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++ (GCC 9.2.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
#define rep(i,n) for (int i = 0; i < (n); ++i)
#define sz(x) int(x.size())
#define ALL(c) (c).begin(), (c).end()
#define SUM(x) std::accumulate(ALL(x), 0LL)
#define MIN(v) *std::min_element(v.begin(), v.end())
#define MAX(v) *std::max_element(v.begin(), v.end())
#define EXIST(v, x) (std::find(v.begin(), v.end(), x) != v.end())

using namespace std;
using ll = long long;

template<class T> inline bool chmax(T& a, T b) { if (a < b) { a = b; return true; } return false; }
template<class T> inline bool chmin(T& a, T b) { if (a > b) { a = b; return true; } return false; }
template<class T> inline void dump(vector<T> v) { for (auto& x : v) cerr << x << " "; cerr << endl; }
template<class T> inline void dump(vector<pair<T, T>> v) { for (auto& p : v) cerr << p.first << " " << p.second << endl; }
template<class T> inline void dump(vector<vector<T>> vv) { for (auto& v : vv) {for (auto& x : v) cerr << x << " "; cerr << endl;} }

constexpr int INF = 1e9 + 5;
constexpr long long INFLL = 1LL << 62;
constexpr double eps = (1e-9);

constexpr ll mod = 998244353;

ll dp[3005][3005][2];

int main() {
  cin.tie(0);
  ios::sync_with_stdio(false);

  int a, b, c, d;
  cin >> a >> b >> c >> d;

  dp[a][b][0] = 1;

  for (int i = a; i <= c; i++) {
    for (int j = b; j <= d; j++) {
      dp[i+1][j][0] = (dp[i+1][j][0] + dp[i][j][0] * j) % mod;
      dp[i][j+1][1] = ((dp[i][j+1][1] + dp[i][j][0] * i)) % mod;
      dp[i+1][j][0] = (dp[i+1][j][0] + dp[i][j][1]) % mod;
      dp[i][j+1][1] = (dp[i][j+1][1] + dp[i][j][1] * i) % mod;
    }
  }

  cout << (dp[c][d][0] + dp[c][d][1]) % mod << endl;

  return 0;
}