Potato Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⢀⣤⠖⠛⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠓⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠛⠛⠛⠛⠛⠛⠳⠶⣦⣄
⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡇
⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣸⠃
⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⣤⣤⣤⣾⡄
⠀⢹⡄⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠊⠉⠀⠀⠈⠙⢟⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠁
⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢀⣀⣵⣄⣈⡉⠙⠒⠒⠒⠛⠛⠉
⠀⠀⠀⠈⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠀⠀⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣦⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⣰⠊⠉⠀⠀⠀⠀⠀⠈⠉⠓⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠠⣄⠀⢻⣄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣤⣄⣀⣀⣀⣀⣤⣿⡿⠳⣮⣳⣤⡙⣷⡄⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠻⢿⣿⣿⣿⣦⣀⠀⠠⣻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠳⠾⠿⢿⣝⡒⠄⠀⠀⠀⠀⠀⠀⠀⣠⠞⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢶⣤⣤⣤⣤⣭⣭⡴⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠋⠑⠲⣄⠀⡤⠖⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠏⠀⠀⠀⠀⠀⠈⢻⠀⢀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠉⣿⠀⠀⠰⠆⠀⠀⠀⢸⠀⠈⠀⠀⢠⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⢀⡾⠓⠶⠴⠞⠁⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣰⠂⠀⠀⠀⠀⠙⢷⣤⣤⣤⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠸⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠟⠀⠀⢹⣧
⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⡿⠋⠉⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢀⡤⣀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠸⣀⡸⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠙⢿⡿⠟⠻⢿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠈⠀⠀⠀⠀⠈⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⡀⠀⠀⠀⢀⣀⣈⣿⡗⠀⠀⠀⠀⠀⠰⠒⢆⠀⠀⠀⢻⣇
⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠣⢀⠜⠀⠀⠀⠈⢿⡆
⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⢀⣿⠁⠀⠀⠀⠀⡠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡄⠀⠀⠀⠀⠹⣷
⢸⣿⠀⠀⠀⠀⡎⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠋⠀⠀⠀⠀⠀⢻
⠸⣿⠀⠀⠀⠀⡆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⠀⠀⠀⠀⠙⠦⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠁⢆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣙⠄⠀⠀⠀⠀⣼⡟
⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⢣⡀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠈⠻⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣶⠿⠟⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠈⣳⣄⣀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠒⠲⢤⣼⠃⠀⣠⠖⠋⠉⠀⢀⡟⠉⠀⠙⠲⡔⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠒⣚⣫⣽⣁⠀⠀⢀⣀⣤⣤⣿⣤⣄⣿⡀⠀⢀⣀⣼⣄⡀⠀⠀⠀⣹⣤⣬⣧⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢠⠟⠁⠀⠀⠈⢳⡞⠉⠀⠀⠀⠀⠀⠀⠈⣿⢿⠋⠀⢀⣀⣙⣦⣠⠋⠁⠀⠀⠀⠉⠓⢦⡀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠸⣄⠀⠀⠀⠀⠀⢹⡄⠀⣠⠶⠚⠉⠙⢳⡏⠘⡤⠚⠉⠀⠈⠙⣿⠀⢀⣀⣀⡀⠀⠀⠀⢹⡆⠀⠸⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠙⠲⢤⣀⣀⠀⣏⣹⠃⠀⠀⠀⠀⠀⢳⡜⠁⠀⠀⠀⣠⠔⠋⠉⠁⠀⠀⠉⣦⡤⠖⠋⠁⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⠦⣤⣄⡀⠀⠀⠈⠉⠉⠛⠛⠒⠦⢤⣀⣀⣀⣸⣇⣀⣀⣀⣞⣡⡤⠤⠤⠖⠒⠊⠉⠁⠀⠀⣀⡤⣀⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢝⠲⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⣀⣠⡤⠴⠖⠛⠛⠋⠉⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠉⠓⠊⠉⠙⠓⠲⢤⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⡤⠤⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠋⠉⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠔⠃⠊⢣
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⢧⡄
⠀⠀⠀⠀⠀⢠⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⣄⣀⣀⣀⣀⣽
⠀⠀⠀⠀⠀⠀⠓⠦⣀⣀⣀⣀⠀⠠⣤⣤⣀⣀⣀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣴⣠⡤⠶⠾⠴⠒⠋⠉⡤⠞⠛⠛⠛⠛⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠉⠓⠲⠶⠶⠶⠶⠖⠒⠋⠉⠁⠀⣀⡤⠖⠚⠋⠉⠛⠓⠲⢤⣀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠒⠚⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔⠒⠉⠉⠈⠉⠉⠉⠉⠉⠉⠛⠒⠶⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠊⡡⠀⠀⠀⠂⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠾⠉⠻⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣅⢌⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠲⠄⠀⠐⠈⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣥⣙⡋⢘⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝⠿⢏⠵⡹⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣤⡰⣌⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣾⣿⠓⢣⣶⠹⢷⣀⠀⠐⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠐⢀⢀⡚⡽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⢛⣿⣻⡏⡛⢹⣆⣤⡈⠀⠀⠂⠀⠀⢠⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡿⣀⢹⡟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡻⢿⣿⣯⢿⣦⣠⠑⠀⣷⡻⣷⠃⡃⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⢯⣾⡷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡿⣿⣿⣻⣿⣿⣶⣼⢧⡮⣿⢻⣉⣥⠀⠀⠂⠀⠀⠐⣠⡀⠀⠐⠂⠀⠲⣰⣔⣾⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢽⣿⣿⣿⣯⣛⣪⣴⣤⣎⣉⡱⢎⣴⣴⡆⠐⣀⠉⠀⢠⢤⣮⣵⣾⣶⣿⣿⣯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣟⣿⣿⣿⣿⣿⣿⣟⣿⣎⢛⠟⢀⣠⣾⣶⣹⣾⢿⣿⣾⣿⣿⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡟⣿⣿⣟⣶⣿⣿⣿⣿⣿⣿⣯⣵⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣽⡻⣿⣻⣿⣿⡿⣿⣿⣿⣿⢿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠷⠾⠿⠿⠿⠿⠛⠋⠁






⣀⣀⣀⣄⣀⣤⣀⣄⣤⣤⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀
⣿⣻⣻⣛⣟⣫⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣭⣽⣽⣿⣯⣭⣭⣿⣭⣽⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⠿⠿⠿⠿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡾⢿⡿⠛⠁⠀⠀⠀⠀⠒⠈⣉⣙⣻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠿⠟⠛⠋⢉⣥⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠤⠴⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣦
⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢿⣧
⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠠⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡀⢑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⢸⡧⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠀⠀⠀⠀⠀⠸⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠘⠣⠤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠄⠐⠀⠁⠐⢿⡇
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⢺⣿⡀
⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⢤⣿⣇
⠀⣯⣈⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⣿⡇
⣿⠀⡠⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢀⣹⣷
⣿⠌⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⣿
⢹⡆⠀⡰⠁⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⠀⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠸⣷⣰⠁⠀⠀⠀⠀⠀⠀⠺⣄⣀⣀⣀⣀⠤⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⡅⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⢠⠟⣿
⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣿
⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⣼⣿
⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⢀⣴⠟⣹⡿
⠀⠀⠀⠈⢻⣿⡳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⢠⣿⠇
⠀⠀⠀⠀⠀⠙⢿⣦⡉⠛⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣤⣄⡐⠒⠂⠤⠤⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣶⣶⣤⣬⣷⣄⣀⣠⣋⣈⣹⣦⣤⣤⣤⣶⣾⠿⠿⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⢛⠋⠭⠭⠝⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠉⠀⠈⠒⠒⠀⠀⠀⠀⠀⢀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢶⣛⠯⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠑⠀⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠁⠀⠈⠉⠙⠳⠶⣼⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠙⠓⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⣀⣀⡀⠀⠀⠀⣀⠀⠀⢀⣈⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠶⠾⠭⠴⠶⠖⠚⠛⠉⠉⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠋⠉⠉⠉



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣯⣤⣍⣥⣮⣭⣿⣥⣿⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠛⠋⠉⠳⣦⡀
⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⠘⠀⠀⠀⠊⢳⡀
⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⠀⠠⢄⡀
⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣴⣴⣦⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠷⣾⣛⢉⡵⠾⠳⢶⠦⣄
⠀⠀⠀⠙⠦⢤⣤⣤⠴⠿⠾⠛⠛⢁⣠⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⢀⠼⠁⠀⠀⠀⣾⠋⠁⠀⠀⠀⢸⣿⣿⡷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⠘⠉⠀⠈⢉⣱⣦⠀⠀⣼⠰⣠⠀⠀⣰⢳⣶⣀⠀⠀⣀⢈⣧⠟⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠏⢽⡇⠀⢇⠀⢹⡷⢴⣿⠞⠉⠉⠉⠻⣿⣾⢏⡴⢠⠏
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⣀⠳⠦⣤⡄⠀⠀⢀⡀⠀⣠⡼⠃⠀⣾⡀⢀⢳⣤⣿⣤⣤⡤⠴⠶⠟⡣⠞⢡⡇
⠀⠀⠀⠀⢀⣴⣿⠀⠈⠁⢛⠻⢷⡄⠈⠛⠒⠲⠚⠓⠚⠋⠁⠀⢸⠀⠱⡈⠓⠦⠄⠀⠀⠀⠀⠈⠀⣀⡴⣿
⠀⠀⠀⡼⠋⠁⠉⠁⠀⠀⠈⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠈⠓⠶⠶⢶⣤⣤⣶⣚⣩⣅⣤⡿
⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠠⣤⣤⣤⠴⠞⠛⠉⠉⠙⢿⠶⣦
⠀⠀⢹⣄⠀⠀⢀⣰⣇⣀⣠⡶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣄⡀⠀⠈⠀⠀⠀⠀⠀⢀⡤⠖⠒⠉
⠀⠀⠀⠉⠉⠉⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠛⠛⠛⠛⠛⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠤⠤⠤⠤⠤⢤⣤⣼⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⢀⣤⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣀⣀⣀⣠⣴⣟⣁
⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠋⠉⠉⠉⠉⠉⠉⠙⠛⢷⣄
⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⠀⠀⠀⢸⠁
⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠼⠿⠓⠲⣶⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠸⣦⡀⠀⠘⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠉⠀⠀⠀⠀⠀⠀⠑⢌⠙⠲⠦⣄⣀⣀⣀⣠⣤⠶⠛⠁
⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⡴⠖⠛⠓⠰⠤⢤⣀
⠀⠀⠀⠀⠈⠙⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠛⠋⠒⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢀⠀⠈⢧⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⣸⣿⢦⣙⢦⡈⢳⣄⡀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠶⠾⠟⠛⠉⠀⠙⢷⣿⣷⣿⣧⣄⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠶⢬⣽⣿⣗⠦⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣄⣀⣀⣀⣀⣖⣚⣥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠋⠉

⠀⠀⠀⠀⣀⡤⠶⠖⠛⠛⠛⠻⠶⢦⣤⡀
⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠈⠉⠉⢹⡷⠶⠶⣄⣀
⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠙⢳⡄
⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⣿⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗
⣸⠇⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⣿⠀⠀⠈⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⡟
⣿⠀⢰⡚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆⠀⣀⣠⣤⣇⣤⣤⣤⣤⣀⣀
⠸⣧⣌⣿⣢⣀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⡀⣄⣀⣄⣠⣿⣱⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⢷⣯⡿⣿⣿⣿⣷⣶⣾⣵⢠⣤⣖⣴⣤⣿⠗⣤⣀⣤⠀⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠉⠛⢧⣿⣿⣿⢯⣿⣿⣾⣿⢿⣿⣿⣟⣾⣿⣿⣿⣷⣿⠿⠾⠿⠿⠿⠻⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣭⣭⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⠛⠋⠉⠉⠉⠙⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢀⠞⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠚⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⡜⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⢰⠁⠀⠀⠀⠀⠈⠉⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣠⠞⠁
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣠⠴⠋⠁
⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣣⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠦⠤⢄⣀⣀⡤⠤⠖⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠐⠀⠀⠀⠉⠀⠈⠙⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⡌⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠎⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⡀⠀⠌⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠄⠈⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠱⣀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠛⠛⠛⠋⠉⠓⠲⢤⡀⠙⢧⡀⠀⡀⠀⠀⡉⠑⠴⠆⡄⢀⠂⢠⠒⠂⡐⠴⠥⡠⠀⠀⠤⠀⢸⠇
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠉⠳⣾⣁⠄⠀⢦⡀⠀⠈⣈⠠⡄⢰⣈⡢⣄⣤⣀⠠⢀⠉⠀⡼
⠀⠀⠀⠀⠀⢰⠏⠀⠀⢠⠃⠀⠀⠀⠐⠀⣀⡀⠒⢀⣀⢹⠀⠀⠀⠀⠉⠳⢽⣒⠐⠂⠔⣨⡉⠒⠒⠄⢀⡒⠠⣳⡴⠄⣠⠞⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⡞⠀⠀⢀⡴⠚⠉⠀⠀⠀⠠⠀⠉⠳⣄⠀⠀⠀⠀⠀⠈⠛⠾⢿⣿⣶⣾⣮⡷⠖⢒⣉⣂⡤⠞⠁
⠀⠀⠀⠀⠀⣿⠀⠀⢄⡇⠀⠰⠋⠀⠀⠀⠀⠀⣀⡥⠴⠒⠀⠙⠛⠲⠦⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢻⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⢔⠟⠉⠀⠀⠀⠀⠈⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠇⢠⠀⠀⠀⢠⢂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠹⣌⠍⡸⡄⠰⠀⠀⡘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠳⣽⡿⡍⠀⠀⢠⢧⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣆⠀⠸⡌⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡂⢱⡔⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⣆⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⢂⠈⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⣑⠢⣀⠀⢀⠀⣀⡠⢀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠶⠶⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠀⠒⠒⠒⠒⠶⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣞⠍⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠴⠶⠒⠒⠚⠛⠲⠶⠶⠶⠤⠤⠾⠿⢵⣀⣀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⢀⡤⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠸⡇
⠀⠀⡰⠋⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⡇
⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⡀⠀⠀⢀⡠⢀⡀⣊⡾⠃
⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⣿⣭⠭⢐⣂⣵⠞⠋
⠀⠀⠘⢦⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⣰⠟⠛⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠈⠳⣬⡓⠦⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⢀⠠⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⠭⣭⣭⣭⣴⣯⣭⣼⣟⣒⣒⣒⣒⣒⣠⣤⣬⠴⠿⠶⠛⠉





