14. Given $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ $(a>b>0)$, and $P$ is a point on the ellipse. In $\triangle F_{1} P F_{2}$, the external angle bisector of $\angle F_{1} P F_{2}$ is $l$, and the symmetric point of $F_{2}$ with respect to $l$ is $Q$. $F_{2} Q$ intersects $l$ at point $R$.
(1) When point $P$ moves on the ellipse, find the equation of the trajectory of point $R$;
(2) Let the trajectory of point $R$ be curve $C$, and the line $l^{\prime}: y=$ $k(x+\sqrt{2} a)$ intersects curve $C$ at points $A$ and $B$. The area of $\triangle A O B$ is $S$. Find the value of $k$ when $S$ is maximized.
14. (1) Let the moving point $R(x, y)$ and $Q\left(x_{1}, y_{1}\right)$. Then
$$
\left\{\begin{array}{l}
x = \frac{x_{1} + c}{2}, \\
y = \frac{y_{1}}{2}
\end{array} \Rightarrow \left\{\begin{array}{l}
x_{1} = 2x - c, \\
y_{1} = 2y
\end{array}\right.\right.
$$

Since point $P$ is on the ellipse, we have
$$
\left|P F_{1}\right| + \left|P F_{2}\right| = 2a.
$$

From the symmetry of points $F_{2}$ and $Q$ about the line $l$, we get
$$
\left|P F_{2}\right| = |P Q|.
$$

Since points $F_{1}$, $P$, and $Q$ are collinear, we have
$$
\left|F_{1} Q\right| = \left|P F_{1}\right| + |P Q| = 2a,
$$

which implies $\left(x_{1} + c\right)^{2} + y_{1}^{2} = 4a^{2}$.
From the system of equations (1), we get $4x^{2} + 4y^{2} = 4a^{2}$, i.e., $x^{2} + y^{2} = a^{2} (y \neq 0)$.
Thus, the trajectory equation of $R$ is
$$
x^{2} + y^{2} = a^{2} (y \neq 0).
$$
(2) Let $\angle A O B = \alpha$. Then
$$
\begin{array}{l}
S_{\triangle A O B} = \frac{1}{2} |O A| |O B| \sin \alpha \\
= \frac{1}{2} a^{2} \sin \alpha.
\end{array}
$$

When $\alpha = 90^{\circ}$, $S_{\triangle A O B}$ is maximized. At this time, the distance from point $O$ to the line $l^{\prime}$ is $d = \frac{|\sqrt{2} a k|}{\sqrt{1 + k^{2}}}$.
In $\triangle A O C$, $\frac{d}{a} = \cos 45^{\circ} = \frac{\sqrt{2}}{2}$, i.e.,
$$
\frac{|\sqrt{2} k|}{\sqrt{1 + k^{2}}} = \frac{\sqrt{2}}{2} \Rightarrow k = \pm \frac{\sqrt{3}}{3}.
$$