meetforever/GRPO_CL huggingface dataset:
user: You are a member of the Senate of an ancient empire governed by a mighty dictator. You have joined a bipartisan secret committee of the Senate that is plotting to overthrow the dictator.

In order for the plot to succeed, it is crucial that all the states in the empire ultimately support the plan–and to accomplish this, the governors of all the states need to be members of the same Party.

Right now, each state governor is a member of either the Orange Party or the Purple Party. Since you are confident that you can get either party to back the plot, it does not matter which party ultimately prevails.

The secret committee has studied the political situation and determined that two governors will influence each other if they are friends with each other and members of the same Party. To get all the state governors on board, each month a lobbyist will do whatever it takes to get one governor to switch parties. When this happens, all the friends of the governor who are members of the same Party will also switch affiliation, as will the friends of the friends within the party, and so on. To avoid suspicion, the secret committee will alternate Orange/Purple lobbyists each month. They may start the ball rolling with either party in the first month.

The secret committee also knows which governors are friends with each other, that each governor is friends with at least one other governor, and that there are no isolated groups that are only friends with each other.

Your task is to determine the minimum number of months required for all the state governors to be members of the same party. Once this occurs, the next steps in the plot can take place.

-----Input-----
There will be a single test case in the input. This test case will begin with a line with two integers, $n$ ($1 \le n \le 100$) and $m$ ($n-1 \le m \le n(n-1)/2$), where $n$ is the number of governors, and $m$ is the number of known friendships. On the next line will be $n$ integers, either $0$ or $1$, indicating the current party affiliation of the governors, in order ($0=\text {ORANGE}, 1=\text {PURPLE}$). On each of the following $m$ lines will be two integers, $a$ and $b$ ($1 \le a < b \le n$) indicating that governor $a$ and governor $b$ are friends. As in life, friendships go both ways: if $a$ is a friend of $b$, then $b$ is also a friend of $a$. All $m$ $(a,b)$ pairs will be unique.

-----Output-----
Output a single integer, indicating the minimum number of months necessary for every governor to belong to the same party.

-----Examples-----
Sample Input:
4 3
0 1 0 0
1 2
2 3
2 4
Sample Output:
1
user: After The Stig’s identity was revealed, the TV show Top Gear is in dire need of a new, tame racing driver to replace him. And of course you have been asked to take the job. However, you are not very fond of driving quickly, and especially not around the twisting and turning tracks they use in the show. To help you alleviate this problem, one of your algorithmic friends has suggested that you should calculate the roundtrip with the least possible amount of turning required.

The driving track consists of unique, straight lines, and there are always exactly $2$ or $4$ roads heading out from each node. A roundtrip must be an Eulerian circuit, i.e. it must traverse each edge of the graph exactly once, and end up where it started. (Such a circuit is guaranteed to exist in the input graphs.) Thus the total amount of turning is the sum of the turning required at each node, where continuing in a straight line means a turn of $0$. The roads on the track can be driven in any direction.

-----Input-----
One line with $3 \leq N \leq 10000$ – the number of nodes – and $N \leq M \leq 2N$ – the number of edges.

$N$ lines with the integer $x$ and $y$ coordinates of each node, in order. $0 \leq x, y \leq 10000$. The nodes have unique coordinate pairs.

$M$ lines with two space separated integers $i$ and $j$, denoting an edge between nodes $i$ and $j$. The nodes are $0$-indexed.

-----Output-----
The least amount of turning you must do to complete an Eulerian circuit, in radians. The answer must be correct within an absolute or relative error of $10^{-6}$.

-----Examples-----
Sample Input:
3 3
0 0
0 1
1 0
0 1
0 2
1 2
Sample Output:
6.283185
user: A complete binary tree is made of nodes arranged in a hierarchic structure. One of the nodes is the root node, said to be at level 0. The root node has two child nodes, which are at level 1. Each of those has two children at level 2 etc.

In general, a complete binary tree with $N$ levels has $2^N - 1$ nodes, each of which has two child nodes, except those at level $N - 1$.

A number can be written into each node. Write the numbers $1$ to $2^N - 1$ into a complete binary tree with $N$ levels so that, for each node at level $D$, the absolute value of the difference of the sum of all numbers in the left subtree and the sum of all numbers in the right subtree is $2^D$.

