#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using vi = vector<int>;
using vvi = vector<vi>;
using vll = vector<ll>;
using vvll = vector<vll>;
using vb = vector<bool>;
using vd = vector<double>;
using vs = vector<string>;
using pii = pair<int, int>;
using pll = pair<ll, ll>;
using pdd = pair<double, double>;
using vpii = vector<pii>;
using vvpii = vector<vpii>;
using vpll = vector<pll>;
using vvpll = vector<vpll>;
using vpdd = vector<pdd>;
using vvpdd = vector<vpdd>;
template <typename T>
void ckmin(T& a, const T& b) {
  a = min(a, b);
}
template <typename T>
void ckmax(T& a, const T& b) {
  a = max(a, b);
}
mt19937_64 rng(chrono::steady_clock::now().time_since_epoch().count());
namespace __input {
template <class T1, class T2>
void re(pair<T1, T2>& p);
template <class T>
void re(vector<T>& a);
template <class T, size_t SZ>
void re(array<T, SZ>& a);
template <class T>
void re(T& x) {
  cin >> x;
}
void re(double& x) {
  string t;
  re(t);
  x = stod(t);
}
template <class Arg, class... Args>
void re(Arg& first, Args&... rest) {
  re(first);
  re(rest...);
}
template <class T1, class T2>
void re(pair<T1, T2>& p) {
  re(p.first, p.second);
}
template <class T>
void re(vector<T>& a) {
  for (int i = 0; i < (int((a).size())); i++) re(a[i]);
}
template <class T, size_t SZ>
void re(array<T, SZ>& a) {
  for (int i = 0; i < (SZ); i++) re(a[i]);
}
}  // namespace __input
using namespace __input;
namespace __output {
template <class T1, class T2>
void pr(const pair<T1, T2>& x);
template <class T, size_t SZ>
void pr(const array<T, SZ>& x);
template <class T>
void pr(const vector<T>& x);
template <class T>
void pr(const deque<T>& x);
template <class T>
void pr(const set<T>& x);
template <class T1, class T2>
void pr(const map<T1, T2>& x);
template <class T>
void pr(const T& x) {
  cout << x;
}
template <class Arg, class... Args>
void pr(const Arg& first, const Args&... rest) {
  pr(first);
  pr(rest...);
}
template <class T1, class T2>
void pr(const pair<T1, T2>& x) {
  pr("{", x.first, ", ", x.second, "}");
}
template <class T, bool pretty = true>
void prContain(const T& x) {
  if (pretty) pr("{");
  bool fst = 1;
  for (const auto& a : x) pr(!fst ? pretty ? ", " : " " : "", a), fst = 0;
  if (pretty) pr("}");
}
template <class T>
void pc(const T& x) {
  prContain<T, false>(x);
  pr("\n");
}
template <class T, size_t SZ>
void pr(const array<T, SZ>& x) {
  prContain(x);
}
template <class T>
void pr(const vector<T>& x) {
  prContain(x);
}
template <class T>
void pr(const deque<T>& x) {
  prContain(x);
}
template <class T>
void pr(const set<T>& x) {
  prContain(x);
}
template <class T1, class T2>
void pr(const map<T1, T2>& x) {
  prContain(x);
}
void ps() { pr("\n"); }
template <class Arg>
void ps(const Arg& first) {
  pr(first);
  ps();
}
template <class Arg, class... Args>
void ps(const Arg& first, const Args&... rest) {
  pr(first, " ");
  ps(rest...);
}
}  // namespace __output
using namespace __output;
namespace __algorithm {
template <typename T>
void dedup(vector<T>& v) {
  sort((v).begin(), (v).end());
  v.erase(unique((v).begin(), (v).end()), v.end());
}
template <typename T>
typename vector<T>::iterator find(vector<T>& v, const T& x) {
  auto it = lower_bound((v).begin(), (v).end(), x);
  return it != v.end() && *it == x ? it : v.end();
}
template <typename T>
size_t index(vector<T>& v, const T& x) {
  auto it = find(v, x);
  assert(it != v.end() && *it == x);
  return it - v.begin();
}
template <typename C, typename T, typename OP>
vector<T> prefixes(const C& v, T id, OP op) {
  vector<T> r(int((v).size()) + 1, id);
  for (int i = 0; i < (int((v).size())); i++) r[i + 1] = op(r[i], v[i]);
  return r;
}
template <typename C, typename T, typename OP>
vector<T> suffixes(const C& v, T id, OP op) {
  vector<T> r(int((v).size()) + 1, id);
  for (int i = (int((v).size())) - 1; i >= 0; i--) r[i] = op(v[i], r[i + 1]);
  return r;
}
}  // namespace __algorithm
using namespace __algorithm;
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wunused-parameter"
struct monostate {
  friend istream& operator>>(istream& is, const monostate& ms) { return is; }
  friend ostream& operator<<(ostream& os, const monostate& ms) { return os; }
  friend monostate operator+(const monostate& a, const monostate& b) {
    return a;
  }
} ms;
#pragma GCC diagnostic pop
namespace __io {
void setIn(string second) { freopen(second.c_str(), "r", stdin); }
void setOut(string second) { freopen(second.c_str(), "w", stdout); }
void setIO(string second = "") {
  ios_base::sync_with_stdio(0);
  cin.tie(0);
  cout << fixed << setprecision(15);
  if (int((second).size())) {
    setIn(second + ".in"), setOut(second + ".out");
  }
}
}  // namespace __io
using namespace __io;
struct union_find_bipartite {
  struct node {
    int parent;
