Consider a bacteria culture that initially has \( 1000 \) bacteria. The culture grows exponentially such that the number of bacteria doubles every \( 3 \) hours. However, every \( 6 \) hours, \( 20\% \) of the bacteria are eliminated due to a sterilization process. How many bacteria will there be after \( 24 \) hours? Let's think step by step and output the final answer within \boxed{}.
user