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# (Mono-mcs 6.8.0.105) (judge verdict: Accepted)

using static System.Math;
using static System.Console;
using System.Collections.Generic;
using System.Linq;
using System.Text;

namespace AtCoder
{
	class Program
	{
		static void Main(string[] args)
		{
			var n = long.Parse(ReadLine());
			var line = ReadLine().Split(' ').Select(value => long.Parse(value)).ToArray();

			var maxLeaf = line.Max();
			long node = 0;
			var nodes = new long[n + 1];
			for(var i = 0; i <= n; i++)
			{
				if(i == 0)
				{
					node = 1;
				}
				else
				{
					if (node >= (long.MaxValue - 100000000) / 2)
					{
						node = long.MaxValue - 100000000;
					}
					else
					{
						node *= 2;
					}
				}
				node -= line[i];

				if (node < 0)
				{
					WriteLine(-1);
					return;
				}

				if (i != n)
				{
					nodes[i] = node + line[i];
				}
				else
				{
					nodes[i] = line[i];
				}
			}

			for(var i = n; i >= 1; i--)
			{
				if(nodes[i - 1] - line[i - 1] > nodes[i])
				{
					nodes[i - 1] = nodes[i] + line[i - 1]; 
				}
			}

			WriteLine(nodes.Sum());
		}
	}
}