Diploma Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⢿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⠉⠀⠀⠀⠈⠙⠻⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣤⣀
⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣤⣀
⠀⢀⣠⣴⣾⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣤⣀
⣴⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⡆
⠙⠻⢿⣶⣤⣄⣀⣤⣤⣤⣶⣶⣾⡿⠿⠿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⠁
⠀⠀⠀⠈⢹⣿⣿⣿⣯⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠟⠋
⠀⠀⠀⠀⢀⣿⡇⠈⠛⠻⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠟⠋⠁
⠀⠀⠀⠀⢨⣿⡇⠀⠀⠀⢸⣿⠛⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⡿⠟⢻⣿
⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⢸⣿⠀⠀⠀⠉⠛⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠋⠁⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡿⣿⡆⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣦⣄⣀⣀⣤⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣸⣿⠁⢿⣧⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟
⠀⠀⢀⣿⡏⠀⠸⣿⡄⠀⠀⠈⠻⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠋
⠀⠀⢸⣿⣧⣀⣤⣿⡷⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣶⡿⠿⠛⠁
⠀⠀⠀⠉⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠻⠿⠿⠿⠿⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⠿⠟⠛⠋⠉⠀⠀⠀⠈⠙⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⣶⣿⣿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⣶⣶⣾⠿⠿⠟⠛⠛⠋⠉⠉⠛⠿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣾⣿⣿⡿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠛⠉⠁⠀⠉⠙⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⠟
⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⣠⣶⣶⣷⣶⣦⣄⠀⠙⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣾⡿⣿⣦⣠⣤⣶⣶⣶⡿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⣾⡟⠀⣾⡿⠁⣠⣤⣄⡙⢿⣷⡀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏⠈⢹⣯⠀⠈⢻⣿⡉⠉
⠀⠀⠀⠀⠀⠀⣿⣇⢰⣿⡇⢸⣿⠋⢻⣷⣶⣿⣷⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣿⣿⣟⣠⣄⣸⣿⣧⣿⠿⠿⢿⠆
⠀⠀⠀⠀⠀⠀⢹⣿⣾⣿⣧⠘⢿⣧⣀⡉⣁⣿⡟⠀⣼⡿⠀⣀⣠⣴⣶⡿⠿⠛⠋⠉⠀⣼⣿⠿⠛⠻⣿⡿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⢿⣷⣄⠙⠻⠿⠟⠋⣀⣼⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣶⣶⡿⠿⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣲⣖⠀⠠⠤⠤⠤⠤⠤⠤⠤⠄⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢀⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠤⠖⠒⢢⠀⠈⢷⣀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⢣⡤⠚⢯
⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠈⢧
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠄⠒⠳⡀⠘⠦
⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⢈⣁⣤⣤⢶⣖⢟⣻⢿⡿⠀⠢⣠⣷⣄⢤⣀
⠀⢀⠜⠁⠀⠀⠀⠀⠀⣀⣀⣀⡠⡇⠀⠀⠀⣀⠤⠒⢉⣠⣤⣶⣻⣿⣷⣟⣿⡯⣟⡯⠓⠉⠀⠀⠀⠘⡝⢟⠀⠀⠀⣀⠐⠁
⠀⠧⠤⠔⠒⠒⠊⠉⠉⠉⠀⠀⠀⢰⡀⠔⢈⣠⣶⣿⣭⡷⣿⣿⣯⢿⣛⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠳⡀⣁⠀⠀⠀⡀⠄⠉
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢸⣤⣾⣭⣯⠿⢿⣚⡿⠿⠓⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠭⠁
⠀⠀⠀⠀⡴⠋⠀⠉⠒⠤⣀⠀⠀⠈⠓⠒⠒⠋⠉⠁
⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⣠⠒⠢⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⡇⠀⠀⠙⣄⠀⠀⠀⣀⠤⠤⢤
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⢲⠉⠉⣷⡍⠀⠀⠀⢸⡇
⠀⠀⠀⠈⠙⠓⠒⠲⠦⠤⢤⣀⣀⠀⠀⠀⡀⠙⢓⠶⡟⠒⡶⣧⢄⠀⠀⠀⡸⠦⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠟⢁⡤⠾⠥⣤⣇⣀⡃⠂⢳⢴⣤⠔⠁⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠒⢤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢀⡴⠋⠀⠀⠀⠀⠀⠉⠙⡇⢸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢸⠀⠉⠑⠲⢤⣀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡜⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⣸⣦⣰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠠⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣠⠎⠀⠀⠀⠀⣠⠏⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⡴⠁⠀⠀⠀⠀⡴⠃⠀⠀⠀⡠⠿⠿⠿⠷⠒⠒⠀⣀⣀⣠⣤⣤⣐⣒⡀⠰⠀⠀⠄
⠀⠀⠀⠀⠀⠰⠤⠤⠶⠒⣞⠒⠚⠋⠉⠉⠉⠀⠀⢀⡔⠀⠀⠀⠀⠐⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⢇⠀⢿⡷⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣮⡀⠀⠀⠀⣠⠐⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠘⠀⢠⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠈⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠄⠰⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣆⣼⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠀⢇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠭⠭⠭⠭⠭⠭⠭⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡏⣪⣿⣿⣶⣿⣶⣷⣶⣶⣶⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠨⠭⠭⠭⠍⣩⠭⢭⡉
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠁⢀⡻⠷⠿⠿⠿⠿⠿⠿⠿⠿⢿⠿⠶⠶⠒⠂⢀⣀⣉⡅⠸⠿⠭⠽⠅⣇⠀⢀⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣽⡿⠃⠀⠀⣴⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠂⠀⠀⠈⠉⠉⠉⠃⠨⠭⠭⠽⠅⡈⡹⣉⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠈⠛⠛⠻⠻⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠐⣶⣶⣶⣶⣶⣶⣶⣶⣷⣤⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⡻⢍⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢬⡳⠆⣤⡠⠤⠤⠤⠄⠠⠄⠀⠀⢠⠀⠀⠀⠈⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠘⠉⠓⢬⡙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣮⠀⠀⠀⢀⡴⠚⠉⠙⠷⣦⣉⠲⣤⠤⠤⠤⠤⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣈⣳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠚⠀⡾⠤⣤⣀⡀⠀⠈⠛⠿⠷⠶⠶⠶⠶⠶⠶⠖⠶⡶⠶⠶⠦⠤⠤⠤⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠞⠀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠈⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⢀⡠⠖⠊⠉⠁⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡄⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⠈⠡⠴⠖⠀⠶⠶⠶⠆⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠁⠀⠀⢇⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⡀⣦⠀⠀⡀⠀⠀⠀⠀⠐⠂⠁⠈⠑⡤⠊⠀⠀⠀⠀⠀⡿⢀⣉⣀⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠀⢀⠇⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡀⠀⢀⠄⠀⠀⠶⠀⠀⠀⣤⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⡠⠄⠤⢀⡠⠒⠉⠐⠄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐⠀⠈⠉⠁⠀⠀⡠⠄⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠘⠒⢤⡤⠴⠦
⠀⠀⠀⠀⠠⣄⠀⠰⡄⠃⠀⢠⠂⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠠⡍⠀⠀⠀⠀⠠⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣗⠀⣀
⠀⠀⠀⠀⡀⠀⠈⡏⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠛⠁⡍⠉⠁
⠀⠀⠀⠀⠈⠁⢈⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠤⠹⡏⠛⠛⠛⠛⠛⣿⠋⢠⠀⠀⠀⣠⠖⢀⡀⠀⠀⠀⢀⡤⣀⠇⠀⠀⣈⠂⢉
⠇⠀⠀⠀⠈⠙⠅⠀⠴⠓⢦⣀⠀⠀⢠⠋⠀⠑⢤⣀⠀⠀⠀⠈⠑⠒⠒⠐⠒⠒⠉⠀⠘⡄⠐⠋⠀⠀⠀⠑⣄⡤⠚⠉⠀⡘⠀⠀⠀⠈⠙⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡶⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠓⠠⠀⠀⠀⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⣄⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠈⠙⠒⠶⠀⣤⣤⣀⢀⣀⣀⣤⣤⠄⠐⠂⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠃⠀⠒⠲⠦⠤⠄⣀
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡒⣚⠂⠑⠋⠒⠢⠤⠤⠤⠤⢴⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠒⣄
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠛⠓⠲⠦⠤⠤⠤⣤⣤⣤⡤⢤⣤⣤⣄⣤⡴⠏
⠀⢀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⢤⠚⠏⠁⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣉⣭⣭⣤⣤⣭⣭⣭⣉⣑⣒⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣶⣶⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢋⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣶⣤⣬⣍⣙⡛⠛⠿⠿⣿⣿⣿⠿⠛⣋⣥⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣁⣤⣤⡰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣀⣠⣤⣶⣽⢿⣿⣿⡎⣿⣿⣿⣿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣿⡇⣿⡿⣣⣿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⢿⣮⣻⣿⣿⣿⣿⣿⡿⠟⠃⣽⣶⡙⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⣿⡷⣻⡇⣿⡿⠟⠋⠁⠀⠀⢰⣿⣿⡇⠀⠹⣿⡟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠞⠛⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠀⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣠⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣯⠐⠀⣿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠙⠛⠛⠛⠙⠛⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠀⣼⣿⣇
⠀⠀⠀⠀⠀⢀⡤⠞⠉⠈⠻⣷⣶⣶⣾⣿⣿⣿⡀
⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣷⡄
⠀⠀⢸⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡍⢦⡀
⠀⠀⠈⠳⡵⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣴⣿⣦⣄
⠀⠀⠀⠀⠘⢮⢧⡀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⣀
⠀⠀⠀⠀⠀⠈⠳⡳⣄⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠀⠀⠀⠈⠑⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣝⣦⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠉⠓⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠋⠀⢻⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠁⠀⠀⠈⣿⣿⣿⠋⠉⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣟⠁⠀⠀⠀⠀⢹⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡬⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠙⣆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠂⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡔⠒⠉⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠈⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⢳⡀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⡀⢤⠐⠂⠀⠈⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢳⡀⠀⠀⠀⡠⢊⠠⠐⠈⡁⠄⠀⠘⠀⢀⠌⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠳⡄⣠⠾⠂⡁⠔⠈⠀⠀⠀⠀⡀⠔⠁⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⢀⠟⠉⠢⣌⠀⠀⠀⡀⠠⠐⠈⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⠀⢀⣀⡠⠤⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠉⢳⠖⠲⣶⣿⣿⣿⣷⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠸⠄⠈⠿⠾⠿⣿⣿⣷⣆⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣘⠂⠀⣤⣤⣬⣿⣿⣿⣯⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⠉⠉⠉⠉⣺⣿⣩⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠤⠤⠠⢠⣄⣀⠟⠻⣻⡧⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠈⣁⣠⣴⣯⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠉⠉⠀⠀⠀⠀⠹⠶⠞⠛⠿⠝⠋⣄⠀⣠⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠉⠉⠒⠒⠶⠤⣤⣄⣀⣀⠠⢦⢢⣶⣿⢿⡝⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠠⠤⢄⣀⠀⠉⢁⣴⡿⠟⠉⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠿⣛⣻⡭⠭⠭⠭⠭⣝⣛⠿⢿⣿⣿⣦⣄
⠀⠀⠀⠀⣠⣾⣿⡿⢛⡥⠚⠉⠁⠀⣀⣠⣤⣀⡀⠀⠈⠉⠓⠮⣛⢿⣿⣷⣄
⠀⠀⢀⣾⣿⡿⣫⠞⠁⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠈⠳⣝⢿⣿⣷⡀
⠀⢠⡿⠟⣛⣤⣴⠾⠿⠟⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣙⠿⣿⡄
⢀⣿⣿⡆⡍⠻⢶⣶⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⣋⡀
⣸⣿⡿⣸⠁⢻⡀⠈⣝⡻⣷⣮⣝⡛⠿⣿⣿⣿⣿⣿⡿⣿⣻⣿⣿⠿⠛⠋⠁⢦⢻⣿⣧
⣿⣿⡇⠇⣀⣬⡷⠀⣿⣿⣶⣭⣛⢿⣷⣤⣉⣩⣭⣶⡿⢟⣫⣭⣶⣿⡇⠀⠀⢸⢸⣿⣿
⣿⡿⢡⣾⠋⠀⠀⠀⠙⠛⢿⣿⣿⣿⣮⣝⣻⣛⣭⣶⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⣘⣟⣯⠴⠶⠛⠒⢦⣄
⢿⣦⣌⢛⠓⠶⢶⣶⣀⡀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣋⢉⣀⣤⡴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⢹
⠸⣿⣿⡼⡄⠀⠀⣾⣿⣷⠀⠀⠀⠀⠙⢻⣿⡿⠟⠋⣹⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼
⠀⢻⣿⣷⡹⡄⠀⠈⠉⠁⠀⠀⠀⠀⠀⣠⣼⣶⣿⣿⣿⡿⢿⣿⣷⠀⠈⠀⣀⣠⣤⣴⣶⠾⠿⠟⠛⠛⠋
⠀⠀⠻⣿⣿⣜⢦⡀⠀⠀⠀⢀⣠⡴⠞⢻⡿⠟⢩⣿⠋⠀⢸⣿⣿⣶⠾⣛⣛⣭⠅
⠀⠀⠀⠙⢿⣿⡷⢙⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⢈⣡⣴⠾⠛⣉⠵⣪⣾⣿⡿⠋
⠀⠀⠀⠀⠀⣵⠾⠛⢧⡄⠀⠀⠀⠀⢀⣠⣴⠾⣛⡭⠤⣖⣫⣵⣾⣿⡿⠋
⠀⠀⠀⠀⢸⡏⠀⣴⠞⢻⠀⣀⣤⡾⠟⣩⣴⣶⣶⣾⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⢸⣇⠀⠻⢄⣼⣿⠟⠉⠀⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠠⠄⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠒⠀⠁
⠀⣀⠠⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⢄⡀
⣟⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⣴⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠤⣀
⠀⠉⠂⠭⣒⠤⣀⣀⢤⠠⠆⠐⠂⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡲⡄
⠀⠀⠀⠀⠀⠉⢚⠯⣓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠔⠒⠁
⠀⠀⠀⠀⠀⠀⢸⠀⡸⠁⠂⠄⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢤⠐⠂⠈
⠀⠀⠀⠀⠀⢀⠼⡄⡇⠀⠀⠀⠀⠁⠂⢄⡀⠄⡀⠀⣀⠠⠀⢀⠀⠄⠐⠈⠁⠀⠀⢸
⠀⠀⠀⠀⠀⣢⣀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠪⡭⠔⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠊⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠔⣒⠒⠠⠤⠖⠒⡄⠀⠀⠀⠀⠀⡠⠐⠢⡇
⠀⠀⠀⢸⠄⠀⠀⠸⢆⡀⠀⠀⠀⠀⠀⠀⢞⠁⠉⠢⡀⠀⠀⡆⠀⠀⠀⢀⠔⠀⠀⠀⠷⡀⢀⠠⠐⠉⠓⢄
⠀⠀⢀⡨⢂⡀⡠⠃⠀⣉⣢⣄⡀⠀⠀⠀⠀⠁⠂⠤⠝⠤⠗⠓⠒⣤⡐⠡⠄⠀⠀⠀⠀⠏⠀⠀⠀⠀⠐⠊⠉
