Recently, Bobo, having some free time, wants to write a poem. His method of writing poems is straightforward: he carefully selects verses, where each verse is a binary string of length and all verses are distinct. He then considers any concatenation of these verses in any order to form a binary string of length as a good poem.

Unfortunately, Bobo only has a broken typewriter. When Bobo tries to type (where is either or ) on paper, the typewriter has a probability () of typing correctly and a probability of typing . Bobo wants to know, using the optimal strategy, what is the maximum probability that he can write a good poem?

输入描述:

The first line of input contains three integers  (), which represent the number of verses, the length of each verse, and the probability that the typewriter works correctly, respectively. The probability  is guaranteed to have at most six decimal places.

Then  lines follow, with the -th line  containing a binary string of length , representing the content of the -th verse. It is guaranteed that the  verses are distinct.

输出描述:

Output a single number in a line, representing the maximum probability that Bobo can write a good poem using the optimal strategy. Your answer will be considered correct if and only if its absolute or relative error does not exceed . Formally, if the correct answer is  and your output is , then your answer will be considered correct if and only if 
.

1 3 0.5
000

0.125000000000

The only good poem that meets Bobo's requirements is . In this case, every time Bobo presses  or , there is always a  probability of getting the desired . Under the optimal strategy, there is a probability of  of producing a good poem.

2 3 0.9
010
101

0.590490000000

4 2 0.618
00
01
10
11

0.201586731534