#include <bits/stdc++.h>
using namespace std;
template <int MOD_>
struct modnum {
  static constexpr int MOD = MOD_;
  static_assert(MOD_ > 0, "MOD must be positive");

 private:
  using ll = long long;
  int v;
  static int minv(int a, int m) {
    a %= m;
    assert(a);
    return a == 1 ? 1 : int(m - ll(minv(m, a)) * ll(m) / a);
  }

 public:
  modnum() : v(0) {}
  modnum(ll v_) : v(int(v_ % MOD)) {
    if (v < 0) v += MOD;
  }
  explicit operator int() const { return v; }
  friend std::ostream& operator<<(std::ostream& out, const modnum& n) {
    return out << int(n);
  }
  friend std::istream& operator>>(std::istream& in, modnum& n) {
    ll v_;
    in >> v_;
    n = modnum(v_);
    return in;
  }
  friend bool operator==(const modnum& a, const modnum& b) {
    return a.v == b.v;
  }
  friend bool operator!=(const modnum& a, const modnum& b) {
    return a.v != b.v;
  }
  modnum inv() const {
    modnum res;
    res.v = minv(v, MOD);
    return res;
  }
  friend modnum inv(const modnum& m) { return m.inv(); }
  modnum neg() const {
    modnum res;
    res.v = v ? MOD - v : 0;
    return res;
  }
  friend modnum neg(const modnum& m) { return m.neg(); }
  modnum operator-() const { return neg(); }
  modnum operator+() const { return modnum(*this); }
  modnum& operator++() {
    v++;
    if (v == MOD) v = 0;
    return *this;
  }
  modnum& operator--() {
    if (v == 0) v = MOD;
    v--;
    return *this;
  }
  modnum& operator+=(const modnum& o) {
    v += o.v;
    if (v >= MOD) v -= MOD;
    return *this;
  }
  modnum& operator-=(const modnum& o) {
    v -= o.v;
    if (v < 0) v += MOD;
    return *this;
  }
  modnum& operator*=(const modnum& o) {
    v = int(ll(v) * ll(o.v) % MOD);
    return *this;
  }
  modnum& operator/=(const modnum& o) { return *this *= o.inv(); }
  friend modnum operator++(modnum& a, int) {
    modnum r = a;
    ++a;
    return r;
  }
  friend modnum operator--(modnum& a, int) {
    modnum r = a;
    --a;
    return r;
  }
  friend modnum operator+(const modnum& a, const modnum& b) {
    return modnum(a) += b;
  }
  friend modnum operator-(const modnum& a, const modnum& b) {
    return modnum(a) -= b;
  }
  friend modnum operator*(const modnum& a, const modnum& b) {
    return modnum(a) *= b;
  }
  friend modnum operator/(const modnum& a, const modnum& b) {
    return modnum(a) /= b;
  }
};
template <typename T>
T pow(T a, long long b) {
  assert(b >= 0);
  T r = 1;
  while (b) {
    if (b & 1) r *= a;
    b >>= 1;
    a *= a;
  }
  return r;
}
using num = modnum<998244353>;
vector<num> fact, ifact;
void init() {
  int N = 1100000;
  fact = {1};
  for (int i = 1; i < N; i++) fact.push_back(i * fact[i - 1]);
  ifact.resize(N);
  ifact.back() = 1 / fact.back();
  for (int i = N - 1; i > 0; i--) ifact[i - 1] = i * ifact[i];
}
num ncr(int n, int k) {
  if (k < 0 || k > n) return 0;
  return fact[n] * ifact[k] * ifact[n - k];
}
int main() {
  ios_base::sync_with_stdio(false), cin.tie(nullptr);
  init();
  int n, k;
  cin >> n >> k;
  vector<int> l(n), r(n);
  for (int i = 0; i < n; i++) {
    int b;
    cin >> b;
    l[i] = max(0, b - k);
    r[i] = min(n, b + k);
    if (l[i] > r[i]) {
      cout << 0 << '\n';
      return 0;
    }
  }
  vector<vector<num> > dp(1, vector<num>(1, 1));
  int pl = 0;
  int pr = 0;
  for (int i = 0; i < n; i++) {
    int nl = l[i];
    int nr = r[i];
    vector<vector<num> > ndp(i + 2, vector<num>(nr - nl + 1, 0));
    for (int a = 0; a < dp.size(); a++) {
      for (int pmex = 0; pmex < dp[a].size(); pmex++) {
        dp[a][pmex] *= fact[a];
      }
    }
    for (int T = 0 + pl; T < (int)dp.size() + pr; T++) {
      int L = 0;
      int R = 0;
      num cur = 0;
      for (int nmex = nl; nmex <= nr && nmex <= T + 1; nmex++) {
        if (T - (nmex - 1) >= ndp.size()) continue;
        int r = min(pr, nmex - 1) + 1;
        int l = max(pl, T - int(dp.size()) + 1);
        if (l < r) {
          while (L < l && L < R) {
            cur -= dp[T - L][L - pl];
            L++;
          }
          while (R > r && R > L) {
            R--;
            cur -= dp[T - R][R - pl];
          }
          if (L == R) {
            L = R = l;
          }
          while (R < r) {
            cur += dp[T - R][R - pl];
            R++;
          }
          while (L > l) {
            L--;
            cur += dp[T - L][L - pl];
          }
        }
        ndp[T - (nmex - 1)][nmex - nl] += cur;
      }
    }
    for (int a = 0; a < dp.size(); a++) {
      for (num& x : dp[a]) x *= ifact[a];
    }
    for (int a = 0; a < ndp.size(); a++) {
      for (num& x : ndp[a]) x *= ifact[a];
    }
    for (int a = 0; a < dp.size(); a++) {
      for (int pmex = 0; pmex < dp[a].size(); pmex++) {
        int mex = pl + pmex;
        if (mex >= nl && mex <= nr) {
          ndp[a + 1][mex - nl] += dp[a][pmex] * 1;
          ndp[a][mex - nl] += dp[a][pmex] * (mex + a);
        }
      }
    }
    pl = nl;
    pr = nr;
    dp = ndp;
  }
  num ans = 0;
  for (int a = 0; a < dp.size(); a++) {
    for (int pmex = 0; pmex < dp[a].size(); pmex++) {
      int mex = pl + pmex;
      ans += dp[a][pmex] * ncr(n - mex, a) * fact[a];
    }
  }
  cout << ans << '\n';
}

