Vision Program

Each time the robot camera takes a picture, the image is stored as a black and white image in the robot's memory. Each image is an $H \times W$ grid, with rows numbered $0$ through $H-1$ and columns numbered $0$ through $W-1$. There are exactly two black pixels in each image, and all other pixels are white.

The robot can process each image with a program made of simple instructions. You are given $H$, $W$, and a positive integer $K$. Write a procedure that produces a program which, for any image, determines whether the distance between the two black pixels is exactly $K$. The distance between the pixel in row $r_1$, column $c_1$ and the pixel in row $r_2$, column $c_2$ is $|r_1-r_2|+|c_1-c_2|$, where $|x|$ equals $x$ if $x \geq 0$ and $-x$ if $x < 0$.

The robot's memory is a sufficiently large array of cells indexed from $0$. Each cell stores either $0$ or $1$, and once set, its value does not change. The image is stored row by row in cells $0$ through $H \cdot W - 1$. The first row is stored in cells $0$ through $W-1$, and the last row is stored in cells $(H-1) \cdot W$ through $H \cdot W - 1$. If the pixel in row $i$ and column $j$ is black, cell $i \cdot W + j$ holds $1$; otherwise it holds $0$.

A robot program is a sequence of instructions numbered consecutively from $0$. The instructions run one by one. Each instruction reads one or more cells (its inputs) and produces a single output, $0$ or $1$. The output of instruction $i$ is stored in cell $H \cdot W + i$. The inputs of instruction $i$ can only be cells $0$ through $H \cdot W + i - 1$, which hold pixels or outputs of earlier instructions.

There are four instruction types:

• NOT: exactly one input. The output is $1$ if the input is $0$, otherwise $0$.
• AND: one or more inputs. The output is $1$ if and only if all inputs are $1$.
• OR: one or more inputs. The output is $1$ if and only if at least one input is $1$.
• XOR: one or more inputs. The output is $1$ if and only if an odd number of inputs are $1$.

The output of the last instruction must be $1$ if the distance between the two black pixels is exactly $K$, and $0$ otherwise.

• $1 \leq H \leq 200$
• $1 \leq W \leq 200$
• $2 \leq H \cdot W$
• $1 \leq K \leq H+W-2$