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)

#region いつもの
#pragma warning disable
using System;
using System.Collections.Generic;
using System.Linq;
using IEnumerable = System.Collections.IEnumerable;
using IEnumerator = System.Collections.IEnumerator;
using BitArray = System.Collections.BitArray;
using BigInteger = System.Numerics.BigInteger;
using TextReader = System.IO.TextReader;
using static Global;
using static NumGlobal;

static class Global
{
    private class ComparerReverseImpl<T> : Comparer<T> where T : IComparable<T> { public override int Compare(T y, T x) => x.CompareTo(y); public override bool Equals(object obj) => obj != null && GetType() == obj.GetType(); public override int GetHashCode() => GetType().GetHashCode(); }
    public static IComparer<T> ComparerReverse<T>() where T : IComparable<T> => new ComparerReverseImpl<T>();
    public static string AllLines<T>(IEnumerable<T> source) => string.Join("\n", source);
    public static string AllJoin<T>(IEnumerable<T> source) => string.Join(" ", source);
}
static class NumGlobal
{
    public static int Pow(int x, int y)
    {
        int res = 1;
        for (; y > 0; y >>= 1)
        {
            if ((y & 1) == 1) res *= x;
            x *= x;
        }
        return res;
    }
    public static BigInteger ParseBigInteger(string s)
    {
        // MonoのBigInteger.Parseが遅いので自前実装
        var res = BigInteger.Zero;
        var splited = new string[(s.Length + 7) / 8];
        for (var i = 0; i < splited.Length - 1; i++)
        {
            splited[i] = s.Substring(8 * i, 8);
        }
        splited[splited.Length - 1] = s.Substring(8 * (splited.Length - 1));
        foreach (var sp in splited)
        {
            res *= (int)Math.Pow(10, sp.Length);
            res += int.Parse(sp);
        }
        return res;
    }
    public static int BitCount(int x)
    {
        x = x - ((x >> 1) & 0x55555555);
        x = (x & 0x33333333) + ((x >> 2) & 0x33333333);
        x = (x + (x >> 4)) & 0x0f0f0f0f;
        x = x + (x >> 8);
        x = x + (x >> 16);
        return x & 0x3f;
    }
    public static int BitCount(long x)
    {
        x = x - ((x >> 1) & 0x5555555555555555);
        x = (x & 0x3333333333333333) + ((x >> 2) & 0x3333333333333333);
        x = (x + (x >> 4)) & 0x0f0f0f0f0f0f0f0f;
        x = x + (x >> 8);
        x = x + (x >> 16);
        x = x + (x >> 32);
        return (int)(x & 0x0000007f);
    }
    public static int MSB(int x)
    {
        x |= (x >> 1);
        x |= (x >> 2);
        x |= (x >> 4);
        x |= (x >> 8);
        x |= (x >> 16);
        return BitCount(x) - 1;
    }
    public static int MSB(long x)
    {
        x |= (x >> 1);
        x |= (x >> 2);
        x |= (x >> 4);
        x |= (x >> 8);
        x |= (x >> 16);
        x |= (x >> 32);
        return BitCount(x) - 1;
    }
    public static int LSB(int x)
    {
        x |= (x << 1);
        x |= (x << 2);
        x |= (x << 4);
        x |= (x << 8);
        x |= (x << 16);
        return 32 - BitCount(x);
    }
    public static int LSB(long x)
    {
        x |= (x << 1);
        x |= (x << 2);
        x |= (x << 4);
        x |= (x << 8);
        x |= (x << 16);
        x |= (x << 32);
        return 64 - BitCount(x);
    }
}
static class Ext
{
    public static T[] Init<T>(this T[] arr, Func<T> factory)
    {
        for (int i = 0; i < arr.Length; i++)
            arr[i] = factory();
        return arr;
    }
    public static T[] Fill<T>(this T[] arr, T value)
    {
        for (var i = 0; i < arr.Length; i++)
            arr[i] = value;
        return arr;
    }
    public static T[,] Fill<T>(this T[,] arr, T value)
    {
        for (var i = 0; i < arr.GetLength(0); i++)
            for (var j = 0; j < arr.GetLength(1); j++)
                arr[i, j] = value;
        return arr;
    }
    public static T[] Sort<T>(this T[] arr)
    {
        Array.Sort(arr);
        return arr;
    }
    public static T[] Reverse<T>(this T[] arr)
    {
        Array.Reverse(arr);
        return arr;
    }
    public static T[][] Chunk<T>(this T[,] source)
    {
        var len0 = source.GetLength(0);
        var len1 = source.GetLength(1);
        var res = new T[len0][];
        for (var i = 0; i < len0; i++)
        {
            res[i] = new T[len1];
            for (var j = 0; j < len1; j++)
            {
                res[i][j] = source[i, j];
            }
        }
        return res;
    }

