
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 curve $C_1$ is $\left\{ \begin{matrix} x=\sqrt{3}+2\cos \alpha \\ y=2+2\sin \alpha \end{matrix}\right.$ ($\alpha$ is the parameter). The equation of line $C_2$ is $y=\frac{\sqrt{3}}{3}x$. Establish a polar coordinate system with $O$ as the pole and the non-negative half-axis of $x$ as the polar axis.

$(1)$ Find the polar equations of curve $C_1$ and line $C_2$.

$(2)$ If line $C_2$ intersects curve $C_1$ at points $P$ and $Q$, find the value of $|OP|\cdot |OQ|$.

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