Problem F

Statement
Copy Copied
Description:
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 $$$A$$$.

Input Format:
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.

Output Format:
None

Note:
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: