
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 $\begin{cases} x=2\cos \varphi, \\ y=2+2\sin \varphi \end{cases}$ (where $\varphi$ is the parameter), and 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 of circle $C$;

$(2)$ The polar equation of line $l$ is $2\rho\sin \left( \theta+ \frac{\pi}{6} \right)=5 \sqrt{3}$. The ray $OM$: $\theta= \frac{\pi}{6}$ intersects circle $C$ at points $O$ and $P$, and intersects line $l$ at point $Q$. Find the length of segment $PQ$.

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