Ignore Submasks

You are given an array of $n$ integers, $a_1, a_2, \ldots, a_n$. Each integer is between 0 and $2^k - 1$, inclusive.

Let $f(x)$ be the smallest $i$ such that $(a_i \& x) \neq a_i$, or 0 if there is no such $i$. Here $(a \& b)$ is the bitwise AND operation.

Find $f(0) + f(1) + \ldots + f(2^k - 1)$. This value may be very large, so find it modulo $998\,244\,353$.

The first line contains two integers $n$, $k$ ($1 \le n \le 100$, $1 \le k \le 60$).

The next line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($0 \le a_i < 2^k$).

Print $f(0) + f(1) + \ldots + f(2^k - 1)$ modulo $998\,244\,353$.

In the first example, $f(0) = 2$ and $f(1) = 0$.

In the second example, $f(0) = 1$, $f(1) = 1$, $f(2) = 2$, and $f(3) = 0$.