Let $a$, $b$, $c$ be positive real numbers for which:
\[
  \frac{5}{a} = b+c, \quad
  \frac{10}{b} = c+a, \quad \text{and} \quad
  \frac{13}{c} = a+b.
\]
If $a+b+c = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$.

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