
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 $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ ($a > b > 0$), its left and right vertices are $A_{1}$ and $A_{2}$, respectively, and a circle with $A_{1}A_{2}$ as its diameter is tangent to the line $bx - ay + 2ab = 0$. Determine the eccentricity of the ellipse $C$.

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