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 $2 \leq i \leq n-1$,
- $a_{n-1} + a_n = b_{n-1} + b_n$.

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

Let's think step by step and output the final answer within \boxed{}.
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