Fe Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣜⣿⣿⣿⣿⣇⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⣠⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣝⠻⣿⣿⣿⣿⣿⣿⣟⡀⠀⣴⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⣿⣿⡿⠿⠿⠿⢷⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⣼⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⠟⣋⣤⣿⣿⣿⣿⣿⣿⣿⣟⠻⣿⣿⣿⣿⠀⠀⠀⠀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣭⣶⠿⢿⣮⣝⠻⢿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⡇⠀⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⣉⣤⣶⣦⣌⠛⢿⣦⡙⠃⣿⣿⣿⣿⣿⣿⣷⣼⠀⠀⠀⠀⣼⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣉⣴⣾⣿⣿⡿⣿⣿⣷⣄⠙⢿⣦⡻⣿⣿⣿⣿⣿⡇⠛⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢡⣾⣿⣿⣿⠟⢁⣴⣤⡙⠻⣿⣷⣄⠙⣿⣌⠻⣿⣿⣿⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⣰⠏⣴⣿⣿⣿⠟⣡⣾⣿⣿⣿⣿⣷⣌⡙⢿⣷⣌⢻⣧⡈⠻⠇⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⣼⢃⣾⣿⣿⡿⢃⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣷⡙⣷⡀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⡼⣡⣿⣿⣿⡟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣮⣷⡄⢠⣿⠏
⠀⠀⠀⠀⡼⠱⠿⠿⠿⠟⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⣮⡿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⠉⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣵⣶⣮⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠁⠀⠀⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⢶⣿⣿⣿⣷⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠓⠒⠒⢲⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡸⢠⣬⣭⣭⡝⢇⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣗⣒⣶⣒⣺⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢬⣯⣛⣭⡅⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡉⠀⢉⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⢿⣿⡿⢢⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡸⠙⠛⠉⢗⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢧⣶⣭⣶⡼⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡜⠦⣤⠴⢣⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢻⠛⠁⠀⠀⢻⣛⠛⣛⡿⠁⠀⠈⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣦⣠⣴⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠈⠻⠿⠟⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠿⠟⠋⣿⠧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡿⠿⣿⠛⠿⠟⠿⠿⠛⠓⠂⠀⠀⠶⠿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣦⡄
⠀⣀⣀⣰⠃⠀⠈⠛⠉⠐⠲⠶⡿⢿⣯⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣣⣤⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢿⣿⠟⠁⠀⠀⢨⣿⡇
⠀⠀⠢⠂⠀⠀⠀⠀⠀⠴⢲⡾⢟⢾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⡇
⠀⣀⠀⡁⠀⠀⠀⠀⠀⢠⡾⠁⠀⠁⠀⠀⠀⣶⣀⣠⣄⣤⣤⢤⣆⠀⡀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⣿⣷⡀⠀⠀⠀⢀⣴⣯⠒⠀⠀⣴⣠⣤⣤⣭⣽⣿⣿⣿⣯⢀⣯⣵⣴⢶⠓⢦⣀
⠀⣿⡿⠃⠀⢸⣿⣿⣿⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣟⣠⣿⡿⣿⣿⣏⣤⣴⣿⣶⣦⣄⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡀
⠀⣯⡶⣲⣾⣾⡿⣯⡇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣷⣿⣿⣿⣻⣿⣿⣿⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠛
⠀⣯⣵⣿⣿⠃⠀⠁⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡤⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡶⠒
⠀⡿⣿⡿⣿⣦⣀⣀⡀⠀⢸⣿⣿⡿⠻⣿⣿⣿⢿⣿⣿⣷⣷⣿⣿⣿⣯⡙⣿⣿⣿⣿⣿⣿⣿⣟⣛⣡⣤⣠⣀⡀
⠀⣽⣿⣇⣼⡟⠛⠃⠀⠀⢿⣿⣧⠠⣴⣾⣯⣭⣭⣵⣽⣿⡿⢿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣶⣶⣶⣶⡀
⠀⢀⠀⠷⢿⠇⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣧⠻⣼⣿⣿⣿⣿⣦⣝⣍⠁⢻⠻⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⡀⠀⠀⠀⠀⠀⠛⠷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣛⣿⣿⣿⣿⣿⡿⣢⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿
⠀⠠⠀⠀⠀⠀⠀⠰⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠈
⠀⣿⣦⣀⡀⠀⠀⠀⢰⣾⣿⣿⣶⣿⣿⣻⢿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠈⠻⣿⡿⠃⠀⠀⠐⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠺⣽⠆⣠⣄⡈⣀⣀⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣬⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⣷⢀⣶⣿⣿⣿⣿⡿⡟⠁⠀⠀⠀⠘⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣍⣿⣿⣿⡏⠛⣿⠿⣿⣿⠇
⠀⢁⣼⣿⣿⣿⣿⣷⣶⣶⣤⣤⣄⣠⣀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠉⡉⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⡟⠛⠁
⠀⢀⢰⠀⡆⢀⠀⠀⠀⢸⣿⠀⠀⠀⢀⡀⡆⢰⢀⡀⠀⠀⠀⠀⠀⢀⢠⡄⡄⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⡄⣤⢀
⠀⢸⢸⡄⡇⢻⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⡇⣸⢸⠃⠀⠀⠀⠀⠀⢹⣾⣷⣿⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣷⣿⣿
⠀⡼⠎⠇⠻⠘⠀⠀⠀⠈⠉⠀⠀⠀⠘⠱⠃⠟⠼⡤⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣆⣀⠀⠉⠁⢀⣀⣾⣿⣿⣿⣿⡄
⠀⠈⠢⠀⠑⠚⠛⠛⠛⠢⢠⠚⠛⠛⠛⠓⠉⠀⠈⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣶⣮⣝⢆⢞⣭⣶⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠏⠀⠘⠿⠿⠿⠛⠉
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤
⠀⢀⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠤⠖⠒⠒⢒⠊⡩⠀⠠⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠃
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣿⢶⡇
⠀⠀⠀⠀⠀⠂⠀⠀⣀⢀⡀⡀⢀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⠀⠐
⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠁⠀⠀⠀⠀⠀⠀⠐⠦⠤⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈
⠀⠀⢴⣶⣶⣦⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢲⣶⠛⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⢠⡄
⠀⠀⠙⠿⠿⠟⠃⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁
⠀⠀⠀⠀⠀⣀⡀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢳⢓⠏⢩⢣⡆⣀⢎⢀⣵⡇⠀⣀
⠀⠀⠀⢻⡍⣩⡏⠌⣿⠛⠗⠸⡇⠀⠀⠀⠀⠸⢀⡼⠆⣇⡶⠘⢀⠟⠇⣸⠃⠇⠀⣿
⠀⠀⠁⠀⠁⠈⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣷⣄⠀⠀⠀⠄⡐⡀⠀⠀⡀
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⡆⠀⠀⠄⠀⡇
⠀⠶⠾⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠙⠁

⠀⣠⣶⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣶⣄
⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⣿⡇⠀⠀⠉⣿⢢⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣰⠶⣿⠺⠧⣰⢿⡇⣶⡂⠨⠇⠀⠀⠀⢸
⣿⡇⠀⠀⠚⠓⠊⠓⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠋⠘⠓⠛⠀⠀⠉⠛⠃⠚⠛⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣀⣀⣀⣀⡀⠀⣰⣿⣿⣿⡟⠉⠻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣇⣀⣠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣶⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢀⣤⣀⡄⢀⣤⣄⡀⢠⣄⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢨⣿⠋⠁⣿⡇⣿⡇⢸⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠘⠛⠀⠀⠙⠓⠛⠁⠘⠛⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⠀⢀⣠⡀⠀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠳⣦⠀⢸⡷⣯⣰⣻⣇⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠛⠁⠙⠈⠛⠉⠀⠈⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠈⠻⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣾⠟

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠠⡀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄
⠀⠀⠀⣀⠀⢀⢁⠂⠀⠀⠀⠀⢀⠊⠀⡀⢁⡒⠀⠀⠑⡀⠀⠀⢀⡀⣠⣾⣿⡃⣀⠀⠀⠀⠌⠀⡀⣡⣾⡇⣀⠈⠀⠀⠀⣾⣿⣿⠟⠋⣻⣿⣷⡄
⠀⠀⡀⠌⠀⢄⠈⠀⠨⢄⠀⠀⠈⢀⠴⢄⠢⠁⡠⢦⠀⠁⠀⠀⣈⡁⢻⣾⠾⢁⣉⠀⠀⢈⠀⣨⣅⢻⠷⢃⣭⡀⠀⠀⢸⣿⡿⢾⡌⣡⡾⠺⣿⡇
⠀⠀⠁⠀⠀⠄⠀⠆⠀⠀⠁⠀⠈⠀⠀⠀⠁⠈⠀⠀⠀⠁⠀⣾⣿⣿⡎⠁⠀⣿⣿⣷⠀⠈⠜⢿⠿⠃⠀⠻⢿⢟⠄⠀⠈⢿⣀⣠⣿⣿⣄⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠄⢀⣀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⡀⠤⣀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠂⠀⢄⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣤⣤⣀
⠀⠀⠀⠀⢸⢀⣀⡀⢃⠀⠀⠀⢀⠊⠀⠀⢊⢀⠑⡀⠑⠄⠀⠀⠀⠀⢰⡿⡿⢿⡆⠀⠀⢀⠊⠀⠀⣴⣿⢷⡀⠈⠄⠀⢠⣾⣿⣿⢉⢈⠙⣿⣷⡀
⠀⠀⢀⡀⠨⡈⠀⠡⠼⠀⠀⠀⠀⠀⡀⠠⢄⠀⠄⠃⠀⠀⠀⠀⠀⣀⣘⢷⣿⣿⣇⠀⠀⠀⠀⢀⣀⣸⢾⣿⣇⠀⠀⠀⣾⣿⡿⠿⢄⠈⠀⢿⣿⡇
⠀⠀⣏⣀⣀⣈⡄⠀⠁⠇⠀⠀⠀⠐⡒⠒⢒⠆⠀⠀⠀⡀⠀⠀⣾⣿⣿⣷⢹⣿⣿⡄⠀⠠⠀⢼⣿⣿⢸⣿⣿⠀⠀⠀⢹⣿⠀⠀⢐⡇⠀⢸⣿⠇
⠀⠀⠣⢀⣀⠔⢄⣐⠠⠃⠀⠀⠀⠑⠌⠀⠁⠀⠀⡁⠈⠀⠀⠀⠙⠿⠿⠋⠺⠿⠟⠁⠀⠀⠑⠄⡉⠁⠈⠉⠁⠔⠀⠀⠀⠙⢷⣶⣿⣷⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⢸⠀⠀⡟⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢸⠀⠀⡏⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⢸⠀⠀⡇⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⢸⠀⠀⣇⢀⡇⠀⣸⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⠀⠀⣇⢸⡇⠀⣾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⢹⡇⠀⣿⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⢸⠀⠀⡟⠙⡏⠉⢻⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡟⢹⡇⠀⣿⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡇⠀⠀⠀⠀⠀⢸⠀⠀⡇⢸⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸⠀⠀⡇⢸⡇⠀⣾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠘⠶⠶⠇⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠶⠇⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⣤⣤⠀⣶⣶⠀⣀⣀⠀⠀⠀⠀⠀⠀⢙⣿⣿⡇⠀⠀⠀⠀⢀⣴⡶⠛⢻⣿⠉⢙⣿⠶⢄⣠⡾⠋
⠀⠀⠀⢀⣶⡄⠉⠁⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⢀⡞⣿⣿⠁⠀⠀⠀⢰⡟⠛⣿⣶⣿⣿⣿⣿⣿⣦⣼⣿
⠀⠀⠀⡈⠉⠀⠀⠀⠀⣀⣐⠒⠲⠶⠚⠀⠀⠀⠀⣼⣸⣿⡏⠀⠀⠀⠀⠸⣷⣾⣿⣿⣿⣿⠿⣯⡟⠛⠛⠁
⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⠀⠀⢻⡿⠟⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣧⣾⣽⣿⡿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠠⠴⠶⠾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⠉⠉
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⣞⣿⡄⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰⣳⣄⡀
⠀⠀⠀⠈⣿⡄⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠘⣧⠸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡻⣿⣿⣿⣤⡀⣠⣤
⠀⠀⠀⠀⠀⢻⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠶⠶⢶⠶⠶⢶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠋
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠈⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡂⠁⢀⠚⠁⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⠀⠀⠀⠀⠀⠛⠛⠛⠛⠓⠚⠒⠢⢴⠀⠀⠀⠀⠀⠀⠀⠄⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣶⡍⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣽⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠛⡀
⠀⠀⠐⠒⠒⠶⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡇⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠶⠶⠶⠶⠶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⢁⢀⠤⠀⠒⠒⠒⠒⠒⠂⠠⠬⢉⡛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠀⡔⢊⡇⢃⣁⡐⠀⠤⠄⡠⡀⠀⠀⠀⠈⠀⢁⢢⢙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣼⠋⢀⠄⠀⠠⡈⢠⣿⣿⣿⣿⣶⣶⣤⡈⢠⠂⠀⠀⠀⠀⠉⠸⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⣼⠇⢀⠄⠀⠀⢠⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃⡴⠂⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷
⠀⠀⠀⠀⠀⢠⡟⠀⡆⠈⠀⠀⡄⢸⣿⣿⣿⠃⠤⠤⡤⠤⡄⠥⠠⠤⠄⡠⠀⠀⠀⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⠁⣾⣿⣿⣿⣿⣶⡦⢨⠊⣠⣶⣶⣶⣦⡌⠢⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⠀⡀⠀⠀⢸⢀⣿⣿⣿⡏⢉⡛⢁⠄⣸⣿⣿⣟⣉⣹⣿⡄⢓⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣷⠀⠇⠀⢈⠀⠸⣿⣿⣿⠀⡇⠈⠀⡄⢹⣿⣿⡙⠛⠛⠫⠴⣷⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣧⠰⡠⠄⠉⠂⠤⠌⠉⠤⠀⠂⠀⢀⢄⠙⠿⣿⣶⡶⠖⡐⠚⠻⠛⡿⠛⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣅⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⡀⠀⠀⠐⣻⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⣐⣀⣀⣀⡀⠀⠀⠀⢂⣀⣨⣴⣷⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⣄⠀⠀⠀⠀⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄⡻⣦⣄⣀⠀⣽⣿⣿⣷⣰⣦⠀⣠⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣽⣷⣞⡻⢿⣎⠉⠻⣞⠻⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣤⡀⣰⠠⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠿⠯⠛⠓⠈⠣⡀⠠⣉⠀⠠⡙⠻⣀⣄
⠀⠀⠀⠀⠀⠀⢠⡜⣿⣿⠑⢦⡘⣿⣿⣿⣿⣿⣿⣿⣷⡤⠜⠿⣿⣷⣶⡒⠦⢄⣻⣄⣸⣆⠀⠸⣆⡉⠸⢶⡀
⠀⠀⠀⠀⢀⠀⢸⣿⣿⣿⡆⢤⣀⣙⣿⣿⣉⣉⣲⢾⣉⣁⢀⣠⢩⣿⣿⣿⣷⣶⣾⣿⣿⣯⣧⣼⣿⣿⠦⠾⣷⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠈⢿⣾⣿⣿⣿⣿⣶⣽⢿⣿⣿⣿⣽⠃⠈⢉⣿⣋⣡⣔⣻⡇⠀⣒⣛⣩⣿⣿⣿⣟⠛⠿⠓⠒⠦⠈⢿⣯⣼⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣞⢩⣽⠟⣡⣼⣿⢕⣠⣼⠿⠗⢛⡽⢟⣉⣙⣳⣶⣶⣿⣷⣾⣿⣿⣿⡟⡏
⠀⠀⠀⠀⠰⣤⣾⣿⣿⣿⣿⣿⣟⡻⢿⣯⣴⣿⣿⣯⣿⣷⡥⠶⠿⣿⡟⢋⣵⣠⡐⣡⣶⡿⣿⣿⣿⠟⣩⢡⣾⣰⣿⣯⠛⣿⠇
⠀⠀⠀⢠⣠⣿⣿⣿⣿⡿⢿⡿⠿⠛⣿⠿⣿⣿⣿⣿⣿⣿⠟⣻⣷⣦⣁⣾⣿⣿⣿⢻⣤⣧⣿⣿⡟⣴⣿⣿⣿⣿⣟⠁⡌⠁
⠀⠀⠀⠀⡙⠛⣻⣿⣯⠄⠲⣿⠷⠶⣣⣾⣿⣿⢟⠟⣿⣶⡚⠛⣻⣿⣿⣿⠙⣿⢷⡞⠿⣿⣿⣿⣿⢻⣷⣯⣾⡟⢹⡰⡧⠃
⠀⠀⠀⠀⠻⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⡶⠛⢛⡯⢠⣿⡇⢰⣿⣾⡷⣄⠀⠙⣿⣱⣻⣿⡿⠋⠀⠈⠈⠁
⠀⠀⠀⠀⢦⣬⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠗⣘⢿⠀⢸⣿⣏⢃⠘⣷⡀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠉⢛⣿⡿⠗⠂⠻⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⢿⣯⡋⣇⢸⣿⡭⠀⡄⠘⡇⠈⣹⣿⡇
⠀⠀⠀⠀⠶⣾⣛⣉⣬⠭⣭⣽⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠚⠓⠞⢷⢹⣼⣿⡷⡇⢸⡄⡿⠀⣿⠟⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢭⣿⣦⣕⠉⠫⡛⣿⣿⣿⣿⣿⣷⣄⠘⠇⢻⣿⣿⡏⡰⢟⠁⢠⠿⢋⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢿⣯⣝⡓⠀⠀⠈⢧⠛⢿⣿⣿⣿⣿⣦⠈⣿⣿⢻⢁⡃⣠⣿⣶⡟⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⢿⣷⣦⣄⠈⠀⢸⠸⣿⡟⢿⣿⣷⢹⣿⡇⢸⣭⣾⡿⣻⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣭⣄⣤⡀⠀⢀⣈⣥⣤⣽⣿⡈⣿⣧⡼⢻⢿⡿⣿⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⠿⠿⠿⠟⣫⠿⠟⠛⠿⣿⣷⠘⢿⣿⣯⣾⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣮⠛⠛⢛⡉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⢠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⣿⢿⣿⡿⣿⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠻⠀⠈

⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⣄⡄⠀⠀⣤⢠⣤⠀⠀⠀⢀⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠔⠎⡛⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠘⠂⠘⠀⠃⠇⠘⠂⠉⠈⠀⠀⠀⠀⠀⠁⠈⠁⠀⢀⡀⢸⣳⡠⠔⠁⠀⠀⢀⡜⠓⢤⡀⢰⠋⠀⠀⠠⠀⠠⠀⡇⠀⣰⣦⡁
⠀⠀⠸⢇⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣸⣼⣠⠏⠀⠀⠀⠐⠀⠈⠀⠀⠈⠀⠸⡆⢄⠀⢅⠀⣐⠼⠁⠀⡇⡜⠃⠀⠀⠀⣀
⠀⠀⠀⠘⡆⡜⠀⠀⠀⠀⣤⠀⡰⠂⠀⠀⠼⢾⡀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠲⠒⠒⠟⣶⠛⠛⠉⠁⠀⠀⠘⡇⣇⠀⢀⡰⢶⣝
⠀⠀⠛⢄⢻⡇⠀⠀⣠⡞⠀⠙⢶⡄⠀⠀⠀⠀⠀⠀⢉⠦⠤⠤⠴⢠⡤⣧⡄⢼⠆⠀⠀⢰⠉⠂⠌⣐⠀⠁⠀⠈⢳⠀⠀⠀⠀⠀⠀⢳⣈⠉⢉⣀⠜⠋
⠀⠀⠀⠈⢢⡇⠀⠀⢈⣧⡀⠀⣸⡄⠀⠀⡠⡄⠠⠞⠁⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠘⣆⠌⠐⠀⢁⠂⠈⠂⡼⠀⣠⣤⣄⠀⡀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⢳⠀⠀⠁⠘⠛⠙⡍⠀⠀⠀⠉⠁⠀⠀⠀⣄⠀⠀⠀⠀⠀⣴⡽⠀⣀⡠⢤⠤⡬⠷⣤⠤⠤⠤⠖⠋⢀⢮⠞⠁⠀⡴⢹⢀⣤
⠀⠀⠀⠀⠀⢹⢠⠶⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣰⡿⠋⠀⢸⠁⠠⠀⠀⠀⠄⠈⢦⠀⠀⠀⠀⢸⠘⠒⣦⡸⠁⠓⠊⡼⠀⣾⣿⡆
⠀⠀⠀⠰⢆⢸⡏⠀⠀⡠⠤⠤⣀⣀⠴⠊⠳⡀⠀⠀⠀⠀⣀⡠⣤⠄⣀⣀⣀⠘⣦⠈⡀⠀⠀⠄⠀⣘⠇⠀⠀⠀⠈⠓⠊⠡⡷⠒⡆⣸⠁⠀⠙⠛⠋
⠀⠀⠀⠀⠈⢾⡇⠀⠘⣅⠀⠀⡼⠉⠓⢦⣸⠁⠰⠀⠖⠉⠃⠀⠀⢀⠠⠐⠉⠉⠉⢷⠚⠛⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣵⠖⢲
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠙⢦⡞⠁⠺⠇⢠⡏⢢⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⣤⣤⠀⢠⣤⡄⠀⣠⣤⡀⠀⠀⣿⡇⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠠⠂⢼⠙⠲⢄⣠⠋⠀⠂⡇⠀⠀⠀⠀⠀⠀⠀⠐⠶⠖⠚⠉⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⣠⡇⠀⢸⢹⠀⠀⠀⡿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⡴⠞⠁⠉⠉⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢸⣾⠀⢸⣿⡇⠀⢸⢸⠀⠀⠀⣙⠇⢸
⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡀⠠⠍⡇⠀⠀⢀⢏⣌⢧⠀⠀⢸⣿⠀⢸⣿⡇⠀⢸⣿⠀⠀⠀⣧⠄⢾
⠀⠀⠀⠀⠠⠀⡷⡆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠛⠛⠀⡇⠀⠀⢸⢸⣸⢸⠀⠀⠘⠋⠀⠈⠛⠁⠀⠘⠛⠀⠀⠀⠿⠤⠼⠰⠆
⠀⠀⠀⠀⠀⠀⡧⡇⠀⠁⠀⠀⠀⠀⠀⠀⠐⡿⠷⢶⠂⠀⠀⢸⢰⡏⠙⢆⡇⠀⡆⠘⣆⣡⠞⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⢹⡇⠀⢠⠀⠀⠀⠀⠀⠀⢀⠇⠂⠘⡆⠀⠀⠀⠶⠁⠀⠘⠃⠘⠃⠀⠀⠀⠀⠀⠀⢠⣾⡁⢀⣀⣠⠔⠛⠓⢤⣠⠎⠀⠈⠲⣤⡀
