Example 5 In the Cartesian coordinate system $x O y$, let $A B$ be a chord of the parabola $y^{2}=4 x$ passing through the point $F(1,0)$, and let the circumcircle of $\triangle A O B$ intersect the parabola at point $P$ (different from points $O, A, B$). If $P F$ bisects $\angle A P B$, find all possible values of $|P F|$.
$(2018$, National High School Mathematics League Competition (A Volume) $)$
Let $l_{A B}: y=k(x-1), P\left(\frac{y_{P}^{2}}{4}, y_{P}\right)$.
As shown in Figure 4.
By Proposition 6, the equation of the circle passing through points $A, O, B, P$ can be expressed as
$$
\left(y^{2}-4 x\right)+\lambda(y-k x+k)\left(y-\frac{4}{y_{P}} x\right)=0 \text {. }
$$

From the equation of the circle, the coefficient of $x y$ is 0, so
$$
k+\frac{4}{y_{P}}=0 \text {. }
$$

Let the line $A B$ intersect the directrix of the parabola at point $C$, and the projections of points $A, B$ on the directrix are $A_{1}, B_{1}$.

Since $P F$ bisects $\angle A P B$, and $F$ is the focus of the parabola, we have
$$
\frac{P A}{P B}=\frac{F A}{F B}=\frac{A A_{1}}{B B_{1}}=\frac{A C}{B C}
$$
$\Rightarrow P C$ is the external angle bisector of $\angle A P B$
$$
\Rightarrow P F \perp P C \text {. }
$$

Also, $C(-1,-2 k)$, thus,
$$
\frac{y_{P}}{\frac{y_{P}^{2}}{4}-1} \cdot \frac{y_{P}+2 k}{\frac{y_{P}^{2}}{4}-1}=-1 \text {. }
$$

Eliminating $k$, we get
$$
\begin{array}{l}
\frac{y_{P}^{4}}{16}+4 \cdot \frac{y_{P}^{2}}{4}-9=0 \\
\Rightarrow \frac{y_{P}^{2}}{4}=\sqrt{13}-2 .
\end{array}
$$

By the focal radius formula, we have
$$
|P F|=\frac{y_{P}^{2}}{4}+1=\sqrt{13}-1 \text {. }
$$