There is a table with $n$ rows and $18$ columns. Each cell contains either a $0$ or a $1$. The table satisfies the following properties:

1. Every two rows are different.
2. Each row contains exactly $6$ cells that contain $1$.
3. For every three rows, there exists a column such that the intersection of the column with the three rows (the three cells) all contain $0$.

What is the greatest possible value of $n$?

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