Problem:
Score : 600 points

Problem Statement

10^9 contestants, numbered 1 to 10^9, will compete in a competition.
There will be two contests in this competition.

The organizer prepared N problems, numbered 1 to N, to use in these contests.
When Problem i is presented in a contest, it will be solved by all contestants from Contestant L_i to Contestant R_i (inclusive), and will not be solved by any other contestants.

The organizer will use these N problems in the two contests.
Each problem must be used in exactly one of the contests, and each contest must have at least one problem.

The joyfulness of each contest is the number of contestants who will solve all the problems in the contest.
Find the maximum possible total joyfulness of the two contests.

Constraints

2 \leq N \leq 10^5

1 \leq L_i \leq R_i \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
L_1 R_1
L_2 R_2
\vdots
L_N R_N

Output

Print the maximum possible total joyfulness of the two contests.

Sample Input 1

4
4 7
1 4
5 8
2 5

Sample Output 1

6

The optimal choice is:

Use Problem 1 and 3 in the first contest. Contestant 5, 6, and 7 will solve both of them, so the joyfulness of this contest is 3.

Use Problem 2 and 4 in the second contest. Contestant 2, 3, and 4 will solve both of them, so the joyfulness of this contest is 3.

The total joyfulness of these two contests is 6. We cannot make the total joyfulness greater than 6.

Sample Input 2

4
1 20
2 19
3 18
4 17

Sample Output 2

34

Sample Input 3

10
457835016 996058008
456475528 529149798
455108441 512701454
455817105 523506955
457368248 814532746
455073228 459494089
456651538 774276744
457667152 974637457
457293701 800549465
456580262 636471526

Sample Output 3

540049931

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using System.Runtime.CompilerServices;
using System.Text;
using static Template;
using static System.Console;
using static System.Convert;
using static System.Math;
using Pi = Pair<int, int>;
using Number = System.Int64;

class Solver
{
    public void Solve()
    {
        var num = Input.num;
        var ar = new List<int>();
        var lr = new int[num][]; var maxlen = 0;
        for (var i = 0; i < num; i++)
        {
            int ll, rr;
            Input.Make(out ll, out rr);
            ar.Add(ll); ar.Add(rr); chmax(ref maxlen, rr - ll + 1);
            lr[i] = new[] { ll, rr };
        }
        ar.Sort();
        var dic = new Dictionary<int, int>();
        var rev = new Dictionary<int, int>();
        for (var i = 0; i < ar.Count; i++)
            if (!dic.ContainsKey(ar[i]))
            {
                var c = dic.Count;
                dic[ar[i]] = c;
                rev[c] = ar[i];
            }
        Array.Sort(lr, (a, b) => a[1] - b[1]);
        var seg = new SegmentTree<int>(dic.Count + 1, 0, (a, b) => a + b, (a, b) => a + b);
        for (var i = num - 1; i >= 1; i--)
        {
            seg.Update(dic[lr[i][0]], 1);
            seg.Update(dic[lr[i][1]] + 1, 1);
        }
        int l = int.MinValue, r = int.MaxValue;
        var res = 0L;
        for (var i = 0; i < num - 1; i++)
        {
            chmax(ref l, lr[i][0]);
            chmin(ref r, lr[i][1]);
            if (r < l) break;
            var ml = seg.Find(0, a => a >= num - i - 1);
            if (seg.Query(0, ml + 1) != num - i - 1)
            {
                seg.Update(dic[lr[i + 1][0]], -1);
                seg.Update(dic[lr[i + 1][1]] + 1, -1);
                chmax(ref res, r - l + 1); continue;
            }
            seg.Update(dic[lr[i + 1][0]], -1);
            seg.Update(dic[lr[i + 1][1]] + 1, -1);
            chmax(ref res, r - l + 1L + lr[i + 1][1] - rev[ml] + 1);
        }

        chmax(ref l, lr[num - 1][0]);
        chmin(ref r, lr[num - 1][1]);
        var ca = r < l ? 0 : r - l + 1;
        for (var i = 0; i < num; i++)
            chmax(ref res, lr[i][1] - lr[i][0] + 1L + ca);
        WriteLine(res);
    }
}
public class SegmentTree<T>
{
    protected readonly T[] item;
    protected readonly int num;
    protected readonly Func<T, T, T> func;
    protected readonly Func<T, T, T> updateFunc;
    protected readonly T init;

