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: Accepted)

N = int(input())
A = [int(n) for n in input().split()]

if N == 0:
    if A[0] == 1:
        print(1)
    else:
        print(-1)
    exit()

if A[0] != 0:
    print(-1)
    exit()
    
B = [1 for _ in range(N+1)]
for i in range(1, N+1):
    B[i] = B[i-1] * 2 - A[i]
    if B[i] < 0:
        print(-1)
        exit()

ans = 0

n_prev = 0
for a, b in zip(A[::-1], B[::-1]):
    B_low = (n_prev + 1) // 2
    B_high = n_prev

    #print('{} {} {}'.format(B_low, B_high, b))

    if B_low > b:
        print(-1)
        exit()

    n_cur = a + min(B_high, b)
    #print(a, n_cur)
    ans += n_cur
    n_prev = n_cur
    
print(ans)