
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 $\begin{cases} & x=2+2\cos \theta \\ & y=2\sin \theta \\ \end{cases}$, ($\theta$ is the parameter); the standard equation of line $l_{1}$ is $x+1=0$. 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 circle $C$ and line $l_{1}$;

$(2)$ If the polar equation of line $l_{2}$ is $\theta =\frac{\pi }{3}$, ($\rho\in\mathbb{R}$) and line $l_{2}$ intersects circle $C$ at points $O$ and $P$ ($O$ is the origin), and line $l_{2}$ intersects line $l_{1}$ at point $Q$, find $|PQ|$.

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