Let $\mathrm{g}_{\mathrm{i}}:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathbb{R}, \mathrm{i}=1,2$ and $\mathrm{f}:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathbb{R}$ be functions such that $\mathrm{g}_1(\mathrm{x})=1, \mathrm{~g}_2(\mathrm{x})=|4 \mathrm{x}-\pi|$ and $f(\mathrm{x})= \sin ^2 x$, for all $x \in\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right]$ Define $S_i \int_{\frac{\pi}{8}}^{\frac{3 \pi}{8}} f(x) \cdot g_i(x) d x, i=1,2$ The value of $\frac{48 S_2}{\pi^2}$ is $\_\_\_\_$ .