Limit, Continuity and Differentiability: JEE Main 2026 Mathematics question

Let $[t]$ denote the greatest integer less than or equal to $t$. If the function $f(x)=\begin{cases} b^2\sin\left(\frac{\pi}{2}\left[\frac{\pi}{2}(\cos x+\sin x)\cos x\right]\right), & x<0\\ \frac{\sin x-\frac12\sin 2x}{x^3}, & x>0\\ a, & x=0\end{cases}$ is continuous at $x=0$, then $a^2+b^2$ is equal to