
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
Nine balls are numbered $1, 2, \ldots, 9$ and randomly placed at nine evenly spaced points on a circle, with each point holding one ball. Define $S$ as the sum of the absolute differences between the numbers of every pair of adjacent balls on the circle. Determine the probability of an arrangement that minimizes $S$. (Note: If an arrangement can overlap with another arrangement through rotation or reflection, they are considered the same arrangement.)

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