#include <bits/stdc++.h>
int getbit(long long x, int y) { return (x >> y) & 1; }
long long getoff(long long x, int y) { return x ^ ((1ll) << y); }
using namespace std;
struct node {
  int diff, ma, mi;
  node *c[2];
  node() {
    diff = 2e9;
    ma = -1e9;
    mi = 1e9;
    c[0] = nullptr;
    c[1] = nullptr;
  }
};
node *trie = new node;
int n, k, f[2000010];
void calc(node *u, int depth) {
  u->ma = -1e9;
  u->mi = 1e9;
  u->diff = 2e9;
  u->ma = max(u->c[0]->ma, (1 << depth) + u->c[1]->ma);
  u->mi = min(u->c[0]->mi, (1 << depth) + u->c[1]->mi);
  if (u->ma < 0) u->ma = -1e9;
  if (u->c[0]->ma != -1e9 && u->c[0]->ma != u->ma)
    u->mi = min(u->mi, u->c[0]->ma);
  int f1 = u->c[1]->mi;
  if (f1 == 1e9 && u->c[1]->ma != -1e9) f1 = u->c[1]->ma;
  int f2 = u->c[0]->ma;
  if (f2 == -1e9 && u->c[0]->mi != 1e9) f2 = u->c[0]->mi;
  u->diff = min(u->c[0]->diff, u->c[1]->diff);
  u->diff = min(u->diff, f1 - f2 + (1 << depth));
}
void build(node *u, int depth) {
  if (depth == -1) return;
  u->c[0] = new node;
  u->c[1] = new node;
  build(u->c[0], depth - 1);
  build(u->c[1], depth - 1);
}
void add(node *u, int L, int x, int depth) {
  if (depth == -1) {
    u->ma = x - L;
    return;
  }
  int kt = getbit(x, depth);
  if (kt == 0)
    add(u->c[kt], L, x, depth - 1);
  else
    add(u->c[kt], L + (1 << depth), x, depth - 1);
  calc(u, depth);
}
void update(node *u, int depth, int stop) {
  if (depth == stop) {
    swap(u->c[0], u->c[1]);
    calc(u, depth);
    return;
  }
  update(u->c[0], depth - 1, stop);
  update(u->c[1], depth - 1, stop);
  calc(u, depth);
}
int main() {
  ios_base::sync_with_stdio(0);
  cin.tie(0);
  cout.tie(0);
  cin >> n >> k;
  build(trie, k - 1);
  for (int i = 1; i <= n; ++i) {
    int x;
    cin >> x;
    add(trie, 0, x, k - 1);
  }
  vector<int> lvt;
  lvt.emplace_back(0);
  for (int q = 1; q <= k; ++q) {
    int sz = lvt.size();
    while (sz--) lvt.emplace_back(lvt[sz] * 2 + 1);
    for (int i = 0; i < (1 << (q - 1)); ++i) lvt[i] <<= 1;
  }
  vector<int> vt;
  int pos;
  for (int i = 0; i < (1 << k); ++i)
    if (lvt[i] == 0) {
      pos = i;
      break;
    }
  for (int i = 1; i <= (1 << k); ++i) {
    vt.emplace_back(lvt[pos]);
    pos = (pos + 1) % (1 << k);
  }
  f[0] = trie->diff;
  for (int i = 1; i < (1 << k); ++i) {
    int mask = vt[i] ^ vt[i - 1];
    int stop;
    for (int i = 0; i < k; ++i)
      if (getbit(mask, i) == 1) {
        stop = i;
        break;
      }
    update(trie, k - 1, stop);
    f[vt[i]] = trie->diff;
  }
  for (int i = 0; i < (1 << k); ++i) cout << f[i] << " ";
  return 0;
}

### Prompt

Please provide a Cpp coded solution to the problem described below:

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>
int getbit(long long x, int y) { return (x >> y) & 1; }
long long getoff(long long x, int y) { return x ^ ((1ll) << y); }
using namespace std;
struct node {
  int diff, ma, mi;
  node *c[2];
  node() {
    diff = 2e9;
    ma = -1e9;
    mi = 1e9;
    c[0] = nullptr;
    c[1] = nullptr;
  }
};
node *trie = new node;
int n, k, f[2000010];
void calc(node *u, int depth) {
  u->ma = -1e9;
  u->mi = 1e9;
  u->diff = 2e9;
  u->ma = max(u->c[0]->ma, (1 << depth) + u->c[1]->ma);
  u->mi = min(u->c[0]->mi, (1 << depth) + u->c[1]->mi);
  if (u->ma < 0) u->ma = -1e9;
  if (u->c[0]->ma != -1e9 && u->c[0]->ma != u->ma)
    u->mi = min(u->mi, u->c[0]->ma);
  int f1 = u->c[1]->mi;
  if (f1 == 1e9 && u->c[1]->ma != -1e9) f1 = u->c[1]->ma;
  int f2 = u->c[0]->ma;
  if (f2 == -1e9 && u->c[0]->mi != 1e9) f2 = u->c[0]->mi;
  u->diff = min(u->c[0]->diff, u->c[1]->diff);
  u->diff = min(u->diff, f1 - f2 + (1 << depth));
}
void build(node *u, int depth) {
  if (depth == -1) return;
  u->c[0] = new node;
  u->c[1] = new node;
  build(u->c[0], depth - 1);
  build(u->c[1], depth - 1);
}
void add(node *u, int L, int x, int depth) {
  if (depth == -1) {
    u->ma = x - L;
    return;
  }
  int kt = getbit(x, depth);
  if (kt == 0)
    add(u->c[kt], L, x, depth - 1);
  else
    add(u->c[kt], L + (1 << depth), x, depth - 1);
  calc(u, depth);
}
void update(node *u, int depth, int stop) {
  if (depth == stop) {
    swap(u->c[0], u->c[1]);
    calc(u, depth);
    return;
  }
  update(u->c[0], depth - 1, stop);
  update(u->c[1], depth - 1, stop);
  calc(u, depth);
}
int main() {
  ios_base::sync_with_stdio(0);
  cin.tie(0);
  cout.tie(0);
  cin >> n >> k;
  build(trie, k - 1);
  for (int i = 1; i <= n; ++i) {
    int x;
    cin >> x;
    add(trie, 0, x, k - 1);
  }
  vector<int> lvt;
  lvt.emplace_back(0);
  for (int q = 1; q <= k; ++q) {
    int sz = lvt.size();
    while (sz--) lvt.emplace_back(lvt[sz] * 2 + 1);
    for (int i = 0; i < (1 << (q - 1)); ++i) lvt[i] <<= 1;
  }
  vector<int> vt;
  int pos;
  for (int i = 0; i < (1 << k); ++i)
    if (lvt[i] == 0) {
      pos = i;
      break;
    }
  for (int i = 1; i <= (1 << k); ++i) {
    vt.emplace_back(lvt[pos]);
    pos = (pos + 1) % (1 << k);
  }
  f[0] = trie->diff;
  for (int i = 1; i < (1 << k); ++i) {
    int mask = vt[i] ^ vt[i - 1];
    int stop;
    for (int i = 0; i < k; ++i)
      if (getbit(mask, i) == 1) {
        stop = i;
        break;
      }
    update(trie, k - 1, stop);
    f[vt[i]] = trie->diff;
  }
  for (int i = 0; i < (1 << k); ++i) cout << f[i] << " ";
  return 0;
}

```