⠀⠀⠀⠀⠀⠀⡴⣇⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡀⠀⢹⡀⠀⠀⠀⢀⠀⢀⠀⠀⠤⠤⠤⢤⡀⠀⠀⠀⠀⠈⠛⠿⡀⠀⠀⠀⠈⠋⠀⠤⠆⠀⠈⠁
⠀⠀⠀⠀⢀⡞⠤⣈⣆⠀⠀⠀⠀⠀⠀⣸⠃⠠⣕⠀⠀⢧⠀⠀⠀⠉⠀⣿⠀⠀⠀⢰⡆⠀⢇⣤⣤⣀⣀⣀⣀⡀⠘⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡞⠈⠘⠋⢹⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣿⠀⠀⠀⢸⣧⠀⣤⡄⠀⢶⠀⠀⠀⠣⠤⠄⣀⣀⣀⣀⣤⠀⠀⢠⣤
⠀⠀⠀⠀⢣⣀⠀⠀⣸⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠠⠐⣿⠀⠀⠀⢸⢿⠀⣿⡇⠀⣿⠀⠀⡤⡄⠀⠀⠀⣶⠀⠀⡷⡄
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⣠⡀⠙⠿⢿⡿⢿⡿⠿⠿⠿⠿⠛⠀⠀⠀⢿⠀⠀⠀⢸⢸⠀⡇⡇⠀⣿⠀⠀⡇⡇⠀⠀⠀⢸⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢖⣢⣀⣀⣹⣷⣤⡤⠼⢷⣦⠤⠤⠄⠀⠀⠀⠀⠀⠆⢸⠀⠀⠀⢸⢸⠀⡇⡇⠀⡿⡆⠀⡇⠇⠀⠀⢸⢸⠀⠀⣷⡇⠀⠐⢶
⠀⠀⠀⢀⣤⣤⠔⠊⠉⡿⠃⠀⠐⠖⠁⠀⠺⠊⠀⠀⠀⠀⠀⢤⡄⠤⠤⢾⠤⠶⠶⠾⠾⠶⠷⠷⠶⠻⠗⠦⠗⡖⠒⠒⣼⠛⠒⠲⠟⠓⠿⠆⠙
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠘⠈⠀⠈⠉⠁⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⢃⣤⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⡴⢁⠤⠂⠀⠀⠀⠀⠀⠀⠀⢀⢸⠅⠘⡻⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⣉⣧
⠀⠀⠀⠀⠀⠀⡔⠁⣴⠋⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⠂⠀⠝⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⢣⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠐⣠⠞⠡⠂⠁⠀⠀⠀⠀⠀⠀⠀⢠⡞⠃⢠⠨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠐⠢⠜⠋
⠀⠀⠀⠀⣠⣾⡫⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣞⡛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣡⠄⠊⣀⡀
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⠙⠘⠁⣟⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠀⠀⢿⣷⡴
⠀⠀⠀⠈⣆⡀⠸⢰⠄⢸⠂⣃⡈⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡀⣠⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠿⠀⠀⣼⢧⡀
⠀⠀⠀⠀⠀⠑⠀⠍⠀⠠⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠈⠁⣟⡄⠀⠀⠀⠀⠀⠀⠰⡇⠠⠈⠓⢬⠇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⢤⠀⠒⢀⡧⠄⠀⠀⠀⠀⠀⠀⠀⠁⢢⣠⠇⠐⣾⠿⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⣀⡠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⣀⢠⠀⠀⠀⠐⠄
⠀⠀⠀⠠⡄⠄⠽⠞⠛⠒⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⣷⢉⣹⣿⣄⣌⢠⣤⡇⠀⠀⠀⠀⠀⣴⠁⠀⠄⠂⠀⠐⠂⠀⠠
⠀⠀⠀⠀⠁⢠⡖⣤⣀⢢⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣟⠃⠀⢽⡄⠀⠈⠁⠀⠀⠀⠀⠀⠁⢴⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠆⠈⠈⢙⣏⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠔⠐⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⢤⡄⣁⡀⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡧⠤⢤⠝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣤
⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⣂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⣾⡅
⠀⠀⠈⠀⠀⠸⡇⠄⢀⠰⣷⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣆⢰⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⣴⠿⢿⣤
⠀⠀⠀⠀⠀⢠⡇⠀⠨⠄⢻⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡸⠛⢠⠈⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠻⠁⠀⠨⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠠⢥⣝⣀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢰⣶⣛⣛⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⢤⡤⠾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠄⢀
⠀⠀⠀⠀⡠⠈⠁⠁⣀⣠⡶⣆⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠐⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡄⠀⠀⠐⢢
⠀⠀⠀⠀⣇⣈⡤⠞⠋⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣾⣿⣦⣤⣦⣼⣤⣶⢴⡆
⠀⠀⠀⠀⣿⠃⠀⠐⠀⠠⣶⡏⠀⠀⠀⠀⠀⠀⠀⢈⠛⠀⠊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠤⠲⣦⣀⡟⠒⠒⠿⣶⣦⠉
⠀⠀⠀⠀⠈⠀⢀⠀⠀⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠠⠆⠀⠐⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠉⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⣠⣴⣲⡤⠖⠀⠀⢀⣤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⢀⣴⣾⠟⠋⠁⢀⣾⣿⣿⠋⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣤⣀⢿⣿⣽⣿⡏⠀⠀⢀⣾⣿⣿⣿⡀⠀⣀⣼⣿⡟⠀⠀⢀⣠⣴⠇⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠿⣿⣿⣿⣽⣶⣾⣿⣿⣿⣫⣷⣿⣿⡿⠟⠋⠀⣠⣾⣿⠟⠁⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣋⣿⣤⡀⢠⣶⣿⣿⡇⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⣡⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣟⣿⣭⣿⣿⣿⠾⠟⠋⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⣿⢇⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣜⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣙⣡⢤⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣧⣿⣦⣤⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⢟⣽⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠿⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⢟⣿⣭⣫⣷⣾⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⣿⣾⣿⣿⣿⡿⡟⣭⠻⠿⠿⣷⣶⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣧⣴⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢿⣟⠋⠁⠀⠀⠈⠁⠀⠈⣿⣿⡿⣓⣿⣿⣿⣿⣟⣵⣿⣿⣿⣿⢿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠅⠈⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣶⣿⣿⡟⠁⠈⠉⠉⢉⣿⡿⠿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣄⣿⢎⣽⣿⠏⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⢛⣵⣿⣿⡟⠀⠀⠀⠀⠀⠀⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠻⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⠶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣸⡏⢹⡛⠛⠻⡟⠶⠶⢶⠶⠶⠶⡶⠶⠶⣶⠶⠶⢲⡖⠞⠻⡟⠋⢻⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣧⣇⣀⣇⠀⢘⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⣸⠀⠀⢸⡇⠀⢘⡇⠀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣉⣙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠈⠉⠉⠉⠉⠙⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣸⠇⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠟⠀⠀⠀⠀⢸⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⡀⠀⠀⠀⠀⢀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢿⡍⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢹⡟⠀⢸⡏⠀⠀⠀⢸⡀⠀⠀⠈⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡴⠟⠀⠀⠸⠷⣤⣤⠶⠾⠷⠶⠶⠶⠋
⠀⠀⠀⠀⢠⡴⠶⠶⠶⣶⠶⣤⣤⣈⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣴⠶⠲⢶⣄⡀
⠀⠀⠀⠀⣿⠀⠀⠀⢰⠃⠀⠀⠀⠉⢻⡄⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉⠀⠈⣿⣄⠀⣠⠞⠛⢶⣄
⠀⠀⠀⠀⠙⢷⣄⣀⣸⣄⠀⠀⠀⣀⡾⠃⢀⡀⠀⢰⡟⠁⠀⠀⠙⣳⣦⣀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠉⠳⣄⣴⠟⠙⢷⣇⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⣛⣯⡶⠛⠋⠙⢷⡈⠻⣄⠀⣠⠞⠁⠀⠙⢷⣄⠀⠀⠀⢸⡅⠀⠀⢀⣴⠟⠁⠀⠀⠀⠙⠷⣤⣀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠳⡄⠀⢀⣼⠇⠀⠈⠻⣧⡀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠙⠶⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣀⣿⠶⠟⠁⠀⠀⠀⠀⠈⠻⣦⣀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉

⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢦⣄⣀⠤⠤⠤⠤⠤⢤⣤⣔⣒⣳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⡀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠙⠛⠓⠦⡤⠶⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣯⢰⣿⣦⣄⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠴⠒⠀⠀⠀⠀⠐⢦⣌⣢⣀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⣀⣤⣶⠿⠟⠁⠁⠁⠈⠀⠀⠀⠀⠐⣤⠀⠀⠀⠀⠀⠀⣀⡀⠀⠙⠋⠇⠀⠀⠈⠉⠉⠉⠉⠉⠛⠲⠖⢋⠉⣠⡀⡝⡆
⠀⠀⢼⡟⠉⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠈⠉⠛⢶⣤⡴⣶⠟⠁⠀⠀⠀⠀⠉⠉⠙⠛⠢⠀⠙⠛⠀⢻⠁
⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡾⠀⠀⡸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⡇
⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⠃⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡴⣡⣴⣶⣶⣿⣿⣟⠻⣭⣭⡭⠿⠋
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢸⣿⣿⣿⣿⣿⣿⡿⠋⢡⠉⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⣀⡀⢸⢿⣿⡻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡜⣞⣿⣿⣌⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡽⣮⣻⣿⣧⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠸⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠙⠘⢿⣿⡇⢻⣿⡤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠹⢷⣄⠀⠀⠀⠀⠀⠀⠀⢩⠋⠉⠻⢦⣀⠙⠛⠛⢷⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠉⠳⠄⢀⠀⠀⠀⡴⠾⠀⠀⠀⠀⠉⠙⠛⠛⠒⠽⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡿⢤⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣛⣋⣉⣉⣛⣛⣛⣛⣛⣛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣤⠶⣾⠛⠉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣍⣉⣙⣛⠛⡿⠁⠉⠉⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣷⣌⢻⣿⣿⣿⣿⣿⣿⣿⣧⠾⠋⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣧⠹⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⡃⢹⣿⣿⣿⣿⣿⣿⡄⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⣡⣿⣿⠟⠛⠛⠛⢻⣷⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⣿⣿⡿⣿⢿⣿⣷⣖⣒⣂⠺⣿⣿⡿⢋⣴⣿⣿⣿⣶⣶⣶⣶⣾⣿⣇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠀⠈⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣾⣿⣍⢳⣶⣾⣿⣿⣛⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⣆⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠿⠿⠛⠛⠉⠉⠉⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡟⠸⣷
⠀⠀⠀⣀⣤⣶⣿⣿⣭⣤⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠷⠶⠶⠶⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⣿
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠐⠷⠦⠭⣭⣭⣭⣛⣛⣛⣛⣛⡻⠿⠿⠿⠿⠿⠷⢶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠁⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡶⠶⠾⠿⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠛⠛⠛⠻⠷⠶⣦⣤⣄⣀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡾⠿⠛⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⡶⠶⠾⠛⠛⠉⠁⠀⠀⠀⠀⠀⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠋⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠿⠶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠿⠋
⠀⠀⠀⢀⣼⣶⣶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉
⠀⠀⢤⣾⣿⣿⣿⣿⠇
⠀⠀⠰⠋⠙⢻⣿⡏
⠀⠀⠀⠀⠀⠋⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣩⣟⣙⣽⣏⣵⣹⣹⣋⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣯⣽⣿⣯⣿⣿⣿⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠘⠂⠖⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡴⠑⠦⠾⠟⠁⢀⣠⣶⣶⣾⣁⠀⠀⠀⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠤⠤⡄⠀⣠⣮⡙⠛⠉⠉⠙⠋⠀⠽⣾⠛⠋⠰⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡙⣿⠟⠋⠀⢠⣦⡀⠀⢀⣨⡇⠀⣆⢀⠀⠁⠀⡠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡙⢻⣷⡶⠆⠀⠛⠉⠉⣶⣼⡏⠁⠀⢘⠿⠇
⠀⠀⠀⠀⠀⠀⣠⠈⠉⠀⠀⠁⠀⣴⣴⣿⣷⠐⢻⣿⣶⣶⠏⣸⣷⣄⢰⡿⠶⡆
⠀⠀⠀⠀⠀⠘⠋⠁⠀⣀⣶⠆⠸⡿⣯⣉⠻⣦⠘⣿⣿⡟⠴⣿⣿⠽⠒⠈⠀⠙⠲⣄
⠀⠀⠀⠀⢠⡾⢷⣦⠀⢼⠁⠀⢀⣀⠈⠻⣿⣿⠟⣛⠽⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠋⠠⠿⢿⡀⢀⣰⣶⣿⣿⡿⢿⡟⠀⠀⠑⢄⠀⠀⠀⠀⠀⠐⢶⣤⡤⠀⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⣰⠟⣿⣿⣿⣿⣿⡏⠀⠸⡀⠀⠀⠀⠈⠱⢄⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⡴⠟⠛⢛⣩⣿⠉⠀⠀⠘⣖⠃⢤⣀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠂
⠀⠀⠀⠀⠀⠀⠀⠖⠉⣉⡿⠃⢀⡀⢸⣿⡆⠀⣬⠆⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢔⣊⣥⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⠁⠀⢉⡁⠈⣿⣧⣶⡄⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣀⡤⢖⣫⣵⠶⠛⠉⢥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⠀⠀⣀⣠⣌⠁⣤⣼⣿⣿⣿⡀⠄⠂⠀⠀⠀⠀⠓⠤⠖⣩⡵⠾⠛⠁⠀⠀⢀⣀⣄⣷⠾
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠙⠃⠐⠖⠉⠃⠛⠀⣨⣿⠿⠏⠙⠃⠀⣇⠀⢠⠀⠀⢀⡴⠛⠉⠀⠀⣀⣠⣴⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠀⠀⠾⠿⢿⣷⣄⡰⠄⣀⡶⠙⣦⣤⠄⢀⣾⠁⠀⢀⣸⣧⡙⢿⣿⣿⣒⠒⠒⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⡴⠿⠋⠀⠀⡿⢃⣠⠴⠾⢶⡸⣇⣾⣶⣿⣿⣿⣷⣌⣇⠉⠉⠉⠙⠒⠦⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢯⢀⡟⠁⠀⠀⠀⠀⠛⠉⠈⠉⠛⢤⡉⠙⠻⢷⣖⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣀⠀⠈⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣖⢲⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⡐⠟⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣎⣭⡛⠫⠭⠙⠉⠒⢄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠹⣆⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠉⠛⠢⢄⠠⣴⣵⡄
⠀⠀⠀⠀⠀⢸⡷⠲⠶⢶⠒⣿⠀⠀⠀⣀⣹⣄⠀⠀⠘⡿⠿⠿⠿⠶⣶⡶⠶⣤⣧⣀⡏⠉⠁⠒⢲⠀⠈⡆⣿⣿⡇
⠀⠀⠀⠀⠀⣸⣧⣀⣠⣸⠀⢋⡄⠀⣾⡟⠙⠻⣇⣀⣴⠗⣤⠀⠀⢀⣀⣈⣲⡀⠈⣿⣿⠀⠀⠀⢸⠀⠀⡇⣿⣿⡇
⠀⠀⠀⢠⡞⠉⠛⠾⠭⠭⠷⣿⠙⠓⠛⢻⡷⢾⠿⠿⣥⣾⣷⣤⡤⠴⠿⠿⠿⣿⣿⡿⠋⠀⠀⠀⢸⠀⠀⣟⣿⣿⡇
⠀⠀⠀⢸⠉⠓⠒⠒⠀⠐⢺⣿⣿⠛⠉⢹⡇⣧⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⡇⢀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇
⠀⠀⠀⢸⠀⡖⠲⠤⠤⠤⠞⣿⣻⡶⠶⣾⡇⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⡇⠀⠀⠀⠀⠈⠀⠀⠈⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣤⣿⡛⣉⣃⡨⡳⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣃⡉⠉⠀⢀⣲⡄⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⣼⣽⣿⣷⣿⣿⣯⠨⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠉⠉⠉⠉⠁⡇⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⡟⠻⠿⠿⠿⠿⠿⠂⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⣉⡓⣒⣒⣛⣩⡵⡀⠀⠀⢸⢿⣿⣇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⡇⠀⠰⢤⡰⡶⠶⠀⢸⣷⢄⣀⠀⠀⠀⠀⠀⢀⣷⢤⣄⣈⣉⣉⣁⣤⢴⡟⠋⠉⠙⠛⠻⠷⢤⣄⣀⡀
⠀⠀⠀⢸⠀⠧⢄⣀⣀⡇⠀⣴⠋⠀⠈⢲⠄⠀⣿⣿⠟⠻⠿⠿⠿⠿⠓⡏⠙⠓⠒⠶⠒⠒⠊⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢓
⠀⠀⠀⠘⠦⣄⣀⣀⣀⡇⠀⣈⡳⣤⡔⣁⠀⠀⣿⡟⠀⠠⡤⠒⠚⢿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠈⠉⠉⠉⠉⢀⣠⡟⠧⢤⣀⣷⣷⡶⡟⠟⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣤⡴⠊⠁⠀⢀⡤
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⣿⠿⠿⣿⣿⣿⠇⢿⠇⡀⠀⣠⣯⣧⠤⠂⡼⣿⠢⢤⣄⣀⣀⣠⠤⠖⢻⠿⣿⣿⣿⣿⣿⠀⢀⡤⠞⠉
⠀⠀⠀⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⠒⠊⠁⠘⠿⠾⠉⠉⡟⠒⠚⠋⢻⡀⠀⠀⠀⠀⠀⠀⣀⡟⠀⠀⠉⠉⠛⠻⠖⠉
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠼⣧⣤⠴⠠⡄⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠚⠋⠁
⠀⠀⠀⣿⠤⠤⠤⠀⠐⠒⠒⠒⠒⠂⠉⢻⠀⠘⠶⠒⠊⠉⠁
⠀⠀⠀⠹⠴⠶⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉

⠀⠀⠀⠀⠠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠐⠂⠐⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⣀⣀⠀⠀⡥⠀⠀⠀⠀⠤⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⡂⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⢠⡎⠀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⢃⠀⠀⠊
⠀⠀⠀⠀⡀⠀⢄⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠁⠠⠀⠄⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠐⠂⠄
⠀⠀⠀⠀⠈⠂⠀⡀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡜⠀⡃⠀⠀⠀⡤⣖⡧⠀⣂⣀⠔⢒⠄⢀⣠⠀⠀⠀⢠⠀⢀⢢⣄⣐⠄⠀⠀⡅⠀⠀⠀⠀⠀⠀⡐⠀⠀⠁
⠀⠀⡀⠀⠠⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠨⢰⡌⢇⠀⢀⣼⠀⢠⠔⠀⢠⡜⢸⢀⠇⢀⡠⠮⠂⠀⠀⡇⠀⠀⠙⠦⠀⠙⠁
⠀⠀⠃⠃⠉⠁⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠋⠁⠚⠁⠐⠚⠀⠙⠀⠋⠀⠀⠊⠁⠛⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠻⠆⠀⠀⠁⠶⠦⠠⠶⠦⠼⠿⠧⠷⠄⠴⠶⠴⠄⠠⠶⠶⠦⠿⠇⠀⠘⠀⠀⣖⠀⠤⠀⢐⡀⠀⠄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⢀⣀⠀⠀⠈⢠⡄⠀⠀⠈
