
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 that the pole of the polar coordinate system coincides with the origin of the Cartesian coordinate system, the polar axis coincides with the positive half-axis of the $x$-axis of the Cartesian coordinate system, and the coordinate lengths of the two coordinate systems are the same. The parametric equation of line $l$ is $\begin{cases} x=-1+t\cos \alpha \\ y=1+t\sin \alpha \end{cases}$ (where $t$ is the parameter), and the polar equation of curve $C$ is $\rho=4\cos \theta$.

$(1)$ If the slope of line $l$ is $-1$, find the polar coordinates of the intersection points of line $l$ and curve $C$;

$(2)$ If the chord length of the intersection of line $l$ and curve $C$ is $2 \sqrt {3}$, find the parametric equation of line $l$.

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