⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⣯⢿⡽⣽⢽⢽⡽⡽⡯⡯⡯⡯⡯⡯⡯⡯⡯⡯⡯⡯⣯⢯⡯⣯⢯⡯⡯⣯⣻⣽⣽⢽⣝⣯⢯⡿⣽⣻⣯⡷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢯⡿⣝⣷⣻⢽⢽⢯⣻⣽⣫⢯⢯⢯⢯⢾⢽⢽⢽⢯⢯⣻⡺⡯⣟⣞⣯⡯⣟⣞⣞⢾⣺⣟⣞⢾⢽⣽⣳⣟⣾⣻⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⡽⣯⢯⡷⣟⡾⣽⣻⢽⣳⣳⢽⢽⢽⢵⢯⢯⢯⣫⢯⣻⡽⣺⡽⣝⣯⣟⣮⢿⢽⣺⣺⢽⡺⣗⣿⣝⣗⣗⣿⣺⣗⡿⣯⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣟⣯⢿⡽⣯⣟⣗⡯⡷⣯⡻⣞⢾⢽⢽⢵⢯⢯⣳⡻⣮⣻⣪⢿⢵⢯⣗⢷⣻⢾⡽⡯⣗⡯⣗⡯⣟⣾⣳⣗⡷⣳⣻⣞⣿⢽⣟⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣿⡽⣯⢿⡽⣷⣳⢗⣟⣯⢷⣫⢯⢯⣳⢯⢯⣫⣗⣗⡯⣞⣞⣮⣻⢽⢽⣺⢽⡽⣯⡯⣟⣽⢽⣺⢽⢵⣻⣞⣷⢯⣗⡷⣻⢾⣻⣽⡾⣧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣾⣳⡿⣽⣻⣽⣗⡯⡯⡷⣯⢟⡾⡽⣝⡮⡯⣗⢷⣳⣗⢯⣗⣗⣗⣗⢿⢽⡽⡽⣽⣳⢿⣽⣺⢯⣞⡽⣝⣮⢷⣟⣯⢾⢽⢽⢯⣟⣷⣟⡿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡽⣗⣿⣽⣻⢾⣺⢯⢯⡿⣽⢽⣺⢽⢵⣫⢯⣗⢯⣗⡯⣗⣽⣞⣞⢾⣝⣗⡿⣝⣷⣻⡽⣷⢽⢯⣞⡽⣵⣫⣟⡾⣯⡯⡯⣯⢯⡿⣞⣷⢿⣇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣟⡿⣽⣞⣷⣻⡯⣯⡯⣷⣻⣽⢽⣺⢝⡷⡽⣽⡺⣽⣳⢯⣗⢷⣻⢮⣳⣗⣗⣟⣗⣟⡾⣯⢿⡽⣽⣺⣺⢵⣳⣳⣟⣯⣯⢟⡾⣽⢽⣯⢿⣽⣾⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⢿⣽⢷⢯⡷⣿⢽⣳⣟⣞⡷⣯⣗⣯⡻⣮⡻⡮⣟⡮⣷⣻⢮⢿⣽⣳⡳⣗⣿⣪⢷⡯⣿⢹⣯⡯⣷⣳⣝⣗⡯⣞⣾⡽⣯⢯⣻⣺⡽⣾⣻⢷⣻⡅⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⡿⣾⣻⣻⡯⣟⣯⢿⣺⢮⣟⣷⣳⣳⣻⣺⡺⣯⣗⣯⢷⢯⢯⣻⣞⣮⢯⣟⡮⣞⡯⣿⢽⢺⣷⣻⢵⣗⣗⣗⡯⣗⡷⡿⣿⡽⣺⣞⣽⢽⣯⢿⣻⡆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣸⡿⣽⢾⣳⡿⣯⣟⣯⣯⣗⡿⣞⡷⡽⣞⣮⣻⣞⣾⡺⣟⡽⡯⣷⣻⣞⣗⣟⡮⣺⡽⣯⢿⢸⣷⢽⣽⣺⡺⡮⣟⣗⣟⣿⣻⣽⣳⣳⢯⣟⣾⣟⣯⡧⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣸⣟⡿⣯⢿⡽⣗⣯⡷⣷⣳⣟⣯⢿⡽⣯⢾⣺⣞⣗⢿⡽⣯⢯⢷⣻⣞⡮⣷⡫⣺⣽⣻⡏⣺⣽⢪⣗⣗⡯⡿⣽⣞⣞⣷⡿⣗⡷⣽⣳⣻⢾⡽⣷⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣷⢿⡯⣿⣽⣯⡷⣟⣿⣺⣞⢽⡯⣟⣾⣻⣞⣾⢽⡯⣿⢽⣝⣯⡿⣾⢽⣽⢪⣸⢾⣳⠫⣸⡏⠄⣷⡳⡯⣟⣷⣻⢮⡷⣿⡯⣟⣞⡾⣝⣯⡿⣷⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⣯⡗⣟⣷⣻⡾⣯⣟⣷⣻⣞⡧⢻⣯⡷⡷⣗⣿⢽⡯⣿⢽⣳⣻⢸⣏⡿⡞⢍⢸⡯⡺⠨⡼⢁⣅⡷⡯⡿⣽⢾⣽⡳⣿⢿⣻⣳⢽⡽⡯⣷⣟⡿⡖⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⡇⠸⣿⣳⡿⣟⣾⣽⢷⣯⢿⡬⢾⡯⣯⢿⣞⣯⣿⣻⣯⣗⡯⢸⣓⡿⣍⣢⣼⣵⡷⣷⡿⡙⡐⡽⡽⡽⣯⣟⣾⢽⣽⡿⣟⡾⡽⣯⣟⣷⣻⣽⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢯⠀⠹⡗⣿⣿⣻⡾⣯⢺⣿⣳⡀⡻⣧⡫⣟⣷⣻⣞⣷⣗⡏⢸⠰⡏⠫⠨⠿⣳⣟⣷⣿⠀⢀⡯⡯⣟⣷⣻⢾⡽⡾⣿⢿⢽⢽⣳⣗⣯⡿⣞⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⠄⠀⠫⡘⣾⣿⣻⣽⢧⠛⣟⣷⢿⣽⣗⠿⢾⣳⡏⡧⣷⠣⠈⠌⠠⠁⠠⠘⠾⠯⠷⠛⠊⠁⡯⡯⣟⣾⢽⡯⣯⢿⣿⣻⡽⣫⣿⣺⣗⣿⣻⡅⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠑⣿⣽⢿⣽⡿⣧⡐⠘⢛⣷⢿⡉⡂⠑⠷⢹⠸⡇⠐⠀⠂⠈⡀⠄⠠⠀⡀⠂⢁⠀⢯⢯⡿⣞⣿⡽⣯⣟⣯⣷⣻⡽⣾⣳⣻⣾⣻⡂⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⢯⣟⣿⣻⣵⢀⠙⡙⠛⢁⠠⠐⠈⢀⠂⠑⢀⠁⠠⠁⢀⠠⠐⠀⡀⠂⠠⠀⢽⢽⣽⣻⣞⣯⡷⣟⣿⣗⡷⣻⣟⡾⣽⢾⣻⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣟⣯⡷⣟⣯⣿⣆⠂⠀⠂⡀⠄⠀⠂⡀⠄⢁⠠⠐⠀⠂⡀⠄⠐⠀⠄⢈⠀⢂⢸⣟⣾⣳⣟⣾⡽⣟⣯⣷⣻⣽⢾⡯⣿⣻⣟⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⣗⣿⣻⣽⣯⣿⣧⠈⠠⠀⠄⢈⠠⠀⢜⠀⠄⠐⠀⡁⠀⠄⢈⠀⠂⠄⡈⢀⢸⣞⡷⣗⡿⣞⣯⣿⡿⣗⣷⣻⡯⣿⢽⣾⡃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣳⢯⣯⣿⣽⣾⢿⣷⡐⠀⢂⠠⠀⠂⠠⠐⠀⡁⠄⠐⠀⢂⠠⠈⠐⠠⠐⠀⢼⣳⣟⣯⢿⡯⣷⣿⣻⢿⣺⢷⡿⣽⣟⣷⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⢯⣟⣷⣻⡷⣿⣟⣯⣿⣧⣄⠐⢈⠀⡁⠄⠁⠤⢐⠈⡀⠂⠠⠈⠄⠁⢌⢜⢜⡷⣟⣾⣻⣽⢷⣿⣻⢿⡽⣯⣿⣳⢟⡧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡽⣞⡷⣿⣟⣯⣿⣯⣿⡾⣿⢿⣶⣤⣐⠀⠡⠐⢀⠐⡀⠌⠐⢈⢠⠕⢕⢜⢸⢽⣟⣾⣻⣾⣻⣯⣿⣻⣽⢷⣟⡾⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢽⣳⣟⣿⣽⣯⣷⡿⣾⢿⣻⣿⢷⣿⣻⣿⢶⣌⡠⠐⠀⢄⠥⡒⡅⡣⡱⠨⡪⣻⣽⣞⣷⣯⢿⣯⣷⣿⡽⣿⢽⡃⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⣷⣻⣽⣾⣻⣾⣟⣿⣻⣿⣽⣿⢯⣿⣾⡿⡿⠿⣓⠝⠡⡃⠕⢌⠢⡊⢌⢜⣽⣗⣿⣺⣯⡯⣿⣳⣿⣻⡝⢿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣳⢯⣿⣽⣯⣷⣿⣽⣯⣷⣿⡞⢽⡛⠌⠅⢂⠀⠂⢅⠁⠌⠌⣂⠅⢊⢂⢪⣾⣗⣯⣟⣷⡫⢾⣿⡅⣿⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣺⣻⣽⣾⣯⣿⡾⡇⢫⣿⣾⡇⠸⡿⠻⢿⠽⠕⠒⢄⢙⡈⠀⡀⠄⠂⡐⣸⢹⣞⣷⢯⡗⣕⢽⢷⢃⠯⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢯⣯⢿⣗⣿⠑⣿⡇⢰⡿⠋⠃⠍⠐⠈⢀⠐⠠⠐⢀⠂⠌⣃⠀⠀⢂⠐⡎⣯⢷⡟⡾⣱⢣⢟⠹⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣻⣞⣿⣯⣿⣀⣽⡇⡿⠡⠠⠁⠠⠈⠠⠀⡐⠠⠁⡂⠄⠡⢸⠀⠈⠀⢐⢁⡯⡿⡍⣇⢧⢳⠱⡩⠢⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⡷⣯⣿⣞⣯⣷⣻⣟⣧⢈⡇⠐⠀⠂⠐⢀⠂⡁⢂⠂⠌⠄⡇⠀⢌⢐⠔⠜⠱⢉⢎⢎⢎⢎⠊⠀⠁⠁⠡⠑⠠⠠⠀⠀⠀⠀⠀⢀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣺⣻⣽⣿⣳⢿⣽⣻⣾⣽⣿⢝⠵⡕⠀⢁⠈⡀⢂⠐⡐⠠⠨⠐⡸⠀⠊⠀⠀⠀⠐⢀⠜⡘⢌⠊⠀⠀⠐⠈⠀⠠⠐⠀⢀⠁⡐⡀⢶⣳⣯⣯⣯⣯
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⣿⢽⣻⢞⣯⡾⣿⣺⢿⣳⣿⡞⠀⠢⠁⡀⠂⠠⠐⢀⠂⠄⠅⠌⡐⠅⢀⠐⢀⠁⠀⠂⠂⠁⠀⠂⠀⠈⠀⠐⠀⠠⠠⠐⡐⢐⣨⡴⣾⣻⡽⣞⣾⣺⢾