#include <bits/stdc++.h>
using namespace std;
template <typename T, typename U>
std::istream& operator>>(std::istream& i, pair<T, U>& p) {
  i >> p.first >> p.second;
  return i;
}
template <typename T>
std::istream& operator>>(std::istream& i, vector<T>& t) {
  for (auto& v : t) {
    i >> v;
  }
  return i;
}
template <typename T, typename U>
std::ostream& operator<<(std::ostream& o, const pair<T, U>& p) {
  o << p.first << ' ' << p.second;
  return o;
}
template <typename T>
std::ostream& operator<<(std::ostream& o, const vector<T>& t) {
  if (t.empty()) o << '\n';
  for (size_t i = 0; i < t.size(); ++i) {
    o << t[i] << " \n"[i == t.size() - 1];
  }
  return o;
}
template <typename T>
using minheap = priority_queue<T, vector<T>, greater<T>>;
template <typename T>
using maxheap = priority_queue<T, vector<T>, less<T>>;
template <typename T>
bool in(T a, T b, T c) {
  return a <= b && b < c;
}
unsigned int logceil(int first) {
  return 8 * sizeof(int) - __builtin_clz(first);
}
namespace std {
template <typename T, typename U>
struct hash<pair<T, U>> {
  hash<T> t;
  hash<U> u;
  size_t operator()(const pair<T, U>& p) const {
    return t(p.first) ^ (u(p.second) << 7);
  }
};
}  // namespace std
template <typename T, typename F>
T bsh(T l, T h, const F& f) {
  T r = -1, m;
  while (l <= h) {
    m = (l + h) / 2;
    if (f(m)) {
      l = m + 1;
      r = m;
    } else {
      h = m - 1;
    }
  }
  return r;
}
template <typename F>
double bshd(double l, double h, const F& f, double p = 1e-9) {
  unsigned int r = 3 + (unsigned int)log2((h - l) / p);
  while (r--) {
    double m = (l + h) / 2;
    if (f(m)) {
      l = m;
    } else {
      h = m;
    }
  }
  return (l + h) / 2;
}
template <typename T, typename F>
T bsl(T l, T h, const F& f) {
  T r = -1, m;
  while (l <= h) {
    m = (l + h) / 2;
    if (f(m)) {
      h = m - 1;
      r = m;
    } else {
      l = m + 1;
    }
  }
  return r;
}
template <typename F>
double bsld(double l, double h, const F& f, double p = 1e-9) {
  unsigned int r = 3 + (unsigned int)log2((h - l) / p);
  while (r--) {
    double m = (l + h) / 2;
    if (f(m)) {
      h = m;
    } else {
      l = m;
    }
  }
  return (l + h) / 2;
}
template <typename T>
T gcd(T a, T b) {
  if (a < b) swap(a, b);
  return b ? gcd(b, a % b) : a;
}
template <typename T>
class vector2 : public vector<vector<T>> {
 public:
  vector2() {}
  vector2(size_t a, size_t b, T t = T())
      : vector<vector<T>>(a, vector<T>(b, t)) {}
};
template <typename T>
class vector3 : public vector<vector2<T>> {
 public:
  vector3() {}
  vector3(size_t a, size_t b, size_t c, T t = T())
      : vector<vector2<T>>(a, vector2<T>(b, c, t)) {}
};
template <typename T>
class vector4 : public vector<vector3<T>> {
 public:
  vector4() {}
  vector4(size_t a, size_t b, size_t c, size_t d, T t = T())
      : vector<vector3<T>>(a, vector3<T>(b, c, d, t)) {}
};
template <typename T>
class vector5 : public vector<vector4<T>> {
 public:
  vector5() {}
  vector5(size_t a, size_t b, size_t c, size_t d, size_t e, T t = T())
      : vector<vector4<T>>(a, vector4<T>(b, c, d, e, t)) {}
};
template <unsigned int N>
class Field {
  inline unsigned int pow(unsigned int a, unsigned int p) {
    unsigned int r = 1, e = a;
    while (p) {
      if (p & 1) {
        r = ((unsigned long long)r * e) % N;
      }
      e = ((unsigned long long)e * e) % N;
      p >>= 1;
    }
    return r;
  }
  inline unsigned int inv(unsigned int a) { return pow(a, N - 2); }

