
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 origin of coordinates, and a moving point $N$ lies on the circle $C$: $x^2 + y^2 = 8$. A perpendicular line to the $y$-axis is drawn through $N$, with the foot of the perpendicular being $M$. Point $P$ satisfies $\overrightarrow{MP} = \dfrac{1}{2} \overrightarrow{MN}$. Find the equation of the trajectory of point $P$.

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