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

⠀⠀⠀⠀⢀⡴⠉⠀⠣⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠸⡇⣶⡀⠀⠀⠀⠀⠀⠾⠯⣟⣀
⠀⠀⠀⠀⠀⠓⣿⣅⠀⠀⢠⡶⠒⢲⠶⠦⠌⠹⠿⣶⡶⠆
⠀⠀⠀⠀⠀⠀⢹⣿⣆⢀⠛⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠈⢸⣿⣿⠁⠀⢸⠛⠛⠷⠶⢦⣤⣄⣀⠀⠚⠃⠈⣆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠸⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⣄⢹⣄⣠⣤⣄⣀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠸⡆⠹⣿⣿⡈⠛⠉⠉⣿⣻⣿⡷⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠹⣿⣷⠀⠀⠀⠻⣿⣿⣿⣶⡆
⠀⠀⠀⠀⠀⠀⢀⢀⠂⠙⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣷⣀⢹⣿⣤⣤⣤⠘⣷⣿⣿⣿⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠸⣷⣇⢻⣷⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⣿⣿⣿⣛⣛⣄⠛⣛⣛⣿⠧⣤⡤⣤⣬⡤
⠀⠀⠀⠀⠀⠀⠄⠠⡟⠛⣊⡷⠈⠉⠈⠀⠀⠀⠀⠃⠀⠠⠜⢻⣯⣍⣉⣿⣿⣿⣶⣿⡷⠶⠶⢿⣀⡁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋⠉⠇⠀⠀⠳⠀⠰⣿⣿⣿⣿⡿⣿⣷⣿⣿⣻⠿⠏⠉⠉⠿⢴⣿⠺⣷⡙⢻⡆
⠀⠀⠀⠀⠀⠜⠀⠀⠁⠀⠈⠁⠀⠁⠀⠐⡄⠩⣿⣾⡿⢿⢛⣿⣍⣿⣿⣤⣄⣠⣤⣁⣼⣯⣤⣿⣱⣾⣷⣀⣶⣄
⢀⣤⣴⣄⣀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠐⠀⠀⠚⣻⣿⣭⣿⣵⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿
⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⠿⠿⠿⠷⠤⠤⠴⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⢀⠸⡀⠀⠸⠘⡀⣀⢤⠀⠀⠀⠀⠀⠀⠀⢠⣝⡾⢷⣫⣤⣀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠄⠀⣀⡀⠀⠀⠈⠁⠀⠀⣴⣑⣁⠈⠀⠈⢣⣄⠙⡄⠀⢘⡿⣷⡾⢿⡛⠯⠝⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣰⡀⠀⢠⠀⡈⡄⢰⢀⣀⣄⢺⣷⢷⡆⠁⠠⠐⡀⡂⡏⠀⠉⠉⠉⠉⠋⣿⡟⢻⠀⠀⠈⢹⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⠰⢃⢀⡠⠐⠉⠀⠀⠀⢀⠤⠿⠚⠂⠑⡃⣚⣿⡠⢄⡅⠀⠀⠀⠀⠀⠀⣿⡧⢸⠀⠀⠀⠘⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⡶⠤⠀⠀⠀⠠⠄⠀⠀⣀⣀⢀⡤⠖⠉⠀⠀⠉⠑⠫⢧⣤⡀⠀⠀⠠⣻⡅⢸⠀⠀⠀⠚⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⢐⠀⠀⡎⠠⠂⢀⠀⠀⠀⠀⠀⠀⢀⣰⣷⣶⠀⠀⠀⠀⠀⠀⠀⣤⣽⡷⡤⠀⠀⢹⡇⢸⠀⠨⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠂⡋⠠⠁⠀⠐⠀⠀⠀⠀⠐⠂⡰⠁⠛⠻⠀⣾⢒⠶⡲⢦⠀⠿⠿⠙⣽⡀⠀⣸⣇⢸⠀⠀⢠⠒⠟⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⢠⣆⣀⣈⠁⠊⣁⣀⣀⣀⡀⣰⠃⠀⠀⠈⠀⢸⣯⣶⣡⣾⠀⠐⠖⠂⠙⣿⣿⣿⣿⣿⣷⣃⣾⣷⠾⣷⣷⢦⡀
⠀⠀⠀⠀⠐⠀⠀⢀⡠⢒⠘⠊⠉⠉⠭⢙⣳⣧⣍⠀⠀⠈⠉⠻⠀⢰⠖⠒⢿⢳⣶⣴⠖⣾⡗⠒⡂⠑⠒⡛⠋⣛⡛⣛⠛⠓⢈⣕⡺⣿⣿⣞⢧
⠀⠀⠄⠀⠀⠀⠀⠀⣛⠙⠙⠚⠓⠺⢶⠋⠁⠀⢹⢀⠀⠀⠀⠀⠀⢸⠁⠆⢸⣿⢹⡿⣛⢽⠉⡞⢹⠀⠀⡏⠉⣿⡏⣹⠉⠉⠉⠉⠁⠻⢯⡁⠣⡀
⠀⠀⠀⠴⠤⣤⣤⣴⣟⣰⠀⡀⠀⠀⠰⠀⠉⠁⢸⣸⣤⣄⠴⠴⠀⣸⣦⣀⠾⠿⠾⠧⢥⣼⣶⣀⣲⣀⣺⣷⣴⣿⡷⠿⠿⠶⠶⠶⠗⠒⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠀⠁⠉⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⡶⢖⢲⠛⠛⠋⢉⡍⢍⡝⡻⠛⡳⠒⣶⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⡚⠫⣍⠳⢌⠢⡙⢌⠀⠇⠀⡇⠸⠀⠜⠔⣡⡞⠡⢊⡡⠖⣋⠤⠞⢉⠹⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠲⢌⡑⡂⠍⡒⠅⠢⣈⣤⣬⣄⣡⠞⠛⠲⠞⠉⠉⠳⠴⠶⣭⣴⠶⡶⣄⠒⣩⡷⣤⠉⡓⣦⣀
⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠳⡄⠁⠀⣴⣾⣿⣿⣿⣿⣿⡴⢶⢶⠶⣶⠖⣶⡶⠒⢿⠾⠶⢷⣿⣰⡿⠧⣿⡟⠉⠁⠈⠳⢦⡀
⠀⠀⠀⣠⠞⠹⢧⡀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠶⣾⣿⣿⣿⣿⣿⣿⡿⠶⠶⡶⣶⢶⡾⠁⠀⠀⠀⢱⡄⢸⣿⢹⡇⠈⣿⡇⠀⠀⠀⢀⣠⠿⢦⡀
⠀⠀⡼⠃⠀⠀⠀⠁⠀⠀⢄⣠⣤⠞⠁⠀⣀⣷⣿⣿⣿⣿⣿⣿⣿⣈⣓⣙⣙⣈⣻⠇⠀⠀⠀⠀⠀⣧⢸⣿⢸⡇⢰⣿⡇⠀⠀⠀⠈⢡⠀⢈⣷⡄
⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⣿⣿⠿⢿⣿⣟⣏⣙⣁⣛⣘⣃⣸⣤⣤⣤⣀⣀⣠⣯⣸⣿⣼⣅⣈⣿⣇⣀⣀⣀⣀⠀⠀⠈⠀⠻⡄
⣸⣷⣶⣶⣶⣦⣤⣤⣀⣀⣀⣠⣤⠤⠤⠶⠶⣟⣛⣿⡿⠿⡟⢛⣛⣯⠿⠟⠛⣋⣭⡤⠶⢚⣫⡭⠖⠋⠁⣠⡴⠛⠉⢀⣩⠟⠉⡽⠉⣟⠛⣟⠲⣶⢿
⣿⠢⠬⢭⣅⣉⢉⣙⠛⠻⠻⡿⣿⣶⣦⡶⠟⢉⣁⣀⣤⠾⠛⠉⠁⣀⣤⠶⠛⠋⢁⣤⠞⠋⠁⠄⣠⡴⢛⠁⠀⢀⡴⠋⠀⠀⡾⢁⢰⣗⢀⣹⣄⣈⣻
⣏⠑⠮⣄⡠⡉⠙⠓⢦⣴⣠⣠⣴⢛⡩⢀⣤⣞⡋⠉⠉⣀⣤⣶⣻⣥⣤⣤⣤⣞⠋⠀⠀⠀⣠⡾⠋⠀⢁⡀⢠⡟⣁⣤⠶⠟⣩⠉⠉⠉⢉⣉⣁⡤⢾
⢻⢳⢤⣤⣉⣳⣷⣿⣛⣫⣯⣥⣬⣩⣿⣿⣟⣉⣹⣻⣿⣽⣽⣭⣭⡁⠹⠛⠟⢻⢧⣀⣠⡾⠋⠀⠀⠀⡚⣵⠿⠭⠕⠒⠉⠀⠀⠀⠀⠠⢝⣛⣓⣟⣿
⠈⢿⣭⣼⣻⣿⣿⡿⣿⠷⣿⣷⣾⣷⣿⣿⣿⣿⡻⣿⣿⣿⠿⣿⣯⣟⡿⣷⢶⣤⣈⣻⣿⡟⣽⢳⣠⢄⣼⣟⣿⣟⣷⣖⣶⣦⣀⣤⣤⣀⣀⡈⠉⣨⠏
⠀⠈⢷⠦⣠⡠⠍⢿⣟⣷⣾⣅⣇⢉⢻⣷⡿⡻⣿⢤⢿⡿⣶⣿⣻⣿⣿⣿⣯⣽⡟⣛⣮⣏⡙⠾⢯⣿⣿⣽⢧⡿⣿⢻⣭⡟⣽⣛⢷⢶⡭⠽⣽⠏
⠀⠀⠀⠳⣄⠹⣯⠭⠇⡹⢤⢿⠿⣿⣦⣻⣶⣻⣫⢿⣷⣟⡻⣯⣿⣟⣯⣟⠿⡽⣷⢿⣆⡿⣓⣾⢧⣸⡟⣛⣧⠹⡷⣟⣲⡧⠯⣝⡟⣼⣭⡿⠁
⠀⠀⠀⠀⠈⠳⢿⡅⣢⢍⡛⣿⡁⣹⣿⡇⣿⣿⣟⣏⢿⣾⣷⣛⣟⣳⣝⣯⣾⡟⣫⣻⣞⢿⣭⢿⣦⣯⣿⣧⣞⣷⢛⡟⢾⣓⢦⣈⣳⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠳⣴⣐⢢⣟⡙⣿⣿⣿⣿⣼⣿⣭⡑⢿⣚⣿⣴⣿⡷⢿⢯⣿⣫⣿⣿⡛⣾⡷⣾⣩⣿⣝⣿⣟⣿⣟⣽⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣥⣼⣿⣿⣿⡟⡷⢮⣉⠟⣾⡿⡻⣼⣗⡿⢳⣿⣿⣮⣽⣛⣷⣿⣿⣯⣴⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠿⢶⣟⣚⣇⣿⣯⣍⣽⣷⣿⣿⣿⣿⣭⣭⠾⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⣤⣷
⠀⠀⠀⠀⠀⢀⣠⡾⠛⠉⠉⠀⢸⡿⠀⠀⠀⠀⢠⡤⠤⡤⠄⠀⠀⠀⠀⠀⠀⢰⡷⣄⡀
⣤⣶⣶⣴⣶⣤⡀⡁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⣷⢶⡇⣰⣄⠀⠀⠀⠀⠀⣿⡦⣿⢃⣀⣤⡀⣤⣤⣀
⠉⢩⣉⣙⣛⣛⣏⣡⡀⠀⠀⠀⢀⢷⡀⠀⠀⠀⢹⣿⣛⣟⣛⣻⣴⣶⠀⠀⠀⠀⠀⠻⠿⣿⣻⡷⠿⢿⣿⠦⣴
⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⢰⡆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠁
⠀⢠⣶⣶⣾⣷⣦⣤⣤⣄⣄⠀⠀⠀⠀⠀⠀⢀⣄⣠⣄⡀⣄⠀⡇⠀⠀⠀⠀⠀⣠⣴⢶⣶⣦⣿⣿⣿⡶⣤⣄
⠀⠚⢿⣟⣿⢿⣿⣿⣿⣿⡷⣤⡆⠀⠀⠀⠀⠸⣿⣿⣿⣿⣾⣶⣷⠀⠀⠀⠀⠀⣿⡿⡆⢹⣿⢻⣿⣿⣾⢿⡇
⠀⠀⠀⠉⠁⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠉

⣠⣤⣤⡤⣤⡤⣤⣤⣤⡾⡫⢭⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⢠
⠀⠉⠙⣻⣿⣿⣿⣿⣟⣴⣿⣿⣷⣦⠀⠀⠀⠀⢰⠒⠂⠀⣀⡤⠖⠛⠁⠀⠀⣤⣶⣾⣶⣗⣶⣦⡄
⠀⠀⣾⣿⣿⣿⢹⣿⠛⣿⣟⣿⣿⠉⠀⠀⠀⠀⣾⣷⣌⡾⣇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⣋⣻⣿⣷⣆⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⣶⣆
⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠘⣇⠀⠀⠀⠀⠀⠀⣸⠇⠹⣦
⠀⠀⠀⠀⠀⠀⣸⠋⢻⣄⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠹⣄⠀⠀⠀⠀⣰⠏⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠙⢷⣄⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠹⣆⠀⣠⠞⠁⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠳⣄⢸⠃⠀⠀⣴⡀⠀⠀⠀⢹⡾⠃⠀⠀⢠⣦⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⡇⠘⢦⣀⠀⠈⠀⣀⣠⠔⠃⠈⣆⠀⠀⠀⠀⠈⢷⡀
⣤⣄⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠙⠓⠛⠋⠉⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠘⣧
⠈⢿⡛⠶⢦⣄⡈⣇⠀⠀⠀⢀⡀⠀⠀⠀⡼⠁⠀⠀⠀⠀⣴⠒⠒⡶⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇
⠀⠀⢳⡀⠀⠈⠙⠻⠆⠀⠀⠀⡏⠳⢄⣸⠁⠀⠀⠀⠀⢰⠃⠀⡼⠁⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠁⢀⡀⠀⠀⠀⢸⠀⢰⠃⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢿⠲⣄⡀⣸⠀⡟⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⡄⠀⠉⣙⣻⣧⣀⡀⠀⠀⠀⠰⠿⠞⣷⠀⠀⠀⠀⢰⠇
⠀⠀⢰⡇⠀⠀⠀⠸⠓⠦⣤⣸⠀⠀⠀⠀⣠⠖⠛⠶⢤⣄⣠⠟⠉⠙⠳⣄⠀⠀⠀⠀⡏⠀⠀⠀⢠⡏
⠀⠀⢸⡇⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡜⠁⢀⣤⣄⠀⣠⠶⠖⠶⣄⠀⠈⢧⠀⠀⠀⡇⠀⠀⠰⢿⣤⠴⠶⠛
⠀⠀⠈⡇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢰⠁⠀⡏⠠⠎⢻⠁⠀⡶⠀⠈⡇⠀⠘⡆⠀⣾⡁⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⢻⡀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠸⡆⠀⠙⠦⠤⠾⣧⡀⠀⢀⣰⠇⠀⠀⣷⠀⠈⣩⠗⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠁⠢⣀⠀⠹⣄⠀⠀⠀⣀⡈⠉⠛⠉⣀⡄⠀⠀⣽⠖⠋⠁⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠙⠒⠾⠷⢦⡀⠈⠉⣟⠛⡟⠉⠀⠀⠀⡇⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⡀⠀⠀⠀⠀⠀⠹⣄⠀⢸⡖⣇⠀⠀⠀⣼⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠲⠶⠶⠖⢻⡄⠈⣧⢹⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠘⠿⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠠⠤⠆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠛⠲⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣶⠓⠟⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⣄⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠉⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⢦⣤⣤⣤⣤⡤⠶⠞⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⣤⣤⠄⣙⡋⠞⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠰⠀⠛⠿⠁⠈⠁⠀⠈⠉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⢀⠠⠤⠤⠤⠤⣄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠐⠀⠀⠠⠀⣆⠘⣃⠀⠀⠒⡆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⣿⣾⠇⠀⣀⡀⢐⠀⠀⠀⠀⠁⠀⠀⠀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠛⠛⢃⡀⢴⣖⣤⣬⣤⣤⣤⣬⣤⣶⡂⠛
⠀⠀⠀⠀⣶⣶⣦⣆⣀⠀⠀⠀⠀⠀⠁⠀⢀⣿⣇⡀⠀⡀⣠⡼⢽⣿⣿⣿⠉⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣛⣛⣛⣿⣿⠀⠀⠀⣶⠀⣀⣴⣿⣿⣧⡟⡈⢁⣿⣟⣫⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⣿⣿⣿⠿⢿⡀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠀⠉⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⡤
⠀⠀⠀⠈⠋⠙⠉⠃⠉⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⣐⣶⣼⣿⠛⠻⣾⣿⣏⣉⣉⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⢹⣯⣭⣿⣿⣾⣿⣿⣿⣯⣭⡭⠭⠭⠭⢿⣉⣉⡙⣿⣯⣭⣿⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⢤⡉⢉⣀⣀⢀⠀⠁⠛⢻⡇⠐⠒⠒⠒⢚⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⣤⣤⣤⡤⠈⠛⠛⠛⠛⠻⠿⠿⠟⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⠿⢿⣿⠋⠀⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⠿⠿⠶⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣛⣛⡛⠻⠇⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣯⠍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠿⠿⣿⣶⡄⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣋⣥⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⣀⣀⣠⣼⣿⡿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣴⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠋⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋⢹⡇
⠀⠀⠀⠀⠀⠀⢸⣿⢿⣟⠻⣿⣿⣿⠿⠷⣶⣶⣤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡶⠾⠛⠋⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣦⡛⢷⣤⣍⠁⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣍⠻⢶⣬⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣿⡙⢷⣤⣈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠿⣦⣌⡙⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣷⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣮⣝⠛⠿⢶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡾⠟⢻⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣧⡈⠛⠷⣶⡄⠈⠉⠛⠛⠿⠷⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⠶⠶⠶⠿⠿⠿⠿⠟⠛⠛⠋⠉⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣏⠛⠷⢶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠷⣦⣀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣦⣄⡀⠀⠀⠀⢀⣤⣠⣀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣈⠛⠿⠇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣌⠛⠻⠷⠆⠀⣿⣿⣿⣿⠋⠉⢹⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣟⠛⢿⣶⣶⡆⢸⣿⣿⣿⡏⠀⠀⢸⣿⣿⣿⠄⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡿⢷⣦⣤⣀⠀⢸⣿⣿⣿⡀⠀⢀⣾⣿⣿⣿⠀⠀⢀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠉⠙⠀⠸⣿⣿⣿⣷⣶⣿⣿⣿⣿⡟⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇⠀⣀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠻⠿⡶⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣭⣿⣶⣤⠀⠀⠀⠀⠉⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠿⣿⡿⠃⠀⠀⠈⢿⣿⣿⣿⡿⠿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡉⠙⠿⣿⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠷⢶⣦⠈⠉⠛⠛⠿⠷⣶⣶⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⡶⠟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢷⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠻⠿⠿⠷⣶⡶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠟⠛⠛⠉⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣬⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣉⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⠿⣿⣿⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣴⣶⣶⣾⡿⠿⠟⠛⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦
⠀⣤⠖⠒⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡞
⠘⢧⠾⠻⢦⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢷
⠀⠀⠀⠀⠀⠹⣆⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠈⢧
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠙⣦⣄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⠉⠓⠒⠒⠒⠋⠉⢳⡀
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠤⠀⢀⡴⠃⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⡷⣄⣀⣀⣠⣤⠶⠭⣀⡠⠒⠀⠛⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢋⣉⢳⠀⠀⡸⠓⠒⣲⡞⢑⡒⠐⢆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡘⠀⣤⠀⠈⡇⠰⠄⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣰⠋⠈⠉⠀⠀⡇⠀⠀⠉⣇⡈⠀⢀⠞⠀⡀⠸⡇⠠
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠱⡀⠀⠀⣠⠗⡤⡀⠤⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⡏⠀⠀⠀⠀⠀⠘⢲⣴⣚⣉⣉⣉⠥⠔⠊⡇⢠⡇⠄
⠀⠀⠀⠄⠀⠀⠀⠻⡄⠐⢷⠬⢥⣉⣁⣀⣀⡀⡤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢸⡇⠀⠀⠀⢀⠄⠀⠸⣀⢈⠀⠀⢀⣀⠠⢾⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢸⣄⣀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⢹⠉⠉⢉⠀⠀⠐⠉⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡼⠳⢤⣀⡀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢿⡟⠉⠉⠉⢻⠈⠉⢻⠀⠀⢀⡤⠒⠒⠺⣅⠀⣀⣈⣉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⠐⠀⠀⠸⠋⠀⢀⠈⠱⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡆⠀⠀⠀⠀⠀⢸⢀⠀⠘⢄⡀⠀⣀⣧⠀⠀⠀⠀⢠⠃⢠⠏⠀⠑⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠣⣀⠀⠀⠀⡠⠎⠈⠀⠀⠀⠈⣹⡁⠀⠑⠦⠤⠒⣁⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠙⠀⠀⠀⠀⠀⢉⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣈⠁⠉⠉⢉⣀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⢿⡿⠿⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⢠⠋⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡄⠀⠘⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠸⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠚⠁

