meetforever/GRPO_CL huggingface dataset:
user: You have recently been hired as an architect for the BAPC (Bureau of Architecture and Promising Constructions), responsible for top-quality buildings such as the Tower of Pisa. However, in the past couple of weeks, some of the structures that the BAPC have made have collapsed! It is up to you to figure out whether any other structures are in danger.

After some research it seems like the $x$-coordinate of the centers of gravity of some of the structures are off: if the center of gravity is too much to the left or to the right, the structure will fall over. Hence, you decide to check all the blueprints and see whether the structures are stable or not.

Given is a grid of up to size $100$ by $100$, using characters in .#/\_|-. The $\texttt{.}$ characters denote empty space, while each other character represents a completely filled $1\times 1$ box (any difference in symbols used is due to the artistic freedom of the other architects), whose center of mass is at the center of the box.

Every structure forms a single connected component that touches the ground, i.e. the bottom layer of the grid.

The structure falls to the left if the $x$-coordinate of the center of gravity is less than the $x$-coordinate of the leftmost point of the structure that touches the ground, and it falls to the right if the $x$-coordinate of the center of gravity is larger than the $x$-coordinate of the rightmost point of the structure that touches the ground. It is guaranteed that the center of gravity is never exactly above the leftmost or rightmost point where the building touches the ground.

Given a blueprint, is the structure balanced, does it fall to the left, or does it fall to the right?

-----Input-----
 - The first line has $1\leq h \leq 100$ and $1\leq w\leq 100$, the height and width of the grid.
 - Then follow $h$ lines with $w$ characters each. Each character is either ., indicating empty space, or one of #/\_|-, indicating a filled $1\times 1$ box .

-----Output-----
 - Print a single line containing left, balanced, or right.

-----Examples-----
Sample Input 1:
3 3
/-\
|.|
#.#
Sample Output 1:
balanced

Sample Input 2:
3 3
...
___
..|
Sample Output 2:
left

Sample Input 3:
3 3
./\
.\/
.|.
Sample Output 3:
balanced
user: Musicians of a popular band "Flayer" have announced that they are going to "make their exit" with a world tour. Of course, they will visit Berland as well.

There are n cities in Berland. People can travel between cities using two-directional train routes; there are exactly m routes, i-th route can be used to go from city v_{i} to city u_{i} (and from u_{i} to v_{i}), and it costs w_{i} coins to use this route.

Each city will be visited by "Flayer", and the cost of the concert ticket in i-th city is a_{i} coins.

You have friends in every city of Berland, and they, knowing about your programming skills, asked you to calculate the minimum possible number of coins they have to pay to visit the concert. For every city i you have to compute the minimum number of coins a person from city i has to spend to travel to some city j (or possibly stay in city i), attend a concert there, and return to city i (if j ≠ i).

Formally, for every $i \in [ 1, n ]$ you have to calculate $\operatorname{min}_{j = 1} 2 d(i, j) + a_{j}$, where d(i, j) is the minimum number of coins you have to spend to travel from city i to city j. If there is no way to reach city j from city i, then we consider d(i, j) to be infinitely large.


-----Input-----

The first line contains two integers n and m (2 ≤ n ≤ 2·10^5, 1 ≤ m ≤ 2·10^5).

Then m lines follow, i-th contains three integers v_{i}, u_{i} and w_{i} (1 ≤ v_{i}, u_{i} ≤ n, v_{i} ≠ u_{i}, 1 ≤ w_{i} ≤ 10^12) denoting i-th train route. There are no multiple train routes connecting the same pair of cities, that is, for each (v, u) neither extra (v, u) nor (u, v) present in input.

The next line contains n integers a_1, a_2, ... a_{k} (1 ≤ a_{i} ≤ 10^12) — price to attend the concert in i-th city.


-----Output-----

Print n integers. i-th of them must be equal to the minimum number of coins a person from city i has to spend to travel to some city j (or possibly stay in city i), attend a concert there, and return to city i (if j ≠ i).


