vgel/INTELLECT-2-RL-Dataset-Code-Only huggingface dataset:
# Task
 Two integer numbers are added using the column addition method. When using this method, some additions of digits produce non-zero carries to the next positions. Your task is to calculate the number of non-zero carries that will occur while adding the given numbers.

 The numbers are added in base 10.

# Example

 For `a = 543 and b = 3456,` the output should be `0`
 
 For `a = 1927 and b = 6426`, the output should be `2`
 
 For `a = 9999 and b = 1`, the output should be `4`
 
 
# Input/Output


 - `[input]` integer `a`

  A positive integer.

  Constraints: `1 ≤ a ≤ 10^7`


 - `[input]` integer `b`

  A positive integer.

  Constraints: `1 ≤ b ≤ 10^7`

 
 - `[output]` an integer

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
There is a forest that we model as a plane and live n rare animals. Animal number i has its lair in the point (x_{i}, y_{i}). In order to protect them, a decision to build a nature reserve has been made.

The reserve must have a form of a circle containing all lairs. There is also a straight river flowing through the forest. All animals drink from this river, therefore it must have at least one common point with the reserve. On the other hand, ships constantly sail along the river, so the reserve must not have more than one common point with the river.

For convenience, scientists have made a transformation of coordinates so that the river is defined by y = 0. Check whether it is possible to build a reserve, and if possible, find the minimum possible radius of such a reserve.

Input

The first line contains one integer n (1 ≤ n ≤ 10^5) — the number of animals. 

Each of the next n lines contains two integers x_{i}, y_{i} (-10^7 ≤ x_{i}, y_{i} ≤ 10^7) — the coordinates of the i-th animal's lair. It is guaranteed that y_{i} ≠ 0. No two lairs coincide.

Output

If the reserve cannot be built, print -1. Otherwise print the minimum radius. Your answer will be accepted if absolute or relative error does not exceed 10^{-6}.

Formally, let your answer be a, and the jury's answer be b. Your answer is considered correct if \frac{|a - b|}{max{(1, |b|)}} ≤ 10^{-6}.

Examples

Input

1
0 1


Output

0.5

Input

3
0 1
0 2
0 -3


Output

-1


Input

2
0 1
1 1


Output

0.625

Note

In the first sample it is optimal to build the reserve with the radius equal to 0.5 and the center in (0,\ 0.5).

In the second sample it is impossible to build a reserve.

In the third sample it is optimal to build the reserve with the radius equal to 5/8 and the center in (1/2,\ 5/8).

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
The impending doom called final examination is approaching this year’s high school seniors: one of the compulsory tasks is going to be writing an essay in their mother tongue. Mirko is under the impression that certain political parties are going to keep their promises, those regarding informatization and digitalization, which is why he thinks that this year’s essays are going to be marked by a computer, instead of a human.

To test his doubts, Mirko is going to write an essay that doesn’t necessarily make sense at all, but should pass the automatic basic conditions check. The essay passes the check if
 - it contains at least $A$, and at most $B$ words;
 - every word contains at least one, and at most 15 letters;
 - the used vocabulary is large enough, in other words, the essay contains at least $B/2$ different words.

Mirko was up late last night watching Big Brother, so he is asking you to write such an essay instead of him. The essay should be output in a single line, using only lowercase letters of the English alphabet and spaces (therefore, without punctuation such as dots, commas and such). The words used can, but need not be, words from the English (or any) language.

-----Input-----
The first and only line of input contains the integers $A$ and $B$ ($1 \leq A \leq B \leq 100000$) from the task.

-----Output-----
The first and only line of output must contain any essay that meets the rules from the task.

-----Examples-----
Sample Input 1:
2 7
Sample Output 1:
dobar je ovaj marko marulic

Sample Input 2:
26 30
Sample Output 2:
nama je profesor reko da to ne treba za lektiru al onda je bila buka i nisam ga cuo pa jel bi mi mogli dat bodove

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
You are responsible for installing a gas pipeline along a road. Let's consider the road (for simplicity) as a segment $[0, n]$ on $OX$ axis. The road can have several crossroads, but for simplicity, we'll denote each crossroad as an interval $(x, x + 1)$ with integer $x$. So we can represent the road as a binary string consisting of $n$ characters, where character 0 means that current interval doesn't contain a crossroad, and 1 means that there is a crossroad.

