Pointing Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠉⠙⠛⣷⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟⡖⠀⠀⠀⠀⠀⠀⠀⠉⢿⣷⡀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢾⠏⣼⠏⡀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⡄⠀⠀⠀⡘⢦⡀
⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠐⠋⢘⣿⡿⠁⠀⡠⠒⣈⣤⣤⠀⠀⣀⠙⣿⣰⠀⡄⠐⡈⠻⠳⣄
⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⢈⡴⠋⣡⠴⠞⠂⠀⠈⠳⣾⡏⠀⢇⠀⡇⠀⠀⠀⠳⣄
⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠐⠿⣧⣀⣾⡤⠊⠀⠀⠀⠀⠀⠘⢿⣿⣿⠀⢸⣴⠀⠀⠀⠀⠀⠪⣧
⠀⠀⠀⠀⢸⡆⠰⣄⠀⠀⠀⠀⠀⠀⠰⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⢸⠸⡄⣶⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⢀⡼⡷⠀⠘⠆⠀⠀⠀⠀⠀⠀⠸⣿⣧⣾⡀⠀⠀⠀⠀⠀⠀⠀⣿⡥⠀⠀⠀⣇⣸⡦⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⢠⠏⠀⠀⠀⠳⣌⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⣄⡀⠀⣀⣤⣾⣿⡿⠀⠀⠀⣿⣿⠅⠀⠀⠀⠀⠀⢂⠰⡇
⠀⠀⣾⠀⠀⠀⠀⠀⠈⠀⠰⡄⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⢿⣿⣿⣿⡿⠿⢿⣷⣴⣿⡟⠀⠀⠀⠀⠀⠀⠈⢆⣷
⠀⠀⢹⣦⣄⡀⠀⠀⠀⠀⢲⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⠙⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢈⠻⣆
⠀⠀⠈⡇⠉⠛⠂⠀⠀⠀⠀⢛⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠶⠄⢹⣿⡷⠒⠀⠀⠀⠀⠀⠀⠈⠳⣼⡄
⠀⠀⢠⡗⠘⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⢸⣿⡯⠍⠁⠀⠀⠀⠀⠀⠀⠀⠺⣿⡀
⠀⠀⢸⡀⠀⠀⠠⢴⣄⡀⠀⠀⠀⠈⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⣾⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠈⢷⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⠈⢙⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⢀⢌⢦⠰⢄⠪⡟
⠀⠀⠀⠈⢷⣄⡀⠀⠀⠈⠻⣶⡀⠀⠀⠀⠘⡿⣿⣦⣤⡄⠀⠀⠀⠠⠊⣠⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣶⡾⠃
⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠈⠿⣦⡀⠀⠀⠈⠹⣻⣿⣿⣶⣵⣷⣷⣾⠿⣿⠇⠀⠀⠀⢀⣤⠶⣄⠀⢀⣠⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠙⢿⣄⡀⠀⠀⠑⠽⠛⡹⡻⢻⣿⠁⣸⣇⢀⠀⢠⣴⠋⠀⠠⣨⡷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⡀⠙⢿⣶⣄⣀⠀⡠⣀⣴⣿⠏⠀⢻⣿⣯⠀⠘⠁⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀⠉⠛⢿⠿⣾⣿⠟⠁⠀⠀⠈⠻⣿⣦⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠶⠟⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠉⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠋⠀⠀⠀⠀⠀⠙⢦⡀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣊⡽⠏⡉⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⡉⠉⠉⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⣸⡷⠀⠀⠀⠀⠀⠀⠁⠀⢨⢺⡇⠀⠀⢐⢂⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⡿⣱⡀⢀⣴⣖⠒⠒⠦⣄⠈⣷⣿⠀⠀⠐⠐⣅⠀⠀⠄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢺⡇⠀⠀⠀⣷⣿⠇⢿⡽⠿⠶⠶⠤⢬⣧⡘⣿⠀⠀⠀⠀⠙⠀⠀⢠⡎⠓⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠞⠀⠀⠸⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢹⠀⠀⠀⠀⠀⠀⠀⢔⡴⠁⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠰⡀⠂⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠠⡃⠐⠀⠀⣿⣦⡀⠀⠀⠀⠀⡄⣢⡿⠃⠀⠀⠀⠀⠀⣧⠀⠀⠰⠁⠀⡄⠀⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠐⠃⠠⡀⠶⣼⣷⢩⣗⣶⣖⠺⠋⠁⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣰⡿⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠐⢦⡰⡄⠀⣶⣾⣿⢺⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⡴⠁⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠇⠀⠀⠀⠀⠀⠤⣌⣻⣟⣏⣨⣭⣽⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠇⠀⠀⠀⣸⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠐⠒⠮⠝⡏⠛⢡⢿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⢠⡏⠀⠀⠀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⡠⠤⠶⠦⣷⣠⡀⠉⠈⠓⠆⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⡼⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⡴⠋⠀⠀⠀⠀⠹⣿⣿⡂⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⣸⠃⠀⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣻⠃⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠌⠻⣮⣭⠫⠃⢀⠢⢀⣀⣠⡼⠃⠀⠀⠀⣰⠟⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠢⠀⣤⠀⣴⢆⢠⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣤⣴⡖⠋⠋⠁⠀⠀⠀⠀⣰⠋⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡘⠁⠼⢿⣦⡏⣠⡿⠒⠦⣄⡀⠀⠀⠀⢷⠀⠀⠙⣦⣄⣐⠄⠀⢀⠔⠁⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠓⠚⠻⣤⣀⠀⠀⡉⠓⢦⠀⢸⡄⠀⠀⠀⠙⠯⣙⣛⣅⣀⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡐⠀⢸⠂⡾⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠋⠉⠀⣀⣀⣀⢠⡄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠦⣼⠀⣀⣀⣤⣄⣀⣀⣠⣤⠤⠤⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⣿⣿⠟⠃⠀⣰⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢠⣀⠀⠀⠈⢻⣍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠺⢛⣿⡿⠛⠛⡛⠛⢦⡀⠈⠢⠤⣬⡄⠄
⠀⠀⠀⠀⠀⢰⣤⣾⣷⣶⣷⡎⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠦⢤⣽⣀⣀⣉⣛⣛⣿⣯⣤⠞
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⣷⣄⣀⠀⠀⣀⣀⣀⠤⢚⣿⣿⣿⣯⡁⣰⠀⠉⠉⠉⠁⠀⠀⡄⣴
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢻⡎⠉⠙⠈⠉⠀⠴⠞⠿⣿⣿⣿⡈⣹⠓⢦⣄⣠⣤⣤⡤⢤⠶⠏
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⢰⣼⠜⣿⠗⢦⣀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠧⣄⣀⣀⡀⠀⠀⢄⡜
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣴⣶⠀⠀⠛⠀⠀⠉⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢹⣿⠀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣾⠿⠛⠛⠛⢿⣷⣄⣴⡿⠟⠛⠛⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠹⣿⡿⠁⠀⠀⠀⠀⢹⣿⣤⣶⡿⠿⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢀⣶⡿⠿⠿⣷⣦⡀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⣾⣿⠀⠀⠀⠈⠙⢿⣦⣀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠸⣿⣆⠀⠀⠀⠀⠀⠙⢿⣶⡀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠙⢿⣦⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⠿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿

⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣀⣠⡤⣤⠄⠐⠀⠀⠚⠿⠀⠻⢶⣤⣀
⠀⠀⠀⡈⠁⠀⠐⠶⠂⠀⠐⡆⠉⠉⡿⠁⠀⠀⢰⡰⣀⣀⡀⠐⠁⠉⣿⢷⣦⣀⡀⠀⣠⠀⣀⠰⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢤⡔⡣⠂⠀⢠⠞⠿⢿⣏⠂⠀⠀⠀⠀⠊⠻⡟⡿⡿⡿⡏⠉⠀⢹⣻⡿⢹⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠐⠀⠀⠀⠀⢀⡠⠞⢻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢡⠀⠀⠀⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⠤⠒⠉⠀⠀⠀⣨⢿⡁⠀⠀⠀⣿⠄⢀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⠔⠊⠉⠁⡟⠀⠁⠁⠯⠌⠈⠀⠀⡅⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢶⠒⠁⠈⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⣸⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠋⢀⣀⡀⠠⠀⠒⠀⠀⠀⠙⠀⠀⠀⢤⣿⢰⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠛⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡤⠤⠤⠀⠀⠀⠀⠀⠤⠀⠀⠤⠀⣤⡄⢀⠀⠀⣀⣀
⠀⠀⠰⣶⣶⣶⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠉⠉⠉⠉⠛⠻⠿⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⠀⡄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⣿⣦⣀⣤⣶⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠙⠻⣿⣶⣶⣶⣶⣦⣴⣿⣦⣦⣴⣶⣿⣶⣴⣶⠴⠞⠁
⠀⠀⢸⣿⣿⣿⣿⠟⢁⣼⣿⠛⠛⠋⠁⠀⠀⠀⠀⠀⣀⢲⣷⣿⣿⣷⣿⣿⣷⣦⡀⠀⣎⣉⣙⡉⠙⠉⠁
⠀⠀⠘⢟⣿⣿⡏⠀⣾⣿⡇⠀⠀⠀⡀⠐⣔⣴⡷⣿⢟⣿⣿⣿⣿⣿⡿⢉⢹⡿⠿⠿⡶⠿⠟⠓⠠⠀⠀⠄
⠀⠀⠀⣿⣿⣿⠁⢸⣿⣿⣿⡟⣟⣶⣻⢷⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⠟⢻⣿⣿⣀⣀⠀⠀⠀⠀⠀⢀⣤⠇
⠀⠀⢈⣿⣿⡟⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⢿⣦⣿⣾⣿⡿⡿⠿⢷⣷⣦⠾⠛⠁
⠀⠀⢨⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣾⡋⠻⢿⣿⣿⣀⠀⠀⠀⠀⢘⠇
⠀⠀⢠⣿⣿⡇⠃⠀⣿⣿⣿⣿⣿⣮⡛⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷⡄⠀⠙⠻⠿⠿⣿⠿⠾⠋
⠀⠀⢸⣿⣿⣟⠃⠀⢹⣿⡿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣰⠆
⠀⠀⢰⣿⣿⣿⠀⣤⡌⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠉⠙⠻⠿⢿⠷⠾⠋
⠀⠀⢘⣿⣿⣿⣆⠿⣿⠟⣰⠇⠈⠉⠉⠉⠉⠉
⠀⠀⠈⠛⠛⠛⠛⠛⠒⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣟⣟⣛⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀⠶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣶⣬⣻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣯⠈⠉⠛⠻⠿⠋⠀⠀⠉⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣼⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣝⢻⣿⠏⠀⣠⣤⣤⣄⣀⠀⣠⣤⠴⡄⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠙⠁⠈⠑⠶⠖⠛⠉⠀⠟⠛⠉⠁⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠶⠞⠁⡀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠒⠒⠈⣭⡽⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠙⠦⣀⠀⠀⠀⠀⠀⠀⢠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣤⣤⡤⠴⠖⢻⡍⠳⣄⠈⠻⣦⣄⣀⣀⣠⣿⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⣛⣦⣄⠀⠁⠀⠈⠙⢦⡀⠉⣩⣿⣿⠏⢰⡟⠲⣄⡀
⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⣰⡧⠚⠀⣿⣿⡍⠓⠂⠠⢄⠀⠀⠈⠓⢦⣼⣏⠀⢸⡇⠀⠀⠉⠓⠦⢄
⠀⠀⠀⠀⣸⠃⠀⠀⣴⠀⠈⠛⠛⢿⣿⣿⣿⣥⠀⠀⠈⣧⠀⠀⠀⣾⣿⣿⠃⡾⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠠⡏⠀⠀⠀⢸⡇⠢⣄⠀⢀⣿⠁⠀⣿⡇⠀⠀⢿⠀⡀⢸⡇⣿⣇⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⡀
⠀⠀⠀⠀⢿⠀⠀⠀⣿⠀⠀⠀⠙⢾⣿⣤⢶⡟⢀⠆⣠⠟⠀⠙⠾⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⢸⠃⠀⠀⢹⡇⠑⠢⢀⣼⣋⡧⠾⣧⣀⡞⠁⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠘⢷⡀⠀⠘⢷⣄⣀⢀⣿⣏⣠⠰⢿⣿⠁⠀⠀⠀⠀⠀⣸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠈⢻⡀⢀⡀⠻⣄⡀⠹⣿⣀⣤⣾⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠱⣄⣉⣷⣶⣿⣿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢸⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣾⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡊⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡃⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣠⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠈⢢⡀
⠀⠀⠀⢀⣀⣀⣀⣀⡀⠤⠤⠤⠤⠤⠄⣀⣀⠀⠠⠤⠤⢶⣶⡾⢯⠀⠀⠀⠀⠀⠹⣆
⣶⣊⣉⣁⠀⢀⣤⣀⣀⣀⠠⣦⡄⠀⠀⢠⡞⠀⠀⠠⣦⠘⢉⣇⢼⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⠿⢷⣾⣷⣶⣎⣿⣿⣴⣰⣧⣂⣰⣻⡀⠀⠀⠀⠀⠀⠹⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣄⠀⠀⠀⠀⢈⠀⠀⠀⣚⣹⣧⢸⣿⣑⣼⠀⠀⠀⠀⠀⠈⠉⠙⣿⣿⣿⣿⡷⣿⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⣤⣼⣿⣦⣤⠤⠿⠿⣿⣧⠻⣿⣯⣰⡂⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⣙⠛⠛⠛⠉⠉⡀⠀⠠⣦⣾⡻⣶⣙⠻⢿⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣦⣤⣾⡿⠿⠛⢻⣿⣷⠌⠉⠛⠂⠀⠀⠉⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣯⣭⣥⠀⠀⣤⣾⣿⣋⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢽⣿⣿⣾⣿⣿⣿⣿⣮⣾⣶⣾⣆⣀⣠⣄⡸⣀⣔⣴⡏⣿⣿⣿⣿⣿⣿⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠙⣿⢿⣿⣿⣿⣿⣿⡉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣿⣿⣿⣿⣧⣤⣤⣤
