Let the area of the region $\left\{ ( x , y ) : x - 2 y + 4 \geq 0 , x + 2 y ^ { 2 } \geq 0 , x + 4 y ^ { 2 } \leq 8 , y \geq 0 \right\}$ be $\frac { m } { n }$, where $m$ and $n$ are coprime numbers. Then $\mathrm { m } + \mathrm { n }$ is equal to $\_\_\_\_$.