⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⢒⢏⢏⢧⢳⢕⢷⣝⢿⣦⡄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡺⡹⡸⡸⡪⡳⣹⢪⢏⡯⣳⡽⣽⢾⣿⣆⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⢻⢱⢱⢱⢕⢝⡜⣎⢯⡪⡯⣳⢯⣗⣿⡽⣿⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡻⡸⡸⡸⡸⡜⡎⣎⢧⡫⡮⣳⢽⢽⣺⡽⣾⢷⡿⣿⣾⣿⣇
⢀⠴⡊⠍⡉⡉⠑⠓⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⡝⣜⢜⡜⡎⡇⡗⣝⡜⡮⡮⡯⣺⢽⡽⣞⣿⣽⡿⣿⣿⢿⣿⣿
⡎⡇⡢⡁⡂⠄⠡⠁⠂⡀⢀⠁⠓⠲⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢣⡳⡹⡸⡜⣜⢎⡗⣝⢮⡺⣝⡮⣯⢯⣯⢿⣽⡷⣿⡿⣿⣿⣿⣿⣿
⢸⢕⢢⢂⠪⠨⢐⠁⡂⠄⠠⠐⠀⡀⠀⠈⠑⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⢼⡱⡵⣹⡪⣝⢎⣗⣝⢮⣳⢽⣺⡽⣞⡿⣾⣻⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠈⢞⢜⢌⢎⠜⡐⡐⡐⡈⠄⠂⡁⢀⠈⡀⠐⠀⠀⠉⠣⢢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣗⢗⣗⢽⣪⢗⡽⣪⣗⣗⣗⡯⣾⢽⣳⣟⣯⣿⣻⣽⣷⣿⣿⣿⣿⣿⣿⣿
⠀⠈⡳⡕⡕⡕⡱⡨⡐⡐⠨⢀⠂⠄⠂⡀⠂⠐⢀⠠⠀⠀⠈⠲⡄⡀⠀⠀⠀⠀⠀⣴⣻⣺⡽⡾⣽⣺⢽⡽⣞⣾⣺⢾⡽⣯⣿⣽⣯⣷⣿⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⢎⢗⢵⡱⡸⡐⡅⢕⢐⠨⠠⠁⠄⠨⠀⢂⠀⠐⠈⢀⠀⠈⠊⢆⡴⠒⢦⣰⣿⣽⢷⣟⣿⣳⡿⣽⡯⣿⣞⣯⣿⣻⣽⣾⣷⣿⢿⣾⣿⣿⣿⣿⣿⣿⣾⣿⣿
⠀⠀⠀⠈⠓⡵⡝⡮⡪⡪⡢⡱⢨⠨⠨⠨⠠⠡⠐⡈⠠⠁⠠⠀⠌⢀⢸⣪⢪⢢⢪⢻⣾⣿⢯⡻⣳⢫⡫⡪⡲⡸⡰⡢⣣⢲⡱⡭⣝⢟⣟⢿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⢎⢗⣝⢮⢪⡪⡪⡸⡨⡊⠌⢔⠡⢐⠈⠄⡁⡂⠌⠠⢘⡮⣯⡺⡜⣆⢻⢮⢷⢽⣪⢗⣝⢮⡳⡝⣎⢗⣕⢧⡳⣝⡮⡷⡽⡽⣯⣟⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢣⢳⢕⢧⢳⡱⡱⡢⡣⡑⢅⠪⡐⠨⡐⠐⢀⠈⠠⠀⢻⣺⢮⢯⡪⡮⡹⣯⣟⡾⡽⡮⡷⣝⡮⣗⡽⣪⣗⢯⣗⡯⣯⣟⣯⣷⢿⣯⣿⣿⣿⣿⡟⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡹⡸⣕⢵⡹⣸⢸⢸⢨⢪⠨⡊⠠⠈⡀⠠⠐⠈⡘⣽⣽⡳⣝⡮⡳⡵⣻⣽⢿⡽⣯⣗⡯⣗⣯⣗⣯⣟⣾⣽⢷⣿⣽⣾⣿⣿⣿⣿⣿⣿⠃⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣣⢣⢣⢳⡱⡕⣇⢇⢇⢇⠢⢈⠄⢂⢐⠨⢀⠂⢵⢷⣻⣳⢽⢽⣪⢯⣿⢿⣽⢷⣯⢿⣽⡾⣾⢷⡿⣾⣯⣿⣷⣿⣿⣿⣿⣿⣿⣿⢩⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢞⢜⢜⢜⢼⢸⢪⡣⡑⡌⣂⠪⡐⡐⠌⢔⢨⠨⢯⢿⣯⢿⢽⣺⡳⣯⣿⣿⣻⣟⣿⣷⡿⣿⣟⣿⣿⣽⣾⣿⣿⣿⣿⣾⣿⣿⢃⠸⡄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢕⡅⢇⢇⠧⡳⡡⡣⢢⢃⢎⢜⢸⠨⡢⡣⡫⣻⣿⣻⣯⢷⣻⣗⡿⣽⣿⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡃⡂⡢⡃⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡣⠣⡣⡱⢸⢨⢢⠣⡪⡢⡣⡣⡣⡣⡳⡽⣿⣾⣻⣗⣯⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡁⡢⢂⢪⡂⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⢆⢕⠡⡑⢅⠣⡣⡱⡑⡕⢕⢕⢝⢜⢽⢿⣽⣾⣟⣾⡝⡝⡟⡿⢿⢿⣿⣿⣿⣿⣿⢿⠿⡛⢍⠕⡐⠔⡐⡡⡪⠂⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡕⢌⠢⡑⢌⠢⡱⢘⠌⡆⡣⢱⢱⢫⢿⣷⡿⣯⣗⢑⠅⠕⡡⠑⠔⡐⠢⢂⠢⠡⡑⠌⠢⠨⡐⢅⠪⢰⡙⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠱⡘⠤⡑⡌⡢⡱⡐⠎⢎⢊⢎⠺⢽⣿⣿⢞⠄⠅⠅⠄⠅⡁⡂⠅⡂⠅⠅⡂⢅⠕⡑⢌⠢⣑⢵⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣢⠨⠨⡂⠪⡐⡐⢌⢊⠢⡑⡌⢎⢕⢝⢿⡇⢕⢡⠡⡑⠨⡀⡂⡂⡢⠡⡑⢌⠢⡑⢌⢢⢑⣜⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣅⠅⠪⡐⠔⢌⠢⢢⠱⡨⢪⢊⢎⢪⠪⡎⡆⡢⢑⠌⡌⡢⢊⠔⡌⡪⡘⡌⡪⡘⡌⡢⡺⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⡂⠪⠨⡢⡑⡅⢇⢕⢕⠕⡕⢅⢇⠇⠈⠲⢅⡊⠔⢌⠢⡃⡪⠢⠱⡨⠢⡑⣐⠜⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⡮⣈⠢⢱⠸⡸⡸⡸⡸⡘⡌⡎⠇⠀⠀⠈⠘⠪⣄⢅⠢⠨⠨⠨⠠⣁⢖⠊⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⢪⡢⡣⡱⢱⠸⡨⡪⡸⡜⠀⠀⠀⠀⠀⠀⠀⠁⠓⠑⠎⠊⠊⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠪⡎⣎⢎⢼⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