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Algebra Level 4

Positive numbers x,yx, y, and zz satisfy xyz=1081xyz = 10^{81} and (log10x)(log10yz)+(log10y)(log10z)=468(\log_{10}x)(\log_{10} yz) + (\log_{10}y) (\log_{10}z) = 468 .

Find

(log10x)2+(log10y)2+(log10z)2\sqrt {(\log_{10}x)^2 + (\log_{10}y)^2 + (\log_{10}z)^2}

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