
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 $\begin{cases} x=1+\cos{\varphi} \\ y=\sin{\varphi} \end{cases}$, where $\varphi$ is the parameter. A polar coordinate system is established with $O$ as the pole and the non-negative half-axis of $x$ as the polar axis.

$(1)$ Find the standard equation and the polar equation of circle $C$;  
$(2)$ The polar equation of line $l$ is $2\rho\sin(\theta+\frac{\pi}{3})=6\sqrt{3}$, the ray $OM$: $\theta=\frac{\pi}{6}$ intersects circle $C$ at points $O, P$, and intersects line $l$ at point $Q$. Find the length of segment $PQ$.

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