⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠊⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⢰⢲⢾⢿⣿⣿⣿⣷⣾⠟⠁⠀⠀⠰⠀⠀⠂⡰⣠⡀⢐⣦⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣺⣬⣷⣿⣾⣷⣾⡼⣚⣝⣻⢻⣿⣶⣦⣤⠁⠀⠠⠈⡣⠼⣒⣯⡻⢑⡇
⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣾⣖⠽⣿⣿⣷⡀⠀⢤⣥⢁⡊⡀⠀⠹⠇
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠄⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡡⠀⡆⣼⣿⣿⣿⡿⢟⣿⣿⣿⣴⠻⣿⣷⡀⠀⠈⠚⠦⡊⠻⠃⠀
⠀⠀⢠⣿⣿⣿⣿⣿⠟⠋⠉⠈⠙⢿⣿⣿⣿⣿⣿⡏⠲⡇⠀⠠⠓⡴⠀⠀⠀⠀⠀⣰⣿⢟⠋⠀⠀⠀⠠⡏⠈⢹⣿⣿⣯⢉⣾⣿⣿⣿⣿⣿⡜⣿⣇⠀⠠⠀⠀⠀⠈⠀⠀
⠀⠀⣿⣿⣿⣿⣿⠁⠀⠐⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠥⠔⠋⠸⢒⠀⠀⠀⢠⣾⠟⢱⠀⠀⠀⠀⠀⠀⡇⠀⠈⢻⠘⠞⠛⢹⣿⣿⣿⣯⣿⣿⣮⢿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⢠⡆⠈⣀⠁⣐⠟⠁⠀⠆⠀⠀⠀⠀⠀⠠⡧⠀⠀⠘⠒⠀⠀⠈⠐⠻⠿⢹⣿⣿⣿⡞⡆⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟⡦⠀⠀⠀⠰⡎⠀⣠⡞⡏⠀⡠⠎⠀⠀⠀⠀⠂⠠⢠⡵⢡⠀⠀⠀⠀⠀⠠⢰⡰⡑⢸⣿⣿⣿⣿⢷⣄⠀⠀⠀⠀⠀⠀
⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢋⠈⠀⢀⡢⠊⢡⣣⡾⢻⡇⣄⡀⠐⠀⠀⠀⠀⠀⠀⠰⡉⠀⠀⠠⠀⠀⠀⠀⡺⢒⡍⣁⣴⣿⣟⣾⡿⠄⠘⣑⠀⠀⠀⠀⠀
⠀⠈⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢀⡚⡠⢞⣿⠏⢀⢇⣧⠁⠐⢂⠆⠀⠀⠀⠀⠀⢀⠈⠂⠄⠀⠡⠀⠀⣴⣼⣶⣟⣿⣿⣿⣾⠏⡞⣆⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⡠⢲⠚⢈⣾⠏⠐⣴⠂⣿⠀⠀⠸⠀⠀⠀⠀⠀⠀⢸⠀⠀⣁⠈⢂⣿⣿⣿⣿⡿⣽⣿⣿⡿⠛⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠄⠂⠁⠀⡇⠀⣾⣟⠠⠞⠁⠂⣿⡧⢀⠀⡸⠀⠀⠀⠀⠀⢸⡆⠀⡏⠀⣸⢾⣿⣿⣿⣺⣿⣿⣿⣿⣵⡀⠀⠂⠘⡠⠀⠀⠀⠀⠀
⠀⠀⠀⠈⢻⣿⣿⣿⣿⣆⠐⠀⠀⠀⠀⠀⠀⠀⣸⡇⣸⣿⣧⠍⠠⠀⠀⠹⡟⢌⠑⢧⡀⠀⠀⠀⠀⠸⠆⢰⣧⣾⣿⣿⣾⣻⣿⣿⣿⣿⣿⣿⣷⣿⣆⠀⢠⢣⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄⠀⣀⠀⠄⠐⡀⠨⣷⡿⣿⣇⢠⠃⢀⠠⠐⠛⠦⡁⢛⢆⠀⠀⠀⠀⠀⡪⣸⣿⣿⣿⣿⣿⣿⣮⡿⣿⣿⣟⣿⣿⣿⡻⣧⡀⢸⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣯⣿⣽⣿⣦⣤⣁⣀⠀⠀⢹⠿⣿⡿⢂⠴⠉⠀⢀⡄⢩⢉⠃⡈⠆⠀⠀⠀⠀⠰⠜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣅⠈⢻⣾⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⡿⠿⠷⠾⢷⠗⡀⠀⢠⠈⠰⢸⠂⠐⠒⠲⠀⠀⠀⠀⠀⠹⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⠀⠐⡟⢷⡤⠄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠩⢍⠽⢁⠒⠀⡩⠁⠰⠀⠀⠃⣀⠆⢣⠄⠀⣇⢦⠀⠀⠀⠀⠀⠀⢑⠆⠁⣼⣿⣿⣿⣿⣿⣿⣿⢼⣿⣿⣿⡀⢰⠇⠈⠻⡄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⢀⠔⡡⠁⠘⠉⡚⠀⠀⠆⠠⢃⠃⠀⠁⡈⣌⠀⢇⠎⠄⠀⠀⠀⠀⠀⠀⠠⠑⣿⣏⡉⠙⠻⣿⣿⡿⣾⣿⣿⣿⣇⡞⠀⠤⠄⠹⡀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠀⠔⠋⠀⠣⠀⢀⡖⠸⢀⡽⡠⠎⠚⠰⠠⣀⡁⠰⠉⠉⠀⡗⠀⠀⠀⠀⠀⠀⠀⠱⡯⢿⠿⣄⠀⠈⠻⡷⣿⣿⣿⢁⣿⠁⠀⠀⠀⠀⢃⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠰⢀⠶⠃⣜⡘⡀⡀⢸⠀⠀⣸⢸⡶⢠⠀⠀⢀⡼⡃⣠⡄⠶⡾⠀⠀⣀⠀⠀⠀⠀⠀⠀⣇⠀⡷⠻⣆⢠⣤⣸⣿⣿⣧⡿⢻⠰⠀⠀⠀⠀⠘⠀
