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 C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <algorithm>
#include <iostream>
#include <vector>
using namespace std;

#define rep(i, n) for (int i = 0; i < (n); i++)

const int64_t mod = 1000000000 + 7;
int64_t mpow(int64_t a, int64_t x) {
  int64_t res = 1;
  while (x > 0) {
    if (x & 1) { res = res * a % mod; }
    a = a * a % mod;
    x = x / 2;
  }
  return res;
}

int64_t solve(int r, int c) {
  auto n = r + c + 2;
  vector<int64_t> fac(n), inv(n);
  fac[0] = 1;
  for (int i = 1; i < n; i++) {
    fac[i] = i * fac[i - 1] % mod;
  }
  inv[n - 1] = mpow(fac[n - 1], mod - 2);
  for (int i = n - 2; i >= 0; i--) {
    inv[i] = inv[i + 1] * (i + 1) % mod;
  }
  auto binom = [&](int a, int b) -> int64_t {
    if (a < 0 or b < 0) return 0;
    if (a < b) return 0;
    return fac[a] * inv[b] % mod * inv[a - b] % mod;
  };
  auto countPath = [&](int a, int b) { return binom(a + b, b); };
  int64_t res = 0;
  for (int i = 0; i <= r; i++) {
    res = (res + countPath(i + 1, c)) % mod;
  }
  return res;
}

int main() {

  int r1, c1, r2, c2;
  cin >> r1 >> c1 >> r2 >> c2;

  auto ans = solve(r2, c2);
  ans = (ans - solve(r1 - 1, c2) + mod) % mod;
  ans = (ans - solve(r2, c1 - 1) + mod) % mod;
  ans = (ans + solve(r1 - 1, c1 - 1)) % mod;
  cout << ans << endl;

  return 0;
}