Consider the function. \(f(x)=\left\{\begin{array}{cc}\frac{ a \left(7 x-12-x^2\right)}{ b \left|x^2-7 x+12\right|}, &am…

Consider the function.
\(f(x)=\left\{\begin{array}{cc}\frac{ a \left(7 x-12-x^2\right)}{ b \left|x^2-7 x+12\right|}, & x<3 \\ 2^{\frac{\sin (x-3)}{x-[x]}} & , x>3 \\ b & , x=3,\end{array}\right.\)
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:

[JEE Main 2024, 27 Jan (Shift 1)]