Evaluate the following integral.\\
$\int_{0}^{1} \int_{0}^{\pi} \int_{\pi/4}^{\pi/2} 24p \cos^{3} \phi \,d\phi \,d\theta \,dp$ \\
$\int_{0}^{1} \int_{0}^{\pi} \int_{\pi/4}^{\pi/2} 24p \cos^{3} \phi \,d\phi \,d\theta \,dp = \boxed{}$ \\
(Type an exact answer, using $\pi$ as needed.)