vgel/INTELLECT-2-RL-Dataset-Code-Only huggingface dataset:
Title: Mushroom Gnomes

Time Limit: 3 seconds

Memory Limit: 256 megabytes

Problem Description:
Once upon a time in the thicket of the mushroom forest lived mushroom gnomes. They were famous among their neighbors for their magic mushrooms. Their magic nature made it possible that between every two neighboring mushrooms every minute grew another mushroom with the weight equal to the sum of weights of two neighboring ones. 

The mushroom gnomes loved it when everything was in order, that's why they always planted the mushrooms in one line in the order of their weights' increasing. Well... The gnomes planted the mushrooms and went to eat. After *x* minutes they returned and saw that new mushrooms had grown up, so that the increasing order had been violated. The gnomes replanted all the mushrooms in the correct order, that is, they sorted the mushrooms in the order of the weights' increasing. And went to eat again (those gnomes were quite big eaters). What total weights modulo *p* will the mushrooms have in another *y* minutes?

Input Specification:
The first line contains four integers *n*, *x*, *y*, *p* (1<=≤<=*n*<=≤<=106,<=0<=≤<=*x*,<=*y*<=≤<=1018,<=*x*<=+<=*y*<=&gt;<=0,<=2<=≤<=*p*<=≤<=109) which represent the number of mushrooms, the number of minutes after the first replanting, the number of minutes after the second replanting and the module. The next line contains *n* integers *a**i* which represent the mushrooms' weight in the non-decreasing order (0<=≤<=*a**i*<=≤<=109).

Please, do not use %lld specificator to read or write 64-bit integers in C++. It is preffered to use cin (also you may use %I64d).

Output Specification:
The answer should contain a single number which is the total weights of the mushrooms modulo *p* in the end after *x*<=+<=*y* minutes.

Demo Input:
['2 1 0 657276545\n1 2\n', '2 1 1 888450282\n1 2\n', '4 5 0 10000\n1 2 3 4\n']

Demo Output:
['6\n', '14\n', '1825\n']

Note:
none

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
The annual snake festival is upon us, and all the snakes of the kingdom have gathered to participate in the procession. Chef has been tasked with reporting on the procession, and for this he decides to first keep track of all the snakes. When he sees a snake first, it'll be its Head, and hence he will mark a 'H'. The snakes are long, and when he sees the snake finally slither away, he'll mark a 'T' to denote its tail. In the time in between, when the snake is moving past him, or the time between one snake and the next snake, he marks with '.'s.
Because the snakes come in a procession, and one by one, a valid report would be something like "..H..T...HTH....T.", or "...", or "HT", whereas "T...H..H.T", "H..T..H", "H..H..T..T" would be invalid reports (See explanations at the bottom).
Formally, a snake is represented by a 'H' followed by some (possibly zero) '.'s, and then a 'T'. A valid report is one such that it begins with a (possibly zero length) string of '.'s, and then some (possibly zero) snakes between which there can be some '.'s, and then finally ends with some (possibly zero) '.'s.

Chef had binged on the festival food and had been very drowsy. So his report might be invalid. You need to help him find out if his report is valid or not.

-----Input-----
- The first line contains a single integer, R, which denotes the number of reports to be checked. The description of each report follows after this.
- The first line of each report contains a single integer, L, the length of that report.
- The second line of each report contains a string of length L. The string contains only the characters '.', 'H', and 'T'.

-----Output-----
- For each report, output the string "Valid" or "Invalid" in a new line, depending on whether it was a valid report or not.

-----Constraints-----
- 1 ≤ R ≤ 500
- 1 ≤ length of each report ≤ 500

-----Example-----
Input:
6
18
..H..T...HTH....T.
3
...
10
H..H..T..T
2
HT
11
.T...H..H.T
7
H..T..H

Output:
Valid
Valid
Invalid
Valid
Invalid
Invalid

-----Explanation-----
"H..H..T..T" is invalid because the second snake starts before the first snake ends, which is not allowed.
".T...H..H.T" is invalid because it has a 'T' before a 'H'. A tail can come only after its head.
"H..T..H" is invalid because the last 'H' does not have a corresponding 'T'.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Title: System of Equations

Time Limit: None seconds

Memory Limit: None megabytes

Problem Description:
Furik loves math lessons very much, so he doesn't attend them, unlike Rubik. But now Furik wants to get a good mark for math. For that Ms. Ivanova, his math teacher, gave him a new task. Furik solved the task immediately. Can you?

You are given a system of equations: 

You should count, how many there are pairs of integers (*a*,<=*b*) (0<=≤<=*a*,<=*b*) which satisfy the system.

Input Specification:
A single line contains two integers *n*,<=*m* (1<=≤<=*n*,<=*m*<=≤<=1000) — the parameters of the system. The numbers on the line are separated by a space.

Output Specification:
On a single line print the answer to the problem.

Demo Input:
['9 3\n', '14 28\n', '4 20\n']

Demo Output:
['1\n', '1\n', '0\n']

Note:
In the first sample the suitable pair is integers (3, 0). In the second sample the suitable pair is integers (3, 5). In the third sample there is no suitable pair.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Ohana Matsumae is trying to clean a room, which is divided up into an n by n grid of squares. Each square is initially either clean or dirty. Ohana can sweep her broom over columns of the grid. Her broom is very strange: if she sweeps over a clean square, it will become dirty, and if she sweeps over a dirty square, it will become clean. She wants to sweep some columns of the room to maximize the number of rows that are completely clean. It is not allowed to sweep over the part of the column, Ohana can only sweep the whole column.

Return the maximum number of rows that she can make completely clean.


-----Input-----

The first line of input will be a single integer n (1 ≤ n ≤ 100).

The next n lines will describe the state of the room. The i-th line will contain a binary string with n characters denoting the state of the i-th row of the room. The j-th character on this line is '1' if the j-th square in the i-th row is clean, and '0' if it is dirty.


-----Output-----

The output should be a single line containing an integer equal to a maximum possible number of rows that are completely clean.


-----Examples-----
Input
4
0101
1000
1111
0101

Output
2

Input
3
111
111
111

Output
3



-----Note-----

In the first sample, Ohana can sweep the 1st and 3rd columns. This will make the 1st and 4th row be completely clean.

In the second sample, everything is already clean, so Ohana doesn't need to do anything.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Chef has two integers A and B. In one operation he can choose any integer d, and make one of the following two moves :
Add d to A and subtract d from B.
Add d to B and subtract d from A.

Chef is allowed to make as many operations as he wants. Can he make A and B equal?

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

- First line will contain T, number of test cases. Then the test cases follow.
- Each test case contains of a single line of input, two integers A, B.

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

For each test case, if Chef can make the two numbers equal print YES else print NO.

You may print each character of the string in uppercase or lowercase (for example, the strings yEs, Yes, YeS, and YES will all be treated as identical).

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

$1 ≤ T ≤ 1000$
$1 ≤ A,B ≤ 1000$

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

----- Sample Output 1 ------ 
Yes
No

----- explanation 1 ------ 
Test case $1$: Since $A$ and $B$ are already equal, Chef does not need any operations.

Test case $2$: It can be shown that $A$ and $B$ can never be made equal using any number of given operations.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Given two integers $n$ and $x$, construct an array that satisfies the following conditions:   for any element $a_i$ in the array, $1 \le a_i<2^n$;  there is no non-empty subsegment with bitwise XOR equal to $0$ or $x$,  its length $l$ should be maximized. 

A sequence $b$ is a subsegment of a sequence $a$ if $b$ can be obtained from $a$ by deletion of several (possibly, zero or all) elements from the beginning and several (possibly, zero or all) elements from the end.


-----Input-----

The only line contains two integers $n$ and $x$ ($1 \le n \le 18$, $1 \le x<2^{18}$).


-----Output-----

The first line should contain the length of the array $l$.

If $l$ is positive, the second line should contain $l$ space-separated integers $a_1$, $a_2$, $\dots$, $a_l$ ($1 \le a_i < 2^n$) — the elements of the array $a$.

If there are multiple solutions, print any of them.


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

Output
3
6 1 3
Input
2 4

Output
3
1 3 1 
Input
1 1

Output
0



-----Note-----

In the first example, the bitwise XOR of the subsegments are $\{6,7,4,1,2,3\}$.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
=====Function Descriptions=====
.union()

The .union() operator returns the union of a set and the set of elements in an iterable.
Sometimes, the | operator is used in place of .union() operator, but it operates only on the set of elements in set.
Set is immutable to the .union() operation (or | operation).

Example

>>> s = set("Hacker")
>>> print s.union("Rank")
set(['a', 'R', 'c', 'r', 'e', 'H', 'k', 'n'])

>>> print s.union(set(['R', 'a', 'n', 'k']))
set(['a', 'R', 'c', 'r', 'e', 'H', 'k', 'n'])

>>> print s.union(['R', 'a', 'n', 'k'])
set(['a', 'R', 'c', 'r', 'e', 'H', 'k', 'n'])

>>> print s.union(enumerate(['R', 'a', 'n', 'k']))
set(['a', 'c', 'r', 'e', (1, 'a'), (2, 'n'), 'H', 'k', (3, 'k'), (0, 'R')])

>>> print s.union({"Rank":1})
set(['a', 'c', 'r', 'e', 'H', 'k', 'Rank'])

>>> s | set("Rank")
set(['a', 'R', 'c', 'r', 'e', 'H', 'k', 'n'])

=====Problem Statement=====

The students of District College have subscriptions to English and French newspapers. Some students have subscribed only to English, some have subscribed to only French and some have subscribed to both newspapers.

You are given two sets of student roll numbers. One set has subscribed to the English newspaper, and the other set is subscribed to the French newspaper. The same student could be in both sets. Your task is to find the total number of students who have subscribed to at least one newspaper.

=====Input Format=====
The first line contains an integer, n, the number of students who have subscribed to the English newspaper.
The second line contains n space separated roll numbers of those students.
The third line contains b, the number of students who have subscribed to the French newspaper.
The fourth line contains b space separated roll numbers of those students. 

=====Constraints=====
0 < Total number of students in college < 1000

=====Output Format=====
Output the total number of students who have at least one subscription.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Chandler is participating in a race competition involving N track races. He wants to run his old car on these tracks having F amount of initial fuel. At the end of each race, Chandler spends si fuel and gains some money using which he adds ei amount of fuel to his car. 
Also for participating in race i at any stage, Chandler should have more than si amount of fuel. Also he can participate in race i once. Help Chandler in maximizing the number of races he can take part in if he has a choice to participate in the given races in any order.
Input
First line will contain T denoting number of test cases.
Now in each test case first line will contain 2 integers F and N denoting initial amount of fuel and number of tasks respectively.
Next N lines will contain 2 integers si  and ei denoting fuel spent and amount of fuel he adds after participating in race i.

Output
For each test case output maximum number of races in which Chandler can take part.

Input Constraints
1 ≤ T ≤ 10
1 ≤ F ≤ 1000
1 ≤ N ≤ 100
0 ≤ ei  < si  ≤ 1000

SAMPLE INPUT
2
10 2
8 0
4 2
5 1
10 0

SAMPLE OUTPUT
2
0

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Little Tanya decided to present her dad a postcard on his Birthday. She has already created a message — string s of length n, consisting of uppercase and lowercase English letters. Tanya can't write yet, so she found a newspaper and decided to cut out the letters and glue them into the postcard to achieve string s. The newspaper contains string t, consisting of uppercase and lowercase English letters. We know that the length of string t greater or equal to the length of the string s.

The newspaper may possibly have too few of some letters needed to make the text and too many of some other letters. That's why Tanya wants to cut some n letters out of the newspaper and make a message of length exactly n, so that it looked as much as possible like s. If the letter in some position has correct value and correct letter case (in the string s and in the string that Tanya will make), then she shouts joyfully "YAY!", and if the letter in the given position has only the correct value but it is in the wrong case, then the girl says "WHOOPS".

Tanya wants to make such message that lets her shout "YAY!" as much as possible. If there are multiple ways to do this, then her second priority is to maximize the number of times she says "WHOOPS". Your task is to help Tanya make the message.


-----Input-----

The first line contains line s (1 ≤ |s| ≤ 2·10^5), consisting of uppercase and lowercase English letters — the text of Tanya's message.

The second line contains line t (|s| ≤ |t| ≤ 2·10^5), consisting of uppercase and lowercase English letters — the text written in the newspaper.

Here |a| means the length of the string a.


-----Output-----

Print two integers separated by a space:  the first number is the number of times Tanya shouts "YAY!" while making the message,  the second number is the number of times Tanya says "WHOOPS" while making the message. 


-----Examples-----
Input
AbC
DCbA

Output
3 0

Input
ABC
abc

Output
0 3

Input
abacaba
AbaCaBA

Output
3 4

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

Time Limit: None seconds

Memory Limit: None megabytes

Problem Description:
You are given two arrays *A* and *B*, each of size *n*. The error, *E*, between these two arrays is defined . You have to perform exactly *k*1 operations on array *A* and exactly *k*2 operations on array *B*. In one operation, you have to choose one element of the array and increase or decrease it by 1.

