
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 the curve $C$ is:
$$ \begin{cases} x = 2\cos \theta \\ y = \sin \theta \end{cases} $$
With the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis, a polar coordinate system is established.  
(1) If we keep the $x$-coordinates of points on curve $C$ unchanged and stretch the $y$-coordinates to twice their original lengths, we obtain curve $C_1$. Find the polar equation of curve $C_1$.  
(2) The polar equation of line $l$ is $\rho \sin(\theta + \frac{\pi}{3}) = \sqrt{3}$, and it intersects curve $C_1$ at points $A$ and $B$. Find the area of triangle $AOB$.

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