
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 $\begin{cases} x=1+\cos\phi \\ y=\sin\phi \end{cases}$, (where $\phi$ is a parameter). A polar coordinate system is established with the origin $O$ as the pole and the non-negative half-axis of the $x$-axis 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 $O$ and $P$, and with line $l$ at point $Q$. Find the length of the line segment $PQ$.

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