⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠉⠙⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⡀⢺⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⡡⢈⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⡀⢂⠐⢼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢐⣴⠋⡄⣹⣿⣧⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⢀⠐⡄⢂⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠟⣥⢛⣴⣯⣿⣿⣦⣶⣶⣶⣶⣶⣶⣿⣿⣷⣿⣆⡱⢈⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠐⣤⣾⣻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣵⣈⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢻⣿⡿⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠛⣿⡟⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠱⢼⣿⣿⣾⢁⣻⣿⣿⣿⣿⣿⡟⡠⢿⣿⣿⡇⠌⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀ ⢸⣿⣿⣿⠀⠘⣿⣿⠣⠐⢸⣿⣿⣿⣿⣿⡇⠀⠸⣿⣿⠃⠀⣼⣿⣿⣿⠀ ⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣶⣶⣿⣶⣦⣄⡀⣀⣈⣸⣿⣿⣿⣆⠀⠙⠁⠀⣠⣿⣿⣿⣿⣿⣿⣿⣄⠀⠈⠋⠀⣠⣿⣿⣿⣿⣁⣀⠀⣠⣶⣷⣷⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣠⢼⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠘⢿⣿⣿⣿⡿⣿⣿⣿⢹⡳⣎⠿⡼⣭⢏⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡻⣏⠿⣜⢮⡽⡿⣿⣿⣿⡿⣿⣿⣿⣉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⡠⢾⡹⣭⣛⢶⣹⢳⣎⢷⡹⣎⢿⡵⣫⢞⡵⢧⡻⣜⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢻⡜⣧⢻⣜⡻⣼⢣⡟⣭⠿⠝⠉⣰⠤⣄⡬⡘⠵⣄⠀⠀⠀⠀
⠀⢠⢾⡹⢧⣛⢶⡹⣎⢷⣫⠞⣧⢻⡜⠗⢋⣡⡻⣼⢳⡝⣮⠷⠟⠘⢟⣿⣿⣿⡿⣟⣷⡻⣭⢳⡝⣮⢗⣮⢳⣭⢳⡝⣮⠟⠀⣞⢮⣟⡼⣹⠽⡌⢧⣳⠀⠀⠀
⢀⡟⣮⣝⡣⣟⢮⡳⣝⢮⣳⢻⠘⠁⡀⢠⣏⢷⡹⣎⢷⣽⡛⠀⣸⢳⠘⣎⢿⡹⢷⣹⢎⡷⣭⢳⡝⣮⢻⣜⡳⣮⡻⣜⠧⠿⠀⢯⡀⠀⠯⠳⢟⢴⢧⣏⠿⡀
⢸⡹⢶⣭⢳⢮⣳⡝⣾⢳⡻⠁⠀⣼⠁⠷⣎⡷⣹⢞⡿⡼⠀⠀⣏⠿⢠⢞⣧⣛⢧⣛⢮⡳⣭⢳⡻⣜⡳⢮⡽⣹⡟⣇⠮⢡⡄⣯⡟⣵⢲⡜⣯⣛⠾⣼⢻⠀
⢰⢏⡷⣎⡟⣮⢳⠽⣟⣻⡇⠀⠸⡍⠀⡿⡼⠷⣹⣎⠷⡇⠀⢸⢃⠂⡬⡱⣱⢯⡞⡽⣎⠷⣭⢳⡝⣧⣛⢧⡛⠁⢤⡀⢹⣵⡇⡷⣝⢮⡳⣝⠶⣭⢻⡜⣯⠀
⠘⡞⣵⢫⣞⢵⢿⢿⡹⢦⡳⡀⠀⠃⠀⢹⢳⠿⣰⢺⡽⡇⠀⠈⢁⣚⣚⣵⣼⠣⠻⢵⢫⠛⣜⣧⢻⠁⢩⣛⡅⠀⢛⣡⣚⣞⢀⢟⡼⣣⢟⣼⢻⡜⣧⢻⡼⠃⠀
⠀⢹⣣⢟⡾⣿⡹⢮⣝⡳⣽⢺⡢⠀⠀⢰⢲⡝⣮⢳⣭⡇⠀⡶⣝⢮⣳⠳⢆⠸⠯⠄⢁⣆⠘⣎⢿⠀⣀⡥⢤⠀⠛⣶⠹⢂⡼⣫⠷⣽⡾⣏⢷⡹⣎⢷⡋⠀⠀
⠀⠀⢫⣞⡳⣧⣛⡷⠎⠙⠮⢷⠿⠀⠀⢨⢷⡹⣎⢷⡾⡅⠀⢿⡹⣎⢷⡻⠘⢧⡴⠇⠸⣜⡦⠘⠇⢠⣏⡞⣧⣓⢦⣴⣚⢧⡻⣵⡿⢯⠕⢛⣮⢳⡝⡞⠀⠀⠀
⠀⠀⠀⠱⣽⢲⠃⢠⡝⠉⢁⣯⠇⠀⢀⡟⣮⢳⡽⣯⢻⡽⡄⠰⣛⢮⡳⡅⣀⠈⠈⣠⡟⣵⢫⡗⣼⡳⠮⣽⡲⠩⡷⣼⠉⣾⢿⡽⣹⢳⢤⣚⢮⣳⠏⠀⠀⠀⠀
⠀⠀⠀⠀⠘⣭⣄⠀⠘⠒⠙⠀⠀⣠⣾⣹⣾⣯⣻⣜⣧⣻⣜⣦⡉⣓⣋⣴⣫⣽⣹⣮⡝⣮⣳⣏⣷⡄⣷⠐⢁⠀⢳⣽⠀⢙⣧⡟⣵⣏⣾⣹⣮⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠹⣷⣶⣠⣴⣶⣿⣿⣾⣿⡿⣯⢿⡽⣯⢿⣽⢯⡿⣿⣟⣿⢟⠷⢿⡾⣿⠉⢹⡾⡇⢹⣶⠀⣿⡄⠈⠛⣠⠘⠟⠆⣼⣿⣾⠇⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠙⢧⡟⣼⢻⡼⣟⠷⣭⢳⡝⣮⢳⡝⣮⢳⢮⣳⣧⠻⠼⣺⢆⠀⢳⣭⢀⠈⣳⢳⡈⠗⠀⣿⢷⣒⢶⡹⣖⢤⣒⢧⡻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣏⠷⣭⢻⡜⣧⢻⡜⣧⠻⠘⠉⢁⣀⡠⣤⡰⣞⣻⡄⠀⠂⡼⡀⠉⣩⣔⡶⣮⡝⣮⡝⣮⢳⡝⣮⡽⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣜⢧⡻⣜⣧⠋⢀⡤⣖⢯⡳⢧⣛⣖⣝⡷⣫⢞⡴⡺⢉⡄⢸⣷⡻⣝⢶⡹⢶⡹⣩⢖⠪⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡽⢮⡅⢰⣏⢷⡹⢮⡝⢧⢯⡿⣙⢾⡱⣏⡞⡠⢯⡅⣘⢶⡹⣎⢷⣫⢗⣧⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡤⣭⢷⢫⣟⢻⢿⡹⣞⡽⣲⡝⣞⡀⢽⡯⢡⡽⣎⢷⡹⣎⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢎⡷⣜⢯⡞⣵⢫⡞⣵⣛⣶⡝⢶⢦⡳⣝⢮⡳⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⢧⡻⣜⢧⣛⣶⢟⣧⣛⢧⣏⢷⠝⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⠿⣏⡟⣞⢶⡹⠞⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