apaz/codeforces-genesys huggingface dataset:
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 1.0 seconds
Memory limit: 256.0 MB

# Problem
The only difference between this problem and D1 is that you don't have to provide the way to construct the answer in D1, but you have to do it in this problem.

There's a table of $$$n \times m$$$ cells ($$$n$$$ rows and $$$m$$$ columns). The value of $$$n \cdot m$$$ is even.

A domino is a figure that consists of two cells having a common side. It may be horizontal (one of the cells is to the right of the other) or vertical (one of the cells is above the other).

You need to place $$$\frac{nm}{2}$$$ dominoes on the table so that exactly $$$k$$$ of them are horizontal and all the other dominoes are vertical. The dominoes cannot overlap and must fill the whole table.

## Input Format
The first line contains one integer $$$t$$$ ($$$1 \le t \le 10$$$) — the number of test cases. Then $$$t$$$ test cases follow.

Each test case consists of a single line. The line contains three integers $$$n$$$, $$$m$$$, $$$k$$$ ($$$1 \le n,m \le 100$$$, $$$0 \le k \le \frac{nm}{2}$$$, $$$n \cdot m$$$ is even) — the count of rows, columns and horizontal dominoes, respectively.

## Output Format
For each test case:

- print "NO" if it's not possible to place the dominoes on the table in the described way;
- otherwise, print "YES" on a separate line, then print $$$n$$$ lines so that each of them contains $$$m$$$ lowercase letters of the Latin alphabet — the layout of the dominoes on the table. Each cell of the table must be marked by the letter so that for every two cells having a common side, they are marked by the same letters if and only if they are occupied by the same domino. I.e. both cells of the same domino must be marked with the same letter, but two dominoes that share a side must be marked with different letters. If there are multiple solutions, print any of them.

## Examples
```input
8
4 4 2
2 3 0
3 2 3
1 2 0
2 4 2
5 2 2
2 17 16
2 1 1
```
```output
YES
accx
aegx
bega
bdda
YES
aha
aha
YES
zz
aa
zz
NO
YES
aaza
bbza
NO
YES
bbaabbaabbaabbaay
ddccddccddccddccy
NO
```

Now solve the problem and return the code.
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 1.0 seconds
Memory limit: 256.0 MB

# Problem
A tree is an undirected connected graph without cycles.

You are given a tree of $$$n$$$ vertices. Find the number of ways to choose exactly $$$k$$$ vertices in this tree (i. e. a $$$k$$$-element subset of vertices) so that all pairwise distances between the selected vertices are equal (in other words, there exists an integer $$$c$$$ such that for all $$$u, v$$$ ($$$u \ne v$$$, $$$u, v$$$ are in selected vertices) $$$d_{u,v}=c$$$, where $$$d_{u,v}$$$ is the distance from $$$u$$$ to $$$v$$$).

Since the answer may be very large, you need to output it modulo $$$10^9 + 7$$$.

## Input Format
The first line contains one integer $$$t$$$ ($$$1 \le t \le 10$$$) — the number of test cases. Then $$$t$$$ test cases follow.

Each test case is preceded by an empty line.

Each test case consists of several lines. The first line of the test case contains two integers $$$n$$$ and $$$k$$$ ($$$2 \le k \le n \le 100$$$) — the number of vertices in the tree and the number of vertices to be selected, respectively. Then $$$n - 1$$$ lines follow, each of them contains two integers $$$u$$$ and $$$v$$$ ($$$1 \le u, v \le n$$$, $$$u \neq v$$$) which describe a pair of vertices connected by an edge. It is guaranteed that the given graph is a tree and has no loops or multiple edges.

## Output Format
For each test case output in a separate line a single integer — the number of ways to select exactly $$$k$$$ vertices so that for all pairs of selected vertices the distances between the vertices in the pairs are equal, modulo $$$10^9 + 7$$$ (in other words, print the remainder when divided by $$$1000000007$$$).

## Examples
```input
3

4 2
1 2
2 3
2 4

3 3
1 2
2 3

5 3
1 2
2 3
2 4
4 5
```
```output
6
0
1
```

Now solve the problem and return the code.
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 1.0 seconds
Memory limit: 256.0 MB

# Problem
Polycarp doesn't like integers that are divisible by $$$3$$$ or end with the digit $$$3$$$ in their decimal representation. Integers that meet both conditions are disliked by Polycarp, too.