Output the minimum possible value of error after *k*1 operations on array *A* and *k*2 operations on array *B* have been performed.

Input Specification:
The first line contains three space-separated integers *n* (1<=≤<=*n*<=≤<=103), *k*1 and *k*2 (0<=≤<=*k*1<=+<=*k*2<=≤<=103, *k*1 and *k*2 are non-negative) — size of arrays and number of operations to perform on *A* and *B* respectively.

Second line contains *n* space separated integers *a*1,<=*a*2,<=...,<=*a**n* (<=-<=106<=≤<=*a**i*<=≤<=106) — array *A*.

Third line contains *n* space separated integers *b*1,<=*b*2,<=...,<=*b**n* (<=-<=106<=≤<=*b**i*<=≤<=106)— array *B*.

Output Specification:
Output a single integer — the minimum possible value of  after doing exactly *k*1 operations on array *A* and exactly *k*2 operations on array *B*.

Demo Input:
['2 0 0\n1 2\n2 3\n', '2 1 0\n1 2\n2 2\n', '2 5 7\n3 4\n14 4\n']

Demo Output:
['2', '0', '1']

Note:
In the first sample case, we cannot perform any operations on *A* or *B*. Therefore the minimum possible error *E* = (1 - 2)<sup class="upper-index">2</sup> + (2 - 3)<sup class="upper-index">2</sup> = 2. 

In the second sample case, we are required to perform exactly one operation on *A*. In order to minimize error, we increment the first element of *A* by 1. Now, *A* = [2, 2]. The error is now *E* = (2 - 2)<sup class="upper-index">2</sup> + (2 - 2)<sup class="upper-index">2</sup> = 0. This is the minimum possible error obtainable.

In the third sample case, we can increase the first element of *A* to 8, using the all of the 5 moves available to us. Also, the first element of *B* can be reduced to 8 using the 6 of the 7 available moves. Now *A* = [8, 4] and *B* = [8, 4]. The error is now *E* = (8 - 8)<sup class="upper-index">2</sup> + (4 - 4)<sup class="upper-index">2</sup> = 0, but we are still left with 1 move for array *B*. Increasing the second element of *B* to 5 using the left move, we get *B* = [8, 5] and *E* = (8 - 8)<sup class="upper-index">2</sup> + (4 - 5)<sup class="upper-index">2</sup> = 1.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Given a square matrix (i.e. an array of subarrays), find the sum of values from the first value of the first array, the second value of the second array, the third value of the third array, and so on...

## Examples

```
array = [[1, 2],
         [3, 4]]

diagonal sum: 1 + 4 = 5
```
                
```
array = [[5, 9, 1, 0],
         [8, 7, 2, 3],
         [1, 4, 1, 9],
         [2, 3, 8, 2]]

diagonal sum: 5 + 7 + 1 + 2 = 15
```

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
There are N cells arranged in a row, numbered 1, 2, \ldots, N from left to right.

Tak lives in these cells and is currently on Cell 1. He is trying to reach Cell N by using the procedure described below.

You are given an integer K that is less than or equal to 10, and K non-intersecting segments [L_1, R_1], [L_2, R_2], \ldots, [L_K, R_K]. Let S be the union of these K segments. Here, the segment [l, r] denotes the set consisting of all integers i that satisfy l \leq i \leq r.

* When you are on Cell i, pick an integer d from S and move to Cell i + d. You cannot move out of the cells.



To help Tak, find the number of ways to go to Cell N, modulo 998244353.

Constraints

* 2 \leq N \leq 2 \times 10^5
* 1 \leq K \leq \min(N, 10)
* 1 \leq L_i \leq R_i \leq N
* [L_i, R_i] and [L_j, R_j] do not intersect (i \neq j)
* All values in input are integers.

Input

Input is given from Standard Input in the following format:


N K
L_1 R_1
L_2 R_2
:
L_K R_K


Output

Print the number of ways for Tak to go from Cell 1 to Cell N, modulo 998244353.

Examples

Input

5 2
1 1
3 4


Output

4


Input

5 2
3 3
5 5


Output

0


Input

5 1
1 2


Output

5


Input

60 3
5 8
1 3
10 15


Output

221823067

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
You've got a positive integer sequence a_1, a_2, ..., a_{n}. All numbers in the sequence are distinct. Let's fix the set of variables b_1, b_2, ..., b_{m}. Initially each variable b_{i} (1 ≤ i ≤ m) contains the value of zero. Consider the following sequence, consisting of n operations.

