Find the greatest integer $n$, such that there are $n+4$ points $A$, $B$, $C$, $D$, $X_1, \dots, X_n$ in the plane with $AB \neq CD$ that satisfy the following condition: for each $i=1,2,\dots,n$, the triangles $\triangle ABX_i$ and $\triangle CDX_i$ are equal.

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