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# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.Globalization;
using System.IO;
using System.Text;
using System.Linq;

using E = System.Linq.Enumerable;
using System.Threading;

// C# 7.2
public static class UpdateExtension {
    public static bool UpdateMax(this ref int x, int newValue) {
        if (x < newValue) { x = newValue; return true; }
        return false;
    }
}

internal partial class Solver {
    public void Run() {
        var n = ni();
        var a = ni(3 * n).Select(x => x - 1).ToArray();
        //n = 2000; a = E.Range(0, 3 * n).Select(x => x % 10 + 1).ToArray();
        var dp = new int[n, n];
        var inf = (int)1e9;
        var max = E.Repeat(-inf, n).ToArray();
        // A B C1 C2 C3
        Fill(n, dp, -inf);
        dp[a[0], a[1]] = 0;
        dp[a[1], a[0]] = 0;
        max[a[0]] = max[a[1]] = 0;
        int baseNumber = 0;

        int maxValue = 0;
        int cnt = 0;
        var plan = new List<(int x, int y, int value)>(10000);
        for (int i = 0; i < n - 1; i++) {
            plan.Clear();
            void UpdatePlan(int x, int y, int value) {
                cnt++;
                plan.Add((x, y, value));
            }

            int card1 = a[3 * i + 2];
            int card2 = a[3 * i + 3];
            int card3 = a[3 * i + 4];
            if (card1 == card2 && card2 == card3) {
                baseNumber++;
            } else {
                // match case
                if (card1 == card2) {
                    for (int j = 0; j < n; j++) {
                        UpdatePlan(card3, j, dp[card1, j] + 1);
                    }
                }
                if (card2 == card3) {
                    for (int j = 0; j < n; j++) {
                        UpdatePlan(card1, j, dp[card2, j] + 1);
                    }
                }
                if (card1 == card3) {
                    for (int j = 0; j < n; j++) {
                        UpdatePlan(card2, j, dp[card3, j] + 1);
                    }
                }
                UpdatePlan(card1, card2, Math.Max(dp[card3, card3] + 1, maxValue));
                UpdatePlan(card2, card3, Math.Max(dp[card1, card1] + 1, maxValue));
                UpdatePlan(card3, card1, Math.Max(dp[card2, card2] + 1, maxValue));
                // other case
                for (int j = 0; j < n; j++) {
                    // (j,k) c1 c2 c3 --> (k,c2,c3)_(j, c1)
                    UpdatePlan(j, card1, max[j]);
                    UpdatePlan(j, card2, max[j]);
                    UpdatePlan(j, card3, max[j]);
                }

                foreach (var (x, y, value) in plan) {
                    if (dp[x, y] < value) {
                        dp[y, x] = dp[x, y] = value;
                        max[x] = Math.Max(max[x], value);
                        max[y] = Math.Max(max[y], value);
                        maxValue = Math.Max(maxValue, value);
                    }
                }
            }
        }
        int ans = 0;
        for (int j = 0; j < n; j++) {
            for (int k = 0; k < n; k++) {
                if (j == k && k == a[a.Length - 1]) {
                    ans = Math.Max(ans, dp[j, k] + 1);
                } else {
                    ans = Math.Max(ans, dp[j, k]);
                }
            }
        }
        cout.WriteLine(ans + baseNumber);
    }

    private static void Fill(int n, int[,] dp, int value) {
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                dp[i, j] = value;
            }
        }
    }
}

// PREWRITEN CODE BEGINS FROM HERE

static public class StringExtensions {
    static public string JoinToString<T>(this IEnumerable<T> source, string separator = " ") {
        return string.Join(separator, source);
    }
}

internal partial class Solver : Scanner {
    static readonly int? StackSizeInMebiByte = null; //50;
    public static void StartAndJoin(Action action, int maxStackSize) {
        var thread = new Thread(new ThreadStart(action), maxStackSize);
        thread.Start();
        thread.Join();
    }

    public static void Main() {
#if LOCAL
        byte[] inputBuffer = new byte[1000000];
        var inputStream = Console.OpenStandardInput(inputBuffer.Length);
        using (var reader = new StreamReader(inputStream, Console.InputEncoding, false, inputBuffer.Length)) {
            Console.SetIn(reader);
            new Solver(Console.In, Console.Out).Run();
        }
#else
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });
        if (StackSizeInMebiByte.HasValue) {
            StartAndJoin(() => new Solver(Console.In, Console.Out).Run(), StackSizeInMebiByte.Value * 1024 * 1024);
        } else {
            new Solver(Console.In, Console.Out).Run();
        }
        Console.Out.Flush();
#endif
    }

