vgel/INTELLECT-2-RL-Dataset-Code-Only huggingface dataset:
New Year is just around the corner, which means that in School 179, preparations for the concert are in full swing.

There are $n$ classes in the school, numbered from $1$ to $n$, the $i$-th class has prepared a scene of length $a_i$ minutes.

As the main one responsible for holding the concert, Idnar knows that if a concert has $k$ scenes of lengths $b_1$, $b_2$, $\ldots$, $b_k$ minutes, then the audience will get bored if there exist two integers $l$ and $r$ such that $1 \le l \le r \le k$ and $\gcd(b_l, b_{l + 1}, \ldots, b_{r - 1}, b_r) = r - l + 1$, where $\gcd(b_l, b_{l + 1}, \ldots, b_{r - 1}, b_r)$ is equal to the greatest common divisor (GCD) of the numbers $b_l$, $b_{l + 1}$, $\ldots$, $b_{r - 1}$, $b_r$.

To avoid boring the audience, Idnar can ask any number of times (possibly zero) for the $t$-th class ($1 \le t \le k$) to make a new scene $d$ minutes in length, where $d$ can be any positive integer. Thus, after this operation, $b_t$ is equal to $d$. Note that $t$ and $d$ can be different for each operation.

For a sequence of scene lengths $b_1$, $b_2$, $\ldots$, $b_{k}$, let $f(b)$ be the minimum number of classes Idnar has to ask to change their scene if he wants to avoid boring the audience.

Idnar hasn't decided which scenes will be allowed for the concert, so he wants to know the value of $f$ for each non-empty prefix of $a$. In other words, Idnar wants to know the values of $f(a_1)$, $f(a_1$,$a_2)$, $\ldots$, $f(a_1$,$a_2$,$\ldots$,$a_n)$.


-----Input-----

The first line contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the number of classes in the school.

The second line contains $n$ positive integers $a_1$, $a_2$, $\ldots$, $a_n$ ($1 \le a_i \le 10^9$) — the lengths of the class scenes.


-----Output-----

Print a sequence of $n$ integers in a single line — $f(a_1)$, $f(a_1$,$a_2)$, $\ldots$, $f(a_1$,$a_2$,$\ldots$,$a_n)$.


-----Examples-----

Input
1
1
Output
1
Input
3
1 4 2
Output
1 1 2
Input
7
2 12 4 8 18 3 6
Output
0 1 1 1 2 2 2


-----Note-----

In the first test we can change $1$ to $2$, so the answer is $1$.

In the second test:

$[1]$ can be changed into $[2]$,

$[1, 4]$ can be changed into $[3, 4]$,

$[1, 4, 2]$ can be changed into $[2, 3, 2]$.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Colonel has n badges. He wants to give one badge to every of his n soldiers. Each badge has a coolness factor, which shows how much it's owner reached. Coolness factor can be increased by one for the cost of one coin. 

For every pair of soldiers one of them should get a badge with strictly higher factor than the second one. Exact values of their factors aren't important, they just need to have distinct factors. 

Colonel knows, which soldier is supposed to get which badge initially, but there is a problem. Some of badges may have the same factor of coolness. Help him and calculate how much money has to be paid for making all badges have different factors of coolness.


-----Input-----

First line of input consists of one integer n (1 ≤ n ≤ 3000).

Next line consists of n integers a_{i} (1 ≤ a_{i} ≤ n), which stand for coolness factor of each badge.


-----Output-----

Output single integer — minimum amount of coins the colonel has to pay.


-----Examples-----
Input
4
1 3 1 4

Output
1
Input
5
1 2 3 2 5

Output
2


-----Note-----

In first sample test we can increase factor of first badge by 1.

In second sample test we can increase factors of the second and the third badge by 1.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Today Adilbek is taking his probability theory test. Unfortunately, when Adilbek arrived at the university, there had already been a long queue of students wanting to take the same test. Adilbek has estimated that he will be able to start the test only $T$ seconds after coming. 

Fortunately, Adilbek can spend time without revising any boring theorems or formulas. He has an app on this smartphone which contains $n$ Japanese crosswords to solve. Adilbek has decided to solve them all one by one in the order they are listed in the app, without skipping any crossword. For each crossword, a number $t_i$ is given that represents the time it takes an average crossword expert to solve this crossword (the time is given in seconds).