For example, the sum of the left subtree of the root node must differ from the sum of the right subtree by 1. The sums of the left and right subtrees of a node at level 1 must differ by 2. Each number must be used exactly once. The solution need not be unique.

-----Input-----
The first and only line of input contains the integer $N (1 \le N \le 15)$, the number of levels in the tree.

-----Output-----
Output the $2^N - 1$ separated by spaces on a single line, the binary tree in the preorder traversal. The preorder traversal first outputs the number in the root node, then outputs the left subtree (again in the preorder traversal), then the right subtree.

-----Examples-----
Sample Input:
2
Sample Output:
1 2 3
user: Being a traveling salesman is tough work. Per is one such salesman and would like to find an efficient way to visit all the cities in a foreign country exactly once.

Per defines efficiency in a peculiar way as Per hates flying on planes. Even worse, he absolutely refuses to use automobiles. Per’s favorite mode of transportation is trains. He will gladly take trains for as long as possible.

The train system in this country is very peculiar, and very limited. Train lines are all one-way, and once anyone takes a train out of a city, there is no sequence of train lines that return to that city. This is because the country is trying to make money off of the more costly planes. In this country, every city has exactly one airport, so you can travel by plane from any city to any other city.

Per doesn’t just want to know the minimum number of flights he needs. He also wants to know in which cities he can visit the airport during some trip with fewest flights. Per likes airport restaurants, you see, and would like to know which restaurants he can visit, so he can choose his route to visit his favorites. He can visit the airport if he flies in or out of the city. Note that Per can start in any city.

Consider this country with four cities, with the arrows representing one-way train routes:

There are several possible trips Per could take, but he’s going to need to fly at least once. Here are some (but not all) possible routes with fewest flights, with $\rightarrow $ indicating a train trip and $\Rightarrow $ indicating a flight:$1 \rightarrow 2 \rightarrow 4 \Rightarrow 3$

$2 \rightarrow 4 \Rightarrow 1 \rightarrow 3$

$1 \rightarrow 3 \rightarrow 4 \Rightarrow 2$


In this example, every airport is visited on at least one of the routes. Per has the option to choose his route so he can visit any airport restaurant he wishes.



-----Input-----
Each test case will begin with a line with two space-separated integers $n$ ($1\! \le \! n\! \le \! 10^5$) and $m$ ($0\! \le \! m\! \le \! 10^5$), where $n$ is the number of cities and $m$ is the number of train lines. The cities are numbered $1..n$.

Each of the next $m$ lines contains two space separated integers $a$ and $b$ ($1\! \le \! a,b\! \le \! n, a\! \neq \! b$), which indicates that there is a train line from city $a$ to city $b$ (but not back). All train lines will be distinct.

-----Output-----
Produce exactly two lines of output.

On the first line, output a single integer, which is the minimum number of flights Per must take to visit all of the cities.

On the second line, output a list of space-separated integers, which are the cities with airports he can visit. If he can visit an airport on any one of the routes with the minimum number of flights, it should be listed. Output these numbers in increasing order. If no airports are to be visited, output a blank line.

-----Examples-----
Sample Input:
4 4
1 2
1 3
2 4
3 4
Sample Output:
1
1 2 3 4
user: For an undirected, simple graph $G = (V, E)$ we call a subset $V’ \subseteq V$ an independent set if no two elements of $V’$ are connected by an edge. An independent set of $G$ is called a maximum independent set if there is no independent set in $G$ with strictly more vertices. Given a specific kind of connected graph $G$, find the size of a maximum independent set of $G$.

-----Input-----
 - The input starts with one line, containing integers $n$ ($1 \leq n \leq 100$), the number of vertices in the graph, and $m$ ($n-1 \leq m \leq n + 15$), the number of edges in the graph.
 - Then follow $m$ lines, each containing integers $a, b$ ($1 \leq a, b \leq n$) indicating that there is an edge between vertices $a$ and $b$.

The graph given by this input is guaranteed to be both simple and connected: there is at most one edge between each pair of vertices, there are no loops, and there is a path between each pair of vertices.

