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 Java (OpenJDK 11.0.6) (judge verdict: Time Limit Exceeded)

import java.io.PrintWriter;
import java.util.HashMap;
import java.util.Map;
import java.util.Scanner;

public class Main {
    long MOD = 998244353L;
    Map<Integer, Long> cache = new HashMap<>();
    void solve(Scanner sc, PrintWriter pw) {
        int A = sc.nextInt();
        int B = sc.nextInt();
        int C = sc.nextInt();
        int D = sc.nextInt();
        pw.println(func(A, B, C, D));
    }

    private long func(int a, int b, int c, int d) {
        if (c == a && b == d) {
            return 1L;
        }
        int key = (c * (1<< 12)) + d;
        if (cache.containsKey(key)) {
            return cache.get(key);
        }
        long ans = 0;
        if (c > a) {
            long part = func(a, b, c - 1, d);
            ans = (part * d);
        }
        if (d > b) {
            long part = func(a, b, c , d - 1);
            ans = (ans + (part * c));
        }
        if (c > a && d > b) {
            ans = (ans - (func(a, b, c - 1, d - 1) * (c - 1) * (d - 1)));
        }
        ans = ((ans %MOD) + MOD) % MOD;
        cache.put(key, ans);
        return ans;
    }

    public static void main(String[] args){
        Scanner sc = new Scanner(System.in);
        PrintWriter pw = new PrintWriter(System.out);
        new Main().solve(sc, pw);
        pw.flush();
        pw.close();
        sc.close();
    }
}