hackercupai/hackercup huggingface dataset:
You may be familiar with the works of Alfred, Lord Tennyson, the famous
English poet. In this problem we will concern ourselves with Tennison, the
less famous English tennis player. As you know, tennis is not so much a game
of skill as a game of luck and weather patterns. The goal of tennis is to win
**K** sets before the other player. However, the chance of winning a set is
largely dependent on whether or not there is weather.

Tennison plays best when it's sunny, but sometimes, of course, it rains.
Tennison wins a set with probability **ps** when it's sunny, and with
probability **pr** when it's raining. The chance that there will be sun for
the first set is **pi**. Luckily for Tennison, whenever he wins a set, the
probability that there will be sun increases by **pu** with probability
**pw**. Unfortunately, when Tennison loses a set, the probability of sun
decreases by **pd** with probability **pl**. What is the chance that Tennison
will be successful in his match?

Rain and sun are the only weather conditions, so P(rain) = 1 - P(sun) at all
times. Also, probabilities always stay in the range [0, 1]. If P(sun) would
ever be less than 0, it is instead 0. If it would ever be greater than 1, it
is instead 1.

## Input

Input begins with an integer **T**, the number of tennis matches that Tennison
plays. For each match, there is a line containing an integer **K**, followed
by the probabilities **ps, pr, pi, pu, pw, pd, pl** in that order. All of
these values are given with exactly three places after the decimal point.

## Output

For each match, output "Case #i: " followed by the probability that Tennison
wins the match, rounded to 6 decimal places (quotes for clarity only). It is
guaranteed that the output is unaffected by deviations as large as 10-8.

## Constraints

* 1 ≤ **T** ≤ 100
* 1 ≤ **K** ≤ 100
* 0 ≤ **ps, pr, pi, pu, pw, pd, pl** ≤ 1
* **ps** > **pr**