Scarf Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢲⣿⣿⠿⠟⢀⣽⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡀⢠⣿⣿⣿⣿⣻⡿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠉⠀⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⣝⣛⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⣦⣶⢿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠛⠻⠟⢿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣵⣿⣿⣿⣿⣽⣠⣿⣿⣟⣉⣉⠍⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣿⣿⣿⣿⣿⡇⢿⣿⡿⠛⣿⡿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⡾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠘⢿⣸⡿⣿⢻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⡇⠈⢿⣇⡏⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⢺⣷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⣿⣿⣿⡿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣽⣿⠛⠛⡙⠀⠀⠀⠀⠀⠀⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣤⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣝⢯⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠒⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⠉⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⡆⠀⣀⠴⠶⠥⠤⢤⣄⣀⡀⠀⠀⠀⢸⠀⠀⠀⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠹⡀⠀⠀⠸⡾⠁⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀⡞⠀⠀⢠⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠁⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⡄⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠐⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⣀⣀⣠⣤⠴⠛⠀⠀⠀⠐⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠤⣄⣀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠉⠙⠓⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠲⠤⢤⣄⣀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⡴⠋⠀⠀⠿⣯⢽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡿⠄⢠⣠⡄⣈⠉⠲⢄⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢀⡞⠀⣤⡿⢳⣖⠀⢀⡽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠛⠉⠐⣾⣯⣹⡟⠂⣀⣄⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣀⡄⠋⠷⠿⠉⡰⠋⣠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠒⠒⠦⠤⢄⡀⠛⠉⠠⣾⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⣳⡆⠀⠀⡼⢁⣴⣿⡙⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⡶⡶⣦⣄⡈⠓⢄⣀⠉⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠁⢀⡞⢁⡾⡏⡿⢇⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢸⠇⡿⣇⡏⣻⣶⣄⠈⠓⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠔⠋⣠⣿⣳⣿⣿⡈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠇⣊⣠⢠⣷⣷⣏⢃⡟⣿⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣠⣾⢏⣮⡿⢽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠁⠛⢹⡿⡨⣇⡿⡸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⢻⢹⠸⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠓⢹⡟⣃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⢼⣟⠰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠉


⣿⢿⠿⠿⡿⡿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣿⣿⣻⣿

⠀⠀⠀⣠⣶⡾⠿⠿⠿⣶⣦⣄⡀
⠀⠀⣾⡟⠁⠀⠀⠀⠀⣴⣿⠛⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⣿⡇⠀⠀⠀⠀⢰⣿⠃⠀⠀⢈⣽⣿⣿⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣾⣿⡟⠛⠛⠛⢿⣦⡀
⠀⠀⣿⡇⠀⠀⠀⠀⣾⡿⠀⠀⠀⣼⣿⠁⠀⠀⠉⠙⠛⣿⡿⠿⠿⠿⣿⣿⡿⠿⠿⠿⢿⣿⡟⠉⠁⠀⣿⡇⠀⠀⠀⠀⢿⣧
⢀⣾⡟⠁⠀⠀⠀⢀⣿⡇⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⣼⡿⠀⠀⠀⢸⣿⠁⠀⠀⠀⢀⣾⡏
⣾⡟⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⣸⣿⠀⠀⠀⠀⢰⣿⠃⠀⠀⢀⣿⡏⠀⠀⠀⢠⣿⠟
⣿⣇⠀⠀⢀⣾⡟⠁⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⢀⣿⡇⠀⠀⠀⢠⣿⠏⠀⠀⠀⣼⡿⠀⠀⠀⠀⢸⣿
⠘⢿⣷⣄⣼⡿⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⣼⡟⠀⠀⠀⠀⣾⡟⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠉⠻⢿⣷⣤⣀⠀⣿⡏⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⢀⣾⡟⠁⠀⠀⠀⢰⣿⠃⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣷⣤⣄⣀⡀⠀⢰⣿⠁⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡁⠀⠀⠀⣀⣤⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⢿⣿⣶⣶⣤⣼⣿⣧⣤⣀⣀⣀⣀⣸⣿⣄⣤⣤⣼⣿⣷⣶⣿⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠉⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠉⠉⢹⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⣀⣀⣀⠀⠀⠀⠈⣿⡏⢿⣧⣀⣠⣤⣴⣶⣶⠿⠿⠿⠛⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠛⠛⠛⠿⠿⠿⣷⣶⣿⡇⠈⢿⣿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠈⣿⣇⢀⣀⣠⣤⣶⣶⣶⣿⡿⠿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣤⣤⣄⣼⣿⠀⠀⢸⣿⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠉⠉⠛⢻⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣀⣀⣀⣀⡀⠀⠀⠀⢸⣿⠀⠀⠀⢿⣿⣤⣶⣾⠿⠿⠛⠛⠛⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠛⠛⠛⠛⠻⠿⠿⣷⣶⣾⣿⠀⠀⠀⢸⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠀⠀⠀⣼⣿⠀⢀⣀⣠⣤⣤⣶⣶⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⣿⣿⠿⠿⠛⠛⠉⠉⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠿⠿⠿⢿⣶⣶⣶⣤⣤⣀⣿⡇⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢛⣿⡇⠀⠀⣼⣿⢀⣀⣤⣤⣶⣾⠿⠿⠿⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣤⣤⣤⣤⣀⣀⣀⠀⠀⠀⢸⣿⠀⠀⢠⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⠉⠉⠉⠙⠛⠛⠛⠿⠿⣷⣾⡟⠀⢀⣿⡟⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣾⣿⣶⣶⣾⠿⠿⠿⠿⠿⠛⠛⠛⣻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢻⡿⣿⣶⣦⣤⣄⣀⣀⣀⣠⣿⠏⠀⠀⢿⣯⣥⣤⣄⣀⣀⣀⣀⣀⣀⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢡⡟⣼⢇⣿⢹⡟⣿⢻⡟⣿⠟⠋⠀⠀⢀⡿⢹⡟⣿⢻⡿⣿⠟⡿⣿⢿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢃⡿⢱⡟⣼⠇⣾⢡⡟⣼⢇⣿⠀⠀⠀⠀⣾⢣⡿⣸⡏⣾⢃⣿⢸⡇⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠿⢰⡟⢠⡟⣸⠇⣿⢸⡇⠀⠀⠀⠈⠋⠾⢡⡿⢰⡟⢸⡇⣾⢡⡟⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠃⠿⠸⠏⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠘⠃⡿⢸⡇⣿
⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣴⣆⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⣶⡖⢂⣀⣀⣐⣒⡒⠒⠂⠠⠤⣄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⣭⠔⠒⠈⠉⠩⠃⠀⠀⠀⠀⡏⠁⠀⠈⠉⢙⣛⠒⠂⠬⠅⣂⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⡃⣆⠀⠀⠀⢠⠸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢨⢹⠀⠀⠀⠀⠀⣹⡏⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⡈⡳⠤⣀⣀⡼⠴⢦⠴⠶⠶⠷⠶⢖⡒⠀⠼⠬⢀⣀⣀⣀⣠⠟⢟⡄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢨⡟⠢⠄⠀⢉⠙⠑⢤⠀⠀⠀⠀⠀⠈⠣⡉⠉⠙⡏⡏⠁⠀⠀⠘⣧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣞⠀⠀⠀⠀⠘⡄⠀⠀⠱⡀⠀⠀⠀⠀⠀⠱⡀⠀⡇⢁⠀⠀⠤⠊⢳⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢼⣍⠀⠀⠀⠀⢆⠃⠀⠀⠀⢧⠀⠀⠐⢒⣒⡈⢇⠀⠁⢨⠀⠀⢀⠠⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⡸⠒⠒⠒⠒⢺⠈⠉⠁⠀⠀⠀⢘⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠪⡒⢄⡀⠀⢸⡗⠒⠂⠀⠀⠐⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⠑⠬⡒⢤⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⠀⠀⠀⠀⠀⠀⢀⠽⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠢⢕⠢⣀⠀⠀⢀⠎⠀⠿⠿⠿⠿⠋⠓⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠉⠒⠅⣲⠎⠀⠀⡇⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢎⠢⡀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⢧⠤⠀⠀⠤⠤⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠈⠀⠀⠑⠪⣖⢄⡀⢀⠞⠀⠀⠀⠀⠀⢘⠀⠈⠉⠉⠉⠁⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠴⢪⠊⠝⡲⢤⡀⠀⠀⡁⠞⠁⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⢃⢎⠗⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠠⠤⢐⡒⠊⠩⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡁⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⣠⠚⢍⢫⠫⡑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠁

⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡏⠁⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⢃⣹⡴⠇⣹⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢫⠛⢻⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡿⣇⣤⣰⢲⣿⢧⡛⣷⠈⠢⣀⠀⠀⠀⠀⢀⡴⠋⡧⡭⣷⣜⡜⣅⠻⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠙⠤⠒⣴⣯⡇⠀⡯⠹⡄⠀⠈⠓⠲⠶⠒⠋⠀⢠⡏⣭⣉⢻⠑⢧⠰⠆⢸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡟⢳⣦⣰⢿⣏⡇⣠⡟⢵⡿⣄⠀⠀⠀⠀⠀⣀⣴⣿⢷⢜⢯⢿⡇⡤⢍⣊⡉⣷⣏⣿⣿⣿⣿⣿⣷⡀
⠀⢠⣿⣿⣿⣿⣿⣿⡿⢻⡷⢋⣳⡏⠻⣿⣯⠚⢃⣌⡁⣝⢦⡀⢀⡴⢚⣡⡮⣋⠝⡬⢻⡼⡠⢌⠛⢷⠺⡿⣿⣏⡿⣿⣿⣿⣿⣧
⠀⣾⣿⣿⣿⣿⣿⢏⣤⣾⠴⠜⢸⢣⢿⣾⢻⡈⠂⠛⣧⠠⢧⣿⣾⣒⣛⣛⢦⡥⠴⢴⠿⠤⣌⣻⣀⣷⣜⢁⠇⢿⡇⣿⣿⣿⣿⣿⡆
⢀⣿⣿⣿⣿⡟⣙⢧⠗⡗⡐⠈⠹⡟⢸⣷⠋⡟⢺⢒⢃⠖⠀⢌⡏⢿⢽⡁⣘⠄⣍⣁⣜⠒⡢⡜⣀⠘⣊⣾⡗⣼⡷⢋⣿⣿⣿⣿⡇
⢸⣿⣿⣿⡟⢁⡭⣭⠽⢿⣙⠩⣄⣯⡺⠁⠀⢸⡇⣎⡙⡄⢟⠏⠛⡘⠈⢳⡏⠉⡆⣛⠃⡺⠇⣳⣩⣼⠋⣿⣽⡿⣾⢿⡿⣿⣿⣿⣿
⢸⣿⣿⣿⡗⢋⣴⠿⢄⡼⢳⡇⣎⣼⣷⡿⢏⡗⣱⡚⠧⢻⡯⢠⡕⢱⡆⣄⢿⣲⣬⣤⢵⡵⣞⣹⣷⢏⣤⣿⠟⣰⣿⡏⠀⠘⣿⣿⡟
⠈⣿⣿⠻⣗⠏⡎⡎⣉⠒⢡⢱⣍⢶⢙⣧⠛⣰⣅⠟⣮⡤⠖⢉⠒⢁⣤⢣⢸⠙⠀⡉⣛⡾⢾⣿⣿⣿⣿⠿⣛⡿⢿⡇⠀⠀⠘⣿⡇
⠀⢿⡏⠀⢿⡔⡡⢖⡸⣫⡳⡬⢌⡷⣍⣩⣎⠡⢁⠼⣒⣛⠲⣽⠖⣣⢿⢨⡶⠃⣎⢁⡿⣗⢺⠘⢆⡛⣋⡭⢋⣟⡾⠁⠀⠀⠀⢹⠃
⠀⠘⠁⠀⠈⢻⣄⣉⠣⣉⠴⣇⢫⢶⠁⡙⠺⣷⣤⣤⣅⠠⢇⡺⢿⠷⣯⠌⢀⠱⢚⣇⣣⠊⣉⠴⢉⢖⣲⣀⣠⢾
⠀⠀⠀⠀⠀⠀⠙⢾⣌⡡⢋⣒⣊⢜⣀⡰⡴⡡⣩⠟⣻⡷⣶⣛⣿⣻⠋⡿⡾⢿⣵⣂⣴⣁⣘⣃⣯⡿⠷⣫⡜⢺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣊⡤⢲⠃⣠⡇⢰⣥⣧⠏⣏⡴⣫⢵⣿⣮⢸⡰⣷⠿⠓⡩⣯⣔⠼⠏⡏⠶⠤⠂⠯⢺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣋⢡⡼⢃⣺⣿⣥⡿⢯⣴⣧⣿⣋⣻⣽⢊⡑⠠⡒⠂⣈⣀⡟⠊⠀⠤⡄⠒⡌⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠛⠛⠛⠻⣍⣟⠉⣛⣥⣄⣠⢿⠈⢠⢠⡑⢾⠒⢺⢃⠤⢲⣔⠳⣺⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣵⣛⡿⠋⢴⣷⠓⠆⠥⠙⡌⢦⢢⡾⢣⣘⢸⢋⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣨⣆⣠⡾⣀⡒⠑⢢⣱⠿⣥⢰⠢⡇⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣞⡡⢜⡃⠈⣰⠦⢤⠀⠰⡀⢢⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢾⡇⠀⣛⡚⠉⢴⣶⡵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣅⣚⣽⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣖⡶⠖⠒⠒⠒⠦⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⢩⢃⠔⠉⠉⠁⠐⠒⠤⠤⠤⠍⣙⣛⣶⣶⣦⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⢀⠔⠝⠻⣄⡀⠀⣀⣀⣤⠤⠶⢶⣶⡖⠲⢬⡉⠉⠁⠀⠀⠀⠈⠉⢿⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡞⠌⠀⠀⡟⠀⠀⠉⠲⣄⡀⠀⠀⠉⠫⡢⡀⠈⢦⣀⡠⠤⢀⠀⠀⢸⢀⠟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢹⢳⠦⡄⠀⠰⠀⠀⠀⠀⢻⠙⢷⣄⠀⠀⠘⣝⡄⢸⠀⠀⠙⠲⠚⠛⢺⢋⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢅⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⢷⡄⠀⠘⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣙⣿⣶⣄⣀⣀⣈⣷⣀⣀⣾⡄⠀⠀⠈⢿⡄⠀⢹⣿⠀⠀⠀⠀⠀⣀⣰⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢩⡟⠘⡜⡄⠀⠀⠈⣿⡀⠘⡏⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠐⢰⠀⠀⠀⠀⠃⢰⣁⡰⡐⠀⠀⢀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠆⠀⠀⢸⣸⣟⠙⠛⠷⠶⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢠⢀⠀⢀⡜⠁⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⢠⠀⠀⠀⠀⢠⣸⠔⠈⠀⢀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠙⠀⠀⠀⠀⠆⠄⠀⠀⣠⠞⠁⠀⠀⢠⠃⡄⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠠⠰⣀⡾⠉⢧⠀⠀⣠⠃⡜⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡾⡁⠀⠀⠀⠂⠀⠀⠀⠀⠀⡷⠋⠀⠀⠘⣇⡴⢁⠌⠀⠀⠀⡸⢸⣿
⠀⠀⠀⠀⠀⠀⢠⡟⠇⠠⠀⠀⠀⠈⠀⠡⠀⢀⡾⠁⠀⠀⠀⠀⢻⣤⠃⠀⠀⠀⡰⠁⡘⢻⡀
⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⡀⠀⠀⠀⠀⢡⡼⠁⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⡴⠁⡐⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢄⣸⠃⠀⠀⠀⠀⠀⠀⠀⢹⠀⢀⠜⢀⠎⠀⠀⣸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠁⡠⠁⠀⠀⡰⠁⣷
⠀⠀⠀⠀⠀⢸⡇⠑⠄⠀⠀⠀⠀⠈⢄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠊⠀⠀⢀⠜⢀⠜⢻
⠀⠀⠀⠀⠀⢸⡇⠑⠀⠢⡀⠀⠈⢄⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡠⠋⡠⠂⠀⢸
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠂⠀⠀⠲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡠⠊⢀⠔⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢻⡃⠀⠀⠈⠢⡐⢄⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠞⠁⠀⢀⡤⣶⡿
⠀⠀⠀⠀⠀⠀⠈⣿⠘⢔⡀⠀⠈⠢⡑⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠤⢒⣪⡵⡎⣿⢷
⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⠥⠾⢶⣤⣬⣽⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⡏⣿⠀⡇⣷⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠲⡟⠻⡏⢸⡉⣟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡧⠿⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢳⡀⣇⠈⡧⠹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠁⠉

⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠤⠤⠤⠀⠐⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠒⠶⢤⣄⡀
⠀⠀⠀⢀⠔⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠞⠋⠉⠉⠈⠀⠀⠙⠳⣄
⠀⠀⠀⡇⠀⠀⠒⠶⢶⣤⣤⣄⣀⣤⣄⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠈⠳⣄
⠀⠀⠀⢹⡦⠀⠀⠀⠀⠀⠉⠛⠲⠶⢿⣭⣍⣏⣽⣿⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢹⠄
⠀⠀⠀⢸⣿⣦⣄⣀⠀⠲⣀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⣀⣴⡶⣿
⠀⠀⠀⢸⡟⠀⠀⠉⠓⠶⢬⣷⣄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠈⠛⠉⢠⡟
⠀⠀⠀⢸⡧⠀⠐⠀⠀⠀⠀⠙⠻⣦⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⢷⡄
⠀⠀⠀⠀⢿⣐⡑⠢⣄⡀⠀⠀⠀⠈⠙⠧⠤⢄⣀⣀⣀⣤⡴⠒⣶⡖⠒⠶⠶⠶⣶⣾⣿⣿⠋⠀⣸⠇
⠀⠀⠀⠀⠀⠉⠙⠰⢌⣁⡀⠀⠀⠴⠿⠷⠶⠤⠀⠀⠀⠀⠀⢼⣿⣷⠀⡀⠀⠀⠀⢹⣿⣿⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠲⠤⠤⣤⣄⣀⣀⣤⣤⣤⣴⣿⣿⡇⠐⠀⠀⠀⠀⢿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⡇⠀⠀⠀⢀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠜⡄⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣇⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⢸⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⢻⡀⠀⠀⠀⢸⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠸⣇⠀⠀⠀⢸⡁⡆⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠀⢻⡄⠀⠀⠈⣿⣔⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⡆⣷⡀⠀⠀⠸⣿⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠀⠀⢹⣧⠀⠀⠀⢻⡆⠀⠀⠀⢄⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠿⣧⠀⠀⠈⠃⠀⠢⡀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣫⣇⣼⢳⠠⠛⢢⡙⣧⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠻⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢿⠏⠸⠶⠃⠀⠉⠙⣧⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠃⠀⠀⠀⠀⠀⠀⠀⢰⠟⠲⣌⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠀⠀⠀⠀⠑⢄⣧⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢠⠀⠘⠄⡀

