<s>[INST] ### Step 1: Define the problem and identify the given information
The given information states: "We fix two non-empty subsets of $[3n]$ and assume that they have the same size $m$, where $n+1 \leq m < \frac{3}{2}(n+1).$ Let us denote the two non-empty subsets by $A$ and $B$ and suppose $|A|=|B|=m.$"

However, the specific question being asked is not directly stated in the provided snippet. Therefore, I will assume a question based on the information given. Let's assume the question is: Given these conditions, can we make a general statement about the number of elements that are common to both subsets A and B?[/INST]