Score : $1600$ points

Problem Statement

Snuke found a random number generator. It generates an integer between $0$ and $2^N-1$ (inclusive). An integer sequence $A_0, A_1, \cdots, A_{2^N-1}$ represents the probability that each of these integers is generated. The integer $i$ ($0 \leq i \leq 2^N-1$) is generated with probability $A_i / S$, where $S = \sum_{i=0}^{2^N-1} A_i$. The process of generating an integer is done independently each time the generator is executed.

Snuke has an integer $X$, which is now $0$. He can perform the following operation any number of times:

• Generate an integer $v$ with the generator and replace $X$ with $X \oplus v$, where $\oplus$ denotes the bitwise XOR.

For each integer $i$ ($0 \leq i \leq 2^N-1$), find the expected number of operations until $X$ becomes $i$, and print it modulo $998244353$. More formally, represent the expected number of operations as an irreducible fraction $P/Q$. Then, there exists a unique integer $R$ such that $R \times Q \equiv P \mod 998244353,\ 0 \leq R < 998244353$, so print this $R$.

We can prove that, for every $i$, the expected number of operations until $X$ becomes $i$ is a finite rational number, and its integer representation modulo $998244353$ can be defined.

Constraints

• $1 \leq N \leq 18$
• $1 \leq A_i \leq 1000$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$A_0$ $A_1$ $\cdots$ $A_{2^N-1}$

Output

Print $2^N$ lines. The $(i+1)$-th line ($0 \leq i \leq 2^N-1$) should contain the expected number of operations until $X$ becomes $i$, modulo $998244353$.

2
1 1 1 1

0
4
4
4

$X=0$ after zero operations, so the expected number of operations until $X$ becomes $0$ is $0$.

Also, from any state, the value of $X$ after one operation is $0$, $1$, $2$ or $3$ with equal probability. Thus, the expected numbers of operations until $X$ becomes $1$, $2$ and $3$ are all $4$.

2
1 2 1 2

0
499122180
4
499122180

The expected numbers of operations until $X$ becomes $0$, $1$, $2$ and $3$ are $0$, $7/2$, $4$ and $7/2$, respectively.

4
337 780 799 10 796 875 331 223 941 67 148 483 390 565 116 355

0
468683018
635850749
96019779
657074071
24757563
745107950
665159588
551278361
143136064
557841197
185790407
988018173
247117461
129098626
789682908