
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 rectangular coordinate system $(xOy)$, there is a circle $C_{1}$: $(x+\sqrt{3})^{2}+y^{2}=4$, and a curve $C_{2}$ with parametric equations $\begin{cases} x=2+2\cos \theta \\ y=2\sin \theta \\ \end{cases} (\theta \text{ is the parameter})$. Establish a polar coordinate system with $O$ as the pole and the positive half of the $x$-axis as the polar axis.

(I) Write the polar equation of $C_{1}$ and convert $C_{2}$ into a standard form (Cartesian equation);

(II) If the polar equation of the line $C_{3}$ is $\theta =\dfrac{\pi }{3}(\rho \in R)$, $C_{2}$ intersects with $C_{3}$ at points $A$ and $B$, find the area of $\triangle ABC_{1}$ ($C_{1}$ is the center of circle $C_{1}$).

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