
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
Find the smallest positive integer \( n \) such that \(\underbrace{2^{2 \cdot \cdot}}_{n} > 3^{3^{3^{3}}}\). (The notation \(\underbrace{2^{2 \cdot \cdot}}_{n}\) is used to denote a power tower with \( n \) 2's. For example, \(\underbrace{2^{2^{2}}}_{n}\) with \( n=4 \) would equal \( 2^{2^{2^{2}}} \).)

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