Door symbol Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣤⣤⣤⣤⣤⣤⣼⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣈⣛⣛⣛⣛⣛⣛⣛⣛⣛⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣤⣤⡤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⡽⢋⣩⢤⡤⣍⡓⢯⡢⡀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣵⢞⣛⢛⣛⠷⣮⡲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⠏⡴⢹⠀⢸⠀⠀⡟⢦⠱⡹⡀⠀⠀⠀⠀⠀⢰⢳⡟⣴⢸⣿⢸⣿⡇⣮⢻⡜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠸⠧⠼⠦⠼⠦⠴⠧⠼⠆⡇⡇⠀⠀⠀⠀⠀⣿⣿⣜⣛⣘⣛⣈⣛⣃⣛⣩⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢀⣀⣀⣀⠀⢀⣀⣀⣀⠀⡇⡇⠀⠀⠀⠀⠀⣿⣿⠿⠿⠿⢿⣿⠿⠿⠿⢿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⡇⠀⢸⡇⢸⠀⠀⢸⠀⡇⡇⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣾⣿⢸⣿⣿⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⡇⠀⢸⡇⢸⠀⠀⢸⠀⡇⡇⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⢸⣿⣿⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⡇⠀⢸⡇⢸⠀⠀⢸⠀⡇⡇⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⢸⣿⣿⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⣅⣀⣸⡇⢸⣀⣀⣸⠀⡇⡇⠀⠀⠀⠀⠀⣿⣿⠸⠿⠿⢿⣿⠸⠿⠿⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢀⣀⣀⣀⡀⢀⣀⣀⣀⠀⡇⡇⠀⠀⠀⠀⠀⣿⣿⠿⠿⠿⢿⣿⠿⠿⠿⢿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⢣⢤⣼⡇⢸⢠⢤⢸⠀⡇⡇⠀⠀⠀⠀⠀⣿⣿⢸⢟⢻⣾⣿⢸⢛⢻⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⡜⠛⢹⡇⢸⠘⠚⢸⠀⡇⡇⠀⠀⠀⠀⠀⣿⣿⢸⣬⣽⣿⣿⢸⣬⣼⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠈⠉⠉⠉⠁⠈⠉⠉⠉⠀⡇⡇⠀⠀⠀⠀⠀⣿⣿⣦⣭⣭⣼⣿⣮⣭⣭⣾⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⡶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣧⠇⠀⠀⠀⠀⠀⠿⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠥⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡟⢙⣛⣛⣛⣛⣛⣛⣛⡛⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇⠿⢀⣩⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⣿⣿⣿⣿⣿⣿⣿⡿⢃⣴⣿⣿⡿⠛⣉⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⣿⣿⣿⣿⣿⣿⣿⢁⣾⣿⡋⠡⣴⣿⣿⣿⣦⠈⢛⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⠟⢃⣤⣙⠛⣉⣄⠘⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡟⠛⠛⣿⣿⣿⣿⣧⠸⣿⣿⡷⠆⣉⠛⠿⠛⡉⠰⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣧⣿⣿⣿⣿⣿⣿⣿⡆⠻⣿⣦⣌⡛⠿⣶⠿⢋⣠⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡏⣿⣿⣿⣿⣿⡿⢋⣴⡆⣈⠻⢿⣿⣶⣤⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⣿⣿⣿⣿⣯⣴⣿⣿⡇⣿⠀⣦⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠻⠿⠿⠿⠿⠿⠿⠿⠃⣿⠀⣿
⠀⠀⠀⠀⠀⠀⣤⣤⡄⠘⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⠀⣤⣤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⡠⠔⠒⠉⠉⢹⠉⠉⠑⠒⠤⡀⠀⠀⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⡰⢊⠔⠊⠉⠉⡇⢸⠀⢸⠉⠉⠒⢬⡲⡀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢀⠎⢠⠃⠀⠀⠀⠀⡇⢸⠀⢸⠀⠀⠀⠀⠱⡈⢆⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⡜⠀⡎⠀⠀⠀⠀⠀⡇⢸⠀⢸⠀⠀⠀⠀⠀⡇⠘⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⡇⢸⠀⢸⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⡇⢸⠀⢸⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⡇⢸⠀⢸⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⡇⢸⠀⢸⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⡇⢸⠀⢸⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠓⠀⠀⠀⠀⠐⣧⣸⣀⡘⠀⠀⠀⠀⠀⠃⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⣠⢏⠉⠉⠉⡳⡍⢸⠈⢁⢏⠉⠉⠉⡑⡄⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⣷⠀⠀⠀⠀⢸⡇⢸⠀⢸⠂⠀⠀⠀⠈⡇⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⣿⠀⠀⠀⠀⢸⡇⢸⠀⢸⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⣿⠀⠀⠀⠀⢸⠃⢸⠀⢸⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⡿⡀⠀⠀⠀⢸⡇⢸⠀⢸⣄⠀⠀⠀⢀⡇⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠙⠮⠤⠤⠤⠜⠁⢸⠀⠈⠢⠤⠤⠤⠔⠃⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠒⠃

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⣶⣶⣶⣶⣶⢸⣿⡿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣸⣿⣿⡇⠀⠀⣿⡇⣿⡇⠀⢸⣿⢸⣿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣿⣿⣿⣿⣶⣶⣿⡇⣿⣷⣶⣾⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢸⡏⣭⣭⡝⣿⣿⣯⣭⣭⢹⡇⣿⢩⣭⡍⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢠⣀⠀⢸⡇⣿⣿⡇⣿⣿⣿⣿⣿⢸⡇⣿⢸⣿⡇⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠈⠉⠽⠋⠀⢸⡇⣿⣿⡇⣿⣿⣿⣿⣿⢸⡇⣿⢸⣿⡇⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⡇⣿⣿⣿⣿⣿⢸⡇⣿⢸⣿⡇⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⡇⣿⣿⣿⣿⣿⢸⡇⣿⢸⣿⡇⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⡇⣿⣿⣿⣿⣿⢸⡇⣿⢸⣿⡇⣿⢸⣿⡟
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠿⠿⢇⣿⣿⡿⠿⠿⢸⡇⣿⠸⠿⠇⣿⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠋⠙⠛⠛⠛⠛⠃⠛⠛⠛⠛⠛⠘⠛⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⣛⣛⣛⣛⡻⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⠞⢉⡴⠒⢲⢒⡲⣌⠙⢦⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢟⣵⢟⣫⣭⢩⣭⡛⢷⣝⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⡏⣰⣋⢯⢣⣸⣏⡽⣹⢳⡈⣧⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⣾⢡⣎⢿⣿⡎⡟⡴⣣⢻⣆⢿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⠀⣿⡌⢸⠻⣼⢋⠀⡟⣿⡇⢸⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⡇⡸⣿⢰⡙⢰⣾⣇⡿⡀⣿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⢸⠀⣷⣬⠿⣏⣿⣉⡾⣯⣱⡇⢸⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⡇⠿⣆⣙⠷⢸⢟⣀⡾⠇⣿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⢸⠀⣿⠿⠞⡯⣽⡯⡟⠺⢥⡇⢸⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⡇⢛⣫⣵⡖⡔⣶⣭⣛⠃⣿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⢸⠀⡷⠀⠀⠀⣿⡇⠀⠀⠀⡇⢸⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⡇⣿⣿⣿⡷⡰⣿⣿⣿⡇⣿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⢸⠀⠷⠶⠶⠞⠁⠙⠲⠶⠴⠇⢸⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣧⣙⣛⣫⣾⣿⣮⣛⣛⣣⣿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠀⠀⠀⠀⢸⣿⣿⣿⣿⣮⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣾⣿⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣛⣛⣝⣻⣟⣽⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠚⠛⠛⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠙⠛⠛⠓⠶⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠀⠀⠀⢀⣀⣤⣤⠶⠶⠶⠶⡆⠀⢸⡇⠀⢰⠶⠶⠶⠦⣤⣤⣀⡀⠀⠀⠈⠉⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡴⠚⠉⠁⠀⡇⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⢸⠀⠈⠉⠓⢦⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣧⣤⣤⣤⣤⣧⣤⣤⣤⣤⡇⠀⢸⡇⠀⢸⣤⣤⣤⣤⣼⣤⣤⣤⣤⣼⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡏⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⢹⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⢻⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡿⠚⠛⠛⠛⡟⠛⠛⠛⠛⡇⠀⢸⡇⠀⢸⠛⠛⠛⠛⢻⠛⠛⠛⠛⢿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⢹⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣧⣀⣀⣀⣀⣇⣀⣀⣀⣀⣧⣴⣼⣧⣦⣼⣀⣀⣀⣀⣸⣀⣀⣀⣀⣼⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡏⠉⠉⠉⠉⡏⠉⠉⠉⠉⡇⠈⢸⡇⠁⢸⠉⠉⠉⠉⢹⠉⠉⠉⠉⢹⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣇⣀⣀⣀⣀⣇⣀⣀⣀⣀⡇⠀⢸⡇⠀⢸⣀⣀⣀⣀⣸⣀⣀⣀⣀⣼⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡁⠀⢸⡇⠀⢈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢈⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇⠀⢸⡇⠀⢿⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⢸⡟⠛⠛⠛⠛⠛⣿⠀⡇⠀⢸⡇⠀⣿⠀⣿⠛⠛⠛⠛⠛⢻⡇⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠸⠧⠤⠤⠤⠤⠤⠿⠀⡇⠀⢸⡇⠀⣿⠀⠿⠤⠤⠤⠤⠤⠼⠇⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣧⣀⣀⣀⣀⣀⣀⣀⣀⣠⡧⠀⢸⡇⠀⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣼⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⣦⣴⣶⠈⢹⡆⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⢰⣶⠶⠶⠶⠶⠶⠶⣶⣦
⠀⠀⠀⠀⣿⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⢰⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠈⠙⠿⠀⠘⠃⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠈⠉⠛


⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⣀⡀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⡄
⠀⠀⢸⠉⠉⠉⠉⢹⣿⣿⣿⡇⠀⠀⠀⠀⢸⠀⣶⣶⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⣿⣿⡇
⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣯⣴⣦⣼⣿⡇
⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠸⠤⠤⠤⠤⠼⠻⠿⠿⠇⠀⠀⠀⠀⢸⠀⠿⠿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿




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

⠀⠀⠀⠀⠀⠀⠐⢀⠐⡆⣠⠆⠀⢲⡀⡶⠀⠀⠐⣶⠀⢀⠀⠀⡀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠁⠀⢶⠀⠹⣇⠀⠀⣀⡼⣇⠸⣤⠞⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⣄⡀⢻⠉⣉⠙⢿⡀⢹⠴⠚⠣⣄⠘⣄⣣⠀⠀⢹⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢺⡞⠉⠀⠀⠙⣾⠀⡀⠀⣄⠀⢸⣃⣤⠖⠋⠉⠈⠀⠀⢀⡴⠛⢫⣋⠍⠏⣏⡛⠗⣤⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢐⡧⠚⠁⠀⠘⣦⡏⠁⠈⡆⠀⠀⠀⠀⡴⣋⣤⣤⠀⠙⣷⠶⠶⠶⠾⠾⢷⠆⡇
⠀⠀⠀⠀⠀⠀⢰⠤⠴⠖⠷⠶⠿⠷⠾⠧⢤⠤⣿⣤⣾⣀⣴⣈⣓⣾⠓⣶⠾⣇⢹⠠⢸⠀⠀⣿⣿⢻⢹⡆⠀⢸⡶⡇
⠀⠀⠀⠀⠀⠀⠘⠚⠓⠛⠛⠛⠛⠓⠒⠒⠦⠤⣤⡀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠻⠚⠶⠼⢦⣄⣾⣈⢙⠉⠁⠀⢸⣷⡇
⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣄⣀⣄⣀⣀⣄⣀⣀⣀⡀⡀⠀⠀⠀⠈⠉⠋⠓⠒⢼⣍⡇
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠠⠄⠀⠀⠀⠈⠉⠉⠉⠉⠙⠒⠒⠲⠦⠤⣀⣀⣈⡇
⠀⠀⠀⠀⠀⠀⢸⢀⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡤⠤⠤⠤⢤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⡃
⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⡠⢠⠴⣿⢻⡆⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⡀⢀⣀⢀⠉⠉⠉⠙⠒⠒⠶⠤⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⡋⢳⣯⠭⡇⠀⠀⠀⠀⠀⠀⠀⠹⣟⠿⠿⠉⠩⠉⠍⠉⠙⢛⣺⡶⠦⣤⡀⠀⡅
⠀⠀⠀⠀⠀⠀⠐⠀⢀⣤⠶⣄⠀⢹⡉⡾⠴⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡒⠂⠀⠂⠀⠂⠾⣍⣽⠁⠂⡌⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠻⡞⢩⠷⣤⣳⢷⡊⠀⢀⣤⠞⢻⢉⣯⣿⠃⠀⠠⠀⠀⣽⢷⣮⡀⡂⠐⠀⢾⡆⢄⣅⣠⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠲⢷⠋⣰⢻⠻⣾⢇⡴⡇⢸⣤⠞⠛⡶⠃⠀⠀⠀⠀⠀⣳⠾⠃⢹⣭⣅⣁⣹⡗⠻⣤⡿⠦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣟⠲⠼⡞⠁⡞⠀⣏⣾⣠⠷⢯⣬⠽⠋⠀⠀⠀⢀⠀⠀⠀⣇⣤⡀⠀⣿⡏⠁⢽⡏⣁⣽⡉⡒⡇
⠀⠀⠀⠀⠀⠀⢀⠀⠈⠙⠦⣧⢞⡁⠘⢿⠛⠛⠉⠉⠀⠀⠀⠀⡀⣹⡛⠻⡆⢸⠿⣼⠇⠀⣿⡧⠰⢾⡇⠦⢿⠀⣤⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠶⢾⡿⠛⢷⣮⣧⣶⣤⡦⠀⠀⠸⠀⠙⡹⢿⡴⠃⢸⠀⢿⠀⠀⡟⡇⠘⣸⡗⢐⢸⠒⠂⡇
⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠧⠤⠤⠤⠵⠿⠩⠤⠦⠤⠾⠤⠴⠥⠤⠧⠠⠼⠦⠼⠤⠤⠧⠯⠩⠽⠍⠍⠽⠭⠭⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⢠⠙⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣷⣼⡄⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠐⣶⣿⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣺⣿⣿⣿⣿⣽⣿⣿⣯⡄⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠐⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢃⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣍⣉⠉⠉⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣟⡂⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣷⣶⣶⣦⠄⠀⠀⠈⠩⠽⠿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣛⣛⣀⠀⠈⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢇⣀⠀⠀⠀⢤⢀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⢀⣔⣉⣉⣀⣀⣀⠤⠚⠉⠒⠠⣄⠀⠀⠀⠀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠃⠀⢀⣤⡴⠚⠉⠁⠀⠈⣀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣦⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠲⢤⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢨⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣾⠁⠀⠀⠀⠀⠈⢳⣰⠟⣿⠲⡀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠳⡴⠶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆⢀⣼⣿⠉⠳⡄⠀⠀⠀⢀⡴⠋⣸⡏⢳⡀⠀⠀⣠⠖⠶⠞⠃⠀⢿⡑⠉⠢⣄⠀⠀⠀⠀⡴⠃⠀⠀⠀⢀⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠙⣦⣠⣴⠟⠀⣾⢧⣙⡢⠙⠲⢮⣁⣀⣀⠀⠀⠀⠈⢿⠂⣀⣈⡷⢾⣧⡾⠁⠀⠀⠀⠀⢺⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣇⢀⡴⣾⠉⠀⠀⠀⠀⠻⣄⠀⠉⠂⠀⠀⠀⠉⠉⠑⠄⠀⠀⠈⠋⠉⠁⠀⢾⡈⠛⠶⢤⣤⣤⠞⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠁⠀⣿⡄⠀⠀⠀⠀⢠⡘⣯⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⢸⡀⠀⠀⣸⠃⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡆⢸⣿⡇⠀⠀⠀⠀⠈⢿⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⡀⢻⡄⣿⡇⠀⢠⢿⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣾⣿⣿⡀⠀⠀⠀⡄⠘⣧⠈⠓⠦⢄⠀⣀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⣿⠀⢹⣿⣿⣠⣯⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣷⠀⣿⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⣿⣿⣧⠀⠀⢸⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣏⢢⡄⣸⣿⡄⠙⢆⠀⠀⠀⢸⣿⡇⠀⠀⢀⣿⣿⣿⠀⠀⣾⣿⣧⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠁⠀⠀⣾⣿⣿⠀⠀⢾⣿⣿⣿⣦⣠⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⡉⢻⠋⢲⠾⣿⣿⣿⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⢀⡀⠀⠠⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⡄⠀⠃⠀⠀⡀⠀⠀⠀⠐⠐⠀⠀⠠⠀⠀⠀⠀⡄⠀⠀⠙⢆⠀⠀⠀⠀⠀⡴⠋⡂⠔⠸⠀⠀⠤⠼⢦⠀⠿⠃⠀⠀⠀⠀⠈⣀⡶⠒⢡⡲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⣤⠀⠀⠀⠀⠀⠀⠤⠀⠒⠀⠀⠰⠀⠀⠈⠃⠀⢐⠀⡸⠏⠀⠀⠀⠀⠀⠀⢠⡤⠀⣶⣶⠐⠿⠂⠀⠀⢯⡈⢇⠀⣠⡏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⡀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠠⢰⡇⠀⠀⠀⠀⠀⠀⠀⠰⠂⣤⠈⠀⣦⠀⠀⠀⠀⠙⠵⡎⡘⡳⠈
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⡄⠀⢨⠀⠀⢀⡀⠀⠀⢃⠀⠁⠀⠓⠀⡠⠀⠀⠀⠁⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠖⠈⠀⠀⠀⠀⢠⡀
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠃⢀⠀⠀⠇⠸⠃⠀⠀⠀⠑⠤⢀⢀⣀⣀⠤⠂⠀⠀⠀⠀⠀⡈⡠⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠰⠄⠤⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣰⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠆⠀⠘⢰
⠀⠀⠀⠀⠠⠀⠀⠠⠐⢤⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⡀⠀⠀⠀⢀⠃⠁⣟⡀⠈⠀⠀⠀⠀⢀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠠
⠀⠀⠀⠀⠄⠠⠀⠀⡀⡄⠀⠀⠀⠀⢆⠄⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠉⠄⠀⠀⠈⠁⢸⣏⡀⠀⠀⠀⣀⠀⡀⠀⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠤⠀⢐⢤⠼⠔⠉⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⢀⠀⠀⢀⡀⢰⠀⠀⠀⠀⣶⠋⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢀⠄⠠⢄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠠⢠⠀⠀⠀⠐⠈⠈⠀⠀⠈⢁⡏⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⠞⠀⠀⠀⠀⢇⢒⠆⠀⢱⠀⠀⡈
⠀⠀⠀⠁⠀⠀⠀⠀⡄⢠⡀⠃⠀⠀⠀⠀⠀⣇⠀⠀⠄⠀⠳⠀⠄⠀⠀⢠⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠊⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢳⡄⠀⡠⠃⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢀⡀⠀⠈⠫⡀⢀⣠⡰⡶⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡃⠄⠁⠀⠀⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣽⠁⠀⠀⠀⠀⠰⢁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢠⠄⠘⢋⣉⠘⠑⠢⡀⠀⠀⠀⠀⠀⠤⠈⠀⡀⠄⠋⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠘⠛⡠⠀⢀⠜⠀⠀⠀⠀⣀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠔⠤⠄⠀⠀⠀⠀⠀⠀⠀⠈⠒⠲⠦⠴⠖⠚⠁⠀⠀⠀⠀⠀⠁⠙⠀⠀⠀⠀⠀⠀⠰⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠐
⠀⠀⠀⠀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢠⠞⠁⣈⡙⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠚⠀⠄⠰⠀⠀⠄⡡⠆⠀⠀⠀⠀⠀⠀⣤⠀⠈⠒⠀⠀⠑⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠃⠚⠐⠂⢀⠈
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠚⠁⠀⠉⠑⠲⢼⡀⠀⠀⣆⠲⠀⡠⠠⢄⠀⠀⠀⠰⠞⠀⠀⢰⣤⣼⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠘⠄⡃⠀⠚⠃⣀⠨⠵⠤⣀⠉⠀⡟⠀⢶⡸⠀⠀⠀⠀⠀⠀⠘⣿⣭⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠔⡀⠀⠀⠑⢆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡰⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⣀⠀⢇⠀⠲⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⢀⠀⠱⣄⠊⠀⠘⢀⡤⠀⠀⠀⠀⠀⠀⡸⢁⣀⠀⠀⠱⢸⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠢⣌⣁⣄⢸⣇⣀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠰⢈⡁⠀⠀⠀⣿⡰⠊⢁⠑⡄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠈⠙⠷⢶⡷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠐⢣⠃⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣴⣶⣒⣒⣒⣒⣒⣒⣒⣒⣠⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡏⠀⠀⠀⠀⣀⣀⣀⠤⠤⠔⠒⠚⠋⢹⡇⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡏⠉⢁⣀⣀⣠⠤⠤⠤⢦⠀⢸⡇⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡅⢸⠉⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠆⡇⠀⠀⡇⠘⠒⠒⠒⠒⠒⠈⠉⠉⠀⢸⡇⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇⢠⠤⠤⡄⠀⣤⠐⠒⢲⠀⢸⡇⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇⢸⠀⠀⡇⠀⢳⠀⠀⢸⠀⢸⡇⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇⢸⠀⠀⡇⠀⢸⠀⠀⢸⠀⢸⡇⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇⢸⠀⠀⡇⠀⢸⠀⠀⢸⠀⢸⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⣧⣼⠀⠀⡇⠀⢸⠀⠀⢸⠀⢸⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡏⢹⠀⠀⡇⠀⢸⠀⠀⢸⠀⢸⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇⢸⠀⠀⡇⠀⢸⠀⠀⢸⠀⢸⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡇⠀⠀⡇⢸⠀⠀⡇⠀⢸⠀⠀⢸⠀⢸⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⡇⠀⠀⡇⢸⠀⠀⡇⠀⢸⠀⠀⢸⠀⢸⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇⢸⠀⠀⡇⠀⢸⠀⠀⢸⠀⢸⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇⠸⠤⢄⡇⠀⣺⠀⠀⢸⠀⢸⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠉⠋⠀⢸⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠓⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠸⡗⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠲⠤⢤⣼⣷⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⡼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣠⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠀⠉⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⢰⣾⠿⠛⠟⠛⢻⡇⠀⢸⡿⠻⠚⠛⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠽⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠄⠀⠀⠐⣿⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠆⠀⠀⠀⣾⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠄⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡯⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠈⣧⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡧⠀⣀⢀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠎⠀⡈⡇⢸⡄⠀⠀⠀⠀⢘⠀⠀⣿⠀⠀⠀⠀⡗⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⠀⢿⣂⡭⣭⣭⣍⠇⢀⠀⢀⠀⠀⣿⠀⠀⠀⠄⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⡀⢰⣟⣆⢼⣇⣤⣤⣭⣤⣾⠀⠀⣿⣒⣒⣒⣰⠷⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠒⡎⢛⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠉⠂⠁⢰⡶⠻⠿⠟⠻⠿⠀⠀⣿⠛⠛⠛⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⠀⠀⣿⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆⠀⠀⠀⢾⡇⠀⠀⠀⠀⢰⠀⠀⣿⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡆⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⣯⡇⠀⠀⠀⢸⣇⠀⠀⠘⠂⣸⠀⠀⣿⡀⠀⠀⠁⣀⡆⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⡃⠀⠀⠀⠘⠋⠉⠉⠛⠋⠛⠀⠀⠛⠋⠑⠂⠒⠛⠁⠀⠀⠐⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡃⢸⣇⣀⢀⢀⢀⠀⡀⡀⡀⣀⡀⡀⠀⠀⢀⢀⣀⢀⠀⠠⠤⠆⣤⠀⠀⢀⣐⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠨⡭⠑⠚⠛⠛⠛⠛⠛⠙⠛⠛⠛⠹⠷⠿⠼⠦⠴⠶⠶⠞⠲⠳⠚⠟⠻⠶⠶⠆⠦⠾⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣀⣀⣀⡀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⢂⠀⠈⠈⠀⡡⢊⠅⠉⠘⡡⢊⠔⣋⠘⡁⠋⠙⡀⠁⠚⠰⠂⠖⠠⡂⠀⢒⠚⡰⠂⢛⠘⠃⠋⢘⡃
⠀⠀⠀⠀⠀⠀⠀⡀⠜⠶⠦⢦⡤⣤⢈⣀⣡⣊⡔⣡⠐⣁⢊⡄⢈⠐⣀⣪⣀⡔⣁⢀⣑⣀⣈⣀⣘⣁⣀⣀⡐⣀⣂⣄⣤⣼⣇
⠀⠀⠀⢠⡴⣈⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠈⠁⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠈⠁⠉⠉⠉⠉⠁⠁⠀⠀⠈⠑⠠⣀
⠀⠀⠀⠀⡆⠕⡩⢊⠕⡩⢋⠝⡫⢛⠙⢋⠝⠋⠗⡠⠖⠶⡲⢖⠶⠲⠒⠶⠒⠖⠲⠲⠔⠖⠠⠀⠴⢠⠆⠤⠒⠒⠒⠲⠲⠒⠖⠖⠰⠘⠇
⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠈⠐⠁⠐⠀⠂⠀⠈⠐⠠⠊⠀⡡⠂⠐⠠⠀⠔⠡⠀⠄⠠⠀⠔⠠⠀⡀⠀⠀⠀⠁⠂⠀⠐⠀⠀⠂⠂⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢹⡇⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢻⡆⠀⠀⠀⠀⠀⢠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⢀⡼⠁⠀⠀⠀⢰⣿⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣼⠁⠀⠀⠀⠀⠀⢠⡄⠀⠀⣀⣤⣶⣦⡀⠀⠀⠀⣼⣿⣀⡴⠋⠀⣠⠶⠶⠶⣿⡿⠶⣶⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠻⠿⣿⡿⠀⠀⢠⣿⠃⠀⣴⡿⠋⠀⣸⣿⠀⠀⣰⣿⡏⠁⠀⠀⠘⠁⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⢠⣿⠃⠀⢀⣾⡟⠀⢸⣿⠁⢀⣴⠋⣿⣠⠞⢹⣿⢀⡴⠛⣿⡆⠀⠀⠀⣿⡟⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⣴⠟⣿⡇⠀⣸⠿⠶⠟⠁⢀⣿⠁⠀⢸⣿⠏⠀⣸⣿⠁⠀⠀⣰⣿⡇⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⣿⣇⣴⠃⠀⠀⠀⡴⠋⣿⠀⠀⠘⣿⠀⠀⣿⡏⠀⠀⣴⠋⣿⡇⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⢀⡞⠁⠀⣿⠀⠀⠀⠀⠀⠀⢻⣧⣤⠞⠁⠀⠘⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣴⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⢸⠇⠀⠀⠀⠀⠀⠀⣰⡿⠁⢸⠇⠀⢠⣿⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⢀⡞⠀⢀⣀⡀⠀⠀⣰⡿⠀⡠⠋⣀⣠⣿⣇⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣠⠎⠀⣰⡿⠋⣻⠀⢰⣿⠷⠊⠰⠛⢻⣿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠁⠀⢰⣿⡧⠞⠋⢠⣾⡟⠀⢀⣠⢴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⢸⣿⡀⠀⣠⣾⣿⠿⡏⠉⠀⣼⣿⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢠⡟⠙⠿⠛⠉⢸⣿⠀⡗⠀⠀⣿⣇⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⣴⠟⠀⠀⠀⠀⠀⠘⣿⣴⠇⠀⠀⠘⠿⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣀⡀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣼⠃⢱⡄⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⣀⣤⠴⠋⢀⡼⠃⠀⠀⠻⣄⠈⠳⢤⣀⣀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⡏⠀⣠⠔⠋⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⢹⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢠⡾⠛⢷⡄⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠘⢷⣤⡾⠋⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢠⡤⣤⠤⣤⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢸⡿⣿⡗⣿⡁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⡇⠀⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾⠀⢸⠀⢸⡷⠛⢲⡟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼⠀⢸⡇⠀⢸⡯⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣾⣷⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⣴⠒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⠒⡆⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⣿⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣗⡽⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡛⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⣿⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡗⣝⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠙⠒⠒⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠚⠒⠃⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⡿⠿⠿⢿⠿⣉⠿⢿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⢛⣛⣛⣻⣛⣿⣿⠛⣛⢛⣛⠛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⢀⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⣹⣿⣠⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠔⣦
⠀⠀⠀⠀⠸⣦⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⢠⣞⣩⡏⠙⢻⡍⠀⢻⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣾⣿
⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⢠⣢⠀⠀⢸⡇⣿⣿⣿⠃⠀⠀⣷⠀⠈⣿⣿⣿⠀⠀⠀⡐⣆⠀⠀⣸⠃⢸⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⣿⣿⡆⠀⣼⠁⣿⣿⣿⠀⠀⠀⣿⠀⠀⢹⣿⣿⡄⠀⢰⣿⣿⠀⠀⡟⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣄⡸⢻⣿⣇⢠⠇⠀⢻⣿⣿⡇⠀⢠⠏⠀⠀⠀⢿⣿⣧⣀⡜⣿⣿⡀⣸⠃⠀⠘⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠹⠿⠟⠁⠈⠻⠿⠋⠀⠀⠀⠛⠿⠷⠶⠋⠀⠀⠀⠀⠈⠻⠿⠟⠀⠘⠿⠿⠃⠀⠀⠰⣶⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⠀⠀⢀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢼⢸⡆⠀⠀⣷⠀⠀⠀⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⡇⠀⠀⠀⡄⢴⣯⣿⡄⠀⠀⠀⠀⠀⠀⢀⡀⢰⡇⠀⠀⠀⠀⠀⢸⣇⣿⣀⣾⣮⣇⣄⠖⣿⠀⣴⣠⣗⣷⢴⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣇⣸⣷⠸⡀⢀⣇⣿⣿⣿⣷⠂⠀⠀⠠⣄⣀⣸⡧⢸⣧⣰⣆⠀⢀⣈⣿⣿⣿⠟⠻⣿⣿⣿⠤⠿⣞⣻⣷⣹⣿⢉⠿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡂⢰⣿⣿⣿⣿⡈⡒⠘⣿⣿⣟⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠈⢻⠃⠀⠀⠹⢿⢛⣘⣧⣾⠛⣽⣟⠡⣛⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣆⣷⣿⣿⣿⣿⡿⢳⡏⠀⢻⣿⣿⠁⢅⢠⡀⠀⠀⡙⠈⢿⣿⣿⡿⠋⠃⠰⠴⠦⣴⡏⠀⠀⠀⠀⠼⣟⣡⡿⣟⠛⣿⠋⣸⣿⣗⣠⣿
⠀⠀⠀⠀⣠⣴⣿⣿⣻⣿⣿⣿⣻⣿⢋⣽⠀⠀⠘⣿⠟⣀⠈⣟⣰⣤⣴⡁⠈⠙⣿⣿⠀⠀⠀⠀⠗⠀⢈⡧⠀⠘⠐⠦⠀⣸⣿⠀⢸⣷⣾⠿⣗⡼⠳⣄⡀
⠀⠀⠀⣸⣿⣿⣿⣷⣿⣿⣿⣿⠽⢿⣼⠋⠀⠀⠀⣿⠍⠀⠀⢿⣌⣿⠇⠀⠀⢀⠻⣿⣿⣤⣤⣶⣄⡾⠁⠀⠀⠀⢀⠀⠀⢈⡿⣦⣿⢽⣿⠂⣹⣿⣤⡧⠁
⠀⠀⢠⣿⣿⠿⠟⠛⠋⠁⠀⠀⢀⠀⠀⠀⠀⠀⠐⠏⠀⠑⠒⢘⠛⠟⠀⠈⢁⡠⠄⠈⠀⠀⠀⠙⠘⠳⠀⠀⠰⠛⠻⠟⠟⢿⣤⣾⠸⠯⠐⠂⠉⢧⠀⠈⠆
⠀⠀⠈⠉⠀⠀⢺⠇⠀⠀⠐⠚⢻⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠉⠉⠁⠋⠛⠿⠶⠆⠊⢉⣉⡲⠏⣹⣦⡨⠷⢶⣄⠀⠀⠠⢄⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀⢀⡀⠤⠤⣄⠀⠐⠶⣦⡀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠛⠓⠒⠚⠛⠛⠻⠓⠒⠺⠿⠀⠀⣀⣄⣀⣀⣀⠛⣫⣉⣱⡖⠂⠉⣳⣀⣀⣸⡧⠄
⠀⢠⠀⠰⠖⠀⢀⣠⣤⣄⠄⠀⠀⢠⡀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠙⠛⠛⠿⣷⣶⣾⣤⣼⣿⣠⡀⠰⣿⣆⢀⡄⣤⢷⣀⡄⡀⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣄⣩⣿⣾⣧⣽⣧⣿⣷⣞⣻⢻⡄⣿⡆⠀⠄⠀⡀⠀⠀⠀⢰⣧⣦⡀⢂⢰⣄⠀⠀⠀⠘⣆⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⢿⡿⢿⣿⣟⣉⣉⣹⣿⣿⣿⡇⠀⢠⡄⣧⣶⣄⣦⢸⡿⣿⡇⠘⡞⣿⠀⠀⠀⡄⢻⢄⠀⢠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⠏⠛⠛⠛⠾⣓⠀⠠⠅⡽⢳⣿⣿⣼⣿⣿⣷⢀⣦⣿⠄⠀⠀⢸⢸⣿⡆⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣉⣉⣁⣈⡁⠀⠀⢰⣉⣉⣩⡍⣻⣿⣿⣭⣄⣀⣦⠚⠀⢺⣿⢿⡟⠋⠁⢓⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠑⠈⠙⠋⠉⠍⠭⢙⣿⣦⡤⢤⣶⣾⣟⡿⣯⣭⡭⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⢶⠲⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⡇⢿⢹⣿⠾⡵⠆⢀⣴⣶⣀⣶⣿⣿⣿⣿⣦⠀⠀⠉⠙⠋⠉⠛⠛⠛⠛⠓⠒⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠖⠮⠅⢀⠸⢾⣿⣿⣿⣷⣶⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠌⠛⢻⠛⠛⠟⠋⢩⢥⠌⠟⠗⣠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠛⠚⠻⣶⣌⣁⡋⢠⠭⠬⣿⣶⡆⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠹⠇⡉⠲⠒⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀
⠀⠀⠀⠀⢀⡤⠖⠚⢯⢆⠉⠻⣿⡟⠉⠉⠉⠉⢹⠏⠉⢻⡿⠛⠉⠉⠙⠻⣿⠋⠉⣿⣿⠉⠙⠋⠉⠙⠉⠉⠉⣿
⠀⠀⠀⠀⡾⣥⠘⣶⣼⣿⡄⣤⢻⡇⠀⠐⠛⠛⢿⠀⠀⢿⠀⠀⣶⡶⠤⠴⣿⠀⠀⠙⠋⠀⠀⣷⣶⠀⠀⣶⣶⣿
⠀⠀⠀⠀⣿⡕⡽⣿⡿⡏⠱⣙⠈⡇⠀⠀⣶⣶⣾⠀⠀⣸⠀⠀⠻⠷⠀⠀⣿⠀⠀⣶⣦⠀⠀⣿⣿⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠹⠣⡼⢼⡄⠀⠀⠁⣼⣧⣤⣤⣿⣿⣿⣤⣤⣽⣷⣤⣤⣤⣤⣴⣿⣤⣤⣿⣿⣤⣤⣿⣿⣤⣤⣾⣿⣿
⠀⠀⠀⠀⠀⢣⠻⠆⠇⠀⠤⣼⣿⡇⠀⠤⠼⠁⣤⠈⢻⠀⠤⠀⣿⣿⣄⠙⢁⡼⠁⣤⠈⢋⠀⣿⠀⡏⠀⠤⠈⣿
⠀⠀⠀⠀⠀⢸⢈⠴⠀⠉⢷⣿⣿⡇⠀⣶⣾⡀⠛⠀⣼⠀⣤⠘⢿⣿⣿⡀⢸⣧⡀⠛⠀⣼⠀⠛⢀⣧⠀⣄⠘⣿
⠀⠀⠀⠀⠀⢸⠸⠃⠀⠀⢸⣿⣿⡟⠛⠛⠛⠛⣿⠛⢻⡿⠛⠛⠛⠻⣿⠛⢻⣿⠛⠛⠛⠛⠛⠛⢻⠟⠛⠛⠻⣿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⡇⠀⠚⠃⢀⣿⠀⢸⠀⢠⣶⠦⠐⡇⠀⠈⠉⠀⢸⣶⡆⠀⣾⣿⡀⠈⠓⠲⣿
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢸⣿⣿⡧⠀⢰⡄⠘⢿⠀⢸⡆⠀⠛⠁⢀⡇⠀⢸⣿⠀⢸⣿⡇⠀⣿⣿⠈⠙⠃⠀⣿
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠠⢶⠒⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⠀⠂⠀⠁⠀⠀⠠⠀⠐⠐⠦⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠂⢹⣗
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣤⣤⣄⣀⡀⠀⠀⠀⠀⠁⠐⠂⠄⠁⡀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠉⠀⢀⡀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠇⠉⠉⠛⠛⠳⠶⢶⣤⣤⣀⣀⠀⠀⠐⢹⡤⠤⠐⠊⠁⠀⣀⣤⠴⠞⠛⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⡖⠈⠁⣀⣠⡴⠾⠇⠨⡏⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⢰⠛⡋⠀⡀⢀⡇⢨⡇⠀⠀⢀⠄⠇⠀⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢐⡆⠀⡁⠊⡇⢸⣿⠀⡰⠣⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⠇⠐⠁⠀⠀⢨⡇⠈⡠⠁⢰⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢰⡆⢠⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣼⣧⣄⠅⠠⠀⠸⠇⠐⠒⢂⠡⢤⡀⣿⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣤⣤⣤⣄⣀⠀⢠⠤⠤⠴⠼⢐⡼⠃⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠰⢦⠀⠀⠀⡀⢸⠂⠐⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⢸⠐⠀⡐⠀⡇⢸⠄⠀⠀⠐⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡨⠀⠀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠸⢠⠚⠀⠀⠁⢸⠦⠀⣠⢂⡄⠀⠀⣿⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⣶⢀⠐⠂⢀⠀⢸⣖⠋⡁⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠻⠿⠉⢴⣚⢻⠀⢸⠁⠘⠀⠀⠀⠀⠀⡟⠀⠂
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠙⠸⠀⢸⣄⣀⠐⠂⠄⠀⠀⢶⠀⠠⠤
⠀⠀⠀⠀⠀⠠⡀⠀⠀⠄⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠤⠶⠒⠉⣉⠉⠓⠦⣤⣐⠀⠀⠀⠀⠈⠉⠒⠠⠇⠀⢸⣽
⠀⠀⠀⠀⠀⠈⠨⠀⠀⡄⠀⠀⠸⠀⢀⢠⡀⠀⠑⠀⠀⠀⠒⠂⠀⠀⠀⠄⢈⠁⠋⠒⠢⢤⣀⡀⠀⠀⠀⢀⢸⣿⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠠⡂⠠⠣⣸⣿⡶⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠦⠄⠀⠀⠀⠀⢀⣀⠉⠒⠦⢄⡢⢸⣿⡀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠓⣈⡠⠿⠦⠖⠛⠉⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠊⠍⠁⢊⠉⠁
⠀⠀⠀⠤⠶⠽⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿
⣿⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⣿⠀⠀⠀⢸⣿⠋⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣷⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠄⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠐⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢸⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠇
⣿
⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠋⣱⣾⣿⣷⣿⡟⠛⠈⡻⠋⠁⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠓⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡄⢀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠈⠛⠋⠀⣿⡀⠀⢀⠀⠀⠠⣿⣇⠀⠀⣾⣇⠀⠀⠀⠀⠃⠀⠀⠀⠀⣧
⣶⣶⣿⡿⠀⠀⠀⠀⢀⡄⠁⠀⣰⣿⠆⠀⠀⣼⣿⡇⠀⣼⡇⠀⣨⣿⣧⡀⢠⣿⣧⡀⠀⢸⡄⠀⠀⠀⠀⣰⣿
⣿⣿⡿⠁⣀⠀⠀⠀⠈⠁⠀⠀⣿⣿⠇⠀⢘⣻⣿⠷⠄⣿⡷⠀⣸⣿⣷⠀⡠⣿⣿⢀⣴⣿⣷⠀⠀⠀⣴⣿⣿
⣿⣿⡇⣼⣿⠀⠀⠀⠀⠀⠀⠠⢿⡿⣦⠀⣈⣿⣿⣁⠰⣿⡟⠒⢻⣿⡏⣾⣿⣿⡷⣶⣿⣿⠷⠀⠀⠀⠻⢿⣿
⣿⣿⣧⣿⡏⠀⠀⠀⠀⠀⢰⣗⢿⣷⣤⠀⢻⣿⣿⣟⣰⣿⣿⢐⣯⣿⣧⣾⣿⣿⡟⠂⢾⣿⣷⠀⠀⠀⠀⠀⣿
⠻⢿⣿⣿⠃⠀⠀⠀⠀⢠⣿⣿⣿⣿⣤⠀⢾⣿⣿⢛⣇⣿⣏⣸⣿⣿⡯⣏⠹⣿⢷⣆⣹⣿⣟⠀⠀⠀⠀⠀⡟
⠀⣼⣿⡏⠀⠠⠀⠀⠀⢼⣿⣿⣿⣿⣼⡹⡒⣿⣿⠘⣿⣿⣿⡟⣿⣻⡆⣰⣶⣿⣰⣏⣽⣿⣏⠁⠀⠀⠀⠀⡇
⢰⣿⣿⠁⢀⠆⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣷⡆⢿⣿⣿⡿⣿⣠⡮⠸⣾⣿⣽⣿⡏⣍⣽⣿⣟⠃⠀⠀⠀⠀⠃
⣾⣿⡟⠀⡼⠀⠀⠀⠀⢸⣿⡏⣼⣿⣿⣿⣿⣤⡟⣿⣾⣿⣿⣿⣷⣶⡏⡗⢸⣿⣨⣽⣿⣿⡌⠁
⣿⣿⡇⢰⠇⠀⠀⠀⠐⠻⣿⡿⡿⠛⠟⣿⢷⢿⠉⢿⣿⣿⡏⢹⣿⣿⣷⠇⢨⣿⣿⣿⣿⡇
⣿⣿⣷⡟⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⢸⣿⠈⠀⢿⣾⠿⡏⣵⡜⣿⣿⣿⣶⡇⢸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢸⣿⡄⠀⠘⣿⣾⠁⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⣼
⣿⣿⡏⠀⠀⠀⠀⠀⢰⢠⠏⠀⠀⠀⣿⣿⡇⠀⠀⠸⣿⣀⣼⣿⣿⣿⣿⣿⡿⣿⠛⠉⠁⠀⠀⠀⠀⠀⠀⣠⣿
⣿⣿⡇⠀⠀⠀⠀⠀⢛⠿⠻⠿⠶⠶⠿⠿⠳⣤⠀⣀⣿⣿⣿⣿⣛⣋⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⣰⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠐⠿⣿⣷⣲⣶⣶⣶⣷⣶⣠⣶⣶⢶⣶⣼⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠟⠛⡛⠿⠯⠾⠿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢀⣿
⣿⣿⣷⣤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⢤⣤⣤⠤⠤⠀⣾⣧⣤⣤⣿

