
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 an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with three vertices $B_1(0, -b)$, $B_2(0, b)$, $A(a, 0)$, and focus $F(c, 0)$, it is known that $B_1F \perpendicular AB_2$. Find the eccentricity of the ellipse.

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