
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 \( \frac{x^2}{8} + \frac{y^2}{4} = 1 \) with one focus at \( F = (2, 0) \). Determine a point \( P = (p, 0) \) where \( p > 0 \) such that 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}
user