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: Runtime Error)

#include <iostream>
#include <string>
#include <algorithm>
#include <vector>

using namespace std;

#define MOD 1000000009

vector<int> A;
vector<int> candidate(1, -1);
int maxDepth;
int res;

void saiki(int d, int nowP, int count) {

    if (d == maxDepth - 1) {
        if (A[d] == nowP) {
            cout << count << endl;
            exit(1);
        }
        return; 
    }

    int a = A[d];

    for (int i = (nowP - a) * 2; i >= nowP - a; i--) {
        saiki(d + 1, i, count + i);
    }
}

int main () {
    int n;

    cin >> n;

    for(int i = 0; i < n + 1; i++) {
        int a; cin >> a;
        A.push_back(a);
    }

    maxDepth = A.size();

    saiki(0, 1, 1);

    cout << -1 << endl;
}