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;
using ll = long long;

constexpr int MN = 100005;
ll s = 0;
ll n, a[MN];
ll min_p[MN];
ll max_p[MN];

int main() {
  cin >> n;
  for (int i = 0; i <= n; ++i) {
    cin >> a[i];
    s += a[i];
  }
  if (a[0] > 0) {
    cout << -1 << endl;
    return 0;
  }

  min_p[0] = max_p[0] = 1;
  for (int i = 1; i <= n; ++i) {
    max_p[i] = min((ll)1e10, max_p[i - 1] * 2 - a[i]);
    if (i < n && max_p[i] == 0) {
      cout << -1 << endl;
      return 0;
    }
  }
  max_p[n] = 0;

  for (int i = n - 1; i >= 0; --i) {
    if (max_p[i] * 2 < a[i + 1]) {
      cout << -1 << endl;
      return 0;
    }
    max_p[i] = min(max_p[i], max_p[i + 1] + a[i + 1]);
    s += max_p[i];
  }
  cout << s << endl;
}