Let the function, $\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{l}-3{\mathrm{ax}}^2-2,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{x}<1\\ {\mathrm{a}}^2+\mathrm{bx},\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{x}\ge 1\end{array}\right.$ Be differentiable for all $\mathrm{x}\in \mathbf{R}, \mathrm{where} \mathrm{a}>1,\mathrm{b}\in \mathbf{R}\mathrm{. }$
If the area of the region enclosed by y = f(x) and the line $\mathrm{y}=-20\text{ is }\mathrm{\alpha }+\mathrm{\beta }\sqrt{3},\mathrm{\alpha },\mathrm{\beta },\in \mathrm{Z}$, then the value of $\mathrm{\alpha }+\mathrm{\beta }$ is ______.