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠿⣿⣿⣟⣛⠻⠛⠛⠛⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠋⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⣿⣿⣿⣿⡿⠟⠉
⣿⣿⠟⠋
⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⠒⠚⠉⠉⠉⠒⠲⠦⢴⣾⣷⡀⠀⠉⠂⢀
⠀⠀⠀⠀⠀⢠⣿⢿⡞⠛⠃⠀⠀⠀⠀⢸⣧⡀⠿⡿⢿⠂⠀⠀⠀⠐⢄⡀
⠀⠀⠀⠀⢀⣿⣿⣿⠷⣤⣈⡐⠀⠀⠀⢨⣿⣧⠀⠀⠀⠀⠀⠀⠀⣾⣦⡄⠈⠑⠢⡀
⠀⠀⠀⠀⢌⣾⠋⡴⠿⠏⠀⠉⠐⢄⠢⣸⣿⣿⣇⡀⠀⠀⠀⣰⣄⣿⠏⠀⠀⠀⡄⠌⠂
⠀⠀⠀⠀⣿⡟⠠⠛⠛⠓⠂⠀⠀⠘⡇⢻⣿⣿⣗⠀⠀⠀⢰⣿⠏⢀⡒⠀⠀⠀⠁⣴⣀⡀⠀⢤⡀
⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⣿⣿⡟⠀⠀⠀⣿⣿⠾⡋⠀⠀⠀⠀⣸⣿⣯⣿⢷⡆⠹⡄
⠀⠀⠀⠀⠈⢢⣠⣀⠀⠀⠀⠀⠀⢀⣢⢸⣿⣿⡆⠀⠀⢰⣿⠛⠉⠀⠀⠀⠀⠀⣿⣿⣿⡌⠂⠀⠀⢇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣶⣾⣿⣷⣿⣿⡿⠆⠀⢀⣿⣟⣺⠅⠀⠀⠀⠀⣾⣿⣿⣿⠁⠀⠀⠀⡌
⠀⠀⠀⠀⠀⣼⣟⢏⣿⣿⣿⣿⣿⢿⡿⠋⠙⠁⠀⠀⣼⣿⡃⠠⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⢀⠇
⠀⠀⠀⡴⠋⠉⠉⣽⣿⣿⣿⣿⣟⠶⣄⠁⠀⠀⠀⣰⣿⡟⠀⠀⠁⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⢸
⠀⠀⣾⠊⠀⠀⠈⣩⣿⡿⠿⠿⠿⣷⣾⣷⣶⣾⣿⣿⣿⣅⠂⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⢀⠇
⠀⠀⡟⢰⣠⡤⢈⠜⠫⠁⠀⠀⠀⠈⠁⠈⠉⠛⠻⢿⣿⣿⡷⠂⠀⠀⠀⣼⣿⡅⠀⠀⠀⠀⢠⠎
⠀⠀⢷⣿⡿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠤⠈⠈⢻⣶⣤⣶⣾⣿⡷⠀⠀⣀⣠⡴⠋
⠀⠀⠘⢿⣷⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡏⠁⠠⠄⠀⠀⠰⠀⢻⣿⣿⢿⢿⣿⠷⠞⠉⠁
⠀⠀⠀⠈⠻⣇⠀⠀⢸⡄⠀⠀⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⡆⣼⡟⠛⠉⠉
⠀⠀⠀⠀⠀⠘⠦⣀⣼⡟⠷⠧⠿⢧⢷⣷⣤⣾⣟⠋⠀⢰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠒⠲⠾⠭⠭⠭⠵⠞⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠀⠀⠀⠀⠀⠠⠄⣀⣀
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⣶⢶⣶⠶⣾⡉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⣠⣀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡅⠸⠆⠀⠀⠀⠀⠀⠀⠀⠢⣤⣄⣠⡄⣀⡀⠀⠀⠚⠓⢦⠮⠿⠤⠰⠤⠦⠤
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠙⠀⠀⢀⠀⠀⣀⢄⣄⣤⣾⣿⣿⡁⡀⠨⠙⠲⠆⠀⠀⠍
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡤⡄⠀⢬⣀⣽⠷⠿⠿⠛⠈⡟⠉⠅⢮⣀⢀⠀⠀⠀⢀⡀⡰
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠁⠀⠀⠀⠀⠀⡠⠓⠓⣄⠀⠘⠉⠙⠒⠒⠒⠶⡞⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⠁⠀⡀⠀⠀⠀⠀⠀⢱⡐⠀⠀⠛⠦⣤⣄⣀⣀⣀⡠⠁
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠧⠗⠁⠁⣾⣻⣷⣤⣀⣀⣂⣻⣦⣄⣀⣀⢀⠈⠉⠉
⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠁


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

⣠⠄⠒⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠐⠒⠂⠠⠄⠤⠀⠤⠤⠤⠤⠄⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⢒⡦⠄⠀⠠⠤⣀
⠈⠒⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠋⠉⠀⠀⠀⠀⠀⠀⠉⠐⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢒⣦⣄⡀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠙⠂⠒⡶⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠀⠀⢀⣶⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⡠⠊⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠲⠶⠶⡖⡈⠃⢀⣀⡀⠔⠈⠀⠀⠀⢀⡆⠀⠀⠐⠦⣤⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠈⠀⠀⠀⠀⡀⠄⠊⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣀⡤⠔⠒⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠛⠀⠀⠀⠀⠀⠀⠀⣠⡤⠄⠀⠀⠀⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣔⠐⢋⣀⣀⣀⣀⣀⣠⠤⠀⠀⠐⠈⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⡠⠤⠤⠒⠒⠋⠙⠻⢯⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢳⡄
⠀⠀⠀⠀⣰⠋⠉⠉⠉⣽⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣄⣀⣀⣀⣤⣤⡤⠤⡤⠤⠤⠤⠤⠞⠁
⠀⠀⠀⠀⡿⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣶⣤⣠⡄⠀⠀⣔⠉⠓⣦⡄
⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠿⣧⣀⡽⠙⠒⠦⠬⢭⠭⠟⡿
⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠠⠤⠤⠴⣶⡿⠿⣄⠟⠙⠦⠤⠤⠤⠤⢤⠤⠞⠁
⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡤⠛⠦⢤⣀⣀⠀⠀⢀⡼
⠀⠀⠀⠘⡇⠀⠰⠆⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣄⡀⠀⠉⢹⠋⠉
⠀⠀⠀⠀⢿⣄⣀⣀⣸⡟⠙⠒⠶⠤⠤⠤⠴⠶⠒⠚⠉⠉⠉⠛⠚⠁
⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣶⣶⣶⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠟⣽⠟⠁⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣶⣶⣤⡆⢠⣤⣶⣶⣶⡄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠉⠟⢫⣿⡇⠈⠉⢽⠏⠉⠀⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⣴⠟⢧⣤⠔⠳⠦⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠞⠡⠄⠀⠄⠂⠀⠀⠀⣸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣆⠀⠚⠛⠛⠃⠀⠀⣴⠋⠸⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡏⢿⣦⣀⠀⠀⠀⣀⠜⠁⠀⠀⢹⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣆⠙⠻⠿⢿⠛⠁⠀⢀⣄⠀⠈⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣿⣿⡆⠀⢻⣿⣿⡿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠉⢿⣿⣶⣾⣿⣿⣧⣹⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠩⣶⡝⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠲⢯⡉⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣷⣤⡞⠀⢻⣿⣿⣆⠀⠀⠀⠀⠀⢠⣼⣿⡌⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⣿⣯⣶⢆⣼⣿⣿⣿⣷⡀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣈⣽⣧⢸⣿⣿⣿⣿⣿⣷⡄⠀⠀⠈⣿⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠉⠀⣀⣨⡉⣿⣿⣿⣿⣿⣿⣆⠀⠀⣿⣿⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣹⡏⠈⠉⠉⢀⣀⣄⣽⣿⣿⣿⣿⣿⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡟⡇⠉⣼⡆⠘⠛⠋⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇⠃⠘⢋⣠⣠⣤⣴⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⠘⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣇⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢿⣿⣿⣿⣿⠇⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡄⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⠏⠉⣿⣿⣠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⢿⣿⠻⣿⠀⢸⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡏⢻⣿⣿⣿⣿⣿⣦⠙⠷⡘⠇⠈⠎⢿⠀⠙⠀⢸⠻⡟⢿⣿⣿⡟⠀⢀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⢷⡄⢻⣿⣿⣿⣟⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣇⠀⠉⠿⠇⠀⢿⡄⠘⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⡏⢷⡀⢻⣿⡻⣿⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠻⠀⠀⠀⠀⠀⠀⠑⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⠸⠈⠳⠀⢿⣷⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣼⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡄⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣾⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⣿⡇⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢟⣛⣻⡿⠿⣿⣿⣿⠿⣿⠿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣬⣉⣿⣶⣶⣴⣶⣶⣶⣴⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢠⣴⢻⣦⣄⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⠁⠀⠀⠃⠈⠻⠛⠋⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠘⣿⣿⠿⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠉⠀⠀⠠⣀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⣿⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡏⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠰⣄⠀⠀⠀⠀⠀⠀⠀⠙⢷⣅⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠊
⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⡀⠀⠀⠀⢀⡴⡏
⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⡠⠀⠉⠿⣦⣀⣠⡟⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠹⢦⡀⠀⢤⠜⠁⠀⠀⢀⣨⢿⡿⣡⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⡰⠋⠽⠗⣾⠧⣤⠴⠛⠁⠸⣿⣯⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣦⣞⣓⣐⣀⣀⣀⡼⠃⠀⠀⠀⠀⠻⣯⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⡄⠈⠩⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣯⣾⣾⣿⡿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠻⠟⠻⠿⠿⠋⠁⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡎⠿⠀⢠⣾⣿⢿⣶⡀⢠⣶⣿⢏⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⣦⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢠⠀⠀⠀⠀⢀⣀⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⣿⣎⢷⣄⠀⠀⠀⠀⠤⠤⠂⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⠶⠖⠚⠹⣿⣿⣿⣿⣿⡄⠻⣷⣶⡀⠀⠀⠀⣠⡟⢦
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠈⠻⢏⣛⠛⢀⣴⠋⣿⣷⡶⠞⠁⠀⢸⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠈⠉⡴⠀⢈⣵⠏⠉⠀⣴⣆⠀⡄⠀⠀⠈⡢⠉⡏
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⡴⠋⠀⢰⠞⢁⠂⢠⠘⢻⣿⡗⠀⠀⠀⠺⠄⡀
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⡱⠀⠋⠋⠛⠟⠙⠚⠃⠀⣼⣇⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⢖⣶⡶⠟⠋⠀⣼⡁⠀⠀⢶⠶⡶⠶⠤⠀⠀⣿⣿⡄⠀⠀⠀⠀⢸⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠧⣿⠗⡿⠃⠀⠀⡀⢸⣿⡇⠀⠀⣀⣨⣧⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠸⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣠⠃⡼⠃⠀⠀⠀⢱⣿⣟⣿⡂⠀⠁⢸⣇⡀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠸⢿⣆⠀⠀⠀⠀⡠⣼⣿⡙⣾⣿⣶⣿⠛⠋⠀⠀⠀⠀⢻⣿⣿⣷⠀⠀⠀⠀⡎⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⢀⡀⢳⣝⠻⠿⠞⠛⠻⠿⠇⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⡇⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣿⡇⠀⠀⠀⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡏⢻⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣛⣛⣙⣻⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣿⣯⣿⣯⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⢉⣻⣿⣶⣤⣤⣀⣀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⣶⣶⡶⠈⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠁⠀⠀⠉⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠚⠁⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⢃⡶⠴⠾⠿⠧⠄⠒⢬⡣⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠒⠈⠉⠁⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⡼⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡄⠀⠀⠀⠀⠀⣼⣿⣿⣷⠃⠀⠈⠉⠉⠉⠉⠚⠆⠀⣭
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠰⣿⣿⣿⡇⠀⠀⠀⠀⢺⣿⣿⣿⣿⠀⠠⠴⠇⠀⠀⠀⠀⠀⢶⡏
⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣀⣼⣿⣿⣿⡇⠠⠀⠀⠀⠀⠀⠀⡀⠀⢸
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⢀⣿⣿⡿⠀⠀⠀⠀⠈⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣹⠇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠺⢿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣶⣤⣤⣤⠴⣺⣿⠇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢐⣿⠀⠀⠀⠀⠀⢈⡿⣿⠀⠀⠐⠒⠚⠛⠮⠄⠀⢸⣿⣿⠏⠀⠉⡅⠀⣿⣿
