The Films

5000ms

524288K

Description:

In "The Man in the High Castle" world, there are $$$m$$$ different film endings.

Abendsen owns a storage and a shelf. At first, he has $$$n$$$ ordered films on the shelf. In the $$$i$$$-th month he will do:

Empty the storage.
Put $$$k_i \cdot m$$$ films into the storage, $$$k_i$$$ films for each ending.
He will think about a question: if he puts all the films from the shelf into the storage, then randomly picks $$$n$$$ films (from all the films in the storage) and rearranges them on the shelf, what is the probability that sequence of endings in $$$[l_i, r_i]$$$ on the shelf will not be changed? Notice, he just thinks about this question, so the shelf will not actually be changed.

Answer all Abendsen's questions.

Let the probability be fraction $$$P_i$$$. Let's say that the total number of ways to take $$$n$$$ films from the storage for $$$i$$$-th month is $$$A_i$$$, so $$$P_i \cdot A_i$$$ is always an integer. Print for each month $$$P_i \cdot A_i \pmod {998244353}$$$.

$$$998244353$$$ is a prime number and it is equal to $$$119 \cdot 2^{23} + 1$$$.

It is guaranteed that there will be only no more than $$$100$$$ different $$$k$$$ values.

Input:

The first line contains three integers $$$n$$$, $$$m$$$, and $$$q$$$ ($$$1 \le n, m, q \le 10^5$$$, $$$n+q\leq 10^5$$$) — the number of films on the shelf initially, the number of endings, and the number of months.

The second line contains $$$n$$$ integers $$$e_1, e_2, \ldots, e_n$$$ ($$$1\leq e_i\leq m$$$) — the ending of the $$$i$$$-th film on the shelf.

Each of the next $$$q$$$ lines contains three integers $$$l_i$$$, $$$r_i$$$, and $$$k_i$$$ ($$$1 \le l_i \le r_i \le n, 0 \le k_i \le 10^5$$$) — the $$$i$$$-th query.

It is guaranteed that there will be only no more than $$$100$$$ different $$$k$$$ values.

Output:

Print the answer for each question in a separate line.

Sample Input:

6 4 4
1 2 3 4 4 4
1 4 0
1 3 2
1 4 2
1 5 2

Sample Output:

Sample Input:

Sample Output:

494942218
13125
151632

Note:

In the first sample in the second query, after adding $$$2 \cdot m$$$ films into the storage, the storage will look like this: $$$\{1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4\}$$$.

There are $$$26730$$$ total ways of choosing the films so that $$$e_l, e_{l+1}, \ldots, e_r$$$ will not be changed, for example, $$$[1, 2, 3, 2, 2]$$$ and $$$[1, 2, 3, 4, 3]$$$ are such ways.

There are $$$2162160$$$ total ways of choosing the films, so you're asked to print $$$(\frac{26730}{2162160} \cdot 2162160) \mod 998244353 = 26730$$$.

Informação

Provedor Codeforces

Código CF1017H