Matryoshka

Matryoshka is a famous Russian folk craft doll. Matryoshka can be divided into upper and lower parts, and when opened, there is another smaller doll inside. The nesting structure is such that when the small doll that appears is opened, a smaller doll is contained.

<image>


You found an unusually shaped matryoshka doll on your trip and bought N dolls. The shape of the i-th doll is a rectangular parallelepiped of xi × yi × zi.

After watching Matryoshka for a while, you are about to put away Matryoshka. Before that, I want to reduce the space required by storing some dolls in another. When storing a doll, one other doll can be stored only in the doll that has not yet stored any doll. However, only the dolls that are stored directly are counted, and the doll that contains the doll can be stored in another doll.

The stored doll becomes invisible from the outside. However, the following conditions must be met.

* The doll may rotate, but each side of the rectangular parallelepiped is parallel to any side of the other rectangular parallelepiped.
* After rotation, the length of the doll on the side to be stored is shorter for each of the lengths of the corresponding sides.
* At most one doll can be stored directly in one doll



Since the volume of the closet is limited, we want to minimize the sum of the volumes of the dolls that can be seen from the outside. Your job is to create a program that finds the minimum sum of the volumes of the dolls that are visible from the outside, which can be achieved by repeating the operation of storing the dolls any number of times.

Input

The input consists of multiple datasets. The maximum number of data sets does not exceed 50. Each dataset is represented in the following format.

> N
> x1 y1 z1
>::
>::
> xN yN zN
>

Each dataset consists of N + 1 rows, and the first row of the dataset is given the integer N, which represents the number of dolls. In the i-th line of the following N lines, three integers xi, yi, and zi representing the size of the i-th doll are given, separated by a half-width space. These integers satisfy 1 ≤ N, xi, yi, zi ≤ 100.

> The end of the input is represented by a single zero line.

> ### Output

For each data set, output the minimum value of the sum of the volumes of the dolls that can be seen from the outside in one line.

> ### Sample Input


2
one two Three
4 2 3
3
2 5 2
3 3 4
5 5 5
Five
1 1 1
2 2 2
3 3 3
4 4 4
5 5 5
Five
1 1 1
2 1 1
3 1 1
4 1 1
5 1 1
Ten
3 1 4
1 5 9
2 6 5
3 5 8
9 7 9
3 2 3
8 4 6
2 6 4
3 3 8
3 2 7
0


Output for Sample Input


twenty four
145
125
15
864





Example

Input

2
1 2 3
4 2 3
3
2 5 2
3 3 4
5 5 5
5
1 1 1
2 2 2
3 3 3
4 4 4
5 5 5
5
1 1 1
2 1 1
3 1 1
4 1 1
5 1 1
10
3 1 4
1 5 9
2 6 5
3 5 8
9 7 9
3 2 3
8 4 6
2 6 4
3 3 8
3 2 7
0


Output

24
145
125
15
864
#include <bits/stdc++.h>

using namespace std;

#define pb push_back
#define mp make_pair
#define mt make_tuple
#define fi first
#define se second
#define all(a) (a).begin(), (a).end()
#define endl "\n"
#define rep(i, a, b) for (auto i = (a); i < (b); ++i)
#define rrep(i, a, b) for (auto i = (a); i > (b); --i)
#define UNIQUE(v) (v).erase(unique((v).begin(), (v).end()), (v).end())
typedef long long ll;
typedef unsigned long long ull;

