meetforever/GRPO_CL huggingface dataset:
user: You are given a binary string S of N bits. The bits in the string are indexed starting from 1. S[i] denotes the ith bit of S.

Let's say that a sequence i1, i2, …, iK(1 ≤ K; 1 ≤ i1 < i2 < … < iK ≤ N) produces a palindrome when applied to S, if the string S[i1] S[i2] … S[ik] is a palindrome (that is, reads the same backward or forward).

In addition, a sequence i1, i2, …, iK(1 ≤ K; 1 ≤ i1 < i2 < … < iK ≤ N) is said to be exponential, if ij + 1 = p * ij for each integer 1 ≤ j < K and for some integer p > 1. Note, that a sequence of one element is always exponential.

Your task is to count the number of exponential sequences that produce a palindrome when applied to S.

-----Input-----
The first line of input contains an integer T denoting the number of test cases. The description of T test cases follows.

The only line of description for each test case contains a binary string S of N bits.

-----Output-----

For each test case, output a single line containing the number of exponential sequences that produce a palindrome.

-----Constraints-----
- 1 ≤ T ≤ 10
- Subtask 1(20 points): 1 ≤ N ≤ 20
- Subtask 2(30 points): 1 ≤ N ≤ 1000
- Subtask 3(50 points): 1 ≤ N ≤ 5 × 105

-----Note-----

The first test of the first subtask is the example test. It's made for you to make sure that your solution produces the same verdict both on your machine and our server.

-----Time Limits-----

Time limit for the first and the second subtasks is 3s. Time limit for the third subtask is 6s.

-----Example-----
Input:
2
11010
101001011

Output:
9
18

-----Explanation of the first case in the example test-----

The following sequences are counted in the answer: {1}, {2}, {3}, {4}, {5}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}.
user: A permutation of length n is an array of size n consisting of n distinct integers in the range [1, n]. For example, (3, 2, 4, 1) is a permutation of length 4, but (3, 3, 1, 4) and (2, 3, 4, 5) are not, as (3, 3, 1, 4) contains duplicate elements, and (2, 3, 4, 5) contains elements not in range [1,4].
A permutation p of length n is good if and only if for any 1 ≤ i ≤ n, pi ≠ i.
Please find the lexicographically smallest good permutation p.
Definition for "lexicographically smaller":
For two permutations p and q, we say that p is lexicographically smaller than q if and only if there exists a index 1 ≤ l ≤ n such that:
- For any 1 ≤ i < l, pi = qi. Note that if l = 1, this constraint means nothing.
- and, pl < ql.

For example, (2, 3, 1, 4) < (2, 3, 4, 1) < (3, 4, 1, 2). The lexicographically smallest permutation is, of course, (1, 2, ..., n), though this one is not good.

-----Input-----
First line of the input contains an integer T denoting number of test cases.
For each test case, the only line contains an integer n.

-----Output-----
For each test case, output the lexicographically smallest good permutation of length n. It's guaranteed that this permutation exists.

-----Constraints-----
- 1 ≤ T ≤ 10
- 2 ≤ n ≤ 105

-----Subtasks-----
- Subtask #1 (17 points): 2 ≤ n ≤ 9
- Subtask #2 (83 points): Original Constraints

-----Example-----
Input:
4
2
3
5
6

Output:
2 1
2 3 1
2 1 4 5 3
2 1 4 3 6 5

-----Explanation-----
Example case 1. The only good permutation of length 2 is (2, 1).
Example case 2. Consider all permutations of length 3, they are(in lexicographically order):
- p = (1, 2, 3), it's not good since p[1] = 1, p[2] = 2 and p[3] = 3;
- p = (1, 3, 2), it's not good since p[1] = 1;
- p = (2, 1, 3), it's not good since p[3] = 3;
- p = (2, 3, 1), it's good since p[1] ≠ 1, p[2] ≠ 2 and p[3] ≠ 3;
- p = (3, 1, 2), it's good since p[1] ≠ 1, p[2] ≠ 2 and p[3] ≠ 3;
- p = (3, 2, 1), it's not good since p[2] = 2.
Thus the minimum good one is (2, 3, 1).
Example case 3. Consider two good permutations for third test case: p=(2, 1, 4, 5, 3) and q=(2, 4, 1, 5, 3), then p < q. You can check lexicographically condition as follows. Find the first index where the entries of two permutations are different, and compare those entries. For example, in this case, the first position where the entries differ is index 2. You can see that p[2] < q[2], as 1 < 4, so p is lexicographically smaller than q.
user: Nobody knows, but $N$ frogs live in Chef's garden.
Now they are siting on the X-axis and want to speak to each other. One frog can send a message to another one if the distance between them is less or equal to $K$.
Chef knows all $P$ pairs of frogs, which want to send messages. Help him to define can they or not!
Note : More than $1$ frog can be on the same point on the X-axis.

