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 OCaml (4.10.0) (judge verdict: Accepted)

let split_string ?(pattern="") = Str.split @@ Str.regexp pattern
let n = Scanf.sscanf (read_line ()) "%d" @@ fun n -> n
let a = read_line () |> split_string ~pattern:" " |> List.map int_of_string

let rec range res acc = match res, acc with
  | [], _ -> failwith "program error"
  | res, [] -> res
  | (l, r) :: _, x :: xs -> range (((l + 1) / 2 + x, r + x) :: res) xs
let range = function
  | [] -> failwith "program error"
  | x :: xs -> range [(x, x)] xs

let rec build sum prev rs ns = match rs, ns with
  | _, [] -> sum
  | (l, r) :: rs, n :: ns -> let t = min (prev * 2) r in build (sum + t) (t - n) rs ns
  | _ -> failwith "program error"

let () =
  let acc = range (List.rev a) in
  match acc, a with
    | x :: xs, n :: ns -> Printf.printf "%d\n" @@ if fst x = 1 then build 1 1 xs ns else -1
    | _ -> failwith "program error"