#include <bits/stdc++.h>
using namespace std;
const int p = 998244353, inv2 = 499122177;
inline int fpow(int x, int y) {
  int ans = 1;
  while (y) {
    if (y & 1) ans = 1ll * ans * x % p;
    x = 1ll * x * x % p, y >>= 1;
  }
  return ans;
}
const int deg = 17;
struct poly {
  int a[deg];
  poly() : a() {}
  inline int &operator[](const int &i) { return a[i]; }
  inline poly operator+(const poly &b) const {
    poly c;
    for (int i = 0; i < deg; i++)
      c[i] = a[i] + b.a[i], c[i] -= c[i] > p ? p : 0;
    return c;
  }
  inline poly operator-(const poly &b) const {
    poly c;
    for (int i = 0; i < deg; i++)
      c[i] = a[i] + p - b.a[i], c[i] -= c[i] > p ? p : 0;
    return c;
  }
  inline poly operator*(const poly &b) const {
    poly c;
    for (int i = 0; i < deg; i++)
      for (int j = 0; i + j < deg; j++)
        c[i + j] = (c[i + j] + 1ll * a[i] * b.a[j]) % p;
    return c;
  }
  inline poly operator*(const int &b) const {
    poly c;
    for (int i = 0; i < deg; i++) c[i] = 1ll * a[i] * b % p;
    return c;
  }
};
inline void XOR(vector<poly> &a, int lim, int type) {
  for (register int mid = 1; mid < lim; mid <<= 1) {
    for (register int i = mid << 1, j = 0; j < lim; j += i) {
      for (register int k = 0; k < mid; k++) {
        a[j + k] = a[j + k] + a[j + k + mid];
        a[j + k + mid] = a[j + k] - a[j + k + mid] - a[j + k + mid];
        if (type == -1)
          a[j + k] = a[j + k] * inv2, a[j + k + mid] = a[j + k + mid] * inv2;
      }
    }
  }
}
int n, k, c;
int x;
vector<int> v[100086 << 2];
int cnt[100086 << 2];
int pcnt[100086], sum[100086], inv[100086];
vector<poly> build(int x, int l, int r) {
  if (l == r) {
    vector<poly> v(1);
    v[0][0] = 1;
    return v;
  }
  int mid = l + r >> 1;
  vector<poly> L = build((x << 1), l, mid), R = build((x << 1 | 1), mid + 1, r);
  int lim = r - l + 1;
  L.resize(lim), R.resize(lim);
  if (cnt[(x << 1 | 1)] & 1) {
    for (int i = 0; i < lim / 2; i++) {
      for (int j = 0; j <= k; j++) {
        swap(R[i][j], R[i + lim / 2][j]);
      }
    }
  }
  cnt[x] = cnt[(x << 1)] + cnt[(x << 1 | 1)];
  XOR(L, lim, 1), XOR(R, lim, 1);
  for (int i = 0; i < lim; i++) L[i] = L[i] * R[i];
  vector<int> w;
  for (int i = 0; i < v[(x << 1)].size(); i++) {
    int val = v[(x << 1)][i];
    if (val - k >= l)
      w.push_back(val);
    else
      v[x].push_back(val);
  }
  for (int i = 0; i < v[(x << 1 | 1)].size(); i++) {
    int val = v[(x << 1 | 1)][i];
    if (val - k >= l)
      w.push_back(val);
    else
      v[x].push_back(val);
  }
  for (int t = 0; t < w.size(); t++) {
    int val = w[t] & (lim - 1);
    vector<poly> v(lim);
    for (int i = 0; i < lim; i++) {
      for (int j = 0; j <= k; j++) {
        v[i][j] = pcnt[i & (val - j)] & 1 ? p - inv[j] : inv[j];
      }
    }
    for (int i = 0; i < lim; i++) L[i] = L[i] * v[i];
  }
  cnt[x] += w.size();
  XOR(L, lim, -1);
  return L;
}
int main() {
  sum[0] = 1;
  for (int i = 1; i < 100086; i++) sum[i] = 1ll * sum[i - 1] * i % p;
  inv[100086 - 1] = fpow(sum[100086 - 1], p - 2);
  for (int i = 100086 - 2; ~i; i--) inv[i] = 1ll * inv[i + 1] * (i + 1) % p;
  for (int i = 1; i < 100086; i++) pcnt[i] = pcnt[i >> 1] + (i & 1);
  scanf("%d%d%d", &n, &k, &c);
  for (int i = 1; i <= n; i++) {
    scanf("%d", &x);
    v[x + (1 << c)].push_back(x);
  }
  vector<poly> v = build(1, 0, (1 << c) - 1);
  int val = 1ll * sum[k] * fpow(fpow(n, k), p - 2) % p;
  for (int i = 0; i < (1 << c); i++) printf("%d ", 1ll * v[i][k] * val % p);
}

