Sebastián and Sebastian are playing a game.

The game consists of counting the number of materials and submaterials delivered to their factory while unloading a truck. The winner of the game will be the one who remembers most accurately how many materials and submaterials were received. The deal is that the loser has to take the winner out to dinner. Since neither of them wants to lose, both decide to cheat and secretly take notes to ensure their victory.

Each delivery is represented by three integers: the product, the material, and a corresponding submaterial. The full list contains N lines, each one representing a single delivery. But something went wrong. Due to a technical failure, the factory's official system lost all delivery records. Now, the only source of truth is the list that both Sebastián's created even if it contains repeated entries. They need your help to analyze their combined notes and generate a clean summary.

Great! Now that you have a working neural net activation function, Sebastian got another task for you to make. Byte-Pair Encoding ($$$BPE$$$) is an algorithm commonly used in LLM tokenizers (such as GPT-4, Gemini, and other stuff that probably wrote your homework at 11pm last Friday). $$$BPE$$$ encodes large strings into smaller strings by using a translation table, which is based on a huuuge dataset.

However, Sebastian found something interesting in this dataset. Turns out that this entire string $$$s$$$ could be represented as the repeated concatenation of another string $$$t$$$! More formally, this means that there's an integer $$$k$$$ so that $$$s = t\mathbin\Vert t\mathbin\Vert ...\mathbin\Vert t$$$ ($$$t$$$ concatenated $$$k$$$ times). Can you help Sebastian find the smallest string $$$t$$$ where this is true?

In the mystical kingdom of TriDChess, the ancient game of chess has evolved into a thrilling three—dimensional contest of strategy and intellect. The game is played on a board with dimensions $$$A \times B \times C$$$, where each of the three axes represents a spatial dimension. This board forms a cuboid, filled with possibilities for cunning moves and strategic placements.

The piece of focus in this game is the 3D Knight, a nimble entity that combines the powers of traditional chess knights with the freedom of movement afforded by the third dimension. These 3D Knights have the ability to move in an L—shape across the board, jumping over other pieces. Their movement is defined as follows:

1. L—Shaped Movement: A 3D Knight can move in an L—shape from its current position $$$(x, y, z)$$$:

• Move two squares along one axis and one square along another axis:

• $$$(x \pm 2, y \pm 1, z)$$$, $$$(x \pm 2, y, z \pm 1)$$$, $$$(x \pm 1, y \pm 2, z)$$$, $$$(x, y \pm 2, z \pm 1)$$$, $$$(x \pm 1, y, z \pm 2)$$$, $$$(x, y \pm 1, z \pm 2)$$$.

2. The Knight's jump allows it to move over other pieces, but it must land within the bounds of the board, where $$$1 \leq x \leq A$$$, $$$1 \leq y \leq B$$$, $$$1 \leq z \leq C$$$.

Additionally, there are between 1 and 500 blocked cells on the board where Knights cannot be placed. These cells are specified by their coordinates and must be avoided when placing Knights.

The challenge you face is to determine the maximum number of these agile 3D Knights that can be placed on the cuboid board such that no two Knights are capable of attacking each other. This means no two Knights should be in a position where one could reach the other in a single move.

Sergio must leave for an urgent matter, so he invents a game to keep his little brother Javier busy. In the game, Javier is given two kinds of tokens:

• Unique tokens: There are $$$m$$$ tokens labeled from $$$1$$$ to $$$m$$$ (each token can be used at most once).
• Zero tokens: There are $$$n$$$ tokens, all labeled $$$0$$$ (these tokens are identical; using one "$$$0$$$" is indistinguishable from another).

To finish the game Javier needs to create all the possible distinct arrangements.

An arrangement is defined as an ordered sequence of tokens of length $$$L$$$, where $$$1 \le L \le n$$$. In forming an arrangement you must obey these rules:

• You may use a unique token at most once in any arrangement.
• You may use the $$$0$$$ tokens as many times as needed (up to the arrangement's length; since you have exactly $$$n$$$ copies, no arrangement can have more than $$$n$$$ $$$0$$$ tokens).
• Two arrangements are considered distinct if there exists an index $$$i$$$ such that the token in position $$$i$$$ of one arrangement is different from the token in position $$$i$$$ of the other.

Your task is to help Javier figure out how many arrangements he needs to create in order to finish the game. Because this number can be huge, output it modulo $$$10^9+7$$$.

