Systematics Emojis & Symbols

⠀⠀⠀⠀⠀⢀⠀⢀⣤⣤⠀⠀⢠⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀
⠀⠀⠀⠀⠀⢹⡇⠀⠙⠋⠀⠀⠸⢃⠀⠸⠟⠱⠀⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢠⡆⠈⠃⠁⠀⠠⡶⠆⠀⠀⠈⠻⡿⠋
⠀⠀⠀⢠⠀⠘⠀⢀⣾⣷⠀⠀⢠⣿⡇⠀⢀⠀⠀⣿⠀⠘⠇⠀⠀⠀⠀⡀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⢰⡄⢈⣿⣿⡆⠀⢸⣿⡇⠀⠘⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠂⠲⠀⠀⠲⣾⠆⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠃⠈⣿⣿⠀⢠⣾⣿⠇⠀⣾⠒⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢰⡿⠛⠁⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣠⠀⡀⠀⢀⠀⣆⣰⠀⠀⠀⣀⣀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⣠⠀⠀⣾⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠻⠏⠽⠀⠀⠀⠙⢓⠒⠂
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠛⠋⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠄⠀⠀⢀⣤⣿⣹⣶⣤⡀⠀⠀⡀⠀⠀⡿⡐⠁⠀⠀⠀⠘⢾⣷⣦⡏⠀⠀⢠⡤⣀⡀⠀⠀⠄⠀⠴⠀⠀⠈
⠀⠀⠀⠀⣾⣦⡦⠀⠀⣿⣿⣿⢛⡼⠀⠀⠛⠛⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠿⠧⠸⠟⠀⠰⠂⠤⠀⠄⠀⠐⣆
⠀⠀⠀⠀⠘⠛⠁⠀⠐⠉⠾⡿⠋⠁⠀⠀⠀⠀⠀⣸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠘⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⣰⣦⠀⢀⣄
⠀⠀⠀⠀⠀⠟⠃⠀⠀⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠢⠄⠀⠀⠀⠀⠀⠐⠀⢉⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⢠⠀⠈⡻⠀⠀⠀⢸⣦⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠐⢦⠀⠀⣷⣤⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⢠⠬⠀⠸⠉⣏⡹⠀⠘⣷⣠⠀⠀⢀⡄⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠩⠟


⠀⠀⠀⠀⠀⡀⠐⠄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⣤⠐⢄⠀⠀⢰⡄⠀⣼⠇⠐⢷⠆⠀⣤⣤⠀⠐⠆
⠀⠀⠀⠀⠶⡍⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡁⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⠀⠀⢣⡏⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠈⠐⠦⠖⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠰⡶⠶⠆⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡄⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣤⡆⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠘⠛⠏⢀⣤⣀⣰⣂⠀⢲⣿⣿⠆⠀⠂⠀⠐⢻⡿⠃
⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢲⡆⠀⠀⠈⠠⠤⣤⠄⠀⠀⠀⠛⠂⠀⢉⠙⠿⠟⠀⢀⠙⣁⠀⣥⣭⡉⢀⣠⣵⣄⣆
⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠠⢤⣤⠀⠀⠀⠀⣄⣴⡇⠀⠤⠆⣤⡟⠀⠀⠴⠁⠀⢽⣿⠂⠸⠽⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⣀⠀⠀⠀⠀⣶⣶⠀⠀⠀⠀⠸⠏⠇
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣸⡆⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⠲⡄⢸⣟⠀⢺⣀⠀⢠⣀⣄
⠀⠀⠀⠀⠹⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠈⠋⠀⠈⠋⠀⠈⠙⠋
⠀⠀⠀⠀⣀⡀⠀⠀⠀⢠⣀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⣴⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⠀⠀⢰⠆
⠀⠀⠀⣠⣼⣧⡀⠀⠀⠀⠉⠀⠀⠈⠛⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠸⠟⠀⠀⠀⠄⠀⠀⠈⠰⢄⣠⡄⠀⢰
⠀⠀⠀⠈⠹⢿⣇⠀⠀⠂⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠉⠀⠀⣦⠀⠀⠘⢷⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠀⠀⡀⡐
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠰⠆⠘⣿⠃
⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⣶⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇
⠀⠀⠀⣤⡄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣿⣿⣿⣿⣿⠃
⣿⣿⡇⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⣿⣿⡇⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠘⣡⡿⠀⠀⠀⠀⠀⢠⡜⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⣛⣋⢀⡀⠀⠀⡀⣘⣛⡛⣀⡀⠀⠀⣀⣈⢙⣋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠋⠛⠙⣿⠀⢸⡟⠉⠋⠉⠙⣿⠀⢸⡏⠛⠉⠋⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⡤⣤⠀⣿⠀⢸⡇⢠⣦⣤⠀⣿⠀⢸⡇⢠⡴⣤⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⣿⠀⣿⠀⢸⡇⢸⡇⣿⠀⣿⠀⢸⡇⢸⡇⢺⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⣿⠀⣿⠀⢸⡇⢸⡇⣿⠀⣿⠀⢸⡇⢸⡇⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⣦⣿⠀⣿⠀⢸⡇⢸⣧⣿⠀⣿⠀⢸⡇⢸⣧⣾⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⡀⠀⣿⠀⢸⡇⠀⣀⡀⠀⣿⠀⢸⡇⠀⣀⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣏⣿⠀⣿⠀⢸⡇⢸⣏⣿⠀⣿⠀⢸⡇⢸⣯⣿⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠆⢸⡇⠀⠈⠀⠀⣿⠀⢸⡇⠀⠈⠀⠀⣿⠀⢸⡇⠀⠈⠀⠀⣿⠀⠶⢶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠘⠛⠛⠛⠛⠛⠛⠀⠘⠛⠛⠛⠛⠛⠛⠀⠘⠛⠛⠛⠛⠛⠛⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢱⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢒⠓⣮⢻⢠⣤⠄⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣟⣛⣛⣛⣛⣻⣿⣛⣛⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣯⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⣿⢸
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⢸⣿⡷⠶⠾⠿⠷⣶⣶⣶⣶⣾⣿⣿⢸
⠀⠀⠀⠀⠙⠛⠈⠉⠉⠁⣿⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⢸⣿⣟⣛⣛⣛⣛⣻⣿⣛⣛⣿⣿⣿⢸
⠀⠀⠀⠀⢠⣅⠀⠀⠀⠀⣿⢸⣿⣟⣛⣛⣛⣛⣛⣛⠻⠿⠿⠿⡿⢸
⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⣿⢸⣿⣟⣛⣛⡛⢛⣵⢾⣛⣻⣟⣛⡳⣮⡀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⡿⣱⢟⣵⣿⣿⣿⣿⣿⣿⣦⡻⣦⢀⣀⣀⢠⣥⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⢣⡏⣾⡏⣿⣿⢻⣏⢿⠟⣿⣷⢹⡆⠀⠀⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣟⣟⢸⡇⣿⣧⠿⡿⣸⡿⣣⣜⢿⣿⢸⡇⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣇⢿⡜⣿⣿⣾⣿⣿⣿⣿⣿⢋⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣭⣭⣭⣭⣥⡻⢮⣝⡻⠿⠿⢟⣫⡵⢟⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠙⠛⠛⠋⠉⠀⠙⢣⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣫⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⡉⢑⡢⣄
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢹⡉⠰⠌⢂
⠀⠀⠀⣠⠚⡉⣭⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠈⢄⠘⠁⠀⠀⠈⡆
⠀⠀⡐⠁⠲⣬⠔⠣⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⠀⠀⠐
⠀⢠⠱⠆⠀⢸⡀⠀⠀⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⡀⠀⢰⠎⠀⠀⢀⠰
⠀⡌⠀⠀⠀⢻⠁⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⡀⠀⠀⢐⡄⢘⣀⠀⠀⠈⠀⡄
⡰⢡⠔⠂⠀⢸⠀⠀⠀⣰⢄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⠹⠏⠁⠈⢙⠀⠀⡄⠀⠘⡇⠀⠀⠀⠀⢀⣀
⣇⡀⠀⠘⢄⣸⠀⣸⠆⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠙⠀⠀⠀⠀⠈⠀⡰⠋⠚⠀⠇⠀⣀⡴⠊⢀⡀
⡏⠃⠀⠀⠀⢹⠊⠈⠢⠔⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣄⠀⠀⠀⢠⠀⠀⠳⢄⠀⠀⢰⠋⠀⠀⠀⠀⣀⠞⠁⢂⡤⠋⢀
⠃⠐⣀⠄⠀⢨⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⢈⠱⣄⠀⢸⠀⢂⠀⠈⠢⣀⢀⠀⠀⠀⠠⠀⠉⠀⡠⠾⣠⠀⡈
⢰⠀⠈⠢⣀⢸⠀⠀⠀⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠬⣅⠘⣆⠈⡄⠀⠡⡄⠀⠀⢹⠂⠀⠀⢀⡀⠴⢌⠀⠀⠀⠠⠁
⢸⠀⠈⠉⠀⠻⢀⡤⠊⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀⠸⠆⢉⢦⠁⠀⠀⠈⠢⡀⢸⠀⠀⡠⠊⠀⠀⠀⡆
⢸⠀⠀⠀⠀⠀⡋⠀⠀⢀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡋⠒⠙⡎⠀⠀⠀⠀⠀⠀⠀⢹⣥⠊⠀⠀⠀⠀⡰⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠈⠢⠄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⠁⠀⣐⠋
⠀⡄⠀⠀⠀⢀⡷⠊⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣐⣀⠤⠐⠉⠒⢄⡀⠀⠀⡇⠀⠀⡀⠤⠆⢠⠆⢨
⠀⠸⡀⠀⠀⠀⠇⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠙⡀⠀⣀⠀⠀⠙⠢⣀⣁⠔⠋⠀⠀⠀⡌⠀⢸
⠀⠀⠱⡄⠀⠀⡆⢠⠃⠀⠀⠀⠀⠀⠀⠀⣠⠊⣉⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⣠⡀⢳⠀⠀⠐⠀⠀⠀⠨⠁⢀⠀⠀⠀⠔⠀⠀⡀
⠀⠀⠀⠹⣆⠀⠇⡌⠀⠀⠀⠀⠀⠀⠀⢰⠃⠐⣭⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠈⠉⠣⣠⠖⠉⠐⠂⠤⡇⠉⢀⡠⠊⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠂⢂⡉⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⡀⠀⠙⢄⠀⠀⠀⠀⣇⠐⠀⠀⠀⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠈⢃⡔⠁⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡁⠀⠀⠈⠱⡀⠀⠀⡏⠀⠀⠀⠠⠂⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⢢⡉⠈⡤⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠒⠉⠄⠀⡇⢀⠔⠈⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠁⠀⢁⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠘⠀⣡⠂⠀⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⢸⢀⡘⢠⡄⠀⢨⠖⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⢻⠃⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢠⡈⠒⠊⢰⠆⠈⠃⢠⡁⡣⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠸⠂⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠈⠒⠠⣐⠀⠀⡷⠀⢱⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠖⠛⢄⠀⠃⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠃⠀⠀⠹⡇⢠⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢲⠀⠀⠀⡗⠁⠐⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢀⡙⠢⢠⡇⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢈⠀⠀⠈⢃⠀⠌⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢢⡄⠀⢼⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢸⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠸⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢳⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⠀⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢀⠈⣈⣀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⣉⠑⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠹⠊⡡⠌⢣
⠀⠀⠀⠀⡴⠞⢻⡛⠓⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠡⠀⠘⡋⠀⠀⠁⢇
⠀⠀⠀⡜⢡⡄⢚⣹⡆⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⡄⠀⠙⢦⡁⠀⢀⠴⣼
⠀⠀⢸⣀⠀⠙⠋⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢃⠀⠀⠀⣀⠔⠁⠀⠀⣇
⠀⢠⠁⠋⠀⢠⡴⠀⢀⡀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡜⠀⠀⠘⠄⠀⠘⠁⠀⠀⢀⠀⣿
⠀⡾⠀⢀⠀⠀⡂⠀⠀⠀⠀⢱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠊⠐⠤⣀⡀⠸⠦⣠⡗⠀⠀⡘⠀⣩
⡌⠀⠋⢡⠀⠀⡇⠀⠀⢀⡼⠇⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢰⡄⠘⠀⠀⠀⠀⡅⠀⢀⢁⡌⢡⠀⠀⠀⠀⢀⡴⠶⠶
⣧⣄⠀⠀⠳⣄⡇⢀⣶⠋⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢡⠀⠀⠀⠀⠀⡄⠠⠉⠉⡆⣴⠀⢀⡠⠖⠉⢠⡄⢀
⡇⠉⠀⠀⠀⠈⠗⠁⠈⠳⠞⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢄⡀⠀⠀⠸⡄⠀⠘⠳⡄⠀⠀⠀⡏⠀⠀⠀⡇⣿⣠⠏⠀⢺⡤⠋⢀⡍
⠇⠀⢂⠀⠁⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⢆⡙⢆⠀⠠⡇⠐⡄⠀⠘⠢⣀⠀⡇⠀⠀⠀⡇⠈⠁⠀⡀⠾⣀⡄⣠
⢰⠀⠈⠣⣀⠀⠀⠀⠀⠀⡌⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠣⢸⣅⠘⢆⠀⢡⠀⠈⠢⡀⠀⠀⠙⡇⠀⠀⠀⣇⣀⢴⡁⠀⠀⠀⢠⠁
⢸⡄⠀⠂⠉⠙⠆⠀⡠⠊⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠙⡄⠙⢦⡼⠀⡇⠀⠑⢄⠀⠀⡇⠀⢀⠔⠋⠀⠀⠘⡀⠀⠀⡌
⢸⠀⡀⠀⠀⠀⢰⠊⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠗⠤⠛⣤⠀⠀⠀⠃⠀⠀⠀⠑⠚⡇⡠⠋⠀⠀⠀⠀⢠⠁⠀⢀⠃
⠘⡀⠰⡀⠀⠀⢸⠀⠀⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠙⢄⢀⠄⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢠⠁⠀⢀⡾
⠀⣇⠀⠙⢦⡀⢸⢀⡠⠊⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢂⠀⠀⠉⣢⣄⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡆⠀⡔⠻⡇
⠀⢻⡀⠀⠀⠉⢾⠁⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠴⢶⠞⠁⠀⠀⠉⠲⢄⠀⠀⡇⠀⢀⠤⠘⠉⠁⠸⠀⠀⡇
⠀⠀⢧⠀⠀⠀⢸⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⢢⡷⠈⠀⠀⠀⠀⢠⠃⠀⢀
⠀⠀⠈⢧⠀⠀⢸⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⢀⠎⠁⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠀⠀⡇⢀⡀⠀⠀⠀⠁⠀⠀⢾
⠀⠀⠀⠈⢳⡄⠸⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡎⠄⠀⣤⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢄⡀⠈⠉⠉⠓⠲⡏⠀⠀⡠⠊⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⣂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣇⡴⠊⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⠀⠘⣇⡴⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠈⠢⡀⠀⠀⡟⠀⠀⠀⢀⠔⠁⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠳⢀⡏⠀⣀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠄⠐⠳⡀⠀⡇⠀⡠⠒⠉⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢠⠀⠘⡇⠀⠉⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠙⠀⣇⡔⠁⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⢢⢸⡇⠀⠀⣤⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⢠⡟⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⢉⠳⢄⣠⠆⢐⠀⠸⡇⢀⠘⢃⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠘⡇⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠘⠳⣄⠁⠀⠙⠀⠀⣇⠚⠄⠘⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠁⠈⢰⡆⠀⠘⡇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢘⠃⠙⢦⡀⡇⠀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠉⠀⠀⠀⢹⡇⡰⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⣇⠐⠀⢸⡏⠀⠀⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠿⠂⢸⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠘⡇⠀⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠐⣤⠀⠀⣧⠌⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠘⡀⠀⢿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠡⡀⢸⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠁⣼⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢹⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⠘⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣁⣀⠀⠀⣀⡀⣠⣴⠖⢒⠒⢦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠇
⠀⠀⠀⠀⠺⠏⠉⠉⢛⠯⢤⠟⠋⠋⠋⠙⠉⠉⠛⠉⠉⠛⣩⠟⠋⠁⢸⣿⠁⠀⠈⠙⠂⠹⢿⣿⣿⣦⡀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠒⠰⠦⠤⠄⠀⢀⡀⠀⡀⠀⠀⠀⡀⢀⣼⠟⠉⠀⠉⠈⠛⠶⠄⠀⠀⠀⠀⠸⠿⠿⠿⠿⡄⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡖⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋⠁⠀⢀⣀⣀⣀⣀⣠⣤⣤⠀⠀⣀⣠⣠⣋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⢀⣆⡀⠀⠀⡀⠀⢀⠀⠈⠙⠛⠓⠀⠄⠀⠀⣏⠉⢹⣿⠉⠀⠒⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠻⠋⣰⠇
⠀⠀⠀⠀⠀⡜⠱⢄⠈⣠⡶⢼⣿⣧⠀⠠⠰⣷⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⡄⠀⠀⣾⣿⠿⣿⣣⣿⣿⣿⣿⢿⡿⢁⣠⣶⡟
⠀⠀⠀⢀⣼⣷⣄⣼⡀⡿⠤⣂⡛⠛⠀⡠⢁⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠘⣿⡇⠀⠋⠁⠀⠀⠀⠀⠀⢠⣌⡙⠿⣿⣄⠙⠻⣿
⠀⠀⠀⠈⠙⠛⠛⠛⠓⡆⠀⠀⠀⠠⠂⢀⢸⣯⠕⠈⠀⠄⠀⠀⠀⠀⠀⢠⠃⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡀⢽⣷⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣀⢣⠟⠛⠉⠀⢀⣨⣤⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⢸⣿⡆⠀⣀⣠⣄⣀⠀⠀⠀⠀⣿⠂⠈⠉⠃
⠀⠀⠀⠀⠀⠶⠞⠋⠁⠙⠀⠀⠀⠀⠀⣹⣥⣞⢡⣾⣷⣦⣄⡀⠀⡐⠛⠛⡉⠀⢀⣼⣿⣿⣶⠸⣿⣿⣿⠆⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣴⣥⣿⣿⣿⡄⢿⣯⣿⡿⠟⢷⣮⣷⣮⣴⠾⠿⢟⣻⣿⠏⣸⡟⠛⠁⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⡿⣿⣿⣟⠛⠙⢿⣦⡌⢁⣀⣀⣤⣙⡿⣟⢿⣤⣀⡀⠉⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⣰⣶⣖⣾⠝⠟⠩⠀⢸⡟⠋⠀⠀⢀⣿⠑⣿⣿⣟⣿⢸⣏⣿⠘⣿⣿⣿⡿⢹⣏
⠀⠀⠀⢀⣾⣿⣿⣿⣧⣗⡆⣰⣶⡏⡰⠀⠀⠀⢰⣻⣧⡙⠿⠿⢿⠀⠙⠃⢰⠾⣿⠟⢡⣾⣿⠂⠀⠀⠀⠀⠀⠀⠀⠦⡀
⠀⠀⢀⣿⣿⣿⣿⣿⢻⡿⣿⣿⣿⣷⡷⣀⠀⠀⠈⠳⣟⣿⣶⣶⣤⠀⠀⠀⢠⣴⣶⣶⣿⡿⠃
⠀⠀⠘⠻⠿⢿⣿⣿⣿⣿⣿⣯⣿⣿⠞⠒⠛⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⣿⠿⣿⣶⡦⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠊⠉⠉⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⣀⣄⣤⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣰⠞⠉⠀⠀⠘⢦⡀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⠀⠀⡀⠀⣀⠘⢩⡄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣐⣤⠶⠛⠐⠛⢦⡽⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⠀⠀⠀⠀⠀⠀⣵⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠻⠁⢾⠀⠀⠀⠀⠀⢠⠖⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶
⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣾⠁⠀⢚⠀⠀⠀⠀⠀⢸⢉⡄⢛⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣹⠉⠀⠒⠉⠀⠀⠀⠀⠀⠘⣥⠐⠈⣥⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⠆⠀⠉⠁⡹⠀⠀⠀⠀⠀⠀⠷⠊⢁⠤⡆⠀⡀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣷⠐⠂⠀⣌⠁⠀⠀⠀⠀⠀⠀⠐⣫⡀⠒⡓⠀⠅⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡀⢈⡁⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠉⣥⠀⢀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠁⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡄⢰⠇⠨⠃⠀⠀⠀⠀⠀⡄
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⢨⠄⠌⢹⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡖⢺⡇⠀⠀⠀⠀⠀⠀⠀⠃
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⢲⠒⠒⢺⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡃⢘⡃⠀⠀⠀⠀⠀⠀⠀⢰
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⢉⠉⠉⢩⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⢉⣉⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⢩⣅⠄⠀⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠠⡄⠀⠐⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠐⠂⢩⡄⠀⠀⠀⠀⠀⠀⠀⢰⣏⠉⠐⠇⠀⠃⠀⠀⠀⠀⠀⠀⠀⣆
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢠⠐⢁⠄⠙⣄⠀⠀⠀⠀⠀⢀⠓⠀⣠⡛⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⢶⣁⠔⠁⣼⠀⠀⠀⠀⢀⡌⣁⠶⠋⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢬⠉⣶⣁⠈⠀⠀⠀⠀⠸⡗⠋⢀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀⠀⠀⠀⠀⠉⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠛⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣮⣽⢷⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢮⡥⢿⡷⠊⠳⡖⢦⡄⣤⢠⡤⢲⠖⠠⠀⠀⠈⠓⠛⠀⠀⠀⠀⠀⠀⠸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣤⣀⠆⠐⠀⠀⠃⢘⠀⠈⣠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠙⠲⠂⣦⠤⡦⠼⠤⠗⠉⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⠤⠤⠤⠤⠄⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠉⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡴⠚⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣶⣶⣶⣶⣶⣶⣦⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣶⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠉⠀⣰⡟⢻⣆⠀⠉⠉⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠋⠀⠀⠀⠀⠀⣴⠏⠀⠀⠹⣦⠀⠀⠀⠀⠀⠙⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣿⠛⠻⠶⢶⣤⣄⣼⡋⠀⠀⠀⠀⢙⣧⣠⣤⡶⠶⠟⠛⣿⠿⣆
⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⢻⡆⠀⠀⠀⢀⣾⣋⣽⣿⠷⠾⣿⣯⣙⣿⡀⠀⠀⠀⢰⡟⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⣾⠃⢀⣀⣸⣷⡴⠶⢿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⢿⡷⠶⢦⣾⣅⣀⡀⠘⣷
⠀⠀⠀⠀⠀⢸⣿⠛⠛⠉⠁⢹⡆⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⢰⡏⠈⠉⠛⠛⣿⡇
⠀⠀⠀⠀⠀⣿⠹⣧⠀⠀⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⠀⠀⣼⠏⣿
⠀⠀⠀⠀⠀⣿⠀⠘⣷⡀⠀⣰⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣆⠀⢀⣾⠃⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠘⢷⣼⠏⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠹⣧⡾⠃⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢀⣼⢿⡄⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢠⡿⣧⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢿⣄⡾⠃⠀⢻⣄⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⣠⡟⠁⠘⢷⣠⡿
⠀⠀⠀⠀⠀⠀⠈⢿⣷⠶⠶⠾⠿⣷⠻⣷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣾⠿⣾⠿⠷⠶⠶⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠹⣦⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣴⠏⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠙⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣼⣿⣄⠀⠀⠀⠀⠀⠀⣠⣿⣧⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢠⡞⠉⣉⣉⡉⠉⠉⠉⠉⠉⠉⡉⢉⠉⡉⢉⡭⠭⣍⠙⡆⢸⣇⣀⣀⣰⠟⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠈⠛⠛⠋⠀⣿⢸⡏⠉⠉⠙⠦⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⢠⣤⣤⣤⣤⣤⣤⡄⠀⠀⣤⣤⣤⠄⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⣹⠶⠶⠶⠶⠾⡇⢸⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠙⣿⠁⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⣿⢰⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠰⠶⠶⠶⠶⠶⠶⠆⠀⠀⠴⠶⠶⠄⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⣠⣿⡀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⣹⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣤⠴⠛⠛⠛⠛⠛⠒⠶⣯⣸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠶⠶⠶⠶⠶⣶⠞⠋⣀⡴⠖⠛⠛⠛⠓⠲⢤⣀⠙⠷⡄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⣴⠃⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠙⣦⠀⠀⠀⠀⣠⠞⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⣸⠃⢰⠏⠀⣀⠀⠀⢀⡀⢀⠀⠀⠀⢀⠀⠹⡆⠘⣏⠉⠉⠉⢻⣄⣠⠟⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⡏⠀⡟⠀⠀⣿⠀⠀⢸⡇⠈⠳⣄⡴⠋⠀⠀⣿⠀⢿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣇⣀⣀⣀⣀⣀⡇⠀⣇⠀⠀⣿⠀⠀⢸⡇⠀⣠⠟⢧⡀⠀⠀⣾⠀⣼⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡏⠉⠉⠉⠉⠉⢻⡀⢹⡄⠀⠈⠛⠚⠋⠀⠘⠁⠀⠀⠙⠀⢠⠏⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣇⠀⠀⠀⠀⠀⠈⢷⡀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠈⠙⠛⠛⠛⠛⠛⠛⠻⣤⡈⠓⠦⣄⣀⣀⣀⣀⣤⠴⠛⢁⣴⠋⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡶⢤⣄⣈⣉⣉⣁⣠⡤⠖⠻⣄⣠⠞⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣠⠞⠁⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣤⣤⣤⣴⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠛⠿⢿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣶⠶⠶⣶⡄⠀⠀⠀⠀⠀⢀⣸⢧⣤⢰⣿⢟⣁⣀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⣀⣤⣶
⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠁⠀⣾⣩⢏⣳⣀⡶⣶⣀⡾⢧⠺⡍⣿⣱⣋⡽⣍⣳⣰⡯⡷⢮⢷⣂⣀⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣆⠀⠀⢳⣼⣧⠞⠉⣷⢿⡅⠷⣼⣋⣽⠟⢸⣧⣹⣡⣾⠘⣿⣟⣫⠿⠻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⣠⣾⣿⣿⣿⣿⣿⠿⠿⠿⠿⠋⠀⠀⣰⣿⠈⢿⡆⠀⠉⠀⢠⣾⠃⠈⢰⣿⠀⠀⠈⠈⠀⠀⠈⠙⠻⠿⠿⠋
⠀⢠⣿⣿⣿⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠀⠀⢴⣿⣿⣶⡄⢸⣿⡆
⢠⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣼⣿⣿⡅⠀⢸⣿⣿⣿⣯⣼⣿⣷
⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠗⠀⠀⢻⣿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⠶⠶⡄⢠⡶⠶⠶⠶⣦⢀⡴⠶⠶⠶⢶⡀⣴⠶⠶⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⢸⠆⠀⠀⠀⣿⢸⡇⠀⠀⠀⢸⡇⣿⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠚⠃⠘⠓⠒⠒⠒⠛⠈⠛⠒⠒⠒⠛⠁⠛⠒⠒⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⡇⢸⡛⠛⠛⠛⣿⢰⡟⠛⠛⠛⢻⡆⣤⠶⠖⠛⠛⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⡇⢸⣅⣀⣀⣀⣿⠸⣇⣀⣀⣀⣸⠇⢹⡄⠀⠀⣀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠉⢉⣍⠉⠁⠀⠈⠉⠉⠉⠁⠀⠈⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡟⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⣴⠶⣼⢸⡇⣿⣼⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠆⠹⣆⠙⠈⠃⠋⠙⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋




⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⠀⠀⠀⣶⠀⠀⢸⠇
⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣤⠀⠸⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡟⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠉⠁⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠛⠛⠛⢻⣿⣿⣴⣿⡗⣤⣤⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠀⠀⠀⠀⠀⢨⠅⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠀⠘⠁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣶⣶⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⢹⣿⣿⡿⡿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⡔⣶⣿⣷⡆⠀⠀⠀⠀⠀⠀⢨⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣙⣛⡁
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠛⠀⠉⠉⠉⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢸⣿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠈⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⢰⣦⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠇



⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠐⠒⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠰⠶⠶⠶⠶⠶⠶⠆⠀⠘⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠰⠶⠶⠶⠶⠶⠶⠆⠀⠀⡇⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠘⠛⠛⠛⠛⠛⠛⠃⠀⢰⣤⡄⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠧⠤⠤⠀⠀⡃⠀⠀⠀⢀⡠⢴⣿⣶⣶⣿⣿⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢀⡠⠄⠒⡦⠤⠤⠴⠶⠷⠶⠶⠶⠆⠀⢀⠇⠀⠀⠀⣇⠠⠄⠼⡞⠉⠉⠙⢻⣽⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⢻⠋⢉⣉⣉⣙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⣩⣷⡶⠶⠿⣯⡉⠑⡆⠀⣿⣿⣤
⠀⠀⠀⠀⠀⢰⣶⣴⣶⡿⣧⣼⣀⣈⣉⣉⣉⣉⣑⣢⣤⣀⣀⣠⣤⣄⡀⠀⢣⣀⣀⣛⣹⠒⠀⠉⢙⣁⣀⣧⣼⣿⠏⠉
⠀⠀⠀⢀⣀⣾⣷⠟⠉⠉⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠇⠀⠀⠈⡀⠀⠀⡌⡀⠀⡿⠻⣿
⠀⠀⠀⠘⢿⣿⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡇⠀⢃⠃⠀⡇
⠀⠀⠀⠀⢈⣿⣻⣄⣀⣀⣠⣾⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⣼⠋⠀⠈⢳⡄⠘⢸⠀⢸⢸⠀⡇
⠀⠀⠀⠀⠙⠋⠛⢿⣯⡿⠿⢿⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠙⣦⡈⡇⠈⡎⡄⡃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢸⠀⠀⠀⠀⢾⡁⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⢈⣧⢧⠀⢳⢃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⣠⠟⠘⡸⠀⠸⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠅⠆⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢎⠉⠉⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⢢⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻


⣏⠉⣹⣛⣛⣛⣛⣛⣟⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣿⣟⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠛⠛⠛⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣼⠏⠀⠙⠳⢶⣤⣤⣀⣀⣀⣤⣴⠶⣆⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣸⡏⢠⡶⠶⠶⣦⡀⠀⢩⡽⠷⠶⢦⡀⢹⣆⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⣿⠀⠀⠀⢸⡟⠛⢻⡇⠀⠀⢸⡟⠛⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠘⠳⠶⠲⠛⠁⠀⠈⠳⠶⠶⠟⠁⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠶⣤⡀⠀⠀⠀⣀⣤⠶⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⠙⠛⠛⠛⠉⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠟⠙⢷⡄⠀⠀⠀⠀⠀⢠⡾⠋⠻⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠶⠛⠛⠋⠉⠁⠹⣆⠀⠀⠙⢦⡀⠀⢀⡴⠋⠀⠀⣰⠏⠈⠉⠉⠛⠛⠶⢶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢹⣶⡟⠁⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠳⣄
⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣴⣿⣭⣿⣦⣤⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⡆
⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣧
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣷⠶⠛⠛⠷⢦⡀
⠙⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠟⠁⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠹
⣶⣶⣶⣶⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣦⣦⣤⣴⣶⣶⣶
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠘⠀⠁⠀⠁⠀⠀⠀⠀⢠⡞⣟⢷⡀⠀⠀⠀⠀⣠⠴⣶⡶⢤⡀⠀⠀⠀⢠⣶⣾⠒⢲⠀⠀⠀⠀⢀⣀⣀⠴⠦⡀
⠀⠀⠰⠴⠶⠶⠆⠀⠀⠀⠀⠈⢷⣶⣿⡇⠀⠀⠀⠀⠀⠫⠿⠯⠃⠀⠀⠀⠀⠘⡏⠙⣶⡾⠀⠀⠀⠀⠘⠛⠈⠢⢴⡃
⠀⠀⠛⠛⠀⠚⠛⠀⠀⠀⠀⠀⠘⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠤⢤⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀
⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⠀⣸⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⣠⢤⠀⠀⠀⠀⠀⢰⡶⠶⣶⣶⡆
⠀⠀⠻⠿⠿⠿⠟⠀⠀⠀⠀⠀⢸⣿⣽⠀⠀⠀⠀⠀⣽⣟⣿⣿⣿⡅⠀⠀⠀⠀⠸⠀⠝⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⣴⢲⡄⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣾⠛⠻⣷
⠀⠀⠁⢰⠚⡆⠀⠀⠀⠀⠀⣰⠞⢾⣻⣢⠀⠀⠀⠀⢾⣿⣿⠆⡠⠁⠀⠀⠀⠀⢸⣸⣿⡇⠀⠀⠀⠀⠀⣷⣈⣤⣿
⠀⠀⠂⠀⠛⠐⠒⠀⠀⠀⠀⠘⠲⠚⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠿⠶⠾⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣀⣀⢀⡀⡀⣀⢀⠀⠀⠠⠀⣀⢄⡀⠀⢀⠀⠀⠀⠀⢰⣶⣶⠀⠀⠀⠀⠀⠀⡟⠀⠀⡇⠀⠀⠀⠀⢠⣤⡤⠞⠛⠃
⠀⠀⠁⠉⠉⠉⠈⠀⠀⠀⠈⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣷⡤⣤⡇⠀⠀⠀⠀⠈⠋⠓⢦⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⣀⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠈⠁⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡤⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠻⡅⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⢀⣀⡤⠶⠚⠳⢤⠼⠃⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⣹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢸⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⢤⡄⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⢻⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⠦⠤⠤⠤⠾⠀⠀⠘⠧⠤⠤⠤⠴⠇⠀⠀⠿⠤⠤⠤⠤⠟⠀⠀⠸⠦⠤⠤⠤⠾



⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣛⣛⣿

⠀⠀⠀⠀⡤⠤⡤⠀⠀⠀⠀⠀⢀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣦⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⣸⠀⠀⣳⠀⠀⠀⠀⣰⢋⣴⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⢨⡷⡇⡤⡄⠀⠀⠀⠀⠀⣀⣀⣀⡀⡾⠁
⠀⠀⠀⢹⠁⢐⡷⠀⠀⠀⠀⠘⢿⡟⣾⢂⣶⢐⣶⢰⣆⠀⠀⠀⠀⠈⠀⠈⡯⡇⠀⠁⠀⠀⠀⠀⠀⠨⠉⠉⠁⡾⣆
⠀⠀⠀⢩⡣⠊⠀⠀⠀⠀⠀⠀⠀⢬⡯⢨⣯⠨⣿⠬⣿⡄⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠙⠄
⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⢙⣿⠉⠋⠉⠁⠀⠀⢀⣀⣀⣤⣋⣼⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠐⠠
⢤⡀⢀⣴⣦⣾⣷⣤⣄⠀⠀⠀⠀⠀⢀⡀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢳⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⢸⡇⣸⡇⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠿⠉⠀⠹⠇⠀⠀⠀⠈⡿⠿⣿⢸⣿⣿⡏
⠀⠘⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢼⣹⣏⣿⣻⣻⡇⠀⠀⠀⠀⣁⠀⢰⣶⠆⠀⠀⠀⠀⠀⠀⢰⡉⡎⣹⢸⣿⣿⣷
⠀⠀⠉⠾⠎⠛⠱⠿⠊⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠠⠿⠀⠾⠀⠧⠀⠠⠄⠀⠀⠀⠾⠿⠿⠿⠯⠿⠿⠿⠆
⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠰⠛⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠛⠛⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠞⠳⢴⠀⠀⠀⠀⠀⠀⢰⣷⠖⣲⠒⡶⢲⢲⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠒⣇⠘⠇⣸⠒⠀⠀⠀⠀⠀⣼⠏⠀⠁⠈⠀⡿⣼⡇⠀⠀⠀⢀⣀⣰⣺⣲⣀⣀
⠀⠀⠀⢀⣻⣟⡅⠀⠀⠀⠀⠀⠀⢀⡀⣋⢙⡋⣙⣀⡀⠀⠀⠀⢀⣿⢀⡀⠠⠀⢸⣇⣿⡇⠀⠀⠀⢈⣿⣉⢹⠉⣹⣏
⠀⠀⠀⠚⠓⠓⠓⠂⠀⠀⠀⠀⠀⠐⠂⠒⠐⠂⠒⠐⠂⠀⠀⠀⠈⠀⠀⠁⠀⠀⠉⠙⠛⠃⠀⠀⠀⠐⠂⠒⠐⠒⠂⠒
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣀⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⠀⠀⠀⠀⠀⡎⣭⣭⡝⠀⢀⣀⠀⠀⠀⠀⠀⠀⣠⡀⡴⠋
⠀⠀⢰⣿⠀⠀⢰⡇⠀⠀⠀⠀⣀⣤⣄⢴⠟⠻⣇⠀⠀⠀⠀⠀⠀⠀⡇⠶⠶⢮⠀⣼⣿⡀⠀⠀⠀⢀⣴⣿⣿⣦⡀⢀⡴
⠀⠀⠘⣿⠀⢠⣿⡇⠀⠀⠀⠀⢿⣿⣿⣦⣄⢀⣭⣦⣄⠀⠀⠀⠀⠀⣏⣛⣛⣉⠀⠛⣿⠃⠀⠀⠀⢀⣿⣿⣿⣿⣿⣮
⠀⠀⠀⠻⡄⣾⡿⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⢺⣿⣿⣿⡇⠀⠀⠀⠀⡧⠷⠿⢼⠀⠀⠁⠀⠀⠀⢀⣼⣿⠿⠻⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣟⣛⣿⣛⣛⣿⣭⢯⣽⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣀⣀⣠⡤⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⢿⠿⠛⠁⠠⣤⣤⣤⣤⣤⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡆⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣇⣀⣀⠉⠉⣿⣿⠉⠉⠉⢹⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⣿⣿⠿⢿⣿⣿⠀⠀⣿⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠋⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⠇⠀⠀⣿⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⣿⣤⣄⠀⢿⡿⠀⠀⠀⢸⡿



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

⠀⠀⠀⠀⠀⢀⡀⠀⣿⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠈⠉⣤⣶⣦⡉⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⣷⣿⣷⡆
⠀⢰⣶⣶⣶⣷⣿⣿⣿⣿⡇⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠹⣿⡿⠁
⠀⠀⢻⣿⣿⣻⣿⣻⣿⡁⢤⠈⠁⠀⠀⠀⠀⠀⠀⠐⢤⠤⣿⡧⢤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠿⠿⠿⠿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠇⠻⠿⠇

⠀⣀⣠⣴⣦⣤⣤⣴⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⡀⠠
⠀⠛⢛⣉⣉⣉⢩⣭⠡⣤⡄⣶⡄⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢸⣿⢸⣿⠀⣿⡇⣿⡇⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠛⠢⠄
⠀⠀⢸⣿⢸⣿⠀⣿⡇⣿⣇⣿⣇⡀⠀⠀⠀⠀⢸⡇⠀⠀⡀⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⢰⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⡇⠠⣾⣿⣿⡆⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠘⠿⣿⣿⡿⠿⢿⣿⣿⡿⠿⠟⠁⠀⠀⣿⣻⣿⣿⣛⣿⣻⡿⢿⣛⢿⣿⡟⠿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⠿⠶⠿⠿⠿⠷⠾⠿⠛⠃⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⣄
⠀⠀⠀⢀⡒⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠲⠛⠓⠂
⠀⠀⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣤⣤⣀⣀⡀
⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡆⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠙
⠀⠀⢰⣿⢻⡖⣶⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⢀⣀⣀⠘⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐
⠀⢀⣸⣿⣼⣧⣿⣼⣇⣴⣿⣦⣤⣀⠀⠀⠀⣀⣿⣿⣿⣿⣴⣿⣿⣿⣿⣙⣀⡀⠀⠀⠀⠒⣁⡐⠀⠀⠀⠀⠀⠀⠼⠾⠆
⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠰⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠰⣟⣷⣶⣶⣤⣄⠀⡀⠀⠀⠀⠀⣾⣻⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤⡄⢀⠀⠀⠀⠀⠀⣀⣀⣠⣶⣀⢀⡀
⠀⢸⣿⣏⠛⠛⠻⠓⢾⣧⡀⠀⠀⠈⣿⣿⡛⠛⠛⠛⢾⣷⠀⠀⠀⠀⢿⣿⣿⣿⣟⢿⡗⠀⠀⠀⠀⡏⢈⡉⠉⡉⠀⡇
⠀⢀⣿⣿⣦⡀⠀⠀⠘⢆⠇⠀⠀⠀⣿⣿⣧⡀⠀⠀⠘⡇⠀⠀⠀⣾⣿⢻⣧⢠⣿⣏⣿⣷⠀⠀⠀⡇⠨⠀⠤⠄⣠⢧⡀
⢰⢾⡿⣿⢿⢷⡆⠀⠀⠀⠀⠀⠀⡾⣿⢿⢿⡿⣆⠀⣠⣦⠀⠀⠀⠀⣽⣮⣳⣤⣾⣽⣯⠀⠀⠀⠀⡇⠘⠁⠁⡞⣀⡴⠊
⠼⠷⠷⠿⠾⠷⠷⠀⠀⠀⠀⠀⠰⠷⠿⠾⠷⠷⠿⠄⠽⠿⠄⠀⠀⠀⠉⠉⠹⠿⠏⠉⠁⠀⠀⠀⠀⠓⠒⠒⠒⠛⠮⠤⠞
⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⣴⣶⣤⣠⣀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦
⢀⣴⡟⣏⣯⣻⣽⣿⣷⣄⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣠⣭⣅⠀⡀⠀⠀⠀⠀⠀⠀⢠⣭⣥⣀
⢸⡇⠀⠿⣿⠟⠀⠀⠀⣿⠀⠀⠀⠈⢩⣿⣯⣯⣿⣽⣿⡍⠀⠀⠀⠀⢰⢿⣿⡟⠛⣛⡀⠀⠀⠀⠀⢠⠿⣿⣿⠉⢸⣿⡿
⢸⡇⢀⣀⢋⣀⣀⣀⠀⣿⠀⠀⠀⠀⠚⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣞⣿⣿⣻⣻⣟⣟⣿⡷⠀⠀⠰⣾⣟⣿⣿⣻⣿⣟⣟
⣼⣷⣾⣿⣿⣿⣿⣿⣶⣿⡆⠀⠀⠀⠀⠛⢻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠠⠤⠤⠴⠴⡶⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠴⠦

⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⣀⡤⡄⣠⣴⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⢠⠞⣡⣴⣶⣤⡙⢷⡀⠀⠀⠀⢸⠃⠖⠒⣿⣗⣨⣿⣂⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⠀⠀⠀⠀⠀⠰⣿⣷⡿⠻⣷⣿⡇
⠀⡟⠀⠁⠀⣈⣻⡵⠈⡇⠀⠀⠀⢤⣾⣿⣿⣿⠝⢋⠉⠸⡀⠀⠀⠀⠀⣤⣴⣾⣿⣿⡇⠀⠀⠀⠀⣶⣿⡟⠀⡄⢸⣿⣷
⠀⢻⡀⠀⢾⣽⠏⠀⣰⠇⠀⠀⠐⢿⣿⣳⣸⣿⠗⠈⣀⡾⠀⠀⠀⠀⢴⣿⡉⠈⣿⣿⠃⠀⠀⠀⠀⢉⣿⣿⡶⣠⣾⣿⣏
⠀⠀⠙⠶⢤⣤⠤⠞⠁⠀⠀⠀⠀⠙⠻⠿⠿⠛⠁⠙⠁⠀⠀⠀⠀⠀⠀⠛⢿⡟⢿⠿⠀⠀⠀⠀⠀⠀⠙⠉⢻⣿⠋⠛⠁

⠀⢠⠒⣢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣀
⠀⢸⢸⣿⣿⣧⣤⣤⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⡿⣧⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠏
⠀⢸⢸⣿⣿⣿⣿⣿⣷⣄⠀⠀⢠⣶⣴⣦⣶⣸⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⣻⣻⣟⣿⠀⠀⠀⠀⢀⣴⣮⣍⣈⢁⣈⣽⣶
⠀⢾⢻⣿⣿⣿⣿⡿⣷⣿⡆⠀⢸⣿⠛⣿⡟⢻⣿⣿⠿⣶⡀⠀⠀⠀⢀⡯⣩⢩⡍⣍⡀⠀⠀⠀⠀⢸⣿⣿⣿⡌⠛⠋⢹
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣩⣿⣯⣹⣿⣿⣴⣿⠃⠀⠀⢠⣭⢡⣭⢨⣭⣨⣥⡀⠀⠀⠀⠘⠿⣿⣿⡇⣀⡤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠈⡟⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣧⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠶⠶⠶⠤⢴⡷⠀⠀⠀⠀⠸⠾⠷⠶⠶⠶⢶⣄
⠀⠀⠀⠀⣴⡶⠶⠶⠶⠶⠶⠛⠛⠛⠛⠛⠛⠋⠉⠉⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⣆
⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣠⣤⣤⣤⣤⡄⠀⠀⠀⠀⢰⠶⠶⠶⠶⠶⢋⣰⡇
⠀⠀⠀⠀⠀⣷⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠶⠶⠶⠶⠚⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣶⣶⠶⢶⡇⠀⠀⠀⠀⢸⠀⣾⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠹⣧⣤⣀⣠⣤⣤⣤⣤⣤⣶⣶⡶⠶⠶⠶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡶⠶⠶⠶⠞⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⠛⠛⠛⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠷⣤⡀⠀⠀⣠⡾⠋⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠈⠙⢶⡴⠋⠀⠀⠀⠀⠀⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⢀⡟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣞⣡⡾⠋
⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⡶⠶⠶⠶⠶⠶⠶⠶⠿⠛⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢙⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣼⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⢀⡾⢁⣙⠳⢤⡀⠀⠀⢀⡞⣿
⠀⠀⠀⠀⠀⠀⢻⣶⣛⣛⣋⣩⣭⣭⣭⣥⣤⣤⣤⣤⣶⡶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠀⠀⠀⠀⣰⠏⣴⠟⠛⠷⣦⡉⠳⠶⢋⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⣉⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣟⢦⣤⠞⢁⣾⠋⠀⠀⠀⠈⠻⢶⡶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠉⢙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⠒⢛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡈⠀⠀⠀⠀⠀⠀⠈⣇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⡎⢹⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠏⢣⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⢀⠤⠤⠤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠐⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣀⣘⣒⣒⣒⣒⣒⣒⣒⣒⣚⣀⣀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠘⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠈⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⣱⠀⠤⢀⣀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠐⠒⠒⠒⡖⡴⣊⠅⠒⠉⠉⠁⠒⠭⡢⣘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠟⠜⠁⠀⠀⠀⠀⠀⠀⠀⠈⢎⢆⠇
⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄⡼⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡊⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⡄⠀⣤⣤⣤⣤⣤⣤⣤⣦⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⠃⠀⠛⠛⠛⠛⠛⠛⠛⠛⡜⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡜⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⡏⠉⣾⣿⣿⣿⣿⣿⣿⣷⠘⠪⡓⢄⡀⠀⠀⠀⣀⠤⡪⣞⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⡉⢉⣉⣉⣉⣉⣉⣉⣉⡁⠀⠀⠈⠑⠂⠥⠥⠌⠒⠫⣀⠜⢆⡃
⠀⠀⠀⠀⠀⠀⠀⢸⢨⠀⣇⣀⢿⣿⣿⣿⣿⣿⣿⡯⢠⣀⣀⣀⣀⣀⣀⣀⣀⣀⡽⢄⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠰⠀⡖⠒⣴⣶⣶⣶⣶⣶⣶⣦⠰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠆⢘⢣⡀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⢸⢘⠀⠓⠐⠛⠛⠛⠛⠛⠛⠛⠓⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠀⡳⡄⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣀⠇⠈⢆⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠴⠛⠉⠉⠙⠓⠲⢤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⠛⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣸⠃⢀⠽⠥⡀⠀⠂⣠⠼⢥⡀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠇⠀⢠⡇⠀⠁⢶⠆⡀⠀⠀⡁⠶⠆⠇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸⠁⠀⠀⠂⠀⠀⢸⡆⠈⠀⠈⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡆⠀⠀⣼⠀⠀⠄⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⢡⡗⢒⣩⣿⠰⠀⠤⠈⠀⠀⠄⠀⠆⢠⠀⠂⠐⠀⣗⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡚⠛⠻⢼⣿⣟⠝⣿⣄⠀⡘⠀⡄⠀⣠⠀⡇⢸⠀⣤⠠⠄⣿⢬⠊⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⡉⠉⠓⠺⣧⣐⡩⣿⠀⠄⠁⠆⠁⠂⠁⠠⢠⠂⠐⠀⠂⠀⣯⡀⢂⣤⠟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣌⠙⣶⡶⠯⠔⠚⣿⠀⡌⠀⡄⠀⡄⠀⡌⠸⠄⠤⠀⠇⠀⣿⡿⢻⡈⢧⠈⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢹⠀⠀⠀⠀⣿⠤⠀⠆⠀⠂⠀⠂⢀⠃⠈⠀⠉⠘⠁⡏⢹⢄⡱⠬⠓⠁⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠸⠀⠀⠀⠀⣿⡄⠀⡆⠀⡄⠀⡄⠀⡄⠸⠀⠆⡘⠀⣇⡇⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⢸⠀⠃⠀⠁⠀⠉⠀⢱⢰⠁⠈⠀⡁⢠⡟⣧⣒⣒⡊⢋⣉⣗⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⠀⠀⠀⠀⠀⠸⣄⠔⠀⠆⠀⠤⠀⠆⠈⠆⠐⠀⣃⣸⠃⢣⠈⠉⠉⠉⠉⢹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡜⠀⠀⠀⠀⠀⠀⠈⠙⣷⣶⣶⣶⡶⠒⢶⣾⣿⣿⠏⠀⠀⠈⡆⠀⠀⠀⠀⠘⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡗⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠋⠁⠀⠀⠉⣿⠁⠀⠀⠀⠀⢸⡴⡦⠀⠀⠀⡞
⠀⠀⠀⠀⠀⡤⠤⠄⠒⠒⠂⠤⣰⣁⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠤⠂⠀⢀⣻⡤⠚⠀⠀⠀⠀⠓⠿⠤⠤⠒⠃
⠀⠀⠀⠀⠀⢀⢌⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⢒⣀⡄⠉⠝⡻⠋⠀⠐⠊⠽⢫⠋
⠀⠀⠀⠀⠀⠀⣠⠤⢬⡉⡆⠀⠠⢀⣠⡀⠀⠀⠀⠈⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠊⡽⠑⠐⠀⣰⣃⡼⢥⡾⣦⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠡⣈⡡⠌⠀⠀⠀⠀⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣽⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠀⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⠾⠟⠋⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⠶⠶⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡆⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢠⣤⣤⣤⡄⠀⣾⣿⣿⣿⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⠉⠉⠉⠁⠀⠈⠉⠉⠉⠀⠀⠉⠉⠉⠁⠀⠈⠉⠉⠉⠀⠀⠉⠉⠉⠉⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⡯⡄⢠⠀⠙⢿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⢰⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠸⡿⠷⠆⠀⠻⠿⠟⠁
⠀⠀⠀⠀⠀⣿⣿⣟⣛⣛⣛⣛⡃⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⢂⣀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠾⠷⠆⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⣭⣭⡅
⠀⠀⠀⠀⠀⣿⣿⡿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣛⣀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣶⣦⠀⠀⠀⠀⠠⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠴⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢻⣿⣿⡇⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⣿⣿⣯⣭⣭⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣧⠀⠀⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠻⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠰⠄⠀⠀⠀⠁⠀⣤⠀⠀⠀⠁
⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⡌⡇
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣹⢰⠀⠃⡆⠀⢠⣤⣤⡀⠆⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠑⠂⠀⠴⣤⣄⠀⠀⢀⣴⣶⣶⣦⠀⠀⣤⡀⠒⠀⠀⠀⠀⠐⠓⠂⠀⠐⠀⠛⠛⠀⢰⡆⣿⡉⢸⣿⣿⣡⡆⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠈⠙⢯⣽⣽⣿⣛⣁⣀⣉⠋⠐⢲⣶⣶⡀⢀⣀⣀⣀⠀⠀⠀⠈⠉⠙⠉⣩⠄⣾⣿⣿⡦⠀⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠊⠀⠀⠀⠒⠀⠀⠺⠿⠿⠿⠿⠿⠶⠶⠶⠛⠛⠙⠉⠉⠉⠁⠀⠀⠀⠀⠁⠒⠲⠬⢴⣿⣏⣹⣷⣠⣶⣿
⠀⠀⠀⢠⠀⢠⠀⠀⠇⠀⠉⣀⣌⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣏⣩⣉⢀⡒⠂⣰⠄⠐⠢⠀⠠⠤⠄⢀⢩⣙⡛⠿⣿⣿
⠀⠀⠀⠈⠠⣤⡰⣶⡎⠿⣷⢹⣿⡜⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢐⠒⠂⡮⠤⢴⣀⣠⡂⡀⣼⠃⠉⡸⠁⠑⡒⠚⠀⡤⠆⠀⣀⠇⠀⠀⠉
⠀⠀⠀⠀⠀⠙⡇⠀⢱⠀⠸⣆⠈⢷⡀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡠⠀⢀⠌⠀⢠⠃⠀⣰⠏⠈⠉⠋⠑⠓⡖⠐⠥⡤⢰⣀⡀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢿⠀⠘⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠤⢤⣆⣀⣠⠁⠀⣰⡟⠀⠀⡌⠀⠀⡌⠀⠀⢰⠁⠀⠀⡇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⠁⠀⠀⠀⠀⠈⠻⠿⣶⣶⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⢿⣶⠛⠠⠤⠼⣀⣀⠰⠁⠀⢀⡇⠀⠀⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⡖⣾⠖⠤⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡀
⣿⣿⡟⠛⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢦⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠁⠀⠀⠀⠀⠈⠳⣄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡞⠛⠛⠛⠀⣀⠀⡂⠀⠠⠤⡽⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢺⡉⠛⠁⠀⠀⠀⠀⠀⡰⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢀⡤⠶⠢⢤⡀⠀⠀⠻⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠶⠖⢤⡀⠀⠀⠀⢨⠁⠉⠈⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠉⠉⠈⡇⠀⠀⠀⢰⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⢸⠦⠤⠤⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣄⠀⠀⣠⣧⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⢸⣄⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⣀⡀⣼⠁⠈⠁⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠐⠒⠒⠚⣿⡯⠗⠛⢻⡄⠀⠀⢀⡿⠦⠮⣬⣄⣒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⢤⡇⠀⠀⠀⠀⡏⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢀⡀⠍⢛⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡛⠫⠷⣤⣄⣠⡴⠇⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⣊⣩⠴⠞⠋⠁⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠙⠓⠶⠤⣧⣐⡒⠀⠤⢀⡀⠶⢈⣩⡴⠖⠛⠉⣠⡀⢰⠂⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡋⡓⠰⠦⣤⣄⣈⠉⠛⠓⢶⠖⠛⠉⠀⢀⠔⠄⢈⡄⡆⢘⠁⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠁⠘⠲⠦⣤⣉⡙⠓⡆⢘⡇⠀⠋⡁⠀⠆⠇⠀⠄⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣠⣤⣤⣤⠖⢤⡀⠀⠈⠉⠁⠘⡁⠀⡋⠁⠀⠃⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⠿⠟⠛⠫⠁⠀⠀⠂⠉⠓⠦⣀⠀⠸⡇⠀⠀⠀⠀⢀⠤⠔⠂⠁⠠⠒⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⠍⠻⠋⠉⠠⢀⠐⠄⠀⠀⠄⠐⠀⠀⠠⡀⠂⢌⢑⣶⡇⡠⠀⠂⠀⣀⡀⠰⡊⡃⠀⠃⠇⣿
⠀⠀⠀⠀⠀⠀⢰⠈⠉⠉⢁⣀⡀⠈⠲⢄⡀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠂⣀⣠⠤⠖⠊⠁⠀⡃⠀⡤⢄⠐⡇⡆⠀⡁⠇⠀⠁⠀⡟
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸⡧⠭⢉⣳⡄⠉⠲⣄⡀⠀⢀⣀⣠⠤⠚⠋⠁⠀⠀⠀⢀⣀⠀⡇⠀⠇⠾⠀⠄⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠉⠑⠒⠾⠁⠀⠀⠀⠙⡖⠋⠁⠀⠀⠀⠀⠀⡤⠄⠀⠳⠚⠀⡇⠀⠃⠀⠀⠀⢀⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠒⠒⠠⠤⢀⡀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡀⠀⣖⡆⠀⠙⠁⠀⠀⠀⠠⡇⠀⠀⣀⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠈⠐⠠⠄⣀⠀⠉⠐⠒⢤⡀⠀⡇⠘⠧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⠴⠚⠉
⠀⠀⠀⠀⠀⠀⢭⣀⡉⠐⠠⢀⣀⠈⠀⠂⠀⢸⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣄⡀⠁⠂⠀⠄⢸⠀⠀⡇⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠸⠀⠀⡇⠀⠀⣠⡴⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢤⡧⠖⠋⠁

⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣬⣧⣴⣤⣴⣮⣤⣾⣵⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⡀⢀⡀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠠⣿⢾⡇⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏⢸⡇⢿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⠀⠀⢀⣤⣄⠀⣤⣶⣶⣶⣄⠀⢠⣴⣶⣾⣿⣇⣶⡆⠀⣾⣿⠂
⠀⠀⠀⠀⠀⢀⣤⢴⡦⢸⣿⡟⠉⠉⠀⢠⣾⡿⣿⣆⠛⣿⣿⠙⣿⣧⠈⠛⣿⣿⠉⢸⣿⣿⣿⣿⣿⠀⠀⠸⢷⡲⢤⡀
⠀⠀⠀⢀⡼⢫⠖⠉⠀⠸⣿⣷⣶⠀⠀⣼⣿⣤⣿⣿⠀⣿⣿⣰⣿⠏⠀⠀⣿⣿⠀⢸⣿⡏⢸⣿⣿⠀⠀⠀⠀⢹⡆⣷
⠀⠀⠀⠸⣇⢻⡀⠀⠀⠀⣿⡏⠁⠀⠀⣿⣿⠛⢻⣿⡄⢻⣿⠹⣿⣦⡀⠀⣿⣿⠀⠈⣿⡇⠘⣿⣿⠀⢀⣠⡴⢋⡴⠃
⠀⠀⠀⠀⠙⠲⠯⣿⡅⠀⢿⣿⣿⣿⠀⣿⣿⠀⠸⣿⡇⢸⣿⠀⠘⣿⣷⠀⣿⡿⠀⠀⠿⠇⠀⠿⠛⠁⠐⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⣀⠀⠀⠀⠀⣀⣠⣤⡄⢀⣀⠀⠀⠀⠀⣠⣄⠀⣠⣤⣴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⠿⠇⠀⠰⣿⣿⡟⠛⠃⣿⣿⠀⠀⠀⣼⡿⣿⡆⠙⢻⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠙⢿⣦⡀⠀⠀⣿⣿⠿⠿⠀⣿⣿⠀⠀⢰⣿⣧⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠠⣶⣶⣿⠇⠀⠀⣿⣿⠀⠀⠀⣿⣿⣀⣀⢸⣿⠉⢹⣿⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠙⠛⠋⠀⠀⠀⠛⠛⠀⠀⠀⠙⠛⠛⠛⠘⠛⠀⠘⠛⠃⠙⠛

⠀⠀⠀⢠⣤⣴⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣄⣀⣀⣀⣤⣤⣤⡀
⠀⠀⠐⣿⠁⠀⠘⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢉⣿⠼⠇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿
⠀⠀⠀⠛⢷⡤⣾⢹⡶⠷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣰⣿⠿⣿⣄⣀⣀
⠀⠀⠀⢀⡶⠲⣤⢻⣄⢀⣼⠃⠀⠀⠀⠀⠀⠀⣴⣶⣦⣧⣾⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠳⣦⠀⠀⠀⠀⠀⠀⠀⡏⠀⣯⢸⣿⣿⣿⠁⠉⡁
⠀⠀⠀⠈⢧⡤⠞⠈⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠷⠖⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⣒⣿⣏⠉⠉⣻⡏⠉⡇⠀⠀⠀⠀⠀⠀⠰⡿⠋⠉⠙⠻⡗⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⣿⠶⠆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⢸⣿⣿⣯⠀⢠⠏⢳⠀⡇⠀⠀⠀⠀⠀⠰⣿⣿⠀⡿⡀⢻⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⡏⠉⢉⡭⢍⠉⠉⠉⠉⡇
⠀⠀⢸⣿⣿⡇⢀⠎⠀⠀⢧⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣴⠎⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠳⢾⠒⡄⠀⠀⡇
⠀⠀⢸⣿⣿⡿⠚⠒⠒⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠛⡿⠁⠀⠀⠀⠙⢦⡄⠀⠀⠀⠀⠀⠧⠤⠤⠤⠤⠭⠤⠤⠴⠇
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⣠⡄⣨⣧⣤⣶⣤⣤⠄⠀⠀⠀⠀⠀⠀⣠⡞⣭⢻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⠶⠶⣆⠀⡄⠀⠀⠀⠀⠀⠀⢀⣤⣤⠿⠿⣤⣤⡀
⠀⠀⠀⠀⠀⢸⡇⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣭⡦⠮⣿⣿⠀⠀⠀⠀⠀⠀⢰⡇⢸⣤⣤⣤⣀⣟⢰⠇⠀⠀⠀⠀⠀⢀⣸⡟⠔⠁⠈⠦⢻⣅⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠸⡇⢸⣿⣿⣏⣻⣧⡟⠀⠀⠀⠀⠀⠀⠘⢿⡌⢭⢈⠀⣅⢸⡟⠃
⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⠿⠿⠀⠀⠀⠀⠀⠀⠀⢇⢸⣿⣬⣯⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠰⣿⠦⣥⣬⠶⣿⠇
⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋


⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣶⣦⣠⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢤⡀
⠀⠀⠀⠀⢀⣴⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣈⡟⠁⢾⠇⠈⢯⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡚⢲⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡾⠛
⠀⠀⠀⠐⢘⣿⣿⣿⣿⡇⠀⠄⠀⠀⠀⠀⠀⠀⠻⣇⢴⣿⣿⡆⣼⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣉⠇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛
⠀⠀⠀⠉⠈⠛⠟⠟⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⣿⡟⠻⠟⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣟⠛⣻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣇⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣦⣷⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠛⠁⠛⠀⠄⠙⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠣⡀⠀⢻⡀⢠⠄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⣀⣀⣀⡀⠀⡄⠀⡄⠀⠀⠘⠂⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⢱⣦⡀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⣾⣽⣿⣿⣿⡎⠀⢐⡃⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠘⠳⣦⠀⠀⠀⠀⡜
⠀⠀⠀⢿⣿⣿⣿⣿⠁⠀⢸⡇⠀⠀⣀⡆⠀⠀⠀⠀⠀⢀⠀⢠⡆⠂⠀⠀⠓⢀⡈⠂⠀⠦⠀⢰⠃
⠀⠀⠀⠈⢿⣿⣿⣿⠄⠀⠀⠧⡞⠉⠉⠁⠀⠀⠀⠀⠀⠘⠊⠛⣷⣆⠈⠂⠀⠀⠉⠠⠄⠀⢠⠇
⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠐⡀⠀⠀⠀⠁⣀⡀⠠⣤⠄⠀⠟⢻⣾⠃⠀⢠⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⢹⣿⡿⢀⣀⡤⠄⣐⣓⣤⣦⣭⣭⣥⣴⠄⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⣠⠿⣛⣩⠶⠚⠋⠉⠤⠀⣸⣿⡉⠁⣀⣀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠓⣿⠉⣡⣶⣿⣿⣿⣿⢗⣿⣿⣿⣿⣷⣶⣧⣄⠀⠀⠻
⠀⠀⠀⠀⠀⢀⣿⠀⣹⡿⠿⣛⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⣿⣿⣷⡶⣾⣿⣿⣿⣿⣿⣿⡿⣯⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⢿⣿⡟⠋⠉⠉⣱⡋⣸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⡴⠋⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢰⠀⠰⡦⢀⠂
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠐⠂⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠢⠤⠀⢀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠢
⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣃
⠀⠀⣐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇
⠀⠈⡟⢑⠰⠺⡀⢻⣬⢄⣂⣀⣑⠂⢀⡀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣴⣿⣿⣿⣿⣷
⠀⠀⣣⡄⠈⢛⣧⣀⣾⣏⡀⢿⡞⣝⡫⢓⡀⠀⠀⠀⠘⢀⣀⣨⢠⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣁⣿⣿⣿⣿⡿
⠀⠀⣴⣷⠀⠼⡳⠆⢈⣿⢀⠹⣿⣠⠔⠉⠁⠀⠀⠀⠀⠀⠘⠿⠍⠝⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣿⣿⣿⣿⣿⣧
⠀⠀⠸⠟⠂⢾⠻⡏⠀⣙⠋⡈⠉⠁⠀⠀⠀⠀⠀⠀⠀⡃⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣖⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠙⠛⠻⠿⠿⠿⠻
⠀⠀⢸⠀⠂⠈⢛⣴⣬⠁⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣝⢆⡠⠰⠀⠈⠀⠀⣤⣄⢤⣄⠤⡄⣄⣤⣴
⠀⠀⢸⡗⠢⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⡋⠁⠀⠀⠀⣰⢦⣿⣾⡳⢮⡳⢶⣍⢄⢿
⠀⠀⠌⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢀⢠⡟⣬⡷⣌⢻⣽⢶⣝⠳⡭⠻⠾
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⡽⣮⡳⣮⡳⣌⠳⡎⠻⠁
⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣻⣿⣿⣿⣟⣯⠻⣬⠷⣍⢳⣆
⠀⠀⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⢿⣮⠻⣝⢧⡄
⠀⠀⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉

⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⡤⠄⠀⢠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠦⢤⡤⡤⠀⢀⡾⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⣳⠀⠀⠀⠂
⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣸⠀⠀⡀⠀⠀⢀⢔⠏⠀⠀⠀⠀⠀⠀⠉⠙⠂⠀⠴⠋
⠀⠀⠀⠀⠸⠄⠀⠀⣸⠀⠀⡀⣴⠀⠀⣝⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⢸⠀⠐
⠀⠀⠀⠢⠤⠤⠤⠴⠉⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⢐⠈⠀⠀⠀⠀⠀⢤⡀⠀⠠⣀⠀⠀⣀⡞
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⡁⠀⠀⠈⠙⠛⠃⠀⠀⢠⡶⠂⠀⢹⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠈⠀⠀⠀⣠⠀⢸⡇⠀⠀⠤⠤⠦⠀⠀⠀⠄⠀⠀⠈⠉⠉⠁⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣷
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠁⣦⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⢅⠉
⠀⠀⠀⠀⠐⠰⠤⠄⠀⢠⡴⠧⠀⠀⠁⠀⠀⢀⣤⠖⠛⠀⠀⣀⣀⣀⣀⣀⣸⠀⠀⠀⠤⠀⠐⠤⠖⠒⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠐⠤⠾⠀⠀⠀⠀⠀⠀⠀⣈⣁⠀⠠⡀⠀⠀⠉⠀⠉⢉⠉⠁⢀⠀⠀⠀⠀⣐⠀⠒⠀⣠⡟
⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⠤⠀⠑⡀⠀⠀⠀⠯⠉⠁⠀⡿⠀⠀⠀⠀⠐⠖⠐⠀⢐⡇⠀⠀⠀⠙⠒⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢀⠀⠀⡇⠀⠈⠢⢤⣀⡤⠠⠁⠀⠀⠀⠀⠀⡀⡀⠀⢊⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡊⠈⠁⠀⡇⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⡀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⣀⣠⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⢩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠒⠊⠉⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⡦⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⡴⢲⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⠋⠀⠘⡇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⢀⠴⠊⠢⡀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⢀⠔⠉⠀⠀⢠⣤⣴⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⣠⠖⠁⠀⠀⠀⠙⢆⠀⠳⡄⠀⠀⠀⠀⠀⢀⠔⠁⢀⠔⠁⠀⠀⠀⠀⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠈⢦⡀⠀⣠⠔⠁⣠⠖⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠈⢦⡀⠑⠞⠁⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠑⢄⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢠⣤⣤⣤⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⡤⢤⠤⠤⠤⠤⣿⣿⡿⣿⢤⠤⠤⢿⣿⣿⣿⠤⠤⠤⢼⣿⣿⡿⠤⠤⡤⢼⣿⣿⣿⠤⠤⠤⢼⣿⣿⣿⠥⠤⡤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸

⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⡬⠬⣄⠀⠀⠀⠀⢴⣿⣿⣷
⠀⢸⡇⣀⣀⣀⣀⡀⠀⠀⢀⣤⣿⣧⣤⣼⣷⣀⣀⠀⠀⠀⠐⠃⠗⠐⠂⠓⠀⠀⠀⠀⠀⢺⣷⣶⣶⣶⡆
⠀⣿⣭⣯⣽⣭⣯⣽⠀⠀⢨⡭⣭⠭⡭⢭⣭⣭⣭⠀⠀⠀⢠⣔⣏⣉⣗⣤⠀⠀⠀⠀⠀⢸⣟⢣⠚⣿⡅
⠀⣿⠛⠛⣿⣶⣷⣾⠀⠀⢸⣧⣿⣤⣧⣼⣿⣿⣿⠀⠀⠀⢀⣠⣃⣀⣣⣀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣷⡄

⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⢰⡛⠛⠛⢳⠀⠀⠀⠀⠛⠀⠛⠃⠘⠛⠛⣳⡀⠀⠀⠀⣀⣶⠶⠖⠒⠆
⠀⠀⣀⠀⢀⡀⠀⣿⡇⠀⠀⣠⣏⢳⡄⣴⢋⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣷⠀⢀⣀⡀
⢠⣾⣿⣷⣿⣿⣦⣿⣧⠀⠀⣺⣒⢸⡇⣿⢐⣺⡇⠀⠀⣤⣶⣾⢠⣴⣾⡆⣿⡇⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿
⢸⣇⣇⣸⣸⣘⣇⣿⣿⠀⠀⢺⠒⢸⡇⣿⠐⢺⠃⠀⠀⣛⣛⣛⣘⣛⣛⠃⣿⡇⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠁⠉⠁⠀⠀⠛⠛⠛⠛⠁⠛⠛⠛⠁
⠀⠺⣦⠀⣀⢠⣤⠀⠀⠀⠀⢸⣿⠀⠀⣠⣶⣤⠀⠀⠀⠀⠀⣀⣀⠀⡠⠢⡀⠀⠀⠀⠀⢸⡟⠛⠛⢻
⠀⠀⣾⣆⣿⣸⣿⠀⠀⠀⠀⢸⣿⡶⢶⣿⣿⣿⡷⠀⠀⠐⣿⣿⠀⡇⣇⢰⣷⠀⠀⠀⣰⠟⡀⠀⠀⠈⢳⡄
⠀⣴⠿⠿⠿⡿⢿⢧⠀⠀⠀⢸⣿⣟⣻⣿⣿⣿⣏⠀⠀⢸⣿⣿⣦⣿⣿⣽⣿⠀⠀⢸⡇⠀⡷⠶⣿⠀⠀⢹
⠀⣿⠉⠉⠩⡯⢿⢽⠀⠀⢠⣼⣿⣯⣭⣿⣿⣿⣧⠀⠀⣸⣿⣿⣏⣹⣿⣸⣿⠀⠀⢸⡇⢀⡷⠒⣿⠀⠀⣸
⠀⠉⠀⠀⠈⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⢀⣄⣴⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⣀⣛⣛⣛⣛⣛⣛⡀⠀⠀⢀⣀⣤⣷⠒⠻⢿⠀⠀⢸⣿⢰⣿⣿⢀⢸⣿⡗⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿
⠀⢿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣿⠀⣼⢿⡆⠀⢸⠀⠀⢸⣿⣸⣿⣿⡍⠹⣿⠇⠀⠀⢀⣤⣤⣤⣴⣿⣿⣿
⠀⠘⠻⣿⠟⠿⣿⠟⠀⠀⢰⣿⣴⣿⣾⣿⡀⣸⠀⠀⢸⣿⣿⡀⢸⣷⠶⠛⠀⠀⠀⠸⢼⠤⡧⠾⢼⠤⣿
⠀⠙⠛⠛⠛⠛⠛⠛⠋⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠈⠉⠁⠉⠈⠉⠉

⠀⠀⠀⠀⠀⣀⣠⠚⣿⣷⣄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⡠⠊⠁⢸⣧⠀⠙⠛⡿⠀⠀⠀⠀⠀⣿⣿⢀⣀⠠⡤⠀⠀⠀⠀⠀⠀⠀⢠⣭⠀⠀⢀⣀⢸⣿⡇
⠀⠀⠅⠀⢀⣸⣿⣿⣶⠋⠀⠀⠀⠀⠀⠀⣿⣿⠈⣿⠰⣶⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣀⢸⣿⢸⣿⡇
⠀⠀⢳⣾⣿⣯⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⣿⣿⢀⣿⢸⣿⢰⣶⠀⠀⠀⠀⠀⢸⣿⢸⡏⢸⣿⢸⣿⡇
⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠈⠉⠈⠉⠀⠀⠀⠀⠀⠈⠉⠈⠉⠈⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡴⡄⢀⣤⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⣆⢠⣄
⠀⠀⢐⡀⠀⠂⠂⡎⠒⠀⠂⠀⠀⠀⠀⠀⡇⢀⣄⠙⢡⠾⠁⠀⠀⠀⠀⠀⠀⢸⠀⡟⢇⢋⡞⠹⡦
⠀⠀⢸⡇⣤⡀⠀⡇⡂⠉⠀⠀⠀⠀⠀⠀⣇⡞⡋⠳⠃⢐⠀⠀⠀⠀⠀⠀⠀⢸⡼⠀⠘⠿
⠀⠀⠸⠃⣿⡧⣶⠙⠁⠀⠀⠀⠀⠀⠀⠀⠿⠴⠶⠤⠤⠶⠦⠆⠀⠀⠀⠀⠀⠸⠧⠦⠤⠤⠤⠤⠴
⠀⠀⠀⠀⠈⠠⠧⠄

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠅⣼⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠉⣷⡡⠃
⠀⢠⡾⣆⠀⠂⣼⠏⢴⠆⠀⠀⠀⠀⠀⠀⠀⣠⡀⣸⣷⣴⣿⠀⠀⠀⠀⠀⠀⠸⠀⠙⢂⠜⣫⣾⣷
⠀⠀⠀⠹⡆⢠⡏⠈⣠⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣯⣿⣿⠀⠀⠀⠀⠀⠀⠘⢲⣾⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠹⡟⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠤⠿⠿⠿⠿⠿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⣼⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣶⣦⡀⢸⡟⣁⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣦⡀
⠀⠀⠀⠈⠙⠻⠌⠜⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⢠⡀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⠿⠛⠙⢏⡇
⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣾⡆⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠞⢛⣿⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣷⣿⣷⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠸⡆
⠀⠀⠀⠀⠀⣿⡥⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢾⡧⣿⠴⡧⢼⡧⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣸⠓⠒⢻⡀⠀⠀⠀⠤
⠀⠀⠀⠀⣸⡏⢉⡺⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢺⡗⣿⠒⡗⢺⡗⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣷⡖⠒⠒⢧⠀⠀⠀⠶
⠀⠀⠀⠠⣿⣿⣧⣤⣾⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡇
⠀⣀⣀⣀⣀⣤⣤⣴⣶⡞⢠⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⢹⠋⠉
⠀⠻⣿⣿⣿⣿⣿⣿⡟⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⢠⣤⣄⣼⣿⡁⢹⣿⠏⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣽⣄⣀⢀⣀⡀
⠀⠀⠈⢻⣿⠿⠟⠋⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣭⢨⣭⣯⣿⣭⡅⠘⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠊⢡⣏⣺⣯⠽⠛⠉⠁
⠀⢰⡟⢳⡴⠖⠻⣭⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣬⣿⣼⣿⣯⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢳⣿
⠀⠈⠓⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠖⠓⢾⡇


⠀⠀⠀⢀⣀⣤⣄⣀
⠀⣠⡾⠋⣤⣶⣶⠍⠛⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠺⠓⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣯⣭⣤⣦⡔⣠⡀
⢠⡟⠀⠀⠘⢿⠏⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣀⣆⣀⣀⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣵⣿⣷⡀
⢸⡇⣶⣶⣶⠎⢶⣶⣶⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⣽⢹⣿⣿⡗⡀
⠀⢻⣜⠿⠋⠀⠈⠻⢟⣼⠃⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣉⣿⠻⠟⠃⢻⣧⣀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣇⣺⣿⣿⡇⡇
⠀⠀⠉⠳⠶⠴⠴⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠿⠿⠷⠝⠿⣿⡯⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠒⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠄⠴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡇
⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢠⡇⠀⠀⠀⣸⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠁⠀⠀⡞⠀⠈⡄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣜
⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⠀⠀⠀⠀⣀⠀⢀⣾⣿⡟⠁
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠈⠉⠉⠁⠀⠀⠈⠀⠀⠀⠀⠙⢷⣿⣿⣻⣷⡶⠦
⠀⠀⠈⡇⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⠀⣀⠀⠀⢀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⡿⡿⡫⠊
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠇⠀⠀⠊⠁⠀⠐⠰⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠐⠂⠀⠒⠄⠀⣍⣋⡁⣥
⠀⠀⢠⡇⠀⠀⠀⣀⢴⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠄⠀⠈⠋⣿⡏
⠀⠀⢸⠀⠀⠀⠀⡅⠀⠥⠄⠀⠘⠐⠀⢐⡐⣷⠀⠀⠀⠀⠀⠀⠀⢲⡀⠁⠀⠀⠀⠁⠀⠐⠇⢠⠀⠀⢷⣿⡎
⠀⠀⠈⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⣀⣀⣀⣠⣄⣀⣀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠿
⠀⠆⣉⣉⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⠀⠀⢀⣀⣀⠄

⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⠀⠀⠉⠐
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠃⢀⠀⠀⡂⠁⠀⠈⡀⠅⠀⠀⠀⠀⠀⠛
⠀⠀⠉⢀⠈⠉⡁⠀⢉⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠆⠀⠀⠐⠀⠈⠤⠀⠀⠊⠀⠀⠀⠀⠀⡀⡅
⠐⡘⠄⠀⠀⠀⠙⡀⠀⠀⠀⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠶⢂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠻⡁⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⠀⠀⠀⠂⢁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢠⠦⢸⡀⠀⠀⠀⠋
⠀⠘⠁⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡢⣀⠈⠁
⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⠓⣾⠎⠀⠀⠀⠠⠈⠉⠉⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⡄⢠⡆
⡌⠀⡀⠀⠀⠴⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⡀⠀⠠
⠀⠀⠈⠀⠀⠈⠀⣼⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⡆⠀⠀⢀⡄⠠⠀⠃⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⣂⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⠀⠉⠀⠀⢀⠒
⠀⠀⠈⠀⠀⠀⠀⠀⢀⡀⢠⠀⠀⠀⠀⠀⢀⠀⠈⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⡀⠀⢄⠀⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢠⣀⣀⣀⣀⠀⢀⣀⠈⠁⠀⠀⠀⠀⠀⠸⠀⠀⡀
⠀⡀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⢤⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠎⠐⠂
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⠄⢈⢤⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢐⠈⠀⠀⠀⠀⠀⠂⠂⠂
⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠠⡆⠀⠀⠀⠈⠈⠉⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⡆⠂⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠐⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠀⠀⠀⠀⡀
⠀⠀⠈⡁⠀⢀⠀⠀⠠⠄⠐⠀⣠⠀⡀⠀⠞⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⢄
⠐⠀⠁⠢⣤⠋⠀⠀⠀⢠⠸⠈⠉⠀⠩⠄⠀⠐⠐⠚⠀⠀⠒⠀⡀⠀⢀⠘⢩⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡠⠀⠀⢀⠀⠀⠑⡄
⠀⠀⢀⢀⡄⡀⠀⠀⠀⢰⡀⠀⠄⢃⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠂⠄⠠⠀⠀⠈
⠰⠀⠃⠀⠀⠀⠄⠠⠂⠀⠁⠀⠀⠀⢃⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠆⠀⠀⠂⠀⠃⠀⣠⠄
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⢡⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⡀
⠀⠶⢤⣦⠂⠀⠀⠀⡈⠐⠲⠒⠂⢸⠀⠀⠀⠀⠁⠀⡠⠀⢀⡄⡀⢀⠉⠀⠠⠄⠀⠀⠀⠀⠒⠎⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢠⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠁⠀⠐⠀⠀⠀⠀⢀⠄
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠂⠀⠀⠄⠀⠉⠐⠀⠀⠓⠀⣧⡄⠀⠀⠀⠀⠀⠀⠀⣀
⡠⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠐⠀⣠⠀⣀⠀⠀⠀⠀⠩⠕

