⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢠⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⣠⠦⣤⣴⠋⠉⢻⣤⣤⣄⠀⣼⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⠉⠛⠃⡀⠈⠰⠲⠀⠈⠀⠀⠻⠚⠛⠋⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⢉⣡⣴⣶⣾⣷⡀⠀⢠⣧⠀⠀⠀⣼⣿⣶⣶⣤⡉⢻⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣅⣿⣿⡿⠿⢿⣿⣷⣦⣾⣿⣷⣶⣾⣿⡿⠿⢿⣿⣿⣦⡷⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡄⢹⡟⠁⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠈⢿⣿⠠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⣇⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃⢿⣿⣧⣠⡀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⢀⣰⣿⣿⠘⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣏⡴⠘⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠇⠀⢸⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⣿⡆⠀⠈⠛⢻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠁⠀⢠⢿⠏⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠓⠒⠂⠘⠚⢚⠃⠀⠀⠀⠐⠦⠶⠞⠀⢀⠀⠀⢀⠳⠟⣃⡾⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⣶⣤⣭⣥⣶⣄⣀⣀⣀⣠⣶⣤⣭⣥⣴⠞⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⠾⣶⣤⠿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⠿⢿⣤⡿⢶⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣼⣿⣿⣿⣿⣷⡘⢟⢆⠘⠏⠉⠹⠁⠀⠀⠘⠿⢻⠁⠘⢎⣿⠏⢐⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢿⣿⣿⣿⣿⣿⣷⡈⠻⣷⡄⠀⢡⠘⠿⢷⠿⠇⣸⠀⣠⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠿⠿⠿⣤⠹⢟⢦⠈⢳⣤⣼⣤⣶⡋⣴⡽⣡⡞⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢳⣞⠻⣿⢟⡽⢿⢯⡷⣝⣟⣽⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣾⣿⡟⢻⣷⣈⣾⣿⡇⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣇⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣴⣿⣤⠇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣤⠁⠀⠀⢠⣷⣬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠸⣷⣧⠟⠀⠀⠀⠀⠀⠀⠀⠉⠛⢛⠛⠋⠀⠈⠻⠿⠿⠟⠋⠀⠀⠀⠀⡀⠈⠃⠀⠀⠀⣋⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠠⡿⠿⡿⣷⠀⠀⠀⠀⠀⠀⣀⡄⠀⣨⡀⠀⠀⠀⠀⣀⣀⣀⡀⣀⣠⠴⣾⣡⠖⡿⠀⢠⡿⠽⡋⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀
⠀⠀⠀⣠⢀⡼⢟⣠⣤⣿⣠⣤⣴⣿⣇⠓⣻⣶⣿⠿⠿⣿⠿⣿⠟⠛⠛⢛⠛⠛⠛⢿⡟⠛⠿⠛⠛⠋⠹⠿⠿⠿⠷⠶⢿⣯⣃⣳⣥⣧⣤⣀⣀⣀⣤⡄⢉⢰⣾⠆
⠀⣠⣤⣭⣷⣀⣼⣿⣻⠯⠷⠚⠋⠉⠉⠙⠧⣀⠈⠉⠙⠛⠳⠯⠤⣄⣀⣿⠃⠀⣶⠀⠀⡀⠀⠀⠀⠀⢀⣤⣤⡤⠴⠒⣫⠟⠉⠀⠙⠁⠈⠉⠛⠻⢿⣶⣎⣠⣧⡀⠋
⢰⡏⣼⣿⣭⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣿⡶⢾⣿⣶⣞⣋⠉⠁⠀⠀⠀⢀⡔⠁⠆⠃⡀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢟⣤⣿⠀⠀⠀
⢰⣿⢿⣿⠿⣥⡀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⣾⣴⣮⣱⣄⡤⠤⠖⣒⣻⣿⣯⣴⣿⣧⣾⣿⣦⣶⣌⡉⠙⠒⢶⡴⠯⣀⣀⡀⠀⢉⡁⣀⠀⠀⠀⠀⠀⢀⣹⣼⣿⣿⣧⠀
⢸⡏⢃⣉⠽⢛⡇⠀⠀⠀⠀⣀⣰⠧⠴⠖⠚⠉⠉⠀⠀⠀⠙⢦⡘⠻⠿⠿⠿⠿⠿⠿⠿⡿⣿⣿⢛⠁⠀⡠⠊⠀⠀⠀⠀⣈⠉⠛⠛⠿⢦⣄⣀⠀⠀⢺⢧⣼⠀⣤⡿⠀
⠘⠻⠥⢤⡴⣟⡁⠶⠶⠟⠛⠓⠒⠒⠒⠒⠒⠒⠲⠲⠶⠶⠿⠶⠿⢦⡤⠤⠤⠶⣶⣶⡾⡗⠛⠛⢛⡶⠚⠓⠒⠒⠒⠚⠓⠛⠓⠒⠒⠒⠒⠒⠚⠛⠛⣿⣷⣊⣛⣸
⠀⠀⠀⠀⠉⠙⠛⠿⣖⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠙⠳⣤⣤⠿⡿⢿⣧⡀⡴⠋⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢖⣛⡭⠞⠉⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠤⠴⣶⣲⣂⣠⡤⠤⠤⣤⣤⣤⣤⣤⡄⢸⣟⠟⠹⣽⠎⢧⠀⣀⣀⣀⣤⠤⠤⠤⣤⣤⣤⡶⢶⣿⣿⠿⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠉⠉⠉⠙⠋⠉⠉⣟⡉⠀⠀⢰⠞⡟⠒⠒⠒⠒⠒⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠰⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