Histogrami

A histogram is a graphical representation of a statistical distribution of data. In other words, it is a function that assigns a positive integer value to each number in the interval $0, 1, 2, \dots , W - 1$. For this task, we describe a histogram with a series of points in the standard coordinate system which, sequentially, determine the top edge of the area that the histogram encloses.

More specifically, we define a histogram of width $W$ with an even number of points in the form:

\[ (x_0, y_1), (x_1, y_1), (x_1, y_2), (x_2, y_2), (x_2, y_3), \dots ,(x_{N/2-1}, y_{N/2}),(x_{N/2}, y_{N/2}). \]

Therefore, starting from the first point, for every pair of adjacent points it must hold that their $y$ coordinates are equal and that their $x$ coordinates are equal. Thus, the histogram begins and ends with a horizontal segment and in between horizontal and vertical segments alternate.

Additionally, the following holds for the shape of a histogram:

  1. $x_0 = 0$ – the histogram begins on the line $x = 0$.

  2. $x_{N/2} = W$ – the histogram ends on the line $x = W$.

  3. $x_ i < x_{i+1}$, for each $i = 1, 2, \dots , N/2 - 1$ - the histogram is defined from left to right and the length of each horizontal segment is at least one.

  4. $y_ i > 0$ – the height of the histogram is always greater than zero.

  5. $y_ i \not= y_{i+1}$ – the length of each vertical segment is at least one.

For a histogram $H$, let $y_ H(x)$ be the height of the histogram in the interval $\left\langle x, x+1 \right\rangle $. For example, the surface area which the histogram encloses can be calculated by formula $\sum ^{W-1}_{x=0} y_ H(x)$. If two histograms, $H$ and $H’$, are given, with equal width $W$ (which can possibly be defined with a different number of points), then we can measure the difference between these two histograms in various ways. This measure of difference between two histograms is also called inaccuracy of histogram $H’$ with regard to histogram $H$. In this task, we examine two different ways of measuring differences between two histograms and we define them in the following way:

\begin{align*} \operatorname {diffcount}(H’, H) & = \sum ^{W-1}_{x=0} \operatorname {diff}(y_ H(x), y_{H'}(x)), \text {where $\operatorname {diff}(y_1, y_2) = 0$ if $y_1 = y_2$ and $1$ otherwise.} \\ \operatorname {abserror}(H’, H) & = \sum ^{W-1}_{x=0} |y_ H(x) - y_{H'}(x)| \end{align*}

In other words, the first way measures the number of unit segments $\left\langle x, x + 1 \right\rangle $ in which the two histograms differ, whereas the second way is the sum of absolute values of differences on those individual unit segments.

Write a program that will, for a given histogram $H$, set of points $S$ and way of measuring inaccuracy, find and output histogram $H’$ which only uses points from the set $S$ in its definition and has the least possible inaccuracy with regard to the given histogram $H$.

\includegraphics[width=0.9\textwidth ]{histogrami.png}
Figure 1: Illustration of the histogram and the set $S$ in both test cases given below and the solutions for the first and the second way of measuring inaccuracy.

The definition of histogram $H’$ must comply with all aforementioned conditions (for example, there shouldn’t be two continuous horizontal segments in the definition of histogram $H’$) and each point in the definition must be one of the points from the set $S$.

Input

The first line of input contains integers $N, M, G$ ($2 \le N, M \le 100\, 000$, $1 \le G \le 2$, N even), respectively: the total number of points in the definition of histogram $H$, number of points in set $S$ and the number which determines what method of measuring the inaccuracy is used: $G=1$ for diffcount and $G=2$ for abserror.

Each of the following $N$ lines contains integers $X$ and $Y$ ($0 \le X \le 10^6$, $1 \le Y \le 10^6$) – coordinates of one point from the definition of histogram $H$. The histogram’s definition will comply to all the conditions from the task statement.

Each of the following $M$ lines contains integers $X$ and $Y$ ($0 \le X \le 10^6$, $1 \le Y \le 10^6$) – coordinates of one point from set $S$. All the points in the set $S$ will be different from one another, but they can match with points from the definition of the histogram $H$.

Output

The first line of output must contain the least possible inaccuracy $D$. The second line of output must contain an even integer $L$ – the total number of points in the definition of the required optimal histogram. In each of the following $L$ lines, output two integers $X$ and $Y$ – coordinates of the point in the histogram’s definition.

The histogram’s definition must comply to all the conditions from the task.

Please note: The solution may not be unique, but the input data will be such that there is always at least one solution.

Sample Input 1 Sample Output 1
10 12 1
0 2
2 2
2 3
4 3
4 1
5 1
5 5
9 5
9 2
10 2
0 4
0 6
3 3
3 6
9 4
9 1
5 3
5 5
10 5
9 2
10 1
8 5
5
6
0 6
3 6
3 3
5 3
5 5
10 5
Sample Input 2 Sample Output 2
10 12 2
0 2
2 2
2 3
4 3
4 1
5 1
5 5
9 5
9 2
10 2
0 4
0 6
3 3
3 6
9 4
9 1
5 3
5 5
10 5
9 2
10 1
8 5
14
4
0 4
9 4
9 1
10 1