⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠤⠔⣠⠤⠔⠢⠞⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⢠⠏⢀⠂⢰⠀⠀⢢⣢⠰⡲⠤⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠁⡏⠀⢸⠀⡈⣆⡨⠗⠒⠒⠒⠒⠲⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢾⢶⣲⡌⣦⣤⣤⣄⡀⠀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢀⣼⣮⣥⣷⣷⣷⣶⣿⠂⢰⣿⣴⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠋⠐⡆⡉⠛⠿⠟⠋⣆⣸⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⣿⡕⡆⠀⠀⠀⠰⡶⢾⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣙⡾⠁⠀⠀⠲⣿⠽⢾⢶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣼⣿⣿⣶⣶⣶⣤⣭⣭⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣽⣻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠗⠋⠀⠄⠙⠉⣀⡇⠉⢻⣿⠟⣷⣿⣿⣿⣿⣿⠿⡺⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⣶⣠⣀⠀⣴⣶⣿⢹⠆⠀⢁⠠⣿⣻⣸⡿⠿⢧⠀⣿⣭⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣜⢿⣿⣿⣿⣿⠏⠇⢇⢬⣶⣅⡇⠈⠙⠃⠁⢌⠁⢘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠘⣞⢧⡿⡤⠀⠀⠀⢠⠊⢠⣾⣿⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⡐⣈⡉⣽⡇⡀⣠⣆⢀⣴⣾⣿⣿⣶⣦⣿⡆⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡏⡇⣤⠞⢋⣫⣿⣿⣿⣿⣿⣦⢼⣷⣾⣿⢀⢀⣼⣧⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠊⠠⢗⣒⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢖⣺⢽⣿⠻⣄⡿⣙⣿⣿⣿⣼⢻⣿⣷⡝⠛⠁⠀⣸⣟⣿⣣⢯⣽⣿⣯⣟⣿⣿⣿⠿⠻⢹⣝⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢋⣿⠙⡞⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣠⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣂⣀⣦⣬⠭⠔⣨⡷⠀⠐⠒⠂⠈⠀⠀
⠀⠀⠀⠀⠀⠀⣸⡏⠠⢷⢿⣿⣿⢽⢿⣿⠿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣟⣷⣿⡿⠀⠀⢀⡀⠀⢀⣀⠀
⠀⠀⠀⠀⠀⣰⡟⣰⣧⡀⢸⣿⢯⡟⢡⣼⣶⣶⠋⠛⠿⠿⢿⣻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣹⠋⡠⡇⢠⡤⢿⢦⡀⡝⡇
⠀⠀⠀⠀⢠⣿⡟⣿⣿⣿⣿⣿⠴⣿⣿⣿⣿⠃⠀⠀⢀⣤⣽⣿⡻⡚⣧⠿⣿⣿⣿⣿⣻⡿⢿⣿⣿⣿⣷⣿⣚⣡⣿⣿⡇⢾⣷⢏⡗⣡
⠀⠀⠀⢠⡿⠽⣿⣟⣿⣿⣿⠅⢐⣿⣿⣿⣯⣀⣀⣸⣿⣿⣿⡽⣍⠲⣿⣷⣿⣿⣿⣯⠁⢷⠀⠉⠙⢿⣿⣿⣿⣲⣿⠈⠋⠊⠓⢺⠛⠁
⠀⠀⠀⢸⣿⡆⠀⠙⢿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡁⠉⠠⣙⡿⠏⠈⠻⣷⠢⣸⡆⠀⠀⠀⠉⠉⠛⠻⢿⡃⠀⠀⠀⢸⠀⠀
⠀⠀⠀⠘⠻⣷⣆⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣼⡏⠁⢈⠁⠀⠀⠂⠀⠈⢃⣨⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠉⠀⠀
