
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
Selected Exercise $(4-4)$: Coordinate Systems and Parametric Equations

Given that point $P$ is a moving point on the curve $C_1: \frac{x^2}{12} + \frac{y^2}{4} = 1$, and the parametric equation of line $C_2$ is $\begin{cases} x = 3 + \frac{\sqrt{3}}{2}t \\ y = \sqrt{3} - \frac{1}{2}t \end{cases} (t \text{ is the parameter})$, find the maximum distance between point $P$ and line $C_2$, and then find the coordinates of point $P$.

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