-----Input-----
- The first line contains three integers $N$, $K$ and $P$.
- The second line contains $N$ space-separated integers $A_1$, $A_2$, …, $A_N$ denoting the x-coordinates of frogs".
- Each of the next $P$ lines contains two integers $A$ and $B$ denoting the numbers of frogs according to the input.

-----Output-----
For each pair print "Yes" without a brackets if frogs can speak and "No" if they cannot.

-----Constraints-----
- $1 \le N, P \le 10^5$
- $0 \le A_i, K \le 10^9$
- $1 \le A, B \le N$

-----Example-----

-----Sample Input:-----
5 3 3
0 3 8 5 12
1 2
1 3
2 5

-----Sample Output:-----
Yes
Yes
No

-----Explanation-----
- 
For pair $(1, 2)$ frog $1$ can directly speak to the frog $2$ as the distance between them is $3 - 0 = 3 \le K$ . 
- 
For pair $(1, 3)$ frog $1$ can send a message to frog $2$, frog $2$ can send it to frog $4$ and it can send it to frog $3$.
- 
For pair $(2, 5)$ frogs can't send a message under current constraints.
user: After six days, professor GukiZ decided to give more candies to his students. Like last time, he has $N$ students, numbered $1$ through $N$. Let's denote the number of candies GukiZ gave to the $i$-th student by $p_i$. As GukiZ has a lot of students, he does not remember all the exact numbers of candies he gave to the students. He only remembers the following properties of the sequence $p$:
- The numbers of candies given to each of the first $K$ students ($p_1, p_2, \dots, p_K$) are known exactly.
- All elements of the sequence $p$ are distinct and positive.
- GukiZ didn't give more than $x$ candies to any student (the maximum value in the sequence $p$ is not greater than $x$).
- For each student $i$, there is at least one other student $j$ such that $|p_i - p_j| \le D$.
- The professor gave out the biggest possible total number of candies, i.e. $S = p_1 + p_2 + p_3 + \ldots + p_N$ is maximum possible.
GukiZ would like to know the total number of candies $S$ he had at the beginning. However, times change and after six days, the professor is really tired, so it is possible that there is no sequence $p$ which satisfies the constraints. Can you help GukiZ find the number of candies he gave out, or tell him that he must have made a mistake?

-----Input-----
- The first line of the input contains a single integer $T$ denoting the number of test cases. The description of $T$ test cases follows.
- The first line of each test case contains four space-separated integers $N$, $K$, $x$, $D$.
- The second line contains $K$ distinct space-separated integers $p_1, p_2, \dots, p_K$.

-----Output-----
For each test case, print a single line containing one integer — the number of candies GukiZ had, or $-1$ if there is no valid sequence $p$.

-----Constraints-----
- $1 \le T \le 50$
- $3 \le N \le 10^9$
- $1 \le K \le \mathrm{min}(N, 2 \cdot 10^4)$ 
- $1 \le x \le 2 \cdot 10^9$
- $1 \le D \le 10^9$
- $1 \le p_i \le x$ for each valid $i$
- All values $p_i$ from input are distinct

-----Subtasks-----
Subtask #1 (15 points): $1 \leq x, N, D \leq 15$ 
Subtask #2 (35 points): $1 \leq x, N, D \leq 10^5$ 
Subtask #3 (50 points): original constraints

