Given two natural numbers $w$ and $n$, the tower of $n$ $w$'s is the natural number $T_n(w)$ defined by

\[ T_n(w) = w^{w^{\cdots^{w}}},\]

with $n$ $w$'s on the right side. More precisely, $T_1(w) = w$ and $T_{n+1}(w) = w^{T_n(w)}$. For example, $T_3(2) = 2^{2^2} = 16$, $T_4(2) = 2^{16} = 65536$, and $T_2(3) = 3^3 = 27$. Find the smallest tower of $3$'s that exceeds the tower of $1989$ $2$'s. In other words, find the smallest value of $n$ such that $T_n(3) > T_{1989}(2)$. Justify your answer.

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