meetforever/GRPO_CL huggingface dataset:
user: Mr. Turtle loves drawing on his whiteboard at home. One day when he was drawing, his marker dried out! Mr. Turtle then noticed that the marker behaved like an eraser for the remainder of his drawing.

Mr. Turtle has a picture in his head of how he wants his final drawing to appear. He plans out his entire drawing ahead of time, step by step. Mr. Turtle’s plan is a sequence of commands: up, down, left or right, with a distance. He starts drawing in the bottom left corner of his whiteboard. Consider the $6 \times 8$ whiteboard and sequence of commands in the first diagram. If the marker runs dry at timestep $17$, the board will look like the second diagram (the numbers indicate the timestep when the marker is at each cell). Note that it will make a mark at timestep $17$, but not at timestep $18$. 

Mr. Turtle wants to know the earliest and latest time his marker can dry out, and he’ll still obtain the drawing in his head. Can you help him? Note that timestep $0$ is the moment before the marker touches the board. It is valid for a marker to dry out at timestep $0$.

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. The input will start with a line with $3$ space-separated integers $h$, $w$ and $n$ ($1 \le h,w,n \le 1000000, w \cdot h \le 1000000$) where $h$ and $w$ are the height and width of the whiteboard respectively, and $n$ is the number of commands in Mr. Turtle’s plan.

The next $h$ lines will each consist of exactly $w$ characters, with each character being either ‘#’ or ‘.’ . This is the pattern in Mr. Turtle’s head, where ’#’ is a marked cell, and ‘.’ is a blank cell.

The next $n$ lines will each consist of a command, of the form “direction distance”, with a single space between the direction and the distance and no other spaces on the line. The direction will be exactly one of the set $\{ \texttt{up}, \texttt{down}, \texttt{left}, \texttt{right} \} $, guaranteed to be all lower case. The distance will be between $1$ and $1000000$ inclusive. The commands must be executed in order. It is guaranteed that no command will take the marker off of the whiteboard.

-----Output-----
Output two integers, first the minimum, then the maximum time that can pass before the marker dries out, and Mr. Turtle can still end up with the target drawing. Neither number should be larger than the last timestep that the marker is on the board, so if the marker can run to the end and still draw the target drawing, use the last timestep that the marker is on the board. If it’s not possible to end up with the target drawing, output -1 -1.

-----Examples-----
Sample Input 1:
6 8 5
........
...#....
########
#..#...#
#..#####
#.......
up 3
right 7
down 2
left 4
up 3
Sample Output 1:
20 20

Sample Input 2:
6 8 5
........
........
###.####
#......#
#..#####
#.......
up 3
right 7
down 2
left 4
up 3
Sample Output 2:
17 17

Sample Input 3:
3 3 2
...
.#.
...
up 2
right 2
Sample Output 3:
-1 -1
user: It is well known that a set of six unit squares that are attached together in a “cross” can be folded into a cube.

But what about other initial shapes? That is, given six unit squares that are attached together along some of their sides, can we form a unit cube by folding this arrangement?

-----Input-----
Input consists of $6$ lines each containing $6$ characters, describing the initial arrangement of unit squares. Each character is either a ., meaning it is empty, or a # meaning it is a unit square.

There are precisely $6$ occurrences of # indicating the unit squares. These form a connected component, meaning it is possible to reach any # from any other # without touching a . by making only horizontal and vertical movements. Furthermore, there is no $2 \times 2$ subsquare consisting of only #. That is, the pattern

##
##

does not appear in the input.

-----Output-----
If you can fold the unit squares into a cube, display can fold. Otherwise display cannot fold.

-----Examples-----
Sample Input 1:
......
......
######
......
......
......
Sample Output 1:
cannot fold

Sample Input 2:
......
#.....
####..
#.....
......
......
Sample Output 2:
can fold

Sample Input 3:
..##..
...#..
..##..
...#..
......
......
Sample Output 3:
cannot fold
user: Given an input string composed solely of lowercase English letters, find the longest substring that occurs more than once in the input string. The two occurrences are allowed to partially overlap.

-----Input-----
The input is a single line containing a string of lowercase letters. The string contains more than one character, but no more than $10^5$. At least one letter will appear at least twice.

-----Output-----
Print a single line of output: the longest substring that occurs more than once in the input string. If there are multiple longest repeated substrings, print the one the would come first when the longest substrings are sorted in lexicographical (alphabetical) order.

-----Examples-----
Sample Input 1:
abcefgabc
Sample Output 1:
abc

Sample Input 2:
abcbabcba
Sample Output 2:
abcba

Sample Input 3:
aaaa
Sample Output 3:
aaa
user: Johan lives in Stockholm, which means he has to travel a lot by bus and subway. To travel, he needs to pay a number of coupons, depending on which zones his trip goes between.

A ticket is specified by an interval $[A, B]$ of zones it is valid in, and costs $2 + |A - B|$ coupons to purchase. When Johan buys a ticket, it is valid for exactly $10000$ seconds and can be used for any trip which both starts and ends in zones in the interval $[A, B]$.

Johan is now going to make $N$ trips in order. For each trip, you will be given the zone Johan travels to, as well as the time at which he starts his trip. Initially, Johan is located in zone $0$. Help him minimize the number of coupons he must purchase!

Notes: Johan can buy a new ticket at any point, even if he already has a valid ticket. A ticket only needs to be valid at the moment Johan starts a trip, it is OK if the $10000$ seconds expire while Johan is taking the trip.

-----Input-----
The first line contains the integer $0 \le N \le 400000$ – the number of trips Johan wants to make.

The next $N$ lines contains two space-separated integers $0 \le Z \le 10$ and $0 \le T \le 10^9$, the zone and time for Johan’s next trip (in seconds from now). The values of $T$ will be strictly increasing.

-----Output-----
The first and only line should contain a single integer: the minimum number of coupons Johan must purchase to make all his trips.

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

Sample Input 2:
2
1 4
2 10005
Sample Output 2:
6

Sample Input 3:
3
1 4
2 10
0 15
Sample Output 3:
4
user: An LCM tree is a binary tree in which each node has a positive integer value, and either zero or two children. If two nodes $x$ and $y$ are the children of node $z$, then the Least Common Multiple (LCM) of the values of node $x$ and node $y$ must equal the value of node $z$. 

You are given $n$ nodes with positive integer values to be arranged into an LCM tree. In how many ways (modulo $10^9 + 7$) can you do that? Two ways are considered different if there are two nodes $x$ and $y$ so that $x$ is a child of $y$ in one way but not in the other way.

The illustration shows one of the two ways for the first sample case. The other way can be obtained by swapping the two nodes with value $4$. Note that swapping the two leaves with values $2$ and $4$ does not give a different way.

-----Input-----
The first line has an odd integer $n$ ($1 \leq n \leq 25$). The second line has $n$ positive integers no larger than $10^9$, giving the values of the nodes.

-----Output-----
Output the number of ways to arrange the given nodes into an LCM tree, modulo $10^9 + 7$.

-----Examples-----
Sample Input 1:
7
2 3 4 4 8 12 24
Sample Output 1:
2

Sample Input 2:
3
7 7 7
Sample Output 2:
3

Sample Input 3:
5
1 2 3 2 1
Sample Output 3:
0
user: You are a treasure hunter searching for gold in the wild. Luckily you got a map of the area from a black marketeer. The map clearly shows where a box of gold is along with the surrounding terrain. Of course, you want to head to the treasure as fast as possible. However, the trip might not be easy.

