Consider a convex pentagon $ABCDE$. The side lengths are $AB=4$, $BC=4$, $CD=x$, $DE=x$, and $EA=4$. If $\angle A=90^{\circ }$ and $\angle B=90^{\circ }$, and the degree measure of $\angle E$ is $120^{\circ }$, what is the length of $x$?