#include <cassert>// c
#include <iostream>// io
#include <iomanip>
#include <fstream>
#include <sstream>
#include <vector>// container
#include <map>
#include <set>
#include <queue>
#include <bitset>
#include <stack>
#include <algorithm>// other
#include <complex>
#include <numeric>
#include <functional>
#include <random>
#include <regex>
using namespace std;

typedef long long ll;

#define ALL(c) (begin(c)),(end(c))
#define REP(i,n) FOR(i,0,n)
#define REPr(i,n) FORr(i,0,n)
#define FOR(i,l,r) for(int i=(int)(l);i<(int)(r);++i)
#define FORr(i,l,r) for(int i=(int)(r)-1;i>=(int)(l);--i)
#define EACH(it,o) for(auto it = (o).begin(); it != (o).end(); ++it)
#define IN(l,v,r) ((l)<=(v) && (v)<(r))
#define UNIQUE(v) v.erase(unique(ALL(v)),v.end())
//debug
#define DUMP(x)  cerr << #x << " = " << (x)
#define LINE()    cerr<< " (L" << __LINE__ << ")"

class range {
private:
    struct Iter{
        int v;
        int operator*(){return v;}
        bool operator!=(Iter& itr) {return v < itr.v;}
        void operator++() {++v;}
    };
    Iter i, n;
public:
    range(int n) : i({0}), n({n}) {}
    range(int i, int n) : i({i}), n({n}) {}
    Iter& begin() {return i;}
    Iter& end() {return n;}
};

//output
template<typename T> ostream& operator << (ostream& os, const vector<T>& as){REP(i,as.size()){if(i!=0)os<<" "; os<<as[i];}return os;}
template<typename T> ostream& operator << (ostream& os, const vector<vector<T>>& as){REP(i,as.size()){if(i!=0)os<<endl; os<<as[i];}return os;}
template<typename T> ostream& operator << (ostream& os, const set<T>& ss){for(auto a:ss){if(a!=ss.begin())os<<" "; os<<a;}return os;}
template<typename T1,typename T2> ostream& operator << (ostream& os, const pair<T1,T2>& p){os<<p.first<<" "<<p.second;return os;}
template<typename K,typename V> ostream& operator << (ostream& os, const map<K,V>& m){bool isF=true;for(auto& p:m){if(!isF)os<<endl;os<<p;isF=false;}return os;}
template<typename T1> ostream& operator << (ostream& os, const tuple<T1>& t){os << get<0>(t);return os;}
template<typename T1,typename T2> ostream& operator << (ostream& os, const tuple<T1,T2>& t){os << get<0>(t)<<" "<<get<1>(t);return os;}
template<typename T1,typename T2,typename T3> ostream& operator << (ostream& os, const tuple<T1,T2,T3>& t){os << get<0>(t)<<" "<<get<1>(t)<<" "<<get<2>(t);return os;}
template<typename T1,typename T2,typename T3,typename T4> ostream& operator << (ostream& os, const tuple<T1,T2,T3,T4>& t){os << get<0>(t)<<" "<<get<1>(t)<<" "<<get<2>(t)<<" "<<get<3>(t);return os;}
template<typename T1,typename T2,typename T3,typename T4,typename T5> ostream& operator << (ostream& os, const tuple<T1,T2,T3,T4,T5>& t){os << get<0>(t)<<" "<<get<1>(t)<<" "<<get<2>(t)<<" "<<get<3>(t)<<" "<<get<4>(t);return os;}
template<typename T1,typename T2,typename T3,typename T4,typename T5,typename T6> ostream& operator << (ostream& os, const tuple<T1,T2,T3,T4,T5,T6>& t){os << get<0>(t)<<" "<<get<1>(t)<<" "<<get<2>(t)<<" "<<get<3>(t)<<" "<<get<4>(t)<<" "<<get<5>(t);return os;}
template<typename T1,typename T2,typename T3,typename T4,typename T5,typename T6,typename T7> ostream& operator << (ostream& os, const tuple<T1,T2,T3,T4,T5,T6,T7>& t){os << get<0>(t)<<" "<<get<1>(t)<<" "<<get<2>(t)<<" "<<get<3>(t)<<" "<<get<4>(t)<<" "<<get<5>(t)<<" "<<get<6>(t);return os;}

