
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 one ball at each point. The sum of the absolute differences between the numbers of all adjacent balls on the circle is denoted as $S$. Determine the probability of the arrangement that minimizes $S$. (Note: If an arrangement can be made identical to another by rotation or reflection, they are considered the same.)

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