
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 \( p \).

Ellipse \( C_1: \frac{x^{2}}{4} + y^{2} = 1 \) intersects the parabola \( C_2: x^{2} = 2py \) (where \( p > 0 \)) at points \( A \) and \( B \). \( O \) is the origin.

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