⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣧⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣋⣟⠭⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣭⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢮⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣦⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠢⣽⣅⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢤⣿⡇⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⣞⡇⠀⠀⠀⠀⡏⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣿⣷⠀⠀⠀⠀⢻⡈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⣿⡆⠀⠀⠀⠀⢳⣬⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⣿⣧⠀⠀⢠⡄⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹⣿⡀⠀⢸⢧⠟⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣿⡇⣠⠋⢾⣾⢸⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠸⣿⣶⣿⣷⡏⢰⡿⢿⠏⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣴⢋⣿⣿⣿⠇⡟⠁⣏⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡞⣏⢦⠇⢸⡿⢿⠋⢀⣤⣀⡘⢦⡟⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⢀⣴⡆⠀⠀⠈⣹⣿⡷⠆⠀⣧⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⢁⡴⠋⡀⠙⢄⠀⣰⣿⣟⠓⠀⠀⢉⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢰⡏⡠⠊⢀⡴⣇⠀⢀⡞⠉⠛⠀⡀⢀⣄⣩⠌⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢣⠶⠖⠊⢀⣈⠉⣹⡷⢀⣴⡯⠔⣛⡵⠁⣠⡏⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢿⠟⠀⣰⡞⠉⣿⡷⠇⠃⣠⢴⣶⣾⡋⢀⡴⣽⠁⠀⠘⣏⣀⢰⣆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣶⣶⣅⣠⣶⠀⠒⠟⢁⡴⠋⠀⠀⠀⢹⣿⣿⡋⣧⢸⡇⡏⣀⣀⠀⠙⣿⣉⠙⢤⡄⠀⠀⠀
⠀⠀⣠⣴⣺⢿⣿⣿⡛⠛⠿⠿⣯⣷⡲⣶⣟⣻⡀⠀⣠⣿⣿⣖⣸⣨⣿⠿⠛⣻⣿⣶⣾⣾⠇⠀⠻⣄⠀⠀
⠀⣾⢟⠿⠿⢶⣮⡙⢏⢢⡀⢠⡌⣿⣿⡿⠟⡿⢳⣼⣿⣿⣿⣾⣿⣧⣤⣤⣤⣿⣿⣭⣿⠁⠀⠀⣀⣈⣧⠀
⢺⣥⢿⠾⠿⠿⠿⡿⠚⢋⣠⠯⣿⢉⢉⠻⠾⠛⢿⣿⠻⠿⢛⢋⣤⣯⣭⠽⠶⣾⣻⢿⣻⢿⠶⢛⣻⡿⢽⠄

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠸⡿⠋⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣿⠛⠓⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⠟⠋⠉⠙⡿⣿⣿⣿⣿⣿⣿⣯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠋⣿⣿⣿⣿⣿⢇⠀⠀⠀⠀⢻⠈⢢⣽⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣦⣿⣿⣿⣿⣿⠘⡀⠀⠀⠀⠸⡆⢹⣻⡿⡿⣿⣷⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⣿⣿⣿⣿⣿⡄⡆⠀⠀⠀⠀⣧⡿⠈⣿⣿⡞⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⢻⣿⣿⣿⡿⣷⣷⠀⠀⠀⠔⢳⠁⢃⡟⡡⠷⣽⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣼⣿⣿⣿⣻⣿⢿⡀⠀⠀⠀⢻⠀⢽⡟⠂⠀⢻⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣽⣿⣿⣿⡽⠈⢸⡇⠀⠀⠀⠘⠀⡿⣯⡒⠤⢀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠐⢸⣼⣿⣿⣿⠁⠀⠈⠇⠀⠀⠀⠀⡠⠇⠛⢷⡖⢤⣿⣿⣿⣿⣷⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⣸⣿⣿⣿⣿⣄⡀⠀⣰⣠⠀⠀⠀⢧⠣⣸⠀⣳⡇⣿⣿⣽⠟⠻⣿⠟⠒⢒⣢⠖⡚⠛⠚⠛⠋⠿⡷⣶⣦⣤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⢹⢽⣿⣿⣿⣿⣿⣍⢉⢣⢃⠀⠀⣾⠀⠈⡏⢸⣽⣿⣷⡟⡠⠊⠀⢀⡜⡉⢏⠀⠈⠁⠀⠀⠒⠤⡙⣄⠉⢢⠉⠐⢮⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⡈⡿⣿⣿⣿⣿⠇⡿⡟⢿⣾⠀⢠⠃⠀⡰⢳⠿⣿⢿⠇⢰⠃⠀⢰⠛⢰⠇⠘⣆⠀⠀⠀⠀⠀⠀⠈⠺⣦⠄⢣⠀⠀⢣⡈⠺⣆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣩⣿⣿⡿⠃⡇⠀⢸⠱⡌⣷⡀⢫⢀⡾⠃⣸⣲⣿⣻⡀⢸⠀⢀⣯⣧⡌⠀⡄⠊⠦⠀⠀⠀⠀⠀⠀⠀⠘⣼⢾⡆⠀⠀⢻⣎⢎⠢⣄⠀⠀⠀⠀
⠀⠤⣠⣿⠾⢿⣇⣀⢱⠀⡞⡀⢰⡿⣇⣇⠎⠰⠃⠹⣿⡖⠛⠿⣿⣄⣸⠉⣼⡇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣳⢯⡀⠀⠀⢫⡻⣆⠙⣦⡀⠀⠀
⣀⡽⠋⠀⠀⣸⣏⠀⠀⢧⢧⣇⠀⠀⣿⠥⠀⡠⠢⠔⢹⡘⡀⠀⠀⠙⡿⣦⠋⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠉⠀⢸⡳⠙⢦⡀⠀⠳⡬⢦⡈⢳⣄⠀
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠈⢆⢹⠀⠀⢸⡖⠉⢀⡐⠴⢿⣇⠀⠀⠀⢰⢛⠏⢸⠃⠀⠀⠀⡼⡄⠀⠀⠀⠀⢀⡠⠔⢋⣿⢿⡤⠴⠿⢶⣶⣿⣠⣵⠤⡽⠗
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⢿⡄⠀⠘⡷⡀⠀⠀⡷⠋⢦⠀⠀⣰⣣⡎⢀⣻⠀⠀⠠⠊⢀⣔⣒⣶⣶⠾⠛⠉⢙⣾⠟⠁⠀⠀⠀⠀⢹⠀⡨⠔⠋⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢳⡳⡀⠀⡇⠀⡞⠀⡔⡹⡹⠀⠊⡿⣀⣀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠙⡋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣆⠘⢳⠘⢆⢰⠀⣠⣎⡀⠧⡧⣤⠤⠾⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⣸⠀⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣯⢦⡇⠀⠘⠳⣷⡍⠒⠭⣉⢘⡏⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡼⠃⣀⣀⡠⠤⠤⢔⣧⣀⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢿⣫⣷⣀⠀⠀⠈⢣⡀⠀⠀⠀⡟⠉⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⢤⣴⠓⠉⠀⠀⠀⠀⠀⠎⠁⠈⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠻⠽⠦⠤⠾⡇⠀⠀⢰⣧⢖⣒⡈⠁⢉⣉⠐⠒⠤⢄⣀⡼⡁⣠⠴⠒⠀⠀⠀⠀⡈⠉⠲⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡇⢀⠀⠀⢾⡇⠀⠀⢸⡜⠰⣏⣷⢠⠌⡹⣄⠀⠀⠀⡘⠀⢻⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣀⣀⣀⣉⡇⠀⠀⡏⠁⠠⣨⢿⠘⠂⣟⢳⡄⠀⠸⠃⠀⠸⣘⣠⣀⠀⠀⠀⠀⢸⠙⢷⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣗⣈⠻⢿⣦⣀⢧⣶⣾⢿⢺⡧⣞⣅⠃⠀⡰⠃⠀⠀⠀⣇⢌⠩⠿⢷⣧⣶⣺⡆⢸⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣟⠂⣽⠟⣿⠘⢜⣮⣯⡿⣢⣨⡿⠀⡰⠁⠀⠀⠀⠀⠸⡏⢉⠉⣡⢻⢫⢮⣩⢾⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡗⠭⡝⢿⣿⣿⢰⣿⠛⠁⠈⠛⠚⢁⡴⠁⠀⠀⠀⠀⠀⠀⢫⠇⡰⢡⠋⢧⣿⠞⡏⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⣼⡀⠀⠸⣿⠟⣁⣀⠀⠀⠀⡠⡻⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠍⠁⠀⠀⠙⠀⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⡀⢠⣿⣴⠛⣯⠷⠓⣴⢱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣦⢴⠽⡆⡇⢰⢹⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⣿⣿⡟⢺⣿⣧⣣⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡼⣧⡗⡑⡇⠘⣯⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⠟⠛⢯⠳⡽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡴⠉⢻⡷⡀⠀⢳⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣶⣤⣇⣡⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣢⣼⠷⣧⡠⠀⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣫⠄⡠⠥⡝⢉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣢⣟⣂⣼⡉⠴⢳⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⡀⠱⢧⡴⣑⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡽⠉⠈⠀⠇⠀⢸⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣌⣬⣤⠟⡧⠰⠁⣼⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇⠀⠀⠀⣸⡀⡼⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠚⠁⠀⠉⠉⠚⢀⢧⣏⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⡎⡄⢉⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠱⠀⠀⠀⠀⣀⣼⡿⠙⡾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢏⠳⢩⣬⣶⢿⡟⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢈⢿⠝⠱⡀⢰⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣵⣭⠿⣻⡾⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⣶⠀⠀⢠⠫⠋⠀⠀⢃⠈⢟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠚⢁⣴⢶⣳⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣿⠛⠀⡴⡿⠁⠀⠀⠀⠘⡆⠸⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠞⠉⠀⣌⡏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⣿⣿⠀⣼⡟⠁⠀⠀⠀⠀⠀⢰⡀⢳⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⣌⡵⡸⡙⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢿⣇⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⡜⡀⠀⠀⠀⠀⠀⠀⢀⡿⡫⠛⢀⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣠⠘⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢹⢧⠀⠀⠀⠀⠀⠀⣼⡃⠑⠁⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣶⣁⣚⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⡞⡄⠀⠀⠀⠀⢸⡷⣤⡴⢾⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢳⣿⡟⠫⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢱⣁⠀⠀⠀⢀⣿⣿⣯⠟⢁⣨⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠷⢿⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⡿⡄⠀⠀⣾⣷⣱⣧⠼⠛⢹⢦⠹⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠋⠁⢀⡾⣇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⠇⠀⢠⣿⡾⠋⠀⠀⣠⠟⠘⣄⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠴⢒⣉⠠⢖⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣘⣾⣴⡿⢻⡄⢀⢔⣫⠔⠊⣩⠟⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢖⡞⠁⠸⡶⣠⠾⣿⡶⡄⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠹⢹⡟⡠⣎⡿⡱⣛⣽⣟⣤⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣼⡐⢺⠿⣽⠓⣺⣽⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠓⠒⠃⠀⢊⠾⣽⠄⣠⠎⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠦⠄⣀⡠⠾⠬⠭⠗⠏⠀⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠚⠛⠒⠒⠒⠚⠛⠛⠽⠶⠶⠾⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