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠚⠛⠛⠲⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠏⠀⠀⠀⠀⠀⠈⠉⠉⠙⢦⡀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢶⡄⠀⣯⣀⡿
⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣉⣀⣀⡀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠏⠀⢸⡏⠉⢹⡇⠀⠀⠀⡟⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠶⠶⠾⣆⠀⠀⠀⠀⠀⠀⠀⣼⠋⠁⠀⠀⢸⣇⣀⣸⡇⠀⠀⠀⣧⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⣀⣀⣤⠞⠁⠀⠀⠀⠀⢸⡏⠉⠨⡇⠀⠀⠀⡏⠈⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⢴⡇⠀⠀⢰⡷⠶⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⣸
⠀⠀⠀⠀⠀⢤⣤⢤⣤⠤⠤⣤⠤⣤⠤⠀⠀⠀⣶⠒⠒⠒⠒⠒⠒⠒⠛⠓⠒⠚⠛⠒⠒⠚⠓⠒⠒⠛⠓⠒⢲
⠀⠀⠀⠀⠀⠀⡧⠀⡇⠀⠀⣿⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠀⢸
⠀⠀⠀⠀⢀⣀⣟⣀⣇⣀⣀⣿⣀⣿⣀⠀⠀⠀⣿⠀⠀⠀⠀⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠿⠀⢸
⠀⠀⠀⠀⢸⠉⣭⣭⡍⠉⠉⣭⣭⣍⢹⡇⠀⠀⣿⠀⠀⠀⠀⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⡦⠀⢸
⠀⠀⠀⠀⢸⠀⠷⠴⠇⠀⠀⠿⠤⠿⢸⡇⠀⠀⣿⠀⠀⠀⠀⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠗⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⡶⠶⠶⣦⠀⠀⢸⡷⠶⠶⣿⠶⠶⠶⠶⠶⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⣇⣀⣀⣿⠀⠀⢸⡇⢀⡴⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠳⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⢠⣤⣬⣭⣬⣥⣤⡀⢸⡧⣾⠀⢤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠄⢹⡦⣤⡀⠀⢸
⠀⠀⠀⠀⢸⠀⢺⠀⠀⠀⠀⠀⢸⡇⢸⡇⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠈⣷⠀⢸
⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠈⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⣸⠀⣸⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡀
⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢶⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⡉⠁⠀⠀⠈⠌⠉⠛⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⢀⠀⠀⢀⡠⠶⠆⠭⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣥⡄⠀⢀⡴⠋⠠⢔⢤⡀⢾⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣯⡴⡶⡾⠁⠀⢰⠟⠿⠏⠂⠙⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿⢿⣄⡄⠀⠀⡀⠀⠈⠓⢒⡞⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⠿⡩⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥⣤⠖⠉⢴⠐⠤⡀⠂⠖⢯⠿⠟⣳⡀⠀⠀⠀⠀⠀⢀⡏⡞⣮⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠏⡄⢁⣤⡤⠈⠷⠀⣇⡀⠀⠀⣤⠀⠘⣽⠀⠀⠀⠀⠀⢸⠈⠁⡇⢫⣸
⠀⠀⠀⠀⠀⠀⢠⣶⡋⣉⠈⠀⠂⠑⠀⠉⢻⠀⢐⡀⢬⡄⠁⡉⣷⠦⣀⡗⡄⠀⠀⢀⣄⣸⡀⡀⠀⡜
⠀⠀⠀⠀⠀⢠⣿⡏⠁⠀⠁⠒⢄⡀⠀⠀⠀⠉⠖⡂⠀⡀⠀⠙⠣⡁⡎⣦⠀⠀⣀⣨⡝⠛⢍⢃⡜
⠀⠀⠀⠀⠀⢸⡿⡉⡄⠀⠀⠠⠀⠉⢢⣀⣀⠀⠀⠳⡀⠐⢲⡀⠀⣥⠺⣥⣠⠼⠿⢌⠳⠀⡀⢹⡄
⠀⠀⠀⠀⢀⡾⠑⠁⢀⠀⢰⢰⢜⠜⠀⠀⠀⠉⠓⢄⣃⠘⣧⡁⢀⠃⢈⡎⠉⢧⠀⠈⢳⠀⢀⣼⣿⡀
⠀⠀⠀⢠⡞⣀⣀⢀⣀⠠⠀⣯⡀⠀⠀⠀⠀⠀⠀⠀⡌⠣⢸⡧⢘⠀⠈⠀⠀⡬⢀⠀⣼⢹⡿⠋⠛⠁
⢀⣠⣀⠸⡇⠡⠖⠉⠐⠓⢦⡘⣇⠀⠀⠀⠀⠀⡀⠀⢻⠀⠸⡿⢁⢀⠁⠀⠀⡁⢠⡷⣧⡯⠒⢻⣿⣿⡲⣤⡀
⣯⢂⣬⣷⡧⠀⠀⢀⣲⠤⡀⢘⡳⣌⣈⡄⠀⠀⠀⠀⢸⡀⠹⠃⡄⢸⢜⠆⣼⢅⡞⠉⡟⠀⢀⠞⡉⣻⡇⠈⢻
⣿⡘⢋⣯⡅⢀⡰⠛⠁⠊⢠⠋⠀⠉⠉⢰⣠⣤⣠⣄⣸⣏⢐⣴⢀⣜⣻⢹⡿⠛⠙⢾⡄⠀⠘⣄⠈⠉⠀⣠⣿
⣿⠿⠃⢰⡧⡗⠳⠄⠀⠰⠂⠀⠀⠀⠀⠋⠀⠀⠠⣐⠲⢭⣛⣏⢻⣼⡧⢸⠷⠀⠀⢀⠁⠀⠀⢈⣷⢒⣫⢋⡞
⣿⣲⡀⡀⢀⣀⡢⢝⣇⡷⡀⠀⠀⠀⢀⠖⡰⢄⠑⢂⠀⢱⠍⠋⠛⠛⡿⣿⠀⢴⢷⢾⡆⠀⠙⠿⡭⠿⠞⠋
⣘⣿⣷⣷⣘⣯⣿⣌⣙⣂⣀⣀⣀⣀⣫⣜⣣⣠⣔⣁⣀⣁⣓⣦⣤⣘⣢⣼⣿⣾⣿⣶⣾⣾⣖⣊⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡄⠀⠀⠀⠀⠀⢀⡤⠴⠖⠒⠒⠒⢦⡀⠀⠀⠀⠀⠀⠀⢀⡠⣤⢠⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢸⠀⡇⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⡴⠋⠀⢸⢸⠀⣨⠗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡇⢸⢸⠀⡇⠀⠀⠀⡼⠁⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⢳⠀⠀⢰⠃⠀⠀⢸⠈⣫⠴⠊⢹
⠀⠀⠀⠀⠀⠀⠀⣀⣀⢸⠀⡇⡇⢸⢸⠀⡇⠀⠀⢠⢣⡄⠀⠀⡿⡄⢶⣄⠈⢇⠀⠀⡀⡇⠀⠘⣆⣤⡀⠘⠋⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⠀⡇⡇⢸⢸⠀⡇⠀⠀⢸⢸⠁⠀⢸⣷⣟⢼⣎⣗⣾⡄⠀⡇⢸⠀⠀⢿⠀⣗⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠴⠧⠾⠼⠴⠷⠧⠼⠾⠤⠧⠄⠀⢸⢸⡀⢀⣇⣘⣠⠋⢧⣘⣀⣇⠀⡇⢸⠀⣴⣚⣈⣓⡍⠉⠁⠀⠀⠀⠀⢰⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⢧⣸⣆⠀⢀⣐⣀⠀⢀⠿⡀⣇⡞⠀⠀⠀⠀⠀⠀⠀⣰⢦⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠙⡷⣄⠀⠀⠀⠀⠘⢮⡟⣨⡳⡤⣉⡡⡴⢋⣤⣱⡟⠀⠀⠀⠀⠀⠀⠀⡠⠋⠉⠓⣤⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣛⣀⡉⠙⡇⠀⠀⠀⠀⠀⠈⠁⣹⠇⠀⠀⣿⡉⠉⠀⠀⠀⠀⠀⠀⠀⠰⠯⠇⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⡠⠖⠒⠦⣼⠀⠶⠶⠆⠀⡇⠀⠀⣠⠖⠚⠻⣋⢷⠀⠀⠊⢱⢋⡻⠒⠲⠤⣀
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢸⣀⣈⣉⣁⣀⡇⠀⡜⠁⠀⠀⠀⠈⠹⡖⠦⠶⡞⠉⠀⠀⠀⠀⠘⡆⠀⠀⢸⣉⣉⣉⣉⣉⣉⣿⣏⣹
⠀⠀⠀⠀⢀⡹⠀⠀⠀⡆⠀⠀⠈⣇⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢃⠀⢀⠃⠀⠀⠀⣤⠀⠀⢹⠀⠀⢸⠀⠀⠀⡴⠦⡄⠀⠀⢸
⠀⠀⠀⠀⡎⠀⠀⠈⠳⡷⠊⠀⠀⠈⢧⠀⢀⡇⠀⠀⢀⠀⠀⠀⠀⠸⡄⡼⠀⠀⠀⠀⣇⠀⠀⠀⣇⠀⢸⠀⠀⠀⠳⠶⠧⠀⠀⢸
⠀⠀⠀⠀⠱⢄⣀⣀⣀⣀⣀⣀⣀⡠⠏⠀⡼⠀⠀⠀⢸⠀⠀⠀⠀⠀⢳⠃⠀⠀⠀⠀⣿⠀⠀⠀⢹⡀⢸⠀⣒⣒⣒⣒⡒⠒⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠸⣤⣤⣤⣤⣤⣤⣤⣤⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠒⠦⣾⠀⠀⠀⠀⠀⠷⠇⠀⠀⠀⠀⣯⡤⠤⠶⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣲⣿⠭⢬⣣⣴⠃⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⣼⡝⢶⣄
⠀⠀⠀⠀⠀⠀⠈⠉⣽⣷⡾⠋⠀⠀⠀⡵⣝⣯⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠉⠉⠚⠻⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠧⠤⠤⠴⠊⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠠⠀⡠⠀⠀⠀⠄⡠⠄⠒⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠅⠀⠀⢰⢠⣀
⠠⢊⠷⠲⠄⠀⠀⠀⠀⠀⠀⢦⡀⠀⠐⢄⠠⠊⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⡦⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠒⠢⡀⠀⠀⠀⠀⠀⠐⢒
⠉⠊⠛⠛⠅⠔⠅⠲⠄⠀⢀⠐⣠⠞⠁⢀⡀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⢀⠄⠄⣀⣤⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠊⠍
⠀⡠⢤⣄⠀⠀⠀⠘⠀⠀⠈⠘⣅⠀⠔⠁⠙⢦⣀⠔⠀⠀⠀⠀⠄⡀⠈⠀⠊⠁⠀⠁⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠈⣜⠀⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠶⠦⢄⠀⠀⠀⠄⣀⡀⡀⠂⠐⠔⠤⠤⠦⠄
⠀⠈⠁⠐⠈⠀⠀⠀⠀⠀⢄⣀⠀⠀⠀⠀⠀⢲⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡀⠀⡠⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢀⡀
⠀⠠⠀⠐⠤⠤⠐⠒⠈⠉⠈⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠉⠐⠒⠠⠠⢄
⠀⠀⠀⠀⠀⡠⡄⠀⢤⢐⡪⠭⠤⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠡⠀⠃
⠀⠀⠒⠅⠐⠃⠀⠈⠀⠀⠀⠀⠀⠂⠀⠈⠝⠛⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠐⢀⠀⠀⠀⠀⣀
⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⣀⣰⣄⠀⢀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠂⠀⠙⣴⠀⠀⣧⠁⠀⠑
⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠒⠒⠒⠚⠂⠀⠀⠀⠁⡀⠀⠉⠀⠀⠀⠘⠆⢀⠠⠇⡆⠀⠈⠢⡄⠔⠀⠀⡀
⠀⠀⠢⠠⠂⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⡲⠶⢄⠀⠀⠁⠀⠀⠀⠀⠄⠂⠀⠀⠰⠐⠤⠄⠀⠀⠀⠅⡇⠀⠀⠀⠀⠀⠀⠀⠲
⣳⠄⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⡇⠀⠀⠀⠈⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠐⠀⠀⠀⠀⠀⠀⡀
⠀⢀⣄⠀⠁⡀⠀⠀⠀⠐⣆⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢁⠀⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢩⠔
⠂⠁⠈⢧⣀⠠⠀⠀⠀⠢⡀⠀⠁⠑⠂⠤⢄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡰⠀⠀⠐⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠁⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠐⠢⠀⠠⠀⣄⡀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢧⠆
⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠂⡐⠁⠀⠀⠀⢀⠆⢀⡀⠀⠀⠀⠉⠉⠁⠓⠀⠀⠀⢃⠀⠈⡄
⠀⡀⠀⠀⠅⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐⠖⣄⡀⠀⠙⠢⠈⠀⡠⠁⠀⠀⠀⠒⠒⠰⠭⠡⠰⢠⣐⣒⠐⠀⠀⠀⠘⡄⠀⢸
⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠉⢂⠀⠀⠀⠀⠈⠉⡶⣄⠀⠀⠪⢀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠄⠀⡀⠀⠉⠀⠀⠀⠀⠐⠀⠀⢣
⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠠⢊⡔⠁⠀⣄⣄⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠁⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⡀⠀⠊⠀⠉⠓⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⡀
⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣒⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢰⣶⣿⡯⠍⠀⡭⣙⠙⠀⠀⠢⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡽⠾⢿⣿⠛⠁⠀⠀⠴⠀⠈⠁⠲⣄⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢿⠅⢀⣴⣿⣃⣠⣄⣀⡀⠀⠔⠀⠀⠀⠈⠛⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠁⡤⢫⣿⣿⠏⠀⠀⣉⠙⠳⢦⡀⠀⠈⠀⠀⠀⠳⣤⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠫⠐⡴⠃⢠⣿⡏⠀⠀⠘⠛⣆⠀⠈⠻⢶⡀⠀⠀⠀⠀⠈⠹⡄⠀⠀⠠⣀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢟⠀⠀⣸⡟⠁⠀⣿⣿⣄⢘⠂⠀⢠⠃⠀⠀⠃⠌⡿⣶⡇⠂⠐⠲⠀⢷⡀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢻⣆⠀⡼⢻⣿⣿⠁⠀⡴⢦⡞⢉⣾⠄⠀⠒⠈⠻⣶⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣗⠀⣿⠀⢸⣿⠀⣦⠀⣾⣥⣾⣿⢿⣶⣴⢀⣴⣻⠿⣾⣶⡆⠀⣻⣦⠀⢀⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⢏⡀⣿⠀⠸⣿⣿⣇⠀⣇⠈⢉⡡⠾⣿⣿⣧⣸⠁⠸⠟⣿⠃⠠⠃⡆⠀⡏⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠸⠐⣿⣄⡀⠈⠛⢿⣷⣿⣧⣾⣷⣤⣿⣿⣿⣿⣤⣤⣴⣟⣤⣖⣨⠃⣘⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄⠀⠀⠀⠀⠀⢻⣟⣫⣿⣦⡈⣠⣭⣿⣽⣿⣿⣉⢻⢿⣿⣿⣿⣿⣿⣿⡏⣈⢧⡀⠟⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠻⠋⠉⠉⠛⢻⣾⡿⣛⢻⣿⡿⠟⠻⢿⣿⣿⣿⣿⣿⣧⣥⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠈⠛⠿⠿⠟⠷⠷⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⠅⠀⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠄⠀⠀⢘⠟⠋⣿⣧⣿⣿⡿⢽⡿⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢵⡄⠈⠀⠘⣭⡾⢶⣿⡻⠟⠋⣰⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣉⣁⣄⣠⣬⠋⠀⠀⣽⣿⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣟⠋⠁⠀⠀⢀⣀⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉




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

⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⡠⠀⡀
⠀⠀⠀⠀⠀⠀⢄⡐⢠⠤⠁⢄⡖⠘⠀⠂
⠀⠀⠀⠀⣤⠐⡩⠛⢶⣀⡱⠏⠀⢰
⠀⠀⠀⠀⠈⠁⢇⣀⠤⠟⠷⠄⢒⠁
⠀⠀⠀⠀⠀⠀⠈⡣⠤⢀⡀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠟⠻⢿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⣿⠀⠀⢨⣿⣿⡇⠀⠀⣿⣿⡇⢀⡀⠀⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣶⣬⣙⠛⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣷⣦⣌⣙
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⣿⠀⠀⢈⣿⣿⡇⠀⠀⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⠟⠛⠻⣿⣿⡟⠛⠛⣿⣿⡿⠛⠛⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣤⣄⣤⣿⣿⣧⣤⣄⣿⣿⣧⣤⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣧⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉



⣿⠿⠿⠿⡿⠻⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⣿

⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⣿⠉⢙⡛⠛⠲⡆⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠶⠶⠶⠶⠆⣠⠀⠀⠐⠶⠶⠶⠶⠶⠶⠶⠖⠀⠀⢠⡀
⠀⢀⠼⣿⠀⠀⠙⠗⠀⡇⠀⠀⠀⢀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠁⠀⠀⠀⠀⠀⠰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿
⠀⠘⠀⠛⠛⠛⠛⡛⠛⠃⠀⠀⢀⠈⣶⡾⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠂⠀⠀⠀⠀⠈
⠀⠀⢸⣇⠈⠩⠉⣭⣭⣧⡄⠀⣿⠀⠉⣶⣾⢻⡇⠀⠀⢀⠀⠸⠆⢀⣷⣦⣤⣀⢀⣀⠀⠀⣉⠀⠀⠀⠛⠻⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠛⣿⠛
⠀⠀⢸⣿⣷⡄⠐⠊⠳⣟⣿⣦⡘⠂⠀⠉⡿⠾⠆⠀⠀⣿⡗⣴⣶⣮⠉⠉⠉⠉⣉⣿⠀⣀⡀⡀⢀⣀⣀⠀⣀⣀⡀⢀⣀⣀⠀⣀⣀⡐⣾⣿⣿⠖⠛⣷⠄
⠀⠀⢸⣿⣿⣿⡦⢢⡄⡄⣤⢤⠭⠆⠀⠀⠀⠀⠀⠂⠘⣿⡇⣿⣿⣿⠀⠀⠀⠀⣭⣽⡁⣿⣿⡇⢸⣿⣿⠀⣿⣿⡇⢸⣿⣿⠀⣿⣿⡇⠙⠉⠁⠲⠴⡟
⠀⠀⢸⣿⣿⣿⡇⠎⢱⣥⣍⣿⣅⣣⣤⡀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⢠⣶⣶⣶⣶⣶⣶⣿⣿⣷⣼⣿⣿⣶⣿⣿⣷⣾⣿⣿⣶⣿⣿⣷⣶⣶⡆
⠀⠀⢸⣿⣿⣿⡏⠚⠛⠛⠛⠛⠛⠋⠉⠉⣴⣶⣶⣶⣶⣶⣶⣶⡶⣶⡄⣴⣿⢀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣴⣿
⠀⢠⢸⣿⣿⡿⣧⣀⣀⣀⣀⣀⣀⡀⠀⠀⠛⣿⢿⣿⣿⠟⠛⠛⠓⠛⠃⢿⣿⠈⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠙⠛⠛⠛
⠀⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢰⣶⣶⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡄
⠀⠛⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠃⠀⠀⠘⠛⠉⠉⠙⠛⠃⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠘⢿
⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢠⣤⣤⣤⣴⣶⣶⣶⡏⠋⢠⣤⡄⢡⡀⠀⠀⠀⠀⢀⠄⠀⠀⣀⡤⢀⠀⠀⠀⡶⢻
⠀⠀⠀⠀⢸⡿⣿⢿⡇⠀⠀⠀⣿⡿⠿⢿⣿⠿⢿⡆⠀⠀⠀⠰⣿⡷⡹⠿⠿⠿⠿⠿⠛⠓⠂⡈⠋⢁⣼⣷⡀⢀⡠⢂⣭⠐⠿⠂⠧⢆⡸⠋⠉⠀⠛⣾
⠀⠀⠀⠀⠘⠀⠀⠈⠃⠀⠀⠀⣿⡰⡿⢂⡇⢾⣿⡇⠀⠀⠀⠀⣶⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⣽⣷⡡⣶⣿⡿⠛⠀⠀⠀⠀⠀⠁⠀⠀⢾⣸
⠀⠀⠀⠀⠀⡠⠀⢄⠀⠀⠀⠀⣿⣷⣶⣾⣿⣶⣾⠇⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⡝⠋⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡀⢻
⠀⠀⠀⠀⠀⠻⣛⠟⠀⠁⠀⠰⠛⠁⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠘⠀⢀⣶⡀⠀⢰⣆⠀⠀⣰⣆⠀⠹⠋⣴⠶⣦⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⢀⠀⢠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢀⡆⠀⠀⣠⠀⠀⡘⣷⣝⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢾
⠀⢰⠏⠀⣰⠉⠯⠉⠉⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠾⠇⠀⠰⠿⠀⠀⠁⠂⠤⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⣤⣤⣤⣥⠤⠤⠤⢤⣤⣤⣦⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠘⠓⠒⠛⠒⠒⠚⠛⠓⠒⠛⠛⠒⠒⠒⠒⠂⠀⠀⠀⠀⢀⣴⡆⣦⣄⢠⣶⠶⡶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣿⣦⣤⣤⣤⣿⢿⢿⠀⠀⠀⢠⣄⣤⣤⣤⣤⣤⣤⣤⠀⣤⠶⡄⠀⠀⠀⠀⠀⠀⠀⢺⠿⢁⠻⢿⢸⣧⣣⣤⣿
⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⢙⡋⠚⠛⠛⠛⠛⠛⠛⡀⠛⠶⠃⡀⠀⠀⠀⠀⠀⠀⠈⢀⣀⣀⠉⠘⠉
⠀⠀⠀⠀⣿⣿⡿⣿⢿⣿⣿⠀⠀⠀⠀⣰⣿⣿⣿⢿⣿⢿⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡀⣿⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⡙⠷⠟⡀
⠀⠀⠀⠀⣿⣿⠇⠿⢼⣿⣿⠀⠀⠀⠀⣿⣿⣶⣴⣽⣿⣼⡿⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣤⣤⠀⠀⠀⣶⣿⣿⣷⣶⣾⢻⣿⣶⡄
⠀⠀⠀⠀⠛⢛⣛⣛⣛⣻⠛⠀⠀⡄⠀⡹⠛⣻⠛⡟⢻⡋⠠⡀⠠⡀⠀⠊⠑⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣼⣿⣿⣇⡀
⠀⢀⣠⣴⣾⣿⣿⡿⠋⠁⠀⡠⠊⢀⡜⠁⢠⠃⠀⡇⠀⢣⠀⠙⣄⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⠀⠀⢸⣿⡆⠀⠀⢐⣟⠂⠀⠀⠀⣿⣿
⠀⡏⠩⢉⠙⣿⣿⠀⣾⢻⣿⢻⡟⣿⡿⣻⣿⢻⣿⢻⣿⢻⡿⢻⡟⣿⡟⣿⡟⣿⡄⠀⠀⠀⢀⣀⣀⣿⣇⣀⣀⡀⠀⢸⠿⡇⠐⠂⣭⣭⠁⠈⠀⢠⡿⣿
⠀⠓⠒⠒⠚⠉⠛⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠀⠘⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⡠⢴⣖⠓⠒⢤⡀
⠀⠀⠀⠀⠀⢰⢿⠤⣁⣡⡐⢓⡖⢻⣤⡁⠠⣵⡀
⠀⠀⠀⠀⠀⠈⡰⠺⡤⠛⡓⣺⡂⠈⠁⡌⢗⣼⡇
⠀⠀⠀⠀⠀⠘⠧⣑⡧⣼⡏⠈⠃⢰⡞⢩⢈⣗⡃
⠀⠀⠀⠀⠀⠀⠀⠘⢶⡬⣍⠓⢀⣽⢰⠾⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⢓⡠⠀⠓⠤⠊


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

⢠⡄⠂⠀⠀⠤⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⢀⣼⡅⠀⠀⠀⠀⠀⢀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄
⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣮⣽⣿⣿⡇⠀⠀⠀⠀⠀⣿⠀⣾⡇⣧⡄⠀⠀⠀⠀⠀⠀⡛⠛⠛⠒⠢
⢸⡇⢄⢸⡇⢠⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⡇⠀⠀⠀⠀⢀⣿⠠⣿⠇⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠄


⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⣰⡾⠛⠙⠻⣷⡄⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢀⣾⣽⠖⠲⡿⢦⠀⠀⠀⠀⠀⢀⣞⣉⡇⢹⠱
⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⣟⣽⣿⣿⠅⠀⠀⠀⠀⢸⣸⡅⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⣲⢖⡃⡺⢄
⠘⢿⣶⣴⣾⠗⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠻⢯⡁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢩⣷⣻⠗


⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⣤⣤⡀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣼⡆⢀⢈⣿⢀⠄⠀⠀⠀⠀⣠⡀⢁⢂⣠⣶⡤⠢⠀⠀⠀⠀⠀⠀⠀⢀⠐⠢
⣿⣿⣧⣤⣾⣿⡇⠀⠀⠀⠀⠀⢻⡇⠈⠘⡿⢀⠀⠀⠀⠀⠀⠸⠱⣦⡦⢫⢔⠔⠳⠄⠀⠀⠀⣯⣱⣾⣿⣶⣶
⠛⠛⠛⠻⠿⠟⠇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠛


⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⢀⣀⢀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠
⣼⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⡇⣠⠿⢿⡃⠀⠀⠀⠀⢨⣭⢨⣭⠈⠉⠁⠀⠀⠀⠀⠀⢀⡀⣴⢰⣿⢸
⢿⣿⣯⣽⣿⣿⠇⠀⠀⠀⠀⠀⡿⠛⢾⣿⣦⣾⡇⠀⠀⠀⠀⠰⠶⠰⠶⠰⠶⠲⠶⠆⠀⠀⠀⢸⡇⣿⠨⣿⢸
⠀⠏⢀⠻⠛⠁⠀⠀⠀⠀⠀⠀⠿⠶⠿⠿⠿⠿⠃⠀⠀⠀⠀⠘⠛⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠙⠀⠋⠘
⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢗⣻⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢵⢟⣩⢿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⠏⣿⣳⣿⣿⣯⣞⣯⣿⢿⡿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠜⠁⠙⠙⠉⣾⠋⡩⣻⣟⢕⠿⣻⢻⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠺⠿⣷⠿⣻⣷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢊⣠⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣫⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣻⣿⣿⣠⡔⠁⠀⠀⠀⠀⠀⠀⣠⣾⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢿⣿⣻⣿⣾⣿⣟⠕⠀⠀⠀⠀⢀⠀⡺⣫⣽⢟⢽⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⢛⣿⠟⠁⠀⠀⠀⠀⠁⠀⠈⢫⣴⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢇⣬⢠⡔⠀⠀⠀⠀⠀⠀⢀⣤⠚⠁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣧⣾⣿⣿⣻⣷⠛⠔⠀⠀⠀⠠⢛⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⠿⢿⣿⡿⠃⠀⠀⠀⠀⠺⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣡⣴⡷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠁

⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⡇⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢠⠶⡄⠀⠀⠀⣀⠤⠖⠒⣶⠒⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢸⠾⡅⠀⣠⠊⠁⠀⠀⠀⡯⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢸⠶⡇⡰⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢸⣶⣣⡇⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠸⠤⠻⡏⠉⠉⠉⠉⡩⢽⠏⠉⠉⠉⠉⠉⠉⢹⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢣⠀⣀⠔⠋⠀⡞⠀⠀⠀⠀⠀⠀⢀⡞⠀⢠⠒⠒⠻⣦⡀⠀⠀⠀⣀⠴⢦⡀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠻⣅⠀⠀⡼⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⢸⠀⠀⢀⡇⠙⣦⠔⠋⡥⠤⠤⡌⠢⣄⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠥⣀⣀⣀⠤⠔⠊⠁⠀⠀⠀⢸⠉⠉⠉⠗⠋⠉⠙⡆⣇⣀⣀⡇⠀⠈⠛⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠚⠚⡇⠓⠒⠚⡇⣇⣀⣀⡇⣎⣉⣉⣹⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠤⠴⡏⡋⠉⠉⡇⣇⣀⣀⡇⡯⠤⠤⢽⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢀⡎⣉⣉⣉⡇⣇⣀⣀⡇⣟⣐⣂⣻⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠉⠉⠉⠀⠉⠉⠉⠁⠈⠉⠉⠁⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠒⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠁⠉⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⡠⠒⣶⠀⠘⡄⠀⠀⢀⠄⠊⠢⡀⣠⡶⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠉⢉⢤⠾⠋⠄⡰⠁⠀⠀⢿⣦⡀⠀⢈⠁⠀⠀⡠⠂
⠀⠀⠀⠀⠀⢠⡤⠤⠛⠃⠀⠀⠀⠈⠈⠩⠍⡇⠀⠀⠀⢺⡇⠀⠀⢂⣀⠀⠤⠄⠠⠇⠀⠀⠀⠀⣰⣕⠁⢀⡴⠊⠀⠀⠀⠀⠀⣨⡟⢀⠀⠀⠀⢪⡀
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠐⠀⠀⠀⠀⠂⠀⢀⡇⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢙⣣⠤⠌⠀⠀⠀⠀⠀⠠⢾⡁⠀⢀⠤⣦⡀⠀⠑⡄
⠀⠀⠀⠀⠀⠹⣧⡠⠤⣶⡆⠀⠁⢸⠉⠉⠁⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⡀⠈⠀⠜⠀⠀⠀⠀⠻⣁⡂⠆⠀⠀⠀⠀⠀⠀⠀⠙⠒⠁⠀⠈⠛⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠈⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣴⠖⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠈⢣
⠀⠀⠀⠀⠀⢰⡟⠹⢑⡄⣴⣻⠋⠐⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⣸⡇⠀⣶⠶⠒⡄⠀⠀⠀⠀⣿⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠑⠒⢋⣴⡷⠁⠰⢀⠜⠀⠀⠀⠀⠀⠀⢀⣄⡤⢤⠴⠶⠒⠚⠛⠛⠛⠓⡄⠀⠸⢇⣀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⢸⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢠⢎⠞⠀⠀⢠⠊⢀⡀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣼⠟⠀⣠⡐⠑⡄⠀⠀⠀⠀⠀⠈⣷⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡴⡳⠋⠀⠁⡔⠁⠠⡟⠭⠳⡄⠀⠀⠀⠀⠙⠒⠒⠒⠂⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠲⠊⠈⠻⠒⠁⠀⠀⠀⠀⠀⠀⠙⣒⣒⠁
⠀⠀⠀⠀⠀⠀⠻⡁⠀⢤⠎⠀⠀⠀⠙⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠁
⠀⠀⠀⢀⣠⣤⠤⢀⠀⠀⠀⠀⠀⠀⡴⡒⠀⠀⢠⣴⡂⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠙⢆⠀⠀⠀⣾⡏⠁⠂⠤⢀
⠀⠀⢠⢛⠋⠀⣠⡀⠙⡆⠀⠀⠀⢳⡇⠀⡁⠀⠀⠉⢹⡆⢸⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠄⣠⠖⠁⠀⠀⠀⠈⠁⠒⠤⢔⡀⠀⠑⢐⠄
⠀⠀⣇⡎⠀⠌⣸⣿⠀⢸⠀⠀⢀⢾⠋⢐⠁⠀⠀⠀⢅⡄⠸⢄⠀⠀⠀⠀⢠⣾⠟⠁⠀⢠⣋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⡮⠟⢁⡴⠋
⠀⠀⣿⡇⠸⣴⡿⠃⢀⡎⠀⠀⠀⢀⣣⠘⡆⠀⠀⠀⠜⠟⠀⡎⠀⠀⠀⠀⠀⠉⠓⠦⢅⣀⠈⢉⠓⠶⢤⠀⠠⣴⡿⠛⠙⢀⡴⠋
⠀⠀⠈⠳⣄⣀⣀⡤⠊⠀⠀⠀⠀⢸⡟⠰⠇⠀⠀⣤⣤⠖⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣤⠞⠀⠀⠘⢆⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠠⠤⠃⠀⠙⠃⠥⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⣴⡄⣠⣤⣤⣌⣭⣍⣩⡁⠀⠀⠀⠀⢸⣿⢣⣿⡿⣱⣿⡜⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⣰⠖⠒⠒⠒⠀⠀⠀⠀⠀⠀⣼⣿⣷⣿⣷⣿⣿⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣠⣴⣶⣶⣶⣶⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢟⣻⣟⣛⣛⡋⠄⠀⠀⠀⠀⠀⠀⠀⢠⣴⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠠⣴⣿⢻⣽⣿⢗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠿⠃⠀⠀⠀⢰⡟⢛⣦⠻⠓⠈⠁⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡐⣷⣿⢀⣀⡀⠀⠀⠘⠻⢿⣋⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⠾⠶⠿⠿⠿⠆⠀⠀⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⢸⣿⣿⡇⢤
⠀⠀⠀⠀⠀⠀⠠⣿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣿⠀⠀⢸⣿⣿⡇⠼
⠀⠀⠀⠀⠀⠀⠠⠭⠬⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠤⠥⠤⠤⠬⠭⠭⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣖⠻⣳⢶⣾⡻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⣶⣶⣶⣶⣾⣿⣿⣟⣿⣿⣿⣿⣶⣯⣝⣳⠮⣟⡶⣿⣿⣦⣛⢿⣷⣼⡇
⠀⢀⣀⣤⣤⣤⣴⣴⡶⠶⣾⣿⣿⣹⡍⠉⣽⢭⣭⡙⠓⠲⠶⣽⣿⣚⣿⡿⣫⢴⡟⠹⣏⣹⣿⣶⣾⡿⣿⣿⠿⢮⣿⣛⡻⣷⣿⣟⢿⡄
⠀⣿⣯⢠⣿⣿⡿⣭⣿⣛⣻⣿⡿⢿⣤⣶⣶⣛⣛⣛⣻⣿⣦⡹⢿⠿⢃⣼⣧⣤⡴⢳⣀⡟⢿⣽⣿⣶⣴⣿⣿⣻⣿⣿⣿⣷⣿⣿⣯⣷
⠀⣿⣿⣤⡭⣉⣽⣿⡷⠿⣿⣟⣆⡟⢯⣿⣽⡲⢮⣭⣽⣿⣿⣧⢸⠏⣾⡿⣽⠯⣿⢿⣝⣷⣤⠲⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⢿⣿⣿⢀⣿⣶⣧⣶⣾⣿⠉⣾⢃⢈⢷⣽⡿⣿⣿⠿⣛⣛⣡⡞⣸⣣⣿⣷⣼⣯⡾⡋⣤⣽⣆⢚⣽⢷⣶⣦⠩⣷⣾⣿⣿⣟⣿⣿⣿⣷
⠀⢸⣿⣧⣈⣛⣛⣛⣫⣭⣶⠀⣿⣿⢸⣦⣿⠇⣲⣶⣿⣿⣟⣻⡁⣿⣿⢿⣻⢿⣿⢏⣼⣿⣯⣵⣶⢹⣿⣽⣿⡌⠿⣿⣿⣿⣿⣿⣿⢿⣿⡄
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣯⣭⣷⡈⢹⣿⣟⣡⣾⣿⣿⣿⣿⣿⣿⡇⢿⣿⠁⠈⣮⢿⣿⡿⢿⣿⣿⣿⡸⡿⢾⣿⣟⣶⣍⡉⠟⣒⡴⢘⣿⣿⡇
⠀⢹⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⣿⣿⢯⡟⣸⣿⣿⣼⣿⣿⣿⡾⢿⣿⢿⣿⣯⢩⣭⣭⣥⣶⣿⣿⣿⣇
⠀⠀⣿⣿⠿⠛⠟⣛⣉⣛⣿⣿⣻⣯⣿⣷⣿⠿⠿⠿⢿⣿⣿⣿⣽⣆⠹⣾⣾⣹⣯⣾⣿⠿⢿⣶⣯⠻⣶⣮⢷⡋⠁⣶⣿⣿⣿⣿⣿⣷⣏⣻⡄
⠀⠀⣿⣿⣻⣿⣟⣿⣯⣽⣯⣽⣿⣿⣿⡽⣿⣆⠀⡀⠢⢌⣉⠛⠈⢛⣃⡹⣿⣿⣿⣿⣷⣿⡷⠿⢿⣿⣮⣿⡾⠟⢛⣦⣶⡯⣌⣫⣭⣉⣻⣾⣿
⠀⢠⣿⣛⣻⣿⣿⣿⣭⣭⡭⠷⠾⠿⠿⠟⠚⠛⠃⠀⠁⠒⠀⠉⢉⣉⠉⠀⠈⢩⣅⠀⣄⣦⣶⣆⠀⠀⠉⠀⠀⢶⣄⢙⢧⡞⣋⣱⣶⣿⣿⣾⣿⡀
⠀⠈⣿⠉⠉⠉⠉⠁⠀⠀⢀⣀⢀⡤⠀⠤⠄⢄⠀⣖⡀⠀⡾⢧⣸⢿⡄⠀⠀⠎⠙⠂⠋⠛⠙⢋⣄⠀⠀⠀⠀⠀⠉⢧⢻⠀⠛⢛⣉⣩⣍⢹⣿⡇
⠀⠀⣿⠀⢲⣶⢸⡍⠀⠀⢸⣿⠰⡷⠜⠶⠾⠈⠛⡁⠛⠁⡀⣤⣠⡀⠀⠀⢦⠆⢰⣿⡖⣆⣸⢸⢿⠄⠀⠀⠀⠀⠀⠀⣻⠆⣈⢵⣆⣲⣶⣿⣿⣇
⠀⠀⣿⠀⠘⣋⡈⣫⡄⣠⣤⢠⣶⠀⣾⡆⡆⠈⢻⠁⠀⠀⢷⠟⠻⠇⠀⠀⠘⢀⣈⠛⠁⠈⣅⡠⢀⣶⣄⣄⢰⡄⢠⣶⣿⣄⣿⣶⣿⣿⡿⠿⣿⣿⣆
⠀⠀⣿⠀⠸⣿⡗⠿⠧⠸⠿⠁⠯⠀⠋⣃⠙⣃⣄⠀⠀⠰⢶⠂⣄⣤⠀⡾⣆⠀⢹⠀⠀⠀⠈⠇⠘⠿⠏⠈⠋⠀⢸⣿⣿⣿⣜⣻⢟⣽⣦⢻⣿⣟⣿
⠀⠀⣿⠀⢀⠀⢠⢰⡖⢰⡾⠂⠀⠀⠀⣿⡄⢿⡿⠂⠀⠀⠘⠀⠉⠘⢈⣁⢈⠀⢠⡤⢀⣦⣰⠐⡞⠀⠀⠀⠀⠀⠀⢿⣿⡿⢹⠃⣾⣿⣿⡀⣷⢿⣿
⠀⠀⣿⠀⠸⣧⠘⠘⠁⠘⠛⣀⢃⠀⠀⢩⣦⡀⣴⠀⠀⠀⢳⣀⡏⣇⢸⣿⡾⣧⠸⣯⡴⠟⠛⠀⠋⢀⠀⠀⣴⡄⠀⠀⢹⣿⢩⣴⣿⣿⣿⠇⣿⣿⣿
⠀⠀⢿⡀⠀⣦⣠⡆⢰⢿⡀⠈⣏⠀⠀⠈⣿⠆⠯⠄⠀⠀⢀⠉⠀⡉⢠⡉⣠⡄⢰⠀⠀⠀⢸⣸⣷⣾⠀⠀⣿⣿⡄⠀⢸⣿⡟⣷⢿⠿⣯⣾⣿⡿⢽
⠀⠀⢸⡇⠀⠋⠃⠋⢉⢀⠁⣤⡀⠀⠀⢠⣶⡄⣾⡆⢶⣼⣩⡜⡇⣿⣾⢧⠏⠻⠸⠶⠀⠀⠈⠋⠁⠉⠀⣼⣿⣿⣧⠀⣸⣿⣸⣭⣾⠟⣠⡉⠿⠏⣼
⠀⠀⢸⡇⠀⢰⣿⡄⣿⣿⡇⣿⠇⠀⠀⠘⠿⠃⠉⠋⠘⠈⢉⣀⢡⣤⢠⣤⠄⠀⠀⠀⠀⠀⠀⢰⣶⣄⢸⡟⣾⣿⣿⣰⠟⣏⣠⣿⠏⣼⣿⡇⢿⢠⣿
⠀⠀⠈⣷⠀⢉⢀⡁⣠⣌⢠⠀⣠⣴⣦⠀⠀⠐⣦⣿⣼⠁⣿⣿⠸⠿⠜⠛⠆⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⢰⠋⢿⣿⠃⣸⣟⣹⣿⠄⣿⣿⠇⡇⣾⡿
⠀⠀⠀⣿⠀⠈⣿⠀⣿⡿⠸⠦⠇⠛⠛⠀⠀⠀⠘⠁⠁⠀⠈⠀⢀⣤⡀⠀⠀⠀⠀⠀⢴⣶⣦⣄⠘⣿⣿⣟⣠⡾⣿⡴⣋⣶⣿⢿⢐⣿⡿⢰⢸⣏⢏
⠀⠀⢠⣿⠀⠀⠈⠀⠀⢀⣀⣀⣠⡀⢀⣤⣶⢲⣶⡐⣺⣏⡿⢢⢟⣾⣿⡄⠀⠀⠀⠀⠈⢿⣿⣿⣿⡋⢻⣿⣿⡷⢿⡷⢶⣿⣯⡆⣼⣿⠃⠆⡿⡸⣸
⠀⠀⠘⣿⠀⢿⣿⣻⣻⣾⠾⠿⠟⠁⠈⠉⠈⠉⠉⠉⠁⢀⣴⣽⣿⣷⣽⣿⡶⠶⠒⠖⠿⠭⠿⣿⣿⣷⢛⣼⣿⠸⢿⡗⣼⣿⡖⣰⣿⢃⡼⣼⢡⣷⢃

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢤⣀⣀⠀⣠⠤⠶⠤⣄
⠀⠀⠀⠀⣠⠚⠉⠉⠉⠙⠢⠤⢤⡞⠑⠞⠁⠀⢸⠁⠀⠉⡡⠤⢤⡀⠈⢳⠲⠤⣄
⠀⠀⠀⢰⠁⢀⡤⠒⠦⣀⣄⡀⠀⠓⢤⡀⣠⠒⢉⣰⠉⠉⠀⠀⣀⣱⠀⠸⣄⠀⠈⠣⢤⡀
⠀⠀⠀⢸⡀⢺⠀⠀⠀⠉⠀⠃⠀⠀⠀⢱⣇⠀⠏⠀⠀⠀⠀⠘⠁⠀⣳⠀⠈⠳⠤⡀⠀⠙⠦⠲⢄
⠀⠀⠀⠀⢳⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠾⣌⡶⠒⠲⠤⠒⠢⢤⡀⠀⠁⠀⠀⠀⠀⢸⡤⠔⠒⠤⡼
⠀⠀⠀⠀⡾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡔⠙⠚⠉⢦⡀⠙⣄⡀⠀⠀⠀⡴⠲⠔⠉⠉⠳⡘⡆
⠀⠀⠀⠈⣇⠈⠉⠉⠱⣄⠀⠀⣀⢔⣒⣦⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣹⣄⣀⠀⠀⠀⠀⠀⠀⠰⢣⡃
⠀⠀⠀⠀⠈⠑⠒⢤⣀⠴⣍⠉⢁⡜⢲⠚⡖⢻⠒⠒⠒⣿⠒⠒⢺⠓⡞⢲⣾⣦⣀⣤⣄⣀⣀⣀⣀⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠘⡏⣏⢹⠉⠉⠉⣿⠉⠉⢹⠉⡏⣟⣏⣻⣛⣛⣻⣛⣛⣿⣹⣛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣲⣀⣀⣖⣆⣯⢿⢹⠉⠉⠉⣿⠉⠉⢹⢹⢋⣧⣯⢼⠤⠤⣼⠤⠤⣯⣼⡼
⠀⠀⠀⠀⠀⢀⣀⣀⣇⣀⣀⣀⣀⣀⣀⣹⢸⢸⡇⠀⢠⡛⡄⠀⢸⢸⢸⢹⣷⢸⠀⠀⢸⠀⠀⡇⣸⡇
⠀⠀⠀⠀⠀⠘⢦⢤⣤⣤⣤⣤⣤⣤⣤⣼⣼⣸⣇⣠⡻⠋⢞⣆⣼⣸⣼⢸⣿⣸⠀⠀⣸⠀⠀⣇⣻⡇
⠀⣀⣀⣀⣀⣀⣸⢸⣽⢼⢽⣭⡯⡯⣯⡽⡿⢸⡵⣿⣧⠀⣼⣯⢿⠘⡿⣼⡏⢹⠉⠉⢹⠉⠉⡏⢹⣇⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣸⣸⣺⢺⢺⣒⡗⡗⣗⡇⡇⣼⡝⠉⢉⣿⡉⠁⢻⣣⡇⡟⡇⢸⠀⠀⢸⠀⠀⣇⢸⢻
⠀⠀⠀⡏⢉⣉⠉⣉⠉⣉⠉⣉⠉⣉⣹⢡⠿⣼⠷⠶⠾⣿⠷⠶⢾⡧⢧⢹⠃⡎⠀⠀⢸⠀⠀⢸⠘⡜⡄
⠀⠀⠀⡇⢸⣿⠀⣿⠀⣿⠀⣿⠀⣿⡏⡸⠀⡇⠀⠀⠀⣿⠀⠀⠈⡇⠸⡈⣦⣇⣀⣀⣸⣀⣀⣸⣄⣧⣷
⠀⠀⠀⡇⢸⣿⠀⣿⠀⣿⠀⣿⠀⡿⢳⠗⢒⠗⠒⠒⠒⣿⠒⠒⠒⢷⠒⢷⠻⣄⣀⣀⣸⣀⣀⣀⣇⣘⣆⣧
⠀⠀⠀⢧⣸⣿⣤⣿⣤⣿⣤⣿⣴⠧⡼⠤⣼⠤⠤⠤⠤⣿⠤⠤⠤⠼⡤⠼⡦⢵⡤⠤⢼⠤⠤⠤⢼⡤⠽⡼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡾⠒⢲⠗⠒⠒⠒⠒⣿⠒⠒⠒⠒⢷⠒⠺⡖⢻⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣴⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦
⠀⠀⠀⠀⣼⣿⣿⡟⠁⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⠋⠊⠁⡀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧
⠀⠀⠀⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠰⣿⠎⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧
⠀⠀⢠⣿⣿⣿⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⣿⣿⣿⡄
⠀⠀⣾⣿⣿⠇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠸⣿⣿⣷
⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄
⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢩⣶⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠮⠍⠐⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇
⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃
⠀⠀⢿⣿⣿⡆⠀⠀⠀⠀⠀⠸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢰⣿⣿⡿
⠀⠀⠸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠹⡋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣴⣦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃
⠀⠀⠀⢹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟
⠀⠀⠀⠀⢻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠻⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣀⣀⣀⣀⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢄⣀⣀⣀⣀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠈⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠈⠁⠈⡇⠀⠀⠀⠀⡠⠔⠢⣀⠀⣀⠄⠒⡄⢀⡠⠖⠙⢆⠀⡠⠔⠢⡀⠀⡠⠖⠁⠈⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢘⡒⡀⠀⠀⠀⠓⢒⠶⠒⠁⠀⠀⣀⠜⠋⠀⠀⡀⠘⠁⠀⠀⡀⠔⠉⠀⠀⣠⠜⠉⠀⠀⢀⡠⠊⡅
⠀⠀⠀⠀⠀⡤⠖⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⢨⠇⠉⠈⠉⡸⠚⢅⠀⠀⠀⡠⠪⡀⠀⠀⡠⠛⡄⠀⠀⡰⠫⡀⠀⠀⡠⠊⢆⠀⠀⢀⠔⡁⡌⠀⡆
⠀⠀⠀⢀⠎⢀⠔⠀⠁⠉⠘⠒⠠⠤⠤⠄⠒⠊⠁⠀⠀⢠⠎⠀⠀⠈⢣⣤⠊⠀⠀⠘⣤⡎⠀⠀⠈⢦⠎⠀⠀⠙⣤⠎⢀⢀⣀⣧⠔⠁⠠⠀⠀⠀⠃
⠀⠀⠀⠸⠀⢸⠐⠒⠒⠒⠒⠒⠒⠒⠒⢲⠀⠀⠀⠀⣀⡀⠀⢀⠀⠀⠀⠀⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠢⠸⠘
⠀⠀⠀⡇⠀⢼⠘⠀⠀⠀⠀⠀⠀⠀⠀⣸⠤⠒⠚⠋⠐⠤⡀⠀⠈⠂⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢨⠀⣀⣀⣀⣀⣠⠁
⠀⠀⠀⡇⠀⢸⠈⠀⠀⠀⠀⠀⠀⡰⠊⠁⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⢢⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠸⠸⠀⡀⠀⠀⠑
⠀⠀⠀⡇⠀⠸⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢂⠀⠀⠀⠎⣠⠞⠀⡖⢁⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⡠⢒⢠⠀⡇
⠀⠀⠀⡇⠀⣀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⣀⠄⠠⠀⢀⡀⠀⠀⠀⠈⠆⠀⣨⠄⠀⡘⠀⡄⠀⠀⡇⢠⠀⠀⠀⠤⠀⠀⠖⢀⠅⠀⠘⡄⠃⠤⠌⢀⠁
⠀⡠⠤⠃⠀⠸⠀⠀⠀⠀⢸⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠸⠘⠤⠤⠤⢴⠀⡶⠀⠐⡇⢸⠒⠒⠒⠒⠒⠀⠀⠉⠀⠀⠀⢨⠃⠒⠒⠁
⢸⠀⠀⠀⠀⠸⠀⠀⠀⠀⠸⠀⠀⢰⠁⠀⢀⠀⠀⠀⡀⠀⠀⠇⠀⠀⠀⡄⠀⠘⠀⢸⠀⠁⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠸⣄⣀⣀⣀⣠⣀⣀⣀⣀⣆⠀⠀⡊⠀⠀⠈⠀⠀⠀⠁⠀⠀⢸⠀⠀⠀⡇⢀⡴⠒⠺⣄⡰⠒⠒⠧⠠⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠓⠶⠶⠒⠲⠒⠒⠒⠒⠒⠚⠒⠒⠖⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡏⠉⠉⠀⠉⠉⢱⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⢠⠒⠒⠒⡇⠘⠶⠴⠶⠶⠶⠶⠶⠶⠶⠴⠴⠃⡗⠒⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⡆⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠁⠢⡄⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢀⡀⠤⠇⠓⠀⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢏⠉⠉⠉⡏⠉⠉⠉⣇⡤⠒⢞⡄⠀⠀⠀⠀⠀⠀⠀⡼⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠢⡀⠀⡇⠀⣀⡾⠁⠀⠀⠈⢞⠄⠀⠀⠀⠀⢀⡼⠁⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⣀⣄⣀⣀⣀⣀⣍⢉⠉⡔⠀⠀⠀⠀⠀⠈⢾⣄⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢸⠸⠤⠿⠿⠿⠿⠿⠿⠿⠸⢴⠁⠀⠀⠀⠀⠀⠀⠈⢋⣆⢀⡞⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢸⢸⠉⣿⣿⣿⣿⣿⣿⣿⣸⡽⠞⠒⠳⢶⣄⠀⠀⠀⠀⢻⡻⡄⣀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⢸⢠⠄⣤⣤⣤⣤⣤⣤⣼⠋⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠉⠑⠋⠥⢖⡤⠄⣀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⢸⢘⣒⣛⣛⣛⣛⣛⣛⣿⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⡪
⠀⠀⠀⠀⠀⠀⠀⢸⠸⣀⠿⣿⠿⣿⢿⡿⡟⣷⣄⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⢨⡘⠒⠒⠒⠒⣒⠖⢊⡤⠚⠓⠛⠛⠛⠛⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⡡⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠤⢀⣀⣀⡀⠠⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠔⠊

⠀⠀⠀⠀⠀⢸⡯⢉⠉⠉⠉⣿⠀⠀⠀⠀⡯⠭⠭⠭⠭⢹⠁⠀⠀⠀⠀⠀⠀⠀⡀⠐⢀⠐⠂⢀⠂⠀⠀⠀⢀⡀⠐⢀⠐⠂⡀⠂
⠀⠈⢋⣤⡄⢸⡇⠈⠀⣀⣀⡿⠀⠀⠀⣠⣇⣀⣀⣀⣀⣸⠀⠀⠀⠘⠃⣤⡀⠂⠀⠀⠀⠀⠐⠀⠂⠀⢀⣴⠂⣰⠒⠀⠀⠐⠀⠂
⠀⠀⢿⣾⣦⢸⡇⢀⡀⠀⠀⡧⠀⠀⣰⣿⡷⠞⠁⠀⠀⢸⠀⠀⠀⢤⣶⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠚⠁
⠀⠀⢸⣿⣿⢸⡇⠀⠀⠀⠀⡗⠀⠀⣿⣿⡇⠀⠀⠀⠀⢸⠀⠀⠀⠈⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⢸⣿⡇⠸⢧⣤⣬⣤⣤⠧⠀⠀⣼⠏⣷⢠⣤⣤⣤⣼⠀⠀⠀⠀⣽⠋⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢹⡇
⠀⠀⠈⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣀⣀⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠛⠀⣀⣛⣓⡛⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠛⣼⡟⣛⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡤⠲⠀⣤⣦⣿⡇
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣠⠀⣿⣿⣿⡇
⠀⠀⠀⣼⣏⣶⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⣤⣤⣬⣿⣿⣤⣤⣤⣤⣤⣦⣄⠀⠀⠀⠀⢼⣿⣿⢍⣀⡀⢘⡋⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⡿⢿⡼⠿⠯⠼⠯⣿⡇
⠀⠀⢈⣿⢻⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠸⠿⠿⢿⣯⣼⡿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠛⠁⠘⠁
⠀⠀⠸⠃⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛
⠀⠀⠀⢀⣼⣿⡀⠀⠀⠀⠀⠀⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣀
⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢴⡶⣤⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢹⣇⣀⣾⣿
⠀⠀⠀⠈⣿⠉⣿⠀⠀⠀⠀⠀⢸⡟⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠀⠀⠀⢰⡇⣾⣾⣿⣿⣿
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠛⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠘⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⡉⠑⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠶⠒⠂⠀⠀⠙⠒⠀⠄⡀
⠀⠀⠀⢀⡠⠴⠛⠛⠉⠉⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀
⢀⡴⠊⠁⠀⠀⠀⠀⢀⣀⣀⣠⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⡏⠀⠀⠀⣀⠔⢲⣿⠭⠥⠔⠓⠒⠀⠀⠀⠉⠑⡄⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⢳⣀⡤⠖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠃⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⣰⠋⠀⠀⠀⣀⣠⡤⠶⠖⣒⣈⣉⣩⠭⠉⠒⠲⣄⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⢿⡀⠀⣀⡻⠮⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠏⠀⠁⠂⠆⠀⠀⠋⠈⠀⠇
⢈⠟⠋⠁⠀⠀⠀⠀⢀⣀⣠⣤⠤⠤⠰⠶⠶⢋⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀
⣿⠀⠀⣀⠤⠶⣚⣛⣯⠥⠶⠒⠚⠃⠉⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢱⠀⡄⠁
⠙⢦⣀⣀⡴⠋⠉⠀⠀⠀⠀⠀⣀⣀⣀⣀⠤⠴⠋⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⢀⡞⢡⡞⠀⠀⢀⡤⠶⢶⣽⡋⣩⠟⠉⠋⠛⠢⡄⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⢸⠀⠸⡀⠀⢀⡿⢱⠀⠀⠀⠙⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠁
⢸⠀⣀⣈⣩⣭⠤⠼⠤⠤⠤⠤⠧⠤⢤⣤⣀⣀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⢳⠀⠀⠀⠀⠀⠀⠠⠁
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⡀
⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠈⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠱⡀⠀⠀⠀⠀⢰
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣼
⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠼⠀⠀⠀⠀⣸⣄⣀⣀⣀⣴⠁
⠀⠀⠘⢦⠤⠄⠲⠶⠒⠒⣶⣒⣒⣲⡒⠲⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠴⠤⢀⠃
⠀⠀⠀⢸⠀⠀⠀⣀⣀⣀⣀⣀⡀⢀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠃⠀⢀⣀⣀⣀⣀⣂⣉⣭⣽⡇⢈⠆
⠀⠀⠀⢼⠃⠀⢤⣤⣤⡤⠤⠤⠴⠶⡒⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⠀⠀⠀⠀⣀⣀⣈⣭⣤⣵⡶⢈⠆
⠀⠀⠀⣼⡃⠤⠤⠤⠤⠴⠶⠿⠿⠝⡓⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⣐⣢⣭⣭⣥⢈⠆
⠀⠀⠀⢼⡃⠤⠤⠤⠤⠴⠶⠿⠿⠿⡖⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠉⠉⠉⢉⣩⣭⣥⣶⡎⣈⠂
⠀⠀⠀⠘⢧⣀⠀⠐⠒⠒⠒⠺⣋⣗⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠉⠉⡭⢛⠜⠁
⠀⠀⠀⠀⠀⠉⢻⠛⠛⠛⠛⡿⢹⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡛⣛⣷⠞
⣤⣤⣤⣤⣤⣤⣤⣽⣷⣶⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠐⢒⡒⠒⠒⠒⠆⠀⠀⠀⢠⠒⠒⠒⠒⠒⢦⠀⠀⠀⠀⠀⡴⠲⠿⠖⢢⠀⠀⠀⠀⠀⢠⣶⣿⡷⠐⠄
⠀⠀⠀⠻⠷⠀⠀⠀⡆⠀⠀⠀⢸⠀⠤⠤⢾⠉⣛⠀⠀⠀⠀⠀⡟⠿⠀⠀⢸⠀⠀⠀⠀⠀⣿⣿⣿⣣⣄⣈
⠀⠀⠼⠿⠿⠽⠿⠿⠧⠀⠀⠀⠈⠉⠩⠿⠏⠉⠙⠂⠀⠀⠀⠀⣧⣤⣤⣶⣸⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⣠⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⣠⠶⡶⢦⡀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⠆
⠀⠀⡇⠀⣄⢀⣄⡀⣿⠀⠀⠀⠀⣿⠒⣿⢚⡇⠀⠀⠀⠀⠀⠀⢸⣿⠟⠉⣿⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣤
⠀⠀⣇⣺⣛⣟⣟⣇⡿⠀⠀⠀⠀⠈⠓⠛⠛⢧⡀⠀⠀⠀⠀⠀⢸⣿⣿⡃⠋⠀⠀⠀⠀⠀⠀⠺⠛⠒⠛⠻
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠁

⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⣤⣀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣀⡀⠀⠀⠀⠀⠀⣀⣤⣤⣠⡀
⠀⠀⣀⣰⣋⡠⠿⣶⡶⠀⠀⠀⠀⣰⠏⠁⠀⢛⢳⡄⠀⠀⠀⠀⢘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠴⢒⣄
⠀⠀⠉⠉⠻⠥⠴⢇⠀⠀⠀⠀⠀⢿⡠⠶⠴⠿⣸⠇⠀⠀⠀⠀⠸⠴⢤⡤⠦⠄⠀⠀⠀⠀⠀⡟⢖⣛⣜
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠈⠛⠶⠶⠞⠋⠀⠀⠀⠀⠀⠀⠰⠾⠷⠀⠀⠀⠀⠀⠀⠀⠳⠶⠴⠶⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⡿⠿⠿⠿⠿⠿⣟⠀⠀⠀⠀⡞⠉⣩⡿⡆⠀⠀⠀⠀⠀⠀⣿⡿⣿⢿⣿⠀⠀⠀⠀⠀⡠⡼⠩⠍⢧⢄
⠀⠀⡤⣤⣤⣔⣩⣕⣾⠀⠀⠀⠀⢷⣛⣛⣳⡃⠀⠀⠀⠀⠀⠀⣿⣁⣉⣀⣿⠀⠀⠀⠀⠀⠂⡇⠐⡂⡰⠐
⠀⠀⠦⠤⠤⠬⠵⠭⠮⠀⠀⠀⠀⠀⠈⠉⠀⠹⣦⠀⠀⠀⠀⠀⢿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠘⠭⣈⣣⠴⠃

⠀⢀⠀⢀⣀⣀⣀⡀⢀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⡀
⠀⢘⣯⠀⠠⠙⠋⠁⣼⡁⠄⣻⣿⠷⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⣤⣄⣙⠛⠿⣿⡂⠴⢳⡬⠛⠀⠀⠀⠠⣜⠁
⠀⢸⡟⠀⠀⠀⠀⠀⢾⠕⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠸⣿⡇⠀⠈⠑⢣⡐⢇⠀⠀⠀⠀⠀⢹⡅
⠀⠸⣷⠄⠀⠀⣀⡠⠚⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠈⢻⣆⡀⠀⣀⡀⢶⡀
⠀⠰⡿⣉⠶⣆⡁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡧⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠙⣟⡄⠉⢩⣘⠇
⠀⢸⣷⣿⣧⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠐⣋⣹⡇
⠀⢸⣆⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠐⣿⣷⣦⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⣻⡇
⠀⢸⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠲⠋⠂⣈⣽⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⣿⡇
⠀⠸⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡬⠃⠀⡠⠐⠣⠈⠁⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢛
⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⡀⠁⠤⡤⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬
⠀⢸⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⠏⠁⠐⠅⢠⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣯⡄
⠀⢸⣿⠯⠥⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣴⡴⢿⣶⣶⣶⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠋
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡃⠀⠀⢉⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠂
⠀⢠⣄⢀⡁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢠⠞⠰⠳⣿⠆
⠀⢸⣿⠹⣿⣖⠰⣿⠄⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣰⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⢏⣶⢏⣼⠷⣿⡆
⠀⠘⠃⢀⡸⠟⢲⠂⣶⣀⠀⠀⠀⠀⠀⣿⣷⠀⢰⡧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠚⢶⣇⣀⣽⣿⣤⡼⠇
⠀⢸⣷⣾⣶⣶⣶⣿⣿⣿⣧⣄⣀⡀⠀⠙⠿⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠈⠛⠛⠉⠉⠛⠛⠛⠛⠂⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠃⠈⠉⠛⠉
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠
⠀⠀⡗⢺⣧⠄⢸⣀⡀⡀⢀⣾⠀⢀⠀⠀⡇⠀⢰⠀⠀⠀⣀⣀⣀⡀⡇⠀⡇⣇⣂⣶⡀⠀⠀⢀⣠⣸⣸
⠀⠋⠷⠾⠿⠇⠸⠿⠿⠇⠸⠿⠿⠿⠖⠀⠿⠿⠿⠶⠇⠸⠿⠯⠽⠿⠿⠈⠿⠿⠟⠿⠁⠭⠤⠽⠿⠿⠿
⠀⢀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠰⠀⠀⢀⣀⡀⠀⠀⢀⠀⢰⡆
⠀⣾⣷⣤⣄⡆⢀⣠⠶⣷⡆⢀⣦⣠⣼⡄⠀⣤⣿⣞⠃⣴⣾⣵⣶⡄⠀⣼⣿⣿⢩⡏⣽⡄⢘⣻⣿⣇⣠
⠀⠉⠉⠉⠉⠁⠉⠉⠉⠉⠁⠉⠉⠉⠉⠁⠀⠉⠉⠉⠁⠉⠉⠉⠉⠉⠀⠉⠉⠉⠈⡉⡉⠁⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠠⠀⣤⡄⡆⠀⠀⣤⠀⠀⡆⠀⠀⠀⠀⢀⡆⠀⣀⠀⢀⣀⠀⠀⠀⢀⡀⣿⡇⠀⠀⢀⣧⣶⣾
⠀⣾⣾⣶⢸⡄⣿⣿⣧⣄⠀⣿⣤⣶⡇⠀⣾⣶⡗⣼⡇⢰⣿⣦⣼⣿⣦⡄⢠⣿⡗⣿⣧⡄⢠⣾⣶⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠧⠷⠾⠼⠶⠧⠿⠼⠼⠧⠧⠾⠼⠶⠧⠷⠾⠼⠶⠧⠷⠼⠼⠧⠷⠾⠼⠼⠧⠷⠾⠼⠶⠧⠷⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⢻⡹⡏⠉⠉⠋⠙⢟⢻⠉⠉⠉⠉⠛⡟⡟⠉⠉⠉⠉⢻⠻⡏⠉⠉⠉⠉⠛⡟⡟⠉⠛⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⣸⢸⠀⠀⠀⠀⢀⡇⡇⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⣠⡇⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⢠⠞⣽⢸⠀⠀⠀⡠⢋⡇⡇⠀⠀⢀⡔⣹⡇⡇⠀⠀⢠⢞⡵⡇⣿⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⣰⠣⠎⣸⢸⠀⠀⡼⠱⠃⡇⡇⠀⢠⠎⠜⣹⡇⡇⠀⣰⠣⢎⠐⡇⣿⠀⢠⠎⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⢰⠃⢂⠎⢾⢸⠀⡼⠐⣰⠃⡇⡇⢀⠏⠐⡼⢹⡇⡇⣰⠃⢃⠎⠞⡇⣿⢀⠎⠐⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣧⠇⢀⡄⠈⣼⢸⣼⠁⢠⠀⢁⡇⣧⡞⠀⡤⠀⣹⡇⣧⠇⢀⡄⠈⡜⡇⣿⡞⠀⡤⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠏⠀⠀⠀⠘⢹⣸⠃⠀⠀⠀⠊⣇⡟⠀⠀⠀⠀⢻⣇⠏⠀⠀⠀⠘⠁⡇⡿⠀⠀⠀⢸
⠀⠀⠀⡠⢤⣤⢞⡩⣍⠲⡤⣬⢏⡔⣒⢢⣀⡁⡸⠃⢂⣀⡄⠈⢀⡟⠐⠂⢀⠀⠉⢰⠏⠐⠀⡀⠀⠁⣀⡟⠐⠂⢀⠀⢹
⠀⣠⠞⠀⠀⢯⣏⢀⡀⠀⠀⠈⠛⠙⠀⠁⠉⠉⣧⡯⡾⠶⣄⣢⠎⡰⣣⠃⠋⢐⡷⢃⡜⡼⡐⡁⠘⣠⠟⡰⣣⠃⠋⠀⢻
⠀⢧⠀⠀⡔⠋⠁⢏⠙⠋⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⢳⣴⣵⣳⣳⡷⠋⣠⢞⣜⣝⣝⣽⠟⢁⣞⣵⡵⣱⣳⢣⣾
⠀⠀⠭⠭⠭⠭⠉⠉⠉⣭⣭⡍⠉⢉⣉⠉⢉⡉⠉⣉⠩⠭⡉⠉⠹⠿⠏⠉⠉⠉⠩⠭⠭⠭⠉⠭⠉⠩⠭⠭⠭⠉⡩⠭⠉
⠀⠀⠠⠬⠭⠉⠉⠥⠤⠀⠰⠆⠀⠐⠒⠒⠒⠀⠀⠐⠲⠈⠉⠉⠀⠉⠉⠩⠍⠉⠉⠡⠄⠀⠀⠒⠒⠂⠘⠛⠦⠌⠵⠶
⠀⠀⠀⠀⠚⠛⠓⠀⠀⠀⠉⠁⠀⠀⠉⠉⠉⠉⠀⠀⠉⠉⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠂⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣋⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠉⣙⡟⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣥⠀⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⢻⣭⣟⣻⣤⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠇⠀⠀⠀⢀⡀⠀⢠⣾⢿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⠀⠀⠀⠀⠈⠻⣤⡟⣿⡞⢻⣬⢛⡿⣿⣿⣿⣿⠃⠀⠀⠀⠈⠿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠻⣿⡟⣿⠶⡟⠻⢻⣯⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢻⣿⡿⠰⣿⣅⡘⠛⢷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣷⣶⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣛⠧⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢿⣿⠟⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢈⡻⠿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⣴⡄⠀⠀⠀⣰⠖⠚⠿⣽⣼⣿⣿⣶⣤⣀⠀⠀⠀⠀⠸⣿⣥⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣤⡄⠀⠀⠺⣆⣀⠻⢿⣿⣿⣿⣿⣷⡄⠀⠰⢿⠿⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣼⣿⣿⠀⠀⠀⣿⡿⠀⠸⠿⢿⣿⣿⣿⣧⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠟⢸⣿⣿⣒⣆⣠⣖⣒⣲⣿⢻⣿⣿⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠉⠀⠀⢀⣉⣿⣿⣿⣿⣿⣿⠇⠈⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠸⣶⣾⣿⣾⣟⠀⠀⠘⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿⣿⣿⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣯⣭⣭⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣭⣭⣭⠛⠛⢛⣩⠥⣄⣴⣾⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⠿⠛⣛⡟⠁⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠾⠇⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣾⡆⠀⣿⠀⠀⠀⠀⣠⣾⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⣠⣾⠟⣳⠀⠀⣰⡾⣛⣭⣽⡻⣷⡄
⠀⠀⠀⠀⠀⠀⣿⠀⢠⡄⠀⣿⡇⠀⣿⠀⠀⠀⣸⣿⣿⣿⢟⣭⡻⣿⣿⡄⠀⠀⠀⣿⡿⠿⠿⠿⣿⣫⣾⠟⠁⠚⠁⠀⢰⣿⣸⣿⣿⣿⣿⡞⣿
⠀⠀⠀⢰⡆⠀⣿⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⠀⢻⣿⣿⣿⠻⠿⢇⣼⣿⣶⠀⠀⠀⣿⡿⠿⠿⣿⣣⣬⣷⠀⠀⠀⠀⠀⠘⣿⡜⣿⣿⣿⡿⣳⡿
⠀⠀⠀⢸⠇⠀⢿⠀⠸⡇⠀⢿⠇⠀⠿⠀⠀⠀⠈⢿⢟⣵⣿⣿⣿⢟⣱⠏⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠻⢷⣿⣷⠾⢛⣵⣤⡀
⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠈⠙⠓⠲⠚⠋⠁⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃

⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⣈⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡉⡀
⢸⣸⠃⠃⢻⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠃⠃⢻⣸
⣀⣹⣷⢲⣾⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣹⣷⢲⣿⣁
⠛⢻⣵⣰⣾⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣤⣰⣿⠛
⢰⢻⡁⡀⣹⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⡁⡀⣹⢳
⣈⣘⢻⣿⢟⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣶⡆⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣙⣷⣿⢛⣈
⠟⠓⣸⣿⡗⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠒⢀⡾⠦⢀⣉⡓⠠⠾⡶⠰⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⣿⣿⡓⠛
⢸⣸⠋⠋⢻⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠻⢃⡤⣤⣷⡟⠘⠘⣷⣧⡤⣀⠻⠛⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠋⠋⢻⣸
⣀⣹⣷⢲⣿⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠁⣧⡙⠀⢹⣷⣤⣴⣿⠁⠘⢩⣆⠲⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣹⣷⢲⣿⣁
⠟⢻⣷⣰⣼⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢈⠃⣽⠿⠿⣶⣿⣿⣻⣻⣿⣷⡾⠿⢿⡄⢁⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣦⣰⣿⠛
⢰⢻⡁⡀⢹⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠸⣓⠂⢀⣿⣿⣒⣯⣿⣺⣿⣇⡀⣐⡻⠸⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⠁⡀⣹⢳
⣈⡘⣳⣷⡟⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢠⣇⠹⣿⠟⠛⢿⣿⣾⣾⡿⠏⠛⢿⡿⢁⣦⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡘⣳⣷⡛⣈
⠿⠃⢼⣿⠄⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠀⢫⡶⣀⣼⡏⠈⠈⣿⣆⡐⡮⢃⠈⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⢼⣿⠀⠿
⢰⢸⠏⠏⢻⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠂⠨⣉⠛⠿⠸⠸⠟⢋⣩⠐⢺⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠋⠏⢿⢰
⡈⢻⣦⢤⣾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠛⢋⡀⣶⡆⣀⠙⠋⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢻⣦⢤⣾⠋
⠿⢿⣯⢠⣽⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣼⣨⣿⠿
⢠⢾⠋⠉⢹⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢾⠉⠉⢻⢦
⠘⠘⠧⠧⠿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠧⠧⠟⠘
⠀⠀⢱⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠁

⠀⠀⣦⢀⣀⣀⢀⡀⣴⠀⠀⠀⠀⢰⢩⣭⣭⣭⣍⢳⠀⠀⠀⠀⣰⠲⠲⠲⢶⡄⣿⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣼⠭⡭⡭⡝⡆
⠀⠀⣿⣩⣭⣭⣉⣉⣿⠀⠀⠀⠀⢸⢨⢩⠭⢍⣅⢸⠀⠀⠀⠀⣿⠀⣒⣒⢸⡇⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⠀⠉⠭⠇⡇
⠀⠀⣿⢬⡧⠭⢤⡦⣿⠀⠀⠀⠀⢸⢘⣒⣒⣒⣷⢸⠀⠀⠀⠀⣿⠠⠤⠤⢸⡇⠿⠀⠀⠀⠀⢠⣋⣋⠏⠋⣉⣱⠀⠀⠀⠀⠀⣿⠀⢀⡀⠀⡇
⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠂⠀⠂⠂⠐⠋⠀⠀⠀⠀⠙⠒⠒⠒⠚⠁⠋⠀⠀⠀⠀⠘⠛⠛⠒⠒⠛⠛⠃⠀⠀⠀⠀⠘⠒⠒⠒⠒⠃

⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⢀⡞⠉⠉⠉⠳⡄⠀⠀⠀⠀⢀⡞⠋⢩⠉⠳⡄⠀⠀⠀⠀⢀⡴⢋⡉⣉⠳⡄⠀⠀⠀⠀⠀⡴⠋⢩⠍⠻⣆⠀⠀⠀⠀⠀⡴⢋⣉⣉⡙⣆
⠀⠀⢸⡀⠒⠒⠒⢀⡇⠀⠀⠀⠀⢸⡅⠒⢿⠒⠀⡟⠀⠀⠀⠀⠸⡇⢠⠿⢧⠀⡿⠀⠀⠀⠀⠸⡇⠐⢒⡒⠂⣿⠀⠀⠀⠀⠀⣇⠨⡭⣭⠅⣸
⠀⠀⠀⠳⠦⣤⡴⠞⠁⠀⠀⠀⠀⠀⠙⠦⣤⡤⠞⠁⠀⠀⠀⠀⠀⠙⠦⣤⡤⠞⠁⠀⠀⠀⠀⠀⠙⠦⣬⡤⠞⠁⠀⠀⠀⠀⠀⠙⠦⣤⡤⠞⠃


⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢄⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤
⠀⠀⢸⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⢿⣷⡄⠀⠀⠀⠀⠀⠀⢀⡶⣎⠀⠀⠀⠀⠀⠀⢠⡟⢛⣻⣟⠛⢻
⠀⠀⢸⣷⣘⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣾⣿⣷⠀⠀⠀⠀⠀⠀⡘⠁⠺⣆⠀⠀⠀⠀⠀⢸⡇⠘⢧⡼⠃⢸
⠀⠀⠘⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⡀⣀⢀
⠀⠀⢠⢚⠄⢸⠜⠓⡄⠀⠀⠀⠀⠀⣆⣠⠀⣆⢀⡄⠀⠀⠀⠀⠀⢀⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⢊⡽⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⠒
⠀⠀⠸⠈⢢⢸⡰⠁⢇⠀⠀⠀⠀⢀⡜⢳⡀⠘⡾⠁⠀⠀⠀⠀⠀⠈⣿⠁⡀⣀⠀⠀⠀⠀⠀⠀⠙⣧⡟⠁⠛⠂⠀⠀⠀⠀⠀⠛⣦⡼
⠀⠀⠘⢦⡃⢘⢠⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⡼⢇⠀⠀⠀⠀⠀⠀⣴⠋⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⠀⠀⢰⡟⠀⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⢠⣶⡟⠋⠀⠀⠀⢠⡿⠀⠀⠀⠉⠛⣶⣤⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⡀⠀⣾⠃⠀⠀⣠⡴⠟⠋⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⠇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣿⣿⣇⣀⢀⣉⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣶⠶⠶⠾⠷⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠸⠶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠉⠀⠀⠉⢻⣿⣿⣇⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣷⣀⡀⠀⣀⣼⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠙⣿⣿⠏⠁⠈⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡄⠀⠄⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⢀⡤⠒⠀⠀⠀⠀⡄⢠
⠀⠀⠀⠀⢀⠀⢀⠀⠁⠀⠈⠀⠀⠀⠸⠀⠀⠀⠀⠁⠀⠀⠀⠆⠀⠀⠀⠀⠰⠀⠀⠉⠀⠾⠒⡂⠈⠙⠉⠁⠀⠀⢠⠆⠠⠀⠃
⠀⠂⠈⠀⠚⠀⠉⣀⡀⠐⣦⠀⢀⣠⠀⠦⢰⢰⢄⡄⢀⠀⢀⢃⠀⡀⢄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⢀⠈⠀⠂⠀⢰⢛⡇⠀⣾
⠀⠀⢀⠔⠑⣄⠘⣯⣹⠇⣿⠀⠀⡇⠐⠃⠘⠘⠈⠆⠎⢩⠾⢸⢰⠧⠀⠂⠀⠀⠀⡇⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢰⡇
⠀⠀⠎⠀⠀⠸⡆⠈⠉⠀⣿⠀⣤⣄⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠂⡃⠀⠀⠀⡀⠀⠀⠀⣬⠛⣿⠁
⠀⡟⠀⠀⠀⠀⢳⡠⠊⠀⢻⡄⢿⡿⠇⡀⡀⠈⠀⠀⠀⠀⡀⠀⡱⡁⠀⠀⡴⡔⠂⠀⠀⠀⠀⠀⠀⣇⠄⠀⠀⠀⠀⢀⠀⢧⣴⠇
⠀⡇⠀⠀⠀⠀⡜⡇⠀⢀⠘⠃⠀⠠⠄⠑⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠬⠌⠀⢁⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠃⠀⠩⠄⢀⡀⠀⢠
⠀⣆⠀⠀⠂⠼⠀⠐⠐⠖⠀⢰⠌⠠⡄⠀⢄⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡰⠀⠐⢈⠀⠼⠀⢰⡛⠒⠂⠠⠤⡜
⠀⠄⠀⢀⠄⠀⠀⠀⠀⠀⠀⠁⠀⢀⠡⠄⠀⡀⠀⠀⠀⠀⠀⢾⣿⣿⣷⣶⡶⠇⠀⠀⠄⠀⢒⡐⠀⠀⠈⢃⠤⠀⠢⠈⢁⣁⠒⠠
⠀⠋⠀⠀⠰⡄⠀⠐⠀⠀⠀⡴⠀⠈⢀⠀⠀⡀⠀⠄⠀⠀⠀⡀⠐⠃⠀⠀⠀⢢⠆⠀⠄⠑⠀⠀⠀⠀⠶⠁⠀⠀⠓⠐⠀⠀⢀
⠀⠀⠀⠀⡀⠉⠂⠀⠀⠐⠄⠀⠀⠀⣠⣦⠀⡆⠀⢀⣤⣄⢀⣀⣁⠀⠐⠀⠀⣠⣈⠀⠀⣰⣆⠀⠒⠊⠰⠀⡀⠀⣀⡀⠀⠀⠠⠄
⠀⢀⠰⠀⠀⠀⠀⠰⠄⠀⠀⣀⠀⢠⡏⣿⢸⡇⠀⣸⠃⠘⢺⡇⠉⠃⡟⣷⠀⣿⢹⠀⢰⡏⢿⠀⡤⠄⠀⠊⠀⠀⠚⡁
⢀⡀⠁⠀⠀⠀⠰⠀⠀⠀⣰⠃⠀⣾⣠⣿⢸⡇⠀⣿⠘⣶⢸⠦⡄⠀⣷⢯⡀⣿⣏⠀⣾⠷⢾⡄⠰⠎⠀⢠⠖⠀⠐⣁⠤⠀⠐
⠐⠃⠄⠂⢸⠂⠠⠄⠀⠀⢘⠁⠠⡏⠀⣿⢸⣇⣀⢿⣀⡏⢸⠀⠀⠀⡇⠀⣷⣿⠘⣇⣿⠀⢸⡇⠐⡒⡂⠀⠀⠁⢀⠀⠀⠀⠀⠀⢂
⠀⡀⠄⠀⡀⠀⠀⠀⠀⠀⠧⠆⠀⠁⠀⠁⠀⠀⠀⠀⠉⠀⠈⠛⠛⠉⠓⠞⠁⠛⠀⠋⠉⠀⠀⠀⠀⠄⠀⢠⠀⠀⠈⠀⠂⠀⠀⠀⠉
⠀⠂⠒⠒⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⡠⠐⠂⠀⠀⠀⠀⣴⠛⢶⠊⣳⠈⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢠⠀⢰⠀⠀⠘⠿⣅
⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠄⡀⠀⠸⠀⠀⠀⠀⠀⠀⡆⢘⠁⠀⠀⠀⠠⢤⡄
⠀⠀⠠⠀⠀⠐⠋⠣⠀⠀⠨⠀⠀⠀⡐⠁⢀⠀⠀⠁⠀⠈⠩⠁⣆⡼⢧⣠⠖⠀⣄⣈⣡⠤⢄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠠⣄⡀
⠀⠀⠈⠀⠀⠀⠀⠀⠁⠠⠄⠠⠄⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠂⠈⠘⠁⠀⠈⠀⢁⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠔⢢⠀⠡⠊⠈⠀⣠⠖⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⣤⠶⣢⠦⣀⣠⡼⠀⠀⠀⠀⠀⢀⠈⠅
⠀⠫⡉⠉⠉⠀⢀⠀⠀⠀⠈⠔⠂⠀⠠⠃⠀⠀⠀⠀⠊⠃⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡴⠁⠉⠀⠀⠀⠀⠀⠆⢀⠙⠋
⠀⠀⠜⠀⠀⠀⠀⠱⡐⢋⠔⠂⠑⠄⠈⠀⠀⠀⠜⠀⠆⠀⠀⠀⠀⠲⠃⠀⢀⠔⠃⢀⠀⠤⠞⠁⢬⣧⠖⠀⠀⢹⠀⠀⠸⠀⢨⠉⠃
⠀⠒⠀⠀⠠⠂⠀⠀⠐⡘⠜⠠⠰⡄⢀⠀⠀⠀⠀⠀⠀⠀⠘⠉⣉⡷⠂⠀⠘⠠⠢⢁⣠⠀⠀⠀⠀⡀⠀⠀⠀⠈⠣⠠⠃⠀⠀⠉⠒
⠀⠀⠄⣠⠈⠀⠀⠇⠀⠈⢌⣀⠠⠊⠁⠘⠀⠂⠘⠀⡆⠀⣠⣞⠋⢀⠀⢀⢀⠀⠀⢀⡁⠐⠁⡴⠋⠡⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒
⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣁⣀⣀⣀⣐⣁⣉⣀⣀⣀⣘⣉⣙⣓⣚⣃⣀⣈⣀⣀⣐⣃⣀⣀⣈⣁⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠶⠒⠛⠋⠉⠉⠛⠲⣦⣀
⢀⡴⡻⠋⠀⠀⠀⠀⠉⠉⠙⠓⠒⠶⠦⠤⢤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⠤⠤⠴⠶⠒⠒⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢳
⣾⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠈
⣧⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣸
⢹⡄⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⢀⡼⠃
⠀⠹⣆⠀⠈⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠠⠤⠤⠤⠤⠤⠄⠒⠒⠒⠒⠒⠒⠒⠒⠚⠒⠒⠂⠀⠀⣀⣠⠴⠖⠋⠁⠀⠀⣠⡴⠋
⠀⠀⠈⠳⣦⣀⠀⠈⠙⠳⠶⣤⣄⣀⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠶⠒⠋⠉⠀⠀⠀⢀⣠⠶⠋⣽
⠀⠀⠀⠀⠱⡉⠛⠶⣤⣀⡀⠀⠀⠉⠉⠛⠛⠓⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠒⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⢀⣀⣤⠶⠚⠉⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠱⡄⠀⠀⠈⠉⠛⠲⠦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠖⠚⠉⠁⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠒⠒⠲⠶⠶⠶⠶⠶⠒⠒⠒⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⠛⠲⡶⠤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡤⠴⠖⠒⠛⠋⢫⣀⣠⠞
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣶⣾⣥⣤⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡌⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢠⣤⣤⣤⣤⣤⠀⠀⠀⣿⡟⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣌⠻⢿⣿⣿⣿⠀⠀⠀⠋⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣌⣉⣉⣀⣤⡴⠟⠉⢳⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣤⣤⣉⣉⡁⠀⢠⣾⣦⡙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣦⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡄⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠙⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⠀⠘⠛⠛⠛⠛⠀⠀⠛⠛⠛⠛⠃⠀⠈⠛⠛⠛⠛




⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠄⣤⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠁⠀⠀⡸⢇⠀⠀⠁⠀⠀⠀⠀⠀⢀⡲⠀⠀⠜⠀⠆⡀⠀⠀⠀⠀⠀⢀⡀⠀⠴⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠆
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠈⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⡁⠀⠀⢆⡀⠀⠀⠐⠖⠚⠒⠒⠒⠒⠒⠒⠒⠓⡖⠀⠀⠀⠀⠀⠀⢣⡬⠃⠘⣦⠟⠀⢣⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠁⢸⠀⠈⡇⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⡀⢀⣀⢀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠱⣆⠁⠀⠀⡀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠡⣇⣹⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡆⠀⠀⠀⡀⢠⡀⠀⠀⠀⠀⡀⢰⠀⠀⡀⠀⠀⡀⠀⢀⠀⢐⣦⢠⣤⣤⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡂
⠀⠀⢠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠤⠿⡶⠆⠀⣁⠼⣧⠀⠀⠀⢠⠄⢨⠀⢄⠤⠸⢀⡅⠠⠠⢄⢠⡿⠘⠉⢉⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢏⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡿⠗⠚⠇⠀⠀⠀⠈⢻⠦⡟⠁⠙⡷⢼⠋⠀⢻⠇⡟⠃⠘⡿⢺⠋⠁⢓⠀⠀⠀⠀⠀⠀⡀⠘⠢⠜⠁⢀
⠀⠀⠠⡄⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠠⠤⠭⠥⠤⠤⠬⠯⠤⠤⠤⠽⠥⠤⠤⠬⠏⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⣠⠀⠀⣀⡐
⠀⠀⠀⠠⢀⢀⠀⠁⠀⠀⠀⠀⡀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢀⡀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠒⢂⣀⡀⣴⠋⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠦⠀⡀⠀⠀⡤⠀⠄⠀⠀⠀⢀⠄⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠆⠀⠂⠀⠠⠠⠤⠜⠙⢷⣋⣀⣠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠹⠀⠀⠀⠆⠀⢲⡀⠀⠐⠃⠀⡂
⠀⠀⠀⠠⠀⠀⣒⡒⢒⣀⢀⣓⠒⣂⡀⣀⡀⢀⣀⢀⣀⠈⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⣂⡀⣐⡢⢜⣂⢐⣀⠒⣀⡚⢀⡀⢓⣢⠰⣛
⠀⠀⠀⠀⠀⠀⠘⣗⡺⠁⠀⢻⣚⠏⠀⠈⣗⡺⠁⠀⢹⣓⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣗⡻⠁⠈⢻⣚⠏⠀⠙⣗⡻⠁⠈⢻⣚⡏
⠀⠀⠀⠐⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠁⠀⠀⠈⠉⠀⠀⠀⠈⠁⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢤⣤⢤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣥⣷⣥⣿⣌⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠻⣿⡿⢫⣿⣛⣛⣛⣛⣛⣿⣛⣻⣿⣝⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⡧⢿⣿⣮⣿⣭⣭⣭⣭⣿⣭⣿⣯⣿⣵⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣫⣭⣽⣛⣛⡱⢸⡿⢫⣿⣿⢏⣿⣯⡻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣼⣿⣮⣿⣯⣾⣿⣯⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡿⢟⣵⣿⡿⢫⣿⣿⢏⣿⣯⡻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢾⣶⡷⢻⢻⠿⣿⣮⣿⣯⣾⣿⣯⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣭⣶⣿⣿⣿⡌⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡶⣿⣿⣿⣿⣿⣷⣭⣾⣿⣿⣿⣿⣿⢠⡀⠀⠀⠀⣿⣿⣷⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠸⣷⠀⠀⠀⣿⣿⣿⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣭⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣭⣤⣤⣤⣽⣿⣯⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠉⢀⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠁⢀⣴⡿⠋⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠙⢿⣦⡀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⢀⣴⡟⠻⣷⣴⡿⠋⠀⠀⣨⣿⣶⠶⢶⠶⠶⠶⠶⠶⡶⣶⣿⣅⠀⠀⠙⢿⣦⡀⠙⢿⣦⡀
⠀⠀⠀⢀⣴⡟⠉⢀⣴⡿⠋⠀⠀⣠⣾⠟⠙⣿⡄⠀⠀⠀⠀⠀⠀⢠⣿⠉⠻⣷⣄⠀⠀⠙⢿⣦⡀⠙⢿⣦⡀
⠀⢀⣴⣟⠁⢀⣴⡿⠋⠀⠀⣠⣾⠟⠁⠀⠀⣸⣿⣶⣶⣶⣶⣶⣶⣿⣇⠀⠀⣨⣿⣷⣄⠀⠀⠙⢿⣦⡀⠙⣿⣦⡀
⠀⢸⣿⠛⠿⣿⣿⣤⣀⣠⣾⠟⠁⠀⠀⣠⣾⣿⠿⠛⠉⠉⠉⠙⢿⣿⣿⣷⣾⠟⠁⠈⠻⣷⣄⣀⣤⣿⣿⠿⠛⣿⡇
⠀⢸⣿⠀⠀⣿⠀⠉⢻⣿⢷⣶⣤⣠⣾⣿⠟⠁⠀⠀⢀⣤⡄⠀⠀⢹⣿⣿⣿⣷⣄⣤⣶⡾⣿⡟⠉⠈⣿⠀⠀⣿⡇
⠀⢸⣿⠀⠀⣿⠀⠀⢸⡇⠀⠀⠉⣿⣿⠇⠀⠀⠀⠀⠘⠻⠋⠀⠀⣸⣿⣿⣿⣿⣿⠉⠀⠀⣿⡇⠀⠀⣿⠀⠀⣿⡇
⠀⢸⣿⠀⠀⣿⣤⣤⣼⣿⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣤⣤⣤⣿⡇⠀⠀⣿⣤⣤⣿⡇
⠀⢸⣿⠀⠀⣿⠉⠉⢹⡏⠀⠀⠀⣿⣿⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⣿⡇⠀⠀⣿⠉⠉⣿⡇
⠀⢸⣿⠀⠀⣿⠀⠀⢸⣇⢀⣠⣴⣿⣿⣧⠀⠀⣿⣿⣏⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⣿⡇⠀⠀⣿⠀⠀⣿⡇
⠀⢸⣿⠀⠀⣿⣀⣤⣾⣿⡟⠋⠁⠈⠻⣿⣷⣄⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠈⠙⢻⣿⣧⣤⣀⣿⠀⠀⣿⡇
⠀⢸⣿⣤⣾⠿⣿⣉⠀⠙⢿⣦⡀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣷⣄⢀⣴⡿⠋⠀⣩⣿⠿⣶⣤⣿⡇
⠀⠈⠻⣧⣄⠀⠈⠻⣧⡀⠀⣹⣿⣦⡀⠀⠀⣸⡟⠛⠋⢻⡟⠛⠛⢿⣇⠀⠀⢈⣿⣿⣏⠀⣠⣾⠟⠁⠀⣠⣿⠟⠁
⠀⠀⠀⠈⠻⣷⣄⠀⠈⣻⣿⣟⠁⠙⢿⣦⣰⡿⠁⠀⠀⢸⡇⠀⠀⠈⢿⣆⣴⡿⠋⠈⣻⣿⠟⠁⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠈⠻⣷⣾⠟⠉⠻⣷⡄⠀⢹⣿⠿⠿⠿⠿⣿⡿⠿⠿⠿⠿⣿⡏⠀⣠⣾⠟⠁⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠈⠻⣶⣾⣇⣀⣀⣀⣀⣼⣇⣀⣀⣀⣀⣹⣷⣾⠟⠁⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⣼⡟⠛⠛⠛⠛⠛⢻⡟⠛⠛⠛⠛⠋⢻⣧⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣿⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠶⣦⣄
⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢻⣦
⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⢆⣠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⠤⠤⠤⢤⣤⣤⣼⡁⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⣰⠃⠀⠀⠀⠀⠀⢠⠁⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢄⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⢻⣧
⠀⢰⠇⠀⠀⠀⠀⠀⠀⡏⠀⠀⣸⠁⠀⠀⠀⣾⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠈⣇⠀⠀⠀⠈⣧⠀⠀⠀⠀⢳⡀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⣿⣇
⢀⡟⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡿⠀⠀⠀⠀⡟⠀⠀⠀⢰⡇⠀⠀⠀⢸⠀⠀⠀⠀⢿⠀⠀⠀⠀⢹⡆⠀⠀⠀⠈⢷⡀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢹⣿
⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡇⠀⠀⠀⢸⡇⠀⠀⠀⢺⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣷⠀⠀⠀⠀⠘⣷⠀⠀⠀⠈⠇⠀⠀⠀⠀⣼⠇
⣟⠻⣦⣤⣀⣀⣀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠇⠀⠀⠀⠸⠇⠀⠀⠀⠸⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢀⣀⣠⡴⠋
⢿⡀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠓⠒⠒⠒⠲⠶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠶⠖⠒⠒⠒⠛⠛⠛⠋⠉⠉⠁⠀⠀⢀
⠈⢷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾
⠀⠀⠈⠙⠻⣷⠶⠶⠶⠶⠶⣤⢤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡤⢤⡶⠖⠛⠁
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⠇
⣀⣀⣀⣀⣀⣀⣀⣈⣙⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣷⣶⣋⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡖⠒⢲⠀⠀⠀⠀⠀⠀⠀⢀⠤⢒⠂⢐⡂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⢠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠
⠀⠀⢰⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⢠⢁⠞⠁⡠⠄⠈⠢⠑⡄⠀⠀⠀⠀⠀⡀⠤⡀⡀⠿⠸⠸⢂⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⠠⢤
⠀⠀⠀⠀⠀⡂⡯⠭⢽⠀⠒⠒⡆⠀⠀⠀⠀⢘⠁⡆⠀⡠⠄⡇⠀⠀⠀⠀⠀⡼⠤⠂⠘⡈⠖⠉⠉⠁⠀⠀⠀⠀⠀⠀⠸⠠⠠⠤⠠⠤
⠀⠀⠀⠀⠀⠄⣏⠉⣿⢰⣄⣠⡇⠀⠀⠐⡈⢮⡀⠅⠀⣇⡴⢠⠂⠀⠀⠀⠀⠈⣄⣀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡏⠉⠑⠂⠠⣄⠔⡱
⠀⠀⢸⠈⢹⠅⠏⠉⡿⠸⡏⢀⠇⠀⠀⠀⠈⠢⠍⣁⡉⠡⠴⡣⠑⠄⠀⠀⡔⠁⠀⠎⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⢀⠗⠠⠀⣈⣀⠠⠊
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠈⠀⠈⠈
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⠔⢹⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⡒⠀⠀⠀⠠⠂⠀⠀⠀⡠⡀⠈⠆⠀⠀⠀⠀⠀⠀⢀⠄⣒⣂⠠⣀⠆⠸
⠀⠀⠀⠋⠀⠊⠀⡄⡂⣀⢇⠀⠀⠀⠀⠀⠀⢀⠔⣉⠢⠔⡠⠲⠁⠀⠀⠀⠀⢄⡀⠀⠀⠀⣇⣠⠃⠀⠀⠀⠀⠀⡐⡐⢁⢄⢀⠐⡌⠎⠁
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⡆⠆⠀⠀⠀⠀⠴⡡⠊⠀⠑⠊⠀⡇⠀⠀⠀⠀⠀⠀⠀⠜⢌⠀⠃⠁⠗⠀⠀⠀⠀⠀⡁⢇⠔⠢⡥⠊⡀⡀
⠀⠀⠀⠀⠀⠀⠈⠊⠤⠌⣜⢀⠀⠀⠀⠀⠀⡄⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡂⠀⠀⠀⢄⠙⠪⣂⠀⠀⣊⡔⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠀⠀⠀⠤⠤⠧⠠⠤⠸⠤⠤⠧⠤⠀⠀⠀⠀⠀⠀⠀⠈⠄⠠⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠒⠁

⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⢩⢩⡍⣭⠉⠉⠉⠉⠉⡆⠀⠀⠀⠀⠀⠐⠂⠐⠒⡄⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠦⣄⡀
⠀⠀⠀⠀⠀⠀⣔⡄⢀⡢⠈⠀⠀⠀⠀⠰⠒⠒⠒⠒⠖⠶⠲⠖⠲⡁⠀⠀⠀⠀⠀⣀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠰⠳⠣⠇⠶⢰⠐⠞⢆
⠀⠀⠀⠀⠀⠙⠀⠘⠝⠀⢀⠀⠀⠀⠀⢀⣘⣈⣃⣛⠄⠀⠠⡐⣄⠇⠀⠀⠀⢀⣀⣀⡀⠁⣀⣀⣀⠀⠀⠀⠀⠆⠤⠴⠤⢤⢔⠢⢑⠾
⠀⠀⠀⢆⣀⣀⣀⣄⣀⣄⣀⠲⠀⠀⠀⠀⢤⠠⠄⠈⠀⡀⠀⠐⡈⡆⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⢸⠀⠀⠀⠀⠡⡴⡤⡄⣼⢀⠀⠀⡁⠁
⠀⠀⢘⠘⠒⠒⠒⠒⠒⠒⠒⠓⡆⠀⠀⠈⠀⠀⠀⠓⠉⠒⠐⠚⢝⠇⠀⠀⡏⠉⠈⢰⠁⠀⠁⠈⠀⠈⠀⠀⠀⠀⠈⠳⠥⠬⠶⠥⠄⠪⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠁⠁⠁⠈⠁⠈
⠀⠀⠀⢀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠈⠔⠈⠉⠁⢢⠀⠀⠀⠀⠀⠀⠱⠆⠒⠀⠂⠒⠀⠐⠂⠶⠃⠀⠀⠈⡤⠤⠄⠠⠤⠤⢬⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠀⢀
⠀⠀⢠⠤⠥⠔⠠⡀⢰⠠⠖⢄⡀⠀⠀⠀⠀⣀⣀⠉⠀⠀⠔⡄⠀⠀⠀⠀⠀⠀⠀⡄⢀⢄⠀⡰⡀⠀⠀⠀⠀⠀⠀⠀⣽⢛⣙⣫
⠀⠀⠘⠒⠒⠢⠐⠁⢠⠒⠦⠊⠁⠀⠀⠀⠀⣃⣸⣀⣖⣲⣀⡇⠀⠀⠀⠀⠀⠔⡆⡇⠀⣄⡀⢌⠘⠀⠀⠀⠀⡄⠀⠀⢤⠀⠀⠠⠠⠀⠠
⠀⠀⠀⠀⠢⠠⣀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠢⠤⠤⠤⠤⠤⠤⠤⠎⠀⠀⠀⠀⠈⠁⠁⠘⠝⠛⠫⡞⢄⠀⠀⠀⣇⣋⣀⣸⠈⠁⢃⣉⣀⣀
⠀⠀⠀⠠⠄⠀⠤⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠤⠈⠤⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠎⠁⠉⠱⠀⠀⠰⠉⠈⠱

⡿⢿⠿⠿⠿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⡿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣡⠞⠋⠛⢦⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⡟⣰⠋⠀⠀⠀⠈⢷⣘⣧⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⡤⠶⠶⠶⣾⢠⣏⣙⣳⡶⠶⢛⣉⣭⣤⡶⠶⠶⠦⣍⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⢰⡇⣼⠿⢋⣡⠶⢾⣉⠻⢿⣄⣧⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠹⣆⠀⣠⣾⠇⣿⠞⠋⠀⠀⠀⠈⠛⢦⣌⠻⢦⡀⢀⡾⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡙⢿⣡⣾⠀⡏⠀⢀⡴⠖⢦⡀⠀⠘⡏⢻⣦⣹⠟⣱⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢻⣦⡙⢿⠀⣇⠀⠘⢧⣀⣼⠃⠀⢠⣇⣼⠟⣡⣾⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣰⠏⠀⠙⢿⡆⣿⢦⣀⠀⠀⠀⢀⣤⠾⢋⣥⠞⠁⠈⢷⡘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠸⡇⢻⣦⣍⡛⠶⣾⣋⣥⣾⠋⡟⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣌⠳⠶⠶⠶⢿⠘⣟⣩⡽⠷⠦⣬⣙⡛⠺⠷⠶⠶⠶⢋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠚⠛⠛⣇⢹⡅⠀⠀⠀⠀⡽⢩⡟⠛⠓⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡙⢦⣀⣠⠞⣡⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠦⣬⡤⠞⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠅⣿⣼⢒⡯⠁⣯⡷⢼⢫⡌⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠈⠀⠉⠁⠁⠉⠈⠉⠁⠈

⠀⠀⢀⠑⠄⠀⠀⠀⠀⡔⡨⢭⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⢇⠂⡀⠨⢀⣀⠈⠀⠜⡄⠀⠄⠀⠰⣀⠀⢤⡠⠤⠤⠤⣄⠀⠀⠀⢫⠙⡀
⠀⡠⢂⠊⠈⠀⠂⠖⠚⢀⠎⠌⠀⠐⠬⠉⡞⢁⠔⠁⠀⠀⠀⠀⠀⠱⡄⠀⢀⣆⢱⠤⢤⡀
⠔⣑⡃⠠⢄⠀⠉⠁⠀⠄⣈⣀⡀⡤⠀⠀⣇⠌⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠈⠺⠀⠀⠛⢢
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⡤⠤⠤⠤⢤⣴⡋⠀⠀⡞⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⣞⠁⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⢒⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⡤⠞⠁⣶⠉⠉⠉⠉⠉⠉⠹
⠀⢀⣀⣀⣀⣀⣀⣀⣠⠋⠀⡀⢀⠎⠀⠀⠀⠀⠐⠀⠀⠐⠒⠲⢼⠀⠀⠀⠈⣇⣀⣀⣀⣧⣤⣤⣤⣀⣄⣀⣼
⠀⠀⢀⣀⣀⣀⣀⣀⣘⣆⠀⠘⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢻⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡧⢠⠇⡀⠄⠀⠀⠀⢲⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠈⡏⠀⠀⠀⠀⠀⣸⡀⠀⢀⣀⠴⡁⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢧⠀⠀⠀⠀⠐⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠙⠳⠤⣤⣤⡤⢴⠚⠉⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠥⣴⣀⣠⣀⠵⠀⠀⠸⢄⣤⣀⣀⠦⠼

