Problem:
Score : 600 points

Problem Statement

Given is an integer sequence of length N+1: A_0, A_1, A_2, \ldots, A_N. Is there a binary tree of depth N such that, for each d = 0, 1, \ldots, N, there are exactly A_d leaves at depth d? If such a tree exists, print the maximum possible number of vertices in such a tree; otherwise, print -1.

Notes

A binary tree is a rooted tree such that each vertex has at most two direct children.

A leaf in a binary tree is a vertex with zero children.

The depth of a vertex v in a binary tree is the distance from the tree's root to v. (The root has the depth of 0.)

The depth of a binary tree is the maximum depth of a vertex in the tree.

Constraints

0 \leq N \leq 10^5

0 \leq A_i \leq 10^{8} (0 \leq i \leq N)

A_N \geq 1

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_0 A_1 A_2 \cdots A_N

Output

Print the answer as an integer.

Sample Input 1

3
0 1 1 2

Sample Output 1

7

Below is the tree with the maximum possible number of vertices. It has seven vertices, so we should print 7.

Sample Input 2

4
0 0 1 0 2

Sample Output 2

10

Sample Input 3

2
0 3 1

Sample Output 3

-1

Sample Input 4

1
1 1

Sample Output 4

-1

Sample Input 5

10
0 0 1 1 2 3 5 8 13 21 34

Sample Output 5

264

Solution in C++ (GCC 9.2.1) (judge verdict: Wrong Answer)

#pragma GCC optimize ("O3")

#include <iostream>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>
#include <map>
#include <unordered_map>
#include <set>
#include <unordered_set>
#include <queue>
#include <set>
#include <algorithm>
#include <numeric>
#include <list>
using namespace std;

using QWORD  = uint64_t;
using SQWORD = int64_t;
using DWORD  = uint32_t;
using SDWORD = int32_t;
using WORD   = uint16_t;
using SWORD  = int16_t;
using BYTE   = uint8_t;
using SBYTE  = int8_t;
using DOUBLE = double;
using FLOAT  = float;

#define MIN_SDWORD (-2147483648)
#define MAX_SDWORD (2147483647)
#define MIN_SBYTE (-128)
#define MAX_SBYTE (127)

#define MIN_SQWORD (0x8000000000000000)
#define MAX_SQWORD (0x7FFFFFFFFFFFFFFF)

#define MAX_QWORD  (0xFFFFFFFFFFFFFFFF)
#define MAX_DWORD  (0xFFFFFFFF)
#define MAX_WORD   (0xFFFF)
#define MAX_BYTE   (0xFF)

#define MAX_DOUBLE      (1.0e+308)
#define DOUBLE_EPS      (1.0e-12)
#define MIN_DOUBLE_N    (-1.0e+308)

#define ArrayLength(a)  (sizeof(a) / sizeof(a[0]))

static inline DOUBLE MAX(DOUBLE a, DOUBLE b) { return a > b ? a : b; }
static inline QWORD MAX(QWORD a, QWORD b) { return a > b ? a : b; }
static inline DWORD MAX(DWORD a, DWORD b) { return a > b ? a : b; }
static inline SDWORD MAX(SDWORD a, SDWORD b) { return a > b ? a : b; }
static inline DOUBLE MIN(DOUBLE a, DOUBLE b) { return a < b ? a : b; }
static inline QWORD MIN(QWORD a, QWORD b) { return a < b ? a : b; }
static inline DWORD MIN(DWORD a, DWORD b) { return a < b ? a : b; }
static inline SDWORD MIN(SDWORD a, SDWORD b) { return a < b ? a : b; }
static inline DOUBLE ABS(DOUBLE a) {return 0 <= a ? a : -a;}

#define BYTE_BITS   (8)
#define WORD_BITS   (16)
#define DWORD_BITS  (32)
#define QWORD_BITS  (64)

static inline void inputStringSpSeparated(char *pcStr)
{
    char *pcCur = pcStr;
    for (;;) {
        char c = getchar();
        if (('\n' == c) || (EOF == c) || (' ' == c)) {
            break;
        }
        *pcCur = c;
        pcCur++;
    }
    *pcCur = '\0';
}

static inline void inputString(char *pcStr)
{
    char *pcCur = pcStr;
    for (;;) {
        char c = getchar();
        if (('\n' == c) || (EOF == c)) {
            break;
        }
        *pcCur = c;
        pcCur++;
    }
    *pcCur = '\0';
}


static inline SQWORD inputSQWORD(void)
{
    SQWORD sqNumber = 0;
    SQWORD sqMultiplier = 1;
    bool bRead = false;
    for (;;) {
        char c = getchar();
        if (!bRead) {
            if ('-' == c) {
                sqMultiplier = -1;
            }
        }
        if (('0' <= c) && (c <= '9')) {
            sqNumber *= 10LL;
            sqNumber += (SQWORD)(c - '0');
            bRead = true;
        } else {
            if (bRead) {
                return sqNumber * sqMultiplier;
            }
        }
    }
}


static inline SDWORD inputSDWORD(void)
{
    SDWORD lNumber = 0;
    SDWORD lMultiplier = 1;
    bool bRead = false;
    for (;;) {
        char c = getchar();
        if (!bRead) {
            if ('-' == c) {
                lMultiplier = -1;
            }
        }
        if (('0' <= c) && (c <= '9')) {
            lNumber *= 10;
            lNumber += (c - '0');
            bRead = true;
        } else {
            if (bRead) {
                return lNumber * lMultiplier;
            }
        }
    }
}

