⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⢤⡦⣖⡶⡶⡶⡶⡶⣶⣲⢦⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣲⣺⢽⣳⣻⣺⡽⣽⣳⢯⣟⣽⢽⢯⣗⡯⣟⣞⡮⣟⣞⡯⣗⡦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣞⢷⣻⣺⣳⢯⣗⣟⣮⢯⣗⡯⣗⣟⡾⣽⢽⣺⢽⣳⢯⢯⢷⢯⢯⢷⣻⣳⣻⢶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡯⡯⣟⡾⣽⣳⢽⢾⢽⣺⡵⡯⣟⡾⡽⣽⣺⢽⣞⡯⡯⣟⡾⣽⣫⢿⣝⡯⣟⣞⡷⣽⢽⢯⢾⢦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣺⣺⢽⡽⡯⣗⣯⣗⡯⣟⡽⡯⣗⣯⢟⣗⡯⣟⣗⡯⣟⣮⢯⣟⣗⡯⠗⠋⡉⢈⠈⠀⠀⠀⠀⠀⠉⠉⠙⠻⢢⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣞⡾⣤⡀⠀⠀⠀⠀⠀⢀⣖⣷⣫⢾⣫⡯⣯⢟⣞⡾⡽⣽⢽⡽⣳⢯⡯⣗⡿⣽⣺⢽⣳⢯⣻⣺⢞⣮⢿⢽⣫⡯⡿⣽⡳⣗⣶⣢⡤⣀⠀⠀⠀⠈⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡷⡯⣗⣟⣦⡀⠀⠀⣠⣟⢾⣺⢞⣯⣗⣯⢯⡯⣗⡿⣽⢽⣳⣻⢽⡽⡾⡽⡽⣞⡾⣻⣺⡽⣳⢯⣟⡾⣽⣫⣗⣯⢿⢵⣻⢽⣺⣞⡽⡯⣟⡶⣄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢿⢽⡽⣞⡾⣽⣳⢦⣗⡯⣟⣞⣟⣞⣾⣺⢽⡽⣳⢯⢷⢯⣗⣯⢯⡯⣯⢿⢽⣳⣻⢽⣺⡽⣽⣳⢗⣯⣗⡷⡯⡾⣽⣫⡯⣯⣗⡷⣻⢽⣳⢯⣗⣯⢦⡀⠀⠀
⠀⠀⠀⠀⠀⢹⢽⢾⢽⡽⣞⡾⣽⣺⢽⣳⣻⣺⡵⣗⡯⣯⢯⡯⡿⣝⣯⢾⣺⡽⡽⡾⣽⣻⣺⢽⡽⣳⢯⣗⡯⣟⡾⣺⡽⣽⣫⢷⣳⢯⣗⡷⣻⢽⣻⣺⡽⣞⡾⣽⣺⠄⠀
⠀⠀⠀⠀⠀⠈⡯⣟⡽⣞⡷⡯⣗⣯⢯⣗⣟⣮⢯⢷⣻⢽⣳⢯⣟⣽⣺⢽⣳⢯⢿⣝⣗⣗⡯⣯⢯⡯⣟⡮⣟⣗⣟⣗⣯⣗⡯⣟⡾⣽⣺⡽⣽⣻⣺⢵⡯⣷⣻⡳⡯⣷⠀
⠀⠀⠀⠀⠀⠀⠸⡯⣯⢷⢯⣟⡽⡾⣽⣺⣳⡽⣽⣻⣺⡽⡾⣽⣺⣞⣽⢽⣞⡯⣟⣮⣗⡯⣯⢷⢯⡯⣷⣻⢽⣺⣵⣻⣺⣺⢽⣳⢯⢷⣳⢯⢷⣳⢯⣟⣞⣗⡷⣻⡽⣳⡅
⠀⠀⠀⠀⠀⠀⢀⣟⣗⡿⣽⣺⢽⣫⣗⣯⢾⢽⣳⣳⣗⡯⡿⡵⣗⣷⣫⣟⡮⣯⢷⣳⣗⢿⢽⢽⣳⢯⣗⡯⣟⡾⣺⣞⡵⣯⢟⣞⣯⢯⡯⣟⡽⡾⣽⣺⢞⣗⣯⢷⣻⢽⡆
⠀⠀⠀⠀⠀⠀⣞⣗⡷⡯⠗⠯⢯⣗⣯⣞⡯⣟⡾⡵⣗⡿⡽⡯⣷⣳⢗⣷⣻⢽⣽⣺⢞⡯⡿⣽⣺⡽⣞⣽⣳⣻⣳⢗⡿⡽⣝⡓⠫⢟⡾⣽⢽⣫⢷⣫⡯⣗⡯⣟⣞⡯⡇
⠀⠀⠀⠀⠀⠈⣟⡾⡽⠁⠀⠀⠀⠈⠚⢾⣝⡷⣻⢽⣳⢯⢿⣝⡷⣽⢽⣞⡾⣽⣺⢞⡯⣟⣽⣳⢗⣯⣟⣞⣞⡷⣽⣫⡯⣟⡽⡾⣄⠀⠈⠫⣟⣞⡯⣗⡿⡽⣽⣳⢯⣻⠀
