
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
Twelve distinct points, \( A, B, C, \) and \( D \), are to be selected from \(2024\) points evenly spaced around a circle. Points \( A \) and \( C \) are required to lie on the top half of the circle, while \( B \) and \( D \) should be on the bottom half. All such quadruples (following the half-circle condition) are equally likely to be chosen. What is the probability that the chord \( \overline{AB} \) intersects the chord \( \overline{CD} \)?

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