
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 that $F(1,0)$, point $M$ lies on the $x$-axis, point $P$ lies on the $y$-axis, and $\overrightarrow{MN}=2\overrightarrow{MP}$, $\overrightarrow{PM} \perp \overrightarrow{PF}$, find the trajectory equation of point $N$ as point $P$ moves along the $y$-axis.

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