Adilbek is a true crossword expert, but, unfortunately, he is sometimes unlucky in choosing the way to solve the crossword. So, it takes him either $t_i$ seconds or $t_i + 1$ seconds to solve the $i$-th crossword, equiprobably (with probability $\frac{1}{2}$ he solves the crossword in exactly $t_i$ seconds, and with probability $\frac{1}{2}$ he has to spend an additional second to finish the crossword). All these events are independent.

After $T$ seconds pass (or after solving the last crossword, if he manages to do it in less than $T$ seconds), Adilbek closes the app (if he finishes some crossword at the same moment, that crossword is considered solved; otherwise Adilbek does not finish solving the current crossword at all). He thinks it would be an interesting probability theory problem to calculate $E$ — the expected number of crosswords he will be able to solve completely. Can you calculate it? 

Recall that the expected value of a discrete random variable is the probability-weighted average of all possible values — in this problem it means that the expected value of the number of solved crosswords can be calculated as $E = \sum \limits_{i = 0}^{n} i p_i$, where $p_i$ is the probability that Adilbek will solve exactly $i$ crosswords. 

We can represent $E$ as rational fraction $\frac{P}{Q}$ with $Q > 0$. To give the answer, you should print $P \cdot Q^{-1} \bmod (10^9 + 7)$.


-----Input-----

The first line contains two integers $n$ and $T$ ($1 \le n \le 2 \cdot 10^5$, $1 \le T \le 2 \cdot 10^{14}$) — the number of crosswords and the time Adilbek has to spend, respectively.

The second line contains $n$ integers $t_1, t_2, \dots, t_n$ ($1 \le t_i \le 10^9$), where $t_i$ is the time it takes a crossword expert to solve the $i$-th crossword.

Note that Adilbek solves the crosswords in the order they are given in the input without skipping any of them.


-----Output-----

Print one integer — the expected value of the number of crosswords Adilbek solves in $T$ seconds, expressed in the form of $P \cdot Q^{-1} \bmod (10^9 + 7)$.


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

Output
750000007

Input
3 5
2 1 2

Output
125000003



-----Note-----

The answer for the first sample is equal to $\frac{14}{8}$.

The answer for the second sample is equal to $\frac{17}{8}$.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Ruben and Albert are what you can call abnormally smart. They are also both very fond of mathematically inspired games. Their only problem is that most games are too easy for them, and they end up beating everyone who dares challenge them. Because of that, they’re now mostly playing against each other. To make things interesting, they had a professor design a new game for them.

This new game was interesting at first. Nowadays, however, Albert often complains that it is impossible for him to win a particular round. After long discussions, they’ve now decided to take this a step further, and actually figure out who’d win if they both played optimally. They need you to write a computer program that does this for them.

A state in the game consists of one or more $x\times y\times z$ cuboids. A (legal) move is choosing a cuboid, then a value for each of the three axes (basically choosing three planes), and then cutting the cuboid along these (thus removing a $1\times y\times z$, $x\times 1\times z$ and a $x\times y\times 1$, all overlapping, cuboid). In effect you’ve created between $0$ and $8$ (inclusive) smaller cuboids. All three planes cut from the cuboid need to be on the cuboid (you can’t cut away a hypothetical cuboid on the outside of the real one).

An example might be in order. You’ve chosen a $3\times 5\times 4$ cuboid, and are about to cut it. You now need to choose the three planes. This means you need an $x$ between $1$ and $3$, a $y$ between $1$ and $5$ and a $z$ between $1$ and $4$. Say you choose $2$, $1$ and $3$, respectively. The first cut would alone cut the cuboid into two $1\times 5\times 4$ cuboids, the second into a single $3\times 4\times 4$ cuboid, while the third would alone cut the cuboid into a $3\times 5\times 1$ and a $3\times 5\times 2$ cuboid. Put together these cuts produces $4$ new smaller cuboids, of sizes $1\times 4\times 1$,$1\times 4\times 1$,$1\times 4\times 2$ and $1\times 4\times 2$. Note that cutting a cuboid with an axis of size $1$ would remove it altogether.

