
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 $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$, a line perpendicular to the $x$-axis passes through $F\_2$ and intersects the hyperbola at points $A$ and $B$. $G$ is the centroid of $\triangle ABF\_1$, and $\overrightarrow{GA} \cdot \overrightarrow{F\_1 B} = 0$. Determine the eccentricity of the hyperbola.

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