Let the function,
$f(x)= \begin{cases}-3 a^2-2, & x<1 \\ a^2+b x, & x \geqslant 1\end{cases}$ be differentiable for all $x \in \mathbf{R}$, where $\mathbf{a}>1, \mathbf{b} \in \mathbf{R}$. If the area of the region enclosed by $y=f(x)$ and the line $y=-20$ is $\alpha+\beta \sqrt{3}, \alpha, \beta \in Z$, then the value of $\alpha+\beta$ is $\_\_\_\_$ .