
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
Given an ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \ (a>b>0)\) with left focus \(F\). Points \(P_{1}, P_{2}, \cdots, P_{n}\) are \(n\) points arranged in a clockwise order on the ellipse such that \(\angle P_{1}FP_{2} = \angle P_{2}FP_{3} = \angle P_{3}FP_{4} = \cdots = \angle P_{n}FP_{1}\). Find the sum of the reciprocals of the distances of these \(n\) points to the left directrix \(x = -\frac{a^{2}}{c} \ (c=\sqrt{a^{2}-b^{2}})\) expressed in terms of \(a, b, c, n\).

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