
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 $(xOy)$, the parametric equations of the curve $C$ are given by: $\begin{cases} x=2\cos\theta \\ y=\sin\theta \end{cases}$. Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half of the $x$-axis as the polar axis.

(1) If the horizontal coordinate of each point on the curve $C$ remains unchanged and the vertical coordinate is stretched to twice its original length, obtain the curve $C_{1}$. Find the polar equation of $C_{1}$;

(2) The polar equation of the straight line $l$ is $\rho\sin\left(\theta+\frac{\pi}{3}\right)=\sqrt{3}$, which intersects the curve $C_{1}$ at points $A$ and $B$. Calculate the area of triangle $AOB$.

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