Another amazing log

Let $S$ be the set of ordered triples $(x,y,z)$ of real numbers for which $\log_{10} (x+y) = z$ and $\log_{10}(x^2+y^2) = z + 1$ . If $a$ and $b$ are real numbers such that for all ordered triples $(x,y,z)$ in $S$ , we have $x^3 + y^3 = a\cdot 10^{3z} + b\cdot10^{2z}$ . Then the value of $(a+b)$ is:

This is part of the set My Problems and THRILLER