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)]
#![allow(clippy::many_single_char_names)]
#![allow(clippy::suspicious_arithmetic_impl)]

// using Rust(1.42.0)
// My Library Repo is at "https://github.com/Loptall/prcn_lib"
// check available crates on AtCoder at "https://atcoder.jp/contests/language-test-202001"

/*
    AtCoder Language Update 2020204 お疲れ様でした! 有難いです(p.s. 過去問対応もありがとうございます)
*/

// from std...
use std::cmp::{
    max, min,
    Ordering::{Equal, Greater, Less},
};
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};
use std::mem::swap;
use std::ops::*;
use std::usize::{MAX, MIN};
use std::{i128, i16, i32, i64, i8, isize, u128, u16, u32, u64, u8, usize};

// Dep crates are...
use itertools::*;
use itertools_num::*;
use lazy_static::lazy_static;
use maplit::{btreemap, btreeset, hashmap, hashset};
use num_bigint::{BigInt, BigUint};
use num_complex::Complex;
use num_integer::{binomial, gcd, lcm, Integer};
use num_rational::Rational;
use num_traits::{One, Zero};
use permutohedron::Heap;
use petgraph::unionfind::UnionFind;
use proconio::{fastout, input, is_stdin_empty, marker::*};
use rand::random;

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

    if n == 0 && a[0] != 1 {
        println!("-1");
        std::process::exit(0)
    }

    let s = a.iter().sum::<usize>();
    let mut node = 1usize;
    let mut leave = 0usize;
    let mut ans = 0usize;

    for (i, j) in a.windows(2).map(|x| (x[0], x[1])) {
        // dbg!(node, leave, i, j, ans, "\n");
        ans += node;
        if i > node {
            println!("-1");
            std::process::exit(0)
        }
        node -= i;
        leave += i;
        if j > node * 2 {
            println!("-1");
            std::process::exit(0)
        } else {
            node = min(s - leave, node * 2);
        }
    }
    ans += node;

    println!("{}", ans);
}