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⢦⣤⣀⣀⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⡰⢫⠞⢀⡾⣏⠱⡄⠀⠀⠀⠈⢷⠦⢤⢤⠶⠶⠶⢶⠲⡞⠛⠋⠉⢹⡸⡄⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⡼⢡⠏⢠⡟⠀⠈⠳⡙⢦⡀⠀⠀⣸⠀⣾⢸⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢧⢹⡀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⢠⡇⠀⣰⢣⠏⣠⡿⠀⠀⠀⠀⠈⠢⣝⠢⡴⠃⠀⡇⢸⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢸⠈⡇⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠓⠷⠾⣴⢻⢧⡀⠀⠀⠀⠀⠀⢨⡟⠁⠀⠀⡇⡾⠀⠀⠀⣸⣀⣇⣀⣀⣠⣤⣼⡤⠧⠤⠤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡟⠢⡱⣄⠀⠀⠀⣰⠋⠉⠉⠉⠛⠛⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡣⣼⠃⠀⠙⠢⣝⠢⣤⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠉⡿⠀⠀⠀⠀⠀⠉⡟
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢀⡧⡀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⢀⡾⡑⠦⣀⢸⠳⣝⠦⣄⠀⠀⡟
⠀⠀⠀⠀⢀⡞⠀⠈⠓⠮⡟⠀⠈⠓⠬⣙⣺⠃
⠀⠀⠀⠀⡼⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⡿
⠀⠀⠀⢰⠓⣄⠀⠀⠀⢸⠳⣄⠀⠀⠀⢠⡇
⠀⠀⠀⡟⠑⢤⡑⠦⣄⡏⠑⢮⣙⠲⢤⣼
⠀⠀⢠⡇⠀⠀⠉⠓⢺⠃⠀⠀⠈⠙⠲⡏
⠀⠀⢸⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⢸⠃
⠀⠀⣼⠢⣄⠀⠀⢰⠷⢌⡒⠤⢄⣀⡼
⠀⠀⡟⠓⠬⣙⡲⡞⠀⠀⠉⠓⠲⢦⡇
⠀⠀⡇⠀⠀⠀⣹⠁⠀⠀⠀⠀⠀⢸⠁
⠀⢠⠷⣄⣀⢠⠇⠀⠀⠀⠀⠀⠀⡏
⢰⢁⣤⠀⣭⡿⠒⠒⠒⠒⠒⠒⠺⡇
⠈⠉⠈⣹⠋⠀⣶⢀⡞⢇⣶⡀⠀⢣
⠀⠀⠀⠉⠙⠚⠀⠉⠀⠈⠁⠑⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠾⠿⠇⠠⠦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡏⠉⡀⠖⠓⠃⠐⠂⢬⣟⢏⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢹⣧⠈⢀⡼⠶⠀⠤⠀⣰⡽⠛⣿⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡏⠀⡜⢿⠬⢤⡆⣶⣦⢞⡋⠐⡠⣿⣴⣌⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣀⠠⣝⢦⠀⢈⡁⠀⠐⢋⣥⠞⠀⣿⠋⠋⢙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⢠⣹⡗⠪⣷⠀⠀⡀⠀⣠⣿⠁⢠⢀⡏⠈⠀⢀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⢄⣻⠄⠀⠀⣑⠢⠔⠀⢛⣡⠀⣰⠟⣲⢀⢀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⣹⠓⠴⢄⣠⠀⠀⠀⡺⠖⠩⠦⡀⠁⠀⠼⢀⣆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡇⠈⠀⠐⠀⡨⣄⡁⠁⠀⢠⡄⠀⠈⠀⢠⠀⠈⠀⠉⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠃⢠⠀⠀⠠⡈⠁⢠⠴⡗⠠⠐⠋⢺⠀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠊⠀⢠⠈⢢⠴⠛⠁⠀⠑⠼⠐⠀⠈⠀⢸⠀⢠⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠻⢀⡴⠁⠀⠀⢀⠀⠄⣿⠰⡀⠀⠀⢀⠀⠸⠂⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⠛⠀⠀⠀⠀⠀⠀⠘⢷⣀⠀⠓⢤⡀⠀⠠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠦⠀⠀⠀⠀⠀⠀⠞⠀⠑⢇⢀⠘⠂⡄⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠙⠶⢄⡄⠛⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⡄⠀⠂⠀⠉⢻⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢈⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢀⣠⠶⢚⣉⡭⠭⠭⠥⢤⣭⣙⡲⢄⡀
⠀⠀⣠⠜⡡⢞⣽⡾⠭⠭⠭⣭⣭⣝⢲⡌⢇⢿⢦⡀
⠀⢰⠇⡄⣰⡟⡵⣪⡭⠽⠶⠾⣭⣭⠵⠷⠾⣞⠘⣷
⠀⢸⡸⣇⠹⣱⠋⠁⠀⠀⢰⠟⠉⠢⡄⠐⢄⠈⠻⡿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄
⠀⢸⠁⢍⡛⢿⣄⣀⣤⣀⣾⣄⠠⡀⠈⢆⠀⢠⠀⣿⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡗⠁⣀⣀
⠀⠘⣦⠐⠩⢒⡀⠤⠤⠄⠙⢮⡢⡈⢢⠈⢢⠀⣱⠃⡐⠘⢧⠀⠀⠀⠀⠀⠀⠀⣠⠜⠼⠥⡾⠁⠉
⠀⠀⠳⣦⣁⠂⠨⢥⣀⣉⡀⠈⠳⣔⢄⠑⠀⡰⢃⠄⢀⠊⢸⣀⡀⠀⠀⣠⡴⢋⠕⢀⠀⠘⠒⡶⠾⠙⠏
⠀⠀⠀⠀⠉⠓⠶⠥⢯⣭⣤⣬⠴⠛⣆⣣⠞⡱⠃⡰⣡⠂⣨⠏⣙⠛⠛⣉⠔⡡⠊⢁⡠⠐⣀⠝⣛⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⢊⠔⠉⡰⢁⡼⠁⠠⠤⣀⣀⠤⢊⡤⠊⠥⠒⣩⡴⠋⠁⠀⠑⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢱⡔⠡⢁⠔⣊⣴⣟⠡⣀⠒⠠⠤⠤⢖⡡⠂⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⣠⢣⣷⠞⠁⠀⠈⠙⠲⢮⣥⣔⣊⣭⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢧⡟⡼⣿⠀⠀⠀⠀⠀⠀⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠦⣝⠲⠿⣦⣀⣀⣀⣠⣾⢋⡴⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠪⣙⠓⠒⠒⠒⠈⠁⠙⣟⢀⡤⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡠⡉⠒⠂⠠⠔⠒⠀⠉⣯⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣌⠲⠤⠤⠔⢚⣡⠶⠛⠉⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⡱⢛⠻⠙⠓⠒⠢⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢮⣤⣼⢁⡏⢀⠆⠀⠀⠀⢰⠁⡌⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⡆⢘⠈⡘⠀⠀⠀⠀⡎⢰⠁⠀⠀⢀⠛⣧⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⠟⢛⡸⠄⡛⢠⠁⠀⠀⠀⢸⠁⠀⠀⠀⠀⠎⡰⠸⣇⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠸⢀⢃⠆⠀⠀⠀⠀⡏⢠⠁⠀⠀⠸⠀⡐⡁⠸⡀⠀⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⡀⠀⠀⠀⠀⡿⠼⠶⠤⠤⣀⣰⠁⠆⠀⠀⠠⠁⠀⠀⠀⠀⡧⠊⢈⠵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠌⠁⠀⠈⠉⠃⣇⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⡳⠠⢀⠃⠀⣰⡯⠐⠁⠀⢙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠉⠉⠀⠒⠒⠒⠾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⣰⠃⡇⣀⠤⢂⡡⣻⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠅⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⣧⢄⠢⠓⠈⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⡾⠉⠀⠐⠒⠂⠭⠤⠭⠭⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⡠⠐⢧⢣
⠀⠀⠀⠀⠀⠀⢠⠏⠉⠒⠢⠤⠀⠤⢀⣀⣀⣀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠔⠚⢉⠠⠒⠉⠀⣧
⠀⠀⠀⠀⠀⠀⡸⢒⡢⠄⢀⡀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⠒⠉⠀⢀⡠⠔⠁⡸⡄
⠀⠀⠀⠀⠀⢀⡛⠒⠒⠨⠅⢉⣙⠓⠒⠶⠶⣶⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠤⠒⢁⡤⠒⠉⢀⣱
⠀⠀⠀⠀⠀⢸⢉⡓⠚⠒⠒⠢⠤⠄⣈⣉⣈⣉⠁⢙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠈⣀⠤⠊⠁⢀⠠⠒⣡⠼⡇
⠀⠀⠀⠀⠀⡇⠀⠈⠁⠒⠊⠅⠀⡀⢁⣐⣒⣒⣲⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠊⠁⡀⠔⡈⠄⠒⢁⠤⣂⠹⡀
⠀⠀⠀⠀⢠⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢂⠁⠒⢈⠠⢑⠨⠔⠈⠀⠀⢱⡀
⠀⠀⠀⠀⣎⡈⠉⠙⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠌⡢⠕⠊⠁⠀⠀⠀⣠⠔⢋⠵⡀
⠀⠀⠀⣼⠟⠉⣹⠶⣤⣀⡀⠉⠉⠑⠺⠆⣀⣀⣐⠶⠾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠉⠀⠀⠀⢀⠤⠒⠋⡠⠔⠁⠀⠙⡄
⠀⢀⠎⢙⢀⢶⠏⣰⣿⠃⠙⣗⡦⢤⣀⣀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣀⡤⠔⢊⣁⠠⠔⠈⠀⠀⡠⠚⣉⠽⣆
⠀⠀⠀⢂⠎⠀⠔⠋⡐⢠⢾⣿⠁⣞⡊⢰⡏⢸⡍⣿⢛⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠤⠔⠊⠁⠀⠀⠀⢀⠄⣫⠔⠉⠀⣠⡤⢷⣄
⠀⠀⠀⠈⠀⠜⠀⠠⠁⠁⠸⡏⣘⡿⢠⡎⠀⣸⢇⡟⢰⢰⠼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⠠⡴⢊⠡⠋⠀⣠⢴⡙⣆⠛⣤⡘⢷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠹⢰⠁⡇⠈⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡩⡕⣒⠌⠂⠁⠀⢀⣜⠁⢸⡃⠹⣦⡌⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣉⣀⣀⣀⣠⢶⡋⢸⡇⡈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⢉⢿⢌⢷⠄⢻⡄⢷⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠺⡎⡎⢆⠎⠢⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⠶⠶⠶⠶⠶⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠿⠿⠶⠶⠶⢶⣤⣤⣄⣀
⠀⠀⠀⢀⣠⣴⡶⠿⠟⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠷⣦⣄⡀
⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⡤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠴⠶⠖⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⢸⡇⢸⡆⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣿⣿⣿⡿⠀⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⣿⡅⢀⠻⣄⠀⠀⠀⠀⠀⠈⠛⠟⣭⣻⣟⠟⠠⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣿⡇⠀⠀⠹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⠔⣤⡦⠤⠤⠴⡇⣾⠀⢀⣴⡞⢁⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣀
⣿⣷⣦⡄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠣⠾⠤⠾⠵⠛⠻⣧⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⢀⣼⣿⡿⣿
⠹⣿⡼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣎⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⣠⣾⣿⡿⠃⢸
⢀⣾⣷⣿⣿⣿⣶⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢀⣴⣿⣿⠀⠀⠀⢸
⢸⣧⠀⠘⠻⢣⠈⠻⢿⡦⢈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⣶⣿⡿⢿⠋⠀⠀⢀⣾
⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⣄⠀⠀⠂⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⣾⣿⣿⠿⠳⠿⠛⠛⡋⠉⠀⠘⡄⠀⠰⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠱⠀⠀⠀⢡⡄⠀⢷⠀⠀⠀⠘⡇⠀⠀⠀⠀⡴⣣⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰⣷⡆⠘⡆⠀⠀⠀⣿⠀⠀⠀⠀⠁⣿⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠿⠋⠸⣧⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⣿⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⣦⣤⣤⣤⣤⣤⣴⣶⣾⣿⣿⡟⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣹⣿⡟⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠐⠇⠀⠀⠀⡇⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠇⢰⡇⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⢀⠞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⡴⠃⠠⠄⠀⠀⠀⠀⢠⡟⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⢀⡟⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⣰⠀⡜⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣰⠃⣴⠁⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⡏⢠⡏⠀⠀⠀⠸⣿⣀⣀⣀⣀⣀⣀⡤⠤⠤⠖⠒⠏⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢱⡏⠀⠀⠀⠀⠀⠀⡞⢀⠤⠂⠀⠀⠜⠁⣾⠀⠀⠀⠀⠀⠿⠉⡍⣽⢁⡀⠀⣀⡀⢠⡀⠀⢸⣯⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡏⠰⠆⠀⠀⠀⠀⠀⠋⣠⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⡄⠀⢰⠁⡿⢸⡇⠀⠏⣷⠸⡇⠺⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⠱⠀⠘⡄⠀⠀⠀⠀⠉⠀⠀⠂⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣏⠇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⡿⠀⠀⠀⠀⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢸⣿⡃⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠀⠀⠀⠀⠰⡀⢳⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⠀⠀⠀⠀⠀⠹⡆⠳⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢸⠛⣶⢤⣀⡀⠀⠉⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣾⠀⣿⠀⣯⠈⡛⢳⡶⣦⣤⣤⣄⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠛⠀⠻⠀⣇⢸⡇⢸⡇⠸⣿⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢿⡇⠀⠿⠀⢻⣇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠉⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣾⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠄⠄⡀
⠀⠀⠀⠀⠀⢀⠠⠐⡆⠈⠋⠀⠉⠅⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢸⠀⠀⠃⠀⡏⠂
⠀⠀⢀⠠⠊⠈⠄⠀⢰⠀⠀⠀⠀⠠⠀⠁⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢇⠀⠀⢂⠀⠠⠀⠁⠘
⢀⣔⣇⠀⢁⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⢰⠀⡏⡄⠀⠀⠀⠀⠀⠀⡠⠊⡀⠈⢂⠀⠀⠡⠀⠡⠈⢄⠱⡀
⠘⣿⣻⠀⠸⠀⠘⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠑⠠⡀⢠⠄⡎⠡⠀⠑⡀⠀⠑⡀⠀⢁⡠⠇⠲⣦⣼
⠀⣿⣿⠀⠁⠀⠸⡀⠀⢂⠀⢀⣀⣀⠀⢠⠀⠀⠀⠀⠀⡀⠀⡄⠰⡀⠡⡀⠈⢂⠀⢠⠔⠁⠀⠀⠀⠑⠋
⠀⣿⣿⠀⠐⢄⡀⠔⠒⠈⠁⠀⠀⠀⠉⠣⡠⠁⠸⠀⠀⠀⠀⢠⠀⠑⡀⠐⠀⠀⡖⠁
⠀⠛⠛⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠃⢀⠀⠐⠀⠀⡀⠀⠠⠀⢈⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠮⡀⢨⠀⠀⠃⠀⡠⠖⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣿⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢶⣧⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⣾⡴⢀⡀
⠀⠀⠀⠀⠀⢿⣾⣄⡽⣅⣼⣞⠇⠀⠀⠀⠀⠀⢸⡿⣤⣀⣇⣠⠿⣟⠀⠀⠀⠀⠀⠀⣿⠇⣄⡶⣄⡜⢿⡃
⠀⠀⠀⠀⠀⣰⣤⠾⣤⠼⣤⣤⡀⠀⠀⠀⠀⠀⠠⣤⡤⠿⡿⢧⣤⡤⠀⠀⠀⠀⠀⠀⣤⡠⠾⣤⠼⠄⣠⠄
⠀⠀⠀⠀⠀⠛⠙⠠⣾⠦⠘⠙⠃⠀⠀⠀⠀⠀⠘⠹⢀⣤⣧⡀⠻⠙⠀⠀⠀⠀⠀⠀⠋⠇⣠⣬⣄⠸⠏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠠⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⢀⣾⣤⠘⢿⠋⣠⣼⡄⠀⠀⠀⠀⠀⢠⣼⡀⢙⡏⠁⣶⡤⠀⠀⠀⠀⠀⠠⣤⡇⠈⣩⠉⠰⣦⠄
⠀⠀⠀⠀⠈⢐⡋⣻⣾⣞⡋⣓⠉⠀⠀⠀⠀⠀⢘⠉⢙⡏⢙⣟⠉⣙⠀⠀⠀⠀⠀⠀⡋⠈⣳⣿⣞⡃⢉⡃
⠀⠀⠀⠀⠘⢾⠻⢁⣼⣈⠹⢷⠏⠀⠀⠀⠀⠀⠸⢿⠋⢉⡏⠉⣿⠯⠀⠀⠀⠀⠀⠀⠿⡎⠁⠻⠈⢱⡿⠅
⠀⠀⠀⠀⠀⠈⠀⠨⠿⠯⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣀⡦⠙⢺⠚⢱⣄⡀⠀⠀⠀⠀⠀⢀⣄⠀⢸⡟⠀⣀⣄⠀⠀⠀⠀⠀⠀⣠⡆⠙⢻⠛⠱⣤⡀
⠀⠀⠀⠀⠀⠞⠛⢶⠚⢶⠞⠻⠆⠀⠀⠀⠀⠀⠈⠛⠳⢤⣧⠶⠛⠉⠀⠀⠀⠀⠀⠀⠟⠓⢶⠛⢶⠚⠹⠇
⠀⠀⠀⠀⠀⣻⡶⠛⢲⠛⢳⣾⡁⠀⠀⠀⠀⠀⢠⡴⠟⢩⡏⠛⢶⣤⠀⠀⠀⠀⠀⠀⣷⡖⠛⢲⠛⠲⣼⡇
⠀⠀⠀⠀⠀⠀⠋⠶⢺⠲⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠶⣾⠶⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⡆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡌⢸⡷⢁⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠄⢸⡧⠠⠶⠃
⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⠀⠀⡀⢀⠀⠀⢀⠀⣦⠀⠀⣴⢸⡇⣦⠀⠀⣴⠀⠀⠀⠀⡀⢀⡀⠀⡀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣐⠶⣌⡄⠃⢸⠇⠀⠈⢻⡙⣦⣄⠻⡿⢿⠿⣡⣴⢋⡞⠁⠀⠸⡇⠸⢁⣃⡶⢎⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣈⠚⢿⣦⣔⡷⣦⠘⣟⣿⣿⡏⣴⣦⢹⣿⣿⣿⡃⣴⢾⣦⣴⡾⠛⣋⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⡴⠂⣸⣝⣿⡟⢚⣉⠣⡁⢠⢻⡟⡄⢈⠜⣉⡓⢻⣿⣫⣅⠐⢦⡌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣉⣩⣷⣘⠿⢿⣮⠿⠟⠻⢿⣵⡿⠿⢃⣾⣍⣋⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⢮⣽⣿⣿⡃⣈⡵⣿⠀⠰⠖⠀⣿⢞⡉⢘⣿⣾⣟⣱⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣩⣙⡿⢡⣶⣾⢟⣶⣦⣴⣶⡻⣷⣦⡍⢿⣋⣉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⠳⠄⢙⣫⣿⣧⢬⣉⡔⡁⠘⣼⣧⠃⢈⢢⣉⡥⣼⣿⣝⡋⠠⠞⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣩⣤⡾⠟⠹⡷⠟⢠⣯⣿⣿⣇⠻⠟⣸⣿⣿⣽⡄⠿⢾⠝⢿⣷⣤⣭⡥⢀
⠀⠀⠀⠀⠀⠀⠀⣬⡱⠞⢩⢁⡄⢸⡆⠀⢀⡼⣡⠟⢋⣴⣷⣾⣦⠙⠻⣼⣇⡀⠀⠰⡇⢘⠙⡟⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠿⠀⠀⠻⢸⡯⠟⠀⠀⠻⠉⠉⠀⠀⠁⠈⠁⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠂⢺⡗⠐⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢃⢾⡇⡘⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⠇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣄⡄⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⡄⣬⣯⣄⢀⣤⠆⠀⠀⢠⣾⣄⢹⡏⣰⣷⠄⠀⠀⢶⣦⡸⣮⣯⡾⣰⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣏⠉⣿⡯⠁⠀⠀⠀⣭⣉⣹⣿⣿⣍⣉⡍⠀⠀⠀⠴⢿⣿⣿⣿⠷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⠏⠟⣟⠟⠙⠷⡄⠀⠀⠘⢻⢃⣼⣇⠸⡟⠂⠀⠀⣼⠿⢩⡟⡿⣯⠹⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⡀⠀⠀⠀⠸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢦⣏⣻⣘⣤⠿⠋⠀⠀⠙⠳⢤⣄⣗⣤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣣⡼⢿⠳⣼⣃⠀⠀⠀⠀⢀⡠⠞⡿⠾⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠓⣿⡙⠀⠛⠓⠀⠀⠚⠉⠀⠀⣧⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⢶⡆⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢤⣸⣄⠴⠿⠃
⠀⠤⠤⠤⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠖⢻⠛⠢⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢼⡇⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠯⢬⡙⢞⠉⠙⠋⢩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠀⠀⠀⡈⢆⠳⣠⠔⠁⠀⠀⠀⠀⠀⠀⠠⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠀⠀⠀⠀⢠⠟⠉⢢⠱⡀⢰⣄⠀⠀⠀⡠⠊⢠⠃
⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⢀⠴⡻⡀⠀⠀⠣⡑⢄⢢⠙⠦⡔⠁⢠⠃
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠠⣄⡤⠊⠁⠀⢇⢣⠀⠀⠀⠙⢮⣓⡵⠼⠤⠔⢛⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠘⡈⡄⠀⠀⠀⠀⡨⢶⡀⣰⡊⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⢰⠀⣀⠠⠐⠒⠒⡇⣇⠀⠀⠀⣠⠋⠳⡝⡻⣼⢆⠀⠑⢄⣀⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⡞⠋⠀⠀⠀⠀⠀⡇⡏⠙⠢⡾⠢⣱⣼⡶⡷⠃⠀⠑⢄⠀⠑⣀⠀⠁⠑⠤⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠢
⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⣀⠀⠀⣤⠧⠄⡲⠥⢤⣏⣴⡖⠁⠀⠀⠀⠀⠱⣄⠈⠢⡀⠀⠀⢀⡽⠖⠢⠤⣸⡀
⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⡡⠋⠉⠀⠠⡊⠀⠀⢠⣃⠾⡀⠀⠀⠀⠀⠀⠀⡼⠒⠲⣮⢦⡞⠁⠀⠀⠀⠀⠀⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠈⠱⡴⢩⠏⠀⣱⡀⠀⠀⠀⠀⠀⣇⠀⠀⠈⣣⠈⠒⠤⣀⠀⢀⡠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⢀⠜⡾⠧⡀⠀⠸⡱⡀⠀⠀⠀⠀⠘⣄⣠⠊⠀⠀⠀⠀⠀⠻⠉⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⣰⣧⠞⠀⠀⠈⠢⣄⣹⡧⠀⠀⠀⠀⠀⡎⢳⠀⠀⠀⠀⠀⠀⠀⠹⡓⠒⠪⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠁⠀⠀⠀⠀⠀⠀⠀⣈⠜⠉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣧⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠟⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⣦⡀⣿⢀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠈⠻⣿⠟⠁⠀⢠⡄⠀⠀⠸⣧⠀⢠⡶⢶⣦
⠀⠀⠀⠀⠀⠀⣾⠛⢻⣆⠀⢻⡇⠀⠘⢁⡀⠀⠀⣿⠀⠀⠠⡄⠀⠀⠀⢰⣟⣤⡿⢧⡶⠟
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠙⠷⣾⣧⡀⠀⠈⢡⡄⠀⣿⠀⢠⡦⠀⠀⣀⣴⠿⠿⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⠛⠛⠉⠻⢶⣤⣈⢻⣆⣿⣠⣿⣡⡶⠟⠋⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣈⣙⣿⣿⣿⣿⣿⣭⣤⣤⡄⠠⡄⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠀⠉⣉⣭⠿⣿⣿⣿⡻⢦⣄⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⢶⣤⣤⣤⣴⠞⠋⠁⣴⠏⣿⡍⢷⡄⠈⠛⠷⣦⣤⣤⡴⠾⠋
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡾⢻⡟⠀⠀⠀⡘⠋⠀⣿⡇⠈⢋⡀⠀⠀⠀⢹⡿⢶⣤⣠⣤⣄
⠀⠀⠀⠀⠀⠀⣿⣁⣼⠇⠀⣾⠃⠀⠀⣀⠉⠀⠀⣿⡇⠀⠈⢁⡄⠀⠀⢸⣷⠀⠈⢿⣤⡿
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠙⠀⠀⠀⠈⠀⢀⣼⢿⣿⣆⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⢸⡇⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡾⠃

⠀⠀⢀⡶⠚⠋⠙⠛⠃⠂⠀⠀⠉⠉⠀⠉⠁⠈⠀⠘⠛⠛⠛⠉⠛⠛⠃⠀⠀⠈⠉⠀⠀⠉⠀⠀⠐⠊⠛⠛⠋⠙⠛⠙⠂⠀⠀⠈⠉⠁⠉⠉⠁⠀⠀⠒⢦⡄
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠐⠠⠀⠀⡀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢀⠀⠁⠂⠶⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠖⠈⠁⢀⠀⠘⠀⠠⢄⡀⠈⠉⠓⠢⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠊⠁⢀⣀⠄⠊⠁⠀⠀⠀⠀⠀⠈⠁⠒⠤⣀⡀⠀⠉⠓⠢⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣠⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠚⠖⢠⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠰⡀⠠⢀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⠀⢀⠀⢀⢀⠀⠀⠈⠄⣠⡀⢌⣠⣀⣌⣠⣡⠄⢄⡠⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠁⠂⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢠⡎⠀⠀⠀⣴⡀⠀⠀⠀⢀⡴⠀⠀⣠⠎⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠀⠀⠀⠀⠋⠀⠀⠀⠀⠋⠀⠠⠀⠂⠀⠀⠀⠁⠀⠀⠄⠊⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠘
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢠
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⠿⠟⠁⠀⠀⠀⠀⢸
⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠙⢦⡄⢀⣀⡀⠀⠀⠀⠀⡀⣀⠀⣀⣁⣈⣀⣀⣀⣀⣀⡀⣀⠀⢀⠀⠀⠀⢀⣀⢀⣀⣀⣀⢀⣀⣀⣀⡀⣀⡀⠀⠀⠀⠀⢀⢀⣀⣀⢀⣀⣀⣀⢀⡴⠋
⠀⠀⢀⠀⠀⡀⠀⡀⠈⠉⠀⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⠉⠁⠀⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠀⠉⠁⠀⢀⠀⠈⠉⠉⢉
⣿⡟⣉⣻⠿⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡇
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⡴
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠠⠀⠂⠀⠀⢀⡦⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣷⠊⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢭⠀⢲⣄⣀⠀⠀⠀⠤⠀⠀⡄⢀⠀⠀⠀⣀⣠⠄⠀⢩⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠆⠀⠀⢀⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢣⣄⡙⠃⡈⢁⡀⠀⠀⠷⠀⠀⠁⠀⢉⣤⡞⠛⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠈⠂⠄⠀⠀⠈⡀⡀⠈⠁⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢁⠐⠀⠈⠁⠠⠀⠀⠀⢀⡄⢃⡀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⢂⠀⡀⠂⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠒⠂⠀⣠⠴⠂⠀⠐⠀⠀⠀⠙⠒⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⠄⠀⣀⠀⠂⢈⠀⠋⠴⠤⢈⠁⢠⣤⡀
⠀⠚⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡅⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⢃⣀⣾⠆⠠⠀⠀⠙⠀⡆⣀⡤⠂⠐⢶⣤⡙⠃⢀
⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⢀⡀⠀⠁⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠝⠃⠈⠁⠂⠀⠀⠀⠀⣠⠀⣇⠈⠀⠀⠀⠀⠀⠉⠒⠪⡜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢹⡀⣟⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣍⡇



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣀⠀⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠥⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡐⠀⣀⣽⡒⠛⣁⠀⠂⠀⠀⠀⠀⠀⠀⢴
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠈⢁⠀⢵⡆⠈⠀⠀⠀⠉⠉⣶⠎⠈⠁⠀⠀⣀
⠀⠀⠀⠠⡍⢦⣤⠀⠠⠀⠀⠀⢠⡘⣤⠀⠀⠀⠀⠀⢄⣀⣉⠶⠉⠀⠀⠀⠀⠀⠀⠀⠶⠀⠸⠶⣍⡚⠃⠋⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠈⠰⢟⣋⠣⠞⠀⠤
⠀⠀⠀⠀⠀⠀⠉⠐⠂⢴⣆⠀⢀⡀⠁⠀⢠⡶⠀⠴⠈⠫⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣟⠀⢀⠀⠀⠀⠰⣿⠀⠀⣴⡤⠞⠃⠁⠀⣄⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⡉⠛⠀⠘⠉⠀⠚⠓⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡀⠀⠛⣶⣤⣀⡹⠆⠞⠉⠀⠀⠐⠀⠀⣈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡤⠈⣁⠀⢀⡙⠂⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡍⠁⠀⠀⠄⠉⠉⣡⠠⣌⠛⠛⢷⣉⠀⠈⠛⠋⠃
⠀⠀⠀⠀⠀⠠⡀⠡⠜⠫⠄⠀⠈⠀⠈⠀⠐⡶⠥⡄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠛⠀⠀⢀⣠⡴⠾⠃⠀⣿⡆⠀⠀⠉⠀⠄⣸⠁
⠀⠀⠀⠠⠖⡞⠛⠀⠀⠀⠀⠀⠼⣇⠶⠀⠀⠀⠀⠀⡜⠷⢐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⠦⢉⣡⢆⡈⠀⠀⠀⠀⠄⠀⡿⠀⠀⠀⠀⢠⢀⣌⡙⠤⣄⠀⠠
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠀⠀⠉⢀⠾⡀⠀⠀⠀⠇⠀⠀⣶⠄⠋⠀⠉⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠈⠠⠈⠉⣤⠀⣇⠚⠁⡀⠂⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⢄
⠀⠀⢀⡀⠀⢀⣀⠀⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢱⡆⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⣿⠀⣾⡹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣄⠀⠈⣿⡇⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⣠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠛⠦⣄⣿⣷⠀⣿⠤⠃⠀⠀⠀⠀⡞⠓⡄⠀⠀⠀⠈⡷⠞⣀⣠⣤⠾⠓⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣴⠃⠀⠀⠀⠀⠀⠀⢸⡞⠁⠀⢀⣤⣾⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⣠⢤⡀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣠⣴⣾⡿⢿⡋⣉⣀
⣞⣻⣦⣤⣤⣤⣄⣀⣀⠙⠳⡀⠀⠀⠱⣤⣇⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⣠⣾⡿⠿⠋⠁⠀⠀⠀⠻⠼⠂
⠈⠉⠀⠀⠉⠙⡛⢻⣿⣿⣿⣿⣶⣶⣶⣦⣬⣷⣄⣀⣀⠀⠀⢀⡿⣤⣀⣀⡴⠿⠋⠙⠒⠲⠟⣛⡆
⠀⠀⠀⠀⠀⢨⣩⠏⠀⠀⠀⠀⠀⠉⣹⠛⠛⠛⠛⠉⠉⠙⢲⣿⡀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠓⡞⠁⠀⠀⠀⠀⠀⠀⠀⠈⡿⠿⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠃⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠙⢷⣤⣀⠀⠀⠀⠀⢠⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⢤⣾⡏⠀⠀⠀⠀⠀⠀⣟⢿⣷⣶⠒⠛⠳⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⠿⢁⣾⣿⠁⠀⠀⠀⠀⠀⠀⡽⣄⠙⠿⣳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠙⣆⠀⠀⠀⠀⠀⠸⣦⠟⠀⠀⢹⢿⣿⣦⣀⣀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⡤⢠⣾⡿⠁⠀⡼⠒⣤⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⠙⢿⣿⡙⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢠⣾⣿⠃⠀⠀⠉⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠷⠟⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠐⢇⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣧⣾⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠀⠠⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠑⠄⠀⠀⣀⠨⠓⠤⡀
⠀⠀⠀⠐⢄⠈⠢⡀⠀⠀⠰⡈⣖⡉⠔⢂⡠⠒⠉⠢⡀
⠀⠀⠀⠀⠀⠈⠉⠑⡀⠀⠀⢡⠘⢢⣔⣁⣀⣀⣠⠀⠱⡀
⠀⠀⠀⠀⢀⠀⢀⠀⠱⠀⠀⠀⠀⡀⢢⠠⠤⠤⠀⠒⠂⢳
⢐⡀⢢⢄⡂⠌⣐⠊⠩⢧⠀⠀⡄⡇⠈⢧⠤⠤⠄⠀⠀⠈
⢨⠀⠌⠄⠈⡱⢲⠬⣥⣒⣆⠀⠀⠹⠀⠘⡍⠉⠁⠀⠀⢠
⠈⡶⠀⠀⢀⢃⠎⡰⠃⠀⠸⠀⢆⢸⠀⠀⡏⠀⠀⠀⢐⡞
⠀⠐⡄⠀⠀⡜⢠⠁⠀⡀⠀⡆⠈⢆⠣⠀⡇⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠘⢄⠀⠀⢆⢂⠌⠀⠀⡄⠀⠈⠁⠀⡂⠀⡠⠓⠠⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠆⠀⢀⡇
⠀⠀⠀⠀⠑⠢⢏⣈⠀⠀⠀⡆⠀⠀⠀⡠⠔⠋⠀⠀⠀⠙⢄⠑⠄⡀⠀⢀⡠⠊⠁⠀⠀⠀⠈⢀⠠⠔⢂⡡⠄⠂⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠤⠤⠒⠁⠀⠀⡀⠤⣒⡉⠄⠒⠉⠀⢀⣀⣀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤⠤⠀⠀⠒⠒⠒⠒⠂⠚⠉⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⡀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⠭⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢶⠶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⢤⡴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠑⢽⣤⡀⠀⠀⠀⠀⠀⠀⣠⣶⠽⠋⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠓⢝⢦⡀⠀⢀⡤⡺⠛⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢱⣽⣿⡵⠋⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⢔⡫⠚⠛⢮⡢⡀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣠⣾⠗⠉⠀⠀⠀⠀⠙⢮⣢⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣯⣄⡉⠁⠀⠀⠀⠀⠀⠀⠙⠪⡲⣄⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠨⠉⠙⠓⠶⢤⣄⣀⠀⠀⠀⠈⠪⡳⣄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣄⣈⠪⡳⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠾⣇⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣷⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠜⡿⡟⢙⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣶⣤⣶⣵⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠈⠚
⠀⠀⠀⠀⠀⠀⢰⡿⣻⣯⣵⣿⣿⡏⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⣠⣃⣤
⠀⠀⠀⠀⠀⠀⠸⣿⠻⣿⡿⣿⣿⢃⣾⣻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠏⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⡾⠿⠿⠿⠊⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⣶⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠤⠴⠃⠈⠓⢢⠄
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢁⣠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡞⠛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡟⡗⠚⡇⠀⠀⠀⠀⣠⡶⠶⠂⠐⣧⢒⡶⠃
⠀⠀⠀⠀⠀⠀⠀⢄⣏⠸⣇⣇⣀⣙⣲⡄⢸⣾⢟⣶⣄⣀⠚⢉⣀⣀⣀⣀⣀
⠀⣁⡀⠤⠒⠚⠃⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠒⠒⠀⠸⠖⠠⢀
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣿⣿⣿⣿⣿⣿⣷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⡿⠿⠿⠟⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⣤⣤⣤⣶⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⡻⢿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⠉⠉⠱⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠰⠊⠀⢸⠀⠀⠀⢀⠏⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡾⣴⡠⡆⣼⣰⣠⢀⡶⢰⡆⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⠞⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢠⡇⠁⡃⠻⠃⡁⠁⠀⡄⠈⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣯⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⡄⡇⣄⣼⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠧
⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⣿⡇⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⣿⡇⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠻⣷⣿⣧⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠙⣿⠟⠁⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣶⣄⡀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⣶⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⡘⣿⡀⠀⣶⠀⠀⠀⠀⠀⠀⢰⣄⠘⣿⢁⣠⣤⠀⠀⠀⠀⠀⠀⣰⡗⠀⣼⡟⠀⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣭⣿⣿⣷⡀⣿⡇⢀⣤⠀⢀⣀⡀⠙⢿⣿⠿⠋⢀⡀⠀⠀⡀⠀⢠⣿⠁⣼⣿⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠉⠉⢉⣉⣁⣭⣿⣿⣧⣀⣿⡀⠸⣿⣿⠀⢸⣿⠀⢸⣿⣿⠀⣸⡿⢀⣾⣷⡾⠿⠿⢷⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠉⠉⣠⣬⣿⣿⣧⣄⠀⠻⣆⢸⣿⢠⡾⠋⠀⣠⣿⣿⡟⠛⠻⢷⣦⣄⠀⠀⠈⠉⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣀⣀⠈⠙⠿⣶⣿⣾⣿⣿⣥⣶⠿⠛⠉⠉⠛⠿⠆⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠛⠿⠷⣾⣿⣿⣿⣿⡿⠶⠶⠶⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠰⣶⣤⣀⣀⣠⣶⠿⢛⣿⣿⡿⣿⠿⣶⣄⡀⠈⠉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⠀⠀⠀⠛⠻⢷⣦⣤⣼⣿⣿⠋⠀⣠⡿⠃⣿⡇⠹⣧⠀⠙⢻⣿⣿⡛⠋⣀⣀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⢷⣶⣶⡾⢿⡿⠁⣾⡏⠀⣿⣿⠇⠀⣿⡇⠀⣿⣿⡇⠈⣿⠉⢻⣿⣿⣛⢉⣉⣁⣀⣀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⣿⡟⢀⣿⠃⠀⠈⠀⠀⠀⠀⣠⣶⣿⣷⣄⠈⠉⠀⠀⠛⠃⢸⣿⠉⢻⣿⣿⣛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠉⠀⣼⡟⠀⢼⠏⠀⠀⠀⠀⠀⠀⠛⠋⢁⣿⡄⠙⠗⠀⠀⠀⠀⠀⠈⠿⠀⠘⣿⡌⠛⠿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⣿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠈⠙⠿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢀⣴⣿⣄⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢛⣿⢿⣦⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⢸⣿⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⢸⣿⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⠛⠻⣿⣷⣦
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⠀⢀⣀⣬⣿⡿
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⢿⣿⣿⣆
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠁⠀⠀⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⣿⣿⠃⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠳⠓⠟



⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⢿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⣯⣭⣭⣯⣯⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⡟⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡟⠉⠻⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⡀
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠈⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣰⣶⣦⡀⠠⠄
⠰⠶⠲⠤⠤⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⡖⠐⠒⠂⠈⠉⠉
⠀⠤⠀⠀⠀⠀⠀⠠⠤⠶⠦⠄⠀⠀⠀⠀⠀⠀⠀⠢⠽⠿⠟⠁⠀⠀⠀⠀⠀⠠⠤⠮⠤⠄⠀⠀⠀⠀⠀⠀⠤⡤⢤
⣤⣤⣤⣤⣄⣀⣀⡀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢀⣀⡀⢀⠀⠀⣤⣤⡄⠀⣤⣤⣤⣤
⣿⣿⣿⣿⣶⣿⣿⣷⣶⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⣿⠄⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⣤⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠹⣷⣤⣾⢿⠯⠿⠛⣿⣯⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⡄⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⠀⡒⡺⠟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠴⣚⠶⠿⢦⣶⣮⣽⣿⣿⣿⣿⣿⣛⢛⣿⣶⣦⣌⣛⢷⠶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⢤⣯⣛⣻⠿⣿⣟⣛⡛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠐⠒⠶⠒⠶⢶⣦⣴⡶⢦⣍⠉⢿⣿⣿⣿⡿⣛⢛⠉⣙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣛⣫⣾⣻⣿⣿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠤⠥⢤⣄⣿⣷⣮⡥⢶⣶⣤⣬⣹⣿⣿⣿⠷⠖⠂⠙⢿⣿⣷⣶⣮⣝⡛⠿⢿⣿⣧⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣀⣀⣀⣤⣤⣍⣙⣛⣛⡛⠛⠀⠀⠀⣊⣉⣙⡛⠛⢛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⡴⠖⠛⠛⠛⠒⠛⠛⠛⠛⠶⢤⣄
⠀⠀⠀⠀⢠⡾⠛⣁⡤⢖⣚⣻⣭⣭⣭⣽⣟⣛⣓⠲⢮⡻⣦⣄
⠀⠀⢀⣴⠞⣡⠞⣡⣞⡯⠷⢖⣒⣒⣒⣒⣲⢮⣙⢷⡀⢳⢸⡟⣷⡀
⠀⠀⣾⢁⡆⢠⣾⡟⣵⢞⣋⣭⡭⠴⢤⡬⣭⣝⣬⡶⠷⣾⣼⠃⠿⣷
⠀⢠⣿⢸⡇⠟⡿⣨⡾⠋⠁⠀⠀⠀⣤⠖⠛⢍⡀⠀⢄⠀⠈⠻⣶⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣿⠈⢷⣤⣡⣏⠀⠀⠀⠀⠀⢸⡇⠠⡀⠀⠙⣆⠀⠑⢄⠀⠘⣧⠉⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡻⠓
⠀⠀⣯⠈⠲⣌⡙⠿⠦⣤⣴⣦⣤⣾⡷⡄⠑⣄⠀⠘⢆⠀⠈⢆⠀⣿⠀⡜⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡷⠃⢀⣤⢤⡄
⠀⠀⠻⣄⠁⠢⢍⣒⠢⠤⠤⠤⠤⠈⢻⣌⢦⡈⠱⡄⠈⢢⡀⠈⣼⠃⢠⠁⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠎⢴⡡⣴⡯⠞⠉⠁
⠀⠀⠸⣧⣈⠑⠢⢌⣉⠃⠒⠒⠀⠃⠀⠙⣧⡑⣄⠘⢦⠀⠁⡼⠁⡴⠁⣠⠂⢹⠀⠀⠀⠀⠀⠀⢀⣴⠾⣫⠥⠂⠀⠀⠯⠤⣤⡶⣛⣗⡗
⠀⠀⠀⠉⠛⠻⣦⣄⡢⠭⠽⠿⠿⣉⡁⠤⢨⣿⡌⢳⡀⣀⡞⢡⠞⠀⡴⢁⡀⠻⣷⢶⣦⣤⣤⣶⠟⣡⠞⢁⡤⠊⠁⠀⡀⠰⢧⣤⣄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠷⠶⠶⠶⠿⠛⠁⠻⣤⡶⢋⡴⠁⣠⠞⣰⠋⢠⡞⠁⠤⠤⣀⣁⠠⠞⣡⠖⠁⣀⡤⠖⣉⡠⠔⣢⡴⠾⢷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⡠⢋⠔⠋⢀⡴⠁⣰⡟⠀⠐⠤⢄⣀⣀⣠⠞⢁⡴⢚⡡⠖⠋⣁⣴⠞⠋⠀⠀⠀⠱⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⢆⠞⢡⠋⣀⠴⢋⣠⡾⢋⡀⠐⠦⢤⣀⣀⣀⣤⠞⢉⠔⠁⢀⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡼⡎⢠⢃⡼⣡⡶⠛⠉⠛⠶⣭⣑⡲⠤⠤⢤⠤⠴⠚⠁⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⣇⣟⣼⣹⠏⠀⠀⠀⠀⠀⠀⠉⡛⠛⠷⠶⠶⠶⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢷⣹⢱⡿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢦⣙⠳⢧⣽⣦⣀⠀⠀⠀⠀⣤⣷⠞⣡⡴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢉⠓⠶⣤⣭⣉⣉⣛⣛⡛⠉⢿⣴⡯⠊⢁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠑⠦⢄⣀⠀⠀⠀⠀⠀⢀⠀⢿⣤⣴⡫⠍⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡢⡀⠈⠉⠛⠒⠚⠋⠉⢀⠀⢿⣦⣄⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣌⠲⠤⢤⡤⠤⠖⠚⣁⡴⠟⠛⠙⠐⠒⢝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢦⣤⣤⣤⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠨⠀⡰⠃
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢠⠀⠀⠄⠂⠀⡠⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣠⣄⠔
⠀⠀⠈⠎⢡⡀⠂⠀⢣⠀⠀⠁⠀⡀⠀⠈⠀⠀⠇⠀⢀⠴⠃⠂⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠂⠀⢀⠀⢨⠁⡀⠀⠀⠀⡰⢁⠀⠂
⠀⠀⠀⠤⠤⠅⠢⠆⠀⠀⠀⠀⠀⠂⠀⠀⠀⣘⡀⠒⠂⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⣀⠀⠉⠃⣀⡄⠄⠁⠁⡄⢀⢀⡀⠢⠌⠤
⠀⠀⠀⠀⠠⠤⠤⠤⠀⠑⠂⠠⠀⠃⠁⢰⠀⠂⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠓⣤⢙⢥⡄⠈⠣⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡉⠀⠎⣌⠢⢄⠉⢁⠄⠁
⠀⠀⠀⠀⠀⠒⠀⠀⠀⣀⡄⠞⡀⠄⡄⠙⠐⠐⠄⠀⠂⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⣐⠢⠔⠈⠀⠂⡀⠀⠄⡙⠣⢀⠄
⠀⠀⠀⠠⠤⠀⢀⡠⠄⠁⠀⠀⠀⠀⠆⠀⠀⠀⡀⠀⠧⠂⣉⠉⠀⠀⠀⠀⠀⠀⠐⠀⠀⠌⠀⠐⠀⠀⢔⠀⠀⠁⠀⠉⠘⠀⡀
⠀⠀⠀⠀⠠⠔⢀⠀⠀⠆⠀⡠⠂⠀⠁⠀⡀⠀⠰⡀⠀⢄⠀⠢⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠯⠁⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠈⠀⠀⡸⠀⠀⠀⠊⠀⠀⢀⡇⠀⠙⠄⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠆⢀⡃⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⡠⠀⢀⠀⠀⠠⡇⠀⠀⠀⡀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠤⡀⠀⠈⡃⠀⣰⠇⠀⠀⢸⠀⠀⠀⠃⡀⠤
⠀⠀⠀⠀⠀⠀⢴⠂⠀⠀⠀⢄⠀⠍⠔⢀⡀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠀⠀⠙⢶⠆⠣⠀⠘⠀⡎⠠⢇⣤⡴⠂⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠨⠁⠄⡄⠈⠁⢀⠠⠒⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⢆⠀⠠⡠⢀⠀⠀⡡⠁⠀⠀⠀⠀⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠚⠂⠀⠀⠚⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢰⠂⠠⠀⠁⠀⠀⠈⠠⠐⠐⠋
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠈⢰⠂⠀⠀⠰⠂⠀⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢀⡠⠖⠛⢉⠂⢳⠀⠀⡐⠀⢠⠆⡀⠀⠀⣀⣔
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠀⣄⠐⠃⠃⢀⡀⠃⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠘⠁⡄⠀⠀⡮⠐⠁⣠⡀⠁⢄⡀⠀⠙⠰⢀
⠀⠀⠀⠀⠀⠀⠉⠀⠠⠀⠀⠁⠀⠈⠅⡀⠁⠀⡄⠰⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⠀⠈⠀⠀⠀⢸⠇⠀⠀⠑⠀⠀⡆⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⡀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠂⢈⠆⢡⠒⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠈⢳⣤⣤⣤⣤⣤⣄⣴⠾⠿⠿⠿⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠙⠳⣄⠀⠀⠀⠙⢿⣄⣤⣴⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠖⠂⠀⣠⣴⠶⠿⠿⠿⠿⠿⠿⢿⣷⣦⣄⣀⣼⠟⠋⠉⠀⠀⠈⠙⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣿⣧⣄⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠀⠀⠀⠀⠉⢻⡟⢁⣤⠶⣦⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠞⠀⣾⣡⣦⠘⡧⠀⠀⢀⡾⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⠉⠉⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⡼⠿⣿⣇⠀⠀⣸⣵⣶⡀⣸⠃⣀⣀⣀⣀⡀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢀⣠⡶⠶⠶⠶⠶⣤⣿⠿⠛⠳⠶⠤⠀⠉⠙⠻⢏⣰⠿⠋⠉⠉⠀⠈⠉⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡿⠋⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠹⣷⣄⡀⠀⠀⢀⡀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢋⡽⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣄⣤⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⡶⠋⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠒⠶⠦⠤⠤⠤⣤⣤⡿⢱⣶⣶⡄⠀⠀⠀⠀⣠⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⣿⣆⠀⠀⠀⢀⣀⡤⠶⣻⡽⣿⣿⡇⣴⡶⠶⠿⠛⠋⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⡬⠿⣷⣖⣋⣩⡴⠖⣋⣡⢟⡶⡗⠚⠁⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣄⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠒⠒⠒⠚⠛⠋⠉⠉⢿⣆⠀⣀⡬⠟⠛⠛⠟⣻⠟⢁⣞⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠿⣧⣤⣀⣀⢀⣼⣁⣀⡼⠛⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠉⠙⡻⠋⠛⠉⢹⠇⠀⠀⠘⢿⢿⣷⣶⣤⣤⣤⣶⠟⠋⠉⠛⠿⢿⣷⣶⣦⣤⣤⣤⣤⣶⡾⠟⠋
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣾⠀⠀⠀⠀⠈⢣⡈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠹⠄
⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠀⣲⣆⣀⣉⣉⣁⣶⣂⡒⠒⠒⠠⡤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣣⡔⠈⠁⠀⠸⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢹⡏⠐⠒⠀⠡⢢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢋⣶⠀⠀⠀⠀⡜⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢨⢿⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠤⣹⡗⠠⠤⢖⣓⡳⠖⠛⠛⠛⠛⠣⣀⢐⣒⣂⣤⣤⣤⠤⠞⠛⡥⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢠⡟⠉⠒⠀⢀⠈⠈⠑⡄⠀⠀⠀⠀⠈⠣⡀⠀⠸⠸⠀⠀⠀⠀⣻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣼⠁⠀⠀⠀⠸⡆⠀⠀⠘⡄⠀⠀⠀⠀⣀⢱⠀⠀⡆⡆⠀⠐⠉⢸⣇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⠀⣀⣀⣃⣇⠀⠀⠀⣹⠤⠭⠭⠁⠀⠐⣇⣀⣇⣇⠠⠄⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡢⣀⠀⠀⠀⢸⣀⡀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠈⠂⢕⠢⣀⢸⠁⠀⠀⠀⠀⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠈⠂⢽⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢏⠢⣀⠀⠀⠀⠀⡰⠁⣤⣤⣤⣤⢤⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠉⠂⢵⡦⣄⡴⠁⠠⡇⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠁⠀⠀⣀⣀⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠑⢕⡢⡀⠀⠀⣰⠋⠀⠀⠀⠀⢻⠉⠉⠉⠛⠛⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⢒⠦⣄⠀⠀⠈⠒⡵⠊⠁⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠎⠐⡩⢢⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢀⡀⠠⠤⣐⡻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⠀⠉⠁⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⡠⠆⡻⡑⢝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠌⠂⠁

⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢸⡇⣠
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡄⣴⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⣦⠈⠛⢿⡿⠛⠁⣶⡀⣀⡀⠀⠀⠀⠀⢀⡀⢠⡀⠀⠘⢿⣿⠋⠀⢀⡤⢀⡀
⠀⠀⠀⠀⠀⠀⣈⠿⠏⣬⡁⣩⡝⠿⠟⠀⠀⠀⠀⠀⠀⠀⠈⣛⣿⣿⣤⡶⢿⡿⢦⣤⣿⣿⡋⠁⠀⠀⠀⠀⢈⣛⣾⣷⣀⣠⣼⣧⣄⣀⣾⣷⣟⡁
⠀⠀⠀⢀⣴⣾⣿⢏⣤⣝⣧⡟⣡⣄⢻⣿⣷⣤⠀⠀⠀⠀⠀⠛⠉⢸⡟⣻⣿⣿⡟⢻⡇⠉⠛⠀⠀⠀⠀⠀⠈⠉⠉⠉⢹⣿⣿⣿⣿⡏⠉⠉⠉
⠀⠀⠀⠀⠙⠻⣿⠆⠉⣥⡏⣷⡍⠁⠾⡿⠟⠉⠀⠀⠀⠀⠀⢻⣷⣾⣿⡟⢻⣏⣻⣿⣧⣾⠟⠀⠀⠀⠀⠀⠠⢦⣤⣤⡾⢿⣿⣿⡿⢷⣤⣤⡤⠄
⠀⠀⠀⠀⠀⠀⢠⣿⣧⣩⡄⣬⣵⣿⣷⠀⠀⠀⠀⠀⠀⠀⠰⠟⢻⣿⠀⢙⣿⣟⡉⠀⣿⠛⠻⠂⠀⠀⠀⠀⠰⠞⢻⡟⠀⠀⣸⣇⠀⠀⢹⣟⠳⠆
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⠁⠙⠿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠟⢻⡟⠻⠆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣴⢿⡟⢧⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣄
⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⣿⣹⣧⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣧⣿⣾⣳⣄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠘⣶⠏⢀⣴⠖
⠀⠀⠀⠀⠀⠀⢿⣇⣿⣴⣿⠺⣷⡶⣟⣮⠇⠀⠀⠀⠀⠀⠀⠀⣐⢚⡷⢸⡗⢽⢿⠹⡏⣿⣓⣦⠀⠀⠀⠀⠀⢀⠺⣄⠀⠨⣿⡆⣿⠀⣾⡟⠀⣴⠗
⠀⠀⠀⠀⢀⣄⣄⣽⣶⣥⠟⠈⠻⣬⣾⣦⣤⣠⡀⠀⠀⠀⠀⣠⣿⣿⡟⠛⠃⠈⠁⠀⠧⠉⣽⣿⣄⠀⠀⠀⠀⠉⠙⠛⠳⣦⣌⣁⣿⢀⣩⡴⠟⠛⠉⠁
⠀⠀⠀⠀⢨⣯⣼⣯⣥⣧⠀⠀⠀⣼⣼⡼⣧⢿⠇⠀⠀⠀⠀⡹⣽⡷⣿⠆⠀⠀⠀⠀⠀⣞⣞⣿⣿⠁⠀⠀⠰⠷⡿⢾⡷⠆⣙⣿⣿⣏⡁⢶⣾⣶⣦⣤
⠀⠀⠀⠀⠀⠀⢠⡼⣟⢮⣷⣒⣿⣜⡳⢧⡄⠀⠀⠀⠀⠀⠀⠸⣿⣞⣃⡠⡄⢀⠀⢀⡤⣬⣿⣿⠟⠁⠀⠀⠀⣀⣀⣤⡴⢚⣋⠉⣿⢉⡛⢶⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠳⣹⡞⠉⣽⣖⣿⠉⠷⢣⠗⠀⠀⠀⠀⠀⠀⠀⠹⢍⣿⣱⣧⣻⣵⢤⣇⠴⣭⠿⠀⠀⠀⠀⠀⠈⠼⠋⣤⡿⠏⠀⣿⠸⢿⣆⠈⢿⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠭⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⡻⣻⢿⠟⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⠁⠀⣼⢿⡄⠙⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠃


⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣠⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣴⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⣀⣿⡏⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢸⠀⢸⣷⣦⣈⣇⣤⣶⣿⠀⣶⠀⠀⠀⠀⠀⠀⣀⣀⠐⢿⣷⣠⣿⣧⣀⣿⠗⢀⢀⡀
⠀⠀⠀⠀⢰⣷⣄⡾⠋⢹⡇⢸⠏⣹⣶⠾⠿⠟⠀⠀⠀⠀⢀⣠⠼⣄⣸⣿⠉⢻⣿⠋⢹⣿⣠⠿⣤⡀⠀⠀⠀⠀⠘⠿⣷⣿⣿⣿⢿⢿⣿⣿⣦⣿⠟
⠀⠀⠀⠀⠈⠉⣿⠑⠓⢶⣿⣿⣿⡉⠈⣷⠀⠀⠀⠀⠀⠀⠈⢁⣠⣾⡿⠿⣷⡞⠛⢷⡿⠿⣷⣦⣀⠉⠀⠀⠐⢶⣷⣶⣿⣿⠿⠉⠈⠈⠹⢿⣿⣷⣴⣶⠶
⠀⠀⠀⠀⠀⠀⢻⡄⠒⠉⡿⢻⡄⠈⢑⣿⣶⣦⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣿⣦⣀⣼⣷⣴⣾⠿⠋⢁⠀⠀⠀⢠⣀⣾⣿⣯⠅⠀⠀⠀⠀⢭⣿⣿⣋⣁
⠀⠀⠀⠀⣾⣿⠏⠻⣄⣰⡁⠈⣷⣠⡟⠀⠙⠛⠀⠀⠀⠀⠈⠳⢦⠞⢿⣿⢁⣸⣿⣅⣹⣿⠳⣴⠖⠋⠀⠀⠘⠛⠛⢹⣿⣿⣟⣤⢠⢠⣼⣿⣿⡿⠛⠟⠃
⠀⠀⠀⠀⠈⠉⠀⠀⣼⣿⡋⠛⢹⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⠿⠛⡟⠻⢿⣿⠀⣾⠀⠀⠀⠀⠀⠀⣸⣿⠋⢻⣿⢿⣿⣿⣿⣿⠟⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠓⢦⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⠏⠈⣿⡏⠸⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡤⠃⠀⡎⠉⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣠⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⢴⣶⣶⢶⢶⢶⢤⣄⣰⠃⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡎⢏⣧⢃⢷⡟⣱⣟⣕⣵⣩⣾⣿⣦⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⢿⣿⢸⣿⡾⠟⣋⣭⡥⠷⣒⣶⣾⣫⣿⣟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⢿⢯⡻⡼⠞⣡⠦⣕⣾⠷⣛⢟⠿⣫⣵⣾⡱⠾⢋⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣶⠏⣌⣾⢏⣴⠞⡡⡪⡪⢗⢝⠕⣡⣾⣿⣿⡷⢾⣛⣲⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢿⡄⡧⢡⡾⢡⢊⢎⢎⡴⡡⣢⣾⣿⡿⢛⣵⢾⡻⡇⠅⠇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢿⢧⢡⢻⢡⢃⢆⢏⡞⣴⣿⡟⣱⢟⣵⣿⣵⢸⡄⣿⠀⡄⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣼⡏⡆⡎⡘⣬⢎⣾⡿⣱⣿⢣⠯⢙⡻⠿⠟⢶⢷⡞⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⣾⡇⡇⡇⣿⡏⣾⢧⢣⣿⡇⡾⠭⠅⣐⡛⠉⠀⠠⣭⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣇⣷⢸⢻⢹⠸⡡⣧⣷⡫⠄⠠⠄⠒⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠺⣼⡜⡕⠟⡿⣿⣯⡫⣛⠝⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠻⣷⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣸⠁

⠀⢀⡄⢰⡄⣿⡇⣄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡐⠢⠤⢤⠄
⠠⡄⠹⠂⠂⠈⠀⠀⠿⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⠛⠿⠿⣶⣶⣆
⣀⡀⠀⠀⠀⠀⠀⠉⠉⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠘⠻⠿⠿⣶⣶⡾⠛⠃
⠀⢶⠞⡋⢴⢰⣢⣿⠏⠛⢡⡄⢠⠄⢠⡀⠀⠀⠀⠀⠀⣤⢤⠀⣤⣤⢠⣶⡀⢻⣧⡿⢿⣷⣶⣶⣦⡴⢿⣿⡗
⠀⠀⠘⠒⣴⡿⢋⡗⢋⣴⠋⢠⠏⢠⡟⠀⢉⣦⠀⠀⢱⣼⣞⡄⠸⣿⡎⢿⣷⡀⠽⢿⣷⣶⣬⣭⣝⣷⣶⣶⣷⡆
⠀⠀⠀⠀⠀⠐⠚⠀⠾⠁⢰⠃⣠⡟⡱⢠⣾⠋⠀⠀⠈⢀⣭⠸⡀⢹⣿⡌⢿⣿⣶⣤⣤⣉⣉⠛⢋⣀⣤⣍⣉⣼⡄
⠀⠀⠀⠀⠐⠛⠓⠔⠰⠶⠇⠀⠋⠐⠂⠛⠃⠀⠀⠀⠀⠐⡘⢏⣧⠀⣿⠷⠿⢿⣦⣍⣛⣛⠟⠻⠿⠿⠿⢿⡿⢛⣻
⠀⠀⠀⠀⠘⠶⠶⠶⠤⣤⠀⠀⠀⠀⠀⣀⣄⢶⡄⠀⠀⠀⢁⠟⠀⠀⠀⠀⠀⠀⠀⠉⠙⢛⠛⠋⢰⣾⣿⠿⠿⠿⠿
⠀⠀⢀⣐⠀⠀⠀⠀⠀⠄⠀⠀⣠⢤⡻⣾⡿⣷⡹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢸⣿⣿⣿⣿⣷⣶⡆⠀⠀⣿⣳⣷⡹⣿⢻⣧⣏⣀⣠⣀
⠀⠀⠀⠈⠉⠉⠙⠛⠋⠀⠀⠀⠈⢿⣯⣧⣿⢿⣿⣯⣛⢿⣮⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣻⡄⣿⣿⣻⢷⣮⣻⠾⣭⣿⣶⣷⡖⠀⠀⠀⠀⠀⢰⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⣀⢲⣄⡆⣝⣴⢄⠀⠀⠀⠀⠘⢻⣌⢻⣾⡛⣮⣽⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢈⡝⡛⣿⣿⣿⣿⡗
⠀⠀⠀⠀⣤⡜⠠⠃⠉⠩⣞⣯⣥⡄⠀⠀⠀⠀⠹⣷⡙⢟⢮⡻⣿⣿⢿⣇⠀⠀⠀⠀⠀⠀⠰⠈⠿⣿⣿⣿⣶⣶⣄
⠀⠀⠀⠰⢺⠛⣓⠀⠀⢀⢋⠐⠅⣒⣂⠀⠀⠀⠀⠈⢿⣎⢻⣷⣹⣿⣷⣿⣿⣷⠄⠀⠀⠀⢀⣀⣠⣿⡏⠉⣿⡿⠿
⠀⠀⢠⣓⡛⠸⢸⡇⠄⢺⣬⣯⣭⣭⠶⠆⠀⠀⠀⠀⠀⠙⣷⣿⣿⣿⡿⡿⠻⠋⠀⠀⠀⢀⠙⠻⣿⣏⠴⣼⣿⡃
⠀⠀⠤⢭⣭⣛⡻⠷⠛⠊⠍⠶⣞⣘⣿⡥⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠁⠀⠀⠀⣠⣦⣀⣿⣇⣤⣿⣯⣶⣿⣿⡿⠆
⠀⢀⣛⡲⠶⠬⠉⡍⠀⠀⢘⣭⡭⠴⠶⠞⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣀⢀⣴⣿⣿⣁⣠⠁⢹⣿⡁⢸⣿⣆⡀
⠀⢬⣭⣙⣛⠳⠶⠆⠀⠀⠀⠰⢒⣛⣁⣭⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⣾⠚⠉⣹⣿⣿⣴⣾⣿⣷⣿⣿⡷⠘
⢀⡴⠶⣤⢍⣍⣛⠀⠀⠀⠀⢡⣭⡵⠶⠶⠞⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⣦⣨⠔⠋⠛⠉⠁⣯⣏⠀⢸⣼
⢈⣛⡛⠻⠶⠶⡄⠀⠀⠀⠀⠠⠶⠞⣛⡛⠫⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠂⠀⠀⠀⡄⡟⡏⣤⡮⠜⠿
⣤⣤⣬⣭⣭⣶⣤⣤⣤⣤⣤⣤⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣥⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠁⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠐⠒⠁⠂⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠈⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠐⠀⠀⠀⠀⢈⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⢀⡠⠰⢀⠤⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠄⠀⣠⠂⠁⠀⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢌⠀⢠⠃⢜⡂⠀⠀⠠⠀⣠⠖⠊⠀⠂⠀⡀⠈⠠⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠢⠄⣀⣀⡀
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠉⠀⠀⠀⠀⣀
⠀⠀⠀⢠⠄⢀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐
⠀⠀⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡰⡞⣢⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠳⡄⠀⠀⢀⡀⠀⠀⠀⠀⠄⠃⣾⠀⠀⠇⡀⠖⣂⠽⠛⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢦⡀⢳⣀⠀⠀⢉⠲⣤⠤⠄⢲⠆⡻⠗⣊⠥⠒⡿⠚⣀⣴⣾⠿⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠡⠙⢢⡙⡖⢌⠳⣸⠀⢗⡀⠀⠔⣋⠥⢲⡿⡏⢐⣠⣿⡿⠟⢋⣠⣴⠯⠥⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡂⠀⠀⠀⠀⠙⠓⢤⣳⡜⡇⠸⣭⡶⢫⣼⢛⣟⣻⣷⣿⠟⢉⠤⣚⠥⠒⢉⣤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠯⠿⠒⠛⠉⠉⠁⠘⣽⠿⢩⣤⣾⣁⣯⣀⡰⠙⠛⡧⢲⢈⣉⣍⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠰⠊⠀⠀⢛⠀⠀⠀⠀⠀⠈⢹⠱⣽⡷⢱⠀⢋
⠀⠀⢀⣠⠤⣤⠔⠉⠀⠀⠀⠀⠀⠀⠀⢂⣼⡿⠷⣶⣶⡂⠀⠀⠐⠋⠀⠀⢔⠊⠀⠀⠀⡀⠀⠀⢸⣇⡏⡧⣹⠀⠀⣳⠒⠒⣤⠄
⡠⢊⡥⠒⢩⡉⠂⠀⠀⠀⠀⠀⢠⠀⣴⡿⠉⠙⣛⠒⢲⣧⠀⠀⣀⣀⣤⣤⠞⠛⣿⡿⠟⢻⠿⠶⠺⡿⠓⢠⠇⠀⢀⡇⢀⢜⣡⡀
⠛⣌⠁⢀⣠⡹⢶⣶⣀⠀⠀⠀⣠⣾⣿⡧⠐⠉⠈⣇⡜⣻⣿⡿⢛⡏⢹⢹⡝⣉⣟⣀⣤⡇⠀⠰⠀⠗⠶⠏⠀⠀⣼⠋⣍⢻⣷⡳⣀
⠀⠘⠶⡿⠁⠠⠤⣻⡽⣿⣦⢾⡿⠟⣩⠾⠂⢀⠤⠤⣤⣭⠿⣿⡤⢷⠈⣿⠗⠛⢋⣉⣸⠀⠀⠆⢸⡃⠁⢦⡠⠔⣡⢀⡴⢿⣃⠁⡩
⠀⠀⠀⠱⡀⢸⡉⠉⠉⠉⠉⠠⠄⠀⠁⣠⡶⠣⡔⢊⠽⠍⠛⠉⠈⠽⠟⠃⠀⣀⣡⣤⡎⠀⣀⠀⣾⡡⢔⣴⢂⡀⢠⢾⡁⠐⣴⡿
⠀⠀⠀⠀⢻⡘⢳⣄⠀⠀⠀⠀⣠⡴⠛⠁⠀⢀⡏⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⡿⢷⣿⠗⢿⣿⢦⣿⢶⣟⢩⣒⡰⢟⢢⣽⡇⠺⠂
⠀⠀⠀⠀⠀⢳⣼⢿⠓⣤⡴⠞⠛⢂⠀⠀⠀⣀⣽⢶⣾⣛⡉⠛⠄⠈⢉⠿⢥⣴⢾⡿⠷⣤⠶⠿⠗⠈⢸⠃⢻⢳⠈⠻⠓⠁
⠀⠀⠀⠀⠀⠀⠙⢨⠄⠀⠀⠀⠀⠀⠡⠐⠚⠛⠁⠀⠀⠀⠈⠁⠀⠂⠀⠀⣤⡙⢳⣗⠀⠀⠀⢀⠠⠀⠈⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠰⠌⠀⣀⣀⢄⣀⠀⠀⢰⠁⡈⠉⢢⡀⠀⠀⣔⠀⠀⢀⣠⠔⠛⠋⠻⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⡀⠒⢋⢨⠧⠤⠼⠁⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠁

⠀⠀⠀⠐⣶⣦⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠻⠛⣿⠀⠀⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⢘⣷⣿⡿⠀⠀⠀⢛⡿⠤⡄
⠀⠰⣷⣶⣿⡟⠀⠀⠀⠀⠀⠉⣁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠂⠀⠀⠁⡖⠀⠀⠈⠁⠉⠀⠀⠈⠛⠉⠁⠀⠀⠀⠈⠇⠀⠁
⠀⠀⠀⠈⠉⣰⣄⡀⠀⠀⢀⠀⣹⣷⣶⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⠀⠀⠉⠀⠀⠀⡄⠀⢀⠂⠀⠀⠂⣄⣄
⠀⠀⠀⠀⢸⢿⣿⣿⡆⠀⠈⠀⣿⣿⣿⡀⣤⣦⣴⣤⠀⠀⠀⣦⣤⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣤⣸⠆⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⠿⠀⠀⠀⢸⣿⣿⣧⣿⣿⣿⡿⠀⠀⣼⣿⣷⠀⠠⠟⡍⠁⠀⠀⠀⢀⠰⠀⢐⣿⣿⣿⠀⠠⠀⠀⠹
⠀⠀⠀⠀⠀⢼⣿⣿⠀⠀⠀⠀⢘⡿⣿⠋⢻⣿⣿⣷⡆⠀⠹⠏⠥⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠛⠿⢿⡇⠀⠀⠀⠐
⠀⠀⠀⠀⠉⠀⢺⣿⠀⣀⠀⠀⠀⠀⣿⡇⠀⢹⢿⡟⠁⠀⠃⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢲⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡆⣿⠀⠀⠀⠀⠻⣧⠀⠀⠙⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠂⠀⠀⠀⠰⣆



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡼⠀⠀⠀⠀⠄⢀⡀⠐⠃
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠛⠳⠤⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⢀⣀⣀⣁⡀⣀⣀⣀⣀⡀
⠀⠀⠀⢨⠉⢬⣭⡄⢬⡍⠭⠍⠅
⣿⣿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠻⠞⠛⠱⢿⠀⠀⠀⠀⠀⠀⠀⠀⠷⠞⠛⠺⢟⠄⠀⠀⠀⠀⠀⠀⠶⣵⣾⣿⣿⣾⣤⠀⠀⠀⠀⠀⠀⠑⢄⢸⢠⠚
⠀⠀⠀⠀⠀⠀⠛⢿⡀⠘⢀⣛⠛⠀⠀⠀⠀⠀⠀⠚⣟⡘⠟⢀⣛⠋⠀⠀⠀⠀⠀⠀⣩⢿⣿⣿⣿⣿⣒⠀⠀⠀⠀⠀⠒⠂⢂⠛⠣⡁⠉
⠀⠀⠀⠀⠀⠀⠀⠟⠋⣿⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠛⠓⡿⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⡟⣏⠑⠁⠀⠀⠀⠀⠀⠀⠰⠂⠀⡄⠈⠂


⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⣠⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⢤⢀
⠀⠀⠀⠀⠀⠀⠀⠩⣼⣶⣷⡅⣁⠀⠀⠀⠀⠀⠀⠀⠁⠄⣂⠀⠁⡀⠀⠀⠀⠀⠀⠀⡒⡾⣭⣶⣯⢽⠤⠀⠀⠀⠀⠀⠠⠍⠊⣸⡰⠛⡆
⠀⠀⠀⠀⠀⠀⠘⠀⠻⣿⣿⢃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠅⢛⠡⡀⠁⠀⠀⠀⠀⠀⠀⠰⢷⣘⠿⣟⡾⡅⠀⠀⠀⠀⠀⢨⡭⣀⠙⡋⣅⠶
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠏⠫⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠃⠏⠃


⠀⠀⠀⠀⠀⠀⠀⡀⣠⣣⣐⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣄⠂⡠⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⢠⣀⣠⡀
⠀⠀⠀⠀⠀⠀⡀⢲⢃⠤⡘⡷⢀⠀⠀⠀⠀⠀⠀⣐⢠⡋⣭⢛⡆⡄⠀⠀⠀⠀⠀⠀⢀⠨⣿⣠⠋⠉⠀⠀⠀⠀⠀⠀⣀⣬⣿⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠛⣦⣩⣰⢚⠐⠀⠀⠀⠀⠀⠀⠀⢘⠶⠭⠖⢅⠂⠀⠀⠀⠀⠀⠀⠀⢐⢟⡉⣋⡉⠈⠀⠀⠀⠀⠀⠉⢁⡋⡉⡋⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠃⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢭⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣗⠂⠀⠀⠀⠀⠀⠀⠀⠀⢚⡀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠱⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢦⣀⡀⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⡴⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢠⠀⣠⡆⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠮⢉⠛⢀⠁⠰⠇⠈⢠⡇⠀⣏⡉⠳⠢⣤⡀⠀⠀⠀⡴⠋⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠎⠁⠉⠐⠂⣤⠙⠋⢁⡄⠂⠉⣰⠀⣸⡡⠐⠊⠁⠠⠙⠓⣤⠞⢁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⡀⠒⠠⣈⠕⠂⢀⠤⠂⢁⡠⠐⢁⡜⠀⣛⡀⠄⠂⠉⡀⢤⡞⠁⣰⠏⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠄⢀⠀⠲⠖⠀⡠⠀⠀⠀⠔⠈⡠⠐⢈⡇⢠⡏⠀⡀⠔⠋⡰⠋⢠⡞⠈⠂⡝⢦
⠀⠀⠀⠀⠀⢀⡞⠁⠠⠀⣀⠛⠀⠀⠀⠀⡀⠈⢀⡠⠃⠀⡠⢘⠃⢸⠷⢆⠉⢲⠎⠀⣴⠏⠁⡀⠀⠇⠘⢣
⠀⠀⠀⠀⢀⡞⠡⠆⢀⠀⠁⣠⠊⠀⠠⠂⠀⠀⠀⠀⠄⠀⠀⣬⠀⡴⢄⢀⠞⠁⡠⠞⡅⠠⠀⠇⢀⠀⠈⠀⢷⡀
⠀⠀⠀⠀⡼⠀⠀⠠⠀⠀⠔⠁⢀⠚⠀⠀⠀⢀⠔⠁⠀⡠⠎⠉⠠⣇⡠⠋⢀⠜⠁⠄⠸⠀⢠⠀⠸⠀⠀⠀⠈⣧
⠀⠀⠀⢸⡇⠂⠠⠁⢀⠊⠀⡠⠁⡠⠊⠀⠐⠀⠀⡠⠚⠀⠈⠲⣄⠈⠡⡴⠋⠠⡀⠈⠇⠀⡈⢀⡀⢰⠀⠸⠀⣿
⠀⠀⠀⣸⢡⡔⠀⠠⠀⢀⠊⢀⠜⠀⡠⠂⠀⣠⠊⠑⢄⠀⠀⠀⠈⠳⢄⠈⠂⡀⠁⠦⠀⠀⠁⠀⠀⡀⠀⡀⠀⠀⡇
⠀⠀⠀⢿⡾⠀⠠⠁⡀⠂⠠⠈⢀⠄⠁⠀⠐⡙⠂⡀⠀⣴⠀⠀⠢⡀⠈⠃⠀⠰⣀⠀⠀⠉⠀⠃⠀⠃⠀⠀⠄⢸⡇
⠀⠀⠀⢸⠇⠀⠀⠠⠁⠠⠁⢀⠆⠀⣠⠃⠀⠁⠀⣬⠞⢀⡼⢆⠀⠈⢀⠀⠀⣠⠋⠀⡈⠀⠘⠀⠘⠀⠈⢠⠀⣸⡇
⠀⠀⠀⠸⣧⠁⢀⠃⡀⠁⠰⠂⠀⠀⢩⠀⠠⢀⡜⠁⣠⣿⠀⣜⠣⠀⠀⢂⠀⡏⠀⢠⠁⠀⠃⠐⠂⡘⠀⡀⠀⣿
⠀⠀⠀⠀⢹⢀⠀⠠⠀⠠⠁⢀⡜⠀⠐⠀⡼⠊⢀⠜⢁⡟⠀⣿⠀⠁⠀⢀⠗⠀⠀⠀⢀⠉⢀⠃⢀⠁⣀⠁⢰⡏
⠀⠀⠀⠀⠈⢿⡀⠃⠀⠂⢀⠘⡅⠀⣠⠞⠁⡴⠟⠀⢸⠇⢠⠈⢀⠀⠰⡎⠀⢠⠀⠀⠌⠀⡈⠠⠄⢠⠙⣆⡿
⠀⠀⠀⠀⠀⠈⢿⣀⠁⢀⡆⠀⣇⡴⠃⣠⠞⠀⠀⠀⣿⠀⠸⡁⠋⢦⡐⠀⣀⠃⠀⠘⠀⠠⢀⠂⠀⠾⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣄⡘⢇⣰⠋⢀⡜⠉⠀⠀⠀⠀⡿⠀⡾⠁⢀⠊⠀⡐⠀⢀⠆⠀⡌⠀⠌⢠⣌⡰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠟⠁⡴⠋⠀⠀⠀⠀⠀⠀⠇⠀⠇⠀⠂⢀⠔⠀⠀⠂⠀⠐⢀⠈⢠⣡⠞⠉⠉⠉⠉⠙⠉⠁⠒⠲⠶⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⣠⠞⢧⣄⡀⠀⠀⠀⠀⢰⠀⢠⡔⠀⢠⠂⠀⡐⠀⠠⠀⣀⢂⠴⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁⠐⠒⠢⢤⣀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⣠⠊⢀⡜⠁⠀⠀⠈⠁⠒⠦⢤⣄⡸⠀⣺⢀⡄⠁⣀⣈⠀⣴⠆⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄⠀⢣
⠀⠀⠀⠀⢀⠞⠁⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⠋⠉⠁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⣸
⠀⠀⢀⡴⠋⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠞⠋⣀⡴⠃
⠀⣠⠋⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠴⠶⠀⠛⠉⢀⣀⠠⠖⠛⠁
⢰⣇⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠉⠀⣀⣠⠀⠶⠒⠉⠉
⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢀⡤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡈⠙⠓⠠⠶⠦⠤⠤⠀⠀⠀⠠⠤⠤⠤⠤⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠲⠶⠶⠶⠤⠤⠤⠤⠄⠀⠀⠀⣀⣀⣄⠀⠀⠙⠳⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⣄⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠤⠤⠚⠁⣀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣀⣀⣀⣀⡀⠠⠶⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⢀⣠⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠁⠀⠀⠀⠈⠁⠂⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⣦⢄
⠀⠀⢰⠋⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢡⠂⣠⠤⡄⠀⠹⠣⡀
⠀⠀⢠⠀⠀⠀⠠⠤⠞⠀⡠⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⢀⠎⠀⢸⠀⠻⢄⡸⠀⠀⠁⢳
⠀⠀⠀⡱⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠑⢤⡀⠀⠀⠀⠀⢴⡉⠀⠀⠀⠀⠈⠉⠉⠒⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢳⠦⢄⣀⡄⠀⠀⠘⡄
⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠇⠀⠀⢀⠇⠀⠀⠀⠇
⠀⠀⡧⠄⠠⠤⠂⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⢀⣳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠄⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁⣀⡤⡒⠏⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠸⠀⠀⠀⠀⠀⢀⣀⣤⢶⠋⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠐⠒⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸
⠀⠀⠀⢇⢁⡠⠴⠞⠋⠂⠁⠀⠀⠀⠀⠀⡀⠤⠔⠊⠁⠀⠀⠀⠀⡏⠙⠲⣤⣀⣀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠔⠀⢤⠎⠀⠀⠀⠈⡅⠀⠀⠐
⠀⠀⠀⠸⠉⠈⠀⠀⠀⠀⠀⠀⡀⡠⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢙⠀⠉⠉⠑⣖⣶⠇⠀⠀⠀⠀⠸⡀⠀⢨⠀⠀⠀⠀⠸⠀⠀⠱
⠀⠀⠀⠀⠀⡀⠀⡀⡄⠄⠂⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⠑⠲⣴⠊⢁⠂⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠑⠒
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡆⠀⠀⠈⠷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠈⠁⢄⠀⠀⠀⠀⠀⠀⡀⢀⡄⠒⠈⠉⣿⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⣧⠀⢆⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⣆⠈⢢⠀⠀⢈⡆⠀⠀⠀⠀⡠⠊⣩⡄⠀⠀⠀⠀⡈⠓⠀⠀⠀⠄⠀⢀⡾
⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠁⠀⠀⠀⠂⠐⠛⢂⠀⠀⠀⠀⠹⣆⠀⡴⠀⠁⠀⠀⡀⠔⠈⣠⡾⠋⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢸⠁⠀⠈⠁⠀⠐⠀⠀⠐⠂⠀⠈⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠘⣀⣴⠞⠉⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⢸⢐⣒⣒⠒⠀⠠⠤⠤⠤⠤⠀⠒⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠂⠠⠤⠀⠀⠀⣀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠤⠤⠄⠀⣀⣀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⡀⠀⠀⣼⠀⠀⠀⣀⡠⠒⠉⠀⠉⢳⡀⡀⢀⡀⠤⢄
⠀⠀⣀⠠⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠠⠠⠄⠀⠀⠀⠉⠀⠀⢀⡨⠀⢠⠊⠀⡡⠤⠤⡀⠀⠈⠁⠈⡇⠀⠀⠀⠗⠢⡀
⠀⠉⠉⠁⠀⠀⠂⠀⠌⡑⠂⠤⢀⠀⠀⠀⠀⠀⠁⠐⠒⠒⠒⠒⠪⣀⠠⠔⠀⠀⢸⠀⠀⠓⠀⠀⠹⠀⠀⠀⠴⠃⢀⣀⠀⠀⠀⠚⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⢀⠈⠒⠠⠄⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣈⣣⣀⠠⠄⠐⠁⠀⠀⠀⢀⠀⠈⡸⠀⠀⠐⠀⢢⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⣐⠀⠀⠄⢀⡀⠉⠁⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠂⠤⣀⠀⠀⠈⠉⠉⠀⣠⣤⡤⠶⢧⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠒⠒⠤⠤⢌⣁⡀⠒⠒⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣒⡒⠒⠬⣔⠄⡀⠀⠀⢠⠀⠀⠀⠈⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠢⡀⠀⠀⠀⠀⠈⠉⠉⠁⠒⠒⠒⡶⠶⠦⠤⣄⡀⢀⣸⡄⠀⠈⡆⠀⠉⠉⠁⠀⠀⠀⠀⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠐⢤⡀⢀⠤⠶⠦⣄⡀⠀⠀⢱⠄⠀⠖⠀⡹⠁⠈⠳⠤⠖⠁⠠⣄⠀⠀⠈⡧⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠤⠉⠒⠀⠀⠒⢿⡀⠀⡆⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣆⠀⢀⡴⠄⠊⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠴⠦⠀⠤⠞⠉⠉







⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠉⠙⢿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⣠⠶⠒⠒⠲⢤⡀⠀⠀⣀⣤⣄⡀⢘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⠒⠋⠁⠀⠀⠉⠛⠋⠙⠻⣿⣿⣿⣄⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⢤⣀⡞⠙⢦⡀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⡴⠋⠉⠙⠳⣄⡼⠁⠀⠀⠀⠙⠓⠲⡄⠉⠙⠲⣤⠀⠀⠰⠿⣿⣿⡿⠟⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠐⢦⠴⠃⠀⠀⡴⠋⠁⣀⣠⣤⣤⣤⠄⠀⠀⣿⠀⠀⠀⠈⢳⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⡀⠀⠀⣠⠞⠁⣠⠞⠉⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⢧⣠⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿
⠿⣍⣛⣉⣁⡴⠊⠀⣀⣀⣀⣀⣀⣤⠖⠋⠁⠀⡴⠚⠲⣄⠀⢀⣽⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣿⡿⠟
⠀⢀⣨⣭⣤⣤⠴⠚⠉⠀⠀⠀⠀⠈⠙⢦⡀⠈⢧⣀⡀⣈⣩⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⣀⡴⠒⠒⠦⣀⠀⠈⠳⢤⣀⣀
⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠙⢄⣀⣨⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⠀⠀⠈⠳⠦⣤⣤⠤⣴
⠀⢀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠿⠟⠛⠛⠶⠶⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣻⣿⣿⣿⣿⣷⣶⣤⣤⣶⣿⣿
⠀⢸⣷⣶⣶⣶⠾⠟⣋⣡⣴⣶⣶⣿⣷⣶⣶⣤⣄⡈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣿⣿⠿⠛⢉⣡⣤⣤⣤⣤⣤⣌⣉⠙
⠀⠀⠉⣉⣥⣤⣶⣿⣿⣿⣿⣿⣿⣿⢯⣭⣭⣤⣬⣥⣤⣤⣄⡉⠛⠻⠿⠿⠿⠟⠛⠉⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠘⠟⠛⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⡿⠿⠿⠿⠿⢿⣿⠷⠶⠛⣣⣴⣿⣿⣿⣿⣿⣿⣦⠤⠤⠶⠿⠿⠟
⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⣿⣶⣶⣤⣄⣀⠀⠙⠷⣶⣶⣶⣶⣶⣶⠶⠾⠿⠿⠿⠿⣿⠿⠛⣉⣤⣤⣶⣶⣶⣶⣤
⠀⠀⠀⠀⠙⢿⣿⣿⡿⠟⠛⠛⠻⣿⣿⣿⣿⣷⣦⣄⠀⠰⣶⣶⣶⣶⣶⣶⣶⠶⠟⣁⣤⣶⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢳⣴⠶⠶⠶⢶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣬⣭⣭⣭⣭⣤⣶⣿⣿⣿⣿⣿⣿⠟⢛⣉⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣦⣤⣤⣀⡉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠛⠳⠦⠄⠉⠛⠻⠿⠿⢿⣿⣿⠿⠿⠟⢛⣉⣤⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⣶⣾⣿⣷⣦⣀⠛⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠚⠛⠛⠛⠛⠛⠲⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠖⠚⠛⠋⠛⠳⡄⢻⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⠀⠀⡤⠶⠦⡄⠹⡄⢻⣿⣿⣿⣝⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹⠀⠀⢧⡯⣱⢼⠀⢳⡀⢻⣿⣟⣡⣢⣼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠉⠁⠀⠀⠀⢷⠈⢷⣫⢞⣽⣿⣓⡽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⣷⣿⣢⣬⣯⣦⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⠀⠀⠀⠀⢀⣀⠤⠴⠒⢋⣉⡭⠤⠶⠲⣤⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⡤⠴⠒⠋⣁⣠⠴⠖⠋⢉⣠⠤⣄⠀⠀⣼⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠉⢀⣠⠴⠒⠋⠁⠀⠀⠀⢀⢫⡸⢉⢎⢇⡼⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠉⠀⣀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡨⠥⣪⢞⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⣀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣵⡋⠘⡆
⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠁⢀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⡵⠋⠀⣇⠀⢳
⠀⠀⠀⠀⠀⢠⡞⣇⠀⡠⠚⠁⠀⠀⠀⢀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⡴⠋⠀⠀⠀⢻⠀⢸⡆
⠀⠀⣰⠚⠲⠞⠁⢸⠀⣧⠀⢠⠠⡄⢰⠈⢧⠈⢧⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⣡⠴⠋⠀⠀⠀⠀⠀⠸⡇⠀⡇
⠀⠘⠋⠀⠀⠀⢜⠹⡇⢸⡄⠈⢧⠹⡀⠣⠀⠓⠀⠀⠀⠀⠀⣀⡤⠞⢉⡴⠞⠁⠀⠀⠀⣀⠄⠀⠀⠀⣷⠀⡷
⠀⠀⠀⢀⡤⠤⠞⢀⣿⡀⢻⡄⠈⠂⠉⠀⠀⠀⠀⢀⣠⠴⠚⢉⣤⠞⡏⠀⠀⠀⠤⠶⢚⣡⠖⡄⠀⠀⣼⠀⡇
⠀⠀⢀⣸⠇⠀⠀⢸⠈⢧⠀⠙⢦⣄⣀⣀⣤⠴⠚⠉⣀⡤⠞⠋⢹⡄⢳⠀⠀⠀⠀⠈⣉⣤⠞⠁⢀⡴⠃⣼⠁
⠀⠈⠉⠁⢀⡤⠤⠎⠀⢈⢷⣄⠀⠈⠉⢁⣀⣤⠶⠛⠉⠀⠀⠀⠈⣇⢸⡄⠀⠀⠀⠀⠁⠀⣀⡴⢛⡡⠞⠹⣄
⠀⠀⣀⣀⡞⠀⠀⠀⠀⢸⠀⠉⠓⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠓⠦⠤⠤⠶⣞⣫⠥⠖⠻⣄⡀⠀⠈⣷
⠀⠈⠉⠉⠀⠀⡖⠲⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣹⠃⠙⢯⡉⠀⠳⣄⠀⠀⠀⢉⡇⠀⣿
⠀⠀⠀⠀⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⢠⠏⠀⠀⢀⡗⠀⠀⢯⡀⠀⠈⠉⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠆⠀⢻⡀⠀⠀⠺⣄⠀⠀⠀⢹⠆⠀⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⡇⠀⠀⠀⠘⠃⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢿⣿⡆⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠸⣀⠈⣩⣿⠄⠀⠐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄⣀⣤⣵⣴⡆⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⢴⣾⣷⡴⠇⠀⢠⢶⣤
⠀⠀⠀⠀⢤⠀⠀⠘⣄⢠⠈⣧⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠐⠉⠛⠋⢃⠘⠉⣡⡿⠀⢞⣀⠤⠐⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢙⣀⢾⣿⡏⠀⢀⡆⠸⣿⡁⣸
⠀⠀⠀⠀⠸⠆⠋⠈⠉⣶⠛⠛⠑⣶⣮⠉⠀⠀⠀⠀⠀⠀⣤⠶⢒⣻⣾⣧⣴⣿⣷⠛⠋⠉⠀⠀⠀⠀⠀⡀⣸⣧⣾⣀⡄⠀⠁⠸⠛⡋⠛⢶⣼⣷⣿⣿⣷⠃
⠀⠀⠀⠀⠀⢦⡄⢣⡀⡏⠀⠀⢛⣛⣣⣄⡀⠀⠀⠀⠀⠀⠀⢀⡌⠈⠑⠛⠋⣉⠀⠀⠀⢀⣴⣦⠀⠀⠀⢹⠿⣿⣿⣿⣶⣶⣶⣶⣶⣷⣤⣤⣾⣿⣿⣿⡇
⠀⠀⠀⢲⡀⠠⢟⣈⠁⣷⣶⣿⡋⢿⡿⣿⣿⠀⠀⠀⠀⠀⣄⠘⠘⠆⠀⠀⠆⣈⠀⠀⡀⣾⠻⡇⠀⠀⠀⠀⠀⢿⣿⡇⣿⠋⣿⣿⢹⣿⣿⢸⣿⣿⣿⣿⡷
⠀⠀⠀⠸⡏⢤⢸⣿⠶⣾⣿⠟⠴⢿⣽⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣌⠀⠀⠀⣿⠀⡿⠀⠀⠀⠀⠀⢸⣿⠃⡍⢸⡏⠟⢸⣿⡿⢸⣿⡟⣿⣿
⠀⠀⠀⠀⣿⣮⣷⣿⡂⢀⣿⡂⢨⣿⡟⢹⡇⠀⠀⠀⠀⠀⠀⠸⢇⣠⠀⠀⡤⠀⠖⠀⠀⡿⠀⡇⠀⠀⠀⠀⠀⠘⣿⠀⠀⠸⠇⠀⠘⠙⡇⠈⣿⡇⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣬⣷⡮⠸⣧⣿⣿⣷⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⡀⠀⠀⠀⢦⣤⠀⠀⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠠⠀⠀⠁⢸⣿⠇⡟⣿
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⡀⢿⣰⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣦⠀⡀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢠⠀⠀⢘⢿⠀⡇⡏
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠛⢹⠋⢹⣷⣾⣿⠃⢰⡀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢸⠀⢹⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⣿⣽⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⡌⠀⢸⣿⣿⡟⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⡄⠀⠀⢸⠀⣸⠃
⠀⠀⠀⠀⠀⢀⣿⣿⠃⢀⡍⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣰⠇⠀⢸⣿⡟⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠈⡇⠀⠀⢸⠀⢻
⠀⠀⠀⠀⠀⠸⣰⡏⠀⢸⡇⠀⠀⠈⣿⢶⡄⠀⠀⠀⠀⠀⠀⢿⣿⠀⢸⣧⠀⢸⣿⡄⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠂⠀⠀⠀⡆⠀⠀⢸⠂⡄
⠀⠀⠀⠀⣰⡟⢹⠀⠀⢸⡇⠀⠀⠀⣿⣾⡷⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⣿⠀⢸⣿⣷⣶⣶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠔⠀⠀⠀⡇⠀⠀⠸⢀⡇
⠀⠀⠀⢰⣿⡀⢻⠀⠀⢸⡇⠀⠀⢰⣿⡷⢧⡀⠀⠀⠀⠀⠀⠘⣿⣇⠀⣿⠀⢸⣿⣿⡿⣿⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠸⠁⠋⣿⡇⠀⠘⡇⠀⠀⢸⣿⣧⠘⠿⠀⠀⠀⠀⠀⠀⠸⣿⡀⢻⡄⢸⣿⢿⠁⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠁⠀⠀⠀⣼
⠀⠀⠀⠀⠘⠿⣿⣷⠀⠀⡇⠀⠀⣺⣿⠷⠠⢥⠀⠀⠀⠀⠀⠀⠀⢹⣇⠈⡇⢸⣿⠈⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⠀⠀⠀⠃
⠀⠀⠀⠓⠶⢾⢿⣿⡄⠀⠇⠀⠀⣿⣿⣶⣷⡶⠂⠀⠀⠀⠀⠀⠀⠈⣿⠀⡇⢸⡟⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⡇⠀⢸⣿⣇⠀⣄⠀⢀⡿⣿⣿⡏⢷⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡇⢸⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⡇⠀⠈⣿⡟⠀⣿⣤⣾⡇⢹⣿⣷⠸⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⡇⠀⠀⣿⡇⠀⢿⣿⣿⡇⠀⠸⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⣿⣿⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠁⠀⠀⢸⡇⠀⢸⣿⣿⠁⠀⠀⠡⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣷⠘⣿⣿⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠸⣿⠃⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠈⣿⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢠⣿⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠰⠝⠣⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠸⡄⠆
⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⢇⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢳⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠸⣗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢶⡖⢾⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠈⠎⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣧⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⠟⠋⠉⠛⠋⠉⠙⢍⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣺⣃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⡄⣀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠪⢽⡯⠔⠁⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢸⠀⠰⠔⠀⠀⣸⣠⡆⠀⠀⠀⠀⠀⠀⠀⡀⠇⢂⠀⠀⠈⠒⣗⠁⠀⠀⡄⢸⣸⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⣰⠀⢸⢰⠄⣀⠴⠊⠉⠁⠒⠀⠀⠀⠀⠀⠀⢀⡬⠛⡶⢴⣆⡀⢢⣸⡠⢃⡠⠞⢮⡓⢌⡒⢄
⠀⠀⠀⠀⠀⠀⢀⡢⣀⠀⢀⣀⣀⣈⣑⣼⣷⠊⢴⠀⢴⡂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠐⠉⠀⣉⡉⣿⣿⠭⠤⠄
⠀⠀⠀⠀⠀⠀⠘⠫⠋⠉⠉⠀⣤⠜⣹⢫⣄⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠞⠰⡎⢳⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠈⢠⠃⠀⠙⠁⠀⠀⠈⠒⣶⡯⡀⠀⠀⠀⠀⠀⠀⠠⣄⣒⣲⠞⠳⠀⠀⠀⠞⡽⣖⣒⡂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠐⠵⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠙⡐⠀⠀⠀⠀⠐⢡⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡔⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⢾⢊⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠐⠲⡶⠁⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣄⢦⠀⠀⢠⡄⢀⠀⡾⠂⣧⣞⠁⠒⠀⠀⠀⠀⠀⠀⢐⣧⣱⡀⠐⠄⢄⠀⡈⣄⠄⠀⢀⣞⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠩⠞⠋⠑⠒⠤⢝⣮⣠⡇⠊⢠⡶⠀⠀⠒⠒⠀⠀⠀⠀⠀⠔⠚⠉⠒⠬⣙⣆⣾⣁⠤⠒⡉⠉⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠘⢓⡠⢻⠿⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⠔⢺⠿⢏⡉⠙⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠉⢠⡇⠸⠈⠶⠀⠉⡟⢳⠶⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⡴⢁⣤⡀⠙⠢⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⢶⠋⠇⠀⠀⠀⠀⠀⢇⠀⠀⠠⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⢿⠀⠀⢠⠀⠀⠀⡏⣷⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠠⢛⡒⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⡀⢀⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠋⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠲⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡿⢫⣈⠳⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠿⣍⠙⣦⡙⠷
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠳⡀⠙⣦⠘⠟
⠀⠀⠀⠀⠀⢠⡎⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠙⠂⠈
⠀⠀⠀⠀⢀⡟⠀⡾⠀⢸⠏⢛⣿⢻⡟⣿⠛⣇
⠀⠀⠀⠀⠘⠃⠸⠇⢠⡿⠀⣼⡇⣼⠇⣿⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠸⠟⠀⣿⠀⠻⠀⠘⠇

⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⢳⡄
⢰⡟⠉⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠁⠀⠀⠀⣷
⢿⡀⠀⠀⠀⠈⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⣀⡤⠤⢤⣠⠔⠒⠲⣤⣀⣀⣀⣤⠴⠖⠛⠉⠀⠀⠀⠀⠀⣀⡴⢯
⡾⠛⠶⠦⢤⣄⣀⠀⠀⠀⠉⠙⠛⠓⠒⠚⠓⠒⡆⠀⠈⢷⡀⠀⠀⠙⢧⣠⡤⠴⠒⠂⠀⠀⠠⠴⠚⠋⠉⠀⢘
⠳⣤⣀⠀⠀⠀⠈⠉⠂⠀⠀⠀⠀⠐⠒⠒⠒⢲⠇⠀⠀⠀⢧⠀⠀⠀⠈⢳⣤⣤⣤⡤⠀⠀⠀⠀⠀⢀⣀⡴⠞
⠀⠀⠉⠛⠲⠦⣤⣄⣀⣀⣀⠀⠀⠛⠛⠋⣉⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠤⠶⠶⠖⠒⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⢉⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⣷⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠘⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⢠⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢚⠁⠂⢈⠁⠄⡀⠀⠀⠀⠀⠀⠀⠀⣄⠬⠒⡁⠁⠠⠀⣉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡅⠄⠘⠀⠄⠁⠖⡽⠰⢄⣚⠓⢉⢩⣀⠀⡁⠀⠂⠀⠈⠀⠂⢸
⠀⠀⢀⠴⡘⡈⠁⠆⠦⠀⠁⠉⠐⢄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡌⠀⠆⡁⢀⠢⣀⡑⠀⠀⠢⠀⠰⠧⠀⠂⢈⠁⡄
⠀⠀⢏⠣⣻⢖⢄⡤⠒⣰⠀⠀⠀⠀⠡⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⠠⠀⠀⠈⠄⠑⡀⠱⡀⠀⠀⡄⠄
⠀⢸⠜⣲⠡⢡⠆⣍⠠⠒⠇⠀⠀⠀⠀⠐⠄⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⡀⠀⠀⠐⠀⠰⠀⠡⠀⠂⠠⠄⡘
⠀⠀⠣⣃⢠⣁⠬⡀⢉⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠌⣀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢂⠃
⠀⠀⠀⠈⠙⠐⠚⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⠌⠀⠄⠀⠐⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⣇⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠌⠀⠒⠀⠀⠉⠀⢐⠀⢂⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⠤⠤⢤⠀⡈⠀⠀⢀⠁⠀⠄⠀⢀⡡⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢅⡀⡀⠈⠀⠀⠀⠄⠀⠀⠂⠀⠐⠀⠀⠠⠀⠑⠒⠂⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠑⠂⠒⡄⠀⠀⠈⡁⠀⠁⠀⠀⠁⠀⠀⡐⠐⠐⠀⢂⢡⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠀⠐⠀⠀⠀⢀⠟⠒⠲⠂⠪⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣯⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣴⣿⣿⡿⠷⢦⣕⠀⣀⣿⣿⢿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣼⠇⠈⡿⠀⠀⠘⠆⢰⠟⢡⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠀⠰⡀⠀⠀⠈⠀⢸⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⣤⣿⡾⠦⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⢸⠀⠀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠟⠉⠓⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣼⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⢀⣀⣄⣀⣻⣤⣴⣶⣿⣿⡿⣿⣷⣯⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣲⣦⣶⣿⣷⣾⣿⣾⣿⣽⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠁⠉⠲⠿⠿⠦⠬⠭⡍⢉⣉⣉⠉⠙⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠔⠒⠀⠀⢀⣠⣤⡴⠂⠄⠀⠀⢀⣀⡠⠭⠄⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡀⠁⠠⠤⠀⣀⣐⣲⣦⣤⣤⣤⣤⣄⣀⣤⣤⣤⣤⣤⣶⠶⠿⠛⠁⣠⢄⠀⠀⠀⠪⠹⠭
⠀⠀⠀⠀⣀⣀⢀⣀⣀⡠⠭⠵⠶⠶⢚⢛⡛⠛⠽⠟⠛⠋⠉⠉⠉⠉⠈⠉⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⣳⣷⠶⢦⠀⠀⠐⠄
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⢤⣤⣖⣲⡶⡺⠿⠿⠟⠛⠋⠀⢀⡞⠀⠀⠀⠀⠄⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠾⠛⢛⣉⣁⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡴⠞⢫⡀⠀⠀⠀⢀⡠⠔⠋⡀
⠀⠀⠐⠒⢀⣉⣉⣉⣙⣉⣉⣭⠤⠤⠶⠟⠛⠛⠋⠉⠉⠈⠃⠛⣉⣀⢀⣙⣩⠽⠛⠛⠛⠛⢛⣫⠭⠉⠓⠋⠉⠒⠻⠧⠒⣒⠉⠁⠀⢀⣀⣱⠄
⠉⠁⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠴⠖⠒⡋⠉⠅⠐⠊⠉⠀⠀⢉⣠⠴⠒⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⢸⣶⣶⢷⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠒⠀⠀⠀⢀⣠⡤⠤⠒⢈⠄⠙⠂⠀⠀⠀⠀⠀⠀⢀⡤⢮⣱⣎⡿⢟⣓⣿⠉⢀⣉⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠒⠒⠒⠒⠛⠛⠉⠁⠀⣠⡤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠊⠀⠐⠀⠂⠉⠰⠏⣥⠼⠃⠀⠉⠉⠁⠘⢋⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⢢⡀⠀⠀⠀⠀⣠⣀⡀⠀⣴⠇⠀⣠⣀⠀⣠⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣡⠚⡦⠀⠏⠙⠒⣦⠞⠰⠢⢍⣻⠿⣄⣠⠃⢻⣶⠟⠁⠀⢀⣀⣤⢴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠼⠅⠚⠳⡀⠀⠀⡴⠃⠀⢀⣄⣰⡏⠁⠐⠨⠉⠳⣧⡤⢶⣊⠭⣒⣪⣥⡤⠄
⠀⠀⠀⠀⠀⠀⠘⣆⠰⣓⢄⣴⠊⡡⢔⣂⡭⠴⠆⢣⣠⠊⠀⠀⡠⠊⢠⠟⠑⠢⣄⠀⢀⠔⣷⣮⣷⣞⣋⣉⡡⠴⠒⠓⠲⢄
⠀⠀⠲⣟⣛⣛⣂⣈⣧⣘⣓⡏⢂⢋⣉⣥⠴⠚⣛⡘⡅⠀⢠⠎⢀⣠⠯⢔⡶⠊⠉⠳⡄⢀⡿⠿⠥⢄⠉⢣⡀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⢰⠞⣭⠥⠲⡽⢖⠛⡌⢌⢧⠤⠔⠖⣻⣧⢔⣲⠿⠯⠓⣞⡹⠡⢄⡀⠀⢀⡟⠀⠀⠀⠀⠀⢟⡆⠹⡆⡤⡄⠀⠀⠀⡇
⠀⣀⡤⠔⠊⠉⠀⠀⠀⠘⡌⠳⣘⣿⠼⠔⣊⠭⣾⠿⡉⡇⠀⣀⠜⠋⠀⠀⣘⡣⢔⣁⢀⡤⣄⢀⣀⠀⠀⠘⡄⢱⡇⠇⠀⠀⠀⣧⣄
⣏⠀⠀⢀⠤⠄⠐⢒⣀⠴⢓⡪⠭⠶⡎⢻⣱⠼⡋⣏⣉⠽⠋⠀⠀⣀⠴⠯⣤⣔⣡⣤⣤⢤⣬⣍⡀⢧⡀⠀⡶⡘⢰⠀⢀⡠⠜⠁⠈⣆⢀
⢸⠀⠀⢠⠔⣒⠭⠔⠚⣉⠽⢤⣠⠴⠚⠉⠀⠀⠈⡻⡅⠀⢀⡤⠊⡙⢖⣶⠤⠤⠤⠤⠤⠔⢤⣄⡉⠒⠺⢤⣏⣸⣸⠾⠅⠒⠒⠊⠉⠉⠈⣇
⢸⠀⠀⠀⠉⠀⠀⣠⠾⠗⠊⠁⠀⠀⠘⡎⠇⠀⠈⢠⣣⠞⠇⠁⢎⣉⠘⣯⡀⠂⠀⠑⠒⠒⠁⢀⠌⢉⣶⣤⠬⣣⣿⢆⠀⠀⠀⠀⡀⠀⠀⢹
⠈⢣⣀⠠⠀⢀⢼⠉⠐⠂⠠⡄⠀⠀⠀⠃⠀⠀⠀⣘⢉⡆⢸⠎⡹⡆⠉⢳⣧⣤⡉⠀⣀⠈⠈⣡⣄⢧⡤⣿⣵⡷⠃⠀⡗⢀⠴⠊⠉⣲⡄⢼⡀
⠀⣸⠈⣽⡞⡑⢹⢠⢋⠰⠀⠇⠀⠀⢀⡴⠒⢮⡋⡠⣫⠚⠁⢘⣔⣁⣤⠎⠳⣦⣗⠎⠉⢱⠚⣃⣤⡼⣃⢸⣻⡇⣴⡶⠛⠁⠀⡀⠀⣏⣁⣀⠴
⠀⠈⣧⣜⠐⠀⠸⣄⣑⠉⠀⡀⣠⡔⠁⢄⠀⠀⠉⠲⣄⠀⣠⣋⣠⡶⠋⠀⠁⠀⠘⠾⠒⡾⠖⢻⢤⡀⣿⣆⣧⠚⠁⠀⠀⠀⠀⠀⠀⢹⠇
⠀⠀⠀⢸⡀⡄⠀⠓⠍⠍⠛⠁⡠⢹⢦⡀⠑⢄⡀⢀⣨⠽⡇⢀⡜⠁⠀⠀⠀⠀⠀⠀⡜⢀⣠⠴⠛⣾⣿⣿⢹⠀⠀⠃⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠳⣌⡀⠀⠤⠄⢀⣨⡴⢗⡼⠋⠢⡄⠈⢁⣠⡴⠃⡞⠒⠒⠢⢄⣀⠀⠀⣼⣿⣿⡶⠟⢲⣿⢳⣿⢸⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠙⢯⣉⠉⢉⣁⡤⠔⠋⠄⡠⠔⠋⠛⢉⣀⣤⣤⣽⣦⡀⠀⠀⠙⣄⡜⠉⠀⠀⠀⠀⠈⠁⠀⠹⢾⣧⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢠⣤⣒⡶⠾⠛⠛⠋⠉⠁⠀⠀⠈⠑⠒⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣼⣿⣿⣿⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠿⠙⣛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠰⠿⣿⣿⣿⠋⠛⠛⣻⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⢰⡿⠋⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠐⠛⢤⡀⠀⠀⠀⢀⣴⣿⡄⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⢀⣠⡴⠀⠀⠀⠀⠀⠀⢠⣿⣦⣀⣠⣴⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣶⣶⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣶⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡆⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣠⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢟⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣶⠂⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠇⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋
⡿⢿⠿⡿⠿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⡟⣟⣿

