Rational zeros of a function of three variables

For any integer n, consider the three functions

f_1,n(x,y,z) = xⁿ⁺¹ + yⁿ⁺¹ − zⁿ⁺¹f_2,n(x,y,z) = (xy + yz + zx)*(x^n-1 + y^n-1 − z^n-1)f_3,n(x,y,z) = xyz*(x^n-2 + y^n-2 − z^n-2)

and their combination

f_n(x,y,z) = f_1,n(x,y,z) + f_2,n(x,y,z) − f_3,n(x,y,z)

We call (x,y,z) a golden triple of order k if x, y, and z are all rational numbers of the form a / b with 0 < a < b ≤ k and there is (at least) one integer n, so that f_n(x,y,z) = 0.

Let s(x,y,z) = x + y + z. Let t = u / v be the sum of all distinct s(x,y,z) for all golden triples (x,y,z) of order 35. All the s(x,y,z) and t must be in reduced form.

Find u + v.

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