
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 equation of the curve $C_1$ is $$\begin{cases} x=2\cos\alpha \\ y=\sin\alpha \end{cases}$$ ($\alpha$ is the parameter), and the equation of the circle $C_2$ is $(x-2)^2+y^2=4$. Establish a polar coordinate system with the origin $O$ as the pole and the positive semi-axis of the $x$-axis as the polar axis. The polar equation of the ray $l$ is $\theta=\theta_0$ ($\rho\geq 0$).

1. Find the polar equations of the curve $C_1$ and the circle $C_2$.
2. When $0<\theta_0<\frac{\pi}{2}$, the ray $l$ intersects the curve $C_1$ and the circle $C_2$ at points $M$ and $N$ respectively, which are different from point $O$. If $|ON|=2|OM|$, find the area of $\triangle MC_2N$.

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