**"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.