
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 curve $C$ are $\left\{\begin{array}{l}x=\cos\theta\\ y=2\sin\theta\end{array}\right.$ (where $\theta$ is the parameter). Taking the coordinate origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the line $l$ is $2\rho\cos\theta+\sqrt{3}\rho\sin\theta+11=0$.  
$(1)$ Find the Cartesian equations of $C$ and $l$;  
$(2)$ Find the minimum distance from a point on $C$ to $l$.

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