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄


⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡁
⠀⠀⠀⠀⠀⠀⢀⢈⡉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⣉⣉⠁
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃








⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⣀⠘⠒⡀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠹⣄⠀⠈⠂⠀⠀⠀⠀⢠⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢇⠀⠀⠨⣆⠴⠁⠀⠀⠀⠀⠸⡇
⠀⣠⡠⣄⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠀⠈⠻⣆⠀⠀⠀⠀⠀⢄⣡⠄
⠀⠀⠀⠮⠿⢦⣀⠀⠀⣀⠠⠝⠓⠦⣀⠀⠀⠀⠀⣀⣹⣆⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠘⣡⢘⡅⠀⠀⠀⠀⠀⠨⠟⠚⠛⠋⠉⠉⣩⠽⠒⠢⠄⣩⡆⠈⠳⣄
⠀⠀⠀⠀⠶⢦⡁⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠖⠛⠛⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⣨⠐⠻⠶⣦⣄⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢀⡄⠀⠀⢀⠀⠈⠉⠛⠶⠷⢶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠷⣶⣾⣿⣇⠀⠀⠀⠀⠀⠀⢬⠁⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠉⠉⠀⠀⠀⠀⠀⠀⠲⢀⠎⠀⠂⣀⠐⠂
⠀⡀⠀⢸⡄⠀⠀⠀⠀⠐⣄⠀⠀⠀⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠐⠲⠉
⠀⠘⢶⣄⢻⡄⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡷⢸⡿⠿⡗⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠉⠻⣿⣷⣾⣿⣶⣶⡆⠀⠀⠀⠀⠀⣾⣿⣿⣵⠀⣇⣴⣿⣿⣧⠀⠀⠀⠀⠀⠸⣯⠸⠇⣀⣴⣿⠁
⠀⠀⠀⠀⢀⣀⣠⣴⣿⡿⠛⠛⢋⠝⠀⠀⠀⠀⠀⠘⠛⠁⢷⡼⢁⠜⢻⡿⠋⠀⠀⠀⠀⠀⠀⠘⢆⢸⣿⡟⢃⡀
⠀⠈⠙⠛⠛⠛⠛⠋⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣤⣶⣿⢪⣷⠟⢁⡀⠀⠀⠀⠀⠀⠉⠛⠉⠙⠛⠛⠛⠛⠛⠛⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣠⣿⣿⡿⣾⠁⠾⣿⡿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣤⡜⢿⣿⣿⠙⣷⡴⠻⣍⡙⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣯⢟⠻⠛⠀⠀⠀⢠⣿⠀⢀⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠙⢦⣰⣆⠀⢸⣇⠠⣶⣿⣿⣿⡗⠉
⠀⠀⠀⠀⠀⠀⠀⠠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣼⡟⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⡟⠟⠀⠒⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠉⠊⠓
⠀⠀⠀⠀⠀⠸⣶⣄⠸⢆⠀⠀⠀⠈⠉⠋⠀⠀⠀⢹⣤⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⢙⣻⣧⣼⣤⣶⣶⣦⣤⣄⠀⠀⠀⠀⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⠿⠿⠖⠚⠛⠂
⠀⠀⠀⣴⣾⣿⣿⡟⠉⠙⠻⠀⠈⠉⠉⠀⠀⠀⠀⢀⣀⣻⣿⣀⣀⣀⣀⣠⢔⣒⠼⠟⠋⠁
⠀⠀⠰⠟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠴⠖⠚⠛⢉⣉⣉⣉⣹⣿⣿⣯⣉⣉⣉⠉⠙⠛⠛⢷⣶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠙⠿⢿⠷⣶⣤⣙⡛⠿⣶⣶⢶⣦⣤⣤⣄⡀
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣾⣿⣿⣷⣶⣷⣶⣾⣿⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣛⣿⣻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⡲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⢔⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠖⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⠢⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⠀⠀⢹⠀⠀⠀⢸⠁⠀⢸⡇⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⠷⠶⢿⠷⠶⠶⢾⠷⠶⢾⡇⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⠇⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⣤⣤⣼⣤⣤⣤⣼⣦⣤⣼⠂⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⡇⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⣀⡀⣸⡀⣀⣀⣸⣀⣀⣸⡇⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⠁⠉⢻⠃⠉⠉⢹⠃⠉⢹⣇⣀⣸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⡘⠶⠶⠚⠲⠶⠶⠚⠲⠶⠞⣹⡿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⢫⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣇⣀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣸⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣦⣀⣴⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⠟⠉
⣠⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣄
⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣿⣿⣿⡿⠋⠻⣿⣿⣿⣷⣄
⢹⣿⣿⣿⣿⡄⠀⢀⣴⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣻⣟⣛⣻⠛⠀⠀⠀⠈⢻⣿⣿⣿⣷⣄
⠀⢻⣿⣿⣿⣿⣶⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣧⡀
⠀⠀⠙⢿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⢳⡀
⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢶⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢡⡖⣳
⣿⡀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⠜⠿⣱⡟
⠙⢿⣯⠻⢫⢍⢍⢝⠝⠉⡩⡫⢋⠯⠍⠍⠉⡩⢛⠝⡩⠫⡻⢛⢻⡆⠀⠀⠀⠀⠀⠀⠀⣠⢏⠐⠀⢀⡰⠋
⠀⠀⠙⣗⣕⣁⡑⡁⠡⢠⢊⢀⠔⠐⠁⠀⠀⠔⠀⠊⠀⠀⠔⠕⣹⡇⠀⠀⠀⠀⠀⣠⠾⢁⠉⠀⢰⠏
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠒⠒⠚⠒⠒⠶⠶⢼⠃⠀⠀⠀⣠⡞⠋⠀⠀⢠⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢀⡴⣫⠎⠀⢀⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣴⠫⠀⠁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠃⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⠀⠀⣀⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣉⣴⡾⠿⠛⠛⠉⠉⠉⠙⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡄⠘⠟⠉⣰⣿⣷⣄⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣶⣴⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣸⡷
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⣉⣉⡛⠻⢿⣷⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣶⣌⠻⣿⣷⡦⠈⣀⣤⣶⡶⠛⡷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢹⣿⣷⣿⠟⣠⣾⣿⡇⠋⣠⣾⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠻⣿⡏⢰⣿⡿⠋⣠⣾⣿⡿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣇⠸⠋⣠⣤⣤⡼⠛
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣠⣌⣉⠉⠁⠀⠀⠀⢀⣴⣿⣿⠆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣉⣭⣭⣭⣭⣭⣭⣤⣄⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⣠⣿⣿⣿⣿⠛⠛⠛⠛⠛⢛⣠⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠟⠡⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁

⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⡀
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠉⠛⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⡇⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣇
⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡶⢖⣶⣖⡒⣲⣞⣟⣻⣫⡙⢿⡀⠀⠀⣶⠖⠶⠶⠖⢲⡶⠲⣶⠒⠲⣦⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⣿⠉⠉⠉⠉⠈⠀⠀⠀⢣⢸⡇⠀⠀⢹⢸⠋⠉⠉⠉⠉⠉⠉⠉⢳⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡇⠀⠀⣿⢸⡆⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡇⠀⠀⢹⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡇⠀⠀⢸⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡄⠀⠀⠸⡇⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣇⢸⡁⠀⠀⠀⠀⠀⠀⠀⢸⣇⡇⠀⠀⢸⢿⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠇⠀⠀⠸⡇⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⢺⠃⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⢸⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡟⠃⠀⠀⢸⢸⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⠀⢸⢸⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣧⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡷⠇⠀⠀⢸⢸⠄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⢸⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⣏⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄⠀⠀⣼⢸⠂⠀⠀⠀⠀⠀⠀⠀⢨⣿⠁⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⣸⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⢸⢸⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⢀⡀⠀⢸⣸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡆⠀⠀⠀⣿⠀⠀⠀⠀⢸
⠀⠀⢼⠀⠀⠀⠀⢸⡇⢠⠋⠹⡆⢸⣿⣦⣤⣤⣤⣤⣀⣠⣤⣼⣿⠀⠀⠀⢸⠻⣄⣀⣀⣀⣀⣀⣤⣤⣼⣾⡁⠀⠀⠀⣿⠀⠀⠀⠀⢸⡀
⠀⠀⢸⡇⠀⠀⠀⢸⡇⠸⣀⢀⡇⠘⠛⠒⠒⠛⠛⠛⠛⠛⠛⠘⠛⠀⠀⠀⠈⠛⠛⠛⠙⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⢸⡇⡜⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⢸⡇⢳⣄⣠⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⢸⡇⢸⠰⠆⡁⢰⢿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠍⣭⡭⣭⡭⠽⣯⢹⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⢸⡇⠘⡄⢀⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠘⠋⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⢸⠂⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⣼⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣧⡿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠒⠛⢻⣄⣷⠀⠀⠀⢻⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠋⠙⠛⠛⠛⠋⠉⠉⠉⠉⠛⠃⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⣾⣒⣒⠲⣶⣶⣶⣶⣶⣾⣿⣾⣯⣾⣿⣯⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⡄⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⢿⡏⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡇⠀⠀⠀⣾⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡅⠀⠀⠀⣸⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⢹⠀⠀⠀⠀⢸⡇
⠀⠀⢻⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣿⣶⣶⣟⣷⣿⣶⡖⣶⣶⣶⣶⣶⡖⠶⢶⣶⡶⠶⠶⢿⠇⠀⠀⠀⣼⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡅⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⣸⣁⣀⣀⣀⣸⣷⣴⣦⣤⣶⣲⣤⣦⣤⣤⣠⣴⣶⣶⣖⣒⣶⣶⣶⣶⣶⣶⣒⣶⣲⣤⣶⣖⣒⣲⣞⣛⣛⣳⡒⣒⣿⣀⢀⣀⣀⣸⣇
⠀⢿⣭⡭⣭⣭⣥⣤⣤⣴⠤⢤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠷⠾⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠶⠶⠶⠶⠶⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠘⠀⠀⠠⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⠀⠀⠀⠀⠀⠀⠀⠏
⠀⡀⣀⣰⠀⢀⡀⡀⣄⠀⢀⣰⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⢠⣷⣿⣿⣾⣾⣿⣷⣿⣷⣿⣿⣿⡿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡆⠀⠠⠄⠀⠀⠀⠀⠀⢬⡧
⠸⠟⠟⠿⣿⡿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢺⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠩⢍⡧⠄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣣⠀⠳⠪⣽⣷⣂
⢀⣰⣷⣠⣼⣥⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢈⢠⣿⣀⠛⢭⣽⣯⠶
⠀⣿⣿⣿⣿⣿⠛⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡧⠀⠰⢬⣿⠔⠂⠰⣬⣷⠞
⠀⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣥⣷⣄⠨⠽⣿⠭⠄⡼⣿⣿⣟

⠀⢻⣿⣿⠀⠀⠀⠀⠉⠙⣿⣿⣧⠀⢰⣿⣿⠉⠉⠛⠛⠀⠘⠿⠿⠿⠟⠁⠀⣨⣿⣿⠀⠀⣿⣿⠁⢸⣿⣿⡏
⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⣼⣿⡿⠀⢹⣿⣿⣄⠀⠀⠀⠀⣠⣴⣶⣤⡀⠐⠿⠿⠟⠁⠀⠀⣿⣿⡇⠘⢿⣿⠇
⡀⠘⣿⣿⣧⣄⠀⠀⠀⢰⣿⡿⠃⠀⠈⠻⢿⣿⣷⣦⡀⠰⠿⠿⢿⣿⣿⡄⢀⣄⣀⠀⢀⣠⣿⣿⠀⠀⠀⣠⣤
⣷⠀⠈⠻⣿⣿⣿⣿⣿⡿⠟⠁⠀⢀⡄⠀⠀⠈⠛⣿⣿⡆⠀⠀⢸⣿⣿⠁⠈⠻⠿⠿⠿⠿⠛⠁⠀⣰⣿⡿⣿
⠏⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢿⣷⣀⠀⠀⣠⣾⣿⡏⠀⠀⢿⣿⣿⣄⣀⣤⣄⡀⠀⠀⠀⢀⠀⠙⠿⣴⣿
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠘⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣦⣀⣤⣿⣇⠀⠘⣿⣿
⣿⣿⣷⣆⠀⣼⣿⣿⠟⠛⢿⣿⣷⡀⠀⠀⠉⣁⣀⠀⠀⢀⣠⣤⣄⡀⠉⠉⠁⠀⠙⢿⣿⡿⠛⠻⠋⠀⠀⠸⢿
⠀⣸⣿⡟⠀⣿⣿⣿⣀⢀⣼⣿⣿⠇⠀⢰⣿⣿⣿⣷⠀⠘⠛⢿⣿⣿⡄⠀⠀⠀⠀⢀⣀
⣰⣿⣿⠀⠀⠙⢿⣿⣿⣿⣿⣿⠟⠀⠀⠀⣠⣿⣿⡟⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⣄⠀⢰⣿⣿⣷⣆
⠈⠻⣿⣷⡄⠀⠀⠈⠉⠉⠉⠀⠀⠀⢠⣾⣿⣿⠏⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠘⣿⠈⠻⣿⣿⣿⡏⠉⢿⣿
⠀⠀⠀⣿⡇⠀⢀⣤⡄⠀⢰⣦⣤⡀⢸⣿⣿⣷⣶⣄⠀⠀⠘⣿⣿⣿⣀⡀⠀⠀⠐⠺⠇⠀⢉⠻⠿⠃⠀⢸⣿
⠀⠀⠀⢸⣿⣾⣿⡿⠃⠀⠀⣿⡿⠃⠀⠉⢻⣿⣿⣿⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣦⠀⠀⣾⣿⣿⣷⣦⣴⣿⡟
⣀⡀⠀⠀⠉⠉⢀⣤⣶⣶⣿⣿⡇⠀⠀⠀⠘⣿⣿⣿⣄⠀⠀⠀⠀⠉⠉⠛⠻⣿⣿⡇⠀⠿⠏⠙⠛⠛⠛⠋⠁
⣿⣿⣷⣤⡀⠀⣾⣿⣿⠿⠛⠛⠁⣠⣴⣦⣄⠘⠻⣿⣿⣷⣶⣿⣿⠇⠀⠀⢀⣿⣿⠀⠀⠀⠀⢀⣤⣄⡀⢀⣾
⠋⠛⢿⣿⣿⣿⣿⣿⡏⠀⠀⢀⣾⣿⠿⢿⣿⣷⡀⠀⠉⠛⠛⠛⠉⠀⠀⠀⢸⣿⣿⣆⠀⠀⠀⣸⣿⣿⡷⠸⣿
⢀⣤⡀⠉⠛⠛⠛⠉⠀⠀⠀⣿⣿⡇⠀⠀⠁⢹⣷⠀⠀⠀⠀⢀⣠⣤⣀⠀⠘⢻⣿⣿⣷⣶⣾⣿⣿⠟⠁⠀⠉
⣹⣿⣿⠀⢠⣶⣿⣿⣦⣄⠀⢸⣿⣷⣄⠀⠀⠈⢿⣧⠀⠀⣼⣿⡿⠿⣿⣷⠀⠀⠈⠙⠿⠿⠟⠛⠁⢠⣴⣤⣄
⣿⣿⠏⠀⣿⣿⡏⠀⣸⣿⠀⠀⠙⣿⣿⣧⠀⠀⠻⣿⣷⣾⣿⡿⠀⠀⠘⠛⠀⠀⢀⣀⠀⠀⠀⠀⠀⢻⣿⠛⠿
⠀⠀⠀⠀⠸⢿⣿⣟⠋⠁⠀⠀⠀⣸⣿⣿⠀⠀⠀⠈⠛⠛⠋⣠⣤⣶⣄⠀⠀⢰⣿⠿⢿⣷⣤⣤⣴⣿⡿
⣶⣤⣀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠘⢿⡫⠃⠀⠀⠀⠀⠀⠀⣼⡿⠛⠻⣿⣷⠀⢿⣿⡀⠀⠀⠈⠙⠛⠉⠀⣠⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣴⢟⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣷⡞⠁⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠀⣸⢾⣿⣯
⠀⠀⠀⠀⠀⠀⣰⢿⣿⣿⣆⣸⣿⣿⣿⣷⠀⠀⠀⢰
⠀⠀⠀⠀⠀⢠⡏⣺⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢠⣾⡀
⠀⠀⠀⠀⠀⣾⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣸⣿⡧
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴
⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣷
⠀⠀⠀⠀⠀⣿⡻⢿⣿⣿⣿⡿⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢻⡿⠿⢟⣿⣿⣷⣤⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡏⣹⠿⣿⣦⣴⣦⣀⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣷⣶⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⠿⠯⠀⠀⠈⠉⠁⠉⠁⠀⠀⠁⠀⠉⠉⠉⠻⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿⠋⡁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠠⢾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣟⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢻⠇⠀⠀⢀⣀⣤⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⣇⣸⣤⣶⣾⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣷⣄⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣄⣠⣾⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠃⠙⢿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⠟⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⣶⣶⢠⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⢌⣿⣿⡈⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣾⣿⢟⣭⣶⣷⣶⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠱⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠼⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠈⠙⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣷⣶⣤⣌⣉⠉⠉⢁⣤⣾⣿⠇⠀⠀⠀⠀⠈⠙⢷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⢤⣶⣿⣿⣿⣿⣿⣿⣟⣫⣭⣝⣛⣣⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠋⠉⠙⠛⠻⠿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧
⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣅⣾⣤⠉⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠄⠘⣿⣿⣿⡟⠁⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸⡆⠀⣷⠀⢸⠀⣼⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠪⣄⣀⠀⠀⢾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⣿⠀⢸⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢻⣿⡷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣟⣀⣸⣃⣀⣟⣀⠸⠀⢹⣿⣿⡿⠁
⠀⠀⠀⠀⣀⣤⣤⣤⣀⡀⠀⠀⠘⠛⠀⢁⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠛⠉⠉⠉⠙⠉⢀⣠⣴⡎⢻⡿⠃⣦⣤⣀
⢀⣠⣴⣾⣿⣿⣿⣿⡿⠿⠿⠶⢦⠄⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⢨⣧⢸⣿⣿⣿
⠙⢿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠶⣶⣶⣶⣤⣤⣶⣶⣶⠶⠿⠟⠛⠉
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⢳⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⣶⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻
⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⣻⣿⣿⠟⠿⣿⣾⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢠⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣶⣦⣾⣿⣿⣿⣿⣿⣿⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣿⠃⣹⠇⠀⠀⠀⣤⣴⣶⣶⣾⣿⠋⢻⣿⣿⡛⠁⠀⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠃⠰⣿⣷⠀⠀⠀⠀⠈⠉⠙⠋⠹⣷⣴⣿⣿⣷⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⠛⠙⠏⣻⣤⣄⣀⣀⡀
⠀⢈⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠙⢿⣿⣿⣧⣄⣬⣤⣆⣤⣤⣴⣿⣿⡿⠟⠋⠉⠁
⢀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⡟⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣻⡿⣧⣤⠀⠀⠀⠀⠀⠀⠀⡿⢹⣿⡿⠛⣿⣿⠿⠿⣿⣿⣿⣧⠀⢀⣴⣶⣄⣢⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣽⣿⣿⠀⠀⠀⠀⠀⢀⣰⡄
⡟⠁⠉⠛⠀⠀⠀⠀⠀⠀⠐⣿⢿⣿⣲⣀⡀⡻⡆⣀⡼⣿⡿⢿⣿⣏⣨⣿⣩⣿⢋⣿⣿⡿⣻⣉⠀⣌⣩⠿⣜⣿⢟⠀⠀⠀⠀⠀⢨⣿⡀
⡁⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠇⠀⢉⣻⣿⣿⢿⢫⣿⢷⣾⣵⢿⣩⠁⠙⠿⣿⣖⣾⣿⣿⡿⡛⢿⣷⣿⠿⣿⣧⣽⡆⠀⠀⠀⠀⠀⠈⢻⣿⡄
⡶⠂⠠⣷⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣟⣽⡿⣍⡋⠀⠈⢹⣯⣟⣷⣦⣽⣾⣿⣟⡋⣤⣴⣿⡗⣋⣁⣼⣏⣍⣽⡇⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠓⠀⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡻⠭⢩⢩⡗⣈⣻⣿⣷⣶⡿⣿⣿⣿⠿⠯⣽⣿⣾⣿⣻⣍⣥⣹⣿⠭⣿⡵⠤⣷⡄⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⣰⣿⣿⠀⠀⠀⠀⠀⠀⣠⣬⣨⣶⣠⣿⠾⢿⡿⠟⢛⣋⣉⠱⣿⣿⣿⣿⣿⡶⠁⠈⠛⠿⠿⣶⣾⣷⣦⣷⣾⣁⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠹
⢀⣿⣿⡗⣠⣾⡀⠀⠀⠀⠉⣿⣿⣯⣉⣁⣽⣿⠁⠀⢻⡿⠀⣀⣈⣭⣿⣿⣿⣿⣿⣶⡀⣤⣀⣹⣻⡷⣿⠿⣿⣿⣿⣿⠃
⣸⣿⣿⢱⣿⣿⠇⠀⠀⠀⠀⠈⠙⠻⣯⡻⣾⣶⣿⢶⡆⢘⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠁⠨⠭⢁⣈⣠⣨⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⢠⡀
⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢀⡄⠀⠀⠉⢻⣿⣿⣽⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⢠⣶⣶⣶⣶⡿⣿⣽⡋⠋⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣧
⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢿⠟⢀⡀⠀⢸⡟⢻⡆⣿⣿⣿⣿⡿⢿⣿⣾⡶⢾⡿⢓⢻⣿⢡⣿⣿⠟⠀⣀⡙⠛⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡆
⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⡀⠈⠁⠀⢸⣿⠐⠙⣿⣯⣼⣿⡀⡌⢿⢻⠀⣿⣷⣿⣿⣥⣻⢿⣿⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠚⠸⣿⢠⣶⣿⣿⣿⣿⣿⣧⣸⠀⢂⣿⣿⣿⣿⡟⠁⢸⣿⣀⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣠⣍⠛⠟⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡿⢿⠋⠀⢸⣿⠛⣿⠿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⠃⢀⣀⡀⠀⠀⠀⠀⠀⠇⠀⠸⣿⣿⠂⠀⣿⣿⣿⡇⠹⣿⣿⣄⣿⣿⠋⠀⠀⠀⠀⢸⣿⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡿
⣿⣿⣿⣿⣠⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⢁⡄⠀⢸⣿⣿⡇⠀⢻⣿⣿⡟⣧⣤⣤⡴⠂⠀⢸⠟⢀⣤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⣿⡇⠀⣿⡏⠋⠀⣿⠿⣿⡅⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢀⠈⠙⠛⠛⠀⠘⠃⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠿⣿⣶⣽⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢴⣶⡄⢠⣶⡀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠤⡘⠋⠱⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠈⠁⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠖⠋⠀⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢾⠻⢿⣧⠀⠀⡀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠖⠋⠀⠀⠚⣧⠀⢹⣶⡎⢀⠆⠀⠀⠈⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⡀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⢡⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢰⡀⣀⠁⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⢶⣤⣄⣀⣀⣼⣿⣿⡄⠀⠀⠀⠀⢸⡯⠁⠀⠀⣇
⠀⠀⠀⠀⡰⠀⠀⠀⠀⣸⣿⡏⠼⣿⣿⣿⣿⣿⣟⡷⡆⠀⠀⠀⠀⢺⡇⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡠⢄⠀⠐⠂
⠀⠀⠀⢠⠇⠀⠀⢀⣰⣿⡿⢘⢱⣿⣿⢿⣿⣿⣿⢸⣷⠀⠀⠀⠀⣸⢿⡄⠀⠀⠈⠓⠒⠒⠛⠉⠉⠉⢹⡁⠀⢼⡿⠃⡂
⠀⠀⠀⢸⠀⠀⠰⢋⠙⠻⢷⣥⣲⣿⡿⣸⣿⣿⡜⣾⣿⠀⠀⠀⠀⣽⠀⢳⣴⠇⠀⠀⠀⠀⢀⣰⣀⣀⣨⡷⠒⠚⠛⠃
⠀⠀⠀⢸⡢⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠁⠀⢉⠙⢿⡀⠀⠀⠀⣿⠀⠀⠻⠦⠤⠶⠖⠛⠋⠉
⠀⠀⠀⠀⠙⠛⠲⠶⠤⣤⣄⣀⡀⠀⣸⠀⠀⠀⣴⠂⣺⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠿⠟⠿⠶⠤⢴⣾⡿⠁⠀⠀⢤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣶⣶⣶⣶⣦⣀⣉⣰⣶⣾⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡯⢻⣿⢿⣿⣿⣿⣿⣯⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠂⣄⣺⣿⣯⠿⣿⣿⣿⣷⣾⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠀⢸⣿⣿⣿⡟⠀⢻⣿⡿⠷⠶⠦⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠈⠙⢛⡛⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣓⢯⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠓⠘⠀⣦⡀
⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠰⠏⢩⣿⡿⣭⠁⠀⢀⣴⣦⡤⣀⠀⠀⠀⣿⣿⣦⣤⣶⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢰⡷⢶⢀⠙⠁⠀⠄⣤⡄⠀⠀⠙⠞⠉⠁⠀⢰⣯⢞⣻⠿⠞⠁⠀⣀⣿⣿⣿⣿⣿⣿⣿⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣦⣸⣿⠦
⠈⠀⠀⠶⠀⠀⡀⠀⠀⠄⣴⣦⣦⠀⠀⠀⠀⠈⢁⣼⣷⠀⠀⠀⠀⠙⣏⠙⠛⠛⠛⠋⠙⢢⣶⡔⠁⢀⡔⠆⠀⠀⠀⠘⠛⠓⠂
⠀⠀⠀⠀⠀⢸⡿⠀⠲⠖⠀⠈⠛⠀⠀⠀⠀⢠⣟⡵⢫⣦⣵⠂⠀⠐⠁⠀⣤⠀⢶⡀⠰⢟⡟⠻⢄⢾⡷⠤⡀
⠀⠀⠀⠀⠀⠠⢆⡆⠀⠀⣠⣶⡆⠀⠿⠿⠆⣼⠅⠀⢸⡿⡟⠀⠀⠀⢀⢤⣽⣶⠀⡿⠠⠈⣟⠀⣿⣎⠀⠀⣻⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠀⠀⢀⣀⣤⣾⣿⣟⣡⠀⠑⠃⠒⡋⡅⡠⢋⢁⠛⣤⠀⠘⡄⡙⠂⣹⠯⠀⠀⠈⠀⣤⣄
⠀⠀⠀⠀⠋⠿⠀⠀⠀⢸⣦⡀⠀⠈⠻⣿⣿⣿⡿⠋⠀⢀⠤⣨⡖⢫⢣⣡⠀⢇⡅⢣⣴⠇⢿⡾⠛⢷⠀⠀⠺⣄⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⡥⠀⠀⠀⠀⠀⠉⠁⢰⣦⣤⠈⢯⣟⡛⠀⠀⠈⠚⠋⠪⣵⣦⣬⡁⠀⠡⠵⢰⠀⠀⠰⣦⠟⠀⠀⢠⡄⡦⠀⠉⡉⠀⠘
⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⣲⣂⣔⡏⠛⢁⠂⣹⣿⠀⢀⡄⢠⡆⢺⣿⣿⣻⡟⣆⡀⠀⣀⠀⠀⠀⣎⢝⡉⢢⣤⠈⡡⢾⠛⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢸⠢⡤⠜⠪⡿⠏⠁⠀⣈⣄⣻⣿⠵⠛⣇⢸⡇⣸⠉⢉⠟⠉⢹⠙⠯⣾⣷⡶⠀⠉⠀⠉⣹⠲⠤⢥⡤⣀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠘⢧⡝⣈⢄⠐⠀⢁⣴⣿⣿⡜⣱⢦⡄⢸⡸⡇⡇⡴⢁⡤⢞⣵⣿⣿⣍⠛⠉⠀⠀⠀⠀⢸⠂⠀⠈⢽⣭⣷⠄⣸
⠀⠀⠀⠀⠀⠀⠀⡈⣧⣬⠀⢀⠿⢿⢉⣿⣿⣿⣻⠀⣌⢢⢟⡉⡉⠳⡫⠴⢾⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢿⡀⠀⣰⣾⡿⣷⡰⠁
⠀⠀⠀⠀⡈⠂⠀⠀⠀⠊⠀⠸⡆⢸⠎⡼⠁⠐⢙⡀⠀⠉⠈⣶⠈⠂⠟⠿⡍⡫⢯⠉⠀⠀⣰⣶⣤⡀⠀⠀⠁⠀⠉⠺⣿⠛⠝⠟
⠀⠐⢄⠀⠑⠀⠀⠀⠀⠀⠀⠀⣏⢨⡴⡇⠀⣰⠟⣿⣇⠀⠀⠋⠀⠀⠀⠁⠊⣵⢋⣀⡀⢰⡿⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡾⠃⠁⠀⠘⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⢙⣋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⢡⡌⠉⣩⣭⣭⣍⢩⣭⠉⢩⣭⢩⣭⣭⣥⡄
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣬⣿⠁⢸⡇⢸⣿⠉⣤⣅⢸⣿⣤⣼⣿⠀⢸⣿⠉⠁
⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⢿⡄⢸⣷⠘⢿⣤⣽⡿⠀⣿⠁⢸⣿⠀⠘⣿
⠀⠀⠀⠉⠂⠀⢀⣄⡆⠀⠀⠀⠀⠀⠁⠀⠀⠁⠚⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⡇⢴⡄⢠⣤⡀⣠⡄⣤⣶⣤⣤⠀⣠⣴⣶⣄⠀⣴⣦⠀⣴⡄⢀⣤⣶⣶⡄⠀⠀⠀⠑⠢⣀
⠀⠀⠀⠀⡇⢸⣷⣼⢿⣇⣿⠃⣿⣷⣼⡟⠠⣿⡇⠀⣿⡇⣿⡿⣧⣿⡇⣾⡏⢠⣤⡄⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⡇⠀⣿⡿⠈⣿⡿⠀⢿⡏⠙⣿⡄⠻⣧⣴⡿⠃⢹⡇⠹⣿⡇⠹⣿⣦⣿⠇
⠀⠀⠀⠀⠑⠒⠐⠶⠒⠒⠒⠒⠂⠐⠒⠐⠒⠒⠂⠐⠒⠒⠒⠒⠀⠒⠒⠒⠒⠒⠲⢲⠒⠀⠒

⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⡔
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡄⠀⠀⢠⠀⠀⠀⠀⠀⠘⠢⢌⣉⡙⠒⠣⣄⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⢘⣷⡂⠀⢻⠀⣇⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢸⡇⠀⡇⠀⠀⠠⡒⡉
⠀⠀⢀⠎⠀⠀⠀⠀⠀⣤⣀⡀⠠⢈⠑⠀⠀⠀⢀⠘⠀⠀⠃⠄⠀⣠⣑⣒⡮⣧⠠⠀⠀⠏⣸⠀⠸⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠁⠸⣄⠀⠀⠄⣂⡱⠭⠖
⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠁⠀⢄⠀⠀⠈⢀⠀⢉⠉⠿⣷⣐⠀⠐⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠑⠂⠠⠔⠒⣂⢀
⠉⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠈⠈⠰⢀⠄⠀⠀⠀⠀⠀⠀⠡⠈⠀⡀⠀⠀⠤⡀⠣⠴⣀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠑⠐⣆⠐⠈⠉⠈
⠄⠀⠀⠄⣀⠤⠴⠶⠌⠀⡢⢂⡤⢖⣠⠴⠤⠀⠑⠀⠠⠀⠀⠨⡐⠀⠀⠀⠀⠄⠄⠔⠉⣤⠶⠖⠛⠐⢤⡤⢊⠔⣝⡲⢤⣄⡀⠀⠀⠀⠐⠀⢀⠀⠈⢦⠀⢀
⠁⠀⢀⡀⠀⠀⠀⠀⠀⢀⡰⠋⡴⠫⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⢸⠀⠸⠀⢀⠴⠶⡋⠡⣴⡁⡢⣄⣙⠛⠲⣍⠳⠄⠀⠀⠀⠈⢣⡀⠀⠀⠀⠑
⠁⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⠜⡠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢰⢀⠈⠀⠀⠀⠁⠀⠈⠘⠉⠈⡁⠀⡈⠆⠆⠀⠂⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠘⠀⠀⠰⠀⠀⠠⠀⢁⡘⢦⠢⡠⡀⢦⣄⡤⠴⠶⡷⢶⡳⣶⣴⠖⠂⡀⠀⠀⠀⠐⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠠⠀⠂⠀⠀⠈⠀⠠⠂⠀⣀⠤⠶⣤⡀⠄⠀⡆⠀⠀⠀⠀⠀⠀⠈⢿⠀⠂⠙⢦⠤⠠⠤⠀⡊⠠⢫⡾⠃⡀⠐⠄⡀⠀⠀⠀⡀⠀⠀⠀⠠⢀
⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠀⠀⠔⠀⡠⠋⠀⠀⣀⡈⠙⠧⠊⡑⢀⠈⠀⠀⠀⠀⠂⠲⣆⠀⠀⠀⠀⠀⠀⠀⠠⠐⠹⠉⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⡌⢰⢃⠀⠀⠀⠀⠙⠲⢄⠀⢀⢘⡤⠠⠀⠀⠀⠀⠀⢀⡉⠳⠲⠄⠀⠠⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠤⠀⠀⠀⠀⢀⡀⠀⡀⠠⠤⠤⡊⠀⠈⣸⢡⠐⠄⠀⠀⠀⠀⠔⠡⢂⢚⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠈⠀⠀⠑⡀⠀⡀⠀⠀⠀⠀⣤⠔⠊⠑
⣀⣒⠒⠢⣄⡈⠂⠁⠀⠀⠀⣿⠀⠀⠀⡧⠈⠸⣆⠀⠀⠀⣟⣠⡇⢸⣸⡅⠀⠈⠉⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠁⠀⠀⠁⠀⢠⠔⠁⠊⠀⣀
⡁⠅⠀⠀⠀⠑⣄⠀⠎⠀⠁⣈⢧⠀⠀⠈⠀⠀⠌⠙⢦⡄⠈⠸⠳⠈⢳⢳⡒⠂⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⢀⠂⠈⠀⠂⠤⠀⠀⠀⠐⠊⡀
⠀⠀⢀⣸⠀⢠⡏⡀⠀⢄⠈⠁⠀⠣⡀⠀⠀⠀⠀⢀⣒⣛⡷⡀⠄⠀⢸⢀⠇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠀⠄⠀⠊⠀⠀⠀⢠⠀⠐⡀⡀⢰⠏
⠀⢌⢢⠇⠀⠞⠀⢁⠔⠀⠀⠀⠀⠀⠈⠀⠄⠠⢒⣀⠦⢤⠮⢯⣅⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⢠⠆⢀⡎⣐⠋⠂⠀⢀⣀⠈⣆⢣⠀⢠⡞
⠫⠔⠇⣂⠊⡁⠑⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠜⠁⠀⠀⠀⠀⠋⠉⢀⠔⠈⠀⠀⠀⠀⠆⠀⠠⠁⠀⠀⠂⠀⢠⠏⠀⡚⢀⠁⠠⠀⠀⠀⠍⠈⢿⡌⣦⠟⢁⣰
⡥⠬⡂⡈⠺⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠄⠀⢀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⡟⠃⠀⠠⠞
⠖⠋⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⢠⠀⢀⣚⡀⠀⡃⠰⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠤⠤⢖
⠉⠀⠀⡀⠀⠀⣆⠀⠀⠀⠀⠐⠀⠀⣠⡤⢤⢀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡴⠋⠉⠙⠳⣥⣾⡀⠀⠀⠀⡄⠀⠀⠈⠁⠀⠰⠚⠉
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢀⣠⣾⠗⠋⠱⣸⡇⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠐⡇⣾⡠⠐⢉⣛⢶⣤⢫⡟⣦⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣶⣯⢟⣭⣾⠧⣴⣿⣗⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⡞⠠⢫⢷⣎⠂⠀⠛⢿⣻⡝⡞⡅⠀⠁⠀⠰⠄⠀⡀
⠠⠄⠀⠀⠀⠀⠀⠀⠀⣻⠿⣽⣿⠿⠋⢰⡿⡜⢞⡷⠂⠀⠠⠀⠀⠀⠀⠀⠀⢈⠀⠐⠀⡀⠀⢹⡀⠃⠸⣎⡘⠦⣄⡀⠸⠿⣧⢳⣷⡀⠀⠄⡀
⠀⠀⠀⠀⠁⠐⠠⢄⡀⣿⠁⣿⠉⠀⠀⣾⠀⢃⡘⠇⡁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠈⢱⣄⠀⠈⠈⠂⠈⠻⣦⡀⠙⠀⡽⢻⡀⣀⠀⠀⠀⡀
⠀⠀⠀⠁⠀⠀⠀⢀⠱⣏⠀⠫⣀⢚⣿⣃⡼⠡⠑⡌⠈⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠁⡀⠒⢀⣂⠐⣛⣷⡾⢿⣦⡈⢀⠂⠀⠀⠀⠁⠀⠀⢁⣀⣀
⣲⠀⠄⡀⠀⠀⠄⠀⠁⠈⢓⠶⠤⠽⠷⢁⠐⢀⠾⣍⠀⠄⠀⠀⢀⡠⠖⠐⠚⢶⣄⠄⠈⠀⠀⠀⠈⠀⠀⠈⠀⠈⠀⠁⠠⠈⠀⠉⠁⠀⠀⠀⠀⠀⢀⡜⣩⠄⠂
⣿⡄⡀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠋⠉⠙⠉⠛⠉⠀⠋⠱⢀⡜⢡⢟⠀⠀⠄⢤⡁⠙⢷⡐⢂⠀⠰⠀⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣪⠞⢀⡀
⢣⢳⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠃⠀⠞⠀⡔⠋⠀⢁⡶⣀⠀⠀⠀⠙⠢⠀⠕⡡⢀⠂⠀⠀⠀⠀⠀⠀⠌⠀⢀⠀⠀⠀⠀⠀⠀⠒⡩⠚⡁⣴⣏⠀⢐
⠀⠃⠋⠱⠀⠀⠀⠓⠆⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠰⡇⠀⠀⢞⠐⠈⡆⠀⠀⢠⡄⢡⡎⠀⡟⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠈⠀⢠⠴⡑⢌⠠⡁⡈⠉⠀⣼
⠘⠘⠀⠀⡇⠀⠀⢀⠀⡈⠢⣀⠀⠀⠁⠀⠀⠀⢱⠀⠡⠀⠀⠈⠂⠈⠳⣄⠀⢨⡅⣧⠁⡆⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⢠⡎⠀⡇⠘⠀⠀⠀⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⢂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡒⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⠈⢁⡬⠬⠬⣤⣤⠤⠥⠤⢥⣬⡭⠤⠤⣅⢙⠀⠀⡇⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⣾⡀⠀⠀⢸⠈⠀⠀⡇⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⠀⢸⣶⣶⣶⣾⣶⣶⣶⣶⣶⣾⣶⣶⣶⣾⢸⠀⠀⡇⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⠀⢸⠉⠀⠁⢹⡍⠀⠀⠀⠉⣽⠁⠀⠀⢹⢸⠀⠀⡇⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⠀⠀⠀⢸⠸⠀⠀⡇⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⠀⢸⡤⠤⠤⠼⢥⣤⣤⣤⢤⡽⢤⡤⠤⣼⢀⠀⠀⡇⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⠀⢸⠋⠉⠉⢹⡋⠉⠉⠉⠉⣿⠋⠉⠉⢹⢠⠀⠀⡇⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⢸⠘⠀⠀⡇⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⠀⢸⣄⣀⣀⣼⣇⣀⣀⣀⣀⣽⣄⣀⣀⣼⠸⠀⠀⡇⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⠀⢸⠓⠒⠒⢲⡖⠒⠒⠒⠒⣲⠒⠒⠒⢻⠨⠀⠀⡇⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⠀⢸⡀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡀⠀⠀⣸⠀⡾⢻⡇⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⠠⣄⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⡤⢿⢹⡇⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀⠓⠚⡇⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡇⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⡇⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⣇⣸⣼⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣧⣇⣸⣧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠞


⣿⣿⣶⣾⣶⡿⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣿⣷⣾⢶⡿⡿⠿⠾⠿⣶⡿⠿⠿⠷⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣮⣵⣌⡥⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⡿⠀⣾⣿⣿⣍⣧⡄⠀⢠⣄⠧⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠶⠦⢀⣠⡶⢿⣜⣴⢗⠛⠋⠙⠋⠉⠟⠒⣦⡽⣶⣾⣤⣼⡝⢲⡆
⠀⠀⠀⠀⠀⠀⠀⢰⡶⣳⢎⣰⣟⣩⣿⠛⣴⣿⣏⣉⣹⡿⠚⣩⡑⠖⠶⠍⠛⢿⡽⠿⢣⡀⢘⣠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⣿⣷⣿⣿⣿⣿⡾⣿⣿⣿⢛⡟⣿⣛⠻⣿⡀⣴⡤⡎⢱⡄⠑⠉⢠⣆⡇⣨⠷
⠀⠀⠀⠀⠀⠀⣀⡖⣄⡊⢠⣥⣴⣿⢿⣿⣿⡾⣿⡟⠓⢿⠶⠿⠶⠿⡷⠿⢷⢹⣷⡞⠀⠀⠈⠉⣷⡁⠒⢒⡆
⠀⠀⠀⠀⠀⠘⠷⣾⣿⣧⣼⣿⣿⣄⣿⡟⡃⢴⣾⠖⢻⣿⠿⠷⠆⠀⣰⠆⠀⠀⢭⣳⣀⡠⣶⣴⡶⠟⢣⡟
⠀⠀⠀⠀⠀⠀⠀⣹⣹⣿⣿⡿⢷⣤⣽⣷⣿⡿⣿⣿⣷⣷⣆⣸⣆⣾⣿⣦⣴⣏⣤⡾⣻⣿⣯⣭⢈⣿⣦⠸⠆
⠀⠀⠀⠀⢀⣠⣾⣿⣏⣿⣿⣶⣆⣹⡟⣿⣯⣗⣋⣿⣟⣝⠿⢿⣿⡿⣿⠆⡼⣿⢿⣿⣻⣿⣿⠋⠀⢳⡄
⠀⠀⠀⠀⠉⠉⠽⢿⣿⣼⣿⣿⣿⣽⣿⣿⣿⣻⣷⣺⣿⣏⣳⢾⣿⢋⣿⡤⠫⣷⣤⠤⠀⠀⠉⢦⣤⣅⡁⢉⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⣋⠀⢀⣝⢿⣿⠿⠫⠋⠉⠻⣿⣿⣄⠛⡾⠻⣿⢶⣶⣈⢘⡛⠁⠀⠠⡤⢀⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠈⠛⡿⠿⠟⠾⠃⠀⠀⢸⣿⡘⡖⢳⠀⢀⠙⠑⠿⠿⠇⠀⣠⣀⣠⣤⡂⣀⡄
⠀⠀⠐⠖⢀⣶⣞⢢⢄⡠⢀⣴⠶⣄⠙⡃⢀⡀⠐⠂⠀⣼⣿⠃⢠⢸⡄⠛⠃⠠⠂⠚⠛⠨⠉⠩⠉⠉⠉⠉⠁⠂
⠀⠀⠙⢃⢸⣿⡷⢿⢦⡇⢺⡿⣿⣼⢫⢱⣿⣬⣱⠺⣲⡿⣿⠀⣾⠴⣇⠰⡄⣰⣖⣓⠢⠀⣤⠦⡀⢀
⠀⠀⠀⠐⠡⠀⠒⠊⠃⠈⠠⠒⠃⠨⠏⠀⣨⣽⣷⣶⣿⣴⠋⣴⣿⣤⡹⠶⣴⢬⣍⣋⡛⡺⣛⡗⢺⢴⣯⣆⢠⣀⣀
⠀⠀⠀⠀⠀⠐⠒⣒⠛⣛⠻⠖⢺⠿⢻⡟⢛⡉⠀⠋⠉⠀⠀⡀⠀⠀⠉⠛⠲⠶⠚⠲⠯⠙⠋⠉⠑⠒⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠛⠂⠲⠴⠀⠠⡬⠛⠛⠂⠂⠂⠤⠤⠌⠒⠐⠂⠀⠀⠈⠐⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣹⣿⡁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⣛
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠾⠋⠀⢠⣄⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⣇⠀⠀⠀⠀⢠⣤⣬
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⠄⠀⣾⣿⣶⣧⣤⠀⠀⢰⣆⠀⠀⠀⠀⢌⣧⣄⣤⠤⢶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣶⣆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣴⣦⣀⡟⠀⢸⣿⣧⡀⠀⠀⣠⣶⣷⡀⠘⣿⣷⣶⣤⡀⠙⣿⣿⣶⣤⣄⠀⢻⣿⡄⢰⡿⠛⠻⠿⠏⠀⠿⠿⠿⣦
⡟⠉⠉⠳⠶⠟⠉⢹⣟⣿⣿⣿⠁⠉⣿⠏⠁⠀⠀⠙⣿⣿⠟⠉⠙⣿⡟⠀⠀⠙⣷⣾⡄⠀⠀⠀⢀⣤⣄⡀⢀⣏⣸
⠿⣦⣶⣶⡆⢠⡎⠙⣟⣿⣬⣿⠀⣴⡏⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⢸⣧⠀⠀⢀⣿⡿⠛⠒⠲⢾⣿⠀⢈⣛⠉⠉⠙
⣠⣿⠟⣨⣤⣴⣿⠀⣿⠉⣠⠟⠀⢿⣿⣦⣀⠀⠀⠀⣿⣿⣦⣄⣀⣀⡛⠿⠿⢛⣡⣀⠀⠀⠀⠀⢻⣿⡿⠛⠛⠶⣾
⢿⣧⣴⡟⠉⣿⠏⠀⠈⣿⠏⠀⠀⠀⠉⢻⣿⣷⠀⣠⡀⠈⠉⠉⠙⢿⣿⡿⠂⣿⠟⠛⢧⠀⠀⠀⢠⡟⠀⠀⢠⣄⠙
⠀⠹⣿⡇⣸⣿⣄⣀⣠⣿⡀⠀⠀⠀⢀⣴⡿⠁⠀⣿⣿⣄⡀⠀⢀⣾⡏⠀⠀⢸⣷⣤⠈⣷⣦⡶⠟⠁⢀⣴⣿⣿⣿
⣀⣴⣿⠇⠈⠉⠉⠉⠉⠙⣧⠀⠀⢰⣿⣿⣧⣀⣠⠟⠉⣹⣿⠟⢋⣿⣷⣤⣴⣿⣁⣿⣷⠋⠀⡾⣿⠇⠘⠋⠀⠈⣿
⠛⠉⠁⢠⡾⠷⣄⠀⠀⠀⣿⠳⣤⣀⠈⠉⣻⡟⣡⣶⣾⠟⠛⠒⣿⡅⣸⡟⠁⠀⠀⠙⣧⡀⣠⣧⡼⣷⣴⣶⣄⢸⣿
⡀⠀⣠⣿⡇⢀⣽⣶⣤⣾⠿⠟⠳⠿⢿⣟⣉⣿⡿⠟⠁⠀⠀⠀⢸⣧⣿⡅⠀⠀⠀⠀⠘⢿⣿⣏⣀⣠⣌⡉⠛⣾⣿
⠿⠛⢿⣿⠋⢸⣿⡿⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣠⣿⣿⡝⣿⡿⠛⠛⢿⡄⠀⠈⠉⠉⠛⣿⡉⢻⣻⡿
⠀⠀⠈⣿⣦⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣾⡿⠋⠉⠉⠙⢿⣷⠠⣤⣾⡇⠀⠀⠀⠀⠀⣼⣧⣾⣿
⠀⠀⠀⣿⣀⣻⣇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠋⠀⠀⠈⠻⣿⣄⠀⠀⠀⣠⣿⠃⠈⠙⢷⣀⠀⠀⢀⣴⣿⠁⠀⢻⣿
⣀⣤⣤⣀⠉⣻⣿⣦⣄⣀⣀⣀⣠⠴⣿⣿⡇⠀⠀⠀⠀⢠⠛⠉⢀⣴⣾⡿⣇⠀⠀⠀⣸⣿⣋⣻⠿⠿⣿⢧⣴⠋
⠻⣿⡟⠉⠙⣿⠉⣿⡏⠉⠉⣫⣥⣴⣾⣿⣿⣤⣀⠀⠀⣿⣦⣤⠞⠛⠛⢦⣀⠙⣶⣿⡟⠙⢿⡁⠀⣠⣿⡾⠋
⣾⡿⠃⠀⣼⣿⡇⠈⠑⢲⣾⡿⣿⠋⠀⠀⠀⠙⢻⣿⣦⡍⠋⠁⠀⠀⠀⠀⠹⣿⣮⠻⠿⠿⣿⣿⡄⠻⣿⠀⠀⣶⣶
⠋⠤⠄⠀⠘⠛⠓⠀⠀⠀⠛⠀⠘⠀⠀⠒⠒⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠐⠛⠛⠂⠀⠀⠀⠛⠁⠀⠙⠂⠀⠘⠂
⠀⠢⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⠀⠀⣀⡀⠀⠀⢀⣀⣤⣀⠀⢀⣀⠀⠀⣀⡀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠙⣿⣧⠀⣿⡇⠀⣰⣿⠟⠛⠋⠀⢸⣿⠀⠀⣿⡇⠘⠛⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣴⣿⠏⠀⣿⡇⠀⣿⡏⢀⣤⣤⠀⢸⣿⣦⣴⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢻⣷⡄⠀⣿⡇⠀⣿⣇⠈⢹⣿⠀⢸⣿⠉⠉⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣧⠀⣿⣇⠀⠙⢿⣷⣾⣿⠀⢸⣿⠀⠀⣿⡇⠀⠀⣿⣿



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⢠⣤⡄⠀⢀⣤⣤⣄⠀⢠⣤⡄⠀⣤⡄⠀⣤⣤⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⢸⣿⡇⢰⣿⡏⠙⣿⣇⠘⣿⡇⢠⣿⣷⠀⣿⡏⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣿⣸⣿⡇⣾⣿⠀⠀⣿⣿⠀⣿⣇⢸⡟⣿⢀⣿⠇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠹⣿⣿⡇⢸⣿⡄⠀⣿⡿⠀⢹⣿⣿⠃⢿⣿⣿⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠻⣿⠇⠀⠻⢷⣾⠿⠁⠀⠘⢿⡿⠀⠸⣿⠏⠀⠀⠀⠀⢿⠇




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣽⣿⣿⣽⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⣿⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢯⣿⣿⡿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣽⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠙⠛⠛⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⣀⣀⣤⣤⣄⣀⡀⠀⠀⢸⢸⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⢀⢮⠋⢀⠀⠀⠁⠈⢩⢦⠀⢸⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⢸⠆⠀⠀⠀⠀⠂⠀⠀⢸⠀⢸⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⢸⠀⢀⠀⠀⠀⠀⠀⠀⢸⠀⢸⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⢸⡁⠀⠐⠀⠀⠀⠀⠀⢸⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⢸⠀⠈⠀⠀⠀⠁⠀⢀⢼⠀⢸⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⢸⠁⠐⠀⠀⠀⠂⠀⠀⢸⠀⢸⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⢸⠂⠀⠈⠀⠀⠀⠁⠈⢸⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⢸⡁⠀⠀⠀⠀⠀⠀⠐⢸⠀⢸⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠄⣞⣢⣀⣠⣀⣄⣡⣄⣠⣸⠀⢸⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⢟⢏⣃⣀⣀⣀⣀⣀⣀⣀⣀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⢸⠈⠉⠉⠉⠉⠉⠉⠉⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⢸⢀⠄⣄⡄⠀⠀⡠⣠⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠿⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠺⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠾⠿⠿⠿⠿⠿⠿⠶⢶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⢁⠀⠠⠠⠐⠐⠐⠀⠀⠀⠀⠄⠀⡉⠙⠻⢶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁⡠⠔⠈⠀⣤⡰⠒⣄⡎⢹⠄⣤⠀⠀⠀⠀⠀⠁⠢⢀⠈⠛⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⢀⠀⠁⠀⠀⠀⢠⠇⣇⣰⠛⠦⠞⠀⠉⠁⠀⠀⠀⢀⣾⡇⠀⠈⠀⡀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠄⠀⠀⠀⠀⡀⠀⠈⠀⢠⡄⠀⠀⠀⣤⣠⠀⣠⠶⡶⢸⠟⠀⠀⠀⠀⠀⠠⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠃⢀⠈⠀⠀⣼⡇⢸⠇⡤⣴⠌⡟⢡⡇⣸⠃⣯⡟⢰⡏⣼⣃⠿⠤⠃⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣿⣿
⠀⠀⠀⠀⠀⢀⣿⠃⢀⠂⠀⠀⢰⡏⣷⡟⣾⢁⡟⣸⣇⠼⠵⠻⠞⠀⠑⢉⣉⣀⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡿
⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠸⠀⠙⠃⠙⠋⣉⣀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠋⣹⡏⠀⣠⡄⢘⣿⡇
⠀⠀⠀⠀⠠⠿⠀⠐⣀⣀⣠⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣟⣉⠉⠈⢣⡄⠀⣾⣿⡄⠘⣿⡇⠀⣿⣿⠋⢻⣷⡀
⢠⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠟⠉⣉⠉⢻⣿⣿⠀⢠⠀⢻⣿⣿⡀⢰⡄⠙⠀⢻⣿⠇⠀⠛⢻⣄⣈⣁⣠⣾⣿⣿⡄
⠀⠻⣿⣿⣿⡿⠛⠛⠻⣿⣉⡁⢠⣤⠀⢸⣿⠀⢸⡟⠒⠚⢻⡟⠀⣠⡤⠀⠻⠟⠃⢈⣙⣦⣠⣼⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠹⣿⡟⠀⣼⣿⡄⠈⣿⡇⠀⣠⡀⠉⢿⡄⠈⠛⠃⠀⣿⠁⢠⣤⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠋⠉⣭
⠀⠀⠀⢹⣿⡀⠸⠿⠇⢀⣿⠋⣀⣈⣵⣤⣤⣿⣷⣶⣿⣿⣿⡆⣶⣬⡛⢿⠿⠿⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⢀⠁⢰⡿
⠀⠀⠀⣼⣿⣿⣶⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠁⣾⣿⣿⣦⡀⠀⠀⢀⣀⣀⣤⡄⠀⠀⠀⠀⠀⠂⢀⣿⠃
⠀⠀⠀⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⣿⢟⣩⣶⣿⣿⣿⣿⡏⠀⠀⠀⠀⢀⠂⢀⣾⠏
⠀⠀⠀⠉⠉⠉⠀⠐⣶⡄⠀⠄⠀⠀⠀⠀⠀⠀⠤⣶⣶⣿⣿⣿⣿⣿⣶⣌⡻⣿⡿⣿⣿⣿⠀⠀⠀⠀⠠⠂⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠐⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣻⡿⠿⠿⠿⣿⡔⣿⣿⣿⠇⠀⠀⠀⠂⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠁⠢⠄⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣶⣅⡛⠛⠁⠀⠐⠈⢀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣀⡈⠐⠠⠠⢀⢀⢉⠉⠉⠉⠉⠀⠄⠙⠒⠁⣀⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⢶⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⣶⠾⠟⠋⠁
⣶⣶⡶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣷⣿⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⣴⣾⣿⣿⣿⡿⠟⠻⠛⠛⠛⠛⢿⡿⡿⣿⣿⣿⣿⣿⣶⣴⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠟⢾⣛⣋⠉⠉⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠨⠙⠙⢻⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⣧⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⢹⠀⠀⠠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡄⣠⣐⣚⣷⣷⡦⢀⣀⡀⠀⠀⠀⠀⠘⠛⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⣤⣴⣦⣦⣽⣿⣿⣿⣿⣿⣿⣶⣽⣿⣿⣶⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠛⠛⠉⠉⢹⣏⠈⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡄⠀⠀⣀⣴⣿⣿⣀⠀⠀⢠⣤⣄⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣷⣾⣴⣦⣤⣤⣤⣤⣤⣶⣾⣿⡛⠭⠥⢤⠤⣴⣶⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣽⣽⡏⣿⡿⢿⣿⣽⣿⣿⡷⠁⠀⠀⠀⠄⠘⠟⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡻⣧⠌⣷⢻⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣦⣧⠀⡿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⡷⣧⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢴⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣯⣀⡀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⡷⢜⠂⠀⠀⠀⣀⣴⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠈⣿⣀⣀⣤⣶⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣻⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠒⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡒⠲⠒⠒⠒⠀⠐⠉⠉⠉⠉⠀⠉⠀⠀⠀⠈⠙⠋⢲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣀⣀⢀⠀⠀⠀⣀⣀⣀⢀⠀⠀⠀⠀⠀⠀⡀⠀⣈⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠉⢙⡏⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠞⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠋⠉⠉⠉⠚⠉⠙⠉⠩⠍⠩⠭⠉⠉⠛⠛⠙⢛⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠓⠒⠒⠒⣶⠶⣶⡞⠛⠛⠛⠒⠒⠒⠒⠛⠒⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣤⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠄⡇⠀⣀⡠⠤⢒⠊⠉⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠝⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⣀⡢⠤⠖⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⣀⡀⠀⣀⣴⣶⡾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠛⠃⠀⣤⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣤⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⢠⡖⣾⠀⠀⣧⠔⣺⠀⡠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢬⡳⣠⠖⢦⡇⠙⠓⠺⠷⠀⠉⠋⢇⠔⢲⣄⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⠎⠙⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣤⣶⠟⣻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⣄⡀⠀⠀⠀⠇⣴⠛⢋⣀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡝⠻⣷⣄⠀⠈⠥⠾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣷⣦⡙⠦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢀⡀

⠀⠀⢠⠤⢠⡄⣤⢠⠤⠠⣤⠄⣤⡀⣤⢠⡄⡄⢠⣤⠀⡤⡄⢠⠀⢠⠤⠀⢠⠤⣄⣤⠤⢤⢀⡄⣤⠄⣤⠀⣠⠤⡄⡤⣤⢠⡄⣤⢀⡤⠄⣤⢠⡤⣤⠄
⠀⠀⠨⡽⠘⢧⠟⠨⠽⠀⠿⠸⠛⠧⠿⠸⠙⠇⠾⠻⠄⠯⠗⠿⠤⠸⠯⠀⠸⠤⠟⠿⠭⠈⠿⠁⠿⠅⠿⠄⠷⡤⠇⠟⠉⠸⠻⠹⠗⠯⠅⠏⠻⠇⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢤⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⣛⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣛⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠰⣒⣒⣒⣒⣒⡐⡆⢰⢒⣂⣐⣐⣐⣂⠆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡏⠀⠀⠀⠀⡇⡇⢸⢸⠀⠀⠀⠀⢹⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⡇⡇⢸⢸⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⡇⡇⢸⢸⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⡇⡇⢸⢸⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⡇⡇⢸⢸⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢧⣀⣀⣀⣀⡇⡇⢸⢸⣀⣀⣀⣀⡼⢰⠒⢲⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢘⣒⣒⣒⣒⣒⣒⡁⢈⣒⣒⣒⣒⣐⣒⡋⠑⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡖⠐⠒⠒⠒⡆⡇⢸⢰⠒⠒⠒⠒⢲⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⡇⡇⢸⢸⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⡇⡇⢸⢸⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⡇⡇⢸⢸⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⡇⡇⢸⢸⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⡇⡇⢸⢸⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠰⠭⠤⠤⠤⠬⠥⠇⠸⠬⠥⠤⠤⠤⠭⠆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠃⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⢀⡿⠀⠐⢾⣿⣿⣦⡀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣾⣿⡄⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⡀⠰⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⢲⣼⣿⠿⠀⠀⠀⢀⡉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⢠⣇⠀⢀
⢀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣸⡼⠛⠀⠀⠀⠀⠀⠀⠸⣿⣷⡄⠀⠀⠀⢸⣿⣦⡙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⣸⣷⠀⣼
⣿⡀⠀⣼⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠁⠀⠀⣴⣿⠿⣿⣿⣷⣦⡀⠀⠀⠀⠈⠻⠿⠛⣿⣿⠟⠟⢛⣿⣿⣁⣼
⣿⡆⢐⣿⡆⢸⡆⠀⠀⣘⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣴⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⢀⣿⣿⡂⠀⠰⣿⣿⠷⣿
⣿⣗⣸⣿⣇⣼⡧⠀⠀⢽⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⣨⣿⣿⣧⠀⠻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠐⢿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣷⣶⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣶⣶⣶⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⣶⣶⣶⣶⣶⣶⣶⡶⠆⠘⢛⠿⢿⣿⣿⣿⣿⣿⣿⡇⢰⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠘⠀⠈⢻⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠉⣀⣸⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠽⣿⠿⠛⠃⢹⣟⠿⣷⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠠⠄⠒⠒⠀⢘⠛⠛⠛⢿⣿⡯⠛⠹⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣀⣀⣀⣀⡀⢀⣀⣀⣿⣦⣀⣠⣤⣀⣤⡀⢀⣀⣀⣀⢀⣿⣇⣠⣄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠛⡛⠋⠉⠉⠉⠉⠉⠁⠈⠉⠉⢿⣿⡟⠉⠉⠉⠉⠁⠈⠉⣩⣯⣽⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣾⣿⠶⠤⠤⠤⠀⠀⠀⠀⠀⢀⣸⣋⣤⣤⣴⣦⣶⡄⣴⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣯⠥⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣯⣭⣭⣭⣿⡿⠇⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣮⣭⠀⠀⠀⠀⠀⠀⠐⣿⠟⠀⣛⠀⠀⠀⠀⢀⢈⣛⣡⣿⣿⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠿⠀⣬⡭⠅⠀⢠⡀⠙⠉⠉⠉⠉⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣶⣿⣿⣿⣷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⢀⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⣀⣤⣶⣦⣄
⠀⠀⠀⢸⣿⣿⣿⣿⠟⣹⣿⠀⠀⣸⣿⣿⣿⠟⢉⣷⠀⠀⠀⠀⡀⠀⢀⣴⠆⠀⠀⠀⠀⢀⠀⠀⣠⡶⠀⠀⠀⣀⠀⠀⣠⡾⠂
⠀⠀⠀⢸⣿⣦⡉⣡⣾⣿⣿⠀⠀⢻⣷⡈⢁⣴⣿⣿⠀⠀⠀⠈⠻⣶⠟⠁⠀⠀⠀⠀⠀⠙⢷⡾⠋⠀⠀⠀⠀⠻⣷⡾⠋
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠛⠿⢿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⣀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⢸⣿⡟⠻⡿⠛⣿⣿⠀⠀⢠⣿⠻⣿⠟⢿⣦⠀⠀⠀⠀⢠⡀⢀⣄⠀⠀⠀⠀⠀⢀⣄⠀⣠⠀⠀⠀⠀⢠⣦⡀⣠⡆
⠀⠀⠀⢸⣿⡿⠂⡀⠺⣿⣿⠀⠀⣿⣿⠗⣀⠰⢿⣿⠀⠀⠀⠀⢀⣿⢿⡁⠀⠀⠀⠀⠀⠀⣹⠿⣇⠀⠀⠀⠀⠀⣹⡿⣯⡀
⠀⠀⠀⢸⣿⣷⣾⣿⣶⣿⣿⠀⠀⠘⠿⣶⣿⣷⡿⠋⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠈⠋⠀⠈⠃
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢬⠂⡔⢂⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡇⣠⣶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠉
⠀⠀⠀⠀⢀⢶⡀⠀⡴⣺⣫⣟⣗⢦⠀⠀⡶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡰⠰⠰⠔⢆⠀⠀⢠⡄
⠀⠀⠀⠀⡜⣸⠀⢸⠥⣯⢬⡧⢼⠤⡇⠀⣇⣳⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣚⡃⣛⢘⣓⢛⡃⠀⠘⣷
⠀⠀⠀⠀⡇⢧⠣⡈⢦⣷⠲⡗⡾⡴⢃⡜⡹⢹⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠘⣃⢩⢨⣏⡛⠃⢀⣾⣿
⠀⠀⠀⠀⠙⣄⠳⠙⢆⠉⠛⠛⠉⣰⠋⠜⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣆⠀⠀⠀⠈⠀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠈⢧⠀⡜⠀⠀⠀⠀⢣⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠀⠀⠀⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠸⠴⠃⠀⠀⠀⠀⠘⠦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠛⠛⠉⠛⣿⠉⠙⠛⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠈⢻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⢀⣤⠴⠶⠾⠋⠀⠀⠀⠀⣰⡟⠀⠙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠐⢿⡀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀⠀⠀⠀⣠⣴⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⠛⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠀⠀⢰⡟⠀⠀⠀⠀⢰⡶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⢧⡀⢀⣾⠁⠀⠀⠀⠀⢺⡇⢸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠾⠇⠀⠀⠀⠀⠀⣸⠇⠀⠻⠶⠟⠙⣷⡀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢿⣤⣀⡀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠈⢙⣦⢠⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢾⣷⡿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢧⣽⢷⣦⡀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣀⡴⢿⣩⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⠛⠀⠉⠻⢷⣦⣤⣄⣀⣀⣀⣠⣤⣴⣾⡟⠉⠠⣿⡟⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠙⣧⠉⠉⠉⠉⠉⠁⢠⡟⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣻
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣴⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿⣿⡿⠁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣃⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢿⠿⠃⢀⣄⡸⠆⠀⠀⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⠝⠟⣣⠀⢐⣿⣾⣿⣵⣐⣿⣷⣶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣦⣶⡆⠀⢰⠟⠿⠏⢲⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⢠⠛⣿⠀⠋⠈⣴⣤⣾⣿⣦⣾⣿⣵⣿⣿⣿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⢀⠸⠁⠀⡀⣾⣿⣿⠟⢙⡻⣿⣿⠝⠿⢛⠛⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⡴⠉⠀⠿⢋⢛⣿⣿⠋⣰⣿⣦⣝⣿⣶⣾⣿⡖⠀⠰⠖⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⠦⢤⣄⡀⠀⠀⠀⠀⠀⠀⠚⠿⣿⠿⢛⠀⠠⢿⣬⣻⣿⣡⣿⣿⣿⡿⣿⣿⣿⢋⡠⠤⠀⠀⠁⠀⠘⠛⠀⢀
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠈⡻⣶⣄⡠⣄⡄⠀⣠⠔⠃⢁⡀⣹⣿⣶⣿⣇⠀⢉⡋⠙⠛⠛⠛⠚⠳⠍⠀⠀⠄⠀⠀⠀⠀⠀⠐⠊
⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⢀⡔⣿⣿⣼⣿⡼⠿⠝⠉⠀⠀⢹⣿⣿⣿⣿⣿⢛⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⡀⠀⣠⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡄⢀⣀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⣾⣿⣿⣷⣴⣶⣷⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣷⣄⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠊⠀⠉⠈⠀⠈⠻⣿⣿⣿⣿⣟⣻⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⠷
⠀⡔⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠁
⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⢶⣒⠻⣯⡙⠻⣯⡉⠙⢯⡙⢿⠲⡖⢦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣚⠛⠻⠶⣬⣙⠳⠮⣝⡲⢍⡓⠤⣙⠲⣄⠑⢌⠣⡙⠸⠸⠸⢩⢟⡷⢦⣄⡀
⠀⠀⠀⠀⠀⣠⡶⠛⠲⢟⠉⠉⢓⡛⠒⠬⢝⡓⠦⢍⣓⠮⣕⡊⠑⠢⠑⠀⠀⣀⠤⠤⢄⡁⠊⠐⠩⠔⢛⣲⣄
⠀⠀⠀⡴⣛⣟⠋⢀⡤⠮⡃⠀⠺⠀⠐⠃⠀⢹⣷⢶⣮⣍⡒⠨⠁⠀⢀⣠⣴⠁⠀⠀⠀⠙⡄⠀⡴⢯⡭⠵⠒⣛⣦
⠀⢠⣾⣯⣥⣶⣿⣿⣤⡀⡬⠒⣤⡤⠶⣤⠔⠋⠀⢻⣿⠃⠙⠢⢤⣖⡛⠀⣿⣠⠴⠦⣄⣠⣿⡟⠁⣼⡿⢶⣛⣯⠽⠳⡄
⢠⣿⣿⣿⣿⣇⣨⡍⣙⠳⢾⡉⣀⣤⡚⠁⠀⠀⠀⣣⣿⡇⠀⡾⠛⠦⡉⠛⠤⣤⣶⢶⡿⢿⠍⠁⠀⢹⣇⠀⠙⣾⣵⣴⣿
⣾⣿⣿⣿⣿⣞⢿⣿⢿⡿⣶⣌⠛⠿⣿⣷⣤⠀⣼⠟⠁⣠⣾⣵⣤⣰⢃⣀⣠⣬⣽⣯⣄⠁⠀⣀⣤⣿⣷⣴⣿⣿⣻⣿⣿
⣿⣿⣹⣿⢻⣻⣞⣿⣿⣿⣿⣿⣿⠦⢤⣙⠻⢾⣷⣶⣾⣿⣯⣭⡿⣿⣿⣿⣿⣿⣿⠿⠟⣛⣲⣿⣿⢻⣯⢿⣿⣿⣿⣷⣿
⢿⣟⣿⣿⡻⣭⡟⣿⢾⣿⢿⣿⣿⣷⣆⡉⠳⣦⣮⣉⡟⢿⡿⢿⠟⠋⣩⣭⣴⣒⡺⣿⢟⣿⣿⣛⣯⣿⣿⣽⣿⣿⣿⣿⣿
⠘⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣼⣿⣿⣿⣿⣦⡌⠙⠫⣽⣿⡀⠙⠶⢦⣤⣉⡭⣩⣭⣽⣯⣷⣿⣛⣯⡿⣿⣿⣿⣿⣿⣿⣿
⠀⠘⢿⣟⡿⣿⣿⣿⣿⣿⣿⣷⣿⣽⢿⣿⣾⣿⣿⣵⣮⣿⣯⣳⣦⠀⠀⠉⠻⣿⣿⣿⣾⣿⣿⣿⡭⣽⣛⡳⢿⣿⣿⡿⠃
⠀⠀⠀⠻⣿⣷⣿⣻⣿⣿⣿⣿⣿⣻⢿⣶⠭⣒⢭⣻⣿⠿⠛⠉⠀⠀⠀⠀⠐⢸⣿⢿⣿⣿⣷⠾⣿⣛⣻⣯⣭⣭⠟
⠀⠀⠀⠀⠀⠙⠿⣿⣫⢯⣯⣽⣿⡿⣯⣺⣽⠾⠛⠉⠀⠀⠠⠀⠀⠀⠀⠀⢀⣼⣿⣻⠿⢟⣻⡭⢭⣖⣲⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢾⣽⣷⣿⣻⣿⡀⠀⣀⠀⠀⠰⡀⠀⠀⠂⣠⢞⡯⣟⣭⢖⣻⣽⣒⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⢭⣤⣾⣏⣈⣀⣀⣼⣿⡿⠾⠿⠞⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⣠⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢁⢈⣅⡼⣚⡯⣽⡄⣙⠉⢠⣤⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠰⠵⢛⣉⡃⡭⣾⠛⢿⣭⣟⣼⡿⠾⠱⢹⢀⡤⣝⢆⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⠲⣾⣹⣿⣼⣟⠛⡿⢶⣤⣈⣯⣀⣽⣾⢻⣿⣾⠢⢻⣌⠳⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⣿⣀⣷⣏⣯⢼⣝⠿⣏⢁⣀⣰⣛⣿⣏⠉⢻⡟⢩⣿⡾⣽⢁⡰⣄⢘⣎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠮⣂⣸⣷⣿⡵⠿⣥⣽⣿⣤⣟⢯⣾⣿⣿⣿⠛⣿⣿⣿⣾⣿⠿⣿⣏⡹⣿⠻⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣾⣿⢯⣉⡻⣷⣾⣿⢿⣿⣿⡏⣿⣿⡿⢿⣿⢿⣿⠽⢻⣮⡙⣷⣾⢹⡇⡈⠚⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡙⠮⢿⡼⢹⣿⣯⣿⣿⣿⣿⣷⣾⣿⣷⣾⣯⣾⢻⣻⢷⣎⡻⠋⠡⠗⠀⠆⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡛⣷⢶⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣋⣿⡷⠟⣿⣺⣷⡿⢱⠞⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⣿⢿⣾⠟⢻⣿⣶⣶⣿⣿⡷⣾⢪⡿⠟⣃⣽⡞⡍⡋⠂⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣻⣿⣿⣶⣾⣿⢸⣿⣾⣿⣿⢷⠋⢹⠁⠋⠑⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡺⢉⣯⣼⣿⣿⣿⡿⣿⣿⣾⣿⡟⣅⠈⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⣻⣿⣿⣴⣿⣿⡿⢿⣷⣾⣿⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⢿⣟⣿⣭⡿⠷⢟⠀⠚⣩⣿⣿⢦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣯⣗⢷⠐⡐⠀⠉⠸⢿⢟⡻⣤⡿⠀⠀⠀⠀⠀⠀⢀⢀⣔⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⠀⠀⠈⠀⠀⠀⣐⣞⠊⠱⣀⡀⣆⡄⠄⡢⢦⠰⠎⠉⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡰⣭⣨⣦⣏⢰⡛⣾⠎⢅⠈⠒⢊⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠉⠀⢭⣁⠓⠓⢐⣋⣳⠤⠛⣣⠾⣄⡤⠼⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠓⣿⢾⣮⣿⢳⡏⣿⣶⡿⢁⡘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⣼⠃⠈⠁⠁⠙⠛⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⣉⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⢀
⠀⠛⠟⠀⠀⠈⠉⢀⣶⣦⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢉⣙⣾⠛⢦
⠀⠀⠀⠀⡀⠀⠀⠈⠛⠋⢀⡀⠘⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⣤⣀⠀⠀⠀⠘⠉⣻⣿⠰⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠦⠤⢤⣤⣌⣉⣹⣷⣦⣤⣏⣹⣿⢡⢰⡇
⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠁⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠰⠶⣠⣞⣛⢿⠟⣩⣿⣿⣿⣿⠟⣿⡿⠀⢸⡇
⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠶⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⡀⠀⠀⠀⠀⢀⡀⠀⠀⠴⠟⠉⠁⢰⣾⠗⣫⣷⠟⣿⣡⣴⣿⠇⠁⡿⠁
⠀⠀⠀⢀⣾⣷⡄⠀⠀⠀⠠⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⢳⣶⢦⣀⡏⠷⠀⡀⠀⢀⣠⠖⢂⣴⣾⠟⠇⠘⣿⡿⠿⢃⣴⠞⠁
⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠐⠂⠀⠀⠀⠀⣮⣿⡀⠝⠡⠀⢤⠀⠐⠋⠀⠰⡯⠞⠁⠊⠀⠀⠛⠿⠷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⠀⠀⢠⡐⠑⡄⠐⡀⠀⠁⠀⠀⠆⠠⠀⠀⠁⠘⠀⢀⣂⠀⣠⠾⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠂⠀⠀⠀⢄⠀⣄⢀⣿⣼⣷⣿⡇⠈⠙⠀⠇⠈⠀⠀⠀⠀⠲⣦⠀⠉⠉⠀⢀⣠⣼⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⡀⠀⠂⠆⠀⡿⢿⣿⠟⡠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠾⣂⠀⠀⣠⡾⠟⣽⠟⡟⢷⣦⡀⠀⠀⣰⢦⡴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⢀⠀⠈⠶⠁⠘⠿⠀⢠⠀⡷⡀⠀⠀⠀⠀⠠⠎⣡⠄⠉⠷⢶⣄⢀⣾⡟⠰⣿⠀⠙⣣⢀⣴⡾⢋⣌⢇⣼⣥⡤⡀
⠀⠀⣀⣀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢿⠘⠀⠈⠃⡴⡆⠁⢸⢆⡀⠳⠦⣉⡜⣿⣦⣴⣃⣀⡄⠀⡀⢿⠈⣿⠤⠴⢛⣧⢠⢿⡟⣉⣿⣜⡿⣈⣷⡆⣺⣧⣀
⠀⠀⠁⠀⠑⠈⠀⠀⢠⠀⠀⠀⠀⠀⠰⡁⠀⠓⠀⠠⠋⠄⠊⡀⣃⣼⡁⢀⣼⡟⢋⣥⠌⢿⣶⠋⠦⠖⢀⣿⡄⠀⢸⡏⢤⣄⡀⡻⣿⡶⣥⡙⠦⠀⣹⡅⢏⠁
⠀⠀⠀⠀⠀⠰⠄⣄⢀⠋⠀⠠⠀⠘⠀⢁⠀⣠⠄⡀⠈⠤⠀⠀⠁⠆⢁⣿⣿⠀⠟⢤⣤⡰⢮⡴⡦⢠⣾⡿⠀⠢⡈⠳⠟⣹⣿⣷⡌⣴⣻⣙⣧⣀⣤⡴⠎
⠀⠀⠀⠂⠀⠀⣠⠁⣬⣿⡞⠁⠀⠀⠀⠀⠐⠋⠐⠃⢻⡄⠀⠀⣤⠟⣻⣿⣇⠀⠘⠂⠃⣬⡎⠹⠾⠿⠋⠀⠀⣠⣾⣶⣾⠟⠀⠀⠉⠁⢸⣦⢙⣙⡋⠃
⠀⠀⠀⠀⠀⠀⠀⠤⠰⠟⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⡘⣧⠀⠀⢀⡁⠙⣿⣿⠀⡴⣦⠐⣧⠔⠃⠌⢱⡟⠻⠟⡿⠛⡉⠀⠀⠀⠀⠀⠀⠀⠈⠘⠉⠁⢀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⡀⣤⢦⣄⠀⠀⠀⠀⢀⠀⠐⢁⡀⠀⢸⡶⠓⣦⣿⣤⣿⡿⠀⠰⡞⢠⡉⡀⠀⠄⣼⠇⠆⠈⣷⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⢀⠀⠠⠀⢣⡀⠀⠈⠀⣄⠀⣠⣿⣧⣤⣾⡇⠀⢾⣇⠤⣿⡿⣿⣿⣿⣤⢴⣿⠰⢱⣏⣁⣴⠏⢀⠃⣀⠁⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⢀⣼⣿⣿⠟
⠀⠀⠀⠊⢀⡤⠊⠀⠀⠀⢁⡶⠂⠀⠈⠻⣷⠴⢿⣧⠴⠚⣿⡶⠿⣧⠀⢠⠃⢀⡼⣏⠀⢀⣬⣩⡀⠶⠉⣹⠿⠃⢋⠠⢘⣯⠹⠎⠛⢛⠶⠶⣾⠿⠿⠛⠁
⠀⠀⠀⠀⢼⠁⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠙⣷⠛⠻⣷⡞⠙⣷⠶⣿⡇⣸⡀⣼⢀⠉⠃⠀⠈⠿⣁⣴⠛⠛⢠⠢⡌⠀⣸⠃⠀⢬
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢍⠁⢦⠀⠀⠀⠀⣿⠋⠛⣿⡶⢿⡿⢛⣿⠃⠉⠈⢳⣼⠀⠀⠀⡈⢀⠈⠁⠀⠀⠀⡤⡌⠀⢸⡆⡅⠁⠈⠀⠁
⠀⠀⠀⠀⠀⠹⢦⡀⢸⡀⢻⡶⠋⠀⠀⠀⠀⣿⡛⠺⣿⡗⢿⣧⣼⣏⢠⣅⡀⠀⠀⣱⠄⢺⣄⣨⠟⠀⠀⡐⠴⠀⠀⠀⣼⡇⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢼⡗⠀⠀⠔⠲⠉⢉⢿⡏⠻⣧⡀⢻⡿⣧⡌⢤⡙⢀⢢⡙⠶⠛⢡⡀⠀⠀⠀⠈⠆⡄⠁⢀⣿⡟⢀⠁⠀⠀⠀⠀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔⠀⢀⠤⢷⡁⠀⢨⢀⡈⢑⠄⣱⡟⠳⡄⣇⠼⣣⡝⠃⠚⠁⠀⠀⠀⠀⠀⠆⠤⢀⣼⡿⠁⠀⠃⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢐⠇⠀⠀⠀⠀⠀⠀⠃⣸⣏⠁⠈⠁⠰⣤⣼⠁⣈⠃⠀⠖⠉⠁⣀⠀⠀⠒⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡘⠓⠀⠀⠀⢀⠈⠀⠘⠻⡿⠓⠀⠄⠀⠀⠁⠀⠉⠀⠀⠀⠘⠃⠉⠤⠀⠀⠁⠀⠀⠈⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠀⠀⠀⢀⠀⢀⣉⠉⠐⣭⡀⢀⠁⠀⢰⠄⠀⠰⣦⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⢀⠤⠼⠗⠂⠀⠀⠀⠢⡞⠋⢰⡟⡹⠟⠉⠀⠀⠓⣀⠀⠀⢈⡄⠀⠋⠀⠀⠀⠉⠀⠀⠀⠀⠀⣽⣉⡿⣿⣿⡄⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢽⡖⠒⠳⡄⠁⠀⠀⠰⢀⣀⠀⠀⠀⣧⠥⣼⠿⣴⠂⠀⠀⠈⠀⠈⠀⠀⠀⠀⣀⠀⠀⠘⠀⠀⠈⠃⠀⢠⣼⣿⢨⣷⠿⠇⠀⢀⣼⣿⡿⠃
⠀⠀⠀⠀⠠⠀⠈⠃⠀⠀⠈⢢⠀⠀⢠⠘⢃⠀⠀⣠⠀⡀⢶⠂⠀⣤⠀⠀⠛⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠚⠀⠀⠀⠀⠀⠸⣿⣼⣿⣤⡄⣄⣴⣿⣿⡃
⠀⠀⠀⠀⠀⢷⣄⠀⠀⠀⠀⠀⢣⡔⠂⠀⠀⠉⠾⣽⣔⡷⠛⠓⣡⣟⣦⠤⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⢀⡼⡏⠁⠙⠛⠙⠁
⠀⠀⠀⠀⠀⠀⠉⠣⠀⠀⠀⠀⠀⠱⣄⠀⠀⢈⡁⠈⠋⠿⠾⠾⠷⠋⠀⠀⠀⠈⣁⠀⠀⠀⠀⠠⠶⠀⠀⠀⠰⠆⠀⢀⣴⠛⣬⡇⠈⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠲⣆⠀⠀⠑⠀⣾⢀⡀⢀⠀⠀⠤⠒⠊⠉⠉⠁⠠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠝⠛⢋⢸⣧
⠀⠀⠀⠀⠀⣠⠶⢦⣄⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠻⠦⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⢛⣿⡿⠛⠁⠀⢸⣿⠀⠀⠉⠻⣿⣿⡻⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⠀⣴⣿⠋⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠻⣷⣄⠈⠛⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣏⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⢙⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣠⣿⠟⠛⠻⠿⣷⣾⣿⣧⣤⣄⣀⣀⣀⣀⣸⣿⣀⣀⣀⣀⣀⣠⣤⣿⣿⣶⡿⠿⠛⠻⣿⣆
⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⢠⣿⠏⠉⠛⠛⠛⠛⠻⠿⢿⣿⠿⠿⠛⠛⠛⠛⠋⠉⢻⣿⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⢾⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⣤⣤⣤⣤⣄⡀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⢀⣿⡇⠀⠀⠀⢰⣿⠋⠉⠉⠛⠁⠀⣿⣟⠉⠉⢻⣿⡄⠀⣿⡟⠋⠙⠿⠷⠀⠀⢀⣀⣀⣀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⢸⣿⣷⣶⣦⠀⠘⢿⣿⣶⣦⣤⡀⠀⣿⡿⠀⠀⢸⣿⡇⢸⣿⡇⠀⣤⣤⣤⠀⢸⣿⣛⠛⠛⠁⢠⣶⣶⣿⡇
⠀⠀⠀⠸⣿⡏⠉⠁⠀⢀⡀⠀⠈⠉⣿⡧⠀⣿⣇⠀⠀⣸⣿⠇⠸⣿⣇⠀⠉⣽⣿⠀⠈⠛⠛⠿⣿⡆⠀⠉⠉⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠚⠿⣿⣿⡿⠟⠁⠀⢿⢿⣿⡿⠿⠋⠀⠀⠙⠿⣿⣿⠿⠃⠀⠸⠿⣷⣾⠿⠃⠀⠀⢸⣿⠃
⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣀⣀⣀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⣀⣠⣴⣿⣷⣶⣿⠿⠿⠿⠿⠿⢿⣿⠟⠻⠻⠿⠿⠿⣿⣶⣿⣿⣤⣄⡀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠛⠉⠙⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣿⡿⠉⠙⠻⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀⠀⠘⢿⣧⡀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣾⡟⠀⢀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣈⠻⣷⣄⠀⠀⠀⢸⣿⠀⠀⠀⢀⣴⣿⢏⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣶⣤⣼⣿⣤⣤⣶⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⠿⠿⠿⠿⣿⣿⣶⣤⣄⡀
⣴⣿⡿⠿⢿⣿⣿⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣦⣤⣦⣤⣀
⢿⣿⣤⣀⠀⠀⠈⠉⠛⠛⠿⠿⣿⣿⣶⣶⣤⣤⣀⣰⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⠛⠻⢿⣿
⠈⠙⠿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠺⠇⣼⣿
⠀⠀⠀⠀⠉⠛⠿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠟⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⣾⣿⠏⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⡿⠿⠛⠛⠛⠿⣿⣿⡏⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣷⣶⣶⣾⣿⣿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣷⣶⣿⡟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣯⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠋⠁⠀⠀⠑⠂⠄⠀⠐⠖⠠⠀⠀⠀⠀⣠⡤⠀⠀⣠⣶⣶⡤⠖⠋⠀⠀⠠
⣆⣠⣤⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠚⠁⠀⠀⠘⣛⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⣴⡶⠶⠤
⣿⣿⣿⣷⣶⣷⣶⢀⢀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⣠⡶⠚⠃⠀⠀⠀⠀⠀⢀⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣧⣾⣴⣾⣦⡆⣶⢀⠀⣿⡀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣧⣾⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⣿⠟⢿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣠⢸⡇⣦⣿⣿⣷
⠟⠛⠛⠋⠋⠋⠉⠈⠈⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⠀⠀⣾⡀⠀⢀⡄⢀⢠⣀⡀⣶⢀⣆⣿⣿⣤⣦⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⢰⣿⣧⣆⣼⣧⣿⣾⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣾⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣿⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠟⠻⣿⡿⠿⢿⡿⣿⢿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣧⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠙⠉⠀⠁⠙⠙⠋⠛⠻⠻⠛
⣿⡿⣿⣿⣿⣿⣧⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⣯⠀⢻⣿⣿⣿⣿⡆⣼⣿⠀⠀⣀⠀⠀⠀⣼⡂⠀⠀⠀⠀⠀⣾⣧
⠃⠀⢺⣿⣿⣿⣿⣿⣿⣿⣗⢠⣿⡀⠀⢠⣿⣧⠀⢠⡄⠀⢰⣿⣿⠆⠀⠀⣄⠀⠀⠀⢠⡄
⠆⠀⠤⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣾⣿⣯⣄⣼⣧⠀⣺⣿⣿⣶⡄⢸⣿⠀⠀⠀⣾⣷
⠀⠀⠈⢻⣿⣿⣿⣿⡟⣿⣿⣿⣿⣷⣼⣿⣿⡿⢻⣿⣿⣬⣿⣿⣿⣿⣥⣿⠿⠇⠀⢸⣿⣿⠂
⠀⠀⠀⢀⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⢰⣦⣿⣿⣿⣶⠄⠀⢠⣇
⠀⠀⠐⠶⠿⣿⣿⡃⠀⢐⣿⣿⣿⡏⢹⣿⣟⠀⠐⣺⣿⣿⣿⣿⣿⣿⣿⡃⠀⢒⣿⣿⣿⣿⣿⣆⠀⣾⣿⠄⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠉⣛⡻⠯⠀⠀⠀⢸⣿⣿⠀⠨⣿⠥⠀⠀⠈⣿⣿⡿⢻⣿⣿⡯⠀⠀⠀⢹⣿⣿⣿⣿⣿⣼⣿⣿⣦⠀⣦⠀⢀⣿⡇
⠀⠀⠀⠀⣀⡤⠂⠀⠀⠈⠭⢿⠇⠀⠀⠘⠀⠀⠀⠉⠽⣿⠁⠸⣿⡿⠃⠀⠀⠈⠭⢿⣿⡿⠉⣿⣿⡿⢿⣿⣲⣿⡂⣸⣿⣷⠄⡄⠀⢀
⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠨⠄⠀⠀⠿⠛⠀⠀⠀⠀⠀⠤⣿⠃⠀⢹⣿⠇⠘⣿⢻⣿⣿⡿⢹⣿⣿⣿⠀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢉⠉⠀⠀⠈⠋⠀⠀⠃⢘⣿⣿⠃⠈⣿⣿⣿⣷⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⡉⠉⠛⠛⣻⠷⠀⠀⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢻⡇⣀⡴⠛⠁⠀⠀⠀⣀⠶⡄⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢼⠾⠋⠀⠀⠀⠀⠀⡜⢁⢰⠇⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢠⡀⠀⠀⣠⠄⠓⠛⠁⠀⠀⢀⣄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠘⣯⠙⠲⡌⢳⣄⡼⠛⠛⠋⠀⣀⠀⡴⠋⢸⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⢦⣑⡿⠀⠻⣅⠀⠀⠀⠀⣿⠸⣥⡥⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢦⣽⣆⠀⠀⣠⡿⠤⠤⠄⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⣠⠴⠒⣓⣦⡀⣿⡴⠛⠉⠀⠀⠀⣀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠙⠛⠲⠶⠞⠁⣿⠀⠀⠀⠀⣠⠞⣿⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠛⠀⠀⣠⠞⠁⠀⣿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣀⣀⠀⢠⣾⣥⣤⣤⣤⣬⣽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠛⠙⣿⡆⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠟⠋⠁⢀⣠⣤⠀⢹⣿⣀⣀⣤⣴⣾⠿⢛⣋⣭⣶⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠿⠛⠉⠀⠀⠀⠀⣠⣾⣿⣿⡄⠀⣿⣿⣿⣫⣥⣶⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠋⠁⠀⠀⣴⣶⣤⣤⣄⣴⠟⠁⢹⣿⡇⠀⢹⣿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠋⠁⢀⡀⠀⠀⠀⠀⢹⣷⠀⠈⠉⠁⠀⠀⢸⣿⣿⠀⠘⣿⣤⣤⣤⣴⣶⣶⣾⠿⠿⠿⠿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠿⠋⠁⢀⣀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⣿⣭⣥⣶⣶⣶⣾⠿⠿⠿⠟⠛
⠀⠀⠀⠀⠀⠀⣾⣿⠋⠁⢠⡖⠋⣹⠟⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⣿⡟⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡗⠚⢻⡶⠀⠀⣿⠃⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠸⠷⠶⠾⠃⠀⢠⣿⣀⣄⣀⣀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⢸⣿⣿⣭⣭⣭⣭⣭⣭⣭⣭⣍⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠻⠿⣶⣶⣶⣤⣤⣄⠀⠀⠀⠈⠉⠉⠉⠁⠙⠛⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⠀⠀⠀⣠⣾⢿⣶⣶⣶⣤⣤⣤⣀⣀⣀⠀⠈⠛⠋⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⣠⣾⡿⠁⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠿⠿⢿⣶⣶⣶⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢄⣿⣟⡤⠀⠀⠀⠀⢀⡀⠂⠠⠈⠠
⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠾⢿⣿⢿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠀⠀⠀⠀⠀⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣹⣿⡟⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢸⡀⢀
⠀⠀⠀⠀⠹⢻⣦⣾⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣄⣷⠏
⠀⠀⠀⠀⠀⠈⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠲⡄
⠀⠀⠀⠀⠀⠰⣶⣶⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⣠⢲⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠈⠉⡀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠠⣦⣼⣇⣤⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣴⣶
⠀⠀⠀⠀⢛⣿⣯⣿⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⣿⣿⠷
⠀⠀⠀⠀⠘⠩⣿⠙⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠿

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⢴⣶⣒⣶⡀
⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⣒⣚⣯⠭⠵⠖⠒⠛⠉⣿⠛⠶⢭⣓⠲⢤⣀⡀
⠀⠀⠀⢸⢋⡭⠵⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠙⠓⠦⣭⣛⠲⠦⣤⣀
⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢬⣝⣳⣦
⠀⠀⠀⢸⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿
⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⢸⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢹⣆⡇⡇
⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢸⡇⡇
⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⢸⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⠉⠛⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣄⡀⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⣿⢸⣀⣤⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣄⣸⣧⠇
⠀⠀⠀⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡆⠀⣶⣶⣶⡄⢰⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠁⢸⣿⣿⣿⣧⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡏⢀⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠁⠸⠿⠿⠿⠿⠿⠇⠀⠿⠿⠿⠿⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣿⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣴⡾⠟⠁⢸⣿⠀⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠻⣿⣿
⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⢸⣿⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⢀⣛⣃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠐⠻⢿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀⠀⠀⠀⠀⠀⠘⢃⢐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣷⣦⣄⡈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣴⣾⣿⠿⠛⠛⠛⢿⣿⣿⣿⣿⣷⣦⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⢠⣾⣿⣿⠁⣠⣶⣿⣦⠀⣿⣿⣿⣿⣿⣿⣧⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠀⣿⣿⣿⣿⣄⠈⠛⠛⠿⣿⣿⣿⡄⠸⣿⣿⣿⣿⣿⣿⣿⡿⠃⣠⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡄⠘⣿⣿⣿⣿⣿⣿⣷⡀⢹⣿⣿⣷⣄⡈⠛⠿⠿⠿⠿⠟⠁⣴⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣆⡈⠛⠿⣿⣿⣿⠟⠀⣼⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