⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠘⣿⣿⣇⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⡫⠞⣩⣗⠭⠛⣿⡉⠉⠛⠳⣝⢗⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣷⢫⡻⡳⣾⣪⡕⡡⢪⠃⢈⣦⡀⠀⠠⢹⣷⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⢣⢧⢳⢡⡽⣵⣿⡔⠁⢀⠎⡙⣿⣿⣿⣮⣿⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣮⣾⡯⡸⠏⣿⣿⣻⡴⡟⣰⣷⡏⣿⣿⣇⠸⢟⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⣿⣷⡇⢶⣯⣽⠛⣿⣶⣭⣿⡿⣿⣏⣿⣦⣸⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⡆⠀⣿⣻⣾⠀⠁⠀⠈⠻⣿⠟⠁⣿⣿⣟⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠉⢷⡤⢛⣧⣿⢀⣠⣤⣶⣶⣗⡄⠀⠘⠻⣿⡿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠋⠉⠉⠉⠉⠚⠿⣿⣿⣟⢷⣄⠀⠀⠀⠀⠀⠀⠈⢳⣾⠿⢿⣿⣿⣿⣿⡿⢾⣷⡀⠀⢀⣹⣷⡱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡞⠁⠀⢰⡖⢀⣴⣿⣿⣷⣌⢻⣿⣿⣿⣷⡀⠀⢀⣤⣶⣾⠿⠙⠳⢾⣿⢿⠿⣿⣶⣶⣿⣷⣶⣿⣿⣿⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣴⠃⠁⣿⡆⣿⣷⣼⢣⣾⣬⡉⠻⣧⠙⣿⣷⣿⡷⣠⡿⠛⣻⠿⢇⣄⢀⠐⣿⣷⣠⣿⣷⣾⣿⣿⣿⣮⣿⣿⣇⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣸⣷⣶⣦⣿⣿⢿⣿⣷⡿⠿⣿⣷⡌⢿⣧⠸⣷⣻⢿⣯⣴⡞⣡⡞⣿⣿⣼⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣿⡏⣿⣿⣷⡿⠈⠛⠛⠁⠀⠹⣿⣿⣄⣿⡇⢳⡿⡏⣿⣿⣿⣿⣷⣶⣾⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣾⣏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣿⢳⣿⠛⠛⠁⠀⠀⠀⠀⠀⠀⢹⣷⣿⣿⣷⠘⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣷⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠘⣿⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⠀⣿⣽⣽⣿⠿⡿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⣸⠁⠀⠀⢿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣿⡟⢿⡄⠀⠀⠀⠀⢀⣀⣀⣀⣼⡟⢹⣿⣿⢀⣿⣿⣻⣧⡆⢠⣿⣿⣿⣿⠟⠉⠀⠀⠀⢠⠇⠀⠀⢠⠞⣿⣿⣿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢻⣿⡎⢷⡀⢀⣸⣆⣸⣿⣿⣻⣋⣀⡜⣼⡿⢼⢻⣿⣿⡏⡟⡼⣼⣿⣿⣿⣄⣀⠀⠀⠀⡎⠀⢀⣴⠃⢠⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⢿⣟⣎⢷⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣧⣏⣿⣿⣿⡽⣄⡇⣹⣿⣿⣿⣿⣿⣿⣶⣼⣧⣾⡿⠁⠀⣾⣿⣿⣿⣿⣇⢸⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⣿⣯⣢⣈⣾⣿⣿⣿⣿⣿⣶⡿⣿⣟⣿⣞⣿⣿⣿⣿⡏⢠⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⣿⣿⣿⣿⣿⡜⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⢿⣿⣾⣿⣿⣿⣿⣿⡿⣿⡁⣿⣟⣿⣿⣿⣿⣿⣿⣧⣿⣿⡿⣤⣩⣛⣽⡿⣶⣿⡿⠿⡆⠀⠀⢻⣿⣿⣿⣿⣿⣷⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠟⠋⣩⡝⡿⠻⠷⠞⣣⡴⠊⠙⢓⡒⢃⠀⣀⣀⣻⣿⣿⣿⣿⣿⣾⣄⢸⠿⠿⣷⡶⠶⠂⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠈⠁⠀⠀⠀⢇⢈⣯⣗⠀⢈⣁⣀⠀⠀⠀⠀⠀⠙⠣⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⢅⡙⡛⣶⡶⠆⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡞⠁⠀⠀⠀⠀⠀⠉⠐⠢⣄⢀⠀⢰⠈⠛⢿⣿⣿⣿⣿⣿⡹⢻⣗⣬⣹⡿⡧⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣔⠀⡇⠀⠀⠙⣿⠟⠹⣿⣿⡿⣿⡿⠏⠉⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢧⠀⠀⠀⠈⢷⡾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠘⡿⣭⣼⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠄⢀⣤⠀⣰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⡁⠀⠈⠁⠈⢸⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠘⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣶⣶⠟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠀⠀⠀⠀⠻⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡏⠉⠉⠉⠛⠛⠿⣿⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣫⣿⣦⡀⠀⠀⢀⢀⣨⣹⣿⣿⣿⣷⣦⣤⡤⠞⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢀⡟⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣽⣿⣿⣿⣿⣦⣀⣚⣾⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⢠⣿⣿⣿⣿⣿⣾⣿⣶⣠⣶⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣭⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡤⠿⣿⠿⣿⣯⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