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(dead_code)]
fn main() {
    let n = i::u();
    let a = i::uv(n + 1);
    let sum = a.iter().sum();

    //なるべく早くsumまで枝を広げる
    let mut current_branch = 1;
    let mut leaves = 0;
    let mut ans = 0;
    for k in a {
        if current_branch - leaves < k {
            println!("{}", -1);
            return;
        }
        ans += current_branch - leaves;
        leaves += k;
        current_branch = min(sum, current_branch * 2 - leaves);
       // dbg!(&ans, &leaves, &current_branch);
    }
    if sum > leaves {
        println!("{}", -1);
        return;
    }
    println!("{}", ans);
}

#[allow(unused_imports)]
use std::cmp::*;
#[allow(unused_imports)]
use std::collections::{HashMap, HashSet, VecDeque};
#[allow(unused_imports)]
use std::io::*;
#[allow(unused_imports)]
use std::num::*;
#[allow(unused_imports)]
use std::str::*;

#[allow(dead_code)]
mod i {
    use super::*;

    pub fn read<T: FromStr>() -> T {
        stdin()
            .bytes()
            .map(|c| c.unwrap() as char)
            .skip_while(|c| c.is_whitespace())
            .take_while(|c| !c.is_whitespace())
            .collect::<String>()
            .parse::<T>()
            .ok()
            .unwrap()
    }
    pub fn str() -> String {
        read()
    }
    pub fn s() -> Vec<char> {
        str().chars().collect()
    }
    pub fn i() -> i64 {
        read()
    }
    pub fn u() -> usize {
        read()
    }
    pub fn u2() -> (usize, usize) {
        (read(), read())
    }
    pub fn u3() -> (usize, usize, usize) {
        (read(), read(), read())
    }
    pub fn u4() -> (usize, usize, usize, usize) {
        (read(), read(), read(), read())
    }
    pub fn u5() -> (usize, usize, usize, usize, usize) {
        (read(), read(), read(), read(), read())
    }
    pub fn u6() -> (usize, usize, usize, usize, usize, usize) {
        (read(), read(), read(), read(), read(), read())
    }
    pub fn f() -> f64 {
        read()
    }
    pub fn c() -> char {
        read::<String>().pop().unwrap()
    }
    pub fn iv(n: usize) -> Vec<i64> {
        (0..n).map(|_| i()).collect()
    }
    pub fn iv2(n: usize) -> Vec<(i64, i64)> {
        (0..n).map(|_| iv(2)).map(|a| (a[0], a[1])).collect()
    }
    pub fn iv3(n: usize) -> Vec<(i64, i64, i64)> {
        (0..n).map(|_| iv(3)).map(|a| (a[0], a[1], a[2])).collect()
    }
    pub fn uv(n: usize) -> Vec<usize> {
        (0..n).map(|_| u()).collect()
    }
    pub fn uv2(n: usize) -> Vec<(usize, usize)> {
        (0..n).map(|_| uv(2)).map(|a| (a[0], a[1])).collect()
    }
    pub fn uv3(n: usize) -> Vec<(usize, usize, usize)> {
        (0..n).map(|_| uv(3)).map(|a| (a[0], a[1], a[2])).collect()
    }
    pub fn uv4(n: usize) -> Vec<(usize, usize, usize, usize)> {
        (0..n)
            .map(|_| uv(4))
            .map(|a| (a[0], a[1], a[2], a[3]))
            .collect()
    }
    pub fn fv(n: usize) -> Vec<f64> {
        (0..n).map(|_| f()).collect()
    }
    pub fn cmap(h: usize) -> Vec<Vec<char>> {
        (0..h).map(|_| s()).collect()
    }
}