\[\large{2^{126}\prod_{k=1}^{126} \sin{\left(\theta +\frac{k\pi}{127}\right)}}\]
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 \(AB.\)

Inspiration.