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: Accepted)

/*
 *                             UM.
 *                            J@B@1                                                                    iO@1
 *                           Y@@@B@BB.                                                              7B@B@B@
 *                          :@B@i,B@B@O                                                          ,Z@B@B@B@Br
 *                          @B@q   i@B@BS                                                      7@B@@@O5vMB@q
 *                         8@@B      LB@B@i                                                  FB@@@BNjYjLE@B@
 *                        ,B@B:        0@@@Z                                               P@B@BM1JJ125JPB@B
 *                        B@BB          :@B@B                                            XB@B@Z2LuU52F2u2@B@.
 *                       :@B@             @@@B:                                        v@B@B8uJj51F1525uUB@B7
 *                       @B@O              0@@B.               ..::ir7vvYUuU777r::.   B@B@OULU2F2F151F11Y@B@S
 *                       B@B,               8B@B  :ruXMB@B@B@B@B@B@B@B@B@B@@@B@B@B@B@B@B@5Jj1211F1F1F2FUJO@BB
 *                      U@B@                 @B@B@B@B@B@@@B@B@B@MMqPS5JuYL7rq@B@OBB@B@B8Yu211F1515251515YGB@@
 *                      @B@u                 v@@@@MSur:.                    LB@MvvjJuU5YU252F1F1F25251F2uX@@@
 *                      @@@.                                                N@BML2U2UUU12F15252525251515Jk@@B
 *                     r@B@                                                 YB@Bju52121252515252F15251F2u5@B@
 *                     PB@B                                                  @@@PYUF151F25151F152F2F1F15jF@@B
 *                     @@BS                                                  N@@@UJ2F25252F251525151F1F1u5@B@
 *                     @@@7                                                   B@B@5Yj12F152F1F1F25252515jFB@B
 *                     B@Bi                                                    M@B@O2Luu52525212F151121UY1@B@7
 *                    O@B@:                                                     v@B@BMSuYJJuuUu2u2uujjYJJXB@B@M
 *                  7B@B@,                                                        1B@@@B@GPF1uujuu21PNMB@B@B@B@@
 *                 qB@B2                                                            i8B@B@B@B@B@@@@@B@B@B@q: @@@B
 *                MB@B:                                                                 7SBB@B@B@B@B@Zu:      @B@B
 *               ZB@B.                                              ,v.                                        @B@L
 *              LB@B,                         Y7                    @B@Bu                                      7@B@
 *   :B@B@@B2:  @@B7                         @B@Z                   r@B@B@BP:                                   B@BE
 *    BB@@@B@B@B@BE                        r@B@B                       7@B@B@B@Ou:                              iB@B
 *         :uM@@B@@2.           :7::::ivk@B@B@0                           :5B@B@B@B@B@B@G.                       @B@i
 *            BB@@@B@@         :@B@B@@@B@B@@1                                 .i5M@B@B@@@5                       M@@2
 *            B@B ,@B1          L0EZZG0F7:                                            .:,                        uB@MrP@M7
 *           2@B@                                                                                               ,O@B@B@B@B
 *           @B@1                                                     :@B@@@r                                :@@@@B@BL:,,
 *           B@Bi                         :2ZS;                      :@B@B@B@r                               L@B@B@BU
 *           @B@.                        @@@B@B@                     vB@B@B@B5                                   @B@i
 *           B@B                        7B@B@B@BM                     OB@B@B@                                   ,B@B
 *           @B@                         @B@B@@@i                       rL7.                                    B@BM
 *           B@B7.:                       NB@@M.                                                               .@B@.
