
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 two foci $F_{1}(-2,0), F_{2}(2,0)$ and point $P(3,\sqrt{7})$ on curve $C$.

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

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

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