Sum and XOR

Let $f(A, B)$ denote the number of triples of integers $(X, Y, Z)$ satisfying the following conditions: - $0 \le X, Y, Z$ - $X + Y + Z = A$ - $X \oplus Y \oplus Z = B$ It can be shown that $f(A, B)$ is always finite. Here $\oplus$ represents the Bitwise XOR operator. You are given $2$ arrays $A$ and $B$ of length $N$ and $M$. Your task is to compute the value: $$ \displaystyle \prod_{i = 1}^{