⠀⠀⠀⠀⠀⠀⠒⢠
⠀⠀⠀⠘⠢⠐⣄⠀⠀⠀⠴⠢⡄⠀⢀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⣀⡀⢠⣤⠀⠀⣀⠀⠀⣀⣤⣄
⢰⣤⣤⠴⠶⠤⠾⠶⠾⠀⠀⠀⠒⠂⠛⠓⠰⠿⠧⠴⠷⠾⠦⠤⠤⠴⠆⠀⣤⠤⢤⡄⠁⠀⠀⢰⣋⣁⡀⠉⡿⠔⠢⡄⠉⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠋⠙⠁⠛⠋⠁⠠⠾⠿⠿⠿⠶⠤⠤⠤⠧⠄⠘⠛
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠉⠀⠀⠠⣴⣞⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⡀
⠄⡀⠀⡀⢠⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠉⡉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠁⠁⠁⠈⠉⠉⠁⠉
⡆⣄⡷⠀⡼⣄⠀⢀⡀⠀⣸⠡⢛⣀⣧⣻⣽⣇⣀⣀⣀⣀⣀⠀⣀⣀⣴⡷⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡆⠏⡟⢡⣟⣾⣾⢀⣇⣆⣟⠰⠮⡉⠁⣀⠀⠀⠀⠂⠀⠀⠀⢠⡇⠀⠘⣻⢿⣯⡆⠆⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠈⠘⠀⡄
⠟⢰⣰⡧⣩⡎⠙⠞⠒⠈⠀⢀⡠⠔⠈⠀⠀⠀⢀⠐⢀⡠⣤⡚⠀⠀⠈⠻⢿⣷⢣⢸⡌⠆⢠⠀⢹⠀⡅⢀⠀⢀⠀⠰⢀
⣟⣸⡿⠛⠊⠁⠀⢀⣠⡴⠚⢉⠀⠄⠀⠀⠀⠄⠁⢀⠞⡤⠺⠁⠀⠀⠀⢀⣤⣶⢟⡋⢿⢀⢈⡄⡄⣀⢣⡈⠀⣸⠃⠀⣾⠈⡀
⠁⠀⠀⠀⢀⡤⠴⠋⢁⡀⠢⠀⠀⡠⣂⠄⠀⠀⡰⠋⠘⡽⠀⠈⢠⠀⠀⠀⢈⡿⣸⡅⣻⣈⡌⢰⡟⢋⠀⡇⠁⠇⠀⠂⢸⢸⣷
⢀⣠⠴⠚⠁⠀⢀⡸⠁⠀⠀⣐⣄⠣⠀⠀⠀⡜⢡⢠⣼⠃⠀⠀⣾⠀⠀⠀⢈⣸⣛⣛⣋⢿⣃⡘⣧⠄⠀⣷⢠⠀⡼⠀⡖⢨⡾
⠉⠀⠀⣠⣶⠙⠋⠀⢀⢠⣳⠹⠌⠀⠀⢠⠌⠄⣠⡻⠇⠀⠀⠐⠇⠀⠀⠀⠩⠑⠛⠋⢉⢴⡇⣧⢸⣧⠀⠁⡿⢡⣻⠀⡿⡘⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠤⠖⠒⠒⠒⠒⠒⠚⠉⠉⠉⠉⠓⠢⠤⣄⣀⡀
⠀⠀⠀⠀⠀⣠⠒⠋⠭⠭⠅⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠈⠁⠂⣀⣀⠀⠉⠙⠒⠢⣤⣀⡀
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⢀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡆
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢀⡠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡑⠀⠀⠀⠀⡐⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠒⠒⠠⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠡⡀⠀⠀⡀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠆⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠉⢢⠀⠀⠀⠀⠀⠀⡀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚
⠀⠀⠀⠀⠀⠀⡤⠃⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠂⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄
⠀⣀⠤⠔⠒⠊⢳⣀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠐⠒⠒⢤
⡎⠁⠀⠀⠀⠄⠀⢀⣉⠉⠉⠉⠉⠉⠓⠋⠀⠀⠘⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠂⠀⠀⣤⣤⡤⢄⣀⠗⠂⠀⢀⡠
⠈⠓⠢⠤⢄⣀⠀⠀⠀⠈⠉⠉⠁⠐⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠐⠒⠂⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠉
⠀⠀⠀⠀⠀⠉⠛⠒⠶⠶⠤⠤⣤⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠠⣄⣀⣀⣒⡤⠦⠤⠭⠖⠒⠚⠉⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⡄
⠀⠀⠀⠀⣀⣀⣤⣤⢤⣖⣷⣲⣀⣠⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⢀⣴⣿⣿⡏⠋⠉⣍⣋
⠀⠀⠀⠀⠻⠀⠀⠀⠀⠈⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⢿⡿⠷⠶⠟⠿⣿⠾
⠀⠀⢈⡀⢀⢸⠀⠀⠀⠀⢸⠀⢀⡄⣤⣴⣶⣤⣒⣶⡄⠀⠀⠀⠀⢰⣤⣤⣴⣦⣤⣄⣀⣀⡀⢠⡤
⠀⠀⣾⣿⣤⣾⠀⠀⠀⠀⠼⠀⠈⠋⠹⣽⠿⣽⠿⡽⠥⠾⠇⠀⠀⠸⢹⣟⠿⣿⣿⣯⣿⣯⣿⣿⣶⣶
⠀⠀⠉⠉⠉⢉⡀⠀⠀⠀⠽⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠿⠟⠛⠻⠿⠛⠿⠟⠛
⠀⠀⠀⣴⣶⠋⢻⠃⠀⢀⣭⠀⢀⣀⣄⣦⣀⣸⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⡆
⠀⢠⣿⣿⣿⡄⠸⠀⠀⢘⢘⡃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣖⠀⣤⣄⠀⢠⣏⣿⣿⣷⣶⣿⣶⡆⡀⡀
⠀⠀⠉⠉⣩⠥⢀⣄⠀⢐⢐⡂⠀⠈⠈⠻⠟⠉⠉⠛⠛⠿⠛⠒⠻⠿⠀⠀⠉⠙⠞⠉⠉⠉⠻⠛⠛⠻
⠀⠀⣠⣞⠁⠀⠀⠛⣀⣀⣀⣂⣀⠀⠀⣰⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣏⢑
⠀⣸⣶⣃⣷⡀⠀⠀⠀⠀⠀⣀⡄⠀⠀⣿⣽⣠⣄⣄⣠⣤⣄⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣤⣤⣤⣀⡀
⠀⠛⠻⡿⠻⡷⠀⢀⣠⡾⠟⠉⠉⠀⠀⠉⠈⠛⠻⠿⠛⠛⠿⠻⠗⠀⠀⠈⠙⠛⢿⡿⠋⠛⠛⢿⣿⠟⠃
⠀⠀⠀⠀⣀⣤⢞⠛⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⡆⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⣄
⠀⠤⣶⣟⡋⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴⠟⠋⠉⠀⠀⣿⡇⠐⠛⠛⢿⣿⣿⣿⡿⢡⣾⣿⣿⣦⡀
⠀⣤⣤⣽⣿⣿⣤⣾⣷⡆⠀⠀⠰⠿⢿⣿⣦⡄⠀⠀⠀⠀⠀⢯⠀⠀⠀⢰⡟⢻⣿⢿⣿⣿⡟⢻⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡯⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⡿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢼⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠗⠻⠿⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢫⣴⣶⡏⠀⠀⡹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣾⣿⣷⠀⠀⢠⡇⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢘⣿⣿⣯⠀⠀⢸⣇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢰⣿⣿⣗⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢈⣿⣭⣥⣤⣤⣤⣌⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠉⠉⣩⣭⣍⠙⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢀⣾⢿⣿⡿⣆⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢡⡿⢡⡎⠹⣷⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⣟⣱⣿⢾⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣺⠟⣿⡇⣼⣿⣿⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠻⠶⠟⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢯⡷⣦⠀⠶⠀⣿⠆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡯⢸⡇⢿⡆⣴⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡜⢇⡾⠃⠿⠀⠿⠶⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⡄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠃⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠃⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⡿⠿⠿⠿⠿⠿⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠻⣿⣿⣄⡀⠀⠀⠀⠀⡾⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣯⣭⣭⣭⣶⣤⣤⣶⣶⠾⠃
⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣥⣤⣶⣿⣖⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣦⡀⠀⠀⠀⠀⢠⣾⣿⣿
⠀⠀⣤⣤⣴⡄⢀⡀⠀⠀⣀⣀⠀⠀⢠⣤⣿⣧⣤⣤⠀⡀⠀⣤⣤⣴⣾⣷⣄⣤⣤⣴⠦⣠⣾⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣶⣤⢖⣤⣤⣡⣸⣾
⠛⢻⢻⢛⠛⠃⡛⠛⠓⢛⠛⠻⠛⠛⠛⠛⠛⠃⠉⠁⠀⠈⠀⠈⠉⠙⠉⠉⠛⡿⠟⠛⠛⠉⠉⠀⠀⢰⠆⠀⠀⠀⠸⡭⠿⣿⣿⠾⣿⣿⣿⠿⣿
⢠⠁⠀⡀⠀⠢⠈⠀⢈⠀⠀⠐⠂⠀⠀⠀⠄⠀⠀⠀⠒⠂⢀⡀⠈⠁⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠲⢤⡀⠉⠒⠤⣀⠒⠤
⡘⢰⠀⡧⠘⡀⠀⢄⠀⠑⠄⡀⠀⠢⠈⠁⠢⢀⠀⠢⢄⠀⠀⢀⠀⠂⠀⠤⠄⠀⣀⠤⠂⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠐⠦⢀⠙⠲⠤⣀⠙⠲
⠁⢼⠀⢰⠀⢠⠀⠀⢂⠀⠀⠙⢦⡀⠀⠁⠀⡀⠙⠢⡀⠙⠢⣀⠁⠦⣀⠀⠈⠉⠀⠀⠚⠀⠀⠐⠂⠀⠠⠔⠒⠈⣀⡠⠤⠂⢀⠁⠲⣄⠈⠑⢤
⠀⣼⠀⢸⡄⠀⠇⠀⠀⠳⡄⠀⠀⠹⠄⠀⠀⠈⠢⠄⠈⠲⠄⠈⠓⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⠐⠖⠆⠀⠠⢠⠒⠈⠁⢄⠈⠙⢦⡀
⠀⠸⠀⠀⠧⠀⠘⠀⠀⠀⠡⠬⠍⠀⠀⠀⡀⠤⠈⠀⠀⢀⡐⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠶⠆⠀⠀⡤⠤⠀⠳⠀⡀⠃
⠠⠀⠐⠀⠀⠂⠀⣀⠀⠠⡄⠀⠢⠀⠒⠀⠁⠀⠀⡀⠀⠀⠉⢀⠀⠀⠀⠐⠀⢀⡀⠀⠀⠀⠀⠀⠉⠀⠈⠛⠒⠀⠀⠠⠤⠀⠀⠲⠒⢀⠉⢀⠤
⢶⠀⠘⠀⠉⠁⠀⠀⡀⠀⠠⠤⠄⠀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠀⡐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠄⠀⠐⢀⣤⠄⠀⠈⠁⣀⡤
⠀⠀⢀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠰⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣄⣀⡄⠀⠀⢀⣠⠄⠀⠀⠈⠀⠰
⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣀⡀⠀⠈⠀⣠⡦
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠒⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣀⣀⣀⠀⠀⠀⠀⣠⡴⠆
⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⢀⡠
⠀⠀⠀⠀⠉⠙⠒⠤⠀⠀⠀⠀⠀⠀⠘⠓⠂⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⠊⠁⠀⠀⣀⠀⠀⠈
⢾⣶⡄⣤⡄⠀⠀⠀⠀⠀⠀⠀⠰⠓⢤⠄⡀⠀⠀⠀⠀⠀⠄⠠⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠉⠉⠉
⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠋⠉⠁
⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠚⠂⠀⠀⠀⠀⠀⠰⠗⠂⠂⠒⠀⠀⠀⠀⠚⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⢤⡤⠤⠖⠂

⠋⠀⠬⠘⠀⠀⠀⠀⠀⠁⠀⠀⢰⠂⡀⠀⠀⠀⠊⠀⠀⠀⠉⠀⠀⠁⠀⠁⠈⠈⠀⢈⠉⠀⠀⢮⣭⡷⠂⠀⠀⠈
⠏⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠨⡀⠆⠀⠀⠀⠀⠤⢀⡀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁
⠀⠀⣀⡔⢒⠙⠛⠑⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⣀⠂⠀⠀⠀⠀⠀⠀⠀⢲⠄⠀⠀⠀⣠⠂⡸⠂⠀⠰
⠁⠀⠈⢁⡉⠀⢬⠀⡀⠀⠀⠀⡐⠀⢀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠘⠁⠀⠀⠁⠀⠀⠸⠁⠐⠂⠀⠀⠀⠀⠐⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⣑⢶⣄⡀⠀⠀⠀⠀⠘⠁⠀⣀⡀⠀⠀⠀⠀⠘⡅
⠃⠀⠀⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠂⠀⠀⠀⠀⠀⢀⠠⠝⠛⠖⠀⣀⠀⠀⠃⢀⠀⢰⠒⢰⡄⠀⠲
⠁⠠⠂⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢀⠀⠄⢀⠀⠁⠀⠀⠀⠉⣈⠁⠠⡌⠁⠀⠀⠀⠂⠙⠀⠀⠁⠩⠤⠀⠀⠁
⠀⢀⡀⠀⠀⠀⡆⠀⠉⠠⠄⠈⠇⠀⣠⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠠⠓⠀⠀⠀⠀⠠⣤⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢰
⠰⠈⠀⠀⠀⠀⣁⠀⠀⠀⠀⢱⠀⠀⠀⣔⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠋⠙⢠⠁⢐⣀⡄⠀⠀⡀
⠀⠀⠀⠀⢀⡀⡟⠀⠀⠀⠀⠘⠀⠀⠀⠉⠀⠀⢠⠴⣶⣧⠀⠀⠨⠁⠀⠀⠀⠂⠀⠀⠑⠘⠾⡌⠁⠙⠀⠀⠀⠋⠠⡐⠆
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠖⠐⠀⠀⠠⠀⠀⠈⠙⠉⠁⠀⠀⠀⢠⠄⠀⠐⠀⠤⠄⠲⠄⠀⠀⠀⠀⢠⠆⠀⠀⠀⠡
⠲⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣛⡂⡀⠀⠀⠀⣀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠃⠈⡀⠀⠈
⠄⠀⠠⠤⠒⠂⠀⡀⠘⠠⠐⠂⡀⠈⠁⠀⠻⢡⡗⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡅⠀⠀⠈⠀⠀⠀⢃⠀⡅⠀⠀⠀⠀⠶
⣄⠀⠳⠀⠀⠀⠙⠉⠁⠀⠂⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢌⡁⠀⠀⠢⠀⠀⠀⠀⠖⠆⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡄⠀⠘⠳⠀⠀⠀⢠⠁⡸⡃⠀⠀⠀⠀⠀⢠⡄⠘⠁⠀⠀⠐⠀⠀⠀⢀⣀
⠷⠃⠀⠀⣀⣤⠊⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⢲⡀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⢲⡀⠀⠀⠀⠀⡠⡐⢰
⠀⠀⠀⠯⠽⠞⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠑⢀⠀⠀⢀⠀⠀⠀⠀⠆⠀⣀⠀⠀⠀⠀⠈⢄⠸⠀⠀⢱⠈⠁
⠀⣶⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠈⠩⠡⢤⠄⡀⠀⠀⠲⠊⠁⠀⠀⠀⠉⠁⠀⠀⠠⣄⠀⢀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠁
⢠⠋⠠⠂⠀⠀⠸⠸⢰⠋⠁⠀⠄⠀⠀⠰⠂⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⠀⢠⡄⡀⠀⡄
⡮⢄⠢⠀⣤⡴⢒⢉⣢⠖⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠎⠸⠇⠀⠈⠠⠀⡀⠐⠚⢀⣠⠀⠀⠀⠠⡌⠀⠀⠀⠁⠈⠐⠆
⠁⠁⠀⠀⠀⠁⠐⠒⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠂⠛⠠⠁⠀⠈⢃⣘⠆⠀⠦⠄⠀⠀⠀⠁⠀⠀⠄
⢀⠀⠀⠀⠀⢰⠂⣢⡀⠀⠆⠀⠀⠀⠀⠀⡀⠀⡒⡒⡄⢂⠀⠀⠀⠀⠀⠀⠴⡿⠞⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⣀⠁⠀⢀⠀⠀⢈⢉⡅⠁⠀⠁⠀⡇⠀⠀⠟⢋⠄⢁⠪⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣀⣠⠐⠂⠀⠰

⠀⢶⣤⣠⢤⣤⠤⢤⣤⡄⠀⠀⠀⣤⡤⠀⠒⠶⣆⠀⢂⢶⢶⣖⣀⣶⢶⣶⣆⣲⢲⡖⢂⠒⣶⡆⠀⣠⠄⠀⣲
⠀⣆⠁⠀⠀⠉⠛⠋⠀⠻⠦⣤⡶⠏⠀⠀⠀⠀⠈⠙⠓⠁⠀⠘⠙⠁⠈⠳⠯⠵⠋⠙⠃⠎⠀⠙⠲⠋⠀⠸⣿
⠀⡟⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢾
⠀⣷⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡻
⠀⠣⢑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿
⠀⠶⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⣛⡆⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣼⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤
⠀⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠈⠻⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠶⠖⠒⠚⠉⠉⠉⠛⠛⢚⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⢰⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢉⣙⣛⣛⣓⡚⢛⠚⣒⣙⡉⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⢸⠻⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⣯⣻⣏⣏⣿⣹⣹⣏⣿⡇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣦
⢨⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡉⢩⠛⠉⠉⠉⣿⡉⠉⡇⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡶
⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⣇⠐⢳⡐⡂⣰⢻⢁⠄⡇⠐⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣦
⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡧⡬⡥⡾⡾⡶⣿⢷⣶⡇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿
⠀⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠑⠧⢧⣧⣧⣇⣿⣾⠿⠁⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠸⣭
⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠶
⠀⣶⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠒⠒⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢅⢀
⠀⣮⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⢿
⠀⡷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣬
⠀⣿⡄⠀⣠⠦⣄⠀⡠⠠⣄⣠⢖⣲⢦⡀⢀⣄⡄⠀⢀⢤⣄⡀⠀⠀⠀⠀⣰⠾⠛⠲⣦⠀⣠⣤⣀⠀⠀⢀⠹
⠀⠯⠀⠐⠋⠀⠸⠿⠤⠁⠼⠧⠿⠽⠿⠷⠿⠉⠽⠷⠷⠩⠀⠹⠶⠄⠀⠚⠛⠀⠀⠀⠘⠛⠓⠒⠛⠓⠋⠙⠶

⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡤⣀⠉⠀⠀⠀⠀⢀⡀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠈⠀⠀⠁⠈⠈⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠐⠀⠀⣀⠀⠀⠀⢠⠇⠀⠀⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠠⣀⣠⠖⠄⠀⢃⠀⠀⡇⠀⠀⠀⠀⠀⡠⡈⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠔⠀⠀⠀⠀⢳⠈⠀⣰⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠄⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣦⠀⠂⠁
⠀⠀⣀⠀⠀⡀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣦⣤⡐⠀⠃⠀⠁⠀⠀⠀⡀
⠀⠀⠸⢄⠀⣀⣤⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⣀⣙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⡙⠈⠉⠉⠉⠉⠀⠄
⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⣿⣻⣟⣟⣿⣿⣟⣿⣻⣿⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠂
⠀⠀⠀⢐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣠⣬⡉⠐⠒⠀⠀⠤⠠⠄⣀⡀⠀⢀⠁⠀⠈⠛⠻⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⣰⡇⠀⠀⠠⠤⠀⠀⠚⡗
⠀⠀⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠈⠐⠀⠀⢀⣤⣾⣿⡀
⠀⠀⠘⢿⡿⣿⣖⠀⠀⡀⠀⣀⣀⡰⠖⠉⠉⠉⠀⠀⠀⠀⢠⡀⠀⢠⡆⠀⠀⢴⠆⠀⠀⠀⠀⠂⠀⠀⢙⡌⠳⠴
⠀⠀⠀⠀⠳⢌⡊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠤⠻⠥⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠛
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠇
⠀⢠⠂⣈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⡄⠇⠀⠀⡀
⠀⠀⠘⠌⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠢⠠⠀⠃⠀⠀⠀⠈⠢⢀⡀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣟⣿⣟⣛⣻⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢈⡒⢄⠀⡀⡠⠊⡡⠂
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠚⠘⣿⠀⠀⠀⠀⡰⠒⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣴⣃⢾⣠⠊
⠀⠀⠀⠀⠀⢀⠀⡆⠀⣰⣴⡛⠀⠀⠀⢶⡅⠀⢠⠄⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⠟⠉⡫⣿⡇
⠀⠀⠀⠀⠀⢸⠀⡇⢰⠟⣉⠙⢶⠤⠤⠀⠙⡶⢠⠘⠁⠀⠀⠀⠀⠀⠀⠀⢀⠞⡴⢋⢰⠈⠳⡈⢣⡀
⠀⠀⠀⠀⠀⢸⠘⡇⠛⠀⣈⠀⢸⣷⣶⡶⢶⡆⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠤⢾⠀⠀⠀⠛⡶⠃
⠀⠀⠀⠀⠀⠀⠭⣀⣄⣸⡇⣀⣸⣥⠤⠤⠬⠧⠧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⣏⡀⠀⠀⣿⠉⠀⠁
⠀⠀⠀⠀⠀⠊⠄⢀⠤⠚⠉⡉⠀⢄⠀⠠⡀⠢⡐⠄⠑⠀⠀⠀⠀⠀⠀⠀⠒⡉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠈⠀⠀⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢠⢂⡖⣄
⠀⠀⠀⠀⠀⠀⠘⠂⡿⡿⢄
⠀⠀⠀⠀⠀⠀⠀⠉⡐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⢰⢧⠺⢀⣤⠤⠤⠤⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡤⣬⣆⡤⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡼⠆⢰⠏⠈⢳⡤⠤⠤⠤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢙⣀⣹⣴⢯⣏⢳
⠀⠀⠀⠀⠀⠀⣀⡗⣆⠸⠀⡷⢸⡌⡗⢸⡇⡧⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠑⣸⣏⣿⢣⢺⣿⣿⣿⣼⢿⣯⠄
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠐⣀⣀⣼⣧⣧⣀⣠⣇⣀⠀⠠⠀⠀⠀⠀⠀⠀⣀⣸⣯⠥⠼⠿⠾⠿⢿⣬⣿⣯⡟
⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠀⢀⠄⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⢈⢀⡃⠀⠀⠀⠀⠸⢤⡀⠀⠐⡄⢿⠂⠀⠀⠞⠭⠇⠀⠀⢐⢠
⠀⠀⠀⠀⠀⠐⠞⣮⢫⢧⡀⠀⠀⠀⢂⠀⣿⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⠢⢀⡀⠀⠠⣄⠠⠤
⠀⠀⠀⠀⠀⠀⢠⡹⣇⣩⣆⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⡄⠠⣤⠀⠀⠀⠁⠲⠦⠤⠶⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠶⠇⢠⠀⠀⡀⠀⠀⠀⠀⠉⠰⠬⠀⠀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠸⠒⠒⠚⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⠀⠀⠀⠀⠆⠐⢢⡀⠀⡀⠈⢲⠑⢲⣀⡐⠁
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠄⡀⣠⠔⠀⠀⢠⠞⠁⠀⠀⠀⣀⠔⠚⠓⢲⣷⡀⡉⠫⢿⣤⡧⠉⣹
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠹⠇⠀⠀⢠⠟⠘⠁⠀⠀⠆⠀⠀⣰⣦⣆⣿⣿⣿⣠⣴⣿⣤⣤⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠶⢀⠴⣴⣻⠖⠁⠀⠀⢟⣃⠈⠀⠀⠀⠀⢶⣶⣿⠿⠛⠋⠉⠙⠛⠛⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠠⢀⣈⣥⡴⠶⠚⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠚⠻⣦⣀⡀⠀⠀⢀⣄⣹⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡤⣀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢏⣆⢄⣀⡠⠈⠉⠉⠉⣛⣛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⣩⣷⡶⠈⢻⣭⠤⣤⠒⠒⠒⠊⠉⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡟⡁⠉⠁⠚⣡⣾⠛⠉⡠⠖⠊⠉⠋⠈⠁⠀⠄⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠙⢻⡔⠚⠉⢈⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡿⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠀⡄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣠⣀⣀⡀⠤⡰⠁⠀⣾⠗⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡈⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⣿⠀⣿⣴⣿⠉⣠⠃⠀⣸⡏⣠⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣆⣼⣿⣿⢀⠇⠀⢠⣿⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡲⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠈⠓⠚⠛⠿⣿⣿⣿⠏⠀⠠⡞⢋⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡁⠀⢨
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡙⠗⠀⠀⠸⠁⠊⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⠷⠶⠾
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⢻⣄⢄⣀⣰⡦⠜⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠁⢀⠀⢘
⠀⠀⠀⠀⠀⠰⣀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠃⠈⠀⠈⠻⠏⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠀⠳⠄
⠀⠀⠀⠀⢀⣤⣊⡍⠀⣀⣀⠀⠀⢀⠀⠀⢀⣠⠤⠀⠁⢀⠭⠄⠀⠈⢾⣶⠆⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⢠
⢀⡀⢀⡀⠉⢀⠈⠙⡆⢀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠒⠂⠡⠠⠀⠀⣀⣤⠁
⠈⢳⣾⠇⢾⣿⣿⡆⠘⣄⣃⠀⠀⠀⠀⠸⠀⠀⣠⣤⣀⠀⠀⠀⢙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣰
⠀⠈⠁⠈⢀⠉⠛⢃⢀⣿⡟⠁⠀⠀⠀⠀⠉⠈⡻⣿⣿⣧⠀⠠⢄⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠐⠂
⠀⠀⠀⠀⠀⠛⠻⣿⣿⣶⡶⡆⠤⣀⡦⢀⣀⡀⣀⠈⠁⠀⣰⢬⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠛⠂⠀⠸⠙⠚⠹⠿⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠴⠦⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⢤⠄⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⣞⠛⣳⠄⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠋⢁⣠⣎⣉⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣀⠍⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⢯⣁⠡⠝⠃⠀⠀⠀⠀⣐⠲⣤⠤⠀⠒⠈⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣩⣤⡤⣀⡀⠀⠀⣀⣀⡀⠤⠤⠐⠒⠂⠉⠉⠈⢁⣀⣀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⣡⡼⣇⠈⢊⡾⠋⠉⠀⠀⠀⠀⣠⠤⡄⠀⠀⠀⢠⠞⠻⠮⡺⣝⢷⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⣯⠀⠀⢘⡩⣲⡟⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢠⣞⣳⠄⠀⡀⠀⠀⠀⠀⢁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣴⡟⢫⡄⠐⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠉⠁⠀⠀⠀⠀⠀⢀⡞⠛⠥⣀⡝⢵⡄⠀⠀⠘⠀⠀⠀⠈⠈⠑⠈⠁⠀⠀⡠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⢆⠦⠀⠘⣧⡀⠀⠀⢊⡼⠁⠀⠀⠀⢰⠒⣦⠀⠀⠀⠀⠀⠀⠀⠓⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠉⠠⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠋⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠻⠇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠴⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠤⠴⠒⠒⠂⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠒⠒⠃⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠉⠈⠑⠂⠂⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠖⠢⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⣀⣤⣄⢀⣀⡀⠀⠀⠀⢀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠁⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠻⠿⠿⣿⠿⠃⠀⠀⠈⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠶⣶⣶⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⣄⠀⠀⠉⣩⠟⠛⠻⢅⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⢡⣶⡿⠀⠹⣾⣿⣂⣀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⢀⣀⣄⡀⠀⢸⣿⡿⣶⡄⠙⣿⣿⡋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣠⡶⠷⠒⠀⠀⠀⠀⢽⣿⣿⣿⣿⡇⠀⠙⠛⠛⠁⠀⠈⠻⠿⠷⠟⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠻⠏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⡇⠀⠀⠀⠈⠿⠿⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢴⠂⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣮⠀⠀⠀⣸⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠠⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣯⡀⠀⡀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢠⢤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣤⣀⠀⠀⠙⢟⠛⢋⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⡳⠀⠀⠀⢤⣄⠀⠈⣿⣿⣿⣿⡇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⢀⣀⣿⣿⣧⠀⢿⣿⠏⠙⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡰⠁⢻⣿⣿⣿⠏⠀⠈⠙⠀⠀⠖⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⣀⠤⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣠⠴⠋⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠶⡤⠀⠀⣀⣠⠤⠄⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠇⠀⣀⡀⠤⠐⠂⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠁



⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄
⠀⠀⠀⢀⣤⠀⠀⠀⠈⣿⡟⠁
⠀⠀⠀⠈⠉⠁⠄⠀⠈⠙⠛⠓⠀⠀⠀⠀⠀⢀⡴⠾⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠤⠀⠀⠄⠒⠂⠀⠀⠀⠘⠉⠉⠉


⠀⡀⠄⣠⠄⣀⡀⠀⡀⠀⣀⠀⡀
⢦⡈⠲⢄⡉⠒⢭⡑⠂⠤⢀⠋⠥⣂⠢⠤⣐⡂⠤⢀⣀
⠀⠙⢦⡀⠙⠢⣀⠈⠓⢦⣀⠉⠲⢤⣌⡁⠂⠬⣉⠓⠲⠭⣝⡲⠤⠤⣄⡀
⢳⡀⠀⠙⣦⠀⠈⠳⣄⠀⠈⠳⢶⣀⠈⠙⠳⢤⣀⡉⠑⠶⢤⣍⠙⠒⠦⣭⣙⠒⠦⠤⣄⣀
⠈⢻⣤⠀⠈⢳⡄⠀⠈⠻⣤⠀⠀⠉⠳⣤⡀⠀⠈⠑⠶⣤⣀⠈⠙⠲⢦⣀⠉⠙⠲⢦⣄⡉⠛⠲⢦⣄⡀
⠀⠀⢷⡀⠀⠀⠙⣤⠀⠀⠈⢣⣄⠀⠀⠈⠻⣦⣄⠀⠀⠈⠙⠷⣤⡀⠀⠉⠛⠶⣄⡀⠈⠙⠲⣦⣀⠈⠉⠛⠶⣄⣀
⡇⠀⠈⢿⡆⠀⠀⠙⣷⡀⠀⠀⠙⠷⡀⠀⠀⠀⠙⢷⣦⠀⠀⠀⠈⠛⢷⣤⡀⠀⠈⠛⠷⣤⣀⠀⠉⠳⢦⣤⣄⡈⠙⠳⢦
⣿⣶⠶⠾⠿⣶⣶⣶⣿⣿⣶⣶⣶⣾⣿⣷⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣾⣿⣿⣿⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⡤⠶⢤⡀⠉⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⠍⠀⠀⠀⠈⠳⠤⠔⠁⠀⠀⠀⠀⢙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠈⢁⡴⠒⢲⣤⠤⠤⠴⠖⠂⠀⠃⠀⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠾⠷⠖⠒⠒⠋⠉⠉⠳⠤⠼⠁⠀⠀⠀⢀⠤⢤⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⡼⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⢠⠖⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⡼⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠂⠐⠒⠂⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡋⠁⠉⢻⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣶⣾⣿⡿⠿⠿⠟⢛⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠿⠿⠿⠟⠁⠀⠀⢩⣭⣭⣴⣶⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣶⣶⣾⣿⣿⣿⣦⣤⣴⣿⣿⣿⣿⣿⠋⠙⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡟⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⡀⡀⠀⡄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡈⢢⡈⠃⠁⢤⡨⣫⠀⣘⡂⠀⠈⢰⠀⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣿⣆⠴⣼⠹⢻⣖⢛⡗⣎⡷⠏⣶⠃⡜⠃⠁⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠳⠘⢸⣿⢠⡸⠀⢸⡹⢋⠠⠆⠉⠀⡼⣄⡇⣧⡀⠑⢃⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡴⢱⠘⣿⢼⡃⠀⢸⣁⢢⠃⡴⣩⠄⣟⢙⣇⢚⣪⣶⡜⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣡⣧⣿⣯⣅⣄⣸⣇⣘⣞⣠⣋⢈⣿⠰⠏⣶⣏⠾⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⠉⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢠⠄⢲⠀⢐⢰⡀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠘⢴⢸⡀⠅⠀⠀⠀⠀⢸⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⣄
⠀⠀⠀⠀⢰⣿⠭⠵⠖⠲⣌⡷⣄
⠀⠀⠀⠀⢸⢫⠄⠀⠚⠛⢻⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⠀⠞⠂⣉⣉⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⢁⣀⣀⣀⣠⡴⠊
⠀⠀⠀⠀⡇⠀⠀⠐⠻⣭⡼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⡇⢰⡂⠀⠀⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣀
⠀⠀⠀⠀⢷⡈⠙⢛⣯⡵⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠰⣫⠿⠭⠋⠁
⠀⠀⠀⠀⠈⣇⢀⠀⠈⠘⣆⠻⡄⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠙⠛⠾⣷⡶⣄⠀⢤⡀
⠀⠀⠀⠀⠀⣿⠀⢷⡀⠀⠈⠃⠹⣦⡀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⡀⠹⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠈⢧⠀⠀⠀⠀⠈⠻⣦⡿⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡈⠻⣦⣀
⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⠻⣶⣀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠳⣄⠉⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⢻⡰⡀⠘⣽⢦⠀⠀⠀⠀⠀⠀⢹⡍⢷⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⡇⣸⠀⠛
⠀⠀⠀⠀⠀⠘⣧⠫⠀⣿⠙⠿⠟⠂⠀⠀⠀⢨⣷⢾⠠⢤⣤⣤⣤⠞⠋⣠⣴⠄⠀⣧⡏
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣆⡀⠀⠀⠀⠠⠴⠚⠸⡇⠀⠀⠀⠀⣀⢄⡰⡝⠁⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠨⡉⠛⠛⠒⠒⠀⠀⠘⢁⡼⠀⠀⠀⠀⠑⠒⠺⠀⢀⣼⠋⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠘⢦⣀⣀⣀⠀⠔⠋⠉⠀⠀⠀⠀⠀⢀⣀⣤⡴⠟⠁⠀⠈⠙⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣬⣉⢁⣠⡴⠶⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢃⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣵⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣡⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠛⠋⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⢿⠿⢿⣶⣶⣤⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠈⠉⠉⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢸⡇⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢤⣬⣭⠤⠶⢴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡇⠀⠀⠀⠀⢠⣤⣀⣤⡄⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⣄⣀⣉⣠⣄⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢸⡇⠀⠀⠀⠀⣈⣉⣩⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠟⠀⠀⠀⠀⠈⠉⣉⡀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠘⠉⠉⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⡀⠀⠀⠀⠀⠀⠒⠛⠉⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⣼⠃⠀⠀⠀⠀⣘⣋⣉⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣰⠃⠀⠀⠀⢀⣀⣩⣭⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢠⡏⠀⠀⠀⠀⢈⣉⣤⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢀⡟⠀⠀⠀⠀⠀⢬⠍⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⣠⠏⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡟⠁⠰⠋⠀⠀⠀⠛⠓⠚⢒⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⣀⣘⣛⣛⣓⣺⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠤⣤⣉⡉⣩⣭⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠶⢦⣤⣤⣠⣤⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⢀⣀⣀⠙⠛⠶⠦⣴⣾⡿⠋
⠀⠀⠀⠀⣀⣠⣶⠿⠋⢀⡀⠴⢤⣀⣉⣉⣙⣛⣿⣿⠟⠁
⣀⣤⣾⡿⠟⠋⠁⠠⣤⣀⠙⢻⣶⣤⣽⣿⠿⠛⠋⠁
⠛⠻⠿⣶⣶⣴⣿⣿⣿⣿⡿⠿⠛⠛⠉
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠈⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠶⠶⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣉⠉⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣀⠀⠀⠾⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠒⠒⢲⠒⠒⠛⠛⠉⠉⠉⠉⠉⠉⠏⠁⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡉⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢲⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⢆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣸⠀⠀⠀⠀⠀⠸⠒⠒⠚⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠸⠀⠀⠀⡀⠀⡶⠶⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⡠⠔⠦⣄⠀⡤⠤⠤⠤⠤⡄⢰⠒⡇⠀⡼⠉⠚⡃⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡞⠀⢀⠀⠈⢃⠓⠒⡆⢰⡒⠁⢸⠀⢱⢀⡇⠀⠀⢸⣀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢸⡇⠀⠯⠇⠀⢸⣴⠒⠃⠀⠉⢳⠸⡆⠀⠉⠛⠉⢹⠀⡤⡀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠳⢤⣀⣀⠴⠃⠙⠒⠒⠒⠒⠉⠀⠷⠤⠴⠶⠶⠚⠀⠑⠁⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⠒⠶⠶⠤⠤⠤⠤⠶⠶⠦⠤⠤⠤⠤⠴⠶⠒⠒⠒⠚⠋⠉⠉⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠲⠤⠤⢤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠒⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡤⣭⣤⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣠⣀⣀⣀⣀⠐⡆⢸⣰⣇⠀⣿⠁⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣹⣭⣿⣿⣿⣿⣾⣾⣿⣿⣳⣿⠀⠀⠀⠁
⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣶⣶⠆
⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⢹⣿⡿⠿⣷⠋⠉⣭⣭⣭⣭⣭⣭⣭⣭⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⡀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡹⣿⣿⢇⣤⣯⣀⣀⣹⣽⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣏⣱⣿⣿⣿⣿⣿⡈⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣭⣭⣽⣛⣛⣛⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠓⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠚⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⢄⠀⠀⠀⠀⢀⣠⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡀
⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⡀⠉⠲⢄⠑⠀⠐⠚⠛⠛⠛⡃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⣏⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⢀⣀⣀⣤⣼⣧⠤⠤
⠀⠀⠀⠀⠁⠒⠢⢤⣀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠤⠔⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡀⠀⠀⠀⠠⠄⠈⠁⠀⠀⠀⠀⠀⠀⠁⠐⠀⠠⢄⣀⠀⠒⠀⢄
⣀⣀⣀⣀⡀⢀⡀⠤⠤⠤⠤⠄⠀⠀⠁⠒⠒⠒⠒⠒⠒⠐⠒⠒⠂⠙⠛⠲⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠤⠂⠈⠙⠉⠉
⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⣀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠈⠉⠉⠁⠉⠁⠂⠀⠁⠀⠀⠢⠄
⠀⠀⠀⣴⠀⠀⠁⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠄
⠤⠶⠚⠛⠃⠑⠂⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠐⠀⠀⠰⠶⣤⠠⠴⠚⠈⠉⠁⠀⠈⠁⠈⠉⠂⠀⠀⠀⠀⡈⠀⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⠤⠤⠒⠚⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠙⠛⣛⡋⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⣀⠄⠤⠒⠒⠂⠉⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠰⣾⣿⠟⠀⣿⣿⣿⡄⠀⠀⠤⠀⠐⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠁⠠⠀⠛⠛⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣽⣿⣿⣿⣦⡀⢀⣾⣿⣿⣿⣿⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣶⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣻⣿⣯⣵⣶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣽⣿⣿⣷⣝⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣾⢿⣿⣿⣿⣿⣿⢻⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡷⠶⢟⣼⣿⣿⣭⣭⣭⡾⢿⣾⣻⣶⣿⣽⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢏⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⡇⣿⣿⣿⣯⣿⢧⣿⣿⣻⣿⣿⣿⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣻⣵⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⢿⣿⣻⢿⣾⣽⣿⣿⢋⣾⣿⡿⣿⣿⣿⣿⣴⢿⣛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣎⣿⣿⣿⣶⣯⣿⡷⣾⢯⣿⣿⣿⡎⣿⣇⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣿⣷⢿⣧⣟⣻⣿⡿⣯⣾⣷⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣽⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣯⣽⣟⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡟⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⣀⣤⡄⠀⣇⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢐⣀⣀⣀⠀⠀⠀⠀⢀⠔⠒⣤⣄⣴⣾⣿⣿⣿⡧
⠀⠀⣤⡾⣿⣿⣿⣿⣿⣿⡻⣿⡇⠀⠀⣄⠀⣦⣴⣶⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣿⣴⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠷⠽⠟⠀⠀⠘⠿⡵⠏⠀⢀⣼⣿⡷⠋⢿⣿⡿⠉⠉⢿⣿⡟⠉⠁⠀⠀⣿⡿⢻⣿⣿⣿⠻⣿⡟⢿
⠀⠀⠀⣰⣶⣄⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⣀⣀⣉⣀⣀⡤⠤⠤⠤⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠉⠁⠀⢀
⠀⠀⢠⡟⠀⢻⣦⢀⣀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⡿⢰⠋⠉⢣⠀⢀⣤⣶⣶⣤⣤⣤⣤⣤⣴⣶⣶⣶⣦⣰⠛⠢⡀
⠀⣤⠏⠀⠀⢠⡁⢠⣤⣄⠀⠀⠀⠀⣠⣤⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡆⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣶⡇
⠀⠛⠿⠶⠂⢸⣶⣿⣿⣿⠀⠀⠀⢸⡏⢡⠉⣿⡿⠿⠿⣿⡏⣉⠙⣿⠉⠀⠛⣟⡿⣿⡟⠛⠛⠻⣿⠿⣻⠛⠛⢻⣻⠛⠁
⠀⠀⠀⠠⠞⠻⠟⠇⠿⠸⡃⠀⠀⠘⢷⣤⣴⠟⠀⠀⠀⠹⢧⣤⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡀
⠀⠀⠀⢰⠛⠛⡆⣴⠀⠀⠃⠄⢀⠀⡤⠠⡀⠀⢀⠤⢤⣲⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⢀⣾⠋⡇
⠀⠀⣠⣼⠀⠀⢰⣿⠀⠀⠀⣼⣾⣷⣷⣤⡆⠀⣦⣤⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣤⣴⣶⣶⣦⣴⣾⣄⠘⡿⠀⣿
⠀⠀⠿⣿⣷⣶⢾⠿⠧⠀⠀⢹⢿⠿⡿⣿⢧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣶⠀⠀⣿⣿⡿⠿⢿⣿⣿⡿⠿⠇⢸⣿⠇
⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⣀⣈⡉⠉⠉⠉⠉⠀⠀⠈⣹⡆⠀⠈⠋⠈⠉⠁⠉⠋⠀⠀⠈⠋⠁⠀⠀⠙⠛⣁⣤⣤⣤⡀
⠀⠠⠀⣤⣤⣴⠶⠾⣻⠯⡉⠀⠀⠀⠀⠀⢀⠤⣪⡾⠁⡇⠀⠀⣀⣆⣀⡏⠉⡏⠁⠀⠀⠀⠄⢠⡀⣴⣾⣿⣿⣿⣿⡧
⠀⢸⣤⣿⣿⠆⣤⡄⣷⣤⣴⠀⠂⣬⠀⣿⣷⣾⠟⠀⠀⠁⣀⢸⣿⣿⣿⣁⣀⣿⡇⠀⣶⣿⣶⣾⣷⣿⣿⣿⣿⣭⣾⡇
⠀⠿⣿⡿⢿⡿⠿⠿⢿⣿⠿⠀⠀⡩⠿⠿⠿⢇⠀⠀⠀⢀⣿⣬⣿⣿⣿⣿⣿⣿⣧⠀⠛⢿⣿⠋⠉⠻⠟⢻⣿⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠁⠀⠉⠉⠉⠉⠉⠉⠁

