Evaluate the product $\prod\limits_{k=1}^{2^{1999}} (4\sin^2(\frac{k\pi}{2^{2000}})-3)$

Question

Evaluate the product $$\prod\limits_{k=1}^{2^{1999}} \left(4\sin^2\left(\frac{k\pi}{2^{2000}}\right)-3\right)$$ I tried by making it into $\frac{\sin 3\theta}{\sin\theta}$ form but couldn’t proceed like that, so I wrote the series reversed and multiplied both in the hope of some symmetry but couldn’t observe anything. Please give some ideas on how to work on it.