Puppy Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢀⡴⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠰⠤⣀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠀⠀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠐⠊⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠘⣄⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠤⠄⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠈⠉⠓⢦⡶⠒⠓⠒⠂⠀⠁⠀⠋⣀⢤⠀⠀⠀⠘⣤⣤⣤⣤⡤⠤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢀⡀⠀⢈⠠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠁⠀⠀⡴⠛⠂⠘⠀⠀⣤⣬⡁
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠐⠀⠀⠀⠠⠀⠀⠀⠀⠙
⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢠⠏⢸⠀⠀⠀⠀⠀⢀⢠⠀⢀⣄⠀⠀⣀⡀
⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢀⠰⠋⠀⣴⠀⠀⠐⡻⠛⠊⠀⠀⠈⠁⠀⠔⠻⢁⡀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⣼⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⣲⣾⣿⡿⠿⠿⢿⣷⡄
⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣷⣤⣤⣾⣿⣿
⠹⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢲⠞⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠘⠿⣿⣿⣿⣿⣷⣄⡀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠈⠀⠀⢢⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠛⠋
⠀⠀⠀⠈⠙⠻⠿⢿⣿⠿⠋⠁⠀⢹⣿⣿⣶⣶⣶⣶⣶⡿⢻⣧⠀⠀⠀⠡⡀⠀⠀⠀⠈⠑⠒⢀
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠃⠀⠙⢷⡀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠈⠰⡆⠦⠀⣀⡄⠀⠀⠀⠀⣀⠔⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⡞⠀⢀⣴⣿⣿⣿⣶⣄⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠑⠠⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⢸⠁⠀⢾⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣶⣿⣿⣿⣶⡄⠀⠀⠉⠢⢄⠀⠀⠀⠀⠈⠐⠂⠼⠷⢦⡀
⠀⠀⠀⠀⠀⢸⡆⠀⠈⠿⣿⣿⣿⡿⠋⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠁⠒⠄⣀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣰⠖⠢⡀
⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠂
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠐⣆⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠼⠿⡶⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠾⠿⠁⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡘⠀⠀⢄⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠘⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠈⠆⠐⠀⠀⠀⠀⠁⠀⠀⠂⠠⢀⠀⠀⢀⠀⠤⣀
⠀⠀⠀⠀⠈⠓⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑⠀⠑⠄⠙⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⡈⢢⡀⠐⣀⢸⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣤⠖⠊⠉⠁⠀⠈⠉⠉⠒⠶⣤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢇⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⢀⠇⠀⠀⠀⣿⣙⠄⢀⡠⢤⡀⠠⣋⣿⠀⠀⠀⢸⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢸⠤⠤⢤⣀⣉⡩⠔⣉⠤⠄⣈⠳⠭⣩⠤⠤⠤⠬⡇⠀⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⡇⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⢸⠤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠢⡴⠞⠙⡶⠖⠂⠀⠀⠀⠀⠀⢸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠙⠦⠔⠃⠀⠀⠀⠀⠀⢀⡴⢻⠁⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠑⠒⠤⠤⠤⠄⠤⠤⠤⠤⠔⠒⠉⡇⠀⣎⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢹⡎⠀⢉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢹⠀⠀⠑⢄⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⡇⠙⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢇⠀⠀⠀⠉⠒⢦⠔⠒⠉⠀⠀⠀⣸⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠦⠤⢘⣆⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣠⡃⠠⠴⠥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡀⢀⠀⡎⡀⠉⠉⠉⠀⢸⠀⠀⠈⢑⠋⢁⢹⠀⡀⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠼⠤⠟⠧⢄⣧⠤⠴⠚⠑⠢⠤⠼⡤⠼⠞⠤⠧⠜⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⠿⠿⠿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠀⠀⠀⠀⢀⣤⣴⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⠀⠀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠻⢷⡶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣙⣷⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⢀⣴⣾⠟⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠉⠁⠀⠀⠀⠀⠀⠀⠘⠻⣷⣄
⠀⠀⠀⣀⣴⣿⠟⠁⣀⣤⣶⣾⣿⡷⠀⠀⠀⢀⣼⠏⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠙⢷⡄⠀⠀⢰⣿⣷⣤⣀⡀⠈⠻⣷⣦⡀
⢀⣴⣾⣿⣿⣷⡿⠿⠛⠋⠉⠀⣿⡇⠀⠀⠀⣼⣇⣀⡀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣀⠈⣿⠀⠀⢸⣿⡍⠉⠛⠿⣷⣶⣬⣽⣿⣷⣤⡀
⣿⣿⣟⡋⠉⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⣿⣿⣏⣽⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣩⡇⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿
⠀⠙⠛⠿⣿⣶⣶⣤⣤⣤⣤⣾⣿⠁⠀⠀⠀⠙⠿⢿⣿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⡷⠟⠀⠀⠀⣿⣧⣀⣀⡀⢀⣀⣠⣤⣶⣾⣿⠿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠋⠙⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣯⣍⣙⢲⡄⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠈⠛⢿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⣀⣀⣀⣤⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⡀⠀⠀⠀⠀⠈⣽⡿⠛⠻⠿⠿⠿⠿⠿⢿⣿⣥⣬⣿⠿⠿⠿⠟⠛⠛⢿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⢿⣧⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠈⢿⣷⣤⣀⣀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠈⠛⠻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⡀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⡀⠀⢀⣤⣶⡿⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⣶⣶⣶⣶⣶⣶⣾⠿⠟⠁⠛⠿⣷⣶⣶⣶⣶⣶⣶⣿⠿⠿⠛⠋
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠛⠉⠙⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⢤⡀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⡀
⠀⠀⠀⠀⢀⠂⠀⠀⢠⠞⠀⠀⠀⠀⠙⡀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢳
⠀⠀⠀⠀⢸⠀⠀⣰⠋⠀⢀⣀⣠⡤⠤⠴⠤⠧⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢸⡇
⠀⠀⠀⠀⠈⢦⡴⠃⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣤⡴⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⢸
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⣀⣤⠤⠤⠤⠤⣄⡀⠀⠀⡸⠀⡟
⠀⠀⠀⠀⠀⠠⢣⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⣧⠠⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⢀⡽⠀⠀⠹⠞⠁⠀⠀⠀⠀⠀⣀⡀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠟⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣛⡟⠀⢰⠁⠀⠀⠀⣀⣠⣤⣤⣄⡀⠀⠸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⣸⠀⢸⣿⣷⣄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠒⠉⠁⠀⠀⠀⠀⡷⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠘⢄⠘⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⣀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⠀⠀⠀⢘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠌⢳⣤⣄⣀⣀⣤⡴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⢀⠤⠤⠀⠀⣠⠊⠀⠈⣩⠏⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠚⠈⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⡏⢠⠂⠀⢰⠃⠀⢀⡴⠃⠤⠤⠖⠋⠉⠉⠒⠲⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⡇⠀⠀⡈⠳⠶⠋⠀⠀⠀⠠⠤⠤⠤⢤⣀⡀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠈⠢⠀⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⡡⠴⠒⠒⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠘⠢⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠈⠳⢦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠉⠉⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢠⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⡀⠀⢀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⠦⣀⡤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠄⠶⢤⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⣶⡞⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠉⠢⡄
⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⣤⡜⠀⠀⠀⣛⣃⠂⠀⠀⠀⠀⠀⠀⠀⠙⢉⣀⡀⠀⠀⢹⣦⠑⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⣻⠇⠀⠀⢼⣿⣿⡆⡐⠀⠀⠀⠀⠀⢁⡠⣿⣧⣽⠦⠀⠸⣿⣷⠀⠀⡄⠀⠘⢆
⠀⠀⠀⠀⢨⡇⢀⠀⠀⢸⣿⡀⠘⢀⠀⠉⠡⠄⠀⠀⠀⠀⠀⠀⠘⠇⠈⠉⠁⠀⢠⠅⢻⣿⡀⠰⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣮⡴⠟⠀⠀⠈⣿⠃⢀⡚⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠤⢀⣿⣧⠀⠀⠀⠀⠀⠠⣘⣄
⠀⠀⠀⢰⣿⣦⡀⡀⠀⠀⢿⡆⣄⠁⠀⠀⠀⠀⢠⣀⣀⣀⣤⣶⣄⠀⠀⠀⠀⠀⠀⠲⣠⡿⣿⡄⣦⣀⠀⢀⣤⡀⣬⡃
⠀⠀⠀⢸⣟⣿⣿⣧⣦⣦⢸⡇⡇⠊⣕⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠆⠈⡂⠀⢸⣾⡆⣿⣷⣿⡇⣿⢿⣦⣼⡟⣸⠿⠁
⠀⠀⠀⠀⣿⡟⣿⢱⣷⣾⢸⣧⠣⡄⢻⡇⠂⠀⠉⠹⢿⣿⣿⠿⠋⠀⠀⠀⣸⠟⢁⠼⣿⠿⣿⣷⢹⣿⣿⠄⡴⢏
⠀⠀⠀⠀⠀⠹⣿⣿⣿⡁⢸⣿⠀⠀⠀⢡⡀⠀⠀⠀⢠⣈⡠⠀⠀⠀⠀⢠⡏⢀⢀⣀⡁⠠⣹⣿⠸⣿⣫⡠⢣⠁
⠀⠀⠀⠀⠀⠀⠹⢿⣿⣡⣏⣴⡗⠀⢀⠸⣿⢦⣴⣦⠾⢿⣿⣶⣶⣤⡴⣋⠞⠁⠀⠀⠀⠈⢟⢿⡇⢿⣻⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢺⠉⠁⠀⠀⠘⠓⠈⠻⣿⡄⠀⠀⠀⠀⣨⣿⡞⠁⠀⠀⠀⠐⠄⠀⠀⣑⣿⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⠀⠀⠀⠀⠀⠀⠸⣷⣤⣤⣤⣴⡿⠋⠀⠀⢀⠀⠀⢠⡀⠀⢹⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣷⡀⠂⠀⠀⠀⠀⠈⠉⠋⠋⠉⠀⠀⠀⠀⠀⢿⡂⠀⣡⣤⣼⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⠃⠘⠧⠾⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⠁⠀⠋⠛⠋⠈⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠘⢷⣀⡀⠀⣄⡀⠀⠀⠀⠀⠢⠀⣢⡄⠘⠂⠂⠀⠀⠤⢀⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠦⣳⡖⣶⣤⣿⠾⠗⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠿⠿⠿⢿⣶⣤⡀⠀⠀⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄⠀⣠⡞⠉⠉⠛⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣶⣄⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠀⢰⡄⠀⠀⠈⠙⢦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡾⠋⠉⢉⠉⠛⢷⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠈⠉⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣾⠿⠛⠉⠁⠀⠀⠀⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⡙⠻⣿⣆⠀⣿⣷⡄⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⣰⡿⣧⡉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠹⡆⠘⣿⡀⢹⣿⣿⣄⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⣿⠁⠈⣿⡄⠘⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣸⣷⠀⣿⡇⠀⠙⣿⠿⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⢸⠇⠀⠘⣿⣷⣤⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⢀⣿⠁⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⣿⡀⠀⠀⢿⣿⣿⡿⠁⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⣤⠾⢃⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀⡾⠿⣷⣤⡀⠈⠻⣿⣤⡾⠃⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠙⢧⣴⠶⠒⠲⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠙⢿⣷⣤⣠⡄⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⠿⠿⢿⣷⠀⠀⠀⠀⢘⡏⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠉⠑⠛⠻⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⡀⠀⢀⣰⣿⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⠀⠀⠀⠙⢿⣿⠿⠿⠿⠟⠛⠁⠀⣠⣴⣿⠏⠀⠀⠀⠀⠠⠾⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠛⠿⣶⣤⣤⣄⣠⣤⣤⣴⠾⠿⠗⠶⠾⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠀⠀⠀⠀⠹⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣾⠿⠿⠿⠿⣷⣄⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⠿⢿⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣶⣶⠞
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⣤⣤⣶⣿⣿⡇⠀⠀⢀⡇⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⢠⡇⣆⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠏⢸⠀⠀⠀⢸⣧⠀⠘⣿
⠀⠀⠀⠀⠈⣇⠘⢦⡀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠀⠀⡿⠀⠀⠀⢸⣿⠀⠀⠻⣧⣤⡀
⠀⠀⠀⠀⠀⠹⣄⠀⠙⠶⣤⣤⣿⣿⣧⠀⠀⠀⠀⠀⣸⣿⠿⠟⠛⠛⢦⡇⠀⠀⠀⠘⠿⠛⢦⡀⣄⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣀⣀⣉⣉⣁⣀⢀⠀⠀⢀⣉⣄⡀⠀⣾⣇⠘⣧⡀⠀⠀⣤⠀⢰⠘⣧⣼⣦⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⢿⣿⡿⠿⠛⠙⠛⠿⠿⠟⠻⠿⠛⠿⠿⠿⠿⠷⠟⠻⠟⠀⠈⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡖⠛⠛⠲⢄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠃⠀⠀⠀⠈⠣⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠉⢀⠴⠒⠒⠢⢄⠀⠁⠀⠺⡀⠀⠀⠀⠉⢢⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⡤⠋⠀⠀⠀⠀⠀⠁⠀⣰⠶⢸⠀⠰⢶⡄⠀⠙⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠠⡦⠀⠀⡶⠰⠄⡀⠀⠀⣰⣿⣿⣟⠛⠛⠛⢿⣷
⠀⠀⢀⠀⠀⠀⢀⡔⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠐⠀⠀⠒⠀⠐⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠀⠀⡴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⣼⠁⡇⠀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋⠃
⠀⢀⠀⠠⡇⠀⢷⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡄
⠀⠀⣄⠀⢧⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⠇⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠓⢤⣀⠙⢦⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠻⠀⠀⢠⣔⡉⠉⠑⠒⠢⠤⠄⢀⣀⣀⣀⡠⠔⠂⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⣲⠏⠁⠀⠀⢻⣿⣿⣿⣿⣶⣾⡿⠀⠘⢄⠀⠀⠀⠐⠺⢿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠿⠏⠓⠒⠶⠤⠤⠤⢦⠀⠀⡼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠀⠀⠀⠀⠄⣀⠂⠄⣸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡝⠂⠉⠀⠀⠀⠀⠈⠀⠐⠠⣄⠒⣌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠘⠚⢄⣀⠀⠀⠀⠀⠀⠀⠀⠢⡐⢂⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠒⠒⠒⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠂⠐⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠒⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⠀⠀⠀⢀⡈⠆⠐⠒⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⢀⠔⡡⠊⠀⠀⠀⠀⠀⠀⠀⠑⡆⠁⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⡠⠔⠁⡐⠁⠀⠀⠀⠀⢀⠀⢀⠀⠀⠘⡒⡀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⣠⢎⡀⠀⠀⠀⢀⠔⠁⠀⠀⠑⡄⠀⠀⠰⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢱⣿⣟⡀⠃⢀⠊⠀⣠⣄⢤⡀⠐⠀⠀⠀⠡⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠛⠻⠃⠀⠀⠀⢸⣿⣯⡄⡏⠀⡇⠀⠀⠀⠀⠑⠠⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢾⣍⣉⠉⠙⣦⠀⠉⠛⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠻⣿⡿⠿⠛⠀⠤⠀⠤⠔⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠈⠉⠀⠀⠀⠉⠀⠀⡀⠔⣮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡂⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⡶⠀⠂⠒⠂⠀⡈⠅⠂⡩⠂⠁⠂⠄⡀⠀⠀⠀⠀⠀⠀⠐⠀⠈⡔⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢛⡖⠂⠀⠀⢁⠠⠐⠁⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠁⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠻⢶⡒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⢰⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⡇⠀⠀⠠⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡘⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⡴⢇⠀⠀⠀⢣⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠁⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢸⠀⠀⠀⢸⡄⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠸⠀⠀⠀⠘⠃⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠄⠀⠀⢀⠔⠁⠀⠀⠀⣘⠤⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠣⠰⣥⠂⡄⠀⢀⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠂⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣷⣤⣤⡶⠖⠚⠛⠓⠒⠛⠛⠛⠓⠟⠹⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⡟⠀⠀⠀⣀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣏⣻⣿⣿⣿⣿⠀⠀⠀⠃⠀⠀⣾⣮⣿⡆⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠻⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣾⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⣿⠘⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠟⢹⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠒⠒⢦⡄⠀⢠⡇⠀⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⢻⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⣿⣿⡿⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠛⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⡇⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣡⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠒⠤⣀⣀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠹⢶⣶⣎⣉⣠⣤⣴⡶⠖⠊⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⣤⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠉⠀⠀⣄⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⢀⡀⠀⠀⡀⠀⠀⠀⣠⣶⡿⠟⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠒⠛⠛⠲⠿⠛⠹⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡴⠶⠶⡿⠿⠋⠻⠿⠚⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⢷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠆⣀⣴⠧⠉⠉⠓⠢⡀⠀⠀⠀⠀⠀⣀⠤⠔⠒⢒⣦⣄⣀⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⢀⣼⣟⣁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⣳⣶⡀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⣼⡿⡋⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⠀⠀⠀⠈⢻⣿⡀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⢀⣿⡿⠁⠀⠀⣠⣾⣟⣹⣿⣿⣷⠀⠀⠀⠀⣿⣯⣹⣿⣿⣿⣦⡀⠀⢤⣿⡗⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢸⣷⣶⣤⡴⠙⣿⣿⣿⣿⣿⡟⠒⠒⠘⠚⠟⠿⠿⣿⣿⡿⠃⢿⣴⢿⠙⠇⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠈⢹⠟⠉⠁⠀⣺⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡂⠀⠀⢀⡶⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡀⠀⠀⠀⢸⠆⠀⠀⠚⠁⠀⠀⠀⢀⣤⣶⣶⠶⣶⣦⣄⠀⠀⠀⠀⠉⠀⢀⣿⠁⠀⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⠀⠀⠀⠺⣦⣤⠄⠀⠀⠀⠀⠀⣿⣿⣿⣶⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⢼⡏⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣤⣤⣶⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠀⠈⠹⣆⠀⠲⣄⣀⡤⠾⠞⠁⠀⠀⠙⠛⠶⣶⣀⠀⣀⠞⢀⡼⠉⠛⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠳⣄⣀⣀⡀⠀⠀⠀⣀⣠⡴⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠉⠀⠀⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⡠⢤⡤⠚⠉⠀⠈⠉⠐⠢⣀⠀⠀⣀⠀⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⡠⠞⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠁⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⣀⠴⠊⢀⠌⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⢻⠢⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠉⠀⠀⣠⡊⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠙⢄⠀⠀⡄⢡⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣱⣿⡷⠈⢢⠀⢠⠃⠀⣀⡀⡀⠀⠘⡆⠀⡇⠀⢢⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠹⢿⣿⠆⠈⢀⠃⠀⣼⣿⡷⠈⢻⠀⢰⠀⡇⠀⠀⠑⠢⢀⡀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⣤⠶⠦⠤⢼⣀⠀⠙⢿⣿⠧⠊⠀⢸⠀⠇⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢿⣶⣦⣤⣤⣼⡇⠀⠀⠀⠀⠀⢠⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠙⡿⠿⠛⠉⠈⠀⠐⠒⠒⠊⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⢀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠈⠉⠉⠙⠉⠉⠁⠀⠀⣀⠤⢺⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠶⡤⠀⠤⠀⠐⠂⠂⣉⠬⠖⢊⠔⠊⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⢣⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠷⠤⠠⠤⠒⠊⠁⢀⠤⠊⠁⠀⠀⠀⠀⠀⠉⠓⠤⣀⠀⠀⠀⠀⠀⢃⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼⣀⠀⡀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠘⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢰⡹⠀⠀⠀⢱⡀⠀⠀⠀⠀⣀⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⡜⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣤⠾⢳⠀⠀⠀⠀⢇⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠀⠀⠀⠀⢸⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠀⠐⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢠⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢉⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⣎⣀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣎⣀⣠⢃⠄⠀⠀⠀⢀⡜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠤⢇⡠⠔⠃⠙⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠃


⣀⣀⣀⠀⡀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠈⢑⡀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⡀
⠀⠀⠀⠀⢰⣧⠆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁
⠀⠀⠀⠀⢻⢸⣏⠀⠙⠁⠀⠀⠀⠀⠹⠗⠀⠀⣀⡀⠐⠀⠀⠀⠐
⠀⠀⠀⠀⠈⠉⠀⠠⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣟⠃⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⠀⠀⠀⣀⣾⣏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⢩⢠⠋⣼⡆⠸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣷⣴⣾⣿⡻⢋⡀⠀⠤⢄⡀⠀⣆⡀⠠⡀⢠⣧⣰⣿⣨⡷⢙⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣿⠀⠄⠀⠀⣀⠠⠄⠀⠉⢨⣻⠗⣿⠛⢙⣿⣿⢥⣤⣶⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣭⣈⠓⠀⠐⠉⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠉⠻⣫⣶⡶⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡷⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠅⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⣶⣶⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠼⠋⠙⢿⣿⣿⣿⣿⣿⣿⣿⡆⢸⣿⣿⣿⣿⣿⣿⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⡿⠿⠛⠃⠀⠀⠈⠻⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣶⣶⣦⢻⣿⣿⡿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣤⡈⠻⠿⢋⣼⡿⠁⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠈⢻⡟⠛⣻⠋⢠⣶⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠛⠿⠋⠀⣸⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠿⠿⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⢻⠀⠀⠀⠀⢸⡏⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢀⣿⡇⠻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡏⠀⠀⠐⣷⡄⣴⣿⣿⠇⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠙⢿⣿⣿⡟⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠁⢻⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⡿⣂⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠁⠀⠀⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⢹⡄⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠉⠉⢸⣶⣤⣴⣶⣤⣤⣤⣴⣶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠁⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⣩⣶⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⡇⠉⢷⣿⣧⣭⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢨⣿⠟⠛⠁⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀⠀⠠⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠢⢽⡷⢲⡶⠶⠋⠀⠘⣿⣿⣻⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢳⡀⢄⠀⣥⠉⢿⣿⣵⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣫⣀⣡⠀⠀⠈⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣬⣥⣥⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣍⣍⢿⣿⣿⣽⣿⣿⣿⣿⣿⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠋⡿⠘⢻⣿⣿⣿⣿⣿⣿⣿⣟⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⢻⡧⠜⠛⠁⠀⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡾⠀⢿⣆⠀⢀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠸⣷⣌⡻⣿⢛⡄⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡔⣿⣯⠻⣿⣿⣷⠀⠈⢿⣿⣿⣿⡿⠏⢹⠙⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠸⣿⣇⠈⢻⣿⣆⠀⠀⠹⡜⢿⣗⢀⠇⠀⠀⢿⣿⣿⣿⣿⡿⠻⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠹⣿⡄⠈⢻⣿⣆⠀⠀⠀⠀⢳⡜⠀⠀⠀⠈⠉⠝⠹⢿⣤⣀⢐⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠹⣿⣦⡀⠉⣿⡀⠀⠀⠀⢠⡄⠀⠀⠀⠃⠀⣠⣶⣦⣼⣁⣈⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⣾⢿⣿⣿⠰⢿⡇⠀⡆⠸⡎⡇⠀⠀⠀⠀⣨⠞⠉⠈⢀⣨⣭⣿⡿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⡄⢿⣿⠀⠀⢹⠀⠙⢷⣿⡔⠀⢄⠀⣼⠋⠀⠀⣴⣿⣿⣿⣿⣿⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢶⡌⢻⡇⠀⢸⡇⣌⠊⡿⣷⣤⡘⢿⡿⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⣷⣾⣷⠀⠀⡇⢘⣿⣿⣿⣿⣿⣿⣿⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢿⣧⣿⠀⠀⢡⠈⣿⡿⠀⠻⣿⣿⣿⣇⠀⠈⢻⣿⣿⣿⣿⢿⡿⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣘⢻⡟⠀⠀⢸⣴⣿⣇⣦⣄⣉⣿⣿⣿⢦⡀⠀⠈⠻⢿⣿⡟⠀⠈⣟⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡲⢶⡯⢖⣠⣾⠇⠀⢠⡟⢰⢻⣿⣏⣿⣿⣿⡿⡿⢷⣿⣦⠀⠀⠶⣽⣿⣿⣶⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣴⣯⣄⡾⠯⠟⠀⣰⢏⡴⢣⠞⣻⠃⢉⡽⣱⠋⣥⠶⢶⣷⣶⣶⡾⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣿⣶⡿⠀⠀⠿⢿⣾⣧⣰⣿⣿⡉⣀⠀⠤⠤⠤⠤⠶⠾⠿⠟⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠆⠁⠀⠀⠰⡍⠙⠆⠄⡀
⠀⠀⠀⠀⠀⠀⠠⣗⠀⠀⢰⠃⠀⠀⣴⠍⠠⡀
⠀⠀⣒⡈⠀⠀⠀⠀⠀⠀⠘⣆⣠⠾⢫⡆⠀⠈⢠⡀
⠀⠀⠉⠂⢀⣤⠤⣺⣫⠿⠀⠈⠁⠀⢹⠁⠀⠀⣀⡿⠙⣶⡀⢤⣄⣀⣀⣀⣀⣀⣀⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠠⢦⠶⠁⠀⠈⢀⠂⠀⠠⣼⠂⠀⢰⠋⠁⣠⣿⣇⠈⢻⣿⣿⢯⣍⣉⣉⣡⣭⣤⣈⡉⠉⠉⠁⠂⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⢻⠀⠀⠁⠀⠀⠙⣿⡏⠀⠘⡟⠃⠀⣸⡿⠉⠳⠙⠃⠀⠀⠀⢀⡄⠀⠐⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⡜⠀⡿⠃⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⢸⠠⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠶⠾⠋⠀⠀⠀⠀⠀⠀⢢⡀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠀⢹⣇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⣀⡂⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣢⡙⠄⠀⢀⡼⡥⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⡟⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⢙⣦⠀⠀⢰⡧⠤⠤⠄⠀⠀⠀⠀⠀⠀⠈⢷⣀⠙⢿⣷⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣼⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠸⣿⣿⣦⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡝⣿⣿⢷⡄⠐⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⣿⠀⠑⡀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢈⠁⢠⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡇⠀⠀⠃⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠁⠎⢀⡈⡀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠧⣴⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⠉⢴⣭⣍⣡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠛⠋⠀⢦⢠⣍⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣲⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣟⢿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣮⡙⠇⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⡟⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣿⣿⠁⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⢿⣿⠿⣿⣿⠿⠟⠁⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡼⠁⠀⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣼⡏⠈⠙⠻⠿⢿⣿⣿⣧⣤⣤⣤⣤⡄⠀⠀⢻⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠚⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⣤⣤⣤⣴⣾⡿⠛⠋⠉⠉⠙⢲⣶⣶⣶⣄⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣴⠃⢀⡴⢋⣽⣿⣿⢫⣿⣿⡇⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡕⣦⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⣸⠃⣠⣿⠀⢸⣿⣿⡿⣿⣿⣿⣷⡂⠀⠀⠀⢀⣤⣽⣿⣿⣿⣷⢹⣷⠀⠀⠘⣇
