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: Wrong Answer)

import sys
import math


def main():
    N = int(input())
    A = list(map(int, input().split()))

    if A[0] != 0:
        print('-1')
        sys.exit()

    peak = 0
    peak_pre = 0
    max_peak = [0] * (N + 1)
    max_peak[0] = 1
    for j in range(1, N+1):
        max_peak[j] = max_peak[j-1] * 2 - A[j]
        if (j == N):
            max_peak[j] += A[j]
    for i in range(N, 0, -1):
        a = A[i]
        if N == i and a % 2 == 1:
            print('-1')
            sys.exit()
            break
        if 2**i <= a:
            print('-1')
            sys.exit()
            break
        cur_peak = min(math.ceil(peak_pre), max_peak[i]) +a
        peak += cur_peak
        peak_pre = cur_peak
    print(peak+1)


if __name__ == '__main__':
    main()