
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 on nine equidistant points on a circle, with each point containing one ball. Let $S$ be the sum of the absolute values of the differences between the numbers of all pairs of adjacent balls on the circle. Find the probability that $S$ achieves its minimum value, considering that arrangements that can be rotated or reflected to match another arrangement are considered identical.

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