Call a positive integer $n\geq 2$ [i]junk[/i] if there exist two distinct $n$ digit binary strings $a_1a_2\cdots a_n$ and $b_1b_2\cdots b_n$ such that
[list]
[*] $a_1+a_2=b_1+b_2,$
[*] $a_{i-1}+a_i+a_{i+1}=b_{i-1}+b_i+b_{i+1}$ for all $2\leq i\leq n-1,$ and
[*] $a_{n-1}+a_n=b_{n-1}+b_n$.
[/list]
Find the number of junk positive integers less than or equal to $2016$.

[i]Proposed by Nathan Ramesh
1. **Initial Conditions and Equations:**
   We are given two distinct \( n \)-digit binary strings \( a_1a_2\cdots a_n \) and \( b_1b_2\cdots b_n \) such that:
   \[
   a_1 + a_2 = b_1 + b_2,
   \]
   \[
   a_{i-1} + a_i + a_{i+1} = b_{i-1} + b_i + b_{i+1} \quad \text{for all } 2 \leq i \leq n-1,
   \]
   \[
   a_{n-1} + a_n = b_{n-1} + b_n.
   \]

2. **Analyzing the Equations:**
   - From \( a_1 + a_2 = b_1 + b_2 \) and \( a_1 + a_2 + a_3 = b_1 + b_2 + b_3 \), we get:
     \[
     a_3 = b_3.
     \]
   - Next, from \( a_3 = b_3 \) and \( a_3 + a_4 + a_5 = b_3 + b_4 + b_5 \), we get:
     \[
     a_4 + a_5 = b_4 + b_5.
     \]
   - Continuing this pattern, we see that:
     \[
     a_6 = b_6,
     \]
     \[
     a_9 = b_9,
     \]
     and so on. This implies \( a_{3k} = b_{3k} \) for all positive integers \( k \).

3. **Final Conditions:**
   - From \( a_{n-1} + a_n = b_{n-1} + b_n \) and \( a_{n-2} + a_{n-1} + a_n = b_{n-2} + b_{n-1} + b_n \), we get:
     \[
     a_{n-2} = b_{n-2}.
     \]

4. **Characterizing Junk Numbers:**
   - We claim that the junk positive integers are exactly those which leave a remainder of 2 when divided by 3.
   - Suppose \( n \) does not leave a remainder of 2 when divided by 3. Then either \( n-1 \) or \( n \) is a multiple of 3, hence either \( a_{n-1} = b_{n-1} \) or \( a_n = b_n \). However, we also have \( a_{n-2} = b_{n-2} \) and \( a_{n-2} + a_{n-1} + a_n = b_{n-2} + b_{n-1} + b_n \). Thus, it follows that \( a_n = b_n \) and \( a_{n-1} = b_{n-1} \). Since \( a_{i-1} + a_i + a_{i+1} = b_{i-1} + b_i + b_{i+1} \), it's easy to see that we will have \( a_i = b_i \) for all \( i \), which contradicts the binary string's distinctness.

5. **Example for \( n \equiv 2 \pmod{3} \):**
   - We can take:
     \[
     a_1a_2\cdots a_n = 01 \underbrace{001 \, 001 \cdots 001}_{\frac{n-2}{3} \, 001s},
     \]
     \[
     b_1b_2\cdots b_n = 10 \underbrace{010 \, 010 \cdots 010}_{\frac{n-2}{3} \, 010s}.
     \]

6. **Counting Junk Numbers:**
   - The number of \( 2 \pmod{3} \) positive integers less than or equal to 2016 is:
     \[
     \left\lfloor \frac{2016}{3} \right\rfloor = 672.
     \]

The final answer is \(\boxed{672}\).