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)

import sys

def main():
    input = sys.stdin.readline
    mod = 998244353
    
    a, b, c, d = map(int, input().split())
    dp = dict()
    for i in range(a, c+1):
        for j in range(b, d+1):
            if i == a and j == b:
                dp[(i, j)] = 1
                continue
            if j == b:
                dp[(i, j)] = dp[(i-1, j)]*j%mod
                continue
            if i == a:
                dp[(i, j)] = dp[(i, j-1)]*i%mod
                continue
            dp[(i, j)] = (dp[(i-1, j)]*j+dp[(i, j-1)]*i+mod-dp[(i-1, j-1)]*(i-1)*(j-1))%mod
    print(dp[(c, d)])        
    

if __name__ == "__main__":
    main()