⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢚⢫⣽⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠳⣎⢥⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⣹⢾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣆⣆⣶⣶⣶⡶⣶⢶⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢛⡏⠀⠀⠀⣠⡖⢏⠺⣿⣿⣷⣦⣤⣤⣤⠶⠖⠊⠀⠀⣀⣤⣤⡴⣖⡾⠟⠛⠉⠉⠉⠙⣏⡞⣼⣼⣷⣿⡿⠿⠛⠉⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣉⠖⡩⢝⡦⠴⣞⠱⡸⢌⣣⣽⣿⣿⠿⠛⠉⣀⣠⣤⣶⡾⠟⠛⠋⠉⠙⣾⣇⠀⠀⠀⢀⣀⣤⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣴⢋⠖⡱⢊⡕⢪⢔⠫⡔⢫⣱⣾⠿⠛⢉⣠⡴⣶⠿⠟⠋⠁⠀⠀⠀⠀⠀⢀⣠⡿⣹⢳⣻⠻⡽⣹⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣰⠟⣤⢋⡎⢵⡩⢜⢣⢎⣣⣽⠟⢋⣡⡴⣞⣯⠟⠋⠁⠀⠀⠀⠀⣀⣤⢶⢶⣛⢯⣓⢧⣓⡏⣶⢫⣽⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠼⢯⣞⡤⢳⡸⢅⡞⣩⣖⠷⠋⠳⢶⣏⢧⣽⠋⠀⠀⠀⢀⣠⡴⣞⡻⡝⣮⢝⡲⣭⢞⡼⣣⢧⣛⢶⣫⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣵⢊⠷⡸⢥⣧⠀⠀⠀⠀⠉⠳⣾⣦⣠⣤⡞⣯⠳⣝⡲⣭⢳⣣⢏⡷⢳⣎⢷⡹⢮⡝⣮⢳⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣞⣱⡧⠾⢷⣄⡀⠀⠀⠀⠈⠳⣏⡶⣹⢖⡻⣜⡳⣭⢳⡭⣞⡽⣳⢎⡷⣹⢧⣻⠼⣏⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣩⣥⣶⣿⣭⣶⣿⠿⠛⠛⠀⠀⠙⣷⡹⣎⠷⣭⢳⣭⢳⡝⣮⢳⣝⢾⡹⢧⣟⢮⣟⣭⢿⣮⣝⠻⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⡴⣞⡟⣻⣛⢿⡻⣿⣿⣦⣤⣄⡀⠀⠀⠀⢹⡷⣭⣻⢼⡳⣞⡽⣺⡭⣟⢮⣯⢽⣻⡼⣏⡾⣞⡽⣞⡿⣿⣦⣈⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣰⠿⣭⣷⣭⡞⣵⡹⣎⠷⣭⣛⣿⣿⣭⣉⠀⠀⠀⠀⢿⣧⣛⣮⢷⣫⣞⠷⣽⢞⣯⠾⣽⢶⣻⣭⢷⡯⣟⣳⣟⡿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣹⡿⣯⣴⢿⡹⢶⡹⣎⡟⣶⡹⣖⢿⣿⡿⠛⠦⠀⠀⢸⡷⣏⣾⡳⣽⡺⣟⡽⣾⣭⢿⡽⣞⣷⣫⣟⡾⣽⣳⣯⣿⣿⡿⠿⠛⠛⠲⠀⠀⠀⠀
⣾⣤⡶⣿⡹⣞⡵⣫⣞⣽⣣⣟⣵⣻⠷⣟⡾⣭⢿⣿⣦⡀⠀⠀⢸⡿⣽⢶⣻⣳⢿⣹⡽⣶⢯⣟⡾⣿⡷⣟⣾⣽⣳⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠻⣷⣝⡶⠟⣳⡿⠋⠉⠀⠈⠉⠛⡿⣸⢻⣽⡞⣯⣿⡿⣷⠀⠀⣿⢿⡽⣞⡷⣯⣟⣷⣻⣽⣻⢾⣽⣻⢿⣜⢿⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠘⠻⣤⣾⡟⠀⠀⠀⠀⠀⠀⣀⣴⣿⢸⣿⣽⡳⣿⣷⡌⠁⣰⣿⢯⡿⣽⣻⣵⣻⣞⡷⣯⣟⡿⣾⣽⣻⢿⣦⠙⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀⣰⣾⣿⣿⢇⣿⡿⣶⢿⡿⠉⢷⢀⣿⣏⣿⣹⢷⡿⣾⢷⣏⣿⢿⣾⣿⣷⣿⣿⣿⣿⣷⠾⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡿⣣⣿⢿⡽⣯⢿⡇⠀⣠⣿⣟⣾⣳⣯⢿⣽⢯⣟⣯⣿⡟⣿⣿⣿⣿⠿⠿⠿⣿⣷⣅⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⡿⣫⣾⢿⡽⣯⢿⡽⣿⣠⣼⣿⣻⢾⣳⣯⣿⣟⣾⣟⣯⣿⣽⣷⢻⡿⠋⠀⠀⠀⠀⠀⠉⠙⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⣿⣿⣿⣿⡿⣫⣾⡿⣯⢿⡽⣯⢿⣽⣿⣿⣻⢷⣻⣟⣯⣿⢻⣿⣷⣻⣟⣾⣿⣿⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⣿⣿⣿⣿⢏⣾⣿⢯⣿⡽⣯⣿⣻⣟⣾⣽⣾⣻⣟⣯⣿⣽⡟⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣿⣿⣏⣾⣿⡽⣟⣾⢿⣽⡾⣷⡿⣻⣿⣿⣿⣿⣿⣿⡟⠁⣿⠟⠋⠉⠉⠙⠻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠘⣿⣿⣿⢸⣿⣯⢿⡿⣽⡿⣯⣿⢿⣇⠉⠀⠀⠀⠀⠉⠿⠁⠀⠃⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠹⣿⣿⢼⣿⣟⣿⣟⣿⣽⣿⣽⣿⣿⣿⣶⣶⣴⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⢿⣏⣿⣿⣻⣾⣟⣿⣾⢿⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠙⠪⠿⣿⣯⣿⣟⣿⣿⣟⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠯⠿⠿⠿⠿⠿⠷⠿⠿⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣷⣶⣶⡄⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⡿⢀⣿⣿⣿⣿⡿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠁⣼⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣶⣤⣤⣤⣶⣿⣿⣿⣿⣿⠟⠁⠸⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠈⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡄⢹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