Nostril Emojis & Symbols

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

⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠆⣀⣤⡾⠋
⠀⣿⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣿⠀⠀⠀⠀⠀⠸⣆⠀⠀⢀⣀⡴⠞⠋⢁⡤⠖⠋⠉
⠀⠙⠷⣬⣉⠙⠻⢷⣄⠀⠀⠀⠀⢀⣤⠾⠛⢋⣉⣥⠾⠃⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠉⠀⠀⡰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣏⡀⠛⡟⢠⣿⡿⢹⣿⣿⠃
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠁⠘⠻⢣⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣭⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠋⠀⠔⠒⠒⠄⠀⠀⠠⠖⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠿⡉⣧⠀⠠⠾⠯⠀⠀⠀⠀⠚⠓⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣬⣿⣿⣿⣿⣶⡷⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⣇
⠀⠁⠈⠀⠀⠀⢠⣤⣼⡁⣻⢿⣿⣿⠉⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⢛⣧⣀⡀⠀⠀⠀⢠⣿⣶⣶⣾⠁⢰⡋⠁⠀⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠛⠧⢤⡀⠀⠘⣏⠀⠈⡟⠠⠛⠀⠀⠀⢄⠀⡀
⠀⠀⠀⠀⠀⠀⢀⡤⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠁⠐⣄⡙⠦⠴⠧⠀⠂⠀⠀⠀⠀⢐⡡
⠀⠀⠀⢀⣤⠾⠁⠀⢠⡟⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢥⣀⣀⣤⠃⠀⠀⠀⢀⣴⠋
⠀⠀⣴⠋⠀⠀⠀⠀⠈⢷⣿⣿⡿⣿⡿⢿⣦⣤⡀⠀⠀⡠⠀⠙⠃⠀⠀⢀⡄⠉⠙⢧⡀
⣠⠞⠉⠀⢀⣠⣤⣄⣀⠀⠀⠀⠀⠘⢧⡞⠀⠀⠀⠀⠜⠀⠀⠀⢠⠀⠀⠀⢡⠀⠀⠀⢳⡀
⠻⡄⠀⡾⠉⠀⠀⠀⠀⠉⢆⠀⠀⠀⡸⠁⠀⠀⠀⢠⡀⠀⠀⠀⠈⡇⠀⠀⠈⢇⠀⠀⢀⡗
⠀⠙⣆⢻⡀⠀⠀⠀⠀⠀⠘⡀⠀⠠⠁⠀⠀⠀⢠⠃⠱⡀⠀⠀⠀⣿⠀⠀⠀⠈⢆⠀⡼⣧
⠀⠀⠈⠻⡇⠀⠀⠀⠀⠀⠀⢁⡠⠁⠀⠀⠀⢀⡆⠀⠀⠘⣄⠀⢀⡟⡄⠀⠀⠀⠀⢳⡃⢹⡀
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⡜⠀⠀⠀⣀⠘⣆⡜⢀⡇⠀⠀⠀⠀⠀⠳⡄⠃
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠐⠒⠚⢻⠀⠸⠀⠀⢹⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡟⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠁

⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠤⣄⡀
⠀⠀⠀⠀⢀⡴⠋⠀⢀⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠑⢦⡙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⢸⡇⠀⣠⣤⣤⣤⡈⠳⣌⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠁⠀⠹⣦⢻⠀⣀⣀⣀⣄⣠⣤⣄⠀⢀⣀⠸⠄
⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⠀⠈⠃⠈⣿⢧⠉⠃⠁⠈⣧⠴⢶⡄
⢀⡤⠞⠋⠀⠀⠐⠒⠛⠁⠀⠀⠀⣶⠛⠒⠀⡴⠚⣾⡗⠀⠁⡄⢶⡄⢀⣤⠾⠗⠋⢻⡄
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠝⠃⢀⢠⡄⠚⠉⠛⠛⠀⢙⡛⢡⠀⢀⣄⠐⠚⠋⢀
⠈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠷⠶⠆⡈⠋⡀⠀⣰⠀⠐⠤⠞⠷⡼⠃⠘⠾⠇⢠⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣤⣶⣀⣷⣀⣿⠋⠹⣄⡀⠳⠾⢤⣀⣀⡦⣈⣷⣄⣹⠇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⢰⡏⢀⣾⢦⣄⣽⠋⠀⠉⠁⠉⠁⠀⠀⠀⠀⢣
⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢄⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⣇
⠈⠉⠛⠛⠛⠲⣦⡀⠀⠀⠀⠀⠀⠳⢤⣤⣀⠀⠀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢀⠀⣹⢂⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠖⠛⠋⠉⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠘⢦⠿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢨⡇⠙⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣤⠞⠋⠀⣀⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠇⠀⠀⣼⠉
⣿⣿⠟⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⢿⣦⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⣿⣿⡿
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⢿⡄⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⢀⣿⠀⠀⣿⣿⣧⣤
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⡟⠀⠀⠠⣤⡄⢿
⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⠛⠿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⡀⠀⣸
⠀⠀⠀⠀⠀⣾⡟⠀⣴⠶⣦⡄⠈⠙⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿
⠀⠀⠀⠀⢸⣿⡃⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢿⣧⡀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠿⣶⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠉⠉⢹⡟⠙⠻⣷⡀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⢻⡆⠀⠘⣷⠀⠀⣿⣧⠀⠀⠀⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠸⣿⠀⠀⣿⡇⠀⢸⣿⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠹⠇⠀⠈⠉⠀⢸⣿⣶⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⠿⢿⣶⣤⣤⣄⣀⣀⠀⠀⠀⢈⣿⡆⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⠀⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠈⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢿⣷⡀⠀⠀⠀⠀⢻⣿⠀⢀⣠⣴⡿⠟⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⢿⣧⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠙⢿⣿⠋⠁⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠈⣿⣦⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠹⢷⡄⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠟⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⣿⡇
⢀⣴⣶⣶⣶⣶⣶⣶⣾⠿⠟⠃⠀⠀⠀⠀⠀⣠⣾⠟⠋⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣄⣀⡀⣀⣠⣾⠋
⠀⠙⣿⡿⠿⠿⠿⠿⣿⣿⣆⠀⠀⠀⠀⢻⣷⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⡟⠉
⠀⠀⠹⣿⣦⣠⣤⣶⠿⠛⢿⣧⠀⠀⠀⣸⣿⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠈⠻⢿⣿⠿⠟⠁⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠋⠉⠉⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠉⠉⠉⠉⢙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⡟⣿⣏⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⠚⢿⡏⠙⣿⡾⣿⣷⣾⣶⣿⣿⠀⢸⣿⣷⣶⣶⣾⣷⣾⠟⠉⡿⠗⢲⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠛⣽⡷⣠⣿⣦⡾⠁⠀⠀⠀⠀⢹⣿⠀⢸⣿⠀⠀⠀⠀⠀⠹⣦⣼⣧⣴⣾⡿⠛⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣁⣀⣀⣀⣨⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇⣾⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣽⣁⣀⣀⣀⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⡇⣿⣯⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤
⣿⣿⣿⣿⣿⠿⠿⢿⣿⠿⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠛⠛⠛⠛⠛⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠐⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢒⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠙⢦⡤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠻⣄⠈⠙⠳⣤⡀
⠀⠀⠀⠀⢀⣴⠟⠁⢀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠈⢳⣄
⠀⠀⠀⣠⡟⠁⢀⡴⠋⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⢳⡀⠀⠀⠀⠀⠀⠀⠀⢻⡑⣆⠀⠀⢻⡆
⠀⠀⣰⡟⠀⢠⡟⠁⣴⠃⠀⠀⠀⢀⠶⠶⠶⠶⠶⣶⣶⣶⠶⠶⠶⠞⠛⠋⠉⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠘⡇⠘⣧⠀⠀⢿⡀
⠀⢠⡿⠀⢀⡟⢀⡞⣿⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⣷⠀⠘⣇⠀⠸⡇
⠀⣼⡇⠀⢸⠀⡾⠀⡇⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠋⠙⠃⠀⠀⠀⠀⠀⠀⠚⠛⠆⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⣿⠀⠀⢻⠀⠀⣷
⠀⣿⠁⠀⠀⢰⡇⠀⡇⠀⠀⢠⡇⠀⠀⠀⠀⢀⡤⠶⠒⠒⠶⣄⠀⠀⠀⢀⡤⠖⠒⠲⢦⡀⠀⠀⠀⠀⢻⠀⠀⠀⡧⠀⠀⢸⡆⠀⣿
⠀⣿⠀⠀⠀⢸⠀⠀⢿⠀⠀⣸⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠘⡆⠀⡴⠃⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢸⡄⠀⢠⡇⠀⠀⢸⠁⠀⣿
⠀⣿⡂⠀⠀⠈⠀⠀⠘⡆⠀⡟⠀⠀⠀⠘⠀⠀⣠⣤⣤⠄⠀⠀⠃⠘⠁⢀⣴⣶⡶⢄⠀⠈⠀⠀⠀⠀⠘⡇⠀⣸⠁⠀⠀⢸⠀⠀⣿
⠀⢻⣇⠀⠀⠀⠀⠀⢀⣹⣄⡇⠀⠀⠀⠀⠀⢸⣿⠟⠳⣴⡆⠀⠀⠀⠀⢸⡿⠋⢳⣾⠀⠀⠀⠀⠀⠀⠀⣇⣠⣇⠀⠀⠀⠈⠀⠀⡿
⠀⠘⣿⡄⠀⠀⠀⢠⠏⠁⢸⡇⠀⠀⠀⢂⠀⠀⠻⠦⠴⠟⢀⣀⣀⠀⠄⠈⠛⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⢷⠀⠀⠀⠀⢠⡇
⠀⠀⠹⣿⡀⠀⠀⣿⠀⠉⢿⠁⠀⠀⠀⠀⠁⢄⣀⣀⣀⣴⡾⠁⠀⠉⠂⠐⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⣷⠋⠓⡼⠀⠀⠀⠀⣼⠁
⠀⠀⠀⢹⣷⠀⠀⠸⣦⡀⢸⠀⠀⠀⠀⠀⠀⡔⡉⡁⠀⠸⡇⠀⠀⠀⠀⠀⠀⠔⠐⠐⠀⠀⠀⠀⠀⠀⠀⣿⢀⠔⠁⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢻⣧⠀⠀⠈⠉⢹⡄⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⣿⡄⠀⠀⣄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡆⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢠⠀⠀⠀⣿
⢀⣴⠂⠀⠀⣾⠁⠀⠀⢸⡇⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁⠀⡇⠀⠸⣇⠀⠀⢿⣆
⢸⣧⣀⣠⡼⠃⣠⠀⠀⣿⠃⠀⢸⡏⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣿⠀⢸⣧⠀⠀⠻⣷⣄⣈⠻⣷⣶⠞
⠘⢿⣿⣥⣴⣾⠏⠀⣼⠏⠀⢀⣞⡴⠁⠈⠳⣄⠀⠀⠀⠀⠀⣀⠳⠞⠁⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⢻⣧⡀⢿⣦⣀⠀⠈⠙⠛⢛⡟⠁
⢤⣤⣶⣾⣟⣁⣤⠞⢃⣀⣴⠟⠋⠀⠀⠀⠀⠈⠛⢦⣄⡀⠀⠈⠓⠂⠀⠀⢀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠙⠻⠶⠽⠿⠿⠿⠶⠛⠉
⣀⣈⣁⣀⣀⣀⣀⣈⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣻⣶⣶⣶⣶⣟⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⠿⠷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠞⠛⠋⠙⠛⠒⢺⣿⢥⣀⡀⠀⠀⢀⣀⡤⠜⡗⠒⠛⠉⠙⠛⠳⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⡩⠗⠺⢍⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⢣⣄⠀⠀⠀⠀⢰⣿⠧⠊⠁⠀⠀⠀⠀⠈⠑⠤⣀⡆⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣴⣿⡇⠀⢀⡤⢺⣿⠀⠀⠀⠀⠀⢲⣦⣄⠀⠀⠈⡗⢤⡀⠀⣤⣀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠹⣷⣴⠉⠀⢸⣿⠀⠀⣷⣀⣀⣰⣿⣿⡆⠀⠀⡇⠀⠉⣢⣾⠟⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠻⣷⣤⣸⣿⠀⠀⢻⣿⣿⣿⣿⡿⠃⠀⠀⣇⣤⣾⠟⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠈⠛⢿⣿⣄⡀⠀⠉⠛⠛⠋⠀⢀⣠⣶⡿⠛⠁⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠸⣿⡟⢿⣶⣤⣀⣀⣤⣶⡿⠛⢉⠇⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⠀⠀⠀⠀⠀⠀⣿⣇⣠⣼⣿⡿⢿⣿⣥⣄⣀⣸⠀⠀⠀⠀⠀⠀⣀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣷⣶⣶⡾⠿⢿⣿⡋⠉⠁⠀⠀⠀⠉⠙⢻⡿⠿⣿⣶⣶⣾⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣦⡀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣾⣿⡃⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣦⡶⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⠞⣀⢤⡴⠦⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠊⢠⡿⠊⠐⠋⠉⠐⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢀⣐⣛⢪⡃⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠃⠀⠀⢀⣤⡔⡖⠒⠛⠆⠀⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⣔⡊⠁⠸⣤⠤⢄⡀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡩⠋⠀⠀⠀⠀⠀⠀⠺⡆⠀⢿⠀⠀⠙⣦⠊⠁⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⠄⠀⠀⠀⠀⠀⠀⢀⡱⠀⠘⢦⡴⠊⠉⣯⣭⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⢰⣄⢀⠀⠀⠀⡰⣣⡇⠀⠀⠀⠉⣒⡶⠃⠠⣉⠉⠉⠉⠉⠉⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⠛⠓⠋⠉⠓⠚⢱⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⡐⠂⠤⠤⠶⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠕⣄⢠⣶⣧⡀⠀⠀⠀⠀⠀⠀⠀⣀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠹⡭⢴⠊⠉⠀⠙⢄⠀⠀⠐⠊⢁⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢣⠀⡇⢸⠀⠀⠀⠀⠀⠙⠒⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡁⠀⠀⠈⡇⢱⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⡔⢡⠋⠀⠀⢸⠈⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢠⠎⠀⢧⡀⢀⣀⡜⢱⠯⠹⡤⢄⡜⠱⡤⣄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠴⢇⡀⠀⠈⠉⡏⣞⠃⠉⠉⢉⢦⡋⢠⡤⠗⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢀⢀⣉⢵⡦⢤⢡⢹⣇⣀⡠⣜⠸⣙⣏⡀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠼⣢⡀⠇⢀⠬⠀⠁⡎⠉⠸⠬⠝⡼⡉⣷⣈⠗⠚⣥⡍⢦⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠔⢳⢸⠀⠋⡳⠖⠁⢀⡆⠀⢰⢡⢁⢩⠀⠀⠀⠉⠀⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡛⢁⡴⠯⣿⠦⢄⡉⠉⠉⠀⠀⠀⠧⡼⢸⢸
⣤⢲⢷⡾⣣⠀⢀⣀⣀⠀⡀⢀⠔⠁⢀⡨⠋⠀⠀⠀⠀⠀⠈⠉⠒⣲⡶⠒⠒⠙⠋⠁
⠺⠤⠄⠠⠜⠀⣾⢄⠁⠼⢣⠘⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⣎⠤⠤⠄⢀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠈⠁⠐⠉⠘⠛⠣⣄⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠴⠆⠀⠀⠠⠦⠀⠀⠀⠀⢈⠆⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠁⣀⡀⠤⠀⠒⠒⠒⠄⠀⠀⠀⠀⠀⠰⠀⢀⠀⣀⣠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠉⠀⠀⡀⠂⠀⠀⠰⢂⠀⠀⢀⠔⠁⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠀⠉⠹⣀⣀⠀⠀⢀⣴⠣⠄⠳⢄⠁⠈⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡜⠖⠉⠈⠀⠀⠘⠜⠀⠀⠀⠒⠉⠊⠈⠢⢒⠒⠊⠉⠁⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢬⠬⠀⠀⠀⠀⠀⢀⠃⠇⠀⢰⠀⠀⠀⠀⣠⠜⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡒⠉⠁⠀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠵⠱⠤⠄⡠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠴⠊⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠛⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣒⣦⣤⣀⣠⣤⡤⢤⡀⠀⠀⠀⢀⡤⠖⠚⠃⠀⠀⠀⠀⠀⠈⠉⠙⠓⢦⡀
⠀⠀⠀⠀⢀⣴⣪⢏⢔⡀⠲⣾⣿⣿⣟⣋⠉⢢⠉⢢⡀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇
⠀⠀⠀⣠⣿⠿⢗⠕⢡⠞⠋⠉⠉⠉⠛⠻⢷⡄⢣⡀⠙⣆⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⡴⠋⠀⠔⣡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣷⡄⡜⣦⣈⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⢸⡇⣧⣤⣾⢃⣀⣀⡀⠀⠀⠀⠀⢀⣀⡠⠤⡈⣿⣿⣿⡿⡆⠀⠀⠉⠒⠒⠒⠁⠉⠛⠛⠋⠁
⠀⠈⢧⡹⣿⣏⠈⠀⣈⣉⠁⠀⠀⠀⠉⣁⣀⣀⣀⢸⣿⡟⠁⡇
⠀⠀⠈⣿⣿⣿⠘⠿⣩⣯⠀⠀⠀⠀⠀⢹⡯⠝⠁⣿⣟⣜⡰⠁
⠀⠀⣰⡿⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇
⠀⢸⣿⡵⣣⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⢸⣯⡘⣿⣿⣧⠄⠀⠀⠀⠀⣀⣐⠀⠀⠀⠀⢰⣿⡟⣿⡇
⠀⠀⠛⢿⣮⣿⡇⠀⡐⠀⠀⠈⠛⠋⠁⠀⠀⣠⣿⡿⠁⡹⠁
⠀⠀⠀⠀⠙⣿⡇⠀⠃⢠⡄⠀⠀⠀⠀⣀⢴⣿⣿⠇⣼⠁
⠀⠀⠀⠀⠀⣻⡇⠀⠀⠈⢠⠗⠒⠒⠊⠁⠀⣿⣿⣿⣿⡄
⠀⠀⠀⣴⡿⣫⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣿⡛⡀
⠀⠀⢠⣿⠆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⡇
⠤⠶⢌⢿⣦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣛⢹⣿⡿⣉⠴⠦⠤⣄
⢢⠀⠀⠑⢍⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⠖⠁⠀⠀⡜⠁
⠸⡄⠀⠀⠰⠁⠀⠀⠀⠐⠒⠒⠒⠒⠒⠀⠀⠉⠉⠉⠀⠀⠀⠀⡸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⠑⣶⣤⡀⢰⣤⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣻⣿⣷⣼⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⣛⣛⣻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢁⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⣀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⣀⣤⣤⣴⣦⣤⣤⣄⣀
⣿⣿⣿⣿⣿⡇⠀⢀⣀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⣿⣿⣿⣿⣿⡇⢀⢸⣿⠃⠆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⠀⠀⣿⣿⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠈⠹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⣿⣿⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣷⣤⣤⣶⣦⣴⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠀⠠⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠃⡀⢠⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠸⠀⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡏⡸⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀⠀⢸⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⣿⣿⣿⣿⡇⠀⢸⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⡏⠀⢸⣿⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋
⣿⣿⣿⣿⣧⡀⢸⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠈⣿⡇⠀⠈⣿⣿⣿⣿⣿⣿⣭⣤⣤⣄
⣿⣿⣿⣿⡿⣾⣿⣿⣿⣏⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢀⠀⠀⠀⢸⡇⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁
⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠈⠀⠀⠀⢸⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣦⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⡀⠆⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⡼⠃⠀⠀⠀⠀⣾⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣶⣌⡙⠿⢿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠠⠈⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣷⣦⣤⣄⣀⡀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠻⠿⢿⣿⣿⣿⣾⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⠀⢀⣀⠀⠄⡀
⠀⠀⠀⠀⠀⢀⡀⠤⠄⠒⠒⠒⠒⠒⠒⠒⠀⠄⠀⠀⠀⠀⢀⡠⢆⣈⣁⠀⠁⠐⢤⣦⡉⠁⠀⠀⢄
⠀⠀⡀⢐⡉⠀⠀⣀⠀⠀⠀⣄⠀⠀⠀⢠⣤⠀⠀⠀⠠⠞⠉⢀⠄⠈⣉⠀⠈⠻⠇⡙⣄⠀⢠⣀⠀⠀⠀⡀
⠀⠈⠀⠘⣷⡀⡄⣿⢰⢶⣆⣿⣰⣸⠃⣾⠃⠀⠀⠀⢰⠐⡄⠋⠀⠀⠘⢷⣷⡶⠂⢑⠈⣷⣄⢻⢷⣶⡀⠈⢂
⠀⠀⠀⠀⠘⣷⣿⡏⢻⣀⡿⢹⢿⡏⠸⠃⠀⠀⠀⢠⠄⣸⡿⠓⢜⠶⠛⠈⢯⡁⠀⢈⠀⣬⣻⣏⠁⡝⠋⢲⣄
⠀⠀⠀⠀⠀⠘⠇⠁⠈⠉⠁⠀⠀⠐⠂⠀⢀⣠⣾⠟⠀⠀⠀⠀⠀⠑⠶⠴⠖⠕⢺⣿⡷⣿⣭⣍⣓⡮⢲⣬⣿⣇
⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠻⢿⠄⣀⠀⠉⠀⠀⣠⠖⠛⠉⠑⠂⠳⣝⣦⣄⡤⣽⡿⠷⢍⢳⣽⣿⣷⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢠⢀⣻⣶⣧⠉⠛⠋⡀⣷⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⣀⠠⠀⠀⠀⠀⠀⠀⡠⠤⣤⡀⠀⠀⠀⠀⢣⢠⢻⡎⢿⣿⢧⠀⠀⣸⢏⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠎⣐⠀⠀⠀⠀⠈⠠⠄⠀⠁⠀⠀⠀⠀⢈⡀⢸⡇⠀⣿⡷⢲⣮⣽⠎⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢰⡏⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⣞⡆⣆⣿⡿⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠟⠳⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠘⣽⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⡖⣠⣽⡿⢿⣤⡀⢀⡀⠀⠀⣀⡴⠂⠀⠀⠀⠀⢀⡞⢹⣾⠱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⣈⠸⠁⣸⣄⢬⣿⣿⣦⡙⣿⢿⠉⠁⠇⠀⠀⠀⣠⡟⠐⣿⣿⠠⠀⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢤⡿⢻⣦⠀⠛⢁⡹⠽⠿⣹⣷⣌⢛⣒⣁⠄⠀⠀⣴⠏⠀⠀⠿⠿⠦⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣽⠧⠀⠈⣓⣶⡿⠋⠞⢻⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⠦⣀⠀⢘⣴⡤⢤⡶⢦⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⡿⢟⠎⠝⠃⢒⡉⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣼⡇⠀⢸⣀⠐⣿⡇⣿⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣤⡀⢀⠄⢀⡠⠉⠀⣿⡇⣿⡞⠋⠁⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢲⠤⠠⠢⠀⠉⠀⡥⢀⣿⣧⠟⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠸⠀⡀⡄⠀⢄⠀⠘⠀⡅⠛⠁⠀⣤⠔⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠴⠀⠀⠀⢄⠀⠐⠄⢈⡠⠗⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠃⣄⠀⡀⢀⡉⠀⠀⠀⠂⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢸⡼⣻⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠒⠃⠀⠀⠀⠀⠀⠀⣈⠀⠈⠘⣿⣿⡇⠀⠀⠀⠀⢀⠚⠁⠀⠀⠀⠀⢉⢑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣢⡌⢈⡉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⠀⢻⣿⣿⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠓⠚⠋⢸⣿
⠀⠀⡠⠖⢒⣛⣛⣛⠒⠲⡄⠀⠀⠀⣾⣿⡿⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢷⡦⢐⣷⡆
⠀⠀⡓⠶⠤⠭⠭⢭⢤⠔⣻⠀⠀⠈⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡴⡂⠀⠐⠹⠆⠀⠀⠀⠈⠢⡑⢄⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠇
⠀⠀⢏⠒⠦⠤⠤⠤⠀⢚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠾⠗⠷⠚⠛⠒⣄⠀⠀⠀⠀⠀⠀⠈⠲⡕⠄⠀⠀⠀⢸⣿⣶⣿⡿
⠀⠀⠀⠁⠒⠀⠂⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠸⡇⢠⠤⠔⠒⢲⡄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠸⡜⢋⢛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⢹⠀⠀⠀⠀⠀⠀⢻⡈⠉⠁⠀⠈⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠄⠄
⠀⠀⠀⠀⠀⠰⡶⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢀⣀⣰⢶⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠖⠚⠁⠀⠀⠀⠀⠀⠈⠻⣏⣀⡇
⢀⠤⠚⠆⠀⠀⠀⠀⠀⣴⣾⣿⣦⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⢹⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⢠⠑⠁⠀⠈⣳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠰⡒
⠀⠀⠄⠀⢃⠀⠀⠀⠀⠀⠈⠉⠛⢦⣘⣷⠀⡀⣠⣛⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠰⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣰⢾⣿⣿⣿⣦⠀⠀⠀⠘
⢂⠉⣀⡀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⠛⠛⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠂⠁⠀⠀⠀⠀⠀⠀⠀⢷⣺⣿⣿⣿⡿
⠀⢣⠄⢀⠈⢸⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠀⠀⠀⠀⠀⠀⠀⠀⢙⠿⠛⠛
⠀⠀⠳⡤⢤⠊⡧⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠙⠒⠋⠁⠀⠀⠀⠀⢀⠀⠰⠷⣶⣶⡶⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠁⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠘⡝⢶⡆⠀⠀⠀⠀⠀⠀⠁⠔⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠊⠀⠀⠀⢀⢄⠀⠀⠀⠀⠀⠀⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠤⠒⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡉⠢⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠑⠒⠀⠉
⠀⢰⠀⠀⠀⠀⠀⠤⢂⠔⠁⠀⠀⠀⠀⠀⠀⠀⣠⠊⡀⢀⠀⠼⠂⣈⠢⡀⠀⠀⠀⠀⠀⠀⠀⡾⠑⠒⠓⠒⠬⣉⠹
⠀⠀⠓⠀⠀⠀⠀⡀⠙⢄⡠⠀⠀⠀⠀⠀⠀⡔⠁⢠⠀⠈⡀⢀⢠⠀⢑⡬⠂⠀⠀⠀⠀⠀⠀⠁⣀⣠⣶⣒⣓⣓⣶⣀
⠀⠀⠀⠀⢀⡴⠛⠷⡤⠊⠀⠀⠀⠀⠀⢠⠎⠠⠀⠀⠠⠄⠀⢠⡤⠒⠁⠀⠀⠀⣀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠢⡀⠀⠒⢀⡤⠚⠁⠀⠀⠀⠀⢀⣸⡍⢆⠀⠀⠀⠘⣿⣿⣉⣉⡉⠛⠛⣿⣿⡇⠀⠀⠀⢠⣤⡴⢦
⠀⠀⠘⢄⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡒⢈⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠊⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠈⢧⣠⠏
⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣁⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⠀⠘⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⢄⡒⠲⢶⠤⣁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠔⠁⠀⠀⠀⠈⠢⠀⠣⠈⠳⣌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠺⠏⠀⠸⠟⠉⠀⠀⡈⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠐⡂⢂⣀⣑⠒⠂⠀⠀⣇⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡴⠍⡨⠁⠁⠀⠈⠉⠁⠀⠀⡜⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠛⡨⠔⠙⠦⣄⣀⣀⣀⣀⡠⠔⠂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠮⢋⠔⠈⠀⠀⠀⠀⠀⢈⠉⠁⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠧⠄⠀⠀⠀⠀⠀⠀⢀⠀⠄⠺⠢⣄⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠠⡄⢠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠔⠈⠀⠀⠀⡆⠀⠈⠑⢦⡀⠀⠀⠀⢸⠀⠀⢀⠀⠈⠀⠆⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠉⠢⣄⠀⢸⠀⠀⣀⢒⣀⣀⣰⢻⡄
⠀⠀⠀⠀⠀⣀⣀⣄⡈⠤⣤⣤⣀⣀⣀⣀⣂⢶⠒⠾⡀⣀⣀⣀⣀⣀⣈⡳⣼⢀⢌⡇⠉⠁⠀⠉⢨⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡹⠀⠀⣤⠤⠀⠀⠀⠀⠈⠢⢇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢗⣒⢀⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠤⠤⠊
⠀⠀⠀⠀⠀⠤⠀⠠⢵⢖⣀⣐⡋⠐⠃⠀⠀⠀⣀⠤⣄⠀⠀⡠⢆⠒
⠀⠀⠀⠀⠀⡄⣀⡀⠈⡎⠀⠀⢡⠀⠀⠀⠀⠀⠋⠁⠁⠀⠀⠁⢀⢀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠑⠒⠒⠀⠏⠉⠁⠘⠀⢄⣒⠢⠀⠀⠀⠀⠀⠀⠐⠖⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣒⡂⠌⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣠⣤⣀⣀⠀⠠⢀⣄⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⣐⣒⡂⠐⠢⣐⣢⣵⣦⣶⣀⣬⡙⣷⣤⡢⢄⡑⠢⣥⡀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠖⠀⠂⠀⠈⠓⠢⡀⠙⠿⣯⣻⢿⣿⣾⣿⣿⣷⣭⡲⣴⡙⢳⣟⢿⣷⣆⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠈⠛⠛⠿⠿⣿⣯⡻⣟⢾⣝⢿⠻⣿⣧⣿⣷⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠘⠢⠙⢎⢷⢩⠻⣏⣿⣧⢌⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠈⣿⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⠀⠀⠠⡀⠀⣀⠀⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠄⠰⢀⣡⢰⣿⣼⣼⣆⣿⣷⣿⣿⣿⣷⣶⣦⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⣌⢇⣳⣿⢷⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢳⣷⣼⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠿⣿⣿⣿⣿⣻⠿⣿⣿⠛⠻⠿⢿⣿⣿⣿⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣹⣾⣿⣿⣿⣿⣿⣿⣿⡿⢿⡋⠹⠤⠚⣿⣽⣿⣿⣾⠀⠰⣿⠀⠀⠀⠀⠈⠉⠛⢿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣭⣿⣿⣿⣿⠿⠟⠁⠀⠘⣆⠀⠀⠀⠀⠉⠙⠓⠁⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠂⠒⠂⣤⠄⠀⠐⠘⠦⣄⡀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⢀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠠⠂⠔⠂⡀⠀⠈⠉⠛⠓⠒⠛⠓⠒⠚⠛⠉⠉⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠦⣀⡀⣀⠀⠤⢶⠶⣖⣶⣶⣷⣶⡶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠖⣿⡿⣫⣿⠟⠋⣠⢐⣨⣵⣾⣿⣿⣿⣟⣭⣶⣿⣿⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⡾⢛⡟⢡⣾⣯⣾⡟⣡⡨⣶⣵⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⡿⢿⣿⣿⣿⣿⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣾⣿⠀⣾⢁⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⠟⢋⣴⣿⠿⣿⣿⣿⣿⣶⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢘⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⣉⣤⣶⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠟⠛⠛⠛⠛⠿⠻⠟⠻⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣟⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⠀⡀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⣄⣠⣤⣄⣀⣠⣄⡀⠀⠠⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡯⠀⠇⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⡇⢠⣾⣿⠟⠋⠉⠉⠉⢉⣭⣍⡉⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟⠉⠉⠙⠻⣿⣿⡆⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣴⣾⣿⣎⡸⢏⠀⢀⣤⣴⣶⣷⣿⣿⣿⣿⣶⣦⡀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⠀⠙⣧⠀⣿⣿⠋⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡾⣍⣿⡏⠁⠿⣉⣽⢻⡟⢩⣽⣭⡙⢿⣿⡿⠿⠃⠀⠠⣿⡟⣹⣿⡿⠛⣩⣭⡉⠻⣿⣿⣧⠀⠀⠀⢿⣷⠟⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⣽⠿⡇⢸⠀⠉⠳⣾⠇⣼⣽⣯⣥⣤⣿⠆⠀⠀⠀⠀⣿⠘⠙⠿⢠⣜⣛⢻⠃⠀⣿⡿⠛⠃⣤⠀⡼⠁⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢇⠀⠀⠈⠀⠀⠀⠙⠾⢷⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣉⣡⣤⠾⠏⠀⠀⠀⠘⡶⢈⡆⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠈⢲⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣸⠃⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⣘⣇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⡾⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⡅⠘⠀⠀⠀⠀⠀⠀⠀⣠⣀⢀⡀⠀⠀⠀⢠⡀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⠔⢸⣷⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⢻⡟⢿⢿⣦⣤⣼⠿⣿⣼⣿⣿⠀⠀⠀⠀⠀⡀⠃⡜⠀⣽⣛⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⡿⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣹⡿⣿⡿⣿⣿⣟⢿⣏⠀⠀⠀⠀⠠⠎⠀⠀⡼⠀⣤⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠻⣿⡆⠀⠀⠀⠀⢠⣾⣷⢞⣇⢼⣴⠆⣜⡷⢾⣦⡀⠀⠀⠀⠀⢿⡶⠁⣔⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣿⢷⡄⠀⠀⣰⣿⣾⢷⣾⣿⠷⠿⢾⣿⣿⡾⣿⣷⣆⠀⠀⢊⠼⣿⢘⡐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠸⢧⡿⣶⣾⣷⣾⣿⣿⣿⣷⣤⣶⣶⣾⣿⣿⣿⣧⣺⣷⡶⣾⣾⣟⠠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠈⠛⠿⣿⣿⣿⡟⠁⠹⡿⠿⢿⣿⣿⡿⠉⢻⡿⣿⣿⠿⣻⡏⠂⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⢠⣽⣿⣿⣿⡄⠀⠈⠁⠉⠈⠁⠀⢀⣾⠁⠰⠂⣰⠋⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣾⣿⣿⣿⡏⠛⠦⣄⣤⣤⣤⡤⠖⠟⣁⠔⠀⠌⠁⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡟⠀⢀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⡀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡁⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡁⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣀⣀⣀⣂⣀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⣶⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⡖⠲⣯⣍⣉⣩⣭⣩⣉⣨⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣷⣤⣄⣉⣉⡒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⣿⠟⠛⠏⢹⣿⡿⣿⣷⣦⣅⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢫⠟⣹⡿⠟⠁⠐⠁⠚⠀⠀⠈⠁⠀⠀⠈⠉⣉⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠁⣰⣠⢀⣠⡄⡀⠀⢠⠀⠀⠀⢰⣦⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡇⣷⡆⣸⣾⠇⢀⣯⢿⣿⠛⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣿⣆⠀⡼⢀⣿⢿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣴⢸⡏⢸⣿⢀⠀⠹⣿⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⡞⣸⠃⣸⠏⣿⣿⣿⣿⣿⣿⣿⣧⡾⢱⠇⡾⢁⡷⢿⡿⠶⠄⢿⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢱⢃⡟⢀⣿⢸⣿⣻⣿⠿⠏⠙⠁⠋⠀⠋⠘⠁⠈⠀⠈⠀⣀⠀⠀⠀⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠇⣿⣸⠃⣸⡇⣾⠑⠋⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠿⠁⠂⠀⢨⣿⡇⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢠⡏⠀⠐⠛⠱⠃⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠈⠱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡸⡏⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⢠⢣⡇⡀⣼⡦⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⡖⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⡎⣼⠀⠁⣿⢁⣏⣦⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⢿⣿⣿⣿⣿⡟⠇⠀⠀⠌⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⣿⣼⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠻⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣀⡟⢀⠀⣼⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧
⠀⠀⠀⠀⠀⢀⣿⠁⡾⣼⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠈⠢⡘⠿
⠀⠀⠀⠀⠀⢸⡇⣼⣿⣿⣿⣿⣿⣿⠿⠛⡿⠁⠀⠀⠀⠀⠀⢠⠏⡀⠁⠒⠠⢄⣀⣀⠀⣀⣠⡔⠉⠇⠀⠀⠀⠀⠀⠀⠈⢶⡀
⠀⠀⠀⠀⠀⢸⢠⣿⣿⡿⠟⠋⠁⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠈⠀⠈⠑⠲⠤⢀⡀⠈⠉⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠳⡤⣀
⠀⠀⠀⠀⠀⡿⣼⠛⠁⢄⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠋⠳⠶⠌⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠱⡜⣱
⠀⠀⠀⠀⠀⣾⠃⠑⢄⠀⠐⢄⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡛⣄⠀⠀⠀⠀⠁⠀⠀⠀⢀⡈⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠢⡀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄

⠀⠀⠀⠀⠀⠀⠀⠰⠍⠉⠉⠋⠙⠭⡆⠀⠀⡠⠤⠤⣤⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡆⠀⠀⣰⠂⠀⠰⠂⠀⠀⠀⠀⠋⠁
⠀⠀⠀⣠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢲⡐⣀⣿⠀⠀⠈⢹⠁⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠄⠁⠀⣠⢀⡀⠀⠀⠈⠀⠁⠁⠀⠀⠀⢸
⠀⢀⡴⠀⢀⠠⡀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠂⠈⠉⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠠⠆⠀⠀⠀⢸⠀⠀⠀⠁⠀⠁⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠰⠃⠀⠀⠐⠀⠀⠀⠠⣂⠄⠀⠀⢰⠀⠀⠃⠀⠀⠰⠘⡄⠀⠀⣗⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⡠⠄⠈⠁⠀⠀⠀⠘⠀⠀⡀⠀⠀⠀⠹⠁⠀⠀⠀⠀⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠠⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⡀⠀⠀⠀⠀⢀⡠⠒⠉⣠⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄
⠀⠀⠀⠀⠀⡀⠀⠁⠀⠌⠂⠀⠀⠀⣀⠔⠉⢀⡴⠊⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠇⠄⠂
⠀⠀⠀⠄⠈⠀⠠⠀⠀⠀⠀⢀⠤⠊⢀⡠⠞⠁⠀⡠⠔⠋⠀⠀⠀⠀⠀⠀⣀⡠⡠⠁
⠤⠈⠀⡀⠄⠀⠀⠀⠀⡠⠖⠁⣀⠔⠋⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠐⠈⠕⠂⠀⠀⠀⠒⠀⠀⠀⣴⣾⣧
⠘⠁⠀⠀⠀⠀⣀⠔⠊⢀⡤⠚⠁⠀⠀⠂⠀⠀⢀⠔⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠃⠀⠸⣿⣿⡡
⠀⠀⠀⠀⡠⠞⣡⢀⠖⠉⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠰⠋⠁⠀⣠⠺⠁⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⢈⠀⣀⠔⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⣀⡀⡄⠀⣀⠈
⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣤⣤⠤⠔⠒⠈⠉⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣾⣿⣿⣿⣭⡭⢝⣛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣟⢷⡮⡳⢔⠭⣊⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⣰⣿⣿⣿⣿⣕⡭⣊⠕⠊⠔⠪⣦⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⠙⠀⠀⠀⢸⣿⣿⡿⣿⣝⡻⣮⢉⠓⠦⣤⡐⢿⣿⣾⣯⣉⣉⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⣿⣿⡷⣝⢪⡲⣕⢦⡀⡳⣝⣦⣽⣿⡯⠓⣤⣄⡑⣷
⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣮⡳⡱⡌⢳⡑⢈⢻⣿⣿⡟⡠⡺⠛⢽⣿⣾
⠀⠀⠀⠀⢠⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠙⠋⠉⠻⣗⠔⡙⣄⠙⢦⡹⣿⣿⣿⠈⢀⠀⣿⣿⡃
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣦⠘⣎⢪⣳⣽⣿⣿⣿⢸⢨⢦⢫⢸⠃
⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⡳⣌⠿⣾⣿⣿⣿⣿⣿⣿⣊⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⠀⠁⠀⠀⠀⠀⠀⣾⣿⣿⣿⡝⣝⢮⣓⣚⣫⠵⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡻⣝⠮⣕⣒⣲⡴⠞⡽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣕⠮⣟⣒⣒⣦⠴⢋⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡫⠖⢒⣒⣬⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠟⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣟⠻⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣟⣹⡿⠂⠀⠰⠿⠟⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⣫⣿⡿⠟⠉⢸⣿⣥⣶⣦⡄⠀⠀⠀⣸⣿⣿⣿⣧⠲⡌⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⢸⣿⣿⡟⠃⡀⠀⠀⢀⣀⠘⠋⠀⠀⠀⡽⢿⡉⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⡰⢋⣤⠴⠛⠛⠉⠀⠀⠀⠀⠀⣄⣸⡇⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠂⠀⠙⢧⣻⠿⠓⣒⣈⣉⡁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⡔⣡⠞⠏⠀⠀⠀⠈⠙⣦⣀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠃⣠⡇⠀⣀⡴⠚⢩⣼⣿⣧⡀⠀⠀⣀⠀⠀⠀⠻⣤⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠓⠛⠀⢀⣙⣒⣎⡁⠈⠟⠫⡴⢋⡞⢉⡿⠁⠓⠀⠀⣿⣄⠀⠀⠀⠙⢲⠇⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⡀⣈⡷⢀⡴⣾⠟⣽⣿⡿⠿⠀⣤⡀⠀⣟⣠⠞⠁⠀⠀⠀⢀⣿⠇⠀⠀⠀⣠⠏⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢰⡿⢚⡡⢞⣥⢠⣞⣠⣴⢂⡉⢀⡿⠀⠉⠁⠳⣄⠀⠀⣠⠞⢯⣀⡀⠀⣰⠃⡰⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⢛⣙⡶⠋⢐⣎⡼⠟⠛⠻⣯⠉⠁⢾⣁⣀⣀⠤⢤⣈⠙⠛⠁⠀⣠⠟⠉⣿⠃⡰⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣠⠾⢫⡄⢱⢋⡞⠑⢒⡤⣄⡈⠑⠒⠲⢦⡀⠀⠀⢀⠜⠀⠀⠀⠀⠷⠒⢶⡇⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⠷⣾⢸⡇⡸⢏⣸⣿⣥⡤⠼⠿⠀⠀⠀⠘⣷⢠⠞⠋⢀⡶⣶⠄⠀⠀⢀⣼⣀⡀⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡶⠫⣄⣿⣅⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣌⡓⣚⠽⠛⠁⠀⠀⠀⠀⢸⡇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡤⠈⢿⣉⣙⣁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡿⠶⠶⠶⣤⣤⣄⣀⣸⡃⣾⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣭⣭⢀⣀⣤⣤⣤⣤⣀⠀⢠⡟⠀⠀⠀⠀⢠⡟⠉⠉⠉⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣯⠶⠛⠉⠀⠀⠀⠀⠉⠛⣿⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣸⣇⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠃




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣋⣙⣏⣨⣏⣹⣍⣉⣽⣫⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣶⣶⠉⠙⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠟⠛⠛⠛⠛⠛⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢀⡄⠀⠀⠀⡀⠀⠀⣀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⠇⠀⣠⣾⡇⠀⣰⣿⣷⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⡀⠀⠀⠀⣿⣿⣿⣿⠿⠶⠛⠻⠿⢧⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⣀⣀⣀⣉⠉⠛⠇⠀⠀⣿⣿⣏⣤⣶⡾⠿⠿⠿⣷⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⡿⣛⢻⣿⣷⠀⠀⠀⣿⣿⣿⡋⠀⢠⡤⡄⠀⣀⣽⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⠞⠋⠁⠀⠀⠀⣿⣿⣿⣿⣶⣬⣛⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠲⣤⠛⣋⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠒⠭⠭⠭⢭⣛⣛⣒⣒⣯⣽⣿⣿⣿⣿⡿⣫⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⣠⡿⢿⣿⣿⣿⣿⣿⣿⡿⣫⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⢛⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠄⠀⣿⣿⣛⣭⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⢀⠔⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⡠⢲⣄⣴⣭⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⡾⠋⠉⡜⠀⠀⠀⠀⠈⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢸⠃⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠘⠀⠀⣀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠐⠉⢁⠀⠀⠀⢀⡀⠈⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡈⠛⠁⠀⢀⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢀⠴⠒⠛⢦⣀⠤⠶⣤⡠⠤⠶⠦⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⠴⠤⠀⡄⡰⢀⣀⣀⣠⣤⣤⣴⣏⢀⡠⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢀⢊⠀⡈⠀⠀⢀⠀⠀⠀⠀⠀⠈⣉⡹⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⢠⡀⠀⠑⠦⡐⠘⢄⠈⠉⠁⠀⠀⣏⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⠙⡆⠢⡠⠣⠤⠂⠅⢀⣀⡠⠔⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⢹⡉⡔⠙⠻⠋⠸⠉⠉⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⢰⢧⣤⣍⣐⢒⣂⣒⠦⢽⡆⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢴⡶⠒⠲⣄⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⡜⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣾⠉⠙⠆⢀⡤⢊⣀⡠⠤⠔⠒⠚⣏⠉⠙⣆⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⡽⠲⠤⣄⡈⠳⡄⠀⠀⠀⠀⠀⡷⠶⠒⣧
⠈⢸⡶⠞⠓⠀⡡⠛⠒⠂⠳⡉⠉⠉⠁⠀⠀⠀⠀⠀⠈⢻⠒⠚⠀⠀⠀⠀⠀⠀⠀⣧⢞⡻⠚⠋⠀⠉⠁⠀⠉⠛⠻⢦⠉⠉⠁⢸⠃⠀⠀⡟⠈⠉⠉
⠀⢸⣭⣩⡅⠀⣷⡶⠶⠖⣆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⠤⠴⡇
⠀⢼⡀⣸⠀⢸⣏⣀⣀⡈⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⣄⡀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⡀⠠⡄⠄⢀⡀⢀⠀⠀⠀⠈⠢⠤⠄⠃
⠀⠀⠉⠉⠀⠈⠓⠢⠬⠄⠂⠀⠀⣀⣤⣤⡆⠀⠙⠋⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⡠⢄⠶⡖⠾⠛⠋⠉⠀⠀⠀⠦⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣶⣶⣶⢶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠀⠈⠀⢄⠀⠄⠀⠠⠤⡄⠀⠀⠀⡀⠀⠀⠀⠁⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠂⠀⠀⠀⠒⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⣀⣀⣐⣀⣀⣀⣀⣁⡉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠙⠀⠀⠠⠠⢄⠀⠀⠀⠀⠀⡇
⠀⠀⡇⢀⠤⠀⠀⢄⠀⢠⠔⠢⡀⡔⠒⢄⠁⠀⠀⠀⠀⠀⡎⠈⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⡸⠂⠀⠀⠀⠀⠁
⠀⠀⡇⡎⠀⠀⠀⠀⢡⠘⢄⠤⠃⠣⣠⠎⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠶⠲⡄
⠀⠀⡇⠇⠀⠀⠀⠀⠸⢰⠁⠉⡄⡋⠈⠁⠀⠀⠀⠀⠀⠀⡉⡈⣁⠀⠀⣀⣀⡀⠀⢀⠀⢠⠀⠀⠀⠘⢄⡀⡀
⠀⠀⡇⠈⠢⢄⡠⠔⠁⠈⠒⠊⠀⠁⠒⠁⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠉⠈⠉⠉⡉⡉⠁⠀⠀⢀⠀⠠⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠁⠀⠀⠀⠡⠀⡀⠀⠀⠀⠀⠀⠀⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⢩⡋⠽⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⡀⠇⡄
⠀⠀⢰⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠔⠂⠐⢀⠀⠀⣇⠆⠀⢹⢠⠎⡗⠀⡔⠉⠉⠙⢆⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢀⠀⠀⠀⠘⢤⣠⣤⣎⡫⠃⠀⢇⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠉⠢⠈⠁⠁⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⢀⠀⢀⡀⠀⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠸⠀⠀⠆⠀⢡⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡴⠀⠀⠀⢀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠇⠀⢀⠔⠁⠆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠤⠂⠁⢀⠞⠀⠈⠉⢉⣥⠏⡁⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⢀⢀⠀⠒⠃⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⢄⠂⢄⡀⠐⠒⠒⠒⢆⡁⠀⠀⠀⠀⠀⠈⠒⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⡇⠀⠀⠀⠀⠀⠀⢲⢀⠀⠀⠀⡀⠈⠐⠠⠀⠀⠀⣥⠑⠒⠒⠦⣤⢤⠤⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⠀⠈⠐⠀⠐⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⣆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢃⠀⠀⠈⣷⠀⠈⠀⣤⣤⣴⡾⠏⠀⠀⠀⠀⠀⠙⠻⠶⠖⠂⠀⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⢢⠀⠀⢉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢸⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠨⠴⢠⣕⢄⡀⠀⠀⠀⠀⠘⠳⣦⡄⠀⠀⠀⠀⠀⣴⣿⣁⠀⠀⠀⣄⠸⠀⠀⠊⡀⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⡀⠙⠄⢈⠐⠀⠀⠀⠠⠖⠛⠁⠀⠀⣀⣀⡀⠀⠈⠉⠀⠀⠀⡉⠲⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠉⢹⠆⠈⠂⣄⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠈⠉⠀⠀⠀⠈⠁⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠈⠒⢀⡀⠀⠀⠀⠀⠀⢀⠔⠋⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠉⠉⠣⡀⠀⠀⡔⠀⠋⢀⡎⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠈⠒⠦⠀⠀⠀⠀⢀⠤⠖⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣠⠐⠁⡀⠤⠄⢤⣄⠀⠀⠀⠈⢀⠀⣘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣠⣤⠤⠤⠀⠐⠈⠇⠀⠀⠀⠀⠀⡈⢐⡌⠀⠀⢀⣾⢿⡗⠀⠁⠀⣸⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⠤⣀⣀⣤⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠄⠒⠊⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠲⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢂⣴⠶⣿⣿⣿⣅⡀⠀⢀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⡰⠋⠀⣠⣿⢉⣧⡙⡙⢷⡀⢀⠢⡀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⣀⡴⢡⣿⠘⣿⣧⢹⡌⢻⣄⢣⠘⢦⣀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⣀⣤⣤⠶⠛⣁⣴⠟⢹⡆⠙⠛⠦⣷⡀⢻⣎⢧⠀⠙⠷⣤⣌⣙⢳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⣾⡿⠛⢉⣠⣶⠿⠛⠁⠀⠀⢿⡀⠀⠀⠸⣷⠄⠹⣷⡀⠀⠀⠀⠀⠉⣉⠬⠙⢧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠙⣧⡀⢉⠻⢦⣤⡤⠖⡋⠁⠀⠀⢀⣤⡿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡀⠀⣴⣿⣿⣿⠒⠶⣄⠀⠀⠀⣠⡴⠞⢿⣄⠀⠀⠈⠻⢦⣿⣶⡶⢖⡉⠄⠀⢖⣤⣿⣗⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣾⠁⣸⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⢀⡴⢾⣿⣿⣮⢳⣦⣤⣤⣤⣴⣶⠿⣷⣶⣾⣿⣎⢿⣏⠳⠄⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⣿⣿⣿⣿⡏⠉⠉⠀⣰⠀⠀⠀⠈⠈⠉⠉⠘⢿⣽⣿⣿⣿⣿⣿⣷⡘⢆⠹⣷⡹⣟⢿⣷⣶⣤⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣿⣿⡿⢹⡇⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⡜⣿⡿⢹⡟⡞⡄⢹⣷⢹⣶⣽⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠟⢀⣿⣿⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⡿⠸⢧⡿⠁⣸⠧⠀⣿⢸⣿⡇⣿⡆⢿⣆⢙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣾⣿⣿⣷⡄⢠⣶⣶⢶⣤⡄⠀⠀⠀⠀⠀⡸⢁⣴⠟⠁⣰⠏⡀⡰⡏⢸⣿⣷⠘⣿⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣾⢫⣶⣿⣿⣿⣷⡈⠛⠷⠖⠉⠀⠀⠀⢀⣤⣎⣴⠟⠁⣠⡞⠁⣰⠟⣸⡇⣼⣿⣿⠀⡿⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣸⠁⢠⣿⠙⣿⣿⣿⣷⡀⠀⠀⠀⠀⣠⣴⣿⡿⠋⢀⣠⠾⢋⣠⠞⠁⢰⣿⠁⢿⣿⣿⡾⢁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⡏⢸⣌⢻⣿⣿⣿⣷⣶⡾⣿⣿⡿⠋⣀⣴⠟⣡⠶⢋⣤⠶⠖⠛⣸⣢⠸⣿⣿⣱⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠘⣷⠀⠻⣷⣝⣿⣿⣿⣿⠇⣿⠟⢠⣶⢟⡵⠞⠁⣠⡯⠔⣲⣯⡴⠛⠻⣦⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⢳⡀⠘⣦⠑⠸⡏⢻⣿⣿⠏⣰⡏⣰⣿⢣⡞⠁⣤⠞⠁⣠⠞⣽⡟⠀⠀⢀⣼⣦⣿⣿⣯⡳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣸⠇⢸⣷⡀⠘⣧⠀⢰⣼⠟⠁⣴⣿⣀⣿⢣⡟⢀⣾⠁⠀⣴⠏⠘⣿⠀⠀⢠⠟⠁⠀⢿⡄⢧⠻⣮⡙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⡏⢠⣿⣿⣷⠀⠘⣇⠈⠁⣠⣾⢻⣿⣿⣿⠸⡇⣼⡇⠀⢸⡿⠀⢸⡟⠀⠀⠰⠄⠀⠀⠈⢧⠈⢧⡈⠻⣦⡙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⢸⡇⢹⣿⠀⡀⢿⣠⡾⢿⡿⢸⣿⣿⣿⣇⢸⣿⠁⠀⣿⡇⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⢻⣦⡈⠛⢦⡈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⡇⠀⣿⣠⣷⠿⠋⣠⢪⠃⣼⣿⣿⣿⣿⡌⣿⠀⠀⣿⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣿⢦⣄⠙⢦⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⢀⣷⡴⠟⠋⠀⢀⡴⢡⠎⣰⣿⣿⡟⢸⣿⣷⣿⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⠈⢳⡍⠳⢦⡻⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⠟⠁⣀⣀⡤⠔⢁⣴⢋⣴⣿⣿⣿⡇⡄⣿⣿⣿⠀⠀⣿⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡻⣄⠀⠙⢶⣄⡉⠻⠄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠗⠛⣉⣉⣡⣤⡶⢟⣵⣿⣿⠏⣼⣿⢰⡇⣿⣿⣿⠀⠀⣿⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣍⠳⣦⣀⠉⠻⢶⣤⡿
⠀⠀⠀⠀⠀⠀⢠⠞⣁⣤⣶⣿⣿⡿⡿⠿⠿⣿⣿⣿⠋⣼⣿⠇⣼⢡⣿⠏⣿⠀⠀⣿⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⣿⣿⣷⣌⠙⠿⣿⠛⢹⠁
⠀⠀⠀⠀⠀⠀⣿⣿⡿⢛⣭⣥⣴⠾⡷⢶⣶⣞⣋⣡⣾⣿⠟⠠⢣⣾⡟⢰⣿⠀⠀⣿⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⣻⡿⣿⠈⣷⣀⡿⢀⠆
⠀⠀⠀⠀⢠⡾⢋⠏⢰⣿⣿⡿⣡⢫⡆⢾⣿⠉⣿⣿⡿⢋⣀⣴⠿⡿⢁⣿⡇⠀⢀⡇⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⣀⣴⣾⠟⠀⠿⣳⡿⢋⡴⠃
⠀⠀⠀⠀⢻⣅⠈⢧⠘⣿⣿⡇⣿⢸⣧⡈⠉⢶⣮⣛⣻⣛⣋⣥⡞⢁⡾⣿⠁⠀⣸⣷⠟⢧⣀⣀⢀⣀⣠⠴⠒⠋⠉⢉⠾⠋⠁⢀⡰⣶⣯⠖⠋
⠀⠀⠀⠀⠀⠙⠳⣤⣙⠺⣿⣿⣼⣆⠻⠦⣭⣜⠛⢿⣟⣿⡿⠋⢀⠞⣽⠃⠀⣰⣿⡇⠘⣷⣬⣉⣉⣁⣠⣤⣤⣀⣤⣤⡤⠶⢚⣩⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣍⡙⠛⠿⠶⠦⠽⠷⠾⠛⠉⢀⠴⣋⡾⠁⢀⣴⡿⣡⣯⡾⠛⠛⠛⠋⠉⠀⠀⣽⣿⣿⠙⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⣭⣙⠲⠤⣀⡀⠀⠀⠀⣈⣤⠞⢋⣠⡴⠻⣿⠀⣿⠁⠀⣀⠔⠒⠒⢒⣒⣚⡋⠭⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⡋⠛⠻⢿⡿⣷⣶⠞⠋⠹⡇⠰⠙⠦⠬⢉⣉⠴⠖⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠛⠛⠉⠀⠀⠀⠀⠈⠑⠒⠻⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠛⠛⠛⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠇⠀⠘⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣦⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠛⠉⠀⠀⠀⠉⠛⢷⣦⡀⠀⣤⣤⣤⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠘⠇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠻⠶⠶⠾⠟⠋⠉⠁⠛⠀⢰⡧⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⠶⠶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡈⢻⣿⡟⣁⣀⣀⣀⣀⣀⣀⣀⣼⡏⢀⣀⣀⣀⣈⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠛⠛⠛⠛⠛⠛⠛⠃




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢸⣿⣿⣷⠀⠀⠀⠀⠀⢄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢋⣿⣿⡏⣠⣝⠿⠋⠀⠀⠀⠀⠀⠈⢷⡄⠉⠒⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠘⣿⣷⣿⣾⡇⠀⠀⠀⢀⡀⠀⢀⣲⣆⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣠⡀⠀⠀⢿⣿⣿⠁⠀⠀⠀⠀⠀⠑⠂⣼⣿⡏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣿⣇⡀⢠⣿⠟⢧⡑⣭⣛⣋⣀⣤⣤⣌⠙⢻⡁⠀⢀⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠘⢿⣷⣼⡇⠠⣀⠀⠉⠛⠛⠛⢛⠿⠿⢷⣄⢃⢠⣾⣿⣿⡇⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠹⣿⣡⡶⢖⠮⠀⠀⠀⠀⠴⢶⠶⣤⣹⣿⡿⠟⠉⠁⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡀⠀⠀⠀⠀⣸⠋⠒⠈⠁⠀⠀⠀⠀⠀⠘⠒⠉⡟⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⢷⣶⣦⣷⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣶⣦⣤⣶⣿⣷⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠙⠻⠟⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠈⠛⠛⠛⠉⢉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⢠⠀⢰⣼⣧⠀⠀⣠⣄⣤⡀⠀⠀⠀⢸⣧⣤⣧⣶⣶⣰⣿⢋⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⡀⠀⠈⠛⢃⡀⠈⠻⣛⠋⠋⠁⠀⠀⣸⠏⠛⠉⠛⢻⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡓⣀⣤⣿⣷⡀⠀⠠⠄⠀⠀⢀⣼⣿⠷⠦⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠀⠀⠤⠠⠤⠞⠋⠁⢹⡶⡀⡀⠲⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⡰⠀⠀⠀⠀⠻⢏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠑⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠈⡹⢉⠝⠀⠉⠁⠀⠐⠒⢤⡀
⠀⠀⠀⢀⠚⠉⠀⠀⠉⠂⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢧
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠨⠑⠂⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⠀⠤⠤⠄⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⢻⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⠇⠀⢘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠿⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠀⠉⠀⠀⠞⠲⢤⣀⠀⠀⠀⣀⡠⠤⠒⡶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢤⡴⠟⠋⠁⢀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠐⣶⣤⣀⠀⠈⠓⢶⣉⣥⣤⣄⣸⡷⠖⠛⠉⠉⠑⠲⢄⡀⣀⣠⡟⠀⢀⡴⠿⢿⣿⣄⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣀⣠⠞⠋⢀⡽⢳⣤⠶⠾⠅⠀⣴⣤⡞⠖⠀⠀⠀⠠⠞⠀⠀⠹⣍⠉⠛⠷⣏⠀⠀⠀⠈⠙⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢀⡴⢋⣴⠟⡁⠀⠀⠀⠀⠉⠀⡞⠉⠳⡄⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⢷⡀⠀⠀⠀⠀⠙⢶⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣀⡾⢃⣼⠏⢀⡤⢤⣀⠀⠀⢧⠠⢀⡇⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⢈⡧⠀⠀⠀⠀⠀⠀⠙⢦⣰⠋⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⡜⠁⠀⠘⠇⠀⢸⠀⢀⡈⣧⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠛⠛⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣰⠀⣏⠀⠀⠀⠀⠈⠳⣬⣁⡾⠀⠀⢀⠀⠀⠀⠀⢀⠀⠋⢸⡀⢀⡴⠋⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⡼⠁⠀⠀⠀⣠⠞⠁
⠀⣠⠶⢶⡄⠀⠀⠀⢠⡟⠀⣰⢿⠀⢈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⠷⠤⢊⡿⠋⠀⠀⠀⠀⠀⣠⡾⠻⢾⣷⡀⢠⡞⠁⠀⣾⣧⠀⣸⠇
⢸⡃⠀⠀⠙⠲⣤⠀⡾⢀⣀⣏⡤⠞⠉⠈⠁⠀⢰⡠⠀⠀⠀⠀⠀⠀⢀⠠⣼⡟⠓⠋⢀⡴⠋⠀⠀⠀⠀⠀⣠⡾⠋⠰⡆⠀⠈⠻⢏⠀⢀⠞⠁⢿⠙⠉
⠀⣻⠀⠀⠀⣞⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⠚⠛⠛⠀⢀⡴⠋⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠈⣆⠁⠀⠀⠸⡇
⣸⠃⠀⠀⠀⣹⠓⣶⠶⢤⡴⠊⠁⠀⠈⠁⠀⠀⠀⠀⠘⣆⣠⠶⠶⠚⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣠⣾⠿⠾⠷⠶⠶⠶⠶⠶⣄⡀⠀⢹⣄⠀⠀⠀⣿
⠘⠲⠶⠶⠞⠋⠀⣿⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣠⡾⠋⢀⣠⠤⠶⠀⠀⠠⣀⡀⠈⣷⠀⡿⠙⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣹⠶⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⣧⢀⡴⠋⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠁⠀⢀⣠⣄⣀⠀⠈⠁⣰⠏⠀⠀⠀⢻⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣴⠃⠀⢀⠀⠀⠀⠙⠁⠀⠀⠀⠃⠀⢀⡽⠋⠀⠀⠀⠀⠀⣠⡾⡿⠁⢀⣴⡶⠲⠶⢿⠝⠉⠉⠛⠶⠞⠁⠀⠀⠀⠀⣸⠀⠀⣿
⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⣠⢾⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠁⠀⠀⠀⠀⣠⡾⠋⠀⡇⠰⣾⣹⡄⡏⢳⠃⠀⢰⠃⠀⠀⠀⠞⠀⠀⠀⢠⠿⣄⣰⡏
⠀⠀⠀⢀⡴⠋⠀⠀⢠⡞⠁⠈⢧⡀⠀⠀⠀⠀⢀⣴⠏⠁⠀⠀⠀⠀⣀⡶⠋⠀⢠⣀⣿⠲⣧⠽⠓⠚⠁⠀⠀⢸⠀⢦⠀⠀⠀⠀⢀⣴⠋⠀⠈⠻⣧
⠀⠀⣠⣾⠀⠀⠀⢠⡟⣧⠀⠀⠀⠉⠳⠦⣤⣴⠟⠁⠀⠀⠀⠀⢀⣴⡏⠀⠀⠀⠈⣯⢻⡿⠁⠀⠀⠀⠀⠀⠴⠚⢀⠇⠀⣠⣤⠶⠋⠈⠻⣦⡀⠀⠈⣧⠖⠛⣦
⢀⠞⠁⠀⠀⠀⠐⠚⠦⣜⣧⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⢀⡴⠋⠀⢧⡀⠀⢰⠗⠚⠁⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣸⢷⠀⠀⠀⠀⠺⢧
⢠⠆⠀⠀⠀⣀⣄⡀⠀⢸⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠠⠴⠚⠀⡰⠀⠀⢀⣠⠾⠁⠀⠀⠀⢀⣼⠋⣨⠟⠃⠀⠀⠀⣠
⢇⠀⠀⠀⠀⢁⣼⠛⠲⠋⠀⠈⠻⣄⠀⠀⠀⠀⢀⣤⢿⡀⠀⠀⠐⠾⠃⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⣠⡞⠁⠰⣇⢀⣤⣤⡴⠛⠁
