Before giving you the problem statement, we will define $$$dist([x_1, x_2, \dots x_k], [y_1, y_2, \dots y_k])$$$ as the number of indexes $$$i$$$ ($$$1 \leq i \leq k$$$) such that $$$x_i \neq y_i$$$.

You are given $$$n$$$ and two arrays $$$a_1, a_2, \dots a_n$$$ and $$$b_1, b_2, \dots b_n$$$. Find the maximum $$$dist$$$ when choosing any two subarrays of your choice (one from $$$a$$$ and one from $$$b$$$) that have the same length. A subarray of $$$c$$$ is a contiguous part of an array $$$c$$$, i. e. the array $$$c_i, c_{i+1}, \dots c_j$$$ for some $$$1 \leq i \leq j \leq n$$$.

You finally trapped Lavutz and his $$$K - 1$$$ friends in a room (in total there are $$$K$$$ people). The room is very interesting, as it is split in $$$n$$$ rows and $$$m$$$ columns, and for each row $$$i$$$ and column $$$j$$$ ($$$1 \leq i \leq n$$$ and $$$1 \leq j \leq m$$$) there is a cell which is at $$$a_{i,j}$$$ units of height (distance in units from the ground). A person can move from cell ($$$i$$$, $$$j$$$) to one of the cells ($$$i-1$$$, $$$j$$$), ($$$i+1$$$, $$$j$$$), ($$$i$$$, $$$j-1$$$), ($$$i$$$, $$$j+1$$$) (if that doesn't go out of the boundaries of the room) in exactly $$$1$$$ second, or they can just stay in cell ($$$i$$$, $$$j$$$).

You have a device which can raise the level of the lava, which is initially raised at level $$$0$$$, but you don't want to hurt Lavutz or his friends. A person is hurt if the lava is raised at a level greater than the height of the cell where the person is at that moment.

Now, you want to find for each $$$L$$$ from $$$1$$$ to $$$n \cdot m$$$ what is the minimum time you need to wait such that you can raise the lava to level $$$L$$$ and nobody gets hurt, or $$$-1$$$ if for any amount of time you will wait, you can't raise the lava to level $$$L$$$ without hurting anybody.

Andrei is taking basketball throws. He wants to score $$$n$$$ points.

As he doesn't have a lot of time, he wants to reach $$$n$$$ points from as few throws as possible. As you know, in basketball we can score using throws which get either $$$2$$$ or $$$3$$$ points.

If Andrei can't reach $$$n$$$ points, we should print 'No' (exactly as it is written). Otherwise, we will print two integers $$$a$$$ and $$$b$$$, which represent the minimum number of two point, respectively three point throws Andrei is going to use.

Alice and Bob go to edenland. Edenland is a famous adventure park where people can play various games which take place inside of a forest.

At edenland, you have numerous tracks, each consisting of several games. We will only focus on one track. This track consists of $$$n$$$ games, separated by platforms. Alice takes $$$a_i$$$ time to finish the $$$i^{th}$$$ game, while Bob takes $$$b_i$$$ time. We consider that it takes no time to pass a platform. Alice goes on the track first, followed by Bob.

However, at edenland there is a special rule: you can't overtake the person in front of you, that is, if you start behind someone, you will always be behind that person. To complete a track, you first start by playing the first game, then the second, and so on. As Bob will always be behind Alice, this rule will not change, and Alice will start the track before Bob.

There is another very important rule at edenland: no two people can play the same game at the same time, as it is not safe. Thus, if Bob is on the platform before the $$$i^{th}$$$ game, he will wait on that platform until Alice has finished the $$$i^{th}$$$ game.

Now, you are curious, for $$$q$$$ intervals $$$[l, r]$$$, what is the amount of time Bob will finish the track after Alice, if we consider only the track consisting of games $$$l, l+1, \dots, r$$$.

Consider the following integer sequence $$$f$$$:

$$$f_1 = f_2 = 1$$$, $$$f_k = |f_{k-1} - f_{k-2}| \oplus (f_{k - 1} + f_{k - 2})$$$, for $$$k \geq 3$$$.

