
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
$$
\begin{cases}
x=-2+\frac{1}{2}t \\
y=\frac{\sqrt{3}}{2}t
\end{cases}
(t \text{ being the parameter}).
$$
Using the origin as the pole and the positive $x$-axis as the initial line, a polar coordinate system is established. The polar equation of curve $C_1$ is $\rho = \sqrt{6}$.

(1) Write down the standard equation of line $l$ and the parametric equation of curve $C_1$;
(2) If the $x$-coordinate of each point on curve $C_1$ is reduced to $\frac{\sqrt{6}}{6}$ times its original length, and the $y$-coordinate is reduced to $\frac{\sqrt{2}}{2}$ times its original length, yielding curve $C_2$, let point $P$ be an arbitrary 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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