RMO is really insane

Geometry Level 3

Let \(ABC\) be a triangle.Let \(D\) and \(C\) be points on segments \(BC\) 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 area of the triangle \(APQ\) to that of the quadrilateral \(PDEQ\).

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