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: Runtime Error)

use proconio::input;

fn main() {
    input! {
        n: usize,
        a: [u64; n+1],
    }
    let mut nodes = vec![0; n + 1];
    for (i, &ai) in a.iter().rev().enumerate() {
        if n - i < 30 && nodes[n - i] + ai > 2u64.pow((n - i) as u32) {
            nodes[n - i] = 2u64.pow((n - i) as u32) - ai;
            if nodes[n - i + 1] + a[n - i + 1] > nodes[n - 1] * 2 {
                nodes[n - i + 1] = nodes[n - 1] * 2 - a[n - i + 1];
            }
        }
        if i < n {
            nodes[n - i - 1] = nodes[n - i] + ai;
        }
    }
    if nodes[0] == 1 {
        println!(
            "{}",
            nodes
                .iter()
                .zip(a.iter())
                .fold(0, |sum, (n, a)| sum + n + a)
        );
    } else {
        println!("{}", -1);
    }
}