
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 equation of circle $M$ is given by $x^2+y^2-4x\cos \alpha-2y\sin \alpha+3\cos^2\alpha=0$ (where $\alpha$ is a parameter), and the parametric equation of line $l$ is $\begin{cases}x=\tan \theta \\ y=1+t\sin \theta\end{cases}$ (where $t$ is a parameter).

$(I)$ Find the parametric equation of the trajectory $C$ of the center of circle $M$, and explain what curve it represents;  
$(II)$ Find the maximum chord length cut by the trajectory $C$ on line $l$.

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