
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 with the equation $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ ($a > 0$, $b > 0$) and an eccentricity of $2$, the distance from the focus to the asymptote is $\sqrt{3}$. The line $l$ passing through the right focus $F\_2$ intersects the hyperbola at points $A$ and $B$, and $F\_1$ is the left focus.

(1) Find the equation of the hyperbola.

(2) If the area of $\triangle F\_1AB$ is equal to $6\sqrt{2}$, find the equation of the line $l$.

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