Score : $1600$ points
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:
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.
Input is given from Standard Input in the following format:
$N$
$A_0$ $A_1$ $\cdots$ $A_{2^N-1}$
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