
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
Consider the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \ (a > b > 0)$ with its major focus at $F$. Points $P_1, P_2, \ldots, P_n$ are positioned sequentially in a clockwise direction on the ellipse such that $\angle P_1FP_2 = \angle P_2FP_3 = \angle P_3FP_4 = \cdots = \angle P_nFP_1$. Find the relationship involving these points.

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