⢫⣵⢤⣤⣤⣤⣤⠀⣠⣬⡙⠿⢀⠙⠿⣷⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡠⠀⠀⣀⠐⠟⠁⠀⢠⠊⠁⠀⠀⠡⣀⣀⣸⡿⠀⢠⣤⠈⠉⠀⠀⠀⠈⠨⢀⢁
⠀⡝⢟⣿⢵⣿⣿⣿⣿⣷⡶⠀⠀⢰⣄⠘⠃⠍⠑⡀⠈⠇⠀⠀⠀⠀⢤⠀⠁⢀⠇⠀⡩⠤⠄⠄⠀⢀⠠⠀⠸⢦⠀⠈⣉⠉⠀⢠⣶⠿
⢦⣀⣀⠈⠙⠇⢻⣷⠈⣿⣿⡦⠀⠀⢑⡙⠄⠸⡀⠀⠀⠀⠀⢀⠀⠀⡈⠃⢠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⡐⢒⠉⠀⣀
⠻⠿⠟⠁⠀⣀⠀⠈⠀⠙⢋⠀⠶⠀⠈⢴⣤⠄⣀⣈⠀⠐⣤⡀⠀⡼⢇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢠⠘⢇⣴⡀
⢀⡆⠰⡄⠺⠿⢣⠀⠀⠀⠘⠁⠀⠀⢴⡄⠙⣀⣴⠋⠀⠀⠹⣤⣶⣤⣮⠤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠑⠊⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣡⠆⠚⠛⣦⢾⣿⣿⡛⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⢯⠿⣦⣀⡠⠤⠄⠀⠀⠀⠀⠀⠀⣠⣾
⣶⣲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⠞⠫⠰⠀⠚⠋⠈⣿⡿⠁⠲⠷⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⣽⡿⢷⣿⢋⣠⡤⡠⠀⢤⡶⠄⣀⣴⣿⢋
⡧⣷⣞⡷⠀⠀⠀⠀⠀⠠⠀⣸⣿⡆⠥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠀⠂⠀⠁⠀⠀⡀⡈⠙⠋⠁⠈⠛⠀⠀⠘⠉⠀⠈⠉⠉⠀⢸
⠛⠚⠋⢀⣀⣠⣤⠶⠆⠁⠀⢈⢉⢁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣶⣾⡇⠀⣤⡀
⠚⠛⠛⠛⠉⠁⣠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⠀⠀⠀⠑⠂⠀⠀⠀⠈⠉⠀⢠⣽⠗
⠀⠰⠂⣀⡴⠟⠉⢀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠰⠃⠀⠀⠀⠉⠢⠀⠀⠀⠈⠁
⠀⠒⠛⠉⢀⣠⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠈⢻⠓⢄⡛⠁⠐⠒
⠰⠵⠀⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⣀⢀⡐⣷⠄⠒⠲⠈⡆⠀⠉⢀
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣦⠉⠀⠀⠀⠀⠀⠀⠔⠂⢀⣈⣿⣿⣤⣿⣿⢷⣟⢷⢛⠁⠀⢠⠃⡀
⠀⠀⠒⠰⠟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠚⠀⡀⠀⣄⠀⣠⣽⣿⢋⣽⡿⢾⠏⠋⠁⠘⣠⢀⠠⠂⠀⠀⠁⠀⠰⠆⠐
⠀⠂⠀⠀⠘⠒⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠃⠀⠀⠀⡈⠆⠙⠻⠟⡘⠋⠁⠀⠀⠐⢾⣶⡷
⠒⠀⠀⠀⠀⠀⠰⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠠⠖⠂⠀⠀⠷⠀⡰⣧⠀⠀⢀⡀⠀⠀⠉⠀⠀⠠
⠀⠀⣉⣀⡀⢀⣤⡄⠈⠁⠀⠀⠀⠲⠶⠤⠰⢂⠀⠀⠠⡤⠀⠀⠠⣤⠆⠀⠀⠀⣤⣀⠀⠀⠀⠐⢶⠶⢤⣇⣤⣔⣨⠟⠀⢆⡀⢧
⣿⣿⢿⣿⣷⣿⣽⣶⡖⠀⠧⡀⠇⢀⡈⠑⠚⠁⠀⠚⠀⠀⠐⠈⠢⡀⠀⠀⠀⠢⠛⠛⠀⡯⢸⡆⢀⠔⠚⣿⣿⣿⣁⣤⣄⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⡠
⠚⠙⠽⣟⢿⣷⡌⣿⣿⡗⠀⠀⠀⠈⠘⠐⣶⣄⣠⠞⠀⠀⡄⠈⢢⢑⡄⠀⢠⡄⠐⠂⠀⢤⡍⠐⠛⠓⠈⣿⣟⠏⢿⡌⠁⠐⡦⠀⠀⠀⠐⠦⢤⣤⡴⠶⠚⣁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠶⠒⠒⠒⠒⠒⠒⠒⢶⣶⣒⣒⡲⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⡁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠉⠉⠀⠀⠀⠀⡼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠂⢤⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⡏⠀⠀⠀⠀⠈⠉⠒⡤⣄⡀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠁⠂⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠉⠑⠢⢄⡀⠀⠀⠀⠀⠀⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⡂⠀⠀⠀⠁⠒⢤⡜⢀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠄⢀⣀⠀⠀⠀⠘⠓⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⡇⣿⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠰⣇⣵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠀⠀⡇⠠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡌⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁⠀⡇⠀⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣧⣶⣷⠀⠀⠀⠀⡀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⡤⢷⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡁⠰⠀⠀⠀⡀⠀⠀⠉⠀⡤⠀⠀⠀⠀⠀⠀⢧⣾⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⢈⠂⢄⠀⠀⠾⠃⠀⠀⠀⠀⠐⠀⡄⠀⠠⡿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣌⡀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠘⠇⡇⠀⠀⢠⠃
⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣤⣤⣀⣀⠀⠈⠙⠓⠤⣐⡠⠐⢶⡄⠀⠀⠀⣣⠀⡠⠃
⠀⠠⣾⣿⣿⡟⣿⢻⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠉⠑⠒⠤⢤⣠⣼⣿⠔⠁
⠀⠀⣿⣿⣿⣏⣿⣣⣃⣕⣩⣼⣿⣿⣿⠇
⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠂⠁⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣀⡀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣬⣵⣶⣶⣶⣤⡀⠀⠉⠋⠙⠿⠟⠉⠀⠀⠀⠀⢀⢀⣠⣠⣀⣀⣤⣴⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠴⠶⠿⢟⣛⣫⣿⣿⣿⠛⠻⡹⣄⠀⠀⠀⠀⢢⡄⣀⣀⡤⠼⠛⣻⣿⣯⠿⠯⠽⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⣿⣍⣿⠶⠶⡶⠲⢶⡿⠿⠏⠁⠘⠿⠀⠀⢹⡘⣦⠀⠀⠀⢸⣿⣿⣽⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⣀⣶⡿⠁⠀⡃⠀⢂⠀⢀⠊⠀⠀⣸⠀⠀⡀⠀⠀⠟⠻⣿⠀⠀⡾⢸⠟⠉
⠀⠀⠀⠀⠀⠀⢠⠋⠀⢀⡼⠛⠋⠁⠠⡄⠀⠠⠈⠀⠐⠀⡀⠰⢧⠀⠀⠇⠀⣤⡠⠖⠉⠴⢛⣵⠋
⠀⠀⠀⠀⠀⠀⡀⠀⣰⠏⠀⠀⢠⡀⡀⠳⠄⣀⡦⠠⠀⠐⠀⢀⠠⠤⠐⠂⢉⣀⣠⣤⣴⣾⠿⠁
⠀⠀⠀⠀⠀⠀⠃⠸⡅⠀⠝⠐⠅⠀⠈⢀⠀⣘⣂⡀⠔⠋⡩⠔⠲⠾⠞⢙⣩⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⢠⠀⠻⣶⣶⠐⠀⠀⠀⠐⢁⠜⣁⣴⠂⠀⣠⣠⡤⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⣤⠶⢋⣤⣚⣥⠴⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠐⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠈⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠡⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠌⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠉⠈⠉⠟⠀⠀⠀⠀⠈⠀⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢀⣤⣤⠀⡆⠀⢀⠔⠀⠀⠀⠀⠔⣠⣤⣄⢠⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⡑⠛⠀⢀⣤⣾⣦⣠⣶⣦⡀⠀⠙⢂⣿⡆⠄⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠙⠻⠶⢿⠟⠋⠉⢈⡁⠉⠙⠻⡿⢶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠂⠄⠜⠣⠀⠄⠊⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠘⠛⠛⠛⠢⠀⠀⠀⠀⠀⠀⠀⡀⠘⠛⠛⠛⡄⠀⠀⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣀⣀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⠛⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠸⠉⠉⢹⠀⠯⠉⠉⠽⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠠⡀⠀⡈⠀⠰⠀⠀⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣠⣄⣠⣄⣀⣀⣥⣀⣇⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠸⠟⠋⠁⠀⠀⠀⠈⠙⠻⠷⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁

⠀⠄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠃⠓⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠂⠀⣒⣒⠲⣂⠀⠓⣀⣒⡂⠐⣖⣚⡀⠛⣂⣙⣂⡛⣆⠐⢂⣘⣳⠀⣲⣘⠃⠠⠓⠈⢻⠂⠛⢂⡼⠏⢻⡧⠛
⠂⢀⡒⠀⠀⢹⡆⠀⠘⢳⠀⠀⠉⣷⠆⠀⠈⠳⡀⠀⠒⣃⡀⠚⠃⡀⠘⠁⢠⡘⠃⠀⣬⡋⠀⢀⣍⠀⠀⣍
⣀⠘⠉⠀⠀⡈⠉⠀⠀⢘⣳⣀⠀⠉⢉⣥⡀⠈⠙⢀⣄⣉⠉⠀⢈⣹⡁⠀⠈⠍⠀⠀⠉⣁⣀⠀⠈⢀⣴⣤
⡛⢓⠄⠦⠀⢹⡓⠤⠒⠀⠍⢉⡁⠀⠈⣁⡈⠀⣀⠭⠉⢉⠁⡀⠈⢙⠋⡄⠈⠙⠣⠀⠀⢙⣙⡂⠄⠠⠉⠉⡄
⣿⣿⣾⣷⣶⣾⣿⣷⣾⣾⣾⣿⣿⣿⣿⣿⣷⣶⣿⣷⣿⣾⣿⣿⣷⣾⣿⣷⣿⣿⣿⣶⣿⣷⣿⣿⣿⣿⣿⣾⣿
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⢤⠤⠀⠀⠀⠠⠤⣤⣀⣤⣤⡀⠀⠀⠄⠀⠀⠀⠤⠄⠀⢀⣀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣦⣤⣶⣷⣿⠶
⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣟⣽⣭⣒⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⡿⡍
⠀⢀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣷⣿⣘⣨⣿⣊⣻⣽⣍⣡⣤⡄⣠⡶⢾⣷⣀⠀⠀⠨⡯⠃⠀⠀⠀⠀⣀⡀
⠀⢈⣤⣀⣀⡈⣉⠛⢿⣿⠿⣿⣿⣿⡏⣽⡯⣻⣿⣿⡿⣛⣛⣛⣶⠮⠵⢿⣿⠋⢉⣐⣿⣿⡗⠂⠈⠉⠉⠛⠃
⠀⢠⣤⣤⣤⣉⣩⣉⣒⣶⣾⣿⠿⠿⠇⠻⠓⣻⣟⠛⠛⠛⠛⠋⠛⠂⠀⠸⠿⠿⠿⢿⣻⣭⣽⣤⣤⣤⣄⣠⡄
⠀⠘⢟⡛⠋⡉⠉⠉⠀⣤⣴⣶⠀⠀⠀⠀⠀⢹⣿⣥⣤⣴⢤⠶⠒⣶⣤⠖⢠⣤⡄⢠⣽⡿⠛⠿⠿⠟⠛⠛⠃
⠀⢠⣾⣷⣶⣷⣴⣷⣾⣌⠻⣇⡀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡈⣿⣿⡇⣶⠀⣿⣷⡆⢁⣾⣶⣮⣠⣤⣠⣦⡆
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠦⡤⢼⠄⠀⠀⠀⠙⠻⣿⣷⣇⡷⢹⣿⣤⣿⣭⣿⣬⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣄⣀⣀⣹⣿⣀⣤⣾⡄⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣾⣿⣿⣿⣾⣿⣿⣿⣟⣿⣻⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠙⠛⠋⠉⠉⠉⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠆⠙⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠀⠀⠀⠀⢀⡠⠚⣩⣇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⢋⣡⡴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠓⢀⢐⡉⠚
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠵⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⣴⠁⣀⠔⠈
⠀⠀⠀⠀⠀⠀⢠⣾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡿⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠠⠴⠞⠋⠀⠀⠀⠀⠀⠀⢀⣒⣼⢦⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠳⠇⠿⠇⠟⠁⠀⠀⠀⠀⠀⠀⢠⡠⠀⣐⡤⠤⡂⠀⠀⠀⠀⠀⢀⠀⣀⡀⡀⣀
⠀⠀⠘⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢌⣐⠀⠀⠀⠁⣈⠀⠀⠀⠀⠰⠈⠀⠁⠀⠁⠀⠁⠐⠦⢀
⠀⠀⠀⠀⠢⡀⠀⠀⠀⠱⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⠉⠚⣻⠧⡀⠀⠀⣈⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⢿⠋⠀⠀⠀⢣⡌⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⣯⠏⠁⡀⠀⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠁⠨⣯⣆⠹⣆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⡆⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⠀⠹⣧⡙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀⠠⢄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠀⠙⢿⣾⣆⢄⢀⠀⠠⠖⠂⠈⠀⠀⠀⠀⠰⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⡠⠈⠛⠣⣂⣀
⠀⠀⠀⠀⠀⠄⠐⠁⠁⠐⠆⡀⠀⠀⠀⠙⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠫⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠛⢿⣷⡀⠀⠀⠀⠀⢠⠀⠠⠀⠄⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠀⠀⠀⠀⠤⣀⠀⠈⠉⡁⡀⣀⣹⠀⠀⠀⠰⠇⠀⠀⠀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⠩⠈⡅⠱⠀⠀⠀⠀⠀⠀⠰⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠦⠀⠀⠀⠠⠠⠧⠚
⠀⠀⠀⠀⠀⣰⠤⠀⠀⠀⠀⠀⠀⢰⣰⡶⠄⠁
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⢰⠋⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⡈⠒⠄⢶⣦
⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢄⣀⠀⠀⠀⠀⠀⠀⠀⢠⣬⡁⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠥⠬⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠛⠒⠾⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠒⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠖⠀⠀⠀⢀⣿⡇⠀⠈⣷⠴⠦⡀⠀⠀⠀⠀⢰⣿⡀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢲⣶⣶⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣷⣾⣷⣄⡀⠀⠀⠀⠀⠐⢢⢠⣼⣷⣤⣀⠀⢢⣐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⣿⡿⠿⠝⠛⢷⣦⣄⣀⢀⠀⠀⠹⡏⠈⠙⢿⣿⣆⠹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⣀⣀⣀⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠉⠛⠛⠿⢶⣤⡈⠂⡀⠀⠙⣿⣷⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠈⠻⢿⣷⡤⡀⠀⠈⠙⠛⠁
⠀⠀⠀⡔⠀⣀⣠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠴⣢⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢀⣠⣶⣄⡁
⠀⠀⠀⠀⠀⠈⢩⣽⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠠⠄⠀⠐⠄⠐⠛⢉⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠏⠠⠁
⠀⠀⠀⠀⠂⠈⠙⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠂⠹⣿⣿⣿⣿⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠠⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠂⠿⠧⠴⢶⡶⣷⣶⣿⣿
⠀⠀⠀⠀⠀⠀⢈⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣼⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠠⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡜⠁
⠀⠀⠀⠈⠐⠒⠒⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠰⠶⠒⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⣿⣷⣠⣾⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣈⠋⠈⢛⣛⣋⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠋⠀⠈⢻⣿⠟⠛⢿⣄⡀
⠀⠀⠀⠀⠠⣷⠒⣠⣄⣀⠀⣀⡤⣢⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠄⠀⣀⣠⡤⠀⠀⠀⠀⢨⣿⣅⣀⡀⣈⡿⠗
⠀⠀⠀⠀⠀⠘⠆⠉⠊⢙⠛⠻⠿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠀⠐⠛⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠚⡸⢛⣧⣴⣧⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠄⠀⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⠀⢀⡀⣀⣠⣴⢶⡝⠉⢫⠝⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⢶⣦⣔⢾⣿⣶⣷⣾⣷⣾⣿⣿⣿⣿⣷⣶⣿⣿⣿⡿⣿⣽⡧⢠⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠻⣙⠖⢴⠉⠻⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣽⠟⠓⡇⠀⠸
⠀⢀⣀⣀⣀⣀⣀⣴⣄⣠⣼⣦⣾⣦⣤⣿⠿⣻⡿⣿⣿⣿⣿⣿⣿⡏⢸⢿⢿⣿⡿⣿⣟⣳⣒⣒⣷⣠⣰
⠀⠈⠁⠈⡉⠉⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⣀⣀⣀⣀⣀⣀⣤⣤⣼⣾⣿⣽⣯⣿⡏⢽⣿⡿⣿⣻⣿⣻⣷⠂
⠀⠀⠀⠀⠐⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⡆⠾⠿⢿⡿⠶⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣧⡜⠻⣿⢿⡿⣿⠙⡿⠛⠛⠀⠁
⠀⠀⠀⠀⠀⠈⢾⡏⣨⣿⣿⣿⣿⣿⣿⣿⡇⢉⣉⣉⣉⡉⣩⣭⣭⣌⣹⣶⠖⣰⣾⣿⣿⣿⣶⣷⣿⣀⣿⠏
⠀⠀⠀⠀⠀⠀⢿⣿⠋⡟⢷⡀⢸⣿⣿⣿⡇⣻⣿⣿⣿⣿⣿⣦⣶⡛⢷⣾⣶⣿⣿⣿⣿⣿⢿⡟⠛⠉⣿
⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣾⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿
⠀⠀⠐⢲⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡇⠿⠿⠿⠛⠛⠛⢋⣩⡉⡭⠥⠧⣗⣛⣛⣛⠩⠭⠿⠿⠛⠛⠃
⠀⠀⢲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢀⣀⣀⣤⣤⣤⢶⠖⠚⠋⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⡤⠤⠤⠶⠶⠒⠚⠋⠉⠉⣡⠞⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠤⠴⠶⠶⠶⠒⠚⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢸⡿⠥⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⠉⠛⠲⠦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠿⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⢦⣤⣄⣀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡴⠋⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡇⠀⠀⣾⡷⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣞⣱⠆⠀⠀⠀⠀⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⣅⣠⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠺⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠲⠶⢤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠦⣤⣤⣀⣀⣀⣿⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⢀⠄⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠉⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠄⠐⠤⠀⠴⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⡌⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠐


⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢰⠄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⡊⠀⠈⡆⠀⠀⠀⠀⠀⢀⠀⢤⠈⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡏⠢⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⣀⣄⣀⢤⠔⠄
⠀⠀⠀⠀⠇⠀⠀⠐⢠⡤⢐⠂⠉⠀⠀⠀⠒⠀⠀⠁⠂⠄⠀⠀⣀⡀⣠⠔⠈⠔⠠⠠⡓⠄⠈⢄⡠⠄⠀⣀⢄⣠⠊⠒⠦⠠⠀⠀⠓⠊⠀⠀⠉
⠀⠀⠀⠀⢠⠀⠐⠀⢨⠑⠢⠌⡁⠒⠀⠄⠀⠀⠀⠀⠀⠄⠂⠈⠤⠿⠃⠀⠁⠸⠀⠀⠈⠢⠀⠀⠑⠀⠲⣁⣀⣠⣤⣤⣤⡤⠖⣶⢒⡲⣖⣲⣒⠒⠐⠄
⠀⠀⠀⠀⠀⠒⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠚⠀⠖⠈⠀⠀⠀⠀⠀⠈⠢⣀⠂⠀⠀⣀⡠⠔⣂⡥⠙⡊⠕⠋⡁⠔⠊⢁⠠⠐⡠⠒⡁⠔⡡⠔⠉⠩⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠤⠀⠘⠁⠀⠐⠳⠆⢉⡅⠊⢁⠠⢰⣍⣠⠔⢁⡠⠐⠈⡠⠐⢁⠄⠊⡠⠊⠀⢀⠔⠁⡐⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠬⢑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢺⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢰⠃⠀⢰⠆⣁⡠⠒⢁⠠⠂⠁⠔⠁⠀⡠⠂⠁⡠⠈⢀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠑⡀⠁⠂⠄⡀⠀⠀⡀⠀⠈⠉⣨⠄⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⣠⣰⠀⠈⢉⡘⡧⢤⡊⠁⡀⠔⠁⠀⡠⠊⢀⠔⠁
⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠂⠒⠒⡈⠁⢀⠀⠈⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠠⠑⠋⠈⡁⠠⠀⠂⣉⣙⠈⠱⠒⡰⠣⡄⣠⠂⡠⠔
⠀⠀⠀⠀⠀⠈⠛⠓⠒⠒⠒⠃⠀⠀⢀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠈⢉⡁⠠⠂⡰⠀⠀⠀⠀⠀⢀⠠⠒⠉⢄⡒⠢⣌⣉⠒⠩⢴⡒⢬⠉⡑⢄⡈⣑⠒⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⢀⡧⠀⠒⡿⠀⠀⠀⠀⠀⠀⡀⠄⠒⠈⠀⠠⠒⠈⠀⠀⠐⠀⣠⢊⡀⠎⡉⠒⠠⢍⣉⠒⠒⠅⣓⠢⠬⠢⣒⠆⢅⠙⠠⡁
⠀⠀⠀⠀⠀⠘⡗⠀⠁⠀⠀⡇⢀⣠⣿⠀⢀⠤⠐⠀⠁⠀⠄⠂⠁⠀⠠⠀⠀⠀⡠⢒⢊⠣⣈⠦⡑⡂⠚⢷⡤⢀⡈⠓⠢⢤⡁⠒⢄⠈⠣⢄⠈⠓
⠀⠀⠀⠀⠀⠀⡇⠀⢀⡠⠰⡏⢁⡠⠚⠀⠁⠀⠠⠂⠁⠀⠀⠀⠀⠀⠀⢀⣤⣊⣀⠛⠩⣉⠲⠤⡀⡉⠂⠤⣍⠢⢉⡐⠢⠄⢈⡃⠂⢌⡐⠤⡁⠒⢀
⠀⠀⠀⠀⢀⡠⡇⠀⡁⣀⠤⠊⠁⠀⢀⠠⠂⠀⠀⠀⢀⣀⡄⠂⠀⠰⠐⢈⣘⠆⠁⡑⠦⠄⢉⡒⢄⢉⠒⠤⢈⠐⠢⠬⡉⠒⢢⡈⠒⠠⢈⠑⠢⢕⡂
⠀⠀⠀⠀⠀⢀⣡⠆⠙⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢤⠌⠀⠀⣠⠟⢁⡘⠉⡀⠱⡄⣈⠑⠂⣤⠜⣠⢄⠉⠲⢄⡙⠢⠀⡈⠢⢄⡉⠢⢀⡀⠉⠠⢀⡉
⠀⠀⠀⠀⠴⠛⠁⠀⠀⡀⠈⠐⠀⠀⠀⠀⠀⠀⠘⠀⢰⣤⠞⠃⢾⡁⠈⠢⣌⠒⢄⢀⠙⠂⢄⠀⠘⢄⠑⠢⠤⡈⠢⣄⠉⠢⡀⠈⠉⠂⢌⠠⢄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡞⠀⡎⠉⠦⡀⠑⠠⡀⠑⠆⣀⠙⠢⡀⠙⢤⡀⠉⠒⠢⣈⠒⠤⠉⠒⣌⡒⠤⡀⠀⠈⠢⣉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠒⠀⠉⠒⠀⠘⠂⠀⠀⠑⠀⠈⠐⠀⠈⠂⠀⠀⠈⠀⠀⠂⠀⠀⠁⠀⠉⠂⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣯⠤⠤⠤⢤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣯⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣶⣶⣭⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣁⣀⣀⣠⣟⣷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⠉⠉⠉⣿⣿⠷⠛⠛⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⢼⣧⠞⣉⣙⠲⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⣿⣷⠊⠁⠈⢳⡿⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⣼⣿⠀⠀⠀⢸⣷⠃⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠙⣿⣦⡀⠀⣼⠟⡴⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⣶⣿⡙⠛⢛⣡⠞⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⣿⣿⠙⠻⠿⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣸⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡟⠀⠀⠀⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡇⠀⠀⠀⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢻⠃⠀⠀⠀⢸⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢣⡟⠀⠀⠀⠀⠀⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡴⠋⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⠅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠤⢷⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢉⡟⠉⠁⠒⠒⠒⠒⠒⠒⠒⠒⠂⠉⠉⢉⡉⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⢀⣠⠤⣒⣬⠿⢛⣻⡿⠁⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡅⠀⠀⢀⣤⣾⡭⠖⠋⣉⣤⣶⣿⣯⣤⣤⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣠⣾⣫⣥⣶⣾⡿⣿⣟⣉⡭⠞⠋⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⣿⠟⠹⣿⡻⡄⠀⢸⣿⡇⠀⠀⠀⠀⣿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡏⢧⠀⠻⣿⣷⠶⢚⡼⠃⠀⠀⠀⠀⣿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⢿⣌⣳⣶⡿⠉⠉⠉⠀⠀⠀⠀⠀⠀⡏⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡷⠶⠶⣶⣶⣶⣶⣶⣶⣖⣒⣖⠶⠶⠿⣽⠟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢧⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠴⠞⣃⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛⠛⠓⠒⠚⠛⠛⠛⠃⠒⠒⠒⠒⠚⠉
⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣠⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⢠⡶⠛⠛⠛⠻⡷⠶⠚⠛⠛⢶⣤⠶⠶⣦⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠉⢿⠀⢠⡶⠚⠳⣾⡇⠀⠀⠀⠀⣇⠀⠀⠀⠀⢸⠅⠀⠀⣿⡟⠉⠛⠛⢶⡄⠀⠀⣠⣄⣀
⠀⠀⠀⠀⠀⣤⠶⠛⠳⣄⢹⡆⠀⠀⠈⣇⢸⡇⠀⠀⢹⣇⠀⠀⠀⠀⣿⠤⠒⢤⠤⠼⢦⡄⡤⠿⠤⠤⠄⠀⢸⠇⢀⡾⠁⠈⢹⠇
⠀⠀⠀⠀⠀⢹⡀⠀⠀⢹⣆⣷⠀⠀⠀⠘⣦⡇⠀⠀⠈⣿⠤⠴⡄⠠⡄⠀⠀⢸⠀⠀⠀⡇⡇⠀⠀⠀⡃⣠⣾⣷⣾⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠘⣇⠀⠀⣠⠟⠛⠛⣇⠀⠀⢹⣿⠀⠀⢸⠁⠀⠀⢿⠚⠓⠒⣷⠸⠀⠀⠀⢸⡇⠀⠀⠀⠇⡇⠀⠀⠘⠀⠀⡟⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⢹⡀⠀⡇⠀⠀⠀⢸⢰⠟⠋⠛⢳⠀⢸⠀⠀⠀⢸⠀⠀⠀⡏⠀⡀⠀⠀⢸⡇⠀⠀⠀⢀⡇⠀⠀⡀⠀⢰⠁⠀⠀⣸⠃
⠀⠀⠀⣠⠖⠻⡆⢧⠀⡇⠀⠀⠀⢸⡾⠀⠀⠀⢸⢀⣠⣀⣀⡀⢸⠀⠀⠀⡇⠀⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⡁⠀⡸⠀⠀⢠⡏
⠀⠀⠘⣇⠀⠀⢻⣸⣄⣇⠀⠀⠀⢸⡇⠀⠀⠀⢸⡟⠁⠀⠈⡇⢸⡇⠀⠀⡇⠀⡇⠀⠀⠈⡇⠀⠀⠀⢸⡇⠀⠀⡇⠀⡇⠀⠀⡼
⠀⠀⠀⢻⠀⠀⠘⡏⠉⠉⢧⠀⢀⡼⠷⡆⠀⠀⢸⡇⠀⠀⠀⡟⠋⠙⡇⠀⡇⠀⢰⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⡇⢰⠁⠀⢰⠇
⠀⠀⠀⠘⡇⠀⠀⢃⠀⠀⠸⠀⢸⠀⠀⠁⠀⠀⢸⡇⠀⠀⢐⠃⠀⠘⠁⠀⡇⠀⢸⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⡇⡼⠀⠀⡾⠷⠶⢶⣶
⠀⠀⠀⠀⢹⡀⠀⢸⡀⠀⠀⡆⢸⠀⠀⢸⠀⠀⢸⡇⠀⠀⢸⠀⠀⢸⠀⠀⡇⠀⢸⡀⠀⠀⢸⠀⠀⠀⢸⠃⠀⠀⣿⡇⣀⡴⠷⠚⠉⠁
⠀⣀⣀⣀⣈⣧⣀⣈⣧⣀⠀⠇⢸⠀⠀⢸⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⣄⡤⠷⠴⠾⠷⠶⠒⠛⠓⠚⠛⠛⠛⠛⠛⡟⠉⠁⠀⠀⠀⠀⠀⢰
⢻⡉⠉⠉⠉⠉⠀⠀⠀⢸⠀⢸⣸⠀⠀⢸⠀⠀⢸⠁⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠘⣿⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠉⠀⠀⠀⠀⢧⠀⣿⠀⠀⢸⡀⠀⢸⠀⠀⠀⢸⡆⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠈⠣⣼⠀⠀⠈⡇⠀⢸⠀⠀⠀⢸⣇⡼⠁⠀⠀⠀⠀⠀⠀⠀⠐⠄⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠈⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠴⣧⣄⣼⣄⣤⠴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡴⠋⣿
⠀⠀⢿⠀⠉⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⣠⠞⠀⠀⡇
⠀⠀⢸⡄⠀⠀⠀⠈⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡼⠁⠀⠀⣼⡇
⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢾⡄⠀⢠⠞⢹⠇
⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠸⡉⠑⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠊⠀⢠⠏⠀⢸
⠀⠀⠀⢻⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡇⠀⠀⠀⠀⠈⠙⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠃⠀⠀⣹⠇⠀⣾
⠀⠀⠀⠸⡇⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣀⡀⠀⠀⠀⠀⢐⡀⠁⠀⠀⡰⠃⠀⠀⡇
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀⢸⠃⠀⠀⡐⠁⠀⠀⢸⡇
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠈⠓⠤⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡏⠀⠀⡐⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⠀⠀⢏⠉⠉⠒⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣰⠁⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢾⡄⠀⠀⠀⠀⠀⠈⠙⠒⠤⣄⡀⠀⠀⠀⠀⢰⠀⡴⠁⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⢸⡔⠁⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⢀⣠⠶⠛
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⣤⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡴⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣛⣛⣛⣻⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣞⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠋⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⢰⠀⠀⣿⠀⢰⠇⠀⣰⠂⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠙⢦⠀⠹⡀⠀⠘⡇⠀⡏⠀⡼⠀⢠⠃⠀⢀⡞⠁⠀⠀⢀⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⣄⠀⠙⢦⡀⠳⡀⠘⠀⠀⠃⠀⠃⠀⠃⢀⠃⠀⡠⠋⠀⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⡀⠉⠢⢄⠙⠦⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠔⠁⢀⠴⠊⠀⠀⠀⠀⣀⣤⠴⠾⣭⡀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡽⠖⠒⠲⣤⣀⡈⠁⠀⢠⡴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠤⠔⠚⠩⠋⢀⡀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠹⠞⠙⢆⢸⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠖⠂⠀⢼⣧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⣦⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣐⣶⣲⣆⣾⣾⣿⣦⣼⣇
⠀⠀⠀⠀⠀⠀⠀⢠⡀⢼⣆⠀⠀⠀⠀⠐⢰⠑⢺⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣇⣀⣀⣀⣼⣿⣫⣿⣿⣿⣿⣿⣿⣷⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣿⣶⣶⣤⣄⣠⣯⡴⣯⠷⡶⢶⠗⡶⠲⡶⢦⡤⣤⣤⣦⣼⡀⢸⣯⣿⡽⠷⠿⣟⣛⣛⣿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⡟⣿⢩⠇⣿⢁⡏⢠⡏⣼⢁⡇⣸⠁⡾⢠⠏⠴⠡⠞⣱⠯⢛⣫⠥⠖⣂⣩⡭⠤⠔⠒⣒⣒⣒⣒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇⣸⢰⡇⣸⠁⠾⠀⣋⣨⡤⢤⠲⡖⢦⠒⡺⠖⣉⠔⢊⡩⠔⣚⣩⠥⠒⢒⣉⣉⡭⠭⠭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠀⣉⣴⢲⡙⣏⠳⡜⢆⠳⠈⡠⢊⡠⢊⡠⠚⣩⠔⢋⣡⠴⠚⣋⡩⠤⠶⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠂⠷⠘⡆⢱⡘⢠⠎⡠⢋⡴⢋⡴⢋⡤⠚⢉⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠉⠀⠉






