Tight Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⢀⠄⠠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠠⣄⠀⠀⢀⣤⠞⠉⠉⢧⠀⠀⠈⣈⡰⡆⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⢀⡧⢌⢢⣠⠞⠀⠀⠀⠷⢸⠖⠋⠉⠉⠳⡜⡀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢸⠁⠀⣹⠃⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢀⡇⠰⡀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠐⠷⠋⠁⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⡏⠀⠀⠀⠀⢠⡾⠣⣍⣀⣀⡤⠞⠀⠀⠘⠷⡀⠀⠀⠀⡾⠀⢤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⢸⡀⠀⠀⠀⠀⠀⣴⡼⠀⠀⠀⠙⢄⣀⡤⢞⠟⠁⠀⠈⠻⡁⠀⠀⠀⠀⠀⠀⠈⠲⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠘⠚⠁⠀⠈⠳⣄⡀⠀⣀⠞⠁⠀⠀⠀⠀⠀⣸⠀⠀⠁⠀⠀⠀⠀⠀⠹⣄⣀⣀⣀⣀⣀⣀⠀⠀⠀⠰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡥⠶⠒⠒⠒⠒⠲⠟⠃⠀⠀⠀⠀⠀⢀⣠⠔⠊⠉⠁⠀⣀⣀⣀⣈⣹⡆⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣾⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⡽⠋⠀⠀⠀⣤⠖⢶⠒⠲⡦⣤⣄⡀⠀⠀⠈⠉⠉⠉⢻⠁⠀⠀⠀⢀⡀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠙⣦⠤⢷⠊⢙⣅⣤⠽⠷⢦⣀⠀⠀⢀⡞⠀⠀⢀⣠⠞⠳⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠘⢆⡬⠓⠉⠀⠀⠀⠀⠀⠈⠓⠒⠛⠒⠒⠚⠉⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⠒⢲⡒⠒⠒⠛⠛⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠸⡄⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠻⠀⠀⠀⠀⠹⣄⠀⠈⠳⣄⢀⡤⢦⡀⠀⠀⣴⠆⠰⠃⠀⠘⡇⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠉⠀⢤⣟⣿⣰⡟⣸⠃⢀⠀⠀⢳⠚⠀⠴⠃⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠳⡜⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⢀⠀⠀⣹⣼⠁⢠⠎⢀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠶⡶⠒⠒⠉⠁⠈⠓⠤⣄⣠⣄⣈⡿⠁⠈⠉⣻⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⡈⡸⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⢿⡖⠀⠀⠀⠔⣶⣶⠓⢂⣴⠁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⡩⠉⠳⠜⠧⣄⡀⠈⠉⢸⠁⣀⡽⠚⠉⡁⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠋⠀⣠⠂⠀⠀⠀⠉⠓⠢⣿⣊⣁⠀⠀⠀⢹⠀⠘⡄⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠿⣤⠤⠒⠒⠋⠉⠉⠉⠁⠈⠉⠓⠒⠛⠲⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠂⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⠀⠀⠀⠀⢀⣤⠶⠓⠶⡄⠀⠀⠀⠓⡘⢀⣄⠀⠀⠉⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⣌⠙⢢⡀⠀⢀⡴⠋⠀⠀⢀⡀⣿⢀⣠⣤⣤⣧⣏⠸⡀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⡟⠑⢦⡹⣴⠟⠀⠀⠀⠀⠙⠋⣿⠋⠁⠀⠀⠀⠈⢷⠱⡀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⣀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣹⠃⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⣿⠀⢳⡀⠀⠀⠀⠀⠀⠀⢹⡆
⠾⠭⠛⠃⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣠⡿⣦⡛⠁⠀⠀⣠⡾⠁⠀⠀⠱⣶⡀⠀⠀⠀⠀⢸⡇⢀⣠⢤
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⣠⢰⠇⠀⠀⠀⢧⡀⠀⠀⢀⣠⢾⡟⠁⠈⠙⢿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⣠⠟⠀⠀⠉⠙
⠀⠀⣠⢄⡶⠂⠀⠸⣇⠀⠀⠀⠀⠀⠀⣴⠻⠋⠀⠀⠀⠀⠀⠉⡷⠚⠩⠴⠋⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠉⠉⠀⠀⠀⠀⠙⠷⣤⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⡀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⣠⣤⡤⠤⠤⢤⣤⣀⡀⠀⠀⠀⠻⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣡⠴⠖⠒⠚⠓⠒⠒⠶⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠀⠀⠀⠀⣀⣀⣀⣈⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢓⣶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠒⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣤⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣩⡿⠁⠀⠀⠀⠀⣶⠛⠛⢻⡛⠛⢻⣓⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣠⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠘⢧⣠⡼⢷⠚⠉⢯⣀⣠⣬⠿⠶⣤⡀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣠⡼⠋⢧⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢘⡷⠖⠋⠁⠀⠀⠀⠀⠀⠙⠳⠦⠴⠾⣤⡤⠤⠶⠛⠉⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣀⣀⣤⣤⣤⣤⣤⠤⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠦⠤⠤⢤⣤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣦⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡏⠃⠀⠀⠀⠹⣆⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠈⣟⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡴⠞⠉⠀⠃⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠙⢦⣀⡴⠛⠻⣄⢀⡀⠀⣿⠉⣁⠃⠀⠀⠀⢿⠀⣿⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⠀⠰⢶⣟⣹⣇⣸⠋⣩⠇⠀⢀⠀⠀⠘⡷⠋⠀⠶⠛⠀⠰⠃
⠀⠀⠀⠀⠀⠀⢰⡀⢳⡈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣴⠋⠀⣠⠃⢀⡄⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠳⠦⢤⣤⣀⣀⣀⣠⣤⠤⠴⠒⠋⠳⣄⡀⠀⠄⡀⠈⠒⢶⡖⠋⠙⠲⠶⣧⡤⠾⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠛⠻⢶⡟⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣸⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢿⣗⠀⠀⠀⠠⠖⢒⣿⣾⠓⠒⣠⣟⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⢉⡉⠻⢦⡸⢧⣄⠀⠐⠒⠊⠁⣼⠀⠀⣈⡵⠛⠋⠉⠁⠈⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣡⡾⠋⠀⢀⠀⠀⠀⠈⠛⠶⣤⣀⣀⣯⠴⠞⠁⠀⠀⠀⢳⡀⠀⢰⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⢰⡏⠀⠀⠀⣀⣀⣀⣀⣈⣽⠧⠶⠦⢤⣄⡀⠀⣸⡃⠀⣸⣇⡼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠶⠖⠒⠚⠛⠉⠉⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡴⠁⣠⣴⠃⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠂⠀⠀⠀⠀⠀⠘⢦⣸⠁⡇⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠙⢷⡀⠀⠀⠀⠀⠀⠉⠀⠳⢤⣀⡀⠀⠀⠀⠀⠳⣤⠴⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠘⣇⠀⠀⠀⠈⢦⡀⠀⠀⢀⣼⣿⣿⣆⠀⠀⠀⠈⠀⠶⢦⣼⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣿⣀⣤⣤⠤⠤⢤⣤⡶⠋⠁⢀⡏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⣠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠙⣦⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⣤⢦⢿⣄⣀⠀⡀⠀⠀⢀⣀⣤⠾⠁⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⡴⠚⠚⠛⠛⠛⠛⠛⠛⠛⠋⠉⢉⣴⠖⠀⠙⠯⣛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⣿⠈⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⢦
⠹⣤⣐⠀⡄⠀⣀⣀⣠⣤⠶⠚⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣀⡀⠀⠀⠀⠀⢹⡄⠀⢳⣀⣠⠜⠃
⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢍⣉⡧⠀⢠⡇⢸⡿⡄⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⣦⣟⣹⡴⠫⣧⡼⠇⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢒⣚⣛⣩⡭⠭⠿⠿⠛⠿⠷⠶⣶⠒⠒⠋⠉⠳⣤⣀⣀⠀⠀⠀⢀⡴⠛⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢤⡀⠀⢀⣠⠖⣋⠭⠖⠊⠉⠁⠀⠀⠀⢀⡤⠴⠒⠒⠒⠒⠦⣍⠲⢤⡀⠀⠘⣇⠀⠉⠉⠒⠚⠇⠺⠝⠋⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣠⢖⡫⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣍⣳⣀⣿⠀⠀⢀⣀⡀⠀⢀⡸⠭⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠞⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⠹⣏⡉⠀⠀⣹⠃⣏⠙⣦⣼⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠂⠀⠉⠛⠶⣼⠶⠚⢉⣉⣀⣄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣀⠀⠀⠀⠀⣀⣀⣤⣤⠶⣒⠿⠋⠉⠉⠒⣲⢟⣁⠀⠈⢉⣀⣀⣀⣰
⠀⠀⠀⠀⠀⠀⢀⡴⠺⡇⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠋⠹⢯⣛⣓⡻⣭⠽⠿⢦⡙⢯⠉⣀⣠⠤⣄⡀⡹⢫⡿⠛⠋⠉⢿⡉⠉
⠀⢠⡞⢦⣠⠞⠉⠀⠀⢷⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣀⡤⠞⠋⠁⢀⣀⣧⣤⣤⣀⠀⠀⠀⠐⡀⣀⣠⠷⣌⢻⡁⣀⣠⡤⣿⢡⡏⢠⡤⣄⠀⠀⢳
⢀⣈⣹⠆⠀⠀⢀⡠⠶⠚⢧⡀⢀⡴⡿⠀⠀⠀⢀⡤⠞⠁⠠⣀⡴⢞⡩⠖⣋⣉⣡⠟⠀⣴⠚⠛⠉⣁⣀⣴⣇⡼⠋⠉⠀⢀⡟⣿⢀⠞⣴⠋⠀⠀⠈⣧
⠘⠷⠶⣷⠀⡤⢾⣛⢛⡆⠀⠙⠁⢠⡇⠀⢀⡴⠏⠀⠀⢀⡴⠋⡴⠋⠀⢺⠶⢤⣍⠲⣤⠇⢀⡖⠋⠁⠀⠈⢀⣀⣀⠀⠀⣼⠋⣡⣿⠋⠐⠀⠀⠀⠀⢹
⠀⠀⠸⢥⠼⠁⠀⠉⠉⠀⠀⠀⠀⢸⠁⣠⠟⠀⠀⡀⠀⣼⠀⠀⠀⠀⣰⠃⠀⠐⠀⠙⣮⡳⢾⡆⠀⠀⠀⢴⠻⡁⠈⣧⣼⣥⡴⣿⠃⣠⠴⣶⣆⣤⣄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡃⠀⢀⠠⠘⡀⠈⠛⠶⠦⠶⠃⠀⠀⠀⣤⢴⠋⢱⠾⠃⠀⠀⣀⣨⡆⠱⡄⠘⣿⠓⠚⣩⢞⡵⠶⡞⡿⡲⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣧⠀⠀⠀⠀⠐⠀⠀⢀⠀⠲⡂⠀⠁⠀⠀⡈⠳⢬⠷⠞⠛⠉⠉⠀⠘⣆⣹⡤⢼⡦⢞⡵⠋⠀⢸⣤⣽⢵⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠤⠻⠀⠀⠀⠀⠠⠀⠀⠀⠀⢡⠀⠀⠀⢀⡖⣒⣛⣒⣛⡋⣥⠚⡙⢫⡞⠀⠀⠀⡀⠀⣠⠟⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⢈⣀⡀⢀⡼⣛⣿⡼⣧⠈⠛⠛⠛⢛⣯⣝⣲⣤⠎⢁⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⢳⠐⠀⠀⠈⠃⠀⠀⠀⠩⠀⠈⢳⡽⣭⠶⠋⠀⠀⠀⠀⠘⣷⣯⡟⣧⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠢⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⣀⣠⠅⠠⣤⠂⠂⠀⣚⡡⠈⢿⡯⢙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡦⠀⠀⠲⠀⠒⠀⠘⠀⠀⠁⠈⢥⢀⡀⠀⠒⠚⠘⠉⠉⠉⣀⣠⣼⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣤⣤⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠴⠶⣿⠿⠛⠋⠟⢻⠉⠙⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣬⠿⠉⠉⠛⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣃⣀⠠⣤⣖⡯⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⡁⠀⠘⢦⣀⣀⣠⠤⠝⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠦⣄⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠑⢦⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⢶⣄⡈⠁⠹⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣇⣼⠫⠳⠱⠀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀⠁⠂⣤⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⠒⠺⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡍⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠢⠢⠂⠠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠄⠀⠒⠒⠉⡁⠀⠘⡄⠀⡀⠁⠀⡠⢒⡓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⠀⣀⠤⠋⠀⠀⠀⠀⠀⠀⠀⢘⠩⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠌⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠠⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠠⡀⠀⠸⡄⠀⠀⠉⠛⢽⡓⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⢀⣠⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠔⠁⠂⠒⠨⢑⠢⠜⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⠒⢁⡀⠱⡀
⠀⠀⠈⡢⠄⢀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠲⠌⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⡏⢙⠢⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠔⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⢠⠂⠐⡢⠀⠈⠁⠒⠀⠒⠒⠒⠉⠉⢹⡍⣭⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠦⠤⣀⣀⣀⣈⣐⣈⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡸⢧⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡠⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠘⠉⠠⣏⡒⠒⠒⠂⠠⠶⡆⠐⣖⢺⠍⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⣧⡍⣴⡒⢺⢋⣘⣆⡠⠌⠑⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠢⣀⡠⠤⠀⠒⠒⠃⠰⡀⠀⠉⠉⢸⠛⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣖⠆⣤⡀⠀⢀⡠⠤⠐⠒⠁⠀⠀⠀⠸⢄⠀⠀⠈⠉⠒⢒⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⢄⣠⠄⣀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠂⠲⠀⠀⠐⠰⠐⠰⠀⠀⠂⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠂⠁⠐⠠

