A point \(X\) is drawn on the diagonal of a parallelogram \(ABCD\). The lines parallel to the sides through \(X\) are constructed, and quadrilaterals \(P\) and \(Q\) are formed, as shown in the diagram.
Given that \(Q\) has base \(12\) and 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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