⠀⠀⠀⠀⠀⠀⡄⠀⠀⣸⣿⡀⠀⠀⠀⠀⡾⠓⣿⠀⠀⠀⢀⣀⠀⠛⠒⠲⢾⣿⡟⠀⣠⡾⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠱⡀⣤⣾⣿⣇⠀⠀⠐⣶⣶⣶⣿⡀⣤⣴⣿⣿⡷⠀⣀⣤⣿⣿⣷⣾⢛⣷⠆⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣄⣠⣿⣿⣿⣿⣿⣇⣸⣿⣿⣿⣷⣾⣿⠿⠛⠛⠛⠣⠄⢀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⣠⠚⠉⠛⠒⢦⠀⠀⠀⠀⠀⠀⠈⡇⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣼⠃⠀⠀⠀⠀⣾⠀⠀⡰⠀⠀⠀⢰⣿⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⢧⣀⣀⣀⣀⡼⠛⠂⢰⠁⠀⠀⢠⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⢯⣀⣀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⡟⣡⣾⣿⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠈⠉⠁⠉⠉⠀⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⠀⠲⡿⠿⠭⠀⠻⢫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠉⠁⠀⡀⠀⠀⠀⠄⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠢⠴⠊⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢶⡃⠐⠋⠀⠀⣼⣿⢳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠱⡀⠉⠀⠀⣠⡾⡿⠃⢸⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⣑⣽⠗⢢⣼⠛⡟⢛⡭⠊⠀⢀⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⣩⡝⣢⣽⣯⣴⣟⡍⠀⣠⠖⢰⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠉⠉⣹⣷⣹⣿⡿⢙⣿⡞⠐⠁⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⠀⠉⣵⣎⣿⣿⡇⢸⣷⡷⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏⠳⡒⢻⣿⣿⣿⡇⣟⣥⡾⠀⣼⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠠⣿⣿⣧⠳⣄⠀⢾⣿⣿⣿⣷⡿⢋⣄⣼⣿⣿⣿⣿⣿⠿⣛⣩⣤⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⡈⠛⢘⣿⣿⣿⣿⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⢋⣼⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⠿⠛⠟⠁⠀⠙⠋⠙⠃⠀⠀⠛⠋⠈⠟⠻⢿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠘⡆⠀⢲⠀⢸⡇⠀⡶⠀⢠⠇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣠⠞⡟⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⢧⠀⢹⡄⠸⠆⠘⠇⠠⠇⢀⡏⠀⣰⠃⠀⡴⠋⠀⠀⠀⠀⢀⣀⠀⣀⡼⡿
⠀⠀⠀⠀⠀⠀⠀⠙⠫⣍⠁⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣠⠞⠁⠀⠀⠀⠀⠀⠙⠻⣍⠉⠀⠳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣠⠖⠒⠻⠗⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣠⠴⠒⠛⠛⠛⠓⠶⣤⣄⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠖⠒⠺⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⣠⢶⣿⣤⠞⠋⠳⣤⡀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⢀⣴⡞⣿⣹⣭⢯⣿⣉⡛⠶⣄⠀⠀⠀⠀⢸⡇⠀⣰⢏⣡⣬⡁⠀⠀⢀⣤⣄⡙⢦⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠶⠞⢛⣛⡛⠳⠦⣄
⠀⠀⠀⢀⣴⣏⡶⠋⠁⢰⣶⣀⣀⠀⠉⠳⣌⠳⣄⠀⠀⠸⡇⠸⠃⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠈⠃⢠⡇⠀⠀⠀⠀⣠⠟⠡⠶⠛⠉⠉⠉⠛⠂⠈⠳⡄
⠀⠀⠀⣾⣳⠏⠀⢠⣾⠛⣿⡏⠙⣿⠄⠀⠈⢧⠘⣆⠀⣼⢳⠀⠀⠀⣠⣄⠀⠀⠀⠀⣠⣄⠀⠀⠀⠚⣧⠀⠀⠀⢰⡏⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀⢹⡄
⠀⠀⢸⢯⡟⠀⠀⢸⣿⣷⣿⣷⣤⣄⡀⠀⠀⠈⡇⢹⠀⣧⠀⣤⠀⠀⠻⠟⠀⢠⠀⠀⠻⠟⠀⠀⣠⠀⣼⠀⠀⠀⢸⠁⠀⢀⡴⠛⠁⠀⠀⠀⠰⡇⢹⢘⡇
⠀⠀⢸⠒⡇⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣦⠀⠀⡇⢸⠃⠘⠳⠿⡄⠀⠀⠀⡶⠶⠶⢶⠀⠀⠀⢠⠿⠞⠃⠀⠀⠀⢸⡇⠀⠘⣳⠀⠀⢀⡾⠇⠀⠉⠉⣸⠇
⠀⠀⠘⣯⢿⡀⠀⠀⢰⣦⠀⣿⡇⢹⣿⡟⠀⣸⢃⡾⠀⠀⠀⠀⠙⢦⣀⠀⠙⠛⠛⠋⠀⣀⡴⠋⠀⠀⠀⠀⠀⢠⡟⠀⢠⠞⣡⠎⠳⠋⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠹⣆⠳⣄⠀⠈⢻⡷⢿⣿⣟⡉⢀⡴⢃⡾⠁⠀⠀⠀⠀⠀⠀⣨⣿⠃⠀⠀⠐⣿⣍⠀⠀⠀⠀⠀⠀⢀⣾⣇⠀⠁⢞⣡⣀⡴⢦⣄⣤⠴⠋⠁
⠀⠀⠀⠀⠈⠳⣌⡙⢶⠏⠀⣸⠋⠀⣿⠯⣴⠋⠀⠀⠀⠀⢀⣠⠶⠛⢧⣙⣳⠖⠲⣞⣋⡼⠛⠶⣄⡀⠀⢠⣿⣿⣿⣷⠶⣾⠀⠹⣆⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠈⠙⡿⠀⢰⡇⢀⡾⠁⢠⠿⣄⠀⠀⠀⣴⠋⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠀⠀⠀⠈⠙⣦⠘⢿⣿⣿⢿⡄⠈⢷⠀⢸⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠺⠀⢸⠁⢠⡏⢀⡟⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⢳⡄⢹⡀⠈⡇⠀⠿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⠀⠸⡇⠀⣸⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⣇⠀⢸⠇⠀⠋⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⡿⢠⡏⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⣿⠀⠀⠀⠹⡄⢿⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⢤⣤⠴⠞⢻⡟⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⢻⡟⠳⠦⣤⡤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⣧⠀⠀⢀⡞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⡀⠀⠀⡼⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠸⠆⠀⡾⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢷⠀⠰⠇⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⡼⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢧⡀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠴⠞⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠳⠦⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠒⠲⠦⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠒⠲⠒⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡀⠀⠀⣈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠀⠀⠀⠀⠻⠷⠶⢶⣶⣾⣆⠀⠀⡄⢿⡄⣠⣿⣤⡦⣤⣴⣴⣦⣿⣶⠿⠿⠿⠚⠋⠉
⠀⠀⠀⠀⠀⠤⣤⡤⠤⠤⠶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣦⣄⣳⣬⠿⠟⠛⠛⠉⠙⢯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠋⠩⡄⠀⣰⡄⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠷⣤⣼⣧⠦⠿⠷⢶⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⣀⣀⣤⣶⣿⣿⣿⣿⣤⣄⣾⣾⣇⣀⠀⡀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢠⢠⣬⠉⠙⢛⠩⠝⠻⠙⢻⣿⣀⣀⡄⢉⡍⠉⠉⠙⠿⡟⠋
⠀⠀⠀⠀⠀⣀⠀⣤⡠⠡⠸⠂⢉⣈⣉⣀⢷⣦⣄⣤⡀⠀⣁⣿⣿⣿⣷⣾⣤⣤⣤⣄⢀⠆
⠀⠀⠀⠀⠀⠀⠲⠚⠛⠛⠛⠛⠋⠉⠉⠙⠻⠿⣭⣛⣟⣿⣿⣿⣿⠿⠟⠛⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣇⣀⠐⣻⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⠿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠹⠿⠿⠿⠿⠿⠟⠋⠁⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣌⢿⣿⠀⠀⢀⣤⣤⣀⣀⠀⠀⠀⢀⣠⣄⠈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢹⠆⠛⠀⠀⠿⠿⠿⢿⣿⡇⠀⣾⠿⠿⠢⢤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢃⠁⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⢀⣶⡀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⣆⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠸⣦⠀⠀⡀⠀⠛⡒⠒⠒⢛⡊⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⢿⣿⢻⣿⡆⠹⣷⣄⠀⡀⠀⠈⠙⠛⠁⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠶⠚⠛⠛⠉⠀⠑⠏⢸⣶⡆⣿⠇⣆⠀⠹⠓⣷⡀⠀⠀⠀⠀⠘⠃⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢈⡀⠚⠁⣠⠊⠉⢫⢾⡆⠚⠃⠀⠀⠸⡆⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠉⠀⣀⠀⠀⠀⣴⠃⠘⠀⢠⣦⠀⠀⠀⠁⠀⠀⠙⣌⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⣠⠞⠁⠢⣴⡟⡁⠄⢀⠐⢿⣿⣿⡆⠁⠀⠀⢤⣖⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡁⠀⢦⣼⣧⣶⣶⣤⣼⣆⠀⠙⢿⡀⠀⠀⠀⠀⠈⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣀⣠⣼⠃⠀⣰⠇⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⠀⠀⠀⢡⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⡴⢚⣻⡿⠟⠛⠁⢀⣼⣧⡀⠀⠀⠻⠿⣿⡟⠓⠒⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⢸⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣿⡾⣻⠟⠀⠀⠀⠀⢞⣿⣿⠃⠀⠀⣀⣀⣽⣄⠀⠀⠂⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⢸⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⠏⣰⠏⠀⠀⠀⠀⢱⣾⣿⣻⣷⠀⠀⠉⠉⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⢀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠉⠀⠇⠀⠀⢀⣸⣿⡿⣿⣿⣶⣤⣴⣶⡿⠟⠁⠀⠀⠀⠈⣿⣿⣿⣷⡀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣷⠀⠀⠀⠀⠘⣌⢿⣿⣷⢬⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⣽⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠇⠀⠘⢦⡖⣽⢧⡍⠙⠊⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢿⡀⠀⠀⠀⠃⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⡏⢹⡄⠁⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣷⣤⣤⣶⠶⠶⠞⠋⣉⣉⢭⡉⢉⡭⣉⣉⠙⠳⠶⠶⣶⣤⣤⣾⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⢭⣽⣶⣦⣤⡈⢉⣓⣬⡳⠄⠠⢞⣥⣛⣉⢁⣤⣴⣶⣮⡭⣠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢟⡥⠒⠭⠙⠻⢦⠙⢿⢳⡼⣧⡞⡿⠋⡴⠟⠋⠍⠒⢬⡻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡵⠋⠀⢠⣵⣤⣀⠈⣇⣧⣼⣱⣎⣇⣼⣸⠁⣀⣤⣮⡄⠀⠙⢮⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⢺⡏⣠⣤⡌⢳⢸⡼⣿⣿⣿⣿⢧⡇⡞⢡⣴⣄⢹⡓⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢁⠀⠀⢀⣄⢧⡙⠛⣃⡾⠀⢷⣽⣿⣿⣯⡾⠀⢷⣘⠛⢃⡼⣠⡀⠀⠀⡈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣷⠏⣀⠀⠘⠿⠷⣍⠿⠋⠀⡀⢸⡿⠉⠉⢿⡇⢀⠀⠙⠿⡩⠾⠿⠃⠀⣈⠹⣾⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢯⠻⡜⣴⠋⣠⡀⠀⠈⠀⢀⡄⢀⣯⣾⡇⠀⠀⢸⣷⣿⡀⢠⡀⠀⠁⠀⢀⣄⠙⣶⢣⠟⡽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣦⡑⠝⠜⣦⡏⠀⢀⡶⢀⣼⣷⣿⣿⣿⣷⢠⡄⣾⣿⣿⣿⣾⣧⡀⢶⡀⠀⢹⣴⠣⠫⢊⣴⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣤⡈⠻⠶⠿⣿⣿⣿⣿⣿⣿⣿⣿⡾⢷⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⠟⣁⣤⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣶⣄⡙⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠹⢋⣠⣶⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠠⣤⣷⢛⣿⣂⣀⡠⢄⣤⣼⣿⠆⠀⠀⠠⠀⠤⠀⠀⠀⠀⢿⣶⣚⣁⣀⠀⣠⣦⣤⡄
⠀⠀⠀⠀⠀⠀⡇⢠⣶⣦⣿⣿⣶⣶⣿⢷⣆⣀⣀⣀⣀⣀⣀⣠⣄⣴⣾⣿⣿⣿⣿⠾⢿⡏⢀⣦
⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⠟⢁⣼⣿⣟⣿⡍⢹⣽⣿⣿⠿⠿⡿⠿⠋⠀⠀⠀⣠⡄⠀⡿⡇
⠀⠀⠀⠀⠀⣸⡇⢸⣿⣿⢿⠗⠀⢠⣾⡿⠛⢃⣬⣩⣿⣟⣿⠀⠀⠀⠀⢠⣶⣦⣄⣀⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⡷⠀⢸⠟⠁⠀⠀⢀⡿⠟⢁⣶⣤⣿⣿⣿⡉⣿⡇⠀⠀⠀⠻⢿⣿⣿⣿⣿⡇⠀⣿⢿⡇
⠀⠀⠀⠀⠀⠇⡆⢸⠀⠀⢀⠀⢸⠗⢀⣸⣿⣿⣿⣿⠟⠛⢿⡇⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠘⣿⣼⡇
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠓⠈⠘⠀⠚⠙⢿⡿⣯⣧⠄⣸⣷⠀⠀⠀⠀⠀⠉⠙⢻⣿⡇⢰⡟⡉⠁
⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠈⠁⠀⠂⠀⠀⢠⣿⣿⠍⢲⣿⡇⠀⣷⠀⠀⠀⠀⠀⠐⠛⠃⢈⡿⣧
⠀⠀⠀⠀⠀⢠⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⠟⠉⣀⠀⢻⣿⣇⣸⡆⠀⢠⣄⣠⣤⣶⡆⢸⣟⡿
⠀⠀⠀⠀⠀⠸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠒⢺⣓⡼⠛⠁⠈⣿⢿⣿⣷⣶⣿⣿⣿⣿⣿⣯⢸⡿⠁
⠀⠀⠀⠀⠀⢀⡁⢸⠀⠀⠀⣀⣴⣶⣶⣶⢰⣶⡶⠀⣆⢰⡄⣿⣿⠋⡽⢿⣿⡿⣿⣿⣿⣿⢸⣅
⠀⠀⠀⠀⠀⢨⡇⢸⣶⣿⣾⠿⢿⣿⢿⣿⢸⣿⣓⠘⠿⠻⣿⣧⠃⢀⡿⢘⣿⢃⣿⣿⣿⣿⠸⣿⠁
⠀⠀⠀⠀⠀⠐⡇⢸⠟⠀⠁⣠⢸⣿⡿⣿⣼⣿⣿⠦⢠⡄⢾⡿⠀⢠⠀⢸⣿⣾⣿⣿⣿⣿⢠⣿
⠀⠀⠀⠀⠀⠀⡇⢸⣰⣿⠀⣿⣿⣿⣇⠸⣏⣿⣾⣿⣿⣿⣼⡿⠀⠏⠠⢼⣿⣧⣉⠿⣿⣿⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠇⢸⣿⣿⠀⣿⠛⣩⣿⡄⣸⣿⣿⣿⣿⣿⣷⡤⠁⣀⠀⠀⢿⣿⣿⣷⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠁⢸⣿⠁⠰⡇⠘⠁⢿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠾⣿⠀⢰⣾⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠆⢸⣿⣀⣴⡇⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣸⣿⣿⣿⣿⣿⣿⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣠⣿⣿⣿⢟⣻⣿⡟⠘⠛⠄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢀⣼⣿⣿⣿⣿⡿⠛⠛⠻⠿⠿⢋⠁⠉⠙⠛⢻⣿⣿⣿⣁⢠⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⠇⠀⠿⡏⢸⡿⠋⣁⣿⣶⣦⡀⡀⢌⣳⣤⠲⣄⣤⣿⣿⣿⣟⢸⣿⣁⡀
⠀⠀⠀⠀⠐⠀⡀⢸⣽⠀⠠⠾⣷⣆⣛⣛⣷⣿⠿⠛⠻⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⠁
⠀⠀⠀⠀⠀⢰⡇⢸⣷⣶⣶⣾⣯⣉⣉⣉⣵⣶⣶⣶⣿⣿⣿⣏⣩⣤⠵⠤⡼⢿⣿⣿⣿⣧⠈⣿⡏