⢀⣶⣶⣦⣤⣴⣶⣤⣤⣶⣶⣦⣤⣤⣴⣶⣶⣶⣶⣤⣶⣶⣶⣦⣤⣤⣤⣴⣶⣦⣶⣶⣶⣶⣶⣶⣿⣦⣽⣿⣶⣶⣄
⢸⣿⣿⠉⠉⠙⠛⠛⠛⠛⠉⠙⠉⠉⠉⠉⠉⠙⠋⠙⠉⠛⠩⠛⠉⠉⠙⠉⠋⠉⠉⠉⢛⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⡀⠸⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠇⠀⢣⠘⣄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿
⣀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⢤⣀⠀⠀⠀⠀⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠙⠲⠤⡀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢡⣀⣟⣁⠘⠛⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠀⠀⠀⣨⢅⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧
⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢰⠀⠸⠀⠀⠀⠰⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⡀⠀⠀⠀⡄⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢁⡅⠀⠀⠷⣄⢀⡤⠁⠰⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿
⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃⠒⠀⠀⠀⠀⡦⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢘⠃⠀⠀⠀⠀⡍⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢸⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡆
⢼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⠀⠂⠄⠀⠀⠀⠀⠀⠀⠐⠈⡄⢀⣵⠀⠀⠀⠀⠀⠀⠀⢘⡟
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠂⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⢄⢠⠀⠸⡆⠀⠀⠀⠀⠀⠀⠈⡇
⢸⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠴⠀⠀⠀⠀⠀⠀⢰⡇
⢸⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⢀⡛⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⡇
⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⠁⠀⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⣼⡇
⢼⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠈⠀⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠺⠀⠀⠀⠀⠀⠀⢻
⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡂⠀⠀⠀⠀⠀⠀⣾⠃
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡃⠀⠀⠀⠀⠀⠀⠸⡇
⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⣄⠠⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠰⠁
⢸⣿⡗⠀⠀⠀⠀⢀⠤⠤⠤⠒⠁⣇⡀⠄⠉⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠶⠄⠩⠈⢃⠀⠀⠀⠀⠀⠀⢨
⠘⣿⡇⠀⠀⠀⠀⠻⡄⠶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠴⠡⣤⢀⣀⠀⢈
⠀⣿⣿⣦⣤⣤⣤⣤⣤⣤⣭⣤⣤⣴⣦⣤⣴⣶⣦⣀⣤⣤⣤⣶⣴⣦⣤⣤⣤⣦⣤⣤⣾⣶⣥⣤⣤⣤⣤⣤⣤⣤
⣀⣉⣛⣙⣛⣛⣛⣛⣛⣋⣛⣛⣛⣛⣛⣛⣙⣛⣛⣛⣋⣉⣉⣉⣉⣉⣉⣋⣉⣙⣛⣛⣉⣉⣙⣛⣉⣉⣉⣉⣛⣋⣀
⣿⣿⡟⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠂⠀⢀⠀⠀⠀⠀⠀⣠⠖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⣴⣿⠁⠀⠀⠀⠀⣧⠀⠀⣠⡾⠃⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⢰⡟⢀⣼⡿⠁⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣄⠀⠀⣾⣿⠃⠠⢂⣤⡴⠖⡄⠀⣀⣤⠶⢦⠀⣠⣤⡤⢠⣤⡀⠀⠀⠀⣀⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⢸⣿⡏⡐⣱⣿⠏⠀⣰⢇⣾⡿⠁⣠⡟⢀⣿⡟⠀⢀⣿⡧⠀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠏⠀⣿⣿⠤⠚⠁⢸⣿⡷⠞⠁⠀⣼⣿⠁⠀⣸⣿⠃
⠀⠀⠀⠀⠳⠤⠀⠀⠀⠀⢸⣿⡿⠹⣿⡄⠀⠙⠿⠦⠤⠴⠚⠛⠿⠦⠴⢚⣿⡟⢀⡴⢟⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠙⢧⡀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⣼⣿⠁⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠠⠠⠤⠤⠶⠛⣋⣁⣀⣀⡀⠀⠀⣭⣤⠀⠀⣀⡀⠀⠀⠀⠀⢠⣿⡏⠀⣴⡟⠀⢀⠀⠀⣠⣿⡿⠀⠀⠀⠛⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⡴⠛⠉⠉⠉⠛⠛⠛⣾⣿⣏⣀⠸⣿⠇⠀⠀⠀⠀⣾⣿⠁⣸⣿⢱⠋⠁⠉⢙⣿⣿⠿⣿⣶⣶⣤⣤⣤⠴⠃
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣴⠀⠀⢰⣿⡿⠀⢀⣴⡦⠀⣠⣴⠖⣤⣼⡅⢰⣿⡧⢶⣦⡀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⣿⣿⠃⠀⣸⣿⠃⣰⣿⠇⠀⣿⡿⠀⣿⡿⠁⣾⣿⠃⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⢀⣿⡿⠀⣿⡿⠀⣸⣿⠇⣸⣿⡇⠰⠶⡆⠀⠿⢿⠃⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠤⠚⠻⠷⠤⠛⠷⠴⣿⡿⠀⠿⠟⠀⠘⠿⠿⠊⠙⣿⡇⠀⠀⠀⠀⡼⠋⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠒⠒⠶⠶⢶⣾⣿⣧⣄⣀⣀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠉⠛⠛⠿⠿⢿⣶⣶⣦⣤⣤⣤⣀⣀⣀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠇⠀⠘⡏⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣄⠀⠀⠀⠑⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⡄⠀⡴⠺⠋⠷⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⢦⡟⠀⠀⠀⣿⣼⣋⢰⡋⣷⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⢀⣀⣀⣀⠀⠀⢯⣤⡉⣥⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠚⠃⠀⡴⠋⣴⠲⡌⠙⣆⠀⠺⠿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⢀⡽⠻⣧⠀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣶⠆⠈⢧⡈⠧⠤⠎⣠⣮⠤⣴⠋⠉⢹⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⡉⠛⠛⠋⢹⡅⠀⣨⠤⠦⢬⡁⠀⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⠞⡇⠀⠀⢸⠋⠉⣼⠁⠀⠀⠀⢹⡌⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠛⢦⡤⢻⡄⠀⠀⠀⣸⠣⣤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⠀⣩⠓⠒⢫⡁⢀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⢤⣤⠾⠉⠋⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⡄⠀⠀⠀⠀⠀⣰⡶⠛⠀⣤⣤⣤⣤⣤⡄⠀⠀⠀⢀⣼⠋⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣿⢶⣤⠾⠋⠀⠀⠀⠈⠳⢦⣴⢶⣟⠁⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣴⠟⠁⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣷⠀⠀⢀⡴⠟⠉⠁⢴⠄⠉⠙⠷⣄⠀⠀⢰⡏⠀⠀⠀⢰⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⣤⡾⠇⠀⢠⡟⠁⠛⠀⠀⣴⠀⠀⠘⠃⠙⣧⠀⠀⠿⣦⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⣾⠁⣤⠀⠀⠀⢿⡀⠀⠀⢠⡄⢹⡆⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⣿⣀⠀⠀⢻⡄⠁⠀⠀⠀⠈⠛⠂⠀⠀⠀⣾⠁⠀⢀⣸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⠹⣧⠀⠈⢿⣄⠺⠀⠀⢀⡀⠀⠸⢂⣼⠏⠀⢠⡿⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⢶⣟⠀⠀⠀⠙⠳⣦⣤⣍⣥⣤⡶⠛⠁⠀⠀⢘⣷⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣴⠟⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠙⣧⣀⣀⣀⣠⡟⠉⠉⠛⣿⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢄⠀⠀⢀⠀⠀⢠⠀⠀⠀⢀⠀⣀⠠⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠊⠸⠁⣇⡱⢪⣸⠀⣄⡇⢇⡀⢇⠸⠸⡜⢸⠸⡀⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠶⠶⠶⠶⠶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢶⣤⣀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⢀⣤⣶⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠈⠙⠿⣶⡶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢶⣶⡾⠛⠉⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⡶⠶⠶⠶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣄⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡆⠀⠀⠙⢷⣄⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠂⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡏⢙⣿⠀⠀⠀⠀⠻⣧⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠾⠃⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣱⡟⠛⠀⠀⠀⠀⠀⠹⣧⠀⠈⠿⢶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⢰⣿⠋⠉⠁⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠋⢀⣀⣀⡀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠉⠉⢻⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠰⠶⣶⣶⡾⠿⠟⢋⣁⣀⣀⢸⡏⢹⡇⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⢸⡏⢹⣿⢸⡇⢸⡇⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⣿⣄⣀⡀⠀⠀⠀⠀⣿⡄⠀⠀⠀⢠⣶⢶⣶⢸⡏⠉⣿⢸⡇⢸⣿⢸⡇⢸⡇⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⣀⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢶⡄⠀⠸⣧⠀⠀⠀⢸⣿⠀⣿⢸⡇⠀⣿⢸⡇⢸⣿⢸⡇⢸⡇⠀⠀⠀⣰⡟⠀⢀⣶⠾⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠄⠀⠹⣷⡀⠀⢸⣿⠀⣿⢸⡇⠀⣿⢸⡇⢸⣿⢸⡇⢸⣧⠀⢀⣴⡿⠀⠀⣼⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠈⢿⣦⡘⠿⠶⠿⠿⠿⠾⠿⠿⠷⠾⠿⠿⠷⠾⠇⣠⡬⠉⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠿⠃⠀⠀⠀⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⠶⠶⠶⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⢀⣠⣶⠿⠷⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠾⠿⢷⣤⣀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠾⠛⠉⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠈⠛⠿⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠶⠶⠿⠃

⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠤⠤⢤⣄⣀⡀
⠀⢸⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⣀⡤⠚⡩⠚⠁⠀⠀⠀⠀⠀⠈⠉⠛⠲⠤⣄⡀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶
⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⣠⠒⠒⠲⢤⣀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⢻⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡟⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⢀⡤⠖⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⠖⠒⡞⠉⠀⣀⡴⠚⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢳⡤⠴⠒⢾⣿⣿⣿⣿⣿
⠀⠈⠉⠉⠉⠉⠉⠀⠀⠙⢲⠞⠁⠀⢀⣠⠾⠳⡄⠀⠀⠀⠀⠀⢀⠀⠀⠈⢷⠤⠞⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⡴⠋⠀⢀⡠⠟⢦⡀⣀⠀⠀⠈⣷⣄⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⡴⠋⠀⢀⡴⠃⠘⣦⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⡴⠋⠳⠤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣶⣶⣶⣶⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣤⣤⣴⣿⣿⣿⣿⣦⣤⣤⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣼⣿⣿⡿⠿⠛⠛⠋⠉⠉⠙⠛⠛⠿⢿⣿⣿⣧⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣾⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⣿⡿⠃⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⠀⠀⠘⢿⣿⡄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠑⠿⠀⠀⠀⠈⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠐⠻⢿⣿⣿⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⣶⣄⠀⢀⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⡾⠋⠀⠙⢷⠟⠁⠀⣀⣀⣠⡀⠀⠀⠀⠀⠀⢿⣿⡆
⠀⠀⠀⠀⠀⠀⢠⣤⣼⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⣠⡾⢋⣀⣀⣀⠀⠀⠀⠀⠀⣿⠉⣿⡇⠀⠀⠀⠀⠀⢸⣿⣧⣤⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⣿⢠⡾⠋⠀⢸⡏⠉⣿⢀⣀⣀⣀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠋⠉⣿⣿⡄⠀⠀⠀⠀⣿⢰⣶⣶⣦⢸⡇⠀⣿⢸⣿⠉⣿⠀⣿⠀⣿⡇⠀⠀⠀⠀⢠⣿⣿⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣷⡀⠀⠀⠀⣿⢸⡇⠀⣿⢸⡇⠀⣿⢸⣿⠀⣿⠀⣿⠀⣿⡇⠀⠀⠀⢀⣾⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣷⡄⠀⠀⣿⣸⣇⣀⣿⣸⣇⣀⣿⣸⣿⣀⣿⣀⣿⣀⣿⣇⠀⠀⢠⣾⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠿⠛⢿⣿⣦⡀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠋⢀⣴⣿⡿⠛⠿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⡿⢿⣿⣷⣶⣤⣤⣄⣀⣀⣠⣤⣤⣶⣾⣿⡿⢿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⢹⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⡏⠀⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠁⠀⠀⠈⠿⠿⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣔⣊⡭⠭⣍⣀⠀⢀⣈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠇⢸⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠘⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠼⠀⠀⠀⠘⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⣀⡠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

