
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 $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, respectively, if there exists a point $P$ on the right branch of the hyperbola that satisfies $|PF(_2)| = |F(_1)F(_2)|$, and the distance from $F(_2)$ to the line $PF(_1)$ is equal to the length of the real axis of the hyperbola, determine the eccentricity of the hyperbola.

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