Vika and Wiki
Description
Recently, Vika was studying her favorite internet resource - Wikipedia.
On the expanses of Wikipedia, she read about an interesting mathematical operation bitwise XOR, denoted by $\oplus$.
Vika began to study the properties of this mysterious operation. To do this, she took an array $a$ consisting of $n$ non-negative integers and applied the following operation to all its elements at the same time: $a_i = a_i \oplus a_{(i+1) \bmod n}$. Here $x \bmod y$ denotes the remainder of dividing $x$ by $y$. The elements of the array are numbered starting from $0$.
Since it is not enough to perform the above actions once for a complete study, Vika repeats them until the array $a$ becomes all zeros.
Determine how many of the above actions it will take to make all elements of the array $a$ zero. If this moment never comes, output $-1$.
The first line contains a single integer $n$ ($1 \le n \le 2^{20}$) - the length of the array $a$.
It is guaranteed that $n$ can be represented as $2^k$ for some integer $k$ ($0 \le k \le 20$).
The second line contains $n$ integers $a_0, a_1, a_2, \dots, a_{n-1}$ ($0 \le a_i \le 10^9$) - the elements of the array $a$.
Output a single number - the minimum number of actions required to make all elements of the array $a$ zero, or $-1$ if the array $a$ will never become zero.
Input
The first line contains a single integer $n$ ($1 \le n \le 2^{20}$) - the length of the array $a$.
It is guaranteed that $n$ can be represented as $2^k$ for some integer $k$ ($0 \le k \le 20$).
The second line contains $n$ integers $a_0, a_1, a_2, \dots, a_{n-1}$ ($0 \le a_i \le 10^9$) - the elements of the array $a$.
Output
Output a single number - the minimum number of actions required to make all elements of the array $a$ zero, or $-1$ if the array $a$ will never become zero.
4
1 2 1 2
2
0 0
1
14
8
0 1 2 3 4 5 6 7
2
0
1
5
Note
In the first example, after one operation, the array $a$ will become equal to $[3, 3, 3, 3]$. After one more operation, it will become equal to $[0, 0, 0, 0]$.
In the second example, the array $a$ initially consists only of zeros.
In the third example, after one operation, the array $a$ will become equal to $[0]$.