⠀⠠⠀⠀⠀⠀⣀⢎⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⣀⠀⠀⠀⠈⠐⠁⠘⠃⣦⠉⡱⡄⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢄⡀⣴⠎
⠀⠀⠀⠀⠀⠰⠊⠈⠀⠀⠁⠈⠧⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣡⢻⠀⠉⠀⡸⣖⣂⠀⠀⠀⠉⠀⠀⠀⠀⣠⣰⠘⣫⣛⢶⢄
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡲⡏⠀⠀⠀⠀⠑⠩⢷⡀⠀⠀⠀⠀⠀⠉⠉⠙⠿⡟⠻⣛⣺⠛
⠀⠀⠠⠀⠀⠀⡀⠀⡠⠀⠀⠀⠤⠀⠀⠀⢤⣀⠀⠀⠀⢀⡄⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⢀⡀⠀⠀⠈⠁⠀⠀⠀⢀⠀⠀⠀⠀⣾⠎⠛⢵⣇
⠀⠀⠀⠀⠀⠤⠌⡤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠱⡀⠟⠀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠚⠀⠀⠀⠀⢱⡀
⠀⠀⣀⠀⠀⢠⠆⠈⢢⠀⠀⠀⠀⠀⡀⡀⡙⢢⣸⣬⣆⠔⠊⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⡀⠀⡀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⣩⠼⢹⠿⣏⠭⣌⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡀⠹⣗⡼⠁⠘⣆⣾⠗⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈
⡀⠀⢀⡀⢀⢠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⢁⣤⠁⡇⠀⣆⠈⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠙⢛⣥⠀⣰⣼⠛⠒⠒⠀⠀⠀⠀⠉⠀⠀⠀⠀⠠⡀⡀
⠀⠀⠀⢧⡨⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠀⠁⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⡙⣄⠀⡀⡀⢻⣅⡞⢁⢀⠀⣠⢞⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣀
⡴⠀⣄⡸⠛⠋⠉⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⢀⠀⠀⠀⠀⠛⠛⠛⡻⠃⠛⠟⢹⠿⣇⠛⠛⠘⢿⡛⠛⠘⠃⠀⠀⠀⠀⠐⠒⠚⠛
⣄⣄⡼⠓⠂⡄⢀⡞⢀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢨⡟⠀⠻⣧⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠐⣄⠀⡈⠉
⢽⡾⣽⠤⢾⠤⢾⡖⢾⠒⠀⠀⠀⠀⠀⠈⡏⠑⢄⠀⢠⡀⠀⣀⠴⢺⠃⠀⠀⠀⠀⠀⠒⢚⣻⢧⠀⠀⢀⡞⣿⠒⠂⠀⠀⠀⠀⠀⠀⢴⡤⢸⡦⣽⠤
⠟⠞⢏⠀⠈⠓⠀⠙⢦⠀⠀⠀⠁⠀⠀⠀⠱⡀⠈⡆⢹⡃⡰⠁⢀⡞⠀⠀⠀⠀⠀⠀⠀⣼⠋⠈⠇⠀⠚⠀⠙⢷⠀⠀⠀⢀⡀⠀⠀⠈⢀⡞⠐⠃⣀
⢧⠀⠞⣯⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠐⠳⢦⡌⢢⣥⠬⢤⡧⢔⣩⡴⠆⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣩⣥
⡀⠀⣰⠋⣦⡄⠀⠀⠀⠀⢠⣤⠒⠒⠒⠒⠢⣤⡟⠀⠀⠀⢹⣎⠥⠥⠤⢤⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⢏
⠀⠀⠃⠐⠙⠆⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⢊⣩⠷⣀⠀⣀⡼⢉⡒⡲⠤⠤⠤⠛⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠰⠟⠏⡱⠚⡏⣽⢸⡝⢦⡙⠓⠂⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡞⠁⣰⠇⣺⠆⢳⡀⠹⡄⠀⠀⠀⠂⠀⠀⠘⣆⠄⠀⠀⢠⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⢀⠀⠀⠀⠀⠀⠀⢦⠀⣠⠂⠀⠀⠀⠀⠀⣇⠜⠁⠀⠉⠁⠀⠙⠢⠇⠀⠀⠀⠀⠀⠀⠀⠈⢶⠄⣵⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠃⠀⠀⠀⠀⠠⠤⠀⣶⡀⢤⠤⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠒⠰⠢⠆⠵⢆⠀⢈⠶⡄⠶⠐⠆⠆⠀⠀⠐⠂⠀⠀⢘⣧⢀⠀⡀⢀
⠀⠀⠐⠀⠀⠀⢀⠔⠀⢦⠀⠀⠀⠀⣀⡀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠟⠉⡝⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣎⠤⣰⠮
⢂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠳⡀⢂⡞⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠈⣤⠀⠀⠀⠀⠀⢀⣄⣄⣌⣄⣘⡍⠉⠀⠛⡖
⡹⠟⠛⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠠⢟⣟⠶⠰⠤⠄⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠁⠉⠑⠁⠰⢥⡀⣀⣈⡍
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⠋⡜⢢⡁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⢏⠙⢉⣦⠄
⠀⢠⡄⠀⠀⠀⠰⢀⠆⠀⢰⡰⠏⠀⠀⠈⠀⠀⠐⠃⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠳⣆⢀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠼⠃⠀⠀⠈⠺⣇
⠀⠄⠀⠀⠀⠀⠁⢉⡀⠀⡤⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠹⣆⠌⠀⠀⢂⡴⠏⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⢠⠠⡌⣄⢠⠀⡈⠈⠢⢴⠠⡎⠤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣉⣦⠞⢦⣞⡉⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠐⠁⠀⢀⡔⠐⣃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠤⠶⡶⡷⢼⠤⡷⢰⣾⣶⢠⡧⣼⣴⢤⢤⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣒⣦⣠⡄
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠜⠁⠀⠀⢀⣀⣤⣾⣿⣷⡚⠓⠂⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢈⡤⠊⠁⢈⠂⠀⠀⠙⠒⢲⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⣴⡧
⠀⠀⠀⢀⠀⠉⠘⠒⠤⠀⡤⠀⡀⠴⠦⠀⠀⠀⠀⠀⠀⠸⣄⠀⡇⠀⠀⠈⠳⡄⠀⣀⡴⠋⠀⠀⠀⢀⠀⠀⠘⢛⣟⠃⠟⠙⠛⠛⢻⣿⣦⣄⣀⣀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠷⠶⠶⠶⠓⠚⠉⠀⠀⠀⠀⠚⠰⣦⠀⠀⢹⡶⣤⣀⣀⢢⣄⣮⣵⣿⣷⠶⠶⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣀⣀⣠⣀⣾⣿⣾⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠉⠛⠆
⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⢿⣿⣯⡉
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢠⣇⢁⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠌⢻⢠⣏⠂⡴⢐⢸⡇⣴⠀⡇⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⠤⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠴⠋⠀⠙⠊⠋⠉⠉⣇⠘⢺⡗⠊⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡾⠋⠡⠊⡀⠀⢸⠙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠁⠀⠀⠀⠘⢷⣶⣆⡘⠁⠀⠸⠀⠈⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠙⠻⢿⣤⣴⣿⣾⣶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢤⣰⡄⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣒⣂⣩⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠘⠶⠚⠙
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⢿⢻⣽⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⢀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⣀
⠑⠃⠀⡰⠁⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⢸⡞⠋⠁⡢
⠀⠀⣸⡇⠀⠀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⢰⡞⠂
⠀⠀⢿⡇⠀⡞⠀⢀⡔⠀⠀⢠⡄⠀⢡⠀⣧⠀⢠⠰⡟⣁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠁⠀⣻⡇⢠⡇⢀⡿⠁⠀⠀⠀⠀⠘⡉⣇⣻⠀⡆⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢹⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠄⠀⣿⣇⡼⠁⣼⠇⠀⠀⠀⠀⠀⠀⢹⣿⡟⣸⣏⣴⣿⢃⡀⠀⠀⣀⣀⡤⠘⢿⣼⣸⠟⠶⣶⣶⣦⣤⣀⠐
⠀⢠⣿⡟⢁⣼⠇⠀⠄⠀⠀⠒⠀⠀⠀⠙⠷⠟⠿⠿⠛⠚⣠⣴⣾⣷⣶⣶⣆⣀⣿⡗⠇⢤⣼⡾⡿⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣠⣤⣀
⠀⣸⠟⡴⠟⢁⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⡏⢀⣹⣧⡷⣀⢷⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⣰⠟⣭⣷⣦⣽⣧
⢰⣿⣫⠔⠚⠋⠉⠀⠈⠉⠙⢶⡀⠀⡀⠉⠉⠉⠉⠁⠀⠀⠈⠉⠛⠒⣶⣿⣶⣿⡿⠙⢿⣮⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣿⣘⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⡖⠔⡄⠀⠀⠐⠄⠀⠀⠀⠈⠀⠀⠐⠀⠀⠠⠀⣾⣿⣿⠟⠫⠁⠈⠲⡟⢿⣿⣷⡄⠀⠀⡄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠿⠿⠋⠈⠉⠁⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠀⠀⠀⢀⣀⣀⡀⠀⢤⢀⣀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡁⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣀⠐⠀⠀⢀⣀⠤⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣛⣋⣭⣹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠀⠙⠻⣭⠭⠿⠿⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠁

⠀⠀⠀⢀⣤⣶⣾⣶⣄⠀⠀⠀⠀⠀⣴⠂⠐⣦⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤
⠀⠀⠀⢸⣿⡄⢀⣿⡟⠀⠀⠀⠀⢠⣿⣄⣀⣿⡆⠀⠀⠀⣠⣿⣿⣿⣿⣿⣧⠀⠀⠀⢰⣿⣿⣿⣿⠟⠦
⠀⠀⠀⢠⣿⠇⠸⣿⣀⠀⠀⠀⠀⢼⡿⣿⣿⡿⡇⠀⠀⠀⠹⢿⣿⣿⣿⣿⠟⠁⠀⠀⢼⣿⣿⠋⠁
⠀⠀⠀⠿⠃⠀⠀⠈⠋⠀⠀⠀⠀⠈⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠙⠋⠁

⠀⠀⠀⠀⠀⠀⣠⣶⣤⠀⠀⠀⠀⠀⣴⠶⢷⣄⠀⠀⠀⠀⠀⠠⣶⣶⣶⡆⠀⠀⠀⠀⠀⢀⣶⣶⣶⡄
⠀⠀⠀⠀⣴⣿⣿⣿⡿⠃⠀⠀⠀⠀⣿⣄⢈⣿⠄⠀⠀⠀⠀⠀⠘⢿⣏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⢠⣾⣿⢡⣾⣿⠁⠀⠀⠀⠀⢠⣿⠛⢻⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿
⠀⠀⠀⠛⠁⠈⠙⠃⠀⠀⠀⠀⠀⠸⠿⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠










⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡶⢶⡿⣿⣿⠿⣿⡶⡶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⡻⣯⣿⠁⢾⣤⣿⣤⡿⡿⢾⣟⣷⣣⣪⣽⠿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⣧⡀⢙⣽⡿⠧⣌⠙⠿⣴⠶⢣⣴⢈⡽⡟⣉⢢⡀⣄⡀⡙⢷⣄
⠀⠀⠀⠀⠀⠀⢀⡞⢡⣿⣟⣿⣣⣶⠿⠡⣶⡯⠛⣀⢿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⠶⡛⣾⣿⣷⡄
⠀⠀⠀⠀⠀⢠⠏⠀⢊⣙⢛⠉⠿⣯⣾⣷⢈⣤⣼⢿⣷⣿⣿⣿⣿⣿⣧⣤⣭⣿⡿⢾⣿⣘⣿⡎⢿⣦
⠀⠀⠀⠀⠀⠀⢀⣠⣼⣟⡙⣿⣄⠙⢯⣁⡹⢿⣿⣳⣾⣿⣿⣿⣿⣿⠉⠉⢉⣠⢔⣫⢿⡄⠀⣽⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⣼⣿⢹⡏⣹⡏⣹⣷⣠⡿⣿⣾⣷⣿⣿⣿⣿⣿⡿⠉⠉⠉⢉⣴⠟⠁⠈⣿⣿⠛⢻⣿⣿⣷
⠀⠀⠀⠀⠀⢰⣏⣟⠢⣽⠿⡤⠛⣎⣙⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠐⠟⠁⠀⢀⣴⠟⢿⣶⣼⣿⣿⣿⣇
⠀⠀⠀⠀⢀⣿⡈⠡⣼⣻⣲⢙⡿⣽⠏⣾⣿⣿⣿⡿⣋⣤⣴⠶⠶⠀⠀⠀⠀⠀⠚⠉⠁⠈⡄⢿⣿⠈⢛⣿⣿⡄
⠀⠀⠀⠀⢸⡿⡿⣆⠻⣝⠐⢫⢿⣯⡈⣿⣿⣿⠟⠋⠉⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡿⠀⠆⠸⣿⣇
⠀⠀⠀⠀⣾⣧⡒⢁⠀⠘⣶⢠⣶⣌⡇⢿⣿⣿⣦⡀⠰⠶⠶⠞⠁⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⢸⣿⢦⡈⢄⣿⣿
⠀⠀⠀⣼⢿⣷⣁⡈⠛⠿⣻⢷⣌⠻⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣟⢿⣽⣿⣇
⠀⢀⠞⣙⣟⣛⠻⡍⡐⣽⣟⢦⡙⢷⣬⡜⢿⣿⣿⣿⣿⣿⣿⡷⣤⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⣿⣿⣝⣆⡈⠻⣿⡄
⢠⠋⣸⣿⣬⣿⣿⣧⡾⣦⡙⢷⣝⢦⡙⢿⡦⢿⡛⠟⠛⢿⣿⣿⣦⠙⢦⠀⠀⠀⠀⠂⡩⠃⢰⣿⡿⢋⣿⡼⣧⡀⠈⢳⡄
⡄⢠⡿⣋⣃⣠⡕⡸⢷⣮⡻⣦⡹⠗⣽⣦⣹⣾⣷⣄⠀⠈⢻⣿⣿⣆⡀⢣⠀⠀⠀⠀⠀⠀⣼⣿⣷⣾⣃⣾⣏⣹⣄⠀⠱
⠀⣿⣾⣿⣍⡈⢋⢡⣤⡙⣿⣌⡛⠨⢣⠙⢿⡟⣿⡷⠙⠦⢌⣿⣿⣿⣿⣿⣷⣶⣦⣤⣴⣾⣿⣿⣿⡿⣻⣿⣿⣧⡙⢢⠀⠃
⠸⢻⠟⣿⠲⡀⠙⡈⠆⠁⣎⠛⠃⢠⠋⣐⠰⣽⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣹⣿⣿⣿⡛⣦
⠀⣿⣭⣯⣿⣿⠃⣴⣞⠄⡠⢆⠀⣿⡟⣴⣟⣯⣿⣿⡘⣄⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣡⡿⢻⣿⣀⠺
⠀⢛⣹⢠⣏⢠⣬⢿⣡⣳⣽⣟⣤⣽⠓⣯⣽⣿⣿⣿⣏⠢⣹⣿⣿⣿⣿⣏⡛⢿⣟⠻⣿⣿⣷⣿⣹⣿⣿⣿⣿⢷⣿⣿⡏⠑⡆
⠀⠈⡟⢿⣿⣄⣷⠒⣧⣼⣏⣱⣾⢹⣿⢄⣿⣿⣿⣿⣿⣧⣸⣿⣿⣿⠟⡿⢿⣆⠈⢳⣿⣿⣿⣿⡿⣿⣿⣿⣧⣾⠏⣿⣷⡌⢁
⠀⠀⠐⣀⡄⢿⡇⣼⡉⣹⠦⣿⡄⠈⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⡀⠊⢳⣼⣿⣿⣿⣿⣶⣟⣹⣿⣿⣿⣿⣿⣇⠘⢡
⠀⠀⠀⠹⣷⡈⢿⣇⣹⣟⡾⠟⡁⢻⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⢻⡟⣻⣿⣿⣿⡿⢻⣿⣿⣏⡼⢿⡟⣤⡤
⠀⠀⠀⠀⠹⣷⣤⣅⡛⣉⣤⠾⢃⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⢘⣿⣏⣹⣿⣿⣷⣿⣩⣿⣿⣧⡟⢟⣠⡇
⠀⠀⠀⠀⠀⠈⢻⣿⣯⢿⡷⣀⠸⣧⠐⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⢻⣿⣿⣿⣿⢟⣿⣷⣊⣤
⠀⠀⠀⠀⠀⠀⠀⠙⢾⣜⡹⣿⣧⣈⣆⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣍⣿⣿⣷⣿⠿⠿⠷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⡝⢉⣼⡄⠀⢀⠈⠙⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣙⣋⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠷⣌⣀⣈⣁⣐⣚⣒⣶⣶⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡙⠦⣘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣽⣿⣿⣿⣿⣿⣯⣿⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠓⠶⢤⡀⠀⠀⠙⢿⡿⠀⠀⠀⢴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⣤⠅⠀⠀⠀⠀⠻⠐⣤⣀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠓⠀⠀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣏⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠹⠏⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⠀⢀⣤⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡛⢛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢤⡴⠃⢉⣟⢫⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⢉⣴⠏⠀⠀⠀⠋⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⣠⣶⡿⠁⡄⠀⠀⠀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠞⣠⣾⣿⠟⢀⣾⠃⠀⣾⠀⠀⠀⡟⢻⠁⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡼⠁⣼⣿⣿⡏⣰⣿⠏⠀⢸⣿⠀⠀⠀⡇⠈⠀⠹⡷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣴⠋⠀⢠⣿⣿⣿⣿⣿⡟⠀⡀⢸⣿⠀⠀⠀⠁⠀⢠⡄⢸⣷⠙⠃
⠀⠀⠀⠀⠀⢀⡾⠋⣠⣾⢃⠀⢠⣿⣿⣿⣿⣿⣿⢁⣿⡇⣼⣿⠀⢰⠀⢱⠀⢸⠁⠘⣿⡇
⠀⠀⠀⠀⠀⠻⢅⣘⡛⠣⡟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠀⣾⠀⢸⡆⢀⢠⡤⠘⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡇⣿⠀⢸⡇⠸⣄⣤⡴⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⠿⠿⠙⠻⣿⣿⣿⣿⡿⢿⣿⣅⣀⣸⣷⠀⢘⡇⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣛⣉⣱⣤⣉⣙⣋⣉⣉⣤⣬⣯⣈⣀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠃⢻⣿⣿⠈⠛⠉⠁⠘⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠃⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⢸⣿⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠈⠻⣿⣿⡿⠋⢠⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⢠⣦⠀⠀⠀⠀⠈⠻⣷⣄⢸⣿⢀⣴⡟⠁⠀⠀⠀⠀⢰⣦⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⣄⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣠⣾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣬⣿⣿⣇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⣿⣿⣥⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣉⣉⣉⣿⣿⣿⠀⠀⠀⠀⣤⡀⣼⠇⢸⣿⢸⣆⢠⡤⠀⠀⠀⠀⣸⣿⣿⣉⣉⣉⣉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠻⣷⣄⠿⢶⣼⣷⡟⠀⢸⣿⠀⢻⣿⣷⠾⠟⣠⣾⠟⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡈⠻⣷⣄⠈⢹⣧⣤⣾⣿⣤⣼⣏⠀⣠⣾⠟⢡⡦
⠀⠀⠀⠀⠀⠀⣀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⣠⣘⣷⡀⠈⣻⣿⠟⠉⠀⠀⠀⠀⠉⠙⢿⣿⠁⢀⣾⣣⣤⠀⠀⠀⠀⠀⠀⢀⣴⡿⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠻⣷⣄⠈⠻⣷⣄⠀⠀⠀⢀⣨⣽⡿⠷⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠟⠻⢿⣤⣄⠀⠀⠀⢀⣴⡿⠋⣠⣾⠟⠁
⠀⠀⠀⠀⠀⢀⣤⣌⣿⣷⣤⣌⣿⣷⣄⣤⣈⣉⣤⣀⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣤⣀⣤⣍⣤⣠⣴⣿⣯⣤⣾⣿⣡⣄⡀
⠀⠀⠀⠀⠀⠈⠉⢉⣽⡿⠋⢉⣿⠿⠋⠉⢉⡉⠉⠉⠉⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠉⢉⣉⣭⠉⠉⠻⣿⣍⠉⠻⣿⣉⠉⠁
⠀⠀⠀⠀⠀⠀⣴⡿⠋⣀⣴⡟⠃⠀⠀⠀⠈⠛⣻⣶⣶⠿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠷⣾⣿⣉⠁⠀⠀⠀⠈⠻⣷⣄⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠉⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠛⢋⣿⠃⣠⣾⠿⣦⣄⡀⠀⠀⠀⣀⣤⡾⣿⣄⠀⢹⣏⠉⠁⠀⠀⠀⠀⠀⠈⠻⣷⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣣⣾⠟⠁⠀⣨⡿⠛⢻⣿⠛⠻⣯⣀⠈⠻⣷⣄⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣾⠟⠁⢴⠶⢟⣿⣷⡀⢸⣿⠀⣰⡿⣿⠻⠶⠌⠻⣷⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣉⣉⣉⣩⣿⢿⣿⠀⠀⠀⠀⠘⠃⠘⡷⢸⣿⠠⠿⠀⠙⠀⠀⠀⠀⢸⣿⣿⣍⣉⣉⣉⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣛⣿⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣟⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠈⠻⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠘⠛⠀⠀⠀⠀⢀⣴⡿⠋⣸⣿⠈⠻⣷⣄⠀⠀⠀⠀⠘⠋⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⣠⣾⢿⣿⣷⣄⠈⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⢸⣿⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⢠⠐⠀⢀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣶⡂⣄⠀⠀⣸⠀⡈⠁⠀⠁⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡟⠁⠀⠀⣀⠀⠀⠀⢀⡶⠀⠀⡄
⣀⡄⠈⣿⠀⠀⠀⠀⠀⠀⠤⠀⡰⣄⠀⠸⠇⣀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠈⠐⢶⠗⠊⠁⠀⠀⢸⠀⠀⠀⠀⠀⢀⡀⠀⠰
⠶⠦⠛⠉⠹⣋⢛⣴⣦⣤⡆⠄⣐⠀⢣⠰⠛⠁⠀⠀⢠⠁⠀⠀⠀⠀⠀⠔⠊⢁⠀⠀⡀⠀⠀⠈⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋
⠀⠀⠀⠀⠀⠙⠂⠘⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠮⠡⡀⠀⠀⠈⠉⠉⠻⠂⠀⢠⣄
⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠢⠄⣀⣀⡄⠔⠂⠈⠀⠂⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⣀⢀⣀⡀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠁⠉⠑⠖⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⡦⠙⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠠⠠⡀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎
⠰⠿⡓⠡⠆⠂⠤⣥⣖⡂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠑⠦⡖⠀⠀⠀⠀⠀⠀⠀⠃⠐⢠⡼⠉⢝⠲⠀⢀⣶⠠⠔⠀⠁⠀⠀⠆⠈⠁
⠀⠀⢠⢤⡤⣶⢏⡌⠈⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⢲⠱⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠒⠙⢤⠤⣶⢺⡛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠂⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⢄⠚⣶⡄⠀⠀⠀⣆⡁⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠙⢤⡆⡀
⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⣀⠌⠠⠋⠀⠉⠀⠀⣰⢃⡃⠀⠀⠁⠀⠀⠀⠁⠀⡀⠀⠀⠀⠠⠀⠀⠀⠘
⣤⣤⣦⣜⣀⡛⠀⠈⠈⠀⠐⢀⠊⠀⠀⠀⢰⡾⠞⠁⠎⠀⠀⠐⠀⠀⠀⡆⠐⠀⠈⠐⠁⠒⠀⠉⠀⢀⣠⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠤
⠈⠁⡉⠀⠐⠃⡤⠀⠤⠠⠞⠁⠀⠀⠀⠀⠛⠁⠈⠂⠀⠀⠀⠀⠀⠀⠍⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⣀
⠠⠐⢀⠄⠀⠀⡡⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⣠⡀⢄⠀⠠⠁⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⢰⡆⡀⠀⠘⠿⠃⠂⠀⢈
⠟⠀⢁⠀⠐⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠈⠀⠈⠀⠀⠀⠀⠈⡴⠀⢀⡀⠀⠀⠀⠀⠀⠲⣀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⢀⡀⠢⠠⠈⠓⠠
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣹⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⡄⠀⠀⢀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⡄⠀⡀⢄⠀⠀⠀⠀⠀⠀⡂⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢡⠆⠀⠀⠀⠀⠀⠘⡂⠀⠀⠠⢾⡿⠈
⠀⠀⠀⠀⡨⠣⠄⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⢣⠹⠌⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠁⠈⠀⠀⣠
⠀⠀⠀⠘⠀⠀⠂⠀⠄⠀⠀⠀⢀⣰⠆⠀⠀⠀⣀⠀⠀⢀⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠄⢀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⣀⡔⠁
⠈⠀⠀⠀⠀⠄⠤⠊⠈⠀⢣⡀⠠⠁⡆⣀⠖⠉⠀⠁⠀⠈⢋⡋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠔⠈⠀⢀
⠀⠈⠐⣄⠀⠀⠀⠀⠀⢈⠄⠈⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⢰⡵⠂⠉
⠺⠃⠀⠁⠐⠀⠂⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠒⠠⠀⠉⠊⠉⢳⠂⠀⢄⡰⢾⠀⠀⠚
⢂⠒⠀⠀⠀⠀⠀⡈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠈⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠈⠺⠻⠖
⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢖⡖⠀⢀⠾⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢴⡖⢲⡒⣖⠒⢲⠤⣀⠀⡰⢋⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⣉⡄⠀⢰⠀⣸⢠⢳⠀⡈⢀⡈⠙⠷⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⢉⣀⠤⠤⠤⡜⢀⠇⡜⠸⢠⠇⢸⡇⠀⠀⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠘⠔⠒⠒⠒⣲⠃⡜⢰⠃⣀⡿⢀⢿⠁⠀⠀⡌⠗⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠡⡀⠹⡀⠉⣩⠃⡼⢁⠇⣰⢟⠇⡜⡌⠀⢰⡐⢸⠃⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠳⡐⢄⠁⡴⢁⡜⢡⠊⣰⠏⢈⡰⢀⠀⠀⣼⠀⡜⠁⠐⢸⠀⠀⠀⠀⢀⣠⠶⠛⠉⠉⠈⠉⠉⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢦⡙⣤⠟⣠⠎⡠⠃⣰⠃⢀⡞⢁⠎⡸⢀⠃⡸⠉⣉⡁⢸⠀⠀⠀⣠⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡠⠞⣡⠞⢁⡜⠁⡴⠁⢀⡞⠀⡜⢀⠃⠌⡰⠤⠤⠤⠤⡍⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠚⢁⡴⠋⢠⠎⠀⡰⢫⠀⠸⠀⠎⡜⢰⣓⣒⠒⠂⣖⣡⠾⠋⠀⠀⠀⠀⠀⣶⠃⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡷⡎⢀⠔⠁⡠⠊⠀⡆⠀⠀⠀⡰⣁⠋⠤⠄⣒⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠻⢶⣄⣴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⡡⠊⠀⠈⠳⣤⡊⠀⠀⡜⠀⢀⠀⠜⡵⠁⣈⡁⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠓⠚⠦⠶⠧⠦⠖⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠠⠊



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠚⠛⠛⠓⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠴⠶⠒⠲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠙⢳⢏⣠⡄⠀⢠⣶⣦⡀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⠞⣉⠁⠀⢀⣤⣉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠉⠀⠀⠀⢀⡎⠈⡇⢠⡏⠀⠘⡆⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣠⠖⠒⠚⢿⣆⡇⢸⣦⣄⡠⠟⠋⠉⠁⠀⠈⠉⠂⠀⠀⠀⠐⠋⠉⠀⠀⠀⠈⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠉⠀⣇⠀⠀⠀⠀⠩⠤⢬⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠘⢦⣀⡀⠠⢤⡀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠒⠀⠲⠤⠤⠀⠄⠀⡀⣾⣀⡀⠀⠀⣰⠛⠓⠚⠛⠃⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠨⣉⣡⡽⢀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡚⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠤⠤⢶⡀⠈⠁⠈⠳⠶⠛⠦⣤⣀⡀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠂⠁⠁⠀⠀⠀⡴⠛⢦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠙⠲⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣶⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⢦⢤⣤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⣸⠷⣄⠀⠀⠀⠀⠀⣀⡤⠟⣆⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⠤⠶⠛⠁⠀⠈⠙⠲⠶⠒⠛⠉⠀⠀⠈⠓⠶⠤⠤⠶⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⣾⡿⣿⡿⣻⠟⢿⣿⣽⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣶⡿⠿⢟⠛⢉⢥⣴⣻⡿⠻⢽⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢡⣿⡟⢡⣲⣾⣿⣮⣿⠟⠉⠁⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣠⣾⣟⡥⢙⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⣀⢻⣿⣿⣿⣿⣀⣤⡶⡾⠒⠄⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣧⡄
⠀⠀⠀⠀⠀⠀⡠⠂⣀⣤⣿⣿⠿⠀⢐⣾⣿⣿⡿⠟⠟⠉⢁⣀⣴⣀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠐⠉⠀⠀⠛⠉⢋⡀⣄⣶⣾⣿⣿⢯⠙⠐⠊⠉⠛⠿⢿⣿⣦⣀⣀⣠⣴⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢺⣿⣧⣧⣾⣿⣿⣿⣿⣿⡻⠆⠀⠀⣠⣴⣄⣻⣿⣿⣿⣿⠟⠋⠀⠋⠙⢻⠟⡁
⠀⠀⠀⠀⠀⣠⣶⣾⣿⣛⣛⣿⣯⣭⣿⠉⢂⠁⠀⣀⣸⣿⣿⣿⣿⣿⡿⠛⠂⠀⠀⢀⡤⣴⣾⠿
⠀⠀⠀⠀⠀⢻⡿⠿⢿⣿⣿⣿⡿⣯⡉⠉⠀⠀⢺⣿⣿⣿⣿⣿⣏⠉⠀⠀⠀⠀⠀⠀⠀⠈⡁
⠀⠀⠀⢠⣶⣦⣄⡀⠀⠅⠑⠀⠀⢀⣠⡀⣀⣦⣾⣿⣿⡟⠎⠙⠻⠷⠀⠀⠀⠀⢀⣤⣶⡿⠃
⠀⠀⠀⣿⣿⣿⣿⣷⣢⣦⣿⣼⣷⣾⣿⣷⣿⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⣀⣚⣫⠭⠁
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠉⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠢⡀⠀⠀⠀⢀⡤⣀⠀⢀⣾⠀⢀⣄⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣡⠎⣳⠀⠇⠉⠓⣦⠋⠐⠢⢍⣿⠧⣄⣜⠘⣷⡟⠁⠀⢀⣠⢤⣖⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠵⠓⠊⠙⣄⠀⢀⠜⠁⠀⢠⢄⣼⠃⠀⠐⠈⠉⣻⠤⣖⡯⣕⣲⣭⡵⠶
⠀⠀⠀⠀⠀⠀⠹⣆⠫⣷⢴⡎⢥⢒⡨⠤⠶⣃⢹⡴⠋⠀⢀⠔⠁⡸⠋⠁⢲⣄⠀⠀⢹⣮⣶⣟⣍⡡⠤⠒⠊⠉⠓⢦⡀
⠀⠀⠙⠭⠭⢭⣥⠭⢦⡬⣽⡐⡘⣄⠤⠒⠋⣋⡄⠆⠀⣰⣃⠠⢴⣓⡼⠋⠁⠈⢳⡀⠼⠟⠓⠢⢀⠑⣄⡀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⣸⡱⠖⠂⠱⡛⢍⠱⣱⠰⠖⢘⣊⢽⣷⡪⡍⠛⠊⢩⡯⠞⠐⢤⠀⢀⠼⠀⠀⠀⠀⠀⠪⢥⠘⣇⡖⡆⠀⠀⠀⡇
⣠⠤⠒⠉⠀⠀⠀⠀⠀⢹⢈⣣⠿⢒⣫⡥⠒⣽⣟⣄⣧⣀⠔⠋⠀⠀⡠⠮⢺⠵⠤⠔⠲⣄⠤⢄⠀⠀⠆⢸⢇⠁⠀⠀⢰⠗⢆
⢧⠀⠀⠐⠒⣈⠩⢄⣒⡭⡕⠒⢈⣻⣸⠼⠓⠠⣦⡤⠊⠀⠀⣠⣔⡛⠒⠒⠣⠖⠲⠒⡛⠲⠦⣙⢦⡀⣯⢦⢸⣀⠤⠒⢁⣀⠼⠧⢤
⢸⠀⠀⠰⠭⠒⠋⢁⡠⣒⠭⠖⠚⠡⣄⡄⡀⢠⠜⡇⣀⡴⡏⣲⠮⢫⡟⠛⠛⠹⣉⡉⡹⠒⠦⣄⣉⠑⠷⡜⣾⠉⠈⠁⠀⠀⠀⠀⠘⡇
⠸⡄⠀⠀⠀⠀⡤⢟⡉⠀⠀⢀⠀⠀⡁⠉⠀⠀⢸⠾⡍⢁⡠⢧⡺⢸⣳⠀⠁⠄⣀⡁⠀⠤⠂⢰⠚⣹⢏⡝⡟⠱⡄⠀⢀⡤⢄⡀⠀⡇
⠀⢸⠢⢌⡠⠂⠇⢠⠤⡀⢙⠀⠀⠀⠀⣀⡠⠤⠖⠰⣤⠈⠠⢪⠇⠀⡽⣛⢱⣀⠴⠦⣀⣴⠓⠛⣾⢹⡽⡟⣀⣀⡧⠖⠁⠀⢀⡿⠂⠃⣠
⠀⢻⡄⠻⠱⠀⠀⠠⠀⠄⠈⠀⢀⠴⠊⠀⠉⠣⣌⠘⠀⢀⠔⢓⣊⠟⠀⠉⠙⢆⣄⢀⣇⡴⡋⠉⢶⡌⡟⡡⠟⠉⠀⠀⠀⠀⠀⢷⠒⠊
⠀⠀⠉⢇⠀⠀⠘⡑⠶⠦⠴⠀⠹⣤⡈⠢⡀⠀⠈⠙⢶⡋⠊⡹⠃⠀⠀⠀⠀⠀⠁⡞⠀⢀⡯⢵⣾⣷⢺⠀⠀⡄⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠘⣆⠃⠄⢀⡀⡀⠀⢒⣴⢈⡿⢦⡈⠢⠒⠉⣠⠇⣼⠁⠀⣀⠀⠀⠀⠀⣼⣴⣮⣭⠄⣼⡟⣿⢸⠄⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠈⢷⡒⠂⠒⠒⠊⣉⠵⠋⢀⣠⠕⠶⠖⠋⠁⣀⣣⡀⠀⠀⠉⢆⢀⡜⠛⠋⠉⠀⠈⠛⠀⢻⣸⢆⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠉⠙⠒⠋⠉⠀⠀⣀⢭⣤⣤⠶⠾⠟⠛⠛⠉⠉⠓⠦⠤⡬⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⠃⠀⠀⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣴⣆⠀⢀⡠⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠺⠐⢃⣤⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠾⠜⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣩⣍⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡷⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⠠⣤⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢿⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠉⣶⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⣉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣽⠃⠀⢀⣉⡉⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣻⠏⠀⠀⢸⣿⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢷⡏⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⡿⠀⠀⠀⠀⣛⠛⠛⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡾⠁⠀⠀⠀⢠⣿⣿⣿⣷⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠛⠻⠽⢟⣻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⢠⣿⣷⣶⣦⣦⣤⣤⣭⣭⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠁⠀⠀⠀⠀⠀⢠⣶⣶⣶⣤⣤⣬⣉⡉⠉⠙⠛⠋⠉⠉

⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠠⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠰⢦⣄⣀⣛⣙⣢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣠⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣭⣶⣶⣦⣤⡄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢸⣿⣿⣟⡽⠚⠋⠉⠀⠈⠁⠀⠀⠀⠠⣄⠀⠰⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣇⣠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣶⣶⣶⣶⣶⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠻⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣽⣧⣧⣧⣧⣧⣿⣼⣾⣼⣒⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⡶⠓⠀⠀⠀⠒⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠃⠀⢸⣿⠄⠀⠀⠀⠀⠉⠉⣪⣿⣲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣤⣤⣄⣀⠀⠀⡤⠀⣠⡄⠀⠀⣸⠁⠀⢀⣤⢴⣖⡿⠋⢹⣿⣽⣻⣿⣆⠀⠀⠀⠀⠀⣾⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣾⣷⣜⡿⣿⡿⠂⠀⢾⣽⠃⠀⠀⣿⠶⡿⢟⠀⣁⠲⣅⠀⣰⣿⣿⣿⠟⣋⣴⠀⣀⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⡔⠀⠈⢲⢿⣝⣿⠃⠀⠀⠊⣿⠁⠀⠀⢹⠀⠀⡀⠃⠬⡱⣿⣿⣿⠿⣋⣴⠞⣉⡴⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⢱⠄⠙⠒⣶⡆⣰⡷⣠⢾⣿⡟⠀⠀⠀⠾⢿⡀⣆⠀⡘⡀⠘⠁⢀⣶⣷⠞⣋⣵⠿⣋⣵⣾⣿⣷⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⡈⠀⠀⠀⣻⣴⣟⣺⣷⣌⣻⣧⠀⢀⠂⠀⠄⡀⢿⡀⠰⢹⣼⣷⡟⣫⣴⠿⣫⣴⣾⣿⣿⣿⣿⣿⢫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢋⣀⣘⣉⣥⣶⣿⣿⠿⠛⠛⢿⣄⣾⣌⠂⠂⣧⣷⣥⠀⢷⣻⣷⣿⣯⣷⡿⣿⣷⣿⣿⣿⣿⣿⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡙⣿⣿⣿⣿⣿⠛⠀⢀⠀⠀⠠⠂⠙⢿⣿⣷⣾⣿⣿⣇⠘⢿⣿⣿⣿⣯⣷⣫⣿⣿⣿⣿⣿⣿⡇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣯⣿⣷⣿⠁⠣⠀⢡⣀⣠⣶⣄⣀⡠⠟⣫⡴⢛⣽⣿⣿⣾⠿⠛⢻⡿⠛⣛⠛⢻⣿⣿⣛⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠇⠠⣄⡲⠄⢉⣛⣠⠿⣋⡴⢚⣩⣶⣿⣿⡿⠋⠀⠀⠀⠀⠀⠘⡏⣘⣿⣿⣯⡹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⢷⡤⢋⣥⠞⣩⣶⠛⢿⣿⣿⣿⢑⠰⠀⢂⣤⡄⢀⡀⡼⢿⣿⣟⣟⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣀⠾⢋⠴⡯⠞⠃⠀⠀⠈⠙⠛⠋⠁⠐⠁⢇⠀⠈⠘⢡⢏⣟⣻⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣛⣶⣶⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠾⠄⣼⣿⣿⢏⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣻⢿⣛⣿⡶⢟⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠾⠛⠛⠋⠉⠁⠙⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠛⢛⣛⠛⠋⠉⠛⠲⠦⣄⡀
⠀⠀⠀⠀⣰⠟⢉⡤⢖⣻⣭⠭⠭⠭⠭⢭⣭⣟⡓⢮⡙⣦⣄
⠀⠀⢠⡶⠋⠾⣫⣾⡿⣓⣚⣛⣻⠿⣯⣷⣲⣬⢻⡀⣳⢸⡝⣷⡀
⠀⢀⡟⢠⡆⣴⡿⢡⢞⣭⠴⠶⠶⠶⠶⣶⣮⠶⠖⠛⠾⣾⡁⢹⡇
⠀⢸⡇⢸⡘⢱⣇⡾⠋⠀⠀⠀⠀⣴⠛⠉⠰⣄⠀⠢⡀⠀⠙⣿⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⢀⡟⢈⠻⢶⣼⡇⠀⠀⠀⠀⢠⣇⠈⢄⠀⠈⢣⠀⠈⢢⠀⠘⣇⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠏⠁
⠀⠘⣇⠀⢙⠦⣌⡙⠛⠛⠛⠛⠻⣯⡳⡈⠳⣄⠀⠣⡀⠀⠣⢰⡇⢠⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢺⠏⢀⡴⡓⠿
⠀⠀⢸⡇⠀⡉⠒⠭⣍⣉⣉⣉⣁⠈⢷⣝⢆⠈⢣⡀⠑⠄⣠⠏⢀⠆⢀⠹⡇⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠛⠚⡟⠋⠀⣀⣤⣀⡀
⠀⠀⠘⢿⣦⣄⡉⠒⠦⠤⣄⣀⡠⠄⠀⠹⣦⠳⣄⠑⠀⣰⠋⡴⠁⣠⠋⠠⣧⣀⡀⠀⠀⢀⣴⠞⣫⠖⠉⡠⠀⠈⠋⢹⡟⠛⠒⠻
⠀⠀⠀⠀⠀⠉⠛⠶⣬⣭⣭⣉⣑⣒⣭⣴⠟⣧⠘⣠⡾⢡⠞⠀⡴⢡⠔⢀⡽⠋⠛⠛⠛⠛⢁⠔⣁⠔⠉⠀⣠⠔⠂⣈⠟⣛⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⣨⡟⢋⠔⣡⠔⠊⡰⠃⣠⡟⠀⣀⠉⠉⠀⠈⣡⠞⢁⡤⠖⣉⠤⠒⣉⡴⠞⠉⠉⠫⡹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⡫⡠⢋⠜⠁⣀⠜⢁⣴⠟⠀⣀⡈⠉⠙⠒⠊⣡⠖⢉⠔⠉⣀⣴⠞⠋⠀⠀⠀⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢰⡱⢁⠎⡠⢊⣥⠾⠻⢧⣝⡢⠤⣉⡉⠉⢉⣉⡥⠞⠁⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡏⡇⡞⣸⣳⠏⠁⠀⠀⠀⠈⠛⠻⠶⣦⣭⣭⣤⣤⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢷⣹⢳⣿⣿⠀⠀⠀⠀⠀⠀⠀⢨⢶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣌⡛⢮⣇⣻⣄⠀⠀⠀⠀⢀⣰⡵⠟⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢙⠲⣤⣍⣉⣛⣛⡓⠒⠛⢹⣇⣾⠕⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣎⠓⠤⣀⡉⠉⠉⠉⠁⠀⠀⢸⣥⣤⣞⠭⠽⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠢⡀⠉⠙⠒⠒⠒⠋⠁⠀⠸⣧⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣌⠲⠤⠤⠤⠴⠒⢋⣥⠾⠛⠋⠓⠒⢽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣤⠴⠞⠋⠁

⠀⠀⠀⠀⢀⣴⠟⢿⣶⠶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣄
⠀⠀⠀⢠⡿⠁⠀⠀⠙⢷⣄⠈⠉⠛⠻⠶⠶⣦⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⡶⠶⠿⠛⠛⣻⠟⠋⠀⠈⢳⣄
⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠙⢷⣦⣤⣄⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⢹⡆
⠀⣰⠟⠙⣷⡀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣍⡁⠀⠀⠀⢀⣤⣴⠶⠶⣶⣤⠴⠦⣤⠀⠀⠐⣲⡿⠛⠁⠀⠀⠀⠀⠀⠀⣠⡾⠳⣦⡀
⣼⠏⠀⠀⠈⠛⠦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢛⣷⡾⠋⠁⠀⣠⠞⠉⠀⠀⣀⣸⠷⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⢻⡄
⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⢰⠃⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠐⠶⣤⣤⣴⠟⠀⠀⠀⠀⠀⠋⠀⠀⠀⢰⡇⠀⣀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠈⠙⠻⢶⣤⣄⣀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠶⣴⡟⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣶⠶⠶⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⢿⡄⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣧⣀⠀⠀⠀⠀⣀⣠⣤⣶⠶⠶⠿⠟⠛⠻⠿⣶⣶⡿⠁⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠟⠛⠋⠉⢸⡟⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⡶⠶⠶⠶⠶⠶⣶⣶⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣴⣿⢿⡿⠷⢦⡄⠀⣨⣿⡿⢿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠇⠀⡿⠀⠀⠘⠀⠸⠋⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠐⡀⠀⠀⠀⠀⡀⠀⠠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⣶⣿⡟⠂⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠞⠀⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣧⠀⣆⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⢦⣄⢠⣤⣤⣄⣿⢶⣴⠟⣽⣿⣿⣿⣿⣶⣿⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣽⣦⠾⠿⠿⠿⡿⣿⠷⣿⣿⠿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠒⠒⠒⠂⠀⠤⠄⠀⢉⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠒⠒⠃⠁⠀⣀⣤⡤⠒⠚⠐⠂⠀⣀⣤⠤⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⣤⣤⣤⣭⣽⣶⣶⠶⠶⠶⠤⠶⠶⠶⠶⠾⠖⠛⠋⠁⠀⠰⢀⣀⡀⠀⠑⠘⠂
⠀⠀⠀⠀⠀⠀⠤⠤⠴⠖⠒⠚⠛⠉⠭⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣄⣤⡽⠟⠉⡧⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⠶⠶⠶⠒⠒⠚⠒⠒⠛⠛⠛⠉⠤⠰⠖⠉⠉⠉⠉⠁⠀⠀⣀⡤⠃⠀⠀⠀⠀⠀⢀⡠⠚
⠀⠀⠀⠀⠀⠀⠤⢤⣤⣀⣠⠤⠶⠶⠛⣛⣩⣤⣤⣶⠶⠶⡶⢶⠿⠶⠖⢾⡶⣦⣤⣤⣤⣠⣤⣠⡤⠴⢶⡾⣏⣁⣲⡄⠀⣀⠠⠄⠂⠁⠠
⠀⠠⠀⡤⠠⠤⠤⠤⠴⠖⠒⠒⠒⠂⠉⠉⠁⠁⠀⠀⠀⣀⣀⡬⠤⠄⣒⡤⠖⠒⠉⠀⠀⣀⠠⠒⠋⠁⠀⠀⠀⠈⠉⠉⠀⣶⡀⣀⡀⣤⡤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠒⠚⢋⡉⠁⠀⠀⠀⠀⠀⢀⡀⠀⠠⡖⠋⠁⠀⠀⠀⠀⠀⠀⡠⢀⢠⣠⣀⠸⣿⠏⠿⠷⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡄⣀⣀⣀⣀⣠⡤⠶⠛⠉⠀⢀⣠⡤⠖⠐⠉⠁⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠤⠊⡴⠾⢷⡛⠒⠾⡧⠀⣲⡾⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠁⠙⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣽⣭⣯⣯⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠿⡗⣦⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⢻⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡾⡗⠏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸⢻⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄⠀⠀⠀⠑⣷⣤⣔⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢶⣲⠿⠁⠀⠀⠀⢠⣄⣠⣀
⠀⠀⠀⠀⢀⣴⣼⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧⣦⡀
⠀⠀⠀⣸⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣧
⠀⠀⣼⣿⣿⣿⣿⡿⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡻⢿⣿⣿⣿⣿⣧
⠀⢰⣿⣿⣿⣿⣟⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠖⠃⢠⡄⣠⠄⠘⠲⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⡇
⠀⢹⣿⣿⣿⣿⣾⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⢀⣽⣿⣿⣿⣿⣧⡀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⢿⣿⣿⣿⡏
⠀⢸⣿⣿⣿⣷⣿⣿⣿⡿⠛⣄⡀⠀⠀⠀⠀⠀⠀⢸⡧⠀⣨⣿⣶⣿⣷⣾⣿⡁⠀⢼⡇⠀⠀⠀⠀⠀⠀⢀⣤⠛⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⣄⣿⣿⣷⣶⣤⣄⣀⣀⠈⠻⢶⣿⣿⣿⣿⣿⣿⣿⣿⡶⠟⠁⣀⣀⣤⣤⣶⣾⣿⣿⣠⣼⣿⣿⣟⣾⣿⣿⡿
⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣯⣿⡿⣿⡿⣾⡽⣻⣻⣿⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⢻⣿⣟⣟⣯⣷⢿⣿⢿⣿⣼⣿⣿⣯⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣻⣿⣿⣷⣿⣿⣷⣫⢿⠵⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⢿⡾⡿⣟⣾⣿⣿⣾⣿⡿⣟⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⣿⣿⡿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⠿⠿⠛⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡿⢿⠏⣼⣿⣿⣿⣿⣿⣿⣧⠹⡿⢿⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠻⢿⣇⠈⢰⣿⣿⣿⣿⣿⣿⣿⣿⡆⠁⣸⡿⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡅⠀⣠⡿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢿⣄⠀⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠯⠉⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⠽⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠋⠛⠋⠋

⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢀⣴⣿⣿⡿⠆⠀⠀⠀⣸⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆⠀⠀⠀⠀⠀⠀⢐⣫⣭⣤⡄
⠀⠀⠀⠈⠉⢿⡿⠁⠀⠀⠀⠠⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠋⢿⣿⠋
⠀⠀⢀⣤⣤⣿⣅⣀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⣿⣧⣀⣀⠀⠀⠀⠀⢀⣸⣿⣤⣤⣄⡀
⠀⠀⣾⣿⣿⣿⣿⣿⡇⠀⠀⢀⣿⣿⡟⣠⡄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡇⠀⠀⣼⣿⣿⣿⣿⣿⠻⣿⣄
⠀⢀⡿⠹⣿⣿⣿⣿⠁⠀⠀⠘⠈⠉⠁⢿⡇⠀⠀⠀⠀⠀⢀⣾⠋⢿⣿⣿⣿⠻⣧⠀⠀⣿⠿⣿⣿⣿⡏⠀⠈⣻⣷
⠀⠸⣇⠀⠻⠿⣿⣿⡄⠀⠀⠀⡄⠀⡜⢸⡇⠀⠀⠀⠀⠰⠿⠷⠺⢾⣿⣿⠏⠀⣿⡄⢰⡟⠠⡯⣭⣭⣀⣠⠼⠋⠁
⠀⠀⢹⡆⢠⣿⣿⣿⡇⠀⠀⠘⠁⠀⠀⢸⡇⠀⠠⣀⠀⠀⠀⠀⣠⣾⣿⣧⣠⠾⠋⢠⡿⠃⠀⠀⠀⠙⠋⠁
⠀⠀⢸⣧⠀⣹⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠙⢶⡦⣼⣿⣿⣿⣿⡇⠀⢀⡞⠁
⠀⠀⠀⠉⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠦⡀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠙⠂⠀⢹⣿⣿⣿⠃⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⣤⣼⡇⣀⣀⠤⠂⠀⠀⠀⢸⣿⣿⠏⠀⠀⠀⢰⣿⠇⠠⣶⣶⡄
⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⣾⣿⡄⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠛⠀⠀⣿⣿⡄
⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⣼⢿⣿⠀⠀⠀⠀⢸⡟⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿
⠀⠀⠀⢠⠏⣸⡇⠀⠀⠀⠀⡾⢀⣿⡆⠀⠀⠀⠀⠀⠀⠀⣸⠏⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⢰⡟⢰⣿⠀⠀⠀⠀⢸⡇⣼⡏⠁⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⢰⡿⠁⠁⣿⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠛⠇⠀⠀⠛⠃⠀⠀⠀⠉⠁⠀⠀⠰⣶⠄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣠⢾⡿⠱⢁⠀⢀⡘⠹⠟⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠅⣠⣶⡐⠎⠀⠁⠂⠀⠁⢈⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⢆⠼⠋⠀⠀⠄⠒⢀⣈⣬⣤⣛⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣻⠆⠂⠀⣀⣰⡖⠂⠜⠙⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⣠⣴⡏⠉⠈⠉⠀⢀⠀⠀⣐⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢄⣠⣄⡀⠀⠀⠀⠀⠈⢻⣾⠇⠉⢆⠀⠀⠀⡤⡐⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⡀⠀⠀⠀⠀⠀⣠⣶⣿⣶⠶⣟⡿⢿⡷⣤⠀⠀⠀⠀⠈⠙⠶⠦⣤⣤⡤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⣾⠝⣯⣿⠿⠁⠭⠟⠘⠧⣉⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢠⣧⣢⠞⠁⠀⠀⠒⢈⠥⣴⣾⡍⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠈⣷⠽⠋⢀⣠⡴⣿⡁⠊⠈⠠⠤⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢹⣄⣶⡿⡅⠀⠀⠁⠠⢀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠙⠻⢄⣈⡐⢶⢶⢋⡲⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠛⠉⠙⠛⠛⠛⠓⠶⠶⠶⠶⠶⠶⠶⠛⠛⠉⠉⠁⠀⠉⠙⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⢸⣇⠀⠀⠀⠀⠠⠤⠄⠀⠀⣀⣀⣀⣀⣀⡀⠻⠆⠀⠘⣧⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠛⠶⠦⣤⣤⣀⣠⣶⣟⠉⠉⠉⠉⠉⠙⢦⣄⣤⠴⠋⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⡀⠦⠀⠀⠀⠀⠈⠉⠉⣿⡄⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠁⠀⠀⣀⠀⠀⠀⠀⠀⣿⠃⠀⠘⣣⠀⠀⠀⢸⠀⠤⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⡀⠀⠀⠀⠉⠰⠖⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⢂⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣄⣀⡀⠀⠀⢠⠏⣿⠀⠀⠀⠀⠀⠀⠀⣿⡾⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣿⣿⠏⢰⡇⠀⠀⠀⡀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⢸⠁⠀⠀⠀⠋⠶⠀⠀⠀⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣈⠳⢦⡀⠀⠀⠀⠀⣤⠀⠘⠂⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠓⠀⠀⠀⣹⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⢀⣀⣤⣿⣿⡇⠀⠀⢀⣠⣤⣶⠺⣿⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢻⡟⠛⣿⠙⣯⠙⣿⡷⢶⣟⠙⣿⣀⣿⣄⣿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣿⣴⣿⣷⠿⠿⢿⣇⣼⣿⣶⣿⠿⠛⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠁⠀⠀⠈⠛⠋⠉⠉


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣷⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⣭⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⠀⠙⡿⠗⠲⢲⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⢹⣿⠷⢾⣶⣶⣾⣿⣿⡆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡘⣷⡀⠀⠀⠀⠀⠀⠰⢄⡲⣭⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⢷⡀⠀⠀⢀⣴⠟⠲⣽⣿⣿⣿⡄⣿⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢽⣦⣀⣾⣿⣦⣀⡀⠑⠫⣛⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣷⣶⣤⣙⡻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢳⣏⡿⠸⡷⢈⠛⠛⢻⣟⠿⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡿⣿⠁⠄⠀⠘⠀⢀⠏⠉⢳⣌⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢀⡾⣰⠃⢀⣤⣴⣶⣶⣴⣄⠀⠀⠙⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠞⠁⠏⢠⣾⢻⣿⣿⠟⢿⣿⣷⣤⡀⢿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡤⠚⠀⡼⠀⣿⣿⠀⢿⣿⠀⢰⣿⣿⣿⡽⠜⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣴⠋⠀⠀⢠⠀⢠⣿⣿⡇⠘⣿⡄⢸⡿⣿⣿⣧⡚⢿⣿⣿⡿⣿⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠘⠇⠀⠀⢠⡄⠀⣸⣿⣿⣿⡄⢸⡇⣾⣿⡏⢿⣿⡇⠀⠹⣿⣧⡹⣧⠀⠙⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡇⠀⢠⠟⠀⣰⣿⣿⣿⣿⠷⢼⣷⣿⣿⡀⢸⡟⣧⡀⠀⢻⣿⢿⢿⣷⡀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠃⠀⠂⠀⣾⢱⣿⣿⣿⡟⠀⢸⣷⣿⢻⡇⠘⣧⣿⣦⡀⢸⣏⠈⢂⡙⣿⣆⢹⣿⡄
⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⢸⣿⡿⢜⣧⠀⣿⣿⡙⢷⡌⣿⡄⠀⠱⡸⢿⣿⡿⣧⡀
⠀⠀⠀⠀⠀⠀⡸⠀⠀⢀⠁⠀⠀⢀⣾⣿⣿⣿⣿⣿⡇⠀⣾⣿⠇⠘⣿⠀⣿⢿⡇⠀⢻⠛⢷⣤⠀⠘⣼⣿⡅⠹⣷
⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⢰⣿⣱⣿⣿⣿⣿⣿⣷⠀⣿⣿⠀⠀⣧⢀⣿⠘⣧⠐⠘⡀⠀⠙⣷⡄⠘⣿⣇⠀⣿⡆
⠀⠀⠀⠀⠀⠛⣧⠀⠀⠀⠀⣠⡾⢋⣿⣿⣿⣿⣿⣿⣿⡆⢸⡇⠀⢰⠇⢸⡟⠀⢹⡆⠀⢁⠀⠀⠈⣿⡄⢸⣿⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠘⠛⠶⢦⣾⣏⣴⣿⣿⡿⢹⣿⣿⣿⣿⢷⣸⡏⠀⢿⠀⠀⡇⢠⠀⣷⠀⠈⣄⣀⣀⣘⡇⢀⣷⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣷⣿⣿⣹⣿⡏⣼⠉⣷⠀⠀⠀⠀⣷⣶⣾⣿⣧⡀⢻⡟⠛⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠻⠿⠟⢻⣄⡿⠂⢹⡷⠶⠾⠾⠟⠉⠉⠛⠛⠛⠛⠁⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢴⠾⠇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣋⣉⣋⣉⣉⣩⣏⣉⣽⣫⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⢄⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⡤⠈⠀⠀⠄⠀⠇⠀⠀⠀⠠⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⢠⠀⠀⠀⠠⠀⠤⢲⠀⠀⠆⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢠⠦⠄⠈⡀⡂⠀⠀⡄⠈⠜⡅⢄⡄⠀⢀⠈
⠀⠀⠀⠀⠀⠀⢀⡀⢠⡀⡀⢀⣰⠊⠁⡴⠁⠀⢀⡋⠛⢞⡳⣄⣦⡠⠊⠀⠼⢄⣴⠆⠀⠀⠀⠀⠠⠦⠠⠄⣀
⠀⠀⠀⠀⠀⠸⠁⠒⠀⠠⣤⠃⠃⠀⠰⠂⠉⢀⡀⠀⠉⠘⠂⠘⡟⠁⠈⠁⠀⠈⠁⠐⢄⠘⠁⠀⠀⠀⠀⣀⢈⠃⣀
⠀⠀⠀⠀⣄⣤⠒⡠⠶⠞⠛⠛⠲⢤⣁⠀⠀⠀⠄⠙⠲⠄⡀⢀⣧⠀⠀⠀⠀⠄⠠⠤⡏⢃⢀⠀⠀⠀⠀⠘⡸⠀⠉⡆
⠀⠀⠀⠀⠨⢦⠞⢀⠀⠀⠀⠀⢀⠀⡜⢣⡂⢀⠀⠀⠀⠀⠏⠳⣿⠀⠂⡠⠀⠀⠀⠀⠀⠙⢺⣆⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⡏⠁⣥⠖⠀⠀⠀⠚⢳⢇⠀⣧⠈⠢⠴⠀⠀⣰⢖⣾⠂⠀⠁⠀⠀⠁⠀⠍⠀⠀⢾⡡⣄⠰⠤⠀⠀⢠⠃
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⣫⣾⡾⢻⣇⣀⡆⠀⣠⡦⠊⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡝⠎⠓⢦⣀⠔⠁
⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⣠⢴⡷⠿⠋⠀⠺⠆⠞⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠋⠛⠒⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠦⢔⣀⠄⠋⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠢⠄⠘⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣿⣷⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠈⣿⡏⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣇⠀⣿⠀⠀⠀⠀⠰⣦⣄⠀⣿⡏⢀⣤⡶⠀⠀⠀⠀⢸⡇⢀⣿⣿⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣿⣿⣿⠀⠀⣿⠀⠀⠈⠻⢿⣿⣿⠟⠋⠀⠀⢸⡇⠀⢸⣿⣿⣟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠿⣷⣴⣿⠀⢸⣷⣄⡈⣿⡏⣀⣴⣿⠀⢸⣿⣴⡿⠟⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⣿⣿⣿⣄⣾⡏⠙⢿⣿⣿⠟⠉⣿⣆⣼⣿⣿⣷⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣹⣿⣿⠀⠀⠀⠈⠀⠀⠀⢹⣿⣿⣁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢶⣄⠀⠘⢷⣄⠀⠻⣿⣏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣿⡿⠃⢀⣴⠟⠀⢀⣴⠆⠀⢀⡄
⠀⠀⠀⠀⠀⠀⢀⣀⣼⣷⣤⣨⣿⣶⣤⣬⣿⣷⣤⣸⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣤⣴⣿⣯⣠⣴⣾⣯⣤⣶⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠈⠉⢹⡿⠛⢙⣿⠟⠋⢙⣿⡿⠛⢹⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠙⠻⣿⣟⠉⠛⢿⣏⠙⠻⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠾⠁⠀⢠⡾⠋⠀⣴⣿⣇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣷⡄⠈⠻⣦⠀⠀⠹⠆⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠉⢉⣹⣿⣿⠀⠀⠀⣀⡀⠀⠀⣼⣿⣿⣉⠉⠉⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣿⠁⢻⣇⣤⣾⣿⣿⣦⣄⣾⠋⢻⣿⣿⡿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣴⣾⡟⢻⣿⠀⢸⡿⠋⢀⣿⣇⠈⠻⣿⠀⢸⣿⠛⣿⣶⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣿⢿⣿⠀⠀⣿⠀⠀⢀⣴⣿⣿⣿⣷⣄⠀⠀⢸⡇⠀⢸⣿⢿⣧⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⡇⠀⣿⠀⠀⠀⠀⠰⠟⠉⢀⣿⣇⠈⠙⠷⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠸⠇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⢀⣿⣇⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⣿⡿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃



⣿⠿⠿⠿⡿⢻⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣛⣿⣿⣿⣿⣿⣿

⠠⠱⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡔⡂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢠⡶⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠸⠋⠃⠀⠠
⠀⠂⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠂⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠄⣀
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢧
⠀⠀⠀⠀⠀⢣⣥⡆⠀⠀⠀⠀⠀⠀⠄⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⠀⠈⠀⠲⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠘⡃⡔⠀⠀⢸⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠄⠀⠈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠃⠢⢞⠄⠐⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠤⠘⠁⢐⠀⠀⠀⠀⠀⠀⢀⠈⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⡀⠁⠐⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢁⡀⢀⠀⠀⠀⡀⠀⠐⠋⠘⠀⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠀⠠⠀⠀⠀⠀⠀⠀⠊⠉⠑⠀⠀⠈⠃⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠖⢰⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠰⠀⢀⡂⠀⠀⠀⣀⣀⠔⢠⠀⠀⠀⠀⠀⠀⠀⠀⣅⠉⠀⢀⠛⡔⠀⠁⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣶⣌⣠⠌⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠘⠃⢋⡄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠂⠀⠀⢠⢷⡾⠋⠁⠄⠸⡷⠆⠀⠐⠃⠀⠰⠆⠈⠁⠀⣄⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣼⣃⣿⣽⣿⣷⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣯⠻⣶⣄⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢺⣴⢿⠷⠶⢿⡇⣾⠟⣿⠛⣷⡎⠟⣱⠍⠿⠋⠘⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣰⣿⣄⠈⠣⠘⢿⡀⣶⡘⢖⠩⢿⣿⡛⠉⣦⠰⢤⣛⣰⣼⠃⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡡⠋⠀⠀⠀⠈⣿⣿⣿⣾⡗⣼⢬⣷⠾⡤⢃⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠟⢿⡷⣌⣶⣻⠶⣡⠞⣽⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣬⣿⣷⡞⢸⣿⣌⢿⣿⣿⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟⣧⣼⠷⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡘⣻⠿⡟⣣⣿⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢻⣄⣿⣧⣬⣉⢾⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣉⢉⣻⣧⣸⡛⠛⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠻⠀⣹⡟⠁⠁⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡝⠱⠶⡠⢶⣶⡆⠺⡦⠕⢺⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⣭⣖⡚⢓⣚⢟⡟⠒⣑⡎⠩⢾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢟⣙⢂⣩⢉⣉⣻⣏⠉⣉⣭⢍⡓⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣼⠃⠦⡿⢮⣹⣭⣉⡛⣯⣠⢬⣅⠠⡆⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⠒⠹⡏⢰⡞⢻⣿⣿⡿⡋⠒⠆⠀⠒⠒⢋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠙⣿⠁⠛⠚⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⢀⡴⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⢠⣷⣿⣳⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠰⣿⣿⣿⣿⣿⣿⣇⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣿⣿⣿⣿⣟⣿⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣹⠇⢩⡿⡏⠻⠛⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠸⢼⠇⠀⠀⠀⠀⢠⣶⣦⡀⠀⢰⠀⠀⠁⠀⠀⠀⢘⣈⠁⠠⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠧⠀⠀⠀⢸⡇⠈⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠹
⠀⠀⢠⠀⠀⠀⠀⠀⠚⡀⠁⠀⢠⡥⢿⣄⠀⢻⣶⡄⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⢸⡀⠀⠀⠀⠀⠠⠁⡄⠀⠈⠀⠈⠙⣧⢸⡆⠀⠤⣀⢒⣾⣿⣀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⢀⣀⣀
⡿⠋⠉⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣈⠈⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⢀⣿⡥⣾⡿
⠃⠀⠀⣸⠏⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⣧⠀⢀⡀⠀⠀⠰⣶⣦⣼⡄⠀⠀⠀⠀⠈⢧⠀⠈⢿⡇⠀⠃
⠀⠀⠀⢫⠾⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡏⠁⠐⠂⠀⠀⠀⠀⠈⠐⠀⠀⠀⡀⠈⢇⠀⠀⠈⢀
⠀⡠⠀⠀⠀⠀⠉⠀⣀⣀⣤⣶⠛⠛⡐⠂⠀⠀⢤⣀⣉⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⡼⠶⣶⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⢀⡽⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠽⠏⠉⢛⡷⣾⣛⠉⠉⠀⠀⠀⠈⠈
⠀⠀⢤⡀⢀⠠⠖⠉⠀⢀⠈⠉⠀⠀⡠⠀⢤⡀⠀⠀⠀⠁⠀⠀⠱⡍⠢⢭⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠐⠓⠀⠀⠐⠒⠀⠀⠀⠀⠀⢰⠀⠀⠀⠁⠀⠀⠀⠀⠀⡶⣧⡇⠀⠀⠀⠉⠉⠛⠛⢷⡴⠞⠁⠀⠀⢠
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠇⠀⠀⠀⠀⡀⠀⠀⠸⠇⠀⠀⠀⣀⡘
⢙⣀⢀⠀⠀⠀⢀⡦⠀⠀⠀⢀⣀⣀⡦⠀⠀⢀⠀⠀⠀⠀⢀⠎⠲⣀⠀⠀⠀⠀⠹⡄⠀⠀⢀⠐⣾⠐⠉⠳⠒
⠋⡏⣸⠧⠤⠖⠋⢀⣶⢟⡺⢯⣍⣭⡄⠀⠀⠠⠀⠀⠀⢠⡟⣗⠀⠈⠑⠂⠀⠀⠀⢻⠀⡔⠻⠆⡠⠚⠐⠟
⢀⡁⠀⠀⠀⠀⠀⢸⡟⢸⣋⣐⡤⠄⠀⠀⠀⠀⠳⡀⠀⠈⡤⠚⣀⠀⠀⠀⠀⠀⠀⢈⣾⣱⡁⢀⡷⠤⣴⠋⢰
⠘⠀⠀⠀⠀⠀⠀⣠⡿⢛⠿⡿⣿⢯⣧⣠⡀⣰⠀⡗⠀⠀⣹⡉⠱⠆⠀⠀⠀⠀⠀⢸⣬⡉⠙⣿⡀⠠⣟⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠏⠐⣾⠚⠇⠈⣷⡦⣤⣴⡇⠘⠀⠀⠀⢹⡖⠆⢛⠀⠀⠀⢀⡜⡀⠠⠿⣤⣯⣤⣟⣁⣠⣀
⢶⡄⠀⠀⠀⢀⡀⠀⠀⢩⠀⠀⠀⣿⣼⠛⢿⡸⡆⠀⠀⠀⠘⣷⠬⠋⠀⠀⠀⠘⠀⠐⠀⠀⠀⠀⠉⠈⠉⠉⠉
⠉⠀⢀⡴⡞⢯⠉⠳⠄⠈⠀⠀⢀⣿⡇⠀⠰⣇⠁⠀⠀⠀⠀⠿⡂⠴⣢
⠀⠀⢸⠀⠀⠀⠀⣄⡗⠀⠀⠠⠞⡿⠁⠠⠴⠋⠀⠀⠀⠀⠀⠈⠣⠿⠟⠀⠀⠀⠀⠀⢀⡔⠛⠛⠐⠒⢠⡀
⣀⣀⣰⣛⣆⣀⣂⣀⣀⣀⣀⣀⣘⣃⣀⣀⣀⣀⣀⣀⣐⣒⣄⣀⣀⣀⣀⣄⣀⣀⣀⣀⣸⣀⣘⣛⣀⣀⣀⣘⣄
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡘⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⣿⠀⠀⠀⠀⠀⠀⠘⢞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢧
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠱⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣇⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡅⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡇
⠠⠴⠐⠐⠚⠁⠉⠉⠉⠁⠉⠉⠁⠀⠐⠒⠀⠀⠤⠤⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠠⠠
⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄⠀⠀⠀⠁⠈⠐⠀⠒⠒⠐⠂⠒⠒⠋⠉⢉⠅⠤
⠀⠀⠀⠀⠀⠀⢀⣀⠐⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠚⠚⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣞⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠉⠉⠛⠻⢶⢶⣶⣶⣿⠆⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣿⣿⣥⠀⠀⣜⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠥⠐⠒⢉⣩⠭⢭⣉⡭⣷⡦⣄⡀⢢⣤⣥⣿⣟⠉⠀⠰⡄⣞⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⢀⠔⣊⣹⠁⠀⠀⠉⠀⠀⠹⣶⣽⣿⠏⠙⠋⠋⠀⠀⠠⡂⢨⠆⡏⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⢀⣴⠟⠋⢹⡄⠀⠀⢀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⡀⠀⣠⢃⡎⠰⣧⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣠⡶⠛⠁⡾⠉⠉⠀⠦⠴⠃⠀⠀⢀⡴⠴⠃⣀⣀⠜⠉⠉⠉⠁⢸⠆⢀⢤⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⣠⠾⠋⢠⠞⠉⠁⠀⠀⠀⠀⠀⣠⠤⣌⠉⠀⠤⠞⢁⣠⠜⠓⠒⠲⠖⢋⡠⡌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⡵⠊⠀⠀⠴⠊⣡⢄⡤⠚⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠸⣄⣠⠤⠒⠋⠀⠀⢐⡏⠉⡇⠈⠳⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠚⠁⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡤⠦⣄⣀⡴⠚⢹⠀⠀⠀⠀⢸⣿⡤⣏⣷⠀⡴⠉⠀⢀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠤⠞⠁⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠈⣿⣿⣌⠁⠀⢇⡀⣸⠁
⠀⠀⠀⠀⠀⠀⣠⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣤⣉⣀⣀⡀
⠀⠀⠀⠀⣠⣾⣭⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢬⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠾⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣷⣶⣶⣿⣿⣿⣿⠿⠿⠛⠋⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⠶⠶⠾⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠶⣶⣦⣤⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢾⣿⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠈⠻⣦⣤⡴⠶⠶⠶⠶⠶⠦⠤⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⠶⢶⣶⠾⠋⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⠀⠀⠀⠈⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⣀⣴⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠈⠻⣧⡀⠀⠀⠀⠀⠈⠛⠻⠶⣶⣟⣛⠛⠳⣶⣋⠉⠉⠛⠷⣿⡁⢀⣤⠞⠀⠀⠴⠛⠋⠁⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠤⣤⣤⣙⣿⣟⠂⠀⠙⣧⠀⠀⠀⠈⠻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠉⢿⣆⠀⠀⠘⠇⠀⠀⠀⠀⠹⣷⡶⠶⠖⠂⠀⠀⢀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣬⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣀⣠⣤⣶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠛⠛⠛⢻⡟⠛⢻⡿⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢀⣿⠀⠀⣾⠁⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⢸⡇⠀⣸⡏⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⣿⠀⠀⣿⠃⠀⠀⣸⡇⠀⠀⠀⠀⡀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⡇⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣿⠁⠀⠀⠀⠀⣿⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠘⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡏⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠘⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⢹⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠈⣧⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⢹⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⡆⠀⠸⣷⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡾⠟⠻⣿⡛⠛⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣤⣶⡾⠿⠛⠛⠛⠛⠛⠛⠛⠿⠿⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⢀⡄⠘⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠻⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣦
⠀⠀⠀⠀⠀⠀⢸⡟⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⢠⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠥⠠⢿
⠀⠀⠀⠀⠀⠀⠈⠡⢘⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣂⣾
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣟⣟⣿⡆
⠀⠀⠀⠀⣆⠀⠀⠀⣀⠀⠀⠀⢰⡆⠀⠀⠀⠀⡋⠀⢀⡀⣸⣯⣿⣿⣗⡆⠀⠀⠀⠀⠀⢰⣤⣿⣿⣏⠏⣿⣿⣶⡄
⠀⠀⠀⠀⢸⣦⣴⣾⣿⣷⣦⣠⣿⡇⠀⠀⠀⠀⡈⣿⠋⠀⠀⠀⠙⣻⡇⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⢠⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⢿⣿⡇⠀⠀⠀⠀⠃⢸⣦⣴⠶⢶⣶⡿⣧⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠙⢿⠋⠡⠈⣿⡄⠀⠀⠀⠀⠀⠀⢻⡿⠶⠼⣿⡇⢿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣿⡇
⠀⠀⠀⠀⣿⡇⠀⣠⣿⣧⡘⠀⣿⡇⠀⠀⠀⠀⡄⠀⠘⠀⠀⠀⠘⡇⢾⡄⠀⠀⠀⠀⠀⢸⣿⡏⣿⣿⣿⣿⣧⣿⡇
⠀⠀⠀⢰⣿⢡⣾⣿⣿⣿⣿⣾⡽⣷⠀⠀⠀⠀⠀⢰⠑⢄⣀⣤⣴⣿⡄⢳⡀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⠿⣿⡇
⠀⠀⠀⣾⠇⣾⣿⣿⣿⣿⣿⣿⣇⢻⡆⠀⠀⠀⠀⢸⣎⠀⠠⠄⠈⢿⣧⠀⢳⡄⠀⠀⠀⢸⣿⡗⣿⣿⣿⣿⠳⣿⡇
⠀⠀⢀⡟⢠⣿⣿⣿⣿⣿⣿⣿⣿⠈⣷⠀⠀⠀⠀⣹⠃⠀⠀⠀⠀⢸⣿⡆⠀⢳⡀⠀⠀⢸⣿⡇⢹⣿⣿⣿⠀⣿⡅
⠀⠀⣼⡇⠈⠻⣿⣿⣿⣿⣿⡟⣿⠀⢿⣇⠀⠀⠀⠻⠇⠀⢸⣦⠂⠸⠚⡇⠀⢸⣷⠀⠀⢸⣿⣿⡈⠛⠉⣿⠀⣿⡇
⠀⠀⠻⠦⠀⠇⠈⠹⣿⣿⠋⠀⡇⠀⠸⡏⠀⠀⡄⢲⠃⠀⡇⠹⣧⠀⠀⡇⠀⠀⣿⠁⠀⢸⣿⣿⡇⠀⠀⣿⢀⣿⡇
⠀⠀⠀⡀⠀⢠⠀⢀⠉⠹⢀⠐⠇⠀⢠⠇⠀⠀⢃⠈⠀⠀⠃⠀⢿⠀⠀⢧⠀⠀⣿⠀⠀⠘⣿⣿⡇⠀⠀⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠘⠂⢘⡆⢰⠈⠀⠀⠀⠈⠃⠀⠀⠸⣤⡆⢠⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⢿⣿⡇⣴⠀⡿⢸⣿⠁
⠀⠀⠀⠀⠀⠀⡄⠘⡇⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⠀⠀⠸⡄⠀⢸⠀⠀⠀⠀⠀⠀⠘⢻⣗⠙⠀⡗⣼⠋
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠈⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⠀⣿⢻
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⠂⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡗⣇⠀⣿⢸
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡀⠀⠀⠀⠀⡇⠀⠀⣧⠀⠀⠀⠀⠀⠀⢸⡟⠃⠀⣿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠁⠀⠀⠀⠀⡇⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⣿⡇⠀⡟⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠃⠀⠀⠀⢰⡇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣿⢡⠀⡇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⢹⡜⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⡠⠀⠀⡿⠐⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢸⣿⠀⣧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⢰⡇⠀⢰⡇⠀⠀⠀⠰⡷⡀⠀⠀⠀⠀⠀⢸⣿⠀⡿⡇
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⣾⣅⠀⠀⠀⠀⠋⢰⠂⢸⡇⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⣼⢹⢀⡇⡇
⠀⠀⠀⠀⠀⠀⣀⣀⡀⣄⣠⠀⠀⠀⠿⣼⣉⢹⠀⠀⠀⣿⡄⠘⣿⣏⣿⣺⡇⠀⢀⡇⠀⠀⠀⠠⣿⣸⣿⣇⣹⡄

⠀⠀⠀⢀⢠⠄⠐⠶⠀⠈⢈⡠⠊⠁⠀⠐⠃⠸⠃⠦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⢟⠏⠉⠀⠈
⠀⠀⢰⢁⡾⠀⠀⠀⣀⡼⠈⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠰⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠂⡔
⠀⠀⠈⠠⢤⣀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⢠⠄⠁⠀⠶⠘⢲⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⡤⠀⠀⠀⠶⠁⢀⡴⡁⠈⠀⠉⠀⠀⠈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢶⣶⢶⢲⠒⡆
⠀⠀⠠⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠐⡷⣮⡏⠿⠘⢰⠃⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⢤⠀⡀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⣤⣴⡖⡖⡆⠀⠀⠀⠚⠳⠧⠆⠎
⠀⠀⠀⠾⠀⢠⠆⠀⠀⠰⣄⠀⢸⣴⡇⠀⠀⠀⠀⠤⠔⠀⠀⠀⢲⣾⡏⡇⠀⡇
⠀⠀⡀⠀⠰⠈⠀⠀⠀⠐⢂⢼⣿⣷⠯⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⣰⡾⣠⠀⠀⠀⠀⡤⣡⠻⡟⠻⡕⠀⠀⠀⠀⢠⢿⣷⠀⣠⡤⣶⣶⡶⠀⠀⠀⣄⠒⣄⠀⢠⠀⠀⡆⠂
⢠⠀⠀⢐⣿⠁⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⠩⠬⠱⡰⠉⠁⠀⠀⠀⠀⠈⠁⠘⢀
⢸⡄⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡽⣆⠁⠀⠀⠀⠑⣃⠀⠀⠀⠀⠀⠀⣀⡝⠀⠄⠀⣤
⠀⢳⡄⢰⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⢸⣿⠁⠝⠀⠠⠄⠀⠄⡑⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠃
⠇⣺⡄⣾⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⡄⠚⠃⠀⠈⠠⠢⠁⡀⠒⠬⣆⠀⠄⠀⠀⠀⠶⠄⠀⠁⠈
⣇⣟⣼⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡇⠈⠀⠀⠀⠀⠀⠸⠾⠛⠋⠈⠙⠀⣆⠀⠀⠀⠀⠀⡀
⠘⠋⠁⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣺⡇⠀⠀⠈⠢
⣤⠶⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠴⠀⠀⠀⠀⠈⣆⠘⡇⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢄⡀⠀⠀⠀⠘⣀⠀⠀⠀⠀⠀⠨⡇⠠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡯⣰⠏
⠀⠀⠀⠀⠀⠀⠈⢆⣰⡂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡄⢀⠀⢀⠄⠀⠀⢸⠀⠀⠀⠀⠀⢰⠏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⠄⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠘⢀⡃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⡀⠀⣳⣿⣴⣿⡌⠴⣶⣦⣄
⠀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠤⠒⠒⠚⢻⡟⣿⣧⠀⠉⠿⢿⣆
⠐⠿⠛⠛⠿⠿⢿⣿⣿⣷⣶⣶⣤⣽⠇⣀⣴⠶⠶⠶⢾⡇⣾⣿⡶⠶⠟⠛⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠉⠐⠀⠀⣔⣻⣯⡟⣿⣧⣴⣶⣶⣿⣶⣶⣦⣤⢄
⠀⠀⠀⠀⠸⣟⣿⣿⣿⣿⣿⣿⣿⣿⠉⣒⠈⡙⢿⣿⣿⣜⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣰⣦
⠀⠀⠀⠀⠈⣿⣻⣿⣿⣿⣿⣿⣾⣿⣧⣛⣠⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢉⠉⠉⠙⠁⠉⠁
⠀⠀⠀⠀⠀⣻⢿⣿⣿⣿⣷⣽⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣯⣿⣇
⠀⠀⠀⠀⠀⢺⡽⣻⣿⣿⣿⣯⣿⣿⣽⣿⡿⡽⡻⣳⡿⣺⣾⡿⡿⣫⣛⢕⡽⣿⣿⢿⠽⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠈⣯⣭⣿⣽⣽⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣟⣯⣾⣧⣥⣷⣷⣟⣍⣿⣿⣮⢽⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠻⠻⠻⠛⠿⢻⣻⣿⣿⣿⡾⢾⣗⣿⣿⣿⣿⣿⣽⣽⣿⣿⣿⣿⡟