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: Wrong Answer)

import sys
input = sys.stdin.readline
def main():
    N = int(input())
    A = list(map(int, input().split()))
    node_count = [0] * (N+1)
    for i, a in enumerate(A[::-1]):
        if i == 0:
            node_count[N] = a
            continue
        node_count[N-i] = node_count[N-i+1] + a
    can_build = True
    for i, a in enumerate(A):
        if i == 0:
            if a > 0:
                can_build = False
                break
            node_count[0] = min(node_count[0], 1)
            continue
        if i < N and a >= 2**i:
            can_build = False
            break
        node_count[i] = min(node_count[i], node_count[i-1]*2-A[i-1]*2)
    if can_build:
        ans = sum(node_count)
    else:
        ans = -1
    print(ans)
if __name__ == "__main__":
    main()