
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
We denote an arbitrary permutation of the integers  $1$ ,  $2$ ,  $\ldots$ ,  $n$  by  $a_1$ ,  $a_2$ ,  $\ldots$ ,  $a_n$ .  Let  $f(n)$  denote the number of these permutations such that:

(1)  $a_1 = 1$ ;

(2): $|a_i - a_{i+1}| \leq 2$ ,  $i = 1, \ldots, n - 1$ .

Determine whether  $f(1996)$  is divisible by 3.

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