GAIC Math 2024 Problem 83

Given set: $A=\left\{x+y \left\lvert \frac{x^{2}}{9}+y^{2}=1\right., x+y \in \mathbf{Z}_{+}\right\}, B=\{2 x+y \mid x, y \in A, x<y\}$, $C=\{2 x+y \mid x, y \in A, x>y\}$. Then the sum of all elements in $B \cap C$ is