
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 $\begin{cases} x=1+\cos t \\ y=\sin t \end{cases}$ (with $t$ as the parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive $x$-axis as the polar axis.  
$(1)$ Find the polar equation of circle $C$;  
$(2)$ The polar equation of a line is $2\rho\sin (\alpha+ \dfrac {\pi}{4})=2 \sqrt {2}$, and the polar equation of curve $C_{1}$ is $\theta=\alpha_{0}$, where $\alpha_{0}$ satisfies $\tan \alpha_{0}=2$. The intersection points of curve $C_{1}$ with circle $C$ are $O$ and $P$, and with the line are $Q$. Find the length of segment $PQ$.

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