⢠⠔⠒⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢆⠀⠀⠀⠀⠞⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠒⠂⠀⠀⢸⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢰⠎⠀⠀⠀⠀⠀⠉⠀⡘⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠠⡰⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡇⠀⢀⣤⠦⢄⡘⠀⠀⠀⠀⠀⠀⠈⠉⠝⣙⠍⠉⠀⢀⡠⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡇⠇⠀⡌⠂⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⢀⠔⠉⢀⠀⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠇⢀⠀⠱⣀⠀⠀⣧⠀⠀⠀⢀⡤⠞⠀⠀⠀⠄⢠⡎⠀⠀⠆⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢰⠸⠀⠀⣐⢛⠀⠀⠀⠀⠀⠀⡸⠆⠀⠘⢸⡀⠄⢠⡂⡁⠔⠒⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢰⠀⡆⠀⠀⠀⡌⡆⢧⠀⡝⠘⢦⡀⠀⠀⠀⡐⢄⠀⠀⡐⠁⠀⡠⠪⢋⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠀⠸⡀⠀⠀⡇⢁⠘⣵⠣⢁⠀⠀⠙⠉⠂⡇⠐⠀⢰⠋⠀⣼⠑⠉⠀⠀⠈⠉⠉⠁⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠇⠀⢡⠀⠘⠇⠘⡀⠸⠀⠈⢦⠀⠀⠀⠀⢸⡀⡆⣸⠄⢸⢥⠎⠀⢀⠠⢀⣀⡀⢀⠀⡀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⠀⠀⢢⠀⡆⠀⠐⣀⣧⠀⠈⡥⡀⠠⠀⠘⣀⡅⣭⠄⣾⢃⢠⡤⠃⢓⡤⠒⠈⠉⠐⡄⠱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠡⣡⠀⠀⠈⢇⠀⠀⡘⡜⢅⢘⣼⢽⡉⢰⣸⣇⡶⠍⠀⢠⠋⠀⠀⠀⠀⢄⡄⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠈⠣⡀⠀⠘⡌⠀⠈⠹⣷⡿⣵⡌⣇⣚⣺⠯⠐⠠⡐⡇⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠥⠀⠀⠀⠀⢉⡤⠺⠷⣀⠲⢮⠈⢍⠻⣾⡎⠁⠀⢀⡂⢦⢃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⢕⠤⡐⠁⠀⠀⠐⢘⣮⡬⣷⢶⣻⡟⠇⠀⣀⣶⢎⠘⠘⡀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢺⠷⢂⡠⢶⡉⢙⡷⢞⢿⡿⣿⠁⢡⡞⡏⠈⠀⢂⢡⢡⠀⠀⠀⢠⠕⠛⠃⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡐⡝⠀⢀⠜⢝⡰⠃⠹⢼⡿⢂⡟⢗⡀⠀⡀⠀⣂⢞⠀⠀⢠⡏⠀⠀⡄⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⣰⣾⠀⢠⠃⠀⣜⡇⠀⡇⡇⡇⠀⡖⣄⠈⢃⠪⠂⣌⣾⠀⠀⣷⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⠉⠀⠒⠚⠀⠸⠀⢃⠀⡆⢃⠰⢄⡬⠉⠀⠈⠉⠉⠉⠳⡄⠀⢟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⡀⠀⠀⢸⡀⠁⢀⡜⠑⢧⠸⡀⠈⠆⠀⠀⢠⣮⠁⠢⡀⠱⡀⡿⡜⡄⠀⠀⠀⠀⠀⡠⠮⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣠⡴⠈⠀⠀⠈⠀⠣⠀⡔⠀⠀⠀⠁⠀⠀⠘⢦⠉⠣⠑⡘⢄⠀⠀⠀⢠⠹⠀⢀⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣯⠁⠀⠀⠀⠀⠁⠄⣈⠲⢤⡤⠤⠀⠀⠀⠀⠀⢓⢄⠀⠙⠺⡦⠀⠀⡦⠁⣶⡻⠑⠁⢀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⣢⣴⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠑⠠⡀⠈⢳⡄⠇⠀⢸⡀⡰⢊⡏⢩⠁⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠠⡹⣶⠀⢸⢠⠄⠀⠇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢀⣴⠒⠾⡢⡆⢒⠂⠠⠤⠄⡀⣧⡤⡤⠖⠒⠀⠀⠈⠫⣧⢸⠋⡀⠀⠀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠚⠃⠀⠀⠀⠁⠂⠤⠉⠈⠭⠍⡟⠁⠀⠀⠀⠀⠀⠀⠀⠘⡎⡄⡅⠀⠀⠇⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡠⠒⠐⠂⠤⠀⠄⣷⠀⢸⢠⡆⠀⣟⠀⢀⡠⠀⢄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣒⢹⠪⣐⠒⠀⠀⠀⢾⠧⠀⠀⠀⠀⠀⠀⡄⢀⠔⡠⠏⡀⡈⠁⠀⡄⡀⠘⡤⣸⡘⡃⢠⠁⢀⠊⠀⡸⣫⣑⡄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⠉⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠘⠞⠀⠀⠈⡅⠀⠑⢬⡻⢌⢧⢶⣷⣧⠃⢀⢆⠌⠀⣇⠏⠨⠗⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣠⠵⢢⠀⠀⢽⣎⡏⢸⣏⣽⣯⠂⠂⠠⠚⡝⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢀⠔⠁⠀⠀⠀⢨⠤⣴⠝⣺⣿⣵⣻⣿⡅⢀⢃⢌⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠥⠄⠀⠐⠐⠛⡻⢟⡿⠋⠺⣿⣟⠿⡺⣳⠩⠁⠈⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