Right-angled triangle

Geometry Level 4

\(\triangle ABC\) is a right-angled triangle, where \(AC=3\) and \(BC=4\). Let \(P\) be a point on the hypotenuse of the triangle. We drop perpendiculars from \(P\) to \(AC\) and \(BC\), the footpoints of which are \(X\) and \(Y\).

If the minimum value of \(\overline{XY}\) is \(\dfrac{m}{n}\), where \(m\) and \(n\) are coprime integers, then find the value of \(m+n\).

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