⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⣶⣿⣶⣴⣿⣿⣿⡆⠀⠀⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢄⡀⣀⠀⠀⠀⢗⣹⣯⣽⣿⣟⣿⣿⠿⣿⣿⣷⣔⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⡿⠋⣶⣔⠈⢥⣴⣾⣮⣷⣷⣿⣺⣞⣥⣽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣿⣿⣿⣫⡙⠔⢟⡜⡣⠀⣻⡿⠿⠟⣉⣀⣀⣉⣛⣓⠋⠝⣼⣿⣿⣿⡿⣿⣿⣟⢿⡿⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣩⣽⣭⣿⢇⡳⡦⠍⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣥⣴⣲⣶⣶⣦⣅⡘⠩⠽⠟⢋⡉⡴⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⣻⠟⠋⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢇⠲⢫⢰⣾⣟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣿⣿⡁⠀⠀⢀⣬⣍⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢺⣯⣦⣿⣧⣿⡏⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⢯⠏⠀⢠⡾⡿⣿⣿⣾⣿⠿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠻⣿⡟⠈⢿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠻⣿⠀⡀⠀⠻⠋⣸⡿⠋⠀⠀⠀⠀⠀⠚⡿⢿⣿⣿⣿⣿⠟⠉⠉⠉⠛⠿⣿⡿⣿⣿⣧⢽⣧⡴⣟⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣴⡏⠃⠀⢡⢠⠉⢰⡳⠁⠀⠀⠀⠀ ⠀⢨⢏⣯⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠻⣮⢻⣿⡼⢿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⠻⣯⣿⣿⡀⠀⠘⡧⡆⣼⠅⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⢑⠉⠻⣿⡀⠀⠀⠀ ⠀⠀⣿⣎⣿⡇⣶⣹⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠹⠯⠀⠀⠀⣻⠛⣿⣧⣀⣀⣀⣀⣀⣤⣾⠃ ⠀⢸ ⠀⠘⣿⣶⣤⣄⣀⣀⣄⣤⣿⣿⢿⡇⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠠⡟⠀⣿⣿⣟⣿⣿⣿⣿⣿⣿⢀⣀⣀⣾⣀ ⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⢿⢿⣿⣿⣿⣛⢿⡷⡶⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡆⢸⠲⢁⣸⣿⣿⣿⡿⣟⠋⠅⣠⡻⣿⢻⣿⢼⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢈⡌⢩⡏⡿⡟⠿⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⢛⠻⢐⡺⠀⣈⢭⡉⠉⠉⠉⠉⠙⢞⠿⢃⣽⣴⣶⣴⣾⣨⡅⣛⢾⣽⣿⠏⠀⡀⣠⢒⣻⠣⣿⣶⡶⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⡟⢺⣗⡿⡇⠰⣰⣻⢈⠀⠀⡀⠀⠼⠘⡰⠘⠈⠉⠉⠀⠀⠉⠁⠋⠢⠌⠏⡇⢀⠞⢻⡿⠃⣼⡟⣿⣷⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢷⣿⣿⡇⠀⠘⠤⠳⣀⢠⡁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⢡⡆⢤⣴⡅⢳⣿⡀⣻⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣏⡿⣶⡀⠀⡩⣿⠄⠀⢀⡐⠄⡀⠰⠀⠀⠀⠀⠀ ⠀⠀⢀⣴⡵⡠⣛⣿⣹⣶⣞⣽⣿⣿⠿⠏⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠑⠉⠀⠀⠐⣠⡶⢗⣀⣄⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣛⣿⡽⠋⢻⣿⣿⡛⠙⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣿⠀⠠⠚⠉⠀⠀⠀⠙⣷⠂⠆⠀⠀⠀⠀⠀⠀⠲⢄⡚⡿⣻⣿⡅⢠⣾⣿⣿⡩⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠠⣤⠀⢾⡀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢾⣿⣿⣧⣿⣷⣿⡿⢩⣼⣴⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣎⣒⠠⠭⠖⠒⠉⠀⢠⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠏⢿⣿⡱⣬⣾⣧⣛⡫⣟⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⡀⠀⠀⣼⣧⡀⢮⠝⡀⠀⠀⠀⠀⠀⠀⠀⠀⣥⡌⣿⠗⡤⣘⣱⡿⢋⣿⠿⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⢐⢜⢕⡀⠀⢹⣷⡕⢄⠙⢥⣀⡀⠀⠀⠀⠀⢠⡜⣂⣾⣯⡞⣼⣿⣯⣽⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⢀⠀⠀⠠⠾⢈⡀⣥⣄⡀⠙⢧⡈⢳⣤⣘⠙⢃⠶⣘⣣⣫⡾⣿⣿⡿⣹⣿⣿⣯⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠃⠁⠀⠈⠋⢿⣿⣾⣿⠿⠱⢠⡙⠶⢿⣿⣿⣿⣿⣭⣵⡾⠛⠑⠝⡨⠊⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠑⠒⠋⠀⠐⠂⠀⠤⠬⠉⠉⠡⠐⠔⠁⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