
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
Consider the permutation of  $1,2,...,n$ , which we denote as  $\{a_1,a_2,...,a_n\}$ . Let  $f(n)$  be the number of these permutations satisfying the following conditions:
(1) $a_1=1$ (2) $|a_i-a_{i-1}|\le2, i=1,2,...,n-1$ what is the residue when we divide  $f(2015)$  by  $4$  ?

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