⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⡽⠛⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⠝⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢯⡟⠋⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠖⠀⣴⡷⣦⡾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⡯⠛⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣦⠝⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢁⣾⢿⣽⣳⢿⣽⣳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⣾⢿⡝⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⡿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⡿⣼⣻⢿⣼⣻⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣤⡴⣟⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢹⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣾⣻⡽⣯⢿⡾⣽⢯⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣽⢿⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣞⡷⣯⢿⡽⣯⢿⣽⣻⢾⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣽⡿⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⣦⣄⠀⠀⠀⢀⡀⣀⣠⣄⣀⢀⠀⠀⠀⣿⣞⡿⣽⢯⡿⣽⣻⢾⣽⣻⠀⠀⠀⣀⣀⣀⣰⡿⣯⡟⠁⠟⠀⠀⠀⠀⠀⢀⣀⡠⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃⣴⣶⣻⢯⣿⣻⢷⣯⣟⣿⣻⢶⣆⢻⡾⣽⢯⡿⣽⢯⣟⡿⣞⣗⣤⣶⢿⣯⢿⡽⣯⢿⡽⣯⣟⡶⣦⠀⠀⠀⠀⡾⡏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢷⣻⣟⡾⣽⣻⢾⡽⣾⣽⣻⢾⣻⣽⢯⡿⣽⢯⡿⣞⡿⣽⣻⢾⣽⣻⣞⣯⢿⡽⣯⣟⡷⣯⣟⣷⡄⠀⢀⣤⡳⠄⠀⠀⠀⠀⠀
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣟⡷⣯⢿⡽⣯⢿⣽⣳⣯⣟⣯⣟⣾⣻⣽⢯⡿⣽⣻⣽⢯⣟⡿⣞⡷⣯⣟⣯⣟⡷⣯⣟⣷⣻⡞⠃⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀
⠸⣷⣤⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡽⣯⢿⣽⣻⣞⣷⣻⣞⡷⣯⡷⣯⢿⡝⠻⢿⣽⢾⣯⢿⡽⣯⣟⣷⣻⢾⣽⣻⢷⣻⣞⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠃⠙⠋⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠯⣿⣞⣷⣻⣞⡷⣯⣟⡷⣿⡽⣿⣉⠀⣈⣯⡿⣞⣯⢿⣳⣟⣾⣽⣻⢾⣽⣯⠗⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠛⢚⣽⣷⣻⡽⣷⣻⣽⣻⢿⣻⢷⣻⢯⣟⣯⢿⡞⡓⠋⠛⠉⠀⠟⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⡽⣞⡷⣿⣽⣳⣯⣟⣯⣟⣯⣟⣯⣟⡾⣯⣟⡷⣆⡀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⡿⠂⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣯⢿⣽⣻⡽⣷⢯⣷⣻⣞⡷⣯⣷⣻⢾⣽⣻⢷⣯⣟⣯⢷⡀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡽⣯⣟⡾⣷⣻⣽⣻⣞⡷⣯⡿⡷⣯⣟⡿⣾⣽⣻⢾⡽⣾⣻⢷⡀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣽⣳⣯⣟⡷⣯⡷⣟⣾⣻⡗⠀⢻⣳⣯⣟⡷⣯⣟⣯⣟⣷⣻⢯⡅⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡷⣯⢷⣯⣟⡷⣟⣯⡷⠋⠀⠀⠀⠙⣾⣽⣻⢷⣻⢾⡽⣾⣽⣻⠇⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣞⣿⣽⣻⢾⣽⣻⠝⠃⠀⠀⠀⠀⠀⠀⢸⡗⢫⣿⢯⡿⣽⢷⣯⣟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⣾⡝⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠐⠈⠙⠋⠛⡆⠀⠀⠀⠀⠀⠀⠀⢀⣦⡔⠀⠐⣦⢸⡟⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢯⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠍⠀⠀⠀⠉⠻⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠽⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⠘⠀⠀⠀⠀⠀⠀⠀⢠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⠯⠛⢹⠂⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡿⡿⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⢠⣾⠋⠀⠈⢾⣿⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠈⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