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

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

⣾⡟⢻⣦⢰⣿⠛⣿⡆⢠⡾⠛⠻⣦⠀⣿⠛⠳⣦⡀⣿⡇⠀⣿⠀⣴⡞⠛⠳⠘⢻⣿⠛⠃⢻⣿⠃⢻⣆⠀⣾⠇⠹⣿⠃⠘⢻⣿⠛⠉⢿⣄⣼
⣿⡟⠛⠋⢸⣿⠻⣿⡁⢸⣧⡀⣠⡿⠀⣿⣀⣀⣿⠃⢿⣇⢀⣿⠀⢿⣆⠀⣀⠀⢸⣿⠀⠀⣸⣿⠀⠈⢿⣼⡏⠀⢀⣿⡀⠀⢸⣿⠀⠀⠈⣿⡇
⠉⠁⠀⠀⠈⠉⠀⠈⠉⠀⠉⠙⠉⠁⠀⠉⠉⠉⠁⠀⠈⠉⠋⠁⠀⠀⠉⠋⠉⠀⠈⠉⠀⠀⠉⠉⠁⠀⠈⠉⠀⠀⠉⠉⠉⠀⠈⠉⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣄⠀⠀⠀⠀⠀⣴⠟⠉⠉⠉⠙⢷⡀⠀⠀⠀⢠⡞⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠏⠁⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢻⡀⠀⢰⣿⡄⠀⣼⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢧⣄⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣀⠀⠀⠇⠀⠀⢹⣆⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⣾⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⠉⠀⢀⣶⣄⠀⠀⢙⣦⣄⣀⣠⣴⡟⠁⠀⠀⣰⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⢧⣙⣷⠟⠋⠙⠷⠿⠶⠛⠉⠛⢳⣾⠋⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡞⠋⠻⢦⡽⢿⣄⣤⠶⢛⣋⣙⣛⠳⢶⡟⠁⠀⠀⣼⠻⣆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁⣾⠀⠀⠀⠀⠀⣠⠟⣷⣞⣉⣭⣭⣍⠳⣦⠹⣆⢀⡾⠁⠀⢻⡈⠙⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠁⠀⠀⢠⡏⠀⠀⣸⡇⠀⡟⢸⠇⣼⢻⡴⢦⡌⢷⡘⣧⢹⡏⠀⠀⠀⢸⡇⠀⠀⠀⠉⠳⢦⣄
⠀⠀⠀⠀⢠⡴⠞⠋⠀⠀⠀⠀⠀⠀⢸⠇⠈⠋⠉⠙⣻⡷⠾⠶⣿⡘⠷⠾⠃⣼⢁⡟⢸⠇⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠒⠖⢋⣡⠞⣡⡟⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠻⣤⣤⣤⣤⡤⠿⢦⣤⣤⣴⣏⣛⣛⣉⣥⠾⠋⠀⠀⣷⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⣀⣀⣀⡀⣀⡀⠹⣦⠀⠀⠀⢀⠀⠀⠀⠀⣈⣀⣀⣀⣀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⣠⡞⠁⠀⠀⢀
⠀⠀⣿⠉⠉⠉⢹⡇⠀⢰⡞⠛⠶⠟⢳⣦⠀⢸⡏⢉⣉⣉⣽⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣠⠟⣦
⣤⠤⠿⠤⢤⡄⢸⡇⣴⠾⠁⢀⣤⡀⠈⠻⣶⢸⡇⢸⡏⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡈⠁⠀⠀⠀⣰⠏⠀⠘⢧⡀
⣿⠀⠀⠀⢸⡇⢸⡇⠻⢦⡀⠈⠻⠉⠀⣴⠿⢾⡇⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡼⠋⠻⡄⠀⢀⣼⣃⠀⠀⠀⢈⣷⡀
⣧⠀⠀⠀⢸⡟⠛⠃⠀⠸⣧⣤⡤⣤⣼⡗⠀⠈⠛⢻⣇⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⢠⡟⢷⡀⠀⠀⢀⡾⠁⠀⠀⠙⣆⠈⠉⣽⠀⠀⠀⣿⠉⠁
⣿⣤⣤⣤⣼⡇⠀⠀⠀⠀⠀⠉⠀⠈⢸⣧⣤⣤⣤⣤⣭⡟⠉⠉⠉⠀⠀⠀⠀⠀⣰⠏⠀⠈⢳⡄⠠⠿⢶⠀⠀⠀⢰⠾⠧⠀⢺⠀⠀⠀⣿
⠀⠀⠀⠀⢀⣴⠖⠒⠒⠂⠀⢠⡶⠶⠶⠶⠶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣏⡀⠀⠀⢀⣹⣆⠀⢸⠀⠀⠀⢸⠀⠀⠀⢹⠀⠀⠀⣿
⠀⠀⠀⠀⢿⡁⠀⠀⠀⣴⠛⠋⠛⠳⣄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇⠀⠀⢸⡏⠉⠀⢸⠀⠀⠀⢸⠀⠀⠀⣻⠀⠀⠀⣿
⠀⠀⠀⢀⡈⣷⡀⣀⢸⡇⠀⠀⠀⠀⣿⡛⠛⠛⣻⡆⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⣠⡤⠶⠶⠦⣤⡀
⠀⠀⢸⣏⢉⣉⣉⣉⣹⡇⠀⠀⠀⠀⣿⣉⣉⣉⣉⣉⡉⢹⡇⢲⣄⠀⠀⠀⢀⡴⢸⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠘⢠⡞⠉⠐⠀⠃⠀⠌⠙⣦
⠀⠀⠀⣿⢸⡏⠀⠀⣸⣧⣤⣤⣤⣤⣿⡅⠀⠀⠀⢸⡇⣾⠀⠀⠙⢧⡀⣰⠟⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⣾⢁⡄⠀⠀⣿⡄⠀⣅⢹
⠀⠀⠀⣿⢸⡇⠀⠀⠙⠶⠶⢶⡶⠶⠾⠃⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠛⠁⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⢻⡄⠤⠀⠀⠀⠀⠠⠀⣼
⠀⠀⠀⣿⠸⡇⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⢸⡇⣻⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⡀⠀⠂⠘⣡⡾⠃
⣦⣶⣦⣿⣿⣷⣴⣶⣶⣴⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣦⣴⣦⣧⣤⣦⣯⣦⣴⣼⣶⣤⣬⣦⣴⣼⣧⣦⣤⣤⣿⣿⣿⣿⣥⣶⣶
⣿⣿⣏⣀⣉⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣦⠀⠀⠀⠀⠈⢳⡞⠁⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠿⣄⡀⠀⠀⠀⢸⡇⠀⣴⠶⣄⠀⢰⡟⢯⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⢸⡇⠀⠳⡴⠟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢀⣀⣸⣧⣤⣠⣇⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠋⠁⢀⣀⣀⡀⠈⠉⠓⢾⣇⠀⠀⠀⠀⠀⠀⣾⠉⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⣠⠞⠁⠀⠀⣰⢞⣭⠷⠮⣭⡳⣄⠀⠀⠈⢳⣄⠀⠀⠀⠀⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠸⣅⣼⠛⠛⠛⣿⠋⠀⠀⠀⣸⢣⡏⠀⠀⠀⠀⢹⡜⡇⠀⠀⠀⠙⣦⠤⠤⠤⠤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣼⠧⣄⣀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⣿⣸⣀⣀⣀⣀⣀⣸⣇⣗⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣷⠀⠀⠀⠀⡿⠲⠲⠒⠒⠲⠲⣯⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢶⣠⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣶⣦⠀⠀⠀⠀⣿⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⣸⠏⠉⠉⠉⠙⠳⠦⣄⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠶⠶⠿⣄⠀⠀⣿⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣿⠀⠀⣰⣏⣀⣀⣴⠞⢦⠀⠀⠀⢷⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⠀⠀⠀⠀⠀⠘⢦⣀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⣀⡼⠋⠀⠀⠀⠙⠲⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⢹⡆⠀⠀⠀⠀⠀⠀⢹⡷⢦⣄⣀⡀⠀⠀⢀⣀⣠⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢹⠉⠙⠙⡏⠉⠁⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣼⡀⠀⠀⡇⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢾⣀⣽⠀⠀⡇⠀⠀⠀⠙⢦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⣄⢸⡇⠀⠀⠉⠀⢠⡞⠙⡆⠀⠀⠀⠀⣿⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠴⠟⠉⠁⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠈⠉⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⡆⠀⠀⢰⠀⠀⢠⠀⠀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⢸⡇⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⠀⣹⠀⠀⢸⠀⠀⠀⠸⡇⠀⠀⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡇⢸⠃⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⡀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄⠀⢸⠀⡇⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡄⢸⠀⡇⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⡃⠀⡀⠀⢳⠀⠀⢸⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢸⢰⡇⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⡃⡀⢀⠀⢺⠀⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡇⣼⢸⡇⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⡇⣿⢸⡇⢸⠀⠀⢀⡇⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠘⠀⠀⠘⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⡇⢻⢸⠁⣸⠀⠀⢸⡇⠀⠀⢸⠀⠀⢸⠀⠀⠀⣿⠀⠀⢠⡄⠀⠀⠀⢰⠀⠀⠀⡇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠈⢸⢰⡇⢸⢸⠀⣿⠀⠀⢸⡇⠀⠀⢸⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣄⠘⢸⡇⡏⢸⠀⡟⠀⠀⢸⡇⠀⠀⡏⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠟⠀⢸⡇⡇⣼⠀⡗⠀⠀⢸⡇⠀⠀⡇⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⣾⠀⡇⠀⠀⢸⠃⠀⠀⡇⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠁⡇⣿⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⣾⠀⠀⠀⡟⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠇⠃⢨⠀⡇⡟⢸⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⢸⠇⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡇⢸⡇⠂⠀⣿⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⢰⠇⡇⢸⠃⠀⠀⣿⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢸⢀⡄⢸⠀⠀⠀⣿⠀⠀⠀⠃⠀⠀⣿⠀⠀⢘⡇⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠈

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

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⢔⡼⠓⠒⢉⣩⢝⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⡰⠶⠈
⠀⠀⠀⠀⢀⡎⡄⠁⠐⢖⠒⠛⠁⠀⠁
⠀⠀⠀⠀⠈⠐⢄⠀⠀⠑⡄⠀⠀⠠⠇⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠛⣰⣄⠤⣱⡀⠀⠐⠁⢈⡄⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⡼⡧⠀⠠⣤⠞⠁⠀⠁⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⣀⢀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡖⠓⢦⣀⠀⠀⠀⠀⠀⡸⠀⢠⠁⠀⠀⠀⠀⠈⠌⠗⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢡⠒⠋⡠⠡⢳⡀⠀⠉⠀⠀⡈⠀⠀⠀⠀⡄⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠄⠳⠤⠁⠀⠀⠹⣄⠶⢄⢀⡁⢀⣀⡀⠠⡇⠀⠐⢯⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠯⡬⠀⠀⠉⠉⠩⠇⠏⠷⠌⠰⠯⡉⢂⡐⠚⠃⠖⠀⠀⠣⠼⣗⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⢠⠆⠠⡤⡤⠤⡤⢆⡸⠠⠆⠆⠠⡀⠀⡀⣠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⢰⠅⣇⠀⠙⠋⠥⣘⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢀⡴⡃⡐⠋⠀⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠙⡴⡈⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣶⠏⠣⠒⠒⠚⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⠟⠒⠦⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⢀⣀⣀⠀⣠⣤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠛⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⣀⡀⢀⣀⠀⡀⡀⢀⣀⣀⣀⢀⡀⡀⢀⢀⣀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠚⠸⠺⡝⢇⡽⢸⡼⠛⣇⠟⢯⡠⠸⠇⢸⡅⢳⠏⢸⠀⠿⠀⢻⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠀⢀⡔⠉⠙⢦⠀⣰⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠖⠸⡀⠀⠀⢸⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⣶⣤⠽⢦⡶⠧⣤⠃⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢰⠉⠛⢳⣾⣿⣿⢝⢅⡰⠙⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⡏⠐⢣⡷⢯⡻⢎⣿⣿⠀⠀⡇⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠧⢠⡼⠤⡤⣟⣟⠿⠋⠄⠀⢱⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⢲⢩⠖⠖⣤⢰⢒⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⢠⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⢹⣸⢫⡀⠃⣨⢿⣸⠀⢸⠀⠀⠀⢀⠀⠀⢠⠛⡄⢰⡃⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠼⣀⡀⠉⣉⣧⣤⠬⠟⠉⠀⠀⣠⠋⠳⡐⢧⠀⡼⠂⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠀⢰⠊⠳⣄⣨⡆⠀⠀⠀⠀⢹⠀⢸⠁⢸⠀⡇⠀⣇⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢩⠯⣯⣀⣀⡿⠯⢭⡽⠲⡀⣠⢺⠀⢸⠀⢸⠀⢡⠊⠀⡄⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣺⠀⠙⢲⠒⠁⠀⢸⡇⠀⠈⢁⣘⣀⣘⣀⣘⣀⣘⣄⠀⠁⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠒⠈
⠀⠀⠀⠀⠀⢰⠊⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀
⠀⠀⠀⠀⠀⢘⠀⠀⠠⡤⠤⠤⠤⠤⠤⢤⣄⠀⠀⠀⡠⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⢲⡄
⠀⠀⠀⠀⠀⠈⡇⠀⠀⡤⠤⠤⠤⠤⠤⣀⠀⠀⡠⠊⣠⠶⠖⠶⠶⠶⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢃⠀⠀⣤⣤⣤⣤⣤⣤⣤⡴⠎⣠⠞⡡⠒⠶⠶⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⣀⣀⣀⣀⣀⣀⣤⠞⡡⠊⠀⠀⠀⠀⢀⠐⠂⠉⠉⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⣤⣀⣤⣤⣤⣤⡤⠚⢠⠞⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢠⣤⠤⢤⠤⣤⠤⠞⠁⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠰⠶⠶⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⢻⣚⠛⠛⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠈⣏⣉⣉⣉⣓⡀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠘⣄⣀⣀⣀⠑⢆⡀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠹⣤⣤⣌⠳⣄⠑⢄⠀⠀⠀⣎⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠘⣄⣈⠳⢌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠈⢯⠳⣌⠳⢌⠉⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠑⢌⠳⣌⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣠⠊⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠈⠣⡈⠓⠚⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠈⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⢠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠑⠦⣀⢀⠴⠋⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠈⠁⠀⢀⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⡠⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⢦⣤⠴⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⢿⡏⠀⠀⠀⠙⡶⠚⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢧⡀⠀⠀⢺⡟⠃⠀⠀⢹⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⠀⠙⠒⠚⠹⣄⠀⠀⣠⢟⣀⡞⠐⠀⠀⠀⠀⣠⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢯⠙⣄⣣⣀⣀⣀⣀⡬⠟⠋⠁⠀⠉⠀⠀⠀⠀⠀⢠⡇⠀⢻
⠀⠀⠀⠀⠀⢀⠄⠒⠒⠚⠳⠤⢿⠯⠄⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠀⠈⣇⠀⠀⢀⡤⠴⠶⠦⢤⣀
⠀⠀⠅⣀⢰⠃⠀⠀⠀⠀⠀⠀⡧⢤⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⡇⠀⠀⠸⡄⠀⢯⠀⢠⡄⠀⠀⢈⡇⠀⢀⡤⠤⠤⢤⡀
⠀⠀⠀⠈⠈⢧⠀⠀⠀⠀⠀⣸⠃⣠⣧⠜⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢻⡀⢹⡀⠀⠀⢳⡤⢾⡗⠺⢯⣀⡤⢞⡁⠀⠉⢲⡆⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⠚⠁⠘⢧⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢸⡇⠀⠱⣤⠀⡸⡇⠈⠋⠀⠀⠹⡄⢀⠀⠀⢠⠏⠐⠋⠒⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⢣⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⣞⡗⠀⠀⠀⠀⠁⢻⡀⠀⠀⠀⠀⣿⠀⠀⠀⠘⢦⡀⠀⠀⠀⣀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⢣⠀⠀⠀⠀⢸⡇⢀⡴⠯⣬⠇⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣠⡏⠀⠀⢀⡤⠞⠁⢀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢣⠀⠀⠀⠘⡇⠈⠙⠛⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢺⠉⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⢳⣠⠀⠀⡇⠀⠀⢀⣠⡤⠄⠤⢤⠆⢀⢹⡖⠲⢤⡴⠖⠶⢤⣥⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣰⣿⣍⡿⠀⣠⣞⠉⠸⣆⠀⠀⢾⢠⠏⣸⠁⡤⠤⢶⡄⠀⠀⠸⠀⢀⡀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⡵⠃⣼⣡⡴⠧⠀⣹⡆⣸⠇⢀⣈⠓⠒⠁⢰⠇⠀⠀⢳⡀⠀⠀⠀⢸⡏⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣠⠎⠀⣈⡠⠜⠳⣄⡰⣅⠀⢧⠘⢧⡀⠉⠓⠒⠁⠀⠀⠀⠈⢷⡀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⢁⡼⢗⠊⠉⠀⠀⣠⠶⢬⡀⠻⣄⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢦⡟⠢⠀⠁⠀⡴⡏⠁⠀⠀⠙⣆⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⠀⠀⣰⠇⠈⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠘⣆⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡔⣽⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠹⡀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠉⠀⠒⠒⠒⠒⠒⢲⠚⠀⠀⠀⠀⠀⠀⠙⢲⡶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠰⡒⠋⠁⠈⠉⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⣤⣄⣀⣀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢠⠀⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣿⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣼⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠸⡄⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⢃⣵⣌⣑⡶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠅⠨⠿⠛⠉⠉⠁⠀⠠⠼⠆

