meetforever/GRPO_CL huggingface dataset:
user: A string is called double free if no two adjacent letters are the same.

A string is called $k$-incremental if for all values of $j$ in the range $[1,k]$, there exists exactly one character with $j$ occurrences, and the string’s length is $1+2+3+\ldots +(k-1)+k$. For example, if $k=3$, then a $3$-incremental string should have one character appear once, another twice, another three times, in any order, for a total string length of $6$.

A string is both $k$-incremental and double free if it meets both these criteria. Now consider examining all such strings of lowercase letters for a given $k$ in alphabetical order. Consider the following examples.

$k=2$: aba, aca, ada, …, aya, aza, bab, bcb, bdb, …, zxz, zyz

$k=3$: ababac, ababad, …, ababay, ababaz, ababca, …, zyzyzx

What is the $n^\mathrm {th}$ string in an alphabetized list of all $k$-incremental, double free strings?

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. There will be exactly one line of input. It will contain two integers, $k$ and $n$ ($1 \le k \le 26, 1 \le n \le 10^{18}$), which is asking for the $n^\mathrm {th}$ string in the alphabetically sorted list of all $k$-incremental, double free strings.

-----Output-----
Output the $n^\mathrm {th}$ $k$-incremental, double free string in the alphabetized list. If no such string exists, output $-1$.

-----Examples-----
Sample Input 1:
2 650
Sample Output 1:
zyz

Sample Input 2:
2 651
Sample Output 2:
-1
user: As an employee of the world’s most respected political polling corporation, you must take complex, real-world issues and simplify them down to a few numbers. It isn’t always easy. A big election is coming up and, at the request of Candidate X, you have just finished polling $n$ people. You have gathered three pieces of information from each person, with the values for the $i^\text {th}$ person recorded as:
 - $a_ i$ – the number of digits of $\pi $ they have memorized
 - $b_ i$ – the number of hairs on their head
 - $c_ i$ – whether they will vote for Candidate X

Unfortunately, you are beginning to wonder if these are really the most relevant questions to ask. In fact, you cannot see any correlation between $a$, $b$, and $c$ in the data. Of course, you cannot just contradict your customer – that is a good way to lose your job!

Perhaps the answer is to find some weighting formula to make the results look meaningful. You will pick two real values $S$ and $T$, and sort the poll results $(a_ i, b_ i, c_ i)$ by the measure $a_ i \cdot S + b_ i \cdot T$. The sort will look best if the results having $c_ i$ true are clustered as close to each other as possible. More precisely, if $j$ and $k$ are the indices of the first and last results with $c_ i$ true, you want to minimize the cluster size which is $k-j+1$. Note that some choices of $S$ and $T$ will result in ties among the $(a_ i,b_ i,c_ i)$ triples. When this happens, you should assume the worst possible ordering occurs (that which maximizes the cluster size for this $(S, T)$ pair).

-----Input-----
The input starts with a line containing $n$ ($1 \leq n \leq 250000$), which is the number of people polled. This is followed by one line for each person polled. Each of those lines contains integers $a_ i$ ($0 \leq a_ i \leq 2000000$), $b_ i$ ($0 \leq b_ i \leq 2000000$), and $c_ i$, where $c_ i$ is $1$ if the person will vote for Candidate X and $0$ otherwise. The input is guaranteed to contain at least one person who will vote for Candidate X.

-----Output-----
Display the smallest possible cluster size over all possible $(S, T)$ pairs.

-----Examples-----
Sample Input 1:
6
0 10 0
10 0 1
12 8 1
5 5 0
11 2 1
11 3 0
Sample Output 1:
4

Sample Input 2:
10
6 1 1
0 2 0
2 1 1
6 1 1
8 2 0
4 4 0
4 0 0
2 3 1
6 1 0
6 3 1
Sample Output 2:
8
user: Today your friends are playing a game of Exploding Kittens. On a player’s turn, she must perform the following actions in order:
 - Draw a single card from the top of the deck.
 - If the drawn card is an “Exploding Kitten”, she must leave the game unless she has a “Defuse” card in her hand, in which case she must discard both the “Exploding Kitten” card and the “Defuse” card, so that these two cards will never be drawn again in the same game. Otherwise, she can choose to do nothing.
 - If she has more than five cards in his hand, she must discard cards until he has no more than five cards.

