Extension: Logarithmic Differentiation

Let $\omega$ be a primitive $2007$th root of unity, i.e. a complex number such that $\omega^{2007} = 1$ but $\omega^k \ne 1$ for $k=1,2,\dots,2006$. It turns out that $\sum_{j=1}^{2006} \frac{1}{2 - \omega^j}$ is a real number (in fact a rational number). Find the largest integer less than or equal to $\sum_{j=1}^{2006} \frac{1}{2 - \omega^j}$.

Source: HMMT February 2007