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bitmasks

combinatorics

data structures

divide and conquer

dp

math

*3000

No tag edit access

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H. Keep XOR Low

time limit per test

1 secondmemory limit per test

256 megabytesinput

standard inputoutput

standard outputYou 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 $$$998\,244\,353$$$.

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 $$$998\,244\,353$$$.

Examples

Input

4 2 0 1 2 3

Output

8

Input

3 6 4 2 2

Output

7

Input

4 0 1 1 2 2

Output

6

Codeforces (c) Copyright 2010-2023 Mike Mirzayanov

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