
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 curve $C_1$ is $\begin{cases}x=2\cos \theta \\ y=2+2\sin \theta\end{cases}$ ($\theta$ is the parameter). Taking the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, 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 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 $\Delta MAB$.

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