Players alternate turns in a single round, and the relative orders in which players take turns must be identical between rounds. When a player leaves the game, the relative draw order of other players will stay the same. The last player left in the game is the winner, and the game will stop at that point.

You don’t want to watch the game because there’s no skill involved, but you still would like to know the result. You somehow know the location of all “Exploding Kittens” and “Defuse” cards in the deck. Assuming everyone starts with an empty hand and plays optimally (i.e. discard “Defuse” cards only when absolutely necessary), who will win the game?

-----Input-----
The first line consists of three integers $N$, the number of players in the game, $E$, the number of “Exploding Kitten” cards, and $D$, the number of “Defuse” cards. Note that $2\leq N\leq 1000$ and $2\leq D, E\leq 10000$.

The next line of the input contains $E$ integers $e_ i$ ($0\leq e_ i\leq 2\cdot 10^9$). This represents the number of cards between a “Exploding Kitten” card and the top card of the deck.

The final line of the input contains $D$ integers $d_ i$ ($0\leq d_ i\leq 2\cdot 10^9$). This represents the number of cards between a “Defuse” card and the top of the deck.

It is guaranteed that all “Exploding Kittens” and “Defuse” cards given in the input are located at distinct locations within the deck.

-----Output-----
If there is a winner, print the ID of the winner, defined as the number of players who took their turn before the winner on the first round. Otherwise, print $-1$.

-----Examples-----
Sample Input 1:
2 4 3
3 4 5 7
1 2 10
Sample Output 1:
0

Sample Input 2:
3 5 2
1 4 7 9 11
2 3
Sample Output 2:
2
user: You are an employee of the Boxing And Processing Company and you are tasked with distributing boxes in one of the company’s enormous warehouses. At BAPC Ltd., boxes travel by conveyor belt. Two conveyor belts can be merged into one by letting one drop its content onto the other. On the other hand, a belt can be split into two by using a splitter, which sends a specific portion of its input to its first output and the rest to its second output. Normally your splitters are adjustable, being able to distribute its input over its output belts in any conceivable ratio, but in an attempt to cut costs your boss has ordered cheap knock-off splitters which can only distribute its input in a fixed ratio $a:b$. Instead of arguing with your boss about how you really need some $c:d$ splitters, you decide to make them yourself.

Of course, with your frugal boss, you have to watch your costs. You cannot build your splitter system in such a way that there are boxes that never leave the system. Nor can you use splitters that never receive any boxes. Finally, you cannot use too many $a: b$ splitters in total.

Given the ratios $a:b$ and $c:d$, construct a network of belts and at most $200$ knock-off $a:b$ splitters that has a single global input belt and two global output belts over which the global input is distributed in a ratio $c:d$.

Note that a splitter of ratio $x : y$ sends exactly $\frac x{x+y}$ of the incoming boxes to the first output and exactly $\frac y{x+y}$ of them to the second output.

-----Input-----
 - The first line contains two integers $1\leq a,b\leq 10^9$ with $a+b\leq 10^9$ denoting the ratio $a:b$ of your knock-off splitter.
 - The second line contains two integers $1\leq c,d\leq 10^9$ with $c+d\leq 10^9$ denoting the ratio $c:d$ of your desired splitter.

-----Output-----
 - The first line contains an integer $1\leq n\leq 200$, the number of $a:b$ splitters used.
 - Then follow $n$ lines, the $i$-th of which contains two integers $-2\leq l_ i,r_ i<n$.

Here $l_ i$ is the index of the splitter connected to the left output of the $i$-th splitter, where it deposits $a/(a+b)$ of its input. Similarly $r_ i$ is the index of the splitter receiving $b/(a+b)$ of the input of the $i$-th splitter. The splitters are indexed starting from $0$. The global input is connected to splitter $0$. The special values $-1$ and $-2$ for $l_ i$ and $r_ i$ correspond to the first and second global output, which need to receive $c/(c+d)$ and $d/(c+d)$ of the global input respectively.

Note that you cannot place a splitter in such a way that no box ever reaches it, nor in such a way that a box that passes through it will never reach the output.

If there are multiple possible solutions, you may output any one of them.

