Score : $1100$ points

Problem Statement

Given is an integer $N$.

Snuke is going to do the procedure below.

• Let $x$ be a variable and initialize it with a random integer between $0$ and $N$ (inclusive). For each $0 \leq i \leq N$, $x=i$ is chosen with probability $A_i/10^9$.
• Repeat the following operation $K$ times.- With probability $x/N$, decrease the value of $x$ by $1$; with probability $1-x/N$, increase it by $1$. (Here, note that $x$ will still be between $0$ and $N$ after the operation.)
• With probability $x/N$, decrease the value of $x$ by $1$; with probability $1-x/N$, increase it by $1$. (Here, note that $x$ will still be between $0$ and $N$ after the operation.)

For each $i=0,1,\cdots,N$, find the probability, modulo $998244353$, that $x=i$ after the procedure.

Constraints

• $1 \leq N \leq 100000$
• $0 \leq A_i$
• $\sum_{0 \leq i \leq N}A_i = 10^9$
• $1 \leq K \leq 10^9$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $K$

$A_0$ $A_1$ $\cdots$ $A_N$

Output

For each $i=0,1,\cdots,N$, print the probability, modulo $998244353$, that $x=i$ after the procedure.

2 1
0 1000000000 0

499122177 0 499122177

First, $x$ is always initialized with $x=1$. In the subsequent operation, the value of $x$ is decreased by $1$ with probability $1/2$ and increased by $1$ with probability $1/2$. Eventually, we will have $x=0,1,2$ with probabilities $1/2,0,1/2$, respectively.

4 2
200000000 200000000 200000000 200000000 200000000

723727156 598946612 349385524 598946612 723727156

10 100
21265166 263511538 35931763 26849698 108140810 134702248 36774526 147099145 58335759 4118743 163270604

505314898 24510700 872096939 107940764 808162829 831195162 314651262 535843032 665830283 627881537 696038713