
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
The numbers \( p \) and \( q \) are chosen such that the parabola \( y = p x - x^2 \) intersects the hyperbola \( x y = q \) at three distinct points \( A, B, \) and \( C \). The sum of the squares of the sides of triangle \( ABC \) is 324, and the intersection point of its medians is at a distance of 2 from the origin. Find the product \( pq \).

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