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)

# -*- coding: utf-8 -*-
N = int(input())
A_list = list(map(int, input().split()))

if N == 0:
    if A_list[0] == 1:
        print(0)
    else:
        print(-1)
elif A_list[0] != 0:
    print(-1)
elif N == 1:
    if A_list[1] == 1 or A_list[1] == 2:
        print(A_list[1] + 1)
    else:
        print(-1)
else:
    ans = 1
    deathable_node = 1
    sum_rest_A = sum(A_list)
    for i in range(1, N + 1):
        now_A = A_list[i]
        deathable_node = min(sum_rest_A, deathable_node * 2) - now_A

        if deathable_node < 0:
            print(-1)
            break

        sum_rest_A -= now_A
        ans += deathable_node + now_A
#         print(i, now_A, sum_rest_A, deathable_node, ans)
#         if deathable_node == 0 and i != N:
#             print(-1)
#             break
    else:
        if deathable_node == 0:
            print(ans)
        else:
            print(-1)