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

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

⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⣶⣿⣿⣿⣿⣿⠟⢛⡛⠻⣿⣿⣿⣿⣿⣶⠀⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢁⣾⣿⣿⣷⡈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣿⣿⣿⣿⣿⡈⢿⣿⣿⡿⢁⣿⣿⣿⣿⣿⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠺⣿⡿⠐⠛⠂⢿⣿⣿⣿⣿⣿⠆⣬⣥⠰⣿⣿⣿⣿⣿⡿⠐⠛⠂⢿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⢠⣿⣿⡀⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠶⠶⠦⠘⢿⣿⣿⡏⠘⠛⠛⠃⢹⣿⣿⡿⠃⠴⠶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⣠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⣶⠀⠀⠸⣧⣼⠇⠀⠀⣶⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢷⠀⠀⠛⠾⠻⣄⠀⠀⢸⡇⠀⠀⣠⠟⠷⠛⠀⠀⣠⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠟⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠹⣦⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢙⣷⠤⢤⣄⠀⠙⢦⡀⠀⠀⠀⣸⣧⣤⣼⣧⣤⣼⣇⠀⠀⠀⠀⣠⡾⠋⠀⣠⣤⣤⡞⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠙⢷⣄⠀⠙⣦⡶⠛⠉⠁⠀⠀⠀⠀⠉⠉⠛⢶⣤⡾⠋⠀⣠⠞⠁⠀⠈⠛⠞⠋
⠀⠀⠀⠀⠀⠀⣠⡤⣤⠀⠀⠀⠀⠀⠀⠀⠙⣷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠞⠁⠀⠀⠀⠀⠀⠀⣤⢤⣄
⠀⠀⠀⠀⠀⠀⢿⣤⣾⡃⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⣠⡶⠶⠶⢶⣄⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⢘⣷⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠶⠶⠶⣾⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣷⠶⠶⠶⠶⠶⠋
⠀⠀⠀⠀⠀⣴⠶⣦⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⣰⠞⠛⠛⠛⠛⠛⠛⠳⣆⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣴⠶⣦
⠀⠀⠀⠀⠀⠻⠶⠟⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢾⡷⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠻⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠶⠶⠶⢿⡇⠀⠀⠀⠀⣿⠀⠀⠀⠸⠇⠀⠀⠀⣿⠀⠀⠀⠀⢸⡿⠶⠶⠶⠶⠶⣄
⠀⠀⠀⠀⠀⠀⣾⠛⢿⡅⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠙⠷⠶⠶⠶⠶⠶⠶⠾⠋⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⢨⡿⠛⣧
⠀⠀⠀⠀⠀⠀⠙⠓⠛⠀⠀⠀⠀⠀⠀⢀⡴⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⣄⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⡀⠀⢀⡴⠋⠀⣠⡾⠛⠷⣤⣀⣀⠀⠀⠀⠀⣀⣀⣤⠾⠻⣄⠀⠙⢷⣄⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡼⠛⠛⠋⠀⣠⡾⠋⠀⠀⠀⠀⢹⡟⠛⢻⡟⠛⢻⡏⠀⠀⠀⠈⠳⣄⠀⠙⠓⠒⢿⣅⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠻⣆⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⣴⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠋⠀⠀⣤⢶⣴⠋⠀⠀⢸⡇⠀⠀⠙⣦⡶⣤⠀⠀⢷⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⡿⠀⠀⢰⡟⢻⡆⠀⠀⠿⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⢀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⢀⡴⠋⠀⠀⠈⠱⣆⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠎⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠦⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⠀⠀⠀⠀⠀⢀⠤⠊⠿⠤⠖⢀⠴⠚⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⢸⡄⠀⠀⠄⠊⠁⠀⠀⢀⡴⠚⠁⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⣷⠈⠀⠀⠀⢀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡇⠀⠀⣻⡄⢀
⠀⠀⠀⠀⠀⠀⠀⢰⣮⣅⣀⠙⠛⣋⠵⠊⠀⠀⠀⠀⠀⣴⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠉⠑⢿⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⢻⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣽⡦⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣄⡻⣦⣀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠷⠶⠒⠛⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣤⡶⠶⠆⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠋⠉⠻⠛⠀⠀⠀⠀⠀⠙⢻⡦⠀⠀⠀⢀⣤⣿⠙⠋⠙⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠟⢀⣠⡶⢾⣛⣛⣛⠶⣦⡀⠘⠷⠶⣦⠀⣸⡷⠀⣴⠟⠻⣦⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡀⠀⣠⡞⠁⣰⠟⠉⠉⠙⣷⡈⠹⣦⡀⠀⣸⡇⢻⡦⠀⢿⣤⣠⡿⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⠇⢠⡟⠀⠐⣿⡀⠀⠀⠀⣸⡇⠀⠸⣧⠈⠿⢦⡌⠻⣶⣀⡈⢁⣀⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠘⢷⡄⢀⣴⠏⠀⠀⠀⣿⠀⠀⢸⡇⠀⠈⠋⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⡇⠈⣷⡀⣠⡶⠟⣃⣀⡛⠷⣦⡀⣰⡏⠠⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠈⠷⠟⠀⠀⢹⡏⠀⠀⠸⠷⠋⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⣷⠀⠀⠀⠀⠸⡏⠀⠀⠀⠀⢰⡟⠛⢋⣤⣤⡶⢶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⡶⣶⠀⠀⢰⡶⣦⣤⠾⢃⣴⠿⢁⣬⣴⣦⣭⡀⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁⠀⠀⠀⣸⡧⢰⡟⠁⠀⢤⣌⢻⣆⢸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⢻⡄⠀⠀⣠⡿⢀⡿⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣈⠻⣦⣀⣉⣤⠾⣁⣸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣤⣤⣍⣩⣤⣤⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠋⠁⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠁⠠⠀⠀⠀⡄⠠⠀⠆⠀⠀⠄⠠⠀⠀⠲⠀⡄⢀⠀⠀⠀⠒⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠈⠀⠁⠈⠐⠀⠀⠀⠃⠀⠃⠀⠀⠀⠀⠸⠁



⣀⣀⣀⢀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⡒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⣊⣤⣶⣾⣿⣿⣿⣿⣥⠈⠉⠒⠚⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⣫⣴⣿⣿⠿⣻⡿⠛⠋⠉⠉⠉⠛⠢⠀⠀⠀⠈⠐⣶⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢥⣾⣿⡶⠛⠁⠞⠁⠄⣒⠒⠒⠂⣀⣀⢀⠀⠀⠀⠀⠀⠈⠃⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢠⣾⣿⣟⠁⠀⠀⣠⡴⠟⣛⣻⣶⣜⢿⣿⢎⠳⡷⠀⠰⠶⣷⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⡀⢸⣿⣿⠃⠀⣠⣿⠋⣰⠟⠉⠉⠉⠉⠐⠀⠀⠀⠀⠀⠀⠀⠈⣿⣣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⣇⠈⠐⣶⣆⣰⠃⢡⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⢷⣿⣿⣿⠋⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⠀⠈⠻⠻⠷⣾⡷⠀⠀⠀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿⣿⣷⣄⣷⣦⣾⣟⠁⠀⠀⠈⣀⣀⣀⣉⠙⠛⠃⠀⠀⠀⠴⣛⣉⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣮⣿⣿⡇⣙⢿⣷⣪⠂⠀⠀⠙⠻⣟⣿⠏⠳⠄⠀⠀⠀⡴⠻⡟⠉⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⡿⡟⠱⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣅⢻⣄⠁⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠙⣿⣿⣷⣦⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⡠⠞⠙⣿⣿⣿⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠿⠶⠾⡦⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⢉⡁⠀⠀⠀⠀⠀⣿⢸⣿⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠩⠷⠶⠖⠀⡰⠁
⠀⠀⠀⠀⠀⠀⣸⣿⡄⢀⡴⠞⠀⢰⣯⣾⣿⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠘⣿⣏⡀⣨⣿⠁⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢄⣻⣿⠃⢀⣨⣽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡿⢷⢾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⠟⡀⠤⠤⠄⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢠⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠛⠋⡀⠀⠀⠁⠘⠢⠤⣀⠀⠈⠂⠀⠈⠻⠮⡳⣌⠐⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠘⠻⠧⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠑⠀⠀⠈⠪⡓⡙⠛⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣋⡀⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣋⡴⠚⠁⠈⠲⢄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⡥⠚⠁⠀⠀⠀⠀⠀⠀⠑⢦⡈⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣫⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣩⣞⡁⣀⢀⣀⠀⠆⠠⠔⠂⢠⠀⠁⠀⠀⠀⠀⠀⠉⠓⢤⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⡡⢺⠋⠡⠀⠀⢀⠜⠙⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⠊⠀⣧⣤⣄⣲⡶⠾⠶⠾⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣌⠳⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⣠⡎⠀⠀⠀⢐⣀⣀⣀⣈⣀⣤⣄⣀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠑⢦⡀
⠀⠀⠀⠀⠀⠀⡴⢋⠄⠊⢸⡇⠀⠀⢰⢸⣿⣿⣿⣿⣿⣿⣿⣿⢸⡆⠀⠀⠀⠀⠀⣆⠀⠀⣀⣀⣲⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢦⡈⡶
⠀⠀⠀⠀⠀⠘⠒⠁⠀⠀⢸⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠸⠃⠀⠀⠀⠀⠀⢈⣤⠤⠤⠤⠤⣄⠀⠀⠀⠀⠀⠀⡀⢹⡇⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡌⣿⣿⣿⣿⣿⣿⣿⣿⡆⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢂⡄⠀⠇⣿⡿⠿⠿⠿⠿⠛⠛⠇⡄⢀⠀⠥⢠⠄⣾⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠀⠀⠀⠀⢤⡀⠀⠠⠤⠤⠤⠄⠂⠃⠀⠀⠈⠀⠁⢻⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠛⢤⡿⠒⠢⢄⡐⠀⠀⠒⠀⠀⠀⠐⠂⠒⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⣿⠠⠀⠀⠀⠀⠀⠀⣗⢢⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀⠉⢻⠟⢶⣬⡓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠎⠂⠀⠀⠀⠀⣿⠀⠁⠀⡠⠔⣊⣭⡿⠋⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⡓⠧⣄⡀⢨⠻⢦⣽⣤⣄⣀⡠⠤⣀⡀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢿⡀⡠⢊⡴⠟⠉⠻⣶⡾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣦⣍⡛⢶⣤⣈⠉⠛⠻⢶⠶⢶⣮⣝⡲⢄⠀⠀⢸⡧⠖⠚⣛⡒⠦⣸⣧⣷⠟⢳⣄⡠⢔⣩⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⡈⠉⠙⠛⠁⠀⠉⣿⠶⢎⣀⡀⠀⣽⠻⠬⣧⣒⣞⣠⣾⠛⠛⠛⡷⠞⠋⣡⡴⠿⠷⠶⠛⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠘⠳⢶⣤⣉⡉⠛⢿⡶⢶⣤⣬⣭⣽⠀⠀⠀⠳⢾⡿⢫⠀⢀⣀⡤⠞⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠁⠀⡏⢹⣿⠃⠀⠀⠀⠀⢸⣿⠀⣏⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠉⠉⠛⠓⠒⠒⠒⠒⠛⠛⠋⠉⠉⡇⢸⣿⡀⢀⡀⠤⠒⢹⣿⡀⢸⢣⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⠸⣿⣏⠀⠀⠀⢀⣤⣿⣷⣫⠞⣦⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠙⠿⠿⠿⠾⠛⣛⡿⠛⠋⠀⠀⢹⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠒⠶⠶⠦⠶⠶⠼⢷⣤⣤⠤⠶⠒⠚⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡜⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⣧⣄⣀⣀⣠⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠈⠈⠀⠀⠀⠈⠀⠉⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠒⠚⠾⡦⣀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠀⠹⡞⠳⣄⠀⢀⠀⠀⢄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣰⠁⠀⠈⠳⣄⠡⡀⠀⠐⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠡⠀⠀⠀⠀⡀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠙⢲⣤⣤⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠔⠀⠀⠀⠀⡐⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠢⢾⠅⠀⢀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠹⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡀⠀⠀⠤⠤⠤⢬⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠄⣠⠞⠁
⠀⡜⠉⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠙⢻⣍⠁⣠⠾⠛⢮⡑⢦⡀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣡⠞⠁
⠀⠻⡴⣠⠲⣄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡴⠋⠈⠛⠻⣶⣄⠀⠉⠢⡉⠓⠦⠤⠔⠋⠀⠂⠁⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠈⠳⣌⠳⡄⠀⠀⠀⢀⡔⠋⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⠳⣧⣄⠀⠈⠉⠒⠒⠂⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠈⠳⣬⠖⠉⠉⠳⡄⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣑⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠈⢦⡀⢀⡀⠳⣌⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⣀⣀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡂⢀⡼⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠳⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠳⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣖⣤⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠋⠉⢀⣠⣾⣶⣄⡀⠉⠙⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⣶⣯⣾⣿⣿⣿⣿⣷⣦⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⢰⣟⣿⠿⣿⡝⠛⠻⡿⣿⡄⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⣿⣭⡉⣻⡿⠋⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣁⡀⠀⢀⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⠀⠀⠇⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡤⠶⢺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠛⢿⣿⠀⢀⠞⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⣠⣴⣖⢿⣿⣿⣿⡆⢀⡿⣷⣏⢠⠟⣛⡲⣦⣤⣀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⢠⡞⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⢹⣿⠟⣄⠈⢛⣽⣿⡿⣷⡄⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡟⣰⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⣩⢿⣤⣼⣭⣿⣿⣿⣷⠸⣿⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠙⠳⠞⢚⣿⠛⠛⠛⢻⣿⠀⣹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠿⣿⠁⠀⠀⠀⢀⣒⣂⣿⣿⣶⣶⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢾⣿⣶⣿⣦⣤⣴⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⡠⠚⠉⠙⠢⡀⠀⠀⠀⠀⠀⢿⡏⢝⡿⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠇⠀⠀⠀⠀⠀⣴⡻⢽⣿⣆
⠀⠀⣗⣇⢾⢆⣗⡇⠀⠀⠀⠀⠀⠸⢷⡶⠇⠀⠀⠀⠀⠀⠀⣀⣤⣀⣤⣆⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠙⢿⣷⣿⠟⠁⠀⠀⠀⠀⠀⠀⣾⣷⡀⠀⠀⠀⠀⠀⠀⢩⣁⣁⣡⠅⠀⠀⠀⠀⠀⠀⣿⣿⡟⠛


⠀⠀⠀⠀⠀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣛⡛⠀⠀⠀⠀⠀⠀⣴⣟⠛⣿⣆
⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⠀⢠⣤⠤⣤⡤⡄⠀⠀⠀⠀⠀⠀⠈⠟⡁⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿
⠀⠀⢰⣾⣿⣿⣿⠀⠀⠀⠀⠀⠘⣻⣶⡿⠋⠁⠀⠀⠀⠀⠀⠀⣷⣶⣦⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠛


⠀⠀⣠⣶⣶⣶⡄⠀⠀⠀⠀⠀⢀⠖⣫⣍⠓⣄⠀⠀⠀⠀⠀⡴⠲⠿⠖⡆⠀⠀⠀⠀⠀⣴⠋⢹⣿⣆
⠀⠀⣿⣿⣽⣿⣿⣀⠀⠀⠀⠀⣾⠀⠹⠟⠀⢸⠀⠀⠀⠀⠀⡏⡍⠈⠀⡇⠀⠀⠀⠀⠀⢻⣦⣤⣾⣿
⠀⠀⠈⠻⠿⠇⠉⠉⠀⠀⠀⠀⠈⠿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢷⣂⣀⣀⡇⠀⠀⠀⠀⠀⠀⣿⣿⡟⠛


