This was easier than I though. I'll be using the Newton-Girard Formulas like I did in
A PowerSum Puzzle.
Then from the given quantities S(1) = P(1) = 0, P(4) = 1, and S(3) = 1983.
Substitute into S(3) - S(2)*P(1) + S(1)*P(2) - 3*P(3) = 0 to get 1983 - S(2)*0 + 0*P(2) - 3*P(3), which simplifies to P(3) = -661.
1/a + 1/b + 1/c + 1/d can be expressed as P(3)/P(4), which by direct substitution equals -661.
Solution