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)

import sys
from functools import lru_cache
 
#read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines
sys.setrecursionlimit(10**6)
def read():
  return int(readline())
def reads():
  return map(int, readline().split())
def mp(arg):
  return map(int,arg.split())

n=read()
a=list(reads())
if a[0]>1:
  print("-1")
  exit()
elif a[0]==1 and n>0:
  print("-1")
  exit()
res=0
node=[0 for _ in range(n+1)]
node[-1]=a[-1]
#node[-1][0]=node[-1][1]=a[-1]
sub=[0]*n
for i in range(n-1,-1,-1):
  #node[i][0]=(node[i+1][0]+1)//2+a[i]
  node[i]=min(node[i+1]+a[i],1<<i)
#print(node)
for i in range(n):
  sub[i]=node[i]-a[i]
  node[i+1]=min(node[i+1],sub[i]*2)
#print(sub)
#print(node)
for i in range(n+1):
  if (node[i]<=0 and i<n) or node[i]<0:
    print("-1")
    exit()
  res+=node[i]
print(res)