Sum of Function Values

Problem Description

You are given $n$ integers $a_1, a_2, \dots, a_n$ and a positive integer $k$ with $0 \le a_i \le 2^k - 1$.

Define the function $f(x)$ to be the smallest $i$ satisfying $a_i \mathbin{\&} x \neq a_i$; when no such $i$ exists, $f(x) = 0$.

Compute $\sum_{i=0}^{2^k-1} f(i)$. Since the answer may be large, output it modulo $998244353$.

Input Format

The first line contains two integers $n$ and $k$.

The next line contains $n$ integers $a_1, a_2, \dots, a_n$, the given numbers.

Output Format

One integer on a single line: the answer.

Sample Input 1

2 1
0 1

Sample Output 1

2

Sample Input 2

2 2
2 1

Sample Output 2

4

Sample Input 3

5 10
389 144 883 761 556

Sample Output 3

1118

Sample Explanation

For sample 1, $f(0) = 2$, $f(1) = 0$, so the answer is $2$.

Constraints

For all data, $1 \le n \le 100$ and $1 \le k \le 60$.