⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣉⠉⠉⠀⠈⠙⠛⠛⠛⠂⠒⠲⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⢏⡂⠠⢀⠀⠀⣀⠤⠒⢉⢃⠼⠈⡝⣟⡟⠒⠤⠤⠀⣐⡀⠀⠈⠻⣠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⠰⣨⠕⠁⡠⣶⣿⣦⣒⣲⣿⣿⡔⠂⣀⣔⣽⣭⢿⣆⣀⣴⢟⠩⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢤⠾⠷⢛⡁⠹⣻⣷⣿⡗⢒⣿⣿⡿⡟⢸⣮⢻⣫⣔⣕⣭⣎⠘⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡚⣿⣿⣿⣼⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⢿⣟⡯⠉⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡾⣽⣳⡀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠍⠜⣱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⡿⣧⣗⡯⣄⠹⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠗⠁⡴⠺⢇⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠶⠟⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡆⣷⣍⠹⢷⣻⢦⠘⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣤⢺⠷⠤⢄⣉⣉⣶⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣽⠿⠛⠁⢸⣿⡿⣦⣟⣯⣷⡈⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢠⡛⣄⠢⠄⢸⣯⣷⣟⣻⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢹⡿⣹⢿⣿⡏⣙⣿⣿⣧⠀⣿⠈⢿⣟⡾⣽⢷⣻⠆⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣼⠀⢰⡍⣯⡷⠾⣿⣤⣿⢯⣇⠐⣠⡤⣴⢴⣛⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⢿⣿⣿⣿⡿⣿⢻⣿⢻⣿⣇⠀⠘⣷⠁⠚⡿⠀⣼⢿⣿⢏⠀⣃⣀⡈⢿⣷⣭⢻⡽⣻⣅⡸⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠔⢿⠀⠀⢹⣆⣛⣛⣯⣿⡗⠉⠘⠊⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠅⢡⡦⡿⣿⣾⣷⣿⣿⡷⣯⢿⣿⣧⣤⢿⣦⢶⣿⣴⣯⣼⣷⣃⣀⣹⣿⣿⣿⣷⣾⣯⣾⣵⣯⣶⣸⣦⣀⣽⣤⣤⣄⣀⣀⣀⣄⣀⣀⣤⣤⣀⣀⢀⡀⣠⣿⠀⠀⠂⠸⣆⠀⢨⡏⠀⡖⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⢷⡻⣝⣮⢳⣽⢎⣷⢫⣟⢶⣹⢮⣻⢼⣫⢶⣫⢞⣳⡝⣯⣛⣧⢟⣼⣣⠿⣜⣳⡽⣺⣜⢧⡟⣾⡹⢯⣝⣯⢻⡽⣛⣯⢻⡝⣯⣝⣻⠁⠇⡏⠀⡿⠟⠀⠀⠰⠀⠈⠈⠛⠁⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣷⣏⣾⡹⠳⠮⠿⠼⠿⠾⠽⢾⠿⠮⠷⡝⠊⠓⠋⠞⠭⠷⠽⠧⠿⠾⠯⠿⡾⢿⠽⠶⠽⠳⠾⠽⠾⠵⠿⠽⠾⠯⠷⠯⠽⠮⠷⠯⠷⠾⡇⠀⣀⠃⢀⣿⢀⠒⠚⣁⣀