Problem:
Score : 600 points

Problem Statement

Snuke is standing on a two-dimensional plane. In one operation, he can move by 1 in the positive x-direction, or move by 1 in the positive y-direction.

Let us define a function f(r, c) as follows:

f(r,c) := (The number of paths from the point (0, 0) to the point (r, c) that Snuke can trace by repeating the operation above)

Given are integers r_1, r_2, c_1, and c_2.
Find the sum of f(i, j) over all pair of integers (i, j) such that r_1 ≤ i ≤ r_2 and c_1 ≤ j ≤ c_2, and compute this value modulo (10^9+7).

Constraints

1 ≤ r_1 ≤ r_2 ≤ 10^6

1 ≤ c_1 ≤ c_2 ≤ 10^6

All values in input are integers.

Input

Input is given from Standard Input in the following format:

r_1 c_1 r_2 c_2

Output

Print the sum of f(i, j) modulo (10^9+7).

Sample Input 1

1 1 2 2

Sample Output 1

14

For example, there are two paths from the point (0, 0) to the point (1, 1): (0,0) → (0,1) → (1,1) and (0,0) → (1,0) → (1,1), so f(1,1)=2.

Similarly, f(1,2)=3, f(2,1)=3, and f(2,2)=6. Thus, the sum is 14.

Sample Input 2

314 159 2653 589

Sample Output 2

602215194

Solution in Rust (1.42.0) (judge verdict: Accepted)

#![allow(unused_imports)]
#![allow(dead_code)]

use proconio::input;
use proconio::marker::{Bytes, Chars};
use std::collections::*;

fn main() {
    input! {
        r1:i64,
        c1:i64,
        r2:i64,
        c2:i64,
    }
    let modulo = 1e+9 as i64 + 7;
    let comb = CombinationMod::new((r2 + c2) as usize + 2, modulo);
    let square_sum = |r, c| (comb.query(r + c + 2, r + 1) + modulo - 1) % modulo;
    let mut ans = 0;
    ans = (ans + square_sum(r2, c2)) % modulo;
    ans = (ans - square_sum(r1 - 1, c2)) % modulo;
    ans = (ans - square_sum(r2, c1 - 1)) % modulo;
    ans = (ans + square_sum(r1 - 1, c1 - 1)) % modulo;
    if ans < 0 {
        ans = (ans % modulo + modulo) % modulo;
    }
    println!("{}", ans);
}

struct CombinationMod {
    fac: Vec<i64>,
    inv: Vec<i64>,
    inv_fac: Vec<i64>,
    modulo: i64,
}
impl CombinationMod {
    pub fn new(size: usize, modulo: i64) -> CombinationMod {
        let mut fac = vec![1; size + 1];
        let mut inv = vec![1; size + 1];
        let mut inv_fac = vec![1; size + 1];
        for i in 2..=size {
            fac[i] = fac[i - 1] * i as i64 % modulo;
            inv[i] = modulo - inv[modulo as usize % i] * (modulo / i as i64) % modulo;
            inv_fac[i] = inv_fac[i - 1] * inv[i] % modulo;
        }
        CombinationMod {
            fac: fac,
            inv: inv,
            inv_fac: inv_fac,
            modulo: modulo,
        }
    }
    fn query(&self, n: i64, r: i64) -> i64 {
        self.fac[n as usize] * self.inv_fac[r as usize] % self.modulo
            * self.inv_fac[(n - r) as usize]
            % self.modulo
    }
    fn inverse(&self, n: i64) -> i64 {
        self.inv[n as usize]
    }
}