
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 $l$ as an asymptote of the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \ (a > 0, b > 0)$, and $l$ intersects the circle $(x-c)^2 + y^2 = a^2$ (where $c^2 = a^2 + b^2$) at points A and B, if $|AB| = a$, find the eccentricity of $C$.

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