Problem Statement

Happy New Year! When it comes to indoor play on New Year's, it's sugoroku!

There is a sugoroku board consisting of $N+1$ squares: squares $0$, square $1$, $\dots$, square $N$. There is an $M$-sided die (dice) that produces positive integers $A_1, A_2, \dots, A_M$ with equal probability. Here, $A_1, A_2, \dots, A_M$ are distinct. Person $1$, person $2$, $\dots$, person $L$ will play a game using this board. The game proceeds as follows:

• Initially, place $L$ pieces, piece $1$, piece $2$, $\dots$, piece $L$, on square $0$.
• Turns come around in numerical order starting with person $1$. Strictly speaking, after person $i$'s turn, the turn goes to person $(i \bmod L) + 1$. Each person performs the following operation on their turn:
• Let $i$ be the number of the person whose turn it is. Roll the die. Let $x$ be the square where piece $i$ is located and $y$ be the integer that came up on the die, and move piece $i$ to square $\min(N, x+y)$.
• The first person to place their numbered piece on square $N$ wins, and the other people lose.

For $i =1,2,\dots,L$, find the probability, modulo $998244353$, that person $i$ wins.

What is probability $\text{mod }998244353$? It can be proved that the probability to be found is always a rational number. Also, under the constraints of this problem, when the value is expressed as $\frac{P}{Q}$ using two coprime integers $P$ and $Q$, it can be proved that there exists exactly one integer $R$ that satisfies $R \times Q \equiv P\pmod{998244353}$ and $0 \leq R \lt 998244353$. Find this $R$.

Constraints

• $1 \leq N \leq 2.5 \times 10^5$
• $1 \leq M \leq N$
• $2 \leq L \leq 2.5 \times 10^5$
• $1 \leq A_1 \lt A_2 \lt \dots \lt A_M \leq N$
• All input values are integers.

Input

The input is given from Standard Input in the following format:

N M L
A_1 A_2 \dots A_M

Output

Output $L$ lines. The $i$-th line should contain the probability that person $i$ wins the game, computed modulo $998244353$.