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