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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.BufferedOutputStream;
import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.io.PrintWriter;
import java.util.Arrays;
import java.util.StringTokenizer;

/**
 * Created by darshan on 5/29/19.
 */
public class Main {
    public static void main(String[] args) throws IOException {
        BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
        PrintWriter out = new PrintWriter(new BufferedOutputStream(System.out));
        StringTokenizer st;

        int n = Integer.parseInt(br.readLine());
        int size = 1 << n;
        st = new StringTokenizer(br.readLine());
        int[][] maxPair = new int[size][2];
        for (int i = 0; i < size; i++) {
            maxPair[i][0] = Integer.parseInt(st.nextToken());
        }

        for (int i = 0; i < n; i++) {
            int twoPowerI = 1 << i;
            for (int j = 0; j < size; j++) {
                if ((j & twoPowerI) != twoPowerI) {
                    merge(maxPair[j], maxPair[j | twoPowerI]);
                }
            }
        }

        int max = 0;
        for (int i = 1; i < size; i++) {
            max = Math.max(max, maxPair[i][0] + maxPair[i][1]);
            out.println(max);
        }
        out.close();
    }

    private static void merge(int[] from, int[] to) {
        int[] temp = new int[4];
        temp[0] = from[0];
        temp[1] = from[1];
        temp[2] = to[0];
        temp[3] = to[1];
        Arrays.sort(temp);
        to[0] = temp[3];
        to[1] = temp[2];
    }
}