
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
At ARML, Santa is asked to give rubber duckies to  $2013$  students, one for each student.   The students are conveniently numbered  $1,2,\cdots,2013$ , and for any integers  $1 \le m < n \le 2013$ , students  $m$  and  $n$  are friends if and only if  $0 \le n-2m \le 1$ .

Santa has only four different colors of duckies, but because he wants each student to feel special, he decides to give duckies of different colors to any two students who are either friends or who share a common friend.  Let  $N$  denote the number of ways in which he can select a color for each student.  Find the remainder when  $N$  is divided by  $1000$ .

*Proposed by Lewis Chen*

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