
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
\[
\begin{cases}
x = 2 + 2\cos \alpha \\
y = 2\sin \alpha
\end{cases}
\]
with $\alpha$ as the parameter. Taking the origin $O$ as the pole and the non-negative half-axis of $x$ as the polar axis, we establish a polar coordinate system in which the polar equation of line $C_2$ is $\rho \sin \theta = \sqrt{3}$.
(1) Find the polar equation of curve $C_1$.
(2) Let the intersection points of $C_1$ and $C_2$ be $A$ and $B$. Calculate the area of $\triangle AOB$.

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