⠀⠀⠀⠁⠉⠁⠀⠠⢋⠊⠀⠉⠈⠉⠁⢀⡀⠐⠒⠒⠒⠒⠓⢲⡞⢍⠉⠀⠦⠄⠉⠥⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢸⠀⠀⠀⠀⣡⠀⠈⡇⠀⠈⠉⠉⠁⠀⢀⡇⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠈⡈⠁⠉⠁⠀⠘⠀⠀⡄⠀⠀⠀⠀⠀⣠⠊⡄⠀⠥⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣠⠥⠔⠒⠉⠉⡸⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠚⠋⠁⠀⠀⠀⠀⠀⠀⡁⣀⡀⠇⠀⠀⠑⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠁⠙⠂⠑⢄⣀⣙⡢⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣷⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⠿⢛⣫⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⢿⡿⠿⣛⣩⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⡛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⠀⣿⣿⣶⣦⣍⣙⠻⠟⣋⣩⣴⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⣿⣿⣿⣿⣷⣶⣶⡶⢆⡐⢶⣶⣶⣿⣬⣭⣭⣽⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⡿⢨⣿⣿⣿⣿⣷⣿⣿⣿⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⣿⣿⡿⠿⠿⠿⠿⠟⣴⡆⢰⣦⡙⠿⠿⠿⠿⣿⣿⣿⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢿⠇⠈⠏⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠒⠂⠤⠤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠢⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⢒⣂⡩⠭⠛⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠒⠤⢤⣀⣀⢀⡠⠔⣊⡩⠄⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠊⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠔⢻⠃⠤⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠉⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡏⠀⡜⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠚⡄⠀⠘⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢳⣶⣇⠀⠀⠀⠈⠙⠒⠒⠤⠤⠤⠤⠤⠤⠤⠤⠤⠐⠒⠒⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡾⠀⡜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⢰⡇⣧⠸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⢸⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣥⣀⣇⣸⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠛⡖⠒⠚⠉⠉⢀⣀⠤⠜⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠚⠛⢦⠤⣔⣒⠊⠭⠝⠊⠁⣀⠼⠒⠒⠋⠉⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠔⠒⠉⢹⠊⠉⠙⣾⣒⠒⠒⠲⢖⡊⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⠤⠔⠒⣉⣉⣤⠤⠤⠔⠒⢊⣹⠟⡦⢰⠖⣧⠀⠉⠲⣄⠀⠈⠓⠦⣄⣀⡤⠤⠔⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡰⠒⢳⠀⡏⠀⠀⠀⠀⢀⡠⢖⡽⢉⠜⢁⠎⠀⢰⠀⢀⣀⠤⠷⢤⣀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⢠⠁⠀⠀⢀⠖⠉⡴⢋⡴⠋⣠⠿⠤⠤⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠓⠴⣳⣎⣀⣀⠤⢇⣀⡠⠗⠉⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠙⠒⠒⠊⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣋⣉⣉⣉⣉⣉⣉⣉⣉⣉
⠀⠀⠀⠀⠉⠒⠤⣄⡤⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠉⠉⢉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢠⡇⣽⣷⣦⣄⡀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡀⠀⠀⠚⠁⠈⠁⠌⠲
⠀⠀⠀⠀⠀⠀⢸⣇⣿⣿⣿⣿⣿⣿⣶⣴⣾⣿⣿⣿⣿⣿⡇⠀⠇⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⣸⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣷⠀⡆⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠰⢲⣷⡞⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠋⠀⣦⡀⣀⣀⢀⣴
⠀⠀⠀⠀⠀⠀⠈⣻⣛⣏⣉⣉⣉⣛⣛⣋⣉⣉⣉⣙⣛⣋⣙⣀⢠⣿⣿⡟⢻⣿⣿⡄⠀⠀⢀⣀⣀⣀⣀⣀⡨
⠀⠀⠀⠀⠀⣀⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⡄⠁⠞⠛⣿⠁⠈⣿⠛⠳⠀⣼⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⢉⣸⣿⣿⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠘⠓⠓⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣺⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠚⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⣬⣍⣛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣛⣩⣥⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣶⣦⣭⣙⣛⣛⣩⣥⣶⣶⣿⣿⣿⣿⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⣿⣿⠿⠛⠋⠉⠉⠉⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⣀⣀⣀⣀⠀⣀⣉⣉⢁⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡉⠉⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢻⣤⣿⣿⠿⠿⠿⠿⠿⠇⣿⣿⣿⡯⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠶⠖⠒⠚⠛⠛⠛⠚⣛⣛⣛⠑⠒⠶⠶⠶⠦⠬⢭⠍⠉⠀⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠉⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⡀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠛⢲⠀⠀⣾⠋⠈⢳⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣇⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠁⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠈⠁⠀⢀⣤⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠙⠶⠃⠀⠀⠀⠖⠂
⠀⠀⠉⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣿⠀⡸⠃⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠈⠈⠁⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢠⡶⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠚⠁⠀⣤
⢀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⢀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠠⠞⠁⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄
⠉⠀⠀⠀⠘⢁⡏⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣇⠀⣠⠴⠂⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⢤⠀⠀⠀⡀
⠀⣄⡀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠛
⠀⠈⠁⠀⠀⢤⠀⠀⠀⠀⠐⠃⠀⠀⠈⢿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⢀⠄⣤⠀⢠
⠀⠀⠀⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢿⡇⠀⠀⠀⠀⠀⢀⣴⠋⠀⣟⣰⠋
⢠⡄⠀⠀⠀⠀⠀⠀⢠⠂⠀⠁⠐⠄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣿⣿⠀⠀⠀⠀⠀⠘⠃⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⢻⣾⣿⣿⡿⠿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠻⢿⣿⠀⠀⣿⣿⠀⠀⠀⠀⢀
⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣠⣭⣭⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠉⠀⠀⠀⠀⠗
⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢒⣶⣾⡿⠛⠶⠖⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠈⠀⠀⣠⡀⠐⠚⠀⠀⠀⠀⠀⠀⢸⠃⢠⡟⣡⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠖
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠑⠢⡀⠀⠀⠀⠀⠀⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢄⠀⠀⠀⠀⠏⠀⠀⠀⠀⠡⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠀⠈⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢀⠤⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣖⣺⣭⡿⠶⠶⢿⡟⢻⡷⢶⣾⣯⣽⣒⡦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣺⣵⣿⠋⠙⢿⣷⣆⢸⣾⡇⠸⠇⣾⡇⢰⣿⣿⡿⠷⣯⣳⢄⡀
⠀⠀⠀⠀⠀⠀⣠⢞⣵⠟⡛⢿⣿⠄⢅⣨⣽⣷⠶⣿⠿⢶⣾⢿⣦⣭⣼⠋⡑⠀⣾⣿⢷⣝⢦⡀
⠀⠀⠀⠀⢀⡼⣵⠿⣷⣦⠁⢤⣼⣿⠿⠛⠉⠈⠀⠀⠀⠀⠀⠈⠈⠙⠹⢿⣧⣤⣿⡷⢂⡙⣷⣝⣄
⠀⠀⠀⢠⢞⣾⡏⢴⡄⢙⣷⡿⠋⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⣿⡿⠏⢻⣮⢧
⠀⠀⢠⢯⡿⠻⢷⣤⣤⣿⠍⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣶⣤⡀⡀⠀⠀⠀⠈⢿⣷⣰⠾⠛⠻⣯⢧
⠀⠀⡞⣾⠋⠁⣠⣼⣟⠃⣀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣥⣐⠠⢀⡀⠻⣿⡐⠞⠃⣿⡞⡆
⠀⢸⢻⣿⠿⠷⣤⣿⠇⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⠴⠚⠁⠀⢻⣿⠲⠟⠛⣿⢻
⠀⣼⣼⡇⠶⠦⢸⣯⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠰⣿⠞⠁⠔⢻⣿⡆
⠀⣿⣿⡿⢶⡶⣿⡇⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡄⠀⠀⠀⠀⠀⣿⡟⢿⠿⣿⣿⡇
⠀⢹⢻⡇⠺⠃⣸⣏⠀⠀⠀⠀⠀⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠫⠅⠂⠀⠀⠀⠀⠐⣿⠑⢁⠆⣼⢿⠃
⠀⠸⡾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠰⣷⠀⠈⠙⠿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣷⣿⣼
⠀⠀⢷⢻⣿⣿⣿⣿⣿⣆⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⢷⠇
⠀⠀⠈⢯⢿⣿⣿⠛⣿⣿⡦⡀⠀⠨⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣺⣿⡟⠻⣿⣿⢯⡏
⠀⠀⠀⠈⢯⣻⣿⣶⣿⣿⣿⣾⣢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣪⣾⣿⣿⣿⣾⣿⢯⠏
⠀⠀⠀⠀⠀⠳⣝⢿⣿⣿⣿⣿⠿⠛⢷⣆⣤⣀⢀⠀⠀⠀⠀⢀⡀⣄⡶⠿⣿⣄⢹⣿⣿⣿⢟⡵⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣝⢿⣿⡏⠠⡾⠆⣼⡏⢉⡛⢿⣿⡟⠛⢻⣿⣅⠐⣦⠀⣿⣯⣽⢟⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠯⣟⠦⣄⣰⣿⠇⢠⠁⢾⣿⡇⢘⠀⠻⣿⣆⣈⣴⢟⣫⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠯⢽⣛⣺⣶⡾⠿⠶⢾⣶⣒⣻⠭⠗⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⢀⠄

⠀⢀⠄⠄⠀⡦⢀⠄⠆⠠⠀⠀⡀⠀⠀⠀⠀⠀⡀⢀⠀⠠⠀⠂⠀⠠⢀⠀⣀⢀⡀⠄⣠⡐⠂⠐⠢⠀⠠⡀⡀
⠀⠈⠂⡛⡌⢔⠄⢸⢀⢱⠀⡀⠀⠙⢀⠰⠁⠇⠂⢁⠆⠀⠀⠀⢠⠂⠁⠀⠦⡬⠆⠾⠓⠉⠁⡐⠂⡨⠍⠀⠤⢀
⠀⠀⡁⢨⠀⢂⠨⣉⣠⠀⠄⠁⠁⠃⡀⢄⠰⠀⡀⢀⠂⠃⡬⠂⠈⢃⡄⠀⠘⠱⠚⠄⠄⠀⠀⠀⡀⡛⡡⠀⡠⠋
⠀⢀⡌⠺⠂⠀⠀⢀⠀⣀⢀⡠⠀⡀⠀⢄⢀⢄⠀⢀⡠⢄⠠⠀⠠⡀⢠⠠⡀⡄⠀⠀⣄⢀⠀⠠⡀⢁⢤⡘⠠
⠀⠀⠀⡖⢙⡄⢰⢡⠞⠁⡎⡔⢢⢸⠀⠈⣾⢸⠰⢁⠀⠁⡀⢆⠇⣰⡇⡆⢣⠉⡇⡇⠁⡅⢖⡈⠁⠀⠀⠁⠐
⠀⠈⢃⠀⠄⠀⠸⣜⠦⡀⢇⠣⠜⣸⢸⠙⢌⢸⠰⡘⠴⡂⡇⣤⠫⣠⠁⢲⢸⠀⡇⡇⢀⠈⠆⢈⡆⠨⣆⠃⠈⠁
⠀⠰⠀⠊⠉⠐⠄⠈⠉⠁⠀⢁⠉⡁⠘⠃⠈⠊⠀⠈⠀⠁⠙⠁⠙⠉⠊⠀⠛⠀⠑⠁⠈⠀⠒⠉⠀⡔⠒⠄
⠀⠀⠂⠠⠀⢀⡬⠂⠈⠂⢀⠄⠂⢈⠆⠈⠀⢒⠀⡤⠄⠘⠀⠡⠀⠧⠁⠄⡒⠀⡋⠨⡑⡌⠀⠈⠐⠖⠀⢠⠀⠂
⠀⠀⠀⠇⠠⠡⠘⡄⠉⠀⠋⢄⡈⠧⠀⠀⠁⠈⠈⠈⠀⠒⠴⣐⡄⠀⡠⠊⠀⠁⠱⡢⠬⠾⠀⠢⠀⠠⠐⠱⠀⠄
⠀⠀⠀⠀⠃⠰⠁⠆⠀⠂⠄⠶⠃⠀⠀⠀⠀⠀⠀⠹⠁⠘⠇⠈⠀⢴⠀⢈⠠⠐⠢⠈⠄⢰⠀⠒⠂⠂⠀⠰⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠤⠤⠤⠤⠤⠄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠒⡂⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠑⢒⡒⠒⠶⠤⠤⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣾⣛⣭⢥⢔⢒⣒⣒⠀⠤⠤⠤⠄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⢔⣒⣲⣲⣯⠽⠇
⠀⠀⠀⠀⠀⠀⠀⢠⢃⡞⠛⠀⠀⠈⠉⠈⠀⠈⠉⠙⠓⠓⠶⠶⠶⠶⢶⣍⣩⣬⣭⣐⣲⣶⢿⡿⡿⢽⢾⠾⡻⢛⠝⡩⠫⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡼⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣒⣲⡤⠤⣀⣀⠀⠀⠀⠀⢀⡿⣀⢕⠠⡎⢊⠕⡡⡪⢖⡥⣨⣈⣜⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠷⣶⣶⣶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠉⠀⠀⢸⡎⢐⠅⡪⢘⡕⣩⣊⣴⡥⡾⣟⣾⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠉⠈⠉⠋⠙⠛⠹⠻⠿⠲⠤⡤⣤⣽⣃⣐⣒⣀⣽⣧⣾⡷⣯⣿⠷⠷⢟⡛⠛⠽⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣕⢄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⢀⣁⣨⣨⣽⣿⣩⣟⡷⣶⣶⡶⣿⢽⢿⡿⡾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢚⢣⠀⠈⣮⣧⣭⣤⣤⣒⣖⣶⣶⡽⢯⢿⢟⡗⡻⢋⢟⠝⡹⣫⢫⢋⠜⡱⢣⢎⠔⡡⢫⢊⠔⡔⡹⣷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡱⡄⠸⣌⠙⣎⠝⡡⡡⣣⢊⠔⡱⡡⢊⠜⡴⡡⢋⢞⠕⡕⡡⢋⠌⡰⡡⢊⠜⡰⣡⢊⢎⢌⢌⣽⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠷⡜⡄⠉⠀⠰⢊⢎⠔⡡⡣⢊⠔⡱⢣⢊⠜⡱⡣⢪⢎⠔⡡⣫⢊⠔⣱⢋⣞⣔⣱⣡⣫⣮⢟⡿⠧⣬⣽⡷
⠀⠀⠀⠀⠀⡠⢔⡊⠅⠀⠀⠙⣾⣆⣠⣷⡥⡥⣮⢴⢥⣯⣾⣶⣷⣿⣺⣻⣷⣿⡿⠿⢿⣿⡽⠿⠯⢿⡏⠟⠿⠟⡛⠿⣛⣿⣿⡿⠋
⠀⠀⠀⡠⣛⣁⣈⣷⣒⣒⣤⠤⠤⢍⣛⣛⠛⠛⠛⠫⠩⠭⠭⠅⣉⣩⠬⠤⠶⣤⡤⠔⢀⡒⠠⠠⠀⠀⠐⠈⠍⢓⣶⣿⡿⠩⠃⣇
⠀⠀⣼⣿⠍⠉⠀⢉⡉⠉⠛⠉⠭⠷⠖⠧⠬⣭⣙⣶⡶⠶⠢⠤⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢿⢟⢡⢁⢇⢣⣛⣦⡀
⠀⠀⣧⣿⠂⠀⠀⣎⣠⡏⠀⠀⠀⠀⠀⠀⠤⠤⢀⣀⣉⠉⠙⠛⠛⠶⠶⣖⣬⠭⣅⣒⡒⠤⠤⢀⣂⣴⣿⠏⡍⠎⡌⡆⣌⣸⣾⣝⣓⠉⠭⠒⠢⢄⡀
⠀⠀⠈⠪⠷⣦⣤⣭⡿⣆⣀⣈⠃⠐⠐⠒⠠⠥⢁⣒⡒⠀⠤⠤⢀⣀⣀⠀⠀⠈⠁⠉⠘⠙⢳⢿⡻⠹⡞⢠⠡⡜⣰⢺⡽⢞⣓⣀⠠⠀⠀⣀⣴⣾⠇
⠀⠀⠀⠀⠀⠀⢀⣤⢿⣿⣿⠿⠿⡭⣗⣶⣤⣤⣄⣀⠀⣉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢂⢃⢱⢁⢣⡧⣿⠾⠉⢅⣀⠀⢀⣤⣴⠿⡛⣿⠁
⠀⠀⠀⠀⢀⡔⣽⢟⣻⠾⣭⣗⡒⠤⢀⣀⠉⠉⠚⠛⠻⠛⡶⠿⠶⣶⣶⣤⣄⣀⡀⠀⠀⠀⢯⡆⡎⣦⡼⠿⠯⠥⠤⢤⣤⣴⣿⢿⢋⢏⢎⢎⣏⡀
⠀⠀⠀⠀⢸⢸⡃⠁⠀⠀⠀⠈⠙⠓⠶⣂⡭⣑⠒⠤⢀⡀⠀⠀⠁⠀⠀⠉⠙⠛⠻⠿⣷⡶⢮⣿⣿⠯⣅⣠⣧⢼⡻⡿⡻⠩⠳⢁⢊⢎⢎⠌⢿⣀⠨⠥⠄
⠀⠀⠀⠀⠘⣽⡓⠀⠀⠀⠀⠀⠀⠐⠢⠤⣉⠑⠋⠶⠦⣬⣑⡒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣤⣞⣯⢵⢞⠏⡐⡴⡡⣡⢣⢃⢊⣎⣦⣞⡿⠓⠒⣈⣉⣉⡀
⠀⠀⠀⠀⠀⠜⣽⣒⡄⠀⠀⠂⠤⢀⡀⠀⠈⠚⠵⣀⠀⠈⠁⠊⠥⣒⡬⣉⡒⠨⢅⣪⣥⣶⢯⡗⡿⢡⢃⢂⠊⡜⡐⡰⡱⢡⣧⢾⣚⣷⣋⣉⡉⠉⠤⠤
⠀⠀⠀⠐⠀⠁⠒⠨⠿⢧⣢⢄⡀⠀⠘⠭⠴⣤⠀⠀⠈⠑⠒⠀⠀⠀⠈⠉⠚⡿⢺⢛⢫⢋⠌⡜⡰⠡⢣⢃⠌⣔⡱⢖⣏⠵⠾⠬⠭⠄⠀⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠛⠶⢯⣖⣠⠀⢀⠛⠓⠦⠤⣄⠀⠀⠀⠀⠀⢰⡅⡕⡱⢡⢊⠎⡰⡱⣡⣥⠾⣎⣯⠗⠛⡛⢀⠀⠠⠄⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⢪⢅⣒⡠⠄⣉⠐⠂⠀⠀⢸⠏⡔⡡⢣⢊⣼⢴⣻⣿⠞⠛⠑⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠲⣭⢙⡒⠤⢄⣻⣴⡵⣟⣯⠿⠟⠉⠩⠁⠀⠠⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠑⠦⠄⣿⣟⡩⠉⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠄⠒⠒⠂⠉⠉⠩⠔⠒⠒⠢⠤⠤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠔⠒⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠉⠉⠛⠒⠲⠶⠤⢤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⢰⠋⠉⠁⠒⠒⠶⠤⠤⣉⣉⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣭⣿⡞⠃
⠀⠀⠀⠀⣏⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠲⠶⠤⣬⣍⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⠉⠁⠀⡾
⠀⠀⠀⠀⠹⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠲⠶⢤⣤⣤⣀⣀⣀⣤⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⢀⣀⣉⣷⣶⠦⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠛⠓⠒⠠⣤⣀⡀
⠀⠀⠀⠈⢹⡿⠶⢦⣄⣀⠀⠀⠈⠉⠙⠓⠒⠦⠤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠚⠷⣶⣄⠀⢀⣠⣤⣾⣿⠯⠥⠴⠒⠒⠋⠉⠉⠀⠀⢣
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠉⠛⠲⠦⣄⣀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠶⠶⢤⣤⣀⣀⡀⠀⠀⢸⡅⠐⣀⣠⣽⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⣤⣄⣀⣀⣀⣤⣤⡤⠶⠶⠚⠛⠉⠉⠛⠻⠶⢿⠿⠛⠉⠁⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣠⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣃
⠀⣠⣤⢶⡛⡛⠿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠤⠶⠖⢚⣫⣉⣿⣯⡿
⣸⡿⠉⠛⠛⠿⢷⣶⣭⣛⡿⣷⣦⣄⡀⠀⠀⠀⠀⠀⢰⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠴⠶⠒⠚⣋⣉⣩⣥⣤⠶⠶⠞⠛⠛⠉⠉⢈⡿
⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣿⣿⣷⣦⣄⣀⠀⡼⢡⡅⠀⠀⣀⣀⣠⣤⣴⠶⠶⠒⠛⣋⣉⣩⣤⣤⠶⠶⠞⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣶⣶⣾⣛⣋⣉⣭⣤⣴⣶⠶⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠵
⠀⠻⠿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣀⣀⣤⣤⠾⠶⠞⠛⠋
⠀⠀⠀⠀⠈⠉⠛⠻⢷⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⠶⠾⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢶⣦⣤⣀⡀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⠶⠾⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢶⣶⣖⣓⣀⣠⣤⣶⣶⠶⠿⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⡸⠁⣾⠃⠀⢀⣤⠀⠀⠀⠀⣀⡤⠤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠉⠀⠀⠀⠀⠀⢀⡈⠉⠓⠲⠤⢤⣀⡀⢀⡿⠁⡇⠀⠀⠀⢀⠃⢀⣿⢀⣀⣾⣹⠀⠀⠀⠸⢧⣀⡠⠆⣠⠟
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠿⠛⠛⠻⣮⠓⠤⢄⣀⣀⡤⠼⠋⢀⣠⠤⢤⣄⠀⠉⣿⡟⢺⠷⠖⠒⠀⢸⠀⢸⡗⢯⢼⡇⢹⠉⠉⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⢀⣴⡿⠋⡡⠂⢀⠄⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⢹⣧⠀⣿⠀⡼⠀⠀⠀⠀⡇⢀⣿⠁⠀⣾⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦⡀
⠀⠀⠀⣠⣿⠏⠀⠐⠀⠠⠊⠀⣠⠞⠀⠀⠀⠀⠠⠤⡀⠈⠁⠀⠀⣀⣀⣸⡿⢸⡟⢠⠇⠀⠀⠀⢠⠁⣸⠇⠀⠀⡟⢠⠁⣤⠂⠀⠀⠀⠀⠀⠀⠀⣇⢈⡇
⠀⠀⢰⣿⠏⠀⠀⠈⠲⠶⢒⢋⡁⣠⣤⣤⡆⡀⣤⠀⠠⡄⢀⣴⡿⠛⢛⣿⡇⢸⣧⡞⢠⡄⢀⡄⢸⣰⠏⠀⢠⡀⡇⡜⠀⡤⢀⡤⢤⡠⣤⢠⡄⣀⣈⡾⣄
⠀⠀⣿⡏⠀⠀⠀⠀⢀⡞⠧⢶⡏⣰⠋⡞⢸⢁⣟⠀⢰⣡⣿⡏⠀⠀⣾⣿⠀⠀⣿⣀⣼⣀⣼⣃⣼⠋⡴⢋⣿⠀⣿⠀⣼⢃⣞⠀⡹⢰⠇⡞⣰⣵⠃⢹⢸⡆
⠀⢸⣿⣇⠀⠀⠀⠀⠸⢤⡤⠞⠐⢋⠈⠉⠴⠃⣤⡀⢈⣿⣿⠀⢀⣾⠟⡏⢀⡜⠈⠉⠉⠉⠀⠉⠈⠉⠙⠋⠙⠊⠈⠉⠈⠁⠙⠋⠁⠉⠈⠋⠑⣜⠒⢁⡼⠃
⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⠀⣠⠔⣡⣶⣀⣠⣴⣿⡿⠀⠻⣿⡿⠟⠁⠀⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠈⠻⣿⣶⣤⣤⣶⣶⠟⠁⣼⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣻⣛⣛⣿⣛⣛⣻⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣽⣽⣭⣽⣿⣭⣿⣭⣭⣭⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⠀⠀⠤⠤⠤⠄⣀⣀⣀⣀⣀⣀⠀⢀⡄⠀⠀⠀⣠⠀⢀⣀⢀⡄⠰⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⢡⡶⣄⣴⣴⢀⣴⣶⠰⣦⢀⡴⣆⢀⡟⣰⢂⡄⣸⢃⡴⣆⢀⡟⣰⢀⡴⣄⣤⣴⠄
