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
Cengiz recently learned Fibonacci numbers and now he is studying different algorithms to find them. After getting bored of reading them, he came with his own new type of numbers that he named XORinacci numbers. He defined them as follows:

- $$$f(0) = a$$$;
- $$$f(1) = b$$$;
- $$$f(n) = f(n-1) \oplus f(n-2)$$$ when $$$n > 1$$$, where $$$\oplus$$$ denotes the bitwise XOR operation.

You are given three integers $$$a$$$, $$$b$$$, and $$$n$$$, calculate $$$f(n)$$$.

You have to answer for $$$T$$$ independent test cases.

## Input Format
The input contains one or more independent test cases.

The first line of input contains a single integer $$$T$$$ ($$$1 \le T \le 10^3$$$), the number of test cases.

Each of the $$$T$$$ following lines contains three space-separated integers $$$a$$$, $$$b$$$, and $$$n$$$ ($$$0 \le a, b, n \le 10^9$$$) respectively.

## Output Format
For each test case, output $$$f(n)$$$.

## Examples
```input
3
3 4 2
4 5 0
325 265 1231232
```
```output
7
4
76
```

## Note
In the first example, $$$f(2) = f(0) \oplus f(1) = 3 \oplus 4 = 7$$$.

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: 2.0 seconds
Memory limit: 256.0 MB

# Problem
You are given an array $$$a_{1}, a_{2}, \ldots, a_{n}$$$. You can remove at most one subsegment from it. The remaining elements should be pairwise distinct.

In other words, at most one time you can choose two integers $$$l$$$ and $$$r$$$ ($$$1 \leq l \leq r \leq n$$$) and delete integers $$$a_l, a_{l+1}, \ldots, a_r$$$ from the array. Remaining elements should be pairwise distinct.

Find the minimum size of the subsegment you need to remove to make all remaining elements distinct.

## Input Format
The first line of the input contains a single integer $$$n$$$ ($$$1 \le n \le 2000$$$) — the number of elements in the given array.

The next line contains $$$n$$$ spaced integers $$$a_{1}, a_{2}, \ldots, a_{n}$$$ ($$$1 \le a_{i} \le 10^{9}$$$) — the elements of the array.

## Output Format
Print a single integer — the minimum size of the subsegment you need to remove to make all elements of the array pairwise distinct. If no subsegment needs to be removed, print $$$0$$$.

## Examples
```input
3
1 2 3
```
```output
0
```
-----
```input
4
1 1 2 2
```
```output
2
```
-----
```input
5
1 4 1 4 9
```
```output
2
```

## Note
In the first example all the elements are already distinct, therefore no subsegment needs to be removed.

In the second example you can remove the subsegment from index $$$2$$$ to $$$3$$$.

In the third example you can remove the subsegments from index $$$1$$$ to $$$2$$$, or from index $$$2$$$ to $$$3$$$, or from index $$$3$$$ to $$$4$$$.

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
Let us define a magic grid to be a square matrix of integers of size $$$n \times n$$$, satisfying the following conditions.

- All integers from $$$0$$$ to $$$(n^2 - 1)$$$ inclusive appear in the matrix exactly once.
- Bitwise XOR of all elements in a row or a column must be the same for each row and column.

You are given an integer $$$n$$$ which is a multiple of $$$4$$$. Construct a magic grid of size $$$n \times n$$$.

## Input Format
The only line of input contains an integer $$$n$$$ ($$$4 \leq n \leq 1000$$$). It is guaranteed that $$$n$$$ is a multiple of $$$4$$$.

## Output Format
Print a magic grid, i.e. $$$n$$$ lines, the $$$i$$$-th of which contains $$$n$$$ space-separated integers, representing the $$$i$$$-th row of the grid.

If there are multiple answers, print any. We can show that an answer always exists.

## Examples
```input
4
```
```output
8 9 1 13
3 12 7 5
0 2 4 11
6 10 15 14
```
-----
```input
8
```
```output
19 55 11 39 32 36 4 52
51 7 35 31 12 48 28 20
43 23 59 15 0 8 16 44
3 47 27 63 24 40 60 56
34 38 6 54 17 53 9 37
14 50 30 22 49 5 33 29
2 10 18 46 41 21 57 13
26 42 62 58 1 45 25 61
```

## Note
In the first example, XOR of each row and each column is $$$13$$$.

In the second example, XOR of each row and each column is $$$60$$$.