The players take turns making a move. The winner is the player that removes the last cuboid.

-----Input-----
The first line of input is a line containing either RUBEN or ALBERT, the name of the player who starts that particular round.

 Then follows a line containing $N$, the number of cuboids that particular game starts with.

$N$ lines follow, each describing a cuboid. A cuboid description consists of three numbers, $x$, $y$ and $z$, the size of that particular cuboid.

-----Output-----
Output the name of the player that wins the game (either RUBEN or ALBERT).

-----Limits-----
 - $1 \le N \le 100$
 - $1 \le x,y,z \le 30$

-----Examples-----
Sample Input 1:
RUBEN
1
4 1 7
Sample Output 1:
RUBEN

Sample Input 2:
ALBERT
2
4 4 4
2 2 2
Sample Output 2:
RUBEN

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
For skilled programmers, it is very easy to implement a sorting function. Moreover, they often avoid full sorting to reduce computation time if it is not necessary. Here, we consider "rough sorting" which sorts an array except for some pairs of elements. More formally, we define an array is "$K$-roughly sorted" if an array is sorted except that at most $K$ pairs are in reversed order. For example, '1 3 2 4' is 1-roughly sorted because (3, 2) is only the reversed pair. In the same way, '1 4 2 3' is 2-roughly sorted because (4, 2) and (4, 3) are reversed.

Considering rough sorting by exchanging adjacent elements repeatedly, you need less number of swaps than full sorting. For example, '4 1 2 3' needs three exchanges for full sorting, but you only need to exchange once for 2-rough sorting.

Given an array and an integer $K$, your task is to find the result of the $K$-rough sorting with a minimum number of exchanges. If there are several possible results, you should output the lexicographically minimum result. Here, the lexicographical order is defined by the order of the first different elements.



Input

The input consists of a single test case in the following format.


$N$ $K$
$x_1$
$\vdots$
$x_N$


The first line contains two integers $N$ and $K$. The integer $N$ is the number of the elements of the array ($1 \leq N \leq 10^5$). The integer $K$ gives how many reversed pairs are allowed ($1 \leq K \leq 10^9$). Each of the following $N$ lines gives the element of the array. The array consists of the permutation of $1$ to $N$, therefore $1 \leq x_i \leq N$ and $x_i \ne x_j$ ($i \ne j$) are satisfied.

Output

The output should contain $N$ lines. The $i$-th line should be the $i$-th element of the result of the $K$-rough sorting. If there are several possible results, you should output the minimum result with the lexicographical order.

Examples

Input

3 1
3
2
1


Output

1
3
2


Input

3 100
3
2
1


Output

3
2
1


Input

5 3
5
3
2
1
4


Output

1
3
5
2
4


Input

5 3
1
2
3
4
5


Output

1
2
3
4
5

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Takahashi is going to set a 3-character password.
How many possible passwords are there if each of its characters must be a digit between 1 and N (inclusive)?

-----Constraints-----
 - 1 \leq N \leq 9
 - N is an integer.

-----Input-----
Input is given from Standard Input in the following format:
N

-----Output-----
Print the number of possible passwords.

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

-----Sample Output-----
8

There are eight possible passwords: 111, 112, 121, 122, 211, 212, 221, and 222.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
A + B is often used as an example of the easiest problem possible to show some contest platform. However, some scientists have observed that sometimes this problem is not so easy to get accepted. Want to try?


-----Input-----

The input contains two integers a and b (0 ≤ a, b ≤ 10^3), separated by a single space.


-----Output-----

Output the sum of the given integers.


-----Examples-----
Input
5 14

Output
19

Input
381 492

Output
873

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Like all other good drivers, you like to curse, swear and honk your horn at your fellow automobile drivers. Today you’re at the rear of a long line, brooding over the others’ inability to keep proper distance to the car in front. But are you really keeping your own distance?

You have calculated that in order to never have to use your breaks, you must keep a distance to any car $x$ in front of you at least $p(n+1)$ where $n$ is the number of cars between you and $x$, and $p$ is an integer constant determined by which of your cars you are currently driving.

