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 C++ (GCC 9.2.1) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>
using namespace std;

long long a[1 << 20], R[1 << 20];

int main() {
	ios::sync_with_stdio(0);
	cin.tie(0); cout.tie(0);
	int n; cin >> n;
	for (int i = 0; i <= n; i++) cin >> a[i];
	if (n == 0) {
		if (a[0] == 1) cout << 1 << endl;
		else cout << 0 << endl;
		return 0;
	}
	R[0] = 1;
	for (int i = 1; i <= n; i++) {
		R[i] = (R[i - 1] - a[i - 1]) * 2;
		if (R[i] > 1LL << 60) {
			for (int j = i + 1; j <= n; j++) R[j] = R[i];
			break;
		}
	}
	long long ans = 0;
	for (int i = n; i >= 0; i--) {
		if (a[i] > R[i]) {
			cout << -1 << endl;
			return 0; 
		}
		ans += a[i];
		a[i - 1] = min(a[i - 1] + a[i], R[i - 1]);
	}
	cout << ans << endl;
	return 0;
}