By considering the binomial expansion of \(\left(z-\frac{1}{z}\right)^{6}\), where \(z=\cos \theta+\mathrm{i} \sin \theta\), express \(\sin ^{6} \theta\) in the form
\(\frac{1}{32}(p+q \cos 2 \theta+r \cos 4 \theta+s \cos 6 \theta),\)
where \(p, q, r\) and \(s\) are integers to be determined.

Hence find the exact value of \(\int_{0}^{\frac{1}{4} \pi} \sin ^{6} \theta \mathrm{~d} \theta\).