
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 $\begin{cases} & x=2+2\cos \theta \\ & y=2\sin \theta \end{cases}$, where $\theta$ is the parameter. Let $M$ be a moving point on curve $C_1$, and moving point $P$ satisfies $\overrightarrow{OP}=a\overrightarrow{OM}$ ($a > 0$ and $a\ne 1$). The trajectory of point $P$ is curve $C_2$.

$(1)$ Find the equation of curve $C_2$ and explain what kind of curve $C_2$ is;

$(2)$ In the polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar coordinate of point $A$ is $(2,\dfrac{\pi }{3})$. The ray $\theta =\alpha$ intersects curve $C_2$ at point $B$ different from the pole. Given that the maximum area of $\Delta AOB$ is $4+2\sqrt{3}$, find the value of $a$.

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