CF1826F.Fading into Fog

This is an interactive problem.

There are $n$ distinct hidden points with real coordinates on a two-dimensional Euclidean plane. In one query, you can ask some line $ax + by + c = 0$ and get the projections of all $n$ points to this line in some order. The given projections are not exact, please read the interaction section for more clarity.

Using the minimum number of queries, guess all $n$ points and output them in some order. Here minimality means the minimum number of queries required to solve any possible test case with $n$ points.

The hidden points are fixed in advance and do not change throughout the interaction. In other words, the interactor is not adaptive.

A projection of point $A$ to line $ax + by + c = 0$ is the point on the line closest to $A$.

这是一个交互式问题。

平面上预先固定了 $n$ 个互不相同的隐藏点，其坐标为实数。每次查询中，你可以指定一条直线 $ax + by + c = 0$，系统将返回这 $n$ 个点在该直线上的投影（以某种顺序给出）。注意：所给的投影并非精确值，请参阅“交互说明”部分以获得更清晰的解释。

你需要用尽可能少的查询次数，猜出全部 $n$ 个点的坐标，并以任意顺序输出它们。此处“最少查询次数”是指：对任意含 $n$ 个点的合法测试用例，均能保证求解成功的最小查询数。

所有隐藏点在交互开始前即已固定，且在整个交互过程中保持不变。换言之，交互器是非自适应的（non-adaptive）。

点 $A$ 到直线 $ax + by + c = 0$ 的投影，定义为该直线上距离 $A$ 最近的点。

The first line contains a single integer $t$ ($1 \leq t \leq 50$) — the number of test cases.

The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($2 \leq n \leq 25$) — the number of hidden points.

For each test case, it is guaranteed that for any pair of hidden points, their $x$ coordinates differ by at least $1$. Analogously, $y$ coordinates of any pair also differ by at least $1$.

Coordinates $x$ and $y$ of all hidden points do not exceed $100$ by absolute value.

第一行包含一个整数 $t$（$1 \leq t \leq 50$）—— 测试用例的数量。

接下来是测试用例的描述。

每个测试用例的第一行包含一个整数 $n$（$2 \leq n \leq 25$）—— 隐藏点的数量。

对于每个测试用例，保证任意两个隐藏点的 $x$ 坐标之差的绝对值至少为 $1$；同理，任意两个隐藏点的 $y$ 坐标之差的绝对值也至少为 $1$。

所有隐藏点的坐标 $x$ 和 $y$ 的绝对值均不超过 $100$。

In the sample the hidden points are $(1, 3)$ and $(2.5, 0.5)$

A picture, which describes the first query:

A picture, which describes the second query:

在样例中，隐藏的点为 $(1, 3)$ 和 $(2.5, 0.5)$。

描述第一次查询的图片：

描述第二次查询的图片：