Background

Translated from ROIR 2023 D1T4。

There are $n$ points on the plane, namely $P_1,P_2,P_3,\dots,P_n$。The coordinates of the $i$-th point are $(x_i,y_i)$。

Choose two points $P_i,P_j$ ($i\ne j$)。Let $P_i$ be the starting point and $P_j$ be the ending point。With the ending point $P_j$ as the center, starting from the direction of the vector $\overrightarrow{P_iP_j}$, sort all points except $P_j$ in counterclockwise order (if angles are the same, sort by increasing distance to $P_j$)。Suppose the $t$-th point in this order is $P_k$。Then repeat the operation: let $P_j$ be the starting point and $P_k$ be the ending point, reorder in the same way, and compute the index of the new ending point。This process repeats in a loop。

In the figure below, initially $n=6,t=4,i=1,j=2$。Sort all points except $P_2$ in order, and the result is $P_3,P_5,P_1,P_6,P_4$，so the next ending point should be $P_6$，and the starting point is $P_2$。

Now $n=6,t=4,i=2,j=6$。Continue sorting by the rules. As in the lower-left figure, the result is $P_4,P_3,P_2,P_1,P_5$。So the next ending point is $P_1$，and the starting point is $P_6$。

Now $n=6,t=4,i=6,j=1$。Continue sorting by the rules. As in the upper-right figure, the result is $P_5,P_6,P_4,P_2,P_3$。So the next ending point is $P_2$，and the starting point is $P_1$。Now we return to the initial situation and enter a cycle。

Problem Description

We paint each of the $n$ points with one of three colors。The color of point $i$ is determined as follows：

• If there exists a point $j$ such that, choosing point $i$ as the starting point and point $j$ as the ending point, in the process above point $i$ will become the starting point infinitely many times, then paint point $i$ green。
• If point $i$ is not painted green and there exists a point $j$ such that, choosing point $i$ as the starting point and point $j$ as the ending point, in the process above point $i$ can still become the starting point at least once, then paint point $i$ blue。
• If point $i$ is neither green nor blue, then paint point $i$ red。

For each point, determine which color it should be painted。

Input Format

The first line contains two integers $n$ and $t$。

The next $n$ lines each contain two integers $x_i$ and $y_i$。It is guaranteed that no two points coincide。

Output Format

Output one line containing $n$ characters。The $i$-th character of the string indicates the color of the $i$-th point。

Green points are denoted by the letter G，blue points by B，and red points by R。

6 4
-1 -1
1 -2
4 -2
2 -4
2 3
-4 -5

GGBBRG

2 1
1 1
2 2

GG

Hint

Explanation for sample $1$：

In the example given above, we already know that $P_1,P_2,P_6$ form a cycle. These three points will definitely be visited infinitely many times, so they should be painted green。

When the starting point is $P_3$, we can take the ending point as $P_1$。After sorting, $P_3$ is exactly the fourth point, so $P_3$ returns to being a starting point again, as shown below. Therefore it should be painted blue。

When the starting point is $P_4$, we can choose $P_5$ as the ending point. After sorting, $P_4$ is exactly the fourth point. Therefore $P_4$ can be painted blue。When $P_5$ is the starting point, it is easy to see that it can be painted neither green nor blue, so it should be painted red。

This problem uses bundled testdata。

Constraints for all data: $2 \le n \le 1 000, 1 \le t \le n−1, −10^9 \le x_i, y_i \le 10^9$。

Translated by ChatGPT 5