Infinite factorials

\[ \sqrt{6n \times \sqrt{1\sqrt{1\sqrt{1\ldots}}} \times \sqrt{2\sqrt{2\sqrt{2\ldots}}} \times \ldots \times \sqrt{n\sqrt{n\sqrt{n\ldots}}} } \]

For a particular positive value of \(n\), the expression with infinitely nested radicals above can be simplified to \(12n\). What is the value of \(n\)?

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