-----Example Input-----
2
4 3 5 3
2 1 5
3 2 8 2
3 8

-----Example Output-----
12
-1

-----Explanation-----
Example case 1: There are four students. We know that the first student got $p_1 = 2$ candies, the second student got $p_2 = 1$ and the third got $p_3 = 5$ candies; we don't know the number of candies given to the last student. The maximum possible amount of candies given to some student is $x=5$. The best possible option is giving $p_4=4$ candies to the last student. Then, the fourth constraint (with $D=3$) is satisfied for all students. Only the pair of students $(2, 3)$ have numbers of candies that differ by more than $3$, but still, for each student, there are at least two other students with close enough numbers of candies.
Example case 2: GukiZ made some mistake in distribution and there is no valid sequence $p$. The answer is $-1$.
user: You are given an integer $D$. Find an integer sequence $A_1, A_2, \ldots, A_N$ such that the following conditions are satisfied:
- $1 \le N \le 10^5$
- $1 \le A_i \le 10^5$ for each valid $i$
- $\sum_{i=1}^N \sum_{j=i}^N \left( \mathrm{min}(A_i, A_{i+1}, \ldots, A_j) - \mathrm{GCD}(A_i, A_{i+1}, \ldots, A_j) \right) = D$
It can be proved that a solution always exists under the given constraints.
Note: $\mathrm{GCD}(B_1, B_2, \ldots, B_M)$ is the greatest integer which divides all the integers $B_1, B_2, \ldots, B_M$.

-----Input-----
- The first line of the input contains a single integer $T$ denoting the number of test cases. The description of $T$ test cases follows.
- The first and only line of each test case contains a single integer $D$.

-----Output-----
For each test case, print two lines. The first of these lines should contain a single integer $N$. The second line should contain $N$ space-separated integers $A_1, A_2, \ldots, A_N$.
If there are multiple solutions, you may find any one of them.

-----Constraints-----
- $1 \le T \le 10$
- $0 \le D \le 10^9$

-----Example Input-----
4
2
5
200
13

-----Example Output-----
3
3 3 2 
5
2 8 5 1 10 
7
12 10 15 11 19 13 15
4
5 4 4 10
user: Pankaj likes to eat Ice cream when he is working late into the night. Today has been yet another long day for Pankaj. So, he wants to eat ice cream now. He opens the fridge and sees that he has 2 types of containers holding the ice cream.
The first container is a cone with radius r1 and height h1. There is also a hemisphere on the top of the cone which has the same radius. The other container is a cylindrical with radius r2 and height h2. Pankaj wants to know the amount (volume) of ice cream in both the containers. Since Pankaj is tired after coding all day, you have to help him with this task.

-----Input-----
- The first line of the input contains an integer T denoting the number of test cases. The description of T test cases follows.
- Each test case consists of a single line having the r1, h1, r2 and h2. Each value is given upto 2 decimal places. See example for more information.

-----Output-----
- For each test case, output a single line containing the volumes of the two containers separated by space. The answer is considered correct if it is correct upto 6 decimal places.

-----Constraints-----
- 1 ≤ T ≤ 100
- 0 <  r1, h1, r2, h2 ≤ 100

-----Example-----
Input:
2
1.00 1.00 1.00 1.00
3.02 7.23 5.20 6.00

Output:
3.141592654 3.141592654
126.739919445 509.691992118
user: Given an Integer N, write a program to reverse it.

-----Input-----

The first line contains an integer T, total number of testcases. Then follow T lines, each line contains an integer N. 

-----Output-----
For each test case, display the reverse of the given number N, in a new line.

-----Constraints-----
- 1 ≤ T ≤ 1000
- 1 ≤ N ≤ 1000000

-----Example-----
Input
4
12345
31203
2123
2300
Output
54321
30213
3212
32
user: Tanish is the president of the chemistry club at his school. He considers everyone who doesn't like chemistry as his enemy. After years of research, he has invented a lethal poison, which he named success. Now Tanish plans to kill all his enemies with his success. Success is extremely deadly and is untraceable in small concentrations.
Tanish has $\text{N}$ flasks lying on the shelf of his lab. All of these flasks contain the same volume of poison solution. The concentration of poison in the $\text{ith}$ flask is $\mathbf{a_i}$. In one move - Tanish picks up any two flasks, pours half of the contents of each of these flasks into a third initially empty flask, discards the two flasks and places the third flask (which now has the same volume of solution as all other flasks) on the shelf. He performs N-1 such moves to end up with only one flask of poison on the shelf. He wishes to perform these moves such that the concentration of poison in this flask is the minimum possible. What is the minimum concentration of poison that he can achieve?

