
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 a hyperbola $E: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ $(a > 0, b > 0)$ whose right vertex is $A$, and a parabola $C: y^2 = 8ax$ whose focus is $F$, find the range of the eccentricity $e$ of $E$ if there exists a point $P$ on the asymptote of $E$ such that $PA \perp FP$.

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