Triple

SK 酱送给你了一份生日礼物。礼物是 $n$ 个三元组 $(a_i,b_i,c_i)$ 和四个正整数 $x,y,z,k$。 你利用这 $n$ 个三元组填充了 $n$ 个数组，其中第 $i$ 个数组中有 $x$ 个 $a_i$，$y$ 个 $b_i$，$z$ 个 $c_i$（所以第 $i$ 个数组长度为 $(x+y+z)$。 对于 $i=0,1,\cdots,2^k-1$，回答以下询问： - 从每个数组中选择**恰好一个**数，使得这些数的 $\mathrm{xor}$ 和为 $i$，方案数是多少？ 你只需要输出方案数对 $998,244,353$ 取模后得到的结果。 对于 $100\%$ 的数据，保证： - $1\le n\le 10^5$，$1\le k\le 17$； - $0\le x,y,z\le 10^9$； - $0\le a_i,b_i,c_i\lt 2^k$。 $\text{Statement fixed by Starrykiller.}$

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 $ .

1 1
1 2 3
1 0 1

2 4 

2 2
1 2 1
0 1 2
1 2 3

4 2 4 6 

4 3
1 2 3
1 3 7
0 2 5
1 0 6
3 3 2

198 198 126 126 126 126 198 198 

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 $ .