Problem:
Problem B: Cleaning Robot

Dr. Asimov, a robotics researcher, loves to research, but hates houseworks and his house were really dirty. So, he has developed a cleaning robot.

As shown in the following figure, his house has 9 rooms, where each room is identified by an alphabet:

The robot he developed operates as follows:

If the battery runs down, the robot stops.

If not so, the robot chooses a direction from four cardinal points with the equal probability, and moves to the room in that direction. Then, the robot clean the room and consumes 1 point of the battery.

However, if there is no room in that direction, the robot does not move and remains the same room. In addition, there is a junk room in the house where the robot can not enter, and the robot also remains when it tries to enter the junk room. The robot also consumes 1 point of the battery when it remains the same place.

A battery charger for the robot is in a room. It would be convenient for Dr. Asimov if the robot stops at the battery room when its battery run down.

Your task is to write a program which computes the probability of the robot stopping at the battery room.

Input

The input consists of several datasets. Each dataset consists of:

n
s t b

n is an integer that indicates the initial battery point. s, t, b are alphabets respectively represent the room where the robot is initially, the battery room, and the junk room.

The input ends with a dataset which consists of single 0 for n. Your program should not output for this dataset.

Output

For each dataset, print the probability as a floating point number in a line. The answer may not have an error greater than 0.000001.

Constraints

Judge data includes at most 100 data sets.

n ≤ 15

s, t, b are distinct.

Sample Input

1
E A C
1
E B C
2
E A B
0

Output for the Sample Input

0.00000000
0.25000000
0.06250000

Solution in Ruby (judge verdict: Accepted)

require 'matrix'

loop do
    n = gets.to_i
    break if n == 0
    s, t, b = gets.chomp.split.map{|c| c.ord - ?A.ord}

    d = [2, 1, 2,
         1, 0, 1,
         2, 1, 2]
    d[b-3] += 1 if b > 2
    d[b-1] += 1 if b % 3 > 0
    d[b+1] += 1 if b % 3 < 2
    d[b+3] += 1 if b < 6

    m = Matrix.build(9, 9) do |i, j|
        if j == i
            d[i] / 4.0
        elsif j == b
            0
        elsif i / 3 == j / 3 && (i - j).abs == 1
            0.25
        elsif i % 3 == j % 3 && (i - j).abs == 3
            0.25
        else
            0
        end
    end

    a = [0] * 9
    a[s] = 1.0
    v = Matrix[a]
    p (v * m ** n)[0, t]
end