D. Koxia and Game
time limit per test
2 seconds
memory limit per test
256 megabytes
standard input
standard output

Koxia and Mahiru are playing a game with three arrays $$$a$$$, $$$b$$$, and $$$c$$$ of length $$$n$$$. Each element of $$$a$$$, $$$b$$$ and $$$c$$$ is an integer between $$$1$$$ and $$$n$$$ inclusive.

The game consists of $$$n$$$ rounds. In the $$$i$$$-th round, they perform the following moves:

  • Let $$$S$$$ be the multiset $$$\{a_i, b_i, c_i\}$$$.
  • Koxia removes one element from the multiset $$$S$$$ by her choice.
  • Mahiru chooses one integer from the two remaining in the multiset $$$S$$$.

Let $$$d_i$$$ be the integer Mahiru chose in the $$$i$$$-th round. If $$$d$$$ is a permutation$$$^\dagger$$$, Koxia wins. Otherwise, Mahiru wins.

Currently, only the arrays $$$a$$$ and $$$b$$$ have been chosen. As an avid supporter of Koxia, you want to choose an array $$$c$$$ such that Koxia will win. Count the number of such $$$c$$$, modulo $$$998\,244\,353$$$.

Note that Koxia and Mahiru both play optimally.

$$$^\dagger$$$ A permutation of length $$$n$$$ is an array consisting of $$$n$$$ distinct integers from $$$1$$$ to $$$n$$$ in arbitrary order. For example, $$$[2,3,1,5,4]$$$ is a permutation, but $$$[1,2,2]$$$ is not a permutation ($$$2$$$ appears twice in the array), and $$$[1,3,4]$$$ is also not a permutation ($$$n=3$$$ but there is $$$4$$$ in the array).


Each test consists of multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 2 \cdot 10^4$$$) — the number of test cases. The description of test cases follows.

The first line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq {10}^5$$$) — the size of the arrays.

The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \leq a_i \leq n$$$).

The third line of each test case contains $$$n$$$ integers $$$b_1, b_2, \dots, b_n$$$ ($$$1 \leq b_i \leq n$$$).

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$${10}^5$$$.


Output a single integer — the number of $$$c$$$ makes Koxia win, modulo $$$998\,244\,353$$$.

1 2 2
1 3 3
3 3 1 3 4
4 5 2 5 5

In the first test case, there are $$$6$$$ possible arrays $$$c$$$ that make Koxia win — $$$[1, 2, 3]$$$, $$$[1, 3, 2]$$$, $$$[2, 2, 3]$$$, $$$[2, 3, 2]$$$, $$$[3, 2, 3]$$$, $$$[3, 3, 2]$$$.

In the second test case, it can be proved that no array $$$c$$$ makes Koxia win.