#include <bits/stdc++.h>
using namespace std;
const int maxn = 1111111, mod = 998244353;
template <typename T>
inline void read(T &x) {
  x = 0;
  bool f = 0;
  char ch = getchar();
  while (ch < '0' || ch > '9') f |= ch == '-', ch = getchar();
  while (ch >= '0' && ch <= '9') x = x * 10 + ch - '0', ch = getchar();
  if (f) x = -x;
}
int n, k, a[maxn], rt, cnt, ch[maxn * 20][2];
void insert(int &x, int v, int dep) {
  if (!x) x = ++cnt;
  if (dep == -1) return;
  insert(ch[x][(v >> dep) & 1], v, dep - 1);
}
struct wtf {
  vector<int> s, mx, mn;
};
wtf dfs(int x, int dep) {
  if (dep == -1) {
    wtf w;
    w.s = vector<int>(1, 1e9);
    w.mx = w.mn = vector<int>(1, 0);
    return w;
  }
  int lim = (1 << dep);
  vector<int> s(lim * 2, 1e9), mx(lim * 2, 0), mn(lim * 2, 1e9);
  wtf l, r;
  for (int _ = (0); _ <= (1); _++)
    if (ch[x][_]) {
      wtf w = dfs(ch[x][_], dep - 1);
      if (_)
        r = w;
      else
        l = w;
      for (int i = (0); i <= (lim - 1); i++) {
        s[i] = min(s[i], w.s[i]);
        s[i + lim] = min(s[i + lim], w.s[i]);
        mx[i] = max(mx[i], w.mx[i] ^ (_ * lim));
        mx[i + lim] = max(mx[i + lim], w.mx[i] ^ ((_ ^ 1) * lim));
        mn[i] = min(mn[i], w.mn[i] ^ (_ * lim));
        mn[i + lim] = min(mn[i + lim], w.mn[i] ^ ((_ ^ 1) * lim));
      }
    }
  if (ch[x][0] && ch[x][1]) {
    for (int i = (0); i <= (lim - 1); i++)
      s[i] = min(s[i], r.mn[i] - l.mx[i] + lim);
    for (int i = (0); i <= (lim - 1); i++)
      s[i + lim] = min(s[i + lim], l.mn[i] - r.mx[i] + lim);
  }
  return (wtf){s, mx, mn};
}
int main() {
  read(n);
  read(k);
  for (int i = (1); i <= (n); i++) read(a[i]);
  for (int i = (1); i <= (n); i++) insert(rt, a[i], k - 1);
  vector<int> ans = dfs(rt, k - 1).s;
  for (int i = (0); i <= ((1 << k) - 1); i++) printf("%d ", ans[i]);
}

### Prompt

Please create a solution in cpp to the following problem:

You are given an array a consisting of n distinct elements and an integer k. Each element in the array is a non-negative integer not exceeding 2^k-1.

Let's define the XOR distance for a number x as the value of 

$$$f(x) = min_{i = 1}^{n} min_{j = i + 1}^{n} |(a_i ⊕ x) - (a_j ⊕ x)|,$$$

where ⊕ denotes [the bitwise XOR operation](https://en.wikipedia.org/wiki/Bitwise_operation#XOR).

For every integer x from 0 to 2^k-1, you have to calculate f(x).

Input

The first line contains two integers n and k (1 ≤ k ≤ 19; 2 ≤ n ≤ 2^k).

The second line contains n integers a_1, a_2, ..., a_n (0 ≤ a_i ≤ 2^k-1). All these integers are distinct.

Output

Print 2^k integers. The i-th of them should be equal to f(i-1).

