
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 the tower function of twos, defined recursively as $T(1) = 2$ and $T(n + 1) = 2^{T(n)}$ for $n \ge 1$, let $A = (T(2009))^{T(2009)}$ and $B = (T(2009))^A$. Express the largest integer $k$ for which $\underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k\text{ times}}$ is defined.

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