vgel/INTELLECT-2-RL-Dataset-Code-Only huggingface dataset:
Chef calls a pair of integers (A, B) *equivalent* if there exist some positive integers X and Y such that A^{X} = B^{Y}.

Given A and B, determine whether the pair is *equivalent* or not.

------ Input Format ------ 

- The first line of input will contain a single integer T, denoting the number of test cases.
- Each test case consists of two space-separated integers A and B, as mentioned in statement.

------ Output Format ------ 

For each test case, output YES if (A, B) is an *equivalent* pair, NO otherwise.

The output is case-insensitive. Thus, the strings Yes, YES, yes, and yeS are all considered identical.

------ Constraints ------ 

$1 ≤ T ≤ 10^{5}$
$2 ≤ A, B ≤ 10^{6}$

----- Sample Input 1 ------ 
3
2 3
8 4
12 24

----- Sample Output 1 ------ 
NO
YES
NO

----- explanation 1 ------ 
Test case $1$: There are no positive integers $X$ and $Y$ which satisfy $A^{X} = B^{Y}$.

Test case $2$: Let $X = 2$ and $Y = 3$. Thus, $A^{X} = 8^{2} = 64$ and $B^{Y} = 4^{3} = 64$. Thus, the pair $(8, 4)$ is *equivalent*.

Test case $3$: There are no positive integers $X$ and $Y$ which satisfy $A^{X} = B^{Y}$.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
You are given a string s containing lowercase letters and an integer k. You need to :

First, change some characters of s to other lowercase English letters.
Then divide s into k non-empty disjoint substrings such that each substring is palindrome.

Return the minimal number of characters that you need to change to divide the string.
 
Example 1:
Input: s = "abc", k = 2
Output: 1
Explanation: You can split the string into "ab" and "c", and change 1 character in "ab" to make it palindrome.

Example 2:
Input: s = "aabbc", k = 3
Output: 0
Explanation: You can split the string into "aa", "bb" and "c", all of them are palindrome.
Example 3:
Input: s = "leetcode", k = 8
Output: 0

 
Constraints:

1 <= k <= s.length <= 100.
s only contains lowercase English letters.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Title: A Determined Cleanup

Time Limit: None seconds

Memory Limit: None megabytes

Problem Description:
In order to put away old things and welcome a fresh new year, a thorough cleaning of the house is a must.

Little Tommy finds an old polynomial and cleaned it up by taking it modulo another. But now he regrets doing this...

Given two integers *p* and *k*, find a polynomial *f*(*x*) with non-negative integer coefficients strictly less than *k*, whose remainder is *p* when divided by (*x*<=+<=*k*). That is, *f*(*x*)<==<=*q*(*x*)·(*x*<=+<=*k*)<=+<=*p*, where *q*(*x*) is a polynomial (not necessarily with integer coefficients).

Input Specification:
The only line of input contains two space-separated integers *p* and *k* (1<=≤<=*p*<=≤<=1018, 2<=≤<=*k*<=≤<=2<=000).

Output Specification:
If the polynomial does not exist, print a single integer -1, or output two lines otherwise.

In the first line print a non-negative integer *d* — the number of coefficients in the polynomial.

In the second line print *d* space-separated integers *a*0,<=*a*1,<=...,<=*a**d*<=-<=1, describing a polynomial  fulfilling the given requirements. Your output should satisfy 0<=≤<=*a**i*<=&lt;<=*k* for all 0<=≤<=*i*<=≤<=*d*<=-<=1, and *a**d*<=-<=1<=≠<=0.

If there are many possible solutions, print any of them.

Demo Input:
['46 2\n', '2018 214\n']

Demo Output:
['7\n0 1 0 0 1 1 1\n', '3\n92 205 1\n']

Note:
In the first example, *f*(*x*) = *x*<sup class="upper-index">6</sup> + *x*<sup class="upper-index">5</sup> + *x*<sup class="upper-index">4</sup> + *x* = (*x*<sup class="upper-index">5</sup> - *x*<sup class="upper-index">4</sup> + 3*x*<sup class="upper-index">3</sup> - 6*x*<sup class="upper-index">2</sup> + 12*x* - 23)·(*x* + 2) + 46.

