I love being inspired!

Algebra Level 4

\[ \Large \sqrt{2+ \frac{2}{ \sqrt{2- \frac{2}{ \sqrt{2+ \frac{2}{ \sqrt{2- \frac{2}{ \sqrt{ \ldots }}}}}}}}}\]

If the value of the infinitely nested expression above equals \(x\), find the value of \(1000x\).