Usually, we can install the pipeline along the road on height of $1$ unit with supporting pillars in each integer point (so, if we are responsible for $[0, n]$ road, we must install $n + 1$ pillars). But on crossroads we should lift the pipeline up to the height $2$, so the pipeline won't obstruct the way for cars.

We can do so inserting several zig-zag-like lines. Each zig-zag can be represented as a segment $[x, x + 1]$ with integer $x$ consisting of three parts: $0.5$ units of horizontal pipe + $1$ unit of vertical pipe + $0.5$ of horizontal. Note that if pipeline is currently on height $2$, the pillars that support it should also have length equal to $2$ units.

 [Image] 

Each unit of gas pipeline costs us $a$ bourles, and each unit of pillar — $b$ bourles. So, it's not always optimal to make the whole pipeline on the height $2$. Find the shape of the pipeline with minimum possible cost and calculate that cost.

Note that you must start and finish the pipeline on height $1$ and, also, it's guaranteed that the first and last characters of the input string are equal to 0.


-----Input-----

The fist line contains one integer $T$ ($1 \le T \le 100$) — the number of queries. Next $2 \cdot T$ lines contain independent queries — one query per two lines.

The first line contains three integers $n$, $a$, $b$ ($2 \le n \le 2 \cdot 10^5$, $1 \le a \le 10^8$, $1 \le b \le 10^8$) — the length of the road, the cost of one unit of the pipeline and the cost of one unit of the pillar, respectively.

The second line contains binary string $s$ ($|s| = n$, $s_i \in \{0, 1\}$, $s_1 = s_n = 0$) — the description of the road.

It's guaranteed that the total length of all strings $s$ doesn't exceed $2 \cdot 10^5$.


-----Output-----

Print $T$ integers — one per query. For each query print the minimum possible cost of the constructed pipeline.


-----Example-----
Input
4
8 2 5
00110010
8 1 1
00110010
9 100000000 100000000
010101010
2 5 1
00

Output
94
25
2900000000
13



-----Note-----

The optimal pipeline for the first query is shown at the picture above.

The optimal pipeline for the second query is pictured below:

 [Image] 

The optimal (and the only possible) pipeline for the third query is shown below:

 [Image] 

The optimal pipeline for the fourth query is shown below:

 [Image]

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Vlad found two positive numbers $a$ and $b$ ($a,b>0$). He discovered that $a \oplus b = \frac{a + b}{2}$, where $\oplus$ means the bitwise exclusive OR , and division is performed without rounding..

Since it is easier to remember one number than two, Vlad remembered only $a\oplus b$, let's denote this number as $x$. Help him find any suitable $a$ and $b$ or tell him that they do not exist.


-----Input-----

The first line of the input data contains the single integer $t$ ($1 \le t \le 10^4$) — the number of test cases in the test.

Each test case is described by a single integer $x$ ($1 \le x \le 2^{29}$) — the number that Vlad remembered.


-----Output-----

Output $t$ lines, each of which is the answer to the corresponding test case. As the answer, output $a$ and $b$ ($0 < a,b \le 2^{32}$), such that $x = a \oplus b = \frac{a + b}{2}$. If there are several answers, output any of them. If there are no matching pairs, output -1.


-----Examples-----

Input
6
2
5
10
6
18
36
Output
3 1
-1
13 7
-1
25 11
50 22


-----Note-----

None

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Given are integers a,b,c and d.
If x and y are integers and a \leq x \leq b and c\leq y \leq d hold, what is the maximum possible value of x \times y?

-----Constraints-----
 - -10^9 \leq a \leq b \leq 10^9
 - -10^9 \leq c \leq d \leq 10^9
 - All values in input are integers.

-----Input-----
Input is given from Standard Input in the following format:
a b c d

-----Output-----
Print the answer.

