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Problem Description Let \(\overline{AB}\) be a chord of a circle \(\omega\), and let \(P\) be a point on the chord \(\overline{AB}\). Circle \(\omega_1\) passes through \(A\) and \(P\) and is internally tangent to \(\omega\). Circle \(\omega_2\) passes through \(B\) and \(P\) and is internally tangent to \(\omega\). Circles \(\omega_1\) and \(\omega_2\) intersect at points \(P\) and \(Q\). Line \(PQ\) intersects \(\omega\) at \(X\) and \(Y\). Assume that \(AP=5\), \(PB=3\), \(XY=11\), and \(PQ^2 = \tfrac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m+n\).

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