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 $x$. Find the sum of all possible values of $f(1)$.

Let's think step by step and output the final answer within \boxed{}.
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