    int size = 1, root_status = -1, root_size = 1;
    bool root_path_parity = 0, consistent = 1;
    node(int id) : parent(id) {}
    int choices() const { return !consistent ? 0 : root_status == -1 ? 2 : 1; }
    int minimum_size() const {
      if (root_status == 1)
        return root_size;
      else if (root_status == 0)
        return size - root_size;
      return min(root_size, size - root_size);
    }
  };
  vector<node> uf;
  int components, bipartite_components, rank, minimum_size;
  union_find_bipartite(int N = 0)
      : components(N), bipartite_components(N), rank(N), minimum_size(0) {
    for (int v = 0; v < N; v++) {
      uf.emplace_back(v);
    }
  }
  int rep(int v) {
    if (uf[v].parent == v) return v;
    int root = rep(uf[v].parent);
    uf[v].root_path_parity ^= uf[uf[v].parent].root_path_parity;
    return uf[v].parent = root;
  }
  void before_modification(int u) {
    components--;
    if (uf[u].consistent) {
      bipartite_components--;
      rank -= uf[u].root_status == -1;
      minimum_size -= uf[u].minimum_size();
    }
  }
  void after_modification(int u) {
    components++;
    if (uf[u].consistent) {
      bipartite_components++;
      rank += uf[u].root_status == -1;
      minimum_size += uf[u].minimum_size();
    }
  }
  bool set_status(int u, bool u_status) {
    int u_root = rep(u);
    if (!uf[u_root].consistent) return false;
    before_modification(u_root);
    const bool implied_root_status = u_status ^ uf[u].root_path_parity;
    if (uf[u_root].root_status == -1) {
      uf[u_root].root_status = implied_root_status;
    } else if (uf[u_root].root_status != implied_root_status) {
      uf[u_root].consistent = false;
    }
    after_modification(u_root);
    return uf[u_root].consistent;
  }
  struct result {
    bool added_connectivity, is_consistent;
  };
  result unite(int u, int v, bool edge_parity) {
    int u_root = rep(u), v_root = rep(v);
    if (u_root == v_root) {
      before_modification(u_root);
      if (edge_parity != (uf[u].root_path_parity ^ uf[v].root_path_parity)) {
        uf[u_root].consistent = false;
      }
      after_modification(u_root);
      return {false, uf[u_root].consistent};
    }
    edge_parity ^= uf[u].root_path_parity ^ uf[v].root_path_parity;
    u = u_root;
    v = v_root;
    if (uf[u].size < uf[v].size) swap(u, v);
    before_modification(u), before_modification(v);
    uf[v].parent = u;
    uf[v].root_path_parity = edge_parity;
    uf[u].size += uf[v].size;
    uf[u].consistent &= uf[v].consistent;
    if (edge_parity)
      uf[u].root_size += uf[v].size - uf[v].root_size;
    else
      uf[u].root_size += uf[v].root_size;
    if (uf[v].root_status != -1) {
      bool implied_u_status = uf[v].root_status ^ edge_parity;
      uf[u].consistent &=
          (uf[u].root_status == -1 || uf[u].root_status == implied_u_status);
      uf[u].root_status = implied_u_status;
    }
    after_modification(u);
    return {true, uf[u].consistent};
  }
  result constrain_to_be_same(int u, int v) { return unite(u, v, 0); }
  result constrain_to_be_different(int u, int v) { return unite(u, v, 1); }
  bool is_consistent(int u, bool u_status) {
    int u_root = rep(u);
    if (!uf[u_root].consistent) return false;
    if (uf[u_root].root_status == -1) return true;
    return u_status == (uf[u].root_path_parity ^ uf[u_root].root_status);
  }
  bool is_consistent(int u, int v, bool edge_parity) {
    int u_root = rep(u), v_root = rep(v);
    if (!uf[u_root].consistent || !uf[v_root].consistent) {
      return false;
    } else if (u_root == v_root) {
      return edge_parity == (uf[u].root_path_parity ^ uf[v].root_path_parity);
    } else if (uf[u_root].root_status == -1 || uf[v_root].root_status == -1) {
      return true;
    } else {
      return (uf[u].root_path_parity ^ uf[u_root].root_status ^
              uf[v].root_path_parity ^ uf[v_root].root_status) == edge_parity;
    }
  }
};
int main() {
  setIO();
  int N, K;
  re(N, K);
  vi init(N);
  for (int i = 0; i < (N); i++) {
    char c;
    re(c);
    init[i] = c == '1';
  }
  vvi tog(N);
  {
    for (int k = 0; k < (K); k++) {
      int S;
      re(S);
      for (int i = 0; i < (S); i++) {
        int e;
        re(e);
        tog[--e].push_back(k);
      }
    }
    for (int i = 0; i < (N); i++) assert(int((tog[i]).size()) <= 2);
  }
  union_find_bipartite uf(K);
  for (int p = 0; p < (N); p++) {
    int CON = int((tog[p]).size());
    if (CON == 0) {
      assert(init[p]);
    } else if (CON == 1) {
      assert(uf.is_consistent(tog[p][0], !init[p]));
      assert(uf.set_status(tog[p][0], !init[p]));
    } else if (CON == 2) {
      assert(uf.is_consistent(tog[p][0], tog[p][1], !init[p]));
      assert(uf.unite(tog[p][0], tog[p][1], !init[p]).is_consistent);
    }
    ps(uf.minimum_size);
  }
  return 0;
}

