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.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace AtCoder2 {
	class Program {
		static void Main(string[] args) {

	
			var inputArray = retreiveLongArray();
			var inputCount = 1;

			// 不正な入力は処理しない
			if (inputArray.Length != inputCount) {
				Console.WriteLine("invalid input");
				return;
			}
			long N = inputArray[0];
			var Aarray = retreiveLongArray();

			//最初の枝が生えてないと無理
			if (Aarray[0] > 0　&& N>0) {
				Console.WriteLine(-1);
				return;
			}

			//Nが0の時の例外処理
			if (N == 0) {
				if (Aarray[0]==1) {
					Console.WriteLine(1);
					return;
				}
				Console.WriteLine(-1);
				return;
			}

			checked {
				var maxChoutens = new Dictionary<long, long>();//可能な限り枝を伸ばした場合
				var minChoutens = new Dictionary<long, long>();
				maxChoutens.Add(0, 1);
				for (int i = 1; i <= N; i++) {
					var kouho = (maxChoutens[i - 1] - Aarray[i - 1]) * 2;
					if (kouho <= 0) {//頂点がマイナスになる場合、不可能
						Console.WriteLine(-1);
						return;
					}

					kouho = Math.Min(kouho, long.MaxValue/2);
					maxChoutens.Add(i, kouho);

					//N番目のチェックをしていない
					if (i==N) {
						if (kouho<Aarray[i]) {
							Console.WriteLine(-1);
							return;
						}
					}

				}

				long eda = 0;
				for (var i = N; i >= 0; i--) {
					eda += Aarray[i];
	
					eda = Math.Min(eda, long.MaxValue / 2);
					minChoutens.Add(i, eda);
				}

				//答えを求める
				long choutenCount = 0;
				for (int i = 0; i <= N; i++) {
					choutenCount += Math.Min(minChoutens[i], maxChoutens[i]);
				}

				//答えの出力
				Console.WriteLine(choutenCount);
			}
		}

		/// <summary>
		/// スペースで区切られた文字列配列の入力を受け取る
		/// </summary>
		/// <returns></returns>
		private static string[] retreiveStringArray() {
			var inputArray = Console.ReadLine().Split(' ');
			return inputArray;
		}

		/// <summary>
		/// スペースで区切られた整数配列の入力を受け取る
		/// </summary>
		/// <returns></returns>
		private static long[] retreiveLongArray() {
			var inputArray = Console.ReadLine().Split(' ');
			var longArray = inputArray.Select(s => long.Parse(s));
			return longArray.ToArray();
		}
	}
}