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 $*n*$ ($1 ≤ *n* ≤ 10^{5}$) — the number of numbers given from Alice to Borys.

The second line contains $*n*$ integers $*a*_{1}, *a*_{2}, ..., *a*_{*n*}$ ($1 ≤ *a*_{*i*} ≤ 10^{12}$; $*a*_{1} ≤ *a*_{2} ≤ ... ≤ *a*_{*n*}$) — 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.

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 $[3, 3, 3]$.