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: Memory Limit Exceeded)

# 深さdの上限を先に決める
N = int(input())
# Aは，葉として残す必要があるものの数
A = list(map(int, input().split()))

# 深さdのノード数の上限
limit = [1]
for i in range(1, N+1):
    if i == N:  # 一番深いところの上限はA[N]
        limit.append(A[i])
    else:
        limit.append((limit[-1]-A[i-1])*2)

# Nが0の時の処理
if N == 0:
    if A[0] != 1:
        print(-1)
        exit()

# 一番深いところから構築して最後にsumをとる
ans = [A[-1]]
now_leaf = A[-1]  # 一番深いところから構築しており，今現在の葉の数
for i in range(N-1, -1, -1):
    # print(A[i])
    ans.append(min(now_leaf+A[i], limit[i]))
    now_leaf = ans[-1]
# 構築したものに対してジャッジする, 葉を確保できているかどうか
for i in range(N):
    # ans[N-i]とすることで，今は一番浅いところから見ている
    if (ans[N-i]-A[i])*2 < A[i+1]:  # 葉が確保できているか
        print(-1)
        exit()
print(sum(ans))