Deadlift Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠛⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⢠⣶⣿⣿⣿⣶⡀⠀⠀⠀⠀⢸⡿⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⡀
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⣼⣧⣾⣿⣿⣿⣿⠟⠹⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣇
⠀⠐⠶⢾⣿⣿⣿⣿⣿⣿⣿⡗⠒⠒⠒⢿⣿⣿⣿⡿⠟⠛⠒⠒⠻⣿⣿⣿⡗⠒⠒⠲⢾⣿⣿⣿⣿⣿⣿⣿⣿⣶⠆
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡏⠉
⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠁⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⡀⠀⠀⢻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡶⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⡤⡤⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢵⠄⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⢨⣇⠀⢀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣦⣘⣡⣾⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠿⠿⠿⢿⡿⠟⣻⠿⠑⠀⣀
⠀⠀⠀⠀⣀⢀⡄⣰⣶⣦⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠈⢻⣿⣾⡿⠁⠀⠀⡠⠋⠀⠀⣠⠀⠀⠀⢠⣶⣦⡀⣄⢀⡀
⠀⠀⠀⣼⠃⣾⢀⣿⣿⣿⣧⠀⠀⠀⠶⠶⣴⣦⠸⠀⠀⣀⠀⢻⣿⠁⡀⠀⠀⠀⣰⣶⠶⠦⠀⠀⢰⣿⣿⣿⡇⢹⡄⢿⡀
⠀⠀⠀⣿⠀⣿⢸⣿⣿⣿⣿⡇⢀⠀⠀⠂⠈⣃⠁⠀⢀⡌⠀⠈⠉⠈⠩⡉⠉⡈⢑⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⢸⡇⢸⡇
⠀⠀⠀⣿⠀⣿⠀⣿⣿⣿⣥⣤⠀⠙⠓⠶⠂⠀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠐⠖⠛⠁⢠⣬⣽⣿⣿⡇⢸⡇⢸⡇
⠀⠀⠀⢻⡆⢻⡆⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⡟⠻⣿⡇⠀⠀⠀⠀⢀⣿⡟⢻⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⣿⠁⣿⠁
⠀⠀⠀⠘⣷⠘⣷⡈⢿⣿⣿⣿⠀⢀⠤⠴⠲⣾⣿⡿⡙⣅⠀⠀⠀⠀⢠⡋⡿⣯⣳⡦⠠⢄⡀⢸⣿⣿⣿⠇⣸⡏⣸⡏
⠀⠀⠀⠀⠙⣧⡘⢷⣌⠻⠿⠃⠀⢇⡀⢸⣿⣶⣶⠶⠃⠋⠀⠀⠀⠀⠘⠛⠷⢶⣶⣾⠗⣀⠝⠀⠻⠿⢋⣴⠟⣰⡟
⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠉⠉⠀⠀⠀⠀⠀⠈⠀⠈⠉
⠀⠀⠀⠀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⠀⠀⢀⣀⡀⠀⢀⣀⣀⣀⠀⠀⣀⣀⣀⠀⠀⣀⣀⣀⢀⣀⣀⣀⣀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠙⣿⠋⢻⣧⠈⣿⣏⡭⢿⠀⠀⣼⠿⣷⠀⠈⢻⡟⠙⣿⡆⢹⣿⠉⠀⠀⢹⣿⠉⠉⣿⣏⡏⢿⢸⠏⣿⡏⣿
⠀⠀⠀⠀⣀⣿⣄⣼⡿⢀⣽⣏⣋⣾⣆⣸⣿⢛⣿⣇⣀⣸⣧⣠⣿⠇⣸⣿⣀⣸⡇⣸⣿⣀⣀⣿⣏⡃⠀⠀⣀⣿⣇⡀
⠀⠀⠀⠀⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠈⠉⠉⠀⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠉⠉⠉⠈⠉⠉⠁⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢠⣤⣤⣤⣶⣶⣦⡀⠀⠀⠀⢸⣿⣿⠛⠉⠉⣿⡏⠀⠀⠀⠀⠀⠀⠀⢻⣿⠉⠙⢻⣿⣿⡄⠀⠀⠀⠀⣰⣶⣶⣶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣾⣿⡿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⢀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⠤⠶⠶⢿⣿⠷⠒⠒⠚⠿⠿⠒⠒⠒⠒⠒⠒⠒⠻⢿⠗⠒⠲⠾⣿⣿⠶⠦⠤⢴⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⣀
⠀⠀⠈⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠋
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡏⢀⣠⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣄⡀⠀⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠙⠛⠉⠉⠉⠉⠀⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠃⠈⠛⠛⠻⠿⠿⠟⠁






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

⠀⠀⠀⠀⣴⣿⡿⠆
⠀⠀⠀⠀⠻⡏⡁⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤
⠀⠀⠀⠀⢠⠋⢨⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⡄
⠀⠀⠀⠀⡆⡆⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⢤⠇
⠀⠀⠀⠀⠄⡃⠄⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢊⠠⢙⡂
⠀⠀⠀⠀⠱⢛⡃⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⡸⠄⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿
⠀⠀⠀⠀⢀⠈⢇⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀⠀⢷⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡿⠻⡉⠐
⠀⠀⠀⠀⢸⠓⠲⣄⣇⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⢀⡀⠀⠀⢴⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠲⠂⠁⠀⡀⠤⣈⠄⠓⠂
⠀⠀⠀⠀⢀⠀⣴⠛⠛⠒⠒⡦⡀⠀⠀⠀⠀⡇⠀⠈⠛⠻⣁⡎⢎⠁⠀⠀⠀⠀⠀⢀⠞⠛⠤⣄⡀⠀⠈⠀⠸⡇⠀⢹
⠀⠀⠀⠀⠈⡀⡏⠀⣴⢲⠀⠘⡻⠀⠀⠀⠀⠉⠲⡀⠀⠀⣿⣧⣼⣘⡆⠀⠀⠀⠀⢈⠀⠂⠘⠋⠉⠛⠅⡀⢉⡅⢱⠊
⠀⠀⠀⠀⠀⢈⠷⡀⠈⠁⢀⡼⠃⠀⠀⠀⠀⠀⠀⠈⢄⣾⠇⠉⠀⠘⢹⡄⠀⠀⠀⠀⠑⠤⠀⢀⠀⡀⢀⡸⣻⠁⡄
⠀⠀⠀⠀⠀⠈⠀⡙⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡹⠀⢾⢿⠆⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢼⢷⠉⢸⡁
⠀⠀⠀⠀⠀⠃⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣄⣤⣀⣴⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠇⢸⢠⠎⣹
⠀⠀⠀⠀⠀⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣋⠀⠛⠺⣤⠟
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠎⠀⣀⡀⠈⠜⡄
⠀⠀⠀⢀⣀⣷⠺⣟⣖⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠺⣿⠂⣲⠄⠀⠀⠀⠀⠀⠀⠀⠀⢟⠄⠈⠳⠋⢀⠃⠃
⠀⠀⠀⠀⠺⠶⠤⠬⠥⠭⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠶⠴⠬⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠐⠛⢢⣅⣀⣀⡥⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠁⠀⡄⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠠⠇⠀⡂
⠀⠀⠀⠀⡀⠀⢄⠀⠂⠀⠀⡀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡤⢀
⠀⠀⠀⠀⠉⢀⢄⠀⡀⠀⢢⣀⣂⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⢀⠀⡠⢼⠀⠸
⠀⠀⠀⠀⠀⠈⠀⠈⠳⠀⠀⠝⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⡎⢀⣀⣄⠘⢰⢇⡀⠀⠃⡀⠀⣠⠀⡀
⠀⠀⠀⠀⠀⠀⠘⠀⡠⠉⠀⠀⡠⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⡀⠙⠓⠻⢿⣄⣶⣴⠇⡴⠁⡀⢀
⠀⠀⢠⠊⠀⡀⠀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⠁⠙⠑
⠈⠙⡇⠀⠤⠧⣄⢀⣀⡀⠁⠂⠈⠀⡤⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠂⠂
⠀⠀⠣⠀⠄⠀⠉⠉⠈⠷⠶⠓⢲⠾⠀⠤⣠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⢰⡀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠴⢡⡤⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠰⠃⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⡴⠶⣾⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠹⡇⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢻⣿⣿⣿⠀⠐⠀⠛⠿⠁⣠⡇
⠀⠀⠀⢀⣀⣀⣠⣤⣼⣯⣂⠀⠀⠀⠀⠀⢿⣿⣿⢃⣸⣿⣿⣿⣥⣴⣾⣀⠠⠀⠊⠀⠀⠀⠀⠀⢠⣪⣿⣤⣤⣄⣀⣀
⣶⣿⣿⣿⣿⠿⠿⠿⢻⣿⣧⠆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢦⣿⣿⠻⠿⠿⢿⣿⣿⣿⣷
⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿⣦⣼⠖⠒⠒⠒⠛⠻⣿⣿⣿⣿⣿⡿⣿⡋⠉⠁⢀⢠⠀⠓⠒⠒⠒⢾⣤⣾⣿⠀⠀⠀⠀⠀⠀⠉⠉
⢀⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠋⠀⠈⠳⣤⣀⣽⢄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤
⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⠫⣑⠀⠒⠀⠀⠀⠀⢲⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠘⣿⣿⠿⠿⠟⠛⠛⠋⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⠀⢀⡀⢸⡄⢠⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⣽⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠂⠀⠲⠠⠔⠀⠄⠴⠐⠐⠀⠀⠰⠀⠄⠆⠸⠠⠸⠈



⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⢿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣧⣀⠀⠀⠀⠀⠙⢿⣿⣿⢿⣿⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢻⣿⣿⣿⡆⠀⠀⠀⠀⠀⠉⠁⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣠⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⠉⠛⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢰⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣧⡀⢀⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⠄⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠏⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣶⣿⣿⣷⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣳⣿⣿⡿⠁⠀⠀⢀⣤⣌⠻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢡⣿⣿⡄⡸⣿⡿⢀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣦⡙⣛⢉⣯⢬⣴⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢹⣷⣚⣫⡿⠇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡽⣬⣉⠉⠅⢂⣾⢡⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣼⣵⣮⣿⡟⢳⣿⣽⣯⣧⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢧⢹⡿⣷⣿⡇⢸⣿⣾⢿⡏⡼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⡄⢸⡿⢿⡿⣇⣸⢿⡿⢿⡇⢠⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡻⠈⣗⡪⠺⡇⢸⠗⢕⣺⠃⠓⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣞⠉⡺⠇⠸⢗⠈⣳⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠒⢒⣶⡄⠀⠀⠀⠀⠀⣎⣀⠀⢸⠀⢶⡀⢀⡶⠀⡇⠀⣈⣹⠀⠀⠀⠀⠀⢠⣶⡒⠂⠐⠄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⡀⢸⣿⣞⡀⠀⠀⠀⢠⢏⢈⣷⡃⣼⣻⡻⢏⣟⣧⢘⣞⡃⡹⡄⠀⠀⠀⢀⣻⣿⡇⢀⠀⢀⠀⢸
⠀⠀⠀⠀⠀⢀⠀⠼⣿⣆⣿⣶⣿⣾⣯⣭⣭⣽⡳⠶⣟⣦⣯⠍⠡⢭⠍⠡⠊⠩⠍⡌⠩⣽⣵⣷⠶⢞⣯⣭⣭⣽⣷⣿⣷⣿⣰⣿⠅⠀⡀
⠀⠀⠀⠀⠀⠈⠀⠁⣿⣿⣿⣿⣿⣿⣿⣿⡎⠀⠀⠀⠀⠀⠁⠀⠀⢂⢡⠀⠀⡜⡘⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡼⢄⡅⢨⠠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜⠡⠊⠀⠈⠄⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡔⠁⠀⠀⠀⠘⡜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢯⡌⠀⠀⠀⠀⠀⠀⢰⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠅⠀⠀⠀⠀⠀⠀⢸⠁⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣶⣟⠂⠀⠀⠀⠀⠀⠀⠀⢧⣦⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⣿⣿⣿⣿⣿⣿⡏⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠐⣿⣿⣿⣿⣿⣿⡇⢹⣿⡇
⠀⠀⢀⣠⣤⣤⣤⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇⣸⣿⣿⣿⣿⣿⣿⣷⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣶⡄
⠀⠀⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⢀⣀⣿⣿⣿⣿⣿⣿⣿⣧⠤⠤⠴⠶⠶⠖⠒⠺⠿⠿⣿⣿⣿⣿⢻⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠒⠒⠒⠒⢺⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠛⠛⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡏⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠈⠁
⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠙⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣟⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡀⠀⠀⠀⢸⣿⣆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣯⣤⣤⣤⣤⣬⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣇⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠔⡃⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠉⡱⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠰
⠀⠀⠀⠀⠀⠀⠊⠀⡀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⢠⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠖⠀⠀⠀⠀⠀⠄⢢⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⠀⠠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⡁⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⣿⣀⣀⣀⣀⣀⡀⠀⠀⠀⣿⣿⣿⣿⡇⢀⣀⣀⣀⠀⠀⣀⣀⣄⠀⠉
⠀⠀⠀⠀⠀⣠⡀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣿⣿⠀⠀⢰⣿⣿⣿⣿⢠⣿⣿⡿⡿⠀⠀⠀⢀⣴⠿⠂
⠀⠀⠀⠀⠀⣿⠿⠀⢿⣿⣿⣤⣤⣀⠀⠀⣿⣿⡇⢸⣿⣿⣶⣿⣿⣿⠇⣿⣿⣷⠆⢻⣿⣿⡁⠀⠈⢿⣿⣶⣤⣄⡀⠀⠁
⠀⠀⠀⠀⠀⠁⠀⠀⢀⠈⠉⢹⣿⣿⣷⡀⣿⣿⡇⢸⣿⣿⢿⣿⡏⠁⠀⣿⣿⡇⠀⠀⠙⢿⣿⣶⡄⠀⠉⠉⠛⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣿⣿⣿⣿⠃⣿⣿⡇⢸⣿⣿⠘⣿⣿⡄⠀⣿⣿⣷⣶⣶⡄⠈⢿⣿⣿⣠⣶⣴⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠛⠁⠀⠙⢿⡇⢸⣿⣿⠀⠸⣿⣿⣦⣈⣉⣉⣉⣉⣁⣤⣿⣿⣿⢻⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠉
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠦⣤
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠘⠙⡀⠀⠀⠀⠀⠀⠀⠀⠼⣿⣶⣶⡶⠀⠀⢀⠄⠀⡤⠀⠀⠀⠀⠀⠀⠀⣸⣤⠀⠀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠁⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⣽⡿⠇⠀⠀⠀⠀⡀⠀⠀⠔⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠋⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⡀⠒⠀⠀⠀⠀⠀⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠊⠁⣒⢄⠀⠀⠀⠀⠀⠀⢀⡔⠂⠉⠉⠐⠠⡀⠀⠀⠀⠀⠀⢀⡠⣒⠉⠐⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣤⠞⢉⣒⣒⣛⣃⣹⣽⠇⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠸⢿⣏⣘⣛⣒⣖⡉⠓⢤⡀
⠀⠀⠀⢀⣴⠿⠗⣱⠛⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⢟⣎⠚⠿⣦⡀
⠀⠀⢠⡾⠁⡴⢫⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠟⠯⠍⠩⠽⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡝⢦⠈⢷⡄
⠀⢠⢿⢃⠎⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡔⠐⠤⠤⠂⢦⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠱⡘⣿⡆
⢠⠇⣿⡼⠀⣄⡼⢷⣦⡒⢶⡤⠴⠶⢒⢶⠦⢄⡀⠀⠀⡸⠁⢀⡄⢠⡀⠈⢇⠀⠀⢀⡠⠴⢶⡒⠶⠦⢤⡶⢒⣴⠾⣧⣀⠀⢣⣿⠸
⣿⣤⣿⣧⣤⠴⠧⠀⢈⣻⠁⠀⠀⠀⠀⠉⣀⢠⡿⠞⠛⣁⠀⢩⣀⣀⡌⠀⣈⡛⠓⢿⡆⣠⠁⠀⠀⠀⠀⠈⣟⡁⠠⠼⠦⣤⣼⣿⣤
⠈⠳⣦⣄⠀⢀⠀⠀⠀⠙⢐⡄⠀⠀⠀⠉⢿⡘⣀⠀⠀⠀⠉⢢⣝⣫⡴⠋⠁⠀⠀⣀⢃⡟⠉⠀⠀⠀⣀⡠⠋⠀⠀⠀⡀⠀⣠⣴⠞
⠀⠀⠈⠻⣄⢠⣙⣦⣤⣤⣿⢷⣄⢄⣀⠀⣪⡷⣌⡢⠄⠀⡀⠀⣿⣿⠁⣀⠀⠠⢔⣡⢾⣑⡂⢀⡠⣺⡿⣧⣤⣀⣀⣋⡄⣡⠟⠁
⠀⠀⠀⠀⠀⠙⠻⠿⣯⣭⣴⢾⡿⠛⠿⣿⠋⠀⠀⠙⣿⠋⡅⠀⠛⢻⠀⢨⠙⣿⠋⠀⠀⠙⣿⠿⠛⢿⡷⣦⣭⣽⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⡀⠀⠀⠉⠀⠀⣠⠟⣴⡇⠀⡆⣠⠀⢸⣶⠻⣄⠀⠀⠉⠁⠀⢀⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠒⠲⠶⠒⠊⠁⠴⠿⣷⠄⣿⣿⣤⣿⠿⠦⠈⠑⠒⠶⠖⠒⢁⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣷⣤⣀⣀⡀⠀⠀⠀⠈⡟⣿⣿⢻⠁⠀⠀⠀⢀⣀⣀⣤⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡇⠀⠀⢀⠀⢹⢿⣿⡏⠀⡀⠀⠀⠘⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⢧⡀⠀⣼⠀⣾⢸⡏⣷⠀⣇⠀⢀⡼⡫⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢿⠋⢹⠀⢻⢸⡇⡟⠀⡏⠙⣿⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣀⣘⣆⣘⣾⣷⣃⣰⣃⣀⣧⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠺⣷⣾⠷⠿⣷⣾⠖⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⢠⣀⠀⠀⢠⡀⠀⠀⣀⡇⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠘⣧⠐⢛⣁⣈⠛⠂⣽⠃⠀⠀⠀⠙
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣾⣿⣿⣷⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⡀⠀⠀⣀⣀⡤⠀⠀⢤⣄⣀⠀⠀⢀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⣠⠈⠀⠀⠀⠀⠀⠀⠀⠠⠿⣿⣿⠿⠿⠛⠋⠁⠀⣀⣀⠀⠈⠉⠛⠻⠿⢿⣿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠁⣤
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡀⠀⠀⠲⠶⠶⠾⠛⠁⠈⠛⠳⠶⠶⠒⠂⠀⢀⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠸⡆⠀⠀⠀⠀⠀⡴⠀⠀⠀⠁⠀⠀⠀⢀⣀⣠⣤⠖⠲⢦⣤⣀⡀⠀⠀⠀⠈⠀⠰⠀⣦⠀⠀⠀⠀⠀⣸⠃
⠀⠀⣤⡈⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣀⣬⣽⣿⣿⣿⣿⣿⣶⣾⣤⣀⠀⠀⠀⠀⠀⠀⠘⠷⡀⠀⠀⠀⢃⣤
⠀⠀⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣫⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠈⠀⢦⡀⠀⠀⠀⠀⠀⠀⣠⡿⣋⣼⣿⡿⠛⠉⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢀⡴⠀⠁
⠀⠀⠀⠀⣈⠛⠀⠀⠀⡜⠀⣰⣿⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣆⠀⢣⠀⠀⠀⠛⡁
⠀⠀⠀⠀⠙⣷⢰⣦⡾⠁⢀⣿⣿⣿⡿⠏⠀⢀⣀⣀⣀⣠⣀⣀⣀⡀⠀⠀⠸⣿⣿⣿⣿⡆⠘⢷⣴⡇⡾⠋
⠀⠀⠀⠀⠀⠘⡇⠛⠀⠀⢸⡿⠋⣡⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣌⡙⠻⣿⣧⠀⠈⠛⢸⠁
⠀⠀⡇⠀⠀⠀⡇⠀⢀⠄⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠛⢰⡀⠀⠘⠀⠀⠀⢰
⠀⠀⣧⠀⠀⠀⠃⢀⣾⣸⣿⣿⢿⡿⠿⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⣷⡀⢘⠀⠀⠀⣾
⠀⠀⢻⡀⠀⠀⢀⣿⣧⣿⣿⣿⡞⢀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⡀⠀⠀⢠⡟
⠀⠀⣾⣿⣿⣿⢿⡿⢿⣿⣿⣿⣷⣾⠁⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⣿⣿⣾⣷
⠀⠀⢹⣿⡿⠿⣿⠁⠀⠈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⠅⠀⠈⣿⡿⢿⣿⡏
⠀⠀⠀⣿⣿⣿⣭⣧⣠⣄⠈⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠁⣠⣄⣾⣭⣿⣿⣿
⠀⠀⠀⢾⣿⡿⠛⢻⣿⣿⣷⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣾⣿⣿⡟⠛⢿⣿⡯
⠀⠀⠀⠘⠉⢠⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⡄⠙⠃
⠀⠀⠀⠀⠚⠈⠃⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠘⠁⠓
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁
⣿⣿⣿⣿⣿⣿⢻⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⡿⠿⢿