#ifdef LOCAL_DEFINE
struct InitInput
{
  InitInput()
  {
    FILE *stream1;
    stream1 = freopen("in.txt", "r", stdin);
    assert(stream1 != nullptr);
    cerr << "This problem is not interactive" << endl;
  }
} LOCAL_INPUT;
struct LOCAL_OUTPUT
{
  LOCAL_OUTPUT()
  {
    FILE *stream2;
    const char *outputfile = "out.txt";
    stream2 = freopen(outputfile, "w", stdout);
    assert(stream2 != nullptr);
    cerr << "output [ " << outputfile << " ]" << endl;
  }
} /*LOCAL_OUTPUT*/;
#define show(x) cerr << #x << " = " << (x) << " (line " << __LINE__ << ")" << endl
#define showA(a, n) \
  do{for(int i=0;i<(n);i++)cerr<<"("<<i<<" = "<<(a)[i]<<") ";cerr<<endl;}while(0)
#define showA2(a, n, m) \
  do {for(int i=0;i<(n);i++){for(int j=0;j<(m);j++){cerr<<"("<<i<<", "<<j<<" = "<<(a)[i][j]<<") ";}cerr<<endl;}cerr<<endl;}while(0)
#else
#define show(x)
#define showA(a, n)
#define showA2(a, n, m)
#endif

struct InitAim
{
  InitAim()
  {
    ios::sync_with_stdio(false);
    cin.tie(nullptr);
    cout.precision(12);
    cout << fixed;
#ifdef LOCAL_DEFINE
    cerr << "This problem is not interactive" << endl;
#endif
  }
} aim_cpo;

///////////////////////////////////////////////////////////////////////////////////
// TEMPLATE(data structure)
///////////////////////////////////////////////////////////////////////////////////

template <typename T>
bool chmin(T &a, T b) { return a > b ? (a = b, true) : false; }
template <typename T>
bool chmax(T &a, T b) { return a < b ? (a = b, true) : false; }
template <typename T>
void ln(T i, T n) { cout << (i == n - 1 ? "\n" : " "); }
template <typename T, typename S>
ostream &operator<<(ostream &out, const pair<T, S> &pair1)
{
  out << '(' << pair1.fi << ", " << pair1.se << ')';
  return out;
}
template <typename T>
ostream &operator<<(ostream &out, const vector<T> &vector1)
{
  out << '[';
  for (unsigned int i = 0; i < vector1.size(); i++)
  {
    out << vector1[i];
    if (i == vector1.size() - 1)
      out << "]";
    else
      out << ", ";
  }
  return out;
}

#define GCD(a, b) __gcd(a, b)
template <typename T>
T LCM(T a, T b)
{
  return a / GCD(a, b) * b;
}
template <typename T>
T EXTGCD(T a, T b, T &x, T &y)
{
  T d = a;
  if (b != 0)
  {
    d = EXTGCD(b, a % b, y, x);
    y -= (a / b) * x;
  }
  else
    x = 1, y = 0;
  return d;
}
template <typename T>
bool is_prime(T a)
{
  for (int i = 2; i * i <= a; i++)
    if (a % i == 0)
      return true;
  return false;
}
template <typename T, typename S>
T Pow(T a, S b)
{
  T res = 1, now = a;
  while (b)
  {
    if (b & 1)
      res *= now;
    b >>= 1;
    now *= now;
  }
  return res;
}

