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: Runtime Error)

n=int(input())
a=list(map(int,input().split()))
if n==1:
  if a==[0,1]:
    print(2)
  elif a==[0,2]:
    print(3)
  else:
    print(-1)
  exit()
subt=[0]*(n)
if a[0] == 0:
  subt[0]=2
elif a[0]== 1:
  subt[0]=1
else:
  print(-1)
  exit()

for i in range(n-1):
  if a[i+1] > subt[i]-1:
    if i<n-1:
      print(-1)
      exit()
    elif a[i+1] > subt[i]:
      print(-1)
      exit()
  else:  
    subt[i+1]=(subt[i]-a[i+1])*2
if a[n]> subt[-1]:
  print(-1)
  exit()
if a[n] > subt[-2]:
  subt[-1]=a[n]
else:
  subt[-1]=a[n]
  for j in range(2,n):
    if (subt[-j]-a[-j])*2 < subt[1-j]:
      print(-1)
      exit()
    if (subt[-j]-a[-j]) <= subt[1-j]:
      pass
    else:
      subt[-j]=subt[1-j]+a[-j]
print(sum(subt)+1)