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.

What is rational number modulo 998244353?

We can prove that the sought probability is always a rational number.
Moreover, under the Constraints of this problem, when the value is represented as $\frac{P}{Q}$ by two coprime integers P and Q, we can prove that there is a unique integer R such that $R \times Q \equiv P\pmod{998244353}$ and $0 \leq R \lt 998244353.$ Find such R.

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.

Constraints

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

Sample Input 1

1 1 3 1

Sample Output 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.

Sample Input 2

0 0 0 0

Sample Output 2

1

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

Sample Input 3

0 2 8 9

Sample Output 3

744570476

Sample Input 4

47717 21993 74147 76720

Sample Output 4

844927176

见题面 Input 部分

见题面 Output 部分