/* MOD */
ll MOD = 1000000000L + 7L;
#define Madd(a, b) (((a) % MOD) + ((b) % MOD)) % MOD
#define Mmul(a, b) (((a) % MOD) * ((b) % MOD)) % MOD
#define Msub(a, b) (((a) % MOD) + MOD - ((b) % MOD)) % MOD
template <typename T, typename S>
T ModPow(T a, S b)
{
  assert(b >= 0);
  T res = 1, now = Msub(a, 0);
  while (b)
  {
    if (b & 1)
      res = Mmul(res, now);
    b >>= 1;
    now = Mmul(now, now);
  }
  return res;
}
template <typename T>
T ModInverse(T a, T mod, bool prime)
{ // if mod is prime, "prime" is true.
  if (prime)
    return ModPow(a, mod - 2);
  else
  {
    T x, y;
    EXTGCD(a, mod, x, y);
    return (mod + x % mod) % mod;
  }
}
template <typename T>
T EulerTotient(T a)
{
  vector<pair<int, int>> v;
  for (T i = 2; i * i <= a; i++)
  {
    int cnt = 0;
    while (a % i == 0)
    {
      cnt++;
      a /= i;
    }
    if (cnt != 0)
      v.emplace_back(i, cnt);
  }
  if (a != 1)
    v.emplace_back(a, 1);
  //showV(v, (int) v.size());
  T res = 1;
  for (int i = 0; i < (int)v.size(); i++)
  {
    if (v[i].se == 1)
    {
      //res *= v[i].fi - 1;
      res = Mmul(res, v[i].fi - 1);
    }
    else
    {
      //res *= Pow(v[i].fi, v[i].se) - Pow(v[i].fi, v[i].se - 1);
      res = Mmul(res, Msub(ModPow(v[i].fi, v[i].se), ModPow(v[i].fi, v[i].se - 1)));
    }
  }
  return res;
}
#define Mdivide(a, b) Mmul(((a) % MOD), (ModInverse((b), MOD, true))) % MOD
ll comb(ll a, ll b)
{
  chmin(b, a - b);
  ll res = 1LL, now = a;
  for (ll i = 1; i <= b; i++)
  {
    res = Mmul(res, now);  //res *= now;
    res = Mdivide(res, i); // res /= i;
    now--;
  }
  return res;
}

template <typename T>
class BIT
{
public:
  BIT(int size)
  {
    BITTable.assign(++size, 0);
  }
  T sum(int k)
  {
    T res = 0;
    for (++k; k > 0; k -= k & -k)
    {
      res += BITTable[k];
    }
    return res;
  }
  T sum(int l, int r)
  {
    if (l == 0)
      return sum(r);
    return sum(r) - sum(l - 1);
  }
  void update(int k, T x)
  { // b[k] += x;
    for (++k; k < (int)BITTable.size(); k += k & -k)
      BITTable[k] += x;
  }

private:
  vector<T> BITTable;
};

template <typename T>
class IntervalTree
{
  using F = function<T(T, T)>;

public:
  IntervalTree(int n, const F func, const T init) : func(func), init(init)
  {
    size = 1;
    while ((int)size < n)
      size <<= 1;
    table.assign(2 * size, init);
  }
  void set(int k, T &x)
  {
    table[size + k] = x;
  }
  void build()
  {
    for (int i = size - 1; i >= 0; --i)
    {
      table[i] = func(table[i * 2], table[i * 2 + 1]);
    }
  }
  void update(int k, const T &x)
  {
    k += size;
    table[k] = x;
    while (k >>= 1)
    {
      table[k] = func(table[k * 2], table[k * 2 + 1]);
    }
  }
  T query(int a, int b)
  {
    T L = init, R = init;
    for (a += size, b += size; a < b; a >>= 1, b >>= 1)
    {
      if (a & 1)
        L = func(L, table[a++]);
      if (b & 1)
        R = func(table[--b], R);
    }
    return func(L, R);
  }
  T operator[](const int k) const
  {
    return table[k + size];
  }

private:
  unsigned int size;
  vector<T> table;
  const F func;
  const T init;
};

class UnionFind
{
public:
  explicit UnionFind(int _n) : n(_n)
  {
    par.resize(static_cast<unsigned long>(_n));
    rank.resize(static_cast<unsigned long>(_n));
    sizes.resize(static_cast<unsigned long>(_n));
    for (int i = 0; i < _n; i++)
    {
      par[i] = i;
      rank[i] = 0;
      sizes[i] = 1;
    }
  }
  int find(int a)
  {
    if (par[a] == a)
      return a;
    return par[a] = find(par[a]);
  }
  bool same(int a, int b)
  {
    return find(a) == find(b);
  }
  void unite(int a, int b)
  {
    link(find(a), find(b));
  }
  int size(int a)
  {
    return sizes[find(a)];
  }
  void view()
  {
    for (int i = 0; i < n; i++)
    {
      cout << " par"
           << "[" << i << "]=" << par[i] << ((i == n - 1) ? "\n" : ",");
    }
    for (int i = 0; i < n; i++)
    {
      cout << "size"
           << "[" << i << "]=" << sizes[i] << ((i == n - 1) ? "\n" : ",");
    }
    cout << endl;
  }

