⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⢻⢰⣮⢹⠀⠶⣾⡰⢾⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⣺⣘⣫⣼⣀⣛⣿⣟⣠⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠻⠿⢿⣿⠿⣿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠐⠀⠈⣿⣿⠟⠻⠿⢿⣿⣿⣷⢤⣀⣀⠀⠀⠀⠀⠀⠀⣀⠔⠛⠻⢶⣭⣶⡤⢄ ⠀⠐⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠉⢰⣿⣯⣄⡠⢠⠤⢽⣿⣿⡆⠀⠀⠈⠉⠁⠒⠲⠮⠥⠄⣀⣀⡀⠀⠉⠛⠿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠿⠇⠻⠿⠟⢿⣿⢗⠶⠶⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠉⠉⠁⠐⠒ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣀⠿⢿⣿⣿⣿⣿⣿⣄⠰⢧⣽⠦⠀⣼⣯⡞⡀⠄⠀⠢⠉⠙⠻⢛⠗⡶⢶⣄⣤⣀⡀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡿⠃⠀⠀⠀⠂⡉⠙⢻⢧⡘⠿⠿⠋⢀⣼⣿⠳⠈⠠⡂⠀⠀⠀⢈⠁⠤⢀⠈⢼⣾⣶⣿⣿⠛ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⡀⠀⠈⠢⣘⠂⠙⠳⠴⢚⢽⠟⠛⢴⣦⢀⠉⠀⣀⣁⡴⠄⠒⠒⠒⠚⠚⠛⠛⠛⠚ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠨⣴⣁⣀⡀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⣰⠛⠀⠀⠀⠀⠀⠈⠀⠀⢚⡵⠃⠀⠀⠀⠈⣿⣿⣾⣿⢛⢒⣶⣦⣄⠀⠀⠀⠀⠀⠀⢠ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡲⠊⠁⠀⠀⠀⠀⠀⠉⠉⠓⠶⣤⣀⣀⣴⠃⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⠽⠿⠗⠸⠐⣿⣿⣿⣷⣆⡀⡠⠓⠊⠁ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡙⢛⣟⠁⠀⠀⣼⠀⢀⢨⣝⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣽⣿⣾⠄⠀⢀⠐⣢⠉⡛⣿⣿⣿⣿⠠⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣰⣿⣿⡞⠃⠀⠀⣰⠉⢦⡀⠓⠋⠀⠀⣀⠞⠀⠀⠀⠀⠀⢀⣾⣷⣿⣿⣿⣿⣿⠁⣵⣾⣾⣿⣿⣿⣿⣿⠇⠀⠀⠀ ⡀⠀⢀⣀⣤⣀⣀⠤⠤⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣼⡿⣿⣿⠇⠂⠀⣰⠃⠀⠐⠋⠓⠒⠒⠛⠁⠀⠀⢀⣠⣴⣿⣥⣼⡿⣿⣿⣿⡿⠛⣀⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⠀⠀⠀⠀⢰⠇⣼⣿⣵⣾⠏⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠁⠀⠀⠀⠉⠛⠻⣿⣟⡛⠙⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⣠⠞⠁⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣿⣿⣿⣿⠏⢀⡤⣾⡀⠀⣀⣠⠤⠖⠚⠉⠐⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⠿⣿⢿⣿⣿⠟⠁⠀⠀⠀⠁⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣅⠸⣿⣿⣿⣯⣾⣿⣶⠶⠵⣟⡚⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠈⠙⠛⠛⠒⠦⡀⠀⠀⠀⠀⠀⠀⠀ ⠋⠀⠉⠉⠒⠲⠦⣤⣀⣀⣀⣀⣀⣀⣼⢹⢀⣿⣷⣿⣿⣳⣿⣉⣁⣀⣤⣆⠏⠝⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢥⣀⣠⣤⣤⣄⣵⡂⠀⠀⠀⠀⠹⡀⠠⢤⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⢧⠀⡀⠙⠆⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⠿⢿⠟⠻⠻⠋⠑⠀⠀⠈⠐⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡯⠀⠀⠀⠀⠀⣸⣠⡁⠀⠠⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⢈⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣾⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣇⣈⣙⣂⣁⣀⡄ ⣧⣤⣤⣦⣤⣦⣶⣾⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣶⣿⣿⣿⣿⣿⣦⣤⣤⣀⣀⣀⣠⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⡿⠿⢿⣿⣿⡇

