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: Time Limit Exceeded)

def INT(): return int(input())
def MAP(): return map(int, input().split())
def LIST(): return list(map(int, input().split()))


N = INT()
A = LIST()

init_nodes = [0]*(N+1)
stems = [0]*(N+1)
init_nodes[0] = 1
stems[0] = 1


def solve():
    if N > 0:
        if A[0] > 0: return -1
    else:
        if A[0] == 1: return 1
        else: return -1

    if A[-1] == 0: return -1


    for i in range(N+1):
        if A[i] > 2**i:
            return -1


    for i in range(1, N+1):
        init_nodes[i] = 2 * stems[i-1]
        stems[i] = init_nodes[i] - A[i]
        if stems[i] < 0: return -1

    ans = sum(init_nodes)

    dif = init_nodes[-1] - A[-1]
    ans -= dif
    for i in reversed(range(N+1)):
        if init_nodes[i] - dif < 0: return -1
        dif = (dif-(init_nodes[i] - dif))//2
        if dif <= 0: break
        ans -= dif
    return ans

print(solve())