Ada Function

Ada has a set of $N$ non-negative integers $S = \{x_1, x_2, \ldots, x_N\}$. She defined the function $W$ as follows: - $W(\{\}) = 0$. - If $S = \{a\}$, then $W(S) = a$. - If $S = \{a, b\}$ (so $|S| = 2$), then $W(S) = (a \oplus b) - 1$. - If $|S| \ge 3$, then to calculate $W(S)$, find any two elements of $S$ that are congruent (have the same remainders) modulo the highest possible power of $2$