀⠀⠀⠈⠆⠀⢠⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡏⠀⠀⢰⠁⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠘⡤⢃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢇⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⡄⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠙⢄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠙⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢄⣀⣀⣀⣀⢤⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠳⡀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢇⠀⠀⠀⠀⢰⠃⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠳⠤⠤⠔⠁⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠒⠒⠚⣙⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⢏⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠼⢿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠔⠒⣏⡉⠉⣒⣻
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⣴⣿⠿⠿⣿⣦⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠻⠿⢻⣦
⠀⠀⣿⣿⡀⢀⣿⡿⠀⠀⠀⠀⢰⠺⣧⠀⢿⣿⠀⣀⣀⢀⣇⡤⣤⣠⢶⠄⠀⠀⠀⣿⡿⢀⡀⢸⣿⠃
⠀⠀⠈⠛⢫⣿⡿⠁⠀⠀⠀⠀⠸⣧⣬⣷⣼⣿⢰⡏⣿⣼⣹⠇⣿⣿⡅⠀⠀⠀⠀⠈⠻⢿⡿⢿⣽⠇
⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠈⠉⣰⣿⠃⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠆⢸⣿⠆
⠀⠀⢰⣴⣶⣶⣾⠁⠀⠀⠀⠀⠀⠰⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣾⣿⡾⠋
⠀⠀⠈⠛⠋⠙⠙⠂


⠀⠀⢀⣠⣄⣠⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⣤⣤⡀
⠀⢰⢿⡿⠛⢻⣿⣧⠀⠀⠀⠀⠀⢰⡿⠋⣁⠀⠀⠀⢀⣀⡄⡄⡶⠂⠀⠀⠀⠀⠀⣼⢯⣿⠾⣷⣎⣻⡆
⠀⠸⣹⣧⣠⣸⣿⣿⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢸⣾⣿⣿⣿⡇⡟⠃⠀⠀⠀⠀⠀⢿⣻⡇⠀⠈⣿⣽⡇
⠀⠀⢀⡉⠛⣩⣿⡇⠀⠀⠀⠀⠀⠘⠇⠈⢋⠀⣸⢹⠿⠟⠃⠃⠉⠁⠀⠀⠀⠀⠀⠈⠻⠇⠀⣸⣿⡿
⠀⠀⠫⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⢰⡟⢢⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡇⢀⣶⣻⠞
⠀⠀⡶⣴⣶⢶⣶⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠝⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠆⠀⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠐⡔⠉⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⠀⠀⡀⡁⠀⡄⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠈⠢⢀⠠⠣⢀⡀⠔⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⢀⠊⢈⡁⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠈⠳⡀⠀⠀⠠⡤⠈⠆⠀⠀⠀⠀⠀⠀⢢⠀⣀⠠⡘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠘⡀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⡠⠃⠀⠀⣰⡇⠀⠀⠃⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⠓⠒⠒⠊⠁⡀⠠⣾⣿⠁⠀⠀⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠟⠀⠀⠀⠰
⢠⠀⠄⣪⠘⢄⠀⠀⠀⠀⣀⡠⠜⠓⠤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠁⠦⠈⠀⢀⡳⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠁⢒⠞⣠⠀⡎⠀⠀⠀⠀⠀⠀⠖⠢⢄⣀⠀⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠒⠁⠦⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⢤⠤⠤⠤⠤⢴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠘⡄⠀⢸⠀⣀⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠶⠖⣶⣖⣲⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⣤⢞⣻⣿⣶⣶⣶⣬⣍⡛⠳⢮⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣽⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡝⢦⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡾⢁⣾⣿⣿⡿⡿⠛⠛⠻⣟⢿⣿⣿⣿⣿⡎⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣼⠇⣼⣿⣿⢋⡞⠉⠁⠒⢄⠈⢧⣹⣿⣿⣿⢿⡸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⣿⣿⣿⢸⣇⠀⠀⠀⠀⠐⢨⣿⢸⣿⣿⣼⡇⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⠀⣿⣿⣿⡜⢌⢦⡀⠀⢀⣠⢞⣷⣿⣿⣿⣿⡇⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠸⣿⣿⣿⣦⣙⠛⣾⣉⣵⣾⣿⣿⣿⣿⢸⡇⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡇⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣉⠛⠻⠿⠿⠛⠛⠏⢸⣿⣿⢷⣿⢱⠃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⡼⣠⣿⣿⡟⣾⡏⡘⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢛⣶⣦⣭⣭⣤⣴⣯⣾⣿⣿⡿⣽⡟⡰⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣼⢏⡔⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡙⠿⠿⣿⣿⣿⡿⠟⣑⣫⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠒⠀⠀⠀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄⠀⣶⡆⣶⡆⣶⣦⡀⢰⣶⢰⣶⡶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣦⣿⡇⣿⡇⣿⡿⣷⣼⣿⢸⣿⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠹⣿⡇⣿⡇⣿⡇⠈⢻⣿⠸⣿⣶⣶⡦






⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⢶⡶⣶⡶⠶⠶⠶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠆⠀⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠑⡄⠁⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠀⠀⠤⠀⠀⠄⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠑⠤⠄⠓⠤⠤⠊⠀⠙⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⡠⠤⠤⢄⡀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⡎⠀⠌⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠙⡄⠀⠀⠰⠂⠘⡄⠀⠀⠀⠀⠀⢸⠆⠤⠐⠨⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠀⠀⢰⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠈⠢⣀⡀⢀⣀⠜⠀⠀⣠⣾⠀⠀⠀⡄⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠘⢻⡿⠀⠀⠀⡃
⠀⠀⠀⡄⢄⠀⠀⠀⠀⠀⠀⢑⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⠀⠃
⠘⠄⡁⠲⠈⠠⣄⣀⠠⠔⠂⠉⠀⠈⠑⠒⠒⠒⠒⢢⠀⠀⠀⠀⠀⠀⠀⢰
⠆⠀⡤⠀⠀⠔⠁⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠠⠫⣀⠀⠀⠀⠀⠀⢀⠆⠀⠈⠑⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠤⢀⣀⣀⣀⣀⣠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠘⡆⠀⢸⠀⣀⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⢀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠾⠽⡇⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠺⠷⠖⢺⠁⠀⠀⠀⠀⣇⠈⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⢴⡞⠿⣶⠶⠶⢾⠀⠀⠀⠀⠀⠈⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⡄⠀⠀⠀⠀⡠⠚⣡⡔⣄⠀⣀⣉⠒⠤⠄⣀⣼⡀⠀⠀⠀⠀⢠⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢳⠀⠀⣠⠎⠀⠀⠙⠧⣚⠸⣧⣨⠂⠀⠀⠀⠀⠈⢆⠀⠀⠀⡸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠳⣴⠃⠀⠀⢠⠞⠉⠉⠑⣄⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢠⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡇⠀⠀⢀⠇⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢸⡴⢃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⣥⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠘⢆⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠈⠑⠦⡄⠠⡤⠞⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠂⠄⠒⠒⢺⡆⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠈⢣⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⠀⠀⠘⡄⠀⠀⠀⠀⣰⠃⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠈⠢⠤⠤⠖⠁⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⢲⠶⠆⠲⠚⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡾⠀⠀⠀⢸⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠒⠂⠞⣷⣀⡀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠚⠻⣉⣉⣉⣉⡩⠇
⣀⡀⣀⣀⡀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⣠⠖⠋⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠙⢦⡀⢠⡞⠁⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣤⡄⢸⡇⠀⠀⠀⢀⣶⣆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⢿⡇⠀⠀⠀⠀⠀⣧⣿⢸⡇⠀⠀⠀⠸⣾⣿⠀⠀⢀⣿⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⢻⡄⠀⠀⠀⠀⠙⠋⣰⢿⡀⠀⠀⠀⠙⠁⠀⠀⣼⠃⠈⠙⠷⣤⡀
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣠⡴⠃⠈⠳⣄⡀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⠛⠋⠉⠉⠙⠛⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⡀⠀⠀⠘⣧
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⠀⠹⣆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢻
⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠸
⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⢀⣀⣀⣀⣤⡤⠄⢶⡏⠁⢸
⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣤⣄⣀⣀⣀⣀⣠⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⣸
⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡶⠾⢧⡏
⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠛⠒⠛
⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⣦⣤⣤⣤⣤⣤⣴⣶⡶⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⢀⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⣾⠃⠈⠉⠻⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠶⠶⠶⠾⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠉⠛⠷⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⠶⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣛⣛⣓⣶⣶⣶⣶⣶⣶⣶⣖⣛⣋⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡤⠤⠴⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⠲⣄⠀⡴⠋⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠘⣾⠁⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⢶⡄⡿⠀⠀⠀⠰⣷⡀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠋⠹⡄⠀⠀⠀⠘⣿⠇⣿⠀⠀⠀⠸⡿⠃⠀⢰⠟⢶⣄⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠹⣆⡀⠀⠀⢀⡴⠋⢧⡀⠀⠀⠀⠀⣠⠏⠀⠀⠈⠛⢦⡀
⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉⠀⠀⠀⠙⠲⠦⠶⠚⠁⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠉⠉⠙⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠒⡶⠀⠀⠈⣷
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⣇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢀⣰⡇⠀⠀⠀⠀⢻⡀
⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠠⠄⠤⠤⠤⠄⢲⡏⠀⡇⠀⠀⠀⠀⢸⡇
⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⢤⣤⣤⣤⡴⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⢀⡇⠀⠀⠀⠀⠈⣧
⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠲⢏⣉⡿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠶⠶⠶⠶⠶⠖⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⣼⠋⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠈⠉⠛⠓⠶⠶⠶⠶⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⢠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠙⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡘⠀⠈⢃⠠⠀⡃⢶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣾⣿⣷⠈⠁⠿⠀⠀⠐⠀⣩⣄⠂⠠⠃⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠙⡛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⠆⠴⠈⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡿⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢰⣿⣿⡆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠍⠉⢵⡆⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⣈⠛⠛⠡⡦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⢈⣁⠰⠆⡰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠀⠰⠀⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣁⠺⡿⢃⣈⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡘⠟⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡙⢡⣶⣶⡄⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⣶⠌⣉⠡⡦⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠘⠿⠟⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡛⢁⣤⣤⡀⣴⡄⠀⠀⠀⠠⠄⠨⠄⠀⠘⠃⡘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣿⣿⠇⠬⠀⠀⣿⣿⠆⠀⢰⣿⣿⡆⠴⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠁⠀⠀⡌⠛⢛⡁⣠⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⠰⠂⢠⠌⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡿⠂⡐⣀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⠐⠀⠡⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢰⣿⣿⡆⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡘⠂⠊⡙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡉⠐⢀⠀⠁⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠁⠛⠈


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⡛⡛⢛⡻⣻⢿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠛⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠿⣧⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠃⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠙⣿⡶⠶⠛⠛⠛⠛⠶⣤⣀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠷⠶⢿⣿⡉⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⣠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡷⠀⠀⠀⣿⡀⠀⠐⢿⠿⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠻⣷⣀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠙⠻⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠉⠀⠀⠀⠻⣧⣀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠈⠛⠷⣶⣦⣴⡶⠿⠛⠁⠀⢀⡀⠀⠀⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⢀⣠⡄⠀⠀⠀⣈⡛⢷⣶⣤⣤⣴⣶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⣤⠀⣠⡿⠿⠀⠀⠀⠀⠙⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠊⠻⢶⣄⣰⠏⠉⠉⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⣀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄
⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⣤⣟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡾⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄
⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⠿⠿⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠈⠙⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏
⣿⡇⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⣀⣀⠀⣀⣀⣠⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃
⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠙⢷⡀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇
⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟
⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁
⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠇
⠀⠀⠀⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣶⣤⣤⣄⣀⣀⣀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣷⣶⣤⣀⡀⠀⠀⠀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣶⣾⣿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠖⠛⠛⠛⢛⣛⢛⡒⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⢉⠀⠐⣄⡤⠖⠚⠉⠉⠉⠉⠉⠙⠛⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠐⢆⢀⡵⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠑⢄⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⢄⠀⢂⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⠀⠑⡾⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠘⣏⢙⣲⡄⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠑⢺⠃⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣷⠢⣈⢹⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⡈⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡟⠢⣄⢹⡇⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⠲⡇⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣸⢇⠑⢌⢻⠇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⠢⣷⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢀⣰⠋⡀⠣⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢼⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⡉⢆⠰⣀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡉⠓⠶⢤⣄⣀⣀⣀⣀⣀⣤⡤⠖⢻⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠖⡛⡙⠉⠙⢷⣄⠀⠀⠀⠈⠉⠁⣽⠁⠈⢆⡿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⡀⠀⠀⢀⣠⣤⠴⠞⠳⡄⠀⢀⡠⠞⠃⠂⠡⣸⠃⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⢹⡓⠛⠉⠁⠀⠀⠀⠀⠙⠻⣍⣀⠂⠀⣠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⢤⣤⣄⣀⣀⣠⣤⡤⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣏⣉⣙⠛⠿⢭⣝⡲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⣿⢚⣭⣾⣿⣿⣿⣿⣿⣶⣄⣙⡳⣍⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢯⡞⣡⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣷⡝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⡿⢠⣿⣿⡟⡱⠖⠠⢄⠈⢳⡙⣿⣿⣿⣿⡸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡇⢸⣿⣿⠸⡇⠀⠀⠀⠈⠀⣻⡏⣿⣿⣿⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⡇⢸⣿⣿⡜⣝⢆⠀⠀⢀⣰⢿⣽⣿⣿⣿⡇⣿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⣿⣿⣿⣬⣛⠳⣖⣋⣴⣿⣿⣿⣿⢿⡇⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡇⡇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⠍⠛⠛⠛⠛⠛⠻⢀⣿⣿⣏⣿⢃⢃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣠⢃⣾⣿⡿⣽⡟⡘⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣿⣷⣦⣴⣶⣾⣶⣿⣿⡿⣽⡟⡑⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣟⡽⣫⠞⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢬⣍⣛⣛⠻⠛⢋⡥⠽⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡈⣉⡀⣀⡀⠀⣀⡀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⣿⡇⣿⡇⣿⣷⡄⣿⡇⣿⣏⣉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡍⠻⣿⡇⣿⡇⣿⡋⢻⣿⡇⣿⣏⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠉⠁⠙⠁⠉⠀⠀⠉⠁⠉⠉⠛⠉





