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)


n = int(input())
tree = [int(i) for i in input().split()]
top = []
bot= []
sub=0
leave = 0
mult = .5
nub=0
splice =  0
if tree[0]>0 and n>0:
  leave=1
elif tree[0]>1 and n==0:
  leave = 1
elif tree[-1]<1:
  leave=1

for i in range(n+1):
  if tree[i]>mult*2:
    leave = 1
  else:
    pass
  top.append((mult-sub)*2)
  mult = top[-1]
  sub = tree[i]
  bot.insert(0,(tree[-(i+1)]+nub))
  nub = bot[0]

for d in range(n):
  if top[d]==bot[d]:
    splice=d
  elif top[d+1]==bot[d+1]:
    splice=d+1
  elif top[d]<bot[d+1]<top[d+1]:
    splice=d
  elif bot[d]<top[d+1]<bot[d+1]:
    splice=d+1

top=top[0:splice+1]
bot=bot[splice+1:]
if leave ==1:
  print(-1)
else:
  print(int(sum(top)+sum(bot)))