
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 $(xOy)$, curve $C$ is defined by the parametric equations $\begin{cases} x=1+\cos t \\ y=\frac{1}{2}\sin t \end{cases}$, where $t$ is a parameter. Curve $C_1$ is obtained by keeping the $x$-coordinate of each point on curve $C$ constant and doubling the $y$-coordinate. In the polar coordinate system with the origin $O$ as the pole and the positive $x$-axis as the polar axis, curve $C_2$ has the polar equation $2\rho\cos (\theta-\frac{\pi}{6})=3\sqrt{3}$.

1. Find the polar equation of curve $C_1$.
2. Given point $M(1,0)$, the polar equation of line $l$ is $\theta=\frac{\pi}{3}$. The intersection points of line $l$ with curve $C_1$ are $O$ and $P$, and the intersection point with curve $C_2$ is $Q$. Find the area of triangle $MPQ$.

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