There is a rectangle \(ABCD\) with side lengths 2 and 3 as shown on the right.

Point \(P\) is on \(\overline{BC}\) and point \(Q\) is on \(\overline{CD}.\)

\(M\) is the midpoint of \(\overline{AD}.\)

Let \(m\) be the length of the shortest path from \(A\) to \(M,\) while visiting \(P\) and \(Q\) sequentially.

Find the value of \(m^2.\)

*This problem is a part of <Shortest Path> series.*

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