
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 \(C: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)\) with foci \(F_{1}\) and \(F_{2}\), a line \(l\) passing through \(F_{2}\) intersects the right branch of the hyperbola at points \(A\) and \(B\) such that 
\[ \overrightarrow{A F_{1}} \cdot \overrightarrow{A F_{2}}=0 \]
and 
\[ \overrightarrow{F_{2} B} + 2 \overrightarrow{F_{2} A} = \mathbf{0}. \]
Find the eccentricity of the hyperbola \(C\).

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