meetforever/GRPO_CL huggingface dataset:
user: You are given a picture consisting of $n$ rows and $m$ columns. Rows are numbered from $1$ to $n$ from the top to the bottom, columns are numbered from $1$ to $m$ from the left to the right. Each cell is painted either black or white. 

You think that this picture is not interesting enough. You consider a picture to be interesting if there is at least one cross in it. A cross is represented by a pair of numbers $x$ and $y$, where $1 \le x \le n$ and $1 \le y \le m$, such that all cells in row $x$ and all cells in column $y$ are painted black.

For examples, each of these pictures contain crosses:

 [Image] 

The fourth picture contains 4 crosses: at $(1, 3)$, $(1, 5)$, $(3, 3)$ and $(3, 5)$.

Following images don't contain crosses:

 [Image] 

You have a brush and a can of black paint, so you can make this picture interesting. Each minute you may choose a white cell and paint it black.

What is the minimum number of minutes you have to spend so the resulting picture contains at least one cross?

You are also asked to answer multiple independent queries.


-----Input-----

The first line contains an integer $q$ ($1 \le q \le 5 \cdot 10^4$) — the number of queries.

The first line of each query contains two integers $n$ and $m$ ($1 \le n, m \le 5 \cdot 10^4$, $n \cdot m \le 4 \cdot 10^5$) — the number of rows and the number of columns in the picture.

Each of the next $n$ lines contains $m$ characters — '.' if the cell is painted white and '*' if the cell is painted black.

It is guaranteed that $\sum n \le 5 \cdot 10^4$ and $\sum n \cdot m \le 4 \cdot 10^5$.


-----Output-----

Print $q$ lines, the $i$-th line should contain a single integer — the answer to the $i$-th query, which is the minimum number of minutes you have to spend so the resulting picture contains at least one cross.


-----Example-----
Input
9
5 5
..*..
..*..
*****
..*..
..*..
3 4
****
.*..
.*..
4 3
***
*..
*..
*..
5 5
*****
*.*.*
*****
..*.*
..***
1 4
****
5 5
.....
..*..
.***.
..*..
.....
5 3
...
.*.
.*.
***
.*.
3 3
.*.
*.*
.*.
4 4
*.**
....
*.**
*.**

Output
0
0
0
0
0
4
1
1
2



-----Note-----

The example contains all the pictures from above in the same order.

The first 5 pictures already contain a cross, thus you don't have to paint anything.

You can paint $(1, 3)$, $(3, 1)$, $(5, 3)$ and $(3, 5)$ on the $6$-th picture to get a cross in $(3, 3)$. That'll take you $4$ minutes.

You can paint $(1, 2)$ on the $7$-th picture to get a cross in $(4, 2)$.

You can paint $(2, 2)$ on the $8$-th picture to get a cross in $(2, 2)$. You can, for example, paint $(1, 3)$, $(3, 1)$ and $(3, 3)$ to get a cross in $(3, 3)$ but that will take you $3$ minutes instead of $1$.

There are 9 possible crosses you can get in minimum time on the $9$-th picture. One of them is in $(1, 1)$: paint $(1, 2)$ and $(2, 1)$.
user: An agent called Cypher is decrypting a message, that contains a composite number $n$. All divisors of $n$, which are greater than $1$, are placed in a circle. Cypher can choose the initial order of numbers in the circle.

In one move Cypher can choose two adjacent numbers in a circle and insert their least common multiple between them. He can do that move as many times as needed.

A message is decrypted, if every two adjacent numbers are not coprime. Note that for such constraints it's always possible to decrypt the message.

Find the minimal number of moves that Cypher should do to decrypt the message, and show the initial order of numbers in the circle for that.


-----Input-----

The first line contains an integer $t$ $(1 \le t \le 100)$ — the number of test cases. Next $t$ lines describe each test case.

In a single line of each test case description, there is a single composite number $n$ $(4 \le n \le 10^9)$ — the number from the message.

It's guaranteed that the total number of divisors of $n$ for all test cases does not exceed $2 \cdot 10^5$.


-----Output-----

For each test case in the first line output the initial order of divisors, which are greater than $1$, in the circle. In the second line output, the minimal number of moves needed to decrypt the message.

If there are different possible orders with a correct answer, print any of them.


-----Example-----
Input
3
6
4
30

Output
2 3 6 
1
2 4 
0
2 30 6 3 15 5 10 
0



-----Note-----

In the first test case $6$ has three divisors, which are greater than $1$: $2, 3, 6$. Regardless of the initial order, numbers $2$ and $3$ are adjacent, so it's needed to place their least common multiple between them. After that the circle becomes $2, 6, 3, 6$, and every two adjacent numbers are not coprime.

