polyhedra – Size of integer variables around mean in an integer program

Consider the linear integer program

$$underline xinmathbb Z^n$$
$$Aunderline xleq b$$

where number of rows in $A$ is $m$ and number of bits to represent $A$ and $b$ is $L$.

  1. Is there a bound on $|underline x|_infty$?

Denote $x_{mean}$ to be $frac{sum_{i=1}^nunderline x_i}n$.

  1. Is there a bound in $|underline x-x_{mean}|_infty$?