Let \( f : \mathbb{R} \to \mathbb{R} \) be a function such that \( f(x) + 3 f\!\left(\frac{\pi}{2} - x\right) = \sin x, \; x \in \mathbb{R} \). Let the maximum value of \( f \) on \( \mathbb{R} \) be \( \alpha \). If the area of the region bounded by the curves \( g(x) = x^2 \) and \( h(x) = \beta x^3, \; \beta > 0 \), is \( \alpha^2 \), then \( 30\beta^3 \) is equal to ________.