Polycarp starts to write out the positive (greater than $$$0$$$) integers which he likes: $$$1, 2, 4, 5, 7, 8, 10, 11, 14, 16, \dots$$$. Output the $$$k$$$-th element of this sequence (the elements are numbered from $$$1$$$).

## Input Format
The first line contains one integer $$$t$$$ ($$$1 \le t \le 100$$$) — the number of test cases. Then $$$t$$$ test cases follow.

Each test case consists of one line containing one integer $$$k$$$ ($$$1 \le k \le 1000$$$).

## Output Format
For each test case, output in a separate line one integer $$$x$$$ — the $$$k$$$-th element of the sequence that was written out by Polycarp.

## Examples
```input
10
1
2
3
4
5
6
7
8
9
1000
```
```output
1
2
4
5
7
8
10
11
14
1666
```

Now solve the problem and return the code.
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 1.0 seconds
Memory limit: 256.0 MB

# Problem
Some number of people (this number is even) have stood in a circle. The people stand in the circle evenly. They are numbered clockwise starting from a person with the number $$$1$$$. Each person is looking through the circle's center at the opposite person.

A sample of a circle of $$$6$$$ persons. The orange arrows indicate who is looking at whom.

You don't know the exact number of people standing in the circle (but this number is even, no doubt). It is known that the person with the number $$$a$$$ is looking at the person with the number $$$b$$$ (and vice versa, of course). What is the number associated with a person being looked at by the person with the number $$$c$$$? If, for the specified $$$a$$$, $$$b$$$, and $$$c$$$, no such circle exists, output -1.

## Input Format
The first line contains one integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases. Then $$$t$$$ test cases follow.

Each test case consists of one line containing three distinct integers $$$a$$$, $$$b$$$, $$$c$$$ ($$$1 \le a,b,c \le 10^8$$$).

## Output Format
For each test case output in a separate line a single integer $$$d$$$ — the number of the person being looked at by the person with the number $$$c$$$ in a circle such that the person with the number $$$a$$$ is looking at the person with the number $$$b$$$. If there are multiple solutions, print any of them. Output $$$-1$$$ if there's no circle meeting the given conditions.

## Examples
```input
7
6 2 4
2 3 1
2 4 10
5 3 4
1 3 2
2 5 4
4 3 2
```
```output
8
-1
-1
-1
4
1
-1
```

## Note
In the first test case, there's a desired circle of $$$8$$$ people. The person with the number $$$6$$$ will look at the person with the number $$$2$$$ and the person with the number $$$8$$$ will look at the person with the number $$$4$$$.

In the second test case, there's no circle meeting the conditions. If the person with the number $$$2$$$ is looking at the person with the number $$$3$$$, the circle consists of $$$2$$$ people because these persons are neighbors. But, in this case, they must have the numbers $$$1$$$ and $$$2$$$, but it doesn't meet the problem's conditions.

In the third test case, the only circle with the persons with the numbers $$$2$$$ and $$$4$$$ looking at each other consists of $$$4$$$ people. Therefore, the person with the number $$$10$$$ doesn't occur in the circle.

Now solve the problem and return the code.
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 1.0 seconds
Memory limit: 256.0 MB

# Problem
Polycarp has found a table having an infinite number of rows and columns. The rows are numbered from $$$1$$$, starting from the topmost one. The columns are numbered from $$$1$$$, starting from the leftmost one.

Initially, the table hasn't been filled and Polycarp wants to fix it. He writes integers from $$$1$$$ and so on to the table as follows.

The figure shows the placement of the numbers from $$$1$$$ to $$$10$$$. The following actions are denoted by the arrows.

The leftmost topmost cell of the table is filled with the number $$$1$$$. Then he writes in the table all positive integers beginning from $$$2$$$ sequentially using the following algorithm.

First, Polycarp selects the leftmost non-filled cell in the first row and fills it. Then, while the left neighbor of the last filled cell is filled, he goes down and fills the next cell. So he goes down until the last filled cell has a non-filled neighbor to the left (look at the vertical arrow going down in the figure above).

After that, he fills the cells from the right to the left until he stops at the first column (look at the horizontal row in the figure above). Then Polycarp selects the leftmost non-filled cell in the first row, goes down, and so on.

A friend of Polycarp has a favorite number $$$k$$$. He wants to know which cell will contain the number. Help him to find the indices of the row and the column, such that the intersection of the row and the column is the cell containing the number $$$k$$$.

## Input Format
The first line contains one integer $$$t$$$ ($$$1 \le t \le 100$$$) — the number of test cases. Then $$$t$$$ test cases follow.

