For every positive integer \(n\), let a sequence \(\{a_n\}\) satisfy:

\[\large \begin{cases} a_{2n+1}=a_{2n-1}+2n-1 \\ a_{2n+2}=na_{2n} \end{cases} \]

Find the value of \(\displaystyle \lim_{n\rightarrow\infty}\frac{12a_{2n}a_{2n+9}}{a_6\cdot(n+1)!}\).

*This problem is a part of <Inconsequences!> series.*

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