⠀⠀⠀⠀⠀⠀⠀⢿⣀⡴⠻⠞⠙⠛⠷⠛⣻⠇⠐⠷⠙⠛⠷⠻⠞⠛⠻⠞⠷⠛⠛⠷⠻⠾⠷⠛⠞⠙⠟⠷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠒⠛⣋⠟⠛⠒⠶⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠦⠴⠚⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠤⠤⢤⡤⠤⠤⠶⠖⠚⠋⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⡟⠀⠀⠈⠛⠛⠒⠒⠒⠒⠒⠒⠒⠲⠶⠤⠤⠤⠤⠤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠈⢀⡶⢶⢀⣴⣠⡶⢀⣴⣶⣶⢲⢿⡄⢀⣴⢶⡄⠀⣼⠃⣴⢾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⡰⢿⣥⠟⢿⡟⣿⣡⢾⡟⣿⠃⠋⣾⣡⠺⡷⢿⣧⢾⣏⡄⢙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠟⠁⠀⠀⠀⣀⣀⣬⣥⣤⣼⣋⠀⠀⠈⠁⠀⠀⠈⠁⠈⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠉⠁⢀⣴⠟⠉⠉⠛⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⣤⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠶⠒⠛⠁





⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡤⠴⠶⠖⠒⠒⠂⠀⠀⠀⠀⠀⠐⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒⠢⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⣉⠒⠬⢍⡛⠶⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠄⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠤⢍⡒⠦⢌⣑⠂⠈⠙⠛⠶⣤⡀⠀⢀⣀⡤⠴⠖⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣐⠨⢍⡒⠬⢍⡂⠌⠉⠒⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠈⠙⠳⢮⣝⠒⠤⠁⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⢶⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠈⠙⠛⠛⠛⠷⢦⣤⣀⡀⠀⠀⠀⣀⡾⠁⠀⣀⣠⣤⡴⠶⠞⠛⠛⠉⠋⠉⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠓⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⢶⣶⣤⣤⣤⣤⣤⣤⡶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠶⠶⡶⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡸⠱⣎⠻⣶⠤⠶⠶⠶⠶⠶⠶⠶⠶⠟⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⣀⡤⠴⢶⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠁⠀⠹⣦⡿⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⡤⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⢶⡿⢀⣠⣤⣤⣤⣽⣿⣿⣿⣿⣿⣛⣛⣉⣉⣉⣉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⣀⣴⠶⠛⠋⠉⠀⠀⢸⣿⠟⠉⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠭⢭⣭⣍⣉⣉⣙⣛⣛⣛⣛⠛⠓⠶⠶⠶⠶⠶⠶⠶⣶⠿⠇
⢀⡾⠋⠀⢀⡀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠌⠁⠐⠒⠓⠒⠒⢒⡒⠒⡒⠶⠶⠿⠯⠭⠭⠅⠀⠀⠀⢰⠛⠲⢤⣀
⣾⠁⠀⠀⠀⠈⠙⠒⠄⠀⢸⡇⠀⠀⠀⠀⠤⠤⠄⢀⣀⣀⣈⣉⣛⣒⣒⣒⠒⠒⠢⠤⠤⠭⠭⠭⠍⠉⠁⠀⠀⠀⠀⣿⠀⠀⠀⠈⠛⠶⣤⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣂⠤⠤⠤⠤⣤⣤⣤⣤⣄⣀⣀⣀⣈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢻⣀⣀⣀⣤⣤⣴⡾⠟
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠓⠶⠶⠶⠶⠾⠿⠿⠿⠿⢿⣿⣿⣯⣥⣤⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⡏⠁⠀⠀⡾
⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⠀⠀⠀⠀⠀⣀⣤⠴⠶⠖⠛⠛⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢠⡇
⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠀⠀⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠉⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣄⡀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣩⣤⣤⠤⠴⠶⠒⣒⣀⣤⣤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⣤⣈⡻⣝⡒⠀⠀⠀⠀⠀⠒⠚⢛⣋⣉⣩⣭⡤⠶⠶⠖⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⣴⠖⢐⡆⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⡀⠀⠀⢠⡀⠀⣀⡀⣠⠄⡀⠀⠀⣰⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⣼⡇⠀⢚⡥⢰⣴⢄⣤⣠⠖⣠⣬⡷⣠⠖⣤⢠⠏⣴⢀⡤⣰⢏⡴⢊⡄⣸⠏⣰⢍⡔⢰⣴⢀⡄⢦⠄⠀⡾⠁
⠀⠀⠀⠀⠘⠃⠐⠘⠃⠋⠋⣾⠁⠛⣸⠏⠘⠐⠃⠘⠃⠛⠘⠃⠘⠁⠛⠘⠃⠛⠈⠻⠐⠃⠘⠑⠋⠃⡾⠁⠞⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠀⠙⠛⠛⠒⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⣀⠠⠔⠒⠀⠀⠐⠒⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣮⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⢀⠔⣊⠄⠂⢀⡠⠔⠂⠀⠀⠀⠀⢣⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄
⠀⢀⠔⡡⠊⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠂⣸⡄⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⠀⠀⢀⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⢠⠃⠜⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢷⣄⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠑⣒⡒⠒⠛⠉⠉⣼⡡⠤⢒⣞⠀⠀⢀⡠⠔⠃⢰⠀⡆
⡌⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⣴⠃⠀⠈⢦⠢⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡜⠀⠈⠁⠐⠒⠒⠒⠒⠊⠁⠈⠀⠈⠁⠀⠀⠸⠈⢳⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢉⠔⠞⠁⠀⠀⠀⠀⠱⣌⠐⣆⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠁
⢡⠀⠀⠀⠀⠀⠠⢐⣉⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢾⠓⠇⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠘⡄⢠⠄⠐⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⢸⣡⡆⠀⠤⠔⠖⡀⠀⢸⠁⠇⠀⠀⠀⠸⠏⠀⠀⠀⠠⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⣯⡀
⠀⠐⡌⡄⠀⠀⠀⢀⣄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢡⢀⠞⠢⣬⠇⠀⠀⠑⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡸
⠀⠏⢻⣧⠀⠀⠀⠀⠁⠀⠀⠀⡠⠀⠀⡆⠀⠀⠀⠀⢀⡜⡇⣠⠜⠁⠈⢰⠁⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠈⠒⠊⠁⠀⠀⠀⠀⠀⠀⡠⠋⠉
⠀⠈⠣⡘⢣⡀⠀⠀⠀⠀⠀⠀⠈⠠⠠⠂⠀⠀⢀⣴⣫⠵⠚⠁⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠄⣀⡀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠉
⠀⠀⠀⠀⣸⢹⠢⣄⣀⠀⠀⠀⠀⢀⡀⠤⠔⢪⠋⡇⠑⣄⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣏⠙⠬⣉⡭⢛⣯⡀
⠀⠀⠀⠀⡟⠈⡄⢱⠀⠉⠉⠉⢩⣟⡒⠒⢚⠇⠊⠇⠀⠈⢧⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠁⠹⡂⠾⡲⠛⠁⠘⠲⢄
⠀⠀⠀⠀⢣⠀⠀⠈⡄⠀⠀⡠⠫⡉⠛⠛⠁⠄⠚⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠑⠤⢠⠇⠀⡐⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢄⡠⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡔⠂⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢠⣼⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠃⢤⣀⠀⠤⠤⠤⠼⠄⡀⠀⢈⠁⠁⠗⠐⠒⠒⠀⠀⠀⢰⠏⠙⢅⣀⡠⠓⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠰⢟⣧⣀⠀⠀⠉⠉⠋⠉⠕⠀⠀⠀⠀⠀⠀⢀⡀⣉⣲⢄⡀⠀⠀⠀⠀⠉⠉⠓⠘⢀⣀⣀⣠⣤⣤⡤⠬⠓⠓⠚⠃⠀⠀⠀⠀⠀⠉⠑⠂⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠫⢅⡐⠒⠓⠃⠂⠵⠂⠀⠀⠂⠛⠉⠙⣨⡤⣤⠤⣤⣆⣀⣀⣀⢤⣄⣀⣀⣛⣉⠀⠠⠤⠤⠤⠤⠒⡖⠒⠒⠒⠒⠂⠀⢱⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡴⡖⡶⠶⠶⠶⠶⠛⠛⠉⠉⠉⠙⢢⡀⠈⠀⠀⢀⣀⣀⣀⡆⠀⠀⠀⠀⢩⣽⠉⠉⠈⠁⠈⠒⢄⣀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠄⠤⠤⢧⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢏⠉⠀⠀⠀⠀⠀⠑⣄⠀⡀⠀⠀⢣⠉⠒⢤⡀⠀⠀⠀⠙⡄⠉⠒⢺⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⣦⣀⠀⠀⠈⡇⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣸⡀⠀⠀⢀⣀⣀⣀⡈⡗⠀⠀⠀⠘⡆⠐⠂⡟⠁⠐⠀⢠⠓⠒⠂⢸⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠈⠙⠢⠴⡇⡿⢤⠀⠀⠀⣤⠤⠤⠔⠒⡖⠉⠁⡸⠉⠉⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⡆⠀⠀⠇⠀⠀⢀⠮⣀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢅⠀⠀⠀⠀⡇⡇⠈⢆⠀⢀⣸⠀⠀⠀⠀⠟⠈⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⡇⠀⠀⠦⣤⡀⣜⠑⠢⣑⠢⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡑⢄⡀⠀⡇⡇⠀⠀⠻⣄⠈⢇⠀⠀⠀⢰⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠫⠩⠇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠑⠢⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢍⠢⠇⡇⠀⠀⠀⠈⠲⢜⡇⠀⠀⠈⠓⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣴⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣶⣿⣿⣷⣶⣶⣶⠄⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⢀⣴⡾⠛⣋⣙⣛⡻⢿⣦⡀
⠀⠀⠀⣼⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⡆⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣼⣿⣴⣿⣿⡿⣿⣿⣦⣻⣷
⠀⠀⠀⣿⣿⣿⡏⠀⠀⠉⠻⢿⡿⣿⡇⠀⠀⣿⣿⢟⣁⠉⠈⠙⢋⡉⠹⣿⣿⡄⠀⢟⣿⣿⣿⠋⠀⠘⢿⣿⣿⣿
⠀⠀⠀⡜⡏⠆⢀⣦⠀⠀⠀⣰⣶⢘⠷⠀⢀⣿⡇⠀⢀⡄⠀⠀⠀⣠⡀⣿⣯⠀⢠⠘⢿⣯⠀⠀⠀⢀⠉⠹⣿⣇⠀⠐⣶⣦
⠀⠀⠀⠀⣡⠀⠘⠛⠀⠀⠀⠛⠃⣨⡔⠀⠀⠀⡃⠀⠚⠋⠀⠀⠘⠛⠁⢇⠆⠀⠈⢰⡀⠛⠃⠀⠀⠛⠇⠀⣘⠉⠀⠙⠄⠁
⠀⠀⠀⣰⣿⡄⠀⠀⠀⠀⠀⠀⢀⣿⣷⡤⠀⠀⠐⠀⠀⢠⣀⣠⠄⠀⡀⠀⠀⣀⠀⣿⣇⠀⠀⢤⣤⠆⠀⢀⣿
⠀⠀⠀⠛⠿⠿⣶⢤⠀⡀⡤⠶⠛⠛⠉⠀⠀⠀⠀⠀⠀⣀⠈⢁⠀⠀⠀⠀⡤⠈⡠⢙⠛⠳⢤⡀⢀⣠⡴⠿⠿⢗
⠀⠀⠀⠀⠠⠒⠀⠀⠀⠀⠁⠃⠒⠀⠀⠀⠀⠀⢠⣴⠆⠑⠀⠀⠁⣶⡁⠀⠀⠸⠁⠀⠸⠂⠁⠀⠂⠑⠀⠂⠀⠀⠄
⠀⠀⠐⠁⠀⡠⠂⠀⠀⠀⠀⠀⢂⡀⠀⠦⡀⡀⢠⡟⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⢈⠃⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠓⠛⢀⠉⠀⠀⠀⠀⠉⠰⠃⠚⠂⡰⠁⣾⠃⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⢙⠌⠉⠀⠀⠀⠀⠩⠁⠍⠀⠀⠀⡘⡇⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⢄⠨
⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⣤⣀⣀⠀⠀⠀⠀⣀⣀⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣤⣀⠀⠀⠀⠀⢀⣀⣸
⠀⠀⠀⠀⠿⠷⣶⣦⣤⣤⣴⣶⠾⢿⠀⠀⠀⠀⢿⣿⣿⡿⠁⢿⣿⣿⣿⠇⠀⠀⠜⠈⠙⠛⠛⠛⠛⡛⠛⠍
⠀⠀⠀⠀⠈⠁⢐⠂⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠨⠀⠉⠀⠀⠀⠉⠉⠠⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⡄⡄⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⢰⠠⡄⠀⠀⠀⠀⡄⠠⠀⠀⠀⠐⠂⠠⡁⠈⢆⠀⡄⠠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠤⠉⠀⡆⠄
⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠘⠠⠔⠋⠀⠀⠀⠀⠙⠦⢔⠃⠀⠀⠋⠰⠚⠁⠀⠀⠀⠀⠙⠲⠨⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣶⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠉⠉⠉⠙⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠩⠅⠈⠭⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠄
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠉⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡁⠐⡶⢆⣀⡀⠀⠀⠠⠜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠩⠭⡉⠉⠛⠿⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⣿⣿⣿⣿⣿⣿⣿⣯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⡀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿
⣿⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠒⢻⠟⠛⠋⠓⠀⠀⠀⠀⠀⠀⠛⠛⠛⠿⣿⣿⣿⡇⣿⡾⣆⡟⡏⣽⣿⣿⣿⣿⣿⣿⣿⡖⠀⠀⠐⠒⠀⠐⣾⣿⣿⣿⣿
⣿⠇⠤⠄⠀⠀⠀⠀⠀⠀⠀⠒⡇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠏⠉⠉⠉⠉⠁⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠔⠒⠂⠀⢸⣿⣿⣿⣿
⣿⢤⡄⠀⠀⠀⠠⠄⠀⠀⠀⠒⠋⠀⠀⠂⠀⠀⠀⠀⢠⡤⠀⠀⠐⠐⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢀⣄⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⡼⠤⠤⠤⠤⠤⠦⠤⠤⠤⠶⠦⠤⠀⠀⠀⠀⠠⢴⠾⠤⠤⠤⠤⠤⠡⠤⠠⠄⠑⠤⠤⠤⠤⠴⠉⠍⠍⠉⣿⣿⣿⡆⠀⠈⡁⠀⠀⠀⠀⣤⣹⣿⣿⣿
⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠏⡀⠀⠀⡄⠀⠀⠸⠉⠀⠈⠉⣹⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⠀⠀⠀⠀⢰⣀⣀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠋⠀⠙⠢⢴⠁⠀⠀⡀⠀⠀⠀⢠⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⠻⠼⠇⠘⠘⠃⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⣀⣉⠒⡄⠀⠀⠀⠀⣧⡀⠀⠀⣾⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣴⣿⠀⠏⢧⡄⡄⣄⣀⣘⡦⣸⣿⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠈⠈⠀⠺⠉⣴⠃⣿⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣷⣶⠂⠀⠀⠀⠂⢰⣶⣶⣶⣶⡂⠀⠀⠀⠀⢰⣶⠂⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⡆⠐⠀⠀⠀⠀⣶⣿⠀⠀⠀⠀⠈⠐⠈⠀⠀⠐⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣧⣤⣄⣠⣠⣼⣿⣦⣠⣄⣠⣠⣾⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⣿⣿⣶⣤⣤⣤⣤⣴⠀⠀⠀⠀⣀⣀⣤⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠠⠤⠄⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠠⠠⠂⠀⠀⠀⠀⠀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠒⠲⠦⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠲⠦⠤⢤⣀⣀⡀⠒⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠒⢛⡏
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠶⠦⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠞⠛⠉⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢲⡞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣧⠦
⠀⠀⠀⠀⠲⣤⢉⡉⠉⠉⠉⠙⠓⠲⠦⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠤⣦⣤⡀⠀⠀⣀⣠⣴⣶⣟⣋⡩⠤⠄⠀⠀⠈⠁⠆
⠀⠀⠀⠀⠀⣧⠀⠀⠉⠛⠶⣤⣄⡀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠶⣤⣄⣀⣀⠀⠀⠀⠀⠘⣯⠀⠀⢀⣹⣿⣶⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢤⣄⣀⠀⠀⢀⣀⣀⣀⡤⠤⠽⠛⠛⠓⠶⠦⣼⣶⠾⠛⠋⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⢀⣠⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⢀⣤⡤⠟⠛⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠄⠐⢒⣚⣭⣻⡗
⠀⣾⡟⠙⠛⠷⢷⣤⣈⡙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠔⠒⠒⢉⣉⣁⣤⠤⠴⠒⠒⠛⠋⠉⢉⡿
⢸⣿⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠾⠿⣿⣶⣦⣀⡀⢰⠃⡄⢀⣀⣀⣤⣤⣴⠶⠶⠞⢛⣛⣉⣭⣤⡤⠶⠖⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢷⣾⣿⣟⣫⣭⣥⣴⣶⠶⠿⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣹⣀
⠀⠈⠛⠷⢶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡶⠶⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⣴⣦⣀⣀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⠴⠶⠞⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⣯⣤⣶⠶⠾⠛⠋⠉⠉






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢰⡀⠀⢠⡀⠀⣿⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⡿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠸⡇⠀⡟⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⣧⠀⠀⠀⡄⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⠀⠀⢠⡇⠀⠀⡰⠃⠀⠀⠀⣠⠴⠂⠀⢀⡀⠀⢾⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⠀⠀⠀⠇⠀⠈⠀⠀⡐⠁⠄⠀⡠⠞⠁⣀⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠦⣄⣀⠀⠘⠶⣄⡁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⢀⣀⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠤⠒⠚⠉⠀⡀⠀⠀⠀⠶⠖⠂
⠀⠀⠀⢠⡖⠦⣠⠶⣄⢶⣄⡆⣴⠲⡎⣷⣶⠐⢠⣦⠠⢶⠶⢰⡆⢰⢰⡆⠀⢠⣦⠠⢶⡶⢰⡆⣴⢶⡄⣶⣄⡆⣴⣶
⠀⠀⠀⠈⠓⠂⠙⠓⠃⠚⠈⠃⠙⠚⠃⠛⠙⠃⠛⠙⠃⠘⠃⠈⠓⠊⠘⠓⠂⠋⠙⠃⠘⠃⠘⠃⠙⠚⠁⠓⠙⠃⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠚⠛⠉⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠂⠤⣄⣀⠈⠉⠛⠒⠲⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣠⠔⠊⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠈⠙⠶⠄⠈⠙⠓⠶
