CF1101G.(Zero XOR Subset)-less

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.

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 $0$ .