
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
A point P is arbitrarily chosen on the circle $x^2+y^2=4$. A perpendicular line segment to the x-axis is drawn through point P, with D being the foot of the perpendicular. Point M is on the line segment PD, and $|DP| = \sqrt{2}|DM|$. Point P moves on the circle.  
(Ⅰ) Find the equation of the trajectory of point M;  
(Ⅱ) A line passing through the fixed point C(-1, 0) intersects the trajectory of point M at points A and B. Is there a point N on the x-axis such that $\overrightarrow{NA} \cdot \overrightarrow{NB}$ is constant? If it exists, find the coordinates of point N; if not, explain why.

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