⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠐⠐⠐⠈⠈⠈⠈⠈⠁⠀⠀⠀⠁⠁⠐⠐⠀⠄⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠆⢄⢀⢀⢀⢀⢀⢀⡀⣀⣀⠀⠀⠀⠀⠤⣠⢄⢄⣄⢄⡄⣄⢤⢼⢼⢜⡒⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⢜⢌⢎⢕⢕⢱⢨⠢⡣⡱⢸⢐⢕⢲⠰⠤⠤⠤⠰⢀⢠⢣⢱⣁⡣⡣⡱⡱⡑⡌⡎⢕⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡣⡣⡊⡆⡕⢜⠔⡕⢅⢇⢕⠜⡜⢌⢆⢇⢍⢎⠭⡪⡩⡪⢢⢑⢆⢆⢣⢱⢘⢌⢎⢪⠸⡨⢪⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡣⡱⡨⡕⡕⡸⡘⡌⢎⢢⢑⠆⡇⢎⢪⢢⠣⡪⢪⠪⡊⡆⡣⡱⡡⡃⢎⢪⠢⡣⡱⡘⡔⡕⡱⣑⡕⡥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢱⢑⢜⠬⡪⡢⡣⡊⡎⡪⡸⡨⠪⡸⡘⠁⠣⠅⠡⢑⡱⡑⡅⡣⢊⡂⢥⢑⠅⡊⡐⡰⡱⡘⡜⡸⢰⠸⡨⡪⢢⢄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡣⡣⡑⡕⡍⡎⡦⠕⠊⠈⠀⠀⡀⣀⢠⡠⢤⢢⢲⢱⠱⡑⡕⡜⡔⡕⢥⠣⠣⢪⡒⢮⣶⢃⣷⣿⣾⡝⡎⣎⢳⢱⢫⢦⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢎⢆⢇⢇⢇⠫⠤⠴⢔⢲⠱⡹⡘⡌⡆⡇⢇⢮⡖⠑⡽⠈⠊⠐⠐⠐⡀⠌⠠⠁⢇⠈⡿⡄⢙⣷⣿⣗⡭⡦⣣⢗⢧⠇⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠴⡣⡋⡎⡪⡊⡆⢭⢃⣏⣎⣦⣷⠇⠅⣡⡨⢔⠄⢟⠀⠌⠪⠀⠁⠁⠁⠁⠀⠈⠀⠈⠀⢈⠘⠒⣄⠸⣿⣽⠪⠋⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⢠⠣⡣⡱⡸⣨⣦⣧⣿⣾⣿⣟⣿⣟⡯⢢⠏⢑⣴⣴⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣷⡈⢇⢻⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⢎⢆⡇⡗⡜⣿⣿⣯⣿⣯⣷⣿⣟⣯⣿⢥⠃⢀⣿⣿⣽⣿⣻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣷⣿⢿⣿⡗⢘⢝⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⢗⡣⡣⣓⢧⡓⡿⣿⣾⡿⢸⣿⣯⣿⣿⣻⣝⠀⠘⣿⣷⡿⣯⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣽⡿⡯⠁⠀⢘⣯⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠈⠙⠙⠊⠃⠉⠈⢿⣷⣿⢌⣷⣿⣷⣿⢿⡕⠀⠀⠈⠑⠟⠛⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣼⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣿⣷⢿⣻⣾⡿⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⡿⣟⣿⣿⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣿⡿⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⢰⢸⡱⣝⣿⣿⢿⣻⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠘⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⢕⣹⢽⢸⢸⣯⢿⣻⣿⡿⣯⡗⢶⣀⡀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠁⣀⣠⠤⠒⠍⠈⠀⠀⠈⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⠣⣸⢸⣸⡾⡺⡽⣿⣽⣿⣟⣿⣷⢽⡻⣸⢸⢙⡓⡳⢔⣰⢐⢰⡢⡗⡟⣽⣵⣪⡪⣆⢄⡀⠀⠀⠀⢿⡿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡝⢎⢾⣸⢮⠧⡏⣾⢿⣻⣗⣿⣿⣽⣺⢇⢇⢇⢇⢇⢕⠕⣝⠈⢐⢧⢫⡪⣪⣾⣷⢝⢮⡪⡪⡂⠀⠀⢹⣿⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡜⢌⢎⢢⠪⡙⡕⣗⣜⣿⣿⣿⣮⡳⣿⡮⡻⡚⡎⡓⡕⡱⡣⢯⢐⢀⢏⢧⡧⡷⣻⡿⣷⡑⡌⡎⢎⢆⠀⢘⡏⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡰⣱⢹⢬⠢⡣⡃⢧⢱⢸⢺⣿⣾⢿⣷⡍⢗⠕⡕⢜⢌⢎⢌⢎⠪⡹⣹⡫⣎⡇⡏⡪⣿⣿⢷⡱⡸⡸⢪⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⢇⣇⢇⢷⢑⠜⡜⡜⣜⢬⠪⣿⣾⣿⣿⣽⣎⡪⣊⡪⡂⡇⡪⡂⡇⢕⣪⣏⢇⡗⡕⢍⢿⣻⣿⣟⡜⢜⠸⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⡮⣚⢼⡸⡑⡽⡜⡜⣞⢮⢳⡹⣷⣿⣷⡿⣿⣮⢊⢎⡣⡫⡪⢫⢋⢇⡇⢝⢎⡮⣞⢎⢻⣟⣯⣿⣧⢣⠹⡸⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠪⠚⠼⠪⡴⢝⠜⡜⢎⢪⠢⡣⢻⣷⡿⣿⣻⣽⣯⡕⢜⢌⢎⠜⡌⡆⡏⡆⡣⡪⡎⡕⢅⢿⣿⣯⣿⣇⢇⠎⠮⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡫⡊⡮⡊⡎⢜⢸⢻⣿⢿⣿⣟⣿⣿⣨⢢⠣⡱⡱⡸⡸⡜⢔⢅⢇⢎⢎⢺⣿⣾⣟⣿⣎⢪⢱⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⡆⡇⣇⡇⢇⠕⢜⠼⣿⡿⣯⣿⣿⣾⣿⡜⢍⢎⢞⢌⢎⢎⢆⢣⢣⢣⢺⢘⢷⣿⣻⣯⣷⡇⡣⢣⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢪⡪⡊⡖⡜⢔⢍⢪⢘⢾⣿⡿⣿⣾⣷⣿⡗⡇⡫⡹⡨⡢⣳⠡⡣⡹⡪⡸⡪⢞⣿⣟⣿⣽⢷⢡⢣⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