Triangle \(ABC\) has two sides \(AB\) and \(AC\) such that \(\frac{AB}{AC} = \frac{4}{3}\). Locate point \(X\) on the third side \(BC\) such that \(BX = 2CX\). Now, let \(P\) be a point on segment \(AX\) and extend \(BP\) to meet \(AC\) at point \(Y\). Similarly, extend \(CP\) to meet \(AB\) at point \(Z\).

If points \(B, Z, Y, C\) all lie on a circle, as shown in the diagram, the sum of all possible values of \(\frac{AP}{PX}\) can be written in the form \(\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers.

Find \(p + q\).

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