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)

#define rep(i, n) for(int i = 0; i < (int)(n); i++)
#define all(x) (x).begin(),(x).end()
typedef long long ll;
// #include<C:\Users\J\Desktop\AC\athead.h>
#define showa(v) for(int iii = 0; iii < v.size(); iii++){ cout << v[iii] << " ";} cout << endl;
#define show(iiii) cout << iiii << endl; 
#define shows(iiiii) cout << iiiii << " ";
#include<iostream>
#include<string>
#include<vector>
// #include<stdio.h>
#include <cmath>
#include <cstdlib>
#include <cassert>
#include <algorithm>
#include <set>
#include <deque>
#include <numeric>
#include<tuple>
#include <utility>
#include <map>
#include <cassert>
#include<unordered_set>
using namespace std;

// vector<long long int> memo(100000);


// long long int rec_depth(int depth, vector<int> a, int cap, int n){
//     if(depth == n){
//         if(a[depth] > cap) return -1;
//         else{
//             // memo[depth] = a[depth];
//             result += a[depth];
//             return a[depth];
//         }
//     }
//     if(a[depth] >= cap) return -1;
//     else{
//         long long int res = rec_depth(depth+1, a, (cap-a[depth])*2, n);
//         // cout << res << ", " << result << endl;
//         if(res == -1){
//             return -1;
//         }
//         else{
//             if(res+a[depth] > cap) {
//                 result += cap;
//                 return cap;
//                 }
//             // cout <<  cap << endl;
            
//             else{ 
//                 result += res + a[depth];
//                 return res + a[depth];
//                 }
//         }
//     }
// }

 
int main(){
    int n;
    cin >> n;
    vector<long long int> a(n+1);
    // vector<int> memo(n+1);
    vector<long long int> cap(n+1);

    long long int intmax = 100000000;
    
    rep(i,n+1) {
        if((i!=0 && a[i-1] >= cap[i-1])){
            cout << -1 << endl;
            return 0;
        }
        cin >> a[i];
        if(i == 0){
            cap[i] = 1;
        }
        else{
            if((cap[i-1] - a[i-1]) > intmax){
                cout << -1 << endl;
                return 0;
            }
            cap[i] = (cap[i-1] - a[i-1])*2;
        }

    }
    long long int result = 0;
    long long int res = 0;
    // int cap = 0;
    for(int i=0; i<n+1; i++){
        int depth = n-i;
        if(depth == n){
            if(a[depth] > cap[depth]){
                cout << -1 << endl;
                return 0;
            }
            else{
                result += a[depth];
                res = a[depth];
            }
        }
        else{
            // long long int res = rec_depth(depth+1, a, (cap[depth]-a[depth])*2, n);
            // long long int res = rec_res;
            if(res+a[depth] > cap[depth]) {
                result += cap[depth];
                res = cap[depth];
                }     
            else{ 
                result += res + a[depth];
                res = res + a[depth];
                }
        }
    }

    // long long int kekka = rec_depth(0, a, 1, n);
    // if(kekka == -1)    cout << -1 << endl;
    cout << result << endl;
    return 0;

}