Now, you are wondering, for $$$q$$$ triples of integers ($$$l, r, M$$$), what is the sum of all $$$f_i$$$, $$$l \leq i \leq r$$$, modulo $$$M$$$. That is, find $$$\displaystyle(\sum_{i = l}^rf_i)$$$ and print it's remainder modulo $$$M$$$.

By $$$|x|$$$ we denote the absolute value of $$$x$$$ and by XOR we denote the bitwise XOR operation.

Suceava city consists of $$$N$$$ neighborhoods, indexed from $$$1$$$ to $$$N$$$, connected by $$$N - 1$$$ bidirectional streets. It is widely recognized that you can travel between any two neighborhoods by using one or more streets.

Suceava is home to $$$M$$$ gangs, all of which are in conflict and indexed from $$$1$$$ to $$$M$$$. Initially, each street is occupied by a gang.

Over the course of $$$T$$$ days, some gang may conquer a street occupied by another gang.

We define the insecurity level of a gang as the maximum number of streets you can traverse that are occupied by the gang, while ensuring each street is traversed only once.

Being given $$$Q$$$ queries $$$(g, t)$$$, you need to determine the insecurity level of gang $$$g$$$ on day $$$t$$$.

Grigorutz is a simple man, he doesn't like to do homework. Today, his english teacher decided to test Grigorutz's intelligence by playing a game with him. There are $$$n$$$ essays which might be in Grigorutz's homework. Let $$$a_i$$$ be the time Grigorutz will spend to write the $$$i$$$-th essay (in minutes). The game is played as follows:

1. There will intially be a deque containing the element $$$0$$$ and a variable $$$T = 0$$$.

2. The teacher will go through the essays from $$$1$$$ to $$$n$$$. At the $$$i$$$-th step, Grigorutz can do one of the two operations:

$$$a)$$$ $$$T := T + deque.front(), \ deque.push$$$_$$$front(a_i)$$$

$$$b)$$$ $$$T := T + deque.back(), \ deque.push$$$_$$$back(a_i)$$$

After the $$$n$$$-th element is added to the deque, $$$T$$$ will be the amount of time to spend to solve his homework. Obviously, Grigoutz wants to minimze this time. Can you help him do so?

Grigorutz asks you to write a program which given $$$n$$$ and the number of minutes to write each of the essays will print the minimum value of $$$T$$$ after the $$$n$$$ operations.

As you all know, AI use is on the rise. As our friend Gigel dislikes this, he came up with a plan of sabotaging the growth of AI.

An AI brain works in intricate ways. First lets consider an infinite grid where there are $$$n$$$ neurons placed. We know for each neuron its position in the grid and its color, given as a tuple $$$(x, y, col)$$$. An AI Thought is described by:

• The length of the neural sequence $$$m$$$, meaning that our thought is created by a list of $$$m$$$ neurons.
• The colors we want to use for our list of neurons, $$$col_1, col_2, \dots \ col_m$$$. This means that the $$$i^{th}$$$ neuron in our thought list should have the color $$$col_i$$$. As an important observation, we are allowed to use the same neuron multiple times in our list.
• The time we need to generate this thought is $$$manh(i_1, i_2) + manh(i_2, i_3) + \dots + manh(i_{m-1}, i_m)$$$, where $$$manh(i, j)$$$ is the Manhattan distance between points (neurons) $$$i$$$ and $$$j$$$ in the grid($$$|x_i - x_j| + |y_i - y_j|$$$).

Now, depending on how we choose the neurons, the time needed for generating a thought can differ. As Gigel hates AI, he wants to prove the world how inefficient it is, so he asks for your help. In order to do this, you need to compute for $$$T$$$ thoughts the maximum possible amount of time for generating each thought.

