Find the smallest positive integer $n$ such that $\underbrace{2^{2^{\cdots^{2}}}}_{n} > 3^{3^{3^3}}$. 

(The notation $\underbrace{2^{2^{\cdots^{2}}}}_{n}$ is used to denote a power tower with $n$ $2$’s. For example, $\underbrace{2^{2^{\cdots^{2}}}}_{4}$ would equal $2^{2^{2^2}}$.)

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