⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠉⡽⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣛⡄⠙⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢣⠀⢘⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠈⡇⢰⡿⠀⠀⣀⡴⢞⣴⡦⣀⣴⣾⠀⠘⣷⡀⠀⠈⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡅⠀⠠⠀⣸⣧⡶⢛⠥⠚⠉⣁⠼⠟⠫⠿⠟⠛⠋⠁⠀⠠⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢀⡀⠀⠉⠥⠊⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠁⠠⣄⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣮⣶⣿⡿⠋⠁⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠙⠿⣿⣿⣶⣿⣷⣶⣤⣦⣀⢠⡀⠀⠀⢸⣿⣾⣿⠿⠿⠛⠉⠿⡽⡀⠀⠀⠀⢀⠑⢍⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠁⠀⠀⠀⠀⢰⡿⠁⠉⠉⠛⣿⡿⢿⡾⠿⠀⠐⠛⠛⢹⣷⠀⠀⠀⠀⢸⣷⡇⠀⠀⠀⠀⠑⠢⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⡎⣸⡇⠀⠀⠀⢀⣿⣥⠀⠀⠠⢔⣒⠀⠀⠘⣷⣣⣤⣀⣤⣿⣿⠕⠁⠀⠀⠀⠀⠀⠈⠻⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠀⠀⠀⠀⢈⣧⡹⢿⡦⣤⣴⣚⡿⠃⠀⢀⣀⣥⣤⣤⣤⣤⣈⠛⠻⠛⠻⠯⠉⠂⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣷⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢼⡟⠀⠀⠀⠀⠀⠀⠈⠛⠿⠶⠿⠟⠋⢁⣠⣴⣾⢿⣿⣿⣿⣿⣿⣿⠟⠻⣷⡄⠀⠠⡀⠀⠀⠐⣤⣄⠀⠀⠀⠀⠀⠀⡈⢻⣷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠿⠾⠯⢽⣿⣿⣽⣿⡟⠉⠀⠀⠉⠀⠀⠹⡞⡀⣄⠀⣸⠋⢳⡀⠀⠀⠀⠀⠘⢶⣝⣿⣆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠛⡏⠀⠀⠀⠀⠀⠀⠀⠁⠀⠜⠛⠛⠏⠀⠘⣿⠂⠀⠀⠀⠱⣄⡻⣿⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢾⡟⠀⠀⠀⠀⠀⠀⣰⠞⢽⠁⢠⢿⠆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⡀⠀⢀⢹⡇⠀⠀⠀⠀⠘⣿⡿⠏⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⢰⠟⠀⠻⣤⠾⠎⠀⠀⢀⢂⠀⠠⠁⠔⠀⠀⣴⣿⣄⠀⠀⠓⠄⠐⢄⠑⠀⠀⠁⠢⢀⠀⠀⣸⡇⠀⠀⠀⠀⢶⣾⣿⡀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⣤⠀⠀⠀⠀⠀⣾⠊⠀⠀⠀⠀⠠⠀⠠⠃⠂⠀⠁⠈⢀⡠⠞⠁⢹⡆⠉⠒⠄⠤⣄⣄⣁⣤⣀⣠⣄⣠⡅⣠⠛⠁⠀⠀⠀⣤⣼⣿⡟⢃⠀⠀⠀⠀
⠀⠀⠀⠀⠁⣿⣽⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠔⠁⠀⡀⠁⠀⠀⣠⣖⠋⠀⠀⠀⢸⡆⠀⠀⠀⠀⠎⠇⠀⠈⠉⠉⠙⠛⠉⠁⠀⠀⠀⢀⢀⣿⡟⠁⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠟⢹⡆⢀⠀⠀⠀⠸⣷⡀⢀⠊⠀⠀⢄⣀⣴⠖⠋⠀⢻⡆⠀⠀⠀⠀⡇⠀⠀⠀⢰⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢾⡟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⣿⣸⣧⡀⠀⠀⠙⣷⣤⣀⣤⠴⠛⠉⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠓⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢀⣤⣮⡶⠟⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⠿⣷⡹⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠷⠶⠶⣶⣶⣀⣰⡿⠀⠀⠀⠀⠀⠀⡠⣶⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠙⠀⠙⠷⡤⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⣠⣴⡵⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⢀⣀⣥⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣀⠀⠀⣀⣀⠀⢀⣀⣀⣴⡾⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢴⣬⡽⢿⡍⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠉⡽⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣛⡄⠙⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢣⠀⢘⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠈⡇⢰⡿⠀⠀⣀⡴⢞⣴⡦⣀⣴⣾⠀⠘⣷⡀⠀⠈⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡅⠀⠠⠀⣸⣧⡶⢛⠥⠚⠉⣁⠼⠟⠫⠿⠟⠛⠋⠁⠀⠠⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢀⡀⠀⠉⠥⠊⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠁⠠⣄⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣮⣶⣿⡿⠋⠁⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠙⠿⣿⣿⣶⣿⣷⣶⣤⣦⣀⢠⡀⠀⠀⢸⣿⣾⣿⠿⠿⠛⠉⠿⡽⡀⠀⠀⠀⢀⠑⢍⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠁⠀⠀⠀⠀⢰⡿⠁⠉⠉⠛⣿⡿⢿⡾⠿⠀⠐⠛⠛⢹⣷⠀⠀⠀⠀⢸⣷⡇⠀⠀⠀⠀⠑⠢⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⡎⣸⡇⠀⠀⠀⢀⣿⣥⠀⠀⠠⢔⣒⠀⠀⠘⣷⣣⣤⣀⣤⣿⣿⠕⠁⠀⠀⠀⠀⠀⠈⠻⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠀⠀⠀⠀⢈⣧⡹⢿⡦⣤⣴⣚⡿⠃⠀⢀⣀⣥⣤⣤⣤⣤⣈⠛⠻⠛⠻⠯⠉⠂⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣷⣄⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⢼⡟⠀⠀⠀⠀⠀⠀⠈⠛⠿⠶⠿⠟⠋⢁⣠⣴⣾⢿⣿⣿⣿⣿⣿⣿⠟⠻⣷⡄⠀⠠⡀⠀⠀⠐⣤⣄⠀⠀⠀⠀⠀⠀⡈⢻⣷⡀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠿⠾⠯⢽⣿⣿⣽⣿⡟⠉⠀⠀⠉⠀⠀⠹⡞⡀⣄⠀⣸⠋⢳⡀⠀⠀⠀⠀⠘⢶⣝⣿⣆⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠛⡏⠀⠀⠀⠀⠀⠀⠀⠁⠀⠜⠛⠛⠏⠀⠘⣿⠂⠀⠀⠀⠱⣄⡻⣿⡀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢾⡟⠀⠀⠀⠀⠀⠀⣰⠞⢽⠁⢠⢿⠆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⡀⠀⢀⢹⡇⠀⠀⠀⠀⠘⣿⡿⠏⠀⠀⠀⠀ ⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⢰⠟⠀⠻⣤⠾⠎⠀⠀⢀⢂⠀⠠⠁⠔⠀⠀⣴⣿⣄⠀⠀⠓⠄⠐⢄⠑⠀⠀⠁⠢⢀⠀⠀⣸⡇⠀⠀⠀⠀⢶⣾⣿⡀⠀⠀⠀⠀ ⠀⠀⠀⠀⣼⣿⣤⠀⠀⠀⠀⠀⣾⠊⠀⠀⠀⠀⠠⠀⠠⠃⠂⠀⠁⠈⢀⡠⠞⠁⢹⡆⠉⠒⠄⠤⣄⣄⣁⣤⣀⣠⣄⣠⡅⣠⠛⠁⠀⠀⠀⣤⣼⣿⡟⢃⠀⠀⠀⠀ ⠀⠀⠀⠀⠁⣿⣽⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠔⠁⠀⡀⠁⠀⠀⣠⣖⠋⠀⠀⠀⢸⡆⠀⠀⠀⠀⠎⠇⠀⠈⠉⠉⠙⠛⠉⠁⠀⠀⠀⢀⢀⣿⡟⠁⠃⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠟⢹⡆⢀⠀⠀⠀⠸⣷⡀⢀⠊⠀⠀⢄⣀⣴⠖⠋⠀⢻⡆⠀⠀⠀⠀⡇⠀⠀⠀⢰⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢾⡟⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠘⣿⣸⣧⡀⠀⠀⠙⣷⣤⣀⣤⠴⠛⠉⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠓⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢀⣤⣮⡶⠟⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠸⠿⣷⡹⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠷⠶⠶⣶⣶⣀⣰⡿⠀⠀⠀⠀⠀⠀⡠⣶⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠙⠀⠙⠷⡤⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⣠⣴⡵⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⢀⣀⣥⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣀⠀⠀⣀⣀⠀⢀⣀⣀⣴⡾⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢴⣬⡽⢿⡍⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