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 C (judge verdict: Time Limit Exceeded)

/*
AizuOnline A1020
Cleaning Robot
*/
#include <stdio.h>
#include <float.h>
int  n;
char s,t,b;
//                A B C D E F G H I
int   k_t[9][9]={{2,1,0,1,0,0,0,0,0}, //A
                 {1,1,1,0,1,0,0,0,0}, //B
                 {0,1,2,0,0,1,0,0,0}, //C
                 {1,0,0,1,1,0,1,0,0}, //D
                 {0,1,0,1,0,1,0,1,0}, //E
                 {0,0,1,0,1,1,0,0,1}, //F
                 {0,0,0,1,0,0,2,1,0}, //G
                 {0,0,0,0,1,0,1,1,1}, //H
                 {0,0,0,0,0,1,0,1,2}  //I
};

int k_t_a[9][9];
init()
{
  n = 3;s = 'A';t = 'H';b = 'G';
}
unsigned long kakuritu(int bat,char from,char to)
{ unsigned long ret,w;
  char next;
  int  prob;
  

  // printf("%c %c \n",from,to);
  if(bat==0)
    if(from==to)
      return(1L<<31);
    else
      return(0L);

  ret=0L;
  for(next='A';next<='I';next++)
    {  prob= k_t_a[from-'A'][next-'A'];
      if(prob)
	w = (kakuritu(bat-1,next,to)) >> 1;
      if(prob  & 2)  //0.5
	 ret += w;
       if(prob & 1)  //0.25
	 ret += w >> 1;
	 
    }
  return(ret);

}
mod_k_t(char b)
{
  int i,j;
  for(i=0;i<9;i++)
    for(j=0;j<9;j++)
      k_t_a[i][j]=k_t[i][j];
    
  for(i=0;i<9;i++)
    if(k_t_a[i][b-'A'] > 0)
      {  k_t_a[i][b-'A']=0;
	 k_t_a[i][i] += 1;
      }
  
}

main()
{
  //init();
  while(EOF!=scanf("%d\n",&n) && n!=0)
  { scanf("%c %c %c",&s,&t,&b);
    //printf("%d %c %c %c\n",n,s,t,b);
    mod_k_t(b);
    printf("%f\n",(float) (kakuritu(n,s,t))/65536/32768);
    }
  return(0);
}