# Functional Equation along the Rationals

Let $f$ be a function defined along the rational numbers such that $f(\tfrac mn)=\tfrac1n$ for all relatively prime positive integers $m$ and $n$. The product of all rational numbers $0 such that $f\left(\dfrac{x-f(x)}{1-f(x)}\right)=f(x)+\dfrac9{52}$ can be written in the form $\tfrac pq$ for positive relatively prime integers $p$ and $q$. Find $p+q$.

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