⠀⠀⠀⠰⣾⣷⠆⠉⠁⠀⠀⠀⡴⠞⠉⢀⠀⠀⢀⠜⠁⠀⣠⠀⠀⠀⢰⠀⠀⢳⡈⢢⡀
⠀⠀⠀⠀⠋⠙⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⢀⡴⠃⠀⠀⢰⠇⠈⠀⡇⠈⡆⠀⠀⠹⠄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠰⠋⠀⣰⠂⠀⠀⠀⠀⠀⣷⠀⣧⠀⢀⠀⠀⠀⠀⠻⠆⠀⠀⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⣿⠀⢸⡄⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⠃⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣠⣤⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣿⣿⣶⣦⣿⣷⣶⣶⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣟⠿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⣙⠿⣿⣄⡈⠛⠿⣿⣿⣿⣇⣹⣿⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣷⣶⣎⠛⠻⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣶⣴⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣭⣤⣤⣤⣽⣛⡻⢿⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣶⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡟⢿⣟⣷⣟⣿⣿⣿⣿⢹⡟⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣯⡻⢤⣈⡙⠻⣯⡻⣿⣿⣘⣶⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡷⣮⡙⠲⢜⣷⡘⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⣿⣿⣷⣾⡅⠄⢿⣷⡘⣿⣿⣿⣽⣿⣿⣿⣧⣶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⣤⠠⣽⣷⡈⣿⣿⣿⢹⣿⣿⣿⣿⢛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢯⣽⣿⣿⣿⣦⣬⣿⣷⡘⣿⣿⣸⣿⣿⡿⢁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⡛⠻⣷⡹⣿⣿⡿⢉⠁⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣛⣿⣿⣿⣿⣿⣦⣾⣷⠙⣿⣧⣾⡜⣿⣷⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⣿⣿⠁⢹⣿⣿⣿⣿⡟⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡛⠛⠛⠉⠄⠙⠿⣧⢹⣿⣄⡎⠹⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣮⣻⣿⣿⣿⣿⣿⣷⣶⣿⣧⢻⣿⣇⣰⡟⠈⠹⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣯⣍⣉⡉⠋⢛⣿⣿⣇⢻⣿⣿⠀⠀⠠⣼⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⢻⣿⣿⣟⣻⣿⣟⣿⣿⣿⣿⣆⢻⣿⣾⠆⣰⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣿⣭⣭⣭⣿⣶⠶⣌⣻⣿⡎⣿⣯⣴⣷⠋⢩⣿⣿⣟⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠁⠲⢆⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡥⠤⠀⢿⣿⣿⡌⢿⣿⡟⠉⣈⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠉⠙⠛⠋⠙⢶⣤⡀⠀⠀⠀⠀⠀⠀⠘⠻⠛⠿⠿⣿⢷⣶⣶⣤⣉⣻⣿⡜⣿⣷⡞⢿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⣠⡄⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⣼⠇⠈⠙⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣯⣥⣉⠻⣿⡜⣿⣿⠻⣿⣿⡛⠉
⠀⠀⠀⠀⣰⣿⣿⣿⣷⣁⣄⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣧⡉⢻⣿⣿⣴⠟⡟
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣾⣿⠧⣠⡄⣀⡀⠀⠀⠠⣾⡏⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠟⠛⠛⠶⣯⣿⠻⣿⡹⣿⡟
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣤⠄⣾⡟⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠸⠛⠓⠿⣷⡿⣏
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⠏⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⣿⣆
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣏⡶⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⢠⣿⣿⣿⣿⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣧⣿⢿⡇⡀⢸⡇⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠸⣿⣿⣿⣿⣟⠀⠨⠴⢨⣿⢛⡩⢉⢻⣿⣿⣿⡟⠻⣸⢃⣿⡇⠀⠀⠀⠈⠛⠷⣶⣄⠀⠀⠀⠀⣀⣛⡠⠤⢾⠋⠀⠀⠐⠲⣆⣀⠀⠀⠀⠀⠀⠀⢃
⠀⠈⠉⠙⠛⠻⣦⣴⣾⣾⣇⠀⠙⠊⢠⣿⣿⣏⣀⣠⡶⠛⠫⣥⣆⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⡙⠁⠀⠀⠈⠁⠀⠘⠳⣤⢈⡋⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠉⠙⠲⠦⣤⣌⣛⠳⠾⠷⠿⢿⡿⢟⡿⠁⠀⠉⠀⠊⠉⠑⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⠀⠂⢤⣤⠀⠀⠀⠈⠻⡿⣤⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⢶⣤⡞⣁⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠃⠤⠀⢀⠀⡀⢟⡀⠄⠀⠀⡄⠁⠀⠀⠀⠀⠀⠘⠈⠻⠦⣄⠐⢀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢄⡀⠀⠄⠀⠴⠀⠀⠀⢠⡦⠀⠀⢠⠟⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠠⠀⠐⠚⠋⠉⠀⠐⠉⠀⠀⠀⢸⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣉⣀⣉⣿⡀⣸⣉⣉⣧⣉⣸⣿⣋⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡩⠉⠉⠉⠁⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠤⠤⠀⠀⠀⠠⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠤⠄⠀⠀⠤⠤⠀⡄⠀⠠⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡤⠤⣀⡀⠀⢄⡀⢀⡄⠀⠠⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠒⠁⠀⠀⠀⠀⠁⠐⠀⠤⠄⡀⠠⠤⠀⠀⠀⠀⠀⢀⠄⠐⠂⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⠤⣀⠀⢀⢧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁⠈⢏⢄
⠀⠀⠀⠀⠀⡀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠪⠅⠀⠀⠀⠀⠀⢠⠛⠰⠆⠈⠆
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠄⢀⠀⠀⢀⡤⠖⠋⠁⠈
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⢄⡀⠀⠀⠀⠀⠀⢠⡔⠀⡌⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡸⣆⡀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐⠀⠀⠀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣒⣠⣤⣌
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⢠⡾⠋⠀⠀⢨⠇⠀⠀⠀⠀⠀⢸⠀⢹⠀⠀⠀⠀⠒⢤⡿⠄⠀⠀⢀⡾⠁⠀⠀⠤⣴⠯⠄⠀⢀⡀⠀⠀⠀⠰⣄⡇
⠀⠀⠀⢰⡟⠀⠀⠀⠔⢋⡖⢦⢠⡄⡀⢠⡬⠓⠞⡤⠀⣠⠄⠀⡾⠀⣴⢀⡄⣼⠃⢠⡤⠀⠀⡟⠀⣬⢀⡧⢹⢀⣶⣤⠀⠘⢳⡄
⠀⠀⠀⢸⡇⠀⠀⠀⡠⢾⣡⠏⠾⢨⡇⢏⢴⡇⢸⡃⠸⠧⢾⠀⢧⠔⠇⠺⠀⠻⠄⠻⠀⠧⠈⠧⠘⠇⠸⠧⠏⠸⠃⢸⠀⢀⠞⠁
⠀⠀⠀⠀⠙⠒⢒⣋⡤⠤⠀⠀⡠⠐⠃⠉⡽⠙⠛⠒⠦⢤⣀⠀⠠⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠒⠂⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⢧⣀⠤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢠⣤⣶⣾⣿⣶⣦⠤⠀⠀⣀⠀⠀⢀⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢳⡄⠐⠆⠀⠀⠈⣽⣿⣟⣯⡅⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⣼⡆
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⣾⠇⠀⠀⠘⠀⠀⠉⠙⠛⠉⠁⠘⠀⢠⡶⠀⠀⠀⠘⠷⠶⠾⢿⣿⠋⠉⠓⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠚⢋⣴⡶⡄⢰⡄⡠⣶⡄⣠⣴⢦⣴⠀⢨⢷⡄⢀⣴⠶⣄⡀⢠⣿⠃⢠⣿⠂⣦⠀⣿⠃
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣿⠟⢶⢷⣿⡟⢱⡿⢱⣿⠁⣠⡿⢀⢎⣾⠃⣾⡏⢀⣿⢁⣿⡏⡰⢙⣿⠴⠋⠘⠟
⠀⠀⠀⠀⠀⠀⠹⣇⣀⡠⠚⣿⡠⠃⢺⡟⠀⢿⡣⠺⣯⠴⣿⣇⠎⢸⣏⠜⢿⣠⢾⣯⠞⣿⡰⢱⣩⡿⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⣸⡟⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣇⣠⠟⠁⠀⠀⠶⠄⠀⣀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⢸⣿⠘⣦⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣶⢸⣿⡇
⠀⠀⠀⠀⠀⠘⣿⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣇⢸⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⣿⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠸⣿⣾⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣼⡟⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢨⡗⣿⣿
⠀⠀⠀⢀⡀⠀⠠⢻⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢸⡇⣿⣿⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠈⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⢀⣀⢀⣀⣀⡀⠀⠀⡀⣠⣤⠿⠀⢻⡧⣿⣿⠤⠄⠐⠲⠶⠖⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠿⠋⠀⠀⠀⠊⠋⠀⠀⠀⠀⠀⠐⡶⣆⣠⣀⣀⣀⣸⣷⣿⣿⣠⣦⣴⢶⠶⡤⠿⠿⠿
⠀⠀⠀⠀⣰⣶⣶⣄⡈⢿⣿⠟⢀⡀⠀⠀⠀⠂⢘⠛⢚⣛⣛⣿⣏⣭⣭⣭⣭⣬⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⠶⣿⣾
⠀⠀⠀⢀⣻⣿⣿⣿⢿⣿⣷⣾⣿⣿⠟⠃⠘⠛⢻⣿⠛⠛⠛⠙⠛⠉⠉⠉⠉⠉⠉⠀⠀⠀⣿⡇⠈⠁⠀⠀⠀⠀⠀⠀⢹
⠀⠀⣀⠘⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣠⡆⠀⠀⢻⠄⠀⢰⣦⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⣿⠃⠓⠟⠁⠀⠀⠀⠀⠀⢀
⠀⠀⣻⡀⢛⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⡇⠀⣄⠧⠡⠄⠀⠙⠀⠈⣁⣀⣀⣀⣀⣀⣀⣀⣛⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⣽⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠅⠸⠾⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠍⣭⠉⠁
⠀⠀⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡿
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡃⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⡤⠤⠤⠤⠤⠤⠶⠶⠖⠒⠒⠚⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣿⣿⠛⠿⢛⠿⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉
⠀⠀⠀⠀⠀⢀⣀⡠⠤⠒⠊⣁⣉⣣⠀⣀⣀⠀⠀⠀⠀⠀⠀⠤⠀⠂⠈⠁⠀⠀⠀⣠⠖⡠⠖⠑⠢⢄⡀
⠀⣠⠴⠒⠋⢉⣀⠤⣖⣚⠭⠕⠒⠉⣁⠄⠀⠀⠀⠀⢀⠔⠀⠀⢀⣀⣠⠤⠖⠚⠉⠑⢜⠀⠀⠀⠀⠀⠈⠑⠦⣀
⡎⠈⢿⠖⣫⡕⠚⠉⢁⣠⠤⠖⠚⠽⡁⠀⠀⠀⢀⠔⢁⡴⣞⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀
⡇⠀⢸⡏⠀⠉⡖⠋⠉⠀⠀⣠⠴⠂⡉⠆⣠⣐⠁⠀⡘⡄⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⡇⠀⢸⡇⡄⢀⡇⠀⠀⢠⠎⡡⠚⠓⣿⢟⠅⡀⠠⠁⠢⣉⢆⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⣀⡴⢺⡁
⡇⡆⡟⡇⡇⢸⠃⠀⣠⠼⣟⡠⠐⠒⠯⠷⢤⣀⠑⢄⠀⡅⠊⠳⡀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⡤⠖⠋⣁⠔⣪⢿⣀
⠧⠦⡇⡇⡇⢸⠀⢈⣷⢤⡀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⢹⣐⡤⠟⢆⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠿⢂⠤⣊⠕⠋⠀⠀⠈⠓⣦
⠀⠀⠋⣧⢯⡿⣞⠙⢯⡢⣈⠓⠦⣄⡀⠀⠀⠀⢀⡴⠛⢿⠑⢄⡀⠈⠳⡀⠈⢣⡤⠴⠖⠚⠋⠉⣉⡠⢔⣻⠮⣵⠊⠁⠀⠀⢀⣠⠴⢻⡁
⠀⠀⠀⠀⠀⢳⡿⣙⠲⢭⡢⢕⠤⡀⠉⡲⣤⠚⠁⣠⢔⡽⠦⡀⠙⠢⣄⠙⢦⡏⠌⠠⠄⣒⣪⠭⠒⠊⠉⠀⣀⣬⠛⣀⡤⠞⠋⢀⡴⣊⠽
⠀⠀⠀⠀⠀⠀⠉⠪⣕⢤⡈⠙⠿⣦⣄⣐⣿⠴⣋⣴⠙⢦⡀⠈⠲⢄⠈⠑⢦⡉⠒⠋⠉⠀⠀⠀⢀⣠⠴⠛⣡⢜⣿⠁⢀⡠⢚⡥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠮⣒⠀⢐⢽⢀⡤⠚⠁⠘⢦⡀⠈⠳⣄⡀⠙⠦⣀⠈⠓⢤⡀⣀⡴⠚⠉⢀⠴⣪⠔⣋⡼⢟⡩⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠛⠁⠀⠀⠀⠀⠀⠈⠲⣄⠀⠙⠲⣄⠈⠳⢤⡀⣼⠕⠀⠤⣊⣵⣪⠖⠋⣁⠿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠈⠙⠦⣄⠉⠛⠦⣶⡾⠛⠉⢀⡤⣊⡵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠈⠑⡶⡫⠅⢀⡠⢚⡥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣼⣧⠆⣀⡥⠖⠉

⠀⠀⠀⠀⠀⡠⠀⠠⡀
⠀⠀⠀⢠⠎⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠆⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡄
⠀⠀⠀⡏⠀⠀⠐⠊⢠⢶⢀⢄⡴⢀⡔⡦⠰⣥⠀⣠⢖⡄⡾⠓⢀⠄⣴⢠⠏⢀⠖⡆⢲⠓⢂⡌⢠⢶⢠⢄⡶⢀⢾⠁⠀⡞
⠀⠀⠸⡇⠀⠀⢀⠄⡏⠄⣾⢸⢡⡏⣰⠃⠁⠃⣰⠃⡞⢰⠃⡠⡏⣸⠃⡎⢠⠏⢼⢡⠇⢀⠞⣠⣃⠄⡿⢸⠡⣆⣸⠔⣈
⠀⠀⠀⠙⠒⠒⠁⠀⠁⠀⠁⠈⠁⢈⠏⠀⠀⠁⠀⠁⠈⠀⠉⠀⠉⠈⠁⠈⠀⠉⠀⠀⠉⠀⠈⠀⠁⠀⠁⠈⠁⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⢟⣽⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⢯⣾⣿⣿⢟⣼⠿⣿⣿⡿⣿⡿⢿⣿⣻⣿⣿⡿⣿⡿⢻⣿⣿⣿⣿⢃⣾⡿⣿⡿⠻⣿⣟⣿⠿⣿⣿⣿⣿⡟⣻⣿⡏⣽
⣿⣿⣿⣼⣿⣿⣿⣿⢏⣦⠏⡞⣰⢏⡾⢱⣷⢰⣿⢫⡞⣼⣲⣿⢏⡾⢡⢏⣾⢏⡾⣸⢧⣾⡏⣼⢣⣦⠏⡞⣰⢿⣗⣿⣿⣾⣿
⣿⣿⣿⡸⢿⡿⣿⣿⣘⣿⣠⣇⣻⣼⢥⣿⣿⣻⣧⣻⣜⣧⣿⣿⣺⣇⣿⣼⣧⣺⣥⣯⣺⣯⣼⣧⣻⣿⣰⣇⣿⣜⣩⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠙⠉⣿⣿⣿⣿⠻⠏⠙⠋⠻⣿⣿⣿⡿⣿⣿⣿⡿⣻⣿⣿⣿
⣿⣿⣿⠿⠟⠛⠛⠉⠉⠁⠀⠀⢺⣿⣿⣿⠋⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠿⣭⡀⠀⠀⠀⠀⠈⣿⡟⠁⠀⠀⠀⠀⠀⠠⢿⣿⣧⡈⠻⣿⠃⡿⢟⣽⣿
⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣟⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣏⣠⣀⡈⢓⣤⣀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡄⠀⣴⣿⣿⣿
⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠋⠉⠉⠛⢿⣿⣿⣷⡾⢿⣿⣅⠀⠀⠀⠀⠀⣠⣾⣿⡱⢎⣵⣶⢶⣮⡛⢿⣿
⣿⣿⣿⣿⡀⠀⠀⢀⣀⣀⣤⣤⣤⣾⣿⣿⣆⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠻⣉⣠⣴⣾⣿⣿⣷⣶⣶⡼⠗⠿⣿⣿⣿⣿⣿⣇⠶⣹⣿⣶⣽
⣿⣿⣿⣿⡷⠛⠛⠟⠛⠋⣹⣿⣿⣿⣿⣿⣿⣷⣦⣤⣴⣾⣿⡟⠀⣀⣀⣤⡄⠀⠀⠀⣽⣿⣿⡟⠉⠉⠉⠉⠉⠁⠀⣠⣿⣿⡿⠛⠉⠛⣿⣿⣿⣿⣿
⣿⡛⠉⠉⠀⠀⠀⠀⣀⣴⡟⠛⠋⠉⠁⢨⣿⡿⢿⡿⣿⡿⢿⣿⠿⠿⣿⣿⣷⣿⣿⣿⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠁⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣷⣦⣴⣤⣤⣾⣿⣿⣷⡄⠀⠀⠀⠀⠙⣷⣧⣯⣾⣷⣽⣿⣥⣿⣽⣼⣿⣴⣧⣷⣶⣾⣿⣧⣀⣀⣀⣀⣀⣀⣠⡿⣠⡦⠀⠀⠀⠀⠸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢀⣰⣿⣟⢹⣿⠁⣿⣿⣿⣿⢆⣿⣿⣿⣿⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣿⢦⣿⣿⣿⣿
⣿⣿⠟⠋⠁⠀⠀⢸⣿⣿⣿⣿⣷⣶⣦⣿⣿⣿⣿⢰⣼⣦⣇⣽⣿⣿⣮⣯⣼⣴⣧⣇⣴⣙⣿⣿⣿⡿⡿⢩⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿
⣿⠏⠀⠀⣠⣴⡆⢸⣿⠛⠛⠻⣹⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡯⢾⡻⣉⠹⡋⣿⠉⣯⡝⣹⣿⠇⠀⠀⠈⠉⠉⠉⠛⠛⠻⣿⠏⠀⠈⠁⠈⣿⣿
⣿⣦⣠⣾⣿⣿⡧⢸⣿⣿⣶⠶⣛⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢠⣤⣀⣀⣀⠀⠀⠀⠠⣿⣤⠀⠀⠀⢈⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠘⠛⠙⢷⣠⣿⣿⣿⣿⡿⣻⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀⠀⣺⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⡆⢴⣶⢿⠿⢿
⣿⣿⣿⡿⠋⠀⠹⣿⣧⣄⣀⡀⢀⣠⣿⣿⣿⣿⣿⣾⣿⣿⣷⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⠟⠛⠿⠿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿
⣿⡿⠋⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣈⡿⣿⢿⡽⠿⠿⢿⣛⣛⠋⠀⠈⠈⠛⠿⢿⣿⡿⠋⠙⠓⠆⠠⢤⣼⣿⣿⡟⠉⠙⠉⠉⠛⢻⣿
⣿⡇⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⠛⣻⡏⠀⣼⣿⣿⣿⡏⠻⣿⣷⠶⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣁⣄⠀⠀⠀⠀⠴⣿⣿
⣿⡇⠀⠀⠀⠀⠀⣴⣿⣿⠟⠃⠐⣿⠀⢰⠛⠻⢿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⣀⠀⠀⣨⣿⣿
⣿⣧⣄⡀⠀⢀⣾⣿⣿⣯⣄⣀⠀⣡⣠⡏⠀⠀⣼⣿⣿⣿⣿⣿⡀⠀⠀⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣀⣀⣈⣿⣷⣤⣾⣿⣴⣶⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⣄⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣿⡀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⠛⢿⡇⠀⢦⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡸⡿⣷⠀⠘⡇⠀⢀⠂⠀⢀⣀⣿⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⡷⡏⠀⠀⠑⣠⣄⣴⣾⠟⠁⠀⢀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣷⣿⣦⡀⠀⣿⢿⣿⣿⡷⠉⢄⣸⣟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣭⣿⣿⣿⠙⠐⠋⠉⠉⠉⠃⠀⢸⣿⡿⢯⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣿⣹⣿⣿⣿⣿⣦⢠⣾⠟⣿⣿⣿⡿⢿⡇⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡟⣿⣉⡟⠉⠉⠙⢻⢸⠋⠙⠉⢻⣿⡷⢾⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⠛⣙⢿⡄⠀⣠⡛⠘⣣⡀⠀⣿⣯⡁⢱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠘⢿⡏⣷⠀⠏⣭⣭⡌⠃⣸⣿⡷⢠⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣠⡌⣇⣿⣷⡀⠈⠛⠀⣰⢿⣻⠄⣘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠂⢨⣭⢩⡄⡁⡅⠰⠛⠓⠒⠻⡅⠀⢠⠀⠀⠀⠶⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⡿⠋⠻⣿⣿⣿⣿⢿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⠴⠫⣄⣠⠦⠦⡭⠛⢋⡼⠁
⠀⠀⠀⠀⠀⡠⠒⠉⠁⠀⠀⠈⠉⠐⠠⡀⠘⡖⠚⠁⠈⠒⠚⠁⢀⢸⡁
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠈⠦⣀⠀⠀⢀⠴⠁⠘⡷⠡
⠀⠀⠀⠀⠀⠈⠐⠠⠤⠀⠀⢀⡀⠔⠊⠀⢀⡤⠊⠉⠐⢲⡍⠠⢆⠁⠀⡰⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⠣⣀⢀⣀⠴⣚⠉⡆⢸⠠⠊⠀⠀⠉⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⢉⢹⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡣⠁⠁⠙⠆⠀⠀⠀⡇⠀⠀⠀⠀⢰⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠅⠛⠀⠀⠀⢼⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡀⡤⠄⣌⢀⡀⠀⠀⣀⣘⣀⠀⢀⣸⡤⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡑⠘⠂⠈⠂⠘⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠃⠀⠒⠒⠲⠒⠠⠤⠤⠄⠠⡄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠋⠀⠀⠀⠙⠛⠃⠤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⣤⣤⠔⠛⠁⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⢴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠙
