Let \(\triangle ABC\) be a triangle with side lengths \(BC=10, CA= 12, AB= 8.\) Let \(M\) be the midpoint of \(BC,\) and let \(D,E\) be the feet of perpendiculars from \(B,C\) to \(CA, AB\) respectively. \(P, Q\) are the midpoints of \(MD\) and \(ME\) respectively. Let \(\ell \) be the line passing through \(A\) parallel to \(BC.\) \(QP\) intersects \(\ell \) at point \(X.\) Given that \(\cos \angle CMX = \dfrac{x}{y}\) for some coprime positive integers \(x,y,\) find \(x+y-31.\)

**Details and assumptions**

- This problem is not entirely original.

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