Orientation Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠻⠿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⣶⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠿⠛⠛⠋⠉⠉⠉⠉⠙⠛⠻⠿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⠉⠀⠀⠀⠠⠀⠀⠜⠡⠀⠀⠄⠀⠀⠀⠉⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠄⠐⠀⢀⣶⠐⠂⠠⡀⠀⠀⠀⠀⠀⠙⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⣤⡊⠀⠐⠈⠀⠀⢀⣿⠀⠈⠰⠀⠈⠀⣀⡀⠁⠀⠀⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠠⠂⡈⠋⢀⡀⠀⠀⠀⢸⠏⠀⢀⠃⠀⢀⠸⠛⠁⠀⠀⠀⠀⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠠⠃⠄⡀⠀⠈⢻⡢⢄⡀⣿⡀⡆⣀⠤⡺⠋⠀⠈⠄⢀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠂⠀⠀⠈⠉⠒⠠⣹⡄⣹⠿⡧⣟⣡⠞⠁⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⠀⠀⠀⠀⠶⠶⢶⣶⣯⣭⣭⣤⣤⡇⣶⣶⢹⠿⠿⠷⠶⠶⣦⣤⣤⡄⠀⠄⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠉⠀⠀⡀⠀⠀⠀⠀⠉⠉⢻⡟⣳⢤⣤⡞⠲⡂⠀⠉⠁⠀⠀⠀⡀⠀⠁⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠠⠀⠆⠂⠁⠀⣰⠿⠊⠉⢸⣿⠛⠲⢼⣆⠀⠁⠂⠶⢠⠁⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⠀⠀⠀⠀⠡⡈⢂⡀⠈⠁⠀⠀⠀⣸⡟⠀⠀⠀⠙⠃⣀⠊⠠⠀⠀⠐⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⠀⠸⠿⡁⠢⠀⢀⠀⠀⢻⡇⠀⠀⣀⠠⠀⠹⠟⠀⠀⠂⠀⢀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡀⠀⠀⠀⠀⠀⠐⠠⢌⠀⠀⢹⠃⠀⢀⠠⠀⠈⠀⠀⠀⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠨⠂⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣶⣶⣾⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⡶⠶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣤⣶⣶⣾⠱⣶⣶⣤⣦⠀⠀⠀⠀⠀⠀⢠⡾⠛⠁⠀⢀⣌⡀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⢻⣿⣷⣿⠀⠀⠀⠀⠀⢠⡿⠁⠀⢰⣞⡭⠯⣭⣳⡂⠀⠈⢿⡄
⠀⠀⠀⠀⠀⢀⣀⣛⣛⣉⣉⣻⣤⣾⣿⣿⣿⣄⡀⠀⠀⠀⠾⡇⠀⣀⣾⡁⢀⠀⠀⢈⣷⣀⠀⢸⠷
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠉⣷⣤⣤⣴⣶⠁⠀⠀⠀⠀⢻⡆⠀⠀⠈⠧⠀⠀⠀⠾⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⢿⣄⠀⠈⠳⠦⣦⠴⠞⠁⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢣⡟⠛⠛⠉⠋⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⣀⠀⠀⠀⣀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠀⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣿⣀⡀
⠀⠀⠀⠀⠀⠉⠙⣿⠁⠀⠀⠁⣿⡆⠀⠈⣿⠏⠉⠁⠀⠀⠀⢸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠄⠀⣿⣷⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⠀⠀⠀⢴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⠀⠀⠀⠀⠉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⢀⣠⣴⣄⣠⡇⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣄⣠⣦⣄⡀
⠀⠀⠀⠀⠈⠙⠻⠋⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⣛⣛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠙⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣆⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠖⠚⠛⠛⠓⠂⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠉⠀⣀⣤⣤⣤⣤⣤⣄⣀⠀⠉⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠶⠋⢉⠀⠀⢸⠀⠀⠀⠉⠛⢦⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢠⡞⠁⠀⠀⣿⠳⣄⡀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⢠⡟⠀⠀⠀⠀⢸⡄⠈⠻⣦⡀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⠁⠀⠀⠀⠀⢸⣧⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠒⠒⠂⠀⠀⠀⠁⠀⠸⠇⣀⣤⡀⠀⠀⠐⠒⢺⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢻⡄⠀⠀⠀⠀⠀⠀⢠⣶⠛⠋⢸⡇⠀⠀⠀⠀⢸⠇⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⢷⡀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠈⣇⠀⠀⠀⣠⡟⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⠻⣄⠀⠀⠀⠀⠀⢀⠀⠈⠳⡿⠀⢀⡴⠋⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠛⠶⣤⣀⣀⣸⣃⣀⣠⣤⠞⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⠀⠉⠉⠉⠉⠉⠁⠀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠶⠶⠚⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⡀⠀⢀⡀⡀⡀⡀⠀⠀⡀⠀⠀⠀⡀⡀⠀⠀⢀⠀⢀⢀⡀⠀⡀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠑⠀⠀⠁⠂⠐⠂⠀⠀⠐⠀⠁⠁⠈⠀⠀⠂⠀⠀⠀⠓⠂⠐⠂⠐⠀⠂





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠈⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠿⠛⠛⠛⠛⠛⠻⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⢀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⢀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣶⣤⣤⣤⣤⣤⣴⣶⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠰⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣶⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠖⠒⢦⣤
⠀⠀⠀⠀⢹⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⢻⣿⣿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠚⠛⣛⡛⠛⠂⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣆⣀⣀⣠⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⢻⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠛⣻⣿⣿⠈⣿⡛⠛⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⣉⡤⠗⠚⠉⣽⣿⣿⠀⠸⣏⠉⠓⠲⢌⡙⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢏⡤⠚⠁⠀⠀⠀⢰⣿⣿⣿⠀⠀⢻⡄⠀⠀⠀⠉⠲⣌⡹⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⣠⠊⠰⣤⣀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠘⣧⠀⠀⢀⣠⣴⠎⠳⡌⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⡼⠁⠀⠀⠹⣿⣿⣶⣼⣿⣿⣿⣿⠀⠀⠀⢹⣦⣶⣿⣿⠃⠀⠀⠘⣆⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⡸⠁⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⡃⠀⠀⠀⠀⠘⡏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢠⣇⣠⣤⠶⠞⠛⠉⠉⠀⠀⠙⢿⣿⣿⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣶⣤⣄⣹⡀⣿
⠀⠀⠀⢶⠀⣦⢠⡆⢀⣀⣤⡴⠾⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⣟⡃
⠀⠀⠀⠘⡾⠹⡾⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠾⠛⠋⠁⠀⣏⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠙⡟⠛⠿⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣿⣿⣦⡀⠀⠀⢀⣀⣠⣤⠶⠞⠛⣹⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣳⠀⠀⠀⠀⠈⣹⣿⣿⣏⠀⠀⠀⠀⣿⣿⣿⣿⣶⣿⣿⡍⠁⠀⠀⠀⢠⡏⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⢧⡀⠀⠀⣰⣿⣿⡿⢿⡄⠀⠀⠀⣿⣿⣿⣿⡿⢿⣿⣿⡄⠀⠀⢀⠞⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠱⣄⠰⠟⠋⠁⠀⠈⣷⠀⠀⠀⣿⣿⣿⡿⠀⠀⠈⠙⠿⠄⡰⠋⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣎⠳⣄⡀⠀⠀⠀⢸⣇⠀⠀⣿⣿⣿⠃⠀⠀⠀⢀⡤⠊⣱⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡙⠲⡤⣄⣀⣿⡀⠀⣿⣿⣏⣀⣠⡴⠚⣉⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣤⣬⣹⣧⠀⣿⣿⣉⣥⣴⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣆⣿⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⣀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣭⣭⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⢸⡇⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠿⠶⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠶⠿⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⣦⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠀⠀⠀⠀⡴⠛⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣴⠟⠀⢀⡀⢀⡀⣠⣤⠀⡀⢀⡀⢀⡀⠀⠛⢶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠙⢾⡇⣿⣒⠀⢹⡼⢱⡜⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡀⠀⡀⢀⣈⡉⢀⣀⡁⢀⣀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡷⠒⡇⢸⠒⠂⢸⠤⡞⢸⡖⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣁⣀⣁⣈⣉⣁⣈⣀⣀⣀⣉⣁⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢹⡏⠉⢹⡏⠉⢹⡏⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣸⣇⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠈⠛⠉

⠀⢸⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⢸⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⢸⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋
⠀⢸⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣷⣄⠀⠀⠀⠀⠀⣤⣤⣤⣤⣼⣿⣶⣄⠀⠀⠀⠀⠀⣤⣤⣤⣤⣼⣿⣶⣄
⠀⠉⠉⠉⠉⠉⠉⣩⠟⠁⠀⠀⠀⠿⠿⠿⠿⠿⠿⢿⣿⡿⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢹⡿⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠹⡿⠋⠁


⠀⢠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀
⠀⢸⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠢⡀
⠀⢸⠀⠀⠀⠀⠀⠉⢲⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠈⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣪⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡌⢢
⠀⢸⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠔⠁
⠀⢸⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣎⠔⠁


⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀
⠀⣀⣀⣀⣀⣀⣀⣹⡷⡀⠀⠀⠀⡠⠤⠤⠤⠤⠬⠦⠈⠢⡀⠀⠀⠀⢰⠒⠒⠒⠒⠋⠀⠈⠲⢄⠀⠀⠀⢠⠒⠒⠒⠒⠋⠀⠈⠳⢄
⠀⠋⠉⠉⠉⠉⠉⣹⡶⠃⠀⠀⠀⠓⠒⠒⠒⠒⠒⡶⢀⡴⠋⠀⠀⠀⢸⣄⣀⣀⣀⣀⠀⢀⡠⠚⠁⠀⠀⠸⣀⣀⣀⣀⡀⠀⢀⡠⠞
⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠔⠋
⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠛⠉⢿⣆⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⣿⡇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⣍⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠁⠀⣀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠈⠳⣄⠀⠀⢀⡴⠚⠁⠀⢀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠀⠈⣣⡴⠋⠀⠀⣠⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⡀
⠀⠀⠀⠀⢀⣴⡿⠉⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⡆⢀⣴⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⣷⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣼⡿⠃⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠻⣦⡀⠀⠀⠀⠀⠐⠁
⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠈⢻⣄
⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⡜⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⡼⠁⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⢀⡼⠁⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠋⠲⣄⡀⠀⠀⣠⠟⠀⠀⣠⡟
⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠀⠀⠀⠙⢢⡴⠋⠀⠀⣰⡏
⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⣰⠏⣅⠀⢀⣼⠟
⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⢿⣠⣿⠋
⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠷⠶⠶⠾⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⣾⡟⠁
⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⣴⣿⠋
⠀⠀⠀⠀⠀⠹⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣯⡳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠖⠋⠁⠀⠀⣀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣬⣉⡛⠓⠶⠶⠶⠶⠶⠶⠶⠖⠛⠋⠉⠀⠀⠀⢀⣠⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣶⣤⣤⣄⣀⣀⣀⣀⣀⣤⣤⣤⣴⡾⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠌⣧⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⡧⠀⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣜⠮⠁⠀⠀⢀⣴⠖⢀⡠⠐⢀⠀⣄⡴⡀⠀⠀⠀⣷⠀⠀⠀⠀⡄⡀⠀⠠⢀⠀⠶⣄⡀⠀⠀⠈⢕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⡁⠐⢁⣄⠀⠀⠄⢉⠈⠀⠁⠀⠀⣿⡀⠖⠀⠉⠀⠓⠀⠀⠀⠈⠢⡈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠛⢦⣌⠡⠘⠁⠡⣀⡄⠆⠒⢙⡂⠀⣿⡓⠃⠓⠬⣤⣀⠎⠀⠐⢏⠦⣠⠏⡈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠔⢁⣀⠀⡙⢦⣐⡤⠃⡡⡔⢀⠧⠌⠃⠀⣿⡗⠣⠽⡒⢠⣉⠒⢤⠃⣠⣞⠁⠀⠈⢄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡌⠞⠀⢈⡢⠙⢷⣾⡔⠉⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠉⠢⠓⣠⡾⢟⠁⠀⠺⢃⡀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠀⢑⠞⢡⣂⠄⠙⢿⣦⣀⠄⠂⠀⠀⣿⣿⠐⢠⠀⠀⣠⣾⢯⠁⡈⢖⡁⠀⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢄⡄⠉⠒⠌⢠⣁⠃⠀⠀⠀⠙⢿⣷⣄⠀⠀⣿⣿⠀⠀⣠⣾⡟⠁⠀⠩⣑⡈⢇⠒⠉⠀⡀⠀⠀⢀⡀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⠀⡖⠂⠀⢨⠁⣦⡄⠀⠀⢠⠁⠀⠀⠙⢿⣷⣶⠋⠁⣠⣾⡿⠃⠐⠀⠀⠀⢃⣠⢨⡅⠀⢩⣣⠀⡄⢸⡇
⣉⣽⠿⠆⠀⠀⣀⣀⣀⣿⣠⣤⣤⣬⣥⣴⣾⣶⣶⣷⣶⣾⣿⣿⣿⣿⣄⠀⠙⢿⣠⣾⣿⠏⠀⠀⠀⠀⠀⠀⠈⠈⠀⠋⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⢰⠶⡶
⣉⣻⣶⠆⠀⠀⠀⠀⠀⣶⠀⠀⠠⢤⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠉⣧⡀⠈⠻⣿⣿⣿⣿⡿⠿⢿⣿⢿⡟⠛⠻⠿⠛⠛⢙⡏⠉⠉⠉⠁⠀⠻⠀⠇
⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠠⠀⡒⣃⠈⠨⡆⢢⢥⠀⠀⠈⢆⣠⣿⠿⠁⢀⣴⠻⣿⣧⡄⠀⢀⠐⠀⠀⠀⣆⡌⠸⠅⠁⡰⡒⠀⠃⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠹⡌⢤⢆⠀⢠⣾⠟⠁⡀⠰⣿⣿⠀⠈⠛⢿⣦⡀⠀⠀⢀⢌⡔⠠⠏⠐⠢⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢚⣥⠀⠒⠱⢌⢲⣷⠟⠁⠀⠀⠁⠘⣿⣿⠀⠀⠂⠀⠙⠿⣦⡀⠢⠄⡠⠉⠂⣀⠐⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⢠⠰⢀⠔⣺⠟⣁⠠⣑⠦⢀⠀⠀⣿⣿⠀⠀⠀⢀⡀⢔⡉⠿⣦⡈⠑⢄⠠⣪⠊⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠐⢄⠀⣠⠞⣁⠚⠁⢓⠤⡁⠚⠩⠱⣿⣿⠀⠸⠈⠝⠐⣈⢴⠚⠈⠛⢦⣀⠀⠀⡐⢁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡴⢋⠘⢠⠐⠀⠎⠀⠀⠋⠋⠲⢸⣿⠀⠒⠂⠋⠹⠈⠀⠳⠰⣊⠓⠈⡷⣌⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠛⢷⣄⠑⠀⡀⠀⠸⢱⡄⡠⠀⠂⣸⣿⠀⡀⠁⢠⠄⢶⠰⠄⠀⡀⠄⢉⣠⠾⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡞⡠⡀⠀⠀⠈⠛⠀⠀⠈⠐⠀⠄⢀⡀⠈⠚⣿⠀⠃⠀⢀⡁⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⢀⠤⡙⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⣉⣿⢈⣉⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣍⣻⠆
⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠿⡿⠿⢿⣿⠿⠿⠿⠿⣿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠠⠄⠀⢀⣤⣶⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⡠⠒⠉⣉⠭⠝⠳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡠⠊⠀⠀⣀⡀⠀⠈⠳⡀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠟⠛⢻⣿⣿⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠰⠁⢀⠔⠛⢢⠀⠑⡀⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡏⠀⠀⣼⣿⣿⣿⣿⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢠⠃⠀⡞⠀⠀⡸⠀⢀⠃⠀⡇
⠀⠀⠀⠀⠀⠁⣼⣿⣿⣿⣿⡟⠀⠀⢰⣿⣿⣿⣿⡿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡸⠁⢀⠎⠀⡸⠀⠀⢠⠃⠀⡘⠀⣸⣁
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠁⠀⢠⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣃⢀⣜⠀⢠⠃⠀⢀⠏⠀⢰⠁⢠⠇
⠀⠀⠀⠀⢠⠏⠀⣼⣿⣿⠇⠀⠀⣾⣿⣿⣿⣿⡏⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⢠⠇⠀⠰⠀⢑⡏⠀⠀⡞⠀⢀⠇⠀⡞⠀⠀⠀⠘
⠀⢰⠀⢀⡏⠀⣰⣿⣿⡟⠀⠀⣼⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡎⠀⢀⠃⠀⡼⠀⠀⡸⠓⠒⡾⣄⡸
⢇⠈⠄⠸⢶⣶⣿⣿⡿⠁⠀⣰⢿⣿⣿⣿⣿⠁⣀⠀⠀⠀⠀⠀⠀⠀⢢⠈⠆⠘⢆⠒⠚⢄⢰⠃⠀⢰⠃⠀⠰⠁⢠⠃⢀
⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⣯⣿⣿⣿⣿⠇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⢧⠤⠤⢇⠀⡎⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠨⡇⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣠⠜⠳⣄⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠋⠀⣤⣳⡌⠳⣄⡏⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠂⢀⡠⠟⢷⡄⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡖⠉⠀⠀⠀⠙⢦⡀⠈⠳⣄
⠀⠀⠀⠀⣀⡤⠖⠛⡿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠙⢦⡀⠘⢦
⠀⠀⠀⠀⡇⠀⠀⠀⢷⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠈⠳⣄⠈⢳⡀
⠀⠀⠀⠀⡇⣀⡤⢞⠋⢢⡀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⢧⡀⢧
⠀⠀⠀⠀⢻⡉⠰⠯⢤⡀⠙⢦⡾⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠙⢼⡆
⠀⠀⠀⠀⠀⠙⢄⠀⠋⣠⠖⠋⠙⢄⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠈⠳⡏⠀⠀⠀⠀⠈⠳⣄⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⢹⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⠇⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⠳⣄⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢠⠞⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠳⣤⣀⣀⣀⣀⠤⠚⠁⡰⠋⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⠓⢦⣄⡀⠈⠉⠉⢀⣠⡴⠋⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠉⠁⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠶⠶⠶⠶⠒⠛⠉