#pragma warning disable IDE0052
    private readonly TextReader cin;
    private readonly TextWriter cout;
    private readonly TextWriter cerr;
#pragma warning restore IDE0052

    public Solver(TextReader reader, TextWriter writer)
        : base(reader) {
        cin = reader;
        cout = writer;
        cerr = Console.Error;
    }

    public Solver(string input, TextWriter writer)
        : this(new StringReader(input), writer) {
    }

#pragma warning disable IDE1006
#pragma warning disable IDE0051
    private int ni() { return NextInt(); }
    private int[] ni(int n) { return NextIntArray(n); }
    private long nl() { return NextLong(); }
    private long[] nl(int n) { return NextLongArray(n); }
    private double nd() { return NextDouble(); }
    private double[] nd(int n) { return NextDoubleArray(n); }
    private string ns() { return Next(); }
    private string[] ns(int n) { return NextArray(n); }
#pragma warning restore IDE1006
#pragma warning restore IDE0051
}

#if DEBUG
internal static class LinqPadExtension {
    public static string TextDump<T>(this T obj) {
        if (obj is IEnumerable) return (obj as IEnumerable).Cast<object>().JoinToString().Dump();
        else return obj.ToString().Dump();
    }
    public static T Dump<T>(this T obj) {
        return LINQPad.Extensions.Dump(obj);
    }
}
#endif

public class Scanner {
    private readonly TextReader Reader;
    private readonly CultureInfo ci = CultureInfo.InvariantCulture;

    private readonly char[] buffer = new char[2 * 1024];
    private int cursor = 0, length = 0;
    private string Token;
    private readonly StringBuilder sb = new StringBuilder(1024);

    public Scanner()
        : this(Console.In) {
    }

    public Scanner(TextReader reader) {
        Reader = reader;
    }

    public int NextInt() { return checked((int)NextLong()); }
    public long NextLong() {
        var s = Next();
        long r = 0;
        int i = 0;
        bool negative = false;
        if (s[i] == '-') {
            negative = true;
            i++;
        }
        for (; i < s.Length; i++) {
            r = r * 10 + (s[i] - '0');
#if DEBUG
            if (!char.IsDigit(s[i])) throw new FormatException();
#endif
        }
        return negative ? -r : r;
    }
    public double NextDouble() { return double.Parse(Next(), ci); }
    public string[] NextArray(int size) {
        string[] array = new string[size];
        for (int i = 0; i < size; i++) {
            array[i] = Next();
        }

        return array;
    }
    public int[] NextIntArray(int size) {
        int[] array = new int[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextInt();
        }

        return array;
    }

    public long[] NextLongArray(int size) {
        long[] array = new long[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextLong();
        }

        return array;
    }

    public double[] NextDoubleArray(int size) {
        double[] array = new double[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextDouble();
        }

        return array;
    }

    public string Next() {
        if (Token == null) {
            if (!StockToken()) {
                throw new Exception();
            }
        }
        var token = Token;
        Token = null;
        return token;
    }

    public bool HasNext() {
        if (Token != null) {
            return true;
        }

        return StockToken();
    }

    private bool StockToken() {
        while (true) {
            sb.Clear();
            while (true) {
                if (cursor >= length) {
                    cursor = 0;
                    if ((length = Reader.Read(buffer, 0, buffer.Length)) <= 0) {
                        break;
                    }
                }
                var c = buffer[cursor++];
                if (33 <= c && c <= 126) {
                    sb.Append(c);
                } else {
                    if (sb.Length > 0) break;
                }
            }

            if (sb.Length > 0) {
                Token = sb.ToString();
                return true;
            }

            return false;
        }
    }
}