# It's not necessarily an altitude!

Geometry Level 5

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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