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⠇
⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠶⠶⠤⠀⠀⠀⠀⠀⠶⠶⠦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⢀⣠⠶⠆⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⠶⣀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠉⠳⣦⡀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⣿⠀⠀⠀⢀⣴⠋⠀⠀⣀⣶⠾⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠷⣦⡀⠈⠻⣄⠀⠀⠀⠙
⠀⠀⠀⠀⣿⠀⠀⠀⡞⠁⠀⠀⣀⣁⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣈⣡⣀⠀⠉⣷⠀⠀⠀⣾
⠀⠀⠀⣠⣿⠀⠀⢀⣧⣰⠶⠟⠉⠉⠉⡙⠋⠙⢿⡉⠷⢶⣄⣀⣀⣀⣴⠶⠎⢻⡟⠉⠻⠏⠉⠉⠙⠷⢦⣾⣤⠀⠀⣿⣄⡀
⠀⢠⠶⠉⡿⠀⠀⢸⣿⠃⠀⠀⠀⠀⠘⠇⠀⠀⠸⡇⠀⠈⢻⣿⣿⣿⡇⠀⢀⡘⠁⠀⠀⣸⡆⠀⠀⠀⠀⣿⣿⡃⠀⣿⠉⠙⣦
⠀⣿⠀⠀⠀⡄⠀⢸⣿⠆⠀⠀⠀⠀⠀⢠⣶⣶⣤⠆⠀⠀⣾⡏⠉⢿⡇⠀⠘⠁⣰⣶⣦⣼⠃⠀⠀⠀⠀⣼⣿⠇⠀⣿⠀⠀⠈
⠀⣿⠀⠀⠀⣷⠀⢀⣿⣷⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⢰⡿⠀⠀⠸⣇⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⠀⢀⣿⡏⠀⢀⠁
⠀⢙⠀⠀⠀⠙⠓⠈⡙⠻⣶⡀⠀⠀⠘⣿⣿⣿⡿⢃⣴⡟⠁⠀⠀⠀⠻⣧⡄⢹⣿⣿⣿⡟⠀⠀⠀⣠⡿⠟⠁⠀⡿⠀⠀⠀⣤
⠀⠀⠿⡀⠀⠀⠀⠀⠷⠀⠈⠹⠿⠲⠆⠸⠿⠻⠿⠟⠉⠀⢀⡀⠀⠀⠀⠈⠙⠿⠿⠿⠿⠇⠠⠾⠿⠋⠀⢰⣀⣾⠀⠀⠀⣠⠟
⠀⠀⠀⠻⣤⡀⠀⢀⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠐⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠁⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠠⣤⣤⣤⣤⠠⢤⣤⣤⣤⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡅⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣤⢤⣤⣶⡖⠙⠛⢻⡇⠀⠀⠀⠀⠀⢸⡟⠙⠛⠻⣶⣦⡠⢤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠁⠀⢀⣤⠋⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠿⣄
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠈⠳⣦⣀⣀⡀⠴⠇⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠈⠷⣄
⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠠⠴⠶⣶⣾⠀⠀⠀⠀⠀⠀⠙⣶
⠀⠀⢀⣴⠉⠀⠀⠀⠀⠀⠀⠸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⡆⠀⠉⠉⣉⣀⣀⣤⣤⡀⢿⡇⠀⠀⠀⠀⠀⠀⢀⣷
⠀⣠⠞⠉⠱⢦⣀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣷⡄⢰⣿⣿⣿⣿⣿⣿⣷⡄⣠⣀⠀⠀⠀⣀⡶⠋⠉⣶
⣄⠋⠀⠀⠀⠀⠉⣶⣄⠴⠶⠙⡏⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣷⡄⣿⣿⣿⣿⡿⠿⠿⠃⠙⠋⠁⣴⣶⠋
⣿⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡏⠉⠻⣿⣿⣧⡈⠉⠉⠀⠀⠀⠀⣀⡀⠿⠇⠉⠁
⡿⠀⠀⠀⠀⠀⠿⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀⠀⠉⠉⠻⢦⣀⣀⡀⠶⠎⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⡴
⠀⠷⣀⠀⠀⠀⠀⠛⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠿⠟⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠶⠋
⠀⠀⠻⣄⠀⠀⠀⠀⠻⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠰⠶⠾⣏⠉
⠀⠀⠀⠈⠷⣄⠀⠀⠀⠁⣄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢏⡀⠀⠀⠀⠀⠀⠀⣠⡖⠂⠈⠉⠉⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠉⣤⣠⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⣀⣤⣤⣾⣿⣷⡄⠀⠀⠀⠀⣠⡄⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠙⠋⠉⠀⠀⠀⡀⠒⠛⢦⣄⣠⣤⣠⣾⣿⣿⣿⣯⣿⣿⣿⣷⣤⣴⠾⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⣴⠏⠀⠒⠒⠋⠉⠀⠀⠀⠀⠘⣿⣿⣿⣿⠿⠛⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⣰⡇⠀⠀⠀⠀⠀⠀⠀⢰⡆⠈⢡⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠀⠀⠀⢠⡌⠁⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠉⠙⠛⠋⠉⢻⡟⠁⠀⠀⠀⠸⠟⠉⠋⠛⠛⠉⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠄⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠀⠀⣸⡅⠀⠀⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿⠶⠶⠎⠙⢇⠀⠀⠀⠀⠀⠀⢰⡏⠁⠶⠶⠶⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠶⠶⠈⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡈⠁⠀⠀⠀⠀⠀⠈⠑⠶⢄⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠈⠁⠀⠀⣀⣀⣀⣀⣄⣀⡀⣸⡇⠀⠀⠀⠀⠸⣷⣀⣀⣀⣀⣀⣀⣀⠀⢀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣄⣀⣠⣤⣤⣠⣤⣀⣀⣈⠉⠛⠋⠁⠀⠀⠀⠀⠀⠙⠛⠋⢉⣀⣠⣄⣠⣄⣀⣤⣀⣠⣽⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠟⠛⠉⠻⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡏⢸⢹⠋⠙⠀⡟⠉⠉⢛⣹⣿⣿⠙⡏⠀⠀⢻⢿⡇⡆⢠⠐⠀⢰⠒⠖⢐⣖⠂⢰⢶⠒⡓⡖⣾⣿⣿⡀
⠀⠀⠀⠀⡗⠴⣰⡆⠀⡆⣇⢀⣴⣿⣿⣿⣿⣾⣇⣤⣀⣠⣬⣧⣏⠸⠜⠀⠈⢁⠃⡶⡏⣾⠠⣾⣿⠉⣼⣿⣿⣿⠁
⠀⠀⠀⠀⣧⣤⣼⣧⡎⢠⣿⣿⣽⣯⣾⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣷⣇⠀⠀⠈⠠⢠⡇⠘⢰⣟⣯⣾⣿⣿⣿⣿
⠀⠀⠀⠀⡿⣿⣿⣿⡿⠿⣿⢿⣏⣵⢟⣵⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣶⣾⣠⣶⣾⣿⣿⣿⣿⣯⣿⣿
⠀⠀⠀⠀⣿⣿⡏⣻⢃⣴⣏⣾⣿⢟⣿⣿⣿⣿⣿⣿⠟⠰⣾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣷⣿⢾⣿⡟⡽⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣰⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣷⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡿⢡⣿⡟⣿⠱⣻⣿⣿⣽⣿⣿⡿⣻⣿⣀⣴⣿⣿⣿⣿⣿⣿⣿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡟⢠⣏⡾⣿⣿⣿⣿⠉⣿⣿⣿⣞⣽⣿⣿⣿⡿⣽⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣷⣿⣽⡱⣿⣿⣿⣿⣇⣟⣽⣟⡿⢻⠟⣹⢋⣽⢿⣿⣿⣿⡏⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠿⠯⠿⠵⠿⠿⠿⢽⣿⡿⢛⣿⣴⡿⢸⡯⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣶⣬⡉⠁⠉⠙⠻⠧⠿⠿⠽⠿⠟⠋⠛⠛⠁⠀⠀⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣶⣄⠚⠛⠛⠛⠿⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢷⡀⠀⠀⣠⡏⠀⠀⠈⣳⡀⠀⠀⣰⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢳⡀⠀⠉⣿⠀⠀⠀⡏⠁⠀⡼⠃⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡅⠀⠀⢀⣿⠄⠀⣿⠀⠀⠀⡇⠀⢼⡅⠀⠀⢀⣽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⣃⠀⠀⢀⡖⠒⢲⡄⠀⠀⠃⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢴⠋⠉⠳⠤⠞⠁⠀⠀⠓⠦⠤⠞⠙⢲⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢘⠇⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣼⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠓⠒⠋⠀⠀⣠⠖⠉⠁⠀⠈⠉⠑⢦⡀⠀⠈⠳⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⡀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢰⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠸⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⡀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢦⠀⠀⠉⠲⢤⣀⣀⣀⣠⠴⠋⠀⠀⢠⠞⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠞⠉⠙⢲⡀⠀⢀⡖⠚⠛⢦⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⡿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⡿⠿⠛⢋⣉⣤⣶⠿⠟⢻⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⠿⠟⠛⠉⣁⣠⣤⣶⠾⠟⠋⣁⣤⣴⣾⣿⠿⠿⠟⢻⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⠿⠟⠛⠉⠁⣀⣤⣴⡶⠿⠛⠋⣁⣠⣴⣶⠿⠟⠛⠉⣁⣠⣤⣶⣾⣿⣿⠿⠿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⠿⠟⠛⠉⠁⢀⣀⣤⣶⡾⠿⠛⠉⣁⣤⣴⡶⠟⠛⠉⢁⣀⣤⣴⣾⣿⠿⠿⠛⠋⠉⠁⠀⠀⠀⣿
⠀⠀⢀⣤⣶⡿⠟⠛⠉⠁⠀⣀⣠⣴⣶⡿⠿⠛⢉⣁⣤⣶⠾⠛⠋⣉⣀⣤⣴⣶⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⣿⣀⣀⣀⣤⣶⣾⡿⠿⠛⣉⣡⣤⡶⠿⢛⣋⣡⣤⣶⣾⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⡟⠿⢿⣿⣿⣯⣵⣶⣾⣿⣿⣿⣷⣶⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢠⣿⡇⠀⠀⠈⠉⠛⠛⠛⠋⠉⠉⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⠿⣿⣶⣤⣤⣤⣤⣴⣶⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⠀⠀⠉⠙⠛⠛⠛⠛⠛⠉⠉⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⠇
⣿⣿⠃⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⡿⠿⠛⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⡿⠟⠛⠉
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⠿⠟⠋⠉
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⠿⠟⠋⠉
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣀⣠⣤⣶⣿⣿⠿⠟⠋⠉
⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⡿⠟⠛⠉
⠀⠈⠻⣿⣷⣶⣤⣶⣶⣶⣿⣿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⢀⢴⣾⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢫⣿⣿⣿⣧⠀⠀⠀⠀⠀⣇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣿⣿⣿⡏⠀⢠⡴⠂⠀⠹⣿⣿⣿⡟
⠀⠀⣤⣶⣶⣦⡀⠐⠿⡧⠀⢀⣼⡿⣿⡆⠀⠙⢿⡿⠋⠀⠠⣿⡯⠄⠀⠀⠙⡿⠋⠀⠀⣠⢄⠀⠀⠀⣀⣀⡀
⠀⣼⣸⣿⣿⣿⡇⠀⠈⠁⠀⠘⢻⣷⡾⠿⠀⠀⠘⢅⠀⠀⠀⠈⠀⢀⣀⣀⠀⢑⠀⠀⢸⣄⠄⡇⠀⣶⡟⢻⣯⡀
⠀⠸⣿⣿⣿⡿⠁⠀⠀⠀⣳⣄⠀⠉⠀⢀⣀⣤⡀⢂⠀⢾⣻⡄⠀⢨⣆⣢⠇⠈⠆⠀⢠⠏⠉⣀⠀⠙⠿⠟⠛⠁
⠀⠀⠘⢿⠟⠀⠰⣯⠧⠀⢰⢀⢢⡀⠀⢀⣚⡟⠀⠀⠀⠈⠁⢀⠔⠉⠈⠀⠀⠀⠀⠀⠏⠀⠺⡿⠃⠀⣠⡀
⠀⠀⠀⠉⡆⠀⢀⣤⣤⡀⢘⣠⣤⣄⠀⣤⡀⣠⡀⢀⣠⣤⡀⠀⣤⣄⠀⢠⣤⡀⣤⣤⣤⣤⢀⣤⡄⠀⠻⢿
⠀⠀⠀⠀⡆⠀⡿⠁⠈⠁⣼⠋⠈⣹⡆⣿⢷⣼⡇⣾⠃⠀⣯⢰⣧⣿⠃⣿⣭⣷⠀⢸⡇⠀⠸⣦⣍⠀⢀⣁⣤⣀
⠀⠀⠀⠀⠀⠀⠿⢶⠾⠃⠻⢷⡶⠟⠀⠿⠘⠿⠃⠻⠶⠾⠿⠈⠟⠹⠇⠹⠇⢿⠀⠸⠇⠀⠻⣤⠟⢠⢫⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣦⣴⡄⠠⣤⡀⢠⢔⢤⠀⠀⠀⢠⣀⠀⠀⠀⢤⡒⠀⠀⢀⣀⠀⠀⣀⣠⡄⠀⢀⣀⣀⢸⣾⣿⣿⣿⡇
⠀⠀⠀⠸⣿⣧⣽⣷⠈⠛⡗⠸⢄⠼⣅⠀⠀⠻⠞⠂⠀⠸⣿⠯⠀⠀⠻⠿⠂⢀⣺⡟⠃⠰⣿⣭⣷⠀⠻⣿⣿⠋
⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⢰⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⠀⠀⠀⠸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣍⣿⣦⠀⡔⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠁⢀⡿⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⣑⢢⣀⣤⡤⠖⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⢡⠞⡄⢪⠈⠰⠁⢈⡁⠖⡆⠀⢸⣄
⠀⠀⠀⠀⠀⠀⢀⣠⠤⠐⠲⠶⠶⠂⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠛⢶⣌⢂⣠⢢⡠⣤⢍⣉⡀⠀⢀⠞⡛⢦
⠀⠀⠀⢀⡴⢊⡁⠤⠐⠒⠒⠂⠠⠤⢀⡀⠉⠲⣄⠀⠀⠀⠀⠀⠳⣠⠤⢤⡾⠛⡿⠿⢍⠉⠣⣌⠙⣦⠘⠤⠜⡸
⠀⠀⢠⢿⣚⣁⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠁⠒⠈⣛⡶⠀⠀⢀⡜⠣⠆⣞⠁⠀⡇⢸⣀⠑⢄⢱⡘⠋⠉⢳⡔⠁
⠀⢠⠃⢀⡿⠳⠦⠠⢤⣌⣉⣉⡉⠛⠛⠛⣛⣛⣭⠛⣇⠀⢠⠋⠀⠀⣠⣌⠳⣄⣣⣼⡶⠶⠆⠋⠀⠀⠀⠈⣳
⠀⠘⠀⢾⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠹⡇⢿⠀⢠⣈⡰⠀⢹⠶⠒⠉⠉⠀⠙⠒⢶⠐⠭⡄⠀⡓⡆⢳
⠀⡟⢁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣼⠀⠎⠀⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠓⢎⡀⢀⢌⢀⡞
⠀⣇⣾⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢟⣘⣄⡰⢁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣘⣢⠃⣧
⠀⢹⣾⠀⠀⠀⠀⠀⠀⠙⠁⠀⠰⠀⠈⠁⠀⠀⠀⠀⠸⠉⠙⣇⣉⠄⠀⠀⠘⠁⡐⠂⠀⠺⠂⠀⠀⢈⠭⣙⣦⣟
⣰⠋⢹⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⣠⠀⠀⢣⡀⠀⠀⡀⠀⠀⠀⠉⠁⢸⠈⠃
⠳⣄⣐⡄⠀⠀⠀⠀⠀⠛⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣰⠉⠣⡀⠀⠀⠀⠈⢦⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⢀⡗⣤⠆
⠀⠈⠉⠙⠦⣀⠀⠀⠀⠀⠈⠢⠠⠔⠁⣀⣀⣠⡤⠚⠁⠀⠀⠈⠲⢤⣀⠀⠀⠑⠲⠒⠉⠀⠀⠀⠀⢀⡠⠊⠉⠁
⠀⠀⠀⠀⠀⠈⠙⠓⠒⣲⣄⡀⠀⣀⣴⡍⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⢀⣴⠃⠋⠁
⠀⠀⠀⠀⠀⠀⢀⠴⠋⡜⢯⣬⣝⡽⢷⡈⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⣠⠞⡹⢿⢿⠒⢾⡛⣦⠙⢆
⠀⠀⠀⠀⢀⠔⠁⠀⡼⢄⣀⠈⠉⠀⡠⣧⠀⠀⠀⠙⠦⣄⠀⠀⠀⣰⠋⢰⢢⠃⠀⠉⠀⢷⠘⡆⠈⢧
⠀⠀⠀⡰⠁⠀⠀⡜⠙⠲⠬⠥⠤⠷⠚⠉⡆⠀⠀⠀⠀⠈⡷⠀⣴⠃⢠⣏⡎⢘⠳⣴⢚⣿⡄⠹⡀⠈⢧
⠀⣠⢎⠀⠀⠀⣼⢍⠂⠄⣀⣀⣀⡀⠤⢊⠽⣄⡀⠀⠀⡰⠀⢰⠃⠀⡜⠹⠁⠘⡀⠀⢸⠁⠁⠀⢳⠀⠈⢆
⢀⡟⠲⠦⠤⣴⠁⠀⠈⠒⠒⠀⠐⠒⠉⠀⠀⢷⠈⠲⢾⣀⣠⠇⠀⢸⠁⣦⠀⠀⠰⣀⠇⠀⣿⠀⠈⣇⠀⠈⡄
⠘⡄⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢈⡿⠀⠀⠀⢶⣧⠀⠀⠀⠉⠀⠀⣸⠀⠀⢸⡀⠀⢸⡄
⠀⠙⢦⠀⠀⠀⠈⠳⡤⢄⣀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⡼⠁⠀⡸⠀⠰⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣇⠀⠀⢳⡀
⠀⠀⠀⠳⣄⠀⠀⠀⡹⢀⠁⠙⢦⡀⠀⠀⠀⠀⠈⣄⢴⣅⡁⢀⠇⠀⠀⡄⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⠀⠀⠈⣧
⠀⠀⠀⠀⣿⢳⡶⢞⠁⠁⠀⠀⠀⠑⢄⡀⠀⠀⢈⡇⠀⠉⠛⢾⠀⠀⠶⡄⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢸⡶⠛⠚⠉⠑⢄
⠀⠀⠀⠀⣟⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠓⠈⣧⣀⣠⣄⣻⠦⢦⣦⠿⠀⠀⠀⠀⠀⠈⠓⠓⠳⠺⣟⢀⣀
⠀⠀⠀⠀⣿⠀⠃⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢹⠁⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠉⠓⠶⠶⠖
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠸⡄⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢀⡃⢱⠀⠀⠀⠀⠀⠰⡇⠀⠀⢷⡥⠄⠤⠀⣀⣀⣤⣠⣀⣀⣀⣤⡀⢤⡴⠋
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⠁⠀⡇⡀⠀⠀⠀⢠⡇⠀⠀⢸⡭⠭⠥⠤⡤⣼⠁⠀⣧⠉⠖⠀⢀⣸⡇
⠀⠀⠀⠀⢈⠷⠒⠺⠛⠛⠒⠲⣄⢠⠗⠒⠒⠒⠒⠒⠷⣄⠀⠈⡏⠀⠀⠀⠉⣿⠀⠀⣻⠀⠀⠁⠀⢸⠁
⠀⠀⠀⠀⣿⠀⡰⠖⢺⣻⠲⣄⢩⢸⣠⢖⣮⡿⠒⢦⡈⣸⠆⢀⡷⠤⢶⠖⠦⣿⠀⠀⣹⠔⡶⢖⣦⣼
⠀⠀⠀⠀⣏⢞⡤⠔⠚⠓⠈⠉⡷⢸⠁⠈⠑⠦⣀⠀⢫⡜⠀⢋⡀⢰⣻⡄⠓⠚⡇⡸⠍⠚⠁⣉⣉⠎⣇
⠀⠀⠀⠀⣸⠯⠀⢀⣀⣀⣀⣀⣳⣜⣂⣐⣒⣒⣈⣑⣻⠄⢸⣀⣄⣸⣏⣠⡤⣴⡇⣧⣠⣤⣤⣼⣃⣠⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣌⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣷⡔⣌⢷⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣾⠉⠉⠙⢛⠿⣧⣽⣯⠉⢉⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣠⣴⣿⣷⣤⠤⠀⠀⠀⠉⠛⠿⣿⣇⡀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⠄⠒⠉⠀⣴⣿⠿⠛⠛⠿⢷⡇⠀⡀⠀⠀⠀⠀⠀⠈⠙⠳⣤⣄⠙⠠
⠀⠀⠀⠰⠖⠚⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣴⣟⠀⠀⠀⠀⠀⠁⠨⣿⣷⣦⡄⣀⠀⠀⠀⠀⠀⠙⢦⣔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣲⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠉⢻⣿⣮⣱⠷⠡⠂⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⡛⠛⡛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡙⠓⢶⣤⡀⠀⠀⠀⠈⣳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣾⣿⠃⠀⠐⠀⠀⢲⣤⣀⡀⠀⠀⠀⠀⠀⠙⠳⠶⣽⣿⣶⣄⠀⡎⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠿⣿⣿⣦⣠⣤⡀⢈⠌⢹⠛⢿⣧⣤⡀⣄⡄⡀⠀⠀⠀⠉⢿⣿⣿⣦⣤⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠁⢀⡈⡉⠙⠿⠇⢿⣆⣄⣆⠀⡄⢨⡟⠻⣿⣮⣡⣀⠀⠀⠀⡼⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣦⣜⣿⠀⢟⢻⣿⣯⠀⠀⠀⠀⠟⣿⣿⣾⣤⣲⣀⣀⢦⣿⣿⣿⣿⣿⣿⣷⣶⣤⡾⠿⣷⣄⠑
⠀⠀⠀⠠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣙⣿⠿⣿⣤⡄⣈⠁⠛⣿⣽⣿⣿⣷⣾⣿⣿⡿⠿⠛⠛⠋⠉⠀⠀⠀⠈⠻⢷⣄
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡿⢭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣶⣾⣥⣧⣼⣵⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀
⠀⠀⠀⠈⠡⠶⠶⠶⠶⠲⠟⠳⠯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠉⠉⠉⠙⠛⠛⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⠿⢛⣋⣥⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢛⣉⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⣉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣿⣿⣷⣶⣤⣉⡙⠻⠿⠿⠟⢛⣉⣤⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⣿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣿⣿⣷⣶⣶⣶⣶⣦⣤⣤⣤⠄⠠⣤⣤⣤⣴⣶⡄⣒⠚⢛⢻⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣶⣶⡌⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢈⠻⠟⡁⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣿⠿⠿⠿⠟⠛⠛⠛⠛⣠⣿⡇⢸⣿⣦⡙⠛⠛⠛⠻⠿⠿⠿⣿⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣿⠁⠀⠟⠁⠁

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

⠀⠀⠀⢀⠄⠀⠀⠀⠀⢀⡔⠀⠀⣠⠄⠀⠀⡀⠀⠀⠀⠀⡴⠁⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⢄
⠀⣠⠄⠀⡞⠁⠀⠀⠀⢸⠀⢀⠴⡇⠀⠀⢰⡇⠀⠀⠀⣼⠃⠖⡖⠀⣼⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⠁⢠
⣸⠃⠀⠀⠑⠠⢴⠄⠀⠘⣇⠐⠀⠑⠀⠀⢸⠅⠀⠀⢀⡟⠀⢀⠀⠀⡏⠀⠐⠀⠀⠀⠀⢤⠀⢀⣸⡄⢸
⣿⠀⣤⢰⢸⡆⣼⠀⡞⢵⠈⢡⡆⣴⢱⡆⣸⢀⡞⣰⢸⡇⢰⠃⣷⢸⡇⣼⢸⠇⠐⢹⠃⡟⠈⢠⡇⡇⣼
⢻⡐⠫⠀⠈⠃⣿⠐⠧⣿⠁⣾⠀⠯⠸⡇⢿⡘⠇⠙⠘⠧⠈⠂⠛⠉⠃⠙⠙⠀⠀⠈⠀⡇⠘⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠻⠄⠀⢹⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⣡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡉⠳⢦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣤⣴⣶⠶⣤⣭⣉⡀⠀⠀⠀⠀⠀⣀⣀⣀⡀⢀⣀⣀⣀⠀⠀⠀⠀⠀⣾⠿⠿⣶⣤⣈⣉⡙⠃⠀⠀⠀⢀⣠⣴⣶⢶⣶⣤⡀
⠀⠀⠀⠀⠀⢰⡏⢀⡀⠀⠀⢀⠈⣷⠀⠀⠀⡀⠀⢻⠉⡀⠀⠀⢀⠈⡿⠀⠀⠀⠀⠀⣿⠀⡀⠀⠀⢈⠉⡇⠀⠀⠀⢠⢼⠉⡀⠀⠀⢀⡈⢹⢤
⠀⠀⠀⠀⠀⢸⣧⡀⠀⢠⠀⠀⣠⣿⠀⠀⠈⣽⡖⠈⠀⠀⠠⠄⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠄⠀⠀⠉
⠀⠀⠀⠀⠀⠸⢿⣿⠇⣠⡐⢿⣿⠿⠀⠀⠀⠀⠈⠀⠀⣠⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⡀⠀⠀⠀⠀⠀⠆⠀⠆⢀⣄⣀⣄
⠀⠀⠀⠀⠀⠀⠀⣼⠀⣿⠇⢸⡆⠀⠀⠀⠀⠀⠘⡃⣰⡛⢻⡛⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠃⠀⠀⠀⠀⠀⠀⡄⢰⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠘⠛⠀⠛⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⠓⠀⠀⠀⠀⠀⠀⠃⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠯⠉⠉⠽⣿⣿⡄⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣵⠶⠾⠿⠿⠷⠶⡌⠁⠀⠀⠀⠀⡿⠿⠿⠟⠛⠉⢻⡏⠀⠀⠰⣿⣿⡿⠿⠿⠟⠋⠉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠻⠐⠀⠀⠀⠰⠀⠇⠀⠀⠀⠀⠀⠃⠰⠀⠀⠀⠂⠈⠁⠀⠀⠀⢻⣿⡇⠰⠀⠀⠀⠆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠠⠺⣦⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣽⠆⢷⡀


⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⠟⠛⠛⠋⠁⠘⠃⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠙⡇⠀⠀⠀⠀⢸⡟⠛⠛⠛⠉⠈⢻⠇⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⢻⡀
⠀⠀⠀⠀⠀⢰⣿⠀⠃⠀⡀⠘⠀⣼⡇⠀⠀⠀⠀⠙⠀⠀⢀⠀⠘⠀⠃⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠈⠃⠘⠀⠀⠀⠃⠘
⠀⠀⠀⠀⠀⠀⠉⠁⢠⡌⢁⡄⠀⠉⠀⠀⠀⠀⠀⡆⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⣤⠉⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⣶⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣈⣁⠀⣴⣧⠒⣼⣆⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡧⢰⣿⣿⠀⠀⠀⠀⠀⠀⠃⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⢀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠯⠭⢸⡇⠀⠀⠀⢸⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡞⠫⣾⣿⠟⠓⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⢉⡴⢋⣾⣷⣬⠏⣀⠄⠀⠀⠈⠉⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢧⢀⡴⢋⣴⣿⣿⣿⣿⣴⣧⠞⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⢠⣾⢋⣴⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀
⠀⠀⠀⠀⠀⢀⡜⠁⠀⡠⠊⣠⠞⠋⡡⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢦⡄⣠⣄⣀⣀⡀
⠀⠀⠀⠀⠀⡎⠀⢻⠟⢥⠊⠁⠀⠀⠑⠛⣀⣀⣀⣤⣤⣾⣵⣷⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢀⣿⣿⣿⣿⣿⣿⣷⣷⡄
⠀⠀⠀⠀⢰⠁⣠⡎⠀⢀⡱⠀⣀⣠⣴⣾⣿⣿⣿⡿⢿⣿⣧⣌⠛⠛⣧⣤⡀⠀⠀⠀⠀⠀⠀⡀⠀⠎⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⡠
⠀⠀⠀⠀⣅⡼⣡⡿⣶⡶⡾⢿⣿⣿⣿⣿⣿⣿⣿⣏⠸⣿⣿⣿⣷⣶⣤⣄⣙⣳⣂⣤⢀⣠⣾⣷⡶⠀⢰⣿⣿⣿⠛⡿⣿⢿⣿⡿⡏⡇
⠀⠀⠀⣠⣿⡾⠋⠀⠻⣧⡄⢤⠙⢯⣽⣿⣿⣿⣟⡛⠲⢿⣝⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⠯⣼⣿⠃⠀⢸⣿⣿⣿⣇⣁⡇⣿⣿⡇⠇⠃
⠀⠀⠐⠿⠋⠀⠀⠀⠀⠈⠙⢦⣀⣼⠁⠈⠹⣷⠀⠈⠉⣦⠈⠻⠦⣤⣽⣿⣟⣻⣷⢤⣬⣀⣻⡿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⠀⢸⠟⠒⠚⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⣿⣿⣿⣿⡇

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠠⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣠⣴⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⢷⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⡀⢀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣿⣿⡿⠿⠛⠁⠀⠈⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠙⠁⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠆⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠐⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠤⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡔⠂⠀⠀⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡰⠀⠘⢌⣇⠀⠐⣄⠀⢠⡆⢿⠦⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠑⠠⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⡏⠀⠀⢁⠀⠈⠀⠸⡇⢀⠐⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠃⠌⣢⣀⠗⠐⠂⢄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⠀⠀⠀⠄⠀⠀⣠⣧⡀⠆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠊⠉⠂⠀⠀⠀⠀⠀⢛⠀⠀⠀⢈⠑⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⣅⡃⠀⠰⠀⣼⠏⠀⢷⠘⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⢁⠙⡀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⣀⣀⣀⣀⣀⠀⠀⠀⢰⠀⣸⣈⣵⡀⠀⡆⣿⣀⠀⣸⢀⣇⣀⣀⣀⣸⣀⣀⣀⣀⣀⡇⠀⠀⣀⠀⠀⠀⠘⠁⠀⠈⠁⠀⢀⡀⢀⣀⣀⣀⣀⣀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠀⠀⠀⣀⠑⠄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣠⣤⣶⡾⠿⠛⠁⠈⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡠⠘⠀⣀⢀⡔⠁⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠴⠤⠒⠈⠉⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠠⡀⠀⡠⠂⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⢤⠊⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⢎⠉⠁⠂⠐⠊⠠⣄⢀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠒⠀⠂⣿⠁⠀⠀⠀⠀⡇⢹⡦⣄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠔⠃⠀⢠⠀⠀⠀⠑⢠⠀⠀⡄⠈⠀⠀⢡⠀⠸⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⡀⢻⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡄⠀⡁⠀⠀⢸⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⢸⡀⠘⠀⠀⠀⠀⣠⣿⣄⠀⡀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠂⠀⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢁⠀⡧⠀⠀⢸⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠨⣇⣀⠀⠀⠀⢰⡿⠉⢻⠀⡇⠀⠀⠀⡇
⠀⠀⠀⢸⡀⡀⣀⠁⠀⠀⢀⢀⡀⠀⠀⠀⢀⡀⢀⡃⠀⠀⠀⠀⣸⢀⡇⡀⡀⣘⣀⢀⣀⢀⡊⠀⠀⠀⣴⡀⠀⣻⣋⣹⡂⠀⠀⣸⣀⠀⣸⡇⣃⢀⡀⢀⡇
⠀⠀⠀⠀⠁⠁⠉⠀⠀⠀⠈⠈⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠈⠀⠁⠁⠁⠈⠁⠈⠈⠈⠁⠀⠀⠀⠉⠀⠀⠈⠁⠈⠀⠀⠀⠈⠁⠀⠈⠀⠉⠁⠀⠈
⣀⡀⣀⣀⣀⣀⣀⣀⡀⣀⣀⢀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠦⠀⠀⢈⢸⣄⠁⠠⢧⠀⠀⠀⠆⠈⠀⢀⢀⠼⡀⠉⠀⢀⠀⠂⠰⠀⡀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠈⠂⠀⠀⠙⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠀⠂⠑
⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂



⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⠓⢰⢓⣶⣦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠉⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠶⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⣠⣤⠴⠦⣤⣀⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠀⠀⣼⠋⢠⡾⠋⠀⣰⣆⠀⠉⢷⡄⠘⣧⠀⠀⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠀⡟⠀⣿⠀⢶⡟⠋⠙⢻⡶⠀⣿⠀⢻⡀⠀⠀⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⣷⠀⢻⡄⠀⣿⣤⣤⣿⠀⢠⡿⠀⣾⠀⠀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠈⠻⣦⣉⣀⣀⣉⣴⠟⠁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠳⣤⣀⠉⠉⠉⠉⣀⣤⠞⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠉⠛⣻⣟⠛⠉⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣤⣤⣄⠀⠀⣰⠏⠹⣆⠀⠀⣠⣤⣤⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠹⣆⣴⢿⣄⣠⡿⣦⣰⠏⠀⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠹⠃⣸⠏⠹⣇⠘⠏⢀⣀⣀⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⣉⣉⣉⣹⡇⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢹⡆⠀⠉⠉⠉⠉⠉⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⠶⠶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡄⠃⠀⠀⠀⠀⠀⢠⠀⠀⢀⡀⠀⣠⠤⡀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠀⠋⠀⢸
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠂⠠⡀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠆⠀⠀⠀⠒⠀⢀⡀⢀⡆⠀⠀⠀⠀⠘⠧⠃⠀⠠⠄
⠀⠈⠀⠀⠘⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⡇⠞⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⢀⡄⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠠⠊⢀⡀
⡄⠀⠀⡰⡆⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠈⠀⠊⠁⠠⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠓
⠀⠀⠀⠻⠠⠐⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠈
⢀⡀⠀⠀⠀⠀⠤⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠈⠀⠀⠆
⠀⠀⠀⠐⠀⠀⠀⠃⠀⠀⠛⠛⠛⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠛⢛⠛⠛⠛⠛
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣿⡀⠀⣿⠀⠀⠀⠀⣠⠎⣼⡰
⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠘⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⡇⠐⣿⡇⠀⠀⠀⠋⠀⠉⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣘⣛⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣿⡇⠀⠀⠂⠀⠀⠤
⠀⠠⠀⠀⠀⣀⠀⠀⠀⠀⢠⠖⡲⢾⡟⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠉⠈⠓⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠠⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡤⡄⠀⠀⠀⢀⣀⣠⣤⣤⣤⣦⣄⠀⠀⠀⠻⠃⠀⢀⡀⢠⣲⠀⠀⠀⠀⠀⡤⡀
⠀⠀⠀⠀⠀⠀⠸⠝⠀⠀⣴⡀⠈⠁⠀⢰⠆⠈⠙⢳⣾⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠘⠃⠀⠀⠀⡰⡿⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢄⠀⠀⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠛⠉⠉⠀⢠⠀⠘⠀⢠⠊⠀⠀⢀⠤⡀
⠀⠀⠀⠀⠀⠀⢠⢢⠀⠀⢠⠀⠀⠡⠀⡄⠀⠁⠀⠁⠀⠀⠀⠈⠻⠟⠀⠀⠀⢀⠀⠀⠰⢠⠁⠀⠐⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠀⠀⠀⠀⢤⣿⠄⠀⠀⠀⡖⠀⠁⣀⣴⢀⡀⠀⠀⠀⠀⡤⡄
⠀⠀⠀⠀⠀⡴⢫⡿⠀⣠⣄⢠⡴⡄⢠⠖⡖⠾⣶⢠⠞⡿⣸⠁⡼⢠⠇⣸⢡⠖⡖⣸⠉⡬⣡⣤⡀⣀⠀⠈⠁
⠀⠀⠀⠀⢸⠃⠙⢁⡼⣇⡼⢻⠀⠧⠾⢾⡇⠀⠓⠋⠛⠛⠉⠛⠙⠛⠚⠓⠛⠚⠗⠧⠾⠳⣇⡼⣿⠋⡽⢰⡖⠆
⠀⠀⠀⠀⠸⣄⡤⠞⠀⠈⠀⠀⠀⠀⠲⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠪⠕⠀⠀⢀⣤⡀⠀⠀⠀⣀⡅⠈⠛⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠯⠆⠀⡖⠋⠙⠒⢦⣀⣠⣤⡤⢤⡀⠀⠀⠈⠉⣀⣤⢶⡻⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢦⣄⡀⠀⠀⠉⠓⠚⠛⠉⠉⠈⠢⡀⢀⡼⠀⠀⢠⣟⣯⣼⡶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠦⢤⣤⣤⣤⣄⣤⣤⣬⡥⠶⠖⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⢉⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣫⢟⠅⠠⠀⠉⠒⠄⡀⠉⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠵⠞⠑⠁⠀⢀⠀⠀⠀⠀⠀⠁⠂⢄⡈⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⣠⣤⣁⠢⢌⠀⠀⡀⠈⠀⠀⠀⠀⠈⠂⠀⡈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠤⡞⠋⠀⠈⠙⠲⣌⡂⢄⡀⠀⠀⠀⠀⠀⠠⠠⠈⠑⠢⣀⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⢀⡴⠋⠀⠀⠉⠂⡄⠀⠄⢦⠉⠲⢌⡀⢀⡀⠠⠎⠐⠠⠀⡀⠂⠀⢁⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⢀⡴⠋⠀⠀⠀⠲⠶⠄⠀⠀⠀⠈⠀⠀⠀⠉⠲⢌⡐⠤⡀⠀⠀⠀⢀⠔⣡⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠔⠁⣴⢟⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠀⠀⠀⠀⠉⠂⣈⠑⠀⠰⢃⣾⢟⣽⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠁⣠⢞⢡⡾⠀⡄⣀⣰⣇⡀⢀⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⢀⣴⠟⡵⣫⣾⣿⣦
⠀⠀⠀⠀⠀⡠⢾⡿⠋⣠⠞⠉⠾⠄⢁⡀⣙⠂⢀⠛⢑⠀⡀⠊⠁⣭⡉⠁⠄⠀⡀⠀⢀⣠⠞⢁⣴⠟⡡⣪⣾⡛⣮⣽⠋
⠀⠀⠀⠀⠀⣷⣍⡀⠀⠳⢦⡀⠀⢄⠈⢥⠎⡖⢃⡀⣠⡄⣥⡜⠁⠀⠘⠉⠂⢸⡤⣫⠞⢁⣴⠟⡡⣪⣾⠿⡿⣽⣿⢿
⠀⠀⠀⠀⠀⣿⣬⡙⠶⣤⡐⢝⡳⣌⠁⠋⢹⣁⢾⡟⠀⠸⣏⢳⠀⠓⢠⠀⡠⣪⠞⢁⣴⠋⠠⣪⣾⡻⢧⣍⣾⠟⢑⣼⣷⠄
⠀⠀⠀⠀⠀⣿⣄⠬⣑⠬⡙⠷⣬⡚⢅⡂⠹⠉⣼⡇⣠⣲⠦⡐⠈⠀⠁⣨⠞⢁⣴⣿⠟⣠⣾⣟⣷⣪⣶⣿⢟⢔⣽⠟⠁
⠀⠀⠀⠀⠀⠈⣻⣷⣮⣑⠬⣒⢌⡙⠷⢬⡑⢌⡁⢓⣈⠉⠀⠏⠀⠀⡼⢃⣴⣿⠏⣡⣾⢿⢷⣿⣿⣿⠟⠁⣵⡿⠋
⠀⠀⠀⠀⠀⢜⡻⢿⣿⣿⣷⣦⡑⠪⠢⠄⠈⢗⠬⡀⠹⡳⢦⡀⠠⠊⣰⢟⠉⣠⣾⣟⢕⡓⢃⣾⡿⠋⣠⣾⠋
⠀⠀⠀⠀⠀⢻⡟⠷⣬⣻⢏⣻⢿⣦⣀⠈⠢⢄⠈⡛⠷⣬⡑⠉⣠⢞⡵⢩⣾⠯⣙⢖⣅⣶⣿⢏⢴⣿⠟⠁
⠀⠀⠀⠀⠀⢸⡟⢉⠢⢍⠛⢿⣟⡮⣝⡷⣶⣦⢍⠒⢕⡌⢹⠟⡱⢋⣼⢟⠡⡂⠑⣵⣿⢟⢕⣵⡟⠁
⠀⠀⠀⠀⠀⠀⠻⣶⣬⡒⠍⡂⢌⠛⠾⣽⣖⠿⢷⣯⣒⠄⠸⢊⣴⠟⡡⢕⠋⢢⣿⠋⠀⣡⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣬⡑⠬⣒⢌⡈⠙⡿⣮⣝⠿⠶⠟⣙⣿⣿⣄⣴⡿⠃⣪⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣀⠁⠀⢄⡈⠙⠻⣿⣖⣾⠿⠿⣿⣿⢏⣔⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣤⣌⡑⠭⣂⢍⡛⢶⣾⠟⣡⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢾⣕⠠⣉⠂⣞⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣥⣩⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠿⣇⡀
⠀⠀⠘⠛⠛⠿⣯⡿⠛⠛⠓⠀⠀⠀⠘⠛⠛⢿⣼⠟⠛⠛⠀⠀⠀⠀⠘⠛⠛⢿⡟⠛⠛⠀⠀⠀⠀⠘⠛⠿⢥⡀⡬⠿⠛⠃
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⣀⣠⠾⠁⠷⣤⣀⠀⠀⠀⠀⠀⠀⣿⠰⠶⣟⣹⠶⠆⠀⠀⠀⠀⢀⣀⡾⣧⣀⡀⠀⠀⠀⠀⠀⠛⢶⣶⠿⣶⡶⠋
⠀⠀⠈⠙⠳⢦⠀⣤⠞⠋⠁⠀⠀⠀⣤⣼⣛⣦⡤⢸⡇⠀⠀⠀⠀⠈⠉⠉⠛⣦⡾⠛⠉⠉⠁⠀⠀⠀⣀⣴⢿⣶⡿⣦⣀
⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⣤⡀⢸⡆⢀⡄⠀⠀⠀⠀⠀⢀⠘⣦⡀⢀⡴⠁⡀
⠀⠀⢀⣀⣨⣿⠛⢿⣇⣀⡀⠀⠀⠀⠀⠀⣹⣿⢿⣿⡁⠀⠀⠀⠀⠀⣀⣈⣻⡾⣷⣏⣀⡀⠀⠀⠀⠀⠈⠙⢾⠟⢿⡷⠋
⠀⠀⠀⠈⠉⠿⣶⠾⠍⠁⠀⠀⠀⠀⠀⢀⣼⠿⡾⢿⡄⠀⠀⠀⠀⠀⠈⢉⡽⢿⡿⣯⡉⠁⠀⠀⠀⠀⢀⡴⢿⡷⢾⡿⢦⡀
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠉⠀⠸⠇⠈⠁⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠙⠄

⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠰⠆⢀⣿⡄⠰⠆⠀⠀⠀⠀⠀⠈⠃⣸⢷⡀⠋⠀⠀⠀⠀⠀⠐⢦⡀⢸⡆⣠⠶⠀⠀⠀⠀⠀⠠⣦⠀⠛⠀⣴⠄
⠀⠀⢠⣤⣴⠛⠁⠙⣦⣤⡄⠀⠀⠀⠈⠻⣏⠁⠈⢉⡿⠋⠀⠀⠀⠀⣤⣤⣴⡟⣷⣤⣤⣄⠀⠀⠀⠀⣤⠀⠛⠀⠛⠀⣤
⠀⠀⠀⠀⡈⠻⣤⡞⢉⡀⠀⠀⠀⠐⠛⠀⣿⡤⢦⣼⡆⠘⠓⠀⠀⠀⠀⣠⠌⢿⡏⢤⡀⠀⠀⠀⠀⠀⢀⣤⠛⠀⠛⣠
⠀⠀⠀⠈⠁⠀⠿⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⡆⠈⠁⠀⠀⠀⠀⠀⠈⠁⠀⢸⠁⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠛⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⠈⠳⠋⠙⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠄⡐⠠⢆⢀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷⠊⠔⠁⠀⠐⠉⢉⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠀⠀⠐⠀⠁⢀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠶⠁⠀⠀⠀⠁⡠⣀⡀⠀⠀⠀⣠⣶⣙⠀⡾⢋⠀⡠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠐⠠⠀⠀⢹⣿⣿⡆⠀⠀⠛⠿⠋⠀⠋⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢶⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠈⠛⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠄⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠠⣀⣀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠐⣨⣄⠀⠀⠀⠀⠀⠀⢠⠴⠦⣄⠀⠀⠀⠀⠀⠀⢸⠀⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢈⠑⠶⠀⢟⡀⠀⠙⢓⡀⠀⠀⠀⡘⠁⠙⠌⠆⠀⣠⡀⠀⠀⢸⠴⠾⢯⡀⠀⠀⠀⠀⠀⣿⠀⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢀⠲⡤⣥⠆⠈⡁⠁⠀⠐⠈⠀⠰⠀⠀⠀⣰⡟⣧⠀⠀⠈⠠⠤⠞⠁⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣅⠠⡀⡷⡾⠶⠛⢔⠠⠄⠊⠀⠀⠀⢰⠋⠉⠉⣧⠀⠀⠀⠀⠀⣀⠠⠖⠶⠀⠀⣿⠀⡇⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⠶⠓⠻⣿⠀⠉⠀⡠⠆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⣿⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡌⠒⢤⡤⠀⠀⠉⢀⣤⠀⠃⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠲⠃⠀⣿⠀⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠚⠁⢠⠄⠐⠋⠀⢀⡀⠰⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠤⣄⡒⠀⣀⠀⠐⠂⢀⡀⠀⣐⣢⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠀⠃⠀⢀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡉⠉⠛⠛⠛⠛⠛⠙⠋⠁⠀⠀⠈⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⢸⡀
