Product of 126 sines

Geometry Level 5

$\large{2^{126}\prod_{k=1}^{126} \sin{\left(\theta +\frac{k\pi}{127}\right)}}$ The previous product of sines can be expressed in the form $\frac{\sin{(A \theta)}}{\sin {(B\theta)}},$where $$A,$$ and $$B$$ are positive integers. Find the product $$AB.$$

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