A square pyramid with base \(ABCD\) and vertex \(E\) has eight edges of length 4. A plane passes through the midpoints of \(AE\), \(BC\), and \(CD\). The plane's intersection with the pyramid has an area that can be expressed as \(\sqrt{p}\). Find \(p\).

(第二十五届AIME1 2007 第13题)