⠀⠀⠀⠀⠀⢠⡇⠀⢻⣿⣆⢸⣿⣿⡇⣿⣿⣿⣿⣿⡆⠀⠀⢸⣿⣿⣿⣿⣿⣿⢸⡟⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠻⣿⣧⣿⣿⣧⣿⣿⣿⣿⡿⠃⠀⠀⠈⠻⣿⣿⢻⣿⣿⠟⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠛⢿⣿⡇⣿⣿⣿⣿⠃⠀⠀⢀⣀⠀⠹⡟⠈⣿⠏⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠏⠀⠀⢾⣿⣿⠿⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣯⠀⠀⠀⠀⢙⡟⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣶⣶⡶⠛⠋⠈⠀⢰⡄⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠛⠋⠀⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢸⣿⣿⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣈⠻⢿⣧⡙⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢬⣙⣓⠂⠀⠀⠀⠀⠀⣀⣀⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠒⠒⠲⣄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢰⢠⠀⢠⠀⣌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠐⢂⡰⢯⡁
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⠀⡀⠹⣤⣶
⠀⠀⠀⠀⠀⠀⣾⠀⡇⢀⣤⠀⠀⣠⡄⢸⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⡏⠀⡇⠀⠛⡼
⠀⠀⠀⠀⠀⠀⠘⠶⣇⠀⠁⣀⣀⠈⠀⣸⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠙⠶⠞⠳⠶⠳⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢦⣀⠉⠉⣀⡴⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⠉⠉⠀⠀⠀⠙⣦⠀⢀⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⣀⣀⠀⢸⡇⠀⠘⣇⡼⢻⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢈⡏⠀⠀⢸⡇⠀⠀⠹⠁⡾⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣰⠠⠄⠠⠄⣆⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⣸⣧⡀⠀⣸⣇⠀⠀⣀⡼⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢆⣘⣃⡰⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉⠉⠁⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⡄⠙⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠈⡇⠀⡇⠀⠉⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣾⣷⣶⣷⣶⣾⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠊⠉⠉⠁⠀⠀⠀⠉⠓⠤⢤⡀
⠀⠀⠀⠀⠀⢳⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢦⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⢣⡈⠢⣀⠀⢀⣀⠠⠤⠤⠤⣀⡀⠈⢆⢀⡜⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠙⢢⣄⡉⠉⠀⠀⠀⠀⠀⠀⠉⠲⡖⠋⠀⢤⡀⠀⠀⠀⣸⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠂⠀⢰⠙⣆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡼⠀⡰⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⢇⠉⠒⠒⠒⢲⠋⢻⠀⠀⡼⠑⡺⠁⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡇⠀⠀⠀⠸⣀⡇⠀⠘⢢⠀⡇⠈⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠁⠀⠀⠀⠀⠀⠈⠒⠒⠋⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠤⠄⠤⠤⠀⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⢠⠆⠀⠀⠀⠀⠀⠀⣀⢰⡀⠉⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠰⠂⠀⡏⢰⣿⡆⠀⠀⠀⣾⣷⠈⡇⠀⠀⠌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⡄⢀⡁⠀⠀⢀⣾⢲⣦⠀⠀⠀⣿⠀⡄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠔⣼⠁⠀⠀⢀⣿⣿⣏⠀⢀⠀⢸⣄⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⣸⡀⡾⣦⡜⠛⠟⠻⣤⣾⠀⢰⢩⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠦⢐⢈⡻⢦⣤⠾⢋⢈⡤⠉⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣤⣀⣨⠀⣉⠁⠀⠈⠁⠀⠀⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠐⠺⣼⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣾⡏⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠙⣿⣿⣿⢿⣿⣿⣿⡿⣻⣿⠋⠶⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢰⡄⠘⣿⣿⣿⣿⣿⣿⡟⠱⣿⠈⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢀⠘⣘⡧⠀⠈⣿⢿⣿⡿⠃⠀⠀⣏⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⡙⠁⠀⠀⠀⠀⠙⠃⠀⠀⠀⠈⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣲⠞⣇⠀⠉⠀⠀⣃⢀⠀⠀⣤⠀⠀⠀⠠⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠙⢳⣄⠞⠈⠁⠛⠂⠴⠸⠀⠀⡔⠓⠀⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠐⣆⠀⠀⠀⢠⡞⠀⠀⠀⡇⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡇⠀⠂⠀⠘⢣⡦⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢀⡴⢹⢇⠀⠀⠀⠀⣸⠿⠁⠀⡅⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠉⠀⢸⠀⠙⠒⠒⠋⢩⠀⠀⠀⡇⠉⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⠖⠋⠀⠀⠀⠀⠀⢸⠀⠀⠀⡟⠳⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣄⡠⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⡟⡛⢛⡻⣻⢿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡆⠀⠰⣄⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⢀⡟⠁⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡸⣿⡃⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣷⠚⠁⠀⠀⣀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢤⣿⣿⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⣤⡞⣣⣤⢬⣹⣶⣷⣴⣀⣤⣾⣠⣴⣦⣶⣴⣞⣁⣁⠰⡄⠀⢻⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣾⣿⣾⣯⡛⣓⣿⣙⠿⠛⠛⠙⠙⠛⠛⢿⣿⢻⣿⠾⠟⣿⣾⣦⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⢸⣿⢿⠟⠧⠉⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⡉⠙⠛⢁⣽⣿⣿⡏⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣖⢸⣿⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⣿⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⣼⣼⢸⣿⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠴⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⢈⣿⠀⠀⠠⠀⢄⣾⣶⠁
⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⣶⣿⢇⣼⣿⠁⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⢀⣀⠀⠈⣶⡆⠀⠀⠀⠀⠀⢸⣿⣷⡜⣿⣿⣿⢸⡴⣺⡾⠁
⠀⠀⠀⠀⠀⣴⠟⠳⢦⣄⣠⣤⣾⣿⣿⡅⣠⠃⠀⠀⠀⠀⠀⢂⣿⣿⢿⣿⣿⣾⡴⠀⠀⠀⠀⠀⠸⣄⣾⣿⣿⣿⣦⣌⣈⣠⡴⠫⠤⠄
⠀⠀⡀⣠⣼⡇⠀⢀⣦⣿⣿⣟⠙⢻⣿⡿⠉⠠⠊⠠⢄⡤⠀⠀⠧⣴⣾⣿⣮⠌⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡟⢻⣿⣿⣧⡄⠀⠙⢀⣄
⠀⠀⡴⠟⢻⣷⠓⠛⠿⣿⣿⠿⣷⣿⣿⠅⠀⠀⠀⠀⡀⠠⠀⠀⠀⠈⢻⡿⠃⠀⠀⠰⠄⠄⠀⠀⠀⠀⢸⣿⣿⣷⣿⢾⣿⣿⠟⠒⣦⡾⠻⢧⡀
⠀⠀⠘⢶⣾⣿⣶⣶⣦⣿⣯⢴⣿⣿⣿⣧⡀⠀⠀⠀⠀⢐⡓⣃⣤⣴⣿⣿⣦⣄⣀⠀⠂⠀⠀⠀⠀⣠⣼⣿⣿⣿⣶⣈⣿⣤⣴⣶⣾⣳⣶⠎
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠉⠙⠉⠛⠛⠛⠛⠉⠁⠉⠉⠉⠛⠛⠛⠋⠉⠉⠙⠙⠛⠋⠉⠀⠀⠀⠀⠙⠋⠙⠛⠛⠉⠛⠉⠛⠛⠋⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣍⣫⣭⣉⣽⣭⣿⣻⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⣠⡾⠋⠁⢀⠀⠀⠀⠁⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠈⢁⢄⣴⠏⢀⣴⣷⡎⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⡞⢡⣾⠏⠀⡟⢻⡟⢁⠀⠀⠀⠀⠀⢠⡀⢈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⢇⠘⡏⠀⡀⠀⣀⠀⢀⠈⠁⠆⢀⣴⣆⡀⠀⣿⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠅⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠈⢰⣯⡄⠀⣾⣿⣿⠦⡀⠀⠰⠟⠿⠿⠁⣸⡟⠸⠙⣆⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠀⡿⠓⣀⣾⣿⣿⣿⡷⠃⢠⠀⠰⠂⠁⢠⡿⠁⠀⠀⠸⠀⢀⣤⣶⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠠⢁⡞⣠⠐⢻⣿⣿⣿⠏⠀⣤⠀⠀⠀⠀⢀⣾⠁⡸⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢀⣴⣿⣧⣽⣄⡀⠉⠻⡄⠀⠴⠋⠀⠔⠉⣠⣾⣇⠀⢣⠀⠀⠀⢻⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⣿⡿⠋⠉⣻⣿⣶⣶⡾⠀⣰⣿⣶⣿⣿⣿⣿⣿⡄⠸⠄⠀⢠⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⣿⠟⠋⠉⠘⠋⠀⠀⠀⣴⣿⣿⣿⣿⣷⠸⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠻⠇⠀⠀⠋⠀⠀⠀⠀⢀⡞⣁⣤⠾⣿⣿⣿⣿⡇⠀⠀⢠⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠉⠀⠰⣿⣿⣿⣿⣿⡀⢠⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⢀⣠⠖⠀⠀⢸⣿⣿⣿⣷⣤⣽⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⢠⣾⠀⠰⠀⠀⠖⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣧⢸⡎⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠁⠀⠀⡘⠁⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠸⠇⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⢀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⢋⣞⠰⡆⠀⠀⠸⣷
⠀⠀⠀⠀⢀⠞⠀⠀⠀⢠⠾⠆⠁⠀⠀⠀⠿⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠿⢁⣾⣿⠀⠹⡆⠀⠀⠈⣧
⠀⠀⢀⠴⠋⢀⡤⠀⠈⠀⠈⠀⣻⣄⡀⢀⠀⠀⠀⠀⠀⠘⠀⢰⡄⠀⠀⠀⠀⣰⠿⠟⠛⢉⣴⠋⠁⠈⠳⣤⡀⠀⠀⠀⢹
⠀⡜⠁⠀⠀⢡⢡⣤⡴⠖⠐⠉⠉⠉⠉⠈⠉⠲⣤⣀⣀⣴⡖⠈⠛⢠⠀⢀⣼⠁⠀⣀⣴⠞⠁⠀⠀⠀⠀⠘⢷⡀⠀⠀⢸⡄
⠀⡇⠀⠀⡀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⡹⠀⠀⠀⠀⠀⢠⡿⢶⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠸⡇
⠀⡇⠀⠀⠐⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠁⠀⠀⠀⢀⢠⡿⡀⠀⠀⠂⢻⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠠⢡⡟⠀⢇⠀⠀⢀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠘⡄⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣰⠏⠀⠀⡎⠀⠀⠈⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠸⡀
⠀⠸⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⢀⡴⠃⠀⠀⢠⠃⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⢣
⠀⠀⠀⠀⢠⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⢀⡞⠀⠀⠀⠠⠎⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠈⢧
⠀⠀⠙⣄⠀⡀⠀⣶⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢀⡞⢀⡤⠤⠘⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠀⠗⠄
⠀⠀⠀⠈⠓⠹⠸⠁⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢀⢂⡾⠀⢸⣢⠀⠠⠄⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣀⢠⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⢁⡞⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠂⠛⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡐⠁⡌⠀⣀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢇⡾⡄⠀⠋⣤⠏
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠟⠟⠃⠐⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⠇⠀⣀⠀⠀⠀⠀⠀⠐⠰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣞⠛⢨⣲⣿⡀⠠⠿⠃⠀⠤⢤⡀⢀⣙⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⣭⣥⡆⠀⠜⠋⠀⣾⠶⠟⠃⠀⠀⣀⠀⢀⡐⠊⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣗⣻⣧⣦⠈⢹⣃⠽⠋⠀⢀⣛⣁⠀⠘⠀⠰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣾⠛⢿⡟⣻⡙⠛⡀⣤⣦⣾⣿⠿⠆⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣽⣿⣷⡻⣡⣿⡁⠄⢧⠀⠈⠉⠁⠀⠀⠀⠀⢰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣆⣡⣴⣿⣿⣿⣷⣦⡘⢧⠀⠀⠀⠀⠀⠀⣴⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣭⣿⣿⣿⣻⣿⡿⡜⣧⠠⡁⠀⠀⠀⠈⠻⣟⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢿⣿⣿⢻⣶⣽⣿⣿⣿⡛⣿⣿⠏⣿⣦⣄⣀⠀⣀⣪⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡞⠿⣷⡄⠘⣟⢁⠉⡻⢿⣽⢿⣏⣬⣻⣿⡄⣿⠉⠉⡅⠿⠹⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⣁⠀⠈⠐⠈⡷⡃⠀⠀⠀⠉⣙⣿⡿⠙⠩⠰⠇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣋⠀⠀⠊⠀⠀⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠑⠁⠅⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠐⢊⣹⣦⣀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣴⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣴⣶⠄⠉⠋⠻⣿⣧⣄⡐⠀⣉⣤⡄⠀⠀⠈⠀⠀⢀⠱⠶⣦⢷⡟⠀⠘⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⡅⡀⠘⢠⠐⠄⠉⢯⣛⢷⣤⣋⡤⠀⠃⠀⠀⡰⠟⠀⠀⠙⡏⢡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣇⠙⢀⠀⠁⠀⠐⠈⣿⣷⣭⠁⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠙⣇⣀⠀⡀⠀⠀⣠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⣀⣈⠁⠀⢠⠀⠀⣘⣿⣿⡄⠇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠉⠁⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣭⣠⠂⠀⠀⠀⠘⢿⡻⢶⡞⢀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢖⣄⠁⡀⠀⠀⠉⠙⠓⢆⡀⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢶⡤⣀⠀⠀⠇⠐⠆⢻⣇⣀⣄⣊⣷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠠⠾⠶⠗⠂⠉⠀⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠉⠉⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⢀⢤⠄⠊⠀⠁⠈⠐⠠⣀⡀⠤⠠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⡀⠔⡡⠁⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⠠⠐⠊⢀⠔⠀⠀⠀⠀⢀⠔⠂⠐⢄⠀⠀⠈⡀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢻⣷⡄⠠⠀⡠⠁⢀⠀⠀⠀⠂⠀⡇⠱⡀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠘⠿⠷⠀⢀⠁⢠⣿⣗⠈⡇⠈⠀⡇⠀⠈⠂⢄⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⣶⡉⠒⠚⢤⡀⠙⠻⠃⠀⢠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠙⢿⣿⡿⠟⠣⠀⣀⠀⠄⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠉⠁⠀⠀⠉⠀⣀⠠⢔⢇⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢲⡂⠀⠀⢀⣈⠥⠐⣈⠔⠈⠁⠒⠤⣀⠀⠀⠀⠀⠀⠰⠀⠸⠖⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡙⠀⠉⢀⡠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⢁⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠙⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⣹⠀⠀⠘⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢾⠀⠀⠀⣆⠀⠀⠀⣀⠤⠒⠀⠀⠀⠀⠀⠀⢉⣀⠉⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠖⠉⠀⠀⠀⠸⡀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠂⠀⠀⠀⡇⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠀⠀⢀⠔⠀⠀⠀⠰⢀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠴⣀⣰⡡⢀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠀⠊⠀⠑⠂⠀⠀⠀⠐⠒⠒⠒⠒⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄


⠀⠀⠀⠀⠀⢀⣤⣴⠒⠻⢻⡿⢷⣦⡀
⠀⠀⠀⠀⠤⣿⡷⢬⣦⣤⣼⣿⣆⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣤⡖⠀⠀⠀⠉⢡⡀⠤⣭⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠈⠧⡀⠀⢀⡄⠯⠀⠿⢿⣿⢿⣿⡟⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⣠⠋
⠀⠀⠀⠉⠀⠀⠀⠀⠀⢲⣾⣷⠿⣣⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠛⢻⣿⣿⣷⣶⣶⣶⣶⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢍⢇⣰⡀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡿⠁⠀⢀⣴⣻⣿⣿⣿⣿⣿⣿⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠹⢓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢸⣿⣾⣿⣿⣿⣻⣿⢻⡃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⠃⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠙⠛⢿⣿⠋⢉⣀⣿⣿⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢨⡀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠀⠀⠀⠀⠀⠀⡀⡸⠋⠀⠀⠰⠀⠀⠀⠀⠊⠛⣿⣷⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⠀⠻⠀⠀⠀⠀⠄⠁⢸⠸⠀⢀⣀⠀⣀⠠⠼⠃⡄⣐⠒⢦⣀⡀⠂⠀⠀⠀⢸⠟⠋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⢣⣁⣠⠀⠀⠀⠀⠀⣮⠀⠚⠉⠀⠀⠈⠀⠀⠂⠀⠹⣤⣶⣿⣇⠀⠀⠂⠀⡘⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⡿⠃⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠋⢿⣿⣦⣄⠈⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠈⠁⠲⣄⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠈⠑⠂⡀⡀⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠈⠂⠐⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣈⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⢀⢀⣰⣦⣀⣤⠀⠀⠀⡀⠀⠀⠀⠈⣀⣰⣶⣆⣮⣦⠄⠀⠀⢀⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣠⡄⡀⠠⠆⢀⠇⣀⠀⠀⠀⠆⢀⢤⡤⢶⣆⣜⠿⠿⢿⣿⣷⣡⣎⣈⠁⡀⠐⠀⠀⠈⠻⠿⣿⣻⠋⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠜⠿⠿⢿⣴⣤⣤⣬⡞⠥⠂⠀⡄⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠛⠛⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠋⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠉⠀⠀⠈⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣀⣀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⣤⠶⠛⠉⢹⡄⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣴⠖⠃⠀⠀⠀⠀⢀⣤⣤⣤⡤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠁
⠀⠀⠈⠓⠶⠦⠤⠴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀
⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉
⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣸⡇
⠀⠀⠺⠿⠿⠇⠀⣀⣀⡴⠾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣾⠇⠟
⠀⠀⠀⠀⠀⠀⠛⠛⠉⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢀⣼⠗⠚⠛⠀⢠⡄⠀⠀⣤⣀⠀⠀⠀⣀⡀⢀⣾⠁⠀⠀⠀⠀⢀⡾⢱⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⣀⣤⡶⢿⡃⠀⠀⠀⢰⡟⠁⠀⠀⠈⠉⠛⠈⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⣸⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⢀⣴⠛⠉⠀⠀⠾⢷⣀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⢠⡾⠟⠁⠀⠀⠀⠀⠀⠀⢹⡤⠀⠀⠙⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⠀⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠙⠳⡄⠀⠀⠀⠀⠀⠀⠀⢻⡇⡀⠀⠀⠀⠹⣆⠀⠀⣨⣳⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠶⠛⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠉⠳⢶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣤⣄⣀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⣀⢈⣷⣼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠓⠠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠙⠒⠢⢄⣀⡠⠖⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠤⠠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣤⡀⣿⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⢸⢀⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣶⣷⣦⡀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⣶⣿⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣿⢿⣿⣧⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣟⡼⠁⢻⡄⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠢⣛⣿⣿⡿⠃⠀⠀⢀⣠⡤⠤⠤⣤⡀⠀⠈⠉⠉⠀⠀⠀⠀⢻⡄⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣶⣶⣶⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⢠⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠙⠷⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣠⠏⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠁⠀⠀⠀⠈⢹⠶⢤⣀⡀⠀⠀⠀⠀⠀⠔⠃⠄⠀⠀⣀⣀⡤⠴⠚⠙⣄⠀⠀⠀⠀⠀⠀⠘⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠈⠈⠉⠉⠑⠤⣄⣀⣀⣠⠤⠊⠈⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠇⠈⠐⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⢦⠀⠀⠀⠀⢸⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠳⣄⠀⣠⠏⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡁⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠑⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢼⠀⠀⠀⠀⠀⠀⠀⡸⠓⢤⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⠄⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⠁⡀⠀⠀⠀⠀⠀⠀⢰⠤⡴⠋⠀⠀⠀⠀⠀⠀⠀⢠⠇⢠⠃⠀⠀⠀⠀⠀⠀⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣇⠀⢀⠀⠀⠀⣠⠏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠙⢄⠀⠀⡆⠀⢰⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠀⠀⠀⠙⠦⠤⢄⣀⣀⠴⠤⠔⠃⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣴⣶⣶⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠈⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠈⢉⣤⣤⣶⣤⣤⣀⣀⠀⣀⣤⣶⠶⠶⣮⣝⢿⠁⠀⡸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠁⠈⠹⣶⣄⡾⠋⠁⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠈⢻⣧⡐⠛⠛⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠹⠛⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣿⣷⠀⠀⠀⠈⢿⣿⣿⡿
⠀⠀⠀⠀⢠⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢉⣉⡉⠻⣿⠀⠀⠀⠀⠀⢀⣀⣈⠙⢷⣿⣿⡆⠀⠀⠀⠀⠻⠿⠃
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣻⣯⣿⠆⠟⠀⠀⠀⠀⠀⢿⣯⣿⡷⠌⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠉⠀⠀⠀⠀⣀⡀⠀⠀⠠⢭⣉⣀⣀⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁⠀⠀⠀⠠⣾⢿⣿⣿⡿⣶⠀⠀⠀⠉⠙⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⣴⣿⣿⡦⠉⠀⠀⠀⠀⠀⠀⠈⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡉⠀⠀⠀⠀⠀⠀⠀⡀⢀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⠀⠀⠀⢀⣠⣾⣿⣤⣀⠀⠀⠀⣀⣼⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠛⠿⠿⠿⠟⢻⣻⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⣠⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠛⠁

⠀⠀⣄⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠉⠊⠉⠁⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣶⣾⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⣿⣿⣶⣶
⠀⠀⠀⠀⠖⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠂⠀⠀⠀⠀⢰⣿⡿⠿⠛⢿⣿⡀⠀⠀⠀⠂⠀⠀⠀⠘⠀⠀⠀⠀⠀⠘⠍⠀⠀⠈⠛⠋⠋
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠘⠟⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣷⣀⣀⣂⣆⠀⠀⠀⠀⠀⠀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠐⠀⠘⠿⠧⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡯⠀⠀⠀⠀⠀⠀⠈⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣧⠀⢀⣾⣿⣆⠀⠀⠀⠀⡞⠃⠌⢁⠀⠀⠀⠰⠟⡿⠃⠀⢰⡏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⡄⠀⠀⠰⢦⣖⣰⠞⠀⠀⠀⠀⠀⠀⣘⣃⡈⠃⠀⠀⠀⣆⠀⠀⠀⢠⠀⠀⠈
⠀⠰⠀⠐⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠸⠇⠀⠂⠀⠀⠘⠟⠛⠛⠃⠀⠀⠀⠀⠙⠙⠀⠀⠀⠀⠀⠀⠘⠽⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡀⠤⡤⠀⢀⠄⠀⠀⡀⠀⣺⣦⣄⡖⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠰⠆⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⠁⠈⠋⠀⠀⠀⠳⠂⠀⠀⠀⠀⠲⠞⠁⠀⠀⢠⣶⣿⠛⣧⣹⣎⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣤⡄⠀⠐⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠹⣋⠠⡆⢀⡈⠀⠀⠀⠀⡸⢧⡏⣤⠀⣧⡛⠀⠀⠀⠀⠀⠀⠀⠄⢠⡄⡇⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⠳⢶⠖⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⢀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢤⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⣤⣦⡦⣣⠔⣶⣄⠀⠀⠈⣷⣿⡿⣿⣿⡇⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠰⣇⣽⠛⢫⣻⡃⠀⠀⠀⠀⣀⠀⠀⢀⠉⢡⡟
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⠿⣿⡿⠄⢠⡾⠯⠙⠂⠀⠀⠟⠋⣤⠝⠛⠃⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠰⣿⣿⠲⢾⣟⣿⠀⠀⠀⠘⠿⠀⣄⢙⣁⠄⠡
⠀⠀⠀⠀⣓⠈⠀⠀⠀⠀⠀⠀⠀⠞⠻⠏⢻⡟⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠐⠀⠉⠁⠀⠀⠀⠀⠀⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠢⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢶⣶⣤⠀⠀⠀⠀⠀⠀⠀⢩⠁⠀⠈⠋⠀⢹⠿⠃⠀⠀⠀⠀⠈⠈⣡⣴⠀⠁⠀⠀⠀⠘⠀⠀⠂⠀⠀⠀⠰⣶⡆⣶
⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠂⣥⣄⢀⡹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣆⠀⢀⠀⠀⠀⠀⠀⠀⡖⠁⠈⡏⠀⢀⠀⠀⠀⠂⠀⠀⠘⠀⠀⠀⠀⠿⣁⠙
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢿⠟⠘⠁⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠲⣴⣼⠬⡵⡃⠀⠀⠀⠀⠀⠀⡇⠀⡂⠈⠀⠠⠀⠀⠀⠐⠐⠗⠈⠀⠀⠀⠀⠀⠉
⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠙⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠟
⠀⠀⠘⠙⠁⢸⡿⣶⣶⡆⠀⠀⢠⢀⠐⣦⡴⡆⠀⠀⠀⣟⣿⡄⣤⣾⡿⠂⠀⠀⢀⣠⣸⣶⣿⣧⡤⠀⠀⠀⠀⠀⠀⠀⣤⠞⠚⢷⣦⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⠘⠃⠀⢐⣿⡿⠀⠸⢦⡟⠀⠀⠀⣼⣻⠇⣟⣿⠀⠀⠀⠀⠀⢩⠿⠛⠛⠛⠄⠀⠀⠀⠀⠀⢀⠀⠿⠛⠓⠚⠂⠀⠀⠀⠀⣰⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣙⢿⡿⠀⠀⠀⠸⣿⠴⣿⢷⣴⡿⠀⠀⠀⠙⠁⣀⠈⠃⠀⠀⠀⠂⠀⠃⠀⣶⡆⠀⠀⠄⠀⠀⠀⠠⠆⢀⣶⢀⣀⢰⣤⢀⡄⠀⠀⠛⢿⠿⠿⡟
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠛⠒⠿⠾⠟⠁⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠘⠀⠀⠼⢄⣀⡠⠀⠀⠀⠀⠀⠓⢺⣿⣬⣋⣼⡷⠜⠁⠀⠀⠀⠘⠦⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠤⠔⠂⠈⠉⠉⠉⠉⠉⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠐⠒⠀⠀⠀⠀⠰⢄⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⠁⢠⣲⡶⠀⠀⠀⠠⢴⠄⠀⠀⠈⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡘⠀⠀⠀⡠⠂⠀⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠂⡇⢀⡀⠀⠣⡀⢠⠄⠀⠀⠀⢀⠀⢠⡄⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⢻⡈⢷⡀⣀⡤⠤⣤⣤⡤⢶⠃⢠⢾⠳⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠙⢆⢱⡏⠀⠀⠀⣧⠖⡡⠊⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠈⢳⠀⠀⠀⠀⢣⠔⠁⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠸⡀⠀⠀⢽⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⢳⡀⠀⠈⠢⠤⠤⠊⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⠇⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣼⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⢰⠁⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠈⠢⡀⠀⠀⠀⠘⡄⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⣇⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⡴⠁⢱⡀⠀⠀⠀⢸
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⡄⠀⠉⠀⠀⠀⢹⠀⠀⠀⠈⡇⠀⠀⠀⢠⠞⠁⠀⠀⠳⡄⠀⠀⠈⡇
⡰⢹⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡄⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⢧⠀⠀⢰⠛⡆⠀⠀⠀⠀⠹⣄⠀⠀⢻
⡇⢸⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠡⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠸⡄⠰⠃⢀⡇⠀⠀⠀⠀⠀⠘⡆⠀⠘⡄
⣇⠈⢆⠀⠀⠀⠀⠀⠀⢹⠀⠀⡆⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢧⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢱
⠘⢦⠈⠓⠤⣄⣀⣀⣤⠼⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠘⣯⠵⠶⠤⡀⠀⠀⠀⠀⠀⠀⠱⡄⠈⢧⡀
⠀⠀⠙⠦⣄⣀⠀⠀⠀⣀⣹⡆⠀⠀⠈⠙⢣⣤⣀⣀⣀⣀⣀⣀⣀⣀⣼⠀⠀⠀⠸⢄⣀⣑⣮⡇⠀⠀⠀⠀⠀⠀⠘⡆⠀⠉⠒⠢⣄
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠳⠦⠷⢤⠴⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⣰⠀⠰⡙⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣀⣸⡀⣈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣭⣷⣾⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠙⢿⡗⠶⣤⣄
⠀⠀⠻⣆⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠛⠋⠉⠙⠛⠛⠶⣤⣄
⠀⠀⠀⠘⢧⡀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄
⠀⠀⠀⠀⠈⢳⡄⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠹⣦⣀⡀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⢀⣠⠠⠠⠀⠀⠀⠀⠀⠀⢠⢤⢀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠛⠿⠟⠀⠄⠀⠀⠀⠀⠀⠸⡾⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⠈⣯⢻⣄
⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⢹⡄⠉
⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠁
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠀⢀⣀⣴⠟⠁
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢀⡶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣼⠃⠀⠘⢷⣄⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠈⠳⣿⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣰⡷⠶⠴⠶⠞⠛⠉⢻⡄⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣾⠁
⠀⠀⣴⠟⠀⠀⠀⠀⠀⠁⠈⢷⡄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢿
⠀⢸⡇⠀⡀⠀⠀⢠⠀⠀⠐⢀⡿⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⡠⠼⢧⣄⡀
