NOI '19 P5 - Landlords

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Points: 25 (partial)
Time limit: 1.0s
Memory limit: 512M

Problem type

Kevin is playing cards and now he needs to shuffle cards m times. Now let's describe the rule of the i^{th} shuffle.

For the i^{th} shuffle:

  1. Kevin will take out A_i cards from the top and make it a new pile. Now there are two piles of cards. One is the original top A_i cards and the other is the remaining n-A_i cards. The relative order in these two piles remains unchanged. Note that when A_i is n or 0, there is one pile which has no card at all.

  2. Now let's merge those two piles of cards into a new pile. Suppose the first pile has X cards and the second pile has Y cards. With probability \frac{X}{X+Y}, we select the bottom card of the first pile and put the selected card to the top of the new pile. Then, with probability \frac{Y}{X+Y}, we select the bottom card of the second pile and then put the selected card to the top of the new pile.

  3. Repeat 2 until both piles are empty.

Now we have Q questions for you. You have to tell us after m times of shuffles, what's the expected score of some specific positions' card. Note that for card i, let's denote its score as f(i). In this problem, f(i) equals either i or i^2.

Input Specification

The first line contains three integers n,m,type. When type = 1, f(i) = i. When type = 2, f(i) = i^2.

The following line contains m integers A_1, \dots, A_m.

The following line contains an integer Q.

In the following Q lines, each line contains an integer c_i (1 \le c_i \le n), indicating that Kevin wants to know the expected score of the c_i position from the top.

Constraints

For all test cases, 3 \le n \le 10^7, 1 \le m,Q \le 5 \times 10^5, 0 \le A_i \le n, type \in \{1,2\}.

Test CasenmTypeAdditional Constraints
1\le 10\le 11None
2\le 80\le 80
32
4\le 100\le 5 \times 10^5All A_i are equal
5\le 10^71None
6
7
82
9
10

Output Specification

For each query, output a single integer on a single line as the answer. If the answer is \frac{A}{B}, please print C (0 \le C < 998\,244\,353) where A \equiv C \times B \pmod{998\,244\,353}.

Sample Input 1

4 1 1
3
1
1

Sample Output 1

249561090

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