#include <bits/stdc++.h>
enum class LampConstraint { Distinct, Same };
enum class LampSetState { On, Off, PhantomA, PhantomB };
bool isLampSetStateKnown(LampSetState st) {
  return st == LampSetState::On || st == LampSetState::Off;
}
typedef struct Edge {
  int nodeId;
  LampConstraint cons;
  Edge(int nodeId_, LampConstraint cons_) : nodeId(nodeId_), cons(cons_) {}
} Edge;
bool compareEdges(const Edge &a, const Edge &b) {
  if (a.nodeId < b.nodeId) return true;
  if (a.nodeId > b.nodeId) return true;
  return a.cons < b.cons;
}
typedef struct LampSetInfo {
  LampSetState state;
  int ownerId;
  int componentSize;
  int onColorSize;
  int offColorSize;
  explicit LampSetInfo(int id)
      : state(LampSetState::PhantomA),
        ownerId(id),
        componentSize(1),
        onColorSize(1),
        offColorSize(0){};
} LampSetInfo;
typedef std::set<Edge, decltype(compareEdges) *> Edges;
int componentOnBestSize(std::vector<LampSetInfo> &nodes, int nodeId) {
  LampSetInfo &s = nodes[nodeId];
  int ownerId = s.ownerId;
  if (s.state == LampSetState::On || s.state == LampSetState::Off) {
    return nodes[ownerId].onColorSize;
  } else {
    return std::min(nodes[ownerId].onColorSize, nodes[ownerId].offColorSize);
  }
}
LampSetState deducedState(const LampConstraint &cons, LampSetState st) {
  switch (cons) {
    case LampConstraint::Same:
      return st;
    case LampConstraint::Distinct:
      switch (st) {
        case LampSetState::On:
          return LampSetState::Off;
        case LampSetState::Off:
          return LampSetState::On;
        case LampSetState::PhantomA:
          return LampSetState::PhantomB;
        case LampSetState::PhantomB:
          return LampSetState::PhantomA;
      }
  }
}
void propagateLampState(const std::vector<Edges> &graph,
                        std::vector<LampSetInfo> &nodes, int startNodeId,
                        LampSetState learnedState) {
  std::set<int> visited;
  std::stack<std::pair<int, LampSetState>> nodeIds;
  nodeIds.push(std::make_pair(startNodeId, learnedState));
  int onCount = 0;
  while (!nodeIds.empty()) {
    std::pair<int, LampSetState> currentNode = nodeIds.top();
    nodeIds.pop();
    if (visited.find(currentNode.first) != visited.end()) continue;
    nodes[currentNode.first].state = currentNode.second;
    onCount += ((currentNode.second == LampSetState::On ||
                 currentNode.second == LampSetState::PhantomA)
                    ? 1
                    : 0);
    visited.insert(currentNode.first);
    for (const Edge &e : graph[currentNode.first]) {
      if (visited.find(e.nodeId) == visited.end())
        nodeIds.push(
            std::make_pair(e.nodeId, deducedState(e.cons, currentNode.second)));
    }
  }
  int ownerId = nodes[startNodeId].ownerId;
  nodes[ownerId].onColorSize = onCount;
  nodes[ownerId].offColorSize = nodes[ownerId].componentSize - onCount;
}
void switchOwner(std::vector<Edges> &graph, std::vector<LampSetInfo> &nodes,
                 int startNodeId, int newOwner) {
  std::set<int> visited;
  std::stack<int> nodeIds;
  nodeIds.push(startNodeId);
  while (!nodeIds.empty()) {
    int currentNode = nodeIds.top();
    nodeIds.pop();
    if (visited.find(currentNode) != visited.end()) continue;
    visited.insert(currentNode);
    nodes[currentNode].ownerId = newOwner;
    for (auto &e : graph[currentNode]) {
      if (visited.find(e.nodeId) == visited.end()) nodeIds.push(e.nodeId);
    }
  }
}
int main() {
  int n, k;
  long long result = 0;
  std::string lamps;
  std::vector<LampSetInfo> nodes;
  std::cin >> n >> k;
  std::cin >> lamps;
  std::vector<std::vector<int>> setsForLamp(n);
  std::vector<Edges> graph(k, Edges(compareEdges));
  for (int i = 0; i < k; i++) {
    int c;
    nodes.push_back(LampSetInfo(i));
    std::cin >> c;
    for (int j = 0; j < c; j++) {
      int a;
      std::cin >> a;
      setsForLamp[a - 1].push_back(i);
    }
  }
  for (int i = 0; i < n; i++) {
    LampSetState learnedState;
    LampConstraint cons;
    int startNode;
    int a, b, ownerA, ownerB;
    switch (setsForLamp[i].size()) {
      case 0:
        if (lamps[i] != '1')
          throw std::runtime_error("Conflicting enforced states of lamps!");
        break;
      case 1:
        startNode = setsForLamp[i][0];
        learnedState = (lamps[i] == '1' ? LampSetState::Off : LampSetState::On);
        if (!isLampSetStateKnown(nodes[startNode].state)) {
          result -= componentOnBestSize(nodes, startNode);
          propagateLampState(graph, nodes, startNode, learnedState);
          result += componentOnBestSize(nodes, startNode);
        } else {
          if (learnedState != nodes[startNode].state)
            throw std::runtime_error("Conflicting enforced states of lamps!");
        }
        break;
      case 2:
        a = setsForLamp[i][0];
        b = setsForLamp[i][1];
        ownerA = nodes[a].ownerId;
        ownerB = nodes[b].ownerId;
        cons =
            (lamps[i] == '1' ? LampConstraint::Same : LampConstraint::Distinct);
        if (nodes[ownerA].componentSize > nodes[ownerB].componentSize) {
          std::swap(a, b);
          std::swap(ownerA, ownerB);
        }
        if (ownerA != ownerB) {
          result -= componentOnBestSize(nodes, ownerA);
          result -= componentOnBestSize(nodes, ownerB);
          if (isLampSetStateKnown(nodes[a].state) &&
              !isLampSetStateKnown(nodes[b].state)) {
            propagateLampState(graph, nodes, b,
                               deducedState(cons, nodes[a].state));
          } else if (isLampSetStateKnown(nodes[b].state) &&
                     !isLampSetStateKnown(nodes[a].state)) {
            propagateLampState(graph, nodes, a,
                               deducedState(cons, nodes[b].state));
          } else if (!isLampSetStateKnown(nodes[b].state) &&
                     !isLampSetStateKnown(nodes[a].state)) {
            propagateLampState(graph, nodes, a,
                               deducedState(cons, nodes[b].state));
          }
          nodes[ownerB].onColorSize += nodes[ownerA].onColorSize;
          nodes[ownerB].offColorSize += nodes[ownerA].offColorSize;
          nodes[ownerB].componentSize += nodes[ownerA].componentSize;
          result += componentOnBestSize(nodes, ownerB);
          switchOwner(graph, nodes, a, ownerB);
        }
        graph[a].insert(Edge(b, cons));
        graph[b].insert(Edge(a, cons));
        break;
      default:
        throw std::runtime_error("Malformed input data!");
    }
    std::cout << result << std::endl;
  }
  return 0;
}

### Prompt

Your challenge is to write a Cpp solution to the following problem:

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>
enum class LampConstraint { Distinct, Same };
enum class LampSetState { On, Off, PhantomA, PhantomB };
bool isLampSetStateKnown(LampSetState st) {
  return st == LampSetState::On || st == LampSetState::Off;
}
typedef struct Edge {
  int nodeId;
  LampConstraint cons;
  Edge(int nodeId_, LampConstraint cons_) : nodeId(nodeId_), cons(cons_) {}
} Edge;
bool compareEdges(const Edge &a, const Edge &b) {
  if (a.nodeId < b.nodeId) return true;
  if (a.nodeId > b.nodeId) return true;
  return a.cons < b.cons;
}
typedef struct LampSetInfo {
  LampSetState state;
  int ownerId;
  int componentSize;
  int onColorSize;
  int offColorSize;
  explicit LampSetInfo(int id)
      : state(LampSetState::PhantomA),
        ownerId(id),
        componentSize(1),
        onColorSize(1),
        offColorSize(0){};
} LampSetInfo;
typedef std::set<Edge, decltype(compareEdges) *> Edges;
int componentOnBestSize(std::vector<LampSetInfo> &nodes, int nodeId) {
  LampSetInfo &s = nodes[nodeId];
  int ownerId = s.ownerId;
  if (s.state == LampSetState::On || s.state == LampSetState::Off) {
    return nodes[ownerId].onColorSize;
  } else {
    return std::min(nodes[ownerId].onColorSize, nodes[ownerId].offColorSize);
  }
}
LampSetState deducedState(const LampConstraint &cons, LampSetState st) {
  switch (cons) {
    case LampConstraint::Same:
      return st;
    case LampConstraint::Distinct:
      switch (st) {
        case LampSetState::On:
          return LampSetState::Off;
        case LampSetState::Off:
          return LampSetState::On;
        case LampSetState::PhantomA:
          return LampSetState::PhantomB;
        case LampSetState::PhantomB:
          return LampSetState::PhantomA;
      }
  }
}
void propagateLampState(const std::vector<Edges> &graph,
                        std::vector<LampSetInfo> &nodes, int startNodeId,
                        LampSetState learnedState) {
  std::set<int> visited;
  std::stack<std::pair<int, LampSetState>> nodeIds;
  nodeIds.push(std::make_pair(startNodeId, learnedState));
  int onCount = 0;
  while (!nodeIds.empty()) {
    std::pair<int, LampSetState> currentNode = nodeIds.top();
