Integral Calculus: JEE Main 2026 Mathematics question

Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that $f(x)+3f\left(\frac{\pi}{2}-x\right)=\sin x$, $x\in\mathbb{R}$. 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 ______.