Recurring Styles #15 - Continued Product!

Calculus Level 5

\[\Large{L = \lim_{n \to \infty} \dfrac{ \left| a_{n+1} - a_n \right| }{\sqrt{n}}}\]

Let \(\{ a_n \} \) be a sequence such that \(a_1=1, \ a_n a_{n+1} = n\) for every \(n \in \mathbb Z^+\). If \(L\) can be expressed as:

\[\large{ \sqrt{\dfrac{\pi^A}{B}} - \sqrt{C\pi^D}}\]

where \(A,B,C,D \in \mathbb Z\), submit the value of \(ABCD\) as your answer.

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