SMO 2012 Open Round 1 Question 10

Let $\alpha_n$ be a real root of $nx^3+2x-n=0$, where $n$ is a positive integer. If $\beta_n=\lfloor(n+1)\alpha_n\rfloor$ for $n=2,3,4,\ldots$, find $\displaystyle\frac1{1006}\sum_{n=2}^{2013}\beta_n$.

Here $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$, as defined in the source paper.