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 (1.15.1) (judge verdict: Accepted)

type Number = i64;

const M : Number = 1_000_000_007;

fn main()
{
    let mut line = String::new();
    std::io::stdin().read_line(&mut line).unwrap();

    let data: Vec<&str> = line.split_whitespace().collect();
    let r1 = data[0].parse().unwrap_or(0);
    let c1 = data[1].parse().unwrap_or(0);
    let r2 = data[2].parse().unwrap_or(0);
    let c2 = data[3].parse().unwrap_or(0);

    let n = r2 + c2 + 2;

    let mut facts : Vec<Number> = vec![1; n];
    let mut finvs : Vec<Number> = vec![1; n];
    let mut invs  : Vec<Number> = vec![1; n];

    invs[0] = 0;

    for i in 2..n
    {
        facts[i] = ((i as Number) * facts[i - 1]) % M;
        invs[i]  = M - (invs[(M as usize) % i] * (M / (i as Number))) % M;
        finvs[i] = (finvs[i - 1] * invs[i]) % M;
    }

    let ncr = |n: usize, r: usize| -> Number
    {
        (facts[n] * ((finvs[r] * finvs[n - r]) % M)) % M
    };

    let sum = |x: usize, y: usize| -> Number
    {
        (1 .. (x + 2)).fold(0, |val, index| (val + ncr(y + index, index)) % M)
    };

    let a = sum(r2, c2);
    let b = sum(r1 - 1, c2);
    let c = sum(r2, c1 - 1);
    let d = sum(r1 - 1, c1 - 1);

    let ans = (a + d - b - c + 4 * M) % M;
    println!("{}", ans);
}