Simply Simplify!!

Algebra Level 2

If x=422+1x=\frac{4\sqrt{2}}{\sqrt{2}+1}

Then the value of

12(x+2x2+x+22x22)\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 k\sqrt{k} where kk is a positive integer.Find kk

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