
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
On the circle $x^{2}+y^{2}=4$, take any point $P$. Draw a perpendicular line segment from point $P$ to the $x$-axis, let $D$ be the foot of the perpendicular, and point $M$ is on the line segment $PD$, with $|DP|=\sqrt{2}|DM|$. Point $P$ moves on the circle.

$(1)$ Find the equation of the trajectory of point $M$;

$(2)$ A line passing through the fixed point $C(-1,0)$ intersects the trajectory of point $M$ at points $A$ and $B$. Determine whether there exists a point $N$ on the $x$-axis such that $\overrightarrow{NA} \cdot \overrightarrow{NB}$ is a 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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