
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 circle $C$ are $\left\{\begin{array}{l}x=2+2\cos\theta\;,\;\;\\ y=2\sin\theta\end{array}\right.$ ($\theta$ is the parameter), and the parametric equations of line $l$ are $\left\{\begin{array}{l}x=1+\frac{1}{2}t\;,\;\;\\ y=\frac{\sqrt{3}}{2}t\end{array}\right.$ ($t$ is the parameter).<br/>$(1)$ Establish a polar coordinate system with the coordinate origin as the pole and the positive x-axis as the polar axis. Find the polar coordinate equation of circle $C$;<br/>$(2)$ If point $P(1,0)$, 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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