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

Sample Input 2:
1 2
3 4
Sample Output 2:
3
-1 1
2 1
0 -2
user: Teacher Herkabe has decided to rank his students again. This time, he wants his list to also be aesthetically pleasant, so he has decided that similar names (those beginning with the same letter or sequence of letters) must be close to one another on the list. Therefore, he has devised the following rule:

For every two names on the list that begin with the same letter sequence, all names between them on the list must also begin with that letter sequence.

For example, consider the names MARTHA and MARY (characters from a beautiful story). They both begin with the sequence MAR, so names beginning with the same sequence (like MARCO and MARVIN) can appear in between (but not MAY, for example).

Notice that the lexicographically sorted ordering always satisfies this rule, but it is by no means the only valid ordering. Your task is determining how many different orderings satisfy the rule, i.e. how many options teacher Herkabe has for his ranking list.

-----Input-----
The first line of input contains the positive integer $N$ ($3 \leq N \leq 3\ 000$), the number of names. Each of the following $N$ lines contains a single name: a sequence of between $1$ and $3\ 000$ (inclusive) uppercase English letters. The names are distinct and given in no particular order.

-----Output-----
The first and only line of output must contain the required number of possible ranking lists, modulo $1\ 000\ 000\ 007$.

-----Examples-----
Sample Input 1:
3
IVO
JASNA
JOSIPA
Sample Output 1:
4

Sample Input 2:
5
MARICA
MARTA
MATO
MARA
MARTINA
Sample Output 2:
24
user: Sleazy Bob has happened upon a vending machine. After watching enough people buy tasty snacks, Bob has realized that the vending machine is broken!

Here’s what Sleazy Bob observes:
 - A person tries to buy a snack
 - The vending machine then checks to see if there are any left of that snack
 - If there are any left, the machine charges the person for that snack
 - If the machine successfully charges the person, it then gives the person a different snack! Possibly no snack at all, if the machine is out of the different snack!

Sleazy Bob notices that, although the machine is broken, it is at least consistent. Whenever a customer buys a snack from position $i$, the machine vends from position $f(i)$, where $f$ is some predictable function.

Now, Bob wants to make a profit from the machine. He wants to buy some snacks from the machine, and then turn around and sell the snacks he gets for the market price of the snack. This may be different from the vending price. If a cheap snack is at $i$, and an expensive snack is at $f(i)$, he could rake in the cash! Assuming Bob can always find buyers for his snacks, what is the maximum net gain that Bob can obtain by buying some number, possibly zero, of snacks and turning around and selling them later? You may assume that Bob has enough money from other shady activities to cover the cost of buying any number of snacks from the vending machine.

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. Each input begins with a line with a single integer $n$ ($1 \le n \le 100000$), which is the number of snack positions in the machine. Each of the next $n$ lines contains 4 space-separated integers, $f$ $p$ $m$ $s$, which describe a snack position in the machine, in order from 1 to $n$, where:
 - $f$ ($1\le f \le n$) is the value of $f(i)$. That is, it is the position from which the machine will vend when Bob buys from this position
 - $p$ ($1\le p \le 1000000$) is the price Bob must pay to buy from this position
 - $m$ ($1\le m \le 1000000$) is the market price of the snack at this position
 - $s$ ($1\le s \le 1000000$) is the number of snacks at this position

-----Output-----
Output a single line with a single integer, indicating the maximum profit Sleazy Bob can get from his nefarious abuse of the broken vending machine.

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

Sample Input 2:
3
2 2 3 8
3 1 5 6
1 9 4 7
Sample Output 2:
39
user: AcmeCorp is always looking to expand its drilling operations.

Their latest target is a largely uninhabited area in the North of the country, known as Crossland. Since the pocket of oil is quite deep already, and new drill bits are expensive, Crossland’s oil company would like to minimise the distance drilled—which may not be vertical—to get to the valuable spot.

The elevation-contoured survey map we have obtained marks the oil pocket at $(0,0)$, $0$ metres above sea level.

-----Input-----
 - One line containing one positive integer $N$, the number of polygon-shaped contour lines.
 - $N$ more lines each containing two integers, $H_0$ and $H_1$ ($0 \le H_0, H_1 \le 10^6$), the height of land in metres above sea level outside and inside the contour respectively, followed by a positive integer $M_ i$ representing the number of vertices of this contour.
 - The remaining $2 \cdot M_ i$ integers on each line are distinct coordinate pairs ($x_ j, y_ j$) ($-10^6 \le x,y \le 10^6$) denoting the $j^{th}$ point on the contour. No two contour lines touch, nor does a contour line touch the point $(0,0)$.

