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
Alyona has recently bought a miniature fridge that can be represented as a matrix with $$$h$$$ rows and $$$2$$$ columns. Initially there is only one shelf at the bottom of the fridge, but Alyona can install arbitrary number of shelves inside the fridge between any two rows. A shelf is two cells wide, does not occupy any space but separates the inside of the fridge to the lower and upper part.

An example of a fridge with $$$h = 7$$$ and two shelves. The shelves are shown in black. The picture corresponds to the first example.

Alyona has $$$n$$$ bottles of milk that she wants to put in the fridge. The $$$i$$$-th bottle is $$$a_i$$$ cells tall and $$$1$$$ cell wide. She can put a bottle on some shelf if the corresponding space above the shelf is at least as tall as the bottle. She can not put a bottle on top of another bottle (if there is no shelf between them). Two bottles can not share a cell.

Alyona is interested in the largest integer $$$k$$$ such that she can put bottles $$$1$$$, $$$2$$$, ..., $$$k$$$ in the fridge at the same time. Find this largest $$$k$$$.

## Input Format
The first line contains two integers $$$n$$$ and $$$h$$$ ($$$1 \le n \le 10^3$$$, $$$1 \le h \le 10^9$$$) — the number of bottles and the height of the fridge.

The second line contains $$$n$$$ integers $$$a_1$$$, $$$a_2$$$, ..., $$$a_n$$$ ($$$1 \le a_i \le h$$$) — the heights of the bottles.

## Output Format
Print the single integer $$$k$$$ — the maximum integer such that Alyona can put the bottles $$$1$$$, $$$2$$$, ..., $$$k$$$ in the fridge at the same time. If Alyona can put all bottles in the fridge, print $$$n$$$. It is easy to see that Alyona can always put at least one bottle in the fridge.

## Examples
```input
5 7
2 3 5 4 1
```
```output
3
```
-----
```input
10 10
9 1 1 1 1 1 1 1 1 1
```
```output
4
```
-----
```input
5 10
3 1 4 2 4
```
```output
5
```

## Note
One of optimal locations in the first example is shown on the picture in the statement.

One of optimal locations in the second example is shown on the picture below.

One of optimal locations in the third example is shown on the picture below.

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
Pavel has several sticks with lengths equal to powers of two.

He has $$$a_0$$$ sticks of length $$$2^0 = 1$$$, $$$a_1$$$ sticks of length $$$2^1 = 2$$$, ..., $$$a_{n-1}$$$ sticks of length $$$2^{n-1}$$$.

Pavel wants to make the maximum possible number of triangles using these sticks. The triangles should have strictly positive area, each stick can be used in at most one triangle.

It is forbidden to break sticks, and each triangle should consist of exactly three sticks.

Find the maximum possible number of triangles.

## Input Format
The first line contains a single integer $$$n$$$ ($$$1 \leq n \leq 300\,000$$$) — the number of different lengths of sticks.

The second line contains $$$n$$$ integers $$$a_0$$$, $$$a_1$$$, ..., $$$a_{n-1}$$$ ($$$1 \leq a_i \leq 10^9$$$), where $$$a_i$$$ is the number of sticks with the length equal to $$$2^i$$$.

## Output Format
Print a single integer — the maximum possible number of non-degenerate triangles that Pavel can make.

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

## Note
In the first example, Pavel can, for example, make this set of triangles (the lengths of the sides of the triangles are listed): $$$(2^0, 2^4, 2^4)$$$, $$$(2^1, 2^3, 2^3)$$$, $$$(2^1, 2^2, 2^2)$$$.

In the second example, Pavel cannot make a single triangle.

In the third example, Pavel can, for example, create this set of triangles (the lengths of the sides of the triangles are listed): $$$(2^0, 2^0, 2^0)$$$, $$$(2^1, 2^1, 2^1)$$$, $$$(2^2, 2^2, 2^2)$$$.

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: 256.0 MB

# Problem
You have received your birthday gifts — $$$n$$$ triples of integers! The $$$i$$$-th of them is $$$\lbrace a_{i}, b_{i}, c_{i} \rbrace$$$. All numbers are greater than or equal to $$$0$$$, and strictly smaller than $$$2^{k}$$$, where $$$k$$$ is a fixed integer.

One day, you felt tired playing with triples. So you came up with three new integers $$$x$$$, $$$y$$$, $$$z$$$, and then formed $$$n$$$ arrays. The $$$i$$$-th array consists of $$$a_i$$$ repeated $$$x$$$ times, $$$b_i$$$ repeated $$$y$$$ times and $$$c_i$$$ repeated $$$z$$$ times. Thus, each array has length $$$(x + y + z)$$$.

You want to choose exactly one integer from each array such that the XOR (bitwise exclusive or) of them is equal to $$$t$$$. Output the number of ways to choose the numbers for each $$$t$$$ between $$$0$$$ and $$$2^{k} - 1$$$, inclusive, modulo $$$998244353$$$.

## Input Format
The first line contains two integers $$$n$$$ and $$$k$$$ ($$$1 \leq n \leq 10^{5}$$$, $$$1 \leq k \leq 17$$$) — the number of arrays and the binary length of all numbers.

The second line contains three integers $$$x$$$, $$$y$$$, $$$z$$$ ($$$0 \leq x,y,z \leq 10^{9}$$$) — the integers you chose.

Then $$$n$$$ lines follow. The $$$i$$$-th of them contains three integers $$$a_{i}$$$, $$$b_{i}$$$ and $$$c_{i}$$$ ($$$0 \leq a_{i} , b_{i} , c_{i} \leq 2^{k} - 1$$$) — the integers forming the $$$i$$$-th array.

## Output Format
Print a single line containing $$$2^{k}$$$ integers. The $$$i$$$-th of them should be the number of ways to choose exactly one integer from each array so that their XOR is equal to $$$t = i-1$$$ modulo $$$998244353$$$.

## Examples
```input
1 1
1 2 3
1 0 1
```
```output
2 4
```
-----
```input
2 2
1 2 1
0 1 2
1 2 3
```
```output
4 2 4 6
```
-----
```input
4 3
1 2 3
1 3 7
0 2 5
1 0 6
3 3 2
```
```output
198 198 126 126 126 126 198 198
```

## Note
In the first example, the array we formed is $$$(1, 0, 0, 1, 1, 1)$$$, we have two choices to get $$$0$$$ as the XOR and four choices to get $$$1$$$.

In the second example, two arrays are $$$(0, 1, 1, 2)$$$ and $$$(1, 2, 2, 3)$$$. There are sixteen $$$(4 \cdot 4)$$$ choices in total, $$$4$$$ of them ($$$1 \oplus 1$$$ and $$$2 \oplus 2$$$, two options for each) give $$$0$$$, $$$2$$$ of them ($$$0 \oplus 1$$$ and $$$2 \oplus 3$$$) give $$$1$$$, $$$4$$$ of them ($$$0 \oplus 2$$$ and $$$1 \oplus 3$$$, two options for each) give $$$2$$$, and finally $$$6$$$ of them ($$$0 \oplus 3$$$, $$$2 \oplus 1$$$ and four options for $$$1 \oplus 2$$$) give $$$3$$$.

Now solve the problem and return the code.