-----Examples-----
Input
4 2
1 2 4
2 3 7
6 20 1 25

Output
6 14 1 25 

Input
3 3
1 2 1
2 3 1
1 3 1
30 10 20

Output
12 10 12
user: You are given a string, s, and a list of words, words, that are all of the same length. Find all starting indices of substring(s) in s that is a concatenation of each word in words exactly once and without any intervening characters.

Example 1:


Input:
s = "barfoothefoobarman",
words = ["foo","bar"]
Output: [0,9]
Explanation: Substrings starting at index 0 and 9 are "barfoor" and "foobar" respectively.
The output order does not matter, returning [9,0] is fine too.


Example 2:


Input:
s = "wordgoodstudentgoodword",
words = ["word","student"]
Output: []

Answer: {"fn_name": "findSubstring", "inputs": [["\"barfoothefoobarman", ["\"foo","\"bar\""]], ["\"wordgoodgoodgoodbestword\"", ["\"word\"","\"good\"","\"best\"","\"word\""]], ["\"barfoofoobarthefoobarman\"", ["\"bar\"","\"foo\"","\"the\""]]], "outputs": [[0,9], [], [6,9,12]]}
user: Compare two version numbers version1 and version2.
If version1 > version2 return 1; if version1 < version2 return -1;otherwise return 0.

You may assume that the version strings are non-empty and contain only digits and the . character.
The . character does not represent a decimal point and is used to separate number sequences.
For instance, 2.5 is not "two and a half" or "half way to version three", it is the fifth second-level revision of the second first-level revision.

Example 1:


Input: version1 = "0.1", version2 = "1.1"
Output: -1

Example 2:


Input: version1 = "1.0.1", version2 = "1"
Output: 1

Example 3:


Input: version1 = "7.5.2.4", version2 = "7.5.3"
Output: -1

Answer: {"fn_name": "compareVersion", "inputs": [["\"1.01\"", "\"1.001\""], ["\"1.0\"", "\"1.0.0\""], ["\"1.0.1\"", "\"1\""]], "outputs": [0, 0, 1]}
user: Given an array of integers nums sorted in ascending order, find the starting and ending position of a given target value.

Your algorithm's runtime complexity must be in the order of O(log n).

If the target is not found in the array, return [-1, -1].

Example 1:


Input: nums = [5,7,7,8,8,10], target = 8
Output: [3,4]

Example 2:


Input: nums = [5,7,7,8,8,10], target = 6
Output: [-1,-1]

Answer: {"fn_name": "searchRange", "inputs": [[[5,7,7,8,8,10], [8]], [[5,7,7,8,8,10], [6]], [[], [0]]], "outputs": [[3,4],[-1,-1],[-1,-1]]}
user: Given a string containing digits from 2-9 inclusive, return all possible letter combinations that the number could represent.

A mapping of digit to letters (just like on the telephone buttons) is given below. Note that 1 does not map to any letters.



Example:


Input: "23"
Output: ["ad", "ae", "af", "bd", "be", "bf", "cd", "ce", "cf"].


Note:

Although the above answer is in lexicographical order, your answer could be in any order you want.

Answer: {"fn_name": "letterCombinations", "inputs": [["\"23\""], ["\"\""], ["\"2\""]], "outputs": [[["\"ad\"", "\"ae\"", "\"af\"", "\"bd\"", "\"be\"", "\"bf\"", "\"cd\"", "\"ce\"", "\"cf\""]],[[]],[["\"a\"","\"b\"","\"c\""]]]}
user: We have a grid with H rows and W columns of squares.
Snuke is painting these squares in colors 1, 2, ..., N.
Here, the following conditions should be satisfied:
 - For each i (1 ≤ i ≤ N), there are exactly a_i squares painted in Color i. Here, a_1 + a_2 + ... + a_N = H W.
 - For each i (1 ≤ i ≤ N), the squares painted in Color i are 4-connected. That is, every square painted in Color i can be reached from every square painted in Color i by repeatedly traveling to a horizontally or vertically adjacent square painted in Color i.
Find a way to paint the squares so that the conditions are satisfied.
It can be shown that a solution always exists.

