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.


-----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.


-----Examples-----
Input
4
5 5 7 2

Output
2

Input
3
1 2 3

Output
-1

Input
3
3 1 10

Output
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 $0$.

Source: codeforces | Difficulty: HARD
Tags: Matrices

Mathematics


Solutions:
import sys
from math import *
from fractions import gcd
readints = lambda : map(int, input().strip('\n').split())
n = int(input())
arr = list(readints())
pref = 0
basis = []
for v in arr:
	pref = pref ^ v
	for b in basis:
		v = min(v, v ^ b)
	if v > 0:
		basis.append(v)
if pref == 0:
	print(-1)
else:
	print(len(basis))