⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣺⢽⡽⡽⡽⡽⡽⡽⡽⡽⡽⡽⡽⡽⡽⡽⡽⣽⢽⢽⣫⡯⣟⡶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡾⣽⣺⣺⢽⣺⢽⣫⢯⢯⢯⢯⢯⡯⡯⡯⣟⣽⢽⣫⣗⡿⣝⣗⡯⣷⣻⢽⡻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡯⣯⣻⣺⣺⣺⢽⣺⢽⣞⡯⡿⣝⣯⣻⣺⢽⢯⣻⣺⢽⣳⣳⢯⣗⡯⡯⡷⣽⢽⣝⣗⣟⣦⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡿⡽⣽⣺⢞⣞⣞⡾⣽⣺⢽⢮⡯⣯⣗⡷⣳⢯⢯⣟⢾⢽⢽⣺⢵⣻⣺⢽⡽⣫⡷⣻⣺⣺⣺⢵⢯⣆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⢽⡽⣳⢽⢽⣺⡳⡯⡷⡽⣝⣗⡯⣗⣗⡯⣯⢯⣻⣺⢽⡽⣽⣺⢯⣗⡯⣟⣞⣗⡿⣽⣺⣺⢽⢽⢽⣺⢧⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡳⣯⣻⣺⢽⢽⣫⢾⢽⣫⢯⡯⣗⡯⣯⣻⣺⢽⡳⡯⣗⡯⣗⣯⣗⡯⣗⣷⣫⢷⡻⣮⣻⣺⣺⣺⡽⣽⣫⢾⢽⣣⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣳⢯⣗⣗⡯⣯⣻⣺⢽⣻⣺⢽⣝⣗⡯⣗⣗⡯⡯⡯⣟⣵⣻⣳⣳⣳⢯⣗⣗⡯⡯⣟⣞⡾⣵⡻⣮⣻⣺⢞⣯⢯⣗⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣗⡯⣗⣗⢷⣝⣗⢷⢽⢽⣺⣺⢽⣺⣺⣽⣳⢽⣞⡯⣟⣗⣗⡯⡾⡵⡯⣗⣷⣫⢯⡯⣗⣟⣞⣗⡯⣗⡯⣞⣟⡾⣽⣺⢽⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡵⡯⣗⡯⣗⡷⡽⡽⡽⣽⣺⣺⢽⣺⢵⡷⣻⢽⣻⣞⣗⣗⡷⣯⣻⢽⡽⣳⣳⢯⣻⣺⢽⣺⣞⣵⣻⢽⢽⣳⢯⣞⡷⣽⣻⡂⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢾⢽⣫⢷⣫⣗⡯⡯⡯⣟⢾⣺⣺⢽⣞⣏⠊⠊⠔⠸⣗⣷⣳⣻⣺⢾⢽⢽⢽⣺⢽⣺⣞⣟⣞⡾⣺⣞⡯⣟⡾⣽⣺⢽⣺⢮⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⢯⣟⢾⢽⣺⣺⢽⢽⢽⢽⢯⣗⡯⣟⢮⡧⠀⠀⠀⠨⣻⢮⣗⡷⣫⢿⡽⣽⣻⣺⢽⣳⣳⣳⣳⢯⣗⡷⣫⢷⢯⢷⢽⢽⣺⡽⡯⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⣟⡮⡯⣟⣞⢾⢽⢽⢽⢽⣹⣞⣞⠇⢸⡊⠀⠀⠀⠀⣺⡇⢗⣯⢯⡯⢯⣗⣗⡯⣟⣞⣞⡾⣽⢽⣺⢽⢽⣫⡯⣟⡽⡯⣗⡯⣟⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣞⢾⢽⣝⢷⣝⡯⡯⡯⣟⡗⢸⣞⡞⠀⠸⡂⠀⠀⠀⠀⢸⠃⠘⡽⣯⠯⢸⣳⢯⢯⣗⡯⡾⡽⣯⣻⣺⢯⣟⡮⣯⣗⡯⣟⣵⣻⢽⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣺⢽⣳⢽⣳⡳⣯⣻⣽⣻⢌⠐⣷⠁⠀⠀⠂⠀⠀⠀⠀⠘⠀⠀⢹⢝⠇⠸⠝⠝⠽⣞⡽⡿⣝⣞⣾⢽⣗⣗⣯⣗⡷⡯⣷⣻⣺⣓⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣞⣟⡾⣽⣺⢽⣺⡎⡞⡍⠙⠚⠾⡤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⠖⠖⠚⠉⠋⠑⢸⡞⢝⣾⣳⣳⢗⣯⢯⣷⣳⣟⡆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢻⣺⢵⣳⣻⣺⢇⢕⣨⣴⠴⣦⡴⣬⣑⠍⠩⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠨⣷⡳⣯⣻⣺⢽⣗⣷⢯⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⢹⣞⣟⡮⣗⣿⡸⡐⢉⠄⠨⣗⡿⣽⢽⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡽⣻⣻⣻⠛⠲⣦⣄⡀⡿⣝⡷⣽⣺⣻⣗⡏⢸⡃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⣮⣷⣻⣳⡟⣆⠂⠐⠨⣄⡉⠫⠋⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⣝⣗⡗⠁⠀⡈⠍⠢⣯⡳⣟⡞⢊⣵⡟⠀⠘⠄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠨⡷⣯⡇⣧⠁⠀⠀⠀⠈⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⠡⠤⠤⠎⠐⠀⢕⣯⠗⣈⢴⣻⢾⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⡿⣷⢼⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⡧⣞⡷⡦⡙⡏⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠸⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⢯⣟⣗⡯⣟⣽⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣗⣵⣟⡮⣯⢗⡯⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣟⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⣿⣽⢿⢮⣻⣺⢽⠇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠸⣿⣻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣷⣿⡻⣺⣿⣫⣗⣯⣻⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠝⠳⢿⢳⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⡟⠀⠁⡾⣳⣳⣗⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡣⢍⠕⢤⢀⠀⠀⠀⠀⠀⢀⡠⡲⡹⡸⠁⠀⠀⠁⠀⠀⠀⣷⣳⢽⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠑⢌⠢⢑⢑⢲⠰⡔⡎⡇⡇⡇⢵⠁⠀⠀⠀⠀⠀⠀⢐⣟⢾⢽⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠄⠐⢔⡐⡄⡄⣀⢀⡀⠎⠀⠀⠐⡈⠢⠡⡱⢱⠱⡱⡑⡕⡜⡅⠀⠀⠀⠀⠀⠀⠀⢸⣞⡯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⢀⠠⠠⠠⠐⠈⠀⠂⡀⠀⠀⠈⠪⡊⠌⠈⠀⠀⠀⠀⠀⠀⠀⠡⠑⢌⢢⢣⢣⢣⢣⢣⢃⠀⠀⠀⠀⠀⠀⠀⣞⣗⣯⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠐⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣨⢵⢤⣤⢤⡤⣤⢤⡤⣤⢬⣬⣢⢵⣜⡬⡮⡼⡬⣮⣲⢵⣁⠀⠀⠀⠀⠛⠚⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