-----Input-----
First line of the input contains a single integer $\text{T}$ - the number of test cases. 
For each test case,
The first line contains a single integer $\text{N}$, the number of bottles. 
The second line contains N space-separated integers, the concentrations. 

-----Output-----
For each test case, print one line containing the lowest final concentration. The output will be accepted if the absolute error is $\leq 10^{-6}$.

-----Constraints-----
$1 \leq \text{T} \leq 10$
$2 \leq \text{N} \leq 10^4$
$0 \leq \mathbf{a_i} \leq 10^9$

-----Sample Input-----
2
2
9 3
3
3 2 9

-----Sample Output-----
6.00000000
4.00000000

-----Explanation-----
In test 1, Tanish mixes flasks 1 and 2. In test 2, he first mixes flasks 1 and 3 to get a flask of concentration 6 and then mixes this flask with flask 2 to get a flask of poison concentration 4.    

-----Note-----
The concentration is given in the unit ppb i.e. parts per billion.
1 gram poison in $10^9$ ml solution is 1 ppb.
Concentration in ppb = (Number of grams of poison / Volume of solution in ml) x $10^9$
user: "Ring Ring!!"  
Sherlock's phone suddenly started ringing. And it was none other than Jim Moriarty..
"Long time no see ! You miss me right ? Anyway we'll talk about it later . Let me first tell you something. Dr.Watson is with me . And you've got only one chance to save him . Here's your challenge:.
Given a number N and another number M, tell if the remainder of N%M is odd or even. If it's odd, then print "ODD" else print "EVEN" 

If Sherlock can answer the query correctly, then Watson will be set free. He has approached you for help since you being a programmer.Can you help him? 

-----Input-----
The first line contains, T, the number of test cases..
Each test case contains an integer, N and M

-----Output-----
Output the minimum value for each test case 

-----Constraints-----
1 = T = 20
1 <= N <= 10^18 
1 <= M<= 10^9 

-----Subtasks-----
Subtask #1 : (20 points)
1 = T = 20
1 <= N <= 100 
1 <= M<= 100 

Subtask 2 : (80 points) 

1 = T = 20
1 <= N <= 10^18 
1 <= M<= 10^9 

-----Example-----
Input:
2
4 4
6 5

Output:
EVEN
ODD
user: Xorgon is an extremely delicious treat formed by the sequence $S$ of binary integers $s_1, s_2,..,s_N$. A really interesting property found in a Xorgon is that the xor of all elements in any contiguous subsequence of length $K$ in $S$ will result in $1$.   
Chef has been asked to prepare a Xorgon. However, he has at his disposal a binary sequence $X$ containing the binary integers $x_1, x_2, ...x_N$. To prepare a Xorgon, the chef may flip the value of as many binary digits in $X$ as required.(i.e. turn a $1$ to a $0$ and $0$ to a $1$). Unfortunately, flipping a digit takes a lot of time and the chef has to serve many orders. Can you help the chef calculate the minimum number of flips required to cook a Xorgon from the given $X$?

-----Input:-----
- The first line will contain two space-separated integers $N, K$.
- Next line contains N space-separated integers $x_1, x_2, ...,x_N$. 

-----Output:-----
Output in a single line minimum number of flips required to turn $X$ into a Xorgon.

-----Constraints-----
- $1 \leq K \leq N \leq 3*10^6$
- $0 \leq x_i \leq 1$

-----Sample Input:-----
7 5
1 0 0 1 1 1 1

-----Sample Output:-----
1

-----EXPLANATION:-----
Flip the last bit from 1 to 0 to obtain a Xorgon.
user: Naman owns a very famous Ice Cream parlour in Pune. He has a wide range of flavours with different pricing. 