In the second example, *f*(*x*) = *x*<sup class="upper-index">2</sup> + 205*x* + 92 = (*x* - 9)·(*x* + 214) + 2018.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
There are $n$ players sitting at the card table. Each player has a favorite number. The favorite number of the $j$-th player is $f_j$.

There are $k \cdot n$ cards on the table. Each card contains a single integer: the $i$-th card contains number $c_i$. Also, you are given a sequence $h_1, h_2, \dots, h_k$. Its meaning will be explained below.

The players have to distribute all the cards in such a way that each of them will hold exactly $k$ cards. After all the cards are distributed, each player counts the number of cards he has that contains his favorite number. The joy level of a player equals $h_t$ if the player holds $t$ cards containing his favorite number. If a player gets no cards with his favorite number (i.e., $t=0$), his joy level is $0$.

Print the maximum possible total joy levels of the players after the cards are distributed. Note that the sequence $h_1, \dots, h_k$ is the same for all the players.


-----Input-----

The first line of input contains two integers $n$ and $k$ ($1 \le n \le 500, 1 \le k \le 10$) — the number of players and the number of cards each player will get.

The second line contains $k \cdot n$ integers $c_1, c_2, \dots, c_{k \cdot n}$ ($1 \le c_i \le 10^5$) — the numbers written on the cards.

The third line contains $n$ integers $f_1, f_2, \dots, f_n$ ($1 \le f_j \le 10^5$) — the favorite numbers of the players.

The fourth line contains $k$ integers $h_1, h_2, \dots, h_k$ ($1 \le h_t \le 10^5$), where $h_t$ is the joy level of a player if he gets exactly $t$ cards with his favorite number written on them. It is guaranteed that the condition $h_{t - 1} < h_t$ holds for each $t \in [2..k]$.


-----Output-----

Print one integer — the maximum possible total joy levels of the players among all possible card distributions.


-----Examples-----
Input
4 3
1 3 2 8 5 5 8 2 2 8 5 2
1 2 2 5
2 6 7

Output
21

Input
3 3
9 9 9 9 9 9 9 9 9
1 2 3
1 2 3

Output
0



-----Note-----

In the first example, one possible optimal card distribution is the following:  Player $1$ gets cards with numbers $[1, 3, 8]$;  Player $2$ gets cards with numbers $[2, 2, 8]$;  Player $3$ gets cards with numbers $[2, 2, 8]$;  Player $4$ gets cards with numbers $[5, 5, 5]$. 

Thus, the answer is $2 + 6 + 6 + 7 = 21$.

In the second example, no player can get a card with his favorite number. Thus, the answer is $0$.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Given a matrix you need to find the submatrix with the largest number of elements, where the GCD (Greatest Common Divisor) of its elements is greater than one. A submatrix of the matrix is a sub-section composed of contiguous rows and columns of the original matrix.

Input
Two numbers n,m in the first line. Followed by n lines with m numbers in each line.

Constraints  

1<=N,M<=200

1<=numbers<=10000

Output
Just a largest area where GCD is greater than 1.

Sample Input  

3 3
2 6 8
4 8 3
6 9 4

Sample Output  

4

If you observe the following submatrix:  

2 6  
4 8  

The GCD is 2.
There is no matrix larger than this with a GCD > 1.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Title: Sereja and Subsequences

Time Limit: None seconds

Memory Limit: None megabytes

Problem Description:
Sereja has a sequence that consists of *n* positive integers, *a*1,<=*a*2,<=...,<=*a**n*. 

First Sereja took a piece of squared paper and wrote all distinct non-empty non-decreasing subsequences of sequence *a*. Then for each sequence written on the squared paper, Sereja wrote on a piece of lines paper all sequences that do not exceed it.