//input
char tmp[1000];
#define nextInt(n) scanf("%d",&n)
#define nextLong(n) scanf("%lld",&n) //I64d
#define nextDouble(n) scanf("%lf",&n) 
#define nextChar(n) scanf("%c",&n)
#define nextString(n) scanf("%s",tmp);n=tmp

// values
template<typename T> T INF(){assert(false);};
template<> int INF<int>(){return 1<<28;};
template<> ll INF<ll>(){return 1LL<<58;};
template<> float INF<float>(){return 1e10;};
template<> double INF<double>(){return 1e16;};
template<> long double INF<long double>(){return 1e16;};

template<class T> T EPS(){assert(false);};
template<> int EPS<int>(){return 1;};
template<> ll EPS<ll>(){return 1LL;};
template<> float EPS<float>(){return 1e-8;};
template<> double EPS<double>(){return 1e-8;};
template<> long double EPS<long double>(){return 1e-8;};

template<typename T,typename U> T pmod(T v,U M){return (v%M+M)%M;}


namespace EGraph{
    typedef ll Cost;Cost CINF=INF<Cost>();
    struct Edge{
        int from,to;Cost cost;
        Edge(int from,int to,Cost cost)
        : from(from),to(to),cost(cost) {};
        bool operator<(Edge r) const{ return cost<r.cost;}
        bool operator>(Edge r) const{ return cost>r.cost;}
    };
    typedef vector<vector<Edge> >  Graph;
}
using namespace EGraph;

namespace ShortestPath{
    using namespace EGraph;

    struct Task{
        int prev,pos;Cost cost;
        Task(int prev,int pos,Cost cost)
            :prev(prev),pos(pos),cost(cost){};
        bool operator>(const Task& r) const{ return cost > r.cost;}
    };

    //verified by codoforces 144D http://codeforces.com/contest/144/submission/4976825
    // // 負の辺がない
    // // O(E*logV)
    vector<Cost> dijkstra(const Graph& g,const int s,vector<int>& prev){
        const int V=g.size();
        vector<Cost> d(V,CINF);d[s]=0;
        fill(ALL(prev), -2);
        
        priority_queue<Task,vector<Task>,greater<Task> > que;que.push(Task(-1,s,0));// [ ,e,,f, ] <=> e.cost < e.cost
        vector<bool> visited(V);
        while(!que.empty()){
            Task task=que.top();que.pop();
            if(visited[task.pos])continue;
            visited[task.pos]=true;
            prev[task.pos]=task.prev;
            EACH(e,g[task.pos])if(d[e->to]>d[e->from]+e->cost){
                d[e->to]=d[e->from]+e->cost;
                que.push(Task(e->from,e->to,d[e->to]));
            }   
        }
        return d;
    }
    vector<Cost> dijkstra(const Graph& g,const int s){
        vector<int> prev(g.size());return dijkstra(g,s,prev);
    }
}
using namespace ShortestPath;

class Main{
    public:

    int N;
    vector<vector<int>> cs;
        
    map<vector<short>,int> enc;
    // map<int,vector<int>> dec;
    
    void run(){
        cin >> N;
        cs=vector<vector<int>>(N,vector<int>(N));
        REP(y,N)REP(x,N)cin >> cs[y][x];

        vector<short> per(N);iota(ALL(per),0);
        int ind=0;
        do{
            enc[per]=ind;
            // dec[ind]=per;
            int te=0;REP(i,N)for(int j=i+1;j<N;j++)if(per[i]>per[j])te++;
            ind++;
        }while(next_permutation(ALL(per)));
        
        Graph g(ind);
        int code=0;iota(ALL(per),0);
        do{
            REP(i,N)for(int j=i+1;j<N;j++){
                swap(per[i],per[j]);
                int ncode=enc[per];
                g[code].push_back(Edge(code,ncode,cs[per[i]][per[j]]));
                 swap(per[i],per[j]);
            }
            code++;
        }while(next_permutation(ALL(per)));
        
        vector<Cost> ds=dijkstra(g,0);
        ll Mv=0;REP(code,ind)Mv=max(Mv,ds[code]);
        cout << Mv <<endl;
    }
};

int main(){
    cout <<fixed<<setprecision(20);
    cin.tie(0);
    ios::sync_with_stdio(false);
    Main().run();
    return 0;
}
### Prompt

