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# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Linq;
using System.Numerics;
namespace ConsoleApp1
{
    class Program
    {
        static void Main(string[] args)
        {
            Console.ReadLine();
            var a = Console.ReadLine().Trim().Split(' ').Select(BigInteger.Parse).ToArray();
            var len = a.Length;

            if (a[0] == 1 && len == 1)
            {
                Console.WriteLine(1);
                return;

            }
            if (a[0] != 0)
            {
                Console.WriteLine(-1);
                return;
            }

            var b = new BigInteger[len];
            var c = new BigInteger[len];
            b[0] = 1;
            for (int i = 1; i < len; i++)
            {
                b[i] = (b[i - 1] - a[i - 1]) * 2;
            }
            c[len - 1] = a[len - 1];
            for (int i = len - 2; 0 <= i; i--)
            {
                c[i] = a[i] + c[i + 1];
            }
            BigInteger r = 0;
            for (int i = 0; i < len; i++)
            {
                var v = BigInteger.Min(b[i], c[i]);
                if (v <= 0)
                {
                    Console.WriteLine(-1);
                    return;
                }
                r += v;
            }
            if(b[len-1] < c[len-1])
            {
                    Console.WriteLine(-1);
                    return;
            }
            Console.WriteLine(r);
        }
    }
}