meetforever/GRPO_CL huggingface dataset:
user: Celery — the legendary Pokenom has been spotted in Alexa Forest.

To become the best Pokenom trainer, Bash has arrived at Alexa Forest to capture Celery. After lots of information gathering, Bash was able to draw a map of Alexa Forest, and noted down $K$ sightings of Celery.

Alexa Forest’s map is a convex polygon $A$ with $N$ vertices on the Cartesian plane. $K$ sightings of Celery can be considered as $K$ points — all are strictly inside Alexa Forest.

Bash is ready to search Alexa Forest to find Celery. However, Bash realized that Alexa Forest is simply too big. It would take decades to search the entire forest. But Bash is smart. Based on his research, Bash knows that Celery can only be found inside a polygon $Z$, where the vertices of $Z$ are a subset of $A$, and all $K$ sightings of Celery must be strictly inside polygon $Z$.

Of course, there can be multiple polygons $Z$ satisfying the above conditions. Your task is to help Bash find the polygon $Z$ with smallest number of vertices.

-----Note-----
A point $P$ is strictly inside Polygon $A$, iff $P$ is inside $A$ and $P$ does not lie on the border of $A$.

-----Input-----
 - The first line of input contains a single positive integer $N$$(3 \le N \le 2 \cdot 10^5)$.
 - The next $N$ lines each contain $2$ integers $x_ i$, $y_ i$ — the coordinates of the $i$-th vertex of Alexa Forest $(-10^9 \le x_ i, y_ i \le 10^9)$. The vertices are listed in either clockwise or counterclockwise order. It is guaranteed that Alexa Forest is convex.
 - The next line contains a single positive integer $K$ $(1 \le K \le 10^5)$.
 - The next $K$ lines, each line contains $2$ integers $x_ i$, $y_ i$ — the coordinates of a sighting of Celery $(-10^9 \le x_ i, y_ i \le 10^9)$. All points are guaranteed to be inside Alexa Forest and no points are on the border of Alexa Forest.

-----Output-----
Output a single integer — the smallest number of vertices of polygon $Z$.

-----Sample Clarification-----
 - In the first example, the only valid polygon satisfied is the whole Alexa Forest.
 - In the second example, there are two possible solutions with $4$ vertices:

-----Examples-----
Sample Input:
4
0 0
0 3
3 3
3 0
2
1 1
2 2
Sample Output:
4
user: You are the kingpin of a large network of criminal hackers. Legend has it there has never been a richer criminal than you. Not just because you are the smartest, but also because you are the stingiest.

The police have been after you for years, but they have never been able to catch you thanks to your great set of escape routes. Whenever they want to catch you in one of your many hideouts, you quickly get away through your network of tunnels, back alleys and speakeasies. Your routes are set up so that from every hideout you have in the city you can get to any other hideout by following only your secret passageways. Furthermore, because you are such a penny-pincher, your network is the smallest possible: from every hideout to every other hideout there is precisely one route through the network, no more and no fewer.

Yesterday, your mole in the police force has informed you of an unfortunate fact: the police are on to you! They have gotten wind of your secret network, and will attempt to catch you. They are planning to block some of your escape routes, and catch you in the act. They will start blocking your secret passageways one by one, until you have nowhere left to go.

Fortunately, your headquarters are absolutely safe. If you can just get there, you are always fine. Furthermore, your mole in the police force can inform you immediately as soon as the police start blocking passageways, so that they only have time to block one of them before you get notified. If you get back to your headquarters before they block any more routes, you’re safe.

You want to add some passageways to the network so that whenever at most one of them is blocked, you can still get to your headquarters from any other hideout. Since the news has not changed your frugality, you want to extend your network as cheaply as possible. Can you figure out the least number of passageways you need to add, and which ones you need?

-----Input-----
 - The input starts with two integers $2 \leq n \leq 10^5$, the number of hideouts in the network, and $0 \leq h < n$, the location of your headquarters.
 - Then follow $n - 1$ lines, each with two integers $0 \leq a, b < n$, signifying that there is an escape route between location $a$ and location $b$.

