⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣴⣯⣥⣶⣾⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣠⣴⣾⣿⣿⣿⣽⣿⢿⣿⣦⠹⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⣾⣟⣭⣥⣤⣈⢙⣎⠙⢫⣿⣿⡇⢻⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣟⣿⣿⣿⣿⡿⠋⣿⣿⣶⣾⣿⣿⣼⣿⣼⢩⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣼⣾⣿⣿⢻⡏⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣷⠟⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⣿⠨⣿⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢹⣇⣿⣿⡜⡇⠀⣠⣤⠄⠀⠁⠀⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⣿⣿⣆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠸⣿⣿⡿⣿⣻⣤⣿⣿⠀⠀⢀⣤⣌⠉⠁⠀⢠⣶⣤⣀⡀⠀⠀⠀⢸⣿⣿⣿⣟⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢻⣿⢿⡘⢷⣽⣏⢿⣿⣇⢸⣿⣉⣻⣷⣤⣼⣿⣴⣿⣷⣾⣶⣤⣀⡹⣿⣿⣿⡿⣦⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠻⣧⣙⢦⡙⢾⣷⣽⣿⣷⣽⠿⣟⣛⣿⣯⣭⣭⣽⣿⣿⣻⡿⠿⣿⣾⣷⣝⢿⣿⢿⡱⣄⠀⠀⢀⣤⣴⡶⠷⢶⣾⡿⠿⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠙⠻⣷⣬⣳⣽⡿⠛⣥⣾⣿⣿⣿⣿⣳⡾⠽⠶⠶⣿⣿⣽⣿⣶⣭⣛⠿⣷⣝⣶⣵⣌⣢⣾⣿⣟⠉⢀⡴⠋⠀⠀⣠⣽⠿⠿⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣀⣀⣩⡿⠋⢀⣾⣿⣭⣛⠿⢿⣭⣷⣲⢤⣀⠀⣿⣄⠀⠉⠛⠿⣿⣿⣦⣝⠿⣿⣿⣿⣿⣿⢋⢀⣾⣀⣠⣾⣿⣯⣴⣤⢄⠚⠛⢿⣿⣄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠙⠻⡿⡕⢀⣯⣯⣽⠛⠿⢿⣷⣾⣭⣟⡿⢾⣽⣿⡷⠿⠶⢤⡀⠀⠙⠻⣮⣷⣝⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠙⠛⢿⣿⣶⣦⣍⠻⣆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⠟⠀⠀⣀⣠⣭⣿⣿⣛⣷⢾⣽⣻⢦⣐⢢⣍⡳⢄⠀⠈⠻⣿⣿⣿⣿⣿⣿⡟⠃⠉⠁⠀⠀⠀⠀⠀⣀⣙⣿⣿⣿⣦⢹⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠿⢿⣿⡿⠃⠀⣀⣠⣿⣿⠿⠿⠿⠾⢿⣷⣮⣟⡿⣾⣿⢮⣟⠦⣙⢦⡀⠙⣿⣿⣿⣿⣟⣷⣦⠀⠀⠀⠀⠀⠀⠺⠿⣿⣿⣧⣿⣿⡇⣧⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠿⠿⠿⠶⠷⠿⣷⣯⣿⣿⣷⣝⣷⣌⠻⣮⠳⡑⣄⠘⣿⣿⠙⣿⣿⣇⠀⠀⣠⣾⡿⢷⣦⡀⠘⣿⣿⣾⣿⡇⣹⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠋⠀⣠⣿⠧⣄⣴⣦⣀⠀⠈⠻⡘⢿⣿⣯⡻⣷⣌⢷⣜⢞⢆⢹⣿⣇⢸⣇⣿⠀⢾⣟⠻⢿⣿⣿⣿⡄⣿⣿⣿⢿⡇⡟⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠹⣿⣶⣦⣭⣿⣒⣲⡭⠙⣆⠀⠈⠻⣿⣮⢿⣷⡹⣆⠉⣆⣿⢹⠀⣿⢿⡆⠀⠀⠀⣸⣿⣿⣿⣇⣿⣿⣿⣻⣧⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⢄⣰⣦⡤⠴⢚⣉⣭⣤⡤⠖⣂⡉⠙⢻⣷⣶⠆⠈⠻⣷⣮⢷⡹⣆⠘⣾⠘⡇⣿⢸⡇⠀⠀⣠⣿⣿⣿⣿⣿⢿⣿⣯⠙⡜⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⢿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠭⢭⣙⡺⠽⣶⡄⠀⠘⢯⣻⡷⣻⡄⣼⡆⣷⠁⣼⡇⠀⠀⢸⣿⣿⣿⡿⣣⣿⣿⡟⡼⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⠏⣴⣿⣿⡿⣫⣿⣿⠿⠛⠋⠉⠙⢒⣭⣓⢮⡹⣦⡀⠈⢯⣿⡘⢿⣻⣿⡟⢸⢻⡇⠀⢀⣿⣿⡿⣿⣾⣿⣿⣟⣝⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣾⣿⣿⣿⣿⣿⡿⠁⠀⠀⢠⣾⡿⠿⣟⣛⣷⡹⣌⢿⣆⠈⣿⣇⠘⣿⣿⡇⠀⢸⠁⣠⣾⣿⣿⣄⠈⢻⡻⣿⣿⡷⠫⠭⢭⣹⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣿⣿⣿⣿⣿⢿⣿⣿⠆⠀⠀⠸⣿⠓⠻⢿⣿⣿⣿⡜⣿⣿⡎⣹⣿⡀⣿⡇⠀⣆⣼⣾⣿⢿⠿⠿⣿⠀⣾⣻⠽⠾⣶⣴⣦⣤⣿⣿⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣿⣿⣿⣿⣧⣤⣄⡀⢀⣀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣸⢻⣧⠸⣿⡇⣿⣳⣾⣿⠿⢛⣵⡿⡷⢰⢏⣴⡧⠞⠛⠛⠻⣿⡿⣿⣿⡿⠃⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣮⡟⢋⣿⣷⣼⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⢀⢹⣼⣧⣾⣿⢀⣿⣿⡿⠟⣋⣥⣾⣿⠷⣿⣣⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⣻⡛⢛⣫⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠸⢿⣿⣿⣽⣿⣿⣿⡽⠛⠀⢀⣽⠟⣁⣴⣾⣿⠟⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠋⠁⠀⣠⣴⣿⣒⡯⠭⠥⠤⣭⣉⠓⢮⣷⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢟⠝⠋⢀⣠⣔⣿⡻⠭⠓⠒⢒⡒⠦⣍⠉⣺⣝⢦⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⠕⢁⣴⢞⣫⣿⣿⣷⡖⠋⠛⣿⡛⢿⣿⣼⣷⣹⣿⡎⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⡕⢁⣼⣿⣷⣿⠟⣩⣾⣾⣷⣿⣿⣿⠃⣸⣿⣿⣿⣿⣿⣧⣹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢏⡴⣻⠽⠒⠚⢯⣿⣿⣿⣿⣷⠁⠀⠀⠀⠈⠛⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣣⠪⠊⢀⣀⣤⣶⣿⣿⠿⠟⡻⠋⠀⠀⢀⡴⣶⡄⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠁⢀⡼⢫⣿⣿⣿⠿⢁⣖⡔⠁⠀⢀⣼⣽⣼⣿⠇⠀⠀⠀⣀⠀⠀⢀⡤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠃⠀⢀⣴⣿⣿⣿⡗⡡⠫⢊⣤⣶⡾⠟⠛⠛⠛⠓⠒⠒⠒⢛⣿⣤⣀⣈⣷⡌⢷⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢠⣾⣿⡿⣿⣿⠟⣵⣿⣿⣿⢿⠟⢺⣶⣶⠶⢶⣖⣂⡤⢭⣙⣛⣶⣛⣿⢧⡾⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠀⢀⣿⣿⡟⣼⠋⢂⣾⣿⣿⣯⣟⣬⣶⣾⣅⣐⡿⣶⣍⡻⠮⢝⡲⢔⢾⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⢹⡇⣠⣿⣻⣽⢯⣛⣁⣉⣉⣉⣓⡢⣍⠓⣯⣿⣦⠀⠙⣄⠙⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⢸⡿⡇⣽⣷⣿⣿⣿⣟⣯⠷⠼⣿⡏⠙⢿⣿⣿⣿⣿⣿⣿⣧⠀⢺⡆⢸⣽⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠸⣏⣿⢿⢻⣯⣾⣿⣿⢦⣤⡀⠀⠀⠀⢾⣿⣿⡿⣹⣳⣿⣿⡇⣼⣧⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⢻⡹⣯⣷⣿⣤⣌⢿⣆⢿⣧⠀⠀⠀⠀⠀⠙⠛⢁⣽⣷⡞⢐⡿⣿⢰⠇⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡟⣿⣿⣷⣝⢿⣿⣿⣿⣟⣼⣿⣾⠇⠀⠀⠀⠀⠀⠀⠀⣿⡿⢏⡤⣞⣴⣷⡟⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣸⣷⣿⣻⣿⣿⣷⡮⠯⢿⣚⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⢾⣿⣖⣽⣾⣿⡭⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣿⣟⣿⣿⡏⡟⣉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