Let \(ABC\) be a triangle. Let \(D, E\) be a points on the segment \(BC\) such that \(BD = DE = EC\). Let \(F\) be the midpoint of \(AC\). Let \(BF\) intersect \(AD\) in \(P\) and \(AE\) in \(Q\), respectively.

Determine the ratio of the area of the triangle \(APQ\) to that of the quadrilateral \(PDEQ\).

If this ratio can be expressed as \( \dfrac pq\), where \(p\) and \(q\) are coprime positive integers, submit your answer as \(p + q\).

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