⠀⠀⡰⡪⣈⢵⢢⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⡇⠀⠀⠀⠀⠀⣷⣷⠶⣤⡆⠀⠀⠀⠀⠀⣴⣿⢻⣿⣆
⠀⠀⡸⡨⣲⣅⡁⠀⠀⠀⠀⠀⣀⣶⣀⡍⠃⠀⠀⠀⠀⠀⠀⡟⠙⠚⠿⠃⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠑⠩⠤⠻⡇⠀⠀⠀⠀⠀⠸⠟⢿⠗⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠸⣿⡇
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠉⠁
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣀⣀⣀⣀⣤⣴⣶⣶⣿⡿⠋⢻⡇⠀⢀⣤
⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣿⡇⠀⢾⣿
⠉⠛⠿⠿⣿⣿⣿⣿⣿⣷⣽⣟⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⢸⣟⣀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠓⠒⠒⠒⠒⠒⠒⠻⠿⠿⢿⣿⣏⡁⠀⠈⢻⣷⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣦⣾⣃⣔⣀⣼⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠉⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⣀⣀⣀⣤⣤⡤⠤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠻⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡠⠄⠂⠀⣀⣤⣤⣀⡀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡠⠞⠁⠀⢀⠀⠉⠙⠻⢿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠥⣀⡀⠠⠔⡪⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢇⠀⠀⠁⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣣⠰⡀⠀⠀⠙⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢢⣄⣀⠒⠠⠑⢦⣤⣀⣀⣤⣤⣾⣷⣄⣀⣄⣴⠃⠈⠁⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⠎⠛⠻⠦⣄⣀⣼⣷⣝⢿⣿⡿⠿⠿⠿⠿⡿⠃⠀⠀⢀⠀⠀⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠉⠛⠳⡙⠃⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⠾⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣠⠵⣤⣀⠀⠀⠀⠐⠁⠀⢠⠘⡦⡀⢀⢂⣾⡇⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢰⠁⠀⠀⠉⠙⠓⠲⠦⠤⣤⣄⣠⣧⠘⠁⠉⣿⣧⠀⠀⠀⠀⠘⣱⡿⡽⣹⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⡀⠀⢀⠈⠋⠀⠀⠀⠀⣿⣿⠀⠀⠀⣈⡴⣫⢎⡔⣡⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢠⠘⠉⠉⠙⠒⠶⡤⠂⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣺⣯⢞⡵⢋⡴⠋⠀⠉⠛⠿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⢹⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⡤⠊⡁⢁⡤⠋⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠙⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢶⡯⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢲⣤⣲⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠉⢀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⢿⣿⣿⣶⡂⠀⠀⡀⠀⣿⣿⣿⣷⢀⠎⡘⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠲⡛⠋⣯⠙⢿⣿⣾⣴⡇⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣥⣀⣛⠿⣿⣇⠀⠀⠀⢄⠀⠀⠀⠈⠐⠬⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣇⠁⣠⠀⠀⠠⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⣄⡀⠀⣀⠀⢹⠁⠀⠀⢠⣚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡇⠸⣭⠃⢸⠀⠀⡴⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠀⠀⠀⠀⠀⣇⡠⠷⠴⠾⠤⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡞⠁⠀⠀⣠⠞⠁⢀⣤⠶⠶⣦⡀⠈⠳⣄⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⣶⠁⠀⠀⣾⠁⠀⠀⠈⣷⠀⠀⠈⢏⠉⠙⠶⠃⢠⠖⡄
⠀⠀⠀⠀⠀⠀⠠⣏⡷⠤⠤⠤⠴⡇⠀⠀⣰⣿⣶⣶⣶⣶⣿⣦⠀⠀⢸⠒⠒⠒⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡤⡤⡇⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⠀⠀⢸⠶⠶⠶⠶⠶⣎⡱
⠀⠀⠀⠀⠀⠀⢮⡹⠉⠀⡤⢄⣀⣹⡄⠀⢿⣿⣿⣷⣼⣿⣿⡿⠀⢠⠯⠤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠘⠦⣈⠉⠉⠉⠉⠉⠉⣁⡴⠃⠀⠀⠀⣠⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠓⡖⠒⣶⠉⢹⠀⠀⠀⠀⠀⠙⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⡇⢠⠾⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣻⠏⠀⠀⣴⢳⡈⠙⠁⢸⣀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠓⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⡶⢿⣭⣿⣽⡿⢶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡓⠀⢼⣿⠿⠿⠿⠟⠛⢶⣿⣻⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠶⢀⠇⠃⠀⠀⠛⠀⠀⠀⣀⣀⠀⠻⣯⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢛⢿⡄⠚⠙⢶⠤⠐⡿⠟⢿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠍⠀⠀⠀⠀⠀⢐⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣘⣃⠀⠀⠀⠀⣻⣦⣿⠁
⠀⠀⠀⠀⡄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣀⣤⠞⣿⢇⡽⣿⣦
⠀⠀⠀⡈⠀⠀⠀⢠⡇⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠙⣾⠿⠋⠀⠀⠸⣼⣛⣯⡟
⠀⠀⠀⡧⠀⠀⠐⠀⠉⠉⠉⠉⠉⠉⠉⠛⠻⣝⣟⠁⠈⢹⣷⣾⣿⣄⠀⠀⠀⠀⠙⣿⢯
⠀⠀⠀⣧⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⡆⠀⠈⠻⠿⠉⠛⢦⠀⠀⢀⡴⠃⣰⠷⣄
⠀⠀⠀⠀⠉⠙⠲⠤⣄⣀⣀⠀⠀⠀⢠⣴⣾⡟⡼⠁⠀⠀⠠⣤⠴⢻⡞⣷⣞⣉⡀⢰⡇⠀⠈⠙⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⣷⣶⣿⣿⣦⡆⣸⠇⠀⠀⠀⠀⠘⢻⡿⠁⠀⠳⣸⠁⡠⠂⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠻⠛⠿⢾⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠠⣿⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠇⠀⠘⠟⠒⠒⣤⠶⣦⣶⣶⣿⣷⢔⠤⣄⣄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡄⠀⢚⠃⠀⠀⠀⠀⠀⠐⠟⣿⢛⣻⣿⣿⣯⣗⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠙⠄⢻⡅⠀⠀⠀⠀⠉⠉⣼⢣⣏⣭⣵⣾⣯⣟⠭⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠠⡄⠸⡘⡇⠀⠀⠀⠀⠀⢸⣧⠜⠋⠁⠀⠀⠀⠙⢷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡈⠣⠐⢧⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠉⢂⣀⡨⠤⠖⠋⠁⠀⠀⠀⠀⢀⣠⡴⠖⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⣀⠀⠤⠤⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⣠⠆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣩⡽⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡴⠁⢈⠴⠶⢦⣤⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⡜⠁⠀⠃⠀⠀⠀⣉⠎⠑⠢⣀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⣰⠏⢀⡞⠀⠀⠀⢆⠢⣀⠘⠁⠀⠀⠀⠀⠙⢢⡀⠁⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⡼⠃⢀⡞⠀⠀⠀⠀⣼⣧⡀⠀⡀⠀⠀⠀⢠⠆⠀⠙⣶⠂⠠⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⢸⡇⢀⣾⡀⠀⠀⢠⠞⠋⠉⠉⠢⣄⠉⠀⠐⠁⡀⠀⠀⠈⠳⣤⡤⢤⡀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⢘⣧⡞⢱⠇⠀⣠⠊⠀⠀⠀⠀⠀⢀⡙⠢⡈⠈⡒⠀⠀⠀⠀⠀⠙⠲⢅⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⢸⡏⡿⡬⠔⠚⠁⠀⠀⠀⠀⠀⠀⠀⡴⣱⡌⣦⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠘⢧⡼⠡⠎⠀⠀⠀⠀⠀⠀⠐⣐⣈⣠⡃⠄⡏⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⢸⠗⡒⣒⠦⠤⠤⣄⣀⡀⠀⠀⣀⣼⠇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠈⢷⣕⡀⠡⣄⠀⠀⠀⠎⠙⠢⢄⡠⠐⠀⠸⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣥⣒⡤⠀⠐⠰⢤⠠⡤⠍⠆⠂⠀⠹⡄⠀⠂⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⢤⣳⠄⠤⡀⠑⢄⠂⠘⣦⡈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣌⠈⠊⢣⣰⠏⠻⣦⡀⠁⠅⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠏⠀⠀⠈⠙⠶⣤⣀⠁⠠⠂⡪⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣧⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⠶⣤⡀

⠀⠀⢀⣀⣤⣤⣤⡤⢄⣀⣀⠀⠀⠀⠀⣀⣤⣴⣶⠶⢶⣀⣀⡀⠀⠀⠀⣠⣴⣶⣿⣿⣛⣻⣧⣤⡄⠀⠀⣀⣠⠤⣄⣀⡀
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣯⠟⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣭⠟⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⠟⠁⣠⣾⣿⣷⣿⣿⣿⡅
⠀⢸⣿⣿⣿⣿⣿⡿⠋⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⡿⠋⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣾⣿⣿⠋⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⣿⡿⠁⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣴⣷
⣸⡿⠁⠰⢟⣫⡿⣟⣯⣿⣿⣿⣿⣿⠃⢠⣿⠿⣻⣿⢿⣫⣿⣿⣿⣿⣿⣿⠇⢀⡿⢟⣿⡿⣟⣽⣿⣿⣿⠟⠀⣴⣿⣿⣿
⠋⠁⠀⠘⠹⢷⣿⣿⣿⣿⣿⣿⠿⠟⠀⠘⣶⣯⣽⣾⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠚⠭⣷⣿⣿⣿⠿⠿⠋⠀⠘⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠙⠻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣄⣀⣿⣿⡿⠟
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣴⠶⣲⠖⠚⡗⠲⣖⠶⣦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⣡⠊⠁⣰⠃⠀⠀⡇⠀⠘⣆⠈⠱⣌⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⡼⠁⠀⢠⠇⠀⠀⠀⡇⠀⠀⠸⡄⠀⠈⢦⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣸⠁⠀⠀⣼⠀⠀⠀⠀⡇⠀⠀⠀⣇⠀⠀⠈⡆⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢠⣏⠀⠀⠀⡇⠀⣀⣀⡀⣇⣀⡀⠀⢹⠀⠀⠀⢹⡀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣸⡇⠀⣠⠖⡗⢺⡁⠀⢹⡏⠀⢙⠶⢾⢦⡀⠀⢻⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣇⡇⣼⠁⠰⣧⡴⠃⠐⢶⡷⠃⠋⢀⣸⣀⡷⣄⢸⢹⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡏⢹⡒⠀⣯⢁⣠⠆⢠⡇⠐⣆⠀⣽⠉⢀⣸⣾⠜⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠙⠲⣏⠀⢨⠇⢀⣇⡆⣈⠀⢿⠉⠈⣹⢿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⡏⠙⠯⣄⣨⣿⣀⣻⣄⣼⠿⠚⠁⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⠀⣿⠁⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡤⠤⡷⠴⠖⠒⡗⠒⠒⣺⠞⣿⣓⠒⠒⢺⠒⠒⠶⢾⠤⠤⣤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⢤⣀⣇⣀⣀⠀⣇⠀⠀⠀⢀⡇⠀⠀⠀⣸⠀⢀⣀⣸⣀⣤⠤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠻⠿⠯⠭⠭⠭⠭⠭⠭⠭⠭⠭⠿⠿⠓⠒⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟⠛⠛⠿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡁⠀⠀⠀⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⣿⣿⣤⣤⣠⣤⣤⣼⣿⣿⣿⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⠀⠀⠀⠀⠀⢀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣷⡀⠀⠀⠀⡀⠀⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠋⠀⠿⠟⠛⠓⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠛⠿⠿⠂⠙⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣶⣤⣴⣶⡤⢸⡏⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣼⡇⢠⣤⣴⣶⣤⣴⣿
⠀⠀⠀⠀⠀⠀⢀⣶⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⡀⢸⡧⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢹⡇⢀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠉⠉⠉⠉⠉⠉⠉⠉⠁⢸⣇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⡇⠈⠉⠉⠉⠉⠉⠉⠉⠉⠻⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠃⢸⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⡇⠙⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣤⠠⣾⣷⣶⠆⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⠴⢶⣷⠀⣠⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠀⠀⠀⠀⠀⠈⢿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡿⠁⠀⠀⠀⠀⠀⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠒⠛⠢⣄
⠀⠀⠀⠀⢀⣠⡤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⢀⡞⠉⢀⢰⣶⣠⡀⠀⠀⡰⢅⠀⠀⢀⠀⠀⠀⠀⠈⠳⢤⣀
⠀⠀⣼⠀⠀⠀⠀⠀⢙⣿⣀⣤⣶⣀⠲⡠⠁⣠⣢⠤⠀⠀⠀⠀⠘⠳⢦⣄
⠀⠀⠘⣄⠀⠀⠐⠋⠋⠉⠉⣙⣻⣿⡷⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣙⠶⣄
⠀⠀⢀⡞⠱⠦⠤⠒⠊⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡓⠝⠳⡄
⠀⡰⡯⠐⠲⣆⠀⠀⠀⠾⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⢰⡄⠀⢀⡃⠒⠀⠀⠀⣈⣴⡶⠶⠶⠾⢤⡄⣠⡤⣴⢂⠠⠲⠆⢀⠀⠀⠀⠀⠀⠀⠃⠠⡌⢳⡶⠾⠟⠒⠦⠴⢿⡓⣦⣄⣀
⠀⠙⣺⠛⠳⠶⢒⣋⣉⣉⣀⣀⢀⣤⣴⠿⣿⣏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠘⣷⣆⢸⠀⠀⠀⠬⢷⠁⠍⠏⠉⠓⠲⠦⢤⣤⣤
⠀⢀⣇⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⡔⠀⠘⣉⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣋⢸⠀⠀⠀⢀⣸⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈
⠀⠀⠻⣧⣸⣠⣄⣀⢀⣐⠀⡉⠠⣩⣴⡼⠟⢫⣤⡙⢿⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡧
⠀⠀⠀⠈⢹⡋⠛⠛⠛⠛⠒⠛⠛⠛⢩⠀⠀⣼⡯⠔⠉⠉⠓⠆⣤⣶⣤⣀⡀⠀⠄⢄⡀⢀⠀⠀⠀⡞⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⢷⣦⣀⡀⡀⠀⠀⣄⣀⣷⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠰⠀⠀⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠉⠉⠁⠈⠙⠿⢼⣐⡜⣁⢀⠀⠀⠀⠀⠀⠀⠉⠀⠁⣠⣀⠀⠀⢠⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠚⠿⢧⣤⣥⣶⠾⠶⠛⠛⢶⣙⡀⡏⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⠀⠀⠀⠀⠀⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠃⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢢⣄⣀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣻⣿⡟⡇⡀⡄⠀⠂⠀⠂⠀⡐⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⣱⣹⡡⡐⣐⡀⡀⢀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣵⣿⣿⣿⣟⡙⠐⢡⢃⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠾⠮⢼⣔⣐⣃⣩⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⡛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣵⣾⠿⣖⠛⢷⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣏⣤⡴⠟⠛⠙⠛⢿⣮⣙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⡄⠀⣀⡤⣴⣒⣿⣭⠭⠭⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢳⣿⣯⠿⣚⡭⠖⠛⠉⠘⢻⣿⡆⠀⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣫⡵⠋⠁⠀⠀⠀⠀⠀⠀⢿⡇⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⡾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⣨⡇⣿⣾⠿⠷⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢮⣿⣶⣘⣄⠀⠀⠀⠀⢀⣿⢿⣿⣿⠟⠁⠀⠀⠀⠀⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣝⣿⣿⣷⣶⣦⣤⣼⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠉⠙⢿⡿⠛⠉⠁⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢸⡗⣆⠀⠀⠀⢀⣠⡤⠖⠋⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢶⡦⣿⢤⣤⣤⠔⠋⠁⠀⠀⠀⠀⣠⠶⠆⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣮⣼⣿⣄⡀⠀⠀⣀⣤⠖⠋⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣟⢻⣿⠶⠛⠉⠀⠀⠀⢀⡠⠂⠀⠀⢀⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⢻⣶⣤⣀⢀⣠⠖⠋⠀⢀⣤⣾⣯⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠿⣿⣿⣭⣄⣀⣀⠴⠛⠈⠙⠄⠈⣹⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⡟⠀⠉⠇⠀⠀⠀⠀⢀⡼⣳⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠃⠀⠀⠀⠀⠀⣠⡴⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣶⣦⠴⠤⠤⠤⣖⣛⡵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⠀⠙⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡄⠀⠙⢿⣷⣦⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠙⠻⣾⠟⠻⡿⢿⣷⡈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢷⣄⡀⢸⠗⠂⠀⣜⠛⢇⣀⠘⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢶⠀⣠⢟⣴⣿⠟⠁⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣵⣿⠋⠁⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠉⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠀⠀⠈⢿⠀⢠⡄⠀⠀⠀⠐⣆⠀⠀⠀⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡃⠀⠀⠀⢀⠁⣿⣿⡄⠀⠀⠀⣿⣇⠀⠀⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⠘⣷⡜⢿⣿⡀⠀⠀⣿⣿⠀⠀⠙⠉⠉⠉⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⣭⣝⣈⢿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⣿⣿⣿⡜⠇⠀⢰⣿⣿⠐⣿⣿⡇⣶⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⠛⠛⠛⠛⢸⣿⣿⣿⡇⣦⠀⠈⠁⠀⡄⣿⣿⣇⣿⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢇⣿⡇⠀⠀⢸⡇⣿⣿⢸⡇⠀⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⠀⠀⢀⣀⠀⠀⢸⣿⣿⣿⢸⡟⠀⠀⢀⣿⡇⣿⣿⣸⡇⠀⠀⠀⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠇⠀⠀⠀⢸⣿⣷⡦⢸⣿⣿⡇⡾⠁⠀⢀⣾⣿⢧⣿⡇⣿⢁⣠⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⠃⣾⣿⣿⡇⠁⠀⣠⣾⣿⡏⣾⣿⣧⣿⠈⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠏⠀⠀⠀⢀⣼⣿⣿⠃⠀⢿⣿⣿⣴⣶⠆⣨⣭⣭⣼⣿⣿⢸⡟⠀⠀⠹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⡿⠁⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠸⣿⣿⡈⠛⠛⠿⠿⣿⣿⠼⠇⠀⠀⠀⠈⢿⣇
⠀⠀⠀⠻⢿⣷⣶⣤⣝⡋⠀⠀⣀⣴⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⡄⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⣀⣤⣦⣤⣤⡀
⢰⣶⣶⣶⣶⣶⣶⣶⣿⣋⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀⠀⠀⠀⠀⢻⣷⣄⠀⠀⠀⠚⠿⠿⣿⣿⣿⣿⣆
⣾⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠋⠛⠙⠛⠛⠀⠀⠀⠀⠀⢻⣿⣷⣄⠀⠀⠀⠀⠀⠈⠙⠿⣿⣧⡀
⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠙⢿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠀⠀⠙⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠙⠄
⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠙⢿⣿⣿⣿⣦⣀⠀⠀⠀⣠⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠀⠙⢿⣿⣿⣿⣷⣦⡀⠹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠻⣿⣿⣿⡿⠃⢹⣿⣿⣿
⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠛⠁⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⣿⣶⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢠⠀⠀⢠⠄⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⡿⠀⠀⢸⠀⠀⢸⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡆⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⢸⡀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠂⡇⢸⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠇
⠀⠀⠀⠀⠠⢰⠀⠀⡇⣸⠀⡇⠀⠀⣸⠀⠀⢸⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⢀
⠀⠀⠀⠀⢸⢸⠀⠀⡇⣿⠀⡇⠀⠀⣿⠀⠀⢸⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⠀⠄⠄⠀⡇⠀⠀⣇⠀⠀⢸
⠀⠀⠀⠀⢸⢸⢠⠀⡇⣿⠀⡇⠀⠀⣿⠀⠀⢸⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠘⠀⢸⠀⡇⣿⠀⡇⠀⠀⣿⠀⠀⢸⠀⠀⢸⠀⠀⠀⣿⠀⠀⠸⠀⠀⠀⠀⠁⠀⠀⢹⠀⠀⢸
⠀⠀⠀⠀⡇⡄⢸⠀⡇⡿⢰⡇⠀⠀⣿⠀⠀⢸⠀⠀⢸⠀⠀⠀⣿⠀⠀⢰⠀⠀⠀⠀⣇⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⡇⡇⢸⠀⡆⡇⢸⡇⠀⠀⡿⠀⠀⢸⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⡇⠃⣼⢸⠀⡇⢸⠀⢀⠐⡇⠀⠀⢸⠀⠀⢸⠀⠀⠀⡏⠀⠀⢸⠀⠀⠀⠀⠋⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠇⠀⡿⢸⢀⡇⢸⠀⠀⠈⡇⠀⠀⢸⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⣷⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠁⡷⢸⢸⡇⢸⠀⠀⢸⡇⠀⠀⢸⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⡇⢸⢸⡇⣿⠀⠀⢸⡇⠀⠀⣼⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⢀⠀⠀⡇⢸⢸⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⢸⢀⠀⡇⢸⢸⠀⡏⠀⠀⢸⠃⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠘⠸⢀⡇⣿⢸⠀⡇⠀⠀⢸⠂⠀⠀⡇⠀⠀⢸⠀⠀⢰⡇⠀⠀⢸⠀⠀⠀⠀⡿⠀⠀⢸⠀⠀⠀⡇
⣀⣀⣀⣀⣀⣀⣀⣉⣈⣠⣧⣀⣤⣼⣄⣤⣀⣇⣀⣀⣘⣀⣀⣘⣃⣀⣀⣘⣀⣀⣀⣀⣃⣀⣀⣘⣀⣀⣀⣃⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣉⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠹⣿⡟⣿⡽⣌⠻⢷⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⣀⣀⣀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⡇⠀⠈⢿⡌⠻⡆⠀⠀⠁⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⢠⣤⡉⠉⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠘⠀⣿⣿⠃⠀⠀⠀⢿⣦⠑⠀⠀⠀⢀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣠⣾⣿⣦⡀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⢀⣸⣿⣤⣀⠀⠀⠈⣷⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣷⢹⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠰⠞⣋⣩⣧⣄⡈⠁⠀⠀⢹⡀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⡾⠟⣿⡇⠭⡗⠄⠀⠀⢸⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⡆⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢆⠐⠈⠁⠐⠀⠀⠀⠀⠸⠈⠀⠀⠀⢀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⠀⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠛⠉⢻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣏⡉⠃⠀⣀⡀⠀⠀⠀⠀⠀⠀⠁⠀⣼⡟⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⢟⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⠶⠎⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠁⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠟⣡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣶⣦⣤⣤⣤⣄⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠁⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠛⠉⠉⠀⠀⢀⣴⣿⣿⣿⠀⠀⠀⠀⠀⣸⣿⡿⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠸⡿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠀⠀⣠⡴⢫⣿⣿⣿⡇⠀⠀⠀⠀⢠⠟⡉⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠻⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣶⠟⠋⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⣌⡀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠉⠀⠀⠒⠒⠒⠀⠀⠀⠀⠤⠤⠤⠤⠤⠄⠀⠀⠀⣻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠁⠀⠀⠀⠸⣿⣿⣿⣿⣧⡀⠀⠀⢹⣿⣷⡶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡙⢆⠀⢸⡟⠻⠿⣷⣦⣄⡀⣤⠤⠶⢤⣴⣶⣶⣒⣶⣒⣛⣛⠉⠉⠉⠉⠉⠉⠉⡉⠁⠈⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⣤⣴⢟⣿⣿⣿⣷⣿⣟⠷⠀⣾⣧⢀⣀⠀⠉⠛⣿⣶⠦⢤⣤⣤⣤⣤⣦⣤⣿⣭⣽⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠊⠀⠈⢻⣿⢹⣿⣢⡝⠂⣾⣿⣿⣯⣄⠀⠀⠈⢻⣷⣟⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣷⣷⣶⣷⣿⣿⣿⣾⣾⣧⣤⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⣾⣿⣿⣿⣿⣿⣿⣷⣤⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⢿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢀⡀⠀⠀⣸⣿⡟⠀⠘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⢀⣴⠿⣷⣠⡾⠋⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡀⣴⡿⠁⠀⠙⠟⠁⠀⠀⠀⠀⣸⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠹⣿⣿⣿⣿⣿⣿⠏⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⣠⡶⡄⡴⢦⣠⠶⣄⡤⢠⢠⠴⠄⢶⠆⡄⢤⢠⢠⠀⣴⠠⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠘⠉⠂⠓⠚⠘⠓⠃⠙⠚⠈⠓⠂⠘⠀⠃⠈⠃⠘⠀⠛⠀⠘





