Advanced Number Theory · Superchallenge problem

Compute the product $\sin\dfrac{\pi}{8}\,\sin\dfrac{2\pi}{8}\,\sin\dfrac{3\pi}{8}\cdots\sin\dfrac{7\pi}{8}$. It equals $\tfrac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$.