vgel/INTELLECT-2-RL-Dataset-Code-Only huggingface dataset:
Problem Statement
Any 2 dimensional space can be tiled (covered using tiles) using a fractal pattern called Rauzy Fractal. This is a very simple tiling pattern which follows a simple recurrence. We start with 3 tiles. In the first 3 steps, we lay down 1st tile, the 2nd tile, and the 3rd tile. Then in each consecutive step, we lay down whatever tiles were laid down in the previous three steps. Below is a simple illustration of the tiling process in each step where a, b and c denote the first 3 tiles.

a

b

c

abc

abc c b

abccb abc c

...

Now given the size of the first three tiles can you calculate the total area of the floor covered after N steps of tiling?

Input Format
First line gives the number of test cases T. Then each line after that has the size of the three tiles (a, b and c) followed by N (each separated by a space).

Constraints
1 ≤ T ≤ 5;
0 ≤ a, b, c ≤ 1000003;
0 ≤ N ≤ 10^9

Output Format
Print T lines, each line containing the total area covered, modulo the prime 1000003.

SAMPLE INPUT
5
1 1 1 1
1 1 1 2
1 1 1 4
1 2 3 4
1 2 3 6

SAMPLE OUTPUT
1
2
6
12
43

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Bleatrix Trotter the sheep has devised a strategy that helps her fall asleep faster. First, she picks a number N. Then she starts naming N, 2 × N, 3 × N, and so on.
Whenever she names a number, she thinks about all of the digits in that number. She keeps track of which digits (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) she has seen at least once so far as part of any number she has named. Once she has seen each of the ten digits at least once, she will fall asleep.
Bleatrix must start with N and must always name (i + 1) × N directly after i × N.
For example, suppose that Bleatrix picks N = 1692. She would count as follows:

N = 1692. Now she has seen the digits 1, 2, 6, and 9.
2N = 3384. Now she has seen the digits 1, 2, 3, 4, 6, 8, and 9.
3N = 5076. Now she has seen all ten digits, and falls asleep.

The purpose of this kata is to return the last number Bleatrix Trotter sees before falling asleep.
Input
Will always be positive integer or zero

Output
The last number Bleatrix Trotter sees or "INSOMNIA" (-1 in Rust and C++) if she will count forever

Please note, this challenge is not my idea. It's from Google Code Jam 2016

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
On her way to ChefLand, Marichka noticed $10^K$ road signs (numbered $0$ through $10^K - 1$). For each valid $i$, the sign with number $i$ had the integer $i$ written on one side and $10^K-i-1$ written on the other side.
Now, Marichka is wondering — how many road signs have exactly two distinct decimal digits written on them (on both sides in total)? Since this number may be large, compute it modulo $10^9+7$.
For example, if $K = 3$, the two integers written on the road sign $363$ are $363$ and $636$, and they contain two distinct digits $3$ and $6$, but on the road sign $362$, there are integers $362$ and $637$, which contain four distinct digits — $2$, $3$, $6$ and $7$. On the road sign $11$, there are integers $11$ and $988$, which contain three distinct digits — $1$, $9$ and $8$.

-----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 $K$.

-----Output-----
For each test case, print a single line containing one integer — the number of road signs with exactly two digits, modulo $10^9+7$.

-----Constraints-----
- $1 \le T \le 10^5$
- $1 \le K \le 10^9$

-----Subtasks-----
Subtask #1 (20 points): $1 \le T, K \le 5$
Subtask #2 (80 points): original constraints

-----Example Input-----
1
1

-----Example Output-----
10

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

Time Limit: 2 seconds

Memory Limit: 256 megabytes

Problem Description:
You are given a *n*<=×<=*m* field consisting only of periods ('.') and asterisks ('*'). Your task is to count all right triangles with two sides parallel to the square sides, whose vertices are in the centers of '*'-cells. A right triangle is a triangle in which one angle is a right angle (that is, a 90 degree angle).

Input Specification:
The first line contains two positive integer numbers *n* and *m* (1<=≤<=*n*,<=*m*<=≤<=1000). The following *n* lines consist of *m* characters each, describing the field. Only '.' and '*' are allowed.

