9. (16 points) $F$ is the left focus of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ centered at the origin $O$. A line $l$ with a non-zero slope passing through $F$ intersects the ellipse at points $A$ and $B$. The line segment $AO$ is extended to intersect the ellipse at point $C$. Find the maximum value of the area of $\triangle ABC$.
When the semi-focal distance of the ellipse is $c$, and the inclination angle of the line $l$ is $\theta(0c$ is considered, the function
$$
f(x)=\frac{b^{2}}{x}+c^{2} x(0c$ is considered, the maximum area of $\triangle A B C$ is $\frac{2 b^{2} c}{a}$.