
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 its two foci at $(-2, 0)$ and $(2, 0)$, and point $P(3, \sqrt{7})$ lies on the hyperbola $C$.

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

(Ⅱ) A line passing through point $A(0, 2)$ intersects the hyperbola $C$ at two distinct points $E$ and $F$. If the area of the triangle formed by the origin $O$ and points $E$, $F$ is $2\sqrt{2}$, find the equation of line $l$.

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