A Sudden Dip

Algebra Level 3

Let \(f(x)\) be a polynomial function of minimal degree such that \[ \begin{align*} f(1) &= 1 \\ f(2) &= 2 \\ f(3) &= 3 \\ &\vdots \\ f(2016) &= 2016 \\ f(2017) &= 1. \end{align*} \] Find the value of \(\big|f(2018)\big| \! \pmod{1000}.\)

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