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)

#![allow(unused_imports)]
use itertools::Itertools;
use num::clamp;
use proconio::{fastout, input, marker::*};
use std::cmp::*;
use std::collections::*;
use superslice::*;

#[fastout()]
fn main() {
    input! {
        n: usize,
        a: [isize; n+1],
    }

    let mut cum = vec![0; n + 1];
    cum[0] = a[0];
    for i in 1..n + 1 {
        cum[i] = cum[i - 1] + a[i];
    }

    let mut ans = cum[n];
    let mut b = 0;
    for i in 0..n + 1 {
        let sum_a = cum[n] - cum[i];
        b = if i == 0 { 1 - a[0] } else  { min(2 * b - a[i], sum_a) };
        ans += b;
        if b < 0 {
            println!("-1");
            return;
        }
    }
    println!("{}", ans);
}