Intersection points, part 2

Algebra Level 5

\[f(x)=-\frac{1}{10}\left(x^3+3x^2+4x-18\right)\]

Given that the graphs of \(f\) and its inverse \(f^{-1}\) intersect at 5 points, denoted as \((a_i,b_i)\) for \(i=1,2,3,4,5\).

Let \(\displaystyle S=\sum_{i=1}^{5} |a_i|\). Find \(\displaystyle \left \lfloor 1000S\right \rfloor\).

\[\] Notation: \( \lfloor \cdot \rfloor \) denotes the floor function.

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