
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 circle $C$ is $\begin{cases} x=3+2\cos \theta, \\ y=-4+2\sin \theta \end{cases}$ ($\theta$ is the parameter).

 $(1)$ Establish a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis, and find the polar equation of circle $C$;

 $(2)$ Given $A(-2,0)$, $B(0,2)$, and any point $M(x,y)$ on circle $C$, find the maximum value of the area of $\triangle ABM$.

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