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)

import sys


def print_(ans):
    print(ans)
    sys.exit()


if __name__ == '__main__':
    n = int(input())
    a = list(map(int, input().split(' ')))
    if n == 0:
        if a[0] == 1:
            print_(1)
        else:
            print_(-1)
    if a[0] != 0:
        print_(-1)
    max_rest = [0 for _ in range(n + 1)]
    max_rest[0] = 1 - a[0]
    for i in range(1, n + 1):
        max_rest[i] = max_rest[i - 1] * 2 - a[i]
    for i in range(n + 1):
        if max_rest[i] < 0:
            print_(-1)
    rest = [0 for _ in range(n + 1)]
    for i in range(n - 1, -1, -1):
        req_rest = (a[i + 1] + rest[i + 1] + 1) // 2
        if max_rest[i] < req_rest:
            print_(-1)
        rest[i] = min(max_rest[i], a[i + 1] + rest[i + 1])
    ans = sum(a) + sum(rest)
    print_(ans)