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 proconio::{input, marker::*};

#[allow(unused_imports)]
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};

#[allow(unused_imports)]
use std::io::Write;

#[allow(unused_macros)]
macro_rules! debug {
    ($($a:expr),*) => {
        #[cfg(debug_assertions)]
        writeln!(&mut std::io::stderr(), concat!("[DEBUG] ", $(stringify!($a), "={:?} "),*), $($a),*).unwrap();
    }
}

macro_rules ! min { ( $ x : expr ) => ( $ x ) ; ( $ x : expr , $ ( $ z : expr ) ,+ ) => { { std :: cmp :: min ( $ x , min ! ( $ ( $ z ) ,* ) ) } } }

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

    if n == 0 && a[0] != 1 {
        println!("-1");
        return;
    }

    let mut vec = vec![0; n + 1];
    vec[0] = 1;
    for i in 0..n {
        if vec[i] <= a[i] {
            println!("-1");
            return;
        }
        vec[i + 1] = (vec[i] - a[i]).saturating_mul(2);
    }
    if vec[n] < a[n] {
        println!("-1");
        return;
    }
    let mut sum = 0;
    let mut s = 0;
    for i in (0..n + 1).rev() {
        s += a[i];
        sum += min!(vec[i], s);
    }
    println!("{}", sum);
}