-----Output-----
 - Output the number of vertices in a maximum independent set of the input graph.

-----Examples-----
Sample Input:
2 1
1 2
Sample Output:
1
user: Mr. Lucky has a store that sells numbers. These numbers have an interesting property: each number formed by its first $k$ digits is evenly divisible by $k$, for $k$ from $1$ to $n$, where $n$ is the number of digits in the number. The numbers do not have leading zeroes.

Mr. Unlucky wants to open a competing store. Price for lucky numbers is driven by demand and supply, and given by the formula\[ \mbox{price} = \frac{\mbox{demand}}{\mbox{supply}} \]

while demand for numbers with $n$ digits is given by the formula\[ \mbox{demand} = \mbox{citySize} \cdot \mbox{dayOfMonth} - n^e \]

where $e$ is the base of the natural logarithm. Supply for lucky numbers with $n$ digits is simply the number of lucky numbers with $n$ digits. Help Mr. Unlucky calculate the supply for $n$ digit lucky numbers.

-----Input-----
The input is a single integer $n$.

-----Output-----
Output the supply for $n$ digit lucky numbers.

-----Limits-----
 - $2 \leq n \leq 1000$

-----Examples-----
Sample Input:
2
Sample Output:
45
user: Rookie Bunny is helping the motor vehicle department to renew licenses for its customers. She works at the ticket machine and gives tickets to customers to tell them at which counters they will be served. There are two counters providing service in the office today. The two counters will be open for at most $s_1$ and $s_2$ minutes respectively.

There are $n$ customers waiting in a line in front of the ticket machine. Every customer can be served at either counter. The $j$th customer needs $t_ j$ minutes at a counter. If neither counter has $t_ j$ minutes till its closing time, then the customer has to leave and come back tomorrow. Besides, all customers in the line behind the $j$th customer have to leave to respect the order of the line and avoid customer complaints.

Rookie Bunny may assign each customer in the line to either counter of her choice. She would like to maximize the number of customers that can be served today.

-----Input-----
The first line has three positive integers $n$, $s_1$, and $s_2$ ($n \leq 1000, s_1 \cdot s_2 \leq 10^{7}$). The second line has $n$ integers. The $j$th integer is $t_ j$ ($1 \leq t_ j \leq 10^6$).

-----Output-----
Output the maximum number of customers that can be served today.

-----Examples-----
Sample Input:
5 20 20
7 11 9 12 2
Sample Output:
4
user: A couple of computer science students have decided to start a book circle, in which they will be able to share the passion of their lives, namely reading. Finally, the first season of reading is coming to its end, and the students in the book circle are going to present their fantastic books to the rest of the computer science students. However, there is a problem. To attract audience to the presentation, one of the book circle members (without mentioning any names) has promised the audience that a book circle member will sing in between each two presentations. The person who made this promise doesn’t know how to sing, so he or she must have been thinking that someone else in the group could do the singing. But in fact, no one in the group wants to sing. The book circle always sticks to its promises, which means there is some singing to do. However, they have also promised to present all of the books that they have read. Someone in the group has figured out that if they minimize the number of presentations – they will minimize the number of times they have to sing, since if a student has read several books, he or she can present a bunch of them in the same presentation. But how should this be carried out?

Your task is to help the book circle calculate the minimum number of presentations needed such that all books that have been read will be presented. You will be given the number of boys in the group, and the number of girls in the group. You will also be given the book titles read by each student. Each book that was read was read by exactly one girl and one boy.

Students always present books alone, and they can present only the books that they have read.

-----Input-----
The first line consists of two integers $1 \le B,G \le 1000$, the number of boys and girls respectively. Then follow $B$ lines, each starting with the name of the $i$:th boy. Then follows an integer $N_ i$, the number of books that the $i$:th boy has read. Then follow $N_ i$ book titles on the same line. Then follow $G$ lines, each starting with the name of the $i$:th girl. Then follows an integer $N_ i$, the number of books that the $i$-th girl has read. Then follow $N_ i$ book titles on the same line.

The book titles and student names consist solely of alphanumeric characters, and they don’t have spaces in them. They both have between $1$ and $20$ characters. Also, no two students will share names.