-----Output-----
The output consists of:
 - An integer $m$, the least number of escape routes you need to add to make the network safe again.
 - Then, $m$ lines with two integers $0 \leq a, b < n$ each, the hideouts between which one of the escape routes has to be added.

In case there are multiple solutions, any of them will be accepted.

-----Examples-----
Sample Input:
4 0
0 1
0 2
0 3
Sample Output:
2
3 2
3 1
user: Each Halloween, the $N$ witches of Sweden all gather high up in the air with their brooms to perform a magical dance. The dance consists of taking one’s broom and rotating clockwise around one of the ends of the broom. Each broom is exactly $1\text { m}$ long, and everyone rotates at the same speed. The $i$’th witch originally starts rotated $r_ i\text { radians}$ clockwise. $0$ radians means that the broom points to the right, i.e. positive $x$-direction; $\pi /2$ radians points in the positive $y$-direction. All the witches are located at the same height in the air, at positions $(x_ i, y_ i)$ for $1 \le i \le N$. Their brooms are always parallel to the $x-y$ plane.

This dance is very beautiful, but has in recent years been plagued by a certain problem. It turns out that nobody had choreographed the dance properly. This caused some of the witches to crash into each other’s brooms. As a result, their brooms would break and they would fall down helplessly, being forced to use their parachutes – can you imagine a greater embarrassment for a witch?

This year, a coalition of witches are tired of their brooms breaking. They have thus produced a choreography beforehand, consisting of the original locations and rotations of the witches’ brooms. Can you check if the dance will cause any brooms to crash into each other?

-----Input-----
The first line of input consists of a single integer $N$ ($1 \le N \le 200000$), the number of witches. The next $N$ lines contains a description of each broom. Each line consists of the real numbers $x_ i, y_ i$ ($-10^{9} \le x_ i \le y_ i 10^{9}$), and the real number $r_ i$ ($0 \le r_ i < 2\pi $), all with at most $20$ digits after the decimal point. The coordinates are measured in meters, and give you the end of the broom that the broom rotates around.

The input will be such that the answer does not change even if the length of the brooms were $1 + 10^{-6}\text { m}$ long.

-----Output-----
Output ok if no witches will crash into each other, and crash if at least one pair of witches will crash.

-----Examples-----
Sample Input:
2
0 0 0
0 1.5 0
Sample Output:
ok
user: The new semester at Hogwarts has just started, but something is not quite right – the staircases are not cooperating with the school administrators! Hogwarts has a single room of movable staircases connecting the $N$ floors. There are $M$ staircases in total, and no two staircases connect the same pair of floors (and, obviously, a staircase would not connect a floor to itself – these are smart magic staircases with dignity). The only way to manipulate the staircases is by pressing red and green buttons located on each floor. The floors are labeled in some manner by the integers $0$ through $N-1$. Pressing the red button on floor $i$ ($0\leq i \leq N-1$) has the following effect on the staircases. Any staircase that is currently not connected to floor $i$ does not move. Suppose a staircase connects floors $i$ and $j$ ($j \neq i$). After pressing the red button on floor $i$, it will instead connect floors $i$ and $j+1 \bmod N$ – unless $j+1\bmod N = i$, in which case it will connect floors $i$ and $j+2 \bmod N =i+1 \bmod N$ instead. Pressing the green button is simply the inverse operation of pressing the red one on the same floor (or, equivalently, the same as pressing the red one $N-2$ times).

While alone, the staircases got all messed up. The school adminstrators have presented a plan for how they would rather like the staircases to be placed.

You, a low-ranking house elf, have been given the task to fix this.

-----Task-----
Find some sequence of at most $250000$ button presses that will change the staircase room from its current state to the desired state.

-----Input-----
There is a single test case. On the first line $N$ and $M$ are given ($3 \leq N \leq 50$, $0 \leq M \leq N(N-1)/2$).

Then follow $M$ lines with pairs of integers $i$, $j$ ($0 \leq i,j\leq N-1$), describing the current state of the room of staircases. Each such line means that there is a staircase connecting floors $i$ and $j$. After this follow another $M$ lines with pairs of integers $i$, $j$, describing the desired state of the room of staircases.

You may assume that neither the current state of the room nor the desired state of the room contains two staircases between the same pairs of floors (but some pairs of floors may appear in both the current and desired states of the room).

