12.6 (CMO 2001) Given $a \sqrt{2}<a<2$. A convex quadrilateral $ABCD$ is inscribed in a unit circle 2 and satisfies the following conditions:
(1) The center of the circle is inside the convex quadrilateral;
(2) The longest side is $a$, and the shortest side is $\sqrt{4-a^{2}}$.
Tangents $L_{A}, L_{B}, L_{C}, L_{D}$ are drawn to circle 2 through points $A, B, C, D$ respectively. It is known that $L_{A}$ intersects $L_{B}$, $L_{B}$ intersects $L_{C}$, $L_{C}$ intersects $L_{D}$, and $L_{D}$ intersects $L_{A}$ at points $A^{\prime}, B^{\prime}, C^{\prime}, D^{\prime}$ respectively. Find the maximum and minimum values of the ratio $\frac{S_{\text {quadrilateral } A^{\prime} B^{\prime} C^{\prime} D^{\prime}}}{S_{\text {quadrilateral } A B C D}}$.
12.6 Let the center of $\odot_{2}$ be $O$, and denote $\angle A O B=2 \theta_{1}, \angle B O C=2 \theta_{2}, \angle C O D=2 \theta_{3}, \angle D O A=2 \theta_{4}$. Then $\theta_{1}, \theta_{2}, \theta_{3}, \theta_{4}$ are all acute angles, and $\theta_{1}+\theta_{2}+\theta_{3}+\theta_{4}=\pi$. It is not difficult to find that
$$
\begin{array}{l}
S_{A^{\prime} B^{\prime} C^{\prime} D^{\prime}}=\tan \theta_{1}+\tan \theta_{2}+\tan \theta_{3}+\tan \theta_{4} \\
S_{A B C D}=\frac{1}{2}\left(\sin 2 \theta_{1}+\sin 2 \theta_{2}+\sin 2 \theta_{3}+\sin 2 \theta_{4}\right) .
\end{array}
$$

The area ratio can be expressed using $\theta_{1}, \theta_{2}, \theta_{3}, \theta_{4}$. Suppose $A D=a, C D=\sqrt{4-a^{2}}$, then $\sin \theta_{4}=a / 2, \sin \theta_{3}=\sqrt{4-a^{2}} / 2$. Therefore,

we can calculate $\cos \left(\theta_{3}+\theta_{4}\right)=0, \therefore \theta_{3}+\theta_{4}=\pi / 2, \theta_{1}+\theta_{2}=\pi / 2$. Let $\alpha=\tan \theta_{3}+\tan \theta_{4}=\frac{4}{a \sqrt{4-a^{2}}}, \beta=$ a decreasing function of $t$, the maximum value of $t$ is 1.
Thus, $\min \frac{S^{\prime}}{S}=\frac{4}{a \sqrt{4-a^{2}}} t$'s minimum value $=2 \sin \theta_{3} \cdot \sin \theta_{4}=\frac{1}{2} a \sqrt{4-a^{2}}, \max \frac{S^{\prime}}{S}=\frac{8}{a^{2}\left(4-a^{2}\right)}$. Here, $S^{\prime}$ represents the area of $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$, and $S$ represents the area of $A B C D$.