
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 a line $l$ passing through point A(0, 1) with a slope of $k$ intersects the circle $C: (x-2)^2+(y-3)^2=1$ at points M and N.  
(1) Find the range of values for $k$;  
(2) Determine whether there exists a real number $k$ such that $\overrightarrow{OM} \cdot \overrightarrow{ON} = 12$ (where O is the origin of the coordinate system). If it exists, find the value of $k$ and calculate $|MN|$. If it does not exist, explain why.

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