Mathematics Question from JEE Main 2024 · 27 January, Shift 1

Consider the function

$$f(x)=\begin{cases} \dfrac{a\left(7x-12-x^2\right)}{b\left|x^2-7x+12\right|}, & x<3\\[6pt] 2^{\frac{\sin(x-3)}{x-[x]}}, & x>3\\[6pt] b, & x=3 \end{cases}$$

where $[x]$ denotes the greatest integer less than or equal to $x$. If $S$ denotes the set of all ordered pairs $(a,b)$ such that $f(x)$ is continuous at $x=3$, then the number of elements in $S$ is: