Cookie Farms

You start with 0 cookies. Clicking a giant cookie earns you 2 cookies per second. Whenever you hold at least $C$ cookies, you may buy a cookie farm. Each farm costs $C$ cookies and raises your production by $F$ cookies per second.

You win the moment you hold $X$ cookies that you have not spent on farms. Compute how long winning takes if you play the best possible strategy.

Cookies accumulate continuously. After 0.1 seconds you have 0.2 cookies, and after $\pi$ seconds you have $2\pi$ cookies. Buying a farm takes no time, and you may buy any number of farms.

The first line contains the number of test cases $T$. Each of the next $T$ lines contains three space-separated real numbers $C$, $F$ and $X$, with the meanings given above.

Each of $C$, $F$ and $X$ is written as at least one digit, then a decimal point, then one to five digits. No number has a leading zero.

• $1 \le T \le 100$
• $1 \le C \le 500$
• $1 \le F \le 4$
• $1 \le X \le 2000$

For each test case print one line in the form Case #x: y, where $x$ is the test case number starting from 1 and $y$ is the minimum number of seconds needed to hold $X$ cookies.

Print $y$ rounded to exactly seven decimal places, keeping trailing zeros. The judge compares the output string exactly. Every answer in the input data sits far from a rounding boundary, so double precision arithmetic produces the same digits.

Whether one more farm pays off depends only on your current production rate. As the rate grows, the time needed to earn a farm shrinks slowly, while the time the farm saves shrinks faster. Once buying stops paying off, it never pays off again.