
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
In the Cartesian coordinate system $xOy$, the line $y=x+4$ intersects with the parabola $C: x^2=2py$ ($p>0$) at points $A$ and $B$, and $OA \perp OB$.
(1) Find the equation of $C$;
(2) Is there a point $D$ on the positive x-axis such that the circumcenter of $\triangle ABD$ lies on $C$? If it exists, find the coordinates of $D$; if not, explain why.

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