A complicated circle inside a square

Geometry Level 4

Given a circle is inscribed in a square \(ABCD\) with sides of length \(12\). Point \(M\) is on \(AB\) such that \(AM=5\). Let \(P\) and \(Q\) be points of intersections of \(MD\) and the circle, with \(P\) closer to \(AB\). If the length of \(MP=\frac{a-b\sqrt{c}}{d}\) in simplest form, find \(a+b+c+d\)

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