Output Specification:
Output a single number — total number of square triangles in the field. Please, do not use %lld specificator to read or write 64-bit integers in C++. It is preffered to use cout (also you may use %I64d).

Demo Input:
['2 2\n**\n*.\n', '3 4\n*..*\n.**.\n*.**\n']

Demo Output:
['1\n', '9\n']

Note:
none

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
There are N towns in the State of Atcoder, connected by M bidirectional roads.
The i-th road connects Town A_i and B_i and has a length of C_i.
Joisino is visiting R towns in the state, r_1,r_2,..,r_R (not necessarily in this order).
She will fly to the first town she visits, and fly back from the last town she visits, but for the rest of the trip she will have to travel by road.
If she visits the towns in the order that minimizes the distance traveled by road, what will that distance be?

-----Constraints-----
 - 2≤N≤200
 - 1≤M≤N×(N-1)/2
 - 2≤R≤min(8,N) (min(8,N) is the smaller of 8 and N.)
 - r_i≠r_j (i≠j)
 - 1≤A_i,B_i≤N, A_i≠B_i
 - (A_i,B_i)≠(A_j,B_j),(A_i,B_i)≠(B_j,A_j) (i≠j)
 - 1≤C_i≤100000
 - Every town can be reached from every town by road.
 - All input values are integers.

-----Input-----
Input is given from Standard Input in the following format:
N M R
r_1 ... r_R
A_1 B_1 C_1
:
A_M B_M C_M

-----Output-----
Print the distance traveled by road if Joisino visits the towns in the order that minimizes it.

-----Sample Input-----
3 3 3
1 2 3
1 2 1
2 3 1
3 1 4

-----Sample Output-----
2

For example, if she visits the towns in the order of 1, 2, 3, the distance traveled will be 2, which is the minimum possible.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Chef taught his brother Chefu about right angled triangle and its properties. Chefu says that he has understood everything about right angled triangles. Chef wants to check learning of his brother by asking the following question "Can you find a right angled triangle whose length of hypotenuse  is H and its area is S?"
Chefu is confused how to solve it. I hope you are not. Please solve this by finding a right angled triangle with hypotenuse H and area S. If it not possible to do so, then output -1.

Input
The first line of the input contains a single integer T denoting the number of test-cases. T test cases follow.
For each test case, there will be a single line containing two space separated integers H and S.

Output
Output the answer for each test-case in a single line. If it is not possible to find such a triangle, output -1. Otherwise print 3 real numbers corresponding to the lengths of the sides of the triangle sorted in non-decreasing order. Please note that the length of the triangle sides should not differ by more than 0.01 in absolute value from the correct lengths.

Constraints

1 ≤ T ≤ 10^5
1 ≤ H ≤ 10^6
1 ≤ S ≤ 10^12


Example
Input:
4
5 6
6 10
258303 89837245228
616153 77878145466

Output:
3.00000 4.00000 5.00000
-1
-1
285168.817674 546189.769984 616153.000000

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Celery — the legendary Pokenom has been spotted in Alexa Forest.

To become the best Pokenom trainer, Bash has arrived at Alexa Forest to capture Celery. After lots of information gathering, Bash was able to draw a map of Alexa Forest, and noted down $K$ sightings of Celery.

Alexa Forest’s map is a convex polygon $A$ with $N$ vertices on the Cartesian plane. $K$ sightings of Celery can be considered as $K$ points — all are strictly inside Alexa Forest.

Bash is ready to search Alexa Forest to find Celery. However, Bash realized that Alexa Forest is simply too big. It would take decades to search the entire forest. But Bash is smart. Based on his research, Bash knows that Celery can only be found inside a polygon $Z$, where the vertices of $Z$ are a subset of $A$, and all $K$ sightings of Celery must be strictly inside polygon $Z$.

Of course, there can be multiple polygons $Z$ satisfying the above conditions. Your task is to help Bash find the polygon $Z$ with smallest number of vertices.

-----Note-----
A point $P$ is strictly inside Polygon $A$, iff $P$ is inside $A$ and $P$ does not lie on the border of $A$.