⠀⠀⠀⠀⠀⠈⡃⠀⣭⠉⠉⠩⠉⠭⠭⠍⠭⠭⠭⠭⠭⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⣉⡉⠉⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠳⠖⢉⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠟⠡⡴⠁
⠀⠀⠀⠀⠀⠀⠢⡀⠂⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠈


⣶⣶⣶⠶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⡿⣿⣿⣿⡿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣟⢱⣿⣿⣿⣧⡀⠉⠉⠀⠀⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠟⣿⡿⠀⠈⣿⡿⢀⣀⣀⡀⣿⣿⣿⣰⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠰⣿⠇⢸⣿⣿⣇⠛⠋⠁⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⣾⣿⣷⡸⠛⠉⢸⠁⢀⣠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠿⠋⢹⡇⠀⣀⡘⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠸⠟⠉⣿⠀⢀⣘⢳⣾⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⢹⡇⠀⢀⠛⣶⣿⣧⢸⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢳⣾⣿⠀⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠃⠘⠟⠿⠀⠻⠿⠗⠘⠿⠛⠃⠻⠿⠟





⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣶⣾⣭⣭⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡏⠛⠛⠉⠁⠈⠙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠘⡿⡿⢿⠋⠀⠀⣰⡶⢶⠀⡴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢀⠀⠀⠀⠀⠀⠘⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢦⡀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠁⠀⠙⠳⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠢⡀⠀⠀⢤⣈⠉⣃⡀⣄⡀
⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠘⢆⠀⠀⠈⢣⡛⢿⡸⡈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄
⠀⠀⠀⠀⠀⣼⢄⣤⠄⠀⠀⠀⠑⠀⠠⠀⠱⣿⣿⣧⠐⡳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠸⡄⠧⣀⠀⠘⢿⣿⡆⠀⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⣠⡈⠃
⠀⠀⠀⠀⣾⣿⡿⠇⠀⠀⠀⠀⠀⠈⢇⠈⠑⢤⡈⢿⣿⡀⠀⠀⠀⠀⠀⠀⢸⡁⠀⢀⡠⠚⠉
⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠑⣄⠻⡇⠀⢀⡴⠀⠀⠀⠈⡇⠘⠁
⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠈⠂⣀⠔⠉⠀⠀⠀⠀⠀⢑⡆
⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣠⢘⣷⠖⠚⠁⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣤⡀⠀⠀⢠⠎⣿⣿⢹⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠈⢿⣎⢿⣿⣿⣦⡀⠈⠀⠟⠛⠀⠀⠀⠀⢀⣠⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢷⠁⠙⢿⣿⣿⣶⣄⡀⠀⠀⠀⢀⣴⣿⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⠀⠙⠻⣟⠉⠛⠓⢦⣴⣿⡟⠁⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠈⠙⢀⡀⠀⠈⠉
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠿⠿⠿⠿⠋⣡⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠈⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠁⢚⡓⠀⠀⠶⠦⠀⠀⠿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠁⠀⠀⠒⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢦⣈⡀⢀⡄⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠘⢦⡤⠛⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⣶⣿⠳⣤⣤⣤⣴⠾⠡⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⢾⡿⠿⠿⣷⣾⣿⠀⠀⠀⠈⠉⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠚⠉⠀⠀⠀⠀⢰⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠶⢤⣴⣿⡟⠛⠋⠉⠀⠀⠀⠀⠀⠀⢀⠴⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠙⠶⣤⣀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣀⣠⠴⠚⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠙⠻⣶⣤⠼⣣⠞⠉⠉⠉⠉⣿⡀
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⣠⠾⢯⣷⡶⠶⠞⠛⠉⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⣀⣀⠀⠀⢐⢺⡇
⠀⠀⠐⠇⠰⢄⡤⠤⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠏⠉⠛⠶⢬⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠲⠶⠶⠶⠶⠖⠚⠋⠁⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠈⠉⠙⣦⡀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⣿⡟⠶⣄⠀⠀⠀⠀⠀⠹⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⢰⠟⣇⠀⠈⠻⣦⡀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠈⠀⠘⣆⠀⠀⠈⠻⠷⢶⣾⣿⣿⡿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣠⣤⣴⠟⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠟⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠽⣏⣡⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⣡⠾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢀⠰⣶⠚⠃⠀⠀⠀⠀⠀⣀⣀⣀⣸⡏⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⣧⠘⣇⠀⠀⣀⣀⣀⣸⠉⠉⠉⢉⡤⣦⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⡟⢘⣷⣏⣀⣼⠉⠉⠉⠉⠀⠀⢸⡏⢀⣹⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠭⠤⠼⠃⠀⠀⠀⣀⣀⣠⡤⠶⠚⣩⡴⠋⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⣞⣛⣛⣉⣉⣩⣭⠤⠶⠚⠋⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠶⠾⠿⠯⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠇



⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢟⣵⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠛⠿⠇⠸⠛⠋⠁⢹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢀⣠⣤⠤⠈⠹⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣞⠿⣷⡀⠉⠟⠙⠛⠂⠀⠋⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⢤⡀⠀⠀⠀⢀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠲⠶⠛⠁⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠐⢶⡋⠕⠛⠁⠀⢀⣿⣿⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢣⡀⠉⠁⠀⢀⣴⣿⡿⠁⣸⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠚⢉⣷⠶⢄⣱⠶⣶⠶⢛⡫⠚⠀⢀⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠴⠞⣛⠿⡀⣻⣿⢀⣽⣖⠋⠀⢀⠔⢸⣿⣿⣿⣿⡿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠲⠤⠼⣿⡖⢿⣿⣿⢿⣿⢫⡄⡰⠁⢀⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠤⠴⢻⠷⣾⣿⡟⢠⣟⣻⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣇⠠⢴⣿⣶⣿⣿⡇⢸⡿⢛⡀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣋⠈⠩⠁⣾⣿⣿⣿⡇⣾⣾⡿⠃⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡘⢧⣄⠈⣿⣿⣿⣿⣿⠟⣡⣴⣰⣿⣿⣿⣿⣿⣻⣟⣫⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣄⠈⠃⣸⣿⣿⣿⣿⣾⡿⣿⣿⣿⣷⣿⣟⣽⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣟⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⢟⣱⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⢳⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⢿⡿⠉⠁⠻⠟⠻⠟⠁⠀⠙⠿⠟⠹⡿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠛⠀⠀⠀⠀⠉⠉⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣿⣿⣿⣟⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣻⣿⣿⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠿⠛⠛⠋⠙⠻⡿⡿⠷⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠁⠀⠀⣀⣤⣤⣤⣤⡄⢻⣶⣧⣄⠀⠀⠉⠉⠒
⡛⠂⠈⠁⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠉⠙⠛⠿⣷⣦⣄⠀⠀⠀⠀⢦⣤⣤⣤⣀⡀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠗⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠳⣤⣀⡀⠀⠀⠁⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣄⠀⠀⠀⣴⣿⡿⠀⢀⣴⣿⣆⠀⠀⠀⠀⠈⡀⠀⠀⠈⠻⣿⣷⡄⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠛⠛⠛⣿⣿⣿⣷⣶⣾⣿⣷⣤⣴⣿⣿⡏⠁⠁⡀⠀⠀⡹⠛⠿⠒⠂⠀⠙⢿⣿⣷⠂⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⠀⠠⣧⠠⠤⠐⢦⠲⠤⠰⠿⠿⣿⠙⠛⠛⠛⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠙⠳⡖⠒⢸⣿⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣼⣦⣀⠀⢸⠇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠻⠿⠿⠟⠛⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡴⠖⢤⡀⢀⣀⣠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠒⠲⢤⣴⣿⣿⣧⠀⠀⢻⣿⡛⠀⢀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠶⠲⣶⣾⢿⣵⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⠖⠈⢣⣄⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠽⣿⣿⣿⡇⠀⡀⠀⣻⣿⣿⠀⣠⣶⣿⡇⡌⡤⠻⠟⠛⠁⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⣿⡷⠀⠋⢸⣿⣿⣿⣻⣿⣿⣿⣿⣿⠋⢀⣶⣿⣿⢿⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⣿⣿⢏⡾⠁⠀⣼⣿⢫⣿⠿⠃⡿⢝⣿⡿⢶⣽⡿⣵⣶⠈⠉⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡅⠀⠀⣿⣿⡓⢜⠁⠀⢾⡿⠛⡿⠀⠀⠠⠈⢿⡖⠊⢻⡸⣿⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⣿⡿⠁⠀⠁⣠⣿⡁⠞⠉⠀⠀⠀⠀⢸⡿⠀⠀⠙⠿⠵⠶⠶⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡃⠰⠐⠿⣷⠀⠀⠉⠀⣿⣷⣶⣿⣶⣒⡠⠤⣸⡁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣄⠀⠀⠈⢣⡀⠀⠀⢸⣿⣿⠃⠀⠀⠉⢳⡄⣉⠒⠮⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣶⣿⣿⣷⡒⠒⠟⠻⢦⣄⣼⡿⣇⠀⠀⠀⢀⣼⠳⠁⠀⣰⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⡿⠋⠙⠛⠒⣶⠦⠘⣿⣷⣬⡉⠙⠒⠛⠁⠀⠀⠰⠋⠀⢠⡗
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠏⣿⣿⣿⡆⠀⠀⠀⠀⠇⠁⠀⠈⠻⢍⠻⠿⠶⠶⠶⠶⠄⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣟⠋⢀⣾⣿⠋⠐⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠙⣲⠆⠀⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣴⡿⠗⠁⠀⠀⠀⠀⠀⠈⠢⢀⡀⠀⠀⠀⠀⣸⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣟⠉⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⠾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢦⣤⣀⣀⠀⠀⠀⠀⣠⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣽⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⡿⣿⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⣾⡿⠿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢳⢟⣯⣭⣝⢻⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⣿⣿⣿⣿⣸⣬⣶⣶⣮⣵⢸⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⢹⣿⣿⣿⣧⡻⢿⣿⠿⣫⣶⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣧⢿⣿⣿⣿⣷⡰⣶⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⢻⣿⣿⣿⡎⢿⠿⣿⣿⣿⣾⣝⠻⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⢻⡿⢿⣿⣮⣛⠿⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣬⣛⠿⠿⠛⢀⣬⣽⣶⣶⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁⢀⣾⡿⠿⢿⣿⣿⣿⡟⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢱⣾⣿⠸⠿⠿⠿⠘⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣦⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣯⣴⣶⣤⣠⣤⣬⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠉⠉⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣏⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠑⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣠⣶⣦⣤⣀⠀⢀⣀⣤⣶⣾⡶⠀⢸⣡⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠘⠩⠙⣟⣿⡏⠘⠿⠯⠝⠝⠉⠀⠘⢼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠛⢻⠇⠀⠀⠀⠀⠀⠀⡀⠀⣦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣧⠀⠀⣰⣷⣄⢀⣤⠶⣄⠀⠀⠁⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠂⠸⠀⠈⠙⠋⣁⢤⡘⡄⢀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⣧⠀⠶⠶⠾⠯⠵⠾⠃⠀⢸⣾⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠷⣤⢸⡆⠀⠀⠒⠒⠂⠀⠀⠀⣼⠏⠀⣹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣛⣿⠀⣷⣀⣀⣀⡀⠀⠀⢀⡴⠁⠀⠀⢸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢟⣿⣦⡍⠛⣿⣿⠛⠛⠛⠁⠀⡀⠀⠀⢸⣿⣿⣿⣿⡛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⡦⢾⣿⠿⡇⠀⠈⠛⢳⣤⣤⣤⠉⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⣮⣳⢶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠁⣤⣤⣾⣿⣤⡿⢤⠄⢠⢾⣿⣿⣿⡆⠀⢀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⣝⠛⢶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⣠⠶⠴⣾⣿⣿⣿⣿⠋⠀⠉⣹⠏⠉⢀⣠⣴⠀⠾⢠⣿⣿⡟⠀⠄⠀⠀⣿⣿⣿⣟⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣇⣤⣿⣦