⡇⣿⣿⢿⠿⡿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣠⣀⣀⣀⣤⣤⣦⣿⣿⣿⣿⣿⣶⣿⣿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⡿⣿⣿⣿⣾⣶⣦⣤⣦⣤⣤⣧ ⡄⣀⣁⣂⣙⣈⣇⣤⣤⣤⣤⣤⣧⣿⣿⣿⣿⣿⣾⣶⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⣶⣿⢈⠀⡄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⡿⣿⣾⣶⣤⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠠⠀⡁⣠⣸⠀⠀⠀⠀⠀⡯⣿⣿⣿⣿⣼⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⠦⠐⠈⠀⠀⠑⠋⠻⠻⠟⢿⠿⣿⣿⣿⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠆⠙⡀⠀⢧⢠⠀⠀⠀⠀⣿⣿⣿⣿⣿⢻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢤⠠⡀⠹⠀⠀⠀⠀⡂⣵⣄⣤⣤⣠⣀⢥⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠝⠏⣆⣤⣀⣁⣉⣿⣳⣿⣿⣷⣿⢀⢹⣼⣀⣀⣀⣀⣀⣀⣤⠦⠲⠒⠉⠉⠀⠋ ⠀⠀⠀⠀⠀⠀⠀⡀⠦⠒⠛⠛⠙⠈⠀⠀⠀⠀⠀⡄⣶⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⡚⣟⠵⠶⣶⣿⣾⣯⣿⣿⣿⠸⣅⠞⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠁⠀⠀⠀⠁⠟⣿⣿⢿⣿⠿⠺⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢦⠐⠉⠚⠖⠤⣠⣀⠀⡀⣾⡤⢀⠏⣿⣿⣿⣿⢿⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠁⠞⣠⠃⠟⡿⣿⣿⣿⣿⣿⣿⣿⠙⡛⣟⣿⠻⠛⠉⠀⠀⠀⠁⠉⠚⠴⣀⠀⠀⠀⠀⠀⠀⠃⡴⢀⠀⠏⣾⣵⣿⣼⠇⢰⠀⠀⠀⠀⠀⠀⠀⠄⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠏⣿⣿⣿⣿⣿⣿⣿⣀⠛⡿⣿⣿⣿⡿⣼⣥⣿⣴⣠⢀⠀⠀⠁⠛⠒⠒⠓⠋⠐⠀⠃⣰⠀⠂⠇⣿⣿⡿⣼⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⣀⣀⣤⣀⢀⠀⡀ ⠀⠀⠀⠇⣿⣿⣿⣿⣿⣾⣾⣵⠁⣿⣿⣿⣿⣿⣷⣾⢀⠀⠀⠀⠀⠀⠞⣀⠀⠀⠋⠓⡀⢦⠉⣰⠀⠀⠃⡞⣿⣿⣰⠀⡞⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠠⣿⣿⣿⣿⡛⠉⣢⠐⢀⠀⠄⣾⣿⣽⣼⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣝⢨⢀⠀⣼⠀⠀⠁⣟⢛⡙⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠁⠊⠓⡠⡀⣆⣷⣿⣿⣿⠐⠸⠗⠿⠽⣿⣸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡞⠀⠀⠀⠃⣴⣀⣀⣤⠶⠓⠉⠉⠀⠀⠀⠀⠀⠁⠊⡲⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢠⠀⠀⠀⠀⠀⠀⣄⣦⣶⢒⢛⣿⣾⣿⣿⠈⠀⠀⠀⠃⡵⢚⠀⠀⠈⠀⠀⠀⠀⠀⠛⣰⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⡀⣀⣁⣴⠨⠀⠀⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠚⠛⠛⠛⠚⠚⠒⠒⠒⠄⡴⣁⣀⠀⠉⢀⣦⢴⠛⠟⢽⢚⠴⠳⠙⠂⣘⠢⠈⠀⡀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠛⢹⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠛⣿⣿⣶⣾⢼⠈⢀⠤⠁⢈⠀⠀⠀⡂⠠⠈⠳⣿⣼⢀⠋⠿⠿⡘⢧⢻⠙⡉⠂⠀⠀⠀⠃⡿⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⡀⣀⣤⣄⢶⡶⠗⢛⠻⠙⠉⠢⠀⠄⡀⡞⣯⣼⠀⠦⣽⢧⠰⣄⣿⣿⣿⣿⣿⢿⠿⣀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠒⠐⠁⠉⠉⠀⠀⠀⠀⠀⡀⣀⣀⣄⣤⠶⠶⢗⣿⢿⠟⠿⠻⠇⠿⠿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠿⠛⠉⠀⡀⣀⣀⠄⠥⠮⠲⠒⠁⠉⠈⠀⠀⡆⣿⣿⢽⠤⢠⡠⣄⣯⣿⢰⠉⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠂⠐⠀ ⢄⡤⣶⣭⢶⠻⠛⠔⣀⠀⠀⠀⠀⠀⠀⣀⣀⢤⣷⣿⣿⢿⠿⠻⠟⣿⣿⠈⠀⠐⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⣿⠿⣿⢿⠿⠻⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⡀⣄⣦⣶⣿⣶⣴⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣠⣟⣿⣛⣀⣼⣫⣘⣺⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢾⡰⣾⠶⠀⢹⣮⢰⢻⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⠀