⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡭⣷⠴⠖⠒⠒⠒⠒⠒⠒⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣿⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢈⡽⠧⠤⠤⠤⠤⣤⣄⣀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠙⣲⢋⣩⣭⣷⡄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⡴⠚⠁⠸⣿⣿⡿⠀⠀⠀⠈⠉⠳⢆⣀⠘⢿⡿⠀⠀⠀⠀⠀⠓⢯⡀⣀⣻⣹⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣵⠋⠀⠀⠀⠀⠈⢏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣫⠇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⢀⣤⠤⠶⠶⠷⠶⠦⠤⠤⣤⣀⣀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣡⡶⣻⣉⣀⣀⠀⠀⣀⣠⠤⠤⠔⠚⠓⠛⠿⢦⣄⡸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⣰⠃⠀⠀⣴⣍⡉⢁⣤⢾⡆⣼⡆⠀⠀⠀⠀⠀⠛⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠅⢰⠃⠀⠀⢸⠛⡟⠛⠛⠛⢿⠀⣿⠃⠀⠀⠀⢸⢆⠀⣈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⡟⣿⢿⣀⣿⢰⠀⠀⠀⠀⢸⢿⣿⣤⣤⣸⣿⡋⡾⡇⢟⡜⣇⠀⠀⠀⠀⠀⠀⠀⠀⡜⡜⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣼⡄⠀⡇⠟⠉⠁⣿⡘⠀⠀⠀⠀⠸⣽⡇⢧⠀⠸⣽⠙⢷⣗⠈⠀⢻⡄⠀⠀⠀⠀⠀⠀⢰⣳⠃⢹⡄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣏⡁⢦⣷⣶⣰⣦⣿⡇⠀⠀⠀⠀⠀⣿⢃⣼⣧⣠⣻⣆⠘⣟⡄⠀⠈⣧⠀⠀⠀⠀⠀⠀⣼⡧⠚⠁⢷⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣷⣇⣾⣽⣿⣿⣿⢿⡷⠀⠀⠀⠀⠀⠹⡄⣻⣿⣿⣿⣿⣦⣻⡇⠀⠀⣿⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠘⣇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠙⣽⡿⣟⡏⢿⣿⣾⡏⠀⠀⠀⠀⠀⠀⠙⢸⣿⣯⣽⠛⢿⡿⢧⣀⣰⣻⡆⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⣰⣿⣧⣿⡘⠼⠬⠼⠁⡔⣱⠀⠀⠀⠀⠀⠸⠯⢭⣿⣀⢾⡿⠖⢚⣋⣽⣇⡠⠔⢻⠏⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢠⢿⣿⣿⣿⠰⠷⠕⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⣴⡔⡰⠚⢻⢟⠏⠉⠳⣼⠇⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡟⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⢼⠀⠀⢰⣿⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⢸⠁⠀⠀⠙⢿⣿⣿⣷⣦⣄⠀⠈⠉⠉⠉⠁⠀⢀⣀⣤⣴⣶⡟⡸⠀⠀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣈⢧⣀⣀⡀⠀⠙⢿⣿⣿⣿⣿⣦⣤⣤⣴⣶⣿⣿⢿⣿⣿⣿⣧⠇⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣻⠋⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇⡉⠉⠛⠶⣄⡙⠻⣿⣿⣿⢿⣿⣿⡿⢟⣡⠼⠛⠋⣹⡏⠀⢀⣟⣁⠀⠀⠀⠀⠀⣀⡠⠴⠊⣡⠞⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣦⡀⠀⠈⠛⠉⠉⠉⠛⠉⠿⠛⠾⢭⣤⣤⠴⠖⠛⠋⠙⡛⣷⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢿⣟⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣠⣴⣧⣇⣀⣠⠤⠶⠒⣛⡭⠷⣄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠴⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⠏⠈⠀⠀⢀⠔⢊⡇⠀⠀⠹⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⢛⡥⠀⠀⠀⠀⣀⣤⣤⣄⠄⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⣄⠀⡴⠋⣰⠋⡀⠀⠀⠀⠘⣆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣿⡚⠀⠀⠐⠙⠋⠚⢿⠞⠁⢠⠀⡈⠰⣿⣿⣻⠟⠦⡀⠀⠀⠀⠀⠀⠒⠻⡾⠁⣼⢣⠞⠁⠀⠀⠀⠀⠸⡆⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠘⣷⡀⠀⠠⣾⡛⠛⠉⠘⡆⢿⠷⠀⠀⠈⠛⠗⣤⣄⠀⠀⠀⠀⠀⠀⡘⢛⣳⢰⣳⠋⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⠞⠉⠛⢤⣠⢌⣀⠀⠀⠀⠙⠓⠒⠒⠛⠀⠓⢮⣟⢿⣷⠀⠀⠀⠀⠠⠍⣯⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⡴⠋⠀⠀⠀⠐⣂⣀⠀⠉⠁⠒⢀⣀⣤⡀⠀⢀⠀⠀⠈⠁⢀⠀⣤⣬⣭⠉⠀⠘⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⢰⣷⠀⠀⠀⠀⠀⠀⠈⣛⢦⡀⠀⠈⠘⠋⠙⠛⠉⠀⠀⠀⠀⠈⠉⢱⠆⠀⠀⠀⠀⠀⢳⡄⠠⠤⣄⠀⠀⠀⠀⠀⠀⠘⣇⠀
⠀⠀⠀⣠⣴⣖⣶⣦⡤⢤⣀⠀⠀⢠⡇⠀⠛⣇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣴⠖⠶⠦⠤⠤⣤⡀⠀⠀⠀⢀⣤⠾⠀⠀⠀⠀⠀⠀⠐⣷⣿⠭⢭⣿⡒⠶⣤⡀⠀⠀⢻⠀
⣀⡴⠿⠿⠴⢤⡀⠀⠉⠀⠀⠙⠲⣼⠁⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⢠⠤⠽⠦⠴⣶⡿⠒⠂⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢇⢹⡀⠈⠃⠀⠀⢸⡀
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠈⣷⢠⣤⢏⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⡼⠀
⢷⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⢀⣿⣿⠀⢸⢣⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀
⠀⠳⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⣀⣴⣿⣿⡿⠀⠈⠾⣰⣿⡀⠀⠀⠀⠀⠀⠀⢀⣠⠎⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀
⠀⠀⠀⠉⠛⢶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠹⣧⣄⡀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣻⠿⡃⡇⠀⠀⠀⠐⠿⠿⣶⣤⡤⠄⠰⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠉⠛⢦⣀⠀⠀⠀⠀⠀⡛⣅⡌⠉⠓⠶⢤⣀⡀⠀⠀⠘⢿⣖⣛⣉⡇⠀⠀⢀⠤⡄⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢶⣄⠀⠀⠀⠙⡿⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⡀⠉⠀⢉⣇⡀⢠⠏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢾⡁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠿⠿⢿⡄⠀⠈⠲⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⡻⠀⠈⣇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣫⠵⠀⠀⠀⠈⠷⢤⣀⣀⣀⣀⣉⣁⣠⠤⠴⠶⠛⠛⠉⣏⣧⠀⢠⠃⠀⠀⢹⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣧⠄⢸⡼⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