A12422. Triple

You have received your birthday gifts — $n$ triples of integers! The $i$ -th of them is $\lbrace a_{i}, b_{i}, c_{i} \rbrace$ . All numbers are greater than or equal to $0$ , and strictly smaller than $2^{k}$ , where $k$ is a fixed integer.

One day, you felt tired playing with triples. So you came up with three new integers $x$ , $y$ , $z$ , and then formed $n$ arrays. The $i$ -th array consists of $a_i$ repeated $x$ times, $b_i$ repeated $y$ times and $c_i$ repeated $z$ times. Thus, each array has length $(x + y + z)$ .

You want to choose exactly one integer from each array such that the XOR (bitwise exclusive or) of them is equal to $t$ . Output the number of ways to choose the numbers for each $t$ between $0$ and $2^{k} - 1$ , inclusive, modulo $998244353$ .

The first line contains two integers $n$ and $k$ ( $1 \leq n \leq 10^{5}$ , $1 \leq k \leq 17$ ) — the number of arrays and the binary length of all numbers.

The second line contains three integers $x$ , $y$ , $z$ ( $0 \leq x,y,z \leq 10^{9}$ ) — the integers you chose.

Then $n$ lines follow. The $i$ -th of them contains three integers $a_{i}$ , $b_{i}$ and $c_{i}$ ( $0 \leq a_{i} , b_{i} , c_{i} \leq 2^{k} - 1$ ) — the integers forming the $i$ -th array.

Print a single line containing $2^{k}$ integers. The $i$ -th of them should be the number of ways to choose exactly one integer from each array so that their XOR is equal to $t = i-1$ modulo $998244353$ .

In the first example, the array we formed is $(1, 0, 0, 1, 1, 1)$ , we have two choices to get $0$ as the XOR and four choices to get $1$ .

In the second example, two arrays are $(0, 1, 1, 2)$ and $(1, 2, 2, 3)$ . There are sixteen $(4 \cdot 4)$ choices in total, $4$ of them ( $1 \oplus 1$ and $2 \oplus 2$ , two options for each) give $0$ , $2$ of them ( $0 \oplus 1$ and $2 \oplus 3$ ) give $1$ , $4$ of them ( $0 \oplus 2$ and $1 \oplus 3$ , two options for each) give $2$ , and finally $6$ of them ( $0 \oplus 3$ , $2 \oplus 1$ and four options for $1 \oplus 2$ ) give $3$ .