### Prompt

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

There are n lamps on a line, numbered from 1 to n. Each one has an initial state off (0) or on (1).

You're given k subsets A_1, …, A_k of \{1, 2, ..., n\}, such that the intersection of any three subsets is empty. In other words, for all 1 ≤ i_1 < i_2 < i_3 ≤ k, A_{i_1} ∩ A_{i_2} ∩ A_{i_3} = ∅.

In one operation, you can choose one of these k subsets and switch the state of all lamps in it. It is guaranteed that, with the given subsets, it's possible to make all lamps be simultaneously on using this type of operation.

Let m_i be the minimum number of operations you have to do in order to make the i first lamps be simultaneously on. Note that there is no condition upon the state of other lamps (between i+1 and n), they can be either off or on.

You have to compute m_i for all 1 ≤ i ≤ n.

Input

The first line contains two integers n and k (1 ≤ n, k ≤ 3 ⋅ 10^5).

The second line contains a binary string of length n, representing the initial state of each lamp (the lamp i is off if s_i = 0, on if s_i = 1).

The description of each one of the k subsets follows, in the following format:

The first line of the description contains a single integer c (1 ≤ c ≤ n) — the number of elements in the subset.

The second line of the description contains c distinct integers x_1, …, x_c (1 ≤ x_i ≤ n) — the elements of the subset.

It is guaranteed that: 

  * The intersection of any three subsets is empty; 
  * It's possible to make all lamps be simultaneously on using some operations. 

