Hint to puzzle 134: Sum of reciprocal roots

Firstly, note that

a and b are roots of x⁴ − x³ + x + 1 = 0

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

• a' + b' + c' + d',
• a'b' + a'c' + a'd' + b'c' + b'd' + c'd',
• a'b'c' + a'b'd' + a'c'd' + b'c'd',
• a'b'c'd'.

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