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<iostream>
#include<algorithm>
#include<vector>
#include<cstdint>
#include<cstdlib>
template<typename T>
void fin(T const& t){ std::cout << t << std::endl; exit(0); }

int main() {
  std::cin.tie(0); std::ios::sync_with_stdio(false);
  int N; std::cin >> N; 
  std::vector<int64_t> A(N+1), B(N+1); 
  for(int i = 0; i <= N; ++i) std::cin >> A[i];
  B[N] = 0;
  for(int i = N-1; i >= 0; --i) B[i] = B[i+1]+A[i+1];
  B[0] = 1 - A[0];
  for(int i = 0; i < N; ++i) {
    // B[i+1] <= 2B[i] - A[i+1]
    B[i+1] = std::min(B[i+1], B[i]*2-A[i+1]);
    if(B[i+1] < 0) fin(-1);
    if(B[i] <= 0) fin(-1);
  }
  int64_t ans = 0;
  for(int i = 0; i <= N; ++i) ans += A[i]+B[i];
  fin(ans);
  return 0;
}