
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 \( C \): \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) with eccentricity \( \sqrt{3} \), and the point \( (\sqrt{3}, 0) \) is a vertex of the hyperbola.
1. Find the equation of the hyperbola.
2. A line \( l \) passing through the right focus \( F_2 \) of the hyperbola at an inclination of \( 30^\circ \) intersects the hyperbola at distinct points \( A \) and \( B \). Find the length of \( AB \).

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