-----Output-----
Output should start with a single number $P$, the minimum number of presentations needed.

-----Examples-----
Sample Input:
2 2
kenny 1 harrypotter1
charlie 1 lordoftherings
jenny 1 harrypotter1
laura 1 lordoftherings
Sample Output:
2
user: Goo is working in a well-known Slovak antivirus company which unfortunately cannot be named. In addition to antivirus software, they are developing a spam filter. Recently, Goo has made a few improvements to the filter and he wants to demonstrate his progress to his boss. As you can imagine, demonstrating low-level ideas of your implementation is not a good way to impress your boss, so Goo instead decided to make a presentation with plenty of graphs showing filtering results. The company has a huge database of e-mails and each e-mail is marked as a spam or ham (i.e. not spam). These e-mails were all correctly marked by people – every time someone in the company receives an e-mail, he marks it as either spam or ham and adds it to the database.

The success of Goo’s program can be measured in a simple way. Goo ran his program on all e-mails in the database. For each message he noted if his program correctly decided whether the message was spam or ham. The messages were processed in order from the oldest to the newest one. To impress the boss, Goo wants to select e-mails from a period of time and calculate the success rate only for this period. Of course, a period containing only one e-mail won’t impress anyone, so Goo wants to choose a period which is long enough.

-----Task-----
You are given a sequence of test results and a number $k$. Your task is to find a continuous subsequence of length at least $k$ which has the highest possible success rate among all such subsequences. The success rate of a subsequence is defined as the number of successfully classified e-mails divided by the length of the subsequence.

-----Input description-----
On the first line there is an integer $k$ ($1\le k \le 100$) denoting the minimal subsequence length. The second line contains a string consisting of characters 0 and 1, denoting answers of the program for each e-mail in the database. Number 1 indicates that Goo’s program gave a correct answer and 0 that it failed. The length of the string will be at least $k$ and at most $100000$ characters.

-----Output description-----
The first and only line of output should consist of two integers $f$ and $\ell $, separated by a single space. The integer $f$ is the 1-based index of the first element of subsequence with the best success rate and $\ell $ is its length. If there are multiple optimal solutions, you can output any one of them.

-----Examples-----
Sample Input:
1
01
Sample Output:
2 1
user: You may be under the impression that treasure maps originate from pirates, who create maps after burying their treasure so that they will later have a way to retrieve their stolen fortunes. In reality, it is extremely rare for pirates to bury treasure; there are only a few such documented cases.

As it turns out, the vast majority of treasure maps are published by the Mapping Association of Plunder and Spoils (MAPS), a secretive group of people responsible for curating reports of lost or stolen treasure and then making and selling corresponding treasure maps.

Over the years, MAPS has become increasingly creative with the structure of its treasure maps, so these maps no longer resemble a stereotypical treasure map. In fact, they no longer come in one piece; treasure hunters must assemble a map from multiple pieces before being able to use it to locate buried treasure.

Formally, an assembled treasure map is a rectangular grid of squares, where each square indicates the rectilinear distance (sum of horizontal distance and vertical distance) to the square containing the treasure, modulo $10$.

Given a list of smaller rectangular map pieces, you must rearrange the pieces to reconstruct a valid rectangular map, rotating (but not flipping) the pieces as needed. Every map piece must be used exactly once in the reconstructed map, no two map pieces may overlap, and the reconstructed map must not contain any gaps.

-----Input-----
The first line contains an integer, $N$ ($2\leq N\leq 8$), indicating the number of rectangular map pieces. This is followed by a series of lines describing each map piece. The description of the $i$-th piece ($1\leq i\leq N$) begins with a line containing two space-separated integers, $W_ i$ and $H_ i$, giving the $i$-th piece’s width and height, respectively, where $1\leq W_ i, H_ i\leq 10$. This is followed by $H_ i$ lines of $W_ i$ characters each, where each character represents one square of the map piece (using a digit from $0$ to $9$, inclusive). It is guaranteed that one of the map pieces contains a square corresponding to the location of the treasure.