⠀⠀⠀⠀⠘⠃⠀⠇⠀⡀⠀⠀⠁⠀⠀⠉⠉⠁⠀⠀⠀⠘⠈⠁⠀⠀⠈⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠁⠈⡇
⠀⠀⠀⠐⠀⠀⠄⠄⠀⠃⠀⠈⠀⠀⠸⡂⠠⢄⠀⠀⠀⠠⡄⠀⠂⣄⠀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠁⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠹⣷⡀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣆⠀⢠⣾⠃⠀⠀⠀⠘⢿⣄⠀⢠⣾⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠻⣷⡿⠁⠀⠀⠀⠀⠀⠈⢻⣦⡿⠃⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠰⣿⣶⣶⣶⡄⠀⠀⠀⠀⣼⡿⠿⠿⠿⠀⠀⠀⣀⣈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⣿⠀⠀⠀⠀⣶⠀⢸⡇⠀⠀⠀⠀⣿⡇⠀⣷⠀⠀⠀⠀⣿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⣠⣦⣄⠀⣿⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⠀⢿⣀⡀⠀⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠹⣷⣶⡿⢨⣿⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⣿⡇⠐⣿⠀⠀⠀⠰⣿⡛⣷⣴⣿⠁⣿⣅⣀⣤⣄
⠀⠀⠀⠀⠀⠀⣼⡿⠷⡿⠋⠀⠀⠀⠀⠈⠻⣶⠾⣷⡀⠀⠀⠛⠀⢸⢇⣠⣤⣤⡀⢿⡇⠀⠋⠀⢀⣤⣤⣽⠗⠀⠀⠀⠀⠈⠙⠛⣩⣿⠆
⠀⠀⠀⠀⠀⠀⠙⢿⣶⠀⠀⣴⠟⠛⣷⡄⠀⢠⣶⠟⠃⠀⣠⣤⣀⠀⢸⡏⠉⢹⣿⠀⠀⣀⣄⡀⢼⣯⣍⡉⠀⢠⡾⠛⢿⣦⠀⠈⣿⣀
⠀⠀⠀⠀⠀⠀⣶⠾⠟⠀⠀⢿⣦⣤⣾⠇⠀⠸⣷⣄⡀⣾⡋⠉⠻⡶⠿⠋⠀⠘⠻⢶⣾⠟⠙⢿⣦⠉⣻⡇⠀⠸⣷⣄⣼⡿⠀⢀⣉⣛⣿⠂
⠀⠀⠀⠀⠀⠀⠻⡶⠿⣷⡀⠀⠈⠉⠀⢀⣴⣶⣤⣿⠃⠘⣿⡆⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⣴⡿⠁⣾⣛⣡⣄⡀⠈⠉⠁⠀⢠⣾⠟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⣴⡶⢶⣆⢹⣯⠀⢈⣷⠶⠶⠿⠁⠀⣴⡟⠋⠉⠉⠻⣷⡀⠀⢹⣧⣤⣘⡛⠋⠙⣿⠂⣶⠶⢿⣆⣻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠀⠈⠿⠟⠛⠀⢸⣿⣀⣀⡀⠀⢰⣿⠀⠀⠀⠀⠀⢹⡇⠀⠀⠉⠉⢹⡇⠀⠀⠿⠶⠟⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣷⠀⠀⢻⣦⡀⠀⢀⣠⡿⠃⠀⣰⡷⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠁⠀⠀⠉⠛⠛⠛⠋⠀⠀⠰⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣴⡿⢷⣤⡀⠀⣠⣤⣶⣦⡀⢈⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⣿⡇⠀⣿⡏⠀⠘⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠒⠒⠒⠒⠒⠒⠚⠒⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠒⠓⠒⠒⠶⠶⠼⠤⠄⢀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣄⣀⣀⣀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣠⣤⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⡻⠛⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣠⣤⠀⠀⠀⠀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠹⠿⠿⠿⠉⠀⠰⠃⠀⠀⠀⠀⠈⡏⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⡶⢒⣲⡖⣠⠖⠖⠖⠒⠒⠛⠒⢶⠷⢶⠶⠶⠶⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣛⣉⣿⣁⣀⣭⡀⣭⠀⢀⣩⡅⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢏⣙⠻⣿⣿⣿⣿⣿⣷⣿⣿⣿⣾⣿⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠛⠟⠻⠿⣍⣿⣿⣟⣢⣤⣄⣠⣤⣤⣤⡤⠤⠄⢐⣒⣀⠠⠤⠴⠛⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⡾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠉⠀⠀⠀⠀⠀⢠⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠳⠶⠶⠶⡶⠟⢡⠄⠂⠁⠀⠀⠀⠀⠀⣀⣼⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠒⠢⠤⠤⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠒⠲⠦⠤⠤⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⢀⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠐⠒⠲⠤⠤⡔⠊⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣦⣀⢀⣴⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠘⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡏⠀⠀⣠⣶⣦⣠⣾⣿⣿⠟⠀⠀⢹⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⠀⠀⠙⢿⣿⣿⣿⠟⠁⠀⠀⠀⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⣦⣀⡀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣆⠈⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⠁⠰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣄⠙⠿⠛⢁⣠⠄⡈⠛⠿⠋⣠⣯⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣶⣶⣿⠋⠘⣿⣶⣶⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠉⣿⣿⣿⡿⠃⠀⠀⠘⢿⣿⣿⣿⠉⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠁⠀⠀⠀⠀⠈⢿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⡤⠤⢤⠔⠂⠉⠉⠉⠉⠁⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⡠⠔⠂⠉⠩⢭⣤⡆⠚⢤⠤⢄⣼⣄⣀⣀⣄⣀⡀⠀⢰⡆⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⢱⡀
⠀⠀⢠⠊⠀⠀⠀⠀⠀⠐⠉⠖⣀⡘⠀⠀⠀⢀⡤⠂⢀⠠⣄⠉⠉⣀⡤⢈⣛⡻⣍⡉⠉⠉⠙⢷⠉⢠⣛⠘⠋⠉⠒⠦⣄
⠀⢠⠇⠀⠀⠀⠀⠀⠀⢀⠃⣾⠉⠇⠀⠀⠐⡏⢸⣿⣏⣡⠈⠀⠀⠛⢰⣿⣍⣷⠈⡇⠀⠀⠀⣼⠻⡏⠙⣇⠀⠀⠀⠀⠈⠳⡄
⠀⣆⠀⠀⠀⠀⠀⠀⢀⡎⠀⠙⢤⣴⠀⠀⠀⢡⡘⠻⠿⠋⡰⣀⣀⠳⣈⠻⠟⢋⣴⠁⠀⠀⠀⣟⣰⠃⠀⢹⡄⠀⠀⠀⠀⠀⠹⡄
⠀⡇⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠈⡄⠀⠀⠀⠙⠓⠚⠋⠀⠉⠉⠂⠈⠙⠓⠉⠀⠀⠀⠀⣸⠋⠁⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢳⡀
⢰⠁⠀⣀⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⢲⠀⠠⠀⠀⠀⠀⠈⢷⣄
⠸⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠓⠒⠛⠁⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠐⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⣉⣁⣀⣠⠤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠠⣤⣤⡒⠟⠀⢸⡆⢤⡤⠤⠄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⣠⠴⠎⠁⠁⠀⣨⠷⠙⢢⡀⠀⠁⠒⣄⠀⠀⠀⠈⠙⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠃⠘⠀⠀⠂⠸⠇⠀⠀⠀⠹⣆⠀⢠⠀⠇⠀⠀⠀⠀⠀⡇⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠲⠀⠀⠀⠀⠀⠷⠀⠀⠀⠈⠃⠘⠆⢸⠐⣤⡄⠀⠀⠀⠀⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠐⠂⠒⠀⠀⠀⠀⠈⠄⠀⠘⠀⠀⢠⠉⠈⠈⠑⡀⠀⠀⠀⡤⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡆⠀⠀⠀⢱⠀⠀⠀⠀⠐⢁⠃⠀⠀⢸⠂⠀⠀⢀⡇⠀⠀⠀⣻⡀⢰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⢃⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠀⠀⠀⢘⣄⡀⢀⠜⠇⠀⠀⠀⡟⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⠈⢱⠖⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠹⣅⠀⢰⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠳⠚⠁⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠳⠞⠀⠀⢀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡼⠧⢤⣀⣀⣀⣀⣀⣀⣀⡤⠞⡄⠀⠀⠀⠀⠀⠀⢰⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠥⡒⢠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣀⣀⣀⣀⠠⠎⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣏⡉⠁⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠮⠥⠄⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢒⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡟⠛⠓⠶⡴⠦⢤⣤⣤⡤⠦⠴⡶⠶⠖⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⢀⡀⠈⢸⠆⠀⠀⠀⣦⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠳⡤⠀⣱⠏⠀⠀⠀⢠⠃⠠⡄⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⡴⠃⠀⠀⠀⠀⡘⠀⠀⠈⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⡏⠀⠀⠀⢶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⣀⠜⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⣀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⣼
⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡅
⠀⠀⠀⣠⠔⠋⠁⠀⠠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇
⢀⠶⠋⠀⠀⠀⢠⠔⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⡀⠙⢦⡀
⡎⠀⠑⠄⣀⠖⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⠤⠀⠀⠙⠢⣄
⢣⣄⣀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡑⠠⠄⠒⠉⢱
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⢀⡴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⡀⠈⡉⠉⠁
⠀⠀⠀⢀⣾⠀⠀⠠⠤⠤⠤⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⡨⠿⠟⠛⠒⠀⠤⠀⢄⡀⠀⠀⠈⠀⠀⠀⠀⠠
⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⡁⢄⠠⠄⠒⣂⣉⠙⠲⢤⡁⠀⣀⠀⠀⠀⠠⠀⠀⠄
⠀⠀⠀⠘⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠼⢷⠐⠒⠉⠁⠀⠀⠀⣀⣀⣉⠳⣄⠑⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣏⣷⠀⠀⠀⠀⠀⠀⠀⠈⡀⢀⠇⠀⢞⣁⣠⣤⠀⡄⡶⣶⢦⡙⣏⣏⣻⡇⠈⢷⡀⠠⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡘⠈⠉⡇⢳⢰⡇⣇⠸⡇⣳⢹⣨⡿⣧⡄⠘⣇⠀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⢸⠈⡆⣇⢰⡀⢻⢸⠊⡏⣇⢹⣾⠀⠁⢸⠀⠀⢉⠀⠀⢀
⠀⠀⠀⠀⠀⢻⣯⡂⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⢰⡛⣆⢇⣸⡌⠲⠵⠾⠋⠀⠓⠛⠛⠁⠀⠀⡼⠀⠠⠀⠒⠂⠀⢆
⠀⠀⠀⠀⠀⠈⣿⡛⡀⠀⠀⠠⠀⠀⠀⠈⣷⡐⠀⠙⠛⠛⠋⠀⢀⠀⢀⡀⠄⠠⠒⢀⡙⡤⣰⢅⡆⠀⠁⠐⠦⠀⠈⠄
⠀⠀⠀⠀⠀⠀⢸⣷⣱⠀⠀⠀⠀⠐⠀⠀⢈⢿⣄⠠⢔⢒⣂⠀⠘⣧⠀⠒⣀⠩⢁⡴⢚⡵⣝⢻⣅⠀⠀⠀⠀⠀⠀⠛⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠿⣿⡱⡀⠀⠀⠀⡀⠀⠀⠷⠝⢳⣦⣉⡛⠒⠦⠼⠦⠭⠤⢚⣩⡴⡟⠧⠺⣯⣫⠛⠲⣤⣀⠀⠀⠀⠀⠀⠈⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠰⣹⡷⡱⡀⠀⠀⠀⠀⠀⠀⡀⠀⠃⠉⠻⠻⠶⠶⠶⠺⠃⠛⠋⠀⠀⠀⠀⠀⠀⠐⠐⢱⡠⣿⣶⢤⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡔⢃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⠄⠥⠈⠑⠲⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣷⣎⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⣿⣥⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⢀⢀⣀⣀⣀⣤⠠⠤⠤⠤⠤⠤⠤⠴⠦⠦⠆⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⢀⣀⣀⠠⠄⠤⠔⠒⠒⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣤⠊⠀⠑⡲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆⠎⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣶⡄⠀⠀⠘⠀⠀⢀⠀⠉⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠃⢠⡀⠀⠀⠀⠀⠀⣿⢷⡀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠔⠀⠀⠆
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠓⠀⠀⠀⢀⡿⠘⣷⡀
⠀⠀⠀⣄⡀⠴⠉⢆⡀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠈⠻⢦⣤⣄⣀⣀⣤⣤⣄
⠀⠀⠀⠀⠑⢄⢠⠖⠉⠁⣀⣤⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠛⠉⠀⠀⠀⠐⣶⠤⣠⠤⡾⠁
⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠙⠛⠛⠷⢦⣄⡀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠤⠤⠤⢽⣄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢿⣀⡿⠁⠀⠀⠀⠐⢤⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠉⠀⡀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠘⣿⠇⠀⠀⠀⠀⠀⠀⠙⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡀⠀⠀⠀⠀⠿⠀⠀⠀⢸⡆⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠣⡀⠀⠀⠀⠀⠀⠀⠀⣼⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢅⠀⢈⠝⠉⠀⠀⣀⡤⠞⠁⠀⠈⠙⠒⣲⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠃⠀⠀⠀⠈⠉⠛⠲⢄⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢄⠦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠒⠀⠀⠈⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢉⣤⡀⠀⠀⠀⢹⡿⠀⠐⠘⡃⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⣤⠀⠀⠀⠀⣤⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⡀⡀⠀⠁⠿⠇⠀⠐⠂⠀⠀⠀⠐⠀⢩⣥⢄⠀⠀⠀⠀⠈⠀⠀⠉⠋⠀⠘⠠⠀⠛⠃⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⠀⠀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠀⠀⠀⠀⠀⠀⡀⠀⢀⠐⢠⠀⠀⠀⠀⠀⢴⠃⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠚⠇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠄⠀⠰⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠆⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸⠇⠀⠀⠀⠀⣀⠰⠂⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠄⢠⡌⠀⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠔⢀⠀⢀⣠⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠤⢄⡠⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠄⢠⡤⠠⣤⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠠⡤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⠀⠸⡇⠀⣿⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢻⠓⠚⠟⠚⣿⠒⠚⠀⠀⠚⠂⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠟⠛⣿⠛⠄⠀⠀⠛⠛⠢⠀⠀⠛⠛⠉⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣀⣛⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⣀⠀⣀⠀⢀⠀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠏⠐⠊⠀⠐⠁⠀⠧⠄⠘⠤⠒⠠⠌⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⣤⡶⢋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⡰⠛⢋⣥⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣰⠞⠉⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡾⠟⣡⣴⡾⣁
⠀⠀⠠⠲⠶⠶⠶⠶⣤⣾⣠⣞⡡⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⣧⠾⢛⡥⠞⣣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠯⠥⠤⠤⠦⠴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢉⣴⠞⢋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠋⠥⠶⠶⠆

⣿⣿⣿⣿⡿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠿⠿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠋⠛⢿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠿⢿⣿⣿
⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻
⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡴⠁⠀⢀⣾⠋⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣆⣠⠄⢠⡞⢁⣿⣶⣾⡶⣿⣉⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣾⠁⣠⣿⣟⣿⣿⣿⣿⡿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⣿⣿⣿⣿⢈⣿⢸⣏⣿⣿⣿⠃⣾⡷⣣⣔⣠⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣿⢻⣷⠿⠛⢸⢿⣿⠛⣿⣸⣿⣼⣿⣿⣥⣤⡼⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣼⢿⣛⣿⡆⠀⢸⣿⠟⣴⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣝⣿⠁⣸⣷⣴⢿⣿⣻⣿⣿⠛⣻⡟⣫⣽⣿⣿⣿⣿⣿⣀⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠞⠛⣿⣶⡟⠿⠿⠋⠉⠉⣵⣿⡿⢿⣾⣻⣿⣿⣿⣿⢿⡟⢋⣸⡷⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣴⣿⢛⠁⠀⠀⠀⢤⣿⣿⣯⣴⣿⣿⣿⣿⠿⠟⣻⡿⠛⣻⢟⡴