⠀⠀⠀⣶⡶⠀⠀⠀⠀⣰⣶⢄⠀⠀⠀⢠⣶⡀⠀⠀⠀⢀⣶⣄⠀⠀⠀⠀⣴⣦⡀⠀⠀⠀⣠⣶⢆⡀⠀⠀⠀⣴⣦⡀
⠀⠀⢰⣿⣿⡆⠀⠀⢰⣿⣿⠚⠂⠀⠀⣿⣿⠛⠂⠀⠀⣾⣿⡟⠃⠀⠀⢺⣿⣿⠓⠀⠀⠀⠿⣿⡿⠃⠀⠀⢸⣿⣿⠛
⠀⠀⢻⣿⣿⡗⠀⠀⢀⣿⣿⡄⠀⠀⢸⣿⣿⡆⠀⠀⠀⣿⣿⣇⠀⠀⠀⢀⣿⣿⡄⠀⠀⠀⣸⣿⣧⠀⠀⠀⢼⣿⣿⡀
⠀⠀⠈⣿⣿⠁⠀⠀⠈⣿⣿⠁⠀⠀⠀⢹⣿⠁⠀⠀⠀⢹⣿⡏⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠩⣿⡏⠀⠀⠀⠈⣿⣿⠁
⠀⠀⠀⠛⠛⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠛⠋
⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⣤⡀
⠀⠀⠀⣴⣯⡿⠃⠀⠀⣿⣯⡀⠀⠀⠀⣠⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣯⡀⠀⠀⠀⣼⣿⡅⠀⠀⢸⣿⣧
⠀⠀⢀⣿⣿⢧⡀⠀⠀⢳⣿⠁⠀⠀⣠⣾⣿⡀⠀⠀⢀⣤⣤⣾⣿⣶⠀⠀⣸⣿⡏⠙⠦⠀⠀⣿⣿⡇⠀⠀⢸⣿⣟
⠀⠀⠛⢻⡇⠀⠀⠀⠀⢿⡿⣆⠀⠠⡿⠿⢿⡇⠀⠀⠺⠿⠿⣿⠁⠀⠀⠀⢻⡿⡇⠀⠀⠀⠘⣿⣿⠃⠀⠀⠺⣿⡿
⠀⠀⠀⠸⡇⠀⠀⠀⠠⡞⠁⢻⠄⠀⠃⠐⠌⡧⠀⠀⠘⠀⠐⠻⠀⠀⠀⠀⢸⠇⡷⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⡿⡇

⠀⠀⠀⣶⣦⠀⠀⠀⠀⣠⣴⣆⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⣴⡦⠀⠀⠀⠀⢀⣠⣶⠀⠀⠀⢀⣴⣦⠀⠀⠀⢀⣾⣇
⠀⠀⢰⣿⣿⠇⠀⠀⢰⣿⡿⡇⠀⠀⠀⣼⣿⣿⠀⠀⠀⣞⣿⣿⣳⠄⠀⠀⣻⣿⣷⠀⠀⠀⠸⣿⣿⠇⠀⠀⠈⣿⣟
⠀⠀⢸⣿⣿⡄⠀⠀⢸⣿⣧⡙⠂⠀⠀⣿⣿⣏⠀⠀⠀⢨⣿⣿⡅⠀⠀⠈⣿⣿⡄⠀⠀⠀⢠⣿⣿⡄⠀⠀⢰⣿⣿⡄
⠀⠀⠀⣿⣿⠀⠀⠀⢈⡏⢹⡅⠀⠀⠀⢩⡏⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣾⠈⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠉⠉⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣿⠿⠿⠷⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⡾⠀⣀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠙⡁⠀⠈⠉⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠂⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣤⣄⣀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⡄⠈⣻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡔⠁⠀⠀⠀⠈⠀⠂⠱⣸⠃⠻⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠠⠀⠀⠀⠀⠢⠀⠽⡀⠀⠀⠿⠢⣄⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠈⠳⣄⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣧⠀⢢⣤⣴⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠈⣦⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠈⠓⠄⠉⠉⢹⣧⡀⠠⠄⠀⠀⠀⢀⣰⠿⢍⣀⡀⠀⢠⣼⡏⢻⠻⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣿⠻⢿⣿⣿⠿⠛⢋⠉⠀⢠⡤⠈⣉⣿⣿⣿⡇⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠎⠀⠀⠀⠁⢀⣾⣧⡀⢶⣽⣄⡀⠈⢀⡀⠀⢀⡄⢀⣨⢿⡿⣿⣇⠀⠀⢂⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢶⡶⠟⠈⠛⢿⣿⣿⠉⠛⣦⠀⡉⠉⠁⠀⠉⠋⢺⠇⠈⠻⣷⣶⡏⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠹⡄⠀⠠⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⡁⠛⠒⠐⠃⠚⡑⢿⠀⠀⠀⢸⣇⠀⠀⠸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢐⡆⠀⠀⠀⠀⠀⡿⢦⡀⠀⢇⠀⠀⠠⠀⠀⡇⢸⠀⠀⠀⢸⡇⠦⠀⢀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠳⡀⢰⠎⡀⠀⠀⠀⠀⢸⡟⠀⠿⡅⠀⠁⠀⠀⠀⣰⣃⠨⡆⠀⠀⠸⣇⠀⠁⡜⢱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠑⣌⢰⠃⠀⠀⠀⢀⣾⠖⠀⠀⠄⣶⠄⠀⠀⠀⢩⣱⡆⠵⡀⠀⠀⠙⣇⠀⠀⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠘⠀⠀⠀⠀⠀⢸⣿⠁⠀⠉⣚⠈⣷⡀⠀⠀⠟⠁⡈⢔⣇⠀⠀⠀⠈⢃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⣼⣯⡂⠀⠀⠉⢀⢏⠳⣄⠀⠀⣠⠁⠀⣸⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⣿⠀⠀⠀⠀⠀⠤⠢⢷⡘⣠⠞⠋⠀⠀⠻⡇⠀⢀⠠⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠬⢿⠀⠀⠀⠀⠀⠀⠀⢀⡟⠠⠴⠒⠀⠀⠀⣧⠀⠀⣤⣤⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡿⠿⠖⠀⠘⠀⠀⠀⠀⠀⠀⠰⠻⠃⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢠⣿⣿⣿⣿⣿⡇⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢸⣿⣿⣿⣿⣿⣿⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⢁⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣆⣀⣀⣀⣀⣀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠤⠤⠤⠤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣤⣄⡀
⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠇⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠛⠋⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠠⣿⣧



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠈⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠉⠀⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡤⠀⠤⠖⠀⠀⠀⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⢀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡐⠀⠑⠄⡀⠉⢉⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠓⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠤⠤⠤⠤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠤⠀⠠⠤⠤⠤⠤⠀⢀
⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠠⠄⠐⠂⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⡆⠀⠀⠀⠀⠈⢀⡀⠠⠀⢂⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠁⠀⠐⠂⠆⡓⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠠⠼⠤⠄⠀⠒⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⢄⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡤⠄⠀⠀⠀⠀⠀⡀⠼⠄⠈⠁⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⢊⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠒⠈⠀⠀⠠⠀⠀⠁⠀⠄⠀⣀⠀⠀⠀⠐⠆⠤⠀⠀⠀⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠤⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠢
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤
⣿⣿⣇⣀⣀⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢠⡴⣦⠀⣶⠀⢠⡄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⣷⡀⢿⡶⠀⣿⢻⡄⣿⡀⣸⠃⣾⠯⠁⢺⡶⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡈⠳⣿⡿⡧⠘⠛⠃⠉⠈⠀⠉⠁⠉⠈⠃⠀⢰⡟⢀⡾⢃⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠻⣮⣻⡆⠈⠁⠀⠀⠀⣠⡶⠒⢶⡤⢤⣄⠀⠀⠀⠀⠀⠛⠱⢿⣿⠏⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣽⡃⠈⠉⠀⠀⠀⠀⣠⣾⣏⠉⢳⢎⣽⡛⢛⣿⣄⡀⠀⠀⠀⠀⠈⢡⡞⣏⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠄⠈⠛⠀⠀⠀⠀⠀⣠⣿⡁⠈⡴⠿⣄⣿⡷⣌⠀⣹⣧⡄⠀⠀⠀⠀⠀⠙⠁⠠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣠⣾⡧⣀⠙⠁⡠⣿⡦⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓
⠀⠀⠀⠀⠀⠀⢠⡄⣼⡏⠀⠀⠀⠀⠀⢠⡖⢋⢹⣿⣼⡁⣹⣿⣾⠀⢸⣷⣿⠃⣹⣤⣿⡏⢙⢲⡄⠀⠀⠀⠀⠀⢹⣷⢠⣄
⠀⠀⠀⠀⢠⡆⣿⡇⣿⣧⠀⠀⢀⣤⡶⢿⣆⡿⠆⢹⣿⣿⠟⣿⠀⠓⠃⢸⡟⢿⣿⣿⡏⠰⠿⣰⡿⢶⣤⡄⠀⠀⣸⣿⢸⣿⢰⡄
⠀⠀⠀⠀⢸⣧⢻⣷⠸⣿⡄⠀⠀⠁⢀⣾⡏⢁⡶⣿⣿⣿⣾⣮⣷⣤⣤⣿⣾⣿⣿⣿⣿⢷⡀⢹⣷⡀⠈⠀⠀⢠⣿⡏⣾⡟⣼⡇
⠀⠀⠀⣠⣄⢿⣎⢿⣧⡻⣿⣄⠀⠀⣾⣿⠃⢸⣾⣿⣿⣿⠈⡟⠿⠿⠿⠿⢻⠋⣿⣿⣿⣷⡇⠈⣿⣷⠀⠀⣠⣿⠟⣼⡿⢱⡿⢡⣄
⠀⠀⠀⠈⠉⠈⢿⣎⠻⣷⣝⠿⣷⠄⣿⣉⢔⣭⣵⣤⣿⣿⣇⢳⣤⡀⢀⣤⡾⣸⣿⣿⣥⣮⣭⡲⣍⣾⠠⣾⠿⣫⣾⡟⣱⡿⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠃⠈⠉⠁⠀⠀⠘⠃⠋⠀⠀⠈⢻⣿⣿⣬⣛⣻⣟⣛⣥⣿⣿⡿⠉⠀⠀⠈⠘⠃⠀⠀⠈⠉⠁⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⣥⣤⣀⣀⡀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⢶⡀⠀⠀⠀⠉⣻⣿⡿⠀⠈⠙⠛⠋⠙⠛⠋⠁⠀⢻⣿⣿⠋⠀⠀⠀⢀⡴⢄⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠙⠛⣡⣤⡀⠀⠀⣿⠻⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠿⣻⠇⠀⢀⡰⣌⠻⢮⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣋⡾⢃⡆⣠⠈⣉⡉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣉⣉⡀⣦⠈⢿⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣼⣿⣯⠞⢱⡶⠄⢠⣀⡀⢀⠀⠀⣀⠀⣤⠆⠘⢻⡅⠈⢷⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠋⠀⣿⡛⠀⣼⢿⡇⢸⡇⠀⢻⡀⢻⡟⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠋⠘⠃⠸⠷⠂⠘⠃⠈⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣝⢷⣄
⠀⠀⠀⣴⢿⣿⣿⣿⣿⣿⣧⠹⣷
⢀⠀⣰⣿⣿⣿⢿⣿⣿⣿⣿⡄⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡳⣦⡀
⠙⠒⢤⣬⣭⡅⢀⢈⣿⣿⣿⡇⢸⡿⠉⢢⢤⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣠⣯⣿⣿⣿⣿⣷⡘⣷
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⡇⠀⠇⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠷⢾⠇⠙⡿⠛⢻⣿⣿⡇⣿⣆⡀
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⢏⣼⡟⠑⠀⠰⡀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠈⢷⠀⢹⣶⣾⣿⣿⣇⣿⠛⠛
⠀⠀⠀⠈⠻⢿⣿⣿⣛⣵⡿⠋⠀⠀⡆⠀⠱⡄⠀⠀⠀⣼⣿⡿⢋⡙⢇⠀⠀⠀⢸⠆⠀⣿⣿⣿⢟⣾⠏
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢹⡄⠀⠹⣄⠀⠀⣿⠛⠀⠈⠀⢸⠀⠀⠀⣎⡀⠀⡿⠿⠷⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠱⣄⡙⠂⢀⣀⠀⢸⣄⡤⠋⠀⠁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⣠⣀⠉⠑⠢⣌⠙⠻⢦⡀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠙⠉⠂⠀⠀⢹⣧⡀⠀⢳⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠠⢄⣸⣿⣿⣷⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⣼⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣠⣼⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠟⠛⠋⠷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⠉⠁⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠁⠀⠀⠀⠀⡤⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣰⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⣠⣿⡇⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡆⢢⡄⣀⣀⠀⢠⠋⠙⠊⠉⠁⠐⡇⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣦⣻⡀⠀⠈⠓⠁⠀⠀⠀⢀⡠⠚⢏⣁⣩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⣾⣁⣀⣉⣳⣤⣀⠀⠀⣠⣴⣋⣀⣀⣼⣿⣿⣷⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠁⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣺⢻⣿⣷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢎⡵⣻⣣⡏⣩⢻⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⢳⣿⡟⣿⣿⣾⣿⣿⣿⡿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠉⢿⢸⣿⠷⠛⠿⠿⠛⠙⣿⠇⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠜⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠟⠁⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠏⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠉⠉⠉⠉⠉⠉⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠿⠳⠄⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⢤⡆⠠⠒⠋⠀⠀⡌⢧⠀⠀⠀⠀⠀⠀⢀⣠⠂⠀⠐⠦⢤⣄⣀⣀⠀⠀⡀⠀⣀⠴⠃
⠀⠀⣀⠀⠠⠶⣯⡀⠃⠀⠀⠀⠀⠀⢳⠀⠀⢀⣀⣤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣉⡤⠞⠁
⢠⠏⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠈⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠉
⡟⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠐⣄⠀⠠⠚⠈⢷⡀⡆⠀⠠⠤⠒⠚⠉⠁
⣷⠀⠀⠀⠀⠀⠀⠀⠐⣾⣷⡇⠀⠀⠀⢳⠀⠀⠀⠀⠀⢻⡁
⠘⣆⠀⠀⠀⠀⠀⠀⠀⢹⡜⡇⠀⠀⠀⠘⣇⠀⠀⠀⠀⠈⣷⡀
⠀⢹⢧⡀⠀⠀⠀⠀⠀⠀⢳⡇⠀⠀⠀⠀⢻⣀⣀⣀⣠⡴⠋⢷⡀
⠀⠈⠃⠱⣄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠸⡏⠉⠉⠁⠀⠀⠈⣷
⠀⠀⣾⠛⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢹
⣀⣀⣘⣂⣑⣈⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⣀⣔⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣇⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⡀⠀⠀⠀⠀⠀⠀⣠⣀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⡇⠀⠀⠀⠀⢠⡾⡟⠛⣻⣿⣿⣴⣶⡄⠀⠀⠀⠀⠀⡈⠉⢙⡲⠠⣄
⠀⠀⠀⠀⠀⠀⢠⣿⢺⣸⠿⡇⠀⠀⠀⢸⠀⡇⢐⣿⡿⠆⢚⡿⠃⠀⠠⠊⠀⠉⠀⠐⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠸⣿⢻⣿⣺⠃⠀⠀⠀⠈⡇⢻⠛⣿⣀⠄⠁⠀⠀⠀⠀⠀⠀⠀⣦⠀⢸⠇⡀⣿⠓⡀
⠀⠀⠀⠀⠀⠀⠀⣷⣾⣿⠿⠀⠀⠀⠀⠀⠹⣬⣾⡿⡏⡇⠀⠀⠀⠀⠀⠀⠀⢈⢩⣀⡿⠂⡗⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠠⠙⣿⠻⠀⠀⠀⠀⠀⠀⠀⢹⠙⣿⣿⣆⠀⠀⠀⠀⠀⠒⠀⣣⠘⣿⡏⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⣷⡇⠆⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣇⣿⠀⠀⠀⠀⠀⠠⣾⣿⠇⢹⡇⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⡿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⠀⡆⡇
⠀⠀⠀⠀⠀⠀⢀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢿⠋⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣦⣸⠇
⠀⠀⠀⠀⠀⢀⢢⣦⣾⡄⠀⠀⠀⠀⠀⠀⠀⢸⡸⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠘⢻⣘⡍⠇⠀⠀⠀⠀⠀⠀⠀⠈⣆⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⠶⡶⠛⢾⢿⡿⡿⣻⡟⣻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣴⣿⣰⣦⡛⣸⣸⣇⣇⣻⣇⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣻⣿⣿⣿⣿⣿⠛⠛⠉⠉⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⢿⣿⣿⣇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡃⣀⣀⣀⣀⠀⠀⠀⠀⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⣿⣿⡿⠟⢁⡶⠏⠿⠿⠠⣤⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢫⣼⣷⣶⣿⣬⣭⣿⣝⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⣮⠛⣈⣿⢀⡔⠋⠱⠴⠆⠀⠿⣶⠓⣦⣥⣿⣤⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⣫⣦⣛⣿⣽⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠀⠐⠁⣾⣿⡃⠛⣤⠁⢾⣦⠀⣈⣺⠏⣿⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠊⢹⠿⠁⠹⢿⣿⣿⠋⠉⠙⠂⠂⡤⣀⠀⠀⢸⡀⠀⣀⠀⠛⠹⡿⠷⠈⠀⢈⣿⣷⣬⠏⠻⠟⠛⣻⠿⢿⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣶⣜⡰⠞⡰⠀⣤⣄⡰⣾⡇⣠⣤⣤⣄⠈⠀⠉⢈⣳⠾⣿⣦⣠⣌⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⡏⠙⠀⠀⡇⠀⢸⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⠗⠀⠠⣿⣿⣿⡏⠑⠂⣀⠀⠁⠀⣥⡄⠀⠀⠀⠶⢒⡄⢨⠕⠛⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀⠀⠀⣇⠀⣸⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢠⠙⠓⠀⠆⠉⡛⡁⠰⣦⣼⢿⣿⣷⣶⣶⡔⢷⣄⣠⣗⣿⠿⢁⠀⢠⢀⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣹⣿⣿⣿⣿⣿⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⡆⢀⡀⢀⡼⠃⣇⠙⠀⣠⡇⢸⣿⣿⣍⣉⣉⣁⡙⣿⣿⣧⣤⣈⣣⣄⡼⢡⡜⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠼⠉⠞⠀⠘⣿⣿⣿⣿⣾⣿⡇⣸⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⢋⠉⠻⣿⠷⢶⠉⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣆⠀⠀⠀⠀⠉⠙⢋⣿⣿⠛⢃⣽⡷⠾⠿⣿⣿⣿⠿⠿⠿⣿⣿⣷⣾⠟⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⢿⣿⣁⡀⣤⣤⣤⣾⣿⣟⣹⣿⣟⠁⢀⡀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠻⠿⠿⠿⣿⣿⣷⢾⣿⠀⡿⠤⠀⠚⠁⠀⠀⠲⠒⠒⠀⢻⠉⢻⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠀⠈⢻⠀⡇⠀⠀⠀⠀⠀⢀⣿⣶⣿⣿⣿⡦⠈⢿⣿⢷⣴⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠃⠀⠀⢸⡀⡷⠤⢤⠀⠀⠀⠐⣿⣿⣿⢿⣿⣵⣶⢾⡇⢸⣿⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⢾⢶⣤⡔⢻⣧⠠⢤⣤⣤⣤⣤⣿⣿⣿⣿⣃⠀⠀⣾⡇⣼⠷⠢⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠒⠒⠚⠛⠛⠯⠾⣿⡿⠛⢿⣿⣿⣶⣿⠱⠇⢤⠛⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⢀⢄⣠⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⢀⠐⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⢼⡌⢿⣼⣦⣶⠀⠀⠀⠀⠀⠀⠀⠀⠠⣸⡿⠀⠀⠀⢠⣿⣿⠁⠠⡶⠀⠀⠘⠛⠈⠛⣿⣿⣿⣿⡿⠿⠟⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⢉⠀⣿⠏⠉⢣⣤⣀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⢰⣿⣿⣿⣿⣆⢀⡌⠀⠀⠀⠀⣈⣳⣬⣿⣥⡴⠶⠖⠶⠿⠻⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⡀⠀⡀⠀⠀⢸⠁⠘⣿⣿⣿⠁⠀⠀⠀⣼⡟⠉⠹⢿⣿⣠⠀⠀⠀⠀⠀⠈⠙⢻⡟⠋⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⢣
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠟⠈⡵⠤⢎⡉⠃⢤⣸⣿⣿⣄⣴⣾⣸⡿⠃⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠐⣀⠀⢡⣴⠀⠀⠀⠻⣧⣀⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠉⠁⠈⠀⢸⣄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡁⠀⠀⠠⠀⠘⣿⣿⣿⣿⣿⣿⢏⠇⠀⠀⠀⠀⠀⠀⠈⢿⠆⠀⠀⣀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵⣦⠤⠀⠶⢶⣼⣿⣿⡿⠿⢁⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢀⣤⣍⣉⣀⡘⢲⡆⠀⠀⠀⠀⠀⠀⠀⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠿⢿⣿⣤⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡶⠉⠉⠉⢹⣿⣿⣶⣶⠶⠋⠐⠒⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠢⠤⠤⠔⡆⢰⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠛⠀⠀⠀⠀⣿⣿⡿⠉⠀⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣀⣀⠀⠀⢹⡅⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠁⠀⠀⠀⠀⠀⣿⣿⡇⢈⠛⠋⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⠁⠀⢀⠀⢀⣀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢠⣼⠇⣿⡏⠙⣶⣤⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⢠⣴⡏⠁⠻⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢀⣶⠒⠊⠃⡴⢻⣷⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠾⠋⠁⣠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣈⣁⣀⣀⣀⣧⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⠐⠾⠿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠛⣻⡟⢹⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣰⠿⢳⣦⣤⣄⣈⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣭⣽⣿⣿⣿⣿⠄⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣋⣿⣿⣶⡞⠁⠈⠋⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠿⠻⣦⣾⠟⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡟⢹⡟⠈⢾⠛⣿⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⢿⡇⢟⣧⡌⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡞⠃⠙⠟⢀⣈⠤⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣋⠛⢹⣿⣷⠈⠳⣶⣶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⣤⣤⣾⣡⣤⣴⣶⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣋⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠦⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⢀⣀⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡉⠛⠁⠶⠶⠗
⢠⠶⠾⠃⠉⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣠⣀⣀
⠘⣠⣴⣶⣦⣾⣿⣿⣤⡤⠶⠶⠀⠀⣂⠀⠒⠐⠒⣶⣶⣤⣴⡇⠐⢆⠂⠀⢙⠃⠂⠒⠲⠶⣾⣿⣿⣿⣯⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⡌⠄⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠒⠀⠩⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠈⢿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠠⠃⠀⢠⡄⢀⣀⠀⠠⠀⠢⢤⣤⣀⢰⠀⠀⠑⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠘⣿⣿⣿⡿⠿⠟⠃⠀⠀⠀⠀⠀⠾⡇⠀⠀⠀⢀⠀⡀⠀⠀⠀⣼⢆⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠇
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠁⠈⠀⠠⠋⠀⠘⠂⠀⢈⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣲⣦⣤⣤⣤⣶⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠙⢿⣿⣿⣿⠿⠋⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢙⡿⣧⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡛⠀⢋⠠⡀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠈⠆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠐⠁⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠸⠀⢡⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣋⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⢦⣶⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣰⡯⠙⠛⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣸⣿⣿⣿⣏⣤⣿⣆⠀⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣏⢷⡿⠏⠲⣤⠙⣆⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⢺⠿⠛⣿⣿⣿⣿⣿⡷⠀⠀⠀⠈⠃⢹⠂⠄⢠⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄⠀⠀⣀⣴⠶⠋⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠸⢿⢸⠀⣼⣿⡟⠁⠙⠿⢆⣄
⠀⠀⠀⠀⠀⢰⣿⣿⠛⢿⣿⠻⣿⠋⠀⠀⠀⠀⠠⠿⠿⢿⣟⣛⣂⣠⣤⡤⠤⠴⣾⢸⡆⢻⣿⡇⠘⠟⣿⡟⠋
⠀⠀⠀⡄⠀⢸⣿⡇⢰⣂⢥⣼⠷⠶⠒⠒⠚⢛⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠈⣧⣷⡜⣿⣿⣦⣴⣿⠇
⠀⠀⠀⢣⢠⠸⣿⣿⣄⣴⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠣⣧⣙⢿⣿⣿⡿⠃⠀⠀⠀⠀⠐⢿⣿⣿⡿⣟⠛⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢠⠘⠁⣿⠇⢻⡄⠘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣸⡏⠀⠀⣇⡰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢹⣷⠀⠀⠈⠃⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⢄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⣇⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣾⣿⣇⠀⢛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠉⠉⠁⠀⠓⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠆⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣍⣿⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠶⣿⠀⣸⠷⠦⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣿⢿⣝⣃⠈⣴⡿⢿⣿⡏⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠃⣸⡆⢸⡆⠛⢻⣏⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣽⡿⣿⣴⣮⣿⡿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⣶⣶⣇⣷⣾⣿⣶⣾⣟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢁⣿⣿⡹⠟⢳⠛⢿⣿⣿⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣿⣿⡟⢻⡇⣸⠛⣿⣿⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢸⠏⣿⣿⠀⢹⠂⢺⣿⠋⡿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢸⠃⣿⣿⠞⠙⠙⢾⣿⡇⢻⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⡼⠀⣿⢽⡉⠸⠁⣙⢽⠀⢰⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡾⠀⣿⣿⠀⠀⠀⢸⣿⡇⠘⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣩⡿⠃⠀⣿⡿⡁⠉⠀⠁⢿⡇⠀⢻⣍⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⠁⠀⣻⣧⣄⠀⢀⣠⣽⡇⠀⠹⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠰⠿⠁⠀⢸⣿⣄⢳⣴⣶⢀⣾⣇⠀⠀⠻⠧⠀⠀⠀⠀⠀⠀⢸⢻⠁⠀⢠⣿⡟⠻⣷⡿⠛⢿⣧⠀⠀⠹⡿
⠀⠀⠀⠀⠀⠀⡦⣦⡄⠀⠀⣾⠋⢻⣿⠉⣻⡿⠋⢿⡀⠀⠀⡄⠐⠆⠀⠀⠀⣴⢆⡄⠀⠀⢈⣿⣄⣠⣿⣷⣀⣼⡏⠀⠀⠀⣦⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡖⡄⢻⡇⣿⢁⢠⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠀⠀⠀⢾⣼⣿⣿⡏⣿⣿⣯⣸⡇⠀⠀⠀⠚⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣿⣸⠃⣿⣾⣄⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡟⣿⣿⡇⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇⠹⠀⢸⢸⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠹⡇⠀⢸⡿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠸⠃⠀⠀⠇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠄⠀⠀⠸⣿⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡄⠀⠀⣦⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣷⡄⠀⠀⣠⣿⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⢹⡇⠀⠀⣿⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⡇⠀⠀⣷⡿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⠁⠀⠀⠏⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢈⠃⠀⠀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠶⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⠀⠀⠀⢻⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠁

⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠷⠾⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⠆
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠹⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⢿⣿⣿⣦⡀⠀⠀⠀⠀⣴⣿⣿⣦⠀⠀⠀⠀⢀⣠⣿⣿⡿⠏⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⣀⡀⢸⣿⣿⣿⣿⠀⠀⢀⣴⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣶⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠁⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠈⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠘⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⡲⣄⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠈⠪⣢⣹⢿⣦⠀⣶⣤⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠄⠠⣤⢀⠌⠀⠀⠀⡰⡘⠀⢸⠈⠀⠸⢸⠀⡇
⠀⠀⠐⠀⠒⠂⠀⠠⠀⠀⠂⠁⠀⠀⠀⠈⠁⠈⢼⠌⠀⠀⠀⠀⠱⡇⠀⢸⠀⠀⠀⢈⠀⡇⠀⠠
⠀⠀⠀⣀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠔⠒⠂⢁⠎⠀⠀⠀⠠⠡⢡⢰⠀⠸⠀⠀⠀⠈⠀⡇⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⢁⠁⡄⡈⡀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣾⡋⠙⢦
⠀⠀⠀⠀⠀⢀⣴⣷⠋⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠈⠳⠒⠶⣷⣷⣷⣾⣶⣴⣶⣿⡼⣾⠧⠄⠀⠀⢿⣿⢂⡿
⠀⠀⠀⠀⢠⢦⠊⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⢸⠐⠀⡠⠞⠁⠀⠀⠈⠉⠉⠉⠉⠭⠉⠉⣉⠉⠉⠉⢉⡉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠩⠉⠩⠄⠀⠈
⠀⠀⠀⠀⠀⠱⠎⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⣶⣶⠒⠂⠀⠀⠀⠐⠀⠀⠀⠐
⠀⠀⠀⠂⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿
⠀⠀⠀⠁⢄⡀⠁⢒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⣿⡏⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡟⡇⠀⡄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⡇⠀⠃⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⡇⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⢀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⣿⡿⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣀⡀⣀⡘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⡀
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢙⡃⡛⠁⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⡶⠲⠛⣏⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡹⣟⡟⣸⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣏⡟⠳⠲⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢷⣷⡷⠇⣸⠏⠉⠁⠀⣠⠴⠒⣚⣯⣛⣿⣶⣾⣟⣫⣟⡒⠶⢦⡀⠀⠉⠉⢿⡀⠷⢧⣷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠘⣷⢲⠋⣧⠀⠀⢀⣾⠃⢠⡄⠀⠀⠈⠉⢻⠋⠉⠁⠀⠀⣤⠀⢻⣆⠀⠀⢠⡏⢳⢲⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⡈⠀⠘⢶⡋⠉⣻⣤⢾⣆⣀⠀⠀⠀⣸⡀⠀⠀⢀⣀⣾⢦⣼⡋⠉⣳⠞⠀⠈⣸⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠘⡇⠀⠈⢪⢳⡼⠋⠀⡠⣽⣷⣶⣤⠶⠿⠷⢦⣴⣶⣿⡥⡀⠀⠿⣴⢫⠂⠀⠀⡟⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠈⠻⠿⣿⣯⣿⣿⡿⠁⠀⠀⢳⡀⠀⠈⠀⢀⣤⣼⢿⣿⠿⢫⡿⠗⢿⠖⠿⣯⠻⢿⣿⢿⣤⣀⠀⠀⠀⠀⣰⠃⠀⠀⠹⣿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠳⢤⣠⠖⠛⠋⠁⠀⢸⣦⣠⣤⣤⣴⣤⣤⣤⣠⣾⠀⠀⠉⠛⠓⢦⣠⠶⠋⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠛⢧⣄⣸⣀⡤⠿⠻⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣰⣄⠈⠹⠉⢀⣴⣠⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢿⣿⣽⡆⠀⠀⣾⣽⣿⠟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠈⣿⣿⣿⣿⣿⣿⣿⡏⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣆⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⣿⣤⣼⣯⣿⡿⠃⠻⣿⣯⣿⣤⣼⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⢀⣴⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠉⣹⣿⣦⣄⠀⠀⠀⠀⠊⠀⠀⠀⢐⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⣠⣾⠟⠁⠉⠻⣷⡄⠀⠀⠀⠀⠀⠔⠋⠁
⠀⠀⠺⢷⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠉⣠⣾⠟⠁⢀⣀⣀⡀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣶⣆
⠀⠀⠀⠀⠀⠉⠙⠻⠿⣶⣦⣄⣀⣴⣾⠟⠋⣠⣴⡿⠛⠁⠀⣴⡿⠛⠛⢿⣆⠀⢻⣇⠀⠀⠀⠀⠒⠒⠚⠿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣯⣤⣶⠿⠛⠉⠀⠀⠀⠀⣿⣇⠀⢀⣼⡿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣦⣤⣄⡀⠀⠀⠈⠛⠿⠿⠛⠁⠀⢸⣿⠀⠀⠀⠲⢤⣀⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠈⠋⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢷⣶⣦⣤⣄⣀⣀⣀⣀⣀⣤⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣦⣬⣍⣉⣙⣛⣛⣛⣉⣩⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠈⠉⠙⠛⠛⠛⠛⠛⢻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟

⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇
⠀⠀⠀⠀⣾⣿⣧⡉⣱⣿⣿⡄⠀⠀⢠⣾⣿⣹⠀⣏⣿⣷⡄⠀⠀⠀⠀⢀⣀⢼⡿⢀⣀⠀⠀⠀⠀⢀⣀⣼⠙⣄⣀
⠀⠀⠀⠀⣿⣿⠿⠇⠿⣿⣿⡇⠀⠀⢨⣷⣿⣿⠀⣿⣿⡿⡇⠀⠀⠀⠀⣿⠉⠁⡄⠉⢻⡆⠀⠀⠀⣾⣿⣿⠸⣾⣿⡇
⠀⠀⠀⠀⣿⢻⣄⡀⣄⡸⢻⡇⠀⠀⢰⡘⢿⡟⠀⢻⡿⢃⡄⠀⠀⠀⠀⣿⢰⣤⣷⣴⢸⡇⠀⠀⠀⠛⢹⣤⠀⣴⠈⠃
⠀⠀⠀⢸⡇⢸⣿⣷⣿⣿⠘⣧⠀⠀⢸⡇⡸⠁⠀⠀⢧⢸⡇⠀⠀⠀⢠⡟⣸⣿⣿⣯⠸⡇⠀⠀⠀⡷⢸⠏⠀⠻⡄⣾
⠀⠀⠀⢸⡁⣿⣿⣿⣿⣿⡇⣽⠀⠀⢘⠀⣵⣦⣀⣴⣾⠀⠋⠀⠀⠀⢸⢳⣿⣿⣿⣿⡆⢿⠀⠀⢸⡇⣾⣶⣤⣶⣇⢹
⠀⠀⠀⠘⠀⡿⣿⡍⣿⡟⠃⠈⠀⠀⠸⠠⢿⣿⠿⣿⡿⠆⠙⠀⠀⠀⠴⢸⣻⣯⢹⣿⡇⠰⠀⠀⢰⠀⢿⣿⢻⣿⠟⠠⡄
⠀⠀⠀⠀⠀⣧⣿⠃⣿⣷⠆⠀⠀⠀⠀⠀⣾⣿⠀⣿⣷⠀⠀⠀⠀⠀⠀⠐⣿⡿⢸⣿⡇⠀⠀⠀⠀⠀⣾⣿⠘⣿⣷
⠀⠀⠀⠀⠀⠈⢻⠀⢹⠋⠀⠀⠀⠀⠀⠀⡿⣿⠀⢿⢿⠀⠀⠀⠀⠀⠀⠀⠉⡃⢨⠛⠁⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⣧⡟⠀⢸⣼⠀⠀⠀⠀⠀⠀⣿⡏⠀⢸⣾⠀⠀⠀⠀⠀⠀⠀⣆⡇⠈⣇⡆⠀⠀⠀⠀⠀⣼⡟⠀⣿⡇
⠀⠀⠀⠀⠀⢹⡇⠀⠈⣿⠀⠀⠀⠀⠀⠀⡿⡇⠀⢸⢿⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣿⠃⠀⠀⠀⠀⠀⣿⡇⠀⠿⡇
⠀⠀⠀⠀⠀⢘⠃⠀⠀⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠻⠀⠀⠀⠀⠀⠀⢀⠂⠀⢀⡄
⠀⠀⠀⠀⠐⠋⠀⠀⠀⠈⠂⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠋⠀⠀⠘⠂⠀⠀⠀⠀⠀⠊⠀⠀⠀⠃
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠒⠉⠉⠉⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠒⢤⡀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⢀⣿⠀⠈⣷⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⡄⠘⠿⠋⠀⢠⡿⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⢦⣄⣠⣴⠟⠁⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣰⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⠋⢿⡄⠈⠻⣿⣦⡀⠀⠀⢀⣠⣤⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣶⣿⣿⣿⠿⠛⠉⢿⣿⠀⠈⠛⠀⠀⠀⠙⢿⣶⣜⠉⠀⠀⠀⠀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣆⠀⠀⢠⠀⢀⣴⣶⣿⡟⠉⠛⠳⡄⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠀⠀⠀⢱⣾⣿⣿⣿⣿⠀⢀⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢹⣿⣿⣿⠈⠉⠀⠀⠁⢀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠀⠉⣯⣈⡅⡀⠀⠀⠠⠆⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡍⠉⣀⣤⠀⠀⠉⢤⣷⣶⡶⠶⠀⢀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠸⣿⠀⠀⢸⡆⢹⣿⡿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣿⠀⠀⠀⠃⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⢀⠁⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⢀⠔⠀⠀⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠿⣷⣦⣤⠴⠖⠁⠀⠀⠀⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣀⣀⣀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣶⣿⠟⠋⠉⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠐⠢⣄⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠟⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠂⠀⠀⠀⠀⢈⣷⣄⠀⠀⠀⠀⣸⣿⠋⠁⠀⠀⠀⢀⣠⣴⡿⠁⠀⠀⡠⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⣠⣿⣷⣦⣤⣶⣶⣿⣿⣿⣿⡇⠀⠀⢰⢣⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢀⠀⠀⠀⢀⣼⣿⣿⠃⣀⣠⣴⡿⠿⠿⠟⠛⠛⠋⠉⠀⢸⣿⡇⠀⠀⠉⠆
⣠⣴⣦⡤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠊⠀⢠⣿⣾⣿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠸
⣿⣿⣿⠀⣤⣀⣀⠀⠈⠑⣦⣤⡤⢾⡇⢀⡅⠀⣰⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠇
⢹⣿⣿⣷⠟⠛⠿⣷⣤⣤⡇⠀⠀⠈⢻⣏⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠃⠀⠺⠇
⢻⣿⣿⡇⠀⠀⠀⠀⠙⢿⡇⣀⠀⠀⠀⠙⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣠⡄
⠘⠿⢿⣷⠄⠀⠀⠀⠀⠀⠻⠼⠗⠢⠴⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣦⣄⣀⣀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣶⣶⣶⣾⣾⣷⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣶⣶⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠀⠘⢿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⣠⣤⣤⣿⣿⣿⣿⣿⣿⣿⡦⠤⠴⠶⠶⠶⠒⠒⢾⣿⣿⡿⠛⠛⠛⠛⠛⠻⣿⣿⣿⡛⠓⠒⠒⠒⠶⠶⠶⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤
⠛⠛⠉⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠉⠉⠛
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣤⡀

⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⣶⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⣄
⠀⠀⢀⣴⣶⣶⣮⠁⠀⠀⠀⠀⠀⣴⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣀⣀⢰⣿⣿⡆⠀⠀⠀⠀⠀⢀⣤⣤
⠀⠀⢸⣷⣿⣿⠿⣦⡤⠀⠀⠀⢸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⠀⠙⠋⠀⠀⠀⠀⠀⣠⡸⢿⡿⠃
⢀⠀⠀⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠂⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏
⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠂⠀⡉⠀⠀⠀⠀⣿⣿⣷⣶⣦
⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⣿⡟⢸⣿⠀⠀⠀⠀⠀⠀⢀⣻⣿⣇⣽⣿⠀⠀⠁
⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠘⠟⠙⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠙⠟⠀⠀⠀⠀⠀⠀⠛⠋⠀⠙⠟⠀⠀⠀⠀⠀⠀⠻⠿⠛⠋⠻⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡧⠀⠀⠀⠀⠀⢠⡶⢸⣿⣿⠄
⠀⠀⠀⠀⢠⣾⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣠⣬⣭⣤⠀⠀⠀⠀⠀⠀⣠⣬⣭⣥⡀⠀⠀⠀⠀⢸⣧⣤⣭⣥⡄
⠀⠀⢠⣴⣮⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠛⣿⣿⣿⡇
⠀⢰⣿⣿⣿⣀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠀⠀⠀⠠⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠄⠀⠀⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⢀⣠⣰⣿⣿⣿⣿⣷⣠⣀⠀⠀⠀⠀⠀⢸⣿⣻⣿⠀⠀⠀⠀⠀⠀⢸⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⡇
⢠⣬⣭⣭⡙⣿⡆⠉⠋⠀⠀⠀⠀⠀⣾⣿⢹⣿⡟⠛⣻⣿⠛⣿⡆⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⣼⣿⣿⣿⣇⢻⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢿⣷⢠⣿⡏⠀⣿⡇⠀⠀⠀⠀⢸⣿⢾⣿⠀⠀⠀⠀⠀⠀⢸⣿⢼⣿⠀⠀⠀⠀⠀⠀⠀⣿⡷⣿⡇
⠀⠀⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢃⠀⠀⠀⠀⠀⠠⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢰⣾⣷⡄⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⣀
⠀⢀⣼⣿⣿⣥⡀⠀⠀⠀⠀⠀⢾⣥⣾⣿⣷⣬⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠸⣷⣦⣿⣶⡿⠀⠀⠀⠀⠀⠀⢻⣿⣿⣾⡿
⠀⢸⣿⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢩⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡅
⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⠇
⠀⠀⣿⡟⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣽⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿
⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿⠀⠀⠀⠀⠀⠀⠀⢸⡿⢸⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣤⣀⠀⢠⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣿⣿⣷⠀⢸⣿⣿⠙⢟⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣿⣽⠀⢸⣿⠃⠀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⠁⠀⢸⣿⣄⠀⢙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠏⠈⠀⠀⢸⣿⣿⣦⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⢸⣏⠿⣿⣿⣿⢡
⠀⠀⠀⠀⢀⣀⣤⣄⣀⡀⠀⠀⠀⢀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠉⢿⣿⡆⢠⡀
⠀⢶⠿⣛⣻⣟⣻⡛⠛⠉⠐⠖⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣹⣷⣮⣿⣳⣄⢷⣤⣤⣤⣤⣄⢀⣠⣴⣶⠾⠃⠉⢠⢤⡀
⠀⢾⠿⠛⣹⣿⣫⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣧⣽⣿⡎⠻⣷⡿⣦⡯⡭⠽⢚⣽⣿⡿⠛⠁⠀⣠⠲⠀⢼⣷
⠀⢺⣶⢿⣿⣏⠉⠙⢿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣝⣿⠇⢀⢼⣿⣿⣷⢲⣾⣿⠿⠋⣀⠀⡠⠊⠁⠀⠀⢾⣿
⠀⠀⠀⠈⢻⣯⣤⠀⠀⠹⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿⣿⡆⢈⣿⣿⢇⣿⠛⢋⣡⣶⣾⣿⣿⣏⠀⠀⠀⢌⣿⣿
⠀⠀⣤⣤⣾⣿⡏⢀⣀⣀⣀⣽⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠙⣧⣿⣿⣷⣤⣝⣿⣶⣿⣧⣸⣿⢿⠟⣿⠟⠻⣶⣄⡲⣷⣿⣿
⠀⠀⠈⠉⠉⠀⠀⠈⢿⣿⡿⠻⢿⠿⣧⡈⠙⠋⠉⠉⠉⠉⠙⠟⠛⠀⠘⣽⣿⣿⡿⣿⣿⣿⡇⣿⣿⣿⣟⣦⡾⠋⠀⣠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡑⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣿⣷⢯⣿⣿⡇⢿⣿⣋⣿⠁⠀⣠⡶⣻⣿⣿⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⢸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠛⣠⣿⣿⡌⣼⡿⢸⣧⣿⣿⣿⣷⣴⣾⣩⣾⣿⠛⠀⠁⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣠⣄⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠻⣿⢸⣿⣱⣿⠇⠘⣿⣿⡿⣿⡧⠹⣿⣿⡿⠃⠀⠀⢠⣿⣸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣟⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠚⡇⣼⡏⣧⣿⠀⣄⢀⣿⣿⣟⠻⣾⠇⠘⣷⠀⣤⣿⣿⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣌⢶⠘⢛⢿⡻⣷⡀⠀⠀⠀⠀⠀⠖⣿⣿⣾⣧⣿⠀⠛⣻⣿⣿⠿⢟⣿⠀⠀⢻⣿⡟⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢳⡀⠢⠙⣝⢿⡄⠀⠀⠀⠠⠾⣿⠿⠋⠁⢀⡴⠛⢫⣽⡿⣿⢻⡇⠀⠀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠣⠀⠙⢌⠃⠙⣦⠀⠀⠘⠇⢸⡆⠀⠀⣀⣴⣾⣿⣿⡷⢻⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠈⠂⠀⢨⣧⡀⠐⠆⢸⠄⠀⠀⣻⣽⣿⣿⡖⣿⠹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠆⠀⠀⠀⠀⠀⠀⡀⢻⣇⠘⠇⢸⡀⠀⠈⣩⣿⣿⡿⡇⢸⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠉⠄⠀⢙⡧⢸⣀⣴⡾⠛⠏⣏⢰⣧⢻⡇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠄⠀⠀⠀⠸⠃⡿⢿⡋⠀⠀⢠⡿⢸⣿⡟⣇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣧⡄⢼⡄⠀⠀⠀⠀⠀⠰⠀⡷⠘⢧⢠⣿⢸⡇⢸⣿⣧⢿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⡇⠐⠈⠳⣿⣿⠇⠘⣻⣿⣜⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣯⣿⣯⣥⣌⠀⠀⠀⠀⠀⠀⠀⠈⢢⣽⣤⣼⣿⣂⣲⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠛⣿⡿⠟⣻⣻⡄⠀⠀⠀⠀⠒⠀⠀⠀⣭⠛⣿⡿⣯⣍⠉⠉⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣯⣠⡾⠋⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⡜⢡⡾⠛⠝⠶⠉⠻⣦⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠀⠀⠀⠀⠀⠀⠘⣻⣦⣆⠀⠀⠀⣰⣿⣦⡔⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⡇⢆⣰⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠂⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠠⡄⣵⣺⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡈⡄⠀⠀⠀⠀⠀⠀⢠⢫⣿⣿⣽⣿⣿⣿⣿⣆⡠⡀⠀⢠⣄⣶⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢀⣰⡌⠀⠀⡴⣡⢿⡿⣯⣿⢣⣿⣿⣿⣿⣿⣿⣵⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣿⣿⣿⣾⣿⣴⣿⢿⡽⣻⣴⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⢾⡿⢿⡿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⢻⠛⢻⡟⠙⡿⠻⣟⠿⢻⣻⠟⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣾⣾⣿⣾⣷⣶⣿⣶⣿⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣠⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡉⣁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣦⡀
⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣽⡌⣅⡍⣼⣥⣄⣀⣀⣀⣀⣀⣀⣀⡀⠀⣾⣿⣿⣿⣿⣿⣷
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣮⣭⣼⠚⡟⠛⠛⢛⣻⢽⣟⢷⣍⣵⢟⣿⢽⣛⠛⠛⠛⡟⢺⣬⣭⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⡿⣿⣿⣿⣿⡏⢳⢶⢺⠀⣀⣴⠃⡠⠀⠈⠙⢻⠛⠉⠀⠠⡈⢳⣄⡀⢸⢲⢶⠋⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠈⢿⣿⣧⢙⣿⣿⣿⡇⢸⡀⠀⡻⡁⡸⠖⢳⣄⣀⢀⣸⣀⢀⣀⣴⠓⠾⡀⡹⡃⠀⣸⠀⣿⣿⣿⡟⣠⣿⣿⠏
⠀⠀⠈⠻⢿⣷⣿⣿⡿⠁⠀⣇⠀⠘⠙⣁⣰⣾⡿⢿⡯⢽⠭⣿⠿⣿⣶⣀⡙⠘⠀⢀⡇⠀⠹⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠢⠤⠚⠋⠁⠈⣧⣤⠤⢶⠤⢤⣤⡏⠀⠉⠛⠢⠤⠚⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢉⠳⢾⠶⠋⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢿⣿⣧⠀⢠⣿⣿⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⣿⣿⣿⣿⣿⡏⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣰⣿⣿⣿⣿⣿⣷⡀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⣸⣿⣿⠏⢿⣿⣿⣠⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠋⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠓⡆⠀⠀⠜⣀⣠⠤⣤⠶⠒⠋⠉⠉⠉⠉⠓⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠛⠤⡄⢀⡸⠔⠒⢉⣡⠖⠛⠒⠒⠒⠒⠒⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢲⡠⠗⠛⢤⠴⠚⠉⡏⠀⢐⣁⡀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡮⠀⠀⠀⠙⣅⠀⠀⢻⣰⠋⠈⠹⣄⠀⠈⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠸⣀⡶⢀⣠⣄⣵⠀⠀⠀⣿⠀⠀⠀⢹⡶⠋⠉⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠙⠀⢋⣀⣸⣿⡆⠀⠀⣿⠀⠀⠀⠀⡷⠖⠉⠉⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⡀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣠⠞⣩⠥⣌⢻⠃⠀⠀⢹⡀⠀⠀⢀⡟⠒⠶⣏⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠙⠀⠉⠃⠀⠀⣿⣿⣿⠁⠙⢿⣿⣿⣷⣄⠀⠀⠀⠙⠒⠤⣤⠙⠈⠀⠀⠀⠀⢙⠲⣾⡟⢡⡄⠀⠀⠀⠉⠉⠉⠉⠙⠛⠒⠒⠢⢤
⠀⠀⠀⠀⠀⠀⠀⠈⠚⠀⠀⠀⣠⠔⠲⠤⠼⢿⣿⡟⠀⠀⠈⠻⣿⣿⣿⣷⣶⣤⣦⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠙⠦⣄⡀⠀⠀⠀⡀⠀⠀⢀⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢋⠔⢛⣥⣤⠄⢀⣈⣶⠤⠴⠒⠏⠛⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣠⠴⡖⠲⣤⡀⠀⠀⠀⠀⠀⠀⠉⢹⠖⠶⠦⠤⠥⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠳⠤⠋⠸⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡇⠀⠀⠀⣠⢺⠇⢸⠁⣠⡃⠉⠢⣄⠀⠄⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠳⠾⠔⠛⠉⠁⠈⠀⠀⠈⠋⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡷⢤⣴⣶⡞⠛⠉⠉⠁
⠀⠀⠀⠀⢀⣀⣀⣀⡆⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⡏
⠀⢀⡴⡺⢷⣛⡟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡦⢄⡀⠀⠀⠀⠀⠀⠀⣴⠋⠁⢸⣿⣿⠁
⣠⢋⠊⠠⣶⣼⡇⠃⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⠏⠈⠉⠒⠢⠤⣄⣠⠞⠀⠀⣾⣿⣿
⡇⡞⠉⠉⠳⡽⡇⠇⠃⠀⢀⢀⠀⠀⠀⢀⡴⢋⡴⠁⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⢹⣿⣿
⠳⣧⡀⠀⣰⠇⠙⢄⠀⠀⠀⢉⣹⠖⠒⠿⡴⢏⣀⡀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠸⣿⢿⣣⠤⡄
⠀⠀⠉⠉⠁⠀⠀⠀⠙⠒⠚⠉⠀⠀⢀⡞⠁⠀⠙⠩⠙⠂⠀⡝⠀⠀⠀⠀⠀⠀⠀⢠⢛⠊⠱⠯⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣗⡒⢦⣤⡤⣤⡶⢳⠞⠛⠓⣄⠀⠀⡠⠚⢹⠃⢠⣼⢮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠻⠿⢿⣶⡫⠤⠧⣤⣤⣖⡟⠀⠸⣀⠄⠧⣀⣈⣉⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠻⣤⣠⡋⠉⣉⣠⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠳⣜⢦⡲⡄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠈⠣⣙⣌⢪⡧⣑⣄⠀⢰⡶⣤⠀⢰⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⡎⡹⢩⠁⡏⠛⠀⠀⢹⢹⠀⢸⡏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠤⠤⠒⣎⠉⠀⡹⢳⢀⠎⠀⠀⠀⠀⠀⡜⣰⡁⡈⠀⡇⠀⠀⠀⢸⢸⠀⢸⡇⢸
⠀⠀⠔⠊⠉⠉⠉⠐⠒⠒⠤⣄⠠⠐⠊⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢀⢟⠎⠀⠀⠀⠀⠀⡰⢠⢫⠁⠁⠀⡇⠀⠀⠀⢸⢸⠀⢸⡇⢸⠀⢢⡀
⠀⠀⠀⠈⠁⠀⠀⠀⠄⠂⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠔⠚⢡⠎⠀⠀⠀⠀⠀⡰⢁⠆⡎⢸⠀⠀⡇⠀⠀⠀⢸⢸⠀⢸⡇⢸⠀⠀⠳⡄
⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠠⠔⠊⠀⠀⠀⠀⡀⢯⡀⠀⠀⠀⠀⡰⠁⠎⢠⠁⡜⠀⠀⠃⠀⠀⠀⢸⢸⠀⢸⡇⢸⠀⠀⠀⠘⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⢦⡀⠀⠠⢁⠌⠀⡘⢠⠀⡀⠀⡀⠀⠀⠄⢸⠀⠀⢸⡇⢸⠀⠀⠀⣼⣿⡁⠀⠈⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡶⡒⠋⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠤⠧⣪⡬⣲⠧⣼⣴⣝⣢⣇⣴⣶⣜⣾⣀⣷⣾⣇⣸⠀⠀⠀⢿⣿⣿⡅⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⡰⢋⡹⠛⠓⠒⠂⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉⠉⠙⠛⠋⠉⠛⠈⠁⠘⠋⠀⠀⠀⠀⠀⠀⠙⠿⠶⠞⢡
⠀⠀⠀⠀⠀⠀⢰⠓⠏⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣱
⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⢀⡴⠋⠀⠀⠀⠀⠈⠉⠉⠩⠉⠉⠉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⢉⠉⣭⠀⠀⢀⠠⠀⡠⠀⢈
⠀⠀⠀⠀⠀⠀⠀⢧⣀⠔⠋⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢠⡄⠤⠤⠤⠤⠀⡸
⠑⠄⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣾⣿⣿⡇⠀⠀⢠
⠀⠀⠑⠄⠚⠀⠀⠀⠀⠀⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡇⠀⠀⠆
⠀⠀⠀⠀⠑⢄⠀⠀⠐⠒⠤⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣩⣿⣻⣿⡇⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠑⠲⠤⠀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣿⡿⠜⡇⠀⡆⠀⠆⢰⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡿⣇⢹⠇⠀⣥⠀⠾⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣤⣾⠇⠀⠸⠀⠀⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⣿⣯⢿⠄⠀⠀⠈⠙⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⢸⣿⡏⣿⠀⠀⠀⠘⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢿⠏⣿⠀⠀⠀⠀⢀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠂⠀⠀⠀⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣹⡛⣉⣝⣛⣿⣫⣽⣫⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡍⢿⠛⠒⠶⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣘⣇⠀⠀⠀⠀⠀⠉⠉⠛⠒⠦⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣩⠟⠓⠲⠦⢤⣄⣀⡀⠀⠀⠀⠀⠀⠈⣉⣙⣳⣶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⢾⣅⣀⠀⢀⣀⣀⣤⣬⠽⠿⠶⠒⠛⠋⠉⠉⠁⠀⠶⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠤⠶⠖⠚⠛⠉⠉⠉⠁⠀⠀⣀⠀⢠⣄⢰⡆⠀⠀⠀⠀⣶⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢉⠉⠉⠀⠀⢀⡀⣠⡀⣤⠀⡶⠻⡆⢸⠇⣸⣿⡆⣼⠙⢿⠁⠀⠀⠀⠀⠤⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⢸⠁⣿⠻⣿⠠⡧⠾⠃⠘⠐⠃⠀⠁⠀⠀⣀⣀⣀⣤⣤⠤⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⠀⠀⠀⠀⠉⠀⠁⠀⠀⣀⣀⣀⣤⠤⠤⠶⠶⠾⢯⣍⣉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣈⣠⣤⠤⠴⠶⠒⠛⠛⠛⠿⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠲⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⠶⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠉⣉⣛⣲⣶⠤⠴⠶⠖⠚⠛⠉⠉⣁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣨⣭⠽⠿⠖⠒⠚⠋⠉⠉⠉⠀⢀⣀⠀⣠⠤⠄⣶⣚⠀⣤⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠚⠛⠋⠉⠉⠀⠀⢀⣀⡀⠀⢠⡶⠆⣴⠛⢻⠄⣿⣿⠀⣯⣠⠀⢿⠥⠀⣉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⢀⣶⡀⢸⡇⣿⣼⡃⠀⢸⡏⠁⠻⠶⠋⠀⠃⠈⠀⠈⠁⢀⣀⣀⣠⣭⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠀⡾⠛⠇⠸⠀⠋⠈⠁⠀⠀⣀⣀⣀⣠⡤⢤⣶⣖⣒⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣀⣀⣀⣤⡤⠴⠶⠶⠶⢿⣉⣉⡉⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⠶⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⠤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠲⢦⣤⣤⣤⣤⠤⠶⠖⠒⠛⣋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣛⣲⣶⡤⠴⠶⠒⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⠶⠒⠛⠋⠉⠉⠁⠀⢀⣠⣤⡀⢠⣶⠀⢷⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⢇⡾⠻⠇⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢀⣀⣀⣀⣤⠤⠴⠶⠒⠚⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠀⢀⣀⣀⣤⡤⢤⣶⣖⠒⠛⠋⠉⠉⠈⠉⢹⡗⠲⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⣿⣉⣉⠉⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠦⣤⣴⣋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠤⣤⣀⣀⠀⠀⠀⠀⠀⢻⡉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⢤⣬⣧⡘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⣠⡤⣖⣛⣙⢧⡀
⠀⠀⠀⠀⡿⣉⠉⠤⠼⣿⢹⡆
⠀⠀⠀⢸⠇⠵⠄⡒⠒⢻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⢋⡉⠉⣁⣠⠔
⠀⠀⠀⢸⠀⠀⠐⠻⣭⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⢸⠀⣆⠀⢀⣷⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⡀⠀⣀⡀
⠀⠀⠀⠘⣆⠈⠩⢽⣞⢧⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣾⣀⢘⡝⠻⠊
⠀⠀⠀⠀⢹⠀⡄⠀⠘⢦⠳⡄⠀⠀⠀⣠⠞⠉⠁⠀⠈⠉⠙⠿⡷⣆⠐⢦⡀
⠀⠀⠀⠀⢸⠀⠸⡄⠀⠀⠀⠙⢦⡀⣼⠃⠀⡀⠀⠀⠀⠀⠀⠀⠙⠘⢧⡈⠳⣄
⠀⠀⠀⠀⢸⡄⠀⢹⡀⠀⠀⠀⠀⠙⢧⣆⢸⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠙⠲⢌⠛⠶⠦⣤⣤⠄
⠀⠀⠀⠀⢸⡇⡀⠸⣧⡀⠀⠀⠀⠀⠈⢿⠻⡄⠀⠀⠀⠀⠀⣰⠃⠀⠀⡇⢰⠈⠇
⠀⠀⠀⠀⠀⢷⠹⠀⡏⠻⠶⠖⠀⠀⠀⢸⣶⡇⢤⣤⣤⣤⠞⠁⣠⡆⠀⣧⡏
⠀⠀⠀⠀⠀⠘⣧⠀⠻⣄⣀⠀⠀⠀⠖⠋⢹⠀⠀⠀⠀⢀⢄⠼⡋⠀⢀⣿⡅
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠨⡉⠉⠉⠉⠀⢀⣈⡴⠃⠀⠀⠀⠉⠉⠃⢀⡼⠋⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣈⠲⠤⠤⠀⠉⣀⣀⣀⣀⣀⣤⣤⠴⠞⠋⠀⠀⠀⠈⠙⠛⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠈⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⠀⠀⢻⣿⡿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇⠀⠀⠹⠿⠃⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣼⣿⣷⠀⠀⣠⣄⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣷⣾⣿⣿⣧⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⠀⣀⠀⠘⢿⣿⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⠉⠀⠀⠘⣿⣿⣿⣿⡇⠀⠀⠈⠄⢨⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⠊⠁⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠀⠠⢸⣿⣿⠀⠀⠈⠙⠿⣿⠗⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⡿⠀⣴⣶⣶⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢴⣿⣿⠀⠀⠀⠀⠀⠖⠀⠀⣠⣶⡄⠹⣧
⠀⠀⠀⠀⢀⠞⢤⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⣿⡆
⠀⠀⠀⠀⠰⣤⣶⣇⠘⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢿⡇
⠀⠀⠀⠀⠀⢹⣿⣿⡀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣷⡄⠀⠀⠀⢻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀⠀⢻⣿⣿⣿⣿⡄⠀⠀⢀⣿⣿⠃⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀⠀⠙⢿⣿⣿⡇⠀⠀⢸⣿⡟⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⡠⠙⣿⣿⣄⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⠛⣿⣿⣿⣿⡇⠀⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠀⠀⠀⠀⠈⠛⣿⠆⠸⡿⢷⡄⠀⠀⠀⢸⣿⣿⡏⠀⠘⣿⣿⣿⡇⠀⠀⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠈⢿⣿⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⣾⣿⡟⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣧⣠⣄⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⢰⣿⣿⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠙⠿⣿⡿⢄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣾⣿⡟⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣤⣄⣙⣠⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⡆⢠⣿⣿⡇⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⡇⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣧⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣾⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢲⣷⠂⠀⠀⠀⢰⣷⠆⠀⠀⠀⢰⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠰⣾⡆⠀⠀⠀⠰⣾⡖
⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⡿⠿⠿⠿⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠋⠁⠀⢀⡄⠀⠀⠉⠻⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢀⣀⣀⣾⣿⣆⣀⣀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠩⣭⣭⣭⣭⣭⣿⡇⠀⠈⠻⣿⣿⣿⣿⡿⠋⠀⠀⣿⣯⣭⣭⣭⣭⡭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⢻⣷⡀⠀⠀⣿⡿⠿⣿⡇⠀⠀⣸⣿⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣄⠀⠁⠀⠀⠀⠁⢀⣴⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣶⣶⣶⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠶⠝⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠁⠀⠀⠀⠀⠀⠈⠙⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⢐⡇⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣀⣤⡂⢄⢀⣀⡼⢳⡄⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣼⣀⡀⠀⢰⣿⣿⡊⠄⣀⡁⢜⣾⣿⠆⣸⡟⣿⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣠⢾⣀⠀⣿⡷⠉⠂⠉⡀⠉⠙⡦⢤⡴⠞⠋⠛⣼⣿⠋⠃⠀⠀⣰⡷⠤⣀
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠻⠀⢘⣧⣼⣠⡾⠁⢀⡤⠀⠀⠀⢀⡀⠀⣿⡗⢦⡀⢎⣶⠟⠀⠀⠈⣧
⠀⠀⠀⠀⠀⡸⣷⡴⠇⠀⢀⡏⢻⣿⣿⣤⣬⣥⣤⠤⠸⠂⠀⠈⠁⣿⣳⡆⠉⠉⣀⠀⠀⠰⢄⣿⣣
⠀⠀⠀⠀⠀⣧⡏⠀⠐⠶⣏⣀⠀⣻⣷⠈⢿⣦⣤⡤⢞⠳⢴⡶⠆⠿⠇⠀⠀⠀⢸⠏⠀⠀⠀⠈⣻⡀
⠀⠀⠀⠀⠀⣸⣄⣤⡤⠄⠻⠟⠀⠈⠁⠀⣴⣼⠖⠒⣨⡙⠒⠦⢀⣄⣀⠀⠈⠁⠒⠒⠚⢷⣤⣴⠟⣳
⠀⠀⠀⠀⠈⢿⣿⡛⠲⢶⣒⠂⠀⣀⡀⠠⡀⣿⣿⢿⣿⣿⡿⠦⣐⣀⣀⠀⠀⠀⠲⠒⠚⣋⣥⢾⡿⠃
⠀⠀⠀⠀⠀⠈⢿⣍⡛⢶⣖⣚⣷⣶⣶⣶⣿⣿⣿⡀⠀⠀⠀⠻⣦⣴⣶⣦⣈⣩⣭⣽⣿⣭⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠉⠉⠉⠉⠉⠁⠈⠋⠀⠀⠀⠀⠈⠉⠻⣏⠉⠈⠉⠙⠋⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⡏⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹
⣀⣀⣠⡇⠘⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠃⢸⣄⣀
⠛⠛⠻⡇⠀⠃⠀⠠⢾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡗⠄⠀⠘⠀⢸⠟⠛
⠀⠀⠀⣇⢀⡇⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢸⡀⣸
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠘⢆⠀⠰⢄⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⡠⡄⢀⡴⠃⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣦⡈⢲⡶⢄⡠⠀⠀⢄⡠⢶⡖⢁⣴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⠀⠓⢀⡀⠚⢠⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠿⠿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⣀⠟⢲⠿⣅⡀⢀⡨⢿⡾⠃⢀⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢼⡟⢦⠀⠀⠈⠉⠉⠀⠀⠀⡴⢻⡧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠸⠓⠦⠄⠀⠀⠠⠴⠚⠇⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠀⡂⠀⠀⠀⠀⢐⠀⠿⢦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤
⠀⠀⠀⠀⠀⢠⣤⣾⠛⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣶⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢛⣷⣤⡄
⠀⠀⠀⠀⠀⢸⡇⢸⠀⣿⢸⡇⠀⠀⢠⣤⣤⣤⣤⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠻⣮⠙⠷⣤⠀⠀⠀⠀⢀⣤⣤⣤⣤⡄⠀⠀⢸⡇⣿⢸⡇⢸⡇
⠀⠀⠀⠀⢸⣿⡇⢸⠀⣿⢸⣷⣶⣶⣾⣇⡀⢀⣼⣷⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣿⣶⣶⣶⣶⣾⣧⡀⢀⣸⣷⣶⣶⣾⡇⣿⢸⡇⢸⣿⡇
⠀⠀⠀⠀⠀⢹⡇⢸⠀⣿⢸⡇⠀⠀⠀⠉⢻⣏⠙⢧⡀⠀⠀⠀⢹⣣⡟⠳⠶⣤⣤⣤⣤⣤⠀⣿⠀⠀⠀⢠⡾⠉⣹⡟⠁⠀⠀⠀⢸⡇⣿⢸⡇⢸⡏
⠀⠀⠀⠀⠀⠸⠷⣾⣀⣿⢸⡇⠀⠀⠀⠀⠀⢹⣄⠈⢻⣄⠀⠀⢹⣿⡃⠲⠶⠀⠀⠾⠆⢸⡆⣿⠀⠀⣰⠟⠀⣰⠏⠀⠀⠀⠀⠀⢸⡇⣿⣨⡷⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⠾⠇⠀⠀⠀⠀⠀⠀⠻⣆⠀⠹⣦⣄⠘⢿⣇⠀⢀⣀⣀⠀⢀⣾⣿⠃⣠⡼⠋⠀⣰⠏⠀⠀⠀⠀⠀⠀⠸⠷⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠙⠳⣌⠙⢦⣀⣉⣉⣠⠾⢋⣵⠞⠉⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠘⢷⣴⣟⠉⠉⣿⣤⡾⠁⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣠⡟⠁⠙⠛⠛⠋⠈⢷⣄⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢀⣤⡄⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢹⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠉⠉⠉⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣤⣤⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⡇⢹⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢦⣾⠀⠈⣷⠴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⣿⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢰⡏⠀⠀⢻⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⢸⡇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣤⣼⠁⠀⠀⠘⣧⣤⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠁

⠀⢀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⡟⣉⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡰⠃⣇⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⡼⠁⠀⢹⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⡤⠀⠀⠀⣸⠁⠀⠀⠀⢧⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠘⠇⠀⠀⢰⠃⠀⠀⠀⠀⣸⠂⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⠃⠀⠀⣏⠀⠀⠀⡇
⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢰⠃⠀⠀⠀⠈⠁⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠀⠀⠀⢸⡇⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢸⠁⠀⠀⠀⠀⠈⠓⠦⣤⣀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⠳⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣄⣀⣃⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣈⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⡣⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡷⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣹⠋⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣐⠋⠐⢿⠀⠐⢉⠭⠓⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡐⢞⣻⣿⣠⣀⠈⠀⡄⠊⣶⣄⡀⠠⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣣⣼⣿⠞⢻⠁⠀⢰⠀⣹⣿⣿⡦⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣹⠃⠀⡿⠋⠀⠀⣿⠀⣴⠇⣠⣽⣿⡟⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠏⠀⠘⣿⡧⠀⠀⣧⣼⡏⢴⠤⣾⣿⠃⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣴⠏⠀⠀⠀⣧⣇⢀⣴⣿⠁⢀⣤⣴⣿⡟⠀⠀⠀⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣤⣴⠏⠀⠀⠀⢠⣸⠟⠋⢹⡿⠀⣸⠉⣽⡿⠀⠀⠀⠀⢱⡀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⡇⢸⣿⡇⠀⠀⠀⠀⠈⣿⣷⠿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣾⡿⠃⠀⠀⠀⠀⠀⠀⢸⠇⠀⡸⢠⡟⠀⠘⣿⡇⠀⠀⠀⠀⠀⠘⣿⡠⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠊⡱⠋⠀⠀⠀⠀⠀⠀⢠⣴⣿⣷⠀⣧⠊⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠈⢷⣸⣧
⠀⠀⠀⠀⠀⠀⠀⢾⡆⠀⠀⠀⠀⠀⠀⠐⣿⣤⣉⣻⡷⠍⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡇
⠀⠀⠀⠀⠐⠀⠄⠀⠃⠀⠀⠀⠀⠀⠀⠁⠈⢻⣿⣿⡇⠀⣄⣀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⢿⣿⠀⠀⠀⢻⣿⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⢸⡏⠀⠀⠀⠘⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⡾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣿⠀⠀⠀⠀⠸⣿⣷⡸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⣿⠀⠀⠀⠀⠀⢟⢹⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢩⡇⠀⠀⠀⠀⠀⠘⢶⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣾⣇⠀⠀⠀⠀⠀⠀⠸⡆⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⡇⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⣿⡇⠀⠀⠀⠀⠀⠹⡿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⠀⠀⠀⠀⠀⠀⠀⢣⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡏⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠓⠚⠛⠛⠉⠁⠀⠀⠀⠀⠈⠋⠛⠒⠶⠦⠄⠶⠖⠒⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣶⣶⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣸⣿⣿⣋⣁⣸⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⡄⠉⣻⣿⣿⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡗⠛⠛⠋⢹⣿⣿⡿⠉⠉⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠉⠉⢹⣿⣿⣿⠓⠒⠲⠶⢶⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⡀
⠀⠀⠈⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠛⠛⠻⠃
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⠀⠀⠉⠛⠛⠛⠛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁






⣿⣿⡿⠛⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢳⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⣰⠃⢻⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢸⡆⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢸⠃⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣰⠃⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡴⠃⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠈⠲⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⣀⢢⡐⣅⡦⡵⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⣤⣶⣚⣋⣭⣬⠹⣆⡀
⠀⠀⠀⠀⠀⣿⠏⠉⠡⠤⠬⣿⡏⣿
⠀⠀⠀⠀⢸⡇⢋⣀⠠⠤⠤⣼⣷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠖⠒⠒⠒⠀⢀⣠⠄
⠀⠀⠀⠀⢸⠀⠈⠀⣐⣶⠤⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠒⠶⠶⠶⠞⠋⠁
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⡶⢋⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⢿⠀⣯⣀⣀⣠⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⣀⣀⣀⡠⣄
⠀⠀⠀⠀⠘⣆⠀⠉⠭⡵⡞⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⠀⠰⣡⠚⠷⠚
⠀⠀⠀⠀⠀⢻⡀⢄⠀⠈⠙⣆⠹⣆⠀⠀⠀⠀⠀⣠⡶⠛⠉⠉⠉⠉⠙⠛⢿⣿⢶⡄⠠⣤⡀
⠀⠀⠀⠀⠀⢸⡇⠘⣆⠀⠀⠈⠃⠙⣦⡀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠹⣆⠈⠻⣆
⠀⠀⠀⠀⠀⢸⡇⠀⠘⣆⠀⠀⠀⠀⠈⠛⢦⣼⢇⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⠙⢷⣤⣀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠻⣮⣆⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⢀⠉⠙⢦⡈⠙⠛⠳⠶⠶⠶
⠀⠀⠀⠀⠀⠸⡇⢄⠀⢸⡳⣄⠀⠀⠀⠀⠀⠀⠘⣟⢻⡄⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⢸⠀⡆⠀⠇
⠀⠀⠀⠀⠀⠀⢿⡨⣂⠸⡟⢾⣶⣲⠤⠀⠀⠀⠀⣻⣤⡇⣀⣀⡀⢀⣀⣴⠟⠁⣠⠄⠀⢸⣤⡇
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣷⡀⠀⠀⠀⠀⢀⣀⣠⠟⢿⠑⠀⠉⠉⠉⠉⡁⠀⣞⠟⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⠻⠶⢤⣤⣤⣤⠅⠀⠀⠸⠀⣤⠀⠀⠀⠐⢬⣊⣸⠄⠀⠀⣾⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠘⢦⡀⠀⠀⠀⠀⠀⣠⠤⠖⠃⠀⠀⠀⠀⠀⠀⠀⠁⣠⡾⠃⠈⠻⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⡙⠲⠤⠴⠀⠈⢀⣀⣠⣤⣀⣀⣤⣤⣤⡴⠾⠛⠉⠀⠀⠀⠀⠀⠉⠛⠛⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣤⡴⠞⠛⠉⠉⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⣰⣶⡄⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡶⢦⣄⠀⠀⠀⠀⢀⣀
⠀⠀⣀⣤⣴⣾⣿⡅⠀⠀⠀⠀⠀⢀⣴⣾⣿⣦⡀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⡷⠶⠟⠃⠀⠀⢀⣿⣿⣇
⠀⠈⢿⣄⣿⣿⣿⢿⡀⠀⠀⠀⠐⢿⣽⣿⣿⣏⣿⠆⠀⠀⠀⢠⣾⣿⣿⣆⠀⠀⠀⢀⣴⠟⣿⣿⣿⠀⠀⠀⠀⠀⣴⣿⣿⣿⣟⣷⡀
⠀⠀⠀⠉⣿⣿⣿⡈⢿⠄⠀⠀⠀⠀⢹⡿⣿⡏⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠈⠁⠀⣿⡿⣿⠀⠀⠀⠀⠀⠈⠙⣿⣿⡏⠉
⠀⠀⠀⠀⢸⣷⠹⣷⡀⠀⠀⠀⠀⠀⣿⠇⢹⣧⠀⠀⠀⠀⠀⢘⣿⠃⢻⡟⠀⠀⠀⠀⠀⠀⣿⡇⢿⡇⠀⠀⠀⠀⠀⢨⣿⣿⡇
⠀⠀⠀⢀⣾⠏⠀⣿⡇⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⣆⣿⠇⠀⠀⠀⠀⠀⣴⡟⠁⢸⣧⠀⠀⠀⠀⢀⣾⠏⠘⣿⡄
⠀⠀⠀⠈⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⢀⣠⣴⢦⣤⡴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣋⡀⠀⠀⠀⠘⠻⣦⣘⣿⣣⣴⠟⠁⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⢀⠀⠀⢀⣹⣿⣁
⠀⠀⢀⣼⣿⣯⡀⠀⠀⠀⠀⣀⣤⡾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⡅⠀⠀⠀⠘⢷⣶⣿⣿⣿⡏
⠀⢀⣾⣿⣿⣿⣷⡄⠀⠀⠈⠉⠁⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⡇⠀⠀⠀⠀⠀⠀⠉⣿⣿⡇
⠀⠀⠹⣿⡟⣿⡟⠀⠀⠀⠀⠀⠀⢀⣿⢻⣿⠀⠀⠀⠀⠀⠀⢠⣿⠉⣿⡄⠀⠀⠀⠀⠀⣿⡟⣿⣾⠃⠀⠀⠀⠀⠀⠀⢸⡿⢸⡇
⠀⠀⢠⣿⠃⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⡇⠀⠀⠀⠀⢠⣿⠇⢹⡇⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⣷
⠀⠀⠿⠃⠀⠹⠇⠀⠀⠀⠀⠀⠀⠘⠟⠘⠿⠀⠀⠀⠀⠀⠀⠸⠇⠀⠸⠇⠀⠀⠀⠀⠿⠃⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠿⠂

⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⣿⣿⣩⡿⠇⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢴⣿⡆⠀⠀⠀⠀⠀⠀⠐⣿⡿⢠⣄⠀⠀⠀⠀⠀⠀⠘⣿⡟
⠀⠀⣰⣿⣿⣟⠋⠀⠀⠀⠀⢀⣴⣿⣿⣿⢷⣄⠀⠀⠀⠀⠀⣰⣿⣿⣿⣄⠀⠀⠀⠀⠀⢠⣾⣿⣿⡟⠷⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠈⠻⢿⣿⣿⠀⠀⠀⠀⠀⣾⠃⣿⣿⣿⣠⡿⠂⠀⠀⠀⠘⣯⣿⣿⣿⣿⠂⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟
⠀⠀⠀⣿⡏⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⢘⣿⢻⣿⠁⠀⠀⠀⠀⠀⠸⢻⡿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠂
⠀⠀⠀⣿⡇⣿⡇⠀⠀⠀⠀⠀⢀⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠇⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⣿⠁⢻⠇⠀⠀⠀⠀⠀⢸⠏⠸⡟⠀⠀⠀⠀⠀⠀⠀⠈⢿⠼⡿⠀⠀⠀⠀⠀⠀⠀⢿⠇⢿⠇⠀⠀⠀⠀⠀⠠⡿⠃⠹⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠊⠉⠉⢫⣍⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠈⣷⡌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡀⠀⠀⠘⢿⡇⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⢤⣄⢻⣿⣿⡗⠒⠀⠀⢨⣿⠀⢊⡷⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠿⠀⠀⠈⠁⠀⠀⢰⡟⠁⠀⠈⡿⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠆⠐⠀⠂⠀⠀⠲⣼⡇⠀⠀⢸⣶⣯⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢀⣤⣤⢤⣤⠀⠀⠘⡇⠀⠀⢸⣿⣹⠟⢁⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠠⠶⠆⠀⠀⠀⠀⠀⢀⡴⢛⣽⠃⣠⡾⠝⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⣀⣠⣤⣴⣯⣴⠾⢳⣿⠟⣛⠏⡠⣤⣶⠾⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢿⣿⣿⣿⡟⢋⣀⡴⢋⠀⢀⣩⣷⠟⣀⣽⣷⣤⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣴⡾⢋⣥⡾⠒⣋⡟⡠⢃⣴⡿⠛⠀⠉⠀⠀⠀⠀⠀⠈⠉⠉⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢺⣷⣏⡠⢚⠛⠀⣼⣻⣾⡿⠃⡘⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣒⣭⣴⠟⡩⠋⣰⣿⣄⠀⠁⣼⠟⢁⡴⠃⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⣴⣿⡿⣋⣼⢃⣼⣿⡏⠋⠀⣼⣷⠶⠟⠁⠘⣄⠻⣄⡀⠀⢀⡤⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠐⢩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣶⣿⡟⣰⠏⣻⢼⡿⢋⣜⣦⣼⢿⠁⠀⠀⠀⠀⢻⣦⡀⠙⢦⡆⠀⣀⠆⠀⠀⠀⠀⠃⠔⠂⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠞⢼⡟⢹⠁⡿⠚⣋⣼⢡⠿⣿⣿⣿⠈⡇⠀⠀⠀⠀⢸⣿⣿⣶⡾⠿⠛⠛⠛⠲⢦⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⡼⢃⢐⣋⡴⠛⢶⡇⠀⣿⡇⢸⢠⠤⣿⠇⢀⣇⠀⣀⣠⠄⢸⣿⠟⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢸⠃⣸⣯⣽⣅⣰⣿⡇⠀⣿⣧⢸⠃⢀⡌⣠⣿⣟⣋⠉⠀⣠⣾⡇⣸⡀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⣀⢸⣄⣿⣯⣤⠀⠙⢿⣧⣶⡟⣿⡿⠀⣾⣿⣿⣯⣭⣥⣶⣾⣿⣿⡇⢿⣧⠀⠀⠀⠈
⠀⠀⢀⠾⢴⣿⠿⠋⠉⠀⠀⠀⠀⠙⣿⣇⣿⡇⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⣿⣦⣀⣤⡴
⡐⢊⣡⣶⣼⠁⠖⠲⠶⠛⠋⠁⠐⠒⠸⠿⣧⣼⣤⣿⣄⢹⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢠
⡇⢈⣡⢴⡏⠀⠀⣒⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣛⠛⠻⢿⠟⠉⠁⠀⠀⠀⠐⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⣆
⢱⠉⢀⣬⣷⠀⠉⠁⠀⠀⠈⠉⠉⠉⣂⡴⠂⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⣀⣀⠈⢂⠀⠀⠀⠀⠀⢀⣾⠁⠀⡇⠀⢀
⠈⣦⠈⣁⣸⡼⣧⣴⣛⣁⣤⣴⣶⣿⣿⣦⣄⣠⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⣿⣿⣿⣿⣿⠿⠁⠀⢰⠃⠀⣼
⠀⠙⠲⢾⣿⣷⠌⣉⣭⣭⠕⢲⣶⡶⠭⣛⡻⢿⣿⣷⣶⣶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⣟⠋⠁⠀⠀⢀⡞⠀⣸⠇
⠀⠀⠀⢰⠁⣾⠀⣿⡏⢷⠶⡾⣿⡇⠀⢿⣿⣷⣬⣍⡛⠿⢿⣿⣷⣶⣶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣡⣤⣴⠟⠀⣰⠏
⠀⠀⠀⣿⠘⡟⢻⠏⣷⣄⣳⡄⠝⡄⠀⡿⢁⣾⠃⣼⡟⠳⢦⣬⣉⣛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣇⣀⣰⠏
⠀⠀⠀⢻⡀⠳⠞⣠⢿⣹⣿⣷⠀⡇⢰⡇⢸⡟⢻⣿⡇⠀⠀⠈⠙⠻⠿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠿⠿⠿⠿⠿⠟⠉
⠀⠀⠀⠀⠙⠒⠋⠁⣸⠋⣹⡟⢠⣧⠈⣇⢸⡷⠾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣿⢶⣿⢸⠀⣿⠈⣷⡄⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿⣼⣿⢸⢠⣿⣿⢿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣿⡀⢹⡟⢸⡏⣿⢸⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣾⡇⠈⡇⠘⣇⠻⠿⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⢰⡏⣿⢹⡇⢠⡇⠀⠙⠦⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣇⢻⣾⠇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣁⣠⠟
⣶⣶⣶⣶⣶⣶⣶⡶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣯⣭⣭⣽⣭⣴⣭⣽⣯⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣴⣶⣶⣶⣿⣿⣧⣯⣤⣽⣶⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣦⢱⡀
⠀⠀⠀⠀⢀⣤⣸⣿⣿⣿⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⢢
⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⢸⡏⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣿⣿⣿⣿⡎⡇
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢹⣿⣿⣿⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠈⠻⣿⣆⡀⠀⠀⠀⢠⣿⣿⣷⡄⠰⠃⡤⢿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣜⠦⢀⣀⣸⣿⣿⣿⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣾⣯⣿⣿⣿⣧⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢿⣿⣿⣿⣿⣇⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣡⣿⣿⣿⣿⣿⣿⡿⣻⣽⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠋⠉⠉⠉⢉⣼⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⢰⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣟⡽⢛⡵⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣽⣿⣿⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣫⣞⡿⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣿⣿⣿⣯⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡞⢡⡞⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⢿⢏⠏⠔⣹⠋⣿⣇⠀⠙⣿⣯⠻⣿⣿⣿⣿⣿⣿⣿⣿⣏⡴⢋⡼⣣⠞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡾⢩⢋⠊⠎⠊⠀⠀⠉⠀⠈⢻⣦⡘⠘⣿⣷⣿⢿⣿⣿⣿⣿⣿⣿⣿⣱⢏⡼⣣⠞⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣹⠻⠃⠃⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢘⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣶⢋⡜⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⡆⠽⣿⡇⢸⣿⣼⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⢃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢾⢸⢰⢠⠹⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣃⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠘⠷⠆⠀⠀⠀⠀⢀⠀⡄⣴⣼⢠⢸⢸⢸⣻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠾⣁⣴⣿⣧⣤⣼⣼⣾⣼⣿⣿⣿⣿⣿⡿⢺⣿⣿⣿⣿⣿⣿⡿⠿⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣡⡏⠀⢸⣿⢿⣏⣩⣿⣿⣿⡿⢁⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⡿⢻⣿⣧⠀⠈⢿⣿⣮⣉⣉⣻⣿⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠁⣾⠧⣻⣦⠀⠀⠉⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠄⠀⠀⠀⠏⢀⢧⣛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢀⡾⠀⠀⠀⠀⠀⠠⣿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠘⣽⡀⠀⠀⠀⠀⢠⣟⣿⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠁⠀⠀⠀⠀⣸⣿⡟⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢹⡄⠀⠀⢀⡜⡻⣻⣱⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠃⣰⡠⢊⠞⡵⠁⣵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⢀⡏⠰⢣⠊⡜⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢸⠇⠀⣡⠊⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⣰⢟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⢰⠀⡯⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⠀⡇⠀⡰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠆⢰⠁⢘⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋⠀⠐⠀⠋⠀⣾⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⣫⠓⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⢹⣧
⠀⣀⣀⣀⣠⣴⣾⡟⠻⠋⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠰⠂⠙⡃⠘⣿⡆
⠘⣯⣉⣉⠉⠉⠁⠀⠀⣀⣀⣤⣶⣿⡷⠿⢿⣯⣽⣷⣦⣤⣌⣀⣰⣿⡧
⠀⠀⠉⠉⠛⠻⠷⠷⠾⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠋⠁

⠀⠀⠀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⠀⠀⠀⠀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⣤
⠀⢀⣸⣿⣿⣿⣷⣀⣄⣀⣀⣀⣀⣠⣤⣤⣤⣤⣀⣠⣄⣰⣿⣿⣿⣿⣀⡀⢀⣰⣿⣿⣿⣿⣠⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⢤⣿⣿⣿⣿⣄⡀
⠀⠈⠹⣿⣿⣿⣿⠈⢿⣤⡀⠀⠀⣀⣀⠀⠀⢀⣤⡾⠃⠸⣿⣿⣿⣿⠀⠀⠀⠈⣿⣿⣿⣿⠀⠻⣦⣄⠀⠀⣠⣄⠀⠀⢀⣴⡾⠃⠀⣿⣿⣿⣿
⠀⠀⠀⠉⠉⠉⠁⠀⠀⠙⣿⣦⣄⣿⣿⣀⣴⣿⠋⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⢿⣷⣤⣿⣿⣤⣴⣿⠏⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⠁⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠅⠀⠀⠀⠀⠋⠁

