
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
For how many permutations  $(a_1, a_2, \cdots, a_{2007})$  of the integers from  $1$  to  $2007$  is there exactly one  $i$  between  $1$  and  $2006$  such that  $a_i > a_{i+1}$ ? Express your answer as  $a \** b^c + d \** e^f$  for integers  $a$ ,  $b$ ,  $c$ ,  $d$ ,  $e$ , and  $f$  with  $a \nmid b$  and  $d \nmid e$ .

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