A sequence of positive integers *x*<==<=*x*1,<=*x*2,<=...,<=*x**r* doesn't exceed a sequence of positive integers *y*<==<=*y*1,<=*y*2,<=...,<=*y**r*, if the following inequation holds: *x*1<=≤<=*y*1,<=*x*2<=≤<=*y*2,<=...,<=*x**r*<=≤<=*y**r*.

Now Sereja wonders, how many sequences are written on the lines piece of paper. Help Sereja, find the required quantity modulo 1000000007 (109<=+<=7).

Input Specification:
The first line contains integer *n* (1<=≤<=*n*<=≤<=105). The second line contains *n* integers *a*1,<=*a*2,<=...,<=*a**n* (1<=≤<=*a**i*<=≤<=106).

Output Specification:
In the single line print the answer to the problem modulo 1000000007 (109<=+<=7).

Demo Input:
['1\n42\n', '3\n1 2 2\n', '5\n1 2 3 4 5\n']

Demo Output:
['42\n', '13\n', '719\n']

Note:
none

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Title: Broken checker

Time Limit: None seconds

Memory Limit: None megabytes

Problem Description:
"This problem is rubbish! There is not statement, and there are only 5 test cases. The problemsetter took liberties with this problem!" — people complained in the comments to one round on Codeforces. And even more... No, wait, the checker for the problem was alright, that's a mercy.

Input Specification:
The only line of the input contains an integer between 1 and 5, inclusive. All tests for this problem are different. The contents of the test case doesn't need to be equal to its index.

Output Specification:
The only line of the output contains an integer between 1 and 3, inclusive.

Note:
This problem has no samples, since there so few test cases.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Consider two non-negative long integers, $a$ and $\boldsymbol{b}$, where $a\leq b$. The bitwise AND of all long integers in the inclusive range between $a$ and $\boldsymbol{b}$ can be expressed as $a\ \text{&}\ (a+1)\ \text{&}\ \ldots\ \text{&}\ (b-1)\ \text{&}\ b$, where $\text{&}$ is the bitwise AND operator. 

Given $n$ pairs of long integers, $a[i]$ and $b[i]$, compute and print the bitwise AND of all natural numbers in the inclusive range between $a[i]$ and $b[i]$.

For example, if $\boldsymbol{a=10}$ and $b=14$, the calculation is $10\:\text{&}\:11\:\text{&}\:12\:\text{&}\:13\:\text{&}\:14=8$.  

Function Description  

Complete the andProduct in the editor below.  It should return the computed value as an integer.  

andProduct has the following parameter(s):  

a: an integer  
b: an integer  

Input Format

The first line contains a single integer $n$, the number of intervals to test. 

Each of the next $n$ lines contains two space-separated integers $a[i]$ and $b[i]$.

Constraints

$1\leq n\leq200$  
$0\leq a[i]\leq b[i]<2^{32}$

Output Format

For each pair of long integers, print the bitwise AND of all numbers in the inclusive range between $a[i]$ and $b[i]$ on a new line.

Sample Input 0
3
12 15
2 3
8 13

Sample Output 0
12
2
8

Explanation 0

There are three pairs to compute results for:

$\boldsymbol{a=12}$ and $\boldsymbol{b=15}$ 

$12~\text{&}~13~\text{&}~14~\text{&}~15~=12$, so we print $12$ on a new line.
$\boldsymbol{a}=2$ and $b=3$ 

$2\&3=2$
$\boldsymbol{a}=8$ and $b=13$ 

$8\:\text{&}\:9\:\text{&}\:10\:\text{&}\:11\:\text{&}\:12\&\:13\:=8$

Sample Input 1
2
17 23
11 15

Sample Output 1
16
8

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Chef's love for Fibonacci numbers helps him to design following interesting problem.
He defines a function F for an array S as follows:

where
- Si denotes a non-empty subset of multiset S.
- sum(Si) denotes the sum of all element of multiset Si.
- Fibonacci(x) denotes the xth Fibonacci number.
Given an array A consisting of N elements. Chef asked you to process following two types of queries on this array accurately and efficiently.
- C X Y: Change the value of Xth element of array to Y i.e AX = Y.
- Q L R: Compute function F over the subarray defined by the elements of array A in the range L to R, both inclusive.

