
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 equation of circle $C$ is $\begin{cases} & x=4\cos \theta \\ & y=4\sin \theta \\ \end{cases}$ ($\theta$ is the parameter), and line $l$ passes through point $P(1,2)$ with an inclination angle $\alpha= \frac{\pi}{6}$.  
$(1)$ Find the parametric equation of line $l$;  
$(2)$ Suppose line $l$ intersects circle $C$ at points $A$ and $B$, find the value of $|PA|\cdot|PB|$.

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