Eggplant Summoning Magic

The Korean Eggplant Association is studying many ways to supply high-quality eggplants at low prices. Eggplant summoning magic is a recently developed method that sacrifices a small number of eggplants to summon a large number of them.

To use eggplant summoning magic, you must float $N$ eggplants in the air and chant a spell. Represent each eggplant's position as a point in three-dimensional coordinate space, and apply a suitable function $f(P)$ to each point. The function $f(P)$ must satisfy the following conditions.

• The domain and codomain of $f(P)$ are the set of all points $(x, y, z)$ whose coordinates are all rational numbers.
• For any two distinct points $A$, $B$, $f(A)$ and $f(B)$ are different.
• Three distinct points $A$, $B$, $C$ lie on one line if and only if $f(A)$, $f(B)$, $f(C)$ lie on one line.
• For two distinct points $A$, $B$ and two distinct points $C$, $D$, line $\overleftrightarrow{AB}$ is parallel to line $\overleftrightarrow{CD}$ if and only if line $\overleftrightarrow{f(A)f(B)}$ is parallel to line $\overleftrightarrow{f(C)f(D)}$. If $\overleftrightarrow{AB} = \overleftrightarrow{CD}$, then lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ are parallel.

After applying $f(P)$ to all $N$ points, choose $M$ of the $N$ points so that the polygon with those $M$ points as vertices is a regular $M$-gon. Then you can summon $N \times M$ eggplants. ($M \ge 3$) If no suitable $f(P)$ exists, or no suitable set of $M$ points can be chosen, the summoning fails. Given the positions of the $N$ eggplants, find the maximum number of eggplants you can summon by choosing one suitable function $f(P)$ and one set of $M$ points, exactly once.

The first line contains the number of eggplants $N$. ($3 \le N \le 500$)

Each of the next $N$ lines contains the position of one eggplant as a point. The $i$-th line contains three integers $x_i$, $y_i$, $z_i$ separated by spaces. ($0 \le x_i, y_i, z_i \le 10^8$)

All eggplant positions are distinct.

Print on the first line the maximum number of eggplants you can summon by choosing one suitable function $f(P)$ and one set of $M$ points, exactly once.