Every flavour costs ₹ X per gram and quantity of each flavour in the parlour is indefinite. Now, Naman has 

received an order for a party wherein he is asked to prepare each Ice Cream with N number of flavours. 

He needs to prepare each Ice Cream of W grams and at the same time has to keep the cost minimum. 

Can you help him out?

Note: If it is not possible to prepare such an Ice Cream, print “Not Possible”.

-----Input:-----

First line contains an integer T denoting the number of test cases.

First line of every test case contains an integer N denoting the number of flavours.

The second line of every test case by N space separated integers denoting the cost per gram of each flavour. 

Second line of every test case contains two space separated integers W and Y denoting the weight 

of Ice Cream in grams and the number of flavours to be used respectively. 

-----Output:-----

For each test case, print the total cost of the Ice Cream in a single line.

-----Constraints:-----
1 ≤ T ≤ 100
1 ≤ N,W,Y ≤ 40000

-----Example:-----Input:
2
5 
4 6 8 1 10
10 3
2
1 2
1 2Output:

18

Not Possible

-----Explanation:-----

In the first test case to achieve the minimum cost, the Ice Cream will consist of 8 grams 

of flavour costing ₹1, 1 gram of flavour costing ₹4 and 1 gram of flavour costing ₹6.

In the second test case it is not possible to make an ice-cream weighing 1 gram using 2 flavours.
user: You are given two points $P$ and $Q$ and an opaque sphere in a three-dimensional space. The point $P$ is not moving, while $Q$ is moving in a straight line with constant velocity. You are also given a direction vector $d$ with the following meaning: the position of $Q$ at time $t$ is $Q(t) = Q(0) + d \cdot t$, where $Q(0)$ is the initial position of $Q$.
It is guaranteed that $Q$ is not visible from $P$ initially (at time $t=0$). It is also guaranteed that $P$ and $Q$ do not touch the sphere at any time.
Find the smallest positive time $t_v$ when $Q$ is visible from $P$, i.e. when the line segment connecting points $P$ and $Q$ does not intersect the sphere.

-----Input-----
- The first line of the input contains a single integer $T$ denoting the number of test cases. The description of $T$ test cases follows.
- The first and only line of each test case contains 13 space-separated integers.
- The first three integers $P_x, P_y, P_z$ denote the coordinates of $P$.
- The next three integers $Q_x, Q_y, Q_z$ denote the initial coordinates of $Q$.
- The next three integers $d_x, d_y, d_z$ denote the components of the direction vector $d$.
- The last four integers $c_x, c_y, c_z, r$ denote the coordinates of the centre of the sphere and its radius.

-----Output-----
For each test case, print a single line containing one real number — the time $t_v$. Your answer will be considered correct if its absolute or relative error does not exceed $10^{-6}$. It is guaranteed that $t_v$ exists and does not exceed $10^9$.

-----Constraints-----
- $1 \le T \le 10^5$
- the absolute values of coordinates of all points do not exceed $2\cdot10^9$
- $1 \le r \le 10^9$

-----Subtasks-----
Subtask #1 (25 points): $P_z = Q_z = d_z = c_z = 0$
Subtask #2 (75 points): original constraints

-----Example Input-----
1
3 0 0 -10 -10 0 0 10 0 0 -3 0 3

-----Example Output-----
1.0000000000
user: Spring is interesting season of year. Chef is thinking about different things, but last time he thinks about interesting game - "Strange Matrix". 
Chef has a matrix that consists of n rows, each contains m elements. Initially, the element aij of matrix equals j. (1 ≤ i ≤ n, 1 ≤ j ≤ m). 
Then p times some element aij is increased by 1. 
Then Chef needs to calculate the following: 

- For each row he tries to move from the last element (with number m) to the first one (with the number 1). 
- While staying in aij Chef can only move to aij - 1 only if aij - 1 ≤ aij. 
- The cost of such a movement is aij - aij - 1.
- Otherwise Chef can't move and lose (in this row).
- If Chef can move from the last element of the row to the first one, then the answer is the total cost of all the movements. 
- If Chef can't move from the last element of the row to the first one, then the answer is -1. 

