
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 curve $C$ is defined by the parametric equations $\begin{cases} & x=\sqrt{3}\cos \alpha \\ & y=\sin \alpha \end{cases}$ (with $\alpha$ as the parameter). In the polar coordinate system with the origin $O$ as the pole and the non-negative half-axis of $x$ as the polar axis, the polar equation of line $l$ is $\frac{\sqrt{2}}{2}\rho \cos (\theta +\frac{\pi }{4})=-1$.

$(1)$ Find the standard equation of curve $C$ and the Cartesian equation of line $l$;

$(2)$ A line $l_{1}$ passes through point $M(-1,0)$ and is parallel to line $l$, intersecting curve $C$ at points $A$ and $B$. Find the product of the distances from point $M$ to points $A$ and $B$.

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