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: 2.0 seconds
Memory limit: 256.0 MB

# Problem
An array of integers $$$p_{1},p_{2}, \ldots,p_{n}$$$ is called a permutation if it contains each number from $$$1$$$ to $$$n$$$ exactly once. For example, the following arrays are permutations: $$$[3,1,2], [1], [1,2,3,4,5]$$$ and $$$[4,3,1,2]$$$. The following arrays are not permutations: $$$[2], [1,1], [2,3,4]$$$.

There is a hidden permutation of length $$$n$$$.

For each index $$$i$$$, you are given $$$s_{i}$$$, which equals to the sum of all $$$p_{j}$$$ such that $$$j < i$$$ and $$$p_{j} < p_{i}$$$. In other words, $$$s_i$$$ is the sum of elements before the $$$i$$$-th element that are smaller than the $$$i$$$-th element.

Your task is to restore the permutation.

## Input Format
The first line contains a single integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^{5}$$$) — the size of the permutation.

The second line contains $$$n$$$ integers $$$s_{1}, s_{2}, \ldots, s_{n}$$$ ($$$0 \le s_{i} \le \frac{n(n-1)}{2}$$$).

It is guaranteed that the array $$$s$$$ corresponds to a valid permutation of length $$$n$$$.

## Output Format
Print $$$n$$$ integers $$$p_{1}, p_{2}, \ldots, p_{n}$$$ — the elements of the restored permutation. We can show that the answer is always unique.

## Examples
```input
3
0 0 0
```
```output
3 2 1
```
-----
```input
2
0 1
```
```output
1 2
```
-----
```input
5
0 1 1 1 10
```
```output
1 4 3 2 5
```

## Note
In the first example for each $$$i$$$ there is no index $$$j$$$ satisfying both conditions, hence $$$s_i$$$ are always $$$0$$$.

In the second example for $$$i = 2$$$ it happens that $$$j = 1$$$ satisfies the conditions, so $$$s_2 = p_1$$$.

In the third example for $$$i = 2, 3, 4$$$ only $$$j = 1$$$ satisfies the conditions, so $$$s_2 = s_3 = s_4 = 1$$$. For $$$i = 5$$$ all $$$j = 1, 2, 3, 4$$$ are possible, so $$$s_5 = p_1 + p_2 + p_3 + p_4 = 10$$$.

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: 4.0 seconds
Memory limit: 256.0 MB

# Problem
You are given $$$n$$$ arrays that can have different sizes. You also have a table with $$$w$$$ columns and $$$n$$$ rows. The $$$i$$$-th array is placed horizontally in the $$$i$$$-th row. You can slide each array within its row as long as it occupies several consecutive cells and lies completely inside the table.

You need to find the maximum sum of the integers in the $$$j$$$-th column for each $$$j$$$ from $$$1$$$ to $$$w$$$ independently.

Optimal placements for columns $$$1$$$, $$$2$$$ and $$$3$$$ are shown on the pictures from left to right.

Note that you can exclude any array out of a column provided it remains in the window. In this case its value is considered to be zero.

## Input Format
The first line contains two integers $$$n$$$ ($$$1 \le n \le 10^{6}$$$) and $$$w$$$ ($$$1 \le w \le 10^{6}$$$) — the number of arrays and the width of the table.

Each of the next $$$n$$$ lines consists of an integer $$$l_{i}$$$ ($$$1 \le l_{i} \le w$$$), the length of the $$$i$$$-th array, followed by $$$l_{i}$$$ integers $$$a_{i1}, a_{i2}, \ldots, a_{il_i}$$$ ($$$-10^{9} \le a_{ij} \le 10^{9}$$$) — the elements of the array.

The total length of the arrays does no exceed $$$10^{6}$$$.

## Output Format
Print $$$w$$$ integers, the $$$i$$$-th of them should be the maximum sum for column $$$i$$$.

## Examples
```input
3 3
3 2 4 8
2 2 5
2 6 3
```
```output
10 15 16
```
-----
```input
2 2
2 7 8
1 -8
```
```output
7 8
```

## Note
Illustration for the first example is 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
You are given two integers $$$n$$$ and $$$k$$$.

You need to construct $$$k$$$ regular polygons having same circumcircle, with distinct number of sides $$$l$$$ between $$$3$$$ and $$$n$$$.

Illustration for the first example.

You can rotate them to minimize the total number of distinct points on the circle. Find the minimum number of such points.

## Input Format
The only line of input contains two integers $$$n$$$ and $$$k$$$ ($$$3 \le n \le 10^{6}$$$, $$$1 \le k \le n-2$$$), the maximum number of sides of a polygon and the number of polygons to construct, respectively.

