#include <bits/stdc++.h>
#pragma GCC diagnostic ignored "-Wunused-result"
#pragma GCC diagnostic ignored "-Wunused-function"
using namespace std;
namespace {
namespace shik {
template <class T>
void _R(T &x) {
  cin >> x;
}
void _R(int &x) { scanf("%d", &x); }
void _R(int64_t &x) { scanf("%" SCNd64, &x); }
void _R(double &x) { scanf("%lf", &x); }
void _R(char &x) { scanf(" %c", &x); }
void _R(char *x) { scanf("%s", x); }
void R() {}
template <class T, class... U>
void R(T &head, U &...tail) {
  _R(head);
  R(tail...);
}
template <class T>
void _W(const T &x) {
  cout << x;
}
void _W(const int &x) { printf("%d", x); }
void _W(const int64_t &x) { printf("%" PRId64, x); }
void _W(const double &x) { printf("%.16f", x); }
void _W(const char &x) { putchar(x); }
void _W(const char *x) { printf("%s", x); }
template <class T>
void _W(const vector<T> &x) {
  for (auto i = x.begin(); i != x.end(); _W(*i++))
    if (i != x.cbegin()) putchar(' ');
}
void W() {}
template <class T, class... U>
void W(const T &head, const U &...tail) {
  _W(head);
  putchar(sizeof...(tail) ? ' ' : '\n');
  W(tail...);
}
template <class T, class F = less<T>>
void sort_uniq(vector<T> &v, F f = F()) {
  sort(begin(v), end(v), f);
  v.resize(unique(begin(v), end(v)) - begin(v));
}
template <class T>
inline T bit(T x, int i) {
  return (x >> i) & 1;
}
template <class T>
inline bool chkmax(T &a, const T &b) {
  return b > a ? a = b, true : false;
}
template <class T>
inline bool chkmin(T &a, const T &b) {
  return b < a ? a = b, true : false;
}
template <class T>
using MaxHeap = priority_queue<T>;
template <class T>
using MinHeap = priority_queue<T, vector<T>, greater<T>>;
template <int _MOD>
struct ModInt {
  static const auto MOD = _MOD;
  template <class T>
  using integral_only = typename enable_if<is_integral<T>::value>::type;
  int x;
  constexpr ModInt() : x() {}
  template <class T, integral_only<T> * = nullptr>
  ModInt(T _x) {
    x = _x % MOD;
    if (x < 0) x += MOD;
  }
  ModInt operator-() const { return {x == 0 ? 0 : MOD - x}; }
  ModInt &operator+=(ModInt rhs) {
    x += rhs.x;
    if (x >= MOD) x -= MOD;
    return *this;
  }
  ModInt &operator-=(ModInt rhs) {
    x -= rhs.x;
    if (x < 0) x += MOD;
    return *this;
  }
  ModInt &operator*=(ModInt rhs) {
    x = (long long)x * rhs.x % MOD;
    return *this;
  }
  ModInt &operator/=(ModInt rhs) { return *this *= rhs.inv(); }
  ModInt operator+(ModInt rhs) const { return ModInt(*this) += rhs; }
  ModInt operator-(ModInt rhs) const { return ModInt(*this) -= rhs; }
  ModInt operator*(ModInt rhs) const { return ModInt(*this) *= rhs; }
  ModInt operator/(ModInt rhs) const { return ModInt(*this) /= rhs; }
  ModInt inv() const {
    int a = x, b = MOD, u = 1, v = 0;
    while (b != 0) {
      int t = a / b;
      a -= t * b;
      u -= t * v;
      swap(a, b);
      swap(u, v);
    }
    return u;
  }
  template <class T, integral_only<T> * = nullptr>
  ModInt pow(T e) {
    ModInt r = 1, p = *this;
    while (e) {
      if (e & 1) r *= p;
      p *= p;
      e >>= 1;
    }
    return r;
  }
  bool operator==(ModInt rhs) const { return x == rhs.x; }
  bool operator!=(ModInt rhs) const { return x != rhs.x; }
  bool operator<(ModInt rhs) const { return x < rhs.x; }
  bool operator<=(ModInt rhs) const { return x <= rhs.x; }
  bool operator>(ModInt rhs) const { return x > rhs.x; }
  bool operator>=(ModInt rhs) const { return x >= rhs.x; }
  friend string to_string(ModInt i) { return to_string(i.x); }
  friend ostream &operator<<(ostream &os, ModInt i) { return os << i.x; }
};
const int MOD = 1e9 + 7;
using mint = ModInt<MOD>;
const int N = 110;
struct M {
  mint x[N][N];
  const mint *operator[](int i) const { return x[i]; }
  mint *operator[](int i) { return x[i]; }
};
int n;
M operator*(const M &a, const M &b) {
  M c;
  for (int i = 0; i < int(n + 1); i++)
    for (int j = 0; j < int(n + 1); j++) {
      mint s;
      for (int k = 0; k < int(n + 1); k++) s += a[i][k] * b[k][j];
      c[i][j] = s;
    }
  return c;
}
int m;
char s[N];
M dp[N];
void main() {
  R(n, m, s);
  dp[0][0][0] = dp[0][n][n] = dp[1][0][0] = dp[1][n][n] = 2;
  for (int i = (1); i <= int(n - 1); i++) dp[0][i][i] = dp[1][i][i] = 1;
  for (int i = 0; i < int(n); i++)
    if (s[i] == '0') dp[0][i][i + 1] = 1;
  for (int i = 0; i < int(n); i++)
    if (s[i] == '1') dp[1][i][i + 1] = 1;
  for (int i = (2); i <= int(m); i++) dp[i] = dp[i - 1] * dp[i - 2];
  mint ans = dp[m][0][n];
  W(ans);
}
}  // namespace shik
}  // namespace
int main() {
  shik::main();
  return 0;
}

