smc

Let $S$ be the set of ordered triples $(x,y,z)$ of real numbers for which $${\log }_{10}(x+y)=z\text{ and }{\log }_{10}(x^2+y^2)=z+1.$$ There are real numbers $a$ and $b$ such that for all ordered triples $(x,y,z)$ in $S$ we have $x^3+y^3=a\cdot 10^{3z}+b\cdot 10^{2z}.$ What is the value of $a+b?$