3. For the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$, the right focus is $F$, and the right directrix is $l$. Points $P_{1}, P_{2}, \cdots, P_{24}$ are 24 points arranged in a counterclockwise order on the ellipse, where $P_{1}$ is the right vertex of the ellipse, and $\angle P_{1} F P_{2}=\angle P_{2} F P_{3}=\cdots=$ $\angle P_{23} F P_{24}=\angle P_{24} F P_{1}$. The sum of the reciprocals of the distances from these 24 points to $l$ is $\qquad$
3. $6 \sqrt{5}$ Analysis: Let $\angle P_{1} F P=\theta, P$ to $l$'s distance be $d$, then $e d \cos \theta+d=\frac{a^{2}}{c}-c=\frac{4}{\sqrt{5}}, e=\frac{\sqrt{5}}{3}, \frac{1}{d}=$ $\frac{\sqrt{5}}{4}\left(\frac{\sqrt{5}}{3} \cos \theta+1\right)=\frac{5}{12} \cos \theta+\frac{\sqrt{5}}{4},$ the sum of the reciprocals of the distances from 24 points to $l$ is
$$
\frac{5}{12}\left(\cos 0+\cos \frac{\pi}{12}+\cos \frac{\pi}{6}+\cdots+\cos \frac{23 \pi}{12}\right)+\frac{\sqrt{5}}{4} \times 24=6 \sqrt{5} .
$$