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




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠶⠒⠒⠚⠉⠉⠉⠉⠉⠉⠙⠒⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠊⠉⠀⠀⠀⠈⠉⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢇
⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡄
⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⠤⠔⣇
⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢀⣀⡤⠴⠒⠚⢉⣹⠀⠀⠀⢻
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⠙⢿⣄⡀⠀⣀⣀⣸⣤⣅⣀⣀⢸
⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡴⡹⠻⢦⣌⣩⡿⠉⠉⠉⠉
⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣟⠜⠁⠀⠀⣽⣟⠢⣄⡠⠤⠖⠚
⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢻⣀⣠⠤⠾⠛⠊⠙⠺⢕⣦⡀
⠀⠀⠀⠀⠀⠘⡆⠀⢀⡀⠀⢀⣀⡤⢽⡗⠚⠉⠁⣀⣸⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦
⠀⠀⠀⠀⠀⠀⠸⣤⡿⠛⠉⠁⠀⠀⢸⢹⣤⡶⠿⣟⢇⢹⡀
⠀⠀⠀⠀⠀⢀⣴⠿⣕⠢⣀⠀⠀⠀⣘⣘⣄⣀⣀⠘⡌⡎⢇
⠀⠀⠀⠀⠀⠚⠃⠀⢻⡙⠺⢷⣟⡛⠛⠉⠉⢉⣁⣠⠼⢿⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣀⣨⡿⠶⠒⠋⠉⠀⠀⠀⠈⢷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⢰⣿⣿⠀⡀⠀⣾⣿⡆
⠀⠀⠀⠀⠀⣯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢠⡀⣠⠀⣿⡄⠀⠀⣠⣼⣿⣿⣰⣿⡄⣿⣿⡇⠀⠀⠀⠀⢰⡆
⢠⣄⡀⠀⠀⡷⣇⣄⠀⣀⢀⡄⠀⠀⣀⢠⣄⣼⣿⣿⣧⣼⢸⣿⣿⣿⣿⡇⢼⣷⣿⣿⣿⣿⣿⣿⣧⣿⣿⣧⢸⣷⠀⠀⢸⡇⢀⡀
⠈⢿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣧⣿⣷⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⡷⠗
⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⠿⢃
⠀⠀⠀⠀⠀⠻⠿⠛⠛⠁⠀⢸⣿⠿⡿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠌⠁⠀⠀⠘⠻⠿⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠈⠀⠁⠀⠀⠀⠈⢹⡿⡟⠻⢿⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠃

⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠁⠰⠄⠉⠁⠰⠄⠈⠀⠀⠀⠀⠀⠀⠉⠀⢀⡀⠀⠀⡀⢀⣁⣀⣀⡀⢀⠀⠀⢀⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣈⣠⣄⣁
⠀⠀⠀⠀⢴⠄⠀⠀⢼⡄⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⠁⠀⡐⡔⣸⠂⠐⠚⢧⢂⢂⢀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠷⢸⡇⠿⣆
⠀⠀⠀⠀⠄⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⢀⠀⠀⠀⢸⠀⠀⠀⡄⠀⠄⠀⢈⠀⠀⠠⠀⠀⠀⠀⠀⠰⠭⠭⢸⡇⡭⠭⠇
⠀⠀⠄⠀⠒⠂⠀⠀⠓⠂⠄⠤⠀⠀⠀⠀⠀⠀⠆⠀⠘⠀⠀⠀⠸⠀⠐⠐⠂⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⡬⣭⣁⣘⢸⡇⣇⣈⣯⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠂
⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣀⣏⣉⡉⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⢘⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡵
⠀⠀⠀⠙⠘⠙⠙⠁⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠈⠁⣉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠈⠋⠛⠛⠺⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈
⠀⠀⠰⠶⠶⡦⣀⠀⠀⠀⠀⠀⠐⠘⠐⠂⠀⠀⠀⠀⠀⢰⠎⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢐⣶
⠀⠀⢠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⢀⣈⢉⠉⠁⠀⠀⠀⠀⠀⢘⠀⠙⠀⠀⠀⠀⠀⠀⢀⠒⠛⠂⠀⠀⠀⠀⠀⠀⠀⢔⣀⣀⠐⠀⠀⠀⠀⠀⣤⣤⣤⡼⠓⠓⠻⣤⣤⢤
⠀⠀⠅⡆⡍⡍⡍⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡘⠛⠛⠂⡀⠀⠀⠀⠀⠠⣀⠋⠀⠀⠙⣀⠄⠀⠀⠀⠀⠀⠀⠨⡝⠙⠯⠀⠱⡆
⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⡶⠀⢀⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡀
⠀⠀⠀⠀⣠⡀⣠⣀⠀⠀⠀⠀⠀⢀⡀⣀⣄
⠀⠀⠀⠰⠂⠄⠂⠰⠀⠀⠀⠀⠀⠆⠐⠀⠐⠂⠀⠀⠀⠀⠀⠀⠠⠏⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠁⠈⠀⠁
⠀⠀⠀⠀⠊⠀⠘⠐⠒⠒⠒⠒⠐⠆⠀⠀⠒⠒⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⢀⠀⡀⡁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⡀⢠⠀⣁⠀⡀

⠀⠀⠀⠀⣄⣠⠀⠀⠀⠀⠀⢠⣐⡄
⠀⠀⠀⠀⣿⣿⠄⠤⠤⠤⠄⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⡸⣿⣿⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢐⠐⠀⠐⠐⠀⠐⠐⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠐
⠀⠒⠁⢠⠝⠿⡀⡠⠀⠀⡀⡨⠟⠧⠀⠙⠂⠀⠀⠀⠀⠀⠄⢠⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⡀⠀⠀⠀⢄
⠀⠀⠀⠴⠶⠲⠃⠀⠀⠀⠀⠳⠶⠶⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠒⠀⠀⠀⠒⠀⠀⠀⠂⠐
⣶⣶⡾⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⠠⠄⠉⠉⠉⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄
⠀⠀⠀⠀⠁⡆⣶⣤⣶⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠄⠐⡂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠁⠁⠙⠛⠋⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠾⠤⠤⠷⠤⠄
⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⣠⡴⡆⡀⡀⠀⠀⠀⠀⠀⠀⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⢀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣯⣭⣭⣍⣿⡆⠀⠀⠀⢈⡁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢠⡆⠀⡀
⠀⠀⠀⠀⠀⢠⠀⢠⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡄⠙⢿⠀⡀⠽⠃
⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠘⡟⡀⠀⠀⠀⠀⠂⢀⣇⠠⠘⠃⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⣠⡇⣇⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠃⠘⠀⠃⠀⠀⠰⠋⢹⢿⠋⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠈⠚⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣫⠵⠮⣝⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⣀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣻⣿⣿⣟⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠾⠛⠛⠛⠛⠷⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⣁⠤⠀⠒⠂⠀⠀⠐⠒⠀⠤⣈⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠔⠉⢀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠉⠢⢀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢠⠎⠀⠀⠀⣸⣷⠎⠁⠀⢸⡇⠀⠈⠁⠂⢄⠀⠀⠀⠱⡀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⡰⠁⠀⠀⡠⠊⠀⠀⢂⠀⠀⢸⡇⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠆⠘⣦
⠀⠀⠀⠀⠀⠀⠀⢰⠇⢠⠁⠀⠀⠌⠀⠀⠈⢢⣄⣠⠄⠚⠓⠠⢄⣠⡔⠁⠀⠀⠣⠀⠀⠈⡄⠸⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡆⠀⠀⠘⠀⠀⠀⠀⢀⠋⣴⣿⣿⣿⣷⣦⠙⡄⠀⠀⠀⠀⠂⠀⠀⢰⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣤⣤⣤⣤⣴⡆⢸⣿⣿⣿⣿⣿⣿⡇⢰⣶⣤⣤⣤⣤⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣧⠀⡄⠀⠀⠁⠀⠀⠀⠀⢡⠘⣿⣿⣿⣿⣿⡿⠃⡌⠀⠀⠀⠀⠀⠀⠀⢠⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢿⡀⢡⠀⠀⠈⡄⠀⠀⠀⢠⠷⠌⡙⠛⠛⢛⡤⠾⡄⠀⠀⠀⢠⠁⠀⠀⡌⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢂⠀⠀⠘⢄⠀⠐⠁⠀⠀⠀⢹⡏⠀⠈⠄⠈⠂⠀⡠⠃⠀⠀⡰⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠈⠢⡀⠀⠀⠑⢄⡀⠀⠀⠀⢸⡇⠀⠀⠈⣦⣤⠊⠀⠀⢀⠔⠁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠐⠢⡀⠀⠀⠈⠁⠂⠀⠸⠇⠀⠐⠘⢿⠟⠆⢀⠄⠂⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣄⡈⠐⠠⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠂⢁⣠⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢧⣂⡄⠀⣉⠁⠀⠀⠈⢉⠀⢠⣐⡼⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⠲⠖⠒⠚⠛⠉⠁

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

⠀⠀⠀⠀⠀⣇⡴⠪⡀⣨⡶⢋⡵⠊
⠀⠀⠀⠀⠀⣿⠗⣠⢟⡡⢞⠍⢀⠂⠀⠀⢈⠤⠀⢀⣤⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢺⣷⢷⣷
⠀⠀⠀⠀⠀⣿⣾⡫⠋⠈⣀⣤⡶⠞⢁⠆⢀⣤⠔⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢾⣿
⠀⠀⠀⠀⠀⣿⣿⣴⠞⢛⣉⣵⣶⣤⠾⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸⣿⣺⣿
⠀⠀⠀⠀⠀⣿⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿
⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣄⠀⠀⠀⠀⠀⢻⣿⣼⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃⠀⠀⠀⠀⢹⣿⢺⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣿⣿
⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡏⢣⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⠶⠲⠶⠶⢶⣶⡶⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠈⢧⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢀⡖⠋⣠⡞⠁⠀⠀⠈⠉⠻⣆⠀⠀⠀⠀⠘⡇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢠⠏⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣷⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣾⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣀⣀⣧⡤⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠓⠲⠦⣤⣼⡤⠶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠋⠉⠉⢹⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢀⣀⣸⣦⡴⠇⠀⠀⠀⠀⠀⠀⠀⠘⠷⢦⣤⣄⣀⣸⣡⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⢀⣠⠶⣦⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⣿⣄⣠⡄
⠀⠀⠀⠀⢛⣿⢀⣽⠗⠀⣼⢿⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⠘⢷⡄⠀⢸⣧⣌⠹⡇⠀⣶⢶⡆
⠀⠀⠀⣠⣾⠿⠋⠀⠀⣼⣯⡄⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣠⡾⠋⠀⠈⠁⠘⣧⣧⠀⢷⠶⣽⣆
⠀⠀⠘⠛⠁⠀⠀⠀⠀⠈⢸⢇⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠞⠉⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠈⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⣭⢮⠹⣿⣿⣷⣝⡻⢿⣿⣿⣮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣝⣣⣜⢿⣿⣿⣿⣿⣷⣶⣯⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡷⠝⣨⣭⣑⢿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡏⣭⣿⡿⢸⣿⣿⣿⢶⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡇⣿⣿⣿⣷⣙⣛⢫⣶⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠿⣿⣿⣿⣿⣿⣿⡿⠛⠻⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣶⣯⣭⣽⣛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢰⡄⠀⠀⠀⣇⠀⠀⢠⡆⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣀⡤⠖⠚⠋⠉⠀⣿⠉⠙⠓⠲⢤⣀⠐⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠙⠲⣄⠠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣀⠀⠀⢸⡇⠀⣿⣧⢰⣶⣤⡀⠀⠀⠀⠈⠳⣄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠄⡼⠁⠀⠀⠈⠲⣾⡿⠂⠀⠀⠀⠀⣿⣿⠈⠉⠋⠀⠀⣠⠂⠀⠀⠘⢦⠠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⡀⠀⠈⠳⣦⣀⠀⠀⠀⣿⣿⡇⠀⠀⣠⡞⢿⣷⠄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢰⠃⠀⠀⢠⣿⡿⠀⠀⠀⠈⠻⣧⡀⠀⣿⣿⠃⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠘⡆⠛
⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⣾⣀⣀⣀⣤⣬⣷⣶⣶⣾⣿⣦⣌⡳⡀⣿⣡⠞⠋⠁⠀⠀⠀⠐⠛⠇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢠⠇⡧⠀⠀⠀⠐⠚⠛⠛⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⢛⢶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⠶⠒⠂⠀⠀⢠⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠸⠖⠂⠀⠀⠀⣠⣴⠞⣡⣿⠀⢳⣌⠙⠻⠛⠛⠛⢹⣍⡉⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠘⡆⠀⠀⠀⠀⣀⡄⢀⣼⠟⠁⢰⣿⣿⠀⠀⠹⣷⣄⠀⠀⠀⢿⡿⠀⠀⠀⢰⠃⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⠻⣿⠟⠁⠀⠀⠸⣿⣿⠀⠀⠀⠀⠙⢳⣄⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠃⠙⣆⠀⠀⠐⠁⠀⢀⣶⣦⡄⣿⣿⠀⢀⡀⠀⠘⣿⠟⠓⠂⠀⠀⣰⠋⠘⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⣄⠀⠀⠀⠀⠉⠛⠃⢹⣿⠀⠸⠗⠀⠀⠀⠀⠀⠀⣠⠞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠓⠦⣄⡀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠂⠀⠉⢛⡒⠶⠤⣿⠀⠤⠶⢒⡛⠉⠀⠐⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢸⠀⠀⠀⠈⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠈⢣⡄⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⡰⠁⠠
⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠀⠄⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠈⠀⠀⡂⢀⠢⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠂⠀⠀⠁⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠑⠠⢐⠠⠂⢁⠠⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡤⠀⠠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠔⡀⠀⠈⠀⠀⠀⡐⠀⠀⢀⠔⠁
⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠂⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠠⠀⠀⡀⠐⠀⠀⢀⠔⠁
⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠌⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠐⠠⠀⠑⠈⠀⢀⠔⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢎⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⢒⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠐⠀⠈⠀⡠⠊⠀⠀⠀⠀⠀⠁⠐⠀⢀⠠⠐⠈
⠀⠀⠁⠂⢀⠀⠀⠀⠠⠀⠀⠁⢄⠀⠀⠀⠀⠡⡂⠀⠁⠀⠀⠁⠀⠘⠀⡄⠀⡈⠀⠀⢀⠇⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠘⢆⠠⠄⠀⠄⠠⠠⠄⠁⠠⠀⠀⣠⠃⠀⠀⠀⡠⠈⠀⠀⠀⠀⠂⢀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⢠⠄⣑⢄⠠⠀⠈⠁⠀⢀⠀⠁⠀⠀⠉⠀⠒⠤⢄⠀⠜⡢⠈⠀⠀⠠⠀⡀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈
⠐⠀⠀⠤⢀⠀⠀⠂⠀⠀⠀⠀⠂⢄⠈⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡢⡀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠀⠤
⠀⠄⠀⠀⠀⠀⠈⠐⠠⠄⣈⠀⠠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⢀⣀⠀⠀⠀⠐
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠂⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡄
⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠠⠠⠤⠤⠠⠠⠤⠄⢄⢤⢠⢀⢠⢠⠠⣠⠀⡄⣤⣦⠧⠤⠤⠤⠤⠤⠤⠸⠠⠠⠤⠤⠤⠦⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠒⠒⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠈⠁
⢀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀
⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂

