Problem:
Score : 700 points

Problem Statement

There is an integer sequence of length 2^N: A_0, A_1, ..., A_{2^N-1}. (Note that the sequence is 0-indexed.)

For every integer K satisfying 1 \leq K \leq 2^N-1, solve the following problem:

Let i and j be integers. Find the maximum value of A_i + A_j where 0 \leq i < j \leq 2^N-1 and (i or j) \leq K.
Here, or denotes the bitwise OR.

Constraints

1 \leq N \leq 18

1 \leq A_i \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_0 A_1 ... A_{2^N-1}

Output

Print 2^N-1 lines.
In the i-th line, print the answer of the problem above for K=i.

Sample Input 1

2
1 2 3 1

Sample Output 1

3
4
5

For K=1, the only possible pair of i and j is (i,j)=(0,1), so the answer is A_0+A_1=1+2=3.

For K=2, the possible pairs of i and j are (i,j)=(0,1),(0,2).
When (i,j)=(0,2), A_i+A_j=1+3=4. This is the maximum value, so the answer is 4.

For K=3, the possible pairs of i and j are (i,j)=(0,1),(0,2),(0,3),(1,2),(1,3),(2,3) .
When (i,j)=(1,2), A_i+A_j=2+3=5. This is the maximum value, so the answer is 5.

Sample Input 2

3
10 71 84 33 6 47 23 25

Sample Output 2

81
94
155
155
155
155
155

Sample Input 3

4
75 26 45 72 81 47 97 97 2 2 25 82 84 17 56 32

Sample Output 3

101
120
147
156
156
178
194
194
194
194
194
194
194
194
194

Solution in C++14 (GCC 5.4.1) (judge verdict: Time Limit Exceeded)

#include <bits/stdc++.h>

#ifdef DEBUG
#include "debug.h"
#endif

using namespace std;

#define mp make_pair
#define mt make_tuple
#define all(a) (a).begin(), (a).end()
#define rep(i, a, b) for (int (i) = (a); (i) < (b); ++(i))
#define sz(a) ((int)(a).size())
#define make_unique(a) (a).resize(unique(all((x))) - (x).begin())

typedef long long ll;

template<typename T> inline bool mmin(T& a, const T &b) { return a < b ? (a = b, true) : false; }
template<typename T> inline bool mmax(T& a, const T &b) { return a > b ? (a = b, true) : false; }

const int maxn = 1 << 19;
int a[maxn];

int main() {
    std::ios_base::sync_with_stdio(false);
    cin.tie(nullptr);
    
    int n;
    cin >> n;
    for (int i = 0; i < (1 << n); ++i) {
        cin >> a[i];
    }
    int res = a[0] + a[1];
    cout << res << '\n';
    for (int i = 2; i < (1 << n); ++i) {
        vector<int> cur = {a[0]};
        for (int mask = i; mask > 0; mask = (mask - 1) & i) {
            cur.push_back(a[mask]);
        }
        sort(cur.rbegin(), cur.rend());
        //cout << cur[0] << ' ' << cur[1] << '\n';
        res = max(res, cur[0] + cur[1]);
        cout << res << '\n';
    }
    
    return 0;
}