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)

def main():

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

    C1 = [[1, 1]]
    B1 = [[1-A[0], 1-A[0]]]
    for i in range(1, N+1):
        if 2*B1[-1][1] < A[i]: return -1
        x, y = max(B1[-1][0]-A[i], 0), 2*B1[-1][1] - A[i]
        B1.append([x, y])
        C1.append([x+A[i], y+A[i]])
    if A[N] < C1[-1][0] or A[N] > C1[-1][1]:
        return -1
    C1.append([A[N], A[N]])
    # print(C1)

    C2 = [A[N]]
    for i in range(N-1, 0, -1):
        x, y = (C2[-1] + 1) // 2, C2[-1]
        z = 0
        if y < B1[i][0] or x > B1[i][1]: return -1
        if B1[i][0] <= y < B1[i][1]:
            z = y
        else:
            z = B1[i][1]
        C2.append(z + A[i])
    # print(C2)

    return sum(C2) + 1

    # return ans

if __name__ == '__main__':
    print(main())