-----Input-----
 - The first line of input contains a single positive integer $N$$(3 \le N \le 2 \cdot 10^5)$.
 - The next $N$ lines each contain $2$ integers $x_ i$, $y_ i$ — the coordinates of the $i$-th vertex of Alexa Forest $(-10^9 \le x_ i, y_ i \le 10^9)$. The vertices are listed in either clockwise or counterclockwise order. It is guaranteed that Alexa Forest is convex.
 - The next line contains a single positive integer $K$ $(1 \le K \le 10^5)$.
 - The next $K$ lines, each line contains $2$ integers $x_ i$, $y_ i$ — the coordinates of a sighting of Celery $(-10^9 \le x_ i, y_ i \le 10^9)$. All points are guaranteed to be inside Alexa Forest and no points are on the border of Alexa Forest.

-----Output-----
Output a single integer — the smallest number of vertices of polygon $Z$.

-----Sample Clarification-----
 - In the first example, the only valid polygon satisfied is the whole Alexa Forest.
 - In the second example, there are two possible solutions with $4$ vertices:

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

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Ayrat is looking for the perfect code. He decided to start his search from an infinite field tiled by hexagons. For convenience the coordinate system is introduced, take a look at the picture to see how the coordinates of hexagon are defined: 

[Image] [Image] Ayrat is searching through the field. He started at point (0, 0) and is moving along the spiral (see second picture). Sometimes he forgets where he is now. Help Ayrat determine his location after n moves.


-----Input-----

The only line of the input contains integer n (0 ≤ n ≤ 10^18) — the number of Ayrat's moves.


-----Output-----

Print two integers x and y — current coordinates of Ayrat coordinates.


-----Examples-----
Input
3

Output
-2 0

Input
7

Output
3 2

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
You are given a sequence a1, a2, ..., aN. Find the smallest possible value of ai + aj, where 1 ≤ i < j ≤ N.

-----Input-----
The first line of the input contains an integer T denoting the number of test cases. The description of T test cases follows. 
The first line of each description consists of a single integer N.
The second line of each description contains N space separated integers - a1, a2, ..., aN respectively.

-----Output-----
For each test case, output a single line containing a single integer - the smallest possible sum for the corresponding test case.

-----Constraints-----

- T = 105, N = 2 : 13 points.
- T = 105, 2 ≤ N ≤ 10 : 16 points.
- T = 1000, 2 ≤ N ≤ 100 : 31 points.
- T = 10, 2 ≤ N ≤ 105 : 40 points.
- 1 ≤ ai ≤ 106

-----Example-----
Input:
1
4
5 1 3 4

Output:
4

-----Explanation-----
Here we pick a2 and a3. Their sum equals to 1 + 3 = 4.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Roy bought a Ring for Alfi, his girlfriend :p. But Alfi will only accept the Ring if Roy played a game with her and answered her queries correctly during the game.

Alfi places a ring at origin of a Cartesian Plane. Both Alfi and Roy move the ring alternatively.
Alfi takes odd steps and Roy takes even steps. 

Direction of Movement:
Alfi always moves either left or right alternatively, i.e. if she moved left in her last step, she will move right in her current step. Initially she moves right. Roy always moves up or down alternatively, i.e if he moved up in his last step, he will move down in his current step. Initially he moves up.

Displacement of Ring:
Initially both of them displace the ring by 1 unit.
During Alfi's turn(other than her first step), she sums up her last displacement and Roy's last displacement, and moves the Ring by the value thus obtained.
Similary, during Roy's turn(other than his first step), he sums up his last displacement and Alfi's last displacement, and moves the Ring by the value thus obtained.

Sample example of movements:

Now that the direction and displacement of Ring's movements are clear. Its time to answer Alfi's weird queries.

Given the step number N, what is the perpendicular distance of Ring from Y-axis after that step.
Now Alfi is such an impatient girl, she wants him to answer as quickly as possible, otherwise she won't accept the Ring.

Input:

First line of input will contain integer Q, number of queries.
Next Q lines each will contain integer  N, step number of the game.

Output:

Print the result of each query in a new line.
Since the result can be very large, print it modulo 1000000007

Constraints:

1 ≤ Q ≤ 10000

1 ≤ N ≤ 1000000000

SAMPLE INPUT
4
1
2
3
4

SAMPLE OUTPUT
1
1
1
1

Explanation

From the figure its clear that perpendicular distance of Ring from Y-axis  after steps 1, 2, 3 and 4 are all 1.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Read problems statements in mandarin chinese, russian and vietnamese as well. 