Output

You must output n lines. The i-th line should contain a single integer m_i — the minimum number of operations required to make the lamps 1 to i be simultaneously on.

Examples

Input


7 3
0011100
3
1 4 6
3
3 4 7
2
2 3


Output


1
2
3
3
3
3
3


Input


8 6
00110011
3
1 3 8
5
1 2 5 6 7
2
6 8
2
3 5
2
4 7
1
2


Output


1
1
1
1
1
1
4
4


Input


5 3
00011
3
1 2 3
1
4
3
3 4 5


Output


1
1
1
1
1


Input


19 5
1001001001100000110
2
2 3
2
5 6
2
8 9
5
12 13 14 15 16
1
19


Output


0
1
1
1
2
2
2
3
3
3
3
4
4
4
4
4
4
4
5

Note

In the first example: 

  * For i = 1, we can just apply one operation on A_1, the final states will be 1010110; 
  * For i = 2, we can apply operations on A_1 and A_3, the final states will be 1100110; 
  * For i ≥ 3, we can apply operations on A_1, A_2 and A_3, the final states will be 1111111. 



In the second example: 

  * For i ≤ 6, we can just apply one operation on A_2, the final states will be 11111101; 
  * For i ≥ 7, we can apply operations on A_1, A_3, A_4, A_6, the final states will be 11111111. 