The area is described by a grid of $N$ rows and $M$ columns. You are currently located at the cell labeled ‘S’. The treasure is located at the cell labeled ‘G’. Despite your eagerness for the treasure, you cannot keep moving tirelessly. The surrounding area has different landforms: plain (‘.’), forest (‘F’), mountain (‘M’) or river (‘#’). Rivers are impassable since no boats are currently available. You can navigate in four directions (up, down, left, right) between neighboring cells.

At the beginning of your trip you have $K$ stamina points. Moving onto a plain, forest, and mountain costs you $1$, $2$, and $3$ points of stamina respectively. Once you run out of stamina and can no longer move to nearby cells, you must camp and rest. Your stamina is replenished to the full $K$ points on the second day so that you can start moving again.

Plan wisely with the map and determine the minimum number of days you need to reach the treasure.

-----Input-----
The first line has three integers $N$, $M$ ($1\leq N, M \leq 50$), and $K$ ($1 \leq K \leq 100$). Each of the next $N$ lines has $M$ characters describing the grid shown by the treasure map. The grid contains only characters ‘.’, ‘F’, ‘M’, ‘#’, ‘S’, and ‘G’. It is guaranteed that ‘S’ and ‘G’ appear exactly once in the grid. You may assume your starting point (‘S’) and the treasure (‘G’) are both on a plain.

-----Output-----
Output the minimum number of days you need to reach the treasure. If it is impossible to get to the treasure, output “-1”.

-----Examples-----
Sample Input 1:
2 5 4
S#.F.
.MFMG
Sample Output 1:
3

Sample Input 2:
1 2 1
GS
Sample Output 2:
1

Sample Input 3:
2 2 10
S#
#G
Sample Output 3:
-1
user: The Manhattan Positioning System (MPS) is a modern variant of GPS, optimized for use in large cities. MPS assumes all positions are discrete points on a regular two-dimensional grid. Within MPS, a position is represented by a pair of integers $(X,Y)$.

To determine its position, an MPS receiver first measures its distance to a number of beacons. Beacons have known, fixed locations. MPS signals propagate only along the $X$ and $Y$ axes through the streets of the city, not diagonally through building blocks. When an MPS receiver at $(X_ R,Y_ R)$ measures its distance to a beacon at $(X_ B,Y_ B)$, it thus obtains the Manhattan distance: $|X_ R - X_ B|+|Y_ R - Y_ B|$.

Given the positions of a number of beacons and the Manhattan-distances between the receiver and each beacon, determine the position of the receiver. Note that the receiver must be at an integer grid position (MPS does not yet support fractional coordinates).

-----Input-----
The first line contains an integer $N$, the number of beacons ($1 \leq N \leq 1000$). Then follow $N$ lines, each containing three integers, $X_ i$, $Y_ i$, and $D_ i$, such that $-10^6 \leq X_ i, Y_ i \leq 10^6$ and $0 \leq D_ i \leq 4 \cdot 10^6$. The pair $(X_ i, Y_ i)$ denotes the position of beacon $i$, while $D_ i$ is the Manhattan distance between receiver and beacon $i$.

No two beacons have the same position.

-----Output-----
If there is exactly one receiver position consistent with the input, write one line with two integers, $X_ R$ and $Y_ R$, the position of the receiver.

If multiple receiver positions are consistent with the input, write one line with the word “uncertain”.

If no receiver position is consistent with the input, write one line with the word “impossible”.

-----Examples-----
Sample Input 1:
3
999999 0 1000
999900 950 451
987654 123 13222
Sample Output 1:
1000200 799

Sample Input 2:
2
100 0 101
0 200 199
Sample Output 2:
uncertain

Sample Input 3:
2
100 0 100
0 200 199
Sample Output 3:
impossible
user: In the recently proposed Visual Python++ programming language, a block of statements is represented as a rectangle of characters with top-left corner in row $r_1$ and column $c_1$, and bottom-right corner in row $r_2$ and column $c_2$. All characters at locations $(r, c)$ with $r_1 \leq r \leq r_2$ and $c_1 \leq c \leq c_2$ are then considered to belong to that block. Among these locations, the ones with $r = r_1$ or $r = r_2$ or $c = c_1$ or $c = c_2$ are called a border.

Statement blocks can be nested (rectangles contained in other rectangles) to an arbitrary level. In a syntactically correct program, every two statement blocks are either nested (one contained in the other) or disjoint (not overlapping). In both cases, their borders may not overlap.

Programmers are not expected to draw the many rectangles contained in a typical program – this takes too long, and Visual Python++ would not have a chance to become the next ICPC programming language. So a programmer only has to put one character ‘$\ulcorner $’ in the top-left corner of the rectangle and one character ‘$\lrcorner $’ in the bottom-right corner. The parser will automatically match up the appropriate corners to obtain the nesting structure of the program.

Your team has just been awarded a five-hour contract to develop this part of the parser.

-----Input-----
The first line of the input contains an integer $n$ ($1 \leq n \leq 10^5$), the number of corner pairs. Each of the next $n$ lines contains two integers $r$ and $c$ ($1 \leq r, c \leq 10^9$), specifying that there is a top-left corner in row $r$ and column $c$ of the program you are parsing. Following that are $n$ lines specifying the bottom-right corners in the same way. All corner locations are distinct.

-----Output-----
Display $n$ lines, each containing one integer. A number $j$ in line $i$ means that the $i^{\text {th}}$ top-left corner and the $j^{\text {th}}$ bottom-right corner form one rectangle. Top-left and bottom-right corners are each numbered from $1$ to $n$ in the order they appear in the input. The output must be a permutation of the numbers from $1$ to $n$ such that the matching results in properly nested rectangles. If there is more than one valid matching, any one will be accepted. If no such matching exists, display syntax error.

-----Examples-----
Sample Input 1:
2
4 7
9 8
14 17
19 18
Sample Output 1:
2
1

Sample Input 2:
2
4 7
14 17
9 8
19 18
Sample Output 2:
1
2

Sample Input 3:
2
4 8
9 7
14 18
19 17
Sample Output 3:
syntax error
user: Theta likes to play Lure of the Labyrinth, which is an online game that uses a compelling graphic novel storyline to engage middle grades students in mathematical thinking and problem-solving. To find lost pets, students have to infiltrate a world of monsters and solve puzzles! Implemented by a professional game studio, these puzzles are quite engaging and challenging.

In the manager’s cafeteria, students are asked to practice proportions by serving $5$ monsters sitting at the table: Salamander, Yeti, Golem, Imp, and Kraken. Each monster wants some amount of burger, slop, sushi, and drumstick.

The amount of each item each monster wants on their plate is proportional to the amount of each of the other items that monster wants, and the proportionality ratio is the same for all monsters.

For instance, as shown in sample input $1$ and in the accompanying figure, if Golem (center) wants $6$ units of burger on his plate and $12$ units of drumstick, and Salamander (left) wants $40$ units of drumstick, then Salamander will want $20$ units of burger. Students practicing proportions will program the dispenser machine to release $20$ units of burger. Similarly, Kraken (right) wants $36$ units of drumstick because it has $81$ units of slop on its plate and Salamander has $40$ units of drumstick and $90$ units of slop. If the students compute all proportions in time, the monsters eat and a pet can be saved!

As students progress in the game and reach more difficult levels, fewer and fewer amounts are given, requiring more intuition and thinking to solve the puzzle.

Give a set of partially filled plates, write a program that computes the number of distinct solutions that are consistent with it!

-----Input-----
The input consists of $2$ lines of $10$ entries each describing the partially filled plates. The first line describes the top row (burgers and slop), the second line describes the bottom row (sushi and drumstick). On each line, the first $2$ entries describe Salamander’s plate, the next $2$ Yeti’s, then Golem’s, Imp’s, and finally Kraken’s. Each entry is either the underscore character _ describing an empty slot or a positive integer number $a$ ($0 < a \le 200$) if it is already known. Entries are separated by single spaces. You may assume that each arrangement has at least one possible solution (that is, the partial information is not inconsistent).

-----Output-----
Output the number $n$ of distinct solutions for the puzzle given in the input! If there are infinitely many solutions, output “many”! Note that although any given entries are guaranteed to be less than $200$, inferred entries must be positive integers, but they are not subject to a maximum.

-----Examples-----
Sample Input 1:
_ 90 22 _ 6 _ _ _ _ 81
_ 40 _ _ _ 12 60 _ 90 _
Sample Output 1:
1

Sample Input 2:
85 55 _ 99 51 _ _ _ _ _
_ _ _ _ _ _ _ 85 63 153
Sample Output 2:
1

Sample Input 3:
160 _ _ 136 _ _ _ _ _ 170
_ _ _ _ 120 _ _ 144 _ _
Sample Output 3:
8640
user: The Kanto region has $N$ junctions and $N - 1$ bidirectional roads. Junctions are numbered from $1$ to $N$, inclusive. All roads have the same length and each of them connects two different junctions.

At any moment, a Pokenom can appear in any of these junctions.

To become the best Pokenom trainer, like no one ever was, Bash is studying the behavior of the Pokenom in Kanto region. Bash has found that, when trying to capture a Pokenom at junction $u$, the Pokenom can run away to junction $v$, iff the optimal path between $u$ and $v$ has length exactly $2$.

More formally, a path of length $K$ from junction $s$ to junction $t$ is an ordered sequence of junctions $v_{0} \rightarrow v_{1} \rightarrow v_{2} \rightarrow \cdots \rightarrow v_{K}$, where $v_{0} = s, v_{K} = t$ and for each valid index $i$, $v_{i}$ and $v_{i+1}$ are connected directly by some road. A path is called optimal iff there is no shorter path with the same starting and ending junctions. Two paths $v_{0} \rightarrow v_{1} \rightarrow \cdots \rightarrow v_{k}$ and $w_{0} \rightarrow w_{1} \rightarrow \cdots w_{l}$ are different iff either $k \neq l$ or there exist some valid $i$ such that $v_{i} \neq w_{i}$.

A Pokenom can use an optimal path of length exactly $2$. Help Bash count the number of such paths.

-----Input-----
 - The first line contains one integer $N$ $(1 \leq N \leq 3 \cdot 10^{5})$ — the number of junctions.
 - Each of the rest $N - 1$ lines contains two integers $u$ and $v$ $(1 \leq u, v \leq N, u \neq v)$ — two endpoints of a single road.

-----Output-----
 - The only line contains exactly one integer — the number of optimal paths of length $2$.

-----Explanation for the first example-----
 - There are two optimal paths of length $2$: $(2 \rightarrow 1 \rightarrow 3)$ and $(3 \rightarrow 1 \rightarrow 2)$.
 - The path $(1 \rightarrow 2 \rightarrow 1)$ is a valid path of length $2$ from $1$ to $1$ but it is not optimal since there is a path of length $0$ from $1$ to $1$.

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

Sample Input 2:
5
2 1
1 5
3 1
4 3
Sample Output 2:
8

Sample Input 3:
10
1 2
2 3
2 4
1 5
3 6
2 7
7 8
5 9
5 10
Sample Output 3:
24
user: The Swedish Military Intelligence and Security Service (Militära underrättelse- och säkerhetstjänsten, MUST) is trying to locate a submarine in the Stockholm archipelago. They have been intercepting encrypted messages that they believe contain information on the location and route of the submarine. Now all that remains is to decrypt these messages. 

From previous intelligence operations (the details of which we are not at liberty to reveal to you), it is known that the messages are likely encrypted with a simple substitution cipher. In other words, every letter is replaced with another letter from the alphabet (a letter could also remain unchanged). So for instance, it could be that every ‘a’ gets turned into a ‘k’, every ‘b’ remains a ‘b’, every ‘c’ becomes an ‘a’, and so on. Obviously, different letters in the original message become different letters in the encrypted one, otherwise decryption would not be unique.

Alas it is not known which letters are substituted for which other letters. However, a suspicious person in scuba gear has been apprehended and was found carrying a note with what appears to be a fragment of a decrypted message. If this fragment could be matched to an encrypted message, the code can be broken!

Can you help MUST find out if the message fragment could be part of a given encrypted message, and if so, in how many positions?

-----Input-----
The input consists of:
 - one line with a string consisting of at least $1$ and at most $250000$ lowercase letters, the encrypted message;
 - one line with a string consisting of at least $1$ and at most $250000$ lowercase letters, the decrypted fragment.

-----Output-----
If there is a unique position in the encrypted message where the message fragment could occur, output the substring of the encrypted message that could correspond to the fragment. Otherwise, output the number of positions in the encrypted message where the fragment could occur.

-----Examples-----
Sample Input 1:
secretmessage
boot
Sample Output 1:
essa

Sample Input 2:
treetreetreetree
wood
Sample Output 2:
3

Sample Input 3:
oranges
apples
Sample Output 3:
0
user: Many airports have moving conveyor belts in the corridors between halls and terminals. Instead of walking on the floor, passengers can choose to stand on a conveyor or, even better, walk on a conveyor to get to the end of the corridor much faster.

The brand new Delft City Airport uses a similar system. However, in line with the latest fashion in airport architecture, there are no corridors: the entire airport is one big hall with a bunch of conveyor lines laid out on the floor arbitrarily.

To get from a certain point $A$ to a certain point $B$, a passenger can use any combination of walking on the floor and walking on conveyors. Passengers can hop on or off a conveyor at any point along the conveyor. It is also possible to cross a conveyor without actually standing on it.

Walking on the floor goes at a speed of $1$ meter/second.

 Walking forward on a conveyor goes at a total speed of $2$ meter/second.

 Walking in reverse direction on a conveyor is useless and illegal, but you may walk on the floor immediately next to the conveyor. (Conveyors are infinitely thin.)

How fast can you get from $A$ to $B$?

-----Input-----
The first line contains four floating point numbers, $X_ A$, $Y_ A$, $X_ B$, and $Y_ B$. They describe the coordinates of your initial location $A = (X_ A,Y_ A)$ and your final location $B = (X_ B,Y_ B)$.

The second line contains an integer $N$, the number of conveyors in the hall ($0 \leq N \leq 100$). The following $N$ lines each contain four floating point numbers, $X_1$, $Y_1$, $X_2$, and $Y_2$, describing a conveyor which starts at the point $(X_1,Y_1)$ and ends at the point $(X_2,Y_2)$, running in a straight line from start to end.

All coordinates are floating point numbers in the range ($0 \leq X, Y \leq 1000.0$), expressed in units of meters, given with at most $6$ decimals after the decimal point.

Conveyors are at least $1$ meter long. Conveyors do not intersect or touch. Your start and destination are not on any conveyor.

-----Output-----
Write one line with a floating point number, the minimum time (in seconds) needed to get from $A$ to $B$ in seconds.

Your answer may have an absolute error of at most $10^{-4}$.

-----Examples-----
Sample Input 1:
60.0 0.0 50.0 170.0
3
40.0 0.0 0.0 0.0
5.0 20.0 5.0 170.0
95.0 0.0 95.0 80.0
Sample Output 1:
168.7916512460

Sample Input 2:
60.0 0.0 50.0 170.0
3
40.0 0.0 0.0 0.0
5.0 20.0 5.0 170.0
95.0 0.0 95.0 100.0
Sample Output 2:
163.5274740179

Sample Input 3:
0.0 1.0 4.0 1.0
1
0.0 0.0 4.0 0.0
Sample Output 3:
3.7320508076