
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Consider a board with  $n$  rows and  $4$  columns. In the first line are written  $4$  zeros (one in each house). Next, each line is then obtained from the previous line by performing the following operation: one of the houses, (that you can choose), is maintained as in the previous line; the other three are changed:
* if in the previous line there was a  $0$ , then in the down square  $1$  is placed; 
* if in the previous line there was a  $1$ , then in the down square  $2$  is placed; 
* if in the previous line there was a  $2$ , then in the down square  $0$  is placed; 
Build the largest possible board with all its distinct lines and demonstrate that it is impossible to build a larger board.

Present the answer in LaTex format: \boxed{Your answer}
user