The first operation is assigning the value of a_1 to some variable b_{x} (1 ≤ x ≤ m). Each of the following n - 1 operations is assigning to some variable b_{y} the value that is equal to the sum of values that are stored in the variables b_{i} and b_{j} (1 ≤ i, j, y ≤ m). At that, the value that is assigned on the t-th operation, must equal a_{t}. For each operation numbers y, i, j are chosen anew.

Your task is to find the minimum number of variables m, such that those variables can help you perform the described sequence of operations.


-----Input-----

The first line contains integer n (1 ≤ n ≤ 23). The second line contains n space-separated integers a_1, a_2, ..., a_{n} (1 ≤ a_{k} ≤ 10^9).

It is guaranteed that all numbers in the sequence are distinct.


-----Output-----

In a single line print a single number — the minimum number of variables m, such that those variables can help you perform the described sequence of operations.

If you cannot perform the sequence of operations at any m, print -1.


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

Output
2

Input
3
3 6 5

Output
-1

Input
6
2 4 8 6 10 18

Output
3



-----Note-----

In the first sample, you can use two variables b_1 and b_2 to perform the following sequence of operations.  b_1 := 1;  b_2 := b_1 + b_1;  b_1 := b_1 + b_2;  b_1 := b_1 + b_1;  b_1 := b_1 + b_2.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Alice and Bob are very good friends and they always distribute all the eatables equally among themselves.

Alice has A chocolates and Bob has B chocolates. Determine whether Alice and Bob can distribute all the chocolates equally among themselves.

Note that:
It is not allowed to break a chocolate into more than one piece.
No chocolate shall be left in the distribution.

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

- The first line of input will contain an integer T — the number of test cases. The description of T test cases follows.
- The first and only line of each test case contains two space-separated integers A and B, the number of chocolates that Alice and Bob have, respectively. 

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

For each test case, output on a new line \texttt{YES} if Alice and Bob can distribute all the chocolates equally, else output \texttt{NO}. 
The output is case insensitive, i.e, \texttt{yes}, \texttt{YeS}, \texttt{yES} will all be accepted as correct answers when Alice and Bob can distribute the chocolates equally.

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

$1 ≤ T ≤ 1000$
$1 ≤ A, B ≤ 10^{5}$

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

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

----- explanation 1 ------ 
Test case $1$: Both Alice and Bob already have equal number of chocolates, hence it is possible to distribute the chocolates equally among Alice and Bob. 

Test case $2$: If Bob gives one of his chocolates to Alice, then both of them will have equal number of chocolates, i.e. $2$. So, it is possible to distribute the chocolates equally among Alice and Bob.

Test case $3$: There are total $3$ chocolates. These chocolates cannot be divided equally among Alice and Bob. 

Test case $4$: Alice and Bob cannot have equal number of chocolates, no matter how they distribute the chocolates.

Write Python code to solve the problem. Present the code in 
```python
Your code
```
at the end.
Chef has an array A consisting of N elements. He wants to find number of pairs of non-intersecting segments [a, b] and [c, d] (1 ≤ a ≤ b < c ≤ d ≤ N) such there is no number that occurs in the subarray {Aa, Aa+1, ... , Ab} and   {Ac, Ac+1, ... , Ad} simultaneously. 
Help Chef to find this number.

-----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 test case contains a single integer N denoting the number of elements in the array.
- The second line contains N space-separated integers A1, A2, ..., AN. 

-----Output-----
- For each test case, output a single line containing one integer - number of pairs of non-intersecting segments. 

-----Constraints-----
- 1 ≤ T ≤ 5
- 1 ≤ N ≤ 1000
- 1 ≤ Ai ≤ 109

-----Subtasks-----Subtask 1 (7 points)
- 1 ≤ N ≤ 20Subtask 2 (34 points)
- 1 ≤ N ≤ 300Subtask 3 (59 points)
- Original constraints

-----Example-----
Input:
2
3
1 2 3
4
1 2 1 2

Output:
5
4

-----Explanation-----
Example case 1.
All possible variants are correct: {[1, 1], [2, 2]}, {[1, 1], [2, 3]}, {[1, 2], [3, 3]}, {[2, 2], [3, 3]}, {[1,1], [3, 3]}.

Example case 2.
Correct segments: {[1, 1], [2, 2]}, {[1, 1], [4, 4]}, {[2, 2], [3, 3]}, {[3, 3], [4, 4]}.

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