⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣦⣦⣶⣶⣶⣶⣴⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⢿⣻⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣴⣶⣿⣿⣿⡿⣯⣿⣿⣿⣯⣷⣿⣷⣿⣾⣿⣿⣿⣽⣿⣷⣿⣿⣾⣷⣿⣿⣻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣾⣿⢿⣷⡿⣿⣾⣿⣾⣿⣻⣷⣿⢿⢯⣷⣿⣿⣽⢿⡾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠹⡨⢊⢌⣟⣿⣾⣿⣿⣻⡽⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣗⣜⢖⢕⢿⢿⣿⣿⣽⣯⡧⠘⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣯⣿⣞⢇⡇⣎⠪⡢⣱⣿⣿⣿⣿⣯⣿⡀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡟⢵⣳⡵⡧⡟⡞⡽⣿⣿⣿⣷⡿⣷⣧⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣿⣿⣿⡿⣿⣻⣿⣿⡿⡫⢃⢑⠘⣗⣝⢜⢜⢌⢎⢟⣿⣿⣾⣿⣿⣻⣦⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠨⢐⢐⢐⢈⠘⠮⡳⢽⢺⢪⣻⣿⣿⣿⣿⣿⣿⢿⣧⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡟⣿⣿⣷⢿⢟⡟⡎⠌⢌⠐⠄⡂⡐⡈⡄⣆⡢⡈⠠⠉⠙⠛⠟⣿⣿⣿⣿⣿⣿⡇⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢯⡿⡟⢌⠢⡑⠌⢌⢂⠅⢅⢂⢂⡊⠌⠄⢌⠪⢢⡈⠄⠁⠄⣺⣿⣿⣿⣿⡿⡿⠀⠀⠑⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣓⢇⢫⠪⣊⢪⢨⢊⢢⢑⠌⡂⡢⡱⣸⣸⣮⣶⣼⣤⠨⠑⢁⠀⢾⣿⣿⢟⢝⠨⠠⡁⠌⡀⢀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡵⡪⡪⡪⢢⢱⢨⢊⢢⠱⡨⡊⢔⢑⠽⣅⣾⣿⡻⡘⠅⠅⠄⢐⠈⢹⢺⣵⣣⡣⡱⢐⢐⠐⠠⠐⡀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⡪⡪⡪⢪⢢⢣⠱⡡⡣⢪⢘⢔⢡⢑⠌⢌⠢⢁⠂⠌⡐⢈⠠⠈⢜⠀⠘⢺⣺⣪⡪⣂⢊⠌⡐⠠⠁⠄⠁⠅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⢧⢣⢣⢣⢣⢪⢪⢪⢸⢘⢌⢆⢕⠔⡅⢕⠨⠂⠅⠅⡂⡂⡂⠡⢹⠀⠀⠀⠑⢗⣯⢮⡢⢕⠨⠨⡈⠌⠄⠡⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡣⡳⡱⡕⣕⢕⢕⢕⢕⢕⢕⢕⢕⢅⢇⠪⡢⡑⠅⢕⠡⡊⡔⡌⣬⢸⠀⠀⠀⠀⠀⠙⢷⢽⣕⢝⢌⢔⠨⠨⢐⢁⠂⠄⡑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣺⢳⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⣎⢇⡏⣎⢮⢪⢎⢎⢎⢎⢎⢆⢇⢇⢣⢪⠨⡊⡢⡱⡘⡌⣎⠂⠀⠀⠀⠀⠀⠀⠀⠈⠹⢾⣝⢮⢆⢇⠕⡐⡐⠨⢐⢀⠂⡑⠄⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⢗⡽⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣪⡺⡜⣎⢮⡪⡎⣇⢏⢮⢪⢎⢮⢪⢪⢢⠣⡣⣵⢾⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