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 

Solution (in CPP):

Solution:
#include <bits/stdc++.h>
const int P = 998244353;
inline int mul(const int &a, const int &b) { return 1ll * a * b % P; }
inline int sub(int a, const int &b) {
  a -= b;
  return (a < 0) ? a + P : a;
}
inline int add(int a, const int &b) {
  a += b;
  return (a >= P) ? a - P : a;
}
int qsm(int a, int b) {
  int ans = 1;
  while (b) {
    if (b & 1) ans = mul(ans, a);
    a = mul(a, a);
    b >>= 1;
  }
  return ans;
}
int n, k, c, a[1000001], sum, f[17][32][2], b[31][31], g[17][17][32][2];
int h[17][65536], lim, tem[17][65536], ttem[17];
int invmul[100001], _mul[100001];
int C(int n, int m) {
  if (n < m || m < 0) return 0;
  return mul(mul(_mul[n], invmul[m]), invmul[n - m]);
}
void FWT(int *a) {
  for (int i = 1; i < lim; i <<= 1)
    for (int j = 0; j < lim; j += (i << 1))
      for (int k = 0; k < i; k++) {
        const int Ny = a[i + j + k];
        a[i + j + k] = sub(a[j + k], Ny);
        a[j + k] = add(a[j + k], Ny);
      }
}
int main() {
  scanf("%d%d%d", &n, &k, &c);
  _mul[0] = _mul[1] = invmul[0] = invmul[1] = 1;
  for (int i = 2; i <= 100000; i++) invmul[i] = mul(P - P / i, invmul[P % i]);
  for (int i = 2; i <= 100000; i++) invmul[i] = mul(invmul[i - 1], invmul[i]);
  for (int i = 2; i <= 100000; i++) _mul[i] = mul(_mul[i - 1], i);
  for (int i = 1; i <= n; i++) {
    scanf("%d", a + i);
    sum ^= a[i];
    int tem = a[i], cnt = 0;
    for (; tem; tem -= tem & -tem) {
      cnt += tem & -tem;
      if (cnt > k) {
        cnt -= tem & -tem;
        break;
      }
    }
    sum ^= cnt;
    int L = 0;
    while ((1 << L) <= (tem & -tem)) ++L;
    if (!L) L = c;
    ++b[L][cnt];
  }
  tem[0][0] = 1;
  int fac = 1;
  for (int i = 0; i <= c; i++) {
    memset(f, 0, sizeof f);
    f[0][0][0] = 1;
    for (int j = 0; j <= k; j++) {
      memset(g, 0, sizeof g);
      memcpy(g[0], f, sizeof f);
      memset(f, 0, sizeof f);
      for (int l = k; l; --l)
        for (int m = k; ~m; --m)
          for (int sum = k; ~sum; --sum)
            for (int x = 0; x < 32; x++)
              for (int bit = 0; bit < 2; bit++)
                for (int use = 1, Mul = invmul[l];
                     use <= m && use <= b[i][j] && use * l <= sum;
                     use++, Mul = mul(Mul, invmul[l])) {
                  int nx = x, nb = bit;
                  if (use & 1) {
                    if (l <= j)
                      nx ^= j - l;
                    else
                      nx ^= l - j - 1, nb ^= 1;
                  }
                  g[m][sum][x][bit] =
                      add(g[m][sum][x][bit],
                          mul(g[m - use][sum - use * l][nx][nb],
                              mul(Mul, C(b[i][j] - (m - use), use))));
                }
      for (int m = k; ~m; --m)
        for (int sum = k; ~sum; --sum)
          for (int x = 0; x < 32; x++)
            for (int bit = 0; bit < 2; bit++) {
              if (m > b[i][j]) continue;
              int nx = x;
              if ((b[i][j] - m) & 1) nx ^= j;
              f[sum][nx][bit] = add(f[sum][nx][bit], g[m][sum][x][bit]);
            }
    }
    lim = (1 << (i));
    for (int i = 0; i <= k; i++) FWT(tem[i]);
    for (int i = 0; i <= k; i++) memset(h[i], 0, (lim) << 2);
    for (int sum = k; ~sum; --sum)
      for (int x = 0; x < 32; x++)
        for (int bit = 0; bit < 2; bit++) {
          h[sum][x ^ (bit ? ((1 << i) - 1) : 0)] =
              add(h[sum][x ^ (bit ? ((1 << i) - 1) : 0)], f[sum][x][bit]);
        }
    for (int j = 0; j <= k; j++) FWT(h[j]);
    for (int j = 0; j < lim; j++) {
      memset(ttem, 0, sizeof ttem);
      for (int l = 0; l <= k; l++)
        for (int m = 0; l + m <= k; m++)
          ttem[l + m] = add(ttem[l + m], mul(h[l][j], tem[m][j]));
      for (int l = 0; l <= k; l++) tem[l][j] = ttem[l];
    }
    for (int i = 0; i <= k; i++) FWT(tem[i]);
    fac = mul(fac, qsm(lim, P - 2));
  }
  fac = mul(fac, mul(_mul[k], qsm(qsm(n, k), P - 2)));
  for (int i = 0; i < lim; i++) printf("%d ", mul(tem[k][i ^ sum], fac));
}