⡤⠰⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣺⡻⣾⠣⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣈⣥⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣊⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡆⡇⠀⣿⣟⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⡿⣝⣳⡏⠀⠃⠠⣀⠀⢀⡤⢿⠟⣼⣟⣿⣿⣻⠉⠀⠀⠈⢣⡟⠳⣦⣴⠞⢋⡄⣀⠂⠡⠉⡉⢻⡏⣽⡟⠉⠙⠉⠉⠉⠋⠉⠉⠛⠛⠓⠒⠖⠒⠒⠚⠏⠉⠋⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢴⢮⡟⣵⡟⠀⡈⢀⢰⠁⣨⠁⣸⠁⢰⣯⣻⡿⠿⠶⠀⠀⠀⠀⣀⣸⠿⠋⡀⠄⣾⡿⣏⠆⡁⠂⠄⣿⡙⣴⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢬⢯⣯⠽⣾⠁⢀⠁⠀⡆⢀⡤⠊⠛⢀⣿⢿⣽⡉⠀⠀⠀⠀⢠⡾⠛⠁⢄⡂⣼⣿⠋⡴⡝⠠⢀⠅⣈⡿⠜⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣠⢏⡷⣞⡽⡗⠀⡄⠀⠸⠀⠰⠀⢀⠀⣼⢯⣿⠷⠀⠀⠀⣤⡴⠋⠄⠁⣈⣸⣿⣿⠎⢠⢳⠁⢂⠄⢂⢸⣏⢳⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣞⡾⣝⣮⡽⠀⠠⠀⠀⠇⠀⡆⠀⠀⢠⡞⣧⣿⡄⠀⣀⣴⠟⠀⠡⢈⡰⢾⡏⣿⠇⡀⣏⠇⠌⠠⠈⠄⣿⠎⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣸⡽⣳⣝⣶⠇⠀⠀⠀⡸⠀⢰⠀⠠⠀⢶⡻⣼⣷⢀⣶⠛⠁⡀⠌⣰⡎⠃⠄⣰⡟⢀⢰⡹⠀⠌⠠⢁⢸⡟⣌⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣯⣝⡳⣞⡽⠀⣀⣄⢤⠧⠤⡆⠀⠆⠠⣟⣵⣻⣻⠞⠀⠠⠁⣤⡾⠋⢀⠁⢂⣿⠃⢀⢧⠃⠡⢈⠐⡀⣿⠣⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣼⣋⡾⣵⢫⡆⠀⢁⠀⠸⠀⢰⠀⠰⠀⢰⣯⣶⠟⡁⠠⠈⣤⣿⠋⠄⠡⠀⠌⣸⡟⠠⢸⡚⠈⠄⢂⠐⣸⡟⡥⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡿⣹⢳⣭⢻⠇⠀⡀⠀⡇⠀⡌⠀⠇⣀⣼⡿⠃⠠⠐⢠⡝⠓⠀⡀⠆⠠⢁⢢⣿⠇⠀⣧⠗⡈⠄⠂⢄⣿⠓⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⡿⣧⢿⣼⣻⣄⠀⠴⣴⣽⣻⠶⣾⣝⣻⠟⠀⠠⢁⣼⠗⡁⢈⠐⠠⢈⠐⣠⢳⣿⢸⣶⣹⣼⠃⠠⠁⣼⡏⠓⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⢳⡞⢉⣡⠤⣒⡮⢝⣓⣮⣹⡿⣶⡼⠃⠠⠈⣴⠛⡉⢀⠐⠠⢈⠐⢠⡜⣱⣾⡟⣼⡿⣼⡿⢀⠁⠂⣿⠁⠂⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣯⠵⣒⣭⣶⠿⣛⡽⠚⣡⣴⠫⠀⠄⣡⡾⠃⠂⠄⠠⢈⠐⠠⢈⡚⣴⣿⣿⠃⠛⣷⠿⠇⠀⠌⢰⣿⠀⣂⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢱⣾⠓⢉⡴⢏⢷⡤⠻⢧⣋⡇⢀⣾⠣⢀⠁⠂⠌⡐⠠⢈⠰⣉⣾⣿⣿⡏⠻⣿⣷⣤⡄⠈⣄⣨⣿⢊⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢸⡿⠚⢁⡠⠞⣷⣿⡟⣦⣿⣧⠞⡁⠂⢀⠂⡁⠂⠄⡁⣂⣳⣿⣟⣽⠋⣄⣰⠯⢭⡤⠶⠛⠋⣿⣥⣶⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