⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⢀⢄⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⠸⡘⡔⡕⢕⢱⢱⢨⢹⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⡕⡕⡕⡕⡜⡜⢜⢔⢕⢕⢕⢝⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⡢⡣⡇⡇⡣⡱⡸⡘⡜⡔⡕⢕⢍⢞⢾⣽⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡧⡻⡸⡘⡜⣜⢜⡜⡜⡌⡮⡎⡎⡎⡎⣷⡿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡯⡣⣣⡯⠪⡊⣞⢞⢼⡸⣸⣽⢜⡜⡜⡜⣿⡝⠌⠈⠑⢕⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣶⣶⣷⢱⢝⢏⠢⣷⡓⠕⠇⢳⢇⡷⣽⢺⣣⢇⡗⡿⡕⢱⢸⢠⠱⡁⡪⢐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣜⣯⢿⢽⣞⡜⣮⠢⢀⠁⠀⠄⠐⠳⡏⡳⡽⣪⡾⣽⣯⣿⣮⡗⣗⣶⣖⣬⢴⢥⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣷⣿⡾⣿⣽⢷⡯⣮⣇⠄⠀⡂⠠⠕⢀⢐⣾⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣅⠣⢑⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⡙⢅⠣⢅⢍⠻⢿⣷⣻⣆⣁⣄⣦⣶⣷⣿⣟⡿⢻⣿⣿⣿⢟⢏⠻⣽⣿⣿⣿⣞⠌⡂⢕⢐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠪⡈⢆⢑⠔⡌⢜⠸⡸⢷⡯⣟⣿⣽⡿⣷⣿⠅⣶⢄⢹⣿⡗⢀⣶⣢⠈⣿⣿⣿⣇⡪⢐⠅⡢⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⡂⠕⢌⢂⠢⡁⡊⢆⡓⣬⡱⣻⣿⣞⡾⣿⣻⣿⡌⢿⢏⠎⡫⠹⡘⢿⠿⢀⣿⣿⣿⣗⡯⣗⡵⡴⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠡⡊⡪⡂⠢⠡⡂⠪⡐⡑⢦⡻⣼⣻⣗⡽⣽⢿⢿⡏⡆⢕⠨⠂⠅⡂⠅⡪⡨⣿⣿⣿⣿⣝⣷⡽⣽⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡑⠌⢶⣪⣎⠌⡊⢌⢂⠢⡊⢔⠩⠳⢿⣗⡯⣻⣿⣿⡷⡪⣐⠅⡅⢥⢢⢣⠱⠘⣿⣯⡛⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠢⠡⡑⡘⡪⠣⢌⢂⠢⡁⡢⢂⠅⢕⠡⡑⠝⢿⢽⣿⡏⠂⠕⠕⢕⠕⡑⠄⠁⠁⠙⣿⣷⣿⣿⣿⣯⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡾⣜⣞⣞⢦⠡⡑⢄⢑⠐⢌⢐⠌⠢⡈⠢⡑⡐⢍⠻⡀⠈⠨⠈⠢⠈⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⣗⢷⢽⣽⡳⣮⡦⡵⡵⣥⡂⢅⢑⠌⡂⡢⡈⡢⡈⡂⡢⠠⡀⠀⠀⠀⠀⠀⠀⢀⠀⣻⣿⣿⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡐⡐⢍⢟⠺⣹⣿⣿⣿⣿⣮⣯⣷⣔⢅⢂⢂⠢⡂⡂⡢⠨⠢⡈⡢⠢⠠⡀⡁⠌⠐⢐⢈⢻⣿⣻⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣺⢶⢦⣗⣾⣷⣿⣟⣿⣿⡿⣿⡿⣷⣧⣢⡑⡐⡐⠌⢌⢂⠢⠨⡨⠨⡼⡀⠀⠀⠀⠀⠄⢻⣿⢷⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣝⣗⣷⣿⢿⣾⡿⣯⣿⣿⣿⢿⡿⣿⣟⣿⣾⢶⣵⢔⢔⠡⡑⣨⢬⢬⢆⠀⠀⠀⠀⠀⠀⣿⣿⣷⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣺⣞⣞⠜⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣻⣯⣿⡿⣟⣯⣶⡵⣽⣺⢽⢽⢽⣂⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⢷⠇⡣⣾⣻⣯⣷⣿⣿⣿⣯⢷⡃⠛⣿⣽⣿⣿⣿⢿⣟⣷⣷⣯⣿⣽⢪⠄⠀⠀⠀⠀⢽⣿⡿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⣀⡠⢾⡻⣮⣟⠇⡲⣟⣿⣿⣟⣿⡏⢿⠿⡻⣿⡀⠀⠀⠙⢷⣿⣾⣿⣿⣿⣿⣿⣿⣿⡝⠄⠀⠀⠀⠀⣽⣿⣿⣿⣟⡿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢔⠝⣌⢜⣕⣿⡽⣞⡷⡽⣯⣟⣿⣽⣟⢀⢆⠑⠄⢍⢻⣦⡀⠀⠀⠙⢿⣽⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⡅⠅⣽⣿⣿⣿⣿⡿⢍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⣵⢿⢵⢸⣲⢜⢕⢪⢾⡫⡸⡐⡂⡊⡂⡢⠡⡡⢑⢐⠹⣿⣷⣄⠀⠈⠻⣿⣿⣿⣯⣿⣿⣿⣧⠀⡂⠬⡨⢪⠿⡿⡟⡏⢇⠢⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠋⠃⠧⣷⣿⣾⣳⣯⡮⣟⢇⢆⠪⠐⢌⠐⢅⠂⢕⠠⡡⠹⣿⣿⣷⠀⠀⠈⢿⣿⣿⣿⢿⣻⣿⣷⣐⠕⢌⢂⠣⡑⢜⠨⡐⠌⡂⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠱⡸⡸⡹⡘⡌⡆⠪⡈⠢⡑⠄⠕⢄⠑⠄⢕⢘⢛⠋⠀⠀⠀⠈⣿⣿⣾⣿⣿⣿⣿⣿⣷⣕⡐⠅⡊⠄⢕⠠⡑⡨⢐⠡⡑⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢕⢪⢪⢪⠨⡂⡊⠢⠨⠨⡈⠢⠡⠡⡑⡐⡑⣦⣀⡀⡀⢀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡊⢌⠢⡈⡢⡈⡢⣱⠈⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠱⠡⢣⢢⢪⢘⢌⢌⢌⠌⡊⢌⠢⢨⢸⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣱⢐⠦⢿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠘⠊⠎⠮⣪⡢⣕⢵⢵⠝⠫⡩⡣⡫⡪⡣⡫⡻⡻⡻⡫⡻⡿⣿⣿⣿⣿⣿⣿⣿⠌⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠣⠣⠪⠊⠘⠈⠈⠈⠈⠈⠈⠊⠋⠻⠾⡿⢯⢷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