 public:
  inline Field(int first = 0) : v(first) {}
  inline Field<N> pow(int p) { return (*this) ^ p; }
  inline Field<N> operator^(int p) { return {(int)pow(v, (unsigned int)p)}; }
  inline Field<N>& operator+=(const Field<N>& o) {
    if ((long long)v + o.v >= N)
      v += o.v - N;
    else
      v += o.v;
    return *this;
  }
  inline Field<N>& operator-=(const Field<N>& o) {
    if (v < o.v)
      v -= o.v - N;
    else
      v -= o.v;
    return *this;
  }
  inline Field<N>& operator*=(const Field<N>& o) {
    v = (unsigned long long)v * o.v % N;
    return *this;
  }
  inline Field<N>& operator/=(const Field<N>& o) { return *this *= inv(o.v); }
  inline Field<N> operator+(const Field<N>& o) const {
    Field<N> r{*this};
    return r += o;
  }
  inline Field<N> operator-(const Field<N>& o) const {
    Field<N> r{*this};
    return r -= o;
  }
  inline Field<N> operator*(const Field<N>& o) const {
    Field<N> r{*this};
    return r *= o;
  }
  inline Field<N> operator/(const Field<N>& o) const {
    Field<N> r{*this};
    return r /= o;
  }
  inline Field<N> operator-() {
    if (v)
      return {(int)(N - v)};
    else
      return {0};
  };
  inline Field<N>& operator++() {
    ++v;
    if (v == N) v = 0;
    return *this;
  }
  inline Field<N> operator++(int) {
    Field<N> r{*this};
    ++*this;
    return r;
  }
  inline Field<N>& operator--() {
    --v;
    if (v == -1) v = N - 1;
    return *this;
  }
  inline Field<N> operator--(int) {
    Field<N> r{*this};
    --*this;
    return r;
  }
  inline bool operator==(const Field<N>& o) const { return o.v == v; }
  inline bool operator!=(const Field<N>& o) const { return o.v != v; }
  inline explicit operator unsigned int() const { return v; }
  inline static vector<Field<N>> fact(int t) {
    vector<Field<N>> F(t + 1, 1);
    for (int i = 2; i <= t; ++i) {
      F[i] = F[i - 1] * i;
    }
    return F;
  }
  inline static vector<Field<N>> invfact(int t) {
    vector<Field<N>> F(t + 1, 1);
    Field<N> X{1};
    for (int i = 2; i <= t; ++i) {
      X = X * i;
    }
    F[t] = 1 / X;
    for (int i = t - 1; i >= 2; --i) {
      F[i] = F[i + 1] * (i + 1);
    }
    return F;
  }

 private:
  unsigned int v;
};
template <unsigned int N>
istream& operator>>(std::istream& is, Field<N>& f) {
  unsigned int v;
  is >> v;
  f = v;
  return is;
}
template <unsigned int N>
ostream& operator<<(std::ostream& os, const Field<N>& f) {
  return os << (unsigned int)f;
}
template <unsigned int N>
Field<N> operator+(int i, const Field<N>& f) {
  return Field<N>(i) + f;
}
template <unsigned int N>
Field<N> operator-(int i, const Field<N>& f) {
  return Field<N>(i) - f;
}
template <unsigned int N>
Field<N> operator*(int i, const Field<N>& f) {
  return Field<N>(i) * f;
}
template <unsigned int N>
Field<N> operator/(int i, const Field<N>& f) {
  return Field<N>(i) / f;
}
struct Ring {
  template <typename T>
  static T div(T p, T q, T N) {
    T t = 0, nt = 1, r = N, nr = q;
    while (nr != 0) {
      T q = r / nr;
      t -= q * nt;
      r -= q * nr;
      swap(t, nt);
      swap(r, nr);
    }
    t = (t < 0) ? t + N : t;
    r = (r < 0) ? r + N : r;
    if (gcd(p, N) % r != 0) {
      return 0;
    }
    return (t * p / r) % N;
  }
};
class TaskF {
 public:
  Field<1000000007> z(const vector<int>& a, int k) {
    if (a.size() < k)
      return 0;
    else {
      vector<int> B;
      Field<1000000007> ans = 0;
      for (int i = 0; i <= a.size() - k; ++i) {
        int temp = a[i];
        for (int j = i; j <= i + k - 1; ++j) temp = max(temp, a[j]);
        B.push_back(temp);
        ans = ans + temp;
      }
      return ans + z(B, k);
    }
  }
  Field<1000000007> z2(const vector<int>& a, int k) {
    if (a.size() < k)
      return 0;
    else {
      vector<int> B;
      Field<1000000007> ans = 0;
      for (int i = 0; i <= a.size() - k; ++i) {
        int temp = a[i];
        for (int j = i; j <= i + k - 1; ++j) temp = max(temp, a[j]);
        B.push_back(temp);
        ans = ans + temp;
      }
      cerr << B;
      return ans + z2(B, k);
    }
  }
  Field<1000000007> z3(const vector<int>& A, int K) {
    int N = A.size();
    vector<int> S;
    vector<int> L(A.size()), H(A.size(), A.size());
    for (int i = 0; i < A.size(); ++i) {
      while (!S.empty() && A[S.back()] < A[i]) {
        H[S.back()] = i;
        S.pop_back();
      }
      if (S.empty() || A[S.back()] < A[i])
        L[i] = -1;
      else
        L[i] = S.back();
      S.push_back(i);
    }
    Field<1000000007> ans = 0;
    for (int i = 0; i < N; ++i) {
      int l1 = i - L[i];
      int l2 = max(H[i] - i - 1, K);
      int lo = min(l1, l2) - (min(l1, l2) - 1) % K;
      int hi = max(l1, l2) - (max(l1, l2) - 1) % K;
      int up = min(N, H[i] - L[i]) - (min(N, H[i] - L[i]) - 1) % K;
      Field<1000000007> cur =
          Field<1000000007>{lo + K + 1} * Field<1000000007>{(lo - 1)} +
          2 * Field<1000000007>{min(H[i] - i, i - L[i])} *
              Field<1000000007>{(hi - lo)} +
          Field<1000000007>{2 * H[i] - 2 * L[i] - hi - up - K} *
              Field<1000000007>{(up - hi)};
      ans += cur * A[i];
    }
    return ans / 2 / K;
  }
  void solve(istream& cin, ostream& cout) {
    int N, K;
    cin >> N >> K;
    vector<int> A(N);
    cin >> A;
    cout << z3(A, K - 1) << endl;
  }
};
int main() {
  ios_base::sync_with_stdio(false);
  cin.tie(nullptr);
  cout.tie(nullptr);
  TaskF solver;
  std::istream& in(std::cin);
  std::ostream& out(std::cout);
  solver.solve(in, out);
  return 0;
}

### Prompt

Your task is to create a cpp solution to the following problem:

Given an array a of n integers and an integer k (2 ≤ k ≤ n), where each element of the array is denoted by a_i (0 ≤ i < n). Perform the operation z given below on a and print the value of z(a,k) modulo 10^{9}+7.
    
    
      
    function z(array a, integer k):  
        if length(a) < k:  
            return 0  
        else:  
            b = empty array  
            ans = 0  
            for i = 0 .. (length(a) - k):  
                temp = a[i]  
                for j = i .. (i + k - 1):  
                    temp = max(temp, a[j])  
                append temp to the end of b  
                ans = ans + temp  
            return ans + z(b, k)  
    

Input

The first line of input contains two integers n and k (2 ≤ k ≤ n ≤ 10^6) — the length of the initial array a and the parameter k.

