Problem: Alice has a computer that operates on w-bit integers. The computer has n registers for values. The current content of the registers is given as an array a_1, a_2, …, a_n. 

The computer uses so-called "number gates" to manipulate this data. Each "number gate" takes two registers as inputs and calculates a function of the two values stored in those registers. Note that you can use the same register as both inputs.

Each "number gate" is assembled from bit gates. There are six types of bit gates: AND, OR, XOR, NOT AND, NOT OR, and NOT XOR, denoted "A", "O", "X", "a", "o", "x", respectively. Each bit gate takes two bits as input. Its output given the input bits b_1, b_2 is given below:

\begin{matrix} b_1 & b_2 & A & O & X & a & o & x \\\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\\ 0 & 1 & 0 & 1 & 1 & 1 & 0 & 0 \\\ 1 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\\ 1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 \\\ \end{matrix} 

To build a "number gate", one takes w bit gates and assembles them into an array. A "number gate" takes two w-bit integers x_1 and x_2 as input. The "number gate" splits the integers into w bits and feeds the i-th bit of each input to the i-th bit gate. After that, it assembles the resulting bits again to form an output word. 

For instance, for 4-bit computer we might have a "number gate" "AXoA" (AND, XOR, NOT OR, AND). For two inputs, 13 = 1101_2 and 10 = 1010_2, this returns 12 = 1100_2, as 1 and 1 is 1, 1 xor 0 is 1, not (0 or 1) is 0, and finally 1 and 0 is 0. 

You are given a description of m "number gates". For each gate, your goal is to report the number of register pairs for which the "number gate" outputs the number 0. In other words, find the number of ordered pairs (i,j) where 1 ≤ i,j ≤ n, such that w_k(a_i, a_j) = 0, where w_k is the function computed by the k-th "number gate".

Input

The first line contains three integers: w, n, and m~(1 ≤ w ≤ 12, 1 ≤ n ≤ 3⋅ 10^4, 1 ≤ m ≤ 5⋅ 10^4) — the word size, the number of variables, and the number of gates.

The second line contains n integers a_1, a_2, …, a_n (0 ≤ a_i < 2^w) — the value of variables stored in the registers.

Each of the next m lines contains a string g_j~(|g_j| = w) with a description of a single gate. Each character of g_j is one of "A", "O", "X", "a", "o", "x".

Output

Print m lines. The i-th line should contain the number of ordered pairs of variables for which the i-th gate returns zero.

Examples

Input

4 3 1
13 10 6
AXoA


Output

3


Input

1 7 6
0 1 1 0 1 0 0
A
O
X
a
o
x


Output

40
16
25
9
33
24


Input

6 2 4
47 12
AOXaox
AAaaAA
xxxxxx
XXXXXX


Output

2
3
0
2


Input

2 2 2
2 0
xO
Ox


Output

2
0

Note

In the first test case, the inputs in binary are 1101, 1010, 0110. The pairs that return 0 are (13, 6), (6, 13), and (6, 6). As it was already mentioned in the problem statement, 13 ⊕ 10 = 10 ⊕ 13 = 12. The other pairs are 13 ⊕ 13 = 11, 10 ⊕ 10 = 8 and 10 ⊕ 6 = 6 ⊕ 10 = 4. 

Solution (in CPP):

Solution:
#include <bits/stdc++.h>
#pragma GCC optimize("O3", "unroll-loops")
using namespace std;
namespace {
const int MOD107 = 1000000007;
const int MOD998 = 998244353;
using ll = long long;
using ull = unsigned long long;
using pii = pair<int, int>;
using pll = pair<ll, ll>;
template <class T>
using minheap = priority_queue<T, vector<T>, greater<T>>;
template <class T>
using maxheap = priority_queue<T, vector<T>, less<T>>;
mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());
struct Modular {
 private:
  long long MODVALUE = 0;
  long long val = 0;
  Modular initModular(long long val) const { return Modular(val, MODVALUE); }
  inline void validate() const {}
  inline void validate(const Modular& other) const {}

