
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]


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Given $F_1$ and $F_2$ are the left and right foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (where $a > 0, b > 0$) respectively. A perpendicular line is drawn from $F_2$ to the asymptote of the hyperbola, with the foot of the perpendicular being $P$. If $|PF_1|^2 - |PF_2|^2 = c^2$, find the eccentricity of the hyperbola.

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