You are implementing a vision program for a robot. Each time the robot camera takes a
picture, it is stored as a black and white image in the robot's memory. Each image is an
grid of pixels, with rows numbered through and columns numbered
through . There are **exactly two** black pixels in each image, and all other pixels
are white.

The robot can process each image with a program consisting of simple instructions.
You are given the values of and a positive integer . Your goal is to write a
procedure to produce a program for the robot that, for any image, determines whether
the **distance** between the two black pixels is exactly . Here, the distance between a
pixel in row and column and a pixel in row and column is .
In this formula denotes the absolute value of , which equals if and equals
if .

We now describe how the robot works.

The robot's memory is a sufficiently large array of cells, indexed from . Each cell can store either or and its value, once set, will not be changed. The image is stored row by row in cells indexed through . The first row is stored in cells through and the last row is stored in cells through . In particular, if the pixel in row and column is black, the value of cell is , otherwise it is .

A robot's program is a sequence of **instructions**, which are numbered with
consecutive integers starting from . When the program is run, the instructions are
executed one by one. Each instruction reads the values of one or more cells (we call
these values the instruction's **inputs**) and produces a single value equal to or (we
call this value the instruction's **output**). The output of instruction is stored in cell
. The inputs of instruction can only be cells that store either pixels or
outputs of previous instructions, i.e. cells to .

There are four types of instructions:

- : has exactly one input. Its output is if the input is , otherwise its output is .
- : has one or more inputs. Its output is if and only if
**all**of the inputs are . - : has one or more inputs. Its output is if and only if
**at least one**of the inputs is . - : has one or more inputs. Its output is if and only if an
**odd number**of the inputs are .

The output of the last instruction of the program should be if the distance between the two black pixels is exactly , and otherwise.

#### Implementation details

You should implement the following procedure:

```
void construct_network(int H, int W, int K)
```

- : dimensions of each image taken by the robot's camera
- : a positive integer
- This procedure should produce a robot's program. For any image taken by the robot's camera, this program should determine whether the distance between the two black pixels in the image is exactly .

This procedure should call one or more of the following procedures to append instructions to the robot's program (which is initially empty):

```
int add_not(int N)
int add_and(std::vector<int> Ns)
int add_or(std::vector<int> Ns)
int add_xor(std::vector<int> Ns)
```

- Append a or instruction, respectively.
- (for
`add_not`

): the index of the cell from which the appended instruction reads its input - (for
`add_and`

,`add_or`

,`add_xor`

): array containing the indices of the cells from which the appended or instruction reads its inputs - Each procedure returns the index of the cell that stores the output of the instruction. The consecutive calls to these procedures return consecutive integers starting from .

The robot's program can consist of at most instructions. The instructions can
read at most values in total. In other words, the total length of arrays in
all calls to `add_and`

, `add_or`

and `add_xor`

plus the number of calls to `add_not`

cannot
exceed .

After appending the last instruction, procedure `construct_network`

should `return`

. The
robot's program will then be evaluated on some number of images. Your solution
passes a given test case if for each of these images, the output of the last instruction is
if and only if the distance between the two black pixels in the image is equal to .

The grading of your solution may result in one of the following error messages:

`Instruction with no inputs`

: an empty array was given as the input to`add_and`

,`add_or`

, or`add_xor`

.`Invalid index`

: an incorrect (possibly negative) cell index was provided as the input to`add_and`

,`add_or`

,`add_xor`

, or`add_not`

.`Too many instructions`

: your procedure attempted to add more than instructions.`Too many inputs`

: the instructions read more than values in total.

#### Example

Assume . There are only two possible images where the distance between the black pixels is .

- Case : black pixels are and
- Case : black pixels are and

A possible solution is to build a robot's program by making the following calls:

`add_and([0, 5])`

, which adds an instruction that outputs if and only if the first case holds. The output is stored in cell .`add_and([2, 3])`

, which adds an instruction that outputs if and only if the second case holds. The output is stored in cell .`add_or([6, 7])`

, which adds an instruction that outputs if and only if one of the cases above holds.

#### Constraints

#### Subtasks

- ( points)
- ( points)
- ( points)
- ( points)
- ( points)
- ( points) Pixel in row and column is black in each image.
- ( points)
- ( points) No additional constraints.

#### Sample grader

The sample grader reads the input in the following format:

- line :
- line :
- last line:

Each line excepting the first and the last line represents an image with two black pixels. We denote the image described in line by image . One black pixel is in row and column and the other one in row and column .

The sample grader first calls `construct_network(H, W, K)`

. If `construct_network`

violates some constraint described in the problem statement, the sample grader prints
one of the error messages listed at the end of Implementation details section and exits.

Otherwise, the sample grader produces two outputs.

First, the sample grader prints the output of the robot's program in the following format:

- line : output of the last instruction in the robot's program for image ( or ).

Second, the sample grader writes a file `log.txt`

in the current directory in the
following format:

- line :

The sequence on line describes the values stored in the robot's memory cells after the robot's program is run, given image as the input. Specifically, gives the value of cell . Note that the value of (the length of the sequence) is equal to plus the number of instructions in the robot's program.

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