The second line of input contains n integers a_0, a_1, …, a_{n - 1} (1 ≤ a_{i} ≤ 10^9) — the elements of the array a.

Output

Output the only integer, the value of z(a,k) modulo 10^9+7.

Examples

Input

3 2
9 1 10


Output

29


Input

5 3
5 8 7 1 9


Output

34

Note

In the first example: 

  * for a=(9,1,10), ans=19 and b=(9,10), 
  * for a=(9,10), ans=10 and b=(10), 
  * for a=(10), ans=0. 



So the returned value is 19+10+0=29.

In the second example: 

  * for a=(5,8,7,1,9), ans=25 and b=(8,8,9), 
  * for a=(8,8,9), ans=9 and b=(9), 
  * for a=(9), ans=0. 



So the returned value is 25+9+0=34.

### Response

```cpp
#include <bits/stdc++.h>
using namespace std;
template <typename T, typename U>
std::istream& operator>>(std::istream& i, pair<T, U>& p) {
  i >> p.first >> p.second;
  return i;
}
template <typename T>
std::istream& operator>>(std::istream& i, vector<T>& t) {
  for (auto& v : t) {
    i >> v;
  }
  return i;
}
template <typename T, typename U>
std::ostream& operator<<(std::ostream& o, const pair<T, U>& p) {
  o << p.first << ' ' << p.second;
  return o;
}
template <typename T>
std::ostream& operator<<(std::ostream& o, const vector<T>& t) {
  if (t.empty()) o << '\n';
  for (size_t i = 0; i < t.size(); ++i) {
    o << t[i] << " \n"[i == t.size() - 1];
  }
  return o;
}
template <typename T>
using minheap = priority_queue<T, vector<T>, greater<T>>;
template <typename T>
using maxheap = priority_queue<T, vector<T>, less<T>>;
template <typename T>
bool in(T a, T b, T c) {
  return a <= b && b < c;
}
unsigned int logceil(int first) {
  return 8 * sizeof(int) - __builtin_clz(first);
}
namespace std {
template <typename T, typename U>
struct hash<pair<T, U>> {
  hash<T> t;
  hash<U> u;
  size_t operator()(const pair<T, U>& p) const {
    return t(p.first) ^ (u(p.second) << 7);
  }
};
}  // namespace std
template <typename T, typename F>
T bsh(T l, T h, const F& f) {
  T r = -1, m;
  while (l <= h) {
    m = (l + h) / 2;
    if (f(m)) {
      l = m + 1;
      r = m;
    } else {
      h = m - 1;
    }
  }
  return r;
}
template <typename F>
double bshd(double l, double h, const F& f, double p = 1e-9) {
  unsigned int r = 3 + (unsigned int)log2((h - l) / p);
  while (r--) {
    double m = (l + h) / 2;
    if (f(m)) {
      l = m;
    } else {
      h = m;
    }
  }
  return (l + h) / 2;
}
template <typename T, typename F>
T bsl(T l, T h, const F& f) {
  T r = -1, m;
  while (l <= h) {
    m = (l + h) / 2;
    if (f(m)) {
      h = m - 1;
      r = m;
    } else {
      l = m + 1;
    }
  }
  return r;
}
template <typename F>
double bsld(double l, double h, const F& f, double p = 1e-9) {
  unsigned int r = 3 + (unsigned int)log2((h - l) / p);
  while (r--) {
    double m = (l + h) / 2;
    if (f(m)) {
      h = m;
    } else {
      l = m;
    }
  }
  return (l + h) / 2;
}
template <typename T>
T gcd(T a, T b) {
  if (a < b) swap(a, b);
  return b ? gcd(b, a % b) : a;
}
template <typename T>
class vector2 : public vector<vector<T>> {
 public:
  vector2() {}
  vector2(size_t a, size_t b, T t = T())
      : vector<vector<T>>(a, vector<T>(b, t)) {}
};
template <typename T>
class vector3 : public vector<vector2<T>> {
 public:
  vector3() {}