 public:
  explicit operator long long() const { return val; }
  template <class T>
  Modular& operator=(const T& other) {
    val = (other % MODVALUE + MODVALUE) % MODVALUE;
    return *this;
  }
  template <class T>
  void setMod(const T& MOD) {
    MODVALUE = MOD;
  }
  Modular& operator+=(const Modular& other) {
    validate(other);
    val += other.val;
    if (val >= MODVALUE) val -= MODVALUE;
    return *this;
  }
  Modular operator+(const Modular& other) const {
    Modular tmp = *this;
    return tmp += other;
  }
  template <class T>
  Modular& operator+=(const T& other) {
    return *this += initModular(other);
  }
  template <class T>
  Modular operator+(const T& other) const {
    return *this + initModular(other);
  }
  template <class T>
  friend Modular operator+(const T& other, const Modular& m) {
    return m.initModular(other) + m;
  }
  Modular& operator++() { return *this += 1; }
  Modular operator++(int) {
    Modular tmp = *this;
    ++*this;
    return tmp;
  }
  Modular operator-() const { return initModular(-val); }
  Modular& operator-=(const Modular& other) {
    validate(other);
    val -= other.val;
    if (val < 0) val += MODVALUE;
    return *this;
  }
  Modular operator-(const Modular& other) const {
    Modular tmp = *this;
    return tmp -= other;
  }
  template <class T>
  Modular& operator-=(const T& other) {
    return *this -= initModular(other);
  }
  template <class T>
  Modular operator-(const T& other) const {
    return *this - initModular(other);
  }
  template <class T>
  friend Modular operator-(const T& other, const Modular& m) {
    return m.initModular(other) - m;
  }
  Modular& operator--() { return *this -= 1; }
  Modular operator--(int) {
    Modular tmp = *this;
    --*this;
    return tmp;
  }
  Modular& operator*=(const Modular& other) {
    validate(other);
    val *= other.val;
    val %= MODVALUE;
    if (val < 0) val += MODVALUE;
    return *this;
  }
  Modular operator*(const Modular& other) const {
    Modular tmp = *this;
    return tmp *= other;
  }
  template <class T>
  Modular& operator*=(const T& other) {
    return *this *= initModular(other);
  }
  template <class T>
  Modular operator*(const T& other) const {
    return *this * initModular(other);
  }
  template <class T>
  friend Modular operator*(const T& other, const Modular& m) {
    return m.initModular(other) * m;
  }
  Modular pow(long long power) const {
    Modular m = *this, ans = initModular(1);
    while (power) {
      if (power & 1) ans *= m;
      m *= m;
      power >>= 1;
    }
    return ans;
  }
  Modular pow(const Modular& other) const { return pow(other.val); }
  Modular inv() const { return pow(MODVALUE - 2); }
  Modular& operator/=(const Modular& other) {
    val *= other.inv().val;
    val %= MODVALUE;
    if (val < 0) val += MODVALUE;
    return *this;
  }
  Modular operator/(const Modular& other) const {
    Modular tmp = *this;
    return tmp /= other;
  }
  template <class T>
  Modular& operator/=(const T& other) {
    return *this /= initModular(other);
  }
  template <class T>
  Modular operator/(const T& other) const {
    return *this / initModular(other);
  }
  template <class T>
  friend Modular operator/(const T& other, const Modular& m) {
    return m.initModular(other) / m;
  }
  friend istream& operator>>(istream& in, Modular& m) {
    in >> m.val;
    m.val %= m.MODVALUE;
    if (m.val < 0) m.val += m.MODVALUE;
    return in;
  }
  friend ostream& operator<<(ostream& out, const Modular& m) {