⠈⠙⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⢠⠏⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⢸⡇⠀⠀⢀⣤⠶⠚⠁⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⢄⣨⠿⠋⠀⣠⠎⠉⠙⠢⠤⣀⣠⣤⠄⠀⠀⠈⡄⠘⣇⡤⠐⠋⠀⠀⠀⠀⠀⣀⣴⠞⠉⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠈⠁⠀⣄⡴⠻⢦⣤⣀⠀⠀⠀⠀⠈⠙⠒⠒⠚⡇⠀⢹⣤⠶⠶⡄⣀⣤⡴⠞⢻⡄⢀⣤⠘⣆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⠀⠀⠀⢻⡄⠀⠀⠈⠉⠛⠳⠶⢦⣤⣤⣤⣤⣤⣷⡶⠏⠀⠀⣴⠛⠉⠁⠀⠀⢀⣿⠊⠀⠀⠈⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢈⡗⠀⠀⠀⠀⡞⠀⢠⡄⠀⠀⠘⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠈⠛⢿⣀⣀⣠⠼⠃
⣶⣶⡶⠶⡶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⢀⣤⢄⡄⠀⠀⠀⠀⠀⠀⡀⠀⣠⠋⠀⣠⡞⠣⡈⢢
⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⣷⣄⠬⠌⠘⠘⠳⠀⠀⠀⠒⠢⡤⠄⢣⠴⣎⠀⢴⠀⠘⠺⡄
⠀⠀⠀⠀⣿⣿⠟⣋⡀⠆⠈⢉⠳⡀⠰⠀⡀⠠⠐⠒⠒⣒⡂⠀⠘⡴⢷⡇⠀⠢⢀⣶⠃
⠀⠀⠀⠀⢙⠋⣆⠬⠌⠊⢘⣸⡷⠈⡂⠉⢡⠀⠀⠀⠀⠰⠄⠀⡠⠘⠤⣨⡡⠔⡋⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⣿⣷⣵⡶⠥⢺⡟⠁⠀⠇⠀⠀⡀⠀⠠⠄⢀⠄⡪⠊⡠⠊⠀⠑⠒⠁⠀⠀⠢⡀⠈⢂
⠀⠀⠀⠀⢀⠅⠋⠁⠑⢠⠿⣷⠀⠀⠀⠀⢀⠇⠀⠀⠔⠁⠀⠀⢰⠁⠀⡰⠄⠀⠀⠀⠀⢀⡇⠀⡸
⠀⠀⠀⠀⡎⠀⠠⠔⠂⢫⠀⢸⠀⢐⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⣤⠒⠁⢺⠀⠀⢠⣎⣁⢈⠕⠁
⠀⠀⠀⠀⢃⠀⠀⠀⣀⠼⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠛⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠒⠒⠒⠒⠶⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣀⠴⠒⠛⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠒⠲⢤⣀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⡠⠊⢁⣠⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣄⡈⠓⢄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠠⠊⣠⡶⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⣶⣄⠑⠄⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠠⠚⠉⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠈⠛⠆⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣠⣷⠶⠛⠓⠾⢷⣄⡤⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⡾⠷⠛⠻⠶⣾⣤⡄⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢈⡿⠋⠀⠀⠀⠀⠀⠀⠈⠻⡦⡄⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢰⣿⠀⠀⠀⢀⣴⡟⠁⣀⠀⠀⢱⠘⠀⠀⠀⠀⠋⡎⠀⠀⣠⣶⡖⠂⡀⠀⠀⠀⣿⡆⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢿⠀⠀⠀⠘⣿⣻⣿⡿⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⢿⣛⣷⣾⠇⠀⠀⠀⡿⠁⠀⠀⡇
⠀⠀⠀⠀⠀⣠⠖⠒⡇⠀⠀⠀⠘⢆⠀⠀⠀⠈⠉⠉⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠉⠉⠁⠀⠀⠀⡰⠃⠀⠀⠀⡗⠛⢦
⠀⠀⠀⠀⢀⢃⢤⡀⡇⠀⠀⠀⠀⠈⠓⠤⣀⣀⣀⣀⡠⠖⠁⢀⡤⠂⠀⠀⠠⢄⡈⢮⡢⢄⣀⣀⣀⣀⠤⠊⠁⠀⠀⠀⠀⡇⣠⠄⡇
⠀⠀⠀⠀⠘⡇⢰⡛⢳⠀⠀⠀⠀⠀⠉⠒⠦⠤⠄⠀⠀⠀⠀⣏⠀⡠⠀⠀⢀⠀⣳⠀⠉⠓⠶⢤⣤⡤⠶⠊⠁⠀⠀⠀⢀⠟⣳⠀⡇
⠀⠀⠀⠀⠀⠹⣄⠁⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠈⠒⠚⠁⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠁⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣰⣿⣤⣈⣷⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⣿⣿⣿⣿⣧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁⠙⠁⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣤⣄⣠⠟⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡑⢄⡀⠀⠈⠉⠁⠀⠀⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠻⠶⣦⣤⡤⠤⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢩⢏⠴⣲⣞⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠰⠟⠁⠈⠁⠀⠀⣙⣒⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢤⠭⠥⢬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁⠀⠀⠠⢴⣮⢻⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣋⠀⠀⠀⠀⠀⠲⢁⠀⠈⢧⠖⠒⢄⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡵⠁⠀⠀⠀⠀⠀⠀⠐⢷⠀⠸⡄⠀⠀⣱⣎⢀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣠⠂⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠹⣤⠊⠀⢸⠭⣜⡁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠰⣤⢀⠀⠀⠀⡠⢫⣧⠀⠀⠀⠀⢙⡳⠟⠀⠠⣀⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠛⠉⠉⠉⠑⠚⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢇⠀⠒⠀⠀⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⡝⣂⡠⣴⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠈⣏⠤⠒⠉⠀⠈⢢⡀⠀⠀⠂⠉⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⢣⠀⢁⠈⠀⠀⠀⠀⠀⠈⠒⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⠇⢸⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⡔⢡⠋⠀⠀⠸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢀⠞⠀⣆⠀⠀⠀⡰⢣⣿⠙⣆⠀⣠⠒⣤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⢠⡋⠀⠀⠈⠉⡞⡵⠇⠘⠓⠒⢦⠘⠁⣀⣸⠈⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠈⠓⣢⣤⣀⠀⢻⡇⠀⠀⣨⠇⡏⣶⠁⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠇⠀⣊⠉⠋⡜⠓⠈⣀⣿⢹⠚⢳⡈⡦⠹⣏⡓⢢⡘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡹⡄⠄⠸⠴⢄⡤⠃⠀⡆⠀⠀⠆⠇⠈⠉⠁⠀⠘⠋⠠⠗⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠗⠋⣠⣳⣦⣀⠈⠢⠄⠀⠒⠁⠀⢘⣠⢰⢆
⢀⢠⣄⣤⢄⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠓⢲⠎⠀⠉⠉⠈⠙⠒⠤⠤⣀⣀⣀⣀⣰⡾⠸
⢱⠈⣙⡋⢱⠄⢠⡂⠒⢈⣆⡀⢏⡀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⡛⠁
⠈⠉⠀⠀⠁⠀⠚⠁⠢⠴⠲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣭⣤⡛⠉⠉⠁⠀⠒⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⢀⣀⣀⣤⡤⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣫⠖⠉⡐⠒⢲⣿⣿⣮⣍⠉⠛⠶⣬⣣⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢠⣫⣠⠞⠛⠉⠭⢭⣿⣿⣶⣬⣿⣿⣄⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣵⢿⡀⢰⣏⢄⣢⡴⠶⠛⠉⠉⢹⣿⣿⣿⣿⣿⣷⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡮⣪⢏⣼⡇⢸⣛⠉⠀⠀⠀⠀⠀⠀⠸⣿⣿⣝⢿⡻⡀⠑⠝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡾⣷⢿⣿⣇⡌⣉⠁⠀⠐⠚⠒⠲⢤⠀⠹⣿⢿⡀⠹⣌⠳⡌⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⡞⣵⣿⣿⠟⠁⠙⠁⠀⠀⠀⠾⠳⠄⠀⠀⢻⣷⣝⢦⡈⢗⢌⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⡜⡜⣟⣿⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡼⢸⡧⡠⡙⠢⢭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⢸⢸⠸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡋⢇⣹⣌⡛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠈⢀⢂⣿⠀⠁⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡆⡆⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡏⣰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢥⣙⡻⣿⣿⣦⣤⣤⡤⠠⠀⠂⠀⠀⠀⠀⢸⠀⢧⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣺⠟⣿⣿⣿⣿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣜⠌⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⣡⢀⠜⠋⠻⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣦⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠛⠁⠀⠠⣺⢿⠈⣿⣹⢿⡆⠀⡐⠁⠀⠀⠀⠀⠀⠀⠉⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⣡⡞⡱⣾⠀⣼⢋⡞⣠⢣⢟⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠻⣧⡁⢳⡀⣯⠞⡰⡱⣣⣣⣿⡇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⣻⣿⡇⠁⡸⠐⣱⠿⡫⢾⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⣇⡇⡇⡇⡏⠞⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⢻⣌⣄⠡⡀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⣪⠿⣿⡴⠃⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠋⠀⠀⠉⠀⠙⠛⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⡼⢦⣤⠶⠶⢦⣤⣤⡀⠀⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣶⣶⣶⣶⣶⢶⡾⣇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣋⣥⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣭⠉⠉⠉⢹⡷⣏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣛⣩⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠄⠀⠀⡀⢸⡇⠙⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠾⠿⠷⠷⠶⠶⠶⠶⠶⠶⠶⣶⠀⠀⠀⠀⢸⠶⠶⠀⣿⣹⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠶⠶⠶⣶⢶⡶⠶⠶⠶⠶⡞⠁⠀⠀⠀⣠⠾⠛⠂⠀⠀⠀⠘⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⢤⣤⣹⣎⣧⠀⠀⠤⢴⡇⠀⠀⢠⡞⠁⠀⠀⢀⣤⣄⠀⠀⠀⣈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⣿⠂⢙⠛⠛⠛⠋⠩⢿⡇⠀⠀⣾⠀⠀⠀⠀⠘⠧⠟⠀⠀⠀⣏⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⣏⣹⠆⠀⠀⣤⣴⡇⠀⠀⣿⡤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠉⠁⠀⠀⠀⠀⣸⠇⠀⠀⠹⡄⢈⡗⠀⠀⠀⣴⠛⣷⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⢤⣤⣤⠤⣤⠴⢾⡏⠀⠀⠀⠀⠙⢿⣁⠀⠀⠀⠈⠛⠉⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠶⠶⠶⠶⠶⠛⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠖⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⡇⢾⢸⠅⢸⣤⡇⡇⡎⠁⣼⣆⡇⠀⢸⢭⠂⡏⢀⣿⣰⡉⠀⠉⢸⢰⡋⡇⡿⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠁⠈⠀⠀⠀⠀⠈⠀⠀⠈

⡀
⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣾⣿⣿⣿⣶⣦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢟⣫⣵⣶⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣻⢦⣄⠀⠀⣠⠊
⠀⠀⠈⠱⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢓⣽⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣗⢻⣷⠉
⠀⠀⠀⠀⠀⠈⠉⠁⠒⠀⠀⠀⠀⠀⠀⠀⣸⣷⣿⣿⠇⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣼⣿⣿⣿⡿⠉⢁⣀⡚⢥⣀⣀⣠⠶⠶⢶⣤⡀⠀⣿⣿⣹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢱⠀⠀⠀⢼⣿⣿⣿⣿⠃⠀⠛⠉⠛⡸⠃⢻⠁⢴⡟⢶⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⣀⡀⢊⣾⣿⣿⢹⡟⠓⠦⠤⠤⠚⠁⠀⠘⢆⡀⠀⠀⣀⡴⠁⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⡨⠂⢿⣿⣿⢰⡇⠀⠀⠀⠀⠀⠂⠘⠀⠀⠈⠉⠉⠁⠀⠀⣿⣿⣿⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠀⠁⣹⣿⣧⣾⢀⡀⠀⠀⠀⣴⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠁⠈⠁⠀⠀⢸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠼⠀⠋⣿⣿⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⠀⠀⢰⠀⠀⢸⣿⣿⣿⡟⠀⠀⢰⡀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⢿⣇⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣮⣿⡜⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠟⣼⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⣨⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣫⡖⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⡉⡿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⣱⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣯⣻⣿⣿⣷⠃⣿⠇⠀⠀⠀⢀⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⢣⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⠏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⣿⡿⣿⡿⣿⣿⢟⣾⣿⣿⣿⣿
⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣾⡿⣿⣿⠷⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡛⠲⢤⣀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡒⠶⠶⠦⠤⠤⣤⢽⣄⡀⠉⠳⠏⠰⢴⡟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠒⠒⠛⠀⠀⠀⠀⢀⣀⡤⠤⠴⠶⢶⡄⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢄⡤⠚⠉⠓⢦⡴⠚⠉⠁⠀⠀⠀⠀⠘⣷⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠤⢤⡀⠀⠀⠀⢀⠴⠒⠲⠄⠀⠀⣟⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠠⣏⠀⣞⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠛⠉⢉⣕⣬⠿⠀⠀⠀⠀⢶⡋⠀⠈⠀⠀⠀⢰⠃⡼⣫⠍⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣄⡰⠚⡇
⠀⠀⠀⢠⣠⠶⠋⠁⠀⠰⡀⠀⠀⠀⡟⠒⣄⢲⣤⣾⠤⠤⣄⠀⠀⠀⠀⠀⠸⠶⢡⣌⡠⠟⠐⠒⠒⠤⠤⠤⣄⣀⣀⣀⡀⢀⣀⣀⣀⣀⡃⠛⠀⣰⠗⢚⡆
⠀⠀⠀⢽⠃⠀⠀⠀⠀⠉⢉⠒⠒⠲⢹⡆⠉⠀⠀⠠⠤⡴⠋⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠐⠢⠀⠀⣀⠀⠀⠉⠉⠉⢉⡍⡇⠀⠈⡇⠀⠞⢁⣠⠞
⠀⠀⠀⠈⠳⣄⡀⠰⠤⠈⠉⣀⠀⠀⡈⣇⠀⠀⠐⢒⡿⠤⠤⣤⣤⡤⣴⡞⠋⠀⠀⠈⠉⠛⠲⠤⢄⣀⡀⠉⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡇⠀⠰⢯⣀
⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠤⣥⣀⣀⣳⣈⠲⣉⡿⢁⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀⠀⠛⠃⢸⠷⠤⠖⠓⠢⢄⣀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡽⠉⠙⠉⠁⣿⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣼⠁⠀⠀⠀⠀⠁⣆⡀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⢠⠒⠠⠊⡟⠀⢠⠃⠀⠀⠀⢰⠒⠀⠈⢱⣀⠇
⠀⠀⠀⠀⠀⠀⢱⠄⠀⠀⠉⡇⡎⠀⠀⠀⠀⠈⠲⣀⡴⠋⡌
⠀⠀⠀⠀⠀⠀⠈⠛⠲⠚⠉⠀⣧⣤⣤⣀⣀⣀⣀⣠⡤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⠶⡤⠤⣤⡼⣼⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢳⠀⠀⠑⠾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣇⠀⣠⠴⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠸⡄⠙⠶⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠱⡀⠈⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢳⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣷⣤⢤⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣃⢹⡎⢣⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⣁⡤⠽⢿⣌⢇⠈⡆⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⣾⠁⠀⠀⠀⠈⢾⡰⠁⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡳⢤⣿⡀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣆⠉⠛⠒⠲⠒⠚⢛⣉⣡⠶⠋
⣀⣀⡀⢀⣀⢀⣀⢀⣀⣀⣀⣀⢀⣀⢀⣈⣉⣉⣛⣛⣛⣋⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢸⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡆
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⢀⡄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣹⣦⠷⠾⠚⠷⠾⠷⢾⣧⣶⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⠑⠒⢚⡟⣶⣶⡶⢶⠶⢭⣿⣯⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣿⢦⣼⣇⡘⠛⢁⣿⣠⣤⣶⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠋⠛⣿⠛⢿⠙⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢰⣆⣀⣶⢀⣴⢀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣠⢿⣿⠟⠛⠿⣛⠿⣿⣥⠶⠀⠀⠀⠀⡄⢴⢰⣆⣰⠀⡤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠙⢷⡞⠁⢸⡇⢼⣿⣷⢸⡆⢀⣿⠂⠀⠀⠀⣦⣽⠞⣻⣿⣿⡿⡷⢾⣁⡤⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡈⢷⣈⢉⣡⣾⣡⡾⠁⠀⠀⠀⠀⢺⣽⡀⢻⡟⠛⣱⠏⠀⠈⣻⡁⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠷⠶⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢠⡀⣦⣰⣆⣰⡇⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣀⣀⠀⢀⣀⣤⣴⠞⢷⡀⠀⠀⠀⢀⡰⣦⡿⠛⡽⣛⡻⣍⠛⠶⣯⡶⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢼⠋⣿⠛⣿⢹⡇⠘⠃⠀⠀⠀⠀⠀⠀⢹⡏⠀⠸⡿⣿⡿⢸⠀⠀⢈⡿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⣶⣤⣙⡂⠐⣋⣠⣴⡟⠳⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠞⠛⠷⠶⣦⣄⠀⠀⠀⠛⠀⡿⠙⣿⠉⠿⠈⠋⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣀⣀⣠⣤⣤⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡷⠶⠶⣿⠻⢿⡿⠃⣿⣤⡞⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⣶⣦⣽⣷⣤⣴⠾⢿⡉⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠁⠼⠁⣿⠀⠻⠆⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠸⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠷⠶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⡀⡴⠙⠁⠙⣨⠀⠀⣺⠒⠚⠛⠳⠶⣆⠀⢈⣒⣁⠈⠀⠐⠂⣡⣌⡁⢊⠀⠀⠦⢡⠢⡄⡂⠀⠀⠀⠀⣠⠠⠗⢆⠀⢄
⠀⠀⢆⡀⠀⠀⡬⠐⠀⢿⠀⠐⠑⠀⠀⡇⠀⠸⠭⠼⠀⠘⠀⠀⠣⡤⠃⠈⣀⠀⠘⡈⠩⠁⠄⠀⠀⠀⠀⡐⠀⢠⠏⡔⣀⠤
⠀⠀⠈⠋⠘⠓⠀⠀⠀⠸⣄⡈⠂⢀⡼⠃⡀⠀⠐⠂⡀⠀⠀⠁⠠⠀⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⡁⠅⠀⠀⠈⠚⠈⠂
⠀⠠⠀⠀⠀⢐⡀⠈⠁⡀⠈⠛⠞⠋⠁⠀⠀⡤⠀⠄⠠⠀⠀⠀⣀⠀⡀⠐⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠠⢢⠀⠀⠀⠃⠘⠀⠀⠀⠲⠀⠀⠁⠰⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⠠⠄
⠀⠀⠀⠀⡠⠀⡄⠀⠘⠂⠄⢀⣀⡀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠄⠀⠀⠄⠀⠂⠈⠐⠁
⠀⠔⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⣠⣄⠁⡄⢠⣤⡀⠠⠀⠀⠘⠀⠀⢙⡀⠀⠀⠀⠃⠀⠀⠑⠀⠀⠈
⠀⣜⡁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣦⡀⠀⢹⣿⣄⠀⣸⣿⡀⠀⣰⣿⠆⠀⠀⠉⠓⠀⠀⠊⠀⠀⢰⣤⣤⣤⣰
⠀⢠⢬⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠐⢆⣹⡟⠓⢛⡟⢻⣿⡟⢻⡟⢿⣿⠁⠀⠃⠀⠀⠀⠈
⠀⠀⠀⠈⠁⢀⢐⡀⠀⠐⠀⢀⠔⢀⠂⠘⢛⠻⣿⡇⢛⣛⡇⣼⣿⡇⢸⡇⢸⣿⡿⠿⠿⠀⠀⠀⠀⠀⠿⢿⠀⠀⠀⠄
⠀⠀⠀⠀⠎⠁⠀⠀⠠⠀⠀⠑⠒⠁⢀⣤⣴⣾⣿⣇⣹⣿⣇⣉⣩⣧⣘⣁⣾⣿⣄⣠⡘⠁⠀⠀⠀⠌⠀⠀⡀
⠀⠀⠀⠀⠀⠤⠔⠢⠀⠠⠃⠀⡌⠀⠈⠩⠄⠀⣹⣿⢹⠛⡿⢹⣏⢹⣿⣿⡟⠉⢉⠙⠁
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠠⠀⠀⣶⡿⠛⢻⣿⡿⣿⣿⠟⢿⣿⠛⠿⣶⣦⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢆⣀
⠀⠄⠀⠁⠀⠀⢨⠀⠀⡀⠀⠀⢀⠤⠒⠒⠈⠀⠀⠸⣿⠄⠘⠿⠠⠀⣿⣦⠀⠉⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⡄
⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠐⠒⠋⠲⠁⢀⣈⡉⠀⢡⣤⠸⡆⠀⠀⠈⣷
⠀⠂⠀⠀⡃⠀⠃⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⢲⠀⠀⠂⠀⠀⠉⠁⠙⠁⢀⠟⢟⠷⢷⣤⣁⠀⠛⠶⠖⠊⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠀⠀⢀⠀⠣⡀⠀⠐⠀⢀⠀⠠⠀⠰⠉⡇⢀⠰⠀⠀⠀⡠⠐⠀⠀⠢⣀⠐⠠⠀⠈⠑⡀⠈⠁⠘
⠀⠰⠤⠤⠆⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡠⢀⡤⠳⠤⠤⠔⠈⠀⠂⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠤⠒⠀⠀⢰⡤⡄