⠀⠀⠀⠀⡀⠀⠰⡞⠃⢀⢣⠟⠀⣜⣎⠃⣇⠃⢸⠀⡆⣣⡀⣇⠇⣇⣴⠏⢻⡟⢋⢀⡜⠁⢀⡴⡝⠀⢠⠃⠀⠀⠃⢸⣆⣞⠀⡼⠎⢰⣿⣿⠛⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⡜⡵⠀⡰⠁⠀⣬⠈⠀⣜⢎⠙⠀⡽⠀⢸⢰⢰⢛⠁⢸⣰⣟⠀⠀⡇⣤⢫⠋⣠⡾⠈⢀⠃⠀⢦⠀⠀⢠⠀⢸⣽⡎⣷⣦⢀⣾⠟⠁⢠⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀
⠀⡆⠉⠀⠀⢔⠁⠀⢠⠀⠀⢠⠌⢸⠆⠀⡅⠀⡌⠈⢨⣺⠠⡸⠙⡏⠢⢰⣰⠇⠆⡠⠉⠀⠀⠘⠀⠠⡀⠆⠀⣸⡄⢸⠑⠢⣑⣴⠨⡇⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀
⠀⢇⡔⡊⢊⠂⠀⠀⡇⠀⠀⠌⠀⡼⠀⢀⠀⢠⣁⠷⠸⠲⡇⡗⡄⠸⠀⢀⠃⠀⠀⠀⠀⠀⠀⠆⢀⠃⠌⠀⠐⢼⠃⠘⠀⠓⡐⡈⡦⢷⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀
⠀⢔⠞⠁⡄⢀⠜⢰⠀⠀⡈⠀⢠⠀⠀⢈⠀⡈⠇⢀⠋⢲⣃⢕⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⢀⠂⢌⡊⠀⠈⠈⡼⠀⡊⠀⢀⠏⣴⡴⠁⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀
⠠⠊⠀⠜⠠⢠⠁⡆⠀⡐⠀⠀⡆⠀⠀⡆⠀⢃⡧⡡⠈⠀⠀⠀⠉⡄⣀⡇⠀⠀⠀⠀⠀⠀⠈⠎⠸⡔⠑⠀⠀⢰⣴⠃⠀⣜⠠⡟⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀
⢈⠄⠌⡓⠁⠂⢠⠃⠠⠀⠀⡈⠁⠀⢠⡧⠎⡈⣐⠀⠀⠀⠀⠀⠀⠰⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⢧⡞⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡘⢄⠔⠁⠀⣘⢠⠃⠀⠠⠰⠀⠀⣸⠇⡀⡎⠘⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⣘⢧⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠆⠁⠀⠀⠀⠏⠆⠀⠀⡄⡄⠀⠀⠋⡙⢁⠃⠄⠃⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡖⢌⢚⠎⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠢⣀⠀⠀⢐⣀⠀⢀⢀⠅⡍⠒⡥⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣈⢼⠢⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠊⠔⢧⠒⡈⢺⠌⡂⡇⣈⠅⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⠂⠄⠸⠘⠀⠘⠎⠠⢫⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠊⠄⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡈⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡡⠆⠆⡄⠀⠀⠀⢰⠀⠀⠀⠀⡀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠀⠀⣇⠀⠀⢠⠋⠀⠀⠀⠐⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡎⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠇⠀⢰⡑⠀⠀⠀⠀⠀⠃⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠆⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡰⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