
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
How many ways are there to arrange the  $6$  permutations of the tuple  $(1, 2, 3)$  in a sequence, such that each pair of adjacent permutations contains at least one entry in common? 
For example, a valid such sequence is given by  $(3, 2, 1) - (2, 3, 1) - (2, 1, 3) - (1, 2, 3) - (1, 3, 2) - (3, 1, 2)$ .

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