### Prompt

Develop a solution in Cpp to the problem described below:

You are given a binary string s (each character of this string is either 0 or 1).

Let's denote the cost of string t as the number of occurences of s in t. For example, if s is 11 and t is 111011, then the cost of t is 3.

Let's also denote the Fibonacci strings sequence as follows:

  * F(0) is 0;
  * F(1) is 1;
  * F(i) = F(i - 1) + F(i - 2) if i > 1, where  +  means the concatenation of two strings.



Your task is to calculate the sum of costs of all subsequences of the string F(x). Since answer may be large, calculate it modulo 109 + 7.

Input

The first line contains two integers n and x (1 ≤ n ≤ 100, 0 ≤ x ≤ 100) — the length of s and the index of a Fibonacci string you are interested in, respectively.

The second line contains s — a string consisting of n characters. Each of these characters is either 0 or 1.

Output

Print the only integer — the sum of costs of all subsequences of the string F(x), taken modulo 109 + 7. 

Examples

Input

2 4
11


Output

14


Input

10 100
1010101010


Output

553403224

### Response

```cpp
#include <bits/stdc++.h>
#pragma GCC diagnostic ignored "-Wunused-result"
#pragma GCC diagnostic ignored "-Wunused-function"
using namespace std;
namespace {
namespace shik {
template <class T>
void _R(T &x) {
  cin >> x;
}
void _R(int &x) { scanf("%d", &x); }
void _R(int64_t &x) { scanf("%" SCNd64, &x); }
void _R(double &x) { scanf("%lf", &x); }
void _R(char &x) { scanf(" %c", &x); }
void _R(char *x) { scanf("%s", x); }
void R() {}
template <class T, class... U>
void R(T &head, U &...tail) {
  _R(head);
  R(tail...);
}
template <class T>
void _W(const T &x) {
  cout << x;
}
void _W(const int &x) { printf("%d", x); }
void _W(const int64_t &x) { printf("%" PRId64, x); }
void _W(const double &x) { printf("%.16f", x); }
void _W(const char &x) { putchar(x); }
void _W(const char *x) { printf("%s", x); }
template <class T>
void _W(const vector<T> &x) {
  for (auto i = x.begin(); i != x.end(); _W(*i++))
    if (i != x.cbegin()) putchar(' ');
}
void W() {}
template <class T, class... U>
void W(const T &head, const U &...tail) {
  _W(head);
  putchar(sizeof...(tail) ? ' ' : '\n');
  W(tail...);
}
template <class T, class F = less<T>>
void sort_uniq(vector<T> &v, F f = F()) {
  sort(begin(v), end(v), f);
  v.resize(unique(begin(v), end(v)) - begin(v));
}
template <class T>
inline T bit(T x, int i) {
  return (x >> i) & 1;
}
template <class T>
inline bool chkmax(T &a, const T &b) {
  return b > a ? a = b, true : false;
}
template <class T>
inline bool chkmin(T &a, const T &b) {
  return b < a ? a = b, true : false;
}
template <class T>
using MaxHeap = priority_queue<T>;
template <class T>
using MinHeap = priority_queue<T, vector<T>, greater<T>>;
template <int _MOD>
struct ModInt {
  static const auto MOD = _MOD;
  template <class T>
  using integral_only = typename enable_if<is_integral<T>::value>::type;
  int x;
  constexpr ModInt() : x() {}
  template <class T, integral_only<T> * = nullptr>
  ModInt(T _x) {
    x = _x % MOD;
    if (x < 0) x += MOD;
  }
  ModInt operator-() const { return {x == 0 ? 0 : MOD - x}; }
  ModInt &operator+=(ModInt rhs) {
    x += rhs.x;
    if (x >= MOD) x -= MOD;
    return *this;
  }
  ModInt &operator-=(ModInt rhs) {
    x -= rhs.x;
    if (x < 0) x += MOD;
    return *this;
  }
  ModInt &operator*=(ModInt rhs) {
    x = (long long)x * rhs.x % MOD;
    return *this;
  }
  ModInt &operator/=(ModInt rhs) { return *this *= rhs.inv(); }
  ModInt operator+(ModInt rhs) const { return ModInt(*this) += rhs; }
  ModInt operator-(ModInt rhs) const { return ModInt(*this) -= rhs; }
  ModInt operator*(ModInt rhs) const { return ModInt(*this) *= rhs; }
  ModInt operator/(ModInt rhs) const { return ModInt(*this) /= rhs; }
  ModInt inv() const {
    int a = x, b = MOD, u = 1, v = 0;
    while (b != 0) {
      int t = a / b;
      a -= t * b;
      u -= t * v;
      swap(a, b);
      swap(u, v);
    }
    return u;
  }
  template <class T, integral_only<T> * = nullptr>
  ModInt pow(T e) {
    ModInt r = 1, p = *this;
    while (e) {
      if (e & 1) r *= p;
      p *= p;
      e >>= 1;
    }
    return r;
  }
  bool operator==(ModInt rhs) const { return x == rhs.x; }
  bool operator!=(ModInt rhs) const { return x != rhs.x; }
  bool operator<(ModInt rhs) const { return x < rhs.x; }
  bool operator<=(ModInt rhs) const { return x <= rhs.x; }
  bool operator>(ModInt rhs) const { return x > rhs.x; }
  bool operator>=(ModInt rhs) const { return x >= rhs.x; }
  friend string to_string(ModInt i) { return to_string(i.x); }
  friend ostream &operator<<(ostream &os, ModInt i) { return os << i.x; }
};
const int MOD = 1e9 + 7;
using mint = ModInt<MOD>;
const int N = 110;
struct M {
  mint x[N][N];
  const mint *operator[](int i) const { return x[i]; }
  mint *operator[](int i) { return x[i]; }
};
int n;
M operator*(const M &a, const M &b) {
  M c;
  for (int i = 0; i < int(n + 1); i++)
    for (int j = 0; j < int(n + 1); j++) {
      mint s;
      for (int k = 0; k < int(n + 1); k++) s += a[i][k] * b[k][j];
      c[i][j] = s;
    }
  return c;
}
int m;
char s[N];
M dp[N];
void main() {
  R(n, m, s);
  dp[0][0][0] = dp[0][n][n] = dp[1][0][0] = dp[1][n][n] = 2;
  for (int i = (1); i <= int(n - 1); i++) dp[0][i][i] = dp[1][i][i] = 1;
  for (int i = 0; i < int(n); i++)
    if (s[i] == '0') dp[0][i][i + 1] = 1;
  for (int i = 0; i < int(n); i++)
    if (s[i] == '1') dp[1][i][i + 1] = 1;
  for (int i = (2); i <= int(m); i++) dp[i] = dp[i - 1] * dp[i - 2];
  mint ans = dp[m][0][n];
  W(ans);
}
}  // namespace shik
}  // namespace
int main() {
  shik::main();
  return 0;
}

```