  vector3(size_t a, size_t b, size_t c, T t = T())
      : vector<vector2<T>>(a, vector2<T>(b, c, t)) {}
};
template <typename T>
class vector4 : public vector<vector3<T>> {
 public:
  vector4() {}
  vector4(size_t a, size_t b, size_t c, size_t d, T t = T())
      : vector<vector3<T>>(a, vector3<T>(b, c, d, t)) {}
};
template <typename T>
class vector5 : public vector<vector4<T>> {
 public:
  vector5() {}
  vector5(size_t a, size_t b, size_t c, size_t d, size_t e, T t = T())
      : vector<vector4<T>>(a, vector4<T>(b, c, d, e, t)) {}
};
template <unsigned int N>
class Field {
  inline unsigned int pow(unsigned int a, unsigned int p) {
    unsigned int r = 1, e = a;
    while (p) {
      if (p & 1) {
        r = ((unsigned long long)r * e) % N;
      }
      e = ((unsigned long long)e * e) % N;
      p >>= 1;
    }
    return r;
  }
  inline unsigned int inv(unsigned int a) { return pow(a, N - 2); }

 public:
  inline Field(int first = 0) : v(first) {}
  inline Field<N> pow(int p) { return (*this) ^ p; }
  inline Field<N> operator^(int p) { return {(int)pow(v, (unsigned int)p)}; }
  inline Field<N>& operator+=(const Field<N>& o) {
    if ((long long)v + o.v >= N)
      v += o.v - N;
    else
      v += o.v;
    return *this;
  }
  inline Field<N>& operator-=(const Field<N>& o) {
    if (v < o.v)
      v -= o.v - N;
    else
      v -= o.v;
    return *this;
  }
  inline Field<N>& operator*=(const Field<N>& o) {
    v = (unsigned long long)v * o.v % N;
    return *this;
  }
  inline Field<N>& operator/=(const Field<N>& o) { return *this *= inv(o.v); }
  inline Field<N> operator+(const Field<N>& o) const {
    Field<N> r{*this};
    return r += o;
  }
  inline Field<N> operator-(const Field<N>& o) const {
    Field<N> r{*this};
    return r -= o;
  }
  inline Field<N> operator*(const Field<N>& o) const {
    Field<N> r{*this};
    return r *= o;
  }
  inline Field<N> operator/(const Field<N>& o) const {
    Field<N> r{*this};
    return r /= o;
  }
  inline Field<N> operator-() {
    if (v)
      return {(int)(N - v)};
    else
      return {0};
  };
  inline Field<N>& operator++() {
    ++v;
    if (v == N) v = 0;
    return *this;
  }
  inline Field<N> operator++(int) {
    Field<N> r{*this};
    ++*this;
    return r;
  }
  inline Field<N>& operator--() {
    --v;
    if (v == -1) v = N - 1;
    return *this;
  }
  inline Field<N> operator--(int) {
    Field<N> r{*this};
    --*this;
    return r;
  }
  inline bool operator==(const Field<N>& o) const { return o.v == v; }
  inline bool operator!=(const Field<N>& o) const { return o.v != v; }
  inline explicit operator unsigned int() const { return v; }
  inline static vector<Field<N>> fact(int t) {
    vector<Field<N>> F(t + 1, 1);
    for (int i = 2; i <= t; ++i) {
      F[i] = F[i - 1] * i;
    }
    return F;
  }
  inline static vector<Field<N>> invfact(int t) {
    vector<Field<N>> F(t + 1, 1);
    Field<N> X{1};
    for (int i = 2; i <= t; ++i) {
      X = X * i;
    }
    F[t] = 1 / X;
    for (int i = t - 1; i >= 2; --i) {
      F[i] = F[i + 1] * (i + 1);
    }
    return F;
  }

 private:
  unsigned int v;
};
template <unsigned int N>
istream& operator>>(std::istream& is, Field<N>& f) {
  unsigned int v;
  is >> v;
