// #define _GLIBCXX_DEBUG // for STL debug (optional)
#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>
using namespace std;

#define debug(...) fprintf(stderr, __VA_ARGS__)
#define int long long int
 
template<typename T> void chmax(T &a, T b) {a = max(a, b);}
template<typename T> void chmin(T &a, T b) {a = min(a, b);}
template<typename T> void chadd(T &a, T b) {a = a + b;}
 
typedef pair<int, int> pii;
typedef long long ll;
 
int dx[] = {0, 0, 1, -1};
int dy[] = {1, -1, 0, 0};
const ll INF = 1001001001001001LL;
const ll MOD = 1000000007LL;

map< vector<int>, int > memo;

struct Elem {
    vector<int> vec;
    int cost;
    Elem() {}
    Elem(vector<int> v, int c) : vec(v), cost(c) {}
    bool operator<(const Elem &e) const {
        return cost > e.cost;
    }
};

int cost[10][10];
signed main() {
    int N; cin >> N;
    for(int i=0; i<N; i++) {
        for(int j=0; j<N; j++) {
            cin >> cost[i][j];
        }
    }

    priority_queue<Elem> que;
    vector<int> init(N);
    iota(init.begin(), init.end(), 0);
    memo[init] = 0;
    que.push(Elem(init, 0));
    
    while(que.size()) {
        auto e = que.top(); que.pop();
        auto vec = e.vec;
        auto orig = e.cost;
        if(orig > memo[vec]) continue;

        for(int i=0; i<N; i++) {
            for(int j=i+1; j<N; j++) {
                int vl = vec[i], vr = vec[j];
                int c = cost[vl][vr];

                swap(vec[i], vec[j]);
                if(!memo.count(vec) or memo[vec] > orig + c) {
                    memo[vec] = orig + c;
                    que.push(Elem(vec, orig + c));
                }
                swap(vec[i], vec[j]);
            }
        }
    }

    int ans = 0;
    for(auto v : memo) ans = max(ans, v.second);
    cout << ans << endl;
    return 0;
}


### Prompt

Please provide a CPP coded solution to the problem described below:

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
// #define _GLIBCXX_DEBUG // for STL debug (optional)
#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>
using namespace std;

#define debug(...) fprintf(stderr, __VA_ARGS__)
#define int long long int
 
template<typename T> void chmax(T &a, T b) {a = max(a, b);}
template<typename T> void chmin(T &a, T b) {a = min(a, b);}
template<typename T> void chadd(T &a, T b) {a = a + b;}
 
typedef pair<int, int> pii;
typedef long long ll;
 
int dx[] = {0, 0, 1, -1};
int dy[] = {1, -1, 0, 0};
const ll INF = 1001001001001001LL;
const ll MOD = 1000000007LL;

map< vector<int>, int > memo;

struct Elem {
    vector<int> vec;
    int cost;
    Elem() {}
    Elem(vector<int> v, int c) : vec(v), cost(c) {}
    bool operator<(const Elem &e) const {
        return cost > e.cost;
    }
};

int cost[10][10];
signed main() {
    int N; cin >> N;
    for(int i=0; i<N; i++) {
        for(int j=0; j<N; j++) {
            cin >> cost[i][j];
        }
    }

    priority_queue<Elem> que;
    vector<int> init(N);
    iota(init.begin(), init.end(), 0);
    memo[init] = 0;
    que.push(Elem(init, 0));
    
    while(que.size()) {
        auto e = que.top(); que.pop();
        auto vec = e.vec;
        auto orig = e.cost;
        if(orig > memo[vec]) continue;

        for(int i=0; i<N; i++) {
            for(int j=i+1; j<N; j++) {
                int vl = vec[i], vr = vec[j];
                int c = cost[vl][vr];

                swap(vec[i], vec[j]);
                if(!memo.count(vec) or memo[vec] > orig + c) {
                    memo[vec] = orig + c;
                    que.push(Elem(vec, orig + c));
                }
                swap(vec[i], vec[j]);
            }
        }
    }

    int ans = 0;
    for(auto v : memo) ans = max(ans, v.second);
    cout << ans << endl;
    return 0;
}


```