-----Output-----
On the first line of output, write a single integer $Q$ ($0 \le Q \le 250000$), the length of your sequence of button presses. Then print $Q$ lines, each being either “R $i$” or “G $i$” for some $i$ ($0 \leq i \leq N-1$), meaning that the red or green button on floor $i$ should be pressed.

-----Examples-----
Sample Input:
5 4
0 1
0 3
1 2
2 4
0 2
0 4
2 3
2 4
Sample Output:
2
R 0
G 2
user: You have just been hired by Amalgamated, Inc. in the country of Acmania to oversee the transportation of raw materials to the company’s factories. Each supplier of raw materials and each factory resides in one of Acmania’s states. No state has both a factory and a supplier (and never more than one of either) and there are arcane laws governing which transportation companies can transport materials across state lines. Because of the fierce competition between factories and between suppliers each transportation company handles the output of at most one raw material site and delivers to at most one factory (or to another transportation company). Each supplier can produce enough material to contract with at most one factory and no factory will contract with more than one supplier. Your job is to determine the maximum number of factories that can be supplied with raw materials.

For example, suppose that there are three suppliers in states A, B and C, and three factories in states D, E and F. Let’s say you contract three transportation firms: firm $1$ can transport between states A, E and G; firm $2$ can transport between states A, C and E; and firm $3$ can transport between states B, D and F. In this case, you can supply at most two factories (for example, factory E can be supplied from supplier A using firm $1$, and factory F can be supplied from supplier B using firm $3$). If you find a fourth firm that transports between states G and F then you can supply all three factories: factory D can be supplied from B using firm $3$, factory E can be supplied from C using firm $2$, and factory F can be supplied from A using firms $1$ and $4$.

-----Input-----
The input will start with four positive integers $s$ $r$ $f$ $t$ indicating the number of states, raw material sites, factories and transportation companies, where $1 \leq r,f \leq 200$, $r+f \leq s \leq 600$ and $1 \leq t \leq 1000$.

Next will follow a line containing $r$ state names, one for each raw material site.

The next line will contain $f$ state names, one for each factory site.

Finally there will be $t$ lines, one for each transportation company. Each of these lines will start with an integer $n$, $1 \leq n \leq s$, indicating the number of states the company is allowed to work in, followed by $n$ state names. No state will contain both a raw material site and a factory site.

All state names will be alphabetic strings with no blanks.



-----Output-----
Output the maximum number of factories that can be supplied with raw materials.

-----Examples-----
Sample Input:
7 3 3 3
A B C
D E F
3 A E G
3 A C E
3 B D F
Sample Output:
2
user: Welcome, ladies and gentlemen, to Aperture Science. Astronauts, War Heroes, Olympians — you’re here because we want the best, and you are it. That said, it’s time to make some science.

Now, I want each of you to stand on one of these buttons. Well done, we’re making great progress here. Now let’s do it again. Oh, come on - don’t stand on the same button! Move, people! No, no, that button’s only for the Astronauts, you know who you are. What?! You say you can’t do everything I ask? Ok let’s start over. You there, the Programmer, figure out how many times we can do this. And make it quick, we have a lot more science to get through…

-----Input-----
There will be a single test case in the input. The first line of this case will contain $n$ ($2 \le n \le 80$) giving the number of people (and the number of buttons) in the experiment. The next $n$ lines will contain $n$ characters each. If the $j^{th}$ character of the $i^{th}$ line is Y it indicates that the $i^{th}$ person can stand on the $j^{th}$ button (it is N otherwise).

-----Output-----
Output $k$, the maximum number of times everyone can be standing on buttons such that nobody stands on the same button more than once (This might be $0$). After that, output $k$ lines. Each line should contain $n$ integers separated by single spaces, where the $i^{th}$ integer describes which person is standing on the $i^{th}$ button. All of the lines should be valid and none of them should put the same person on the same button as a previous line of the same test case. Note that correct outputs might not be unique.

-----Examples-----
Sample Input:
3
YYY
NYY
YNY
Sample Output:
2
1 2 3
3 1 2
user: It is often heard that the apple doesn’t fall far from the tree. But is that really so?

