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

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