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++ (GCC 9.2.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
using namespace std;
int n,dp[2003][2003],a[6003],ans,tg,MX[2003],cur;
vector<int>A,B,C;
void add(int x,int y,int z){A.push_back(x),B.push_back(y),C.push_back(z);}
int main(){
	cin>>n;
	for(int i=1;i<=3*n;i++)cin>>a[i]; 
	memset(dp,-1,sizeof(dp));
	memset(MX,-1,sizeof(MX));
	dp[a[1]][a[2]]=dp[a[2]][a[1]]=MX[a[1]]=MX[a[2]]=cur=0;
	for(int i=3;i<3*n;i+=3){
		if(a[i]==a[i+1]&&a[i]==a[i+2]){tg++;continue;}
		A.clear(),B.clear(),C.clear();
		sort(a+i,a+i+3);
		if(a[i]==a[i+1])for(int j=1;j<=n;j++)if(dp[j][a[i]]>=0)add(j,a[i+2],dp[j][a[i]]+1);
		if(a[i+1]==a[i+2])for(int j=1;j<=n;j++)if(dp[j][a[i+1]]>=0)add(j,a[i],dp[j][a[i+1]]+1);
		if(dp[a[i]][a[i]]>=0)add(a[i+1],a[i+2],dp[a[i]][a[i]]+1);
		if(dp[a[i+1]][a[i+1]]>=0)add(a[i],a[i+2],dp[a[i+1]][a[i+1]]+1);
		if(dp[a[i+2]][a[i+2]]>=0)add(a[i],a[i+1],dp[a[i+2]][a[i+2]]+1);
		add(a[i],a[i+1],cur),add(a[i+1],a[i+2],cur),add(a[i],a[i+2],cur);
		for(int j=i;j<i+3;j++)for(int k=1;k<=n;k++)if(MX[k]>=0)add(a[j],k,MX[k]);
		for(int j=0;j<(int)A.size();j++)if(dp[A[j]][B[j]]<C[j])
			dp[A[j]][B[j]]=dp[B[j]][A[j]]=C[j],MX[A[j]]=max(MX[A[j]],C[j]),MX[B[j]]=max(MX[B[j]],C[j]),cur=max(cur,C[j]);

	}
	
	for(int i=1;i<=n;i++){
		for(int j=i+1;j<=n;j++)if(dp[i][j]>=0)ans=max(ans,dp[i][j]);
		if(dp[i][i]>=0)if(i==a[3*n])ans=max(ans,dp[i][i]+1);
	}
	printf("%d",ans+tg);
}