A pentagon $ABCDE$ is formed by three triangles: $△ABC$, $△ACD$, and $△ADE$. Given that $AB=BC=4$, $AE=ED=4$, and $CD=2$. The diagonal lengths $AC$ and $AD$ are such that $△ABC$ and $△ADE$ are equilateral. The area of pentagon $ABCDE$ can be written as $\sqrt{m}+\sqrt{n}$ where $m$ and $n$ are positive integers. What is $m+n$? Refer to the image, 078b0dc9-1f41-47f8-9e37-97b3b862504a_1.png