### Prompt

Construct a Cpp code solution to the problem outlined:

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>
using namespace std;
const int p = 998244353, inv2 = 499122177;
inline int fpow(int x, int y) {
  int ans = 1;
  while (y) {
    if (y & 1) ans = 1ll * ans * x % p;
    x = 1ll * x * x % p, y >>= 1;
  }
  return ans;
}
const int deg = 17;
struct poly {
  int a[deg];
  poly() : a() {}
  inline int &operator[](const int &i) { return a[i]; }
  inline poly operator+(const poly &b) const {
    poly c;
    for (int i = 0; i < deg; i++)
      c[i] = a[i] + b.a[i], c[i] -= c[i] > p ? p : 0;
    return c;
  }
  inline poly operator-(const poly &b) const {
    poly c;
    for (int i = 0; i < deg; i++)
      c[i] = a[i] + p - b.a[i], c[i] -= c[i] > p ? p : 0;
    return c;
  }
  inline poly operator*(const poly &b) const {
    poly c;
    for (int i = 0; i < deg; i++)
      for (int j = 0; i + j < deg; j++)
        c[i + j] = (c[i + j] + 1ll * a[i] * b.a[j]) % p;
    return c;
  }
  inline poly operator*(const int &b) const {
    poly c;
    for (int i = 0; i < deg; i++) c[i] = 1ll * a[i] * b % p;
    return c;
  }
};
inline void XOR(vector<poly> &a, int lim, int type) {
  for (register int mid = 1; mid < lim; mid <<= 1) {
    for (register int i = mid << 1, j = 0; j < lim; j += i) {
      for (register int k = 0; k < mid; k++) {
        a[j + k] = a[j + k] + a[j + k + mid];
        a[j + k + mid] = a[j + k] - a[j + k + mid] - a[j + k + mid];
        if (type == -1)
          a[j + k] = a[j + k] * inv2, a[j + k + mid] = a[j + k + mid] * inv2;
      }
    }
  }
}
int n, k, c;
int x;
vector<int> v[100086 << 2];
int cnt[100086 << 2];
int pcnt[100086], sum[100086], inv[100086];
vector<poly> build(int x, int l, int r) {
  if (l == r) {
    vector<poly> v(1);
    v[0][0] = 1;
    return v;
  }
  int mid = l + r >> 1;
  vector<poly> L = build((x << 1), l, mid), R = build((x << 1 | 1), mid + 1, r);
  int lim = r - l + 1;
  L.resize(lim), R.resize(lim);
  if (cnt[(x << 1 | 1)] & 1) {
    for (int i = 0; i < lim / 2; i++) {
      for (int j = 0; j <= k; j++) {
        swap(R[i][j], R[i + lim / 2][j]);
      }
    }
  }
  cnt[x] = cnt[(x << 1)] + cnt[(x << 1 | 1)];
  XOR(L, lim, 1), XOR(R, lim, 1);
  for (int i = 0; i < lim; i++) L[i] = L[i] * R[i];
  vector<int> w;
  for (int i = 0; i < v[(x << 1)].size(); i++) {
    int val = v[(x << 1)][i];
    if (val - k >= l)
      w.push_back(val);
    else
      v[x].push_back(val);
  }
  for (int i = 0; i < v[(x << 1 | 1)].size(); i++) {
    int val = v[(x << 1 | 1)][i];
    if (val - k >= l)
      w.push_back(val);
    else
      v[x].push_back(val);
  }
  for (int t = 0; t < w.size(); t++) {
    int val = w[t] & (lim - 1);
    vector<poly> v(lim);
    for (int i = 0; i < lim; i++) {
      for (int j = 0; j <= k; j++) {
        v[i][j] = pcnt[i & (val - j)] & 1 ? p - inv[j] : inv[j];
      }
    }
    for (int i = 0; i < lim; i++) L[i] = L[i] * v[i];
  }
  cnt[x] += w.size();
  XOR(L, lim, -1);
  return L;
}
int main() {
  sum[0] = 1;
  for (int i = 1; i < 100086; i++) sum[i] = 1ll * sum[i - 1] * i % p;
  inv[100086 - 1] = fpow(sum[100086 - 1], p - 2);
  for (int i = 100086 - 2; ~i; i--) inv[i] = 1ll * inv[i + 1] * (i + 1) % p;
  for (int i = 1; i < 100086; i++) pcnt[i] = pcnt[i >> 1] + (i & 1);
  scanf("%d%d%d", &n, &k, &c);
  for (int i = 1; i <= n; i++) {
    scanf("%d", &x);
    v[x + (1 << c)].push_back(x);
  }
  vector<poly> v = build(1, 0, (1 << c) - 1);
  int val = 1ll * sum[k] * fpow(fpow(n, k), p - 2) % p;
  for (int i = 0; i < (1 << c); i++) printf("%d ", 1ll * v[i][k] * val % p);
}

```