
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 $C$: $x^{2}=2py (p > 0)$ with a focus at $F$, a line passing through $F$ with a slope of $1$ intersects the parabola at points $M$ and $N$ such that $|MN|=8$.
1. Find the equation of the parabola $C$.
2. Let line $l$ be a tangent to the parabola $C$ such that $l$ is parallel to $MN$, and let $P$ be a point on $l$. Find the minimum value of $\overrightarrow{PM} \cdot \overrightarrow{PN}$.

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