private:
  void link(int a, int b)
  {
    if (same(a, b))
      return;
    if (rank[a] > rank[b])
    {
      par[b] = a;
      sizes[a] += sizes[b];
      sizes[b] = 0;
    }
    else
    {
      par[a] = b;
      if (rank[a] == rank[b])
        rank[b]++;
      sizes[b] += sizes[a];
      sizes[a] = 0;
    }
  }
  int n;
  vector<int> par;
  vector<int> rank;
  vector<int> sizes;
};

template<typename T>
class Dinic{
public:
  Dinic(int SIZE) :SIZE(SIZE) {
    v = vector<vector<tuple<int, T, int>>>(SIZE + 10);
  }

  void AddEdge(int from, int to, T cap) {
    v[from].push_back(make_tuple(to, cap, v[to].size()));
    v[to].push_back(make_tuple(from, 0, v[from].size() - 1));
  }

  void bfs(int s) {
    level = vector<int>(SIZE + 10, -1);
    queue<int> q;
    level[s] = 0;
    q.push(s);
    while (!q.empty()) {
      int now = q.front(); q.pop();
      for (int i = 0; i < (int)v[now].size(); i++) {
        int next, nextrv; T nextc; tie(next, nextc, nextrv) = v[now][i];
        if (nextc > 0 && level[next] < 0) {
          level[next] = level[now] + 1;
          q.push(next);
        }
      }
    }
  }

  T dfs(int now, int t, T f) {
    if (now == t) return f;
    for (int &i = iter[now]; i < (int)v[now].size(); i++) {
      int next, nextrv; T nextc; tie(next, nextc, nextrv) = v[now][i];
      if (nextc > 0 && level[now] < level[next]) {
        T d = dfs(next, t, min(f, nextc));
        if (d > 0) {
          get<1>(v[now][i]) -= d;
          get<1>(v[next][nextrv]) += d;
          return d;
        }
      }
    }
    return 0;
  }

  T max_flow(int s, int t) {
    T flow = 0;
    for (;;) {
      bfs(s);
      if (level[t] < 0) return flow;
      iter = vector<int>(SIZE + 10, 0);
      int f;
      while ((f = dfs(s, t, INT_MAX)) > 0) {
        flow += f;
      }
    }
  }
private:
  int SIZE;
  vector<vector<tuple<int, T, int>>> v;
  vector<int> level, iter;
};

template< typename flow_t, typename cost_t >
struct PrimalDual {
  const cost_t INF;

  struct edge {
    int to;
    flow_t cap;
    cost_t cost;
    int rev;
    bool isrev;
  };
  vector< vector< edge > > graph;
  vector< cost_t > potential, min_cost;
  vector< int > prevv, preve;

  PrimalDual(int V) : graph(V), INF(numeric_limits< cost_t >::max()) {}

  void add_edge(int from, int to, flow_t cap, cost_t cost) {
    graph[from].emplace_back((edge) {to, cap, cost, (int) graph[to].size(), false});
    graph[to].emplace_back((edge) {from, 0, -cost, (int) graph[from].size() - 1, true});
  }

  cost_t min_cost_flow(int s, int t, flow_t f) {
    int V = (int) graph.size();
    cost_t ret = 0;
    using Pi = pair< cost_t, int >;
    priority_queue< Pi, vector< Pi >, greater< Pi > > que;
    potential.assign(V, 0);
    preve.assign(V, -1);
    prevv.assign(V, -1);