Dr senku

Dr Senku, the scientist, has discovered a new kind of cell! They are a really interesting kind of organism that interacts with electricity in weird ways. Each cell can either be red colored or blue colored. When zapped with electricity, the cell transitions from its color to the contrary, that is, a red cell will become blue and a blue cell will become red.

Senku has set the cells in a rectangular matrix of size $$$N\times M$$$. He can zap a whole column or row of the matrix, changing the color of all its cells. He can also zap an individual cell by hand, but this takes a lot of work, so he tries to avoid it.

This morning, he got a request from his university director to make it so all cells are red by the end of the day. Since he is very lazy, he first wants to know what the minimum number of times he will have to zap an individual cell to achieve this. Help him find this number!

While working at Meta AI, Sebastian developed low-latency translation models for deployment on edge devices such as mobile phones or his pretty garbage ThinkPad L390. While developing these models, he got the task of programming the popular $$$softmax$$$ function, but there's a problem: Since these low-power devices lack floating-point support, all operations must be done with integers under a large prime $$$p$$$. The softmax function is defined as: $$$$$$softmax_{int}(x_i)=\frac{2^{x_i}}{\sum_{j=1}^{n} 2^{x_j}}$$$$$$ where $$$x$$$ is a vector of (in this case) integer inputs, the denominator is the sum of all the inputs in our batch, and $$$x_i$$$ is the input to our softmax function, which is also in our input batch with an index $$$i$$$. Since Sebastian's AI model will be deployed on thousands of edge devices, he needs an extremely efficient algorithm to help him compute this function. Can you help him?

Remember that, since we need only integer outputs, we can represent $$$softmax(x_i)$$$ as a rational fraction $$$\frac{P}{Q}$$$ with $$$Q \gt 0$$$. To give this answer, you should print a single integer $$$P\times Q^{-1} \mod p$$$. Assume $$$p = 10^9 + 7$$$.

While having a Discord meeting, Sebastian and Sebastián discovered they have the same name! Any time another person called them out, they would both try to answer at the same time, generating confusion between everyone. To prevent this, they planned out a game so that the winner could keep the name, while the loser would change their name to Notbastian (Not Sebastian). The game works as follows:

• $$$n$$$ marbles are placed on the ground, and Sebastian picks out the marbles with an odd number written on them.
• On the contrary, Sebastián will pick the marbles with an even number written on them.
• The winner will be the person with the larger amount of marbles between the two.
• If there's a tie in the amount of marbles Sebastian and Sebastián picked, Sebastian will win, since he has a greater Codeforces rating, granting him anything he wants.

The beginning of the age of pirates

Gol D. Roger, the pirate king, conquered everything this world has to offer. Before being executed, he hid his treasure, the "One Piece" on an island on the Grand Line. The Grand line is a group of $$$N$$$ islands in the sea numerated from $$$1$$$ to $$$N-1$$$. Due to its weather conditions, the Grand line cannot be traversed as any typical sea; any travel on the Grand line has to start on the Red line, which is represented by the number $$$0$$$, and follow the sea routes. There are $$$N-1$$$ unidirectional sea routes that connect two islands. From each island, it is also possible to go back directly to the Red Line. It is known that every island can be reached from the Red line following sea Routes.

Roger will hide the $$$m$$$ maps to his treasure on islands throughout the Grand line, and each map will have to be recovered in order and then brought back to the Red line before you can go to the next one. To make sure the journey to the One Piece is as fun as it should be, Roger wants to know $$$\sum b_i$$$ where $$$b_i$$$ represents how many times a pirate crew that started on the Red line, and has followed the shortest path that follows the conditions (Get the maps in order, always return to the Red line after recovering a map), has passed through the island $$$b_i$$$ before collecting map $$$i$$$.

Paulino, while writing new problems, realized that he suffers from integer dyslexia. Paulino's integer dyslexia is very unusual because when he writes a sequence of numbers, he often starts with a different digit instead of the first one and then appends the skipped digits at the end. For example, he might write 123 as 231 or 312.

Given the correct number and the number that Paulino wrote, can you determine if they represent the same sequence?

Appa and Winnie are dogs with very different passions. While Winnie loves drawing, Appa is fascinated by mathematics.

Seeing her love for drawing, Winnie decides to join the Public Art Classes, where the first lesson is to create geometric figures using only '.', '/', and '\', as shown below:

Winnie's drawing.

