March is visiting the beautiful city of Belobog!

The city of Belobog can be modeled through a coordinate system. March is currently at the point $$$(x_1,y_1)$$$ and she would like to get to the point $$$(x_2,y_2)$$$.

Due to technological advancements, most people prefer to teleport instead of walk. March can perform the following operation any number of times:

• Choose two integers $$$x$$$ and $$$y$$$. Then, she will teleport to the location that is the reflection of her current location across the point $$$(x,y)$$$.

Is it possible to reach $$$(x_2,y_2)$$$ through any number of operations?

Note: You may want to use $$$64$$$-bit integers instead of $$$32$$$-bit integers in this task. For example, in Java, you may want to use long instead of int. In C++, you may want to use long long.

Yunli is writing numbers on the blackboard!

She first writes the number $$$1$$$ one time, then the number $$$2$$$ two times, then the number $$$3$$$ three times, and so on.

Then, Yunli gives you an integer $$$x$$$, and she wants you to tell her how many numbers she must write in this fashion until the sum of all numbers is at least $$$x$$$.

Sampo is hiding inside a tunnel system. The tunnel can be modeled as a tree$$$^{\text{∗}}$$$ with $$$n$$$ nodes and $$$n-1$$$ edges. There is an exit situated at every node with exactly one neighbor.

Sampo wants to decide in which node to hide in. Obviously, he should hide in a node that is the furthest distance away from any exit. Please help Sampo determine which node is the best one to hide in. If there are multiple nodes that are equally as far from an exit, output the node with the least index.

Note: You may want to use $$$64$$$-bit integers instead of $$$32$$$-bit integers in this task. For example, in Java, you may want to use long instead of int. In C++, you may want to use long long.

Serval has an undirected unweighted connected graph of $$$n$$$ nodes and $$$m$$$ edges.

To help you guess the structure of the graph, Serval tells you the distance from node $$$1$$$ to node $$$i$$$ for each $$$2 \leq i \leq n$$$ (distance from node $$$1$$$ to node $$$i$$$ is the minimum number of edges in any path from $$$1$$$ to $$$i$$$). Additionally, Serval tells you that the number of edges in the graph is minimized. That is, there does not exist a graph that also satisfies the distance requirements with less edges.

Despite this, there can still be a lot of possible graphs! Please count the number of graphs with minimum number of edges that satisfies the distance requirements. Since the answer can be massive, output it modulo $$$998\,244\,353$$$.

Note: You may want to use $$$64$$$-bit integers instead of $$$32$$$-bit integers in this task. For example, in Java, you may want to use long instead of int. In C++, you may want to use long long.

Sushang has a multiset $$$S$$$ containing $$$n$$$ elements and $$$k$$$ empty arrays $$$a_1,a_2,\ldots,a_k$$$. She will repeat the following operation until $$$S$$$ is empty:

• Choose an element $$$x$$$ in $$$S$$$ and an integer $$$1\leq i\leq k$$$
• Remove $$$x$$$ from $$$S$$$ and insert $$$x$$$ at the back of $$$a_i$$$

After that, Sushang calculates the score of each array as follows: if $$$a_i=[x_1,x_2,\ldots,x_m]$$$, then score$$$(a_i)=\sum_{i=1}^mx_i\cdot i=x_1\cdot1+x_2\cdot2+x_3\cdot3+\ldots+x_m\cdot m$$$. The score of an empty array is $$$0$$$.

Sushang wants to minimize the sum of scores over all arrays.

The value of $$$k$$$ is not yet known, so please find the minimum sum of scores possible for $$$k=1,2,\ldots,n$$$ independently.

Tingyun gives you an array $$$a$$$ of size $$$n$$$.

Given an integer $$$x$$$, define functions $$$f_0(x),f_1(x),\ldots,f_n(x)$$$ as follows:

1. $$$f_0(x)=x$$$
2. $$$f_k(x)=f_{k-1}(x) \text{ mod } a_k$$$ for any integer $$$k \gt 0$$$.

In other words, $$$f_n(x) = (\ldots((x \text{ mod } a_1)\text{ mod } a_2)\ldots \text{ mod } a_n)$$$.

Note for positive integers $$$a$$$ and $$$b$$$, $$$a \text{ mod } b$$$ is the unique integer $$$0\leq c \lt b$$$ such that $$$a-c$$$ is divisible by $$$b$$$. For example, $$$5 \text{ mod } 3 = 2$$$.

Tingyun then gives you $$$q$$$ values of $$$x$$$. For each $$$x$$$, please output the value of $$$f_n(x)$$$.

Note: You may want to use $$$64$$$-bit integers instead of $$$32$$$-bit integers in this task. For example, in Java, you may want to use long instead of int. In C++, you may want to use long long.

Given an array $$$a$$$ of size $$$m$$$, let $$$f(a)$$$ be the number of numbers $$$1\leq i\leq m$$$ where $$$\min(a_1,a_2,\ldots,a_i)=a_i$$$.

For example, if $$$a=[2,4,3,2,1]$$$, $$$f(a)=3$$$ because $$$a_1=\min(a_1)$$$, $$$a_4=\min(a_1,a_2,a_3,a_4)$$$, and $$$a_5=\min(a_1,a_2,a_3,a_4,a_5)$$$.

Seele gives you two integers $$$n$$$ and $$$k$$$.

There are $$$k^n$$$ length-$$$n$$$ arrays such that each element is a positive integer from $$$1$$$ to $$$k$$$, inclusive. Over all such arrays, Seele asks you to output the sum of $$$f(a)$$$ over all such arrays $$$a$$$.

Since the answer may be huge, output the answer modulo $$$998\,244\,353$$$.

You may want to use C++ for this problem if you want full credit.

Bronya has $$$n$$$ cards ($$$n$$$ is even), with the $$$i$$$th card having the number $$$a_i$$$ written on it. Given an integer $$$k$$$, she plays the following game:

1. Initially, Bronya's score is $$$0$$$.
2. While Bronya has at least two cards in her hand, she picks two cards
3. She discards the two cards and adds $$$1$$$ to her score if the sum of the numbers on the card is exactly $$$k$$$.

Naturally, Bronya wants to maximize her score for any given $$$k$$$.

Bronya has not yet determined what value of $$$k$$$ she will use. Therefore, your task is to find out what her score will be for each value of $$$k$$$ from $$$1$$$ to $$$2\cdot n$$$.