
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 $C$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$ with one of its foci being $F(2,0)$ and an eccentricity of $e = 2$.

(1) Find the equation of the hyperbola $C$;

(2) If a line $l$ with a slope of $1$ intersects the hyperbola $C$ at two distinct points $M$ and $N$, and the area of the triangle formed by the perpendicular bisector of the segment $MN$ and the two coordinate axes is $4$, find the equation of the line $l$.

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