Functional Equation along the Rationals

Number Theory Level 3

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 $0<x<1$ 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 $p+q$ .