As shown in the figure, three cylinders with radius $\sqrt{3}$ and different heights are mutually externally tangent and placed on a plane $\alpha$. Let $\mathrm{P}$, $\mathrm{Q}$, $\mathrm{R}$ be the centers of the bases of the three cylinders that do not meet plane $\alpha$. Triangle $\mathrm{QPR}$ is an isosceles triangle, and the angle between plane $\mathrm{QPR}$ and plane $\alpha$ is $60°$. If the heights of the three cylinders are $8$, $a$, and $b$ respectively, find the value of $a + b$. (Given: $8 < a < b$) [4 points]
As shown in the figure, three cylinders with radius $\sqrt{3}$ and different heights are mutually externally tangent and placed on a plane $\alpha$. Let $\mathrm{P}$, $\mathrm{Q}$, $\mathrm{R}$ be the centers of the bases of the three cylinders that do not meet plane $\alpha$. Triangle $\mathrm{QPR}$ is an isosceles triangle, and the angle between plane $\mathrm{QPR}$ and plane $\alpha$ is $60°$. If the heights of the three cylinders are $8$, $a$, and $b$ respectively, find the value of $a + b$. (Given: $8 < a < b$) [4 points]