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: Wrong Answer)

#include<bits/stdc++.h>
using namespace std;
#define REP(i,n) for(int i=0;i<(n);++i)
typedef long long ll;

int main(){
    ios::sync_with_stdio(false);
    cin.tie(nullptr);
    int N; cin >> N;
    vector<ll> A(1<<N);
    REP(i,1<<N) {
        cin >> A[i];
    }
    vector<vector<ll> > dp(1<<N,vector<ll>(2,0LL));
    REP(i,1<<N) dp[i][0] = A[i];
    for(int j=0; j<N; ++j) {
        for(int i=0; i<(1<<N); ++i) {
            if(i&(1<<j)) {
                int pi = (i & ~(1<<j));
                if(dp[pi][1] >= dp[i][0]) {
                    dp[i][0] = dp[pi][0];
                    dp[i][1] = dp[pi][1];
                }
                else if(dp[pi][0] >= dp[i][0] && dp[pi][1] < dp[i][0]) {
                    dp[i][1] = dp[i][0];
                    dp[i][0] = dp[pi][0];
                }
                else if(dp[pi][0]<dp[i][0] && dp[pi][0] >= dp[i][1]) {
                    dp[i][1] = dp[pi][0];
                }
                else continue;
            }
        }
    }
    vector<ll> res;
    for(int i=1; i<(1<<N); ++i) res.push_back(dp[i][0]+dp[i][1]);
    for(int i=1; i+1<(1<<N); ++i) res[i+1] = max(res[i+1], res[i]);
    for(auto x:res) cout << x << '\n';
    return 0;
}