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H. Triple
time limit per test
1.5 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

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 $$$998244353$$$.

Input

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.

Output

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 $$$998244353$$$.

Examples
Input
1 1
1 2 3
1 0 1
Output
2 4 
Input
2 2
1 2 1
0 1 2
1 2 3
Output
4 2 4 6 
Input
4 3
1 2 3
1 3 7
0 2 5
1 0 6
3 3 2
Output
198 198 126 126 126 126 198 198 
Note

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 $$$3$$$.