A14681 | Keep XOR Low

You are given an array $a_1, a_2, \ldots, a_n$ and an integer $x$ .

Find the number of non-empty subsets of indices of this array $1 \leq b_1 < b_2 < \ldots < b_k \leq n$ , such that for all pairs $(i, j)$ where $1 \leq i < j \leq k$ , the inequality $a_{b_i} \oplus a_{b_j} \leq x$ is held. Here, $\oplus$ denotes the [bitwise XOR operation](https://en.wikipedia.org/wiki/Bitwise_operation#XOR). As the answer may be very large, output it modulo $998\,244\,353$ .

The first line of the input contains two integers $n$ and $x$ ( $1 \leq n \leq 150\,000$ , $0 \leq x < 2^{30}$ ). Here, $n$ is the size of the array.

The next line contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $0 \leq a_i < 2^{30}$ ): the array itself.

Print one integer: the number of non-empty subsets such that the bitwise XOR of every pair of elements is at most $x$ , modulo $998\,244\,353$ .