⠀⠀⠒⠀⠀⠀⠀⠄⠀⠀⠀⢠⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠇⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠣⠀⠀⠀⠀⠀⠀⠀⠃⠀⣀⠉⢀⣀⣀⡀⠊⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠁⠒⠤⢤⡀⠀⣀⣀⣰⡿⢯⡙⠃⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠶⠞⠀⠀⠀⠁⠀⡀⠀⠈⠜⢇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠉⡆⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⢀⠃⠀⢠⡇⡟⠁⠀⠀⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢀⡴⠫⣤⣤⡀⠀⠀⠀⡎⠀⠀⣸⠁⠩⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠩⠀⠀⠈⠁⠂⢀⣀⣼⡄⠀⠒⠋⠀⠀⠈⠁⡄⠀⠀⠀⢀⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠑⠀⠀⠀⠁⠀⢀⡰⣄⠀⣀⡔⣇⠀⠀⠀⠈⠀⡐
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡤⠈⠁⢏⠲⢶⡶⠸⢇⠀⣀⡀⠀⠸⠋⠙⠉⠀⡞⠓⠀⠣⠔⠊
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⡁⠀⢀⣨⡷⡾⠀⠀⠀⢢⡿⠿⠶⣾⣦⣶⠖⠛⠛⠂⠀⠀⠄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠢⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠧⣤⠼⠷⣶⣿⡏⠀⠈⢤⠞
⠀⠀⠀⡀⣠⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⣾⣶⣿⢶⡄⠀⠈⠓⡄⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠈⠀⠀⠀⡀⠈⠀⠀⢻⠀⠒⠀⠄⠤⠄⠑⠒⠊⠘⠉⡉⠹⣭⣠⡦⠴⠉⠒⠈⠓⠀⠈⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡨⠀⣠⡆⠀⠀⠂⠈⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡠⠖
⠀⠀⠀⡠⠄⠀⠀⠐⢾⣆⠀⠀⠀⠀⠀⠀⡘⠁⠀⠃⠁⠀⠀⠈⣀⠄⠀⠀⠠⠄⠀⠀⠀⠀⠠⠊⠀⢄⠀⠈
⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠜⠋⠔⠀⠀⡀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣸⠶⠞⠋⡄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢧⣸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠙⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡁⢰⣿⣛⣿⠃⠀⣀⡀⠀⠀⠀⠀⠈⠉⢻⡟⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠙⠛⠋⠀⢸⡏⢙⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠛⠈⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠉⣳⢉⡵⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⣼⠄
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⢀⣠⡿⠀⠀⠀⠀⠀⢀⣀⡀⠛⠃⠰⠏⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠹⢶⡾⠟⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⡇⠀⠀⠀⢰⣶⠀⢠⣤⣤⣾⣽⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣿⣿⣿⠀⠀⠀⣾⡇⠀⠀⣻⡏⠀⠀⢈⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⢹⣿⣿⠀⠀⠀⢻⣿⡀⠰⠷⠿⣶⢶⡾⠾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⠀⠉⠛⠀⠘⣿⣿⠀⠀⠀⢸⣿⣷⣤⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⢀⣤⡀⠀⠀⢿⣿⠀⠀⠀⢸⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠀⠀⠀⠉⠀⠀⠀⢸⡟⠀⠀⠀⠸⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠄⠀⠀⠀⠘⠛⡇⣀⣰⠀⣠⣆⠀⠀⢄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣎
⠀⠀⠀⠀⢤⡘⡣⣷⢶⡍⢠⣴⣶⣯⣟⣛⠠⠼⢟⢤⣷⣥⣔⣒⠺⡟⠀⠀⠀⠀⠀⠀⣠⣴⠛⠋⠛⠘⠀⠁⠀⠀⠈⣨⡟⠈⠉⠉
⠀⠀⠀⠀⣶⢿⢷⠉⣱⡿⢧⣬⠙⡞⠓⠎⠓⠒⠾⠳⠿⠿⠓⣟⢳⠟⡋⠁⠀⠀⢀⡾⠛⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⣀⣀
⠒⠓⠒⣰⠻⡈⣧⡶⣫⠚⣍⣁⡈⠀⠴⠆⣤⢤⠔⠦⠤⡤⣤⣼⢿⢏⡉⠂⠀⠀⡞⠠⠆⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁⠀⣴⣿⠟⢻⡇
⢠⣀⡀⠀⠙⣤⢡⡾⠃⢀⠾⠷⣴⡟⣋⣭⢭⣌⣙⣓⠘⠖⣦⠹⠏⠉⠋⠂⠀⠀⡇⠀⠀⠀⣀⣠⣴⣾⠟⠋⠁⠀⠀⠀⢳⣿⡀⠩⡇⠀⣠⣲⣶
⠀⠈⣹⠟⠴⠿⣄⠴⠭⢒⡿⠗⠶⠶⠶⠒⡶⠁⠈⠋⠙⠋⢹⣄⠞⠙⠋⠁⠀⠀⠒⠒⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⢷⡾⠟⠉⠈
⠀⠀⠀⠀⠀⡰⡻⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣯⠿⠀⠈⠀⣀⠴⠚
⠀⠐⠒⠒⠯⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠖⠖⠻⠻⠈⠈⠹⣧⠀⠀⠀⠀⠀⠀⠀⣤⣶⡾⠿⠋⠁⠀⢠⡀⢸⡀
⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡖⢶⣄⠀⠀⠀⢧⢻⣷⠀⠀⢀⠴⣿⡇⠈⡇
⠀⠀⠀⠀⣠⡶⠻⠿⢶⣤⡟⠋⠀⠀⠀⠙⢷⡄⠀⠀⠀⢸⠀⠀⠀⠀⣀⣤⡴⠖⠖⠖⢦⣄⠀⠈⢻⣧⠀⠀⠘⡆⣿⣆⡴⠃⠀⡿⡇⠀⢹
⠀⠀⠀⣼⠁⠀⡴⡄⠈⣿⠃⢰⠟⢶⡀⠀⢘⣷⠀⠀⠀⠈⡆⠀⢀⡾⠋⣠⣶⣟⡿⣷⣦⢘⡷⢖⣿⠟⠛⢦⡀⠳⠾⠋⠀⠀⠀⣿⣷⠀⠈⡇
⠀⠀⠀⡇⠀⠘⣧⣄⣴⠟⢧⣈⢃⣸⠃⠀⢠⣿⠀⠀⣠⢾⣧⠀⡼⢠⣾⣿⣿⣯⣴⣿⣿⢋⣴⠟⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢹⡀
⠀⠀⠀⣿⠀⠀⠀⠉⠀⠀⠀⠉⠉⠁⠀⠠⣽⠇⠀⣼⠁⠀⠉⠀⠃⣿⣿⣿⣿⣿⣿⣿⡏⣼⠏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢈⣧
⠀⠀⠀⠘⣧⡄⢀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⡏⠀⠀⠀⠀⡀⢻⣿⣿⣿⣿⣿⣿⡇⢿⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⣿⣿⣄⣠⡿⠋
⠀⠀⠀⠀⠈⠛⠿⠶⣧⣶⣴⣠⡄⡀⢸⠃⠀⠀⠀⣷⠀⠀⡀⠀⣧⠈⠛⠻⠿⠿⠛⢋⢳⣬⣳⣄⣀⣀⣴⡏⠀⠀⠀⠀⠀⠀⠀⣷⣿⠿⠋
⠀⠀⣔⡒⣤⣄⡀⠀⠀⠀⠀⠉⠙⠻⣿⢠⡖⢦⡀⢹⣦⡦⠷⠶⠿⠿⣮⣦⣴⡶⠾⠾⠛⠋⠉⠉⠛⠛⠋
⠀⠀⣯⠹⣎⢫⡹⣄⠀⠀⠀⠀⠀⠀⠈⠧⣭⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠤⣤⣄⣀⡀⠀⣀⣀⣀⠀⠀⠀⣀⣤
⠀⠘⣾⠳⡎⢷⡱⢌⠳⣔⡟⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠉⣉⡩⣤⠀⢲⠂⢠⡏⣹⡷⠚⢩⡉⠉⣍⣉⠩⡃⢸⣶⣄
⠀⢠⢏⣧⠻⡄⠻⡌⢧⡘⢧⡸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢁⡆⠀⠉⠉⠋⠁⣈⡀⢸⡀⠨⣇⠀⠘⠋⠀⠛⠙⢂⠁⢸⡏⡏⢻⡆
⠀⣾⢸⡟⣇⠘⠦⠙⠆⠉⠀⠙⠎⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠑⢘⠃⠀⠈⢛⠀⠀⠒⠒⢸⡇⢹⡇⠀⠈⠐⠆⠀⠈⠐⠀⠈⣇⡇⢸⡇
⠀⢹⡄⠻⡼⣇⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⠀⠀⠈⠐⠀⠀⠐⠒⠀⣏⢸⡧⠀⠀⠤⠅⠀⠀⠈⠀⠀⣿⡇⠈⡇
⠀⠀⠻⡄⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⡌⠀⠀⡀⠀⠀⠀⠀⠀⢀⡇⣸⡯⠀⠀⠐⠐⠀⠀⠀⠂⠀⣿⡇⠀⡇
⠀⠀⠀⢧⠀⠘⣷⣂⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢸⣧⣠⠇⠀⠀⠠⠀⠀⠀⠒⠒⢸⠇⢸⣟⠁⢀⣁⣀⣀⣀⠈⢁⣠⠿⣇⠀⡇
⠀⠀⠀⠘⣧⠀⠈⢷⣖⡀⠀⠀⠀⠀⠀⠈⢙⡦⠀⠀⠀⠀⠀⡿⣾⣿⣷⣦⣤⣤⣶⣶⣾⣿⣿⣿⣶⣬⣷⣾⣿⣿⣷⣿⣿⣿⣯⣤⣤⣾⣿⡇
⠀⠀⠀⠀⠈⢧⠄⠀⠈⢿⣧⣄⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣟⣿⣿⣫⠞⣿⣿⣿⣯⣿⣿⣿⣯⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢼⡁⢀⣠⡾⠋⠈⢷⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⣿⠿⠿⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠘⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⣿⢿⠿⠿⡿⡟⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣛⣿⣟⣿⣿⣻⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣽⡇⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⡿⣾⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠐⠀⠁⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⣀⠤⠴⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⡇⠀⡆⣀⣠⠤⠔⠒⠉⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠈⢷⠚⠉⠉⣤⠀⠃⠀⠀⠀⠀⠀⠀⣄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢶⠚⠋⠀⡞⠋⠋⡀⠀⡄⢀⠨⠀⠁⠠⢀⣀⣠⣤⣴⣷⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⣾⠛⡏⢸⠀⢸⠀⠀⠀⡅⠀⠀⠀⡴⣃⣨⣴⣶⣾⣿⣿⣿⣿⣿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⣿⢮⣀⣃⠠⡇⠈⠀⢩⢀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡿⠀⣯⣏⣿⣿⣾⣷⣿⣿⣿⠀⠄⠀⠈⣸⡟⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⣿⠀⠀⠀⠀⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⢀⡏⣀⡀⠤⢄⣸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣾⣅⣤⣴⣀⣀⠉⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣋⣿⣿⣿⣿⣶⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠛⡣⢄⡙⣿⣿⡄⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡷⠾⠶⣤⠇⣿⣿⡧⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⣤⣶⣤⡇⡿⣿⣿⡤⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣥⣷⣿⠃⣃⡇⢹⣿⣿⣶⡦⠀⠀⠛⠰⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⡇⠀⠳⠇⠸⣿⣿⣿⡟⠀⠀⠀⠀⣾⣧⢻⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠑⠀⠀⠃⠀⣿⡏⠈⠳⣄⠀⠀⠀⡏⣿⡜⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⡿⢿⣶⣄⠈⠣⣲⣼⠇⣿⡇⡇⠀⠀⠀⠀⢋⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⡇⠀⠉⢹⣿⣦⣨⣿⠀⣿⡇⠑⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⠀⠇⠀⠀⢸⣿⣿⣿⡏⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢸⣿⣿⣿⣶⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢿⣤⡼⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⠟⠀⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣴⣟⣠⣄⠀⣤⣤⣤⢰⢠⣤⣤⡄⠀⠀⢠⣾⠿⢿⣷⣦⡀⠀⠀⠀⢠⠞⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄
⠀⠀⠀⠀⢾⠿⢿⡎⢙⣯⣅⠀⠈⠃⢄⢺⣆⠄⠀⠀⠀⠀⠘⢿⣶⣾⣿⣿⣷⠀⢻⣹⠏⠉⠙⢏⡽⠁⠀⠀⠀⠀⠠⢴⣿⡆
⠀⠀⠀⣀⠀⢿⣿⣵⠿⠃⢘⡇⠀⠀⠀⣸⡇⠀⠂⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⠀⣰⠳⡀⠀⢀⡜⢧⠀⠘⠶⣶⢶⣶⣾⣿⣷⡶⡶⣶⡶⠃
⠀⠀⠀⠸⣦⣀⣠⡿⢻⣶⣾⠃⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠻⠃⢠⡿⠃⠀⠉⠉⠹⣍⡟⠉⠉⠁⠀⠀⠀⠀⠀⢠⡯⡆⠀⠀⡀
⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉
⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⠀⢠⣴⣷⡤⠀⠀⠀⣀⡤⠴⡷⠤⣀⡀⠀⠀⡠⠔⠉⡹⡉⠑⠢⡀
⠀⠀⠀⣼⣿⠃⠀⠀⢀⣰⣤⠄⠰⠶⢶⣷⠶⠶⠀⢠⠋⠳⣤⢧⡴⠋⢇⠀⢰⣥⠤⢤⠥⢵⠤⢤⣵⠀⠀⡠⢖⡉
⠀⠀⠀⣿⣿⡀⠀⠀⠀⠽⠛⠄⠀⢀⠀⡇⠀⠀⠀⢻⠓⢛⣧⣤⣟⠛⡻⠃⢠⠀⠑⡦⣀⡠⠖⠁⢀⠀⢾⡀⢸⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠈⠛⡿⠦⠀⠀⠀⠷⣏⣀⣇⣈⡷⠇⠀⠈⢆⣸⠔⠋⠑⢼⣄⠞⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡴⡄⢀⡀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠲⣶⣤⣤⣤⣤⣤⣤⣶⡖⠀⠀⠀⠀⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡏⠉⠀⠉⠩⣇⡀⠀⢰⢣⠀⣿⠀⠁⣦⠀⠀⢨⣽⣿⣽⣯⣽⣯⣭⠀⠀⠀⠀⡀⣀⢉⢀⡀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⢣⠄⠀⠀⠀⠀⣴⠋⠀⠸⣧⡳⣿⠖⣡⡟⠀⠀⠀⢰⡇⠀⠀⢸⡇⠀⠀⠀⢰⣼⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠰⢛⣋⠓⠀⠀⠀⠿⢶⡄⣀⡀⡴⠾⠂⠀⠀⠈⣛⢾⢞⠋⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠙⢻⣟⣻⣻⣟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠙⠃⠀⠀⠀⠀⠀⠈⠺⠈⠀⠀⠀⠀⠀⠘⠇⠀⠀⠘⠇⠀⠀⠀⠀⠀⠋⠛⠛⠛⠙
⠀⠀⠀⠀⠀⠀⠂⣀⠂⠄⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠐⣄⠀⠀⢀⣤⣤⣤⣤⠄⠀⠀⠀⠀⡄
⠀⠀⠠⡴⠤⠤⣴⣙⡀⠄⠐⣦⠀⠐⠒⣿⠗⠒⠀⢠⣾⠀⠀⠀⠀⠀⢸⡄⠀⠀⣿⡆⠦⢸⡏⠹⠅⠀⠀⠀⢀⣀⣰⣷⣀⣀
⠀⠀⠀⠀⠀⢰⢃⣉⣳⠶⢎⡇⠀⠀⠀⣿⠀⠀⠀⢸⣿⡆⠀⠀⠀⢀⣾⣿⠀⠀⠻⠷⠶⣿⣿⠾⢷⣦⠀⠀⠈⠉⠹⡟⠉⠉
⠀⠀⠐⠓⠒⡟⠉⠉⢲⡀⢠⠛⠀⠀⠀⣿⠀⠀⠀⠈⢿⣧⣶⡶⣶⣦⣿⠏⠀⠀⠀⢀⠓⣸⡇⠘⠸⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠙⠦⠤⠤⠒⠁⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠉⠋⠻⠋⠋⠁⠀⠀⠠⠚⠛⠛⠛⠁⠀⠀⠙⠆⠀⠀⠀⠈⠃

⣀⣀⣠⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣠⣀⣤⣀⣀⣠⣀⣀
⣿⣿⣏⣉⣩⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⢀⠀⠀⢻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⡀⠀⠈⠀⠀⠀⣷⡘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⡄⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣧⠀⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⣿⠀⢸⡟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⣇⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⡆⠀⣿⢸⡇⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⠇⠀⣿⢸⡇⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣿⠀⠀⣿⠀⣿⡀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⢿⠀⠀⣿⠀⣷⡙⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⣿⠀⠀⣿⠀⣧⢹⣇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣿⢠⡟⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣿⠁⠀⢸⡇⢰⡇⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⣾⠀⢸⡇⣼⡇⢘⣇⣤⠴⣖⢛⣛⣟⣛⡷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠆⢶⣲⣾⣿⣻⣷⣦⣄⣀⠀⠀⠀⠀⠀⢀⣿⣀⣤⠾⠃⠀⣹⢏⠟⠁⢸⡧⠾⠖⠛⠋⠉⠉⠉⠙⡧⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣋⣭⠷⠖⠛⠉⠉⢹⣺⣾⣿⣿⣿⣿⣿⣿⣶⣶⣤⣾⣿⣿⡇⠀⠀⠀⣾⠁⠀⠀⡿⠀⠀⠀⠀⠀⠀⣀⣠⠾⢙⡼⠁
⠀⠀⠀⠀⠀⢀⡴⢋⡴⠞⠉⠀⠀⠀⠀⠀⠀⣏⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⠀⢀⡏⠀⣠⣎⣁⣠⣤⡤⠴⠖⠞⣉⣤⠴⠋
⠀⠀⠀⠀⠀⣿⠰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⠃⡀⡤⣸⡷⠾⠥⠬⠴⠤⠶⠶⠟⠚⠋⠉
⠀⠀⠀⠀⠀⢻⡌⢳⣤⣀⣤⠤⠄⣀⢀⣀⣀⠀⠀⠀⠀⠈⠙⠳⠦⢄⣉⡙⠛⠛⠿⣿⣀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠙⠳⣼⠯⣄⣠⢞⡵⠋⡠⠘⠛⢢⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣀⠀⡞⠀⡞⠀⠰⣿⣟⣓⡢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡇⢸⢡⡶⢂⢤⣠⡟⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠮⣦⣉⣻⠶⠤⠽

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠻⢿⠛⡿⠿⣿⠿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⡔⠉⠀⠈⠑⣦⡖⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠲⣬⣢⣀⠔⠢⣀⣞⣵⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣤⣶⡿
⠀⠀⠀⠀⠈⠒⠤⠔⠊⠁⠉⠀⠀⠀⠰⡿⠀⠀⠀⠀⠀⠀⠒⡺⠛⠱⣾⡷⠎⠻⣗⠂⠀⠀⠀⠀⠀⠀⠰⣶⣶⣾⣿⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣧⣤⣤⣤⡀⠀⠀⠚⠒⠒⢲⢲⡖⡖⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣼⣿⣿⣷⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠉⠁
⠀⠀⠀⠀⠀⣰⢟⡻⣧⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡗⠐⣷⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣴⣾⡦⠀⠀⣶⣷⣦⠀⠀⠀⠀⣿⡤⠀⣼⡇⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⣰⡟⠀⠁⢻⣦⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⢸⣿⢿⣷⡆⢶⣿⠿⣿⠀⠀⢰⣾⠿⠁⠀⠙⢿⣦⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⣿⢳⡆⣿⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠻⠿⠋⠀⠀⠻⠿⠛⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⣄
⠀⠀⠀⠀⢹⣧⣾⣷⣼⡟⠀⠀⠀⠀⠀⢀⣀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⢹⡿⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⣀⣤⣶⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢿⣿⣿⡟⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⠇⠀⠀⢀⣤⣶⣿⣿⣿⣵⣶⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⡇⢸⣿⡇
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⠻⢿⣿⣿⠿⠋⡡⠐⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⣷⣿⣿⡇
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣈⠡⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡠⠄⠀⠐⣒⡐⢀⣴⣶⡉⢂⣚⠂⠀⠤⢄⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠶⠖⠦⣄⣀⠀⠹⣮⣿⣿⣵⠏⠀⣀⣠⠴⠲⠾⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠶⢀⡡⠄⠀⠐⠒⠦⠀⠀⠈⠲⣽⣿⣿⣯⠖⠁⠀⠀⠔⠒⠀⠀⠠⢈⠀⠖
⠀⠀⠀⠀⠀⠀⠐⡰⣚⣥⣤⣤⣤⠤⠶⠞⠶⢦⣦⠀⢹⡟⣿⡇⠀⣤⡴⠶⠓⠶⠤⣤⣠⣤⡬⡒⢖⠂
⠀⠀⠀⠀⠀⢠⡞⢁⣤⣤⠠⠛⠁⢤⣴⣿⣶⣶⣤⣳⣼⣧⣼⣧⣞⣤⣶⣶⣷⣦⡤⠉⠋⢀⣤⣤⡘⢷⡀
⠀⠀⠀⠀⡐⢸⠀⢸⣿⣿⢟⣡⣴⡾⢋⣉⠀⠨⡙⠟⢿⢿⣿⡟⠟⡫⠀⢀⣉⠻⣷⣤⣜⡻⣿⣿⠆⠈⡇⠄
⠀⠀⠰⠲⣷⣿⣜⣇⣿⠗⠁⢚⣿⣷⣿⣿⣷⣄⠈⠀⣸⣈⣘⣄⠈⢀⣴⣿⣿⣿⣿⣿⣂⠉⠺⣿⣼⣰⣿⣾⠂⠆
⠀⠀⢠⢊⣾⣿⣿⠿⠒⠉⣠⣿⡿⠁⠈⢿⣧⣿⣿⣟⠛⠛⠙⢛⣿⣿⣯⣼⠿⠁⠙⣿⣿⡈⠑⠢⢽⣿⣿⢧⡱⡀
⠰⠄⡡⠀⣾⣿⠕⠀⢀⡔⡫⣿⣿⣦⣴⡶⠛⠋⠑⠀⠀⠀⠀⠀⠐⠉⠛⠛⣶⣤⣴⣿⡯⡙⢄⠀⠀⢿⣿⣧⠀⣌⠢
⠀⡰⠁⣰⣿⣏⡖⡠⠋⠀⠔⡹⣿⣿⠉⠙⠆⢘⣦⣄⠠⠄⠤⢀⣠⣴⠂⠘⠋⢉⣿⣿⢍⠢⠀⠱⡄⢦⣿⣿⣅⠈⠆
⠀⣷⡿⣿⣿⣿⠁⢁⠀⠀⠀⣸⣿⣿⣷⣀⠘⢦⠙⣿⣿⣆⣶⣿⡿⢃⠼⠀⣰⣿⣿⣿⡆⠀⠀⠀⠈⠘⣿⣿⣟⠷⡾
⠀⠉⢰⢯⣿⣿⣧⠟⠀⠀⠰⠿⢿⣿⣿⣭⣷⣶⣖⣀⣻⣿⣿⣏⣐⣶⣶⣯⣵⣿⣿⡿⠿⠂⠀⠀⢳⣰⣿⣿⡻⡄⠃
⢀⢃⠏⢸⣿⣿⣟⡎⢀⡞⣠⠞⢟⣿⣿⣯⡉⣹⡿⢿⢿⣿⣿⡿⠟⡿⡏⢩⣿⣿⣿⠯⠳⣄⢣⠀⢦⣿⣿⣿⡇⢱⠰
⠉⢘⠀⠘⢹⣿⣿⣧⣾⠀⡏⠀⡼⢫⠿⣿⣇⢱⣷⣾⣾⣾⣶⣶⣷⣿⠀⣾⣿⠿⡛⢧⠀⢹⠈⣧⣼⣿⣿⡏⠂⠀⡇
⠀⢨⡄⠐⢸⣿⣿⣿⡟⣸⠀⠀⠀⠀⢰⣿⣿⣾⣿⡟⠛⢿⠿⠛⢿⣿⣾⣿⣿⡄⠀⠀⠀⢸⣄⢿⣿⣿⣿⡇⡄⣦⠇
⠀⠀⢁⢠⠈⢿⣿⣿⣾⣿⣆⣾⢠⡆⠚⣻⣿⣿⣟⠁⠀⢨⠁⠀⠘⣿⣿⣿⣏⠃⢰⡀⣧⣸⣿⣶⣿⣿⡇⠁⡇⡏
⠀⠀⠘⡞⣄⡸⣿⡏⣿⣿⣿⣷⣿⣇⡜⢹⣿⠹⣿⣷⣶⣶⣶⣶⣿⡿⢻⣿⡉⢦⣸⣿⣿⣿⣿⣟⢻⣿⢣⡰⢰⠃
⠀⠀⠀⠁⠘⢧⠈⡅⢻⡿⣿⢧⣿⣿⣿⣽⠃⠠⠘⢛⣤⣬⣥⣌⡚⠠⠆⢻⣽⣿⣿⣿⣹⣿⢻⡟⡘⢀⠾⠃⠉
⠀⠀⠀⠀⠀⠀⠑⠐⠈⣇⢻⢸⣿⣿⣿⣿⣶⣒⣫⣷⣾⣿⣿⣶⣯⣟⣚⣾⢿⣿⣿⢿⢏⠛⡾⢀⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠰⠗⠙⡄⢆⠀⢿⣿⣿⡙⣿⣿⡿⣿⣿⣿⣿⣿⣿⠿⣻⣿⣿⡿⠈⡘⡰⠁⠺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢦⡌⢻⣿⣿⣧⣸⡷⠙⣿⡿⠣⣿⣠⣾⣿⣿⡏⣡⡴⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢻⡻⠿⣿⢷⣷⣿⣵⣷⣿⣿⡟⣽⣿⠑⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⣈⠫⢻⣿⣿⣿⣿⡿⠁⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⢧⢸⣿⠟⠿⣋⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠈⠀⠀⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣰⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣠⡘⣄⠀⠀⢀⠀⠀⠀⢀⠤⠤⢱⣄⠒⠈⠀⠀⣀⠀⠀⠀⠀⠀⠰⣄⡀⢠⢿⡄⢀⣤⠂
⠀⠀⠀⠀⠈⡉⠀⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡤⠡⠵⡈⢯⡍⣩⠋⠀⠀⠀⠀⠀⠂⡑⠀⠁⠀⠈⠉⠙⠄⠀⠀⠀⠀⠉⠉⠋⠀⠛⠉⠃
⠀⠀⠀⠀⠀⢹⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠡⠈⡀⠀⠀⠀⠀⡞⣰⠃⠀⠀⠀⠀⠀⡤⠚⡀⠀⢁⡀⠀⠀⠑⢰⠀⠈⠛⠧⣀⠀⠀⠀⠀⠀⣠⠾⠛
⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡘⢄⡀⢀⡌⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⡁⢄⠀⡀⠘⠀⠀⢀⣴⣁⣄⠀⠀⠀⣀⣙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠌⠒⠒⠖⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡄⢸⠳⣈⣉⠠⠀⠀⠀⠉⠁⠀⣼⡄⠀⢤⡇⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢍⡡⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⠃
⠀⠀⠀⢀⠤⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠦⠤⣄⡀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⣀
⠀⢀⡜⠁⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠣⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⡾⠀⢸⡇⠀⠀⠀⠠⢀⠀⣈⠢⠀⠀⠀⠀⠀⠈⡄⠀⠀⡐⠀⠀⠀⠀⠀⢰⠇⠀⠈⠈⠲⡄⠀⠀⠀⠀⠈⣧⠀⠀⡀⠘⠀⠀⠀⠀⠀⠃⠘
⠰⡇⠀⠘⣇⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠄⠀⠉⠀⠀⠁⠀⡄⠀⠀⠀⢸⠀⠀⠊⠀⠀⢲⠀⠀⠀⠀⠀⣼⠀⠀⠈⠀⠀⠀⠉⠉⠁⠀⠀⠈⠁
⠀⢿⡀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠈⠦⣀⣀⣠⠼⠁
