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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡟⣡⠛⣴⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣵⡘⡔⣋⢦⢋⠖⣱⠱⣊⠵⣊⠼⡑⢦⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠳⡌⣥⣿⣿⣿⠿⡿⠿⢿⠻⠿⠿⠿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣄⠫⡜⢎⠥⡓⢥⡚⡜⣢⠝⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⢡⢣⢟⡻⣉⠶⣈⠶⡱⡙⡆⢏⢳⣉⠞⣌⠖⣢⠍⡭⢛⠿⢿⣿⣿⣿⣿⣷⡘⢎⡣⢝⡢⢕⢣⡒⡍⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⣡⢏⡒⢎⡴⢃⠮⣑⠎⡵⢱⠪⣍⠖⣌⠳⣌⣚⣤⣫⣔⣫⣜⡲⡌⡝⡻⣿⣿⣿⣎⡱⢎⠖⣩⠖⡱⠜⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⣿⣷⡀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⡸⡔⣪⠜⡥⢪⣑⢣⢣⠚⡥⢣⡓⢬⣚⣴⣿⣿⠿⠟⠛⠛⠛⠿⠿⣿⣷⣷⣬⡙⢿⣿⣞⡌⢞⡡⢞⡡⢏⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⢬⡱⢜⡰⢍⡒⢧⡘⡆⣇⢫⠜⣡⣾⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣶⡘⠿⣯⢦⡙⢦⢱⢊⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣧ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⣉⠖⣡⠳⣌⢣⡙⢦⠱⡚⢤⠓⣮⣿⡿⠋⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣕⢪⡱⢜⡢⢧⡉⠖⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣼⣿⢻⣿⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠓⣬⡙⢆⡳⢌⠖⣩⠲⣩⠱⣍⢺⣿⡟⠀⠀⠀⣸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡧⢜⢣⡜⢦⡩⢓⡜⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⣼⣿⢏⠲⣹⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣏⠱⢦⡙⢬⢒⠭⣊⢥⠓⡥⡓⢬⣿⣿⠀⠀⠀⣼⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡜⢢⢞⡰⡱⢋⡴⢊⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣾⣿⠫⣌⠳⣉⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⡤⢛⡤⡙⣆⠫⣜⢢⢣⡹⢰⡙⢮⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⠒⣧⡘⢥⢋⡴⢋⠦⣙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⡟⡰⣋⠴⡙⣤⢻⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢏⠴⣋⠴⡓⣌⠳⣌⡒⣣⠜⣱⡘⢫⣿⡇⠀⠀⠀⢿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⢫⡔⣩⠎⡼⣐⠫⡜⣌⠖⣩⠿