Refined Product Optimality
Contest: Good Bye 2024: 2025 is NEAR
Difficulty rating: 1700
Tags: binary search, data structures, greedy, math, schedules, sortings

Although Iris occasionally sets a problem where the solution is possibly wrong, she still insists on creating problems with her imagination; after all, everyone has always been on the road with their stubbornness... And like ever before, Iris has set a problem to which she gave a wrong solution, but Chris is always supposed to save it! You are going to play the role of Chris now:

- Chris is given two arrays $$$a$$$ and $$$b$$$, both consisting of $$$n$$$ integers.
- Iris is interested in the largest possible value of $$$P = \prod\limits_{i=1}^n \min(a_i, b_i)$$$ after an arbitrary rearrangement of $$$b$$$. Note that she only wants to know the maximum value of $$$P$$$, and no actual rearrangement is performed on $$$b$$$.
- There will be $$$q$$$ modifications. Each modification can be denoted by two integers $$$o$$$ and $$$x$$$ ($$$o$$$ is either $$$1$$$ or $$$2$$$, $$$1 \leq x \leq n$$$). If $$$o = 1$$$, then Iris will increase $$$a_x$$$ by $$$1$$$; otherwise, she will increase $$$b_x$$$ by $$$1$$$.
- Iris asks Chris the maximum value of $$$P$$$ for $$$q + 1$$$ times: once before any modification, then after every modification.
- Since $$$P$$$ might be huge, Chris only needs to calculate it modulo $$$998\,244\,353$$$.

Chris soon worked out this problem, but he was so tired that he fell asleep. Besides saying thanks to Chris, now it is your turn to write a program to calculate the answers for given input data.

Note: since the input and output are large, you may need to optimize them for this problem.

For example, in C++, it is enough to use the following lines at the start of the main() function:

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

The first line of each test case contains two integers $$$n$$$ and $$$q$$$ ($$$1 \leq n \leq 2\cdot 10^5$$$, $$$1 \leq q \leq 2\cdot 10^5$$$) — the length of the array and the number of operations.

The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \leq a_i \leq 5\cdot 10^8$$$) — the array $$$a$$$.

The third line of each test case contains $$$n$$$ integers $$$b_1, b_2, \ldots, b_n$$$ ($$$1 \leq b_i \leq 5\cdot 10^8$$$) — the array $$$b$$$.

Then $$$q$$$ lines follow, each line contains two integers $$$o$$$ and $$$x$$$ ($$$o \in \{1, 2\}$$$, $$$1 \leq x \leq n$$$), representing an operation.

It's guaranteed that the sum of $$$n$$$ and the sum of $$$q$$$ over all test cases does not exceed $$$4\cdot 10^5$$$, respectively.

Output format:
For each test case, output $$$q + 1$$$ integers in a line, representing the answers that Chris will calculate, modulo $$$998\,244\,353$$$.

Constraints:
Time limit: 3.0 s
Memory limit: 512.0 MB

Note:
In the first test case:

- Before the modifications, Chris can rearrange $$$b$$$ to $$$[1, 2, 3]$$$ so that $$$P = \prod\limits_{i=1}^n \min(a_i, b_i) = 1 \cdot 1 \cdot 2 = 2$$$. We can prove that this is the maximum possible value. For example, if Chris rearranges $$$b = [2, 3, 1]$$$, $$$P$$$ will be equal $$$1 \cdot 1 \cdot 1 = 1 < 2$$$, which is not optimal.
- After the first modification, Chris can rearrange $$$b$$$ to $$$[1, 2, 3]$$$ so that $$$P = 1 \cdot 1 \cdot 3 = 3$$$, which is maximized.
- After the second modification, Chris can rearrange $$$b$$$ to $$$[2, 2, 3]$$$ so that $$$P = 1 \cdot 1 \cdot 3 = 3$$$, which is maximized.
- After the third modification, Chris can rearrange $$$b$$$ to $$$[2, 2, 3]$$$ so that $$$P = 6$$$, which is maximized.
- After the fourth modification, Chris can rearrange $$$b$$$ to $$$[2, 2, 4]$$$ so that $$$P = 6$$$, which is maximized.

Example input:
4
3 4
1 1 2
3 2 1
1 3
2 3
1 1
2 1
6 8
1 4 2 7 3 5
7 6 5 6 3 3
2 5
1 6
1 5
1 5
1 5
2 3
2 3
1 6
13 8
7 7 6 6 5 5 5 2 2 3 4 5 1
1 4 1 9 6 6 9 1 5 1 3 8 4
2 2
2 11
2 4
2 4
1 7
1 1
2 12
1 5
5 3
10000000 20000000 30000000 40000000 50000000
10000000 20000000 30000000 40000000 50000000
1 1
2 2
2 1

Example output:
2 3 3 6 6
840 840 1008 1344 1680 2016 2016 2016 2352
2116800 2646000 3528000 3528000 3528000 4233600 4838400 4838400 4838400
205272023 205272023 205272023 264129429

Editorial: