$S_n=x^n+y^n+z^n$. If $S_1=0$, find $m$ such that $S_pS_q=mS_{p+q}$.

Question

$S_n=x^n+y^n+z^n$ $(x\neq y\neq z\neq 0)$. If $\space S_1=0, \space$ find $\space m \space$ such that $$S_pS_q=mS_{p+q}$$ I thought of this problem while solving a particular case in which $\space p=2 \space$ and $\space q=5$. I got $\space m= \frac{10}{7}. \space$ I tried for $\space p=2 \space$ and $\space q=3 \space$ and got $m= \frac{5}{6}. \space$ My question is how can we get a General expression for $\space m \space$ (in terms of $\space p \space$ and $\space q$)?