    public static Tuple<TSource, TMax> MaxBy<TSource, TMax>
        (this IEnumerable<TSource> source, Func<TSource, TMax> maxBySelector)
        where TMax : IComparable<TMax>
    {
        TMax max;
        TSource maxByItem;

        var e = source.GetEnumerator();
        e.MoveNext();
        maxByItem = e.Current;
        max = maxBySelector(maxByItem);
        while (e.MoveNext())
        {
            var item = e.Current;
            var next = maxBySelector(item);
            if (max.CompareTo(next) < 0)
            {
                max = next;
                maxByItem = item;
            }
        }
        return Tuple.Create(maxByItem, max);
    }
    public static Tuple<TSource, TMin> MinBy<TSource, TMin>
        (this IEnumerable<TSource> source, Func<TSource, TMin> minBySelector)
        where TMin : IComparable<TMin>
    {
        TMin min;
        TSource minByItem;

        var e = source.GetEnumerator();
        e.MoveNext();
        minByItem = e.Current;
        min = minBySelector(minByItem);
        while (e.MoveNext())
        {
            var item = e.Current;
            var next = minBySelector(item);
            if (min.CompareTo(next) > 0)
            {
                min = next;
                minByItem = item;
            }
        }
        return Tuple.Create(minByItem, min);
    }
    public static IComparer<T> Reverse<T>(this IComparer<T> comparer) => Comparer<T>.Create((x, y) => comparer.Compare(y, x));
    public static Dictionary<TKey, int> GroupCount<TSource, TKey>(this IEnumerable<TSource> source, Func<TSource, TKey> keySelector) => source.GroupBy(keySelector).ToDictionary(g => g.Key, g => g.Count());
    public static Dictionary<TKey, int> GroupCount<TKey>(this IEnumerable<TKey> source) => source.GroupCount(i => i);
}
public class ConsoleReader { private string[] ReadLineSplit() => textReader.ReadLine().Split(Array.Empty<char>(), StringSplitOptions.RemoveEmptyEntries); private string[] line = Array.Empty<string>(); private int linePosition; private TextReader textReader; public ConsoleReader(TextReader tr) { textReader = tr; } public int Int => int.Parse(String); public int Int0 => Int - 1; public long Long => long.Parse(String); public double Double => double.Parse(String); public string String { get { if (linePosition >= line.Length) { linePosition = 0; line = ReadLineSplit(); } return line[linePosition++]; } } public class SplitLine { private string[] splited; public SplitLine(ConsoleReader cr) { splited = cr.ReadLineSplit(); cr.line = Array.Empty<string>(); } public int[] Int => String.Select(x => int.Parse(x)).ToArray(); public int[] Int0 => String.Select(x => int.Parse(x) - 1).ToArray(); public long[] Long => String.Select(x => long.Parse(x)).ToArray(); public double[] Double => String.Select(x => double.Parse(x)).ToArray(); public string[] String => splited; } public SplitLine Split => new SplitLine(this); public class RepeatReader : IEnumerable<ConsoleReader> { ConsoleReader cr; int count; public RepeatReader(ConsoleReader cr, int count) { this.cr = cr; this.count = count; } public IEnumerator<ConsoleReader> GetEnumerator() => Enumerable.Repeat(cr, count).GetEnumerator(); System.Collections.IEnumerator System.Collections.IEnumerable.GetEnumerator() => GetEnumerator(); public string[] String => this.Select(cr => cr.String).ToArray(); public int[] Int => this.Select(cr => cr.Int).ToArray(); public int[] Int0 => this.Select(cr => cr.Int - 1).ToArray(); public long[] Long => this.Select(cr => cr.Long).ToArray(); public double[] Double => this.Select(cr => cr.Double).ToArray(); } public RepeatReader Repeat(int count) => new RepeatReader(this, count); }
public class Program
{
    [System.Diagnostics.DebuggerBrowsable(System.Diagnostics.DebuggerBrowsableState.Never)] private ConsoleReader cr;
    public Program(ConsoleReader consoleReader) { this.cr = consoleReader; }
    static void Main() => Console.WriteLine(new Program(new ConsoleReader(Console.In)).Calc());

    #endregion
    public object Calc()
    {
        var r1 = cr.Int;
        var c1 = cr.Int;
        var r2 = cr.Int;
        var c2 = cr.Int;


        var factors = Mod.CreateFactor((int)3e6);

        Mod sum = factors.Combine(r2 + c2 + 2, r2 + 1);
        sum -= factors.Combine(r2 + c1 + 1, c1);
        sum -= factors.Combine(r1 + c2 + 1, r1);
        sum += factors.Combine(r1 + c1, r1);
        return sum;
    }
}
struct Mod : IEquatable<Mod>
{
    public const long mod = 1000000007;
    public readonly long val;
    public Mod(long val, bool isValid) : this(isValid ? val % mod : val) { }
    public Mod(long val) { this.val = val; }
    public override bool Equals(object obj) => (obj is Mod) ? this == ((Mod)obj) : false;
    public bool Equals(Mod obj) => this == obj;
    public override int GetHashCode() => val.GetHashCode();
    public override string ToString() => val.ToString();
    public static implicit operator Mod(long x) => new Mod(x);
    public static Mod operator +(Mod x, Mod y) => (x.val + y.val) % mod;
    public static Mod operator -(Mod x, Mod y) => x.val >= y.val ? (x.val - y.val) % mod : (x.val - y.val) % mod + mod;
    public static Mod operator *(Mod x, Mod y) => (x.val * y.val) % mod;
    public static Mod operator /(Mod x, Mod y) => x * y.Inverse();
    public static bool operator ==(Mod x, Mod y) => x.val == y.val;
    public static bool operator !=(Mod x, Mod y) => x.val != y.val;
    public Mod Inverse()
    {
        long a = val, b = mod, u = 1, v = 0;
        while (b > 0)
        {
            long t = a / b;
            var b2 = a - t * b;
            a = b; b = b2;
            var v2 = u - t * v;
            u = v; v = v2;
        }
        u %= mod;
        if (u < 0) u += mod;
        return u;
    }
    public static Mod Pow(Mod x, int y)
    {
        Mod res = 1;
        for (; y > 0; y >>= 1)
        {
            if ((y & 1) == 1) res *= x;
            x *= x;
        }
        return res;
    }

    public static Factor CreateFactor(int max) => new Factor(max);
    public class Factor
    {
        private readonly Mod[] fac, finv;
        public Factor(int max)
        {
            ++max;
            var inv = new Mod[max];
            fac = new Mod[max]; finv = new Mod[max];
            fac[0] = fac[1] = 1;
            finv[0] = finv[1] = 1;
            inv[1] = 1;
            for (var i = 2; i < max; i++)
            {
                fac[i] = fac[i - 1] * i;
                inv[i] = mod - inv[mod % i].val * (mod / i) % mod;
                finv[i] = finv[i - 1] * inv[i];
            }
        }

        // 二項係数計算
        public Mod Combine(int n, int k)
        {
            if (n < k) return 0;
            if (n < 0 || k < 0) return 0;
            return fac[n] * finv[k] * finv[n - k];
        }

        public Mod Factorial(int n) => fac[n];
        public Mod FactorialInvers(int n) => finv[n];
    }
}