
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
Consider a hyperbola given by the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ with one of its foci as point $F$ and one endpoint of its conjugate axis as point $B$. If line segment $BF$ intersects with one of the hyperbola's asymptotes at point $A$ such that $\overrightarrow{FA} = 2 \overrightarrow{AB}$, calculate the eccentricity of the hyperbola.

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