⡿⢿⠿⠿⠿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣰⣄
⠀⢀⣿⡿⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⡗⡆⠀⢀
⠀⠀⣿⡇⠀⠙⢿⣦⠒⢲⢲⠀⡀⠀⠀⠀⠀⣀⠀⣠⣄⣀⣇⣌⣀⣿⣰⣦⣄⠀⠀⣀⣠⣤⣶⡶⠂⠠⣤⣾⠒⠲⣷⣷⣶⣿⡆
⠀⠀⣿⠁⠀⠀⠈⡿⠤⢾⣼⣿⣿⣿⠄⠀⠀⣿⢯⣿⣿⣿⣿⣧⣤⣿⣿⣿⣟⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⡧
⠀⠀⣿⣿⡟⣠⣤⣴⣶⣾⣿⣿⣿⣽⠀⠀⠸⠿⠾⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠈⢿⣿⠟⠉⠙⠋⠻⣿⡿⠋⠁
⠀⠀⣈⣉⣀⣙⣛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⢀⡀⣀⣀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⡤⡄
⠀⠀⢀⣌⣉⣋⣿⣦⢤⠤⡤⡤⡤⢤⢤⠀⠀⠀⢸⠀⣸⢀⣿⣇⠀⡀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣀⣸⣧⣧⣀⣰
⠀⠀⣾⣿⣿⣿⣿⣿⣀⣆⣥⣴⣼⡞⠛⠁⠀⠀⠘⠉⢻⣿⣿⣉⣏⣿⣹⣹⣏⡟⠀⠀⠀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢨⣶⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠘⠿⠿⠽⢿⠿⠿⠯⠿⣿⣿⣿⡿⠿
⠀⠀⠉⠁⠻⣿⣿⠟⠀⠀⠻⣿⣿⠟⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉
⠀⠀⠀⠀⣠⢤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢺⠛⠙⠙⢹⡇⠀⠀⠀⠀⠀⠀⢰⣹⠆⡷⢪⡗⢸⠖⡂
⠀⠀⢰⣶⣿⣸⣧⣀⣼⣹⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⣾⣏⣿⡴⠄⢀⣾⡇⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡁
⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣟⣉⣉⡟⠉⢹⡧⠀⠀⠀⠀⠀⣿⣿⣿⣅⣸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠘⠉⠉⠉⠉⠉⠉⠉⠁
⠀⢸⡿⣿⣿⡿⠇⠹⠿⣿⣿⡿⣿⠟⠀⠀⠀⠀⠀⣀⣀⣠⢿⣉⡿⠿⠿⠿⠿⢿⣻⠇⠀⠀⢠⡄⠀⠀⣆⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠈⠉⠀⠀⣀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣀⣠⣬⣥⡤⠀⠀⣾⣧⠀⢐⡳⠀⢸⣯⣉⣉⣹⡇
⠀⠠⡆⠐⠒⢲⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠴⠶⢶⣆⠀⣠⣾⣿⣿⣿⣿⣿⡅⠀⠈⠉⠉⠀⠉⠉⠁⠈⠉⠉⠉⠉⠁
⠀⠀⢱⡾⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢳⡦⢤⣼⣧⣤⣿⣿⣿⣿⠿⢋⣿⡅⠀⠀⠀⠀⣄⠀⣀⢀⡀⢀⡀⢀⡄
⠀⠀⢸⣿⣿⣿⣿⣭⣼⣭⣿⣿⣿⣿⣿⠇⠀⠀⣸⣷⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢠⣿⡀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠈⠛⠁⠀⠁⠀⠈⠉⠁⠙⠋⠀⠀⠀⠉⠉⠙⠛⠁⠉⠉⠁⠙⠛⠈⠛⠃⠀⠀⠀⠀⠚⠛⠃⠀⠀⠀⠀⠀⠚⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢈⣙⠻⢦⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣾⣶⣟⣉⣈⣭⣿⣿⣿⣿⣯⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⠟⠉⢭⣽⣿⣿⣿⣿⣿⡇⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠃⠀⠀⠐⣺⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣯⠁⠀⠀⢤⣼⣿⣿⣿⣿⡭⠪⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠀⠀⠀⢒⠺⢿⣿⣿⡋⠐⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠀⠀⠀⠀⡀⣸⣿⣿⠿⡉⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠈⣙⣿⡿⠓⠚⠀⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢀⣀⣂⣈⡉⠉⠁⠀⠀⢴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣴⣿⡿⠇⣠⠄⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠛⠉⢀⣤⡾⠃⠀⣠⣴⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⣀⣤⣶⡿⠟⣡⣶⣿⣿⠏⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠁⢀⣾⡿⠛⢉⣐⣿⣿⣿⠿⣡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠚⠉⠀⠀⢀⣤⠞⠋⠀⣀⣼⣿⣿⣿⣣⠞⠁
⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠖⠒⠋⠉⠀⠀⠀⢀⣠⣤⡤⠈⠉⠀⠠⢀⣨⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⣤⣾⡿⠋⠉⣀⣠⣤⣶⣶⣾⣿⣿⣿⣟⢛⣋⣀⣤⣶⣾⡿⠟⠋⠁
⠀⠀⠀⠰⣿⣏⠠⠚⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠋⣉⡉⠛⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠺⠟⢀⣝⠁⠺⠗⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⠤⠤⠾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠶⠃⠹⣀⠀⣀⣄⡀⣺⠙⡄⠀⡠⣼⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠉⠛⠂⠀⠉⠁⠀⠙⠒⠃⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⠖⢒⣲⠀⡔⢲⡀⢠⠶⡄⡔⠒⠒⠲⡄⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠘⠦⢌⠓⣤⠁⠇⢣⢸⠀⣇⡉⡇⢸⠉⠁⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣴⣒⣚⡠⣏⡰⢧⣈⣾⣀⣀⡙⣇⣸⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠤⠤⢤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠤⠄⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢰⣇⠀⣴⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⣿⣄⡀⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡀⠀⢠⣶⣾⣿⣿⡍⡅⠀⠀⠀⢠⡀⣼⣟⣩⣁⡀
⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠀⠀⢿⣿⣿⣿⣧⣗⠀⠺⣿⣄⠀⢹⣿⣟⣋⣠⡐⢀
⠀⠀⣴⡈⢤⠄⠀⠀⠀⠀⠈⠁⠀⠉⠀⢻⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣟⣁⣀⣿⡟⢩⢹⣿⣟⣟⠩⢉⣯⡤⠆
⠀⠶⠿⣭⡤⠶⢖⠉⠉⠉⠁⠈⠃⠋⠉⠙⡛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⠛⠋⠀⠈⠙⠛⠛⠐⠊⠐⠀⠉⠀⣀
⠀⠀⠀⡔⣤⠡⢄⠀⣤⣀⡠⣤⠀⡠⠤⢠⢄⢄⠀⠀⠀⢀⣠⡄⢠⢀⠀⠀⠀⠠⣀⢀⡀⢀⡀⡀⡀⢀⠀⠀⡀⠀⠀⠀⠁
⠀⠀⢹⡶⡌⢱⢛⠀⣿⡌⠉⣢⡽⢇⣦⡏⢀⢜⣮⣆⠀⢇⢁⠡⣉⣬⢠⣴⡧⣌⡅⣚⡤⣡⡶⠗⠠⠄⠀⠵⢹⣅⡠⡄
⠀⠀⣹⣽⣇⣷⢡⠲⣦⡆⢼⡣⡇⣻⡀⣧⠨⠞⢸⡟⠻⣤⡘⢹⣫⡍⣊⡡⢿⢳⣼⠽⢿⠾⠃⢜⡋⠦⠟⠃⢨⣏⣀
⠀⠀⢸⡇⡏⢾⢸⡟⢫⡇⢻⡟⣇⢺⣆⡟⢻⡯⢱⡏⠿⡳⢻⢾⣷⡄⣿⢠⣬⡀⢸⡷⣺⣧⡴⣦⢶⡖⡐⢺⣫⠌⠐
⠀⠀⣹⣏⡉⣺⠘⠒⢸⡇⢤⠮⣯⣼⠇⣷⢸⣋⣸⠎⠑⡟⡴⠛⢻⠂⠿⠠⠾⠉⣸⣏⠟⣻⣐⡗⣻⡿⠆⣶⠡⡆
⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠚⠷⠤⠤⣭⣤⣄⣹⡅⠉⠈⣿⠀⠉⠃⣛⠄⠣⠀⢸⡇⢀⣹⣎⡟⣹⠛⠁⡿⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠛⠒⠒⠦⠿⣀⣀⡀⢸⡇⠀⠀⢸⡏⠈⠐⠋⡟⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠓⠒⠓⠠⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠒⣾⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⡆⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣴⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣶⣶⣶⣶⣿⣇⠀⠀⢶⣶⡶⠶⠶⠶⠶⠶⠲⢶⣥
⠀⠀⠀⠀⠀⢠⠾⠿⢿⣿⣿⣿⣟⣟⣿⣻⣿⣿⣿⠛⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⣟⠛⣻⣟⠛⣛⠿⣷⣦⣝⣿⣦⡀⣀⣀⣠⣤⣬⣿⣆
⠀⠀⠀⠀⢠⠃⣀⣴⣿⡿⠿⠿⢷⣶⣶⣾⣿⣿⣿⠀⠀⠀⠀⢀⣾⣿⡿⢋⣤⣼⣿⣿⣿⣿⣿⣿⣧⣬⡝⢿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⣾⣿⢟⣩⣾⣶⣿⣷⣾⣦⣙⠻⣿⣿⣿⠀⠀⠀⠀⣾⣿⡿⣡⣼⣿⣿⠟⠉⠭⠬⠍⠛⢿⣿⣷⣄⢻⣿⣿⣦⣤⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢸⡿⣡⣿⣿⠟⡋⠉⠭⡛⢿⣿⣧⡜⣿⣿⣤⣤⣤⣼⣿⣿⠃⣹⣿⣿⢡⠋⠀⠀⠀⠀⠘⡎⢿⣿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⣿⣾⡇⣽⣿⡇⠈⠀⢀⡀⠀⠆⢿⣿⣧⢸⣿⣶⣶⣶⣾⣿⣿⠘⣿⣿⣇⢠⠀⠀⠘⠃⠀⠀⡀⣸⣿⣿⠇⠀⠀⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠿⠿⠏⢻⣿⣇⠃⠀⠈⠀⢀⠀⣾⣿⡏⠸⠿⠶⠶⠶⠾⠿⠿⠀⢿⣿⣿⣎⠣⠀⣀⣀⠀⠜⣡⣿⣿⣿⠀⠀⠀⠀⠀⠰⣦⣄⣤⡿⠛⢻⡟
⠀⠀⠀⠀⠀⠙⢿⣿⣦⣝⣂⣀⣭⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣶⣬⣥⣶⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠿⠿⢿⡿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠿⠿⢿⣿⠿⠿⠏⠉

