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Algebra Level 4

\[\large { { \left (\frac { 2\omega +1 }{ \omega } \right ) }^{ 1000 }-{ \left (\frac { { \omega }^{ 2 }-1 }{ { \omega }^{ 2 } } \right ) }^{ 1000 } = \ ? }\]

Details and Assumptions

\( w\) denote the cube root unity, which is the root of \(x^3 = 1 \) other than \(1\).

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