Brutal-Xors

You are given an integer $N$. Find the number of distinct XORs it is possible to make using two positive integers no larger than $N$. Formally, let $S$ be the **set** $$S = \{x\oplus y \mid 1 \leq x, y \leq N\}$$ where $\oplus$ denotes the [bitwise XOR](https://en.wikipedia.org/wiki/Bitwise_operation#XOR) operation. Find $|S|$ (where $|S|$ denotes the size of set $S$. Note that a set, by defin