
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 $C: \dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$ where $a > 0$ and $b > 0$ is tangent to the circle $O: x^{2} + y^{2} = 3$. A line passing through one focus of $C$ with a slope of $\sqrt{3}$ is also tangent to circle $O$.

1. Find the equation of the hyperbola $C$.
2. Let $P$ be a point in the first quadrant on circle $O$. A line $l$ passing through $P$ and tangent to circle $O$ intersects the right branch of hyperbola $C$ at points $A$ and $B$. If the area of $\triangle AOB$ is $3\sqrt{2}$, find the equation of line $l$.

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