 *  .;JEB@@@B@B@B@B@.                                                                                       .  @B@u
 *@@@B@B@B@B@@@B@18U                                                                                      :B@B@B@BU,
 *7@BOui.    ,@@B                                                                                          SP@B@B@B@B@Or
 *            @@@U                                                                                           B@BJ.YO@B@B@i
 *            r@B@                                                                                         :B@Bk     .k@B@
 *             B@B@                                                                                       LB@@k         2i
 *              B@BM                                      .7jXEGqF7:                                     OB@@L
 *              .B@BM                                   .B@B@B@B@B@B@.                                 :@B@B:
 *               .B@B@                                   @@MYr::ivG@B                                .M@B@G
 *                 B@@@S                                                                           ,MB@B@,
 *                  v@@@BF                                                                      .1B@B@Br
 *                    2@@B@BL                                                                ,FB@@@B8,
 *                      r@B@B@BF,                                                        :YBB@B@B@B
 *                         L@B@B@B@P7,                                           .ivXB@B@B@B@B@M@B@
 *                            ,1B@B@B@B@@@BOP2L7i:,.              ..,:i7LSNB@@B@B@@@B@B@B@Z5v;.LB@@
 *                              @B@OEB@B@@@B@B@B@B@B@B@B@B@@@B@B@B@B@B@@@B@B@B@B@BM0SJ7i::::i:,u@B@
 *                              B@Bu ::i;7vu2XNGOMB@B@BMB@B@B@B@B@B@@@B1UFuj77ii:::::::iir;r;i.YB@B
 *                              @B@L.:i:i:i::::::::::..Y@B@BMYi:i;SB@B@N:.::i:iirir;r;rii::::ivO@B@
 *                              B@@X::,::::iirir;riri:E@B@1         ,@B@Br:;;r;rii:i::::i7JEB@@@@@B
 *                              @@@B@BBq5v7ii:::::::.2@@@i  ..,..     @B@@,,:::::irv2XMB@B@B@B@2@B@:
 *                             .B@BBB@@@B@B@B@BMNP5u7@B@1 .,,:,,  :.   @B@P50MB@B@B@B@B@@@BS:   @@B1
 *                             E@B@   ijGB@B@B@B@B@B@B@Bi .,:,,..@@B@7 B@B@B@B@B@B@BM57.        kB@B
 *                            .@B@:          .,ivu5Nq@B@u  ..,.. SB@B@@@B@PL7i,                 ,@B@
 *                            @@@8                   i@B@:    .     :B@B@@                       B@@2
 *                          i@@@@                     0@B@u          B@@B.                       vB@B
 *                         ,@B@G                       L@B@BOv:.:iFB@B@M                          @B@Bi
 *                          vNi                          S@@B@B@B@B@BM:                            MB@N
 *                                                          758BMqJ,
 *
 *                 .  YO.               vq                            :G       Z:
 *        SqOMBB@B@Br @@r rBE           @B     B@@@@@B@ONX8k    i::::.OB1.:::.u@O.::::i           @B@B@U:@@B@@BPEBu
 *        B@@NB@k.    5@i  uB@E.        BM     1U2uUJvirB@@Z   r@@B@B@@@B@B@B@B@B@@@B@Bi   LB@B@1 BX :@k uLLLvr@BJ:
 *            iB      iBi    7@     .@M8@BGMZZ         @@F            ,B       Pi          v@  Bq @i v@        B@
 *       vuL7r8@S7vJL7N@Z7LLri;72.   F7@Bvvv@@       @BX         @@@B@B@@@@@B@@@B@B@B      7@  @F Bi @q  @B@Bu @B
 *       N@B@G@@@8@BBOMB@G@BMNXG@,     B@   @@      .Bk          .:u;    i@:      Zv       7@  Bk @,;@  ,BY @B B@
 *            r@       @G     5.      ,@v   BZ :::,.r@E .::i,      @B     B@    .@BL       7@  @F B:i@. .@  @M @B
 *            7B: ,vO, @@   iB@:      @B   7@:MB@B@B@@@B@B@BM       @@.    B:  2@q         7@  BS @i 0@  B. @O B@
 *       ,r2EBB@B@B@Bi G@  @BB        B@   @B        @S              :    r@   ..          7B  @F @7  B7 @  @B @B
 *       E@B@UOBr       @B@Bi          L@0PB        .BZ        .@B@B@B@B@B@B@B@B@B@B@B@,   r@  BF @i  @G B@B@B B@
 *            7@,      kB@U    ;r        @@@.       .@Z                GBuL@iBBi           vB@B@q BP:5@7 @u,.  @B
 *            LBi   YB@BrB@    @@       @B:L@Br      BM             .M@B  rB  rB@J         v@. Pi @XZ8r  .     B@
 *        .   G@i B@BM.  ,B@,  @B    iB@B    N, 7r..q@k         ,LB@B8    J@,   i@B@B1r           Br           @@
 *        MB@B@B  ,i       B@B@B,    B@:        @B@B@F         .@BB:      P@i      :OBZ          .@U       B@B@B:
 *                          .ll                                           rB.                     :
 */
/*****************
 *   I<3   C++   *
 *  IWANT ANYAC  *
 * ILOVEPROGRAM! *
 *IT'SINTERESTING*
 * ILOVEPROGRAM! *
 *  INCONTESTS   *
 *   GETSCORE    *
 *    ACCODE     *
 *     LETS      *
 *      GO       *
 *Written by:    *
 *    charlieyan *
 *****************/
#include <assert.h>
#include <ctype.h>
#include <errno.h>
#include <float.h>
#include <fstream>
#include <iomanip>
#include <iostream>
#include <limits>
#include <locale>
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <time.h>
#include <wchar.h>
#include <wctype.h>
#include <algorithm>
#include <bitset>
#include <cctype>
#include <cerrno>
#include <clocale>
#include <cmath>
#include <complex>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <ctime>
#include <deque>
#include <exception>
#include <fstream>
#include <functional>
#include <limits>
#include <map>
#include <iomanip>
#include <ios>
#include <iosfwd>
#include <istream>
#include <ostream>
#include <queue>
#include <set>
#include <sstream>
#include <stack>
#include <stdexcept>
#include <streambuf>
#include <string>
#include <utility>
#include <vector>
#include <cwchar>
#include <cwctype>
#include <numeric>
#include <list>
#define LL long long
#define pb push_back
#define pob pop_back
#define mp make_pair
#define FILL(a,v) memset((void*)a,v,sizeof(a))
#define FILL0(a) FILL(a,0)
#define FILL1(a) FILL(a,-1)
#define FILLBIG(a) FILL(a,0x7f)
#define ALL(a) (a).begin(),(a).end()
#define Inf ((int)0x7f7f7f7f)
#define INF ((long long)0x7f7f7f7f7f7f7f7f)
#define Mod 1000000007
#define foreach(it,x) for(__typeof(x.begin()) it=x.begin();it!=x.end();it++)
#define FOR(i,x) for(int i=0;i<x;i++)
#define RFOR(i,x) for(int i=x-1;i>=0;i--)
#define FFOR(i,a,b) for(int i=a;i<b;i++)
#define FRFOR(i,a,b) for(int i=a-1;i>=b;i--)
#define SZ(a) ((int)(a).size())
#define FI first
#define SE second
#define OVER(...) return printf(__VA_ARGS__),0
#define fastio ios::sync_with_stdio(0)
#define filei(f) freopen(f,"r",stdin)
#define fileo(f) freopen(f,"w",stdout)
#define eprintf(...) fprintf(stderr,__VA_ARGS__)
#define MASK(a) (1<<(a))
#define MASKX(x,a) ((x)<<(a))
#define BIT(a,b) (((a)>>(b))&1)
#define cddx(a) priority_queue<a >
#define cxdd(a) priority_queue<a,vector<a >,greater<a > >
#define deval(a) {cerr<<#a<<" = "<<a<<endl;}
#define dearr(be,en) {cerr<<#be;cerr<<" = {";for(auto i=be;i!=en;i++)cerr<<' '<<*i;cerr<<" }\n";}
#define reg register
#define prtime eprintf("\n\nTime used: %dms\n",clock());
#define PI 3.1415926535897932384626
#define TODEG(a) (a*180/PI)
//抗击母体，给地球一片净土
//消灭徐毒，还人类万年安康
using namespace std;
typedef pair<int,int> pii;
typedef unsigned long long UINT;
typedef unsigned int uint;
typedef long long INT;
typedef long double ldouble;
typedef pair<INT,INT> PII;
typedef stringstream sst;
typedef istringstream iss;
typedef vector<int> vint;
typedef vector<INT> VINT;
typedef vector<pii> vpii;
typedef vector<PII> VPII;
typedef priority_queue<int> pqi;
typedef priority_queue<INT> PQI;
typedef priority_queue<pii> pqp;
typedef priority_queue<PII> PQP;
typedef priority_queue<int,vint,greater<int> > xqi;
typedef priority_queue<INT,VINT,greater<INT> > XQI;
typedef priority_queue<pii,vpii,greater<pii> > xqp;
typedef priority_queue<PII,VPII,greater<PII> > XQP;
inline int read()
{
	int num=0;
	char c=getchar();
	if(c=='-')return -read();
	while(c>='0'&&c<='9')
	{
		num=(num<<3)+(num<<1)+c-'0';
		c=getchar();
	}
	return num;
}
inline INT READ()
{
	INT num=0;
	char c=getchar();
	if(c=='-')return -READ();
	while(c>='0'&&c<='9')
	{
		num=(num<<3)+(num<<1)+c-'0';
		c=getchar();
	}
	return num;
}
//Coded by Charlieyan
//版权所有，翻印必究

int N,n;
pii a[MASK(19)-1],b[MASK(19)-1];
int mxans=-Inf;

int main()
{
	N=read();
	n=MASK(N);
	for(int i=0;i<n;i++)
	{
		a[i].first=read();
		a[i].second=i;
		b[i]=a[i];
		for(int j=0;j<19;j++)
		{
			if(a[i].first<a[i&~MASK(j)].first)
			{
				a[i].first=a[i&~MASK(j)].first;
				a[i].second=a[i&~MASK(j)].second;
			}
		}
		if(a[i].second==i)b[i].first=-Inf;
		for(int j=0;j<19;j++)
		{
			if(b[i].first<a[i&~MASK(j)].first&&a[i].second!=a[i&~MASK(j)].second)
			{
				b[i].first=a[i&~MASK(j)].first;
				b[i].second=a[i&~MASK(j)].second;
			}
			if(b[i].first<b[i&~MASK(j)].first)
			{
				b[i].first=b[i&~MASK(j)].first;
				b[i].second=b[i&~MASK(j)].second;
			}
		}
	}
	for(int i=1;i<n;i++)
	{
		mxans=max(mxans,a[i].first+b[i].first);
		printf("%d\n",mxans);
	}
	return 0;
}