Tutz likes math problems; this time he is trying to solve the following problem:

You are given $$$K$$$, $$$S$$$, $$$T$$$ and are asked how many sequences of $$$K$$$ integers (more formally $$$a_1$$$, $$$a_2$$$, $$$a_3$$$, $$$\cdots$$$, $$$a_K$$$) follow these rules:

• $$$a_i \gt 0$$$, for any $$$i$$$ ($$$1 \leq i \leq K$$$)
• $$$a_1 + a_2 + a_3 + \cdots + a_K = S $$$
• $$$a_1 \cdot a_2 \cdot a_3 \cdot \ \dots \ \cdot a_T = a_2 \cdot a_3 \cdot a_4 \cdot \ \dots \ \cdot a_{T+1} = \dots = a_{K-T+1} \cdot a_{K-T+2} \cdot a_{K-T+3} \cdot \ \dots \ \cdot a_K$$$

You should find the number of such arrays modulo $$$10^9 + 7$$$ ($$$10^9 + 7$$$ is a prime number).

Bob has $$$n$$$ sticks. The $$$i$$$-th stick has length $$$a_i$$$. He needs to choose a tuple of $$$4$$$ sticks ($$$i$$$, $$$j$$$, $$$k$$$, $$$p$$$) such that:

• $$$1 \leq i \lt j \lt k \lt p \leq n$$$;
• by moving and rotating the sticks any way you want (without breaking them) you can obtain a parallelogram with sides of lengths $$$a_i, a_j, a_k, a_p$$$.

Bob is interested if there is such a tuple. He asks you to print "YES" if you can choose $$$4$$$ sticks satisfying the above properties, and "NO" otherwise.

After hearing enough nonsense discussions, X decided to be short with the next problem's statement.

Given an array of length $$$n$$$, find out how many subarrays exist such that the difference between the maximum and minimum value is at least equal to the difference between the first and the last position of the subarray.

More formally, if we have the subarray $$$[L, R]$$$, whose maximum value is $$$A$$$ and whose minimum value is $$$B$$$, we want $$$A - B \geq R - L$$$.

Sannu went on a trip to the mountains together with his friend group. Altogether there were $$$N$$$ people gathered in a cabin in order to watch all of One Piece, and they decided to go eat at a nearby restaurant on the third day of their trip. Although their rented cabin had enough bathrooms, only one shower was usable. From Sannu's keen observations during the second day, they discovered that that shower has running water just at some moments during the day, and you cannot control the temperature of the water. Sometimes there is hot water, sometimes lukewarm, sometimes only cold water and during the rest of the day there is no water running from the shower. Knowing the preferred water temperature for each person, as well as how much each person stays in the shower, please find out what is the earliest moment of the day in which everyone has showered (and is ready to leave for the restaurant). Do note that, due to the size of the shower cabin, only one person can take a shower at a given time.

You are playing a game in which you must defend your village from a dragon.

The village can be represented as a tree (a connected acyclic graph) with $$$N$$$ nodes, indexed form $$$1$$$ to $$$N$$$, each node representing fortifications of varying heights. The height of fortification $$$i$$$ is denoted as $$$h_i$$$.

The dragon has a power level of $$$P$$$ and starts flying at the base of fortification $$$u$$$ (i. e. its initial height is $$$0$$$). Its goal is to attack fortification $$$v$$$. It flies along the path from $$$u$$$ to $$$v$$$ and when it encounters a fortification $$$x$$$ with a height $$$h_x$$$ greater than or equal to its current height, it loses $$$h_x$$$ points from its power and continues flying at a height of $$$h_x$$$.

Unfortunately, there's a bug in the game: if the dragon's current power $$$P_{crt}$$$ becomes less than $$$0$$$, it immediately becomes $$$-P_{crt}$$$. You have the ability to shuffle the fortifications along the path from $$$u$$$ to $$$v$$$. Your objective is to find an arrangement such that the dragon's power is reduced to $$$0$$$ when it reaches fortification $$$v$$$, preventing the dragon from attacking it.

Being given $$$Q$$$ scenarios $$$(u, v, P)$$$, you should find if it's possible to shuffle the fortifications along the path from $$$u$$$ from $$$v$$$ so the dragon won't attack the tower $$$v$$$.