
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
The right focus of the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) is \(F\), and the right directrix is \(l\). Points \(P_{1}, P_{2}, \cdots, P_{24}\) are 24 points arranged counterclockwise on the ellipse, with \(P_{1}\) being the right vertex of the ellipse. The angles \(\angle P_{1}FP_{2}=\angle P_{2}FP_{3}=\cdots=\angle P_{23}FP_{24}=\angle P_{24}FP_{1}\) are all equal. Find the sum of the reciprocals of the distances from these 24 points to \(l\).

Present the answer in LaTex format: \boxed{Your answer}
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