The total number of points across all contours will not exceed $10^5$, and it is guaranteed that the outer height of a contour is always equal to the inner height of its containing contour, if one exists.

-----Output-----
The first and only line of output should contain one real number indicating the closest slanted distance from the surface to the target. Your answer should be correct up to an absolute or relative precision of $10^{-6}$.

-----Examples-----
Sample Input 1:
2
5 8 4 -2 0 0 -3 2 0 0 3
8 10 4 -1 -1 1 -1 1 1 -1 1
Sample Output 1:
5.2696518641

Sample Input 2:
1
8 9 3 -14 -14 14 -14 0 20
Sample Output 2:
9
user: Mirko and his older brother Slavko are playing a game. At the beginning of the game, they pick three numbers $K$, $L$, $M$. In the first and only step of the game, each of them picks their own $K$ consecutive integers.

Slavko always picks the first $K$ integers (numbers $1, 2, \ldots , K$). Mirko has a special demand—he wants to choose his numbers in a way that there are exactly $L$ happy numbers among them. He considers a number happy if it meets at least one of the following requirements:
 - the number is smaller than or equal to $M$
 - the number is prime

Out of respect to his older brother, $L$ will be smaller than or equal to the total number of happy numbers in Slavko’s array of numbers. They will play a total of $Q$ games with different values $K$, $L$, $M$. For each game, help Mirko find an array that meets his demand.

-----Input-----
The first line of input contains $Q$ ($1 \leq Q \leq 100000$). Each of the following $Q$ lines contains three integers, the $i$-th line containing integers $K_ i$, $L_ i$, $M_ i$ ($1 \leq K_ i, M_ i \leq 150$, $0 \leq L_ i \leq K_ i$) that determine the values $K$, $L$, $M$ that will be used in the $i$-th game.

-----Output-----
Output $Q$ lines, the $i$-th line containing an integer, the initial number of Mirko’s array in the $i$-th game. If an array with the initial number being smaller than or equal to $10000000$ does not exist, output $-1$. If there are multiple possible solutions, output any.

-----Examples-----
Sample Input 1:
3
1 1 1
2 0 2
3 1 1
Sample Output 1:
1
8
4

Sample Input 2:
3
4 1 1
5 2 3
5 0 3
Sample Output 2:
6
4
24
user: You may have seen puzzles, sometimes called cryptarithmetic puzzles, or simply cryptarithms, that look like this:SEND+MORE=MONEY

The goal of this kind of puzzle is to replace each letter with a (base-$10$) digit so that the resulting arithmetic equality holds. Here are the rules:
 - A given letter must always be replaced by the same digit.
 - Different letters must be replaced by different digits.
 - The leading (leftmost) letter of any word cannot be replaced by 0.

For example, if we try to solve the above puzzle, we find that the following replacements work:S$\rightarrow $9, E$\rightarrow $5, N$\rightarrow $6, D$\rightarrow $7, M$\rightarrow $1, O$\rightarrow $0, R$\rightarrow $8, Y$\rightarrow $2

This gives us a correct equality:9567+1085=10652

Your task is to write a program to solve such puzzles. Note that some cryptarithmetic puzzles are impossible to solve. An example is:A+A=A

Here the only way to make the equality hold is to replace A with 0, but this violates the rule that the first (and in this case only) letter of any word cannot be replaced by 0.

Also note that some puzzles may have multiple solutions. In this situation, find the solution in which the alphabetically lowest letter is replaced by lowest possible digit, then the second lowest letter is replaced by the lowest possible unused digit, then the third lowest letter is replaced by the lowest possible unused digit, etc. An example is:C+B=A

Clearly this has more than one solution, but the minimal solution, as described above, is:2+1=3

-----Input-----
The input consists of a line containing a single puzzle. A puzzle is a string of characters of the form $w_1$+$w_2$=$w_3$, where $w1$, $w_2$, $w_3$ are words, and ‘+’ and ‘=’ are the usual “plus” and ”equals” characters (ASCII values $43$ and $61$, respectively). A word is a nonempty string of uppercase letters (A–Z). The maximum length of any puzzle is $100$.