Given the value of $p$ and the current distances (in random order) to each of the cars ahead of you, calculate the minimum distance you should be keeping to the car directly in front, in order to not have to use your breaks.

-----Input-----
First a line with $1 \leq n \leq 100000$ – the number of cars ahead of you – and $1 \leq p \leq 20$ – the deceleration constant.

Then a single line with $n$ unique integers denoting the current distance to each of the cars ahead of you. Each of these integers are in the interval $[1, 10^7]$



-----Output-----
The minimum distance you must keep to the car directly in front, in order to not have to use your breaks.

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

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
There is an old adage about opening a can of worms. A lesser known adage is one about shooting a can of exploding worms with a BB gun.

Imagine we place some cans of exploding worms on a long, straight fence. When a can is shot, all of the worms inside will explode. Different types of worms have different blast radii. Each can contains only one kind of worm.

When a can explodes, if another can is in the blast radius, then that can will also explode, possibly creating a chain reaction. Each can explodes only once. This process continues until all explosions stop.

For each can, suppose that it is the only can shot. How many cans in total will explode?

-----Input-----
There will be a single test case in the input. This test case will begin with a line with a single integer $n$ ($1 \le n \le 100000$) representing the number of cans on the fence. Each of the next $n$ lines will have two integers $x$ ($-10^9 \le x \le 10^9$) and $r$ ($1 \le r \le 10^9$), where $x$ is the location of the can on the fence and $r$ is the blast radius. No two cans will occupy the same location.

-----Output-----
Print $n$ integers on a single line separated by single spaces. The $i^{th}$ integer represents the number of cans that will explode if the $i^{th}$ can is the one that is shot.

-----Examples-----
Sample Input:
3
4 3
-10 9
-2 3
Sample Output:
1 2 1

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Arya and Aryan live in a country called Nadaca. Nadaca consists of $N$ cities numbered $1$ through $N$, which are connected by some bidirectional roads. Each road has a positive (not necessarily integer) length. Arya lives in city $1$ and Aryan lives in city $N$.
Arya claims that for each city $v$, the shortest path from city $1$ to city $v$ has length $a_v$. Aryan does not disagree, but claims that for each city $v$, the shortest path from city $N$ to city $v$ has length $b_v$. You are the wisest person they know, so they asked you to tell them if it is possible for their claims to be true, i.e. if a road network which fully satisfies their claims exists. Help them!

-----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 a single integer $N$.
- The second line contains $N$ space-separated integers $a_1, a_2, \ldots, a_N$.
- The third line contains $N$ space-separated integers $b_1, b_2, \ldots, b_N$.

-----Output-----
For each test case, print a single line containing the string "Yes" if Arya's and Aryan's claims can be true or "No" otherwise.

-----Constraints-----
- $1 \le T \le 10^3$
- $2 \le N \le 10^6$
- $0 \le a_i, b_i \le 10^9$ for each valid $i$
- the sum of $N$ over all test cases does not exceed $10^6$

-----Subtasks-----
Subtask #1 (50 points): the sum of $N$ over all test cases does not exceed $10^3$
Subtask #2 (50 points): original constraints

-----Example Input-----
2
3
0 1 1
1 1 0
3
0 1 5
5 1 0

-----Example Output-----
Yes
No

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Gas stations in the White Tiger service area have $ N $ lanes assigned numbers from $ 1 $ to $ N $. The first car in each lane can refuel.

Cars entering the gas station will choose the lane with the fewest cars in line and line up at the end of the line. If there are multiple such lanes, choose the one with the lowest number. After refueling, the car will leave the lane and the car behind it will refuel. Once you have selected a lane, you cannot move to another lane. Also, the order of the cars in the lane does not change.

When given information on the number of lanes, the cars that have entered and the lanes that have been refueled, create a program that outputs the numbers of the cars that have been refueled in order. The entrance information is given as the number of the car entering the stand, and the information on the end of refueling is given as the lane number where the refueling of the first car is completed. However, it is assumed that no cars are lined up in any lane at first.



input

The input is given in the following format.