-----Constraints-----
 - 1 ≤ H, W ≤ 100
 - 1 ≤ N ≤ H W
 - a_i ≥ 1
 - a_1 + a_2 + ... + a_N = H W

-----Input-----
Input is given from Standard Input in the following format:
H W
N
a_1 a_2 ... a_N

-----Output-----
Print one way to paint the squares that satisfies the conditions.
Output in the following format:
c_{1 1} ... c_{1 W}
:
c_{H 1} ... c_{H W}

Here, c_{i j} is the color of the square at the i-th row from the top and j-th column from the left.

-----Sample Input-----
2 2
3
2 1 1

-----Sample Output-----
1 1
2 3

Below is an example of an invalid solution:
1 2
3 1

This is because the squares painted in Color 1 are not 4-connected.
user: Consider a sequence $A$ of integers, containing $N$ integers between $1$ and $N$. Each integer appears exactly once in the sequence.

A subsequence of $A$ is a sequence obtained by removing some (possibly none) numbers from the beginning of $A$, and then from the end of $A$.

Calculate how many different subsequences of $A$ of odd length have their median equal to $B$. The median of a sequence is the element in the middle of the sequence after it is sorted. For example, the median of the sequence $(5, 1, 3)$ is $3$.

-----Input-----
The first line contains two integers, $N$ ($1 \le N \le 100000$) and $B$ ($1 \le B \le N$).

The second line contains $N$ integers separated by spaces, the elements of sequence $A$.

-----Output-----
Output the number of subsequences of $A$ whose median is $B$.

-----Explanation of Sample Input-----
In Sample Input 3, the four subsequences of $A$ with median $4$ are $(4)$, $(7, 2, 4)$, $(5, 7, 2, 4, 3)$ and $(5, 7, 2, 4, 3, 1, 6)$.

-----Examples-----
Sample Input 1:
5 4
1 2 3 4 5
Sample Output 1:
2

Sample Input 2:
6 3
1 2 4 5 6 3
Sample Output 2:
1
user: Mirko’s great grandmother Katica is an avid mathematician. She likes to torment Mirko with math games. This time she wrote down a sequence of numbers on a piece of paper and told Mirko he may do the following:
 - Choose any two numbers in the sequence (let’s call them $A$ and $B$) and a prime number $X$ such that $A$ is divisible by $X$. After that, Mirko erases $A$ and writes $A/X$ in its place. In the end he erases $B$ and writes $B\cdot X$ in its place.

Mirko may perform this operation as many times he wants. His goal is to obtain the maximum possible score, because he gets candy from his great grandmother if he does so. The score for one sequence is the greatest common divisor of all the numbers in the sequence.

He is not very good at this, and he likes his candy so he has asked you to help him. Write a program that will calculate the maximum possible score. Since you are such a nice person, you should also print the smallest number of times Mirko must perform the operation to obtain the maximum possible score.

-----Input-----
The first line of input contains one integer $N$, ($1 \leq N \leq 100$), the number of elements in the starting sequence.

The second line of input contains $N$ positive integers smaller than or equal to $1000000$, the sequence Katica gave to Mirko.

-----Output-----
The one and only line of output should contain two integers. The first integer is the maximal possible score Mirko can obtain. The second integer is the smallest number of operations Mirko needs to perform to obtain it.

-----Examples-----
Sample Input 1:
3
4 4 1
Sample Output 1:
2 1

Sample Input 2:
3
8 24 9
Sample Output 2:
12 3
user: Gregory Brandon Sr. works at some startup in Redmond that owns a huge campus with a lot of office buildings. Since Gregory likes to work in different environments, he will work in exactly two buildings every day. On the $i$-th day, Gregory walks from his home to building $a_ i$ in the morning, walks from building $a_ i$ to building $b_ i$ at noon, and walks from building $b_ i$ to his home in the evening.

