
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 $C: \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} - 1 = 0$ with $a > 0, b > 0$, and its two foci located at $F_1: (-2,0)$ and $F_2: (2,0)$, and the point $P(3, \sqrt{7})$ lies on the curve $C$.

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

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

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