A bag contains $8$ green candies and $4$ red candies. You randomly select one candy at a time to eat. If you eat five candies, there are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probability that you do not eat a green candy after you eat a red candy. Find $m+n$.

Let's think step by step and output the final answer within \boxed{}.
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