
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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.

Present the answer in LaTex format: \boxed{Your answer}
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