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

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

If x = 1 and y = 1 then x \times y = 1.
If x = 2 and y = 1 then x \times y = 2.
Therefore, the answer is 2.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
One day, Tanya was studying graph theory. She is very inquisitive, so the following problem soon came to her mind.
Find the number of undirected unweighted connected simple graphs with $N$ vertices (numbered $1$ through $N$) and $M$ edges, such that for each $i$ ($2 \le i \le N$), the shortest path from vertex $1$ to vertex $i$ is unique and its length is equal to $A_i$. In other words, for each $i$ ($2 \le i \le N$), there should be exactly one path with length $A_i$ between vertices $1$ and $i$, and there should be no paths with smaller length between these vertices.
Two graphs with $N$ vertices are distinct if we can find two vertices $u$ and $v$ such that there is an edge between these vertices in one graph, but not in the other graph.
Since the answer could be very large, compute it modulo $1,000,000,007$ ($10^9 + 7$).

-----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 two space-separated integers $N$ and $M$.
- The second line contains $N - 1$ space-separated integers $A_2, A_3, \ldots, A_N$.

-----Output-----
For each test case, print a single line containing one integer ― the number of graphs modulo $10^9 + 7$.

-----Constraints-----
- $1 \le T \le 1,000$
- $2 \le N \le 10^5$
- $N-1 \le M \le \mathrm{min}\left(2\cdot 10^5, \frac{N(N-1)}{2}\right)$
- $1 \le A_i \le N-1$ for each valid $i$
- the sum of $N$ over all test cases does not exceed $2 \cdot 10^5$
- the sum of $M$ over all test cases does not exceed $2 \cdot 10^5$

-----Subtasks-----
Subtask #1 (50 points): $M = N-1$
Subtask #2 (50 points): original constraints

-----Example Input-----
3
4 3
1 2 1
4 6
1 2 1
3 2
2 2

-----Example Output-----
2
0
0

-----Explanation-----
Example case 1: The two graphs which satisfy all conditions are:

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
The Physical education teacher at SESC is a sort of mathematician too. His most favorite topic in mathematics is progressions. That is why the teacher wants the students lined up in non-decreasing height form an arithmetic progression.

To achieve the goal, the gym teacher ordered a lot of magical buns from the dining room. The magic buns come in two types: when a student eats one magic bun of the first type, his height increases by one, when the student eats one magical bun of the second type, his height decreases by one. The physical education teacher, as expected, cares about the health of his students, so he does not want them to eat a lot of buns. More precisely, he wants the maximum number of buns eaten by some student to be minimum.

Help the teacher, get the maximum number of buns that some pupils will have to eat to achieve the goal of the teacher. Also, get one of the possible ways for achieving the objective, namely, the height of the lowest student in the end and the step of the resulting progression.


-----Input-----

The single line contains integer n (2 ≤ n ≤ 10^3) — the number of students. The second line contains n space-separated integers — the heights of all students. The height of one student is an integer which absolute value doesn't exceed 10^4.


-----Output-----

In the first line print the maximum number of buns eaten by some student to achieve the teacher's aim. In the second line, print two space-separated integers — the height of the lowest student in the end and the step of the progression. Please, pay attention that the step should be non-negative.

If there are multiple possible answers, you can print any of them.


-----Examples-----
Input
5
-3 -4 -2 -3 3

Output
2
-3 1

Input
5
2 -3 -1 -4 3

Output
1
-4 2



-----Note-----

Lets look at the first sample. We can proceed in the following manner:

  don't feed the 1-st student, his height will stay equal to -3;  give two buns of the first type to the 2-nd student, his height become equal to -2;  give two buns of the first type to the 3-rd student, his height become equal to 0;  give two buns of the first type to the 4-th student, his height become equal to -1;  give two buns of the second type to the 5-th student, his height become equal to 1. 

To sum it up, when the students line up in non-decreasing height it will be an arithmetic progression: -3, -2, -1, 0, 1. The height of the lowest student is equal to -3, the step of the progression is equal to 1. The maximum number of buns eaten by one student is equal to 2.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Write a function called `sumIntervals`/`sum_intervals()` that accepts an array of intervals, and returns the sum of all the interval lengths. Overlapping intervals should only be counted once.

### Intervals

Intervals are represented by a pair of integers in the form of an array. The first value of the interval will always be less than the second value. Interval example: `[1, 5]` is an interval from 1 to 5. The length of this interval is 4.

### Overlapping Intervals

List containing overlapping intervals:

```
[
   [1,4],
   [7, 10],
   [3, 5]
]
```

The sum of the lengths of these intervals is 7. Since [1, 4] and [3, 5] overlap, we can treat the interval as [1, 5], which has a length of 4.

