
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 the ellipse \( \frac{x^2}{4} + y^2 = 1 \) with a focus at \( F = (\sqrt{3}, 0) \), find the point \( P = (p, 0) \) such that \( p > 0 \) and for any chord \( \overline{AB} \) passing through \( F \), the angles \( \angle APF \) and \( \angle BPF \) are equal.

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