#include <bits/stdc++.h>
const int LIM = 12, LOG = 16, N = 40, MOD = 998244353;
int f[N], finv[N];
int addmod(int x) { return x < MOD ? x : x - MOD; }
int add(int x, int y) { return addmod(x + y); }
int submod(int x) { return x < 0 ? x + MOD : x; }
int sub(int x, int y) { return submod(x - y); }
int mul(int x, int y) { return 1LL * x * y % MOD; }
int pow(int x, int y) {
  int res = 1;
  while (y > 0) {
    if (y % 2 == 1) res = mul(res, x);
    x = mul(x, x);
    y /= 2;
  }
  return res;
}
int inv(int x) { return pow(x, MOD - 2); }
struct Data {
  int len, mat[N][N], tmp[N][N];
  void init(int num) {
    len = num;
    for (int i = 0; i <= LOG; ++i)
      for (int j = 0; j < len; ++j) mat[i][j] = 0;
    mat[0][0] = 1;
  }
  void hadamard() {
    for (int i = 1; i < len; i *= 2) {
      for (int j = 0; j <= LOG; ++j)
        for (int k = 0; k < len; ++k)
          if (k & i) {
            tmp[j][k ^ i] = add(mat[j][k ^ i], mat[j][k]);
            tmp[j][k] = sub(mat[j][k ^ i], mat[j][k]);
          }
      std::swap(mat, tmp);
    }
  }
  void invert() {
    for (int i = 0; i <= LOG; ++i)
      for (int j = 0; j < len; ++j) mat[i][j] = mul(mat[i][j], inv(len));
  }
  void hadamardInv() {
    hadamard();
    invert();
  }
  void update(const Data &d) {
    for (int i = 0; i <= LOG; ++i)
      for (int j = 0; j < len; ++j) {
        tmp[i][j] = 0;
        for (int k = 0; k <= i; ++k)
          tmp[i][j] = add(tmp[i][j], mul(mat[k][j], d.mat[i - k][j]));
      }
    std::swap(mat, tmp);
  }
  void print() {
    hadamardInv();
    for (int i = 0; i <= LOG; ++i) {
      for (int j = 0; j < len; ++j) printf("%d ", mul(mat[i][j], f[i]));
      printf("\n");
    }
    hadamard();
  }
} prep[N], tmp[N], dp[2][N], res[1 << LIM + 1];
void precalc() {
  f[0] = finv[0] = 1;
  for (int i = 1; i <= LOG; ++i) {
    f[i] = mul(f[i - 1], i);
    finv[i] = inv(f[i]);
  }
  for (int i = 0; i < LOG; ++i) {
    prep[i].init(LOG << 1);
    int num = i + LOG;
    for (int j = 1; j <= LOG; ++j) prep[i].mat[j][num ^ num - j] = finv[j];
    prep[i].hadamard();
  }
  for (int i = 0; i <= LIM; ++i) {
    tmp[i].init(LOG << 1);
    tmp[i].hadamard();
    dp[0][i].init(LOG);
    dp[1][i].init(LOG);
  }
}
void recurse(int mask, int h, Data dat) {
  for (int i = 0; i <= 1; ++i) {
    Data tmp = dat;
    tmp.update(dp[i][h]);
    if (i == 1) mask ^= (1 << h + 1) - 1;
    if (h > 0)
      recurse(mask, h - 1, tmp);
    else {
      tmp.hadamardInv();
      res[mask] = tmp;
    }
  }
}
void solve() {
  Data dat;
  dat.init(LOG);
  dat.hadamard();
  recurse(0, LIM, dat);
}
int main() {
  precalc();
  int n, k, c;
  scanf("%d%d%d", &n, &k, &c);
  int xum = 0;
  for (int i = 1; i <= n; ++i) {
    int a;
    scanf("%d", &a);
    xum ^= a;
    tmp[__builtin_ctz(a / LOG + (1 << LIM))].update(prep[a % LOG]);
  }
  for (int i = 0; i <= LIM; ++i) {
    tmp[i].hadamardInv();
    for (int j = 0; j <= LOG; ++j)
      for (int k = 0; k < LOG; ++k) {
        dp[0][i].mat[j][k] = tmp[i].mat[j][k];
        dp[1][i].mat[j][k] = tmp[i].mat[j][k + LOG];
      }
    dp[0][i].hadamard();
    dp[1][i].hadamard();
  }
  solve();
  int coeff = mul(f[k], inv(pow(n, k)));
  for (int i = 0; i < 1 << c; ++i) {
    int num = i ^ xum;
    printf("%d ", mul(coeff, res[num / LOG].mat[k][num % LOG]));
  }
  printf("\n");
}

### Prompt

Please create a solution in cpp to the following problem:

You are given a positive integer k and an array a_1, a_2, …, a_n of non-negative distinct integers not smaller than k and not greater than 2^c-1.

In each of the next k seconds, one element is chosen randomly equiprobably out of all n elements and decreased by 1.

For each integer x, 0 ≤ x ≤ 2^c - 1, you need to find the probability that in the end the bitwise XOR of all elements of the array is equal to x. 

Each of these values can be represented as an irreducible fraction p/q, and you need to find the value of p ⋅ q^{-1} modulo 998 244 353. 

