Lettuce Emojis & Symbols

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣠⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠮⢤⡀⢈⠖⠉⠁⠈⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠁⠀⢠⡇⠸⣇⠀⠀⠀⠀⠀⠣⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠞⠋⠀⠀⠀⠀⠀⢇⠀⠙⢧⢰⣠⠞⠉⠉⢻⠉⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⠀⠀⠀⠰⠀⠀⠈⠣⣴⠛⠋⠓⠀⠀⠀⠘⢗⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢽⡄⠀⠀⠀⠀⠀⠈⠘⠓⢄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⡀⣡⠊⠀⠀⠀⠀⠀⢷⠀⠀⠀⢠⠀⠀⠀⠀⢸⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢇⠃⠀⠀⡀⠀⠀⢀⡞⠀⠀⠀⢸⠀⠀⠀⠀⠘⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⣾⠘⢀⡴⠋⠀⠀⠀⣸⠀⠀⠀⠀⠀⣧⢠⠄⠐⣿⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠳⡀⡇⣴⠊⠀⢀⠀⠀⠀⢱⡇⠀⠀⡄⠀⢳⠇⠀⠀⠘⡆⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠱⡇⠃⢀⡔⠕⠈⠀⠀⢾⡀⠀⠀⠃⠠⠘⢀⠆⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠐⡄⡇⣇⠎⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⠆⠖⠉⠀⢀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡀⠘⡇⡿⢀⠔⠋⠀⠀⠀⢯⡀⢰⣄⠎⠌⠀⡀⠸⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⡄⣧⢱⠎⠀⡀⠀⠀⡤⠚⠃⢠⢏⠌⡠⠎⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⢸⠀⡰⠁⠀⠀⢳⠀⡠⢃⠞⢉⡶⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠸⢸⡜⠁⠀⢠⣢⢞⠟⢡⠋⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⣴⠘⡇⠀⠰⡿⣠⠊⡰⠛⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⠀⡇⢶⢺⠟⠁⣼⠕⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡏⠀⢇⣸⠏⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘⣇⣠⣺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣮⣀⣸⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠉⠉⠑⠲⢤⣀⣀⣀⣀⠀⠀⢀⡴⠒⠒⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡖⠉⠀⠀⠀⠀⠀⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⢄⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠐⠈⠀⣀⡀⣈⡦⣄⣀⣠⣄⡀⠀⠀⠐⡀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠓⠦⠤⣄
⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⢀⠴⠒⠒⠚⠋⣁⡤⠤⣥⣀⣀⣉⠛⠋⠉⠙⠳⣄⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠢⢈⡼⠋⠀⠀⣠⠖⠋⠀⠀⠀⢰⠊⠉⠙⠦⣄⣀⠀⠀⠈⢧⠀⠀⠀⠀⢹⠙⢦⡀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⢧⠀⠀⠀⠀⢀⡠⠖⠉⠀⢀⡴⠋⠁⠀⠀⠀⣠⠤⠊⠀⠀⠀⠀⠐⠉⢳⡄⠀⠸⡆⠀⠀⠀⠘⡄⠀⠻⡄⠀⠀⠸⣥⣀
⠀⠀⠀⣠⠞⠀⠀⠀⢠⠏⠀⠀⠒⣲⠋⠀⠀⠀⠀⠀⢰⠃⠀⠀⠑⠀⠀⠀⠀⠀⠀⢳⡀⠀⢳⡀⠀⠀⠀⠙⠢⢤⡘⣆⠀⠀⠀⠈⠳⡄
⠀⣠⠞⢁⡴⠤⣄⠀⡏⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠘⠤⡀⠀⠀⠀⡀⠳⡄⠀⠙⠲⣄⡀⠀⠀⠀⡇⢹⠀⠀⠀⠀⢠⠇
⠀⡇⠀⢸⠀⠀⠈⠳⣷⣠⠤⣴⡃⠀⠀⠀⠀⡤⠒⠉⠀⠄⠀⡀⠀⠀⠀⠈⠂⠀⠀⠰⠀⠙⣆⠀⠀⠀⠉⠳⣄⠀⠱⣸⠀⠀⠀⠀⣾
⠀⢣⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠸⡁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠘⡖⠀⠀⠀⠀⠘⡆⠀⢸⠀⠀⠀⠀⢻
⠀⠈⣆⡼⠀⠀⠀⠀⡀⠔⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠘⡀⠀⠀⠀⠀⣷⠀⠀⠀⢀⡴⠃⠀⢸⠀⠀⠀⠀⠈⢣⡀
⢠⡞⠉⠀⠀⢀⠂⢁⠀⠀⠀⢀⡟⠀⠀⢀⠴⠁⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠡⡐⠀⠀⠀⠀⡏⠀⠀⣴⠋⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠹
⣸⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⡟⠀⠀⢰⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢠⡐⠀⢰⠧⡀⠀⢧⠀⠀⠀⠀⠀⡇
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⠀⠠⠍⢢⡟⠀⠈⠢⢘⡆⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣠
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣄⠀⠈⢣⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠓⢀⡴⠚⠀⠀⠀⠀⡞⠔⡁⠊⠀⢠⠏
⢹⠀⠀⡇⠀⠀⠀⠠⡀⠀⠀⠀⠂⠈⠛⠲⢤⣷⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠞⠀⠀⠀⠀⢀⣾⠁⠈⠀⠀⠀⢸
⠘⡇⠀⢠⠀⠄⠀⠀⠀⠀⠂⠠⠄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠒⠒⠒⠒⠲⠦⠤⣄⣀⣇⣰⠃⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⣸
⠀⢱⡀⠀⢂⠈⡀⠀⠀⡀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠤⠄⠀⡀⠀⠀⠀⠀⢀⠠⠂⠉⠙⣇⠀⠀⠀⠀⣠⠟⠂⠈⠀⠀⠀⠀⢀⡇
⠀⠀⠳⡄⠀⠡⡈⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠁⠒⠀⠐⠁⠀⠀⠀⠀⡟⠀⠀⢀⠴⡉⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠙⢦⡀⠈⠀⠀⠀⠀⡇⢀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠀⠀⢀⡤⠞⠛⠛⠉⠁⠀⠁⠀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠇⠀⠠⢀⡀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⠀⠀⠀⡰⠋⠀⣀⣀⠀⠀⣀⣠⠤⣶⠶⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠘⠀⠀⠀⠀⠁⠒⠒⢂⡴⠒⠲⢤⡀⠀⠀⠀⣠⢞⡴⠚⠉⠉⠉⠉⠉⢉⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢬⣤⣤⣤⡤⠤⠖⠋⢠⠏⠉⠲⣝⠲⠴⠚⡵⠋⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⣏⠀⠀⠀⠈⠓⠶⠚⠁⠀⢀⣀⡤⠖⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣲⣦⣤⣤⣴⣶⣒⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢤⠲⠀⠀⢠⣶⣦⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠈⠀⠘⣶⣦⠀⣼⢹⠁⠠⡆⠀⡰⠷⠰⢀⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⢻⠀⠀⠀⠘⠁⠀⡿⠞⠀⠀⠹⠞⠃⠀⢀⠄⣀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠉⡦⠀⠀⢹⡆⠀⠀⣶⠄⠀⠀⠀⠕⠀⠀⢰⢀⡋⠀⠻⣷⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⠄⠁⠂⠀⠀⣿⠀⠠⢿⣦⠀⠀⠠⠀⣆⢰⡼⠊⣀⠔⠘⢧⠞⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠸⠀⢀⠘⠒⠀⠀⠈⠀⠘⠃⠀⠀⠂⡄⠙⡌⠀⠊⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡴⠂⠀⠀⡇⡼⠁⠀⢀⢴⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠉⠀⢤⠀⠀⠸⠃⢀⡤⠟⠋
⠀⠀⠀⠀⠀⠤⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠏⠀⠀⠀⡀⠀⢀⠄⢀⣬⡆
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠠⠀⠠⠒⠀⡀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠄⠀⠀⡄⣠⠂⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡞⡧⠀⠀⠈⠀⣀⢤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡅⠄⠘⠀⠌⠁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠐⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠠⡄⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢻⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⣇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⠈⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠠⠄⠀⠀⠂⠀⠀⠂⠀⠀⠀⠉⠈⠙⠉⠉⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠄⠨⠀⠀⠀⠀⠤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⠀⠘⡦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⢤⣀⠜⠀⢰⠂⠁⢸⣆⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣄⠀⡏⠉⠁⠀⠀⠉⠈⠉⠀⢠⠒⠉⠋⠀⢠⡃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⡟⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠄⠀⢀⠃⠀⠀⠀⡉⠉⠓⠋⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢼⠀⠀⠒⠇⠀⠀⠀⠀⠀⠈⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠁⠀⡁⠉⠀⠀⠀⠂⠀⠀⠀⠀⠸⠀⠀⠎⠘⡀⠀⠀⠠⠀⠀⠀⡀⠂⠀⢀⠀⣹⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠉⠲⠇⠀⠀⠀⠀⠀⠀⠀⠐⠦⠤⠄⡀⠀⠠⣇⡀⠀⠀⠀⣠⠀⡄⠀⠀⢠⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠄⠀⠀⠀⠸⠀⠠⠀⠀⠒⢤⠠⡀⢈⠀⡄⠐⠂⠘⢂⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⠀⠀⠆⠘⠀⡀⠀⡀⢀⢀⠈⢀⠂⢀⠒⠀⠀⠀⠠⠀⠀⠀⠈⠊⡠⠀⢠⠀⠀⠀⢀⣈⣙⠆
⠀⠀⠀⠀⠀⠀⠈⢧⢀⡀⠀⠀⠀⠀⠀⢙⡋⠒⠁⠈⠁⡆⠀⠊⠀⠀⠹⡀⠀⡐⠀⠀⠀⠀⠀⠀⠁⡀⢸⣯⠵⠚⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠧⡄⠀⠀⠀⠀⣯⣄⠀⠀⠀⡠⠔⠀⠀⠀⠀⠴⡁⠰⠃⠀⠀⠐⠀⠀⠃⠁⠀⢨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⠀⢰⠶⠞⢺⡉⠀⠀⢱⠄⢀⠄⠀⠀⡠⢠⡀⠀⠀⠀⢄⠀⠀⢀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣧⠀⠀⠘⡂⠠⡂⡤⢈⢀⠀⠣⡀⢀⡤⠊⠐⠀⠀⠐⣏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣤⡀⠉⠂⢘⠀⠀⠀⣀⡴⠁⠀⢀⠐⠀⠀⣀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⣄⠀⠀⠉⠞⢀⠁⠑⡀⠀⠒⠀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠻⣄⢀⣠⡀⣑⠀⠁⣠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⡇⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⠄⢸⡛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⠙⢀⡵⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠀⣴⣀⣴⢂⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣴⠎⢹⢧⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⡀⠀⠠⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠈⡆
⠀⠀⢀⠀⠀⠓⠒⠢⡀⢑⣀⣇⠀⠀⠀⠀⠀⠀⠀⢠⢃⡿⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠀⠈⣅⠀⠑⡄⠆
⠀⠀⢀⠵⣴⠦⣀⡂⠘⠮⠿⣈⡿⣄⠀⡄⠀⠀⠀⠈⢸⣇⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢽⠢⢀⡿⡆
⠀⠀⠁⢄⠀⠂⠙⢛⣾⡀⡀⠈⠻⢉⢄⡈⠀⠀⠀⣧⢈⣿⠟⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡄⠸⠛⠸⠑
⠀⠀⠀⠀⠻⢦⣀⠀⠠⡟⠛⠀⡾⢉⠀⢈⠠⡀⠀⡿⣅⢃⣼⡿⠀⠀⠀⠰⣆⠈⠀⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠈⠑⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⣀⠈⠈⠉⠋⠀⠀⠀⠈⠀⡙⣦⡀⠂⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠘⠷⠁⠀⠀⢀
⠀⠀⠀⠀⢀⠀⢀⢀⠀⣠⠀⣴⠀⠀⠀⠀⠀⠀⠀⠠⠤⠷⠀⠤⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⠒⠒⠁
⠀⢀⠀⡀⠹⣄⡻⠰⡔⠃⠀⠞⡇⠀⠀⠀⠀⢀⡀⢀⡔⠀⠤⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠈⠠⡇⠰⠈⢇⢸⡃⠀⡀⢪⡅⠀⠀⠀⠘⠀⠀⠐⡇⢀⠀⠀⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠟⢀⠄
⠀⠀⢀⣇⠀⢠⠈⡯⠱⠞⢁⡯⠁⠀⠀⠀⡀⢥⠀⠀⡙⠀⠤⢤⣤⣴⡴⠏⡜⠀⠀⠀⠀⠄⣷⠞⢉⠧
⠀⠀⠈⢻⢀⠘⡙⠓⠀⣼⣿⠃⠀⠀⠀⠀⠳⡨⢠⠞⠁⢠⠖⠋⠁⠀⠀⢰⡀⠀⠀⢀⣴⠞⠯⠀⣄⡀
⠀⠀⠀⠸⡏⣼⡡⠌⢀⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⣉⣤⣾⣀⣆⡀⠐⣴⠟⠀⠀⢠⠊⠁⠀⠀⠀⠈⠈⣿
⠀⠀⠀⠀⡏⡬⠩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠈⠉⠛⠛⠚⠋⠁⠀⠀⠀⠂⠀⠀⠀⢠⠁⠹⣶⡃⡀
⠀⠀⠀⠀⠳⢣⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠂⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠡⡋⡁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡐⠀⣌⠏⢰⠿⣭⡠⠆⠀⠀⠀⠈⢡⣀⣄⡐⠀⠤⡀⠀⠀⠀⠀⠀⡀⠰⢁⡆⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡻⣆⠠⣔⠰⢖⡼⢐⢒⡀⠀⠀⠀⢀⡈⠙⢻⢢⣤⡠⠀⠀⠀⠀⠨⠆⠀⢸⡦
⠀⠀⠀⢀⡃⠀⢤⣞⡈⣩⡏⣁⣖⠒⠒⠊⢠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⣿⠛⠤⡀⠀⠀⠀⣷⢀⡔⢀⡇
⠀⠀⠀⠀⠂⠀⠀⠃⠀⠀⢂⣀⠀⠐⠒⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠀⠀⠀⠀⠀⠀⠀⠛⢊⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢁⣀⠹⢿⣒⠖⣒⢦⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡖⠒⢿⣿⠾⠋⠉⢳⣤⣿⣏⢁⣉⠋⠉⣉⣉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠉⣹⡇⠀⠀⠀⠀⠀⠐⡏⠁⠀⠈⠙⠉⠙⠛⠛⠉⠻⡛⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠽⠾⠁⠀⠀⡠⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡁⣳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡇⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡁⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⣶⠫⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢡⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡄⢀⣴⢋⠄⠀⢰⠒⠋⠁⠀⡽⠀⢠⠁⠀⠀⠀⠀⠀⣀⣴⡆⢸⡁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⡼⢁⡎⠀⠀⢸⡇⠀⠀⠀⠁⡠⠃⠀⠀⠀⣰⡾⠛⠉⠁⠀⠈⡛⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣄⡇⣼⠀⠀⠀⣸⡇⠀⠀⢀⡜⠁⢀⣀⣀⣠⡏⠀⠀⠀⠀⠀⢠⣿⠂⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⡇⡏⠀⢯⠈⠉⠀⠀⣠⠎⠀⠀⠁⠀⠀⢈⣀⡀⠀⠀⢀⣠⣿⠟⠋⣹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠇⠷⣠⠎⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⢠⠽⠛⠶⠟⠋⠀⠀⠀⣏⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⠀⣸⠀⢀⣴⠟⠀⢀⣤⠴⢶⣤⣄⡀⠀⣸⠆⠀⠀⠀⠀⠀⡖⠛⣉⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠠⠟⠀⠞⠏⠀⠀⠀⠀⢀⡾⠀⠀⠉⠛⠁⠀⡀⠀⠀⠀⠳⣽⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣤⣀⠀⠀⣀⡀⣠⣤⠖⠚⣛⣀⣀⣀⣠⣴⠞⠋⠀⠀⠀⢀⣸⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⣷⣈⣩⡽⠿⠛⠛⠓⠛⠿⠿⠿⠿⠻⠟⠛⠲⢦⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠚⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠻⣇⡤⠞⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠁⠀⠀⣿⠁⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⢠⡏⠀⡀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⣠⡾⠛⠋⠀⠀⠀⢸⡀⠀⠀⠀⢀⣤⠶⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⣠⣾⠁⠀⢀⡶⠛⠁⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠀⠀⢹⣶⠶⠚⠃⠀⢠⡾⠋⠻⢷⣶⠋⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣰⠋⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣾⡄⠀⠀⠀⢸⡇⠀⠀⢀⣮⡤⠴⠄⠀⢰⠟⠀⣠⠀⣼⠃⠀⠀⠀⠀⠀⣾⣿⣦⡀⠀⣠⡶⠛⠁⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⡏⢳⣀⣤⡀⠈⣧⠀⢀⣞⣁⣀⡀⠀⢠⡟⢰⢋⣿⢾⠃⠀⠀⠀⠀⠀⣤⣿⠿⠿⠟⠉⠁⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⠃⢰⠀⠉⠈⠀⠀⠿⣦⣻⠀⢸⠋⠉⠁⠀⢀⣼⠁⣾⠟⠁⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⢸⡿⠀⠘⠷⠶⢶⡄⠀⠀⠀⠹⣦⢸⣷⠞⠁⣠⣾⣿⢠⡟⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢸⠇⣿
⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⠀⠙⣦⡀⣶⡄⠿⠸⠃⠀⣰⣿⣿⣿⣼⡇⠀⠀⠀⠀⢀⡶⠛⠇⠀⠀⠀⠀⠀⢰⡾⠔⠀⠀⠈⣴⣿⣤⣄
⠀⠀⠀⠀⠀⠀⣿⠉⠀⠀⠀⠀⠀⠀⠈⢿⡏⠀⢀⣠⡶⠟⢉⡴⠋⢉⡇⠙⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⡿⠉⢹⡇
⠀⠀⠀⠀⠀⠈⣿⡁⠀⠀⠀⠀⢦⣄⡄⠸⣿⣶⠟⢁⣠⣾⣿⠃⣾⠋⠁⠀⠀⠀⢠⡟⠀⠀⠀⢀⠀⢀⣀⣿⣯⣤⠀⠄⠀⠀⣸⣡⣴⠿⣧⡴⠶⠶⠶⣦⣤
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⡀⠹⣷⠀⢻⣷⣴⡿⢿⣿⣿⣠⠿⠀⠀⣤⣤⣤⣾⠃⠀⠀⠀⢸⠀⣼⠟⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⡟
⠀⠀⠀⠀⠀⠀⣾⡿⡄⠀⠀⠀⠈⢳⣜⡇⢸⣿⠋⠀⣼⢻⣉⡿⠂⠀⡼⠁⢠⠀⠈⠀⠀⢀⠀⣸⡿⠥⠤⠄⠀⠀⠀⠀⣰⠃⠀⢀⠀⠀⡐⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠱⣦⣝⣿⠸⣿⠀⢰⢏⣿⣇⠀⢀⣰⡇⢀⣸⡇⠀⢀⠀⢸⡾⠇⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠂⢀⡴⠛⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠈⢦⡘⢿⣿⠀⣿⣠⡏⡜⢸⡟⠀⣾⠛⠃⣼⠏⠁⠀⢸⣦⠎⠀⠀⠀⠀⢀⣴⠿⢶⠟⠁⠀⠀⣸⡴⠋⠀⠀⠀⠀⣀⠀⢀⣿
⠀⠀⠀⢠⡾⠋⣿⡀⠀⠀⠲⣄⠈⢿⣌⣿⢰⣿⠋⡐⠀⡘⠁⢀⣿⠀⠀⡇⠀⠀⣶⣞⣉⠀⠐⠀⠀⢀⡾⠁⠀⠀⠀⢠⣠⡾⠋⠀⠀⠀⠀⠀⠀⠈⢻⣾⠋
⠀⠀⣰⠏⠀⡜⣿⠗⠀⠀⠀⠹⣧⠘⣿⡟⢸⡟⠀⠀⠠⠁⢀⣿⠁⠀⣤⡇⠀⠀⠈⠉⠉⠉⢀⣠⣴⠟⠀⢰⣀⣤⠶⠛⠳⠄⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⢀⡟⠀⢸⠇⠹⣆⠀⠀⠀⢰⡹⣷⣽⠃⣼⠇⠀⠀⠀⠀⣸⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠋⣹⣡⡴⠖⠻⠯⠄⢀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣰⡎
⠀⢸⠁⢀⠏⠀⠀⠙⣧⡀⠀⠀⢿⣿⠏⢠⡿⠀⠀⠀⠀⢀⡏⠀⣲⠶⠀⠀⠀⠀⢰⣶⣫⠵⠚⠻⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠟⢳⡆
⢀⡇⢀⡏⢀⡀⠀⣼⠋⢻⠀⠀⢸⠟⠀⣼⠇⣠⠂⠀⣠⠟⢁⠞⠁⠀⠀⣠⣶⣄⣸⡟⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣟⣠⡴⠟⠁
⣼⠀⢸⠁⢸⣅⢠⡏⠀⢸⡇⠀⠈⠀⢠⣿⡴⠃⢀⣾⠋⠀⠀⠀⠀⢀⡾⠋⠀⠉⠉⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠛⠛⠉⠉⠁
⣿⠀⠸⠀⠀⠙⣿⠁⠀⣼⠂⡠⠀⢀⣾⣉⣠⡶⠟⠁⠀⠀⠀⠀⢠⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣤⣤⣤⣴⡾⠟⠋
⣿⡄⠀⠀⢀⡼⠟⠀⠀⣿⡾⠁⢀⣼⣿⠟⠋⠀⠀⠀⠀⠀⠀⣰⣿⡿⢿⡿⠿⠶⣶⣦⣤⣤⣤⣤⣤⣤⣶⣿⠟
⢸⣧⠀⠀⠸⣧⡄⠀⢸⣿⡷⠾⠛⠉⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⢉⣴⣿⣀⣶⡶⠿⠃⠀⠉⠉⠉⠉⠁
⠀⢿⣆⠀⢰⡟⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣦⣼⣇⣠⣴⠿⠋⠉⠉⠙⠁
⠀⠈⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣠⡴⠒⠒⠚⠋⠀⢀⣽⣿⠟⠁
⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠟⠋
⠀⠀⠙⠿⣷⣤⣤⣤⣴⣾⡷⣶⡶⠿⠿⠟⠛⠉
⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠛⣷⡀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⢀⣴⠟⠋⣉⡉⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣻⣧⣀⣠⡾⠋⠀⠈⢻⡄⠀⠀⢸⡇⠠⡏⠉⡟⠀⠀⠘⠛⠶⠶⠞⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠧⠀⠊⠀⠈⠉⠉⠃⠀⠀⠀⢸⡿⣦⣤⠾⢃⣀⡿⠸⣄⠀⢀⣀⡀⠀⣀⣄⡀⠀⠸⣇⣠⣀⣀⣠⡴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢸⡅⣰⠋⠉⠀⢠⡇⠀⢸⠏⣇⣠⡏⢠⡷⢁⡞⠋⠀⠉⠉⠁⠀⢹⡇⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣦⠀⠀⠀⠀⠀⠤⢦⠀⣰⠃⠀⠀⢸⣀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠈⠁⠀⠸⢦⡿⢀⠀⢰⠚⢶⠂⠀⡈⠷⠶⠛⠁⠀⠈⠛⣦
⠀⠀⠀⠀⠀⠀⠀⣿⠁⢀⡾⠁⠀⠘⠢⣝⡄⠁⠀⠘⠶⢤⡄⠀⠉⠙⢦⠀⠐⢄⡀⢀⢀⡄⢀⡴⠶⠶⠶⢃⡞⠛⠋⠀⠙⠲⡾⠃⢠⣀⠀⢰⡟⠛⢲⣏
⠀⠀⠀⠀⣀⣤⠾⠃⠀⠘⢦⡀⠀⠀⠀⠈⢻⡀⠀⠠⠀⣠⠏⠀⠀⠀⠈⠛⠲⢶⣍⠲⣮⠀⡞⠀⣠⠶⠶⠛⠀⠀⠀⠀⢠⠎⠀⣰⠋⠹⠤⠼⠃⠀⠀⠙⣷
⠀⠀⠀⣼⠋⠀⠀⠀⣤⣦⣴⠃⠀⠀⠀⠀⠸⡇⣠⢃⠀⠙⠶⣎⠀⣀⣤⣄⠀⠀⠙⡇⢁⣠⠇⠀⣏⠀⠀⠀⠀⠀⠀⠀⠳⠤⠴⠋⠀⠀⠀⢀⣠⠾⠶⠶⠋
⠀⠀⠀⠻⣦⠀⠀⠀⠈⢧⢨⠀⠀⠀⢢⡳⣄⣧⢡⡎⠀⠀⢰⠇⠀⣿⠀⠈⣷⠀⢸⠁⡿⢁⠀⢀⣽⠀⠀⡀⠀⠀⢠⠀⠀⢀⠄⠀⠀⠀⠀⠈⣿⣀⣀⡀
⠀⠀⠀⢀⡼⠇⠀⠠⠴⡞⠈⣏⠀⠀⠀⠑⢌⠻⡆⢃⠀⠀⠘⠧⠴⠋⠀⠘⣧⠀⠈⠛⠧⡜⠉⠉⠁⠀⠀⢣⠀⠀⣸⠃⡴⠃⠀⠀⠀⠀⠀⠀⠈⢉⣩⡗
⠀⠀⠀⣿⠁⠀⣰⣄⠀⡟⠀⠙⢦⣤⡀⠀⠀⠑⢿⡌⠄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢶⣀⡟⠀⠀⠀⠀⠀⢸⡇⡼⢃⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⡍
⠀⠀⠀⠻⣄⣰⡟⠉⠻⣇⣀⠀⠀⢀⡧⠀⠀⠀⠀⢳⡈⠀⠀⠀⠀⢀⠃⠀⠀⠀⡴⠋⡭⠀⠀⠀⠀⠀⠀⢸⡿⠁⡾⠀⠀⠀⠀⡀⠀⠀⠠⣀⡀⠀⣿
⠀⠀⠀⠀⠈⠉⠀⠀⠀⢩⡟⠀⠀⣿⡀⠀⠠⡀⠀⠀⢧⠀⠀⠀⣠⢎⠀⠀⠀⠠⡇⠀⠳⡄⠀⠠⠀⠀⠀⢘⠇⣸⠃⠀⢀⣴⠋⠀⠀⠀⢸⡟⠳⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡄⠀⠘⠦⢦⡄⠈⢓⠦⣌⡄⢢⡜⣡⠃⠀⠀⠀⢠⡟⠀⣤⠃⠀⠀⡇⠀⠀⢸⠀⡏⠀⢠⡿⢹⣆⣀⣸⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⡷⠀⠀⠑⢌⡃⠈⠰⠁⠀⠀⠀⠀⣿⠀⡞⠀⠀⠀⢰⡃⠀⠀⡌⢀⠇⠀⢸⣿⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠸⡇⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⢆⠀⠀⠀⢿⡀⢠⠁⣸⡀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⠀⢻⠀⠀⠀⠀⢠⠀⠀⠀⠀⣠⢀⡞⠁⠀⢀⡎⠀⠀⠀⠘⢰⡆⢠⡯⠂⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡗⠀⠈⢦⣄⡀⠠⣌⠀⠀⢀⣰⠃⠘⣧⠀⢸⡁⠀⠀⠀⠀⢠⡎⢠⠏⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠈⢳⠀⠈⡃⠀⠈⣇⠀⠀⢘⡇⢀⡷⠀⢰⠀⢀⠎⡰⠃⠀⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣄⠀⢧⡀⠘⢆⠀⢻⠀⠀⠈⢧⡞⠁⠀⢸⡴⠃⡰⠁⣴⠷⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠙⢦⠈⢂⠈⡆⠀⠀⣠⠃⢀⡼⠋⣠⠞⠁⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡘⣇⠀⠣⠀⠀⠸⡇⠀⠞⠀⣸⣥⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠖⠛⠓⠀⠀⠓⠶⢤⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣶⣶⠶⠶⣤⣤⣀⣀⣨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉


⣶⣶⣶⣶⣶⡾⢷⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⣾⢷⣶⡿⠿⠿⠶⣶⠶⠾⠷⡶⢿⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠈⢉⡏⡈⠟⠻⠶⠶⣚⡍⢙⠷⢌⠘⢆⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⣶⠄⠀⠋⠀⣠⡶⠠⠯⠀⠐⠀⢁⣁⠀⢀⣶⣶⢦⡉⠉⢦
⠀⠀⠀⠀⠀⠠⣶⣤⣴⡟⠀⠀⠀⠠⡀⢠⢤⠚⠀⢄⠀⠀⠀⠨⠶⠫⡀⠅⠈⢧⡀⢸⠃⡄
⠀⠀⠀⡠⠒⠁⠨⣿⣿⠢⡀⠀⢇⠀⠀⠘⣎⠃⠀⠉⠲⠉⠀⠀⡤⠛⣁⡴⠥⢼⣷⡟⠀⠸⢆⠐⠀⡤
⠀⠀⠀⠀⠀⠠⣠⣿⢿⠀⠰⡀⢹⣄⠀⠀⠈⢢⠀⠀⠤⠠⢄⣾⣿⡀⡀⢞⣨⠿⢋⣀⣴⣷⣿⣾⣀⡟⣶
⠀⠀⡄⡄⠀⢀⣽⠃⠈⡆⠀⢱⡄⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠙⡆⠉⢉⣿⣿⣿⣿⣿⣧⢻⡟⠻⣶⡂⡞
⠀⠀⡇⠀⠀⢺⣿⠁⠀⠹⢶⡄⠀⠀⠀⠀⠀⡆⠀⠀⠀⡀⠀⠀⡘⣆⣼⢿⣿⣿⣳⡿⠋⠀⣽⣶⣄⠉⠂
⠀⠀⠃⠀⠀⠰⢿⡄⠀⠀⢼⡁⠀⠀⢀⠀⠈⠴⠃⣠⠔⢋⣄⣸⣅⢿⣿⣿⠟⠋⠁⠀⢀⣾⣿⡞⠟⠁⠀⠃
⠀⠀⠀⢆⠀⣠⠎⠉⣶⠀⠲⠷⡶⣤⣌⣅⣘⣗⣸⣟⣱⣯⣿⣷⠶⠛⠋⠙⠲⠾⣿⣶⣿⣏⠏⠀⠈⠀⣈⣼
⠀⠀⠀⠀⠘⠁⠀⠀⠐⢳⣦⣤⣀⣀⡛⠘⢘⡿⣷⠞⠟⠉⠉⢱⣄⠀⠀⠀⠀⠀⢌⣼⣿⡥⠽⠢⠤⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠛⠛⣿⣿⡖⠾⠛⠋⠀⠀⠀⠃⡀⠙⠒⠌⣁⣤⣶⡿⠋⠁⠀⠀⢠⠤⢂⡤⠂
⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠉⢹⢿⣧⣀⣀⣠⢀⣈⣀⣀⣀⡀⢤⣾⠛⠙⠋⠀⠀⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣀⡤⠯⠿⠛⠁⠀⠀⠉⠁⠀⠀⠀⠈⠁⠀⠐⠒⠂⠀⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢀⠔⠋⠀⠀⠐⡀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⢻⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠈⡀⠀⠀⠀⢀⣹⡩⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⣿⢿⣧⢻⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠑⠚⠁⠒⠢⠀⠀⠀⡀⠘⣹⡅⣺⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⣛⣛⣿⣿⣿⣘⣿⣿⣿⣿⡿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠢⢴⡱⣶⡙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣟⣿⣿⣿⣼⣿⣯⣟⣿⣿⣜⢿⣿⣷⡀
⠀⠀⠀⠀⠀⡖⠈⠀⢅⠀⠀⠀⢀⠀⠀⠀⠄⡀⠀⠀⠀⠉⢎⢻⡌⢧⡀⠀⠀⠀⠀⠀⣴⣺⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⡻⣿⣷⡄
⠀⠀⠀⠀⠀⡇⠀⠀⠈⡆⠀⠀⠈⠛⢆⡀⠀⠈⢄⠀⠀⠙⠒⠷⡝⢦⢹⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⡄
⠀⠀⠀⠀⠀⠣⣄⡤⠴⠶⣄⡀⠀⠓⠀⢭⡢⣄⠀⠈⠙⣏⢳⣄⠛⣎⢻⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⡇
⠀⠀⠀⠀⢀⠊⠁⠀⠀⢤⣀⣯⣍⣳⣦⡀⠈⠻⢷⣲⢄⡈⠲⣌⡙⠻⠌⣧⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⢿⣿⣿⣿⣿⣿⡇⢳
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠐⠋⠙⠿⠟⠷⠿⠶⣮⣵⣦⣬⣉⡛⠒⠀⣹⡄⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⣉⡛⠛⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠉⠁⠒⠤⠤⠒⠒⠢⠔⠒⠉⠓⠒⠶⠴⠴⠶⠛⠋⠑⠲⠚⠁⠀⠀⠀⠀⠉⠉⠙⠿⠿⠟⠛⠻⠷⠛⠋⠙⠛⠻⠿⠿⠿⠛⠋⠙⠒⠛⠃

⠀⠀⠀⠀⠀⣸⣟⠻⣦⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⣰⣿⣿⠀⣿⣿⠿⣿⣶⣾⣿⠖⠛⣹⣿⣧⣠⣶⣾⣿⣿⣶⣶⣶⣿⣷⣆
⠀⠀⠀⠀⣟⢹⣿⠀⣿⣿⣧⣤⣼⣿⣿⠖⠀⢿⡟⠿⠟⣩⣷⣾⣿⣿⣿⣛⣿⣦⡘⢿⣷⣶⣾⣿⣷⣶⡀
⠀⠀⠀⠀⢻⣌⣿⠀⣿⡎⠻⣿⡻⣿⣿⣿⣿⡿⢿⣶⣾⣿⠟⠑⢾⠋⠈⣿⡿⣿⣷⡀⠈⣩⣿⣿⠿⠿⣿⣿⠿⠿⣶⡄
⠀⠀⠀⢠⣾⣿⡟⠀⣿⡇⠀⣿⣧⢹⣏⡀⣀⢄⡀⠉⠈⠀⠀⠀⠀⠀⢸⣿⡾⠋⢹⡿⣶⡳⣿⡇⠀⣠⣶⣶⣄⠀⠘⣿⣿⡿⣶⡄
⠀⠀⠀⣼⣿⣿⡇⢸⣿⣿⡀⠸⣿⡎⣿⣿⣿⣟⣴⢃⠄⣠⠀⠀⠀⠀⠘⣿⣤⣶⡞⠿⠿⠿⠻⠿⠟⠻⠋⠀⢻⣿⣶⣄⠀⠀⢿⣿⣿⣷⡆
⠀⠀⣾⣿⣿⣿⠁⢸⣿⣿⡇⠀⠙⣿⣽⣿⣿⣿⣿⣿⣾⠋⠀⠀⠀⡸⡄⠀⠘⠋⠀⣀⠀⠀⠀⠀⠀⠀⢀⠄⣀⣴⣾⣿⣷⠀⣲⣿⣿⣿⠃
⠀⠀⣿⣿⣿⣿⠀⣾⣿⣿⣷⢀⠀⠈⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢱⣻⡄⠀⠛⠫⠗⠀⡴⠁⢠⠆⣠⠏⢐⣿⣿⣿⠋⡿⢰⣿⠋⢠⣿⠀⢀⣴⣶⣦⡀
⠀⣿⣿⣿⣿⡇⢠⣿⣿⣿⣿⣾⡇⠀⢸⣿⣿⣿⣿⣿⡟⠀⠀⠀⢳⣿⣷⡹⣆⠀⠀⠀⡼⠃⢠⣿⡞⠁⠀⣻⣿⡟⠀⢸⡇⢠⡟⠀⢸⣿⣶⡿⠋⠀⠻⠟⣿
⠀⢻⣿⣿⣿⠃⢸⣿⡿⢹⣿⣿⠃⢀⣼⣿⣧⢻⣿⣿⢟⠀⠀⠀⠈⢿⣿⣿⣾⠇⠀⡼⠁⠀⣼⣏⡠⠀⢘⣿⣿⠃⠀⢸⡇⠈⢷⣶⡿⠛⠉⠀⠀⡰⠀⢰⣿
⠀⣸⣿⣿⣿⠀⣿⠟⣰⣿⣿⠏⠀⣾⣿⣿⣿⢸⣿⣷⣏⠀⠀⢦⣆⣸⣿⣿⡟⠀⣼⠃⠀⣰⠟⠋⠀⠀⢨⣿⣿⣄⠀⢸⡇⠀⠀⠀⠀⣰⠃⢀⣼⠃⠀⢸⣿
⣿⣿⣿⣿⡿⠀⠏⣼⣿⣿⠏⠀⠀⣸⣿⣿⡟⣼⣿⣿⠏⠀⠀⠘⣿⣿⣿⡿⠀⢰⡟⠀⢠⡏⠀⠀⠀⠀⠠⢿⣿⣿⣇⢸⡇⠀⣷⠀⣰⣏⣴⠟⠁⠀⣀⣀⣼
⠸⣿⣟⣿⡇⠀⢸⣿⣿⠋⢀⠀⠀⢿⣿⣿⢣⣿⣿⣿⠃⠀⠀⠀⠘⣿⣿⠃⠀⣿⠃⣠⡟⣀⣤⠖⠀⠀⢠⣿⣿⡟⠁⣼⠇⢀⣿⣰⡿⠋⠁⠀⠀⣴⣿⣿⠟
⣰⣿⣿⣿⡇⠀⣿⣿⢃⣴⡏⠀⠀⢈⣿⣿⢸⣿⣿⠃⢀⠀⢦⢠⡀⠘⠟⠀⣸⣿⣴⡿⠛⠉⠀⠀⠀⠀⠠⢿⣿⡇⢀⣿⠀⠸⣿⠟⠀⠀⡠⠀⠸⣿⡟⠁
⣿⣿⣿⣿⡇⠀⣿⣧⣾⣿⠁⠀⣾⣿⣿⣿⡇⣿⣷⠀⢸⡀⡘⢷⣷⡀⠀⢠⣿⡿⠋⠀⠀⠀⣠⠖⠀⠀⠀⠛⣿⡇⢸⡇⠀⣼⡏⠀⣠⡞⠁⠀⠀⢹⣧
⠸⣿⣿⣿⡇⢠⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⢸⣿⡷⠀⠀⢻⣿⣿⣿⡇⠀⣼⡿⢁⣤⣶⣶⣟⣥⠔⠀⠀⠀⠀⠸⣿⣾⠁⣼⣿⣴⠾⠋⠀⠀⠀⣴⡿⠋
⠀⢿⣿⣿⠇⢸⣿⣿⣿⣿⠀⠀⠈⢿⣿⡟⣿⣿⡇⠀⠐⣄⢻⣿⣿⠇⢀⣿⡿⠋⠁⠀⠀⠀⢀⣤⠎⠀⠀⠀⢀⣿⠏⢰⣿⠟⠁⣀⣤⠄⠀⢀⣿
⠀⠸⣿⣿⠀⣼⣿⡿⢡⣿⠀⠀⣾⣿⣿⡇⣿⡏⠀⠀⠀⢸⣾⣿⣿⠀⢸⡿⠀⠀⠀⣠⣤⠞⠋⢀⣠⡔⠀⢠⣾⠏⠀⣼⣿⣶⠟⠋⢀⣴⣾⡿⠏
⠀⠀⢿⣿⠀⣿⡿⠁⢸⡇⠀⠀⢹⣿⣿⡇⣽⡃⠀⠀⡆⠘⣿⣿⡟⠀⣸⡇⠀⢀⣾⣯⣤⣶⠾⠛⠁⠀⢀⣾⡟⠀⢠⣿⡏⠀⠀⠀⢸⣿
⠀⠀⢸⡿⠀⡿⠁⡀⢸⡇⠀⠀⣾⣿⣿⡇⢺⠁⠀⠀⢹⣄⣿⣿⡇⠀⣿⠀⢀⡾⠟⠉⠀⠀⡀⠀⠀⢀⣼⠟⢰⠀⣼⡿⠀⠀⣤⡾⠟⠃
⠀⠀⢸⡇⠀⠀⣰⡇⢸⡇⠀⣆⠘⢿⣿⣧⠈⢀⠀⡀⠈⢻⣿⣿⠃⠀⣿⣶⠋⠀⣀⣤⠶⠋⠀⣠⣶⡿⠋⣠⠟⢠⣿⠃⢠⡾⠋
⠀⠀⢸⡇⠀⣼⣿⡇⢸⡇⠀⣿⢠⠈⣿⣿⠀⢾⡆⠹⣆⠀⠙⡿⠀⢠⡿⠁⢠⡾⠉⠀⠀⠀⢰⣿⠋⠀⢰⡟⢠⣿⠃⢠⣿⠃
⠀⠀⢸⣿⠀⣿⣿⡇⢸⡇⠀⣿⢸⠀⣿⡇⠀⢸⣿⣴⣿⣧⠀⠀⠀⠸⠁⣰⠟⠀⠀⠀⠀⠀⣿⡇⠀⢀⣿⣡⣿⡁⢀⣾⠏
⠀⠀⠈⣿⠀⣿⣿⡇⢸⣷⠀⢸⢼⠀⣿⡇⠀⠀⢻⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⢠⣿⠃⠀⣸⣿⣿⠟⢡⡾⠃
⠀⠀⠀⣿⡄⢻⣿⠇⠘⣿⡀⢸⣸⡆⢻⣷⣳⡀⠈⣿⣿⣿⡟⠀⢠⠀⣀⣤⡴⠋⠀⢀⣴⡿⠃⠀⢀⣿⡿⢃⣴⡟⠁
⠀⠀⠀⢸⡇⢸⣿⣿⡀⣿⡇⠸⣿⣿⡈⢿⣿⣿⣦⠸⣿⣿⡇⠀⣾⡾⢋⣁⡤⠀⠀⣾⡏⠀⠀⠀⣸⣿⣷⡿⠋
⠀⠀⠀⠈⣷⠈⣿⣿⡇⢿⡇⠀⣿⣿⣧⠈⣿⣿⣿⣷⣿⣿⠀⢠⣿⠟⠋⠁⢀⠄⢸⡿⠀⢠⡇⢀⣿⡟⠁
⠀⠀⠀⠀⢻⡄⢿⣿⣧⢸⣇⠀⢹⣿⣿⠀⢻⣿⣿⣿⣿⠇⠀⣾⠃⠀⢀⡴⠋⢠⣿⠃⢀⡾⠀⣼⡟
⠀⠀⠀⠀⠘⣷⠘⣿⣿⡆⣿⡀⠸⣿⣿⠀⢸⣿⣿⣿⡟⠀⣸⡏⠀⣠⡟⠀⢀⣿⠏⠀⣼⡇⣰⡟
⠀⠀⠀⠀⠀⠹⣧⢻⣿⣷⢹⣧⠀⢿⣿⠀⣾⣿⣿⡟⠀⣰⣿⡇⣰⡟⠀⢠⣾⠋⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠹⣧⣿⣿⣧⣿⣆⠸⣿⣇⣿⣿⠟⢀⣼⣿⣿⡿⠋⢀⣴⠟⢁⡞⢀⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣆⣿⣿⡿⢋⣰⣿⣿⠿⠋⣀⡴⠟⠁⣠⡿⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣭⣭⣬⣁⡰⠿⠿⢋⡥⠴⠛⠉⢀⣴⣾⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠿⠷⣤⣀⣠⣤⣴⡾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣴⣶⣿⠿⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣯⣿⡶⢶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣶⣺⣾⢉⣷⣾⣿⣿⣿⣟⣿⣿⡟⣆⢀⡤⣄⣀⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣄⢸⣿⣿⣿⣿⣿⢿⣽⣿⣿⣿⣿⣿⡿⠛⢁⠆⣿⣿⡏⣽⣷⠏⢑⣶⣴⡆
⠀⠀⠀⣀⣀⣰⣿⣿⣃⢠⡍⡇⣸⣿⣟⠯⠁⡈⠋⠛⠻⣜⢻⠏⠀⠀⢸⢰⣿⣿⣇⣿⣿⣆⣼⣿⡿⠁
⠀⠀⣸⣿⣿⣿⣿⣿⡿⢠⣾⣹⣾⣿⡅⢠⡼⠁⠀⢢⡀⠻⣿⠀⠀⡀⠠⢼⣿⣿⣿⣿⠿⢋⠻⠟⡽
⠀⠀⠘⢿⣽⢡⣿⡿⠃⣠⣹⣽⠿⣃⠀⠀⠀⠀⠂⡭⢰⠀⡇⠀⠀⠈⢄⢸⣿⣟⣿⡇⣈⠄⢧⡼⠁
⠀⠀⠀⠀⢳⣿⣿⡇⢀⠐⣼⡿⡏⠀⠖⠁⠀⢠⡃⢡⠗⠸⡄⠀⠀⠀⡘⢸⣿⣿⢿⣇⣍⣸⡏
⠀⠀⠀⠀⢈⣿⣿⡇⢰⡞⣿⣿⣧⢐⠃⠀⡙⢢⠉⠘⣴⠻⣦⠀⠀⠀⠃⢸⣿⣧⣾⣿⡟⢁⣧
⠀⠀⠀⠀⢸⣿⣿⡇⠨⠇⣿⣿⡯⣿⠀⠀⠐⢌⠁⠀⠿⢀⣯⡀⠀⠈⡆⢻⣿⢹⣿⣿⠁⣼⠋
⠀⠀⠀⠀⢾⣿⣿⣇⠀⠐⢸⣿⣿⢉⠀⢆⠀⣀⠁⠈⣴⠻⣿⡟⠀⣀⠀⣿⡿⣰⡿⢃⡼⠋
⠀⠀⠀⠀⠸⣿⣿⣿⣷⣆⠀⢻⣿⡽⡀⠠⡑⠄⠇⢀⠀⢠⣿⣷⣄⣿⢠⣿⠿⣟⡴⠋⠁
⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⠈⡎⣿⣿⡄⠀⠐⣰⠄⣸⠀⢸⣿⡟⢨⡏⣼⣿⡇⠠⣁⣦⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠣⠀⠸⣿⣻⡄⢠⣺⠁⢊⠀⣾⣿⡿⠿⣰⣿⣿⡅⠀⠘⡅
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣁⠀⢀⠀⠙⣏⡃⢠⣿⠐⠧⢠⣿⡏⠀⣰⣿⣿⣿⠇⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⣿⣥⠀⠈⣥⣿⡃⢸⣟⣾⠋⢀⣼⣿⣿⠟⡠⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣿⣶⣦⣹⣿⣿⣏⣯⣿⣷⣏⠟⡛⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠿⠿⣿⣿⣿⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠶⠶⣤⣀⣀⣀⡤⠶⠚⠋⠉⠉⢛⠉⠓⠲⠦⢤⣴⣏⣉⡉⠉⠉⠉⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⣀⣤⠖⠋⠉⠉⠉⠛⠒⢦⡄⠀⢠⢠⠀⠀⠀⠀⢀⣀⣠⠞⠛⠳⢦⡀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⣀⣤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡆⠘⠀⣠⠞⠛⠉⠉⠁⠀⠀⠀⠀⠙⠳⣄⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⢸⠀⢀⡾⠛⠛⠙⠛⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⢸⠃⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠢⢄
⠀⠀⠀⠀⠀⡠⠃⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀⠀⠀⠀⠀⢸⠀⠘⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⢀⣀⣴⣾⣧⠀⠀⠀⠀⠁⠒⣄
⠀⠀⠀⣠⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⢸⠀⠀⠀⠀⠀⣠⣶⠾⠟⠛⠛⠛⠛⠛⠛⠛⠁⠈⠉⢳⡄⠀⠀⠀⠀⣸
⠀⠀⠀⣇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⡟⠀⠀⠀⠀⢼⡟⠁⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⡒⠶⠦⠽⢦⣀⠀⠀⡇
⠀⠀⣤⠟⠒⠚⠃⠀⣀⣤⣴⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢻⣆⡇⠀⠀⠀⠀⢸⣇⣠⣴⣶⢟⠋⠉⠉⢉⣉⠉⠉⠉⠙⠓⠢⠀⠀⠉⠲⣇
⠀⢼⡁⠀⠀⠈⠁⠘⠋⠀⠀⠀⠀⠀⠈⠙⠻⢶⣦⣤⣄⣀⡀⠀⢻⡇⠀⣠⣠⣴⠿⠛⠋⡡⠀⠈⠉⠓⠢⢄⠀⠉⠀⠀⠀⠀⠀⠀⣀⠤⠀⠈⡧
⠀⠀⠉⣿⢩⠗⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⠿⠟⠂⡼⠛⠁⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⢀⡠⠖⠋⣨⠶⠒⠤⠤⠤⠞⠁⠀⣠⣾⠃
⠀⠀⠀⣿⢡⠃⠀⠀⠈⡢⢄⡀⠀⠀⠀⠀⣀⢤⠞⠉⠀⠀⠀⡈⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⡴⠞⠉⡴⡲⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⡱⠃
⠀⠀⠰⣿⢸⡄⠀⢠⡞⠀⠀⠙⠢⣄⠁⠐⠁⠜⠀⠀⠀⠀⢠⠇⣀⣤⠶⠛⠛⠉⠙⠛⠛⠋⠁⠀⠀⠀⠃⠀⠀⠀⣀⣀⣤⣴⣒⠐⠍⣰⠃
⠀⠀⠀⢿⡀⢧⠀⠘⣇⠀⠀⢀⡞⠁⢹⠀⠀⠀⠀⠀⠀⣴⡿⠞⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣋⠽⠛⠉⠀⠀⢀⣤⣾⠃
⠀⠀⠀⠈⢷⡌⢧⢀⡿⠀⠀⢸⠇⠀⢸⠀⠀⣠⣴⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣣⠞⠁⠀⠀⡤⠒⠈⠈⠁⡆
⠀⠀⠀⠀⠀⠙⢦⠻⡁⠀⢀⣿⠀⠀⠈⣧⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⡸⡃⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠳⣌⠲⡞⠃⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠐⠀⠀⢀⡠⣊⠔⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢌⠳⣄⠀⣠⣿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⣠⢔⠅⠀⠀⠀⣀⢄⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⣙⢏⡀⠘⣄⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡴⢊⡵⠀⠀⣀⠠⠚⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠪⢤⡘⢆⠀⢅⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠆⠀⠰⠖⢉⣔⠼⠋⠁⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⠀⠉⠉⠐⠒⠢⢅⡂⠠⣀⠌⠀⣠⠶⢐⠉⣁⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠤⣀⣀⠀⠀⡨⠛⠀⠀⠠⠩⠈⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁


⣿⣿⠿⠻⡿⢻⠿⢿⠟⣿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶⣦⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣍⡛⠿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠘⠿⢶⣤⣝⠻⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠉⠻⣆⠙⢿⡙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣭⣟⡻⠶⣦⣄⣀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢻⣆⠀⠙⢷⣄
⠀⠀⠀⠀⢀⣼⡿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣭⣛⠿⣦⣄⡀⠀⠀⢧⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠛⠶⠦⣿⡆⠀⠀⠙⢷⣄
⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⢀⡤⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣍⠻⢶⣄⡀⠃⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠹⣦
⠀⠀⢠⣿⡇⠀⠀⠀⠀⠒⠁⠀⠀⠀⣀⡠⠤⠄⠒⠒⠒⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡉⠻⣦⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⢻⡇
⠀⢀⣿⡏⢿⡀⠀⠀⠀⠀⢀⠠⠔⠋⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣆⠈⠻⣦⡀⠀⢸⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⣿⠀⠀⢸⣿
⠀⣸⣿⡇⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠈⠻⣆⠈⠀⠀⠀⢀⡾⠀⠀⠀⠀⢰⡟⠋⠀⠀⢸⣿
⠀⣿⣿⣷⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠈⢷⡄⠀⠹⣧⠀⠀⠀⡜⠁⠀⠀⢀⣤⠟⠁⠀⠀⠀⢸⣿
⠀⣿⠆⣿⠀⠀⠀⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠸⣧⠀⠀⡇⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⢻⡆⠀⠀⠀⠈⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠠⠄⠶⠶⠶⠶⠒⠀⠄⠀⣀⣠⣀⠀⠀⠀⢿⡄⠀⢹⡆⠀⡇⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⣸⣿
⠀⢸⣇⢸⣧⠀⠀⠀⠀⠀⠙⢿⣿⣦⣀⠀⠀⠀⠀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠘⣿⠀⠀⣿⡀⢋⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⣿⡇
⠀⠘⣿⡀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⣧⠈⠀⠀⠸⣏⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⢻⣧⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠀⠤⠀⠀⣀⡀⠀⠀⢸⣷⠀⠀⠘⣧⠀⠀⠀⠈⢻⡆⠀⠀⠀⣼⠇
⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣄⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⢹⣇⠀⢀⣴⠞⠁⠀⠀⣼⠏
⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⢿⡄⠈⠻⣦⡀⠀⣸⡏
⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠈⠉⠻⢶⣶⠞⠶⢦⣤⣴⠞⠋⠛⠛⠶⣦⡹⣷⡀⠀⠈⣷⠀⠀⣸⠇⠀⢿⡄
⠀⠀⠀⠀⠀⠀⢻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠉⠉⠉⠙⠷⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⣿⣦⣤⣼⡷⠞⠋⣀⣠⣬⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠹⣿⡛⢶⣤⣀⣀⢀⣀⣀⣠⡴⠞⠋⠁⠀⠀⠀⢀⣤⠶⠞⠉⠻⣦⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣼⠋⢀⣠⣴⠞⠋⠉⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⢛⣿⣂⣠⣤⣄⠀⠀⠀⢸⣧⣴⠟⠋⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⣉⣨⣿⡀⠀⠀⣼⠏⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⣀⣠⣤⣴⠶⠾⠛⠛⠉⠉⠉⠈⠙⣛⢻⡇⠀⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⢀⣀⣠⣤⠶⠶⠛⠛⠛⠛⠿⠶⣶⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣦⣤⣤⣤⣤⣤⣤⣤⣤⡴⠶⠶⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⢀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀⠀⠀⠄⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠐⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠄⢰⠀⠀⠀⠀⠀⠈⠋⠉⡇⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠧⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀⠀⠀⠂⠀⢲⠞⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣠⠀⠀⠀⠀⠄⠀⠀⠀⢀⡂⠬⣄⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⠄⡐⠂⠀⠐⠺⠊⠀⠁⠀⠀⠀⠀⢄⡤⠂⠁⠀⠀⠈⠒⠔⠒⠋
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡅⠀⠤⠂⠀⡀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠤⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠐⠠⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠑⡄⠀⡀⠀⠈⠢⢄⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⡀⠀⠐⠂⠒⠠⢄⡀⠀⠀⠀⠀⠀⡀⢉⠋⠐⠊⠳⡀⢩⣦⠀⠀⠀⠉⠓⠲⠦⣄⣀⢀⠀⠀⠀⠀⣆⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⢢⠀⠀⠀⠀⠉⠁⠙⠒⠛⠉⠉⡹⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠡⠤⠤⠚
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠐⠀⠀⢀⣀⡀⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⣀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠉⠑⠂⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠈⠐⣆⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⣇
⠀⠀⠀⠀⠈⠓⢖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠤⠀⠊⠀⠀⡠⠂⠀⠀⠀⠀⢰⠀⠀⠱⣀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⣀⣀⠀⠔⠊⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠙⠤⡠⠀⠂⡤⠶⠬⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠈⠙⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⣀⣤⠴⠤⠞⠁⠀⠉⠉⠉⠙⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠄⠀⠹⣆⡾⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠀⠀⠀⡇⠀⠈⠀⠀⢀⣿⠤⠤⠖⠒⠦⠤⣤⡒⠁⠀⢀⣀⡴⠚⠋⠙⠳⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠁⢀⣠⠞⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡞⠉⠁⠀⠀⢀⠀⠀⠀⠀⠈⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠈⠀⠀⠀⣀⠤⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⠆⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠀⢀⠤⠒⠈⠁⠀⠀⠀⢸⡆⡀⠄⠊⠁⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠰⠀⠀⢀⠔⠊⠀⣀⠀⠀⠀⠀⠀⠀⣼⣈⣀⣀⠀⠀⠤⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢀⠀⠀⢸⠁⠀⠀⡀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠈⠑⠲⢤⡀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢂⠀⢸⠀⠀⠀⠆⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠙⢲⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠘⡇⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠀⢀⣀⣀⡴⠃⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢀⣠⡞⠁⠀⠀⢀⣀⠀⠀⠁⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡴⠋⣡⠞⢁⡤⡶⠛⠓⠲⠦⠤⠤⠶⠶⠦⠶⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢋⡞⣡⠞⢁⡴⠋⣰⠃⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢟⣱⠿⠋⣁⠴⢋⡤⠚⢁⡤⠶⢒⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠿⠖⠋⣁⣤⣚⡥⠚⢁⡤⠞⣉⡴⠞⠉⠀⠈⠙⠲⢦⣄⣀⠀⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⢸⣅⣠⣤⡤⠶⠿⠛⣉⣥⣴⡾⠋⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣤⣤⣀⣀⣤⣤⡤⠤⣤⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⣿⣀⣀⣤⡤⠶⠚⣋⡭⠞⢉⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠛⣁⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣀⡤⠂⡋⠉⠐⡲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠁⠀⠀⠀⠀⠀⠒⠒⠊⠙⣦⠔⠚⠀⠀⠼⣟⡯⠭⢤⠈⠀⠀⠀⠀⠈⠀⠘⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣉⣆⣠⣄⠤⢎⣅⣀⠀⠀⠁⠒⢆⡀⠀⠀⠀⠀⠈⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢀⡰⠁⠴⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠓⢤⣀⠀⠉⠑⢲⡞⠉⠉⠉⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁⠀⢠⠀⡠⢊⠴⠁⠀⠄⠀⡀⣀⠤⢄⡤⢤⡸⠦⢄⣀⠀⠀⠀⡵⠚⣓⡟⠁⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣼⡕⠁⠀⠀⣠⠤⢔⠉⣁⣀⡦⠤⢴⠖⠉⠉⠉⠙⠉⢺⡁⠐⠈⠳⡄⠀⠀⠀⠀⡴⠊
⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⢸⡿⠀⠀⣀⠌⡠⠔⠚⠉⠁⣀⣠⠞⠡⡀⠀⠀⠀⠀⠀⢸⠀⡀⢄⡸⠻⡀⠀⠀⠀⠙⢲
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⡞⢳⠀⢠⡰⠋⠀⠀⠀⠀⡴⢁⠀⠀⠀⠢⠀⠀⠀⢠⢎⠝⠋⠁⠀⠀⠀⠷⡄⠀⠀⠀⠈⢣
⢠⠒⠂⠤⠖⢠⡀⠀⢇⠀⠀⠜⡄⠑⠻⠤⡤⠤⢤⠈⣱⠃⠀⠀⠀⠀⠀⠀⠀⣀⣼⠾⡄⠀⠀⠀⠀⠀⢀⡇
⢸⠠⠀⠤⠀⠀⠑⠢⠌⢧⠘⠀⣇⠀⢀⠀⠀⠀⠀⠀⣠⡬⠥⢶⠦⠄⣀⣀⣤⠷⠁⡀⠉⠀⠀⠀⠀⠀⡿⠁⠀⣠⣶⠞⠉
⣞⡠⠔⠀⠀⠀⠀⠀⠀⠀⠉⢧⠠⠉⠉⠉⠢⠴⢖⠋⠁⡔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢸⣱⠃
⢇⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠈⠳⣀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣵⠃⠀⠀⢠⠝⡄
⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢦⣀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠴⠊⠀⡀⣠⠔⠋⢀⠇
⠀⢸⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢀⡄⠀⠀⠀⠈⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⢁⣠⢖⡿⠙⠂⠒⠊⠁
⠀⠸⡀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠠⠤⠞⠓⠒⠒⠒⠒⠒⠊⠉⣍⣠⠞⡱⠊⠉
⠀⠀⠑⠢⣀⡐⠀⠀⣀⢀⡄⣸⠔⠲⣄⠸⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠛⠷⣀⠀⠀⢀⠉⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⡜⣻⠀⠀⣹⡆⠀⠐⣀⠘⠃⣐⠃⠀⠀⢠⡆⠀⢱⠒⢛⢳⠦⠐
⠀⠀⠀⠀⠊⠀⠐⢬⣒⢥⡬⣟⠁⠀⠉⠁⠀⠀⠈⠘⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠨⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠊⠑⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠐⠠⠀⠀⠀⠀⠀⠀⢸⣀⠆
⠀⠀⠀⠀⠀⠀⠖⠰⠆⠀⠒⠘⠃⠀⠀⠀⠀⠀⠀⡼⣆⠀⠠⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢈⠀⠀⠀⠀⢄⠂⠓⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡴⡒⢃⠈⡐⠀⠀⠀⠠⡄⡄⠀⠁⠀⠀⠀⠀⠀⠄⠎⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⣀⠥⢀⡐⡀⠐⣀⠈⡀⠀⠀⠀⢡⡄⠀⠀⠀⠀⠀⠀⠀⢔⠁⡔⠀⠀⢀
⠀⠀⠀⠀⠀⠒⠒⢀⠃⢐⡈⠀⢩⡀⢡⡄⠀⠀⠀⡆⠀⣤⡆⡀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣀⠀⡘⢃⡄⠈⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠈⡋⠈⠃⠀⡀⠀⠀⠀⠂⠀⠰⠄
⠀⠀⠀⠀⡠⠋⠈⠒⠋⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠐⠙⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠖⠒⠚⠉⠛⠋⠋⠋⠀⠠⠦⠤⠆⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠺⣝⠛⠷⠤⢤⣈⠉⠉⠒⢤⡀⠀⠀⣠⠴⠒⠲⢤⡀⠀⠀⠀⢀⣠⠤⠤⠶⠒⠒⣒⣶⢶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠉⠲⢄⠀⢱⡤⠚⠛⠒⠦⣄⡀⠙⢦⡠⠚⠉⠀⣀⡤⠖⠋⠉⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠱⣾⠀⠀⠀⠀⠀⠀⠉⠢⡀⢳⣀⡤⠊⠁⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢀⡿⡄⠀⠀⠀⠀⠀⠀⠀⠈⢦⣻⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⠤⠔⢻⣆⢙⣦⣄⣀⣀⣀⣀⣠⠤⢤⣝⡇⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⢀⣀⡤⠴⠖⠒⠲⠤⣀⠀⠀⠀⠀⠈⣿⣾⠏⢀⡴⠛⠉⠉⣙⣦⣼⢦⣁⠀⠀⠀⢀⣠⠞
⠀⢀⣤⣾⠭⠤⠶⠶⠶⠤⢤⣀⡘⡆⠀⠀⠀⠀⣸⣿⣠⠋⢀⡤⠚⠉⠁⠀⠈⠳⡌⠉⠉⠉⠉
⠀⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠉⢣⠀⠀⠀⠀⣿⣿⢃⡔⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢰⣿⡷⠋⠀⠀⠀⠀⠀⠀⠀⢀⡼⣁⡤⣄⡀
⠀⠀⠀⠀⠀⠈⠳⢄⣀⡀⣀⣠⣴⣃⣴⣊⠉⠉⠛⢾⣿⡓⠒⢶⣤⠤⠴⢒⠟⠛⠓⠦⣄⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢀⡞⠁⠀⠈⠉⠓⢤⡀⠙⣇⠀⣸⡇⠀⠀⣼⠀⠀⠀⠀⠈⠳⣌⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠉⠢⡈⢦⣿⠁⠀⣸⡟⢦⡀⠀⠀⠀⠀⠈⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⠀⢀⣿⠀⠀⠉⠙⠓⠚⠋⠉⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡤⠤⢤⡤⠖⠒⣿⠲⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣗⠀⠀⣿⡇⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠋⠀⠀⣿⡇⠛⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⡴⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠞⠛⠋⠉⠉⠉⠉⠉⠉⠉⠁⠀⢈⣷⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠉⠙⢷⣴⠖⠲⣆
⠀⠀⠀⠀⠀⣰⠞⣁⣤⡴⠶⠶⢤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡏⠀⠀⢀⣤⣤⣤⣿⠁⠀⠀⠹⣆
⠀⠀⠀⢠⡾⢡⡞⠉⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⢸⡇⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⢰⡏⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⢠⡴⠋⠀⢀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⣿⣀⣠⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⣤⣼⠃⣴⠛⠉⠉⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⢻⡄
⠀⢠⡟⠉⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠈⢿⣤⡿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢻⡄⠀⠀⠀⠀⣿⡀
⠀⢸⠁⠀⠀⠀⣦⠀⠀⢸⡃⣀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢤⣤⣿⡄⠀⢠⡄⢸⡇
⠀⣸⠇⠀⠀⠀⠸⣆⠀⢸⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣧⠀⣸⠃⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠙⢧⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡾⠋⠀⢸⡇
⠀⠈⣧⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢠⣄⠀⠀⢸⡇⠀⠀⣾⠁
⠀⠀⠘⣧⠀⠀⢷⡀⠀⠀⢹⡄⠀⢀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⢶⣄⣼⠃⠀⣼⠃
⠀⠀⠀⠙⣦⡀⠀⠻⣦⣀⠈⢿⡛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠈⣿⣠⡾⠁
⠀⠀⠀⠀⠈⠳⣤⡀⠀⠙⠓⠾⢷⡀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢀⣸⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠈⠻⣄⣀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣽⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠙⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣅⢾⡽⠛⠟⢹⣿⠀⠀⠀⠀⠀⢾⣥⣑⢦⡄⠀⣦⣶⠛⠡⠆⠁⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠙⣷⠘⣶⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠳⣿⠀⠘⣷⠀⠀⠘⣇⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⠿⢿⣟⣉⣠⣴⣟⣼⠃⠈⣿⡀⠀⠀⠀⣠⣴⣬⣥⣠⢼⡟⢀⣠⡿⠀⠀⠀⣼⠏⣰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡼⠃⠀⢀⣴⡿⢋⣉⣉⠙⠳⣤⣠⣾⣧⡀⠀⠀⢿⡅⣴⠖⢂⣬⠖⠋⢡⡤⠐⠚⢡⣟⢸⠃
⣿⣿⣿⣿⣿⣿⡿⠛⠻⣿⣿⣿⣿⠈⠀⠀⣾⡏⣼⠋⣨⣽⣷⣤⡀⠙⠛⠇⠀⠀⠀⢘⡷⢹⡆⠘⣿⡀⠀⢿⡀⠀⠀⠀⠻⣌⢆
⣿⡿⠟⠻⠿⠟⠀⠈⠀⢸⣿⣿⣿⡆⠀⠀⣿⣿⣯⣾⣯⠀⠀⠈⠛⠦⣄⡀⠀⠀⣀⡾⠁⠀⣷⠀⠈⢻⣦⠨⠙⣆⠀⠀⠀⠙⣿
⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣮⡀⠹⣿⣯⠈⢀⣀⣀⠀⠀⠀⠀⠩⣿⠛⢻⠁⠀⠀⡿⠀⠀⢀⣿⠀⠀⠘⡇⠀⣠⣣⡿
⠁⠀⠀⠀⢀⣤⡤⢀⠀⡄⠻⠿⠿⣿⠏⢿⡄⠹⣿⣷⠛⠙⠻⣷⣶⣶⣄⣚⣁⠴⠃⠀⠀⣼⠃⡀⢀⡾⠃⠀⢰⣾⠃⠸⣿⣏⠛
⠀⡀⠀⣴⠟⠁⠀⠀⢷⣬⣼⡶⣶⣶⣒⣠⣇⣾⡿⡡⣲⡖⠁⣉⡉⠉⠁⠀⣀⣄⣀⣤⣞⣡⡭⠞⠋⠀⣀⠀⠉⣧⠀⠀⣿⣿⡄
⠀⢇⣸⠃⠀⠀⠀⡀⢀⣾⡣⠀⠀⠈⠙⠛⠛⠋⠀⢶⣿⢠⡾⠛⠛⠛⠻⠟⠛⠋⣩⡿⠿⠷⠶⠶⠶⣿⠟⠛⣋⣁⣤⣶⣿⠋⢹
⢀⢌⡏⠀⠀⠀⠀⠈⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⢀⡤⠤⠤⠄⠤⠤⠒⠁⠀⠀⠀⠀⠀⠀⣿⠀⢀⣬⣴⣾⠟⢅⣡⣼
⢸⢸⡇⠀⠆⠀⠀⠀⢸⣇⢀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣿⡿⣃⠍⠀⢸⣯
⠀⢦⣿⡀⡄⠀⠎⠀⠀⠘⢷⣇⠀⠀⠀⢀⣴⣿⡙⡆⢿⡆⢻⠀⠀⠄⠤⢤⡀⠀⠀⠀⢠⠖⠋⠀⠀⠀⢸⣿⣽⣶⣾⣤⣾⣿⠃
⣴⣌⣘⣧⠀⠀⠀⡀⠀⡄⠈⠹⣄⠀⢀⢼⣇⡿⠸⣇⠀⣿⢸⠠⣤⣤⣤⡀⠀⠀⠀⠀⠀⣤⣶⣶⠟⠀⣤⣿⣿⠀⠙⠛⠛⠁
⡟⠻⢿⣃⢀⠀⠀⠀⠀⠟⠆⠀⣿⠈⠀⠀⢻⡇⠀⠟⠀⣿⢸⠀⠀⠉⠉⠉⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⣼⣿⢿⣦⣄⣀⣀⣀⡀
⠃⠀⠀⠔⠀⠀⠁⠀⠀⢀⣣⣾⣧⡀⠀⠀⠀⠹⢦⠀⣼⡷⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣶⣭⡛⠓⠒⠢⠘
⠀⠀⠀⠔⢉⠁⢅⣵⠿⠛⠉⢙⣛⣍⣭⠄⠀⠀⣸⣿⠏⢠⠇⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⣿⣿⠟⠁⠀⠀⠀⣠
⠀⠀⣆⢱⣠⠶⣻⠏⠀⠀⠀⠀⠀⢀⣈⣉⣁⣬⣵⣿⠃⣄⠠⠀⠘⢦⡀⠀⠀⢶⣿⣿⣿⠗⠀⣰⡿⢋⣼⡟⠁⣠⡤⠴⠒⠋⠁
⠀⠀⣩⣾⠁⠀⠉⠠⠀⠀⠀⠀⢠⣶⠿⠿⠛⠿⠿⣷⣤⡈⠳⣔⠄⠘⢧⡳⣄⡀⠈⠉⠁⣠⣾⡟⢻⣿⣿⣆⠀⠛⠷⡔⢀⡰⢆
⠀⠉⠸⣷⡀⠀⠀⠀⠂⠀⠀⣰⠏⠔⡪⠀⠀⠈⠉⠙⠻⣿⣦⣄⠙⢆⠀⠻⣄⠙⣷⢶⣾⣿⣿⣇⡈⣉⣛⡿⠿⠾⣶⣶⣤⣹⢸
⣀⡠⡢⠘⣿⡆⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠈⠉⠉⠛⠑⠺⣿⣿⣸⣸⡇⠁⢹⡀⢉⣏⣿⣿⣿⣿⣿⣿⣿⣷⡅⠂⠀⢹⣿⣯⣼
⡽⠿⣤⠾⠋⣀⣠⣀⡀⠀⢸⡇⠀⠢⢀⣠⣤⣄⡀⠀⡀⠀⠹⣟⠿⢛⡤⣶⣸⠃⠀⡿⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣾⣿⣿⡿
⣀⣀⡈⠲⠶⢟⣡⣯⣽⡱⣼⠆⢸⡇⣿⣿⣿⣿⣿⣦⡀⠀⠀⠩⠛⠋⠀⠀⠁⢀⡤⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⠛⡟⣿⣿⣏⣀
⠉⠙⣷⣶⡾⠋⢹⠋⠉⠻⠝⠀⠀⢳⠘⣿⣿⣿⡏⠛⢿⣶⣌⠲⣤⣄⣈⡉⠉⣡⢞⣼⣿⣿⣿⣿⣿⣿⢟⣥⡿⠀⢱⠩⣛⢿⣿
⠀⠀⢈⣽⢽⠀⢤⢰⠋⠉⠒⠀⣀⣘⠄⣹⣿⣿⣿⡶⡟⠛⣿⣿⣷⣿⣿⣥⣴⣿⣿⣿⣿⣿⣿⣿⡿⣧⡟⠩⠀⠀⠈⠀⠈⠱⣿
⠁⡰⣦⣀⠸⢀⣾⠏⣠⣴⣾⣯⣭⣶⣾⣿⡿⠋⡙⠁⠱⣎⣼⣮⡛⢿⣿⣿⣿⣿⡿⠿⣿⡟⠯⠋⢠⢿⡇⠀⠀⣤⣤⣄⣀⣠⣽
⢋⠽⠏⣀⠠⢞⣫⣾⣿⡿⠿⠿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠹⣷⠀⣉⣳⣌⠙⠛⠶⠴⠋⠀⠀⢀⡠⠊⢸⣧⠀⠀⢻⣿⣿⡋⠛⠛
⡁⠐⠈⣁⢨⣿⣿⣿⡟⠀⠀⠀⣂⣀⠀⢀⣀⠀⠀⠀⣀⡴⣿⣿⠉⠉⠙⠻⢷⣆⣀⣀⣀⣠⣭⣤⣤⣾⠏⠀⠀⠀⡹⣿⣷⣄
⣡⣶⣶⣿⣿⣿⣿⣿⠃⠀⠀⡘⠁⢀⡔⠉⠀⣠⠍⠛⠰⠚⠁⣿⠀⠠⡎⠒⠰⠩⢝⣛⠛⠛⠛⢛⣋⠡⠔⠀⠀⢀⢤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣛⣁⣀⡁⣀⢀⢀⡜⣀⣀⣠⣇⢀⣀⣀⣠⣴⣿⣀⣀⣄⣴⣛⣓⣂⣤⣤⣤⣄⣀⣠⣤⡄⣀⣢⣵⣿⣿⡟⣁⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠔⠊⠉⠙⠓⢆⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⠈⠓⠒⠤⣀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠢⣄⡀⠀⢀⣴⠏⠈⢣
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡴⠃⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⣸⡇
⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⡀⠀⣀⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⠴⠒⠒⠀⠀⢀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠹⢆⡀⢠⠞⠁⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠉⡏⠀⠀⠀⠀⠀⠸⣧⣀⠘⠿⡟⣏⠁⠀⢹⠦⣄⣀⣠⠔⠋⢀⢜⠃⠳⣄⠀⠀⠀⢀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⢹⡀⠈⠛⠿⣶⣳⠘⢦⣤⠏⢧⣈⡭⣴⣒⣯⠗⠋⠀⠀⠀⠙⠲⠚⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣄⣈⣻⡦⣄⠀⠀⠉⠛⣦⡀⢠⢠⣶⠛⢉⣁⣤⣶⣤⣀⣀⣀⡀⠤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠖⠋⠈⣧⡅⢸⠉⠙⠛⠋⠁⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣥⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢻⢏⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣙⣈⠔⢀⣙⣄⣭⣓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠉⠉⠹⠦⠏⠉⢻⡽⠉⠑⠾

⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⠿⠿⠿⠿⠿⠷⢶⣦⣄⡀
⠀⠀⠀⣠⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⣉⠛⢷⣤⡀
⢀⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠙⢷⣆
⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠳⡄⠻⣧⡀
⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⣶⣦⣄⡀⠀⠀⠀⠀⢀⢻⣇
⠀⠀⠈⠛⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣀⠀⠈⠀⣿⡄
⠀⠀⠀⠀⠀⠉⠛⠿⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⡀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⣶⣦⣽⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠶⠶⠶⠶⠶⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⢀⣤⠄⠀⠀⠀⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⣰⡿⠁⠀⠀⠀⣠⣾⠟⠉⠀⠀⣴⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⢰⡿⠀⠀⠀⣠⡾⠋⠁⠀⢀⠀⡐⠁⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣿⠃⠀⣠⣾⠟⠁⠀⠀⢀⣠⠞⠁⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡿⠀⣴⡟⠁⠀⠀⠀⠒⠉⣀⣤⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⣼⠏⣀⣤⣤⣶⡶⠾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡶⠾⠿⠿⠷⢶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠛⠉⠁
⠀⠀⢀⣠⣤⡶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢤⣈⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋
⠀⠠⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣌⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠰⢶⣦⣤⣀⠀⠀⠀⠀⠀⠈⠳⣌⠻⣦⡀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣦⣄⡀⠀⠀⠈⠀⠹⣷⡀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⢶⣦
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣤⡀⠀⠀⠙⣇⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄⠀⢸⣦⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡈⣿⡄⢠⣿⠁⠀⠀⠀⢀⣠⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⣶⣦⣤⣤⣤⣄⣀⣀⣀⣀⣹⣷⣾⣇⣸⡏⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠛⢿⣿⣿⣿⠃⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⡸⠂⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠀⣰⡿⠁⠀⠀⠀⠀⠀⢠⣼⠟⠁⠀⠀⠀⡴⠃⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⣰⡟⠁⠀⠀⠀⢀⣠⣾⠟⠁⠀⠀⠀⣠⠞⢁⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣰⡿⠁⠀⠀⣠⣴⠿⠋⠀⠀⠀⢀⠀⠀⢁⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⢀⣴⡾⠋⠁⠀⠀⠀⠀⠀⢁⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣴⡟⠁⠀⠀⢀⣀⣠⣤⣶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣤⣶⠶⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠾⠿⠿⠿⠷⣶⣦⣄⡀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣤⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⠞⠁⠀⣀⣤⣴⡶⠿⠿⠷⢶⣦⣤⣀⠙⢿⣾⣿
⠀⠀⠀⠀⠀⠀⣠⣾⠋⢠⠞⠁⢀⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⣹⣿⡀
⠀⠀⠀⠀⠀⣰⡟⠁⠴⠃⠀⠰⠟⠁⠀⠀⠀⠀⠀⣠⣴⠶⠿⠛⠛⠿⠾⣿⣿⣿⡇
⠀⠀⠀⢀⣾⠟⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿
⠀⠀⣠⣾⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⣴⣿⣁⣤⣤⣤⣤⣤⣤⣴⣶⡶⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠁⠈⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣾⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⢤⣤⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⣀⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡉⠹⢿⣿⣿⢿⣭⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠈⢿⣿⣿⠉⠙⠛⠻⡿⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠐⠛⠛⠛⠉⡛⠛⠛
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⣶⣤⣤⡀⠀⠀⠀⢼⣿⣿⣟⠛⠛⠻⣿⣿⣿⡁⠈⠻⣿⡷⢬⡛
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠃⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⡗⠁⠤⣨⣿⣿⣿⠄⠀⢩⣝⠿⣶
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⣴⠿⠛⣻⠿⠿⠿⠿⢿⢹⣿⣿⣿⣿⣯⠀⠀⣸⣿⣿⣿⡃⠀⠀⣉⣧⣴
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣠⣴⣤⣴⣿⣦⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣧⠀⢠⣿⣿⣿⣿⡅⠀⠀⠀⣩⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢁⣶⣿⣿⣿⣿⡿⢻⣿⠏⠙⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⢽⣿⣿⣿⡇⠀⠀⠀⢀⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⢠⣿⣿⠛⠿⠿⠋⣠⣾⠋⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠘⣿⣿⣿⡇⠀⠀⠀⢠⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣶⣾⠿⠛⠁⢀⣴⣿⣿⣿⣿⣿⡿⢊⣿⣿⣿⠶⠦⢻⣿⣿⡇⠀⠀⠀⢸⣿⠃
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣿⣿⣿⡟⠉⠉⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡛⠛⢤⣿⣿⡟⠀⠀⢿⣿⣿⠅⠀⠀⠀⢸⣿⠄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣻⣹⣰⣿⡇⠀⣹⠆⠀⣸⣿⡟⠀⠀⠀⠀⠐⣿⡄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠈⠠⠀⠀⣿⣿⠃⠀⠀⠀⠀⠨⣿⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⣐⣿⠄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⢲⣶⣾⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⠛⠛⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠁⠀⠀⠀⠀⠀⠀⢿⢿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣦⡑⠀⠀⠀⠀⠀⣀⣴⡾⠟⢻⣿⣿⣿⣀⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡀⠀⠀⢰⡾⠋⠀⠀⠀⠘⢻⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠿⣶⣄⠘⠇⠀⠀⠀⠀⠀⢀⣿⣿⡿⠇⠀⠀⠀⠀⠀⢀⣄⣴
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠈⠋⡀⠀⠀⠀⢀⢠⣴⣾⡿⠋⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⢀⣤⣶⡿⡛⢛⣩⡄⠀⠀⣀⡀⠀⠀⣾⣿⡿⢟⠻
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣆⠀⠀⠻⠋⢩⣽⡟⣱⣿⣿⠇⠀⢠⣻⣿⣤⣾⣿⡿⠃
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⢿⡽⢿⣿⣿⠿⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⢀⣤⣿⣛⣫⠬⠉⠉⢛⠀⠙⠻⠛⠛⠋⠁⠙
⠀⠀⢸⣍⣁⣾⣿⣿⣿⣞⣣⣴⣾⡿⠋⠀⠐⢿⣿⣿⣿⣿⠟⠛⠁⠉⠀⠀⠒⠈⠝⠛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⢸⣿⣿⡿⣫⣿⣿⣿⡿⠟⠋⠀⠀⠀⢰⣾⣿⠟⠉⠃⠀⠀⣀⡤⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡉
⠀⠀⣸⣿⣿⣼⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠈⠷⠄⠀⣠⣤⣶⠾⠋⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛
⠀⠀⢿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠐⠟⠛⠋⠉⠉⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠒⠄
⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⡘⠻⡆⠀⠀⠀⢥⡀
⠀⠀⠻⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠈⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠶⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⡴⠤⠦⣄⠀⠉⠓⢦⣀⡴⠚⠛⠓⢦⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⡟⠒⠦⣄⡀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠉⠲⣿⡀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠐⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡼⢁⣀⣤⣤⠤⠤⠤⢤⠀⠀⠀⠀⢧⠀⠀⠀⠈⢃⠀⠀⠈⢳⡀⠀⠆⠀⠀⢸
⠀⠀⣠⠤⠖⠚⠛⠉⠉⠙⠛⠒⠶⢤⡀⠁⠄⠀⠀⠈⡇⠀⠀⠀⢸⡄⠄⠀⠀⠱⣠⠀⠀⠀⢸
⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⡟⠐⠀⠀⠀⡧⠀⠀⠂⠀⢻⠀⠠⠀⣾⠦⣤⣀
⠀⣿⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣗⠀⠀⠀⢸⠁⠔⠀⠀⢀⡇⠀⢰⣀⣀⡈⡆⠀⠀⡿⠀⢀⣈⣙⣢⡀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⠆⠀⠀⢀⣼⠗⠋⠉⠀⠈⠙⡟⣠⠴⠛⠉⠉⠉⠉⢉⡷
⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⣇⠀⠀⡼⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢀⠄⣠⠋
⠀⢸⡀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠸⡤⠼⠓⠋⠀⠀⠀⠐⠀⠀⠀⠠⠀⠀⢸⠀⠀⠐⠀⣠⠞⠁
⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡁⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣤⠞⠁
⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⣁⡴⠃⠀⠁⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢓⠀⠀⠀⢠⡇⣠⠞⠁
⠀⠀⠈⣇⠀⠀⢀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡤⠀⠀⠰⠀⠀⣆⠄⠀⢀⡾⠞⠁
⠀⠀⠀⠘⣶⠶⢛⠁⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⢰⠇⠀⣠⠎
⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠻⢦⣀⣿⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⡀⠈⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠶⠦⠤⠤⠤⠤⠶⠖⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠢⡄⠀⠀⠀⣠⠖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠦⣀⡜⡇⠀⢞⣂⣸⣀⣠⠞⢣⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢹⡄⠸⡄⠀⠀⢀⡼⣇⠀⠀⡾⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠂⠀⠀⢸⣧⡀⠙⣄⠀⣞⠀⠈⠓⢾⠁⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⢣⡱⣄⠈⠓⢬⣙⣲⠤⢃⣀⣀⡤⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠁⠀⠙⢮⣙⠲⠤⣀⣉⣉⣩⠥⠴⢒⣋⡷⠖⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣠⣤⢄⡀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠒⠛⠉⠉⠉⠁⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠁⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣀⣀⡀⠀⠀⠀⠀⠐⠆⢠⣦⠀⠀⠀⠁⠀⠈⢳⠄⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠉⠑⢤⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⡴⠊⠁⠀⠀⠀⠙⢢⡀⠀⠀⠀⠀⣠⠞⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⣀⡠⠞⢹⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣹⡇⠀⢸⠀⣟⢩⣍⠓⢦
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣸⠀⠹⣇⠀⠸⡄⠈⠒⠚⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠉⠙⠒⠃⠀⠀⠘⣆⠀⠳⡀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⣤⣽⠦⠖⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠤⣀⣀⠀⠀⠀⠀⢸⣃⡀⠀⠰⠹⢮⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⡞⠉⠉⠉⠙⠋⠁⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⢀⣠⣴⣋⣀⣀⣀⡠⠀⠀⣀⣠⠴⡛⠢⠤⣤⣠⡤⠤⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠉⠉⠉⠀⣸⠈⠉⢩⠀⠀⠀⠀⠀⡟⣄⠈⠓⠦⢤⣀⣀⡤⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄⣀⡴⠃⠀⠀⠸⣆⠀⠀⢀⣴⠁⠈⠳⢤⣀⣀⣀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠰⠶⠤⢀⣀⡀⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠐
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⡀⠀⠀⡠⠶⠖⠀⠄⠀⠀⠀⠀⠀⢨⣥⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠁
⠀⠀⠀⠀⠀⢀⡶⠋⠉⠉⠻⡄⠘⠟⠐⠁⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⠀⠀⠀⠀⠀⠈⠇⠈⠉⢿⣿⣿⠃
⠀⠐⠚⠿⢛⠻⣤⣀⠀⠀⠀⢁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⣿⠢⡀⠀⠀⠀⠀⠀⠈⠀⠀⠾⠿⠃
⠀⠀⠀⠀⠀⠀⠹⣿⣾⡇⢀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠰⣿⡄⠘⣆⠀⠀⠀⢀⠄⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠿⠃⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢛⣿⣏⠄⠘⡄⠀⠀⠈⢄⠀⢾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣤⣠⣴⣴⡶⠿⠟⠓⠤⠀⠀⡇⢠⣭⣿⡷⠂⠀⡇⠀⠀⠀⠀⠀⢈⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠰⠿⠥⠒⠁⠀⠀⠀⠀⠀⠀⠀⠠⠠⠬⣿⣟⣁⢀⠇⢀⠀⠀⢠⣆⣀⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⡯⢀⠎⠀⠈⠻⣶⡋⠀⠀⣽⣿⣿⠋⡨
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠈⠻⣶⣾⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠋

⡷⣤⠤⠀⠤⠤⠄⠒⠒⠒⠀⠤⠤⣀⣀
⠱⣄⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠂⠤⠤⢠⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠳⣶⡀⠈⠉⠀⡶⠒⠒⠒⠒⢲⠖⠒⠒⠒⠒⡲⠆⠀⠤⢤⠤⠈⣉⣑⠒⠠⠄⣀⢎⠂⠀⠀⠀⣀⠀⠀⠀⠠⢤⠤⣒⠢⣄⣀⣤⠤⠤⠤⠤
⠀⠀⠀⠈⠒⠍⡒⠮⢵⣀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡇⠀⠀⠀⠆⠀⠀⠀⠘⠉⠉⠉⢲⣿⣷⣄⣀⣄⠏⠀⠀⠀⠀⣀⣁⠶⠚⢋⡟⠁⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠬⠭⠍⣉⣓⡒⠒⠒⠂⠰⠤⠤⠤⠿⠤⠤⠤⠾⠦⠶⣖⣛⠽⠟⠁⠀⠀⠀⠀⢀⣠⣼⠀⢈⡇⡠⡞⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠁⠀⠀⠀⠈⠁⠉⠁⠀⠀⠀⡴⠀⠀⠀⠀⣠⠁⠀⠀⣟⡸⢋⠞⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠉⢳⠴⡴⢋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠚⠣⣀⣀⣴⢊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⢾⠀⠀⠐⣁⢔⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡎⠀⠀⠀⡗⢒⣟⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣲⠃⠀⠀⢱⠤⢠⡴⣟⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⢀⠖⠂⠫⡍⠀⠀⠈⣦⡀⢀⣬⠔⡨⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠲⢼⡀⠀⠀⡐⠤⢀⣔⠠⢘⡩⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠖⠚⠟⠒⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠤⠤⠤⠤⠄⣀⣀⣀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡠⠤⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣓⣄⣀⢹⣶⢤⣤⣤⣤⠞⠁
⠀⠀⠈⠫⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠝⡻⠁
⠀⠀⠀⠀⠀⠈⠙⠒⠢⠤⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠤⠆⠐⠒⠒⠒⠒⠲⠶⠶⠆⠐⠒⠂⠉⢡⡷⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣿⣿⣿⡈⣿⡉⠉⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢻⣿⣿⣿⡿⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⣤⣻⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢤⣼⣾⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠀⣴⡎⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣭⡙⢷⡀⠀⠀⠀⠀⣀⠔⢺⣿⣤⣴⠀⠀⠉⠋⠁⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠃⢸⡇⠀⢀⠄⠊⠀⠀⣷⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⡀⠀⣠⢞⣵⠖⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠤⠀⠠⠐⠮⢀⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⠀⠀⠀⠀⠀⢿⡟⣿⣧⠞⣱⠟⠁⠀⠀⠠⠀⠈⢉⠉⠛⠛⠻⠗⡒⠤⣀⠀⠀⢴⣿⣴⣿⣿⣷⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⡿⠀⠀⠀⠀⠀⠀⣰⢃⡾⢁⡴⠋⠀⠀⣀⡄⠈⠟⠀⠀⠀⠀⠀⠈⠑⠦⡙⠲⣌⠙⠻⠃⠉⠉
⠀⠀⠀⠀⠀⠀⢠⣄⣀⡀⠈⠀⠀⠀⠤⣶⣄⣠⢇⡞⣴⢋⣀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠈⠱⢄
⠀⠀⠀⢠⣔⣒⠐⠀⠰⠁⠀⠀⠀⠀⠀⠺⡛⠛⢿⣡⠟⠁⠀⢀⠀⠈⢢⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢇
⠀⠀⠀⠙⠈⠁⠐⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡎⠀⠀⢰⠿⣻⣶⣿⠗⠤⠀⠀⠀⠀⠀⠀⢰⠂⠡⠘⡠⠎⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⢀⣀⣴⡾⡄⢀⠔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠻⠿⠟⣿⣗⣾⡆⠀⠀⠀⠀⠀⠀⢡⡀⢌⣶⠀⠀⠀⠀⠀⢠⠠⢘⠿⣿⣿⠂
⠀⠈⠛⠻⡿⢿⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠉⢻⣿⡀⠀⠀⢤⣄⠂⠀⠈⠶⠞⠀⢀⠀⠀⠀⠀⠀⠃⢿⣷⡿⣿⣿⡆
⠀⠀⠀⠀⠀⣼⣶⣲⣖⢀⠀⠀⠀⠀⠀⠀⠄⣀⣀⣈⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡇⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠽
⠀⠀⠀⠀⠀⠀⠋⢍⣧⡀⠀⠐⠦⠀⣠⣿⣿⠿⠿⠟⠻⠿⢿⣿⣷⣤⡀⠀⢀⣀⠐⣿⣟⣻⣿⡇
⠀⠀⠀⠀⢨⠀⢀⣠⡀⠀⠀⠀⢀⣤⠿⠋⠀⠀⠀⠀⠀⠀⠀⠉⢫⣭⡅⢐⠋⠉⠑⢄⣈⡝⠋⠉⠉⠙⢶⡀⠀⠀⢴⣿⢧⠐
⠀⠀⠀⠀⠈⠧⣈⣛⠝⠀⠀⠀⢸⡇⠀⠀⣀⢴⣿⣶⣦⠀⠀⠀⠀⠙⣧⡀⠙⠋⢀⡾⠋⠀⠀⣠⢰⣦⠀⣣⠀⠰⡿⠿⠟⡏
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⣾⡿⠿⣻⣧⠀⠀⠀⠀⢻⡭⢿⣿⡟⠀⠀⠀⠀⢷⣸⠿⣠⠟⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣶⡗⣿⣿⣿⣿⡏⠀⠀⠀⠀⠘⣿⣿⣿⣿⣧⠀⢇⠀⠀⠉⠉⠁⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠈⠓⠤⣤⣤⡤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⣠⣴⡖⣶⠤⠀⢤⣄⢀⡀
⠀⠀⠀⠀⠀⠀⢀⡄⢀⣴⣾⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣶⣖⣠⠛⣿⢁⣀⣀
⠀⠀⠀⠀⠀⣰⣽⣿⣿⣿⣿⠋⠉⠉⢉⡀⣀⣀⣀⣉⣻⣿⣿⣿⣿⣾⣿⣿⣽⠛⢷⣤
⠀⠀⠀⠀⣨⣿⣿⣿⠿⠉⠀⢐⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣟⢲⣆⠆⠹⣧
⠀⠀⠠⣾⣿⣿⣿⠁⢰⣶⣽⣿⣿⣿⣿⣬⣍⠉⠀⠉⠀⠀⠀⠉⣿⣿⣿⣷⣿⣿⣟⣠⢈⢷⡀
⠀⣀⣾⠟⣿⣿⠃⢀⣾⣿⠟⠉⠁⠈⠽⣿⣿⣳⡀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⣿⣿⡜⠀⣧⣀
⠀⢹⣿⢰⡿⠁⠲⣿⣿⡟⢀⠁⠀⠀⠀⠠⣿⠟⠟⠉⠉⠲⠆⠀⠀⠰⠈⠁⢿⣿⣿⣿⣷⠀⣿⣿⡄
⠀⢸⡇⡼⣧⢠⣸⣿⣿⣿⠃⣄⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⣿⣿⣿⣁⠀⢸⣿⣁
⠀⢸⡇⠘⣙⣦⡹⣿⣿⣿⠷⠁⠀⠀⠀⠀⣇⠃⠀⠀⠀⠀⠀⠈⢻⢷⣄⠀⠘⠻⣿⣏⡀⢠⣿⣿⡇⠂
⠀⠨⣿⠀⠨⣻⣿⣾⣿⣽⡦⣄⠀⠀⠀⠀⣯⣡⠀⠀⠀⠀⠀⠀⠘⢞⣿⣷⡀⠀⣿⡏⢠⣿⣿⣿⣿⡆
⠀⢠⣷⢀⣠⣿⠿⣿⣿⣿⠿⠖⠓⠲⠶⠛⢛⣋⠉⠛⢾⠿⠓⠦⣤⣄⣽⣿⡇⢈⣿⡇⣸⣿⣷⣿⢿⡷
⠀⢸⣿⣆⠚⣻⣶⣿⣏⡱⣾⣿⡔⡿⣿⣴⣼⣿⡌⠓⢶⣷⡀⠄⠀⠹⣿⣿⣿⠿⣿⠄⣿⣿⣿⣿⠘⡇
⠀⢸⣿⣿⣷⡌⠿⣿⣿⣿⣮⣮⠁⢋⠉⠉⠉⣿⡇⡀⢸⣿⣿⣦⣤⣄⣿⡿⡇⣰⣶⣿⣿⣯⣝⣹⣻⡗
⠀⠀⡈⣿⣿⣿⣦⣀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣧⣿⣿⣿⣿⣿⣿⣭⣴⣶⣍⡈⢹⣯⣭⠙⠲⢹⠃
⠀⠀⠹⣿⣟⣿⣿⣿⣷⣦⣬⣍⣻⡟⡿⠛⠿⣛⣿⣿⣿⣿⠛⠻⣿⠿⣿⡏⣿⠻⣏⠢⢉⡫⠈⣠⠞
⠀⠀⠀⢿⣿⣿⣽⡿⣿⣿⣿⣿⣿⣿⠟⣼⡿⢿⢿⣿⠏⡸⣦⠈⣹⣄⠙⢿⠈⢦⢉⣠⣴⣶⠟⠋
⠀⠀⠀⠀⠈⠻⣿⣾⠏⣽⠋⡽⠏⣿⣄⠉⠾⠁⠛⣿⠀⠁⠘⠃⠸⠙⠗⢈⣀⣤⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠼⠿⠌⠀⠠⠿⣿⣿⣷⣦⣤⣀⣄⣀⣀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠛⠛⠉⠙⠋⠉⠉
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⣾⠋⠉⠉⢙⡷⠟⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠙⠁⠀⣠⠋⠀⠄⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠀⠀⠀⠀⢠⡴⠚⠉⠙⠀⢨⣴⠿⢛⢛⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠛⠀⠀⢀⡴⢫⠀⠀⠀⠀⠀⠀⠙⣄⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⡞⠁⠊⠀⠀⠀⢚⣁⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠾⠿⠦⠀⣸⠣⠀⠀⠀⠀⢰⣿⡁⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⢰⠟⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡇⢀⠀⠀⠀⠀⠀⠙⠁⠀⣀⡤⠒⠋⠻⣆
⠀⠀⠀⠀⠀⠀⠸⣧⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⠀⠀⠀⠸⡄⠰⠂⠀⢠⠞⠁⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣶⡾⢅⠑⠄⠀⠀⢹⣤⡀⢠⠏⠀⠀⠞⠁⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⡶⠞⠋⡄⠀⠁⠁⠀⠀⢸⡏⠛⢾⣤⣀⣀⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⢻⣌⢷⡴⠞⠛⠉⠉⠀⠀⡇⠈⠀⠀⠇⠀⠀⠈⠄⠀⠀⠀⠈⠈⢁⢸⡇⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡾⠁⠀⠀⢂⠀⡀⠀⠃⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠠⠂⠀⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⠐⠀⠀⡀⢁⠀⠀⠑⠀⠀⠀⠀⠀⠀⠑⠠⢀⡟⠛⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⠀⠀⠀⠀⠁⠁⠀⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠓⠦⠤⣏⣀⣠⡁⠀⢘⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⡀⠠⣀⠀⠀⠀⠀⠈⠐⠉⠙⢦⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠄⠂⠀⠀⠀⢀⣞⠀⠁⠄⠀⠈⠀⣀⣢⣴⠾⠻⠶⠦⣤⣤⣈⠀⡀⠈⣧⠖⠋⢸⡇
⠀⠀⠀⠀⠀⣰⠟⠋⠁⠀⠀⠀⢀⣠⠶⠋⣨⡿⠛⠺⠗⠚⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠁⠸⣦⡶⠟⠁
⠀⢀⡴⣤⡴⠏⠀⠀⠀⡖⣻⠏⠉⣀⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠤⠴⠚⢻⡀
⠀⢸⡇⠀⢧⠀⠀⢀⡼⡟⠁⠉⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣄⣠⣤⠞⠁
⠀⠘⣧⡀⠈⢧⣠⣾⡀⠙⠢⣄⣰⡟
⠀⠀⠈⠛⠚⠋⠁⠈⠻⣦⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠠⡀⡠⠔⡀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⣧⡀⢀⣴⣶⣶
⠀⠀⠀⠀⢀⣀⠀⡌⠀⠇⠀⠘⡀⣌⡀⠔⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣼⣹⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⡅⢤⡈⡢⢐⢲⠂⣀⡜⠁⡰⠀⢁⡠⠂⡆⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⣶⡆
⠀⠀⠀⠀⠡⠐⠘⢇⢀⡳⢉⠀⠀⢡⡧⠀⢈⢰⣀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠱⢄⠀⡏⠠⣨⠄⣄⡌⣀⡀⣸⠋⢰⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⠠⠦⡄⡀⢙⠀⢸⠟⢁⠈⠁⠠⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠢⡀⠥⡈⠢⡔⠀⠈⡼⠠⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢦⢸⢂⠀⣸⣮⣔⡡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⣤⣿⣯⣶⣽⣤⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣥⡴
⠀⠀⠀⠀⠀⠀⠀⢘⡒⢺⢛⠏⣿⠉⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠁⡋⠂⡃⠙⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠐⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⣶⣤⣀
⠀⠀⠀⠀⢹⣿⣿⣿⣦⡀⠀⠀⣀⣀⣤⣶⣿⠃⠀⠀⠀⠠⣤⣤⣄⡀⠀⠀⢀⣀⣠⣶⣶⣿⣿⠇⠀⠀⠀⠀⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣧⣶⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⣰⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⣠⣴⣶⣿⣿⡶⠂
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠘⠿⣿⡿⠟⠿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠙⢹⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣸⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⣶⣦⣄⠀⣠⣶⣿⣿⠁⠀⠀⠀⠀⠀⣰⣿⣿⣧⡄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⢿⣧
⠀⠀⠀⠀⠸⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⡿⠉⠂


⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦
⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⠀⠀⠀⠀⠀⢰⣀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⣴⡀⠀⣠⣾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣿⣦⢀⣴⣾⣿⣿
⠀⠀⠀⠀⢰⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠈⣿⣿⣷⣶⣤⡀⠀⣾⣿⣿⠀⠀⠀⢸⣿⣿⣶⣿⣻⣿⣿⣿⠃⠀⠀⠀⠀⠀⢰⣿⣷⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡆⢿⣿⠟⠀⠀⠀⠀⠻⠿⢏⡜⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣻⣿⣿⡿⠁
⠀⠀⠀⠀⠈⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠲⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛⠋⠁


⠀⠀⠀⠀⠀⣠⣶⣶⣷⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⢠⣾⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣾⣿⡟⠁
⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠘⢻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⡿⠏⠀⠀⠀⠀⠀⠀⠀⠸⢻⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠂⡀⠤⠀⠚⠛⠲⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⣠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⢹⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣤⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣧⡀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣶⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣧⣿⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢰⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠆⠀⢀⠄⠀⠀⠀⠿⠃⠀⣀⠤⠙⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣼⠏⠈⠀⢀⡴⠋⠀⠀⠀⣀⣤⠖⠋⠁⠀⠀⠉⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⠀⠀⠀⢱⠀⠀⠀⠀⢀⣤⣿⠋⠀⣀⡴⠋⢀⣠⣴⠞⠋⠁⠀⢰⣦⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠆⠀⣀⣴⣿⠟⣡⣴⣾⡥⠶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⠿⠿⠛⠛⠋⠀⠉⠛⠉⠉⠀⠒⠒⠂⠠⢄⣀⣀⣠⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⢰⡇⢀⣴⣶⡿⠿⠛⠉⠤⠀⠒⠒⠒⠲⠦⠤⣄⣀⡀⠀⠀⠈⠛⠀⠀⠀⠈⢻⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⠇⠀⠀⠀⠀⣠⣾⡿⠛⣁⣤⣶⣶⣾⣷⣾⣿⣿⣿⣶⣶⣦⣤⣉⠛⠧⣄⡀⠀⠀⠀⠀⠀⠘⢻⡿⠹⠿⢿⣷⡄
⠀⠐⠲⣄⡀⠀⠀⠀⠀⠀⠀⣻⣿⠀⠀⠀⣠⣾⠟⣩⣴⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⠉⠉⠉⠛⠿⢿⡻⢦⣌⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⡀
⠀⠀⠀⠙⣿⣦⣄⠀⣀⠀⠀⠙⠋⠀⠀⣴⡿⢋⣼⣿⣿⠟⠛⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣝⢷⣄⠁⠀⠐⠷⠆
⠀⠀⠀⠀⠙⣿⣿⣾⣿⠀⠀⠀⠀⢀⣼⡟⣳⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡹⣆
⠀⠀⠀⠀⠻⣿⣿⣿⡟⠁⠀⠀⠀⣾⣿⣿⣿⠋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⣇⠀⢠⣶⡆
⠀⠀⠀⠀⠀⠈⠈⠙⣇⠀⠀⠀⣸⣿⣿⡿⠁⢰⣿⣿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠲⠄⠀⠀⠀⠀⠀⠀⢡⢹⡆⠈⠉⠳⡄
⠀⠀⠀⠀⢤⣀⣀⠀⠀⠀⠀⢇⣿⣿⣿⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⠛⠁⠀⠃⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣆⢸⣿⣿⠀⠀⠀⢠⣴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠘⡈⣿⣿⡀⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡤⠋
⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⢡⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡆⠀⠈⢣⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣿⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠙⠓⣾⠁⠀⠀⠀⠹⡄
⠀⠀⠀⠀⢀⡞⠉⠀⠀⠈⠳⡞⠁⠀⠀⠉⢧⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⠞⠛⠲⣤⣤⣼⠃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠀⠀⢀⣀⣸⡀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⣇⠀⠀⠀⣸⠀⠈⠹⡄
⠀⠀⠀⠀⣨⠧⠤⢤⣠⠞⠉⠉⠉⠻⣏⠁⠈⠉⠳⡄⠀⠀⠀⠀⠀⣠⡏⠁⠈⠓⢶⡚⠁⠀⠀⢠⠇
⠀⠀⠀⡾⠁⠀⠀⠀⢻⡀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣿⠀⠀⠀⣠⠞⠁⠹⣄⣀⣀⣠⠟⠲⠶⠖⠋
⠀⠀⠀⢧⠀⠀⠀⠀⣼⠓⠦⠤⠴⢞⡀⠀⠀⢀⣠⠋⠀⢀⡼⠁⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠈⠓⠦⠤⠶⣧⠀⠀⠀⠀⠀⣹⠻⣏⠉⠀⠀⢀⡞⠀⠀⠀⠀⣀⣤⠴⠖⢒⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣤⣤⣤⠴⠃⠀⠘⢧⠀⠀⣼⠁⠀⢀⡴⠚⠁⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⣠⠤⠶⠒⠒⠲⠶⢤⣄⡀⠀⠘⣇⢠⡇⢀⡴⠋⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠓⠲⢤⣀⠀⠀⠀⠀⠀⠉⠳⣄⢸⣼⣡⠏⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣤⣀⣀⡀⠀⠈⣻⣿⠋⠀⢀⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠉⠉⠉⠁

⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⢞⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢀⡴⠒⡇⠇⠀⡀⡹⠀⠀⠀⠀⠀⠀⡀⠀⠠⠠⠤⣀⠀⠀⠀⠀⢀⠀⠤⢇⠹⡧⣔⢀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠈⠒⠢⡀
⠀⠀⠀⠀⡏⠀⠀⠁⠀⠀⠇⣇⣀⠀⠀⠀⡖⠠⠂⠑⡀⠤⢄⠀⢱⠀⠀⠀⠴⢣⣈⡎⠼⡴⠤⣖⢃⠄⠀⠀⠀⠀⠁⠀⡘⢀⡀⢀⠀⠈⢢
⠀⠀⠀⠈⡜⠆⠀⠀⠀⠀⠈⡁⡎⠀⠀⡐⠃⡆⢀⣬⠀⠀⡌⠆⠀⠁⠀⠀⠑⠱⢒⠌⠚⢇⣈⡃⠙⡀⠀⠀⠀⠀⠀⠀⡇⠀⠳⠂⠀⠀⠀⡇⠄
⠀⠀⠀⡏⢦⠀⠀⢠⠃⠀⠀⢧⣇⠀⠀⠀⡊⠀⡜⠐⠄⠀⠃⢠⠀⡅⠀⠀⠀⠈⠁⠐⣩⡀⣈⡇⠉⠁⠀⠀⠀⠀⠀⠀⢄⠀⠄⣃⡠⠂⠀⠇
⠀⠀⢘⠅⠀⠀⠀⠂⡆⠀⢊⠁⡜⠀⠀⢄⠘⠄⢹⣄⠀⡄⡠⠁⠀⡄⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠈⠂⠄⣈⠠⠤⢊
⠀⠀⠘⠒⠀⢳⠄⠰⠀⠀⠈⢫⠀⠀⠀⠀⠐⠀⢸⠈⠐⡘⠁⣀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠢⠐⠀⠈⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠂⠑⠈⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡄⠀⢠⠠⠀⠀⠀⠐⠀⠀⡠⡀⠀⠀⡀⠘
⠀⠀⠀⠀⠀⠀⠸⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠦⢤⣄⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠲⠀⠀⠀⢀⠧⣀⠈⢣⠀⠀⠀⠀⠀⠔⠉⠂⠈⠀⠙⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⡇⠰⠀⠀⠀⠀⠀⠀⡀⢧⠀⠀⣠⠶⣄⠀⠀⠘⣇⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣀⡤⣄⡀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢧⠈⠀⢧⣀⠀⠀⡤⢦⡶⢃⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⣄⠀⠀⠀⠈⢧⡠⠤⠅⠄⠒⡅⡆⠉⠉⠄⠀⠀⢀⠔⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⢀⡟⠀⠈⣃⣘⠲⡄⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⡀⡁⠀⠀⠀⡰⠀⠸⡄⠀⠀⠀⢀⣀⠴⠀⠀⠀⠀⠀⠀⡇⠀⠀⢣⡀⡀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠑⢦⣌⠃⢸⡢⢣⡀⠀⠀⠀⠀⠀⠀⠘⠀⡀⢧⠓⠀⠀⠀⢰⢠⠀⡇⠀⠀⠤⠊⢀⡸⠀⠀⠀⠀⠀⢰⠀⠀⡀⠀⠋⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠸⡍⠉⡝⠀⡼⠓⠺⢭⠓⡄⠀⠀⠀⠀⠀⠐⠃⠂⠀⠀⠀⠀⢇⠸⢠⠃⢠⠜⠔⣀⠃⢀⡀⠀⠀⠀⠀⡸⠀⠀⠙⢄⡠⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠹⢄⡀⢀⣁⡀⠀⣚⡩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡐⡸⡤⢂⢊⡸⢉⡉⠄⠀⡀⡠⠓⠀⠐⠀⢦⠀⢸⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠈⢀⡴⠂⠈⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢡⡣⠎⡰⣋⠥⠊⣁⡖⠒⠊⠈⠀⠀⠀⠘⢦⣀⠁⠼⠂⢀⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢟⣀⢠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⡶⠯⠭⠤⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠐⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⡼⠀⡀⠀⠀⠀⠀⠀⠀⠀⡭⠖⡋⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇⡇⠅⠀⠀⠀⠀⠀⠀⠘⠒⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢠⡄⣀⣾⠳⢽⠛⢣⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣶⣿⠿⠉⢘⡇⣿⣶⠖⢀⡈⠰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢻⡹⠀⠀⢿⢿⣿⣿⡆⢸⠁⢆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⣴⣷⣶⢀⣼⣷⡦⠇⠀⢸⣿⣿⣷⣤⣤⠀⢻⡇
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⢻⡿⠿⢫⣏⣿⡟⣧⣾⣿⡿⣿⣿⠃⢻⠁⢼⡇
⠀⠀⠀⠀⣿⠀⠀⢀⣧⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠐⡌⢏⢸⣿⣣⢹⣿⣿⠉⣿⣾⠄⠀⢸⡇
⠀⠀⠀⠀⣿⠁⠀⣾⣿⠀⠀⠀⠀⡜⡿⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣷⣢⣿⣦⣽⣿⣿⣾⣿⣿⢸⡿⠶⢄⡀⢸⡇
⠀⠀⠀⠀⠈⠀⠀⣿⡟⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡠⠀⠸⣿⢈⡥⠿⣿⡿⢿⣿⣿⣿⣻⡿⣻⣿⢷⣖⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠⠆⢻⠷⢟⠠⣧⣥⡾⣿⣿⣿⣯⣧⣿⣟⣳⠈⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡌⠁⠀⠀⠀⠐⢻⢓⡟⣩⣷⣹⡿⢿⣿⣉⣾⣿⣽⣦⣡⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣇⣿⣿⣿⠺⢿⣿⣿⣿⡿⣿⠟⣆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢉⣼⣿⡏⠻⣿⡻⣿⡿⢹⡇⣽⠿⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⡀⠉⠙⣇⢰⣿⢧⣿⣧⣿⣿⠻⡟⠆⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⢃⣤⠜⢻⡿⡋⠉⣿⣿⡄⣿⢀⢀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢷⣶⣾⣶⣦⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠴⣦⢹⣹⣴⡿⢿⣧⣻⡟⢳⡝⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⢻⢹⣷⠘⣿⣿⣇⢠⡉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⢿⣿⣿⣶⣶⣤⣤⣄⣀⣄⣀⣈⣿⣿⡀⢹⣿⣿⠰⡟⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣆⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⡠⡤⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡼⠞⠂⠒⠾⠛⠀⣀⢠⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡶⠉⠀⠀⠀⠀⠀⠀⠀⣀⡀⣬⣥⣄⣤⣤⣤⣄⠈⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢰⠊⠀⠀⠀⠀⠀⠀⠀⢀⡤⠟⠛⠉⠀⠀⠈⠉⠽⠟⡛⢿⣟⣈⡥⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⡠⢄⣀⡀⢀⡸⠟⢀⣤⠶⠖⠚⠋⠙⠛⠶⣤⡳⠲⢄⠙⠿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢿⠳⢛⡻⢤⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⣆⣸⣿⣾⠋⠀⠀⠀⡆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡀⠀⠀⠀⢸⡆⢸⠙⡶⢍⢣⠀⠀⠀⡀⠀⠀⠀⢀⠐⠀⠀⠈⡻⢿⠏⠀⢰⡄⢀⠀⢀⠇⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠡⠀⣰⠒⡻⠀⠀⠀⠀⠻⣤⠳⡄⠀⠈⠀⢛⠔⠁⠀⢀⠾⢫⣵⠇⠀⠀⠈⢧⠀⣀⠾⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠄⠉⢰⠇⠀⠀⠀⠀⠀⠙⢦⣈⡢⢔⡀⢸⡄⣠⡞⠁⢀⡝⠃⠀⡀⡐⠀⢸⣿⡯⣂⡀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡄⠀⠀⠀⠄⠀⠀⠘⠁⠙⠞⢏⣲⡷⠋⠀⣠⠞⠁⠀⠀⠀⠀⠀⢠⣿⣧⡟⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠸⡇⡄⠃⡬⠄⢀⡀⢤⡤⠖⠚⢋⣁⠤⣤⡞⢻⠄⠀⠀⠀⠀⠀⠀⣼⠏⣡⠆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣞⢀⣶⣒⣬⠝⣋⣴⠦⠒⠋⠉⠀⠀⠀⡄⢀⠂⠀⠀⠐⠀⢰⣣⠂⠀⢀⡀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢉⡶⠟⠛⠛⠋⠀⠀⠀⠀⢂⠀⠀⠀⠠⡙⠈⡄⢀⠂⠀⢰⡿⠀⢀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠒⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠃⠀⣠⡾⠁⢀⡾⣱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢦⡀⠀⠠⢀⠀⠀⠀⠁⢄⡀⠀⢀⣠⠶⠛⠁⣠⠞⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠙⠷⣤⣄⣉⣀⣀⡤⢤⡄⠠⠛⡁⠀⣠⠞⢡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠁⠀⠀⠈⠁⠈⠁⡀⠄⠂⠀⣡⡴⠞⠙⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠒⠀⣤⡖⠋⠔⠂⠀⠀⠀⠚⣀⣁⣤⣤⣴⣊⠥⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠛⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢦⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⠋⠉⠉⠁⠀⠀⠈⠀⠀⠉⠉⠓⠒⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠳⠎⠉⠉⠉⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠓⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠙⠛⢦⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⣿⣀⣠⣄⡠⣤⣀⡴⠤⣤⢿⠒⢾⢋⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠎⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⢹⠆⠀⠀⠀⠀⠀⠀⠀⣰⠖⠶⠋⠈⠛⠋⠻⠿⣏⡷⠋⠈⠧⣤⠟⢹⡶⠾⠞⠋⠓⠊⠙⢦⣤⣀⣤⠶⠞⠁⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⢀⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠖⢦⡀⢠⠖⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢆⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣇⠀⣸⣿⢛⣷⠶⢆⣠⡄⣠⡀⢀⣀⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠓⠚⠒⠚⢷⣞⡛⠛⢻⡿⣷⣿⣄⡧⣿⡾⡟⢺⡾⠏⠈⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⣷⠹⣼⠻⠃⠀⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠀⠿⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⡏⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣧⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠁⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢸⣿⣝⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣀⡀⠀⠀⠀⣼⣿⣿⣌⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣩⠴⠋⡝⢁⡤⠚⠋⠛⠻⠿⣿⣿⣶⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠖⠁⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠈⠛

⣷⣷⣾⣿⣶⣿⣿⣿⣷⣿⣶⣾⣿⣷⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣾⣿⣶⣷⣾⣷⣾⣾⣿⣷⣾⣿⣷⣶⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣻⣟⣿⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⣀⡈⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣾⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣷⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣿⣿⣿⣿⣿⠏⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡆⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣧⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿
⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠄⠀⠈⠀⠀⠀⠀⠐⠀⡀⢀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠈⠈⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠈⠀⠁⠈⠙⠃⠢⠄⡀⠀⠀⠀⠐⢠⣤⠂⠒⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠐⠀⠀⠀⠀⠀⠀⠑⡀⠈⠂⠀⠀⠀⠃⠙⠀⠀⠀⡰⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⢀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠇⠀⠀⠀⠀⠀⠀⠀⣄⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢀⠀⢠⠎⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠐⠀⠙⢀⠠⠂⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠆⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⡠⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠂⠄⠂⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢁⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠈⠆⠀⠉⠀⠀⠀⠀⠶⠷⠠⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⡈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠆⡀⣀⠤⠀⠀⢠⠀⠀⠀⠀⠀⢂⠀⠐⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡶⣏⠀⠀⠀⠀⠀⠀⢶⠀⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠁⠠
⠀⠀⡀⠐⠀⠀⠀⠀⣀⠤⠐⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⢀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠂⡀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⢀⠠⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠿⠛⡟⢻⡿⢛⡿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠀⣤⡄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠐⠛⠓⣤⡀⠀⠀⠀⠹⢟⣹⣷⠖⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⣓⠻⠶⠤⣄⣀⡤⠒⠛⣽⠛⢆⠶⣐⠛⢍⡁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⣻⣿⢿⢿⢷⡆⢸⠀⠀⠨⢸⡟⠳⣮⡹⡄⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠉⠹⢺⣯⣶⣿⡳⠶⢰⣶⣅⣙⣟⠂⢿⣾⡁⠒⢤⡴⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⣠⠛⢻⣾⣿⣦⣄⣩⣼⠿⢷⡿⠋⢿⣶⣿⡟⠻⠙⢿⣄⠉⠙⠃⠀⠈⠁⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠴⠴⣶⣦⢬⣝⣻⣷⣮⣄⠉⣨⡷⠋⣡⣴⣾⣷⠀⣈⣠⣿⣅⣰⡾⢊⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣀⣷⣷⣦⣽⣟⣋⠉⣹⠷⠋⠀⠠⠏⢡⣴⣿⣧⢛⣿⢷⣿⡄⠀⢻⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⠙⠛⠁⣽⣟⠋⠉⣹⠟⠁⠀⠀⠀⢀⡰⠿⢿⣿⠟⢩⠟⣻⠟⡿⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠤⣿⠿⣟⣉⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠠⣿⣿⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠘⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⡐⠀⠀⢀⡜⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠈⠛⠢⢄⡀⠀⠈⢦⡆⠀⠀⠀⠀⠔⠈⠈⠁⢀⠠⠀⠀⠉
⠀⠀⠀⠳⠌⡙⠲⢤⡀⠀⠀⠀⠀⠀⠀⡅⠐⠀⠀⢠⠎⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣸⡆⠀⠀⠀⠀⠀⣀⠔⠊
⠀⠀⠀⠀⠑⡈⠢⣀⠈⠳⣄⠀⠀⠀⠀⡀⠀⢀⠔⠁⠀⠀⢀⠔⠀⠌⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⢠⠃⡠⠊
⠀⠀⠀⠀⠀⢱⠀⠀⠑⢄⠈⠡⡀⠀⠀⢣⣰⡎⠀⠀⠀⡠⠁⠀⠌⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⠈
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠑⢄⠘⢄⠀⠀⣿⡇⠀⠀⠜⠀⢀⠌⠀⢠⠀⠀⠀⠲⠦⠀⡀⠀⠀⠒⠤⡀⠀⠀⢸⣇⣼⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠳⡌⢆⠀⢹⡇⠀⠎⠀⠠⠂⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⢄⠈⢢⠀⢸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣄⡘⢎⣆⢸⣷⢰⠀⡰⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠔⠀⠂⢸⡿
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣾⣿⠘⢀⠃⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣸⡇⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠈⠙⢄⠠⢁⠐⠀⠀⡀⠀⠀⠀⠈⣿⣿⠀⡟⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⢣⠀⠁⠢⡀⠀⠲⡀⠀⠀⢹⡿⢸⢇⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠠⠂⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠈⠢⡀⠘⡄⠀⢸⣧⡿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢠⢃⠤⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣈⠢⡘⡄⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢧⣥⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠀⢀⣴⣦⡀⠀⠀⠀⠄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡃⠰⠂⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡤⠖⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⠀⠀⠰⠆⠘⠛⠛⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣿⠀⠀⠿⠥⠦⠶⠤⠴⠧⠴⠀⠀⣿⠂⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠶⠶⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠶⠶⠶⢶⣤⣄⡀⠘⠛⠃⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠐⠆⠀⣰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠉⠻⣦⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⢠⡏⠀⢸⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠇⠀⠀⠀⠀⢸⣿⠀⢻⡄
⠀⠀⠀⠀⠠⡆⣋⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣛⢰⠄
⠀⠀⠀⠀⠐⠆⣡⠀⢸⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣌⠰⠂
⠀⠀⠀⠀⠀⣓⣠⠄⢸⡇⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠠⣄⣚
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⡀⠀⠄⠀⠴⠂⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⣤⣤⣀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠼⠇⠀⠀⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣄⣴⣾⣠⣄⡔⣡⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣯⠎⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣗⣨⣽⣻⣿⣿⡿⠥⢤⡀⠀⢹⡟⣠⡩⣡⠀⠀⠔
⠀⠀⠀⠀⠀⠀⢀⡞⢠⣤⣶⣿⣾⣿⡟⡉⠛⢉⣠⣬⣵⡶⠚⠋⠉⠉⠉⠛⠛⠿⣿⠟⡀⣸⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣷⠟⠛⢛⡻⣿⠁⢘⣿⣿⣿⡟⠁⠀⠈⠸⣘⠀⠀⠀⠀⠀⠀⠙⢩⣿⣶⣞⣃⣀⠠
⠀⠀⠀⠀⠀⠀⢾⡿⠛⢀⡲⣿⣿⡟⠀⠲⣿⠃⠃⠁⣴⢣⠖⠋⠁⠀⠀⠀⠀⠀⠀⢀⣤⣀⠙⣿⢿⣿⠁
⠀⠀⠀⠀⣀⠀⠋⠀⠀⢰⣿⣿⡟⣿⡀⢀⣹⣧⣤⣟⣵⣯⣄⣤⣄⣀⠀⠀⠀⠀⢢⡠⡝⢿⣷⣿⣿⡷
⠀⠀⠀⠊⠀⠀⠀⠐⣰⣿⣿⢿⡇⠘⠻⠿⠋⠀⠈⢿⣅⠀⠀⠀⠀⠉⠛⢶⣤⡀⠀⣅⠘⠊⢈⠿⠟⢒⡀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠈⢀⣀⠠⣴⣾⣿⡟⢸⣇⣤⢰⣾⣦⣤⡀⠀⠉⠀⠀⠀⠀⠀⢀⣤⣬⣟⠢⠬⠗⠀⢸⠀⢻⣫⠆⢀⣠⣾⠉⠀⠰⠃
⠀⢸⠀⡓⠀⠈⠙⢿⣿⣿⡇⢸⠛⠈⠸⣿⠹⣿⢩⣿⣿⡶⢤⣶⡿⢋⡽⠻⠻⣿⣦⣤⢶⡶⠋⠀⢸⡿⠀⣼⡿⠃⠀⠀⡠
⠀⠀⠁⢉⡉⠈⣻⣻⣿⣿⡇⠘⡆⢰⠀⠹⠀⡈⠈⠹⠿⠐⣽⠁⠀⡄⢀⠀⠀⠙⢻⡄⠀⠹⣀⠀⣼⣷⣾⣿⡁⠀⠒⠢⣇
⠀⠀⠀⠀⠀⢀⡭⣽⣿⣿⣿⡄⠀⠀⠀⠀⢶⡇⠀⠀⠀⠀⠃⠀⠸⠀⠎⠐⠁⠀⠀⣿⡄⢠⠈⣰⣿⣿⣟⠐⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠘⣫⣽⣿⣿⣿⡄⠀⠀⠀⡆⢧⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡆⠀⣿⠇⣸⣴⣿⣿⣯⣩⣓⡀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⢈⡩⠽⠿⣿⣿⣦⡀⠀⠘⠀⢷⠀⠀⠀⢀⡠⠊⢀⠀⠀⡴⠀⢸⣿⣾⣿⠟⠻⣌⠁⠈⠉⠉⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠘⠁⠤⠤⠉⠁⣿⡿⣻⣶⣄⡀⠘⠃⠶⢛⣉⣤⡾⠛⠀⢈⣠⣶⣿⡿⢗⠙⠇⠱⢸⡆⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠐⠉⠉⠙⠒⠄⠀⠀⠉⠈⠋⠉⠙⢻⡿⠛⠲⢶⣶⣶⣶⣶⣿⡿⡿⣿⠫⡀⠀⣠⠞⠛⠟⠁⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠞⠋⠀⠀⠀⠀⠙⠫⣌⡙⠃⢀⣠⡴⠢⠔⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣅⣭⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠈⠃⠠⠀⠐⠀⠀⠀⠀⠳⣆⠀⣠⠀⠀⠀⠀⠀⠀⠶⢀⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⣤⣴⢄⠀⠀⠀⢀⠀⠀⠠⣤⡠⠟⢀⠀⠀⠀⠀⠶⠀⢉⡁⠀⣄⠀⠉⠀⠀⠀⠀⠀⢿⣷⡄⠑⠀⠈⠠⡄⠓⠓⠂⠀⠀⠀⢈⡏⠁
⠀⠀⠐⢿⠿⠋⢠⣹⣶⣏⡀⠀⠀⠈⠓⠀⠇⠀⠀⠀⠠⠀⠀⠘⠁⠀⡀⠀⠙⠸⠃⠀⠀⠀⢦⣌⠻⠀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⡀⠀⠀⠀⠈⠛⣿⠅⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠈⠙⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⢹⣾⡷⡄⠀⢠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⢀⣀⡀⠀⠐⢈⢨⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠂
⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⣀⡀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡤⠀⠀⡀⠰⠐⠃⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⢀⢸⡗⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⡀⢠⠀⠉⡞⢀⣀⠀⠀⠀⠈⣿⢀⡞⠀⢀⣀⠀⠀⠀⢸⡄⢀⡠
⠀⠀⠀⢼⡯⠁⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠀⠀⠀⣀⡬⠉⠁⠀⠀⠀⣿⡜⡇⣠⡟⠁⠀⠀⠲⣾⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠘⢇⡡⡿⠀⠀⠀⠀⠀⠉⠁
⠀⠄⠀⠀⠀⠐⠓⣾⠂⠸⢿⡆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⢠⢀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⢠⣦⣤
⠀⠀⠀⠀⠚⠛⠡⣾⣟⡳⢄⣻⡼⣃⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠳⠊⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠘⠃⡴⠂⢀⡀
⠀⠀⢠⣄⠀⠀⠘⠉⠀⠉⣵⡾⠉⠉⠁⠀⠀⢾⣧⣷⡄⠀⠀⠚⠃⠠⠾⠲⡀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⢿⣿⣿⠃⠄
⠀⠀⠻⡿⠃⠀⠀⠀⠶⠀⠉⠀⠀⠀⠀⠀⠀⢿⡽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡿⠗

⠀⡀

⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢰⢶⠂
⠀⠀⢀⡿⠐⡱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⢀⠶⠁⣾
⠀⠀⣼⡇⠀⢧⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢀⠀⣸⠷⠂⢀⣿
⠀⢰⣻⠀⠀⠀⢧⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡟⠁⠀⠀⢸⢿
⠀⡌⣿⠀⠀⠀⢸⡀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢧⠀⢸⣅⠒⠤⣀⠀⠀⠀⣰⠁⠀⠱⣼⡴⠊⠀⠀⣼⢸
⠀⡇⣿⠀⠀⠀⠀⢣⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡘⠀⠘⣧⠘⡎⢻⡀⡈⠳⣄⠀⡇⠀⢠⢰⠃⠀⠀⠀⣸⠃⠎
⠀⢰⢻⠀⠀⠀⢀⠈⡄⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠒⠊⡡⢫⡇⠀⠀⠀⠀⠀⠃⠀⠈⢿⢣⣇⠀⠳⣶⠀⠘⣆⡇⠀⠀⣛⡠⠀⢀⡼⠁
⠀⠸⡸⡇⠀⠀⠀⠑⣧⢀⠀⣸⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⡠⠞⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣣⣿⢦⠀⠬⣦⠀⠸⣿⠀⠀⡏⠀⣠⠞
⠀⠀⢃⢻⡀⠀⠀⠀⠘⡎⠀⡏⠀⠀⠀⣠⠞⠁⠀⠀⢀⣴⠋⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⣇⠀⠀⢄⠹⡌⡆⠑⢤⡈⠳⡆⡇⣇⢸⣧⠊⠁
⠀⠀⠀⠀⠳⡄⠀⠀⠰⡇⢀⠃⠀⠀⡴⠁⠀⠀⣠⠞⠉⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠙⢿⣱⠀⠀⠉⠳⢵⣾⣸⡟⠁
⠀⠀⠀⠀⠀⠙⢦⡀⠀⢷⢸⠀⠀⣸⠁⠀⣴⠞⠁⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⣀⠤⠖⠚⠉⠳⣄⠈⠢⣿⡄⠀⠀⠀⠀⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣸⡆⠀⠀⡏⢠⡞⠁⠀⠀⠀⢀⣠⠾⠋⠀⠀⠀⡠⠔⢋⣀⡀⣠⣀⠀⣐⣄⡙⠢⢌⠻⣄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣇⢰⣰⣧⠏⠀⣀⣤⠴⠞⠋⠀⠀⠀⠀⠐⠛⠷⢯⣉⡁⠈⠉⠀⠊⣉⣉⣭⡭⠬⢽⣿⣷⣄⠀⢈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣾⣿⡿⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠒⠂⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠶⠁

⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣆⠀⠀⠀⠀⣿⣷⣶⣶⣦⡄⠀⠀⠀⠀⣠⣴⣾⣿⣷⣶⣄
⠀⠀⠀⠀⠠⠛⠻⠿⣿⣿⣿⠇⠉⠂⠀⠀⠹⣿⣿⣿⣿⣏⣀⠀⠀⠀⠈⠙⠿⠿⣿⡿⠿⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⣀⣄⣠⣄⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣦⣤⡤
⠀⠀⠀⠀⢈⠿⢿⡿⠿⠋⠀⠀⠀⠀⠀⠈⠹⣿⣿⡿⠟⠛⠛⠀⠀⠀⠀⠉⠂⠹⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠠⣀⣄⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⢀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠐⢿⣿⣷⣿⣧⣶⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡆⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠉⠽⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢿⣿⣿⡿⢛⠁⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠈⠂⠀⠀⠀⠀⠈⠉⠉⠁⠀⠉

⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀
⠀⠀⠈⢀⠠⡄⠴⠠⠀⠀⠀⢀⠤⣀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⡄⠉⠁⠀⠀⠀⠀⠀⢰
⠀⠀⠈⠈⠀⡀⠀⣀⢀⡀⠀⠀⠀⠀⠁⠀⠀⠄⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠄
⠀⠀⠀⠀⠀⠀⡶⠛⢯⠛⣢⠀⠀⠠⠁⠀⠀⠀⣀⢂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠁⠈⠀⠀⠀⠀⡐⢀⠀⠀⠄⠀⠀⠣⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣇⣀⠀⠀⠠⣾⠇⠀⠈⠈⠉⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠠⠀⠀⠀⠀⠀⠀⠛⠉⠑⠀⠀⠈
⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠥⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠴⠈⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠋
⠀⠀⠀⠀⠁⣄⠀⠀⠁⠀⡆⠀⠀⠀⠀⠸⠀⢀⠀⠀⠦⠄⠀⠀⠀⠢⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⡀⠀⠈⠋
⠀⠀⠀⠀⡰⠉⠉⠀⠀⠀⠁⢄⣀⣤⠰⠊⠀⠄⠀⠸⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢀⠀⠀⠰⠶⠁⠀⠀⠀⠀⠀⠉⠀⠛⣉⠄
⠀⠀⠀⠀⠀⠐⠄⠀⢖⡀⠀⠀⠀⠀⠀⠤⠈⠀⠐⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡄⢸⠃⠀⠒⠀⠀⠠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⢠⠀⠆⠀⠀⠢⠀⠠⠀⠀⠀⠀⠀⠀⡀⠁⠀⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡤⠆⠀⣇⡼⠀⠀⠀⢀⢄⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠘⠁⠃⠀⠀⠀⠀⠀⢠⡄⠀⢀⡠⠤⠐⠒⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠉⠀⠀⠀⠆⠘⠈⠴⢃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⡂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠔⠛⠀⠄⠀⢀⠀⠈⠂⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⡀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⡴⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠈⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠠⢀⡀
⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⢚⣄⠀⢨⡆⠘⠊⠀⠠⠁⠀⢀⢠⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⣀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⢤⡄⠀⠀⠀⠁⠵⡋⠀⠀⠀⠳⢄⠀⠀⠀⢀⡠⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⣾⡯⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⣀⡁⠀⠀⠀⠂⠐⠀⠀⠂⡀⠀⠀⠀⠁⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⣀
⠀⠀⠀⠀⠀⠌⢄⠀⠀⠀⠀⢩⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠠⠀⠃⠀⠀⠁⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠁⠂⢀⠂⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⣄⠀⠀⠀⠀⠀⡀⠈⠀⠑⠴⠃⢛⠆⠤⠌⡀⡀⡀
⠀⠀⢀⢄⣀⠀⠀⡀⢀⠤⠀⢤⠄⡄⠀⠀⠠⣤⡊⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⠀⠃⠀⠀⠀⠀⢀⠀⠀⠂
⠀⠀⠐⡐⡉⣆⣀⠀⠀⠀⠋⠀⠄⠀⠠⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢜⢀⠁⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠐⠁
⠀⠀⠀⠉⠉⠈⠀⠀⠀⠀⠒⠀⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀
⠀⠀⠐⠀⠀⠀⢀⡀⠆⠀⠠⠘⠃⢀⠀⠀⢀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠂⢠⠀⠀⠠⠀⠉⠀⠰⠀⡄
⠀⠀⠀⡄⠀⠀⠈⠀⠀⠀⠀⠂⠠⠄⠐⠀⠀⠀⠈⠀⢰⠀⠠⠈⢀⠀⠁⠀⠀⠂⠐⠄⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠣⠀⠀⢀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⠋⠉⠉⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡟⠁⠀⠀⠀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠏⠀⠀⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠏⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⢀⣤⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏⠀⠀⠀⠀⣴⣿⠏⠀⠀⣀⣤⣿⣟⠁⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⠀⢼⠟⠁⠀⢠⣾⣿⡿⢻⣇⠀⠀⣸⡇⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠁⠀⠙⠻⢶⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢠⣶⡶⢶⣄⣀⣿⣿⡿⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠙⠛⢻⣿⢹⣿⣿⠇⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣼⠇⢸⣿⣿⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⣼⡟⠀⠀⢿⣿⡀⠀⢀⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠀⠀⢀⣾⡟⠀⠀⠀⠈⠻⣷⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣶⡿⠋
⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⢰⣿⠛⠶⡄⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⣦
⠀⠀⠀⢀⣾⣿⡿⠛⢻⣦⠀⠈⠻⣄⣀⣿⡄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⢷⣄⠀⠀⠀⣠⣾⣿⣿⠿⢷⡄⠀⠀⣰⣷⠀⠀⣰⡿⠋⢠⣿⣿
⠀⠀⠀⣾⣿⡿⠁⠀⢸⡿⠀⠀⠀⠈⠙⣿⡿⢿⣿⣷⠀⠀⣼⣿⣿⠟⠁⠀⢸⣿⡆⠀⣼⣿⣿⠟⠁⠀⣰⣿⠀⢠⣿⡟⠀⣼⡟⠁⢠⣿⣿⠇
⠀⠀⢰⣿⣿⠃⠀⠀⣿⠃⠀⠀⠀⠀⢠⣿⢡⣿⣿⠃⠀⣼⣿⣿⠏⠀⣀⣴⣿⠟⠀⣸⣿⣟⠁⠀⢀⣴⡿⠃⢀⣾⣿⢃⣾⠟⠀⢠⣾⣿⡟
⠀⠀⢸⣿⡿⠀⠀⢠⣦⠀⠀⠀⠀⢠⣿⢣⣿⣿⠏⠀⠀⣿⣿⡟⠛⠛⠛⠉⠀⠀⣠⣿⣿⡟⠲⠾⠟⠋⠀⣠⠿⣿⣿⣾⠏⠀⠀⣾⣿⣿⠁
⠀⠀⠈⣿⡇⠀⣠⣿⣿⡆⠀⢀⣴⡟⠁⣾⣿⡟⠀⠀⣴⢿⣿⡇⠀⠀⠀⢀⣠⡾⠋⢹⣿⣇⠀⠀⣀⣤⠞⠁⠀⠻⠿⠋⠀⠀⣼⣿⣿⠃
⠀⠀⠀⠙⢿⣾⠟⢻⣿⣇⣴⠟⠋⠀⠀⢻⣿⠁⣠⡾⠋⠀⠙⢿⣦⣤⣶⠿⠋⠀⠀⠈⠻⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⣴
⠀⠀⠀⠀⢀⣶⠏⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⢀⣾⠏
⠀⠀⠀⣴⡿⠁⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢀⣴⡿⠁
⠀⢀⣾⡟⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉
⢀⣾⡟⠀⠀⠀⣼⣿⣿
⣼⣿⠁⠀⠀⣰⣿⣿⡇
⣿⣿⠀⠀⣰⣿⣿⡟
⠹⣿⣤⣾⣿⡿⠋
⣤⣬⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣄⡀⣀⢠⣀⣀⡄⠀⠀⠀⠀⠀⢄⣄⣀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⠠⠤⠤⠤⠀⠀⠀⡀⠀⢠⣠⣀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⣀⡄


⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡄
⠀⠀⠀⠀⠃⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⢙⠃⠀⠁⠂⠀⠀⠀⠀⠀⠀⠉⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣤⠆
⠀⠀⠀⠀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡄⠀⠀⠀⠀⠀⢰⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⠀⠀⠀⠀⢀⡄
⠀⠀⡶⠚⣿⣿⣿⠳⡆⠀⠀⠀⠀⢰⠒⣿⣿⣿⠒⡆⠀⠀⠀⠀⢰⠖⢻⣿⣿⡗⢶⠀⠀⠀⢰⡞⣿⣧⣾⡜⢲⠀⠀⠀⠀⠈⣼⣷⣄⠀⡀⠀⣿⠁
⠀⠀⡇⢀⡿⠿⠿⠀⡇⠀⠀⠀⠀⢸⠀⠻⠿⣿⠀⡇⠀⠀⠀⠀⢸⠀⢸⠿⠿⠇⢸⠀⠀⠀⢸⡅⠹⠟⠻⠗⢸⠀⠀⠀⠀⢠⠛⠿⠟⠀⠁⠀⠈⢀⣤
⠀⠀⠛⠒⠒⠒⠒⠚⠃⠀⠀⠀⠀⠘⠓⠛⠒⠒⠒⠃⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠛⠀⠀⠀⠘⠓⠒⠒⠒⠒⠛⠀⠀⠀⠀⠘⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠂⠐⠀⠄⠀⠤⠴⠦⠄⠀⠀⠀⠀⠠⠤⠐⠴⠤⠦⠴⠀⠀⠰⠀⠶⠦⠆⠰⠀⠶⠶⠀⠀⠀⠂⠰⠦⠤⠀⠂⠠⠴⠂⠀⠀⠂⠀⠐⠀⠤⠲⠴


⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈
⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣀
⠀⢸⢡⣤⡀⢲⣿⣷⣿⠁⠀⠀⢸⠋⢡⣤⡀⢻⣿⣷⡇⠀⠀⠀⡏⢡⣄⡀⢺⣿⣷⡏⠀⠀⠀⣿⠁⡀⠀⣤⣶⣦⣿⠀⠀⠀⢸⠉⠀⠀⢠⣴⣿⣿⡁
⠀⢸⣈⠙⣿⣿⠛⣋⣹⡆⠀⠀⢸⣇⠈⠙⠻⡿⠋⢁⡇⠀⠀⠀⣇⡈⡙⣻⣿⡛⣁⣿⠀⠀⠀⣏⣘⣻⣷⣼⣿⢛⣹⠀⠀⠀⢸⣄⣛⣻⣿⣿⡟⣹⡆
⣤⣬⣭⣭⣭⣭⣭⣭⣭⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⣭⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣤
⣿⣿⣋⣝⣛⣩⣛⣻⣻⣽⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⢐⠒⢤⡀⠀⡠⠀⠀⠀⠀⠘⠿⣿⡀⣴⣶⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⣠⠀⠀⠀⠈⠢⣠⣁⠈⠀⠂⠀⠀⠀⢠⣶⣾⣿⡿⢻⣴⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣰⢋⣠⣄⠀⠀⠀⠀⣇⠀⢸⠏⢀⣾⠃⠀⠀⠀⠀⠀⠈⠙⣼⠆⠀⠀⠀⠀⠀⢉⣿⣧⣽⣿⠟⢁⡀
⠀⠀⢰⣦⣤⣄⡀⠹⡟⢀⣴⣾⣿⠁⠀⠀⠀⣠⣉⠛⡟⠻⡿⠟⠀⠀⠀⠀⣿⡀⣨⡴⠟⢁⡤⠀⠀⠀⠀⢃⠀⣢⡸⡄⠀⡠⠀⠀⠀⠀⠙⣻⣉⣱⣴⣿⡟
⠀⠀⠀⠙⢿⣿⣿⣄⣷⣿⡿⠟⠁⠀⠀⠀⠀⠿⠿⢷⡷⣿⣿⡇⠀⠀⠘⣆⠘⢷⡇⠀⣠⡟⠀⠀⠀⠀⠀⠀⠉⠈⠙⣧⣈⠀⠀⠀⠀⠀⠈⠻⠿⠿⡏⣡⣤⡄
⠀⠀⢠⣤⣄⣀⠀⠀⡇⢀⣤⣶⣿⠃⠀⠀⢰⣶⣶⣼⣃⣤⣤⣄⠀⠀⠀⢿⡄⠀⣥⠾⠛⢀⡔⠀⠀⠀⠀⠀⠤⢀⠀⢻⠟⠁⠀⠀⠀⠀⠀⢿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠻⢿⣿⣷⣄⣧⣾⣿⠿⠁⠀⠀⠀⠈⡙⠛⢹⡟⠛⠿⠋⠀⠀⢳⡌⠻⣤⠀⠀⣰⡟⢀⠀⠀⠀⠀⠀⠢⣠⡂⣸⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⣧⣶⣿⡷
⠀⠀⣀⣀⣀⣉⡉⠉⡏⠁⣀⣤⣤⣤⠄⠀⣾⣿⣿⣾⣧⣶⣿⣷⠀⠀⠀⢿⣄⠨⡴⠞⢋⣴⠃⠀⠀⠀⠀⠀⡀⠀⠈⣿⠠⠊⡈⠀⠀⠀⠀⣈⣉⣉⣏⣩⣵⣤
⠀⠀⠈⠻⢿⣿⣿⣦⣥⣾⣿⡿⠟⠋⠀⠀⠈⠉⠉⠁⡏⠙⠛⠋⠀⠀⠀⠀⠙⠷⢀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠄⠑⠄⡿⠛⠈⠀⠀⠀⠀⠀⠻⠿⠿⡟⢛⣛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇



⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣆⢸⡇⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣝⡇⠀⠀⠀⠀⠀⢸⡇⠀⣰⠦⠀⠀⠀⠀⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢠⡀⣿⣼⡇⣾⠃⢀⣀⡄⠀⠀⠀⣠⣀⣀⠀⡇⡜⣰⠃⠀⠀⠀⣀⡀⠘⣧⠞⢁⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⣀⠀⠀⠀⠐⢆⠀⠀⠀⠀⢀⢴⠃
⠀⠀⠀⢀⠈⣷⣸⣿⣟⣥⡾⠟⠁⠀⠀⠀⠀⠰⣛⢦⣳⣼⠟⣓⣶⡤⠀⠀⣿⠀⢀⡏⣠⠞⠃⠀⠀⠀⠀⣴⡈⠻⡟⣠⣾⡿⠀⠀⠀⢀⡁⠁⠀⡇⢀⠊⢆⡀
⠀⠀⠀⠈⢷⡜⣿⢿⣩⣴⡶⠟⠋⠁⠀⠀⣤⣠⢬⣁⢸⠛⣭⡕⠚⠀⠀⠀⠻⡀⡼⠋⠁⢀⡀⠀⠀⠀⠀⢹⣿⣦⣽⡿⠟⢁⠀⠀⠀⠀⠗⢣⢀⣇⡄⠂⠺⠁
⠀⠀⠀⠳⣌⠻⣞⣯⠴⠾⣿⣛⣉⠀⠀⠀⠘⠮⣓⢾⣾⣯⠤⢿⠟⠀⠀⠀⠀⢙⡇⠀⣠⠟⠃⠀⠀⠀⣶⣄⠙⠛⣇⣤⣶⡿⠀⠀⠀⠰⡄⢂⠀⠛⠡⠁⢡⠆
⠀⠀⠘⢦⣝⢷⣝⠷⠚⣛⣛⣉⣥⠀⠀⢴⢶⡒⠢⣄⣋⡩⠯⠥⣤⠀⠺⡆⠀⢸⠤⠞⠁⠀⠀⠀⠀⠀⠘⢿⣿⣦⡿⠿⣋⣵⠀⠀⠀⠀⢊⠠⠐⣾⡇⡉⡈
⠀⠀⠐⢦⣝⢷⣌⠛⣛⣻⣯⣭⡤⠀⠀⠈⠳⢌⣙⢾⣷⣒⣛⡽⠋⠀⠀⠙⢦⣸⠀⠀⣠⡀⠀⠀⠀⠀⣷⣦⣉⢹⣴⣾⣿⠏⠀⠀⠀⠸⢄⠡⠀⢸⡐⡰⠺
⠀⠀⠲⢦⣝⡿⣮⣉⣩⣭⣷⡶⠶⠀⠀⢺⣽⣝⡲⢼⡤⢞⣓⡶⡶⠀⢿⡂⠀⢹⣀⡴⠋⠁⠀⠀⠀⠀⡈⠿⣿⣿⢉⣩⣥⣶⠇⠀⠀⠀⠂⠡⠈⠿⠋⠁⠁
⠀⠀⠀⠀⠉⠛⠶⣭⠤⡾⠾⠿⠛⠁⠀⠀⠑⠮⠭⢿⠯⠭⠵⠚⠁⠀⠀⠳⣄⣸⠁⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣬⡿⠿⠟⠁⠀⠀⠀⠀⠀⠡⣀⣰⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠐⠀⠁⠄⠀⠀⠀⠐⠀⠀⠐⠈⠌⠎⡁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠩⠈⠀⠀⠀⠀⢄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⢂⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠆⠀⠀⠀⠀⠀⠈⠃⠀⠀⢊⣶⠿⠁⠀⡁
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠘⡀
⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠰⠄⠢⠀⠂⠀⠀⠀⠀⡠⢐⠀⠁⠀⠀⠀⠀⠃⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠄⠀⠀⢠⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⡀⠴⢆⣂⡤⠀⠀⢈⠀⠀⠃⠀⠀⠀⠀⠀⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠃⢌⡠⠂⠀⠀⠀⠈⠢⠐⠁⠀⠀⠀⡀⠀⠀⠠⠈⠀⠀⠀⠀⠈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠡⠁⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡊
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡠⠀⠀⠀⣰⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄
⠀⠀⠀⠀⠉⡀⠀⣀⠐⠁⠀⠀⡀⢀⡕⠀⠀⠀⠀⠀⠁⡀⠐
⠀⠀⠀⠀⠘⠡⠈⠠⡀⠂⡠⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⢀⠘⠀⠈⠁⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠂⠀⠀⠀⠐⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠄⠀⠀⠀⠀⡐⠁⠀⠁⡀⠀⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢄⠀⠙⠀⠀⠀⠸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀
⣿⣿⣿⣿⣛⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣻⣿⣿⣛⣛⣿⣟⣛⣛⣻

⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠗⢨⣉⣣⡀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⠤⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣮⣙⣷⣤⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⣄⢠⠂
⠀⠀⠱⠬⣾⣷⣶⡀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⣿⣄⠀⠀⠀⠀⢠⣿⢃⣀⣀⡀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠶⠤⠀⡤⠚⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠶⢶⡶⠞⣱⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣳⠊⠉⠀⣰⣁⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿⡟⢋⢀⣰⣧⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⣷⣶⣿⠻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣾⣿⣶⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠴⠒⠲⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠖⠶⢦⣄
⠀⠀⠀⠀⡠⠙⢎⠒⠒⠒⠞⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣿⡛⠲⠶⠾⠿⣦
⠀⠀⠀⠀⣄⠀⢠⠀⠀⡀⠀⠐⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡍⢢⣀⠤⠴⢞⢷⣄
⠀⠀⠀⠀⠈⢦⣼⡀⢀⠉⠒⣄⣨⣌⣻⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣿⣇⣰⣝⠻⣦⣾⣿⣿⣷⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠛⠹⠠⠿⠷⠿⠿⠿⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⣦⠀⠀⣴⠋⠀⢠⡏⢀⡴⠶⢶⡄⠀⠀⠀⠀⠀⠀⣴⠶⢦⣄⠸⣇⠀⠈⢷⡄⠀⢸⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣿⢀⣾⡇⢀⣠⣾⣷⡏⠀⣀⡼⠃⠀⠀⠀⠀⠀⠀⢳⣄⠀⠘⣷⣿⣦⣀⠀⣿⣆⢸⡄⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢴⣯⣿⠏⠁⢀⣿⠋⠁⠀⣸⣯⠞⢻⡆⠀⠀⠀⠀⣾⠳⣮⣧⡀⠀⠈⢻⣅⠀⠈⢿⣿⣷⠀⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⣿⡉⠀⠐⢻⡏⠀⠀⢾⡿⠃⠀⣼⠃⠀⠀⠀⠀⢻⡄⠈⠻⣿⡆⠀⠈⢻⠁⠂⠀⣽⣿⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠹⣿⠁⢀⣀⡾⠁⣀⣠⣿⠁⠀⢶⠇⠀⠀⠀⠀⠀⠀⢳⡄⠀⢸⣧⣀⣀⠸⠧⢀⡀⢸⣟⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠘⠻⠚⠁⠀⠀⠀⠉⠉⠉⠀⢤⣼⠀⠀⠀⠀⠀⠀⠀⢸⣇⠤⠈⠁⠈⠁⠀⠀⠀⠈⠛⠋⠂⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⣀⣀⣀⣀⣸⣧⣠⣶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠤⠤⠴⣿⠀⠀⠀⠀⠀⢀⣿⠤⠤⠄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣿⣠⣤⡤⢤⣤⡀
⠀⠀⢠⡞⠉⠉⠉⠉⠙⢿⡟⠛⠛⣽⣿⣏⣉⠒⠂⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠒⢉⠉⠁⠀⠀⠀⠈⣽⠟⠁⠀⠀⢀⣀⣿
⠀⠀⠀⠙⠛⠶⣦⡀⠀⠀⠙⠶⣤⣬⣭⣶⣿⡇⡢⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠶⠟⠁⠀⠀⣠⡞⠋⠉
⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⡆⣄⡀⢀⡀⠀⢿⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢱⣿⣷⡀⠀⠀⠀⠀⠀⠀⠈⠉⠾⣿⠷⣹⣿⡆⣾⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠋⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢿⣇⣤⡀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢫⣭⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡟⠻⢿⠃⠀⢹⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣩⣤⡙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢿⣿⠇⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢺⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢟⣃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⡧⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠁⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣶⠄⢀⣀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⣿⣿⡯⣽⣿⣤⣭⣄⣈⣽⡶⣾⡻⢷⣦⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⣿⠃⠀⢠⠟⢠⣿⣻⠿⣓⡊⣹⣿⡟⣤⣫⢻⣉⡃⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⠄⢾⣷⣶⣿⡈⡟⣿⣇⢻⣿⠘⣿⡇⢘⣧⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠸⡧⠀⠀⠀⠀⠀⣀⠀⡀⠻⠿⠿⠓⡟⣥⢠⠊⢈⠘⣷⣥⠘⣿⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠗⢠⡀⠀⠀⡀⠀⠢⠀⠉⠛⠀⠀⠸⠀⠠⣼⣀⡄⠸⠃⠀⡼⣟⣏⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠃⠀⠀⡇⠀⠀⠀⢤⣀⡀⠀⠀⠀⠠⣧⠛⠀⢹⢇⣼⣇⡸⠿⠋⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠙⠿⠉⠁⠀⠀⠉⠀⠠⣌⣿⡟⢋⠀⠀⠀⢸⣷⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠅⢀⣀⣠⣷⣄⣾⣿⣿⠛⠁⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠷⠄⡄⣀⡉⢩⡾⠃⠀⠀⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡉⠻⣶⣾⣶⣾⣷⣿⠣⣾⣡⡤⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣓⠂⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⣿⢿⣿⣧⡿⠋⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣿⡴⢁⣠⣶⡫⠀⠲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠀⠄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠚⠛⠛⠛⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠊⠹⠋⢾⣶⢶⠖⠛⢛⣿⣟⡁⠀⠈⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢉⡙⠟⠳⡈⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⠿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠴⢲⣴⡘⠗⢻⠾⠗⡖⡀⣿⡦⡆⠀⠀⠀⠀⣿⣿⣿⡏⡍⠋⢱⣄⣄⣈⢠⡅⢿⡇⢘⢹⣿⣿⣿
⠀⠀⣿⣴⡔⠋⡻⣶⡯⣗⣥⢸⢷⡮⠐⣾⡇⠀⠀⠀⣿⣿⡑⠛⠛⡤⡦⡉⢡⠪⠅⡏⢠⠁⣮⡍⠘⣿⣿
⠀⠀⠠⣯⣱⣬⣼⢼⠹⣉⡾⢽⣠⡶⢹⢳⡗⣤⡄⠀⣿⣿⣿⠐⠾⠛⠘⢃⣾⠰⢊⡀⠟⢃⡷⢈⢠⡹⢻
⠀⠸⣿⡙⠭⡧⡽⢤⣁⢻⣱⡏⢠⢏⡰⣦⣔⣎⠀⠀⣿⣿⠀⠙⣒⣀⢃⡙⠲⡅⠘⢱⡛⡀⢞⡁⠋⢡⣿
⠀⠀⠉⠱⣶⠦⠓⠒⠻⣿⣯⠤⠿⠟⠉⠉⠙⠁⠀⠀⣿⣿⣷⣮⡈⣙⣦⣭⣅⡀⠀⢛⣃⣨⣴⣶⣦⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣁⣀⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡐⡀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢂⢀⢤⠊⠀⠩⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢿⠀⢀⠀⠀⣠⠐⠱⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠹⡨⠄⠉⠁⠀⠀⠀⠄⢠⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠆⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠨⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠠⡀
⠀⠀⠀⠀⠀⢀⡠⡀⠀⠀⠀⠀⠀⠃⠌⠠⠀⠀⣀⣀⣀⢠⡀⠀⠤⠆⢒⠀⠀⠀⠀⠔⡀⠌⠀⠀⠁⠘
⠀⠀⠀⢀⡌⡄⠀⠀⠀⠀⠂⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀
⠀⠀⡠⠁⠐⠤⠀⠀⠀⠂⠀⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠄⠀⠊⠐⠠⡀
⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠘⠀⠀⢂⣠⡀⢠⣀⠂⠀⢸⠀⡐⠀⠀⠄⠀⠀⢀⠈⢂
⠀⢜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠇⠐⢩⣆⠀⠀⣶⡙⠀⡈⠀⠡⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⢃⠀⠀⡀⠀⠀⠀⠰⡦⠀⠀⠀⠀⢀⠀⠈⠉⠀⠀⠉⠁⢀⡃⠐⢤⠄⣀⣀⣀⣀⣀⣀⡠
⠀⠀⠀⠆⠈⠛⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠑⢤⣀⣀⣀⣀⠴⢋⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠑⠒⠒⠀⠀⠉⠀⠀⠀⠀⠀⠐⠤⠤⠤⠤⠤⠔⠁⠀⠀⠀⠉⠁⠀⠀⠀⠈

⡇⡤⠤⠀⠀⠀⠒⠒⠂⠀⠠⠄⣀⡀
⠘⢄⡒⠤⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠂⠀⠤⠄⢀⣀⠀⠀⠀⠠⡀
⠀⠈⠙⢇⠤⣀⡠⠇⠈⠉⠁⠹⠁⠀⠀⠀⡙⠉⠁⠙⠒⠒⠂⢠⠭⠅⢒⣤⣄⠀⠀⠠⠄⠀⠀⠈⠻⢓⣈⡦⣶⠒⠀⠀⠣
⠀⠀⠀⠀⠈⠐⠩⠍⣒⡒⠠⠬⠄⠤⠤⠀⢣⣀⣀⣠⣄⠀⡀⣠⣀⣀⡤⢽⣽⠛⠋⠈⠀⠀⠀⡠⠒⢣⠀⡰⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠐⠒⠐⠒⠀⠀⠀⠐⠂⠂⠀⠉⠁⡠⠀⠀⠀⢀⡄⠉⠙⡆⡾⡪⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠳⠤⡴⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠛⢢⠠⡴⡪⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠻⣀⢀⣡⡲⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠻⡀⠀⢀⡧⡨⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡠⠠⢔⠒⠉⠻⡀⠀⢀⣋⢝⠫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀⡀⠀⠈⠄⠀⣀⡉⢔⡫⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠒⠚⠓⠀⠁


⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⠂⠀⠉⠉⠉⠀⠀⠀⠐⠒⠒⠂⠀⠢⠤⠄⠀⠀⠠⠤⠤⠤⢆⠀⠀⢀⡀⠀⠀⠀⡰⠄
⠀⠀⠠⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡴⠺⡷⠛⠋⠉
⠀⠀⠀⠀⠉⠒⠢⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠤⠄⠀⠀⠤⠤⠤⠤⠤⠀⠐⠒⠈⡿⠔⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔⠀⠀⡆⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠎⡰⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡸⠊⠁⠀⠀⠀⠇⠀⢀⠇⠀⠀⠀⠀⠸⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣕⡁⠀⠀⠀⠀⠀⢰⡠⠂⠀⢰⠀⠀⠀⡮⠀⠈
⠀⠠⢼⠀⡆⠀⠀⢄⢀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠈⠦⡀⠀⢸⠀⠀⠀⠈⢂⠀⠈
⠀⠀⠈⢎⠀⠀⡀⠀⠉⢣⡇⠀⠀⠀⣸⠀⠀⠀⠈⠀⠀⠱⡀⢸⠐⠊⠀⠀⠀⡆
⠀⠈⠉⠉⠳⣠⠁⠀⠠⠤⣥⠀⡠⠄⢡⡔⠉⠀⠀⠀⠀⠀⠈⢸⠀⠀⠀⢀⣜⣀⠤⠀⠀⠀⠂⠀⠀⠀⠐
⠀⠀⠀⠠⣀⡘⣄⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣠⠀⠀⡠⠎⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⢦⠄⠒⠢⠼⡄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠐⠲⠆⡀⠄⠀⠘⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⠐⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡀⠎⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢰⠆⠁⠀⢾⠆⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡁⠈⠘⡄⡀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠌⠄⠀⠸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢀⣀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⣀⣀⡀⠱⡸⢦⡀⡾⣷⠀⡠⠔⠊⢹⣋⠴⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡛⣄⡏⠉⠳⣧⣀⣷⡷⣿⠞⣁⡦⢾⣋⣁⡴⠁⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⠽⠶⢦⣇⣹⣦⣻⣏⣾⣋⣀⡠⠔⠋⠀⠀⠀⠀⠀⠀⣠⢪⢿⠀⠀⠀⠀⠀⠀⠀⠀⡠⡺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣴⣋⣛⣿⡿⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡿⢻⠀⠀⠀⠀⠀⠀⡠⢺⣸⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⠀⠀⠀⠀⠀⡰⠁⢸⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⣧⢾⠀⠀⠀⠀⣰⠁⠀⣞⡠⠖⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⡇⡼⠀⠀⠀⠀⣷⠀⢤⠏⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣱⠃⠀⠀⠀⢸⠇⢣⣞⠠⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⡀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⢸⠀⡞⣁⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣽⣄⢸⣎⡇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣼⡞⡩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡻⣼⣼⠃⠀⢹⣵⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⢿⣛⡿⠿⢿⣿⠀⠀⣜⠺⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⢧⠀⣇⠠⡗⢻⠀⠀⠀⠀⠀⠀⢷⣤⡀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣿⢳⡇⢸⠀⠀⠀⠀⠀⠀⠘⡷⣝⣦⡀⠀⠸⣇⠀⠀⠀⣠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⣾⣓⣲⣤⡀⢿⣺⡵⡿⠀⠀⠀⠀⠀⠀⠀⢷⠘⣄⠱⡄⣸⢿⠀⢠⢾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣏⡴⠶⠯⣽⣿⣮⣿⡴⠃⠀⠀⠀⠀⠀⠀⠀⠸⡍⠙⣆⣷⢿⢸⡶⢻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣟⡅⠀⠀⠀⠉⠛⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣙⣇⣿⣿⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠠⠀⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⡇⢨⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⣀⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿
⠀⣿⣿⣿⡿⠻⠟⠛⠃⡚⣻⣿⣿⣧⣴⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣩⣽⣻⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣷⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉
⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠉⠉⠋⠉⠉⠉⠉⠉
⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⣿
⠀⢸⣷⣄⠀⠀⠀⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⣏⣸
⠀⢸⣿⣿⣷⡀⠀⢸⣿⣷⣕⠆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣅⣿⢉⣿⠀⠀⠀⠀⠀⢀⣤⠆
⠀⣾⣾⣿⣿⢹⡄⠀⠻⣟⡛⣷⡔⠄⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣿⠏⣩⡿⠀⠀⣠⢤⣾⣿⡟
⠀⢹⣆⣈⣿⣿⢻⣆⡀⠈⠣⣌⣿⡼⢠⡿⠁⠀⠀⠀⠀⢀⡤⣤⠔⣶⣸⠃⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⢋⣿⠁⢀⣼⣿⣾⣿⡿⠁
⠀⠀⢿⣌⣈⣿⣿⡟⣷⡄⠀⠈⠙⣿⡟⠀⠀⠀⣠⣴⣄⣾⣴⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣏⣽⠏⣴⣼⣿⢛⡳⠋
⠀⠀⠘⣯⠉⢙⣿⣿⡏⣿⣆⠀⠀⢸⠁⠀⣰⣾⣿⣿⡿⠑⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠉⠋⣠⣿⡿⠛⠋⠀⠀⢀⣀⡀
⠀⠀⠀⠈⢁⠉⠉⣻⣿⣯⣼⢧⠀⢀⡤⠞⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢋⠄⠊⠁⠉⠀⠀⣠⣴⣶⠟⣟⢉⣽⣇
⠀⠀⠀⠀⠹⣿⣏⣉⣹⣿⣯⣸⣧⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣾⠁⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢠⣾⣿⠟⣿⣄⣿⡯⢾⡿
⠀⠀⠀⠀⠀⠀⠉⠹⣥⡼⢿⣯⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⣩⠏⣽⣴⠿⢿⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⣠⣶⣿⡇⣸⣄⣼⣿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⣷⡇⠀⠀⠀⠀⠀⡀⣠⣶⢀⣿⣿⣟⣛⢙⡟⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣿⢿⣿⣿⣿⣿⡿⡷⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⣿⣇⣹⣾⣿⡉⠉⣩⡟⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⣿⣧⣾⣿⣿⣟⣛⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣼⠟⠉⣿⣿⣏⣁⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣿⣿⡿⠿⣍⣍⡻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⣼⠁⣨⣾⣏⣀⣀⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡿⠿⠉⠳⠶⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣭⣶⣿⣀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⡿⠧⠄⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠻⣦⠀⠀⣙⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⢠⡇⠐⡇
⠀⠀⠀⠚⠛⠛⠀⣇⠀⠀⠀⢸⡇⠘⠛⠛⠀⠀⠀⠀⢠⡟⠛⠛⠛⠛⠛⠛⢻⡆⠀⠀⠀⠀⠀⠀⢸⡀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⣠⠘⠳⠶⠶⠋⣤⡀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠙⠳⢦⣤⣤⠄
⠀⠀⠀⠀⠀⠞⠁⠀⠀⣾⠀⠀⠈⠻⠆⠀⠀⠀⠀⢰⡏⢷⡀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠈⠻⣄⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠷⣄⡀⠈⠛⢶⡋⠁⠀⠀⠀⢀⡴⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠈⠉⠛⠶⢤⣽⣆⠀⠀⢀⡟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣤⡼⠃⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠶⠶⠶⠶⠖⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠰⠶⠶⠶⠶⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⢩⡟⠀⠀⠀⠀⣠⡴⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠘⣏⠉⠉⠉
⠀⠀⠀⠀⠀⠚⠛⠛⠛⡿⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠸⡟⠛⠛⠛
⠀⠀⠀⠰⠶⠶⠶⠶⣾⠃⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠃

⡆⢑⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡊
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠁⡞⠑⢆⣶⠉⡏⢀⠶⡀⣾⣱⠀⡆⠀⣎⡁⠀⢰⣉⠈⢻⠁⣞⣱⢠⠊⠑⡄⣆⠖⢰⣉
⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣧⣤⣃⣟⣀⣇⣼⣉⣃⣻⣼⣃⣧⣄⣧⣄⣀⣠⣼⣂⣘⣀⣏⣹⣘⣦⣴⣃⣇⣱⣘⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣋⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⠀⠀⢀⠠⣠⠆⠙⢌⠨⠀⢻⠶⡏⢴⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠚⠁⠁⠀⠈⠁⢄⣙⡛⠓⠈⡀⢀⣦⣄⣶⣮⣀⣄⠈⢀⡳⢸⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣦⢙⣛⣢⣈⣋⣉⣽⣶⣾⣿⣿⠿⢿⣿⡿⠿⣟⢟⣛⣿⣿⢶⣯⣤⣞⡿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢎⣡⣤⣨⢉⣡⡾⠿⠟⠛⠅⠉⢉⣀⣤⣬⣁⡀⠤⠀⠉⠉⠠⣭⣀⣈⣝⣻⣿⣯⣹⣧⢀⠤⠂⢊⠜⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠨⠖⢀⣠⣶⣿⣭⣶⡤⣤⣤⠐⠒⢉⣈⣁⣁⠉⠓⣦⡁⠀⣀⣠⠽⠗⠛⠿⣻⣿⢻⣿⣿⣯⠎⡔⢁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠥⠀⢐⣻⣿⡉⠁⠁⠀⠀⠀⠈⠉⠉⠉⠉⠛⠙⠿⢲⡀⠻⣅⠀⣀⠤⠂⢡⡞⠛⠋⢻⠿⣿⣿⠌⠔⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠐⠂⢸⣧⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠦⡬⢛⢷⡂⡕⠜⠀⠀⠠⠃⠘⢿⣿⡠⢊⣡⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⣾⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⡿⢬⣟⣰⠄⠀⠀⠀⡄⠀⣾⡟⣴⠋⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡓⣤⢹⣿⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣷⡄⡈⠀⢀⡟⠛⠁⠀⠠⢾⣇
⠀⠀⠀⠀⠀⠀⢀⡤⠁⢈⣿⢟⠀⠈⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡽⣿⣿⣶⡟⠀⠀⠠⠾⣶⣿⡘⡆
⠀⠀⠀⠀⢀⡖⠠⢌⡳⣾⣿⠸⡀⠀⠉⠐⠠⣄⠀⠀⠀⡠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢫⣾⣁⠀⠐⠀⢀⣶⢿⡼⠇⡁
⠀⠀⠀⠀⠈⠃⠚⣠⡀⣿⢿⣀⠙⠄⠀⠀⠀⠀⣐⠀⠀⠁⢀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠠⡻⠭⢀⠨⣹⣶⡿⠻⡆⢱⣤⡇
⠀⠀⠀⠀⢸⣻⡍⣭⠍⠛⣆⠙⠧⠀⠀⠀⠀⠀⠀⠁⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡟⠂⣥⣾⢿⠀⢙⣄⠀⠈⠿⠃
⠀⠀⠀⠀⠈⢯⣧⠔⣋⢉⣻⣦⠀⠲⢶⣬⣀⠀⠐⢦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠋⢋⡟⣤⣿⡿⠿⠆⠀⠀⠝⢉⢤⣀⠆
⠀⠀⠀⠀⠀⠘⣥⡦⣡⠂⠚⠛⣷⣶⣤⡽⡿⠿⣶⣤⣈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣟⠋⠉⠀⠀⠀⠈⢁⣄⠀⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠐⠇⣞⢸⠿⢿⡿⢿⡻⢶⣦⠈⠠⠚⠋⣠⣼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⠉⠛⠀⠀⠠⣈⣼⢋⣱⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠙⠀⢻⠷⣆⠀⢄⠈⠙⠹⡷⢶⡤⢤⣠⣀⡀⠀⠀⠈⣹⣿⠐⢀⠤⠴⢦⣼⡧⠝⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠰⡜⢿⡶⡶⠶⠖⠻⠿⠟⠛⠻⠛⣛⣦⣤⣐⣡⣾⣠⣴⠾⠏⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠚⠛⠛⢿⣛⣻⣍⡿⠤⣤⡁⢲⠿⠟⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠁⠐⠒⠂⠐⠒⠀⠀⠒⢄⠀⡴⠉⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⡀⠠⠔⠁⠀⠀⠀⠀⠀⣀⣀⣀⠀⠤⢀⡠⠋⠀⠀⠀⠘⡏⠉⠉⠉⠑⢄
⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠓⠉⠁⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠘⢄⣀⠀⠀⠀⠉⠐⠒⠒⠒⢄
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡀⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠐⠀⠀⠈⠑⠤⡀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠁⠀⠀⡇
⠀⠀⠀⠀⡠⠁⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠐⠤⣀
⠀⠀⢀⠜⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⡠⠤⠄⣀⢱
⠀⠀⠈⠀⠀⠀⢤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣀⣠⠤⠖⠁⠀⠀⠀⡸⠸⡀
⠀⠀⠘⡄⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠙⢆
⠀⠀⣠⠬⠲⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢄⣀⣀⡠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⡘
⠀⢰⠁⡠⠤⠧⢄⣀⣀⣠⠤⠄⠒⠒⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠡
⠀⡌⡘⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⢀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄
⡎⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠀⠠⠼⡄⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠰
⢣⠀⡇⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠲⢄⠀⠀⡘⠀⢠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠄
⠀⡇⠈⠢⠄⣀⣀⡀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢠⠁⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠤⠊⠁⠀⠀⠀⠀⠀⡇
⠰⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣄⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠸⡀⠀⠀⠙⢦⣆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠇⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⢀⣀⠤⠤⠒⢲⠋⠁
⠀⠈⠢⡀⠀⠀⠈⠑⠤⢄⡀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢠⠊⠀⢀⡠⠒⠊⠁⠀⠀⠀⢀⠆
⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⠤⢀⣑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠰⠁⢀⠔⠁⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠈⠢⢄⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠈⠉⠒⠢⠄⣀⠀⠀⠀⠀⠀⠱⡀⢠⠃⡰⠁⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⢄⡀⠀⠀⠀⠀⠉⠉⠁⠒⠒⠤⢄⣉⠒⠦⣀⠀⠀⡇⡌⡔⠀⠀⠀⠀⠀⢀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢕⣶⡀⡷⠀⠀⢀⣀⠤⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⠤⢀⣀⣀⣀⣀⣀⣀⡀⠬⠽⠗⠒⠈⢁⣀⠔⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠤⣤⣄⣤⠖⠛⠛⠛⠲⣤⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠷⣤⣄
⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⢹⣄⠀⠀⠀⣠⣄⣀⣴⠶⠶⠛⠳⠦⣤⣤⠾⠛⠲⠶⣄⣀⣠⡀⠀⠀⠀⠀⣸⡇⠀⠀⠉⠛⢶⣄
⠀⢀⡴⠋⠁⠀⠀⠀⠀⢀⣤⣿⠛⠛⠛⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠶⠤⠴⢾⣯⣄⠀⠀⠀⠀⠀⠉⠳⣄
⢠⡟⠀⠀⣀⣠⣤⠶⠞⢋⣴⡟⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠸⣿⣌⡛⠶⢦⣤⣀⡀⠀⠙⣧
⣿⠀⠐⠛⠉⠀⣀⣤⠾⠋⠀⣇⠀⠀⠀⠀⠀⠈⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⢰⡇⠈⠛⢦⣄⣀⠈⠙⠓⠀⢸
⠈⠛⠓⠒⠛⠛⠉⠀⠀⢀⡴⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡀⠀⠀⠈⠉⠛⠛⠒⠛⠋
⠀⠀⠀⠀⠀⠀⣠⠞⠛⠛⠁⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡀⠀⠀⣀⡽⠃⠀⠀⠀⠀⠀⠀⠀⢠⡟⠈⠛⠛⠳⣆
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⣀⢸⡇⣀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠘⠳⣄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⡀⠀⠀⠉⠛⠋⠁⠀⠀⣀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⣸⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⠶⠶⠶⠚⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣄
⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠈⠛⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢷⣄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣰⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠉⠉⢷⠶⠴⢾⣦⣀⣀⣀⣴⣦⣀⣀⣀⡴⠶⠴⣶⠋⠉⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸⡇⠉⠉⠉⠀⠀⠉⠉⠉⡇⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣤⣿⣤⣤⣤⡗⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣿⣤⣤⣤⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣄⣘⣷⣤⣄⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⢿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠀⠀⠉⠉⠉


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