Expand the expression? No way!

Algebra Level 5

Let \(f(x) \) denote a monic \(2015^{\text{th}} \) degree polynomial with integer coefficients and with roots \(r_1, r_2, \ldots , r_{2015} \)

And \(S_n = (-1)^n n^3 \), where \(S_n \) denote the \(n^{\text{th}} \) symmetric sum of numbers \(r_1, r_2, \ldots , r_{2015} \).

What is the last three digits of

\[ \left| \prod_{k=1}^{2015} (r_k + 1) \right| ? \]

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