
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 holding one ball. Let $S$ denote the sum of the absolute values of the differences between the numbers of all adjacent pairs of balls on the circle. Find the probability of the placement of the balls that minimizes $S$.

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