
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 hyperbola \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) with the left vertex \(A\) and the right focus \(F\), let \(P\) be an arbitrary point on the hyperbola in the first quadrant. If it is always true that \(\angle PFA = 2 \angle FAP\), find the eccentricity of the hyperbola.

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