A chemical process requires the concentration of a reactant in a vessel to be modeled by the function $C(t) = \frac{at^2 + bt + 12}{t^2 - 4}$, where $t$ represents time in minutes. Given that the concentration levels off to a value of $3$ as $t \to \infty$ and there is a removable discontinuity at $t = 2$, determine the value of the constant $b$.