Unifying Divisibility

Let $m$ be the number of integers $n$ with $1 \leq n \leq 2013$ such that the polynomial $x^{2n} + 1 + (x+1)^{2n}$ is divisible by $x^2 + x + 1$ . Find $m \pmod {1000}$ .

**Problem credit goes to Artofproblemsolving foundation.