## Output Format
Print a single integer — the minimum number of points required for $$$k$$$ polygons.

## Examples
```input
6 2
```
```output
6
```
-----
```input
200 50
```
```output
708
```

## Note
In the first example, we have $$$n = 6$$$ and $$$k = 2$$$. So, we have $$$4$$$ polygons with number of sides $$$3$$$, $$$4$$$, $$$5$$$ and $$$6$$$ to choose from and if we choose the triangle and the hexagon, then we can arrange them as shown in the picture in the statement.

Hence, the minimum number of points required on the circle is $$$6$$$, which is also the minimum overall possible sets.

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: 5.0 seconds
Memory limit: 256.0 MB

# Problem
You are given a tree of $$$n$$$ nodes. The tree is rooted at node $$$1$$$, which is not considered as a leaf regardless of its degree.

Each leaf of the tree has one of the two colors: red or blue. Leaf node $$$v$$$ initially has color $$$s_{v}$$$.

The color of each of the internal nodes (including the root) is determined as follows.

- Let $$$b$$$ be the number of blue immediate children, and $$$r$$$ be the number of red immediate children of a given vertex.
- Then the color of this vertex is blue if and only if $$$b - r \ge k$$$, otherwise red.

Integer $$$k$$$ is a parameter that is same for all the nodes.

You need to handle the following types of queries:

- 1 v: print the color of node $$$v$$$;
- 2 v c: change the color of leaf $$$v$$$ to $$$c$$$ ($$$c = 0$$$ means red, $$$c = 1$$$ means blue);
- 3 h: update the current value of $$$k$$$ to $$$h$$$.

## Input Format
The first line of the input consists of two integers $$$n$$$ and $$$k$$$ ($$$2 \le n \le 10^{5}$$$, $$$-n \le k \le n$$$) — the number of nodes and the initial parameter $$$k$$$.

Each of the next $$$n - 1$$$ lines contains two integers $$$u$$$ and $$$v$$$ ($$$1 \le u,v \le n$$$), denoting that there is an edge between vertices $$$u$$$ and $$$v$$$.

The next line consists of $$$n$$$ space separated integers — the initial array $$$s$$$ ($$$-1 \le s_i \le 1$$$). $$$s_{i} = 0$$$ means that the color of node $$$i$$$ is red. $$$s_{i} = 1$$$ means that the color of node $$$i$$$ is blue. $$$s_{i} = -1$$$ means that the node $$$i$$$ is not a leaf.

The next line contains an integer $$$q$$$ ($$$1 \le q \le 10^5$$$), the number of queries.

$$$q$$$ lines follow, each containing a query in one of the following queries:

- 1 v ($$$1 \le v \le n$$$): print the color of node $$$v$$$;
- 2 v c ($$$1 \le v \le n$$$, $$$c = 0$$$ or $$$c = 1$$$): change the color of leaf $$$v$$$ to $$$c$$$ ($$$c = 0$$$ means red, $$$c = 1$$$ means blue). It is guaranteed that $$$v$$$ is a leaf;
- 3 h ($$$-n \le h \le n$$$): update the current value of $$$k$$$ to $$$h$$$.

## Output Format
For each query of the first type, print $$$0$$$ if the color of vertex $$$v$$$ is red, and $$$1$$$ otherwise.

## Examples
```input
5 2
1 2
1 3
2 4
2 5
-1 -1 0 1 0
9
1 1
1 2
3 -2
1 1
1 2
3 1
2 5 1
1 1
1 2
```
```output
0
0
1
1
0
1
```

## Note
Figures:(i) The initial tree (ii) The tree after the 3rd query (iii) The tree after the 7th query

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: 1.5 seconds
Memory limit: 512.0 MB

# Problem
Andrew was very excited to participate in Olympiad of Metropolises. Days flew by quickly, and Andrew is already at the airport, ready to go home. He has $$$n$$$ rubles left, and would like to exchange them to euro and dollar bills. Andrew can mix dollar bills and euro bills in whatever way he wants. The price of one dollar is $$$d$$$ rubles, and one euro costs $$$e$$$ rubles.

Recall that there exist the following dollar bills: $$$1$$$, $$$2$$$, $$$5$$$, $$$10$$$, $$$20$$$, $$$50$$$, $$$100$$$, and the following euro bills — $$$5$$$, $$$10$$$, $$$20$$$, $$$50$$$, $$$100$$$, $$$200$$$ (note that, in this problem we do not consider the $$$500$$$ euro bill, it is hard to find such bills in the currency exchange points). Andrew can buy any combination of bills, and his goal is to minimize the total number of rubles he will have after the exchange.

Help him — write a program that given integers $$$n$$$, $$$e$$$ and $$$d$$$, finds the minimum number of rubles Andrew can get after buying dollar and euro bills.

