
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
Someone has arranged a 10-volume collection of works in a random order. We define a disorder to be a pair of volumes where the volume with the larger number is positioned to the left. For a given arrangement of the volumes, the number $S$ of all disorders is counted. What values can $S$ take?

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