⠀⠀⣿⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣸⡏⠀⠀⠀⠉⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⡰⠋⢙⣿⣦⡀⠀⠀⠀⠀⠀
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣦⣮⣤⡀⣸⣿⣿⣿⣆⠀⠀⠀⠀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⣿⢟⣫⠟⠋⠀⠀⠀⠀
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣷⣿⡁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣧⣿⣿⣆⠙⢆⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣤⣿⣿⣿⡟⠹⣿⣿⣿⣿⣷⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⠏⢧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⠏⠀⠀
⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⢀⣼
⠀⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠖⠒⠛⣿⣽⡟⠛⠓⠒⠶⠶⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⣯⠏
⠀⠀⢿⣯⣿⠷⣶⣤⡴⣶⣶⣤⠤⠤⣤⣄⡀⠀⣀⡤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠴⠿⣼⠿⠦⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣄⣀⣤⣴⠒⣻⣿⣿⢻⣿⣿⣿⠋⣩⣴⠋⠀
⠀⠀⠀⠙⠿⣦⣼⣟⢷⣷⢹⣿⣌⣿⡟⢺⣿⠛⠻⣄⠀⠀⠀⠀⢀⣠⣤⠤⠖⠒⠒⠛⠒⠒⠒⠦⠤⣤⣀⠀⠀⢀⣤⠖⠛⢿⣇⠐⡾⣷⣿⡟⢚⣿⣷⣿⠶⠋⠁⠀⠀
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠾⢿⣾⣮⣗⣸⣿⣆⠄⠀⠙⣦⡖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣷⡟⠀⡀⢨⣽⣿⣽⣶⢿⡿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⠙⠻⢿⣷⣶⡂⣸⡟⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⣧⣄⣿⣾⣿⡋⠉⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⣠⠾⣿⣿⣿⣿⡁⠀⠈⢳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⣿⣿⣿⣫⣶⠟⢦⡀⠀⠀⣀⠹⣆⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⡼⢿⡷⣾⠀⢀⡞⠁⠀⠹⡄⢻⣿⣿⡆⠀⠘⣿⣦⣤⣀⣀⠀⠀⣀⣀⣤⣶⣿⡯⠀⢀⣾⣿⡿⠋⢀⡞⠀⠀⠙⢆⣀⣿⣻⣯⣷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⡾⠷⠛⢳⡞⣻⠋⠀⠀⠀⠀⢳⡀⢻⣿⣿⣦⣠⣿⡯⣷⡉⣽⠿⠿⠟⠉⣹⡯⣿⣷⣤⣾⣿⣿⠁⠀⣼⠃⠀⠀⠀⠈⠻⡍⠏⠁⠉⢷⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠈⡱⠁⠀⠀⠀⠀⠀⠈⢷⠈⣿⣿⣿⠟⣿⣇⡝⣮⡈⠀⠀⠀⣴⡟⠀⢿⡟⢿⣻⣿⣇⠀⣰⠇⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⢿⡀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⡼⠃⠀⠀⠀⣀⣀⣠⣤⣼⠿⣿⢿⠃⣰⠋⠈⠁⢻⡙⢶⣴⠟⣹⠃⠀⠀⠱⡄⢹⣿⣟⠲⢿⠤⠤⣤⣤⣀⡀⠀⠀⠸⡆⠀⠀⠈⣧⠀⠀⠀
