Simply Simplify!!

Algebra Level 2

If \(x=\frac{4\sqrt{2}}{\sqrt{2}+1}\)

Then the value of

\(\frac{1}{\sqrt{2}}(\frac{x+2}{x-2}+\frac{x+2\sqrt{2}}{x-2\sqrt{2}})\)

can be simplified to a number of the form \(\sqrt{k}\) where \(k\) is a positive integer.Find \(k\)

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