
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 $M$ is given by $\begin{cases} & x=1+2\cos \beta \\ & y=1+2\sin \beta \end{cases}$ (where $\beta$ is the parameter). Taking the origin as the pole and the positive $x$-axis as the polar axis, we establish a polar coordinate system. The polar equation of line $l_1$ is $\theta =\alpha$, and the polar equation of line $l_2$ is $\theta =\alpha +\frac{\pi }{2}$.

$(I)$ Write the polar equation of curve $M$ and identify what kind of curve it is;

$(II)$ Suppose $l_1$ intersects curve $M$ at points $A$ and $C$, and $l_2$ intersects curve $M$ at points $B$ and $D$. Find the range of the area of quadrilateral $ABCD$.

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