
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 $\begin{cases} x=1+\cos \phi \\ y=\sin \phi \end{cases}$ (parameter $\phi$). Establish a polar coordinate system with $O$ as the pole and the non-negative half-axis of $x$ as the polar axis. The polar equation of line $kl$ is $\rho(\sin \theta+ \sqrt {3}\cos \theta)=3 \sqrt {3}$.  
$(1)$ Find the polar equation of circle $C$;  
$(2)$ The ray $OM$: $\theta=\theta_{1}(\theta < \theta_{1} < \frac {\pi}{2})$ intersects circle $C$ at points $O$ and $P$, and intersects line $Ll$ at point $Q$. Find the range of $|OP|\cdot|OQ|$.

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