Let \(S\) be the set of ordered triples \((x,y,z)\) of real numbers for which  

\(\log_{10}(x+y) = z\) and \(\log_{10}(x^{2}+y^{2}) = z+1.\)  
There are real numbers \(a\) and \(b\) such that for all ordered triples \((x,y,z)\) in \(S\) we have \(x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.\) The original answer is in \(\frac{m}{n}\) format, please find the value of \(m+n\).

Let's think step by step and output the final answer within \boxed{}.
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