⠀⠀⠀⠀⢠⡏⠀⠀⠀⣰⣃⣤⠖⠋⣉⡡⢆⣠⠟⠁⣼⣿⡿⢸⣇⠶⠊⠀⢸⣷⠛⠉⢳⣿⠀⠀⠐⢶⠹⡌⣿⡿⣆⠈⠱⢦⡐⠦⣄⣉⣙⣶⣄⣹⡀⠀⠀⢸⡇⠀⠀
⠀⠀⠀⠀⣾⠀⠀⢰⣶⣿⣿⣿⡿⢿⣥⣶⣟⣁⣠⣞⣽⠟⡇⢸⣿⡀⣀⡴⠋⢹⡄⠀⣸⣉⣻⣦⣄⣸⣰⡇⣿⢹⣮⣷⣤⣤⣿⠿⠞⣛⣿⣿⣿⣿⡿⠂⠀⠀⢿⠀⠀
⠀⠀⠀⠀⡟⠀⠀⠀⢹⠋⠳⢿⣿⣷⡶⠦⢭⣽⣾⣿⡟⠰⢷⣘⣿⠁⠿⠋⠉⠙⣿⠉⡿⠉⠉⠉⠏⢩⣿⢠⣟⣐⣿⣿⢷⣾⣷⣒⣩⣿⠿⠟⠉⠀⢱⠀⠀⠀⢸⠀⠀
⠀⠀⠀⢠⡇⠀⠀⠀⢸⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⡟⣠⣾⣳⣽⣷⣦⢠⠄⣖⢹⣿⠃⠃⠠⠂⣰⣿⣿⢿⣧⣄⣻⣿⣿⣛⠟⠛⠋⠀⠀⠀⠀⠀⢸⡄⠀⠀⢸⡇⠀
⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣸⠿⠋⢻⣿⣿⣿⣿⡽⣽⣾⣿⣦⣬⠞⠏⠀⢤⣼⣿⣿⣿⢱⣿⣿⣿⣿⡆⠈⠙⠲⣤⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀
⠀⠀⠀⠀⡇⢰⣄⣤⣾⠀⠀⠀⢀⣠⠶⠋⠁⠀⢀⣾⣿⢿⣿⣾⣇⢹⡏⣻⣿⠞⠀⠀⠀⠰⣿⣏⣸⡇⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⠳⢦⣀⠀⠀⢸⢳⣦⡞⢸⡇⠀
⠀⠀⠀⠀⣷⡼⣯⡽⢿⣆⣤⣞⣋⣀⣀⣀⣀⣀⣸⣿⣿⣧⠬⠹⣯⣬⣿⠉⠹⣄⠀⠀⠀⣰⠏⠉⣿⢤⣿⠟⠲⣾⣿⣻⣧⣤⣤⣤⡤⠤⠤⠽⠿⢦⡼⠛⣷⠛⢿⠀⠀
⠀⠀⠀⠀⢻⡄⠘⠃⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣄⠀⢻⣿⠏⢦⠀⠈⠐⠀⠸⡁⠀⡟⠙⣿⠟⠀⣠⣾⣿⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠙⢀⡿⠀⠀
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣄⠿⣄⠈⢿⡆⠀⠀⠀⢴⡿⠀⣠⠟⣠⣾⣿⢿⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⣸⠇⠀⠀
⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⢿⣶⡈⢦⢸⡇⠀⢠⠀⢸⡇⠐⢁⣼⣿⢿⣯⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢠⡏⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⣿⣳⠀⠀⡇⠀⣼⠀⢸⡇⠀⣜⣿⣹⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⢀⡿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣯⡃⢸⡇⠀⢹⠂⠈⡇⠀⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠷⣼⠃⠀⠀⠀⠀⢷⡰⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⢿⣴⣿⣦⣀⣠⣀⣤⣿⣧⢾⠆⠀⠀⠀⠀⠀⠀⠀⣠⠶⠃⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣆⣉⣷⢦⣀⠀⠀⠀⠀⢸⡜⠿⣷⣿⣿⠿⣽⡿⠛⡞⠀⠀⠀⠀⢀⣠⣴⢊⣁⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣠⢿⣩⡷⡄⠈⠙⠓⠤⢤⣀⣙⣦⣈⣻⣦⣾⣁⣠⣞⣁⣀⠤⠴⠚⠋⣀⣿⣻⣧⡀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣟⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⡿⠷⣿⢿⡯⠉⠉⠀⠀⠀⠀⠀⠉⠉⣿⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣄⣀⡀⠀⠀⠀⠀⠀⠙⣶⡿⢸⠇⠀⠀⠀⠀⣀⣠⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠒⠒⠲⢾⣟⡥⠿⠒⠒⠛⠛