    while(f > 0) {
      min_cost.assign(V, INF);
      que.emplace(0, s);
      min_cost[s] = 0;
      while(!que.empty()) {
        Pi p = que.top();
        que.pop();
        if(min_cost[p.second] < p.first) continue;
        for(int i = 0; i < graph[p.second].size(); i++) {
          edge &e = graph[p.second][i];
          cost_t nextCost = min_cost[p.second] + e.cost + potential[p.second] - potential[e.to];
          if(e.cap > 0 && min_cost[e.to] > nextCost) {
            min_cost[e.to] = nextCost;
            prevv[e.to] = p.second, preve[e.to] = i;
            que.emplace(min_cost[e.to], e.to);
          }
        }
      }
      if(min_cost[t] == INF) return -1;
      for(int v = 0; v < V; v++) potential[v] += min_cost[v];
      flow_t addflow = f;
      for(int v = t; v != s; v = prevv[v]) {
        addflow = min(addflow, graph[prevv[v]][preve[v]].cap);
      }
      f -= addflow;
      ret += addflow * potential[t];
      for(int v = t; v != s; v = prevv[v]) {
        edge &e = graph[prevv[v]][preve[v]];
        e.cap -= addflow;
        graph[v][e.rev].cap += addflow;
      }
    }
    return ret;
  }

  void output() {
    for(int i = 0; i < graph.size(); i++) {
      for(auto &e : graph[i]) {
        if(e.isrev) continue;
        auto &rev_e = graph[e.to][e.rev];
        cout << i << "->" << e.to << " (flow: " << rev_e.cap << "/" << rev_e.cap + e.cap << ")" << endl;
      }
    }
  }
};


///////////////////////////////////////////////////////////////////////////////////
// MAIN
///////////////////////////////////////////////////////////////////////////////////
// 735134400 約数が1344個ある高度合成数(<= 1e9)
// 897612484786617600 約数が103680個ある高度合成数(<= 1e18)
// 苦手分野 重複順列
// LLの数値をつかう時は最後にLLをつける癖をつけよう

void solve(int &n) {
  int x[101], y[101], z[101];
  int sum = 0;
  rep(i, 0, n) {
    cin >> x[i] >> y[i] >> z[i];
    sum += x[i] * y[i] * z[i];
  }
  int s = 2 * n;
  int t = s + 1;
  PrimalDual<int, int> F(2 * n + 2);
  rep(i, 0, n) {
    rep(j, 0, n) {
      if (i == j) continue;
      auto check = [&](){
        if (x[i] > x[j] && y[i] > y[j] && z[i] > z[j]) return true;
        if (x[i] > x[j] && z[i] > y[j] && y[i] > z[j]) return true;
        if (y[i] > x[j] && x[i] > y[j] && z[i] > z[j]) return true;
        if (y[i] > x[j] && z[i] > y[j] && x[i] > z[j]) return true;
        if (z[i] > x[j] && x[i] > y[j] && y[i] > z[j]) return true;
        if (z[i] > x[j] && y[i] > y[j] && x[i] > z[j]) return true;
        return false;
      };
      if (check()) {
        F.add_edge(i, n + j, 1, -x[j] * y[j] * z[j]);
      }
    }
  }
  rep(i, 0, n) {
    F.add_edge(s, i, 1, 0);
    F.add_edge(n + i, t, 1, 0);
    F.add_edge(i, t, 1, 0);
  }
  cout << sum + F.min_cost_flow(s, t, n) << endl;
}

int main()
{
  int n;
  while (cin >> n, n) solve(n);
#ifdef LOCAL_DEFINE
  cerr << "Time elapsed: " << 1.0 * clock() / CLOCKS_PER_SEC << "s." << endl;
  show(MOD);
#endif
  return 0;
}

///////////////////////////////////////////////////////////////////////////////////
// NOTE
///////////////////////////////////////////////////////////////////////////////////
/*
*/