⠀⠀⠙⠶⢦⣄⣀⣼⣤⣤⡴⠟⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠊⠀⠀⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⢦⣀⣀⣠⣧⣀⣀⣧⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⡤⠾⣥⣀⣀⣴⢦⣤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠶⣿⡟⢿⡿⢻⣿⠗⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠚⠛⠣⡄⢰⡜⠋⠛⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢠⡀⠀⢈⣷⣄⠀⣹⣿⠀⣠⣾⡃⠀⠀⣄⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣠⡀⠈⣿⠉⢿⡟⢻⣿⠋⣿⠁⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⣿⡇⠀⢸⠄⠸⠁⠰⠇⠀⡏⠀⢰⣿⣿⣓⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⢋⣿⣿⣟⡀⣈⠀⠀⠀⠀⠀⠀⢡⢀⣸⣿⣿⡍⣧⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣷⢀⣿⠟⢿⣿⣿⡿⠏⠀⠀⢀⡀⠀⠀⠘⢿⣿⣿⡟⠛⣿⡆⢸⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⣿⠸⣿⡔⢿⢿⠏⠠⠀⡀⠀⠘⠃⠀⠀⡀⠦⠘⠟⠿⠀⣿⠇⣿⣸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠃⣿⡆⣿⡏⠃⠀⠀⠀⠛⣧⠀⠀⠀⠀⢸⡍⠀⠀⠀⠀⢠⣿⢀⣿⡇⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⡏⠘⣿⡿⣿⡇⠀⢨⡇⠀⢸⡏⣘⢻⣟⣃⢘⣧⠀⠸⡇⠀⢸⣿⢸⣿⣇⣹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠈⣴⣿⣿⡇⢸⣿⠀⠈⠀⣾⣿⣿⣾⣿⣿⣿⣾⣿⣷⡄⠁⠀⢸⡏⢸⣿⣿⣿⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠈⣿⣿⣿⣷⠘⣿⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⢻⠇⢸⣿⣿⣿⡟⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡎⢹⣿⣿⣿⠀⢹⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠃⠀⠀⡜⠀⣿⣿⣿⡏⠀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⢠⣿⣿⣿⡇⠸⣧⠀⠀⣁⣻⣿⣿⣿⣿⣿⣿⣿⣉⠀⠀⢸⡇⢠⣿⣿⣿⡇⢀⣯⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⣿⠀⢿⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⢸⠃⣾⣿⣿⣿⡟⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⣿⣿⠀⣿⣿⣿⣿⣧⠈⠀⠀⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠀⠀⠈⣰⣿⣿⣿⣿⠁⣼⣿⡟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⣿⣧⣿⣿⣿⣿⣿⡄⠀⠀⠛⣿⢿⣿⣿⣷⣌⠛⢿⠛⠀⠀⢠⣿⣿⣿⣿⡿⣰⣿⣿⣷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡞⣿⣿⣿⣏⣿⣿⣿⣿⣧⡆⠀⢸⣿⣿⣿⣿⣿⣿⣓⢛⡇⠀⢰⣾⣿⣿⣿⣿⣣⣿⡟⢹⢉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⡟⣋⣟⠀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⠃⠄⠀⣸⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⢠⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠿⢿⠻⡿⢿⣿⣿⣿⣿⣿⣿⠟⠷⠉⠛⠙⠻⣿⠿⡿⢿⠟⠋⠛⠉⠳⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣰⡾⢛⣃⣺⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢹⣀⡇⡾⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠛⡿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣤⣾⣿⣿⣼⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢸⢿⡿⣇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⢤⠰⣦⣤⣠⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣛⣿⣿⡿⠋⠉⠉⠁⠐⠚⠛⠁⠀⠀⠀⠀⠀⠉⠙⠓⠉⠉⠉⠉⢻⣯⣁⢀⣀⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡤⠄⠀⣀⡀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⠁⠀⠀⠐⠉⠉⠀⠀⠈⠁⠀⠀⠐⠀⠒⠒⠂⠶⡶⢄⠱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⢀⣿⡏⠠⠂⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⢀
⠀⠀⠀⠀⡀⢀⠀⠀⠀⢰⡆⠀⠀⣤⠏⡤⠀⠀⠀⠐⠂⡰⠦⠀⠀⠀⠄⠂⠀⠀⣤⠂⠀⠀⠀⠀⠀⠀⠀⢹⣷⠈⠛⠂
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⡄⢀⣴⣏⣜⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢳⢦⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣏⣡⡄⠀⠀⣀⣤⣔⢄⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡆⢿⠘⢠⠁⠀⠀⠀⣿⠀⡄
⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⣼⣟⣵⠶⠓⢠⣿⣿⣧⣿⡌⠹⠃⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⣿⢷⣶⡀⠀⠈⠛⢾⡇⠀⡇⠀⠀⠀⢉⠀⡿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⠉⣿⣿⡿⣀⠀⠀⠑⠘⠋⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠺⠻⣟⡽⠳⠀⣸⠸⣿⠁⠀⠀⠀⠀⠀⠈⢸⠃
⠀⠀⠀⠀⠰⡟⡀⠀⠀⠀⡟⢰⣿⣿⠀⠈⠹⣿⣶⠀⠀⠀⠀⢀⣤⡄⠠⠀⡀⣀⠀⠀⠀⠣⡀⠀⢀⣤⠀⠘⡓⢿⠀⠀⠀⠀⠀⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠈⠻⣆⠀⢀⠁⣾⠏⡿⠀⠀⣀⢹⣿⡀⠀⠀⣰⡿⣋⡀⠀⠌⠁⠈⠻⣄⠀⠀⠐⣶⣿⡏⠀⠀⣿⣿⢸⣇⠀⠀⠀⢰⢧⠇
⠀⠀⠀⠀⠀⠀⠀⠘⢇⡜⢴⣿⢸⡇⠀⠀⠛⡿⡭⠁⠀⢠⣿⣶⣿⣿⠁⡀⢺⣿⣷⣼⡄⠀⠀⠻⣿⠀⠀⠀⣽⣿⡘⢿⡀⠀⠀⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⣼⡇⣸⠗⠋⠁⠈⠀⠃⠀⠀⠀⠑⣧⡈⢡⣼⣗⣠⠉⣻⠿⠇⠀⠀⠀⠉⠲⣀⢀⡿⣹⣇⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⢃⣿⢀⡴⣶⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⢠⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⣀⠀⢸⡇⡿⣿⠐⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⣻⡯⠸⠰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾⢄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠈⣧⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣩⣿⠚⣧⣿⣿⣦⡀⠀⠀⠀⠀⠀⢀⣤⣿⣦⣡⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⡼⢏⣼⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠸⡇⢻⣿⣿⣤⣄⣀⣠⣴⡿⣿⣟⣙⣧⣀⠀⠀⠀⠀⢠⠎⠁⠔⠃⠛⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⣇⣿⣿⣿⣿⠗⠒⠂⢺⣿⡋⠀⠿⣷⣶⣶⣶⡿⠐⠀⠀⠛⠰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⡷⠞⣨⠛⠁⠀⠀⠀⠙⠋⠁⠀⠀⠸⣯⣽⣟⠓⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠹⠿⣿⡿⠿⠛⣷⡶⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠃



⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠁⠀⠀⠀⠁⠈⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣧⣾⣿⢹⣿⣟⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣛⣷⣼⣿⢸⣯⣽⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠇⣾⣿⣿⣷⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠟⠁⠀⠈⡻⣷⣿⣿⡟⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡏⣠⣴⣶⣦⣍⡹⣿⣿⠁⣿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠀⢻⣷⣾⣿⣿⡇⢿⡿⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠙⡿⠋⠈⠁⢸⣷⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⣴⣾⣿⣆⠀⠀⣿⡟⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣦⣞⠉⠉⡟⠉⢧⣼⣿⠛⠸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡏⠈⣱⣧⣍⠀⣿⡏⠀⢰⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠻⠿⠟⠁⠀⠏⠛⡁⠀⢿⣷⣴⣿⠟⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠁⠀⠀⠀⠀⠁⠠⠊⠉⠉⠁⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢷⣄⠀⠠⡤⠀⠦⠄⠀⠀⠴⣶⣾⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣟⠢⠹⡷⠈⣣⠀⠀⠀⠚⠁⢴⠟⠛⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⡆⢷⡜⢧⣤⢠⠞⠋⢤⣿⣾⠺⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⠺⠂⣾⣿⢹⣇⠰⣿⣷⡏⠀⢸⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⣿⣿⣷⣜⣿⣿⡏⣿⣦⣾⣿⠁⠀⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢷⡌⢿⣿⢿⣿⣿⢿⣿⣿⣿⣿⠇⣇⣠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣃⣈⣿⠈⣿⣿⢿⣿⣿⣿⣿⡀⢿⣿⣿⡿⢋⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠘⣿⠀⠙⠻⠻⡯⠃⢨⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⣿⣿⡇⠀⠈⠈⢓⣦⣼⣁⠀⣾⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⢿⣿⡇⠀⠀⠀⠉⠀⠺⠁⠉⡙⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⢠⢀⡍⠰⢿⠀⠀⠀⠀⠀⠀⠀⢠⡆⢹⠙⠉⠻⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡇⢀⣨⡧⢼⢀⠀⠀⠀⠀⠀⠀⢸⡸⢆⡼⢀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠐⠓⠈⠁⠀⠀⠀⠀⠀⠀⠈⠛⠺⠷⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡾⠿⣿⡛⠛⠿⣛⠛⠛⠛⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⣠⣤⠴⠶⠒⠛⠋⠉⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠈⠙⣧⠀⠀⠀⠀⠀⣠⢞⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⠞⢡⡾⠁
⣠⠖⠒⠒⠒⠒⠚⠋⣁⣠⣴⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣤⣀⠀⠀⣿⠛⠶⣤⣤⡤⠿⠃⠀⣴⠃⢀⡿⠁
⠉⠓⠶⢤⣄⡉⠉⠛⠛⠋⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡏⠉⣿⡇⠀⣿⡀⠀⠀⠀⠀⠀⢀⡼⠁⠀⣼⠇
⠀⠀⠀⠀⠈⠙⠷⣤⣀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣯⣤⣤⣉⣷⠀⢻⡏⠉⠙⠛⠒⠲⠞⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣾⡏⠀⣠⣤⣴⣶⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠿⢛⣿⣿⣧⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠙⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠛⠁⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄⠀⠀⠀⢀⣤⣤⣤⣤⣶⣦⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣿⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠙⠿⠿⣿⠟⠁⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⣀⣰⣇⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠤⠤⣤⣿⣦⡶⠞⢠⡞⠁⠀⠀⠀⠀⠀⠀⢸⡇⣀⡠⠋⠀⠀⠹⣆⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⢀⣴⠶⠦⣤⣄⣀⣀⡤⢶⡋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢻⠁⠀⠀⠀⠀⠘⠲⢤⣄⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣠⠞⠁⠀⢰⡇⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⢸⠃⠈⣧⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠟⠃⠀⠈⣻⡄⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠀⠀⠀⠀⠈⢀⣀⣤⡴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⣟⣠⡤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡃⠀⠀⠀⠀⠀⠀⣴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢠⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣤⠤⠤⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣰⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⡆⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⢀⡖⠈⠁⠀⠀⠀⠀⠀⠀⠅⢀⠠⠒⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡀⢀⣤⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⠃⢸⣷⡼⠀⠀⠀⠀⠀⣿⣏⣷⠀⠀⠀⠈⠒⠢⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠌⢠⣾⣍⣉⣉⣁⣳⣄⠀⠙⠟⠁⠀⠀⠀⠀⠀⢀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠈⠈⠂⠀⠀⠀⠀⠀⠀⠀⠊⠁⠠⠁⠆⠀⠀⠀⠀⣀⢀⢀⡀⠀⠀⢀⡀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠒⠒⠀⠀⠀⠀⠀⢠⠔⢁⠌⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢀⠀⠂⢀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠔⠈⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⡕⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠁⠐⠤⡀⠀⠀⠀⠀⠀⠀⢲⠠⠴⢈⠙⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠗⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠟⠀⠀⠀⠂⠀⠈⠁⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠀⠀⠀⠠⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠐⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠂⢀⠀⠀⠀⡠⠁⠀⠀⢀⡞⠀⠀⠀⠀⡇⠄⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠰⢀⠀⠠⠊⠀⠀⢀⠔⠈⠀⠀⠀⠀⡘⠃⣈⠀⣅⢀⠔⡀⠐⠁⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⡀⠀⠀⠐⠁⠀⠀⠀⠀⠀⢈⢀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠊⠁⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠒⠒⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠦⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⡀⠄⢣⠀⠀⠀⠠⠅⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡃⠤⠀⠀⠀⠀⠀⠠⠀⡀⠓⢄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠚⢦⠀⠀⠀⠀⡠⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠈⢂⡀⠀⠁⠀⠀⠰⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠀⠀⢠⢤⡤⣀⡍⠀⠐⢲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠆⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠂⠀⠀⡀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠠⠀⠂⠀⠀⠰⠀⠸⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⠐⠠⠄⡀⢀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⡖⠀⠈⠁⠀⠀⠂⠀⠀⠀⠀⠀⠁⢀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣠⠧⢄⠀⠀⠀⠠⠠⠄⠈⠀⢀⠈⠈⠁⠈⠁
⠀⠀⠐⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠆⠄⠀⣴⠇⠀⢀
⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠁⠀⠀⢀⠐⠔⠛⠡⠁⠢⠦⢄⣀
⠀⠀⠰⠀⠀⠀⠐⠒⠒⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡴⠀⠈⠁⠁⠀⠀⠀⠀⢠⠀⡤⠀⠠⠃
⠀⠀⠀⠈⠄⠀⠤⠀⠠⠤⠀⠀⠑⠀⢠⠄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⡴⠀⠀⡸⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⢀⠠⠀⠊⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠉⠀⠀⠁⡀⠀⢀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠈⠀⠁⠀⠉⠈⡉⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠄⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠉⠁⠈⠀⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢄⡀⠀⠀⠀⢀⣀⣀⣀⡀⠀⢀⠴⡊⠉⠓⠢⢤⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠞⠉⠀⠀⡠⠀⡩⠶⠶⠞⠁⠀⠘⠓⠒⠶⢥⡊⠈⡀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⣧⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡇⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⠶⠅⠤⢄⠀⠀⠀⢀⣠⠱⣄⣷⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡴⠁⢠⣴⡆⠀⠱⡄⠀⡜⢠⠀⠈⣿⢧⡀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⣇⠀⢻⣿⣷⡆⢠⡇⠀⢹⣿⣦⡆⣿⡈⠳⣄⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣰⡿⠀⠀⣈⡦⠤⣌⣁⣤⠿⢒⠖⠚⠛⠻⡒⠃⠉⢢⡈⠳⢦⣀⣀⡷
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢀⡾⢻⡅⠀⠀⠁⢠⢖⠀⠀⠀⠀⠀⠓⠦⡤⠋⠀⢀⣄⢸⡇⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢧⣀⣤⠞⠋⠀⠀⢷⠀⠀⠀⠀⠀⠑⢦⣀⣀⣀⣀⣠⣄⣀⠴⠃⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠤⣀⣀⢀⠀⠈⠫⡉⠉⠛⣿⠟⢀⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠈⢏⠙⠳⠤⣄⣀⣈⣉⣠⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠌⠀⡙⡄
⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠰⡖⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢙⢧
⠀⠀⡇⣇⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⠁⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠉⢸⡆
⠀⠀⡇⠹⡀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⣾⠃
⠀⠀⢻⡀⠙⢦⣀⣀⡴⠶⠾⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⡏⠀⠀⠀⢠⠖⠀⣇
⠀⠀⠀⠻⣦⣀⢀⡏⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⠀⡐⠁⠀⢸⠈⣧
⠀⠀⠀⠀⠀⠉⠹⡇⠀⠀⠀⠀⠙⡄⠀⢺⠀⠀⠀⠀⢸⠖⡟⠉⠀⠀⠀⡞⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢹⠀⡿⠀⠀⠀⠀⢸⣰⡇⠀⠀⠀⢠⠃⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠸⢤⡇⠀⣀⣀⢀⣀⣏⡇⠀⠀⠀⢸⣰⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡇⠈⠁⠀⠀⠀⠀⢱⠀⠀⠀⠀⣆⠙⡌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣰⣄⣠⣦⣤⣇⡀⠀⢀⡆⠀⣠⣸⣧⣤⣀⣀⣨⣀⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠒⠚⠉⠉⠑⠒⢤⣀⡠⠔⠒⠊⠉⠉⠉⡛⠙⠒⠶⠤⢄⡀
⠀⢀⡴⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠔⠒⠛⠛⠲⣄
⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣠⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣾⣿⠋⢹⡇⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢀⢻⣶⣿⡿⠃⠀⠀⠀⠀⠀⠀⣿⠿⡁⢈⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠈⢣⡀⠀⠀⢀⡾⣿⠀⠀⠀⠀⠀⠀⠁⣀⣀⣀⣀⣀⡀⠀⠀⠻⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢳⣀⣴⠟⠀⢿⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⠀⠀⠉⠓⠦⣕⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠙⣷⡀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⢀⣠⣴⠏⠀⣨⡖⠀⡀⠀⠀⢀⡾⠁⠀⠈⢷⡀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠟⠛⠋⠁⣀⡴⠋⡇⠀⠰⠀⣠⣿⠀⠀⠀⠀⠀⢻⡄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢻⡄⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣤⠴⠚⠉⣠⡾⠀⠀⢀⡶⠋⡏⢳⡄⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠶⠿⠟⢁⣀⡴⠋⠀⠀⡇⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠇⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⠀⠀⠀⡰⠀⠀⠀⠸⡆
⢰⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠠⠀⠀⠀⠀⠀⠀⠉⠛⠛⠷⠶⠶⠖⠒⠛⠉⠁⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⢹⡄
⡇⡆⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⣧
⡇⠸⡀⠀⠀⠀⡰⠊⣿⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢸⡀
⢣⠀⠙⠦⢤⡾⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠄⠒⠒⠂⡄⠀⠀⠀⠈⡇
⠈⠳⢤⣀⡼⠁⠀⠀⣿⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⣇
⠀⠀⠀⢀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⣧⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢀⡴⠛⠉⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⡁⡠⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠡⠀⠀⢸
⠀⠀⠈⠳⠿⣤⣶⣀⣸⠀⠀⠀⠀⠀⠀⠀⢸⡷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠤⠤⣷⠀⠀⠀⠀⠀⠑⢄⢸
⠀⠀⠀⠀⠀⠀⠀⠈⢹⡀⠀⠀⠀⠐⠀⠀⢸⠀⠀⠈⠛⠲⠶⠤⠶⠶⠞⠛⠉⠁⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠺⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠸⣧⣀⢠⡏⠀⠀⣠⠀⠀⠀⠀⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠈⠉⠛⠻⣤⣀⣿⣀⣀⣀⣠⣾⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢾⣄⠀⠀⢸⠀⠀⠀⢀⣧⣤⠞
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣛⣛⣳⣶⣶⣛⣁⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠉⠠⡉⠉⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⣀⣀⠀⠀⠈⢢⢀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢣⣾⣿⣿⣿⣿⣶⣄⠀⠱⡀⢤⡘⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠱⡘⠿⠀⠈⠉⠉⢹⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⣸⠁⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠛⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢡⠀⠀⠀⡀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⢿⠀⠀⠀⠈⠳⠦⣤⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣿⣿⣿⣿⣿⣦⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠿⠤⢤⣤⣤⣤⠽⠶⠚⠉
⠀⠀⠀⠀⠘⢦⡀⢀⠠⠐⠁⠀⠀⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠈⠙⢻⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣄⣀⠀⢸⣿⣿⣿⡿⠿⣿⣿⣿⣿⣷⡄⠈⠙⠛⠻⠟⠛⠉⠀⠀⠀⡠⠒⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠹⠿⠛⠉⠀⠀⠀⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠐⠉⠉⠐⠠⢀⠀⠀⠀⠀⣀⠔⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⡀⠀⢀⣤⣀⡀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠑⠒⠊⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⢀⠔⢀⡴⠋⢀⡼⠉⠛⠒⠲⠶⠶⠶⠶⠖⢺⡇⠀⠀⠀⡴⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠂⠁⢀⡴⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣸⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⢣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠙⢦⣀⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⠳⣝⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁

⠀⠀⠀⠀⢈⣯⠀⣦⡶⣶⡶⣶⢾⡟
⠀⠀⠀⠀⠐⢶⣶⡖⣶⡆⣶⡇⢰⣞⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠘⠉⠛⠉⠛⠛⠋⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠈⠙⠖⣄
⠀⠀⠀⠀⠀⠀⠠⣒⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠌⠀⠀⠀⠀⢀⠀⠈⠂⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣴⣦⣄⠀⠀⠀⠀⠀⠈⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠠⣃⠌⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡐⢩⣁⠀⠉⢣⠀⠠⠀⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢀⡀⣇⢀⡩⠚⠀⠈⠀⡄⠀⡠⠐⠈
⠀⠀⠀⠀⠀⠠⢄⣀⡀⢀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠘⢤⠈⠢⠉⠀⠔⠋⠁⠝⠈
⠀⠀⠀⢠⢠⣠⢈⡍⠠⠈⣉⢌⡀⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⡀⠀⠀⠀⢀⡠⠪⠀⠀⠀⠘⠝⢄
⠀⠀⠀⠈⠀⠉⠚⠒⠐⠒⠙⠊⠛⠑⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡂⠀⠈⠢⠀⡠⠂⠀⠀⡄⢀⠀⠀⢠⣘
⠀⠀⠀⠀⠀⠀⠘⠛⠀⠋⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠉⠀⢠⠀⠈⠀⠀⡇⢀⠅⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠸⠀⠀⠀⠀⢀⠀⠀⠀⢀⡣⠊⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⠀⠀⠀⠀⢰⠀⠀⠀⠎⢀⠔⠁
⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⢿⣷⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠀⠁⠔⠈⠀⠀⡄⠀⠈⠳⠀⠈⠙⡅
⠀⠀⠀⠀⠀⠀⠺⡿⠻⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠑⠠⠦⠐⠑⠂⠓⠋⠐⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⠐⠀⠰⠀⠀⢠⣄⣀⠀⠀⠠⠴⠠⠀⠴⠀⠀⠀⠀⠀⠐⠀⠤⠀⠀⠌⠀⠘⠐⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⠀⠀⠐⡀⠀⠻⠟⠋⠀⠀⠀⣈⣤⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠘⠟⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠰⠂⠰⡆⠀⠀⠀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠰⠮⠖⠁⠐⡄⠀⣿⠒⠧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠁⠀⠛⠃⠀⠀⠈⠉⠀⠐⠊⠰⠃⠑⠂⠠⢠⡀⠀⠀⠀⠀⠀⢠⠠⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠐⠀⡄⠀⡀⠀⠈⠀⠁⠀⠀⠠⠀⠀⠀⠰⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⡷⣦⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠛⠙⠿⠀⠄⠀⠀⠁⠀⠁⡀⠀⠁⠀⠀⠀⡀⠉⡀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⠰⠦⠈⠓⠒⠀⢈⢉⢉⠻⠀⢈⡄⠀⢤⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣣⡔⠢⠤⢤⠀⠀⠀⣠⠠⠀⡄⠀⠀⠀⠀⠈⠛⠛⠗⠠⠀⠀⠀⣠⠄⣈⣀⣀⠀⠀⠀⠀⢀⢃⠰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⢀⣀⡀⡀⠀⠀⠀⡀⣠⣤⣤⣤⣄⠁⡀⠀⠁⠀⠀⠀⠀⢆⠀⠀⠈⠁⠘⠷⠲⠖⠒⠐⠏⠃⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠈⠁⠀⠀⠚⠇⡙⠛⠛⠻⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠢⠦⠀⠀⠀⠀⠈⢠⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣹⣷⠄⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠐⠀⠀⠉⠏⠀⠀⠘⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠦⠄⠀⢐⠀⠀⠀⠀⠀⠀⠀⠘⠊⠀⠂⠀⠀⠀⠀⠀⠀⢀⣤⡀⠒⠒⠂⠀⠂⠤⠄⠀⠴⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠒⢦⡀⠀⠀⠀⠀⠀⠀⢘⣿⣧⠀⠀⢄⠀⠀⢀⠀⠀⢀⠒⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢠⡿⢿⣀⣀⠐⠂⠀⠀⠀⠄⡀⠀⠰⠀⠀⠘⣿⣿⣷⣶⠄⠈⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠠⠀⠀⠀⠐⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠀⡀⠀⡟⡏⠀⠀⠛⠀⣀⠀⠈⠀⠀⠃⠀⡀⠀⠀⠀⠐⠉⠀⠀⠀⢰⡾⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠩⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⡆⡄⠈⠁⠈⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⢀⠀⡆⠀⠀⠀⠐⠀⠀⠀⢈⠁⠁⠀⠀⢠⠙⠁⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⣲⣿⡛⠈⣽⣿⣧⡐⠻⡏⣝⢛⣝⣿⣇⢻⣶⡤⠄⣴⣆⣶⠠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣴⠾⣿⣿⠿⠋⠀⢈⣹⠍⣰⣿⠟⠋⠀⠈⠂⠉⠋⠫⣠⢋⠇⢉⠖⠈⠿⣿⣷⡀⠾⣏⠙⢶⣄
⠀⠀⠀⠀⠀⠀⢸⣿⠃⢈⣩⠁⠀⠀⢐⠾⢣⣾⡟⠛⠂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠉⠻⣿⣎⠛⣦⡀⠙⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢿⡅⣾⡏⢀⡴⣽⠋⣰⣿⣿⠉⠀⠀⠀⠀⠄⣦⠀⠀⠀⣀⠀⢀⠄⡀⠀⠀⠀⠠⠍⢿⣦⠘⢣⡀⠈⠫⡉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⡇⣾⠀⠁⣴⣿⡟⠁⠀⠀⠀⠉⠀⣶⠻⡇⡀⠀⡿⠇⠈⣜⠣⠂⠀⠀⠀⠀⠘⣿⣧⠈⠁⠀⡐⣅⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⢣⣶⣿⣿⠟⠀⠀⠀⠀⠹⢶⡴⠾⠴⣿⣧⠀⡤⡄⣼⣷⠶⠻⡧⠤⠀⠀⠀⠸⣿⣷⡀⠀⣉⣿⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣟⡁⣀⠀⠀⠰⠶⠶⠦⠀⠀⠘⣿⣿⡷⣿⣿⠁⠀⢰⠟⠂⠀⠀⠀⢰⣿⣿⠿⣦⡸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣉⣀⡤⢀⣠⣴⢼⣆⢀⣼⡏⠛⢷⠿⣿⣤⡴⡿⡶⣤⠀⢡⡀⢰⣿⠋⠀⠘⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⡿⠟⠛⢻⠉⠋⣙⣛⢋⣹⠋⠀⠀⠋⠀⠉⢿⣴⣙⣿⠌⠱⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠘⢷⣀⠀⠁⡼⠋⠀⠀⠀⠀⠀⠀⠈⠻⡈⣁⣠⠞⠀⠘⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣿⣯⠀⠀⣴⠈⠉⢁⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠉⠃⣤⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡦⢙⣦⡈⢿⡅⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⢀⣿⢀⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠯⣤⣬⣷⡜⢷⡰⠟⠀⠀⠀⠠⣾⡟⠉⠙⣷⡄⠀⠀⢻⣿⢃⣼⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁⠀⢨⣿⣶⣾⣶⠂⠀⡀⠀⠀⢻⣻⣿⣿⡛⠁⠀⠀⢰⣽⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠷⠀⠀⠉⠙⣩⣷⣿⣦⡀⠀⠀⢀⣤⢙⠻⣏⡅⠀⠀⠀⣾⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⠀⠀⠀⠀⢈⡹⢿⣿⣿⣿⣶⣤⣀⡀⠁⠀⢀⣠⣤⣴⡿⡿⠙⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠂⠀⠀⠀⠀⠀⠀⠭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠠⠔⠀⠀⠀⠀⠀⠈⠁⣼⠃⣯⡈⢏⡿⡿⠿⠿⠁⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡟⣩⣶⣦⣀⠀⠀⠀⠀⠈⠉⠀⠈⢇⠈⠠⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢟⡾⠷⣿⡶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠖⠛⠿⢿⣿⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⠀⠉⠉⢴⣷⣿⣿⣦⣀⡀⠀⠀⠀⠀⣠⡾⢻⣿⡿⠧⢀⠀⣠⣙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢀⡀⠐⠀⠏⠕⠙⠃⠙⢷⣀⢠⡆⣼⡿⠀⠀⠀⠈⠀⠈⠁⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡦⠀⠀⢀⡠⠀⡀⠀⠀⠀⢿⣾⣿⡟⠀⢀⠄⢰⣠⠂⠀⢀⠈⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠈⠁⠀⠀⠁⠀⠀⠈⢻⣿⡇⠀⠈⠀⠀⢸⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣠⣾⠀⠀⢠⡞⠀⠀⡇⠘⠛⠦⣦⣇⡀⠀⣤⠀⡟⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠾⠷⠶⠾⠿⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⣠⣶
⠀⠀⠀⠀⠀⠠⢣⠛⠛⡀⠄⠀⠀⢀⣀⣀⣤⣄⣀⣀⠀⠀⢠⣿⣿⣿⣿⡀⠀⠀⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢼⡿⢆⠺⡶⠀⠀⠈⠉⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⠟⠿⠀⢀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠛⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⢸⠟⠉⠛⠿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠉⠻⠭⣀⠁⠁⠀⢀⡀⠀⠀⠀⠀⠄⠁⠀⠀⠀⢀⠐⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⣆⠀⠀⠈⠈⠙⠋
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⢀⠀⠀⠀⣴⣅⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠈⠛⠰⠿⢷⣦⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⣀⢰⠦⡀⠀⠀⠀⠀⠀⠀⣼⣬⡖⣠⠈⠆⠀⠀⣽⣿⣿⡿⣾⠟⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠛
⠀⠀⣶⣾⣿⣷⣶⣶⣦⣤⣤⣴⣌⣣⣾⣿⣻⠀⠀⠀⠀⠀⣠⣿⣿⣦⡀⠰⠆⠀⠀⠙⣿⡿⠃⠀⠦⡀⠀⠐⠀⠈⠀⡀⡀⠄
⠀⠀⣿⠻⢿⣿⣿⣿⣿⣿⣿⠟⠏⠇⠀⣠⣉⡀⠦⠀⢠⠊⠙⠉⠛⠛⠉⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠁
⠀⠀⣠⡀⠐⠯⠽⠿⣿⣿⣧⡐⠀⡀⠐⠁⠀⠁⠀⠀⡆⠳⣤⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠒
⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠁⠑⠊⠉⠀⠀⠀⠀⠀⠀⠑⣄⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉
⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣶⣶⣶⣦⠀⠀⠀⠰⠤⠂
⠀⠀⢿⣷⡀⠀⣼⡿⣷⠀⣠⣴⣶⣿⣿⣆⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡀⠀⣿⣿⣿⣿⣿⣿⣦⢤⣴⣶⣶⣦⣤⣀⣀
⠀⠀⠈⣿⣿⣿⣿⣧⣿⢿⣿⣿⣿⣿⣿⣿⠀⠀⣾⣯⣼⢯⣔⣆⣀⣀⣀⣀⣸⣿⣫⠟⠁⠀⣿⣝⡻⣿⣼⣿⡿⢸⣿⣿⣿⣿⣶⡼⣟⠛⠛⠛⠲⠦
⠀⠀⠸⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⢿⣿⠏⠀⢸⣟⣅⢀⣙⣿⣿⣿⡿⣿⢝⡻⡟⠉⠀⠀⠀⠈⠉⠻⡏⣿⣿⣷⠘⢻⡿⣿⣿⡿⣿⣿⡀
⠀⠀⠐⣿⣿⣿⡿⣿⣿⣿⣿⡿⢃⣼⠃⠀⠀⠀⠀⠉⠻⠛⠛⠿⢯⡅⠈⢂⠀⠀⢸⠃⠀⠀⠀⠀⠀⣿⡟⠛⠻⣶⡎⠀⠘⢿⣧⠙⢿⣧⡀
⠀⠀⠀⠀⠉⠉⠴⠟⠈⠙⣿⡇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⠂⠀⠀⠈⠢⠀⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⡇⠀⢀⣴⠏⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠙⠐⠁⠀⠀⠀⠀⠀⠾⠏⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠿⠆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣒⣯⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣦⣀⣀⡀⠀⠀⣀⣤⣤⣤⣤⣴⠶⠶⠶⠶⠶⢶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠶⢦⣤⣤⣾⠟⠁⠘⠋⠹⠿⢿⣿⣿⡾⠋⢉⣤⠴⠒⠒⠒⠛⠋⠉⠉⠉⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⠟⠃⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠶⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⢻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣤⡀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣋⡵⠒⠋⢹⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⢽⣶⡄⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⣴⢿⡿⠋⠁⠀⢀⣴⣯⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⣙⣦⣀⣴⠿⠃
