
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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Elective 4-4: Coordinate System and Parametric Equations

In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases}x=3+t \\ y= \sqrt{3}t\end{cases} (t$ is the parameter$)$, 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 general equation of line $l$ and the Cartesian coordinate equation of circle $C$;
$(2)$ Point $P$ is on line $l$, find the Cartesian coordinates of point $P$ such that the distance from point $P$ to the center of circle $C$ is minimized.

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