In the second test case $4$ has two divisors greater than $1$: $2, 4$, and they are not coprime, so any initial order is correct, and it's not needed to place any least common multiples.

In the third test case all divisors of $30$ greater than $1$ can be placed in some order so that there are no two adjacent numbers that are coprime.
user: While doing some spring cleaning, Daniel found an old calculator that he loves so much. However, it seems like it is broken. When he tries to compute $1 + 3$ using the calculator, he gets $2$ instead of $4$. But when he tries computing $1 + 4$, he gets the correct answer, $5$. Puzzled by this mystery, he opened up his calculator and found the answer to the riddle: the full adders became half adders! 

So, when he tries to compute the sum $a + b$ using the calculator, he instead gets the xorsum $a \oplus b$ (read the definition by the link: https://en.wikipedia.org/wiki/Exclusive_or).

As he saw earlier, the calculator sometimes gives the correct answer. And so, he wonders, given integers $l$ and $r$, how many pairs of integers $(a, b)$ satisfy the following conditions: $$a + b = a \oplus b$$ $$l \leq a \leq r$$ $$l \leq b \leq r$$

However, Daniel the Barman is going to the bar and will return in two hours. He tells you to solve the problem before he returns, or else you will have to enjoy being blocked.


-----Input-----

The first line contains a single integer $t$ ($1 \le t \le 100$) — the number of testcases.

Then, $t$ lines follow, each containing two space-separated integers $l$ and $r$ ($0 \le l \le r \le 10^9$).


-----Output-----

Print $t$ integers, the $i$-th integer should be the answer to the $i$-th testcase.


-----Example-----
Input
3
1 4
323 323
1 1000000

Output
8
0
3439863766



-----Note-----

$a \oplus b$ denotes the bitwise XOR of $a$ and $b$.

For the first testcase, the pairs are: $(1, 2)$, $(1, 4)$, $(2, 1)$, $(2, 4)$, $(3, 4)$, $(4, 1)$, $(4, 2)$, and $(4, 3)$.
user: You are given three strings $a$, $b$ and $c$ of the same length $n$. The strings consist of lowercase English letters only. The $i$-th letter of $a$ is $a_i$, the $i$-th letter of $b$ is $b_i$, the $i$-th letter of $c$ is $c_i$.

For every $i$ ($1 \leq i \leq n$) you must swap (i.e. exchange) $c_i$ with either $a_i$ or $b_i$. So in total you'll perform exactly $n$ swap operations, each of them either $c_i \leftrightarrow a_i$ or $c_i \leftrightarrow b_i$ ($i$ iterates over all integers between $1$ and $n$, inclusive).

For example, if $a$ is "code", $b$ is "true", and $c$ is "help", you can make $c$ equal to "crue" taking the $1$-st and the $4$-th letters from $a$ and the others from $b$. In this way $a$ becomes "hodp" and $b$ becomes "tele".

Is it possible that after these swaps the string $a$ becomes exactly the same as the string $b$?


-----Input-----

The input consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 100$)  — the number of test cases. The description of the test cases follows.

The first line of each test case contains a string of lowercase English letters $a$.

The second line of each test case contains a string of lowercase English letters $b$.

The third line of each test case contains a string of lowercase English letters $c$.

It is guaranteed that in each test case these three strings are non-empty and have the same length, which is not exceeding $100$.


-----Output-----

Print $t$ lines with answers for all test cases. For each test case:

If it is possible to make string $a$ equal to string $b$ print "YES" (without quotes), otherwise print "NO" (without quotes).

You can print either lowercase or uppercase letters in the answers.


-----Example-----
Input
4
aaa
bbb
ccc
abc
bca
bca
aabb
bbaa
baba
imi
mii
iim

Output
NO
YES
YES
NO



-----Note-----

In the first test case, it is impossible to do the swaps so that string $a$ becomes exactly the same as string $b$.

In the second test case, you should swap $c_i$ with $a_i$ for all possible $i$. After the swaps $a$ becomes "bca", $b$ becomes "bca" and $c$ becomes "abc". Here the strings $a$ and $b$ are equal.

In the third test case, you should swap $c_1$ with $a_1$, $c_2$ with $b_2$, $c_3$ with $b_3$ and $c_4$ with $a_4$. Then string $a$ becomes "baba", string $b$ becomes "baba" and string $c$ becomes "abab". Here the strings $a$ and $b$ are equal.

