XOR Determinant

You are given two arrays $b$ and $c$ of length $n$, each containing non-negative integers. Build the $n \times n$ matrix $A$ with $A_{ij} = b_i \oplus c_j$. Find the determinant of $A$ modulo $998244353$.

Each test contains multiple test cases. The first line contains an integer $t$ ($1 \le t \le 1000$), the number of test cases. Each test case starts with a line containing $n$ ($1 \le n \le 5000$). The next line contains $b_1, b_2, \ldots, b_n$ ($0 \le b_i < 2^{60}$), and the line after that contains $c_1, c_2, \ldots, c_n$ ($0 \le c_i < 2^{60}$). The sum of $n$ over all test cases does not exceed $10000$.

For each test case, print the determinant of $A$ modulo $998244353$ on its own line.

First test case: $\begin{vmatrix} 6 & 3 \\ 1 & 4 \end{vmatrix} = 6 \cdot 4 - 1 \cdot 3 = 21$. Second test case: the matrix is $1 \times 1$, so $\begin{vmatrix} 23792195055071677 \end{vmatrix} = 23792195055071677$. Modulo $998244353$, this is $214139910$. Third test case: $\begin{vmatrix} 0 & 3 & 2 & 5 \\ 3 & 0 & 1 & 6 \\ 2 & 1 & 0 & 7 \\ 5 & 6 & 7 & 0 \end{vmatrix}$ expands to $3 \cdot 3 \cdot 7 \cdot 7 - 3 \cdot 1 \cdot 7 \cdot 5 - 3 \cdot 6 \cdot 2 \cdot 7 - 2 \cdot 3 \cdot 7 \cdot 6 + 2 \cdot 6 \cdot 2 \cdot 6 - 2 \cdot 6 \cdot 1 \cdot 5 - 5 \cdot 3 \cdot 1 \cdot 7 - 5 \cdot 1 \cdot 2 \cdot 6 + 5 \cdot 1 \cdot 1 \cdot 5 = 441 - 105 - 252 - 252 + 144 - 60 - 105 - 60 + 25 = -224$ Modulo $998244353$, $-224$ becomes $998244129$.