
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 ellipse \( C_{1}: \frac{x^{2}}{4} + y^{2} = 1 \) and the parabola \( C_{2}: x^{2} = 2py \) (where \( p > 0 \)), suppose \( C_{1} \) and \( C_{2} \) intersect at points \( A \) and \( B \), with \( O \) as the origin of coordinates.

(1) If the circumcenter of \( \triangle A B O \) lies on the ellipse, find the value of \( p \).

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

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