Tasty Logarithm #2

Geometry Level 3

\[\begin{eqnarray}\log_{10} (\sin x )+ \log_{10} (\cos x) &=& -1 \\ \log_{10} (\sin x + \cos x) &=& \dfrac{1}{2} (-1 + \log_{10} n)\end{eqnarray}\]

Given that \(x\) and \(n\) satisfy the system of equations above, find the value of \(n\).

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