
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 $\Gamma$:

$$
\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \quad (a > 0, b > 0)
$$

with left and right foci $F_{1}$ and $F_{2}$ respectively. Suppose a line $l$ passing through $F_{2}$ intersects the right branch of the hyperbola $\Gamma$ at points $A$ and $B$, such that $\left|A F_{1}\right| = \left|F_{1} F_{2}\right|$ and $\left|B F_{2}\right| = 2 \left|A F_{2}\right|$. Determine the eccentricity of the hyperbola $\Gamma$.

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