⠀⠈⠳⣄⠀⠀⠙⠛⠒⠒⢒⡡⠂⠀⠀⠀⠀⠀⠀⢰⠀⠰⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠈⠛⠲⠶⠶⠶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠶⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⡄⠀⠀⢠⠖⠒⠤⠀⠀⠀⢢⠀⠀⠀⠀⠀⢠⢰⡃⠀⡀
⠀⠀⢈⠕⢀⣠⡤⣤⣄⡐⠪⡀⠀⠀⠀⠀⢀⣀⡤⢤⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠈⠀⡶⠒⠒⠒⠒⠁⠀⠀⠀⠀⣀⣸⢸⡄⡄⢹
⠀⢠⠏⣰⠏⠂⠀⠀⠈⠝⡆⠱⡀⠀⠀⠔⠋⠀⠉⠉⠉⠁⠓⠬⡓⢄⡀⠀⢠⠀⣯⠐⠂⠐⠀⠀⢐⣀⣀⣀⠀⠀⠀⠀⠀⡇⣸⢸⡇⡇⢸⢸⡟
⠘⢹⠀⣯⠀⠰⣀⣈⠆⠀⡽⠀⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⢋⡄⠀⠺⣄⣈⣉⠁⠀⠀⠀⠀⢤⣄⠈⣷⠀⠀⠀⠀⡇⢉⣨⠥⣅⡚⢸
⠀⠈⣆⠘⢆⡄⠉⠀⢀⡰⠃⡼⠀⠀⠀⠑⠦⣂⣤⠤⠤⣔⣚⡭⠞⠁⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⠐⠂⢸⠀⠃⠀⠀⠀⠀⣇⡏⣴⠛⣦⢹⣸
⠀⠀⠈⠓⢤⣀⡁⢀⣀⡤⠚⠙⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⡠⠔⠒⠒⠲⠞⠁⣸⠀⠀⢈⠀⠀⠀⠀⠀⠀⢿⡳⣝⢒⣋⢜⡿
⠀⠀⠀⠀⠀⠈⠫⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠒⠒⠚⠃⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠙⠺⠿⠗⠋
⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⢰⠄⢴⠀⢦⠀⢰⠄⠠⡆⢰⠄⢴⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⠤⠀⢠⠀⢠⡄⢠⠀⢠⠀⠀⠀⠀⢠⡄⠀⣀⣠⠀⠐⠊⢡⠀⠀⠀⢀⢀⡄⡄⢀⠀⡀⢠⢠⡠⡀
⠀⢠⠄⡤⣴⠄⠀⠀⠠⢦⢤⠠⡄⠀⠘⡇⠘⣄⠈⠂⢸⠀⠊⢀⡜⢀⡞⠀⠀⠀⠀⢁⣀⠀⢩⠁⠀⠀⠀⠈⠦⠀⠀⡇⢸⡼⡇⢸⠀⠇⢸⢸⡇⢰
⠀⠸⠀⡇⡆⡰⠀⠈⡆⢹⠸⢰⡇⠀⠀⠘⢦⡈⠓⠄⢸⠀⠐⠋⣠⠞⠀⠀⠀⠀⠀⢢⣼⠀⠀⠀⡀⡇⠰⡆⠃⠀⠀⣇⠘⣗⠁⠈⠀⠀⠈⣺⠁⠸
⠀⠀⢆⠣⡇⢣⡀⢀⡜⢸⠗⡸⠁⠀⠀⠀⠀⠉⠐⠀⢸⡄⠀⠈⠁⠀⠀⠀⢠⠀⠀⠀⠇⠃⠀⠀⠛⠀⠀⠀⠀⠀⠀⠈⢆⠈⠒⢤⠀⡤⠒⠁⡔⠁
⠀⠀⢢⣆⠀⢀⠉⠉⡀⢀⡰⡖⠀⠀⠀⠀⠀⠀⠀⣀⣾⣇⡀⠀⠀⠀⠀⠀⠀⠁⠚⢀⠀⠀⠚⡘⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁⡔⠠⠀⠄⢢⠈
⣀⣀⣈⣘⣃⣀⣁⣈⣁⣈⣃⣁⣀⣀⣀⣀⣀⣐⣋⣁⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣈⣓⣒⣚⣘⣒⣒⣁⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣈⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠓⣤⠤⠤⠤⠤⠶⢍⡀⠀⠀⠻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡔⠃⠀⠀⠘⡆⠀⠀⠀⠀⠀⠈⠓⢄⠀⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠀⡤⠃⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⢚⠦⣀⢀⠓⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢸⠀⠀⠁⠀⢱⠀⠑⠂⠔⠂⠤⠤⠤⠴⢂⠀⡖⠊⠁
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⡇⠀⠀⣀⣂⣀⡀⢀⡤⠐⠒⢷⡦
⠀⠀⠀⠀⠀⠀⠘⠴⡀⢠⢧⡄⢆⠰⠤⣀⠀⡷⠶⡏⠁⢠⠀⣹⠹⣄⠀⠂⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠳⡳⠈⠉⠀⠀⠘⠦⠤⠔⠣⢀⠌⠁⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⡀⠀⠀⠀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠂⠽⣝⠉⣝⢉⡎⡃⠀⢤⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈⠛⢆⠈⣷⡱⣤⠜⣦⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⢄⠀⠀⠀⠀⠉⠀⠀⡇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠙⡦⠤⣀⠀⠀⣀⣰⠖⠉⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠤⠇⠀⠀⠀⠀⠘⡇⠀⠀⠀⣸⡑
⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠜⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣠⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠠⣇⠀⠀⠀⠀⢸⠀⠀⣀⣀⠤⠄⠃⠀⠰⡤⠤⡴
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⡩⢷⠈⠉⠉⣹⠁⠀⣷⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠈⠀⠀⢀⠔⠊⢆⡀⠑⠒⢶⡁⠉⣉⠽⣄⠀⠀⠀⢀⠄⠀⢇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠡⠀⠈⠀⠀⠀⢋⠉⠉⣷⠀⠙⠤⠤⠋⠳⡀⠐⠁⠀⠀⠀⠑
⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠆⢸⠀⠀⠀⠀⠑⠊⠁⢇⠀⠀⠀⠀⡰⠁⠃⠀⠀⠀⠀⢀⠇⠐
⠀⠀⠀⠀⠀⠈⠢⠤⠤⠤⠤⠤⠊⠀⠑⢄⡀⠀⠀⠀⡠⠊⠒⠒⠒⠉⠀⠀⠈⠢⠄⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠉⣭⠙⠒⠒⠒⠒⢉⠙⠒⠒⠒⢻⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣻⠉⠉⠙⢢⣄⣀⣤⠬⣁⣀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⢃⡼⠁⡏⠀⠀⡀⠈⢻⢶⣧⠀⠀⢀⣠⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠋⠩⡀⠀⣸⣤⣀⣉⡰⠃⠸⣄⡤⠖⣉⡽⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡤⠀⠀⠀⠀⠑⠦⣝⡋⠀⠀⠀⠀⠀⠀⢀⣀⣤⢚⡡⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⣞⠣⣄⠀⠀⠀⠀⢀⡎⠉⣿⣒⢒⣺⠋⡹⠃⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡆⠗⠀⠏⠑⠠⡔⠉⡇⢀⠂⠀⢀⠃⣞⠁⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠁⠳⠞⠀⠀⠀⠀⠀⡿⢸⠀⠀⠀⡘⢠⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢸⠒⠤⢀⡀⢴⢰⠃⠀⠀⠀⡇⠀⡄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⡠⠴⠋⠀⠀⠀⠀⡆⡃⠀⠀⠀⠀⢠⠔⠋⠂⠀⠀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡕⠀⠀⠀⣀⠴⠥⡀⠀⠀⠀⡠⠃⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠈⡶⢄⣈⡷⠀⠀⠈⣢⠤⠒⠁⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡸⠀⠀⠈⢳⠧⠶⢊⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣰⠃⠀⠀⠀⣴⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⡟⠀⢦⣀⣀⠀⠀⠀⣀⣀⠠⠜⢄⠀⠐⡠⠀⠀⠀⢢⠉⠉⡱
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⢠⢧⣀⠀⠐⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢸⠀⢀⡇⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇⠀⠈⠉⠒⠒⢲⠁⠀⠀⠀⠠⠀⢀⠀⠀⡇⠀⠀⢸⠀⠈⡇⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⡞⠒⠦⢄⣀⠀⠀⠀⠀⠀⢱⠀⡠⠊⠀⠀⠑⢄⠀⠀⠌⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢸⠀⢀⠂⠀⠐⠒⠩⡉⠉⠉⠉⢣⡀⠀⠀⠀⠈⠉⠉⠙⣳⠃⠰⡀⠀⠀⠀⠀⠀⡆⢠⠂⠀⠀⠐⢣
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠸⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠒⠒⠒⠒⠒⠊⠉⠁⠀⠈⢆⠀⠀⠀⠀⡰⢁⠆⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉⠀⢨⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠁⢡
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢠⡘⢿⣿⣿⣿⣶⣄⠀⠀⣀⡤⠖⠛⠋⢉⣉⣭⣍⣉⣉⣩⣭⣉⡉⠙⠛⠲⢤⣀⠀⠀⣠⣶⣾⣿⣿⡿⢃⡄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⢳⣤⡉⠻⣿⣿⣿⡧⠚⠁⠀⠀⠀⠒⠋⠉⠀⠈⠉⠉⠉⠀⠉⠙⠂⠀⠀⠀⠈⠓⢼⣿⣿⣿⠟⢉⣠⡞⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠀⠙⠻⣆⢸⡿⠋⠀⠀⢀⣠⣤⣤⣀⣀⣤⣤⠖⠁⠈⠲⣤⣤⣀⣀⣤⣤⣄⡀⠀⠀⠙⢿⡏⣰⠟⠋⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠈⠀⠀⠀⠀⠚⠉⠉⠉⠉⠉⠉⠉⠀⠀⡀⢀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠓⠀⠀⠀⠀⠁⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠴⠶⢶⣶⣤⣤⣶⡞⠁⠈⢷⣦⣤⣤⣶⡶⠶⠦⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⡔⠀⠀⠸⣦⣀⠀⠀⠀⢀⣉⠉⠁⢀⡀⢀⡀⠈⠉⣉⡀⠀⠀⠀⣀⣴⠇⠀⠀⢢⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⢠⡞⠀⠀⠀⠀⠈⠛⠿⣶⣄⠈⢿⣿⡛⠉⠀⠀⠉⢛⣿⡿⠁⣠⣶⡿⠛⠁⠀⠀⠀⠀⢳⡄⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⣴⠏⠀⡄⠀⠀⢳⣤⣤⣄⣈⠻⣿⣦⣌⡓⠂⠀⠀⠐⢋⣡⣴⣿⠟⣁⣠⣤⣤⡞⠀⠀⢀⠀⠹⣦⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⢠⢾⡏⢠⡞⢠⡆⢠⡈⢿⡇⠀⢾⡵⢢⡌⠩⣭⣅⠀⠀⣨⣭⠍⢡⡔⢮⡷⠀⢸⡿⢁⡄⢰⡄⢳⡀⢹⣷⡄⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣏⡀⢠⣟⠀⣼⡇⠸⣧⡘⣷⣤⡙⠢⠌⡀⠼⣿⠀⠀⠙⠇⠸⠋⠀⠀⣿⠧⣀⠡⠔⢋⣤⣾⢃⣼⠇⢸⣷⠀⣻⡄⢀⣹⣧
⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⣾⡏⠀⢹⣿⣆⠈⣿⠘⠙⠿⠛⠓⠀⣠⠸⠃⠀⣧⡀⢀⣼⠀⠈⡇⣄⠀⠚⠛⠿⠋⠁⣿⠁⣰⣿⡏⠀⢹⣷⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠘⢹⡇⠀⢹⠃⢰⠀⢻⣿⠀⢀⠀⢀⠴⠶⠶⠞⢁⡄⢀⣀⣉⠻⠟⣉⣀⠀⠠⡈⠳⠶⠶⠦⡀⠀⡀⠀⣿⡟⠀⡆⠘⡏⠀⢸⡏⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⠂⣼⠀⢸⣿⡀⢿⣄⠀⢀⣠⣤⠴⢋⣄⠀⠤⣍⡳⢞⣩⠤⠀⣠⡙⠶⣤⣀⡀⠀⣠⡿⢀⣿⡇⠀⣧⠐⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⣿⡄⠈⠻⢷⡌⢻⠀⠉⠀⠀⠶⠻⢿⣷⣶⣶⣭⣭⣴⣶⣾⡿⠟⠶⠀⠀⠉⠀⡟⢡⡾⠟⠁⢠⣿⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠋⣿⡀⠀⢻⡿⣷⣦⡀⠙⠀⢀⠄⠀⣠⣤⣤⣈⠙⢻⣿⣿⣿⣿⡟⠋⢁⣤⣤⣄⡀⠠⡀⠀⠋⢀⣴⣾⢿⡟⠀⠀⣿⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣰⡄⠀⠘⢿⠻⢆⠀⣟⠀⢉⣥⣤⣀⠙⠂⠀⠉⢻⡟⠋⠀⠐⠋⣁⣤⣬⡁⠀⣻⠀⡰⠟⡿⠃⠀⢠⣆⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⢿⣦⣄⡀⠀⠀⠀⢿⣄⠈⢀⣀⡈⠙⠓⠂⠀⢸⡇⠀⠒⠚⠋⢉⣀⡀⠁⣠⡿⠀⠀⠀⢀⣠⣴⡿⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⠀⠻⣿⠻⢷⣦⣄⠀⠙⣿⣿⣟⠛⢳⡖⠶⠶⠿⠛⠶⠶⢶⡞⠛⣻⣿⠟⠉⠀⣠⣴⣾⠟⣿⠟⠀⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⣦⡀⠀⡀⠉⠙⢷⣄⠈⢿⣿⡄⢸⣧⣾⣿⣿⣿⣿⣷⣼⡇⢠⡿⠁⠀⣠⡾⠋⠉⢀⠀⢀⣴⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠙⠿⣷⣿⣶⣄⢈⢻⣆⠈⣿⣿⣼⣿⣿⠿⢿⡿⠿⣿⣿⣧⣿⠃⠀⣰⡟⡁⣠⣴⣿⣾⠿⠋⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠏⠻⣿⣷⣿⣆⠸⣿⣿⣿⣿⣄⣘⣃⣠⣿⣿⣿⣿⠀⣰⣿⣿⣿⠟⠹⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⢿⡄⠻⣷⡙⢏⡛⠛⠛⢛⡹⢋⣾⠇⢠⡿⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⠴⠖⠒⠒⠲⠦⠴⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⢀⠴⢤⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡗⠀⠘⠤⡼⠁⠀⢴⣹⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠉⠉⠁⢀⣀⣀⣀⣀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠁⠀⠀⣼⠟⠉⠉⠉⠙⢿⣷⣦⡀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣄⠀⠀⠀⠀⠀⠠⣾⣆⣄⣐⠲⣶⠀⠩⢿⡷⣄⠀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⣋⡄⠀⡀⠀⠀⠀⣿⣿⣿⣱⣧⡉⠀⠀⠀⢻⣿⡀⠀⠤⠾⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠁⠈⠀⠀⡁⠀⠀⢸⣿⣿⣿⣿⡟⠁⠀⠀⠀⠘⣿⡇⠀⠀⠘⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⢻⡟⠛⠛⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⡄⠀⠀⠀⠀⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⢾⣿⠟⠀⠀⠀⠀⠀⠀⠉⠓⠆⣀⣤⣤⣀⣀⠀⠀⠀⠘⢻⡟⠁⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠿⠋⠀⠀⠀⠀⠘⠃⠀⠀⠀⣀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣎⣿⣿⣦⡀⠀⠀⠀⣤⡄⠀⣀⣤⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣴⣾⣿⣫⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡻⣿⣿⣦⡀⠀⠈⠀⢸⡇⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⣷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣷⡂⠀⠀⠀⠀⠐⠀⠂⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣷⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡧⠀⠸⣿⣿⣿⣿⣿⡿





⣿⣿⡟⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠺⡶⠿⠿⡷⠀⢠⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⢠⠄⠤⣀⡤⠔⠢⡀⠀⠀⠘⡷⠤⠒⠉⠀⡀
⠀⠀⠀⠈⠁⡀⠀⡿⠀⠀⡇⢀⡀⠀⠀⠀⢀⡎⠉⠑⡍⠀⠀⠀⢀⣧⢠⡼⠵⢤⣀⣱⠀⠄⠊⠀⠀⢘⠉⢵⠈
⠀⠀⠀⠀⠀⠁⡰⠥⠀⠀⢣⠀⠁⡠⢤⣴⠟⠻⠀⢀⣼⠀⠀⠐⣅⠈⢲⡀⠀⣤⠑⣾⠉⠛⢆⠀⢠⢬⠉⢁⣤⡀
⠀⠀⠀⠀⠀⢠⠃⣠⣀⣠⣌⣆⠀⠣⡴⡅⠀⠀⠙⡿⠿⠀⠀⠀⢈⡉⢶⠬⠦⣄⣠⡏⣦⠤⠚⠀⡈⢊⡴⣋⠜⠁
⠀⠀⠀⠘⢁⣿⣿⣿⣿⣿⣿⣿⣆⠀⣷⠹⡄⠀⣐⣒⣒⡄⠀⠀⢀⣸⠈⢆⣀⡤⠛⠒⠊⠀⠀⠀⡤⣋⠜⢁⣀⡀
⠀⠀⠀⠀⢎⢻⣿⣿⣿⣿⣿⡿⢈⠀⢸⠖⠚⠚⠒⠒⠒⠚⣆⠀⠀⠙⠀⠀⠀⠀⣠⣴⠶⢒⡚⠿⣿⠁⠀⠀⢸⣏
⠀⠀⠀⠀⣈⠑⠒⠂⡒⠒⠒⠈⠁⠉⠘⠒⠦⠤⠤⠦⠆⠒⠃⠀⠀⣤⡆⠀⠀⡾⢋⠖⡒⠒⡌⢢⠙⡄⠀⠀⢸⣍⠁
⠀⠀⠀⠈⠚⠁⠀⠀⠻⠯⠀⠀⠀⠀⠀⠄⠀⠀⠀⠔⠒⠄⠴⠀⠀⠈⠁⠀⠀⡇⠸⣀⡛⠘⡜⡸⠀⡇⠀⡆⢸⠮
⠀⠀⠀⠀⢴⢀⡶⠚⢲⡀⠀⠓⠺⠰⠀⠀⠀⠀⠀⠊⠒⠤⠴⠂⠀⢠⣤⠀⣄⠹⣤⠈⢉⠉⠐⣡⠞⠀⠀⠧⠿⠁
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠁⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠉⠀⠀⢠⠴⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠙⠓⢦⡀⠀⣠⠞⠛⠳⡄⠀⢰⡆⠉⠂⣴⠶⠶⠄⠈⢀⡀⠀⢀⡄⠀⣠⠤⢤⡒⠀⢸⣂⣀⣀
⠀⠀⠀⠀⠐⣆⠀⠀⢀⡇⢰⡇⠀⠀⠀⠉⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⣾⢧⠀⢸⡇⢰⡇⠀⠈⠁⠀⢸⠉
⠀⠀⠀⠀⠀⡌⠛⠒⠋⠀⢸⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⣟⠛⠛⠂⠀⡟⠈⣇⢹⠃⢺⢁⠀⠀⢀⡀⠸⠶⠖⡒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⢧⣀⣀⣠⠄⠀⣸⡇⠈⠁⣿⣤⣤⡄⠸⠃⠀⠘⠟⠀⠈⠦⠤⠖⠋⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠉⠉⠀⠀⠀⡉⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⢒⣤⣤⡄⢀⡴⣊⠥⢄⠉⢆⠲
⠀⠀⠀⠀⢠⣄⠈⠉⠀⠀⠠⡆⠀⠀⠘⠀⠈⢫⡂⢠⠛⠒⠠⢼⡀⠛⠑⠙⠀⠀⣀⠁⡀⡄⢠⣾⡔⠁⠀⢀⠃⣼⢠⣤
⠀⠀⠀⠀⠀⠁⢰⠭⡆⢰⠯⡆⠀⡟⡇⠀⠀⠀⡸⠡⠀⠀⠠⠀⠈⠟⢠⣄⣤⠀⠳⣄⢀⡗⠋⠀⠀⠀⢴⣧⡴⠃⢸⠏
⠀⠀⠀⠀⢶⣾⣯⡿⡽⠽⠯⡿⠿⠿⣿⠟⠀⠘⢇⠀⠀⠀⠀⠀⠀⡆⠈⠻⠃⠀⢂⡬⠛⠬⡀⠀⠀⠀⠔⠈⠀⢻⣮⡄
⠀⠀⠀⠀⢸⣿⠀⣶⡇⠀⣿⡇⢠⣿⣿⠀⠀⢀⡠⠣⡀⠀⠐⣂⣀⡇⠀⠀⠀⡰⠉⠀⠀⠀⠉⢦⠀⠀⠀⠀⣦⣠⠟⠁
⠀⠀⠀⠀⢸⣿⠀⣿⡇⠘⠿⠇⠀⠿⣿⡆⠀⡉⣀⢀⠙⠉⠀⠀⠀⠀⠐⠀⠙⠁⠀⠀⠀⠀⠀⠸⠟⠄⠀⠀⠸⠛⠋
⠀⠀⠀⠰⠿⠿⠿⠯⠿⠿⠯⠿⠽⠽⠽⠿⠃⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡤⠴⠖⠚⠋⠉⠉⠉⠈⠉⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣞⡵⠊⠀⠀⣀⡤⠴⠶⠒⠲⢦⡄⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⡴⠋⠁⠀⠀⠀⢀⣠⠜⠉⠉⣛⡆⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠘⠤⠾⠿⣿⡿⠟⠋⢀⣀⠴⠚⠁⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⣀⡤⣄⣀⣤⠤⠖⠒⠉⢁⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠷⠚⠉⠁⠀⠀⠀⣀⣠⠤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠚⠋⠙⢷⡄
⠀⠀⠀⠀⠀⠀⢰⠃⢀⣀⣠⠤⠖⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠉⠀⠀⠀⠀⠀⢨⡟
⠀⠀⠀⠀⠀⠀⢿⣛⣿⣶⣶⣶⣶⣶⣖⣒⣒⣒⣒⣒⣚⣛⣛⣛⣭⡭⠿⠛⠀⠀⠀⠀⢀⣤⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⣀⣠⣤⡴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠶⠶⠞⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠶⠦⢤⡞⢠⢠⢸⣿⣒⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠼⡿⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⣠⠄⠈⡁⠀⠘⠀⠾⠂⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣦⡠
⠀⠀⠀⠀⡤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣷⡀⠀⠀⠐⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣝⠷⠾⢓⣆⠀⠂
⠀⠀⠀⢠⡤⠆⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⣦⠖⣾⣯⡉⠀⠀⡀⠀⣴⣷⣿⣼⣿⠗⠀⠀⡀⠀⠀⠀⠀⠀⠟⣷⢿⡏⠉⠂
⠀⠀⠀⢀⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠋⠐⣴⣷⣏⣩⡅⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠠⠀⣄⠀⠀⠀⠀⠀⢤⠿⠙⠍⠋⠈⠀⠀⠀⠀⣠⣤⢰⣦⣄⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⡀⢰⡤⠀⡀⠀⠀⠀⠀⢺⡃⢰⡆⠀⠈⡀⠀⠀⠀⠻⠀⠀⡅⢻⠋
⠀⠀⠀⡀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠁⠰⠆⠀⠀⠈⠁⠀⠙⣦⡀⣿⠀⠀⠀⠘⢶⡰⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⡤⠀⠁⠀⢠⡄⠀⠀⠀⠀⠀⠿⠿⠋⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠚⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠁⡀⠀⢠
⠀⠀⠀⠰⠂⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢶⣴⣇⣀⠀⠀⠀⡀
⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⠀⠀⣾⣭⣻⢻⡷⡔⠚⠁
⠀⠀⠀⡀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠈⠀⠀⠐⣟⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⣼⣩⡄⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⢿⡿⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⠀⠀⠀⠄⠀⠀⠀⠁⣼
⠀⣀⠀⠀⠀⠰⣄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠐⢀⣠⣞⠁⠀⠀⠀⢀
⠀⠁⠀⠀⠀⠀⠘⠀⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠿⠀⠀⠀⠀⠄⠉⠙⠛⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⢈⠀⠀⠀⠀⠰⠦⠠⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⠀⠈⢘⠀⠀⠐⠀⠀⠀⠘
⠀⠀⠀⠀⠁⡄⠃⡀⠀⠀⠀⠀⠀⠀⠀⠂⠐⣲⣾⣿⣞⢲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠈⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡚⢲⡄⠀⠀⠘⠟⢻⣿⠛⠃⠀⠀⠂⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠎⣰⠃
⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠉⠳⡖⠀⠀⢠⠆⠀⠈⠀⠂⠀⠀⠀⡀⠠⢥⣮⣿⣿⣷⡀⠀⠠⠄⠀⣇⠀⠉⢸⠀⠈
⠀⠀⠀⠀⠂⠀⠀⠀⢠⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠛⠻⠻⠟⠑⠦⠀⠀⠀⠉⠀⠀⡀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⣴⠆⠁⠀⢀⣠⠾⠋⠀⠀⠀⠀⠀⠒⠀⠀⠠⣤⠉⠀⠀⠄⠀⢨⠀⠀⡀⠀⢼⡴⠁⠀⣀⣀
⠀⠀⠀⠀⠁⠀⠀⡼⠀⠀⠀⠀⠀⣀⡴⠒⣫⡟⠁⠀⠀⠀⠀⢰⡀⣠⡔⠀⠀⠀⠛⠀⠀⠀⠀⠀⠗⠀⠞⠀⠀⠈⠁⠐⣴⠛⠃
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣠⠞⣁⡴⠚⠁⠀⠀⠀⠀⠀⠐⠘⢿⣿⣆⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⡾⠁⠀⠀⢄⣸⣿⣯⠤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⣸⣧⠖⠚⠙⠋⠉⠁⠀⠀⠀⠀⠀⠙⠲⠦⠤⠴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢴⣶⡄
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢫⣭⣙⠳⠒⠶⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⣦⠴⠒⠀⠀⣀⠤⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠟⠉⠀⠀⠀⠀⠊⣀⣤⡴⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠟⠁⠀⠀⠀⠀⠀⠀⠀⡨⣱⣿⣷⣤⣠⠉⡙⠻⣝⢿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⡀⡄⡴⣠⣾⣿⣿⣿⢿⢿⣿⣾⣷⣧⣏⣗⠶⠭⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⢸⢸⣧⣿⣿⣿⣿⣷⣷⣒⠺⠾⠼⢽⣏⡟⡭⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⢸⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣭⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣄
⠀⠀⠀⠀⠀⡠⢂⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠿⠿⣿⣿⣿⣿⣆
⠀⠀⠀⢠⣞⣴⠿⠋⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢟⣛⣯⡽⠖⠛⠉⠀⠀⠀⠀⠉⠻⢿⣿⡄
⠀⠀⣰⡿⠛⠁⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣫⡭⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧
⠀⡼⠋⠀⠀⠀⢰⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠈⠀⠀⠀⠀⠀⠘⣇⢿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣇⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣦⡻⣦⣀⠀⠀⢀⣀⣀⣤⣴⣶⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⡶⠶⠂⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⣷⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠘⡛⣳⠳⢭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣏⢹⡏⣹⢷⡀
