Rikka with Stable Marriage

People in love always feel humble. Sometimes, Rikka is worried about whether she deserves love from Yuta.

Stable marriage problem is an interesting theoretical model which has a strong connection with the real world. Given $n$ men and $n$ women, where each person has ranked all members of the opposite sex in order of preference. We use a permutation $p$ of length $n$ to represent a match that the $i$th man gets married with the $p_i$th woman. A match is stable if and only if there are no two people of opposite sexes who would both rather have each other than their current partners, i.e., $\forall i \neq j, \neg (r_i(p_j,p_i) \wedge r_{p_j}(i,j))$ where $r_a(b,c)$ represents whether person $a$ loves $b$ more than $c$.

Rikka wants to resolve the confusion in her mind by considering a special case of the stable marriage problem. Rikka uses a feature integer to represents a person's character, and for two persons with feature integers equal to $x$ and $y$, Rikka defines the suitable index of them as $x \oplus y$, where $\oplus$ represents binary exclusive-or.

Given $n$ men with feature integers $a_1, \dots, a_n$ and $n$ women with feature integers $b_1, \dots, b_n$. For the $i$th man, he loves the $j$th woman more than the $k$th woman if and only if $a_i \oplus b_j > a_i \oplus b_k$; for the $i$th woman, she loves the $j$th man more than the $k$th man if and only if $b_i \oplus a_j > b_i \oplus a_k$.

Rikka wants to calculate the best stable match for this problem, i.e., let $\mathbb P$ be the set of all stable match, she wants to calculate $\max_{p \in \mathbb P} \left(\sum_{i=1}^n \left(a_i \oplus b_{p_i} \right) \right)$. Since $n$ is quite large, this problem is too difficult for Rikka, could you please help her find the answer?

The first line of the input contains a single integer $T(1 \leq T \leq 50)$, the number of test cases.

For each test case, the fisrt line contains a sigle integer $n(1 \leq n \leq 10^5)$.

The second line contains $n$ integers $a_1, \dots, a_n(1 \leq a_i \leq 10^9)$ which represents the feature number of each man.

The third line contains $n$ integers $b_1, \dots, b_n(1 \leq b_i \leq 10^9)$ which represents the feature number of each woman.

The input guarantees that there are no more than $5$ test cases with $n > 10^4$, and for any $i,j \in [1,n], i \neq j$, $a_i \neq a_j$ and $b_i \neq b_j$.

For each test case, output a single line with a single integer, the value of the best stable match. If there is no stable match, output $-1$.

2
4
1 2 3 4
1 2 3 4
5
10 20 30 40 50
15 25 35 45 55

20
289

2019 Multi-University Training Contest 9