Let $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. The value of the sum
\[ \sum_{(a,b,c) \in S} \frac{1}{2^a 3^b 5^c} \]
can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.

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