Squares, Midpoints, and Cosine

Geometry Level 2

\(ABCD\) is a square. Let \(M\) and \(N\) be the midpoints of \(BC\) and \(CD\), respectively. If \(\cos \angle MAN\) can be expressed as \(\frac{m}{n}\) where \(m\) and \(n\) are relatively prime positive integers, what is \(m + n\)?

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