In the vibrant town of Numerica, where numbers ruled the marketplace, a clever mathematician named Alaric presented a challenge. Traders were tasked with finding the longest stretch of numbers in an array, but here's the catch – they could only use up to three different numbers in that stretch. It was like a numerical puzzle, a quest to uncover the magic of the longest subarray with just three types of numbers.

As traders delved into Numerica, each subarray they discovered was a chapter in their adventure, revealing the delicate dance of numbers within Alaric's constraints. The goal was simple yet intriguing: for an array $$$A$$$ find the longest subarray with at most three different numbers, and earn the admiration of the entire marketplace.

You are given two arrays $$$A$$$ and $$$B$$$ of length $$$N$$$. For each position $$$i$$$, we need to choose one element either $$$A_i$$$ or $$$B_i$$$ and the goal is to choose values in such a way that the sum of the values is as small as possible. Note that we choose elements in increasing order of $$$i$$$.

However, there is a catch as the problem would be way too easy otherwise. Every time we take a value from array $$$A$$$, the cost of every subsequent value we choose will double.

For example, if we have the arrays:

$$$[4, 1, 3, 4]$$$ $$$[7, 2, 1, 3]$$$

If we choose $$$4$$$ from the first array, this means that for the next $$$3$$$ positions, the true costs will be $$$(2, 6, 8)$$$ and $$$(4, 2, 6)$$$ respectively which can be further doubled as well.

Note that if we take an element from $$$B$$$ this doubling process does not happen.

Given an array $$$a$$$ of $$$n$$$ elements, you want to partition $$$a$$$ into $$$2$$$ subsequences of size at least $$$2$$$ such that every element in the array belongs to exactly one subsequence and the difference between the maximum and minimum element in each subsequence is the same. Print the maximum possible difference or $$$-1$$$ if a partition cannot be found.

For each test case, you are given arrays $$$a$$$ and $$$b$$$ with $$$n$$$ integers. For each $$$a_i$$$ determine whether you can multiply $$$a_i$$$ by any subset of the previous elements in the array (subset may be empty, meaning you do not multiply) such that the ones (unit) digit of the product is the digit $$$b_i$$$.

When working with negative numbers, the units digit still refers to the last digit when writing down the number. (The units digit of $$$-19$$$ is $$$9$$$)

You have an $$$n$$$ by $$$n$$$ square garden which can be thought of as a grid. In each cell, there is either good planting land, a well, or bad planting land. You only want to focus on one gardening strip because it's much easier to maintain. A gardening strip is a $$$2$$$-tall strip that is always horizontal (but can have a variable length), where if you cut out just the gardening strip, all planting land must be at most $$$k$$$ away from the closest well, and all land in the strip is either good planting land or wells. Distance between two cells is defined by: $$$|a.x - b.x| + |a.y - b.y|$$$ (Manhattan distance). Now when trying to choose a gardening strip, you ask yourself, how many gardening strips are in my garden square?

In the country of Farmer John, there exist $$$N$$$ cities and $$$M$$$ roads connecting some of these cities. We assume that the roads are directional and that the road going from city $$$u$$$ to city $$$v$$$ has a non-negative length $$$L(u,v)$$$.

A new road is to be built, and therefore, you are given a list of $$$K$$$ pairs of cities that this road could connect. Each proposed road from city $$$u$$$ to city $$$v$$$ is accompanied by its corresponding length $$$L(u,v)$$$.

You are also given two cities, $$$s$$$ and $$$t$$$. Your goal is to choose at most one of the proposed roads to achieve the minimum distance from city $$$s$$$ to city $$$t$$$ and print that distance.

The soccer league started and our friend, B.B.C. wants to show the entire world he's the best soccer statistician in the world. Soccer is played as a game between two teams and the team who scores more goals wins. In case of equality, we will consider the game to end in a draw.

For this reason, he gave you the results of $$$T$$$ teams playing in different leagues and he wants you to tell him if for a given tuple $$$(W, D, L, G_f, G_a)$$$, where $$$W$$$ is the number of wins, $$$D$$$ is the number of draws, $$$L$$$ is the number of losses, $$$G_f$$$ is the numbers of goals scored and $$$G_a$$$ is the number of goals conceded, there is a unique set of results that corresponds to the tuple given in the input.

For example, if we have the tuple $$$(1, 1, 1, 1, 1)$$$, we have a unique set of results that lead us here $$$(1-0, 0-1, 0-0)$$$, the order these results show isn't important, however if we have $$$(1, 1, 1, 2, 2)$$$, we can have $$$(2-0, 0-2, 0-0)$$$ or $$$(1-0, 0-1, 1-1)$$$, so the set of results is not unique.

Now you need to show B.B.C. what you're made of and conquer this challenge!

Take note that if there is no valid way of assigning scores, you should also print "Better luck next time".

In math class, the teacher is trying a new grouping system. Every day he gives each of his $$$n \cdot 10^{100}$$$ students a card with some number from $$$1$$$ to $$$n$$$, to create $$$n$$$ random groups of equal size $$$10^{100}$$$. This means that he hands out $$$10^{100}$$$ cards with the number $$$1$$$, $$$10^{100}$$$ cards with the number $$$2$$$, ...., and $$$10^{100}$$$ cards with the number $$$n$$$. Each pair of students are either best friends, friends, or not friends. The issue is that students refuse to work in a group without all their best friends (they are okay working without their normal friends), and so they start trading their cards.

The students are socially awkward and only trade their cards with their normal friends and best friends. You are given $$$m$$$ pairs of students, each pair denoting two best friends who must be in the same group (students do not start with any normal friendships). Your job is to compute the minimum number of normal friendships that must be made before the students know their cards such that no matter what cards the students get, after trading any number of times, each pair of best friends has the same cards. Note that if two students are best or normal friends, they can trade infinitely many times.

Hori-san is eating a yummy cake baked for her by Miyamura. The cake is a strange cake that can be represented by a collection of line segments. Hori will take one or more bites in order to eat the cake.

In any bite, Hori can choose any segment and eat that segment as well as any segments that are fully contained within the segment.

It is guaranteed that all endpoints of segments are distinct. In addition, if two segments intersect, it is guaranteed that one of them will fully contain the other.

How many different sequences of bites are there for Hori to eat the entire cake (no segments left) modulo $$$10^9+7$$$?

Scientists recently invented a new game called CRP to test the IQ of people, as well as push them past their limits. CRP is a single player game that can be played using $$$2$$$ boards and a pen.

Assume that the two boards are named $$$A$$$ and $$$B$$$. You are given a sequence of $$$N$$$ numbers, each one belonging to the set $$$\{0,1,2,3\}$$$. The sequence is currently written on board $$$A$$$. In order to win the game, you need to move all numbers from board $$$A$$$ to board $$$B$$$ with the smallest amount of moves, such that all numbers with equal values are adjacent to each other on board $$$B$$$.

To move a number from $$$A$$$ to $$$B$$$, you are allowed to make use of the following operations described below:

1. Operation $$$C$$$: This operation is denoted by the letter $$$c$$$. In this operation, you can replace every number on board $$$A$$$ with its complement. For example, $$$x$$$ becomes $$$3-x$$$.

2. Operation $$$R$$$: This operation is denoted by the letter $$$r$$$. In this operation, you can reverse the order of the numbers on board $$$B$$$. For example, $$$\{1, 3, 0, 3\}$$$ becomes $$$\{3, 0, 3, 1\}$$$.

3.Operation $$$P$$$: This operation is denoted by the letter $$$p$$$. In this operation, you can take the first number from $$$A$$$ and push it to the end of $$$B$$$. For instance, if $$$A = \{1,3,3\}$$$ and $$$B = \{1,2,2\}$$$ before, then after performing the operation $$$P$$$ one time, $$$A = \{3,3\}$$$ and $$$B = \{1,2,2,1\}$$$.

The scientists decided to hold a tournament to popularize the game and find the people with the highest IQ. They have also agreed to give a big prize to each person of the tournament that successfully wins a game.

The tournament consists of $$$T$$$ CRP games. In each of the $$$T$$$ games, only one player is chosen and allowed to play the game. If he wins the game, he gets a big prize and a chance to meet with the scientists; otherwise, he doesn't get anything.

Let the string $$$S_i$$$ denote the sequence of operations $$$(c,r,p)$$$ with the smallest length, such that after performing the respective operations, the player currently chosen to play wins the $$$i$$$-th game.

Your task is to find and print $$$S_i$$$ for each $$$1 \le i \le T$$$.

If there exists more than one possible $$$S_i$$$ with the same length, then you need to find the lexicographically smallest one.

In the quest of trying to be the best student ever, you pose your teacher an interesting question. You begin with two numbers, $$$x$$$ and $$$y$$$ both equal to $$$1$$$. You can do two operations, operation $$$1$$$ is to set $$$x := x + y$$$, and operation $$$2$$$ is to set $$$y := x + y$$$. The task is to count how many ways (modulo $$$10^9 + 7$$$) you can do operation $$$1$$$ or $$$2$$$ until $$$ax + by \gt c$$$. Two ways are different if either the number of operations is different or in one way you do operation $$$1$$$ instead of operation $$$2$$$. Because your teacher cannot solve the problem, you decide to help him out and solve it on your own.

You are given an array $$$A$$$ of size $$$N$$$ and $$$Q$$$ queries.

In each query, you will be given three integers: $$$l$$$, $$$r$$$, and $$$x$$$. Find the value of

$$$$$$

(A_l \text{ mod } x) + (A_{l + 1} \text{ mod } x) + \dots + (A_{r - 1} \text{ mod } x) + (A_r \text{ mod } x)

$$$$$$

Note that $$$(x \text{ mod } y)$$$ denotes the remainder when $$$x$$$ is divided by $$$y$$$.