
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 equations of line $l$ are given by $\begin{cases} x=-2+\frac{1}{2}t \\ y=\frac{\sqrt{3}}{2}t \end{cases} (t \text{ is the parameter})$. Establish a polar coordinate system with the coordinate origin as the pole and the positive $x$-axis as the polar axis. The polar equation of curve $C_1$ is $\rho=\sqrt{6}$.
(1) Write the Cartesian equation of line $l$ and the parametric equations of curve $C_1$;
(2) If each point on curve $C_1$ has its abscissa reduced to $\frac{\sqrt{6}}{6}$ times the original and its ordinate reduced to $\frac{\sqrt{2}}{2}$ times the original, a new curve $C_2$ is obtained. Let $P$ be any point on curve $C_2$. Find the minimum distance from point $P$ to line $l$.

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