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 Rust (1.42.0) (judge verdict: Accepted)

use std::io::*;

fn main() {
    let mut s: String = String::new();
    std::io::stdin().read_to_string(&mut s).ok();
    let mut itr = s.trim().split_whitespace();
    let n: usize = itr.next().unwrap().parse().unwrap();
    let a: Vec<usize> = (0..n + 1)
        .map(|_| itr.next().unwrap().parse().unwrap())
        .collect();

    // 1. 下から必要な分裂数を決定(ある深さより下にある葉の数)
    // 2. 上から条件を満たしながら分裂させる
    let mut s = vec![0; n + 2];
    for i in (0..n + 1).rev() {
        s[i] = s[i + 1] + a[i];
    }

    let mut ans = 0;
    let mut now = 1;
    for i in 0..n + 1 {
        if a[i] > now {
            println!("-1");
            return;
        }
        ans += now;
        // 生やせる最大の枝
        let eda_max = (now - a[i]) << 1;
        // これ以上生やすと下がおかしくなる
        let eda_min = s[i] - a[i];
        now = std::cmp::min(eda_max, eda_min);
    }
    println!("{}", ans);
}