
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 permutations of the numbers $1, 2, 3, \ldots, n$ are there in which exactly

a) once,

b) twice,

it occurs that the number on the left is greater than the number on the right among two adjacent numbers?

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