⠀⠀⠀⠀⠀⡼⢣⣿⣿⣿⣿⣿⣿⣿⣶⡶⢶⣾⣶⣶⡿⠛⠁⠀⣀⡀⠈⣿⣿⠀⠀⠀⢀⣿⣿⣿⣿⣿⢫⣿⣿⣿⣿⣿⣿⢟⣥⣾⣿⣿⣿⡏
⠀⠀⠀⠀⣼⠡⣾⣿⣿⣿⣿⣿⣿⣿⡙⠳⢦⣈⠙⠻⠿⠶⠖⠋⠁⠀⠀⠈⢿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠙⠷⢤⣄⣀⡀⠀⠀⠂⠀⠀⠈⠛⠦⢤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢸⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⣀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣌⣻⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣾⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⢦⣤⣀⠉⠁⢀⡀⠀⠀⠀⠀⢠⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣶⣝⠦⢙⠻⢶⣯⣭⣭⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢸⠀⢉⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆⠉⠻⠤⢤⠼⠶⠛⢻⡟⠻⠿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣦⣌⠛⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣤⠀⠀⣿⣿⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣝⣷⣶⣿⣿⣿⣿
⠀⠀⠀⢰⡇⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣰⣿⡆⠀⣿⣿⡆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⣾⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⡇⠀⣿⣿⣧⣀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⢀⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣾⣿⣿⣿⠀⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢿⣄⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣾⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏
⠀⠀⠀⠀⠙⢷⣮⣿⣿⣿⣿⣿⣿⡿⢟⣭⣾⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⢿⣿⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣁⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣤⠂⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣣⠀⠀⠀⠀⠀⠀⠻⣿⣛⡁⠠⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡜⠋⠀⠀⠀⠂⠀⠳⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢯⡽⠁⠀⠀⠀⠰⣦⢀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣥⡾⠁⠀⠀⠀⠀⠀⠐⠛⠛⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠁⠀⠀⠈⠀⠀⠀⣀⡴⢟⣵⣿⣿⣿⣷⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⢠⣾⣿⡿⠟⠋⡼⠃⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⢀⣾⣿⣿⣿⢆⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⣼⡿⠛⠉⢀⠘⠋⠉⠛⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠁⠘⢃⣤⣤⣥⢤⡞⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣧⣟⣴⠀⠄⠀⠀⠀⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⢉⣽⣍⠻⠛⠁⠈⠀⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣅⠙⠛⠀⠠⠀⠀⠀⠀⠀⠁⢀⠾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠦⠀⠾⠛⠋⠰⣶⣦⠄⠈⠀⠀⠀⢠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⡀⠀⠠⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠘⢻⡤⠀⠀⠀⣠⡄⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠼⠃⠀⠀⠀⠀⠋⠀⠀⠀⠀⢈⣵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⣆⠀⠻⠆⢠⡆⠀⠀⠀⠈⠀⠀⢻⣾⠂⢠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⠿⣦⣽⣷⣄⣠⣤⣎⣠⣿⣤⣾⣁⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣻⣶⢏⣍⣠⣿⣿⡆⠀⣍⣿⣿⣧⡙⠛⣳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⣻⣿⣿⣷⣶⣿⣿⣿⢿⣇⠀⢿⡿⢟⠁⣸⣷⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠁⠻⣿⣿⣿⣿⣷⣾⣧⡉⢿⣿⡆⠸⣏⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⣾⣿⡿⣿⣿⣿⣿⣿⠙⣶⣽⣇⠀⣟⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⠀⠙⢾⣿⣿⣿⡟⠀⢿⣿⣿⡄⠸⢷⡔⢈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣾⡟⢿⡟⠋⠀⠀⣨⣿⣿⣿⣷⣦⡴⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢨⡇⣼⡇⠀⠀⢸⣿⣟⠿⣿⣿⠀⠙⣆⠀⣠⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⣸⠃⢸⡷⡄⢰⣿⣿⡇⠀⣽⣿⠀⠀⠘⠟⠋⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡉⢰⣯⠀⢸⣷⣇⣼⡇⢹⣧⠐⠻⡎⠟⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠈⠡⠀⠀⡟⢻⣿⣷⣾⠁⠀⠀⠠⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠇





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⠟⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣀⣤⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠛⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠔⠚⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡍⠉⠀⢋⡽⠃⠀⠀⣰⠓⠲⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠷⠀⠻⢚⡟⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⣠⠏⠀⠀⢲⣴⠇⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣟⠁⠀⠀⠀⢸⠀⠈⠛⣲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢸⡆⠀⠀⠀⢿⡤⠖⠋⠀⣀⠀⡰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠛⠦⡄⠀⠀⠀⠀⠀⢠⡏⠀⠀⢸⠃⢠⠞⠙⣦⠀⠀⠀⢠⠏⠻⢁⡤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢹⠀⠀⠀⠀⢀⡞⠀⠀⢠⠏⢀⡎⠀⢠⡇⠀⠀⠀⣼⡤⠴⠚⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡞⠀⠀⠀⢀⡞⠀⠀⣠⠋⢠⠎⠀⢀⡞⠀⣠⠖⣦⠀⠀⠀⢠⣄⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠁⠀⠀⣠⠏⠀⢀⡴⠁⣰⠋⠀⢀⡞⢀⡴⠃⢠⠏⠀⠀⢀⡎⠈⠁⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⢧⣠⠞⠁⠀⠀⢾⣤⠜⠁⠀⣠⠏⣠⠋⠀⡰⠋⠀⡀⠀⠞⠒⠒⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⣴⡧⠞⠁⣤⠞⢁⡴⠋⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⠴⠋⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⣀⣀⣀⣤⠴⠞⠉
⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⠀⠀⣠⠞⠉⠉⠁
⠀⠀⠀⠴⠚⠁⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠐⠛⢛⢻⠷⢶⣦⡤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠔⣁⠀⠀⠉⠁⠀⠀⢩⣺⣷⡰⡄⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⣀⡠⠴⢚⡟⣏⡞⢁⠄⢠⠔⣊⡩⠉⠉⠛⢿⣿⣿⡔⡀⢂⠘⣦⣀
⠀⠀⠀⢀⡴⠋⠀⠀⠈⠋⠐⣿⡰⡃⢀⡔⢉⡤⠤⢄⠐⠦⣭⣻⣿⣧⣀⠀⠂⠈⠻⡷⣄
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢥⣾⣷⠁⢋⠞⢁⡴⠞⠉⠁⠀⠤⣜⣿⣇⢹⡰⣶⠀⠀⠉⢸⣷⡄
⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣨⣺⣯⢳⢏⣴⣫⠆⠀⠀⠀⠀⢐⣿⢸⣗⠘⡿⡏⢠⡔⠀⠀⠈⠻⣄
⠀⠀⢹⣦⠑⢄⠀⠀⠀⠀⠚⢴⢿⡼⣿⠋⠀⠀⠀⠀⠀⠀⣶⣿⣸⣷⢸⡇⢳⠈⣇⠄⠀⠀⠀⡘⠑⢄
⢀⡴⠻⢧⢷⡈⠆⠀⠀⠀⠀⠐⢽⣿⣷⡀⠀⠀⢀⠀⡀⣿⣿⣣⣿⣟⡈⠁⢺⡄⣿⡩⠀⠀⠀⠑⠄⢘⡇
⡾⠀⠀⠀⠘⠻⡌⠂⡀⠀⠀⠀⠀⠋⠿⣝⣳⣶⣷⣞⣚⣭⣿⣿⣿⣽⡃⠀⢸⣿⡷⡖⠀⠀⠀⠀⠘⣆⣧
⣷⡔⠀⠀⠀⠀⠁⢀⡇⠀⠀⠀⠀⠀⠀⠘⡜⡽⡽⡹⠹⠻⡻⣿⣛⠋⠙⣷⣾⣿⣷⡀⠀⠀⠀⠀⠀⠐⢿⡄
⢹⣿⣿⣦⠀⠀⠈⠭⢙⣾⢀⠀⠀⠀⠀⠀⠈⠝⡱⢣⣷⢰⣷⣼⡯⠤⢤⡘⣿⣿⢥⣒⡀⠀⠀⠀⠀⢙⣷⣷⡀
⢨⣇⠈⠻⣇⠀⠀⠀⠙⠿⢿⡆⠀⠀⠀⠀⠀⠈⢡⣿⢃⣿⠻⠋⡿⡎⠁⠈⠊⣿⣿⣤⠠⠤⡀⠀⠀⠀⠛⠿⣧
⢸⠾⠟⢄⣺⣇⡀⠀⠀⠈⢉⢳⣴⢂⠀⠀⠀⠀⠐⣣⠋⠃⠁⠐⠃⠀⠀⢀⣳⣿⣗⣂⠀⠀⠀⠀⠈⠉⠉⢛⣚
⠸⡄⠀⠀⠉⠻⣿⣄⠀⠀⠀⠠⢹⣾⡔⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡀⣠⣿⠯⠷⠦⠀⠀⠀⠀⢀⠀⡄⠀⡀⢀
⠀⠹⣗⡄⠀⠀⠙⣿⣧⠀⠀⠀⠑⠽⣷⣶⠇⠀⢀⠀⠀⢀⢀⣶⣿⣧⣿⣫⠥⠀⠀⠀⠀⠀⠰⠪⠓⠊⠆⢨⡠
⠀⠀⠈⠻⣵⣂⠄⠘⢝⣿⣄⠀⠀⠈⣪⢿⣿⣦⡮⠶⠴⣧⣿⣿⣾⡿⣓⡚⠃⠀⡄⣠⣤⠴⠀⠠⣵⡾⠖⠊⠁
⠀⠀⠀⠀⠈⠑⢿⣢⣀⠪⢿⣧⣄⠀⠐⢵⣿⣿⣿⣿⣿⢿⡿⠉⢰⣏⡖⠀⢠⡴⠋⠁⠈⣲⡽⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠳⢶⣀⠈⢿⣾⣀⢀⡉⢀⡽⠟⣡⢞⠃⠀⠘⣗⠏⠀⠋⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣬⡋⢿⣿⣿⣿⣶⠟⡡⠀⠀⠀⠀⠹⣭⡆⠀⠀⠀⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣲⣾⣿⣽⣟⣋⣀⣀⣀⣀⣀⣀⣈⣙⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠦⡀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠤⠤⠤⠤⠤⠄⠠⠤⣀⠀⢀⡠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢀⡀⠤⠌⠛⠉⠉⠉⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡤⠒⠀⠉⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠉⠁
⠀⠀⡷⠂⠐⠒⠂⠉⠉⠁⠀⠀⢠⠔⢊⣱⡁⠀⠁⠀⠁⠀⠀⠈⠉⠉⠉⠒⠪⣉
⠀⠀⠑⢤⠀⠒⠀⠒⠒⠀⠀⠀⢢⡀⠀⠈⠛⠶⠤⠤⠤⠤⠒⠁⠀⠀⠀⠀⠀⠈⠣⣀
⠀⠀⠀⠈⠑⢖⠒⠒⠒⠂⠀⠀⢸⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠱⣄⠈⠁⠒⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢆⠀⠢⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠙⢦⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠣⡈⠳⣄⠀⠀⠀⠀⠀⠈⠑⠠⢅⡀⠀⢀⣀⠤⠄⠒⠊⠉⠁⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠛⠉⠉⠉⠙⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⣀⣴⣶⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⢀⣼⡿⠋⠁⠈⠙⢿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠿⠿⠿⣿⣿⣶⣄
⣸⣿⠁⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠈⢿⣿⣿⣇
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟
⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃
⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏
⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣀⡀⢸⣿⣄
⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠉⠙⠛⠿⣶⣤⣀⡀⢼⣿⣿⡟⠋⠉⠙⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣧⡀
⠀⠀⠀⠀⣸⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡀
⠀⠀⠀⢰⣿⡏⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡄
⠀⠀⠀⣿⡿⠁⠀⠀⠈⠙⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠻⣿⣿⣿⣷⣶⣤⣤⣤⣤⣴⣶⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣷⣶⣶⣶⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠙⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠇
⠀⠀⠀⠀⠈⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏
⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣷⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣿⣿⠿⠛⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠿⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⢀⣶⠀⢠⣦⠀⢀⡈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣿⠁⢠⣾⠃⢰⡿⠃⢠⣿⠃⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⣀⣴⣶⣾⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣧⣤⣿⠁⢠⣿⠃⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⠏⣠⣶⣶⣦⡙⣿⣿⣿⣿⣿⣿⣿⣷⣿⠃⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣷⣌⣙⣛⣁⣴⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠿⣿⣿⡿⠿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢹⣆⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⡾⠀⠀⠀⠛⠛⠛⠋⠉⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⠟⢷⡀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣠⡞⠁⠀⠀⠙⢦⣌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠋⠀⠀⠀⠀⠀⠀⠙⠋⡀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢿⡻⣆⢻⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡙⢷⣌⣷⡘⢿⣷⠘⣧⡾⢻
