
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 $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ ($a>0$, $b>0$), let $c = \sqrt{a^{2}+b^{2}}$, and $P$ be a point on the line $x = \frac{a^{2}}{c}$ such that $PF_1 \perp PF_2$ and $|PF_1| \cdot |PF_2| = 4ab$. Determine the eccentricity of the hyperbola.

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