Let $f(n)=\varphi(n^3)^{-1}$, where $\varphi(n)$ denotes the number of positive integers not greater than $n$ that are relatively prime to $n$. Suppose
\[ \frac{f(1)+f(3)+f(5)+\dots}{f(2)+f(4)+f(6)+\dots} = \frac{m}{n} \]
where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.

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