You are given integers $$$N,M,K,R,C,$$$ and $$$V$$$. Find the number of $$$N$$$ by $$$M$$$ integer matrices $$$a = (a_{i,j})$$$ that satisfy all of the following conditions, modulo $$$998244353$$$.

• $$$1 \leq a_{i,j} \leq K$$$ for all $$$1 \leq i \leq N,\, 1 \leq j \leq M$$$.
• $$$a_{i,j} \leq a_{i,j+1}$$$ for all $$$1 \leq i \leq N,\, 1 \leq j \leq M-1$$$.
• $$$a_{i,j} \leq a_{i+1,j}$$$ for all $$$1 \leq i \leq N-1,\, 1 \leq j \leq M$$$.
• $$$a_{R,C}=V$$$.

You are given an integer array $$$A$$$ of length $$$N$$$, consisting of $$$0$$$'s and $$$1$$$'s. Let $$$a$$$ be initially the array $$$A$$$. You are going to perform the following operation $$$N-1$$$ times.

• Let $$$n$$$ be the current length of $$$a$$$. Choose an integer $$$i$$$ ($$$1 \leq i \leq n-1$$$) and delete the $$$i$$$-th and the $$$(i+1)$$$-th elements of $$$a$$$. Then, by letting $$$x$$$ and $$$y$$$ be the deleted elements, insert either $$$x\mathbin{\&}y$$$ or $$$x\mathbin{|}y$$$ to the position of the deleted elements. Here $$$x\mathbin{\&}y$$$ and $$$x\mathbin{|}y$$$ denote the bit-AND and bit-OR operations, respectively.

There are $$$2^{N-1} \times (N-1)!$$$ ways to perform the operations. Count the number of ways that result in a single value of $$$1$$$, modulo $$$998244353$$$.

You have $$$R$$$ red balls, $$$B$$$ blue balls, and one green ball. You are going to arrange the balls in a row. The score of an arrangement is defined as follows:

• Let $$$l_{\mathrm{R}},l_{\mathrm{B}},r_{\mathrm{R}},r_{\mathrm{B}}$$$ be the number of red/blue balls to the left/right of the green ball, respectively. Then, the score is the maximum integer $$$x$$$ such that $$$l_{\mathrm{B}} \times x \leq l_{\mathrm{R}}$$$ and $$$r_{\mathrm{B}} \times x \leq r_{\mathrm{R}}$$$.

Find the sum of scores of all possible arrangements, modulo $$$998244353$$$. Note that balls of the same color cannot be distinguished, thus two arrangements are considered different if and only if there is such an $$$i$$$ that the color of the $$$i$$$-th ball in the first arrangement differs from that of the second.

You are given integer sequences $$$A$$$, $$$B$$$, and $$$C$$$, each of length $$$N$$$. For each $$$k=1,2,\ldots,N$$$, find the following value modulo $$$998244353$$$.

$$$$$$ \sum_{1 \leq i \leq N} \left( C_i \times \prod_{1 \leq j \leq k} (A_i+B_j) \right) $$$$$$

You are given integers $$$A,B$$$, and a simple undirected graph of $$$N$$$ vertices and $$$M$$$ edges. The vertices are numbered from $$$1$$$ through $$$N$$$, and the edges from $$$1$$$ through $$$M$$$. The edge $$$i$$$ connects the vertices $$$U_i$$$ and $$$V_i$$$. Here, it is guaranteed that $$$V_i-U_i=A$$$ or $$$V_i-U_i=B$$$.

Find the number of matchings of the graph, modulo $$$998244353$$$. Note that a matching of the graph is a subset of edges whose end-points are all distinct.

You are given an integer sequence $$$A_1,A_2,\ldots,A_N$$$. You'll make a rooted tree with $$$N$$$ vertices numbered from $$$1$$$ through $$$N$$$. The vertex $$$1$$$ is the root, and for each vertex $$$i$$$ ($$$2 \leq i \leq N$$$), its parent $$$p_i$$$ must satisfy $$$p_i \lt i$$$.

You define the score of a rooted tree as follows:

• Let $$$x$$$ be the lowest common ancestor of the vertex $$$N-1$$$ and the vertex $$$N$$$. Then, the score is

  $$$$$$ \prod_{v \in (\text{subtree rooted at $x$})} A_v $$$$$$

  Note that we consider $$$x$$$ itself is in the subtree rooted at $$$x$$$.

There are $$$(N-1)!$$$ ways to make a tree. Find the sum of scores of all possible trees, modulo $$$998244353$$$.

You are given an integer sequence $$$A_1,A_2,\ldots,A_N$$$ and an integer $$$K$$$.

You'll prepare $$$K$$$ piles of stones. Each pile should contain exactly $$$A_i$$$ piles for some $$$i$$$. All piles are distinguishable; there are $$$N^K$$$ different configurations.

You and Mike will play a game with the piles. You and Mike alternately do the following operation, with you going first.

• Choose at most $$$6$$$ piles (choosing $$$0$$$ piles is not allowed) and remove an arbitrary positive number of stones from each of the chosen piles. Note that the player can remove different numbers of stones from different piles.

The player who cannot make a valid move loses. Assuming both players play optimally, count the number of initial configurations that result in your loss, modulo $$$998244353$$$.

Posted on a blog are $$$N+M$$$ harsh comments. You made $$$N$$$ comments, and the $$$i$$$-th of them has $$$A_i$$$ downvotes. The $$$i$$$-th of the other $$$M$$$ comments has $$$B_i$$$ downvotes.

Mike is going to delete the comments, one by one, by repeating the following operation:

• Choose a comment randomly and delete it. More precisely, let $$$x_1,x_2,\ldots,x_k$$$ be numbers of downvotes the remaining comments have. Then, he will choose the $$$i$$$-th of them with the probability $$$x_i/\left(\sum_{1\leq j \leq k}x_j \right)$$$ and detele it.

Note that the choices in the operations are independent.

Find the expected number of operations Mike will do until he deletes all of your comments. The answer is a rational number, so print it modulo $$$998244353$$$ as usual. We can prove that such representation is always possible under the constraints of this problem.

You are given an integer $$$N$$$ and an integer sequence $$$X$$$ of length $$$M$$$. Count, modulo $$$998244353$$$, the number of permutations $$$P = (P_1,P_2,\ldots,P_N)$$$ of $$$(1,2,\ldots,N)$$$ that satisfy the following condition:

• The lexicographically smallest subsequence of $$$P$$$ of length $$$M$$$ coincides with $$$X$$$.

This problem might be well-known in some countries, but how do other countries learn about such problems if nobody poses them?

You are given a non-decreasing positive integer sequence $$$A = (A_1, A_2, \ldots, A_N)$$$ of length $$$N$$$. For each $$$k=0,1,2,\ldots,N$$$, count the number of non-decreasing non-negative integer sequences $$$x = (x_1, x_2, \ldots, x_N)$$$ of length $$$N$$$ that satisfy following conditions, modulo $$$998244353$$$:

• $$$x_i \leq A_i$$$ for all $$$1 \leq i \leq N$$$.
• The number of indices $$$i$$$ with $$$x_i=A_i$$$ is exactly $$$k$$$.