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⢀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣽⣿⣷⣤⡎⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣏⢹⣼⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣴⣿⣿⣿⣿⠟⠛⣸⡿⠛⠙⠻⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠶⠶⢤⣀⠀⠀⠀⢠⣾⣷⣿⣿⣿⣿⡿⠛⠁⣤⣶⡉⣶⣄⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⡡⠀⠀⡰⠋⠀⠀⠀⣈⣭⠓⢒⣾⠏⢉⡿⠛⡋⠁⠀⢠⣾⣿⡇⠉⣦⣀⣱⣀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⡈⠀⠀⣰⡔⠀⠀⠀⠀⠀⠁⠐⠋⠁⠀⠜⠀⡞⠀⠀⠔⢋⡀⢸⣇⣴⣯⡿⣏⠃⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⢸⡄⢀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣴⡟⠀⣿⡟⠛⢦⣏⠈⢀
⠀⠀⠀⠀⠀⠒⠶⣾⣿⣿⣿⣿⠀⣇⢸⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠀⠁⠀⣼⡟⢀⣼⠏⠀⠀⠀⠙⣦⣛⣷⡞
⠀⠀⠀⠀⠀⠀⠞⣿⣿⣿⣿⣿⣧⡈⠀⣿⣧⠈⠻⢦⡐⢤⣄⣠⣾⠃⠀⣧⠀⠀⠀⣼⡿⢡⡟⢁⠀⠀⠀⠀⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠙⠛⠋⠉⠀⠁⠁⠘⣧⢠⠾⠿⢷⣄⠀⠈⠑⠈⠉⣿⠀⣠⡞⠳⣤⡀⠹⠵⠋⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣱⣶⠇⠈⢳⣶⣶⣤⣤⣿⣾⢿⡇⣰⡿⠧⢀⣐⡺⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢰⣿⠋⠀⠀⡼⡿⢻⠋⠙⢻⠁⠀⠹⡿⠷⠻⢿⣟⠓⢤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣻⡇⠀⠀⠀⠹⣼⢿⡀⠀⠘⣧⡄⠐⡇⠀⠀⠀⠉⠻⣆⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠈⠻⡹⣄⡀⠘⣿⡄⢃⠀⠀⠀⠀⠀⠘⢿⡦⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡽⣆⠀⠀⠀⠀⠀⠙⣿⣷⡀⢸⡅⠈⡆⠀⠀⠀⠀⠀⠀⠹⡜⣦
⠀⠀⠀⠀⠀⠀⠉⠁⢀⡀⠀⠀⠀⠀⠀⢷⣿⣇⠀⠀⠀⠀⠀⢹⣿⠛⣸⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⢻⡈⣷
⠀⠀⠀⠀⠈⠙⠓⠀⣈⣛⠛⠛⣓⠒⠒⣾⡟⠛⡆⠀⠀⠀⠀⢠⡾⣏⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉⠙⠊⠉⠛⠁⠀⠀⠀⠀⢸⣿⠟⠉⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠘⠶⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠙⠛⠛⠓⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⢀⣠⣿⠀⢹⣇⣀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢺⣏⠉⠛⠋⠉⠀⠀⠀⠈⠉⠛⠋⢉⣿⠂⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠛⠛⠶⠶⠶⣤⣤⡟⠀⠀⢀⣿⠃⠀⠀⣠⡴⠶⢶⣤⡀⠀⠀⢻⡇⠀⠀⠀⢿⣤⣤⠶⠶⠶⠛⠛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⣠⣤⠾⠇⠀⠀⣾⠋⠀⠀⠀⠈⢻⡄⠀⠀⠿⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢿⣤⣄⡀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣤⣿⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠙⢷⣄⣀⣠⣴⠟⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⠧⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠾⣧⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡶⠟⠳⢶⣤⠀⢠⡴⠾⠛⠷⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⠉⠉⠻⣦⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠉⠙⢳⣄⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠉⠛⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⣶⠏⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠹⣶⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠀⠀⠀⣿⣤⡀⠀⠀⠀⠀⢀⣠⣼⡇⠀⠀⢸⣷⣄⡀⠀⠀⠀⠀⣀⣤⣿⠀⠀⠀⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠀⢿⣤⣤⣤⠿⠈⠙⠳⣦⣤⠶⠛⠉⠸⣧⣤⣤⣼⠇⠈⠛⠶⣤⣴⠞⠋⠁⠿⣤⣤⣤⠿⠀⠙⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⢀
⣿⡟⡋⣻⠿⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣒⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠉⠛⠻⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡏⠀⠀⠀⠀⠉⠛⣳⣦⣤⣄⠀⠀⠀⣀⣤⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠇⠀⠀⠐⠿⠿⣿⣿⣿⣿⠿⠟⣰⣾⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣟⠿⢿⣷⡄⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⢿⠉⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⣌⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣄⣠⣿⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣬⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠻⣷⣤⡀⠀⠀⠀⠀⠀⠱⢶⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣇⠀⠀⠈⢿⣿⣶⣤⣀⡀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠘⣿⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⢻⡄⠀⠀⣰⠟⠈⠻⢿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠃⠀⢠⡟⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠉⠛⠶⣤⣀
⠀⠀⠀⢀⣿⣿⠿⠿⠛⠻⠿⠁⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠛⠀⠀⠀⠀⠀⠤⠄⠀⠀⠐⠒⠒⠛⠛⠛⠂⠉⠉
⠀⠀⠐⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠚⠓⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⣿⡿⠿⠿⠿⠿⠿⠿⣿⡿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⣠⣶⣶⣄⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⢻⣿⣿⡟⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⣿⣿⣿⠿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠻⠟⠻⠟⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⡔⠦⣠⠲⡄⣦⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠳⠖⠙⠶⠉⠇⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠟⠛⠿⢶⣤⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⢠⣄⠀⠀⠈⠻⣦⠀⠀⢀⣾⠛⠛⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⢸⣿⠀⢀⣴⠶⢿⡷⠾⠛⠁⠀⠀⠀⠈⠛⠷⠶⠶⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⢀⣼⠏⢰⣟⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣻⣆⣀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠛⠁⠀⠀⣿⠀⠀⠀⠀⣠⡶⠟⠛⠛⠛⠻⢷⣤⡀⠀⠀⠀⢙⣿⠍⠉⠉⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⡀⠀⣀⣼⠏⠀⠀⢀⣾⣋⣤⡶⠟⢷⣦⡀⠀⠈⢿⡄⢀⣴⠟⠁⠀⢀⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⠏⠁⠀⠀⠀⣼⠟⠋⠁⠀⠀⠀⠈⠻⣦⣄⢈⣿⠟⠁⠀⢀⣴⣿⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⢿⣄⣠⣴⠾⠿⣦⡀⠀⠈⠛⠟⠁⠀⢀⣴⠟⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄⠀⠀⠘⣿⡉⠀⠀⠀⠈⠻⣦⣄⠀⠀⢀⣴⠟⢁⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠈⠻⣦⣄⣀⣀⣀⣨⣿⢷⣴⠟⠁⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠉⠀⠀⠀⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣀⣹⣷⠶⠟⠛⠛⠛⠛⠛⠛⣷⡀⠀⢀⣠⣴⠾⠛⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡏⠉⠀⠀⠀⠀⠰⠶⠶⠶⠶⠟⠛⠛⠛⠉⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣿⡷⠶⣶⣦⣤⣤⣤⣀⣀⣀⣀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣥⢲⠴⠶⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⠾⢻⢾⣾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⠦⠤⠴⠤⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠲⠾⢶⠲⠆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⠶⠾⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣦⢰⢢⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠠⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣒⠆⠀⠀⠀⢑⢒⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠄⠀⠀⠀⠀⠀⠢⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢈⢡⢸⠖⣓⣶⡈⢁⡀⡀⢀⡤⡞⠻⡦⠲⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⠠⠸⡿⢿⠹⢿⠻⠹⠿⠿⠿⠿⡟⢿⠇⠠⢒⡔⠀⡰⠹⠷⡿⠿⠾⠾⢾⡷⠇⠠⠤⠄
⠀⠀⠀⠀⠀⠀⢠⣶⣺⣶⣶⢲⢶⣶⣷⣶⣶⣿⢸⢼⡷⣇⡇⣟⢻⢼⠼⢿⡿⢧⢤⠤⣦⢤⣦⡄⣬⡥⠐⠒⠒⠂
⠀⠀⠀⠀⠒⠒⠒⠶⠀⠂⠶⠆⠲⠒⠶⠶⠶⠠⠶⠖⠶⠶⠦⠆⠶⠶⠲⠶⠒⠈⠰⠶⠖⠰⠶⠠⠶⠖⠶⠆

⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⣀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠚⣩⡤⠶⠦⣬⡙⠂⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⢀⡀⠒⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠁⠴⢚⠲⠄⠁⠀⠀⠀⠀⠀⢀⠂⠀⠋⣹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣇⣀⣠⣿⡇⠀⠀⠀⢀⠀⠀⠤⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠽⠿⠿⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠼⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠇⠀⠀⠽⠿⠿⠀⠒⠐⠃⠐⠂⠿⠿⠿


⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⣒⣒⠴⠶⣂⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠐⣲⠁⢈⠆⢘⣒⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣍⣉⣁⣀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢔⣀⡢
⠀⠀⠀⠼⠷⠤⢤⡾⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣧⣤⡆⠀⠀⠀⠀⠀⢸⣿⣟⣉⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠪⠭⣕
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠟⠿⠛⠛⠃⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⣶⣤⠚⣿⠳⣴⣶⠀⠀⠀⠀⣀⣤⣶⣶⣄⣀⠀⣤⣴⣾⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀
⠀⠀⠀⣶⠖⣿⣿⡿⢶⡶⠀⠀⠀⠘⠿⠿⢻⠟⠿⠟⠈⠛⠛⢻⡟⠛⠛⠀⠀⠀⠀⠠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠸⠛⣿⣧⣠⣄⣀
⠀⠀⠀⠀⢠⣤⣭⣄⡄⡀⠀⠀⠀⠀⠀⢀⣐⢴⠀⠀⠀⠀⠀⣀⣲⠀⠀⠀⠀⠀⢸⣦⣍⣉⣵⣿⠀⠀⠀⠀⠀⠀⢸⡟⠿⠿⢿⣿
⠀⠀⠀⠄⢤⢤⡤⡤⡌⠄⠀⠀⠀⠀⠀⠀⠀⡭⠀⠀⠀⠀⠀⠁⠭⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⡩⡀⠀⠀⠀⠀⠀⠾⡷⠤⠤⢬⣷⡄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⣄⡟⠙⢂⣄⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣙⢒⠂⣾⣿⠈⢳⠀⠀⠀⠀⠀⣖⢂⣐⠁⢤⠤⠐⢂⡁⠇⠀⠀⠀⠀⢸⠐⠩⠃⡇
⠀⠀⢀⣴⣆⣿⣿⣿⣿⣴⣦⠀⠀⠀⠀⠀⠀⠲⣒⣆⢇⠀⣿⡟⠀⠀⠀⠀⠀⢄⠀⡶⠔⠒⠀⢀⠁⡦⠆⠀⠀⠀⠀⣸⢖⠤⡖⠇
⠀⠀⠀⠙⠁⠈⠙⠋⠁⠙⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠓⠒⠁⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⠈⠒⠚
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⢀⣠⣤⣿⣿⣿⣿⣿⣷⣤⣄⡀⠀⢀⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠉⠉⠉⠉⠛⠿⠿⠿⠿⠿⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⢿⣿⣿⡇⢰⣶⣶⣶⣶⠀⢸⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇⢸⣿⣿⣿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⢸⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠸⠿⠿⠿⠿⠀⠸⠿⠿⠿⠿⠇⠀⠿⠿⠿⠿⠇⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣶⣶⣶⣶⣶⣦⣄⣀⣀⣀⣀⣠⣴⣶⣶⣶⣶⣶⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠀⠈⠉⠛⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⠈⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⣄⣀⡀
⠀⠀⠀⡠⠖⠉⣉⣉⣩⣷⣦⣄⠀⢀⣀⡀⠀⠀⣾⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⢠⡞⠀⠴⠋⠁⢀⣠⡤⣄⣈⡷⠋⠀⠙⡆⠀⡟⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⡄
⠀⣼⠀⠀⠀⠀⡴⣿⠁⠀⠀⠀⠴⠂⠀⠀⡇⢰⠃⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿
⠀⣿⠀⠀⠀⠀⣇⠻⣄⣀⣀⣤⣄⣀⡤⠞⣡⡇⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⠀⠀⣴⠛⠳⣆⠀⠀⠀⠀⠀⠀⢸⠀⢸⡆
⠀⢻⠀⠀⠀⠀⠈⠑⠊⠉⠀⣠⠏⠀⠀⣰⡟⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡼⠀⠀⣟⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸⡇
⠀⠘⡇⠀⠀⠀⠀⣀⠴⠒⠉⠀⠀⢀⡜⡹⠁⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⣰⠃⣠⣄⠘⡆⠀⡾⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇
⠀⠀⣿⠀⠀⠀⣰⠃⠀⠀⠩⠂⢠⠞⡼⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⣰⠃⠘⢧⣨⠇⣹⣾⡄⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇
⠀⠀⠛⠀⠀⠀⠟⠀⠀⠀⢀⠔⠁⣸⠁⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣰⠃⠀⠀⠀⠀⢰⠇⠀⢻⡄⠀⠀⠀⠀⠀⣿⠹⡏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⢠⠇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⣰⠃⠀⠀⠀⠀⠀⣼⠀⠀⠈⣇⠀⠀⠀⠀⣠⢏⡀⣇⠈⢧⡀
⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⣼⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠀⢰⠃⠀⢀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣻⠀⠀⢀⡼⢡⠛⡇⣿⢰⠶⣽⣆
⠀⠀⠀⣠⣴⣋⠤⠤⠤⠤⣄⡀⡇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢸⠁⠀⣤⠒⢦⠀⠀⣿⠀⠀⠈⠙⢧⠀⢸⠁⣾⠀⡇⢿⢸⠀⢹⢹
⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⣰⠋⠀⠀⢀⡟⠀⠀⢻⡤⠞⢦⠀⢸⡄⠀⠀⠀⠘⡆⠸⣆⠘⠦⠇⣿⠸⣦⢞⡾
⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡦⢄⣀⢀⡀⠀⡀⣀⡠⠔⠊⠀⠀⠀⢀⡼⠀⠀⠀⢸⡅⠀⠈⢧⡀⢷⣶⣲⣶⡯⠗⠀⠘⠲⠤⠞⠙⠦⠖⠋
⠀⠀⠀⠀⢧⠸⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠉⠉⠉⠉⠀⠀⠀⠀⢀⣀⠄⠊⠀⠀⠀⠀⠘⣇⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⢀⠴⡄
⠀⢠⣶⣦⠘⡖⠲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠺⠔⠁⢀⡤⢤⡀
⠀⢸⡧⣾⠀⣟⠚⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⣿⡆⠀⢸⠀⠀⢳
⠀⠀⡗⡇⠀⣿⠀⣿⠀⠀⠀⠀⠀⣀⣤⠶⡶⠛⠛⠛⠉⠉⠉⠉⢹⠉⠉⠉⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣠⣴⣾⡯⠓⠀⠀⣸⠛⣧⠀⠸⠤⠤⠼⠃
⠀⠘⣇⣷⠀⢸⠋⢻⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠋⠁⠀⠀⠀⣰⠉⠉⠉⢧
⠀⠰⡏⢻⠀⢸⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠖⠒⠒⠲⣤⠖⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⠖⠒⠒⠾⢦⡀
⠀⠀⡇⢸⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠒⠒⠚⢋⣉⣉⣭⣥⣤⣤⣀⡀⠀⠀⠀⠘⡇⢼⡃⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⡇⢸⡆⢸⡇⢸⡇⠀⠀⢀⣀⡴⠁⠀⠀⠀⠀⢀⠔⠚⠉⠁⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀⣧⠠⣇⠀⠀⠀⠀⠀⠀⡾⠀⢀⡞⠉⠉⠉⢧⠀⠀⢸⡄
⠀⠀⣇⢸⡇⢸⡇⢸⡇⠀⠀⢹⡂⢻⡀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡀⠸⡄⠀⠀⠀⠀⢸⠃⠀⣼⠀⠀⠀⠀⠸⡆⣤⠀⣧
⠀⠀⣿⢸⡇⠀⡇⢸⡇⠀⠀⠀⡇⠀⢳⡀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠀⠀⠀⠈⣧⠐⡇⠀⠀⠀⠀⣼⠀⠀⡇⠀⠀⠀⠀⠀⣧⢸⠀⢸⡀
⠀⠀⣿⢸⡇⠀⡇⢸⡇⠀⠀⠀⢹⡀⠀⢷⠀⠀⠀⠈⠓⠒⠶⠶⠶⠶⠖⠒⣊⣉⣤⠀⠀⠴⠀⠀⢹⡀⢳⠀⠀⠀⠀⡏⠀⢸⡇⠀⠀⠀⠀⠀⢿⢀⠀⢸⡇
⠀⠀⣿⠈⡇⠀⡇⠸⡇⠀⠀⠀⠀⢳⠄⠘⣇⠀⠀⠀⠐⠒⠒⠒⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⡃⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⠈⠀⢸⠃
⠀⠀⢿⠀⡇⠀⣿⠀⡇⠀⠀⠀⠀⠘⢶⡀⠙⣆⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⠤⠤⠤⠖⠒⠉⠉⠉⠉⠀⠀⠀⠀⢿⠀⠀⠳⠦⠤⠤⠤⠤⠞⠀⠀⡼
⠀⠀⢸⣀⡇⠀⣿⢀⡇⠀⠀⠀⠀⠀⠈⠿⠴⠚⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⠤⠤⠤⠤⠤⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣴⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠴⠋⠒⠁⠀⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣡⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡍⠐⡊⢝⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠋⢙⣷⢴⣍⠢⢄⠀⠀⠀⡀⡀⠀⠀⠀⣼⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⣠⣿⠿⣿⣿⣄⣂⡑⠮⡢⡊⠈⢐⢑⣵⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⣼⢃⣀⣙⣪⣛⢿⣿⣇⠓⠈⠳⣔⣵⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠘⣿⠋⢁⠀⠉⠛⢷⣭⡿⠋⢻⣗⣈⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡼⢆⠀⠘⢿⡉⠉⠳⢤⡀⠈⢧⣴⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢒⡵⠉⢀⣼⠓⢄⠀⠉⢳⡶⠾⠷⡄⠀⢻⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⣈⢬⠳⢤⣿⣿⡳⢄⠓⡶⠁⢀⣤⢤⠇⠀⢸⡟⠿⣶⡮⢹⠦⣄
⠀⠀⠀⠀⢀⠔⠁⡠⡺⠹⠕⠝⢦⡹⢻⣿⣷⣹⠁⢠⡟⡠⠋⠀⣠⣿⢧⣶⣼⡟⣸⡄⢠⠉⠲⢄⡀
⠀⠀⢀⠔⠁⣠⠊⠈⠀⠀⠀⠀⠀⠙⢮⡋⢿⣿⠀⠸⡏⠀⣀⡾⠃⡏⢱⣷⣿⠇⡇⠚⠈⠀⠀⠀⠉⡲⡄
⠀⡴⠁⣀⠞⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⡻⣆⠀⠙⢶⡋⠀⢸⢁⣿⣯⡫⢸⠃⠀⠀⠀⠀⠀⠀⠱⣹
⠀⠳⡸⡫⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠪⣻⣷⣄⡀⠉⠲⢯⣼⣿⣿⠄⡞⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠙⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣑⣷⣿⣿⠟⠙⠳⣤⣀⠉⢻⣿⢢⢷⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⠀⢸⠙⠛⠟⢁⣜⢢⠡⠑⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⡠⣲⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣦⣡⢟⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡑⣕⣾⣟⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⣤⣄⣄⢌⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠔⠉⡱⢀
⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠐⢉⠀⠀⠀⠀⠀⠀⠈⠠⡁⠀⠘
⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢩⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢊
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣲⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒
⣀⠀⠀⡄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⡀⠁⢦⠤⣀⡀
⠀⡄⠀⢰⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠣⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈
⣃⣀⣀⣐⣁⣀⣀⣀⣀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⡶⠟⢋⣡⣉⠻⠷⣦⣤⣄⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡏⢉⣉⣉⣉⣉⣉⣤⣴⠾⠛⠉⠙⠻⢶⣤⣬⣉⣉⣉⣉⣉⠉⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡏⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⢸⡇⠀⠀⠀⠀⠀⣠⣶⠿⠛⠻⢷⣦⡀⠀⠀⠀⠀⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⠀⠀⣴⣾⣿⣶⣶⣶⣶⣶⣾⣿⣶⡄⠀⠀⠀⣿⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⡇⠀⠀⠀⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⣿⡇⠀⠀⠀⣿⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⣧⠀⠀⠀⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡇⠀⠀⢠⣿⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢻⣆⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡇⠀⢀⣼⠃⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⢻⣆⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⢀⣾⠃⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠛⢷⣄⠀⠀⠀⠀⠀⢀⣴⠞⢁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠙⠷⣦⣀⣠⡶⠟⢁⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣌⠙⢉⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠛⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠾⠶⢘⣿⣀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⡠⣦⢀⣸⣿⣧⡀⠀⠀⢀⡴⠾⠗⠒⢳⣦
⠀⠀⠀⠐⢶⡄⠀⢸⣿⣿⣧⡀⠀⢠⣤⣤⣿⣿⣿⠶⣤⠀⠀⢠⡞⢋⣿⠉⢷⡄⠀⠀⢸⣿⣿⣾⣿⠛⢿⡶⠄⠀⢈⡿⣰⣺⡆⢸⡇
⠀⠀⠀⠀⣾⣵⠃⣿⢯⣅⡀⠀⠀⠀⠀⢘⢿⣿⣿⣀⡀⠀⠀⠈⣷⠟⠁⢀⣾⠇⠀⠀⣼⠿⠿⢸⣿⣿⣿⣧⠀⠀⢸⡇⠻⠿⠇⢸⣿
⠀⠀⠀⠀⠻⠛⠛⠁⠀⠙⠿⠆⠀⠀⠀⠿⢷⣭⣽⡯⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⠛⠁⠀⣿⠀⠀⠀⠸⠇⠀⠀⠰⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠴⠒⠒⠒⠛⠃⠈⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠈⠙⡇⢀⣠⣤⣤⣤⣤⣴⡶⠶⢶⣦⠄⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣶⣶⣿⣶⣶⠾⠿⡇⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠋⠁⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠘⢆⣶⡄⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣠⠴⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠉⠀⠀⠀⠀⠀⠀⠘⠃⢸⡆⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢠⡏⠀⣤⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠛⠉⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠘⣧⣀⣀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡶⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼⣃
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⣀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⣇⣼⠏⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠘⠛⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠓⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡷⢦⣄⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣧⡀⢀⣀⣀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠙⠻⣷⣶⣤⣤⣀⣀⡀⠀⠀⠀⣀⣀⣀⣤⣴⣾⣿⣿⡿⠋⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⣀⣠⣀⠀⢿⡛⢿⣿⣿⡿⣿⣿⣿⢿⣿⣿⠿⠿⠛⠛⠉⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⡞⠉⠀⠼⡇⠀⠙⠿⣧⣴⣟⡩⠴⠶⠚⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡀⠀⠀⣈⣷⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⡀⠈⠳⣤⣤⣤⣿⡇⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⡄⠀⠀⢸⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⢹⣿⠟⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣆⠰⠚⢧⡍⠁⢙⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣀⠀⠀⣙⡋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣩⣆⣤⣾⡡⠖⣩⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠙⠛⠿⣿⣿⣶⣶⣶⣶⣶⠶⠿⠛⢻⣿⣿⣿⡿⠃⠈⠋⣠⠴⠋⢀⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠧⣤⣾⣿⣿⠿⠿⠿⣿⣿⣷⣄⡤⠿⣿⣿⣿⣧⣀⡴⠚⠁⠀⢀⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠴⠿⠛⠁⠀⠀⠀⠀⠈⠙⠿⣿⣷⣦⣿⡿⠟⠁⠙⢦⣄⣀⣠⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠀⠉⠛⢻⣿⣷⣿⢿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⡿⣛⣵⣴⣦⣀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⠿⠟⣷⣿⣿⣿⣿⣦⣀⡀⠀⠀⢽⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⣿⣿⠅⠒⠄⣶⣍⡻⢿⣿⣿⣿⣿⡄⠀⠀⠹⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⣿⣿⠀⠀⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⢼⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣧⠀⠀⠐⡟⢡⡿⣻⣿⣿⣿⣿⣿⠶⣄⣀⡲⢿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠆⠀⢘⢆⡾⣱⣿⣿⣿⠻⣿⢿⠑⠈⠹⣙⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣏⡿⠃⡭⡸⣿⣿⡿⠋⠙⣿⡝⠧⡄⣰⡀⣻⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣧⡞⣹⡇⢠⡶⠐⠻⣿⣿⣧⣶⢹⣼⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣹⣿⣿⣿⠟⠁⢹⣿⠜⣸⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⡟⠉⠀⠘⣸⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠸⠯⠀⠀⠀⠀⠿⠼⠻⠿⠿⠿⡿⠿⠿⠿⡿⡿⠿⠇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠷⠶⠶⢶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠛⢉⣉⣀⣤⠤⠴⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠦⠤⣌⣉⠙⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⣁⡤⠖⠋⠉⣁⣠⡤⠴⠖⠒⠒⠛⣛⣛⣛⣛⣋⣛⣛⣛⡛⠛⠓⠒⠶⠤⣄⣈⠙⠓⢦⣈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣠⠞⠁⣠⠴⠚⢋⣡⣤⠶⠶⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠻⠷⠶⣦⣍⡙⠲⣄⠈⠳⣄⠙⣷⡀
⠀⠀⠀⠀⠀⠀⢠⡟⢀⡜⠁⣠⠞⢁⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡈⠳⣄⠘⢦⠈⢿⡄
⠀⠀⠀⠀⠀⢀⡿⠀⡾⠀⣰⠃⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠹⡆⠘⣇⠸⣧
⠀⠀⠀⠀⠀⢸⡇⢸⡇⢀⡇⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢻⠀⢹⠀⣿⡀
⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⣤⣾⠇⣸⠀⢸⠀⣿
⠀⠀⠀⠀⠀⢸⣧⠘⣇⠀⣧⠘⣷⡀⣀⣴⠿⠿⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡶⠟⠛⠻⣿⣟⣉⣡⡤⠟⢀⡟⢠⡿
⠀⠀⠀⠀⠀⠀⢿⡄⠹⣄⢘⣧⡼⠟⠋⣁⡤⣤⣀⡈⠉⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠉⣀⣠⡤⠶⠦⣄⠙⠻⢷⣤⣄⣸⡇⠸⣧
⠀⠀⠀⠀⢀⣀⣬⣿⠶⠟⠋⣁⣤⠴⠛⠁⠀⠀⠈⠉⠛⠒⠶⠶⠤⠤⢤⣤⣤⣤⡤⠴⠶⠖⠚⠋⠉⠀⠀⠀⠀⠈⠙⠲⣤⣈⡙⠛⠿⢶⣿⣧⣤
⠀⠀⠀⠀⠻⣯⠁⣠⠴⠖⠛⢩⡶⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠻⣿⡍⠙⠒⢦⠀⢸⡟
⠀⠀⠀⠀⠀⢿⡆⢻⠀⠀⠀⢸⣧⣤⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣴⣿⠇⠀⠀⢸⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⢸⡆⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢻⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢻⡇
⠀⠀⠀⠀⠀⣿⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⢛⣋⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠸⣇
⠀⠀⠀⠀⢀⣿⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⣾⣿⣿⣿⣦⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢿⡀
⠀⠀⠀⠀⢸⡏⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⣿⣿⣿⣿⣿⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣶⣦⠘⡇⢸⡇
⠀⠀⠀⠀⣿⠁⡏⠀⣠⡶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠙⣿⣿⣿⢁⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣀⣸⣿⡇⣧⠘⣧
⠀⠀⠀⠀⣿⠀⡇⠐⣿⣀⣨⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⣿⣿⣿⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⠀⣿⠀⡿
⠀⠀⠀⠀⢹⣇⢳⡀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⣿⣿⣿⣇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢰⡇
⠀⠀⠀⠀⠈⣿⡌⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄⠀⠀⠀⢀⡇⣾⣿⣿⣿⣿⡸⡆⠀⠀⠀⣠⠶⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢀⡾
⠀⠀⠀⠀⠀⠈⢿⣆⡙⢦⣀⠀⠀⠀⠀⠀⠀⣾⣇⣀⣽⣿⠆⠀⠀⣸⠿⠿⠿⠿⠿⠿⠇⢷⠀⠀⠀⣿⣄⣸⣿⡇⠀⠀⠀⠀⢀⣠⠶⠋⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣉⠓⠦⣤⣀⣀⡀⠘⠻⠿⠿⠋⠀⠀⠀⠻⠶⠶⠶⠶⠶⠶⠶⠞⠀⠀⠀⠉⠛⢛⣋⣀⣠⡤⠶⠚⠉⣁⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣦⣤⣍⣉⣙⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠋⢉⣉⣉⣁⣤⣤⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠟⠻⣷⣶⣤⣄⡀⠀⠀⠀⣀⣠⣴⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠈⠻⡏⠙⢿⣆⢀⡴⠋⠁⠀⠈⠉⠛⠻⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢹⡇⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⡾⠛⠛⠛⠳⢤⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣤⡴⠾⠋⠉⠉⠻⢦⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠈⠙⣄
⠀⠀⠀⠀⢀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⣤⣁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⢷⣄
⠀⠀⠀⣠⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠉⠙⠛⠿⠶⣤⡀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣿
⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣁⣀⣀⡀⠀⣀⣴⣿⠃⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣤⠦⣤⣄
⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⢻⣿⠁⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⢠⣿⣧⣄
⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠁⠀⡹
⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣤⣤⣀⣀⣠⣤⡶⠟⠋⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣯⠀⢹⣿⡏⠁⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢿⡿⣿⣷⣿⣿⣿⣄⡀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⡿⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠀⢸
⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣦⣤⣀⠀⠀⢰⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⠀⣿
⢹⣿⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⣧⡈⠙⢿⣿⣶⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⣴⠿⠻⠿⠁⠀⠀⣿
⠈⢿⣎⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡈⠃⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⣤⣄⣠⣍⣷⠶⠀⠀⠀⠀⢠⣿
⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢻⣿⣿⣿⡌⠻⣿⡙⠛⠛⠿⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⢿⡟⣿⡉⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣇⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢿⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡰⠶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠒⠒⠓⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠓⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣾⠏⠀⠀⠀⠠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡉⠃⠀⠉⠿⠁⢀⡞⠉⠉⣹⡅⠛⢐⣧⡿⢋⡿⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠀⠀⠀⠀⣰⣿⠋⠉⠉⢻⣷⡀⠀⠀⠀⠐⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠨⠂⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠪⠂⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⢀⢄⣀⠀⠂⣀⣀⢀⣴⣿⣷⣤⣤⣤⣴⣿⣷⣄⣀⣀⡀⠂⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠕⣈⡀⠚⢂⣿⣿⣿⣿⠉⢹⣿⣿⣿⣇⠘⠃⡈⠐⠅⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⠔⣤⡐⣿⣿⣿⣿⠃⢻⣿⣿⣿⣗⣠⡔⠄⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⢀⣄⡠⠒⠒⠒⠀⠒⣿⣿⣿⣿⣤⣼⣿⣿⣿⡗⠒⠒⠒⠒⠢⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠛⠋⠉⢉⠉⠉⠛⠻⢿⣿⣷⣄⠀⠀⠀⣤⣴⣶⣿⣿⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠉⣠⣴⣾⣿⡏⢻⣿⣶⣦⡀⠙⢿⣿⣷⡀⠀⠀⠉⠛⠿⣿⣿⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⣼⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣆⠈⢿⣿⣧⠀⠀⠀⠀⠀⠈⠛⠇⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢸⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡆⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢸⣿⣿⣿⣿⣿⣇⣈⣉⣉⣉⣹⣿⠇⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⣀⠉⠙⠛⠛⠋⢉⣀⣴⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣶⣾⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡄⠀⠀⠉⠙⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⢠⣾⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣄⠀⢷⣦⡀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣾⣿⣿⣷⣦⣄⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣶⣶⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠿⠟⠋⠉⠀⠀⠀⠈⠿⠟⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⢀⡀⢀⣀⠀⣀⡤⢀⢀⠀⣀⡀⣄⢠⢀⠀⡀⡄⣠⡀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠸⠁⠯⠼⠺⠧⠇⠻⠼⠸⠧⠄⠯⠸⠈⠷⠁⠇⠹⠄⣳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⡿⠉⠙⡟⠛⠟⡟⠟⠻⡟⠻⠟⠿⠿⠟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣛⣛⣿⣿⢿⣿⣿⣿