⠀⠀⠀⣧⠀⠙⡷⠖⠛⡇⠀⠀⠀⠀⠀⢀⠄⠀⠀⣤⣾⣿⢿⣿⣶⣄⠀⠀⠀⠀⠀⢸⡧⠬⣿⣿⠥⢼⡇
⠀⠀⠀⠈⠛⠚⠐⢶⠶⠃⠀⠀⠀⠀⠀⠀⠀⢀⣼⣵⣄⣩⣹⣿⣿⣿⣷⡄⠀⠀⠀⠀⠳⣎⢸⠈⣢⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃⠘⠇⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠀⠀⠈⠉⠉⢿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣄⠀⡀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⢸⣿⣿⡟⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠲⣶⠟⠛⢧⡴⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠿⠷⣤⣶⠄
⠀⠀⠐⠺⣯⣀⢀⣼⠷⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢯⠀⠀⣹⣏
⠀⠀⠀⠀⠛⠙⡟⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⡿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢱⢉⡿⢣⢀⠀⠀⠀⠀⠀⠀⠊⠀⠐⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡘⣿⡧⢊⡾⠀⠀⠀⠀⠀⠈⡄⢠⣴⣦⡔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣻⢗⢾⠟⠁⠀⠀⠀⠀⠀⠀⠠⡸⣷⣾⢇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⡶⠋⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠤⠤⠤⠤⠤⢤⣤⣤⣄⣀⡤⠞⠀⠀⣠⠿⢤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠛⢲⣤⠶⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡷⠞⠉⡇⠀⠀⠀⠀⢀⣠⡤⠴⠖⠒⠒⠒⠲⠶⢤⣀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣔⣊⣻⠀⠀⢰⣇⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠐⣅⠀⠀⣀⡼⠀⠀⠀⠀⢙⡾⠁⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠉⠁⠀⠀⠀⠀⢀⣼⣥⣤⡴⠋⠉⠉⠉⠉⠳⣦⣤⡤⠤⠴⠖⠋⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡀
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⣀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠒⠋⠉⠉⢳
⠀⠀⢠⣞⣀⣀⣤⣤⣴⣖⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠤⠤⠤⠶⠖⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣼
⠀⣰⠿⠶⠦⠤⠤⠤⠤⢬⣭⣭⣍⣉⣉⣉⣉⣉⣉⣉⣉⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠉⠉⠉
⣴⣧⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡶⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡄⠀⢰⡖⠤⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⡶⠋⠀⢀⢸⠀⡠⠛⠛⠛⠛⠛⠉⠉⠀⠀⢀⡴⠿⠛⠛⢛⠛⠛⠋⠈⡇⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡠⠖⠉⢰⡴⠚⠁⠀⠀⠀⠘⠾⠊⠀⡠⣶⡄⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠐⡀⣰⠁⠀⠀⡠⡀
⠀⠀⠀⠀⡀⠄⢠⣤⠔⠉⢀⡤⠞⠁⠀⠀⠀⠀⠀⣀⠀⢠⡀⠀⠳⠿⠥⠀⠀⠶⠃⠸⠀⠀⠈⠣⣀⠀⠀⢀⣨⠔⠁⠀⠀⠀⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣼⠋⠀⠀⠀⠀⠶⠂⠀⢠⠏⠂⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⡤⡸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠿⠀⠀⠀⠀⠸⢦⣾⣥⢤⣄⠀⠀⢆⠠⢴⢀⢀⠎⠀⠉⠒⠢⢄⣀⠀⢀⣀⠀⠀⠀⠀⠙⣿⡤⣤
⠀⠀⠀⠀⠀⠀⠋⠀⢰⣖⡒⢤⡀⠀⣄⢠⠀⠀⠀⣾⢁⣀⣠⣟⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢰⠉⠁⠀⠟⠢⠀⣠⠀⡮⠖⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡮⠉⢹⠗⠋⡀⠐⠂⠂⠀⠀⠉⠛⢡⠏⢀⣽⠂⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⡿
⠀⠀⠀⠀⢀⠤⠐⠋⠁⢀⠏⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠋⡿⠋⠉⢳⡄⠀⠀⠈⠢⡐⠀⠀⠠⢀⡸⠀⠀⠀⠀⢀⠀⣹⠧⢒⠆
⠀⠀⠀⢰⣿⣶⣤⣄⡀⠘⢄⠀⠀⡆⢀⠀⠀⢠⡄⠀⠀⠀⠸⣅⠀⠀⠚⠀⠀⠀⠀⠀⠘⡀⠒⠊⠉⠀⠀⠀⠀⠀⢺⡷⠛⠊⠁
⠀⠀⠀⠘⣿⣿⣿⣿⣷⣦⡼⠀⠀⡏⠙⠀⠓⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡞⠙⡆⠀⠼⠁
⠀⠀⠀⠀⠈⠙⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⢄⣀⠀⢠⣤⡌⢀⡄⠀⠀⠀⠸⡿⠀⠀⠀⢀⣼⠙⡣
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣠⠖⠛⠃⡀⣀⠀⠀⠀⣞⣽⣿⣿⣿⢶⠚⠁⠘⠀⠉⠀⠁⠀⠀⠀⠀⢿⠀⢀⣴⠟⠊⢹⡅
⠀⠀⠀⠀⠄⠸⣔⡽⠀⠀⠠⢀⣘⣧⣤⠤⠴⠞⠁⠀⠀⠀⠀⢹⢹⠁⣀⣌⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠭⡟⠁⠀⠀⢸⠅
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢳⣏⠉⠐⠳⣄⠀⠀⠀⠀⠀⠀⠀⠘⡞⣿⣇⢸⡇⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠉⠊⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢰⠹⣧⠀⠀⠂⢉⣻⣦⣤⡼⠀⠀⣀⡀⠀⠀⠀⠀⢃⢿⣿⠀⣿⠀⠀⣀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢄⡀
⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⠸⣍⣀⠈⡧⠀⠀⠘⠉⠀⠃⠀⠀⠄⠸⡸⣿⡇⢻⡇⠘⠋⠉⠻⡁⢘⡦⠀⠀⠀⠀⠀⠀⠀⠀⠧⡡⠃
⠀⠀⠀⠀⣀⡀⠀⠀⢈⡴⠒⠛⠻⣏⠃⠀⠀⠀⣀⡠⢠⡀⠀⠠⠀⡇⢿⣿⣘⣿⠀⠀⠀⠀⢸⢎⠀⠀⠀⠀⠑⠁⠈
⠀⠀⠀⠀⠛⠁⠀⠀⠾⠀⠀⠀⠀⣸⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠘⢼⣟⡩⠋⠀⠀⠀⠰⠃⠀⠳⣄⠀⠀⣠⣐⣒⣒⢒⡄
⠀⠀⠀⠀⢞⣹⠀⢴⡦⠀⠀⠐⠋⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⢠⠆⠀⢀⡠⠖⠉⠀⠀⠈⡟⠛⠛⡟
⠀⠀⠀⠀⠀⣀⣶⠶⠦⢤⣤⡤⢤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠙⠙⠈⠂⠾⠃⠀⠀⢰⣹⠆⠀⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⡆⠀⠸⡀⠀⠀⣀⣤⣤⣤⣤⣖⠚⠉⠉⠉⠉⠉⣷⠴⠶⠶⠶⠶⠚⠃⢀⠀⠀⢠⢊⣁⣀⣨⣌⣲
⠀⠀⠀⠀⡀⠸⣄⣀⣀⣠⠇⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠛⠛⠛⠃⠁⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠸⢿⣿⣿⣿⣿⠟⠰⠄
⠀⠀⠀⠰⠀⠀⠀⠀⠘⠒⠒⠒⠛⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠓⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢦⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡀⠀⠀⠉⠉⠛⢶⣄⡀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢻⡇
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢳⡀⠀⠀⠀⠀⠀⠉⠳⢌⠀⣠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠹⠃
⡼⠁⠀⠈⢆⠀⠀⣀⡀⠀⠀⠀⢶⠘⠻⣆⠀⠀⢀⡤⠂⠀⠀⠙⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠂⠀⠀⠀⠁
⢷⡀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠓⠀⠁⠀⠀⠀⠀⠀⠀⠈⢄⣤⣀⣄⣦⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒
⠈⢷⡀⢦⠀⣈⣆⠀⠀⠀⠀⠀⢰⣲⡀⠀⠀⢠⣤⣄⠀⠀⢄⡀⠀⠀⢘⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⢀⣠⣤⡶⠏⠀⠀⠀⠀⠀⣄
⠀⠈⣷⣾⣇⠹⣿⣧⠀⠀⠀⢀⣿⢿⠆⠔⠛⠃⠊⠚⠻⣶⣿⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣴⡾⠛⢉⣡⣴⠀⠀⠀⣀⡤⠞⠃
⠀⠀⠘⣿⣿⣆⠹⣿⣆⠀⠒⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠉⠉⠉⠫⠶⡀⠀⠀⠀⠀⠀⢷⣠⣶⣾⣿⣿⣿⠶⠚⠋⠁
⠀⠀⠀⠘⣿⣿⣦⡟⡿⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠛⠛⠃⠀⠸⣍⠀⠀⠀⠠⣙⣦⣄⠀⠀⠀⢀⣀⡀⢠⠤⠖⠁⠀⠀⣠⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠈⠳⢦⣄⣠⣜⠻⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠹⡇⢸⣧⡀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⡀⠀⠀⠀⠀⢀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠙⠻⣿⣶⣌⠙⠿⣿⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢣⠀⠈⠻⢿⣷⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠹⢦⣄⣠⣀⡤⡤⢒⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠙⠾⣿⣷⣴⠿⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠑⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢣⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠾⢦⠀⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣐⣦⠤⠖⠚⢉⡉⠤⠄⠒⢄⠈⠳⡀⠀⠢⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⡁⠤⠐⠂⠉⠀⠀⠀⠀⠀⠈⣆⠀⠙⣄⠀⠀⠀⠐⠕⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠘⣆⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠠⡀⠀⠂⠀⡜⠁⠀⠀⡎⠀⠀⠀⡀⠤⠄⠀⢤⡀⣀⠤⠊⢁⠇⠀⠀⠘⣆⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠄⠀⢀⠎⠀⠙⠳⣄⣇⣀⣀⡮⣄⡀⠤⢴⡺⢃⡠⠔⠊⠁⠀⢀⡤⠊⢹⡀⠀⠈⢞⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠞⠖⣯⠚⣉⢻⠧⠤⠤⠭⠭⠟⠛⠛⠉⠁⠀⠀⣀⠤⠚⠁⠀⣠⠜⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠀⠀⡌⠘⠼⢿⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉⠀⠀⢀⡴⠊⠁
⠀⠀⠀⢀⡠⠔⠚⠂⠤⠤⠤⠭⠭⢭⡤⣤⣤⠀⠒⠒⠉⠁⠀⠀⠀⠀⠀⡰⠋
⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⡅
⡜⢅⡀⠀⠀⠀⠀⠀⣀⣀⣀⡠⠴⠫⠤⠋⠈⠀⠀⠀⠀⢀⣀⠤⠖⣋⠴⠃
⠈⠐⠨⢑⡒⠤⠤⢤⣀⣁⣀⣀⣀⣲⣶⠥⠤⠤⠒⢒⣉⡩⠴⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⡄⣀⣠⣴⣶⣿⣿⠿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⣀⣤⣶⣿⡿⢿⣿⣿⠁
⢠⣴⣿⡿⠟⠋⠁⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⢿⡟⠁⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢻⣿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⢀⣾⡿⠉⣷⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⣾⣿⠁⠀⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢀⣠⣶⣿⡷⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠰⠶⠶⣿⣿⣤⣤⠀⠀⣸⣿⡇⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠇⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⣿⣿⠇⣴⠟
⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⠉⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⣿⣿⡿⠃
⠀⠀⣴⣾⣿⠿⠋⢹⣿⡇⠀⠀⢀⣴⣿⣿⠿⠿⢿⡆⠀⠀⣼⣿⠃⠀⠀⠀⣿⣿⠇⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠉⠋⠁⠀⠀⣿⣿⡇⠀⠀⣾⡿⠋⠀⠀⢠⡆⠀⠀⢸⣿⡟⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠸⣿⣿⠋⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢰⣿⠁⠀⠀⣰⣿⠃⠀⢀⣾⣿⡇⠀⠀⠀⣸⣿⡇⠀⠀⠀⢀⣼⣿⣿⡀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢿⣦⣴⣾⢿⣧⣠⣴⠏⠹⣿⣧⣤⣴⠞⠉⢻⣷⡀⣀⣠⠞⠁⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠈⠙⠉⠀⠈⠛⠋⠁⠀⠀⠉⠋⠉⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠈⠛⠃⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡿
⣿⣿⠛⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠄⠀⠀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠂⠠⢄⣀⣀⠠⠤⠶⢀⠀⠤⡄⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠟⠒⠻⢤⣀⠀⣀⣐⡀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠰⢠⠏⢿⠁⠀⠉⠙⣶⣦⡀⠀⠀⣀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⠊⢅⠀⢘⢦⣀⣁⡼⠁⢇⣠⠔⣉⢽⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡰⠀⠀⠀⠈⠑⠬⣙⡁⠀⠀⠀⢀⣀⢠⠤⠎⠡⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡾⣝⠢⣀⠀⠀⢀⡼⠀⡝⠭⠭⡟⣁⠋⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡇⢈⢀⠃⠈⠐⠁⢰⠀⠃⠀⠰⢀⠇⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢱⡀⠀⠀⠀⡘⡠⠀⠀⢠⡆⢠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢀⡜⠀⠈⠁⢉⠁⠁⠀⠀⠀⠃⡠⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢈⠄⠀⠀⢀⣠⡊⠀⠀⠀⢀⠎⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⢄⡀⡯⠀⠀⠑⣄⠠⠊⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠏⠀⠈⢳⠤⠖⡊⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⡄⠀⠣⠄⠀⠀⠀⢠⠄⠒⠁⠤⠀⠤⠀⠀⠀⠳⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠓⠤⢀⡀⢀⡇⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⢸⢄⡀⠀⠀⠀⠀⠀⠀⢇⠀⢀⠀⠀⠀⡀⠀⠀⡈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠆⠀⠠⠀⠀⠉⠒⠒⠠⠐⢦⠀⠈⠉⠒⠒⠂⠤⢤⡄⠐⠁⠀⠀⠀⠈⠀⢀⠄⠀⠀⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠈⠓⠤⠤⠤⠤⠤⠒⠉⠀⠨⡀⠀⠀⠀⢀⠂⠂⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠂⠁⢠⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⣟⣟⣿⣻⣛⣟⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣛⣿⣟⣻⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⠴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⢀⣠⠤⠖⠚⠋⠉⠉⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⢠⣤⢤⣀⣾⢸⡇⠀⠀⢀⡀⠀⠀⢀⣾⡿⠁⢀⡇
⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⣀⣠⡤⠤⢬⣿⣼⣹⣻⢸⠃⠀⢠⣿⡇⠀⢀⣾⡿⠁⢀⠞
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣁⡀⠀⠀⢸⡟⠁⠀⢀⠀⠀⠈⠻⣿⠟⠓⢲⣿⣿⣄⣀⣼⣿⠥⠚⠁
⣧⠀⠀⠀⠀⢀⣀⠤⠴⠒⣻⣿⠏⠉⢻⡆⢀⣴⡶⢢⣰⣿⠃⠀⢀⣴⡏⠀⠀⣼⣿⠁⠀⢠⣿⢇⣴⣶⡆
⠙⢦⡀⠀⠀⠉⢉⠉⠓⢦⣿⡟⠀⠀⠀⠀⣾⡟⠀⢸⣿⡟⠀⠀⣸⣿⠇⠀⠀⣿⡏⠀⠀⣼⣿⡞⣸⣿⠃
⠀⠀⠙⠳⠶⡶⠟⠀⠀⢸⣿⢧⠀⠀⠀⠀⣿⡇⣠⢻⣿⠃⠀⢠⣿⡿⠀⠀⣼⣿⠀⠀⢠⣿⡟⠀⣿⡟⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠘⡆⠀⠀⠀⠈⠉⠁⢸⣿⠀⡰⠃⠹⣧⣠⠞⠘⣿⠀⡰⠃⠙⠁⠀⣿⡇⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⢠⠇⠀⠀⠀⠀⠀⠀⠀⠙⠊⠁⠀⠀⠀⠀⠀⠀⠙⠊⠁⠀⠀⠀⠀⠈⠓⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠛⠛⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋⠀⣠⣴⣦⣄⠀⠹⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠁⣀⣼⣿⣿⣿⣿⣧⣀⠈⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠉⢻⣿⣿⣿⣿⡟⠉⢀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣤⣀⠀⠀⠙⣿⡇⠀⢸⣿⣿⣿⣿⡇⠀⢸⣿⠋⠀⠀⣀⣤⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⣠⣾⡿⠋⠉⠀⠀⠉⠛⢿⣷⣤⣤⣿⡇⠀⢸⣿⣿⣿⣿⡇⠀⢸⣿⣤⣤⣾⡿⠛⠉⠀⠀⠉⠙⢿⣷⡄
⠀⠀⠀⣠⣾⡿⠟⠀⢠⣾⣿⣿⣷⡄⠈⠉⠉⠉⠉⠁⠀⢸⣿⣿⣿⣿⡇⠀⠈⠉⠉⠉⠉⠁⢠⣾⣿⣿⣷⡄⠀⠻⢿⣷⣄
⠀⠀⢰⣿⠏⠀⣠⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣄⠀⠹⣿⡆
⠀⠀⣿⣿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣿⣿
⠀⠀⢸⣿⣆⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⣰⣿⡇
⠀⠀⠀⠹⢿⣷⣄⠀⠘⣿⣿⣿⡿⠃⠀⣀⣀⣀⣀⡀⠀⢸⣿⣿⣿⣿⡇⠀⢀⣀⣀⣀⣀⠀⠘⢿⣿⣿⣿⠃⠀⣠⣾⡿⠏
⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠉⠁⢀⣠⣾⡿⠛⠻⣿⡇⠀⢸⣿⣿⣿⣿⡇⠀⢸⣿⠟⠻⢿⣷⣄⡀⠉⠉⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⠿⠟⠋⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⡇⠀⢸⣿⠀⠀⠀⠙⠻⠿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⣸⣿⣿⣿⣿⣇⠀⣸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢈⣛⣛⣛⣋⣿⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣛⣛⣛⣻⣟⡏⠀⢻⣛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⢹⣿⣿⣿⣿⡏⠀⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⢸⣿⣿⣿⣿⡇⠀⢸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⣀⣼⣿⣿⣿⣿⣧⣀⠀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠉⢻⣿⣿⣿⣿⡟⠉⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄⠀⠙⠻⠟⠋⠀⣠⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣤⣤⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣢⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠠⠄⠂⠀⠀⣐⡒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣗⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⣠⢠⡀⠀⠀⣐⠃⠈⠀⢀⠐⠀⠖⢀⣀⣀
⠀⠀⠀⠀⠀⢀⡞⢱⣜⡉⠴⠈⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⣀⠀⠀⠀⢀⠀⠉⡿⢲⣤⡞⣠⣶⢲⣶⠦⠥⠒⠛⠉
⠀⠀⠀⢀⠠⠾⠶⠯⠤⠤⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠸⠋⠀⠈⠉⡀⢄⠀⠀⢀⣴⡌⡟⠉⠉⡉⠛⠟⢛
⠀⠀⠀⠀⠀⠀⡀⠀⣴⡶⡖⡄⠀⢿⠀⠀⠀⠠⡔⠂⣤⠀⠀⠀⠀⢳⡆⠙⢃⣠⣿⣆⠀⠀⠙⢛⣷⣶⣤⠖⠒⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠧⠴⣾⠇⢱⡱⡶⣾⠀⠀⠶⡶⠁⠀⣿⠤⠤⠀⠀⠀⠙⠛⠉⠉⠁⠙⠷⣦⣄⡾⠣⠄⠀⢀⣀⡀⠀⠀⠒⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆⠀⠑⠈⠛⠀⠴⠊⢀⡁⠀⡟⠀⠀⠀⠀⠀⠀⢠⡞⠆⠀⠀⠀⠈⠉⠓⠦⣤⣤⣀⣀⣉⣀⡁⠂
⠀⠀⠀⠀⠀⠀⠀⢸⢿⢿⡄⠱⡀⠀⠀⠐⠒⣉⣞⡌⡅⠘⡇⢠⠄⠀⢀⣿⡞⠈⢀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⢰⢶⡆⠀⢸⠄⠀⠹⣦⡀⡙⠒⠂⠚⢫⠎⠀⣷⠀⠹⡀⠀⠀⠉⠉⢧⡀⢨
⠀⠀⠀⠃⣰⠁⠀⠘⠂⠀⠰⠃⠁⣾⠿⢿⠀⠘⡤⠰⠇⠀⠀⠑⢼⣄⠀⣑⣦⡙⠞⠀⠀⢀⡴⠚
⠀⠀⠀⠀⣿⢆⠀⡁⠀⠀⠀⠀⠀⡿⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠟⠋⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⣎⢴⣦⣤⣴⡶⢿
⠀⠀⠀⠀⣡⠤⢌⡱⣄⠂⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⢖⣴⣦⡄⢠⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⣷⣆⠀⡀⠀⡀⢼⠃⠀⠁
⠀⠀⠀⠈⢁⣴⣦⣸⠏⠀⡄⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡀⠥⠀⣢⠏
⠀⠀⠀⠀⠀⠀⠀⢣⠀⣰⠃⠀⠀⡗⠀⠸⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠈⣿⣟⣻⠁
⠀⠀⠀⠀⠈⠆⠀⠈⠛⣿⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣤⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠛⣂⠀⠀⠀⠀⠀⠘⣿⠅
⠀⠀⠀⠀⡐⠆⢀⠀⢠⣿⣀⠀⠀⡆⠀⠀⠀⠀⢀⣲⢮⣽⣛⣿⣷⣦⡀⠀⠀⢀⡠⠒⠉⠤⠶⠛⠿⡆⠀⠀⠀⢀⣠⡆⣠⣄⡀
⠀⠀⠀⠀⠛⡶⠦⠍⣛⠟⠈⠀⠀⡇⠀⢀⠀⠀⣸⡯⠤⣶⣶⣝⣻⣛⣛⠂⡐⠁⣀⣀⣴⣶⣿⣚⣻⣿⡄⠀⠀⠈⣑⣀⠈⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠁⠀⠚⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡀⣀⣀⠀⠀⢀⡀⠀⠀⡀⠀⠀⢀⡀⠀⢀⡀⠀⣀⣀⠀⢀⣀⡀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⢀⡛⠀⠈⠁⠈⠁⡉⠉⠉⠉⠉⠉⠃⠀⠈⠉⠀⡀⠉⠁⠀⠈⠉⡉⠉⠉⠀⠈⠉⠃⡀⠉⠈⠉⠉⠉⠉⠉⠁⠉⠉⠀⠉⠉⠉⠉⠉⢋⠀⣀⠘⡆
⠀⠀⣄⠀⠰⡶⠠⠒⠛⠐⠒⠚⠓⠚⠛⠛⠒⠐⠚⠑⠚⠷⠊⠒⠚⠋⠀⠀⠔⠋⠚⠒⠛⠐⠛⠂⠁⠈⠚⠋⠓⠓⠚⠓⠒⠋⠙⠒⠀⠚⣿⡇⢸⠇
⠀⠀⡍⠀⠠⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠘⡇
⠀⠀⢧⡀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⢉⣤⣁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠤⡄
⠀⠀⠆⠀⠠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡁⠂⠀⠃⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠟⢠⡟⢚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⡅
⠀⠀⡗⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠰⢇⡄⠠⠀⠈⠂⢄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⢘⡀⢤⡄⠋⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠸⡇
⠀⠀⣃⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠀⡠⠁⢤⠀⢀⠈⠀⡀⡄⠀⠀⠐⠁⠈⢔⠗⠥⠀⠀⠐⠖⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⣾⠀⠠⡁
⠀⠀⣿⠂⢠⡇⠀⠀⠀⠀⠀⠀⠘⣶⡀⠀⠀⠄⢁⠀⠀⠈⡄⠀⠱⠁⠀⠀⠙⡆⠀⠀⠀⠀⠀⠐⠑⠦⠄⠀⠀⠘⠂⠀⢀⠀⠀⠀⠀⠀⣿⡂⢀⡆
⠀⠀⡴⠀⠈⡇⠀⠀⠀⠀⣄⠀⠀⠘⠁⠀⠸⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⢀⠠⠀⠀⠀⠀⢿⠄⠀⡁
⠀⠀⢯⠀⠐⡇⠀⠀⠀⠀⠋⠀⠀⢀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠃⢙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⠀⠀⠀⣏⠀⠐⠇
⠀⠀⣾⠂⣈⡝⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⢀⡇
⠀⠀⣿⠀⠀⢱⠀⠀⠀⠀⢀⠌⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣉⢙⠧⡉⣉⠈⠃⠀⠐⠂⠀⠀⠀⠀⠀⠈⠀⠀⣘⠀⠀⠀⠀⠀⠀⢿⠄⠀⡀
⠀⠀⠟⠀⠐⡏⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢹⡇⠀⢁⣎⠁⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⣮⠀⠀⠃
⠀⠀⡟⠀⣀⡅⠀⠀⠀⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠊⢀⠀⠈⠙⠀⠀⠀⣀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠤⠀⠈⠀⠀⠀⠀⣳⠆⠀⡇
⠀⠀⣷⠀⠘⡇⠀⠀⠀⠘⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠦⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⣸⠀⠀⠀⠀⣿⡃⠈⠃
⠀⠀⡿⠀⠲⡇⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡁⠶⠀⡀⠀⢃⠀⠀⠀⠄⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡏⠀⢀⡆
⠀⠀⣸⠦⢀⡼⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⢞⠃⠄⣠⣅⡠⠂⡸⠂⢀⢠⣀⠀⢂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣧⠀⠈⠃
⠀⠀⢹⡀⠈⣿⠀⠀⠀⠌⢀⠀⢁⡀⠀⠀⠸⣇⠀⠀⠐⠄⢠⢊⣵⡎⢻⡔⢧⠞⢁⣴⢦⡐⡄⢂⢀⣠⡏⠀⠀⠊⠀⠀⠀⣀⠁⠀⠀⠀⡿⠀⠰⡄
⠀⠀⣾⠁⠰⡞⠀⠀⢀⠄⠁⠀⠀⠁⠀⠀⠀⠛⢿⠛⣿⣳⡆⠛⠀⠈⣧⢷⡟⠠⠎⠀⠀⠈⠠⠮⡍⠛⠁⠀⠀⠀⡀⠀⠐⠃⠀⠀⠀⠀⣿⠀⢀⡇
⠀⠀⣰⡀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠⢴⠷⠾⡃⠚⠂⠤⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠀⠨⡇
⠀⠀⢻⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⢁⠰⠆⠈⠀⢠⡄⠈⠀⠠⠦⢤⣀⣀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠠⡇
⠀⠀⡜⠑⠤⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠉⠂⠭⠩⠀⠀⠆⣠⠀⠀⢀⡀⠀⠀⠐⠐⠀⠁⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢀⡃
⠀⠀⠰⠄⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⡋⠑⠆⡀⡀⠀⠀⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀⢀⠒⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠀⠊⡇
⠀⠀⢧⠀⠈⢻⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠘⠤⡀⠠⡄⠀⠈⠆⠀⠀⠂⠑⡀⠀⠀⠂⠰⠀⠀⠀⠊⠀⠂⠀⡀⠁⠀⠀⠀⠀⠀⠀⢀⡋⠀⠠⠃
⠀⠀⣞⠂⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠀⠀⡇
⠀⠀⣱⡀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⡇
⠀⢠⣿⠀⠐⠿⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠁⠀⡠⠋⠀⡄⠀⠀⠀⠀⠀⠀⠀⡆⠀⢘⠇
⠀⠀⣿⠄⢀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⣁⢲⣄⢰⣀⢀⢠⡀⣀⣠⢦⡄⠘⠋⠁⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣧⠂⢨⡄
⠀⢀⣽⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⡴⠞⠺⢥⠟⠷⢻⡏⢻⠿⠳⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⢘⡄
⠀⠘⣳⠀⠘⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⡃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⢸⡃
⠀⠰⣝⠃⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠒⠩⠩⠍⠯⠍⢠⠄⠱⠀⠀⠀⠤⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⠀⡆
⠀⢸⡧⡀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢠⠁
⠀⢸⠁⠀⣲⣿⠟⠛⠓⠛⠓⠐⠒⠛⠛⠘⠉⠙⠓⠲⠗⠒⠒⠏⠀⠒⠚⠛⠓⠐⡛⠙⠓⢾⠁⠀⠐⠃⠀⠐⠋⠉⠈⠓⡉⠀⠒⠆⠀⠚⢷⡆⠀⢁
⠀⠨⠁⠘⠏⠈⠙⠦⣀⠀⠤⠤⠤⠔⠂⠤⠄⠦⠄⣀⠰⠀⠀⠀⠆⠀⠀⠲⠤⠄⠈⠦⠄⠠⠦⠀⠀⠀⠦⠤⠀⠴⠤⣀⠶⠤⢄⣼⠦⠀⠈⠳⠖⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠄⠀⠁⠐⣤⡀⠀⠐⡆⠀⢰⣧⠠⠤⠄⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⢰⣶⣶⠚⠀⠀⠁⠀⠸⣿⣆⢀⠂⠠⠈⢃⡀⠀⠀⠀⠀⠙⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⡀⢰⣿⣿⣀⠀⡠⠋⠀⡀⠀⣿⣿⠀⡀⠰⡇⠈⣅⠀⠀⠀⢦⣄⣤⣿⡿⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠢⣄⢘⣿⣿⣿⣿⡟⢹⠃⢀⡶⠀⠀⣿⣿⡄⠀⠊⣿⠀⠸⠀⠀⠀⣼⣿⣿⣯⠀⢠⣿⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢰⣿⣿⣿⣿⣿⡿⠅⣸⠀⡆⢶⠀⠀⣿⣿⡌⠀⠀⣿⠀⢠⠠⡆⠸⣿⣿⣿⣷⣠⣿⡟⢐⡟⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡏⠈⠉⢟⠋⢻⣿⣿⣷⡾⠁⠘⡇⢸⡆⢸⣿⣟⡟⠃⠈⣷⠀⠼⢠⡇⠀⣿⣿⣿⣿⣿⣿⠃⣾⡇⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠓⠀⠀⠀⣠⠄⠸⠷⣌⡇⠀⠀⣁⣸⡄⣼⣿⣿⡇⠃⠀⢿⡀⠘⢺⠁⠀⡿⠟⠻⣿⣿⣿⣷⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣄⡴⠶⠞⠁⠲⢲⣀⠈⠳⣤⡀⠰⣾⡇⢹⣿⣯⣇⠀⠀⡿⠀⠀⠃⠀⡘⠁⠀⣹⣿⣁⣉⢛⣻⢿⣾⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠻⣦⣀⠑⠀⣯⣇⣼⣿⣿⣿⣥⣀⡇⠀⠀⡇⠠⠁⢀⡴⠋⠁⠀⠙⠟⠛⠛⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉⢳⣀⡗⢿⣿⡟⢯⡟⢿⣽⣷⡀⠀⡇⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⡾⠧⣀⠀⠀⡠⣠⡄⠀⣸⣿⣇⣬⠅⠻⢿⣥⣿⢟⣯⢟⡀⣷⡏⠀⢠⠀⠀⠀⠰⠃⠀⠀⢀⠀⠀⠛⣷⣄
⠀⠀⠀⠀⠀⢰⠂⠀⠸⣁⠀⠋⠀⠀⠀⠐⠆⠀⠀⠀⢀⣾⣿⡏⠠⠀⣹⣲⣿⣿⡒⡳⣾⣿⣾⡇⠀⠘⣄⠀⠀⠀⠀⠀⠀⠈⠁⢄⡠⢾⣿⣇
⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠀⠀⡀⠀⠀⠀⣤⠀⠀⢺⣿⣿⠃⡠⢾⡟⠉⠉⣉⠛⠿⢄⠙⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡙⣧
⠀⠀⠀⠀⡀⠀⠀⠀⠔⠻⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⣿⣿⣿⠊⠀⠌⣷⡇⡟⠶⣶⡣⠄⠡⠈⡆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣼⢿⣿
⠀⠀⠖⠀⠡⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⣀⠙⢶⣤⠐⢬⣅⡀⠀⠀⣸⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣟⡫⣧
⠀⢐⠃⠀⠀⠀⠁⠀⠀⠆⠀⠀⠐⠀⠀⠀⡆⠀⠀⣾⣿⣵⣿⡤⣿⣯⠀⠠⣿⠀⠀⣸⣿⣼⢶⣽⣿⡗⢄⠀⠀⠐⠀⣀⠀⠀⠀⡇⠀⠈⠈⢰⠹⡇
⠀⠒⠰⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠁⠀⣸⠗⢿⡧⠘⢰⠏⠀⠀⠀⣿⠀⠒⢈⣿⡿⠀⠈⠛⠀⠀⠀⢀⡀⢀⣼⠂⠀⠘⠣⠀⠀⠀⣾⠁⠻⡄
⠀⠂⠀⡄⠀⠀⢀⠀⠀⠀⠀⢀⣀⣠⠴⠃⣠⠴⢡⣴⡿⠏⡤⠊⠀⠄⢠⠀⣿⠀⠀⠀⣄⡙⢄⣸⣧⣤⡗⢐⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⡟⠆⠀⡿
⠀⡉⠀⠀⠀⠀⡆⠀⠀⣠⠖⠁⠀⠀⠀⠀⣴⠞⠋⠁⠐⠊⡁⠀⠀⠀⡿⠀⢛⠀⠀⢠⣿⡿⣾⣿⣿⣌⡛⠿⣄⢰⡀⠀⠀⠠⣶⠀⠀⢸⡿⢃⠀⣹⠇
⠀⠀⠀⠇⠀⠀⢹⡀⠀⠀⠀⠀⢀⣤⣾⠛⣉⣀⡁⣀⣰⣯⡊⣠⠴⢶⠎⡉⡛⠁⠑⠾⣗⠙⠃⣈⣷⡙⠣⠁⠋⠻⣷⣄⠀⠀⠀⢨⣘⠿⣯⣧⣽⣿
⠀⠀⠀⠈⢄⠀⣸⠃⠀⣠⣶⣿⡿⡟⡽⡟⡏⢹⢿⡿⣿⣍⠻⠇⣠⣤⡇⠀⠦⠀⢤⣀⣿⡷⠊⠁⢨⣿⣶⠈⢱⢴⢮⣍⣷⣄⠀⠀⠈⣻⣿⣿⣿⡿
⠀⠀⠀⠀⠘⡗⠁⢀⣴⢿⣿⡟⠁⠀⠠⡏⠇⠀⠀⠐⡘⡮⡁⢤⡩⣤⣼⠀⠁⠀⣌⣀⣿⡥⢒⣥⣾⠏⡞⢠⠀⠀⠀⡜⠁⠋⠻⣷⣾⣿⢟⣽⣿⠁
⠀⠀⠀⠀⠀⠹⣦⣾⣳⣿⣿⠁⠀⠆⠄⠈⠊⢄⠀⢶⣲⣑⣿⣆⠳⡉⠙⠀⡅⢐⢋⣿⡟⣰⣿⡿⠋⣠⠁⠎⡃⡄⡀⣅⠀⠀⡧⢹⣭⣽⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠄⠀⡴⣤⠀⠠⠘⣦⣀⣽⣿⣿⣿⣇⢥⠀⠀⢇⠨⣸⡿⣼⣟⣛⣥⣾⣯⠈⡰⡹⠃⢇⢅⠀⢀⡟⢸⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣍⣿⣿⡆⡄⢷⡘⢗⡀⠄⢀⠈⢻⣟⣿⣿⣿⣠⡄⠀⡀⢠⣿⣿⣿⣿⡿⠿⠟⢁⣸⡟⠁⢠⡾⡀⠀⣾⣧⣘⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⡌⠿⣟⢿⣿⣷⣦⣆⣠⢬⣙⠿⢿⣿⡇⢀⣲⠺⣿⣿⣿⣿⣿⠷⠛⣫⡏⣠⣠⠞⠁⡁⣰⣿⢳⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣟⡳⢮⣽⣿⣿⣿⣿⣿⣿⣾⣿⣷⣶⣴⣶⣿⡿⣿⣿⣿⣷⡾⠷⠾⠟⡉⣠⢠⣷⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣬⣍⣛⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⢁⠀⣠⢂⣔⣴⣧⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣝⡛⣿⣿⣿⣿⣿⣿⣿⠟⠘⣿⣿⠻⢿⣿⣿⣿⣾⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠛⠉⠀⠀⠈⠛⠷⠿⠿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣷⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣾⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣰⣿⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣆⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣞⣯⣾⣿⠟⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠻⣿⣷⣽⣳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⣴⣿⠟⠋⠁⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠈⠙⠻⣿⣮⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⣸⡿⢁⣤⣤⣄⣀⣀⣀⣀⣽⣷⣤⠤⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⣤⣾⣍⣁⣀⣀⣀⣠⣤⣤⡈⢿⡇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠇⣼⣿⣿⣿⡿⠟⣛⣍⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠈⣩⣙⠻⢿⣿⣿⣿⣧⠸⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡟⢠⣿⠿⣋⣵⣾⡿⠋⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⢿⡿⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⢿⣷⣮⣙⠿⣿⡄⢻⣿⣿⣿
⠀⠀⠀⠀⠀⡇⠀⣿⣿⢋⣴⠟⣡⣾⣿⣿⡟⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣌⠻⣦⡙⢿⣿⠀⢰
⠀⠀⠀⠀⠀⡇⢰⡟⣡⣿⢋⣼⢿⣿⣿⣿⠶⠚⠃⠀⠀⠘⠁⠉⠉⠉⠙⠃⠀⠀⠀⠀⠘⠋⠉⠉⠉⠉⠛⠀⠀⠈⠓⠶⣿⣿⣿⡿⣧⡙⣷⣌⢻⡄⢸
⠀⠀⠀⠀⢰⣿⣿⣾⠟⠁⢋⣵⣿⠟⠋⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠰⠦⡄⠀⠀⣠⠴⠦⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠙⠻⣿⣮⡙⠈⠻⣷⣿⣾⡀
⠀⠀⠀⣰⢿⣿⡟⠁⣠⣾⣿⡿⠁⢀⣠⣴⠞⠛⠋⠉⠉⠙⠛⠦⠀⢤⣄⠀⠀⠀⠀⠀⢀⣠⠄⠀⠔⠚⠋⠉⠉⠙⠛⠻⣦⣄⠀⠈⢻⣿⣷⣄⠈⢻⣿⣿⣆
⠀⠀⢸⡇⣺⡟⢀⣾⣿⣿⡟⢁⣴⣿⡿⠿⠛⠛⠿⢶⣄⡀⠀⠀⠀⠀⠙⣿⣦⣼⣧⣴⣿⠋⠀⠀⠀⠀⢀⣤⡾⠟⠛⠛⠿⢿⣷⣤⠀⢻⣿⣿⣷⡀⢻⣇⢸⠂
⠀⠀⠀⠹⣿⠁⣿⣿⣿⣿⣷⡿⠋⠁⠀⠀⠀⠀⠀⠀⠉⢿⣶⣤⣄⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⣠⣤⣶⠿⠁⠀⠀⠀⠀⠀⠀⠈⠙⢷⣾⣿⣿⣿⣿⠈⣿⠃
⠀⠀⠀⠀⣿⢨⣿⣿⣿⣿⠋⠀⠀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⣠⣤⣄⣀⠀⠀⠙⣿⣿⣿⣿⠃⣿
⠀⠀⠀⠀⢻⣧⣿⣿⣿⡇⣠⣶⣿⠿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡿⣿⣶⣄⢹⣿⣿⣿⣼⡏
⠀⠀⠀⢠⣿⣿⣿⣿⣏⠀⠛⣯⣾⣿⣿⠛⢿⡟⣿⣦⣀⣀⡀⣀⣴⣶⣿⣿⣿⠶⠶⢿⣿⣿⣶⣦⣀⣄⣀⣀⣴⣿⢻⡿⠛⣿⣿⣷⣽⠛⠁⣹⣿⣿⣿⣿⡄
⠀⠀⠀⠟⠁⠸⣿⣿⣿⡆⡀⠈⠻⣿⣿⣄⣀⣽⣎⣻⠟⣋⣵⣿⣿⠿⠟⠛⠄⠀⠀⠠⠟⠻⠿⣿⣿⣮⣙⡻⢟⣵⣯⣀⣰⣿⣿⠟⠁⢀⠠⣿⣿⣿⠏⠈⢳⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⡇⠀⠀⠙⠿⣬⣭⣿⠿⣿⣿⡿⠟⠛⠛⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⠛⠛⠻⢿⣿⣿⠿⣿⣭⡥⠞⠃⠀⠀⣸⡇⣿⣿⣿
⠀⠀⠀⠀⠀⣸⣿⣿⠇⢼⣿⣆⠀⠀⠀⠀⢉⣡⣾⠿⠯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡽⠿⣷⣌⡉⠀⠀⠀⠀⣰⣿⡇⢸⣿⣿⣇
⠀⠀⠀⢀⣼⣿⣿⣿⠀⠘⢿⣿⣿⣶⣶⣾⣿⣿⠃⠀⠀⠈⣀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠀⣀⠁⠀⠀⠈⣿⣿⣶⣶⣶⣿⣿⡿⠁⠀⣿⣿⣿⣧
⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠹⣿⣍⣽⣿⣿⠋⠀⠀⠀⠀⠈⢀⣤⠶⠒⠒⠺⠿⠿⠗⠒⠒⠶⣤⡀⠁⠀⠀⠀⠀⢻⣿⣿⣯⣩⣿⠏⠀⠀⠀⣿⣿⣿⣿⣧
⡆⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⠀⢹⡿⣡⣾⢿⡀⠀⠀⠀⣀⣴⣿⣶⣶⣶⣶⣶⣤⣤⣶⣶⣶⣶⣶⣿⣦⡀⠀⠀⠀⢨⡿⣿⣏⢿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆
⣧⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠈⣱⡟⠁⠘⢇⠀⠀⠐⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡟⠙⠆⠀⠀⡼⠃⠈⢻⣌⠃⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇
⢷⡄⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⣿⣯⣿⣿⣿⣿⣹⣿⣿⣾⠟⠁⠀⠀⠀⠈⠀⠀⠀⠀⢻⣆⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠸⣧⠈⢿⣿⣿⣿⣿⣿⣿⣧⣴⡿⠃⠀⠀⠀⠠⢄⡀⠀⠀⠀⢀⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⡀⠀⠀⠀⢀⡠⠄⠀⠀⠀⠘⢿⣇⣼⣿⣿⣿⣿⣿⣿⡿⢀⣿
⠀⠹⣧⡸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠳⡀⠀⠈⢷⣌⡻⢿⣿⡏⢻⣿⡿⢟⣡⡿⠁⠀⢀⠞⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⠇
⠀⠀⠹⣷⣿⣿⣿⣿⣿⣿⣿⡏⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡏⣹⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣹⣿⣿⣿⣿⣿⣿⣿⣾⠋
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡌⢿⣷⣾⣦⣄⣀⠀⣀⣀⣤⠤⠀⢀⣙⣻⣿⣿⣿⣿⣟⣋⡁⠀⠤⣤⣀⣀⢀⣀⣠⣴⣷⣾⡿⢡⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠻⣿⣿⡇⢿⣿⣿⣿⣦⡉⠻⣿⣿⡿⠟⣋⣭⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣭⡙⠿⣿⣿⣿⠟⢋⣴⣿⣿⣿⡟⢸⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠹⠇⠸⣿⣿⡞⢿⣿⣶⣶⣶⠞⠉⢉⣿⣿⡿⢿⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⠟⢿⣿⣿⡉⠉⠳⣶⣶⣶⣿⡟⣻⣿⣿⠃⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣄⢻⣿⣿⣿⡆⠀⣼⢃⣼⠀⠀⣿⠀⣿⠋⢻⡋⠙⣿⠀⣿⠀⠀⣧⡙⣧⠀⢸⣿⣿⣿⠏⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⣿⡏⢿⣷⠀⠀⢸⣿⣷⣼⣿⣶⣿⣦⣾⣷⣴⣿⣶⣿⣧⣾⣿⡇⠀⠀⣿⠟⣻⣷⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣿⣿⣟⢿⡀⣿⣆⠀⣿⢟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⠀⣰⡟⢰⡟⣿⣿⣿⣷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣾⣷⣿⣿⣶⣯⠸⣿⡿⠿⠿⢿⣷⣿⢻⣿⠿⠿⠿⢿⣿⢃⣿⣶⢿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⢻⣿⣷⡉⠀⠀⠀⠀⢌⣹⣈⡡⠀⠀⠀⠈⢡⣿⣿⡏⢸⣿⣷⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣼⡇⣼⣿⡿⠃⠀⠀⠀⠀⠀⢻⠏⠀⠀⠀⠀⠀⠸⢿⣿⣇⢸⣭⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣿⣷⣿⣿⠁⠀⠀⠀⠀⢀⣀⣴⣤⣀⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣿⣦⡀⠢⣄⠀⠀⠀⠈⠁⠀⠀⠀⣠⠄⢀⣴⡿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢧⡰⣏⡻⣷⣤⣭⣛⣉⣉⣉⣉⣉⣥⣴⡾⣟⣹⠀⢸⡟⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠰⢷⡘⣷⠀⠉⠙⢻⠛⢻⠟⠻⠛⠉⠉⢀⡟⢡⡾⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠈⠛⢿⣿⣿⣦⣿⣄⣰⣄⣰⣦⣼⣷⣿⠿⠛⢁⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣔⡦⣄⣈⡉⠛⠛⠛⠛⠛⠛⠛⠋⢉⣁⣤⢔⣲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣮⣭⣙⣛⣟⠿⣿⣿⣟⣿⣛⣛⣭⣵⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣛⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⣻⣽⠟⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⣦⣤⣤⣤⣤⣶⡶⠚⠉
⣿⣿⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣅⠟⠰⣿⣥⣀⣉⣭⣭⣁⣤⣉⣡⣌⣥⣤⣥⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡗⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠻⣧⠀⠙⠻⣶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⣠⡞⠁⣀⡀⢻⡆⠀⠀⠀⠉⠙⠛⠛⠛⢀⡀
⠀⠀⠀⠀⢀⣾⠋⠀⠀⠛⠁⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⢠⣿⠃⢀⡄⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⣀⣀⡀⠀⠈⢿⣿⣦
⠀⠀⢠⣿⠃⠀⠘⠁⠀⠀⢀⡀⠘⣿⡆⠀⠰⡆⠈⠉⠻⣦⠀⠈⣿⡏
⠀⠀⣾⡏⠀⡀⠀⠀⠀⠀⠈⠁⠀⣿⣷⠀⠀⢻⡀⠀⠀⢨⡄⠀⢸⣷
⠀⣸⣿⠀⢸⠟⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⣧⠀⠀⠈⣿⠀⠈⣿⡀
⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⣿⡇⠀⠀⣿⣤⡀⠀⣿⡄⠀⣿⡇
⢸⣿⠁⣀⡀⠀⠀⠀⠀⠀⠈⠛⠀⠈⣿⡇⠀⠀⣿⡟⣧⠀⣿⡇⠀⣿⡇
⣾⡟⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇⣿⠀⣿⡇⠀⢸⣷
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⣿⡇⠀⠀⣿⡇⣿⠀⣿⡇⠀⢸⡇
⣿⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⢱⣿⠀⣿⡇⠀⣼⡇
⣿⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⡇⠀⠀⣿⠀⠀⠀⣿⠇⠀⣿⡇
⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⢰⣿⠃⠀⢰⡇⢀⣀⣠⡿⠀⠀⣿⡇
⠀⠙⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠛⠛⠛⠉⠀⠀⢀⣿⠁
⠀⠀⣷⠈⠙⠛⠿⢿⣶⣶⣶⣶⣶⣾⡿⠀⣀⣀⡀⠀⠀⠀⠀⠀⢸⣿⠶⣛⣛⣛⣓⡲⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⠁⠀⣾⠏⢹⣿⠀⠀⠀⠀⠀⠾⠷⠋⠉⠁⠈⠉⠉⠛⠶⣭⣛⠻⠶⣶⣶⣶⣾⠿⢋⣤
⠀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠻⠿⠟⠁⠀⠀⢀⣠⣴⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⣤⣤⣴⠶⠛⠁
⠀⠈⠛⢦⣄⡀⠀⠀⠀⠀⠀⠈⡿⠀⠀⠀⠀⣀⣠⡴⠾⠛⠉
⠀⠀⠀⠀⠉⠻⠷⣶⣦⣤⣤⣤⣥⣤⡶⠖⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠇⠀⢠⠀⠄⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⡜⢠⠀⠀⠊⠀⣴⠀⠀⠀⠀⠀⠀⠈⢡⣿⢷⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⡘⢠⠏⢀⠤⢤⡾⠋⠀⠀⠀⠀⠀⠀⠐⣻⣿⢻⣆⠀⠀⠀⠀⡇⠀⠀⢾⣡⣄
⠀⠀⠀⠀⢰⢀⡿⠀⠇⢠⡾⠃⠀⠀⠀⠀⠀⢀⠀⠐⠀⢿⠸⠙⢷⣄⠀⠀⠸⣦⠶⠀⠋⠙⣆
⠀⠀⠀⠀⡎⠈⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⢧⠢⠙⢧⡀⠀⠈⢶⡙⡀⠀⠈
⠀⠀⠀⠀⡇⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⢀⣿⠀⢘⡇⠀⡀⣨⠷⣤⠈⠙⠃⠀⠐⡀
⠀⠀⠀⠀⠁⣾⣿⡇⠀⠀⠀⠀⡀⠰⢿⣤⣦⠀⢀⠀⢼⣿⡆⡌⠻⣿⣿⣷⡀⢸⡇⠀⠀⠈⣿⡗
⠀⠀⠀⠀⠁⠿⠛⠁⠀⢀⠀⢀⠀⠀⠀⢙⠟⠀⣸⠙⣿⣿⣇⠀⠒⠛⠻⣿⣷⣼⡇⠀⠀⠀⠛⠃
⠀⠀⠀⠀⡆⠀⠀⠀⠀⢈⣧⣏⠀⠀⠉⠛⠀⠀⢆⠀⠈⢋⣿⡇⢀⣵⡄⢸⣿⣿⡟
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⢸⣇⠀⠀⡸⠀⠀⠀⠠⠈⠠⠀⢹⠤⠤⠏⠀⠘⣿⣿⡏⠁
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⣿⠀⠀⠁⠀⠀⠀⢰⠀⠀⠀⢁⠀⠀⠀⡀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⡄⠀⠀⠀⢀⣏⡀⠰⠒⠿⠀⠀⣼⣿⢠⣿⣿⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣛⠀⢸⡇⢀⠀⠈⠀⣄⣸⠏⠁⣤⣤⡴⠗⠚⣫⣡⣿⣿⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢸⠀⠀⢰⠀⠀⠉⠙⠓⠺⣭⣤⣤⡤⠾⠟⠟⠻⢿⣿⠀⠀⡗⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢸⣧⡀⠈⠁⡸⠀⠀⢠⣔⠻⣿⣶⣖⠂⠁⣀⠀⠀⣿⠀⠀⠂⠀⢰⡧
⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⣸⣿⡇⠀⣧⡇⡀⠀⠈⢿⣦⠘⣻⣉⣤⡾⠃⠀⣿⣿⠃⠀⠀⠀⢸⡗
⠀⠀⠀⢠⠀⡀⠀⠀⣾⡎⢸⣻⡇⠠⣤⢇⡁⠀⠀⠘⣿⣷⣼⣿⠟⠀⠀⢸⣿⣿⡇⠀⠃⠀⢼⡇
⠀⠀⠀⠠⠀⡀⠀⢀⣿⣷⣾⢲⡇⠐⣿⢸⠀⡆⠀⠀⠹⣿⣿⡏⠀⠀⠀⣼⣿⣿⣧⣰⠀⠀⣿⡇
⠀⠀⠀⢰⠈⢣⠜⢻⣿⣿⣿⡾⢴⣤⡏⣿⣣⡿⡀⠀⠀⠹⢿⠁⠀⠀⢀⣿⣿⣿⣿⣿⡀⣄⣿⡇
⠀⠀⠀⢨⠀⠤⠂⠤⠿⠛⠛⠀⠀⠙⡇⠔⠉⠀⡀⠀⠐⠢⠌⢷⠒⡀⠚⠿⣿⡿⠟⠁⠉⠉⠙⠃
⠀⠀⠀⢸⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠓⠂⠤⠂⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣶⣦⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠰⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⢰⢋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠠⠁⠠⠬⠐⢂⠀⠀⠀⠀⡦⠀⡐⠉⢂⢀⡀⠀⠀⠀⣀⣧⡵⠖
⠀⠀⠀⠀⠀⣥⠜⠁⢢⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠸⡀⠁⠐⠀⠂⢉⡜⠀⠀⠀⠈⢢⡠⣯⡟⣍⣈⠏⠀⠀⢼⠁⢈⡇
⠀⠀⢀⠀⡀⠀⠀⠰⢀⠀⠀⠘⡁⣤⠁⠠⡀⢀⠀⠀⠑⠤⡄⠀⡠⠎⠀⠀⠀⠀⠐⣁⣸⠨⢵⡋⠑⡀⢀⡴⠚⢯⠉
⠀⠀⠈⡉⠅⠀⠠⢂⠌⠀⠀⠀⠐⢄⠀⠀⠐⣄⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠄⠈⠉⠀⠀⠈⠉⠚
