hackercupai/hackercup huggingface dataset:
A rich, retired programmer has decided to invest his (possibly) massive
savings into constructing custom yachts for himself.

Initially, he has a real amount of dollars, chosen uniformly at random from
the range [**A**, **B**]. Constructing a yacht consists of **N** sequential
steps, with the **i**th step requiring **Ci** dollars.

The programmer blindly pays for the steps in order, until he's either
completed all of them, or can't afford the cost of the next step. If the
former occurs, he puts his completed yacht aside and restarts the process from
the first step with his remaining money - he wants as many yachts as possible!
Otherwise, in the latter case, he immediately stops his project entirely,
without spending any additional money on other steps.

What's the expected amount of money which the programmer will be left with
once he stops spending it on yachts? Your output should have at most 10-6
absolute or relative error.

### Input

Input begins with an integer **T**, the number of times the programmer embarks
on a yacht-creation spree. For each spree, there is first a line containing
the space-separated integers **N**, **A**, and **B** in that order, then a
line containing **N** space-separated integers, the **i**th of which is
**Ci**.

### Output

For the **i**th spree, print a line containing "Case #**i**: " followed by the
expected amount of money the programmer will have left.

### Constraints

1 ≤ **T** ≤ 50
0 ≤ **A** < **B** ≤ 1,000,000,000
1 ≤ **N** ≤ 100,000
1 ≤ **Ci** ≤ 1,000,000,000

### Explanation of Sample

In the first case, the programmer starts with between 5 and 8 dollars. The
programmer's initial amount of money will fall into one of the ranges [5, 6),
or [6, 8), or [8, 8]. If it falls into the first range, the programmer will
end with [1, 2) dollars. If it falls into the second range, the programmer
will end with [0, 2) dollars. If the programmer starts with exactly 8 dollars,
he'll end with 0 dollars.