⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠣⢀⣀⣂⣭⣥⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣈⣔⣸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢢⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⢿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠪⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠺⠛⡿⣿⣿⡿⣿⣿⣿⣿⡇⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠹⢸⢿⢵⢿⡏⢹⡏⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡜⠘⠈⠃⠻⠃⢚⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⡀⠀⠀⠀⢸⣷⠇⠺⣿⢹⠹⣿⢻⣿⠾⣿⣷⣌⣰⠏⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⡨⠘⠀⣨⠏⠀⠳⠿⡾⠓⠈⢑⣬⠿⡞⠍⢻⠑⣦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠤⠞⣳⣧⡼⣉⠀⠀⢀⣀⣥⠀⠀⠈⠛⠁⠈⢾⠻⣑⣟⣉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢒⡿⠟⠛⠑⠂⠁⠀⠀⠓⠉⠀⠀⠀⠀⠀⢀⢄⠇⠊⠀⠀⠈⠙⠲⠦⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⣒⣫⣿⠗⠋⠀⠀⠀⠀⠀⠀⠓⠂⡶⣤⣄⠀⠀⠀⠐⣾⠂⠁⢠⣄⣲⠋⠀⠀⠀⢶⡶⡖⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠉⣨⠎⠁⠶⢯⡐⠔⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⡀⠀⠉⠀⣐⡷⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⡀⢴⢾⠋⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡀⠀⠀⣀⡈⠀⠈⠒⠄⠀⠨⠱⠧⡑⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠞⠁⠀⠈⠓⠀⠤⠴⠚⠃⠀⣺⣿⡿⠀⠀⠀⢘⣣⡐⠍⠁⠀⠀⠀⢠⠠⠂⠀⠀⢘⡦⠀⢀⡄⠀⠀⠐⢔⢬⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⣀⣠⠤⠒⠀⠀⠀⣰⢟⡣⠂⠀⠀⢀⣒⡊⡿⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⢃⡯⢰⣯⠞⢀⡔⠀⡘⠁⢀⠀⢼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠤⣾⠟⠁⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⡜⡁⠀⠀⠀⠀⠀⢊⣴⠀⠀⣨⡪⠂⠀⠋⠀⢸⠘⠓⠉⠀⠀⠘⠏⠉⢧⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠉⠀⠀⠀⠀⠀⠀⢀⣠⡤⠀⣀⣤⠀⠀⠠⣚⡅⠀⠀⠀⠀⠀⠀⢀⢼⢠⡯⢚⠂⠀⠀⠀⠀⡌⢠⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣠⠚⢏⡱⠂⠉⠳⠮⣑⠀⠀⠀⠘⢀⠤⠀⠀⠀⠀⠀⢘⠀⠀⢘⢤⡀⡀⠀⢸⠁⠨⣇⠀⠀⠀⠀⠢⢢⢆⠦⡘⡆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⣐⠢⢍⡞⠀⠀⠀⠀⠀⢒⣭⠂⠀⠊⢰⡠⠀⠀⠀⠀⠀⠋⠀⠐⣡⢏⠽⡵⠚⣟⠀⠀⠈⠢⣔⢠⠀⠀⠑⠅⠉⠀⠘⡅⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡶⣆⠡⡤⢀⢀⠀⠤⠤⠤⠼⠂⠀⠀⡉⠔⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⢀⡟⣆⠀⠀⠀⠫⠳⣠⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢨⡿⢑⠈⢈⡶⣆⣒⡍⠂⠀⠀⠀⠐⢸⠁⠀⠀⠀⠀⠀⠐⠀⠀⠴⢚⠃⠀⠀⡼⠁⠘⢦⡀⠀⠀⠁⠜⡆⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠺⣕⠡⠀⠉⢉⡓⠬⠐⠀⠀⠀⣀⠸⠀⠀⠀⠀⠀⢠⣼⠁⠀⠀⠘⠀⢀⣼⠃⠀⠀⠀⠙⠦⣄⠀⠈⠁⠀⠀⠀⢀⡀⢠⢸⣖⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠐⣫⢊⠁⠀⠀⠐⠙⣤⡐⠀⠈⠀⠤⢑⠡⠀⠀⢀⠫⠊⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⣱⡷⠀⠀⠸⣡⠨⠚⠙⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠹⠂⠘⣳⠂⠀⠀⠀⠀⠼⡂⡀⠀⠀⠉⢄⠺⠀⠀⣤⠎⢀⡤⡖⢷⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠁⠀⠀⢯⠟⠁⣂⠀⣸⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢿⡇⠀⢠⠊⠀⠀⠀⠠⠕⠚⠁⠀⠀⠈⠡⠀⢀⣘⠔⠀⠀⡱⢞⢅⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣔⠀⠀⠀⠘⢠⢹⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⣗⠄⠀⢗⠁⡂⠀⠐⡏⠠⢂⠀⣠⣆⣻⡴⡿⠓⣚⠁⠀⢨⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢸⠰⡀⠀⠀⠀⡟⠸⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢰⡷⡀⠈⠙⠩⠐⡠⠀⠕⠮⢮⠯⠁⢒⡋⠴⠍⢒⡚⢭⡵⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠘⠊⢃⢈⠂⠘⡓⡄⠀