⠀⠀⠘⠱⠖⠓⠆⠳⠖⠖⠟⠲⠶⠞⠓⠆⠰⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣄⠀⠀⠀⠀⠀⣠⣧⡀⠀⠀⠀⣀⣀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣤⣾⣷⣤⣀⣤⠀⠀⠀⣃⠋⣘⠀⠀⠀⣴⣄⣭⢩⣥⣶⠀⢸⡏⢸⠀⢸⠏⢸⠀⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠴⠿⠿⠿⠿⠿⠿⠿⠿⡆⣴⣦⣩⠀⣩⣴⡄⢸⣗⢻⠛⠚⣻⣓⣷⠠⣏⣹⢯⡽⡍⡹⠀⠲⡏⠉⢹⠶⡂
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣼⡆⠀⢠⡄⠀⢠⢶⡶⡶⡶⣶⢶⢶⠶⡄⢾⣽⣶⠒⣴⣧⡇⢸⣶⣾⠒⠒⢺⣶⣿⠰⠇⢹⠆⢲⠆⢲⠀⣤⡇⠀⢸⣶⠇
⠀⠀⠀⠀⠀⢾⣿⠆⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⢹⠀⠀⠀⠀⣹⡃⠀⢸⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣴⣾⣶
⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⣹⡅⠀⢸⡇⠀⢺⣇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣿⠀⠀⠀⠀⠀⠀⣯⣯⣿⡀
⠀⠀⠀⠀⢠⡟⢸⣇⠀⠀⠀⣨⡁⠀⣸⡇⠀⣸⣿⣆⣁⣀⣀⣀⣀⣐⣀⣀⣂⣀⡀⢀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣁⣼⣿⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⡇⡀⣿⠀⠀⢀⣿⣏⠀⣿⡇⠀⠿⠿⠉⠯⠘⠀⠃⠋⠈⠈⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠈⠁⠃⠘⠸⠏⠻⠿⠇⠀⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⢀⡾⠉⠉⢹⣆⠀⠈⠉⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⣼⠃⠀⠀⠀⢻⡄⠀⠀⣴⢦⠀⠀⠀⠀⢀⡴⣤⢀⣉⣴⣁⣀⢠⢶⣄⠀⠀⠀⠀⣴⣦⠀⢀⣀⠀⠀⠀⠀⠀⠀⢰⣾⣷⡄⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⢰⡶⠚⠉⠉⠛⠿⣷⠀⠀⣿⢻⣷⣶⣶⣶⣿⣻⣻⣿⣿⣿⣿⣿⣞⣿⣟⣶⣶⣶⣶⣿⣿⠀⠫⠯⢧⣤⣤⣤⣠⡤⣴⠲⠶⣦⣤⣤⣤⣤⣤⡾⠼⠟
⠀⢠⡿⠀⠀⠀⠀⠀⠀⢹⣧⠀⣿⣽⣿⣿⣿⣿⣿⡽⣿⣿⣯⣿⣿⣯⣿⢿⣯⣿⣿⣿⣿⣿⣽⠀⠀⠅⣭⡯⣿⠿⢽⠿⢿⠀⠈⠫⡿⡿⢽⢿⠿⣏⠈⡁
⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠛⠃⠛⠘⠋⠛⠙⠛⠛⠀⠘⠛⠛⠀⠛⠛⠋⠀⠛⠛⠙⠋⠛⠋⠘⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⣄⡀⠀⠀⠀⣠⣾⣿⠿⠟⠛⠛⠿⢿⣷⣦⣀⠀⠀⠀⣀⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⡗⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⡰⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠁⢹⣿⡄⠀⣴⠖⠲⣤⠀⢀⡴⠞⠶⣄⠀⣸⣿⠃⠙⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡸⣇⡀⠀⣿⡀⢸⡁⠀⢠⡟⣰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡗⠈⢻⣷⣌⠙⠛⣻⠛⢻⠛⠛⢉⣼⣿⠏⠐⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠋⠁⠀⠀⠀⠹⣿⣆⠀⢹⠄⢸⡆⢀⣾⡿⠁⠀⠀⠀⠉⠛⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⢻⠀⢸⠄⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⣶⣾⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣬⣿⣯⣽⣿⣭⣽⡅⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠉⣩⣭⣭⣭⣭⡉⠁⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢀⣤⠀⣤⣤⠀⢠⣤⠀⣤⡴⣤⡀⣤⡄⠀⣠⣤⣤⣤⡀⣤⡄⠀⢠⡄⢠⣤⣤⣤⡄⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠀⣿⡿⣷⡀⣿⠀⢿⣦⣄⠀⣿⡇⢸⡟⢀⣠⣤⠀⣸⣧⣤⣼⣏⠀⠀⣿⠀⠀⠀⠀⢸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠈⣿⠀⣿⡄⠹⣿⣿⢀⣀⣈⣿⠇⣾⡇⠸⣧⣀⣉⣿⠁⢻⡇⠀⢹⡗⠀⢀⣿⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠉⠀⠉⠁⠀⠈⠉⠀⠉⠉⠁⠀⠉⠁⠀⠈⠉⠋⠉⠀⠈⠁⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠀⠈⠉⠀⠈⠉⠀⠈⠉⠀⠈⠉⠀⠉⠁⠀⠉⠁⠀⠉⠁⠈⠉⠁⠈⠉⠀⠈⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢸⠁⣔⡦⣖⢰⣰⢰⡆⢀⣶⢰⢲⢢⣲⢤⣤⡄⢠⢴⡂⣔⡦⡖⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⠄⣀
⠀⣠⡴⠞⠛⠳⢦⣿⣿⣿⠟⠁⠀⠀⠀⣠⠖⢋⣉⣉⡿⠁⢈⡽⠟⠀⠀⠀⠀⠀⡰⢉⠔⢚⡒⠠⡑⠄
⢰⠏⣴⠟⠛⡳⢟⠑⣆⠀⠀⠀⠀⠀⢰⠃⡼⢋⣤⣿⣵⡞⢻⠀⠀⠀⠀⠀⠠⠤⡧⡇⠌⢸⠌⡦⠥⢼⠤
⢿⡀⣿⡘⠯⠅⣼⠀⣿⠀⠀⠀⠀⠀⢸⡀⢧⠘⠾⠟⣠⠇⣸⠀⠀⠀⠀⠀⠀⠀⢣⠑⢌⣐⣂⡠⢃⠌
⠈⢷⣌⠛⠒⠛⣡⡾⠃⠀⠀⠀⠀⠀⠀⠳⣌⡙⠒⠚⣁⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢼⠄⠒⠁
⠀⠀⠉⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣳
⢀⠞⣋⣍⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢚⣉⣉⡓⢦⡧⠞⠁⠀⠀⠀⠀⢀⠔⠋⢉⡉⠙⣲⡗⠚⠁
⡞⣼⢡⠶⡵⠇⠀⠀⣀⣀⡀⠀⠀⠀⢰⠃⡞⣡⠴⢦⡝⢧⢱⡀⠀⠀⠀⠀⢠⠇⢠⠞⠉⣩⠿⡅⠘⡄
⡇⣧⢻⣘⣟⣛⠛⠛⠿⠿⠇⠀⠀⠀⢸⠸⡄⢇⠘⢃⡇⢸⢀⡇⠀⠀⠀⠀⠸⡄⠸⡄⠘⠃⢀⠇⢀⡇
⠹⡜⢦⡭⢎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡙⠦⣍⡩⠔⢃⠞⠀⠀⠀⠀⠀⠀⣱⣄⠈⠓⠚⠉⣠⠞
⣤⣽⣶⣶⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣶⣶⣶⣮⣥⣤⣤⣤⣤⣤⣤⣼⣥⣤⣭⣽⣯⣭⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡦⠤⡀
⠀⠀⠀⠀⠀⠀⠀⡤⣄⠀⠀⠀⠨⡼⣌⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠅⡇⠀⠀⠀⡇⠘⡩⡇⠀⠀⠀⢀⠂⠉⠠⠀⠀⠀⠈⠉⠢⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠠⠄⢀⠀⠄⠆⠁⠀⠁⡇⠂⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠠⢠⠀⠄⠄⠀⠀⡇⠀⡔⠓⠀⠀⠤⠀⠀⠀⠀⢛⠧⡀⠀⡇⠋⠀⠀⠁⡌⠀⠀⠀⢘
⠁⠏⡇⠀⠀⠀⢸⣹⠧⠇⠶⢄⠀⡇⠀⡇⠀⠐⠗⠾⣷⡄⠀⠀⢸⠠⣧⠀⡇⣁⠀⠀⠀⣇⣘⣿⠀⢸
⠀⠚⡗⠤⠀⠀⢸⡇⠀⠀⠀⢸⣀⢧⣬⡇⠀⠀⠀⡃⡸⠧⣄⡀⢸⡴⠓⠒⢻⠷⣀⠀⠀⠁⠸⠙⢳⢸
⠀⠀⠃⠠⡆⠀⠀⡇⠀⠀⠀⢸⡇⢸⠀⡇⠀⠀⠀⠄⠇⠀⡷⡇⠘⠆⠀⠀⢨⠀⣹⣇⣰⣀⣨⠀⣾⣸
⠨⠀⠠⠘⠇⠀⠸⠷⠀⠀⠀⠤⠅⠠⠀⠆⠀⠀⠠⠢⡆⠀⡇⡗⡄⡆⠀⠀⢠⠠⠀⠀⠀⢰⠉⣷⣾⠁⠀⡀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠂⠉⠉⠉⠉⠐⠒⠒⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⣕⢔⠕⠗⠃⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⢍⡻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⡰⠋⡸⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⠛⣷⣄
⠀⠀⠀⠀⠀⠀⡜⠀⠈⠀⠀⠀⠀⠀⢀⡤⡖⣞⣟⣛⣻⣲⣶⣤⡀⠀⠀⠀⠀⠀⢵⣼⣟⢷⡀
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣰⢟⡼⠛⠉⠁⠀⠀⠀⠉⠉⠙⢷⡀⢀⡀⣰⡿⣫⣾⣕⢷
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢠⢏⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⡾⢋⢴⢟⠵⡷⣽⠇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠢⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢖⠕⡱⣫⢞⡟⢌⡇
⠀⠀⠀⠀⠀⡇⢀⢀⢀⠄⡀⣠⡴⢙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠑⡧⡪⢚⢕⠵⡏⢬⡇
⠀⠀⠀⠀⠀⢃⢐⢕⠅⡨⡪⡫⢿⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡪⣪⢔⠕⡡⣺⠌⢮⠃
⠀⠀⠀⠀⠀⠸⡠⡪⣪⢊⢄⢔⢽⡗⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⣞⢕⡥⡪⣪⣏⢎⡞
⠀⠀⠀⠀⠀⠐⣯⠼⠵⠵⠵⠧⢾⣯⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣓⣵⣧⣮⣮⣾⣇⢪⠃
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⡇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡛⠉⣺
⠀⠀⠀⣀⠄⠀⠸⡀⠀⠤⠤⠢⠀⡇⣿⣀⡀⢄⠀⠀⠀⠀⢀⣀⡀⢀⠀⠀⣀⣠⡀⡇⣀⠏
⢀⡠⠿⠀⢰⣏⣱⣟⢟⣓⡒⠶⢾⡟⡿⡖⠋⠲⡃⠀⠀⠀⣼⣰⣧⢼⣦⣤⢤⣤⡶⣳⡿⠧⠀⡐⢤⡀
⠁⣖⣒⡶⣋⢿⣿⡧⣜⣿⣯⣽⣿⣿⣧⣁⣠⢤⡙⠀⠀⠀⡩⣟⣩⣯⠿⣾⣿⣿⣶⣿⣟⡛⡿⡁⠀⠈⠲⣄
⠀⠀⠻⠟⠉⣈⠯⣹⣿⣷⡖⣿⡘⡛⠓⠒⣧⣜⡸⠀⠠⢎⡀⠀⠀⣸⣶⢾⢅⣿⢱⣟⣿⡿⢷⠇⠆⠀⠚⠤⣀
⠀⠀⠀⠀⠜⠓⠚⠋⠉⠒⠻⣸⣧⡷⠀⠀⠈⠉⠁⠀⢴⠏⠠⠄⣴⡝⠁⢸⣼⠛⠋⠙⢯⢧⠀⠀⠀⠀⠀⠀⠐⣭⡒
⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠉⠁⠀⣶⣆⠀⠀⠀⠀⡹⠂⠀⠀⠀⣀⣀⠀⣠⣄⠀⠀⠈⠻⠇⠈⠀⣠⢄⠀⠀⠙⢄
⠀⠀⢀⠜⡜⠀⠀⣀⢤⣠⣤⣢⠆⠀⢸⣜⣦⡀⠀⠀⠀⠀⢶⣦⣬⣽⣿⠆⠘⢯⣦⡀⠀⠀⣖⣶⠿⠿⣾⠄⡀
⠀⡼⣽⣾⠀⠀⠀⡏⣶⡟⠁⠀⠀⠀⠸⣞⠿⠃⠀⠀⠀⠠⠀⠀⠀⠈⠵⣒⣶⠽⡾⢗⡲⣄⠀⠀⠀⠀⠀⠈⠈
⠀⠈⠛⡋⠀⠀⠀⠈⢩⢢⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠼⣋⠀⠻⠾⠗⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢟⠓⠒⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⢠⢖⣲⣤⣤⣄⠀⠀⠀⢀⡤⣤⣖⣮⠇⠀⠀⠀⠀⠀⠀⠀⣖⣶⡮⠞⠁⠀⢀⣀⠀⠀⢀⡀
⠀⠰⠈⠛⠋⠈⠉⠁⠀⠀⠀⠺⣑⡾⠡⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠛⠁⠀⠀⠀⠘⠒⠲⠥⣯⡼⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⠅
⠀⠀⠀⠀⠀⢀⠀⢀⡠⢀⠀⢀⡀⢀⠀⢀⠀⣀⠀⣀⠀⣀⠀⢀⡀⣀⠀⡀⠀⡀⠀⡠⠀⡠⠀⡠⢀⠄⣀⠀⡀⡀
⠀⠀⠀⠀⠂⠵⢊⠵⢊⠵⢊⠥⢂⠥⣊⠕⣊⠔⢊⠕⢊⡠⢊⡥⢊⠥⢊⠔⢊⠔⣉⠔⣉⠔⣫⠖⡡⢎⡴⠫⠊⠞⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠿⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣁⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠒⠲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⢀⣴⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠒⠲⡆⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠋⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁⠀⠙⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⡀⠀⠀⠀⠀⢀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⢀⣀⡀
⠀⠀⠀⠀⠌⣠⣤⣤⡙⣆⠀⡖⣡⣴⣤⠌⢳⠀⠀⠀⠀⠀⠀⢠⠶⣈⠘⠛⠓⠀⠈⠉⠛
⠀⠀⠀⢸⡀⣿⣿⣯⡇⡶⠸⡇⠻⣿⣿⠇⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⣩⣤⣤⣤⣆
⠀⠀⠀⠀⠱⣌⣛⡉⡴⠃⠀⠐⠤⡉⢁⠰⠊⠀⠀⠀⠀⠀⠀⠸⠒⠊⠥⣤⣿⣟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⣀⣀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⢿⠟⠋⢉
⠀⠀⠀⠀⠀⣀⣤⡾⢿⣿⣹⣭⣷⢶⣄⡄⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⣸⡼⢄⣁⣀⣷
⠀⠀⠀⠠⠾⣿⠷⠟⠁⠉⠀⠀⠙⠲⠿⡿⠶⠀⠀⠀⠀⠀⠀⠸⢿⣟⣛⠱⠒⢉⡥⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠋⡰⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠳⠶⠖⠓⠶⢄⡀
⠀⠀⠀⠀⢠⣏⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰⣾⣷⡖⣶⣲⡒⣶⡒⣞⣲⠒⣭⣶
⠀⠀⠀⠀⢸⣏⢇⠎⠙⠒⠒⢾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣌⠳⠿⠿⠿⡿⢿⠿⣿⣾⣷⠏
⠀⠀⠀⠀⠘⢳⣿⠀⠀⢀⡤⠞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣤⣬⣿⣿⣽⣿⠟⠁
⠀⠀⠀⢀⣀⣼⡼⡄⢀⣀⣀⡀⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡄
⠀⠀⠀⠘⢿⣿⢹⣇⢸⡟⠛⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣆⣈⣭⣥⣤⣤⣤⣤⣬⣡⣶⠆
⠀⠀⠀⠀⠀⠹⣼⢿⣶⠀⣇⡠⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣠⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⢠⣿⠛⠁⠀⣌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠸⠄
⠀⠀⠀⠀⢀⣺⣷⠀⠀⠀⢿⡧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠐⠂
⠀⠀⠀⠀⢹⣿⠇⠀⠀⢀⠀⠀⢠⣶⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢈⠁
⠀⠀⠀⠀⢸⣿⠀⠰⣿⡿⠃⠀⣻⣗⣹⠋⣻⡆⠀⠀⠀⠀⠀⠀⣴⣿⣧⠈⠑
⠀⠀⠀⠀⢸⣿⠀⠀⠀⣰⣾⣿⣿⣷⠘⢍⣿⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠘⢈⣶⣶⣤⠀⣄
⠀⠀⠀⠀⠈⠛⣢⣷⣿⡿⠿⠛⠁⠙⠳⠞⠁⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⠘⡌⣿⣿⣿⠈⣍⡀
⠀⠀⠀⠀⠀⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠏⠀⠀⠉⠁⠒⠒⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠐⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠸⣆⢀⣀⣀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠛⠛⠛⠉⠛⠛⠋⠩⢕⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠑⠌⡢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⡀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⡀⠈⢊⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠈⠀⠀⠀⠀⡀⠀⣾⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠑⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣿⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⢐⣤⣄⣀⡘⣿⣿⠌⣀⣤⡤⠖⠂⠀⠀⠀⠀⠀⠀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⠀⠀⢧⣧⠠⠤⢬⣤⣬⣭⣴⣾⡋⣻⠿⠿⠿⠶⠶⠖⠀⠀⡧⡃⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠈⠈⠀⠀⢈⣠⡴⢿⡟⢋⢽⣿⡇⠈⠛⠲⠤⠀⠀⠀⠁⠁⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢱⠀⠀⠀⠀⠀⠀⠐⢦⡛⢀⣿⢸⡟⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣧⣥⠤⠤⣤⣄⣀⡀⠀⠉⠜⠣⢸⡏⠈⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⡉⠁⣦⡀⣶⠀⢀⣶⡀⣉⠙⠒⢒⠶⢶⣶⠒⢒⡚⠛⣿⠟⢛⠒⣲⡾⢥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣯⠳⣿⠀⣾⣿⣧⢹⣠⠃⢸⠰⡏⢬⡄⣾⣷⡄⢸⠀⣼⠸⣅⡅⢰⣎⠙⣦
⠀⠀⠀⠀⠀⠀⠀⣀⣈⣇⣙⣠⣬⣤⣅⣀⣉⣘⠃⠀⢘⣀⣙⣯⣤⣿⣶⣻⡿⠟⠋⠛⣳⣶⣥⡋⢰⠃
⠀⠀⠀⠀⠀⠄⠘⢛⣿⣿⣿⣿⣿⣿⡆⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠸⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣰⠟⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡟⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠂⠀⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠃⠠⠄⠀⠂⠀⠀⠀⠀⠈⡐
⠀⠀⠀⠀⠀⠖⠀⠀⠠⣞⡉⠀⠉⠁⠀⠀⠀⠀⢀⡄⠐⠀⢉⣀⡠⡆⠀⠀⠀⡀⠐⡌⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⠖⠢⢤⡀⠀⠀⠀⠀⠀⢨⠉⢠⠾⢋⠁⠀⢀⠁⠒⠀⠀⠈⡁
⠀⠀⠀⠀⠀⠆⡀⠀⠀⠉⠀⢠⠐⠊⠁⡟⠓⠠⠠⢄⡀⢸⢀⣺⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣸⠀⠀⣘⠩⠀⠀⠀⢀⠀⠙⠋⠀⠧⢀⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠼⠤⠒⠓⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠀⠁⠀⠀⢀⡠⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⢀⣀⣀⣀⣀⠤⠖⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢐⡀⠀⠀⡀⠀⠌⠀⠀⠀⠓⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠣⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⣀⡀⠄⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠖⠒⠒⠒⠶⠶⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣀
⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠒⠒⠒⠛⠓⠒⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣨⠟⠲⢤⣀
⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⡀⣠⠞⠁⠀⠀⠀⠈⢻⡄
⠀⢸⠁⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠷⢤⡀⠀⠀⢀⡴⠋⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⣦⠶⠒⠒⠒⠚⠛⠛⠛⠛⠒⠲⠶⠤⣤⣄⣀⠀⠀⠀⠀⠀⢸⠇⠀⠙⢦⡴⠋⠀⠀⡇
⠀⡿⣆⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⢤⣼⡀⠀⠀⢰⠀⠀⠀⢸⠇
⠀⣧⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣼⢀⣠⠴⠋
⠀⠸⡄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠹⣄⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣄⣀
⠀⠀⠀⠈⠦⣀⠀⠀⠉⠳⢤⣀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠉⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡝⠛⠬⠍⣉⡐⠒⠒⠒⠒⠒⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠙⠲⠤⣀⡴⠋⠀⠀⠀⠀⠀⠀⣠⠞⢻⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⡶⠶⠒⠉⢀⣀⡤⠤⠂⠒⣊⡩⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⢸⠏⠙⠒⠤⣀⣠⠔⠊⠁⠀⣾⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⠘⠛⠻⠭⠭⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⢸⡄⠀⠀⠀⠀⡇⠀⠀⠀⢀⡇⠀⠀⠀⠀⡴⠟⠡⠤⠤⠤⠤⠤⠤⠤⠤⢄⣀⣀⡀⠀⠉⠉⠒⠲⠦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⣄⡀⠀⠀⣿⠀⣠⠖⠉⠀⠀⠀⠀⠀⠙⠂⠉⠉⠉⠉⠉⠉⠀⠒⠒⠒⠒⠒⠚⠩⠽⠯⠭⠭⠗⣒⣒⣛⣛⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠲⢶⠞⠁







⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⣶⣶⣶⣶⡇⠀⠀⠀⠀⠀⠀⠼⠧⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⢸⣿⣀⠀⠀⠀⠀⠀⠀⣯⣭⡟⠉⢯⣩⡇⠀⠀⠀⠀⠀⠀⣿⣽⠀⠀⠀⠀⠀⠀⣿⣿⣏⣤⣤⣄⣀⡀
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣇⣨⠇⠀⢸⣀⡇⠀⠀⠀⠀⠀⠀⢚⣖⡂⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠺⠿⠇⠀⠀⠀⠀⠀⠀⠛⠚⠃⠀⠘⠚⠃⠀⠀⠀⠀⠀⠀⠨⠹⠄

⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⣤⣤⠄
⠀⠀⠙⢳⠀⠀⠀⠀⠀⠀⠀⡾⣿⢿⢿⡿⡿⡶⠀⠀⠀⠀⠐⢿⠿⠿⣿⠇⠀⠀⠀⠀⣤⣤⣤⣶⣶⣤⣤⣄
⠀⠀⢈⢈⡷⠀⠀⠀⠀⠀⠀⡇⣿⢸⢸⡇⡇⡇⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠬⠤⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠛

⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣲⠄⠀⠘⣿⠀⠀⠀⠀⠀⠀⢤⣽⣿⣧⠄⠀⠀⠀⠀⠀⠀⣀⣲⣶⣀
⠀⠀⠀⡜⡆⠀⠀⠀⠀⠀⠀⠀⢽⠀⠀⠀⣯⠀⠀⠀⠀⠀⠤⣤⣷⣾⣿⣤⠄⠀⠀⠀⣀⣒⣒⣺⣿⣒⣒⣀
⠀⠀⠼⠛⠻⡄⠀⠀⠀⠀⠀⠰⠿⠷⠀⠼⠿⠇⠀⠀⠀⠀⠀⠸⠽⠨⠽

⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡞⡃⠀⠀⠀⠀⠀⠀⠀⣄⠀⣜⣢⠀⣄⠀⠀⠀⠀⠀⣴⣄⡼⣇⣠⣆⠀⠀⠀⠀⠀⠀⣀⠐⢦⡀
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣟⣿⣿⣿⠀⠀⠀⠀⢀⢤⣜⣷⡌⣿⣾⠋
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠸⠽⠯⠯⠿⠿⠽⠀⠀⠀⠀⠀⣿⣭⡏⢹⣭⡿⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠿⠿⠿⢶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠉⣀⣤⣴⣦⣤⡀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⣼⠟⣉⣤⣤⣍⠀⣠⡀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠠⣿⡁⠈⣿⡆⣿⡇⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠈⢿⣆⡙⢿⡿⠋⣰⡿⠁⢰⣿⠛⠿⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠙⠻⢿⡿⠿⠋⢀⣴⣿⣃⠀⠀⣀⣈⣙⣿⣦⣄⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢿⡆⠀⢸⡇⠀⢠⣿⠛⠛⠛⠻⢷⣿⡏⠉⣉⡉⠉⢉⣉⣉⣉⠉⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣼⡿⠿⠿⠿⠿⠿⣧⠀⠀⠀⠀⠀⣿⡇⠙⠛⠛⠀⠛⠛⠛⠛⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⣿⣶⣶⣶⣶⣶⣶⣿⠿⠿⢷⣦⡀⠻⠷⠶⣶⣶⠶⠶⢶⣶⠶⠶⠾⠿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢰⣿⣦⣤⣤⣤⣤⣴⡿⠀⠀⠀⠘⣿⡆⠀⠀⠸⣿⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠸⣿⣀⣀⣿⠇⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⣿⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠸⣿⠀⠈⠛⢻⣯⡀⠀⠀⠀⠀⠀⠿⠃⠀⠀⢰⣿⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⣤⣤⣤⣤⣿⣦⣤⣤⣿⣧⣤⣤⣄⠙⠻⢷⣶⣶⠶⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⣿⠁⢀⣀⣀⠀⣀⣀⣀⣀⡀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⠁⠀⠀⠀⢀⣀⣀⣀⣿⠇
⠀⠀⠀⠀⣿⡀⠉⠛⠉⠈⠙⠛⠛⠛⠁⢸⣿⢷⣦⣤⣤⣤⣤⣶⠿⠛⠁⠀⠀⠀⠀⠀⣿⠛⠛⠋⠁
⠀⠀⠀⠀⠛⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠿⠛⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⠶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠯⡀⠀⠀⠀⠴⠚⠛⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣿⣿⣿⣟⣯⣥⣤
⠀⠀⠀⢀⡞⠳⣤⢦⡀⠸⣶⡋⠀⠀⣠⠞⠁⢹⣆⠀⠀⠀⠀⢀⣦⡀⣀⠀⠀⠙⠵⣿⣿⣿⢟⣾⣷⣄
⠀⢀⡴⠋⠀⢢⡈⠻⡀⠀⠀⢈⣦⡞⠁⠀⢀⡏⠸⡆⠀⢀⣴⣿⣯⣻⣯⡀⠀⠀⠀⠘⢟⣵⣿⣿⢧⣿⡄
⠀⠘⢧⡀⠀⠀⣹⠖⠃⢀⡴⣫⠎⠉⢢⣠⠞⠀⢀⡇⠀⠘⢿⣿⣿⡷⠉⠁⠀⢀⡴⠋⠀⠈⢿⣫⣿⣿⠇
⠀⠀⠀⠙⠒⢾⢳⢀⡴⣫⠞⠁⠀⠀⠈⠛⠢⡀⡈⠀⠀⠀⠀⠙⠉⣦⠀⢀⣴⠏⠀⠀⠀⠀⠀⠉⢻⡟
⠀⠀⠀⠀⢀⡞⣯⡻⠞⠁⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⢀⣼⣦⡛⠁
⠀⠀⠀⠀⠈⢧⣛⡵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠕⠁
⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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




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




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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⢠⣤⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⢸⡇⠀⣤⣶⣿⡇⣀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⢸⣿⣧⣿⡆⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⣀⠀⠀⠀⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣴⣶⣶⣶⣿⣿⣶⣶⣶⣦⠀⠒⠶⣿⣿⣿⣿⣿⢽⣿⣿⣿⡇⠀⡀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣦⣠⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡟⢻⣿⣿⡿⠀⢸⣿⡏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⡏⠈⢸⡏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⡇⢸⣿⠁⠀⠀⠘⠉⠃⠀⡿⠿⠿⠿⠛⠛⠿⠿⠿⢿⣿⠿⠿⠏⠉⠀⠀⠀⡇⠀⠈⠁⢸⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠏⠁⠈⠙⣿⣿⢻⠇⢸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠓⠒⠀⠀⠀⠀⠀⠃⠀⠀⠀⠸⠛⠻⠻⠿⠇⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣇⣀⣀⣂⣀⣀⣛⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣽⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠦⣄⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠘⢦⣄⡀⠀⠀⠀⠀⣻⠀⠀⠀⠀⣸⠇⠀⠀⠀⢠⡴⠚⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠠⠔⠃⠀⠀⢀⠴⠋⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡆⠀⠀⠀⠉⠉⠛⠛⠛⠶⠀⠀⠈⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠚⠛⠛⠛⠳⠀⢀⡤⠶⠛⠋⠉⠉⠉⠉⠙⠛⠷⣦⣀⠀⠘⠉⠀⢀⡴⠋⠀⠙⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⠀⠀⢠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠰⠋⢀⣠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⣠⢤⣄⠀⠀⣠⠞⠁⠀⠰⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⡴⠋⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠁⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠒⠒⠒⠒⠒⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠓⠒⠛⠒⠒⠒⠲⠶⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⡤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠖⠚⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣀⣀⣤⣤⠤⠶⠶⠖⠒⠛⠛⠛⠛⠛⠛⠛⠛⢛⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠦⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠶⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢢


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠂⠉⠉⠉⠉⠉⠀⠁⠀⠐⠂⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢤⠞⠂⢀⢠⢤⢲⣐⣰⣐⣒⣒⣦⡤⣄⣀⠀⠀⠀⠀⠁⠒⠤⣀
⠀⠀⠀⠀⠀⢀⠔⠁⢁⠤⣞⣭⠶⠛⢋⠍⠩⣭⣭⣭⠭⠽⢿⣷⣿⣷⣦⣤⣀⠀⠀⠀⠑⠠⡀
⠀⠀⠀⢀⠔⠁⢀⡔⣭⠾⢋⣤⣶⡯⠷⠒⠊⣍⠍⠉⠉⠉⡽⣛⡿⠭⣙⣻⣿⡿⣦⣄⠀⠀⡈⢂
⠀⠀⢠⠂⠀⡠⣱⠟⢁⠔⡡⠚⠁⠀⠀⠀⠀⠟⠓⠀⠀⣴⢰⠍⠃⠀⢀⠉⠛⠯⣟⠿⡷⣄⣊⢄⠙⢦
⠀⢠⠁⠀⡘⡵⠋⣰⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡼⠼⢸⡃⠀⠀⠀⠀⠀⠠⠈⠳⣌⣢⡳⡅⡀⡈⢷⣄
⠀⠇⠀⣘⣽⠉⡔⡷⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⣷⠎⣪⠅⡾⢀⡀⠀⠀⠀⠀⠀⠈⠀⠈⠻⣷⣵⡨⡪⠈⢯⣆
⠘⠀⢀⢫⡇⣰⣴⠁⠀⠀⢀⠤⡀⠔⠀⠀⠀⢀⠏⣜⣦⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⡎⢧⢢⢕⠘⡚⡆
⡀⠀⠸⣹⠂⠿⡇⠀⠀⠀⠀⠈⠀⠀⣦⣤⣠⢮⢚⠳⣝⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢶⠘⣴⡀⢕⠀⠧⣳
⡇⠀⢛⣗⢘⢸⠀⠀⠀⠀⠀⠀⠀⠀⣈⠿⡏⠱⣬⣴⠿⣼⣭⣭⣽⠿⠟⠛⠉⠀⠀⠀⠀⠁⠀⠀⣯⣇⠕⡂⣑⢮
⡇⠀⢸⡇⢸⡿⠠⢄⡀⠀⠠⠴⠾⠿⠶⢿⠁⠀⣠⠄⡴⣿⣯⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠈⡁⢕⠀⠮⡙
⠇⠀⠸⡇⢸⢻⠐⡿⠇⠀⠀⠀⠀⠀⠀⡜⠀⢀⠀⣼⡏⠋⠛⠿⢦⡀⠀⠀⢀⢄⠄⠂⠀⠀⠐⠀⣴⠇⢕⢠⡑⠇
⠸⡀⠀⢻⡸⡀⡆⢀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⡾⠉⢿⣰⠀⠀⠀⠀⠠⠂⠑⠁⠀⠀⠀⠀⠂⢠⢿⡲⠅⡶⣸
⠀⢧⠀⠀⠳⣳⡷⡄⠀⡀⠀⠀⠀⠀⢻⠁⢠⡞⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢠⣦⡬⣎⣜⢣⡇
⠀⠘⡳⡀⠀⠈⠻⣭⠢⡀⠢⠀⠀⠀⡼⢠⡟⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠰⠿⠿⠊⠽⣍⡟
⠀⠀⠘⢞⠢⡀⠀⠈⠻⣾⣶⣄⠊⠠⣷⡣⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⣠⢞⢟⡰⣊⠶⡑⡞
⠀⠀⠀⠈⠣⣍⠢⣄⠀⠀⠉⠳⠍⣢⣥⣀⠀⠐⠀⠀⠠⠀⠘⣯⠄⠐⠀⠀⣀⡤⣾⡷⢞⠕⢨⡎⢣⠝
⠀⠀⠀⠀⠀⠈⠙⢬⣙⢢⣄⡂⠠⡠⢉⠙⠛⠫⠵⠒⣲⣶⣶⣶⣖⠒⠭⠝⢛⢟⣫⡪⣃⢵⠱⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠮⣌⡑⠒⡤⣥⣁⡊⠒⠔⠢⠂⠄⠠⢂⠕⡪⢚⣕⢥⢼⠺⣙⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠓⠴⠤⣱⣩⣹⡹⡙⡙⡆⢋⢋⢩⣹⣈⠦⠖⠋⠁
⠀⠀⢐⢄⡢⣐⠄⡂⢔⡠⢄⠄⡠⢀⠄⡠⢀⢄⠄⡠⢈⠉⡩⢉⠉⣉⠁⡠⢀⠀⡠⢀⠄⡠⡀⢄⢔⢄⠠⡠⡠⢀⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠟⠙⢦⡀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⡇⠀⠸⣄⠀⠀⠀⠀⠀⢠⣾⠂⠀⠀⠉⠢⣄⠀⠀⣴⠚⠉⠉⠉⠙⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⡴⠁⠀⠀⠈⠑⠲⠤⠤⠖⠋⠁⠀⢀⣶⣄⠀⠀⠙⢾⣿⡄⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢀⣠⠴⠋⢀⡤⠲⢤⡀⠀⠀⣠⣶⠒⠂⠀⠀⣾⡿⠁⠀⠀⠀⠀⠻⣿⣶⡦⠤⠔⠚⠁
⠀⠀⠀⠀⠀⠀⣀⡠⠤⠴⠒⢻⣿⠁⠀⠀⠀⣪⣤⣤⣀⣧⠴⢺⣿⠇⠀⠀⠀⢸⣿⠁⠀⣠⣶⡖⠀⠀⠈⠻⣿⣦
⠀⠀⠀⢠⠞⠉⠀⠀⠀⢀⡠⠬⣿⣄⠤⠒⠉⢸⣿⡏⢁⡇⠀⠸⣿⡆⠀⢀⡆⠈⠿⠤⠚⠁⣿⠁⢀⠄⠀⠀⠈⢿⡇
⠀⠀⠀⠻⣶⣶⡄⠀⠀⠘⠒⠋⠹⣟⢆⠀⠀⠘⣿⢁⡜⠀⠀⠀⠙⢿⠶⠞⠀⠀⠀⠀⠀⠀⣿⠀⡏⠀⠀⠀⠀⢸⠇⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣶⠀⠀⠀⠀⠹⣌⡆⠀⠀⠈⠉⠀⠠⠤⠄⣾⡏⡇⣾⣷⡜⠠⠤⠀⠀⢸⡀⠙⠦⠤⠤⢖⡉⠀⢀⣾⡟⠁⢸
⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠁⣉⣁⣀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠈⡇⣼⣿⠃⠀⢸
⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⣤⣄⢰⡉⠉⠳⡄⠀⠀⠀⣞⠉⠑⣄⠀⢠⡞⠁⠀⠀⠳⠀⣠⡤⣄⠀⣰⡟⠀⠀⡰⢁⣿⡿⠀⢀⠞
⠀⠀⠀⠀⠀⠸⣿⡁⠀⠀⡰⠻⣿⣦⠙⣿⠗⡟⢉⣿⡷⠘⣶⣄⣸⠀⢸⣷⡀⠀⠀⠀⣼⡟⠀⣸⠀⣿⡇⣠⠞⠁⢸⣿⣇⡠⠋
⠀⠀⠀⠀⠀⠀⠹⣧⣠⠔⠁⠀⢹⣿⣇⢿⣰⠃⣾⡟⠁⣠⣈⠛⣿⡆⠀⢻⣿⣦⣀⣀⣿⡷⠚⠁⣠⣿⡏⠀⢀⣀⣼⣿⡏⢀⣀⣠⠴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡈⠁⠀⣿⢀⠜⣿⠏⠀⣿⡇⠀⡴⠻⣿⣿⠀⠙⢿⣄⡴⠁⢻⣷⠀⠈⢉⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠉⠁⠰⣿⠀⡼⢻⠀⠘⢇⣀⣸⡿⠀⠀⠀⠀⣀⠀⠈⠻⣇⢀⡜⢹⣿⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠴⢾⣿⠧⣀⣀⠀⠀⠀⠿⠜⠁⢸⡄⠀⠀⠀⠀⠀⡠⠒⠉⠉⠀⠉⠙⠲⣌⠉⠀⠸⣿⢰⠃
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⢀⣿⡟⠀⠀⠈⠉⠓⠦⣄⡀⠀⠀⠻⢄⡀⣸⠁⠀⣧⣀⣀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢻⡾
⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⣠⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢤⣀⠈⠁⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣄⣀⣀⣀⣀⣀⡤⠚⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣹⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⣀⣀⣀⠀⢀⡀⠀⢸⣷⢸⣟⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⢺⡗⣿⢸⣿⡿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⢿⠿⣏⣿⣹⡿⣿⢹⡆⢸⣷⣺⣗⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠛⣿⣿⡟⢻⣿⣧⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠠⣿⣿⠿⡏⣿⣽⡿⣿⢿⡇⢸⡯⢼⡧⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣿⣿⡿⢾⣿⣿⣿⢸⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣀⣐⣯⣽⣭⣿⣿⣿⡟⣯⣽⡇⢸⣯⣹⣯⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠽⣿⣿⡯⢽⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢰⣿⣿⣿⣟⣻⣛⣇⣿⣻⣟⣿⣻⡇⣸⣗⣺⣗⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣭⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⢻⣟⢻⡖⢲⡞⣿⠛⣿⣿⣿⣶⣷⣿⣻⣟⣷⣾⣿⣿⣿⣾⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⣶⣿⣾⣿⣿⠭⣿⣿⡯⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠭⣿⠭⡯⢽⡯⢽⡯⢽⠭⣿⣯⣽⣬⡷⣿⢸⡷⣿⢿⣿⣿⣿⣽⣿⣿⡇⠀⢴⣶⢶⣶
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠭⣿⣿⡯⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠭⣿⠭⡯⢽⡯⢽⡭⢽⠭⣿⡯⢽⣭⣿⣿⢿⣯⡿⢿⣿⡟⣿⣾⣿⣿⣧⣤⣾⣽⢿⣿
⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⠭⢽⣿⡯⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠭⣿⠭⡯⢽⡯⢽⠭⢽⠭⣿⣟⣻⣽⣯⣿⣻⣏⣿⣿⣿⡗⣿⣿⣿⢿⣿⣿⣿⣽⣿⣽
⠀⠀⢸⣿⣿⣿⡟⣿⣿⢻⣿⣿⠭⢿⣿⡯⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠭⣿⢭⡯⢽⡯⢽⡯⣽⠭⣿⣷⣾⣶⣿⣿⣿⣿⣷⣾⣿⡗⣿⣿⡯⣿⢽⡯⣿⣽⢿⣿
⠀⠀⢸⣿⣿⣿⣷⣿⣿⣿⣿⣿⠭⢽⣿⡯⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠩⣿⢹⣿⢹⡏⢹⡇⣿⠁⣿⣟⢻⣶⡇⣿⣻⣟⣷⣾⣿⡧⣿⣿⡇⣿⢸⡇⣿⣿⢿⣿
⣾⣿⣿⣿⣿⣿⣇⣿⣿⣼⣿⣿⠭⣿⣿⡯⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣺⣟⣺⣗⣻⣗⣿⣓⣿⡿⢿⠿⣯⣿⢽⡿⣿⢽⣿⡇⣿⣿⡇⣿⢺⡇⣿⣿⣻⣿
⣟⣿⣿⣟⣿⣿⡷⣿⣿⢸⣿⣿⠭⢽⣿⡯⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣒⣿⣚⣗⣺⣗⣺⣗⣻⣒⣿⣿⣿⠿⣯⣿⢹⡿⣿⢿⣿⣇⣿⣿⡧⣿⢼⡧⣿⢾⣻⣿
⣯⣷⣿⣧⣿⣿⣷⣿⣿⣻⣿⣿⠭⣿⣿⡯⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣒⣿⣒⣗⣺⣗⣺⣓⣺⣒⣿⣿⣾⣾⣿⣿⣿⣟⣿⣾⣿⡇⣿⣿⣇⣿⣸⣇⣿⣾⣻⣿
⣟⣟⣟⣿⣿⣿⣿⣿⣿⣻⣿⣿⠭⣽⣿⡯⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣒⣿⣒⣗⣺⣗⣺⣓⣺⣒⣿⣟⣻⣿⣗⣿⣻⣟⣿⣻⣿⡇⣿⣿⡗⣿⢸⡇⣿⣿⣻⣿
⣟⣿⣟⣷⣿⣿⣏⣿⣿⢿⣿⣿⣭⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⣷⣾⣷⣾⣷⣾⣶⣿⣿⣿⣶⣧⣿⣹⣏⣿⣾⣿⣏⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣦⣄⠀⠀⠀⣤⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⣿⣿⠟⢁⣤⣀⣴⣿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⢁⣤⡈⢁⣴⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢁⣴⣿⡿⣿⣿⡿⠋⣀⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠁⣴⣿⡿⠋⣠⡈⠋⣠⣾⣿⣿⠇⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠈⠉⣠⣾⣿⣿⣾⣿⣿⡿⠋⠀⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⣤⣤⣬⣤⣬⣥⡄⠀⠀⠀⠀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣄⠀⠀⠀⠘⠻⣿⣿⡿⠛⠁⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣶⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠋⠛⠿⠟⠙⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⡿⠋



⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡶⠶⠿⠿⠛⠛⠛⠛⠛⠛⠻⠶⠶⢶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠋⠉⠁⠀⠀⠀⠈⠉⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠿⠟⠛⠛⠛⠛⠛⠻⢷⣦⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠤⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻
⠀⢠⡿⠉⠛⠓⠲⠦⢤⣄⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣰⠃⢸
⢠⡿⠁⠀⠀⠀⠀⠀⠀⢀⡿⣇⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⢸
⣿⡟⠶⣤⣀⡀⠀⠀⢀⠞⠁⠘⣆⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⣸
⠸⣧⠀⠀⠈⠉⠓⠲⣿⠀⠀⠀⢹⡄⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⣿
⠀⢻⡄⠀⠀⠀⠀⠀⠸⡆⠀⠀⠈⣧⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⣾⠇
⠀⠈⣷⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠙⠓⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠸⣇⠀⠀⠀⠀⠀⠈⣇⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢦⣀⠀⠀⠀⠀⠉⠛⠲⢤⣀⡀⢀⡠⠊⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠸⣤⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠉⠳⣦⡀⠀⠀⠀⠀⠀⢈⣽⡟⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠛⠿⠛⠛⠛⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⠶⣄⠀⣠⠴⠋⠀⣇⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⣿⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣿⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀⠀⠀⣧⠀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⣿⡾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠒⢀⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢘⠄⠀⠀⠈⠁⠐⠢⢄⠀⠐⠀⠐⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠚⠃⢱⡄⠀⠀⠀⠀⢀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡠⠄⠀⠀⠀⠀⠀⡀⠀⢁⠀⠀⠀⡀⠇⠀⢀⡀⠀⠀⠀⠀⣨⠆⠀⠀⠀⠀⠳⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢠⡿⠀⠁⢀⠉⢀⣀⠴⠞⠛⢢⡈⠓⠈⡄⢱⡎⠀⢀⡀⠀⠀⠀⢦⡈⢻⡄⠀⠀⠀⢀⡀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠘⠃⡌⠀⣡⠞⠋⠀⠀⠀⠀⠁⠀⠋⠀⠁⠃⠈⠁⠊⠉⠳⠄⠀⠈⢱⡈⢷⣄⡀⠀⠀⠀⠀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡷⠒⣀⡁⠰⢎⡌⠁⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⠛⠦⠯⡝⣦⡀⣠⠞⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡾⢁⠔⠈⠀⠀⠐⡉⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⣠⣤⣶⢿⡏⠐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⣷⣯⠄⠀⢐⡉⢤⠄⠒⡛⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠂⢀⠀⠀⠠⠀⠈⠻⣆⠀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣿⡋⠠⡔⠁⠀⠀⠀⠈⡀⠀⠘⠋⠗⣉⣉⡉⢉⡳⡄⠀⢠⡾⠛⠉⣀⣈⣙⠒⠀⢠⠀⠀⠄⡀⠀⠀⠀⠀⠀⢹⠀⠉⠑⠂
⠀⠀⠀⠀⠀⡠⠄⠚⣿⡇⠀⠀⠀⠀⣀⠀⣴⠇⠀⠀⠐⠪⠿⠿⠇⠐⠀⠀⠀⠸⠇⠐⠹⠧⠿⠉⠛⠀⠀⠀⢀⠀⠀⠀⠀⢀⠈⣼⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣎⠦⠀⠀⠫⣄⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⠓⠀⠛⠀⠀⠀⠘⠀⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠄⢣⡀⠈⣷⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠔⠀⢈⠀⡠⠃⢀⣼⠃⠀⠠⠀⠀⠸
⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢸⣐⣴⠇⠀⢻⢾⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⢁⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢰⢃⠎⠀⣴⠋⠁
⠀⠀⠀⠀⠀⠆⠀⠀⢠⡔⠶⠛⠻⠋⠚⠻⠋⢀⠇⠀⠀⠀⠀⢀⡤⠀⠈⠁⢤⡔⠁⠠⠀⠀⠀⠀⠀⠃⠀⠐⠀⠐⠸⡦⢤⣜⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠲⣶⣤⣴⠾⠳⢄⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠞⠃⠀⠀⠀⡄⠐⠒⠿⠋⠙⡆⠀⠈⠀⠀⢸⢠⡖⠺⢋⣉⣉⠝⢳⠀⠀⠀⠀⠀⠀⢸⡆⠀⠤⠤⢤⠤⠀⢈⡙⢦⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⢠⠲⡃⠀⠀⠀⣆⡄⠀⠈⠛⠀⠁⡄⠀⠈⠀⠁⠀⠸⡀⠀⠀⠀⠄⠀⠘⠤⡀⠀⠈⠉⠉⠉⠀⠀⢠⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠘⢆⠀⢀⡣⠀⠀⠀⠈⢥⡀⠀⠀⡀⢠⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⢀⡀⠀⢲⡄⠀⠙⢠⣴⡥⠈⠄⠀⠙⠀⠀⠐⢃⠜⠀⢀⡴⠾⠛⠁⠀⠀⠀⡀⠄⣐⠭⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡛⣀⠈⠱⠀⠁⢱⡀⠀⠀⠈⠁⠀⢸⡧⡾⠂⠸⣀⣤⣿⠶⠊⠁⠀⠀⠀⠀⠁⠘⠀⠃⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠁⠀⠀⢸⡟⠁⠀⠀⣀⠄⢠⠎⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠚⣗⠣⠠⠤⠴⠊⠠⠄⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠴⠀⢀⡴⣍⣩⣿⣾⠿⠋⠉⠒⠀⠤⢀⡁⠀⠀⠀⠀⠐⠤⠤⠉⢀⡀⠨⠅⠀⠀⠀⠈⠳⣤⣴⡾⣷⠀⠀⢀⡼
⠀⠀⠀⠀⠀⢰⣶⠁⠠⠤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠘⣤⡀
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢷⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠻⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣶⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠈⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⡿⣋⣽⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢴⣏⣷⣄⡀⠀⠀⠀⠀⠀⢿⣶⣿⣯⣽⠂⠀⠀⠀⠀⠀⠀⠀⢀⡿⠻⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣎⠃⡄⠙⣆⠸⠀⠀⠀⠀⠻⡵⠟⠁⠀⢀⣀⡀⠀⠀⠀⠀⣾⣿⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣧⠀⠏⠛⢇⠀⠀⠀⠀⠁⠀⣴⠛⠉⠁⢻⡀⠀⠀⢠⣿⣦⣿⠁⠉⠛⠷⣄⠀⠀⠀⠀⠀⠀⢀⣴⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⠀⢨⣷⡀⠀⠀⠀⢸⠇⠀⣀⣤⣤⢷⡴⣤⢾⣿⣽⡃⠀⠀⠀⠀⠘⣧⠀⠀⢀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇⢘⣿⡿⣄⣤⢚⡿⠈⠈⠁⠉⠉⠈⢷⠉⣦⡤⠀⠉⢻⣆⠀⠀⠀⢿⣤⢶⡛⠳⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⣴⠖⢛⣼⠽⢛⣴⣿⠛⠙⠙⠛⠺⠷⠷⠼⠾⠿⠷⠦⠤⡄⢸⣇⣹⠴⢾⡿⣵⣦⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠢⣄⣿⢧⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣿⣸⣖⡄⠳⢿⡟
⠀⠀⠀⠀⠀⠀⠀⣠⠄⢀⡀⡤⠤⡤⣤⣰⣿⡎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⢈⣿⣦⣦⣬⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢿⣅⣻⡷⢶⣮⣥⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢹⡼⠄⢸⠋⠉⠑⣒⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⡿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⡇⢸⠀⢨⣛⣾⣿⠛⠠⠤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡞⢀⡘⠛⠻⠿⣦⣀⠷⠚⠋⠉⠁
⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠠⡀⡉⢙⠿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⠘⢡⣤⣶⣴⢿⡁
⠀⠀⠀⠀⢁⣧⠀⠀⣀⣠⣤⠄⠛⠛⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣄⡰⠶⠟⠻⠶⠃
⠀⠀⢀⣠⡤⢭⡄⢀⢠⣉⣍⡤⠤⠄⠒⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⣉⡀
⠀⠀⠈⠉⠑⣴⡦⠀⠋⠉⠀⢀⣶⣨⣶⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠻⠎⣶⣷⡟⠁
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣠⣬⡾⠟⠉⣀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣗⣦⡀⡰⣿⠏
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠉⠁⠀⢠⡔⣽⠿⣿⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⣇⡉⠛⠹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣡⠞⠁⣸⠿⣿⣿⣶⣤⣀⣀⠀⠀⠀⠀⠀⣀⣀⣠⣶⡿⢿⠻⣾⣿⣧⡄⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠈⠁⡤⢝⣛⣩⡼⢛⡿⢻⡟⣛⡿⢿⣿⢻⣿⣿⠗⢸⣆⡈⠻⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⠀⠀⠀⠀⠀⠈⠝⢳⠶⠾⠋⣠⡊⢉⣯⣹⡄⠹⣿⡽⢟⣋⠭⠟⢉⠀⠈⠻⣯⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠈⡨⣤⠀⠀⠞⠑⢪⣭⢬⡒⣿⠓⠀⠀⠀⠈⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢈⣐⢧⣰⡄⠀⠈⠋⢹⣿⣷⠏⠛⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠈⠉


⣿⡟⣛⡻⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⠐⡔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⢁⣠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⢀⣤⡀⢀⣀⠀⠀⠀⠀⢠⡄⠀⠈⣿⡇⠀⠀⠀⣿⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦
⡀⠀⠀⠀⠀⣤⣼⣿⣿⣿⣿⠀⠀⠀⠀⣾⣷⠀⠀⣿⡇⡇⠀⢠⣿⣿⡇⢨⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠄⠀⣷
⣧⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠁⠀⠀⣀⣿⣿⠀⢺⣻⡇⡇⠀⣺⣿⢿⡇⢸⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⠠
⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣤⠀⣧⣿⣿⠀⣼⣿⣷⣿⠀⣯⣿⣰⣿⣼⣿⣤⠾⣧⣾⣿⣆⢀⣄⠀⠀⠀⢀⡄⠀⠀⠀⠀⣿⠀⢸⣿⠂
⣿⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣤⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣯⣜⣣⣤⠌⣧⣼⡇⠀⠀⡇⠀⣿⡄⣼⣿⡄
⣿⡇⠀⠀⣿⣿⢚⣿⣿⣿⣿⣿⣇⣀⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⣿⣿⣿⣦⣷⣴⣽⣷⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⠿⠋⢻⡿⣥⣮⣿⣿⣿⣿⣿⡿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣍⣉⣿⣯⣿⣿⣿⣿⣿⣿⡷⠝⣿⣻⣿⣿⣿⣿⣟⣛⠿⠿⠿⠿⠿⠿⣿⣿⡿
⠀⠀⠀⠈⠀⠀⠛⠇⠿⣿⣿⠋⠁⠀⢹⣿⣿⣿⣿⣿⣿⡏⠉⠉⠛⢉⣟⣻⣏⠉⠉⠻⣷⣿⡧⢤⣾⣿⡿⠛⠻⠿⠛⠻⠿⠿⠲⠷⠾⠾⠷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠂⠀⠀⠈⠉⠉⠛⠙⠛⠙⠃⠀⠀⠀⠀⠀⠋⠁⠀⠀⠈⠀⠀⠀⠀⠨⢽⠿⠛⠙⠶⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣸⡷
⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠂⠀⠛⢸⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣧⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⠧⠾⠿⠀⠹⠦

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

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠂⠁⠀⢴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣆⠤⠂⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣀
⠀⠀⠀⢀⡠⠴⠚⠋⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⣀⡀
⠀⠀⠈⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡝⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠽⣶⣄⠀⠈⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠆⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢺⣿⣟⠒⠊⢩⡆
⢀⣤⣶⣾⣿⣿⣿⣿⣿⠀⠀⢸⡇⠀⠀⠒⠦⢄
⢮⣉⠉⠛⠛⠛⠛⠟⠛⠂⠀⠈⠋⠀⠀⠀⠀⣠⡧
⠈⠛⠻⠷⢶⣶⣶⣤⣀⣀⣀⣀⣠⣤⣤⠴⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⡋
⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡅⠀⠀⠀⠀⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⡀⠀⠀⢀⣿⣿⠀⠀⠀⢀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⣾⣿⣶⣾⠿⠿⠿⢿⣷⣶⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢏⣵⣿⣿⣿⣿⣮⡻⣿⣇
⠀⠀⠀⠀⠀⠀⢠⣤⡀⣀⣀⡀⣤⣴⣿⡇⣿⣿⡿⣫⣝⣿⣿⣿⢹⣿⣤⣤⢀⣀⣀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠁⠈⠉⠀⠛⠛⣿⣇⣿⣿⣷⣽⣯⣿⣿⣿⣸⣿⠛⠛⠈⠉⠁⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣮⡻⣿⣿⣿⣿⢟⣵⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐⢿⣿⠿⠿⠸⠟⠻⠇⠿⠿⣿⡿⠂⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠸⠯⠭⠅⠀⠀⠈⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠈⢩⡍⠁⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡃⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠳⠏⡧⠗⢯⡦⠗⠇⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣤⣤⣄⣴⣴⣷⣶⣴⣶⣆⣄⢠⣄⡠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⣹⣿⣿⣿⡿⠋⠁⠀⣀⣀⣠⣄⡀⠀⠉⠙⠙⣿⣿⣿⣦⠈⠉⠛⢿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⢰⣿⣿⣿⡁⠀⠀⣴⣾⣿⠟⠛⣻⣿⡄⠀⠀⠀⢀⣿⣿⣿⡆⠀⠀⠀⠈⠙⠻⣷⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠸⣿⣿⣿⣤⣴⣿⣿⡏⠀⠀⠀⠀⣻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⡰⢹⡯⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⠀⠀⠂⠿⢿⣿⣿⣿⡿⠿⠀⠠⠐⠚⠐⢯⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦
⠀⠀⠀⠀⢀⣠⠜⠀⣟⡀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣀⣀⢀⣠⣤⣤⡝⠇⠤⣠⣶⢤⡄⡀⣠⣽⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿
⠀⠀⢠⡏⠁⠀⠀⠀⣿⢨⠄⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠿⠻⠾⠾⠧⡬⡉⠴⠛⠛⠛⠋⢋⣻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠁
⠀⠀⠈⠓⠤⠤⠤⣄⡘⣖⡖⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣦⡀⠀⠀⠀⠃⠀⠀⢀⣤⣤⣽⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⢯⡃⣀⡀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣶⣶⣿⣾⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⣀⣤⣶⣿⠟⠋⣉⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣶⣴⣀⣀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⠿⠿⠛⠋⠁⠀⣀⣤⣴⣾⠿⠟⠛⡉⠠⠶⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⣿⡛⠻⢷⢶⣦⣤⣤⣄⣀⣀⣀⣀⣤⣤⡤⠶⠾⠟⢛⣩⡥⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠋⠂⠒⠤⠤⠠⣌⠉⣍⢩⣥⠤⠤⠤⠒⠞⢛⠉⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⡷⢞⡁⣶⠶⡆⠀⣠⢦⣤⣴⡆⠀⢴⡖⢺⣽⢸⠍⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠃⣀⡙⣞⡹⣙⠇⠀⠘⠃⠛⠛⠘⠓⠀⠀⠀⠀⠄⠀⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡼⣏⡿⣟⡇⣿⣳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⡶⠿⠿⠿⠿⠿⠿⠿⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠚⠛⠉⠉⠉⠉⠉⠉⠛⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣀⣤⣾⠿⠟⠛⠛⠛⠛⠻⢷⣦⡀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⢀⣤⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡘⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀
⠀⠀⠀⣾⡟⠐⠒⠒⠲⠦⠤⠤⣾⡁⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢸
⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⢠⡏⢳⡀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣿
⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢳⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⢰⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠤⣄⡀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢀⣾⡟
⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠈⠉⠓⠦⣄⣠⠟⠀⠀⠀⠀⠀⣼⡟
⠀⠀⣿⡧⠤⠤⠤⠤⠤⠠⢤⡄⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⣿⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠹⡄⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⡀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⢸⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠹⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠙⠒⠤⡀⠀⣠⠞⠁⠀⠀⠀⢸⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠶⠶⣶⣶⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⢸⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠘⡃⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⣷⠀⠀⠀⣴⡿⠁
⠀⠀⠀⢰⡇⠀⠀⢠⠇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀⠀⠀⢹⠀⣠⡾⠋
⠀⠀⢠⡟⠀⠀⠀⠸⣆⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣬⣶⠟⠁
⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉
⠀⣴⠏⠀⠀⠀⠀⣸⠇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⣸
⠀⡽⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣼⠃⠀⠀⠀⢠
⠈⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢠⡏⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⢀⡾⠃⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⢀⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣦⡀⠀⠀⣠⡾⣻⣥⣬⣟⢷⣄⠀⠀⢀⣴⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⢀⣀⣟⢸⣏⣷⣾⣹⡇⣻⣀⡀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠃⠀⠀⢷⡼⣷⣽⣯⣾⢧⡿⠀⠀⠘⢿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠻⢷⡇⢸⠾⠟⠁⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣽⣶⣿⣿⣿⣿⣿⣿⣶⣯⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠀⠀⠐⠒⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠀⠀⠀⡀⠀⠀⠀⠄⠤⢤⠀⣀⠀⠀⠀⠈⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⢀⠀⠒⣡⡱⠔⠒⠈⠈⠈⣉⠙⠒⠒⡦⢭⣐⠤⡀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⢊⣠⠶⠏⠑⠀⠀⠀⠁⠀⠀⠀⠀⠸⠗⠀⠀⠀⠙⠲⢶⣀⠀⠈⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡰⠋⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢡⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡗⢺⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⡜⢀⣲⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠭⠉⠉⠀⠀⢀⠀⠀⠀⠁⠀⡀⠀⠾⠰⢤⠁⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡂⡩⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢄⣂⡼⢥⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢪⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠅⣎⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠸⠱
⠀⠀⠀⠀⠀⠀⠄⠀⠑⣵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⢀⠄⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠠⡅⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⢀⣠⣦⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠡⣄⠠⠀⠀⠀⠀⠀⠀⠀⢠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⢀⠔⡴⢧⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⢠⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠀⠀⠄⣤⠰⠄⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠢⢄⠀⠀⠁⠂⠠⠀⣈⠀⠀⠀⠀⡀⠠⠄⠘⠂⠉⠀⠀⠀⣠⠐⡈⠄⢠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠁⠂⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠠⠒⠉⢂⢀⡠⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠈⠑⠐⠰⠀⠀⠂⠒⠂⠂⠉⠈⠀⠀⢁⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠦⠤⠤⢀⣠⣀⣀⡀⠤⠤⠐⠘⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⢐⠀⡂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⣵⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⣿⠅⠀⠀⠀⠀⢀⡀⠂⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠀⡀⠆⠸⣿⡆⠀⠈⠀⠀⣀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠱⡘⢄⠀⢻⣿⡀⠆⠀⠀⠉⢀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⢎⢂⠀⢻⣧⠀⠀⠀⠀⠠⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡉⠉⠐⠢⢕⡀⠱⣕⢄⠻⣦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠳⠶⣤⣌⡐⠌⠓⠤⡙⠆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠈⠙⠓⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⡈⡀⡌⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠆⠁⣥⠁⠀⠀⠀⢀⣀⣀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢷⢠⠇⠀⣠⣴⢿⣿⣿⣿⣿⣿⢧⣠⣤⡀⠀⠀⠀⠀⠀⢴⡿⢿⣶⣾⣿⣿⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠈⢸⢠⣾⣿⣿⣿⣿⣿⣻⣷⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⠀⢸⣿⣿⣷⣿⣿⣿⣿⡿⠟⢋⣉⠀⠀⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠯⣙⣿⠿⢟⡋⠵⠒⠉⠁⠀⠀⠀⠀⠈⠳⣎⠉⠉⠛⠛⠋⠉⠉⠀⢀⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢼⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠈⢷⡐⢄⠀⠀⠀⠀⢀⣮⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡢⠀⠘⣸⡸⡀⠀⠀⠀⠀⢀⡠⣈⠔⠀⠀⡼⠀⢰⠀⠀⠀⢣⠈⢷⡈⠳⠄⢀⣀⡾⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠰⠁⠀⢂⠁⠳⣍⣢⣤⢔⡢⠕⠊⠀⠀⠀⡰⠁⠀⢨⠀⠀⠀⠈⡆⠘⣧⠀⠀⠛⠉⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢀⡀⠄⠈⢄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡜⠁⠀⢸⣴⠀⠀⠀⠀⠰⡀⠸⡆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠓⠋⠉⠉⠉⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⡄⡁⠀⠀⠀⠀⠀⠡⠀⠹⡀⠀⢀⣪⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠡⣀⠠⡈⠳⣄⠀⠀⣀⠠⠔⠊⠀⠀⠀⠀⡇⠇⠀⠀⠀⠀⠀⠀⠁⠀⠱⣶⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢌⡒⢄⡉⠐⠂⠐⠀⠀⠀⠀⢠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣤⣀⣀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢄⣀⠀⠀⠈⠒⢬⡓⠤⣈⠁⠒⠒⠒⠁⢸⠀⠀⢈⠉⠐⠒⠂⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠤⣀⡈⠑⠢⣑⠢⣄⠀⠀⠀⠣⠔⠉⠑⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣺⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠑⠀⠙⠢⢍⡒⠄⠀⠀⠀⠘⠀⠀⢀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⢁⣀⣤⣤⣤⣤⣴⣶⣶⣦⣤⣤⣤⣤⣀⣈⠉⠛⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⣴⣿⡿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠿⠿⢿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠂⣸⣿⡿⠀⠈⠁⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠏⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠛⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⣤⣈⠙⠿⣿⣿⣦⣄
⠀⠀⠀⢀⣠⣾⣿⠿⠋⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣇⡀⢀⣼⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠙⠛⢁⣴⣄⡀
⠀⠀⠀⠀⠙⠟⢁⣤⣄⡀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣠⣶⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣶⣄⡉⠛⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⣰⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⢀⡀⠀⣾⣿⡏⠀⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣶⣶⣾⣿⠟⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣀⡉⠉⠛⠛⠛⠛⠻⠿⠿⠟⠛⠛⠛⠛⠉⢁⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣄⠀⣸⣿⡀⢠⣿⣧⢸⣿⡇⢰⣷⠀⠀⣶⣶⠀⣀⣀
⠀⠀⠀⠀⢸⣿⡇⢀⣀⣀⢸⣿⣿⢸⣿⣿⣇⢸⣿⣿⢸⣿⣷⣸⣿⣶⡆⣿⣿⠀⣿⣿⡀
⠀⠀⠀⣾⣿⣿⡇⣾⣿⣿⣼⣿⣿⣸⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⢠⣿⣿⣧⣰⣿⣿
⠀⢐⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⣀⡄⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⢀⣀⢸⣿⡇⢸⣿⣷⣆⢸⣿⣿⠀⣴⣶⡀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⢰⣿⣆⣾⣿⡇⣿⣿⢾⣿⡧⣼⣿⣿⣿⠈⣿⣿⣶⣿⣿⣧⣴⣦⢸⣿⣿⠀⣀
⠀⢀⣤⣸⣿⣿⣼⣿⣿⣿⣿⣷⣿⣿⣸⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⣄
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣻⣿⣷⣦⣄⣀⡀
⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⡿⣶⣶⣿⣿⣿⠿⠿⠿⠿⣯⣍⠫⠭⠍⠹⠯⠽⠿⠿⠿⠿⠛⠟⠶⠶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢢⣿⣿⣿⣦⡠⠤⠒⠀⠀⢀⣀⣀
⠰⠶⠴⠤⠤⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠨⣿⢿⣿⣶⡖⠒⠒⠚⠂⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢐⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠺⢿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠂⠀⠀⠉⠁⠀⠀⠨⠥⠀⠀⢀⢀⣈⣉
⠐⠛⠋⠉⠉⠉⠀⠀⠀⠀⠉⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠈⠉⠀⠀⠀⠀⠐⠃⠙⠛⠛⠛
⢶⣶⡶⠶⢶⠦⢤⣤⣤⡤⠤⠤⠤⠤⠤⢤⣤⣀⣀⠀⠀⠀⢀⣀⣠⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⢠⣤⣤⣄⡀⢄⣀⣐⣲⢶⣶⣖⡀⢰⣶⣶⣶⣶⡶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⢀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⣿⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⠿⠛⣛⣛⠻⠿⢿⣿⣽⣿⣿⣭⣿⡂⠀⠈⠉⠻⢿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⠿⠿⠿⠿⠻⠟⠛⠋⠁⠙⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠂⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣟⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣀⠀⠙⠋⡿⠟⠿⡿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣤⠴⢶⣦⣶⣄⣠⣤⣝⣻⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣯⣥⣤⣀⠘⠾⣽⣬⣤⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⢿⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠁⠀⠈⠐⠉⠙⠛⣿⢿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣷⣾⣯⣭⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠒⠒⠶⣶⣭⣭⣟⣻⣯⣵⣶⣤⣻⣝⢿⣿⣿⣿⣿⡻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣻⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿
⠐⠒⠒⠒⠚⠲⠚⣿⣶⣦⣽⠷⢶⣤⡈⠙⢽⣿⣿⣿⣿⣿⢟⣉⠉⠀⢈⡛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣾⣋⣉⣡⣾⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣚⣁⣀⡀⣾⣿⣦⣌⣉⣤⣤⣤⣀⣀⡉⢿⣿⣿⣿⣿⣷⣦⠤⠭⠹⢿⣿⣿⣯⣤⣤⣌⡛⠿⢿⣿⣾⣿⣿⣥⣤⣶⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠛⠛⠛⠋⠋⠀⠙⠛⠉⠻⡿⢿⣿⣿⣿⣿⣶⠤⠤⠀⠀⡤⠿⠿⠿⣿⣿⣿⣿⣷⣶⣶⣾⣭⣟⣛⣛⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⢦⣶⣶⡿⣿⣿⣶⣤⣮⣥⣤⣤⣀⠀⠀⠀⢠⣤⣶⣶⣦⣬⣤⣤⣤⣭⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣀⡀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⡈⠙⡄⠰⠀⠀⠹⠋⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⡴⡊⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣨⣛⡳⠇⠀⠑⠅⠠⠈⠀⡠⠀⣀⠀⠀⠀⠀⡀⠀⢠⠀⠈⢉⡐⠠⠀⠐
⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⡼⠉⠳⢦⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠀⠀⢠⠄⠀⠀⣙⠃
⠀⠀⠀⠀⠀⢀⡟⠀⠀⢑⠳⠁⠀⡀⣼⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⣷⡐⠇
⠀⠀⠀⠀⠀⠸⠈⠀⠢⠅⠀⢀⠊⡨⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⡄⠀⠈
⠀⠀⠀⠀⡰⠲⡈⡓⢚⣦⣤⣖⠫⡷⣇⠀⠀⠐⠀⠀⠀⠀⠀⠐⡀⠠⢸⠀⠀⠀⣴⣀⡹⣰
⠀⠀⠀⠼⠄⢀⣀⡈⠁⠀⠓⠠⠊⡹⢁⠀⠀⠀⠠⠀⠀⠀⡚⢄⣇⡀⣐⣐⠖⢀⢚⡛⢃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠂⣴⡀⠀⠀⠀⠀⠂⠈⠁⠀⠘⢄⠑⢄⠈⡅⠀⣀⠼⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀⠈⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⡐

