Let $\mathbb{N}$ be the set of all integers, $$ \begin{aligned} & A=\left\{(x, y) \in \mathbb{Z} \times \mathbb{Z}:(x-2)^2+y^2 \leq 4\right\} \\ & B=\left\{(x, y) \in \mathbb{Z} \times \mathbb{Z}: x^2+y^2 \leq 4\right\} \text { and } \\ & C=\left\{(x, y) \in \mathbb{Z} \times \mathbb{Z}:(x-2)^2+(y-2)^2 \leq 4\right\} \end{aligned} $$ If the total number of relations from $A \cap B$ to $A \cap C$ is $2^p$, then the value of $p$ is.