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…

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