
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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Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$, respectively. Point $P$ is on the right branch of the hyperbola, and $|PF_{2}| = |F_{1}F_{2}|$. The distance from $F_{2}$ to the line $PF_{1}$ is equal to the length of the real axis of the hyperbola. Find the eccentricity of this hyperbola.

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