Input

The first line of input contains three integers n, k, c (1 ≤ n ≤ (2^c - k), 1 ≤ k ≤ 16, 1 ≤ c ≤ 16).

The second line contains n distinct integers a_1, a_2, …, a_n (k ≤ a_i ≤ 2^c-1).

Output

Print 2^c integers: the probability that the bitwise XOR is equal to x in the end for x in \{0, 1, …, 2^c-1\} modulo 998 244 353.

Example

Input


4 1 3
1 2 3 4


Output


0 0 0 748683265 0 499122177 0 748683265 

### Response

```cpp
#include <bits/stdc++.h>
const int LIM = 12, LOG = 16, N = 40, MOD = 998244353;
int f[N], finv[N];
int addmod(int x) { return x < MOD ? x : x - MOD; }
int add(int x, int y) { return addmod(x + y); }
int submod(int x) { return x < 0 ? x + MOD : x; }
int sub(int x, int y) { return submod(x - y); }
int mul(int x, int y) { return 1LL * x * y % MOD; }
int pow(int x, int y) {
  int res = 1;
  while (y > 0) {
    if (y % 2 == 1) res = mul(res, x);
    x = mul(x, x);
    y /= 2;
  }
  return res;
}
int inv(int x) { return pow(x, MOD - 2); }
struct Data {
  int len, mat[N][N], tmp[N][N];
  void init(int num) {
    len = num;
    for (int i = 0; i <= LOG; ++i)
      for (int j = 0; j < len; ++j) mat[i][j] = 0;
    mat[0][0] = 1;
  }
  void hadamard() {
    for (int i = 1; i < len; i *= 2) {
      for (int j = 0; j <= LOG; ++j)
        for (int k = 0; k < len; ++k)
          if (k & i) {
            tmp[j][k ^ i] = add(mat[j][k ^ i], mat[j][k]);
            tmp[j][k] = sub(mat[j][k ^ i], mat[j][k]);
          }
      std::swap(mat, tmp);
    }
  }
  void invert() {
    for (int i = 0; i <= LOG; ++i)
      for (int j = 0; j < len; ++j) mat[i][j] = mul(mat[i][j], inv(len));
  }
  void hadamardInv() {
    hadamard();
    invert();
  }
  void update(const Data &d) {
    for (int i = 0; i <= LOG; ++i)
      for (int j = 0; j < len; ++j) {
        tmp[i][j] = 0;
        for (int k = 0; k <= i; ++k)
          tmp[i][j] = add(tmp[i][j], mul(mat[k][j], d.mat[i - k][j]));
      }
    std::swap(mat, tmp);
  }
  void print() {
    hadamardInv();
    for (int i = 0; i <= LOG; ++i) {
      for (int j = 0; j < len; ++j) printf("%d ", mul(mat[i][j], f[i]));
      printf("\n");
    }
    hadamard();
  }
} prep[N], tmp[N], dp[2][N], res[1 << LIM + 1];
void precalc() {
  f[0] = finv[0] = 1;
  for (int i = 1; i <= LOG; ++i) {
    f[i] = mul(f[i - 1], i);
    finv[i] = inv(f[i]);
  }
  for (int i = 0; i < LOG; ++i) {
    prep[i].init(LOG << 1);
    int num = i + LOG;
    for (int j = 1; j <= LOG; ++j) prep[i].mat[j][num ^ num - j] = finv[j];
    prep[i].hadamard();
  }
  for (int i = 0; i <= LIM; ++i) {
    tmp[i].init(LOG << 1);
    tmp[i].hadamard();
    dp[0][i].init(LOG);
    dp[1][i].init(LOG);
  }
}
void recurse(int mask, int h, Data dat) {
  for (int i = 0; i <= 1; ++i) {
    Data tmp = dat;
    tmp.update(dp[i][h]);
    if (i == 1) mask ^= (1 << h + 1) - 1;
    if (h > 0)
      recurse(mask, h - 1, tmp);
    else {
      tmp.hadamardInv();
      res[mask] = tmp;
    }
  }
}
void solve() {
  Data dat;
  dat.init(LOG);
  dat.hadamard();
  recurse(0, LIM, dat);
}
int main() {
  precalc();
  int n, k, c;
  scanf("%d%d%d", &n, &k, &c);
  int xum = 0;
  for (int i = 1; i <= n; ++i) {
    int a;
    scanf("%d", &a);
    xum ^= a;
    tmp[__builtin_ctz(a / LOG + (1 << LIM))].update(prep[a % LOG]);
  }
  for (int i = 0; i <= LIM; ++i) {
    tmp[i].hadamardInv();
    for (int j = 0; j <= LOG; ++j)
      for (int k = 0; k < LOG; ++k) {
        dp[0][i].mat[j][k] = tmp[i].mat[j][k];
        dp[1][i].mat[j][k] = tmp[i].mat[j][k + LOG];
      }
    dp[0][i].hadamard();
    dp[1][i].hadamard();
  }
  solve();
  int coeff = mul(f[k], inv(pow(n, k)));
  for (int i = 0; i < 1 << c; ++i) {
    int num = i ^ xum;
    printf("%d ", mul(coeff, res[num / LOG].mat[k][num % LOG]));
  }
  printf("\n");
}

```