⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⠋⡌⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣼⢇⢜⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢐⢼⣻⣭⡺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢌⢢⠡⣱⣿⡿⣯⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢺⣎⢎⣗⠼⣿⡿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣗⢷⣿⣟⢿⣻⣴⢰⢐⡐⡔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⣿⣿⣿⡿⣷⣯⡿⡟⠏⠓⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢎⢪⢹⢿⣽⢟⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢆⢕⢕⡵⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠨⡊⠢⡱⣕⣿⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡈⠢⡈⡪⣺⣺⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠡⡈⠢⡪⣪⣗⣟⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠅⠕⠌⢎⢮⢣⣳⢷⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣌⡂⡢⠠⠨⠨⠨⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⢑⢑⢕⢕⣵⣳⣿⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡼⣮⡹⡌⠌⢌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡸⣰⢱⢝⢾⣳⣿⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠉⣯⣪⣢⠑⢌⢂⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⡝⡮⡪⡮⡻⣿⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⣢⣺⣽⡗⣕⢐⢐⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡢⡣⡈⡊⡣⡑⡝⢟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣰⣠⣷⣕⣴⡰⣴⡆⡀⠠⣷⣿⠝⢯⡆⡣⡢⡑⠄⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢽⢸⢐⢐⢐⠌⢌⠢⡡⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣦⢯⢛⢛⢝⢛⠻⢷⣟⣷⡿⡝⠈⠈⠀⠀⠀⠹⡜⡌⡌⡂⠌⠢⡑⢄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⡨⡢⡢⡑⡐⡑⡕⡕⡅⠀⠀⠀⠀⠀⠀⠀⠀⢼⡩⣐⢔⢽⡩⡡⡣⡣⣿⣯⣿⡅⡀⠀⠀⠀⠀⠀⠈⢎⢆⠌⢌⢐⢈⢂⠢⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⠀⠀⠀⠀⠀⠀⠀⢌⠮⡫⠩⡊⠢⠩⠩⢪⢪⡾⡷⣯⣿⢀⠀⠀⠀⠀⠀⠀⠀⢣⡃⡂⡂⡂⡂⡣⡪⡂⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠸⡸⡸⡸⡸⡸⡸⡸⡽⠀⠀⠀⠀⠀⠀⢨⢨⢪⢊⢊⠪⡨⠨⠨⡢⡷⣿⡿⣿⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⡷⣕⢕⢔⢕⢕⢕⢕⢅⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡪⡪⡪⡪⡪⡪⡪⡪⡪⣯⠠⠀⠀⠀⠀⢀⡸⣸⡴⣦⢧⣇⡎⣎⢪⢪⣫⢟⣯⣟⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⣣⢣⢣⢳⡱⡱⡱⡡⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