
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 parameter equation in the Cartesian coordinate system xOy:
$$
\begin{cases} 
x=1+ \sqrt{3}\cos\theta \\
y=\sqrt{3}\sin\theta
\end{cases}
$$
($\theta$ as the parameter). Establishing a polar coordinate system with O as the pole and the non-negative half-axis of x as the polar axis,

(1) Find the polar equation of the curve C;
(II) The polar equation of line l is $\rho\cos(\theta-\frac{\pi}{6})=3\sqrt{3}$, and the ray OT: $\theta=\frac{\pi}{3}$ ($\rho > 0$) intersects curve C at point A and line l at point B. Find the length of segment AB.

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