-----Output-----
For each puzzle, output a single line containing the minimal solution if the puzzle is solvable, or “impossible” if the puzzle is not solvable. Note that a solution must contain exactly the same number of characters as the corresponding puzzle, with ‘+’ and ‘=’ characters in the same positions as in the puzzle, and all letters replaced by digits (0–9).

-----Examples-----
Sample Input 1:
SEND+MORE=MONEY
Sample Output 1:
9567+1085=10652

Sample Input 2:
A+A=A
Sample Output 2:
impossible
user: You have recently been promoted to lead scientist at NASA, the National Association for Sorting Algorithms. Congratulations! Your primary responsibility is testing the sorting algorithms that your team produces. Fortunately, NASA has a large budget this year, and you were able to buy some state-of-the-art integers you can use to test the sorting algorithms.

As the lead scientist, you are well aware that algorithms are tested by their behaviour on worst case inputs. So, to test sorting algorithms, you need sequences that are as unsorted as possible.

Given a sequence of numbers $(a_1,\ldots ,a_ n)$ we say that an element $a_ k$ is sorted if for all indices $j$ such that $j > k$, $a_ j \geq a_ k$ and for all indices $j$ such that $j < k$, $a_ j \leq a_ k$. For example, in\[ (1, 3, 2, 3, 4, 6, 5, 5) \]

the sorted elements are the $1$, the second occurrence of $3$, and the $4$. Note that a sequence is sorted if and only if all its elements are sorted. A sequence is called entirely unsorted if none of its elements are sorted.

Given a sequence of integers, what is the number of entirely unsorted sequences you can make by permuting its elements? Two sequences $(b_1, \dots , b_ n)$ and $(c_1, \dots , c_ n)$ are considered to be different if there is some index $i \in \left\{ 1, \dots , n \right\} $ for which $b_ i \neq c_ i$. Because the number of permutations may be very large, please give it modulo $10^9 + 9$.

-----Input-----
The input starts with an integer $1 \leq n \leq 5000$. Then follows a single line with $n$ integers $a_1, \ldots , a_ n$, with $0 \leq a_ i \leq 10^9$ for all $i$.

-----Output-----
Print a single integer: the number of entirely unsorted sequences you can make by permuting the $a_ i$, modulo $10^9 + 9$.

-----Examples-----
Sample Input 1:
4
0 1 2 3
Sample Output 1:
14

Sample Input 2:
5
1 1 2 1 1
Sample Output 2:
1
user: Consider an $n \times m$ checkerboard. On each cell of the checkerboard, place a positive integer. The values in each column of the checkerboard must be in strictly increasing order from top to bottom, and the values in each row of the checkerboard must be in strictly increasing order from left to right.

1

2

3

4

3

4

5

6

5

6

7

8

7

8

9

10

A Magic Checkerboard has an additional constraint. The cells that share only a corner must have numbers of different parity (Even vs Odd). Note that the following checkboard is invalid, because 2 and 4 share only a corner and have the same parity:

1

2

4

6

The first $4 \times 4$ example is a valid Magic Checkboard. Given a partially filled magic checkboard, can you fill the remaining locations on the checkboard, so that the sum of all values is as small as possible?

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. Each input starts with a line with two space-separated integers $n$ and $m$ ($1 \le n, m \le 2000$), representing the number of rows ($n$) and the number of columns ($m$) of the checkerboard. Each of the next $n$ lines will contain $m$ space-separated integers $c$ ($0 \le c \le 2000$), representing the contents of the checkerboard. Zero is used for cells without numbers that you must fill in. You may use any positive integers to fill in the cells without numbers, so long as you form a valid Magic Checkerboard. You are not limited to numbers $\le 2000$, and the numbers are not required to be unique.

-----Output-----
Output a single integer representing the minimum sum possible by replacing the 0 cells with positive integers to form a valid Magic Checkerboard. Output $-1$ if it is not possible to replace the 0 cells to meet the constraints of a Magic Checkerboard.

-----Examples-----
Sample Input 1:
4 4
1 2 3 0
0 0 5 6
0 0 7 8
7 0 0 10
Sample Output 1:
88

Sample Input 2:
4 4
1 2 3 0
0 0 5 6
0 4 7 8
7 0 0 10
Sample Output 2:
-1