⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⢀⡤⢚⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⢏⠠⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⣄⠤⢄⣀⣀⠀⢄⣀⣀⠀⠙⠛⠒⠤⠤⣀⣠⢄⣈⡼⣾⠸⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠈⠙⠂⠒⠂⠀⠀⢀⣀⣀⠀⣀⣀⠀⠉⠛⠆⡄⠘⢆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠟⠋⠉⠉⠉⢀⠔⠋⠉⢀⣀⡀⠤⣀⣀⠀⠀⣠⠆⠘⠢⡄⠁⣀⡈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠞⠃⠀⠀⢀⠀⠀⢈⣁⠀⣠⠖⠁⠀⠀⡜⠀⠀⢀⡜⠀⠀⡜⠁⠀⢀⡀⠀⠙⢦⠻⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠐⠶⢤⣤⡴⠖⠊⠁⠀⠀⠀⠐⠈⠁⡠⠔⢉⠁⢠⡞⢀⠀⠀⠀⡜⠀⡄⢠⢎⠎⠀⢰⠁⠀⡆⠀⠘⣄⠀⠀⠙⡽⣆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⠋⠑⠒⢒⡒⠀⣀⡔⠚⠉⣠⣴⠃⠀⢰⠁⡞⠀⠀⢠⠁⣰⠀⠇⡜⠀⠀⡇⠀⠀⢱⡀⠀⠘⡆⠀⠀⠠⣻⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁⠀⠊⠁⡠⡢⠚⣱⣁⠀⠠⡏⡸⠀⠀⠀⢸⢀⠇⠀⣾⠃⠀⢀⠇⢸⠁⢸⠹⢀⠀⠙⠀⠀⠀⠱⡽⡄⠀⠀⠀
⢀⣤⡤⠤⠈⠉⠀⠀⠀⣠⡦⠂⢺⠏⣐⡰⠃⡇⠀⠀⢧⠃⢀⡀⠀⢸⡞⠀⡴⡝⠀⠀⡞⢠⠀⠀⠈⡇⣾⠀⠀⢀⠀⠀⡀⠙⣽⡄⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⡸⠁⢠⢁⠀⣰⡇⠀⡄⡜⠀⣾⣃⡴⢋⠃⢰⡵⡇⢀⢮⠀⢸⡆⢀⠖⢷⡜⠀⠀⢸⠀⡄⠿⡆⢟⠮⡶⠀
⠀⠀⠀⠀⣀⡠⠊⠋⠀⡠⡊⢠⠃⠎⢀⣾⢿⠁⣰⠃⠃⡜⠙⡵⢁⣸⣧⠟⠁⡧⡞⡘⠀⣸⢱⢾⣤⠚⢽⡣⣄⡘⠀⡇⢺⡜⣼⡄⠀⠀
⠀⠀⠀⠘⠛⢆⣀⣠⠖⣸⠃⠃⡴⣢⣺⠇⡸⣠⢿⠀⢰⢇⢀⠧⠒⠚⠛⠒⠒⣧⢠⣡⣴⢿⣀⣾⢡⠈⢒⢻⣯⢧⠀⠱⡀⢱⣯⣧⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣧⢶⡇⠀⢸⣵⣧⢧⣀⣷⠛⢸⣰⠟⣹⣿⢠⠒⡆⢀⣴⡴⢿⢟⡿⠁⢸⢹⢻⠈⣓⡊⠋⢹⢈⢳⡀⣳⣄⡇⠘⠆⠀
⠀⠀⠀⠀⠀⠀⠺⠁⣼⢁⡠⣿⠏⢧⢠⣾⣿⡄⡘⢉⣹⣿⣁⣼⡿⠟⠛⣡⠄⡤⠟⠀⠀⠸⠼⠸⠰⡉⢉⠀⢠⠏⢠⢳⡟⢏⠻⠄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⢋⣾⡾⣸⠀⡼⣿⣅⣤⣃⣀⣼⣿⣿⣏⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⠏⢰⠀⢿⣇⣈⡃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢀⣏⠰⣽⣿⣿⣿⠋⠛⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⠴⠃⡄⠸⡄⣬⣿⣿⣿⡄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⣿⣿⣯⡀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠀⡿⠟⠀⠀⠁⢀⣽⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠻⣼⢹⡉⠀⠀⠀⠀⣀⡤⠶⠒⠋⢉⣫⠏⠁⠀⠀⠀⠀⠀⢀⡓⡴⠡⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢳⡀⠠⡖⠋⢀⣠⡶⣋⠝⠋⠀⠀⠀⠀⠀⠀⣀⠖⠉⠈⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠉⢶⡿⠟⠉⠀⠀⠀⠀⠀⠀⢀⡴⡿⠁⠀⠀⢀⣾⣿⣿⠟⠙⢻⣷⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠁⠀⠀⠀⠀⢀⣠⠞⣡⠊⠀⠀⠀⣰⣿⣿⡿⠁⠀⠀⠀⣻⣿⣿⣿⣷⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢀⡔⠋⠀⠞⠁⠀⠀⢀⣾⣿⣿⠋⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢀⣀⠥⠔⠒⢲⣆⡀⠀⣠⣿⣿⣿⣯⠀⠀⢠⣾⣿⣿⣿⡿⠋⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠉⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣟⣿⣿⣶⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣯⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣍⡛⠛⠛⠿⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠳⣄⠀⠀⠀⠀⠀⠀⢸⡄⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠘⠳⢤⣄⡀⠀⠀⢸⣿⡄⠀⢿⡄⠀⠀⠀⠀⣀⣠⠤⢴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⡀⠀⠈⠉⠙⠛⠛⠻⠄⠀⠈⠿⠴⠒⠛⠉⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠈⠩⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⢦⡤⠤⠤⠴⠖⠋⠉⠀⢀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠈⠉⠒⢤⣀⠀⠉⠉⠉⠉⣉⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⣠⠏⣠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⡈⠒⠲⠶⠶⢤⡶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⠀⣰⣣⢞⠟⣠⠀⠀⠀⠀⢠⡄⠀⣶⠀⠀⡀⡀⠀⢠⡀⠈⠀⡀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡽⣡⣰⡿⠃⡜⢠⠇⢀⣠⠏⠀⣸⠀⢠⣧⠀⡀⣇⣷⠀⠀⣇⠀⠀⢧⠀⠀⡀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⢷⠏⠃⢃⣾⠇⡾⠀⡾⣼⠀⢰⡇⠀⣸⡇⠀⣧⣿⣼⡆⠀⢾⣦⠀⢸⣿⠀⣷⡀⠀⣷⣂⣭⣷⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣨⡏⠀⢀⣸⡼⢠⠃⢸⠁⡇⢀⣿⡇⢠⡟⡇⢠⣿⣿⣿⣵⣄⣸⣯⣿⠀⣯⡆⢿⢻⣦⡈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣼⡷⠞⢉⡇⢀⠏⣾⡇⢸⣠⡿⠴⢇⣾⣻⣠⠞⢶⣷⡸⣿⠈⢿⡼⠋⠁⠀⠙⣷⣜⢣⠘⡾⣿⣝⣒⣭⣳⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠟⠋⠀⠀⣼⢀⡟⠀⡿⡇⢸⣿⡇⣰⠛⠁⠀⣉⣳⣶⣿⣇⣼⣤⡿⢀⣀⣤⣶⣿⣿⣿⣾⣆⣹⣽⣿⣌⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⢁⣼⡇⡇⢸⣿⣷⣃⣠⣶⣾⣿⣿⣏⣿⡏⠁⠙⢿⣿⣿⣿⣿⣿⣿⡿⠉⣿⢩⡏⣟⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⣯⢾⣼⢁⣧⣴⡿⠛⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠻⢿⣿⣿⣿⠟⠁⠀⢸⣞⣿⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⠟⢸⠇⣸⣿⣏⣷⡆⠈⠻⠿⠿⢿⠿⠃⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡞⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠀⣾⡿⣿⣿⢟⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠤⠤⣄⠀⠀⠀⢰⣟⣰⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣇⠈⢾⡈⢳⠀⠀⠀⠀⠀⣠⠴⠛⠉⣀⣀⣠⣤⣤⣤⣬⣷⠀⠀⢸⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠻⠾⣷⠀⠀⠀⢸⣇⣤⣶⣿⣿⡿⠟⡿⢻⠻⣿⣿⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢿⣿⠟⣹⠰⠐⠃⠙⢋⣸⠟⠁⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢦⡀⠈⠙⠶⠷⠶⠒⠒⠉⠉⠀⣠⢾⣿⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠘⡇⢽⠳⢤⣀⣀⣀⣀⣀⣠⣴⢻⠅⣿⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣯⠙⠳⣴⣻⢏⣿⣿⣿⡇⣿⢸⣴⡿⢸⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣄⣀⣼⣿⠿⣿⣿⣿⣿⣿⡀⠀⠀⠙⢾⣏⣹⣿⣏⣿⡞⠛⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⣿⣿⣿⣿⣦⡘⢿⣿⣿⣿⡇⠀⢆⠀⠀⠈⠉⠒⠉⠁⣰⡀⠀⠿⢙⣿⣰⣿⣿⣿⣟⣶⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣿⣿⣿⣿⣿⣿⣦⣻⣿⣿⣇⠀⠀⢿⡆⠀⠀⠀⠀⠀⡟⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢻⠀⠀⠀⠀⠸⣇⠀⠀⣸⣿⣿⣿⣿⣿⣿⣟⣁⣀⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣶⣮⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠁⠀⠀⠀⠀⠉⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠓⡶⣤⣄⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⣿⡀⢻⡀⠀
⠀⠀⠀⠀⠀⣰⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⠀⠀⢰⠞⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⣿⣿⣯⣧⡀