Problem Statement

There are $2$ circles $1,2$ on the $xy$-plane, which consists of $2$ orthogonal axes (the $x$-axis and the $y$-axis). These $2$ circles have no intersection points, and neither of them lies inside the other.

Circle $1$ has center $(C_{1x},C_{1y})$ and radius $r_1$, and circle $2$ has center $(C_{2x},C_{2y})$ and radius $r_2$.

Find all $4$ lines that are tangent to both of these $2$ circles simultaneously, apply a specific process to them to obtain $1$ real number, and output that real number.

For the specific process, refer to the output format below.

Note that it can be shown that, under the constraints, exactly $4$ such lines always exist. (See the image in Sample 1.)

You are given $T$ test cases; find the answer for each of them.

Constraints

• $1 \leq T \leq 2\times 10^{5}$
• $-100 \leq C_{1x},C_{1y},C_{2x},C_{2y} \leq 100$
• $(C_{1x},C_{1y}) \neq (C_{2x},C_{2y})$
• $0 < r_1 \leq r_2 \leq 100$
• $\sqrt{(C_{2x}-C_{1x})^2+(C_{2y}-C_{1y})^2} > r_1+r_2$
• All inputs are integers.

Input

$T$
$\text{case}_1$
$\text{case}_2$
$\text{case}_3$
$\vdots$
$\text{case}_T$

Each test case is given in the following format.

$C_{1x} \ \ C_{1y} \ \ r_1$
$C_{2x} \ \ C_{2y} \ \ r_2$

Output

Output the results for all $T$ cases.
For each case, first represent the $4$ lines in the following forms, using a total of $12$ real numbers $a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4,c_1,c_2,c_3,c_4$.

• $a_1x+b_1y+c_1=0$
  
• $a_2x+b_2y+c_2=0$
  
• $a_3x+b_3y+c_3=0$
  
• $a_4x+b_4y+c_4=0$

• $a'_1x+b'_1y+c'_1=0$
  
• $a'_2x+b'_2y+c'_2=0$
  
• $a'_3x+b'_3y+c'_3=0$
  
• $a'_4x+b'_4y+c'_4=0$

$X_1$
$X_2$
$X_3$
$\vdots$
$X_T$

Sample

3
1 1 1
4 -2 2
16 -34 14
-76 99 29
0 -16 8
0 16 8

3.0923292192132454
4.8406596632442525
4.0000000000000000