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.

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