  f = v;
  return is;
}
template <unsigned int N>
ostream& operator<<(std::ostream& os, const Field<N>& f) {
  return os << (unsigned int)f;
}
template <unsigned int N>
Field<N> operator+(int i, const Field<N>& f) {
  return Field<N>(i) + f;
}
template <unsigned int N>
Field<N> operator-(int i, const Field<N>& f) {
  return Field<N>(i) - f;
}
template <unsigned int N>
Field<N> operator*(int i, const Field<N>& f) {
  return Field<N>(i) * f;
}
template <unsigned int N>
Field<N> operator/(int i, const Field<N>& f) {
  return Field<N>(i) / f;
}
struct Ring {
  template <typename T>
  static T div(T p, T q, T N) {
    T t = 0, nt = 1, r = N, nr = q;
    while (nr != 0) {
      T q = r / nr;
      t -= q * nt;
      r -= q * nr;
      swap(t, nt);
      swap(r, nr);
    }
    t = (t < 0) ? t + N : t;
    r = (r < 0) ? r + N : r;
    if (gcd(p, N) % r != 0) {
      return 0;
    }
    return (t * p / r) % N;
  }
};
class TaskF {
 public:
  Field<1000000007> z(const vector<int>& a, int k) {
    if (a.size() < k)
      return 0;
    else {
      vector<int> B;
      Field<1000000007> ans = 0;
      for (int i = 0; i <= a.size() - k; ++i) {
        int temp = a[i];
        for (int j = i; j <= i + k - 1; ++j) temp = max(temp, a[j]);
        B.push_back(temp);
        ans = ans + temp;
      }
      return ans + z(B, k);
    }
  }
  Field<1000000007> z2(const vector<int>& a, int k) {
    if (a.size() < k)
      return 0;
    else {
      vector<int> B;
      Field<1000000007> ans = 0;
      for (int i = 0; i <= a.size() - k; ++i) {
        int temp = a[i];
        for (int j = i; j <= i + k - 1; ++j) temp = max(temp, a[j]);
        B.push_back(temp);
        ans = ans + temp;
      }
      cerr << B;
      return ans + z2(B, k);
    }
  }
  Field<1000000007> z3(const vector<int>& A, int K) {
    int N = A.size();
    vector<int> S;
    vector<int> L(A.size()), H(A.size(), A.size());
    for (int i = 0; i < A.size(); ++i) {
      while (!S.empty() && A[S.back()] < A[i]) {
        H[S.back()] = i;
        S.pop_back();
      }
      if (S.empty() || A[S.back()] < A[i])
        L[i] = -1;
      else
        L[i] = S.back();
      S.push_back(i);
    }
    Field<1000000007> ans = 0;
    for (int i = 0; i < N; ++i) {
      int l1 = i - L[i];
      int l2 = max(H[i] - i - 1, K);
      int lo = min(l1, l2) - (min(l1, l2) - 1) % K;
      int hi = max(l1, l2) - (max(l1, l2) - 1) % K;
      int up = min(N, H[i] - L[i]) - (min(N, H[i] - L[i]) - 1) % K;
      Field<1000000007> cur =
          Field<1000000007>{lo + K + 1} * Field<1000000007>{(lo - 1)} +
          2 * Field<1000000007>{min(H[i] - i, i - L[i])} *
              Field<1000000007>{(hi - lo)} +
          Field<1000000007>{2 * H[i] - 2 * L[i] - hi - up - K} *
              Field<1000000007>{(up - hi)};
      ans += cur * A[i];
    }
    return ans / 2 / K;
  }
  void solve(istream& cin, ostream& cout) {
    int N, K;
    cin >> N >> K;
    vector<int> A(N);
    cin >> A;
    cout << z3(A, K - 1) << endl;
  }
};
int main() {
  ios_base::sync_with_stdio(false);
  cin.tie(nullptr);
  cout.tie(nullptr);
  TaskF solver;
  std::istream& in(std::cin);
  std::ostream& out(std::cout);
  solver.solve(in, out);
  return 0;
}

```