
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
Let $O$ be the coordinate origin, and let $M$ be a moving point on the ellipse $C$: $\frac{x^{2}}{2}+y^{2}=1$. Draw a vertical line through $M$ that intersects the $x$-axis at point $N$. Point $P$ satisfies $\overrightarrow{NP} = \sqrt{2}\overrightarrow{NM}$. Find the trajectory equation of point $P$.

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