Interesting Parallelogram

Geometry Level 4

A point \(X\) is drawn on the diagonal of a parallelogram \(ABCD\). The lines parallel to the sides through \(X\) are constructed, Quadrilaterals \(P\) and \(Q\) are formed as shown on the diagram.

Given that \(Q\) has base \(12\), perpendicular height \(4\) and that the area of \(ABCD\) is \(200\), find the sum of all possible values of the base of \(P\).

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