Example 6 Given a regular tetrahedron $S-A B C$ with height $S O=3$, and the side length of the base is 6. Draw a perpendicular from point $A$ to the opposite face $S B C$, with the foot of the perpendicular being $O^{\prime}$. Take a point $P$ on $A O^{\prime}$ such that $A P: P O^{\prime}=8$. Find the area of the section passing through point $P$ and parallel to the base.