Please see the Note section if you need details regarding Fibonacci function.

-----Input-----
First line of input contains 2 space separated integer N and M denoting the size of array A and the number of queries to be performed. Next line of input contains N space separated integers denoting the elements of array A. Each of next M lines of input contains a query having one of the mentioned two types.

-----Output-----
For each query of type Q, output the value of function F for the given range of array A.

-----Constraints-----
- 1 ≤ N, M ≤ 105
- 1 ≤ Ai, Y ≤ 109
- 1 ≤ L, R, X ≤ N
-  type = {'C', 'Q'} 

-----Subtasks-----
- Subtask 1 (20 points) : 1 ≤ N, M ≤ 1000, 1 ≤ Ai, Y ≤ 106, type = { 'Q' } 
- Subtask 2 (20 points) : 1 ≤ N, M ≤ 50000, 1 ≤ Ai, Y ≤ 109, type = { 'C', Q' } 
- Subtask 3 (30 points) : 1 ≤ N, M ≤ 105, 1 ≤ Ai, Y ≤ 109, type = { 'Q' } 
- Subtask 4 (30 points) : 1 ≤ N, M ≤ 105, 1 ≤ Ai, Y ≤ 109, type = { 'C', Q' }

-----Example-----
Input
3 5
1 2 3
Q 1 2
Q 2 3
C 1 2
Q 1 2
Q 1 3

Output
4
8
5
30

-----Explanation:-----
- Q1 : F = Fibonacci(1) + Fibonacci(2) + Fibonacci(1+2) = 4 % 1000000007 = 4
- Q2 : F = Fibonacci(2) + Fibonacci(3) + Fibonacci(2+3) = 8 % 1000000007 = 8
- Q3 : A = {2, 2, 3}
- Q4 : F = Fibonacci(2) + Fibonacci(2) + Fibonacci(2+2) = 5 % 1000000007 = 5
- Q5 : F = Fibonacci(2) + Fibonacci(2) + Fibonacci(3) + Fibonacci(2+2) + Fibonacci(2+3) + Fibonacci(2+3) + Fibonacci(2+2+3) = 30 % 1000000007 = 30

-----Note-----
FibonacciK denotes the Kth Fibonacci number. Fibonacci series is defined as follows:
- For 1 ≤ K ≤ 2, FibonacciK = 1
- Otherwise, FibonacciK = FibonacciK-1 + FibonacciK-2
Please check this link for more details about Fibonacci numbers.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Aditya can  always be seen eating bananas. On his birthday he decides to give his friends a Banana treat.
As we know Aditya is a miser ,he  bought only N bananas .K  hungry friends came to his party.

Now,its party time and Aditya is too busy with his birthday preparations.His friends will be happy only if all of them get equal number of bananas.You need to tell if Aditya will be able to make all his friends happy or not.

If he is able to distribute all bananas equally among friends print "YES" (Quotes are for understanding)
 also state number of bananas each friend will get else print "NO" :(

Note: Aditya will not waste any bananas.

Constraints:

1 ≤ N ≤ 10^9

0 ≤ K ≤ 10^9

Input Format:

Two space separated integers N & K

Output Format

Print YES if all bananas can be distributed equally among friends.Also output number of bananas each friend gets in this case.
Print NO if bananas cannot be  distributed equally.

SAMPLE INPUT
24 8

SAMPLE OUTPUT
YES
3

Explanation

There were 8 friends at the party .Each of them got 3 bananas.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
We have three boxes A, B, and C, each of which contains an integer.

Currently, the boxes A, B, and C contain the integers X, Y, and Z, respectively.

