
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
Let  $W = \ldots x_{-1}x_0x_1x_2 \ldots$  be an infinite periodic word consisting of only the letters  $a$  and  $b$ . The minimal period of  $W$  is  $2^{2016}$ . Say that a word  $U$  *appears* in  $W$  if there are indices  $k \le \ell$  such that  $U = x_kx_{k+1} \ldots x_{\ell}$ . A word  $U$  is called *special* if  $Ua, Ub, aU, bU$  all appear in  $W$ . (The empty word is considered special) You are given that there are no special words of length greater than 2015.

	Let  $N$  be the minimum possible number of special words. Find the remainder when  $N$  is divided by  $1000$ .
*Proposed by Yang Liu*

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