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: Wrong Answer)

n = int(input())
a = list(map(int, input().split()))
if n == 0:
    print(a[0])
    exit()
if a[0] > 0:
    print(-1)
    exit()
l = [0] * (n + 1)
u = [0] * (n + 1)
l[n] = a[n]
u[n] = a[n]
for i in range(n, 0, -1):
    l[i - 1] = -(-l[i] // 2) + a[i - 1]
    u[i - 1] = u[i] + a[i - 1]
if l[0] > 1:
    print(-1)
    exit()
u[0] = 1
itr = 0
while True:
    update = False
    for i in range(n):
        if u[i] < l[i]:
            print(-1)
            exit()
        n_u = 2 * (u[i] - a[i])
        if n_u < u[i + 1]:
            update = True
            u[i + 1] = n_u
    if not update:
        break
    update = False
    for i in range(n, 0, -1):
        if u[i] < l[i]:
            print(-1)
            exit()
        n_u = u[i] + a[i - 1]
        if n_u < u[i - 1]:
            update = True
            u[i - 1] = n_u
    if not update:
        break
print(sum(u))