
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
$15(1,0)$ is a fixed point, $P(0, b)$ is a moving point on the $y$-axis, and point $M(a, 0)$ satisfies $\overrightarrow{P M} \cdot \overrightarrow{P F}=0$. Given that point $N$ satisfies $2 \overrightarrow{P N}+\overrightarrow{N M}=\overrightarrow{0}$.

1. Find the equation of the trajectory curve $C$ of point $N$.
2. Find the trajectory of the intersection of any two mutually perpendicular tangent lines to the curve $C$.

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