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⠄⠄⣀⣤⣴⣾⣿⣷⣭⣭⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠄⠄
⠄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣧⠄⠄
⠄⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢻⣿⣿⡄⠄
⠄⢸⣿⣮⣿⣿⣿⣿⣿⣿⣿⡟⢹⣿⣿⣿⡟⢛⢻⣷⢻⣿⣧⠄
⠄⠄⣿⡏⣿⡟⡛⢻⣿⣿⣿⣿⠸⣿⣿⣿⣷⣬⣼⣿⢸⣿⣿⠄
⠄⠄⣿⣧⢿⣧⣥⣾⣿⣿⣿⡟⣴⣝⠿⣿⣿⣿⠿⣫⣾⣿⣿⡆
⠄⠄⢸⣿⣮⡻⠿⣿⠿⣟⣫⣾⣿⣿⣿⣷⣶⣾⣿⡏⣿⣿⣿⡇
⠄⠄⢸⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⡇
⠄⠄⢸⣿⣿⣿⡇⠄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⠄
⠄⠄⣼⣿⣿⣿⢃⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⡇⠄
⠄⠄⣿⣿⡟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿ ⣿⡏⣿⣿⣿⡇⠄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢤⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠘⣀⡠⠔⠀⣼⣦⡈⠻⣿⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠠⠒⢊⡍⡆⡆⠀⠀⢸⣿⠁⠀⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠷⠀⠀⠀⠓⢛⠃⠀⣤⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠐⠊⠁⠀⣴⡇⠀⠀⢻⠀⢀⢿⡆⠸⣷⠀⠀⢸⣿⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣧⠀⠀⡇⠀⢰⢻⡇⢰⡀⢸⡇⠦⠘⣷⠀⣿⠀⠀⣼⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⡄⢸⣇⠀⡾⢸⡄⠀⠀⡼⣿⣄⢀⣹⠀⡹⠀⢠⣿⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⣿⣦⠻⠿⠃⠈⠻⣿⠿⠃⠈⠻⠟⠋⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠟⠋⠀