⠀⠠⠀⠀⡀⡀⠀⠠⠀⠀⠠⠀⠀⠢⠄⠂⠀⠚⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠆⠀⠀⠀⠀⢀
⠀⣀⠀⡄⠀⢠⡀⠀⠀⡀⠀⠀⠀⡀⠀⠁⡇⠀⠀⠀⣰⠑⠈⠘⠂⡀⠀⢰⣦⣤⣤⢼⡂⠀⠀⠀⠀⠠⠁⡜⠀⡀
⠀⢀⠀⠄⠀⠀⠀⠂⠐⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠐⣄⠀⠀⠀⠀⡃⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠠⠀⠌⠀⠒⠀⠀⠀⠀⠀⠠
⠀⠑⢕⠠⠠⡰⠀⠀⠀⠐⠀⠀⢤⡤⠦⠀⠀⠀⠀⠀⠐⠶⠠⠤⠘⠀⠀⢀⠠⡂⢟⠄⠀⠀⠙⠢⠤⠊⠀⡀⢀⠐⣅⡁⠀⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠴⠀⠀⠀⠀⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣭⣛⡛⣻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡟⣿⠉⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠟⠷⢣⡖⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣏⠏⠉⠈⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠘⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⡍⠿⣶⣅⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⠈⢢⡀⠁⠈⠉⠻⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⢢⡀⠉⢲⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢰⠃⠀⠀⠈⣇⠙⢦⡀⠰⠧⡀
⠀⠀⠀⠀⠀⠀⠀⡰⢱⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡆⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠰⠁⣿⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⡇⢯⠸⡀
⠀⠀⠀⠀⠀⢠⠃⢸⣿⠀⠀⠀⠀⠀⣸⣠⣀⣤⠖⡀⠈⡇⡇
⠀⠀⠀⠀⠀⡜⢠⣿⣿⠀⠀⠀⠀⢠⣭⣽⡧⠔⣹⠇⣠⡇⢇
⠀⠀⠀⠀⢀⡇⠘⢻⣿⠃⠀⠀⠀⠀⡿⠛⣞⣿⣵⡿⠁⢿⣾
⠀⠀⠀⠀⠘⡇⣦⢸⣿⡀⠀⠀⠀⢠⣇⠀⠀⣼⡯⠃⠀⢸⢻
⠀⠀⠀⠀⠀⣷⣿⡄⣿⠗⣉⡑⠛⢦⣹⠀⢠⣿⣷⡀⢀⣼⠆
⠀⠀⠀⠀⠀⡘⢻⣿⣿⣾⣯⠍⠙⢦⡻⠀⢸⡈⠓⠞⠒
⠀⠀⠀⠀⢀⡇⠀⠹⣿⣿⢹⡆⠀⠈⠁⠀⠈⡇
⠀⠀⠀⠀⠈⢧⣳⣤⣹⣿⣆⣇⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠄⠀⣶⣾⡆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡄⠀⢻⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢂⠈⠻⣿⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠃⠀⠈⠁
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡇⠁⠀⠐⣦
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣦⠆⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⡻⢿⣿⣿⣧⣤⡙⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡳⣍⠀⠉⠙⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣠⠎⠉⠉⠱⢤⡄⠶⠀⠀⠰⠦⣄⠀⢠⠔⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠁⠀⠈⠳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠹⠖⠈⠉⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠉⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠙⢦⡀⣀⡀⣤⠒⢤⠀⢀⣀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⠀⠀⠐⠒⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡽⠀⠀⠀⠀⠀⠀⢠⠤⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠤⠤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⢐⠀⢀⠄⠒⡤⠄⠚⠁⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠷⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⢀⣀⠀⠀⠀⠀⠈⢶⡉⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠈⠀⠀⠓⠀⠀⢸⡆⣰⠀⠀⠀⠀⣀⡀⠀⠀⠀⠘⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠙⢂⠀⠀⠀⠀⠀⠀⠘⡅
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢹⠀⠀⢀⠊⠀⠀⢓⡄⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠛⠃⠈⠀⠀⠀⠀⢃⠀⢸⠀⠀⠛⡀⠀⠰⠋⠙⠢⠤⠀⠀⠤⠤⠀⠒⠒⠛⠛⠂⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⢀⠀⠀⠀⠘⢆⠸⠀⠀⠀⠈⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀⠀⠀⠀⠀⠰⠿⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠳⢄⠠
⠀⠠⠄⠚⠁⠀⠀⠀⠀⠀⠁⠷⠀⠀⠀⠀⠇⠀⠀⢀⡳⠆⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠈⢡⠔⠀⠢⡄⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⠀⢣⡀⠀⠀⠰⠀⢠⠋⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⡀⠈⠉⠀⠀⠰⠇⠀⠀⠀⢸⠀⠀⠀⠀⣧⣀⠤⢿
⠀⠀⠀⠀⢀⡴⠁⢀⠎⠁⠀⠀⠀⠙⢄⡴⠃⠀⠆⠀⠀⠀⠀⡴⠃⠉⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠈⠦⡀⠀⠀⠀⡞⠢⠤⠔⠃⠀⠀⠀⡸⠁⠀⢀⡌
⠀⠀⠀⠀⠸⣇⡠⠛⠀⠀⠀⠀⠀⠀⠸⠆⢤⣄⣸⣄⠀⠀⠀⢧⡀⠀⢀⣀⠀⠀⠀⠀⠑⢄⣹⡀⠀⠀⠀⠱⡄⢀⡜⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⠉⡿⠶⠄⢀⡀⢹⠀⠀⠀⠀⠀⠉⠑⢦⡀⠀⢱⠄⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⣠⠞⣠⠎⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠼⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⢣⣀⡀⠀⠀⠀⠀⠀⠘⠶⠶⠀⠀⠀⣼⡀⢰⠀⠀⠀⣀⠰⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢰⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⠋⠘⠋⠙⢧⠀⠐⢲⠀⠀⠀⠀⠀⠀⠹⣶⠆⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠇⠀⠇⠀⠀⠀⠉⠀⠘⢁⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠶⢤⡀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢠⡄⠀⠀⠀⣠⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠆⠀⠙⢤⣤⠀⠀⢀⣀⠀⠀⠀⢀⣀⡀⠀⠀⠤⠤⠄⠒⠂⠉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠸⠀⠀⠀⠀⡄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⠀⠸⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡌⠉⠀⠑⠒⠒⠀⠠⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠑⠒⠠⠤⠤⠤⠤⡤⠋⠀⠀⠀⠀⠀⣄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠠⠤⠤⠀⠀⠤⠼⡄⠀⠀⠀⠀⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠘⡶⠠⠤⠄⠘⠃⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⢀⣰⣆⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠉⠋⠀⠈⠛⠁⠀⠈⣦⠀⠀⠀⣀⠔⠉⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠙⠙⠋⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠴⠦⠀⠀⠋⢹⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⣀⡀⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⠿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠋⠁⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣀⡀⠀⠀⠀⠉⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋⠁⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠙⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁⠀⠀⠀⣠⣴⣾⣿⠿⠛⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠙⠿⣿⣿⣦⡀⠀⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⣀⣤⣶⣿⠿⠛⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠛⢿⣿⣦⡀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⣀⣴⠿⠋⠀⣀⣤⣴⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣈⠻⢿⣿⣶
⠀⠀⠴⠞⣋⣡⣤⣶⡿⠿⠿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣇⠀⠈⣿⣿⡿⣿⡿⢻⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠈⠉⠛⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣯⣷⣷⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣶⣶⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠿⠟⠛⠉⢉⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢷⣶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣶⡿⠿⠛⢋⣁⣤⣴⣶⣿⠿⠿⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠿⠟⠛⠛⠛⣛⣉⣩⣥⣤⣶⣾⠿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⠿⠿⠿⠿⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠋⠀⠀⠀⠀⠠⣤⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠟⠉⠈⠳⣄⠀⠀⠀⣠⠞⠛⠻⢿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠙⣷⣠⡾⠃⠀⠀⠀⠀⠈⠙⠛⠿⣷⣦⣄⣀
⠀⠀⠀⢀⣠⣤⣶⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢻⣶⠶⠤
⠀⠀⠋⠉⠉⢸⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡤⠤⠤⣿⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡷⠖⠒⠛⠉⠉⠹⣏⠉⣹⣍⣿⠟⠀⠀⠀⠀⠀⢸⡇⠀⠈⠻⣤⣼⣭⠿⠁⠀⠉⠉⠉⠉⠒⠒⣿
⠀⠀⠀⠀⠀⠸⣿⠀⢀⡠⠄⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⢹⣷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣿⣷⠀⠁⠀⠀⢠⠀⠀⠀⠀⠐⠲⣄⡀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⢀⣹⠀⠀⠀⠀⠀⠀⠀⢽⣠⡾⠁⠘⠷⣄⡀⠀⣮⠁⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣬⠇⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⠈⠙⠛⠧⠤⠤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠖⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠋⠉⠙⠻⣿⣿⣓⠾⣯⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣈⣷⣮⣝⣷⢯⣝⡳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠉⠛⠻⢧⣍⣷⠮⣝⣓⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢾⣯⣙⢲⣭⣛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣮⣿⡒⢬⣝⡲⢤⣄⣀⣤⠶⠶⠶⢤⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢧⣽⣗⠮⣭⣀⣤⣶⠦⡄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠀⢿⣿⣷⢤⣿⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠈⠻⠿⠿⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣦⣄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣽⣿⣿⣶⣶⣾⣭⣝⡲⣬⡓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⣝⣛⡻⠿⣷⣯⡛⢦⡙⢎⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠢⠀⠉⠛⠿⣷⣮⣻⣿⡴⡍⡊⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⢸⣦⡙⣿⣿⣷⢻⡜⣆⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢷⡘⣿⣿⢸⣿⢸⣿⢸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⣿⡏⣾⣿⢸⣿⢠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣾⣿⣯⣿⢋⣾⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣵⠟⠡⠞⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢫⣾⣻⠇⣼⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠸⡏⣿⣼⣯⠁⠐⣃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡸⣯⡘⢿⣿⣄⢠⣞⣚⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣮⡳⠌⠻⣶⣝⢻⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠠⣾⣿⣦⡈⣷⡝⠧⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢉⣵⣿⠿⠃⠐⢵⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢠⢤⠤⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠰⠯⠖⠾⠶⠿⠼⠶⠮⠽⡶⠀⠰⠿⠴⠾⠾⠯⠀⢰⠵⠾⠧⠗⠾⠷⠦⠾⠶⠾
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠴⠶⠶⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠤⠤⠴⠒⠋⠙⢢
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠴⠒⠲⣄⣀⡴⠋⠀⠀⠀⠀⠀⠀⠉⣿⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⢸⠉⠛⠋⠀⠀⣀⣈⠁
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⡷⢄⡀⠀⠀⢀⣀⡼⠀⠀⠀⠀⠀⠀⠉⠁⠀⠓⠲
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠙⠲⠄⢀⡀⠀⠃⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠓⠀⠀⠢⣄⠀⠀⢀⡉⠒⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⢠⡀⠁⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⠂⠀⠐⠆

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⣿⠉⣿⣿⡝⠻⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⠈⢿⣿⡆⢘⢏⣠⣤⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⣿⡦⠜⠛⠁⠀⠈⠉⠉⠉⠙⠛⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣬⣽⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠒⢦⡀⠀⠀⠀⠴⠖⠒⢄⠀⠙⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⡖⠦⠈⠀⠀⠀⢐⣶⣦⠀⠀⢠⠟⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⡄⠀⠀⠀⠀⠀⠉⠈⠀⠀⢸⣬⣷⣭⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠐⠛⠂⠁⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⠤⠤⣾⡇⠀⢸⡿⠛⠛⢶⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣟⣡⣠⣿⣟⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠂⠀⠀⢿⡀⠈⣿⣿⡿⣿⡇⠀⠀⠀⠀⢠⣤⣾⣏⣉⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⠀⠠⠀⠈⠱⠀⣿⠀⠀⣸⡇⠀⠀⣠⠔⠾⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠰⡐⠀⠀⠀⠀⠀⠀⢆⢠⠻⣦⡴⠟⣀⡄⠀⢹⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢳⣀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⡅⠀⠈⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠓⢄⠀⠀⠀⠀⢹⠟⠉⢏⠀⠀⠀⠀⠀⣠⣦⣾⣿⣿⣿⣿⣿⣴⠏⠀⠀⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢸⠃⠀⠀⠀⡄⠀⠀⠈⢢⠀⠂⠈⠉⠉⢧⢈⣿⠃⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠹⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢀⡏⠀⠀⠀⣰⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⡾⠁⠀⠀⠀⢀⡤⠶⠚⠛⠛⠳⢦⡀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⢀⡞⠀⠀⠀⢀⡿⠀⠀⠀⠀⢀⠞⡀⠀⠀⠀⠀⠸⡄⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠏
⠀⠀⠀⠀⠀⠀⢠⡟⡆⢀⠞⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠎⠀⢱⡀⠀⠀⠀⠀⠱⡀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢠⡏⡴⠋
⠀⠀⠀⠀⠀⠀⡼⠀⢹⠏⠀⠀⠀⠀⠀⡟⢧⠀⠀⣠⠏⠀⠀⠀⢣⠀⠀⠀⠀⠀⠱⣀⠆⠀⠀⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠃⡰⠃⠀⠀⠀⠀⠀⢀⡇⠈⢆⣰⠃⣀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⠋⠁⠈⡇⠀⣿⠓⠶⠤⠤⠼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢁⠀⢻⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣟⣋⠁⠀⠀⠀⠀⢀⣀⣀⣠⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⣠⣀⡀⠠⠴⠋⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣿⣿⣿⣿⡿⢋⣿⣿⡿⠿⠋⣽⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⠼⠛⠉⠀⠀⣼⣿⡿⠟⠋⠉⠀⣼⣿⣿⡿⠟⠋⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⣠⣼⣍⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠐⡏⠀⣨⠀⠀⡀⠀⠀⠀⡾⠛⠻⠆⠀⠀⠀⠀⠀⠈⠹⣇
⠀⠀⠀⠀⠀⣠⡴⠶⣤⡀⠀⠀⠑⠲⡄⠀⠀⠀⠀⠀⠈⠛⠁⠀⡼⠃⠀⠀⠀⠱⢤⠾⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⡇⠀⠀⠈⢷⠄⡀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠶⠚
⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⣠⠇⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⣤⣠⠟⠉⠁⢰⡄
⠀⠀⠀⠀⠀⠘⣧⡀⢀⣀⣿⣆⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢹⠈⠏⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠸⡟⠋⠉⠀⢹⡆⠀⠀⠀⠀⠀⠙⠷⣤⣄⣈⠻⣿⣿⠟⠀⠀⢀⣀⣤⡶⠟⠁⠀⠀⠀⢠⡇⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢀⠀⢉⡟⠛⠛⠷⠾⣿⣿⣿⣉⡁⠀⠀⠀⠀⠀⣰⠏⠉⠙⠒⢴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⡇⠀⢀⣠⢴⡞⠁⠀⠀⠑⠲⠖⠒⠚⠋⠁⠀⠉⣿⢄⠀⠀⢀⡾⠃⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠳⣴⠟⠁⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣠⣤⡴⠶⠾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠻⢶⣴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠠⠴⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⢻⠿⠋⠀⠀⠀⠀⠀⠔⠀⠀⠀⠠⠂⠀⡠⠶⠖⠀⠖⠀⠆⠀⠆⠀⠛⠒⠠⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸
⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⣀⣈⠁⠀⢠⠂⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⣀⠀⢐⠀⠀⠁⠀⠀⠄⢀⣠⡆⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠐⠄⣀⠀⠀⠀⠀⠀⠀⠀⡄⢀⢀⡯⠖⠀⠀⢹⠁
⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⢸⠉⠃⠀⠀⠀⠀⠀⠀⠀⠓⠀⢒⠒⠐⠂⠀⠀⠀⠈⠉⠓⠒⠒⠒⠒⠓⠋⠉
⠀⠀⠀⠀⠠⡀⢰⠀⠀⡆⡆⠘⠀⠀⡇⠀⠀⡏⠀⠀⡔⢸⣽⢲⡄⢁⣈⠀⠀⢀⡀⠐⠃⠀⢀⠉⠀⠀⠘⡁⠀⠒⠒⢀⡀
⠀⠀⠀⠀⠀⠁⠈⢀⠟⠛⢻⡁⠀⢠⡇⠄⢠⠃⠀⠸⣚⣚⣗⣰⡇⢀⠀⠀⠀⠈⠀⠀⠀⣰⠛⢿⡄⣴⠀⠀⠀⠀⣄⢸⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⢸⡇⠀⠈⠁⠀⠀⠙⢋⠏⠃⠄⢀⠁⠀⡀⢠⠀⠀⠀⠘⠳⠾⠃⢸⠀⠀⠀⢰⡟⠊⠃⠀⢠
⠀⠀⠀⠀⠄⠀⠀⢸⠀⠀⠀⣿⠀⠘⠓⠀⠀⢸⢠⡖⠀⠸⠀⢰⠀⢸⠀⢀⣇⢀⠀⠀⢰⠛⠉⠀⠀⠁⠀⠀⠀⢸⡇⠀⠀⠈
⠀⠀⠊⠀⠀⠀⠀⠈⠆⠀⢰⠏⣀⠯⠀⠀⠀⠈⢸⡇⠀⠀⠀⢸⠀⠈⠀⠐⠶⢺⡀⢀⡀⠧⠔⠰⠶⠆⠀⠀⠐⠀⡇⠀⣷
⠀⠀⠀⠀⠀⠐⠀⠀⠀⢒⡋⣀⢹⠀⠀⡄⠀⠀⠠⠁⠆⡏⠀⢸⠀⠀⢠⡆⠀⠀⠀⠈⢱⢸⠀⠀⢆⠉⠀⠀⠀⢰⡇⠀⠁
⠀⠀⠛⠒⠋⠁⠀⠀⠀⠸⡝⠹⢸⡀⠀⠳⠀⠀⡄⠀⠀⠁⠴⠞⠀⠤⢹⠀⠘⠌⠀⠀⠈⢘⠀⠀⢸
⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠑⠃⠀⢡⣤⣤⡄⢰⠁⠀⠀⠀⢠⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠇⠀⠀⠀⣀
⠐⠒⠒⠒⠒⠒⠒⠀⠒⠒⠓⠒⢢⣤⣤⣤⡄⢸⠀⠀⠀⠀⠘⡇⠀⠁⠀⠀⠀⠃⠀⠀⠀⠀⡄⠀⢀⡃⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠘⠀⠀⠂⠀⠀⠏⠀⠀⠀⠀⠀⠋⠀⠀⠄⠀⠃⠀⠀⠳⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢿⣦⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⣿⣿⡿
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⢿⡄⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⢀⣿⠀⠀⢿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⡟⠀⠀⠠⣤⡄⢿
⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⠛⠿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⡀⠀⣸
⠀⠀⠀⠀⠀⣾⡟⠀⣴⠶⣦⡄⠈⠙⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿
⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢿⣧⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠿⣶⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠉⠉⢹⡟⠙⢻⣷⡀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⢻⡆⠀⠘⣷⠀⠀⢿⣧⠀⠀⠀⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠹⠇⠀⠈⠁⠀⢸⣿⣶⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⠿⢿⣶⣤⣤⣄⣀⣀⠀⠀⠀⢈⣿⡆⠀⠀⠀⠀⠀⠀⠀⣸⣿⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⠀⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠈⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢿⣷⡀⠀⠀⠀⠀⢻⣿⠀⢀⣠⣴⡿⠟⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⢿⣧⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠙⢿⣿⠋⠁⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠈⣿⣦⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠹⢷⡄⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⣿⡆
⢀⣴⣶⣶⣶⣶⣶⣶⣶⠿⠿⠋⠀⠀⠀⠀⠀⣠⣾⠟⠋⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡁⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⣀⠀⣀⣠⣾⠋
⠀⠙⣿⡿⠿⠿⠿⠿⣿⣿⣆⠀⠀⠀⠀⢻⣷⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⡟⠉
⠀⠀⠹⣿⣦⣠⣤⣶⠿⠛⢿⣧⠀⠀⠀⣸⣿⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠈⠻⢿⣿⠿⠟⠃⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠉⠉⠉⠉⢙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⡟⣿⣏⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣿⠇⣿⣿⣤⣤⣤⣤⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⢙⣿⡆⢸⣿⠉⠉⠉⠉⠉⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⠚⢿⡏⠙⣿⣶⣿⣷⣾⣷⣿⣿⠀⢘⣿⣷⣿⣶⣾⣷⣾⠟⢉⣾⠗⢲⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠛⣽⡷⣠⣿⣦⡾⠁⠀⠀⠀⠀⢹⣿⠀⢸⣿⠀⠀⠀⠀⠀⠹⣦⣼⣷⣤⣾⡿⠛⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣁⣀⣀⣀⣨⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇⣾⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣽⣀⣀⣀⣀⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⡇⣿⣯⣉⣉⣉⡉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤
⣿⣿⣿⣿⣿⠿⠿⢿⣿⠿⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠉⠓⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⢀⣠⣤⣤⣄⡉⠓⠢⣄
⠀⠀⠀⠀⡰⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡴⠞⢉⡥⠖⠒⠈⠙⢷⣄⠀⠙⢤
⠀⠀⢀⡞⠀⣠⠾⠿⠿⠿⣿⣶⣶⣶⣶⣤⣄⣀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢀⡞⠀⢀⠏⠀⠀⢦⣀⠀⣀⢹⡆⠀⠀⠳⡀
⠀⠀⡞⠀⠈⠁⠀⠀⠀⠀⠀⢀⠀⠉⠉⠙⠻⢿⣿⣦⡀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠸⠁⢐⡵⠚⠛⢶⡄⠀⠀⢸⠀⡇⠀⠀⠀⢱
⠀⢰⠃⠀⠀⠠⣄⣰⣶⣴⣶⣾⣿⣶⣦⣄⡀⠀⠀⠉⠁⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⢳⡄⠀⠸⢠⡇⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠒⠶⢿⣿⠋⠁⣿⢯⣤⣿⡟⣿⠛⢦⡀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⣂⠀⣠⡤⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠘⡆⠀⠀⠀⠀⠹⣦⡀⠈⠳⢭⡭⠞⠁⠀⠈⠻⣦⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⡏⣠⠋⠰⠃⡔⠀⣸⠃⠀⠀⠀⢠⠃
⠀⠀⢹⡀⠀⠀⠀⠀⠀⠙⠓⠶⠦⠤⠶⠚⠛⠀⠀⠁⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⡼⠃⠀⠀⠀⢀⡎
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢀⣠⠴⠒⠋⠉⠉⠉⠉⠙⠒⠦⣄⡀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢁⡴⠋⠀⠀⠀⣇⠀⠀⠀⢰⠄⠀⠀⠀⠙⢦⡈⠳⣄⡀⠀⠀⠠⢄⣀⡴⠟⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⠤⣤⡤⠤⠴⠒⠉⠀⣰⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⣆⠀⠉⠓⠶⠤⠤⠤⠤⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⢹⠀⠀⠰⣄⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠛⠛⠛⠒⠲⠦⣄⡀⠹⣄⠀⠀⢿⡀⣠⣤⣄⠀⠐⣫⣤⣄⢀⡟⠀⠀⣠⠏⢀⣠⠴⠖⠚⠛⠛⠛⠒⠲⠤⣀
⠀⠀⠀⠀⢀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⡌⠳⣄⠀⠀⠈⠀⠈⠀⠀⠁⠀⠀⠀⠀⣠⠞⣡⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠙⢦⡀
⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠙⠲⠤⣄⣀⣀⣀⣀⣠⠤⠖⠋⢁⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠙⢆
⠀⠀⡼⠁⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠐⠀⠀⠀⠈⢧
⠀⢠⠃⠀⠀⠀⣠⡴⣟⠋⡉⠛⠛⠛⡉⠉⣛⢶⣄⡀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠘⡄
⠀⢸⠐⠶⢶⠫⣥⣀⣩⣤⣾⣤⣄⣼⣿⣦⣬⣥⣬⠽⡳⠶⠆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠱⣄⠀⠀⠔⠀⠭⠉⠭⡄⠄⠄⠀⢀⠞⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡇
⠀⠀⢧⠀⠀⠀⠈⠳⠦⠤⡤⠤⠤⠤⠤⠤⠤⠖⠉⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⡼
⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡠⠎
⠀⠀⠀⠀⠀⠀⠙⠢⢤⣀⡀⠀⠀⠀⠀⣀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⣀⠀⠀⠀⠀⢀⣀⡥⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⣾⡥⣄⠀⠀⢄⠀⢀⣠⠀⠀⠉⠁⠈⠉⠉⠉⠉⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠁⠀⠀⢦⠀⠀⠱⠂⠀⢀⣀⣀⣨⠀⠀⠀⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠱⠶⠖⠁⠀⠙⠛⠁⠀⢀⣀⠁⠊⡠⠂⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢿⣸⡇⠈⠀⠀⠋⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠖⠉⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢀⣼⡇⠀⠀⠐⠒⠉⡄⠀⠸⣠⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠸⠿⠇⠀⠀⠀⠀⠀⠸⠆⠀⠛⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠸⠯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠚⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠠⠀⠁⠀⠀⠀⠡⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⡀⠀⠀⠀⠀⠀⡀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠚⠛⠃⠀⠀⠀⠈⠀⠀⣀⡜⠁⣀⠆⠀⠀⠀⠀⠀⠘⣄⡀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣠⠆⠘⠀⠀⠀⠀⠀⢄⡀⠀⠌⠀⣶⡄⠀⠀⠀⠀⠀⢀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠋⠉⠀⠀⠀⠉⠉⠒⠢⣀⣀⣀
⠀⠀⠀⣠⠴⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⡄⠠⠤⠔⠒⠺⡁
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠐⠒⠂⢀⡸⢖⡪⢦
⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣧⠃⠀⢀⡠⢔⣪⠕⠚⢯⠳⡄⠇
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢀⡡⠔⣊⠥⢺⠋⠀⠀⠀⠀⢣⠘⢾
⠀⢸⠀⠀⠀⠀⠀⣠⠴⠋⡕⣊⠥⠒⡩⢃⣀⣸⠀⠀⠀⠀⠀⢸⠀⢸⢧
⣎⠽⡄⠀⠀⠀⠀⠠⢗⡜⣷⠳⡀⠈⢰⠃⠀⠈⢳⢀⣦⠀⢀⠎⣠⣃⡀⢳
⠀⠀⢳⡤⠖⠊⠈⠢⡜⡰⢍⢆⠳⡀⠈⢦⡀⢀⣻⠉⡳⠾⠓⢫⠏⠀⡽⣸
⠀⠀⠀⠀⠀⠀⠀⣸⣵⣱⣌⣎⠣⣑⠀⠀⠈⠙⣡⠞⠀⠀⠆⠀⠒⠈⢠⠇
⠀⠀⠀⠀⠀⢠⠊⣡⣒⣲⠤⠀⠀⠀⠉⠐⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠄⠀⠄⢀⣀⣐⡀
⠀⠀⠀⠀⠀⠘⢤⣀⣈⣀⠤⠖⢳⠤⣀⠀⠀⠀⠀⠀⠀⡾⠥⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠑⠒⢤⠤⠤⠔⢒⣉⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡆⢀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠝⠀⠠⣁⠠⡽⡜⠂⠀⠀⠀⣀⠀⠈⢰⠃⠂⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⢴⠀⠀⠀⢱⣀⣀⣀⠤⠶⠜⠒⠒⠊⠉⠉⠀⠀⠆⡎⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠼⠶⠚⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⢌⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡠⠤⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠸⠀⠐⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢢⠠⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⠋⠀⠀⠢⣦⡀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡌⠀⠀⠉⠢⡈⠲⣌⣦⡖⠠⢥⡒⠷⠠⠀⠀⠈⣶⠏⠀⠀⠀⠀⢁⠀⠁⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠡⠀⠀⠀⠀⣻⣋⠐⠓⢠⣐⠁⠈⠆⠀⠂⠀⡀⠻⠀⠀⠀⠀⠀⠘⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠀⠀⢸⡁⠑⢌⡀⣈⠬⠘⠂⢬⠀⢀⡼⠀⠐⣇⠀⠀⠀⢸⡔⡀⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⠀⠌⢀⠄⡞⠀⠒⠀⠠⠤⠂⠀⠀⠊⠉⢀⠛⠀⠀⣤⠆⢀⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠤⠈⡔⢀⡀⠛⠀⠀⠀⠀⠀⠀⢀⠒⠂⠐⠁⠀⠡⡐⠈⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢷⠊⠉⠁⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠉⠀⢀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠈⠐⠒⠒⣲⠚⣯⡀⠁⠱⠒⠁⠁⠺⠀⠂⠀⠀⠠⠠⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡔⠓⠒⠭⠧⡐⠀⠀⢃⣘⡀⢬⠈⠀⠀⠀⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⢀⣀⢪⠁⠀⠔⠢⠤⠅⠀⠈⠙⡛⠁⢘⣀⠤⡐⠐⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠓⠒⠚⠦⠤⠄⠓⠶⠤⣲⡄⢠⠄⠀⠄⠠⡤⣀⡉⠀⣀⣡⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠈⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⠄⣰⡃⠠⠀⠉⠐⠺⡁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠄⡀⠀⠀⠈⠸⡁⠈⠈⠀⠤⠆
⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠀⠀⠈⢄⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠛⢱⣿⣿⣿⡿⠋⠀⠀⠀⠈⠻⣿⣿⣿⣷⠈⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠏⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠈⢻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⡁⠀⠀⠀⢸⣿⣿⣿⣷⡀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠀⠀⠀⠀⣹⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄⠀⠀⠀⢻⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⠃⠀⠀⢀⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣄⡀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⣠⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣦⣄⣈⡉⠛⠛⠛⠋⣉⣀⣤⣶⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢻⣛⣟⣙⣛⣟⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣤⣤⣤⣤⣴⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣤⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠋⠉⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⠿⠋⠈⠙⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⣿⣿⠇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⢻⣿⣄⢀⣀
⠀⠀⣴⣿⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⣿⠀⠸⢿⣿⣿⣿⣷⡀
⢠⣾⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠟⠀⠀⠁⢹⣿⣿⣿⣿
⠘⢿⣿⣿⣿⡟⠁⠶⠦⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟
⣴⣾⣿⣿⣿⣷⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠙⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿
⠀⠈⠛⠋⢻⣿⣿⣿⠿⠿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠛⢻⡿⢶⡄⠀⠀⠀⢠⣿⣿⣿⠏⠉⠁
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⡇⠈⣇⠀⣷⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣀⡀⠀⠀⠀⠀⠸⣧⠀⠑⠀⠉⠀⢹⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠛⠛⠶⣶⣶⣶⣿⠀⠀⠀⠀⣠⠟⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙⣷⠀⠀⠀⠀⠀⠀⠉⠉⠻⣦⡀⠀⠘⢯⣠⡴⠟⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⢹⣧⡴⠛⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠙⢦⠀⠀⠈⣷
⠀⠀⢀⣤⣤⣤⣤⣤⠾⠋⠀⠀⢀⣴⠟⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠠⣿⠁⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⣾⠁
⠀⠀⠙⣷⠶⠶⢶⣶⡄⠀⠈⢿⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢲⡶⠛⠁
⠀⠀⠀⠘⠳⠶⠟⠉⠻⣦⣤⡾⠁⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⣶⣶⠂⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣼⡇⣿⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠒⡞⢳⡶⠶⠶⣾⡇⢹⡶⠶⠶⢶⠛⡖⢲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⣀⣙⣑⣉⣉⣀⣀⣀⣸⡇⣿⣀⣀⣀⣈⣊⡑⣟⣁⣈⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠇⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡟
⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⣤⣠⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢻⡿⠿⠿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⣿⣿⡇⠀⢰⡇
⠀⠀⠀⢸⣷⣶⣶⣶⣾⠃⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠅⠀⠀⠀⣿⣿⣷⠀⠸⠇⠀⠀⠀⠀⢀⣀⡀⡀
⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⣤⣾⠀⠀⣶⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⣿⠀⠀⡇⠀⠀⠀⠀⢸⣿⡇⡏
⠀⠀⠀⠀⣿⣿⣿⡏⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣾⡞⠂⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡷⣿⠀⠀⡇⠀⠀⠀⠀⢸⣿⡇⡇
⠀⠀⠀⠀⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⣴⣧⠀⣿⡇⠀⠀⠀⠀⣿⣇⡀⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣷⣿⢰⣾⣷⡆⠀⠀⢀⣘⣿⣧⣇⡀
⠀⠀⠀⠀⢿⣿⣿⠃⡇⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇⠀⠀⠀⢸⣿⡇⡇⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢸⣿⣿⢠⡇⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇⠀⠀⠀⢸⣿⡇⡇⢸⣿⢈⡇⠀⠀⠘⣿⠂⠀⠀⠀⣿⣿⡇⠀⣿⣿⡇⠀⠀⠘⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠈⣿⣿⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⣿⣿⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣠
⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⣿⣿⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⣀⣤⣄⣀
⠀⠀⠀⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⢹⣶⣾⣿⣿⣿⣿⣷⣶⣶
⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢀⡾⠋⠀⠀⠙⢷⡄⠀⠀⠀⠀⣀⣙⣛⣛⣛⣛⣛⣋⣠⠀⠀⠀⠈⠻⠟⠁⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⣾⠁⠀⠀⠀⠀⠘⣷
⠀⠀⠀⣿⣿⢶⣶⣶⠀⠀⣿⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠉⠉⠉⠉⠋
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢹⡄⠀⠀⠀⢀⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠻⠲⠒⠓⠓⠚
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠙⢶⣤⣾⡷⠏⠀⠀⢀⣤⣤⣤⡄
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠾⠿⢿⣿⣿⣿⠿⠷⠆⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⣿⣿⢚⣛⣾⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⡆⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢀⣈⡇⠀⠀⠀⠀⠀⣾⠉⠙⠻⣿⣿⠟⠋⠉⣷
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠘⡆⠀⠀⠀⢠⣾⣿⣿⣧⡀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⠦⣄⣰⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠘⢷⣤⣀⣀⣀⣀⣴⡾⠃
⠀⠀⠀⠿⠿⠛⠛⠛⠁⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠙⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠒⠒⠒⠂⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠁
⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠙⠦⠀⠀⠘⠃⠐⠀⠒⠀⠀⠀⠀⠀⠀⠄⢈
⠀⠀⠀⠀⠀⠳⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢟⡿⠷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢔⣆⠀⢀⠐⢦⠄⣀
⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢈⠠⠈⢁⠀⢧⠀⠙⠒⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠠⠤⣤⠒⣉⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠀⠀⠐⠒⠀⠑⢄⠀⠀⠀⢂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢢⡀⢀⡀⠀⠀⠀⢃⣼⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢣⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠅⠤⠤⠀⡀⣑⠪⢠⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠐⡄⠀⡇⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠈⠑⢄⡑⢄⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠘⣎⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣀⠵⢄⠱⡤⠂⠀⠁⠉⠟⠂⠀⠤⢀⠀⠀⠀⢀⠀⡈⣢⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⡀⠀⠄⠈⢀⠀⠁⠈⠀⠀⠁⠀⠜⠁⠀⠈⠁⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡜⢻⡾⣦⣾⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣶⣶⣶⣶⣶⣶⣿⣷⣤⠀⠀⠀⠀⠀⠀⢎⠠⣄⠒⢶⠶⣶⣦⣄⡀⢀⠀⣠⣴⠆⠀⢳⣾⢧⠻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣗⢀⠀⠀⠀⠈⠀⠀⠸⠀⠑⠉⠢⣹⠈⠁⢸⣿⡁⣀⡤⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣿⣿⣿⣿⣿⣿⣿⠿⢛⡛⠿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠿⠿⠏⠉⠻⠋⠉⠁⠀⠀⠈⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠒⠉⠀⠀⠀⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠠⠎⡥⠀⠀⠠⠀⣀⠛⣀⠀⠒⠂⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⣑⣀⡐⣠⣬⣄⣀⣀⣀⣀⣈⣵⣶⣦⣤⣤⣴⣶⣶⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠀⠀⠀⠠⠀⠠⠠⠀⠠⠐⠀⠀⠠⠠⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⡀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢠⠖⠲⡄⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠈⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⠇⠀⠀⢹⠀⠀⠀⠀⠀⢀⠔⠚⠓⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⠀⠀⢸⣿⠀⠀⠀⠀⢠⣇⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠘⣆⠀⢀⠞⠀⠀⠀⠀⢸⠿⠃⠀⠀⡸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠁⠊⠉⠑⠀⠀⠘⢤⣀⣀⡴⠁⠀⠀⠀⠀⠀⠀⠛⢿⠛⠉⠉⠙⢿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠔⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢰⠉⠣⠤⢤⣀⣈⣙⣢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢦⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠘⢄⡀⢀⡠⠤⠤⠤⣀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⡀⠈⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠉⠉⠓⠲⢤⣤⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⣀⣀⣀⣀⣀⣠⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠃⠀⡀⠀⠀⠀⣀⡀⠀⠀⠀⢀⡰⠄⠘⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡴⢲⣌⣱⠒⣰⡇⠉⠠⠀⠄⠮⡔⠛⠳⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠔⠝⡧⠀⠀⠠⣀⠆⠀⠄⠈⠀⢈⡆⠀⠈⠢⡠⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡄⠀⠀⠀⠠⠀⡀⠀⢀⡀⣄⡴⠋⠀⣠⣄⠀⡘⢀⢉⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠂⠤⠇⠬⠀⠀⠀⠀⠀⢀⣀⡖⢘⡴⠋⠀⢠⠞⠉⠙⢊⡠⠹⡞