⣺⢸⢸⢸⢸⢸⢸⡺⡣⡡⡑⢄⠀⠀⠲⣫⣿⢯⣿⡽⣷⣿⡾⣷⡷⣿⣽⣻⣿⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡪⡳⣕⢕⢕⢕⢕⢕⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⣵⣣⢣⢣⢣⢳⢕⢐⢐⢌⠢⢡⠀⠀⢹⣾⢿⣷⠿⢿⢾⡿⣿⣻⣿⣻⣿⡷⡏⠀⠀⠀⠀⠀⠀⠀⡀⡀⢄⢜⢮⡳⡵⡫⡎⣎⢮⣞⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡾⣾⢳⢱⢱⢱⢑⢐⢐⢐⢅⢕⢕⠀⡎⡎⡻⡸⡸⡸⡕⡏⡏⣿⣽⡿⣯⡿⠀⠀⠀⠀⠀⠠⡠⡱⡰⢸⢸⢸⢱⢹⢘⢜⡼⣽⣿⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡯⡪⡪⡎⢎⢆⢂⢂⢂⠢⠱⡱⡹⡸⡨⠨⠪⡪⡪⡺⡸⣸⣳⢿⣿⡿⠃⠂⢂⢀⠠⡨⡪⣢⢱⢌⢆⠇⡇⡵⣱⣵⣿⣻⡿⡾⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠪⡊⡊⡪⡢⡂⡂⠢⠡⡑⡘⢜⢌⠌⢌⢎⠪⡊⣎⢞⡮⡾⣿⣫⢬⢢⠱⢠⢱⢱⡸⡸⣪⡳⡱⡱⣳⡹⣮⣿⣾⣿⣻⡏⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡑⡐⡰⡱⡱⡐⠌⢌⢂⠢⡈⠢⡑⢅⢕⠕⠅⢕⢕⢝⢝⠕⡕⢜⢜⢔⢱⢱⣣⣳⢽⢪⡣⡫⣪⢺⣸⣽⣿⣻⣽⣿⠝⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⢐⢐⢘⢜⢜⢜⢌⢂⠢⡑⢌⢢⢪⢪⠢⡡⡑⡐⡐⡐⡐⡑⡘⢜⢜⢜⢜⢼⣾⢨⣪⢮⣞⣞⣼⣺⣾⣿⣾⣿⠟⠅⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⡆⡢⡢⡣⡣⡣⡣⡱⡱⡨⡨⡨⡪⡪⡂⡪⡂⡂⡂⡢⡂⡊⡪⡢⡣⣻⣺⣿⣷⡷⣟⣿⣽⣾⣿⣯⣷⣿⠟⠑⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢎⢌⢎⢎⢎⢎⢎⢎⢌⠪⠪⡪⡂⡂⠢⠨⠨⡂⡪⡪⡪⡪⡪⡮⣻⣟⣯⡷⣻⣺⣾⢿⣽⣾⢟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡣⡑⢌⢎⢞⡼⡸⡐⢕⢐⠅⢕⠔⠔⠌⢌⢌⢪⢪⢪⢪⢪⡪⣳⢽⣿⣟⡏⡇⣗⢿⡿⡿⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡕⡌⡆⢕⢕⢝⢜⢜⢔⢔⢕⢕⠕⢕⢅⢕⢕⢕⢕⢵⣱⡣⡯⣺⣽⢿⡳⡱⣕⢮⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡕⡕⡕⢕⢕⢕⢕⢕⢕⢕⣕⢕⢕⢕⢕⢕⢕⢵⢝⡾⡺⡸⡵⡿⡟⣜⢮⣗⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢝⣜⢌⢎⢆⢕⢕⢕⢵⡱⡫⠪⡪⡪⡪⡪⡪⡪⡪⡣⡣⡫⣪⢫⢣⢇⣗⡿⣞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣵⢸⢱⢑⢕⢕⢕⠅⢅⠑⢌⠂⠌⠎⠎⢎⢎⢎⡎⡮⣪⡻⡪⣗⢽⢵⢷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢎⢇⢳⢱⢱⢱⢱⢱⢑⠥⡕⡕⡅⡅⡥⡱⡱⡣⡏⡯⡳⣳⡽⡾⣯⣟⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢵⡱⣝⡕⡇⡇⡇⡇⡝⡜⡜⡜⡜⡜⡎⣞⣜⢮⣪⣞⢷⢿⣿⣿⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣳⢵⢵⣝⢼⡸⣜⡜⡼⡸⣸⢸⡸⡸⡸⡸⡪⡯⣺⣪⣗⣯⣷⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡽⣽⣿⣮⣗⡯⡳⣫⢯⢞⣞⢗⢻⢹⣪⡳⡽⣽⢾⣺⣳⢯⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣷⣻⣷⣻⡺⣜⢜⢵⡹⣜⢜⡜⣜⢮⡻⡿⡽⣜⡾⣞⣟⣞⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡾⣯⣿⣟⣿⣮⢯⡳⣝⡮⡷⡽⣮⣳⡽⣽⣝⣗⢯⡳⡕⡇⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⣽⢾⣟⣯⢟⢽⢹⢺⢹⢹⢪⢳⢝⡽⣳⡳⡕⡧⣳⢱⢱⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣺⡽⣽⣯⢯⡳⣕⢝⢏⡎⡮⣪⢗⣽⣺⣪⡳⣝⢎⢮⢪⣪⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