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 PyPy3 (7.3.0) (judge verdict: Accepted)

#C
N=int(input())
A=[int(x) for x in input().split()]

P=[0 for i in range(N+1)]
P[N]=[A[N],A[N]]
for i in range(N-1,-1,-1):
    MIN,MAX=P[i+1]
    mi=int(A[i]+(MIN+MIN%2)/2)
    ma=A[i]+MAX
    P[i]=[mi,ma]
    
    
if P[0][0]==1:
    Q=[0 for i in range(N+1)]
    Q[0]=1
    cnt=1
    for i in range(N):
        Q[i+1]=min((Q[i]-A[i])*2,P[i+1][1])
        cnt+=Q[i+1]
    print(cnt)
else:
    print("-1")