In the fourth test case, it is impossible to do the swaps so that string $a$ becomes exactly the same as string $b$.
user: A permutation of length $n$ is an array consisting of $n$ distinct integers from $1$ to $n$ in arbitrary order. For example, $[2,3,1,5,4]$ is a permutation, but $[1,2,2]$ is not a permutation ($2$ appears twice in the array) and $[1,3,4]$ is also not a permutation ($n=3$ but there is $4$ in the array).

Let $p$ be any permutation of length $n$. We define the fingerprint $F(p)$ of $p$ as the sorted array of sums of adjacent elements in $p$. More formally,

$$F(p)=\mathrm{sort}([p_1+p_2,p_2+p_3,\ldots,p_{n-1}+p_n]).$$

For example, if $n=4$ and $p=[1,4,2,3],$ then the fingerprint is given by $F(p)=\mathrm{sort}([1+4,4+2,2+3])=\mathrm{sort}([5,6,5])=[5,5,6]$.

You are given a permutation $p$ of length $n$. Your task is to find a different permutation $p'$ with the same fingerprint. Two permutations $p$ and $p'$ are considered different if there is some index $i$ such that $p_i \ne p'_i$.


-----Input-----

Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 668$). Description of the test cases follows.

The first line of each test case contains a single integer $n$ ($2\le n\le 100$)  — the length of the permutation.

The second line of each test case contains $n$ integers $p_1,\ldots,p_n$ ($1\le p_i\le n$). It is guaranteed that $p$ is a permutation.


-----Output-----

For each test case, output $n$ integers $p'_1,\ldots, p'_n$ — a permutation such that $p'\ne p$ and $F(p')=F(p)$.

We can prove that for every permutation satisfying the input constraints, a solution exists.

If there are multiple solutions, you may output any.


-----Example-----
Input
3
2
1 2
6
2 1 6 5 4 3
5
2 4 3 1 5

Output
2 1
1 2 5 6 3 4
3 1 5 2 4



-----Note-----

In the first test case, $F(p)=\mathrm{sort}([1+2])=[3]$.

And $F(p')=\mathrm{sort}([2+1])=[3]$.

In the second test case, $F(p)=\mathrm{sort}([2+1,1+6,6+5,5+4,4+3])=\mathrm{sort}([3,7,11,9,7])=[3,7,7,9,11]$.

And $F(p')=\mathrm{sort}([1+2,2+5,5+6,6+3,3+4])=\mathrm{sort}([3,7,11,9,7])=[3,7,7,9,11]$.

In the third test case, $F(p)=\mathrm{sort}([2+4,4+3,3+1,1+5])=\mathrm{sort}([6,7,4,6])=[4,6,6,7]$.

And $F(p')=\mathrm{sort}([3+1,1+5,5+2,2+4])=\mathrm{sort}([4,6,7,6])=[4,6,6,7]$.
user: Being tired of participating in too many Codeforces rounds, Gildong decided to take some rest in a park. He sat down on a bench, and soon he found two rabbits hopping around. One of the rabbits was taller than the other.

He noticed that the two rabbits were hopping towards each other. The positions of the two rabbits can be represented as integer coordinates on a horizontal line. The taller rabbit is currently on position $x$, and the shorter rabbit is currently on position $y$ ($x \lt y$). Every second, each rabbit hops to another position. The taller rabbit hops to the positive direction by $a$, and the shorter rabbit hops to the negative direction by $b$.

 [Image] 

For example, let's say $x=0$, $y=10$, $a=2$, and $b=3$. At the $1$-st second, each rabbit will be at position $2$ and $7$. At the $2$-nd second, both rabbits will be at position $4$.

Gildong is now wondering: Will the two rabbits be at the same position at the same moment? If so, how long will it take? Let's find a moment in time (in seconds) after which the rabbits will be at the same point.


-----Input-----

Each test contains one or more test cases. The first line contains the number of test cases $t$ ($1 \le t \le 1000$).

Each test case contains exactly one line. The line consists of four integers $x$, $y$, $a$, $b$ ($0 \le x \lt y \le 10^9$, $1 \le a,b \le 10^9$) — the current position of the taller rabbit, the current position of the shorter rabbit, the hopping distance of the taller rabbit, and the hopping distance of the shorter rabbit, respectively.


-----Output-----

For each test case, print the single integer: number of seconds the two rabbits will take to be at the same position.

If the two rabbits will never be at the same position simultaneously, print $-1$.


-----Example-----
Input
5
0 10 2 3
0 10 3 3
900000000 1000000000 1 9999999
1 2 1 1
1 3 1 1

Output
2
-1
10
-1
1



-----Note-----

The first case is explained in the description.

