
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, the parametric equation of curve $C_1$ is given by the system of equations:
$$
\begin{cases}
x = 2\cos\theta \\
y = 2 + 2\sin\theta
\end{cases}
(\theta \text{ is the parameter}).
$$
Using the origin $O$ as the pole and the positive $x$-axis as the initial line, establish a polar coordinate system. The polar equation of curve $C_2$ is $\rho = 4\cos\theta$.

(1) Find the polar equation of curve $C_1$.
(2) The ray $\theta = \frac{\pi}{3} (\rho \geqslant 0)$ intersects the curves $C_1$ and $C_2$ at points $A$ and $B$ (different from the origin $O$) respectively. Given the fixed point $M(2,0)$, find the area of triangle $MAB$.

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