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠒⠒⡒⢲⠐⠠⢄
⠀⠀⠀⠀⠀⠀⠀⣸⣤⣤⠀⣉⣈⣃⣀⣤⣶⡇
⠀⠀⠀⠀⠀⠀⠀⢦⣤⣤⡄⢀⣀⣠⡩⢵⣤⡇
⠀⠀⠀⠀⠀⠀⠀⢘⣻⣿⠧⠤⡤⠼⣿⣾⣻⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣹⠋⠁⠀⠀⠀⠀⠉⢭⣇
⠀⠀⠀⠀⠀⠀⢠⠓⡇⠀⠀⠀⠀⠀⠀⠀⠘⢟⣆
⠀⠀⠀⠀⠀⢠⠏⠀⠀⡀⠀⠀⠘⣷⡦⣤⡁⠐⠿⢄
⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⢉⠉⠈⠩⢷⣦⡘⡄
⠀⠀⠀⠀⢸⣷⣤⡀⠀⠀⠀⠀⡀⡞⠀⠀⣨⣒⣸⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠞⢼⢢⡦⢀
⠀⠀⠀⠀⠘⣿⠙⠟⠱⠶⣶⣟⣾⣟⣵⣿⣿⣿⣟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⠌⠀⠫⡀⠄⢠⣀
⠀⠀⠀⠀⢠⢗⠀⠀⠀⠀⠀⠀⡉⣛⢠⠮⣹⣿⡏⢹⣿⠀⠀⠀⠀⠀⠐⠀⠊⠁⠀⠀⠀⠀⠀⠁⠁⡀⠰⢦⣄⠀⡂⢠⡀
⠀⠀⠀⠀⠘⡜⣴⠀⠀⠈⠀⢀⣈⠉⠙⠳⣿⣻⢉⣾⠏⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠨⠨⣆⣄⠁⡀
⠀⠀⠀⠀⠀⠘⢿⣿⣆⣀⣀⣼⠡⠀⣠⣿⣿⣻⡾⠋⠀⠀⠀⠈⠀⠁⠀⠀⠄⡄⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠌⠒⡍⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡯⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠂⠠⢠⠠⠁⠀⠀⠀⠈⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣶⣶⣶⣿⡟⠛⠛⠛⣿⣸⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠤⠴⠟⠛⠛⠛⠛⠒⠋⠉⢹⣻⡟⢻⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⢹⡇⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⡇⠀⠉⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⣿⠀⠀⠀⠀⠀⣿⣷⣀⣀⠀⠀⢸⣿⣧⡼⠛⠛⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡟⠀⠀⣀⣀⣠⣿⣿⣿⣿⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠾⠇⠻⠿⠟⠛⠛⠛⢻⠀⣾⢿⣸⣿⡇⠀⢀⣀⣀⣀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⢛⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣾⣦⣿⢸⣺⣿⣇⣀⣸⣏⣭⣭⣤⣿⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣿⡏⢉⣩⣸⢿⣿⣿⣛⡿⠋⣼⣿⣿⣿⠏⣻⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⢹⡇⣿⣿⣿⢺⣿⣿⡿⠁⣼⣿⣿⣿⠏⣴⣿⣿⣷⣤⣨⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠶⠶⠶⠶⠶⠶⠾⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀⢸⣇⣿⣿⣻⣾⣿⡟⠀⣼⣿⣿⣿⠏⣼⣿⣯⣶⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣽⣿⣿⣿⣿⣟⣻⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⣿⣼⣿⠁⣸⣿⣿⣿⡟⢰⣿⣯⣶⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣠⣾⣞⣯⣽⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣩⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣴⣿⣿⣿⣿⣁⣾⣿⣿⣿⣿⣿⣷⡄⠹⣿⣷
⠀⣀⣾⣿⠿⣻⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣷⠼⠟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣻⣿⣠⣟⢿⣯⣾⣿⣿⣿⣿⣥⡈⣿⠀⢻⣿
⢰⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣭⣭⣭⣍⣿⣿⣿⣿⡍⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⣿⣿⣿⣷⢲⣾⣿⣿⣿⣿⡿⠟⣻⣿⣿⣿⣿⡇⣿⡇⢸⣿
⣸⣿⣿⣫⣽⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣛⣭⣿⣿⣿⣿⣿⡿⠛⢻⡆⠈⣿⣿⣿⣿⣟⣼⣿⣿⣿⣟⢿⣶⣿⣿⣿⣿⣿⣿⡇⣿⠇⣸⣿
⢿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢸⣿⠀⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⡿⢛⣽⣿⣿⣿⠋⣰⡿⢀⣿⡏
⢘⣿⣿⣥⣾⣿⣿⡄⠹⣿⣿⣿⣿⡟⠀⢠⣿⡿⠿⠿⣿⣿⣾⠟⣋⣽⣿⣿⣿⣿⣿⣿⡇⢸⣿⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⢷⣿⣿⣿⡿⢿⣿⣿⣿⡟⣁⣾⡿⠃
⠀⢿⣿⣿⣟⣽⣿⣿⣦⡈⠉⠉⠁⣀⣴⣿⡿⠀⠀⠀⢠⡙⢷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣼⡏⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⣿⠿⣿⣿⣿⣿⡿⠁
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠸⣿⣾⡿⢛⣿⣿⣿⣿⣿⣿⣀⣴⡿⠀⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣧⣾⣿⣿⡿⠟⠁
⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⡿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠸⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣋⣠⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣟⣥⣾⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣥⣾⣿⠿⠿⠟⠁
⣿⣿⡟⠻⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣼⠥⠀⠀⠀⠈⠑⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣿⠾⠃⠀⠀⠀⠀⠀⢀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣀⣴⣶⣶⡶⠶⢶⢶⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⣼⢸⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠃⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠛⠛⠻⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⣴⠶⠿⠛⠷⠶⣦⣀⡏⠈⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⢰⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠞⠛⢧⣄⣀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⢀⡴⠞⠛⠒⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠺⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⠊⠁⢢⢀⠤⢀⠀⠀⢀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠃⠀⡨⠖⠸⡄⠀⠈⡇⠀⢠⠃⠀⢺⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠠⠈⡏⠀⠀⡆⠀⠀⡇⠀⠀⡇⠀⢸⠀⠁⢸⡄⠋⡈⠁⡄⠂⠒⡄
⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⡁⣀⢠⡗⠊⠒⡏⠑⠄⡗⠊⠚⡄⠀⠸⡷⠦⣀⠸⠇⠀⠀⠃
⠀⠀⠀⠀⠀⠸⠀⠆⠑⣆⠀⠀⡏⠠⢰⡇⠀⠀⠃⠃⢰⡇⠀⠀⡧⠤⢴⠀⠂⢸⠐⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠱⡀⠠⠈⢆⠀⡃⠀⢸⢀⠒⠼⠀⠀⢸⠁⠀⢀⡀⠀⠀⠀⠀⡆⠃⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⡈⢣⡁⣀⡟⠇⠠⠀⡇⠚⢉⠑⡶⠊⠑⠆⢀⣀⣠⣥⣁⣄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢩⠉⣁⣈⣀⠈⢇⠁⠀⠀⠁⠀⢠⠀⡇⠘⠀⠀⡸⠁⠄⠠⠤⠄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠳⣄⠀⢸⠀⠈⠀⠃⠀⣠⠞⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⠢⣀⠉⠚⠓⠒⠚⠓⠉⣀⠔⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠀⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⡤⠖⠚⢉⡠⠤⠒⠲⠆⢹⠶⠚⠛⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡤⠖⠋⣁⣤⠖⠊⠁⢀⣠⠴⠚⠀⢰⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠚⢉⣠⠤⠒⠉⠀⢀⣠⣤⣄⡈⠀⣀⣠⡴⠷⠶⠦⣄⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⣀⣤⠤⠖⠒⢉⣉⣠⠤⠖⠛⠁⠀⣤⠔⠒⠀⠈⠀⣀⡤⠜⠋⠉⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡠⠖⠋⣁⠠⠔⠒⠋⠉⢀⠀⠀⣠⠴⠤⢀⠀⢀⣀⠤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣠⠎⣠⠔⠋⠠⠀⠀⠠⠶⠚⠋⠀⣠⣁⡠⠤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⡥⠶⠒⠉⠢⢄⣀⡤⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠖⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣤⣄⡤⠤⠴⠒⠚⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠛⠳⢤⣀
⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀
⠀⠀⣠⣤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢤⣄⣀⠀⠉⠛⠶⢤⣀
⠀⠀⡇⠈⠉⠓⠲⢤⣄⣀⡀⠀⠀⠀⠀⠀⠙⠲⠦⢤⣽⠗⠀⠀⠀⠀⠉⠛⠶⣤⣀
⠀⠀⣇⠀⠀⠀⠀⠀⠀⠈⠉⠙⣳⠒⠒⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠉⠛⠶⣄⡀
⠀⠀⡿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠤⠴⠛⠒⠶⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣄⡀
⠀⠀⣇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⠖⠒⠛⠛⠒⠦⡀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⣀⣀⣽⠒⠒⠒⠲⠶⠦⠤⠤⢽⡇
⠀⠀⡇⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠋⠛⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⣸⠁
⠀⠀⡇⠀⠀⠀⢀⡴⠛⠉⠳⡄⠀⠀⠀⠀⠸⣄⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡏
⠀⠀⡇⠀⠀⢰⠋⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⣿⠀⠀⠈⠳⢦⣤⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⡏
⠀⠀⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠓⠶⠤⣤⣄⣀⡀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠐⠃⠀⠀⠀⠀⣰⠋⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠿⣅⣀⡀⠈⠉⠳⡄⠀⠀⠀⡀⠀⠀⠙⢦⣄⣀⣠⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠦⣿⣀⡀⠈⠛⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⣰⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⡀⠀⠀⠀⠉⠙⠛⠲⠦⣤⣄⣀⢸⠀⠀⠀⠯⠭⠽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠈⠉⠓⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣊⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠚⠃⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠘⠤⠤⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠤⠤⠤⠚⡆
⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⣀⣀⣀⣀⠤⠤⠤⠖⠒⠒⠒⠉⠉⡩⠓⠦⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢰⠚⠒⠒⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⡴⠊⠲⡄⠀⠀⠸⡄⠀⢀⡗⠀⠀⠀⠀⢠⠊⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠳⢄⡤⠃⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠈⠉⠢⡀⢰⠁⠀⠀⢳⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⡇⠈⠢⠤⠤⠃⠀⠀⡔⠉⠀⠈⠱⡀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠤⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠶⣏⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡹⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⡴⠒⠀⠒⢦⡀⠀⣉⣒⣋⣀⠤⠤⠤⠃
⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⢀⣀⣀⣀⠠⠤⠤⠤⠒⠊⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠘⠒⠒⠂⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⡀⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣄⠀⢤⡆⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣿⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣾⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⢿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠉⠉⠉⠙⠿⠋⠓⠒⠒⠐⣀⠀⣀⡀⢸⢦⠀⢀⡀⣀⡀⣀⡀⣠⡄⣀⡀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⣿⠀⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠐⠒⠘⠚⢻⣾⣿⠐⠒⠐⠒⠒⠓⢒⣓⠘⠂⠓⠀⠈⠛⠋⠀⠀⠀⠈⠛⠋⠈⠛⠛
⠀⠀⠀⣸⣿⣄⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡟⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⣇⠀⢠⣤⣤⣤⡄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠢⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣷⣤⣾⣿⣿⣿⡇
⠀⠀⠀⠻⣿⣿⣿⠟⢿⣿⣿⡟⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠁⠀⠀⠈⠉⢉⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠉⠉⠙⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⣴⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⢄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠉⠀⣠⣴⡾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣼⣿⣿⣿⣿⣉⡀⠀⢀⣀⣀⡀⠀⢸⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⢸⣿⣿⣿⠟⠁⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⡇⠀⢸⠑⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣤⣤⣾⣿⣿⣿⠀⠀⢀⣀⣘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠉⠀⠉⠉⠀⠀⣿⣿⣿⣇⠀⣸⣙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠈⠻⠿⠛
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠽⠿⠿⠯⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿⣿⣿⣶⣶⣶⣶⣦⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⣶⣾⣷⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⢶⡦⢰⣶⣶⣶⣶⠀⠉⠲⢼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣀⣀⣀⣈⣛⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⢿⡿⣿⣿⣿⣿⣿⡉⠉
⠀⠀⠀⠀⠀⠘⣿⣿⡿⠛⠛⠛⣿⣿⡿⣿⣿⡿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠀⠈⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠃⠈⠉⠉⠉⠈⠻⠿⠿⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣽⣭⣿⣵⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣷⣶⣷⣿⣷⣷⣿⣿⣿⣿⣷⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣼⢻⠀⠀⢠⠇⢠⠇⡶⠘⢀⡞⠁
⠀⠀⠀⠀⢀⣀⠂⢳⣦⡀⠀⠀⠀⠻⠀⠁⠀⠏⠀⠞⠰⠇⠀⠸⠀⠀⠀⠀⠉⠉⢲⣷⡄
⠀⠀⠀⠶⣻⣿⡇⠈⢻⣷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡛⠚⣛⣛⣛⠃
⠀⠀⠀⢀⣿⡿⠁⢀⣤⣽⣿⡆⠀⠀⡀⠀⠐⢲⣄⠀⠀⠀⠀⡀⠀⠀⠀⠋⠉⠉⠉⠉
⠀⠀⠘⠿⠛⠁⢀⠚⠻⣿⣿⠻⣧⠀⠿⣷⣿⣿⠿⠃⠀⠀⣼⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⣀⣤⣄⠐⣀⣻⣿⡇⠙⠃⠀⠀⠀⠀⠀⠀⢠⣔⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠈⢹⣿⣿⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⠀⠀⠀⠀⢀
⠀⠀⠠⣤⣿⡿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣴⠶⢦⣤⣤⣤⣤⣶⣦⡀⠁
⠀⠀⠀⠙⠋⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢀⡾⠁⣰⠿⠿⠿⠿⠿⢿⣿⣷⣶
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⢹⠞⠀⠀⢠⠏⠀⠈⠀⠀⠀⠀⠀⠤⣯⣽⣿⣧⡀
⠀⠀⠀⠀⠀⢸⣾⠀⠀⠀⠀⠀⠀⣠⢾⡿⡳⢠⡏⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠹⣧
⠀⠀⠀⠀⠀⢸⢻⠀⠀⠀⠀⢀⡾⣽⡟⡁⠀⣸⠀⣠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠄
⠀⠀⠀⠀⠀⠈⠞⠀⠀⠀⢠⣿⠞⠋⠊⠀⠀⢹⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢀⡴⠀⠀⠀⠸⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢣⣿⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⢴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⣟⣿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⣷⡿⡿⣿⣿⣿⣻⢿⣻⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠀⡤⠤⠴⢖⠛⠛⢋⠙⣙⠛⠓⡂⠳⠾⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠋⠀⣠⣄⣃⣀⣡⣈⡂⣐⣂⣀⣀⡒⣀⣡⣀⠀⠶⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢀⣐⣈⣉⣭⠉⠋⠙⠉⠉⠭⠉⠹⠉⠀⣉⡀⠀⠀⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⡏⣿⣿⡟⣿⣿⣻⣿⣿⣻⣿⣟⣿⣿⡇⠀⠀⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣷⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⡏⠁⣴⠟⠛⠯⠉⠉⠉⢩⣿⣭⡍⠉⢹⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⡇⠀⣘⣷⣤⡄⣠⡀⠀⣟⡿⣿⠖⠀⣼⠀⠈⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⡀⣿⣾⡉⢀⣿⣼⢡⡟⣰⡟⢡⣏⢰⠇⣿⠀⡀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠂⣿⣈⣩⣯⣥⣬⣯⣽⣯⣽⣯⣽⣯⣤⣿⠀⠁⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢹⣿⡏⣿⣿⣇⢸⣿⣿⢸⣿⣿⢻⣿⣿⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠛⠳⠾⢿⣿⣼⣿⣿⣼⣿⣿⣼⠿⠋⠀⠀⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠃⠀⠀⣀⠀⠀⠀⠀⡄⠀⠈⡀⠠⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣄⣀⠀⠀⠀⠓⠀⠀⠀⡀⠒⠀⠀⢀⣀⣤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠒⠒⠒⠒⠒⠚⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣿⡄⠀⠀⢀⣼⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⣹⡿⠟⠿⢦⣾⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣶⣶⣿⠟⠀⠁⠀⢉⡟⠉⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡛⠂
⠀⠀⡆⢸⠀⡆⡀⠀⠀⠚⠛⠛⠛⠛⠉⠙⢻⣿⣿⠋⠀⠀⠀⠀⣞⣀⣀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠒⠂⠤⣄⣠⣼⣿⣿⣧⠀⠀⠒⠒⠶⠛⠀⣿⠀⢸⢿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣹⣿⣿⣿⡀⢰⡄⡦⠀⠀⠀⠉⢠⢜⣿⣿⣿⣯⣼⣿⣿⣿⠟
⠀⣀⣤⡤⠤⠤⢐⣒⣄⡨⣤⠤⠖⢒⣣⣬⣼⡭⢽⠛⠛⣿⣶⡶⠶⣶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣧
⢉⢩⣀⣤⢴⠞⠍⠉⢠⣦⣴⡶⠟⠉⣉⣡⣴⣶⡾⠟⢋⣭⣰⣤⣄⣼⣧⣯⣹⡟⣟⠛⠛⠉⠉⠉⠁
⠛⠛⠉⣀⣤⣤⠼⠚⠛⠥⣤⣴⣶⣿⠛⠙⡋⣠⣦⣼⣿⣿⠏⠛⠋⠉⠉⠀⠀⠀⢳⡝⡄
⣶⡞⡿⠿⠷⠶⠶⠶⢿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣾⣷⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠖⠒⠊⠙⠒⠦⡄⠀⢀⠏⠑⠒⠒⠶⠤⠤⣄⡀⠀⢠⠶⠶⠦⠤⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⣼⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡎⠀⠀⠀⠀⠀⠀⠀⠀⠙⡦⢠⣄⣀⣀
⠀⣠⠤⠒⠉⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⠇⠀⠀⠈⠉⠉⠑⡲⢤
⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡼⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠏⠀⠀⠀⠀⠀⠀⠂⠀⡜
⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠂⢠⠇
⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⣀⣀⡼⠦⠴⠒⠒⣒⣚⣋⡉⠉⠉⠙⠒⠒⠒⠒⠦⠧⢤⣄⣀⡀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⣀⣼⡶
⠀⠀⠀⠀⠀⣠⣾⡿⠗⠚⠉⠁⠀⠀⠀⣠⡶⠟⠋⠉⠉⠁⠀⠀⠀⠀⠀⠈⠛⠛⠶⣦⣀⠀⠉⠙⠒⠦⢤⣀⣼⠃⠀⠀⢀⣠⠤⠒⠋⠁⢰⠇
⠀⠀⠀⠀⢾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠰⠋⣠⠒⠉⠀⠉⠒⢄⠀⠀⡠⠐⠈⠉⠐⠠⡀⠙⢦⠀⠀⠀⠀⠀⠈⠉⠓⠖⠋⠉⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠈⢂⡐⠁⠀⠀⠀⠀⠀⠐⡀⠈⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⡅⠀⠀⠀⡀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠔⢉⣉⣑⠂⠀⠀⠀⠀⣼⠿⡇⠀⠀⠀⡀⠀⠀⣿⢿⡆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠐⣻⣿⣇⠀⠀⡀⠀⠀⠈⠛⠁⠀⠀⢀⠃⠀⠀⠙⠚⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⣿⣿⣿⡄⠀⠐⡀⠀⠀⠀⠀⠀⢀⠎⢡⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⣿⣿⣿⣦⡀⠈⠢⠀⠀⡀⠔⠁⠀⠀⠑⠄⢀⣀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠸⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠣⡀⠀⠀⠈⠙⡍⠁⠀⠈⠙⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⠒⠄⡀⠀⠀⠀⠀⠀⠀⡨⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠓⠒⠶⠶⠤⢤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⢸⠀⣠⠖⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣿⣶⣾⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠲⣾⣴⡶⠀⠀⠀⠀⠀⠀⠀⢀⣤⠒
⢺⡟⡿⠿⡷⠀⠀⠀⠀⠀⠠⠞⠛⠒⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⣾⣿⣟⣤⣰⣾⣦⠛⣶⣿⣿⠢⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣚⣳⣶⣶⣦⣾⣿⣿⣷⣶⡶⠀⣠⠀⢶⣶⣾⣾⣿⣶⣆⣀⣀⣤⣠⣄⡀⣀⣤⡀⢰⣷⣶⣶⣂⡀
⠻⣿⠿⠿⣿⠿⠿⠳⠾⠿⠿⠤⠁⠀⠀⠀⠀⠸⠆⠀⠀⠉⠉⠉⠉⠉⠿⠏⠁⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠉⠉⠉⠉⠛⠉⠏⠋⡉⠁⠉⠉⠉⠉⡉⡍⡛⢛⠋
⡋⠤⠚⣁⠤⠖⠊⠉⣀⠴⠒⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠠⠉⠈⢉⡉⠀⠒⢀⠀⠀⠀⠀⠀⠀⠠⠐⠊⠡⠀⢈⠀⠈⠀⠂⠠⢀⠀⡀⢆
⣠⠔⠋⢁⡠⠴⠒⢉⡠⠔⠂⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠠⢄⡀⠀⠠⣄⠀⠐⠀⢀⠀⠀⢀⠄⠂⠀⠠⠄⠈⠀⠀⠀⠀⠀⠄⠁⠀⠐⠀⢀⠀⢸⠀⠇⠸
⢀⣠⠔⠉⠀⡠⠀⠉⠀⠠⠄⣀⡀⠑⠒⠤⠄⡀⠁⠐⠀⠀⠀⠐⠂⠀⠉⠋⠀⠀⢀⡤⠀⢀⠠⠊⠁⡀⠔⠁⠀⠀⠊⠀⠀⣠⠞⠀⠀⠀⠌⠀⢀⠄⠀⡀⠀⢷
⠋⠁⢀⠔⠋⠀⡀⠚⠀⢤⠀⡀⠉⠠⠶⡶⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠖⠉⢀⡠⠊⠁⣀⠄⠁⠀⠀⢀⠞⠀⠀⠀⣠⠋⠀⠀⠐⠀⢀⡇⠀⣸
⠀⠐⠍⠀⣠⠎⠀⣀⣀⠀⠁⢀⣄⡄⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢉⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠂⡀⠂⠀⠀⠀⡄⠀⣼⠀⠀⡿
⠀⣀⠐⠢⠀⠠⣄⡈⠉⢀⡀⠀⠐⠀⠠⣤⡖⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠃⠀⠀⠀⠁⠀⠚⠀⠤⠀⢀⠀⠀⡐⠀⠀⠄⠀⠠⠀⠀⣀⠀⢀
⠣⠈⠁⠀⠐⠦⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⡀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⣀⠀⠁⠀⠘⠃⠀⠰⠖⠀⡤⠀⢈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠠⠒⣄⠀⠀⠀⠀⠀⠀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠈⠀⡀⠀⢺⠀⢀⠄⠀⠀⠀⠀⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠘⠄⠊⠀⠠⠀⠀⠀⠸⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡌⠀⠀⠁⠀⠀⠀⢀⠂⠀⢦⠀⠀⠀⠀⠀⣀⠀⡆⠀⠈⡇⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠤⠒⡠⠀⢠⠇⠀⠀⢰⠈⠀⢀⢿⠐⠀⠈⣠⠊⡁⠉⠰⡇⠀⠀⢱⠀⡇⠀⠀⡀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡋⠀⠎⠁⠀⢠⠒⠀⢄⠀⠀⠁⠀⢸⠀⠀⠜⠁⠀⠁⠐⣠⠁⠀⠀⠂⠀⡇⠀⠀⠀⠀⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢈⡇⠀⠀⠀⠠⠀⠀⢀⠆⠀⠀⠀⡇⠘⡧⠃⠀⠀⡀⠀⢰⡿⠀⠀⢰⠀⣀⣇⠀⠀⠀⠀⠇⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢠⡇⠰⠀⠆⠀⠀⠀⠀⠇⡶⠁⠀⠀⡐⠡⠀⢈⠠⠀⠀⢸⠁⢀⠀⠃⠀⠀⢠⡆⠀⠀⠀⡤⠆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠃⠸⠀⠀⠀⠀⠀⠀⡄⠀⠀⡐⢰⠀⣄⠁⠀⠀⠠⠘⠀⢀⠀⡇⠀⠀⠸⠁⠀⢀⠜⠁⠔⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠸⠀⡆⠀⠀⠀⠀⠀⠀⡇⠀⢠⠇⠈⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⢠⠁⠀⡠⠉⢀⠈⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁⠆⠀⠀⠀⠀⠀⠀⠠⠀⠇⠀⡇⠀⠀⡿⠀⠀⠀⡄⠀⢸⠀⡀⠰⠀⠀⠆⢀⡔⢁⡠⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠢⠤⣧⣠⣤⣤⠴⠐⠒⠚⠛⠠⣄⠁⠀⠀⠇⠀⠀⠀⢃⠔⠋⠉⠉⠉⠛⠛⠖⠛⠉⠁⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣴⣀⣀⣠⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠴⠦⠄⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠠⠤⠀⠚⠓⠈⠁

⢀⣀⣀⢀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠷⠤⣄⠀⠀⠀⠀⠈⠉⠉⠓⠒⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢴⣎⣀⣤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⣀⣀⣠⠤⠴⠚⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡎⠀⠀⠀⢹⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠚⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⠒⠚⠉⠉⠀⠀⠙⢤⣀⡤⠋⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⣀⣀⡤⠤⠶⠒⠊⡟⠉⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡶⠓⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠳⠤⠖⠃⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⡴⠋⠙⢦⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡞⠁⠈⠑⡄⠀⠀⠀⠀⠀⠀⠀⠈⢧⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠹⢤⣠⣞⣠⠇
⠀⠀⠀⠀⠀⠉⢲⡀⠀⠀⠣⣀⣀⡰⠃⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠔⠒⠋⠉⠁
⠀⠀⠀⠀⠀⢀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢹⠀⣀⣀⡤⠤⠒⠚⠉⠉
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠿⠔⠒⠋⠉⠁
⠀⠀⠀⠀⠀⡇⠀⣀⣀⠤⠴⠒⠚⠉⠉
⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠆
⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠄⠤⢤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣤⠎⠩⢞⠀⠅⠉⠒⠠⢬⣃⡉⠉⣹⠋⠐⠤⡀
⠀⠀⠀⠀⣚⠿⠦⠁⠡⠀⠋⠓⢒⠒⠃⠐⠌⠉⡁⠀⣤⡀⠈
⠀⠀⠀⢼⡛⣠⢡⠰⡀⡔⣲⠀⠀⢀⠬⣤⠀⠀⠀⠀⠊⠀⠀⠐
⠀⠀⠀⠈⠁⠁⠉⠷⠃⠂⠠⢸⣆⣁⢠⠰⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡠⠄⣠⠖⢀⡴⠋⣠⠞⠁⣠⠞⠁⣠⠞⠀⠀⡠⠂
⠀⠀⡀⠂⠀⠀⠀⠀⠀⢤⠀⠀⠁⠀⠀⠉⠛⠁⠀⠀⠀⠀⢀⣠⣀⠀⠀⠉⠘⠁⠚⠀⠞⠀⠴⠋⢀⡜⠁⢀⡔⠁⢀⠔⠁⠀⢀⠘⠁⠀⢀
⢀⠀⠀⠠⣀⣠⡄⣤⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⢦⠛⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠜⠀⠀⢠⠂⠀⠀⣠⠋
⠈⠐⠱⣶⠟⠁⠀⠹⠛⣶⠖⠲⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣰⠋
⠀⠀⠠⠿⣧⣄⣀⣾⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⢣⡿⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠂⠀⠉⠉⠉⠛⠃⠦⠤⢯⣄⣀⡀⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦⢀⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠛⠃
⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢀⣤⣶⣿⣠⣴⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