⠀⠀⠀⠘⣷⣿⢃⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠠⣴⣮⠉⠀⠀
⠀⠀⠀⠀⠘⣿⣯⡣⢀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣠⣾⠏⠂⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢽⣿⣖⡢⣀⠀⠀⠀⠀⠀⢀⣘⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡼⢃⣀⣀⣠⣤⣶⣶⣿⠟⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⢛⢯⣏⣧⠂⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠊⢀⣰⣻⣷⠞⠛⠛⠛⠟⠛⠙⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠤⣀⡀⠀⢒⠯⠓⢀⣴⣤⣀⣢⠀⠽⠄⢀⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⠛⠁⠄⢀⠂⡂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠛⠃⠀⢟⡚⠜⠫⠓⠈⢤⣦⠝⠉⠁⣤⢶⣶⡤⡀⠀⠀⠠⠀⠀⠀⣓⠀⠀⠀⠀⠀⠀⠀⠘⣀⠴⢹⠇⠅⠒⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢑⠪⠕⢿⣭⣿⣲⣤⢠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢋⠂⠋⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠹⣞⣷⡿⣕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢈⢗⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠼⡿⡽⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠁⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⢀⢣⠀⠁⠈⣰⡤⣀⠀⠀⠀⠀⠀⠠⠐⡀⠃⠇⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡍⠀⠄⢃⠖⠉⠘⡼⣿⣟⣮⣷⣄⡀⢀⣀⣨⣧⣖⣆⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⣿⣦⡔⣠⣤⣴⡜⣿⣟⣿⣿⣿⣾⣿⡻⣿⣿⣿⣿⣿⢹⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣮⣿⣿⣿⣿⣧⣻⣿⣽⣿⢿⣿⣿⣿⣿⣿⣿⣷⡾⣿⣿⣿⣿⣿⣢⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠘⢿⣿⣿⣿⣿⣿⣿⣿⡶⢿⣿⣿⣿⣿⣿⣿⡗⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠁⢀⣼⠇⠀⠀⠣⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠎⣿⢻⣿⣿⢿⣿⣻⢻⣾⣼⣿⣿⣿⣿⣿⣏⡾⣮⣟⢻⠀⠀⠀⠀⠀⠀⡀⠂⠀⠈⠃⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣞⡫⠋⠀⠉⠉⢯⠛⢰⣿⣿⣿⣿⣿⣿⣿⣦⣯⣿⣏⢾⠂⡀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠡⣾⠅⠀⠀⠀⠀⠀⠀⠳⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡄⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣯⣾⡯⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⡆⠀⠀⠀⠀⠀⠀⣸⣌⢿⣿⣿⣿⣿⣿⡛⠹⣭⢟⡿⣿⡫⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠻⢷⣄⣀⡀⠀⠀⡿⣷⠀⡻⣛⣿⣾⢋⣝⡈⡿⣯⣰⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣽⢿⢡⣾⣉⠗⢧⠋⡈⠻⢅⣾⣿⣿⣙⣿⣗⠀⡠⠋⠀⡠⢤⣡⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣚⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢹⣾⣿⣿⣵⠭⡁⣤⠘⡣⢣⢘⣿⣿⣿⡟⣿⣻⠃⢀⢴⣽⣶⣿⡟⢻⡟⢯⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣴⣻⣿⢟⣽⣍⣭⡔⢤⣀⣭⠐⡌⡿⣷⣿⣧⢱⡟⢀⠵⠋⠈⠡⠀⠉⠉⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⢼⣾⣾⣿⣿⣿⡌⠛⠶⢈⢂⡝⣿⣷⣿⢸⡡⢀⣲⣲⣦⣷⣶⣶⣤⣤⣴⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⡬⣾⣿⣿⣿⣿⣾⡿⣽⣷⡄⣸⣿⣾⣿⣿⣑⣧⠀⠚⢹⡛⠛⡿⠟⠍⠟⣛⠳⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⡑⣿⡿⡿⢿⣿⣿⣤⣟⣿⣿⣼⣿⡸⣿⣯⡇⡼⡀⠙⠶⣴⡶⠖⣯⣶⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡱⡍⠦⠀⠀⠀⠀⠀⠀⢢⡊⣪⢧⢈⢻⣽⢿⣿⣿⣟⣿⣷⢻⣿⡇⢞⠀⢉⠚⣛⡛⡌⡉⠀⢰⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡿⣣⠄⠀⠪⣵⡙⢂⠀⢀⢀⠀⠀⠀⠈⠀⠂⢘⠋⢌⣿⣷⣝⢿⣷⡿⠂⠙⠻⡇⠊⢇⠀⠉⢃⢒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⡖⠐⣊⢂⡀⠉⠁⠂⣔⢖⡋⢃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⡌⢫⠀⠀⢃⠀⢀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠄⠛⢓⠆⠈⠡⡴⡂⠀⠘⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡻⠀⠀⠀⠀⠀⠒⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢀⠀⠀⠀⠀⠀⠀⠀⠀
¹⁸⠀⠀⠀⠀⠀⠀⠀⠢⡴⣄⢀⠀⣠⣤⡀⠀⢹⢿⠀⠀⢠⢦⠃⣀⡵⠀⠄⠐⠀⠂⠀⠀⠀⢻⣷⢀⡀⠂⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣰⡆⠀⠀⠀⠀⠀⠀⠀