
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 $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (where $a > 0$, $b > 0$), its right focus is at point F, the focal distance is 8, and the left vertex is at point A. There is a point B (0, b) on the y-axis such that the dot product of $\overrightarrow{BA}$ and $\overrightarrow{BF}$ equals $2a$. Find the eccentricity of this hyperbola.

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