⠀⠀⠀⠀⠀⣸⡏⣼⣀⣀⠀⠀⣨⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠻⣿⠋⠁
⠀⠀⠀⠀⠀⣿⣵⠟⠉⠙⠷⣦⣹⠟⠀⠀⢀⡴⣫⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⢨⡿⠃⠀⠀⢸⢸⣿⣿⢛⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠛⣿⣿⡏⣧⠀⠀⠀⠀⠀⠀⠘⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠈⠿⣿⣿⣿⠟⢁⣄⡀⠀⠀⠀⠀⠸⣿⣿⣿⡿⣣⠇⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃
⠀⠀⠀⠀⣠⣤⣴⣤⣄⣀⣸⣿⠀⠀⠀⠀⠀⢠⣤⠴⠿⠉⠳⠿⠱⡄⢀⡴⣾⣦⠬⠭⢍⡉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡸⢿⣦⡀
⠀⠀⣠⡾⠛⢛⠁⣬⣿⡿⠟⠻⣄⠀⠀⠀⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠰⢿⣇⠀⠀⠀⠀⠀⢀⡴⠋⢠⣤⣌⠻⣶⣄
⠀⣸⡟⠀⢰⣿⣷⡸⣿⢇⣶⣆⠘⡆⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠛⠛⢶⣦⠰⠆⠀⠀⠈⠉⠛⠢⣄⠀⠀⠀⡏⣰⣦⡸⣿⡏⣠⣾⣿⣆
⠀⣿⣧⡀⣶⣮⡍⣴⣶⣮⣛⠋⠀⣿⠀⠀⠀⠀⠀⠈⠅⠸⣿⣶⣶⣶⣿⠟⠀⠒⠀⠀⠀⠀⠀⠀⠈⠁⠀⣸⡇⢙⣫⣥⣶⣜⠿⡿⢸⣿⡀
⠀⠹⣿⣅⠈⠋⣴⣿⣿⣿⠿⠃⠀⠿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡟⠃⠛⠿⢿⣿⣿⣴⣶⡦⣿⡇
⠀⠀⠈⠻⣿⣀⠉⠉⠀⠀⠀⠀⠀⠀⠋⣧⠀⠀⠀⠀⠀⠀⠠⠚⠉⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠿⠏⢋⣼⡟
⠀⠀⠀⠀⠈⠙⢿⡶⢤⡄⠀⠀⠀⠀⠀⣿⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⣠⡤⢖⡿⣹⣆⠀⠀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠈⣿⡄⣤⣀⠀⠀⠀⠀⢻⡈⠉⠉⠛⠻⠶⠦⠖⠳⣤⠦⠤⢀⣈⣻⠶⠛⠃⠉⠉⠀⡼⠁⢻⠀⠀⠀⠀⠀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣸⡷⠀⠉⠉⠉⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠘⠂⠀⠀⠀⣸⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠸⠟



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣼⣿⣷⠄⠀⠀⠀⠀⢠⣾⣿⣦⣷
⠀⠀⠀⢀⣴⣿⡇⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⠇
⠀⠀⣠⣿⣿⣿⠃⣠⣤⡀⠀⢻⣷⣄
⠀⠀⢿⣿⣿⡟⢰⣿⣿⡇⠘⣸⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡄⠀⠀⠀⠀⡤⠀⠀⠄
⠀⠀⠈⠙⠛⠀⠈⠙⠋⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠈
⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣆⠺⠃⠀⠀⡏⠳⣄⡎⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠻⠿⠿⠃⠀⠀⠀⠀⣷⣿⣿⣿⣶⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⣹⡉⠉⣿⡟⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠠⠀⠀⠈⠙⠛⠛⠛⢺⡿⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣡⣤⣤⣀⣄⣤
⠀⠀⠀⠀⠀⣤⡶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣿⣦⠀⠀⢠⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠏⠁⠙⠻⣿⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠘⠋⠉⠀⣴⣶⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠄⢀⢀⠀⠀⠐⣿⣄
⠀⠀⠀⠀⠀⠀⠘⠿⠿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡀⠀⠴⣦⡄⠀⠻⠟⠀⠀⠀⠈⠋⠀⠀⠀⠘⠿⠃
⠀⠀⠀⠀⢀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣄⣠⠤
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠙⠒⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⣋⡉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⢉⡴⠛⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡋⣉⣉⡛⠻⠷⣶⣤⣄⣀⠀⠀⠀⣠⣴⡶⠿⠿⠿⢶⣦⣴⡾⠟⠉⠀⢠⡟⠁⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠉⠉⠳⢦⣀⠈⠉⠛⣻⣿⠿⠻⣅⠀⠀⠀⠀⣸⠁⠉⠛⠂⠀⢠⣟⣤⣄⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⣙⣷⣄⠀⠉⠀⢀⡄⠘⣆⠀⠀⠀⡟⠀⢶⣄⠀⠀⠛⣽⠋⠁⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠘⠛⢿⣍⠀⠀⣠⠟⡔⢲⣸⡄⠀⠀⣧⣆⣹⡟⠷⠀⠀⠡⣄⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⠀⢀⡀⠀⠀⠉⠀⠛⠛⣹⠃⠀⠀⠘⢯⡉⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡿⠁⠀⠀⠀⠀⢀⡴⠃⣴⣒⣲⣦⠀⠙⢦⡀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⣠⠏⠀⠀⠘⠻⡿⠋⠀⠀⠀⠙⢦⠀⠀⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⣀⠀⠀⠀⣇⠀⠐⢦⣀⡴⠛⠦⡤⠏⠀⠀⣼⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣶⣬⡳⠄⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⠳⣦⠀⠀⠀⠀⠀⠀⠐⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢸⡇⠀⠀⢼⠃⠀⠀⠀⠀⠀⠀⣠⠀⠀⢻⡌⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢸⡅⠀⠀⢘⢧⡀⠀⠀⠀⠀⣰⣋⠀⠀⠈⡇⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠘⣇⠀⠀⠸⣆⠙⠀⠀⠀⠘⢡⡏⠀⠀⢰⡇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⡹⣦⠀⠀⠙⣦⡀⠀⠀⢀⡾⠁⠀⢀⡾⣀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⠀⠈⢷⡀⠀⠨⣝⢷⡞⣫⠀⠀⢠⡞⠁⠈⠉⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⠶⠶⠾⠿⣦⣤⣿⣾⢷⣿⣤⣴⡿⠷⠶⠶⠶⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉

⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⢰⣿⣿⣿⣿⣿⣿⡿⢋⣴⣿⡇⠀⠿⠛⠻⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣤⣄⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⣟⣀⡄⢀⠀⠀⢴⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠁⣠⢿⣿⡻⠿⠇⠘⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⡟⠳⠲⣻⠏⠀⡐⠿⢠⣾⡆⢿⣿⣯⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⢉⣀⡄⠀⡀⠺⣿⣷⣾⣿⣿⢿⣟⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠤⠀⣿⣿⣷⣀⣿⣿⣾⣿⡿⠁⠀⠙⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣠⠀⣤⣍⠿⡏⢿⣿⣿⣿⣿⣧⣄⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡉⣿⣸⣿⣽⣶⡟⣈⢿⣿⣟⣉⣹⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢹⣿⣿⡿⠂⢀⣤⣀⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠻⣿⠟⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣄⠀⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣯⣍⣀⣠⣤⣤⡄⣤⣤⣤⣀⣈⣭⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠏⠁⠀⠈⠙⠻⣿⠀⢼⡿⠛⠉⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠘⠃⠛⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣰⡏⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠈⣷
⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿
⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⠈⢿⣿⣿⣧⡀⠀⠀⠀⠀⣠⣿⣿⣿⠋⢀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠈⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⢿⠆⠉⠛⠿⣿⣦⠀⢠⣾⡿⠟⠋⠁⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣡⣤⣤⣀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣿⣿⣿⣿⣷⡀⠈⠧⠑⠒⠒⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠙⡇
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⡋⠘⣿⣿⣿⣿⣿⣿⡏⢀⠀⠠⣀⡠⠎
⠀⠀⠀⢧⡀⠀⠈⠉⠑⣲⣶⣶⣶⣶⣿⡿⣿⣿⣿⣿⣿⡿⠀⠈⠛⢻⠟⠛⠁⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠈⠑⠒⢤⠀⣿⡿⠟⠛⠿⢿⡿⠀⠉⠉⠋⠉⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⢠⠖⣤⢄⣀⣀⣀⣀⣀⡀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⢸⠀⠀⢠⠞⢡⠞⠁⠀⠀⠀⠀⠀⠀⡇⠀⢀⣷⠁
⠀⠀⠀⠀⠀⠘⡁⠀⢇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠸⣸⡀
⠀⠀⠀⠀⠀⠀⠑⠒⠒⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠐⠠⢤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⢤⡀
⠀⢀⣀⠀⠤⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠒⠀⠀⠖⠒⢄⠙⠐
⡜⠁⠀⠀⠀⠀⠀⠀⠀⠿⠂⠀⢤⢄⡀⠀⢠⡄⠈⠄⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠢⡁⠂⠐⢀⠀⠉⠀⠀⠀⢀⠀⢣
⠇⠀⠀⠀⠀⡴⠂⠀⠀⠀⣠⢤⣤⣀⡈⠂⠈⠃⠀⠈⡆⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢴⠀⠀⠀⠐⠓⠀⢱⠀⠀⠀⠢
⠘⣄⠀⠀⢸⠀⠀⠀⠀⠘⠀⣰⣿⣿⣿⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠒⣄⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⣠⣤⡀⠀⡼⠀⠁
⠀⠘⣆⠀⠸⠀⠀⠀⠀⠈⠢⣉⣻⡿⠿⠃⠀⠀⠠⠸⡄⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⠲⠀⠈⠁⠀⠀⠃⠠⡀⠠⣼⡿⠁
⠀⠀⡙⠀⠀⠀⠀⠠⡀⠀⣠⠤⢄⡀⠀⠀⢀⠀⠀⢰⡇⠀⠀⠀⣀⣠⡴⠞⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠁⠚
⠀⠀⡇⢠⠀⠀⠀⠀⠈⠍⠀⠆⠀⠙⠒⠒⠁⠀⠰⢣⠇⠀⢠⠞⠋⠁⠀⠀⠀⠀⠠⢸⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠠⠀⠀⠀⢀
⠀⠀⠑⠞⠀⣰⠀⠀⢼⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡼⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⢀⡏⠀⠀⠸⡀⠀⠀⢀⡼⠁⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠙⠒⠖⠋⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⢀⠀⠀⢀⡇⠀⠀⡠⠃⠀⠀⠀⡄
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀⠟⠉⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠦⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠄⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠈⠉⠉⠒⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠈⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⠀⠉⠒⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⡀⠀⠀⠀⠀⠈⠢⡙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡀⠀⠀⠾⡇⠀⠀⠀⠀⠀⢿⣇⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⣀⡀⠐⠒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠉⠉⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⢧⠀⠀⠈⠢⡀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡠⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣠⣤⣀⠀⠀⡏⢀⠀⠀⠳⣄⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⣠⣤⣼⣿⣷⠀⠣⣸⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀
⠀⠀⠀⠀⠀⢮⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠇⡀⠀⠀⠀⠀⠻⣿⣿⣾⠏⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠹⣄⠀⠀⠀⠀⠈⣏⠀⢀⣀⡤⠊⠀⠁⠉⠉⠉⠉⠉⠉⠉⠉
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣾⣿⣶⣶⣶⣷⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠄⠤⠂⠀⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠐⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣾⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠈⡆⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢧⣿⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣳⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣤⡀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣴⣾⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⢿⣿⡆⠀⠀⠀⠀⠀⢀⡞
⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡞⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣟⠔⠀⢿⡀⠀⠀⠀⠀⡞
⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⡸⠀⠐⠉⠿⠿⠟⠀⠀⢀⡤⠔⠒⠲⣤⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢸
⠀⠀⠀⠀⠈⠲⡄⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⣄⠀⠘
⠀⠀⠀⠀⠀⠀⠘⡄⠀⢀⡞⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠉⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠣⡠⠊⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡠⢴⠉⠀⠀⠀⠀⠀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠈⠑⠒⠒⠒⢄⡀⠀⠀⠀⡀⠄⠊⠉⠁⠀⠀⠱⡀⠀⠀⠀⠀⠀⡄⠙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⢀⠎⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢅⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠘⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠈⠇⠀⠀⠀⠀⠀⢀⠁⠈⡇⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠁⠀⠀⠀⠀⠀⠀⣘⣀⡔⠂⠀⠀⠀⠀⠀⠀⡸⠀⠔⠁⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⣀⠀⢰⠀⠀⠀⣀⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⡗⠚⡀⠀⠀⠀⠀⡀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠈⠁⠀⠈⠢⠀⢄⣀⣀⠤⠠⠞⠀⠀⠀⠁⠈⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠶⣿⢾⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⡩⠀⡠⠀⠀⠀⠀⠀⠀⠐⠒⠄⢱⡀⠈⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⡄⡇⠈⣠⣤⣄⠀⠀⠀⠀⣾⣷⣦⣸⢁⡀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣧⠹⡞⢻⣿⣿⠀⠀⠀⠐⠿⣿⠃⡎⠘⡇⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠃⠀⡇⠀⡼⠁⠀⢾⣿⡷⠀⠈⠳⣷⡀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠆⡸⠳⢄⠀⠀⠀⠀⠈⠀⠀⠀⠰⠃⠓⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡏⠰⠒⠂⢈⡁⠤⠈⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⢈⡅⠀⠀⠁⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⡀⠀⠳⠞⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠀⠀⠀⣆⡀⠀⢀⡄⠀⠀⢸⡧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢸⠀⠀⠀⢩⠙⠶⠋⡝⠀⠀⢸⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠘⡇⠀⠀⠸⡆⠀⢀⡇⠀⠀⢸⢁⣾⠛⠉⣉⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠾⠷⣄⣧⠀⠀⠀⢷⠀⢸⠁⠀⠀⣾⠞⠓⠲⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢰⠀⠀⢀⠟⠀⠀⠀⠘⣧⡏⠀⠀⠀⡈⢢⣰⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⢦⣼⣄⣠⠤⠞⠳⠤⢧⣠⠧⠼⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠉⠁⠀⠀⠀⠀⠀⢤⣶⣤⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢝⣆⠀⠀⠔⢢⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⠔⢚⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣰⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⡛⠙⠛⢿⣿⣿⣿⣿⣿⣿⣏⣿⣿⡟⢲⣻⢿⣿⣿⣿⡽⠛
⠀⠀⠀⠀⠀⢀⣴⣦⣤⣴⡟⠋⠉⠃⠘⢷⣤⡄⠾⢿⣿⣿⣿⣿⣿⣿⢿⣿⣇⠀⢓⣼⣿⣿⣿⣦⠠⠗
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢻⡇⠀⣨⡝⣿⣿⣿⢿⠈⠀⢦⠉⠀⣾⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣤⣬⣉⠀⡘⠦⠀⠀⢀⣧⠀⠀⢸⣿⣦⣹⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢀⣿⣿⣾⣿⣷⣦⣀⣘⣐⣿⡏⠀⢀⣔⣳⣼⣿⣿⣿⣿⣿⣿⣿⣟⠛
⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⠿⣼⣿⣿⣿⡀⣻⣿⣿⣿⣿⣿⣿⡇⢠⣿⣿⡿⠹⢹⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣧⣸⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⢃⣸⣿⣿⡇⠀⣼⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠿⢿⣿⣘⡿⣇⠀⠉⠙⣿⢿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⡇⠀⠀⠀⠀⠉⠛⠈⠲⠀⠀⠘⢶⣬⡹⣿⠉⠛⢿⣎⣠⡄⠙⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⠬⠃


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠓⣦⠚⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠾⢛⣉⠹⠟⠋⠓⠒⠶⣤⣀
