
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 a parabola $C$: $y^{2}=-2px (p > 0)$ with a focus at $F$, there exists a point $M$ on the parabola $C$ such that the point symmetric to $F$ with respect to $M$ is $M'( \frac {2}{5}, \frac {8}{5})$, and $|MF|=1$.
(1) Find the equation of the parabola $C$;
(2) If the other intersection point of the line $MF$ and the parabola $C$ is $N$, and the circle with $MN$ as its diameter passes through a point $P$ on the $y$-axis, find the coordinates of point $P$.

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