102206: [AtCoder]ABC220 G - Isosceles Trapezium

Problem Statement

In the $xy$-plane, we have $N$ points, each assigned a weight.
The $i$-th point has the coordinates $(X_i,Y_i)$ and the weight $C_i$.

We will choose four of the $N$ points to form an isosceles trapezoid whose vertices are the chosen points.
What is the maximum possible total weight of the points chosen here?

If it is impossible to form an isosceles trapezoid, print -1.

We remind you that an isosceles trapezoid is a quadrilateral that satisfies all of the following conditions.

• It is a trapezoid.
• For one of the two parallel sides, the two angles at its ends are equal.

Constraints

• $4 \leq N \leq 1000$
• $-10^9 \leq X_i,Y_i \leq 10^9$
• $1 \leq C_i \leq 10^9$
• $(X_i,Y_i) \neq (X_j,Y_j)$ if $i \neq j$.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$
$X_1$ $Y_1$ $C_1$
$X_2$ $Y_2$ $C_2$
$\vdots$
$X_N$ $Y_N$ $C_N$

Output

Print the answer.

Sample Input 1

5
0 3 10
3 3 10
-1 0 10
2 0 10000
4 0 10

Sample Output 1

40

We can choose Points $1, 2, 3, 5$ to form an isosceles trapezoid, with the points having a total weight of $40$.
Any other way to choose points would not form an isosceles trapezoid.

Sample Input 2

6
0 1 1
1 4 20
2 7 300
5 6 4000
4 3 50000
3 0 600000

Sample Output 2

650021

Note that a square and a rectangle are also isosceles trapezoids.

Sample Input 3

7
-3 0 1
-2 0 1
-1 0 1
0 0 1
1 0 1
2 0 1
3 0 1

Sample Output 3

-1

We cannot form an isosceles trapezoid.