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 <bits/stdc++.h>
#define FAST ios_base::sync_with_stdio(false);cin.tie(NULL);cout.tie(NULL);
typedef long long ll;
typedef long double ld;
#define pb push_back
#define mp make_pair
#define ff first
#define ss second
#define mod 1000000007
#define inf 1000000000000000000
#define bpc(x) __builtin_popcountll(x)
#define autoit(x,it) for(auto it = x.begin(); it != x.end(); it++)
#define rep(n) for(ll i = 0; i < n; i++)
#define repi(i,n) for(ll i = 0; i < n; i++)

#include <ext/pb_ds/assoc_container.hpp> 
#include <ext/pb_ds/tree_policy.hpp> 
using namespace __gnu_pbds; 

#define ordered_set tree<ll, null_type,less<ll>, rb_tree_tag,tree_order_statistics_node_update> 

using namespace std;

char get(ll n)
{
    if(n == 1)
        return 'P';
    return 'D';    
}

int main()
{
   FAST/**/
   
   ll n;
   cin>>n;
   
   ll arr[n+1];
   rep(n+1)
        cin>>arr[i];
   if((n == 0 && arr[0]!=1) || (arr[0] == 1 && n>0))
   {
       cout<<-1;
       return 0;
   }
   
   if(n == 0)
   {
       cout<<1;
       return 0;
   }
   
   ll curr = 1;
   ll sum = 1;
   
   ll post[n+1];
   post[n] = arr[n];
   for(ll i=n-1;i>-1;i--)
        post[i]=post[i+1]+arr[i];
        
   for(ll i=1;i<=n;i++)
   {
       ll maxi = post[i];
       ll l = curr - arr[i-1];
       ll r = 2*l;
       
       if(curr<arr[i-1])
       {
           cout<<-1;
           return 0;
       }
       if(maxi<l)
       {
           //cout<<"hehe\n";
           //cout<<-10<<" at "<<i<<"\n";
           cout<<-1;
           return 0;
       }
       
       ll ele;
       if(maxi<=r)
            ele = maxi;
       else ele = r;
       
       if(i == n && r<maxi)
       {
           //cout<<-100<<" at "<<i<<"\n";
           cout<<-1;
           return 0;
       }
       
       curr = ele;
       sum+=ele;
   }
   
   cout<<sum;
   
   
   return 0;        
}