⠀⠀⠀⠀⠀⢀⣀⣀⡤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⢀⣀⣀⣀⠀⠀⠀ ⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠒⠲⣤⣄⣠⣤⢶⡶⢾⣻⣛⠿⣹⠭⣏⡽⢭⣫⠽⣭⢫⡽⢭⣛⡟⡶⢶⢦⣤⣤⣤⠴⠒⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠀ ⠀⠀⠀⣼⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠾⠛⣩⡵⣞⡝⡮⢞⣣⠗⣮⢝⣣⠟⡼⡜⣧⡓⢯⢖⠯⣜⢧⡳⣚⡭⢏⡾⣜⣭⢻⡟⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄ ⠀⠀⡀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠋⢁⣠⢶⣛⢧⡝⡶⣩⢗⣫⣜⠻⣜⣎⢧⡻⣱⣛⡴⢫⣏⠾⣙⢾⡸⡕⡯⣜⣯⢾⡽⣞⡿⣞⡷⣾⣽⣻⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡏ ⠀⠀⠀⣿⢆⠀⠀⠀⠀⢠⣴⡿⡟⠋⢀⡴⡺⡝⡮⡵⣎⡳⣭⠳⣎⡳⣜⡻⣜⡼⢎⡵⣓⢮⣙⠷⣸⢫⡝⡶⣭⣟⢾⡽⣞⣯⣽⣳⢯⣟⢾⣳⡞⣷⡽⣻⢿⣦⡀⠀⠀⠀⠀⠀⣰⢾⡇ ⠀⠀⠀⣿⣊⠷⣄⣀⣼⡿⣣⠟⢀⣰⣏⢾⡱⢏⡵⡳⣍⣳⡵⣛⡼⠉⠉⣞⠶⣹⢎⡳⡝⣮⢹⢞⣱⣏⣾⢳⡷⣞⣯⡽⣞⣳⢾⡽⢾⣭⣟⡾⣽⣳⣻⡽⠛⠫⠽⢛⠲⣤⡤⢞⡡⣿⡇ ⠀⠀⠀⣿⣇⢚⣴⡿⣫⡼⠁⠊⠁⢀⠀⠹⣭⡻⠜⠃⠉⠀⢀⡽⣚⡵⣺⡍⣟⣲⢫⡵⣛⣬⢷⡯⣷⢻⡼⣯⢷⣛⣮⡽⣏⣯⢷⣻⣻⡼⡾⣽⢶⡟⠓⣤⡀⠡⢀⠂⣸⠑⢝⢮⡱⣿⡅ ⠀⠀⠀⢹⣎⣾⣿⠝⠁⠀⣀⣤⣶⣻⠀⠀⠀⠀⢀⣀⣤⡖⣯⢞⣵⣹⠶⣽⢮⢷⢯⣻⣽⢫⣟⡾⣝⣯⢟⡾⣝⣻⣼⢻⡽⣞⡯⢷⣳⡟⣽⣳⡟⠀⠀⠀⠙⠶⣄⡼⠃⠀⠀⢩⣷⣿⠁ ⠀⠀⠀⢸⣷⣿⣥⡄⠂⣱⣟⡷⣫⣽⣻⢶⣟⡿⢯⣟⡾⣽⣛⣾⢳⡯⢿⣭⡟⣯⣻⡵⣞⣿⢺⡽⣏⡾⣏⡿⣽⡳⢯⣟⣳⢯⣟⡯⣷⢻⣳⢿⣧⣄⠀⠀⠀⠀⠈⠁⠀⠀⠀⢰⢏⣿⠀ ⠀⠀⠀⠈⣿⣟⣾⠃⢰⣟⡾⣽⣳⣯⣷⢿⠟⠻⠿⣾⢿⣧⣿⣞⣯⣽⣻⢮⣟⣳⡽⣽⣫⣞⣯⡽⢯⡽⣯⡽⣶⣻⢟⡾⣭⢷⣛⡾⡽⣏⡿⣞⣿⣿⣶⡦⣄⡀⠀⠀⠀⠀⢠⠏⣸⡇⠀ ⠀⠀⢀⣾⣿⣟⡎⠀⣾⢯⡽⣾⣿⣟⠕⠁⠀⠀⠀⠀⠈⠉⠚⠙⠫⠟⠻⢿⡾⢿⣿⣷⣿⣞⣷⣻⣯⢿⣼⣳⢯⡷⣯⣟⣽⣫⢯⣽⡻⣝⣷⢻⣞⣿⡿⣿⢾⣍⡛⢧⢒⢶⣿⣿⣿⠁⠀ ⠀⠀⣼⢯⣿⣿⡇⠀⣿⣯⣿⣿⡿⠁⠀⠀⠀⠀⢀⣠⣄⡤⢤⣤⣀⣀⠀⠀⠀⠈⠁⠐⠉⠉⠯⠛⠛⣟⡻⢻⠿⢿⣷⣿⣾⣿⣿⣾⣿⢿⢾⡿⠾⢿⣿⣷⣯⣟⡷⣦⣩⣾⣿⡿⣧⠀⠀ ⠀⢰⣿⣟⡾⣿⣇⠐⣿⣿⣿⣿⠁⠀⠀⠀⢠⡶⠛⠉⢀⣠⣤⣤⡄⡉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⣀⣀⣀⣀⠀⠈⠛⢦⡄⠀⠀⢘⣿⣟⣿⣻⣟⣟⠹⡷⣯⢿⡆⠀ ⠀⣿⢿⣺⡽⠀⣿⣦⣿⡿⣾⡟⠀⠀⠀⣴⠋⠀⣠⠞⠉⠁⡀⢀⠀⡉⠳⡘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣠⣶⡿⣟⣯⠿⣽⢿⣦⠀⠀⠹⣆⠀⠀⣿⣿⣳⢯⡾⣽⠀⢿⡽⣞⣿⠀ ⢰⣟⣯⢷⡏⢠⣿⣻⡽⣛⣿⡇⠀⠀⣸⠃⠀⢰⠏⠀⠌⠐⡀⠂⡐⠠⢀⠻⡌⣷⠀⠀⠀⠀⠀⠀⠀⢠⡟⣰⣟⡷⣻⡽⣞⡿⣽⠾⣽⣗⠀⠀⢹⡄⠀⢼⣿⣳⢯⡽⢯⡇⢸⡿⣽⣻⡄ ⣸⣿⣞⣻⠄⢸⣷⢯⣽⣻⣿⠀⠀⠀⡷⠀⠀⢾⠐⢈⠠⠁⡐⠠⠐⡀⢂⠈⡇⢹⡆⠀⠀⠀⠀⠀⠀⣸⠇⣿⢾⣽⡳⣿⣹⢾⡽⣻⢧⣿⠃⠀⠀⡇⠀⢸⣿⣽⣫⣟⣯⡇⢸⣿⣳⢿⣇ ⣿⣳⢯⣟⡂⢸⣯⣟⣶⣻⣿⠀⠀⠀⣧⠀⠀⢸⣠⠒⢒⠉⢉⡉⠁⠚⠰⣤⡇⢸⠃⠀⠀⠀⠀⠀⠀⢺⡅⣿⠻⠖⡟⢓⡉⢛⠛⠽⣟⣾⠁⠀⠀⡇⠄⢸⣿⣞⣧⣟⢾⠃⣸⡷⣯⣻⣯ ⣿⣿⣷⢏⡇⢸⣷⣏⣶⢿⣿⠀⠀⠀⢸⡆⠀⠀⢁⠰⠆⠰⠎⠀⠀⠀⠀⡰⠁⡾⠀⠀⠀⠀⠀⠀⠀⠸⣇⢀⡆⣆⠈⡉⠆⠏⠀⡀⠈⡿⠀⠀⢰⠇⠀⢸⣿⡾⢷⡾⣿⠀⣿⣹⢷⣹⣷ ⢹⣿⣯⣿⣯⠘⣷⡿⣾⢿⣿⠀⣀⠠⡄⢳⡄⠀⠈⠢⢄⣀⠀⠀⣀⠠⠂⠁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠑⢤⡁⠁⠄⢂⠐⣀⠶⠁⠀⢠⣞⣀⡀⢸⣿⣿⣯⣿⠇⣰⣿⡿⣟⣿⡷ ⠸⣿⣷⣻⢾⣇⢹⡿⣽⣻⣿⡇⣼⢃⡼⢃⡜⠲⠤⣀⡀⠀⠀⠀⣀⡠⠔⠊⠀⠀⠀⣄⠀⠀⠀⠀⣀⠀⠀⠈⠓⠤⣀⠉⠁⠈⠉⠀⣀⡤⡖⢭⡐⢢⡉⣽⣿⣷⣻⡞⢠⣿⣯⣟⣯⣿⠇ ⠀⠻⣷⣯⣟⣾⣦⣻⡷⣯⢿⣷⣆⠾⡑⣞⠩⣝⠢⠷⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠦⠚⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⣒⠮⣱⢋⢴⡋⣴⣿⡿⣞⡟⣰⣿⣟⡾⣽⣾⠏⠀ ⠀⠀⠙⠻⣾⣳⢯⡿⣽⢯⣟⣷⠉⠛⠾⣤⣓⡌⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆⢣⣎⣧⣾⢿⡿⣽⢿⣻⣟⡷⣯⣿⠟⠃⠀⠀ ⠀⠀⠀⠀⠈⠙⠻⠿⢯⣿⣾⣽⣷⡀⠀⠀⠉⠘⠙⠒⠖⠶⠤⢤⡤⣤⣀⣀⣀⣀⣀⣀⡀⣀⢀⢀⣀⣀⣀⣀⣀⣀⣤⠤⢤⡤⠖⠶⠒⠛⠋⠉⠁⣠⣿⣿⣽⣯⣷⠿⠛⠉⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣹⣿⣿⣿⡿⢿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⣰⣿⣻⣟⣿⡱⣫⣼⣻⣟⣭⡚⢽⣿⡿⣽⢿⡆⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢠⣿⢷⣳⣻⡇⠙⢦⡀⠀⠀⢀⡴⠋⣿⣽⣫⣟⣷⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢯⡄⢀⡴⣿⣟⡯⣷⣿⠀⠄⠂⠙⢦⡴⠋⠀⠄⢻⣷⣻⡼⣿⢦⡀⢀⡼⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠙⠻⢾⣿⡽⣞⣷⡏⠀⡐⠈⠄⡀⠠⢀⠁⢂⢸⣟⣧⣟⢿⡧⠿⠋⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣼⣧⣿⡽⣾⠃⠠⢀⠁⠂⠄⡁⠄⡈⠄⢸⣿⣞⣾⣹⣧⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⣠⣠⣏⠀⠈⠛⢿⣄⠁⠄⡈⠐⠠⢀⠂⠄⣐⣬⠟⠉⠀⠀⢻⣄⣀⣀⣤⠞⠀⠀⠀⠀⠀⠰⠉⠉⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡇⠀⠀⠀⠈⢻⢷⣄⠠⠁⠂⠄⢂⣴⡿⠃⠀⠀⠀⠀⢸⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡘⠧⡤⢤⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣓⠲⠦⠤⠤⠬⠗⠻⡷⠚⠲⠒⣿⠳⠥⠤⠤⡤⠤⠖⢺⡏⠢⣀⠀⠀⠀⠀⠀⣀⠸⠥⢧⣭⣓⣌⡿⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡿⣷⢶⣶⢿⣻⢿⡇⠀⠀⠀⣿⢿⣻⢿⡶⣶⢿⣻⢻⡇⠀⠀⠉⠐⠒⠂⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢿⣭⣟⠾⣯⡽⢯⡗⠀⠀⠸⣟⣯⣽⡳⢿⣭⢿⣹⢯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⣾⢞⣟⣳⣟⣿⠇⠀⠀⠀⣿⣾⣵⡻⣟⡾⣽⣻⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⠿⠏⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