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 <algorithm>
#include <cstdint>
#include <cassert>
#include <cmath>
#include <cstdio>
#include <deque>
#include <iomanip>
#include <iostream>
#include <map>
#include <queue>
#include <set>
#include <stack>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>
#include <cfloat>

using namespace std;

using lint=int64_t;
using uint=uint32_t;
using ulint=uint64_t;

template<class T>
using vector2d=vector<vector<T>>;

template<class T>
bool UpdateMax(T &a, const T &b){
  if(a<b){
    a=b;
    return true;
  }else{
    return false;
  }
}

template<class T>
bool UpdateMin(T &a, const T &b){
  if(a>b){
    a=b;
    return true;
  }else{
    return false;
  }
}

template <class T>
void OutVec(const vector<T>& vec){
  for(int i = 0; i < vec.size() - 1; ++i){
    cout<<vec[i]<<" ";
  }
  cout<<vec.back()<<endl;
}

template <class T>
void OutVec2d(const vector2d<T>& vec){
  for(auto v:vec){
    OutVec(v);
  }
}

int main(){
  cout << std::fixed << std::setprecision(16);
  cin.tie(nullptr);
  std::ios::sync_with_stdio(false);

  lint n;
  cin>>n;
  vector<lint> a(n+1);
  vector<lint> cum(n+1);
  for(int i=0;i<=n;i++){
    cin>>a[i];
  }
  cum[0]=a[0];
  for(int i=1;i<=n;i++){
    cum[i]=cum[i-1]+a[i];
  }

  if(n==0){
    if(a[0]==1){
      cout<<1<<endl;
    }else{
      cout<<0<<endl;
    }
    return 0;
  }

  if(a[0]==1){
    if(n==0){
      cout<<1<<endl;
      return 0;
    }else{
      cout<<-1<<endl;
      return 0;
    }
  }

  lint result=1;
  lint prev=1;
  for(int i=1;i<=n;i++){
    lint upper_lim=cum[n]-cum[i-1];
    lint lower_lim=a[i];

    if(prev>upper_lim||prev*2<lower_lim){
      cout<<-1<<endl;
      return 0;
    }else{
      result+=min(prev*2, upper_lim);
      prev=min(prev*2, upper_lim)-a[i];
    }

  }
  cout<<result<<endl;


  return 0;
}