⠀⠀⠀⢰⣶⣿⣷⣷⣀⠀⠀⠀⠀⠈⠰⢤⡾⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠘⠤⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠆⠀⠑⢀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣾⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠉⡈⢀⠀⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠆⠁⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠃⠀⠐⠃⠀⠀⠀⠠⠘⠀⢀⣀⣀⠀⡀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⡠⢊⠆
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠐⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠉⡇⠠⠀⠀⠀⠀⠀⢻⣷⠄⠀⠠⠊⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣿⡦⢄⠤⠖⠁⠀⠀⠀⠀⠀⠠⣠⣶⣶⡀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⠢⠄⠄⠊
⠀⠀⠀⠀⠀⠠⢁⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⡡⠀⠀⠀⠀⠈⢤⡐⢀⠆
⠀⠀⠀⠀⢠⠁⠏⠀⠀⠀⠄⢀⠴⠀⠀⠀⢠⠿⢿⠿⣿⣿⡿⠷⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠁⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⢀⡠⠂⠐⠔⠁⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠈⠒⢀⠸⠈⠂⠀⠀⠀⠀⠀⠀⢂⡀⠀⡰⢀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⣈⠁⢈⣤⣤⡗⠀⠀⠀⠀⠀⢠⠊⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡍⠀⠀⠀⠀⠀⠀⠀⠠⠠⠂
⠀⠀⠀⠀⠀⡄⢠
⠀⠀⠀⠀⠀⡆⠀⠄⣀⠀⠀⠀⠀⠀⠀⡠⢐⣤⡄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠠⠐⠁⠀⠑⢩⠁⠀⠀⠀⠀⠀⠀⠈⢿⡇⣀⣀⠀⡄⠀⠀⠀⢰⡸⠏⠟⢀⢰⠀⠀⠀⠈⡀⣬⢢⠀⠀⡠⡂⠥⠀⠠⠁
⠀⠀⠀⡁⡄⠀⠐⠒⠄⠅⠀⠀⠀⠀⠀⣠⣴⡾⠛⠻⢿⠃⠀⠀⠀⠀⠀⠃⠈⠄⢉⡀⠂⠀⠀⠀⠐⠠⢙⢠⣌⠜⢁⠠
⠀⠀⠀⠑⠈⠀⠒⠨⠈⠀⠀⠀⠀⠀⠠⡈⠛⠃⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⣀⣰⠘⠋⠈⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠴⠓⠂⠈

⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢠⣶⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣼⣄⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⣷
⠀⣄⠀⣘⣿⣿⣟⡀⢠⡀⠀⠀⠀⠀⠀⢸⠿⡇⠀⠀⠀⠀⠀⢠⣀⣼⣟⡋⣛⣿⣄⣠⠀⠀⠀⢠⣾⣿⣦⣼⣿⣿
⢸⠿⣿⣿⣿⣿⣿⣿⡿⢿⠀⠀⠀⠀⣶⣾⣿⣿⣾⠀⠀⠀⠀⢸⣿⣿⣿⣯⣯⣿⣿⣿⠀⠀⢀⣾⣿⠻⣿⣿⣿⣿⣦
⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠿⠿⠀⠻⠿⠀⠀⠀⠀⠺⠃⠘⠿⠿⠿⠟⠀⠿⠀⠀⠐⠿⠟⠀⠻⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⣠⠞⠛⢦⡀⠀⠀⠀⠀⠀⠀⣤⠘⣿⠃⢘⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⡶⠀⢀⡀
⠀⠀⡞⠁⠀⡄⠀⠻⡄⠀⠀⠀⠀⠀⠙⣻⡛⣶⣾⠀⠀⠀⠀⠀⣶⣤⣶⢠⣾⣿⣷⡼⠋⠀⠀⠀⣘⣟⡀⠀⢀⣻⡃
⠀⠘⠁⠀⠂⡅⠂⠀⠃⠀⠀⠀⠀⠀⠀⢿⣥⣿⠀⠀⠀⠀⠀⠀⢹⣿⡏⠛⣿⣿⡇⠀⠀⠀⠀⢠⣿⣿⣷⠀⣾⣿⣿
⠀⠀⢠⠀⠀⠇⠀⢠⠀⠀⠀⠀⠀⠀⠀⢼⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢿⣿⡇⠀⠀⠀⢀⣄⣼⣿⢀⣠⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠘⠛⠃⠀⠀⠀⠀⠉⠛⠛⠀⠈⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠤⡀
⠀⠀⣿⣶⣶⣶⣾⣦⡄⠀⠀⠀⠀⢀⣀⣤⣿⣦⡀⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢙⠀⠀⠀⢰⣤⣄⣠⣴
⠀⠀⣿⣿⣿⠟⢿⣿⡇⠀⠀⠀⠀⠙⠋⠷⣶⡶⠋⠋⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠖⠀⠀⠀⠈⢿⣿⣿⡟
⠀⠀⣿⣿⣿⣇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠎⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⢀⣀⡀
⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠟⠂⠀⠀⠀⠀⠀⣠⣿⣧⡼⣽⣟⣔⡟
⠀⠀⠀⠁⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢀⢀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡗⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⡰⠡⠉⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⢠⣤⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠊⢁⡄⠀⠀⠀⠀⠀⠀⠀⠛⠛⢂⠍⠀⠀⠀⠀⠀⠀⠄⠐⠒⢒⡄
⠀⠀⠻⣷⣄⠔⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⠄⠀⠀⠁⠀⠀⠀⠀⢿⣆⡀⠀⠀⠛⠀⠀⠀⠀⢸⡇⠀⠠⢶⣦⠇
⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠈

⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣟⡛⠛⠛⠷⢶⣤⡀
⣾⣁⣠⣴⣿⣿⣿⣿⣿⣶⣄⠀⠀⣤⡤⠀⠀⠀⠀⠀⣀⡴⢟⣋⣭⣿⣿⣿⠿⠷⢦⡀⠀⠉⢻⣦⣄
⠉⠛⠫⠛⠛⠉⠛⢿⣿⡌⢻⣷⣼⣿⢷⣄⣀⣀⣀⣠⣴⣾⡿⠿⠯⣥⡀⠀⠉⠁⠀⠀⠀⠀⠀⢻⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠙⠻⢯⡀⠙⠛⠛⠛⠉⠁⠀⠀⠀⠐⠷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠙⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠆⠙⠿⠿⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠓⠒⠀⠐⠢⣤⣄⠀⠀⠀⠀⠡⠀⠀⠀⠈⢀⡴⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣠⣴⣶⣶⡶⣶⣤⣤⣼⣿⣷⡄⠈⠀⠀⠀⠠⠆⠌⠹⣆⣀⠠⣄⠂⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠽⠿⠛⠛⠿⢦⣭⡉⠉⠿⠏⢿⡠⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠊⠀⠂⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⣀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠻⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠦⣄⡀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠤⠂⠄⢀⠀⠀⠀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠤⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣶⣶⣶⣦⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣶⣾⣿⣶⣶⣶⣶⣶⣷⣶⣶
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠘⠀⢿⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⢣⠸⣇⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⡄⢿⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⡇⠘⣦⠀⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢀⡀⠀⠀⠀⠀⠀⢰⡀⣿⠀⠀⠀⠀⠈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢰⡿⠁⠀⠀⠀⣶⡄⢸⡇⢸⡇⠀⠀⠀⠀⠀⢹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠈⠀⠀⠀⠀⠀⠹⠇⠀⢆⠸⡇⠀⠀⠀⠀⠀⢠⡉⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢃⡄⠀⠀⠀⠀⠘⣧⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣰⡿⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⠀⢸⠇⠀⠀⠀⠀⠀⠈⠷⣄⡹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠙⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠘⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡄⢸⠀⡀⠀⠀⣰⣶⣤⡀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢀⠃⠀⢠⣿⠁⠈⠙⢦⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⣧⠀⣼⠀⠀⢸⡇⠀⠀⠀⠀⠳⡄⠀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⣯⠀⣯⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣦⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡿⠀⠀⢸⡇⠀⠀⠀⠀⣦⡀⠈⢇⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣯⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⢹⣧⡀⠈⢧⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⣏⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⣷⠀⠀⡇⠘⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡇⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⢹⣧⠀⢸⡄⠃⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠈⢿⡄⠀⢠⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣆⠀⠀⢠⠇⢰⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⢸⣧⠀⠘⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⢸⠀⢸⠀⠀⠀⣼⡇⠀⠀⠀⠀⣿⣇⠀⠀⣿⠀⠀⠄⢀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢰⣿⠁⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⡇⠀⣿⠀⠀⠀⠀⠀⣿⢻⠀⠀⢿⡇⠀⡇⠈⠀⢻⡇
⠀⠀⠀⠀⠀⠀⣼⡟⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣿⠀⢸⠀⡇⢠⣿⠀⠀⠀⠀⠀⢻⡌⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠆⠀⠀⡟⠀⡘⠀⠁⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⠀⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⡇⠀⠀⢸⡇⠀⠀⠀⠀⢐⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣸⡏⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡇⢸⡇⠀⠀⣿⠁⠀⠀⠀⠀⢸⡟⠃⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⣼⠁⢸⠃⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣿⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣿⠀⣾⠀⠀⢰⡟⠀⠀⠀⠀⢠⣿⠀⠀⢀⣠⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣸⡏⠐⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⡟⢠⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⣼⠁⢰⠃⠸⠀⣿⠃⠀⠀⠀⠀⣾⠃⢀⣾⠃⠀⠀⠀⠀⠀⠂⠀⢸⡇
⠀⠀⠀⠀⣿⠁⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⣿⠀⣿⠀⡄⢰⡿⠀⠀⠀⠀⢠⡿⢀⣼⡟⠀⠀⠀⣠⠄⠁⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢀⡟⠠⠇⢸⡇⠀⠀⠀⠀⣼⣿⠟⠋⠀⠀⠀⠚⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢸⡇⢸⡇⠀⠀⣾⠇⢀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢀⡿⠀⣼⠀⡇⠠⣿⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣠⣾⠋
⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⢠⣾⢿⣿⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣸⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠐⠚⠁⠀⠀⣾⡇⠈⢿⣷⡄
⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡿⢠⠆⠀⠀⠀⠀⢀⣀⠀⠂⠉⠁⠀⠀⠀⠀⠀⢰⣿⠀⠀⠈⠻⣿⣶⣤
⠀⠀⠀⠈⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢰⡇⣀⢀⣀⠴⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠈⢻⡿⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣜⠁⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣧⣄⡀⠉⠛⠻⠷⣶⣦⣤⣤⣤⣾⣁⣼⠟⢿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣶⣤⣀⠀⠈⠉⠉⠉⠉⠉⠁⠀⣼⠀⠈⠀⠀⠀⡀⠠⠀⣀⣠⣶⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⣀⠀⠀⠀⠀⠀⡿⢰⡧⠄⠀⣀⣤⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣦⣄⣸⡇⣀⣤⣶⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⢉⣭⣭⠉⠉⠉⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠉⢹⣿⣿⠉⠉⠉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⡀⠀⠀⠑⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⠛⠛⠒⠒⠒⠲⠦⢤⣀⣀⠀⠀⣀⣀⠴⠚⠁⠉⠉⠙⢛⣓⡲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠠⢤⣤⣴⠶⠾⡿⠋⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣌⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢀⠖⢐⣈⣙⡋⢉⡓⠒⠒⠤⠭⠿⣶⣶⣦⠤⠀⠀⠀⠀⡀⠀⠀⣨⠻⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢠⠏⣴⡟⠉⠉⠙⠻⠶⠽⣤⣥⣤⣠⣤⣠⣤⠄⠀⠀⠀⠢⣌⠳⡄⠀⠱⠙⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢠⠏⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢹⠆⠀⠀⠀⠀⠘⣆⠹⡄⠀⠀⡱⢘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠓⠀⠀⠀⡞⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⡄⢳⠀⠐⢃⢎⣺⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠁⠀⠀⢸⢇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠈⠀⠀⠀⠀⠀⠀⣇⢸⡀⠀⡠⡠⡇⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢸⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠂⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠈⠞⢰⠏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠈⡇⢿⠀⠀⣀⣀⣠⣤⣤⣤⣴⣖⣒⣊⣉⡉⠳⣅⣂⠤⠤⠤⠤⠤⣤⣬⣤⣭⣤⣤⣤⠤⢤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⠀⠀⠀⠀⠀⠀⠁⠀⢆⠈⠉⣡⣾⠟⣻⡟⢹⣿⣿⣿⣟⣿⣦⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣉⣉⣠⠤⠝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠑⠀⠀⠀⢀⣀⣀⣠⣤⣤⣬⣳⡄⠈⠛⠛⠿⠿⠿⠿⠿⠿⠛⠛⠁⠀⠀⣀⣀⣀⣼⡯⠛⠋⠉⠉⠉⠀⠀⡀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⣀⠴⢿⡿⠿⠿⠿⠿⠛⠓⠛⢉⣀⣀⡤⠤⠤⠤⠤⠶⠶⠒⠒⠚⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⢀⡀⡠⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⣤⢶⣾⣧
⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠞⠁⣃⡵⢨⡉⠉
⠀⠀⠀⠀⠀⢀⡞⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣩⠤⣴⣶⡾⠿⠟⠒⢹⡇
⠀⠀⠀⠀⢀⣾⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣔⣒⡯⠭⠗⠚⠉⠉⠀⡠⠔⢜⣊⣽⡇
⠀⠀⠀⢀⡼⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⣤⣴⣶⣤⣶⡶⠿⠿⠛⠛⠋⠉⠉⠀⠀⣠⠆⠔⡀⣀⣰⡵⠞⠋⠉
⠀⠀⣰⠃⠳⢀⣤⣤⡤⠤⠤⣤⡤⠤⠴⣒⣒⣒⣚⣛⣻⠭⠭⠽⠛⢛⠋⠉⠉⢁⣄⠄⠀⠀⠠⣤⠀⠄⣁⣠⣄⡤⠤⠷⠚⠛⠉⠉
⠀⢰⡋⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢀⠀⠀⡠⢠⠂⣠⡶⢋⣐⣁⣤⣼⣥⠾⠶⠶⠞⠓⠊⠉⠉
⠀⢨⣀⣕⣀⣂⣀⣈⣰⠞⡁⣀⣊⣐⣁⣁⣤⣥⣤⠶⠾⠷⠟⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠠⣤⣤⣤⠄⢠⡄⢠⣤⠀⠀⠀⠀⢠⢤⠄⠀⠀⠀⠀⠀⠠⢤⡀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠠⣤⣤⣴⡄
⠀⠀⢐⣀⣄⣿⠶⠎⠹⠾⠉⠀⠀⠐⠃⡜⠻⠀⢤⠀⠀⠀⠀⠀⠃⢳⣄⠀⠀⠤⠉⠠⠄⠉⢢⡀⠉⠠⠴⠀⠈⣠⠿⢿⠃
⠀⠀⠈⠒⢶⣶⢀⣀⣀⣀⠀⠀⢀⠋⣼⣛⠁⣈⠀⢀⠀⠀⠀⠘⢷⡀⠆⠀⠀⠀⠀⠀⢘⡛⣛⡛⠀⠀⠀⠀⠀⠙⠖⢿⣷⡀⠤⢠⠤⡄
⠀⠀⢠⣖⠛⢁⡾⠄⠷⠬⠄⠀⡈⢠⣛⣃⣀⡈⢠⠃⠄⠀⠀⢠⡃⠉⡙⠀⠀⠀⠀⠀⢀⣇⣇⣛⠀⠀⠀⠀⠀⣾⣻⣿⡿⠱⠞⠹⣤⣘⣆
⠀⠀⠰⣎⣏⠿⠀⠀⠀⠀⠀⠤⢠⣎⣹⣿⣿⣿⣃⠛⡴⡄⠀⠀⡝⠶⠶⠀⠀⠀⠀⠀⣿⠉⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⡏⢈⣿⡆
⠀⠀⠘⣽⣬⡵⠀⠀⠀⠀⠐⢂⣾⣉⠋⠉⠉⠉⠁⡿⢣⢇⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢈⣄⣰⣿⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⣷⣿⡉⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠂⠀⠀⢀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⢠⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠡⡀⠀⠀⢀⠀⠀⠈⢀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠂⠄⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠁⠠⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠄⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⣈⡀⠀⠀⣀⣀⣀⣀⣸⣿⣿⣄⣀⣀⣀⡀⠐⠀⠀⢀⣀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⢀⠀⠠⠀⠐⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⠾⠿⠿⠿⠿⢶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢆




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⠉⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠚⠁⠒⠀⢄⠀⠀⢀⡬⠀⠀⠀⢠⣠⣀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⣠⠄⠄⠀⠠
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠡⡀⠘⠀⠀⠀⠀⡘⠉⠃⠀⠀⠀⡴⠀⠀⠀⢀⡴⢋⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠴⠂⠀⠀⠀⠢⠀⠀⣷⠀⠀⠀⡀⣦⡄⢀⣻⣀⡑⠀⠿⠀⢐⠀⠀⠀⠀⠀⠀⣀⠃⡠⠀⠀⠈⠁⠀⠈⢢⡀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠰⠀⠠⢍⡩⠅⠈⠄⠉⠀⠑⠀⠀⢀⣀⢀⠀⣈⡀⠀⠡⠀⠀⠀⠀⠈⠋⠀⠈⠀⠃⠀⠆
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⠀⠀⠄⠠⣦⡀⣰⠚⠆⠀⠀⠤⠀⠈⠀⡄⢁⠀⣨⠼⠋⠀⠀⠀⠀⠰⠀⠼⢦⡀⠀⠌⠀⠀⠤⠐⢠⠖
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠠⠈⠀⠘⢠⠖⠀⠀⠐⠀⠰⠂⡠⠀⢠⠆⠀⠀⠀⠀⠰⠃⠀⠈⢀⡀⠀⠑⠦⢦⡀⠠⠤⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⣤⠋⡀⠀⠀⡀⢳⡆⠀⠀⠀⢦⠖⠀⠀⢡⡀⠂⠀⡟⠀⠀⠀⠁⠀⠀⠀⠀⠄⠈⠀⢀⠔⠉⡀⠁⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢹⠀⠀⠚⣀⠇⠀⠀⠀⠀⠇⠠⡆⠀⠀⢀⢀⠀⣰⢇⣤⠂⠀⠡⠄⠀⠄⠱⠀⢠⡲⢆⣤⣤⣁⡀⠀⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠸⠀⠀⠐⣿⠀⠀⠈⠀⠄⠙⠊⠀⠀⠀⠀⡇⠀⡟⠁⢠⢢⠄⠀⠀⠀⠂⢀⠐⢡⠊⠀⠄⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠴⡃⠀⡀⠀⠀⠀⠀⠑⡇⠀⣆⠀⡟⠀⡇⠀⠄⣀⡀⠈⠀⠀⡰⠃⠠⠃⠀⠑⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠘⢢⡀⠀⠀⠀⠀⠆⡃⠀⠹⡀⢾⠂⠈⠡⠆⠈⠉⠉⠀⠀⢀⠞⠀⠀⠀⢀⡠⠈⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠎⠀⢈⠧⡀⠀⢰⠂⠂⠘⠆⣘⠀⣨⠀⠀⡸⠃⠀⠄⣠⠌⠀⠨⡀⣠⣴⣷⠆⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁⠀⠉⠈⠃⠤⣄⡆⠀⢹⣶⠁⠀⠀⣧⡠⠔⠂⣀⡀⠔⠐⢠⡏⠀⢸⡇⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠇⠠⠄⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠎⠁⠀⠀⠀⣾⠀⠀⣾⠁⠀⠀⠀⠀⠖⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠰⢄⣀⡀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⡀⠀⠀⣀⠆⠀⠀⠀⠀⢠⡏⠀⡼⠃⠀⠀⠀⠂⠀⠠⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⣀⠙⢆⠆⠀⠀⡀⠀⣶⣶⠶⡶⣶⣤⣤⡀⠐⠇⠀⠀⠂⢀⣤⠟⠀⠀⠙⢳⠀⠀⠀⠆⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⢠⡌⠁⢠⠀⠈⠑⠲⠞⠉⠉⢳⣀⠀⠀⠀⠈⢙⠻⡿⠒⠶⠖⠒⠋⠁⠀⡄⡀⠀⢾⣀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⣬⠀⠀⠀⠀⠁⠂⠈⠀⠀⠀⠀⠀⠀⡄⠀⢹⡀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠋⢠⠙⠂⠀⢸⡀⠠⠀⠀⠀⠀⠀⠀⡞⠀⠀⠘⠃⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠐⠀⠀⠃⢀⣤⣴⣶⠟⠛⠛⡇⠀⠀⠀⠀⠀⠀⠀⠘⠦⠘⠀⠀⠁⠀⠀⠁⠙⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠞⣧⠀⠀⠇⠠⠄⠀⠀⠀⠀⠀⢠⡀⢀⠈⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠓⠈⠠⠂⠀⠀⠀⠀⠠⣤⠄⠀⢀⠀⠀⠂⠈⠂⠀⠀⡾⠂⠀⠰⢿⡶⠒⠚⣦⣠⣄⠐⣴⡆⠤⣶⣻⣦⡲⣐⠔⠢⣶⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠊⢁⠀⠀⠀⢀⣠⠃⠸⠀⠀⠀⠤⡀⠀⡄⠀⠀⡅⣄⢀⣀⣤⠀⠀⢀⣈⡁⠀⠀⠀⠀⠈⣹⢏⣾⢻⡜⢿⠓
⠀⠀⠀⠀⠀⠀⢀⠄⠼⠆⠈⠀⢀⡔⠈⠁⠠⠊⠀⠐⠂⠀⠐⠄⠀⠠⠐⢖⣀⣀⣈⣀⡄⢀⣀⣤⣀⠀⠀⣀⡀⠀⢡⠻⠟⠋⠻⢦⡑⣤⣦⣀⢀
