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)

def solve(n, a_list):

    # judge_if_possible
    b_list = [0] * (n + 1)  # nodes with child

    for d in range(n + 1):
        if d == 0:
            n_nodes = 1
        else:
            n_nodes = b_list[d - 1] * 2
        if a_list[d] > n_nodes:
            return -1
        b_list[d] = n_nodes - a_list[d]
    # print(a_list, b_list)
    # backward
    b_list[n] = 0
    for i in range(n - 1, -1, -1):
        b_list[i] = min(b_list[i], a_list[i + 1] + b_list[i + 1])
    res = sum(a_list) + sum(b_list)
    # print(a_list, b_list, res)
    return res


def main():
    n = int(input())
    a_list = list(map(int, input().split()))
    res = solve(n, a_list)
    print(res)


def test():
    assert solve(3, [0, 1, 1, 2]) == 7
    assert solve(4, [0, 0, 1, 0, 2]) == 10
    assert solve(2, [0, 3, 1]) == -1
    assert solve(10, [0, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34]) == 264


if __name__ == "__main__":
    test()
    main()