
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 a hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ with two asymptotes intersecting a parabola $y^{2} = 4x$ at two points $A$ and $B$ distinct from the origin $O$. $F$ is the focus of the parabola $y^{2} = 4x$, and $\angle AFB = \frac{2\pi}{3}$. Determine the eccentricity of the hyperbola.

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