⣿⢿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⠀⠀⠀⠀⠀⢀⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⣠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⢻⣶⠶⠖⠛⠲⠶⣤⡀⠀⠀⠀⠀⣀⣤⠶⠶⠶⢾⣟⠁⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢠⡞⠉⠀⠀⠀⠀⠀⠈⢷⣄⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⢸⡇⠀⢠⣶⡄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⣼⣧⠀⠀⠉⠀⠀⠀⠀⠀⢀⣾⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠙⣧⣀⠀⠀⠀⠀⠀⣠⡾⠋⠙⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠋⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠈⠙⠻⠶⠶⠾⠛⠋⠀⢀⣀⠀⠈⠙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⢀⣴⣖⠀⢀⡤⠬⢽⡟⠷⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⢠⠖⠊⠙⢆⣠⠾⠙⠀⠀⠀⠀⠙⣿⣷⡀
⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠙⠳⢾⣄⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠿⠿⠿⠿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⡏⠀⠀⠀⠀⠀⢀⡠⢤⣀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⣿⡇⠀⠀⠀⠀⠀⡏⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿
⣿⡇⠀⠀⠀⠀⠀⠣⣄⣀⣤⠋⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⣿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⣤⣀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣤⣤⣤⣴⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇
⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡀⣈⠇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟
⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁
⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠈⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣶⣦⣤⣀⡀⠀⠀⢰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣶⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⣄⡀⠀⠀⠀⣠⠖⠋⠉⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠙⢦⣀⡾⠁⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣇⠀⠀⠀⠀⢠⢻⡆⣿⠀⠀⠀⢰⣻⡆⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠁⢻⡀⠀⠀⠀⠘⠿⠇⣻⡀⠀⠀⠈⠿⠇⠀⢀⡟⠛⠶⣄⡀
⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⢀⡴⠋⢳⣄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠻⢦⡀
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠀⠀⠀⠉⠓⠶⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⠉⠉⠛⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠲⣤⠀⠀⠘⣧
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⣇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢿
⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⣀⣀⡀⣤⠞⠹⡇⠀⠀⠀⠀⢸
⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣄⣀⡀⠀⠀⣀⣠⡴⠟⠁⠀⠈⠉⠉⠀⠀⠀⠘⣆⢠⡇⠀⠀⠀⠀⠈
⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⠶⠿⣼⠁
⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠛⠚⠁
⠀⠀⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠛⠛⠛⠉⠉⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⣰⠏⠈⠙⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠓⠒⠚⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠙⠻⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠉⠛⠳⠶⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⠀⠀⠀⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⣀⠀⠀⣀⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠈⢣⡜⠁⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢴⠄⢸⡇⠀⠀⠀⠶⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⡄⠀⠀⠀⠀⢀⡼⠳⣄⠀⠀⣀⡼⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣤⠀⠙⠲⠤⠤⠶⠋⠀⠀⠈⠉⠉⠁⣠⠖⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠈⠓⠦⠤⣄⣀⣀⣀⣀⣠⡤⠤⠖⠋⠁⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠉⠁⠈⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⢾⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣽⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⣨⠟⠛⠋
⣠⣶⣷⣶⠚⠉⠁⠀⠀⠀⠀⠉⠻⢶⣤⣄⣀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⣀⡤⠞⠁
⢻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠛⠛⠋⠁
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏
⠀⠀⠀⢀⣀⣤⣶⣾⡿⠟⢻⡀⠀⠀⠀⠀⠰⣧⣄⡀⠀⠀⠀⠀⠀⠀⣼⡿⠶⢤⣄⣀
⠀⠰⠾⠿⠿⠛⠋⠁⠀⠀⠘⡇⠀⠀⠀⢀⣠⠞⠉⠛⠻⢶⣦⣄⣀⣰⣿⠁⠀⠀⠀⠈⠉⢓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠃⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⢿⣿⣿⣷⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠳⣆⣠⠴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡏⠀⠐⢶⡞⠁⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠙⡇⠀⠀⢸⠡⠦⠀⠀⢀⡟⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠁⠀⠀⠀⠈⠓⠒⠊⠳⣄⣀⡤⠎⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣠⠶⠛⠋⠉⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⣧⣀⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⣀⠀⠀⠀⠀⢀⣀⣠⣴⠏⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠉⣠⠏⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢳⣶⣦⣤⣤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⣀⠀⠀⣧⢸⡆⢷⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣅⡀⠀⠀⠀⠈⠉⢙⣋⣨⣇⠘⠋⠉⠁⠉⠉⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠉⠉⠑⠒⠲⠤⠤⠤⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⡠⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠉⠉⠀⠀⠀⠀⠉⠉⠓⠦⠞⠀⠀⠀⠈⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢀⣠⠤⠖⠒⠒⠒⠒⠦⣄⠀⠀⠀⠀⠀⢀⡇⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡏⢀⡴⠒⠋⠉⠉⠉⠉⢹⡆⠀⠀⠀⠀⣼⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠳⣸⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⢀⡇⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠈⠓⠲⠤⠤⠶⠒⠉⠀⠀⠀⠀⠀⢸⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣄⣀⣀⣀⣀⣀⣀⣠⣤⠀⠀⠀⠀⡏⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠭⠭⠥⠤⠤⣿⠀⠀⠀⠀⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠈⡇⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⣹⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⡤⠔⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⣤⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⠺⠭⠀⣀⠀⠀⣀⠙⢽⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢠⣵⡆⠈⠉⠀⠀⠀⠀⠈⠻⣶⣿⣦⣾⣿⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠙⠈⠀⠈⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣛⠁⠀⣤⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢧⡠⠐⣲⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡿⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣟⣿⣿⣿⣿⡆⢀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣧⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⡿⣿⢓⢀⠘⠂⠀⠀⠀⠸⣿⠟⡠⢿⡟⠛⠿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠛⣿⣼⣿⠻⠶⣾⣿⣶⣶⣿⠆⠀⠀⣿⢿⣿⡻⢏⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠂⠀⠘⠋⠉⠉⠁⠀⠀⣴⠀⣸⣿⣷⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣿⣿⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣫⠄⠀⠾⣶⣤⣀⣀⠀⠀⠀⢀⣠⣾⣶⣿⣿⣿⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡂⣩⣿⣿⣬⣿⠗⢾⣿⢿⢿⣿⣿⣟⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣶⣿⠏⠛⣿⠿⣧⠼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠁⠛⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠆⠀⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠐⡔⠉⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢴⠀⠀⡀⡅⠀⡄⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠈⠢⢀⠠⠣⢀⣀⠔⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡀⠔⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⢀⠊⢈⡁⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠈⠳⡀⠀⠀⠠⡤⠈⠄⠀⠀⠀⠀⠀⠀⢢⠀⣁⠠⡘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠘⡀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⡠⠃⠀⠀⣰⡇⠀⠀⠃⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⠓⠒⠒⠊⠁⣀⠠⣾⣿⠁⠀⠀⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡟⠀⠀⠀⠨
⢠⠀⠄⣊⠘⢄⠀⠀⠀⠀⣀⡠⠜⠓⠤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠁⠦⠈⠀⢀⡳⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠁⢐⠞⣠⠀⡎⠀⠀⠀⠀⠀⢀⠖⠢⢄⣀⠀⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠒⠁⠦⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⢤⠤⠤⠤⠤⢴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠘⡄⠀⢸⢀⣀⣀⣀⣀⡀

⠀⠀⠀⢀⣴⣾⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣻⣶⣦⡀
⠀⠀⢀⣿⣿⡿⠛⠛⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡉⠛⠛⠉⣿⣿⡄
⠀⠀⠸⣿⣿⣇⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⢸⣿⡀⠀⢻⣿⣷⠀⠀⠀⠀⠀⢸⠃⣀⣠⣄⢀⣤⢦⠀⠀⠀⠀⠀⠸⣯⣿⠇⠀⠀⠸⣿⣽⡇
⠀⠀⠀⠹⢿⣿⡷⣶⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⡀⠙⢿⣆⢸⣿⣿⠀⣾⠋⣿⢣⣿⢸⡟⢸⡟⣾⠧⠋⠀⠀⠀⠀⠀⠀⠻⣿⣶⣿⢿⣶⣿⢿⡧
⠀⠀⠀⠀⠀⠈⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣿⠟⣼⣿⡟⢸⡿⠠⣿⣼⣧⡿⠃⠸⠷⠛⠓⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⣻⣿⡋
⠀⠀⠀⠀⠀⠜⠛⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣼⣿⡟⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⠿⠷⠾⣏⠸⣿⣽⡧
⠀⠀⠀⢠⡄⣤⡤⣄⡄⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣤⣶⣶⣤⣿⡿⠁
⠀⠀⠀⢸⢿⣿⣿⡿⡇⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠾⠛⠋
⠀⠀⠀⠈⠈⠉⠉⠁⠁⠉⠁



⠀⠀⠀⠀⠀⡠⠤⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣤⣤⣀⣀
⠀⠀⢠⡶⣿⣿⣿⣿⣷⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢻⣯⣡⣭⣛⡻⣷⣄
⠀⠀⣿⢹⣿⡝⠀⠈⣹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⣤⡄⠀⠀⡀⢐⢈⡄⣴⡆⣾⢸⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⣿⠟⠛⠻⣿⣟⡌⣿⡂
⠀⠀⢻⢸⣿⡱⢄⣀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⢸⣷⢸⢸⡇⣿⢿⣿⢸⡟⠃⠀⠀⠀⠀⠀⠀⠀⠠⣿⢿⣿⠀⠀⠀⢸⣿⣃⣿⡇
⠀⠀⠀⠣⠙⠿⣿⠿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠸⠿⠃⠀⢸⠹⣿⢸⡇⣿⠘⣿⠸⠷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣻⡄⠀⠀⣼⣿⢻⣿⠁
⠀⠀⠀⣠⣤⣀⣀⣤⣾⣿⠏⠌⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⣠⡆⠀⠸⠀⠛⠈⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⣀⠀⣀⡉⠣⠀⣰⡿⢻⡟⠃
⠀⠀⠀⠣⣻⠿⠿⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡟⠛⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿⣏⡷⠠⢔⣼⣋⣾⠛⠁
⠀⠀⠀⣤⣀⣤⣤⣄⢠⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠁⠁⢉⠴⣫⠖⠋
⠀⠀⠀⠟⠙⠿⠿⠋⠻⠻⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠟⠋

⣿⣿⣿⢿⣿⠿⢿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣷⣦⣶⣶⣴⣧⣾⣿⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣷⢠⣤⠄⠀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠃⠀⠀⠀⠀⠀⠈⠁⠚⠘⠁⠀⢀⣉⣶⣙⣃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⠆⠀⣠⣶⣀⣤⣤⣾⣶⣠⡁⠀⣘⣙⢽⠟⣻⣿⣦⣤⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⢠⣼⣀⠟⢛⠿⣟⠻⠿⢿⡿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣄⢰⣼⣿⣤⣤⣤⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣶⢶⡀⠀⠂⠀⣠⠟⢉⣾⣾⢡⣟⡙⠛⣻⣿⡟⣿⣿⣿⣿⣽⣿⣿⢿⡯⠿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢈⣯⣤⣤⠼⠆⠓⠃⣰⡶⠦⠠⢉⣻⣿⣿⣿⣄⠀⠀⠁⠰⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣠⣴⣿⡾⠟⠋⠙⠛⠛⠺⣟⠃⠈⠈⠒⢊⣭⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣦⠐⡚⠀⠀⣤⣸⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⢛⣻⡈⠛⡻⠄
⠀⠀⠈⢋⣍⣀⠀⠶⠷⠀⣥⡔⠀⠷⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠠⣿⡿⣽⠶⣷⣆⡈⣷⣆⢀⠀⠀⣰⣴⡟⠀⠀⠀⠀⢤⢤⠀⠀⢠⡄⢈⠄
⠀⠀⠀⠀⠛⠀⢸⠀⢈⣽⣿⢀⣏⢡⣤⣿⣿⣿⣿⠓⠀⢀⣀⣤⡾⠓⠛⠁⠈⠉⠀⣀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠈⠧⢰⣿⠇⢹⡿⠛⢹⠿⡷⠄⣈⣰⣀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⢀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡧⠀⠀⠀⣀⡀⠀⣠⡾⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⣿⠇⣀⣶⡶⠿⠿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣼⡿⢺⣿⠋
⠀⠀⠀⠀⢀⠄⣫⡵⠟⠁
⠀⠀⠀⠀⠀⣘⡿⠁
⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⡄⠀⣠⡶⠛⠛⠷⣤⠀⠀⣿⠀⣿
⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⢸⡇⠘⣇⢰⡏⠀⠀⠀⠀⢸⡇⢸⡇⢰⡏⠀⠀⠀⣠⣤⣤⣄⡀
⠀⠀⠀⢠⡟⠁⠀⠀⠙⣷⠀⠀⠀⣿⠀⢿⡈⢷⡀⠀⠀⢀⣼⠃⣾⠀⣾⠁⠀⠀⣾⠋⠀⠀⠈⢻⡄
⠀⠀⠀⠸⣧⠀⠀⠀⢀⡿⠀⠀⠀⢹⡆⢸⣇⣀⣙⣛⣛⣋⣁⣰⡏⢠⡟⠀⠀⠀⢿⡀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠈⠻⠶⠶⠛⠁⠀⠀⠀⠈⣷⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⣸⠇⠀⠀⠀⠈⠛⠶⠶⠟⠁
⠀⢀⣀⣤⣴⠆⠀⠀⠶⣤⣤⣀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣤⣤⠶⠀⠀⠰⣦⣤⣀⡀
⣰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻
⣿⠀⢠⡄⠀⠀⠀⠀⠀⠀⢠⠀⢈⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡁⠀⡄⠀⠀⠀⠀⠀⠀⢠⡄
⢻⡆⢸⡇⠀⠀⠀⠀⠀⠀⣾⠀⣼⠇⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⠸⣧⠀⣷⠀⠀⠀⠀⠀⠀⢸⡇⢰
⠀⠻⣾⡇⠀⠀⠀⠀⠀⠀⣿⡾⠋⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇⠀⠙⢷⣿⠀⠀⠀⠀⠀⠀⢸⣷⠟
⠀⠀⢸⡇⠀⠀⢠⡀⠀⠀⣿⠀⠀⠀⠸⡇⠀⠀⢠⡶⢶⡆⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⢀⡄⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣷⢸⡇⠀⠀⣸⠇⠀⠀⠀⢿⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⣿⠀⠀⢸⡇⠀⢀⡗⠀⠀⠀⠀⢻⡄⠀⠀⣿⣼⠁⠀⠀⣿⠀⠀⠀⠀⢸⡀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⢿⣿⠀⠀⢰⡏⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⣿
⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⣷⠀⠀⢸⡿⠀⠀⣸⠇⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⣿
⠀⠀⠀⢸⡆⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢿⡀⠀⠸⠇⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢰⡇
⠀⠀⠀⢸⣇⣀⣸⣇⣀⣼⠇⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣠⡏⠀⠀⠀⠀⠀⠸⣧⣀⣸⣇⣀⣸⡇
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣀⣠⢀⣻⢿⣿⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡂⠁
⠀⠀⠀⠠⠋⣹⣿⣿⣵⣿⡏⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣿⠠⣦⠁⠁⢠⣾⣏⢤⣴⡶⠄
⠀⠀⠀⠀⠀⡟⠀⠉⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠅⠀⡀⠈⠀⠙⢄⠀⠁
⠀⠀⠀⢀⠌⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⡀⠑⠢
⠀⢀⡤⠋⣀⡔⠁⠀⠀⠀⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠐⢤⡄
⠀⠈⠠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⢀⣠⡼⠾⣿⢻⠷⢷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠿⠛⠿⣤⠄⠀⠀⠀⡀⣤⣷⡾⢿⣷⢷⣾⣤⢀
⠀⠀⢀⣾⡋⠀⢰⣇⣼⣆⣀⣙⣿⡄⠀⠀⠀⠀⠀⠀⣠⠀⣀⢈⣿⡐⠿⢀⣿⡄⠀⠀⠠⣽⣿⡁⢀⣸⣇⡀⢈⣻⣿⡄
⠀⠀⣾⠧⣍⣹⡟⠁⠈⠹⣏⣩⠽⣿⠀⠀⠀⠀⣺⣶⠿⠿⢿⣮⡯⢿⠟⠿⠇⠀⠀⠀⣿⡏⠙⢻⡿⠉⠉⢻⡟⠋⢹⣿⡀
⠀⠀⢿⡄⠈⢹⣧⣀⣀⣴⣯⠀⢀⣿⠀⠀⠀⢻⣿⠃⣠⣦⡀⢹⣷⠦⠀⠀⠀⠀⠀⠀⢾⣧⣠⣼⣷⣄⣀⣼⣷⣄⣸⡿⠂
⠀⠀⠈⢷⣴⢋⡬⠟⠙⢧⡘⣦⣾⠃⠀⠀⠀⠛⣿⣆⠈⠋⢀⣼⡿⠆⠀⠀⠀⠀⠀⠀⠉⢿⣿⡁⠀⢹⡏⠁⢈⣽⡿⠍
⠀⠀⠀⠈⠛⠿⣶⣤⣤⢶⠿⠛⠀⠀⠀⠀⠀⠈⠋⢻⡿⢿⡟⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⡷⣿⣿⢾⡟⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣀⣤⡀⣶⣶⡀⣤⣄⠀⠀⠀⠀⠀⠀⢲⡆⠀⠀⠀⠀⠀⠀⣰⠖⠀⠀⠀⠀⠀⠀⢀⣠⣴⣴⢴⣤⣄
⠀⠀⢠⣤⣼⣿⣿⠿⠿⣿⣿⣯⣠⣄⠀⠀⠀⠙⠺⠿⣄⡶⠼⠦⣶⣠⠿⠗⠋⠀⠀⠀⠀⣼⣟⡿⢾⡋⠙⣻⠾⡟⣶⡀
⠀⠀⠙⣿⣿⠋⠀⠀⠀⠀⠙⣿⣿⡋⠀⠀⣠⢤⢤⢶⣿⣶⡶⣶⢦⡿⣶⣤⣤⣄⠀⠀⢼⣯⢯⣄⣈⣷⣴⣇⣀⣽⣟⣗⡀
⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠿⢦⣼⣼⣬⣿⣤⣿⣬⣧⣿⡧⡿⠟⠀⠈⣿⣽⣀⣨⣿⣁⢈⣿⣉⠀⣟⣿⠂
⠀⠀⢠⣿⣿⣦⡀⠀⠀⢀⣠⣿⣿⣦⠀⠀⠀⠀⠀⣠⠿⣦⣤⣤⣴⠿⣄⠀⠀⠀⠀⠀⠹⣼⣏⠁⣸⠟⢻⡅⠉⣻⣿⡿
⠀⠀⠈⠉⣹⣿⣿⣿⣿⣿⣿⣯⠉⠁⠀⠀⠀⢄⢹⠋⠀⠀⠈⠁⠀⠀⠹⡏⡢⠀⠀⠀⠀⠹⢾⣟⣿⡤⣤⡗⣾⣽⠛⠁
⠀⠀⠀⠀⠈⠉⠀⠻⠿⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠛⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢭⣷⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⣼⠏⠹⣧⣠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢿⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠈⣠⣤⣤⣄⠁⢠⡿⠁
⠀⠀⠀⣠⣤⠶⣤⣄⢰⣗⣾⣿⡖⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠛⠀⣾⠋⠀⠀⠈⣷⠀⠻⢶⡆
⠀⠀⣼⣇⡽⣶⠿⣨⣿⡷⣾⣹⡼⠃⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⠀⢻⣄⠀⠀⣠⡿⠀⣴⠾⠃
⠀⠀⣧⡼⣟⣻⣻⢧⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣾⠃⣀⠉⠛⠛⠉⣀⠈⣷⡄
⠀⠀⠘⠷⣾⣉⣿⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⢻⣆⣰⡟⠉⠻⠋
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⣤
⠀⠀⠐⠒⡂⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⡀⠀⠀⣾⠢⡀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢀⠤⠷⡀
⠀⠀⠈⠊⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡧⠈⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠗⠒⠀⠀⠀⠀⢆⠊⣀⡠⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠭⠄⠒⢈⡢⠀⠀⠀⠫⡩⠃⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠈⠣⢄⠀⠠⠁
⠀⠐⠆⠀⠀⡀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠁⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡥
⠀⠀⠀⠀⢸⡇⠀⠄⢠⡄⠀⣇⡠⢃⣀⠀⢣⠀⡀⠀⢾⠉⠂⠀⣽⠀⡖⢲⠀⢹⡏⢡⠀⠀⢿⠀⠄⠀⠀⣆⠀⡀⠀⠈
⠀⢀⡀⠀⢸⠁⢈⢀⡟⠰⠀⣷⠀⣿⠀⡁⠈⡗⠀⠀⢸⠉⠑⠀⡇⠀⡟⢅⠀⠸⠇⣸⠀⠄⢸⢀⠄⣰⠇⠘⣦⠀⠀⢠
⠀⠈⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡿⠈⠀⠀⠃⠀⠀⠘⠇⠊⠀⠀⠀⠃⢀⡀⠀⠀⠛⠀⠃⠈⠁⠰⠋⠁⠀⠿⠀⠀⠂
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠈⣎⠀⠀⠀⠀⢀⡀⠀⠀⠀⠠⠄⡄⠀⠀⡇
⠀⢀⠀⠀⠀⠘⢲⠤⣀⠘⠀⠀⠂⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⣄⣄⠤⢤⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠈⠀⠀⠀⢰⠈⠀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠒⢒⡲⠭⠙⡄⠀⠀⠀⡒⠢⠔⠒⡄
⠀⢀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⣀⠀⠀⠀⠢⢀⣪⡂⠀⠀⠀⠀⠀⠀⢋⣁⡀⣤⢤⣰⠷⠀⠀⠀⠑⡀⢀⠜⠀⠀⠀⠀⠔
⠊⠀⠈⠂⠐⠊⠀⠀⠉⠁⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠈⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠛⣠⡿⠽⢧⣘⠋⠻⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣠⠏⠁⠀⠀⠀⠈⠑⣤⣶⡌⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⢸⠃⠀⠀⠀⠀⠀⠀⠀⠘⡄⠁⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢶⡄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡴⢿⡆⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠟⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠃⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠃⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠶⢶⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠶⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢟⠷⠀⠘⠛⠇⢀⣿⠿⠚⠞⠛⠇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉⠀⠀⢠⡀⠀⣠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⣦⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠊⠀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠛⣧⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⡛⣡⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠛⠉


⣿⣿⠟⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠐⠒⠂⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠃⠀⠀⡄⠀⠀⠠⠀⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠐⠀⠐⠀⠣⠤⠔⠀⠀⠘⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢘⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠀⡀
⠀⠀⡠⠂⠀⠀⠀⡀⠀⠀⠠⡀⠐⠦⠒⠉⠀⠀⠐⠁⠀⠀⠀⠀⠀⠒⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⡔⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⢆⡔⠀⠀⠤⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣐⢀⣀⣀⠀⢀⡀⠀⠀⠀⠀⡀⠀⡠⠀⠉⠉⠀⠠⠐⠒⠠⢧⡀⠀⠀⠈⢆
⠀⠀⠀⠀⢀⡤⠒⠺⠃⠈⠀⠢⠄⠀⠄⠀⠀⡀⠀⠠⠂⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠂⠀⠀⠀⠀⠀⡀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⡔⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⡌⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠠⡀⠀⠀⠀⠈
⠤⠀⡧⠀⣀⠈⠀⠚⠿⠀⠀⠀⠀⠉⠐⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⣴
⡔⠀⠡⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠑⢄⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡀⠤⠀⠈⠐⢄⠀⠀⠀⠀⠀⠤⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠈⠄⠀⠐⠊⠀⠀⡄⠠⠔⠋⠁⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠤⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⠀⠀⢀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠂⠉⠁
⠀⠈⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⢠⠎
⠀⠀⠈⠄⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠐⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠰⠂⠀⠘⠀⠀⠀⢄⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠁
⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠄⠀⠀⠈⠢⡀⠀⡇⠀⠀⠸⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠁⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠘⠦⠞⠃⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡠⢀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⢀⣀⡠⠆⠀⠐⠀⠀⠀⠀⡠⠤⠀⠀⠀⠀⠀⠀⠒⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠂⠠⠄⠀⢀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⡰⠀⠀⠀⠀⠀⠀⠁⠀⠚⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣟⠀⠀⠀⠐⠀⠀⡀⠢⠊⠀⠀⠀⠄⠀⠂⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠤⡀⠀⠀⠀⠀⠀⡀⠂⠐⠀⠄⠀⠀⠘⠂⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠃⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠐⠀⠀⠀⠀⠢⠖⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡴⠚⠓⢦⠀⠀⠀⠀⠃⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠋⠀⠐⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⠃⠀⠀⠐⠦⠄⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠁⠉⣉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠿⣿⠿⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡷⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣶⡶⠶⠆⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡗⢻⣟⡀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⠆⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠄⠀⢀⣀⢀⠘⠀⠀⠀⠀⠀⢠⣿⣿⠍⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣶⣶⣿⠟⠃⠀⠀⠀⠀⠀⢸⣟⣿⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⠁⠀⠀⠀⠀⡆⠀⠀⠀⢸⣿⣿⡅⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⡀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡇⣿⣧⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⣥⠀⠀⠀⠀⠈⠀⠀⠀⢸⣇⣾⣿⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣬⣷⠀⠀⢀⣀⠐⢦⣤⣼⣿⣿⣿⣿⠷⡤⢾⣇⠀⠠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⣶⣿⣶⣿⠿⠈⠀⠀⠀⠀⣰⣬⣭⣴⣶⡞⠁⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⢉⡀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⡏⣀⡀⠀⠀⠀⠀⣤⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⡗⠙⠁⠀⠀⠉⠉⠙⠋⠉⠉⠀⠀⠙⠀⠀⠀⢰⣿⣿⠿⠀⠲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⡀⠄⠀⣰⣦⢀⣴⣿⠟⠁⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣧⠼⠋⠀⠀⠀⠀⠀⠀⠀⡠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠻⣦⣶⣦⣴⣶⣶⣾⣿⣿⣟⠛⠁⠀⠀⠀⠀⠀⠀⢀⣀⣼⠇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠛⣋⡉⠉⠉⣾⠏⠀⠀⠀⠀⣸⣦⣶⣾⣿⡋⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⠃⠈⠉⢳⢀⣼⠋⠀⠰⠀⠀⣠⣿⣿⣿⣿⣿⡿⠓⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⡇⠀⢀⡴⠋⣸⣇⣠⠾⠅⢀⣴⣿⣿⣿⠻⠿⠏⠀⠀⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡄⠀⠐⠒⠀⠀⠀⣠⣼⣿⣿⣿⣿⣦⣭⣤⣾⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠛⠛⢻⣶⣿⣿⣿⣿⠟⠹⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⡁⢸⡿⠿⠟⠋⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⡿⠿⠋⠁⠀⠀⠉⠙⠛⠛⠛⠋⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠋⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠘⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢞⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣿⣻⣿⣗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⡄⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⢽⣿⣿⣿⣽⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢷⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡪⠽⣿⣿⣿⣯⡔⣀⡀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣸⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡐⠻⡿⣿⣽⠾⣋⣴⠆⡀⠀⠀⠀⢀⣤⣿⢡⡄⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⢻⣷⣾⣯⣵⡾⣋⣠⡾⢀⣼⡣⢁⣾⠃⠀⠀⠀⣈⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣦⡿⣿⣿⣷⣿⣿⣯⣶⣿⣿⣥⡿⣯⣴⣶⣾⠷⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣏⣹⣿⣿⣿⣿⣿⣿⠿⠋⠙⢻⣿⣿⡯⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠙⠋⠿⠿⣟⣋⠁⠀⠀⠀⠈⢱⣬⣁⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⣶⣶⡾⢿⡏⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠾⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⣿⣿⡷⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣻⣿⣿⣿⣿⣃⠀⠀⠀⠀⠀⢬⣿⣿⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣿⣭⣽⡯⠅⠀⠀⠀⠀⠀⠈⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠟⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠟⠻⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠀⠀⠀⣀⣴⣾⣶⣦⣀⠀⠀⠀⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⡀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡞⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⢱⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⢿⣿⣿⣿⣿⣿⣿⡿⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣧

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⣲⣿⣿⠇⠈⠈⠙⠀⠀⠂⠀⠀⠐⠁
⠀⠀⢀⠄⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⣿⡟⠀⢀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠾⡿⠁⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⢀⠄⠒⠀⠂⡀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⣿
⡋⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠠⠒⠉⠉⠐⠄⠀⠀⠁⠀⠀⠀⠀⠒⠀⠀⠀⠀⡀⠀⣀⣀⣄⣄⢀⣤⣾⣿⡋
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠒⢦⠀⠊⠙⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⢿⣿⣿⣿⣯⣤
⠦⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⡠⠄⠀⠀⣻⡿⠜⢻⠻
⠀⠀⠀⠀⠈⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠠⣼⣦⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠱⡦⠀⡀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠑⠀⠀⠀⠐⠈⠻⠆⠀⠀⠀⠀⠤⢤⣾⣿⣷⠀⣿⣀⣠⣀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠤⠾⠿⣿⣾⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡄⠀⢀⠃⠀⣀
⠀⡄⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⡅⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠉
⠀⠀⠼⢆⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⡂⡬⠞⡉⠀⠉⢧⡀⠀⠀⠀⠐⠂⠀⠄⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠎⠀⠀⠀⠀⠀⠀⠳⢀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⡀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠷⠶⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠢⠄⠤⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⣤⠀⠀⢀⣴
⠀⠀⠀⡃⠒⠂⠴⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢆⣠⠀⠀⠀⢰⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢻⣤⣿⣶⣿⠿⠁
⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠘⣿⣧⡀⢠⠋⠀⠀⣠⣶⡥⠀⠀⠀⠀⠀⠀⣸⡿⣿⡁
⡀⠀⠤⠀⠄⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠒⠤⡀⠀⠸⣿⣿⣤⡆⢀⣾⣿⠋⠠⠀⠀⠀⠀⠐⢀⠂⠀⠀⠙⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⠀⠀⠀⠀⣀⣈⣠⣤⣤⣙⠛⠠⡘⣿⡿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠠⠒⠒⠉⠭⠛⠛⢛⡿⡿⢉⣤⠚⣽⣯⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⠿⠛⠛⠻⠿⣷⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⡛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⢀⡶⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠸⣧⣀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣀⣀⣀⣀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⢸⠃⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⣿⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡟⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⢀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣤⣀⣀⣀⣠⣴⣿⣴⣶⡾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠛⠋⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣕⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠿⠶⣦⡀
⠀⠀⠀⠀⠀⠀⣠⡶⢶⣤⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⢿⡄⠀⠹⣧⡀⠀⠀⠀⠀⢻⣆⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠈⢷⣄⠀⣀⣀⡀⠉⠛⠛⠛⢋⣠⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠙⠛⠛⠛⠛⠛⠻⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠈⠿⠾⠶⠶⠾⠂⠾⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣶⡶⠶⣦⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠹⣇⠀⠀⣠⣤⣤⣤⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⢷⣾⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠙⣿⡍⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠫⣾⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⠋⠁⠀⠀⠀⣀⣠⣄⠀⠀⠀⠸⣇⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⣴⠶⠛⠋⠁⠹⣦⠀⠀⠀⠹⠷⠾⠛⠛⠋⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⢸⣇⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⣷⠶⠶⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡴⠶⠿⠃




⡶⣶⢶⢶⡶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⣶⡶⣶⣶⢶⠶⣶⡶⣶⣶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⡷⣿⣿⣿⣷⣶⣶⣦⣿⠏⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⣞⡛⢿⡇⢻⡿⣿⣿⠿⡎⠉⢁⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣠⣦⣤⣬⣽⣿⣷⡠⡁⠐⠶⠿⠃⠀⣰⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⣿⣿⣗⡒⠻⢿⣿⣿⣾⣦⣄⣀⣀⣼⣿⣿⣿⣿⣿⣟⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣧⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠉⠉⢠⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡆⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣯⢳⣦⣾⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⣿⣿⡿⡏⠙⢻⣿⣿⣶⣀⡀⡀⣉⠙⠛⠛⣻⣿⠘⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⡇⠁⢀⡀⣿⠿⣿⣿⣿⣿⣿⣧⣴⣿⣿⢿⣸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡟⣹⡿⣿⣟⣿⣿⣿⢿⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⢩⣿⣿⣷⣿⣿⣿⡻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⢸⠛⠛⢛⣿⡟⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣇⠀⠰⣤⣴⡦⡰⠛⠁⠀⠀⠈⠐⠄⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⢀⠀⠀⡀⠀⠀⠀⠀⠎⠀⣿⣿⡷⢲⣶⣿⣷⠁⣤⡀⡀⠀⠀⠀⠀⠈⠹⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢷⣿⣿⣟⣦⣍⣩⣾⣿⣶⣾⣿⣷⣆⡀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢾⣿⡀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡈⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣯⠞⠙⠈⣽⢃⠁⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠸⡟⠿⠿⣿⠟⠟⠙⠃⠀⠀⠀⠀⡁⠈⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢰⣶⣧⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣾⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠼⣿⡃⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣷⠄⠀⠀⠐⠂⠠⠀⠀⠄⠀⠀⠀⡀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣤⢴⠾⠋⠶⣴⣦⣀⣠⠀⠀⠶⣦⣔⣋⣽⣆⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣤⣄⡀⠀⠀⠀⡼⣏⢀⣀⠀⢰⣮⣻⣿⣿⣿⣆⠸⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⣦⡀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⠿⠿⠿⠿⠶⠤⠀⠀⠠⣿⣦⣼⣿⣿⣿⣿⣿⣿⣆⠰⣦⠀⠀⠀⠀⠀⠀⢃⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣦⣄⣀⣀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣌⡃⠀⠀⠀⠀⣴⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠂⠘⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⣦⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠸⣆⠀⠐⢾⣿⣦⡀⣄⠀⣀⠀⢀⡄⠀⢀⢀⣠⣶⡿⠋⠀⢀⡴⠋⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠻⣟⠃⠀⠀⠈⠻⣿⠟⠖⠋⠑⠋⠗⠗⠻⢋⣴⠟⠁⠀⠀⠠⣷⠄⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣧⠀⠘⣯⣁⠀⠀⠀⢀⣿⣤⣤⡀⠀⠀⠀⠀⢠⡭⠷⠶⢦⣤⠄⠰⢺⠏⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⠂⠀⠙⠳⠄⣠⣴⠟⠋⠉⠀⠙⠆⠀⠀⠀⠀⠀⠰⠶⠿⣥⣄⡐⠁⠠⠴⠛⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠈⠧⠀⠀⣀⡤⢞⣫⣤⠶⠂⢠⡈⠲⣤⡀⢀⣠⡾⠃⣰⠀⠀⠀⠈⠙⠛⣶⠖⠀⠈⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣽⡓⠀⢉⣥⣾⡿⠋⣥⣤⣤⣌⣷⡄⠈⢻⡿⠋⠀⣴⣧⣤⣤⣤⡄⠀⠈⠻⣦⡀⠘⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠉⢁⣴⣿⠟⠋⠀⠀⠈⠛⢶⡄⠙⢿⡆⠀⠀⠀⣾⠏⢁⣴⠟⠋⠀⠀⠀⠀⠈⢻⣆⠈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢣⣿⣿⡿⠟⣛⣋⣀⠘⠛⠻⢿⣦⠈⣿⠀⠀⢸⡇⢠⡿⠟⠛⠛⢂⣀⣀⡀⠀⢀⣹⣧⡈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⡾⠛⠁⡀⠀⢉⡼⠋⠰⣞⢟⣦⠈⠠⠃⢰⣆⠀⠃⠉⢀⡶⢖⡆⠈⠻⣄⠁⠚⠉⠉⠉⠓⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣌⠁⣠⠞⠁⣒⡉⠀⢰⠀⠈⠉⠛⠂⠀⢀⣿⣿⡆⠀⠀⠞⠛⠋⠀⢀⡄⠈⠙⠂⠀⠘⠶⣤⣤⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣴⠟⣫⣶⠟⠉⠤⢤⣼⡆⠀⢰⠀⢦⡀⠈⣿⣿⡇⢀⡞⠀⢠⠆⠀⣸⣧⠤⠄⣀⣤⣤⣀⡀⠉⡙⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⣫⣴⢟⡿⠁⠀⠀⠀⠀⠈⠉⣀⠘⣧⠈⣿⡄⠘⠿⠁⣼⡇⢠⡟⢀⡈⠁⠀⡴⠛⠉⠉⠉⠻⣿⣦⣸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⡟⠁⡾⣠⠇⡶⠀⠀⠀⣠⠞⢹⣆⡼⡇⢸⣷⣠⣤⣄⣿⡄⡿⣄⣾⠙⠢⡀⠀⠀⠀⠀⠀⠀⠈⠻⣟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⡷⢋⣼⠃⠀⠀⠘⠁⠀⠈⢿⠁⠀⣸⠿⠛⠛⠛⠻⠇⠀⠸⡇⠀⠀⠈⠀⠀⡀⠀⠀⠐⣆⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⡾⠃⣴⠇⠀⠀⠀⠀⡀⠀⠀⠀⢡⣶⣿⣿⣿⣷⡄⠀⠀⢀⡄⠀⠀⠀⠀⣿⡄⠀⠀⠘⣧⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠞⡉⠀⠀⣿⣇⠀⠀⠀⠀⠹⡆⠀⠀⢹⣯⣤⣴⣤⣹⡿⠀⢠⡿⠁⠀⠀⠀⣠⣿⡇⠀⠀⠀⣿⣷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢰⡇⢀⠀⠻⣿⣧⡀⠀⠀⠀⠻⡀⠀⠀⠙⠿⠿⠿⠛⠁⢀⠜⠁⠀⠀⢀⣴⣿⠟⠁⣿⡀⠀⣿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣸⡄⠀⠈⢿⡻⢦⡀⠀⠀⠉⠢⣔⠶⠶⠶⠶⣢⠔⠁⠀⠀⢀⡾⢋⡼⠃⠀⢀⣿⡇⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣄⠀⠀⠻⠀⠹⡆⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢠⠏⠀⠘⠀⠀⣠⡞⣿⡇⡼⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠳⠦⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⢠⡿⠁⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠤⢀⢀⣀⡀⢀⠀⡀⢀⠀⣀⢀⠀⠤⠄⠀⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠘⣌⢱⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠈⡬⣱⠖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⡅⠄⢉⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠷⡅⠄⣩⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢰⡗⠞⠛⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣸⡏⠏⠙⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⠀⠀⠀⣠⣰⢂⠀⠀⠀⠀⠀⠀⣤⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣐⡳⠟⠀⠀⠙⠖⠧⠉⠈⠀⠀⠠⢁⢀⣱⢆⡨⠅⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⡛⣡⠿⡐⠌⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠠⣤⠄⠀⠀⠀⠀⠀⠧⠈⠝⠋⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠶⠐⠁⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠦⠀⠀⠸⡀⠈⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠘⠃⠀⠲⠂⠀⠀⠤⠆⠰⠂⠀⠀⠀⠀⠀⠀⣚⣃⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⡵⠆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⡑⠦⣄⡀
⠀⠀⠀⠀⠀⠲⠖⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡤⢿⡷⠀⢠⠚
⠀⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠃⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⢀⣿⠇
⠀⠀⠀⠀⠀⠀⣀⠠⢊⣤⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⠈⠀⢰⣦⣭⠼
⠀⠀⠀⠀⢩⣉⠻⣷⣾⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⢛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠛⣿⣷⢠⠄⠀⠀⠀⠀⠈⠝⢷⣤⡀
⠀⠀⠀⠀⠙⢿⣷⣾⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣽⢒⡐⠂⠼⠷⠖⣻⣿⣿⣦⠀⡐⣺⣶⡄⠀⠀⠸⠗
⠀⠀⡀⠀⠀⢢⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠲⢴⣿⣿⣿⡜⡗⠀⠀⠀⠀⠀⠀⠀⠠⣾⠏⢉⣽⣾⡿⠶⣿⣿⣿⣿⡿⠀⠉⡿⠋⣱⣿⣿⣷⡆
⠀⠀⣶⣤⣄⣝⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⣾⣿⣯⣿⠟⠃⠀⠀⠀⠀⢠⣿⣇⢀⣰⣯⣘⣾⣿⠏⠀⠰⣿⣿⡿⠟⠁⠀⠀⠀⠀⣸⠉⢉⣿⡇
⠀⢠⣳⣛⢟⣿⢻⣿⣿⡟⠃⠀⠀⡆⠀⠘⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣿⣻⣿⣿⣿⣿⣿⣿⣿⡀⠀⠘⣿⣟⠀⠀⣠⣶⣶⡟⠛⡏⠀⠸⠟
⠀⣈⣿⣭⣿⣿⣘⣿⣿⣧⡉⠷⠾⣯⣤⠲⣿⣿⣿⣿⡟⠀⠀⠀⢰⣾⣿⣼⣿⣿⡋⠛⢳⣶⣿⡿⠛⢿⣿⣿⣤⡞⢻⣿⠹⣿⣶⡃
⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣎⠀⢀⡄⠀⢻⣿⣿⠁⠀⠀⣆⣨⣿⠀⣿⣿⡿⠁⠀⢸⣿⣿⠗⣸⣿⣿⣿⣿⣿⣾⣇⣾⡏⢻⣷⡄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣶⣿⣿⣷⣄⠙⢿⡀⢸⠈⣿⣹⣽⣷⣿⡿⣁⣺⠿⣇⣿⠟⠉⣻⣾⣿⣿⣿⣿⣿⣿⣏⠃⠄⠙⠻⣄
⠀⠠⠊⠉⠉⠛⠛⢟⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣤⠸⣷⡄⠈⡼⢿⣿⣿⡿⣲⣿⢿⣿⣿⡟⠘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠉⠳⣄⣾⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣨⡾⠋⠁⠀⠀⠀⠀⠀⠘⠛⢻⣦⡉⠁⡤⡴⠛⠛⠛⣀⣼⠃⠘⣿⠋⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀⣀⣸⣿⣿⡗⠸
⣴⣾⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⣀⡈⣳⣟⣲⣿⣿⣷⣾⡟⠁⠀⠀⠘⠉⣡⠟⢃⣠⠴⠞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⢿⣿⣿⣿⠿
⡾⠿⠛⠻⠻⠻⢿⣿⣿⣿⣿⣷⣤⣉⠿⣿⣿⣿⣟⣿⣿⣿⣷⣶⣶⣶⣤⣾⡇⡀⣿⣶⠾⣶⣴⣿⣿⣷⣽⢻⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⡆
⣆⣴⣶⣦⣤⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣄⢾⣽⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡟⢷⠤
⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠠⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢈⣀⠀⠀⠀⣶⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⣛⣠⣿⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆⠀⠀⡀⠀⠀⣀⢀⣴⣶⣿⡿⠄⠀⠀⠀⠀⠈⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠈⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠈⠿⠋⠁⠀⠀⠀⠲⣆⠁⠀⠀⠀⠀⠀⢀⣀⣸⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠀⠀⠀⠀⠀⢰⠄⠄⠀⠀⠀⡄⠀⠀⠀⠀⢀⡀⠉⢹⡏⠉⠁⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠛⢿⣷⣦⡀⠀⠀⢠⣿⠃⠀⠀⡀⠀⠀⢀⡠⠎⠀⠀⠑⠄⠀⠁⠀⠀⠀⠀⠘⠁⠀⢸⡇⠠⠰⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⠤⠄⠼⢤⠦⠤⠤⣤⠄⢠⡀⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⣿⠀⠀⣀⠠⠄⠉⢀⣠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠷⢀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣵⣿⡶⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣷⡀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⡶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡂⠀⠀⠀⠘⣷⡄⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢿⠿⠿⠿⠿⣿⣿⣿⣿⣦⡈⢻⣦⠀⠀⠀⠀⠀⠀⣿⣿⠟⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣿⣦⣀⠀⠀⠀⠀⢻⣿⠀⠀⠀⢠⣾⠁⠀⠀⣀⣤⣶⣦⡴⠄⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣷⣶⣦⣤⣼⣿⡀⠀⠀⢸⡟⠀⠀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋⠉⠀⠀⠀⠈⠉⠙⠛⠻⣶⣦⣼⣷⡀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣬⣿⣿⣷⣾⣿⣆⠀⠀⠀⠀⠀⠀⡀⠀⠈⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠛⠁⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣦⣶⣶⣶⣶⣦⣤⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣿⣿⣦⣄⣠⡛⠳⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠓⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡆⢸⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠰⠊⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡅⠀⠀⠀⢠⣿⡇⠸⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⠀⠀⢀⣠⣤⣄⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢸⠇⠲⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⣀⣼⡁⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡹⠃⠀⢠⣿⣿⣿⣯⡙⠀⠙⢿⣿⣿⣿⣿⠃⠠⠀⠀⠀⠀⢀⡦⠰⣿⠆⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡞⠁⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣟⣥⡴⠋⠁⠀⢀⣿⣿⣿⣿⣿⡇⠀⠐⠀⠉⠻⣿⠛⠓⠀⠀⠂⠀⣤⠎⠁⠂⣽⣶⡆
⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡏⠀⠂⠀⠀⠀⠀⠈⠉⠉⠒⠉⠁⠀⠀⠀⠀⠈⢹⠇
⠀⠀⠀⠀⠈⢻⣦⡀⠠⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠀⠀⠀⠌⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠹⣷⣶⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⣿⠀⠀⠀⢰⠗⢀⠀⢠⡄⠀⠀⠀⠀⠄⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⢹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠉⠈⣧⣔⣀⡀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣶⠈⣻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⡄⢀⣴⣾⣿⣏⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⡃⠈⠿⠿⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⡽⠛⢛⣿⡷⣿⣿⡟
⠀⠠⠤⣴⣤⣄⡀⠀⠀⠀⠀⣾⡟⠀⠇⠀⢰⡄⠀⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⣎⣹⣷⣿⠋
⠀⠀⠀⠀⠉⠻⣿⣦⠀⠀⣸⣿⠁⠐⠀⠀⠸⡇⠀⠁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⣼⠆⢀⣻⡅⣼⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣇⣠⣿⡇⠀⠀⠀⠀⠀⠻⣄⢀⣄⠙⠀⠋⠉⠁⠉⠈⠉⠑⠆⠉⠙⠂⠉⠉⠈⠉⠁⢠⣤⡄⠘⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠀⣀⣴⡆⠀⠀⠀⠀⠛⠈⠿⠓⣶⢴⡆⢠⡀⠀⠀⣀⠀⠀⢘⡀⠠⡧⠴⠆⠁⡸⠏⠃⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡾⠟⠛⠃⠀⠀⠀⠀⠀⠀⠙⠀⡤⢠⠀⠈⠁⡀⠉⠉⠉⠈⢉⠁⠀⠀⢠⡄⠞⠃⠀⠀⠀⠁⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠃⠀⠂⠀⠃⠘⠀⠘⠃⠀⠀⠀⠀⠀⠠⠂⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠹⡆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⠠⠈⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠈⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢀⡀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣦⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