Chef Ada loves trees, she thinks that the beauty of a tree is determined by the distance between its center and centroid. If the tree has multiple centers or centroids, then the beauty is the maximum among every possible pair.

As a reward for her delicious dishes (and awesome coding skills), CodeChef is going to give a present to Chef Ada, and of course it must be a tree! But not any tree, we want to give her one with n nodes, and beauty b. Help find such a tree.

------ Input ------ 

The first line of input contains one number T, the number of test cases.

Each test case contains two integers n and b, the desired number of vertices and beauty.

------ Output ------ 

For each test case, if no tree satisfies the conditions print "NO" in a new line.

Otherwise, print "YES" in a new line and describe any valid tree. Print n-1 more lines, each with two integers u_{i}, v_{i} (1 ≤ u_{i}, v_{i} ≤ n) denoting the indices of the vertices connected by an edge in the tree.

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

1 ≤ T ≤ 10^{4}
2 ≤ n ≤ 100
0 ≤ b ≤ n-1

------ Notes ------ 

A vertex u is a centroid if after removing u from the tree, the size of any of the resulting connected components is at most n/2.

Lef f(x) be the greatest distance from u to any other vertex of the tree. A vertex u is a center if f(u) ≤ f(v)  for any other vertex v.

----- Sample Input 1 ------ 
1
4 0
----- Sample Output 1 ------ 
YES
1 2
1 3
1 4
----- explanation 1 ------ 

Vertex 1 is the center and centroid of the tree.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
One of the first chain emails I ever received was about a supposed Cambridge  University study that suggests your brain can read words no matter what order the letters are in, as long as the first and last letters of each word are correct.

Your task is to **create a function that can take any string and randomly jumble the letters within each word while leaving the first and last letters of the word in place.**

For example,
    
    mixwords('Winter is coming') // returns 'Wntier is cminog' or 'Wtiner is conimg'
    mixwords('Hey, friends!') // returns 'Hey, fierdns!' or 'Hey, fernids!'

* All punctuation should remain in place
* Words smaller than 3 letters should not change
* Letters must be randomly moved (and so calling the function multiple times with the same string should return different values
* Parameters that are not strings should return undefined

The tests do the following things to ensure a valid string is returned:
    
1. Check that the string returned is not equal to the string passed (you may have to revalidate the solution if your function randomly returns the same string)
2. Check that first and last letters of words remain in place
3. Check that punctuation remains in place
4. Checks string length remains the same
5. Checks the function returns `undefined` for non-strings
6. Checks that word interiors (the letters between the first and last) maintain the same letters, albeit in a different order
7. Checks that letters are randomly assigned.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
At the children's day, the child came to Picks's house, and messed his house up. Picks was angry at him. A lot of important things were lost, in particular the favorite set of Picks.

Fortunately, Picks remembers something about his set S:  its elements were distinct integers from 1 to limit;  the value of $\sum_{x \in S} \text{lowbit}(x)$ was equal to sum; here lowbit(x) equals 2^{k} where k is the position of the first one in the binary representation of x. For example, lowbit(10010_2) = 10_2, lowbit(10001_2) = 1_2, lowbit(10000_2) = 10000_2 (binary representation). 

Can you help Picks and find any set S, that satisfies all the above conditions?


-----Input-----

The first line contains two integers: sum, limit (1 ≤ sum, limit ≤ 10^5).


-----Output-----

In the first line print an integer n (1 ≤ n ≤ 10^5), denoting the size of S. Then print the elements of set S in any order. If there are multiple answers, print any of them.

If it's impossible to find a suitable set, print -1.


-----Examples-----
Input
5 5

Output
2
4 5

Input
4 3

Output
3
2 3 1

Input
5 1

Output
-1



-----Note-----

In sample test 1: lowbit(4) = 4, lowbit(5) = 1, 4 + 1 = 5.

In sample test 2: lowbit(1) = 1, lowbit(2) = 2, lowbit(3) = 1, 1 + 2 + 1 = 4.