⠀⠀⠀⠀⠀⠀⢯⢯⣟⠀⠀⠀⠀⠀⠀⠀⠳⣻⢽⣻⣺⢯⣟⡮⣟⡾⣽⣺⢽⣳⢽⢯⢯⣟⣞⣞⣟⣞⣾⣺⣳⢯⣗⡷⡯⣯⢯⢿⢽⣳⡀⠀⠈⢺⡽⣽⢽⡽⣳⡽⣽⢞⠀
⠀⠀⠀⠀⠀⠀⠈⡿⣺⣆⠀⠀⠀⠀⡻⣖⣦⣈⢻⣞⡾⣽⣺⢽⣳⢯⣗⡯⣟⣞⡯⡿⣽⣺⢞⣗⣯⢾⣺⣞⡾⣽⣺⢽⡽⣽⢽⣫⣟⡾⣵⠀⠀⠀⢫⢯⡷⣻⣳⢯⡯⠇⠀
⠀⠀⠀⠀⠀⠀⢀⣾⣳⢯⣦⡀⠀⠀⠈⠙⢾⣺⣽⣺⢽⣳⢽⣻⣺⡽⡾⣽⢽⣺⡽⡯⣗⡯⣯⣗⡯⣟⣞⡾⣝⡷⣽⢽⢽⣳⢯⣗⡷⡯⡷⡇⠀⠀⠈⣯⢯⣟⢾⢽⡝⠀⠀
⠀⠀⠀⠀⠀⣠⣞⡾⣺⣽⣺⢽⣢⡀⠀⠀⠀⠀⠑⠙⣟⣞⣟⡾⡵⡯⣟⡾⣽⡳⡯⣯⢷⣻⣳⣳⣻⢽⣺⢽⣳⢯⡯⣟⡽⡾⣽⣺⢽⣽⣫⣯⠀⠀⠀⡯⣟⡾⡽⡏⠀⠀⠀
⠀⠀⣀⣴⣺⣽⣺⢽⣳⣳⢯⢯⣗⣿⣲⣄⡀⠀⠀⡠⣟⡾⣵⣻⢽⢯⢷⣻⣳⡻⠽⢽⣽⣺⣳⡽⣞⡯⣯⢟⡾⡽⡾⣽⢽⣫⢷⣫⣟⣞⣾⡺⠀⠀⢠⣟⣗⣯⠟⠀⠀⠀⠀
⠰⣞⣗⡷⣳⣗⡯⣟⡾⡽⣽⢽⣞⡾⣺⣞⢷⠀⣴⣻⣳⢯⣗⡯⣟⣽⣻⣺⡵⠁⠀⠀⠳⣳⣗⣯⢷⣻⢽⡽⡽⡯⣯⢷⢯⡯⣯⣗⡷⡯⣞⡇⠀⠀⣼⣺⣞⠎⠀⠀⠀⠀⠀
⠀⠀⠀⠉⠉⠈⠁⠉⠫⠫⠿⠤⡁⠙⠗⡯⣯⢯⢷⣻⣺⢽⣺⡽⣽⣺⠞⠅⠀⠀⠀⠀⠀⢱⣗⡯⣟⡾⣽⢽⣽⣫⢷⣻⢽⣞⣗⡷⡯⣟⡗⠀⠀⣰⣗⠗⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣠⣲⣞⡯⣟⡾⣽⢽⠳⠏⠓⠁⠀⠀⠀⠀⠀⠀⠀⠨⣗⣟⣗⣯⢷⣻⣺⢞⡯⣯⣻⣺⣞⣽⢽⡳⠁⠀⣰⠗⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣗⣟⡞⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣻⣺⣳⡽⡽⣞⣽⢽⡽⣳⢯⢾⣺⢾⠝⠀⠀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣗⡷⡯⡾⣝⡯⣟⡾⡽⣽⢽⡽⣻⡺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣟⢾⢽⢽⣫⡯⣯⢷⣻⢽⡽⡽⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⢾⣺⢾⢽⢯⡯⣗⣯⢯⣟⡾⡽⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣖⣟⣞⣟⡾⣽⣫⣟⡾⡽⠾⠝⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡴⣞⡯⣯⣗⣟⡾⡵⡯⠗⠗⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠲⠮⠯⠏⠟⠚⠙⠙⠑⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