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You are given two non negative integers $N$ and $M$ $(N<M)$. A beautiful sequence is defined as a sequence $S = \{X_1, X_2, \dots, X_K\}$ of distinct non-negative integers, each at most $M$, that satisfies the following properties: - For all $1 \leq i \leq K$, we have $0 \leq X_i \leq M$. - For all $1 \leq i < j \leq K$, $X_i \neq X_j$. - For all $1 \leq i < j \leq K$, $X_i \& X_j = N$, where $\&