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 System;
using System.Collections.Generic;
using static Assistant.Input;
using static Assistant.Debug;
using System.Linq;
using Assistant;

namespace NOMURA2020_C
{
    class Program
    {
        static void Main(string[] args)
        {
            var N = int.Parse(Console.ReadLine());
            var A = Console.ReadLine().Split().Select(long.Parse).ToArray();

            long[] comb = new long[N + 2];
            comb[N] = A[N];
            for (int i = N - 1; i >= 0; i--)
            {
                comb[i] = comb[i + 1] + A[i];
            }

            bool can = true;
            long[] c = new long[N + 2];
            c[0] = 1;
            for (int i = 0; i < N; i++)
            {
                c[i + 1] = Math.Min(c[i] * 2, comb[i + 1]);
                c[i + 1] -= A[i + 1];
                if (c[i + 1] < 0) can = false;
            }

            long ans = 0;
            for (int i = 0; i < N + 1; i++)
            {
                ans += A[i] + c[i];
            }
            if (A[0] != 0) can = false;
            if (N == 0 && A[0] == 1) { can = true; ans = 1; }

            if (can) Console.WriteLine(ans);
            else Console.WriteLine(-1);
        }


    }


}

namespace Assistant
{
    static class Input
    {
        static List<string> line = new List<string>();
        static int index = 0;
        static String RNext()
        {
            if (line.Count <= index) line.AddRange(Console.ReadLine().Split());
            return line[index++];
        }
        public static int RInt => int.Parse(RNext());
        public static long RLong => long.Parse(RNext());
        public static int[] RInts => Console.ReadLine().Split().Select(int.Parse).ToArray();
        public static long[] RLongs => Console.ReadLine().Split().Select(long.Parse).ToArray();
        public static string RString => RNext();
        //以下未テスト
        public static int[] RIntsC(int c) => Enumerable.Repeat(0, c).Select(x => int.Parse(RNext())).ToArray();
        public static long[] RLongsC(int c) => Enumerable.Repeat(0, c).Select(x => long.Parse(RNext())).ToArray();
        public static char[][] RMap(int h) => Enumerable.Repeat(0, h).Select(x => Console.ReadLine().ToCharArray()).ToArray();
    }

    static class Debug
    {
        static public void Draw2D<T>(T[,] map, int mode = 1)
        {
#if DEBUG
            int W = map.GetLength(0);
            int H = map.GetLength(1);

            T[,] map2 = new T[W + 1, H + 1];
            for (int i = 0; i < W + 1; i++)
            {
                for (int j = 0; j < H + 1; j++)
                {
                    if (i == 0 && j == 0) map2[i, j] = (T)(object)0;
                    else if (i == 0) map2[i, j] = (T)(object)(j - 1);
                    else if (j == 0) map2[i, j] = (T)(object)(i - 1);
                    else map2[i, j] = map[i - 1, j - 1];
                }
            }

            for (int i = 0; i < W + 1; i++)
            {
                for (int j = 0; j < H + 1; j++)
                {
                    if (mode == 0) Console.Write((dynamic)map2[i, j] % 10);
                    if (mode == 1) Console.Write(map2[i, j] + " ");
                }
                Console.WriteLine();
            }
            Console.WriteLine();
#endif
        }

        public static void Draw1D(int[] array, int mode = 0)
        {
#if DEBUG
            Console.WriteLine(string.Join(" ", array));
#endif
        }
    }
}