It can be shown that any positive integer x can be uniquely represented as x=1+2+4+...+2k-1+r, where k and r are integers, k \ge 0, 0 \lt r \le 2k. 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!

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

The second line contains n integers a1,a2,...,an (1 \le ai \le 1012; a1 \le a2 \le ... \le an)- the numbers given from Alice to Borys.

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.

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].