We will now do the operations below in order. Find the content of each box afterward.  
 - Swap the contents of the boxes A and B
 - Swap the contents of the boxes A and C

-----Constraints-----
 - 1 \leq X,Y,Z \leq 100
 - All values in input are integers.

-----Input-----
Input is given from Standard Input in the following format:
X Y Z

-----Output-----
Print the integers contained in the boxes A, B, and C, in this order, with space in between.

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

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

After the contents of the boxes A and B are swapped, A, B, and C contain 2, 1, and 3, respectively.

Then, after the contents of A and C are swapped, A, B, and C contain 3, 1, and 2, respectively.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
You are given a permutation of n numbers p_1, p_2, ..., p_{n}. We perform k operations of the following type: choose uniformly at random two indices l and r (l ≤ r) and reverse the order of the elements p_{l}, p_{l} + 1, ..., p_{r}. Your task is to find the expected value of the number of inversions in the resulting permutation.


-----Input-----

The first line of input contains two integers n and k (1 ≤ n ≤ 100, 1 ≤ k ≤ 10^9). The next line contains n integers p_1, p_2, ..., p_{n} — the given permutation. All p_{i} are different and in range from 1 to n.

The problem consists of three subproblems. The subproblems have different constraints on the input. You will get some score for the correct submission of the subproblem. The description of the subproblems follows.

  In subproblem G1 (3 points), the constraints 1 ≤ n ≤ 6, 1 ≤ k ≤ 4 will hold.  In subproblem G2 (5 points), the constraints 1 ≤ n ≤ 30, 1 ≤ k ≤ 200 will hold.  In subproblem G3 (16 points), the constraints 1 ≤ n ≤ 100, 1 ≤ k ≤ 10^9 will hold. 


-----Output-----

Output the answer with absolute or relative error no more than 1e - 9.


-----Examples-----
Input
3 1
1 2 3

Output
0.833333333333333

Input
3 4
1 3 2

Output
1.458333333333334



-----Note-----

Consider the first sample test. We will randomly pick an interval of the permutation (1, 2, 3) (which has no inversions) and reverse the order of its elements. With probability $\frac{1}{2}$, the interval will consist of a single element and the permutation will not be altered. With probability $\frac{1}{6}$ we will inverse the first two elements' order and obtain the permutation (2, 1, 3) which has one inversion. With the same probability we might pick the interval consisting of the last two elements which will lead to the permutation (1, 3, 2) with one inversion. Finally, with probability $\frac{1}{6}$ the randomly picked interval will contain all elements, leading to the permutation (3, 2, 1) with 3 inversions. Hence, the expected number of inversions is equal to $\frac{1}{2} \cdot 0 + \frac{1}{6} \cdot 1 + \frac{1}{6} \cdot 1 + \frac{1}{6} \cdot 3 = \frac{5}{6}$.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Title: Shooting Gallery

Time Limit: 2 seconds

Memory Limit: 256 megabytes

Problem Description:
One warm and sunny day king Copa decided to visit the shooting gallery, located at the Central Park, and try to win the main prize — big pink plush panda. The king is not good at shooting, so he invited you to help him.

The shooting gallery is an infinite vertical plane with Cartesian coordinate system on it. The targets are points on this plane. Each target is described by it's coordinates *x**i*, and *y**i*, by the time of it's appearance *t**i* and by the number *p**i*, which gives the probability that Copa hits this target if he aims at it.

A target appears and disappears instantly, so Copa can hit the target only if at the moment *t**i* his gun sight aimed at (*x**i*,<=*y**i*). Speed of movement of the gun sight on the plane is equal to 1. Copa knows all the information about the targets beforehand (remember, he is a king!). He wants to play in the optimal way, which maximizes the expected value of the amount of hit targets. He can aim at any target at the moment 0.