⠀⠀⠀⠀⠀⠀⠀⠼⣀⡸⢲⠢⠀⠉⠀⠀⠐⢤⠚⢁⡴⠋⠀⣠⠖⠓⠒⠒⠢⡀⣆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡐⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⡼⢁⣀⡤⠤⢌⠔⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⡀⠲⡀⠀⢀⡴⠋⠀⢀⠜⢁⣦⡿⠤⠀⠀⠀⠈⢀⡔⠲⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠚⠀⠰⢳⡀⠈⠑⠋⠀⢀⠔⢃⢀⠜⠋⠀⠀⠀⠁⢀⡖⠁⢈⣆⠈⠡⡄
⠀⠀⠀⠀⠀⠀⠀⢈⣀⣀⠖⠀⠙⢄⠀⢀⠴⠁⠠⠀⠀⠀⢀⠀⠀⠀⠰⠋⠀⣠⠊⢎⠤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⢂⣄⠑⠀⠠⢄⣀⡀⡄⡈⠁⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢃⠈⠙⠒⠲⠦⠤⡖⠁⢸⠋⠁⡸⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁⠀⠀⠀⠀⠀⠈⠢⠔⠋⠀⠀⠀⠣⠔⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠶⠖⠒⠒⠒⠛⠛⠛⠉⠉⠛⠛⠛⠓⠒⠒⠶⠤⢄⡀⢠⠞⠙⠲⣄
⠀⠀⠀⠀⠀⠀⢠⠏⠁⠉⢷⣠⠤⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⣀⠤⠒⡶⠀⢀⣀⡤⠶⠒⠛⠉⠉⠉⠉⣑⣲⠦⠈⠉⠀⣀⣀⡸⠇
⠀⠀⠀⠀⠀⠀⠸⠤⣀⠀⠈⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⡴⠋⠁⠀⣼⠤⠚⠉⠀⠀⠀⠀⠀⢀⣤⣖⡋⠭⠄⣀⡀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢠⠃⣰⡀⠀⠀⠀⠀⢠⠋⠰⢃⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢀⠏⡘⠁⡇⠀⠀⡀⠀⡆⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠄⠀⠁⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⡈⠰⠿⠿⢧⠀⠀⡁⠠⠁⠀⠀⠀⠀⠀⡠⠴⠒⠋⠉⠉⠉⠉⠁⠊⠁⠀⠀⠢⡀⠀⠁⠄⠀⠁⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⣄⣤⣤⣼⠀⠀⣘⡄⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⡠⠔⠒⠒⢮⡓⠀⠰⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⠛⠋⠙⢻⡆⡞⢁⢤⠀⠀⠀⠀⠠⡓⠀⠀⠀⠀⢠⠞⠁⠀⢀⠎⠀⠀⠀⠀⠀⠹⡄⠀⠱⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⡀⣿⣶⣾⣿⣿⡇⠙⠁⡏⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢀⠂⠀⠢⡎⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢧⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠇⣅⣠⣤⣤⣼⡇⠀⠘⠷⠒⠒⣚⢳⡀⢀⡇⠀⠀⠀⠀⠀⠀⣤⠚⡖⢤⣤⣀⡀⠀⠀⡟⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⡇⠀⢠⠖⠉⣉⣄⣈⡉⠉⠀⠀⠀⠀⠅⠀⠀⠈⣰⡇⠉⠁⠀⠈⢉⡾⠁⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⣦⣤⣶⣶⣶⡇⠀⣿⠀⠈⠙⠒⠚⠉⢀⠀⠀⠀⠀⠀⠀⠀⡾⠏⠙⠿⠶⠶⠚⣋⣀⡀⠀⠀⣸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡄⡏⠉⠉⠉⢉⡇⠀⠘⠳⠦⠤⠶⠒⠛⠉⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⡀⠉⠀⠀⣹⠀⢀⠇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡁⣿⠿⠿⠿⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠉⠉⠀⠀⡜⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢳⣡⣤⣤⣤⣶⡇⠀⠀⠀⠀⠀⠀⠀⠘⢀⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠂⠦⠄⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠸⣸⡏⢉⣉⣉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⠲⢄⡀⣀⣀⣤⡄⠀⠀⠒⠈⠁⠈⠒⠁⠀⠀⠀⠀⠐⠿⣿⣷⡖⠢⡄
⠀⠀⠀⠀⠀⠀⢣⢻⡿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠳⣄⠔⠀⢻⡠⠊⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠋⢳⣶⣿⠁⠀⠀⠀⠀⢀⡴⢢⠞⡿⢲⡆⣀⠀⠀⠘⠛⠀⡄⢣⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⣀⡤⠂⠉⠉⠈⠉⠉⢭⠉⠉⠓⠘⠢⠓⠊⠀⠁⠉⠉⠉⠉⠙⠛⠛⢆⣸⠀⡏⢳⣄⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⡀⠔⠋⠁⠀⠰⣖⠒⠀⡔⠁⢸⣤⠄⠤⠐⣦⡤⠤⠤⠤⢀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢆
⠀⠐⠉⠀⠀⠀⠰⠚⠉⠉⠣⠀⡀⠀⠰⠊⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠂⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠉⠑⠒⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⡉⡆⠀⠀⠀⠀⠀⠚⠃⢂⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⢴⡥⠤⢀⣀⠉⠀⠒⠒⠚⠋⡠⠒⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠄⡁⠒⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠉⠁⠉⢐⠉⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠤⣤⣀⣈⠤⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠠⠞⠋⠉⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠉⠛⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣄⣤⣤⣄⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣠⣦⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⣰⣼⣿⣿⣿⣿⡿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⢿⣿⣿⡿⣿⣿⣾⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⡁⠸⣯⡙⠛⣸⡏⠈⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠸⣗⡛⢋⣼⠇⢀⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠮⠿⠿⠭⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠾⠿⠿⠷⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠳⣄
⠀⠀⠀⠀⠀⠀⢠⠿⡀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢻⣀⠀⢠⡶⣤⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡟⠁⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⣰⠟⠁⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠴⠶⠛⠶⠴⠞⠋
⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠁⠀⠉⠉⠛⠲⣄⠀⣀⣀⣀
⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣯⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠉⠉⠉⠙⢶⡀
⠀⣴⠞⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⣼⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⡄⠀⠀⠀⠀⡏⠀⠁⠀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡀
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠹⣶⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷
⢻⡄⠀⠀⠀⡌⠀⠙⢦⠀⠀⢣⡀⠀⠈⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠶⠀⠀⢀⡤⣄⠀⡆
⠀⠙⠲⣦⠀⠈⠊⠀⠈⣧⠀⠀⢣⠀⠀⠐⡀⠀⡀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠁⠸⠀⡇⠀⢠⡇⠀⠸⠀⠀⠀⠀⠂
⠀⠀⠐⣇⠀⠀⠀⠀⣰⠟⢧⠀⠀⠀⠀⠀⠡⠀⠘⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣀⣠⠇⢠⠇⠀⡠⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⢀⣤⡴
⠀⠀⠀⠈⠛⠲⠖⠛⠉⠀⠈⢧⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⡿⠀⡼⠀⢀⠀⠀⠀⢠⣿⠀⠁⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠷⠶⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠁⠀⠜⠀⠀⠀⡾⠉⠳⠶⠦⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢀⣠⠴⠚⠃⠁⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⠿⢤⣄⣀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠘⡷⠀⠀⠀⠀⠉⠙⠒⠦⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⠈⢳⡀⠀⠀⠀⠀⡾⠀⣀⠀⠀⠀⠀⠀⠀⢀⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢀⡠⠤⢄⣀⡏⠀⠙⠋⠙⢦⠀⢳⡀⠀⠀⢠⡇⡞⠀⢇⡴⢲⢀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠏⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢇⠀⢻⡄⠀⣼⢱⠁⠀⠀⠀⠈⢉⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡞⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢦⠀⢻⣤⡟⡆⠀⠀⠒⠀⠀⡘⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠘⠀⠀⠀⠀⠁⠀⠀⠈⢇⠀⢻⡴⠀⠀⠄⠀⠀⢠⠁⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣳⡄⠀⠾⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⢇⠀⠀⠀⠀⠀⠆⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⡀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠘⡄⠘⡄⠀⠀⠀⠸⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢣⠉⠀⢐⠃⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡘⣄⠀⠀⠀⠀⠰⠀⠀⠀⠈⡄⢸⠀⢀⠋⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡜⢦⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡌⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠑⢄⡀⠀⠀⠀⢀⡾⠀⡸⠰⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡙⠓⠒⠚⠋⣀⡴⢣⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠿⣈⡡⠶⣣⡾⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣳⣶⣶⣶⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠟⠻⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣶⣶⣶⣆⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡴⢲⡴⠋⠉⠓⠋⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠙⠞⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠹⡄⠀⠸⡇⠀⢀⣤⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⢻⡀⠀⢘⡞⠁⠀⠉⠉⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠓⠀⢸⡇⠀⠀⠀⠀⠈⢳⡀⣠⢦⡀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠘⣇⠀⣤⠀⠀⠀⢸⣷⣧⡆⣷⣄⠀⠀⢨⣧⡀
⠀⠀⠀⠤⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠈⠀⠀⠀⡾⠀⠉⠀⠀⠀⠈⠛⠡⠴⠃⠈⢧⣠⠏⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣨⠴⠒⠛⠛⠛⠓⠒⠦⠤⣄⣀⡀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣨⠟⡶⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⢤⣧⣀⠀⠀⢀⡴⣣⠞⠁⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠛⠚⠷⠶⠤⣄⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⣏⠉⡿⠉⠉⠉⠀⠀⠀⠙⢦⣀⣀⣤⠰⡄⣶⢼⢧⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡇⣷⣿⡾⢮⢧⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠿⠋⠀⠈⠛⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡿⠛⠛⠓⠒⠒⠲⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠛⠛⠳⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠉⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠇⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠉⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢼⡃⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⢶⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡤⢦⣀⣀⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⠶⠶⠶⠾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⡾⠿⠛⠛⠻⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠛⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠾⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠛⠉⠀⣤⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⣀⡀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠀⢀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣾⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣴⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⠿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄
⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣀
⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦
⠀⣸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠶⠶⠶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠎⠁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⣿⡗⠉⠉⠹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁⠀⠀⠈⠹⣦⡀⠀⠀⠀⢀⣤⣼⡇
⣿⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⢀⠿⠀⠀⠀⠀⠀⠀⠙⣧⡀⢀⣴⠋⠀⢹⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠙⢢⣤⣤⣤⣤⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠲⠖⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⢀⡼
⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁
⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠃
⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠷⢶⣦⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⡶⠾⠿⠿⠛⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠛⠋⠙⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣏⣀⣀⣀⣀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡀⢸⣿⣿⣹⣿⣧⠀⣀⣠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⢻⣿⣿⢻⣿⣿⢸⣿⡿⣻⣿⣿⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⢿⣿⣧⢿⣿⡟⣿⣿⣻⣿⠇⣿⣿⢣⣿⣿⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⣠⡿⣿⠮⠟⠁⡘⠉⠈⠋⠀⠻⢣⣿⠿⣁⢺⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣫⣿⠋⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣟⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠴⠛⠙⠛⠿⣶⣦⠀⠀⠀⠀⣴⣶⠿⠛⠋⠛⠦⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢯⡀⠀⣀⣤⣶⣶⣄⡄⠁⠀⠀⠀⠀⠈⢠⣠⣶⣶⣤⣀⠀⢠⡹⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣥⢺⡇⠀⠙⠣⠴⣿⠭⠿⠆⠀⡁⠈⠀⠰⠿⣭⣿⡦⠜⠋⠀⢸⡗⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣂⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡾⢐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢂⠀⠀⠀⠀⠀⠠⡵⠦⢄⣠⠴⢮⠄⠀⠀⠀⠀⠀⠰⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡀⠀⣨⠀⠀⠀⠀⠀⠀⢀⣠⣤⣠⣄⡀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣆⠀⠀⠀⠰⠶⠫⠤⠤⠤⠤⠽⠲⠆⠀⠀⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⡀⠀⠀⠈⠦⣄⣀⣀⣠⠴⠁⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⠦⣄⣀⣀⣀⣀⣠⡴⠋⠀⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢀⣀⣤⣤⣶⡾⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢷⣶⣤⣤⣀⡀
⢶⡿⠿⠟⢛⡿⣿⣧⣸⣿⣿⣿⣿⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣼⣿⢿⡙⠻⠿⢿
⠘⣷⡘⢃⡞⠺⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⢉⡟⠻⢳⡤⢠⡶⢶⡄⢤⡾⠛⢻⡍⣿⢾⣿⣿⣿⣿⣿⣿⣿⠛⠗⢳⡘⢃⣾
⠀⠈⢿⣌⢡⡍⢉⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠳⣤⣼⢧⣽⡦⢠⣿⡼⣧⣤⠞⠉⠁⢸⣿⣿⣿⣿⣿⣿⣿⡉⢩⡌⣡⡿⠁
⠀⠀⠀⠻⣦⣀⡮⠔⠚⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠈⠑⠚⠁⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⡗⠂⢵⢀⣴⠟
⠀⠀⠀⠀⠈⠻⣄⠛⢀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣀⣠⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇⠛⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣦⣿⣿⣿⣧⣿⡇⠀⢸⣟⡙⢋⢻⣿⡻⢟⣿⡿⡙⠫⣹⣇⡈⢘⣿⣿⣿⣿⣿⣷⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⣧⣿⡏⠠⠤⠃⣤⠈⣌⡕⢁⡈⢪⣡⠁⣤⠈⠲⠅⠙⣿⣯⣽⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠛⠷⠶⣤⣤⣄⣙⣀⣈⣁⣀⣉⣠⣤⣤⠶⠾⠛⠋⠁⠈⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣯⣋⣉⣿⣀⣬⣉⣭⣉⣸⣟⣩⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠶⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⢤⠀⠀⠀⠈⠀⠀⢰⠀⠀⠀⠁⠀⠠⠆⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠐⠄⠀⠀⠀⢀⣨⡀⠀⠀⠀⠀⡄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠘⣗⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠀⢀⣄⠁⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢀⣴⡄⠀⡠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠁⠙
⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⣀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡠⠤⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡀⡂⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣾⡿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠇⠀⠄⠀⠀⠠⠤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠀⠈⠀⢄⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠀⠀⠀⠒⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠘⠗⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠘



