Description

In this problem, you start with $0$ cookies. You gain cookies at a rate of $2$ cookies per second, by clicking on a giant cookie. Any time you have at least $C$ cookies, you can buy a cookie farm. Every time you buy a cookie farm, it costs you $C$ cookies and gives you an extra $F$ cookies per second.

Once you have $X$ cookies that you haven't spent on farms, you win! Figure out how long it will take you to win if you use the best possible strategy.

Suppose $C = 500.0$, $F = 4.0$ and $X = 2000.0$. Here's how the best possible strategy plays out:

1. You start with $0$ cookies, but producing $2$ cookies per second.
2. After $250$ seconds, you will have $C = 500$ cookies and can buy a farm that produces $F = 4$ cookies per second.
3. After buying the farm, you have $0$ cookies, and your total cookie production is $6$ cookies per second.
4. The next farm will cost $500$ cookies, which you can buy after about $83.3333333$ seconds.
5. After buying your second farm, you have $0$ cookies, and your total cookie production is $10$ cookies per second.
6. Another farm will cost $500$ cookies, which you can buy after $50$ seconds.
7. After buying your third farm, you have $0$ cookies, and your total cookie production is $14$ cookies per second.
8. Another farm would cost $500$ cookies, but it actually makes sense not to buy it: instead you can just wait until you have $X = 2000$ cookies, which takes about $142.8571429$ seconds.

Total time: $250 + 83.3333333 + 50 + 142.8571429 = 526.1904762$ seconds.

Notice that you get cookies continuously: so $0.1$ seconds after the game starts you'll have $0.2$ cookies, and $\pi$ seconds after the game starts you'll have $2\pi$ cookies.

Input Format

The first line of the input gives the number of test cases, $T$. $T$ lines follow. Each line contains three space-separated real-valued numbers: $C$, $F$ and $X$, whose meanings are described earlier in the problem statement.

$C$, $F$ and $X$ will each consist of at least $1$ digit followed by $1$ decimal point followed by from $1$ to $5$ digits. There will be no leading zeroes.

Output Format

For each test case, output one line containing "Case #$x$: $y$", where $x$ is the test case number (starting from $1$) and $y$ is the minimum number of seconds it takes before you can have $X$ delicious cookies.

We recommend outputting $y$ to $7$ decimal places, but it is not required. $y$ will be considered correct if it is close enough to the correct number: within an absolute or relative error of $10^{-6}$.

4
30.0 1.0 2.0
30.0 2.0 100.0
30.50000 3.14159 1999.19990
500.0 4.0 2000.0

Case #1: 1.0000000
Case #2: 39.1666667
Case #3: 63.9680013
Case #4: 526.1904762

Hint

Limits

• $1 \leq T \leq 100$.

Small dataset(8 Pts)

• Time limit: 60 3 seconds.
• $1 \leq C \leq 500$.
• $1 \leq F \leq 4$.
• $1 \leq X \leq 2000$.

Large dataset(11 Pts)

• Time limit: 120 5 seconds.
• $1 \leq C \leq 10000$.
• $1 \leq F \leq 100$.
• $1 \leq X \leq 100000$.