Unifying Divisibility

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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.

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