    nodeIds.pop();
    if (visited.find(currentNode.first) != visited.end()) continue;
    nodes[currentNode.first].state = currentNode.second;
    onCount += ((currentNode.second == LampSetState::On ||
                 currentNode.second == LampSetState::PhantomA)
                    ? 1
                    : 0);
    visited.insert(currentNode.first);
    for (const Edge &e : graph[currentNode.first]) {
      if (visited.find(e.nodeId) == visited.end())
        nodeIds.push(
            std::make_pair(e.nodeId, deducedState(e.cons, currentNode.second)));
    }
  }
  int ownerId = nodes[startNodeId].ownerId;
  nodes[ownerId].onColorSize = onCount;
  nodes[ownerId].offColorSize = nodes[ownerId].componentSize - onCount;
}
void switchOwner(std::vector<Edges> &graph, std::vector<LampSetInfo> &nodes,
                 int startNodeId, int newOwner) {
  std::set<int> visited;
  std::stack<int> nodeIds;
  nodeIds.push(startNodeId);
  while (!nodeIds.empty()) {
    int currentNode = nodeIds.top();
    nodeIds.pop();
    if (visited.find(currentNode) != visited.end()) continue;
    visited.insert(currentNode);
    nodes[currentNode].ownerId = newOwner;
    for (auto &e : graph[currentNode]) {
      if (visited.find(e.nodeId) == visited.end()) nodeIds.push(e.nodeId);
    }
  }
}
int main() {
  int n, k;
  long long result = 0;
  std::string lamps;
  std::vector<LampSetInfo> nodes;
  std::cin >> n >> k;
  std::cin >> lamps;
  std::vector<std::vector<int>> setsForLamp(n);
  std::vector<Edges> graph(k, Edges(compareEdges));
  for (int i = 0; i < k; i++) {
    int c;
    nodes.push_back(LampSetInfo(i));
    std::cin >> c;
    for (int j = 0; j < c; j++) {
      int a;
      std::cin >> a;
      setsForLamp[a - 1].push_back(i);
    }
  }
  for (int i = 0; i < n; i++) {
    LampSetState learnedState;
    LampConstraint cons;
    int startNode;
    int a, b, ownerA, ownerB;
    switch (setsForLamp[i].size()) {
      case 0:
        if (lamps[i] != '1')
          throw std::runtime_error("Conflicting enforced states of lamps!");
        break;
      case 1:
        startNode = setsForLamp[i][0];
        learnedState = (lamps[i] == '1' ? LampSetState::Off : LampSetState::On);
        if (!isLampSetStateKnown(nodes[startNode].state)) {
          result -= componentOnBestSize(nodes, startNode);
          propagateLampState(graph, nodes, startNode, learnedState);
          result += componentOnBestSize(nodes, startNode);
        } else {
          if (learnedState != nodes[startNode].state)
            throw std::runtime_error("Conflicting enforced states of lamps!");
        }
        break;
      case 2:
        a = setsForLamp[i][0];
        b = setsForLamp[i][1];
        ownerA = nodes[a].ownerId;
        ownerB = nodes[b].ownerId;
        cons =
            (lamps[i] == '1' ? LampConstraint::Same : LampConstraint::Distinct);
        if (nodes[ownerA].componentSize > nodes[ownerB].componentSize) {
          std::swap(a, b);
          std::swap(ownerA, ownerB);
        }
        if (ownerA != ownerB) {
          result -= componentOnBestSize(nodes, ownerA);
          result -= componentOnBestSize(nodes, ownerB);
          if (isLampSetStateKnown(nodes[a].state) &&
              !isLampSetStateKnown(nodes[b].state)) {
            propagateLampState(graph, nodes, b,
                               deducedState(cons, nodes[a].state));
          } else if (isLampSetStateKnown(nodes[b].state) &&
                     !isLampSetStateKnown(nodes[a].state)) {
            propagateLampState(graph, nodes, a,
                               deducedState(cons, nodes[b].state));
          } else if (!isLampSetStateKnown(nodes[b].state) &&
                     !isLampSetStateKnown(nodes[a].state)) {
            propagateLampState(graph, nodes, a,
                               deducedState(cons, nodes[b].state));
          }
          nodes[ownerB].onColorSize += nodes[ownerA].onColorSize;
          nodes[ownerB].offColorSize += nodes[ownerA].offColorSize;
          nodes[ownerB].componentSize += nodes[ownerA].componentSize;
          result += componentOnBestSize(nodes, ownerB);
          switchOwner(graph, nodes, a, ownerB);
        }
        graph[a].insert(Edge(b, cons));
        graph[b].insert(Edge(a, cons));
        break;
      default:
        throw std::runtime_error("Malformed input data!");
    }
    std::cout << result << std::endl;
  }
  return 0;
}

```