⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⢿⡿⢿⡷⣶⣴⣾⣿⣿⡻⣝⡧⣟⢶⣹⣖⣳⢮⡳⢮⣳⢽⣫⣟⡻⢿⣳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⣠⣤⣶⣶⢿⡟⣯⢳⣏⡾⣹⢧⡻⣵⢫⣟⡿⣖⢯⣳⢽⡺⣝⡶⣹⣎⢷⡻⣝⣮⣛⢿⣿⣯⣍⠉⠙⠛⠛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣟⣯⢻⡼⣎⡷⡽⣭⢷⡺⣵⣛⢮⠷⣭⢷⡺⣵⢫⡟⣼⢧⡻⣵⢫⣗⠾⣭⣳⢏⣶⠿⠻⠾⢽⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣭⣿⣿⣽⣳⡽⣎⡷⣽⢲⣏⢯⣛⡮⢷⣹⢞⡽⣚⣧⢯⣳⡭⢷⣚⣯⢳⣝⢾⣿⣷⣤⣀⠀⠈⠉⠻⣯⣿⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠁⢀⣤⣿⣷⡝⣾⡱⣯⢞⣭⢷⡹⣏⡾⣹⢞⡽⣎⢷⣣⠿⣭⡳⣞⡽⣮⣿⣥⡀⠈⠙⠻⣷⣤⣀⠈⢿⡽⣻⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⢀⣠⣶⣿⠿⠋⠁⣨⣿⢧⣻⡜⣯⢞⣧⢻⡵⣫⢗⡯⣞⡽⣎⢷⣫⢷⡹⢧⢿⣇⠈⢻⡿⣷⣤⣀⣸⣟⢿⣻⢾⣿⢭⣻⢽⣳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣏⡷⣾⡟⣯⢽⣇⣀⣴⡾⣿⡃⢀⣳⡽⣎⠿⣜⣧⢻⡵⣻⢼⣣⢟⣼⡳⣝⣮⢻⣝⡮⣟⢿⡻⣝⣧⢻⡽⣫⢟⣮⢳⣏⠾⣭⡳⣏⡶⣏⡿⣷⣦⣤⣤⠄⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⣼⡿⣝⡾⣵⢳⡽⣺⡝⣾⢻⡭⣗⡳⣏⡟⡶⣽⣚⢯⣳⢞⡽⣺⢵⣫⡞⣽⡲⣏⢷⡺⣝⢮⣳⢽⡺⣝⡳⣞⢧⡻⣵⢫⣞⡳⣞⢯⣳⡝⣧⢿⣿⡚⠓⠛⠋⠁⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⣼⡿⣱⡻⣼⢣⡿⣜⡧⣟⢮⢷⣹⢞⡽⣺⣝⡳⢧⢯⣳⢽⡺⣝⢧⡻⢶⣹⢧⣻⢼⡳⢯⣝⡾⣱⣏⢷⣫⢷⡹⣞⡽⣺⡝⣮⢽⣚⡧⣗⣻⢼⣣⡟⣿⣦⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢰⣿⣿⡷⣏⢷⡻⣜⡧⣟⣼⢫⡞⣧⢯⡳⣗⢾⡹⣏⢷⡹⣎⢷⣫⢞⡽⣫⢞⣧⣛⣮⢽⡳⢮⣳⢏⣞⢧⣛⣮⢽⡺⣵⢳⡽⣣⢯⣳⢽⡺⣵⢫⣶⢻⣜⣻⢷⡀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣾⠟⢸⡿⣼⢫⢷⡹⣞⡵⣞⣭⢻⡜⣧⢟⡼⣣⡟⣼⢫⠷⣭⡳⣝⢮⣳⢭⣛⢶⡹⣎⢷⡹⣏⢾⡹⣎⢯⣳⢮⣳⢻⡜⣯⢞⣭⢷⡹⣮⠷⣭⢷⡺⣝⣮⢳⣻⣷⡀⠀⠀⠀⠀ ⠀⠀⠀⠀⢸⡏⢠⣿⢻⡼⣛⣮⠽⣎⢷⡹⣎⠿⣜⢧⢯⡳⢧⡻⣜⢯⣛⢶⡹⣎⢷⡹⢮⡝⣮⢳⡝⣮⢳⣭⢳⢯⡝⣾⡱⣏⡞⣧⢻⣜⢯⡞⣧⢻⣜⣻⢼⣣⢟⡵⣞⣽⡶⣿⣇⠀⠀⠀⠀ ⠀⠀⠀⠀⠸⣤⣿⡳⣏⢷⣫⢞⡽⣹⢎⡷⣭⢻⡜⡯⣞⡝⢧⢻⡜⢧⣫⠞⡵⣋⡞⣭⠳⣝⢎⡷⡹⢎⡷⢎⡟⣮⢝⡶⣹⢮⡝⣮⢳⡝⣮⡝⣞⢧⡻⣜⢧⣛⣮⠽⣎⣿⡿⣿⣿⡀⠀⠀⠀ ⠀⠀⠀⣠⣾⣟⣧⠿⣜⣧⣛⢮⡳⣝⢮⡳⣭⠳⣭⣓⢮⡹⡹⣮⡽⣎⠵⣋⢞⡱⢮⡱⣛⡜⣎⠶⡹⣍⢞⡭⣚⠵⣎⢳⠭⡶⣹⢞⣧⣛⢶⡹⣎⢷⡹⣝⢮⡳⣎⢿⡹⣞⣷⡈⢿⡇⠀⠀⠀ ⠀⠐⠾⠛⠛⢩⣿⢏⡷⢮⡝⣮⢳⡝⡮⢵⢣⡟⡴⣩⢎⡵⣽⢋⡿⡜⢮⡱⢎⠵⣣⠳⡜⢲⣉⢞⡱⢎⡺⠴⣩⠞⣬⢋⡞⡱⢧⢺⣿⣜⢮⢳⡹⣎⢷⡹⣎⢷⣿⢮⡝⣮⢟⣷⡘⠃⠀⠀⠀ ⠀⠀⠀⠀⢠⣾⡟⣾⡟⣧⢻⣜⡳⢮⡝⢮⣽⢿⡔⣣⣞⣾⣟⣿⣇⣙⣢⣙⣾⣷⣤⡓⡌⢧⠸⣌⡱⣊⣵⣷⠷⢾⣶⣷⣾⣷⣾⠿⠿⢿⢾⣧⣷⣩⣞⡱⣏⢾⡹⣿⣾⣧⣛⣞⠿⣷⣤⣤⠄ ⠀⠀⠀⢠⣿⣧⣿⣿⠹⡎⢷⢪⣵⢳⣾⡿⣿⣾⣿⣿⣯⠽⠯⣭⣿⢿⣿⠿⠿⢭⡏⠛⣾⡔⢫⠔⣾⠿⣷⡞⠻⠛⠭⠋⠝⢩⠋⠟⣿⣯⢻⣿⣶⢷⣭⡟⠻⣾⠱⡏⢿⣿⠿⠿⠿⠟⠋⠁⠀ ⠀⠀⣠⣿⣿⣿⣿⢯⣳⢻⡜⣮⡟⡟⣭⠷⣿⠉⠹⣯⡏⠀⣰⡿⠋⡼⠋⠀⠀⠈⢻⣧⠙⣷⢬⣼⡟⢰⣿⠙⠳⠾⣤⣽⣐⣫⣾⣶⣿⣛⣛⣿⣿⡜⣽⡗⢢⣿⣷⡝⣞⢿⡆⠀⠀⠀⠀⠀⠀ ⠤⠾⠟⠛⠋⠀⣿⣧⡛⣦⣹⣿⣷⡀⣿⡡⢿⡐⠛⠻⠿⠿⢷⣶⣦⣤⣄⣀⠀⠀⠀⢻⣀⣽⣿⡟⠁⢸⣎⢷⠀⠀⣀⣤⣭⣿⠿⠿⠛⠛⠉⢡⣿⣼⢾⠏⣿⢇⣾⣿⣎⢿⡇⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⣿⣧⢻⣴⣿⣿⣿⣧⢼⣷⣽⣧⠀⠀⠀⢀⣀⣤⣬⣭⣿⠿⠛⠁⠀⢸⡿⣿⠟⠛⣧⢻⠛⠛⠛⠿⠿⣿⣯⣤⣤⣤⣀⣀⡐⠛⠉⠀⣾⢸⣏⢞⣾⣿⣿⣺⡇⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠘⣯⣷⣿⣿⣿⡟⣿⣯⣿⡀⠹⠷⠾⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⣾⢫⡟⠀⠀⢻⡝⢧⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠀⣸⣳⡿⣸⣾⣿⠋⣿⡟⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡹⣿⣷⡈⢿⡜⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣫⠞⠁⠀⠀⠀⠛⢦⣽⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⢾⣿⣿⠿⢟⣽⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠋⠻⢦⣿⣾⣿⣷⣖⣲⣶⣶⣶⣶⣶⣿⠯⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠒⠛⠻⠛⠛⠛⠛⠛⠉⢰⣏⣤⠞⠁⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⡄ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠈⠉⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠒⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⢀⡜⠋⠀⠈⢣⠀⠀⡁⠀⠀⠰ ⠀⠀⠀⠀⠀⠀⠀⠊⠉⠓⡆⠀⠀⠀⠀⠀⠀⠀⠙⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⠀⠘⢧⡀⠀⢀⡼⠀⠀⠀⠀⠀⠁ ⠀⠀⠀⡀⠀⠀⢧⡀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣽⡮⣿⣿⣶⣦⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣴⣶⣾⣿⢹⣿⣷⣤⡀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⠟⠛⠛⠛⢻⡿⢿⣿⣍⡉⠉⠉⠉⠉⠋⠉⠉⠉⠉⠉⣰⣿⣿⠿⣿⠟⠛⠛⠛⠻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠠⠤⣤⣀⣷⠀⢸⣿⣿⣷⣶⣶⣶⣶⣖⣶⣶⣿⣿⣿⣿⠃⠀⣿⣀⣤⠤⠄⠀⢸⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢛⣷⠀⠀⠀⠀⢈⣷⡀⢰⣿⣿⣿⣿⡿⣿⢿⡿⣿⠿⣿⣿⣿⣿⡀⠀⣿⠁⠀⠀⠀⠀⣿⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣻⠇⣸⣿⠀⠉⠉⠉⢉⣿⣿⣾⣿⣿⣿⣿⣿⣼⢣⡿⣜⣿⣿⣿⣿⣿⣷⣾⣿⡉⠉⠉⠁⠀⣿⡇⠸⡟⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⣸⡏⠀⣿⣿⣆⠐⠋⠙⣻⣿⣿⣿⣿⣿⡕⢮⣿⣷⣯⣷⣿⣿⣶⢲⣿⣿⣿⣿⣿⣟⠉⠛⠓⣸⣿⡇⠀⢻⣄⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣿⡇⠀⠘⣿⡽⣿⢻⡿⣿⡿⣿⣿⣿⣿⡇⢸⣿⣯⣿⣻⣿⣻⣿⠀⣿⣿⣿⣏⢿⣟⡿⣷⢾⣟⣿⠃⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⣿⣿⣷⡀⠀⠈⠻⠷⣯⡿⢿⣹⡿⣿⣿⣿⡷⢸⣿⣿⢷⡣⣾⢿⣿⡇⢿⣿⣿⣿⡘⡟⣿⣾⠿⠟⠁⠀⢀⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