
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 two fixed points M(0, 1) and N(1, 2), in a plane, the ratio of the distance from a moving point P to M and the distance from P to N is $\sqrt{2}$. The line $y=kx-1$ intersects the trajectory of point P at points A and B.  
(1) Find the equation of the trajectory of point P and identify the shape of the graph;  
(2) Determine the range of values for the real number $k$;  
(3) Determine whether there exists a value of $k$ such that $\overrightarrow{OA} \cdot \overrightarrow{OB} = 11$ (where O is the origin). If such a value exists, find it; if not, explain why.

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