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 Python (3.8.2) (judge verdict: Wrong Answer)

from sys import stdin
def main():
    #入力
    readline=stdin.readline
    n=int(readline())
    a=list(map(int,readline().split()))  #葉の数

    s=[0]*(n+1)
    for i in range(n,-1,-1):
        if i==n:
            s[i]=a[i]
        else:
            s[i]+=a[i]+s[i+1]

    b=[0]*(n+1)  #葉でない頂点数
    res=0
    flag=True
    for i in range(n+1):
        if i==0:
            if a[i]==1: b[i]=0
            else: b[i]=1
            res+=1
        else:
            v_max=min(s[i],2*b[i-1])
            if v_max<a[i]:
                flag=False
                break
            else:
                b[i]=v_max-a[i]
                res+=v_max
    
    if flag:
        if a[-1]==0:
            print(-1)
        else:
            print(res)
    else:
        print(-1)

if __name__=="__main__":
    main()