⠀⠀⣠⡶⣾⣟⢿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⢰⣿⡇⢿⣿⡈⠛⢿⣿⣿⣿⣿⣭⣉⣙⣉⣉⣉⣉⣤⣴⣶⣾⣿⣿⠿⠟⢛⣛⣷⣄⡀
⣰⡘⣿⣷⡜⣿⣿⣤⡀⠈⠙⠛⠿⠿⠿⣿⣿⣿⠿⠿⠿⠛⠛⢉⣡⣤⣶⣿⣿⣿⠿⠿⠿⢆⡀
⣿⣷⡙⢿⣿⣮⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⡿⢟⣛⣩⣥⣴⣶⣶⣾⣿⣿⣷⡄
⢻⣿⣿⣶⣽⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⠿⠛⠛⠋⠉⣉⣉⣉⣉⠛⠛⠻⠿⣿⣿⡆
⠀⠹⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣭⣿⡄
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢀⣠⣴⣿⡿⠟⠋⠉⠉⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡿⠁⠀⠴⠶⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠈⠛⣛⣩⣶⡶⠒⠓⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⣴⣶⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏⠀⠀⠀⠟⠋⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡀⠻⠇⠀⠀⢠⣤⣤⣶⠿⢋⣿⢻⣿⣿⣿⣿⣿⣿⠃
⠀⠘⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⡏⠉⠀⠀⠸⠃⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⣿⣿
⠀⠀⠀⠀⠀⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠸⣇⡀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠛⠛⠛⢿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⢀⣠⣶⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢀⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣰⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢠⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⢀⣰⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⠟⠉⠀⣠⣴⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠟⠉⠀⣠⣶⣿⣿⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠃⠀⠀⣼⣿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠋⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠘⢦⠞⠀⠀⠀⠀⢰⠖⢦⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢲⡀⠀⠀⠀⠀⠰⡶⠀⠀⠉⠀⠀⠀⢸⠀⠀⢠⡦⠀⠘⢶⠋⠀⠘⡟⠀⠀⢠⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠋⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⢄⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⡔⠁⠀⠀⠹⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣶⠶⠶⠷⠶⠶⠶⠾⠶⠶⠾⠶⠶⠶⠷⠶⠶⠾⠶⠶⢷⣆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⣿⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⢶⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠒⠒⠋⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠵⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⢸⣯⠤⠤⠤⠊
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⢸⣷⠤⠤⠴⡦⠀⣀⣀
⠀⠀⠀⠀⠀⠀⢿⣀⡿⠒⠒⠒⠒⢺⣿⠀⠀⠀⠀⢰⡟⠛⠛⠉⢉⠉⠙⠛⠛⣷⠀⠀⠀⠀⢸⣿⠒⠒⠒⠒⠺⣇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⠇⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⠀⢀⠤⠤⢖⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⠀⢘⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⣈⡁⣀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢼⣿⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⢸⣯⣀⣀⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠾⠖⠒⠊⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠈⠱⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢻⠛⠛⡟⠛⠛⢻⠛⢻⠛⠛⠟⠛⢻⠛⠛⡟⠛⠛⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⢀⡼⠀⠀⣇⠀⠀⡸⠀⢸⠀⢸⡇⠀⢸⠀⠀⠑⡄⠀⠀⠀⠀⣠⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⠟⠒⠒⠒⠋⠀⠀⠈⠊⠀⣸⠀⠀⢸⠀⠀⠀⠀⠻⠂⠀⠀⣥⠀⠀⠀⠀⠻⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢰⡋⣳⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣴⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻
⢻⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴
⠀⠀⠁⣿⡏⠉⢹⡏⢹⡏⠉⠉⢹⡏⢹⡏⠉⠉⢹⡏⢹⡏⠉⠉⢹⡏⢹⡏⠉⠉⢹⡏⢹⡏⠉⠉⢹⡏⢹⡏⠈⠉⢹⡏⢹⡏⠉⢹⣿⠉
⠀⠀⠀⣿⡇⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⢸⣿
⠀⠀⠀⣿⡇⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⢸⣿
⠀⠀⠀⣿⡇⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⢸⣿
⠀⠀⠀⣿⡇⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⢸⣿
⠀⠀⠀⣿⡇⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⢸⣿
⠀⠀⠀⣿⡇⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⢸⣿
⠀⠀⠀⣿⡇⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⢸⣿
⠀⠀⠀⣿⠇⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠸⣿
⠀⠀⠀⣿⡷⠷⠿⠷⠿⠿⠾⠷⠿⠿⠿⠿⠾⠷⠿⠿⠿⠿⠶⠶⠿⠿⠿⠿⠿⠶⠿⠿⠿⠿⠾⠶⠿⠿⠿⠿⠾⠷⠿⠿⠾⠿⠶⢾⣿
⠀⠀⠀⣿⡇⠀⢸⡟⣛⠛⠛⠛⠛⠛⢻⡇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢸⣿
⠀⠀⠀⣿⡃⠀⢸⣧⣥⣤⣤⣤⣤⣤⣼⡇⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢸⣿
⠀⠀⠀⣿⡷⠶⣶⡶⣶⣶⢶⡶⣶⣶⣶⣶⢶⠶⣶⣶⣶⣶⢶⠶⣶⣶⣶⣶⠶⡶⣶⣶⣶⣶⠶⡶⣶⣶⣶⣶⠶⠶⣶⣶⢶⣶⠶⢾⣿
⠀⠀⠀⣿⡇⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⢸⣿
⠀⠀⠀⣿⡇⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⢸⣿
⠀⠀⠀⣿⡇⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⢸⣿
⠀⠀⠀⣿⡇⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⢸⣿
⠀⠀⠀⣿⡇⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⢸⣿
⠀⠀⠀⣿⡇⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⢸⣿
⠀⠀⠀⣿⡇⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⢸⣿
⠀⠀⠀⣿⡇⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⢸⡇⠀⢸⣿
⠀⠀⣀⣿⣇⣀⣸⣇⣸⣇⣀⣀⣸⣇⣸⣇⣀⣀⣸⣇⣸⣇⢀⣀⣸⣇⣸⣇⣀⣀⣸⣇⣸⣇⢀⣀⣸⣇⣸⣇⣀⣀⣸⣇⣸⣇⢀⣸⣿⣀
⣴⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻
⠻⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