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Hint to puzzle 134: Sum of reciprocal roots

Firstly, note that

a and b are roots of x4 − x3 + x + 1 = 0
if and only if
1/a and 1/b are roots of x4 + x3 − x + 1 = 0. (1)

Now let a' = 1/a, b' = 1/b, c' = 1/c, and d' = 1/d be the roots of equation (1).  Using Viète's formulas, write down the values of

Then express the above relations in terms of s = a' + b', t = c' + d', p = a'b', and q = c'd'.