
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 given by $\frac{x^2}{4} + \frac{y^2}{1} = 1$ with one of its foci at \( F = (-2,0) \). A point \( P = (p,0) \), with \( p > 0 \), has a unique property such that for any chord \( \overline{AB} \) passing through \( F \), the angles \( \angle APF \) and \( \angle BPF \) are equal. Find \( p \).

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