
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 equation of the parabola is $x^2=2py$ $(p > 0)$, its focus is $F$, and point $O$ is the origin. A line passing through the focus $F$ with a slope of $k$ $(k \neq 0)$ intersects the parabola at points $A$ and $B$. Tangent lines to the parabola at points $A$ and $B$ respectively intersect at point $M$.

$(1)$ Find $\overrightarrow{OA} \cdot \overrightarrow{OB}$;

$(2)$ Suppose the line $MF$ intersects the parabola at points $C$ and $D$, and the area of quadrilateral $ACBD$ is $\frac{32}{3}p^2$, find the slope $k$ of line $AB$.

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