Example: Points $P$, $Q$, $M$, and $N$ are all on the ellipse $x^{2}+\frac{y^{2}}{2}=1$, and $F$ is the focus of the ellipse on the positive $y$-axis. It is known that $\overrightarrow{P F}$ is collinear with $\overrightarrow{P Q}$, $\overrightarrow{M F}$ is collinear with $\overrightarrow{F N}$, and $\overrightarrow{P F} \cdot \overrightarrow{M F}=0$. Find the minimum and maximum values of the area of quadrilateral $P M Q N$.
Analysis: To find the extremum of the area, it is only necessary to first derive the analytical expression of the area and then solve it using functional relationships or inequalities.

Solution: Given that $\overrightarrow{P F}$ and $\overrightarrow{P Q}$, $\overrightarrow{M F}$ and $\overrightarrow{F N}$ are collinear, $\overrightarrow{P F} \cdot \overrightarrow{M F}=0 \Rightarrow P F \perp M F$ $\Rightarrow M N \perp P Q$.
(1) When one of $M N$ or $P Q$ is perpendicular to the $x$-axis, the other must be perpendicular to the $y$-axis.

Without loss of generality, assume $M N \perp y$-axis, then $P Q \perp x$-axis, $F(0,1)$, at this time $|M N|=\sqrt{2},|P Q|=2 \sqrt{2} \Rightarrow S_{P M Q N}=\frac{1}{2}|P Q| \cdot$ $|M N|=2$.
(2) When $M N$ and $P Q$ are not perpendicular to the coordinate axes, let the line $M N: y=k x+1$, then the line $P Q: y=-\frac{1}{k} x+1(k \neq 0)$, combining the equation of line $M N$ with the equation of the ellipse yields:
$$\begin{array}{l}
\quad\left(k^{2}+2\right) x^{2}+2 k x-1=0(*) \Rightarrow x_{1}+x_{2}= \\
\frac{-2 k}{k^{2}+2}, x_{1} x_{2}=\frac{-1}{k^{2}+2}
\end{array}$$