Counting Optimal Arrays

For an array $A$ of size $N$, define $f(A)$ as $\sum_{i = 1}^{N - 1} (A_i \oplus A_{i + 1})$, i.e. $(A_1 \oplus A_2 + A_2 \oplus A_3 + \ldots + A_{N - 1} \oplus A_N)$. Here $\oplus$ represents the Bitwise XOR operator. $f(A)$ is defined as $0$ when $N = 1$. Given integers $N$ and $M$, consider the arrays $A$ satisfying $|A| = N$ and $0 \le A_i \le M$. Let $S$ denote the maximum possible value of