
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 $(x\mathrm{O}y)$, the parametric equations of circle $C$ are given by $\begin{cases}x=1+\cos\varphi\\y=\sin\varphi\end{cases}$, where $\varphi$ is a parameter. Establish a polar coordinate system with $\mathrm{O}$ as the pole and the non-negative semiaxis of the $x$-axis as the polar axis.

(1) Find the polar equation of circle $C$.
(2) The polar equation of line $l$ is $2\rho\sin\left(\theta+\frac{\pi}{3}\right)=3\sqrt{3}$. If the ray $\mathrm{OM}:\theta=\frac{\pi}{3}$ intersects circle $C$ at point $R$ and line $l$ at point $Q$, find the length of line segment $RQ$.

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