A12161. Lost Array

Bajtek, known for his unusual gifts, recently got an integer array $x_0, x_1, \ldots, x_{k-1}$ .

Unfortunately, after a huge array-party with his extraordinary friends, he realized that he'd lost it. After hours spent on searching for a new toy, Bajtek found on the arrays producer's website another array $a$ of length $n + 1$ . As a formal description of $a$ says, $a_0 = 0$ and for all other $i$ ( $1 \le i \le n$ ) $a_i = x_{(i-1)\bmod k} + a_{i-1}$ , where $p \bmod q$ denotes the remainder of division $p$ by $q$ .

For example, if the $x = [1, 2, 3]$ and $n = 5$ , then:

- $a_0 = 0$ ,
- $a_1 = x_{0\bmod 3}+a_0=x_0+0=1$ ,
- $a_2 = x_{1\bmod 3}+a_1=x_1+1=3$ ,
- $a_3 = x_{2\bmod 3}+a_2=x_2+3=6$ ,
- $a_4 = x_{3\bmod 3}+a_3=x_0+6=7$ ,
- $a_5 = x_{4\bmod 3}+a_4=x_1+7=9$ .

So, if the $x = [1, 2, 3]$ and $n = 5$ , then $a = [0, 1, 3, 6, 7, 9]$ .

Now the boy hopes that he will be able to restore $x$ from $a$ ! Knowing that $1 \le k \le n$ , help him and find all possible values of $k$ — possible lengths of the lost array.

The first line contains exactly one integer $n$ ( $1 \le n \le 1000$ ) — the length of the array $a$ , excluding the element $a_0$ .

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $1 \le a_i \le 10^6$ ).

Note that $a_0$ is always $0$ and is not given in the input.

The first line of the output should contain one integer $l$ denoting the number of correct lengths of the lost array.

The second line of the output should contain $l$ integers — possible lengths of the lost array in increasing order.

In the first example, any $k$ is suitable, since $a$ is an arithmetic progression.

Possible arrays $x$ :

- $[1]$
- $[1, 1]$
- $[1, 1, 1]$
- $[1, 1, 1, 1]$
- $[1, 1, 1, 1, 1]$

In the second example, Bajtek's array can have three or five elements.

Possible arrays $x$ :

- $[1, 2, 2]$
- $[1, 2, 2, 1, 2]$

For example, $k = 4$ is bad, since it leads to $6 + x_0 = 8$ and $0 + x_0 = 1$ , which is an obvious contradiction.

In the third example, only $k = n$ is good.

Array $[1, 4, -2]$ satisfies the requirements.

Note that $x_i$ may be negative.