Input Specification:
The first line contains integer *n* (1<=≤<=*n*<=≤<=1000) — amount of targets in the shooting gallery. Then *n* lines follow, each describing one target. Each description consists of four numbers *x**i*, *y**i*, *t**i*, *p**i* (where *x**i*, *y**i*, *t**i* — integers, <=-<=1000<=≤<=*x**i*,<=*y**i*<=≤<=1000,<=0<=≤<=*t**i*<=≤<=109, real number *p**i* is given with no more than 6 digits after the decimal point, 0<=≤<=*p**i*<=≤<=1). No two targets may be at the same point.

Output Specification:
Output the maximum expected value of the amount of targets that was shot by the king. Your answer will be accepted if it differs from the correct answer by not more than 10<=-<=6.

Demo Input:
['1\n0 0 0 0.5\n', '2\n0 0 0 0.6\n5 0 5 0.7\n']

Demo Output:
['0.5000000000\n', '1.3000000000\n']

Note:
none

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Levko loves array a_1, a_2, ... , a_{n}, consisting of integers, very much. That is why Levko is playing with array a, performing all sorts of operations with it. Each operation Levko performs is of one of two types:

  Increase all elements from l_{i} to r_{i} by d_{i}. In other words, perform assignments a_{j} = a_{j} + d_{i} for all j that meet the inequation l_{i} ≤ j ≤ r_{i}.  Find the maximum of elements from l_{i} to r_{i}. That is, calculate the value $m_{i} = \operatorname{max}_{j = l_{i}}^{r_{i}} a_{j}$. 

Sadly, Levko has recently lost his array. Fortunately, Levko has records of all operations he has performed on array a. Help Levko, given the operation records, find at least one suitable array. The results of all operations for the given array must coincide with the record results. Levko clearly remembers that all numbers in his array didn't exceed 10^9 in their absolute value, so he asks you to find such an array.


-----Input-----

The first line contains two integers n and m (1 ≤ n, m ≤ 5000) — the size of the array and the number of operations in Levko's records, correspondingly.

Next m lines describe the operations, the i-th line describes the i-th operation. The first integer in the i-th line is integer t_{i} (1 ≤ t_{i} ≤ 2) that describes the operation type. If t_{i} = 1, then it is followed by three integers l_{i}, r_{i} and d_{i} (1 ≤ l_{i} ≤ r_{i} ≤ n,  - 10^4 ≤ d_{i} ≤ 10^4) — the description of the operation of the first type. If t_{i} = 2, then it is followed by three integers l_{i}, r_{i} and m_{i} (1 ≤ l_{i} ≤ r_{i} ≤ n,  - 5·10^7 ≤ m_{i} ≤ 5·10^7) — the description of the operation of the second type.

The operations are given in the order Levko performed them on his array.


-----Output-----

In the first line print "YES" (without the quotes), if the solution exists and "NO" (without the quotes) otherwise.

If the solution exists, then on the second line print n integers a_1, a_2, ... , a_{n} (|a_{i}| ≤ 10^9) — the recovered array.


-----Examples-----
Input
4 5
1 2 3 1
2 1 2 8
2 3 4 7
1 1 3 3
2 3 4 8

Output
YES
4 7 4 7
Input
4 5
1 2 3 1
2 1 2 8
2 3 4 7
1 1 3 3
2 3 4 13

Output
NO

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Number is a palindrome if it is equal to the number with digits in reversed order.
For example, 5, 44, 171, 4884 are palindromes and 43, 194, 4773 are not palindromes.

Write a method `palindrome_chain_length` which takes a positive number and returns the number of special steps needed to obtain a palindrome. The special step is: "reverse the digits, and add to the original number". If the resulting number is not a  palindrome, repeat the procedure with the sum until the resulting number is a palindrome.

If the input number is already a palindrome, the number of steps is 0.

Input will always be a positive integer.

For example, start with 87:

87 + 78 = 165;
165 + 561 = 726;
 726 + 627 = 1353;
1353 + 3531 = 4884

4884 is a palindrome and we needed 4 steps to obtain it, so `palindrome_chain_length(87) == 4`

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.