Let $a_{1}, a_{2}, \\ldots, a_{n}, b_{1}, b_{2}, \\ldots, b_{n}$ be $2 n$ positive integers such that the $n+1$ products $$\begin{gathered} a_{1} a_{2} a_{3} \\cdots a_{n} \\ b_{1} a_{2} a_{3} \\cdots a_{n} \\ b_{1} b_{2} a_{3} \\cdots a_{n} \\ \vdots \\ b_{1} b_{2} b_{3} \\cdots b_{n} \end{gathered}$$ form a strictly increasing arithmetic progression in that order. Determine the smallest positive integer that could be the common difference of such an arithmetic progression.

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