In the second case, each rabbit will be at position $3$ and $7$ respectively at the $1$-st second. But in the $2$-nd second they will be at $6$ and $4$ respectively, and we can see that they will never be at the same position since the distance between the two rabbits will only increase afterward.
user: Let's look at the following process: initially you have an empty stack and an array $s$ of the length $l$. You are trying to push array elements to the stack in the order $s_1, s_2, s_3, \dots s_{l}$. Moreover, if the stack is empty or the element at the top of this stack is not equal to the current element, then you just push the current element to the top of the stack. Otherwise, you don't push the current element to the stack and, moreover, pop the top element of the stack. 

If after this process the stack remains empty, the array $s$ is considered stack exterminable.

There are samples of stack exterminable arrays:   $[1, 1]$;  $[2, 1, 1, 2]$;  $[1, 1, 2, 2]$;  $[1, 3, 3, 1, 2, 2]$;  $[3, 1, 3, 3, 1, 3]$;  $[3, 3, 3, 3, 3, 3]$;  $[5, 1, 2, 2, 1, 4, 4, 5]$; 

Let's consider the changing of stack more details if $s = [5, 1, 2, 2, 1, 4, 4, 5]$ (the top of stack is highlighted).   after pushing $s_1 = 5$ the stack turn into $[\textbf{5}]$;  after pushing $s_2 = 1$ the stack turn into $[5, \textbf{1}]$;  after pushing $s_3 = 2$ the stack turn into $[5, 1, \textbf{2}]$;  after pushing $s_4 = 2$ the stack turn into $[5, \textbf{1}]$;  after pushing $s_5 = 1$ the stack turn into $[\textbf{5}]$;  after pushing $s_6 = 4$ the stack turn into $[5, \textbf{4}]$;  after pushing $s_7 = 4$ the stack turn into $[\textbf{5}]$;  after pushing $s_8 = 5$ the stack is empty. 

You are given an array $a_1, a_2, \ldots, a_n$. You have to calculate the number of its subarrays which are stack exterminable.

Note, that you have to answer $q$ independent queries.


-----Input-----

The first line contains one integer $q$ ($1 \le q \le 3 \cdot 10^5$) — the number of queries.

The first line of each query contains one integer $n$ ($1 \le n \le 3 \cdot 10^5$) — the length of array $a$.

The second line of each query contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le n$) — the elements.

It is guaranteed that the sum of all $n$ over all queries does not exceed $3 \cdot 10^5$.


-----Output-----

For each test case print one integer in single line — the number of stack exterminable subarrays of the array $a$.


-----Example-----
Input
3
5
2 1 1 2 2
6
1 2 1 1 3 2
9
3 1 2 2 1 6 6 3 3

Output
4
1
8



-----Note-----

In the first query there are four stack exterminable subarrays: $a_{1 \ldots 4} = [2, 1, 1, 2], a_{2 \ldots 3} = [1, 1], a_{2 \ldots 5} = [1, 1, 2, 2], a_{4 \ldots 5} = [2, 2]$.

In the second query, only one subarray is exterminable subarray — $a_{3 \ldots 4}$.

In the third query, there are eight stack exterminable subarrays: $a_{1 \ldots 8}, a_{2 \ldots 5}, a_{2 \ldots 7}, a_{2 \ldots 9}, a_{3 \ldots 4}, a_{6 \ldots 7}, a_{6 \ldots 9}, a_{8 \ldots 9}$.
user: Consider the following process. You have a binary string (a string where each character is either 0 or 1) $w$ of length $n$ and an integer $x$. You build a new binary string $s$ consisting of $n$ characters. The $i$-th character of $s$ is chosen as follows:

  if the character $w_{i-x}$ exists and is equal to 1, then $s_i$ is 1 (formally, if $i > x$ and $w_{i-x} = $ 1, then $s_i = $ 1);  if the character $w_{i+x}$ exists and is equal to 1, then $s_i$ is 1 (formally, if $i + x \le n$ and $w_{i+x} = $ 1, then $s_i = $ 1);  if both of the aforementioned conditions are false, then $s_i$ is 0. 

You are given the integer $x$ and the resulting string $s$. Reconstruct the original string $w$.


-----Input-----

The first line contains one integer $t$ ($1 \le t \le 1000$) — the number of test cases.

Each test case consists of two lines. The first line contains the resulting string $s$ ($2 \le |s| \le 10^5$, each character of $s$ is either 0 or 1). The second line contains one integer $x$ ($1 \le x \le |s| - 1$).

The total length of all strings $s$ in the input does not exceed $10^5$.


-----Output-----

For each test case, print the answer on a separate line as follows:

  if no string $w$ can produce the string $s$ at the end of the process, print $-1$;  otherwise, print the binary string $w$ consisting of $|s|$ characters. If there are multiple answers, print any of them. 


