Example 23 Hexagon $A B C D E F$ is inscribed in circle $O$, and $A B=B C=C D=\sqrt{3}+1, D E=$ $E F=F A=1$, find $S_{A B C D E F}$.
Solve: Divide the known hexagon into six triangles by connecting each vertex with the center of the circumscribed circle (figure omitted). Move these six triangles to the positions as shown in Figure 1.4.25, forming a hexagon with alternating side lengths of $\sqrt{3}+1$ and 1. This hexagon is inscribed in the same circle as the original hexagon, and each of its internal angles is $120^{\circ}$. Therefore, a regular triangle can be added to the side of length 1, thus expanding the hexagon into a regular triangle with side length $3+\sqrt{3}$. Hence, the required area is
$$
S_{A B C D F F}=S_{\triangle P M N}-3 S_{\triangle P A^{\prime} F}=\frac{9}{2}(2+\sqrt{3})
$$