
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 $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \ (a > 0, b > 0)$ with a line drawn from the right focus $F$ to one end of its imaginary axis $B$, and the distance from the right vertex $A$ to the line $FB$ equal to $\frac{b}{\sqrt{7}}$, calculate the eccentricity $e$ of this hyperbola.

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