### Prompt

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

For an array c of nonnegative integers, MEX(c) denotes the smallest nonnegative integer that doesn't appear in it. For example, MEX([0, 1, 3]) = 2, MEX([42]) = 0.

You are given integers n, k, and an array [b_1, b_2, …, b_n].

Find the number of arrays [a_1, a_2, …, a_n], for which the following conditions hold:

  * 0 ≤ a_i ≤ n for each i for each i from 1 to n.

  * |MEX([a_1, a_2, …, a_i]) - b_i| ≤ k for each i from 1 to n.




As this number can be very big, output it modulo 998 244 353.

Input

The first line of the input contains two integers n, k (1 ≤ n ≤ 2000, 0 ≤ k ≤ 50).

The second line of the input contains n integers b_1, b_2, …, b_n (-k ≤ b_i ≤ n+k) — elements of the array b.

Output

Output a single integer — the number of arrays which satisfy the conditions from the statement, modulo 998 244 353.

Examples

Input


4 0
0 0 0 0


Output


256


Input


4 1
0 0 0 0


Output


431


Input


4 1
0 0 1 1


Output


509


Input


5 2
0 0 2 2 0


Output


6546


Input


3 2
-2 0 4


Output


11

### Response

```cpp
#include <bits/stdc++.h>
using namespace std;
template <int MOD_>
struct modnum {
  static constexpr int MOD = MOD_;
  static_assert(MOD_ > 0, "MOD must be positive");

 private:
  using ll = long long;
  int v;
  static int minv(int a, int m) {
    a %= m;
    assert(a);
    return a == 1 ? 1 : int(m - ll(minv(m, a)) * ll(m) / a);
  }

 public:
  modnum() : v(0) {}
  modnum(ll v_) : v(int(v_ % MOD)) {
    if (v < 0) v += MOD;
  }
  explicit operator int() const { return v; }
  friend std::ostream& operator<<(std::ostream& out, const modnum& n) {
    return out << int(n);
  }
  friend std::istream& operator>>(std::istream& in, modnum& n) {
    ll v_;
    in >> v_;
    n = modnum(v_);
    return in;
  }
  friend bool operator==(const modnum& a, const modnum& b) {
    return a.v == b.v;
  }
  friend bool operator!=(const modnum& a, const modnum& b) {
    return a.v != b.v;
  }
  modnum inv() const {
    modnum res;
    res.v = minv(v, MOD);
    return res;
  }
  friend modnum inv(const modnum& m) { return m.inv(); }
  modnum neg() const {
    modnum res;
    res.v = v ? MOD - v : 0;
    return res;
  }
  friend modnum neg(const modnum& m) { return m.neg(); }
  modnum operator-() const { return neg(); }
  modnum operator+() const { return modnum(*this); }
  modnum& operator++() {
    v++;
    if (v == MOD) v = 0;
    return *this;
  }
  modnum& operator--() {
    if (v == 0) v = MOD;
    v--;
    return *this;
  }
  modnum& operator+=(const modnum& o) {
    v += o.v;
    if (v >= MOD) v -= MOD;
    return *this;
  }
  modnum& operator-=(const modnum& o) {
    v -= o.v;
    if (v < 0) v += MOD;
    return *this;
  }
  modnum& operator*=(const modnum& o) {
    v = int(ll(v) * ll(o.v) % MOD);
    return *this;
  }
  modnum& operator/=(const modnum& o) { return *this *= o.inv(); }
  friend modnum operator++(modnum& a, int) {
    modnum r = a;
    ++a;
    return r;
  }
  friend modnum operator--(modnum& a, int) {
    modnum r = a;
    --a;
    return r;
  }
  friend modnum operator+(const modnum& a, const modnum& b) {
