⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣸⣿⡀⠀⠀⠀⠀⠀⣠⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⣻⣿⡀⠀⠀⢀⡔⠿⢧⡋⢹⢃⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠀⢿⣿⡇⠀⣠⡾⠃⠀⡼⠁⣾⠆⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠘⢸⣿⣇⣴⠟⠁⠀⢸⡀⢸⠧⢠⣾⠋⠀⠀⠀⠀⠀⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠗⠀⠀⠸⢋⡚⢉⡤⣴⠞⢰⠁⡜⢤⣿⠇⠀⠀⠀⠀⠀⠀⡐⠈⡄⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁⠀⠀⢰⣯⠀⢫⡌⣿⡆⠘⢰⢳⢸⡇⠀⠀⠀⠀⠀⠀⢰⠈⠂⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢧⠀⢀⣦⢟⡻⡽⡼⣇⠸⠀⠀⣼⠠⣿⠃⠀⠀⠀⠀⠀⠰⠠⢌⠀⠁⢀⢣⠂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣝⡄⢾⣿⣌⢻⠥⢓⡯⠀⠂⠀⡏⣷⣿⡀⠀⠀⠀⠀⠀⡅⠡⠂⢌⠀⠠⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣵⢿⣼⣿⣿⣿⡀⢏⠀⠳⡀⠀⠀⣇⢹⣿⣇⠀⠀⠀⠀⢰⠈⡐⠡⠌⠀⠀⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣿⡇⡟⣿⣞⡬⢿⡈⢆⠐⡁⠀⠀⣿⣸⣿⣿⡆⠀⠀⠀⠀⠁⡐⠠⠈⠀⠀⠀⡁⠠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠸⡿⡷⡈⢻⣿⣿⣣⠙⡄⠂⠀⢁⠀⢿⣼⣿⡿⠃⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⡤⢶⣶⣿⣉⠛⠯⡹⢿⠉⣦⣄⣀⡙⣽⠱⠈⣿⣿⣿⢷⣄⠠⠁⠂⠀⣾⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀
⠀⠀⣀⡴⠾⠛⡁⠀⢿⣯⡳⠜⡠⠁⠌⠂⠙⣿⡌⣟⣾⣑⢢⢿⡿⣙⠿⣬⢢⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠴⠧⢶⢶⣶⡿⠁⢈⡉⠣⢬⠁⠀⢀⠀⠀⠈⠏⠠⠌⡳⠭⡜⢔⢫⠞⠴⠃⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⢾⢿⣿⣆⣆⠐⠀⠤⣀⠀⠀⠀⠠⠀⠙⢦⡀⠀⠁⠀⠀⠁⢈⠨⠳⣶⡁⢃⠒⠀⠀⠀⢹⣿⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠻⣿⣯⣌⢧⠐⠀⠐⠈⠑⣂⠤⢄⡀⠀⠀⠈⡄⣐⡂⠀⠀⠀⠀⠈⢿⠧⠀⣀⠀⠀⢸⣿⣄⠀⢠⣾⣿⣟⡻⢟⡻⠖⡶⢄⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢻⣻⣬⠻⡖⡄⠀⠄⠡⡀⣉⠒⠂⠉⠐⠀⠀⢄⣉⡙⠳⢶⣶⡤⠤⠳⣆⣀⠰⠀⢸⡿⣻⣦⡟⣿⡛⣮⠱⡃⠐⠠⠀⣿⡄⠐⢀⣤⣶⠾⠋⠉⠀⠀
⠀⠀⠀⠀⣿⣿⡥⠉⣖⡘⢦⡀⡘⢥⣂⠐⡀⠀⠀⠈⠀⠈⠉⠀⠂⠠⠉⣉⠁⠒⠒⠒⠒⡈⣽⣿⣯⡇⠡⡙⠦⡙⠔⡠⢀⠁⣿⣿⡄⠀⠞⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⣿⣯⢱⠋⠈⠣⣛⣉⣁⣈⡅⣠⠘⠶⠳⣶⠖⠐⠒⡤⡥⠥⣤⢦⡥⣤⢤⠤⢄⡿⠻⠓⠛⠄⠈⠱⠈⢐⠀⠦⣽⣻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠯⣴⣦⣶⣾⣿⡿⣽⣳⢾⣥⣌⣆⡠⠀⣀⠀⠁⠈⠉⢉⣉⠛⠙⠒⠒⠒⠉⡅⡐⠀⠀⠰⡤⣀⠀⠀⢈⠤⢳⢶⢹⣿⣿⣆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣧⢏⠾⡼⠯⠽⠿⢛⢉⣤⣶⣾⡭⠭⠖⠒⠒⠒⢈⡁⠄⣡⠊⠠⠀⠐⠁⠲⡤⣀⠮⡜⡣⢎⡱⢾⣭⡻⣷⣦⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠎⠾⠱⣿⠾⣟⡵⣿⠟⠋⢀⠠⠀⣀⠨⠔⠊⠁⠀⣰⡿⠀⠀⢄⡤⣼⣷⠏⡀⢭⡲⢍⣣⠆⣓⠺⣽⣾⡻⣷⡄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣏⠱⠇⠠⢀⣐⡥⠖⢋⡀⠀⠀⠀⢀⡿⢁⣾⣶⣟⣮⣽⣿⣿⣿⣦⡁⠪⢍⡒⠩⣑⣻⡴⣯⣝⣻⣿⣆⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠻⠛⠀⠀⠻⣩⣿⣷⣏⠍⠀⠀⠀⠀⢀⣼⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⡦⢵⣧⢛⡭⢳⠶⣿⣼⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⠓⡀⠐⢠⣿⣿⣿⣿⣧⠄⠀⠀⢀⢬⣿⣿⡿⠃⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠿⢦⣭⣿⡬⢿⣿⣿⠿⠃⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣭⣥⠒⠀⠁⢸⣿⣿⣿⠟⡹⡄⣐⠀⢮⣿⣿⣿⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣮⠉⠄⠉⢀⣤⣾⣿⣿⠯⡕⡜⣐⠈⣜⣾⢻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧⠠⡤⢌⣾⣿⣿⣿⣿⡟⡝⡁⣊⠳⢻⢹⢸⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠷⣸⣿⣿⣿⣿⣯⢞⠕⣈⠙⣶⠹⣤⢳⣿⣯⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠈⢻⣿⣿⣿⣿⡣⢌⢰⠀⢷⡹⢴⣧⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠀⠀⣼⣹⣿⢨⣿⣿⣭⡁⠠⢸⣇⠺⢿⡟⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢹⣷⢿⣾⡿⣿⡿⢁⠘⢸⠘⠘⣾⡑⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢠⡜⠁⠘⢿⣳⣿⣹⠄⠘⠈⠂⠁⢚⠄⠘⣜⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⠜⡘⠀⠘⠀⠀⠄⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣴⡀⠂⢡⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⡄⠂⢆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠘⡆⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