static inline DOUBLE inputFP(void)
{
    DOUBLE dInt = 0.0;
    DOUBLE dFrac = 0.0;
    DOUBLE dMultiplier = 1.0;
    DWORD dwFpCnt = 0;
    DOUBLE *pdCur = &dInt;
    bool bRead = false;
    for (;;) {
        char c = getchar();
        if (!bRead) {
            if ('-' == c) {
                dMultiplier = -1;
            }
        }
        if ('.' == c) {
            pdCur = &dFrac;
        } else if (('0' <= c) && (c <= '9')) {
            (*pdCur) *= 10;
            (*pdCur) += (DOUBLE)(c - '0');
            bRead = true;
            if (pdCur == &dFrac) {
                dwFpCnt++;
            }
        } else {
            if (bRead) {
                return dMultiplier * (dInt + dFrac / (pow((DOUBLE)10.0 , (DOUBLE)dwFpCnt)));
            }
        }
    }
}

/*----------------------------------------------*/
/**
 *  mod による操作ライブラリ
 */
#define ANS_MOD (1000000007)

class MODINT {
    static SQWORD MOD;
    SQWORD m_x;

public:
    MODINT(SQWORD val) {
        m_x = (val % MOD + MOD) % MOD;
    };
    MODINT() {
        m_x = 0;
    }
    static void Init(SQWORD sqMod) {
        MOD = sqMod;
    }

	MODINT& operator+= (const MODINT a)
    {
        m_x = (m_x + a.m_x) % MOD; 
        return *this;
    };
	MODINT& operator-= (const MODINT a)
    { 
        m_x = (m_x - a.m_x + MOD) % MOD; 
        return *this;
    };
	MODINT& operator*= (const MODINT a)
    {
        m_x = (m_x * a.m_x) % MOD;
        return *this;
    };
    MODINT pow(SQWORD t) const {
        if (!t) return 1;
        MODINT a = pow(t>>1);
        a *= a;
        if (t&1) a *= *this;
        return a;
    }
	MODINT operator+ (const MODINT a) const {
		MODINT res(*this);
		return (res += a);
	}
	MODINT operator- (const MODINT a) const {
		MODINT res(*this);
		return (res -= a);
	}
	MODINT operator* (const MODINT a) const {
		MODINT res(*this);
		return (res *= a);
	}
	MODINT operator/ (const MODINT a) const {
		MODINT res(*this);
		return (res /= a);
	}

    /* 逆元 */
    MODINT inv() const {
        return pow(MOD-2);
    }

    /* 除算 */
    MODINT& operator/=(const MODINT a) {
        return (*this) *= a.inv();
    } 

    /* 整数版 */
	MODINT& operator+= (const SQWORD a) {*this += MODINT(a); return *this;};
	MODINT& operator-= (const SQWORD a) {*this -= MODINT(a); return *this;};
	MODINT& operator*= (const SQWORD a) {*this *= MODINT(a); return *this;};
	MODINT& operator/= (const SQWORD a) {*this /= MODINT(a); return *this;};

    SQWORD getVal() { return m_x; };
};
SQWORD MODINT::MOD = ANS_MOD;

/*----------------------------------------------*/

static const DOUBLE d_PI = 3.14159265357989;

static DOUBLE getRad(DOUBLE deg)
{
    return (deg * d_PI * 2.0) / 360.0;
}

/*-----------------------------------------------------*/

int main(void)
{
    SQWORD sqN = inputSQWORD();
    vector<SQWORD> vsqA;
    for (SQWORD sqIdx = 0; sqIdx <= sqN; sqIdx++) {
        SQWORD sqA = inputSQWORD();
        vsqA.emplace_back(sqA);
    }

    vector<SQWORD> vsqMin(sqN + 1);
    vector<SQWORD> vsqMax(sqN + 1);

    SQWORD sqSum = 0;
    for (SQWORD sqIdx = vsqA.size() - 1; 0 <= sqIdx; --sqIdx) {
        sqSum += vsqA[sqIdx];
        vsqMax[sqIdx] = sqSum;
    }
    SQWORD sqTmpMax = 1;
    for (SQWORD sqIdx = 0; sqIdx < vsqA.size() - 1; ++sqIdx) {
        vsqMax[sqIdx] = min(vsqMax[sqIdx], sqTmpMax);
        sqTmpMax -= vsqA[sqIdx];
        if (vsqMax[sqIdx] < 0) {
            printf("-1\n");
            return 0;
        }
        sqTmpMax *= 2;
    }

    SQWORD sqCurNode = 0;
    for (SQWORD sqIdx = vsqA.size() - 1; 0 <= sqIdx; --sqIdx) {
        vsqMin[sqIdx] = (sqCurNode + vsqA[sqIdx]);
        sqCurNode = (sqCurNode + vsqA[sqIdx] + 1) / 2;
    }

    SQWORD sqAns = 0;
    for (SQWORD sqDepth = 0; sqDepth <= sqN; sqDepth++) {
        SQWORD sqMax = vsqMax[sqDepth];
        SQWORD sqMin = vsqMin[sqDepth];
//        printf("%lld %lld\n", sqMin, sqMax);
        if (sqMax < sqMin) {
            printf("-1\n");
            return 0;
        }
        sqAns += sqMax;
    }
    printf("%lld\n", sqAns);
    return 0;
}