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)

#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
//cout << fixed << setprecision(17) << res << endl;
 
int main() {
    ll n;
    cin >> n;
    vector<ll> A(n+1);
    for (auto & a : A) {
        cin >> a;
    }
    vector<ll> frees(n+1,0);
    ll p = 1;
    for (ll i = 0; i <= n; ++i) {
        frees[i] = p - A[i];
        if (frees[i] < 0) {
            cout << -1 << endl;
            return 0;
        }
        p = frees[i] * 2;
    }

    ll res = 0;
    ll cur = 0;
    for (ll i = n; i >= 0; --i) {
        if (cur > frees[i]) {
            cur = frees[i];
        }
        cur = cur + A[i];
        res += cur;
    }
    cout << res << endl;
    return 0;
}