### Examples:

```C#
// empty intervals
Intervals.SumIntervals(new (int, int)[]{ });  // => 0
Intervals.SumIntervals(new (int, int)[]{ (2, 2), (5, 5)});  // => 0

// disjoined intervals
Intervals.SumIntervals(new (int, int)[]{
  (1, 2), (3, 5)
});  // => (2-1) + (5-3) = 3

// overlapping intervals
Intervals.SumIntervals(new (int, int)[]{
  (1, 4), (3, 6), (2, 8)
});  // (1,8) => 7
```

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
You are given a string s, a split is called good if you can split s into 2 non-empty strings p and q where its concatenation is equal to s and the number of distinct letters in p and q are the same.
Return the number of good splits you can make in s.
 
Example 1:
Input: s = "aacaba"
Output: 2
Explanation: There are 5 ways to split "aacaba" and 2 of them are good. 
("a", "acaba") Left string and right string contains 1 and 3 different letters respectively.
("aa", "caba") Left string and right string contains 1 and 3 different letters respectively.
("aac", "aba") Left string and right string contains 2 and 2 different letters respectively (good split).
("aaca", "ba") Left string and right string contains 2 and 2 different letters respectively (good split).
("aacab", "a") Left string and right string contains 3 and 1 different letters respectively.

Example 2:
Input: s = "abcd"
Output: 1
Explanation: Split the string as follows ("ab", "cd").

Example 3:
Input: s = "aaaaa"
Output: 4
Explanation: All possible splits are good.
Example 4:
Input: s = "acbadbaada"
Output: 2

 
Constraints:

s contains only lowercase English letters.
1 <= s.length <= 10^5

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

Time Limit: None seconds

Memory Limit: None megabytes

Problem Description:
Reforms continue entering Berland. For example, during yesterday sitting the Berland Parliament approved as much as *n* laws (each law has been assigned a unique number from 1 to *n*). Today all these laws were put on the table of the President of Berland, G.W. Boosch, to be signed.

This time mr. Boosch plans to sign 2*k* laws. He decided to choose exactly two non-intersecting segments of integers from 1 to *n* of length *k* and sign all laws, whose numbers fall into these segments. More formally, mr. Boosch is going to choose two integers *a*, *b* (1<=≤<=*a*<=≤<=*b*<=≤<=*n*<=-<=*k*<=+<=1,<=*b*<=-<=*a*<=≥<=*k*) and sign all laws with numbers lying in the segments [*a*; *a*<=+<=*k*<=-<=1] and [*b*; *b*<=+<=*k*<=-<=1] (borders are included).

As mr. Boosch chooses the laws to sign, he of course considers the public opinion. Allberland Public Opinion Study Centre (APOSC) conducted opinion polls among the citizens, processed the results into a report and gave it to the president. The report contains the absurdity value for each law, in the public opinion. As mr. Boosch is a real patriot, he is keen on signing the laws with the maximum total absurdity. Help him.

Input Specification:
The first line contains two integers *n* and *k* (2<=≤<=*n*<=≤<=2·105, 0<=&lt;<=2*k*<=≤<=*n*) — the number of laws accepted by the parliament and the length of one segment in the law list, correspondingly. The next line contains *n* integers *x*1,<=*x*2,<=...,<=*x**n* — the absurdity of each law (1<=≤<=*x**i*<=≤<=109).

Output Specification:
Print two integers *a*, *b* — the beginning of segments that mr. Boosch should choose. That means that the president signs laws with numbers from segments [*a*; *a*<=+<=*k*<=-<=1] and [*b*; *b*<=+<=*k*<=-<=1]. If there are multiple solutions, print the one with the minimum number *a*. If there still are multiple solutions, print the one with the minimum *b*.

Demo Input:
['5 2\n3 6 1 1 6\n', '6 2\n1 1 1 1 1 1\n']

Demo Output:
['1 4\n', '1 3\n']

Note:
In the first sample mr. Boosch signs laws with numbers from segments [1;2] and [4;5]. The total absurdity of the signed laws equals 3 + 6 + 1 + 6 = 16.

