
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
Given the polar equation of circle $C$ as $\rho=4\cos \theta-6\sin \theta$, and the parametric equation of line $l$ as $\begin{cases}x=4+t\cos \theta \\ y=t\sin \theta\end{cases}$ (where $t$ is the parameter), if line $l$ intersects circle $C$ at two distinct points $P$ and $Q$.

$(1)$ Write the Cartesian coordinate equation of circle $C$, and find the coordinates of the center and the radius of the circle;  
$(2)$ If the chord length $|PQ|=4$, find the slope of line $l$.

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