⠀⢠⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡈⠁⠉
⠀⢈⣵⣷⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁
⠀⠘⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣹⢗⡠⠐⣾⣆⢀⡄
⠀⠀⠘⢿⣿⣻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⣧⣼⣛⣻⣿⣛⡿⠁
⠀⠀⠀⠀⠻⣿⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢻⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣵⣄⡀⢀⣀⣀⣤⣄⣀⣠⣾⣿⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣟⠹⣿⣿⡏⢻⣷⠝⣉⣭⣭⣭⠯⣥⣶⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⣼⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣙⡟⢁⣴⣉⣉⡷⢤⢀⣿⣟⣀⠛⠛⣟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⣽⣿⣧⡿⠀⣾⢧⣨⣿⣷⣿⣿⠛⣿⣿⣿⣧⣿⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃⠛⠦⠰⣿⣽⣿⡟⠻⣾⣻⣿⡿⣟⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢟⣠⣿⡷⣶⣶⣾⣿⣴⣿⣿⣿⣿⣷⠈⢻⣏⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣸⠍⣯⠙⠞⠟⣛⣻⣾⣿⣿⣿⡿⠋⣆⢸⣿⣿⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⣷⡘⣛⣉⡩⣿⡿⣿⡟⠁⠀⠘⢿⣿⡿⢻⣞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠀⢻⣿⣧⡬⢭⣿⡿⠿⠋⠀⠀⠀⠀⠈⠹⣷⠸⣿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣿⠀⢨⣹⣿⣿⣿⣿⠀⡟⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣿⠀⠸⣅⣿⣿⣿⡷⢺⡅⠀⠀⠀⠀⠀⠀⠀⡟⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡆⠛⠀⢷⣿⡿⢛⠉⢀⠛⢸⡀⠀⠀⠀⠀⠀⠀⡇⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠒⠚⣫⡉⣾⡇⠠⡔⠿⣿⡴⢿⣶⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⣷⣿⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣯⣡⣴⣶⣿⣿⡿⠃⡴⠓⠄⠐⢙⡇⢘⣿⢿⠠⠿⣿⡇⠀⠀⠀⠀⠀⣿⠇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠛⠉⣯⣿⠟⠀⠀⠀⠀⠀⠀⠘⠀⡈⢣⡌⠀⠀⣿⡇⠀⠀⠀⠀⠀⠿⠆⠙
⠀⠀⠀⠀⠀⢀⣴⣻⣿⣿⣿⣿⣿⣷⣧⣾⠿⣷⠿⣷⣶⡟⠲⣶⡆⠀⢦⢱⣿⠀⠀⢻⡟⠀⠀⠀⠀⠀⠘⢀⡀⣤⢂
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣋⠿⣿⣿⣧⠈⠓⠉⠀⠀⠀⣿⠁⣠⡿⠉⠉⠠⢠⡟⠀⠀⠀⠀⠀⠀⠰⠋⠻⠾⢻
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣥⡾⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠰⣿⢰⣿⠀⠀⠰⣧⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠘⠻⢟⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡿⢠⠀⢠⣿⡩
⠀⠀⠀⠀⠀⢹⡟⣿⣿⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡟⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠘⣇⢹⣿⡇⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡆⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢻⡘⣿⣧⣆⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⣿⢿⣿⣼⡇⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢯⣿⣿⡿⢟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠟⣸⡧⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠘⢹⢿⣷⠸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣠⡿⢳⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢻⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⠶⢎⣿⣭⣾⡟⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⣠⠖⣚⣻⣿⣾⣿⣿⣿⣿⣍⣾⣦⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟
⣷⣾⠻⠛⠛⠉⠉⠉⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟
⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣭⠁
⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣸⣇⣿
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢶⣈⣣⣤⣿⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠉⢋⣾⣧⠾⠛⠛⠷⠶⠶⠖⠒⠒⠒⠶⢶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠒⠀⠈⠁
⣶⣷⣾⣷⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⣾⣶⣾⣶⣶⣶
⣿⣿⣿⡏⠀⠉⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠉⠉⠉⠉⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⣄⣀⣀⣀⣤⣤⠤⠤⠤⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⣀⡬⠂⣀⠤⠀⠒⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⣠⠔⢊⣡⠔⠋⠀⠀⠀⠀⠀⠀⠈⠑⠒⠢⠤⠤⠤⠤⠤⠤⠔⠒⠒⠒⠒⠀⠐⠒⠒⠢⠤⠤⠇⠀⠿⠀⠀⠀⠀⠀⠀⢀⡇
⡰⠋⠀⡔⠋⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⣸⠁
⡇⠀⡈⡀⠐⠂⠀⠉⠉⠀⠀⠀⠀⠈⠉⠒⠢⠠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⢀⠁
⡇⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠒⠂⠤⠤⢀⣀⡀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⢠⠃
⠘⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⢒⡇⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡎⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢻⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢧⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠒⣄⠳⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢆⠙⢄⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠈⢣⡀⠙⠢⠭⠷⠒⢦
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣙⣢⣤⣤⣤⣴⣂⣁⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⡀
⠀⠀⠀⠀⠀⠀⠀⣘⢙⢙⣃⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠠⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⢛⢙⣘⡃
⠀⠀⠀⠀⠰⠶⠶⣿⣾⢸⣿⣿⣬⡭⠀⠰⠄⠤⠀⠤⠤⣤⠤⠛⠆⡠⠀⠀⠰⠂⠤⠠⠤⠀⠤⠀⠈⠈⣭⣾⣿⣿⣿⣿⣷⠶⠶
⠀⠀⠀⠀⠀⠀⠀⢿⣼⣿⣿⣿⣿⡟⢿⣷⡤⠀⡤⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠑⠀⣀⠀⢺⣿⠛⣿⣿⣿⣿⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠁⠀⠑⢤⣈⣤⡤⠾⣄⠀⠀⢀⣇⡀⠀⢀⡴⠛⠳⠴⠦⠔⠋⠀⠀⠙⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣬⣿⣿⣿⣿⣿⣿⣿⣧⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣻⣯⣻⣿⣷⣿⣼⣿⣭⣾⣭⣿⣻⣙⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣿⣿⣿⣿⣿⡿⠉⠉⠉⠁⠈⠁⠈⠉⠉⠙⣿⣿⣿⣿⣿⡿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣿⣿⣯⣽⡏⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⡿⢀⣄
⠀⠀⠀⠀⠀⠀⣠⡿⠋⠉⠳⡀⠀⠀⠀⣻⣿⣿⡆⠀⢸⡀⠀⠀⢀⡀⣀⣤⠛⠙⠛⢶⣿⣿⢸⣧⣸⣿⣿⠆
⠀⠀⠀⠀⠀⣰⣿⡇⢀⣤⣠⣷⣶⣶⣭⣽⢀⠀⢠⣄⣸⠛⠛⠛⠛⠋⠉⠉⠛⣿⡆⢠⠈⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠘⠛⣿⣿⣾⠏⢨⡇⢀⣴⠋⠁⠀⠀⠀⠹⠧⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⡇⠀⠈⠻⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠛⣴⡿⠃⢀⡖⠀⠀⠀⣀⠀⠀⠀⠀⢦⣀⡀⢀⣼⣷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢠⡁⠀⠀⠀⠉⠉⠃⠀⠀⠈⠣⠄⠙⠃⠈⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⢤⠀⠀⠀⠀⠰⣶⣶⣄⣀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠰⣿⠋⣸⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⡄⠀⠀⠀⢀⡼⠋⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⣀⠀⠉⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣤⣌⣁⣠⣤⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⢿⣿⡿⠟⠛⠛⠛⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠹⡇⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⢠⡇⠀⠀⠀⠀⣈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⣠⠋⢹⣄⡀⠀⢀⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣀⠁⠀⠀⣿⡧⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠏⠀⠀⠀⠇⠀⠀⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⣀⠀⠀⠀⡘⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢰⠇⠀⠀⠀⢷⡔⠆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡾⠀⠀⠀⠀⠈⢿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣵⣄⡀⠀⠀⠀⠸⡇⠀⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠟⠛⠁⠀⠀⠀⠀⣿⣤⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⡇⠐⢸⠻⣿⣿⡙⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⢻⡇⠀⢸⠄⠘⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠃⠹⣿⣾⣇⠀⠀⠀⠀⠘⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢠⣿⡿⣿⠀⠀⡆⠀⠀⢿⠙⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠚⠉⣿⣼⡄⢀⡛⠔⢂⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⡼⠋⠻⠇⠘⠁⠀⠈⠀⠀⢫⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠊⠀⠀⠀⠀⠀⠀⣀⢠⡄⢠⡬⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣏⠀⠀⠀⠀⣠⡴⠞⣅⠀⡃⢘⠃⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣝⠀⡀⠀⡾⠁⣇⠠⢏⠘⠇⢸⣷⠘⢿⣿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⡳⠉⣸⠇⢸⢾⡄⠸⠃⠀⠸⢻⡄⣷⣝⢿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠔⣠⣾⡏⠀⣾⢸⠀⠀⡄⠀⠠⠋⠁⣿⠉⠀⠀⠩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⠁⠈⢠⡿⠟⠀⠀⣿⢸⢰⡇⢰⡖⣧⠀⣾⣿⡇⠀⠀⠀⣨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡗⠁⠀⣸⣿⠟⠀⠀⢰⡿⠀⢾⣇⠀⣷⡿⠀⣿⣿⣇⣈⣶⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⢀⣴⣿⣷⠀⠀⠀⢸⡏⠀⢸⣿⡄⢹⣷⠀⣹⣿⣿⣿⣿⠀⠀⡈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⢠⣿⣿⡿⠃⠀⠀⠀⢸⡇⠀⣸⣿⡇⠈⡷⠀⢿⣿⣿⣿⣿⠃⠀⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠌⠀⠈⢈⣋⠀⠀⢀⣶⣆⢸⡇⠀⣿⣿⣧⠀⢁⠀⢸⣿⣿⠏⠙⢦⠀⠀⡸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⢻⠀⠀⠀⠈⣿⡿⠀⠇⡆⢹⣿⣿⡆⠘⣦⠼⣿⣿⠀⠀⠀⢁⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⣀⢈⣤⠁⠀⠀⣾⠀⡍⠀⠀⠀⡇⢸⣿⠇⠀⠀⣧⠀⢻⡟⠀⠀⠀⠀⠀⠀⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠀⠀⠀⣰⠀⣼⡇⠀⠀⠀⡟⠀⡿⠁⠀⠀⠘⠀⣾⠃⠀⠀⠀⠀⡀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢰⠀⢠⣿⡞⠁⣠⣠⠀⢰⡇⢠⣧⡀⠀⠀⠂⢠⡟⠀⠀⠀⠀⠀⠰⠄⠠⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡈⠀⢸⡿⠀⠀⣿⠡⠀⢸⣿⠏⠀⢻⠀⠀⠹⣾⠁⠀⠀⠀⠀⠀⠀⢠⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⡀⠁⠀⣸⠃⠀⠀⠉⠀⠀⣿⡏⠀⠀⠸⡆⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠘⠀⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⠏⠀⠀⠠⠇⠀⠀⢿⠀⠀⠀⡀⠀⠀⣿⠇⠀⠀⠈⠁⠀⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⢀⡞⠀⠀⠠⠶⠀⢀⡿⡄⠀⠀⠸⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⢀⡟⠀⠀⠀⣧⠀⠀⠀⣱⠂⠀⠀⢠⡀⠀⠀⡼
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⡇⠀⠀⠀⠘⠇⠀⢠⡟⠀⠀⠀⠈⠧⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠌⠀⠀⠀⠀⣼⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⣧⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠰⠄⣼⠃⠀⠀⠀⠀⠀⠈⢀⡇
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢸⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉
⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢢⣈⠋⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠻⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⢀⣴⠟⢡⣾⠟⣹⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠈⢷⡀⢠⡟⠁⣴⠟⢁⡶⢋⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢳⡿⢤⣼⠃⣰⠟⣠⠟⣱⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢿⡀⠈⠙⢦⡉⠛⢧⣴⠏⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⡇⠀⠀⠀⢱⡞⠳⣝⣾⠃⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢸⡇⠀⠀⠀⠸⣇⠀⢻⡄⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢘⡇⣤⡀⠀⠀⠹⣆⠀⢷⡀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠈⠙⠛⠶⠆⣿⠀⠀⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡎⢷⡀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣯⡟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠘⠃⠘⠃⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⡟⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⠤⠚⡄⠀⠀⠀⠀⠀⠀⠉⠐⠢⡀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⣠⠊⠁⠀⠀⠈⠀⣀⣀⣌⡀⢀⣀⢀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⠀⠀⠰⠁⠀⢀⡠⠔⠂⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠰⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠄⠐⠁⠀⠀⠂⡠⠚⠁⠀⣀⡤⠖⠚⠋⠉⠉⠉⠉⠉⠚⠒⠢⢄⡀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠂⠰⡠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠐⠚⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⣠⣴⣒⠘⣿⣿⠿⡷⡒⠚⠉⢸⠈⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡖⢂⠓⡆⠀⠖⠈⡻⠋⠋⢇⠀⢸⡀⠀⠁⠀⠀⠀⠈⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡆⠀⠀⢧⠀⠀⠀⠀⢄⡤⠹⠀⠀⠃⡙⡗⠀⠀⣀⡰⡇⠁⡎
⠀⠀⠀⠀⡀⠀⠄⠐⠁⠀⠈⢑⠦⢀⠘⢆⠀⡘⣆⠀⠀⡀⣊⠧⠲⠄⣀⠜⠁⠁⠙⠀⢸⠀⢥⡤⠅
⠀⠀⢠⡎⠀⠀⠀⡏⡠⢤⣀⠀⢹⠀⠑⡈⢣⠀⢹⠉⢏⠀⠁⢀⠔⠒⠛⢉⣩⣽⣶⡄⠀⠀⢸
⠀⡔⠈⠃⠀⠀⠀⡟⠁⠀⠀⠀⢹⠀⠀⠀⠄⠳⠴⢧⠈⠀⠀⣿⡿⠛⠛⠋⠉⠉⠙⢇⠀⠀⡏
⢀⡀⠀⠀⠀⠀⠀⢣⡀⠀⠀⣠⠏⡀⠀⠀⠘⡀⠈⠈⢦⠀⠀⢋⡁⠠⠀⢀⡀⠀⡠⠙⠀⡾
⡎⠈⠀⠀⢀⠀⠀⠀⠈⠉⠻⡛⣎⢁⠀⠀⠀⢰⠀⠀⠀⠑⢤⡈⠈⠒⠒⠒⣒⡋⠀⠀⠸⠀⠑
⢣⠀⠀⡀⠀⢆⠀⠀⠀⠀⠀⠉⢸⣏⡄⠀⠀⢀⠀⠀⠀⠠⠀⠙⢆⠀⠀⠉⠀⠀⢀⡠⠂
⠀⠳⡀⠱⡀⠈⢄⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠉⠁
⠀⠀⠘⢆⠱⣀⠀⠉⡳⠚⠋⠀⠀⠀⠀⠀⢀⡐
⠀⠀⠀⠀⠑⠺⠙⠉⠀⠠⠀⠀⠀⠒⠀⠉
⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⣄⠀⢀⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣹⣿⣿⡂⢸⣿⣿⣽⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⣿⣿⠃⠈⣿⠃⠛⢯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠀⠸⣿⣇⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠀⠀⠀⠀⣿⣿⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠀⣿⠹⣿⣿⣿⡸
⠀⠀⢀⣀⣤⣄⣠⣀⣀⣀⠀⠀⢀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣦⡈⢻⣿⡇⠠⡄
⠀⢺⣻⣿⣿⣿⣿⣛⠋⠉⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⣌⣿⢶⣿⣾⣄⠳⣦⣤⣴⣶⡆⢠⣴⣶⡿⠛⠁⢀⡴⢦
⠀⢿⠛⣋⣽⣯⣽⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⢾⣿⡇⠙⢿⣿⣷⡷⠴⢖⣩⣿⣿⠟⠋⠀⣠⡔⠂⠀⣾⣧
⠀⠾⠿⣿⣯⡅⠀⠈⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣮⣿⡓⢠⣰⣿⣿⣿⣳⣿⡿⠛⢁⣄⣀⣔⠈⠀⠀⠐⣿⣿
⠀⠀⠀⠀⢹⣿⡶⠀⠀⠙⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⢸⣿⡇⠠⣿⣿⠃⣿⡉⢁⣴⣾⣿⣿⣿⣿⡀⠀⠀⣿⣿⡟
⠀⠐⢶⣶⡿⠿⠦⣤⣤⣤⣤⣿⣿⣄⠀⣀⣀⣀⠀⢀⣀⣠⣀⣀⠀⢈⣽⣾⣿⣿⣶⣦⣿⠿⣿⣷⣿⣿⢿⣻⣿⠟⢹⣷⣤⡛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠟⠙⠏⢻⣦⠀⠉⠁⠀⠀⠈⠈⠉⠉⠀⢀⣿⣿⣿⡟⣿⣿⣿⡆⣿⣿⣿⣿⡿⠟⠁⢀⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠁⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⢻⢫⣿⣿⡇⣽⣿⣿⣟⠀⢀⡼⣫⣾⣿⠿⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢈⣼⡿⣟⡏⣼⡟⢸⣿⣾⣿⣿⡿⢾⣯⣾⣿⠏⠀⠀⠀⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣿⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣈⣿⢡⡟⡜⣿⡇⠈⢟⠿⢻⣿⣇⢹⡿⣿⡇⠀⢀⣀⣼⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢬⠻⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⡉⢼⣾⣷⡽⣿⠃⣤⣠⣾⣿⣧⣹⣿⠁⠸⣧⣰⣿⠿⠟⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⢣⡈⢯⠻⡻⣷⡀⠀⠀⠀⠀⣉⣹⣿⣿⠿⠛⠉⣀⣽⣿⣿⣝⣿⡟⠀⠀⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢱⠈⠣⡜⣎⠳⡀⠀⠀⠀⡉⠻⡋⠁⠀⠀⠛⣩⣴⣿⣟⣿⣹⡇⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠐⠈⠂⠙⣆⠀⠀⣉⠘⡇⠀⠀⣤⣿⣿⣿⡛⣿⣹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⡉⠀⡇⠀⢀⡼⣿⣿⣿⡇⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠳⡈⠻⢄⣧⠀⡇⠀⣠⣶⣿⣿⣿⣷⣿⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠄⠀⠁⠀⢈⡗⢰⣷⡿⠋⠀⡟⣿⢸⣿⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⢠⡄⠀⠈⠀⠀⠀⢀⠁⠊⠹⡇⠀⣷⢡⡿⢸⣿⡟⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣧⣦⣷⠀⠀⠀⠀⠀⠘⠀⠃⠀⠀⢶⣿⣸⡇⠸⣿⣧⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⣿⠰⣿⡿⠛⠄⠀⠀⠀⠀⠀⠀⠸⣄⠢⡈⢻⣿⡅⢠⣾⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⡿⡄⠀⠀⠀⠀⢠⠀⠀⠀⢿⡿⣿⣿⣶⠓⠒⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢁⣼⠟⠉⠙⠸⣿⣆⠀⠀⠀⠀⠀⠀⢠⠎⣠⣾⢿⣟⠳⣤⣀⡨⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡟⠁⠀⠀⠀⠀⠙⢻⣄⡀⠀⠀⠀⢀⣾⣜⡉⠀⠈⠀⠁⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆⣀⢀⣾⣿⣯⡇⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠐⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣯⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠐⣹⣻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣆⠀⠀⠀⠀⠀⠀⠀⡰⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⢀⡀⣤⣦⣻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡼⠀⠀⡠⠀⠀⢀⡔⣱⣿⣿⣾⡟⣿⣿⣿⣿⣷⣷⣦⣪⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣰⣾⣷⣤⣴⡞⣴⣻⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠖⠒⠒⠒⠒⠒⠚⠛⢛⡛⠓⢛⡩⠽⠶⢶⣦⣄⣀⡤⠤⠴⠖⠒⠒⠒⠒⠒⠒⠶⠤⣤⣤⣀
⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣴⣶⣿⡷⠊⠁⠀⢄⡠⡀⠈⡙⢧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠢⡀
⠀⡾⠁⣰⣦⠀⠀⠀⠀⠀⠀⣀⣴⣶⠶⠞⠋⢹⣿⣄⡀⢠⣽⣦⣿⣿⣼⣿⡌⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⢸⡇⢀⣳
⣸⠃⠀⠛⢿⣄⢤⣤⣤⡴⠾⣋⡴⠃⠀⠀⠀⢸⣿⠟⠛⠻⠿⢿⣿⠿⠟⢹⣿⣿⣉⠉⠉⠛⠿⢶⣦⣄⣀⣀⣠⠾⠃⣼⣷⠘⣿
⡿⠀⠀⠀⠘⣿⢷⣶⣶⣶⣿⣧⡀⠀⠀⠀⠀⣼⣿⡀⠀⠤⢤⣀⢀⣠⠤⢐⣿⣿⣿⠀⠀⢀⣴⣿⣿⣿⣍⠉⠀⢀⡾⠛⣿⠀⢸
⣷⠀⢀⠀⠀⠘⣇⠀⠈⠙⢿⣷⡙⣆⠀⠀⠀⣿⣿⡇⢚⣒⣦⡈⠉⣤⣖⣻⡏⣿⣿⠀⠀⣼⣿⣿⠃⠙⠞⠇⠐⠋⠀⠀⣿⠀⣸
⠸⡆⢸⠀⠀⠀⠘⠂⠀⠀⠀⢿⡇⢹⡷⢦⣀⠹⣿⣇⠀⠭⠝⠁⠀⡟⢧⠈⣁⣿⢃⣠⠶⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⡏
⠀⢳⠘⣇⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⣿⣦⡙⢧⣿⣿⠀⢀⠀⢤⣄⡥⠀⣴⣿⣿⡟⣡⣴⣾⣿⣷⠀⠀⠀⠀⠀⠀⢀⡞⢀⡞
⠀⠈⢧⠹⣦⡀⠀⠀⠀⠀⠀⣼⣰⣿⠋⠉⠙⢿⣿⣿⡀⣿⠖⠒⣖⡒⠂⣸⣿⣿⣿⡿⠛⢻⣿⣿⡇⠀⠀⠀⠀⠀⠘⣠⡟
⠀⠀⠈⢳⡘⢿⣶⣤⣀⣀⡀⢛⣿⠇⠀⡄⠀⠀⠙⢿⣿⣿⣆⠈⠉⠁⣠⣿⣿⠟⠉⠀⠀⠀⣿⢹⣧⣠⡴⠂⠀⣴⡾⠋
⠀⠀⠀⠀⠙⢦⡈⠛⢉⣽⣿⣾⣿⠀⠀⢸⡄⠀⠘⢦⠈⠛⠿⣿⣿⣿⠟⢋⡴⠃⠀⠀⠀⣼⠃⢸⣿⡿⠆⠀⢀⡏
⠀⠀⠀⠀⠀⠀⢻⡀⠈⠀⠈⠻⣿⡆⠀⠀⠹⣄⠀⠈⠳⣄⠀⢸⡿⢁⡴⠟⠀⠀⠀⢀⡾⠁⠀⣼⡏⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣿⣿⡀⠀⠀⠘⢦⡀⠀⠘⢦⣸⣧⠞⠁⠀⠀⠀⣰⠏⠀⠀⣰⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠹⣄⠀⠈⣿⡏⠀⠀⠀⠀⣼⢃⣠⠀⢠⣿⣿⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢀⡿⣿⣧⠀⠙⠶⠀⠀⢹⣇⠀⣿⡇⠀⠀⠀⣸⠇⠘⠉⠀⠀⠻⢿⣇⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢤⣘⢿⡆⠀⠀⠀⠀⢀⣿⣾⣿⣷⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠛⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠠⣄⠈⠙⠻⢦⡀⠀⢀⣼⣿⡿⢛⣛⣒⠀⢀⡘⣷⣄⣠⣧⠶⠂⡀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣈⠉⠲⡖⠂⠙⠃⠈⠉⠁⠈⠙⣿⠋⢉⡀⠉⠉⠉⠁⢿⡶⠊⡀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⢉⣿⠛⢦⣀⠀⠀⠀⣴⠁⣠⣿⣀⣀⣿⣦⣀⡀⠀⠉⠉⠿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠙⢦⡀⠉⠛⠒⠒⣛⠛⠛⣿⠉⠉⢻⣉⠁⠀⠀⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠙⠷⣤⣀⠐⣯⣀⣠⣿⣀⣀⣤⣿⣠⠴⠚⠁⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡀⠀⠉⠉⣉⠉⠉⣟⠁⠉⢠⡄⠀⠀⢀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⢦⣀⣐⣧⣄⣴⣿⣦⣤⣼⣿⢀⡴⠋⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⣷⠀⠀⠈⠉⠉⠉⢉⣿⡇⠈⠉⠉⠁⠀⣼⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣤⣄⣀⣀⣀⣠⣾⣿⣷⣄⣀⣀⣤⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣶⣿⣿⢧⣿⣿⣿⣿⣦⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⣼⢱⣿⣿⣿⣼⠟⠉⠙⣿⣿⣷⡀⢻⣦⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⢀⡿⣾⣿⣿⣿⠏⠀⠀⠀⢘⣿⣿⣷⡀⠹⣯⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⢸⡇⣿⣿⠿⠇⠀⠀⣀⠀⠿⠿⠿⣿⣿⡄⠹⣯⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⢀⣼⠟⠛⢀⣀⣴⣶⣿⣿⣿⣶⣄⣀⠐⠛⢿⣄⡹⣏⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣶⣌⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣴⠏⢿⢿⣿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⢿⡏⢿⣌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⡀⠘⣆⢻⣾⣿⣿⣿⣿⣿⣿⠋⠉⠙⢿⣿⣿⣿⣿⣿⣼⠏⡼⠀⠸⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣶⣧⡀⢹⣮⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣾⠃⣠⣾⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⡙⠈⡟⢹⣿⣿⡿⢋⡄⠀⠀⠀⠀⠀⣈⠻⣿⣿⣿⡇⡟⠘⣉⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠼⣼⡼⡿⠿⠟⠃⠀⠀⠀⢀⠁⠀⠁⠆⠘⠿⠿⣴⣳⠇⢻⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣿⢡⣠⣀⡀⠤⡀⠀⠚⠛⠞⠋⠐⢧⡰⢦⡄⣀⡀⣰⡧⠟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⣿⢿⠁⠀⠠⠄⠀⣠⣾⣶⣄⠀⠀⠀⠀⠙⡟⠈⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⡀⠀⠂⢀⡄⢀⣁⣀⠈⣀⠀⣸⣆⣨⡄⠀⡠⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣅⡀⠀⣿⠋⢣⠴⠇⡽⢻⣿⣿⡿⣿⣷⡤⣣⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠉⠙⠿⢿⣈⡿⢿⣼⣟⡹⠛⠡⣄⡈⣛⣴⡏⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠈⣱⡟⠓⣪⡅⠀⠀⠀⠉⠀⢻⣿⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⡀⠀⠀⣠⠾⣿⡇⠀⢿⡇⠀⠀⠀⠀⠀⢸⡿⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⢿⣿⣄⠈⠳⣄⠀⠀⠀⢀⡾⠃⢠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣄⣄⣉⢉⠈⣉⣤⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠉⠉⠁⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⡀⠀⠀⠀⠀⢀⣿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠟⢁⠀⣀⣀⠀⠈⠻⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⣿⡿⠟⠻⣿⣿⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⡀⢀⣀⡀⢀⣈⡑⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠘⡟⠋⠈⣻⣟⠁⠛⢻⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡏⣾⣯⣙⣋⡝⣷⢻⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣿⣷⡘⠟⠛⠛⠿⢃⣿⣿⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠂⠻⠋⠀⢩⣿⣷⣦⣤⣤⣴⢾⣿⡝⠀⠙⠻⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⠮⠛⣻⠿⠛⠷⢦⣤⡖⠲⠾⠿⢿⣿⡘⣿⣿⢃⣿⡿⠿⠷⠖⢢⣤⡴⠾⠿⠿⡟⠛⠷⣶⢄⡀
