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Algebra Level 5

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:

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