The Functional Equation that Trolled the NIMO Participants

Algebra Level 5

Let $f$ be a non-constant polynomial such that $f(x-1) + f(x) + f(x+1) = \frac {f(x)^2}{2013x}$ for all nonzero real numbers $x$ . Let $N$ the sum of all possible values of $f(1)$ . Find the remainder when $N$ is divided by $1000$ .