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: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using static System.Console;
using static System.Math;

namespace AtTest.ABC_154
{
    class F
    {
        static void Main(string[] args)
        {
            var sw = new System.IO.StreamWriter(OpenStandardOutput()) { AutoFlush = false };
            SetOut(sw);

            Method(args);

            Out.Flush();
        }

        static void Method(string[] args)
        {
            int[] rcs = ReadInts();
            long r1 = rcs[0];
            long c1 = rcs[1];
            long r2 = rcs[2];
            long c2 = rcs[3];
            long mask = 1000000000 + 7;
            CaseCalculator calculator = new CaseCalculator(mask, r2 + c2 + 5);
            long res = 0;
            for (long i = r1 + 1; i <= r2 + 1; i++)
            {
                res += calculator.Combination(i + c2, i);
                res += mask - calculator.Combination(i + c1 - 1, i);
                res %= mask;
            }
            WriteLine(res);
        }

        class CaseCalculator
        {
            long mask;
            long[] allPermutations;
            long[] allInverses;

            public CaseCalculator(long mask, long permutationCnt)
            {
                this.mask = mask;
                allPermutations = AllPermutations(permutationCnt);
                allInverses = AllInverses(permutationCnt);
            }

            public long Combination(long n, long m)
            {
                if (n < m) return 0;

                if (n - m < m) m = n - m;

                return Multi(allPermutations[n],
                    Multi(allInverses[n - m], allInverses[m]));
            }

            public long Permutation(long n, long m)
            {
                if (n < m) return 0;

                return Multi(allPermutations[n],allInverses[n - m]);
            }

            long[] AllPermutations(long n)
            {
                var perms = new long[n + 1];
                perms[0] = 1;
                for (int i = 1; i <= n; i++)
                {
                    perms[i] = Multi(perms[i - 1], i);
                }
                return perms;
            }

            long[] AllInverses(long n)
            {
                var inverses = new long[n + 1];
                for(int i = 0; i <= n; i++)
                {
                    inverses[i] = Inverse(allPermutations[i]);
                }
                return inverses;
            }

            public long Multi(long a, long b)
            {
                return ((a % mask) * (b % mask)) % mask;
            }

            public long Inverse(long val)
            {
                long a = mask;
                long b = val % mask;
                long x = 1;
                long x1 = 0;
                long y = 0;
                long y1 = 1;
                while (b > 0)
                {
                    long q = a / b;
                    long r = a % b;
                    long x2 = (mask + x - q * x1) % mask;
                    long y2 = (mask + y - q * y1) % mask;
                    a = b;
                    b = r;
                    x = x1;
                    x1 = x2;
                    y = y1;
                    y1 = y2;
                }
                return y;
            }

            long DeprecatedInverse(long val)
            {
                return Pow(val, mask - 2);
            }

            long Pow(long a, long exp)
            {
                if (exp == 0) return 1;
                else if (exp == 1) return a % mask;
                else
                {
                    long halfMod = Pow(a, exp / 2);
                    long nextMod = Multi(halfMod, halfMod);
                    if (exp % 2 == 0)
                    {
                        return nextMod;
                    }
                    else
                    {
                        return Multi(nextMod, a);
                    }
                }
            }
        }

        private static string Read() { return ReadLine(); }
        private static char[] ReadChars() { return Array.ConvertAll(Read().Split(), a => a[0]); }
        private static int ReadInt() { return int.Parse(Read()); }
        private static long ReadLong() { return long.Parse(Read()); }
        private static double ReadDouble() { return double.Parse(Read()); }
        private static int[] ReadInts() { return Array.ConvertAll(Read().Split(), int.Parse); }
        private static long[] ReadLongs() { return Array.ConvertAll(Read().Split(), long.Parse); }
        private static double[] ReadDoubles() { return Array.ConvertAll(Read().Split(), double.Parse); }
    }
}