
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
In the Cartesian coordinate system $xoy$, given a fixed point $F(1,0)$, point $P$ moves on the $y$-axis, point $M$ is on the $x$-axis, and point $N$ is a moving point in the plane, satisfying $\overrightarrow{PM} \cdot \overrightarrow{PF} = 0$ and $\overrightarrow{PM} + \overrightarrow{PN} = 0$. Find the equation of the trajectory $C$ of the moving point $N$.

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