
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 ellipse C: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, whose left and right vertices are $A_1$ and $A_2$ respectively, and the circle with diameter $A_1A_2$ is tangent to the line $bx - ay + 3ab = 0$, calculate the eccentricity of C.

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