-----Output-----
Output a line containing $W$ and $H$, separated by a single space, where $W$ is the width and $H$ is the height of a valid reconstructed map (such a reconstruction will always be possible). Then output the reconstructed map as $H$ lines of $W$ characters each. (In the event that there are multiple possible maps that can be produced, it suffices to output any one of them, in any of the $4$ possible rotations.) On the following line, output $W$ hyphen (‘-’) characters. Then output $H$ lines of $W$ characters each, where each character is a digit between $1$ and $N$, inclusive, that is the ($1$-based) index of the map piece to which the corresponding square in the reconstructed map belongs. (For example, the digit in the bottom-left corner of the first sample output is $1$ because the corresponding digit in the reconstructed map (which happens to be $2$) belongs to the $1$st map piece.)

-----Examples-----
Sample Input:
3
4 1
2123
2 2
21
10
2 2
23
12
Sample Output:
4 3
2123
1012
2123
----
2233
2233
1111
user: Modern operating system distributions have tools to manage installed software, making it easy to keep the software up-to-date. Putting different pieces of software into ‘packages’ that can be installed separately keeps things simple and avoids duplicate effort and code. But this means that a package may depend on other packages, requiring their installation before that package can be installed. For example, many programs need to use ‘libc,’ which contains standard C-library functions. To keep track of these dependencies, many distributions use some sort of package manager.

When a user wants to install a new package (or set of packages), a package manager takes care of the headache of tracking down which packages are required to install the desired package. Of course, those may also depend on other packages.

For this problem, determine an order (if one exists) that allows a given list of packages to be installed. No package should be installed before all the packages it depends on are installed. You may assume that, at the beginning, no packages have been installed.

-----Input-----
Input consists of up to $10$ test cases. Each test case start with a number $1 \le n \le 1000$, which is the number of packages the user wants to install. This is followed by $n$ lines which describe $n$ packages. Each package description starts with the name of the package and is followed by a space-separated list of the package’s unique dependencies. Each package has at most $20$ dependencies, and each is one of the other $n-1$ packages. Each package name is a string of up to $40$ non-whitespace characters using the English alphabet (a-z, A-Z), digits (0-9), as well as the characters _, -, ., and + (i.e. underscore, minus, period, and plus). Input ends when $n$ is zero.

-----Output-----
For each test case, output the order of package installation that allow them all to be installed after their respective dependencies. If there are multiple possible orderings, then give the ordering that is lexicographically first (using ASCII values for string ordering). If there is some group of packages that are not able to be ordered within the list, output ‘cannot be ordered’ instead of ordering the packages. Put a blank line between each pair of test cases.

-----Examples-----
Sample Input:
14
libattr
vim-X11 vim-common gtk2 libattr
vim-common
gtk2 libtiff atk pango glib2
libtiff zlib libjpeg
atk
pango xorg-x11-libs freetype glib2
glib2
zlib
libjpeg
xorg-x11-libs grep freetype
grep pcre
pcre
freetype
3
emacs xorg-x11 lisp
xorg-x11
lisp emacs
0
Sample Output:
atk
freetype
glib2
libattr
libjpeg
pcre
grep
vim-common
xorg-x11-libs
pango
zlib
libtiff
gtk2
vim-X11

cannot be ordered
user: My birthday is coming up. Alas, I am getting old and would like to feel young again. Fortunately, I have come up with an excellent way of feeling younger: if I write my age as a number in an appropriately chosen base $b$, then it appears to be smaller. For instance, suppose my age in base $10$ is $32$. Written in base $16$ it is only $20$!

However, I cannot choose an arbitrary base when doing this. If my age written in base $b$ contains digits other than $0$ to $9$, then it will be obvious that I am cheating, which defeats the purpose. In addition, if my age written in base $b$ is too small then it would again be obvious that I am cheating.

Given my age $y$ and a lower bound $\ell $ on how small I want my age to appear, find the largest base $b$ such that $y$ written in base $b$ contains only decimal digits, and is at least $\ell $ when interpreted as a number in base $10$.

-----Input-----
The input consists of a single line containing two base 10 integers $y$ ($10 \le y \le 10^{18}$ – yes, I am very old) and $\ell $ ($10 \le \ell \le y$).

-----Output-----
Display the largest base $b$ as described above.

-----Examples-----
Sample Input:
32 20
Sample Output:
16