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 a letter of the alphabet:

The robot operates as follows:

• if the battery runs down, the robot stops;
• if not so, the robot chooses a direction from four cardinal points with equal probability, and moves to the room in that direction. Then, the robot cleans 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 in 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 in 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 runs down. Your task is to write a program which computes the probability of the robot stopping at the battery room.

You are given battery capacity cap and start, bat, junk respectively represent the room where the robot is initially, the battery room, and the junk room.

Input: One integer (int) and three strings (str).

Output: Float (float).

Examples:

assert cleaning_robot(1, "E", "A", "C") == 0.0
assert cleaning_robot(1, "E", "B", "C") == 0.25
assert cleaning_robot(2, "E", "A", "B") == 0.0625

This mission was taken from AIZU ONLINE JUDGE