In the second sample mr. Boosch signs laws with numbers from segments [1;2] and [3;4]. The total absurdity of the signed laws equals 1 + 1 + 1 + 1 = 4.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Manasa recently lost a bet to Amit. To settle the problem, they are playing a game:  

They have N balls in front of them. Each ball, except the 1st ball, is numbered from 0 to 9. The 1st ball is numbered from 1 to 9. 

Amit calculates all the subsequences of the number thus formed. Each subsequence of sequence S is represented by S_{k}. 

e.g. S = 1 2 3  

S_{0} = 1 2 3 , 

S_{1} = 1 2 , 

S_{2} = 1 3 , 

S_{3} = 2 3 , 

S_{4} = 1 , 

S_{5} = 2 , and 

S_{6} = 3 .  

Each subsequence S_{k} also represents a number. e.g. S_{1} represents tweleve, S_{2} represents thirteen.

Now Manasa has to throw S_{k} candies into an intially empty box, where k goes from 0 to ( maximum number of subsequences -  1).

At the end of the game, Manasa has to find out the total number of candies, T, in the box. As T can be large, Amit asks Manasa to tell T % (10^{9} + 7 ). If Manasa answers correctly, she can keep all the candies. Manasa can't take all this Math and asks for your help.  

Help her!

Note: A subsequence can also be have preceding zeros. For example, the sequence 103 has subsequences 103, 10, 13, 03, 1, 0, and 3 (so both 03 and 3 are counted separately).

Input Format 

A single line containing a number having N digits.  

Output Format 

A number contaning the output.  

Constraints 

1 ≤ N ≤ 2*10^{5}  

Sample Input 00

111

Sample Output 00

147

Sample Input 01

123

Sample Output 01

177

Explanation  

The subsequence of number 111 are 111, 11 , 11, 11, 1, 1 and 1. Whose sum is 147.

The subsequence of number 123 are 123, 12 , 23, 13, 1, 2 and 3. Whose sum is 177.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
You are climbing a stair case. It takes n steps to reach to the top.

Each time you can either climb 1 or 2 steps. In how many distinct ways can you climb to the top?

Note: Given n will be a positive integer.

Example 1:


Input: 2
Output: 2
Explanation: There are two ways to climb to the top.
1. 1 step + 1 step
2. 2 steps


Example 2:


Input: 3
Output: 3
Explanation: There are three ways to climb to the top.
1. 1 step + 1 step + 1 step
2. 1 step + 2 steps
3. 2 steps + 1 step

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Unlike Knights of a Round Table, Knights of a Polygonal Table deprived of nobility and happy to kill each other. But each knight has some power and a knight can kill another knight if and only if his power is greater than the power of victim. However, even such a knight will torment his conscience, so he can kill no more than $k$ other knights. Also, each knight has some number of coins. After a kill, a knight can pick up all victim's coins.

Now each knight ponders: how many coins he can have if only he kills other knights?

You should answer this question for each knight.


-----Input-----

The first line contains two integers $n$ and $k$ $(1 \le n \le 10^5, 0 \le k \le \min(n-1,10))$ — the number of knights and the number $k$ from the statement.

The second line contains $n$ integers $p_1, p_2 ,\ldots,p_n$ $(1 \le p_i \le 10^9)$ — powers of the knights. All $p_i$ are distinct.

The third line contains $n$ integers $c_1, c_2 ,\ldots,c_n$ $(0 \le c_i \le 10^9)$ — the number of coins each knight has.


-----Output-----

Print $n$ integers — the maximum number of coins each knight can have it only he kills other knights.


-----Examples-----
Input
4 2
4 5 9 7
1 2 11 33

Output
1 3 46 36 
Input
5 1
1 2 3 4 5
1 2 3 4 5

Output
1 3 5 7 9 
Input
1 0
2
3

Output
3 


-----Note-----

Consider the first example.   The first knight is the weakest, so he can't kill anyone. That leaves him with the only coin he initially has.  The second knight can kill the first knight and add his coin to his own two.  The third knight is the strongest, but he can't kill more than $k = 2$ other knights. It is optimal to kill the second and the fourth knights: $2+11+33 = 46$.  The fourth knight should kill the first and the second knights: $33+1+2 = 36$. 

In the second example the first knight can't kill anyone, while all the others should kill the one with the index less by one than their own.

In the third example there is only one knight, so he can't kill anyone.

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