The National Statistics Department has tracked the falling of apples in a fruit garden for $G$ consecutive years. The fruit garden can be represented as a matrix with dimensions $R \times S$. Each field of the matrix can contain more than one apple tree.

Interestingly enough, each year there was exactly one apple fall, so the Department decided to write down $G$ pairs of numbers $(r_ i, s_ i)$ that denote the row and column of the location where the apple fell during the $i$-th year. Moreover, by next year, a new tree grew at that location.

Your task is to determine the squared distance between the nearest tree and the apple that fell, measured in unit fields of the matrix (we assume it is that tree from which the apple fell).

The distance between fields $(r_1, s_1)$ and $(r_2, s_2)$ in the matrix are calculated as\begin{equation*} d((r_1, s_1),(r_2, s_2)) = \sqrt {(r_1-r_2)^2 + (s_1-s_2)^2} \end{equation*}

-----Input-----
The first line of input contains two integers, $R$ and $S$ ($1 \leq R, S \leq 500$), the number of rows and columns of the matrix.

Each of the following $R$ lines contains $S$ characters ’x’ or ’.’. The character ’.’ denotes an empty field, and the character ’x’ denotes a field with at least one tree.

The fruit garden will initially contain at least one tree.

After that, an integer $G$ ($1 \leq G \leq 10^5$) follows, the number of years the fruit garden has been under observation.

Each of the following $G$ lines describes the falls of the apples. Each line contains a pair of space-separated integers $r_ i$, $s_ i$ that denote the row and column of the location where the apple fell in the $i$-th year ($1 \leq r_ i \leq R$, $1 \leq s_ i \leq S$).

-----Output-----
Output $G$ numbers, the required squared distances from the task, each in its own line.

-----Examples-----
Sample Input:
3 3
x..
...
...
3
1 3
1 1
3 2
Sample Output:
4
0
5
user: João wants to join the robotic football team of his university. However, since he knows little about robotics and mathematics, he decided to build a $2$-dimensional robotic arm to bootstrap his knowledge.

The robotic arm is composed of $N$ segments of various lengths. The segments can form any angle between them, including configurations that make it appear to self-intersect when viewed from above. The robotic arm works great, but it is not trivial to position the arm’s tip as close as possible to given $x$, $y$ target coordinates with so many joints to control. Can you help João?

-----Task-----
Given the robotic arm description and target coordinates relative to the arm’s origin, calculate a configuration that places the arm’s tip as close as possible to the target.

-----Input-----
The first line contains $N$, the number of segments composing the robotic arm. $N$ lines follow, each with an integer $L_ i$ describing the length of the $i$th segment from the fixed point until the arm’s tip. There is one more line with $2$ integers: the $x$, $y$ coordinates of the target point to reach.

-----Constraints-----

$1$

$\leq $

$N$

$\leq $

$20$

Number of segments in the robotic arm

$1$

$\leq $

$L_ i$

$\leq $

$1000$

Length of the $i$th segment

$-20000$

$\leq $

$x, y$

$\leq $

$20000$

Target coordinates to attempt to reach

-----Output-----
The output should contain $N$ lines, each containing two real numbers $x_ i$, $y_ i$ indicating the coordinates of the tip of the $i$th segment.

The length of the $i$th segment computed from the solution and input $L_ i$ may not differ by more than $0.01$. Similarly, the absolute error between the solution’s distance to the target and the minimum possible distance to the target cannot exceed $0.01$.

Note that, in general, there are many solutions. Your program may output any of them.

-----Explanation for Sample Input 1-----
Sample output 1 corresponds to the drawing on the left:



The drawing on the right corresponds to the following, equally acceptable output for sample input 1:

3.923 3.100
1.118 2.037
5.000 3.000


-----Explanation for Sample Input 2-----

-----Examples-----
Sample Input:
3
5
3
4
5 3
Sample Output:
4.114 -2.842
6.297 -0.784
5.000 3.000
user: The Next Wave Energy Research Club is looking at several atoms as potential energy sources, and has asked you to do some computations to see which are the most promising. 

