
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 two foci \(F_1(-2, 0)\) and \(F_2(2, 0)\), and a point \(P(3, \sqrt{7})\) on the curve \(C\):
(Ⅰ) Find the equation of the hyperbola \(C\).
(Ⅱ) Let \(O\) be the origin of the coordinates, and a line \(l\) passing through point \(Q(0, 2)\) intersects hyperbola \(C\) at two distinct points \(E\) and \(F\). If the area of triangle \(\triangle OEF\) is \(2\sqrt{2}\), find the equation of the line \(l\).

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