    return modnum(a) += b;
  }
  friend modnum operator-(const modnum& a, const modnum& b) {
    return modnum(a) -= b;
  }
  friend modnum operator*(const modnum& a, const modnum& b) {
    return modnum(a) *= b;
  }
  friend modnum operator/(const modnum& a, const modnum& b) {
    return modnum(a) /= b;
  }
};
template <typename T>
T pow(T a, long long b) {
  assert(b >= 0);
  T r = 1;
  while (b) {
    if (b & 1) r *= a;
    b >>= 1;
    a *= a;
  }
  return r;
}
using num = modnum<998244353>;
vector<num> fact, ifact;
void init() {
  int N = 1100000;
  fact = {1};
  for (int i = 1; i < N; i++) fact.push_back(i * fact[i - 1]);
  ifact.resize(N);
  ifact.back() = 1 / fact.back();
  for (int i = N - 1; i > 0; i--) ifact[i - 1] = i * ifact[i];
}
num ncr(int n, int k) {
  if (k < 0 || k > n) return 0;
  return fact[n] * ifact[k] * ifact[n - k];
}
int main() {
  ios_base::sync_with_stdio(false), cin.tie(nullptr);
  init();
  int n, k;
  cin >> n >> k;
  vector<int> l(n), r(n);
  for (int i = 0; i < n; i++) {
    int b;
    cin >> b;
    l[i] = max(0, b - k);
    r[i] = min(n, b + k);
    if (l[i] > r[i]) {
      cout << 0 << '\n';
      return 0;
    }
  }
  vector<vector<num> > dp(1, vector<num>(1, 1));
  int pl = 0;
  int pr = 0;
  for (int i = 0; i < n; i++) {
    int nl = l[i];
    int nr = r[i];
    vector<vector<num> > ndp(i + 2, vector<num>(nr - nl + 1, 0));
    for (int a = 0; a < dp.size(); a++) {
      for (int pmex = 0; pmex < dp[a].size(); pmex++) {
        dp[a][pmex] *= fact[a];
      }
    }
    for (int T = 0 + pl; T < (int)dp.size() + pr; T++) {
      int L = 0;
      int R = 0;
      num cur = 0;
      for (int nmex = nl; nmex <= nr && nmex <= T + 1; nmex++) {
        if (T - (nmex - 1) >= ndp.size()) continue;
        int r = min(pr, nmex - 1) + 1;
        int l = max(pl, T - int(dp.size()) + 1);
        if (l < r) {
          while (L < l && L < R) {
            cur -= dp[T - L][L - pl];
            L++;
          }
          while (R > r && R > L) {
            R--;
            cur -= dp[T - R][R - pl];
          }
          if (L == R) {
            L = R = l;
          }
          while (R < r) {
            cur += dp[T - R][R - pl];
            R++;
          }
          while (L > l) {
            L--;
            cur += dp[T - L][L - pl];
          }
        }
        ndp[T - (nmex - 1)][nmex - nl] += cur;
      }
    }
    for (int a = 0; a < dp.size(); a++) {
      for (num& x : dp[a]) x *= ifact[a];
    }
    for (int a = 0; a < ndp.size(); a++) {
      for (num& x : ndp[a]) x *= ifact[a];
    }
    for (int a = 0; a < dp.size(); a++) {
      for (int pmex = 0; pmex < dp[a].size(); pmex++) {
        int mex = pl + pmex;
        if (mex >= nl && mex <= nr) {
          ndp[a + 1][mex - nl] += dp[a][pmex] * 1;
          ndp[a][mex - nl] += dp[a][pmex] * (mex + a);
        }
      }
    }
    pl = nl;
    pr = nr;
    dp = ndp;
  }
  num ans = 0;
  for (int a = 0; a < dp.size(); a++) {
    for (int pmex = 0; pmex < dp[a].size(); pmex++) {
      int mex = pl + pmex;
      ans += dp[a][pmex] * ncr(n - mex, a) * fact[a];
    }
  }
  cout << ans << '\n';
}

```