
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 $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), with its left and right foci denoted as $F_1$ and $F_2$ respectively, a line passing through $F_1$ and perpendicular to the x-axis intersects the ellipse at points A and B. The line $AF_2$ intersects the ellipse at another point C. If $\overrightarrow{AF_2} + 2\overrightarrow{CF_2} = \overrightarrow{0}$, then the eccentricity of the ellipse is ____________.

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