⠀⠀⠀⠀⠀⠀⠰⠂⣶⡄⢴⠆⢠⡆⢰⡄⢰⡆⠲⠀⢰⡆⢲⣦⢰⡆⢰⣦⠰⠶⠀⠐⠆⢠⡦⢴⡆⠲⠀⠰⡆⢰⡦⠰⡆⢴⠀⡆⠀⢶⡶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠸⠇⠸⠀⠈⠀⠟⠀⠽⠂⠙⠀⠇⠈⠟⠀⠀⠸⠂⠀⠟⠀⠿⠀⠿⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡀⢀⣄⠀⣠⣀⣀⡀⢀⡀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠻⣏⠙⠛⣿⠋⠈⣿⠛⠛⣿⠙⡯⢉⣿⡿⢻⠏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣾⣷⣶⣷⣶⣶⣿⣶⣷⣿⣷⣶⣾⣿⣶⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠻⢶⣄⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠈⢀⡀⢸⡟⠀⠀⠀⠤
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢘⣃⣠⣀⠀⠀⠘⠂⠑⠋
⠀⠀⠀⠀⢻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠁⢻⡆⣀⡀⠀⠀⠐⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⠄⣀⡄⠈⠙⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⣠⣤⡀⠀⠀⠘⠁⠀⠀⡄⠁⠀⠀⠁⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠉⠋⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠐⠁
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢰⣿⣿⠃⠀⠐⠀⠀⠄⠄⠀⠁⠀⢀⠀⠠⠀⠀⠀⢀⣠⣤⡀⢀⣤⡆
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸⣿⣿⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⣾⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠚⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢻⣿⢟⡿⢛⣛⣳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡆⠟⣼⣇⣛⣛⢻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣴⣿⣯⣭⣭⣼⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡬⢛⣭⢭⣝⢿⣿⣿⣿⠿⣋⣭⣽⡭⢁⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢛⣩⣶⣿⣿⣦⢻⣧⢻⣿⠏⣾⠿⣛⣭⣶⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⣛⣭⣦⣙⢿⣼⡿⣸⣿⡸⣿⣿⠿⣛⣥⣮⡙⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣴⣿⣿⣿⣿⣿⣷⣬⡁⠿⠿⠇⣩⣶⣿⣿⣿⣿⣿⣷⣬⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣦⣽⣿⣿⣧⣶⣿⠟⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣛⣻⣿⣿⠇⠀⠀⠀⠀⠈⢿⣿⣛⣛⣛⣻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣄⢾⣿⣶⣾⣿⢇⡄⣀⣀⠀⣀⡀⣄⢿⣿⣶⣾⣿⢆⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣌⡳⢭⣍⣭⠵⢚⣴⣿⣿⣷⣌⢳⣜⠲⢭⣭⣭⠵⣋⣼⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠿⠿⠿⠷⠶⠶⠾⠿⠿⠏⣿⣿⣿⡆⠿⠿⠶⠶⠶⠿⠿⠿⠏⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⣾⣾⣶⡾⣛⡛⣶⣾⣶⡆⣿⣿⣿⡧⢰⣾⡞⣛⡻⣶⣿⣾⡆⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⣿⣿⣿⣧⣛⣃⣿⣿⣿⡇⣿⣿⣿⡇⢸⣿⣆⣛⣣⣿⣿⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣬⣭⣭⣭⣭⣭⣭⣭⣭⡅⣿⣿⣿⡟⢨⣭⣭⣭⣭⣭⣭⣭⡄⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠁




⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠞⢛⠛⠛⠛⠛⠛⠷⠶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠁⠀⠀⠄⠀⠁⠀⠂⠀⠘⠀⡛⢄⠀⠀⢉⣛⠷⣤⣀
⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠠⠄⠐⢦⠀⠀⡈⠈⠁⠲⡆⠀⠀⢀⣿⣎⠢⠀⠀⢻⣷⡌⠙⢷⣄
⠀⠀⠀⠀⣠⡾⢋⡄⣶⠸⠀⠐⡤⠈⠀⠀⢲⣦⡄⠀⠀⠄⠀⠚⠛⠛⠇⠱⣌⡠⠄⠈⠀⠰⠝⢷⣄
⠀⠀⠀⣴⠏⣶⡬⠲⢋⠳⢡⣾⡇⠆⢔⣂⠁⣨⡶⠶⠾⠶⢶⣄⠈⠀⠀⠂⠠⠟⢠⠌⠐⠾⣿⣧⠹⣧⡀
⠀⢀⣾⠃⢈⠛⠀⠀⠀⠂⠀⠙⡉⠛⢻⠃⣼⠋⠀⠀⠀⠀⠀⠙⣷⡀⢂⣤⣾⠃⠀⠀⠀⠀⠈⠃⠀⠘⣷⡀
⠀⣾⠃⠰⠀⠄⢀⡆⣶⡠⡄⠠⠀⠰⠈⢸⡏⠀⠀⠀⠀⠀⠀⠀⠘⣧⠙⠛⠿⠀⠀⢀⡶⠄⠰⠀⠐⠠⠘⣷
⢸⡏⢀⠀⠀⠴⣀⣸⠁⣷⣀⡤⠄⡀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠐⡡⡀⠀⠀⠀⠀⠐⠠⣴⣶⠹⣇
⣾⠁⠢⠀⠐⣻⢥⣬⠄⣤⣬⢿⡓⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⠠⢀⢀⠀⠀⢐⠻⠀⠐⢀⢃⠂⢿
⣿⢰⣄⠀⠀⡉⠀⡜⣦⢏⡄⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⣿⠻⣐⡂⢀⢰⣌⢠⠉⣾⡆⢸
⣿⠈⠇⠈⠠⠀⠀⠁⠉⢠⠀⠀⡄⠈⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠅⢀⢰⣍⣠⠮⣿⡧⠋⠁⠙⠈
⢺⡆⠀⠻⣾⣷⠶⠈⢁⠀⠤⣴⡖⣖⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⡖⠠⠀⠀⠡⠿⠡⠀⠀⠀⠀⠰
⠘⣷⠀⠀⠛⢁⠀⠀⠀⠀⠸⣦⣅⠃⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⡈⠡⢝⠻⣿⡇⠀⢁⠠⣤⣲⠀⢰⡀
⠀⠹⣇⠈⠀⠄⠠⣤⠀⠀⢠⡿⠛⠛⠈⠀⢻⣄⠀⠀⠀⠀⠀⣰⠟⣰⡇⠅⢐⣂⣭⠂⠐⢀⠀⠷⠳⣌⠳⢉
⠀⠀⠹⣦⠀⠀⠀⠀⠐⣀⡄⠀⠀⠀⠀⠁⢠⡙⠷⠶⠶⠶⠟⠁⠀⢻⡇⡀⠬⠉⣙⠄⠐⠈⠀⡤⡰⢋⣴⣌⢰
⠀⠀⠀⠘⢷⣄⠣⠀⢡⠀⠀⠀⢶⠀⠀⠀⣼⣷⡜⢷⣤⡄⠈⠀⠀⠈⠀⠁⠐⣨⣴⣿⡇⠀⡄⠐⠋⠭⠀⠩⣸
⠀⠀⠀⠀⠀⠙⢷⣄⠚⠿⣷⠄⠂⠐⠄⠈⠉⠉⠛⠊⠯⢄⡁⢀⠀⠀⠐⠃⠂⠀⠒⡐⣶⡐⠀⠁⠀⠀⠰⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣍⡁⠂⠄⠀⢀⣀⠀⠀⠐⠀⠄⠀⠉⠃⢀⡂⠀⣀⡂⡈⠸⣋⠹⢖⣿⡷⣌⡀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠶⣬⣥⣤⣤⣤⣥⣬⣴⠶⠾⣿⠇⠈⢦⣭⣤⣍⡀⠴⠈⢋⠈⣀⢱⡟
⠀⠀⠀⠀⢀⣤⡶⠶⣛⠛⠳⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⢻⡟⢉⠀⠀⠀⢨⣄⢠⡿
⠀⠀⠀⣴⠟⠡⢀⠈⠹⣍⡴⠤⠈⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠶⠀⢐⠀⠀⠀⢀⠀⠀⠀⢡⡿⠁
⠀⠀⢸⡏⡘⠀⠐⠈⢠⣛⠳⠐⢂⠀⠘⢷⡀⠀⠀⠀⠀⠀⢠⡾⠃⢀⣀⠀⠀⠀⠁⠈⠙⠆⠀⣰⠟
⠀⠀⢸⡇⣀⠀⡀⣴⡷⢣⣾⡀⡂⠀⢠⡌⢷⣤⣀⣀⣠⡶⠋⢀⡐⢛⠛⠛⢃⡀⢤⣄⠀⢀⣾⠋
⠀⠀⠸⣧⠠⠀⢀⡟⣰⠿⠿⠷⠈⢀⠈⠀⢀⣬⣭⢩⠁⠀⢈⢀⠈⠂⣤⢀⠁⠄⡁⢊⣴⠟⠁
⠀⠀⠀⠹⣧⡚⠉⢠⠀⠀⠀⠂⠠⠈⠀⠀⢿⠫⠐⠠⠄⣠⢸⣧⣹⡼⠿⣤⠎⣨⡴⠟⠁
⠀⠀⠀⠀⠈⠻⣦⣄⡁⠀⠀⡶⠁⠀⠀⡀⠚⠀⠣⠹⡷⠼⠚⠛⠃⢂⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣤⣄⣀⣘⣀⣀⣀⣈⣐⣁⣠⣤⡶⠾⠛⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⣷⣫⠟⠉⠛⢦⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠛⢁⣤⣤⡀⠀⠀⠀⢀⡀⣠⠴⠛⠁⡌⠉⢁⡀⠀⠀⠈⠑⢲⠜⠁⣤⠔
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠟⠉⠀⠙⢶⡄⠄⡉⣟⠡⠀⠊⢀⣄⠀⣸⣷⠀⠀⢀⣴⠇⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠀⠀⠀⠀⠀⠉⢶⣄⡤⠀⣠⠄⢈⣿⣀⣿⣿⡖⠊⠀⠁⠀⠠⢇⣼⣿
⠀⠀⠀⢠⣦⣀⡀⢈⡴⠁⠀⠀⠀⠀⠀⢈⣿⠋⠀⠀⠀⠀⢼⣿⣿⣿⣿⣷⠄⠀⢀⢡⡶⢋⢉
⠀⠀⠀⠀⠈⠙⠻⣮⣼⠷⢤⣀⠀⠀⣠⣾⠗⠀⣀⠈⠀⢨⣿⣿⣿⣿⣿⣿⣶⣿⢠⣶⣤⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠲⣬⡁⠀⠀⠉⠷⢶⣿⣃⡀⣀⣿⣧⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⢀⣴⠄⠈⠙⢶⡀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⣉⣀⡠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠛⠋⠲⠦⣀⡀⠉⠲⢦⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠈⠛⠉⠀⠀⠀⣠⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⠷⣶⣤⣽⣿⣿⣿⣿⡿⠿⠟⠿⠛⠋⠀⢀⡄⢀⡀⠀⠀⣠⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⢁⣿⣿⠿⠟⠛⠀⠀⠀⠀⠀⠀⠴⠿⠟⠉⠀⠀⠈⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠠⡞⠀⠀⢰⣞⡟⠀⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⢀⣠⢞⣵⣤⣾⠃⠀⠀⠐⠹⣷⣜⣷⠀⠀⠈⣶⠂
⠀⠀⠀⠀⠀⠀⢀⠾⣧⡀⠀⠠⠀⠉⠛⠿⣿⣧⡀⠀⠀⠀⠀⠈⣙⣿⠀⠀⠀⠸⣆
⠀⠀⠀⣠⢴⡏⠁⠀⠉⠓⢠⣄⢀⠀⠀⠀⠈⠓⢷⡀⠀⠀⠀⠀⠉⣿⡶⡀⠀⠀⠈⠂⠀⠀⠀⠠⣤⠜⠀⣠⡎
⠀⠀⠈⠛⠻⢧⡀⠀⠀⠀⠀⠈⠙⢦⣤⣀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠃⠧⣿⢦⣀⠀⠀⠀⠘⢧⣤⣄⠀⠴⠋⠁
⠀⠀⠀⠀⠀⠈⠙⣆⠄⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⡼⠃⠀⠀⠘⠁⢣⡟⡠⡀⠀⢀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠚⠀⠀⠀⠀⡆⠀⠋⣠⣼⠇⠀⠀⠁⠀⠐
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⢀⣤⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⣤⠀⠀⠀⣰⡀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠠⢺⠏⢀⣬⡾⠃⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⢠⣿⡇⠀⠀⣿⡇⠀⠀⡀⠈⠙⠛⠒
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠾⢿⣧⢀⢠⣿⣇⠀⢸⡗⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠠⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠾⣿⡧⠀⣽⣿⣸⣿⡆
⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠻⠏⢼⣿⡷

