Problem:
Score : 600 points

Problem Statement

We have 3N cards arranged in a row from left to right, where each card has an integer between 1 and N (inclusive) written on it. The integer written on the i-th card from the left is A_i.

You will do the following operation N-1 times:

Rearrange the five leftmost cards in any order you like, then remove the three leftmost cards. If the integers written on those three cards are all equal, you gain 1 point.

After these N-1 operations, if the integers written on the remaining three cards are all equal, you will gain 1 additional point.

Find the maximum number of points you can gain.

Constraints

1 \leq N \leq 2000

1 \leq A_i \leq N

Input

Input is given from Standard Input in the following format:

N
A_1 A_2 \cdots A_{3N}

Output

Print the maximum number of points you can gain.

Sample Input 1

2
1 2 1 2 2 1

Sample Output 1

2

Let us rearrange the five leftmost cards so that the integers written on the six cards will be 2\ 2\ 2\ 1\ 1\ 1 from left to right.

Then, remove the three leftmost cards, all of which have the same integer 2, gaining 1 point.

Now, the integers written on the remaining cards are 1\ 1\ 1.

Since these three cards have the same integer 1, we gain 1 more point.

In this way, we can gain 2 points - which is the maximum possible.

Sample Input 2

3
1 1 2 2 3 3 3 2 1

Sample Output 2

1

Sample Input 3

3
1 1 2 2 2 3 3 3 1

Sample Output 3

3

Solution in C# (Mono-csc 3.5.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using System.Threading;
using System.Text;
using System.Text.RegularExpressions;
using System.Diagnostics;
using static util;
using P = pair<int, int>;

class Program {
    static void Main(string[] args) {
        var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
        var solver = new Solver(sw);
        // var t = new Thread(solver.solve, 1 << 28); // 256 MB
        // t.Start();
        // t.Join();
        solver.solve();
        sw.Flush();
    }
}

class Solver {
    StreamWriter sw;
    Scan sc;
    void Prt(string a) => sw.WriteLine(a);
    void Prt<T>(IEnumerable<T> a) => Prt(string.Join(" ", a));
    void Prt(params object[] a) => Prt(string.Join(" ", a));
    public Solver(StreamWriter sw) {
        this.sw = sw;
        this.sc = new Scan();
    }

    public void solve() {
        int n = sc.Int;
        var a = sc.IntArr.Select(x => x - 1).ToArray();
        var dp = new int[n][];
        var max = new int[n];
        for (int i = 0; i < n; i++)
        {
            dp[i] = new int[n];
            for (int j = 0; j < n; j++)
            {
                dp[i][j] = -M;
            }
            max[i] = -M;
        }
        dp[a[0]][a[1]] = 0;
        dp[a[1]][a[0]] = 0;
        max[a[0]] = 0;
        max[a[1]] = 0;
        int pl = 0;
        for (int i = 0; i < n - 1; i++)
        {
            int[] b = { a[2 + i * 3], a[2 + i * 3 + 1], a[2 + i * 3 + 2] };
            {
                // 0
                if (b[0] == b[1] && b[1] == b[2]) {
                    ++pl;
                    continue;
                }
            }
            var nex1 = new int[3][];
            {
                // 1
                for (int j = 0; j < 3; j++)
                {
                    nex1[j] = new int[n];
                    int o = b[(j + 1) % 3] == b[(j + 2) % 3] ? b[(j + 1) % 3] : -1;
                    for (int k = 0; k < n; k++)
                    {
                        nex1[j][k] = max[k];
                        if (o != -1) {
                            nex1[j][k] = Math.Max(nex1[j][k], dp[o][k] + 1);
                        }
                    }
                }
            }
            var nex2 = new int[3];
            {
                // 2
                for (int j = 0; j < 3; j++)
                {
                    int b1 = b[(j + 1) % 3];
                    int b2 = b[(j + 2) % 3];
                    for (int k = 0; k < n; k++)
                    {
                        nex2[j] = Math.Max(nex2[j], max[k]);
                    }
                    nex2[j] = Math.Max(nex2[j], dp[b[j]][b[j]] + 1);
                }
            }
            {
                // 1
                for (int j = 0; j < 3; j++)
                {
                    for (int k = 0; k < n; k++)
                    {
                        dp[b[j]][k] = Math.Max(dp[b[j]][k], nex1[j][k]);
                        dp[k][b[j]] = Math.Max(dp[k][b[j]], nex1[j][k]);
                        max[b[j]] = Math.Max(max[b[j]], dp[b[j]][k]);
                        max[k] = Math.Max(max[k], dp[b[j]][k]);
                    }
                }
            }
            {
                // 2
                for (int j = 0; j < 3; j++)
                {
                    int b1 = b[(j + 1) % 3];
                    int b2 = b[(j + 2) % 3];
                    dp[b1][b2] = Math.Max(dp[b1][b2], nex2[j]);
                    dp[b2][b1] = Math.Max(dp[b2][b1], nex2[j]);
                    max[b1] = Math.Max(max[b1], dp[b1][b2]);
                    max[b2] = Math.Max(max[b2], dp[b1][b2]);
                }
            }
        }
        int ans = 0;
        for (int i = 0; i < n; i++)
        {
            for (int j = 0; j < n; j++)
            {
                if (i == j && i == a.Last()) {
                    ans = Math.Max(ans, dp[i][j] + 1);
                }
                else {
                    ans = Math.Max(ans, dp[i][j]);
                }
            }
        }
        Prt(ans + pl);

