There are $N$ houses in a city. Each house is numbered from $1$ to $N$. This city is a $2D$ cartesian plane. The location of each house is denoted by the coordinate $(x,y)$. The government wants to establish electric poles to which these houses can be connected. Each house must be connected to exactly one electric pole. The cost of placing a single pole is denoted by the integer $Z$. You can connect each pole to at most $K$ distinct houses. To minimize the overutilization of resources, the number of poles is restricted to a specific number that is denoted by $L$.
The overall cost of this operation is given by $Z \times P + D$, where $D$ is the sum of the Euclidean distance of all the houses to the respective poles they are connected to and $P$ is the number of poles used. Your task is to minimize the cost.

Input format

• First line: Four space-separated integers $N$, $Z$, $K$, and $L$ 
• Next $N$ lines: Location of houses denoted by $(x, y)$

Output format

• First line: Integer $P$ denoting the number of poles used
• Next $P$ lines:
  • Location of each pole denoted as a 2D-integer coordinate,
  • Number of houses that are connected to the respective electric pole
  • Numbers of each house that are connected 

Constraints
$1 \le N \le 10^5$
$-10^7 \le x[i], y[i] \le 10^7$
$1 \le Z \le 10^8$
$1 \le K \le N$
$(N + K - 1) / K \le L \le N$

Scoring
The score is calculated by the formula $Z \times P + D$ and you are required to minimize this score. If any of these constraints is violated, your answer is wrong. The coordinate of the poles should also lie in the range $-10^7$ to $10^7$ .

Test Generation
The test files contain three types of test cases - Points randomly generated across the whole plane (not necessarily uniform), All points in a straight line, All points in the border of a rectangle and the ratio is 60%, 20%, 20% of all the test cases (approximately).