
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 numbered $1, 2, \cdots, 9$ are randomly placed at nine equidistant points on a circle, with each point occupied by one ball. Let $S$ be the sum of the absolute values of the differences between the numbers on every two adjacent balls on the circle. Find the probability of the arrangement that makes $S$ minimal. (Note: If one arrangement can overlap with another arrangement by rotation or reflection, they are considered the same).

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