
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
Call a positive integer  $n\geq 2$  *junk* if there exist two distinct  $n$  digit binary strings  $a_1a_2\cdots a_n$  and  $b_1b_2\cdots b_n$  such that


- $a_1+a_2=b_1+b_2,$
- $a_{i-1}+a_i+a_{i+1}=b_{i-1}+b_i+b_{i+1}$  for all  $2\leq i\leq n-1,$  and
- $a_{n-1}+a_n=b_{n-1}+b_n$ .

Find the number of junk positive integers less than or equal to  $2016$ .

[i]Proposed by Nathan Ramesh

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