Consider the following two sets: $A=\{x:|x-1|<3\}$;$B=\left\{x:x^2-2ax+a^2-4\le 0\right\}$

$\text{ If }A\cap B=\{x:-2<x\le 1\},\text{ then the number of integers in the range of }a\text{ , is }$

$\begin{array}{l}|x-1|<3\\ \Rightarrow -3<x-1<3\\ \Rightarrow -2<x<4-----\left(1\right)\\ \text{ Also, }{x}^2-2ax+a^2-4\le 0\\ \Rightarrow (x-a{)}^2\le 4\\ \Rightarrow |x-a|\le 2\\ \Rightarrow -2\le x-a\le 2\\ \Rightarrow -2+a\le x\le 2+a-----\left(2\right)\\ \text{ Now, for }A\cap B\text{ to be }(-2,1]\text{ , }\\ a+2=1\\ \Rightarrow a=-1\end{array}$