⠀⠀⠀⠀⠘⠀⠒⠑⠚⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠐⠒⠒⠊⠓⠀⠒⠂⠃⠂⠐⠒⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣷⣷⣷⣷⣧⣀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣶⣤⡇⠀⢀⣀⠀⢻⡿⢻⣿
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠒⠀⢰⣾⣿⣽⡿⠴⣿⣯⣯⣶⠀⢰⣤⣶⣶⣶⣾⣯⣽⣽⣧⣤⣼⣿⠀⣬⣷⢾⣭
⠀⠀⠀⣟⡇⠀⠀⠀⠇⠀⠀⠀⠈⠉⠉⠁⠀⠈⠀⠈⠉⠉⠉⠉⠀⠉⠉⠉⠉⠀⠈⠁⠀⠉⠈⠉⠉⠀⠉⠉⠉⠉⠉⠀⠉⠉⠈⠉
⠀⠀⣴⣿⣿⡄⠀⢠⣧⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠒⠀⠀⠀⠒⠀⠀⠀⠒⠀⠀⠀⠒⠀⠀⠀⠒⠀⠀⠀⠀⠶
⠀⠀⣿⠛⢻⣷⠀⢨⣯⠀⠀⢺⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⣀
⠀⠀⣿⣶⣾⣿⠀⢸⣿⡇⠀⣿⣿⡇⠀⠀⠀⠀⡶⠀⠀⠀⠀⢠⣧⠰⠿⢄⣭⡀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢠⡄
⠀⠀⣿⠉⠉⡇⠀⣶⡦⡆⠀⣿⣿⡇⠀⠀⠀⡀⠁⠀⠀⠀⠀⢸⣿⣓⣲⣸⣿⡇⠀⠀⠀⠀⣿⠁⠉⠈⠁⠁⢹⡏⠀⠀⠀⠀⠀⠒
⠀⠀⣿⠛⠛⡇⠀⠸⠇⠗⠀⠻⠟⠃⠀⠀⠠⠁⠉⠀⠀⠀⠀⠸⠿⠿⠿⠼⠿⠃⠀⠀⠀⠀⠿⠆⠀⠀⠀⠀⠺⠷⠀⠀⠂⠀⠰⠿⠦
⠀⠀⣿⠷⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣿⣶⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣵⠀⠀⠀⠀⠀⢀
⠀⠀⣿⣥⣭⡇⠀⣤⣤⣀⣀⣀⣀⣤⣠⣤⣀⣀⣀⣠⣄⣤⠀⣀⠀⢀⡀⠀⡿⢸⠀⠀⢠⣄⢀⣸⡀⠀⣤⠀⠀⠀⠀⠀⡀⣀⡀⢀⡀
⠀⠀⣿⣭⣭⡇⠀⣀⣸⣇⣦⣷⣸⣴⣆⣼⣴⣰⣤⣿⣀⣀⠀⠉⡟⡯⣿⠛⠛⠛⢻⠛⢻⢿⠉⠉⠉⢉⡿⡟⠛⠛⠛⠛⠓⡏⡟⣿⠁
⠀⠀⠛⠛⠛⠃⠀⠛⠙⠛⠋⠙⠛⠛⠀⠛⠛⠃⠛⠛⠋⠋⠀⠙⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠩⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠾⢿⠟⢿⡟⠁
⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣶⣂⣽⠇⠀⠀⠀⠀⠀⠀⣀⠠⠄⡄
⠀⠘⡀⠀⠈⠻⣦⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣶⣚⣿⢯⡏⣶⣾⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠒⠄
⠀⠀⠈⠐⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣽⣯⡧⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠠⢀⡀⠀⠀⠀⡄⠠⠴⣻⠽⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠒⠂⠀⠀⠀⠁⠈⠉⢀⣠⠀⠀⠀⡄⢀⣀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⠰⢀⡄⢂⢀⠀⠀⠀⠀⠀⠀⠠⠀⠐⢊⡩⢖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠤⠀⠔⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠔⠀⠠⠀⠀⠈⠁⠀⠀⠀⠀⠌⠀⠉⠁⣠⠴⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠠⠀⠀⠈
⠀⠀⠀⠂⠁⠢⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⢚⡩⠴⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠁⢀⡀⠄⠊⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⢒⠬⠥⣤⠖⡋⠁⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣯⠀⠀⠀⠀⢰⡀⠀⠀⡀⢐⣈⣽⣾⠇⡀⠰⣽⡦⢠⡂⢛⣙⣆⠀⠀⠀⠈⣸⠃⠀⠀⠀⣹⣟⠄
⠀⠀⠀⠀⠀⠀⠂⠉⠉⠀⠒⠒⠒⠉⣕⠻⢃⢬⣻⡿⠟⢸⡏⢑⡞⠍⢷⠽⣾⠻⡟⠋⠓⣔⠀⠠⠭⠒⠒⠒⠚⠉⠋⠓⠒⠚⠛⠛⠛
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢁⠀⠙⢻⣧⣤⠓⢰⠌⣿⡕⢉⠝⡃⠀⠀⠀⠀⣝⠖⠁⣄⢀⣀⣦⠠⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠄⠀⠈⠁⡔⠱⠺⣂⣠⡇⠘⢠⡇⠀⠀⠀⡴⠁⠀⢀⣯⡑⠋⡉⠂⠀⢠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠁⠈⠛⠋⢒⡊⠀⠘⠛⠒⠉⠁⡉⠁⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⢰⠖⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠲⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢴⡟⢹⡦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣤⠞⠋⢁⣀⣤⡇⢸⣧⣀⡈⠙⠳⣤⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣠⠟⠁⣠⠞⠋⠁⠀⣻⣞⠁⠈⠙⠳⣄⠈⠻⣄⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⡾⠁⠀⣤⠞⠋⠉⠉⠙⠳⣤⠀⠘⢷⡀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣼⠁⠀⡾⠁⠀⣠⠞⠳⣄⠀⠈⢷⠀⠈⣧⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣛⣛⣛⣻⢼⡇⠀⣀⣽⣦⣴⣏⣀⠀⢸⡇⣾⣛⣛⣛⣛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣯⠉⢻⡁⠀⢷⡞⠋⠀⠀⠀⠀⠙⢳⡾⠀⢈⡟⠉⣽⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠈⢷⡀⠈⠻⢦⣄⣀⣀⣠⡴⠟⠁⢀⡾⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠙⣦⡀⠙⢦⣄⡀⠀⣩⣭⠀⢀⣠⡴⠋⢀⣴⠋⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠛⢦⣄⡈⠉⠛⡇⢸⡟⠉⢁⣠⡴⠛⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⣇⢸⡗⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠦⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠆⠱⠸⠊⠸⠀⠀⠃⠸⠸⠺⠀⠇⠇⠱

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠠⠚⡀⠀⢀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠤⠀⠀⠈⠒⠀⠀⠐⠂⠀⠀⠰⡀⠠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠉⠁⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⣨⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠁⠀⠀⠀⠀⠀⠈⢄⠀⣀⠴⠊⢉⣠⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⡋⠁⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠋⠁⢠⣥⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠉⠳⣄⠀⠀⠀⠀⠀⠁⠠⡀
⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⢀⠀⠘
⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⢀⡴⠞⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠘⠁
⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⡴⠋⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⢀⡎⣸⠃⠀⠀⠠⠄⠀⠃
⢀⡟⠀⠀⠀⠀⠀⠀⠀⢰⠃⣰⠞⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⡞⢰⡏⠀⠀⠀⠠
⢸⡇⠀⠀⠀⠀⠀⠀⠀⡜⣰⠋⠀⠀⠀⠀⠀⠀⢠⠀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⢀⣟⢠⡟⠀⠀⠊⠉⠈⠁
⣾⣇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡠⠄⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⢀⡞⢹⣾⠃⠀⠰
⣿⢿⡀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⢀⣼⠏⠀⠀⠀⠀⠀⠀⢠⠞⠀⣼⠃
⢻⠘⢷⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡟⠁⠀⠀⠀⠀⠀⠀⣰⠋⠀⡾⠃
⠸⣦⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⢀⣼⠁⢀⡾⠁
⠀⢻⡄⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠶⠶⠶⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠨⢠⡟
⠀⠀⢻⡄⠀⠘⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣼⠏
⠀⠀⠀⠹⣆⠀⠀⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⡼⠃
⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⣤⠟
⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠀⠉⠓⠶⢤⣤⣄⣀⣠⣤⠤⠖⠛⠁⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⣤⣤⣤⣤⣤⣤⣤⣤⠶⠚⠉
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣏⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡦⢤⣤⣀⡀⣀⡴⠛⠛⠋⠀⠀⠀⠀⡀⠀⢀⣀⠀⠀⠀⠀⢀⡀⠀⢀⢀
⠀⠀⠀⠀⠀⣠⣾⣿⣈⣿⡷⣾⣿⣽⣿⣿⡧⠈⠀⠀⠀⠀⠀⠃⠀⠰⠶⠄⠠⠶⠖⠶⠺⠅⠶⢿⡿
⠀⠀⠀⠀⢰⣿⢿⣭⢋⡀⠐⠀⠧⢢⠙⣿⣿⣿⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠚⢹
⠀⠀⢀⣠⣿⣿⠉⡗⢛⠄⠀⡀⠀⢈⡁⢸⣿⣶⣿⡇⠀⠀⠐⣾⠀⠐⣦⠛⠀⠀⠘⠃⠀⠀⠀⢀⣿
⠀⠠⣿⣿⣿⣟⣶⣧⡀⠄⠚⠡⢀⠀⣡⣾⣿⣿⣟⡇⠀⠀⠀⣿⠀⠀⣿⡇⢠⡀⠀⠀⣠⡄⠀⠸⢻
⠀⠀⡙⠛⢿⣿⢿⣧⡔⠤⣶⢆⣄⡧⠈⢿⣿⠹⠟⡇⠀⠀⡸⡇⢀⠀⣿⡇⠈⠉⠀⠀⠈⠁⠀⠀⠘
⠀⠀⠈⠀⠘⣿⣞⠙⣷⡿⠀⢀⠀⠁⠂⣸⣿⣆⣃⡇⠀⠀⠐⠷⡦⠶⠿⠷⠀⠀⠠⠦⠀⠀⠀⠈⠂
⠀⠀⠀⠀⡀⠹⣮⣗⣻⣧⣀⠐⢂⣇⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤
⠀⠀⠀⠀⣷⡀⠘⢿⣿⣿⣿⣿⣿⣯⣹⠏⢙⢛⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⢿⣟⠠⣀⠀⠉⠙⠛⠋⠉⢀⠀⢸⣋⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⣺⣿⠿⣿⡿⣦⢶⣶⡶⢾⣏⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢴
⠀⠀⠀⠀⢩⣥⣆⢿⣷⣴⣽⣟⢲⡄⢉⠀⢰⣾⣷⡆⠀⠀⡖⢂⣶⢰⠀⡆⡖⡶⣦⣖⠠⡆⠀⢀
⠀⠀⠀⠀⢰⣮⣉⣽⣿⣉⣹⣿⣈⣑⣾⠀⢈⣴⡛⡇⠀⠀⠳⡌⣽⢾⡠⡧⡧⢿⠏⣯⠸⠇⠀⠘⣧
⠀⠀⠀⠀⢸⡿⠿⣿⣷⠿⢻⣿⢿⠋⢠⣤⡀⣿⣿⡇⠀⠀⣀⣀⡀⣀⣀⢀⢀⣀⣀⡀⡀⢀⡀⢠⠹
⠀⠀⠀⠀⢸⡇⣿⢿⣿⣷⣿⣷⢾⡦⠀⣸⠳⢻⣏⡁⠀⠀⣟⣿⣿⡏⣿⢸⢸⣉⣈⡀⠉⠘⠃⠀⢰
⠀⠀⠀⠀⢸⣧⣽⣿⣿⣭⣻⠟⢁⣠⡞⠋⢰⡬⡿⠁⠀⠀⠟⠟⠛⠳⠟⠘⠚⠟⠛⠇⠛⠚⠂⢰⣿⡆
⠀⠀⠀⠀⢹⡟⡿⣿⣷⠟⠁⣴⢿⡿⢆⣀⣰⠟⠁⠀⠀⢠⢤⠄⣤⠤⡤⠠⣤⣤⣤⡤⠀⡤⠀⢸⠻⡇
⠀⠀⠀⠀⢱⢶⣿⣻⡟⠀⣼⠷⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠚⠘⠙⠘⠛⠊⠉⠉⠉⠁⠉⠉⠀⢸⣀⡇
⠀⠀⠀⠀⢸⣷⣶⡿⠇⢀⣅⣀⠁⠀⠀⠀⣲⠆⢐⢒⣖⡒⣲⠒⡒⢲⣶⣶⡖⢲⣂⠀⠀⠰⠆⠾⠟⠃
⠀⠀⠀⠀⢸⡟⣃⠀⠰⠾⠿⠋⣀⡀⠀⣀⣀⡈⡀⠀⠈⠁⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠸⠟⠻⠷⠖⠤⠶⠾⠿⠿⠦⠬⠿⠦⠥⠾⠶⠤⠬⠷⠿⠿⠷⠦⠼⠿⠿⠿⠿⠦⠼⠿⠿⠇

⣿⣿⠟⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠻⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⠟⠉⠁⠀⠀⠈⠙⠻⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡤⠤⠤⠤⠤⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣛⣋⣛⣙⣻⣿⣿⣿⣇⡀⠀⠀⢀⣼⣿⣿⣿⣟⣛⣛⣛⣛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⡏⣿⠉⢻⠉⡏⠡⣽⡇⣿⢉⣤⣬⡏⠹⠏⢹⣥⢡⣼⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⡇⣿⢸⣆⢀⡟⠶⢨⡇⣿⡘⠷⢀⡇⣰⣆⢸⣿⢸⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⣋⣭⣤⡶⠶⣤⣭⣍⡓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣩⡴⠟⠉⠀⠀⠀⠀⠀⠀⠉⠛⢶⣌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠹⣇⠀⠀⠀⠀⠀⢿⣿⣿⡿⠃⠀⠀⠀⠀⣸⡏⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡹⣧⡀⠀⠀⢀⡠⠧⠼⢆⡀⠀⠀⢀⣴⠟⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⡛⢶⣴⠋⠀⠀⠀⠀⠘⣦⡶⠟⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢿⠀⠀⠀⠀⠀⠀⣿⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣄⡀⢀⣀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠾⠤⠴⠖⠲⠦⠬⠷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠋⠉⠀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⠀⠉⠙⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢀⣠⡴⢞⣛⣭⣭⣷⠶⠶⠶⢶⣾⣭⣭⣛⡳⢦⣄⡀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣠⠾⣫⣵⠾⠛⠉⠡⠀⠀⠂⠀⠀⠀⠀⠀⠉⠙⠛⢷⣮⣝⠷⣄⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠃⢀⣴⢟⣵⡿⠛⠁⠀⠀⠀⢀⠀⠤⠀⠀⠀⠀⠠⠀⡀⣠⡄⠀⠈⠙⢷⣮⡻⣦⠀⠘⢦⡀
⠀⠀⠀⠀⠀⢀⠞⠀⢠⡞⣡⡿⠋⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠈⢄⠀⠀⠀⣸⡿⠁⢀⠀⠀⠀⠙⢿⣌⢷⡄⠀⢳⡀
⠀⠀⠀⠀⢀⠎⠀⣰⡟⣼⡟⠠⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⢀⡜⢡⠃⠀⠀⠑⢄⠀⠀⠈⢻⣦⢻⣆⠀⢱⡀
⠀⠀⠀⠀⡞⠀⢰⡟⣼⡟⠀⠀⠀⠌⠀⠀⠀⠀⣄⠀⠀⠀⠀⢸⡇⢠⠞⢰⠏⣠⠀⠀⠀⠀⠂⠀⠀⠀⢻⣧⢻⡄⠀⢧
⠀⠀⠀⢸⠃⠀⣿⢰⡿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢳⣦⡀⢀⣼⡷⠋⢀⣿⡾⠃⠀⠀⠀⠀⠀⢂⠀⠀⠀⣿⡄⣷⠀⠘⡇
⠀⠀⠀⣿⠀⢸⡇⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⢅⡀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⣇⢹⡆⠀⣷
⠀⠀⠀⡏⠀⢸⡇⣿⡇⠀⠀⠰⠂⠀⠀⠤⠤⠶⣶⣶⣿⣿⡏⠀⠀⢹⣿⣿⣶⣤⣤⣤⡤⠐⠐⠂⡅⠀⠀⢸⣿⢸⡇⠀⣿
⠀⠀⠀⣧⠀⢸⡇⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠓⠦⣤⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⡿⢸⡇⠀⣿
⠀⠀⠀⢻⠀⠘⣧⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠃⢀⣼⡟⠋⠻⢷⣄⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣾⡇⣼⠃⠀⡟
⠀⠀⠀⠸⡇⠀⢻⡄⢿⡆⠀⠀⠈⠄⠀⠀⠀⠠⠞⢡⡏⣠⠎⢹⡇⠀⠀⠀⠙⠳⠀⠀⠀⠀⠀⠁⠀⠀⣰⡿⢠⡟⠀⢸⠇
⠀⠀⠀⠀⠹⡀⠈⢷⡘⢿⣄⠀⠀⠀⠂⠀⠀⠀⢀⠟⣰⠋⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⣰⡿⢁⡾⠁⢀⠟
⠀⠀⠀⠀⠀⢳⡀⠈⢷⡌⢿⣦⡐⠀⠀⠈⠢⠀⣜⡞⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢡⡾⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠳⡄⠀⠻⣦⡙⢿⣦⡀⠀⠀⢸⠟⠀⠠⠄⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⢀⣴⡿⢋⣴⠟⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠻⣦⡙⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⢋⣴⠟⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠛⠷⣤⣉⠛⠿⠷⣶⣦⣤⣤⣤⣤⣤⣶⠾⠟⠛⣉⣤⠾⠋⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣀⠀⠉⠛⠷⠶⣤⣤⣤⣤⣤⣤⣤⣤⠶⠾⠛⠉⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣶⣷⣶⣶⣿⣶⣷⣶⣾⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣶⣶⣾⣿⣿⣿⣷⣾⣶⣷⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠒⠒⠚⠑⠚⢒⡗⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⣰⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣯⣿
⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⣼⡟⠛⠛⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⡆
⠀⠀⠀⣾⣿⣹⡀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⣤⡄⣄⢹⣿⣻⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣄⠠⡀⣤⣀⠀⢀⡠⠀⣀⣀⣶⠀⠀⠀⠀⡀⣠⠀⠀⣰⣿⣿⣿⣿⣿⡟⢡⣠⣤⡄⢀⡄⠀⠀⣴⠀⢠⡦
⠁⠅⢻⣸⣿⣉⣿⣿⣿⣿⣿⣿⣿⠀⣲⣦⣀⣤⣿⠾⠧⠀⠂⠰⠆⢷⡄⠙⣽⠙⠲⠶⠤⠰⢷⣿⠁⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⢻⣾⣶⣾⣿⣿⣤⣼⢇
⣇⣴⣾⣿⣼⣋⣿⣿⣿⣿⣿⣿⣿⣚⣟⡤⠷⠿⡇⠰⠄⡻⣂⣲⠒⡶⡇⣰⣿⠀⣿⣿⣀⠿⠨⣏⣁⢠⣼⣿⢻⢿⡏⠉⢸⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠃⣐⣿⣿⣿⣴⡟⣿⣿⣿⣿⣿⣿⢀⠀⠀⠴⠤⡵⣴⣴⠓⠒⠛⠁⠛⢡⠉⢸⠀⠄⠹⢿⢿⣻⣿⣿⣿⣿⣿⠀⠈⠀⠀⣾⣿⣿⣷⢼⣿⣿⣿⣿⣿⣿⠿⠿
⣭⡼⣿⣽⣿⣿⠓⣛⣛⣻⣿⣭⣭⡼⢅⠀⠈⠹⠇⣸⠀⠉⠲⡆⠒⢲⣼⡃⣿⠁⣟⡇⠀⢠⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⡇⠀⣸
⠶⡃⠘⣿⡿⣷⡙⠉⢩⣽⣿⣷⡶⠀⢻⠀⠰⣦⣠⠚⠫⢤⡤⠶⠂⣾⣿⡆⣿⠃⣿⣇⣀⠸⢿⣿⣿⣿⣿⣿⢀⣴⣤⣶⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⠃⣰⣿
⠶⠡⢁⡘⠛⠻⠄⠀⣿⣿⡿⠿⠛⠀⠀⠀⠀⡆⠀⠀⣸⣷⣤⣤⣧⡏⠹⠇⣾⠀⣿⣿⣻⡇⣒⣖⣷⡦⠉⢙⣟⣿⢿⣿⠿⣣⣼⣶⣃⢈⡿⠿⠿⢹⡞⣿⣿
⠛⡀⠃⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠹⠿⠿⡳⠤⢄⡐⠉⠲⡆⢀⢉⠁⠿⠿⢅⡀⠀⠙⠹⣗⠐⠛⠸⠟⣿⣿⣿⣿⣿⣻⣶⣤⣥⣉⠉
⠆⠰⠔⠰⠀⡄⠀⠍⠊⠀⠲⠄⢀⠀⢠⡄⠶⠰⣆⠀⢀⡤⢿⠀⠙⠶⣾⣿⣷⣦⣀⠙⠃⠘⠛⢯⣀⣈⣁⣢⣀⣈⣉⣡⣿⣾⣿⣿⣭⠉⣹⣿⣿⣿⣷⣞⣷
⣿⣄⠒⠂⢂⡿⠜⢲⡀⡀⠀⠀⠈⠀⠀⠰⣶⠀⠶⣖⠂⠀⠀⠒⠀⠀⠈⠛⣿⣿⣿⣿⣆⢒⡒⣯⠁⠨⣥⡼⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⢿⣿⣿
⢿⣿⠄⠒⠛⠈⠖⣾⠵⢰⡀⠒⡀⢀⠀⡄⠙⢣⣴⢿⣷⣦⣀⠒⠙⠐⠈⠀⣿⣿⣿⣿⣗⢒⣲⣆⠸⢸⣿⣿⡿⢾⡿⣾⣿⣿⣿⣧⣿⣿⣿⣿⡿⣿⡟⠿⣿
⣿⣿⣶⣤⡤⢤⠤⣥⡥⣔⠄⢀⣵⣾⣯⢻⣿⣿⣿⡿⣿⣿⣁⣬⡀⠀⠀⠀⢿⣿⣿⣿⣟⣴⣿⣿⣿⣉⣙⣳⣿⣶⣙⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⡀⠀⣰⣯⣿⣯⣫⢢⣿⡿⣿⣾⣷⣼⡿⠷⢿⣿⣿⡿⣷⠠⢠⣀⣾⣮⣿⣿⣿⣿⣿⡷⡿⢿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣿⣷⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⢶⡖⡴⢦⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢲⢺⡐⡇⢸⢂⠇⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣴⡆
⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠑⠳⠞⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛
⠀⠀⠀⣹⣿⡇⠀⢸⣿⠀⢸⡇⠀⠀⢀⣭⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⢰
⠀⠀⠀⣿⣿⡇⠀⢸⣿⡶⣾⣿⠀⠀⣾⢿⢿⡇⠀⢀⠀⠀⠀⠀⠀⣶⣶⣽⣰⣴⡄⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣞⣵⣼⣀⣀⣀⣀⣀⣀
⠀⠀⠰⣿⣿⡟⠀⢸⢿⣇⡿⠃⠀⢸⣟⠋⢻⣿⠀⠈⠁⠀⢠⣜⡟⠁⣋⣉⣉⢛⣿⣧⣄⠀⠉⢹⡏⠍⡭⠉⣭⢉⠍⡍⠭⡭⡭⢹⡏⠍
⠀⠀⠀⠻⠟⠃⠀⠸⠿⠏⠿⠿⠀⢸⣿⣶⣾⣿⠠⠿⠷⠄⠸⠿⠿⠿⠿⠀⠿⠾⠿⠿⠿⠀⠿⠿⠿⠿⠹⠿⠿⠏⠹⠿⠿⠟⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⡏⠉⠉⡏⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⡇⠘⡿⠿⠿⡟⠀⢺⢸⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣶⣾⠀⠀⠀⣤⣤⠀⠀⢘⡃
⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⢸⣭⡇⠰⣧⣤⣴⡗⠀⣸⢽⠀⠀⠀⢀⠀⠀⠀⣦⣀⣴⣶⣶⣴⣶⣿⣧⣿⣟⣽⣀⣀⣀⣿⣿⠀⠀⢸⡅
⠀⠀⠀⠀⠀⡀⠀⢀⠀⢀⡙⠛⠋⠀⣇⣀⣀⡇⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⣽⣇
⠀⠀⠀⠀⢀⣭⢠⣾⡆⢈⡅⠀⠀⠀⡿⠛⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢸⣿⡇⠤⠄⣿⣿⢰⠲⠀⡿⠿⠿⡇⠀⠀⠀⢠⣄⣠⣄⠀⣗⣻⠀⠀⣠⣄⣤⣇⣀⣠⡄⠀⠀⠀⢀⡄⣤⣀⣠⡄⢀⣙⣉⡀
⠀⠀⠀⠀⢸⣻⣗⣿⡓⣿⣿⢸⢘⠀⣷⣶⣶⣿⣶⣶⣖⣀⣻⣹⣺⣉⣉⣉⣉⣉⣟⣏⣀⢈⣅⣟⣫⣉⣉⣉⣉⣉⣗⡏⣗⡂⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠁⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠁⠈⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠉⠁⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣧⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⠛⢁⣼⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠁⠀⢀⣤⣶⣶⣾⣷⣶⣶⡄⢀⣴⣿⣿⠟⢉⣉⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⢀⣴⣿⣿⠟⠛⠛⠛⠛⢋⣴⣿⣿⠟⠁⠀⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⢀⣿⣿⡟⠁⠀⢀⡤⠀⣰⣿⣿⠟⣡⣴⡄⠀⠘⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠀⠀⣴⣿⠠⣾⡿⠟⠁⠀⢿⣿⣧⠀⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠀⠀⠹⣿⣦⣬⣤⠞⠀⠀⣾⣿⡟⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆⠀⠈⣿⣿⣧⡀⠀⠈⠙⠋⠁⠀⢀⣴⣿⣿⠃⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⠈⠻⣿⣿⣶⣤⣤⣤⣤⣶⣿⣿⡟⠁⠀⢠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⡀⠀⠈⠛⠻⠿⢿⠿⠿⠿⠛⠁⠀⢀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁




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

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

⠑⢄⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠘⡄
⠀⠀⠑⢄⣀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⠤⠐⠋⠁⠀⠀⠀⠀⠀⠀⠹⢆
⠀⡠⠔⠊⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠉⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⢀⣀⠤⠖⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤⠤⠀⠀⠠⢄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⢸⠄
⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⣀
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢀⣀⠤⠔⠺⠁⠀⠀⠀⠈⠑⠒⠤⣀⡀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠒⠊⠉⢇
⠀⠁⠀⠐⠀⢀⣀⣀⠤⠤⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠔⠐⠚⠂⠉⠉⠀⠉⠐⠠⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠓⠠⢀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⡀⠀⠀⠐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠢⠔⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⡀⠄⠐⠈
⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⣀⡀⠀⠄⠳⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠍⠄⠊⠉
⠑⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⢀⠁⢐⠂⠀⠤⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂
⢼⠒⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠌⠀⠈⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⡀
⢘⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢤⠀⠀⠂⠀⠀⠁⠀⠀⠂
⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠠⠄⠀⠀⢀⠙⡶⠢⠤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣤⣤⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⣿⣿⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠛⠛⠉⠉⠙⠛⠛⠳⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⢉⣀⣤⣶⣶⣿⣿⡇⢸⣿⣿⣶⣶⣤⣄⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣶⣄⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣿⣿⣿⣿⣧⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⢠⣾⣿⣿⣿⣿⣿⣧⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣉⣀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⡇⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣏⣉⣉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠉⢉⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢿⣿⣿⣿⣿⣿⡿⠃⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢻⣿⣿⣿⣿⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⡟⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣈⠙⠛⠿⢿⣿⣿⡇⢸⣿⣿⡿⠿⠛⠋⣁⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⢦⣤⣤⣀⣀⣀⣀⣤⣤⡴⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