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Algebra Level 4

Positive numbers \(x, y\), and \(z\) satisfy \(xyz = 10^{81}\) and \((\log_{10}x)(\log_{10} yz) + (\log_{10}y) (\log_{10}z) = 468\) .

Find

\[\sqrt {(\log_{10}x)^2 + (\log_{10}y)^2 + (\log_{10}z)^2}\]

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