Each test case consists of one line containing one integer $$$k$$$ ($$$1 \le k \le 10^9$$$) which location must be found.

## Output Format
For each test case, output in a separate line two integers $$$r$$$ and $$$c$$$ ($$$r, c \ge 1$$$) separated by spaces — the indices of the row and the column containing the cell filled by the number $$$k$$$, respectively.

## Examples
```input
7
11
14
5
4
1
2
1000000000
```
```output
2 4
4 3
1 3
2 1
1 1
1 2
31623 14130
```

Now solve the problem and return the code.
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 3.0 seconds
Memory limit: 256.0 MB

# Problem
Polycarp has a string $$$s$$$. Polycarp performs the following actions until the string $$$s$$$ is empty ($$$t$$$ is initially an empty string):

- he adds to the right to the string $$$t$$$ the string $$$s$$$, i.e. he does $$$t = t + s$$$, where $$$t + s$$$ is a concatenation of the strings $$$t$$$ and $$$s$$$;
- he selects an arbitrary letter of $$$s$$$ and removes from $$$s$$$ all its occurrences (the selected letter must occur in the string $$$s$$$ at the moment of performing this action).

Polycarp performs this sequence of actions strictly in this order.

Note that after Polycarp finishes the actions, the string $$$s$$$ will be empty and the string $$$t$$$ will be equal to some value (that is undefined and depends on the order of removing).

E.g. consider $$$s$$$="abacaba" so the actions may be performed as follows:

- $$$t$$$="abacaba", the letter 'b' is selected, then $$$s$$$="aacaa";
- $$$t$$$="abacabaaacaa", the letter 'a' is selected, then $$$s$$$="c";
- $$$t$$$="abacabaaacaac", the letter 'c' is selected, then $$$s$$$="" (the empty string).

You need to restore the initial value of the string $$$s$$$ using only the final value of $$$t$$$ and find the order of removing letters from $$$s$$$.

## Input Format
The first line contains one integer $$$T$$$ ($$$1 \le T \le 10^4$$$) — the number of test cases. Then $$$T$$$ test cases follow.

Each test case contains one string $$$t$$$ consisting of lowercase letters of the Latin alphabet. The length of $$$t$$$ doesn't exceed $$$5 \cdot 10^5$$$. The sum of lengths of all strings $$$t$$$ in the test cases doesn't exceed $$$5 \cdot 10^5$$$.

## Output Format
For each test case output in a separate line:

- $$$-1$$$, if the answer doesn't exist;
- two strings separated by spaces. The first one must contain a possible initial value of $$$s$$$. The second one must contain a sequence of letters — it's in what order one needs to remove letters from $$$s$$$ to make the string $$$t$$$. E.g. if the string "bac" is outputted, then, first, all occurrences of the letter 'b' were deleted, then all occurrences of 'a', and then, finally, all occurrences of 'c'. If there are multiple solutions, print any one.

## Examples
```input
7
abacabaaacaac
nowyouknowthat
polycarppoycarppoyarppyarppyrpprppp
isi
everywherevrywhrvryhrvrhrvhv
haaha
qweqeewew
```
```output
abacaba bac
-1
polycarp lcoayrp
is si
everywhere ewyrhv
-1
-1
```

## Note
The first test case is considered in the statement.

Now solve the problem and return the code.
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 1.0 seconds
Memory limit: 256.0 MB

# Problem
It is a simplified version of problem F2. The difference between them is the constraints (F1: $$$k \le 2$$$, F2: $$$k \le 10$$$).

You are given an integer $$$n$$$. Find the minimum integer $$$x$$$ such that $$$x \ge n$$$ and the number $$$x$$$ is $$$k$$$-beautiful.

A number is called $$$k$$$-beautiful if its decimal representation having no leading zeroes contains no more than $$$k$$$ different digits. E.g. if $$$k = 2$$$, the numbers $$$3434443$$$, $$$55550$$$, $$$777$$$ and $$$21$$$ are $$$k$$$-beautiful whereas the numbers $$$120$$$, $$$445435$$$ and $$$998244353$$$ are not.

## Input Format
The first line contains one integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases. Then $$$t$$$ test cases follow.

Each test case consists of one line containing two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 10^9$$$, $$$1 \le k \le 2$$$).

## Output Format
For each test case output on a separate line $$$x$$$ — the minimum $$$k$$$-beautiful integer such that $$$x \ge n$$$.

## Examples
```input
4
1 1
221 2
177890 2
998244353 1
```
```output
1
221
181111
999999999
```

Now solve the problem and return the code.