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    protected int Parent(int index)
        => (index - 1) >> 1;
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    protected int Left(int index)
        => (index << 1) + 1;
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    protected int Right(int index)
        => (index + 1) << 1;
    public T this[int i]
    {
        get { return item[i + num - 1]; }
        set { item[i + num - 1] = value; }
    }

    public SegmentTree(int num, T init, Func<T, T, T> func, Func<T, T, T> updateFunc = null)
    {
        this.func = func;
        this.num = 1;
        this.init = init;
        this.updateFunc = updateFunc ?? ((T val1, T val2) => val2);
        while (this.num <= num)
            this.num *= 2;
        item = new T[2 * this.num - 1];
        for (var i = 0; i < 2 * this.num - 1; i++)
            item[i] = init;
    }

    public void Update(int index, T value)
    {
        index += num - 1;
        item[index] = updateFunc(item[index], value);
        while (index > 0)
        {
            index = Parent(index);
            item[index] = func(item[Left(index)], item[Right(index)]);
        }
    }

    public virtual void Update(int left, int right, T value)
        => Update(left, right, 0, 0, num, value);
    protected virtual void Update(int left, int right, int k, int l, int r, T value)
    {
        if (r <= left || right <= l) return;
        if (left <= l && r <= right) item[k] = updateFunc(item[k], value);
        else
        {
            Update(left, right, Left(k), l, (l + r) >> 1, value);
            Update(left, right, Right(k), (l + r) >> 1, r, value);
        }
    }

    public void All_Update()
    {
        for (int i = num - 2; i >= 0; i--)
            item[i] = func(item[Left(i)], item[Right(i)]);
    }

    public T Query(int index)
    {
        index += num - 1;
        var value = func(init, item[index]);
        while (index > 0)
        {
            index = Parent(index);
            value = func(value, item[index]);
        }
        return value;
    }

    public virtual T Query(int left, int right)
        => Query(left, right, 0, 0, num);
    protected virtual T Query(int left, int right, int k, int l, int r)
    {
        if (r <= left || right <= l) return init;
        if (left <= l && r <= right) return item[k];
        else
            return func(Query(left, right, Left(k), l, (l + r) >> 1), Query(left, right, Right(k), (l + r) >> 1, r));
    }

    /// <summary>
    /// check(func(item[st]...item[i]))がtrueとなる最小のi
    /// </summary>
    public int Find(int st, Func<T, bool> check)
    {
        var x = init;
        return Find(st, check, ref x, 0, 0, num);
    }
    private int Find(int st, Func<T, bool> check, ref T x, int k, int l, int r)
    {
        if (l + 1 == r)
        {
            x = func(x, item[k]);
            return check(x) ? k - num + 1 : -1;
        }
        var m = (l + r) >> 1;
        if (m <= st) return Find(st, check, ref x, Right(k), m, r);
        if (st <= l && !check(func(x, item[k])))
        { x = func(x, item[k]); return -1; }
        var xl = Find(st, check, ref x, Left(k), l, m);
        if (xl >= 0) return xl;
        return Find(st, check, ref x, Right(k), m, r);
    }
}

#region Template
public class Template
{
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static bool chmin<T>(ref T num, T val) where T : IComparable<T>
    { if (num.CompareTo(val) == 1) { num = val; return true; } return false; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static bool chmax<T>(ref T num, T val) where T : IComparable<T>
    { if (num.CompareTo(val) == -1) { num = val; return true; } return false; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void swap<T>(ref T v1, ref T v2)
    { var t = v2; v2 = v1; v1 = t; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static T[] Create<T>(int n, Func<T> f)
        => Enumerable.Repeat(0, n).Select(_ => f()).ToArray();
    public static void Fail() => Fail("No");
    public static void Fail<T>(T s) { Console.WriteLine(s); Console.Out.Close(); Environment.Exit(0); }
    static void Main(string[] args)
    {
        var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
        Console.SetOut(sw);
        var p = new Solver();
        for (var i = 1; i > 0; --i)
            p.Solve();
        Console.Out.Flush();
    }
}