    return out << m.val;
  }
  Modular() {
    val = 0;
    MODVALUE = 0;
  }
  Modular(long long val, long long MOD)
      : MODVALUE(MOD), val((val % MOD + MOD) % MOD) {}
};
using Mint = Modular;
inline ll binpow(ll b, ll p, ll mod) {
  b %= mod;
  ll ans = 1;
  for (; p > 0; p >>= 1) {
    if (p & 1) {
      ans *= b;
      ans %= mod;
    }
    b *= b;
    b %= mod;
  }
  return ans;
}
inline ll max(ll a, int b) { return (a > b ? a : b); }
inline ll max(int a, ll b) { return (a > b ? a : b); }
inline ll min(ll a, int b) { return (a < b ? a : b); }
inline ll min(int a, ll b) { return (a < b ? a : b); }
template <class T>
inline bool chkmin(T& x, const T& y) {
  return x > y ? x = y, 1 : 0;
}
template <class T>
inline bool chkmax(T& x, const T& y) {
  return x < y ? x = y, 1 : 0;
}
}  // namespace
namespace IO {
template <class T>
inline void _R(T& x) {
  cin >> x;
}
template <class T1, class T2>
inline void _R(pair<T1, T2>& x) {
  _R(x.first);
  _R(x.second);
}
template <class T>
inline void _R(vector<T>& x) {
  for (auto& i : x) _R(i);
}
template <class T>
inline void _R(deque<T>& x) {
  for (auto& i : x) _R(i);
}
inline void R() {}
template <class T1, class... T2>
inline void R(T1& x, T2&... y) {
  _R(x);
  R(y...);
}
template <class T>
inline void _W(const T& x);
template <class T1, class T2>
inline void _W(const pair<T1, T2>& p);
template <class... T>
inline void _W(const vector<T...>& x);
template <class... T>
inline void _W(const deque<T...>& x);
template <class... T>
inline void _W(const forward_list<T...>& x);
template <class... T>
inline void _W(const list<T...>& x);
template <class... T>
inline void _W(const set<T...>& x);
template <class... T>
inline void _W(const multiset<T...>& x);
template <class... T>
inline void _W(const unordered_set<T...>& x);
template <class... T>
inline void _W(const unordered_multiset<T...>& x);
template <class... T>
inline void _W(const map<T...>& x);
template <class... T>
inline void _W(const multimap<T...>& x);
template <class... T>
inline void _W(const unordered_map<T...>& x);
template <class... T>
inline void _W(const unordered_multimap<T...>& x);
template <class Container>
inline void printContainer(const Container& x) {
  bool isFirst = 1;
  for (auto i : x) {
    if (!isFirst) cout << ' ';
    isFirst = 0;
    _W(i);
  }
}
template <class T>
inline void _W(const T& x) {
  cout << x;
}
template <class T1, class T2>
inline void _W(const pair<T1, T2>& p) {
  _W(p.first);
  _W(' ');
  _W(p.second);
}
template <class... T>
inline void _W(const vector<T...>& x) {
  printContainer(x);
}
template <class... T>
inline void _W(const deque<T...>& x) {
  printContainer(x);
}
template <class... T>
inline void _W(const forward_list<T...>& x) {
  printContainer(x);
}
template <class... T>
inline void _W(const list<T...>& x) {
  printContainer(x);
}
template <class... T>
inline void _W(const set<T...>& x) {
  printContainer(x);
}
template <class... T>
inline void _W(const multiset<T...>& x) {
  printContainer(x);
}
template <class... T>
inline void _W(const unordered_set<T...>& x) {
  printContainer(x);
}
template <class... T>
inline void _W(const unordered_multiset<T...>& x) {
  printContainer(x);
}
template <class... T>
inline void _W(const map<T...>& x) {
  printContainer(x);
}
template <class... T>
inline void _W(const multimap<T...>& x) {
  printContainer(x);
}
template <class... T>
inline void _W(const unordered_map<T...>& x) {
  printContainer(x);
}
template <class... T>
inline void _W(const unordered_multimap<T...>& x) {
