⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠤⠰⠒⣂⠢⠤⠤⢄⡀⡀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢊⠩⢀⠈⠐⠀⠀⠀⠀⠂⠐⠈⠀⠐⠈⠁⢀⠘⠤⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⢤⠚⠀⢂⠐⠀⠀⠀⠀⠐⠀⠀⡀⠄⢀⣈⡠⡤⢤⢤⡤⣈⢀⠈⠢⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⡴⣿⠈⠡⡐⠀⠀⠀⠀⠀⠠⠀⠠⠂⢀⢤⢺⡝⡼⣴⣿⣹⡚⣿⣿⣗⣻⣤⣄⠑⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣴⡻⡎⢃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⢃⢐⣫⣾⡿⣹⢣⣿⣿⢿⡥⣿⣿⣿⣱⢿⣯⢷⣄⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣾⡏⣳⠋⠄⣈⠀⠀⠀⡀⠐⠀⠐⠂⣠⠖⣲⣿⣿⣱⡏⣼⣿⣏⢾⡱⣿⣿⣿⣏⡞⣿⣷⢻⣭⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡟⡼⠛⢁⡀⠀⠀⠀⠀⠀⠀⠀⢢⢥⡼⣩⣿⣿⠧⣾⡉⣿⣿⣿⣼⢱⢿⣿⣻⣿⢿⢿⣿⡗⣻⢫⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣽⣷⠈⡀⢀⠀⠀⠀⠀⠀⢀⠁⠘⣣⢶⣽⣿⣿⣴⣏⢳⣿⣿⢏⣿⠸⣿⣿⣿⣿⣸⣽⣿⣿⠻⣏⡞⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣧⠀⡀⠂⠀⠀⠀⠀⠀⠀⣠⣜⡯⢶⣿⣿⣇⣼⠹⣼⣿⣿⣦⣿⡍⣿⣿⣿⣿⣼⣿⣿⣿⣗⣿⣹⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣷⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⡼⡷⣼⣿⣿⢨⡾⣃⣿⣿⣏⢽⣇⡎⣿⣿⣿⣿⢾⣿⣿⣿⣟⣿⡿⣷⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⡓⡎⠀⠀⠀⠀⠀⠀⠄⠀⢾⣗⡧⣿⣿⡟⢣⡷⢩⣿⣿⢻⣾⢻⢼⣿⣿⣿⣾⣸⢽⣿⣿⢿⣾⡟⡧⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡿⢱⡱⡄⠀⠀⠀⠀⠀⠀⢀⣾⣹⢸⣿⣿⣷⢩⠳⣭⡿⡟⣧⡟⡯⣾⣿⣿⣿⡏⡿⣿⣿⣿⡸⣿⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡏⢧⡓⣦⡀⠀⠀⠀⢀⠀⢩⡟⣿⣸⣿⣿⠸⣏⡓⢮⣿⠿⣴⣻⢣⣿⣿⣿⣿⣹⢷⣿⣿⡏⣷⣿⢹⢓⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡹⡤⠛⠀⡠⠀⡄⠀⠀⠀⢻⣏⡧⢿⣿⣿⢂⡧⣍⡷⠯⠳⠯⣿⣼⣷⡽⣿⣏⣟⣾⣿⣿⢻⣼⡟⡟⣦⣏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢇⡆⠠⠞⠀⢀⠧⠀⠘⠀⢀⣽⡇⣿⣿⣿⢢⣽⣧⡀⠀⡀⠀⣀⠙⣭⡻⢾⣞⣿⣿⣿⢯⣳⣿⡻⡱⣼⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣎⡷⣢⠀⠀⡼⣲⠅⠄⢀⡼⣲⣏⣿⣿⣿⢳⢼⣿⡇⡅⠠⢉⠉⣿⣝⣿⣿⣿⣿⣿⡫⢵⣶⡲⡳⣱⣟⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣺⢷⡁⠃⢰⢱⢹⡄⠸⡱⢺⣏⡷⣻⣿⣿⢹⣸⣿⣧⡙⠓⠋⣰⣿⣿⣿⣿⣿⣿⣿⠟⠉⢛⢦⣻⣟⣵⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⢎⠵⠀⢈⡾⣽⣇⠀⠱⣼⡧⡟⣽⣿⣿⢺⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢘⢣⡜⠚⠻⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣟⢮⠂⠀⢠⡇⣧⢿⣄⢀⡴⢻⣇⢿⣿⣿⣾⣧⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣐⠎⡗⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣏⢮⡹⣙⢆⢳⠼⡞⣿⣳⡽⣯⢛⠺⠿⢯⣧⣿⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⢸⡠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣎⢇⡳⢣⡜⣚⡞⡹⣜⢷⡿⣽⣞⠷⣄⡀⠙⢮⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⡿⢋⣆⣹⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡞⢬⡝⣡⢎⢧⢻⡱⣚⢎⠿⣾⣮⢯⣽⣚⠷⣤⡙⠻⣿⣿⣿⣿⣿⣿⣿⠟⠋⠘⢞⠤⢟⡀⠤⠤⢄⣀⣀⠀⡀⢠⣯⡍⣏⢷⡿⡽⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡝⢮⡜⡴⢪⡝⣆⣳⡩⣎⠳⢍⠿⣷⣕⣍⢫⣡⡛⠇⠈⠉⠛⠿⠿⢫⢴⠀⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠩⢙⡴⢧⡓⣿⡠⠗⠇⠘⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢎⡳⢆⢳⢫⡼⣼⢣⠗⣍⢏⠧⣍⢲⢭⢻⢾⣴⡹⠣⢖⣠⠄⠀⠀⠁⠺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⣷⢹⣿⢹⣄⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡼⡑⣎⡽⢪⣽⢣⣏⢼⣣⢝⣊⢭⣀⡀⠈⠘⠪⠝⡿⣮⣌⢺⡝⣾⡂⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡾⣿⣷⣯⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣳⡙⢦⡝⣽⢧⣫⠌⣿⡞⣬⡃⠀⠀⠉⠛⠛⠒⠒⠒⠱⠿⢷⣧⡑⢯⡾⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣅⣞⣓⣾⡟⣎⠶⢡⣿⢳⢥⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠹⢉⠈⠉⠤⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣞⣌⣾⡿⣜⡱⠃⣼⣿⡹⢎⡅⠀⠀⠀⡀⠀⠠⠐⠀⠀⠀⠊⠀⠐⠀⠀⠁⠀⠂⡑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣎⣾⡿⣱⢎⠅⢠⣿⡿⣱⢣⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠂⠀⠀⠀⠀⠂⠀⠄⠀⠐⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⢳⡱⠎⠀⣼⣿⣗⡣⠆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