#include <iostream>
#include <iomanip>
#include <cstdio>
#include <string>
#include <cstring>
#include <deque>
#include <list>
#include <queue>
#include <stack>
#include <vector>
#include <utility>
#include <algorithm>
#include <map>
#include <set>
#include <complex>
#include <cmath>
#include <limits>
#include <cfloat>
#include <climits>
#include <ctime>
#include <cassert>
#include <numeric>
#include <fstream>
#include <functional>
#include <bitset>
#define chmin(a, b) ((a)=min((a), (b)))
#define chmax(a, b) ((a)=max((a), (b)))
#define fs first
#define sc second
#define eb emplace_back
using namespace std;

typedef long long ll;
typedef pair<int, int> P;
typedef tuple<int, int, int> T;

const ll MOD=1e9+7;
const ll INF=1e18;
const double pi=acos(-1);
const double eps=1e-10;

int dx[]={1, 0, -1, 0};
int dy[]={0, -1, 0, 1};

const int MAX_V = 41000;

vector<vector<P>> G(MAX_V);
vector<int> dist(MAX_V, 1e9);
void dijkstra(int s){
    priority_queue<P, vector<P>, greater<P>> que;
    dist[s] = 0;
    que.push(P(0, s));

    while(!que.empty()){
        int ccost, cv;
        tie(ccost, cv) = que.top();
        que.pop();

        if(dist[cv] < ccost) continue;

        for(auto x : G[cv]){
            int nv = x.first;
            int ncost = x.second;
            if(dist[cv] + ncost < dist[nv]){
                dist[nv] = dist[cv] + ncost;
                que.push(P(dist[nv], nv));
            }
        }
    }
}

int main(){
    int n; cin>>n;
    vector<vector<int>> c(n, vector<int>(n));
    for(int i=0; i<n; i++){
        for(int j=0; j<n; j++){
            cin>>c[i][j];
        }
    }

    vector<int> v(n);
    iota(v.begin(), v.end(), 0);

    map<vector<int>, int> node;
    int cnt = 0;
    do{
        node[v] = cnt++; 
    }while(next_permutation(v.begin(), v.end()));

    sort(v.begin(), v.end());

    do{
        int cv = node[v];
        for(int i=0; i<n; i++){
            for(int j=i+1; j<n; j++){
                swap(v[i], v[j]);
                int nv = node[v];
                G[cv].push_back(P(nv, c[v[i]][v[j]]));
                G[nv].push_back(P(cv, c[v[i]][v[j]]));
                swap(v[i], v[j]);
            }
        }
    }while(next_permutation(v.begin(), v.end()));

    dijkstra(0);

    int ans = 0;
    for(int i=0; i<cnt; i++){
        ans = max(ans, dist[i]);
    }

    cout << ans << endl;
}

### Prompt

Please formulate a Cpp solution to the following problem:

Problem statement

There is a permutation with $ 1,2, ..., N $ sorted. I want to select two different numbers $ i $, $ j $ and replace them repeatedly to make them sorted (in the order of $ 1,2, ..., N $). Every time you replace the numbers $ i $, $ j $, you need $ c_ {i, j} $.

Let $ f (p) $ be the minimum cost required to sort $ p $ for the permutation $ p $. Find the maximum possible value of $ f (p) $.

Constraint

* $ 2 \ leq N \ leq 8 $
* $ 0 \ leq c_ {i, j} \ leq 10 ^ 5 $
* $ c_ {i, j} = c_ {j, i} $
* $ c_ {i, i} = 0 $



input

Input follows the following format. All given numbers are integers.


$ N $
$ c_ {1,1} $ $ c_ {1,2} $ $ ... $ $ c_ {1, N} $
$ c_ {2,1} $ $ c_ {2,2} $ $ ... $ $ c_ {2, N} $
$ ... $
$ c_ {N, 1} $ $ c_ {N, 2} $ $ ... $ $ c_ {N, N} $

output

Output the maximum value on one line.

Example

Input

3
0 1 3
1 0 8
3 8 0


Output

5

### Response

```cpp
#include <iostream>
#include <iomanip>
#include <cstdio>
#include <string>
#include <cstring>
#include <deque>
#include <list>
#include <queue>
#include <stack>
#include <vector>
#include <utility>
#include <algorithm>
#include <map>
#include <set>
#include <complex>
#include <cmath>
#include <limits>
#include <cfloat>
#include <climits>
#include <ctime>
#include <cassert>
#include <numeric>
#include <fstream>
#include <functional>
#include <bitset>
#define chmin(a, b) ((a)=min((a), (b)))
#define chmax(a, b) ((a)=max((a), (b)))
#define fs first
#define sc second
#define eb emplace_back
using namespace std;

typedef long long ll;
typedef pair<int, int> P;
typedef tuple<int, int, int> T;

const ll MOD=1e9+7;
const ll INF=1e18;
const double pi=acos(-1);
const double eps=1e-10;

int dx[]={1, 0, -1, 0};
int dy[]={0, -1, 0, 1};

const int MAX_V = 41000;

vector<vector<P>> G(MAX_V);
vector<int> dist(MAX_V, 1e9);
void dijkstra(int s){
    priority_queue<P, vector<P>, greater<P>> que;
    dist[s] = 0;
    que.push(P(0, s));

    while(!que.empty()){
        int ccost, cv;
        tie(ccost, cv) = que.top();
        que.pop();

        if(dist[cv] < ccost) continue;

        for(auto x : G[cv]){
            int nv = x.first;
            int ncost = x.second;
            if(dist[cv] + ncost < dist[nv]){
                dist[nv] = dist[cv] + ncost;
                que.push(P(dist[nv], nv));
            }
        }
    }
}

int main(){
    int n; cin>>n;
    vector<vector<int>> c(n, vector<int>(n));
    for(int i=0; i<n; i++){
        for(int j=0; j<n; j++){
            cin>>c[i][j];
        }
    }

    vector<int> v(n);
    iota(v.begin(), v.end(), 0);

    map<vector<int>, int> node;
    int cnt = 0;
    do{
        node[v] = cnt++; 
    }while(next_permutation(v.begin(), v.end()));

    sort(v.begin(), v.end());

    do{
        int cv = node[v];
        for(int i=0; i<n; i++){
            for(int j=i+1; j<n; j++){
                swap(v[i], v[j]);
                int nv = node[v];
                G[cv].push_back(P(nv, c[v[i]][v[j]]));
                G[nv].push_back(P(cv, c[v[i]][v[j]]));
                swap(v[i], v[j]);
            }
        }
    }while(next_permutation(v.begin(), v.end()));

    dijkstra(0);

    int ans = 0;
    for(int i=0; i<cnt; i++){
        ans = max(ans, dist[i]);
    }

    cout << ans << endl;
}

```