Facebook Hacker Cup 2017 Qualifying Round

"Okay, Wizard, cast your spell!"

But which of your many spells to cast? In the ever-popular role-playing game Dungeons & Dragons, or D&D, you determine a spell's damage by rolling polyhedral dice with $4$, $6$, $8$, $10$, $12$, or $20$ sides. Since there's a lot of dice-rolling involved, players use shorthand to denote which dice should be rolled. XdY means "roll a $Y$-sided die $X$ times, and sum the rolls". Sometimes, you must add or subtract a value $Z$ after you finish rolling, in which case the notation is XdY+Z or XdY-Z respectively.

For example, if you roll 2d4+1, you'll end up with a result between $3$ and $9$ inclusive. If you roll 1d6-3, your result will be between $-2$ and $3$ inclusive.

In D&D, wizards are powerful but flimsy spellcasters. As a wizard fighting a zombie, your best strategy is to maximize the chance that you can kill the zombie with a single spell before it has a chance to retaliate. What spell should you cast?

Input Specification

Input begins with an integer $T$, the number of zombies you'll fight. For each zombie, there are two lines. The first contains two integers, $H$ and $S$, the minimum amount of damage it takes to defeat the zombie, and the number of spells you have prepared, respectively. The second line contains $S$ spell descriptions separated by single spaces. A spell description is simply the amount of damage a spell does in the notation described above.

Output Specification

For each zombie, print a line containing the probability of defeating the zombie if you select your spell optimally.

Absolute and relative errors of up to ${10}^{-6}$ will be ignored.

Constraints

$1\le T\le 1000$

$1\le H\le 10000$

$2\le S\le 10$

Additionally, the following constraints will hold for each spell:

$1\le X\le 20$

$Y\in \{4,6,8,10,12,20\}$

$1\le Z\le 10000$ if $Z$ is specified.

$X$, $Y$, and $Z$ will be integers with no leading zeros.

Sample Input

5
2 2
2d4 1d8
10 2
10d6-10 1d6+1
8 3
1d4+4 2d4 3d4-4
40 3
10d4 5d8 2d20
10 4
1d10 1d10+1 1d10+2 1d10+3

Sample Output

Case #1: 1.000000
Case #2: 0.998520
Case #3: 0.250000
Case #4: 0.002500
Case #5: 0.400000

Explanation of Sample

In the first case, you can guarantee a kill with the first spell, which must always do at least $2$ damage.

In the third case, your first spell is the best. If you roll a $4$, you'll do the requisite $8$ damage. The second spell requires rolling a $4$ on two dice rather than just one, and the third spell requires rolling a $4$ on all three dice.