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 Lua (Lua 5.3.5) (judge verdict: Accepted)

local n = io.read("*n")
local t = {}
local imos = {}
for i = 1, n + 1 do
  t[i] = io.read("*n")
  imos[i] = 0
end
imos[n + 2] = 0
local rem = 1
local score = 0
local stage = 0
for i = 1, n + 1 do
  local v = t[i]
  while rem < v and stage < i do
    score = score + rem * (i - stage)
    imos[stage + 1] = imos[stage + 1] + rem
    imos[i] = imos[i] - rem
    stage = stage + 1
    v = v - rem
    rem = imos[stage]
    imos[stage + 1] = imos[stage + 1] + imos[stage]
  end
  if i <= stage then
    score = -1
    break
  end
  score = score + v * (i - stage)
  rem = rem - v
  imos[stage + 1] = imos[stage + 1] + v
  imos[i] = imos[i] - v
end
print(score)