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 (Clang 3.8.0) (judge verdict: Accepted)

#include <iostream>
#include <iomanip>
#include <vector>
#include <array>
#include <string>
#include <stack>
#include <queue>
#include <deque>
#include <unordered_map>
#include <unordered_set>
#include <set>
#include <tuple>
#include <cmath>
#include <algorithm>
#include <functional>
#include <numeric>
#include <cfloat>
#include <climits>
#include <cassert>
#include <random>


static std::vector<long long int> factorial;
static std::vector<long long int> inverse;
constexpr long long int MOD = 1000000007;
long long int comb(int n, int r) {
	return factorial[n] * inverse[n - r] % MOD * inverse[r] % MOD;
}
long long int count_square_sum(int r, int c) {
	long long int result = 0;
	if (r < c) std::swap(r, c);
	for (auto i = 0; i <= c; ++i) {
		result += comb(r + i + 1, i + 1);
		result %= MOD;
	}
	return result;
}
int main(){
	factorial = std::vector<long long int>(2000100, 1);
	inverse = std::vector<long long int>(2000100, 1);
	std::vector<long long int> inv(2000100, 1);
	for (auto i = 2; i < factorial.size(); ++i) {
		factorial[i] = i * factorial[i - 1] % MOD;
		inv[i] = (MOD - MOD / i) * inv[MOD % i] % MOD;
		inverse[i] = inv[i] * inverse[i - 1] % MOD;
	}
	int r1, c1, r2, c2; std::cin >> r1 >> c1 >> r2 >> c2;
	long long int result = 0;
	if (r1 > r2) std::swap(r1, r2);
	if (c1 < c2) std::swap(c1, c2);
	result += count_square_sum(r2, c1);
	result -= count_square_sum(r1 - 1, c1);
	result -= count_square_sum(r2, c2 - 1);
	result += count_square_sum(r1 - 1, c2 - 1);
	result %= MOD;
	result += MOD;
	result %= MOD;
	std::cout << result << std::endl;
}