If \( A = \{ x \in \mathbb{R} : |x - 2| > 1 \} \), \( B = \{ x \in \mathbb{R} : \sqrt{x^2 - 3} > 1 \} \), \( C = \{ x \in \mathbb{R} : |x - 4| \ge 2 \} \) 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 \( \_\_\_\_\_\_\_\_\_\_ \).
