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)

import sys
sys.setrecursionlimit(10 ** 7)
input = sys.stdin.readline

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

ans=[0]*(n+1)
ans[0]=1

if a[0] != 0:
    if n != 0:
        print(-1)
        exit()
    else:
        print(1)
        exit()

if a[n] == 0:
    print(-1)
    exit()


ans[n]=a[n]

def dfs(fukasa):
    if fukasa == 0:
        if ans[fukasa] == 1:
            return True
        else:
            return False

    mi = a[fukasa] -(-ans[fukasa+1]//2)
    ans[fukasa] = mi
    if not dfs(fukasa-1):
        return False

    tmp = a[fukasa]+ans[fukasa+1]
    tmp = min(tmp,2**fukasa)

    while tmp >= a[fukasa] -(-ans[fukasa+1]//2):
        ans[fukasa]=tmp
        tmpp = dfs(fukasa-1)
        if tmpp:
            ans[fukasa]=tmp
            break
        tmp -= 1
    else:
        return False
    return True


tmp = dfs(n-1)

if tmp:
    print(sum(ans))
else:
    print(-1)