
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
An ellipse is given by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. A line $l$ perpendicular to the $x$-axis passes through the right focus $F_2$ of the ellipse, intersecting the ellipse at points $A$ and $B$. If $A$ and $B$ form an equilateral triangle with the left focus, find the eccentricity of the ellipse.

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