
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
(1) If the circumcenter of triangle \( \triangle ABO \) is on the ellipse, find the value of the real number \( p \).

(2) If the circumcircle of triangle \( \triangle ABO \) passes through the point \( N\left(0, \frac{13}{2}\right) \), find the value of the real number \( p \).

Elliptic equation: \( C_{1}: \frac{x^{2}}{4}+y^{2}=1 \)

Parabolic equation: \( C_{2}: x^{2}=2py \) (with \( p > 0 \))

\( C_{1} \) and \( C_{2} \) intersect at points \( A \) and \( B \).

\( O \) is the origin of the coordinate system.

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