Gregory owns two umbrellas, but he does not like to carry them around. However, it rains a lot in Redmond, and Gregory must use an umbrella if he walks when it is raining (otherwise he will become infinitely unhappy). Gregory currently has both umbrellas at home, but he may choose to leave them in the office buildings for convenience. It will cost one unit of Gregory’s happiness every time he needs to carry an umbrella while walking, regardless of whether he carries one or two umbrellas, and regardless of whether he is using the umbrella. Luckily, Gregory can accurately predict the weather in the morning, at noon, and in the evening for up to $n$ days. Knowing the weather for the next $n$ days, how many units of happiness must Gregory lose from carrying umbrellas?

-----Input-----
The first line of the input contains two space-separated integers $n$ and $k$ ($1\le n \le 10^4$, $2\le k\le 30$), denoting the number of days and the number of buildings Gregory will ever work in. These buildings are numbered from $1$ to $k$.

Each of the next $n$ lines of input contains two space-separated integers $a_ i$ and $b_ i$ ($1\le a_ i, b_ i \le k$, $a_ i\neq b_ i$), denoting the buildings that Gregory will work in during the morning and the afternoon respectively.

Each of the next $n$ lines of input contains a string of three characters $w_ i$, representing the weather on the $i$-th day. Each character is either S, to denote that it will be sunny or R to denote that it will be rainy. The first, second, and third characters in the string $w_ i$ respectively represent the weather in the morning, at noon, and in the evening of the $i$-th day.

-----Output-----
Output the minimum number of happiness units Gregory must lose from carrying umbrellas.

-----Examples-----
Sample Input 1:
1 2
1 2
SSR
Sample Output 1:
3

Sample Input 2:
2 3
3 2
3 1
SRS
SRR
Sample Output 2:
4
user: Ivica and Marica are attending astronomy classes and are observing two unusual stars. They noted the time when each of them flashed. They further noticed that each of the stars flashes periodically, at regular intervals, and now they are wondering: on what day, at what hour will the stars flash at the same minute for the first time?

For example, if the first star flashed today (Saturday) at $02$:$20$ and the second one at $13$:$00$, with the first star flashing every $05$:$50$ (every $5$ hours and $50$ minutes) and the second every $01$:$00$ (each hour), then the first one will flash again at $08$:$10$ and $14$:$00$, and the second one at $14$:$00$. Therefore, both stars will have flashed at the same minute today at $14$:$00$.

Note that the time $00$:$00$ (midnight) is the first minute of a day.

-----Input-----
Four lines containing four timestamps in $HH$:$MM$ format (hours:minutes), with $00 \le HH \le 23$, $00 \le MM \le 59$.

The timestamps are, in order: the time of the first star’s flash, the time of the second star’s flash, the time between consecutive flashes of the first star, the time between consecutive flashes of the second star.

The first two timestamps will differ, and both fall on the same day – Saturday. Flash intervals will not be 00:00.

-----Output-----
If the stars will never flash at the same minute, output “Never” in a single line.

Otherwise, output in the first line the name of the weekday of the first same-minute flash. A reminder on the correct spelling of weekdays: “Sunday”, “Monday”, “Tuesday”, “Wednesday”, “Thursday”, “Friday” and “Saturday”.

In the second line, output the timestamp of the first same-minute flash, in $HH$:$MM$ format (with a leading zero for $HH$ and $MM$ if they are less than $10$).

-----Examples-----
Sample Input 1:
02:20
13:00
05:50
01:00
Sample Output 1:
Saturday
14:00

Sample Input 2:
02:20
23:28
00:40
23:50
Sample Output 2:
Never
user: An instructor may use curved score to grade a student’s performance. Assume a student’s original score is $x$ out of a total of $100$. One of the curving methods is to apply to $x$ the curving function $f(x) = 10\sqrt {x}$. The function may be applied $k \geq 0$ times. For better score readability, the result is rounded to the next integer and the final curved score is $y = \lceil f^k(x) \rceil $. 

