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 Rust (1.42.0) (judge verdict: Accepted)

// -*- coding:utf-8-unix -*-

use num_traits::{pow, One};
use proconio::{fastout, input};
use std::ops::{Add, Div, Mul, Sub};

const MODULUS: usize = 998244353;

#[fastout]
fn main() {
    input! {
        a: usize,
        b: usize,
        c: usize,
        d: usize,
    }
    let mut yarikata: Vec<Vec<ModP>> = vec![vec![ModP(0); d + 1 - b]; c + 1 - a];
    for i in 0..=c - a {
        for j in 0..=d - b {
            if i == 0 && j == 0 {
                yarikata[i][j] = ModP(1);
            } else if i == 0 {
                yarikata[i][j] = yarikata[i][j - 1] * ModP(a);
            } else if j == 0 {
                yarikata[i][j] = yarikata[i - 1][j] * ModP(b);
            } else {
                yarikata[i][j] = yarikata[i][j - 1] * ModP(a + i)
                    + yarikata[i - 1][j] * ModP(b + j)
                    - yarikata[i - 1][j - 1] * ModP(a + i - 1) * ModP(b + j - 1);
            }
        }
    }
    println!("{}", yarikata[c - a][d - b].0);
}

#[derive(Clone, Copy, PartialEq, Debug)]
struct ModP(usize);

impl One for ModP {
    fn one() -> Self {
        return ModP(1);
    }
}
impl Add for ModP {
    type Output = Self;
    fn add(self, rhs: Self) -> Self {
        return ModP((self.0 + rhs.0) % MODULUS);
    }
}
impl Sub for ModP {
    type Output = Self;
    fn sub(self, rhs: Self) -> Self {
        return ModP((self.0 + MODULUS - rhs.0) % MODULUS);
    }
}
impl Mul for ModP {
    type Output = Self;
    fn mul(self, rhs: Self) -> Self {
        return ModP((self.0 * rhs.0) % MODULUS);
    }
}
impl Div for ModP {
    type Output = Self;
    fn div(self, rhs: Self) -> Self {
        if rhs.0 == 0 {
            panic!("Tried to divide by ModP(0)!");
        }
        let rhs_inv = pow(rhs, MODULUS - 2);
        return self * rhs_inv;
    }
}