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣽⣭⡅⠁⠰⠂⠀⠀⠀⣹⣿⣯⣽⡾⠛⠙⢷⣴⠟⢡⣧⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⣐⣿⠀⠀⠀⣠⠊⢁⣰⣶⣿⣄⣠⣶⠾⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣻⡯⠇⠀⠸⠇⢂⣿⡿⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⣿⣻⠀⢠⡄⠼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣯⡻⠇⠹⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣧⠀⡵⣸⣿⣿⣷⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⣀⣀⣤⣴⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⢰⡿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣽⣿⣶⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣾⣷⣿⣿⣿⣿⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣈⣅⣈⡉⠶⠖⡂
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⡿⣿⡿⣿⢹⡰⣾⣿⣿⣿⠿⣧⣈⡻⠿⣿⣷⠾⡥⠭⠅⠀⠀⠉
⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣧⣿⣿⢻⡜⣿⣧⠈⠁⢈⠛⠻⣿⣃⠐⠒⠐⠒
⣿⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡼⢎⡓⠫⢴⣤⡈⠑⠦⣭⠉⠁
⡄⠈⣉⡹⠛⠛⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⠿⣿⣿⣿⣿⢫⣉⡳⣮⣶⡠⢌⠻⠦⠀⠀⠀⠰⠶
⠏⠀⠀⠀⠀⠀⠹⢿⢿⡿⢿⢿⣿⡿⢻⣿⣿⢻⣟⡃⢸⡇⢹⣿⣿⣧⠙⣿⣛⣿⣿⠎⠁
⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠿⠏⠁⣿⡏⢸⣿⡟⡉⢸⣿⢸⣿⣿⣿⣧⠜⠛⠉
⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢋⡉⠙⠻⢈⠙⠁

⠀⠀⠀⠀⣾⡇⣼⣿⢸⣿⣿⣿⣷⣾⡿⠿⣧⠀⢈⠋⢀⡜⢹⣦⠘⣧⢰⣿⣿⢸⡿⢿⣿⣿⣿⡀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⣼⣿⡏⢁⣿⣿⢾⣿⡆⣿⣾⡇⠀⢴⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⠛⣿⣿⣯⡀⠒⠹⣿⣿⣿⣧⡀
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣸⣿⣿⡇⢸⣿⣿⠸⠿⠀⣿⣿⡇⠀⣼⣿⣿⣟⠛⣥⣷⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣆⠠⣿⣿⣿⣿⡇
⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⡇⣾⣿⡿⠁⢸⠀⣿⣿⡇⠀⣿⣿⣿⣹⣿⣿⣟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢈⣿⣿⣿⡋
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡓⢸⣿⣿⡀⠐⣿⣿⣿⡇⣿⣿⣧⠀⣿⣧⣿⣿⡇⢸⣿⣿⣿⣿⣿⢽⣿⣿⣿⣺⢿⣿⣿⠟⣿⣿⣿⣿⡏⢰⣿⣿⣿⡃
⠀⠀⠀⠘⣿⣿⣿⣿⣿⠅⢸⣿⣿⠃⢰⣿⣿⣿⣿⣿⣿⣿⠀⣿⢽⣿⣿⡇⢘⣿⣿⣿⣿⣿⠀⣿⣿⡿⠡⢴⣿⣿⠈⣿⣿⣿⣿⡇⢰⣿⣿⣿⠁
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡃⢸⣿⣿⠀⢸⣿⣿⣿⡟⢻⣿⡇⠀⣿⠄⣽⣿⡇⢸⣿⣿⣿⣿⣿⠆⣿⣿⣯⠀⢸⣿⣿⠀⢿⣿⣿⣿⡇⣾⣿⣿⣟
⠀⠀⠀⠐⣿⣿⣿⣿⣿⠁⣽⣿⣿⠁⣸⣿⣿⣿⡇⢸⣿⠁⢸⣿⡮⣽⣿⡇⢸⣿⣿⣿⣿⣿⡅⣿⣿⣿⠀⣸⣿⣿⠀⢸⣿⣿⣿⡇⣻⣿⣿⣷
⠀⠀⠀⠀⢿⣿⣿⣿⡟⠀⣿⣿⣿⠀⣿⣿⣿⣿⠇⢸⣿⠀⢸⣿⠇⠉⣿⡇⢸⣿⢿⣿⣿⣿⣧⣿⣿⣿⠀⣿⣿⣿⠀⢸⣿⣿⣿⡇⠹⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⣿⣿⣿⠀⣿⣿⣿⣿⠀⢸⣿⠀⣼⣿⣅⠀⣹⡇⢸⣿⠛⠛⠋⠘⠋⠓⡫⣿⠀⣹⣿⣿⠀⢸⣿⣿⣿⡇⢰⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⡦⠀⣿⣿⣿⠀⣿⣿⣿⣿⠀⢸⡟⠀⣿⣿⡏⠀⢸⡇⠜⣿⡿⠀⠒⠀⢀⡄⢁⣿⠀⣿⣿⣿⠀⢸⣿⣿⣿⡇⢸⣿⣿⣿⡄
⠀⠀⠀⠀⣾⣿⣿⣿⡄⠀⣿⣿⣿⡈⣿⣿⣿⣿⠀⢾⣗⠀⣿⣿⡇⠀⢹⡽⠠⢻⡀⠀⠈⠠⠾⠃⠉⢾⠀⣿⣿⣿⠀⢸⣿⣿⣿⠁⢨⣿⣿⣿⣇
⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⣿⣿⣿⡇⣿⣻⠟⣿⠀⢿⣿⠀⣿⣿⡇⠀⢸⠛⠁⠨⠀⠀⠀⠀⠀⠀⠠⣾⠄⣿⣿⣿⠀⣸⣿⣿⣿⠀⢐⣿⣿⣿⡟
⠀⠀⠀⠀⢹⣿⣿⣿⡃⢠⣿⣿⣿⠁⠉⠘⠒⣿⠀⢸⠛⠐⢿⢻⡇⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⢦⣿⣿⣿⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿⡷
⠀⠀⠀⠀⢸⣿⣿⣿⠂⠀⡇⡿⢺⠀⠀⠀⠀⠘⠀⠀⠀⠀⠻⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡙⢿⣿⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿⣗
⠀⠀⠀⠀⣾⣿⣿⣿⡃⠀⡗⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣿⠀⣿⣿⣿⣿⠂⢸⣿⣿⣿⣇
⠀⠀⠀⠀⣿⣿⣿⣿⠆⠀⡇⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣿⣿⠀⣿⣿⣿⣿⠀⢰⣿⣿⣿⣟
⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⣷⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⣿⣿⣿⣿⠀⢼⣿⣿⣿⡏
⠀⠀⠀⠀⣼⡟⣿⣿⡇⠀⣿⡟⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⡟⠀⣿⣿⣿⡟⠀⢸⣿⣿⣿⡇
⠀⠀⠀⢰⣿⡇⢻⣿⡅⠀⣿⡇⠀⠀⠀⠀⠀⠀⠘⣠⠀⠘⠏⠉⠉⠀⠀⣠⠂⠆⠀⠒⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⣻⣿⣿⠁⠀⢸⣿⣿⣿⡇
⠀⠀⠀⣿⣿⡇⠸⣿⠂⠀⢻⡇⢠⠀⠀⡀⠀⠀⠀⢻⣷⣶⣶⠇⢳⣦⡌⣹⡄⠺⠀⣸⡆⠀⠀⠀⠀⠀⠸⡿⠀⡀⠀⠻⣿⡇⠀⠀⠸⣿⣿⣿⣇
⠀⠀⠀⣿⣿⣓⠈⢿⠀⠀⢸⡓⡟⠀⠀⢳⡀⠀⠀⠀⠀⠈⣿⠀⡀⣿⡿⢡⡀⠀⠀⡇⠀⣠⠀⡄⠻⠀⠀⠅⠀⠑⣄⣠⡿⠃⠀⠀⣰⢹⣿⣿⣧
⠀⠀⢀⣿⣿⠿⣦⣼⣄⣴⡿⣀⠃⠀⠀⢸⣿⣿⣦⣤⣤⣼⣿⣦⣤⣴⣶⠟⠷⢀⠀⠳⠀⠡⠤⠴⠀⠀⠀⠘⣦⡀⠬⠛⠀⣤⠄⠀⢻⣶⣿⡿⢎
⠀⠀⢸⣿⣿⠀⣿⣿⡟⢡⣾⠫⠀⠀⠀⢶⠟⠿⠛⠛⠛⠛⣿⣿⣿⣿⡇⠛⣷⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠸⠷
⠀⠀⢸⣿⡿⠀⠀⠈⠘⠋⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣿⣿⠈⣿⣧⢰⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠈⢙⡀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⢿⣿⣉⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠼⠶⠀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⢋⣀⡀⢀⣋⠁⣤⣤⡤⢠⡆⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡤⠂⠘⠷⣄⠀⠀⠀⠉⠙⢻⠅

⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⡿⣿⣷⣶⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢹
⠀⣸⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣄⣨⣿⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣤⣄⣤⡀
⣰⡿⣡⠀⠀⠀⠀⠀⠀⠀⢀⣠⣮⣿⣾⣿⣿⠿⢿⣿⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡿⢀⣞⡄⠀⠀⠐⠻⠿⣿⣿⣿⡅⢙⣿⣿⣅⣀⣈⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠁⢸⣿⡇⠀⠀⠀⠀⠀⠀⠠⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⠛⠻⠟⠻⠁⠀⠁⣿⣷⣶⣴⡶⠤
⠀⣼⢿⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⠋⠀⠀⠀⠀⠹⣿⣿⣿⣾⣿⣿⣾⣿⣿⣿⣿⡏
⣶⣿⡍⠀⠀⠀⠀⠀⠀⠀⣾⢻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣄⠀⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⡀
⡿⠋⠻⠟⠀⠀⠀⠀⠀⠀⣿⣼⣿⢇⠀⠹⣧⠀⢙⣿⣿⣿⣷⡔⢛⣿⢳⣿⢿⣿⣿⣿⡟⠋⠑⠛⣯⡹⣿⡿⠀⠀⠀⠀⠀⣷⡃
⠃⠀⢀⠄⠀⠀⠀⠀⠀⠀⠏⠈⠛⢿⣿⣷⣾⢟⣿⣽⣟⣴⣟⠛⠻⣿⣿⡭⣾⣿⣿⡾⢿⣷⣼⣿⣷⣏⣻⡁⠀⠀⠀⠀⠀⢻⣷⡄
⡧⠀⣨⡆⠀⠀⠀⠀⠀⠀⠨⣾⣾⣿⡿⣿⣾⣙⠃⠉⠹⣿⣿⣷⣬⣷⣿⣿⠿⢋⣡⣼⡶⢿⠋⢰⣿⣛⣻⡇⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠇⠀⣤⡇⠀⠀⠀⠀⠀⠀⠀⢺⣻⠽⣙⢹⠆⣹⢾⣶⣶⣷⣿⣿⣿⠿⢯⣿⣷⣾⣟⣯⣩⣜⣿⡭⣿⣩⢬⣧⡀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⣼⣿⡇⠀⠀⠀⠀⠀⣀⣤⣨⣶⣠⣷⠾⣿⠿⠛⣋⣉⠁⣾⣿⣿⣿⣷⡆⠉⠙⠻⠿⢷⣶⣷⣶⣿⡶⣀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠹
⢀⣿⣿⠃⣴⣷⠀⠀⠀⠈⣽⣿⣭⣭⣤⣿⠯⠀⠈⡿⢁⣀⣠⣭⣿⣿⣿⡿⣿⣦⠠⣤⣬⣿⣟⢿⡟⣿⣿⣿⡟
⣼⣿⡿⣼⣿⡟⠀⠀⠀⠀⠀⠉⠛⢯⣛⣿⣿⢟⣿⣀⣻⣿⣿⣿⣿⡿⠿⠛⠛⠉⠀⠶⢁⣠⣤⣤⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠠⣤
⣿⣿⣷⣿⣿⠃⠀⠀⠀⠀⣠⣦⠀⠀⠈⢻⣿⣷⢿⣿⣿⣿⣿⣿⣛⣁⢀⣀⣶⣾⣿⣿⢟⣻⣿⡇⠁⣾⣷⠄⠀⠀⠀⠀⠀⠀⠀⢶⡀
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠙⠋⢠⡄⠀⢸⡏⠻⣼⣿⡿⣿⡟⠹⣿⣿⠛⣿⢏⣭⡿⢗⣿⣿⡏⠀⠴⠌⡁⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢠⣦⠀⠀⡀⢸⡇⠁⣸⣿⣾⣿⣧⡇⢹⠙⠀⣿⣿⣿⣿⡶⠟⣿⡇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠘⢇⣽⣿⣿⣿⣿⣿⣿⣾⠀⣼⣿⣿⣿⣿⠄⠀⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢰⣿⣧⠀⢸⣿⣿⣿⠻⣿⣿⠀⣿⣿⠟⠉⠉⠀⠀⣿⡇⠙⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣿
⣿⣿⣿⣿⢀⣾⣷⡄⠀⠀⠀⠀⠀⠀⠈⠛⢋⠀⠈⣿⣿⣿⠀⢹⣿⣿⣿⣏⣀⣀⡠⠀⠀⣿⠃⢀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯
⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣿⣿⣿⠀⢸⡟⠟⠀⣿⣿⣿⠀⠀⠀⠃⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢀⠀⠛⠛⠻⠀⠘⠇⠀⠀⠉⠈⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠤⠤⣄⡀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠈⢹⠀⠀⢠⠊⡟⠅⠜⣓⢄⠀⠀⢀⠖⠁⠀⠀⠀⡀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡠⠊⢩⢂⠎⠰⢀⢀⠖⠉⡨⠛⠉⠙⢭⠉⠑⢄⡎⠀⠀⣀⠠⠊⠈⠆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠇⠀⠉⠁⠀⠀⣿⣌⠀⢰⣁⣠⣾⡾⠶⣇⠀⢸⢹⡏⠁⠀⠀⠀⠀⢺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢠⠐⠒⢤⢀⣷⣿⠒⢻⠇⠀⠀⠀⠀⡿⠒⠁⠀⢃⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠾⣤⢦⠾⣾⡿⠃⠀⠀⢣⡀⠀⠀⡰⠁⠀⠀⠀⠀⢑⠤⠤⢴⡊
⠀⠀⠀⠀⠀⠀⠀⠀⡰⡋⠙⠤⠺⠗⠤⠎⢹⠦⢀⣀⣤⠼⣅⡿⣤⣀⡀⡔⠋⢹⢸⠑⠒⠈⡉⠉⡏⢢
⠀⠀⠀⠀⠀⠀⠀⠀⣅⡇⠀⠀⠀⠀⠀⠀⢸⣄⠋⠐⡿⠒⡛⠐⢻⡇⣿⠇⡄⢸⠸⣦⣤⠤⠂⠀⣇⡠⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⢸⡟⠐⡀⡇⠀⡀⠀⢸⡇⠇⠀⡇⡌⢰⠓⠚⠛⡆⢇⡏⢂⠘⡀
⠀⠀⠀⠀⠀⣠⠖⠋⢰⡇⠀⠀⠀⠀⠀⠀⢸⣁⣠⢷⡇⢀⣇⣀⢸⣷⠤⣷⠛⠀⠸⠀⠀⠀⡇⠈⡇⠈⡄⢡
⠀⠀⠀⠀⣼⠁⠀⣂⡼⠧⠤⠤⠤⠤⠤⠤⠾⠶⠾⠾⠷⠼⠷⠯⠼⠿⠤⠿⠶⠤⠴⠶⠶⠶⠧⠤⠷⢤⢰⣘⡀
⠀⠀⠘⢍⠉⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⡩⠁
⠀⠀⠀⠀⢣⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡜
⠀⠀⠀⡠⠃⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣸⠀⠀⠀⠘⢄
⠀⠀⠐⠃⠀⠀⠀⠀⠘⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠛⠁⠀⠀⠀⠀⠘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⠿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣾⡏⢀⣠⣤⣤⣄⠀⠀⠀⠀⢀⠛⠉⢹⡿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢀⣿⢁⣾⠛⠛⢻⣿⠃⢠⣦⣴⣿⡆⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⣠⣾⣿⣿⣿⡆⣿⡇⢸⣿⢸⣿⣴⣾⠿⠋⠀⣿⣿⠏⣿⣇⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⢡⣿⠁⣠⣼⡿⠁⣿⡇⢸⣿⠘⣿⣿⣤⣤⡄⠀⣿⡏⠀⠻⠋⠀⠈⠋⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⡾⢃⣾⠏⠀⠀⠸⣿⣿⠟⠉⠀⠀⣿⡇⢸⣿⠄⠈⠛⠛⠛⢁⡀⠛
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠆⠀⠀⢻⣿⣄⣠⣾⠋⠁⢸⣿⠀⠀⠀⢀⠹⢿⣿⣿⠇⠀⠻⠃⣀⣩⣤⣶⣶⣿⡿⠿⠃
⠀⠀⠀⢀⣤⣶⡿⠿⠛⢉⣉⣄⠀⠀⠀⢹⣿⣿⣅⠀⠀⢸⣿⣷⣤⣶⡿⠀⠀⠀⣀⣤⣶⣾⠿⠟⠛⠋⠉⠁
⢀⣤⣾⠿⢻⣿⣥⣶⣿⠿⠟⠋⠀⠀⣰⡿⠋⠻⢿⣿⡷⠀⠙⠛⠛⢉⣀⣤⣶⡿⠟⠋⠉
⠛⢋⣤⣶⣿⠟⠋⠁⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠋⠁
⠀⠈⠋⣽⡏⠀⠀⠀⠀⠀⠀⢀⣤⠍⠁⠀⠀⠀⠀⠀⣴⡿⠟⠋⠁
⠀⠀⢀⣿⠁⠀⠀⠀⢀⣠⣴⡿⠋
⠀⠀⠘⣿⣷⣤⣶⣾⠿⠛⠁
⣤⣤⣤⣽⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣈⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠒⠈⠉⠒⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠤⢀⡀
⢀⣠⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣄⡀
⣟⠦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡖⠀⠀⠀⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀
⠈⠑⠢⢌⡐⠤⣀⡀⠀⠀⠀⣀⢀⡄⠀⠴⠂⠛⠋⠉⠉⠈⠁⠀⠉⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠈⠑⠢⢭⣱⣺⢍⡋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⢒⣉⠗
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠟⠲⣍⡑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠐⢂⣈⠤⠄⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⠇⠈⠑⠢⢌⡑⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠒⣀⡡⠤⡖⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆⣸⠀⠀⠀⠀⠀⠈⠑⠢⢌⡐⠤⣀⠀⠀⠀⠀⠀⣀⠠⠄⠂⣈⠠⠄⠒⠉⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⡃⠀⢱⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣙⠲⠖⢊⣉⠤⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠤⠷⠲⣯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠎⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣉⣉⠉⠒⠒⠋⠉⠙⡆⠀⠀⠀⠀⠀⠀⠀⡴⠊⠙⢺⡆
⠀⠀⠀⠀⢸⡼⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⠇⠀⠑⢄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠠⢧⠀⠀⢀⡠⠔⠉⠙⠢⣀
⠀⠀⠀⠀⡘⡡⠀⡀⢀⡜⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⡒⠳⢄⢠⡦⠧⠤⣄⠀⠤⠞⠁⢀⣀⣀⡀⠀⠀⡷⠊⠁⠀⠀⢀⣀⣤⠼⠒
⠀⠀⠀⠺⠎⠴⠾⠔⠋⠀⢠⠔⡻⠓⠻⢗⠦⠤⠄⣀⣀⡀⠀⠀⠀⠈⠉⠈⣹⣀⠀⢀⡼⠦⠤⠄⢀⠀⠦⠤⠽⠂⣰⠧⠤⠶⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡜⠀⠀⠀⠀⠡⡀⠀⠀⢠⠀⣀⣀⣈⣉⣉⣁⣀⣀⠩⣇⠙⡶⠒⠒⠚⠛⠓⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⢱⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠙⠒⠂⠉⠉⠁⠈⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠠⠤⠠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⡇⠀⢀⠇⢀⣀⡠⠤⠖⠒⡲⠋⠀⡅⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⢀⣜⣀⠤⠚⠊⠉⠀⠀⠀⠀⡸⠁⠀⠀⡇⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⡀⡿⡀⠀⠀⠈⠑⢒⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡎⠀⠙⠷⠈⠢⣀⠀⠈⠓⢤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⠤⠶⠶⠒⠒⠀⠀⠀⠀⠀⠀⠐⠢⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠑⡒⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠬⣑⠒⠭⣙⠳⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠶⠒⠈⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠭⣒⡢⢍⡒⠦⢌⡀⠀⠉⠛⢦⣄⠠⠐⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠒⠮⣅⡲⠬⣑⠂⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠤⠭⠷⢦⣍⣐⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣀⣀⣤⠴⠶⠿⠯⠭⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⢤⣤⣶⣿⡥⠶⠖⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡝⢯⠻⢯⣭⣭⣭⣥⣤⣤⣤⣤⠴⠶⠶⠞⠛⠛⠉⠉⠁⠀⠀⢀⣀⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠁⠈⢧⣸⠇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⠤⠄⠀⠀⠀⠀⠁⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡿⢀⣀⣤⣤⣬⣿⣷⣿⣿⣿⣛⣛⣛⣉⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⣠⡴⠞⠛⠉⠁⠀⢸⣿⠏⠉⠀⠀⠀⠀⠀⠠⠤⠤⠬⠭⠭⠭⠍⣉⣉⣙⣛⣛⣛⣛⠓⠶⠶⠶⠶⠶⠶⢶⠾⠗
⢠⡞⠁⠀⢀⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠂⠒⠒⣒⣒⣒⣒⣒⣀⠀⠀⠀⠀⢀⡏⠓⠦⣄⡀
⣿⠀⠀⠀⠀⠈⠉⠂⠀⠸⣇⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠭⣭⣍⣉⣛⣓⣒⣒⣒⡒⠶⠦⠤⠤⠤⠄⠀⠀⢸⡁⠀⠀⠀⠙⠳⣦⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣒⣒⣒⣲⠶⠶⣶⠆⠠⠀⢔⠺⣓⠐⣋⣉⣉⣉⠁⠲⠶⠶⠭⠤⠄⠘⣧⣤⣴⠶⠾⣿⠟⠋
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⢍⡉⠙⠛⠓⠒⠓⠒⠚⠶⠶⠿⠿⠿⠿⠷⠶⠶⠿⠿⡿⣭⣶⣶⡷⠀⠀⢠⡏
⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠀⠀⢀⣤⠶⠚⠁⢀⣀⣤⠤⠤⠀⠀⠀⠀⠀⢀⣀⣤⠤⠀⠀⢸⡇
⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠒⠀⠈⠁⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⡀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣈⣉⣤⣤⠤⠴⠂⣀⣀⣠⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣘⢯⣒⠀⠀⠀⠀⠀⠐⠒⠒⣛⣛⣩⣭⡥⠴⠶⠶⠚⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⣿⣿⣿⣿⣿⣿⣿⣿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⡛⠛⠛⠀⠀⠛⠛⢛⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣶⠀⠀⠀⠠⣾⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣠⣶⣶⣄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣿⣿⣿⣿⣿⣿⣿⣿⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠀⠉⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦
⠀⠀⠀⠀⠀⠉⠛⢷⣤⣤⣶⠶⠶⠟⠛⠛⠉⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡿⣷⠀⠀⣿⠀⠀⠀⠈⠙⠻⢶⣤⣀⣤⡶⠟⠋⠁⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⣾⠃⢿⡄⠀⠻⣦⣀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠰⣿⣤⣼⡷⠀⠀⠈⠙⠻⠶⣦⣤⣤⣀⣀⣀⣤⣤⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠾⠛⠛⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡶⢶⣿⣿⡋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⡶⠾⠛⠛⠋⠉⠉⠁⠀⠀⠀⢈⣿⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⣁⣀⣈⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⣸⣧⡀⢀⣀⣀⣤⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢁⣾⢋⣭⣍⠻⣦⠘⣷⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⢹⣏⢘⣿⡟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣧⢸⣇⠹⠟⣿⠀⣿⠀⠀⣀⣤⣴⠶⠟⢻⣿⡶⣾⣿⡿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⣧⣝⡛⠛⣋⣴⣿⠾⠛⠉⠁⠀⠀⠀⠉⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