Generate 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 <cassert>// c
#include <iostream>// io
#include <iomanip>
#include <fstream>
#include <sstream>
#include <vector>// container
#include <map>
#include <set>
#include <queue>
#include <bitset>
#include <stack>
#include <algorithm>// other
#include <complex>
#include <numeric>
#include <functional>
#include <random>
#include <regex>
using namespace std;

typedef long long ll;

#define ALL(c) (begin(c)),(end(c))
#define REP(i,n) FOR(i,0,n)
#define REPr(i,n) FORr(i,0,n)
#define FOR(i,l,r) for(int i=(int)(l);i<(int)(r);++i)
#define FORr(i,l,r) for(int i=(int)(r)-1;i>=(int)(l);--i)
#define EACH(it,o) for(auto it = (o).begin(); it != (o).end(); ++it)
#define IN(l,v,r) ((l)<=(v) && (v)<(r))
#define UNIQUE(v) v.erase(unique(ALL(v)),v.end())
//debug
#define DUMP(x)  cerr << #x << " = " << (x)
#define LINE()    cerr<< " (L" << __LINE__ << ")"

class range {
private:
    struct Iter{
        int v;
        int operator*(){return v;}
        bool operator!=(Iter& itr) {return v < itr.v;}
        void operator++() {++v;}
    };
    Iter i, n;
public:
    range(int n) : i({0}), n({n}) {}
    range(int i, int n) : i({i}), n({n}) {}
    Iter& begin() {return i;}
    Iter& end() {return n;}
};

//output
template<typename T> ostream& operator << (ostream& os, const vector<T>& as){REP(i,as.size()){if(i!=0)os<<" "; os<<as[i];}return os;}
template<typename T> ostream& operator << (ostream& os, const vector<vector<T>>& as){REP(i,as.size()){if(i!=0)os<<endl; os<<as[i];}return os;}
template<typename T> ostream& operator << (ostream& os, const set<T>& ss){for(auto a:ss){if(a!=ss.begin())os<<" "; os<<a;}return os;}
template<typename T1,typename T2> ostream& operator << (ostream& os, const pair<T1,T2>& p){os<<p.first<<" "<<p.second;return os;}
template<typename K,typename V> ostream& operator << (ostream& os, const map<K,V>& m){bool isF=true;for(auto& p:m){if(!isF)os<<endl;os<<p;isF=false;}return os;}
template<typename T1> ostream& operator << (ostream& os, const tuple<T1>& t){os << get<0>(t);return os;}
template<typename T1,typename T2> ostream& operator << (ostream& os, const tuple<T1,T2>& t){os << get<0>(t)<<" "<<get<1>(t);return os;}
template<typename T1,typename T2,typename T3> ostream& operator << (ostream& os, const tuple<T1,T2,T3>& t){os << get<0>(t)<<" "<<get<1>(t)<<" "<<get<2>(t);return os;}
template<typename T1,typename T2,typename T3,typename T4> ostream& operator << (ostream& os, const tuple<T1,T2,T3,T4>& t){os << get<0>(t)<<" "<<get<1>(t)<<" "<<get<2>(t)<<" "<<get<3>(t);return os;}
template<typename T1,typename T2,typename T3,typename T4,typename T5> ostream& operator << (ostream& os, const tuple<T1,T2,T3,T4,T5>& t){os << get<0>(t)<<" "<<get<1>(t)<<" "<<get<2>(t)<<" "<<get<3>(t)<<" "<<get<4>(t);return os;}
template<typename T1,typename T2,typename T3,typename T4,typename T5,typename T6> ostream& operator << (ostream& os, const tuple<T1,T2,T3,T4,T5,T6>& t){os << get<0>(t)<<" "<<get<1>(t)<<" "<<get<2>(t)<<" "<<get<3>(t)<<" "<<get<4>(t)<<" "<<get<5>(t);return os;}
template<typename T1,typename T2,typename T3,typename T4,typename T5,typename T6,typename T7> ostream& operator << (ostream& os, const tuple<T1,T2,T3,T4,T5,T6,T7>& t){os << get<0>(t)<<" "<<get<1>(t)<<" "<<get<2>(t)<<" "<<get<3>(t)<<" "<<get<4>(t)<<" "<<get<5>(t)<<" "<<get<6>(t);return os;}

//input
char tmp[1000];
#define nextInt(n) scanf("%d",&n)
#define nextLong(n) scanf("%lld",&n) //I64d
#define nextDouble(n) scanf("%lf",&n) 
#define nextChar(n) scanf("%c",&n)
#define nextString(n) scanf("%s",tmp);n=tmp

// values
template<typename T> T INF(){assert(false);};
template<> int INF<int>(){return 1<<28;};
template<> ll INF<ll>(){return 1LL<<58;};
template<> float INF<float>(){return 1e10;};
template<> double INF<double>(){return 1e16;};
template<> long double INF<long double>(){return 1e16;};

template<class T> T EPS(){assert(false);};
template<> int EPS<int>(){return 1;};
template<> ll EPS<ll>(){return 1LL;};
template<> float EPS<float>(){return 1e-8;};
template<> double EPS<double>(){return 1e-8;};
template<> long double EPS<long double>(){return 1e-8;};

template<typename T,typename U> T pmod(T v,U M){return (v%M+M)%M;}


namespace EGraph{
    typedef ll Cost;Cost CINF=INF<Cost>();
    struct Edge{
        int from,to;Cost cost;
        Edge(int from,int to,Cost cost)
        : from(from),to(to),cost(cost) {};
        bool operator<(Edge r) const{ return cost<r.cost;}
        bool operator>(Edge r) const{ return cost>r.cost;}
    };
    typedef vector<vector<Edge> >  Graph;
}
using namespace EGraph;

namespace ShortestPath{
    using namespace EGraph;

    struct Task{
        int prev,pos;Cost cost;
        Task(int prev,int pos,Cost cost)
            :prev(prev),pos(pos),cost(cost){};
        bool operator>(const Task& r) const{ return cost > r.cost;}
    };

    //verified by codoforces 144D http://codeforces.com/contest/144/submission/4976825
    // // 負の辺がない
    // // O(E*logV)
    vector<Cost> dijkstra(const Graph& g,const int s,vector<int>& prev){
        const int V=g.size();
        vector<Cost> d(V,CINF);d[s]=0;
        fill(ALL(prev), -2);
        
        priority_queue<Task,vector<Task>,greater<Task> > que;que.push(Task(-1,s,0));// [ ,e,,f, ] <=> e.cost < e.cost
        vector<bool> visited(V);
        while(!que.empty()){
            Task task=que.top();que.pop();
            if(visited[task.pos])continue;
            visited[task.pos]=true;
            prev[task.pos]=task.prev;
            EACH(e,g[task.pos])if(d[e->to]>d[e->from]+e->cost){
                d[e->to]=d[e->from]+e->cost;
                que.push(Task(e->from,e->to,d[e->to]));
            }   
        }
        return d;
    }
    vector<Cost> dijkstra(const Graph& g,const int s){
        vector<int> prev(g.size());return dijkstra(g,s,prev);
    }
}
using namespace ShortestPath;

class Main{
    public:

    int N;
    vector<vector<int>> cs;
        
    map<vector<short>,int> enc;
    // map<int,vector<int>> dec;
    
    void run(){
        cin >> N;
        cs=vector<vector<int>>(N,vector<int>(N));
        REP(y,N)REP(x,N)cin >> cs[y][x];

        vector<short> per(N);iota(ALL(per),0);
        int ind=0;
        do{
            enc[per]=ind;
            // dec[ind]=per;
            int te=0;REP(i,N)for(int j=i+1;j<N;j++)if(per[i]>per[j])te++;
            ind++;
        }while(next_permutation(ALL(per)));
        
        Graph g(ind);
        int code=0;iota(ALL(per),0);
        do{
            REP(i,N)for(int j=i+1;j<N;j++){
                swap(per[i],per[j]);
                int ncode=enc[per];
                g[code].push_back(Edge(code,ncode,cs[per[i]][per[j]]));
                 swap(per[i],per[j]);
            }
            code++;
        }while(next_permutation(ALL(per)));
        
        vector<Cost> ds=dijkstra(g,0);
        ll Mv=0;REP(code,ind)Mv=max(Mv,ds[code]);
        cout << Mv <<endl;
    }
};

int main(){
    cout <<fixed<<setprecision(20);
    cin.tie(0);
    ios::sync_with_stdio(false);
    Main().run();
    return 0;
}
```