
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 of  $1,2,3,\dots,n$ , with consecutive elements  $a,b,c$  (in that order), we may perform either of the *moves*:


- If  $a$  is the median of  $a$ ,  $b$ , and  $c$ , we may replace  $a,b,c$  with  $b,c,a$  (in that order)
- If  $c$  is the median of  $a$ ,  $b$ , and  $c$ , we may replace  $a,b,c$  with  $c,a,b$  (in that order)

What is the least number of sets in a partition of all  $n!$  permutations, such that any two permutations in the same set are obtainable from each other by a sequence of moves?

*Proposed by Milan Haiman*

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