
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
In the rectangular coordinate system $(xOy)$, the parametric equation of line $l$ is given by $\begin{cases} x=3+ \frac{1}{2}t \\ y= \frac{ \sqrt{3}}{2}t \end{cases} (t \text{ is the parameter})$, and a polar coordinate system is established with the origin $O$ as the pole and the non-negative semi-axis of $x$ as the polar axis. The polar equation of circle $C$ is $\rho=2 \sqrt{3}\sin \theta$.
(I) Find the rectangular equation of circle $C$;
(II) Let the center of circle $C$ be $C$ and $P$ be a moving point on line $l$. Find the minimum value of $|PC|$.

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