$ N $ $ M $
$ info_1 $
$ info_2 $
::
$ info_M $


The first line gives the number of lanes $ N $ ($ 1 \ leq N \ leq 10 $) and the number of information $ M $ ($ 2 \ leq M \ leq 10,000 $). Each information $ info_i $ is given in the following $ M $ line. Each $ info_i $ is given in one of the following formats:


0 $ lane $


Or


1 $ car $


If the first number is 0, it indicates that the first car in the lane with the number $ lane $ ($ 1 \ leq lane \ leq N $) has been refueled. A first number of 1 indicates that a car with the number $ car $ ($ 1 \ leq car \ leq 9,999 $) has entered the stand.

The input meets the following constraints.

* The numbers of cars entering the gas station are all different.
* There must always be at least one piece of information whose first number is 0,1.
* No information is given that the first number is 0 for lanes without cars.

output

For each information on the end of refueling, the number of the car that has finished refueling is output on one line.

Examples

Input

2 7
1 999
1 1000
0 2
1 1001
1 1002
0 1
0 1


Output

1000
999
1002


Input

None


Output

None

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

Time Limit: None seconds

Memory Limit: None megabytes

Problem Description:
Vanya got an important task — he should enumerate books in the library and label each book with its number. Each of the *n* books should be assigned with a number from 1 to *n*. Naturally, distinct books should be assigned distinct numbers.

Vanya wants to know how many digits he will have to write down as he labels the books.

Input Specification:
The first line contains integer *n* (1<=≤<=*n*<=≤<=109) — the number of books in the library.

Output Specification:
Print the number of digits needed to number all the books.

Demo Input:
['13\n', '4\n']

Demo Output:
['17\n', '4\n']

Note:
Note to the first test. The books get numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, which totals to 17 digits.

Note to the second sample. The books get numbers 1, 2, 3, 4, which totals to 4 digits.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
# Unflatten a list (Harder than easy)

This is the harder version of Unflatten a list (Easy)

So you have again to build a method, that creates new arrays, that can be flattened!

# Shorter: You have to unflatten a list/an array.

You get an array of integers and have to unflatten it by these rules:
```
- You have to do several runs. The depth is the number of runs, you have to do.
- In every run you have to switch the direction. First run from left, next run from right. Next left...
Every run has these rules:
- You start at the first number (from the direction).
- Take for every number x the remainder of the division by the number of still available elements (from 
  this position!) to have the number for the next decision.
- If the remainder-value is smaller than 3, take this number x (NOT the remainder-Value) direct
  for the new array and continue with the next number.
- If the remainder-value (e.g. 3) is greater than 2, take the next remainder-value-number (e.g. 3)
  elements/numbers (inclusive the number x, NOT the remainder-value) as a sub-array in the new array.
  Continue with the next number/element AFTER this taken elements/numbers.
- Every sub-array in the array is independent and is only one element for the progress on the array. 
  For every sub-array you have to follow the same rules for unflatten it.
  The direction is always the same as the actual run.
```

Sounds complicated? Yeah, thats why, this is the harder version...
Maybe an example will help.

```
 Array: [4, 5, 1, 7, 1] Depth: 2 -> [[ 4, [ 5, 1, 7 ] ], 1]
 
Steps: 
First run: (start from left side!)
1. The first number is 4. The number is smaller than the number of remaining elements, so it is the remainder-value (4 / 5 -> remainder 4).
   So 4 numbers (4, 5, 1, 7) are added as sub-array in the new array.
2. The next number is 1. It is smaller than 3, so the 1 is added direct to the new array.
Now we have --> [[4, 5, 1, 7], 1]

Second run: (start from right side!)
1. The last number (first from other side) is 1. So the 1 is added direct to the new array.
2. The next element is the sub-array. So we use the rules for this.
2a.The last number is 7. There are 4 elements in the array. So for the next decision you have to
   take the remainder from 7 / 4 -> 3. So 3 numbers (5, 1, 7) are added as sub-array in the 
   new array.
2b.Now there is the 4 and only one element last in this array. 4 / 1 -> remainder 0. It is smaller
   than 3. So the 4 is added direct to the new array.
Now we have --> [[ 4, [ 5, 1, 7 ] ], 1]
```

The given array will always contain numbers. There will only be numbers > 0.


Have fun coding it and please don't forget to vote and rank this kata! :-) 

I have created other katas. Have a look if you like coding and challenges.

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