
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 the hyperbola $C$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b < 0)$, its right vertex and right focus are $A$ and $F$ respectively, and the intersection point of its left directrix and the $x$-axis is $B$. If $A$ is the midpoint of segment $BF$, then the eccentricity of hyperbola $C$ is $\_\_\_\_\_\_$.

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