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 C# (.NET Core 3.1.201) (judge verdict: Time Limit Exceeded)

using System;
using static System.Console;
using static System.Math;
using System.Collections.Generic;


namespace AtCoder
{
    public class Program
    {
        public static void Main(string[] args)
        {
            var s = ReadLine().Split(' ');
            long ans = 0;
            long a = long.Parse(s[0]);
            long b = long.Parse(s[1]);
            long c = long.Parse(s[2]);
            long d = long.Parse(s[3]);
            long row = c - a;
            long line = d - b;
            long black = row * line;

            const long MOD = 998244353L;
            long Max = 90000000;

            long[] fac = new long[Max + 100];
            long[] finv = new long[Max + 100];
            long[] inv = new long[Max + 100];

            fac[0] = fac[1] = 1;
            finv[0] = finv[1] = 1;
            inv[1] = 1;
            for (long i = 2; i < Max; i++)
            {
                fac[i] = fac[i - 1] * i % MOD;
                inv[i] = MOD - inv[MOD % i] * (MOD / i) % MOD;
                finv[i] = finv[i - 1] * inv[i] % MOD;
            }

            long n, k;
            
            n = c * d - 1;
            if (row == 0)
            { n -= c - 1; }
            if (line == 0)
            { n -= d - 1; }
            k = row + line;

            ans = fac[n] * (finv[k] * finv[n - k] % MOD) % MOD;


            WriteLine(ans);
            Out.Flush();
        }

    }

    public static class MyMath
    {
        public static long Factorical(long n)
        { return n <= 1 ? 1L : n * Factorical(n - 1); }
        public static long Permutation(long n, long m)
        { return m == 0 ? 1L : (m == n) ? Permutation(n, m - 1) : n * Permutation(n - 1, m - 1); }
        public static long Combination(long n, long m)
        { return n - m > m ? Permutation(n, m) / Factorical(m) : Permutation(n, n - m) / Factorical(n - m); }
        public static long Gcd(long n, long m)
        { return n % m == 0 ? m : Gcd(m, n % m); }
        public static long Lcm(long n, long m)
        { return n * m / Gcd(n, m); }
        public static long Pow(long n, long m)
        { return (m == 1 || m == 0) ? 1L : n * Pow(n, m - 1); }
    }
}