⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠙⠛⠚⠛⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠂⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⣶⣦⣤⣤⣀⣀⣀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠻⣿⣿⡟⢻⣿⣿⣿⣿⡶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢽⣿⠇⢨⣽⠗⢺⣯⡁⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⠒⠒⠉⢰⠿⣻⡇⠈⢙⣛⡛⠋⢀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣿⡶⠛⣏⠉⠀⠀⡄⠘⣇⠀⠻⣶⣄⣀⣀⣠⣾⢁⠳⣌⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠖⠋⠀⣷⠀⢹⡄⠀⠀⡇⠀⢹⡀⠀⠀⠙⠛⠋⣽⣿⠈⠆⢻⣿⡌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠿⢆⡁⠀⠀⠀⢸⣶⣶⣿⡦⠀⣇⠀⠀⢳⡀⠀⠰⢖⣧⣿⡟⠀⠀⢸⣷⣿⡄⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⢀⣠⡄⢳⠀⠀⠀⠘⢿⠛⢻⡇⠀⢿⠀⠀⠀⣻⡲⠶⢋⣿⣼⣃⠀⠀⠀⣧⢸⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⠴⠿⡇⠘⠉⠀⠉⣹⠀⠀⠀⠀⢸⣧⣠⣷⡄⢸⣤⡴⠛⠉⢳⡀⣼⡿⠋⠙⠳⣤⠀⣿⣿⡇⠀⠀⠐⢿⡦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣴⡾⢇⣀⠀⠈⠓⠒⠒⠋⠁⠀⡀⠀⠀⠘⠻⡆⣿⡁⠈⠁⠀⠀⠀⠀⢻⡟⠁⠀⠀⠀⣈⣴⢿⡏⠀⠀⠀⠀⠀⠉⠲⢭⡑⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡰⣻⡏⢀⡀⠈⠳⢦⡀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢀⣿⠋⣿⠙⠛⠲⠶⠦⠤⠬⡥⠴⠶⠞⠛⠉⢿⣼⠁⠆⠀⠀⠀⠀⣀⠀⠀⠈⢁⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢰⣷⠋⠀⠀⠀⠀⠀⠀⠹⣄⠀⣰⣋⣴⣦⣀⠀⠀⣾⠇⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⣿⣶⣿⣿⠋⠀⢰⡶⠋⠁⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢿⣿⡶⠶⠶⠤⣤⣀⡀⠀⣨⣿⠿⠿⢿⣿⣿⣿⢀⣿⠄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠹⣿⣿⣿⣧⣠⡟⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⠀⠀⠀⢻⣿⣯⣼⣿⡍⠀⠁⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⢦⠈⠻⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠙⠒⠤⣀⡀⠀⠀⠀⠀⠀
⣞⠛⠀⠀⠀⠀⠀⠀⠀⠺⣿⣤⡀⠀⠀⠀⣿⣿⣿⣿⣿⣦⡀⠀⠈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡷⠒⠋⠉⠁⢭⡻⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢢⡀⠀⠀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠹⣿⣿⣿⣿⣿⣯⣤⡀⠀⠁⠀⠀⢀⣀⣀⣠⣤⡶⢶⡚⠛⠻⠉⠉⠀⠀⠀⠀⠀⠀⠉⠲⣍⡛⢿⣿⣶⡄⠀⠀⠀⢠⡄⠀⠠⢹⡄⠀
⡇⠀⠀⠀⠀⠠⠤⢄⣀⣀⡀⢺⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⡟⢻⣵⣶⣿⣿⣿⣿⡀⢠⡀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡹⣝⠻⣷⣄⡀⠉⢛⠀⣰⢷⠃⠀
⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⠀⠀⠀⢀⣀⣘⣿⣿⣿⡟⠛⣿⣿⣷⣶⣶⣿⣷⠈⣿⣿⣿⠿⢟⣛⣧⣾⣿⣿⣿⣿⣷⣶⣤⡀⠀⢀⣠⠶⠛⠻⢾⢢⠈⠛⣿⣶⣥⣾⣷⣿⠂⠀
⠸⡆⠀⠀⠀⠒⠒⠒⠒⠶⠤⡇⠀⠀⢰⣿⣿⡋⣻⣿⣿⡀⠀⢿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣫⣥⣴⣶⣾⣿⡟⠀⠀⠀⠉⠙⠛⠛⠁⠀⠀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠘⣿⠛⣿⣿⣿⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠌⠉⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⡼⠁⣼⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠁⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣏⡄⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⣸⠁⢀⠏⠁⡿⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢶⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠛⢿⣧⣄⡀⣀⣄⠀⣰⡟⠁⠀⠀⢠⠇⠀⡾⠀⢠⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⠙⠻⣿⣧⣴⡿⠁⠀⠀⠀⡞⠀⢸⠃⠀⢸⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠈⠉⠉⠉⢻⣿⣿⣿⣷⣍⠀⠠⢸⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢸⣧⡈⡿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣗⡇⠀⢀⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠠⣿⣾⣟⠿⢛⠿⢿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⢠⣿⣿⠃⠀⠛⠁⠶⠙⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡻⣧⣷⣫⣫⣿⣆⠀⠀⠀⠀⣤⢹⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣾⠿⠿⣯⣽⣿⣶⣦⣄⠀⠀⢠⣇⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣧⣌⣿⣧⣾⣿⣿⣧⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣿⣿⣿⣿⣿⣿⢟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⣿⣿⣿⣿⣿⣕⢤⣀⡤⠒⠒⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣌⣉⣵⠟⠂⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣤⣾⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠛⣭⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣪⠴⠒⢲⣄⠀⣮⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⣠⣤⠤⣴⣦⢾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

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