In $△ABC$ medians $\overline{AD}$ and $\overline{BE}$ intersect at $G$. If $△AGE$ is an isosceles right triangle with $\angle AEG=90^{\circ }$ and $AG=EG$, then $\cos (C)$ can be written as $\frac{m\sqrt{p}}{n}$, where $m$ and $n$ are relatively prime positive integers and $p$ is a positive integer not divisible by the square of any prime. What is $m+n+p$?