If $\mathrm{A}=\{\mathrm{x}\in \mathrm{R}:|\mathrm{x}-2|>1\}, \mathrm{B}=\left\{\mathrm{x}\in \mathrm{R}:\sqrt{{\mathrm{x}}^2-3}>1\right\}, \mathrm{C}=\{\mathrm{x}\in \mathrm{R}:|\mathrm{x}-4|\ge 2\}$ and z is the set of all integers, then the number of subsets of the set (A$\cap$ B$\cap$C)^C $\cap$Z is ____________

$\begin{array}{l}|\mathrm{x}-2|>1\Rightarrow \mathrm{x}-2<-1\text{ or }\mathrm{x}-2>1\Rightarrow \mathrm{x}<1\text{ or }\mathrm{x}>3\\ \mathrm{A}\in (-\propto ,1)\cup (3,\propto )\\ \sqrt{{\mathrm{x}}^2-3}>1\Rightarrow {\mathrm{x}}^2-3>1\Rightarrow {\mathrm{x}}^2-4>0\Rightarrow (\mathrm{x}+2)(\mathrm{x}-2)>0\Rightarrow \mathrm{x}<-2\text{ or }\mathrm{x}>2\\ \mathrm{B}\in (-\propto ,-2)\cup (2,\propto )\\ |\mathrm{x}-4|\ge 2\Rightarrow \mathrm{x}-4\le -2\text{ or }\mathrm{x}-4\ge 2\\ \mathrm{x}\le 2\text{ or }\mathrm{x}\ge 6\Rightarrow \mathrm{c}\in (-\propto ,2]\cup [6,\propto )\\ \mathrm{A}\cap \mathrm{B}\cap \mathrm{C}=(-\propto ,-2)\cup [6,\propto )\Rightarrow (\mathrm{A}\cap \mathrm{B}\cap \mathrm{C}{)}^1=[-2,6)\\ \mathrm{z}\cap (\mathrm{A}\cap \mathrm{B}\cap \mathrm{C}{)}^1=\{-2,-1,0,1,2,3,4,5\}\end{array}$
$\therefore$Number of subsets 2⁸ = 256