(Zero XOR Subset)-less
Description
You are given an array $a_1, a_2, \dots, a_n$ of integer numbers.
Your task is to divide the array into the maximum number of segments in such a way that:
- each element is contained in exactly one segment;
- each segment contains at least one element;
- there doesn't exist a non-empty subset of segments such that bitwise XOR of the numbers from them is equal to $0$.
Print the maximum number of segments the array can be divided into. Print -1 if no suitable division exists.
The first line contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the size of the array.
The second line contains $n$ integers $a_1, a_2, \dots, a_n$ ($0 \le a_i \le 10^9$).
Print the maximum number of segments the array can be divided into while following the given constraints. Print -1 if no suitable division exists.
Input
The first line contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the size of the array.
The second line contains $n$ integers $a_1, a_2, \dots, a_n$ ($0 \le a_i \le 10^9$).
Output
Print the maximum number of segments the array can be divided into while following the given constraints. Print -1 if no suitable division exists.
Samples
4
5 5 7 2
2
3
1 2 3
-1
3
3 1 10
3
Note
In the first example $2$ is the maximum number. If you divide the array into $\{[5], [5, 7, 2]\}$, the XOR value of the subset of only the second segment is $5 \oplus 7 \oplus 2 = 0$. $\{[5, 5], [7, 2]\}$ has the value of the subset of only the first segment being $5 \oplus 5 = 0$. However, $\{[5, 5, 7], [2]\}$ will lead to subsets $\{[5, 5, 7]\}$ of XOR $7$, $\{[2]\}$ of XOR $2$ and $\{[5, 5, 7], [2]\}$ of XOR $5 \oplus 5 \oplus 7 \oplus 2 = 5$.
Let's take a look at some division on $3$ segments — $\{[5], [5, 7], [2]\}$. It will produce subsets:
- $\{[5]\}$, XOR $5$;
- $\{[5, 7]\}$, XOR $2$;
- $\{[5], [5, 7]\}$, XOR $7$;
- $\{[2]\}$, XOR $2$;
- $\{[5], [2]\}$, XOR $7$;
- $\{[5, 7], [2]\}$, XOR $0$;
- $\{[5], [5, 7], [2]\}$, XOR $5$;
As you can see, subset $\{[5, 7], [2]\}$ has its XOR equal to $0$, which is unacceptable. You can check that for other divisions of size $3$ or $4$, non-empty subset with $0$ XOR always exists.
The second example has no suitable divisions.
The third example array can be divided into $\{[3], [1], [10]\}$. No subset of these segments has its XOR equal to $0$.