Just Simplify That Power Tower!

\[ \large \text{lcm} (2^1 ,2^2 ,2^3,\ldots, 2^n) = 2^{2^{2^2}} \]

What is the value of \(n\) satisfying the equation above? \[\]

Notation: \(\text{lcm}(\cdot) \) denotes the lowest common multiple function.

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