⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡠⠒⠂⢁⣀⡀⠤⠀⠂⠆⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⢅⢅⡸⡞⢍⢴⠋⡰⢌⢔⠈⠦⠐⠒⡅⠨⡐⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⣽⢿⣿⢏⢢⢮⣯⣔⡳⢁⣗⡾⣉⣯⡆⣝⣂⢠⠑⣐⠐⠤⣀⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⣿⢏⣳⠏⣠⣿⣷⣿⣾⣀⣷⣾⣧⣿⣟⣷⡼⣧⠀⣧⡈⢳⢄⠠⢃⠌⢢⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⣼⠟⣷⢏⣃⣽⣿⣿⣿⣝⣼⣻⣾⣯⣿⠿⣿⢻⠛⡰⣵⢹⣤⡘⢧⠡⢎⠤⠁⡈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢂⢲⡿⣿⡿⠀⣏⢹⣿⡿⣱⣿⣼⣿⢼⣻⣿⣷⣽⡾⡜⣁⡗⢍⣷⣟⢮⣳⢊⠖⡡⠒⠈⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡁⠘⣿⣿⢱⣱⣿⡾⣿⣍⣾⣯⣿⣿⡾⣽⣿⣿⣿⡏⣿⡏⣯⡯⣟⣜⢿⡲⣏⢎⠥⢃⠠⢀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠱⢸⣏⡆⣸⠻⠿⣱⣧⣿⡷⣿⣿⣿⣃⣿⣿⣿⣿⢾⡇⢽⡽⡿⣮⢗⡽⡊⣏⠮⣀⠃⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡱⣢⢸⣧⡃⡏⢻⢇⣳⠮⢝⣃⢿⣿⣿⣏⡻⣿⣿⣿⣇⢹⣷⣻⣟⢿⣷⢎⡒⡮⡪⣡⢊⢆⢠⠂⠄⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣦⢹⣧⣩⠨⠙⠛⠅⡪⣛⠾⣷⣽⣿⢫⣿⢿⣿⣿⢢⡻⡿⣽⡫⡳⣻⣿⣳⢆⡖⢌⡬⠀⠀⡈⢈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡽⣿⣿⣿⣿⣿⣼⣿⣾⣾⣽⣧⣤⡀⠙⠇⣿⢹⣿⣾⣿⣿⣿⣏⣼⣷⣯⣟⣿⢮⡽⣿⡧⡻⢜⠢⢁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣿⣿⣿⣿⣿⣿⣿⣿⣽⡻⢼⡿⣶⡶⠦⡨⣘⠟⣾⠸⣿⣿⢻⡢⡻⡿⣾⣿⣷⣏⡪⡹⢧⡳⢌⠣⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢬⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣧⠂⣿⢭⣿⣿⠺⣿⡄⠹⠝⡟⣾⣿⣿⣿⣿⣕⣤⣉⢅⠃⠀⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⡛⣠⢟⣹⢣⣧⠻⡙⠴⡏⣭⡳⢮⡝⡻⠿⠿⣾⣿⡍⠂⢁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣾⡻⠱⣣⡘⣯⣿⣿⣷⠌⢳⣻⣝⣇⢧⣿⣿⣶⡷⡧⢌⡡⠃⠤⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡧⣿⣿⣷⣿⣿⣿⣿⠈⣼⡿⢁⡝⣾⣿⣭⣥⣞⣮⠕⡐⢂⠤⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣼⢱⣹⣿⢹⣿⣿⣿⣿⢶⢰⡻⣏⡾⢽⣿⣟⡯⡚⠁⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⣟⣛⣿⢿⠿⣿⣿⣿⢿⡿⠺⡵⣿⣿⣿⣼⣿⣿⡿⢸⠽⢠⣽⣫⠯⠭⠕⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢿⣿⡿⠀⣏⣜⣿⣷⠱⣾⣿⢿⣿⣿⡷⣿⣿⣇⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡇⠀⣿⡻⡝⣿⡀⢻⣿⢸⣿⣿⣧⣿⣿⠏⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⡇⠀⡏⣿⣯⢽⡇⣿⣿⠞⣿⢿⣿⣾⡟⠲⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠇⠀⠁⢧⣿⠺⠇⠯⠛⠊⡏⣾⣿⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣯⡻⣀⡤⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣤⣄⢤⡖⣖⣾⠣⡿⠡⣟⣼⣅⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣳⣾⣧⣕⠿⣀⣵⣿⣷⣾⣿⣯⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣵⢿⣯⣷⣿⣿⣿⣷⣯⣿⣿⣿⡿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡿⣟⣷⣾⡓⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣽⣿⡿⣗⡹⣞⢷⡯⣿⣭⣿⢿⡽⣯⣷⢿⣻⡟⣾⡝⡶⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⢫⡟⡼⣱⢫⡞⣵⣛⠾⡽⣯⢟⡷⢯⣛⢧⡻⣵⣻⠖⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡌⢧⡙⢲⢡⡓⣎⢳⢎⡟⡵⣫⢞⡹⢧⡛⢮⡕⢮⡙⠿⡄⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡘⠤⡉⢆⠣⡜⡌⠗⢪⡜⣱⠣⢏⡜⣣⠝⣢⠙⢢⡙⠜⡙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠃⡘⠄⢣⠐⡉⢆⠣⠜⢤⠋⢦⠱⡘⠜⡠⢉⠆⡘⠤⠉⢆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠐⡈⢂⠱⠈⢆⡑⢊⠤⡉⠆⠱⣈⠆⠁⢂⠌⡈⠄⡁⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⡀⠂⠡⠂⡐⢈⠐⠠⠉⠄⡁⠂⠐⡀⠂⠐⡀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