Cruise

A group of computer science students gathers in Athens for the summer holidays. While working out how to see the Aegean sea by sailboat, they invented the following game.

• Every island has one point value assigned to it.
• The cruise starts at the port of Piraeus and returns to it. Piraeus sits at $(0, 0)$, and every island lies strictly east of Piraeus.
• A course leg is a straight segment from one island to another island or to Piraeus. Apart from Piraeus, the cruise may pass through any point of the Aegean sea at most once.
• The team collects the points of every island the course passes through and of every island inside the area the course encloses. An island lying on the course counts as collected.
• The goal is to maximize the ratio $R$ of collected points over the total length of the cruise.

Given the coordinates and the point value of each island, find the ratio $R$ of the optimal cruise.

Suppose there are six islands, A through F, placed as in the figure below. Piraeus is marked P in the figures. For example, island B is at $(4, 2)$ and its point value is $6$.

The next three figures show three courses that start at Piraeus and return there. Below each figure is the ratio $R$ of that course. Of the three, the leftmost course has the largest $R$. The rightmost course also collects the points of island E, which lies inside the enclosed area.

Route P, A, B, C, P	Route P, C, E, F, P	Route P, C, F, P
$R = \frac{5+6+2}{2+\sqrt{8}+2+\sqrt{32}} \simeq 1.041226$	$R = \frac{2+5+6}{\sqrt{32}+\sqrt{5}+\sqrt{2}+\sqrt{17}} \simeq 0.967965$	$R = \frac{2+5+6}{\sqrt{32}+3+\sqrt{17}} \simeq 1.017218$

The courses P, B, E, C, F, E, P and P, B, F, C, P break the rules. The first passes through island E twice, and the second passes twice through the intersection of segment BF and segment CP.

The first line contains the number of islands $N$. Each of the next $N$ lines contains three integers $X_i$, $Y_i$, $P_i$: the coordinates $(X_i, Y_i)$ of the $i$-th island and the point value $P_i$ assigned to it.

Print the ratio $R$ of the optimal cruise on one line, rounded to six decimal places.

• $2 \le N \le 100$
• $0 < X_i \le 10000$
• $-10000 \le Y_i \le 10000$
• $0 \le P_i \le 10000$
• No two islands share the same coordinates.
• At least two islands lie in different directions as seen from Piraeus. No input places every island on a single ray out of Piraeus, so at least one legal cruise always exists.