
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 $xOy$, it is known that the center of the circle $(x^2+y^2-12x+32=0)$ is $Q$. A line passing through point $P(0,2)$ with a slope of $k$ intersects the circle $Q$ at two distinct points $A$ and $B$.

$(1)$ Find the range of values for $k$;

$(2)$ When $\overrightarrow{OA} \cdot \overrightarrow{OB}=28$, find the equation of line $l$;

$(3)$ Determine whether there exists a point $C$ on the $y$-axis such that $\overrightarrow{CA} \cdot \overrightarrow{CB}$ is a constant value. If it exists, find the coordinates of $C$ and the value of $\overrightarrow{CA} \cdot \overrightarrow{CB}$; if not, explain why.

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