Keep XOR Low
Description
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. 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 $998\,244\,353$.
Input
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.
Output
Print one integer: the number of non-empty subsets such that the bitwise XOR of every pair of elements is at most $x$, modulo $998\,244\,353$.
Samples
4 2
0 1 2 3
8
3 6
4 2 2
7
4 0
1 1 2 2
6