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⣀⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠊⠁⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢀⡔⠉⠉⠙⢦⠀⠀⠀⠀⠀⢀⠖⠉⠉⠓⣄⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⡞⠀⢀⣴⣦⡄⣧⠀⠀⠀⠀⡏⣠⣤⣄⠀⠘⡆⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣿⣿⣿⣼⠀⠀⠀⢸⣷⣿⣿⣿⡇⠀⡧⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢧⠀⢿⣿⣿⣿⡟⠀⠀⠀⠈⣿⣿⣿⣿⡇⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠙⣋⡝⠁⠀⠀⠀⠀⠙⣝⠛⠋⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⢻⡿⠛⠿⠿⣿⣿⡿⠿⠿⣿⡟⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣀⣀⣠⠴⠋⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠒⠒⠒⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢍⠑⠦⣄⡰⠇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⡯⠉⠉⠉⠙⠒⠚⣒⣀⣈⡁⠉⠹⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢄⢀⡤⠒⢤⣀⡠⠔⠊⠉⠁⠀⠨⣧⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠠⣄⡀⠀⠀⢀⡤⠖⠦⠀⠀⣽⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠤⡀⠀⠀⠀⠀⢀⡠⣄⠀⠀⠀⡯⢀⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⣠⠤⠖⠋⠉⠁⢸⡴⠛⠁⠀⣀⠘⠟⠀⠀⠀⠀⠀⣇⠞⠥⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⡔⢉⡇⣀
⠀⠀⢠⡶⠋⠀⠀⠠⠧⣄⣀⣸⡅⠱⠜⠛⠛⣋⡳⠀⠀⠀⠀⠀⢉⡴⠒⠋⠀⢈⡉⠉⠙⠒⠒⠶⠦⠤⠶⡖⠒⢲⠀⢀⠞⢉⡼
⠀⠀⠈⢧⡀⠀⣀⡀⠖⠀⠀⢀⢡⠀⠀⠠⢥⣞⡀⣀⣀⣀⣠⠴⠋⠑⠒⠢⢤⣀⡀⠀⠐⠃⠀⠀⠀⠀⠋⡇⠀⢸⡄⠀⣴⠋
⠀⠀⠀⠀⠈⠑⠲⠤⢤⣍⣀⠘⣆⠢⣒⡖⠉⠀⠈⠉⠉⡵⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠢⣄⡀⠐⠖⢠⡧⠤⠼⠢⢄⣈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢩⠏⠙⠋⠁⠇⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠚
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢁⡏⠀⠀⠀⢰⠸⢤⠀⡰⠁
⠀⠀⠀⠀⠀⣜⠉⠉⠾⡄⡸⠀⠀⠀⠸⣉⠀⣠⢇⠃
⠀⠀⠀⠀⠀⠪⠤⡠⠄⠃⣇⣀⠀⠀⠀⠈⢚⣁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣭⣙⣍⣭⣭⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⡆⠈⠑⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢱⠀⣴⠚⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢇⠈⠍⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢆⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣜⣦⣠⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⡎⣧⠱⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣡⠔⠚⠷⡝⡀⡇⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⠁⠀⠀⠀⠈⠗⠁⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠭⣚⠳⠤⣄⣀⣠⠤⠴⢚⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠲⠖⠒⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣾⣿⣿⣷⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⣿⠿⢿⣿⡟⠿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠞⠁⠀⠀⠙⢿⣷⣦⢭⣑⣒⡾⠤⣬⣙⠮⡻⡿⣿⣿⣿⣿⣿⣿⣭⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠃⠀⠀⠀⠀⠀⠀⠈⠛⠳⣶⣤⣍⣙⡓⢮⣿⣾⣾⣾⣿⣿⣿⣿⣿⣿⣿⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⠀⠀⠉⠛⠛⢿⣿⣿⣿⣿⡛⠛⢻⣿⣿⣿⣿⣟⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⣴⡿⠛⠁⠀⠀⠀⠀⠀⣀⠤⠐⠛⣿⣿⣿⣷⡀⡀⣿⣿⣿⣿⣿⡣⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣤⠶⠆⠀⡤⠂⠀⠒⢿⡿⠛⠚⠁⠀⠀⠀⠀⠸⡏⠹⣿⢻⣶⣿⣿⣿⣿⣿⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡤⢤⣀⢤⣠⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣟⡧⠔⠏⢺⠉⢢⡀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢿⢹⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣇⡇⠀⠠⠘⠁⠀⠉⠒⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡱⠢⠤⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⠘⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠢⠀⢀⣠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡆⢟⢆⠀⠀⠀⠑⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠸⢻⣿⣿⣿⣿⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⡈⢧⡀⠀⠀⠀⠠⡀⠀⠀⠀⣠⠐⠁⠀⠀⠀⠀⠀⠀⠀⢻⡀⢘⠻⣿⣿⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢥⠈⠳⣄⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠘⠹⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣣⠀⠀⠀⢸⣂⣀⣀
⠀⣀⠤⠤⠀⢶⡧⡈⣻⣵⠞⠙⠉⠀⠀⣀⣭⣵⣦⣀
⢸⡱⠉⠀⠀⢹⡿⡞⡏⠀⡀⣄⠀⢤⣀⡈⠉⠉⠉⠛⠷⣦
⠀⠙⣆⠀⢀⡞⣁⣻⡷⠶⠿⠿⠾⢦⣬⣭⣥⣤⣄⡀⠀⠈⢳⡀
⠀⠀⠈⢻⡟⢤⣬⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣯⢦⠀⠀⡇
⠀⠀⠀⢼⡌⢾⡏⣠⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡇⠀⡟
⠀⠀⠀⠘⢞⣻⢀⣀⣀⣀⠉⠃⠀⠀⠐⠚⠋⠉⠙⢺⣿⣧⡞⠁
⠀⠀⠀⠀⠘⢹⠈⠛⢍⡿⠁⠀⠀⠀⠀⢐⣟⢻⡶⠀⣿⡏⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢸⠀⠉⠀⠀⠀⠀⣴⠟⠋⠉⠉⠓⠦
⠀⠀⠀⠀⠐⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢆⠀⢠⡴⠒⠚⠃⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⡀⢀⣀⠀⠀⠀⠀⠲⣠⠉⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⣼⣆⠀⠀⠠⠙⠛⢋⠀⠈⢢⠀⠐⠀⠀⠀⠘⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⢀⣼⣟⣿⡇⢄⠀⠀⠀⠀⠠⡄⠘⠀⠀⡀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⣾⠸⠘⡿⣇⠀⠁⠒⠀⠒⠃⠉⠀⠀⠀⠃⠀⠀⠉⠈⠉⠉⠢⢤⣀⣀⠤⠒⠤⠤⠴⠶⠛⠁
⠀⠀⠀⠀⣿⡀⣀⠈⠚⢧⡀
⠀⠀⠀⠀⡈⢿⠂⠙⠳⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠠⠐⠂⠁⡀⠈⣷⡄⠀⣲⡏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠌⠂⠂⠄
⢀⠀⠀⠂⠂⢹⠟⣡⡾⠯⣀⣀⡀⠀⠀⠄⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠄
⠨⠌⠀⠐⠀⠘⠦⡽⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠡⠁⠠⠀⠀⠈⠀⠀⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⢰⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠠⣦⣄⡀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠛⠁⠹⣇⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⢿⣿⣧⡀
⠀⠀⠀⠀⠀⢸⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⡄⠀⢀⡄⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣯⠉⠉⢠⡄⢠⡄⠀⠀⠀⠘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠀⠀⢠⡄⢠⡄⢠⡄⠀⠀⣀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⣄⠀⠀⢠⡄⢠⡄⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣼⡷⠀⠀⠀⠀⢠⡄⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣷⣤⣤⣤⣤⣤⣤⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⠛⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠟⠛⠀⠀⠀⠀⠀⠘⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣶⡀⠀⠀⠀⣠⣤⣀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠆⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣴⡾⠟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠻⣦⣀⣀⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⣧⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⢸⡇⠀⠀⢻⡄⠀⠀⠀⢀⡾⢿⡇
⣰⡟⠛⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⢠⣿⠁⠀⠀⠘⢷⡀⢀⣴⠟⠁⢺⡇
⣿⣀⠀⠀⠀⠸⣷⣤⡀⠀⠀⠀⠀⠀⠀⣿⡌⢷⣄⣾⠇⠀⠀⠀⠀⠸⣷⠟⠁⠀⠀⣾⡇
⢸⡏⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠹⣧⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠘⢷⣄⣀⣤⣶⣤⣠⣼⠏⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢀⣠⣴⡄
⠀⠀⠉⠛⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠾⠛⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢿⡇⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⣸⣇⠀⠀⠀⠀⣼⡿⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⠈⠛⠿⠶⢶⣾⠿⠃⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⣀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠰⣏⣹⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⡟⠀⠀⠀⠉⠉⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠿⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷⣄
⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣷⡄⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠋⠀⠀⠀⠀⠀⠳⠼⠃⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⣶⠒⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠙⠳⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⣾⡇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⢠⡾⠻⡄⠀⠀⠀⢿⡀
⠀⢿⡇⠀⠀⠀⠀⢾⣏⣿⠆⠀⠀⠀⠀⠀⠀⢀⡴⢦⡀⠀⠀⠀⠀⠀⣯⣨⡷⠀⠈⠳⠞⠁⠀⠀⠀⢸⡇
⠀⢸⣇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⢿⣅⣹⡗⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢶⡄⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠷⣶⣶⣶⣶⣶⣶⡶⠶⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠖⠒⠛⠛⠋⠉⠉⠛⠛⠒⠲⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠱⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⣠⠴⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⠻⠒⠉⠷⣤⣀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉⠀⠀⠀⠀⢀⡞⠁⠒⢤⣀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠴⠒⠋⠉⠀⠀⠀⣰⡷⠀⠀⠀⡎⠀⠀⠀⠀⠈⢳⡀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢄⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⣠⡆⢸⠁⠀⠀⠀⠀⠀⠀⠹⡄⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⡤⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠇⠀⠀⠀⠀⠀⠰⠋⢀⡜⠁⠈⠻⡀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢱⢀⣚⣷⠀⠀⠀⠀⢸⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠰⣾⡀⠛⠋⠀⠀⠀⢀⡟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠦⢄⣀⣀⣀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠘⢆⡀⠀⢀⣴⡟⠙⠶⠶⠶⠶⠖⠋⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⣀⣀⣀⡉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⡩⠟⠀⠀⠀⠀⡠⠚⠉⠙⣦⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠔⠛⠶⠶⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠃⠀⠀⡀⣀⡴⠃⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⣀⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⣀⡠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁⢀⡀⠤⠤⠒⠒⠉⠉⠻⣿⣭⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠀⠀⡄⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠸⣴⡞⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠱⢉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣰⣂⠈⠈⣲⣀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⠲⠮⠖⣿⡇⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⠈⠉⢫⠁⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⡤⠤⣤⣾⠶⠛⢿⣉⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠽⠿⠿⠿⠿⠿⠓⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠛⠛⠉⠉⠁⠀⠉⠉⠉⠙⠛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⣫⠅⢀⣀⣠⣤⣤⠤⠤⠤⣤⣤⣤⣄⡀⠀⠀⠀⠈⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠰⠋⢶⣿⠭⠷⠶⠒⠒⠒⠒⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⡉⠙⠀⠈⠉⠐⠚⢛⣛⡛⠲⠶⠶⠶⠦⠤⠤⠤⠒⠃⠀⠀⠀⡀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⢀⣴⠏⢀⡾⢡⢃⢀⣠⢰⣦⣙⠳⣶⣭⣭⣛⠒⠲⠦⠤⠄⠀⠀⠀⠀⢀⡴⠁⣰⠀⡼⢻⣆
⠀⠀⠀⠀⠀⠀⣾⠇⣠⡟⡱⠃⡞⡼⡟⣾⠉⠻⠷⣦⣍⣙⠻⠿⣿⣒⣶⠤⠤⠤⠤⠒⠉⣀⠴⢁⡞⠁⠈⣿⣧⡀
⠀⠀⠀⠀⠀⢰⡏⣴⢏⡞⠁⡼⣼⣿⡽⠃⠀⠀⠀⠀⠉⠛⠿⢶⣦⣤⣍⣉⣉⣛⣛⣋⣩⡴⠞⠁⠀⠀⠀⣿⠸⣷
⠀⠀⠀⠀⠀⣼⣿⠏⡞⠀⡼⣱⣿⣿⠁⠀⠀⠀⠀⠴⠚⠳⣦⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠞⠳⣄⠀⠀⣿⡆⣿⡇
⠀⠀⠀⠀⠀⣿⡟⢸⠁⣸⢡⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣿⣇⢻⣿
⠀⠀⠀⠀⠀⣿⡇⣿⠀⡏⣾⣾⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠹⣅⢻⠀⠁⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⢀⣿⢻⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠙⢿⡄⠀⢸⣿⢹⣿⡄⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⠁⠀⠀⢸⣿⢸⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠋⢧⢻⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢹⣿⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣶⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢻⣿⠀⢸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⡄⠀⠀⠀⠀⠀⠀⠀⣠⣿⣥⣾⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⠦⣄⡀⠀⠀⠀⠀⠸⣇⣠⠇⠀⠀⠀⠀⢀⣤⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡇⠀⠀⠉⠓⠲⢤⣄⣓⣂⣀⣀⣠⣤⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⢷⡀⠀⠀⠀⠀⠀⣿⠯⡈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠙⠷⣤⣄⣀⣼⠏⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢀⣿⠟⢿⣤⡀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣾⠏⠀⠀⠈⠻⣦⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣸⡏⠀⠀⠀⠀⠀⠀⠙⠺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿

⠀⠀⠀⢸⣿⡆⠀⠀⠈⣹⡇⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⠀⠀⡄⠀⢸⡇⠀⡀⠀⢿⣿⠂⠀⣴⣶⣶⣶⣦⣴⣶⣶⣶⡄
⠀⠀⠀⣼⣿⣧⡀⠀⣾⣿⣿⣆⠀⠘⣿⣿⣿⣿⡇⠀⢠⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⣶⠀⠀⡇⠀⢸⡇⢰⡇⠀⢸⡟⠀⠀⡿⠉⠙⡟⠉⠙⡟⠉⠙⡇
⠀⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⠀⠀⢘⣿⡀⠀⠀⠀⠀⠀⢾⣿⠀⠀⣿⡇⠀⣿⠀⠀⡇⠀⢸⡇⢸⡇⠀⢸⡇⠀⠀⠻⠦⠶⠿⠦⠾⠷⠦⠾⠃
⠀⠀⠘⣿⣿⣿⠃⠀⣿⣿⣿⣿⠀⠀⢿⣿⣿⡿⠀⠀⢸⣿⡇⠀⢸⣿⡆⠀⢺⣿⠀⠀⣿⡇⠀⣿⠀⢠⣷⠀⢸⡇⢸⡇⠀⢸⡇⠀⠀⠐⠀⠐⠒⠒⠒⠒
⠀⠀⠀⢻⣿⡿⠀⠀⢿⣿⣿⡟⠀⠀⢸⣿⣿⡇⠀⠀⢸⣿⠇⠀⢸⣿⠇⠀⢸⣿⠀⠀⣿⠇⠀⣿⠀⠐⣿⠀⢸⡇⠸⡇⠀⠀⡇⠀⠀⠚⠂⠚⠛⠛⠛⠒⠒⠒⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⣾⣯⡀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⠀⠀⠀⠀⠀⢀⣿⣿⡟⠃⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⣿⡟⡇⠀⠀⢠⣤⡹⢿⠃⠀⢰⣿⣦⣤⣤⣴⣾⣷⣶⣄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⡄⠀⠀⠀⠀⣬⣬⡉⠀⠀⠀⣿⣿⣿⣿⠿⢿⣿⣿⣿⡇
⠀⠀⢠⣿⣿⡇⠀⠀⠼⠿⠇⠀⠃⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⢸⡇⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣷⠀⠘⢠⣶⣿⣿⣷⣤⠀⠀⣿⣿⣿⣿⣶⣾⣿⣿⣿⡏
⠀⠀⢸⣿⣿⡗⠀⢁⠰⠶⠀⡇⠈⠀⠀⠈⠻⢿⡿⣿⣿⣿⣿⣿⠀⠀⣼⣇⠀⠀⣼⣷⠀⠈⡏⠀⢠⣿⡈⠄⠀⢸⣿⣿⣿⣿⣿⠀⠀⣀⣀⣀⣀⠀⢀⣀⣀⣀⡀
⠀⠀⠐⠿⠿⠇⠀⠸⠦⠤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟⠁⠀⠸⠿⠿⠃⠘⠿⠿⠃⠀⠁⠀⠻⠿⠗⠀⠀⠘⠿⠿⠿⠿⠟⠀⠀⢿⠿⠿⠿⠷⡿⠿⠿⢿⡇
⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⢀⣄⣄⣄⣤⣠⣀⣠⡀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⢹⣟⢸⠠⠀⠆⠀⠀⠀⣿⣿⣧⣤⣼⣿⢿⡃⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣴⡿⢻⡟⡝⠙⢻⣷⡀⠀⠀⠀⠀⣼⣿⣄⠀⠀⠀⠀⠿⠶⠶⠶⠶⠷⠶⠶⠾⠇
⠀⠀⢸⡇⢸⢐⠀⡇⠀⠀⢸⡏⠈⡉⠀⠉⠁⢹⡇⠀⠀⠀⣿⣿⣄⠀⠀⠀⢸⣿⢠⡿⡱⠀⢀⡿⢻⡷⠀⠀⠀⢀⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⣄⡀
⠀⠀⢸⡇⢸⢈⠀⡇⠀⠀⠘⡅⠀⠁⠀⠀⠀⢨⡇⠀⠀⡆⢡⣠⡌⢰⠀⠀⠘⣿⣿⡱⠁⢀⣾⣣⣿⡏⠀⠀⢰⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣞⣩⣶⠟⠋⡲
⠀⠀⢸⡇⢸⠘⠀⠇⠀⠀⠀⡟⠈⠉⣿⣿⣿⣿⠁⠀⠀⣇⠈⠉⠀⣾⠀⠀⢀⠈⢻⣿⣶⣾⣿⣿⠋⣀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡿⠛⠁⢀⣤⡇
⠀⠀⠈⠃⠘⠈⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠛⠛⠒⠛⠋⠀⠀⠈⠛⠛⠓⠒⠒⠚⠛⠛⠋⠀⠀⠈⠛⠛⠛⠛⠛⠁⠀⠀⠀⣀⠴⠚⠳⠦⣴⡋⣸⠁
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢷⣦⣄⠀⣀⡴⠛⠁
⠀⠀⢈⣩⠿⠛⠻⢯⣉⠀⠀⠀⠀⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⣾⣿⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁
⠀⢠⡟⠁⠀⠀⠀⠀⠈⢧⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢰⣿⢿⢏⣤⡄⠀⠀⠀⢸⣿⣿⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣷⣶⡆
⠀⢾⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣰⠋⢋⣼⣿⣧⡀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠘⣧⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣿⣿⣦⣤⣄⣀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠔⠁⠀⠘⣿⣿⣿⠆⠀⠀⣿⣿⣿⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠙⠲⠦⠶⠚⠋⠀⠀⠀⠀⠻⠿⠛⠤⠬⠿⠀⠀⠀⠀⠘⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠀⠀⠀⠿⠿⠿⠃⠀⠀⠀⠙⠻⠿⠿⠿⠿⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡟⡍⣿⠛⠿⠿⠿⠻⠟⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠿⠿⣿⣷⣤⢃⣔⣥⡶⠶⢄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⢈⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠶⢦⡀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣴⣿⣿⣿⡟⣩⡦⠀⠈⣿⣿⣿⡏⢛⣿⣿⣦⡙⠂
⠐⡀⠂⠠⠀⠂⠉⠀⠀⢸⢫⣾⣿⢃⠹⠠⢁⣀⠀⢸⡟⢸⠇⣀⠛⡜⢿⣿⡍⠒⠢⢄⡀
⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠈⠀⠴⣟⠫⠥⡀⠀⠀⠘⠿⣷⠀⠈⣿⣿⡄⠀⠀⠈⠑⠢⢄
⠀⠀⠙⡦⠀⠀⠀⠀⢀⠀⣿⣿⡇⠀⠀⢰⠁⠠⠀⡀⠀⢀⡀⠘⠈⠀⢠⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⣸⠃⠀⠀⠀⢊⢊⠴⡿⢿⠃⠀⠀⠸⡀⠀⠈⠁⣀⠈⢀⡜⠀⠀⣼⣿⣏⠈⡀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⢠⡏⠀⠀⠀⠀⢠⠏⠀⢢⣄⠀⠀⠀⠀⠒⠛⠁⢤⡠⠟⠛⠈⠀⠀⢻⣭⢌⢧⢙⠀⠀⠀⠀⣰⠀⠀⣷
⠀⣸⠁⠀⠀⠀⠀⠘⣆⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⡼⠋⢊⡞⠀⠀⠀⠀⢠⡏⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠈⠲⠶⠾⢧⣀⣀⠀⡴⠖⠒⠒⠚⢁⣀⡤⠖⠿⠤⠶⠋⠀⠀⠀⠀⢀⣿⠁⠀⢀⡇
⠀⣷⠀⠀⠀⠀⠀⠀⢀⣠⣐⠤⠑⣂⡥⠾⢶⣶⠂⢠⠞⠛⢿⣿⠀⠐⠢⢄⡀⠀⠀⠀⠀⢌⠆⠀⠀⢸⠃
⠀⣿⠀⠀⠀⠀⠀⠸⠶⡋⠄⠂⠁⡳⠂⣸⣦⡁⢠⣮⣶⠒⠒⢃⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢀⡟
⠀⢻⡀⠀⠀⠀⠀⠀⣁⠀⠀⡠⢊⠝⣼⣿⠃⣿⡟⠸⣿⡆⠠⡀⠳⡀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢀⡾⠁
⠀⠘⣇⠀⢀⠂⢁⡎⠀⠈⠈⣀⡏⣼⣿⡏⢠⣿⣧⠀⢻⣿⡀⢱⡀⠘⣦⠀⠀⠀⠀⢠⠁⠀⢀⡾⠁
⠀⣀⠿⠶⠺⡀⠈⣇⠀⣆⠀⡏⠉⠉⠉⠓⠺⠿⢟⠒⠉⠉⠉⠉⠉⠑⠚⠢⢤⣀⠀⠘⠀⢠⡟⠁
⠈⠀⠀⠀⠀⠉⠒⠉⠐⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣴⠏
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