
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 curve $C_{1}$ is given by $\begin{cases} x=1+2\cos \theta \\ y=2\sin \theta \end{cases}$ (where $\theta$ 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.  
$(I)$ Find the general equation of curve $C_{1}$.  
$(II)$ The line $l$ with the polar equation $2\rho\sin (\theta+ \frac {\pi}{3})=3 \sqrt {3}$ intersects curve $C_{1}$ at points $P$ and $Q$. Find the length of segment $PQ$.

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