Too Lazy To Move

Time Limit: 5 Seconds    Memory Limit: 32768 KB

As we all know, Hanoi Tower can be solved in 2^n-1 steps if there are n disks. We might adhere to such rules:

1) When the step number is odd, move the smallest disk to the next pillar (in the order A->B->C->A).

2) When the step number is even, move one disk between the pillar without the smallest disk.

Here is an example:

Step 0 (Initial Step)
A: 3 2 1
B:
C:

Step 1
A: 3 2
B: 1
C:

Step 2
A: 3
B: 1
C: 2

Step 3
A: 3
B:
C: 2 1

Step 4
A:
B: 3
C: 2 1

Step 5
A: 1
B: 3
C: 2

Step 6
A: 1
B: 3 2
C:

Step 7
A:
B: 3 2 1
C:

End.

Now it is your task to find out the m-th configuration for a n-disk Hanoi Tower with the rules defined above.

Input

One test on each line, with two integers n (1 <= n <= 63) and m (0 <= m <= 2^n-1). Input is terminated with end of file.

Output

Print a configuration in three lines. Conform to the format in the sample. There should be one space after each colon. Each test is followed by a blank line.

Sample Input

3 0
3 1
3 2
3 7
63 9223372036854775807

Sample Output

A: 3 2 1
B: 
C:

A: 3 2
B: 1
C:

A: 3
B: 1
C: 2

A: 
B: 3 2 1
C:

A: 
B: 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 
38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 
12 11 10 9 8 7 6 5 4 3 2 1
C:

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Source: ZOJ Monthly, March 2003