
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
Determine with proof, the number of permutations  $a_1,a_2,a_3,...,a_{2016}$  of  $1,2,3,...,2016$  such that the value of  $|a_i-i|$  is fixed for all  $i=1,2,3,...,2016$ , and its value is an integer multiple of  $3$ .

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