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

N = int(input())
line_data = input()
data = list(map(lambda x: int(x), line_data.split(' ')))

if N == 0:
    if data[0] == 1:
        print(1)
    else:
        print(-1)
    sys.exit(0)

if N != len(data) + 1:
    print(-1)
    sys.exit(0)
    
tmp_sum = sum(data)
tmp_data = []
for i in data:
    tmp_sum -= i
    tmp_data.append(tmp_sum)
#print(tmp_data)

num = 1
prev_leaf = 1
for i, d in enumerate(data[:-1]):
    tmp = (prev_leaf - d) * 2
    top = tmp_data[i]
    #print("((prev:{}) - (d:{})) * 2 = {}".format(prev_leaf, d, tmp))
    #print("top = {}".format(top))
    if tmp <= 0:
        num = -1
        break
    if tmp < top:
        prev_leaf = tmp
        #print('add tmp = {}'.format(tmp))
    else:
        prev_leaf = top
        #print('add top = {}'.format(top))                                                   
    num += prev_leaf
    
print(num)