Sines and pies

$\sin\dfrac{\pi}{14} \sin\dfrac{3{\pi}}{14} \sin\dfrac{5{\pi}}{14} \sin\dfrac{7{\pi}}{14} \sin\dfrac{9{\pi}}{14} \sin\dfrac{11{\pi}}{14} \sin\dfrac{13{\pi}}{14} \quad \quad \small \color{#3D99F6}{\text{As } \sin x = \cos \left(\dfrac{\pi}{2}-x\right) \text{ and } \cos (-x) = \cos x} \\ = \cos \dfrac{6\pi}{14}\cos \dfrac{4\pi}{14}\cos \dfrac{2\pi}{14}\cos 0 \cos \dfrac{2\pi}{14}\cos \dfrac{4\pi}{14}\cos \dfrac{6\pi}{14} \\ = \left(\cos \dfrac{\pi}{7}\cos \dfrac{2\pi}{7}\color{#D61F06}{\cos \dfrac{3\pi}{7}} \right)^2 \quad \quad \small \color{#D61F06}{\text{As } \cos x = - \cos (\pi - x)} \\ = \left(\color{#D61F06}{-}\cos \dfrac{\pi}{7}\cos \dfrac{2\pi}{7} \color{#D61F06}{\cos \dfrac{4\pi}{7}} \right)^2 \\ = \left(- \dfrac{\sin \frac{\pi}{7} \cos \frac{\pi}{7} \cos \frac{2\pi}{7} \cos \frac{4\pi}{7}}{\sin \frac{\pi}{7}} \right)^2 \\ = \left(- \dfrac{\sin \frac{2\pi}{7} \cos \frac{2\pi}{7} \cos \frac{4\pi}{7}}{2 \sin \frac{\pi}{7}} \right)^2 \\ = \left(- \dfrac{\sin \frac{4\pi}{7} \cos \frac{4\pi}{7}}{4 \sin \frac{\pi}{7}} \right)^2 \\ = \left(- \dfrac{\color{#D61F06}{\sin \frac{8\pi}{7}}}{8 \sin \frac{\pi}{7}} \right)^2 \quad \quad \small \color{#D61F06}{\text{As } \cos x = - \cos (\pi - x)} \\ = \left(\dfrac{\color{#D61F06}{\sin \frac{\pi}{7}}}{8 \sin \frac{\pi}{7}} \right)^2 \\ = \left(\dfrac{1}{8} \right)^2 \\ = \frac{1}{2^6}$

$\implies n = \boxed{-6}$