
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
 $(GDR 3)$  Find the number of permutations  $a_1, \cdots, a_n$  of the set  $\{1, 2, . . ., n\}$  such that  $|a_i - a_{i+1}| \neq 1$  for all  $i = 1, 2, . . ., n - 1.$  Find a recurrence formula and evaluate the number of such permutations for  $n \le 6.$ 

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