10138_PartyLamps

To brighten up the gala dinner of the IOI'98 we have a set of N ( 10 ≤ N ≤ 100 ) colored lamps numbered from 1 to N.

The lamps are connected to four buttons:

• Button 1: When this button is pressed, all the lamps change their state: those that are ON are turned OFF and those that are OFF are turned ON.

• Button 2: Changes the state of all the odd numbered lamps.

• Button 3: Changes the state of all the even numbered lamps.

• Button 4: Changes the state of the lamps whose number is of the form 3×K+1 ( with K ≥ 0 ), i.e., 1, 4, 7, ...

A counter C records the total number of button presses.

When the party starts, all the lamps are ON and the counter C is set to zero.

You are given the value of counter C ( 0 ≤ C ≤ 10000 ) and the final state of some of the lamps after some operations have been executed. Write a program to determine all the possible final configurations of the N lamps that are consistent with the given information, without repetitions.

No lamp will be listed twice in the input.

Multiple test cases. Each case has 4 lines:

Line 1: N

Line 2: Final value of C

Line 3: Some lamp numbers ON in the final configuration, separated by one space and terminated by the integer -1.

Line 4: Some lamp numbers OFF in the final configuration, separated by one space and terminated by the integer -1.

For each case, output lines with all the possible final configurations (without repetitions) of all the lamps. Each line has N characters, where the first character represents the state of lamp 1 and the last character represents the state of lamp N. A 0 (zero) stands for a lamp that is OFF, and a 1 (one) stands for a lamp that is ON. The lines must be ordered from least to largest (as binary numbers).

If there are no possible configurations, output a single line with the single word 'IMPOSSIBLE'

Output a blank line after each case.

10
1
-1
7 -1

10
0
-1
1 -1

0000000000
0101010101
0110110110

IMPOSSIBLE

In first case, there are 10 lamps and only one button has been pressed. Lamp 7 is OFF in the final configuration.

In this case, there are three possible final configurations:

• All lamps are OFF

• Lamps 1, 4, 7, 10 are OFF and lamps 2, 3, 5, 6, 8, 9 are ON.

• Lamps 1, 3, 5, 7, 9 are OFF and lamps 2, 4, 6, 8, 10 are ON.