Since Appa does not have the opportunity to join any course, he decides to take Winnie's drawings and calculate the area of the shapes. However, he knows nothing about mathematics, except that each '.' represents a 1×1 unit. So he asks for your help in calculating the area of the shapes Winnie draws.

Sebastian is working on optimizing a system for Automatic Speech Recognition (ASR), and he's faced with a challenging task in improving the efficiency of speech data segmentation. Given a sequence of integer tokens representing features extracted from speech, his goal is to partition this sequence into the smallest number of contiguous subarrays such that each subarray contains at most $$$k$$$ unique elements.

In the context of ASR, each subarray represents a segment of speech, and the number of unique tokens in a segment must be limited to prevent overwhelming the system's memory and computational resources. Sebastian needs to minimize the number of such segments required while ensuring that each segment remains within the constraints for efficient processing.

Given an array $$$a$$$ of size $$$n$$$ (where $$$a_i$$$ is the token representing a feature in the ASR system) and an integer $$$k$$$ (the maximum number of unique tokens allowed in a segment), Sebastian must determine the minimum number of subarrays such that each subarray contains at most $$$k$$$ unique tokens.

German is a passionate fan of "Los Ratones" and has just finished watching all their matches from last year. This year, however, he hasn't seen any matches yet. The list a represents the number of times Bausffs died in each match this year.

German is particularly intrigued by Bausffs's unique playing style. He wants to watch matches that start in a specific way, represented by the list b, which details how he prefers to see Bausffs die each day. Additionally, he wants these matches to end in a way that makes him happy, represented by the list c. He calls this way of watching the games Gerbausffs's Law.

German is eager to start watching the matches but doesn't have the time to figure out all the different ways he can watch matches that begin in the form b and end in the form c.

Please help German because his friend Wiritos didn't want to help him.

The GOAT.

Professor S is teaching his students A, T, and R the arcane arts of mathematical magic (otherwise known as mathgic). Part of this is learning to draw geometric figures, so they have to complete the following exercise.

They were be given a 2D plane, first A drew $$$n$$$ vertical lines $$$X=x_0, X=x_1,\ldots,X=x_{n-1}$$$. Afterwards J drew $$$m$$$ horiontal lines $$$Y=y_0,Y=y_1,\ldots, Y=y_{m-1}$$$. Finally R drew a diagonal line $$$D$$$ from point $$$(h,0)$$$ to $$$(h+l,l)$$$.

The challenge that was then given to them is:

For each point that the line $$$D$$$ has integer coordinates, a ray is drawn from there perpendicular to the line and to the right. i.e. from point $$$(x,0)$$$ the ray goes in direction $$$(x-1,1)$$$. Then answer, which line will it intersect first?

The Malinche mountain, in Puebla

Sergio is an avid hiker who loves exploring mountain trails and deep valleys. On his latest adventure, he recorded the elevation at three consecutive points along his path. Now, he wants to determine whether he just climbed over a peak or descended into a valley.

Given three integers $$$a, b, c$$$ representing the elevations at three consecutive points along Sergio's hike, determine whether these points form a mountain (a peak) or a valley:

• A mountain occurs if $$$a \lt b \gt c$$$ (i.e., the middle elevation is the highest).
• A valley occurs if $$$a \gt b \lt c$$$ (i.e., the middle elevation is the lowest).

Can you help Sergio determine if he climbed a mountain or a valley?

In the near future, an elite team of cybersecurity researchers known as the Digital Sentinels discovers that a clandestine group has developed a sophisticated obfuscation technique to hide messages within sentences. This technique allows critical information to go unnoticed, camouflaged among seemingly harmless data.

During an undercover operation, the Sentinels intercept a text file that, at first glance, appears to contain only random characters. However, they suspect that this file hides a deactivation code essential to stop an out-of-control artificial intelligence threatening to take over global infrastructures. The string of text they have detected is 45 4c 20 43 4f 44 49 47 4f 20 44 45 20 44 45 53 41 43 54 49 56 41 43 49 4f 4e 20 45 53 3a 20 22 43 4f 44 45 3a 52 55 53 48 3a 54 45 43 22.

As a member of the Digital Sentinels, you are tasked with decrypting this code to stop the artificial intelligence that is about to cause disruption in the global network. Given this message, output whatever the message requests from you.