    }
}

class pair<T, U> : IComparable<pair<T, U>> {
    public T v1;
    public U v2;
    public pair() : this(default(T), default(U)) {}
    public pair(T v1, U v2) { this.v1 = v1; this.v2 = v2; }
    public int CompareTo(pair<T, U> a) {
        int c = Comparer<T>.Default.Compare(v1, a.v1);
        return c != 0 ? c : Comparer<U>.Default.Compare(v2, a.v2);
    }
    public override string ToString() => v1 + " " + v2;
    public void Deconstruct(out T a, out U b) { a = v1; b = v2; }
}
static class util {
    public static readonly int M = 1000000007;
    // public static readonly int M = 998244353;
    public static readonly long LM = 1L << 60;
    public static readonly double eps = 1e-11;
    public static void DBG(string a) => Console.Error.WriteLine(a);
    public static void DBG<T>(IEnumerable<T> a) => DBG(string.Join(" ", a));
    public static void DBG(params object[] a) => DBG(string.Join(" ", a));
    public static void Assert(params bool[] conds) {
        if (conds.Any(x => !x)) throw new Exception();
    }
    public static pair<T, U> make_pair<T, U>(T v1, U v2) => new pair<T, U>(v1, v2);
    public static int CompareList<T>(IList<T> a, IList<T> b) where T : IComparable<T> {
        for (int i = 0; i < a.Count && i < b.Count; i++)
            if (a[i].CompareTo(b[i]) != 0) return a[i].CompareTo(b[i]);
        return a.Count.CompareTo(b.Count);
    }
    public static bool inside(int i, int j, int h, int w) => i >= 0 && i < h && j >= 0 && j < w;
    public static readonly int[] dd = { 0, 1, 0, -1 };
    // static readonly string dstring = "RDLU";
    public static IEnumerable<P> adjacents(int i, int j)
        => Enumerable.Range(0, dd.Length).Select(k => new P(i + dd[k], j + dd[k ^ 1]));
    public static IEnumerable<P> adjacents(int i, int j, int h, int w)
        => adjacents(i, j).Where(p => inside(p.v1, p.v2, h, w));
    public static IEnumerable<P> adjacents(this P p) => adjacents(p.v1, p.v2);
    public static IEnumerable<P> adjacents(this P p, int h, int w) => adjacents(p.v1, p.v2, h, w);
    public static IEnumerable<int> all_subset(this int p) {
        for (int i = 0; ; i = i - p & p) {
            yield return i;
            if (i == p) break;
        }
    }
    public static Dictionary<T, int> compress<T>(this IEnumerable<T> a)
        => a.Distinct().OrderBy(v => v).Select((v, i) => new { v, i }).ToDictionary(p => p.v, p => p.i);
    public static Dictionary<T, int> compress<T>(params IEnumerable<T>[] a) => compress(a.SelectMany(x => x));
    public static T[] inv<T>(this Dictionary<T, int> dic) {
        var res = new T[dic.Count];
        foreach (var item in dic) res[item.Value] = item.Key;
        return res;
    }
    public static void swap<T>(ref T a, ref T b) where T : struct { var t = a; a = b; b = t; }
    public static void swap<T>(this IList<T> a, int i, int j) where T : struct { var t = a[i]; a[i] = a[j]; a[j] = t; }
    public static T[] copy<T>(this IList<T> a) {
        var ret = new T[a.Count];
        for (int i = 0; i < a.Count; i++) ret[i] = a[i];
        return ret;
    }
}

class Scan {
    StreamReader sr;
    public Scan() { sr = new StreamReader(Console.OpenStandardInput()); }
    public Scan(string path) { sr = new StreamReader(path); }
    public int Int => int.Parse(Str);
    public long Long => long.Parse(Str);
    public double Double => double.Parse(Str);
    public string Str => ReadLine.Trim();
    public string ReadLine => sr.ReadLine();
    public pair<T, U> Pair<T, U>() {
        T a; U b;
        Multi(out a, out b);
        return new pair<T, U>(a, b);
    }
    public P P => Pair<int, int>();
    public int[] IntArr => StrArr.Select(int.Parse).ToArray();
    public long[] LongArr => StrArr.Select(long.Parse).ToArray();
    public double[] DoubleArr => StrArr.Select(double.Parse).ToArray();
    public string[] StrArr => Str.Split(new[]{' '}, StringSplitOptions.RemoveEmptyEntries);
    bool eq<T, U>() => typeof(T).Equals(typeof(U));
    T ct<T, U>(U a) => (T)Convert.ChangeType(a, typeof(T));
    T cv<T>(string s) => eq<T, int>()    ? ct<T, int>(int.Parse(s))
                       : eq<T, long>()   ? ct<T, long>(long.Parse(s))
                       : eq<T, double>() ? ct<T, double>(double.Parse(s))
                       : eq<T, char>()   ? ct<T, char>(s[0])
                                         : ct<T, string>(s);
    public void Multi<T>(out T a) => a = cv<T>(Str);
    public void Multi<T, U>(out T a, out U b) {
        var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]);
    }
    public void Multi<T, U, V>(out T a, out U b, out V c) {
        var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]);
    }
    public void Multi<T, U, V, W>(out T a, out U b, out V c, out W d) {
        var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]); d = cv<W>(ar[3]);
    }
    public void Multi<T, U, V, W, X>(out T a, out U b, out V c, out W d, out X e) {
        var ar = StrArr; a = cv<T>(ar[0]); b = cv<U>(ar[1]); c = cv<V>(ar[2]); d = cv<W>(ar[3]); e = cv<X>(ar[4]);
    }
}