⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣲⣽⣿⣿⣶⡭⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣻⣿⣿⡿⣿⣟⠿⡳⣀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢿⡿⢟⣿⣿⣿⣧⡀⢹⣿⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⣰⣿⣿⣿⣿⣿⣷⡘⢿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣶⡘⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⢢⣿⣿⣿⣿⣿⣿⣿⠟⠀⡘⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣿⣦⡙⡿⢿⠿⠿⠛⠁⣀⣴⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣴⡟⣿⣮⠿⢾⡀⣠⠶⠑⠡⠈⠡⣀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣊⣡⣶⣿⣿⡟⡳⣜⡸⣻⣎⡻⣧⠖⢀⡔⠀⢀⠔⢡⣶⣿⣷⣎⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢿⣷⣿⣿⣿⢥⣿⣧⣌⣉⠛⠻⣶⠾⠛⠉⢁⣀⣱⡀⢿⣿⣿⣿⣿⣧⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⠸⣿⣿⢟⣱⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠙⠿⣿⡟⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⡿⠟⣳⣾⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡶⣲⠾⠿⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡣⠋⠀⠀⢠⡿⣿⣮⣿⣢⣄⣀⣤⣤⣿⣤⣄⣀⣀⣠⣾⣿⣿⣻⣷⣅⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡍⡇⠀⠀⣠⣿⡟⠘⣻⣝⣿⣿⣿⣿⣿⣿⣍⠙⡿⢟⣫⢝⣿⡇⠱⢻⣿⣦⣀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣻⠿⠿⢷⣶⣾⡿⠏⠀⠀⠑⣿⡗⡼⠼⣿⣹⣿⣿⡷⠁⠀⠈⢫⠞⠁⠀⠑⢿⣿⣿⣿⠛⠻⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢅⠔⠁⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⢿⣟⠁⠀⠀⠻⣟⠗⠁⠀⠀⠀⡼⠂⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⠁⡌⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠈⠀⠐⢠⣼⣿⡇⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⣷⠀⠀⠀⠀⡴⢻⡀⠀⠀⠀⠀⠈⣿⣿⣵⡐⠀⠁⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⢰⡽⡇⠀⠀⠀⠀⠘⠀⠀⠀⠈⢀⠂⢡⠀⠀⠀⠀⠀⠈⢻⢿⣮⡱⡀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⣀⠀⢀⣀⣜⠀⠀⠀⢀⣴⢿⠟⠋⠀⠀⠀⠀⠀⠀⠀⢸⡧⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠫⢷⣌⢢⠀⠀⠀⣡⣀⡀⠀⢀⣀⠀⠀
⠀⠀⠓⠤⢤⡄⠀⠉⣿⢦⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢱⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢘⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣷⣳⣴⣾⠁⠀⢀⡬⠤⠶⠃⠀
⠀⠀⠀⢀⠎⢇⣀⣶⠁⠈⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢾⠀⠀⠈⢆⠀⠀⠀⢀⠆⠀⠀⠀⠀⠆⠈⠆⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡏⢈⣶⣀⡸⠱⡀⠀⠀⠀
⠀⣠⠐⢅⣵⢀⠉⠮⠅⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢤⠀⠀⠀⠈⣷⣶⣶⠎⠀⠀⠀⠀⠀⢸⠈⢠⠈⠐⢄⡀⠀⠀⠀⠀⠀⠀⢃⠨⠭⠁⠀⣮⣨⠐⢄⠀
⣔⣠⠞⠁⠆⢴⢇⢰⢫⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠘⡄⠀⠀⠀⢿⣼⣿⠀⠀⠀⠀⠀⠀⢘⠀⠄⠀⠀⠀⠈⡆⢄⠀⠀⠀⢀⠂⡜⡆⣸⠦⠰⡈⠙⢤⣱
⠀⠀⠀⡘⢉⠇⢀⠘⡄⠣⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡇⢳⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣢⣄⠀⠀⠀⠇⠀⠑⠢⡀⢈⡸⠀⠃⠔⠘⡌⢡⠀⠀⠀
⠀⠀⠀⠑⠚⠀⠘⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢿⡸⡄⠀⠀⢸⣿⡇⠀⠀⢀⠀⠀⠀⠸⢝⠿⣷⣭⣶⣄⡀⠀⠀⠀⠈⠢⡀⠀⠢⠗⠀⠘⠺⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⠐⠒⠚⣿⣞⣧⢚⠦⣄⣿⣿⠇⠀⠀⣾⠀⠀⢀⡆⠀⠈⠐⠹⡻⣷⣿⣶⣤⡀⠀⠀⢈⢄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣂⣡⣤⣶⣶⣶⡶⣶⣾⣿⣿⣮⣳⣼⣿⣿⣌⠒⢠⣿⢀⣔⣟⠇⠀⠀⠀⠀⠀⠙⠹⣿⣷⣾⡦⣄⠞⠀⢆⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⣽⣿⣯⣽⣿⣿⣿⣿⣬⣽⣧⣆⣤⠛⡄⣀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣎⡧⡀⢸⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣻⣿⣿⣿⣯⣿⣿⣿⠋⠉⣻⣧⣌⡀⠀⠀⢝⠂⠒⠂⠀⣰⣶⣿⣿⣿⣳⣷⠸⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣾⣾⣿⢿⣿⣦⣴⠋⠈⡵⢫⠟⣟⡶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⠣⠿⠓⠢⡀⠀⡇⠀⠈⠁⠛⠩⠿⠛⠛⡛⠛⠛⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣾⣿⡆⢄⠀⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣻⣿⣿⣿⣿⢧⠀⢠⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠉⢉⣿⣿⣿⡛⣿⢿⣻⠅⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⣹⣿⣿⣼⣿⣀⡠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣡⣶⣿⡋⠉⣟⠉⠈⠝⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢿⡷⠟⠛⣤⠓⢠⠶⣦⡀⢈⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⣿⡆⡆⠀⠀⠉⠳⢮⣛⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢏⡇⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