
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 curve $C$ with the polar coordinate equation $\rho^{2} - 4\rho\cos(\theta - \frac{\pi}{3}) - 1 = 0$. Establish a Cartesian coordinate system with the pole as the origin and the polar axis as the positive half of the $x$-axis. The parametric equations of line $l$ are $\begin{cases} x = t\cos\alpha \\ y = \sqrt{3} + t\sin\alpha \end{cases}$ (where $t$ is the parameter).
(I) Convert the polar coordinate equation of curve $C$ to a Cartesian coordinate equation;
(II) If line $l$ intersects curve $C$ at points $A$ and $B$, and $|AB| = 3\sqrt{2}$, find the value of the angle of inclination $\alpha$.

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