3. Given the hyperbola $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a, b>0)$ with left and right foci $F_{1}$ and $F_{2}$, respectively, and the hyperbola $C$ intersects the circle $x^{2}+y^{2}=r^{2}(r>0)$ at a point $P$. If the maximum value of $\frac{\left|P F_{1}\right|+\left|P F_{2}\right|}{r}$ is $4 \sqrt{2}$, then the eccentricity of the hyperbola $C$ is $\qquad$
3. $2 \sqrt{2}$.

Let $P(r \cos \theta, r \sin \theta), \cos \theta \geqslant 0$. Then
$$
\begin{array}{l}
\frac{\left|P F_{1}\right|+\left|P F_{2}\right|}{r}=\frac{a+e r \cos \theta+e r \cos \theta-a}{r} \\
=2 e \cos \theta .
\end{array}
$$

Therefore, by the maximum value of $\frac{\left|P F_{1}\right|+\left|P F_{2}\right|}{r}$ being $4 \sqrt{2}$, we get
$$
e=2 \sqrt{2} \text {. }
$$