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢰⡏⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢷⡴⠿⠟⢛⠀⠤⠁⠀⠁⠀⠀⠈⠁⠀⠁⠀⠈⡋⢁⠀⠱⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢉⠛⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⢈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠄⠀⠀⠐⢂⠀⠰⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠠⠄⠂⠀⠀⠀⠀⠀⠒⣨⠀⠂⠀⠀⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠽⣇⠈⢷⡀⠆⠘⠆⠀⠰⢃⠤⠀⠀⠀⡀⠀⠀⠄⠈⢤⣀⣀⠤⠤⠀⢉⠀⠉⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠒⠋⠳⣤⠛⠾⢆⡀⠀⠐⠃⠴⠀⠀⠀⠃⠈⢁⠚⠀⠀⠀⡀⠀⠀⠀⠊⠈⠛⠉⠓⢄⠀⠀⢰⠛⡉⠑⠠
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⢁⠤⠄⢨⡧⠄⣀⡀⣔⡀⢀⠀⠀⢤⠀⠀⠈⠁⠚⠉⠀⠀⠀⠆⠀⡈⠀⠀⠀⡄⢀⡆⢀⠀⠀⠆⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣈⠀⢠⠄⠁⠀⠀⠀⠀⠀⠉⡁⠀⠓⠀⠠⠘⠀⠑⡀⠀⠀⠠⠐⠒⠒⠄⠀⠁⠈⠂⠸⠀⠀⠙⠁⡀⠀⠀⠠⢉⡘⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣈⢀⠘⡁⠤⠌⠻⠀⠀⠉⠁⡈⠉⠀⡄⠀⠰⠃⢠⠀⢠⡐⠂⠀⠂⣄⠀⢠⡅⠀⠀⡜⠀⠃⠀⠀⠀⠀⠄⠀⡘⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠜⠁⠀⠀⠀⠀⢀⠀⠀⠐⠒⢢⠈⠉⠐⠒⠛⡄⣠⠕⠚⠉⠉⠓⠛⠁⠀⠀⡈⠉⠀⠀⠐⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠤⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠂⠉⠂⠁⠀⠤⠀⠀⠀⠤⠂⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣫⠼⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣮⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣼⢠⡴⢲⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⠧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢁⡽⠛⠋⠴⠋⠀⣀⣀⣀⣀⣠⣄⣀⣀⡀⠀⠀⠀⢀⡀⢀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠗⠋⠀⠀⠂⣠⠖⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠓⣦⠀⠘⢦⢦⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⡀⠐⠀⣠⣾⣿⡀⠀⢀⣀⢀⡀⡀⡀⡀⠀⠀⣠⡟⠀⠀⠀⠛⠃⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠀⠊⢀⡴⠋⠀⠧⢉⣛⣶⣾⠿⢶⣷⣷⣶⡿⢶⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⢀⢰⠗⠀⠀⠀⣰⠋⠀⠀⠀⠀⣒⡽⡃⠀⠀⠈⠁⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⣀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⢱⡿⠀⠀⠀⠀⠻⠦⢤⣤⣤⣤⣤⣬⣥⠤⠴⠶⠶⠞⠛⠛⠉⠁⠀⠀⠀⢀⡠⠆⠀⠀⢀⡀⠤⠾⡄
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠉⠀⠀⠔⠈⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⢠⣿⠄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠸⠿⠍⢐⣐⣂⣒⠒⠀⠀⠀⠀⠄⠀⠀⠤⠤⠤⠤⠤⠄⠀⠠⠤⠠⠤⠠⠶⠆⠒⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡟⠻⠯⠤⠤⠤⠤⠭⠭⠭⠽⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣤⣭⣙⣒⣒⣒⣒⣒⣒⣒⣒⣚⣩⣬⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⠁⡏⢹⠉⡏⢹⠉⡏⢹⠉⡇⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⠾⣦⣧⣼⣤⣧⣼⣤⣧⣼⣤⠷⣾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠻⠶⠶⠶⠶⠶⠶⠶⠶⠟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠛⠿⠿⠿⠿⠿⠿⠛⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠶⠶⢶⡶⠶⠶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠻⠶⠶⠶⠂⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠀⠀⠀⠀⢀⡴⡖⢶⡄
⠀⠀⠀⠀⠀⢀⡴⠶⣤⣤⣄⡀⠀⠀⣿⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⣿⡀⠀⠀⣴⠟⢮⣁⣼⡇
⠀⠀⠀⠀⠀⢸⣄⣀⣸⠐⠚⣻⠀⠀⣿⣦⣄⣀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⠆⠘⣧⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⠿⠋⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⢶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣉⣸⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠋⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠶⢦⣤⡏⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠈⠀⣰⠋⠉⠛⢶⡇⣠⠴⢤⣄⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⡀⠀⠀⣠⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠛⠁⠀⠀⠈⣷⠀⠀⠀⠀⣰⣿⣿⠟⠉⠀⢹⣦⣾⣿⡟⠋⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣀⣈⣳⣄⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⣸⣿⣿⠏⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠈⠳⠤⠤⠖⠋⠁⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⣸⠇
⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⢸⣿⣿⠇⠀⠀⣠⡾⣿⣿⡏⠀⠀⣀⡼⠋
⠀⠀⣼⣿⣿⡇⠀⢀⣤⣄⠀⠀⠀⢸⡇⠀⠀⣀⣤⣤⣤⣄⣼⣿⣿⣿⠀⠀⣿⣿⣿⠶⠖⠛⠁⠀⣿⣿⡷⠶⠛⣩⣶⣶⣦
⠀⠀⣿⣿⣿⠀⢠⣿⣿⣿⠀⠀⠀⣸⠁⠀⣼⣿⣿⠟⠉⢹⡟⣿⣿⡏⠀⠀⣿⣿⣿⠀⠀⠀⠀⠰⣿⣿⡇⢀⡾⠁⣽⣿⣿
⠀⠀⢻⣿⡏⠀⣼⣿⣿⡿⠀⠀⠀⡿⠀⠀⣿⣿⡏⠀⢀⡿⠀⣿⣿⠇⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⣿⣧⣟⣀⣼⣿⡿⠃⠀⣤
⠀⠀⠈⠻⢿⠶⠋⣿⣿⡇⠀⠀⢰⡇⠀⠀⢿⣿⡇⢀⡾⠁⠀⣿⣿⠀⠀⠀⣾⣿⣿⡆⠀⠀⠀⢸⣿⣿⡿⣟⠛⠛⠉⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⢀⡾⠀⠀⠀⠈⠙⠓⠛⠁⠀⠀⢻⣿⠀⠀⢠⡇⠈⠻⣷⣄⣀⡴⠏⠀⠉⠀⠹⣄⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣠⠟⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠄⠒⠃
⠀⡎⠈⢢⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠠
⠀⠡⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⢠⠋⠁⠀⠙⡄⠀⠀⠀⠀⢸⣿⡟⠛⠻⣷⡆⠸⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⢠⠋⠉⠉⢳⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠓⠺⠀⠀⠀⢰⠁⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⡿⠀⢿⣷⣄⣼⣿⠏⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠋⠁⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⢸⣿⡇⣀⣼⡿⠃⠀⠀⠉⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⢀⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠖⠒⠒⠋⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠿⣷⡄⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣾⠿⠃⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⢠⣶⣄⠀⠀⠀⠀⠀⣀⣀⡀
⠀⢀⣠⣤⠤⠤⢤⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⢀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⣠⣿⣿⠛⠛⢷
⣼⣿⣿⠁⠀⠀⠀⠀⣴⣤⡀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠶⠦⠤⣿⣿⣿⣀⣀⣠⣿⣿⠟⠀⠀⢸⡇
⢿⣿⣿⡀⠀⡿⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢁⣀⡀⠀⠀⣿⣿⡇⠀⠀⣽⣿⡇⠀⠀⠀⣼⠁
⠈⠛⠿⠿⠞⠁⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⣿⣿⡇⠀⠀⣿⣿⠃⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡆⠀⠀⣴⣶⡄⠀⠀⣿⣿⡇⠀⠀⣿⣿⠗⠛⢁⣴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣤⠶⢾⣿⣷⠤⠤⠤⠶⠃⢀⣴⣿⡿⠷⣾⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡀⢠⡞⢰⣿⣿⠇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⣾⣿⡟⠀⢰⡇⢿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⣧⡞⠀⢸⣿⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⢻⣿⡇⠀⡾⠀⢸⣿⡇⠀⠀⣿⣿⣇⠀⣰⢿⣿⡇⠀⠀⡟⠻⠟⠀⠀⠈⣿⣿⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⠀⠛⠷⠞⠁⠀⠈⣿⡇⠀⢰⠇⠈⠛⠛⠋⠈⢿⣧⠀⣸⠃⠀⠀⠀⠀⠀⢿⣿⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⠟⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠘⢿⣄⣠⠏
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣟⣿⣻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣾⣵⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣤⣀⣤⡀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢠⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣻⣿⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣾⣀⣀⠀⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢀⣿⡿⠟⠋⣠⡾⣢⡗⠀⢰⣿⠇⠐⢿⡟⠉⠉⣼⡟⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣸⠏⠀⠀⠰⣿⡴⢻⡇⠀⣾⡯⠀⢀⣿⠁⠀⠀⣿⠃⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣾⣿⣿⣿⠀⠀⠀⠀⠟⠀⠀⠀⠀⠈⠀⠀⠁⠀⠈⠁⠀⠈⠋⠀⠀⠀⠈⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢹⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠈⢳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠁⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⡀⢀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣿⡇⠉⠙⢿⣷⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠠⠴⠆⡀⢰⡯⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠄⠀⠠⠶⠦⠘⠃⠀⠀⢀⣴⠠⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⢦⠀⠀⠀⠁⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣧⠀⠀⠀⠀⠀⠀⣠⣴⡃⠀⠈⠆⠺⣧⡀⠀⠀⠀⠘⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⠙⣷⣤⣤⡤⠖⠛⠛⣛⡳⡀⠈⠁⠉⠛⠃⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠀⠀⠻⣿⡏⠀⠐⠻⠋⣩⠴⢛⣦⣄⡀⠀⠀⠀⠀⢄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠈⠀⠀⡁⠔⠉⠀⠀⠉⣩⡿⣿⣿⠂⠂⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠆⠁⠀⠀⠀⠀⠀⠤⠶⣻⣿⠀⠙⡟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠃⠀⠀⠇⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠄⠀⠀⠀⠀⡠⠂⠀⡠⠊⠀⠀⠀⠀⠀⡆⠀⠈⢀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣠⠞⠙⠆⠀⠀⠀⠀⠰⠇⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢱⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⢸⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠋⠉⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠈⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣼⠧⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡾⠁⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⡞⠁
⠀⢀⣤⠶⠶⠶⠶⠶⣤⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟
⢠⡟⠉⠢⡀⠈⠠⡀⠀⠈⠙⠶⣄⣿⠛⠒⠒⠒⠓⠛⠛⠋⠉⠉⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⣿⢧⠀⠀⠈⠳⣄⠈⢦⡀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⣿⠘⣆⠀⠀⠀⠈⠳⣄⠙⢦⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠸⣆⠈⢢⡀⠀⠄⣀⠈⠳⢄⠙⢦⣘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠂⠁⠀⢀⡾⠁
⠀⠘⢦⡀⠙⢦⡀⠀⠀⠀⠀⠑⢄⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣸⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠙⢦⣀⠈⠓⠶⢤⣤⣀⣤⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠈⠛⠲⠦⣤⣤⣤⠶⠟⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠉⢉⣽⠟⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⣠⠾⠋⠁⠠⠄⠀⢀⡤⠞⠉⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⠀⢀⣀⣠⠴⠛⠁⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⡠⠚⠁⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⠦⠤⢤⣄⣀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣸⡃⠀⠀⣠⠖⠁⠀⣠⠖⠉⠀⠐⠀⠀⣠⠎
⠀⠀⠀⠀⣠⣴⡛⠛⠲⢤⣀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⢴⣟⡁⢀⣠⠞⠁⠀⣀⣀⡤⠖⠋⢁⡼
⠀⠀⠀⣰⠏⠈⠙⠲⢦⣀⡈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠈⠉⠉⠉⠉⠉⢉⣀⣠⡤⠶⠋
⠀⠀⠀⣿⣄⠀⠀⠀⠀⠈⠙⠲⢤⣀⠈⠙⠲⢤⣀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⢹⣏⠑⠦⣄⡀⠀⠂⠀⠀⠈⠙⠲⢤⣀⠈⠙⠲⢦⣤⢿⡆
⠀⠀⠀⠀⠻⣆⠀⠀⠉⠓⠒⠤⠤⣀⣀⣀⣀⣬⠽⠶⠒⠉⠀⢀⡿
⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⣀⠀⠀⣀⣀⣀⣤⡴⠞⠋
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠈⣹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢜⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂
⡒⠶⢶⠤⢄⠀⠀⠑⠊⠀⠀⠀⠀⠀⢀⠔⠂⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠠
⠠⢀⠀⡇⠀⠀⠀⡠⢤⣶⣆⢀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠉⠁⠀⠸⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡾⠀⠀⠀⠀⠱⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡸
⠀⢀⠆⠀⠀⠀⠀⠀⠉⢄⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠙⠋⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠸⠀⠀⠠⠂⠀⠀⠀⠀⠑⠇⡀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠐⠄⠉⠉⠉⢠⠁⠀⠀⠀⠈⠳⣤⣀⣀⢴⢣⠀⠀⠀⠀⠀⠠⠎⠉⠑⡄⠀⠀⠀⠀⠸
⢀⠇⠀⠀⠐⠄⣀⡀⠀⠀⢀⡀⠈⡀⠤⣄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⡼⠶⣄⠀⠰⠋⠙⠳⡄⠀⠀⠈⠧⣀⡴⠁⠀⠀⠀⠀⠠⡀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢺⡀⢀⡸⠀⠘⢤⣀⠴⠃⠀⠀⠀⠀⠀⡶⠖⠢⡀
⠀⡠⠶⢤⠀⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⡞
⠀⣇⠀⣠⡦⣀⣀⢄⡀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠒⠚⠁
⢀⣈⠉⠁⡸⠋⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠁⠈⢳⠄⣇⠀⠀⠀⢠⠆⠀⠀⠀⢀⠄⢤⡀⠀⠀⠀⠀⠀⠀⡠⠋⠀⡠⠀⠈⠀⠀⠀⠘⠀⠙⢄⠀⠀⠀⠀⠀⢀⡀⠀⢸⠀⠀⠀⠀⣀⠠
⢤⠴⠋⠀⠈⠲⠤⠖⠃⠀⠀⠀⠀⠹⢄⣀⠇⠀⠀⠀⠀⢀⠴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⢣⡀⠀⠀⠆⠻⠟⠀⠚⠂⣤⠄⠀⠀⠀⠀⠠⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠁
⢀⣤⠆⢱⠀⠀⠀⠀⡀⠤⡀⠀⠀⠀⣠⠤⢤⡀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡤⠃⠀⠀⠀⠀⠀⠀⠠⠉⠐
⠉⠀⠀⠀⢢⠀⠀⠘⢅⣀⡼⠓⠢⢼⠁⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠐⠄⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⠾⢁⡤⠦⠠⣄⠀⠀⠀⡠⠄⢄⠀⠀⠀⡠⠲⠶⠟⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠈⢄⠀⠀⠀⠀⠤⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠈⡧⠀⠘⢅⣀⡼⠀⠀⠀⠈⡂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠙⠢⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠢⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡞⠁⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢁⠀⠀⠀⠀⠀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣠⠁
⠀⠀⠘⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠑⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢠⠋⠀⠀⠀⠉⢣⠀⠀⠀⠀⢠⠀⠲⡎⠉
⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠈⡄⠀⠀⠘⢄⡀⠀⠀⣠⡎⠀⠀⠀⠀⠸⡀⡀⠆
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣠⠃⠀⠀⠀⣸⣙⠉⠉⢱⠼⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠲⢴⠘⠣⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⢱⠀⢣⡀⠀⣨⠃
⠀⠀⠀⠀⠀⢠⠒⠒⠄⠀⠀⠀⠀⠀⣇⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠡⠀⢀⠌⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠀⠀⡄⠀⠀⠠⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣴⣶⣶⡶⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣦⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣶⣶⡶⠶⠷⠿⠛⠛⠛⠛⠉⣹⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⣿⣷⣾⣶⡾⠿⠿⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠤⠀⠀⠀⣤⣤⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡿⠿⠿⠿⠛⣖⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠖⠶⠿⠿⠛⠛⠁⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⡶
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢼⣶⢤⣤⡄⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⢟⠋⠙⠷⠀⠀⠠⠶⠿⠿⡿⣿⢿⣿⣿⣿⡿⢷⠓⣻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣀⣰⠀⠀⠀⠀⠀⠀⠀⣯⣿⣿⣿⣿⣿⡛⣏⢸⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⠉⠉⠀⠀⠀⠀⠀⠀⠃⣿⣿⣿⣿⣿⣿⡇⢹⠘⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡁⣿⣿⣿⣿⢻⣿⣿⢸⡆⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⢹⣿⢻⢸⡇⢘⣸⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠀⡀
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⣿⣿⣿⣾⣿⠸⡏⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠋⠉⠁⠀⢀⣀⣠⠴⠞⠋
⠀⠀⠀⠀⠴⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠘⠛⠛⠛⠛⠛⠘⠋⠁⠈⠉⠉⠉⠉⠁⠁⠀⠀⠀⣀⣀⣤⣤⠴⠶⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠂⠠⠤⠤⠤⢤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠴⠶⠒⠒⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠒⣉⣩⠤⠤⢄⠈⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣰⣊⣁⡾⠋⠀⠀⠀⠀⢸⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢴⡖⠒⠉⠉⠉⣉⠀⠀⠀⠀⠈⢫⡀⠀⠀⠀⠀⠀⡌⠘⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠧⠶⠶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠘⠢⠤⠬⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⠀⠀⣀⠤⠔⠒⠚⠋⠉⠉⠉⠉⠓⢦⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⣠⠋⢀⠊⢠⢾⡃⠀⠀⢀⠤⠐⠒⠒⠒⢦⣀⣀⣀⠤⠚⠆⠀⠀⠀⢱
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⡎⠀⠉⠙⠃⠻⣷⣐⣒⣂⣸⠿⠃⠄⠀⡈⠄⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠁⠀⠠⠤⠤⠤⠤⠤⠤⠄⠒⠒⠛⠈⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⣶⣶⣿⡿
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠛⠓⠁