⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠾⢤⡀⡤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣒⣒⣋⣛⣁⣀⡉⢉⢆⡇⡠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⡰⡆⠀⠀⢸⢁⠤⠤⠤⠤⠤⠤⡄⢸⢸⢸⠘⠟⠳⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⡠⢤⣄⠇⡇⡟⠀⡇⢸⠀⡠⡢⣖⢢⠀⡇⠀⠠⠈⣀⢲⡄⠈⢎⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⡇⡟⠮⠴⢱⢀⠀⡇⣸⠀⡡⣅⠜⣰⠀⡇⢸⠀⢠⢣⣞⢔⡆⠈⢇⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⡇⣧⡒⠒⠊⢸⠀⠁⡇⠀⠙⠒⠚⠁⠀⡇⢸⠀⣇⣾⢆⡎⢸⣿⠜⡜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠠⣈⡓⣛⡄⡟⢸⢰⠀⠇⠀⠀⡰⢆⠀⠀⠇⢸⡀⠛⠸⡘⠢⡄⠁⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢔⠀⠀⠀⠀⣠⣶⠿⠿⢷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⣿⡍⡇⡇⡏⡸⢠⠀⠀⠀⡇⢸⠀⠀⢰⠈⢓⣂⡔⢡⣖⠥⣄⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢐⣆⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣾⠟⠁⠀⠀⠀⢻⡇⢄⣞⠀⢠⣀⣀⠀⠀ ⠀⠀⢀⡽⠁⠣⡌⢀⠇⡞⠀⠀⠀⠁⡏⠀⠀⢸⠀⣏⡎⡰⡆⢀⠔⠁⢀⢇⠇⠀⠀⠀⠀⠊⡥⠀⢠⣿⣿⣆⠀⠀⠁⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⣸⡟⠙⣡⣴⠿⠛⠻⣷⣄ ⠀⠀⠸⠒⠉⠉⢁⡞⣴⣀⣀⣀⣀⠀⣇⣀⣀⣀⣇⢠⡜⠻⠴⠃⠀⠀⡜⡜⠀⠀⠀⠀⠀⢠⣿⣦⣿⡏⠈⠻⣷⣶⣶⣾⠿⠿⣷⣾⣿⣤⣤⣄⡀⠀⢠⣿⠧⣴⡿⠁⠀⠀⠀⠘⣿ ⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠤⠤⢤⠀⡠⠄⠠⠀⠀⠀⠅⠀⠀⠀⢀⢜⡠⢵⣄⠀⠀⠀⠀⢸⣿⠻⢿⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⣿⣧⣼⡟⠁⠀⠀⠀⠀⢀⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠚⠀⣉⡉⠉⠷⠰⡒⡆⠀⢀⠴⡪⠊⠀⠀⠁⠀⠀⠀⠀⣸⣿⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄⠀⠀⠀⠀⣼⡏ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⠀⡇⠀⠀⠀⢈⣨⠅⣺⠕⠊⠀⠀⠀⠀⠀⠀⠀⢠⣾⢟⣵⣿⣿⣿⣿⡿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⢀⣼⡟⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠀⠀⠐⠋⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢣⣿⣿⣿⣿⣿⣿⣧⣾⣿⠀⠀⢀⣤⣶⣿⢿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠙⣿⣾⠏⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⣿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣴⣿⣿⣿⣧⣰⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠸⣿⡶⠟⠉⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⡆⠀⠀⠀⠀⠀⢹⡇⠀⠀ ⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣤⣄⣹⣷⡄⠀⣿⣿⡻⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡗⠀⠀⠀⠀⠀⢸⣿⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡛⠛⠉⠀⢻⣧⣝⠦⣍⠛⣿⡿⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀ ⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⣶⠄⠀⠀⠙⠻⠷⠾⠛⠋⢀⣀⡄⠸⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⢏⡿⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀ ⠄⠀⠐⠳⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣥⣀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣦⠹⣧⡲⠀⠀⠉⠉⠉⣹⣿⣫⠿⠁⠀⠀⠀⠀⠀⠀⢻⣇⣀⡀ ⠀⠀⠈⣣⣴⢙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠐⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⡈⠛⠷⣶⣤⣤⣾⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠟⠁ ⠀⠀⢠⡀⡴⠹⡦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣧⡀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣷⣴⡀ ⠀⠀⠀⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠻⣶⣄⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣾⠿⣟⠉ ⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠛⠛⠛⠶⠤⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣬⣿⣷⢤⢈⢝⠃ ⠀⠀⢭⣺⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣻⣷⣤⡀⠀⠀⣀⣴⣶⣶⣶⣄⡀⠀⠀⠠⣀⣀⣀⣠⣤⣴⣶⣾⠟⢉⣀⢁⠠⡟⠓⡹⠉⠀ ⠀⠀⠀⠀⢹⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣯⣅⣀⣠⡾⠛⠁⠀⠀⠈⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠐⢪⡁⠀⠋⢠⠇⠀⠀ ⠀⠀⠈⠓⢌⣗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣽⣿⠟⢁⡿⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡂⢀⣿⢿⣦⡀⡸⠀⠀⠀ ⠀⠀⠀⠀⠀⠉⠻⠴⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⠀⠀⠀⠘⠠⢤⣾⣿⠋⠀⣼⠇⣼⡏⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣠⣿⠏⠀⠙⢷⡇⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢫⣀⠀⢀⢸⡄⣆⠹⣆⢀⠀⠈⠻⢷⣦⣀⣀⣵⣿⡿⠁⠀⠀⣿⠆⣿⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠀⠀⠀⢸⠁⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠂⢘⣎⣿⠼⢤⣻⡜⡄⠀⠄⢀⡠⠝⠒⢿⡿⠁⠀⠀⢀⡟⠀⢿⡆⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀ ⠀⠀⢀⣠⠤⠤⠐⠒⠚⠉⠉⠀⠀⣤⢤⠀⠑⢟⡵⠔⠋⣁⣀⠀⠐⠛⠛⠑⠠⣀⢸⠇⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀ ⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⠨⠗⢋⡡⠴⠂⠉⠀⠀⠉⢲⡤⠿⣶⣄⠀⠉⠲⢄⢀⣈⣻⣷⣄⣀⣀⠀⠀⣀⣀⣤⣾⣋⣀⣀⡀⣀⣘⣀⡀⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⠀