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 Java (OpenJDK 11.0.6) (judge verdict: Wrong Answer)

import java.io.IOException;
import java.io.InputStream;
import java.util.Arrays;
import java.util.HashMap;

public class Main {
    public static void main(String[] args)throws Exception {
        StringBuilder sb = new StringBuilder();
        FastReader in = new FastReader(System.in);
        int i, j;
        boolean ok=false;
        int t=ok?in.nextInt():1;
        while(t-->0){
            int n=in.nextInt();
            long arr[]=new long[n+1];
            int f=0;
            for(i=0;i<=n;i++) {
                arr[i] = in.nextInt();
                if(i<40 && arr[i]>(1l<<i)){
                    f=1;
                }
            }
            if(n==0){
                sb.append("1\n");
                continue;
            }
            if(arr[0]==1){
                for(i=1;i<=n;i++){
                    if(arr[i]!=0){
                        f=1;
                        break;
                    }
                }
                if(f==0)
                    sb.append("1\n");
                else
                    sb.append("-1\n");
                continue;
            }
            long ans=1;
            long max[]=new long[n+1];
            max[0]=1;
            for(i=1;i<=n;i++){
                long z=max[i-1]*2;
                z=z-arr[i];
                if(z<0)
                    f=1;
                max[i]=(long)Math.min(z,1e9);
            }
            long cur=arr[n];
            for(i=n;i>=1;i--){
                if(i<40 && cur>(1l<<i)){
                    f=1;
                    break;
                }
                ans+=cur;
                if(max[i-1]*2<cur){
                    f=1;
                    break;
                }
                else if(max[i-1]<cur){
                    cur=max[i-1];
                }
                cur+=arr[i-1];
            }
            if(f==0)
                sb.append(ans).append("\n");
            else {
                sb.append("-1\n");
            }
        }
        System.out.print(sb);

    }
}

class FastReader {

    byte[] buf = new byte[2048];
    int index, total;
    InputStream in;

    FastReader(InputStream is) {
        in = is;
    }

    int scan() throws IOException {
        if (index >= total) {
            index = 0;
            total = in.read(buf);
            if (total <= 0) {
                return -1;
            }
        }
        return buf[index++];
    }

    String next() throws IOException {
        int c;
        for (c = scan(); c <= 32; c = scan()) ;
        StringBuilder sb = new StringBuilder();
        for (; c > 32; c = scan()) {
            sb.append((char) c);
        }
        return sb.toString();
    }

    int nextInt() throws IOException {
        int c, val = 0;
        for (c = scan(); c <= 32; c = scan()) ;
        boolean neg = c == '-';
        if (c == '-' || c == '+') {
            c = scan();
        }
        for (; c >= '0' && c <= '9'; c = scan()) {
            val = (val << 3) + (val << 1) + (c & 15);
        }
        return neg ? -val : val;
    }

    long nextLong() throws IOException {
        int c;
        long val = 0;
        for (c = scan(); c <= 32; c = scan()) ;
        boolean neg = c == '-';
        if (c == '-' || c == '+') {
            c = scan();
        }
        for (; c >= '0' && c <= '9'; c = scan()) {
            val = (val << 3) + (val << 1) + (c & 15);
        }
        return neg ? -val : val;
    }
}