A Sinful Sinfonia!

Calculus Level 5

\[\displaystyle \int_0^{\pi/3} {\sum_{n=1}^{\infty}{\left[\sin(x) \sin\left(x + \frac{\pi}{n}\right) \sin\left(x + \frac{2\pi}{n}\right) \cdots \sin\left(x + \frac{(n-1)\pi}{n}\right)\right]}\, dx} = \ln a\]

Find \(a\).