| **"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. **X**d**Y** 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 **X**d**Y**+**Z** or **X**d**Y**-**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 | |
| 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 | |
| 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 1e-6 will be ignored. | |
| ### Constraints | |
| 1 ≤ **T** ≤ 1,000 | |
| 1 ≤ **H** ≤ 10,000 | |
| 2 ≤ **S** ≤ 10 | |
| Additionally, the following constraints will hold for each spell: | |
| 1 ≤ **X** ≤ 20 | |
| **Y** ∈ {4, 6, 8, 10, 12, 20} | |
| 1 ≤ **Z** ≤ 10,000, if **Z** is specified. | |
| **X**, **Y**, and **Z** will be integers with no leading zeros. | |
| ### 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. | |