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)

n = int(input())
a_list = list(map(int, input().split()))

flag = 0
node_sum = 0
now_node = 0
max_node = []
max_itr = 27
if n < 27:
    max_itr = n+1
for i in range(max_itr):
    if i == 0:
        max_node.append(1-a_list[0])
    else:
        max_node.append((max_node[i-1]-a_list[i-1])*2)
# print(max_node)

for d, a in enumerate(reversed(a_list)):
    max_now = 100000000
    d = n-d
    if d < max_itr:
        max_now = max_node[d]
    if (now_node+1)//2 + a > max_now:
        flag = 1
        break
    elif now_node + a > max_now:
        now_node = max_now
    else:
        now_node += a
    node_sum += now_node
    # print(f"{d} {now_node}")

if flag == 1:
    print(-1)
else:
    print(node_sum)