Let the function, $f ( x ) = \left\{ \begin{array} { l l } - 3 a x ^ { 2 } - 2 , & x < 1 \\ a ^ { 2 } + b x , & x \geqslant 1 \end{array} \right.$ 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 $\_\_\_\_$