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: Wrong Answer)

using System;
using System.Diagnostics;
using System.IO;
using System.Text;
using System.Linq;
using System.Collections.Generic;
using System.Text.RegularExpressions;
using System.Security.Cryptography;

class Program
{
    const string Yes = "Yes";
    const string No = "No";
    const char White = '.';
    const char Black = '#';
    const long Mod = 1000000007;

    static void Main(string[] args) {
#if DEBUG
        MyDebugger.Test(process, "../TextFile1.txt");
#else
        object ret = process(new StreamReader(Console.OpenStandardInput()));
        Console.WriteLine(ret);
#endif
    }

    static object process(TextReader input) {
        var n = int.Parse(input.ReadLine());
        var aa = ToIntArray(input.ReadLine(), n+1);

        if (n == 0) {
            return aa[0] == 1 ? 1 : -1;
        }
        if(aa[0] != 0) return -1;

        long[] maxnode = new long[n+1];
        maxnode[0] = 1;
        for(int i=1; i<=n; i++) {
            var y = (maxnode[i - 1] - aa[i - 1]) * 2;
            maxnode[i] = y;
        }


        long sum = aa.Last();
        long nodecount = aa.Last();

        if(nodecount > maxnode[n]) return -1;

        for(int i=n-1; i>=0; i--) {
            var a = aa[i];

            var max = maxnode[i];
            if(nodecount > max * 2) {
                return -1;
            }

            nodecount = Math.Min(max, nodecount + a);
            sum += nodecount;
        }

        return sum;
    }

    static long Pow(long x, long n) {
        long ans = 1;
        while (n > 0) {
            if (n % 2 == 1) {
                ans = ans * x;
            }
            n /= 2;
            if (n == 0) break;
            x = x * x;
        }
        return ans;
    }

    static long[] ToIntArray(string strs, int n) {
        var ret = new long[n];
        var spl = strs.Split(' ');
        for (int i = 0; i < n; i++)
            ret[i] = long.Parse(spl[i]);
        return ret;
    }
}