H. Light the Street
time limit per test
1 second
memory limit per test
256 megabytes
input
standard input
output
standard output

Mandy, a possessor of great fortune, bought $$$T$$$ streets recently. However, the brightness of the streets doesn't make Mandy happy, who is in love with great brightness. So she decides to rearrange the streetlights in her streets.

As Mandy has enough faith in Brz, she entrusts the project to him. Brz firstly represents a street as a segment with length $$$n$$$. Then he finds out that he can set $$$k$$$ streetlights on it with the funds of Mandy through some calculation. Each streetlight has the same brightness coefficient $$$d$$$, which means it can provide $$$\frac d {r^2}$$$ brightness for a place at distance $$$r$$$. When a place is brightened by multiple streetlights, its brightness is the sum of brightness each streetlight provides.

Specially, the brightness that a streetlight provides for its own place can be regarded as infinite.

Besides, to make her business empire more famous, Mandy will hang huge billboards on each streetlight, which means the light from a streetlight can't travel across other streetlights.

On this basis, Mandy makes her most important request: make the darkest place as bright as possible. On hearing the request, Brz makes his streetlight arrangement in 0.001s. Now he wants to give you a test: What is the maximal possible brightness of the darkest place?

Input

The first line contains one integer $$$T$$$ $$$(1\leq T\leq 10^5)$$$, representing the number of streets that Mandy has bought.

Each of the following $$$T$$$ lines contains three integer $$$n,k,d$$$ $$$(1\leq k\leq n\leq 10^9,1\leq d\leq 10^9)$$$, representing the length of the street, the maximal number of streetlight that can be placed and the brightness coefficient respectively.

Output

Output $$$T$$$ lines. Each line contains one real number, representing the maximal possible brightness of the darkest place of each street.

The answer will be considered correct if the absolute or relative error doesn't exceed $$$10^{-4}$$$.

Example
Input
2
1 1 1
2 2 2
Output
4.000000
11.656854
Note

For the first street, put the only street light in exactly the middle of the street. Then the darkest places are the leftmost and the rightmost place, whose brightness is $$$\frac 1 {0.5 \times 0.5}=4$$$. It can be proved that it is the most optimal arrangement.