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# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using System;
using System.Collections.Generic;
using System.Linq;

namespace _154F
{
    internal class Program
    {
        private const int p = 1000000007;

        private static void Main(string[] args)
        {
            var input = Console.ReadLine().Split(' ').Select(x => int.Parse(x)).ToList();
            var r1 = input[0];
            var c1 = input[1];
            var r2 = input[2];
            var c2 = input[3];

            int result = 0;
            for (int i = r1; i <= r2; ++i)
                for (int j = c1; j <= c2; ++j)
                    result = (result + Combi(i + j, i)) % p;

            Console.WriteLine(result);
        }

        private static readonly IDictionary<string, int> memo
            = new Dictionary<string, int>();

        private static int Combi(int n, int k)
        {
            if (k * 2 > n)
                return Combi(n, n - k);

            if (k == 0)
                return 1;
            if (k == 1)
                return n;

            var key = $"{n}:{k}";
            if (memo.ContainsKey(key))
                return memo[key];

            memo[key] = (Combi(n - 1, k) + Combi(n - 1, k - 1)) % p;

            return memo[key];
        }
    }
}