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)

### constant

N = 3
NN = N * N

DIRS = [[1, 0], [0, -1], [-1, 0], [0, 1]]

### main

ch2xy = {}
for y in (0...N)
  for x in (0...N)
    id = y * N + x
    ch2xy[('A'.ord + id).chr] = [x, y]
  end
end
#p ch2xy

zmtx = NN.times.map{0.0}

loop do
  n = gets.to_i
  break if n == 0

  s, t, b = gets.split

  sx, sy = ch2xy[s]
  tx, ty = ch2xy[t]
  bx, by = ch2xy[b]
  #p [sx, sy, tx, ty, bx, by]

  pmtx = zmtx.clone
  pmtx[sy * N + sx] = 1.0

  n.times do
    pmtx0 = pmtx
    pmtx = zmtx.clone

    for y in (0...N)
      for x in (0...N)
        for dir in DIRS
          x0 = x + dir[0]
          y0 = y + dir[1]
          if x0 >= 0 && x0 < N && y0 >= 0 && y0 < N && (x0 != bx || y0 != by)
            pmtx[y0 * N + x0] += pmtx0[y * N + x] * 0.25
          else
            pmtx[y * N + x] += pmtx0[y * N + x] * 0.25
          end
        end
      end
    end
  end

  printf("%.8f\n", pmtx[ty * N + tx])
end