Cutie pi!

Geometry Level 5

$\large \sin\left( \frac \pi{14} \right) \sin\left( \frac {3\pi}{14} \right) \sin\left( \frac {5\pi}{14} \right) \sin\left( \frac {7\pi}{14} \right) \sin\left( \frac {9\pi}{14} \right) \sin\left( \frac {11\pi}{14} \right) \sin\left( \frac {13\pi}{14} \right)$

If the expression above can be simplified to the form $\frac pq$ for coprime positive integers, find $p+q$.

×