
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 $xOy$, given the parametric equations of circle $C_{1}$ as $\left\{\begin{array}{l}x=2+2\cos\alpha\\ y=1+2\sin\alpha\end{array}\right.$ ($\alpha$ is the parameter), establish a polar coordinate system with the coordinate origin as the pole and the positive x-axis as the polar axis.<br/>$(1)$ Find the polar equation of $C_{1}$;<br/>$(2)$ If the polar equation of line $C_{2}$ is $θ=\frac{π}{4}$ ($ρ∈R$), and the intersection points of $C_{2}$ and $C_{1}$ are $P$ and $Q$, find the area of $\triangle C_{1}PQ$.

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