Score : $600$ points

Problem Statement

On a number line are person $1$, person $2$, and person $3$. At time $0$, person $1$ is at point $A$, person $2$ is at point $B$, and person $3$ is at point $C$. Here, $A$, $B$, and $C$ are all integers, and $A \equiv B \equiv C \pmod{2}$.

At time $0$, the three people start random walks. Specifically, a person that is at point $x$ at time $t$ ($t$ is a non-negative integer) moves to point $(x-1)$ or point $(x+1)$ at time $(t+1)$ with equal probability. (All choices of moves are random and independent.)

Find the probability, modulo $998244353$, that it is at time $T$ that the three people are at the same point for the first time.

Constraints

• $0 \leq A, B, C, T \leq 10^5$
• $A \equiv B \equiv C \pmod{2}$
• $A, B, C$, and $T$ are integers.

Input

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

$A$ $B$ $C$ $T$

Output

Find the probability, modulo $998244353$, that it is at time $T$ that the three people are at the same point for the first time, and print the answer.

1 1 3 1

873463809

The three people are at the same point for the first time at time $1$ with the probability $\frac{1}{8}$. Since $873463809 \times 8 \equiv 1 \pmod{998244353}$, $873463809$ should be printed.

0 0 0 0

1

The three people may already be at the same point at time $0$.

0 2 8 9

744570476

47717 21993 74147 76720

844927176