⠀⠀⠀⠀⠀⠘⢿⣶⣶⣿⣿⢿⣿⣿⣿⣿⣿⣿⣻⡿⡟⠚⢹⣿⣞⣿⠀⠀⠀⠀⢿⣟⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢿⣿⣿⢻⣿⣟⣿⣿⣿⣿⠿⠟⠂⠀⠀⠀⢸⣿⣿⠋⡀⠀⠠⡖⣠⠽⣿⣟⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢈⡙⡟⠾⠇⣹⣾⣽⣶⣤⣤⣤⣶⣷⣿⣾⣿⣿⣿⣤⣷⣭⣤⣿⡞⣷⣟⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣾⣿⣉⢀⣻⣿⣿⡿⣿⣿⣿⣿⡿⠟⠛⠛⠉⢻⣿⣿⣿⣿⠿⢿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⢿⣶⣻⡭⣭⣩⣿⢿⣟⣜⣿⣿⣿⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡄⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⡿⣿⣧⣯⡯⣏⣿⣿⡟⣷⣵⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡄⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⢹⣿⣙⠻⢿⢿⢛⣯⡿⣿⣸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡆⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠸⣾⣾⣶⣿⣮⣭⣭⣥⣿⣿⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣆⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⣿⣽⡋⠈⠉⠉⢙⣿⣿⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢘⣷⣶⡇⠀⠀⠀⠈⢹⣿⣾⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢆⡆⠀⠀⢀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⣿⢾⣧⠀⠀⠀⢰⢸⣯⣻⠣⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣈⡢⠤⡤⠚⢳⡀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣫⠄⠀⠀⢈⣡⣿⢟⠱⡈⠁⣰⣿⣶⣤⠤⠤⣴⣾⣧⠀⠀⠀⠀⢣⡀⠱⡘⡌⢷⡀⠀⠀⢀⡠⠠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢳⣷⣻⠆⠀⠀⠀⢿⣿⡿⣦⡘⠠⣿⡿⡍⢡⣶⣯⡆⠻⢿⡆⠀⠀⠀⠀⠑⣆⠱⡜⡄⢳⢄⠀⠒⢾⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣿⣏⡇⠀⠀⢀⣞⣿⡗⠀⠈⡱⠉⣁⡰⣜⣿⣏⣀⡀⠠⡽⡆⠀⠀⠀⠀⠘⣦⡙⠘⣌⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢻⣿⡺⣄⠀⡼⢜⣻⣌⢧⡀⠀⠀⠰⣾⣿⣿⣿⣿⠶⢛⣗⢾⠀⠀⠀⠀⠀⠈⢻⠦⢸⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢷⣏⣸⣲⡿⣻⠋⣩⠦⣿⣆⣤⣾⣿⣿⣿⣿⣿⣿⡯⢤⣹⡀⠀⠀⠀⠀⠀⠘⢦⡁⣇⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠾⠟⠁⠀⠇⠁⢺⡤⡽⣿⣿⣿⣿⣿⣿⣿⣿⢿⠋⠳⡄⠀⠀⣠⣾⡟⠀⠁⠘⡌⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⢼⠏⠀⢻⠟⣻⠿⡳⡻⠋⠉⢀⣰⢰⣾⣄⠀⣿⣾⣷⣄⣠⣤⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠿⠿⠷⡟⣁⠀⠃⠀⠀⠐⣵⢆⠐⣾⢘⢿⢛⢇⡜⡄⠹⣿⣿⣿⣿⣿⣿⠟⠋⢷⣦⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⠶⣄⣾⣿⣽⣿⣿⣿⡏⢀⠞⠀⠀⡀⢈⠏⡀⡾⡷⢻⢐⣸⣼⡾⢷⠀⠈⠻⣿⣿⣿⣇⣴⣾⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣿⣍⠉⠀⠈⢙⣷⠋⠚⡄⣇⠃⢨⠀⢧⣿⢃⣼⣾⠟⠃⠀⣼⡀⠀⠀⠈⠻⣿⣿⣿⣿⡿⠋⢿⣷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣖⣷⣿⣿⠃⢀⣿⣿⡆⠈⡄⣽⣿⣿⡿⣥⣸⣾⢻⠻⡇⠀⠀⠀⠀⠈⠻⢿⣿⣿⣞⣿⡿⠀⠀⠀⠀
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢩⠜⠀⢉⣿⣿⠃⣰⡿⠋⠹⢥⣬⡷⠟⠉⠙⢋⡁⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⠉⠀⠀⠀⠀⠀