  printContainer(x);
}
inline void W() { cout << '\n'; }
template <class T>
inline void W(const T& x) {
  _W(x);
  W();
}
template <class T1, class... T2>
inline void W(const T1& x, const T2&... y) {
  _W(x);
  _W(' ');
  W(y...);
}
}  // namespace IO
using namespace IO;
namespace outputFormat {
inline string Case(int tc) { return "Case #" + to_string(tc) + ':'; }
inline string YN(bool b) { return b ? "Yes" : "No"; }
}  // namespace outputFormat
using namespace outputFormat;
namespace std {
template <class T1, class T2>
struct hash<pair<T1, T2>> {
  std::size_t operator()(pair<T1, T2> _) const {
    return hash<T1>{}(_.first) ^ hash<T2>{}(_.second);
  }
};
}  // namespace std
int main() {
  int w, n, m;
  R(w, n, m);
  vector<int> a(n);
  R(a);
  const int MAXW = 12;
  const int FULL = 1 << MAXW;
  vector<int> cnt(FULL, 0);
  for (int i = 0; i < n; i++) cnt[a[i]]++;
  vector<int> pow3(MAXW + 1, 1);
  for (int i = 1; i <= MAXW; i++) pow3[i] = pow3[i - 1] * 3;
  vector<vector<int>> bin_bits(FULL, vector<int>(MAXW));
  for (int mask = 0; mask < FULL; mask++)
    for (int i = 0; i < MAXW; i++) bin_bits[mask][i] = (mask >> i) & 1;
  const int MFULL = pow3[MAXW];
  assert(MFULL == 531441);
  vector<int> mcnt(MFULL, 0);
  for (int mask = 0; mask < FULL; mask++) {
    int mmask = 0;
    for (int i = 0; i < MAXW; i++) mmask += bin_bits[mask][i] * pow3[i];
    mcnt[mmask] = cnt[mask];
  }
  for (int mmask = 0; mmask < MFULL; mmask++) {
    vector<int> bits(MAXW);
    int tmp = mmask;
    for (int i = 0; i < MAXW; i++) {
      bits[i] = tmp % 3;
      tmp /= 3;
    }
    int high2 = MAXW - 1;
    while (high2 >= 0 && bits[high2] != 2) high2--;
    if (high2 >= 0) {
      assert(mcnt[mmask] == 0);
      mcnt[mmask] = mcnt[mmask - pow3[high2]] + mcnt[mmask - pow3[high2] * 2];
    }
  }
  vector<vector<int>> allowed_mask(128, vector<int>(2, 0));
  vector<char> gates = {'A', 'O', 'X', 'a', 'o', 'x'};
  for (auto& c : gates) {
    for (int a = 0; a <= 1; a++) {
      for (int b = 0; b <= 1; b++) {
        if (c == 'A' && (a & b) == 0) allowed_mask[c][a] |= (1 << b);
        if (c == 'O' && (a | b) == 0) allowed_mask[c][a] |= (1 << b);
        if (c == 'X' && (a ^ b) == 0) allowed_mask[c][a] |= (1 << b);
        if (c == 'a' && (a & b) == 1) allowed_mask[c][a] |= (1 << b);
        if (c == 'o' && (a | b) == 1) allowed_mask[c][a] |= (1 << b);
        if (c == 'x' && (a ^ b) == 1) allowed_mask[c][a] |= (1 << b);
      }
    }
  }
  while (m--) {
    string s;
    R(s);
    reverse(s.begin(), s.end());
    while ((int)s.size() < MAXW) s += 'A';
    vector<int> reqMask(FULL, 0);
    vector<bool> can(FULL, 1);
    for (int i = 0; i < MAXW; i++) {
      int bit = (1 << i);
      for (int mask = 0; mask < bit; mask++) {
        if (!can[mask]) {
          can[mask | bit] = 0;
        } else {
          reqMask[mask | bit] =
              reqMask[mask] + (allowed_mask[s[i]][1] == 0
                                   ? can[mask | bit] = 0
                                   : (allowed_mask[s[i]][1] - 1) * pow3[i]);
          reqMask[mask] += (allowed_mask[s[i]][0] == 0
                                ? can[mask] = 0
                                : (allowed_mask[s[i]][0] - 1) * pow3[i]);
        }
      }
    }
    int ans = 0;
    for (int mask = 0; mask < FULL; mask++)
      if (can[mask]) ans += cnt[mask] * mcnt[reqMask[mask]];
    W(ans);
  }
  return 0;
}
static auto __init__ = []() {
  ios::sync_with_stdio(false);
  cin.tie(0);
  cout << fixed << setprecision(10);
  return 42;
}();