
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 $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) that is tangent to the circle $O: x^2 + y^2 = 3$, and a line that passes through one focus of the hyperbola $C$ with a slope of $\sqrt{3}$ that is also tangent to the circle $O$.  
(Ⅰ) Determine the equation of the hyperbola $C$.  
(Ⅱ) Point $P$ lies on the circle $O$ in the first quadrant. A line $l$ passing through $P$ and tangent to the circle $O$ intersects the right branch of the hyperbola $C$ at points $A$ and $B$. If the area of $\triangle AOB$ is $3\sqrt{2}$, find the equation of the line $l$.

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