Functional Equation along the Rationals

Let ff be a function defined along the rational numbers such that f(mn)=1nf(\tfrac mn)=\tfrac1n for all relatively prime positive integers mm and nn. The product of all rational numbers 0<x<10<x<1 such that f(xf(x)1f(x))=f(x)+952f\left(\dfrac{x-f(x)}{1-f(x)}\right)=f(x)+\dfrac9{52} can be written in the form pq\tfrac pq for positive relatively prime integers pp and qq. Find p+qp+q.

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