
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]


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In the rectangular coordinate system, the parametric equations of circle $C$ are given by $\begin{cases} x = 1 + \cos \alpha \\ y = \sin \alpha \end{cases}$, where $\alpha$ is the parameter. Establish a polar coordinate system with the coordinate origin as the pole and the positive half of the $x$-axis as the polar axis.

(I) Find the polar equation of circle $C$;

(II) The polar equation of line $l$ is $2\rho \sin\left( \theta + \frac{\pi}{3} \right) = 3\sqrt{3}$. The intersection points of ray $OM:\theta = \frac{\pi}{3}$ with circle $C$ and line $l$ are $O, P$ and $Q$ respectively. Find the length of line segment $PQ$.

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