
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
Given the Cartesian coordinate system $xOy$, with $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, a polar coordinate system is established. The polar coordinates of point $P$ are $\left(\begin{matrix}2 \sqrt{3}, \frac{\pi}{6} \end{matrix}\right)$. The polar coordinate equation of curve $C$ is $\rho^{2}+2 \sqrt{3}\rho\sin \theta=1$.

$(1)$ Write the Cartesian coordinates of point $P$ and the Cartesian coordinate equation of curve $C$;

$(2)$ If $Q$ is a moving point on curve $C$, find the minimum distance from the midpoint $M$ of $PQ$ to the line $l$: $\begin{cases} x=3+2t, \\ y=-2+t \end{cases}(t$ is a parameter$)$.

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