
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 Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $$\begin{cases} x=3+ \frac {1}{2}t \\ y= \frac { \sqrt {3}}{2}t \end{cases} (t \text{ is the parameter}),$$ and a polar coordinate system is established with the origin as the pole and the positive half-axis of $x$ as the polar axis. The polar equation of circle $C$ is $$\rho=2 \sqrt {3}\sin\theta.$$  
(1) Write the Cartesian coordinate equation of circle $C$;  
(2) Let $P$ be a moving point on line $l$. When the distance from $P$ to the center of circle $C$ is minimized, find the Cartesian coordinates of $P$.

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