Diabetes Emojis & Symbols

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣝⣿⠇
⠀⠀⠀⢸⣿⣿⣿⣿⡿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⢟⣭⣯⣭⣭⣭⣭⣭⣭⣭⣍⠉⠉⠉⠉⢉⣍⠁
⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣶⣶⣶⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⠿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣾⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠞⠻⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣭⡟⣿⣿⣿⣿⡿⣭⣯⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⢇⣿⣿⣿⣿⣇⠿⠿⢷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡯⠀⠀⢀⣀⠀⠀⢀⣀⡀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⡇⠀⢸⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠤⠤⡀⠀⠣⠤⢌⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠛⠋⠀⠃⠘⠛⠁⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢰⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢱
⠀⣼⠿⣀⠀⠀⠀⢀⠀⠀⢠⠂⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡼⢱⡀⠀⠉⠐⢠⠃⠈⠑⠊⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠳⢻⠍⠐⠂⢤⠃⠀⠀⠀⢠⠉⠒⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠸⢄⡀⠀⢠⠃⠀⠀⠀⠀⠈⠁⠒⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠨⢀⠋⠉⠒⠠⢄⡀⠀⠀⠀⠀⠐⠉⠑⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠣⠊⠀⠀⠀⠀⠀⠀⠉⠒⠤⣤⠁⠀⠀⡄⠀⠈⠙⠒⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⣤⣀⡀⠂⠀⠀⠀⠈⠉⠳⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠤⠐⠒⠂⠉⠉⠉⠁⠀⠈⠉⠳⡂⠤⣀⠀⠀⠂⠀⠀⠘⠙⠒⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⡠⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⠀⠀⠘⡄⠀⠉⠐⠂⠄⡀⠁⠀⢀⠀⠀⠈⠙⠲⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⡜⠀⠀⠀⣀⠤⠄⠰⠒⣒⠂⠈⣭⠀⠀⠀⠀⠀⢣⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠀⠂⠄⡀⠀⠂⠀⠀⠈⠉⠓⣶⣤⠦⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⡆⠀⠀⡎⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢀⠀⠈⡀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠀⡁⠀⢰⣧⠇⣜⠑⠢⢄⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⢡⠀⠀⠇⠀⠀⠀⠀⠀⢀⠤⠶⢤⠀⠀⠋⠉⢳⠀⠇⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⣏⡸⢀⡉⠓⠎⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠘⡀⠀⢸⠀⠀⠀⠘⡇⠸⠄⠀⠘⠂⢀⡴⠶⠚⠀⢀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⠄
⠀⢸⠀⠀⠀⠀⠀⠇⠀⠀⡄⠀⠀⠀⣦⠀⡆⠀⠀⡇⠈⢇⣀⣤⠀⢸⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⢁⠀⠀⠀⠘⠀⠙⠛⠛⠁⣀⣀⡀⠤⠤⠊⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠈⡄⠀⠈⠲⠤⠤⠆⠒⠒⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⡠⠊⠉⠑⡄⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢎⠀⠀⠈⠑⡄⠀⠸⡀⠀⢀⠔⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠈⠑⠢⠤⠔⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⢢⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⡄⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠀⠀⠀⠀⠓⠤⠒⠀⠀⠀⠀⣠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⠀⢀⢴⠉⠉⢹⡶⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡄⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠐⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣰⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⠤⠼⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢀⡀⠀⢀⡀⢀⡀⣀⠀⣀⠀⡀⢀⡀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢠⡈⠠⠘⠊⢗⠆⣓⠀⠄⠸⡂⢐⠆⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠛⠛⠛⢿⣿⠛⠛⢿⣿⠛⠛⠛⠛⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⣀⣀⣀⣼⣿⣀⣀⣽⣷⣀⣀⣀⣀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡇⠛⢇⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇⠀⢘⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠇⠀⢸⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⠸

⠀⠀⣠⠐⢒⠒⢠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢸⠀⣿⡆⢸⠀⠀⠀⠀⠀⠐⠶⠠⠀⠀⠀⡅⠶⠆⠀⠀⠀⢠⠊⠀⣈⣏⡀⠈⢆⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠇
⠀⠀⢸⡦⣬⡤⡆⠀⠀⠀⠀⠀⠀⢀⢶⣶⣶⣶⡶⡀⠀⠀⠀⠀⢸⠀⠀⢻⣿⠇⠀⢸⠀⠀⠀⠀⠀⢤⣾⢟⠟⠁
⠀⠀⠀⠛⣿⠟⠁⠀⠀⠀⠀⠀⠀⠸⠀⠈⠿⠁⠀⠇⠀⠀⠀⠀⠀⠳⢄⣸⣿⣀⠴⠃⠀⠀⠀⠀⢠⡞⠙⠋
⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⢶⠠⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⣌⠀⢠⡀⠀⠀⠀⠀⠀⠀⣠⡟⣠⣴⠟⣃
⠀⠀⢸⠀⠀⠂⡘⠀⠀⠀⠀⠀⣠⣶⣾⣷⣼⣿⣀⡀⠀⠀⠀⠀⢰⠁⠀⣴⣷⡄⠀⢱⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⡀⠉
⠀⠀⠘⣯⣿⣯⡇⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠘⡄⠀⢹⣿⠁⠀⡜⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⣿⢷⠄⠀⠀⠀⠀⠀⠘⠛⠿⠟⠛⠃⠛⠁⠀⠀⠀⠀⠀⠈⠒⠼⠿⠔⠊⠀⠀⠀⠀⠀⠈⠿⠟⠛⠋⠁

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⣼⣿⡄⠀⠀⠀⠀⠀⠀⠛⢻⠀⠀⢠⡄⠀⣀⠀⠀⠀⠀⠀⣿⣿⣀⣀⣽⣿⡇⠀⠀⠀⠀⠈⢷⠖⠀⠀⢠⣾⠝
⠀⠀⣼⣏⢨⣿⡀⠀⠀⠀⠀⠀⠀⠘⣶⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⠉⣹⣿⠉⢻⠇⠀⠀⠀⠀⣠⣼⣦⣤⣶⡶⢧⡀
⠀⠀⢻⣟⣛⣿⠃⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⡄⢻⣿⠁⣾⠀⠀⠀⠀⠀⢿⣿⣽⣿⣿⣄⣠⠇
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠁

⠀⠀⣠⢤⣶⣦⣤⡀⠀⠀⠀⠀⢀⡴⠒⠒⠒⠒⠒⠶⣄⠀⠀⠀⣠⠖⠒⡒⠒⠒⠒⢦⡀⠀⠀⢀⡴⠒⠒⠒⢒⠒⠲⣄
⠀⠀⡇⠤⠉⠁⠀⡇⠀⠀⠀⠀⢸⡇⣼⣶⣿⣷⣯⠀⣿⠀⠀⠀⣿⢠⣿⣷⣻⣿⣅⢸⡇⠀⠀⢸⡇⠈⢳⣾⣿⡟⠀⣿
⠀⠀⡇⠒⠐⠂⠀⡇⠀⠀⠀⠀⢸⡆⢿⣷⣯⣶⣿⠃⣿⠀⠀⠀⣿⠸⣯⣼⣽⣬⡿⢸⡇⠀⠀⢸⡇⢰⣿⣿⡿⣄⠀⣿
⠀⠀⢧⣉⣄⣀⣿⣿⠂⠀⠀⠀⠘⢧⣬⣤⣤⣤⣬⣤⠟⠀⠀⠀⠻⣤⣭⣭⣭⣭⣥⡼⠃⠀⠀⠘⢧⣬⣬⣭⣤⣬⣤⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⢦⡄⠲⡶⠂⢀⣶⡄⠀⣶⠶⣦⠀⣶⠶⠆⠶⣶⠶⢠⣶⠶⠆⣴⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣸⠇⢠⣧⡀⣾⠷⢿⡀⣿⣛⣷⠀⣿⣉⡁⠀⣿⠀⢸⣟⣉⡁⣉⣛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠀⠁⠀⠈⠀⠈⠉⠀⠀⠉⠉⠁⠀⠈⠀⠀⠉⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⡠⠤⠄⠄⠀⠀⠀⢀⣄⡄⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⡀⣠⠀⠀⠀⠠⡤⡄⡠⡀⣤
⠀⠀⠀⠀⠀⠀⠀⠵⠂⠖⢅⠀⠀⠀⠀⣑⡔⢉⡁⢢⠀⠀⠀⠀⠀⠹⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⢉⣉⢉⣉⠀⠀⠀⠀⠀⣀⣀⣄⠦
⠀⠀⠀⠀⠀⠀⠀⣾⡰⣺⡆⠀⠀⠀⢐⣃⢧⣀⡀⣴⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⢫⣩⡋⠳⠀⠀⠀⠀⡎⢁⡄⠋⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠥⠁⠀⠀⠀⠀⠀⡇⠂⠄⠉⠈⠁⢀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠀⣀⣽⣜⢁⠀⡐⡉⠀⠀⠀⠈⠉⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⢤⠀⠀⠀⠀⠀⠤⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⢂⢈⢞⡀⠀⠀⠀⠀⠀⠀⡀⠸⢗⣎⣧⠀⠀⠀⠀⠀⢠⠦⡤⠀⠀⠀⠀⠰⠦⢀⠎⡱⢄
⠀⠀⠀⠀⠀⢤⠈⡇⢸⠂⠀⠁⠀⠀⠀⠀⠹⢝⡄⠛⢁⠀⠀⠀⠀⣄⣀⢀⡀⣉⡁⠀⠀⠀⠊⠄⠀⠀⠁⣠⠀⠀⠀⠀⠂⠀⡚⠘⠂⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠠⠉⠉⠋⠉⠉⠃⠀⠀⠀⠀⠀⠰⠀⠡⠓⠉⠀⠀⠀⠄⠀⠉⠂⠀⠀⠆

⠀⠀⠀⠀⠀⠀⡤⢀⣈⡷⢫⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠊⠑⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠘⠐⢠⠞⢮⣧⡻⠄⠀⠀⠀⠀⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢆⡰⠀⡀⠀⠀⠀⠀⢀⠐⢂⠀⠀⠀⠀⠀⠀⠂⠀⡀⢊⢀⠒⡤
⠀⠀⠀⠀⠀⢐⠦⣸⡀⠀⡤⠃⠀⠀⠀⠀⢌⠀⠀⠀⠀⠁⠀⠀⢀⠠⣀⣦⢴⠀⠀⠀⠀⠀⠌⠐⢣⡐⡸⡄⠆⠀⠀⠀⢀⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠒⠒⠀⠀⠀⠀⠠⠂⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠓⠒⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠋⠉⠰⠈⠖⠢⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⡇⠀⠀⠀⠀⡞⢩⠑⡄⠀⠀⠀⠀⠀⢘⢀⠂⢀⣐⡀⠂⣘
⠀⠀⠀⠀⠀⢀⣀⡦⣠⣌⣠⠀⠀⠀⠀⠀⠀⠰⠴⠂⠀⠀⠀⠀⠀⢸⠀⠔⠇⠳⠀⠀⠀⠀⠙⠒⠊⡈⠁⠀⢡⠀⠀⠀⠀⠂⠀⠀⠆⠀⠄
⠀⠀⠀⠀⠀⠀⢑⠂⠄⠠⠀⠀⠀⠀⠀⠀⣄⠀⣦⠀⡄⠀⠀⠀⠀⠘⠀⠱⠆⠠⠄⠀⠀⠀⠀⠀⠋⠉⠉⡁⡷⠀⠀⠀⠀⠀⠀⠀⡇⠀⡃
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠐⠃⠐⠂⠀⠀⠀⠀⠀⠑⠈⠀⠃⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣠⣿⣆⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠻⠿⠟⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠈⣿⣿⣿⡏⠀⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⣤⣤⣴⣿⣿⣿⣧⣤⣤⣤⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣟⢛⣛⣛⣛⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⢿⣿⣿⠿⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣠⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠉⠀⠀⠀⠀⣀⣀⣀⡤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠉⠐⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢡⠶⠶⠤⢤⣀⡀⠀⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠴⠶⠚⠉⠀⡼⠷⢤⣄⣀⣸⠏⠉⠓⠲⠦⠤⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠓⢄⡀⠀⠀⠀⠀⠀⢀⡷⢤⣄⡀⢹⠇⠀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⢤⣀⡀⠈⠳⠤⠄⢹⠁⠀⠀⠀⢀⡞⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⡿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠒⠋⠀⠀⠀⠀⠈⠳⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠖⠒⠛⠓⢦⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢰⡇⠀⠀⠀⣾⠉⢉⣡⡤⠤⠖⠒⢦⠀⣄⠸⣦⠴⠖⠚⢻⣉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠁⠀⠀⠀⢸⡀⢹⠀⠀⠀⠀⠀⠸⡆⢀⡀⢳⠖⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠞⠀⠀⠀⠀⠀⣧⠈⡇⠀⠀⠀⢀⣀⡷⠈⠁⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⢲⡄⠀⠀⠀⠀⢸⡄⠹⠴⠖⠛⣉⣡⡤⠴⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠼⠁⠀⠀⠀⠀⠀⠳⠤⠖⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⢠⠀⠠⠀⠄⢠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠀⠀⠀⠈⠀⠀⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡔⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣏⣤⣌⣥⡄⡷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣶⡆⠀⣿⣿⣿⣟⠉⠠
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠋⢹⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⠋⣼⡆⠀⠀⠀⠀⠀⠀⢰⠋⢡⣄⠀⠉⢳⠀⠀⠀⠀⠀⢨⠟⠁⠀⢿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⣟⠻⡷⣿⢇⡧⠀⠀⠀⠀⠀⢰⣶⣴⣤⣄⠀⠀⠙⠁⠀⠀⠀⠀⠀⢀⠏⣤⠙⠋⠤⢠⡌⣇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢹⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⣍⠉⠀⠀⠀⠀⠀⠀⠀⠻⠻⢿⣯⣭⣭⠖⠛⠉⠀⠀⠀⠀⡜⡰⢹⠂⠰⠂⢋⠡⣹⡀⠀⠀⠀⠈⠲⣤⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉⠉⠉⠀⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣝⣷⠀⠀⠀⠀⠀⢀⣄⣠⣴⣾⣿⣧⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣧⡀
⠀⠀⠀⠀⠀⣴⣾⣷⣦⣝⢿⡿⠃⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⡏⠉⠉⠀⠀⠀⠀⠀⠀⠐⣾⢹⣨⣿⠄⣠⡀⠀⠀⠀⠀⢠⡾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⠀⢀⣤⡄⠀⠀⠀⠘⠖⠛⠻⠿⠿⠃⢀⣾⣧⠀⠀⠀⠀⠀⣿⣿⣿⣭⣿⣶⠟⠀⠀⠀⠀⠀⢸⡇⢿⣿⣿⣿⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣯⡿⠁⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠃⠀⠀⠀⠀⠹⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠈⠳⣾⣿⣿⡿⠃⢰⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⢒⣟⣚⣗⠲⡄⠀⠀⠀⠀⠀⠀⠀⣴⣷⢠⠏⢀⣠⠌⡆⠀⠀⠀⠀⢀⡀⠀⢀⣠⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣋⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢾⡏⢹⡀⠅⠀⠀⠀⠀⠀⢠⣾⣿⣿⡜⢆⡀⠁⣠⠇⠀⠀⠀⠀⢺⠏⣰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣀⡀⠀⠘⠛⠟⠋⠸⠿⠀⠀⠀⠀⠀⠀⢀⢘⣻⣿⣿⢀⡂⠀⠀⠀⠀⠀⠻⡟⢻⣿⠿⠲⣭⠉⠁⠀⠀⠀⠀⢠⢇⡜⠁⠙⢻⢻⠛⠛⠁
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⠶⣶⣀⣤⠦⠀⠀⠀⠀⠀⠈⠺⠭⡭⠭⠊⠀⠀⠀⠀⠀⠈⣧⣼⣆⠋⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⢻⢄⣀⣠⠏⡜
⠀⠀⠀⠀⠀⠈⠋⠉⠛⠻⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⣿⠟⢦⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣼⣾⡿⠠⠊⠀⢈⣛⡋⠀⠀⢀⣀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⢸⡟⠋⠀⠀⠀⠀⠀⠙⠛⠁⠐⢸⣿⡇⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠉⠉⣽⠁⠀⣿⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣧⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣼⠃⢀⠀⣍⢈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⡀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠁⠀⠈⠀⠈⠉⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡏⠘⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣷⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⡇⠀⣿⡇⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠃


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⢾⣿⣿⠖⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⡀⠀⠈⠛⠋⠀⠀⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠤⠤⠤⠤⠄⠠⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⡇⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠀⠀⠈⢀⠀⡠⠀⠀⠀⠀⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⣡⠘⢠⢊⠄⡌⢢⠨⢀⡑⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠠⠴⠀⠢⠀⠠⠂⠰⠤⠘⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠉⠒⠀⠤⠤⠤⠄⠐⠊⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠒⠒⠈⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠶⣶⠛⣦⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⣴⣿⣷⣿⣿⣍⣿⣾⡛⣻⣿⣉⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣸⠀⠐⢿⡿⠛⠛⠛⠋⠉⠉⠻⣿⣥⣿⣉⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⠻⠇⠀⠗⠋⠀⠀⠀⠋⠀⠀⠀⣀⣤⡤⠀⠙⣿⣾⣏⣹⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢿⣆⠘⢋⠕⣶⡤⠀⢿⠿⠻⣿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠈⡹⠀⠀⠀⠀⠀⠀⠁⠖⢰⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡐⠠⠦⠀⠀⠀⠀⠀⢰⣏⡹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣒⠒⠂⠀⠀⠀⣠⣾⣿⠟⣿⣴⡀
⠀⠀⠀⠀⡰⠃⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⣀⣀⣤⣴⠞⠋⠸⡏⣴⣩⡷⣿
⠀⠀⠀⢠⠁⠀⠀⠀⠀⣾⣀⣀⣀⣠⣤⣤⣄⣀⠀⢀⣀⢀⣠⣤⣤⣀⠀⠀⣽⡿⠛⠁⠀⠀⠀⢳⣾⣭⣷⠏
⠀⠀⠀⢸⡆⠀⠀⠀⠁⠈⠉⠉⠉⠀⠀⠀⠀⠉⠙⠻⢍⣻⡁⠀⠀⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠙⡿⣷
⠀⠀⠀⢸⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣷⠀⠀⠘⠿⣻⡋⠛⠻⣄⠀⠀⠀⢠⡾⠁⣸⠷⣄
⠀⠀⠀⠀⠉⠛⠿⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⢠⡟⠀⠀⠀⠀⣀⣀⣴⣶⣸⣆⡴⠞⠋⠀⢰⠏⠀⠈⠓⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⢶⣤⣤⣤⣹⣻⣿⡃⣘⠀⣶⠀⠀⠀⠉⠉⠀⢸⢧⣿⠞⠉⢻⡄⢾⠀⠀⡠⠀⠀⠀⠉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⣟⣴⡟⠀⠀⠀⠀⠀⠀⠀⠹⠋⠀⠀⠀⠑⠿⣠⠞⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⢹⡟⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⣻⣦⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠘⠄⠀⠀⢿⡃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡷⠄⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠙⠀⡀⢸⣿⣀⣠⡤⠶⠶⣄⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢱⡞⠹⡿⠓⠚⣛⣓⠢⢄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⠀⠈⢢⠘⣧⠀⠀⠀⠀⠀⠀⠀⢰⡟⢸⣷⣤⠶⠿⠿⣿⣯⡒⢤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢲⡀⠘⡇⣿⠀⠀⠀⠀⠀⠈⠀⣻⡧⠞⠋⠀⠀⠀⠀⠀⠈⢻⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠧⢄⠘⢦⠘⢾⡀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⠛⠄⠀⠀⣃⣠⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠋⠉⠀⠀⠀⠀⢀⣀⣀⣤⠴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⢀⡠⠤⠖⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢶⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠉⠉⠉⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠓⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣀⣀⡽⠋⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⢤⣈⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⣀⠤⠴⠒⠒⠚⠛⠉⠀⠛⣆⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⠔⠋⠉⠉⠉⠉⠒⠄⠀⠀⠀⠀⠀⠀⠙⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠤⢄⣀⡀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠑⠲⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠴⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⢂⠈⡀⠀⠀⠀⠁⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣾⣼⠁⣶⣶⣆⡄⠀⠂⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣿⣶⣿⡿⢿⣿⣖⡀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇⠀⠀⠠⠙⠄
⠀⠀⢀⣠⣴⠶⠆⠀⠉⠑⠲⢤⡀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠠⠀⠀⠀⢠⡇⠀⠀⠀⠀⣂⠈⡄
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠉⠳⠖⠚⠉⠀⠀⠀⠀⠀⠐⢦⡀⠀⢸⡇⠀⠀⠀⠀⣿⡆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣿⣶⡾⣷⡀⠀⠀⣴⣿⣿⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⣦⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⢻⣦⡀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⡀⢿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⡿⣿⡿⠃⡄
⠀⠀⠀⠀⢀⣠⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠚⠛⠹⠛⠉⢁⣄⠸⣿⣿⣿⣿⣷⡼⠁
⠀⠀⠀⢠⣿⣿⣿⠟⠁⠐⢻⣿⠋⠀⠀⠈⠙⠻⢿⣿⣷⣤⣀⣀⣠⣶⣿⣿⡿⠿⠾⠿⠟⠋
⠀⠀⠀⠉⠁⠛⠋⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣷⣷⣿⣿⣿⣷⣶⣾⣾⣿⣿⣷⣿⣿⣷⣾⣾⣿⣶⣾⣷⣿⣿⣿⣷⣷⣿⣷⣶⣷⣾⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⣀⠀⠀⠀⢀⠤⢄⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀
⠀⠀⣴⢟⡿⣶⡔⠉⢶⡆⡇⠀⠀⢀⣤⣄⢈⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡗⢠⠱⣶⡎⡆
⠀⠀⠛⣿⣿⣯⠀⠁⠂⠊⠀⢀⣴⣾⣿⣏⠀⠀⠀⠀⣴⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⢀⣾⣿⣿⣆⠀⠀⠐⠁
⠀⠀⠀⣻⣿⣿⡀⠀⠀⠀⠀⠘⣿⣿⣿⠛⠇⠀⠀⠀⡟⣯⣯⡽⣧⠀⠀⠐⢤⢸⣿⡉⠁⠀⠀⠀⢸⣿⣿⣿⠉
⠀⠀⠀⣿⡏⣿⡇⠀⠀⠀⠀⠀⠹⣿⣿⣦⡀⠀⠀⠀⠃⣿⣿⡇⠋⠀⠀⢠⡞⢛⠙⢷⣶⣆⠀⠀⠘⢿⣿⣿
⠀⠀⣰⡿⠀⣿⡇⠀⠀⠀⠀⠀⣠⣿⢗⣿⠃⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢿⡒⠿⠟⢸⠇⢿⣆⠀⠀⢸⣿⣿
⠀⠀⠛⠁⠀⠛⠀⠀⠀⠀⠀⠘⠛⠁⠘⠏⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠈⠛⠲⠞⠋⠛⠋⠉⠀⠀⠘⠛⠛
⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢴⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡦⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠈⠠⣾⣶⠀⠀⠀⠀⠀⠀⣴⣶⡄
⠀⠀⠀⢀⣼⣿⡆⠀⠐⠒⠀⠀⢀⣾⣿⣷⡄⠀⠀⠀⢠⣾⣿⣿⣷⠀⠀⠀⠀⠀⣾⣿⣿⣆⠀⠀⠀⢠⡴⣾⣿⣷⡆
⠀⠀⠀⣾⣿⣿⡿⣦⣤⠀⠀⠀⢸⡿⢯⢻⣿⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⠿⢿⣿⠀⠀⠀⡿⠀⣿⣿⣿
⠀⠀⢠⣿⣿⣿⣧⣄⠈⠁⠀⠀⠘⣿⣶⣿⡏⠀⠀⠀⠈⢹⣿⣿⠉⣥⣀⣀⠀⠀⢻⣦⣾⠟⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⣾⣿⣿⣿⡿⠿⠀⡇⠀⠀⢸⣿⣿⣿⡧⠀⠀⠀⠀⢸⣿⣿⠀⠿⠿⠟⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠛⠋⠛⠛⠃⠀⠀⠃⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠻⠟⠟
⠀⠀⠀⠂⠐⠐⠂⠀⠀⠀⠀⠀⠀⠂⠂⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣾⠂⠀⠃⣼⣧
⠀⠀⠀⠀⢀⠞⢻⡇⠀⢹⣿⣇
⠀⠀⠀⠀⣿⣷⣾⡇⠀⠸⣿⣿
⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⢻⡟
⠀⠀⠀⠀⠀⠙⠿⠇⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣁⡌⠙⣦⠀⠀⠀⠀⢀⣴⠿⣷⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠘⣧⠀⠀⠀⡼⢋⣀⠀⠀⠋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠸⠲⢄⠀⠂⠀⠉⠀⠚⠒⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣷⣶⣬⠀⠀⠈⢧⣀⠘⠟⠻⢆⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣈⣿⠀⠀⠀⠈⢻⣧⣤⣴⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠉⡿⠿⠟⠒⠙⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢹⣿⣇⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡎⢿⣿⣦⣀⣀⡴⠁⠀⠀⠀⠀⠀⢂⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠈⣽⡯⠰⣿⣦⣄⣀⣀⣭⣿⣦⡐⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠐⠀⠀⢀⣀⣀⡈⠙⠿⣿⣿⣿⣿⠃⠻⣦⣀⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⠉⠀⠉⠻⢶⣦⣶⡿⣿⠃⠀⠀⠈⠙⠿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣩⣿⣿⡾⠃⠀⠀⠀⠀⠀⠀⠙⠻⠷⠶⢶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠲⢦⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠊⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠰⠈⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡎⢦⣤⣤⣤⣤⡰⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠐⣿⣿⣿⣟⢥⡹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣧⣘⣿⣿⣿⣿⣶⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠹⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠙⢿⣿⣿⣟⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠙⠉⣻⣿⣿⠃⠀⠀⠀⠀⠀⠻⢿⠈⠈⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⣶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣧⣃⣻⣤⣭⣭⣭⣬⣥⣭⣥⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠒⠒⠒⠒⠒⠒⠲⠶⠦⠤⠤⠤⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣖⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡁⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⣶⣮⣭⣥⡬⣱⡦⣢⠄⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢻⣿⣿⢫⣼⣿⡿⠁⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠈⡿⣱⣿⣿⣿⡇⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢹⡆⢸⣿⣿⣿⣿⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⣇⣽⣿⣿⣿⡿⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠛⠒⠂⠀⠐⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣓⠀⣿⣿⣿⡛⡃⠀⠋⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⡥⢠⣤⣬⣦⣤⣄⠀⠀⣽⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠾⠿⠿⠟⠿⠑⠶⠾⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡵⠦⠶⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠇⡴⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠉⠀⠀⠀⠀⠈⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢼⣫⣾⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⣰⣿⣿⢐⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⢀⡄⠀⢀⣀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣾⡄⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣤⣷⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣶⣿⣿⣿⣏⠧⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠙⣿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⣿⣿⣿⣿⣇⣷⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣶⡤⢶⣿⣶⡿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⣿⣿⡿⣿⣿⡆⢿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣿⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣴⣶⣶⣶⣶⣶⠀⠘⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⡃⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⣿⣿⣿⣿⠇⣷⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠁⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣾⣿⢇⢀⡀⣛⠻⠤⠐⠒⠂⠈⠉⠏⠈⠁⣀⠀⠈⣙⠉⠍⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡭⢭⠭⠍⠉⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠈⠫⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢀⠀⠁⢈⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣀⣐⣀⣀⠲⡒⣶⣶⣢⣾⣿⣧⣿⣿⣿⠃
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⠬⠭⠭⠭⠭⠽⠿⠿⠟⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣸⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣻⠋⠀⠀⠀⠀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁⠀⠀⣀⣴⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣴⣿⣿⣿⠟⠁⠀⠀⢠⣾⣿⣯⣙⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣻⣿⣿⠟⠁⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢿⡉⠀⠀⠀⢰⣿⣟⡻⢿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣾⠟⠁⠀⠀⠙⠀⠀⠀⠀⠛⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢸⣿⠀⣿⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣶⣶⣴⣶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⣶⣿⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣦⡀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⣣⣿⣿⣿⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣾⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠛⢛⠻⠷⢯⢭⣥⣶⣾⠿⡟⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⡿⣿⣿⣿⣿⣿⡿⠛⣹⡿⡿⣿⣧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⢿⣗⣾⣿⣿⣿⣿⣿⣿⣧⣤⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣯⣿⣿⣯⣿⠟⠛⠉⠉⠛⠛⠛⠻⠿⣿⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠋⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⣩⣄⣠⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣥⣤⣴⣶⣶⣶⣾⣷⣶⣦⣤⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⣝⣻⣿⠉⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣿⡏⣿⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠋⠀⢹⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠈⣷⣸⣆⠈⢦⡀⠀⠀⠀⠀⠈⢩⠁⣀⡀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣆⠀⠳⠀⠀⠀⠀⢀⣀⣤⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⢻⣷⣄⡀⠀⠀⢰⡿⠛⠉⠁⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⣷⣴⣶⣿⡁⠀⣲⣶⡖⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡀⠈⠻⣿⡻⢷⣍⠓⠀⠉⠙⠛⢛⡍⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⢧⣿⣿⣿⣿⡎⢇⠀⠀⠀⠙⢦⡈⠁⠀⠀⢀⣴⣿⠃⠀⣿⣷⣄⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠉⢦⣀⣰⣿⣿⠏⢀⡀⢸⣿⣿⣧⡘⢿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣴⣄⠀⠘⣿⣿⣿⡟⢠⣾⣷⢸⣿⣿⣿⣿⣦⡙⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢰⠛⠻⣦⡀⠘⣿⡿⠀⢸⠟⢻⡆⣿⣿⣿⣿⣿⢟⣬⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡞⠀⠀⠈⠳⢠⡿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠉⠀⢰⠀⡠⢀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡄⠀⠀⠀⠀⠀⠀⢠⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⡀⣀⠀⡆⠀⠉⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠀⠀⢄⣉⣡⣍⣁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠿⠷⠂⠐⠾⠿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣈⣡⣌⣉⡠⠀⠀⢻⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⡀⢀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢻⣿⣿⣿
⣿⣿⣿⡟⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣰⠢⠧⠴⠜⡆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢻⣿⣿
⣿⣿⡿⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⠛⢦⣼⣧⡶⠛⣿⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢿⣿
⣿⣿⠃⠀⠀⠀⠀⠀⣿⠿⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠈⠁⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠿⣿⠀⠀⠀⠀⠀⠘⣿
⣿⣿⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⣿
⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⣿⣿⣿⣷⣶⣶⣆⣀⠀⠀⠀⠀⠀⢀⣠⠀⣀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣀⠀⣄⡀⠀⠀⠀⠀⠀⣀⣰⣶⣶⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⡇⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠰⠀⢸⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠰⠤⣀⡀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⣀⠠⠆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠠⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢸⡆⣇⣿⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣾⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢦⡀⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠂
⠀⠀⠀⠀⠀⢀⣴⠟⢁⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡆
⠀⠀⠀⠀⣠⠞⠁⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⡷⠂⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠠⣴⣾⣿⣿⣿⣿⣶⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣯⣥⠀⣄⣤⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣿⠀⣿⣯⣟⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣧⣶⣶⣿⣽⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣿⣿⣿⣯⣽⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡿⣷⣿⣾⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠈⣿⣿⠿⠟⠿⣿⣿⠿⠻⠿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠐⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠂⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃





⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⣤⣴⢿⣿⣿⣦⣤⡶⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⠙⢦⠀⠀⢀⡖⠋⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠻⣤⣄⠘⢿⣿⣿⣿⣿⠋⢠⣄⡿⠀⠀⠀⠀⠀⠀⠀⢀⢹⣄⣀⠼⠀⠀⠘⢧⣀⣤⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠻⣷⡟⠁⢘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠛⠁⠈⠻⠆⠀⣿⠇⠀⠿⠃⠈⠛⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣤⠀⣤⣿⣿⣿⣿⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣄⠀⣠⣿⣿⣿⣿⣿⠁




⣿⣿⣿⣻⣛⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣛⣛⣛⣻⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠲⠤⠶⣤⠴⠒⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢲⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⠴⠂⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣡⠖⠁⠀⠐⠒⢤⡁⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣁⠴⠖⠒⠚⠓⠶⣄⠀⠀⠀⠉⠒⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⠞⠁⠀⠀⠀⠀⠀⠀⠘⡆⢀⠐⠒⠦⠄⠀⠀⠙⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢸⠀⠀⠀⠀⠀⠀⢠⣤⢠⠟⠉⠉⢢⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠘⣆⠀⠀⠀⠀⠀⠘⠋⣿⠀⠀⠠⣼⠀⠀⠀⠀⠀⠀⣨⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠬⣗⠦⣄⣀⣀⣀⠴⠛⢢⠤⠐⠃⠀⠀⠀⠀⠀⠀⡿⠴⠒⠒⠲⣤⠀⢀⣴⠛
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠻⢄⣀⡅⠀⠰⢀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠶⢍⡲⠦⢸⡷⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠘⠂⠐⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⡠⢄⣀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠠⠊⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢓⠦⠤⢀⣀⣀⣀⣀⣀⣀⣀⡤⠴⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢁⡠⠂⠀⠀⠀⠉⠉⡁⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡀⠘⠓⢭⡉⠐⠲⠤⠚⠳⢤⣀⡜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠉⠉⠀⠉⠓⢤⡀⠀⢀⠀⠉⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⠤⠤⣄⣀⣀⣀⡈⢐⣉⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⢿⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠠⠧⣼⣄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢖⢳⢤⣀⣹⡀⠀⠀⠀⠀⠀⣀⡼⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⣏⢙⣩⣅⠊⠿⡩⣉⣉⡙⠟⠿⢚⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠑⠒⠲⠬⠭⠭⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⠀⠆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⣷⣾⣾⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⣿⡿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡻⠋⠀⠀⠀⠀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠏⠀⠀⠀⢀⣤⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣿⣿⠟⠁⠀⠀⢀⣴⣟⣟⡛⢿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⢾⣿⣿⣿⠟⠁⠀⠀⢀⣴⣿⣯⣿⣿⣟⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢯⡟⠁⠀⠀⠀⣰⣿⡿⢿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡷⠂
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠈⠙⠆⠀⠀⠀⠙⢿⣿⣶⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠙⠛⠛⠚⠛⠛⠉
⠀⠀⠀⠀⠙⢦⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠙⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠆⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⣇⣸⣷⣻⣿⡇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⢸⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⢸⠀⢳⠈⡏⠀⢸⡄⠸⣦⢸⠃⠘⡃⢸⠃⢘⠁⠀⠀⡇⠚⡂⢸⠃⠀⣾⠀⢻⠀⣶⠀⢠⡆⣾⠂⣶⡀⢰⠀⡟⠰⠋⠀⡇
⠀⢸⠸⡆⢸⠀⡇⠀⡇⡇⢸⣇⢸⠂⠀⡇⢸⠂⠘⡄⠀⠀⡇⠸⡇⢸⠂⢀⢹⡆⢸⠀⡏⣇⡞⡇⣽⠂⡇⢱⢸⠀⢰⠀⢣⠀⡇
⠀⢸⠀⡧⠃⠀⣇⢠⠃⡇⢿⡟⢸⡀⠀⡇⢸⣀⢀⡏⠀⠀⡇⢀⢣⢸⣀⢸⠉⡇⠸⠀⡇⠸⠁⡇⢿⡀⠇⠀⣾⠀⢸⠀⣸⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⢀⠐⡒⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⢤⢠⣠⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠖⠔⠀⠒⠂⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⢒⠒⣒⡒⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠀⡇
⠀⢸⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠄⠠⠀⠀⠤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠠⠤⡤⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢖⣲⣴⡆⡲⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠈⠀⡀⣀⣀⣀⠀⢀⡀⣀⠠⠀⠐⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣀⣀⠆⠀⠀⠀⠣⠀⡩⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠋⠙⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⡳⠾⠇⢄⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠃⠄⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡦⠤⠆⢀⣀⠀⢀⠀⢀⠀⡀⡀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠡⠀⠢⠐⢺⠀⠀⠃⠐⠁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠑⡀⠀⠤⡤⢤⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠉⠙⢩⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡠⠉⠴⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠂⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡾⠿⠿⠛⠛⠛⠛⠛⠛⠿⠿⢷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠉⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠉⠛⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋⠁⠀⢀⣠⣴⣶⠿⠿⠛⠛⠛⠛⠛⠛⠿⠿⣶⣦⣄⡀⠀⠈⠙⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⣠⣶⡿⠛⠉⡀⠀⣠⣤⣤⣴⡿⣷⣦⣶⣶⣄⠉⠛⢿⣷⣄⠀⠀⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⣠⣾⠟⠁⠀⣴⡿⠿⣿⡟⠉⢻⣿⠀⢸⣿⠁⠈⣿⡇⠀⠀⠈⠻⣷⣄⠀⠀⠹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⢀⣼⡿⠁⠀⠀⠀⣿⡇⠀⢸⡇⠀⢸⣿⢺⣿⡿⠿⠿⠿⠿⣷⣦⠀⠀⠈⢻⣷⡀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢀⣾⡟⠀⠀⠀⠀⠀⣿⡇⠀⢸⡧⠀⢸⣿⠀⢻⣿⣶⣶⡆⠀⠈⣿⡇⠀⠀⠀⢻⣷⡀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⣿⣷⣤⣿⣿⣤⣾⣿⣦⣾⣿⣧⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⢻⣧⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⣿⡏⠉⠀⢠⣽⡿⠛⠉⢁⣤⣾⣿⠇⠀⠀⣿⡇⠀⠀⠀⠀⠘⣿⡆⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣰⣿⠋⠀⠀⣴⡿⠋⠁⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠿⠇⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠙⣿⣦⣤⡀⠀⠀⠈⠉⠀⠀⢀⣀⣠⣴⣿⠋⠀⠀⠀⠀⠀⢰⣿⠃⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠁⠀⠀⠀⠀⠀⠀⢨⣿⡟⠉⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⢻⣷⡀⠀⠀⠀⠀⣀⣠⣼⣿⠃⠀⠀⠀⠀⠀⠀⠘⣿⣥⣄⣀⡀⠀⠀⠀⢀⣾⡟⠁⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠻⣷⣄⠀⢰⣿⣟⣋⢹⣿⡀⠀⠀⠀⠀⠀⠀⢀⣿⡋⣙⣻⣿⡆⠀⣠⣾⠟⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠈⠿⣷⣄⠉⠛⠻⢿⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠟⠛⠉⣠⣾⠿⠁⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⡀⠀⠈⠻⢿⣶⣄⣸⣿⠀⠀⠀⠀⠀⠀⠀⠐⣿⣇⣠⣴⡿⠟⠁⠀⢀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠿⠿⣷⣮⡻⢿⣦⣀⠀⠀⠈⠙⠻⠿⢷⣶⣶⣶⣶⣶⣶⡾⠿⠛⠋⠁⠀⠀⣀⣴⡿⢟⣥⣾⠿⠿⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣾⡟⢡⣶⡶⣶⣄⠹⣿⡄⠙⠻⢿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡿⠟⠋⢠⣿⠏⣠⣶⢶⣶⡌⢻⣷
⠀⠀⠀⠀⠀⠀⠠⣿⡇⢿⣏⡀⣨⣿⠀⣿⡇⠀⠀⠀⠈⠉⠛⠿⠿⢷⣶⣶⠀⠀⣴⣶⡾⠿⠿⠛⠉⠁⠀⠀⠀⢸⣿⠀⣿⣇⢀⣹⡿⢨⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢻⣷⣌⠛⠿⠟⢋⣴⡿⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⣸⣿⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡙⠻⠿⠛⣡⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠟⠀⠀⠻⠿⠿⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⣿⡟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⢿⣿⠀⠀⣿⡿⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠒⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠅⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢹⣧⡀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣜⣛⠁⠀⣀⣀⣀⡀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⢿⣷⠀⢻⠿⣿⠅⠀⠀⣼⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⢀⡟⠀⠀⠀⠀⠀⠀⠀⣉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡼⣿⣧⣶⣶⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡷⠦⠤⠛⠟⠀⢀⣾⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠉⠉⠁⠀⣰⣷⣿⠏⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣶⣶⣤⣾⡿⣯⡝⠀⠀⠈⠒⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⢿⣿⣿⣿⣿⣿⡟⣹⣿⠁⠀⠀⢀⣠⣶⠊⣡⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠇⣿⣋⣩⣽⣿⣿⣇⣿⣿⡶⠤⠴⠟⢻⠃⢰⠇⠀⠀⠀⠙⠓⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠉⠀⠉⠩⠀⢸⠀⠀⠈⠉⠻⣿⣿⠁⠀⠀⠀⢀⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⠀⠀⠀⠑⡯⠀⠀⠀⠀⠀⡸⠀⢰⠀⢤⡆⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⢠⡏⠀⠀⡆⠸⣿⡇⠀⠀⠀⠘⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠃⠀⢼⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣾⡷⠀⠀⠁⠘⢿⣷⣀⡅⠀⠀⢣⡀⠀⠀⢠⣷⡀⠀⠀⠀⡜⠀⠀⣿⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠩⣿⣿⣿⡄⠀⠀⠀⠱⢤⣀⣼⣿⡃⠀⣠⠞⠀⠀⢀⣽⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠀⣀⣀⣤⣿⠟⠉⠛⠋⠁⣀⣀⣴⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⢰⣿⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠛⠉⠙⠛⢧⣄⡀⠀⣠⣴⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⣿⡏⠀⠀⣠⣾⣿⣿⣿⣿⣵⣦⣤⣤⣤⣄⡤⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⣿⠃⠀⠀⠁⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣍⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠀⠀⠀⠀⢀⠠⠃⡸⠁
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡅⡼⠁
⠀⠀⠀⢿⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⡾⠋⡾
⠀⠀⠀⢸⣇⢰⣶⣶⣄⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⣀⣀⣉⣱⣒⣛⣳⣾⠃
⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠚⠉⠉⠉⠀⣿⣿⡿⠟⠛⠉⠁⢠⠉⠻⠟⠻⢿⣿⣿⣿⣿⣿⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠶⠓⠒⠛⠚⠚⠓⠒⠒⠒⠒⠢⠤⠅⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⣠⣶⣷⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠀⠀⠀⣴⣿⣿⣿⣶⡀
⠀⠀⠈⢉⣿⣿⣿⣿⣿⡄⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⠃⠀⠀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⡤⣄⠀⢀⣆⢠⠂⢠⡄
⠀⠀⠀⠀⢿⣿⣿⣿⣿⠁⠀⠀⢰⣾⣷⡆⠀⠀⠀⠀⣰⣾⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⡏⠀⠀⠀⠀⠉⠁⠉⠈⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠈⠙⠛⠛⠁⠀⠀⠀⠈⠿⠿⠃⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠘⠛⠿⠟
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢠⣴⣶⣄⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠨⣼⣾⡀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⣿⣧⠀⠀⠀⢀⣤⣶⣶⣶⣤⣤⡄⠀⢀⣤⣶⣶⣦⣀⠀⠀⣹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⠳⣿⣿⠇⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢹⣿⡏⠀⠀⠀⠸⠿⠿⠛⠋⠉⠉⠀⠀⠛⠛⠿⠿⠿⠛⠁⠀⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠹⣿⠆⠀⠀⠀⠀⠿⣿⣿⠿⠄⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⠿⣿⣿⣿⣿⠇
⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣤⣤⡀
⠀⠀⠀⠾⢿⣿⠀⠀⠀⠀⠀⠀⣶⣾⣷⣷⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⣀⡀⣿⣿⣧⠀⠀⠀⠀⢴⣦⣴⣼⡇⠀⠀⠀⠀⣻⣿⣿⡇
⠀⠀⠀⢉⣀⣈⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⠞⣡⣤⠀⠀⢰⣿⣷⠈⠙⢿⣦⡀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⢹⣿⣿⠃
⠀⠀⠀⠛⠛⠻⢤⣄⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢠⠞⠁⠺⠟⠁⠀⠀⠀⢹⣿⠀⠀⠀⠉⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣠⣼⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠴⣿⣿⠦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⡁⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⠇⣊⡆⢶⡴⣉⠆
⠀⠀⠀⠀⠣⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠲⣯⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡏⠉⠻⠶⡄⠀⠀⠀⠀⠀⠀⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣟⠀⠀⠀⠀⠀⠹⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⢰⣦⣴⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⡀
⠀⠀⠀⠹⠟⠛⠀⠀⠀⠀⠀⠀⢀⠩⠭⣽⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠆⡀⢀⠀⢂⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣚⠛⠛⠀⠀⠀⠀⠀⠀⠀⠘⠙⣻⠀⠀⠀⠀⠀⠠⠋⢀⡀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠁⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⣀⣦⡠⠀⠀⠀⠀⠈⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠦⠤⠤⠶
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡴⠂⠐⢢⠀⠀⠀⠀⠀⠀⢠⣒⣂⣐⡆⠀⠀⠀⠠⡦⠦⠴⠤⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⣏⠉⠉⢹⠆⠀⠀⠀⠀⠀⣎⢁⠀⡈⣑⠀⠀⠀⢀⣳⠨⡍⠑⢋⠃⠀⠒⡄⠀⠀⠀⠀⢹⡝⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠔⠒⠂
⠀⠀⠀⠀⠀⠀⠀⣋⣩⠢⣉⠀⠀⠀⠀⠀⠀⡠⡃⠿⢈⠄⠀⠀⠀⠀⣉⠉⠉⠉⠁⠀⠀⠤⠂⠀⠀⠀⠀⢸⣦⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘
⠀⠀⠀⠀⠀⠀⠀⣯⡬⢛⢶⠀⠀⠀⠀⠀⠀⣅⣘⣈⣃⣠⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠄⠠⡀⠀⠀⠀⠀⢸⣉⡇⠀⠀⠀⠀⢘⣃⣈⣀⠕⠋⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠑⠌⠲⠋⠀⠀⠀⠀⠀⠀⠸⠇⠀⠸⠇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠻⠝⠠⠆⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠸⠧⠰⠠⠲⠄⠲⠤⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⢤⠀⠀⠀⠀⢀⠤⠤⠤⠤⠤⠤⠤⡀⠀⠀⠀⢀⣀⡤⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠆⠀⠀⠀⠀⠀⠀⠀⡱⠦⠴⢎⠀⠀⠀⠀⢸⣲⣒⣖⣒⣒⣲⣲⡇⠀⢀⣜⠝⢉⠜⡫⣈⣫⢢
⠀⠀⠀⠀⠘⡭⠩⠭⠭⠭⠭⠍⢭⠃⠀⠀⠀⠀⢸⠦⡆⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⢸⠀⠀⠀⠀⠀⣁⣀⣸⢩⠉⣉⣿⠀⠀⢨⡇⢀⣮⣥⣴⣻⡎⢱⠃
⠀⠀⠀⠀⠀⠘⡤⠠⣀⢀⡀⢀⠆⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡶⢶⢺⠀⠀⠀⠀⠀⢉⠀⢹⣶⠆⠀⡍⠀⠀⠠⡷⣺⡗⠋⠉⠹⢁⣜⠂
⠀⠀⠀⠀⠀⠀⠹⣖⣉⣆⣅⡌⠀⠀⠀⠀⠀⠀⢸⣴⡆⠀⠀⠀⠀⠀⠀⠀⣏⢂⡒⣺⠀⠀⠀⠀⠀⢨⣀⣈⣀⣈⣉⡇⠀⠀⠀⠙⠾⣧⣤⣤⣶⠙⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⣦⢸⣷⠀⣰⣿⡆⠀⣿⣛⣷⠄⣿⣟⣛⠒⠛⣿⠛⠂⣿⣛⣛⠠⣾⣛⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⡟⢸⣿⢰⡿⠾⣿⡀⣿⣭⣿⠆⣿⣯⣭⡄⠀⣿⠀⠀⣿⣭⣭⠰⣮⣽⡿


⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⣀⡀⠀⡀⠀⠀⣤⡺⠙⠛⠋⢷⣤⠀⠀⠀⠔⠉⠈⢈⠋⠉⠉⠃⠀⠀⣸⣌⣁⣡⠿⣌⣉⣨⣇⠀⠀⢠⢲⠝⠛⡛⢫⠒⡄
⠀⠀⠀⠀⠈⣸⣇⡍⣸⣇⠉⣸⣇⠁⠀⡜⢿⣌⠋⣛⡽⣻⡵⢣⠀⠀⠂⠀⢀⠘⠂⠀⠀⠀⠀⠀⡗⠏⠠⠹⠲⠋⠤⣽⢚⠀⠀⡇⣘⣒⢞⣓⣚⡀⢸
⠀⠀⠀⠀⠀⡯⠽⡇⡯⢽⠀⡯⢽⡄⠀⢇⣠⢿⡿⠿⠿⢹⣅⣻⠀⠀⠄⠀⢈⠀⠐⠀⠀⠀⠀⠀⡟⡃⡿⠙⠛⣠⡗⢸⢻⠀⠀⡇⠃⠀⠐⠀⠀⡈⠸
⠀⠀⠀⠀⠀⣇⣀⡇⣇⣰⠀⣇⣸⠇⠀⠘⢼⣏⣒⠒⣒⡲⠿⠃⠀⠀⢀⠀⠀⠀⠔⠀⠀⡀⠀⠀⡇⡤⡍⠀⠀⡟⠂⢸⢸⠀⠀⠰⡘⣄⣀⣀⡠⠁⠆
⠀⠀⠀⠀⠐⠒⠓⠛⠒⠓⠚⠒⠒⠃⠀⠀⠀⠉⠙⠒⠋⠁⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠁⠀⠀⠁⠑⠒⠊⠀⠓⠒⠊⠘⠀⠀⠀⠉⠐⠚⠓⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣶⣴⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⡂⠐⠒⢄⠀⢀⣤⢄⡀
⠀⠀⠀⠀⠀⣀⡀⣀⣀⣄⢀⣀⢀⠀⠀⠀⠀⡜⠁⢹⠌⢿⠼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠳⣘⠀⠃⠋⠜⠀⠘⠧⣀⡿⢤⡀
⠀⠀⠀⠀⢸⠄⠀⡀⠀⠀⢀⠀⠠⡇⠀⢰⠒⠓⡖⢺⠖⢺⠒⡆⠀⢀⣰⣉⢉⠤⠀⠈⠉⢗⠆⠀⠀⠘⢄⢅⡝⠉⠉⠁⠀⠀⠀⠀⠀⠈⠒⢿⡤⠠
⠀⠀⠀⠀⠀⣩⣿⣯⠭⢽⣿⣿⣍⠂⠀⠸⠉⠉⡧⢸⡄⢼⠤⡅⠀⠨⣐⡿⢸⡘⠇⠏⠿⢸⠀⠀⡤⠒⠛⡞⠓⠢⡄⠀⠀⠀⢐⣈⣀⣀⣀⣉⣀⣀⣈⡂
⠀⠀⠀⠀⠘⠂⠀⠁⠰⠄⠈⠈⠐⠃⠀⠀⠆⠤⠦⠴⠦⠼⠤⠆⠀⠀⠈⠢⠤⠭⠭⠭⠍⠑⠁⠀⠧⠶⠤⠧⠴⠤⠆⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣿⣿⣿⣿⣿⣷⡄⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⠿⠿⠻⡿⠿⡿⠿⠿⡻⢟⢟⣿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣷⣔⡞⣵⣬⣷⣤⣽⣬⣿⣾⣼⣭⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢏⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢋⡌⣹⡗⣾⣰⣀⣇⣶⣀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣟⢋⢉⡁⡙⢭⡙⠩⡉⢽⣍⣽⣿⣿⣿⣿⣿⣿⣿⡇⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠉⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⢸⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠟⠁⠀⠀⢹⣿⣿⣿⣿⣿⣿⡟⣿⢹⣏⣾⣻⠻⠛⡟⣛⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠋⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣴⣧⣾⣧⣿⣾⣶⣧⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠛⠋⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⡿⠋⢉⣽⡿⠟⣽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠚⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠛⠛⠿⠿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠻⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢣⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⣶⣶⣦⣤⣤⣄⡠⢄⡬⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⠀⡀⠀⠀⢇
⠀⢠⣦⢶⠒⣀⣤⡀⢐⠠⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡯⣶⡵⣂⠉
⠀⠀⠀⠀⠈⠻⣿⣿⣶⣬⣓⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠁⠀⠀⠀⠀⢀⣤⣀⡀⠄
⠀⠀⠀⠀⠀⡐⡰⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣶⣿⣿⠿⣻⠂
⠀⠀⠀⢀⢠⣷⣻⣿⡿⢿⣷⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣽⢷⣶⡔
⠀⠀⠀⠶⠿⠿⠛⣧⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⠻⣿⣿⣿⣿⣯⣟⣿⡿⠻⠛⠛⢉⣥⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⡹⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣴⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠈⠙⢻⣿⣿⡅⠀⠀⠀⠀⢢⣿⣿⣿⣿⣿⣿⣿⣿⣛⣉⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠀⠀⠐⣿⣿⡇⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣿⣿⣿⣿⣿⣿⡿⣿⠏⠀⠀⠀⠀⣽⣿⡷⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⢿⡯⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠭⣽⣿⣿⣿⣿⣟⠭⠊⠁⠀⠀⠀⠀⠀⣽⣿⡗⠀⠀⠀⠀⠀⠀⠀⠉⠚⠩⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠥⣞⡽⣻⠽⡫⠅⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠑⠩⠛⠉⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣴⣶⣶⣿⣿⣶⠄⠀⠀⠀⠀⠀⠸⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡑⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢛⣻⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⡝⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣾⢎⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⡷⡳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠵⠫⠊⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠁⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠐⠂⠐⠋⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠋⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠁⠀⠀⣆⣠⣶⠀⣄⡀⣿⣿⠏⢻⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢁⠀⠠⠄⢸⡿⠋⠅⠀⣿⡇⠀⠀⡦⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⠀⠀⠀⣴⠀⠀⠀⠀⣿⡇⠀⢰⠃⣼⣿⣿⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⣿⡀⢰⠀⠀⣿⡇⠀⡈⢸⣿⣿⡇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢀⣿⠁⠸⠀⠀⣿⣇⠀⠁⣿⣿⣿⢠⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣿⡀⠀⣿⣿⣿⢀⠞⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣀⢀⡴⣿⢿⣇⠀⢸⣿⣿⠆⣐⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠋⠉⠩⠞⠉⣋⣹⡀⠚⢻⣿⠐⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⡛⠦⣀⠐⠛⠛⠛⠛⣧⠀⣾⠃⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⣧⣤⢀⡀⢠⣤⣤⡀⣼⣷⠀⡀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢸⠛⠛⠀⠀⠈⠋⠉⠁⠉⣹⣇⠙⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠿⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⣤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⢠⠄⣀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣉⣿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⣿⣿⠟⠉⠀⠙⢻⡟⠀⠀⠻⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢸⡿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⠀⠀⠀⠀⢀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⠁⠀⠀⠀⠈⠉⠃


⠀⠀⠀⠀⠀⡀⢀⡀⢀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠿⣿⡿⠿⢿⣿⣿⠿⠿⠿⠿⢿⡿⠿⠿⠿⣿
⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⡀⠀⠑⢄
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠠⠔⠁⠂⠤⠀⠑⠄⠀⠠⠄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡗⠒⠒⠒⠒⠒⠲⠖⠒⠒⠶⠦⠤⠤⠤⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢀⡀⠀⠘⠀⣀⣀⠀⠀⠀⢀⠜⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⠠⠂⠀⠀⢀⠁⠠⢀⡈⠀⠀⠈⣰⠂⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡗⠒⠒⠒⠒⠒⢺⠗⠒⠒⠀⠀⠀⠀⢸⠇⠀⡀⠀⠀⢈⠱⣀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⢀⣀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⠀⠑⣧⠀⠀⠀⠀⠀⠀⣈⠀⠀⢠⠂⡄⠀⠀⠉⠉⠉⠀⠁⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣈⠗⢄⡀⠀⠀⠙⠈⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠋⠀⠀⢐⡄⠀⠂⠹⡅⠀⠀⠀⠁⠀⠐⠊⠀⡉⠁⠒⠃⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⡇⠀⣤⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠄⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠉⠉
⠀⠀⠀⠀⠀⡇⠀⣿⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠘⡀⠀⠀⠈⢣⡀⠀⠀⢀⠔⠁⡜⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡌⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠟⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠈⠢⡀⠀⢀⣠⠤⠐⠁⠀⡜⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠓⡀⠁⠐⠒⠚⠓⡲⠂
⠀⠀⠀⠀⠀⡇⠠⡂⣦⠱⢎⡤⡤⠤⡤⢤⡤⠤⠤⡌⠀⠢⣥⢀⠀⠀⠈⠉⠁⠀⠀⠀⢀⡜⠁⠀⠀⠀⡘⠀⠀⠀⠀⠀⠠⠁⡤⠊⠁⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⡇⠀⠙⠁⣠⠔⠁⠀⠀⡇⠊⠀⠀⢰⠁⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢠⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⡇⠀⠀⡔⠁⠀⠀⠀⠀⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⢠⠃⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃
⠀⠀⠀⠀⠀⡇⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢀⠀⠠⡀⠀⠠⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⡴⡙⠀⠀⠀⠀⢠⠃⢀⡜⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⡜⠀⠀⠘⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢡⠃⠀⠀⠀⠀⠃⢀⠞⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⡇⠀⢀⣀⣌⣦⣤⣤⣄⣀⣰⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠃⠀⠀⠀⡠⠀⢠⠎⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⡖⠒⠓⠒⠚⠀⠀⢠⡢⠄⠀⠀⡏⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⡴⠁⠀⠀⠀⡔⠁⣰⢯⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠁⠀⠀⡜⠁⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⠀⣠⠞⠀⠀⠀⢀⠞⠀⡴⠃⠀⠃⠀⠀⣧⣀⠰⠶⠶⠶⠶⠶⠮⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣠⠞⠁⠀⠀⠀⡴⠃⢀⡞⠁⢀⡴⢣⡀⡸⠁
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⣠⠊⠀⣰⢯⠴⠚⠁⠀⠀⣷⠀⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣸⠀⠀⠀⠀⠠⠄⠒⠊⠉⠀⠀⠀⠀⢀⡠⠊⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡐⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣆⣀⣀⣀⣀⣀⣀⣀⣠⣤⣐⣈⣀⣀⣀⣤⣞⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢠⣤⠴⠞⣛⣉⡙⣦⡀
⠀⠀⠀⠀⠀⣾⡟⠛⠉⣉⣉⣻⣾⠹⡆
⠀⠀⠀⠀⢀⡟⠴⠂⠀⣀⣀⣈⣿⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠤⠤⠤⠄⠀⠀⣀
⠀⠀⠀⠀⢸⡇⠐⠋⠡⣤⣀⡈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⠀⠠⣤⣤⣤⣤⡴⠞⠁
⠀⠀⠀⠀⢸⠁⠀⠀⠈⠙⣯⣴⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⢸⠀⣰⠀⠀⠀⣸⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠸⣇⠘⠓⢶⣚⣫⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⠞⣳⣒⡫⠍⠂
⠀⠀⠀⠀⠀⢻⡄⠀⠈⠙⠽⣄⠻⣄⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠖⠺⠿⢶⣤⣾⣤⣄
⠀⠀⠀⠀⠀⢸⡇⠘⣆⠀⠀⠘⢦⠙⣦⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⣌⢳⡄⠙⢶⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠹⣆⠀⠀⠀⠀⠈⢷⣄⠀⣸⠏⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢻⣄⠀⠻⣦⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⡄⠀⠀⠀⠀⠀⠈⠻⣿⢠⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⠳⢦⣈⠻⢶⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠘⣧⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢰⠀⢀⠈⢷⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣿⠰⣄⠈⣿⡷⣄⢀⣀⠀⠀⠀⠀⠸⣇⠹⡆⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠈⡇⣸⠀⠘
⠀⠀⠀⠀⠀⠀⠸⣇⠚⠀⣿⠈⠚⠛⠒⠂⠀⠀⠀⠀⣿⡿⡅⠦⣤⣤⣤⣴⠾⠋⢀⣴⡣⠀⠀⣧⡏
⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠸⣦⡀⠀⠀⠀⠀⠤⠤⠞⠁⣧⠀⠀⠀⠀⠀⣀⢄⣀⢯⠏⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠈⡛⠛⠶⠶⠶⠖⠂⠀⠀⠏⢀⡆⠀⠀⠀⠈⠳⠥⠬⡇⠀⠀⣾⠛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠙⢦⡀⠀⠀⠀⠀⢠⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠃⠀⠈⠻⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣙⠓⠒⠒⠀⢀⣠⣤⣤⣤⣤⣤⣤⣶⠶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆

⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⡀⠤⠠⠀⠀⢠⠀⢄⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⢠⠀⡀⠄⠀⠀⠠⠀⠀⠠⠀⠀⠄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣿⣿⣿⣿⡆⠀⠀⠀⠰⣟⠢⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⣷⠥⡤⢼⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿⣿⣿⣿⡇⠀⠀⠀⠰⢮⠶⣌⣖⠒⠢⢄
⠀⠀⠀⠀⡼⠉⡉⠒⣄⣙⡛⡟⠋⠀⠀⢸⡇⠀⠀⢰⣿⣧⠀⢠⡄⠀⠀⠀⠀⠀⣿⣘⣛⣛⣛⣻⡇⠀⠀⠀⢠⠯⡑⡾⠀⠟⠀⡜
⠀⠀⣤⣤⠃⠀⠋⡩⠥⢤⠤⠥⠃⠀⠀⢸⡇⠀⠀⠈⢛⣩⣴⣿⡇⠀⠀⠀⠀⠀⢻⣯⣭⣯⣭⣿⠇⠀⠀⠀⠈⠳⣏⠀⠀⢀⢜⡀
⠀⠀⣿⣿⠀⠀⠀⡯⢭⡟⠀⠀⠀⠀⠀⢸⡧⣴⣶⣾⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠙⠛⣯⠛⠉⠀⠀⠀⠀⠀⠀⠈⠓⢖⣵⡿⠋
⠀⠀⠙⠉⠓⠒⠚⠓⠚⠁⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⢦⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣄⠀⢸⡇
⠀⠀⠀⠀⠀⣀⢸⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⣿⠠⠇⠀⠡⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠻⠿⠂⢸⡇
⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡟⠛⡟⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⡟⠛⣿⡇
⠀⠀⠀⠀⠀⣿⢿⡇⣿⣿⠁⠀⠀⠀⠀⠀⠀⠘⠻⡿⣻⡿⠟⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⡤⣤⣤⣤⣤⣤⣄⠀⠘⠿⣿⡻⡿⠛
⠀⠀⠀⠀⠀⠻⠾⠃⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠤⠈⠙⠒⠒⠒⠒⠛⠛⠛⠛⠚⠋⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⢩⡍⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣘⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⢿⣿⢺⡇
⠀⠀⠀⠀⢀⣾⣿⣿⢇⠮⢦⠀⠀⠀⠀⠀⠀⡔⠒⠢⢄⠈⠷⣾⡶⠟⠀⠀⢠⣆⠀⠀⢀⣾⣆⠀⠀⠀⠀⠀⠀⣿⢿⣿⣾⣿⢾⡇
⠀⠀⠀⢠⣿⢿⣿⣏⣃⣐⡀⠃⠀⠀⢀⣀⢸⠁⠸⠒⣚⣋⣉⣘⣰⠀⠀⠀⢸⣿⣷⣄⡘⠾⠟⠀⠀⠀⠀⠀⠀⣿⣟⢛⣟⢛⣿⡇
⠀⠀⠀⠘⣷⡈⠻⢿⣿⣿⠃⠀⠀⠀⢸⣿⢸⠀⠀⢠⣓⣒⡏⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠘⠻⠿⡻⠿⠋
⠀⠀⠀⠀⠈⠛⠷⠾⠛⠁⠀⠀⠀⠀⠘⠿⠸⠤⠤⠼⠥⠜⠀⠀⠀⠀⠀⠀⠘⠷⠭⠭⠭⠭⠭⠭⠭⠧⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠞⠛⠛⠛⠳⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⢦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡴⠶⠶⠾⣿⢛⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠛⠋⢹⡇⠀⠀⠀⠀⠀⠿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⣿⡀⠀⠀⠀⠘⠇⠀⠀⠀⠀⣀⣀⣾⣁⣀⠀⠀⠀⣿⣶⣶⣾⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⠀⠀⠀⢈⣁⣀⣠⣤⣤⡴⠶⣶⡾⠛⢉⣿⠋⠉⣿⠀⠀⣼⠋⢩⡿⠛⠛⢻⡆⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠟⠛⠛⠉⠉⠁⠀⠀⠀⢠⡟⠀⠀⣾⠃⢀⣼⢃⣠⠞⠁⠀⠘⣧⠀⣀⣼⣿⡿⠟⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠙⠿⣿⠿⠛⠁⣀⣀⣤⡴⠟⠛⠉⠀⢻⣄⣀⣤⣿⠷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⣶⡶⠾⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⣀⣽⠿⠛⣷⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠉⠻⣤⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢠⣿⠟⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⣀⣠⣤⠾⠛⠉⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⠛⠛⠛⠿
⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢿⡀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠀⣿⣇⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⡶⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡿⠟⠁

⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡀
⠀⣠⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⡀
⣰⣿⣿⣿⡛⣷⡀⠀⠀⠀⠀⣰⣿⣿⣿⡛⣷⡀
⣿⣿⣿⣿⡇⢸⡇⠀⠀⠀⠀⣿⣿⣿⣿⡇⢸⡇
⠹⣿⣿⣿⣴⡿⠁⠀⠀⠀⠀⠹⢿⣿⣿⣴⡿⠃
⣶⣶⡿⠿⠷⠶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣖⣠⠤⠖⠒⠛⣉⣩⠭⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠋⠁⠀⠀⠀⠀⠀⠈⠀⢉⣉⡰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠤⠴⣖⡒⠋⠭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡽⠀⠀⠀⣀⡀⠤⢰⠂⠉⢠⠖⠈⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡶⠀⠀⣴⠎⠉⠀⠔⢡⠔⠓⢦⡇⠀⠀⠀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢀⠀⠀⠘⢧⡀⠀⠀⢻⣀⣠⠤⠛⣶⡏⣁⣴⠡⠒⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠛⡄⠀⠀⠀⠙⢆⠀⠀⠙⠥⣀⠰⠃⠀⠀⣁⣀⣀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡰⢄⠀⢀⣈⣣⠀⠀⣤⡤⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⠋⠴⠶⠂⣠⠀⠐⡻⡄⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠔⠋⠀⠑⡀⠱⣚⣦⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠈⢤⠛⠂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠁⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⣮⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠈⠹⡧⣀⣀⣀⠀⠀⠀⠣⡸⠖⡂⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⢱⠀⠈⢹⡧⠤⠄⢊⣠⠈⡆⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡇⠀⠀⢳⡄⣸⠓⠂⠉⠁⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣤⣴⡶⠧⡤⢤⠄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠆⢠⠃⠀⠇⢸⠀⠀⢣⡀⠀⠀⠀⠀⢀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢠⢸⡤⠢⠁⠈⡦⡤⠒⠣⢀⣀⠀⢰⣿⠧⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠈⠀⢰⠁⢀⢀⣷⡘⣄⣀⠀⠀⠉⢻⣿⣈⡉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⣊⡤⡤⣼⠋⠛⠛⢳⠋⠉⠉⠒⠢⣄⢸⣧⢀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠏⣠⣇⢘⠢⡀⠀⡌⠀⠀⠀⠀⠀⠀⠙⢿⡄⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠝⢺⠆⠆⠄⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣯⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡏⠛⠛⠉⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠛⢁⣀⡀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⡄⠀⠀⠈⠉⠉⠈⠉⠄⡰⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠈⢀⣷⡀⠀⠀⡀⣴⢶⣴⣿⣶⣄⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⢸⣿⡿⠂⣤⣄⢀⣼⣿⣿⣿⣿⣷⣼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⢈⡿⠋⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣰⣿⣿⣿⣿⣿⣯⠀⠈⠉⠛⠉⣿⣹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⡼⣾⣿⣿⣏⣿⣿⣿⠟⠀⠀⠀⢠⣇⠘⠇⠑⠁⠂⠠
⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⣽⣿⢣⣾⣿⣿⣿⠏⠉⠁⠀⠀⠀⢀⠜⣿⣦⢀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠸⢃⣿⣿⣿⢹⡧⢀⣠⡤⣶⣶⡾⠃⠀⠈⠻⣿⡀⠀⠀⠘⢿⣇
⠀⠀⠀⠀⠀⠀⠘⡆⠀⢀⠠⠂⠀⠀⠈⢘⡟⣻⣿⡿⠟⠋⠀⠀⠀⡀⠀⠀⠀⠀⠈⠳⡀⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠙⠀⠀⠀⠀⠠⠞⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢦⢠⡖⠀⣀⣴⡟⢠⡾⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠐⠿⠛⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣾⣷⣶⣶⣴⣴⣶⣾⣷⣶⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣼⣿⣿⣛⣿⣿⣿⣿⣿⣋⣽⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⢋⣙⠉⢿⠿⠹⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢏⣀⢾⣿⣧⣇⢰⣷⣿⣿⣿⣯⢿⡗⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣗⢿⣿⣿⣿⣿⣾⣿⣿⣿⡏⠛⠇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠛⢷⣝⣿⣿⣿⡿⠻⠿⠎⠃

⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠰⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀
⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠄⠀⠀⠀⠀⠀⠀⢀⣠⠔⠁⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣤⠎⠀⠀⠀⠀⢤⢀⡔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⣠⣤⠀⠀⠁⢠⣦⡄⠀⣰⠏⢠⣶⡄⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⣹⣿⣤⣤⣤⣬⣿⣧⣾⣃⡀⠘⣿⡃⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⠉⠙⢻⣿⣿⠁⠉⠻⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣸⣿⣿⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⠀⢠⣿⣿⣿⡄⠀⢀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⣾⣿⣿⣿⡇⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣧⣿⣿⣿⣿⣇⣼⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣿⣒⠀⣤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡿⢟⢷⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣏⠳⠞⢿⡟⠫⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣷⣦⢺⣿⢄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣏⡡⠄⢺⣿⣿⣿⣿⣷⠹⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢠⡀⣀⡀⠀⢀⡾⠿⣿⣷⠀⡆
⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠈⠉⠐⢴⣦⣿⣿⣿⢟⣋⠑⠢⠒⠋⠉⢐⣢⣤⡤⠶⠒⠒⠶⣖⣉⡀⠀⠀⣤⣶⣾⣷⣿⠿⢶⣶⠮⣵⣿⣯⡼⠁
⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⢀⣠⠀⠀⠀⠀⠹⣿⢇⣼⣿⣶⡦⠀⢰⣶⣎⠉⠀⠀⠀⠀⣠⣴⣿⣿⣿⣷⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⢸⣿⡃⠀⠀⣠⣾⣧⡀⠀⠀⠀⣠⠟⡿⢿⣿⡿⣰⠓⣸⣿⣿⡷⠦⢴⣶⡿⠟⠋⠀⠉⠉⠁
⠀⠀⠀⠀⠈⣿⡇⠀⠰⠟⠀⠘⣿⣶⣄⣼⣵⣷⣿⡿⠿⢷⣡⣾⠟⠉
⠀⠀⠀⠀⢸⣿⢧⠀⠀⠀⠀⠀⣿⠿⠋⠀⣨⡝⠉⠀⠀⢠⣿⠏
⠀⠀⠀⠀⢸⣿⣷⣄⠀⠀⠰⠋⠀⠀⠀⣸⡿⢳⣶⣦⠴⠋⠁
⠀⠀⠀⠀⠀⢿⣿⣿⣷⡄⠀⡔⠀⠀⣼⡿⠁⠈⣿⠃⡜
⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣤⣤⣀⣴⠿⠚⠂⢀⣿⡜
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠋⢉⡴⠶⠒⠀⣾⠁
⠀⠀⠀⠀⠀⢈⣿⣿⣿⣯⠀⠀⡰⠶⠦⢤⣏⡟
⠀⠀⠀⠀⢀⣾⣭⣽⣛⣛⣿⠿⢗⠒⣒⣻⡋
⠀⠀⠀⠀⠋⠀⠀⠈⠈⠃⠛⠙⠛⠙⠙⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣟⣿⣻⢹⣿⣷⣿⣿⡟⣽⣿⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡟⣿⣿⡏⣿⣽⣿⡯⣿⣿⣹⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⣧⣽⣿⣧⣾⣿⣿⠇⣿⣇⣿⡏⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣶⣾⣿⣜⢿⣿⣼⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡿⣿⡟⣭⢹⣧⡀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣞⣿⣿⡷⣿⣿⣛⣛⡛⣿⣿⣿⣿⣿⣟⢹⣿⡗⣿⣄⢿⢸⣿⡁
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⡟⣽⣿⡟⣻⣿⣵⣿⢸⣿⡇⣿⣿⣿⢰⣿⣷⣄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⡝⣿⣿⢸⣿⣿⣿⢹⣿⣿⡆⣿⣿⣿⣿⢾⣿⣷⣿⣿⡟⣸⣿⡏⣿⣿⣶
⠀⠀⠀⠀⠐⢹⣿⣿⣿⢃⣿⣿⣷⣿⣿⣿⣼⣿⣿⣿⣟⣿⣿⣿⣾⡿⣿⣿⣏⡴⢿⣿⣷⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠻⢿⣷⣿⣇⣾⡟⢹⣿⣿⠿⣿⣷⢸⣿⢸⣿⡏⣿⢨⣿⣿⣿⡁⣾⣿⢻⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⣿⣇⢹⣿⣿⢸⣿⣿⣸⣿⣼⣿⡇⣿⣜⣿⣿⣿⡇⣿⣿⣌⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠟⣿⣘⣿⠸⣿⣿⣎⣿⢿⣿⢿⢻⣿⣧⣽⣿⣿⢿⣿⣿⡏⣴⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡟⢻⣝⣿⢹⣇⡯⢾⣹⡿⣸⣿⡟⡎⣾⡟⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⢻⣿⡾⣿⡿⣿⣯⣵⡿⢿⣷⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠒⢺⣿⣼⢿⣧⣾⣸⣯⣾⣾⣧⣧⣿⣿⠃
⠀⠀⠀⠀⢀⡀⠀⠀⢈⠀⠀⢰⡄⣠⢡⣿⠣⡞⡿⣫⣿⣿⣸⡗⣿⣫⣾⠇
⠀⠀⠀⠀⠈⣷⡀⠀⠀⠁⠈⢀⡿⠛⣿⣽⠞⡷⠿⣿⣿⡟⢿⣿⣧⣿⠏⠀⠄⢎
⠀⠀⠀⠀⠤⠜⠃⠀⢐⣂⣿⣿⣿⣷⣯⣷⣷⡿⣣⣟⣿⣴⣼⣅⣾⣿⢮
⠀⠀⠀⠀⠀⠀⡠⠖⢸⣿⡏⠟⣿⢫⣿⣿⡟⣾⡟⢼⣿⠋⣾⣩⢙⣻⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣆⢾⣻⡿⡟⡿⣿⢿⣧⣯⣼⡏⣿⢻⡿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⣿⣼⣼⣵⣧⣷⡟⣼⣿⣿⣿⢷⣿⣸⣗⣿⣿⣠⡇⠰⡅
⠀⠀⠀⠀⢀⣀⣀⣿⣷⢿⣟⣿⢹⠋⡯⢿⢻⣿⡟⣽⢃⡿⡿⢫⢛⣟⣻⣿⠛⠛⠁
⠀⠀⠀⠘⠋⠙⠘⠛⠛⠚⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠚⠓⠛⠛⠛⠛⠚⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢸⠀⠀⣿⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⣿⠀⢹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⡀⢸⡀⠀⣿⠀⢀⡇⠀⠀⡇⠀⢸⡀⠀⣟⠀⣸⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠤⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⣿⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡇⠀⣿⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣿⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣿⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⢄⠀⠀⢸⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣸⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣠⠴⠶⣤⠾⠦⣴⠿⠦⡴⠾⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢇⠀⠀⣿⠀⠀⣼⠀⠀⡇⠀⢀⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⢀⡀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢹⡧⡷⢿⢿⡇⠀⡇⣿⠄⠀⠀⠀⠐⠒⠒⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣷⡀⢰⡇⠀⢰⣴⢿⣧⢰⡶⣦⠀⢶⡆⣴⣶⡄⢠⡾⣷⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠛⢻⡿⠛⢿⡿⢸⡿⣿⡄⢹⡇⢸⣿⡏⢸⡇⣿⠃⢻⡆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠁⠀⢸⡇⠀⣸⡇⣼⠇⢹⡇⢸⡇⢸⣿⠁⢸⡇⣿⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢻⡆⠀⢸⡇⠀⣿⡿⠋⠀⢿⡿⣾⡇⢸⡟⠀⣾⡇⣿⡐⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⣾⡟⠀⠘⣧⡀⢹⡟⠲⠀⠘⢧⡿⠉⢸⡇⠀⠿⠇⠸⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠶⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠚⠂⠈⠀⠘⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠤⣄⡀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠰⠂⠀⠀⠀⠤⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢰⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⠀⠀⠀⠀⠀⠀⠑⡤⢎⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠤⡀⠀⠑⠤⣠⢌⠆⠀⣉⢥⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢹⡐⠆⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⡁⢄⠀⠀⠀⠱⡤⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⡄⠀⠀⢀⠀⠀⢀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠸⡆⠀⢸⠂⢀⠾⡄⠈⠁⠐⢄⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⢡⠀⠈⡠⠃⠀⠈⠀⡀⠀⠀⠁⠀⠈⢦⡀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠇⠔⠀⠀⠀⠀⠀⡵⠄⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠈⠁⠀⠁⠘⠤⠀⣀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⣧⡀⠈⢦⡀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡀⢠⠃⠀⠀⠀⠀⢸⠀⠙⠢⠄⢉⡲⠄⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⢸⢠⡀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⡆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣯⣽⣿⠛⣻⣶⣾⣿⣯⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣧⣽⠟⢹⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣄⢁⣤⡈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠈⢿⣿⣿⣿⣅⡈⠉⣤⡈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⣀⢰⡆⣙⢿⣿⣿⣿⣶⣿⣧⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠙⠹⣏⠉⠀⠈⠙⠛⠛⠛⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠟⠀⢠⡖⠀⠀⠀⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠛⠻⣟⣷⣦⠛⠶⠟⠀⠀⣠⡾⠋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢋⣐⠶⣦⣸⣿⣾⢷⣤⣤⣴⠞⣿⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠙⢶⣿⠿⠛⣿⠀⠀⢀⡀⢀⣿⠀⢀⣴⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⣀⣴⠟⠁⣠⡾⠋⠻⠿⡿⠾⠛⣡⣤⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠘⣿⡅⢠⡾⠋⢀⣶⠀⣠⣤⠤⣾⡋⠰⠶⠶⠌⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⢿⣟⠀⢀⣾⣧⣾⣥⣍⣙⣿⣿⠶⠶⠶⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠙⢷⣾⡛⢩⣷⣶⣶⣾⣿⣿⣤⣤⣤⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡄⠀⠀⠀⠙⠻⣾⣿⠋⠉⠉⠀⠀⠀⢀⣾⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⣼⢷⡀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠘⠏⠈⢻⣄⠀⠀⠀⠀⠀⠀⠘⠿⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣠⡿⢦⡀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠙⠷⣄⡀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠊⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠂⠉⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⢀⣤⣶⣿⣿⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣠⡎⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⣶⣶⣶⣦⡀⠀⢀⣈⣩⣭⡣⠀⠀⠀⠀⢻⠏⠁⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡫⠂⠀⠀⠀⢀⡼⠁⠀⠈⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠔⡵⠋⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣉⡛⠀⠀⠀⠀⠉⣷⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⡼⢣⠞⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟⠉⠁⠀⠀⠀⠀⠀⢋⡞⠻⣿⣿⣿⠁
⠀⠀⠀⠀⣠⡟⣰⠏⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣄⠀⠀⠀⠀⠀⠚⠋⠀⠀⠈⠉⠁
⠀⠀⠀⢠⠏⢠⠏⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠴⠆⠈⠀⠀⠀⠀⢀⣠
⠀⠀⢠⡟⠀⠈⠀⠀⠀⠀⠀⢀⣼⡇⣀⡴⠶⠒⠛⠒⢦⡀⣀⠤⠖⠒⠒⠒⠒⠒⠦⢄⡀⠐⣖⢀⣀⡤⠿⠛⠁
⠀⠀⣼⠀⠀⠀⢠⡄⠀⠀⠰⠿⠟⠚⠁⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠐⢶⠤⡽⠖⠋⠁
⠀⠀⣿⠀⠀⠀⢸⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷
⠀⠀⢻⣄⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢁
⠀⠀⠀⠙⠳⢶⣤⣄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠌⠳⢦⣄⣀⡀⠀⠀⢀⣠⣴⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣄⡀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⠶⣶⣶⣶⡶⠶⠟⠛⢳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠐⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠃⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠈⠛⠛⠛⠉⠀⠀⠀⠀⠀⠘⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟

⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣄⣀⡀
⠀⠀⠀⠀⠀⣾⡟⠁⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣤⣄
⣶⣾⣦⡆⠀⢿⣧⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠿⣿⣫⣶⣝⡿⢿⣿
⠻⣿⡿⠃⢀⣼⣿⢿⣶⣶⡾⣿⣧⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣜⣻⣿⣿⣿⣿⡟⣳⣿⣿⡀⢶⣷⠆
⠀⢸⠀⢠⣿⠟⠁⠀⢸⣿⠀⠀⠹⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡻⠿⠿⢟⣽⣿⣿⠛⠀⣈⣈⡀
⠀⠸⡆⣿⡏⠀⢶⣶⣿⣿⣶⣶⠄⢨⡻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣟⣫⡻⢟⣽⢛⣿⠿⠀⠀⣟⣙⡇
⠀⠀⢷⣿⣇⠀⠀⠀⢸⣿⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣮⣵⣮⣽⣿⣿⣶⣶⣶⣿⣹⡇
⠀⠀⠈⠘⢿⣦⡀⠀⠀⠁⠀⢀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣭⣿⣯⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⠻⠿⣶⣶⡾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠟⠻⠟⠛⠛⠛⠗⠿⠾⠇



⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⣾⠟⠉⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢠⡄⠀⠀⣤
⠀⠀⠀⠀⠀⠀⢿⣦⣀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⠿⢷⣶⡼⢇⣶⣆⠿⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⣉⣿⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛
⣴⣿⣿⣦⡀⠀⠘⠛⣿⡟⠃⠀⢀⣴⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⣭⣿⣿⢫⣯⣿⣿⣯⣟⣿⣿
⠙⢿⣿⣿⣿⣦⣀⡀⠉⢀⣀⣴⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣮⡿⣿⣿⣾⣿⣾⣿⣿⣿⣾⣷⣿⣿
⠀⠀⠙⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡷⣿⡷⣿⣿⢾⣷⢟⣿⣤⣀
⠀⠀⠀⠀⠿⣫⣾⣿⣿⢿⢟⣟⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢛⣛⣛⡻⣿⣽⢟⣟⣿⣿⣻⢳⣿⡱⣶⣿
⠀⠀⠀⠀⠸⣻⣿⣿⢵⣝⢷⣝⢗⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢺⣽⡟⡗⢯⣻⣼⣿⣿⣿⣽⣼⣿⣟⢟⣿
⠀⠀⠀⠀⠀⠉⠿⢛⢷⣝⠷⠩⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠁
⣶⣶⠶⠶⠶⣶⣶⣾⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠉⠉⠁⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣦⡀⠀⣀
⠸⡄⠀⠀⠀⢀⠘⠣⣈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⣫⣥⣼⣤⣄⣀
⠀⠘⢦⠠⠐⠌⠳⠀⠈⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⡗⠚⠁⠀⠉⠉⠉⠻⣷
⠀⢠⠈⢧⢵⠀⠚⠀⠀⠀⠙⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⢠⣾⡿
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠖⠛⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⣰⣀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢤⣦⡀
⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⡀⠀⣈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣙⣉⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠻⠋⠘⠿⣫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠻⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂




⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣿
⠀⠀⠀⠀⠀⢠⡀⢸⠀⣿⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢦⣼⣶⠋⡇
⠀⠀⠀⠀⡄⢸⡇⢸⠀⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣀⣀⠤⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⢠⠇
⠀⠀⡄⠀⡇⢸⡇⢸⠀⣿⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⢸
⠀⠀⠋⠀⠃⠘⠃⠘⠀⠛⠀⠛⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢀⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣽⣯⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣶⡆⡃⢸⠇⠄⢻⠄⣒⢸⡇⣒⣺⡆⢰⡇⢐⣺⣅⡒⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣶⣷⣾⣴⣶⣼⣧⣤⣾⣧⣤⣾⣷⣾⣷⣤⣼⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⢔⠀⡀⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⢀⠤⢄⠀⠀⠀⠀⡀⡄⢠⠠⠀⠄⡀
⠀⠀⠀⠀⡇⠘⠔⠠⡇⠀⠀⠂⠉⠁⠑⠂⠠⡄⠀⠀⡦⠠⠤⡖⠀⠀⢀⡜⠀⠀⠘⣄⠀⠀⠀⠧⠨⠈⠤⡇
⠀⠀⠀⠀⠳⢛⣐⠣⠀⠀⠀⠀⡀⠀⢀⠁⣜⡄⠀⠀⡀⠀⠀⣂⡀⠀⠑⡖⠂⠆⠒⠒⠁⠀⠀⠇⠘⣀⠀⠁
⠀⠀⠀⠀⠀⠸⡨⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠢⠤⠤⠾⠆⠀⠀⠁⠀⠀⠀⠃⠀⠀⠀⠰⠠⠈⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⢠⣀⡀⠀⢀⣀⠀⠀⢀⢀
⠀⠀⠀⠐⠀⠀⠄⠃⠀⠀⠀⠐⢠⠀⢐⣝⢵⠀⠀⡇⠠⣘⣃⠇⢨⠀⠏⢠⣀⢀⢸⠆⠁⠀⡔⡖⠀⠉⠉⢀⡒⠆
⠀⠀⠀⠀⡎⠀⠈⠑⠠⣀⠀⠀⢸⠁⢨⠒⠅⠀⠀⠀⡄⠀⠀⠀⠘⠀⠢⠠⠀⠀⠠⠤⠂⠀⠣⠝⢪⣨⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠉⠉⠉⠙⠀⠀⠐⠈⠃⠀⠀⠂
⠀⠀⠀⠀⢶⠻⡶⠀⠀⠀⠀⠀⣀⡤⣤⣀⠀⠀⠀⠀⢀⠔⠢⡀⠀⠀⠀⠀⠀⢀⡔⣲⠀⠀⠀⠀⡰⠲⡀
⠀⠀⠀⠀⡮⠉⣱⠄⢄⠀⠀⢎⠈⠠⠌⠃⢑⠀⠀⠀⡌⠐⠂⢱⠀⠀⠀⠀⣠⣩⠈⠁⠀⠀⠀⠈⢧⣤⠏
⠀⠀⠀⠀⢙⠡⡄⠀⣄⠀⠀⠈⠙⠲⠴⠓⠁⠀⠀⠀⠩⠆⣠⠞⠀⠀⢀⠾⠂⠀⠀⠀⠀⠀⠀⠾⡽⠟⠙⠆
⠀⠀⠀⠀⠀⠉⣁⠁⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⢬⣀⢇⠀⠀⠀⠠⠒⠥⠀⠑⠄⠀⠀⡐⢎⠀⠀⠉⢂⠀⠀⡤⠀⠀⠠⡄⠀⠀⡖⠒⠒⠒⠚⠒⡆
⠀⠀⠀⠀⠡⠤⠤⠤⡈⠀⠀⠐⠠⢤⣰⠥⠂⠀⠀⡀⠽⢉⣬⡯⠀⠀⠀⠷⡄⠀⣴⠇⠀⠀⢸⢨⢨⢨⢠⢹
⠀⠀⠀⠀⠠⡀⠀⣀⠃⠀⠀⠀⠀⠇⠻⠀⠀⠀⠀⠑⠄⣀⣀⠱⠊⠀⠀⠀⠙⡿⠁⠀⠀⠀⠰⠘⠙⠙⠙⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠛⠋⠉⠀⠀⠀⢠⣭⣭⣤⣤⣽⣿⣿⣿⡟⠀⠀⠀⠀⠀⣴⠿⣶⠟⠓⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⣿⣿⣿⣿⣿⡇⠀⢀⣤⣤⣶⣶⣶⣶⣦⣤⣤⣾⣿⣿⣿⣿⣶⣤⣀⡀⠀⠻⠶⠿⢷⡆⠀⠀⠀⡀⠀⠉⣙⠻⣿⣿⣿⣿⣿⣿⣿⣦⡀
⣿⣿⣿⣿⣿⡀⠀⠉⠉⠉⠁⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠞⠀⠀⠀⠀⠀⡀⠀⠀⣼⡿⠀⠀⠛⠀⠈⠉⠻⢿⣿⣿⣿⣿⣷⡀
⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⣠⣴⣤⡀⠁⠀⠀⠀⠀⣠⣶⣶⣿⣇⠀⡀⢀⡈⢻⣿⣿⣿⣇
⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⠋⠀⠀⠀⠀⢿⣄⠈⠻⡆⠀⠀⠀⢰⣿⣿⣿⣿⠟⠘⢃⣸⡟⢀⣨⣿⣿⣿⣶
⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⠀⠀⠸⠛⠛⠋⠀⠀⢤⣄⣽⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⡀⢀⠀⠀⠀⠀⠀⠁⣀⠀⠀⢠⣤⣤⣤⣴⣶⣽⣿⣿⣿⠀⠀⣀⣴⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢻⡆⣰⠃⠀⠀⠟⠀⢠⣿⣿⠟⠛⠛⠉⠀⠉⠻⠃⠀⠈⠉⣿⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⢃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣏⡰⢃⣤⣤⡀⠀⠀⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣄⡉⠉⠁⠈⣛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣾⣧⣤⣾⣿⣿⠟⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠉⠀⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣦⡀⠐⠿⠗⠈⢻⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⣀⣤⣶⣶⣤⣤⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣋⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣶⣿⠇⠀⣀⣼⣿⣿⣿⣿⣿⠿⠃⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠿⢳⣦⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣋⣠⣾⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⣾⠋⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⣿⣿⠇⠠⣒⡆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⠀⠈⠉⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣼⡗⣀⡇⠑⡄⡀⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⣶⣮⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⢧⡿⢿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡐⢶⣿⣦⣁⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⣡⠀⠹⣿⣿⣿⣿⣿⣿⡘⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⢁⣰⣤⣦⣿⡿⠀⢠⠀⠹⢿⣿⣿⡿⢧⡐
⠀⠀⠀⠀⠀⠀⠀⡠⢚⣁⣰⣿⣿⣿⡿⣿⣿⡇⠇⠀⠀⠀⠀⠙⣿⠁⢸
⠀⠀⠀⠀⠀⢀⢊⣴⣿⣿⣿⡿⠟⠁⢹⡼⣿⠁⠀⠀⠀⢀⠠⠀⠳⡄
⠀⠀⠀⠀⠀⢀⡿⢿⣌⠻⢤⡀⠀⠀⠈⣷⠏⠀⠀⡀⠀⠘⣧⡘⢤⡵⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠙⠂⠹⣷⣷⣾⣷⢤⣀⢸⣶⣀⡀⢣⢐⣦⣫⡑⠀⠖⠁⡠⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⢿⡟⠛⠋⠉⠉⠙⠋⠉⣉⣉⣉⣀⣀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢻⣿⣿⣿⣥⣜⢿⣻⣿⣿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⢻⠛⠫⣝⡯⢩⡓⠮⠍⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⡈⠑⠋⠑⠈⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠐⢁⠄⠀⠀⡀⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⢠⠎⠀⠀⠀⡃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢻⡟⠀⠀⠀⠀⢸⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⣽⣿⡿⣿⡿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⢿⣿⣯
⠀⠀⠀⠀⣼⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧
⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧
⠀⠀⣸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⡀⣀⣀⢰⣿⣿⣿⣿⡆⣀⣀⢀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇
⠀⢀⣿⣿⣿⣿⣿⢂⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣤⡐⣿⣿⣿⣿⣿⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠈⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠈⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠍⠁⠉⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⡆⣇⣇⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠰⣄⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠈⢿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠓



⠀⠀⠀⠀⠀⠀⠀⣀⡔⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡴⠋⣠⣄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⢀⡴⠏⢡⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛


⠀⠀⠀⠀⠀⣤⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢻⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠠⠾⢿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣀⣼⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠁

⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠁⢀⡀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⢀⣴⣿⣿⠦⠀⠀⠀⠀⠉⠁⠉⠛⠛⠿⠿⢯⣭⣷⠿⠟⠟⠉⠀⠀⠀⠀⠀⠀⢀⣤⣧⣄
⠀⠀⠀⠀⠐⣶⣿⠟⠁⢀⣴⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣿
⠀⠀⢀⣀⡾⠋⠙⠂⠀⠘⢷⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠟

⠀⠀⠀⠀⠲⢲⡶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠠⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⣀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⣿⠀⠀⠀⠀⠸⠿⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣷
⠀⠀⢰⣿⣿⡄⠛⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣤⣸⣿⣿⣿⠀⠀⠀⠀⠘⠓⠚⠛⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏
⠀⠀⠸⣿⣿⡇⠀⠀⠉⢿⠿⠂⠀⠀⠀⠀⣶⣿⢺⣿⣿⣿⠀⠀⠀⠀⢰⣶⣶⣶⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠟⠻⠛⠛⠟⠟⠀⠀⠀⠀⠙⠉⠙⠙⠧⣴⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⢠⣾⣿⣿⣧⣤⣥⣄⣀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⢉⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⣠⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠉⠉⠛⠛⠉⠁⠀⠀⠀⠀⠉⠉⠋⠉⠉⠙⠛⠋⠉⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠐⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢠⣿⣿⣛⣛⣿⣿⣿⣻⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⣿⠇
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣬⢷⡦⠀⠀⠠⠰⣾⢭⠅⠀⣿⡖⡏
⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡅⠀⠈⠁⠀⠀⠀⠀⠉⠀⠀⠀⣽⢣⣷⣄
⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⠻⠀⢻⣿⣧⡀
⠀⠀⠀⣼⣿⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠴⠤⣄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣶⣄
⠀⠀⢰⣿⣿⠆⠰⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠾⠋⠀⢀⣴⠃⠀⠀⠀⠸⣿⣿⣿⣿⣧⡀
⠀⠀⣼⣿⣿⡆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣐⣿⣇⡀⠀⡔⠓⠀⣿⣿⣿⣿⣿⣷⡀
⠀⠀⣿⣿⣿⣇⠀⠀⠀⢀⠀⡀⠀⠀⠀⡤⠀⠘⠀⠀⢂⠀⢀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠲⢄⠀⠀⠙⢿⣿⣿⣿⣿⣧
⠀⠀⣿⡿⣿⣿⡄⠀⠀⠀⡰⠇⣠⠖⠁⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣿⣿⣿⣿⣿⡆
⠀⠀⣿⣇⠘⢿⣷⠀⠀⠈⠕⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢹⣿⣿⣿⣿⣇
⠀⠀⢻⣿⣦⡐⣷⣤⣤⠆⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠖⠋⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⣦⣝⣛⡛⠛⠃⠋⠀⠀⢀⣀⣀⣍⡙⢿⣦⣤⣤⣤⣤⠐⠀⢹⣿⣿⣿⣿⣿⣿⣿⠛⠋⠉⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿
⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣭⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠃⠀⠀⠀⢠⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⠯⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⢰⡧⠀⠀⠀⠐⠶⠤⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠿⣿⠿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠾⠛⠉⠀⠀⠉⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣶⣶⣶⣶⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⠀⠈⢿⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠟⠻⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠈⠃⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠁⠀⠀⠀⠘⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⣿⡀⠀⠀⠀⠀⠀⠀⣀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠈⠳⡄⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣧⠀⠀⠀⠀⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣼⡇⠀⠀⠀⠻⣿⣿⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠀⠀⠀⠐⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡔⠋⠙⠶⠦⣀
⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⠂⢀⠀⠀⠀⡀⠀⢿⣿⠲⡄⠀⠀⠀⠀⠸⡿⠃⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣴⠚⠀⠀⠀⠀⠀⠀⠉⢹⡄
⠀⠏⢙⣟⡀⠀⢠⠒⠒⢲⠀⠀⠀⣤⣾⣖⣣⣶⣬⣴⠄⠀⡇⢀⣤⣢⣶⣤⡀⠀⠀⣰⣦⣣⣶⣄⠀⣿⣿⣽⣯⣿⡇⠀⠀⠀⠈⠓⢒⠒⠒⡒⠒⣒⠒⠋
⠀⠀⠸⣿⣿⠶⠷⠶⠶⠎⠀⠀⠀⠹⣾⣿⣿⠁⡉⣱⣶⣎⣷⡾⢛⣿⣿⠋⠀⠀⢠⡟⣿⣿⡏⠙⠛⣷⣶⣷⣿⣿⡇⠀⠀⠀⠀⢶⠉⠀⠠⡦⠀⠈⢵⠄
⠀⠀⠀⢻⣿⣶⣶⣤⠀⠀⠀⠀⠀⠀⢾⣿⣿⡏⠉⠙⣿⠋⠉⠁⣾⣿⣿⠀⠀⠀⠈⢱⣿⣿⣷⠀⠀⣿⡿⣿⣿⣿⡇⠀⠀⠀⣼⣶⣶⠀⣾⣾⡆⢰⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⣾⠇⢿⡇⠀⠀⣿⠀⠀⢸⣿⠸⣿⠀⠀⠀⠀⢀⣿⠙⣿⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠸⣿⡇⠀⢸⣿⠀⠈⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠛⠀⠘⠃⠐⠚⠛⠓⠀⠘⠋⠀⠛⠀⠀⠀⠀⠘⠋⠀⠛⠀⠀⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠛⠃⠀⠘⠙⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⢀⠤⠚⠐⠆⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⢐⠀⡂⠀⠀⢐
⠀⠀⠀⠀⣸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣙⡉⠰⣉⣀⢺⡗⡀⣀⡸⠀⠀⢀⡰⠟⠛⠛⠛⠛⠛⠛⠛⠛⣷⠀⠀⠀⠀⠀⠀⠉⢛⠉⠀⠉⡛⠁⠈⢙⡋
⠀⠀⠀⠀⠿⣿⣿⡿⠗⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⠠⠴⠛⠀⠀⠀⠀⠀⣼⣿⣿⣷⠀⠀⡀⡀⢸⣿⣷⠀⣿⠀⠀⠀⣠⣾⣿⣦⢀⣤⣤⣤⣭⣤⣤⣌
⠀⠀⢀⣴⡆⠸⣿⠁⣴⣤⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄⠀⠈⠀⠙⣿⠏⠀⣿⠀⠀⣾⠋⠿⠿⠋⢸⡇⠀⣠⡄⠀⠀⣿
⠀⢠⣿⣿⣿⣿⣿⣷⣿⣿⣧⡀⠀⠀⢀⣤⠈⣿⡿⢁⣄⡀⠀⠀⠀⠀⠀⣠⡈⢿⣿⠁⣄⡠⢤⣤⣤⠿⠤⠤⠿⠀⠀⠋⣰⣿⣿⣷⣬⣥⣾⢟⣁⣀⣀⣿
⠀⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠗⠀⠀⠿⠿⠧⠹⠇⠾⠿⠇⠀⠀⠀⠀⠸⠿⠷⠸⠇⠼⠿⠷⠚⠛⠛⠒⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠛⠛⠡⠼⠿⠯⠉⠁
⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣄⡀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣦
⠀⠀⠀⠀⠴⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⣛⡋⠀⠀⠀⠐⠒⠄⠀⠀⠘⣛⡉⠧⠀⠀⠀⠀⣀⢛⠛⠁⠀⡀⢀⠀⠀⠀⠀⠀⢀⣄⢹⢡⣄⣀⡰
⠀⠀⢀⡴⣿⣄⣤⡀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⣿⣿⡶⠾⠃⣡⣾⠉⣙⣤⣾⣿⡇⠀⠀⠀⠰⣾⢿⣮⣾⣷⣄⣏⠓⠂⠀⠀⠀⠰⣿⣻⠟⣠⠌⠋⠐⠋⠁
⠀⠀⠘⠃⣿⣿⠉⠛⠂⠀⢞⠉⠀⡇⠀⠀⠀⢰⣿⣿⠀⠈⠉⣁⢀⠁⠉⠁⣸⣿⣇⠀⠀⠀⠀⠹⢿⢿⣿⠈⠉⠉⠀⠀⠀⠀⠀⠀⠙⣷⣾⣿⡄⠁
⠀⠀⠀⢰⣿⣿⣆⠀⢀⣠⠞⠀⣀⡇⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⢠⣿⣿⡧⠀⠀⠀⠀⠠⣶⣶⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⡗
⠀⡰⠒⣾⠃⠊⢿⡌⠉⢀⣠⠞⠙⠇⠀⠀⠀⠀⣿⢹⣿⠀⠀⠀⠀⠀⠀⢸⣿⢿⡇⠀⠀⠀⠀⠀⣿⠃⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢿⡇
⠚⠓⠒⠒⠒⠒⠒⠓⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠃⠀⠀⠀⠀⠀⠀⠈⠛⠘⠃⠀⠀⠀⠀⠈⠋⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠂⢀⡐⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⡀⠀⢀⣭⡄⠀⡀⠀⠀⠀⠀⠀⣴⣶⡆
⠀⠀⠤⠀⠀⢻⡿⠀⢀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⢀⣛⡀⠸⡿⠁⠀⠈⣿⠃⣈⣋⡀⠀⠀⠀⠰⢿⣿⠿⠀⠀⠀⢀⡠⢄⡀
⠀⠀⠙⢦⡾⣦⣴⡿⠾⠁⠀⠀⠀⠀⠀⠀⠀⣶⣆⢹⡏⣰⣶⠀⠀⠀⠀⠀⠀⢾⣿⡿⢸⠛⠒⠿⠒⠻⡰⢿⣿⡧⠀⠀⠀⣾⡌⡟⣴⣷⠤⡔⠁⠀⠀⠀⢑
⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢠⡆⠉⠉⠀⠀⠀⠀⠀⠀⢈⣿⣷⠁⢠⣿⣾⣷⡀⢘⣿⣿⡁⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠈⡏
⠀⠀⠀⠀⢸⣿⢿⡇⠀⠀⠀⠀⠀⢰⠂⠒⠒⠒⡴⠛⠛⠦⡆⠒⢲⠀⠀⠀⠀⠈⠛⠻⢤⡈⢼⣿⡇⢁⡬⠛⠛⠁⠀⠀⠀⡖⠒⠑⠒⡆⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠸⠦⣤⣤⠴⠂⠀⠀⠀⣇⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠧⢤⣤⠤⠏⠉⠉⠀⣟⣛⡇

⠀⠀⠀⣠⠖⠚⠉⠓⠢⡀⠀⠀⠀⠀⠀⡠⠒⠋⠙⠒⢦⡀⠀⠀⠀⠀⢀⡴⠒⠋⠙⠒⢤⠀⠀⠀⠀⠀⢀⠔⠚⠉⠓⠲⣄
⠀⠀⡸⠁⠀⠀⣤⠰⠀⠘⡆⠀⠀⠀⡞⠀⠀⠀⠀⠀⠄⠹⡀⠀⠀⢀⠏⠀⠀⢠⡀⠀⠀⢳⠀⠀⠀⢰⠃⠀⠀⣠⣤⠀⠈⢆
⠀⠀⢧⠀⠀⢶⣶⠆⠀⠀⡇⠀⠀⠀⡇⠀⠐⢶⡶⠂⠀⢠⠇⠀⠀⠸⡄⠀⡀⢻⠇⠀⠀⢸⠀⠀⠀⢸⠀⠀⣴⠟⠁⠀⠀⡼
⠀⠀⠈⠢⣀⠀⠋⠀⣠⠞⠀⠀⠀⠀⠘⢦⡀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠑⢄⡀⠈⠁⢀⡴⠃⠀⠀⠀⠀⠳⣌⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⡤⠚⠉⠉⠉⠲⣄⠀⠀⠀⠀⢀⡴⠊⠉⠉⠙⠢⡀⠀⠀⠀⠀⢀⠔⠋⠉⠉⠑⢦⡀⠀⠀⠀⠀⣠⠖⠉⠉⠉⠓⢄
⠀⠀⡼⠀⠀⠆⠀⠘⠀⠈⡆⠀⠀⠀⡎⢀⣤⣄⣷⡀⠀⠸⡄⠀⠀⢠⠇⠀⠀⣰⡀⠀⠀⢱⠀⠀⠀⢰⠁⠀⣰⣧⣞⠱⠀⢧
⠀⠀⢳⠀⠀⠳⣶⠇⠀⢀⡇⠀⠀⠀⣇⠸⣿⣿⡿⣿⠇⢠⠃⠀⠀⠘⡄⠀⠀⢹⡃⠀⠀⣸⠀⠀⠀⢸⡀⢰⣿⣿⡿⠂⠀⡜
⠀⠀⠈⠳⣄⡀⠋⣁⡠⠊⠀⠀⠀⠀⠈⠢⣀⡀⠀⣀⡴⠋⠀⠀⠀⠀⠙⢦⣀⠈⢀⣀⠔⠁⠀⠀⠀⠀⠑⢄⣉⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⢀⡴⠋⠁⡀⠉⠑⢄⠀⠀⠀⠀⢀⠔⠉⠀⠀⠉⠲⣄⠀⠀⠀⠀⣠⠞⣉⣀⣈⡉⠢⡀⠀⠀⠀⠀⡠⠊⠉⠀⠈⠙⢦⡀
⠀⠀⡼⠀⠀⣰⠿⡄⠀⠈⡇⠀⠀⠀⡏⠘⢃⣀⣴⣀⡄⠘⡄⠀⠀⢠⠃⠀⡿⢭⡽⣿⠀⢹⠀⠀⠀⢸⠁⠘⣒⢀⣰⡎⠀⢧
⠀⠀⢳⠀⠀⢿⣚⡗⠀⢀⠇⠀⠀⠀⢇⠀⢸⣿⣿⣿⡇⢰⠃⠀⠀⠘⡆⠀⣧⣺⣇⡾⠀⡸⠀⠀⠀⠸⡀⢾⣿⣿⣏⡱⠀⡞
⠀⠀⠀⠑⢤⣀⣉⣀⠴⠋⠀⠀⠀⠀⠈⠳⢄⣀⣀⣠⠔⠁⠀⠀⠀⠀⠈⠢⣄⣀⣀⡠⠞⠁⠀⠀⠀⠀⠙⠦⣀⣀⣀⡤⠚

⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⢀⠔⠉⢀⣀⠀⠙⢦⠀⠀⠀⠀⢠⠞⣁⣀⣀⣈⡑⢄⠀⠀⠀⠀⡠⢊⣁⣀⣀⣈⠳⡄⠀⠀⠀⠀⡴⢋⣀⣀⣀⣉⠢⡀
⠀⠀⡞⠀⢸⠍⠉⢹⠀⠀⡇⠀⠀⠀⡇⢸⢠⣶⡶⣆⡇⠘⡆⠀⠀⢰⠃⢸⣴⢷⣷⡄⡇⢸⠀⠀⠀⢸⠀⡇⢐⣶⡷⢈⡆⢳
⠀⠀⢱⡀⠸⣄⣀⣴⡆⢠⠇⠀⠀⠀⢧⠸⣈⣛⣋⣃⠇⣰⠁⠀⠀⠈⣆⠸⣘⣟⣛⣁⠇⡼⠀⠀⠀⠸⡄⢧⣙⣛⣑⡰⢃⡎
⠀⠀⠀⠙⠦⢤⣀⡤⠔⠁⠀⠀⠀⠀⠀⠑⠤⣄⣠⠤⠞⠁⠀⠀⠀⠀⠈⠳⠤⣄⣠⠤⠊⠀⠀⠀⠀⠀⠈⠢⢤⣀⡠⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⣤⣤⣶⣦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⠿⠿⠟⠛⢿⠋⢉⣉⣛⠙⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣾⣿⣿⣿⡿⢿⠏⢉⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⢿⠟⠉⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⠿⠛⠁⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡇
⠀⠀⠀⢀⣠⣿⣿⣿⠟⣁⣴⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣧⡀
⠀⠀⠀⣿⣿⣿⣿⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⣴⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⡄⠀⠀⠀⢻⣇
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠀⠀⠀⠀⠀⢿⣇
⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣤⣤⣤⣤⡀⠀⠀⣿⡇⠀⢀⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠾⠛⠛⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⠈⣻⡆⠀⠈⣿⡿⠛⠛⣿⣆
⠀⠙⢿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠙⢿⡟⠛⠛⠃⠀⠀⠹⣷⡄⠈⠀⣿
⠀⠀⠀⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⣀⣀⡀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣀⣀⣀⠾⠃⠀⠀⠀⣿⣇⣀⣴⡿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢹⣇⣀⣀⡴⢶⣶⣶⡟⠋⠁⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠈⣿⡟⠛
⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢸⡟⠉⠉⠀⠀⠈⠛⠁⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⡀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠹⠷⣤⣀⣀⢀⣀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⣰⡿⠋⣉⢉⣉⠙⠻⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⢠⣦⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢰⣿⠑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣤⣤⣆⣀⣤⣴⡾⠟⠀⠀⢠⡜⠋⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⢹⣷⡀⠛⠀⢀⣤⡾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠀⠀⠀⣰⡏⠁⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠛⠉⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠹⠛⢢⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢢⣤⣤⣄⠀⠀⢀⣴⣦⡜⠛⠁⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⠟⠉⠉⣿⣿⣿⣤⣤⣄⠀⠀⢀⣀⣀⣤⣴⠿⠋⠹⣧⡸⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣷⣦⣤⡾⣿⣿⠉⣿⠋⠻⠿⠿⠿⠟⠛⠉⠁⠀⠀⠀⠸⣧⣼⢿⡿⢿⣷⣶⢆⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⢹⠏⣿⡟⠉⠀⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢸⡇⠀⠉⠙⠿⣯⣁⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠸⠇⠀⢹⣷⠀⠀⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⣹⡆⠀⠀⠀⣿⡇⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⢸⡇⠀⠀⢸⣿⠀⠠⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⣴⡟⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⢀⡌⠁⠀⠀⠸⣿⣤⣶⣿⠟⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣏⠀⣠⡿⠃⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⣼⣿⣧⣴⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠛⠛⠁⠀⣿⠃
⠀⠀⠀⢠⣶⠈⠙⠛⢤⣤⣀⣀⣸⠃⠀⢸⣇⣀⣀⡀⠀⠙⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⢠⡿
⠀⠀⠀⣀⠁⠈⣻⣶⠞⠛⠛⠛⠋⠀⠀⠘⠛⠛⠛⠛⠻⣶⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⢀⣀⣠⣿⣇⡀
⠀⠀⠀⡟⢠⣾⠛⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⣠⣿⣿⢿⣿⠿⠶⠾⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⡿⣷⡾⠷⣿⣿⠁⠙⠛⠻⣷⣟⣿⡀
⠀⠀⢠⡇⠸⣿⣀⡶⠃⠀⠀⠀⠀⣰⡟⠛⠟⢛⣷⣶⡶⠿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⠀⠀⠀⠀⠀⢙⣿⠇
⠀⠀⣿⠀⠀⢹⡏⠀⠀⣰⠆⠀⣼⠏⠀⠀⠀⠈⠁⢤⡤⢄⣈⠉⢷⡶⠶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠙⠛⠓⠶⠶⠶⠛⠁
⠀⠀⠀⠀⠀⢸⣧⢀⡼⠃⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⣷⡆⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⣷⠀⠀⠀⠀⢹⣿⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⣼⣿⣆⣿⡇⠀⠀⠀⢰⣶⠆⠀⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠈⣿⡇⢸⣿⡆⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⣿
⣦⠀⠀⠀⠀⠀⠀⠹⠷⠐⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣿⡇⠀⣿⠁⠀⠀⠀⠀⣟⠀⠀⠀⠀⢘⣿
⠁⠐⠒⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣿⡟⠃⠈⠻⠿⢿⣿⠿⢿⣿⠿⢿⣿⡿⠿⠟⠃
⠀⠀⠀⠀⠀⠘⠛⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢀⣿⡇⠀⠀⠀⠀⢸⣿⠆⢸⣿⠀⠘⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⣤⠀⠀⠀⠀⠀⠀⠀⣶⠀⣿⡟⠃⠀⠀⠀⠀⣼⡟⠀⠘⣿⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⣤⠄⠀⢀⣿⡶⣿⣷⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣤⣶⣿⣶⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⣀⣯⣀⣀⣀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣶⣶⣶⣿⣶⣿⣿⣿⠛⠛⠻⣿⠛⠛⠛⢿⣿⣷⣾⣿⣶⣶⣶⣶⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⢿⡏⠁⠀⠀⠀⠈⠿⠿⠿⠇⠀⠀⢸⣿⠀⠀⠀⠸⠿⠿⠿⠃⠀⠀⠀⠀⢙⡿⠛⠷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡛⠶⠋⢳⡶⣶⠙⢳⣶⠀⢰⡖⠐⢶⠀⠙⢿⡆⢸⣿⠳⣶⠀⠶⠶⠀⡴⠇⠀⠶⢆⠙⠃⠐⢂⡀⠾⣶⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠟⠷⠾⠛⠳⠛⠿⠶⠛⠻⠶⠛⠛⠞⠛⠶⠛⠿⠞⠛⠿⠿⠿⠓⠲⠿⠛⠻⠶⠛⠛⠟⠛⠛⠋
⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣷⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣾⣿⣷⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣶⣾⣶⣶⣶⣶⣾⣿
⣿⣿⣿⣿⣿⠟⠛⠙⠿⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠉⠉⠉⠉⠉⠐⠆⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⣀⣀⣀⡠⠤⠄⠀⠁⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣠⣎⣉⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠅⣠⠟⠋⠉⠉⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢤⣄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠆⠘⠆⠘⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠃⣠⣆⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⢺⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⠀⠀⢡⡤⠌⠉⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⢡⡙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⢱⣫⡄⠀⠀⠀⠀⠀⠀⠀⠴⠛⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢰⣿⣫⣍⠛⣄⠀⠀⣠⠞⠃⠀⠆⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠈⠛⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⢻⡼⠟⠁⠀⠘⣠⡜⢁⠃⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⢸⡇⠁⠀⠀⠀⠈⠉⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⠞⠳⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⢸⡇⠀⠱⣄⠀⠀⠀⢠⠁⠀⠀⠀⠃⠀⠀⠀⠀⡠⠞⠁⠉⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀
⠀⣧⠀⠀⠈⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠠⠠⠠⠤⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆
⠀⣉⠁⣀⣀⣠⡴⣿⠀⠀⠀⠀⠀⠀⢠⡀⢀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢰⡿⢿⣯⣽⠷⠟⣇⠀⠀⠀⠀⠀⠀⣠⠟⡙⠁⢀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⢸⠃⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢀⠞⠁⡜⠀⠀⠀⠀⠀⠒⠒⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡎⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⣴⠚⡇⠀⢀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠃
⣿⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⢇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠴⠁⠀⠀⠀⠀⢀⡀⠒⠆⠀⣄⠀⢀⡀⠀⠀⠋⠁⣧
⣿⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡸⡆⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⣠⠴⠋⠀⠀⢀⡔⢁⣠⡤⠤⠶⠀⢠⠀⢹
⢿⡀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣰⠃⠘⣄⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⣀⢯⡴⠋⠁⠀⠀⠀⠀⠠⡄⢸⡄
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠈⠢⣀⠀⠀⠀⠀⣀⡠⠖⠋⠀⠀⠀⠀⠀⢀⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸⠇⠘⣇
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢻
⠀⠀⠹⣦⡀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣇
⠀⠀⠀⠈⠙⠛⠳⠶⠒⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠋⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣧⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣄⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠈⠁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢁⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠐⠋⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⣄⣀⣀⣀⣀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⡿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠟⢉⠈⠁⠀⢀⣋⣀⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠛⣿⣷⣿⣶⣶⣿⣿⣿⡿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⠿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠈⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⣀⣀⠀⠀⠀⠀⠀⣠⣴⠞⠉⠁⠷⢻⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢩⠽⢿⠟⠂⠀⢸⠟⠛⠋⠉⠀⠀⡌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠀⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⣼⡏⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡀⠀⠀⠂⠀⠀⠀⣠⣾⠟⠀⠀⣠⠟⠋⠙⠛⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⠗⠀⠀⢄⣠⡤⠞⠻⠋⠀⠀⣼⠋⠀⠀⠀⠸⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡴⠖⠛⢻⠆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⣀⡤⠊⠁⠐⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⣰⠃
⠀⢀⣤⡖⠚⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⡄⠀⠀⠠⠀⠀⢠⠇⠀⠀⠀⠀⠀⡀
⠀⣾⠉⠓⠐⠀⠀⠀⠀⠀⢠⣤⢤⣀⡀⠀⠐⢻⡏⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣤⣾⠇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠸⡇⠘⡇⠀⠀⠀⠀⠀⠀⠞⠋⠁
⢀⡟⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⢤⡀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠠⠁⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠸⡄⠀⠀⠀⡇⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡇⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣸
⠀⠀⠀⠀⠀⢀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⢀⣇⠀⢀⡏
⣿⣻⣛⣛⣛⣛⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣻⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢠⣄⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢰⣿⣿⣧⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠘⠻⠿⠏⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣤⣤⣤⣿⣿⣧⣤⣤⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⢻⣿⣿⣿⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠶⣦⡀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠉⠁⠀⣀⣤⡿⠃⠀⠀⠈⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣤⣤⣴⡿⠋⠀⠀⠀⢠⣶⠿⠿⠷⠶⠶⠶⠶⠶⣾⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠉⠉⠁⠀⠀⣀⣀⣴⠟⢁⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⠛⠉⠁⢰⣿⠉⠉⠉⠉⠉⠉⠉⣿⡍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠷⠶⠶⠶⠶⠶⢾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣶⣄⠀⠀⠀⠀⠀⠘⢿⣤⣤⣤⣤⣤⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠃⠀⠀⠙⢷⣤⣀⣀⠀⠀⣾⣏⠀⠀⠀⠈⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋





⣿⣿⠿⠻⡿⢻⠿⢿⠟⣿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣧⣤⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⢠⣤⡟⣻⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⡄⠀⠀⠀⠀⠀⠀⢸⣰⣆⣸⠀⠀⠀⠀⠀⠻⠿⠿⣱⣮⠀⠀⠀⠀⠀⣹⣿⣟⢽⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠗⠀⠀⠀⠀⠀⠀⢸⣯⣨⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠘⠛⣿⣽⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠

⠀⠀⠀⣶⠶⠄⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣄⡄⠀⠀⠀⠀⢠⡄
⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠀⠘⠏⠉⠉⠉⢹⡆⠀⠀⠀⠀⣾⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣧⢾⣷⡟⠀⠀⠀⠀⣸⣶⠞⠳⡖⢄
⠀⠀⣿⣟⢸⣻⡆⠀⠀⠀⠀⠀⠿⣿⣗⣿⣿⠇⠀⠀⠀⠸⠿⠿⠗⠀⠂⠀⠀⠀⠀⠘⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠘⠶⠾⠗⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⣴⣶⢰⣶⡄⠀⠀⠀⠀⢀⣤⣄⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⣠⡠
⠀⠀⠂⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣠⠿⠖⠁⠀⠀⠀⠀⠀⣼⣿⣿⣥⠄⠀⠀⠀⠀⠀⠈⠿⣿⡷
⠀⠀⣦⣿⠀⣀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠸⠗⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⢑⣦⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⢨⣿⡿⡇⢠⣤⠀⠀⠀⠀⠀⢠⣾⣿⣧⠀⠀⠀⠀⠀⢰⡣⠀⢠⠈⠀⠀⠀⠀⠀⡐⢣⣾⣷⡄⢣⠀⠀⠀⠀⢠⣿⣿⣿⣿⡆
⠀⢀⣐⣗⣀⢨⣏⠀⠀⠀⠀⠀⢸⡐⡗⢈⠀⠀⠀⠀⠀⠈⢷⠀⠙⣀⠠⠀⠀⠀⠀⢡⡸⣿⣿⢇⡸⠀⠀⠀⠀⠈⠹⣹⣬⠉
⠀⠈⠛⠛⠋⠈⠋⠀⠀⠀⠀⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠀⠘⠃⠉⠀⠀⠀⠀⠀⠀⠀⠉⠒⠐⠈⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⣠⣴⣾⣏⠁⠀⠀⠀⠀⠀⢠⣼⣁⠀⠀⠀⠀⠀⠀⠀⣀⢸⣥⠀⠀⠀⠀⠀⠀⠘⢀⠀⢈⣀⡀⠀⠀⠀⠀⡏⣯⠹⠏⠉⢹
⠀⢰⣿⣿⣿⣿⠛⠀⠀⠀⠀⠠⢤⠿⣄⠀⠀⠀⠀⠀⠀⠰⡁⠈⠛⣳⡄⠀⠀⠀⠀⠀⠈⡍⠀⠻⠃⠀⠀⠀⠀⣧⡍⠒⠀⠀⢸
⠀⠸⠻⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠀⠀⠀⠀⠀⠙⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣈⣿⣦⣶⠿⢿⣿⣦⣿⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⣿⢿⣿⠁⣧⢸⣷⡎⢻⣿⣷⡶⣿⢿⡿⠶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⢫⡝⣿⠘⠏⡆⣿⢘⣋⣥⡟⣿⢿⡇⡇⣯⣇⢿⣸⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠏⢹⢨⣷⣿⢸⣤⡇⣿⣼⡇⣶⣱⠟⡘⢳⣧⣛⣿⢈⢿⡨⣿⣿⡖⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠸⣧⣛⣼⣸⣿⣧⣿⢹⣧⣛⣸⣦⣿⣼⡿⢹⣿⣼⣜⣳⡎⣿⣧⢿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⣘⣿⡏⣿⣏⢹⣿⢸⢟⣉⢻⡇⣻⣿⢱⢾⡏⣤⣹⣟⠻⣶⣶⣿⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠉⣿⡸⢻⣇⠋⣾⣸⣿⢾⡇⣷⣁⣿⢸⣧⢩⣵⣿⢸⣌⠻⣿⢸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣀⠀⣿⣧⢸⣿⣤⣿⣇⣛⣼⣇⡿⠿⢿⣾⢿⣜⣃⣿⣼⣿⣷⣍⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⣿⣿⡧⡶⢹⡏⢿⡿⢹⡇⣿⣷⢸⡎⣿⣿⡇⡇⣾⡏⣿⡝⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡾⠻⣿⡇⣿⡟⡇⣮⣱⢸⡇⢸⢖⣝⢷⢹⠈⢃⡇⣭⡇⡩⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠙⠺⣾⣿⣼⣿⣼⣿⢸⡼⣿⢸⡆⣿⣀⣧⣛⣇⣿⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⢹⣿⣾⢿⣶⡟⣿⠟⡯⣽⡏⠻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⢠⣿⠹⡟⢸⡛⣇⣮⣾⣷⣽⣿⣷⣄⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⣦⠃⡇⣿⢻⣿⠏⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠻⠷⣿⣠⣿⡛⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠧