
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
The tower function of threes is defined recursively as follows: $T(1) = 3$ and $T(n + 1) = 3^{T(n)}$ for $n\ge1$. Let $A = (T(5))^{T(5)}$ and $B = (T(5))^A$. Calculate the largest integer $k$ for which $\log_3\log_3\log_3\ldots\log_3 B$ is defined.

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