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: Time Limit Exceeded)

#include <bits/stdc++.h>
#define ll long long
#define MOD 1000000007
using namespace std;
ll r1, r2, c1, c2, ans, arr[2000020];
ll g(ll, ll);
ll f(int, int);
ll cal(ll, int);
int main()
{
	ios::sync_with_stdio(false);
	arr[0] = 1;
	for(int i = 1; i<=2000010; ++i) arr[i] = arr[i-1]*(ll)i%MOD;
    cin>>r1>>c1>>r2>>c2;
	ans = g(r2, c2)-g(r1-1, c2)-g(r2, c1-1)+g(r1-1, c1-1);
	ans = (ans+MOD)%MOD;
	cout<<ans<<"\n";
}

ll g(ll a, ll b)
{
	ll sum = 0;
	for(ll i = 1; i<=a+1; ++i) 
		//sum+=f(i, b)%MOD;
		(sum+=f(i, b))%=MOD;
	return sum;
}

ll f(int a, int b)
{
	//(a+b)!/a!/b!;
	return arr[a+b]*cal(arr[a], MOD-2)%MOD*cal(arr[b], MOD-2)%MOD;
}

ll cal(ll a, int b)
{
	if(b==0) return 1;
	if(b%2) return a*cal(a*a%MOD, b/2)%MOD;
	return cal(a*a%MOD, b/2)%MOD;
}