Eccoci.

For any integer $n$, consider the three functions

\begin{align} f_{1, n}(x, y, z) &= x^{n + 1} + y^{n + 1} - z^{n + 1}\\ f_{2, n}(x, y, z) &= (xy + yz + zx) \cdot (x^{n - 1} + y^{n - 1} - z^{n - 1})\\ f_{3, n}(x, y, z) &= xyz \cdot (x^{n - 2} + y^{n - 2} - z^{n - 2}) \end{align}

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 $0 \lt a \lt b \le k$ 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 $u + v$.