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G - Yet Another RGB Sequence

Score : $600$ points

Problem Statement

You are given integers $R$, $G$, $B$, and $K$. How many strings $S$ consisting of R, G, and B satisfy all of the conditions below? Find the count modulo $998244353$.

• The number of occurrences of R, G, and B in $S$ are $R$, $G$, and $B$, respectively.
• The number of occurrences of RG as (contiguous) substrings in $S$ is $K$.

Constraints

• $1 \leq R,G,B\leq 10^6$
• $0 \leq K \leq \mathrm{min}(R,G)$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$R$ $G$ $B$ $K$

Output

Print the answer.

2 1 1 1

6

The following six strings satisfy the conditions.

• RRGB
• RGRB
• RGBR
• RBRG
• BRRG
• BRGR

1000000 1000000 1000000 1000000

80957240

Find the count modulo $998244353$.