Let $g_i : [\pi/8, 3\pi/8] \to \mathbb{R}$, $i = 1, 2$, and $f : [\pi/8, 3\pi/8] \to \mathbb{R}$ be functions such that
$$g_1(x) = 1, \quad g_2(x) = |4x - \pi|, \quad f(x) = \sin^2 x,$$
for all $x \in [\pi/8, 3\pi/8]$. Define
$$S_i = \int_{\pi/8}^{3\pi/8} f(x) \cdot g_i(x) \, dx, \quad i = 1, 2.$$
The value of $\frac{48 S_2}{\pi^2}$ is ____.