
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

Given a permutation \(\pi\) of the set \(\{1,2, \ldots, 10\}\), define a rotated cycle as a set of three integers \(i, j, k\) such that \(i < j < k\) and \(\pi(j) < \pi(k) < \pi(i)\). What is the total number of rotated cycles over all permutations \(\pi\) of the set \(\{1,2, \ldots, 10\}\)?

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