For example, if the curving function is applied $k = 3$ times to $x = 80$, it changes the score to\begin{align*} f^3(80) & = f^2(f(80)) = f^2(89.4427) \\ & = f(f(89.4427)) = f(94.5742) = 97.2492. \end{align*}

So the curved score is $y = \lceil 97.2492 \rceil = 98$.

John wants to curve a student’s score $x$ using this method so that the curved score $y$ is between $y_{low}$ and $y_{high}$. How many times should he apply the curving function? In particular, what are the minimum and maximum choices of $k$, so that $y_{low} \leq y \leq y_{high}$?

-----Input-----
The input has a single line with three integers: $x$, $y_{low}$, and $y_{high}$ ($1 \leq x \leq y_{low} \leq y_{high} \leq 100$).

-----Output-----
Output the minimum and maximum possible choices of $k$. If the curving function can be applied an infinite number of times, output “inf” for the maximum. If there is no $k$ that meets the curving requirement, output a single word “impossible”.

-----Examples-----
Sample Input 1:
80 85 98
Sample Output 1:
1 3

Sample Input 2:
98 98 100
Sample Output 2:
0 inf
user: Bees are one of the most industrious insects. Since they collect nectar and pollen from flowers, they have to rely on the trees in the forest. For simplicity they numbered the $n$ trees from $0$ to $n - 1$. Instead of roaming around all over the forest, they use a particular list of paths. A path is based on two trees, and they can move either way i.e. from one tree to another in straight line. They don’t use paths that are not in their list.

As technology has been improved a lot, they also changed their working strategy. Instead of hovering over all the trees in the forest, they are targeting particular trees, mainly trees with lots of flowers. So, they planned that they will build some new hives in some targeted trees. After that they will only collect their foods from these trees. They will also remove some paths from their list so that they don’t have to go to a tree with no hive in it.

Now, they want to build the hives such that if one of the paths in their new list go down (some birds or animals disturbs them in that path) it’s still possible to go from any hive to another using the existing paths.

They don’t want to choose less than two trees and as hive-building requires a lot of work, they need to keep the number of hives as low as possible. Now you are given the trees with the paths they use, your task is to propose a new bee hive colony for them.

-----Input-----
The first line contains two integers $n$ ($2 \leq n \leq 500$) and $m$ ($0 \leq m \leq 20000$), where $n$ denotes the number of trees and $m$ denotes the number of paths. Each of the next $m$ lines contains two integers $u$ and $v$ ($0 \leq u, v < n, u \neq v$) meaning that there is a path between tree $u$ and $v$. Assume that there can be at most one path between tree $u$ to $v$, and needless to say that a path will not be given more than once in the input.

-----Output-----
Print the number of beehives in the proposed colony or ‘impossible’ if its impossible to find such a colony.

-----Note-----
The dataset is huge. Use fast I/O methods.

-----Examples-----
Sample Input 1:
3 3
0 1
1 2
2 0
Sample Output 1:
3

Sample Input 2:
2 1
0 1
Sample Output 2:
impossible
user: Let $x_0, \ldots , x_{n-1}$ denote $n$ boolean variables (i.e., variables taking only values $0$ and $1$). A binary decision diagram (BDD) over these variables is a diagrammatic representation of a boolean function $f(x_0, \ldots , x_{n-1})$ as inputs.

A BDD is a rooted binary tree such that all internal vertices $v$ have precisely two children. The edges connecting an internal vertex $v$ with its children are labelled with a $0$ or $1$ (exactly one of each). Each leaf vertex is also labelled with a $0$ or $1$. We note that a BDD may consist of a single vertex, which is considered to be both the root and a leaf vertex.

Given input $(x_0, \ldots , x_{n-1})$, the boolean function represented by the BDD is evaluated as follows.
 - let $v$ be the root vertex
 - let $i \leftarrow 0$
 - while $v$ is not a leaf do
 - replace $v$ with the child vertex of $v$ by traversing the edge labelled $x_ i$
 - increase $i$ by $1$
 - output the label of leaf vertex $v$

Consider the function $f(x_0,x_1,x_2)$ represented by the BDD above. To evaluate $f(1,0,1)$, we start from the root, we descend along edges labelled $1$, $0$, and then $1$. We reach a leaf vertex labelled $1$, so $f(1,0,1) = 1$.

A BDD is minimal if there is no way to replace any subtree of an internal vertex of the BDD by a single leaf vertex to get a new BDD defining the same boolean function. The BDD depicted above is minimal. It is a fact that for each boolean function $f$, there is a unique minimal BDD that represents the boolean function.

In this problem, you are given an $n$-variable boolean function specified by a list of the $2^n$ different values the function should take for various inputs. Compute the number of vertices in the minimal BDD representing this function.

-----Input-----
The first line of input consists of a single integer $1 \leq n \leq 18$. Then one more line follows that contains $2^n$ values (either $0$ or $1$) describing an $n$-variable boolean function.

We think of these values as being indexed from $0$ to $2^n-1$. The $i$th such value represents $f(x_0, \ldots , x_{n-1})$ where $x_ j$ is the $j$th least-significant bit of the binary representation of $i$. In other words, $x_ j$ is the coefficient of $2^j$ in the binary expansion of $i$.

The third sample input below corresponds to the BDD depicted above.

-----Output-----
Output consists of a single integer $m$ that is the number of vertices in the unique minimal BDD representing the boolean function from the input.

-----Examples-----
Sample Input 1:
2
1 1 0 1
Sample Output 1:
5

Sample Input 2:
2
0 0 0 0
Sample Output 2:
1
user: Mirko found a collection of $N$ toy tanks dating back to the Second World War on his grandfather’s attic. He promptly called his friend Slavko to play with him. They made a battlefield – a wooden board consisting of squares in $N$ rows and $N$ columns.

Each tank can be moved to one of the four neighbouring squares in a single move. A tank can shoot at any square in the same row and column. The tank is said to be guarding the row and column it is in.

Additionally, no two tanks can be in the same square at any time.

After many hours of play and two previous attempts, Mirko’s mom yelled at them to come down for lunch again, and they decided to rearrange the tanks so that each tank guards a different row and column (meaning also that each row and column contains only one tank).

However, they want to do this using the minimum number of moves.

Write a program that finds the minimum number of moves required to rearrange the tanks so that each row and each column contains a single tank, and one such shortest sequence of moves.

-----Input-----
The first line of input contains the integer $N$ ($3 \le N \le 500$).

Each of the following $N$ lines contains two integers $R$ and $C$ ($1 \le R, S \le N$), the row and column of a single tank at the moment of mom’s call. No two tanks are on the same square.

Rows and columns are marked $1$ through $N$, top-down and left-to-right.

-----Output-----
Output the minimum number of moves (call this number $K$) on the first line.

Each of the next $K$ lines should contain the tank being moved and the direction it is moved in, separated by a single space.

Tanks are numbered $1$ through $N$, in the order in which they are given in the input.

The direction can be one of four uppercase letters: ‘L’ for left, ‘R’ for right, ‘U’ for up and ‘D’ for down.

If there are multiple optimal solutions, any one will be accepted.

-----Examples-----
Sample Input 1:
5
1 1
1 2
1 3
1 4
1 5
Sample Output 1:
10
1 D
2 D
3 D
4 D
1 D
2 D
3 D
1 D
2 D
1 D

Sample Input 2:
5
2 3
3 2
3 3
3 4
4 3
Sample Output 2:
8
1 R
1 R
2 U
2 U
4 D
4 D
5 L
5 L