public class Input
{
    public static string read => Console.ReadLine().Trim();
    public static int[] ar => read.Split(' ').Select(int.Parse).ToArray();
    public static int num => Convert.ToInt32(read);
    public static long[] arL => read.Split(' ').Select(long.Parse).ToArray();
    public static long numL => Convert.ToInt64(read);
    public static char[][] grid(int h)
        => Create(h, () => read.ToCharArray());
    public static int[] ar1D(int n)
        => Create(n, () => num);
    public static long[] arL1D(int n)
        => Create(n, () => numL);
    public static string[] strs(int n)
        => Create(n, () => read);
    public static int[][] ar2D(int n)
        => Create(n, () => ar);
    public static long[][] arL2D(int n)
        => Create(n, () => arL);
    public static List<T>[] edge<T>(int n)
        => Create(n, () => new List<T>());
    public static void Make<T1, T2>(out T1 v1, out T2 v2)
    {
        v1 = Next<T1>();
        v2 = Next<T2>();
    }
    public static void Make<T1, T2, T3>(out T1 v1, out T2 v2, out T3 v3)
    {
        Make(out v1, out v2);
        v3 = Next<T3>();
    }
    public static void Make<T1, T2, T3, T4>(out T1 v1, out T2 v2, out T3 v3, out T4 v4)
    {
        Make(out v1, out v2, out v3);
        v4 = Next<T4>();
    }
    public static void Make<T1, T2, T3, T4, T5>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5)
    {
        Make(out v1, out v2, out v3, out v4);
        v5 = Next<T5>();
    }
    public static void Make<T1, T2, T3, T4, T5, T6>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5, out T6 v6)
    {
        Make(out v1, out v2, out v3, out v4, out v5);
        v6 = Next<T6>();
    }
    static Input()
    {
        sc = new Queue<string>();
        dic = new Dictionary<Type, Func<string, object>>();
        dic[typeof(int)] = s => int.Parse(s);
        dic[typeof(long)] = s => long.Parse(s);
        dic[typeof(char)] = s => char.Parse(s);
        dic[typeof(double)] = s => double.Parse(s);
        dic[typeof(uint)] = s => uint.Parse(s);
        dic[typeof(ulong)] = s => ulong.Parse(s);
        dic[typeof(string)] = s => s;
    }
    private static Dictionary<Type, Func<string, object>> dic;
    private static Queue<string> sc;
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static T Next<T>() { if (sc.Count == 0) foreach (var item in read.Split(' ')) sc.Enqueue(item); return (T)dic[typeof(T)](sc.Dequeue()); }
    public const int MOD = 1000000007;
}

public class Pair<T1, T2> : IComparable<Pair<T1, T2>>
{
    public T1 v1;
    public T2 v2;
    public Pair() { v1 = default(T1);v2 = default(T2); }
    public Pair(T1 v1, T2 v2)
    { this.v1 = v1; this.v2 = v2; }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public int CompareTo(Pair<T1, T2> p)
    {
        var c = Comparer<T1>.Default.Compare(v1, p.v1);
        if (c == 0)
            c = Comparer<T2>.Default.Compare(v2, p.v2);
        return c;
    }
    public override string ToString()
        => $"{v1.ToString()} {v2.ToString()}";
}

public class Pair<T1, T2, T3> : Pair<T1, T2>, IComparable<Pair<T1, T2, T3>>
{
    public T3 v3;
    public Pair() : base() { v3 = default(T3); }
    public Pair(T1 v1, T2 v2, T3 v3) : base(v1, v2)
    { this.v3 = v3; }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public int CompareTo(Pair<T1, T2, T3> p)
    {
        var c = base.CompareTo(p);
        if (c == 0)
            c = Comparer<T3>.Default.Compare(v3, p.v3);
        return c;
    }
    public override string ToString()
        => $"{base.ToString()} {v3.ToString()}";
}
#endregion