⠻⡿⣷⣴⣠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠘⠹⢻⢿⣷⣾⣔⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠊⠽⡻⢿⣷⣥⣆⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠐⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠫⢻⢿⣷⣮⣆⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠩⠀⡈⠀⢈⢤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣻⣿⣿⣷⣾⣷⣦⣤⣤⣶⣶⣢⣀⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠅⡀⡤⠌⢀⡘⠈⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣫⠫⠤⢻⡋⠋⡀⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⣰⣶⢶⠶⠶⣒⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣯⣿⣿⢾⡽⣿⣿⣿⣿⣯⢄⠀⠀⠀⠀⠀⢀⣼⣻⠞⠁⠈⣀⡬⢢⡣⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡶⠞⡫⠚⠫⠝⠐⡂⠈⠀⠀⠀⠃⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣟⣿⣾⣿⣿⣯⠆⠀⠀⠀⢠⣳⡽⢁⣶⣤⢅⠇⡢⠊⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⢫⠷⠊⠥⠂⠀⡒⠌⠠⠀⠀⠠⠀⠀⣀⣠⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡽⣾⣿⣿⣿⠀⠀⢠⣳⣿⡵⣿⣏⠞⢌⡎⠁⠀⠀⣀⣠⣄⣶⢜⣿⠕⠎⠝⠚⠈⠀⠀⢈⢀⠀⠖⠍⠁⠀⢠⣃⠤⠚⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢥⠠⣳⣿⣏⣷⠋⡁⣠⠎⢀⣤⣢⣿⡿⣟⠯⠟⠙⢐⠖⠂⠒⠚⠒⠀⠴⠥⢍⣢⠁⠦⡒⡠⠏⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣭⢾⣽⣟⡿⣿⣿⣿⣿⣿⣿⣳⣿⣿⣧⣾⢹⡟⣫⡠⣦⣿⣣⣿⣿⣟⡩⣡⡒⢂⠀⢐⠑⠀⡐⠁⠀⠀⠐⠀⠁⠑⢁⠤⢰⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⡿⣽⣿⣽⣿⣿⣿⣟⣿⣿⣾⣿⣿⣷⣿⡹⣮⣴⣾⣿⣿⣿⡯⡉⠁⠁⠀⠁⠈⠉⡅⠉⠀⠀⣀⠀⠄⠀⠀⡄⣡⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡟⠊⢛⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠁⣀⣀⣀⣃⡀⣌⠌⡒⠢⠀⠐⡈⠀⠀⣔⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠙⠻⠻⣇⢿⣿⣛⠯⢁⡐⢰⡯⢽⣾⣿⢿⢏⣷⣿⣿⢻⣻⣿⣿⡿⠆⡀⠂⠔⣈⡩⣗⠒⠁⠀⠀⣁⣄⡦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⢇⣹⣿⡿⠷⡾⡾⣿⡿⡸⣿⡿⣿⣹⣿⣀⣁⠎⠀⠀⢀⠀⠀⠀⣈⠎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠈⠀⠈⠀⠈⠂⠁⠀⠀⠐⠘⡭⠊⠀⠙⣤⠸⣶⣵⢯⣿⣻⣿⣿⣗⢇⠓⢐⣘⣥⣅⠵⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⢂⠐⡌⠛⢞⣸⠺⣯⣍⣿⣽⣇⡖⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⢑⠛⠀⠀⢠⡀⠀⠌⣡⣿⣿⣿⣿⣿⡟⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⢄⠉⢔⠂⡄⣰⣽⣻⣿⡛⣿⣾⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠂⢄⠀⠀⠀⠀⠂⡠⠀⢀⡉⠄⢑⠢⠃⣙⣵⣿⣯⣿⡾⣹⣿⣿⣿⢿⣻⣶⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠄⠑⠀⠀⠀⢀⠀⠀⠉⠥⠀⠠⠀⡉⢩⡛⢻⣿⣶⣏⡿⣽⣿⣟⢯⣯⡾⣿⣳⣔⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠠⠀⢀⠁⡄⠠⠄⠰⠀⠀⠱⠀⠌⠁⠡⠿⢿⠿⣿⣿⣻⢗⣿⣟⢮⡵⣮⣿⣾⢆⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠈⡄⣾⠀⠀⠀⠁⠀⠀⠠⠐⡂⠀⡙⠻⣭⠿⠯⣟⣿⣧⢏⢝⣾⣹⡿⣽⣮⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⣀⣿⣿⣄⢀⠲⠁⠀⠀⠁⠉⠘⠀⠀⠀⠈⠁⢛⡛⣛⣿⣯⣯⣿⡖⣿⢻⣿⣿⣿⣵⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠻⣛⠧⠥⠆⢀⡄⠂⡄⠀⠠⠠⡠⢀⠀⠀⠁⠀⠈⠉⠽⠿⣟⣾⣿⢿⣯⣿⢷⡟⣷⣵⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠒⠒⠓⠒⠓⠀⠯⠷⠆⠂⣆⣠⡌⠉⠽⣟⣛⣾⢼⣵⠻⣼⣓⣾⣔⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠊⠩⢙⠿⣿⣟⣵⣮⣽⢿⣿⣮⣂⢄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠍⣙⠻⢿⣶⣾⡻⡷⣑⢄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠩⠟⣓⠮⣷⣮⠂⠀