
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
A parabola, given by the equation $x^2 = 2py$ where $p > 0$, has its focus acting as a point on a straight line with a slope of $1$. This line intersects the parabola at points $A$ and $B$. The projections of $A$ and $B$ on the $x$-axis are $D$ and $C$, respectively. If the area of the quadrilateral $ABCD$ is $12\sqrt{2}$, find the value of $p$.

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