### Response

```cpp
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using vi = vector<int>;
using vvi = vector<vi>;
using vll = vector<ll>;
using vvll = vector<vll>;
using vb = vector<bool>;
using vd = vector<double>;
using vs = vector<string>;
using pii = pair<int, int>;
using pll = pair<ll, ll>;
using pdd = pair<double, double>;
using vpii = vector<pii>;
using vvpii = vector<vpii>;
using vpll = vector<pll>;
using vvpll = vector<vpll>;
using vpdd = vector<pdd>;
using vvpdd = vector<vpdd>;
template <typename T>
void ckmin(T& a, const T& b) {
  a = min(a, b);
}
template <typename T>
void ckmax(T& a, const T& b) {
  a = max(a, b);
}
mt19937_64 rng(chrono::steady_clock::now().time_since_epoch().count());
namespace __input {
template <class T1, class T2>
void re(pair<T1, T2>& p);
template <class T>
void re(vector<T>& a);
template <class T, size_t SZ>
void re(array<T, SZ>& a);
template <class T>
void re(T& x) {
  cin >> x;
}
void re(double& x) {
  string t;
  re(t);
  x = stod(t);
}
template <class Arg, class... Args>
void re(Arg& first, Args&... rest) {
  re(first);
  re(rest...);
}
template <class T1, class T2>
void re(pair<T1, T2>& p) {
  re(p.first, p.second);
}
template <class T>
void re(vector<T>& a) {
  for (int i = 0; i < (int((a).size())); i++) re(a[i]);
}
template <class T, size_t SZ>
void re(array<T, SZ>& a) {
  for (int i = 0; i < (SZ); i++) re(a[i]);
}
}  // namespace __input
using namespace __input;
namespace __output {
template <class T1, class T2>
void pr(const pair<T1, T2>& x);
template <class T, size_t SZ>
void pr(const array<T, SZ>& x);
template <class T>
void pr(const vector<T>& x);
template <class T>
void pr(const deque<T>& x);
template <class T>
void pr(const set<T>& x);
template <class T1, class T2>
void pr(const map<T1, T2>& x);
template <class T>
void pr(const T& x) {
  cout << x;
}
template <class Arg, class... Args>
void pr(const Arg& first, const Args&... rest) {
  pr(first);
  pr(rest...);
}
template <class T1, class T2>
void pr(const pair<T1, T2>& x) {
  pr("{", x.first, ", ", x.second, "}");
}
template <class T, bool pretty = true>
void prContain(const T& x) {
  if (pretty) pr("{");
  bool fst = 1;
  for (const auto& a : x) pr(!fst ? pretty ? ", " : " " : "", a), fst = 0;
  if (pretty) pr("}");
}
template <class T>
void pc(const T& x) {
  prContain<T, false>(x);
  pr("\n");
}
template <class T, size_t SZ>
void pr(const array<T, SZ>& x) {
  prContain(x);
}
template <class T>
void pr(const vector<T>& x) {
  prContain(x);
}
template <class T>
void pr(const deque<T>& x) {
  prContain(x);
}
template <class T>
void pr(const set<T>& x) {
  prContain(x);
}
template <class T1, class T2>
void pr(const map<T1, T2>& x) {
  prContain(x);
}
void ps() { pr("\n"); }
template <class Arg>
void ps(const Arg& first) {
  pr(first);
  ps();
}
template <class Arg, class... Args>
void ps(const Arg& first, const Args&... rest) {
  pr(first, " ");
  ps(rest...);
}
}  // namespace __output
using namespace __output;
namespace __algorithm {
template <typename T>
void dedup(vector<T>& v) {
  sort((v).begin(), (v).end());
  v.erase(unique((v).begin(), (v).end()), v.end());
}
template <typename T>
typename vector<T>::iterator find(vector<T>& v, const T& x) {
  auto it = lower_bound((v).begin(), (v).end(), x);
  return it != v.end() && *it == x ? it : v.end();
}
template <typename T>
size_t index(vector<T>& v, const T& x) {
  auto it = find(v, x);
  assert(it != v.end() && *it == x);
  return it - v.begin();
}
template <typename C, typename T, typename OP>
vector<T> prefixes(const C& v, T id, OP op) {
  vector<T> r(int((v).size()) + 1, id);
  for (int i = 0; i < (int((v).size())); i++) r[i + 1] = op(r[i], v[i]);
  return r;
}
template <typename C, typename T, typename OP>
vector<T> suffixes(const C& v, T id, OP op) {
  vector<T> r(int((v).size()) + 1, id);
  for (int i = (int((v).size())) - 1; i >= 0; i--) r[i] = op(v[i], r[i + 1]);
  return r;
}
}  // namespace __algorithm
using namespace __algorithm;
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wunused-parameter"
struct monostate {
  friend istream& operator>>(istream& is, const monostate& ms) { return is; }
  friend ostream& operator<<(ostream& os, const monostate& ms) { return os; }
  friend monostate operator+(const monostate& a, const monostate& b) {
    return a;
  }
} ms;
#pragma GCC diagnostic pop
namespace __io {
void setIn(string second) { freopen(second.c_str(), "r", stdin); }
void setOut(string second) { freopen(second.c_str(), "w", stdout); }
void setIO(string second = "") {
  ios_base::sync_with_stdio(0);
  cin.tie(0);
  cout << fixed << setprecision(15);
  if (int((second).size())) {
    setIn(second + ".in"), setOut(second + ".out");
  }
}
}  // namespace __io
using namespace __io;
struct union_find_bipartite {
  struct node {
    int parent;
