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: Time Limit Exceeded)

use std::io::{self, BufRead};

const M: i64 = 1000000007;

fn fac(n: usize) -> i64 {
  let mut result: i64 = 1;
  let mut m = n;
  while m > 1 {
    result = result * m as i64 % M;
    m -= 1;
  }
  result
}

fn pow(n: i64, p: i64) -> i64 {
  let mut q = p;
  let mut r = n;
  let mut result = 1;
  while q > 0 {
    if q % 2 == 1 {
      result = result * r % M;
    }
    r = r * r % M;
    q >>= 1;
  }
  result
}

fn main() {
  let stdin = io::stdin();
  let mut lines = stdin.lock().lines();
  let xs: Vec<usize> = lines.next().unwrap().unwrap().split(' ').map(|x| x.parse().unwrap()).collect();
  let (r1, c1, r2, c2) = (xs[0], xs[1], xs[2], xs[3]);
  let mut ps: Vec<i64> = vec![0; r2 - r1 + 1];
  let mut qs: Vec<i64> = vec![0; c2 - c1 + 1];
  let mut rs: Vec<i64> = vec![0; r2 - r1 + c2 - c1 + 3];
  let mut temp = 1;
  for i in 0..(r2-r1+1) {
    let l = ps.len();
    ps[l-i-1] = temp;
    temp = (temp * (r2 - i) as i64) % M;
  }
  let mut temp = 1;
  for i in 0..(c2-c1+1) {
    let l = qs.len();
    qs[l-i-1] = temp;
    temp = (temp * (c2 - i) as i64) % M;
  }
  let mut temp = 1;
  for i in 0..(r2-r1+c2-c1+3) {
    rs[i] = temp;
    temp = (temp * (r1 + c1 + i + 1) as i64) % M;
  }
  let coeff = fac(r1 + c1) * pow(fac(r2) * fac(c2) % M, M - 2) % M;
  let mut result: i64 = 0;
  for i in 0..(r2-r1+1) {
    for j in 0..(c2-c1+1) {
      result = (result + (ps[i] * qs[j] % M) * rs[i+j] % M) % M;
    }
  }
  result = result * coeff % M;
  println!("{}", result);
}