Square \(ABCD\) has sides of length \(4\), and \(M\) is the midpoint of \(\overline{CD}\). A circle with radius \(2\) and center \(M\) intersects a circle with radius \(4\) and center \(A\) at points \(P\) and \(D\). Find the distance from \(P\) to \(\overline{AD}\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Let's think step by step and output the final answer within \boxed{}.
user