
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 the ellipse $C$: $\dfrac{x^{2}}{a^{2}}+ \dfrac{y^{2}}{b^{2}}=1(a > b > 0)$ passes through the point $(\sqrt{2},1)$, and its eccentricity is $\dfrac{\sqrt{2}}{2}$.

$(1)$ Find the equation of the ellipse $C$;

$(2)$ Let $M$, $N$ be points on the ellipse, and the product of the slopes of lines $OM$ and $ON$ ($O$ is the origin) is $- \dfrac{1}{2}$. If the moving point $P$ satisfies $\overrightarrow{OP}=\overrightarrow{OM}+2\overrightarrow{ON}$, find the equation of the trajectory of point $P$.

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