⡯⣗⢧⡳⡨⡂⡑⡐⡐⡐⡀⡂⠌⠢⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣻⣿⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠫⡈⡘⠘⢎⢞⢮⡪⡇⣗⢕⢇⢗⡕⣕⢕⢕⢝⡯⡿⡽⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣳⡽⣪⢎⢎⢔⢐⢐⢐⢐⠨⠠⢁⠣⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣻⢼⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⡙⢄⠑⠄⢂⠁⠄⠠⠈⢊⠚⡎⣗⢝⣕⢵⡱⡕⡵⡱⡝⣎⢯⠇⠀⡀⣀⣀⣀⣀⣀⣀⡀⡀⡠⠠⠄⠄⠂⠓⡻⣽⢵⡳⣱⢱⢐⠔⡐⢄⢑⢐⠠⢈⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣞⢿⣽⡻⣿⣿⣿⣿⣿⣿⣿⡯⢌⠢⠡⢑⠐⡈⠄⠡⠨⢀⠢⠐⡈⠜⢜⣮⢺⢜⣎⢗⢽⢜⠍⠊⢅⠩⠈⠄⡂⠄⠄⠄⠄⠂⠄⡂⡐⠠⢁⠂⠄⡙⢯⣟⡮⣺⡸⡌⡢⠡⡂⠢⠨⢐⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢿⣯⣻⢿⣿⣿⣿⣻⣟⣿⡽⣇⠕⢌⠌⡐⡐⡐⠨⠨⢐⠐⠄⠅⡂⠅⡪⡫⡣⠡⡃⢍⢓⠕⢌⠌⡢⠨⡨⢂⠢⠡⡡⢑⠨⡈⡂⡂⡂⠅⡂⢌⢐⠠⡈⡚⣯⣗⣧⡳⡹⡨⣂⠣⡡⡁⠪⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢯⣟⣿⣯⣿⣽⣿⣻⣽⣾⡯⣇⢣⠡⢊⢐⢐⠨⠨⠨⡐⠌⢌⢂⠪⡐⢔⠉⠊⠑⠉⠊⢈⠪⢆⢇⢪⢊⠢⡅⡣⡱⠨⡢⡑⢌⢔⠰⡐⢅⢂⠆⡂⡂⡂⡢⠠⠡⢂⠌⡐⠨⢰⢱⢐⠌⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠾⣽⣿⢿⣻⡳⡽⡕⢅⢊⢢⢑⠔⡡⠣⡑⢌⢌⠢⡢⠱⡨⠪⠨⠐⠀⠂⠈⠀⠄⡈⠊⢎⢎⢕⢕⢕⢌⢇⢆⢇⢕⢔⢕⢜⢔⢱⢨⢪⢨⢢⢪⠸⡘⡔⡑⡌⡪⡸⡘⠐⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣔⡮⡷⣝⢵⣣⢳⡣⡫⢘⠌⠆⢕⢘⠌⡪⠘⠔⠅⠕⡘⠌⠨⠀⠡⠀⠡⠀⡁⠐⡀⠐⠈⡀⠁⠏⢮⡳⣝⣎⢗⡽⣜⢵⢕⢧⡳⣕⠵⢕⠵⠱⠱⠹⠸⠸⠸⠸⠘⠢⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⢯⣫⡳⣝⢞⢎⡧⣃⢣⠨⡐⡠⢈⢈⢀⢂⢐⠀⠁⡀⠁⠀⠄⠐⠀⠌⠀⡈⠀⠂⡀⠂⡀⠂⠁⠠⠈⡀⠂⠈⠑⠳⣟⠞⠓⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣷⣳⣽⣕⣧⢯⣺⣪⢟⢞⠽⠝⠹⠱⡫⡫⢝⢕⢯⡪⡏⣇⢧⢳⠡⢀⠡⠀⠂⠁⡀⡈⠄⢀⠂⠠⠁⠌⠀⠄⠠⠈⠀⠈⠀⠑⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣽⣽⣳⣻⣽⣻⣕⢷⢙⠄⠡⠈⠠⠀⢀⠀⠐⠈⡘⡜⡮⣎⢮⡪⡝⣄⠄⠂⠁⠠⠀⠠⠐⠀⠄⢁⠂⠡⢈⠀⡀⠄⠂⠀⠂⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢽⡺⣝⢗⢟⢾⡪⡑⡐⡈⠐⠈⡀⠐⠀⠀⢀⠐⠠⠘⢜⢎⢧⡳⡹⣜⢦⠐⠈⠀⠂⠠⠀⠁⠐⢀⠐⠀⠂⠠⠀⠀⠀⠀⠀⠄⠈⠀⠀⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢽⣳⢿⣽⡻⡨⢂⠂⡂⠁⠂⠠⠐⠀⡈⠀⢀⠀⠅⠌⣳⡹⣪⢯⡺⣪⡳⡁⠄⠁⠀⠂⠐⢀⠀⠀⠈⠀⠂⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣻⣯⣿⢞⢌⢂⠂⠅⠠⠁⠌⡀⠂⢁⠀⠐⠀⠄⡅⢕⢘⢮⢳⣕⢯⡺⣺⠝⢄⠀⡁⠀⠈⠀⠀⠂⠀⠂⠈⠀⠀⠀⠂⠁⠀⠈⠀⠈⠀⠀⠠⠀⠣⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣯⡯⡣⢊⠄⠅⠡⠁⠅⠂⠄⠡⠐⢈⠠⠁⢜⠨⣂⢪⡫⡮⢮⡳⣝⢾⡍⢆⠅⠄⢀⠠⠀⠀⠀⠐⠀⠀⠀⠄⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣷⣿⡯⡣⠪⡐⠨⢈⠌⠨⠠⠁⠅⠨⠐⡀⡐⡨⢐⢑⢢⢱⢵⡹⣕⢽⡪⣿⠸⡐⠅⠅⠂⢀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⡗⡇⢕⠡⠨⠨⢐⠈⠌⠄⠡⠁⠅⢂⢐⢐⠔⠡⡨⢊⢮⡳⣕⡗⣗⡽⣏⢎⠪⡈⡂⠡⢀⠐⢀⠀⢀⠠⠀⠀⠀⠀⠐⠈⠀⠀⠂⠁⠀⠂⢠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⡱⡑⠅⠅⢅⢑⢐⠨⢐⠁⠅⠡⢈⠄⡆⢕⢌⠪⡐⣕⢯⢯⢞⡽⣪⡟⡎⡪⠨⡐⠠⢁⠐⡀⠂⡐⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡕⡕⡘⢌⠊⠔⡐⠄⢅⠢⠨⢈⠌⢰⢼⣺⢮⣞⡮⡷⡽⣝⡽⡽⣝⠧⡑⢕⢘⠌⢄⢑⠠⠂⠄⠡⢀⠂⡀⢂⠠⠀⠀⠀⠈⠀⠀⡁⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡫⡪⢌⢌⠢⠡⡑⡨⠨⢐⠨⠠⢁⢢⣟⣯⢯⣟⣮⡯⣯⡻⡮⣯⣫⢞⢑⠌⡜⡐⢅⢂⠢⠨⢐⠁⠅⢂⠐⡀⢂⠀⡂⠄⠂⠁⢀⠆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢬⢇⢇⠕⢅⠢⡑⡑⢔⢐⠡⠡⠨⡈⡰⣳⣿⡾⣯⢷⣳⡯⣷⣻⠽⡓⡕⡅⡣⡑⡌⢎⢢⢑⢌⢌⢐⠨⢈⠐⡐⡀⡂⡂⡂⡂⡅⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣰⢏⢇⢇⢍⢆⠣⡑⢌⢂⠢⠡⠡⡁⡢⡯⣳⢵⣫⢟⡟⣗⢟⢝⡜⣜⢎⢎⢎⢎⢜⢜⢸⠰⡑⡢⡱⡐⢕⡐⢅⢢⠢⡊⡆⡕⡜⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣰⡫⡎⡎⢎⢢⠢⡃⣊⠢⡡⠡⡡⢑⢰⡽⡽⣕⡯⡮⡳⣝⢮⡫⡳⡹⡪⡣⡏⡮⡪⡪⡪⡪⡪⡊⡎⡜⡜⡔⡕⡕⡕⡕⡕⡕⢕⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⢗⢵⢱⢱⢑⢅⢕⠜⡐⢅⠪⠨⡐⢼⠝⣯⣟⣗⡯⡯⡯⣞⢮⢮⡳⣝⢮⢳⡹⡜⡎⣇⢇⢇⢇⢇⢕⢕⢕⢕⢵⢱⢕⢕⢕⢕⠕⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⡠⣗⡝⡕⡕⡕⡅⢇⢆⢣⢑⢅⢊⠢⣪⠃⠀⠈⢿⣞⣯⢿⡽⡽⣝⢷⣝⢮⣳⡳⡝⣎⢧⡣⣳⢱⢕⢕⢵⢱⡣⡳⡱⡱⡑⢥⠱⡘⡜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⡻⣜⢎⢇⢇⢇⢎⢎⢢⠱⡐⡢⢑⢜⠂⠀⠀⠀⠀⠻⣞⣯⢿⡽⣽⣳⣝⣗⣗⢽⡪⣗⢧⣫⢺⢜⢎⢇⢗⢕⢕⢕⢕⢕⢱⠡⢣⠱⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⢯⣫⡺⡜⡜⡜⡜⡔⡅⡣⡑⡌⡢⡕⠁⠀⠀⠀⠀⠀⠀⠉⣿⣽⣽⣳⣗⣗⣗⣗⢯⣫⢮⡳⡵⡝⣎⢇⢏⢎⢎⢪⠪⡢⡃⡪⠸⡐⠕⢔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣝⣗⢵⢕⡕⡝⡜⡜⡌⢎⢢⠱⡨⡪⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣺⣗⣷⣻⣺⣺⢽⡺⡵⡫⡞⣎⢮⢪⢪⠪⡊⡆⢇⠪⡢⡑⢕⠡⡃⢇⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢨⣷⡳⡝⡼⡸⡪⡪⡪⡪⡊⢆⢣⠪⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣻⣞⣗⡯⣗⢯⡫⣝⢮⢪⢪⢪⠪⡊⢎⠜⢌⢎⠢⡊⢜⢨⢨⠢⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⡷⡫⡚⠎⠇⠇⠣⠃⡊⠘⡘⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣻⢮⣻⣪⢗⡽⡸⡜⣜⢜⢜⠜⡌⡢⡑⢅⠢⡑⢌⠪⡰⠰⡱⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠨⡪⠢⡈⡂⠅⠨⠀⠂⡀⠄⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣯⢿⡵⡯⣳⢝⢮⡣⣣⢣⢣⢱⢨⢂⠪⡐⢅⠪⡐⢅⠪⡊⡆⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢇⢗⠰⢐⠈⠄⡁⠂⡀⠀⠄⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣽⡯⣯⢯⣗⢯⣳⡹⡜⣜⢜⢜⠔⡅⡣⢊⢆⠕⢜⢰⢑⢕⠜⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠸⡸⡨⡂⢌⢐⢀⠁⠄⠂⠀⠂⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣽⢯⣟⡮⣗⢧⡳⣝⢜⡜⡜⡜⡌⢆⢕⢜⢸⢨⢒⢕⠁⡂⢬⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢹⡸⡐⡐⠄⡂⢐⠐⠈⢀⠐⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣻⢾⡽⡽⣕⢯⢎⢧⡣⡣⡣⡪⡢⢣⢱⢱⢡⢃⠁⡴⣎⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠐⣕⢕⢌⢐⢐⢀⠂⡁⢀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣯⣟⣯⡯⡯⣯⣳⡫⡧⡳⡹⡸⡸⡨⡪⡪⡪⠂⢤⣞⠁⢽⣺⡪⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠪⡎⡆⡂⡂⠔⡀⠂⠄⠐⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣺⡿⣞⡯⡡⡀⡨⠈⠋⡓⠵⢝⢮⡳⡹⡜⡜⡜⡜⠌⠀⡶⢻⠻⣕⢸⡷⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⡧⡳⡐⡐⠡⠠⢁⠂⠁⡀⢘⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣻⣽⢯⣿⣳⢗⡮⣎⣆⣆⢄⠄⢄⠈⠑⠉⠊⠊⠠⢀⢕⡃⢜⣲⢯⢂⣿⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠹⣜⢌⠢⠡⢁⠂⠄⡁⠀⢀⢃⠀⠀⠀⠀⠀⢀⣴⣿⣟⣾⣳⡿⣽⣞⡾⡯⣟⣗⣟⡾⡽⣝⡮⡮⡮⣔⢔⢄⣆⣞⢮⢂⢵⢟⣿⡰⣹⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢗⡇⡇⡑⠄⠅⡂⠄⠈⢀⠰⠀⠀⠀⠀⣴⣿⣿⢷⣻⡾⣷⣻⡷⣯⡯⣟⣽⣳⢯⡯⣟⡾⡽⡽⡯⣗⢿⢽⣺⣺⢽⣎⢔⢝⣞⡮⡪⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢘⢞⡔⢌⠌⡐⠄⠌⠠⠀⢐⠄⠀⣠⣾⡿⣿⣯⡿⣯⡿⣽⣾⣻⣷⣻⣽⢾⣽⣳⣻⣳⢯⣟⡽⡽⡽⣝⣗⢷⣝⣯⡧⡇⢎⡲⣱⡫⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢹⢎⢆⠅⡂⠅⠌⠠⠐⠀⢶⢴⣳⡻⣻⢿⣾⢿⣯⢿⣻⣺⣽⣾⣳⡯⣿⣺⣳⣽⣺⣳⣳⣟⡽⡯⣗⡯⣟⣞⣷⣻⡪⡣⡞⡎⣞⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢯⡢⡑⠄⠅⠅⢂⠐⠀⡙⣯⣞⣝⣗⣟⣮⣻⢽⢿⣽⣻⢾⣳⣯⣟⡷⣯⡷⣗⣷⣻⣺⣺⡽⣽⡽⣽⡳⣯⣞⣗⢷⠽⣕⣝⢮⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠱⣣⠣⡑⡡⠑⡀⢂⠠⠈⣿⢾⣺⡲⣕⢗⢽⢽⢾⡮⣏⡿⡽⣞⡷⣯⢷⡿⣽⢾⣺⣳⢯⡯⣷⢯⢗⣯⢗⣗⢗⢽⡹⣜⢮⣳⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⢗⡕⢅⠢⡁⡂⠄⠠⠀⢻⣿⣳⣻⢮⡳⡳⡕⡧⣫⢳⢫⢯⢮⢏⡗⣟⢽⢯⡯⣷⣻⢽⣽⣺⣞⣿⡺⡏⡎⡮⣳⢝⡾⣽⡞⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣺⢨⢂⠆⡂⠌⢀⠈⠸⣿⣿⢿⣻⣽⣳⣝⡮⣺⣪⡳⡳⡵⡱⡹⡼⡽⣞⣾⣳⢯⣿⣺⣗⣟⢮⢳⡱⡣⣏⢾⢽⣽⣗⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢜⠰⡨⢐⢈⠠⠀⠡⢿⣿⣿⣿⢿⣻⣾⡯⣗⣗⡽⣽⣺⢵⡯⣾⢯⣿⣺⣽⣟⣾⣳⢟⡎⣇⡇⣇⢟⢾⣽⣟⡷⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡵⡱⠨⡂⡐⠠⠈⢰⣿⡿⣿⣾⣿⣿⣯⣿⣿⣽⣟⣷⣟⣯⡿⣯⡿⣾⡽⣞⣷⢯⢯⢳⡹⡪⣞⢵⡳⣽⣾⢏⠫⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡸⡑⢔⠨⠐⡀⣾⣻⣿⣟⣯⣿⣿⣻⣿⢷⠣⡁⠅⠉⠓⠛⠯⢟⢯⢿⡻⡝⡱⡸⣸⡪⣞⣜⣜⢽⣽⡺⡱⣝⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⡣⡑⢌⠐⣼⣿⣿⣟⣿⣟⣿⣾⣿⣻⣿⣷⣦⣵⣸⢈⠨⠘⡔⠐⢀⠁⢠⡣⡯⡺⣪⣳⡳⣗⣟⢮⡚⢙⠸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢮⢨⠂⣮⣿⣿⣾⣿⣻⣟⣯⣷⣿⣿⣽⣾⣿⣽⣟⣷⣶⣽⣦⡨⠀⢠⡳⣝⢮⣫⣞⠞⣜⢵⢳⡝⠆⡀⢨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣜⣾⣿⣿⣿⣿⡟⡟⢟⠻⡻⡻⣾⣿⣽⣾⣯⣿⣽⢿⣷⣟⣿⡟⣗⡽⡮⢟⢪⢢⢣⡣⡳⣝⠪⠐⠀⢜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⡱⡱⡡⡣⡂⢆⢂⢊⠝⢯⣿⣾⢿⣻⣿⣽⣾⡽⡳⡫⣕⢵⡳⡹⣕⣝⠮⡃⠄⠂⠁⡐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣳⢕⡕⡕⡕⡅⡕⢌⣺⣽⣿⣿⣿⣯⢷⢫⢮⡺⣝⢮⡳⡽⡝⢎⠊⠐⠀⠀⡀⠂⢀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⠺⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⣯⣺⣸⣸⣪⣾⣿⣿⣿⣿⣾⡳⡽⣝⡵⣻⢮⢗⢏⠃⠅⠠⠐⠀⠐⠀⠀⠠⠀⢄⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⢞⣵⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢞⣮⡻⡮⡯⡏⠇⡑⠀⡂⠐⡀⠂⠈⡀⠄⠈⠀⢀⡳⣗⠃⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⣾⡿⣿⣟⣿⣺⣗⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣿⡿⣿⣿⣿⣿⣿⡿⣾⡽⡾⡝⡝⠨⠐⡀⠂⡁⢀⠂⠠⠀⡁⠀⡀⠐⡠⡴⣝⡇⠡⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠸⣿⣯⡿⡯⣟⣮⡳⡽⣻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣯⣿⣿⣻⣟⣯⣿⡕⡁⡂⠅⠡⠠⢁⠐⠠⠐⢀⠁⡀⢄⡴⣳⣫⣟⠞⢁⠠⠈⢀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⢻⣿⣿⣷⣗⣯⣯⣞⣗⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⣷⣿⡷⣿⣻⣽⣟⣷⡿⣆⡢⠡⠡⢁⠂⠌⢐⠈⡠⣔⣮⢯⣯⠗⢋⠀⠐⠀⠀⠀⠄⠠⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣻⣽⣿⣿⣮⣌⣬⣐⣌⣌⣔⣮⣟⣷⣿⣙⣄⣌⣀⣄⣂⣀⣂⡠⣐⣀⢄⡸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