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 \(p+q\).