You are given a positive integer $$$n$$$. Your task is to construct an array $$$a$$$ of size $$$|a|$$$ such that $$$|a| \le n$$$.

Let $$$b$$$ be an empty array. For every pair of indices $$$i$$$ and $$$j$$$ such that $$$1 \le i \lt j \le |a|$$$, we add the greatest common divisor (GCD) of $$$a_i$$$ and $$$a_j$$$ to array $$$b$$$.

Your constructed array $$$a$$$ must satisfy the condition that the MEX of the array $$$b$$$ (containing all pairwise GCDs $$$gcd(a_i, a_j)$$$ for $$$i \neq j$$$) is at least $$$n$$$.

The MEX (Minimum Excluded value) of a set of positive integers is the smallest positive integer that is not present in the set. The MEX of an empty set of positive integers is $$$1$$$.

We have an array $$$a$$$ of $$$n$$$ integers, and an initially empty sheet of paper.

We will do the following operations in-order $$$m$$$ times:

• We shuffle the array $$$a$$$.
• We choose a random positive integer $$${k \le n}$$$.
• We choose a random index $$${1 \le i \le k}$$$.
• We write $$$a_i$$$ at the end of the paper.

You are asked to calculate the probability that the integers written on the sheet are non-decreasing. Output the calculated value modulo $$$998244353$$$.

There is a Pizza Man, who sells pizza, called Hosen.Hosen can store no more than $$$m$$$ pieces of pizza and initially he has $$$m$$$ pieces of pizza stored in his store.

There are $$$n$$$ customers; the $$$i_{th}$$$ of them will order $$$a_i$$$ pieces of pizza.

And Hosen has a positive integer $$$d$$$, that is when the number of pieces in the store is less than $$$d$$$, he refills it to $$$m$$$.

Here is a more detailed explanation.

When a customer orders $$$x$$$ pieces of pizza, the following happens:

• If Hosen has at least $$$x$$$ pieces of pizza stored in the store, he sells $$$x$$$ pieces to the customer and now the pieces of pizza Hosen has will decrease by $$$x$$$. In this case, the customer will be happy.
• If Hosen has less than $$$x$$$ pieces of pizza stored in the store, he sells all the remaining pieces of pizza to the customer and now he has zero pieces of pizza. In this case, the customer will be sad.
• After serving each customer, if Hosen has less than $$$d$$$ pieces in the store, he will refill the store to have $$$m$$$ pieces of pizza.

You are given $$$m$$$,$$$n$$$ and the array of customers $$$a$$$; find the minimum value of $$$d$$$ for which all the customers will be happy.

You are the master of an arena with $$$n$$$ fighters under your control, numbered from $$$1$$$ to $$$n$$$. Fighter $$$1$$$ is your favourite—you want them to emerge as the sole champion.

You are given an $$$n\times n$$$ matrix $$$M$$$. For $$$i\neq j$$$:

• $$$M_{i,j}=\texttt{1}$$$ means fighter $$$i$$$ definitely beats fighter $$$j$$$.
• $$$M_{i,j}=\texttt{0}$$$ means fighter $$$i$$$ definitely loses to fighter $$$j$$$.
• $$$M_{i,j}=\texttt{?}$$$ means the outcome is undecided and you may choose it.

It is guaranteed that $$$M_{i,i} = \texttt{0}$$$ for all $$$i$$$, and for all $$$i \ne j$$$, the pair $$$(M_{i,j}, M_{j,i})$$$ is one of: $$$(\texttt{1}, \texttt{0})$$$, $$$(\texttt{0}, \texttt{1})$$$, or $$$(\texttt{?}, \texttt{?})$$$.

You must schedule exactly $$$n-1$$$ one-on-one matches so that in the end exactly one fighter remains undefeated, and that fighter must be number $$$1$$$. A fighter may appear in multiple matches (until eliminated).

Decide any choices for the '?' entries and output a sequence of $$$n-1$$$ matches $$$(a,b)$$$ meaning "$$$a$$$ defeats $$$b$$$". If it's impossible to make fighter 1 the unique champion, print $$$\textbf{No}$$$.

Given a tree of $$$n$$$ vertices, each node is initially either black or white.

A path from node $$$u$$$ to node $$$v$$$ is:

• white if both nodes $$$u$$$ and $$$v$$$ are white.
• black if both nodes $$$u$$$ and $$$v$$$ are black.
• otherwise it is a normal path neither white nor black.

A white path is clean if and only if there is no black path that contains this white path as a subpath of it.

A path is a subpath from another path if the set of vertices of this path forms a subset of the vertices of the other path.

The score of the tree is the number of Clean White Paths.

Note that path $$$(u,v)$$$ is the same as path $$$(v,u)$$$ and has to be counted once, and you have to count paths of the form $$$(u,u)$$$.

Given $$$q$$$ update queries, in each query you are given an integer $$$u$$$, make the node $$$u$$$ black.

After each query, print the score of the tree.

In many parts of the world, when someone absolutely dominates a field—be it sports, coding, cooking, or making memes—we say they're your uncle.

It's not a family thing. It's a respect thing. Like, "Tourist is your uncle" means he owns Competitive Programming, and you just watch and clap.

So today, we're not solving graphs or DP. We're solving respect.

Given a simple question, answer it from your heart.

There are $$$n$$$ cities, numbered from $$$1$$$ to $$$n$$$. City $$$1$$$ is where Alice lives, and city $$$n$$$ is where her rival Bob resides.

The cities are connected by directed roads. Each city has at most two outgoing roads.

Alice wants to move from city $$$1$$$ to city $$$n$$$. She can visit the same city multiple times. In each city she visits, she can choose to buy the pile in that city, but she can buy at most one pile per city during her journey.

When Alice reaches city $$$n$$$, she will play a Nim game against Bob using the piles she has collected. Bob plays first.

Your task is to determine whether Alice can select piles along her journey such that she guarantees a victory over Bob.

Note: Alice must select at least one pile on the path from $$$1$$$ to $$$n$$$, and it's guaranteed there's a path from $$$1$$$ to $$$n$$$.

In Nim, players take turns removing any number of stones from one pile. The player who removes the last stone wins.

Given an array $$$a$$$ of length $$$n$$$ $$$(1 \le n \le 10^5)$$$ $$$(1 \le a_i \le 10^9)$$$. Initially, you are at index $$$1$$$ and you want to go to index $$$n$$$ and visit every index exactly once.

You can jump from your current index $$$i$$$ to another index $$$j$$$ if and only if the absolute difference doesn't exceed 2, that is $$$|i - j| \le 2$$$.

You also have an array $$$b$$$, that initially contains one element $$$a_1$$$. When you jump to a new index $$$j$$$, you mark it as visited and put $$$a_j$$$ at the end of the array $$$b$$$.

Your task is to print the lexicographically minimum array $$$b$$$ that can be obtained after a valid walk, that is after visiting all indices and ending at index $$$n$$$.

You are given an array $$$a$$$ of length $$$n$$$ and a positive integer $$$k$$$.

You will perform the following operation on the array exactly $$$k$$$ times:

• Remove the first element of the array.
• Reverse the remaining array.

After performing the operation exactly $$$k$$$ times, print the resulting array.

For an array $$$a$$$ of length $$$n$$$, we define the function $$$f(a)$$$ as following: $$$$$${f(a) = gcd(a_1, \space a_1) + gcd(a_1, \space a_2) + ..gcd(a_1, \space a_2, \space ..a_n)}$$$$$$ You are given two positive integers $$$n$$$ and $$$m$$$. You need to output the sum of $$$f(a)$$$ over all arrays $$$a$$$ satisfying the following:

• The length of the array $$$a$$$ is equal to $$$n$$$.
• $$${1 \le a_i \le m}$$$ for each $$${1 \le i \le n}$$$.

For two positive integers $$$x$$$ and $$$y$$$, $$$gcd(x, \space y)$$$ equals the greatest integer that divides both $$$x$$$ and $$$y$$$.

You are given an array $$$a$$$ of $$$n$$$ elements and an integer $$$x$$$. You also have a variable $$$v$$$ which is initially equal to $$$0$$$.

You will perform exactly one operation for each element of the array, and the operations must be applied in the order of the array (i.e., from $$$a_1$$$ to $$$a_n$$$). For each $$$i$$$ from $$$1$$$ to $$$n$$$, you can choose one of the following operations:

1. $$$v = v \mid a_i$$$ (bitwise OR)
2. $$$v = v \space \& \space a_i$$$ (bitwise AND)

You must use the second operation (AND) at least $$$x$$$ times throughout the process.

Determine the maximum possible value of $$$v$$$ at the end.

You are given an array $$$a$$$ of $$$n$$$ non-negative integers, and a positive integer $$$m$$$.

You need to answer $$$q$$$ queries. In each query, you are given a non-negative integer $$$x$$$.

You can do the following operation:

• choose two integers $$${1 \le l \le r \le n}$$$ such that $$${r - l + 1 = m}$$$, and for each $$${l \le i \le r}$$$ replace $$$a_i$$$ with $$${a_i \space | \space x}$$$. where $$${x \space | \space y}$$$ denotes the bit-wise OR of $$$x$$$ and $$$y$$$.

For each query, output the minimum number of operations needed to make the elements of the array equal, or output $$$-1$$$ if they cannot be made equal.

You are given an integer $$$n$$$. Your task is to construct a permutation $$$p$$$ of the set $$$\{1, 2, \dots, n\}$$$ such that when the permutation is considered as circular (i.e. $$$p_1$$$ and $$$p_n$$$ are adjacent), the maximum bitwise OR over all adjacent pairs is minimized.

Formally, for a permutation $$$p = [p_1, p_2, \dots, p_n]$$$, define

$$$f(p) = \max_{1 \le i \le n} \Bigl( p_i \; | \; p_{i+1} \Bigr)$$$,

where we define $$$p_{n+1} = p_1$$$ and $$$|$$$ denotes the bitwise OR operation. Your goal is to construct a permutation $$$p$$$ such that $$$f(p)$$$ is as small as possible.

If there are multiple answers, you may output any one of them.