Problem:
Score : 600 points

Problem Statement

We have a grid with A horizontal rows and B vertical columns, with the squares painted white. On this grid, we will repeatedly apply the following operation:

Assume that the grid currently has a horizontal rows and b vertical columns. Choose "vertical" or "horizontal".

If we choose "vertical", insert one row at the top of the grid, resulting in an (a+1) \times b grid.

If we choose "horizontal", insert one column at the right end of the grid, resulting in an a \times (b+1) grid.

Then, paint one of the added squares black, and the other squares white.

Assume the grid eventually has C horizontal rows and D vertical columns. Find the number of ways in which the squares can be painted in the end, modulo 998244353.

Constraints

1 \leq A \leq C \leq 3000

1 \leq B \leq D \leq 3000

A, B, C, and D are integers.

Input

Input is given from Standard Input in the following format:

A B C D

Output

Print the number of ways in which the squares can be painted in the end, modulo 998244353.

Sample Input 1

1 1 2 2

Sample Output 1

3

Any two of the three squares other than the bottom-left square can be painted black.

Sample Input 2

2 1 3 4

Sample Output 2

65

Sample Input 3

31 41 59 265

Sample Output 3

387222020

Solution in PyPy3 (7.3.0) (judge verdict: Time Limit Exceeded)

# coding: utf-8
import sys

sr = lambda: sys.stdin.readline().rstrip()
ir = lambda: int(sr())
lr = lambda: list(map(int, sr().split()))

"""
dp、無理な組み合わせを引く、追加された黒マスの数は決まっている
各マスのdp
下から来たものと左から来たものをdpで保持
下から来たものは次だけ制約あり
dp = [下から、左から]
"""
MOD = 998244353
A, B, C, D = lr()  # Cは縦
dp = [[[0, 0] for _ in range(D+1)] for _ in range(C+1)]   # 1-indexed
dp[A][B] = [0, 1]
for i in range(A, C+1):
    for j in range(B, D+1):
        # iが縦
        if i == A and j == B:
            continue
        if i > A:
            dp[i][j][0] = sum(dp[i-1][j]) * j
        if j > B:
            dp[i][j][1] = dp[i][j-1][0] + dp[i][j-1][1] * i
        dp[i][j][0] %= MOD
        dp[i][j][1] %= MOD

answer = sum(dp[C][D])
print(answer%MOD)