⣷⣤⣤⡖⢫⠰⣹⣾⣿⡟⣿⣿⣿⣿⢸⣿⣯⠿⣷⣿⣭⣿⣧⣸⣿⣿⣿⢿⣆⡕⡮⠄⣲⣤⣴⣿⡟⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⣿⣿⣿⢽⣏⣿⣿⣿⣾⣿⣿⢾⣇⣿⣿⣿⣿⣸⣿⣿⣏⣟⣳⢮⣿⣿⣿⣿⣿⣿⢼⣿⣿⣿⣶⣿⣿⣿⢸⣿⣹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣾⣯⣿⣿⣿⣿⣿⣿⣻⣇⣿⣿⣿⣿⣾⣿⣿⢾⡜⣧⡟⣿⣿⣿⣿⣿⣿⣞⣿⢿⣿⣿⣿⣿⣿⣼⣿⣸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡿⣜⣧⢻⣽⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠺⡁⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⡤⢄⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣤⡀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠉⠲⠤⠤⠤⠤⢄⣀⣀⠀⠀⠈⢢⡱⡄⠀⠀⠀⠀⠐⢿⡿⠉⣿⣿⠗⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠓⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠄⣤⣧⢘⣤⣤⢠⣤⡄⠀⣠⣾⡿⠟⠀⠀ ⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⢀⠤⠂⣀⣀⡀⠉⠉⠐⠢⡀⠀⠀⠹⠋⣨⠛⠉⠈⠙⠁⠀⢛⣿⡁⠀⠀⠀ ⠀⢶⠆⣰⠶⠾⠁⠀⠀⢀⣸⡱⣉⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠱⣀⠀⠀⠸⠿⠁⠀⠀⠀ ⠀⠀⠙⡦⣤⣥⠥⠶⠋⠼⢁⣜⠝⠀⠀⠀⣄⡤⠊⠁⡠⠂⡀⠀⠁⠀⠈⠓⠢⢀⠀⠑⢄⠀⠀⠀⠀⠀⠀ ⠀⢠⠞⢀⡀⠀⠀⢀⠔⠀⠀⡜⡴⣠⢀⣾⠏⠀⠀⡜⠀⠐⡇⠀⠀⠀⠒⢄⠀⠀⣕⠄⢌⠱⢄⠀⠀⠀⠀ ⠀⣮⢴⠃⠀⠀⣠⠃⠀⠀⠀⡟⡃⠇⣸⡧⠀⠀⢸⠁⠀⠀⣇⠀⠀⠀⠀⠀⢳⣄⠈⢟⢎⢗⢦⡭⠥⠶⠀ ⠀⢡⠧⠀⡀⢠⠃⡀⠀⠀⠀⣷⡭⠸⣾⠃⠀⠀⢸⡦⣧⢀⠹⡀⠀⠀⠀⠀⠀⢿⣆⢸⠀⠸⡆⢧⠀⠀⠀ ⠀⢸⢣⣠⢃⣎⡴⢓⠒⠒⡖⠓⢿⡷⣼⡀⠀⠀⠈⢟⣽⣧⣢⡑⡄⠀⠀⠀⠀⠘⣟⣎⢇⠀⢹⡸⡇⠀⠀ ⢀⣜⣼⠏⢸⠜⡱⢃⡴⠎⠀⢀⡰⠙⢿⣇⠀⠀⠀⠀⠑⠈⢩⣷⡯⠦⢀⡀⠐⡄⡞⣽⠈⡆⡄⢧⡷⠀⠀ ⠛⠹⡇⡠⠊⣎⠾⢛⣅⣲⣴⣿⣃⣳⡃⡙⠂⠀⠀⡸⢰⣠⡟⡆⢙⠺⣆⠉⠒⣿⡇⠀⢀⡘⣗⢎⢇⠀⠀ ⠀⢰⢛⢻⣭⣵⣭⣭⣿⡿⢯⣯⣽⣿⣧⡹⠀⠀⢰⡁⣗⣿⣿⣷⣸⣧⡀⢦⠈⢜⡄⢀⢸⠈⡻⡟⠚⠵⠆ ⠀⠘⠙⣼⣏⡏⢩⡟⠇⠀⢹⡉⢫⡽⢱⣣⢇⡰⣣⣣⡯⠟⣿⠿⣿⡽⣿⣮⠗⢨⠺⢾⢰⠀⢐⣽⢆⠀⠀ ⠀⠀⠀⠇⢹⣻⣄⢇⠀⢶⣤⣍⣬⡥⠌⣿⢾⠿⡿⢻⣇⠈⢇⡉⣰⠃⠘⡿⡿⠷⣿⣏⡟⡰⠃⠉⠙⠀⠀ ⠀⠀⠀⠀⠀⡯⡿⡟⢆⠀⠈⠀⠀⠀⠀⠀⠁⠿⣾⡄⠊⠛⠶⣤⣤⡤⠆⡰⣗⡴⣩⡏⡴⠁⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠼⠛⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠴⣻⢯⡽⠋⢱⡁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⡀⠀⠉⢶⡶⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠈⢲⣦⣤⡤⣠⣁⣈⣄⣤⣤⠴⣶⡿⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣦⠛⠁⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣿⣯⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡭⢭⡬⣭⢭⡿⣯⣻⢵⣢⡄⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀ ⠀⠰⡏⠀⠈⠙⠀⠈⢻⣿⣽⡟⢯⡻⢼⢭⣳⢛⣉⡷⣹⢣⡟⡮⣕⡻⣦⢖⡶⣶⣤⡤⠸⠀⠈⠉⢳⠄ ⠀⠀⠘⠤⡀⠀⡠⢴⣥⡞⣧⣛⢧⢏⡯⢞⣭⣛⠶⣳⢼⡽⣺⠵⣫⠵⡫⣞⠙⢿⣥⡀⠀⠑⠒⠒⠉⠀ ⠀⠀⠀⠀⠑⠂⣡⡾⢭⡳⣭⣛⡼⣚⡵⣚⣭⢛⣦⠻⣜⣣⢏⡶⢫⣖⣳⢻⡖⣾⣩⠷⣄⣀⣠⡆⠀⠀ ⠀⠀⠀⠀⢀⣼⠷⣭⢳⣝⡲⡵⢳⡭⢞⡵⢎⡯⢖⡻⡼⢥⡻⣜⡳⢮⢵⡣⣟⡴⣣⢟⣼⣧⠟⠀⠀⠀ ⢀⢀⣀⣤⢞⡯⣛⡼⢳⣬⢳⣹⢣⡝⢾⣘⠯⣜⡫⡵⣋⢷⡹⣜⡭⣳⢣⢟⡴⣛⡬⣏⠶⣧⠀⠀⠀⠀ ⠈⠉⣻⠞⣬⢳⡹⣜⠳⡼⡧⢇⡻⣜⢣⡝⢮⢱⢣⠳⣍⠶⣙⠦⡽⣱⢫⢎⠷⡭⢞⡭⣳⢿⡀⠀⠀⠀ ⠀⣰⣷⡿⣍⡶⢱⡊⢷⡱⣯⡝⡲⣉⠖⣘⣦⣃⢻⡕⢬⢃⢿⡸⣡⠇⣏⢎⠷⣩⣟⡲⢇⡯⣽⠶⠒⠀ ⠀⠿⠁⣷⣿⢌⠣⢜⣹⢧⡏⠳⣱⣸⡜⢠⣇⣹⣶⣯⡷⠾⢾⣿⣦⡭⣖⢌⠣⣵⣿⢳⡭⡚⢭⣆⠀⠀ ⠀⠀⠀⠏⣼⣤⣿⣿⠟⠟⠛⢛⠛⢫⢷⣨⢷⠁⣠⠖⠦⠔⠲⣎⢹⣹⡞⢨⢑⡾⠉⣿⣿⠿⢶⡽⠀⠀ ⠀⠀⠀⢠⢇⣧⣾⣿⠉⠑⠊⠉⣷⠈⣾⠛⣶⡄⠹⢦⡀⢀⡴⠏⢸⣇⡇⢣⡞⣡⠀⣾⣿⠀⢠⠇⣄⡀ ⣀⣀⣐⡛⠙⡿⣿⣿⡗⠤⣠⠞⢁⣰⡟⠀⠘⠬⡦⣤⣭⣭⣤⡴⣿⣫⣷⠛⠉⡁⢜⡟⠋⠀⠈⠢⠴⠟ ⠣⣠⡤⠃⠀⠈⠛⢿⣷⢶⢋⠍⠛⢃⢀⠀⣴⡚⠒⠈⠆⠀⣠⣾⡿⢛⣭⣽⣯⣹⣦⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠘⢃⢨⠀⠀⠉⠉⢣⡈⠑⠂⠀⠀⢸⢰⣿⡏⣴⡿⣻⢼⡹⣯⡟⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢰⡧⠨⢦⡀⠀⠀⢸⣾⢿⣶⣯⢳⡝⣮⢳⣿⠀⢀⣀⡀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⢆⡀⢀⣾⣇⠀⣸⣷⣀⣠⢞⡏⠈⣿⣿⣎⣷⣹⣾⣿⣦⠎⠉⠑⠦⠤⢤⡀ ⠀⠀⠀⠀⠀⠀⢀⣼⠀⠘⣯⠽⣹⣾⢫⢳⣩⢷⡽⣬⡿⠁⠀⠸⣿⣿⣿⠿⠛⠛⠻⣧⡀⠀⠀⠀⣰⠇ ⠀⣀⣀⡠⠒⠒⢾⣧⠀⠀⠈⠛⠿⠿⠞⢰⡉⠺⠟⠋⠀⠀⠀⢰⣿⣿⡇⢀⣾⣿⣦⢹⠈⠒⠒⠉⠁⠀ ⢎⠀⠀⠀⠀⢀⣾⣿⣆⠀⠀⠀⠀⠀⡴⡿⣿⠦⣀⠀⠀⠀⢠⣾⣿⣿⣧⡀⠙⠿⠋⢀⡇⠀⠀⠀⠀⠀ ⠈⠒⠠⠤⠤⠿⠿⠿⠿⠷⠶⠤⠶⠿⠿⠷⠯⠿⠿⠭⠷⠶⠿⠿⠿⠿⠿⠟⠳⠦⠴⠿⠧⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣩⣤⣤⣬⣭⣽⣟⣮⣷⣹⢛⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀ ⠀⠀⠀⣴⠛⠀⠉⠲⠟⠉⠑⢶⣶⠶⣖⣶⠶⣖⡻⢯⡝⡶⣳⡼⠖⢳⣎⠾⣍⢯⡳⢏⡾⣹⢷⣀⣠⣤⣤⣀⣀⡀⠀⡏⠀⠙⠒⠶⣤⠀ ⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⢀⡼⢛⣻⡿⢭⡳⣭⢳⣫⣜⡳⢿⡴⣲⢏⡏⢹⠾⣣⢟⡹⢶⡹⣞⡽⣚⣶⣽⣿⠟⠋⠁⠙⣄⠀⢀⣠⠿⠀ ⠀⠀⠀⠀⠈⢳⡄⠀⢠⡖⢫⡞⣯⢳⣭⢳⡝⣶⠋⣶⢣⠛⣧⡝⢣⣯⡝⢻⡞⣽⣮⡝⢣⠛⢰⠛⣿⡄⠙⣯⣷⡄⠀⠀⠈⠉⠉⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠈⠉⢁⣴⡻⣝⣚⢮⡝⣮⠵⣎⠿⣜⢮⣛⡼⢭⣛⡴⣫⠷⣹⡕⡮⣝⣫⢗⣻⢳⢏⡽⣺⢳⡎⣟⢦⣤⣀⣤⣾⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⣠⣾⢧⣛⡴⣫⢞⡼⣱⠻⣼⣙⢮⢇⡯⣜⢧⠯⣜⢧⡻⣱⠞⣵⣋⢶⣋⢮⢏⡾⣱⣋⢷⣚⠽⣚⢶⣹⣾⠃⠀⠀⠀ ⠀⠀⠀⠀⠀⢀⣰⡿⣝⣎⢧⡽⣱⢎⡷⣍⢟⢦⣛⡬⣏⢶⡹⢮⡝⢮⣣⠷⣙⠾⣱⢎⡷⡹⣎⢯⢖⡧⣛⠶⣭⣛⣭⢻⡏⠁⠀⠀⠀⠀ ⠀⠦⣤⣶⡞⣯⠳⣝⢮⡜⣞⡲⣝⣺⢲⡝⢮⣓⠮⡵⢎⡧⡝⢶⡩⢗⣥⡛⣭⢛⣬⠳⣞⡱⣝⠮⣞⡱⢯⡹⣖⢳⣎⠯⣿⠀⠀⠀⠀⠀ ⠀⠀⠀⢨⡿⣜⡽⣜⠺⡜⣥⠳⣎⢽⣓⢞⡣⢭⢳⡙⣎⠶⡙⢦⢛⡜⢦⡙⢦⣋⢖⡻⣔⠻⣌⢟⡲⣝⡣⢷⣙⢮⡜⢯⡿⣄⠀⠀⠀⠀ ⠀⠀⢠⣟⣾⡿⡱⣎⢳⡙⢦⢛⣼⢊⣿⣌⠳⣍⠶⣉⢆⢳⣍⢆⢻⣎⡲⣉⠖⡹⣎⠶⣉⠳⡜⢮⡱⢎⡵⣛⣮⢧⢫⡳⡽⣭⠿⠶⠂⠀ ⠀⠀⣻⠟⢉⣇⣿⡏⢲⡉⢦⢩⣿⢌⡟⢧⡝⠤⢳⢌⠢⢹⡏⠳⣌⣹⣗⣮⣬⣑⣾⣷⣍⣱⣉⠖⣩⠚⡴⣩⣿⣮⢣⠳⣍⢻⣦⠀⠀⠀ ⠀⠀⠉⠀⢸⡿⢹⡃⢥⣚⣤⣿⣯⣿⣥⣤⣽⣾⢿⣆⡑⢾⣗⡿⠛⠛⠋⠉⠉⣉⣛⠾⣿⣻⡍⣿⠠⣙⢰⡿⠿⣬⣳⣯⣌⣇⢾⡇⠀⠀ ⠀⠀⠀⠀⠘⠃⣾⣿⠿⢿⣿⣏⣉⠃⢀⣠⣀⠈⢻⠛⣶⡿⢸⠀⢰⡟⠋⠙⠋⠉⢙⣷⠈⣷⢛⡇⠒⡄⣾⠇⠀⢻⣿⣟⠛⠻⢿⠃⠀⠀ ⠀⠀⠀⠀⠀⣰⢿⣸⣼⣿⣿⠉⠈⠉⠉⠀⢹⡇⠈⣿⡟⢳⣾⡄⠘⢷⣄⡀⠀⣠⡾⠃⢰⡟⣸⡄⢳⣼⠋⡸⠀⣸⣿⣿⠀⢠⠞⠢⣄⡀ ⠀⠀⠀⠀⠼⠗⠚⣿⣿⣿⣿⡷⢤⡀⢀⣴⠛⠀⣸⢿⠀⠀⠣⣑⣤⣀⣈⣙⣛⣉⣀⣤⣿⡿⣹⡦⠿⠛⠊⢀⡴⣿⠿⠇⠀⠀⢦⣀⣬⡿ ⢰⡏⠙⠎⢱⡆⠀⠘⢾⣿⣿⣷⣀⣭⠟⠢⣶⣾⠛⠁⠀⠀⣀⣤⣏⡉⢻⠉⠉⠉⢉⣭⣿⣿⡿⠷⢶⣴⠶⣭⡘⠁⠀⠀⠀⠀⠀⠉⠀⠀ ⠀⠙⠲⠖⠋⠀⠀⠀⠀⠈⣹⣿⠋⠟⠀⠊⣁⣠⠧⡤⢤⣠⠷⠤⠄⠈⠉⢇⠀⣰⣿⡿⠟⣡⣶⡾⢿⠿⣷⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⡀⠇⠀⠀⠀⠀⠈⠑⣄⠈⠉⠂⠀⠀⠀⢾⢐⣿⣾⠃⣾⣟⡳⣽⢫⡟⣽⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⣠⣎⣀⢱⣆⠀⠀⠀⠀⢸⣾⣿⣿⣸⣟⢮⣝⢮⡳⡝⣾⡿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⣿⣇⠀⠀⢻⣧⠀⠀⢀⣼⣿⠀⢻⣿⣟⡼⡻⣜⢧⣻⣿⣿⡄⡸⠛⠘⠣⣀⣀⣀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⣷⡝⡶⣤⠾⣿⡿⣄⣠⢾⣿⣛⠿⣹⢲⡏⠀⠈⢿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠈⠀⠀⢹⡆ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁⠀⠘⢿⡼⣱⣻⣿⢹⡌⢧⣾⢹⣯⣏⣷⠟⠁⠀⠀⢘⣿⣿⣿⡿⠋⠉⣉⡉⠙⢿⢦⡀⠀⠀⠀⣠⡿⠃ ⠀⠀⢀⣀⣀⡠⠒⠉⠲⣿⣿⠀⠀⠀⠀⠙⠻⠿⠧⠞⠁⣾⠈⠳⠟⠛⠁⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣾⣿⣿⣷⠸⡆⠉⠒⠋⠉⠁⠀⠀ ⠠⡞⠉⠀⠉⠀⠀⠀⢠⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡀⠀⠻⣿⣿⠟⠀⣻⠀⠀⠀⠀⠀⠀⠀ ⠀⠳⣄⠀⠀⠀⠀⣠⣿⣿⣿⣿⣦⣄⣀⠀⠀⣀⣴⣿⣟⣹⡽⣮⣛⡶⠦⣤⣀⣴⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⢀⣴⣿⡄⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠉⠉⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