For any integer $n$, consider the three functions

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

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