
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 $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$) with foci $F_1$ (-2, 0) and $F_2$ (2, 0), and point $P(3, \sqrt{7})$ lies on the hyperbola.
(1) Find the equation of the hyperbola;
(2) A line $l$ through Q(0, 2) intersects the hyperbola at two distinct points E and F. If the area of $\triangle OEF$ is $2\sqrt{2}$, where O is the origin, find the equation of line $l$.

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