
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 $C_1: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) and a hyperbola $C_2: \frac{x^2}{2} - \frac{y^2}{8} = 1$ share a common focus. One of the asymptotes of $C_2$ intersects the circle whose diameter is the major axis of $C_1$ at points A and B. If $C_1$ divides the line segment AB into three equal parts, determine $a^2$.

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