    int size = 1, root_status = -1, root_size = 1;
    bool root_path_parity = 0, consistent = 1;
    node(int id) : parent(id) {}
    int choices() const { return !consistent ? 0 : root_status == -1 ? 2 : 1; }
    int minimum_size() const {
      if (root_status == 1)
        return root_size;
      else if (root_status == 0)
        return size - root_size;
      return min(root_size, size - root_size);
    }
  };
  vector<node> uf;
  int components, bipartite_components, rank, minimum_size;
  union_find_bipartite(int N = 0)
      : components(N), bipartite_components(N), rank(N), minimum_size(0) {
    for (int v = 0; v < N; v++) {
      uf.emplace_back(v);
    }
  }
  int rep(int v) {
    if (uf[v].parent == v) return v;
    int root = rep(uf[v].parent);
    uf[v].root_path_parity ^= uf[uf[v].parent].root_path_parity;
    return uf[v].parent = root;
  }
  void before_modification(int u) {
    components--;
    if (uf[u].consistent) {
      bipartite_components--;
      rank -= uf[u].root_status == -1;
      minimum_size -= uf[u].minimum_size();
    }
  }
  void after_modification(int u) {
    components++;
    if (uf[u].consistent) {
      bipartite_components++;
      rank += uf[u].root_status == -1;
      minimum_size += uf[u].minimum_size();
    }
  }
  bool set_status(int u, bool u_status) {
    int u_root = rep(u);
    if (!uf[u_root].consistent) return false;
    before_modification(u_root);
    const bool implied_root_status = u_status ^ uf[u].root_path_parity;
    if (uf[u_root].root_status == -1) {
      uf[u_root].root_status = implied_root_status;
    } else if (uf[u_root].root_status != implied_root_status) {
      uf[u_root].consistent = false;
    }
    after_modification(u_root);
    return uf[u_root].consistent;
  }
  struct result {
    bool added_connectivity, is_consistent;
  };
  result unite(int u, int v, bool edge_parity) {
    int u_root = rep(u), v_root = rep(v);
    if (u_root == v_root) {
      before_modification(u_root);
      if (edge_parity != (uf[u].root_path_parity ^ uf[v].root_path_parity)) {
        uf[u_root].consistent = false;
      }
      after_modification(u_root);
      return {false, uf[u_root].consistent};
    }
    edge_parity ^= uf[u].root_path_parity ^ uf[v].root_path_parity;
    u = u_root;
    v = v_root;
    if (uf[u].size < uf[v].size) swap(u, v);
    before_modification(u), before_modification(v);
    uf[v].parent = u;
    uf[v].root_path_parity = edge_parity;
    uf[u].size += uf[v].size;
    uf[u].consistent &= uf[v].consistent;
    if (edge_parity)
      uf[u].root_size += uf[v].size - uf[v].root_size;
    else
      uf[u].root_size += uf[v].root_size;
    if (uf[v].root_status != -1) {
      bool implied_u_status = uf[v].root_status ^ edge_parity;
      uf[u].consistent &=
          (uf[u].root_status == -1 || uf[u].root_status == implied_u_status);
      uf[u].root_status = implied_u_status;
    }
    after_modification(u);
    return {true, uf[u].consistent};
  }
  result constrain_to_be_same(int u, int v) { return unite(u, v, 0); }
  result constrain_to_be_different(int u, int v) { return unite(u, v, 1); }
  bool is_consistent(int u, bool u_status) {
    int u_root = rep(u);
    if (!uf[u_root].consistent) return false;
    if (uf[u_root].root_status == -1) return true;
    return u_status == (uf[u].root_path_parity ^ uf[u_root].root_status);
  }
  bool is_consistent(int u, int v, bool edge_parity) {
    int u_root = rep(u), v_root = rep(v);
    if (!uf[u_root].consistent || !uf[v_root].consistent) {
      return false;
    } else if (u_root == v_root) {
      return edge_parity == (uf[u].root_path_parity ^ uf[v].root_path_parity);
    } else if (uf[u_root].root_status == -1 || uf[v_root].root_status == -1) {
      return true;
    } else {
      return (uf[u].root_path_parity ^ uf[u_root].root_status ^
              uf[v].root_path_parity ^ uf[v_root].root_status) == edge_parity;
    }
  }
};
int main() {
  setIO();
  int N, K;
  re(N, K);
  vi init(N);
  for (int i = 0; i < (N); i++) {
    char c;
    re(c);
    init[i] = c == '1';
  }
  vvi tog(N);
  {
    for (int k = 0; k < (K); k++) {
      int S;
      re(S);
      for (int i = 0; i < (S); i++) {
        int e;
        re(e);
        tog[--e].push_back(k);
      }
    }
    for (int i = 0; i < (N); i++) assert(int((tog[i]).size()) <= 2);
  }
  union_find_bipartite uf(K);
  for (int p = 0; p < (N); p++) {
    int CON = int((tog[p]).size());
    if (CON == 0) {
      assert(init[p]);
    } else if (CON == 1) {
      assert(uf.is_consistent(tog[p][0], !init[p]));
      assert(uf.set_status(tog[p][0], !init[p]));
    } else if (CON == 2) {
      assert(uf.is_consistent(tog[p][0], tog[p][1], !init[p]));
      assert(uf.unite(tog[p][0], tog[p][1], !init[p]).is_consistent);
    }
    ps(uf.minimum_size);
  }
  return 0;
}

```