Let the function $f ( x ) = 2 x ^ { 2 } - \log _ { e } x , x > 0$, be decreasing in $( 0 , a )$ and increasing in $( a , 4 )$. A tangent to the parabola $y ^ { 2 } = 4 a x$ at a point $P$ on it passes through the point $( 8 a , 8 a - 1 )$ but does not pass through the point $\left( - \frac { 1 } { a } , 0 \right)$. If the equation of the normal at $P$ is $\frac { x } { \alpha } + \frac { y } { \beta } = 1$, then $\alpha + \beta$ is equal to $\_\_\_\_$.