⠀⠀⠀⠀⠀⠀⢀⡔⠁⠒⠀⠀⠊⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⠐⠻⣷⣾⣷⣾⠟⠓⠀⠀⠀⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢢⡀
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⢀⠇⠀⣀⠀⣠⠀⠀⣀⣴⣿⠏⢀⡀⠀⠈⠉⠉⠉⠉⠙⢻⡟⠉⠉⠉⠉⠉⠁⠀⠀⡀⠹⣿⣦⣀⠀⠀⣄⠀⣀⡀⠸⡀
⠀⠀⠀⠀⢠⠃⡰⣾⣿⣴⣿⣶⣾⣿⣿⣿⣾⣟⣁⠀⠀⠀⠀⠀⠀⠈⢻⡟⠁⠀⠀⠀⠀⠀⠀⣀⣹⣷⣿⣿⣿⣷⣶⣿⣦⣿⣿⢆⠐⡄
⠀⠀⠀⢀⡞⢈⣴⠟⠻⠿⣭⠿⠿⣿⣿⣿⠋⠛⠉⠉⠀⠀⠀⠀⢀⣀⣸⣇⣀⠀⠀⠀⠀⠀⠉⠉⠛⠙⣿⣿⣿⠿⠿⣽⠿⠟⠻⣦⡁⢻⡀
⠀⠀⢠⠟⢃⡾⠁⠀⠀⠀⠴⠃⠀⣿⣿⣿⣦⣠⣞⢶⡀⠀⠀⠀⠀⣹⡟⢻⣏⠀⠀⠀⠀⢀⡶⢲⣤⣴⣿⣿⣿⠀⠺⡆⠀⠀⠀⠈⢷⡘⢿⡄
⠀⠀⡎⢀⣾⣇⠀⠀⠀⠴⠃⠀⣰⣿⣿⢂⣽⣿⣿⣿⠿⠶⢶⣾⣿⠟⣴⣦⠻⣿⣷⡶⠶⠿⣿⣿⣿⣯⡐⢿⣿⣆⠀⠘⢀⠀⠀⠀⣸⣷⡈⢱
⠀⢰⠀⠋⠁⢘⣄⠄⣤⡕⣠⡾⢻⠹⣿⠏⠁⡈⠙⢿⡤⠆⠀⠀⠀⠙⣾⣿⠃⠀⠀⠀⠰⢤⡾⠋⣀⠀⠹⣿⡇⡟⢷⣄⢺⣤⢠⣠⠋⠀⠉⠈⡆
⢠⢧⣀⠠⢥⡮⢻⣿⣿⡿⠋⠀⠈⢧⣿⠘⢧⣽⣶⣤⣤⣴⣶⣶⣶⣾⡿⢿⣷⣶⣶⣶⣦⣤⣴⣾⣯⡼⠃⣿⡼⠁⠀⠙⢿⣿⣿⡿⣿⣤⠄⡠⣼
⡏⡠⠻⡧⢎⠀⢿⣿⠀⠀⠀⠀⠀⠀⢻⡄⢦⡀⢿⣿⠛⠛⠉⠀⠀⣀⣠⣄⣀⠀⠀⠉⠛⠛⣿⡿⢀⡼⢠⡟⠀⠀⠀⠀⠀⠈⢿⡿⠀⡙⢸⠟⢀
⡟⠀⠀⠻⡄⠀⣸⡇⠀⠀⠀⠀⠀⠀⠈⣧⡈⠙⠛⢿⣿⣶⣶⡶⠿⠫⠶⠶⠝⠿⢶⣶⣶⣿⡿⠛⠋⣡⣼⠀⠀⠀⠀⠀⠀⠀⢸⣷⠘⢡⠟
⡇⠀⠀⠀⠘⢂⠟⡇⠀⠀⠀⠀⠀⠀⠀⢿⠻⣄⠐⠦⢼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡧⠴⠂⣠⡟⡻⠀⠀⠀⠀⠀⠀⠀⢸⠫⡐⠃
⢁⣆⠀⣴⠐⠉⠀⡇⠀⠀⠀⠀⠀⠀⠀⣨⠷⠽⣿⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⡿⠯⠾⣥⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠒⡆⠀⣰
⢸⣻⡄⠁⢣⠀⢠⡇⠀⠀⠀⠀⠀⣀⡴⠛⢀⣒⣒⣶⠶⠶⠶⠒⣶⡀⠲⠒⢀⣶⠲⠶⠶⠶⢶⣖⣒⠀⠛⢦⡀⠀⠀⠀⠀⠀⢸⡄⠀⡜⠀⢠⣿
⠈⣿⣧⠀⠀⢣⣸⠀⠀⠀⢀⡴⢚⣉⣄⠈⠉⠉⠉⠡⠶⢶⣦⣄⡉⠻⣷⣾⠟⢉⣠⣴⣶⠶⠌⠉⠉⠉⠁⣀⣉⡓⢦⡀⠀⠀⠈⣇⡜⠀⠀⣼⣿
⠀⢻⡏⢧⠀⢸⡇⠀⠀⠀⣸⣷⠟⠉⡠⠶⢶⣶⣶⣦⣄⡀⠈⠙⠻⣷⣾⣷⣾⠟⠋⠁⢀⣠⣤⣶⣶⠶⠦⠄⠉⠻⣾⣇⠀⠀⠀⢸⡇⠀⡼⢹⡟
⠀⢸⣷⠼⢶⣾⣿⣦⠀⠀⡿⠁⣠⣤⣤⣄⣀⠀⠈⠙⠻⣿⣦⡀⠀⠈⢻⡟⠁⠀⢀⣴⣿⠟⠋⠁⠀⣀⣠⣤⣤⣄⠈⢿⠀⠀⣴⣿⣷⠶⢧⣾⡇
⠀⢸⣧⣸⣿⣿⣯⣿⠀⢸⠁⣠⣤⣤⡄⠀⠉⠛⠶⣄⡀⠀⠙⢿⣦⣄⡀⢀⣠⣴⡿⠋⠁⢀⣠⠶⠋⠉⠀⢠⣤⣬⣄⠈⡇⠀⣿⣽⣿⣿⣆⡼⡇
⠀⢸⣿⠿⠟⢏⡙⠋⠃⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⠹⣿⣿⣿⣿⠏⢀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⠚⠩⠋⡹⢻⠿⣿⡏
⠀⠘⣿⡄⠀⠈⣙⣀⣴⠁⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣌⣿⣿⣡⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠘⣦⣀⣈⠀⠊⢠⣿⠃
⠀⠀⢱⣾⣿⣿⡟⣿⣿⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣿⣿⢻⣿⣿⣷⡎
⠀⠀⠈⠛⠙⠋⠛⠋⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠙⠛⠙⠋⠛
⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⡆⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣧⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣾⠁⢠⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢠⠗⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⣦⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠄⠀⠀⠘⣆⠀⠀⠀⣾⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡓⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣄⣀⡀⠀⠠⣄⠢⡌⢢⣈⣢⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠛⠓⠋⠋⠈⠁⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣖⣮⣭⣭⣭⣒⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠟⠛⠛⠛⠙⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣤⣶⣩⡄⠀⢹⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢿⣿⡟⠉⢙⠛⠋⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣁⣀⣭⠆⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢈⣙⣡⡤⠤⠄⠀⢀⢮⡄⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠛⢡⣤⣶⠀⠀⢠⣃⣾⣧⣦⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⢀⣴⣿⣿⡿⠛⠛⠛⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠚⣹⣿⣿⣿⣿⣿⡛⠁⢀⣀⣀⣀⠀⠘⢿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠈⠀⠀⠀⠀⢀⣽⡿⠛⠋⠉⠙⠿⠂⠈⢇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢀⠀⠀⠀⢈⠙⠣⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣜⢦
⠀⠀⠀⠀⠀⣀⣀⣤⡤⢤⣤⣄⣀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⣠⠏⠀⠀⠀⠙⠦⣤⠚⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠱
⠀⠀⣠⡴⠛⠉⠁⠀⠀⠀⠀⠈⠙⠻⣦⡀⠀⣰⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠓⢐⢃⡒⠊⠖⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⣼⠋⠀⠀⣠⣴⠶⠶⠶⢦⣄⠀⠀⠀⠛⠢⢿⣦⣤⡴⠆⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⢥⡀⠡⢀⡤⠀⠀⠀⠀⠀⠀⠀⣾⡇
⣼⠃⠀⠀⣼⠋⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠁⠀⣻⣿⠃⢠⡄⢀⣀⠀⠀⠀⠀⠀⠈⢳⣵⡾⠃⣀⠚⠁⠀⠀⠀⠀⠀⠀⣰⣿⢃
⡏⠀⠀⠘⠇⠀⠀⡏⢉⡇⠀⠀⣿⡇⠀⠀⠀⣾⣇⡛⠀⣘⣿⣿⣿⣧⡀⠀⣀⣠⣴⣿⣯⠞⠉⠉⠁⠀⠀⠀⠀⠀⠀⣴⣿⠏⡼
⣧⠀⠀⠀⠘⣄⠀⠈⠉⠀⠀⣰⣿⠃⠀⠀⠀⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠃
⠸⣆⠀⠀⠀⠈⠑⣂⣤⣴⡾⠟⠁⠀⠀⠀⣼⣿⣿⣧⣰⣿⣿⣿⣿⣿⣿⠃⠀⠀⡇⣿⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢠
⠀⠹⣦⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣟⠿⣿⣿⣿⣿⣿⣿⣷⣄⠀⣸⣾⡿⠃⠀⠉⠉⢻⡆⠀⠀⠀⠀⣸⡇⣸
⠀⠀⠈⠻⣷⣦⣤⣀⣀⣀⣠⣤⣾⠶⣿⠿⣿⣿⣿⡿⠧⠀⡀⠀⢀⣉⣉⠉⠉⠉⠉⠀⠀⠀⠀⡆⠀⢸⡇⠀⠀⠀⢠⡿⢡⠃
⠀⠀⠀⠀⠀⠉⠛⢟⡛⠛⠋⠉⠀⠀⣨⠟⠢⣝⡳⢤⣀⠀⠀⠉⠋⠉⠁⠘⠋⢶⣀⠀⢀⣀⠀⠇⠀⠀⢠⠀⠀⣰⡟⠡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠚⠋⠁⠀⠀⠈⡙⠶⣬⡙⠲⢦⣤⣄⣀⠴⠲⠦⠉⠑⠛⠀⠀⠀⠀⣠⡟⢠⣾⣟⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠙⠳⢶⣄⠉⠛⠿⠷⠶⢶⣤⣤⣤⣶⣶⣿⡿⣰⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠓⢶⣶⣶⣾⣿⣶⣤⣙⠲⠦⣤⣀⠀⠀⠀⠀⣰⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠉⠉⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⣿⡿⣷⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣠⡟⢸⢹⠇⠘⢿⣿⣿⣷⣄⠉⠻⢿⣿⡆⠀⠀⠀⠙⣿⣿⡿⣿⣿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠳⡝⢧⣿⣸⠀⠀⠈⠻⣷⡈⠻⣷⣦⡀⠉⠛⠀⠀⠀⠀⠈⢿⣧⠘⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣷⢄⡹⡌⣿⣿⠀⠀⠀⠀⠈⠛⠀⠀⠉⠛⠦⠀⠀⠀⠀⠀⠀⠘⣿⠀⢻⡇⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢻⣿⡞⣷⣹⣿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡻⠷⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣼⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠶⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠀⠀⠉⢯⢢
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠙⡜⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣘⠀⢀⡴⣦⡀⠈⡆⢃
⠀⠀⠀⠀⠀⢀⣆⣀⢔⠺⡆⠀⠀⢁⠸⣖⣾⡋⢶⠤⠭⠭⠭⠭⠭⠑⠺⣯⢽⡥⠤⠨⣗⣼⠇⠀⡇⢸⠗
⠀⠀⠀⠀⠀⠙⡖⠂⠻⠶⠃⠀⠀⡈⢠⠀⠀⢣⠈⢆⠀⢀⣠⢤⡴⣤⡠⠋⠻⠘⣄⠀⠀⠀⠀⣰⢁⠆
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⣠⢃⠎⠀⠀⠘⡄⠈⣴⠋⢁⣀⣀⣸⠃⢠⠃⠀⠈⠢⠤⠤⠴⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠴⠞⠓⠉⠀⠀⠀⠀⠘⡄⠹⣦⡏⢙⢷⣿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢹⢿⡻⠿⣇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠁⠀⠙⢒⡃⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⣀⢀⡁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡤⢺⡎⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠺⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⢭⡥⢒⡚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠘⠀⠀⡨⡓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣘⣤⣸⠈⡒⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢻⠑⢺⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣅⣀⠔⠛⢩⠄⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠟⠢⡄⠸⠄⠉⢉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠁⠀⣶⡷⢦⣴⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⣠⣶⣿⢫⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⣰⣿⣔⠂⠛⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠻⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣭⣾⣿⣿⣆⣬⡩⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣈⣡⡬⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣽⢘⡧⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠁⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣹⡟⢁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⢀⠠⠉⢂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠿⠿⣿⣿⡿⢫⠟⢳⣿⣿⣿⣗⣠⣿⣼⣶⣤⣀⣀⣀⣀⠀⠀⠀⠀⢀⣴⠎⣁⡤⠛⠀⠈⠌
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣧⣀⣀⣀⣈⣁⣀⣀⠀⠉⠀⣿⣿⣿⣿⣿⣿⡹⢉⣏⣉⡉⠈⠙⣿⣾⣿⠿⣶⡟⢀⣤
⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⠿⣋⣽⣿⡄⠀⠀⣿⡃⢹⡿⣿⣷⣶⣾⣿⠶⠾⠿⠷⠤⠬⣾⠿⣶⣾⠋
⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢼⣿⣿⠁⠿⠛⠿⢿⡯⠛⣷⣶⣤⠀⠄⠠⠴⠛
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠈⣿⣇⢠⣧
⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠛⣹⣿⣿⣿⣿⣿⣹⣿⣿⣿⢻⣟⡟⠃⠀⢻⠿⢿⡏
⠀⠀⠀⠀⢀⡾⣯⠀⣿⠁⣠⣿⣿⣼⣿⣿⣿⣿⣿⣿⣾⡿⠁⠀⠀⠀⢠⡀⠈⡇
⠀⠀⠀⠀⣼⣷⣿⣠⣿⣶⠟⠀⢿⣿⣿⣿⣿⣿⣿⣻⠿⢇⠀⠀⢀⠀⠀⠹⠆⡇
⠀⠀⠀⣰⣿⢿⡿⣿⣿⠋⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⢷⣦⠀⠀⠀⠐⠂⣼⠂⠁
⠀⠀⢀⣿⣿⣾⣸⡏⠀⠀⠀⠀⠀⠀⠙⣿⣍⣯⣙⠿⣯⡙⠀⠀⡆⠀⠀⠘⣿
⠀⠀⢸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⣼⣿⠀⠀⢿⠆⠀⡀⣻
⠀⠀⣾⣿⣿⣿⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢈⡉⡁⠈⠀⠀⠐⠀⠀⠀⢹⡄
⠀⠀⣿⣿⣿⣿⠘⠁⠀⠀⠀⠀⠀⠀⠀⣠⡟⠻⣶⣤⣿⣤⠀⠀⠀⣧⠀⠀⢸⠁
⠀⠀⢹⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣰⡏⠻⣦⣌⡙⠻⣿⣆⣠⣤⣿⣷⣴⣏⣀
⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣰⡏⢉⣀⠀⠍⢻⣧⠹⣿⣿⣿⣿⣿⣿⣿⡌⠛⠂
⠀⠀⠀⢿⣿⠻⠿⠀⠀⠀⠀⠀⢀⣿⠀⢸⣿⣷⣦⣼⡿⢠⢻⣿⣙⣿⣟⠀⠙⢿⣆⣲⡀
⠀⠀⢀⣿⣿⠖⠷⠀⠀⠀⠀⠀⠸⣿⠀⣾⡿⣤⢉⣿⡇⠘⣿⣿⣿⣿⣈⠃⣀⡻⠻⣿⣧⠀⠐⢦⣄⠀⠀⠀⣀
⠀⢀⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠹⣆⣸⣷⠟⢿⣿⠀⠀⠀⢻⣿⣿⣿⣷⣜⠛⢧⡉⠙⢷⡀⠀⠹⣿⡷⢦⣤⣤⣄
⠀⢸⣿⣿⢿⡟⠻⡗⠀⠀⠀⠀⠀⠀⠘⢿⣿⠀⠈⢳⣆⡼⠀⠈⣿⡟⠉⠺⣿⣷⣦⣉⠓⢾⣿⣇⠀⢸⣿⣶⣈⣿⣿⣷⡀
⠀⠘⠿⠅⠛⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⡟⢀⠀⠀⣿⠁⣠⠀⢻⡇⠀⠀⠀⠙⠻⢿⣷⣦⣭⣻⣿⣦⣿⣿⣿⣍⡙⠛⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡄⠀⠋⠀⠹⣀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡇⠀⡄⠀⠈⡟⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣽⠿⠈⠁⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠇⠀⣿⠀⢸⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢿⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⠀⣿⠰⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⡎⢰⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡇⢿⠂⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⡘⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⢷⠀⠃⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣷⡞⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢛⡇⠸⠂⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢸⡇⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⡿⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢠⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡞⢁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣸⠇⣼⡄⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢀⣿⣷⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡏⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠿⠿⠛⠷⠾⣷⣶⣤⣌⡁⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢸⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠿⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦⣤⣾⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⣿⣿⣼⣿⣿⣿⡿⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠷⢴⣾⡿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠦⠄⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⣦⢀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⣠⡖⠋⠉⠁⠂⠄⢀⣠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⡟⠸⡀⠀⠸⡃
⠀⠀⠀⠀⢠⠖⠉⢸⠁⠀⠀⠀⠀⠀⠀⠀⡤⢤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⣠⣾⡦⢄⠱⡀
⠀⠀⠀⠀⡇⠀⠀⠘⡄⠀⠀⠀⠀⠀⢀⠞⠀⠺⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠼⣥⠏⠀⠀⢰⠇
⠀⠀⠀⡐⢰⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡧⠊⢸⠀⠀⠀⢸
⠀⠀⠠⠃⢸⡄⠀⠀⠈⢣⡀⠀⠀⠀⠀⠑⣄⡀⠀⠀⢀⠇⡀⠀⢨⠀⡴⠐⠒⠤⢤⠠⡏⠀⠀⢸⠀⠀⠀⣸
⠀⠀⠀⡉⠛⣧⠀⠀⠀⠀⠳⣄⠀⠀⡄⠀⠀⡝⠳⢶⣋⣀⠇⠀⠀⠦⢁⣐⡑⠶⣰⣄⡇⠀⣀⣸⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠈⠑⡄⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣼⠉⠉⠉⠉⠀⠀⢀⡼⠱⡀
⠀⠀⠀⠀⣴⣾⣧⡀⢀⣀⠀⠀⣀⠀⣰⣜⢟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⢴⠖⠒⠒⠋⠉⠀⠀⠱⡀
⠀⠀⠀⠀⠀⢹⣾⠧⠼⠿⣤⣼⡿⣿⠛⡿⢞⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣀⣀⣠⣤⠶⠃
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠒⠛⠁⢻⣧⣿⡴⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠉⠁⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢸⣈⣩⣆⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠓⠒⠒⠲⠋
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠏⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⡄
⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⡄
⠀⠀⠀⠀⢠⠀⠀⠀⠀⡿⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⢠⠃
⠀⠀⠀⠀⢸⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸
⠀⠀⠀⠀⠘⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⢈⢦
⠀⠀⠀⠀⠀⣄⠀⢀⣈⡇⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⢀⣠⣪⡜
⠀⠀⠀⠀⠠⠯⠭⠽⢿⡷⣦⡤⠴⠞⠉⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢛⣁⡲⠿⠋⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠚⠛⠛⠛⠛⠓⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⠀⢠⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⠀⠀⠀⠀⢸⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⣼⠃⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⢾⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⠀⠀⠀⠀⢻⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀⠀⠀⠀⢘⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⢠⠶⡿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⠀⠀⠈⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠋⠁⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⡀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡿⠃⠶⠤⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠧⠄⠀⠐⢿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⡟⠕⢁⡴⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⡄⢠⣴⠲⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠺⠵⠷⠤⠤⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠧⠼⠦⠗⠓⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣲⣬⣭⣭⣭⣒⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠟⠛⠛⠛⠛⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⢋⡀⠀⢹⡿⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣰⣶⣾⡟⠛⠛⠿⠟⠁⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠉⠉⠃⢀⣛⠄⠀⠀⠀⠀⠀⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠙⠿⢉⣀⣀⡀⠀⠀⣔⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡔⠞⣛⣉⣭⠈⠀⢀⡖⣸⣟⣠⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠈⠛⠉⠀⢀⣾⣿⣿⡿⠿⢦⠄⠑⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢷⣦⣤⣶⣿⣿⡿⠋⠀⠀⠀⠀⠀⠘⣷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠛⠋⠉⠉⠉⠓⢲⣶⣶⠶⠶⠶⣦⡀⠈⡇⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠲⢰⢀⠀⠰⠟⠉⠀⠀⠀⠀⠈⠀⠀⠈⠢⡘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⢀⡴⠀⠀⠀⣶⡀⢈⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⢣
⠀⠀⠀⢀⣤⡴⠖⠛⠋⠉⠛⠓⠶⣤⣄⠀⠀⠀⡰⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠠⣋⢻⡀⣀⡰⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⢀⡴⠋⠁⠀⠀⣀⣀⣀⡀⠀⠀⠈⠙⢷⣄⢠⣇⠀⠀⢀⢀⠋⠀⠀⠀⠀⠀⠀⢀⠔⠐⠂⠄⠃⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿
⢠⡟⠀⠀⢀⣴⠟⠋⠉⠉⠙⢷⣦⠀⠀⠈⢈⠊⢿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣆⢀⣹⡆⠘⣡⡞⠀⠀⠀⠀⠀⠀⠀⣸⣿
⣾⠀⠀⢀⡿⠁⠀⣠⣤⡄⠀⠀⢻⣧⠀⠀⠀⠈⣠⠋⡇⠀⢺⣄⣴⣦⡀⠀⠀⠀⠀⢀⣽⡿⣋⣥⢤⡄⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇
⡇⠀⠀⠀⢡⠀⠀⠣⢠⠟⠀⠀⣸⣿⠀⠀⠀⢰⣿⣤⠁⢠⣼⣿⣿⣿⣷⣤⣤⣤⣶⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡏⢸
⢷⡀⠀⠀⠀⠣⣀⠀⠀⠀⢀⣴⣿⠃⠀⠀⠀⣸⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⠛⠛⠋⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠈
⠘⣧⠀⠀⠀⠀⠀⠥⣼⠿⠿⠋⠁⠀⠀⠀⣰⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⡀⠀⠀⢇⢿⣄⡀⠀⣤⣄⣀⡀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⣿⣿⣿⣿⣍⠛⠿⠿⠿⢿⣿⣿⣿⣦⣀⣸⠿⠟⠉⠀⠀⠀⢹⡇⠀⠀⠀⠀⢰⣿⢠⠃
⠀⠀⠀⠙⠿⣷⣦⣤⣤⣤⣤⣶⡾⠟⠻⣟⠻⣿⡿⠿⠛⠃⠠⢠⣀⣠⣤⡤⠀⣀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡇⠀⠀⠀⢠⡿⢃⠎
⠀⠀⠀⠀⠀⠀⠉⠻⣉⡉⠉⠀⠀⢀⣠⠾⠛⠢⣍⡳⢦⣄⡀⠁⠈⠁⠀⠈⠈⠁⠷⣤⠀⣀⠄⠀⠋⠀⠀⢠⡄⠀⣠⡿⢁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⢨⠙⠲⣬⡙⠓⠦⢴⣤⣄⠮⠛⠧⠁⠉⠉⠀⠀⠀⢀⣠⡟⣠⣾⣿⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠉⠓⠶⣖⠒⠛⠛⠻⠶⠶⠶⣶⣶⣶⣾⣿⠿⢰⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠳⣶⣶⣶⣾⣿⣶⣬⣙⡓⠶⣤⣀⠀⠀⠀⠀⣠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⡀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢸⣿⡿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⣄⣼⠃⡟⣿⠀⠹⣿⣿⣿⣦⣌⠙⠻⣿⣿⡆⠀⠀⠀⠙⢿⣿⣿⢿⣿⡌⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠙⢮⢻⣤⡇⡇⠀⠀⠘⢿⣧⠙⢿⣷⣄⠀⠙⠳⠀⠀⠀⠀⠈⢿⣿⠈⣿⣇⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⣷⣬⣈⢧⢻⡇⡇⠀⠀⠀⠀⠙⠃⠀⠈⠙⠳⢦⠀⠀⠀⠀⠀⠀⠈⣿⠀⢸⣿⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡸⣿⣷⢻⡎⣿⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢸⣿⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡙⢿⡜⢻⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡆⠀⠀⠀⠀⠀⣶⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⠀⠀⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⢸⣿⣷⣾⣿⠀⠀⢸⣿⣯⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⠀⢀⣄⠀⠸⠟⠁⠸⡏⠀⠀⠿⠛⠉⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⡿⣿⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠁⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣠⣴⣶⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢸⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠘⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠙⠻⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣤⣴⣾⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃



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