-----Example-----
Input
3
101110
2
01
1
110
1

Output
111011
10
-1
user: Alexandra has an even-length array $a$, consisting of $0$s and $1$s. The elements of the array are enumerated from $1$ to $n$. She wants to remove at most $\frac{n}{2}$ elements (where $n$ — length of array) in the way that alternating sum of the array will be equal $0$ (i.e. $a_1 - a_2 + a_3 - a_4 + \dotsc = 0$). In other words, Alexandra wants sum of all elements at the odd positions and sum of all elements at the even positions to become equal. The elements that you remove don't have to be consecutive.

For example, if she has $a = [1, 0, 1, 0, 0, 0]$ and she removes $2$nd and $4$th elements, $a$ will become equal $[1, 1, 0, 0]$ and its alternating sum is $1 - 1 + 0 - 0 = 0$.

Help her!


-----Input-----

Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^3$). Description of the test cases follows.

The first line of each test case contains a single integer $n$ ($2 \le n \le 10^3$, $n$ is even)  — length of the array.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($0 \le a_i \le 1$)  — elements of the array.

It is guaranteed that the sum of $n$ over all test cases does not exceed $10^3$.


-----Output-----

For each test case, firstly, print $k$ ($\frac{n}{2} \leq k \leq n$) — number of elements that will remain after removing in the order they appear in $a$. Then, print this $k$ numbers. Note that you should print the numbers themselves, not their indices.

We can show that an answer always exists. If there are several answers, you can output any of them. 


-----Example-----
Input
4
2
1 0
2
0 0
4
0 1 1 1
4
1 1 0 0

Output
1
0
1
0
2
1 1
4
1 1 0 0



-----Note-----

In the first and second cases, alternating sum of the array, obviously, equals $0$.

In the third case, alternating sum of the array equals $1 - 1 = 0$.

In the fourth case, alternating sum already equals $1 - 1 + 0 - 0 = 0$, so we don't have to remove anything.
user: Berland year consists of $m$ months with $d$ days each. Months are numbered from $1$ to $m$. Berland week consists of $w$ days. The first day of the year is also the first day of the week. Note that the last week of the year might be shorter than $w$ days.

A pair $(x, y)$ such that $x < y$ is ambiguous if day $x$ of month $y$ is the same day of the week as day $y$ of month $x$.

Count the number of ambiguous pairs.


-----Input-----

The first line contains a single integer $t$ ($1 \le t \le 1000$) — the number of testcases.

Each of the next $t$ lines contains three integers $m$, $d$ and $w$ ($1 \le m, d, w \le 10^9$) — the number of months in a year, the number of days in a month and the number of days in a week.


-----Output-----

Print $t$ integers — for each testcase output the number of pairs $(x, y)$ such that $x < y$ and day $x$ of month $y$ is the same day of the week as day $y$ of month $x$.


-----Example-----
Input
5
6 7 4
10 7 12
12 30 7
1 1 1
3247834 10298779 625324

Output
6
9
5
0
116461800



-----Note-----

Here are the pairs for the first test case:  $$
user: Bob is playing with $6$-sided dice. A net of such standard cube is shown below.

[Image]

He has an unlimited supply of these dice and wants to build a tower by stacking multiple dice on top of each other, while choosing the orientation of each dice. Then he counts the number of visible pips on the faces of the dice.

For example, the number of visible pips on the tower below is $29$ — the number visible on the top is $1$, from the south $5$ and $3$, from the west $4$ and $2$, from the north $2$ and $4$ and from the east $3$ and $5$.

[Image]

The one at the bottom and the two sixes by which the dice are touching are not visible, so they are not counted towards total.

Bob also has $t$ favourite integers $x_i$, and for every such integer his goal is to build such a tower that the number of visible pips is exactly $x_i$. For each of Bob's favourite integers determine whether it is possible to build a tower that has exactly that many visible pips.


-----Input-----

The first line contains a single integer $t$ ($1 \leq t \leq 1000$) — the number of favourite integers of Bob. 

The second line contains $t$ space-separated integers $x_i$ ($1 \leq x_i \leq 10^{18}$) — Bob's favourite integers.


-----Output-----

For each of Bob's favourite integers, output "YES" if it is possible to build the tower, or "NO" otherwise (quotes for clarity).


-----Example-----
Input
4
29 34 19 38

Output
YES
YES
YES
NO



-----Note-----

The first example is mentioned in the problem statement.

In the second example, one can build the tower by flipping the top dice from the previous tower.

In the third example, one can use a single die that has $5$ on top.

The fourth example is impossible.