## Input Format
The first line of the input contains one integer $$$n$$$ ($$$1 \leq n \leq 10^8$$$) — the initial sum in rubles Andrew has.

The second line of the input contains one integer $$$d$$$ ($$$30 \leq d \leq 100$$$) — the price of one dollar in rubles.

The third line of the input contains integer $$$e$$$ ($$$30 \leq e \leq 100$$$) — the price of one euro in rubles.

## Output Format
Output one integer — the minimum number of rubles Andrew can have after buying dollar and euro bills optimally.

## Examples
```input
100
60
70
```
```output
40
```
-----
```input
410
55
70
```
```output
5
```
-----
```input
600
60
70
```
```output
0
```

## Note
In the first example, we can buy just $$$1$$$ dollar because there is no $$$1$$$ euro bill.

In the second example, optimal exchange is to buy $$$5$$$ euro and $$$1$$$ dollar.

In the third example, optimal exchange is to buy $$$10$$$ dollars in one bill.

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: 512.0 MB

# Problem
There are $$$b$$$ boys and $$$g$$$ girls participating in Olympiad of Metropolises. There will be a board games tournament in the evening and $$$n$$$ participants have accepted the invitation. The organizers do not know how many boys and girls are among them.

Organizers are preparing red badges for girls and blue ones for boys.

Vasya prepared $$$n+1$$$ decks of badges. The $$$i$$$-th (where $$$i$$$ is from $$$0$$$ to $$$n$$$, inclusive) deck contains $$$i$$$ blue badges and $$$n-i$$$ red ones. The total number of badges in any deck is exactly $$$n$$$.

Determine the minimum number of decks among these $$$n+1$$$ that Vasya should take, so that there will be a suitable deck no matter how many girls and boys there will be among the participants of the tournament.

## Input Format
The first line contains an integer $$$b$$$ ($$$1 \le b \le 300$$$), the number of boys.

The second line contains an integer $$$g$$$ ($$$1 \le g \le 300$$$), the number of girls.

The third line contains an integer $$$n$$$ ($$$1 \le n \le b + g$$$), the number of the board games tournament participants.

## Output Format
Output the only integer, the minimum number of badge decks that Vasya could take.

## Examples
```input
5
6
3
```
```output
4
```
-----
```input
5
3
5
```
```output
4
```

## Note
In the first example, each of 4 decks should be taken: (0 blue, 3 red), (1 blue, 2 red), (2 blue, 1 red), (3 blue, 0 red).

In the second example, 4 decks should be taken: (2 blue, 3 red), (3 blue, 2 red), (4 blue, 1 red), (5 blue, 0 red). Piles (0 blue, 5 red) and (1 blue, 4 red) can not be used.

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: 512.0 MB

# Problem
Petya's friends made him a birthday present — a bracket sequence. Petya was quite disappointed with his gift, because he dreamed of correct bracket sequence, yet he told his friends nothing about his dreams and decided to fix present himself.

To make everything right, Petya is going to move at most one bracket from its original place in the sequence to any other position. Reversing the bracket (e.g. turning "(" into ")" or vice versa) isn't allowed.

We remind that bracket sequence $$$s$$$ is called correct if:

- $$$s$$$ is empty;
- $$$s$$$ is equal to "($$$t$$$)", where $$$t$$$ is correct bracket sequence;
- $$$s$$$ is equal to $$$t_1 t_2$$$, i.e. concatenation of $$$t_1$$$ and $$$t_2$$$, where $$$t_1$$$ and $$$t_2$$$ are correct bracket sequences.

For example, "(()())", "()" are correct, while ")(" and "())" are not. Help Petya to fix his birthday present and understand whether he can move one bracket so that the sequence becomes correct.

## Input Format
First of line of input contains a single number $$$n$$$ ($$$1 \leq n \leq 200\,000$$$) — length of the sequence which Petya received for his birthday.

Second line of the input contains bracket sequence of length $$$n$$$, containing symbols "(" and ")".

## Output Format
Print "Yes" if Petya can make his sequence correct moving at most one bracket. Otherwise print "No".

## Examples
```input
2
)(
```
```output
Yes
```
-----
```input
3
(()
```
```output
No
```
-----
```input
2
()
```
```output
Yes
```
-----
```input
10
)))))(((((
```
```output
No
```

## Note
In the first example, Petya can move first bracket to the end, thus turning the sequence into "()", which is correct bracket sequence.

In the second example, there is no way to move at most one bracket so that the sequence becomes correct.

In the third example, the sequence is already correct and there's no need to move brackets.

Now solve the problem and return the code.