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

Time Limit: None seconds

Memory Limit: None megabytes

Problem Description:
Permutation *p* is an ordered set of integers *p*1,<=<=*p*2,<=<=...,<=<=*p**n*, consisting of *n* distinct positive integers, each of them doesn't exceed *n*. We'll denote the *i*-th element of permutation *p* as *p**i*. We'll call number *n* the size or the length of permutation *p*1,<=<=*p*2,<=<=...,<=<=*p**n*.

You have a sequence of integers *a*1,<=*a*2,<=...,<=*a**n*. In one move, you are allowed to decrease or increase any number by one. Count the minimum number of moves, needed to build a permutation from this sequence.

Input Specification:
The first line contains integer *n* (1<=≤<=*n*<=≤<=3·105) — the size of the sought permutation. The second line contains *n* integers *a*1,<=*a*2,<=...,<=*a**n* (<=-<=109<=≤<=*a**i*<=≤<=109).

Output Specification:
Print a single number — the minimum number of moves.

Please, do not use the %lld specifier to read or write 64-bit integers in C++. It is preferred to use the cin, cout streams or the %I64d specifier.

Demo Input:
['2\n3 0\n', '3\n-1 -1 2\n']

Demo Output:
['2\n', '6\n']

Note:
In the first sample you should decrease the first number by one and then increase the second number by one. The resulting permutation is (2, 1).

In the second sample you need 6 moves to build permutation (1, 3, 2).

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Title: Win or Freeze

Time Limit: None seconds

Memory Limit: None megabytes

Problem Description:
You can't possibly imagine how cold our friends are this winter in Nvodsk! Two of them play the following game to warm up: initially a piece of paper has an integer *q*. During a move a player should write any integer number that is a non-trivial divisor of the last written number. Then he should run this number of circles around the hotel. Let us remind you that a number's divisor is called non-trivial if it is different from one and from the divided number itself. 

The first person who can't make a move wins as he continues to lie in his warm bed under three blankets while the other one keeps running. Determine which player wins considering that both players play optimally. If the first player wins, print any winning first move.

Input Specification:
The first line contains the only integer *q* (1<=≤<=*q*<=≤<=1013).

Please do not use the %lld specificator to read or write 64-bit integers in С++. It is preferred to use the cin, cout streams or the %I64d specificator.

Output Specification:
In the first line print the number of the winning player (1 or 2). If the first player wins then the second line should contain another integer — his first move (if the first player can't even make the first move, print 0). If there are multiple solutions, print any of them.

Demo Input:
['6\n', '30\n', '1\n']

Demo Output:
['2\n', '1\n6\n', '1\n0\n']

Note:
Number 6 has only two non-trivial divisors: 2 and 3. It is impossible to make a move after the numbers 2 and 3 are written, so both of them are winning, thus, number 6 is the losing number. A player can make a move and write number 6 after number 30; 6, as we know, is a losing number. Thus, this move will bring us the victory.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Given a collection of intervals, merge all overlapping intervals.

Example 1:


Input: [[1,3],[2,6],[8,10],[15,18]]
Output: [[1,6],[8,10],[15,18]]
Explanation: Since intervals [1,3] and [2,6] overlaps, merge them into [1,6].


Example 2:


Input: [[1,4],[4,5]]
Output: [[1,5]]
Explanation: Intervals [1,4] and [4,5] are considerred overlapping.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Write a program to obtain length (L) and breadth (B) of a rectangle and check whether its area is greater or perimeter is greater or both are equal.

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

- First line will contain the length (L) of the rectangle.
- Second line will contain the breadth (B) of the rectangle.

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

Output 2 lines. 

In the first line print "Area" if area is greater otherwise print "Peri" and if they are equal print "Eq".(Without quotes).

In the second line print the calculated area or perimeter (whichever is greater or anyone if it is equal).

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

$1 ≤ L ≤ 1000$
$1 ≤ B ≤ 1000$

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

----- Sample Output 1 ------ 
Peri
6

----- explanation 1 ------ 
Area = 1 * 2 = 2 \
Peri = 2 * (1 + 2) = 6 \
Since Perimeter is greater than Area, hence the output is : \
Peri \
6

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