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⢏⡱⡜⡱⡸⢌⠳⢤⢋⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⣊⠼⣡⢚⡱⢌⡳⢤⡙⡤⢛⡤⢍⡣⢿⣿⠀⠀⠀⠀⠙⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢱⢢⡙⣤⢋⡴⢣⡙⡴⢊⡼⢡⡚⠴⣩⢛⠿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⡿⢟⡫⢵⡘⢎⡱⢜⡡⢧⣉⢳⡊⢦⢻ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡗⣡⠓⣆⠳⡸⢜⡰⢣⠜⣡⠳⣘⠦⣙⠺⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢇⣋⠦⡓⣤⠓⣌⠧⡜⢲⡉⢖⢣⡙⡜⣡⠫⣜⢢⠭⣉⠟⡹⢛⡛⣛⠻⡹⣉⢳⢢⡙⢦⠱⣆⡹⠬⡜⣆⠳⣂⢳⠢⡝⣢⢝ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢇⢣⡝⡰⣍⡱⢎⢲⣉⠮⣅⠳⡡⢞⡡⢣⠝⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢋⠦⢥⠓⣍⠦⣙⢆⠳⡸⢡⠞⡩⢆⡹⢔⢣⡓⢬⡒⣍⢲⢩⣑⢣⡕⢣⡹⠔⡭⢲⡑⢮⣡⢓⢦⡱⣋⠴⣪⠵⣩⠖⡹⣜⡡⢞ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢎⡱⢬⡑⢦⢱⢪⠱⡌⢶⣈⢳⢡⡓⠼⣑⠺⣐⠻⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠛⣆⢏⡚⡬⣙⢤⠳⣌⠎⣕⢣⣋⠼⡱⣩⢲⣩⣒⣭⣲⣵⣬⣧⣾⣼⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢚⡥⣚⠵⣢⠝⣎ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⡘⢦⡙⢆⡣⢎⡕⡚⠦⡜⣡⠖⣩⠓⣬⢃⢇⡳⡰⢍⠿⢿⣿⣶⣦⣤⣤⣤⣴⣶⣿⠿⡟⢭⡒⢭⡒⢎⠶⡑⠧⣎⠳⣌⣞⣬⣶⣼⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠶⣩⠞⣥⢛⡬ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢢⡙⢦⡙⢦⠱⢎⢲⣉⠳⢬⡑⡎⢥⣋⠴⣉⢦⠣⡕⡎⢞⡢⡜⢥⢫⠝⣫⠝⣍⠳⣌⠳⣜⢣⠞⣡⢝⣪⣵⣭⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⢉⣿⣿⡿⣽⣻⣿⣿⣿⣯⢞⡱⣛⡬⢧⡹ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢢⡙⢦⡙⢆⠯⣸⢡⢎⢣⡓⠼⣘⠥⣎⢎⡕⣪⢕⡣⢞⣡⠳⣩⠞⣬⢛⡔⣫⢜⡳⣌⠳⣌⢧⣻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡽⣟⣷⣿⣿⣿⣿⣎⡳⢵⢪⢧⡝ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣁⠹⣆⠹⡎⡾⢁⡾⣈⠷⡸⢇⡹⢎⡶⢉⡸⣆⢷⡸⢇⡎⢷⣁⠿⣰⢏⡸⢱⡎⣱⢎⣷⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣏⣷⡿⣏⣿⡾⣷⣏⣿⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣏⡿⣿⡾⣷⣿⣿⣿⣇⢿⣉⠿⣎⡹ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡜⡳⣌⠳⣍⠶⣩⠖⣥⢫⡕⣫⠜⡮⣜⢣⠳⡜⢦⣙⠮⣜⢣⢎⡳⡱⢎⡵⣳⣾⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⣰⣿⣿⣟⣾⢯⣟⣿⣳⡿⣯⣟⣾⣟⡿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢯⣿⢷⣟⡿⣿⣿⣿⣟⢮⣭⢻⣜⡳ ⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⡧⢳⡜⡳⣌⠳⣥⢛⡴⢣⠞⣥⢛⡴⣍⢮⢳⣙⠮⣜⢳⡬⢏⣎⢳⣭⣷⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⢀⣿⣿⣟⣾⢯⣟⡿⣾⣽⣻⢷⣻⡷⣯⢿⣳⡿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣸⣿⣿⢯⣿⣻⢾⣟⣿⣿⣿⣿⢳⣎⠷⡮⣽ ⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣏⠷⡸⣕⢎⡻⣔⡫⣜⢧⡛⣬⡓⡞⡜⣎⡳⢬⣛⢬⡳⣜⣣⣾⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣻⢾⣟⣯⡿⣷⢯⣟⣿⣽⣻⣽⣟⣯⢿⣳⡿⣿⣿⣦⡀⠀⠀⠀⠀⣿⣿⣯⣿⣳⢿⣯⣟⣿⣿⣿⣿⢳⢮⡻⣵⣫ ⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣭⢓⡮⣙⠶⣡⠷⣩⢖⡹⢦⡝⣺⣙⢦⡝⣧⠞⣥⢳⣮⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣽⢿⣞⣯⣿⣽⣻⣟⡾⣷⣻⢷⣯⣟⡿⣽⣻⢷⡿⣿⣿⣄⠀⠀⠀⣿⣿⣟⣾⡽⣿⢾⣽⣿⣿⣿⣿⢫⣗⡻⣖⣿ ⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠯⣜⢧⡻⣥⣛⢶⣫⣝⡳⡞⣵⢫⣞⡹⡖⣯⣳⣿⣿⣿⣿⣿⣿⣻⢿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣯⡿⣯⡷⣟⣾⣽⡾⣿⡽⣯⣿⢾⣽⣻⢯⣟⣯⣿⣻⢿⣿⣷⡀⠀⣿⣿⣯⡷⣿⢯⣿⢾⣿⣿⣿⣯⢷⣫⢷⣫⣿ ⠀⠀⠀⠀⠀⢰⣿⡿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡝⣮⢳⣳⢣⣏⢶⡳⣜⡳⣝⢮⡳⣎⢷⣹⣶⣿⣿⣿⣿⣿⣳⣯⣟⣯⣟⡿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⢾⣻⡷⣟⣿⡽⣾⣽⢷⣟⡿⣞⣿⡽⣯⡿⣯⣟⣾⡽⣟⡿⣿⣿⣴⣿⣿⢷⣟⣯⡿⣽⣻⣿⣿⣿⣯⢷⣫⢷⣻⣿ ⠀⠀⠀⠀⠀⣾⣿⢡⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡜⣧⢳⣏⢮⡳⣝⢮⡳⣝⢮⡳⣝⣮⢳⢿⣿⣿⣿⣿⢾⣽⣞⣯⢿⣞⣿⣻⢿⣿⣷⡄⠀⠀⠀⠀⣼⣿⣿⢾⣻⣯⣟⣯⣷⢿⣻⡾⣯⡿⣽⣿⣳⡿⣯⣟⣷⢿⣽⣻⢯⣟⡿⣿⣿⣿⣟⡿⣾⢯⣟⡿⣿⣿⣿⣿⣞⣯⡽⣾⣿⣿ ⠀⠀⠀⠀⣼⣿⢏⡬⢃⠾⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣟⡼⣳⢎⡷⣝⣞⡳⣽⣚⡧⢿⣜⢮⡟⣞⣿⣿⣿⣿⣿⣳⣯⢿⣯⢿⣞⣯⣿⣻⢿⣿⣦⡀⠀⠀⣿⣿⣿⣻⣽⡾⣽⣳⣯⡿⣷⣟⣯⢿⣯⣿⣿⣿⣷⣻⡽⣿⣞⡿⣯⡿⣽⣷⣻⣟⣾⢿⣽⣻⢯⣿⣿⣿⣿⣟⡾⣵⣻⣿⣿⣿ ⠀⠀⢀⣼⣿⢏⡲⢜⡭⣚⠭⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡷⣭⡻⣞⡵⢾⣹⢖⡯⣞⢯⡞⣧⡻⣞⡼⣿⣿⣿⣿⣿⢾⣻⣽⣯⣟⣷⢯⣟⣿⣻⣿⣷⣄⠀⣿⣿⣟⣷⢯⣟⡿⣽⣳⣿⣳⣯⢿⣻⣾⣿⠉⠻⣿⣿⣟⣷⣻⣽⡷⣿⣻⣞⣷⣯⣟⡿⣞⣿⣻⣿⣿⣿⣿⡽⣽⣳⣿⣿⣿⡏ ⢀⣴⣿⡟⣣⢎⡵⣊⠶⣩⠞⡴⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣳⡽⣞⡽⣳⢯⡞⣷⣛⣾⣹⣳⡽⣳⣻⡼⣿⣿⣿⣿⣿⣯⡷⣟⣾⢯⣿⣻⢾⣽⢯⣿⣿⣦⣿⣿⡿⣾⣻⢯⣿⣻⣽⣞⣯⣟⡿⣽⣿⣿⠀⠀⠀⠙⢿⣿⣯⣷⢿⣳⡿⣽⣾⣳⡿⣽⣟⣷⣿⣿⣿⣿⣳⣟⣳⣿⣿⣿⡟⠀ ⠿⠿⢿⣿⣷⣾⣖⡭⣓⢧⣫⣽⣷⣾⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣳⢯⣽⡳⣯⡽⣞⡽⣶⣫⢷⣽⡳⣷⣛⡾⣽⣿⣿⣿⣿⣿⡽⣯⣿⣳⢿⣯⣟⣯⣷⢿⣿⣿⣿⣟⡷⣿⣻⢷⣯⡷⣿⣽⢾⣟⣿⣿⠇⠀⠀⠀⠀⠀⠈⠻⢿⣿⣯⣟⡿⣞⣷⢿⣻⣾⣿⣿⣿⣿⢯⡷⣾⣽⣿⣿⡟⠀⠀ ⠀⠀⠀⠀⠈⢻⣿⡷⣭⢶⣿⠟⠉⠁⠀⠉⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣟⣶⣻⢗⣻⣭⣟⡶⣯⣟⣶⣻⣳⢯⣟⣳⢯⣿⣿⣿⣿⣿⣟⣾⡽⣿⢾⣽⣻⢾⣟⣾⣽⡾⣽⣻⡷⣟⣿⢾⣽⡷⣯⣿⢾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣽⣻⣷⣿⣿⣿⣿⢯⣟⡾⣽⣿⣿⡟⠀⠀⠀ ⠀⠀⠀⠀⠀⣸⣿⢷⣹⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣾⣽⣫⢷⣻⣼⣻⡵⣞⡷⣽⡳⣟⡾⣽⣛⡾⣽⣿⣿⣿⣿⣿⣿⡽⣯⣷⢿⣻⣾⣻⢾⣽⣟⡷⣿⢯⣟⡿⣾⣽⣟⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣿⣿⣿⣿⡿⣭⣟⣾⣿⣿⣿⠟⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣿⣿⢫⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣾⡽⣯⢷⣳⢯⣽⣛⣾⣳⢿⣹⡽⡾⣽⡽⣳⣞⢿⣿⣿⣿⣿⣿⣿⣾⣻⣽⣾⣻⢯⣷⣻⢿⣽⣻⢯⣿⣳⢿⡾⣽⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⡿⣯⢟⣳⣽⣾⣿⣿⠋⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣿⣿⣻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣽⢯⣽⣛⡾⣽⢶⢯⣻⣳⢟⣽⣳⣻⢗⣯⣟⡾⣽⢿⣿⣿⣿⣿⣿⣷⣯⣟⡿⣽⣯⢿⣳⣿⣻⢷⣻⣯⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣻⡽⣏⣿⣽⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠘⣿⣷⣻⢿⣿⣷⣤⣤⣀⣀⣀⣀⣀⣀⡀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣼⣿⣿⣿⣿⡾⣝⣯⣟⡾⣯⠷⣯⣟⢾⣳⢯⣟⡾⣞⡽⣯⢾⡽⣻⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣻⣾⡽⣟⡿⣾⢿⣿⡇⠀⢀⣀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⡞⣷⣻⣽⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣽⣻⣟⡿⣿⢿⣿⣿⡿⣿⣿⣿⣿⡿⣿⣿⣿⢿⡿⣿⢿⡿⣿⢿⣻⣟⣿⣻⢟⣿⣿⣿⣿⣮⢷⣛⣷⡻⢷⣞⣯⢟⡾⣽⠾⣝⣟⡾⢯⡽⣯⣟⡾⣽⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⢿⣹⢷⣛⣷⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣯⣿⣮⣷⣻⣳⡽⣾⣱⣟⡷⣽⡞⣯⢿⣹⢯⡷⢯⣛⣧⣟⡾⣭⣟⡾⣽⣻⣿⣿⣿⣽⠾⣽⣻⢾⣹⢾⣽⣳⠿⣽⠾⣽⢯⢿⡵⣾⣽⣳⢯⣟⡾⣝⣯⣟⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⢯⣷⣛⣮⣟⡾⣽⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⠿⢿⣿⣷⣿⣾⣷⣿⣽⣯⣽⣷⣿⣯⣿⣮⣷⣟⣧⣿⣽⣞⣷⣯⣿⣿⣿⣿⣷⣯⣟⣽⡻⣶⢯⡿⣽⣻⡽⣞⣯⣿⣾⢶⡯⣟⡾⣽⣻⡼⢯⡽⣞⡿⣼⣳⡽⣞⣳⢯⣻⡽⢾⡽⣞⣳⢯⣽⣳⢯⣟⣶⣻⢾⣵⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣻⣼⣻⡵⣯⢷⣻⣼⢻⣽⣾⣿⣟⣯⢷⢯⣻⣵⣳⡟⣯⢿⡽⣞⡷⢯⡽⣏⣯⣟⣧⣟⣯⣿⣹⢯⣟⡾⣭⢷⣻⢮⣽⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣳⡽⣞⡽⣽⣫⢷⡯⣟⣾⣿⡿⡾⣽⣞⣯⢷⣻⣼⢻⣽⡳⣿⣹⣞⣯⠿⣽⣞⡾⣵⢾⣿⣿⣯⣟⡾⣽⣫⣟⣳⣟⡾⣽⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣦⣤ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣟⣧⣟⣳⢯⣟⣳⢯⣷⢻⣽⣿⣿⣽⣻⣵⢾⡽⣞⣷⣫⣟⡾⣽⣳⢯⡾⣝⡿⣳⢾⣝⣻⣾⣽⣿⣿⣾⣽⣳⡟⣾⣳⢯⡽⣞⣯⢿⣿⣿⣶⣤⣄⠀⠀⢀⣴⣿⣿⡿⣟⣿⣻⣟⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⣾⢷⣹⢏⡿⣾⣹⣏⡾⣏⣿⣿⢿⣿⣿⣿⣿⣿⣾⣶⣿⣾⣿⣷⣏⣷⣿⣿⣾⣿⣿⣿⣿⣿⣿⠿⠿⣿⣷⣏⣹⢷⣏⢿⣹⢿⣸⢿⣎⡿⣹⣿⣿⣿⣶⣿⣿⢿⣷⣿⣿⣿⠿⠿⠿⢿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣤⣶⣿⣿⢯⣟⣳⢯⣯⢿⣹⢷⣳⢯⣻⡽⣿⣿⣶⣾⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠋⠉⠁⠀⠀⠀⠹⣿⣯⣟⣻⣞⣯⡽⣾⢽⣛⡾⣽⣳⢯⣻⣽⣻⠿⣽⣫⣿⣿⠁⠀⠀⠀⣠⣼ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣽⣻⢿⡿⣿⢿⣟⡿⣞⣻⠾⣝⣻⡼⣯⣻⢾⣝⣯⢷⣻⡵⣯⢿⡽⣯⣟⢿⣻⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣬⣶⣿⣿⣞⡷⣽⡞⣽⣳⢯⣟⣽⣳⢯⣟⣧⢷⢯⣟⡷⣯⣻⢿⣷⣤⣠⣾⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠛⠉⠁⠀⠀⠀⣻⣿⡽⣯⣻⡽⣛⡾⣽⣫⣽⣻⣽⡳⣟⣳⢯⣟⣮⢟⣾⣳⣻⡽⡾⣽⣳⣞⣯⣿⡿⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⢿⣻⢯⣷⣛⣾⡳⣟⣳⢯⣟⣞⣧⠿⣽⣞⢾⢯⣟⡾⣽⣳⣯⣿⣿⣿⣿⣿⣿⡿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⢿⣿⣶⣤⣀⠀⠀⣠⣿⣿⣻⡵⣯⣟⣽⣻⣵⡻⣞⣵⣳⢿⣹⢯⣟⣮⢯⣟⡾⣵⣻⢾⣽⣳⣻⣼⣻⣿⡇⠀⠻⣿⣿⣿⣦⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣯⣿⣹⣟⣶⣻⢞⡽⣏⣯⣟⡾⣽⣺⢟⡷⣞⣯⣟⡾⣽⡳⣟⣿⣿⣟⠛⠛⠉⠁⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣽⣻⢿⣿⣿⣿⣿⡿⣯⣗⣿⡵⣞⡷⣽⢶⣻⡽⡾⣭⢿⣭⢷⣻⢞⣻⠾⣝⡷⢯⡷⣞⡷⣽⢶⣻⣿⡇⠀⠀⠹⣿⣿⣿⡇⠀⠀⠀⣰⣿⣿⣿⡿⠙⣿⣷⣻⢞⣧⣟⣯⠿⣽⣞⢾⡽⣳⢯⡿⣹⢯⣞⡾⣽⣳⣿⣽⣾⣿⣿⣷⡄⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣾⣽⣯⣿⣽⣷⣿⣿⣿⡽⣽⣛⣾⡳⣟⡽⣯⢷⣫⢿⣭⠿⣽⣻⣽⣛⣯⣽⣛⡾⣽⣫⢷⣿⣧⠀⠀⠀⣿⣿⣿⣿⠀⠀⢠⣿⣿⣿⣿⠁⠀⢿⣿⣽⣻⢞⣳⣞⡿⣳⢾⣿⣽⣯⣻⡽⢯⣟⣾⣿⡿⠟⠋⠉⠙⢿⣿⣿⣿⡆⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠉⣰⣿⣿⣳⣯⣷⣿⣼⣻⡽⣽⣞⣯⢟⣳⢯⣿⣿⣿⣾⡽⣞⣧⢿⡽⣳⢯⣟⣿⣿⣶⣶⣾⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⡇⠀⠀⢸⣿⡷⣯⣟⣳⢯⡽⣯⣿⡟⠿⣿⣷⣟⣟⣾⣿⣯⣄⡀⠀⠀⠀⠀⢿⣿⣿⣿⡀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⠟⠋⠛⠛⠿⣿⣷⣾⣭⠿⣽⣿⡿⠃⠈⠻⣿⣿⣞⣯⡽⣯⣟⣾⣞⣽⣿⣿⣿⡿⠟⠋⠀⠀⠀⠸⣿⣿⣿⣷⣦⣴⣿⣿⢿⡵⣯⢯⣿⣽⣿⠟⠀⠀⠈⠙⠿⣿⣾⣽⡿⣿⣿⣷⣦⣤⣄⣸⣿⣿⣿⡇⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣀⣿⣿⣞⣿⣿⠟⠁⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣯⣷⣿⣾⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠉⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣤⣶⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