
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 a line passing through point $P(-1,2)$ with direction vector $\overrightarrow{n}=(-1,\sqrt{3})$, and the equation of the circle $\rho =2\cos (\theta +\frac{\pi }{3})$,

$(1)$ Find the parametric equation of the line.

$(2)$ Suppose the line intersects the circle at points $M$ and $N$, calculate the value of $|PM|\cdot |PN|$.

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