\[\left(1- \tan^2 \left(\frac x{2^{2011}} \right)\right)\left(1- \tan^2 \left(\frac x{2^{2010}} \right)\right) \cdots \left(1- \tan^2 \left(\frac x2 \right)\right) = 2^{2011} \sqrt3 \tan \left( \frac x{2^{2011}} \right) \]

What is the value of \(\sin(2x) \) if \(x\) satisfy the equation above?

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