Although an atom is composed of various parts, for the purposes of this method only the number of neutrons in the atom is relevant1. In the method, a laser charge is fired at the atom, which then releases energy in a process formally called explodification. Exactly how this process proceeds depends on the number of neutrons $k$:
 - If the atom contains $k \leq n$ neutrons, it will be converted into $a_ k$ joules of energy.
 - If the atom contains $k > n$ neutrons, it will decompose into two atoms with $i$ and $j$ neutrons respectively, satisfying $i,j \geq 1$ and $i+j=k$. These two atoms will then themselves explodificate.

When an atom with $k$ neutrons is explodificated, the total energy that is released depends on the exact sequence of decompositions that occurs in the explodification process. Modern physics is not powerful enough to predict exactly how an atom will decompose—however, for explodification to be a reliable energy source, we need to know the minimum amount of energy that it can release upon explodification. You have been tasked with computing this quantity.

-----Input-----
The input consists of:
 - One line with two integers $n$ and $q$ ($1 \leq n \leq 100$, $1 \leq q \leq 10^5$), the neutron threshold and the number of experiments.
 - One line with $n$ integers $a_1,\ldots ,a_ n$ ($1 \leq a_ i \leq 10^9$ for each $i$), where $a_ i$ is the amount of energy released when an atom with $i$ neutrons is explodificated.
 - Then $q$ lines follow, each with an integer $k$ ($1 \leq k \leq 10^9$), asking for the minimum energy released when an atom with $k$ neutrons is explodificated.

-----Output-----
For each query $k$, output the minimum energy released when an atom with $k$ neutrons is explodificated.

-----Examples-----
Sample Input:
4 5
2 3 5 7
2
3
5
6
8
Sample Output:
3
5
8
10
13
user: You have all heard of the story of the three little pigs. However, this time they’re facing another predicament yet again!

The three pigs brought a bunch of their pig friends into a tree, which is a connected undirected graph with $V$ vertices and $V-1$ edges. Originally there were $V$ pigs, one on each vertex of the tree. However, $V-P$ big bad wolves suddenly teleported into $V-P$ distinct vertices of $T$ and ate the pigs originally on these vertices. Now there are only $P$ pigs left, and every vertex of $T$ is occupied by a single pig or a single wolf (but not both).

Of course, the surviving pigs now wants to escape. A pig is considered to have escaped if he is on any leaf vertex (a vertex connected to only one edge). A pig currently on vertex $u$ can move to vertex $v$ only if there is an edge connecting $u$ to $v$ and $v$ isn’t occupied by a wolf. However, more than one pig may be on the same vertex at the same time. The pigs want you to help them compute the minimum number of wolves which must be removed such that every pig can escape.

-----Input-----
The first line of the input contains two integers $V$ and $P$, $3\leq P\leq V\leq 200000$. This is followed by $V-1$ lines, each containing two integers $u$ and $v$, where $0\leq u,v\leq V-1$, indicating that there is an edge between vertex $u$ and vertex $v$. It is guaranteed that the graph represented by the input is a tree.

The last line of the input contains $P$ integers, where $p_ i$ ($0\leq p_ i\leq V-1$) denotes the initial vertex occupied by the $i^\mathrm {th}$ pig. It is guaranteed that no two pigs occupy the same vertex.

-----Output-----
Output $W$, an integer denoting the minimum number of wolves to remove such that every pig can escape.

-----Examples-----
Sample Input:
6 3
0 1
1 2
2 3
2 4
1 5
1 2 5 
Sample Output:
0
user: An alphabetical string is a string consisting of $0$ or more capital letters (i.e. [‘A’..‘Z’]). Given an alphabetical string $S[1..N]$, determine the number of palindromic alphabetical strings of length $2N$ that contains $S$ as a subsequence (not necessarily contiguous). A string is palindromic if it is equal to its reverse.

-----Input-----
The first line of input is an integer representing $N$, constrained to $0 \leq N \leq 200$.

The second line of input is an alphabetical string $S$ of length $N$.

-----Output-----
Output the number of palindromic alphabetical strings of length $2N$ containing $S$ as a subsequence. As this could be rather large, output it modulo $10^9+7$.

-----Examples-----
Sample Input:
2
AA
Sample Output:
51