⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⡶⠲⠲⠖⠖⠖⠶⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣀⣤⣤⣶⣿⡟⢉⠉⡁⢐⣠⣤⣤⣄⡀⠠⠀⠀⠀⠉⠙⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⣿⣿⣿⣿⣿⠍⡐⠠⢁⠂⢰⣿⠿⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠑⠊⣝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠿⣙⠢⠉⡄⠱⠀⢆⠀⡀⠀⠀⠀⠀⠈⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢌⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣭⣳⠢⠡⠁⠈⡁⢎⡐⣂⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢿⠋⡀⠀⠀⠀⠀⠰⣬⣾⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣎⢧⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣊⣔⣀⡀⠀⠀⠀⣿⣟⠡⡉⢀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⣆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢛⠛⢛⣿⣧⣄⣸⣿⣿⡿⠿⠿⠿⢿⣤⣤⣀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢻⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡡⠀⢾⣿⣿⣾⣿⣿⢧⡐⡀⠀⠀⣾⣧⣿⣿⡏⠛⢦⡄⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⢿⣷⣄⠈⠻⠿⠟⣻⠻⣷⣲⢡⠀⠀⠻⢿⣿⠿⠃⢀⣼⠇⢐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣷⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⣯⠻⢿⣿⣤⣴⡶⣟⢣⡽⢿⣧⣟⣤⡒⡤⢤⣤⣶⠿⠃⡐⢌⠠⠀⢂⠒⠠⠀⢂⠀⠀⠀⠌⣀⢻⣿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⢯⡄⠀⠀⠀⠀⠀⠀⣿⣿⣯⢿⣿⣮⣝⢯⡹⡱⢎⢣⠛⡷⣭⣛⠿⠿⠿⠿⢛⢁⠢⡑⢨⠀⠆⡉⠤⡘⠄⠡⠀⢀⠐⢀⡘⢠⣿⡿⣤⡀⠀⠀⠀
⠀⠀⣀⣀⣀⣀⣀⣤⣬⣿⣂⠀⠀⠙⣆⠀⠀⠀⠀⠀⢸⣿⣟⣯⡿⣽⣿⣎⠵⡱⢈⠂⢍⠰⡁⠙⠛⠶⢶⠌⠂⠄⡃⠄⠃⠘⡐⠌⠂⠐⠀⠂⠀⡀⢈⠠⣐⣷⠏⠀⠈⣷⠀⠀⠀
⡶⠋⢣⡠⠀⣠⠀⠀⠀⠀⠉⠂⠀⠀⢸⡄⡀⠀⠀⠀⠀⣿⣿⣯⣟⣯⣟⣯⡳⣅⠣⠌⢀⠣⠀⠀⠀⠀⠀⠀⠀⢀⠁⠠⠁⠂⢀⠠⠀⠈⢀⠀⡁⠠⢀⣶⠟⠁⠀⢰⣿⠛⠀⠀⠀
⣧⣤⡼⠥⠤⠓⠶⠒⠈⠉⠉⠒⠢⢄⡈⠛⣇⠀⠀⠀⠀⠘⣿⣾⣽⢯⡿⣽⣳⢮⣱⣎⣴⣨⣡⡄⡀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⠠⡐⢄⠒⠠⣵⡿⠇⢒⠲⣾⠛⠁⠀⠀⠀⠀
⠹⢯⣤⣀⣤⣤⣤⣴⣦⣴⣤⣤⣀⣴⠄⣸⡿⠀⠀⠀⠀⠀⠹⣿⣿⣻⣽⣻⣟⣛⠛⡛⢛⠛⡛⠛⠛⠋⣉⣩⣥⡴⣖⠳⣎⠥⣍⢢⠱⣈⢦⣩⣿⢋⣤⡶⠖⡀⠉⣲⡦⢤⣀⣀⠀
⠀⠈⠉⠉⠉⠉⠉⠀⠈⠉⠙⠛⠻⠿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣯⣷⣿⣿⣷⣯⣦⣳⣴⣳⣮⣿⣿⢷⣧⣟⣼⡻⣬⢳⣎⢧⣻⣴⡿⠟⠛⣿⡏⣠⡼⣫⡾⠋⠰⣦⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣽⡾⣽⢯⣟⣾⣻⣿⣿⣣⣶⣣⠍⠛⠿⣾⣏⠀⠀⠀⣼⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣷⣿⣿⣯⡀⠀⢀⣿⠻⠿⠛⠛⠉⠁⠉⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⢤⡀⠉⢛⠿⠿⠿⢿⣿⣿⡿⠟⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⡇⠀⠀⠀⠀⣀⣀⣤⣴⣒⣀⣠⣀⣀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⣿⣿⣿⣿⡟⢣⡏⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣯⣽⣼⣯⢭⡝⠛⡟⠡⢌⠣⠜⣿⣧⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⡤⠉⡌⠓⠺⢦⡼⠖⣋⠔⣈⠢⢁⣿⠟⣧⣠⢤⣶⣿⣿⣿⣿⣿⡟⢣⠝⣷⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⠿⣶⣥⣤⣀⣄⠠⢌⣳⣬⡾⢾⢛⣩⣿⡿⡿⢿⠿⡿⣗⠺⡏⢙⡿⣄⠂⡜⠠⢮⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣦⣄⣉⠉⠉⠙⠛⣉⣥⣶⣾⣿⣿⡿⣵⢫⣃⢎⡐⡉⠛⠿⣉⠳⡌⢷⣄⠓⣼⡾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠟⠛⠋⠀⢾⣿⣟⠻⠿⢷⣼⣦⣴⣠⣍⣲⣥⣳⣼⡾⠟⠛⣉⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣶⣢⣄⣀⣈⠉⢉⡙⢉⣡⣤⣶⡶⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠓⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