Examples

Input


3 3
6 0 3


Output


3 1 1 2 2 1 1 3 


Input


3 4
13 4 2


Output


2 2 6 6 3 1 2 2 2 2 1 3 6 6 2 2 

Note

Consider the first example:

  * for x = 0, if we apply bitwise XOR to the elements of the array with x, we get the array [6, 0, 3], and the minimum absolute difference of two elements is 3; 
  * for x = 1, if we apply bitwise XOR to the elements of the array with x, we get the array [7, 1, 2], and the minimum absolute difference of two elements is 1; 
  * for x = 2, if we apply bitwise XOR to the elements of the array with x, we get the array [4, 2, 1], and the minimum absolute difference of two elements is 1; 
  * for x = 3, if we apply bitwise XOR to the elements of the array with x, we get the array [5, 3, 0], and the minimum absolute difference of two elements is 2; 
  * for x = 4, if we apply bitwise XOR to the elements of the array with x, we get the array [2, 4, 7], and the minimum absolute difference of two elements is 2; 
  * for x = 5, if we apply bitwise XOR to the elements of the array with x, we get the array [3, 5, 6], and the minimum absolute difference of two elements is 1; 
  * for x = 6, if we apply bitwise XOR to the elements of the array with x, we get the array [0, 6, 5], and the minimum absolute difference of two elements is 1; 
  * for x = 7, if we apply bitwise XOR to the elements of the array with x, we get the array [1, 7, 4], and the minimum absolute difference of two elements is 3. 

### Response

```cpp
#include <bits/stdc++.h>
using namespace std;
const int maxn = 1111111, mod = 998244353;
template <typename T>
inline void read(T &x) {
  x = 0;
  bool f = 0;
  char ch = getchar();
  while (ch < '0' || ch > '9') f |= ch == '-', ch = getchar();
  while (ch >= '0' && ch <= '9') x = x * 10 + ch - '0', ch = getchar();
  if (f) x = -x;
}
int n, k, a[maxn], rt, cnt, ch[maxn * 20][2];
void insert(int &x, int v, int dep) {
  if (!x) x = ++cnt;
  if (dep == -1) return;
  insert(ch[x][(v >> dep) & 1], v, dep - 1);
}
struct wtf {
  vector<int> s, mx, mn;
};
wtf dfs(int x, int dep) {
  if (dep == -1) {
    wtf w;
    w.s = vector<int>(1, 1e9);
    w.mx = w.mn = vector<int>(1, 0);
    return w;
  }
  int lim = (1 << dep);
  vector<int> s(lim * 2, 1e9), mx(lim * 2, 0), mn(lim * 2, 1e9);
  wtf l, r;
  for (int _ = (0); _ <= (1); _++)
    if (ch[x][_]) {
      wtf w = dfs(ch[x][_], dep - 1);
      if (_)
        r = w;
      else
        l = w;
      for (int i = (0); i <= (lim - 1); i++) {
        s[i] = min(s[i], w.s[i]);
        s[i + lim] = min(s[i + lim], w.s[i]);
        mx[i] = max(mx[i], w.mx[i] ^ (_ * lim));
        mx[i + lim] = max(mx[i + lim], w.mx[i] ^ ((_ ^ 1) * lim));
        mn[i] = min(mn[i], w.mn[i] ^ (_ * lim));
        mn[i + lim] = min(mn[i + lim], w.mn[i] ^ ((_ ^ 1) * lim));
      }
    }
  if (ch[x][0] && ch[x][1]) {
    for (int i = (0); i <= (lim - 1); i++)
      s[i] = min(s[i], r.mn[i] - l.mx[i] + lim);
    for (int i = (0); i <= (lim - 1); i++)
      s[i + lim] = min(s[i + lim], l.mn[i] - r.mx[i] + lim);
  }
  return (wtf){s, mx, mn};
}
int main() {
  read(n);
  read(k);
  for (int i = (1); i <= (n); i++) read(a[i]);
  for (int i = (1); i <= (n); i++) insert(rt, a[i], k - 1);
  vector<int> ans = dfs(rt, k - 1).s;
  for (int i = (0); i <= ((1 << k) - 1); i++) printf("%d ", ans[i]);
}

```