hackercupai/hackercup huggingface dataset:
You just landed a job as a machine learning engineer! As a ramp-up exercise, Boss Rob tasked you with modeling the watering wells in his yard, which can be represented on a Cartesian plane.

Boss Rob has a primary well at point \((A_x, A_y)\) and a backup well at a different point \((B_x, B_y)\), each able to water trees within an \(R\) unit radius. Using \(A_x\), \(A_y\), \(B_x\), \(B_y\), and \(R\) (unknown integers to you), Rob plants \(N\) happy little trees at real number points obtained by \(N\) calls to the function:

```
def gen_one_tree(A_x, A_y, B_x, B_y, R):
while True:
r = random.uniform(0, R)
theta = random.uniform(0, 2*math.pi)
x = A_x + r*math.cos(theta)
y = A_y + r*math.sin(theta)
if (x - B_x)**2 + (y - B_y)**2 <= R*R:
return (x, y)
```
Here, `random.uniform(L, H)` returns a real number in \([L, H)\) uniformly at random.

In other words, he picks a point \((x, y)\) in the circular range of the primary well using the special method above. If \((x, y)\) happens to be in range of the backup well, he plants a tree there (else he discards it and tries again with a new \((x, y)\)). This repeats until Rob has planted \(N\) trees.

Given only the planted tree coordinates \((X_1, Y_1), \ldots, (X_N, Y_N)\), you are tasked to predict the exact values of \(A_x\), \(A_y\), \(B_x\), \(B_y\), and \(R\). As you are new, Boss Rob will accept your solution if it correctly predicts at least \(80\%\) of the test cases.

# Constraints

\(1 \le T \le 1{,}000\)
\(500 \le N \le 1{,}000{,}000\)
\(0 \le A_x, A_y, B_x, B_y \le 50\)
\((A_x, A_y) \ne (B_x, B_y)\)
\(1 \le R \le 50\)
The sum of \(N\) across all test cases is at most \(2{,}000{,}000\).

The intersection area of the two circular regions is strictly positive.

Tree coordinates in the data were truly generated using the randomized algorithm as described above. The secret parameters \(A_x\), \(A_y\), \(B_x\), \(B_y\), and \(R\) have also been chosen uniformly at random for each case (rejecting cases where the circles are identical or do not have positive overlap).

# Input Format

Input begins with a single integer \(T\), the number of test cases. For each case, there is first a line containing a single integer \(N\), the number of planted trees. Then, \(N\) lines follow, the \(i\)th of which contains two space-separated real numbers \(X_i\) and \(Y_i\), each given to \(6\) decimal places.

# Output Format

For the \(i\)th test case, print a line containing `"Case #i: "`, followed by the five space-separated integers \(A_x\), \(A_y\), \(B_x\), \(B_y\), and \(R\), in that order.

# Sample Explanation

The first sample case is pictured below, with the primary well's range in red, the backup well's range in blue, and the \(500\) randomly-generated trees in green:

{{PHOTO_ID:6502772429739994|WIDTH:700}}