Consider a board with $n$ rows and $4$ columns. In the first line, there are $4$ zeros (one in each column). Each subsequent line is obtained from the previous line by performing the following operation: one of the columns, which 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 next line, a $1$ is placed.
- If in the previous line there was a $1$, then in the next line, a $2$ is placed.
- If in the previous line there was a $2$, then in the next line, a $0$ is placed.

Build the largest possible board with all its lines distinct and demonstrate that it is impossible to build a larger board.

Let's think step by step and output the final answer within \boxed{}.
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