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: Accepted)

#include<iostream>
#include<vector>
#include<algorithm>
using namespace std;

typedef long long int ll;
const int N_MAX = 100005;
const ll INF = 1000000000000000005;

int n;
int A[N_MAX];
ll nres[N_MAX];


int main(){
    cin >> n;
    for(int i = 0; i <= n; i++) cin >> A[i];
    ll an;
    ll nodenum = 0;
    ll now = 0;
    ll benode = 1;
    nres[0] = 1;
    ll exi = 0;
    if (A[0] > 0)nodenum = -1;
    for(int i = 1; i <= n; i++){
        if (nodenum == -1)break;
        benode = min(2 * benode, INF);
        nres[i] = benode;
        if (A[i] > 0){
            if (benode < A[i]){
                nodenum = -1;
                break;
            }
            nres[i] -= A[i];
            benode -= A[i];

            ll res = A[i];
            while(res > 0){
                if (res <= nres[now]){
                    nres[now] -= res;
                    nodenum += res * (i - now + 1);
                    exi += res;
                    break;
                }
                res -= nres[now];
                nodenum += nres[now] * (i - now + 1);
                exi += nres[now];
                now++;
                exi -= A[now];
                nres[now] -= exi;

            }
        }
    }
    if (n == 0 && A[0] == 1) nodenum = 1;
    cout << nodenum << endl;
}