⠀⠀⠀⠀⠀⠀⣀⢠⠔⠒⠢⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡠⠊⠉⠑⢦⢀⡀
⠀⠀⠀⡀⠔⠉⠀⠁⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠁⠀⠂⠤⠤⢄⠀⠀⠀⠀⠀⠀⠘⠂⠁⠢⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠢⠤⠤⡀⠀⠀⠀⠀⠀⠀⠑⠀⠐⠤⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠁⠑⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠁⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⡆⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⢀⣀⣀⣀⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⡖⠒⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣒⣒⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⣀⣠
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⢀⣀⠀⣀⡀⢀⣀⠀⣀⣀⠀⣀⡀⢠⣶⣶⣶⠀⠀⠀⠀⠀⣃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡆
⠀⠀⠀⢈⣉⠀⡉⡁⢈⣉⡁⣉⣉⢀⣉⡁⢸⣿⣿⣿⠀⡇⠀⡇⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠃
⠀⠀⠀⢨⢨⠀⣍⡅⢠⣩⡄⣨⣥⠀⡍⡄⢸⣷⣶⣾⢀⠀⠀⡇⠀⠀⠀⣒⣒⡆⠀⣖⣒⡂⠀⢰⣒⣲⠀⢰⣒⣒⠀⠀⣒⣒⡆⠀⡇⠀⠀⠀⠂
⠀⠀⠀⠤⠠⠀⠄⠄⠠⠠⠄⠤⠤⠠⠄⠄⢸⣗⣒⣺⠸⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⡀⢀⡇⠀⠀⠀⢰
⠀⠀⠀⠐⠰⠀⠖⠆⠰⠶⠂⠲⠶⠐⠂⠆⠸⠗⠒⠺⣄⠀⠀⡇⠀⠀⣗⣒⣚⡇⠀⠀⣼⠩⠩⢳⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠸
⡔⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣑⣄⣀⣀⣀⣄⣀⣀⣇⣀⠀⠁⠀⠀⠁⠀⠀⠹⡆⠀⠟⠀⠀⡇⠐⡁⢸⢐⡉⢹⢸⡉⢱
⠑⣤⠤⠤⠤⠤⣤⠤⠤⠤⢤⣤⠤⢼⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⣀⣀⣀⣀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠁⠁⠀⠁⠁⠀⠁⠁
⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⠀⢈⠈⡇⣈⢈⡇⣈⢈⡇⢸⠀⠀⠀⢀⡀⠀⠀⠀⡄⠀⠀⠀⠀⡇
⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⠋⠀⠀⢀⣇⣠⣀⠀⠀⡇
⣴⣿⣷⣶⣶⣶⣿⣶⣶⣶⣾⣿⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣾⣶⣶⣶⣶⣶⣷⣶⣶⣾⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣄⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠙⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⣴⡆⣸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠙⠋⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⢸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣧⠀⠀⣾⣿⣿⣿⣿⣿⡄⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⢀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠨⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⡆⠀⣿⣿⣿⣿⣿⣤⣤⣤⣬⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠿⠿⠀⠿⠿⠿⠿⠿⠿⠿⠇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠉⠉⠉⢹⣿⣿⣟⠉⠉⠉⢹⣿⣿⡟⠉⠉⠉⢹⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣀⣀⣀⣸⣿⣿⣿⣀⣀⣀⣸⣿⣿⣇⣀⣀⣀⣸⣿⠀⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣇⣀⣀⣀⣸⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣤⣀⣠⡄⢀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣾⣟⢋⣉⣁⣀⠠⠤⠀⠋⠁⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⣭⣤⡀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢀⣼⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢪⣤⡀⣿⡇⣿⣦⢤⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⣰⣿⣿⣿⣿
⠀⠀⣠⢔⣉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣯⣤⢸⣿⡇⣿⡇⣿⣿⢸⣿⠀⠀⠉⠒⠦⢄⣀⣠⠎⠀⣠⣼⣿⣿⣿⣿⣿
⠀⠈⡁⢸⣿⢀⣤⡕⢄⠀⠀⠀⠀⣀⠔⣥⡄⣿⣿⢸⣿⡇⣿⡇⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⣜⣁⣠⣶⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⢸⣿⢸⣿⡇⣤⣕⢤⡠⢾⣥⡄⣿⡇⣿⣿⢸⣿⡆⣿⡇⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⢸⣿⢸⣿⡇⣿⣿⢰⣶⢸⣿⡇⣿⡇⣿⣿⢸⣿⡇⣿⡇⣿⣿⢸⣿⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⢸⣿⢸⣿⡇⣿⣿⢸⣿⢸⣿⡇⣿⡇⣿⣿⢸⣿⡇⣿⡇⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⢸⣿⢸⣿⡇⣿⣿⢸⣿⢸⣿⡇⣿⡇⣿⣿⢸⣿⡇⣿⡇⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿
⠀⣿⣿⢸⣿⢸⣿⡇⣿⣿⢸⣿⢸⣿⡇⣿⡇⣿⣿⢸⣿⡇⣿⡇⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿
⠀⣿⣿⢸⣿⢸⣿⡇⣿⣿⢸⣿⢸⣿⡇⣿⡇⣿⣿⢸⣿⡇⣿⡇⣿⣿⢸⣿
⠀⠻⠟⠸⠿⠘⠿⠇⠿⠟⠸⠿⠘⠿⠃⠿⠇⠻⠿⠸⠿⠃⠿⠇⠿⠟⠸⠿
⣿⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⣿⡏⢉⣉⣉⠉⣿⣿⡿⣿⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠟⢁⣄⠉⣁⣼⣿⣿⣆⣈⢁⣤⡈⠻⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⣅⠰⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⡿⠂⣼⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠺⠟⠀⣾⣿⡿⠋⣠⣶⣦⣄⠙⢿⣿⣷⡀⠛⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⣿⣿⣿⠁⣼⣿⣿⣿⣿⣧⠈⣿⣿⣿⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣟⢙⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠇⡆⡔⣦⡔⡆⡖⡰⡄⡖⡖⡆⡖⡖⡆⣤⠖⣤⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠙⠃⠁⠉⠁⠁⠁⠁⠁⠁⠁⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣖⠻⠿⣲⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠃⠀⠃⢸⡶⡇
⠀⠀⠀⢀⣠⠤⠤⠤⠤⠧⠤⣤⣀⣀⣾⣈⣁⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⠤
⠀⠀⡞⠉⡇⠀⣴⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣈⠉⢹⡟⠉⠉⠉⠉⠁⠒⠒⠒⠶⡴⠽⠋⠰⠚⠉⢉⣉⣉⣉⣉⣲⣤⡀
⠀⠀⡷⣾⣧⡀⢻⣟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⠇⡇⠀⡄⠀⠀⠀⠀⣤⠱⠞⡡⠄⠐⠊⠉⠓⡒⠋⠉⠙⠛⠳⠈⠉
⠀⢰⠀⠛⢠⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣸⢀⠃⠀⣌⣉⡉⠉⢉⣯⠄⠀⢡⠀⠀⠀⢀⡴⠃
⠴⣇⡀⢀⠾⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠺⠀⡇⠸⠀⢰⠀⠀⠀⠀⠠⢿⣇⠀⠸⡀⠀⠀⡏⠐
⠀⠀⠈⢋⢀⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡆⢸⠁⢸⠀⠀⠀⠀⠀⠀⠉⠉⠸⢳⣴⠃⢀⠔⠓⣤⠤⣄
⠀⠀⠀⢸⣿⢧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠃⡎⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⣿⡆⢸⡀⠀⣸⣉⣈⣇⣀⣀⠠⠤⠤⠤⠤⠤⢄⡀
⠀⠀⠀⠘⠉⠁⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⢰⠃⠀⡇⠀⠀⠀⠀⠀⣀⣀⣰⡏⢩⡙⠇⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡸⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⢸⠀⠀⡇⠀⠀⠀⡴⠋⠁⠀⠀⠱⡈⢉⣠⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠉⣗⡖⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⠀⢰⠀⠀⠀⢸⢁⠀⠀⠀⠀⠀⠈⠉⠀⠱⡀⠀⠙⢏⠒⠠⠤⢤⣀⣀⣀⣀⠤⠤⠐⠉
⠀⠀⠀⠀⣿⣿⠀⠻⠛⠛⠛⠛⠛⠛⠿⠿⠿⠇⢻⠀⢸⠀⠀⠀⢸⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠘⠙⢻⠉⠉⠉⠉⠉⠙⠻⠛⠯⠟⣟⣋⠀⢸⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠞⠲⡄⠈⢆
⢰⢖⠒⠒⢞⢯⡙⢃⡲⣱⢆⡶⡰⢆⠶⣶⣶⡛⢝⠷⣾⠀⠶⠐⠷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠤⢛⣛⠉
⠈⠲⣷⣄⠀⠑⠿⠖⠛⠚⠛⠛⠛⠛⣻⠟⠇⠈⠓⢵⣿⣶⣶⣖⣉⡥⠷⡆⠀⠀⠀⠀⠀⠀⣠⡀⡿⠀⠸⣍⣁⢙⠆
⠀⠀⠈⠻⣍⣉⣉⣉⣉⣁⣀⣀⣐⠒⢺⠦⣸⠆⠀⠈⢻⢻⠛⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⡰⢇⣯⠇⠀⣾⠭⠉⠈⢹⡆
⠀⠀⠀⠀⢸⠿⠀⠀⠀⠀⠀⠀⢠⠚⠛⠢⡬⠂⠀⠀⢸⢸⠈⠁⠀⠀⠀⠁⠀⢀⠖⠒⠊⡇⠀⢏⢀⣴⢣⡤⡄⠀⠀⠁
⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠑⠒⠋⠇⠀⠀⠀⠘⢸⠀⠀⠁⠀⠀⡖⠒⢡⠀⠀⠐⠗⠛⣜⢷⡟⠉⠳⣸⣳⣀⡶⢦
⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣀⠀⠀⡀⠈⠀⠀⠀⠈⢠⡦⢰⠎⡀⠀⠀⠀⠀⠈⠛⠀⢰⡤⣈⡇⠀⢸⠃
⠀⠀⠀⠀⠘⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣅⣥⠴⠒⠊⠉⠉⣱⣆⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡘⣧⡀⢈⠇
⠀⠸⣟⢕⣾⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠖⠈⠀⠀⠉⠉⠙⠛⢛⠒⠒⠒⠚⠋⠛⠁⠀⠈⠉⢱
⠀⠀⠈⠓⢝⠯⡑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠠⢤⣤⠶⠮⣤⡤⠄⠀⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠑⢌⢢⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⡄⠀⠀⠀⠀⠀⠒⢿⡷⡲⢶⠒⢺⠏⠓⠀⠀⠀⠀⢠⣆
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢴⣒⣊⣉⣉⣻⣗⡺⠿⡭⠥⠖⠤⠤⠈⢹⡍⣚⡱⢏⣀⣄⣐⣤⠤⠖⣶⠋⣉⣉⣐⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠽⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠉⠉⠉⠐⠢⠒⠊⠉⠉⠁⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⠀⢸⡀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⡀⠀⢀⡠⠞⠁⠒⠒⡲⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠊⠁
⠀⠀⠀⠀⠀⠀⡏⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⣀⢀⡠⠊⠉⠐⠒⠉⠑⠤⢀⣀⣀⡀⠤⠊
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣷⠀⠀⠀⣶⠒⠲⠶⣦
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢻⠀⠀⠀⡇⠀⠀⠀⢿
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢸⡇⠀⢀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣇⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⣷
⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣸⣄⣼⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀
⣯⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⢻⡇⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⣀⣤⠶⠚⠉⠁⠀⠀⠸⡇⠀⠀⢀⣠⡴⠖⠋⠁⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡧⠶⠛⠉⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⣀⡀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣀⠀⣀⣀⣀⡀⠀⢸
⣿⠀⠀⠀⠀⠀⠰⠒⠒⠒⠒⠒⢲⠒⠒⠒⠂⠐⠒⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⣿⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⢻⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡃⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡀⡀⣀⢀⣀⣸⠀⠀⠀⠀⠀⣸⢀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠆⢸⠸⣷⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣶⡿⣧⠺⡌⢿⠀⡟⣼⢿⣟⣠⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⡿⢷⣍⠳⠈⠀⠀⠀⠠⠟⣫⠿⠷⣾⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣤⣿⣿⠶⡌⠀⠀⠀⠀⠀⠀⠈⠰⢶⣿⣿⣿⠇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣟⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠭⣿⣿⣿⣗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣫⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠺⠿⣟⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣟⣩⣶⠟⣴⢀⡆⣼⠀⣸⣆⣷⣬⣳⣮⣽⣿⠿⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⣿⠟⠋⠈⠘⠗⠠⠄⣾⣿⣿⠯⠁⢿⡿⣛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠠⠆⠰⠗⠐⠃⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⠤⠤⢤⡄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡁⠀⠀⠠⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠈⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠴⠆⠀⢀⡤⣀⣀⣐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⢼⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠳⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⢤⡀
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢤⣬⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠘⠋⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⢀⣀⣠⣾⣿⠏
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣿⣿⣿⣟⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣦
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣤⠶⣶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣹⠆⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡟⠀⠀⣿⠀⠈⢻⡀⠀⠀⠀⣤⠀⠀⠀⠉⠙⢧⡀⠀⠀⠀⣸⣏⣹⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡇⠀⣴⠟⠳⢦⣼⠇⠀⠀⠀⣿⠀⣤⣤⣤⠀⢨⡟⢳⡶⠛⠉⠈⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⢾⣅⣀⣠⡴⠋⠀⠀⠀⠀⣿⠀⣿⠀⣿⠀⢀⣛⡛⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿⠀⣿⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⣿⠀⡿⠀⣿⠀⣿⠀⣿⠀⣾⠛⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣤⣤⣤⣤⡤⠠⣤⣤⣤⡄⠀⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⡽⠋⠉⠉⠉⠉⠉⠉⢿⡍⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠶⠶⠶




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

⠀⢠⠒⠒⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠯⡇⢩⢳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠂
⠀⠨⠒⠒⠒⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⡯⠭⣭⡇⢨⣼⠍⣷⠀⠀⠀⠀⠀⠀⠒⢤⠊⡚⠢⡀⡨⢴⠀⠀⠀⠀⠀⠀⣀⠀⢲⠈⠀⢸
⠀⢠⠤⠤⠤⠤⢄⠁⠀⠀⠀⠀⠀⠀⠀⠹⣫⢵⣢⣳⢮⣓⠏⠀⠀⠀⠀⠀⠈⠑⠃⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⠀⢸
⠀⠘⠒⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠻⠶⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠦⠀⠼⠀⠄⠘⠄



⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣚⣭⣝⣶⣄⠀⠀⠀⠀⠀⠀⢠⢄⢔⢢⠀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠚⠚⠐⠒⠃⠒⠒
⠀⠀⠖⣀⡄⢸⠤⡀⠀⠀⠀⠀⠀⠀⢠⢫⢻⠃⠀⠈⢳⠹⡄⠀⠀⠀⠀⠀⢈⢮⡙⠵⣫⠚⢊⡲⡕⠁⠀⠀⠀⠀⢙⢙⢈⠋⡃⠀⢀⢀
⠀⠤⠇⡇⠀⢸⠀⠉⠃⠀⠀⠀⠀⠀⠘⣜⢾⣄⡀⠀⠈⢀⠅⠀⠀⠀⠀⠀⠘⠂⠈⠉⠡⡢⢪⠛⠺⠀⠀⠀⠀⠀⠉⢉⢈⠉⡅⠉⠉⠈⠉
⠀⣀⣅⢧⣀⣸⣀⣄⠆⠀⠀⠀⠀⠀⠀⠈⠣⠯⡂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡀⠀⠀⠀⠀⠀⣴⣰⠀⠀⠀⠀⠀⢀⠔⠁⠀⡇⠀⠑⢄⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠐⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⣹⢉⠽⣁⣙⠉⡃
⠣⠒⠂⢤⠠⡔⠁⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⡇⠀⠀⢈⡄⠀⠀⠀⠀⠀⡇⡇⢠⡀⠀⠈⡐⠢⠀⠀⠀⠀⠀⠀⠀⡁⣸⣈⣶⢫⠼⣤⡇
⠐⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⡠⠊⡇⠀⠀⡠⠁⠀⠀⠀⠀⠀⢇⠇⠀⠀⢠⡄⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡸⢏⣸⣈⣲⣁⡅
⠠⠀⠀⠀⠧⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠉⠢⠤⡧⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢋⣱⣁⣸⣀⣨⡀⠅



⠀⢀⡀⡆⠄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢁⠸⣈⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡒⢆⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠉⠢⡀
⠀⡇⠀⡇⠀⠤⢸⢸⠀⠀⠀⠀⠀⠀⢰⠁⡜⠁⠀⠀⠑⠀⠁⠀⠀⠀⠀⠀⢰⠉⠉⡆⠀⢀⠀⡇⢨⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⡄⠀⠀⢘
⠀⡇⠀⡆⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠸⣀⠧⡀⠀⠀⡠⢄⡄⠀⠀⠀⠀⠀⢸⠀⠀⠈⠒⠉⠀⡇⢰⠀⠀⠀⠀⠀⠀⠰⣀⢔⢥⠃⠀⠀⡔
⠀⠣⠆⠧⠀⠤⠘⠼⠀⠀⠀⠀⠀⠀⠀⠑⠄⠈⠉⠉⡠⠎⠁⠀⠀⠀⠀⠀⠸⠠⠄⠀⠠⠀⠀⠇⠨⠀⠀⠀⠀⠀⠀⠀⢴⠁⠸⠥⠔⠊
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⣭⣭⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣠⠴⣶⠤⣄⣠⠞⠛⠛⠛⠨⠛⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡞⠁⠀⣿⠀⠞⢳⡀⠀⠀⠀⠈⠉⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡀⠀⠀⠛⠒⠒⢺⠇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠳⣄⡈⠳⣄⣠⠞⠀⠀⠀⣶⣶⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠉⠉⠛⠚⠛⠛⠂⣦⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣽⡉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡖⠉⠙⢦⠀⠀⠀⠀⠀⠀⡴⠊⠉⠉⠲⡄⠀⠀⠀⠀⠀⠀⡤⠤⠞⠙⠤⢦⡀⠀⠀⠀⢀⣀⠘⣆⡞⣀⡀
⠀⠈⢧⣀⣀⡜⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⣠⠏⢠⠞⠓⣆⠘⢤⠀⠀⠀⣀⡴⠛⢯⠋⢣⣀
⢠⠖⠒⠬⠴⠒⢢⡀⠀⠀⠀⠀⠳⣄⣀⣀⡴⣇⠀⠀⠀⠀⠀⠈⣦⠘⠦⠤⠋⢰⠋⠀⠀⠀⠉⣇⠶⡼⠰⢠⠏
⠘⠦⣤⣤⣤⡤⠔⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠳⠄⠀⠀⠀⠀⠙⠛⠢⡴⠚⠚⠀⠀⠀⠀⠐⠏⠓⠲⠒⠉⠆

⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡤⡄⢠⡄⢠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⢤⡀
⠀⠀⠀⠀⢧⡤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠁⠈⠁⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⡴⢭⠇⠀⠀⢸⡇
⠀⢰⠖⠒⠲⡗⠒⠒⣦⠀⠀⠀⠀⠀⠀⠀⢶⠀⢸⠀⢸⡆⢸⣱⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⢀⡇⢄⢸⡇
⢠⠼⢦⠀⡴⠧⡄⢠⠿⡄⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⣠⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⢸⠁⠋⢸⡇
⠘⠓⠚⠀⠓⠒⠃⠘⠒⠃⠀⠀⠀⠀⠀⠀⠛⠀⠘⠀⠳⠋⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠚⠁

⠀⣠⡶⠒⠒⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⢒⠒⠒⠒⢲⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠶⠶⠶⠶⠶⡆
⡾⠷⣃⣀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣖⢸⢰⠄⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡆⠐⡆⠀⠀⡇
⡇⢼⣏⢹⣷⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣋⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠇⠰⠃⠀⠀⡇
⡇⠈⠛⠛⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣹⣿⣿⣉⠉⠉⠀⠀⠀⠀⠀⠀⠼⠭⠭⠭⠭⠭⠭⠭⠭⠷

⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⣰⣿⣿⠀⣾⣿⣷⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣰⣿⣿⡏⢸⣿⣿⣿⡆⢹⣿⣿⡀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⠀⠀⠀⠐⠿⠿⠿⠀⠿⠿⠿⠿⠷⠈⠿⠿⠷⠀⠀⠀⠀⣼⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢀⣴⠟⣿⡟⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠶⠟⠁⠀⣿⡇⠀⠙⢷⠄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣀⡀⠰⢶⣾⣿⣿⣷⣶⣶⣤⣀⡀⠀⠀⠀⠈⠇⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣠⣶⣿⣿⣿⣷⣦⣌⡙⠻⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⣰⣿⡿⢋⣡⣌⠹⣿⣿⣿⣿⣦⠘⣿⣿⣿⣿⣿⣿⣿⣷⠄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁⣾⣿⣿⡄⠻⠿⢿⣿⣿⡆⢹⣿⣿⣿⣿⣿⣿⠋⣰⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣇⠸⣿⣿⣿⣿⣷⡄⢹⣿⣿⣤⡙⠛⠿⠿⠟⢃⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣌⡙⠛⠛⠛⣠⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠖⣂⣀⣀⣀⣀⣀⣐⠢⠀⠀⠐⠒⠖⠂⠀⠀⠀⠀⠄⢀⡂⠠⠄⠀⠀⢰⢈⠉⠭⠭⢉⠒⠊⡉⠭⠭⠍⢰⠀⠀⠀⠀⠙⢮⢄⠀⠀⣀⠀⠀⣀⠀⡇
⠀⠀⡀⣇⣀⣀⣂⣀⣀⡸⠘⠀⠀⠘⡉⠉⠃⠀⠀⠀⠁⠜⣁⣀⠑⡌⠀⠀⢐⠐⢒⣀⡒⠒⢨⠀⠒⠒⠀⠒⢸⠀⠀⠀⠀⡠⣄⡙⢦⡄⠈⠲⡀⢸⠀⡇
⠀⠀⠇⠖⣤⠒⣤⠢⡔⢢⠘⠀⠀⠃⡇⠘⠀⠀⠀⠈⠀⢆⠃⠀⠠⠃⠁⠀⠠⢈⠥⠤⠤⢌⠸⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⠑⢢⠇⠀⠙⢖⡀⠈⠋⠀⡇
⠀⠀⡆⠟⣿⠛⣽⠛⡝⢫⢘⠀⠀⢂⡇⢸⡀⠀⠀⠀⠁⠀⡶⣖⡇⠈⠀⠀⠈⠐⠂⠐⠒⠒⠘⠀⠒⠀⠐⠒⢸⠀⠀⣴⠏⣴⠏⠀⠀⠀⠀⠙⢵⡀⠀⡇
⠀⠀⠇⡛⣿⠛⣿⢛⡇⢸⠠⠀⠀⠘⢭⡭⠃⠀⠀⠀⠀⠀⠛⠒⠁⠀⠀⠤⠘⠮⠍⠭⠭⠉⠾⠏⠭⠩⠭⠭⠜⠀⠸⣟⣾⠏⢀⢠⠤⠤⠠⣄⠀⠙⢇⡇
⠀⠀⠣⠭⠤⠭⠤⠥⠬⠥⠞⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠜⣌⡣⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠉⠀⡔⠃⢀⠤⠤⢤⡀⠙⢆⠀⠁
⠀⢀⣴⠖⡖⠸⠰⢒⢲⣦⠀⠀⠀⡠⠒⠉⣻⡯⢋⢓⣄⠀⠀⠀⠀⠀⡧⣭⢴⠀⠀⠀⠀⠀⠀⡜⢉⢣⠀⠀⠀⠀⠀⠀⣨⠀⢰⠁⠀⠀⠀⢙⡄⠘⡆
⠀⢸⠉⠉⠉⠁⠉⠉⠉⠩⠀⠀⡜⠀⠀⠀⡿⣗⣕⣕⣝⡆⠀⠀⠀⡸⣠⠍⣆⢧⠀⠀⠀⠀⡸⠀⢨⠀⢣⠀⠀⠀⠀⡏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡇⠀⠀⠀⠙⠉⠉⠉⢙⠀⠀⠀⢠⢡⠃⠀⠘⡌⠆⠀⠀⡰⠁⠀⠘⠀⠀⢣⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⣄⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠖⠋⠀⠀⠀⠈⠺⠀⠐⠻⢿⣯⣏⣿⡽⠾⠂⠀⠀⠀⠀⣤⠄⢠⠀⢶⠌⢠⢠
⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠑⠒⠂⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠘⠧⠠⠄⡀⠀⣤⠤⠜⠀⠀⠀⠀⠀⠀⠉⠉⡉⣱⠀⠀⠘⠃⠈⠃⠸⠋⠀⠁⠃⠀⠀⠀⠀⠀⠁⠖⠒⠒⡒⠒⠒⢢⢹⠀⠀⠀⣀⠤⠦⣼⢒⢶⣄
⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢸⠁⠀⠇⠾⠀⠀⠀⠀⢀⣤⠤⡤⢄⡀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠂⠚⢸⠀⢀⠞⠀⠀⠀⣿⢕⢕⠝⣧
⠀⠈⡎⢁⠤⣁⠤⣉⠤⡌⠁⡇⠀⠀⠀⢰⠀⢠⠉⡇⠀⠀⢀⠚⠁⣀⠤⠤⣀⠈⠳⡄⠀⠀⠀⠀⠐⣁⢿⣨⢯⣹⢏⡹⢸⠀⢸⠀⠀⠀⠀⠿⠷⠿⠿⡞
⠀⠀⡏⠻⡠⠻⠄⠟⠄⠏⠙⡇⠀⠀⠀⢈⠀⢸⠠⡇⠀⠀⣌⠀⡜⠀⠀⠀⠀⢱⠀⢙⠀⠀⠀⠀⠸⣥⠿⣬⢯⣼⡇⢹⢸⠀⠘⡄⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⣇⣰⠓⣦⠚⣦⠚⣆⣨⡇⠀⠀⠀⠋⠀⠘⢰⠃⠀⠉⠁⠀⠉⠉⠉⠉⠉⠉⠀⠈⠉⠀⠀⠀⡘⠤⠿⠬⠧⠼⠧⠜⣸⠀⠀⠈⠳⠤⠀⠀⠤⠚⠁
⠀⢀⠣⡘⢒⡉⢒⣉⠒⣁⠜⡇⠀⠀⠀⡆⠀⡆⣹⠀⠀⠉⠉⠁⠉⠉⠉⠉⠈⠉⠁⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢁⣀⣀⣀⡀
⠀⠀⢩⡉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢠⠾⣥⣁⣇⡏⡯⠽⡆⠀⠀⣋⣙⣋⡄
⠀⠀⠀⠘⡦⡀⠀⠀⠀⠀⠀⢀⠀⡠⠬⠄⡀⡀⠀⠀⠒⣉⠭⠤⠍⡁⡒⢈⠉⠤⠭⢉⡂⠀⠀⢀⠞⣘⢢⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢡⠀⡌⡃
⠀⠀⠀⠀⠀⠹⢢⡀⠀⠀⠀⡀⢸⢠⠒⢦⢹⢀⠀⠀⠀⠐⠂⠀⠒⠂⡁⠐⠒⠀⠀⠒⠆⠀⠀⠈⡮⣊⢴⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⡇⡁
⠀⠀⠀⠀⡷⢄⠀⠩⢢⡀⠀⠠⠌⢆⠀⢠⠋⠄⠀⠀⠄⡉⠤⠤⠭⠁⠃⠐⠀⠐⠀⠂⠀⠀⠀⣸⣠⣹⣄⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣸⣀⣇⠃
⠀⠀⠀⠀⣇⣀⣑⡄⠀⠊⢦⡀⠀⢸⡻⢹⠀⠀⠀⠀⠀⠐⠒⠒⠒⠂⡃⠐⠒⠒⠒⠂⠆⠀⢰⢡⠃⠀⠰⡘⡄⠀⠘⢇⣀⣉⡀⢈⣁⡠⠇⠀⠀⠑⠶⠊
⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠬⠼⠦⠀⠉⠁⠀⠀⠀⠀⠻⠭⠭⠉⠩⠅⠿⠨⠍⠉⠭⠡⠟⠀⠗⠃⠀⠀⠀⠑⠧⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⣼⡄
⠀⠀⠀⢀⠤⠐⢶⣟⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡔⠒⠒⢌⠄⠀⠀⠀⡜⢸⠈⡄
⠀⠀⢠⠃⠀⠀⣿⣷⣿⣛⣧⠀⠀⠀⠛⠀⠉⠀⠛⠁⠐⠘⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⣀⢀⡀⣀⡀⠀⠤⢸⠠⠋⠱⢨⠠⠀⠀⡜⠀⢸⠀⠘⡄
⠀⠀⢸⠀⠀⠀⠉⠉⠉⢙⠇⠀⠀⠀⣠⢲⠆⠤⠤⠤⠄⠒⠐⡄⠀⠀⠀⡇⠥⠤⠤⠥⠤⠤⠡⠄⠀⠀⠐⡇⠀⠐⠂⢱⣀⣠⠁⠂⠀⣼⣀⣀⣸⢀⣀⣘⡄
⠀⠀⠀⠳⣄⠀⠀⢀⡠⠊⠀⠀⠀⠀⠙⢼⢂⠂⢒⡒⡂⣠⡀⠇⠀⠀⠀⠧⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠐⠃⠀⠀⠀⠸⢝⡻⠀⠀⠀⠙⠛⠓⠷⠖⠒⠛⠁
⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⢠⡶⢶⡄⠀⠀⠀⠀⠀⠀⢶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣇⠀⠀⠀⠀⠀⠀⢈⣡⣾⡃⠀⠀⠀⠀⠀⢰⣎⣹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⣛⣋⣀⣀⠀⠀⠀⠀⣈⡛⣛⣁⡀⠀⠀⠀⢀⡀⣉⢋⣁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⢻⡇⠀⣿⡟⠛⠛⠛⠛⢻⣿⠀⢸⡟⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⣀⡀⢸⡇⠀⣿⡇⢀⣀⣀⡀⢸⣿⠀⢸⡇⢀⣀⣀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡏⢹⡇⢸⡇⠀⣿⡇⢸⡏⢹⡇⢸⣿⠀⢸⡇⢸⡏⢹⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⢸⡇⢸⡇⠀⣿⡇⢸⡇⢸⡇⢸⣿⠀⢸⡇⢸⡇⢸⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⢸⡇⢸⡇⠀⣿⡇⢸⡇⢸⡇⢸⣿⠀⢸⡇⢸⡇⢸⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠻⠷⠾⠇⢸⡇⠀⣿⡇⠸⠷⠾⠇⢸⣿⠀⢸⡇⠸⠷⠾⠿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣠⣄⠀⢸⡇⠀⣿⡇⠀⣠⣄⠀⢸⣿⠀⢸⡇⠀⣀⣄⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢾⣏⣽⠇⢸⡇⠀⣿⡇⢸⣯⣽⡇⢸⣿⠀⢸⡇⠸⣯⣹⡷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⠀⣿⠀⠀⠉⠁⠀⢸⡇⠀⣿⡇⠀⠉⠉⠀⢸⣿⠀⢸⡇⠀⠈⠉⠀⠀⣿⠀⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠿⠷⠶⠶⠶⠶⠾⠇⠀⠻⠷⠶⠶⠶⠶⠾⠟⠀⠸⠷⠶⠶⠶⠶⠾⠿⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠒⠋⠀⠀⠀⠀⠀⠸⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢀⡖⠛⠓⠒⢦⡀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣀⣀⣀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⣴⡆⠀⠀⠀⣀⣴⡆⠀⠀⠀⣀⣴
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣀⣴⣾⣿⣿⡇⣠⣴⣾⣿⣿⠂⣠⣴⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣟⠛⣿⣿⣿⣿⣟⢛⣿⣿⣿⣿⡏⠉⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣇⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣧⣤⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢛⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⡟⣉⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