⠀⠀⠀⠀⠀⠀⢀⡴⠟⡩⢤⡀⠀⠈⡉⠓⢦⣄⠀⠀⠀⠀⠉⠛⣦⡀⢀⡴⠉⠙⢦⣄
⠀⠀⠀⠀⡤⢤⡏⠀⠌⠀⢨⠁⣀⣠⣅⠀⠀⠈⠙⠒⠦⣄⣀⠀⠀⠑⣾⠁⣠⠴⡇⠈⠑⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡴⢦⡄
⠀⠀⠀⢠⡆⠀⠹⣞⠀⣠⠟⠋⠁⢀⠞⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠂⠘⢿⣡⡶⠗⠀⠀⠀⠀⠀⠀⠉⠉⠐⠂⠤⠀⡴⠋⣄⠀⠙⣆
⠀⠀⡔⠊⠙⢦⠀⠋⠀⢤⣤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣨⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡽⣦⡀⠸⡀
⠀⠈⠳⢶⡶⠺⣥⠔⢧⡀⠈⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠈⣽⠋⠴⠶⢦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡟⠒⠋
⠀⠀⠀⠘⢁⠞⠁⠀⢸⢩⠗⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢴⠏⠀⡤⠀⣾⣯⣭⣍⣛⣷⠶⣤⡀⢀⣀⣀⠀⠀⠀⢠⢿⡄
⠀⠀⠀⠹⣿⠀⠑⠴⢫⡟⢀⠀⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⣠⡷⠛⡟⢀⣾⠁⠀⣿⣿⣿⣿⣿⣿⣶⣾⠃⠈⠉⠙⠓⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠻⣇⠀⢠⡏⠀⡇⠀⠀⠀⣀⣀⡀⠀⠐⠂⠀⠱⠉⠀⠠⡇⢸⡇⠀⠀⠹⣿⣿⣿⣿⣿⣿⠟⠀⠀⠄⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⣠⠚⢁⢇⣀⡀
⠀⠀⠀⠰⠊⠙⢷⣽⣀⣠⡖⢲⣴⠞⠛⠉⠛⠁⠀⢰⣄⠀⠁⠀⡴⡇⢸⣇⠀⠀⠀⠈⣉⠛⠛⠉⠀⠀⠀⠀⡼⠀⠀⠀⠀⢀⡟⠀⠀⢠⠎⠁⠀⠈⣩⠜⠁
⠀⠀⠀⢡⠀⠤⢠⡀⠍⢷⡈⣙⣥⠤⠒⠈⠀⠀⠀⠀⣠⣴⠖⡊⠀⢸⡄⠛⠛⠶⠶⠖⠻⣄⠀⠀⠀⠀⣀⡴⢁⡤⣠⣌⣦⣾⠁⠀⢰⠃⠀⠀⣸⠙⢢⡀
⠀⠀⠀⠀⠳⣄⠈⠳⠀⠀⠙⠟⠀⠀⠀⠀⠐⠛⠀⠺⢯⡥⠞⢃⣠⡾⠿⣦⡀⠀⢀⡀⠀⠈⠙⠒⠛⠋⠁⠀⣸⢦⣌⣈⠙⢳⡀⠀⡾⠀⠀⣐⣁⡤⠚
⠀⠀⠀⠀⠀⠀⠙⡶⢶⠺⣷⣦⣄⣀⣀⣠⡤⢤⢤⣤⣤⣶⣶⣋⣙⣻⣦⡼⣿⣦⣀⣿⣦⡤⠤⠵⣦⣤⣀⣼⣿⣉⡠⠄⠄⢸⣇⡼⠁⡤⠔⠁⢢
⠀⠀⠀⠀⠀⠀⠰⣧⡈⠳⡍⢀⠝⣻⡟⠁⢠⢠⢋⡞⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢈⣷⡦⠀⠀⠀⢠⢾⡇⢠⡤⣼⠠⠴⠊
⠀⠀⠀⠀⠀⠀⢠⠋⠉⢅⡼⠋⠆⢿⠃⠀⢻⣇⣼⣧⣤⣤⣤⣤⣤⣤⣤⢶⢶⡶⡶⡶⠶⠶⠿⠛⠛⠛⠛⠉⠻⠿⠗⢂⣿⣿⡧⢄⣀⡼⠂
⠀⠀⠀⠀⠀⠀⢈⡷⠀⠈⣵⠶⡄⢸⡆⠀⠀⠚⠋⠁⠭⠉⠟⠙⠻⠿⠟⠂⣸⡇⠁⢻⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠘⡷⣄⠈⠀⣰⠃⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣚⣿⣷⠀⠀⢦⠀⢤⡄⠀⠀⠀⠀⠀⠀⣶⠀⠹⣿⣿⠶⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠸⣄⡈⠉⠘⡏⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⣰⠟⠁⠀⡴⠋⠛⠿⠷⠄⠀⠁⠾⠿⠦⠄⠀⠀⠀⢐⡏⠀⠀⠈⠉⠀⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠙⠛⠁⢹⠀⠀⢀⡼⠃⠀⠀⠀⠀⢸⠃⠀⣰⠋⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠴⡆⠀⠀⠼⠀⡰⠋⠀⠀⠀⠀⠀⠀⡇⠀⣼⣡⢀⠄⣠⠖⠻⢷⣶⣶⣤⣤⣤⣀⣠⣤⡶⠛⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⠛⠀⡼⠁⠀⣀⠠⠀⠀⠀⠈⠿⠋⠹⣧⠾⠞⠁⠢⢄⡀⠀⠠⠑⢮⡻⡍⠉⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣿⣄⣀⡷⠚⠉⠉⠒⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⢳⢹⡄⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⢠⡏⡜⠁⣘⠁⢹⡀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠈⢀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⢿⡷⠆⠤⠤⠤⠭⠴⠶⠶⠶⠦⢴⣶⣤⣤⡄⠀⠀⠤⠀⠀⠀⠀⡀⠀⡗⠀⠀⣼⡇⠀⢀⡼⢧⣤⣀⡤⠄⠀⡠⠞
⠀⠀⠀⠀⠀⠀⣀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⣀⠀⠘⡄⠞⠁⣠⣼⣏⣀⣠⡏⣠⠚⡉⠈⠁⠠⠆
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠀⠀⠐⠒⠶⠶⠲⠖⠒⠚⠛⠛⠺⠛⠓⠒⠋⠉⠉⠉⠙⠛⠉⠛⠛⠋⠁⠀⢠⣾⣾⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡁⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡓⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⠎⠓⠀⠙⢿⠻⣿⡟⠈⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠉⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡆⠀⠀⠀⠀⢀⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⠀⠀⠀⠀⠘⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⣿⣿⠝⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠇⣸⣿⣿⢸⡿⠿⣿⣿⡟⠁⠀⠀⠀⠀⠀⢀⣀⡀⣀⡀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣣⣿⣿⠗⠋⠀⠀⠘⠋⠀⠀⠀⠀⠀⢀⠎⣩⣿⣿⣿⣿⣷⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣎⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠈⠈⠛⠋⣡⣩⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⡿⠿⠃⠀⠀⠀⠀⠀⢈⣽⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠐⠢⠄⢶⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣥⡀⠀⠀⠀⠀⠈⠂⠀⠀⠴⠿⠻⠁⠈⠛⣲⡀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣅⣀⡀⠀⠀⠀⠀⠀⠀⠦⣤⣤⣤⣴⡿⠟⠀⠀⠀⠀⠀⠀⣰⣼⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣄⠀⠠⢠⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⡃⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣾⢿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⣹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⢹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠛⢿⢿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⢻⡏⠙⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠸⠟⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠐⢄⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣁⣤⣤⣄⡀⠀⠑⡀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣾⣿⣿⣿⣿⣿⣶⡄⠈⡄⢳⡌⠦⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⡈⠁⠀⠀⠀⠀⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠈⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⡴⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⢀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⡇⠀⠀⠙⠲⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⣿⣷⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣁⣈⣀⠀⣈⣆⣠⠔⠋
⠀⠀⠀⠀⢳⡀⠀⢀⡀⠄⠈⠉⠛⣲⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠛⠿⣿⠿⠛⠋⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠉⠻⠤⣄⣀⡀⠀⢠⣿⣿⣿⡿⣿⣿⣿⣿⣿⣄⠉⠛⠻⠿⠿⠿⠋⠁⠀⠀⡠⠐⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠆⠸⡿⠟⠋⠀⠀⠈⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠐⠉⠁⠂⠀⠀⠀⠀⠀⢀⠠⠀⠠⢀⠀⠀⠀⠌⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⡀⠀⣠⢤⣀⡀⠀⠉⠉⠀⠀⠀⠀⠀⠑⠂⠊⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⡀⠔⣠⠔⠁⣠⠞⠉⠙⠓⠒⠒⠒⠒⠒⠛⡇⠀⠀⠀⡞⡟
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠊⠀⢀⡞⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⢣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⢸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⣈⣑⡦⠽⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣌⣽⣾⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠛⠓⢲⡤⠐⠒⠋⠉⠉⠀⠀⠈⢳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠁⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢧⣸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣠⠶⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠆⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⢀⣴⠋⠀⢠⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡠⠀⣠⡴⢋⡼⠀⠀⠀⠻⠿⠟⠁⠀⠀⠀⠐⠂⠀⠀⠀⣼⣋⣿⣿⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠾⠶⠛⠁⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣦⣄⠀⠀⢿⣿⣿⡟⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠰⡆⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡧⠀⠈⠉⠉⠀⠀⢠⣿⠀⠀⠀⠀⠀⢀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⡀⠀⠀⠱⣄⠀⠀⠀⠀⠈⣻⠛⠛⠉⠀⠀⠀⠀⠀⠀⠰⡏⠘⣇⠀⠀⠀⣠⣎⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⡄⠀⠀⠈⠙⠛⣶⠖⠻⡿⢦⣀⠀⠀⠀⣀⡶⠂⠀⠀⡇⠀⠹⣆⡴⠊⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢦⣸⣄⠀⠀⢿⣆⠀⢀⣰⠏⠙⠛⠛⠁⠀⢀⠀⢠⡏⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠙⠙⢦⡀⠀⠙⠛⠋⠁⠀⢀⣴⠀⢀⣠⣞⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠃⠀⠀⠀⠎⠻⢦⣀⣀⣀⣤⠖⠻⠛⠋⠻⣍⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢒⣒⠫⠝⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠈⠢⢄⣈⡉⠄⠁⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠈⢀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡇⠙⢦⡀⠀⠀⠀⡸⠀⠀⢠⠘⠉⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠘⠁⠀⠀⠹⣆⠀⠀⢷⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢧⡄⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠘⢧⡀⠈⢦⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢳⡄⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⠟⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠷⣤⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⡏⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠻⣤⣀⠀⢀⣖⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠄⠀⠀⠀⠀⢻⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠰⡄⠈⠁⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⠁⠀⠀⠀⠀⠀⠸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢹⣦⡀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⡾⡇⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⣾⣨⠛⠶⣄⣀⣹⠀⠀⠀⠀⠀⠀⠀⢀⡇⢡⣀⣀⣀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠾⠇⠀⠀⠀⠀⠀⠀⢰⠇⠈⠁⠀⠀⠈⡿⠀⠀⠀⠀⠀⠀⠀⢸⡗⠁⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣆⣀⣀⣀⣠⡞⡡⠖⠛⠛⠓⠂⠀⠀⠰⣧⣀⣇⣀⣀⣀⣀⣤⠞⠒⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠸⣿⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠉⠉⠉⣿⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⠶⠶⠤⠶⠶⠶⠚⠁⠀⠀⠀⠀⠀⠈⢷⣤⣄⣀⣀⣀⣨⣄⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠒⠒⠒⠒⠒⠢⠤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⢉⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣍⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡾⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⡇⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⡇⠀⠀⠀⢸⣧⣼⠀⣠⠔⠲⢤⡀⢧⣼⡇⠀⠀⠀⢻⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⢸⠖⠒⠲⠤⠤⠭⠔⢋⣤⡄⢠⣤⡙⠲⠥⠤⠖⠒⠒⠾⡆⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⡟⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⢹⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠐⠦⡴⠞⠙⢶⠶⠊⠀⠀⠀⠀⠀⠀⣸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠳⠤⠴⠋⠀⠀⠀⠀⠀⠀⢀⡴⣿⠁⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠑⠲⠤⢤⣀⣀⣀⣀⣀⣀⣠⠤⠤⠒⢺⡉⠀⡏⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠞⠉⠒⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⡄⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⢀⡏⢦⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢧⠀⠀⠈⠑⠦⣄⣀⣀⣠⠴⠊⠀⠀⠀⡸⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠘⣆⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢰⠃⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠙⢃⠞⠢⢤⡤⠄⠀⢸⠀⠀⢠⣄⣀⡴⠻⡀⠋⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣶⢀⡆⢸⣰⠀⢀⡄⠀⠀⢸⣄⠀⠀⠰⡄⠀⣆⡇⢠⡀⢠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠚⠙⠒⠶⠓⠒⠋⠁⠈⠙⠒⠚⠓⠒⠋⠙⠚⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⠶⠞⠛⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⣠⠴⠒⠒⣀⣿⠶⠶⠶⠶⠶⠶⠶⢦⣤⣄⡀⠀⠀⠀⠀⠀⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣰⠃⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⡆⠀⠀⠌⠉⠙⠻⢦⣴⠾⠋⠉⠙⠉⠉⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⣠⡿⣷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠙⢢⡀⠀⣴⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⣼⠋⠀⠀⠀⣠⣤⣤⠀⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠹⡄⢸⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⣀⠤⠤⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠰⠶⢦⠀⠀⣿⣿⡄⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣸⣡⣶⠂⠈⢧⠀⠀⡘⠀⠀⠇⠀⢀⡤⠒⠢⡄⠀⠀⢸⡏⢿⣄⣠⡾⠋
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠹⣿⣿⡿⢃⠎⠀⡰⠀⠀⠀⠀⠀⢾⣾⣟⡀⢸⠀⠀⠘⣧⠀⠉⠁
⠻⢿⡟⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠉⠉⢁⠠⠊⣴⣶⡒⠒⣦⡢⡘⠿⣛⡡⠎⠀⠀⠀⣏
⠀⠀⣿⠀⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⢀⠂⠁⠀⠀⢠⠀⠀⠙⠿⡿⠿⠛⠀⠈⠐⠠⢀⠀⠀⠀⢀⡿
⠀⠀⣿⠉⠁⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠇⠀⠀⠀⠀⠀⠑⠦⠤⠖⠣⢄⣀⠔⠀⠀⠀⠀⠁⠀⠀⣸⡇
⠀⠀⢹⡆⠀⠀⢹⣆⠀⢀⣀⣤⣴⠶⠶⠛⠛⠛⠍⡳⢤⣀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏
⠀⠀⠀⠻⣆⠀⢀⡽⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠨⢙⠒⠶⠤⠤⢤⣤⣤⣠⣤⣤⣤⣤⡴⠾⠟⠋
⠀⠀⠀⠀⠉⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠲⢄⠠⠔⠈⡤⢠⡶⠚⠁
⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡀⠠⠀⠄⠔⢸⠇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠀⠀⠀⢠⢆⣿
⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⡞⣼⠇
⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠎⢠⡟
⠀⢀⣴⠟⠀⠀⠀⠀⣀⣤⣾⠉⠙⠲⠦⣄⣀⡘⣆⠀⠀⠀⠀⠀⣧⣠⠞⠁⠀⢸⡇
⠀⣾⠃⠀⠀⠀⣰⠟⠉⠀⢹⡆⠀⠀⢀⡿⠉⠙⠻⣆⠀⠀⠀⠀⢻⢿⠀⠀⠀⢸⡇
⠀⣿⠀⠤⠤⢴⣯⠀⠀⠀⠈⣷⠀⠀⠸⢷⣤⡄⠀⠹⣆⠀⠀⠀⢘⣾⣆⠀⠀⠼⠷⠶⣤⡀
⠀⣿⡀⠀⠀⠀⢸⡇⠀⠀⠀⢹⣇⠀⠀⠠⠠⣿⠀⠀⠹⣦⠀⠀⠉⠉⠉⢳⡄⠀⠀⠄⢸⡇
⠀⠈⠙⠛⠳⠷⠟⠃⠀⠀⠀⠀⠙⠛⠲⠾⠟⠋⠀⠀⠀⠘⢷⣤⣤⣤⣤⣼⠟⠛⠛⠛⠛⠁
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⣶⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⢀⡀⠿⠇⣀⡀⠛⠃⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣷
⠀⠹⡿⠻⣿⣿⣿⣿⣷⣶⣿⣿⣷⣿⡁⠀⠀⠀⠀⠀⠀⠸⠇⠀⠈⢿⠟⠀⠀⠀⠀⠀⠀⠙⠻⠷⢦⣄⣀⣀⣀⣀⣠⣤⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠼⠆⢀⡔⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠻⣿⡟⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢻⡿⢹⣿⠀⠀⠙⣿⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠡⣾⠟⠁⠀⠀⠈⣿⠻⣿
⠀⠀⠀⠀⠀⢸⡇⢸⣿⠀⠀⠼⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠿⠄⠀⠀⠀⠀⣿⡀⣿⡀
⠀⠀⠀⠀⠀⠛⠃⠸⠿⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡄⢠⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠹⠸⡈⠠⠹⢸⢘⠸⠰⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠒⣀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠻⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣠⣼⠇⠀⠀⢠⣾⡿⠿⢿⠿⠿⠿⠶⠤⠄