CF1359F.RC Kaboom Show

You know, it's hard to conduct a show with lots of participants and spectators at the same place nowadays. Still, you are not giving up on your dream to make a car crash showcase! You decided to replace the real cars with remote controlled ones, call the event "Remote Control Kaboom Show" and stream everything online.

For the preparation you arranged an arena — an infinite 2D-field. You also bought $n$ remote controlled cars and set them up on the arena. Unfortunately, the cars you bought can only go forward without turning left, right or around. So you additionally put the cars in the direction you want them to go.

To be formal, for each car $i$ ($1 \le i \le n$) you chose its initial position ($x_i, y_i$) and a direction vector ($dx_i, dy_i$). Moreover, each car has a constant speed $s_i$ units per second. So after car $i$ is launched, it stars moving from ($x_i, y_i$) in the direction ($dx_i, dy_i$) with constant speed $s_i$.

The goal of the show is to create a car collision as fast as possible! You noted that launching every car at the beginning of the show often fails to produce any collisions at all. Thus, you plan to launch the $i$-th car at some moment $t_i$. You haven't chosen $t_i$, that's yet to be decided. Note that it's not necessary for $t_i$ to be integer and $t_i$ is allowed to be equal to $t_j$ for any $i, j$.

The show starts at time $0$. The show ends when two cars $i$ and $j$ ($i \ne j$) collide (i. e. come to the same coordinate at the same time). The duration of the show is the time between the start and the end.

What's the fastest crash you can arrange by choosing all $t_i$? If it's possible to arrange a crash then print the shortest possible duration of the show. Otherwise, report that it's impossible.

你知道，如今要在同一地点举办一场拥有大量参与者和观众的演出是非常困难的。尽管如此，你仍未放弃实现“汽车碰撞秀”的梦想！你决定用遥控车替代真实汽车，将这场活动命名为“遥控爆炸秀”，并全程在线直播。

为筹备演出，你布置了一个场地——一个无限大的二维平面。你还购买了 $n$ 辆遥控车，并将它们部署在该场地上。不幸的是，你所购得的遥控车只能沿直线向前行驶，无法向左、向右或掉头转向。因此，你额外为每辆车设定了其初始朝向。

形式化地讲，对每辆编号为 $i$（$1 \le i \le n$）的车，你选定其初始位置 $(x_i, y_i)$ 和方向向量 $(dx_i, dy_i)$。此外，每辆车具有恒定速度 $s_i$（单位：每秒长度单位）。因此，当第 $i$ 辆车启动后，它将从位置 $(x_i, y_i)$ 出发，沿方向 $(dx_i, dy_i)$ 以恒定速度 $s_i$ 运动。

本演出的目标是尽快制造一次汽车碰撞！你注意到：若所有车辆都在演出开始时刻同时启动，往往根本不会发生任何碰撞。因此，你计划在某个特定时刻 $t_i$ 启动第 $i$ 辆车。目前 $t_i$ 尚未确定，有待你选择。注意：$t_i$ 不必为整数，且对任意 $i,j$，允许 $t_i = t_j$。

演出于时刻 $0$ 开始。当两辆编号不同（即 $i \ne j$）的车 $i$ 和 $j$ 发生碰撞（即在同一时刻到达同一坐标点）时，演出结束。演出持续时间即为从开始到结束的时间间隔。

通过合理选择所有 $t_i$，你能实现的最快碰撞是什么？若可安排出碰撞，请输出最短可能的演出持续时间；否则，请报告这是不可能的。

The first line contains a single integer $n$ ($1 \le n \le 25000$) — the number of cars.

Each of the next $n$ lines contains five integers $x_i$, $y_i$, $dx_i$, $dy_i$, $s_i$ ($-10^3 \le x_i, y_i \le 10^3$; $1 \le |dx_i| \le 10^3$; $1 \le |dy_i| \le 10^3$; $1 \le s_i \le 10^3$) — the initial position of the $i$-th car, its direction vector and its speed, respectively.

It's guaranteed that all cars start at distinct positions (i. e. $(x_i, y_i) \neq (x_j, y_j)$ for $i \neq j$).

第一行包含一个整数 $n$（$1 \le n \le 25000$）——汽车的数量。

接下来的 $n$ 行中，每行包含五个整数 $x_i$、$y_i$、$dx_i$、$dy_i$、$s_i$（$-10^3 \le x_i, y_i \le 10^3$；$1 \le |dx_i| \le 10^3$；$1 \le |dy_i| \le 10^3$；$1 \le s_i \le 10^3$），分别表示第 $i$ 辆车的初始位置、方向向量及其速度。

保证所有汽车的起始位置互不相同（即对任意 $i \neq j$，有 $(x_i, y_i) \neq (x_j, y_j)$）。

Print the shortest possible duration of the show if it's possible to arrange a crash by choosing all $t_i$. Otherwise, print "No show :(".

Your answer is considered correct if its absolute or relative error does not exceed $10^{-6}$.

Formally, let your answer be $a$, and the jury's answer be $b$. Your answer is accepted if and only if $\frac{|a - b|}{\max{(1, |b|)}} \le 10^{-6}$.

如果可以通过选择所有 $t_i$ 来安排一次碰撞，则输出该演出可能的最短持续时间；否则输出 "No show :("。

若你的答案绝对误差或相对误差不超过 $10^{-6}$，则视为正确。

形式化地，设你的答案为 $a$，评测机的答案为 $b$。当且仅当 $\frac{|a - b|}{\max{(1, |b|)}} \le 10^{-6}$ 时，你的答案被接受。

Here is the picture for the first example:

The fastest cars to crash are cars $2$ and $4$. Let's launch car $2$ at $0$, car $4$ at about $0.096762$ and cars $1$ and $3$ at arbitrary time. That way cars $2$ and $4$ will crash into each other at about $0.585902$. So here's what it looks like at the moment of the collision:

Here's the picture for the second example:

第一个示例的示意图如下：

碰撞速度最快的两辆车是第 $2$ 号车和第 $4$ 号车。我们可将第 $2$ 号车在时刻 $0$ 发射，第 $4$ 号车在约 $0.096762$ 时刻发射，而第 $1$ 号车和第 $3$ 号车则在任意时刻发射。如此一来，第 $2$ 号车与第 $4$ 号车将在约 $0.585902$ 时刻发生碰撞。下图即为碰撞发生瞬间的情形：

第二个示例的示意图如下：