⠀⠀⠀⠀⠀⢰⣦⡀⣿⣇⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣽⣦⠉⠛⠁⠈⠉⠀⠛⢃⡟
⠀⠀⠀⢠⣦⣌⣿⡻⣿⢿⡿⣿⣴⢻⡆⠀⠀⠀⠀⠀⠈⠻⢦⣤⡀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠹⢮⡛⠃⠀⠀⠀⠙⢣⡞⠀⢀⣀⠀⣤⡀⢠⣄⠀⠈⢿⣄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢠⡏⠀⠀⢸⡝⣿⡇⣿⣿⢹⠀⠀⠀⣿⠀⠀⠀⠀⠸⣧
⠀⠀⠀⢀⣤⡀⣼⢧⢀⡶⣦⢸⡇⠀⣶⠾⣷⠈⠃⠘⠋⢾⡾⢻⠆⣿⠀⠀⢠⣦⡀⣿⡆⢀⣤⡄
⠀⠀⡶⣼⡏⣷⣿⢸⣿⠃⡿⢸⠁⠀⠈⠳⣄⠀⠀⠀⠀⠀⣰⠏⠀⣿⢀⠀⢸⡌⣷⡟⣿⣸⢃⣇⣴⡦
⠀⠀⢷⡘⣿⠘⠟⠈⠛⢸⣧⡿⣦⠀⠀⠀⠹⡆⠀⠀⠀⣴⠃⠀⠀⣿⣏⠻⣾⡇⠻⠃⣿⡟⢸⡟⣹⠇
⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢁⣴⡃⠀⠀⠀⢠⡇⠀⠀⠀⣿⠀⠀⠀⣿⠙⢷⡀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⢀⡇⠀⠀⠀⢠⡴⠋⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢿⡀⠈⠷⣄⣀⠀⠀⠀⢰⡏
⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⢸⠃⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢹⡄⠀⠀⠈⣷
⠀⠀⢀⣿⠀⠀⠀⠀⢸⡇⠀⣼⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠸⣇⠀⠀⠀⠘⣧⠀⠀⠀⢸⣇
⠀⠀⣸⠃⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢻⡆⠀⠀⠀⢻
⣿⣿⠟⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢺⡐⠭⣧⠇⠀⠆
⠀⠀⠀⠀⠀⠀⠀⣸⣥⣤⠖⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢀⣀⡀⠀⠀⠀⠒⡂
⠀⠀⠀⠀⠀⠀⠠⣿⣿⣷⢶⣄⠀⠀⡀⠀⢖⡲⠀⡊⠁⡆⠈⠉⠁⠀⣼⣤⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⡝⠛⣏⣹⠄⠀⣠⡄⠉⠈⠁⠀⠀⠀⠀⠀⡟⠻⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠈⢡⡀⠀⢽⠆⠀⢀⢠⠀⠀⠀⡀⡔⠃⠀⠁
⠀⠀⠀⠀⠀⡀⠖⣀⣀⠀⢠⠖⠒⢄⠀⢿⡀⠀⠀⠀⠀⠄⠂⠀⠠⢇⣹⠃⠀⠄⠒⢌⠂⠀⢀⣼⠃
⠀⠀⠀⠀⣼⣿⠉⠉⠁⠀⠹⣀⣀⠜⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠸⡀⠀⡠⢀⢠⣾⡿
⠀⠀⢀⣴⠟⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠉⠀⠀⢨⡿⣿⡀
⠀⠰⠟⠁⠠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢟⡿⠋⠁⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡃⠈⠁⠀⠀⠀⠀⠀⠸⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⢓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣶⡇⠰⣶⣿⠿⠇⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠁⢻⡇⠀⠀⠃⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⢀⡼⢧⣀⠰⢤⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠚⠀⠤⠥⠀⠀⠀⢸⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄⠘⠛⠛⠁⠀⡰⠋⠘⣿⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣌⠻⣦⣤⣤⣤⠄⠁⠀⠀⢹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣿⣿⣿⣧⡀⠉⠛⠀⠀⢴⣷⡀⠘⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠘⢿⣷⣤⣿⣿⣷⣻⢿⡄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠷⠎⢻⣿⣦⡀⠀⠀⠀⠀⠐⢳⡍⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⣿⢏⡀⣹⣿⣷⣄⠀⠀⠀⠈⣾⣿⡎⢻⣿⣇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠛⢛⣟⢱⣿⣿⣿⣿⣦⠀⠀⠀⣿⣿⣇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠚⠋⢉⣘⠻⣿⣿⣿⣿⣧⠀⠀⣿⣿⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣹⡇⠘⠉⣉⣉⣌⣿⣿⣿⣿⣿⣧⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢻⢡⡿⠀⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣇⠈⢈⣴⣶⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠐⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⡿⢻⣿⡇⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣄⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣸⣿⣿⣿⣷⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⠿⡿⣿⠿⣿⠀⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡈⢿⣿⣿⣿⣿⡌⠛⠈⠂⠁⠸⠀⠈⠀⡏⣿⠙⢿⣿⠃⢀⡀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢿⢷⡈⢿⣿⣿⡟⡟⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⠀⠛⠀⠘⢧⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡞⡎⢧⠘⣿⡟⢧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣌⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⣿⡄⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠻⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⣥⠀⠀⡆⠀⠀⢰⠀⠀⢉⡑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢠⠧⢧⠀⢸⠒⠒⣻⠀⣠⣋⡇⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠘⠀⠈⠃⠈⠃⠀⠋⠘⠁⠈⡇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⣄⣠⡤⠤⠤⠶⠤⡀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⢸⠅⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡀⢡⠤⢄⣌⣀⠤⠒⠀⠀⡠⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠠⡇⠀⡏⢼⠱⡠⠤⠄⡞⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢣⠈⠊⠳⡀⡸⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢧⠀⠀⠙⣇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢠⣏⣧⣀⠀⠈⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣓⡶⠒⠒⠊⣉⠀⠀⠈⢇⠀⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣷⢤⠤⠴⠒⠉⠀⠙⢦⠀⠘⢆⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠫⣍⣀⣠⠤⡄⠀⠀⠀⠳⡀⠈⠃⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡤⠒⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠎⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⣀⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠁
⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠑⠤⣄⣀⣤⠤⠖⠊⠁
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣮⣉⠛⠻⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣤⣈⢻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⣿⣿⣿⣿⠉⠛⠻⠿⢿⣿⣿⠟⠋⠉⠉⠛⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡌⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⢿⣿⣿⡟⠁⠀⢀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠿⣿⠀⠀⠞⣋⣉⣙⣛⣿⠆⠀⣼⣿⣿⠫⡭⢰⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠉⠁⠈⠁⠀⠀⠉⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢦⣴⠟⠀⡀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠊⠀⠀⠠⠤⠤⠀⣲⠇⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠋⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⣀⣀⣀⣤⠖⠛⣿⠙⠳⣄⠀⠉⠳⣦⣀⠀⠀⠀⢀⣼⣿⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠃⠀⠀⠀⠈⣉⠡⠴⣦⡀⠀⠀⠻⠀⠀⠈⠳⢦⡀⠈⠛⠻⠿⣿⣿⣿⡇⠀⣿⢦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⣴⠋⣠⠀⣶⣮⡟⢷⣄⡀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠚⢻⣿⠟⠀⢰⡟⠀⠉⠳⢤⣀⣀
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⡀⢴⣏⠒⠁⢀⣼⣿⣷⡄⠈⠁⠒⠒⠢⡄⠀⠀⠀⠈⠘⣳⣿⣧⡀⠀⢸⡇⠀⠀⠀⠀⠉⠙⠛⠠⡀
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢠⡆⠀⠀⠙⠛⠻⢿⣿⣿⣿⣷⣅⡄⠀⠀⠀⢻⡄⠀⠀⠀⢠⣿⣿⣿⡟⢂⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠈⣿⡀⢀⡀⠀⠀⠀⣹⡟⠁⠀⢻⣿⠀⠀⠀⢘⡧⠀⠀⠀⣼⠉⢿⣿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢠⡟⠀⠀⠉⠳⣄⢐⣿⡇⠀⢀⣼⡟⠀⠀⠀⢈⡟⠘⢦⣄⡿⢠⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣶
⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢹⣿⣿⡟⢻⡿⠀⠐⠁⡴⠊⠀⠀⠀⠉⠃⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣻⡇⠈⠓⠤⣀⣼⣏⣉⠷⠾⣷⣄⣠⠞⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠻⣧⡀⠀⠀⠀⣿⣿⠉⢀⡠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠙⢿⡏⠓⠐⢿⣿⡏⠁⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠴⣄⡈⠿⣄⣄⣀⣿⣷⣤⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣤⣽⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠖⠚⠛⠒⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⣁⣴⠾⣿⠇⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠛⠓⠚⠛⠗⠋⠀⣤⡶⣿⣿⣿⣿⣤⣻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⣤⣶⡾⠃⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢹⣿⣯⣽⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣇⡄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣮⣹⠇⠐⠋⢙⣻⣿⣿⠷⠀⠀⣀⣾⣟⣻⣋⡙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣸⠿⠀⠀⠀⠉⠛⢻⡿⠀⠀⠀⣿⣿⠝⠋⠉⠁⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⣸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⣰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⢿⡄⠀⠀⠀⠀⠈⠃⠐⠿⠛⠁⣤⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⠓⠀⠀⠶⠦⠴⣶⡶⠶⠶⠆⠟⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠰⠶⠶⠀⠀⣸⣿⣿⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⣷⡀⠰⣤⡀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣶⣤⣄⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠀⢀⣿⠹⣄⠈⠙⠷⣦⣤⣤⣶⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⣤⠴⠖⣻⠏⠀⣀⣀⣀⣸⣧⣤⠬⢷⠶⣤⡤⠭⠿⢿⣧⢶⣷⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣞⣛⣉⣉⣉⣁⣀⣀⣀⣀⣴⠏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⡟⣏⠉⠛⠛⠿⠿⠟
⠀⠀⠀⠀⠀⠀⣠⠏⠈⠉⠉⠉⠉⠉⠉⢿⡟⠁⡄⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹⡄
⠀⠀⠀⠀⠀⢰⣇⡀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢰⠇⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠉⢻⣿⠶⠂⠀⠀⠀⠀⢳⣦⡟⠀⢸⡀⠀⠀⢠⣆⣀⣀⣀⣀⣠⣶⣶⣦⣤⡟⢸⡇
⠀⠀⠀⠀⠀⢠⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠧⣤⣸⡇⠀⢠⡿⠶⣍⠉⡿⢹⣿⣿⣿⣿⠀⡧⢸⡇
⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠈⢧⣀⣘⣁⣴⠋⢰⡇⣿⣿⣿⣿⣿⣠⡇⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠉⠉⠁⠀⠈⠀⠙⠛⠛⠛⠛⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡗⣭⢛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣰⡏⣼⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⣿⢰⣟⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣷⣾⣷⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡟⠛⠀⠀⠀⢀⣀⡿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇⠀⠀⣶⣶⠞⠀⠀⠀⢤⣤⡴⠀⠀⣤⡄⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠹⣆⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇⠀⠀⣿⡟⠀⢀⣤⡀⠘⣿⠁⢠⡄⢸⡇⣠⣼⠇⣄⡏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⣿⣿⣿⡇⠀⠀⣿⡇⠀⢸⣿⣷⣾⣿⠀⣿⣿⣿⡇⠙⢻⡇⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⠀⣿⡿⢿⡇⣶⣾⠻⡆⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⣿⡇⠀⠸⣿⡏⠈⣿⠀⠙⠃⣸⡇⠉⠸⠆⠐⠃
⠀⠀⣠⣴⣶⣶⣿⡇⠀⠀⠀⣤⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠸⣿⡿⠃⠀⠠⣿⣿⡀⠀⠀⠀⠈⠛⠃⠀⠀⠉⠁
⠀⠾⠿⠛⠛⠉⠉⣿⣷⣶⣾⣿⡇⠀⠀⠀⠀⠈⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛



⠀⡁⣉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠈⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⡰⠁⠉⠓⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠁⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠐⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠉⣲⡤⡟⠀⠀⠀⠀⡐⣰⣠⡤⣶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⡀⠀⠀⠈⠁⢴⡇⠀⠀⠀⢠⣟⠉⠀⠀⠙⠶⠚⠛⠲⡄
⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⢀⡵⠒⢶⠒⠒⢺⠁⡀⠀⠀⡾⢯⠳⢤⡀⠀⠀⣐⡥⠞⠁
⠀⠀⠀⠀⠀⣸⠀⠀⢀⣀⠴⠋⠀⠀⠘⡆⢀⡏⠀⠀⠀⢸⠃⠘⠳⢦⣉⣉⠉⠁
⠀⠀⠀⠀⢠⡇⠀⠀⢠⡇⠀⠀⠀⠀⠀⢳⣸⣇⣀⠀⠀⣿⣤⣀⠀⠀⠀⠉⠙⣦
⠀⠀⠀⠀⡾⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣰⠃⠙⠻⣝⡒⠶⠤⠤⠞
⠀⠀⠀⢰⠃⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠟⠧⣄⣀⠀⠀⢳
⠀⠀⠀⣏⣀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠚
⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠒⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠂⠀⠀⠀⠰⠶⣶⣦⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢋⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⣿⣿⡉⠉⠙⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢡⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠽⢿⣷⡄⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⢛⡽⢁⣿⣇⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⡄⢂⠠⠀⠠⡈⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠊⠁⢘⣿⣿⠃⠀⢀⡴⠚⣉⣤⣤⡄⠀⠀⣀⡈⠻⣷⢸⠀⢷⠀⢣⠈⠛⠝⢦⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡇⠀⠀⢀⡴⠋⢁⣤⢴⠿⠂⠀⠀⠙⢶⣽⡏⠀⢿⠀⢸⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⢠⡾⣁⠔⠋⠀⠀⠀⠀⠀⠀⣶⣿⣿⣷⠀⠈⣆⡸⠀⠀⠀⠀⠀⠠⣝⣄
⠀⠀⠀⠀⠀⢸⡅⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣿⣷⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠀⠀⡏⣇⠀⣔⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⢸⡗⡀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⠇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠯⠀⠀⠁⢸⡀⢻⣆⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⣠⠚⠷⠀⢀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣖⡁⠀⠀⠘⣇⣼⡟⡂⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⡼⠁⠀⠀⠀⠘⢷⣌⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣴⣄⡠⠀⣀⣤⣾⣿⣿⡿⠂⠀⠀⢰⣿⣿⠇⠀⠀⠀⠀⠀⠀⠰⡀⢻⡆
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⢿⢿⣿⣿⣿⣿⣿⠿⠿⣿⣷⣤⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠱⡘⣧
⠀⠀⠸⣷⣄⠀⠀⠀⠀⠀⠀⠂⢤⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠹⣿⣿⣿⣧⡀⠀⠹⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⢻⠛⠿⣶⡄⠀⠀⠀⠀⠀⢻⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧⣀⡀⠸⣿⣿⣎⣀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⢻⡀
⠀⠀⠀⢸⡇⣀⠀⠈⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢯⠉⠉⠀⣿⣿⣯⡅⠤⠄⠀⠀⠀⠀⠀⠀⠀⠸⢿⣷
⠀⠀⠀⣼⠋⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⢻⡆⠀⠀⠀⣿⣿⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⢿⣧
⠀⠀⠀⣿⠀⠀⠀⠠⢦⣷⣄⠀⠀⠀⠀⠀⠘⡻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠃⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⡇
⠀⠀⠀⠹⣤⠀⠀⠀⠀⠘⢿⣿⣄⠀⠀⠀⠀⠀⠽⢿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⣤⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡱⡄⢢⡀⠠⣽⠇
⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠈⠹⣿⣧⡀⠀⠀⠀⠐⠉⠿⣿⣿⣄⡀⠀⢀⠀⠀⠀⠀⡠⠔⢁⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠑⠁⠑⠀⢬⣠⡿
⠀⠀⠀⠀⠀⠈⢿⣶⡄⠀⠀⠀⠀⠻⢿⣦⠀⠀⠀⠀⠘⡹⢻⣿⣿⣯⣕⣠⢄⣔⣩⣀⣴⣷⣿⡿⠋⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣠⣤⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄⠀⠀⠀⠙⢻⣷⣄⠀⠀⠀⠀⢋⡼⡿⣿⣿⣿⣿⣿⣿⣿⠟⢩⡿⠁⠀⠀⠀⠀⣠⠞⠛⠛⢆⡀⣴⣴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣄⠀⠀⠈⢿⣷⣄⠀⠀⠀⠨⠞⠛⠡⠊⡟⢹⣿⠇⠀⣿⣇⢀⠀⠀⣴⡿⠃⠀⠀⢢⣼⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣄⠀⠘⠿⣷⣧⣄⢀⡀⢀⠠⣪⣴⣿⡟⠀⠀⣿⣿⣿⠀⠀⡏⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡈⠈⠙⠿⣿⣿⣷⣾⣿⡿⠋⠀⠀⠀⠘⢿⣿⣧⡰⣶⣄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⣤⣊⣤⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣈⣛⡛⡻⣿⢷⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠾⠛⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠉⠉⠉⠙⠛⠛⠋⠁⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⡠⠶⠦⣤⡀⠀⣰⢞⣛⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠘⠛⠉⠓⠀⠀⠊⠉⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡇⠀⠀⠀⠈⠤⠄⠠⠂⠐⡄⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠂⢠⣤⡤⣴⠖⣲⠞⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠉⠃⣀⣀⠀⠀⠀⣠⠿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣷⣄⡀⠀⠀⠀⠀⠀⣼⠋⠀⠇⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡇⢿⣿⣿⣶⣤⣤⣴⠾⠃⠀⡜⠀⠀⡟⠙⠓⠦⠤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠫⢸⣿⣷⣌⠻⢯⡿⠛⠋⠀⠀⡠⠎⠀⠀⣠⠃⠀⠀⠀⠀⠀⠉⠉⠛⠶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⠀⠀⠉⠻⣷⡙⠆⠀⠈⠉⢠⣤⣶⣦⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⡼⣿⣿⡿⢧⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡾⠀⣼⣿⣧⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡰⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⡥⠴⠋⠀⠀⠢⠀⠀⣼⠇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢰⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠐⣟⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⣹⡄⠀⡄⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠛⢷⡄⠀⠀⠀⠀⠀⠀⡈⠀⠀⣿⢀⣾⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⣠⡿⠛⠁⠀⠀⠀⢻⣆⠤⠀⣐⣶⠿⠁⠀⠀⣼⣿⠇⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠘⣿⠞⠋⠉⠀⠀⠀⠀⢀⣿⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⢀⡞⢡⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⠀⠀⠀⠀⣰⠟⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣦⠀⠀⢸⡏⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⡈⠁⠿⠁⠀⠀⠀⠀⠀⠀⢀⣴⠟⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣷⡄⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⣿⣿⣿⣿⠟⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⢻⡿⡿⠃⣠⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠶⠞⠉⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠁

⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣃⣀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢠⢤
⠀⠀⠀⠀⠀⢠⢤⡀⠀⢸⡃⠸⡀⠀⢰⠃⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠸⡆⠀⢀⠇⢘⡇⠀⢀⡤⡄
⠀⠀⠀⠀⠀⢻⠀⢳⠀⠘⡇⠀⣇⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⣸⠀⢸⠃⠀⡞⠀⡟
⠀⠀⠀⠀⠀⢸⡀⠈⣇⠀⡇⠀⢸⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⡇⠀⢸⠀⢸⠁⢀⡇
⠀⣏⠙⢦⠀⠀⡇⠀⢸⡀⢹⠀⠘⡄⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢰⠃⠀⡏⢀⡇⠀⢸⠀⠀⡴⠋⣹
⠀⠘⣆⠀⢳⡀⢸⠀⠀⢇⢸⠀⠀⣇⡎⠀⢸⠀⠀⠀⠀⠀⡠⠒⡆⢰⠒⢄⠀⠀⠀⠀⠀⡇⠀⢻⡸⠀⠀⡇⡼⠀⠀⡇⢀⡞⠀⣰⠃
⠀⠀⠘⢆⠀⠱⡀⡇⠀⠈⠾⠃⠀⠈⠁⠁⠈⡆⠀⠀⢀⠞⠀⡰⠁⠈⢆⠀⠳⡀⠀⠀⢰⠁⠀⠈⠁⠀⠈⠷⠃⠀⢸⢀⠞⠀⡰⠃
⠀⠀⠀⠈⢦⠀⠙⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⣠⠎⠀⣰⠁⠀⠀⠈⣆⠀⠱⣄⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠋⠀⡴⠁
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠉⠁⠀⢀⠃⠀⠀⠀⠀⠘⡀⠀⠈⠉⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢀⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⡀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠘⣄⡰⠙⣄⣤⢀⡿⠀⣠⠴⡇
⠀⠀⠀⡀⠀⠀⢠⡄⣽⠑⠾⠀⠈⠃⠀⠙⠈⠃⠰⠞⠁⠐⠇⢀⣠
⠀⠀⠀⠙⢟⠃⠶⠆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠚⢩⠃
⠀⠀⠠⣤⠦⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⣠⠶
⠀⣀⣀⠨⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠏⣀⣀
⠈⠙⠒⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠔⠒⠊⠉
⠀⣠⣔⣋⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠉⠦⠄
⠀⠀⠀⠀⣼⣁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣘⣆
⠀⠀⠀⠈⠁⠀⣯⠴⠆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠹⡄⠁⠀⠀⠀⠀⠀⡴⠈⠀⣠⠛⢩⣶⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣇⠞⠻⢀⡜⢧⡞⣆⡞⠳⣄⣯⠙⠶⣄⡘⢆⠀⠀⠀⢀⣼⡗⠀⠀⢿⠀⠛⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠟⠀⠀⠀⠈⠀⠀⠀⠙⠀⠀⠀⠙⠺⣦⠀⠀⣼⠏⠀⠈⠁⠀⠀⠠⡀⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⢠⡿⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⠀⢸⡇⢀⡖⠂⠀⠀⠐⣒⣚⡇⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠚⠁⠀⠀⠀⣾⢡⡀⠀⠀⠀⠀⠀⠀⠈⣡⠀⢡⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣾⣇⠀⣠⠀⠀⣀⠀⠀⣿⡀⠈⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣆⠹⣤⣤⠋⠀⣴⢟⣧⠀⠈⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣠⠖⠟⠛⠛⣡⣠⠆⣀⣀⣴⡿⠋⠘⣿⣷⣄⠀⠙⣿⡳⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠌⣾⠛⠄⠀⠀⠈⠃⠉⠀⠉⠛⣯⠠⠶⣄⠈⠉⠻⠷⣦⡈⠹⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠼⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣽⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠱⢦⡀⠀⠀⠀⢠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢀⣤⣹⡌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡶⡄⢀⠀⠀⠀⣿⣿⠟⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⠃⠀⡈⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⣀⡀⠀⠀⣦⣀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣠⡤⠀⢀⣴⣿⡄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠻⣿⣿⡄⠀⠀⢿⠇⠀⠀⠀⠉⣿⠨⠲⠕⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠀⠘⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡗⠀⠀⠀⢧⠀⠀⠀⠀⣿⡄⠀⡢⠀⡀⢅⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠈⢣⠀⠀⠀⠸⡇⠂⠁⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇⠀⠀⠀⠀⡏⠀⢦⡄⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠫⠄⠀⠀⠀⠇⢠⠈⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢀⠆⡀⢀⠀⠀⠀⢠⡄⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠣⣈⣀⣴⠆⠀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠉⠑⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⢀⣿⣿⡄⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣶⣶⣤⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⡇⠀⠀⣾⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠉⠀⠀⠁⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⣀⠀⠀⠀⣤⠖⠒⣦⣀⠀⢀⣠⣤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣸⠁⢠⡞⣹⠿⡇⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠶⠀⣰⠇⠀⢸⠀⠹⣤⣧⠀⠀⢀⡴⠛⠉⢩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠀⠀⢸⡆⠀⠀⢹⡄⣠⠞⠁⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠛⠉⠁⠀⠀⠀⠀⣸⣇⠀⠀⠈⣿⠁⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣿⠻⢯⣽⠶⠤⣤⣤⡤⠾⠧⢿⣄⠀⠀⢸⡆⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⣼⢦⣄⣀⣀⣠⣤⣤⣤⡀⠀⠀⢹⡆⠀⢸⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢻⡉⠁⠈⠁⠀⠀⠀⠀⢹⡄⠀⠀⡇⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⡤⠶⠚⠛⠛⠒⠒⢶⠀⠀⠀⠈⣧⠀⠀⣧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠘⡇⠀⠘⠂⢸⡇
⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠴⠛⠉⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣤⣤⣤⣤⡤⠤⠶⠖⠚⠛⠛⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠚⠁
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣉⡤⠴⢤⣠⣠⣤⠄⣠⣤⣤⣀⣤⣠⣄⣄⣀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠒⠚⠉⠀⠀⢀⡄⠙⣿⡿⠛⠻⣍⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⢤⡄⣠⠄⠀⠀⠀⠀⠀⣰⣦⣶⣾⣿⣷⣀⡈⢓⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠘⢹⣿⣿⠁⠀⢰⠃⠀⠀⠀⢀⣤⠿⠉⣽⡿⠤⣼⣭⣭⣭⡯⠠⠐⠏
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⠀⠀⣼⣆⠀⠀⠾⠛⠉⣠⣾⣿⣿⢶⣄⣦⣤⣠⣄⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠸⣿⣿⡧⠀⠀⢿⣿⣿⣤⣤⣶⣾⢟⣽⣿⣿⣶⣶⠠⣀⣄⠀⡱
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⢸⣿⣿⣿⡿⠁⠈⠈⠉⡩⠽⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡀⠤⣼⣿⠛⠛⠛⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡧⠐
⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⠶⠦⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡴⠃⣴⣶⣷⣾⣿⣾⣔⡌⡑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣯⣷⣔⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡙⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣖⣬⠙⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣆⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣠⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠉⠉⢛⠻⠿⡿⠿⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣯⡻⢿⣿⣷⣦⣤⡀⣤⣤⣶⡶⠈⠐⣾⠁
⠀⠀⠀⠀⠀⠀⠸⣷⣿⣿⣿⣿⡿⢿⣿⡟⠓⠿⢻⡃⠈⠉⠁⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠹⠟⠛⠉⠀⠀⡜⢿⣿⠀⠠⣏⡀⣈⠆⠀⠀⠠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣿⣿⡄⠀⢈⣙⣃⣦⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣿⣿⠷⠘⠃⠲⠖⠂⠀⡠⣯⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣧⠼⣃⡀⠀⠀⣀⠀⠀⡐⠐⠴⢻⣿⣦⣀
⠀⠀⠀⠀⢀⣾⡉⠉⣿⣿⣿⣿⣿⠇⠀⡘⣧⡤⢶⡄⠹⡟⠁⠀⠈⢿⣿⣿⣽⣛⢛⣛⣐⣆
⠀⠀⠀⠀⣾⣿⣿⣄⠸⣿⣿⣿⡇⠀⢠⣨⠃⢱⣿⡹⠀⡽⠀⠀⠰⠸⣿⣿⣿⣿⣿⣿⣿⢸⡆
⠀⠀⠀⢸⣿⣿⣿⣿⣷⢻⣿⣿⠁⠀⠐⠋⢤⣼⣽⣇⡞⠀⠀⠀⠉⢋⣿⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣾⣿⣿⣆⢲⣤⣀⣾⣶⣿⡍⡀⠀⠀⠀⠀⠁⣿⣿⣿⣿⣿⣿⣿⠃⡷
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣭⣽⣿⠍⠱⡡⠀⠀⠀⠀⡀⢿⣿⣿⣿⣿⣿⣿⡀⢧
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⢻⣻⡏⢸⣦⠷⠀⠀⠀⠀⠹⢬⣿⣿⣿⣿⣿⣿⠇⣼
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡇⢈⠋⢸⡟⠀⣼⣿⡐⡀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠏⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡈⠀⣸⠁⣼⣿⣿⣿⣦⢁⡀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⡌⢃⣼⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠠⣾⣿⣿⣯⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣧⡍⠀⠀⠀⠀⠀⠁⠿⠉⠹⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡍⠀⠀⠀⠈⠙⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠙⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⢸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⢀⣿⡇⣰⣿⡿⠛⠉⠛⠻⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠘⣿⣷⣶⣾⣿⣿⣶⣦⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⠉⢡⡆⠀⠀⠀⠀⠀⠀⢿⣿⠁⠀⠀⠀⠀⠙⢿⣾⠟⠛⠛⠿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⢠⣀⣿⡇⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠒⠒⠒⠶⣼⣿⡄⢀⣤⣤⣤⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⢠⡀⠀⠈⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠈⣿⡿⠀⠀⠀⠀⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠳⡄⠀⠸⣿⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⢷⣄⠙⣦⣀⣿⡆⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣠⣶⠿⠿⢿⡟⠻⠿⢿⣶⣶⣶⣶⣶⣿⣧⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣾⡟⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⣼⣿⣇⠀⠀⠀⠀⣸⡟⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠃⠀⠀⠀⠀⠀⠰⣤⣀⣀⣀⣀⣀⣠⡶⠿⣷⣄⣀⣀⣤⣾⠏⠙⢿⣦⣀⣠⣴⠏⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡟⠉⠙⠻⢦⣀⠈⠻⠟⠛⠛⢷⣄⠀⠀⠈⠉⠉⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣄⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠙⠳⠦⣀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠙⠷⡀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⠼⠋⣸⠇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣶⣤⣀⡀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠀⠀⠀⢀⣀⣠⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣿⠇⠠⣾⣿⡟⠛⠻⠿⠿⠟⠛⠋⠉⠁⠀⠐⠚⠛⠛⠛⠛⠉⠉⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡀⠀⠈⠙⠛⠷⢶⣤⣄⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠻⠿⢿⣷⣦⣄⡀⠀⠀⠈⠉⠛⠻⠿⢶⣶⣿⣦⣀⣀⡀⠀⠀⠀⠀⢸⡟⠙⠛⠻⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠉⠛⠿⢿⣶⣦⣤⣀⡀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⡿⠟⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⢙⣿⠛⣿⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢀⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⣠⠞⠁⣀⣠⣤⡤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣀⡴⠋⣠⠶⠛⠉⠁⠀⠀⠀⠀⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣡⠞⠋⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣡⠞⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡤⠀⠰⠏⠰⠃⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣶⡶⠺⢻⡟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⡀⠀⠀⣀⢀⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠳⣄⠀⠐⠃⢠⠀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡊⠤⡀⠀⠀⠙⡸⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢀⣣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡲⠶⠶⠶⠶⠋⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠈⠉⢤⣀⠊⠈⠢⣀⣀⣀⣀⣀⣤⣤⠶⠛⡇⠀⠀⠀⠀⠠⣀⡀⠀⢀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⡠⠀⠈⣢⣄⠀⠈⠉⠍⠉⠉⠁⠀⠀⢠⡀⠀⠀⠀⠀⢀⠼⢋⠁⠁⠀⠉⠉⠓⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢧⣀⠀⢀⠞⣀⣠⣴⠟⠈⠣⡀⠀⣠⠜⠢⣄⠀⠀⠸⡏⠑⣒⠶⠊⠁⡴⣣⣖⡥⠒⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠈⢉⡿⠛⠉⠁⠀⠀⠀⠀⠙⢞⠁⠀⠀⠀⠙⠢⣀⠓⠉⠀⠀⠀⣰⠿⠋⠁⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⡼⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠈⠑⠦⠤⠒⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠜⠀⠈⠳⣄⢀⠳⣄⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⢀⣀⣠⣤⣴⣿⡞⣷⠿⠷⣄⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠀⠀⣀⣀⣀⣤⣾⠀⠀⠀⠀⠀⣠⢾⠇
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠉⠁⠀⠀⠙⠛⠋⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠙⢿⣆⢀⣠⠴⠋⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⢤⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠲⠦⠿⠶⢤⣄⣀⣀⣀⣀⣀⣤⣤⡤⠤⠤⢤⣤⣤⣼

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢺⣆⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⡴⠦⠀⣠⣼⣯⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠲⢀⠀⠀⠀⠀⠀⠀⡀⣠⡉⠹⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠀⠁
⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡠⠄⢹⡆⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⢔⢒⢒⠘⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠋⠂⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠁⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠃⠀⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠒
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠐⣶⠧⢀⣀⠀⠀⡎⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠎⠀⠀⠀⣧⠞⠉⠙⡿⠇⠀⣶⠞⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡼⡇⠀⠀⡜⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠴⠋⠀⣇⠀⢰⠃⠀⠀⢀⠄⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠙⠁⢅⣀⡔⠀⢆⠘⢛⠽⡄⢀⡤⠊
⠀⠀⠒⠢⠐⠀⠙⠆⡀⠈⢅⠀⠀⠈⠱⢤⣰⠛⠉⠒⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣱⠀⡰⢀⠆⠐⠲⠤⢄⠉⠀⢇⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠉⠀⠀⠀⠐⠀⠉⠀⠀⠀⠀⠀⠀⠂

⠒⠲⠶⡶⢲⠞⣻⢛⡟⠒⣆⣀⣠⡤⠤⠶⠒⠒⠒⠛⠛⠛⠉⠉⠉⠉⠉⠙⠓⠒⠦⣤⣀
⠀⠀⢰⡇⡾⢠⠇⡞⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣄⡀
⠀⠀⠈⠀⠁⠘⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠉⠓⠲⣤⣄⡀
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡖⠚⠓⠦⣄⣀⣀⣀⡀⠀⠀⠘⠦⠝
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠙⠲⠦⠤⠴⠚
⠀⠀⠀⠀⠀⢸⠁⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⢀⡿
⠀⢀⠀⡆⣾⢸⡗⠒⠲⢿⣦⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠘⠲⣄⠀⠀⠀⢸⠏⠁
⠀⢸⠀⣧⢸⠈⡇⠀⠀⢸⠇⠀⠉⠉⠉⠉⠉⠉⠹⣟⠲⣄⠀⠈⣷⠶⠒⠁
⠒⠚⠓⠛⠾⠷⠿⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣌⣡⠴⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡙⣱⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⡤⣶⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⠇⣿⡉⢧⡤⠀⢀⡞⣿⣷⣆⠀⢀⣴⡟⢫⣵⠞⣾⡏⢵⣺⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾⡋⢿⡗⢮⠻⢦⣼⠟⠉⠻⠏⠀⠁⣟⣨⡼⢫⡽⠁⠐⡿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠈⠷⢈⣯⠜⣿⠔⠋⢀⡀⢛⡀⡰⠤⠈⠻⣴⣷⠰⠞⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠬⠇⠹⠅⠶⠚⠉⠀⠀⠀⠀⠠⡀⠠⠄⠀⠀⠀⠀⠀⠨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⣤⣼⣷⣆⠈⠁⢀⠀⠀⠈⢀⡉⢀⠀⠀⠀⠀⠀⠀⣠⡀⢸
⠀⠀⠀⠀⠀⠀⠀⣴⣦⣀⠘⠇⣀⡱⣿⣦⠀⠀⢀⠀⠀⡔⠃⠀⠀⣠⣶⢶⣿⣯⡗⣠⣦⡀
⠀⠀⠀⠀⠀⠀⢰⡙⠗⣿⣦⠀⠀⠉⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠻⠟⢋⣴⣟⣿⣷
⠀⠀⠀⠀⠀⠀⢨⠽⣿⣿⣁⡒⢦⡤⠈⣍⠙⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢁⡀⢻⣿⣿⡿⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠙⡉⠑⠂⠈⠂⠁⠀⠀⠀⠀⣈⡀⠀⠀⠀⣨⣷⡀⢮⣿⡟⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⠖⠀⠁⢰⠀⠤⣿⠩⣿⡓⠀⣐⠆⢀⣼⡖⠀⣽⣦⢷⡠⣿⡇
⠀⠀⠀⢀⣄⢀⡀⣠⣝⢁⣴⣿⣶⠘⢿⣾⡟⢻⡿⠷⣦⣥⡀⢠⣤⠰⠀⢬⣟⣺⡞⢿⡇
⠀⠀⠀⠀⠷⢶⣾⣻⢿⣯⡽⣭⣻⣯⣄⠈⠀⠙⠻⢿⣯⡿⠿⢟⣉⣓⠻⣄⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢄⡈⢷⡛⣧⣈⢷⣼⣿⠿⠀⠀⠀⢠⡀⠸⢿⣿⡆⠰⣾⣿⣿⣶⣽⣿⣿⣿⣧⡀
⠀⠛⢓⡠⠦⡤⠞⠗⡷⣻⣦⣿⣿⣦⣗⠀⠀⠈⣨⣷⡆⢿⣷⣞⣿⣿⣿⡿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣿⣿⣶⡄⠔⠉⠺⡇⢸⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢛⣟⠛⠛⡟⠻⠛⡛⠛⠋⠀⠀⠀⠛⠟⢻⣟⠛⢻⠛⢻⠛⠛⢿⡁
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠘⡆⢠⡈⢳⠜⢇⠀⢀⠀⠀⠁⣄⡀⣈⡈⠳⢼⠀⢸⡀⠀⡸⠃