Problem Statement

We have 9K cards. For each $i = 1, 2, \dots, 9,$ there are K cards with i written on it.
We randomly shuffled these cards and handed out five cards - four face up and one face down - to each of Takahashi and Aoki.
You are given a string S representing the cards handed out to Takahashi and a string T representing the cards handed out to Aoki.
S and T are strings of five characters each. Each of the first four characters of each string is $1, 2, \dots$, or 9, representing the number written on the face-up card. The last character of each string is #, representing that the card is face down.
Let us define the score of a five-card hand as $\displaystyle \sum_{i=1}^9 i \times 10^{c_i}$, where c_i is the number of cards with i written on them.
Takahashi wins when the score of Takahashi's hand is higher than that of Aoki's hand.
Find the probability that Takahashi wins.

Input

Input is given from Standard Input in the following format:
K
S
T

Output

Print the probability that Takahashi wins, as a decimal.
Your answer will be judged as correct when its absolute or relative error from our answer is at most 10^{-5}.

Constraints

• 2 ≤ K ≤ 10^5
• |S| = |T| = 5
• The first through fourth characters of each of S and T are $1, 2, \dots$, or 9.
• Each of the digit $1, 2, \dots$, and 9 appears at most K times in total in S and T.
• The fifth character of each of S and T is #.

Sample Input 1

2
1144#
2233#

Sample Output 1

0.4444444444444444

For example, if Takahashi's hand is 11449 and Aoki's hand is 22338, Takahashi's score is 100+2+3+400+5+6+7+8+90=621 and Aoki's score is 1+200+300+4+5+6+7+80+9=612, resulting in Takahashi's win.
Takahashi wins when the number on his face-down card is greater than that of Aoki's face-down card, so Takahashi will win with probability $\frac49$.

Sample Input 2

2
9988#
1122#

Sample Output 2

1.0

Sample Input 3

6
1122#
2228#

Sample Output 3

0.001932367149758454

Takahashi wins only when Takahashi's hand is 11222 and Aoki's hand is 22281, with probability $\frac2{1035}$.

Sample Input 4

100000
3226#
3597#

Sample Output 4

0.6296297942426154

见题面 Input 部分

见题面 Output 部分