Help Chef to find answers for all the rows after P commands of increasing. 

-----Input-----

- The first line contains three integers n, m and p denoting the number of rows, the number of elements a single row and the number of increasing commands. 
- Each of next p lines contains two integers i and j denoting that the element aij  is increased by one. 

-----Output-----
- For each row in a single line print the answer after the P increasing commands.

-----Constraints-----
- 1 ≤ n, m, p ≤ 10 ^ 5
- 1 ≤ i ≤ n
- 1 ≤ j ≤ m

-----Example-----
Input:
4 4 6
2 2
3 2 
3 2 
4 3
4 4
4 3

Output:
3
3
-1
4


-----Explanation-----

Here is the whole matrix after P commands:
1 2 3 4
1 3 3 4
1 4 3 4
1 2 5 5

Explanations to the answer: 
- The first line is without changes: 4-3=1, 3-2=1, 2-1=1. answer = 3. 
- The second line: 4-3=1, 3-3=0, 3-1=2. The answer is 3. 
- The third line: 4-3=1, 3-4=-1, Chef can't move to the first number here. Therefore, the answer is -1. 
- The fourth line: 5-5=0, 5-2=3, 2-1=1. The answer is 4.
user: One of Chef's friends offered him a deal: during $D$ days, they are going to exchange money. For each $i$ ($1 \le i \le D$), on the $i$-th day, Chef's friend would give Chef $A$ rupees, while Chef would give his friend $2^{i-1}$ rupees ($1$ rupee on day $1$, $2$ rupees on day $2$, $4$ rupees on day $3$, and so on). Chef's profit from the deal is the total number of rupees he received from his friend minus the total number of rupees he gave his friend.
Chef decided to ask for your advice before accepting the deal. You want to help him by telling him two numbers $D_1$ and $D_2$, where $D_1$ is the maximum value of $D$ such that Chef should accept the deal, i.e. his profit from the deal is positive if $D = D_1$, and $D_2$ is the value of $D$ that leads to the maximum possible profit for Chef. If there are multiple values of $D$ that lead to the maximum profit, $D_2$ is the smallest of these values.

-----Input-----
- The first line of the input contains a single integer $T$ denoting the number of test cases. The description of $T$ test cases follows.
- The first and only line of each test case contains a single integer $A$.

-----Output-----
For each test case, print a single line containing two space-separated integers $D_1$ and $D_2$.

-----Constraints-----
- $1 \le T \le 100,000$
- $5 \le A \le 10^9$

-----Subtasks-----
Subtask #1 (100 points): original constraints

-----Example Input-----
4
5
8
9
1000000000

-----Example Output-----
4 3
5 3
5 4
35 30
user: The chef was not happy with the binary number system, so he designed a new machine which is having 6 different states, i.e. in binary there is a total of 2 states as 0 and 1. Now, the chef is confused about how to correlate this machine to get an interaction with Integer numbers, when N(Integer number) is provided to the system, what will be the Nth number that system will return(in Integer form), help the chef to design this system.

-----Input:-----
- First-line will contain $T$, the number of test cases. Then the test cases follow. 
- Each test case contains a single line of input, $N$. 

-----Output:-----
For each test case, output in a single line answer given by the system.

-----Constraints-----
- $1 \leq T \leq 10^5$
- $1 \leq N \leq 10^5$

-----Sample Input:-----
2
3
5

-----Sample Output:-----
7
37

-----EXPLANATION:-----
Initial numbers for system = [1, 6, 7, 36, 37, 42, 43, 216, …..
For 1) 3rd Number for the system will be 7.
For 2) 5th Number for the system will be 37.
user: If Give an integer N . Write a program to obtain the sum of the first and last digits of this number.

-----Input-----

The first line contains an integer T, the total number of test cases. Then follow T lines, each line contains an integer N. 

-----Output-----
For each test case, display the sum of first and last digits of N in a new line.

-----Constraints-----
- 1 ≤ T ≤ 1000
- 1 ≤ N ≤ 1000000

-----Example-----
Input
3 
1234
124894
242323

Output
5
5
5