
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 the hyperbola $C$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ with a real axis length of $2\sqrt{3}$ and one focus at $(-\sqrt{5}, 0)$.

(1) Find the equation of the hyperbola.
(2) If a line $l$ with a slope of $2$ intersects the hyperbola $C$ at points $A$ and $B$, and $|AB| = 4$, find the equation of line $l$.

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