April's Lagrange Interpolation

Let $P(x)$ be a monic polynomial of degree $2015$ , satisfying, for $x\in \mathbb{Z}^+, x\le 2015$ , $P(x)=\sum_{k=1}^x \dfrac{Q(k)^3+1}{k^3+1}$ where $Q(x)=\left\{ \begin{array}{l}-\sum\limits_{i=1}^{\infty}\dfrac{1}{i^{|x|}}\qquad \text{if the sum converges}\\ -\dfrac{1}{|x|}\qquad \text{if the sum diverges}\end{array}\right.$ Now, let $a_n$ be the coefficient of the $x^n$ term in $P(x)$ . Finally, let $S=\sum_{i=1}^{2015}a_{i-1}$ Find the positive remainder when $\lfloor S\rfloor^{2015-\lceil S\rceil}$ is divided by $1000$ .