
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 $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ (where $a > 0, b > 0$), let $F_1$ and $F_2$ be its left and right foci, respectively. Suppose there exists a point $P$ on the right branch of the hyperbola such that $|PF_2| = |F_1F_2|$, and the distance from $F_2$ to the line $PF_1$ is equal to the length of the hyperbola's real axis. Find the eccentricity $e$ of the hyperbola.

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