Prairie Partition

2000ms

262144K

Description:

It can be shown that any positive integer x can be uniquely represented as x = 1 + 2 + 4 + ... + 2^k - 1 + r, where k and r are integers, k ≥ 0, 0 < r ≤ 2^k. Let's call that representation prairie partition of x.

For example, the prairie partitions of 12, 17, 7 and 1 are:

12 = 1 + 2 + 4 + 5,

17 = 1 + 2 + 4 + 8 + 2,

7 = 1 + 2 + 4,

1 = 1.

Alice took a sequence of positive integers (possibly with repeating elements), replaced every element with the sequence of summands in its prairie partition, arranged the resulting numbers in non-decreasing order and gave them to Borys. Now Borys wonders how many elements Alice's original sequence could contain. Find all possible options!

Input:

The first line contains a single integer n (1 ≤ n ≤ 10⁵) — the number of numbers given from Alice to Borys.

The second line contains n integers a₁, a₂, ..., a_n (1 ≤ a_i ≤ 10¹²; a₁ ≤ a₂ ≤ ... ≤ a_n) — the numbers given from Alice to Borys.

Output:

Output, in increasing order, all possible values of m such that there exists a sequence of positive integers of length m such that if you replace every element with the summands in its prairie partition and arrange the resulting numbers in non-decreasing order, you will get the sequence given in the input.

If there are no such values of m, output a single integer -1.

Sample Input:

8
1 1 2 2 3 4 5 8

Sample Output:

Sample Input:

6
1 1 1 2 2 2

Sample Output:

2 3

Sample Input:

5
1 2 4 4 4

Sample Output:

-1

Note:

In the first example, Alice could get the input sequence from [6, 20] as the original sequence.

In the second example, Alice's original sequence could be either [4, 5] or [3, 3, 3].

Informação

Provedor Codeforces

Código CF773C