
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 a hyperbola \( C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \), where \( A_{1} \) and \( A_{2} \) are the left and right vertices of hyperbola \( C \), and \( F \) is a focus of \( C \). Let \( O \) be the origin, with a circle centered at \( O \) having a radius \( |OF| \). The circle intersects with one of the asymptotes of \( C \) at point \( B \). If \( \angle A_{1}BA_{2} = 60^\circ \), find the eccentricity of \( C \).

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