HMMT 2012 U35

Let $N$ be the number of distinct roots of $\prod_{k=1}^{2012}\left(x^{k}-1\right)$. Give lower and upper bounds $L$ and $U$ on $N$. If $0<L \leq N \leq U$, then your score will be $\left\lfloor\frac{23}{(U / L)^{1.7}}\right\rfloor$. Otherwise, your score will be 0 .