⠀⠀⠑⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠐⠊⠱⡀
⠀⡆⠀⠀⠀⠀⠀⠉⠁⠒⠂⠠⠴⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠐⠒⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠈⢢⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⡠⠐⠺⠁
⠀⠀⠀⢡⠀⢹⠒⠠⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠄⠂⠂⠩⠀⠀⡇
⠀⠀⠀⡿⡰⡀⡇⠀⠀⠀⠀⠀⠉⠐⠀⡄⠀⠀⢢⡀⠀⢠⡀⠀⠀⠀⠀⢀⠊⠱⠀⠀⠁⠀⠀⣦⣆⡇
⠀⠀⠠⠟⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢣⡀⠀⣀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠘⠎⠙
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠒⠄⠀⢸⠀⣠⠐⠆⠲⠈⠦⡀⠰⠋⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠊⡠⠒⠈⠉⠐⠠⡙⢠⠀⠀⠐⠂⠤⣀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⢀⠇⠌⠀⠀⠀⠀⠀⠀⠈⠾⠁⠀⠀⠀⠀⠈⠆⠀⠁
⠀⠀⠀⠀⠀⢠⠀⠀⠑⠀⠀⠀⠀⡠⢘⠀⢀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠇⠀⠁⡈⠛⠉⠉⠉⠋⡠⠅⠀⠀⠀⠀⡠⠊⠀⠀⣀⡴⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡎⠀⢠⠆⣤⣒⣫⠐⢒⣄⡀⡁⠀⠑⠊⠉⡀
⠂⠈⠁⠈⠑⢤⡀⠃⠀⠀⠘⠗⢀⠀⠀⠀⡀⠈⠁⠀⠀⠈⠑⠢⢄⠧⠘⠎⠒⠋⠀⠙⠇⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢠⠀⡆⠰⠀⠒⠳⠓⣺⠀⠂⣀⠄⠀⠀⠀⠈⠂⠠⠏⣀⣀⠀⡖⠴⢀⢀⢠⠂⠈⣄⣄
⠀⠀⠀⠀⠀⠀⠘⢰⠀⢄⠀⠀⠀⠀⠠⠚⠁⠀⠐⠀⠀⠐⠂⠀⠂⠀⠀⠉⢛⠩⣟⠛⢍⣾⠛⠲⡔⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠻⣆⠘⠀⠀⠀⠀⠠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢁⠀⠐⣇⡔⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⢀⡀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠉⢄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢴⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢘⡀⠀⠀⠁⠂⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠀⡈⠢⠈⠑⠢⠄⣀⠀⠀⠀⠀⣀⠠⠀⠀⣀⠔⠀⠈⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠱⠖⠂⠤⠭⠭⠤⠄⠐⠐⠋⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄
⠀⠀⠀⠀⠀⣀⠀⠤⠐⠒⠐⠈⠉⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⢠
⠤⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⣴⠀⠀⠰⠁⠀⠀⠐⢀⠀⠀⠀⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠉⢦⡀⠀⠀⢡
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠐⠊⠁⠀⠀⠉⠃⠐⠃⠀⠀⠀⠀⠀⠀⠄

⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⣀⠀⠀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠙⢷⣦⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣠⡿⠋⠀⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠿⠷⠶⣶⣶⣶⣾⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣧⣤⣤⣬⣤⣤⣤⣤⣄⣈⣁⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⡄⠀⠀⢸⣿⡇⠁⠉⢉⣩⣭⣭⣉⠉⠉⢹⣿⡏⠉⠉⣩⣯⣭⣭⣍⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣾⠟⠁⠀⠀⣾⣿⡇⠀⣴⡿⠋⠉⠉⠻⣷⡀⣸⣿⠇⢀⣾⠟⠁⠀⠈⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢿⣦⡀⠀⠀⣿⣿⠅⢸⣿⠀⠀⠀⠀⠀⣿⡇⣿⣿⠀⢸⣯⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠉⠻⠆⠀⣿⣿⠁⠀⠻⣷⣤⣀⣠⣴⡿⠁⣿⣿⠀⠀⠻⢷⣦⣤⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣿⣿⣦⣤⣤⣬⣿⣿⣿⣯⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣤⣌⠉⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠈⢻⣧⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⣀⣀⣀⣴⡿⠃⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⣰⣿⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⢀⣿⡟⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠸⠋⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡆⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡎⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⢐⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⢻⣽⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡄⠈⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣷⣿⣧⣹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣆⠻⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣈⣛⣷⣾⣻⣿⣿⣟⣶⣤⣀⠀⡀⢀⣀⠁⡀⠀⠈⣻⣬⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣉⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⡃⢂⠀⠈⠀⠉⠀⠀⢀⠀⠈⠬⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⠼⠛⠉⠀⠀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣿⣷⣦⣲⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠠
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣠⡴⠖⠚⣛⣩⣭⣶⣶⣿⣿⣿⡿⠟⠛⠛⠛⠻⠟⠋⠀⠉⠃⢀⡦
⠀⠀⠀⢀⣲⣬⡵⢒⣩⣴⣶⠟⠉⢀⣤⣾⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣉⣀
⠀⢠⠀⢳⡟⣅⠀⠈⠉⠉⠀⠀⠚⠛⠛⢻⣿⠿⠋⠁⢀⡀⠀⠀⠀⠀⢀⣴⠦⡀⠀⠂⠀⠀⠊⢀⢻⣿⣿
⠀⠈⠓⠊⠃⠀⠀⠺⢷⣦⣶⣶⣲⠀⠀⠙⠁⣢⠀⢸⣛⡻⡦⠀⠀⠀⠟⠁⠀⡇⠀⣀⣴⣤⣤⡆⣾⠟⠁
⠀⣀⣀⡀⣬⡛⠏⢑⣶⣶⣶⣶⣦⣤⣄⣀⣒⣉⢠⣘⣫⣜⠁⠈⠀⠀⠀⣀⣾⣵⣿⣿⣿⣿⣿⠟⠁
⣾⣿⣿⣿⣿⣷⣦⣤⣀⣐⣲⣦⣤⣉⠉⠉⠉⠉⠀⠉⠀⠀⠀⠀⠄⠀⠀⣻⢹⣿⣿⣿⣿⠟⠁
⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣤⣤⣤⣶⣧⣄⡄⡶⠃⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢰⣾⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠛⡛⠂⠈⠙⣷⡀⠰⣦⡴⢄⢀⣤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣀⣀⣠⣶⣶⣿⣿⣿⣿⣏⠂⢄⠀⠈⠁⠀⠈⢿⣼⠋⠙⠙⢢⣀⣀⣼⣿⢷⣤⣀⠀⢀⡾⠿⢧⡄
⠀⠀⠈⠻⠻⠌⠻⠿⢉⣽⣿⣿⡄⠀⠀⠀⢀⣀⣤⣤⢀⡀⠁⠀⠀⢀⣁⣈⠉⠀⢀⣥⣾⣟⠣⠀⠀⠀⠐⠳⡤⠴⠤⢦⣀
⠀⠀⠀⠀⠠⡶⠾⣆⠈⠻⢿⠁⠀⠀⢀⣴⣾⣿⣿⣿⣇⠙⠳⠀⠀⠀⠉⠉⠁⠀⠁⠀⢀⣿⠀⠀⠂⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠿⣿⣿⣿⢿⡟⡩⠛⣿⣾⣷⣦⡀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⣼⣿⡁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠐⠂⢀⠀⠀⠀⠀⠀⠙⠯⠀⠰⠁⠀⣫⠿⠛⠙⠁⠀⠀⠀⠀⠲⣿⣿⡌⠀⠀⠀⢠⣿⣿⣇⣀⡄⠸⠁
⠀⠀⢸⣿⣷⣤⢤⣀⡀⠀⠀⠁⠀⢀⡀⠀⠀⠁⠤⠐⠁⢀⣶⣶⣼⣦⡀⠀⠀⠀⠀⠱⠿⢾⠂⠀⢸⣿⣿⣿⣋
⠀⠀⠀⠛⠉⠉⣻⣿⣿⣾⣷⣀⣄⢸⣿⡄⣠⡀⠀⠀⠉⠚⠿⣿⡝⠋⠙⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⡀
⠀⠀⢀⡀⣠⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣀⠀⢠⣶⣀⡄⠉⠙⠦⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣦⣦⣀⣤⡀
⠀⠀⠀⠉⠙⠛⠛⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣠⡄⠀⣠⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣟⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠉⠓⠉⠙⠋⠋⠉⠛⠛⠛⠿⠶⠿⠇⠀⠀⢠⣦⣼⡿⠟⠛⠛⠻⢿⡿⠟⠉

⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠤⠌⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢈
⠀⠀⠀⣠⠞⠃⢿⠀⠀⠀⠀⢀⠔⠉⠙⠑⠛⢦⠀⠀⠀⠀⠀⢸⡁⠁⠚⠊⢩⠏⠀⠀⠀⠀⠀⣠⠃⠈⠒⡇⠀⠀⠀⠀⠀⢸⠉⢈⣋⣏⣈
⠀⠀⡏⣀⡄⠀⢻⠀⠀⠀⠀⠈⠢⠞⠉⡇⠀⢸⠀⠀⠀⠀⠀⠀⣠⠜⠁⢠⡃⠀⠀⠀⠀⠀⡰⠃⡰⠃⠀⡇⠀⠀⠀⠀⠀⢸⠀⠸⢤⣀
⠀⠀⠉⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⡰⠁⢠⠏⠀⠀⠀⠀⠀⠀⠳⠤⢤⠀⠈⡆⠀⠀⠀⡰⠁⠘⠓⠃⠀⠃⡆⠀⠀⠀⠀⠘⠓⠲⡄⠀⢱
⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⢰⠋⠀⠐⠛⠒⢲⠀⠀⠀⠀⢠⠤⠤⠞⠀⠀⡎⠀⠀⠀⠓⠒⠒⠒⡆⠀⡖⠁⠀⠀⠀⠀⠠⠦⠴⠃⠀⢰⠁
⠀⠀⠀⠀⠧⠤⠞⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠾⠀⠀⠀⠀⠺⠤⠤⠤⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠇⠀⠀⠀⠀⠀⠓⠤⠤⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀

⠀⠀⠀⠀⠠⢀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠠⠀⡠⡄⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣠⠴⠶⠦⢓⠄⠀⠀⠀⢰⠒⠒⠢⠾⣒⡴⢦⠀⠀⠀⠀⣀⢤⠼⣤⣆⠀⠀⠀⠀⠀⢀⠔⠄⠿⡂⢦⡀⠀⠀⠀⠀⢀⡤⠲⠟⣆⢅⡀
⠀⠀⡜⠁⢠⣔⡒⠚⠀⠀⠀⠀⠸⠤⠤⣤⠄⠀⢈⡼⠀⠀⠀⢰⠁⢰⡀⡂⠈⡇⠀⠀⠀⢠⡇⠀⡖⠉⢦⢀⣷⠀⠀⠀⢠⠋⢠⠔⠒⢢⡀⢱⡀
⠀⢸⠀⠀⣀⢀⡈⠑⡄⠀⠀⠀⠀⠀⡰⠃⢀⡴⠉⠀⠀⠀⠀⢈⠦⠀⠉⠀⢶⡁⠀⠀⠀⠘⣇⠀⠑⠒⠃⠈⢸⠀⠀⠀⡏⠀⡎⠀⠀⠀⡇⠀⡇
⠀⢸⠀⠠⡁⠀⡱⠀⢹⠀⠀⠀⠀⡼⠁⢀⡞⠀⠀⠀⠀⠀⢠⠃⠀⡖⠁⢢⠀⠘⡀⠀⠀⠀⠈⠑⢒⡖⠀⢀⡎⠀⠀⠀⣇⠀⠱⣄⣀⡠⠃⠀⡇
⠀⠈⠣⡀⠈⠉⢀⡠⠃⠀⠀⠀⢰⡃⠀⡸⠀⠀⠀⠀⠀⠀⠘⢦⡀⠉⠒⠁⢀⡼⠀⠀⠀⠀⣏⠉⠁⢀⡠⠊⠀⠀⠀⠀⠘⢦⣀⠈⠉⠀⣀⠜⠁
⠀⠀⠀⠈⠉⠉⢉⠀⠄⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠁⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠄⠆⠀⠠⡀⣀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠐⠲⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⣀⣤⠀⠚⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⡙⠃
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠋⠻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⣤⣄⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⠔⠛⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡀⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡾⠀⣶⣄⣀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⠀⠈⠉⠉⣿⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⣧⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠲⣴⣿⣆⣀⣾⠃⠀⠀⠀⠀⠀⢸⡿⠀⠀⣠⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠰⠶⠛⠛⠛⠉⠙⠛⠻⠛⠛⠛⠓⠶⠀⣾⡇⠠⣾⠋⠀⠈⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣿⣧⣴⠇⠀⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⣸⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣠⣤⣤⣭⣿⣦⣤⣤⣤⣤⣤⡿⠀⠀⠀⠀⠀⢰⡿⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣠⡾⠛⠁⠀⠀⠈⠉⠉⠉⠉⠙⠋⠉⠉⢳⠀⠀⠀⢠⡿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⣠⡿⠀⠀⣠⡿⠁⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠿⣏⠀⢀⣴⠋⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣷⣾⣿⣿⣿⡏⠁⠀⠈⢠⣾⣇⡀⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣝⠻⣏⡻⣧⡀⠀⢰⠏⠀⠈⠁⢀⣴⣿⠏⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢿⣿⣷⣬⡙⠾⠛⠿⣏⠀⠀⢀⣶⣿⣯⡁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣿⠀⠻⣷⡈⠙⢦⣄⡀⠙⠷⠶⠿⠁⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⣿⠀⠀⢹⣷⠀⠀⠉⠙⠛⠳⠚⠒⠛⠃⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⡄⠀⠀⠀⠀⠀⣠⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢀⣠⣴⡿⠋⣀⣀⠀⠀⠐⠻⠞⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢤⣴⣿⣿⡿⠿⠞⠛⠁⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢨⣮⣝⣛⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠈⢻⡿⠃⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡼⣧⢱⡟⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⡄⠀⢿⣷⠁⠟⣼⣿⠀⢠⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⣿⠀⠀⠸⣩⠞⡳⢮⠻⠀⠀⣿⣿⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠆⠀⣿⣿⠀⠀⢀⡻⣌⣇⡿⡀⠀⠀⣿⣿⠀⠠⢾⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⣶⢰⣾⡇⠀⠀⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⣿⢸⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⣿⢸⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢿⣿⠀⠀⢰⣶⣶⢰⣶⡦⠀⠀⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⣐⣿⣿⡿⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣄
⠀⠀⣮⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣻⠟⠀⣰⣿⠿⠋⠀⠀⡂⠀⡄⠀⠀⠀⠤⠀⠀⠂⠀⢀⣶⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⠿
⡀⢨⡿⡀⠀⠀⠀⠀⢀⣴⡾⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⡾⠋⠀⠐⣻⣿⡂⠀⠀⠀⣶⠀⠀⠀⠀⠣⠤⡀⠀⠀⠀⣿⣿⠆⠀⠀⣀⣴⣦⠀⢤⡄
⡗⣸⡇⠒⠀⣆⠀⠀⢿⣯⡀⠀⠠⠄⠀⠰⠟⠀⠘⣿⣾⠃⢀⣤⣾⣿⠋⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢰⣷⠀⢰⠟⠁⢀⣠⠾⠟⡿⠻⠃⠀⠡⣄
⡛⣿⡗⡀⢼⡂⣀⡟⠈⠛⣷⣄⡀⠀⠀⠀⠀⠀⠀⢃⠥⣾⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠉⠀⠀⠀⠴⠋⢡⠀⠀⠀⣸⠀⠀⢠⡆⢀
⣳⣿⡇⠀⢸⣥⢸⡇⢢⡆⠸⠿⠧⢀⡀⠁⠀⠀⠀⠀⠀⠘⠋⠉⠛⠋⠈⠙⠿⠁⠨⡂⠀⠀⠴⠇⠀⢀⠀⡀⢠⡀⢀⠀⠀⣿⠀⣤⢠⣿⢀⡇⣼⡇⣿
⣿⣿⣿⡀⣨⣷⣿⡇⢹⡇⠀⠀⢠⡮⢷⣄⢇⡄⠀⢀⠀⠀⠀⢀⢰⣀⣀⠀⢀⠀⠈⠁⠀⢀⡄⡆⣄⣏⣶⡇⣾⣷⣿⣼⣾⡟⠺⣿⣾⣯⣾⣷⣿⡇⣿
⠉⠛⠛⠛⠛⠿⣿⣷⣼⣓⣼⣿⢺⡗⢦⠷⠞⠃⠀⠸⠟⠀⠀⠚⠛⠿⠿⠺⠞⢻⡷⠆⠀⠘⠻⣿⢻⣿⣿⣝⣿⣿⣿⠾⣿⣇⢰⣿⣿⣿⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠇⠀⣀⣀⣀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠉⠛⠛⠀⠀⢀⣀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣾⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠄⠀⠀⠉⠁⠐⠒⠲⠾⠿⣿⣷⣶⣶⣦⣤⣤⣤⣤⣤⠤⠶⠶⠿⣿⣭⣿⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠳⢶⣶⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠨⠟⠛⠋⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣥⡷⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣏⣀⠀⠐⠀⣹⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠲⠦⢤⡤
⠀⠀⠀⠒⠀⠀⣀⡀⠘⢷⣦⣄⣀⣼⣿⣿⣷⣾⣿⣿⣿⣿⣷⣶⣷⣀⣴⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣤⣴⣿⣿⣿⣿⣿⣿⠏⠈⠉⠛⢿⣿⣿⣿⣍⠀⠠⠤⠄⠤⠤⣶⣿⣶⣤⣠⡦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⠋⠙⠁⠀⢀⡤⠁⠈⠙⠿⠛⡲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣁⡤⠖⠂⠀⠀⠙⠒⠀⠀⠀⠰⠬⠭⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⣻⣿⣁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠⣠⣹⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⢡⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣾⣼⣷⣿⣿⣿⢿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡧⠋⠀⠀⠚⠀⠀⠀⠀⠀⠘⠀⠀⣴⢠⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⢋⣾⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣻⣋⣽⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⠿⠟⣻⠟⢫⣽⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢠⡏⢽⡿⡏⠉⢀⡖⢼⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⣿⣿⣿⣿⠑⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⡀⢰⣿⣤⣿⢰⠁⠀⠘⠃⣿⣿⣸⣿⣿⣿⣿⣿⠴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣹⣿⠇⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣿⣿⣿⢺⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⠋⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⣿⣼⣿⣿⣸⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣯⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣤⢻⣿⣿⣿⣿⣿⣿⡿⢿⣿⣇⣀⡀⠀⠀⠀⠀⣼⡿⢟⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠐⣹⣿⣿⣿⣷⡟⣿⣿⣿⣶⣾⣿⡷⠿⠟⣋⣁⣀⠀⠀⢠⣭⣽⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠂⢉⢿⣿⣿⣿⣷⣿⡿⣿⣿⣿⠛⠔⠖⢀⣘⠡⠏⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⡂⠐⠒⡀⠙⢧⡤⠨⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⠐⠐⢺⣿⣿⣿⣿⡽⣿⣿⣇⣻⣿⣯⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠻⠟⠛⡛⠛⠿⠿⠿⠿⠶⠮⠉⠉⡁⠡⢬⣅⣤⣾⣿⣿⡿⣯⢿⢽⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣻⣾⣿⣿⣿⣿⠦⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠦⠿⢯⣿⣟⣟⡭⠥⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠒⠂⠢⢝⡭⡔⠜⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡆⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠒⠐⠠⠦⣨⡅⣶⠼⠞⠝⠛⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢄
⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⡴⠼⣭⣷⡚⣯⣦⡶⡶⠶⠶⠦⢄⠀⠀⠀⠀⠀⠀⡀⠀⠐⠂⠠⠄
⠀⠒⠶⣚⣦⣇⢸⡇⣰⡏⠁⣠⡴⠗⠀⠀⠀⠀⠴⠄⡀⠀⠘⡷⢌⢁⡄⣆⢄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠄
⠀⠀⠀⠁⢺⣥⠿⠳⢟⡇⣀⣿⡀⢀⣤⡾⠋⠀⢠⣠⣶⠓⠀⢿⣘⠊⠉⢈⠄⠁⠁
⠀⠀⠀⠰⠊⢠⢬⠿⢸⠛⣇⡭⣄⡺⣿⣿⡀⣬⣟⣋⡀⠂⣾⢆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⡼⠃⠉⣀⣬⡏⣿⢸⡟⢻⡏⢙⢬⣻⣛⣛⡀⢀⠀⡀⣄⣄⣀⠀⠐⠂⠀⠀⠀⠀⠈⠀⠀⠀⠠⣄⠐⣄⠐⣤⣤⣀⡀
⠀⠀⠀⠈⠀⢠⣼⡿⠛⠁⠋⠀⠵⣿⢷⡽⡷⡿⠶⢤⡰⣼⢇⡋⠇⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠻⠐⠻⢾⣾⠏⠉⠁
⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⢤⣾⢻⡎⢿⠹⣦⠒⢦⣿⣚⢣⢤⡀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⣀⣠⣄⣀⠈⠓⠢⠈
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣨⣿⠸⠀⠃⢈⣑⣲⣲⣷⣿⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⡀⠉⠉⠉⠉⠛⢆
⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣿⢈⠀⠀⠀⠀⠤⣞⣻⢿⣿⢍⡛⠀⠀⡄⠀⠀⠀⠀⢦⣇⠀⠀⠱⡐⡁⠁⠀⠠⣄
⠀⢀⡀⠀⠀⠋⣿⣦⣦⣬⠁⡅⢶⢖⡤⡀⠈⣟⣷⠮⡝⠂⠀⢠⠀⠁⠀⠀⠀⠀⡼⠝⣷⡀⣦⣠⣀⣀⢀⣄⣀⣣⣴⣦⣄⡀
⠀⠀⠙⠿⢶⣤⣤⣄⠐⠜⣷⢮⣉⣼⣿⣁⡀⣺⣿⠅⠀⢀⠀⠀⠂⠀⠀⠀⠀⢠⠀⠉⠇⠈⠉⣱⣾⣿⣾⣿⣿⣿⣿⡯⣛⡚⡀
⠀⠐⠒⠐⠺⢫⣛⠂⠀⡸⢿⡳⠀⠚⢻⡟⣉⢸⡏⠀⠀⠈⡄⠠⠌⠈⠀⠀⠀⡈⠀⡄⠀⠀⣸⣿⣯⣿⠟⡵⠩⠋⣐⡿⣿⣟⠂
⠀⡴⢋⠟⢀⣝⠮⠀⢀⡼⣒⠄⠀⠁⡰⢋⡕⣃⠀⠀⠀⠀⢁⠀⠀⠀⠀⠈⢠⠃⠠⠀⢀⣼⡿⠻⠁⠋⠈⠁⠀⠀⠀⠈⠿⣿⡄
⠀⠁⠀⡆⢈⢼⣽⠀⠀⠼⣩⠡⠀⠀⠀⠊⢾⡶⠇⠀⠀⠀⠨⡄⠇⠀⡄⠀⡼⢠⠃⢠⡿⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢀⠀⢆⠀⠚⢻⠄⠀⠀⠫⢯⠄⠀⠀⠀⠀⠙⢤⠀⠈⠄⠀⢇⠀⠀⡇⠀⠁⡌⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃
⠀⣶⠂⡀⠃⠀⠙⠷⡄⠀⠀⠀⠳⡆⠀⠀⠀⠀⠀⠁⠀⠀⡀⠘⡈⡀⡃⠸⣸⡈⠠⠁
⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⣧⣁⣷⣿⡟⣠⡥
⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣶⣶⣄⣴⣾⣿⣿⣿⣿⣿⣿⣷⣶⣆⣀⣤⣤⣤⣤⣀
⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⠾⠓⠀⠚⠛⠻⠿⠿⣿⣿⣿⡿⠿⠟⠛⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠟⣛⣛⣛⣛⡛⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢋⣴⣿⣿⠿⠿⢿⣿⣷⣌⢻⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣿⣿⣿⡏⠀⠀⠈⣿⣿⣿⣷⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⣾⣿⣿⢃⣾⣿⣦⢹⣿⣿⣇⢻⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣿⣿⣇⠀⠀⢀⣿⣿⣿⣿⡆⠀⠀⠀⠀⣿⣿⣿⣿⡇⣿⣿⣿⡌⢿⣿⡟⣸⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣷⣜⢿⣿⣿⣶⣶⡾⢿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⣿⣿⣿⣿⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣬⣭⣭⣭⡶⢸⣿⣿⡟⢸⣿⣿⣿
⠀⠀⠀⠀⠀⢰⣿⣿⣷⣤⣀⣴⣿⣿⣿⠏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠡⣶⣶⣦⣙⢛⣡⣿⣿⡿⢡⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣌⡛⠿⠿⠿⠿⠟⣋⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⠟⠉⠉⠀⠀⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠉⠉⠛⠲⣄
⣿⣿⣿⣿⣿⠁⠀⠀⢀⣴⣶⣄⠀⠀⠀⠹⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⠏⠀⠀⣠⠖⠓⢦⡀⠀⠈⢷
⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⢾⠀⠀⠀⣿⠀⠀⠈⡇⠀⠀⠘⡇
⣿⣿⣿⣿⣿⡀⠀⠀⠈⠻⠿⠛⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠸⣆⠀⠀⠙⢦⣤⠾⠁⠀⠀⠀⣷
⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⣀⣠⠄⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠙⠷⣤⣀⣀⣠⡴⡆⠀⠀⠀⡇
⣿⣿⣿⣿⣿⡿⠟⠛⠻⣿⣿⡿⠀⠀⠀⢀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣤⠤⠶⢮⡉⠀⣸⠇⠀⠀⣸⠃
⣿⣿⣿⣿⣿⣧⡀⠀⠀⠈⠉⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠳⣄⠀⠀⠉⠛⠁⠀⣀⡴⠋
⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠖⠒⠛⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢠⡀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠸⣿⠀⠀⢠⣿⡏⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠆⠗⠈⢳⡌⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣿⡄⠠⡿⠟⢀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠹⣎⣷⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣸⠀⢀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠘⠖
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠤⣼⠀⠀⠀⣿⡿⠖⠛⠉⠁⠀⠀⠀⠀⠀⠉⠉⠓⠦⣿⡄
⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⢲⡀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢤⠀⢠⣶⣤⡄⠀⠀⠔⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⣇⡼⠃⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡏⠉⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣼⠁⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⡀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢹⣄⣤⣄⣀⣠⠄⠀⣤⠄⠀⢀⣀⣤⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⠁⢀⣠⠖⠁⠀⣴⣣⡤⠂⢸⡄⠀⠀⠀⠀⠀⠀⣏⠉⠉⠉⠀⠀⠀⠀⣠⣶⠟⠉
⠀⠀⠒⠒⠶⠶⢶⣶⣤⣄⣀⢸⡇⠀⠀⠀⠀⠀⠀⢿⣠⡟⠁⠀⠀⠀⠐⠉⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⡇⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣮⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠘⡇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⣀⡀⣠⣀
⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⣠⣶⢿⡄⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠳⠂⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠻⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣡⣴⣶⣶⠶⠶⡶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡿⠋⠉⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⢹⠇⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠒⠚⠋⠉⠀⠀⡾⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠢⣀⠀⠀⠀⠀⡄⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⢰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⣰⠇⢀⡤⠞⠀⠀⠀⠀⠀⠀⣰⠋⠻⣷⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⣿⡦⠿⠛⠉⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠈⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣀⣀⣤⡤⠶⠚⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣿⣟⣿⣭⡭⢯⣭⣿⣻⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣯⣓⣿⣿⣿⡿⢷⣾⣿⣿⣿⣿⣿⣯⣽⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⣿⣶⣿⣿⡷⣟⣵⣿⣾⣯⣾⣿⢿⡿⣳⢿⣿⣿⡽⣧
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠓⢄⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⡀⠀⠀⢠⣿⢟⣿⡿⢿⡾⣿⠟⠷⠋⣽⢿⣿⡷⠻⠟⣿⣻⣾⣿⣶⢻⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢷⡀⠀⠀⠀⠀⣸⠀⠀⠀⣷⠀⠀⠾⠏⢈⣁⡀⠋⠀⠁⠀⠀⠘⠁⠀⠀⠀⠀⣾⣯⣿⣷⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢱⠀⢀⣠⡼⣧⠀⠀⠀⠀⣿⠀⠀⠀⡟⠀⠀⢰⢠⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣦⣹⡟⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣦⠈⠁⠀⢹⡀⠀⠀⢠⡟⠂⠀⣸⠁⠀⠀⠸⡆⠠⢤⣤⣤⢤⠀⠀⠀⢀⣀⣀⣠⣀⠈⢻⡟⣛⡻⣮⢿⢼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⢀⣼⣇⠀⠀⣾⠀⠀⢀⡏⠀⠀⠀⠀⡇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠸⠾⣏⠁⢻⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠉⠀⢻⡀⢀⡟⠦⠀⣸⠇⠀⠀⠀⠀⣧⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠈⢧⠾⠁⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⢠⣄⠈⠳⣶⠶⠂⠀⣠⠤⡀⠀⠀⠀⠀⢀⡞⠉
⠀⠀⠀⠀⠀⣤⠔⠒⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⡇⢲⠌⡑⠒⠶⠶⠒⠋⠁⠀⣹⡄⠀⠀⣰⠏
⠀⠀⠀⢀⠼⣇⠀⠀⠀⠠⡄⠀⠀⢀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠹⣌⠢⢄⠀⠈⠀⠈⠀⠈⠀⡼⠁⢀⡼⠋
⠀⠀⠀⣮⠀⠙⣧⠀⢰⡏⠋⣻⠟⠧⣴⠒⠒⠂⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⢷⣄⠈⠡⠴⠶⠒⠂⠉⣀⣴⡿
⠀⠀⠀⠘⢦⡀⠘⢧⠸⡗⠾⠁⠀⠀⡈⢷⣤⣤⡀⠈⡇⠀⠀⠀⣀⣀⣀⣤⠟⠙⠷⣦⣤⣤⣤⠖⠋⠁⠘⡇
⠀⠀⠀⠀⠈⣷⡀⠈⠳⣝⣶⣤⡀⠋⠀⠀⠀⠀⠀⠈⣧⡶⠿⣏⣿⣿⣿⣿⣿⣿⣶⣦⣬⣤⣤⣤⣤⣶⣶⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠹⣟⠢⠤⠞⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣄
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⣸⡷⣄⠀⠳⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣧
⠀⠀⠀⠀⠀⠀⣠⣿⣷⡄⠀⠀⠀⠈⠙⠛⠉⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⣤⣀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢠⡿⣿⣿⡟⢡⣣⢽⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢛⣟⣻⢛⣿⣋⡛⢻⡟⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣻⣻⣿⣛⣿⣟⣛⣛⣻⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡰⠀⠀⠀⠘⡝⣿⣿⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡃⠀⠀⠀⠀⠴⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠫⠁⠽⢧⠀⠀⠀⢀⣂⣿⣅⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢻⠀⠀⠀⠀⠀⢀⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⢸⣿⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣷⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠭⢽⣛⣷⣾⣿⣿⡘⣧⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣩⣥⠠⠤⠤⣤⡄⣴⣾⠩⢿⣻⣛⡛⠡⠌⠳⢶⣶⣶⣶⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠃⠂⠀⠙⠛⠛⠉⠀⠀⠉⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠁

⣿⣿⣷⣄⠀⠈⢛⡜⢦⡛⠓⢚⣣⡆⠍⣳⡽⢿⣧⣟⣿⣽⡽⣿⢉⣻⣯⣭⡝⠃⠀⠸⡟⠁⣦⠈⢻⡇⠙⠿⠈⠟⠿⣷⣶⢠⠟⠀⠱⡿⠁⠀⡛
⣿⣿⣿⣿⣿⣿⣭⣀⣘⠛⠃⠀⠚⠶⡄⠈⡷⢴⣿⢣⣿⢾⣾⡽⢿⢻⣫⡯⣧⣰⠘⠂⣶⠀⠀⠄⢤⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⣆⠈⠃⢀⡰⢿
⣿⣿⣿⣿⣿⣟⠻⣿⣌⢱⣶⡄⣀⡀⠀⣀⠃⣠⣌⡷⠶⣾⣿⠧⣾⢴⣽⣝⠣⠍⢀⣄⣿⣟⣻⣼⡀⠀⢀⡀⠀⠀⡀⠘⠇⠀⠤⠀⠘⣇⠉⣠⣶
⣿⣿⣟⡉⢉⣹⠟⣿⡟⢈⣏⡹⠋⠁⡀⠀⠀⢾⡍⠙⠂⠀⠀⠀⠀⠈⠈⠉⡀⠀⠘⠇⠀⢹⣏⡜⠴⠀⡌⠡⠆⡀⠉⠃⠐⠀⠀⢠⣶⠿⠋⠛⠁
⣿⣿⣿⣿⣿⣿⣴⣿⠿⠟⣛⢥⣶⢾⣄⣀⣛⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢠⡞⠀⠋⠈⠁⠠⣶⠀⣄⠀⠀⠆⣾⠄⠠⠄⠀⠟⠁⠁⠠⢶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⣲⠸⠋⠉⠁⠺⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠋⡓⣸⡀⡼
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⣼⢀⣀⡀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⣀⣼⢟⠏⣽⣷⡷⡀⠀⠀⠘⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣾⣿⣾⣿⠃⣀⠀⣀⠀⠀⠀⠀⢀⣠⡤⠤⡀⠀⠀⠀⠀⠀⣤⠀⠀⠻⣃⡹⣸⠿⢫⣏⣿⡅
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⡀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠆⠀⠀⠀⣄⠉⠉⠀⢸⠇⢁⢸⡍⠁⢠⠀⠀⠀⢀⣤⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣛⢛⣋⣚⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠦⠀⠈⠐⣀⢀⣄⠛⠋⠾⠀⠀⠀⠀⠀⠀⠀⣙⣷⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠟⣫⠅⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠈⢶⢀⡀⢀⠤⠀⠀⠀⠀⠀⠉⢠⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡾⣫⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠉⠁⠙⠁⢠⡀⠠⠶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣣⢜⡵⠀⠀⠀⠀⠐⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣤⣀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣾⣇⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⢴⣿⣅⣀⣣
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⢿⢈⡿⠋⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣀⠀⢀⠐⠀⠀⠀⠀⠀⠰⠞
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢏⠉⠂⣦⣤⠠⠀⠀⠀⣞⢶⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡻⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣾⣿⣷⡤⠤⢹⠍⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠰⡶⣿⠒⣻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢀⠀⠀⠀⣿⡏⠰⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠆⣇⣠⣾⡷⣃⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡡⠒⢦⣤⣿⣻⣄⣽⣿⠖⠢
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡇⠀⠀⠀⠀⠀⠀⠀⠠⢢⢵⣾⡻⣋⣽⣽⣛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠐⢿⠀⠘⢗⢾⡿⠁⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢞⣛⠓⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢸⣭⠇⠀⠀⠀⠀⠀⠀⠀⠘⢍⡩⠁⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠧
⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠒⠢⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⢠⠾⠤⠬⣭⣉⡆⠀⠀⠀⠀⠚⠲⠆⠀⠀⠀⠀⠑⢤⣀⣧⡇⠀⠀⠀⠀⠀⠀⣠⠖⠒⠒⠢⣄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢯⡉⢉⠐⠒⠒⠂⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢰⠁⠀⠀⣠⠤⠜⢧
⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠒⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠔⠉⠁⢀⠤⠼⡄
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢀⡆⠀⠀⠀⠀⠀⢸⡏⠉⠉⢢⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠳⡴⠊⠉⠁⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣸⡇⠀⠀⢸⠀⠀⣰⡷⡀⠀⠸⣷⣀⣠⠜⠁⣀⡀⠀⠀⢹⣆⠀⠀⡜⠀⠀⠀⠀⠈⠲⠤⣀⣠⠏
⠀⠀⠀⢸⠉⠑⠖⠉⠈⡇⠀⠀⠀⣿⠋⠉⠉⡿⠀⢠⡿⠀⣧⠀⠀⣿⠉⠀⠀⣿⠉⠉⠳⡀⠀⢿⣄⡜⠀⠀⠀⠀⠀⠀⠀⠀⠸⣝⠄
⠀⠀⠀⠈⢦⡀⠀⠀⣠⠃⠀⠀⢰⣿⠀⠀⠀⡇⢀⣿⠋⠉⠹⡄⠀⢿⡄⠀⠀⣿⠀⠀⣠⠇⠀⢈⡿⠀⠀⠀⠀⢏⠓⣹⠂⠀⠀⠈⣆
⠀⠀⣀⠀⠀⠙⠢⠔⠁⠀⠀⠀⠀⠉⠀⠀⠀⠁⠸⠇⠀⠀⠀⢧⠀⠈⠁⠀⠀⣿⠛⠋⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠘⠟⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⢴⡏⠉⠳⡄⠀⠀⢸⡇⠀⢰⣾⠉⠳⡄⠈⠉⢹⡏⠀⢀⣀⠀⠀⡀⢸⡇⠀⠑⡄⠀⠀⠀⠀⢶⡄⠀⠀⢠⠀⠀⠸
⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⠇⠀⠀⣾⠃⠀⢸⣟⣀⡴⠁⠀⠀⢸⡇⠀⢸⡇⠀⠀⡇⠨⣷⠀⠀⢹⠀⠀⣀⠀⠘⣷⡀⢠⠇⠀⠀⡇
⠀⠀⠤⣀⡄⠀⢸⣟⠉⠉⠲⡀⠀⣿⠀⠀⢸⡏⠳⣄⠀⠀⠀⣼⡇⠀⣼⠷⠦⢴⠃⠀⣿⠀⢀⡞⠀⣼⠟⡄⠀⠘⣧⡜⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⠇⠈⠉⠁⠀⢸⡇⠀⠈⠀⠀⠀⠈⠀⠀⣿⠀⠀⢸⠀⠀⠻⠶⠋⠀⣼⡯⠤⣧⠀⠀⢽⡇⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠻⠦⠖⠁⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⡀⠀⠀⠀⠉⠀⠀⠻⠀⣠⠀⠀⠀⣴⠟⠀⠀⠟⠀⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠣⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣖⣦
⠀⠀⠀⠀⠀⣠⡀⠀⣀⡁⠀⠀⠀⠀⢀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⡞⠲⠚⡆⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠘⠀⠀⠀⡏⠉⡖⢦⣄⡀⠋⠀⠀⠖⠉⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣘⣄⣧⠤⠖⢲⠀⠀⠘⠦⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡀⡇⣨⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣉⡥⠤⠔⠒⣚⣉⠭⡆⠀⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠰⡶⠀⠀⠀⠀⢠⠀⢡⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠖⠒⠉⠉⠀⠀⢀⣱⡀
⠀⠀⣀⠀⠀⠀⠀⠀⢸⡤⠊⠁⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠀⠀⠀⡧⣤⢤⣖⣲⡭⠿⠜⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠴⠊⠀⠙⠦⣀⣀⡠⠋⠀⠀⠀⠀⠀⠀⠀⢟⠏⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⠋⠁