
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 that $F_1$ and $F_2$ are the left and right foci of the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, a point $P$ on the right branch of hyperbola $C$ satisfies $|PF_1| = 3|PF_2|$ and $\overrightarrow{PF_1} \cdot \overrightarrow{PF_2} = a^2$, find the eccentricity of hyperbola $C$.

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