
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 the line $l$ are $\left\{\begin{array}{l}x=-2+\frac{\sqrt{2}}{2}t,\\ y=-2-\frac{\sqrt{2}}{2}t\end{array}\right.$ (where $t$ is a parameter). Taking the coordinate origin $O$ as the pole and the positive half of the $x$-axis as the polar axis, the polar coordinate equation of the curve $C$ is $\rho \sin ^{2}\theta -4\cos \theta =0$.  
$(1)$ Find the Cartesian equation of curve $C$ and the general equation of line $l$;  
$(2)$ Find the minimum distance from a point on curve $C$ to line $l$.

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